From: Ferruccio Guidi Date: Thu, 22 Jan 2015 15:47:40 +0000 (+0000) Subject: refactoring of \lambda\delta version 1 in matita X-Git-Tag: make_still_working~758 X-Git-Url: http://matita.cs.unibo.it/gitweb/?p=helm.git;a=commitdiff_plain;h=88a68a9c334646bc17314d5327cd3b790202acd6 refactoring of \lambda\delta version 1 in matita --- diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/A/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/A/defs.ma deleted file mode 100644 index 2290c3de4..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/A/defs.ma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/preamble.ma". - -inductive A: Set \def -| ASort: nat \to (nat \to A) -| AHead: A \to (A \to A). - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/C/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/C/defs.ma deleted file mode 100644 index cecc08267..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/C/defs.ma +++ /dev/null @@ -1,45 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/T/defs.ma". - -inductive C: Set \def -| CSort: nat \to C -| CHead: C \to (K \to (T \to C)). - -definition cweight: - C \to nat -\def - let rec cweight (c: C) on c: nat \def (match c with [(CSort _) \Rightarrow O -| (CHead c0 _ t) \Rightarrow (plus (cweight c0) (tweight t))]) in cweight. - -definition clt: - C \to (C \to Prop) -\def - \lambda (c1: C).(\lambda (c2: C).(lt (cweight c1) (cweight c2))). - -definition cle: - C \to (C \to Prop) -\def - \lambda (c1: C).(\lambda (c2: C).(le (cweight c1) (cweight c2))). - -definition CTail: - K \to (T \to (C \to C)) -\def - let rec CTail (k: K) (t: T) (c: C) on c: C \def (match c with [(CSort n) -\Rightarrow (CHead (CSort n) k t) | (CHead d h u) \Rightarrow (CHead (CTail k -t d) h u)]) in CTail. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/C/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/C/props.ma deleted file mode 100644 index 878abb3bc..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/C/props.ma +++ /dev/null @@ -1,139 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/C/defs.ma". - -include "Basic-1/T/props.ma". - -theorem clt_cong: - \forall (c: C).(\forall (d: C).((clt c d) \to (\forall (k: K).(\forall (t: -T).(clt (CHead c k t) (CHead d k t)))))) -\def - \lambda (c: C).(\lambda (d: C).(\lambda (H: (lt (cweight c) (cweight -d))).(\lambda (_: K).(\lambda (t: T).(lt_reg_r (cweight c) (cweight d) -(tweight t) H))))). -(* COMMENTS -Initial nodes: 33 -END *) - -theorem clt_head: - \forall (k: K).(\forall (c: C).(\forall (u: T).(clt c (CHead c k u)))) -\def - \lambda (_: K).(\lambda (c: C).(\lambda (u: T).(eq_ind_r nat (plus (cweight -c) O) (\lambda (n: nat).(lt n (plus (cweight c) (tweight u)))) -(le_lt_plus_plus (cweight c) (cweight c) O (tweight u) (le_n (cweight c)) -(tweight_lt u)) (cweight c) (plus_n_O (cweight c))))). -(* COMMENTS -Initial nodes: 69 -END *) - -theorem clt_wf__q_ind: - \forall (P: ((C \to Prop))).(((\forall (n: nat).((\lambda (P0: ((C \to -Prop))).(\lambda (n0: nat).(\forall (c: C).((eq nat (cweight c) n0) \to (P0 -c))))) P n))) \to (\forall (c: C).(P c))) -\def - let Q \def (\lambda (P: ((C \to Prop))).(\lambda (n: nat).(\forall (c: -C).((eq nat (cweight c) n) \to (P c))))) in (\lambda (P: ((C \to -Prop))).(\lambda (H: ((\forall (n: nat).(\forall (c: C).((eq nat (cweight c) -n) \to (P c)))))).(\lambda (c: C).(H (cweight c) c (refl_equal nat (cweight -c)))))). -(* COMMENTS -Initial nodes: 61 -END *) - -theorem clt_wf_ind: - \forall (P: ((C \to Prop))).(((\forall (c: C).(((\forall (d: C).((clt d c) -\to (P d)))) \to (P c)))) \to (\forall (c: C).(P c))) -\def - let Q \def (\lambda (P: ((C \to Prop))).(\lambda (n: nat).(\forall (c: -C).((eq nat (cweight c) n) \to (P c))))) in (\lambda (P: ((C \to -Prop))).(\lambda (H: ((\forall (c: C).(((\forall (d: C).((lt (cweight d) -(cweight c)) \to (P d)))) \to (P c))))).(\lambda (c: C).(clt_wf__q_ind -(\lambda (c0: C).(P c0)) (\lambda (n: nat).(lt_wf_ind n (Q (\lambda (c0: -C).(P c0))) (\lambda (n0: nat).(\lambda (H0: ((\forall (m: nat).((lt m n0) -\to (Q (\lambda (c0: C).(P c0)) m))))).(\lambda (c0: C).(\lambda (H1: (eq nat -(cweight c0) n0)).(let H2 \def (eq_ind_r nat n0 (\lambda (n1: nat).(\forall -(m: nat).((lt m n1) \to (\forall (c1: C).((eq nat (cweight c1) m) \to (P -c1)))))) H0 (cweight c0) H1) in (H c0 (\lambda (d: C).(\lambda (H3: (lt -(cweight d) (cweight c0))).(H2 (cweight d) H3 d (refl_equal nat (cweight -d))))))))))))) c)))). -(* COMMENTS -Initial nodes: 179 -END *) - -theorem chead_ctail: - \forall (c: C).(\forall (t: T).(\forall (k: K).(ex_3 K C T (\lambda (h: -K).(\lambda (d: C).(\lambda (u: T).(eq C (CHead c k t) (CTail h u d)))))))) -\def - \lambda (c: C).(C_ind (\lambda (c0: C).(\forall (t: T).(\forall (k: K).(ex_3 -K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C (CHead c0 k t) -(CTail h u d))))))))) (\lambda (n: nat).(\lambda (t: T).(\lambda (k: -K).(ex_3_intro K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C -(CHead (CSort n) k t) (CTail h u d))))) k (CSort n) t (refl_equal C (CHead -(CSort n) k t)))))) (\lambda (c0: C).(\lambda (H: ((\forall (t: T).(\forall -(k: K).(ex_3 K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C -(CHead c0 k t) (CTail h u d)))))))))).(\lambda (k: K).(\lambda (t: -T).(\lambda (t0: T).(\lambda (k0: K).(let H_x \def (H t k) in (let H0 \def -H_x in (ex_3_ind K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C -(CHead c0 k t) (CTail h u d))))) (ex_3 K C T (\lambda (h: K).(\lambda (d: -C).(\lambda (u: T).(eq C (CHead (CHead c0 k t) k0 t0) (CTail h u d)))))) -(\lambda (x0: K).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H1: (eq C (CHead -c0 k t) (CTail x0 x2 x1))).(eq_ind_r C (CTail x0 x2 x1) (\lambda (c1: -C).(ex_3 K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C (CHead -c1 k0 t0) (CTail h u d))))))) (ex_3_intro K C T (\lambda (h: K).(\lambda (d: -C).(\lambda (u: T).(eq C (CHead (CTail x0 x2 x1) k0 t0) (CTail h u d))))) x0 -(CHead x1 k0 t0) x2 (refl_equal C (CHead (CTail x0 x2 x1) k0 t0))) (CHead c0 -k t) H1))))) H0))))))))) c). -(* COMMENTS -Initial nodes: 395 -END *) - -theorem clt_thead: - \forall (k: K).(\forall (u: T).(\forall (c: C).(clt c (CTail k u c)))) -\def - \lambda (k: K).(\lambda (u: T).(\lambda (c: C).(C_ind (\lambda (c0: C).(clt -c0 (CTail k u c0))) (\lambda (n: nat).(clt_head k (CSort n) u)) (\lambda (c0: -C).(\lambda (H: (clt c0 (CTail k u c0))).(\lambda (k0: K).(\lambda (t: -T).(clt_cong c0 (CTail k u c0) H k0 t))))) c))). -(* COMMENTS -Initial nodes: 71 -END *) - -theorem c_tail_ind: - \forall (P: ((C \to Prop))).(((\forall (n: nat).(P (CSort n)))) \to -(((\forall (c: C).((P c) \to (\forall (k: K).(\forall (t: T).(P (CTail k t -c))))))) \to (\forall (c: C).(P c)))) -\def - \lambda (P: ((C \to Prop))).(\lambda (H: ((\forall (n: nat).(P (CSort -n))))).(\lambda (H0: ((\forall (c: C).((P c) \to (\forall (k: K).(\forall (t: -T).(P (CTail k t c)))))))).(\lambda (c: C).(clt_wf_ind (\lambda (c0: C).(P -c0)) (\lambda (c0: C).(C_ind (\lambda (c1: C).(((\forall (d: C).((clt d c1) -\to (P d)))) \to (P c1))) (\lambda (n: nat).(\lambda (_: ((\forall (d: -C).((clt d (CSort n)) \to (P d))))).(H n))) (\lambda (c1: C).(\lambda (_: -((((\forall (d: C).((clt d c1) \to (P d)))) \to (P c1)))).(\lambda (k: -K).(\lambda (t: T).(\lambda (H2: ((\forall (d: C).((clt d (CHead c1 k t)) \to -(P d))))).(let H_x \def (chead_ctail c1 t k) in (let H3 \def H_x in (ex_3_ind -K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C (CHead c1 k t) -(CTail h u d))))) (P (CHead c1 k t)) (\lambda (x0: K).(\lambda (x1: -C).(\lambda (x2: T).(\lambda (H4: (eq C (CHead c1 k t) (CTail x0 x2 -x1))).(eq_ind_r C (CTail x0 x2 x1) (\lambda (c2: C).(P c2)) (let H5 \def -(eq_ind C (CHead c1 k t) (\lambda (c2: C).(\forall (d: C).((clt d c2) \to (P -d)))) H2 (CTail x0 x2 x1) H4) in (H0 x1 (H5 x1 (clt_thead x0 x2 x1)) x0 x2)) -(CHead c1 k t) H4))))) H3)))))))) c0)) c)))). -(* COMMENTS -Initial nodes: 295 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/G/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/G/defs.ma deleted file mode 100644 index 23edf5789..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/G/defs.ma +++ /dev/null @@ -1,23 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/preamble.ma". - -record G : Set \def { - next: (nat \to nat); - next_lt: (\forall (n: nat).(lt n (next n))) -}. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/T/dec.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/T/dec.ma deleted file mode 100644 index a088c40e3..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/T/dec.ma +++ /dev/null @@ -1,446 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/T/defs.ma". - -theorem terms_props__bind_dec: - \forall (b1: B).(\forall (b2: B).(or (eq B b1 b2) ((eq B b1 b2) \to (\forall -(P: Prop).P)))) -\def - \lambda (b1: B).(B_ind (\lambda (b: B).(\forall (b2: B).(or (eq B b b2) ((eq -B b b2) \to (\forall (P: Prop).P))))) (\lambda (b2: B).(B_ind (\lambda (b: -B).(or (eq B Abbr b) ((eq B Abbr b) \to (\forall (P: Prop).P)))) (or_introl -(eq B Abbr Abbr) ((eq B Abbr Abbr) \to (\forall (P: Prop).P)) (refl_equal B -Abbr)) (or_intror (eq B Abbr Abst) ((eq B Abbr Abst) \to (\forall (P: -Prop).P)) (\lambda (H: (eq B Abbr Abst)).(\lambda (P: Prop).(let H0 \def -(eq_ind B Abbr (\lambda (ee: B).(match ee in B return (\lambda (_: B).Prop) -with [Abbr \Rightarrow True | Abst \Rightarrow False | Void \Rightarrow -False])) I Abst H) in (False_ind P H0))))) (or_intror (eq B Abbr Void) ((eq B -Abbr Void) \to (\forall (P: Prop).P)) (\lambda (H: (eq B Abbr Void)).(\lambda -(P: Prop).(let H0 \def (eq_ind B Abbr (\lambda (ee: B).(match ee in B return -(\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | -Void \Rightarrow False])) I Void H) in (False_ind P H0))))) b2)) (\lambda -(b2: B).(B_ind (\lambda (b: B).(or (eq B Abst b) ((eq B Abst b) \to (\forall -(P: Prop).P)))) (or_intror (eq B Abst Abbr) ((eq B Abst Abbr) \to (\forall -(P: Prop).P)) (\lambda (H: (eq B Abst Abbr)).(\lambda (P: Prop).(let H0 \def -(eq_ind B Abst (\lambda (ee: B).(match ee in B return (\lambda (_: B).Prop) -with [Abbr \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow -False])) I Abbr H) in (False_ind P H0))))) (or_introl (eq B Abst Abst) ((eq B -Abst Abst) \to (\forall (P: Prop).P)) (refl_equal B Abst)) (or_intror (eq B -Abst Void) ((eq B Abst Void) \to (\forall (P: Prop).P)) (\lambda (H: (eq B -Abst Void)).(\lambda (P: Prop).(let H0 \def (eq_ind B Abst (\lambda (ee: -B).(match ee in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | -Abst \Rightarrow True | Void \Rightarrow False])) I Void H) in (False_ind P -H0))))) b2)) (\lambda (b2: B).(B_ind (\lambda (b: B).(or (eq B Void b) ((eq B -Void b) \to (\forall (P: Prop).P)))) (or_intror (eq B Void Abbr) ((eq B Void -Abbr) \to (\forall (P: Prop).P)) (\lambda (H: (eq B Void Abbr)).(\lambda (P: -Prop).(let H0 \def (eq_ind B Void (\lambda (ee: B).(match ee in B return -(\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False | -Void \Rightarrow True])) I Abbr H) in (False_ind P H0))))) (or_intror (eq B -Void Abst) ((eq B Void Abst) \to (\forall (P: Prop).P)) (\lambda (H: (eq B -Void Abst)).(\lambda (P: Prop).(let H0 \def (eq_ind B Void (\lambda (ee: -B).(match ee in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | -Abst \Rightarrow False | Void \Rightarrow True])) I Abst H) in (False_ind P -H0))))) (or_introl (eq B Void Void) ((eq B Void Void) \to (\forall (P: -Prop).P)) (refl_equal B Void)) b2)) b1). -(* COMMENTS -Initial nodes: 559 -END *) - -theorem bind_dec_not: - \forall (b1: B).(\forall (b2: B).(or (eq B b1 b2) (not (eq B b1 b2)))) -\def - \lambda (b1: B).(\lambda (b2: B).(let H_x \def (terms_props__bind_dec b1 b2) -in (let H \def H_x in (or_ind (eq B b1 b2) ((eq B b1 b2) \to (\forall (P: -Prop).P)) (or (eq B b1 b2) ((eq B b1 b2) \to False)) (\lambda (H0: (eq B b1 -b2)).(or_introl (eq B b1 b2) ((eq B b1 b2) \to False) H0)) (\lambda (H0: -(((eq B b1 b2) \to (\forall (P: Prop).P)))).(or_intror (eq B b1 b2) ((eq B b1 -b2) \to False) (\lambda (H1: (eq B b1 b2)).(H0 H1 False)))) H)))). -(* COMMENTS -Initial nodes: 131 -END *) - -theorem terms_props__flat_dec: - \forall (f1: F).(\forall (f2: F).(or (eq F f1 f2) ((eq F f1 f2) \to (\forall -(P: Prop).P)))) -\def - \lambda (f1: F).(F_ind (\lambda (f: F).(\forall (f2: F).(or (eq F f f2) ((eq -F f f2) \to (\forall (P: Prop).P))))) (\lambda (f2: F).(F_ind (\lambda (f: -F).(or (eq F Appl f) ((eq F Appl f) \to (\forall (P: Prop).P)))) (or_introl -(eq F Appl Appl) ((eq F Appl Appl) \to (\forall (P: Prop).P)) (refl_equal F -Appl)) (or_intror (eq F Appl Cast) ((eq F Appl Cast) \to (\forall (P: -Prop).P)) (\lambda (H: (eq F Appl Cast)).(\lambda (P: Prop).(let H0 \def -(eq_ind F Appl (\lambda (ee: F).(match ee in F return (\lambda (_: F).Prop) -with [Appl \Rightarrow True | Cast \Rightarrow False])) I Cast H) in -(False_ind P H0))))) f2)) (\lambda (f2: F).(F_ind (\lambda (f: F).(or (eq F -Cast f) ((eq F Cast f) \to (\forall (P: Prop).P)))) (or_intror (eq F Cast -Appl) ((eq F Cast Appl) \to (\forall (P: Prop).P)) (\lambda (H: (eq F Cast -Appl)).(\lambda (P: Prop).(let H0 \def (eq_ind F Cast (\lambda (ee: F).(match -ee in F return (\lambda (_: F).Prop) with [Appl \Rightarrow False | Cast -\Rightarrow True])) I Appl H) in (False_ind P H0))))) (or_introl (eq F Cast -Cast) ((eq F Cast Cast) \to (\forall (P: Prop).P)) (refl_equal F Cast)) f2)) -f1). -(* COMMENTS -Initial nodes: 263 -END *) - -theorem terms_props__kind_dec: - \forall (k1: K).(\forall (k2: K).(or (eq K k1 k2) ((eq K k1 k2) \to (\forall -(P: Prop).P)))) -\def - \lambda (k1: K).(K_ind (\lambda (k: K).(\forall (k2: K).(or (eq K k k2) ((eq -K k k2) \to (\forall (P: Prop).P))))) (\lambda (b: B).(\lambda (k2: K).(K_ind -(\lambda (k: K).(or (eq K (Bind b) k) ((eq K (Bind b) k) \to (\forall (P: -Prop).P)))) (\lambda (b0: B).(let H_x \def (terms_props__bind_dec b b0) in -(let H \def H_x in (or_ind (eq B b b0) ((eq B b b0) \to (\forall (P: -Prop).P)) (or (eq K (Bind b) (Bind b0)) ((eq K (Bind b) (Bind b0)) \to -(\forall (P: Prop).P))) (\lambda (H0: (eq B b b0)).(eq_ind B b (\lambda (b1: -B).(or (eq K (Bind b) (Bind b1)) ((eq K (Bind b) (Bind b1)) \to (\forall (P: -Prop).P)))) (or_introl (eq K (Bind b) (Bind b)) ((eq K (Bind b) (Bind b)) \to -(\forall (P: Prop).P)) (refl_equal K (Bind b))) b0 H0)) (\lambda (H0: (((eq B -b b0) \to (\forall (P: Prop).P)))).(or_intror (eq K (Bind b) (Bind b0)) ((eq -K (Bind b) (Bind b0)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq K (Bind b) -(Bind b0))).(\lambda (P: Prop).(let H2 \def (f_equal K B (\lambda (e: -K).(match e in K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | -(Flat _) \Rightarrow b])) (Bind b) (Bind b0) H1) in (let H3 \def (eq_ind_r B -b0 (\lambda (b1: B).((eq B b b1) \to (\forall (P0: Prop).P0))) H0 b H2) in -(H3 (refl_equal B b) P))))))) H)))) (\lambda (f: F).(or_intror (eq K (Bind b) -(Flat f)) ((eq K (Bind b) (Flat f)) \to (\forall (P: Prop).P)) (\lambda (H: -(eq K (Bind b) (Flat f))).(\lambda (P: Prop).(let H0 \def (eq_ind K (Bind b) -(\lambda (ee: K).(match ee in K return (\lambda (_: K).Prop) with [(Bind _) -\Rightarrow True | (Flat _) \Rightarrow False])) I (Flat f) H) in (False_ind -P H0)))))) k2))) (\lambda (f: F).(\lambda (k2: K).(K_ind (\lambda (k: K).(or -(eq K (Flat f) k) ((eq K (Flat f) k) \to (\forall (P: Prop).P)))) (\lambda -(b: B).(or_intror (eq K (Flat f) (Bind b)) ((eq K (Flat f) (Bind b)) \to -(\forall (P: Prop).P)) (\lambda (H: (eq K (Flat f) (Bind b))).(\lambda (P: -Prop).(let H0 \def (eq_ind K (Flat f) (\lambda (ee: K).(match ee in K return -(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])) I (Bind b) H) in (False_ind P H0)))))) (\lambda (f0: F).(let H_x \def -(terms_props__flat_dec f f0) in (let H \def H_x in (or_ind (eq F f f0) ((eq F -f f0) \to (\forall (P: Prop).P)) (or (eq K (Flat f) (Flat f0)) ((eq K (Flat -f) (Flat f0)) \to (\forall (P: Prop).P))) (\lambda (H0: (eq F f f0)).(eq_ind -F f (\lambda (f1: F).(or (eq K (Flat f) (Flat f1)) ((eq K (Flat f) (Flat f1)) -\to (\forall (P: Prop).P)))) (or_introl (eq K (Flat f) (Flat f)) ((eq K (Flat -f) (Flat f)) \to (\forall (P: Prop).P)) (refl_equal K (Flat f))) f0 H0)) -(\lambda (H0: (((eq F f f0) \to (\forall (P: Prop).P)))).(or_intror (eq K -(Flat f) (Flat f0)) ((eq K (Flat f) (Flat f0)) \to (\forall (P: Prop).P)) -(\lambda (H1: (eq K (Flat f) (Flat f0))).(\lambda (P: Prop).(let H2 \def -(f_equal K F (\lambda (e: K).(match e in K return (\lambda (_: K).F) with -[(Bind _) \Rightarrow f | (Flat f1) \Rightarrow f1])) (Flat f) (Flat f0) H1) -in (let H3 \def (eq_ind_r F f0 (\lambda (f1: F).((eq F f f1) \to (\forall -(P0: Prop).P0))) H0 f H2) in (H3 (refl_equal F f) P))))))) H)))) k2))) k1). -(* COMMENTS -Initial nodes: 799 -END *) - -theorem term_dec: - \forall (t1: T).(\forall (t2: T).(or (eq T t1 t2) ((eq T t1 t2) \to (\forall -(P: Prop).P)))) -\def - \lambda (t1: T).(T_ind (\lambda (t: T).(\forall (t2: T).(or (eq T t t2) ((eq -T t t2) \to (\forall (P: Prop).P))))) (\lambda (n: nat).(\lambda (t2: -T).(T_ind (\lambda (t: T).(or (eq T (TSort n) t) ((eq T (TSort n) t) \to -(\forall (P: Prop).P)))) (\lambda (n0: nat).(let H_x \def (nat_dec n n0) in -(let H \def H_x in (or_ind (eq nat n n0) ((eq nat n n0) \to (\forall (P: -Prop).P)) (or (eq T (TSort n) (TSort n0)) ((eq T (TSort n) (TSort n0)) \to -(\forall (P: Prop).P))) (\lambda (H0: (eq nat n n0)).(eq_ind nat n (\lambda -(n1: nat).(or (eq T (TSort n) (TSort n1)) ((eq T (TSort n) (TSort n1)) \to -(\forall (P: Prop).P)))) (or_introl (eq T (TSort n) (TSort n)) ((eq T (TSort -n) (TSort n)) \to (\forall (P: Prop).P)) (refl_equal T (TSort n))) n0 H0)) -(\lambda (H0: (((eq nat n n0) \to (\forall (P: Prop).P)))).(or_intror (eq T -(TSort n) (TSort n0)) ((eq T (TSort n) (TSort n0)) \to (\forall (P: Prop).P)) -(\lambda (H1: (eq T (TSort n) (TSort n0))).(\lambda (P: Prop).(let H2 \def -(f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with -[(TSort n1) \Rightarrow n1 | (TLRef _) \Rightarrow n | (THead _ _ _) -\Rightarrow n])) (TSort n) (TSort n0) H1) in (let H3 \def (eq_ind_r nat n0 -(\lambda (n1: nat).((eq nat n n1) \to (\forall (P0: Prop).P0))) H0 n H2) in -(H3 (refl_equal nat n) P))))))) H)))) (\lambda (n0: nat).(or_intror (eq T -(TSort n) (TLRef n0)) ((eq T (TSort n) (TLRef n0)) \to (\forall (P: Prop).P)) -(\lambda (H: (eq T (TSort n) (TLRef n0))).(\lambda (P: Prop).(let H0 \def -(eq_ind T (TSort n) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | -(THead _ _ _) \Rightarrow False])) I (TLRef n0) H) in (False_ind P H0)))))) -(\lambda (k: K).(\lambda (t: T).(\lambda (_: (or (eq T (TSort n) t) ((eq T -(TSort n) t) \to (\forall (P: Prop).P)))).(\lambda (t0: T).(\lambda (_: (or -(eq T (TSort n) t0) ((eq T (TSort n) t0) \to (\forall (P: -Prop).P)))).(or_intror (eq T (TSort n) (THead k t t0)) ((eq T (TSort n) -(THead k t t0)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq T (TSort n) -(THead k t t0))).(\lambda (P: Prop).(let H2 \def (eq_ind T (TSort n) (\lambda -(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow -False])) I (THead k t t0) H1) in (False_ind P H2)))))))))) t2))) (\lambda (n: -nat).(\lambda (t2: T).(T_ind (\lambda (t: T).(or (eq T (TLRef n) t) ((eq T -(TLRef n) t) \to (\forall (P: Prop).P)))) (\lambda (n0: nat).(or_intror (eq T -(TLRef n) (TSort n0)) ((eq T (TLRef n) (TSort n0)) \to (\forall (P: Prop).P)) -(\lambda (H: (eq T (TLRef n) (TSort n0))).(\lambda (P: Prop).(let H0 \def -(eq_ind T (TLRef n) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | -(THead _ _ _) \Rightarrow False])) I (TSort n0) H) in (False_ind P H0)))))) -(\lambda (n0: nat).(let H_x \def (nat_dec n n0) in (let H \def H_x in (or_ind -(eq nat n n0) ((eq nat n n0) \to (\forall (P: Prop).P)) (or (eq T (TLRef n) -(TLRef n0)) ((eq T (TLRef n) (TLRef n0)) \to (\forall (P: Prop).P))) (\lambda -(H0: (eq nat n n0)).(eq_ind nat n (\lambda (n1: nat).(or (eq T (TLRef n) -(TLRef n1)) ((eq T (TLRef n) (TLRef n1)) \to (\forall (P: Prop).P)))) -(or_introl (eq T (TLRef n) (TLRef n)) ((eq T (TLRef n) (TLRef n)) \to -(\forall (P: Prop).P)) (refl_equal T (TLRef n))) n0 H0)) (\lambda (H0: (((eq -nat n n0) \to (\forall (P: Prop).P)))).(or_intror (eq T (TLRef n) (TLRef n0)) -((eq T (TLRef n) (TLRef n0)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq T -(TLRef n) (TLRef n0))).(\lambda (P: Prop).(let H2 \def (f_equal T nat -(\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort _) -\Rightarrow n | (TLRef n1) \Rightarrow n1 | (THead _ _ _) \Rightarrow n])) -(TLRef n) (TLRef n0) H1) in (let H3 \def (eq_ind_r nat n0 (\lambda (n1: -nat).((eq nat n n1) \to (\forall (P0: Prop).P0))) H0 n H2) in (H3 (refl_equal -nat n) P))))))) H)))) (\lambda (k: K).(\lambda (t: T).(\lambda (_: (or (eq T -(TLRef n) t) ((eq T (TLRef n) t) \to (\forall (P: Prop).P)))).(\lambda (t0: -T).(\lambda (_: (or (eq T (TLRef n) t0) ((eq T (TLRef n) t0) \to (\forall (P: -Prop).P)))).(or_intror (eq T (TLRef n) (THead k t t0)) ((eq T (TLRef n) -(THead k t t0)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq T (TLRef n) -(THead k t t0))).(\lambda (P: Prop).(let H2 \def (eq_ind T (TLRef n) (\lambda -(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow -False])) I (THead k t t0) H1) in (False_ind P H2)))))))))) t2))) (\lambda (k: -K).(\lambda (t: T).(\lambda (H: ((\forall (t2: T).(or (eq T t t2) ((eq T t -t2) \to (\forall (P: Prop).P)))))).(\lambda (t0: T).(\lambda (H0: ((\forall -(t2: T).(or (eq T t0 t2) ((eq T t0 t2) \to (\forall (P: -Prop).P)))))).(\lambda (t2: T).(T_ind (\lambda (t3: T).(or (eq T (THead k t -t0) t3) ((eq T (THead k t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (n: -nat).(or_intror (eq T (THead k t t0) (TSort n)) ((eq T (THead k t t0) (TSort -n)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq T (THead k t t0) (TSort -n))).(\lambda (P: Prop).(let H2 \def (eq_ind T (THead k t t0) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I -(TSort n) H1) in (False_ind P H2)))))) (\lambda (n: nat).(or_intror (eq T -(THead k t t0) (TLRef n)) ((eq T (THead k t t0) (TLRef n)) \to (\forall (P: -Prop).P)) (\lambda (H1: (eq T (THead k t t0) (TLRef n))).(\lambda (P: -Prop).(let H2 \def (eq_ind T (THead k t t0) (\lambda (ee: T).(match ee in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H1) in -(False_ind P H2)))))) (\lambda (k0: K).(\lambda (t3: T).(\lambda (H1: (or (eq -T (THead k t t0) t3) ((eq T (THead k t t0) t3) \to (\forall (P: -Prop).P)))).(\lambda (t4: T).(\lambda (H2: (or (eq T (THead k t t0) t4) ((eq -T (THead k t t0) t4) \to (\forall (P: Prop).P)))).(let H_x \def (H t3) in -(let H3 \def H_x in (or_ind (eq T t t3) ((eq T t t3) \to (\forall (P: -Prop).P)) (or (eq T (THead k t t0) (THead k0 t3 t4)) ((eq T (THead k t t0) -(THead k0 t3 t4)) \to (\forall (P: Prop).P))) (\lambda (H4: (eq T t t3)).(let -H5 \def (eq_ind_r T t3 (\lambda (t5: T).(or (eq T (THead k t t0) t5) ((eq T -(THead k t t0) t5) \to (\forall (P: Prop).P)))) H1 t H4) in (eq_ind T t -(\lambda (t5: T).(or (eq T (THead k t t0) (THead k0 t5 t4)) ((eq T (THead k t -t0) (THead k0 t5 t4)) \to (\forall (P: Prop).P)))) (let H_x0 \def (H0 t4) in -(let H6 \def H_x0 in (or_ind (eq T t0 t4) ((eq T t0 t4) \to (\forall (P: -Prop).P)) (or (eq T (THead k t t0) (THead k0 t t4)) ((eq T (THead k t t0) -(THead k0 t t4)) \to (\forall (P: Prop).P))) (\lambda (H7: (eq T t0 t4)).(let -H8 \def (eq_ind_r T t4 (\lambda (t5: T).(or (eq T (THead k t t0) t5) ((eq T -(THead k t t0) t5) \to (\forall (P: Prop).P)))) H2 t0 H7) in (eq_ind T t0 -(\lambda (t5: T).(or (eq T (THead k t t0) (THead k0 t t5)) ((eq T (THead k t -t0) (THead k0 t t5)) \to (\forall (P: Prop).P)))) (let H_x1 \def -(terms_props__kind_dec k k0) in (let H9 \def H_x1 in (or_ind (eq K k k0) ((eq -K k k0) \to (\forall (P: Prop).P)) (or (eq T (THead k t t0) (THead k0 t t0)) -((eq T (THead k t t0) (THead k0 t t0)) \to (\forall (P: Prop).P))) (\lambda -(H10: (eq K k k0)).(eq_ind K k (\lambda (k1: K).(or (eq T (THead k t t0) -(THead k1 t t0)) ((eq T (THead k t t0) (THead k1 t t0)) \to (\forall (P: -Prop).P)))) (or_introl (eq T (THead k t t0) (THead k t t0)) ((eq T (THead k t -t0) (THead k t t0)) \to (\forall (P: Prop).P)) (refl_equal T (THead k t t0))) -k0 H10)) (\lambda (H10: (((eq K k k0) \to (\forall (P: Prop).P)))).(or_intror -(eq T (THead k t t0) (THead k0 t t0)) ((eq T (THead k t t0) (THead k0 t t0)) -\to (\forall (P: Prop).P)) (\lambda (H11: (eq T (THead k t t0) (THead k0 t -t0))).(\lambda (P: Prop).(let H12 \def (f_equal T K (\lambda (e: T).(match e -in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) -\Rightarrow k | (THead k1 _ _) \Rightarrow k1])) (THead k t t0) (THead k0 t -t0) H11) in (let H13 \def (eq_ind_r K k0 (\lambda (k1: K).((eq K k k1) \to -(\forall (P0: Prop).P0))) H10 k H12) in (H13 (refl_equal K k) P))))))) H9))) -t4 H7))) (\lambda (H7: (((eq T t0 t4) \to (\forall (P: Prop).P)))).(or_intror -(eq T (THead k t t0) (THead k0 t t4)) ((eq T (THead k t t0) (THead k0 t t4)) -\to (\forall (P: Prop).P)) (\lambda (H8: (eq T (THead k t t0) (THead k0 t -t4))).(\lambda (P: Prop).(let H9 \def (f_equal T K (\lambda (e: T).(match e -in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) -\Rightarrow k | (THead k1 _ _) \Rightarrow k1])) (THead k t t0) (THead k0 t -t4) H8) in ((let H10 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 -| (THead _ _ t5) \Rightarrow t5])) (THead k t t0) (THead k0 t t4) H8) in -(\lambda (_: (eq K k k0)).(let H12 \def (eq_ind_r T t4 (\lambda (t5: T).((eq -T t0 t5) \to (\forall (P0: Prop).P0))) H7 t0 H10) in (let H13 \def (eq_ind_r -T t4 (\lambda (t5: T).(or (eq T (THead k t t0) t5) ((eq T (THead k t t0) t5) -\to (\forall (P0: Prop).P0)))) H2 t0 H10) in (H12 (refl_equal T t0) P))))) -H9)))))) H6))) t3 H4))) (\lambda (H4: (((eq T t t3) \to (\forall (P: -Prop).P)))).(or_intror (eq T (THead k t t0) (THead k0 t3 t4)) ((eq T (THead k -t t0) (THead k0 t3 t4)) \to (\forall (P: Prop).P)) (\lambda (H5: (eq T (THead -k t t0) (THead k0 t3 t4))).(\lambda (P: Prop).(let H6 \def (f_equal T K -(\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) -\Rightarrow k | (TLRef _) \Rightarrow k | (THead k1 _ _) \Rightarrow k1])) -(THead k t t0) (THead k0 t3 t4) H5) in ((let H7 \def (f_equal T T (\lambda -(e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t -| (TLRef _) \Rightarrow t | (THead _ t5 _) \Rightarrow t5])) (THead k t t0) -(THead k0 t3 t4) H5) in ((let H8 \def (f_equal T T (\lambda (e: T).(match e -in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) -\Rightarrow t0 | (THead _ _ t5) \Rightarrow t5])) (THead k t t0) (THead k0 t3 -t4) H5) in (\lambda (H9: (eq T t t3)).(\lambda (_: (eq K k k0)).(let H11 \def -(eq_ind_r T t4 (\lambda (t5: T).(or (eq T (THead k t t0) t5) ((eq T (THead k -t t0) t5) \to (\forall (P0: Prop).P0)))) H2 t0 H8) in (let H12 \def (eq_ind_r -T t3 (\lambda (t5: T).((eq T t t5) \to (\forall (P0: Prop).P0))) H4 t H9) in -(let H13 \def (eq_ind_r T t3 (\lambda (t5: T).(or (eq T (THead k t t0) t5) -((eq T (THead k t t0) t5) \to (\forall (P0: Prop).P0)))) H1 t H9) in (H12 -(refl_equal T t) P))))))) H7)) H6)))))) H3)))))))) t2))))))) t1). -(* COMMENTS -Initial nodes: 2821 -END *) - -theorem binder_dec: - \forall (t: T).(or (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: -T).(eq T t (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall -(u: T).((eq T t (THead (Bind b) w u)) \to (\forall (P: Prop).P)))))) -\def - \lambda (t: T).(T_ind (\lambda (t0: T).(or (ex_3 B T T (\lambda (b: -B).(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w u)))))) -(\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t0 (THead (Bind b) w -u)) \to (\forall (P: Prop).P))))))) (\lambda (n: nat).(or_intror (ex_3 B T T -(\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T (TSort n) (THead (Bind -b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T (TSort n) -(THead (Bind b) w u)) \to (\forall (P: Prop).P))))) (\lambda (b: B).(\lambda -(w: T).(\lambda (u: T).(\lambda (H: (eq T (TSort n) (THead (Bind b) w -u))).(\lambda (P: Prop).(let H0 \def (eq_ind T (TSort n) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I -(THead (Bind b) w u) H) in (False_ind P H0))))))))) (\lambda (n: -nat).(or_intror (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: -T).(eq T (TLRef n) (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: -T).(\forall (u: T).((eq T (TLRef n) (THead (Bind b) w u)) \to (\forall (P: -Prop).P))))) (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(\lambda (H: (eq -T (TLRef n) (THead (Bind b) w u))).(\lambda (P: Prop).(let H0 \def (eq_ind T -(TLRef n) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) -\Rightarrow False])) I (THead (Bind b) w u) H) in (False_ind P H0))))))))) -(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t0: T).((or (ex_3 B T T -(\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w -u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t0 (THead -(Bind b) w u)) \to (\forall (P: Prop).P)))))) \to (\forall (t1: T).((or (ex_3 -B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t1 (THead (Bind -b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t1 (THead -(Bind b) w u)) \to (\forall (P: Prop).P)))))) \to (or (ex_3 B T T (\lambda -(b: B).(\lambda (w: T).(\lambda (u: T).(eq T (THead k0 t0 t1) (THead (Bind b) -w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T (THead k0 t0 -t1) (THead (Bind b) w u)) \to (\forall (P: Prop).P))))))))))) (\lambda (b: -B).(\lambda (t0: T).(\lambda (_: (or (ex_3 B T T (\lambda (b0: B).(\lambda -(w: T).(\lambda (u: T).(eq T t0 (THead (Bind b0) w u)))))) (\forall (b0: -B).(\forall (w: T).(\forall (u: T).((eq T t0 (THead (Bind b0) w u)) \to -(\forall (P: Prop).P))))))).(\lambda (t1: T).(\lambda (_: (or (ex_3 B T T -(\lambda (b0: B).(\lambda (w: T).(\lambda (u: T).(eq T t1 (THead (Bind b0) w -u)))))) (\forall (b0: B).(\forall (w: T).(\forall (u: T).((eq T t1 (THead -(Bind b0) w u)) \to (\forall (P: Prop).P))))))).(or_introl (ex_3 B T T -(\lambda (b0: B).(\lambda (w: T).(\lambda (u: T).(eq T (THead (Bind b) t0 t1) -(THead (Bind b0) w u)))))) (\forall (b0: B).(\forall (w: T).(\forall (u: -T).((eq T (THead (Bind b) t0 t1) (THead (Bind b0) w u)) \to (\forall (P: -Prop).P))))) (ex_3_intro B T T (\lambda (b0: B).(\lambda (w: T).(\lambda (u: -T).(eq T (THead (Bind b) t0 t1) (THead (Bind b0) w u))))) b t0 t1 (refl_equal -T (THead (Bind b) t0 t1))))))))) (\lambda (f: F).(\lambda (t0: T).(\lambda -(_: (or (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t0 -(THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: -T).((eq T t0 (THead (Bind b) w u)) \to (\forall (P: Prop).P))))))).(\lambda -(t1: T).(\lambda (_: (or (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda -(u: T).(eq T t1 (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: -T).(\forall (u: T).((eq T t1 (THead (Bind b) w u)) \to (\forall (P: -Prop).P))))))).(or_intror (ex_3 B T T (\lambda (b: B).(\lambda (w: -T).(\lambda (u: T).(eq T (THead (Flat f) t0 t1) (THead (Bind b) w u)))))) -(\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T (THead (Flat f) t0 t1) -(THead (Bind b) w u)) \to (\forall (P: Prop).P))))) (\lambda (b: B).(\lambda -(w: T).(\lambda (u: T).(\lambda (H1: (eq T (THead (Flat f) t0 t1) (THead -(Bind b) w u))).(\lambda (P: Prop).(let H2 \def (eq_ind T (THead (Flat f) t0 -t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort -_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _) -\Rightarrow (match k0 in K return (\lambda (_: K).Prop) with [(Bind _) -\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) w u) H1) -in (False_ind P H2))))))))))))) k)) t). -(* COMMENTS -Initial nodes: 1063 -END *) - -theorem abst_dec: - \forall (u: T).(\forall (v: T).(or (ex T (\lambda (t: T).(eq T u (THead -(Bind Abst) v t)))) (\forall (t: T).((eq T u (THead (Bind Abst) v t)) \to -(\forall (P: Prop).P))))) -\def - \lambda (u: T).(T_ind (\lambda (t: T).(\forall (v: T).(or (ex T (\lambda -(t0: T).(eq T t (THead (Bind Abst) v t0)))) (\forall (t0: T).((eq T t (THead -(Bind Abst) v t0)) \to (\forall (P: Prop).P)))))) (\lambda (n: nat).(\lambda -(v: T).(or_intror (ex T (\lambda (t: T).(eq T (TSort n) (THead (Bind Abst) v -t)))) (\forall (t: T).((eq T (TSort n) (THead (Bind Abst) v t)) \to (\forall -(P: Prop).P))) (\lambda (t: T).(\lambda (H: (eq T (TSort n) (THead (Bind -Abst) v t))).(\lambda (P: Prop).(let H0 \def (eq_ind T (TSort n) (\lambda -(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow -False])) I (THead (Bind Abst) v t) H) in (False_ind P H0)))))))) (\lambda (n: -nat).(\lambda (v: T).(or_intror (ex T (\lambda (t: T).(eq T (TLRef n) (THead -(Bind Abst) v t)))) (\forall (t: T).((eq T (TLRef n) (THead (Bind Abst) v t)) -\to (\forall (P: Prop).P))) (\lambda (t: T).(\lambda (H: (eq T (TLRef n) -(THead (Bind Abst) v t))).(\lambda (P: Prop).(let H0 \def (eq_ind T (TLRef n) -(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow -False])) I (THead (Bind Abst) v t) H) in (False_ind P H0)))))))) (\lambda (k: -K).(\lambda (t: T).(\lambda (_: ((\forall (v: T).(or (ex T (\lambda (t0: -T).(eq T t (THead (Bind Abst) v t0)))) (\forall (t0: T).((eq T t (THead (Bind -Abst) v t0)) \to (\forall (P: Prop).P))))))).(\lambda (t0: T).(\lambda (_: -((\forall (v: T).(or (ex T (\lambda (t1: T).(eq T t0 (THead (Bind Abst) v -t1)))) (\forall (t1: T).((eq T t0 (THead (Bind Abst) v t1)) \to (\forall (P: -Prop).P))))))).(\lambda (v: T).(let H_x \def (terms_props__kind_dec k (Bind -Abst)) in (let H1 \def H_x in (or_ind (eq K k (Bind Abst)) ((eq K k (Bind -Abst)) \to (\forall (P: Prop).P)) (or (ex T (\lambda (t1: T).(eq T (THead k t -t0) (THead (Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead k t t0) (THead -(Bind Abst) v t1)) \to (\forall (P: Prop).P)))) (\lambda (H2: (eq K k (Bind -Abst))).(eq_ind_r K (Bind Abst) (\lambda (k0: K).(or (ex T (\lambda (t1: -T).(eq T (THead k0 t t0) (THead (Bind Abst) v t1)))) (\forall (t1: T).((eq T -(THead k0 t t0) (THead (Bind Abst) v t1)) \to (\forall (P: Prop).P))))) (let -H_x0 \def (term_dec t v) in (let H3 \def H_x0 in (or_ind (eq T t v) ((eq T t -v) \to (\forall (P: Prop).P)) (or (ex T (\lambda (t1: T).(eq T (THead (Bind -Abst) t t0) (THead (Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead (Bind -Abst) t t0) (THead (Bind Abst) v t1)) \to (\forall (P: Prop).P)))) (\lambda -(H4: (eq T t v)).(eq_ind T t (\lambda (t1: T).(or (ex T (\lambda (t2: T).(eq -T (THead (Bind Abst) t t0) (THead (Bind Abst) t1 t2)))) (\forall (t2: T).((eq -T (THead (Bind Abst) t t0) (THead (Bind Abst) t1 t2)) \to (\forall (P: -Prop).P))))) (or_introl (ex T (\lambda (t1: T).(eq T (THead (Bind Abst) t t0) -(THead (Bind Abst) t t1)))) (\forall (t1: T).((eq T (THead (Bind Abst) t t0) -(THead (Bind Abst) t t1)) \to (\forall (P: Prop).P))) (ex_intro T (\lambda -(t1: T).(eq T (THead (Bind Abst) t t0) (THead (Bind Abst) t t1))) t0 -(refl_equal T (THead (Bind Abst) t t0)))) v H4)) (\lambda (H4: (((eq T t v) -\to (\forall (P: Prop).P)))).(or_intror (ex T (\lambda (t1: T).(eq T (THead -(Bind Abst) t t0) (THead (Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead -(Bind Abst) t t0) (THead (Bind Abst) v t1)) \to (\forall (P: Prop).P))) -(\lambda (t1: T).(\lambda (H5: (eq T (THead (Bind Abst) t t0) (THead (Bind -Abst) v t1))).(\lambda (P: Prop).(let H6 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t | -(TLRef _) \Rightarrow t | (THead _ t2 _) \Rightarrow t2])) (THead (Bind Abst) -t t0) (THead (Bind Abst) v t1) H5) in ((let H7 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | -(TLRef _) \Rightarrow t0 | (THead _ _ t2) \Rightarrow t2])) (THead (Bind -Abst) t t0) (THead (Bind Abst) v t1) H5) in (\lambda (H8: (eq T t v)).(H4 H8 -P))) H6))))))) H3))) k H2)) (\lambda (H2: (((eq K k (Bind Abst)) \to (\forall -(P: Prop).P)))).(or_intror (ex T (\lambda (t1: T).(eq T (THead k t t0) (THead -(Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead k t t0) (THead (Bind -Abst) v t1)) \to (\forall (P: Prop).P))) (\lambda (t1: T).(\lambda (H3: (eq T -(THead k t t0) (THead (Bind Abst) v t1))).(\lambda (P: Prop).(let H4 \def -(f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with -[(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) -\Rightarrow k0])) (THead k t t0) (THead (Bind Abst) v t1) H3) in ((let H5 -\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ t2 _) -\Rightarrow t2])) (THead k t t0) (THead (Bind Abst) v t1) H3) in ((let H6 -\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t2) -\Rightarrow t2])) (THead k t t0) (THead (Bind Abst) v t1) H3) in (\lambda (_: -(eq T t v)).(\lambda (H8: (eq K k (Bind Abst))).(H2 H8 P)))) H5)) H4))))))) -H1))))))))) u). -(* COMMENTS -Initial nodes: 1305 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/T/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/T/defs.ma deleted file mode 100644 index 6ddbe6d0d..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/T/defs.ma +++ /dev/null @@ -1,43 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/preamble.ma". - -inductive B: Set \def -| Abbr: B -| Abst: B -| Void: B. - -inductive F: Set \def -| Appl: F -| Cast: F. - -inductive K: Set \def -| Bind: B \to K -| Flat: F \to K. - -inductive T: Set \def -| TSort: nat \to T -| TLRef: nat \to T -| THead: K \to (T \to (T \to T)). - -definition tweight: - T \to nat -\def - let rec tweight (t: T) on t: nat \def (match t with [(TSort _) \Rightarrow -(S O) | (TLRef _) \Rightarrow (S O) | (THead _ u t0) \Rightarrow (S (plus -(tweight u) (tweight t0)))]) in tweight. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/T/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/T/props.ma deleted file mode 100644 index faa9ed95d..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/T/props.ma +++ /dev/null @@ -1,111 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/T/defs.ma". - -theorem not_abbr_abst: - not (eq B Abbr Abst) -\def - \lambda (H: (eq B Abbr Abst)).(let H0 \def (eq_ind B Abbr (\lambda (ee: -B).(match ee in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow True | -Abst \Rightarrow False | Void \Rightarrow False])) I Abst H) in (False_ind -False H0)). -(* COMMENTS -Initial nodes: 34 -END *) - -theorem not_void_abst: - not (eq B Void Abst) -\def - \lambda (H: (eq B Void Abst)).(let H0 \def (eq_ind B Void (\lambda (ee: -B).(match ee in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | -Abst \Rightarrow False | Void \Rightarrow True])) I Abst H) in (False_ind -False H0)). -(* COMMENTS -Initial nodes: 34 -END *) - -theorem not_abbr_void: - not (eq B Abbr Void) -\def - \lambda (H: (eq B Abbr Void)).(let H0 \def (eq_ind B Abbr (\lambda (ee: -B).(match ee in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow True | -Abst \Rightarrow False | Void \Rightarrow False])) I Void H) in (False_ind -False H0)). -(* COMMENTS -Initial nodes: 34 -END *) - -theorem not_abst_void: - not (eq B Abst Void) -\def - \lambda (H: (eq B Abst Void)).(let H0 \def (eq_ind B Abst (\lambda (ee: -B).(match ee in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | -Abst \Rightarrow True | Void \Rightarrow False])) I Void H) in (False_ind -False H0)). -(* COMMENTS -Initial nodes: 34 -END *) - -theorem thead_x_y_y: - \forall (k: K).(\forall (v: T).(\forall (t: T).((eq T (THead k v t) t) \to -(\forall (P: Prop).P)))) -\def - \lambda (k: K).(\lambda (v: T).(\lambda (t: T).(T_ind (\lambda (t0: T).((eq -T (THead k v t0) t0) \to (\forall (P: Prop).P))) (\lambda (n: nat).(\lambda -(H: (eq T (THead k v (TSort n)) (TSort n))).(\lambda (P: Prop).(let H0 \def -(eq_ind T (THead k v (TSort n)) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H) in -(False_ind P H0))))) (\lambda (n: nat).(\lambda (H: (eq T (THead k v (TLRef -n)) (TLRef n))).(\lambda (P: Prop).(let H0 \def (eq_ind T (THead k v (TLRef -n)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort -_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) -\Rightarrow True])) I (TLRef n) H) in (False_ind P H0))))) (\lambda (k0: -K).(\lambda (t0: T).(\lambda (_: (((eq T (THead k v t0) t0) \to (\forall (P: -Prop).P)))).(\lambda (t1: T).(\lambda (H0: (((eq T (THead k v t1) t1) \to -(\forall (P: Prop).P)))).(\lambda (H1: (eq T (THead k v (THead k0 t0 t1)) -(THead k0 t0 t1))).(\lambda (P: Prop).(let H2 \def (f_equal T K (\lambda (e: -T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | -(TLRef _) \Rightarrow k | (THead k1 _ _) \Rightarrow k1])) (THead k v (THead -k0 t0 t1)) (THead k0 t0 t1) H1) in ((let H3 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v | -(TLRef _) \Rightarrow v | (THead _ t2 _) \Rightarrow t2])) (THead k v (THead -k0 t0 t1)) (THead k0 t0 t1) H1) in ((let H4 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow (THead -k0 t0 t1) | (TLRef _) \Rightarrow (THead k0 t0 t1) | (THead _ _ t2) -\Rightarrow t2])) (THead k v (THead k0 t0 t1)) (THead k0 t0 t1) H1) in -(\lambda (H5: (eq T v t0)).(\lambda (H6: (eq K k k0)).(let H7 \def (eq_ind T -v (\lambda (t2: T).((eq T (THead k t2 t1) t1) \to (\forall (P0: Prop).P0))) -H0 t0 H5) in (let H8 \def (eq_ind K k (\lambda (k1: K).((eq T (THead k1 t0 -t1) t1) \to (\forall (P0: Prop).P0))) H7 k0 H6) in (H8 H4 P)))))) H3)) -H2))))))))) t))). -(* COMMENTS -Initial nodes: 461 -END *) - -theorem tweight_lt: - \forall (t: T).(lt O (tweight t)) -\def - \lambda (t: T).(T_ind (\lambda (t0: T).(lt O (tweight t0))) (\lambda (_: -nat).(le_n (S O))) (\lambda (_: nat).(le_n (S O))) (\lambda (_: K).(\lambda -(t0: T).(\lambda (H: (lt O (tweight t0))).(\lambda (t1: T).(\lambda (_: (lt O -(tweight t1))).(le_S (S O) (plus (tweight t0) (tweight t1)) (le_plus_trans (S -O) (tweight t0) (tweight t1) H))))))) t). -(* COMMENTS -Initial nodes: 85 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/aplus/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/aplus/defs.ma deleted file mode 100644 index 4095b163b..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/aplus/defs.ma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/asucc/defs.ma". - -definition aplus: - G \to (A \to (nat \to A)) -\def - let rec aplus (g: G) (a: A) (n: nat) on n: A \def (match n with [O -\Rightarrow a | (S n0) \Rightarrow (asucc g (aplus g a n0))]) in aplus. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/aplus/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/aplus/props.ma deleted file mode 100644 index 94bb9a069..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/aplus/props.ma +++ /dev/null @@ -1,282 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/aplus/defs.ma". - -include "Basic-1/next_plus/props.ma". - -theorem aplus_reg_r: - \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (h1: nat).(\forall -(h2: nat).((eq A (aplus g a1 h1) (aplus g a2 h2)) \to (\forall (h: nat).(eq A -(aplus g a1 (plus h h1)) (aplus g a2 (plus h h2))))))))) -\def - \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (h1: nat).(\lambda -(h2: nat).(\lambda (H: (eq A (aplus g a1 h1) (aplus g a2 h2))).(\lambda (h: -nat).(nat_ind (\lambda (n: nat).(eq A (aplus g a1 (plus n h1)) (aplus g a2 -(plus n h2)))) H (\lambda (n: nat).(\lambda (H0: (eq A (aplus g a1 (plus n -h1)) (aplus g a2 (plus n h2)))).(f_equal2 G A A asucc g g (aplus g a1 (plus n -h1)) (aplus g a2 (plus n h2)) (refl_equal G g) H0))) h))))))). -(* COMMENTS -Initial nodes: 143 -END *) - -theorem aplus_assoc: - \forall (g: G).(\forall (a: A).(\forall (h1: nat).(\forall (h2: nat).(eq A -(aplus g (aplus g a h1) h2) (aplus g a (plus h1 h2)))))) -\def - \lambda (g: G).(\lambda (a: A).(\lambda (h1: nat).(nat_ind (\lambda (n: -nat).(\forall (h2: nat).(eq A (aplus g (aplus g a n) h2) (aplus g a (plus n -h2))))) (\lambda (h2: nat).(refl_equal A (aplus g a h2))) (\lambda (n: -nat).(\lambda (_: ((\forall (h2: nat).(eq A (aplus g (aplus g a n) h2) (aplus -g a (plus n h2)))))).(\lambda (h2: nat).(nat_ind (\lambda (n0: nat).(eq A -(aplus g (asucc g (aplus g a n)) n0) (asucc g (aplus g a (plus n n0))))) -(eq_ind nat n (\lambda (n0: nat).(eq A (asucc g (aplus g a n)) (asucc g -(aplus g a n0)))) (refl_equal A (asucc g (aplus g a n))) (plus n O) (plus_n_O -n)) (\lambda (n0: nat).(\lambda (H0: (eq A (aplus g (asucc g (aplus g a n)) -n0) (asucc g (aplus g a (plus n n0))))).(eq_ind nat (S (plus n n0)) (\lambda -(n1: nat).(eq A (asucc g (aplus g (asucc g (aplus g a n)) n0)) (asucc g -(aplus g a n1)))) (f_equal2 G A A asucc g g (aplus g (asucc g (aplus g a n)) -n0) (asucc g (aplus g a (plus n n0))) (refl_equal G g) H0) (plus n (S n0)) -(plus_n_Sm n n0)))) h2)))) h1))). -(* COMMENTS -Initial nodes: 361 -END *) - -theorem aplus_asucc: - \forall (g: G).(\forall (h: nat).(\forall (a: A).(eq A (aplus g (asucc g a) -h) (asucc g (aplus g a h))))) -\def - \lambda (g: G).(\lambda (h: nat).(\lambda (a: A).(eq_ind_r A (aplus g a -(plus (S O) h)) (\lambda (a0: A).(eq A a0 (asucc g (aplus g a h)))) -(refl_equal A (asucc g (aplus g a h))) (aplus g (aplus g a (S O)) h) -(aplus_assoc g a (S O) h)))). -(* COMMENTS -Initial nodes: 87 -END *) - -theorem aplus_sort_O_S_simpl: - \forall (g: G).(\forall (n: nat).(\forall (k: nat).(eq A (aplus g (ASort O -n) (S k)) (aplus g (ASort O (next g n)) k)))) -\def - \lambda (g: G).(\lambda (n: nat).(\lambda (k: nat).(eq_ind A (aplus g (asucc -g (ASort O n)) k) (\lambda (a: A).(eq A a (aplus g (ASort O (next g n)) k))) -(refl_equal A (aplus g (ASort O (next g n)) k)) (asucc g (aplus g (ASort O n) -k)) (aplus_asucc g k (ASort O n))))). -(* COMMENTS -Initial nodes: 97 -END *) - -theorem aplus_sort_S_S_simpl: - \forall (g: G).(\forall (n: nat).(\forall (h: nat).(\forall (k: nat).(eq A -(aplus g (ASort (S h) n) (S k)) (aplus g (ASort h n) k))))) -\def - \lambda (g: G).(\lambda (n: nat).(\lambda (h: nat).(\lambda (k: nat).(eq_ind -A (aplus g (asucc g (ASort (S h) n)) k) (\lambda (a: A).(eq A a (aplus g -(ASort h n) k))) (refl_equal A (aplus g (ASort h n) k)) (asucc g (aplus g -(ASort (S h) n) k)) (aplus_asucc g k (ASort (S h) n)))))). -(* COMMENTS -Initial nodes: 97 -END *) - -theorem aplus_asort_O_simpl: - \forall (g: G).(\forall (h: nat).(\forall (n: nat).(eq A (aplus g (ASort O -n) h) (ASort O (next_plus g n h))))) -\def - \lambda (g: G).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall (n0: -nat).(eq A (aplus g (ASort O n0) n) (ASort O (next_plus g n0 n))))) (\lambda -(n: nat).(refl_equal A (ASort O n))) (\lambda (n: nat).(\lambda (H: ((\forall -(n0: nat).(eq A (aplus g (ASort O n0) n) (ASort O (next_plus g n0 -n)))))).(\lambda (n0: nat).(eq_ind A (aplus g (asucc g (ASort O n0)) n) -(\lambda (a: A).(eq A a (ASort O (next g (next_plus g n0 n))))) (eq_ind nat -(next_plus g (next g n0) n) (\lambda (n1: nat).(eq A (aplus g (ASort O (next -g n0)) n) (ASort O n1))) (H (next g n0)) (next g (next_plus g n0 n)) -(next_plus_next g n0 n)) (asucc g (aplus g (ASort O n0) n)) (aplus_asucc g n -(ASort O n0)))))) h)). -(* COMMENTS -Initial nodes: 229 -END *) - -theorem aplus_asort_le_simpl: - \forall (g: G).(\forall (h: nat).(\forall (k: nat).(\forall (n: nat).((le h -k) \to (eq A (aplus g (ASort k n) h) (ASort (minus k h) n)))))) -\def - \lambda (g: G).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall (k: -nat).(\forall (n0: nat).((le n k) \to (eq A (aplus g (ASort k n0) n) (ASort -(minus k n) n0)))))) (\lambda (k: nat).(\lambda (n: nat).(\lambda (_: (le O -k)).(eq_ind nat k (\lambda (n0: nat).(eq A (ASort k n) (ASort n0 n))) -(refl_equal A (ASort k n)) (minus k O) (minus_n_O k))))) (\lambda (h0: -nat).(\lambda (H: ((\forall (k: nat).(\forall (n: nat).((le h0 k) \to (eq A -(aplus g (ASort k n) h0) (ASort (minus k h0) n))))))).(\lambda (k: -nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).((le (S h0) n) \to (eq A -(asucc g (aplus g (ASort n n0) h0)) (ASort (minus n (S h0)) n0))))) (\lambda -(n: nat).(\lambda (H0: (le (S h0) O)).(ex2_ind nat (\lambda (n0: nat).(eq nat -O (S n0))) (\lambda (n0: nat).(le h0 n0)) (eq A (asucc g (aplus g (ASort O n) -h0)) (ASort (minus O (S h0)) n)) (\lambda (x: nat).(\lambda (H1: (eq nat O (S -x))).(\lambda (_: (le h0 x)).(let H3 \def (eq_ind nat O (\lambda (ee: -nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True -| (S _) \Rightarrow False])) I (S x) H1) in (False_ind (eq A (asucc g (aplus -g (ASort O n) h0)) (ASort (minus O (S h0)) n)) H3))))) (le_gen_S h0 O H0)))) -(\lambda (n: nat).(\lambda (_: ((\forall (n0: nat).((le (S h0) n) \to (eq A -(asucc g (aplus g (ASort n n0) h0)) (ASort (minus n (S h0)) n0)))))).(\lambda -(n0: nat).(\lambda (H1: (le (S h0) (S n))).(eq_ind A (aplus g (asucc g (ASort -(S n) n0)) h0) (\lambda (a: A).(eq A a (ASort (minus (S n) (S h0)) n0))) (H n -n0 (le_S_n h0 n H1)) (asucc g (aplus g (ASort (S n) n0) h0)) (aplus_asucc g -h0 (ASort (S n) n0))))))) k)))) h)). -(* COMMENTS -Initial nodes: 484 -END *) - -theorem aplus_asort_simpl: - \forall (g: G).(\forall (h: nat).(\forall (k: nat).(\forall (n: nat).(eq A -(aplus g (ASort k n) h) (ASort (minus k h) (next_plus g n (minus h k))))))) -\def - \lambda (g: G).(\lambda (h: nat).(\lambda (k: nat).(\lambda (n: -nat).(lt_le_e k h (eq A (aplus g (ASort k n) h) (ASort (minus k h) (next_plus -g n (minus h k)))) (\lambda (H: (lt k h)).(eq_ind_r nat (plus k (minus h k)) -(\lambda (n0: nat).(eq A (aplus g (ASort k n) n0) (ASort (minus k h) -(next_plus g n (minus h k))))) (eq_ind A (aplus g (aplus g (ASort k n) k) -(minus h k)) (\lambda (a: A).(eq A a (ASort (minus k h) (next_plus g n (minus -h k))))) (eq_ind_r A (ASort (minus k k) n) (\lambda (a: A).(eq A (aplus g a -(minus h k)) (ASort (minus k h) (next_plus g n (minus h k))))) (eq_ind nat O -(\lambda (n0: nat).(eq A (aplus g (ASort n0 n) (minus h k)) (ASort (minus k -h) (next_plus g n (minus h k))))) (eq_ind_r nat O (\lambda (n0: nat).(eq A -(aplus g (ASort O n) (minus h k)) (ASort n0 (next_plus g n (minus h k))))) -(aplus_asort_O_simpl g (minus h k) n) (minus k h) (O_minus k h (le_S_n k h -(le_S (S k) h H)))) (minus k k) (minus_n_n k)) (aplus g (ASort k n) k) -(aplus_asort_le_simpl g k k n (le_n k))) (aplus g (ASort k n) (plus k (minus -h k))) (aplus_assoc g (ASort k n) k (minus h k))) h (le_plus_minus k h -(le_S_n k h (le_S (S k) h H))))) (\lambda (H: (le h k)).(eq_ind_r A (ASort -(minus k h) n) (\lambda (a: A).(eq A a (ASort (minus k h) (next_plus g n -(minus h k))))) (eq_ind_r nat O (\lambda (n0: nat).(eq A (ASort (minus k h) -n) (ASort (minus k h) (next_plus g n n0)))) (refl_equal A (ASort (minus k h) -(next_plus g n O))) (minus h k) (O_minus h k H)) (aplus g (ASort k n) h) -(aplus_asort_le_simpl g h k n H))))))). -(* COMMENTS -Initial nodes: 587 -END *) - -theorem aplus_ahead_simpl: - \forall (g: G).(\forall (h: nat).(\forall (a1: A).(\forall (a2: A).(eq A -(aplus g (AHead a1 a2) h) (AHead a1 (aplus g a2 h)))))) -\def - \lambda (g: G).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall (a1: -A).(\forall (a2: A).(eq A (aplus g (AHead a1 a2) n) (AHead a1 (aplus g a2 -n)))))) (\lambda (a1: A).(\lambda (a2: A).(refl_equal A (AHead a1 a2)))) -(\lambda (n: nat).(\lambda (H: ((\forall (a1: A).(\forall (a2: A).(eq A -(aplus g (AHead a1 a2) n) (AHead a1 (aplus g a2 n))))))).(\lambda (a1: -A).(\lambda (a2: A).(eq_ind A (aplus g (asucc g (AHead a1 a2)) n) (\lambda -(a: A).(eq A a (AHead a1 (asucc g (aplus g a2 n))))) (eq_ind A (aplus g -(asucc g a2) n) (\lambda (a: A).(eq A (aplus g (asucc g (AHead a1 a2)) n) -(AHead a1 a))) (H a1 (asucc g a2)) (asucc g (aplus g a2 n)) (aplus_asucc g n -a2)) (asucc g (aplus g (AHead a1 a2) n)) (aplus_asucc g n (AHead a1 a2))))))) -h)). -(* COMMENTS -Initial nodes: 239 -END *) - -theorem aplus_asucc_false: - \forall (g: G).(\forall (a: A).(\forall (h: nat).((eq A (aplus g (asucc g a) -h) a) \to (\forall (P: Prop).P)))) -\def - \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(\forall (h: -nat).((eq A (aplus g (asucc g a0) h) a0) \to (\forall (P: Prop).P)))) -(\lambda (n: nat).(\lambda (n0: nat).(\lambda (h: nat).(\lambda (H: (eq A -(aplus g (match n with [O \Rightarrow (ASort O (next g n0)) | (S h0) -\Rightarrow (ASort h0 n0)]) h) (ASort n n0))).(\lambda (P: Prop).(nat_ind -(\lambda (n1: nat).((eq A (aplus g (match n1 with [O \Rightarrow (ASort O -(next g n0)) | (S h0) \Rightarrow (ASort h0 n0)]) h) (ASort n1 n0)) \to P)) -(\lambda (H0: (eq A (aplus g (ASort O (next g n0)) h) (ASort O n0))).(let H1 -\def (eq_ind A (aplus g (ASort O (next g n0)) h) (\lambda (a0: A).(eq A a0 -(ASort O n0))) H0 (ASort (minus O h) (next_plus g (next g n0) (minus h O))) -(aplus_asort_simpl g h O (next g n0))) in (let H2 \def (f_equal A nat -(\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort _ n1) -\Rightarrow n1 | (AHead _ _) \Rightarrow ((let rec next_plus (g0: G) (n1: -nat) (i: nat) on i: nat \def (match i with [O \Rightarrow n1 | (S i0) -\Rightarrow (next g0 (next_plus g0 n1 i0))]) in next_plus) g (next g n0) -(minus h O))])) (ASort (minus O h) (next_plus g (next g n0) (minus h O))) -(ASort O n0) H1) in (let H3 \def (eq_ind_r nat (minus h O) (\lambda (n1: -nat).(eq nat (next_plus g (next g n0) n1) n0)) H2 h (minus_n_O h)) in -(le_lt_false (next_plus g (next g n0) h) n0 (eq_ind nat (next_plus g (next g -n0) h) (\lambda (n1: nat).(le (next_plus g (next g n0) h) n1)) (le_n -(next_plus g (next g n0) h)) n0 H3) (next_plus_lt g h n0) P))))) (\lambda -(n1: nat).(\lambda (_: (((eq A (aplus g (match n1 with [O \Rightarrow (ASort -O (next g n0)) | (S h0) \Rightarrow (ASort h0 n0)]) h) (ASort n1 n0)) \to -P))).(\lambda (H0: (eq A (aplus g (ASort n1 n0) h) (ASort (S n1) n0))).(let -H1 \def (eq_ind A (aplus g (ASort n1 n0) h) (\lambda (a0: A).(eq A a0 (ASort -(S n1) n0))) H0 (ASort (minus n1 h) (next_plus g n0 (minus h n1))) -(aplus_asort_simpl g h n1 n0)) in (let H2 \def (f_equal A nat (\lambda (e: -A).(match e in A return (\lambda (_: A).nat) with [(ASort n2 _) \Rightarrow -n2 | (AHead _ _) \Rightarrow ((let rec minus (n2: nat) on n2: (nat \to nat) -\def (\lambda (m: nat).(match n2 with [O \Rightarrow O | (S k) \Rightarrow -(match m with [O \Rightarrow (S k) | (S l) \Rightarrow (minus k l)])])) in -minus) n1 h)])) (ASort (minus n1 h) (next_plus g n0 (minus h n1))) (ASort (S -n1) n0) H1) in ((let H3 \def (f_equal A nat (\lambda (e: A).(match e in A -return (\lambda (_: A).nat) with [(ASort _ n2) \Rightarrow n2 | (AHead _ _) -\Rightarrow ((let rec next_plus (g0: G) (n2: nat) (i: nat) on i: nat \def -(match i with [O \Rightarrow n2 | (S i0) \Rightarrow (next g0 (next_plus g0 -n2 i0))]) in next_plus) g n0 (minus h n1))])) (ASort (minus n1 h) (next_plus -g n0 (minus h n1))) (ASort (S n1) n0) H1) in (\lambda (H4: (eq nat (minus n1 -h) (S n1))).(le_Sx_x n1 (eq_ind nat (minus n1 h) (\lambda (n2: nat).(le n2 -n1)) (minus_le n1 h) (S n1) H4) P))) H2)))))) n H)))))) (\lambda (a0: -A).(\lambda (_: ((\forall (h: nat).((eq A (aplus g (asucc g a0) h) a0) \to -(\forall (P: Prop).P))))).(\lambda (a1: A).(\lambda (H0: ((\forall (h: -nat).((eq A (aplus g (asucc g a1) h) a1) \to (\forall (P: -Prop).P))))).(\lambda (h: nat).(\lambda (H1: (eq A (aplus g (AHead a0 (asucc -g a1)) h) (AHead a0 a1))).(\lambda (P: Prop).(let H2 \def (eq_ind A (aplus g -(AHead a0 (asucc g a1)) h) (\lambda (a2: A).(eq A a2 (AHead a0 a1))) H1 -(AHead a0 (aplus g (asucc g a1) h)) (aplus_ahead_simpl g h a0 (asucc g a1))) -in (let H3 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda -(_: A).A) with [(ASort _ _) \Rightarrow ((let rec aplus (g0: G) (a2: A) (n: -nat) on n: A \def (match n with [O \Rightarrow a2 | (S n0) \Rightarrow (asucc -g0 (aplus g0 a2 n0))]) in aplus) g (asucc g a1) h) | (AHead _ a2) \Rightarrow -a2])) (AHead a0 (aplus g (asucc g a1) h)) (AHead a0 a1) H2) in (H0 h H3 -P)))))))))) a)). -(* COMMENTS -Initial nodes: 977 -END *) - -theorem aplus_inj: - \forall (g: G).(\forall (h1: nat).(\forall (h2: nat).(\forall (a: A).((eq A -(aplus g a h1) (aplus g a h2)) \to (eq nat h1 h2))))) -\def - \lambda (g: G).(\lambda (h1: nat).(nat_ind (\lambda (n: nat).(\forall (h2: -nat).(\forall (a: A).((eq A (aplus g a n) (aplus g a h2)) \to (eq nat n -h2))))) (\lambda (h2: nat).(nat_ind (\lambda (n: nat).(\forall (a: A).((eq A -(aplus g a O) (aplus g a n)) \to (eq nat O n)))) (\lambda (a: A).(\lambda (_: -(eq A a a)).(refl_equal nat O))) (\lambda (n: nat).(\lambda (_: ((\forall (a: -A).((eq A a (aplus g a n)) \to (eq nat O n))))).(\lambda (a: A).(\lambda (H0: -(eq A a (asucc g (aplus g a n)))).(let H1 \def (eq_ind_r A (asucc g (aplus g -a n)) (\lambda (a0: A).(eq A a a0)) H0 (aplus g (asucc g a) n) (aplus_asucc g -n a)) in (aplus_asucc_false g a n (sym_eq A a (aplus g (asucc g a) n) H1) (eq -nat O (S n)))))))) h2)) (\lambda (n: nat).(\lambda (H: ((\forall (h2: -nat).(\forall (a: A).((eq A (aplus g a n) (aplus g a h2)) \to (eq nat n -h2)))))).(\lambda (h2: nat).(nat_ind (\lambda (n0: nat).(\forall (a: A).((eq -A (aplus g a (S n)) (aplus g a n0)) \to (eq nat (S n) n0)))) (\lambda (a: -A).(\lambda (H0: (eq A (asucc g (aplus g a n)) a)).(let H1 \def (eq_ind_r A -(asucc g (aplus g a n)) (\lambda (a0: A).(eq A a0 a)) H0 (aplus g (asucc g a) -n) (aplus_asucc g n a)) in (aplus_asucc_false g a n H1 (eq nat (S n) O))))) -(\lambda (n0: nat).(\lambda (_: ((\forall (a: A).((eq A (asucc g (aplus g a -n)) (aplus g a n0)) \to (eq nat (S n) n0))))).(\lambda (a: A).(\lambda (H1: -(eq A (asucc g (aplus g a n)) (asucc g (aplus g a n0)))).(let H2 \def -(eq_ind_r A (asucc g (aplus g a n)) (\lambda (a0: A).(eq A a0 (asucc g (aplus -g a n0)))) H1 (aplus g (asucc g a) n) (aplus_asucc g n a)) in (let H3 \def -(eq_ind_r A (asucc g (aplus g a n0)) (\lambda (a0: A).(eq A (aplus g (asucc g -a) n) a0)) H2 (aplus g (asucc g a) n0) (aplus_asucc g n0 a)) in (f_equal nat -nat S n n0 (H n0 (asucc g a) H3)))))))) h2)))) h1)). -(* COMMENTS -Initial nodes: 599 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/app/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/app/defs.ma deleted file mode 100644 index 9cf1e37d1..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/app/defs.ma +++ /dev/null @@ -1,31 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/C/defs.ma". - -definition cbk: - C \to nat -\def - let rec cbk (c: C) on c: nat \def (match c with [(CSort m) \Rightarrow m | -(CHead c0 _ _) \Rightarrow (cbk c0)]) in cbk. - -definition app1: - C \to (T \to T) -\def - let rec app1 (c: C) on c: (T \to T) \def (\lambda (t: T).(match c with -[(CSort _) \Rightarrow t | (CHead c0 k u) \Rightarrow (app1 c0 (THead k u -t))])) in app1. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/aprem/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/aprem/defs.ma deleted file mode 100644 index 78b49e920..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/aprem/defs.ma +++ /dev/null @@ -1,23 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/A/defs.ma". - -inductive aprem: nat \to (A \to (A \to Prop)) \def -| aprem_zero: \forall (a1: A).(\forall (a2: A).(aprem O (AHead a1 a2) a1)) -| aprem_succ: \forall (a2: A).(\forall (a: A).(\forall (i: nat).((aprem i a2 -a) \to (\forall (a1: A).(aprem (S i) (AHead a1 a2) a))))). - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/aprem/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/aprem/fwd.ma deleted file mode 100644 index ed48846a8..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/aprem/fwd.ma +++ /dev/null @@ -1,120 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/aprem/defs.ma". - -theorem aprem_gen_sort: - \forall (x: A).(\forall (i: nat).(\forall (h: nat).(\forall (n: nat).((aprem -i (ASort h n) x) \to False)))) -\def - \lambda (x: A).(\lambda (i: nat).(\lambda (h: nat).(\lambda (n: -nat).(\lambda (H: (aprem i (ASort h n) x)).(insert_eq A (ASort h n) (\lambda -(a: A).(aprem i a x)) (\lambda (_: A).False) (\lambda (y: A).(\lambda (H0: -(aprem i y x)).(aprem_ind (\lambda (_: nat).(\lambda (a: A).(\lambda (_: -A).((eq A a (ASort h n)) \to False)))) (\lambda (a1: A).(\lambda (a2: -A).(\lambda (H1: (eq A (AHead a1 a2) (ASort h n))).(let H2 \def (eq_ind A -(AHead a1 a2) (\lambda (ee: A).(match ee in A return (\lambda (_: A).Prop) -with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I -(ASort h n) H1) in (False_ind False H2))))) (\lambda (a2: A).(\lambda (a: -A).(\lambda (i0: nat).(\lambda (_: (aprem i0 a2 a)).(\lambda (_: (((eq A a2 -(ASort h n)) \to False))).(\lambda (a1: A).(\lambda (H3: (eq A (AHead a1 a2) -(ASort h n))).(let H4 \def (eq_ind A (AHead a1 a2) (\lambda (ee: A).(match ee -in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | -(AHead _ _) \Rightarrow True])) I (ASort h n) H3) in (False_ind False -H4))))))))) i y x H0))) H))))). -(* COMMENTS -Initial nodes: 227 -END *) - -theorem aprem_gen_head_O: - \forall (a1: A).(\forall (a2: A).(\forall (x: A).((aprem O (AHead a1 a2) x) -\to (eq A x a1)))) -\def - \lambda (a1: A).(\lambda (a2: A).(\lambda (x: A).(\lambda (H: (aprem O -(AHead a1 a2) x)).(insert_eq A (AHead a1 a2) (\lambda (a: A).(aprem O a x)) -(\lambda (_: A).(eq A x a1)) (\lambda (y: A).(\lambda (H0: (aprem O y -x)).(insert_eq nat O (\lambda (n: nat).(aprem n y x)) (\lambda (_: nat).((eq -A y (AHead a1 a2)) \to (eq A x a1))) (\lambda (y0: nat).(\lambda (H1: (aprem -y0 y x)).(aprem_ind (\lambda (n: nat).(\lambda (a: A).(\lambda (a0: A).((eq -nat n O) \to ((eq A a (AHead a1 a2)) \to (eq A a0 a1)))))) (\lambda (a0: -A).(\lambda (a3: A).(\lambda (_: (eq nat O O)).(\lambda (H3: (eq A (AHead a0 -a3) (AHead a1 a2))).(let H4 \def (f_equal A A (\lambda (e: A).(match e in A -return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead a _) -\Rightarrow a])) (AHead a0 a3) (AHead a1 a2) H3) in ((let H5 \def (f_equal A -A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) -\Rightarrow a3 | (AHead _ a) \Rightarrow a])) (AHead a0 a3) (AHead a1 a2) H3) -in (\lambda (H6: (eq A a0 a1)).H6)) H4)))))) (\lambda (a0: A).(\lambda (a: -A).(\lambda (i: nat).(\lambda (H2: (aprem i a0 a)).(\lambda (H3: (((eq nat i -O) \to ((eq A a0 (AHead a1 a2)) \to (eq A a a1))))).(\lambda (a3: A).(\lambda -(H4: (eq nat (S i) O)).(\lambda (H5: (eq A (AHead a3 a0) (AHead a1 a2))).(let -H6 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) -with [(ASort _ _) \Rightarrow a3 | (AHead a4 _) \Rightarrow a4])) (AHead a3 -a0) (AHead a1 a2) H5) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e -in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead _ -a4) \Rightarrow a4])) (AHead a3 a0) (AHead a1 a2) H5) in (\lambda (_: (eq A -a3 a1)).(let H9 \def (eq_ind A a0 (\lambda (a4: A).((eq nat i O) \to ((eq A -a4 (AHead a1 a2)) \to (eq A a a1)))) H3 a2 H7) in (let H10 \def (eq_ind A a0 -(\lambda (a4: A).(aprem i a4 a)) H2 a2 H7) in (let H11 \def (eq_ind nat (S i) -(\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O -\Rightarrow False | (S _) \Rightarrow True])) I O H4) in (False_ind (eq A a -a1) H11)))))) H6)))))))))) y0 y x H1))) H0))) H)))). -(* COMMENTS -Initial nodes: 500 -END *) - -theorem aprem_gen_head_S: - \forall (a1: A).(\forall (a2: A).(\forall (x: A).(\forall (i: nat).((aprem -(S i) (AHead a1 a2) x) \to (aprem i a2 x))))) -\def - \lambda (a1: A).(\lambda (a2: A).(\lambda (x: A).(\lambda (i: nat).(\lambda -(H: (aprem (S i) (AHead a1 a2) x)).(insert_eq A (AHead a1 a2) (\lambda (a: -A).(aprem (S i) a x)) (\lambda (_: A).(aprem i a2 x)) (\lambda (y: -A).(\lambda (H0: (aprem (S i) y x)).(insert_eq nat (S i) (\lambda (n: -nat).(aprem n y x)) (\lambda (_: nat).((eq A y (AHead a1 a2)) \to (aprem i a2 -x))) (\lambda (y0: nat).(\lambda (H1: (aprem y0 y x)).(aprem_ind (\lambda (n: -nat).(\lambda (a: A).(\lambda (a0: A).((eq nat n (S i)) \to ((eq A a (AHead -a1 a2)) \to (aprem i a2 a0)))))) (\lambda (a0: A).(\lambda (a3: A).(\lambda -(H2: (eq nat O (S i))).(\lambda (H3: (eq A (AHead a0 a3) (AHead a1 a2))).(let -H4 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) -with [(ASort _ _) \Rightarrow a0 | (AHead a _) \Rightarrow a])) (AHead a0 a3) -(AHead a1 a2) H3) in ((let H5 \def (f_equal A A (\lambda (e: A).(match e in A -return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a3 | (AHead _ a) -\Rightarrow a])) (AHead a0 a3) (AHead a1 a2) H3) in (\lambda (H6: (eq A a0 -a1)).(eq_ind_r A a1 (\lambda (a: A).(aprem i a2 a)) (let H7 \def (eq_ind nat -O (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O -\Rightarrow True | (S _) \Rightarrow False])) I (S i) H2) in (False_ind -(aprem i a2 a1) H7)) a0 H6))) H4)))))) (\lambda (a0: A).(\lambda (a: -A).(\lambda (i0: nat).(\lambda (H2: (aprem i0 a0 a)).(\lambda (H3: (((eq nat -i0 (S i)) \to ((eq A a0 (AHead a1 a2)) \to (aprem i a2 a))))).(\lambda (a3: -A).(\lambda (H4: (eq nat (S i0) (S i))).(\lambda (H5: (eq A (AHead a3 a0) -(AHead a1 a2))).(let H6 \def (f_equal A A (\lambda (e: A).(match e in A -return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a3 | (AHead a4 _) -\Rightarrow a4])) (AHead a3 a0) (AHead a1 a2) H5) in ((let H7 \def (f_equal A -A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) -\Rightarrow a0 | (AHead _ a4) \Rightarrow a4])) (AHead a3 a0) (AHead a1 a2) -H5) in (\lambda (_: (eq A a3 a1)).(let H9 \def (eq_ind A a0 (\lambda (a4: -A).((eq nat i0 (S i)) \to ((eq A a4 (AHead a1 a2)) \to (aprem i a2 a)))) H3 -a2 H7) in (let H10 \def (eq_ind A a0 (\lambda (a4: A).(aprem i0 a4 a)) H2 a2 -H7) in (let H11 \def (f_equal nat nat (\lambda (e: nat).(match e in nat -return (\lambda (_: nat).nat) with [O \Rightarrow i0 | (S n) \Rightarrow n])) -(S i0) (S i) H4) in (let H12 \def (eq_ind nat i0 (\lambda (n: nat).((eq nat n -(S i)) \to ((eq A a2 (AHead a1 a2)) \to (aprem i a2 a)))) H9 i H11) in (let -H13 \def (eq_ind nat i0 (\lambda (n: nat).(aprem n a2 a)) H10 i H11) in -H13))))))) H6)))))))))) y0 y x H1))) H0))) H))))). -(* COMMENTS -Initial nodes: 631 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/aprem/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/aprem/props.ma deleted file mode 100644 index fb8321062..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/aprem/props.ma +++ /dev/null @@ -1,76 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/aprem/fwd.ma". - -include "Basic-1/leq/defs.ma". - -theorem aprem_repl: - \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall -(i: nat).(\forall (b2: A).((aprem i a2 b2) \to (ex2 A (\lambda (b1: A).(leq g -b1 b2)) (\lambda (b1: A).(aprem i a1 b1))))))))) -\def - \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1 -a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(\forall (i: nat).(\forall -(b2: A).((aprem i a0 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda -(b1: A).(aprem i a b1)))))))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda -(n1: nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (_: (eq A (aplus g -(ASort h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (i: nat).(\lambda (b2: -A).(\lambda (H1: (aprem i (ASort h2 n2) b2)).(let H_x \def (aprem_gen_sort b2 -i h2 n2 H1) in (let H2 \def H_x in (False_ind (ex2 A (\lambda (b1: A).(leq g -b1 b2)) (\lambda (b1: A).(aprem i (ASort h1 n1) b1))) H2)))))))))))) (\lambda -(a0: A).(\lambda (a3: A).(\lambda (H0: (leq g a0 a3)).(\lambda (_: ((\forall -(i: nat).(\forall (b2: A).((aprem i a3 b2) \to (ex2 A (\lambda (b1: A).(leq g -b1 b2)) (\lambda (b1: A).(aprem i a0 b1)))))))).(\lambda (a4: A).(\lambda -(a5: A).(\lambda (_: (leq g a4 a5)).(\lambda (H3: ((\forall (i: nat).(\forall -(b2: A).((aprem i a5 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda -(b1: A).(aprem i a4 b1)))))))).(\lambda (i: nat).(\lambda (b2: A).(\lambda -(H4: (aprem i (AHead a3 a5) b2)).(nat_ind (\lambda (n: nat).((aprem n (AHead -a3 a5) b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem -n (AHead a0 a4) b1))))) (\lambda (H5: (aprem O (AHead a3 a5) b2)).(let H_y -\def (aprem_gen_head_O a3 a5 b2 H5) in (eq_ind_r A a3 (\lambda (a: A).(ex2 A -(\lambda (b1: A).(leq g b1 a)) (\lambda (b1: A).(aprem O (AHead a0 a4) b1)))) -(ex_intro2 A (\lambda (b1: A).(leq g b1 a3)) (\lambda (b1: A).(aprem O (AHead -a0 a4) b1)) a0 H0 (aprem_zero a0 a4)) b2 H_y))) (\lambda (i0: nat).(\lambda -(_: (((aprem i0 (AHead a3 a5) b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) -(\lambda (b1: A).(aprem i0 (AHead a0 a4) b1)))))).(\lambda (H5: (aprem (S i0) -(AHead a3 a5) b2)).(let H_y \def (aprem_gen_head_S a3 a5 b2 i0 H5) in (let -H_x \def (H3 i0 b2 H_y) in (let H6 \def H_x in (ex2_ind A (\lambda (b1: -A).(leq g b1 b2)) (\lambda (b1: A).(aprem i0 a4 b1)) (ex2 A (\lambda (b1: -A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S i0) (AHead a0 a4) b1))) (\lambda -(x: A).(\lambda (H7: (leq g x b2)).(\lambda (H8: (aprem i0 a4 x)).(ex_intro2 -A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S i0) (AHead a0 -a4) b1)) x H7 (aprem_succ a4 x i0 H8 a0))))) H6))))))) i H4)))))))))))) a1 a2 -H)))). -(* COMMENTS -Initial nodes: 621 -END *) - -theorem aprem_asucc: - \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (i: nat).((aprem i -a1 a2) \to (aprem i (asucc g a1) a2))))) -\def - \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (i: nat).(\lambda -(H: (aprem i a1 a2)).(aprem_ind (\lambda (n: nat).(\lambda (a: A).(\lambda -(a0: A).(aprem n (asucc g a) a0)))) (\lambda (a0: A).(\lambda (a3: -A).(aprem_zero a0 (asucc g a3)))) (\lambda (a0: A).(\lambda (a: A).(\lambda -(i0: nat).(\lambda (_: (aprem i0 a0 a)).(\lambda (H1: (aprem i0 (asucc g a0) -a)).(\lambda (a3: A).(aprem_succ (asucc g a0) a i0 H1 a3))))))) i a1 a2 -H))))). -(* COMMENTS -Initial nodes: 101 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/arity/aprem.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/arity/aprem.ma deleted file mode 100644 index 35e8f58e0..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/arity/aprem.ma +++ /dev/null @@ -1,260 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/arity/props.ma". - -include "Basic-1/arity/cimp.ma". - -include "Basic-1/aprem/props.ma". - -theorem arity_aprem: - \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t -a) \to (\forall (i: nat).(\forall (b: A).((aprem i a b) \to (ex2_3 C T nat -(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c)))) -(\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g -b))))))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: -(arity g c t a)).(arity_ind g (\lambda (c0: C).(\lambda (_: T).(\lambda (a0: -A).(\forall (i: nat).(\forall (b: A).((aprem i a0 b) \to (ex2_3 C T nat -(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) -(\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g -b)))))))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (i: nat).(\lambda -(b: A).(\lambda (H0: (aprem i (ASort O n) b)).(let H_x \def (aprem_gen_sort b -i O n H0) in (let H1 \def H_x in (False_ind (ex2_3 C T nat (\lambda (d: -C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: -C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b)))))) H1)))))))) -(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H0: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (a0: A).(\lambda (_: -(arity g d u a0)).(\lambda (H2: ((\forall (i0: nat).(\forall (b: A).((aprem -i0 a0 b) \to (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: -nat).(drop (plus i0 j) O d0 d)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda -(_: nat).(arity g d0 u0 (asucc g b))))))))))).(\lambda (i0: nat).(\lambda (b: -A).(\lambda (H3: (aprem i0 a0 b)).(let H_x \def (H2 i0 b H3) in (let H4 \def -H_x in (ex2_3_ind C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: -nat).(drop (plus i0 j) O d0 d)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda -(_: nat).(arity g d0 u0 (asucc g b))))) (ex2_3 C T nat (\lambda (d0: -C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d0 c0)))) (\lambda -(d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b)))))) -(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H5: (drop -(plus i0 x2) O x0 d)).(\lambda (H6: (arity g x0 x1 (asucc g b))).(let H_x0 -\def (getl_drop_conf_rev (plus i0 x2) x0 d H5 Abbr c0 u i H0) in (let H7 \def -H_x0 in (ex2_ind C (\lambda (c1: C).(drop (plus i0 x2) O c1 c0)) (\lambda -(c1: C).(drop (S i) (plus i0 x2) c1 x0)) (ex2_3 C T nat (\lambda (d0: -C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d0 c0)))) (\lambda -(d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b)))))) -(\lambda (x: C).(\lambda (H8: (drop (plus i0 x2) O x c0)).(\lambda (H9: (drop -(S i) (plus i0 x2) x x0)).(ex2_3_intro C T nat (\lambda (d0: C).(\lambda (_: -T).(\lambda (j: nat).(drop (plus i0 j) O d0 c0)))) (\lambda (d0: C).(\lambda -(u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b))))) x (lift (S i) (plus -i0 x2) x1) x2 H8 (arity_lift g x0 x1 (asucc g b) H6 x (S i) (plus i0 x2) -H9))))) H7)))))))) H4)))))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda -(u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abst) -u))).(\lambda (a0: A).(\lambda (_: (arity g d u (asucc g a0))).(\lambda (H2: -((\forall (i0: nat).(\forall (b: A).((aprem i0 (asucc g a0) b) \to (ex2_3 C T -nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d0 -d)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 -(asucc g b))))))))))).(\lambda (i0: nat).(\lambda (b: A).(\lambda (H3: (aprem -i0 a0 b)).(let H4 \def (H2 i0 b (aprem_asucc g a0 b i0 H3)) in (ex2_3_ind C T -nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d0 -d)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 -(asucc g b))))) (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: -nat).(drop (plus i0 j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda -(_: nat).(arity g d0 u0 (asucc g b)))))) (\lambda (x0: C).(\lambda (x1: -T).(\lambda (x2: nat).(\lambda (H5: (drop (plus i0 x2) O x0 d)).(\lambda (H6: -(arity g x0 x1 (asucc g b))).(let H_x \def (getl_drop_conf_rev (plus i0 x2) -x0 d H5 Abst c0 u i H0) in (let H7 \def H_x in (ex2_ind C (\lambda (c1: -C).(drop (plus i0 x2) O c1 c0)) (\lambda (c1: C).(drop (S i) (plus i0 x2) c1 -x0)) (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop -(plus i0 j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: -nat).(arity g d0 u0 (asucc g b)))))) (\lambda (x: C).(\lambda (H8: (drop -(plus i0 x2) O x c0)).(\lambda (H9: (drop (S i) (plus i0 x2) x -x0)).(ex2_3_intro C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: -nat).(drop (plus i0 j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda -(_: nat).(arity g d0 u0 (asucc g b))))) x (lift (S i) (plus i0 x2) x1) x2 H8 -(arity_lift g x0 x1 (asucc g b) H6 x (S i) (plus i0 x2) H9))))) H7)))))))) -H4))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda -(c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u -a1)).(\lambda (_: ((\forall (i: nat).(\forall (b0: A).((aprem i a1 b0) \to -(ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus -i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d -u0 (asucc g b0))))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: -(arity g (CHead c0 (Bind b) u) t0 a2)).(\lambda (H4: ((\forall (i: -nat).(\forall (b0: A).((aprem i a2 b0) \to (ex2_3 C T nat (\lambda (d: -C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d (CHead c0 (Bind b) -u))))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 -(asucc g b0))))))))))).(\lambda (i: nat).(\lambda (b0: A).(\lambda (H5: -(aprem i a2 b0)).(let H_x \def (H4 i b0 H5) in (let H6 \def H_x in (ex2_3_ind -C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O -d (CHead c0 (Bind b) u))))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: -nat).(arity g d u0 (asucc g b0))))) (ex2_3 C T nat (\lambda (d: C).(\lambda -(_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda -(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b0)))))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H7: (drop (plus i x2) O x0 -(CHead c0 (Bind b) u))).(\lambda (H8: (arity g x0 x1 (asucc g b0))).(let H9 -\def (eq_ind nat (S (plus i x2)) (\lambda (n: nat).(drop n O x0 c0)) (drop_S -b x0 c0 u (plus i x2) H7) (plus i (S x2)) (plus_n_Sm i x2)) in (ex2_3_intro C -T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d -c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 -(asucc g b0))))) x0 x1 (S x2) H9 H8))))))) H6))))))))))))))))) (\lambda (c0: -C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H0: (arity g c0 u (asucc g -a1))).(\lambda (_: ((\forall (i: nat).(\forall (b: A).((aprem i (asucc g a1) -b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop -(plus i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: -nat).(arity g d u0 (asucc g b))))))))))).(\lambda (t0: T).(\lambda (a2: -A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u) t0 a2)).(\lambda (H3: -((\forall (i: nat).(\forall (b: A).((aprem i a2 b) \to (ex2_3 C T nat -(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d (CHead -c0 (Bind Abst) u))))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: -nat).(arity g d u0 (asucc g b))))))))))).(\lambda (i: nat).(\lambda (b: -A).(\lambda (H4: (aprem i (AHead a1 a2) b)).(nat_ind (\lambda (n: -nat).((aprem n (AHead a1 a2) b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda -(_: T).(\lambda (j: nat).(drop (plus n j) O d c0)))) (\lambda (d: C).(\lambda -(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b)))))))) (\lambda (H5: -(aprem O (AHead a1 a2) b)).(let H_y \def (aprem_gen_head_O a1 a2 b H5) in -(eq_ind_r A a1 (\lambda (a0: A).(ex2_3 C T nat (\lambda (d: C).(\lambda (_: -T).(\lambda (j: nat).(drop (plus O j) O d c0)))) (\lambda (d: C).(\lambda -(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g a0))))))) (ex2_3_intro C T -nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus O j) O d -c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 -(asucc g a1))))) c0 u O (drop_refl c0) H0) b H_y))) (\lambda (i0: -nat).(\lambda (_: (((aprem i0 (AHead a1 a2) b) \to (ex2_3 C T nat (\lambda -(d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d c0)))) -(\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g -b))))))))).(\lambda (H5: (aprem (S i0) (AHead a1 a2) b)).(let H_y \def -(aprem_gen_head_S a1 a2 b i0 H5) in (let H_x \def (H3 i0 b H_y) in (let H6 -\def H_x in (ex2_3_ind C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: -nat).(drop (plus i0 j) O d (CHead c0 (Bind Abst) u))))) (\lambda (d: -C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))) (ex2_3 C -T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus (S i0) j) -O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 -(asucc g b)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: -nat).(\lambda (H7: (drop (plus i0 x2) O x0 (CHead c0 (Bind Abst) -u))).(\lambda (H8: (arity g x0 x1 (asucc g b))).(ex2_3_intro C T nat (\lambda -(d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus (S i0) j) O d c0)))) -(\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g -b))))) x0 x1 x2 (drop_S Abst x0 c0 u (plus i0 x2) H7) H8)))))) H6))))))) i -H4))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda -(_: (arity g c0 u a1)).(\lambda (_: ((\forall (i: nat).(\forall (b: -A).((aprem i a1 b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: -T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda -(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))))))))).(\lambda (t0: -T).(\lambda (a2: A).(\lambda (_: (arity g c0 t0 (AHead a1 a2))).(\lambda (H3: -((\forall (i: nat).(\forall (b: A).((aprem i (AHead a1 a2) b) \to (ex2_3 C T -nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d -c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 -(asucc g b))))))))))).(\lambda (i: nat).(\lambda (b: A).(\lambda (H4: (aprem -i a2 b)).(let H5 \def (H3 (S i) b (aprem_succ a2 b i H4 a1)) in (ex2_3_ind C -T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (S (plus i j)) -O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 -(asucc g b))))) (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: -nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda -(_: nat).(arity g d u0 (asucc g b)))))) (\lambda (x0: C).(\lambda (x1: -T).(\lambda (x2: nat).(\lambda (H6: (drop (S (plus i x2)) O x0 c0)).(\lambda -(H7: (arity g x0 x1 (asucc g b))).(C_ind (\lambda (c1: C).((drop (S (plus i -x2)) O c1 c0) \to ((arity g c1 x1 (asucc g b)) \to (ex2_3 C T nat (\lambda -(d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda -(d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))))))) -(\lambda (n: nat).(\lambda (H8: (drop (S (plus i x2)) O (CSort n) -c0)).(\lambda (_: (arity g (CSort n) x1 (asucc g b))).(and3_ind (eq C c0 -(CSort n)) (eq nat (S (plus i x2)) O) (eq nat O O) (ex2_3 C T nat (\lambda -(d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda -(d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b)))))) -(\lambda (_: (eq C c0 (CSort n))).(\lambda (H11: (eq nat (S (plus i x2)) -O)).(\lambda (_: (eq nat O O)).(let H13 \def (eq_ind nat (S (plus i x2)) -(\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O -\Rightarrow False | (S _) \Rightarrow True])) I O H11) in (False_ind (ex2_3 C -T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d -c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 -(asucc g b)))))) H13))))) (drop_gen_sort n (S (plus i x2)) O c0 H8))))) -(\lambda (d: C).(\lambda (IHd: (((drop (S (plus i x2)) O d c0) \to ((arity g -d x1 (asucc g b)) \to (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: -T).(\lambda (j: nat).(drop (plus i j) O d0 c0)))) (\lambda (d0: C).(\lambda -(u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b)))))))))).(\lambda (k: -K).(\lambda (t1: T).(\lambda (H8: (drop (S (plus i x2)) O (CHead d k t1) -c0)).(\lambda (H9: (arity g (CHead d k t1) x1 (asucc g b))).(K_ind (\lambda -(k0: K).((arity g (CHead d k0 t1) x1 (asucc g b)) \to ((drop (r k0 (plus i -x2)) O d c0) \to (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: -nat).(drop (plus i j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda -(_: nat).(arity g d0 u0 (asucc g b))))))))) (\lambda (b0: B).(\lambda (H10: -(arity g (CHead d (Bind b0) t1) x1 (asucc g b))).(\lambda (H11: (drop (r -(Bind b0) (plus i x2)) O d c0)).(ex2_3_intro C T nat (\lambda (d0: -C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d0 c0)))) (\lambda -(d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b))))) -(CHead d (Bind b0) t1) x1 (S x2) (eq_ind nat (S (plus i x2)) (\lambda (n: -nat).(drop n O (CHead d (Bind b0) t1) c0)) (drop_drop (Bind b0) (plus i x2) d -c0 H11 t1) (plus i (S x2)) (plus_n_Sm i x2)) H10)))) (\lambda (f: F).(\lambda -(H10: (arity g (CHead d (Flat f) t1) x1 (asucc g b))).(\lambda (H11: (drop (r -(Flat f) (plus i x2)) O d c0)).(let H12 \def (IHd H11 (arity_cimp_conf g -(CHead d (Flat f) t1) x1 (asucc g b) H10 d (cimp_flat_sx f d t1))) in -(ex2_3_ind C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop -(plus i j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: -nat).(arity g d0 u0 (asucc g b))))) (ex2_3 C T nat (\lambda (d0: C).(\lambda -(_: T).(\lambda (j: nat).(drop (plus i j) O d0 c0)))) (\lambda (d0: -C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b)))))) -(\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: nat).(\lambda (H13: (drop -(plus i x5) O x3 c0)).(\lambda (H14: (arity g x3 x4 (asucc g -b))).(ex2_3_intro C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: -nat).(drop (plus i j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda -(_: nat).(arity g d0 u0 (asucc g b))))) x3 x4 x5 H13 H14)))))) H12))))) k H9 -(drop_gen_drop k d c0 t1 (plus i x2) H8)))))))) x0 H6 H7)))))) -H5)))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a0: A).(\lambda -(_: (arity g c0 u (asucc g a0))).(\lambda (_: ((\forall (i: nat).(\forall (b: -A).((aprem i (asucc g a0) b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: -T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda -(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))))))))).(\lambda (t0: -T).(\lambda (_: (arity g c0 t0 a0)).(\lambda (H3: ((\forall (i: nat).(\forall -(b: A).((aprem i a0 b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: -T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda -(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))))))))).(\lambda (i: -nat).(\lambda (b: A).(\lambda (H4: (aprem i a0 b)).(let H_x \def (H3 i b H4) -in (let H5 \def H_x in (ex2_3_ind C T nat (\lambda (d: C).(\lambda (_: -T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda -(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))) (ex2_3 C T nat -(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) -(\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g -b)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H6: -(drop (plus i x2) O x0 c0)).(\lambda (H7: (arity g x0 x1 (asucc g -b))).(ex2_3_intro C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: -nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda -(_: nat).(arity g d u0 (asucc g b))))) x0 x1 x2 H6 H7)))))) H5)))))))))))))) -(\lambda (c0: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda (_: (arity g c0 -t0 a1)).(\lambda (H1: ((\forall (i: nat).(\forall (b: A).((aprem i a1 b) \to -(ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus -i j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d -u (asucc g b))))))))))).(\lambda (a2: A).(\lambda (H2: (leq g a1 -a2)).(\lambda (i: nat).(\lambda (b: A).(\lambda (H3: (aprem i a2 b)).(let H_x -\def (aprem_repl g a1 a2 H2 i b H3) in (let H4 \def H_x in (ex2_ind A -(\lambda (b1: A).(leq g b1 b)) (\lambda (b1: A).(aprem i a1 b1)) (ex2_3 C T -nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d -c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc -g b)))))) (\lambda (x: A).(\lambda (H5: (leq g x b)).(\lambda (H6: (aprem i -a1 x)).(let H_x0 \def (H1 i x H6) in (let H7 \def H_x0 in (ex2_3_ind C T nat -(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) -(\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g -x))))) (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop -(plus i j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: -nat).(arity g d u (asucc g b)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda -(x2: nat).(\lambda (H8: (drop (plus i x2) O x0 c0)).(\lambda (H9: (arity g x0 -x1 (asucc g x))).(ex2_3_intro C T nat (\lambda (d: C).(\lambda (_: -T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u: -T).(\lambda (_: nat).(arity g d u (asucc g b))))) x0 x1 x2 H8 (arity_repl g -x0 x1 (asucc g x) H9 (asucc g b) (asucc_repl g x b H5)))))))) H7)))))) -H4))))))))))))) c t a H))))). -(* COMMENTS -Initial nodes: 4526 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/arity/cimp.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/arity/cimp.ma deleted file mode 100644 index c6212ac15..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/arity/cimp.ma +++ /dev/null @@ -1,102 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/arity/defs.ma". - -include "Basic-1/cimp/props.ma". - -theorem arity_cimp_conf: - \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1 -t a) \to (\forall (c2: C).((cimp c1 c2) \to (arity g c2 t a))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: -(arity g c1 t a)).(arity_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda (a0: -A).(\forall (c2: C).((cimp c c2) \to (arity g c2 t0 a0)))))) (\lambda (c: -C).(\lambda (n: nat).(\lambda (c2: C).(\lambda (_: (cimp c c2)).(arity_sort g -c2 n))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: -nat).(\lambda (H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (a0: -A).(\lambda (_: (arity g d u a0)).(\lambda (H2: ((\forall (c2: C).((cimp d -c2) \to (arity g c2 u a0))))).(\lambda (c2: C).(\lambda (H3: (cimp c -c2)).(let H_x \def (H3 Abbr d u i H0) in (let H4 \def H_x in (ex_ind C -(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (arity g c2 (TLRef i) -a0) (\lambda (x: C).(\lambda (H5: (getl i c2 (CHead x (Bind Abbr) u))).(let -H_x0 \def (cimp_getl_conf c c2 H3 Abbr d u i H0) in (let H6 \def H_x0 in -(ex2_ind C (\lambda (d2: C).(cimp d d2)) (\lambda (d2: C).(getl i c2 (CHead -d2 (Bind Abbr) u))) (arity g c2 (TLRef i) a0) (\lambda (x0: C).(\lambda (H7: -(cimp d x0)).(\lambda (H8: (getl i c2 (CHead x0 (Bind Abbr) u))).(let H9 \def -(eq_ind C (CHead x (Bind Abbr) u) (\lambda (c0: C).(getl i c2 c0)) H5 (CHead -x0 (Bind Abbr) u) (getl_mono c2 (CHead x (Bind Abbr) u) i H5 (CHead x0 (Bind -Abbr) u) H8)) in (let H10 \def (f_equal C C (\lambda (e: C).(match e in C -return (\lambda (_: C).C) with [(CSort _) \Rightarrow x | (CHead c0 _ _) -\Rightarrow c0])) (CHead x (Bind Abbr) u) (CHead x0 (Bind Abbr) u) (getl_mono -c2 (CHead x (Bind Abbr) u) i H5 (CHead x0 (Bind Abbr) u) H8)) in (let H11 -\def (eq_ind_r C x0 (\lambda (c0: C).(getl i c2 (CHead c0 (Bind Abbr) u))) H9 -x H10) in (let H12 \def (eq_ind_r C x0 (\lambda (c0: C).(cimp d c0)) H7 x -H10) in (arity_abbr g c2 x u i H11 a0 (H2 x H12))))))))) H6))))) -H4))))))))))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: -nat).(\lambda (H0: (getl i c (CHead d (Bind Abst) u))).(\lambda (a0: -A).(\lambda (_: (arity g d u (asucc g a0))).(\lambda (H2: ((\forall (c2: -C).((cimp d c2) \to (arity g c2 u (asucc g a0)))))).(\lambda (c2: C).(\lambda -(H3: (cimp c c2)).(let H_x \def (H3 Abst d u i H0) in (let H4 \def H_x in -(ex_ind C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) (arity g c2 -(TLRef i) a0) (\lambda (x: C).(\lambda (H5: (getl i c2 (CHead x (Bind Abst) -u))).(let H_x0 \def (cimp_getl_conf c c2 H3 Abst d u i H0) in (let H6 \def -H_x0 in (ex2_ind C (\lambda (d2: C).(cimp d d2)) (\lambda (d2: C).(getl i c2 -(CHead d2 (Bind Abst) u))) (arity g c2 (TLRef i) a0) (\lambda (x0: -C).(\lambda (H7: (cimp d x0)).(\lambda (H8: (getl i c2 (CHead x0 (Bind Abst) -u))).(let H9 \def (eq_ind C (CHead x (Bind Abst) u) (\lambda (c0: C).(getl i -c2 c0)) H5 (CHead x0 (Bind Abst) u) (getl_mono c2 (CHead x (Bind Abst) u) i -H5 (CHead x0 (Bind Abst) u) H8)) in (let H10 \def (f_equal C C (\lambda (e: -C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow x | -(CHead c0 _ _) \Rightarrow c0])) (CHead x (Bind Abst) u) (CHead x0 (Bind -Abst) u) (getl_mono c2 (CHead x (Bind Abst) u) i H5 (CHead x0 (Bind Abst) u) -H8)) in (let H11 \def (eq_ind_r C x0 (\lambda (c0: C).(getl i c2 (CHead c0 -(Bind Abst) u))) H9 x H10) in (let H12 \def (eq_ind_r C x0 (\lambda (c0: -C).(cimp d c0)) H7 x H10) in (arity_abst g c2 x u i H11 a0 (H2 x H12))))))))) -H6))))) H4))))))))))))) (\lambda (b: B).(\lambda (H0: (not (eq B b -Abst))).(\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity -g c u a1)).(\lambda (H2: ((\forall (c2: C).((cimp c c2) \to (arity g c2 u -a1))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c -(Bind b) u) t0 a2)).(\lambda (H4: ((\forall (c2: C).((cimp (CHead c (Bind b) -u) c2) \to (arity g c2 t0 a2))))).(\lambda (c2: C).(\lambda (H5: (cimp c -c2)).(arity_bind g b H0 c2 u a1 (H2 c2 H5) t0 a2 (H4 (CHead c2 (Bind b) u) -(cimp_bind c c2 H5 b u)))))))))))))))) (\lambda (c: C).(\lambda (u: -T).(\lambda (a1: A).(\lambda (_: (arity g c u (asucc g a1))).(\lambda (H1: -((\forall (c2: C).((cimp c c2) \to (arity g c2 u (asucc g a1)))))).(\lambda -(t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c (Bind Abst) u) t0 -a2)).(\lambda (H3: ((\forall (c2: C).((cimp (CHead c (Bind Abst) u) c2) \to -(arity g c2 t0 a2))))).(\lambda (c2: C).(\lambda (H4: (cimp c -c2)).(arity_head g c2 u a1 (H1 c2 H4) t0 a2 (H3 (CHead c2 (Bind Abst) u) -(cimp_bind c c2 H4 Abst u)))))))))))))) (\lambda (c: C).(\lambda (u: -T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H1: ((\forall -(c2: C).((cimp c c2) \to (arity g c2 u a1))))).(\lambda (t0: T).(\lambda (a2: -A).(\lambda (_: (arity g c t0 (AHead a1 a2))).(\lambda (H3: ((\forall (c2: -C).((cimp c c2) \to (arity g c2 t0 (AHead a1 a2)))))).(\lambda (c2: -C).(\lambda (H4: (cimp c c2)).(arity_appl g c2 u a1 (H1 c2 H4) t0 a2 (H3 c2 -H4))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a0: A).(\lambda (_: -(arity g c u (asucc g a0))).(\lambda (H1: ((\forall (c2: C).((cimp c c2) \to -(arity g c2 u (asucc g a0)))))).(\lambda (t0: T).(\lambda (_: (arity g c t0 -a0)).(\lambda (H3: ((\forall (c2: C).((cimp c c2) \to (arity g c2 t0 -a0))))).(\lambda (c2: C).(\lambda (H4: (cimp c c2)).(arity_cast g c2 u a0 (H1 -c2 H4) t0 (H3 c2 H4)))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda -(a1: A).(\lambda (_: (arity g c t0 a1)).(\lambda (H1: ((\forall (c2: -C).((cimp c c2) \to (arity g c2 t0 a1))))).(\lambda (a2: A).(\lambda (H2: -(leq g a1 a2)).(\lambda (c2: C).(\lambda (H3: (cimp c c2)).(arity_repl g c2 -t0 a1 (H1 c2 H3) a2 H2)))))))))) c1 t a H))))). -(* COMMENTS -Initial nodes: 1505 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/arity/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/arity/defs.ma deleted file mode 100644 index 99324dcbc..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/arity/defs.ma +++ /dev/null @@ -1,45 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/leq/defs.ma". - -include "Basic-1/getl/defs.ma". - -inductive arity (g: G): C \to (T \to (A \to Prop)) \def -| arity_sort: \forall (c: C).(\forall (n: nat).(arity g c (TSort n) (ASort O -n))) -| arity_abbr: \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: -nat).((getl i c (CHead d (Bind Abbr) u)) \to (\forall (a: A).((arity g d u a) -\to (arity g c (TLRef i) a))))))) -| arity_abst: \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: -nat).((getl i c (CHead d (Bind Abst) u)) \to (\forall (a: A).((arity g d u -(asucc g a)) \to (arity g c (TLRef i) a))))))) -| arity_bind: \forall (b: B).((not (eq B b Abst)) \to (\forall (c: -C).(\forall (u: T).(\forall (a1: A).((arity g c u a1) \to (\forall (t: -T).(\forall (a2: A).((arity g (CHead c (Bind b) u) t a2) \to (arity g c -(THead (Bind b) u t) a2))))))))) -| arity_head: \forall (c: C).(\forall (u: T).(\forall (a1: A).((arity g c u -(asucc g a1)) \to (\forall (t: T).(\forall (a2: A).((arity g (CHead c (Bind -Abst) u) t a2) \to (arity g c (THead (Bind Abst) u t) (AHead a1 a2)))))))) -| arity_appl: \forall (c: C).(\forall (u: T).(\forall (a1: A).((arity g c u -a1) \to (\forall (t: T).(\forall (a2: A).((arity g c t (AHead a1 a2)) \to -(arity g c (THead (Flat Appl) u t) a2))))))) -| arity_cast: \forall (c: C).(\forall (u: T).(\forall (a: A).((arity g c u -(asucc g a)) \to (\forall (t: T).((arity g c t a) \to (arity g c (THead (Flat -Cast) u t) a)))))) -| arity_repl: \forall (c: C).(\forall (t: T).(\forall (a1: A).((arity g c t -a1) \to (\forall (a2: A).((leq g a1 a2) \to (arity g c t a2)))))). - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/arity/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/arity/fwd.ma deleted file mode 100644 index 31fa35f3c..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/arity/fwd.ma +++ /dev/null @@ -1,1163 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/arity/defs.ma". - -include "Basic-1/leq/asucc.ma". - -include "Basic-1/getl/drop.ma". - -theorem arity_gen_sort: - \forall (g: G).(\forall (c: C).(\forall (n: nat).(\forall (a: A).((arity g c -(TSort n) a) \to (leq g a (ASort O n)))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (n: nat).(\lambda (a: A).(\lambda -(H: (arity g c (TSort n) a)).(insert_eq T (TSort n) (\lambda (t: T).(arity g -c t a)) (\lambda (_: T).(leq g a (ASort O n))) (\lambda (y: T).(\lambda (H0: -(arity g c y a)).(arity_ind g (\lambda (_: C).(\lambda (t: T).(\lambda (a0: -A).((eq T t (TSort n)) \to (leq g a0 (ASort O n)))))) (\lambda (_: -C).(\lambda (n0: nat).(\lambda (H1: (eq T (TSort n0) (TSort n))).(let H2 \def -(f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with -[(TSort n1) \Rightarrow n1 | (TLRef _) \Rightarrow n0 | (THead _ _ _) -\Rightarrow n0])) (TSort n0) (TSort n) H1) in (eq_ind_r nat n (\lambda (n1: -nat).(leq g (ASort O n1) (ASort O n))) (leq_refl g (ASort O n)) n0 H2))))) -(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda -(_: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (a0: A).(\lambda (_: (arity -g d u a0)).(\lambda (_: (((eq T u (TSort n)) \to (leq g a0 (ASort O -n))))).(\lambda (H4: (eq T (TLRef i) (TSort n))).(let H5 \def (eq_ind T -(TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) -\Rightarrow False])) I (TSort n) H4) in (False_ind (leq g a0 (ASort O n)) -H5))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: -nat).(\lambda (_: (getl i c0 (CHead d (Bind Abst) u))).(\lambda (a0: -A).(\lambda (_: (arity g d u (asucc g a0))).(\lambda (_: (((eq T u (TSort n)) -\to (leq g (asucc g a0) (ASort O n))))).(\lambda (H4: (eq T (TLRef i) (TSort -n))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (TSort n) H4) in -(False_ind (leq g a0 (ASort O n)) H5))))))))))) (\lambda (b: B).(\lambda (_: -(not (eq B b Abst))).(\lambda (c0: C).(\lambda (u: T).(\lambda (a1: -A).(\lambda (_: (arity g c0 u a1)).(\lambda (_: (((eq T u (TSort n)) \to (leq -g a1 (ASort O n))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (_: (arity g -(CHead c0 (Bind b) u) t a2)).(\lambda (_: (((eq T t (TSort n)) \to (leq g a2 -(ASort O n))))).(\lambda (H6: (eq T (THead (Bind b) u t) (TSort n))).(let H7 -\def (eq_ind T (THead (Bind b) u t) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H6) in -(False_ind (leq g a2 (ASort O n)) H7)))))))))))))) (\lambda (c0: C).(\lambda -(u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u (asucc g a1))).(\lambda -(_: (((eq T u (TSort n)) \to (leq g (asucc g a1) (ASort O n))))).(\lambda (t: -T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u) t -a2)).(\lambda (_: (((eq T t (TSort n)) \to (leq g a2 (ASort O n))))).(\lambda -(H5: (eq T (THead (Bind Abst) u t) (TSort n))).(let H6 \def (eq_ind T (THead -(Bind Abst) u t) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) -with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ -_) \Rightarrow True])) I (TSort n) H5) in (False_ind (leq g (AHead a1 a2) -(ASort O n)) H6)))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a1: -A).(\lambda (_: (arity g c0 u a1)).(\lambda (_: (((eq T u (TSort n)) \to (leq -g a1 (ASort O n))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (_: (arity g -c0 t (AHead a1 a2))).(\lambda (_: (((eq T t (TSort n)) \to (leq g (AHead a1 -a2) (ASort O n))))).(\lambda (H5: (eq T (THead (Flat Appl) u t) (TSort -n))).(let H6 \def (eq_ind T (THead (Flat Appl) u t) (\lambda (ee: T).(match -ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | -(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) -H5) in (False_ind (leq g a2 (ASort O n)) H6)))))))))))) (\lambda (c0: -C).(\lambda (u: T).(\lambda (a0: A).(\lambda (_: (arity g c0 u (asucc g -a0))).(\lambda (_: (((eq T u (TSort n)) \to (leq g (asucc g a0) (ASort O -n))))).(\lambda (t: T).(\lambda (_: (arity g c0 t a0)).(\lambda (_: (((eq T t -(TSort n)) \to (leq g a0 (ASort O n))))).(\lambda (H5: (eq T (THead (Flat -Cast) u t) (TSort n))).(let H6 \def (eq_ind T (THead (Flat Cast) u t) -(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow -True])) I (TSort n) H5) in (False_ind (leq g a0 (ASort O n)) H6))))))))))) -(\lambda (c0: C).(\lambda (t: T).(\lambda (a1: A).(\lambda (H1: (arity g c0 t -a1)).(\lambda (H2: (((eq T t (TSort n)) \to (leq g a1 (ASort O -n))))).(\lambda (a2: A).(\lambda (H3: (leq g a1 a2)).(\lambda (H4: (eq T t -(TSort n))).(let H5 \def (f_equal T T (\lambda (e: T).e) t (TSort n) H4) in -(let H6 \def (eq_ind T t (\lambda (t0: T).((eq T t0 (TSort n)) \to (leq g a1 -(ASort O n)))) H2 (TSort n) H5) in (let H7 \def (eq_ind T t (\lambda (t0: -T).(arity g c0 t0 a1)) H1 (TSort n) H5) in (leq_trans g a2 a1 (leq_sym g a1 -a2 H3) (ASort O n) (H6 (refl_equal T (TSort n))))))))))))))) c y a H0))) -H))))). -(* COMMENTS -Initial nodes: 1235 -END *) - -theorem arity_gen_lref: - \forall (g: G).(\forall (c: C).(\forall (i: nat).(\forall (a: A).((arity g c -(TLRef i) a) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c -(CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u a)))) -(ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c (CHead d (Bind Abst) -u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a)))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (i: nat).(\lambda (a: A).(\lambda -(H: (arity g c (TLRef i) a)).(insert_eq T (TLRef i) (\lambda (t: T).(arity g -c t a)) (\lambda (_: T).(or (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl -i c (CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u -a)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c (CHead d (Bind -Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a))))))) -(\lambda (y: T).(\lambda (H0: (arity g c y a)).(arity_ind g (\lambda (c0: -C).(\lambda (t: T).(\lambda (a0: A).((eq T t (TLRef i)) \to (or (ex2_2 C T -(\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u)))) -(\lambda (d: C).(\lambda (u: T).(arity g d u a0)))) (ex2_2 C T (\lambda (d: -C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d: -C).(\lambda (u: T).(arity g d u (asucc g a0)))))))))) (\lambda (c0: -C).(\lambda (n: nat).(\lambda (H1: (eq T (TSort n) (TLRef i))).(let H2 \def -(eq_ind T (TSort n) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | -(THead _ _ _) \Rightarrow False])) I (TLRef i) H1) in (False_ind (or (ex2_2 C -T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u)))) -(\lambda (d: C).(\lambda (u: T).(arity g d u (ASort O n))))) (ex2_2 C T -(\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) -(\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g (ASort O n))))))) -H2))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i0: -nat).(\lambda (H1: (getl i0 c0 (CHead d (Bind Abbr) u))).(\lambda (a0: -A).(\lambda (H2: (arity g d u a0)).(\lambda (_: (((eq T u (TLRef i)) \to (or -(ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl i d (CHead d0 (Bind Abbr) -u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a0)))) (ex2_2 C T -(\lambda (d0: C).(\lambda (u0: T).(getl i d (CHead d0 (Bind Abst) u0)))) -(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g -a0))))))))).(\lambda (H4: (eq T (TLRef i0) (TLRef i))).(let H5 \def (f_equal -T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort -_) \Rightarrow i0 | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow i0])) -(TLRef i0) (TLRef i) H4) in (let H6 \def (eq_ind nat i0 (\lambda (n: -nat).(getl n c0 (CHead d (Bind Abbr) u))) H1 i H5) in (or_introl (ex2_2 C T -(\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abbr) u0)))) -(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a0)))) (ex2_2 C T (\lambda -(d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0)))) (\lambda -(d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a0))))) (ex2_2_intro C T -(\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abbr) u0)))) -(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a0))) d u H6 H2))))))))))))) -(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i0: nat).(\lambda -(H1: (getl i0 c0 (CHead d (Bind Abst) u))).(\lambda (a0: A).(\lambda (H2: -(arity g d u (asucc g a0))).(\lambda (_: (((eq T u (TLRef i)) \to (or (ex2_2 -C T (\lambda (d0: C).(\lambda (u0: T).(getl i d (CHead d0 (Bind Abbr) u0)))) -(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a0))))) (ex2_2 C T -(\lambda (d0: C).(\lambda (u0: T).(getl i d (CHead d0 (Bind Abst) u0)))) -(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g (asucc g -a0)))))))))).(\lambda (H4: (eq T (TLRef i0) (TLRef i))).(let H5 \def (f_equal -T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort -_) \Rightarrow i0 | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow i0])) -(TLRef i0) (TLRef i) H4) in (let H6 \def (eq_ind nat i0 (\lambda (n: -nat).(getl n c0 (CHead d (Bind Abst) u))) H1 i H5) in (or_intror (ex2_2 C T -(\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abbr) u0)))) -(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a0)))) (ex2_2 C T (\lambda -(d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0)))) (\lambda -(d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a0))))) (ex2_2_intro C T -(\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0)))) -(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a0)))) d u H6 -H2))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda -(c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u -a1)).(\lambda (_: (((eq T u (TLRef i)) \to (or (ex2_2 C T (\lambda (d: -C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d: -C).(\lambda (u0: T).(arity g d u0 a1)))) (ex2_2 C T (\lambda (d: C).(\lambda -(u0: T).(getl i c0 (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0: -T).(arity g d u0 (asucc g a1))))))))).(\lambda (t: T).(\lambda (a2: -A).(\lambda (_: (arity g (CHead c0 (Bind b) u) t a2)).(\lambda (_: (((eq T t -(TLRef i)) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i (CHead -c0 (Bind b) u) (CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: -T).(arity g d u0 a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i -(CHead c0 (Bind b) u) (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda -(u0: T).(arity g d u0 (asucc g a2))))))))).(\lambda (H6: (eq T (THead (Bind -b) u t) (TLRef i))).(let H7 \def (eq_ind T (THead (Bind b) u t) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I -(TLRef i) H6) in (False_ind (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: -T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: -T).(arity g d u0 a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i -c0 (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 -(asucc g a2)))))) H7)))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda -(a1: A).(\lambda (_: (arity g c0 u (asucc g a1))).(\lambda (_: (((eq T u -(TLRef i)) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 -(CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 -(asucc g a1))))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 -(CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 -(asucc g (asucc g a1)))))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (_: -(arity g (CHead c0 (Bind Abst) u) t a2)).(\lambda (_: (((eq T t (TLRef i)) -\to (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i (CHead c0 (Bind -Abst) u) (CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity -g d u0 a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i (CHead c0 -(Bind Abst) u) (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0: -T).(arity g d u0 (asucc g a2))))))))).(\lambda (H5: (eq T (THead (Bind Abst) -u t) (TLRef i))).(let H6 \def (eq_ind T (THead (Bind Abst) u t) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I -(TLRef i) H5) in (False_ind (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: -T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: -T).(arity g d u0 (AHead a1 a2))))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: -T).(getl i c0 (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0: -T).(arity g d u0 (asucc g (AHead a1 a2))))))) H6)))))))))))) (\lambda (c0: -C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u a1)).(\lambda -(_: (((eq T u (TLRef i)) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: -T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: -T).(arity g d u0 a1)))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i -c0 (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 -(asucc g a1))))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (_: (arity g -c0 t (AHead a1 a2))).(\lambda (_: (((eq T t (TLRef i)) \to (or (ex2_2 C T -(\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abbr) u0)))) -(\lambda (d: C).(\lambda (u0: T).(arity g d u0 (AHead a1 a2))))) (ex2_2 C T -(\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abst) u0)))) -(\lambda (d: C).(\lambda (u0: T).(arity g d u0 (asucc g (AHead a1 -a2)))))))))).(\lambda (H5: (eq T (THead (Flat Appl) u t) (TLRef i))).(let H6 -\def (eq_ind T (THead (Flat Appl) u t) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef i) H5) in -(False_ind (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead -d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 a2)))) -(ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abst) -u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 (asucc g a2)))))) -H6)))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a0: A).(\lambda (_: -(arity g c0 u (asucc g a0))).(\lambda (_: (((eq T u (TLRef i)) \to (or (ex2_2 -C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abbr) u0)))) -(\lambda (d: C).(\lambda (u0: T).(arity g d u0 (asucc g a0))))) (ex2_2 C T -(\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abst) u0)))) -(\lambda (d: C).(\lambda (u0: T).(arity g d u0 (asucc g (asucc g -a0)))))))))).(\lambda (t: T).(\lambda (_: (arity g c0 t a0)).(\lambda (_: -(((eq T t (TLRef i)) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: -T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: -T).(arity g d u0 a0)))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i -c0 (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 -(asucc g a0))))))))).(\lambda (H5: (eq T (THead (Flat Cast) u t) (TLRef -i))).(let H6 \def (eq_ind T (THead (Flat Cast) u t) (\lambda (ee: T).(match -ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | -(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef i) -H5) in (False_ind (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 -(CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 -a0)))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind -Abst) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 (asucc g a0)))))) -H6))))))))))) (\lambda (c0: C).(\lambda (t: T).(\lambda (a1: A).(\lambda (H1: -(arity g c0 t a1)).(\lambda (H2: (((eq T t (TLRef i)) \to (or (ex2_2 C T -(\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u)))) -(\lambda (d: C).(\lambda (u: T).(arity g d u a1)))) (ex2_2 C T (\lambda (d: -C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d: -C).(\lambda (u: T).(arity g d u (asucc g a1))))))))).(\lambda (a2: -A).(\lambda (H3: (leq g a1 a2)).(\lambda (H4: (eq T t (TLRef i))).(let H5 -\def (f_equal T T (\lambda (e: T).e) t (TLRef i) H4) in (let H6 \def (eq_ind -T t (\lambda (t0: T).((eq T t0 (TLRef i)) \to (or (ex2_2 C T (\lambda (d: -C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u)))) (\lambda (d: -C).(\lambda (u: T).(arity g d u a1)))) (ex2_2 C T (\lambda (d: C).(\lambda -(u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u: -T).(arity g d u (asucc g a1)))))))) H2 (TLRef i) H5) in (let H7 \def (eq_ind -T t (\lambda (t0: T).(arity g c0 t0 a1)) H1 (TLRef i) H5) in (let H8 \def (H6 -(refl_equal T (TLRef i))) in (or_ind (ex2_2 C T (\lambda (d: C).(\lambda (u: -T).(getl i c0 (CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: -T).(arity g d u a1)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 -(CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u -(asucc g a1))))) (or (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 -(CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u -a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind -Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a2)))))) -(\lambda (H9: (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d -(Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u -a1))))).(ex2_2_ind C T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d -(Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u a1))) (or -(ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) -u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u a2)))) (ex2_2 C T (\lambda -(d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d: -C).(\lambda (u: T).(arity g d u (asucc g a2)))))) (\lambda (x0: C).(\lambda -(x1: T).(\lambda (H10: (getl i c0 (CHead x0 (Bind Abbr) x1))).(\lambda (H11: -(arity g x0 x1 a1)).(or_introl (ex2_2 C T (\lambda (d: C).(\lambda (u: -T).(getl i c0 (CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: -T).(arity g d u a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 -(CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u -(asucc g a2))))) (ex2_2_intro C T (\lambda (d: C).(\lambda (u: T).(getl i c0 -(CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u a2))) -x0 x1 H10 (arity_repl g x0 x1 a1 H11 a2 H3))))))) H9)) (\lambda (H9: (ex2_2 C -T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) -(\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a1)))))).(ex2_2_ind C T -(\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) -(\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a1)))) (or (ex2_2 C T -(\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u)))) -(\lambda (d: C).(\lambda (u: T).(arity g d u a2)))) (ex2_2 C T (\lambda (d: -C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d: -C).(\lambda (u: T).(arity g d u (asucc g a2)))))) (\lambda (x0: C).(\lambda -(x1: T).(\lambda (H10: (getl i c0 (CHead x0 (Bind Abst) x1))).(\lambda (H11: -(arity g x0 x1 (asucc g a1))).(or_intror (ex2_2 C T (\lambda (d: C).(\lambda -(u: T).(getl i c0 (CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: -T).(arity g d u a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 -(CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u -(asucc g a2))))) (ex2_2_intro C T (\lambda (d: C).(\lambda (u: T).(getl i c0 -(CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u -(asucc g a2)))) x0 x1 H10 (arity_repl g x0 x1 (asucc g a1) H11 (asucc g a2) -(asucc_repl g a1 a2 H3)))))))) H9)) H8))))))))))))) c y a H0))) H))))). -(* COMMENTS -Initial nodes: 3853 -END *) - -theorem arity_gen_bind: - \forall (b: B).((not (eq B b Abst)) \to (\forall (g: G).(\forall (c: -C).(\forall (u: T).(\forall (t: T).(\forall (a2: A).((arity g c (THead (Bind -b) u t) a2) \to (ex2 A (\lambda (a1: A).(arity g c u a1)) (\lambda (_: -A).(arity g (CHead c (Bind b) u) t a2)))))))))) -\def - \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (g: G).(\lambda -(c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (a2: A).(\lambda (H0: (arity -g c (THead (Bind b) u t) a2)).(insert_eq T (THead (Bind b) u t) (\lambda (t0: -T).(arity g c t0 a2)) (\lambda (_: T).(ex2 A (\lambda (a1: A).(arity g c u -a1)) (\lambda (_: A).(arity g (CHead c (Bind b) u) t a2)))) (\lambda (y: -T).(\lambda (H1: (arity g c y a2)).(arity_ind g (\lambda (c0: C).(\lambda -(t0: T).(\lambda (a: A).((eq T t0 (THead (Bind b) u t)) \to (ex2 A (\lambda -(a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t -a))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (H2: (eq T (TSort n) -(THead (Bind b) u t))).(let H3 \def (eq_ind T (TSort n) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I -(THead (Bind b) u t) H2) in (False_ind (ex2 A (\lambda (a1: A).(arity g c0 u -a1)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t (ASort O n)))) H3))))) -(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda -(_: (getl i c0 (CHead d (Bind Abbr) u0))).(\lambda (a: A).(\lambda (_: (arity -g d u0 a)).(\lambda (_: (((eq T u0 (THead (Bind b) u t)) \to (ex2 A (\lambda -(a1: A).(arity g d u a1)) (\lambda (_: A).(arity g (CHead d (Bind b) u) t -a)))))).(\lambda (H5: (eq T (TLRef i) (THead (Bind b) u t))).(let H6 \def -(eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | -(THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t) H5) in (False_ind -(ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g (CHead c0 -(Bind b) u) t a))) H6))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda -(u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind Abst) -u0))).(\lambda (a: A).(\lambda (_: (arity g d u0 (asucc g a))).(\lambda (_: -(((eq T u0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a1: A).(arity g d u -a1)) (\lambda (_: A).(arity g (CHead d (Bind b) u) t (asucc g -a))))))).(\lambda (H5: (eq T (TLRef i) (THead (Bind b) u t))).(let H6 \def -(eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | -(THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t) H5) in (False_ind -(ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g (CHead c0 -(Bind b) u) t a))) H6))))))))))) (\lambda (b0: B).(\lambda (H2: (not (eq B b0 -Abst))).(\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (H3: -(arity g c0 u0 a1)).(\lambda (H4: (((eq T u0 (THead (Bind b) u t)) \to (ex2 A -(\lambda (a3: A).(arity g c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind -b) u) t a1)))))).(\lambda (t0: T).(\lambda (a0: A).(\lambda (H5: (arity g -(CHead c0 (Bind b0) u0) t0 a0)).(\lambda (H6: (((eq T t0 (THead (Bind b) u -t)) \to (ex2 A (\lambda (a3: A).(arity g (CHead c0 (Bind b0) u0) u a3)) -(\lambda (_: A).(arity g (CHead (CHead c0 (Bind b0) u0) (Bind b) u) t -a0)))))).(\lambda (H7: (eq T (THead (Bind b0) u0 t0) (THead (Bind b) u -t))).(let H8 \def (f_equal T B (\lambda (e: T).(match e in T return (\lambda -(_: T).B) with [(TSort _) \Rightarrow b0 | (TLRef _) \Rightarrow b0 | (THead -k _ _) \Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b1) -\Rightarrow b1 | (Flat _) \Rightarrow b0])])) (THead (Bind b0) u0 t0) (THead -(Bind b) u t) H7) in ((let H9 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) -\Rightarrow u0 | (THead _ t1 _) \Rightarrow t1])) (THead (Bind b0) u0 t0) -(THead (Bind b) u t) H7) in ((let H10 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | -(TLRef _) \Rightarrow t0 | (THead _ _ t1) \Rightarrow t1])) (THead (Bind b0) -u0 t0) (THead (Bind b) u t) H7) in (\lambda (H11: (eq T u0 u)).(\lambda (H12: -(eq B b0 b)).(let H13 \def (eq_ind T t0 (\lambda (t1: T).((eq T t1 (THead -(Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g (CHead c0 (Bind b0) u0) u -a3)) (\lambda (_: A).(arity g (CHead (CHead c0 (Bind b0) u0) (Bind b) u) t -a0))))) H6 t H10) in (let H14 \def (eq_ind T t0 (\lambda (t1: T).(arity g -(CHead c0 (Bind b0) u0) t1 a0)) H5 t H10) in (let H15 \def (eq_ind T u0 -(\lambda (t1: T).((eq T t (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: -A).(arity g (CHead c0 (Bind b0) t1) u a3)) (\lambda (_: A).(arity g (CHead -(CHead c0 (Bind b0) t1) (Bind b) u) t a0))))) H13 u H11) in (let H16 \def -(eq_ind T u0 (\lambda (t1: T).(arity g (CHead c0 (Bind b0) t1) t a0)) H14 u -H11) in (let H17 \def (eq_ind T u0 (\lambda (t1: T).((eq T t1 (THead (Bind b) -u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (_: A).(arity g -(CHead c0 (Bind b) u) t a1))))) H4 u H11) in (let H18 \def (eq_ind T u0 -(\lambda (t1: T).(arity g c0 t1 a1)) H3 u H11) in (let H19 \def (eq_ind B b0 -(\lambda (b1: B).((eq T t (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: -A).(arity g (CHead c0 (Bind b1) u) u a3)) (\lambda (_: A).(arity g (CHead -(CHead c0 (Bind b1) u) (Bind b) u) t a0))))) H15 b H12) in (let H20 \def -(eq_ind B b0 (\lambda (b1: B).(arity g (CHead c0 (Bind b1) u) t a0)) H16 b -H12) in (let H21 \def (eq_ind B b0 (\lambda (b1: B).(not (eq B b1 Abst))) H2 -b H12) in (ex_intro2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (_: -A).(arity g (CHead c0 (Bind b) u) t a0)) a1 H18 H20))))))))))))) H9)) -H8)))))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: A).(\lambda -(H2: (arity g c0 u0 (asucc g a1))).(\lambda (H3: (((eq T u0 (THead (Bind b) u -t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (_: A).(arity g -(CHead c0 (Bind b) u) t (asucc g a1))))))).(\lambda (t0: T).(\lambda (a0: -A).(\lambda (H4: (arity g (CHead c0 (Bind Abst) u0) t0 a0)).(\lambda (H5: -(((eq T t0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g (CHead -c0 (Bind Abst) u0) u a3)) (\lambda (_: A).(arity g (CHead (CHead c0 (Bind -Abst) u0) (Bind b) u) t a0)))))).(\lambda (H6: (eq T (THead (Bind Abst) u0 -t0) (THead (Bind b) u t))).(let H7 \def (f_equal T B (\lambda (e: T).(match e -in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow Abst | (TLRef _) -\Rightarrow Abst | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abst])])) -(THead (Bind Abst) u0 t0) (THead (Bind b) u t) H6) in ((let H8 \def (f_equal -T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t1 _) \Rightarrow t1])) -(THead (Bind Abst) u0 t0) (THead (Bind b) u t) H6) in ((let H9 \def (f_equal -T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t1) \Rightarrow t1])) -(THead (Bind Abst) u0 t0) (THead (Bind b) u t) H6) in (\lambda (H10: (eq T u0 -u)).(\lambda (H11: (eq B Abst b)).(let H12 \def (eq_ind T t0 (\lambda (t1: -T).((eq T t1 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g -(CHead c0 (Bind Abst) u0) u a3)) (\lambda (_: A).(arity g (CHead (CHead c0 -(Bind Abst) u0) (Bind b) u) t a0))))) H5 t H9) in (let H13 \def (eq_ind T t0 -(\lambda (t1: T).(arity g (CHead c0 (Bind Abst) u0) t1 a0)) H4 t H9) in (let -H14 \def (eq_ind T u0 (\lambda (t1: T).((eq T t (THead (Bind b) u t)) \to -(ex2 A (\lambda (a3: A).(arity g (CHead c0 (Bind Abst) t1) u a3)) (\lambda -(_: A).(arity g (CHead (CHead c0 (Bind Abst) t1) (Bind b) u) t a0))))) H12 u -H10) in (let H15 \def (eq_ind T u0 (\lambda (t1: T).(arity g (CHead c0 (Bind -Abst) t1) t a0)) H13 u H10) in (let H16 \def (eq_ind T u0 (\lambda (t1: -T).((eq T t1 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u -a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t (asucc g a1)))))) H3 u -H10) in (let H17 \def (eq_ind T u0 (\lambda (t1: T).(arity g c0 t1 (asucc g -a1))) H2 u H10) in (let H18 \def (eq_ind_r B b (\lambda (b0: B).((eq T t -(THead (Bind b0) u t)) \to (ex2 A (\lambda (a3: A).(arity g (CHead c0 (Bind -Abst) u) u a3)) (\lambda (_: A).(arity g (CHead (CHead c0 (Bind Abst) u) -(Bind b0) u) t a0))))) H14 Abst H11) in (let H19 \def (eq_ind_r B b (\lambda -(b0: B).((eq T u (THead (Bind b0) u t)) \to (ex2 A (\lambda (a3: A).(arity g -c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b0) u) t (asucc g a1)))))) -H16 Abst H11) in (let H20 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 -Abst))) H Abst H11) in (eq_ind B Abst (\lambda (b0: B).(ex2 A (\lambda (a3: -A).(arity g c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b0) u) t -(AHead a1 a0))))) (let H21 \def (match (H20 (refl_equal B Abst)) in False -return (\lambda (_: False).(ex2 A (\lambda (a3: A).(arity g c0 u a3)) -(\lambda (_: A).(arity g (CHead c0 (Bind Abst) u) t (AHead a1 a0))))) with -[]) in H21) b H11))))))))))))) H8)) H7)))))))))))) (\lambda (c0: C).(\lambda -(u0: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u0 a1)).(\lambda (_: (((eq -T u0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) -(\lambda (_: A).(arity g (CHead c0 (Bind b) u) t a1)))))).(\lambda (t0: -T).(\lambda (a0: A).(\lambda (_: (arity g c0 t0 (AHead a1 a0))).(\lambda (_: -(((eq T t0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u -a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t (AHead a1 -a0))))))).(\lambda (H6: (eq T (THead (Flat Appl) u0 t0) (THead (Bind b) u -t))).(let H7 \def (eq_ind T (THead (Flat Appl) u0 t0) (\lambda (ee: T).(match -ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | -(TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return -(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind b) u t) H6) in (False_ind (ex2 A (\lambda (a3: -A).(arity g c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t a0))) -H7)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a: A).(\lambda (_: -(arity g c0 u0 (asucc g a))).(\lambda (_: (((eq T u0 (THead (Bind b) u t)) -\to (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g -(CHead c0 (Bind b) u) t (asucc g a))))))).(\lambda (t0: T).(\lambda (_: -(arity g c0 t0 a)).(\lambda (_: (((eq T t0 (THead (Bind b) u t)) \to (ex2 A -(\lambda (a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g (CHead c0 (Bind -b) u) t a)))))).(\lambda (H6: (eq T (THead (Flat Cast) u0 t0) (THead (Bind b) -u t))).(let H7 \def (eq_ind T (THead (Flat Cast) u0 t0) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K -return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) -\Rightarrow True])])) I (THead (Bind b) u t) H6) in (False_ind (ex2 A -(\lambda (a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g (CHead c0 (Bind -b) u) t a))) H7))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (a1: -A).(\lambda (H2: (arity g c0 t0 a1)).(\lambda (H3: (((eq T t0 (THead (Bind b) -u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (_: A).(arity g -(CHead c0 (Bind b) u) t a1)))))).(\lambda (a0: A).(\lambda (H4: (leq g a1 -a0)).(\lambda (H5: (eq T t0 (THead (Bind b) u t))).(let H6 \def (f_equal T T -(\lambda (e: T).e) t0 (THead (Bind b) u t) H5) in (let H7 \def (eq_ind T t0 -(\lambda (t1: T).((eq T t1 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: -A).(arity g c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t -a1))))) H3 (THead (Bind b) u t) H6) in (let H8 \def (eq_ind T t0 (\lambda -(t1: T).(arity g c0 t1 a1)) H2 (THead (Bind b) u t) H6) in (let H9 \def (H7 -(refl_equal T (THead (Bind b) u t))) in (ex2_ind A (\lambda (a3: A).(arity g -c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t a1)) (ex2 A -(\lambda (a3: A).(arity g c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind -b) u) t a0))) (\lambda (x: A).(\lambda (H10: (arity g c0 u x)).(\lambda (H11: -(arity g (CHead c0 (Bind b) u) t a1)).(ex_intro2 A (\lambda (a3: A).(arity g -c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t a0)) x H10 -(arity_repl g (CHead c0 (Bind b) u) t a1 H11 a0 H4))))) H9))))))))))))) c y -a2 H1))) H0)))))))). -(* COMMENTS -Initial nodes: 3365 -END *) - -theorem arity_gen_abst: - \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (a: -A).((arity g c (THead (Bind Abst) u t) a) \to (ex3_2 A A (\lambda (a1: -A).(\lambda (a2: A).(eq A a (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: -A).(arity g c u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g -(CHead c (Bind Abst) u) t a2))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (a: -A).(\lambda (H: (arity g c (THead (Bind Abst) u t) a)).(insert_eq T (THead -(Bind Abst) u t) (\lambda (t0: T).(arity g c t0 a)) (\lambda (_: T).(ex3_2 A -A (\lambda (a1: A).(\lambda (a2: A).(eq A a (AHead a1 a2)))) (\lambda (a1: -A).(\lambda (_: A).(arity g c u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: -A).(arity g (CHead c (Bind Abst) u) t a2))))) (\lambda (y: T).(\lambda (H0: -(arity g c y a)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (a0: -A).((eq T t0 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a1: -A).(\lambda (a2: A).(eq A a0 (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: -A).(arity g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g -(CHead c0 (Bind Abst) u) t a2)))))))) (\lambda (c0: C).(\lambda (n: -nat).(\lambda (H1: (eq T (TSort n) (THead (Bind Abst) u t))).(let H2 \def -(eq_ind T (TSort n) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | -(THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) u t) H1) in -(False_ind (ex3_2 A A (\lambda (a1: A).(\lambda (a2: A).(eq A (ASort O n) -(AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g c0 u (asucc g -a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g (CHead c0 (Bind Abst) u) t -a2)))) H2))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: -nat).(\lambda (_: (getl i c0 (CHead d (Bind Abbr) u0))).(\lambda (a0: -A).(\lambda (_: (arity g d u0 a0)).(\lambda (_: (((eq T u0 (THead (Bind Abst) -u t)) \to (ex3_2 A A (\lambda (a1: A).(\lambda (a2: A).(eq A a0 (AHead a1 -a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g d u (asucc g a1)))) (\lambda -(_: A).(\lambda (a2: A).(arity g (CHead d (Bind Abst) u) t a2))))))).(\lambda -(H4: (eq T (TLRef i) (THead (Bind Abst) u t))).(let H5 \def (eq_ind T (TLRef -i) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort -_) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow -False])) I (THead (Bind Abst) u t) H4) in (False_ind (ex3_2 A A (\lambda (a1: -A).(\lambda (a2: A).(eq A a0 (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: -A).(arity g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g -(CHead c0 (Bind Abst) u) t a2)))) H5))))))))))) (\lambda (c0: C).(\lambda (d: -C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind -Abst) u0))).(\lambda (a0: A).(\lambda (_: (arity g d u0 (asucc g -a0))).(\lambda (_: (((eq T u0 (THead (Bind Abst) u t)) \to (ex3_2 A A -(\lambda (a1: A).(\lambda (a2: A).(eq A (asucc g a0) (AHead a1 a2)))) -(\lambda (a1: A).(\lambda (_: A).(arity g d u (asucc g a1)))) (\lambda (_: -A).(\lambda (a2: A).(arity g (CHead d (Bind Abst) u) t a2))))))).(\lambda -(H4: (eq T (TLRef i) (THead (Bind Abst) u t))).(let H5 \def (eq_ind T (TLRef -i) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort -_) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow -False])) I (THead (Bind Abst) u t) H4) in (False_ind (ex3_2 A A (\lambda (a1: -A).(\lambda (a2: A).(eq A a0 (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: -A).(arity g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g -(CHead c0 (Bind Abst) u) t a2)))) H5))))))))))) (\lambda (b: B).(\lambda (H1: -(not (eq B b Abst))).(\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: -A).(\lambda (H2: (arity g c0 u0 a1)).(\lambda (H3: (((eq T u0 (THead (Bind -Abst) u t)) \to (ex3_2 A A (\lambda (a2: A).(\lambda (a3: A).(eq A a1 (AHead -a2 a3)))) (\lambda (a2: A).(\lambda (_: A).(arity g c0 u (asucc g a2)))) -(\lambda (_: A).(\lambda (a3: A).(arity g (CHead c0 (Bind Abst) u) t -a3))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (H4: (arity g (CHead c0 -(Bind b) u0) t0 a2)).(\lambda (H5: (((eq T t0 (THead (Bind Abst) u t)) \to -(ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) -(\lambda (a3: A).(\lambda (_: A).(arity g (CHead c0 (Bind b) u0) u (asucc g -a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead (CHead c0 (Bind b) -u0) (Bind Abst) u) t a4))))))).(\lambda (H6: (eq T (THead (Bind b) u0 t0) -(THead (Bind Abst) u t))).(let H7 \def (f_equal T B (\lambda (e: T).(match e -in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | (TLRef _) -\Rightarrow b | (THead k _ _) \Rightarrow (match k in K return (\lambda (_: -K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow b])])) (THead -(Bind b) u0 t0) (THead (Bind Abst) u t) H6) in ((let H8 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t1 _) \Rightarrow t1])) -(THead (Bind b) u0 t0) (THead (Bind Abst) u t) H6) in ((let H9 \def (f_equal -T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t1) \Rightarrow t1])) -(THead (Bind b) u0 t0) (THead (Bind Abst) u t) H6) in (\lambda (H10: (eq T u0 -u)).(\lambda (H11: (eq B b Abst)).(let H12 \def (eq_ind T t0 (\lambda (t1: -T).((eq T t1 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3: -A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: -A).(arity g (CHead c0 (Bind b) u0) u (asucc g a3)))) (\lambda (_: A).(\lambda -(a4: A).(arity g (CHead (CHead c0 (Bind b) u0) (Bind Abst) u) t a4)))))) H5 t -H9) in (let H13 \def (eq_ind T t0 (\lambda (t1: T).(arity g (CHead c0 (Bind -b) u0) t1 a2)) H4 t H9) in (let H14 \def (eq_ind T u0 (\lambda (t1: T).((eq T -t (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: -A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g (CHead -c0 (Bind b) t1) u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g -(CHead (CHead c0 (Bind b) t1) (Bind Abst) u) t a4)))))) H12 u H10) in (let -H15 \def (eq_ind T u0 (\lambda (t1: T).(arity g (CHead c0 (Bind b) t1) t a2)) -H13 u H10) in (let H16 \def (eq_ind T u0 (\lambda (t1: T).((eq T t1 (THead -(Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a1 -(AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g -a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t -a4)))))) H3 u H10) in (let H17 \def (eq_ind T u0 (\lambda (t1: T).(arity g c0 -t1 a1)) H2 u H10) in (let H18 \def (eq_ind B b (\lambda (b0: B).((eq T t -(THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq -A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g (CHead c0 -(Bind b0) u) u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g -(CHead (CHead c0 (Bind b0) u) (Bind Abst) u) t a4)))))) H14 Abst H11) in (let -H19 \def (eq_ind B b (\lambda (b0: B).(arity g (CHead c0 (Bind b0) u) t a2)) -H15 Abst H11) in (let H20 \def (eq_ind B b (\lambda (b0: B).(not (eq B b0 -Abst))) H1 Abst H11) in (let H21 \def (match (H20 (refl_equal B Abst)) in -False return (\lambda (_: False).(ex3_2 A A (\lambda (a3: A).(\lambda (a4: -A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u -(asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind -Abst) u) t a4))))) with []) in H21))))))))))))) H8)) H7)))))))))))))) -(\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (H1: (arity g c0 -u0 (asucc g a1))).(\lambda (H2: (((eq T u0 (THead (Bind Abst) u t)) \to -(ex3_2 A A (\lambda (a2: A).(\lambda (a3: A).(eq A (asucc g a1) (AHead a2 -a3)))) (\lambda (a2: A).(\lambda (_: A).(arity g c0 u (asucc g a2)))) -(\lambda (_: A).(\lambda (a3: A).(arity g (CHead c0 (Bind Abst) u) t -a3))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (H3: (arity g (CHead c0 -(Bind Abst) u0) t0 a2)).(\lambda (H4: (((eq T t0 (THead (Bind Abst) u t)) \to -(ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) -(\lambda (a3: A).(\lambda (_: A).(arity g (CHead c0 (Bind Abst) u0) u (asucc -g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead (CHead c0 (Bind -Abst) u0) (Bind Abst) u) t a4))))))).(\lambda (H5: (eq T (THead (Bind Abst) -u0 t0) (THead (Bind Abst) u t))).(let H6 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | -(TLRef _) \Rightarrow u0 | (THead _ t1 _) \Rightarrow t1])) (THead (Bind -Abst) u0 t0) (THead (Bind Abst) u t) H5) in ((let H7 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t1) \Rightarrow t1])) -(THead (Bind Abst) u0 t0) (THead (Bind Abst) u t) H5) in (\lambda (H8: (eq T -u0 u)).(let H9 \def (eq_ind T t0 (\lambda (t1: T).((eq T t1 (THead (Bind -Abst) u t)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead -a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g (CHead c0 (Bind Abst) u0) -u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead (CHead c0 -(Bind Abst) u0) (Bind Abst) u) t a4)))))) H4 t H7) in (let H10 \def (eq_ind T -t0 (\lambda (t1: T).(arity g (CHead c0 (Bind Abst) u0) t1 a2)) H3 t H7) in -(let H11 \def (eq_ind T u0 (\lambda (t1: T).((eq T t (THead (Bind Abst) u t)) -\to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) -(\lambda (a3: A).(\lambda (_: A).(arity g (CHead c0 (Bind Abst) t1) u (asucc -g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead (CHead c0 (Bind -Abst) t1) (Bind Abst) u) t a4)))))) H9 u H8) in (let H12 \def (eq_ind T u0 -(\lambda (t1: T).(arity g (CHead c0 (Bind Abst) t1) t a2)) H10 u H8) in (let -H13 \def (eq_ind T u0 (\lambda (t1: T).((eq T t1 (THead (Bind Abst) u t)) \to -(ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A (asucc g a1) (AHead a3 -a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g a3)))) -(\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t a4)))))) -H2 u H8) in (let H14 \def (eq_ind T u0 (\lambda (t1: T).(arity g c0 t1 (asucc -g a1))) H1 u H8) in (ex3_2_intro A A (\lambda (a3: A).(\lambda (a4: A).(eq A -(AHead a1 a2) (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u -(asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind -Abst) u) t a4))) a1 a2 (refl_equal A (AHead a1 a2)) H14 H12))))))))) -H6)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: A).(\lambda -(_: (arity g c0 u0 a1)).(\lambda (_: (((eq T u0 (THead (Bind Abst) u t)) \to -(ex3_2 A A (\lambda (a2: A).(\lambda (a3: A).(eq A a1 (AHead a2 a3)))) -(\lambda (a2: A).(\lambda (_: A).(arity g c0 u (asucc g a2)))) (\lambda (_: -A).(\lambda (a3: A).(arity g (CHead c0 (Bind Abst) u) t a3))))))).(\lambda -(t0: T).(\lambda (a2: A).(\lambda (_: (arity g c0 t0 (AHead a1 a2))).(\lambda -(_: (((eq T t0 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3: -A).(\lambda (a4: A).(eq A (AHead a1 a2) (AHead a3 a4)))) (\lambda (a3: -A).(\lambda (_: A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda -(a4: A).(arity g (CHead c0 (Bind Abst) u) t a4))))))).(\lambda (H5: (eq T -(THead (Flat Appl) u0 t0) (THead (Bind Abst) u t))).(let H6 \def (eq_ind T -(THead (Flat Appl) u0 t0) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind -Abst) u t) H5) in (False_ind (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq -A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g -a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t -a4)))) H6)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a0: -A).(\lambda (_: (arity g c0 u0 (asucc g a0))).(\lambda (_: (((eq T u0 (THead -(Bind Abst) u t)) \to (ex3_2 A A (\lambda (a1: A).(\lambda (a2: A).(eq A -(asucc g a0) (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g c0 u -(asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g (CHead c0 (Bind -Abst) u) t a2))))))).(\lambda (t0: T).(\lambda (_: (arity g c0 t0 -a0)).(\lambda (_: (((eq T t0 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda -(a1: A).(\lambda (a2: A).(eq A a0 (AHead a1 a2)))) (\lambda (a1: A).(\lambda -(_: A).(arity g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity -g (CHead c0 (Bind Abst) u) t a2))))))).(\lambda (H5: (eq T (THead (Flat Cast) -u0 t0) (THead (Bind Abst) u t))).(let H6 \def (eq_ind T (THead (Flat Cast) u0 -t0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort -_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) -\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u t) -H5) in (False_ind (ex3_2 A A (\lambda (a1: A).(\lambda (a2: A).(eq A a0 -(AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g c0 u (asucc g -a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g (CHead c0 (Bind Abst) u) t -a2)))) H6))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (a1: -A).(\lambda (H1: (arity g c0 t0 a1)).(\lambda (H2: (((eq T t0 (THead (Bind -Abst) u t)) \to (ex3_2 A A (\lambda (a2: A).(\lambda (a3: A).(eq A a1 (AHead -a2 a3)))) (\lambda (a2: A).(\lambda (_: A).(arity g c0 u (asucc g a2)))) -(\lambda (_: A).(\lambda (a3: A).(arity g (CHead c0 (Bind Abst) u) t -a3))))))).(\lambda (a2: A).(\lambda (H3: (leq g a1 a2)).(\lambda (H4: (eq T -t0 (THead (Bind Abst) u t))).(let H5 \def (f_equal T T (\lambda (e: T).e) t0 -(THead (Bind Abst) u t) H4) in (let H6 \def (eq_ind T t0 (\lambda (t1: -T).((eq T t1 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3: -A).(\lambda (a4: A).(eq A a1 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: -A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g -(CHead c0 (Bind Abst) u) t a4)))))) H2 (THead (Bind Abst) u t) H5) in (let H7 -\def (eq_ind T t0 (\lambda (t1: T).(arity g c0 t1 a1)) H1 (THead (Bind Abst) -u t) H5) in (let H8 \def (H6 (refl_equal T (THead (Bind Abst) u t))) in -(ex3_2_ind A A (\lambda (a3: A).(\lambda (a4: A).(eq A a1 (AHead a3 a4)))) -(\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g a3)))) (\lambda (_: -A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t a4))) (ex3_2 A A -(\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: -A).(\lambda (_: A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda -(a4: A).(arity g (CHead c0 (Bind Abst) u) t a4)))) (\lambda (x0: A).(\lambda -(x1: A).(\lambda (H9: (eq A a1 (AHead x0 x1))).(\lambda (H10: (arity g c0 u -(asucc g x0))).(\lambda (H11: (arity g (CHead c0 (Bind Abst) u) t x1)).(let -H12 \def (eq_ind A a1 (\lambda (a0: A).(leq g a0 a2)) H3 (AHead x0 x1) H9) in -(let H13 \def (eq_ind A a1 (\lambda (a0: A).(arity g c0 (THead (Bind Abst) u -t) a0)) H7 (AHead x0 x1) H9) in (let H_x \def (leq_gen_head1 g x0 x1 a2 H12) -in (let H14 \def H_x in (ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq -g x0 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g x1 a4))) (\lambda (a3: -A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) (ex3_2 A A (\lambda (a3: -A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: -A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g -(CHead c0 (Bind Abst) u) t a4)))) (\lambda (x2: A).(\lambda (x3: A).(\lambda -(H15: (leq g x0 x2)).(\lambda (H16: (leq g x1 x3)).(\lambda (H17: (eq A a2 -(AHead x2 x3))).(let H18 \def (f_equal A A (\lambda (e: A).e) a2 (AHead x2 -x3) H17) in (eq_ind_r A (AHead x2 x3) (\lambda (a0: A).(ex3_2 A A (\lambda -(a3: A).(\lambda (a4: A).(eq A a0 (AHead a3 a4)))) (\lambda (a3: A).(\lambda -(_: A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity -g (CHead c0 (Bind Abst) u) t a4))))) (ex3_2_intro A A (\lambda (a3: -A).(\lambda (a4: A).(eq A (AHead x2 x3) (AHead a3 a4)))) (\lambda (a3: -A).(\lambda (_: A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda -(a4: A).(arity g (CHead c0 (Bind Abst) u) t a4))) x2 x3 (refl_equal A (AHead -x2 x3)) (arity_repl g c0 u (asucc g x0) H10 (asucc g x2) (asucc_repl g x0 x2 -H15)) (arity_repl g (CHead c0 (Bind Abst) u) t x1 H11 x3 H16)) a2 H18))))))) -H14)))))))))) H8))))))))))))) c y a H0))) H)))))). -(* COMMENTS -Initial nodes: 4265 -END *) - -theorem arity_gen_appl: - \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (a2: -A).((arity g c (THead (Flat Appl) u t) a2) \to (ex2 A (\lambda (a1: A).(arity -g c u a1)) (\lambda (a1: A).(arity g c t (AHead a1 a2))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (a2: -A).(\lambda (H: (arity g c (THead (Flat Appl) u t) a2)).(insert_eq T (THead -(Flat Appl) u t) (\lambda (t0: T).(arity g c t0 a2)) (\lambda (_: T).(ex2 A -(\lambda (a1: A).(arity g c u a1)) (\lambda (a1: A).(arity g c t (AHead a1 -a2))))) (\lambda (y: T).(\lambda (H0: (arity g c y a2)).(arity_ind g (\lambda -(c0: C).(\lambda (t0: T).(\lambda (a: A).((eq T t0 (THead (Flat Appl) u t)) -\to (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t -(AHead a1 a)))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (H1: (eq T -(TSort n) (THead (Flat Appl) u t))).(let H2 \def (eq_ind T (TSort n) (\lambda -(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow -False])) I (THead (Flat Appl) u t) H1) in (False_ind (ex2 A (\lambda (a1: -A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t (AHead a1 (ASort O -n))))) H2))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: -nat).(\lambda (_: (getl i c0 (CHead d (Bind Abbr) u0))).(\lambda (a: -A).(\lambda (_: (arity g d u0 a)).(\lambda (_: (((eq T u0 (THead (Flat Appl) -u t)) \to (ex2 A (\lambda (a1: A).(arity g d u a1)) (\lambda (a1: A).(arity g -d t (AHead a1 a))))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat Appl) u -t))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) u -t) H4) in (False_ind (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: -A).(arity g c0 t (AHead a1 a)))) H5))))))))))) (\lambda (c0: C).(\lambda (d: -C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind -Abst) u0))).(\lambda (a: A).(\lambda (_: (arity g d u0 (asucc g a))).(\lambda -(_: (((eq T u0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda (a1: A).(arity g -d u a1)) (\lambda (a1: A).(arity g d t (AHead a1 (asucc g a)))))))).(\lambda -(H4: (eq T (TLRef i) (THead (Flat Appl) u t))).(let H5 \def (eq_ind T (TLRef -i) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort -_) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow -False])) I (THead (Flat Appl) u t) H4) in (False_ind (ex2 A (\lambda (a1: -A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t (AHead a1 a)))) -H5))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (c0: -C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u0 -a1)).(\lambda (_: (((eq T u0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda -(a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3 -a1))))))).(\lambda (t0: T).(\lambda (a0: A).(\lambda (_: (arity g (CHead c0 -(Bind b) u0) t0 a0)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u t)) \to -(ex2 A (\lambda (a3: A).(arity g (CHead c0 (Bind b) u0) u a3)) (\lambda (a3: -A).(arity g (CHead c0 (Bind b) u0) t (AHead a3 a0))))))).(\lambda (H6: (eq T -(THead (Bind b) u0 t0) (THead (Flat Appl) u t))).(let H7 \def (eq_ind T -(THead (Bind b) u0 t0) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat -Appl) u t) H6) in (False_ind (ex2 A (\lambda (a3: A).(arity g c0 u a3)) -(\lambda (a3: A).(arity g c0 t (AHead a3 a0)))) H7)))))))))))))) (\lambda -(c0: C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u0 (asucc -g a1))).(\lambda (_: (((eq T u0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda -(a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3 (asucc g -a1)))))))).(\lambda (t0: T).(\lambda (a0: A).(\lambda (_: (arity g (CHead c0 -(Bind Abst) u0) t0 a0)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u t)) \to -(ex2 A (\lambda (a3: A).(arity g (CHead c0 (Bind Abst) u0) u a3)) (\lambda -(a3: A).(arity g (CHead c0 (Bind Abst) u0) t (AHead a3 a0))))))).(\lambda -(H5: (eq T (THead (Bind Abst) u0 t0) (THead (Flat Appl) u t))).(let H6 \def -(eq_ind T (THead (Bind Abst) u0 t0) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow -False])])) I (THead (Flat Appl) u t) H5) in (False_ind (ex2 A (\lambda (a3: -A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3 (AHead a1 -a0))))) H6)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: -A).(\lambda (H1: (arity g c0 u0 a1)).(\lambda (H2: (((eq T u0 (THead (Flat -Appl) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3: -A).(arity g c0 t (AHead a3 a1))))))).(\lambda (t0: T).(\lambda (a0: -A).(\lambda (H3: (arity g c0 t0 (AHead a1 a0))).(\lambda (H4: (((eq T t0 -(THead (Flat Appl) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) -(\lambda (a3: A).(arity g c0 t (AHead a3 (AHead a1 a0)))))))).(\lambda (H5: -(eq T (THead (Flat Appl) u0 t0) (THead (Flat Appl) u t))).(let H6 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t1 _) -\Rightarrow t1])) (THead (Flat Appl) u0 t0) (THead (Flat Appl) u t) H5) in -((let H7 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: -T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ -t1) \Rightarrow t1])) (THead (Flat Appl) u0 t0) (THead (Flat Appl) u t) H5) -in (\lambda (H8: (eq T u0 u)).(let H9 \def (eq_ind T t0 (\lambda (t1: T).((eq -T t1 (THead (Flat Appl) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) -(\lambda (a3: A).(arity g c0 t (AHead a3 (AHead a1 a0))))))) H4 t H7) in (let -H10 \def (eq_ind T t0 (\lambda (t1: T).(arity g c0 t1 (AHead a1 a0))) H3 t -H7) in (let H11 \def (eq_ind T u0 (\lambda (t1: T).((eq T t1 (THead (Flat -Appl) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3: -A).(arity g c0 t (AHead a3 a1)))))) H2 u H8) in (let H12 \def (eq_ind T u0 -(\lambda (t1: T).(arity g c0 t1 a1)) H1 u H8) in (ex_intro2 A (\lambda (a3: -A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3 a0))) a1 H12 -H10))))))) H6)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a: -A).(\lambda (_: (arity g c0 u0 (asucc g a))).(\lambda (_: (((eq T u0 (THead -(Flat Appl) u t)) \to (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda -(a1: A).(arity g c0 t (AHead a1 (asucc g a)))))))).(\lambda (t0: T).(\lambda -(_: (arity g c0 t0 a)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u t)) \to -(ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t -(AHead a1 a))))))).(\lambda (H5: (eq T (THead (Flat Cast) u0 t0) (THead (Flat -Appl) u t))).(let H6 \def (eq_ind T (THead (Flat Cast) u0 t0) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K -return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) -\Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl \Rightarrow -False | Cast \Rightarrow True])])])) I (THead (Flat Appl) u t) H5) in -(False_ind (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity -g c0 t (AHead a1 a)))) H6))))))))))) (\lambda (c0: C).(\lambda (t0: -T).(\lambda (a1: A).(\lambda (H1: (arity g c0 t0 a1)).(\lambda (H2: (((eq T -t0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) -(\lambda (a3: A).(arity g c0 t (AHead a3 a1))))))).(\lambda (a0: A).(\lambda -(H3: (leq g a1 a0)).(\lambda (H4: (eq T t0 (THead (Flat Appl) u t))).(let H5 -\def (f_equal T T (\lambda (e: T).e) t0 (THead (Flat Appl) u t) H4) in (let -H6 \def (eq_ind T t0 (\lambda (t1: T).((eq T t1 (THead (Flat Appl) u t)) \to -(ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t -(AHead a3 a1)))))) H2 (THead (Flat Appl) u t) H5) in (let H7 \def (eq_ind T -t0 (\lambda (t1: T).(arity g c0 t1 a1)) H1 (THead (Flat Appl) u t) H5) in -(let H8 \def (H6 (refl_equal T (THead (Flat Appl) u t))) in (ex2_ind A -(\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3 -a1))) (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 -t (AHead a3 a0)))) (\lambda (x: A).(\lambda (H9: (arity g c0 u x)).(\lambda -(H10: (arity g c0 t (AHead x a1))).(ex_intro2 A (\lambda (a3: A).(arity g c0 -u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3 a0))) x H9 (arity_repl g c0 t -(AHead x a1) H10 (AHead x a0) (leq_head g x x (leq_refl g x) a1 a0 H3)))))) -H8))))))))))))) c y a2 H0))) H)))))). -(* COMMENTS -Initial nodes: 2277 -END *) - -theorem arity_gen_cast: - \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (a: -A).((arity g c (THead (Flat Cast) u t) a) \to (land (arity g c u (asucc g a)) -(arity g c t a))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (a: -A).(\lambda (H: (arity g c (THead (Flat Cast) u t) a)).(insert_eq T (THead -(Flat Cast) u t) (\lambda (t0: T).(arity g c t0 a)) (\lambda (_: T).(land -(arity g c u (asucc g a)) (arity g c t a))) (\lambda (y: T).(\lambda (H0: -(arity g c y a)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (a0: -A).((eq T t0 (THead (Flat Cast) u t)) \to (land (arity g c0 u (asucc g a0)) -(arity g c0 t a0)))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (H1: (eq T -(TSort n) (THead (Flat Cast) u t))).(let H2 \def (eq_ind T (TSort n) (\lambda -(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow -False])) I (THead (Flat Cast) u t) H1) in (False_ind (land (arity g c0 u -(asucc g (ASort O n))) (arity g c0 t (ASort O n))) H2))))) (\lambda (c0: -C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0 -(CHead d (Bind Abbr) u0))).(\lambda (a0: A).(\lambda (_: (arity g d u0 -a0)).(\lambda (_: (((eq T u0 (THead (Flat Cast) u t)) \to (land (arity g d u -(asucc g a0)) (arity g d t a0))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat -Cast) u t))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) u -t) H4) in (False_ind (land (arity g c0 u (asucc g a0)) (arity g c0 t a0)) -H5))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: -nat).(\lambda (_: (getl i c0 (CHead d (Bind Abst) u0))).(\lambda (a0: -A).(\lambda (_: (arity g d u0 (asucc g a0))).(\lambda (_: (((eq T u0 (THead -(Flat Cast) u t)) \to (land (arity g d u (asucc g (asucc g a0))) (arity g d t -(asucc g a0)))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat Cast) u -t))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) u -t) H4) in (False_ind (land (arity g c0 u (asucc g a0)) (arity g c0 t a0)) -H5))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (c0: -C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u0 -a1)).(\lambda (_: (((eq T u0 (THead (Flat Cast) u t)) \to (land (arity g c0 u -(asucc g a1)) (arity g c0 t a1))))).(\lambda (t0: T).(\lambda (a2: -A).(\lambda (_: (arity g (CHead c0 (Bind b) u0) t0 a2)).(\lambda (_: (((eq T -t0 (THead (Flat Cast) u t)) \to (land (arity g (CHead c0 (Bind b) u0) u -(asucc g a2)) (arity g (CHead c0 (Bind b) u0) t a2))))).(\lambda (H6: (eq T -(THead (Bind b) u0 t0) (THead (Flat Cast) u t))).(let H7 \def (eq_ind T -(THead (Bind b) u0 t0) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat -Cast) u t) H6) in (False_ind (land (arity g c0 u (asucc g a2)) (arity g c0 t -a2)) H7)))))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: -A).(\lambda (_: (arity g c0 u0 (asucc g a1))).(\lambda (_: (((eq T u0 (THead -(Flat Cast) u t)) \to (land (arity g c0 u (asucc g (asucc g a1))) (arity g c0 -t (asucc g a1)))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g -(CHead c0 (Bind Abst) u0) t0 a2)).(\lambda (_: (((eq T t0 (THead (Flat Cast) -u t)) \to (land (arity g (CHead c0 (Bind Abst) u0) u (asucc g a2)) (arity g -(CHead c0 (Bind Abst) u0) t a2))))).(\lambda (H5: (eq T (THead (Bind Abst) u0 -t0) (THead (Flat Cast) u t))).(let H6 \def (eq_ind T (THead (Bind Abst) u0 -t0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort -_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) -\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Cast) u t) -H5) in (False_ind (land (arity g c0 u (asucc g (AHead a1 a2))) (arity g c0 t -(AHead a1 a2))) H6)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda -(a1: A).(\lambda (_: (arity g c0 u0 a1)).(\lambda (_: (((eq T u0 (THead (Flat -Cast) u t)) \to (land (arity g c0 u (asucc g a1)) (arity g c0 t -a1))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g c0 t0 (AHead -a1 a2))).(\lambda (_: (((eq T t0 (THead (Flat Cast) u t)) \to (land (arity g -c0 u (asucc g (AHead a1 a2))) (arity g c0 t (AHead a1 a2)))))).(\lambda (H5: -(eq T (THead (Flat Appl) u0 t0) (THead (Flat Cast) u t))).(let H6 \def -(eq_ind T (THead (Flat Appl) u0 t0) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f -in F return (\lambda (_: F).Prop) with [Appl \Rightarrow True | Cast -\Rightarrow False])])])) I (THead (Flat Cast) u t) H5) in (False_ind (land -(arity g c0 u (asucc g a2)) (arity g c0 t a2)) H6)))))))))))) (\lambda (c0: -C).(\lambda (u0: T).(\lambda (a0: A).(\lambda (H1: (arity g c0 u0 (asucc g -a0))).(\lambda (H2: (((eq T u0 (THead (Flat Cast) u t)) \to (land (arity g c0 -u (asucc g (asucc g a0))) (arity g c0 t (asucc g a0)))))).(\lambda (t0: -T).(\lambda (H3: (arity g c0 t0 a0)).(\lambda (H4: (((eq T t0 (THead (Flat -Cast) u t)) \to (land (arity g c0 u (asucc g a0)) (arity g c0 t -a0))))).(\lambda (H5: (eq T (THead (Flat Cast) u0 t0) (THead (Flat Cast) u -t))).(let H6 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda -(_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead -_ t1 _) \Rightarrow t1])) (THead (Flat Cast) u0 t0) (THead (Flat Cast) u t) -H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 -| (THead _ _ t1) \Rightarrow t1])) (THead (Flat Cast) u0 t0) (THead (Flat -Cast) u t) H5) in (\lambda (H8: (eq T u0 u)).(let H9 \def (eq_ind T t0 -(\lambda (t1: T).((eq T t1 (THead (Flat Cast) u t)) \to (land (arity g c0 u -(asucc g a0)) (arity g c0 t a0)))) H4 t H7) in (let H10 \def (eq_ind T t0 -(\lambda (t1: T).(arity g c0 t1 a0)) H3 t H7) in (let H11 \def (eq_ind T u0 -(\lambda (t1: T).((eq T t1 (THead (Flat Cast) u t)) \to (land (arity g c0 u -(asucc g (asucc g a0))) (arity g c0 t (asucc g a0))))) H2 u H8) in (let H12 -\def (eq_ind T u0 (\lambda (t1: T).(arity g c0 t1 (asucc g a0))) H1 u H8) in -(conj (arity g c0 u (asucc g a0)) (arity g c0 t a0) H12 H10))))))) -H6))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda -(H1: (arity g c0 t0 a1)).(\lambda (H2: (((eq T t0 (THead (Flat Cast) u t)) -\to (land (arity g c0 u (asucc g a1)) (arity g c0 t a1))))).(\lambda (a2: -A).(\lambda (H3: (leq g a1 a2)).(\lambda (H4: (eq T t0 (THead (Flat Cast) u -t))).(let H5 \def (f_equal T T (\lambda (e: T).e) t0 (THead (Flat Cast) u t) -H4) in (let H6 \def (eq_ind T t0 (\lambda (t1: T).((eq T t1 (THead (Flat -Cast) u t)) \to (land (arity g c0 u (asucc g a1)) (arity g c0 t a1)))) H2 -(THead (Flat Cast) u t) H5) in (let H7 \def (eq_ind T t0 (\lambda (t1: -T).(arity g c0 t1 a1)) H1 (THead (Flat Cast) u t) H5) in (let H8 \def (H6 -(refl_equal T (THead (Flat Cast) u t))) in (land_ind (arity g c0 u (asucc g -a1)) (arity g c0 t a1) (land (arity g c0 u (asucc g a2)) (arity g c0 t a2)) -(\lambda (H9: (arity g c0 u (asucc g a1))).(\lambda (H10: (arity g c0 t -a1)).(conj (arity g c0 u (asucc g a2)) (arity g c0 t a2) (arity_repl g c0 u -(asucc g a1) H9 (asucc g a2) (asucc_repl g a1 a2 H3)) (arity_repl g c0 t a1 -H10 a2 H3)))) H8))))))))))))) c y a H0))) H)))))). -(* COMMENTS -Initial nodes: 2147 -END *) - -theorem arity_gen_appls: - \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (vs: TList).(\forall -(a2: A).((arity g c (THeads (Flat Appl) vs t) a2) \to (ex A (\lambda (a: -A).(arity g c t a)))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (vs: -TList).(TList_ind (\lambda (t0: TList).(\forall (a2: A).((arity g c (THeads -(Flat Appl) t0 t) a2) \to (ex A (\lambda (a: A).(arity g c t a)))))) (\lambda -(a2: A).(\lambda (H: (arity g c t a2)).(ex_intro A (\lambda (a: A).(arity g c -t a)) a2 H))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H: ((\forall -(a2: A).((arity g c (THeads (Flat Appl) t1 t) a2) \to (ex A (\lambda (a: -A).(arity g c t a))))))).(\lambda (a2: A).(\lambda (H0: (arity g c (THead -(Flat Appl) t0 (THeads (Flat Appl) t1 t)) a2)).(let H1 \def (arity_gen_appl g -c t0 (THeads (Flat Appl) t1 t) a2 H0) in (ex2_ind A (\lambda (a1: A).(arity g -c t0 a1)) (\lambda (a1: A).(arity g c (THeads (Flat Appl) t1 t) (AHead a1 -a2))) (ex A (\lambda (a: A).(arity g c t a))) (\lambda (x: A).(\lambda (_: -(arity g c t0 x)).(\lambda (H3: (arity g c (THeads (Flat Appl) t1 t) (AHead x -a2))).(let H_x \def (H (AHead x a2) H3) in (let H4 \def H_x in (ex_ind A -(\lambda (a: A).(arity g c t a)) (ex A (\lambda (a: A).(arity g c t a))) -(\lambda (x0: A).(\lambda (H5: (arity g c t x0)).(ex_intro A (\lambda (a: -A).(arity g c t a)) x0 H5))) H4)))))) H1))))))) vs)))). -(* COMMENTS -Initial nodes: 341 -END *) - -theorem arity_gen_lift: - \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).(\forall (h: -nat).(\forall (d: nat).((arity g c1 (lift h d t) a) \to (\forall (c2: -C).((drop h d c1 c2) \to (arity g c2 t a))))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (a: A).(\lambda (h: -nat).(\lambda (d: nat).(\lambda (H: (arity g c1 (lift h d t) a)).(insert_eq T -(lift h d t) (\lambda (t0: T).(arity g c1 t0 a)) (\lambda (_: T).(\forall -(c2: C).((drop h d c1 c2) \to (arity g c2 t a)))) (\lambda (y: T).(\lambda -(H0: (arity g c1 y a)).(unintro T t (\lambda (t0: T).((eq T y (lift h d t0)) -\to (\forall (c2: C).((drop h d c1 c2) \to (arity g c2 t0 a))))) (unintro nat -d (\lambda (n: nat).(\forall (x: T).((eq T y (lift h n x)) \to (\forall (c2: -C).((drop h n c1 c2) \to (arity g c2 x a)))))) (arity_ind g (\lambda (c: -C).(\lambda (t0: T).(\lambda (a0: A).(\forall (x: nat).(\forall (x0: T).((eq -T t0 (lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to (arity g c2 x0 -a0))))))))) (\lambda (c: C).(\lambda (n: nat).(\lambda (x: nat).(\lambda (x0: -T).(\lambda (H1: (eq T (TSort n) (lift h x x0))).(\lambda (c2: C).(\lambda -(_: (drop h x c c2)).(eq_ind_r T (TSort n) (\lambda (t0: T).(arity g c2 t0 -(ASort O n))) (arity_sort g c2 n) x0 (lift_gen_sort h x n x0 H1))))))))) -(\lambda (c: C).(\lambda (d0: C).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H1: (getl i c (CHead d0 (Bind Abbr) u))).(\lambda (a0: A).(\lambda (H2: -(arity g d0 u a0)).(\lambda (H3: ((\forall (x: nat).(\forall (x0: T).((eq T u -(lift h x x0)) \to (\forall (c2: C).((drop h x d0 c2) \to (arity g c2 x0 -a0)))))))).(\lambda (x: nat).(\lambda (x0: T).(\lambda (H4: (eq T (TLRef i) -(lift h x x0))).(\lambda (c2: C).(\lambda (H5: (drop h x c c2)).(let H_x \def -(lift_gen_lref x0 x h i H4) in (let H6 \def H_x in (or_ind (land (lt i x) (eq -T x0 (TLRef i))) (land (le (plus x h) i) (eq T x0 (TLRef (minus i h)))) -(arity g c2 x0 a0) (\lambda (H7: (land (lt i x) (eq T x0 (TLRef -i)))).(land_ind (lt i x) (eq T x0 (TLRef i)) (arity g c2 x0 a0) (\lambda (H8: -(lt i x)).(\lambda (H9: (eq T x0 (TLRef i))).(eq_ind_r T (TLRef i) (\lambda -(t0: T).(arity g c2 t0 a0)) (let H10 \def (eq_ind nat x (\lambda (n: -nat).(drop h n c c2)) H5 (S (plus i (minus x (S i)))) (lt_plus_minus i x H8)) -in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (minus x (S -i)) v)))) (\lambda (v: T).(\lambda (e0: C).(getl i c2 (CHead e0 (Bind Abbr) -v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (minus x (S i)) d0 e0))) -(arity g c2 (TLRef i) a0) (\lambda (x1: T).(\lambda (x2: C).(\lambda (H11: -(eq T u (lift h (minus x (S i)) x1))).(\lambda (H12: (getl i c2 (CHead x2 -(Bind Abbr) x1))).(\lambda (H13: (drop h (minus x (S i)) d0 x2)).(let H14 -\def (eq_ind T u (\lambda (t0: T).(\forall (x3: nat).(\forall (x4: T).((eq T -t0 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 d0 c3) \to (arity g c3 x4 -a0))))))) H3 (lift h (minus x (S i)) x1) H11) in (let H15 \def (eq_ind T u -(\lambda (t0: T).(arity g d0 t0 a0)) H2 (lift h (minus x (S i)) x1) H11) in -(arity_abbr g c2 x2 x1 i H12 a0 (H14 (minus x (S i)) x1 (refl_equal T (lift h -(minus x (S i)) x1)) x2 H13))))))))) (getl_drop_conf_lt Abbr c d0 u i H1 c2 h -(minus x (S i)) H10))) x0 H9))) H7)) (\lambda (H7: (land (le (plus x h) i) -(eq T x0 (TLRef (minus i h))))).(land_ind (le (plus x h) i) (eq T x0 (TLRef -(minus i h))) (arity g c2 x0 a0) (\lambda (H8: (le (plus x h) i)).(\lambda -(H9: (eq T x0 (TLRef (minus i h)))).(eq_ind_r T (TLRef (minus i h)) (\lambda -(t0: T).(arity g c2 t0 a0)) (arity_abbr g c2 d0 u (minus i h) -(getl_drop_conf_ge i (CHead d0 (Bind Abbr) u) c H1 c2 h x H5 H8) a0 H2) x0 -H9))) H7)) H6)))))))))))))))) (\lambda (c: C).(\lambda (d0: C).(\lambda (u: -T).(\lambda (i: nat).(\lambda (H1: (getl i c (CHead d0 (Bind Abst) -u))).(\lambda (a0: A).(\lambda (H2: (arity g d0 u (asucc g a0))).(\lambda -(H3: ((\forall (x: nat).(\forall (x0: T).((eq T u (lift h x x0)) \to (\forall -(c2: C).((drop h x d0 c2) \to (arity g c2 x0 (asucc g a0))))))))).(\lambda -(x: nat).(\lambda (x0: T).(\lambda (H4: (eq T (TLRef i) (lift h x -x0))).(\lambda (c2: C).(\lambda (H5: (drop h x c c2)).(let H_x \def -(lift_gen_lref x0 x h i H4) in (let H6 \def H_x in (or_ind (land (lt i x) (eq -T x0 (TLRef i))) (land (le (plus x h) i) (eq T x0 (TLRef (minus i h)))) -(arity g c2 x0 a0) (\lambda (H7: (land (lt i x) (eq T x0 (TLRef -i)))).(land_ind (lt i x) (eq T x0 (TLRef i)) (arity g c2 x0 a0) (\lambda (H8: -(lt i x)).(\lambda (H9: (eq T x0 (TLRef i))).(eq_ind_r T (TLRef i) (\lambda -(t0: T).(arity g c2 t0 a0)) (let H10 \def (eq_ind nat x (\lambda (n: -nat).(drop h n c c2)) H5 (S (plus i (minus x (S i)))) (lt_plus_minus i x H8)) -in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (minus x (S -i)) v)))) (\lambda (v: T).(\lambda (e0: C).(getl i c2 (CHead e0 (Bind Abst) -v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (minus x (S i)) d0 e0))) -(arity g c2 (TLRef i) a0) (\lambda (x1: T).(\lambda (x2: C).(\lambda (H11: -(eq T u (lift h (minus x (S i)) x1))).(\lambda (H12: (getl i c2 (CHead x2 -(Bind Abst) x1))).(\lambda (H13: (drop h (minus x (S i)) d0 x2)).(let H14 -\def (eq_ind T u (\lambda (t0: T).(\forall (x3: nat).(\forall (x4: T).((eq T -t0 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 d0 c3) \to (arity g c3 x4 -(asucc g a0)))))))) H3 (lift h (minus x (S i)) x1) H11) in (let H15 \def -(eq_ind T u (\lambda (t0: T).(arity g d0 t0 (asucc g a0))) H2 (lift h (minus -x (S i)) x1) H11) in (arity_abst g c2 x2 x1 i H12 a0 (H14 (minus x (S i)) x1 -(refl_equal T (lift h (minus x (S i)) x1)) x2 H13))))))))) (getl_drop_conf_lt -Abst c d0 u i H1 c2 h (minus x (S i)) H10))) x0 H9))) H7)) (\lambda (H7: -(land (le (plus x h) i) (eq T x0 (TLRef (minus i h))))).(land_ind (le (plus x -h) i) (eq T x0 (TLRef (minus i h))) (arity g c2 x0 a0) (\lambda (H8: (le -(plus x h) i)).(\lambda (H9: (eq T x0 (TLRef (minus i h)))).(eq_ind_r T -(TLRef (minus i h)) (\lambda (t0: T).(arity g c2 t0 a0)) (arity_abst g c2 d0 -u (minus i h) (getl_drop_conf_ge i (CHead d0 (Bind Abst) u) c H1 c2 h x H5 -H8) a0 H2) x0 H9))) H7)) H6)))))))))))))))) (\lambda (b: B).(\lambda (H1: -(not (eq B b Abst))).(\lambda (c: C).(\lambda (u: T).(\lambda (a1: -A).(\lambda (H2: (arity g c u a1)).(\lambda (H3: ((\forall (x: nat).(\forall -(x0: T).((eq T u (lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to -(arity g c2 x0 a1)))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (H4: -(arity g (CHead c (Bind b) u) t0 a2)).(\lambda (H5: ((\forall (x: -nat).(\forall (x0: T).((eq T t0 (lift h x x0)) \to (\forall (c2: C).((drop h -x (CHead c (Bind b) u) c2) \to (arity g c2 x0 a2)))))))).(\lambda (x: -nat).(\lambda (x0: T).(\lambda (H6: (eq T (THead (Bind b) u t0) (lift h x -x0))).(\lambda (c2: C).(\lambda (H7: (drop h x c c2)).(ex3_2_ind T T (\lambda -(y0: T).(\lambda (z: T).(eq T x0 (THead (Bind b) y0 z)))) (\lambda (y0: -T).(\lambda (_: T).(eq T u (lift h x y0)))) (\lambda (_: T).(\lambda (z: -T).(eq T t0 (lift h (S x) z)))) (arity g c2 x0 a2) (\lambda (x1: T).(\lambda -(x2: T).(\lambda (H8: (eq T x0 (THead (Bind b) x1 x2))).(\lambda (H9: (eq T u -(lift h x x1))).(\lambda (H10: (eq T t0 (lift h (S x) x2))).(eq_ind_r T -(THead (Bind b) x1 x2) (\lambda (t1: T).(arity g c2 t1 a2)) (let H11 \def -(eq_ind T t0 (\lambda (t1: T).(\forall (x3: nat).(\forall (x4: T).((eq T t1 -(lift h x3 x4)) \to (\forall (c3: C).((drop h x3 (CHead c (Bind b) u) c3) \to -(arity g c3 x4 a2))))))) H5 (lift h (S x) x2) H10) in (let H12 \def (eq_ind T -t0 (\lambda (t1: T).(arity g (CHead c (Bind b) u) t1 a2)) H4 (lift h (S x) -x2) H10) in (let H13 \def (eq_ind T u (\lambda (t1: T).(arity g (CHead c -(Bind b) t1) (lift h (S x) x2) a2)) H12 (lift h x x1) H9) in (let H14 \def -(eq_ind T u (\lambda (t1: T).(\forall (x3: nat).(\forall (x4: T).((eq T (lift -h (S x) x2) (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 (CHead c (Bind -b) t1) c3) \to (arity g c3 x4 a2))))))) H11 (lift h x x1) H9) in (let H15 -\def (eq_ind T u (\lambda (t1: T).(\forall (x3: nat).(\forall (x4: T).((eq T -t1 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 c c3) \to (arity g c3 x4 -a1))))))) H3 (lift h x x1) H9) in (let H16 \def (eq_ind T u (\lambda (t1: -T).(arity g c t1 a1)) H2 (lift h x x1) H9) in (arity_bind g b H1 c2 x1 a1 -(H15 x x1 (refl_equal T (lift h x x1)) c2 H7) x2 a2 (H14 (S x) x2 (refl_equal -T (lift h (S x) x2)) (CHead c2 (Bind b) x1) (drop_skip_bind h x c c2 H7 b -x1))))))))) x0 H8)))))) (lift_gen_bind b u t0 x0 h x H6)))))))))))))))))) -(\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H1: (arity g c u -(asucc g a1))).(\lambda (H2: ((\forall (x: nat).(\forall (x0: T).((eq T u -(lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to (arity g c2 x0 -(asucc g a1))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (H3: (arity g -(CHead c (Bind Abst) u) t0 a2)).(\lambda (H4: ((\forall (x: nat).(\forall -(x0: T).((eq T t0 (lift h x x0)) \to (\forall (c2: C).((drop h x (CHead c -(Bind Abst) u) c2) \to (arity g c2 x0 a2)))))))).(\lambda (x: nat).(\lambda -(x0: T).(\lambda (H5: (eq T (THead (Bind Abst) u t0) (lift h x x0))).(\lambda -(c2: C).(\lambda (H6: (drop h x c c2)).(ex3_2_ind T T (\lambda (y0: -T).(\lambda (z: T).(eq T x0 (THead (Bind Abst) y0 z)))) (\lambda (y0: -T).(\lambda (_: T).(eq T u (lift h x y0)))) (\lambda (_: T).(\lambda (z: -T).(eq T t0 (lift h (S x) z)))) (arity g c2 x0 (AHead a1 a2)) (\lambda (x1: -T).(\lambda (x2: T).(\lambda (H7: (eq T x0 (THead (Bind Abst) x1 -x2))).(\lambda (H8: (eq T u (lift h x x1))).(\lambda (H9: (eq T t0 (lift h (S -x) x2))).(eq_ind_r T (THead (Bind Abst) x1 x2) (\lambda (t1: T).(arity g c2 -t1 (AHead a1 a2))) (let H10 \def (eq_ind T t0 (\lambda (t1: T).(\forall (x3: -nat).(\forall (x4: T).((eq T t1 (lift h x3 x4)) \to (\forall (c3: C).((drop h -x3 (CHead c (Bind Abst) u) c3) \to (arity g c3 x4 a2))))))) H4 (lift h (S x) -x2) H9) in (let H11 \def (eq_ind T t0 (\lambda (t1: T).(arity g (CHead c -(Bind Abst) u) t1 a2)) H3 (lift h (S x) x2) H9) in (let H12 \def (eq_ind T u -(\lambda (t1: T).(arity g (CHead c (Bind Abst) t1) (lift h (S x) x2) a2)) H11 -(lift h x x1) H8) in (let H13 \def (eq_ind T u (\lambda (t1: T).(\forall (x3: -nat).(\forall (x4: T).((eq T (lift h (S x) x2) (lift h x3 x4)) \to (\forall -(c3: C).((drop h x3 (CHead c (Bind Abst) t1) c3) \to (arity g c3 x4 a2))))))) -H10 (lift h x x1) H8) in (let H14 \def (eq_ind T u (\lambda (t1: T).(\forall -(x3: nat).(\forall (x4: T).((eq T t1 (lift h x3 x4)) \to (\forall (c3: -C).((drop h x3 c c3) \to (arity g c3 x4 (asucc g a1)))))))) H2 (lift h x x1) -H8) in (let H15 \def (eq_ind T u (\lambda (t1: T).(arity g c t1 (asucc g -a1))) H1 (lift h x x1) H8) in (arity_head g c2 x1 a1 (H14 x x1 (refl_equal T -(lift h x x1)) c2 H6) x2 a2 (H13 (S x) x2 (refl_equal T (lift h (S x) x2)) -(CHead c2 (Bind Abst) x1) (drop_skip_bind h x c c2 H6 Abst x1))))))))) x0 -H7)))))) (lift_gen_bind Abst u t0 x0 h x H5)))))))))))))))) (\lambda (c: -C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H1: (arity g c u a1)).(\lambda -(H2: ((\forall (x: nat).(\forall (x0: T).((eq T u (lift h x x0)) \to (\forall -(c2: C).((drop h x c c2) \to (arity g c2 x0 a1)))))))).(\lambda (t0: -T).(\lambda (a2: A).(\lambda (H3: (arity g c t0 (AHead a1 a2))).(\lambda (H4: -((\forall (x: nat).(\forall (x0: T).((eq T t0 (lift h x x0)) \to (\forall -(c2: C).((drop h x c c2) \to (arity g c2 x0 (AHead a1 a2))))))))).(\lambda -(x: nat).(\lambda (x0: T).(\lambda (H5: (eq T (THead (Flat Appl) u t0) (lift -h x x0))).(\lambda (c2: C).(\lambda (H6: (drop h x c c2)).(ex3_2_ind T T -(\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Flat Appl) y0 z)))) -(\lambda (y0: T).(\lambda (_: T).(eq T u (lift h x y0)))) (\lambda (_: -T).(\lambda (z: T).(eq T t0 (lift h x z)))) (arity g c2 x0 a2) (\lambda (x1: -T).(\lambda (x2: T).(\lambda (H7: (eq T x0 (THead (Flat Appl) x1 -x2))).(\lambda (H8: (eq T u (lift h x x1))).(\lambda (H9: (eq T t0 (lift h x -x2))).(eq_ind_r T (THead (Flat Appl) x1 x2) (\lambda (t1: T).(arity g c2 t1 -a2)) (let H10 \def (eq_ind T t0 (\lambda (t1: T).(\forall (x3: nat).(\forall -(x4: T).((eq T t1 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 c c3) \to -(arity g c3 x4 (AHead a1 a2)))))))) H4 (lift h x x2) H9) in (let H11 \def -(eq_ind T t0 (\lambda (t1: T).(arity g c t1 (AHead a1 a2))) H3 (lift h x x2) -H9) in (let H12 \def (eq_ind T u (\lambda (t1: T).(\forall (x3: nat).(\forall -(x4: T).((eq T t1 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 c c3) \to -(arity g c3 x4 a1))))))) H2 (lift h x x1) H8) in (let H13 \def (eq_ind T u -(\lambda (t1: T).(arity g c t1 a1)) H1 (lift h x x1) H8) in (arity_appl g c2 -x1 a1 (H12 x x1 (refl_equal T (lift h x x1)) c2 H6) x2 a2 (H10 x x2 -(refl_equal T (lift h x x2)) c2 H6)))))) x0 H7)))))) (lift_gen_flat Appl u t0 -x0 h x H5)))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a0: -A).(\lambda (H1: (arity g c u (asucc g a0))).(\lambda (H2: ((\forall (x: -nat).(\forall (x0: T).((eq T u (lift h x x0)) \to (\forall (c2: C).((drop h x -c c2) \to (arity g c2 x0 (asucc g a0))))))))).(\lambda (t0: T).(\lambda (H3: -(arity g c t0 a0)).(\lambda (H4: ((\forall (x: nat).(\forall (x0: T).((eq T -t0 (lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to (arity g c2 x0 -a0)))))))).(\lambda (x: nat).(\lambda (x0: T).(\lambda (H5: (eq T (THead -(Flat Cast) u t0) (lift h x x0))).(\lambda (c2: C).(\lambda (H6: (drop h x c -c2)).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Flat -Cast) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u (lift h x y0)))) -(\lambda (_: T).(\lambda (z: T).(eq T t0 (lift h x z)))) (arity g c2 x0 a0) -(\lambda (x1: T).(\lambda (x2: T).(\lambda (H7: (eq T x0 (THead (Flat Cast) -x1 x2))).(\lambda (H8: (eq T u (lift h x x1))).(\lambda (H9: (eq T t0 (lift h -x x2))).(eq_ind_r T (THead (Flat Cast) x1 x2) (\lambda (t1: T).(arity g c2 t1 -a0)) (let H10 \def (eq_ind T t0 (\lambda (t1: T).(\forall (x3: nat).(\forall -(x4: T).((eq T t1 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 c c3) \to -(arity g c3 x4 a0))))))) H4 (lift h x x2) H9) in (let H11 \def (eq_ind T t0 -(\lambda (t1: T).(arity g c t1 a0)) H3 (lift h x x2) H9) in (let H12 \def -(eq_ind T u (\lambda (t1: T).(\forall (x3: nat).(\forall (x4: T).((eq T t1 -(lift h x3 x4)) \to (\forall (c3: C).((drop h x3 c c3) \to (arity g c3 x4 -(asucc g a0)))))))) H2 (lift h x x1) H8) in (let H13 \def (eq_ind T u -(\lambda (t1: T).(arity g c t1 (asucc g a0))) H1 (lift h x x1) H8) in -(arity_cast g c2 x1 a0 (H12 x x1 (refl_equal T (lift h x x1)) c2 H6) x2 (H10 -x x2 (refl_equal T (lift h x x2)) c2 H6)))))) x0 H7)))))) (lift_gen_flat Cast -u t0 x0 h x H5))))))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda (a1: -A).(\lambda (_: (arity g c t0 a1)).(\lambda (H2: ((\forall (x: nat).(\forall -(x0: T).((eq T t0 (lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to -(arity g c2 x0 a1)))))))).(\lambda (a2: A).(\lambda (H3: (leq g a1 -a2)).(\lambda (x: nat).(\lambda (x0: T).(\lambda (H4: (eq T t0 (lift h x -x0))).(\lambda (c2: C).(\lambda (H5: (drop h x c c2)).(arity_repl g c2 x0 a1 -(H2 x x0 H4 c2 H5) a2 H3))))))))))))) c1 y a H0))))) H))))))). -(* COMMENTS -Initial nodes: 4693 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/arity/lift1.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/arity/lift1.ma deleted file mode 100644 index 4077a8a1c..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/arity/lift1.ma +++ /dev/null @@ -1,44 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/arity/props.ma". - -include "Basic-1/drop1/fwd.ma". - -theorem arity_lift1: - \forall (g: G).(\forall (a: A).(\forall (c2: C).(\forall (hds: -PList).(\forall (c1: C).(\forall (t: T).((drop1 hds c1 c2) \to ((arity g c2 t -a) \to (arity g c1 (lift1 hds t) a)))))))) -\def - \lambda (g: G).(\lambda (a: A).(\lambda (c2: C).(\lambda (hds: -PList).(PList_ind (\lambda (p: PList).(\forall (c1: C).(\forall (t: -T).((drop1 p c1 c2) \to ((arity g c2 t a) \to (arity g c1 (lift1 p t) a)))))) -(\lambda (c1: C).(\lambda (t: T).(\lambda (H: (drop1 PNil c1 c2)).(\lambda -(H0: (arity g c2 t a)).(let H_y \def (drop1_gen_pnil c1 c2 H) in (eq_ind_r C -c2 (\lambda (c: C).(arity g c t a)) H0 c1 H_y)))))) (\lambda (n: -nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c1: -C).(\forall (t: T).((drop1 p c1 c2) \to ((arity g c2 t a) \to (arity g c1 -(lift1 p t) a))))))).(\lambda (c1: C).(\lambda (t: T).(\lambda (H0: (drop1 -(PCons n n0 p) c1 c2)).(\lambda (H1: (arity g c2 t a)).(let H_x \def -(drop1_gen_pcons c1 c2 p n n0 H0) in (let H2 \def H_x in (ex2_ind C (\lambda -(c3: C).(drop n n0 c1 c3)) (\lambda (c3: C).(drop1 p c3 c2)) (arity g c1 -(lift n n0 (lift1 p t)) a) (\lambda (x: C).(\lambda (H3: (drop n n0 c1 -x)).(\lambda (H4: (drop1 p x c2)).(arity_lift g x (lift1 p t) a (H x t H4 H1) -c1 n n0 H3)))) H2))))))))))) hds)))). -(* COMMENTS -Initial nodes: 289 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/arity/pr3.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/arity/pr3.ma deleted file mode 100644 index f8952088e..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/arity/pr3.ma +++ /dev/null @@ -1,635 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csuba/arity.ma". - -include "Basic-1/pr3/defs.ma". - -include "Basic-1/pr1/defs.ma". - -include "Basic-1/wcpr0/getl.ma". - -include "Basic-1/pr0/fwd.ma". - -include "Basic-1/arity/subst0.ma". - -theorem arity_sred_wcpr0_pr0: - \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (a: A).((arity g -c1 t1 a) \to (\forall (c2: C).((wcpr0 c1 c2) \to (\forall (t2: T).((pr0 t1 -t2) \to (arity g c2 t2 a))))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (a: A).(\lambda -(H: (arity g c1 t1 a)).(arity_ind g (\lambda (c: C).(\lambda (t: T).(\lambda -(a0: A).(\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 t t2) \to -(arity g c2 t2 a0)))))))) (\lambda (c: C).(\lambda (n: nat).(\lambda (c2: -C).(\lambda (_: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H1: (pr0 (TSort n) -t2)).(eq_ind_r T (TSort n) (\lambda (t: T).(arity g c2 t (ASort O n))) -(arity_sort g c2 n) t2 (pr0_gen_sort t2 n H1)))))))) (\lambda (c: C).(\lambda -(d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d -(Bind Abbr) u))).(\lambda (a0: A).(\lambda (_: (arity g d u a0)).(\lambda -(H2: ((\forall (c2: C).((wcpr0 d c2) \to (\forall (t2: T).((pr0 u t2) \to -(arity g c2 t2 a0))))))).(\lambda (c2: C).(\lambda (H3: (wcpr0 c -c2)).(\lambda (t2: T).(\lambda (H4: (pr0 (TLRef i) t2)).(eq_ind_r T (TLRef i) -(\lambda (t: T).(arity g c2 t a0)) (ex3_2_ind C T (\lambda (e2: C).(\lambda -(u2: T).(getl i c2 (CHead e2 (Bind Abbr) u2)))) (\lambda (e2: C).(\lambda (_: -T).(wcpr0 d e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u u2))) (arity g c2 -(TLRef i) a0) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: (getl i c2 -(CHead x0 (Bind Abbr) x1))).(\lambda (H6: (wcpr0 d x0)).(\lambda (H7: (pr0 u -x1)).(arity_abbr g c2 x0 x1 i H5 a0 (H2 x0 H6 x1 H7))))))) (wcpr0_getl c c2 -H3 i d u (Bind Abbr) H0)) t2 (pr0_gen_lref t2 i H4)))))))))))))) (\lambda (c: -C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c -(CHead d (Bind Abst) u))).(\lambda (a0: A).(\lambda (_: (arity g d u (asucc g -a0))).(\lambda (H2: ((\forall (c2: C).((wcpr0 d c2) \to (\forall (t2: -T).((pr0 u t2) \to (arity g c2 t2 (asucc g a0)))))))).(\lambda (c2: -C).(\lambda (H3: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H4: (pr0 (TLRef i) -t2)).(eq_ind_r T (TLRef i) (\lambda (t: T).(arity g c2 t a0)) (ex3_2_ind C T -(\lambda (e2: C).(\lambda (u2: T).(getl i c2 (CHead e2 (Bind Abst) u2)))) -(\lambda (e2: C).(\lambda (_: T).(wcpr0 d e2))) (\lambda (_: C).(\lambda (u2: -T).(pr0 u u2))) (arity g c2 (TLRef i) a0) (\lambda (x0: C).(\lambda (x1: -T).(\lambda (H5: (getl i c2 (CHead x0 (Bind Abst) x1))).(\lambda (H6: (wcpr0 -d x0)).(\lambda (H7: (pr0 u x1)).(arity_abst g c2 x0 x1 i H5 a0 (H2 x0 H6 x1 -H7))))))) (wcpr0_getl c c2 H3 i d u (Bind Abst) H0)) t2 (pr0_gen_lref t2 i -H4)))))))))))))) (\lambda (b: B).(\lambda (H0: (not (eq B b Abst))).(\lambda -(c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c u -a1)).(\lambda (H2: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 -u t2) \to (arity g c2 t2 a1))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda -(H3: (arity g (CHead c (Bind b) u) t a2)).(\lambda (H4: ((\forall (c2: -C).((wcpr0 (CHead c (Bind b) u) c2) \to (\forall (t2: T).((pr0 t t2) \to -(arity g c2 t2 a2))))))).(\lambda (c2: C).(\lambda (H5: (wcpr0 c -c2)).(\lambda (t2: T).(\lambda (H6: (pr0 (THead (Bind b) u t) t2)).(insert_eq -T (THead (Bind b) u t) (\lambda (t0: T).(pr0 t0 t2)) (\lambda (_: T).(arity g -c2 t2 a2)) (\lambda (y: T).(\lambda (H7: (pr0 y t2)).(pr0_ind (\lambda (t0: -T).(\lambda (t3: T).((eq T t0 (THead (Bind b) u t)) \to (arity g c2 t3 a2)))) -(\lambda (t0: T).(\lambda (H8: (eq T t0 (THead (Bind b) u t))).(let H9 \def -(f_equal T T (\lambda (e: T).e) t0 (THead (Bind b) u t) H8) in (eq_ind_r T -(THead (Bind b) u t) (\lambda (t3: T).(arity g c2 t3 a2)) (arity_bind g b H0 -c2 u a1 (H2 c2 H5 u (pr0_refl u)) t a2 (H4 (CHead c2 (Bind b) u) (wcpr0_comp -c c2 H5 u u (pr0_refl u) (Bind b)) t (pr0_refl t))) t0 H9)))) (\lambda (u1: -T).(\lambda (u2: T).(\lambda (H8: (pr0 u1 u2)).(\lambda (H9: (((eq T u1 -(THead (Bind b) u t)) \to (arity g c2 u2 a2)))).(\lambda (t3: T).(\lambda -(t4: T).(\lambda (H10: (pr0 t3 t4)).(\lambda (H11: (((eq T t3 (THead (Bind b) -u t)) \to (arity g c2 t4 a2)))).(\lambda (k: K).(\lambda (H12: (eq T (THead k -u1 t3) (THead (Bind b) u t))).(let H13 \def (f_equal T K (\lambda (e: -T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | -(TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) -(THead (Bind b) u t) H12) in ((let H14 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | -(TLRef _) \Rightarrow u1 | (THead _ t0 _) \Rightarrow t0])) (THead k u1 t3) -(THead (Bind b) u t) H12) in ((let H15 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | -(TLRef _) \Rightarrow t3 | (THead _ _ t0) \Rightarrow t0])) (THead k u1 t3) -(THead (Bind b) u t) H12) in (\lambda (H16: (eq T u1 u)).(\lambda (H17: (eq K -k (Bind b))).(eq_ind_r K (Bind b) (\lambda (k0: K).(arity g c2 (THead k0 u2 -t4) a2)) (let H18 \def (eq_ind T t3 (\lambda (t0: T).((eq T t0 (THead (Bind -b) u t)) \to (arity g c2 t4 a2))) H11 t H15) in (let H19 \def (eq_ind T t3 -(\lambda (t0: T).(pr0 t0 t4)) H10 t H15) in (let H20 \def (eq_ind T u1 -(\lambda (t0: T).((eq T t0 (THead (Bind b) u t)) \to (arity g c2 u2 a2))) H9 -u H16) in (let H21 \def (eq_ind T u1 (\lambda (t0: T).(pr0 t0 u2)) H8 u H16) -in (arity_bind g b H0 c2 u2 a1 (H2 c2 H5 u2 H21) t4 a2 (H4 (CHead c2 (Bind b) -u2) (wcpr0_comp c c2 H5 u u2 H21 (Bind b)) t4 H19)))))) k H17)))) H14)) -H13)))))))))))) (\lambda (u0: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda -(_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind b) u t)) \to (arity g -c2 v2 a2)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 -t4)).(\lambda (_: (((eq T t3 (THead (Bind b) u t)) \to (arity g c2 t4 -a2)))).(\lambda (H12: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) -(THead (Bind b) u t))).(let H13 \def (eq_ind T (THead (Flat Appl) v1 (THead -(Bind Abst) u0 t3)) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind -b) u t) H12) in (False_ind (arity g c2 (THead (Bind Abbr) v2 t4) a2) -H13)))))))))))) (\lambda (b0: B).(\lambda (_: (not (eq B b0 Abst))).(\lambda -(v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 -(THead (Bind b) u t)) \to (arity g c2 v2 a2)))).(\lambda (u1: T).(\lambda -(u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 (THead (Bind b) u -t)) \to (arity g c2 u2 a2)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: -(pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead (Bind b) u t)) \to (arity g c2 t4 -a2)))).(\lambda (H15: (eq T (THead (Flat Appl) v1 (THead (Bind b0) u1 t3)) -(THead (Bind b) u t))).(let H16 \def (eq_ind T (THead (Flat Appl) v1 (THead -(Bind b0) u1 t3)) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind -b) u t) H15) in (False_ind (arity g c2 (THead (Bind b0) u2 (THead (Flat Appl) -(lift (S O) O v2) t4)) a2) H16))))))))))))))))) (\lambda (u1: T).(\lambda -(u2: T).(\lambda (H8: (pr0 u1 u2)).(\lambda (H9: (((eq T u1 (THead (Bind b) u -t)) \to (arity g c2 u2 a2)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda -(H10: (pr0 t3 t4)).(\lambda (H11: (((eq T t3 (THead (Bind b) u t)) \to (arity -g c2 t4 a2)))).(\lambda (w: T).(\lambda (H12: (subst0 O u2 t4 w)).(\lambda -(H13: (eq T (THead (Bind Abbr) u1 t3) (THead (Bind b) u t))).(let H14 \def -(f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with -[(TSort _) \Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | (THead k _ _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b0) -\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (THead (Bind Abbr) u1 t3) -(THead (Bind b) u t) H13) in ((let H15 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | -(TLRef _) \Rightarrow u1 | (THead _ t0 _) \Rightarrow t0])) (THead (Bind -Abbr) u1 t3) (THead (Bind b) u t) H13) in ((let H16 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t0) \Rightarrow t0])) -(THead (Bind Abbr) u1 t3) (THead (Bind b) u t) H13) in (\lambda (H17: (eq T -u1 u)).(\lambda (H18: (eq B Abbr b)).(let H19 \def (eq_ind T t3 (\lambda (t0: -T).((eq T t0 (THead (Bind b) u t)) \to (arity g c2 t4 a2))) H11 t H16) in -(let H20 \def (eq_ind T t3 (\lambda (t0: T).(pr0 t0 t4)) H10 t H16) in (let -H21 \def (eq_ind T u1 (\lambda (t0: T).((eq T t0 (THead (Bind b) u t)) \to -(arity g c2 u2 a2))) H9 u H17) in (let H22 \def (eq_ind T u1 (\lambda (t0: -T).(pr0 t0 u2)) H8 u H17) in (let H23 \def (eq_ind_r B b (\lambda (b0: -B).((eq T t (THead (Bind b0) u t)) \to (arity g c2 t4 a2))) H19 Abbr H18) in -(let H24 \def (eq_ind_r B b (\lambda (b0: B).((eq T u (THead (Bind b0) u t)) -\to (arity g c2 u2 a2))) H21 Abbr H18) in (let H25 \def (eq_ind_r B b -(\lambda (b0: B).(\forall (c3: C).((wcpr0 (CHead c (Bind b0) u) c3) \to -(\forall (t5: T).((pr0 t t5) \to (arity g c3 t5 a2)))))) H4 Abbr H18) in (let -H26 \def (eq_ind_r B b (\lambda (b0: B).(arity g (CHead c (Bind b0) u) t a2)) -H3 Abbr H18) in (let H27 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 -Abst))) H0 Abbr H18) in (arity_bind g Abbr H27 c2 u2 a1 (H2 c2 H5 u2 H22) w -a2 (arity_subst0 g (CHead c2 (Bind Abbr) u2) t4 a2 (H25 (CHead c2 (Bind Abbr) -u2) (wcpr0_comp c c2 H5 u u2 H22 (Bind Abbr)) t4 H20) c2 u2 O (getl_refl Abbr -c2 u2) w H12)))))))))))))) H15)) H14))))))))))))) (\lambda (b0: B).(\lambda -(H8: (not (eq B b0 Abst))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H9: -(pr0 t3 t4)).(\lambda (H10: (((eq T t3 (THead (Bind b) u t)) \to (arity g c2 -t4 a2)))).(\lambda (u0: T).(\lambda (H11: (eq T (THead (Bind b0) u0 (lift (S -O) O t3)) (THead (Bind b) u t))).(let H12 \def (f_equal T B (\lambda (e: -T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b0 | -(TLRef _) \Rightarrow b0 | (THead k _ _) \Rightarrow (match k in K return -(\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow -b0])])) (THead (Bind b0) u0 (lift (S O) O t3)) (THead (Bind b) u t) H11) in -((let H13 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: -T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t0 -_) \Rightarrow t0])) (THead (Bind b0) u0 (lift (S O) O t3)) (THead (Bind b) u -t) H11) in ((let H14 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat -\to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n) -\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with -[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u1 t5) -\Rightarrow (THead k (lref_map f d u1) (lref_map f (s k d) t5))]) in -lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (TLRef _) \Rightarrow -((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match -t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef -(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | -(THead k u1 t5) \Rightarrow (THead k (lref_map f d u1) (lref_map f (s k d) -t5))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (THead _ _ t0) -\Rightarrow t0])) (THead (Bind b0) u0 (lift (S O) O t3)) (THead (Bind b) u t) -H11) in (\lambda (_: (eq T u0 u)).(\lambda (H16: (eq B b0 b)).(let H17 \def -(eq_ind B b0 (\lambda (b1: B).(not (eq B b1 Abst))) H8 b H16) in (let H18 -\def (eq_ind_r T t (\lambda (t0: T).((eq T t3 (THead (Bind b) u t0)) \to -(arity g c2 t4 a2))) H10 (lift (S O) O t3) H14) in (let H19 \def (eq_ind_r T -t (\lambda (t0: T).(\forall (c3: C).((wcpr0 (CHead c (Bind b) u) c3) \to -(\forall (t5: T).((pr0 t0 t5) \to (arity g c3 t5 a2)))))) H4 (lift (S O) O -t3) H14) in (let H20 \def (eq_ind_r T t (\lambda (t0: T).(arity g (CHead c -(Bind b) u) t0 a2)) H3 (lift (S O) O t3) H14) in (arity_gen_lift g (CHead c2 -(Bind b) u) t4 a2 (S O) O (H19 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H5 u u -(pr0_refl u) (Bind b)) (lift (S O) O t4) (pr0_lift t3 t4 H9 (S O) O)) c2 -(drop_drop (Bind b) O c2 c2 (drop_refl c2) u))))))))) H13)) H12)))))))))) -(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: -(((eq T t3 (THead (Bind b) u t)) \to (arity g c2 t4 a2)))).(\lambda (u0: -T).(\lambda (H10: (eq T (THead (Flat Cast) u0 t3) (THead (Bind b) u t))).(let -H11 \def (eq_ind T (THead (Flat Cast) u0 t3) (\lambda (ee: T).(match ee in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind b) u t) H10) in (False_ind (arity g c2 t4 a2) -H11)))))))) y t2 H7))) H6)))))))))))))))) (\lambda (c: C).(\lambda (u: -T).(\lambda (a1: A).(\lambda (_: (arity g c u (asucc g a1))).(\lambda (H1: -((\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 u t2) \to (arity g -c2 t2 (asucc g a1)))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (H2: -(arity g (CHead c (Bind Abst) u) t a2)).(\lambda (H3: ((\forall (c2: -C).((wcpr0 (CHead c (Bind Abst) u) c2) \to (\forall (t2: T).((pr0 t t2) \to -(arity g c2 t2 a2))))))).(\lambda (c2: C).(\lambda (H4: (wcpr0 c -c2)).(\lambda (t2: T).(\lambda (H5: (pr0 (THead (Bind Abst) u t) -t2)).(insert_eq T (THead (Bind Abst) u t) (\lambda (t0: T).(pr0 t0 t2)) -(\lambda (_: T).(arity g c2 t2 (AHead a1 a2))) (\lambda (y: T).(\lambda (H6: -(pr0 y t2)).(pr0_ind (\lambda (t0: T).(\lambda (t3: T).((eq T t0 (THead (Bind -Abst) u t)) \to (arity g c2 t3 (AHead a1 a2))))) (\lambda (t0: T).(\lambda -(H7: (eq T t0 (THead (Bind Abst) u t))).(let H8 \def (f_equal T T (\lambda -(e: T).e) t0 (THead (Bind Abst) u t) H7) in (eq_ind_r T (THead (Bind Abst) u -t) (\lambda (t3: T).(arity g c2 t3 (AHead a1 a2))) (arity_head g c2 u a1 (H1 -c2 H4 u (pr0_refl u)) t a2 (H3 (CHead c2 (Bind Abst) u) (wcpr0_comp c c2 H4 u -u (pr0_refl u) (Bind Abst)) t (pr0_refl t))) t0 H8)))) (\lambda (u1: -T).(\lambda (u2: T).(\lambda (H7: (pr0 u1 u2)).(\lambda (H8: (((eq T u1 -(THead (Bind Abst) u t)) \to (arity g c2 u2 (AHead a1 a2))))).(\lambda (t3: -T).(\lambda (t4: T).(\lambda (H9: (pr0 t3 t4)).(\lambda (H10: (((eq T t3 -(THead (Bind Abst) u t)) \to (arity g c2 t4 (AHead a1 a2))))).(\lambda (k: -K).(\lambda (H11: (eq T (THead k u1 t3) (THead (Bind Abst) u t))).(let H12 -\def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) -with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) -\Rightarrow k0])) (THead k u1 t3) (THead (Bind Abst) u t) H11) in ((let H13 -\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t0 _) -\Rightarrow t0])) (THead k u1 t3) (THead (Bind Abst) u t) H11) in ((let H14 -\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t0) -\Rightarrow t0])) (THead k u1 t3) (THead (Bind Abst) u t) H11) in (\lambda -(H15: (eq T u1 u)).(\lambda (H16: (eq K k (Bind Abst))).(eq_ind_r K (Bind -Abst) (\lambda (k0: K).(arity g c2 (THead k0 u2 t4) (AHead a1 a2))) (let H17 -\def (eq_ind T t3 (\lambda (t0: T).((eq T t0 (THead (Bind Abst) u t)) \to -(arity g c2 t4 (AHead a1 a2)))) H10 t H14) in (let H18 \def (eq_ind T t3 -(\lambda (t0: T).(pr0 t0 t4)) H9 t H14) in (let H19 \def (eq_ind T u1 -(\lambda (t0: T).((eq T t0 (THead (Bind Abst) u t)) \to (arity g c2 u2 (AHead -a1 a2)))) H8 u H15) in (let H20 \def (eq_ind T u1 (\lambda (t0: T).(pr0 t0 -u2)) H7 u H15) in (arity_head g c2 u2 a1 (H1 c2 H4 u2 H20) t4 a2 (H3 (CHead -c2 (Bind Abst) u2) (wcpr0_comp c c2 H4 u u2 H20 (Bind Abst)) t4 H18)))))) k -H16)))) H13)) H12)))))))))))) (\lambda (u0: T).(\lambda (v1: T).(\lambda (v2: -T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind Abst) u t)) -\to (arity g c2 v2 (AHead a1 a2))))).(\lambda (t3: T).(\lambda (t4: -T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead (Bind Abst) u t)) -\to (arity g c2 t4 (AHead a1 a2))))).(\lambda (H11: (eq T (THead (Flat Appl) -v1 (THead (Bind Abst) u0 t3)) (THead (Bind Abst) u t))).(let H12 \def (eq_ind -T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) (\lambda (ee: T).(match ee -in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef -_) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return -(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind Abst) u t) H11) in (False_ind (arity g c2 (THead -(Bind Abbr) v2 t4) (AHead a1 a2)) H12)))))))))))) (\lambda (b: B).(\lambda -(_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 -v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind Abst) u t)) \to (arity g c2 v2 -(AHead a1 a2))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 -u2)).(\lambda (_: (((eq T u1 (THead (Bind Abst) u t)) \to (arity g c2 u2 -(AHead a1 a2))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 -t4)).(\lambda (_: (((eq T t3 (THead (Bind Abst) u t)) \to (arity g c2 t4 -(AHead a1 a2))))).(\lambda (H14: (eq T (THead (Flat Appl) v1 (THead (Bind b) -u1 t3)) (THead (Bind Abst) u t))).(let H15 \def (eq_ind T (THead (Flat Appl) -v1 (THead (Bind b) u1 t3)) (\lambda (ee: T).(match ee in T return (\lambda -(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False -| (THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind -Abst) u t) H14) in (False_ind (arity g c2 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) (AHead a1 a2)) H15))))))))))))))))) (\lambda -(u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 -(THead (Bind Abst) u t)) \to (arity g c2 u2 (AHead a1 a2))))).(\lambda (t3: -T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead -(Bind Abst) u t)) \to (arity g c2 t4 (AHead a1 a2))))).(\lambda (w: -T).(\lambda (_: (subst0 O u2 t4 w)).(\lambda (H12: (eq T (THead (Bind Abbr) -u1 t3) (THead (Bind Abst) u t))).(let H13 \def (eq_ind T (THead (Bind Abbr) -u1 t3) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b) -\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow -True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _) -\Rightarrow False])])) I (THead (Bind Abst) u t) H12) in (False_ind (arity g -c2 (THead (Bind Abbr) u2 w) (AHead a1 a2)) H13))))))))))))) (\lambda (b: -B).(\lambda (H7: (not (eq B b Abst))).(\lambda (t3: T).(\lambda (t4: -T).(\lambda (_: (pr0 t3 t4)).(\lambda (H9: (((eq T t3 (THead (Bind Abst) u -t)) \to (arity g c2 t4 (AHead a1 a2))))).(\lambda (u0: T).(\lambda (H10: (eq -T (THead (Bind b) u0 (lift (S O) O t3)) (THead (Bind Abst) u t))).(let H11 -\def (f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) -with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b0) -\Rightarrow b0 | (Flat _) \Rightarrow b])])) (THead (Bind b) u0 (lift (S O) O -t3)) (THead (Bind Abst) u t) H10) in ((let H12 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | -(TLRef _) \Rightarrow u0 | (THead _ t0 _) \Rightarrow t0])) (THead (Bind b) -u0 (lift (S O) O t3)) (THead (Bind Abst) u t) H10) in ((let H13 \def (f_equal -T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T -\def (match t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow -(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) -| (THead k u1 t5) \Rightarrow (THead k (lref_map f d u1) (lref_map f (s k d) -t5))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (TLRef _) -\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T -\def (match t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow -(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) -| (THead k u1 t5) \Rightarrow (THead k (lref_map f d u1) (lref_map f (s k d) -t5))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (THead _ _ t0) -\Rightarrow t0])) (THead (Bind b) u0 (lift (S O) O t3)) (THead (Bind Abst) u -t) H10) in (\lambda (_: (eq T u0 u)).(\lambda (H15: (eq B b Abst)).(let H16 -\def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H7 Abst H15) in (let -H17 \def (eq_ind_r T t (\lambda (t0: T).((eq T t3 (THead (Bind Abst) u t0)) -\to (arity g c2 t4 (AHead a1 a2)))) H9 (lift (S O) O t3) H13) in (let H18 -\def (eq_ind_r T t (\lambda (t0: T).(\forall (c3: C).((wcpr0 (CHead c (Bind -Abst) u) c3) \to (\forall (t5: T).((pr0 t0 t5) \to (arity g c3 t5 a2)))))) H3 -(lift (S O) O t3) H13) in (let H19 \def (eq_ind_r T t (\lambda (t0: T).(arity -g (CHead c (Bind Abst) u) t0 a2)) H2 (lift (S O) O t3) H13) in (let H20 \def -(match (H16 (refl_equal B Abst)) in False return (\lambda (_: False).(arity g -c2 t4 (AHead a1 a2))) with []) in H20)))))))) H12)) H11)))))))))) (\lambda -(t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 -(THead (Bind Abst) u t)) \to (arity g c2 t4 (AHead a1 a2))))).(\lambda (u0: -T).(\lambda (H9: (eq T (THead (Flat Cast) u0 t3) (THead (Bind Abst) u -t))).(let H10 \def (eq_ind T (THead (Flat Cast) u0 t3) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K -return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) -\Rightarrow True])])) I (THead (Bind Abst) u t) H9) in (False_ind (arity g c2 -t4 (AHead a1 a2)) H10)))))))) y t2 H6))) H5)))))))))))))) (\lambda (c: -C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda -(H1: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 u t2) \to -(arity g c2 t2 a1))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (H2: -(arity g c t (AHead a1 a2))).(\lambda (H3: ((\forall (c2: C).((wcpr0 c c2) -\to (\forall (t2: T).((pr0 t t2) \to (arity g c2 t2 (AHead a1 -a2)))))))).(\lambda (c2: C).(\lambda (H4: (wcpr0 c c2)).(\lambda (t2: -T).(\lambda (H5: (pr0 (THead (Flat Appl) u t) t2)).(insert_eq T (THead (Flat -Appl) u t) (\lambda (t0: T).(pr0 t0 t2)) (\lambda (_: T).(arity g c2 t2 a2)) -(\lambda (y: T).(\lambda (H6: (pr0 y t2)).(pr0_ind (\lambda (t0: T).(\lambda -(t3: T).((eq T t0 (THead (Flat Appl) u t)) \to (arity g c2 t3 a2)))) (\lambda -(t0: T).(\lambda (H7: (eq T t0 (THead (Flat Appl) u t))).(let H8 \def -(f_equal T T (\lambda (e: T).e) t0 (THead (Flat Appl) u t) H7) in (eq_ind_r T -(THead (Flat Appl) u t) (\lambda (t3: T).(arity g c2 t3 a2)) (arity_appl g c2 -u a1 (H1 c2 H4 u (pr0_refl u)) t a2 (H3 c2 H4 t (pr0_refl t))) t0 H8)))) -(\lambda (u1: T).(\lambda (u2: T).(\lambda (H7: (pr0 u1 u2)).(\lambda (H8: -(((eq T u1 (THead (Flat Appl) u t)) \to (arity g c2 u2 a2)))).(\lambda (t3: -T).(\lambda (t4: T).(\lambda (H9: (pr0 t3 t4)).(\lambda (H10: (((eq T t3 -(THead (Flat Appl) u t)) \to (arity g c2 t4 a2)))).(\lambda (k: K).(\lambda -(H11: (eq T (THead k u1 t3) (THead (Flat Appl) u t))).(let H12 \def (f_equal -T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) -\Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) -(THead k u1 t3) (THead (Flat Appl) u t) H11) in ((let H13 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t0 _) \Rightarrow t0])) -(THead k u1 t3) (THead (Flat Appl) u t) H11) in ((let H14 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t0) \Rightarrow t0])) -(THead k u1 t3) (THead (Flat Appl) u t) H11) in (\lambda (H15: (eq T u1 -u)).(\lambda (H16: (eq K k (Flat Appl))).(eq_ind_r K (Flat Appl) (\lambda -(k0: K).(arity g c2 (THead k0 u2 t4) a2)) (let H17 \def (eq_ind T t3 (\lambda -(t0: T).((eq T t0 (THead (Flat Appl) u t)) \to (arity g c2 t4 a2))) H10 t -H14) in (let H18 \def (eq_ind T t3 (\lambda (t0: T).(pr0 t0 t4)) H9 t H14) in -(let H19 \def (eq_ind T u1 (\lambda (t0: T).((eq T t0 (THead (Flat Appl) u -t)) \to (arity g c2 u2 a2))) H8 u H15) in (let H20 \def (eq_ind T u1 (\lambda -(t0: T).(pr0 t0 u2)) H7 u H15) in (arity_appl g c2 u2 a1 (H1 c2 H4 u2 H20) t4 -a2 (H3 c2 H4 t4 H18)))))) k H16)))) H13)) H12)))))))))))) (\lambda (u0: -T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H7: (pr0 v1 v2)).(\lambda (H8: -(((eq T v1 (THead (Flat Appl) u t)) \to (arity g c2 v2 a2)))).(\lambda (t3: -T).(\lambda (t4: T).(\lambda (H9: (pr0 t3 t4)).(\lambda (H10: (((eq T t3 -(THead (Flat Appl) u t)) \to (arity g c2 t4 a2)))).(\lambda (H11: (eq T -(THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) (THead (Flat Appl) u -t))).(let H12 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda -(_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead -_ t0 _) \Rightarrow t0])) (THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) -(THead (Flat Appl) u t) H11) in ((let H13 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow (THead -(Bind Abst) u0 t3) | (TLRef _) \Rightarrow (THead (Bind Abst) u0 t3) | (THead -_ _ t0) \Rightarrow t0])) (THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) -(THead (Flat Appl) u t) H11) in (\lambda (H14: (eq T v1 u)).(let H15 \def -(eq_ind T v1 (\lambda (t0: T).((eq T t0 (THead (Flat Appl) u t)) \to (arity g -c2 v2 a2))) H8 u H14) in (let H16 \def (eq_ind T v1 (\lambda (t0: T).(pr0 t0 -v2)) H7 u H14) in (let H17 \def (eq_ind_r T t (\lambda (t0: T).((eq T t3 -(THead (Flat Appl) u t0)) \to (arity g c2 t4 a2))) H10 (THead (Bind Abst) u0 -t3) H13) in (let H18 \def (eq_ind_r T t (\lambda (t0: T).((eq T u (THead -(Flat Appl) u t0)) \to (arity g c2 v2 a2))) H15 (THead (Bind Abst) u0 t3) -H13) in (let H19 \def (eq_ind_r T t (\lambda (t0: T).(\forall (c3: C).((wcpr0 -c c3) \to (\forall (t5: T).((pr0 t0 t5) \to (arity g c3 t5 (AHead a1 -a2))))))) H3 (THead (Bind Abst) u0 t3) H13) in (let H20 \def (eq_ind_r T t -(\lambda (t0: T).(arity g c t0 (AHead a1 a2))) H2 (THead (Bind Abst) u0 t3) -H13) in (let H21 \def (H1 c2 H4 v2 H16) in (let H22 \def (H19 c2 H4 (THead -(Bind Abst) u0 t4) (pr0_comp u0 u0 (pr0_refl u0) t3 t4 H9 (Bind Abst))) in -(let H23 \def (arity_gen_abst g c2 u0 t4 (AHead a1 a2) H22) in (ex3_2_ind A A -(\lambda (a3: A).(\lambda (a4: A).(eq A (AHead a1 a2) (AHead a3 a4)))) -(\lambda (a3: A).(\lambda (_: A).(arity g c2 u0 (asucc g a3)))) (\lambda (_: -A).(\lambda (a4: A).(arity g (CHead c2 (Bind Abst) u0) t4 a4))) (arity g c2 -(THead (Bind Abbr) v2 t4) a2) (\lambda (x0: A).(\lambda (x1: A).(\lambda -(H24: (eq A (AHead a1 a2) (AHead x0 x1))).(\lambda (H25: (arity g c2 u0 -(asucc g x0))).(\lambda (H26: (arity g (CHead c2 (Bind Abst) u0) t4 x1)).(let -H27 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) -with [(ASort _ _) \Rightarrow a1 | (AHead a0 _) \Rightarrow a0])) (AHead a1 -a2) (AHead x0 x1) H24) in ((let H28 \def (f_equal A A (\lambda (e: A).(match -e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a2 | (AHead _ -a0) \Rightarrow a0])) (AHead a1 a2) (AHead x0 x1) H24) in (\lambda (H29: (eq -A a1 x0)).(let H30 \def (eq_ind_r A x1 (\lambda (a0: A).(arity g (CHead c2 -(Bind Abst) u0) t4 a0)) H26 a2 H28) in (let H31 \def (eq_ind_r A x0 (\lambda -(a0: A).(arity g c2 u0 (asucc g a0))) H25 a1 H29) in (arity_bind g Abbr -not_abbr_abst c2 v2 a1 H21 t4 a2 (csuba_arity g (CHead c2 (Bind Abst) u0) t4 -a2 H30 (CHead c2 (Bind Abbr) v2) (csuba_abst g c2 c2 (csuba_refl g c2) u0 a1 -H31 v2 H21))))))) H27))))))) H23)))))))))))) H12)))))))))))) (\lambda (b: -B).(\lambda (H7: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: -T).(\lambda (H8: (pr0 v1 v2)).(\lambda (H9: (((eq T v1 (THead (Flat Appl) u -t)) \to (arity g c2 v2 a2)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda -(H10: (pr0 u1 u2)).(\lambda (H11: (((eq T u1 (THead (Flat Appl) u t)) \to -(arity g c2 u2 a2)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H12: (pr0 -t3 t4)).(\lambda (H13: (((eq T t3 (THead (Flat Appl) u t)) \to (arity g c2 t4 -a2)))).(\lambda (H14: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) -(THead (Flat Appl) u t))).(let H15 \def (f_equal T T (\lambda (e: T).(match e -in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) -\Rightarrow v1 | (THead _ t0 _) \Rightarrow t0])) (THead (Flat Appl) v1 -(THead (Bind b) u1 t3)) (THead (Flat Appl) u t) H14) in ((let H16 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow (THead (Bind b) u1 t3) | (TLRef _) \Rightarrow (THead -(Bind b) u1 t3) | (THead _ _ t0) \Rightarrow t0])) (THead (Flat Appl) v1 -(THead (Bind b) u1 t3)) (THead (Flat Appl) u t) H14) in (\lambda (H17: (eq T -v1 u)).(let H18 \def (eq_ind T v1 (\lambda (t0: T).((eq T t0 (THead (Flat -Appl) u t)) \to (arity g c2 v2 a2))) H9 u H17) in (let H19 \def (eq_ind T v1 -(\lambda (t0: T).(pr0 t0 v2)) H8 u H17) in (let H20 \def (eq_ind_r T t -(\lambda (t0: T).((eq T t3 (THead (Flat Appl) u t0)) \to (arity g c2 t4 a2))) -H13 (THead (Bind b) u1 t3) H16) in (let H21 \def (eq_ind_r T t (\lambda (t0: -T).((eq T u1 (THead (Flat Appl) u t0)) \to (arity g c2 u2 a2))) H11 (THead -(Bind b) u1 t3) H16) in (let H22 \def (eq_ind_r T t (\lambda (t0: T).((eq T u -(THead (Flat Appl) u t0)) \to (arity g c2 v2 a2))) H18 (THead (Bind b) u1 t3) -H16) in (let H23 \def (eq_ind_r T t (\lambda (t0: T).(\forall (c3: C).((wcpr0 -c c3) \to (\forall (t5: T).((pr0 t0 t5) \to (arity g c3 t5 (AHead a1 -a2))))))) H3 (THead (Bind b) u1 t3) H16) in (let H24 \def (eq_ind_r T t -(\lambda (t0: T).(arity g c t0 (AHead a1 a2))) H2 (THead (Bind b) u1 t3) H16) -in (let H25 \def (H1 c2 H4 v2 H19) in (let H26 \def (H23 c2 H4 (THead (Bind -b) u2 t4) (pr0_comp u1 u2 H10 t3 t4 H12 (Bind b))) in (let H27 \def -(arity_gen_bind b H7 g c2 u2 t4 (AHead a1 a2) H26) in (ex2_ind A (\lambda -(a3: A).(arity g c2 u2 a3)) (\lambda (_: A).(arity g (CHead c2 (Bind b) u2) -t4 (AHead a1 a2))) (arity g c2 (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t4)) a2) (\lambda (x: A).(\lambda (H28: (arity g c2 u2 x)).(\lambda -(H29: (arity g (CHead c2 (Bind b) u2) t4 (AHead a1 a2))).(arity_bind g b H7 -c2 u2 x H28 (THead (Flat Appl) (lift (S O) O v2) t4) a2 (arity_appl g (CHead -c2 (Bind b) u2) (lift (S O) O v2) a1 (arity_lift g c2 v2 a1 H25 (CHead c2 -(Bind b) u2) (S O) O (drop_drop (Bind b) O c2 c2 (drop_refl c2) u2)) t4 a2 -H29))))) H27))))))))))))) H15))))))))))))))))) (\lambda (u1: T).(\lambda (u2: -T).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 (THead (Flat Appl) u t)) -\to (arity g c2 u2 a2)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 -t3 t4)).(\lambda (_: (((eq T t3 (THead (Flat Appl) u t)) \to (arity g c2 t4 -a2)))).(\lambda (w: T).(\lambda (_: (subst0 O u2 t4 w)).(\lambda (H12: (eq T -(THead (Bind Abbr) u1 t3) (THead (Flat Appl) u t))).(let H13 \def (eq_ind T -(THead (Bind Abbr) u1 t3) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat -Appl) u t) H12) in (False_ind (arity g c2 (THead (Bind Abbr) u2 w) a2) -H13))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda -(t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 -(THead (Flat Appl) u t)) \to (arity g c2 t4 a2)))).(\lambda (u0: T).(\lambda -(H10: (eq T (THead (Bind b) u0 (lift (S O) O t3)) (THead (Flat Appl) u -t))).(let H11 \def (eq_ind T (THead (Bind b) u0 (lift (S O) O t3)) (\lambda -(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow -(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | -(Flat _) \Rightarrow False])])) I (THead (Flat Appl) u t) H10) in (False_ind -(arity g c2 t4 a2) H11)))))))))) (\lambda (t3: T).(\lambda (t4: T).(\lambda -(_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead (Flat Appl) u t)) \to (arity -g c2 t4 a2)))).(\lambda (u0: T).(\lambda (H9: (eq T (THead (Flat Cast) u0 t3) -(THead (Flat Appl) u t))).(let H10 \def (eq_ind T (THead (Flat Cast) u0 t3) -(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow -(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | -(Flat f) \Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl -\Rightarrow False | Cast \Rightarrow True])])])) I (THead (Flat Appl) u t) -H9) in (False_ind (arity g c2 t4 a2) H10)))))))) y t2 H6))) H5)))))))))))))) -(\lambda (c: C).(\lambda (u: T).(\lambda (a0: A).(\lambda (_: (arity g c u -(asucc g a0))).(\lambda (H1: ((\forall (c2: C).((wcpr0 c c2) \to (\forall -(t2: T).((pr0 u t2) \to (arity g c2 t2 (asucc g a0)))))))).(\lambda (t: -T).(\lambda (_: (arity g c t a0)).(\lambda (H3: ((\forall (c2: C).((wcpr0 c -c2) \to (\forall (t2: T).((pr0 t t2) \to (arity g c2 t2 a0))))))).(\lambda -(c2: C).(\lambda (H4: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H5: (pr0 -(THead (Flat Cast) u t) t2)).(insert_eq T (THead (Flat Cast) u t) (\lambda -(t0: T).(pr0 t0 t2)) (\lambda (_: T).(arity g c2 t2 a0)) (\lambda (y: -T).(\lambda (H6: (pr0 y t2)).(pr0_ind (\lambda (t0: T).(\lambda (t3: T).((eq -T t0 (THead (Flat Cast) u t)) \to (arity g c2 t3 a0)))) (\lambda (t0: -T).(\lambda (H7: (eq T t0 (THead (Flat Cast) u t))).(let H8 \def (f_equal T T -(\lambda (e: T).e) t0 (THead (Flat Cast) u t) H7) in (eq_ind_r T (THead (Flat -Cast) u t) (\lambda (t3: T).(arity g c2 t3 a0)) (arity_cast g c2 u a0 (H1 c2 -H4 u (pr0_refl u)) t (H3 c2 H4 t (pr0_refl t))) t0 H8)))) (\lambda (u1: -T).(\lambda (u2: T).(\lambda (H7: (pr0 u1 u2)).(\lambda (H8: (((eq T u1 -(THead (Flat Cast) u t)) \to (arity g c2 u2 a0)))).(\lambda (t3: T).(\lambda -(t4: T).(\lambda (H9: (pr0 t3 t4)).(\lambda (H10: (((eq T t3 (THead (Flat -Cast) u t)) \to (arity g c2 t4 a0)))).(\lambda (k: K).(\lambda (H11: (eq T -(THead k u1 t3) (THead (Flat Cast) u t))).(let H12 \def (f_equal T K (\lambda -(e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k -| (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) -(THead (Flat Cast) u t) H11) in ((let H13 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | -(TLRef _) \Rightarrow u1 | (THead _ t0 _) \Rightarrow t0])) (THead k u1 t3) -(THead (Flat Cast) u t) H11) in ((let H14 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | -(TLRef _) \Rightarrow t3 | (THead _ _ t0) \Rightarrow t0])) (THead k u1 t3) -(THead (Flat Cast) u t) H11) in (\lambda (H15: (eq T u1 u)).(\lambda (H16: -(eq K k (Flat Cast))).(eq_ind_r K (Flat Cast) (\lambda (k0: K).(arity g c2 -(THead k0 u2 t4) a0)) (let H17 \def (eq_ind T t3 (\lambda (t0: T).((eq T t0 -(THead (Flat Cast) u t)) \to (arity g c2 t4 a0))) H10 t H14) in (let H18 \def -(eq_ind T t3 (\lambda (t0: T).(pr0 t0 t4)) H9 t H14) in (let H19 \def (eq_ind -T u1 (\lambda (t0: T).((eq T t0 (THead (Flat Cast) u t)) \to (arity g c2 u2 -a0))) H8 u H15) in (let H20 \def (eq_ind T u1 (\lambda (t0: T).(pr0 t0 u2)) -H7 u H15) in (arity_cast g c2 u2 a0 (H1 c2 H4 u2 H20) t4 (H3 c2 H4 t4 -H18)))))) k H16)))) H13)) H12)))))))))))) (\lambda (u0: T).(\lambda (v1: -T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (THead -(Flat Cast) u t)) \to (arity g c2 v2 a0)))).(\lambda (t3: T).(\lambda (t4: -T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead (Flat Cast) u t)) -\to (arity g c2 t4 a0)))).(\lambda (H11: (eq T (THead (Flat Appl) v1 (THead -(Bind Abst) u0 t3)) (THead (Flat Cast) u t))).(let H12 \def (eq_ind T (THead -(Flat Appl) v1 (THead (Bind Abst) u0 t3)) (\lambda (ee: T).(match ee in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f -in F return (\lambda (_: F).Prop) with [Appl \Rightarrow True | Cast -\Rightarrow False])])])) I (THead (Flat Cast) u t) H11) in (False_ind (arity -g c2 (THead (Bind Abbr) v2 t4) a0) H12)))))))))))) (\lambda (b: B).(\lambda -(_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 -v1 v2)).(\lambda (_: (((eq T v1 (THead (Flat Cast) u t)) \to (arity g c2 v2 -a0)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda -(_: (((eq T u1 (THead (Flat Cast) u t)) \to (arity g c2 u2 a0)))).(\lambda -(t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 -(THead (Flat Cast) u t)) \to (arity g c2 t4 a0)))).(\lambda (H14: (eq T -(THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Cast) u t))).(let -H15 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (\lambda -(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow -(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | -(Flat f) \Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl -\Rightarrow True | Cast \Rightarrow False])])])) I (THead (Flat Cast) u t) -H14) in (False_ind (arity g c2 (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t4)) a0) H15))))))))))))))))) (\lambda (u1: T).(\lambda (u2: -T).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 (THead (Flat Cast) u t)) -\to (arity g c2 u2 a0)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 -t3 t4)).(\lambda (_: (((eq T t3 (THead (Flat Cast) u t)) \to (arity g c2 t4 -a0)))).(\lambda (w: T).(\lambda (_: (subst0 O u2 t4 w)).(\lambda (H12: (eq T -(THead (Bind Abbr) u1 t3) (THead (Flat Cast) u t))).(let H13 \def (eq_ind T -(THead (Bind Abbr) u1 t3) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat -Cast) u t) H12) in (False_ind (arity g c2 (THead (Bind Abbr) u2 w) a0) -H13))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda -(t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 -(THead (Flat Cast) u t)) \to (arity g c2 t4 a0)))).(\lambda (u0: T).(\lambda -(H10: (eq T (THead (Bind b) u0 (lift (S O) O t3)) (THead (Flat Cast) u -t))).(let H11 \def (eq_ind T (THead (Bind b) u0 (lift (S O) O t3)) (\lambda -(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow -(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | -(Flat _) \Rightarrow False])])) I (THead (Flat Cast) u t) H10) in (False_ind -(arity g c2 t4 a0) H11)))))))))) (\lambda (t3: T).(\lambda (t4: T).(\lambda -(H7: (pr0 t3 t4)).(\lambda (H8: (((eq T t3 (THead (Flat Cast) u t)) \to -(arity g c2 t4 a0)))).(\lambda (u0: T).(\lambda (H9: (eq T (THead (Flat Cast) -u0 t3) (THead (Flat Cast) u t))).(let H10 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | -(TLRef _) \Rightarrow u0 | (THead _ t0 _) \Rightarrow t0])) (THead (Flat -Cast) u0 t3) (THead (Flat Cast) u t) H9) in ((let H11 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t0) \Rightarrow t0])) -(THead (Flat Cast) u0 t3) (THead (Flat Cast) u t) H9) in (\lambda (_: (eq T -u0 u)).(let H13 \def (eq_ind T t3 (\lambda (t0: T).((eq T t0 (THead (Flat -Cast) u t)) \to (arity g c2 t4 a0))) H8 t H11) in (let H14 \def (eq_ind T t3 -(\lambda (t0: T).(pr0 t0 t4)) H7 t H11) in (H3 c2 H4 t4 H14))))) H10)))))))) -y t2 H6))) H5))))))))))))) (\lambda (c: C).(\lambda (t: T).(\lambda (a1: -A).(\lambda (_: (arity g c t a1)).(\lambda (H1: ((\forall (c2: C).((wcpr0 c -c2) \to (\forall (t2: T).((pr0 t t2) \to (arity g c2 t2 a1))))))).(\lambda -(a2: A).(\lambda (H2: (leq g a1 a2)).(\lambda (c2: C).(\lambda (H3: (wcpr0 c -c2)).(\lambda (t2: T).(\lambda (H4: (pr0 t t2)).(arity_repl g c2 t2 a1 (H1 c2 -H3 t2 H4) a2 H2)))))))))))) c1 t1 a H))))). -(* COMMENTS -Initial nodes: 10246 -END *) - -theorem arity_sred_wcpr0_pr1: - \forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall (g: G).(\forall -(c1: C).(\forall (a: A).((arity g c1 t1 a) \to (\forall (c2: C).((wcpr0 c1 -c2) \to (arity g c2 t2 a))))))))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr1 t1 t2)).(pr1_ind (\lambda -(t: T).(\lambda (t0: T).(\forall (g: G).(\forall (c1: C).(\forall (a: -A).((arity g c1 t a) \to (\forall (c2: C).((wcpr0 c1 c2) \to (arity g c2 t0 -a))))))))) (\lambda (t: T).(\lambda (g: G).(\lambda (c1: C).(\lambda (a: -A).(\lambda (H0: (arity g c1 t a)).(\lambda (c2: C).(\lambda (H1: (wcpr0 c1 -c2)).(arity_sred_wcpr0_pr0 g c1 t a H0 c2 H1 t (pr0_refl t))))))))) (\lambda -(t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t4 t3)).(\lambda (t5: T).(\lambda -(_: (pr1 t3 t5)).(\lambda (H2: ((\forall (g: G).(\forall (c1: C).(\forall (a: -A).((arity g c1 t3 a) \to (\forall (c2: C).((wcpr0 c1 c2) \to (arity g c2 t5 -a))))))))).(\lambda (g: G).(\lambda (c1: C).(\lambda (a: A).(\lambda (H3: -(arity g c1 t4 a)).(\lambda (c2: C).(\lambda (H4: (wcpr0 c1 c2)).(H2 g c2 a -(arity_sred_wcpr0_pr0 g c1 t4 a H3 c2 H4 t3 H0) c2 (wcpr0_refl -c2)))))))))))))) t1 t2 H))). -(* COMMENTS -Initial nodes: 213 -END *) - -theorem arity_sred_pr2: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall -(g: G).(\forall (a: A).((arity g c t1 a) \to (arity g c t2 a))))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 -t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (g: -G).(\forall (a: A).((arity g c0 t a) \to (arity g c0 t0 a))))))) (\lambda -(c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(\lambda -(g: G).(\lambda (a: A).(\lambda (H1: (arity g c0 t3 a)).(arity_sred_wcpr0_pr0 -g c0 t3 a H1 c0 (wcpr0_refl c0) t4 H0)))))))) (\lambda (c0: C).(\lambda (d: -C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind -Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 -t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (g: -G).(\lambda (a: A).(\lambda (H3: (arity g c0 t3 a)).(arity_subst0 g c0 t4 a -(arity_sred_wcpr0_pr0 g c0 t3 a H3 c0 (wcpr0_refl c0) t4 H1) d u i H0 t -H2)))))))))))))) c t1 t2 H)))). -(* COMMENTS -Initial nodes: 205 -END *) - -theorem arity_sred_pr3: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall -(g: G).(\forall (a: A).((arity g c t1 a) \to (arity g c t2 a))))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 -t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (g: G).(\forall (a: -A).((arity g c t a) \to (arity g c t0 a)))))) (\lambda (t: T).(\lambda (g: -G).(\lambda (a: A).(\lambda (H0: (arity g c t a)).H0)))) (\lambda (t3: -T).(\lambda (t4: T).(\lambda (H0: (pr2 c t4 t3)).(\lambda (t5: T).(\lambda -(_: (pr3 c t3 t5)).(\lambda (H2: ((\forall (g: G).(\forall (a: A).((arity g c -t3 a) \to (arity g c t5 a)))))).(\lambda (g: G).(\lambda (a: A).(\lambda (H3: -(arity g c t4 a)).(H2 g a (arity_sred_pr2 c t4 t3 H0 g a H3))))))))))) t1 t2 -H)))). -(* COMMENTS -Initial nodes: 151 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/arity/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/arity/props.ma deleted file mode 100644 index 6f131df09..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/arity/props.ma +++ /dev/null @@ -1,434 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/arity/fwd.ma". - -theorem node_inh: - \forall (g: G).(\forall (n: nat).(\forall (k: nat).(ex_2 C T (\lambda (c: -C).(\lambda (t: T).(arity g c t (ASort k n))))))) -\def - \lambda (g: G).(\lambda (n: nat).(\lambda (k: nat).(nat_ind (\lambda (n0: -nat).(ex_2 C T (\lambda (c: C).(\lambda (t: T).(arity g c t (ASort n0 n)))))) -(ex_2_intro C T (\lambda (c: C).(\lambda (t: T).(arity g c t (ASort O n)))) -(CSort O) (TSort n) (arity_sort g (CSort O) n)) (\lambda (n0: nat).(\lambda -(H: (ex_2 C T (\lambda (c: C).(\lambda (t: T).(arity g c t (ASort n0 -n)))))).(let H0 \def H in (ex_2_ind C T (\lambda (c: C).(\lambda (t: -T).(arity g c t (ASort n0 n)))) (ex_2 C T (\lambda (c: C).(\lambda (t: -T).(arity g c t (ASort (S n0) n))))) (\lambda (x0: C).(\lambda (x1: -T).(\lambda (H1: (arity g x0 x1 (ASort n0 n))).(ex_2_intro C T (\lambda (c: -C).(\lambda (t: T).(arity g c t (ASort (S n0) n)))) (CHead x0 (Bind Abst) x1) -(TLRef O) (arity_abst g (CHead x0 (Bind Abst) x1) x0 x1 O (getl_refl Abst x0 -x1) (ASort (S n0) n) H1))))) H0)))) k))). -(* COMMENTS -Initial nodes: 253 -END *) - -theorem arity_lift: - \forall (g: G).(\forall (c2: C).(\forall (t: T).(\forall (a: A).((arity g c2 -t a) \to (\forall (c1: C).(\forall (h: nat).(\forall (d: nat).((drop h d c1 -c2) \to (arity g c1 (lift h d t) a))))))))) -\def - \lambda (g: G).(\lambda (c2: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: -(arity g c2 t a)).(arity_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda (a0: -A).(\forall (c1: C).(\forall (h: nat).(\forall (d: nat).((drop h d c1 c) \to -(arity g c1 (lift h d t0) a0)))))))) (\lambda (c: C).(\lambda (n: -nat).(\lambda (c1: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (_: (drop -h d c1 c)).(eq_ind_r T (TSort n) (\lambda (t0: T).(arity g c1 t0 (ASort O -n))) (arity_sort g c1 n) (lift h d (TSort n)) (lift_sort n h d)))))))) -(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (a0: A).(\lambda (H1: -(arity g d u a0)).(\lambda (H2: ((\forall (c1: C).(\forall (h: nat).(\forall -(d0: nat).((drop h d0 c1 d) \to (arity g c1 (lift h d0 u) a0))))))).(\lambda -(c1: C).(\lambda (h: nat).(\lambda (d0: nat).(\lambda (H3: (drop h d0 c1 -c)).(lt_le_e i d0 (arity g c1 (lift h d0 (TLRef i)) a0) (\lambda (H4: (lt i -d0)).(eq_ind_r T (TLRef i) (\lambda (t0: T).(arity g c1 t0 a0)) (let H5 \def -(drop_getl_trans_le i d0 (le_S_n i d0 (le_S (S i) d0 H4)) c1 c h H3 (CHead d -(Bind Abbr) u) H0) in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop i -O c1 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d0 i) e0 e1))) -(\lambda (_: C).(\lambda (e1: C).(clear e1 (CHead d (Bind Abbr) u)))) (arity -g c1 (TLRef i) a0) (\lambda (x0: C).(\lambda (x1: C).(\lambda (H6: (drop i O -c1 x0)).(\lambda (H7: (drop h (minus d0 i) x0 x1)).(\lambda (H8: (clear x1 -(CHead d (Bind Abbr) u))).(let H9 \def (eq_ind nat (minus d0 i) (\lambda (n: -nat).(drop h n x0 x1)) H7 (S (minus d0 (S i))) (minus_x_Sy d0 i H4)) in (let -H10 \def (drop_clear_S x1 x0 h (minus d0 (S i)) H9 Abbr d u H8) in (ex2_ind C -(\lambda (c3: C).(clear x0 (CHead c3 (Bind Abbr) (lift h (minus d0 (S i)) -u)))) (\lambda (c3: C).(drop h (minus d0 (S i)) c3 d)) (arity g c1 (TLRef i) -a0) (\lambda (x: C).(\lambda (H11: (clear x0 (CHead x (Bind Abbr) (lift h -(minus d0 (S i)) u)))).(\lambda (H12: (drop h (minus d0 (S i)) x -d)).(arity_abbr g c1 x (lift h (minus d0 (S i)) u) i (getl_intro i c1 (CHead -x (Bind Abbr) (lift h (minus d0 (S i)) u)) x0 H6 H11) a0 (H2 x h (minus d0 (S -i)) H12))))) H10)))))))) H5)) (lift h d0 (TLRef i)) (lift_lref_lt i h d0 -H4))) (\lambda (H4: (le d0 i)).(eq_ind_r T (TLRef (plus i h)) (\lambda (t0: -T).(arity g c1 t0 a0)) (arity_abbr g c1 d u (plus i h) (drop_getl_trans_ge i -c1 c d0 h H3 (CHead d (Bind Abbr) u) H0 H4) a0 H1) (lift h d0 (TLRef i)) -(lift_lref_ge i h d0 H4)))))))))))))))) (\lambda (c: C).(\lambda (d: -C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d (Bind -Abst) u))).(\lambda (a0: A).(\lambda (H1: (arity g d u (asucc g -a0))).(\lambda (H2: ((\forall (c1: C).(\forall (h: nat).(\forall (d0: -nat).((drop h d0 c1 d) \to (arity g c1 (lift h d0 u) (asucc g -a0)))))))).(\lambda (c1: C).(\lambda (h: nat).(\lambda (d0: nat).(\lambda -(H3: (drop h d0 c1 c)).(lt_le_e i d0 (arity g c1 (lift h d0 (TLRef i)) a0) -(\lambda (H4: (lt i d0)).(eq_ind_r T (TLRef i) (\lambda (t0: T).(arity g c1 -t0 a0)) (let H5 \def (drop_getl_trans_le i d0 (le_S_n i d0 (le_S (S i) d0 -H4)) c1 c h H3 (CHead d (Bind Abst) u) H0) in (ex3_2_ind C C (\lambda (e0: -C).(\lambda (_: C).(drop i O c1 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop -h (minus d0 i) e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear e1 (CHead d -(Bind Abst) u)))) (arity g c1 (TLRef i) a0) (\lambda (x0: C).(\lambda (x1: -C).(\lambda (H6: (drop i O c1 x0)).(\lambda (H7: (drop h (minus d0 i) x0 -x1)).(\lambda (H8: (clear x1 (CHead d (Bind Abst) u))).(let H9 \def (eq_ind -nat (minus d0 i) (\lambda (n: nat).(drop h n x0 x1)) H7 (S (minus d0 (S i))) -(minus_x_Sy d0 i H4)) in (let H10 \def (drop_clear_S x1 x0 h (minus d0 (S i)) -H9 Abst d u H8) in (ex2_ind C (\lambda (c3: C).(clear x0 (CHead c3 (Bind -Abst) (lift h (minus d0 (S i)) u)))) (\lambda (c3: C).(drop h (minus d0 (S -i)) c3 d)) (arity g c1 (TLRef i) a0) (\lambda (x: C).(\lambda (H11: (clear x0 -(CHead x (Bind Abst) (lift h (minus d0 (S i)) u)))).(\lambda (H12: (drop h -(minus d0 (S i)) x d)).(arity_abst g c1 x (lift h (minus d0 (S i)) u) i -(getl_intro i c1 (CHead x (Bind Abst) (lift h (minus d0 (S i)) u)) x0 H6 H11) -a0 (H2 x h (minus d0 (S i)) H12))))) H10)))))))) H5)) (lift h d0 (TLRef i)) -(lift_lref_lt i h d0 H4))) (\lambda (H4: (le d0 i)).(eq_ind_r T (TLRef (plus -i h)) (\lambda (t0: T).(arity g c1 t0 a0)) (arity_abst g c1 d u (plus i h) -(drop_getl_trans_ge i c1 c d0 h H3 (CHead d (Bind Abst) u) H0 H4) a0 H1) -(lift h d0 (TLRef i)) (lift_lref_ge i h d0 H4)))))))))))))))) (\lambda (b: -B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c: C).(\lambda (u: -T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H2: ((\forall -(c1: C).(\forall (h: nat).(\forall (d: nat).((drop h d c1 c) \to (arity g c1 -(lift h d u) a1))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity -g (CHead c (Bind b) u) t0 a2)).(\lambda (H4: ((\forall (c1: C).(\forall (h: -nat).(\forall (d: nat).((drop h d c1 (CHead c (Bind b) u)) \to (arity g c1 -(lift h d t0) a2))))))).(\lambda (c1: C).(\lambda (h: nat).(\lambda (d: -nat).(\lambda (H5: (drop h d c1 c)).(eq_ind_r T (THead (Bind b) (lift h d u) -(lift h (s (Bind b) d) t0)) (\lambda (t1: T).(arity g c1 t1 a2)) (arity_bind -g b H0 c1 (lift h d u) a1 (H2 c1 h d H5) (lift h (s (Bind b) d) t0) a2 (H4 -(CHead c1 (Bind b) (lift h d u)) h (s (Bind b) d) (drop_skip_bind h d c1 c H5 -b u))) (lift h d (THead (Bind b) u t0)) (lift_head (Bind b) u t0 h -d))))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda -(_: (arity g c u (asucc g a1))).(\lambda (H1: ((\forall (c1: C).(\forall (h: -nat).(\forall (d: nat).((drop h d c1 c) \to (arity g c1 (lift h d u) (asucc g -a1)))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c -(Bind Abst) u) t0 a2)).(\lambda (H3: ((\forall (c1: C).(\forall (h: -nat).(\forall (d: nat).((drop h d c1 (CHead c (Bind Abst) u)) \to (arity g c1 -(lift h d t0) a2))))))).(\lambda (c1: C).(\lambda (h: nat).(\lambda (d: -nat).(\lambda (H4: (drop h d c1 c)).(eq_ind_r T (THead (Bind Abst) (lift h d -u) (lift h (s (Bind Abst) d) t0)) (\lambda (t1: T).(arity g c1 t1 (AHead a1 -a2))) (arity_head g c1 (lift h d u) a1 (H1 c1 h d H4) (lift h (s (Bind Abst) -d) t0) a2 (H3 (CHead c1 (Bind Abst) (lift h d u)) h (s (Bind Abst) d) -(drop_skip_bind h d c1 c H4 Abst u))) (lift h d (THead (Bind Abst) u t0)) -(lift_head (Bind Abst) u t0 h d))))))))))))))) (\lambda (c: C).(\lambda (u: -T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H1: ((\forall -(c1: C).(\forall (h: nat).(\forall (d: nat).((drop h d c1 c) \to (arity g c1 -(lift h d u) a1))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity -g c t0 (AHead a1 a2))).(\lambda (H3: ((\forall (c1: C).(\forall (h: -nat).(\forall (d: nat).((drop h d c1 c) \to (arity g c1 (lift h d t0) (AHead -a1 a2)))))))).(\lambda (c1: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda -(H4: (drop h d c1 c)).(eq_ind_r T (THead (Flat Appl) (lift h d u) (lift h (s -(Flat Appl) d) t0)) (\lambda (t1: T).(arity g c1 t1 a2)) (arity_appl g c1 -(lift h d u) a1 (H1 c1 h d H4) (lift h (s (Flat Appl) d) t0) a2 (H3 c1 h (s -(Flat Appl) d) H4)) (lift h d (THead (Flat Appl) u t0)) (lift_head (Flat -Appl) u t0 h d))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a0: -A).(\lambda (_: (arity g c u (asucc g a0))).(\lambda (H1: ((\forall (c1: -C).(\forall (h: nat).(\forall (d: nat).((drop h d c1 c) \to (arity g c1 (lift -h d u) (asucc g a0)))))))).(\lambda (t0: T).(\lambda (_: (arity g c t0 -a0)).(\lambda (H3: ((\forall (c1: C).(\forall (h: nat).(\forall (d: -nat).((drop h d c1 c) \to (arity g c1 (lift h d t0) a0))))))).(\lambda (c1: -C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H4: (drop h d c1 -c)).(eq_ind_r T (THead (Flat Cast) (lift h d u) (lift h (s (Flat Cast) d) -t0)) (\lambda (t1: T).(arity g c1 t1 a0)) (arity_cast g c1 (lift h d u) a0 -(H1 c1 h d H4) (lift h (s (Flat Cast) d) t0) (H3 c1 h (s (Flat Cast) d) H4)) -(lift h d (THead (Flat Cast) u t0)) (lift_head (Flat Cast) u t0 h -d)))))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda -(_: (arity g c t0 a1)).(\lambda (H1: ((\forall (c1: C).(\forall (h: -nat).(\forall (d: nat).((drop h d c1 c) \to (arity g c1 (lift h d t0) -a1))))))).(\lambda (a2: A).(\lambda (H2: (leq g a1 a2)).(\lambda (c1: -C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H3: (drop h d c1 -c)).(arity_repl g c1 (lift h d t0) a1 (H1 c1 h d H3) a2 H2)))))))))))) c2 t a -H))))). -(* COMMENTS -Initial nodes: 2661 -END *) - -theorem arity_mono: - \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a1: A).((arity g c -t a1) \to (\forall (a2: A).((arity g c t a2) \to (leq g a1 a2))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a1: A).(\lambda (H: -(arity g c t a1)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (a: -A).(\forall (a2: A).((arity g c0 t0 a2) \to (leq g a a2)))))) (\lambda (c0: -C).(\lambda (n: nat).(\lambda (a2: A).(\lambda (H0: (arity g c0 (TSort n) -a2)).(leq_sym g a2 (ASort O n) (arity_gen_sort g c0 n a2 H0)))))) (\lambda -(c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl -i c0 (CHead d (Bind Abbr) u))).(\lambda (a: A).(\lambda (_: (arity g d u -a)).(\lambda (H2: ((\forall (a2: A).((arity g d u a2) \to (leq g a -a2))))).(\lambda (a2: A).(\lambda (H3: (arity g c0 (TLRef i) a2)).(let H4 -\def (arity_gen_lref g c0 i a2 H3) in (or_ind (ex2_2 C T (\lambda (d0: -C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abbr) u0)))) (\lambda (d0: -C).(\lambda (u0: T).(arity g d0 u0 a2)))) (ex2_2 C T (\lambda (d0: -C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0)))) (\lambda (d0: -C).(\lambda (u0: T).(arity g d0 u0 (asucc g a2))))) (leq g a a2) (\lambda -(H5: (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind -Abbr) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 -a2))))).(ex2_2_ind C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 -(Bind Abbr) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a2))) -(leq g a a2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl i c0 -(CHead x0 (Bind Abbr) x1))).(\lambda (H7: (arity g x0 x1 a2)).(let H8 \def -(eq_ind C (CHead d (Bind Abbr) u) (\lambda (c1: C).(getl i c0 c1)) H0 (CHead -x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead x0 (Bind -Abbr) x1) H6)) in (let H9 \def (f_equal C C (\lambda (e: C).(match e in C -return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) -\Rightarrow c1])) (CHead d (Bind Abbr) u) (CHead x0 (Bind Abbr) x1) -(getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead x0 (Bind Abbr) x1) H6)) in -((let H10 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: -C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead -d (Bind Abbr) u) (CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind Abbr) -u) i H0 (CHead x0 (Bind Abbr) x1) H6)) in (\lambda (H11: (eq C d x0)).(let -H12 \def (eq_ind_r T x1 (\lambda (t0: T).(getl i c0 (CHead x0 (Bind Abbr) -t0))) H8 u H10) in (let H13 \def (eq_ind_r T x1 (\lambda (t0: T).(arity g x0 -t0 a2)) H7 u H10) in (let H14 \def (eq_ind_r C x0 (\lambda (c1: C).(getl i c0 -(CHead c1 (Bind Abbr) u))) H12 d H11) in (let H15 \def (eq_ind_r C x0 -(\lambda (c1: C).(arity g c1 u a2)) H13 d H11) in (H2 a2 H15))))))) H9))))))) -H5)) (\lambda (H5: (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 -(CHead d0 (Bind Abst) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 -(asucc g a2)))))).(ex2_2_ind C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 -(CHead d0 (Bind Abst) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 -(asucc g a2)))) (leq g a a2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: -(getl i c0 (CHead x0 (Bind Abst) x1))).(\lambda (_: (arity g x0 x1 (asucc g -a2))).(let H8 \def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c1: C).(getl i -c0 c1)) H0 (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abbr) u) i -H0 (CHead x0 (Bind Abst) x1) H6)) in (let H9 \def (eq_ind C (CHead d (Bind -Abbr) u) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with -[(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return -(\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B return -(\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | -Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead x0 (Bind -Abst) x1) (getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead x0 (Bind Abst) -x1) H6)) in (False_ind (leq g a a2) H9))))))) H5)) H4)))))))))))) (\lambda -(c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl -i c0 (CHead d (Bind Abst) u))).(\lambda (a: A).(\lambda (_: (arity g d u -(asucc g a))).(\lambda (H2: ((\forall (a2: A).((arity g d u a2) \to (leq g -(asucc g a) a2))))).(\lambda (a2: A).(\lambda (H3: (arity g c0 (TLRef i) -a2)).(let H4 \def (arity_gen_lref g c0 i a2 H3) in (or_ind (ex2_2 C T -(\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abbr) u0)))) -(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a2)))) (ex2_2 C T (\lambda -(d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0)))) (\lambda -(d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a2))))) (leq g a a2) -(\lambda (H5: (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead -d0 (Bind Abbr) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 -a2))))).(ex2_2_ind C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 -(Bind Abbr) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a2))) -(leq g a a2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl i c0 -(CHead x0 (Bind Abbr) x1))).(\lambda (_: (arity g x0 x1 a2)).(let H8 \def -(eq_ind C (CHead d (Bind Abst) u) (\lambda (c1: C).(getl i c0 c1)) H0 (CHead -x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind Abst) u) i H0 (CHead x0 (Bind -Abbr) x1) H6)) in (let H9 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda -(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind b) \Rightarrow (match b in B return (\lambda (_: -B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow True | Void -\Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead x0 (Bind Abbr) -x1) (getl_mono c0 (CHead d (Bind Abst) u) i H0 (CHead x0 (Bind Abbr) x1) H6)) -in (False_ind (leq g a a2) H9))))))) H5)) (\lambda (H5: (ex2_2 C T (\lambda -(d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0)))) (\lambda -(d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a2)))))).(ex2_2_ind C T -(\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0)))) -(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a2)))) (leq g a a2) -(\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl i c0 (CHead x0 (Bind -Abst) x1))).(\lambda (H7: (arity g x0 x1 (asucc g a2))).(let H8 \def (eq_ind -C (CHead d (Bind Abst) u) (\lambda (c1: C).(getl i c0 c1)) H0 (CHead x0 (Bind -Abst) x1) (getl_mono c0 (CHead d (Bind Abst) u) i H0 (CHead x0 (Bind Abst) -x1) H6)) in (let H9 \def (f_equal C C (\lambda (e: C).(match e in C return -(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow -c1])) (CHead d (Bind Abst) u) (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead -d (Bind Abst) u) i H0 (CHead x0 (Bind Abst) x1) H6)) in ((let H10 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind -Abst) u) (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abst) u) i H0 -(CHead x0 (Bind Abst) x1) H6)) in (\lambda (H11: (eq C d x0)).(let H12 \def -(eq_ind_r T x1 (\lambda (t0: T).(getl i c0 (CHead x0 (Bind Abst) t0))) H8 u -H10) in (let H13 \def (eq_ind_r T x1 (\lambda (t0: T).(arity g x0 t0 (asucc g -a2))) H7 u H10) in (let H14 \def (eq_ind_r C x0 (\lambda (c1: C).(getl i c0 -(CHead c1 (Bind Abst) u))) H12 d H11) in (let H15 \def (eq_ind_r C x0 -(\lambda (c1: C).(arity g c1 u (asucc g a2))) H13 d H11) in (asucc_inj g a a2 -(H2 (asucc g a2) H15)))))))) H9))))))) H5)) H4)))))))))))) (\lambda (b: -B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c0: C).(\lambda (u: -T).(\lambda (a2: A).(\lambda (_: (arity g c0 u a2)).(\lambda (_: ((\forall -(a3: A).((arity g c0 u a3) \to (leq g a2 a3))))).(\lambda (t0: T).(\lambda -(a3: A).(\lambda (_: (arity g (CHead c0 (Bind b) u) t0 a3)).(\lambda (H4: -((\forall (a4: A).((arity g (CHead c0 (Bind b) u) t0 a4) \to (leq g a3 -a4))))).(\lambda (a0: A).(\lambda (H5: (arity g c0 (THead (Bind b) u t0) -a0)).(let H6 \def (arity_gen_bind b H0 g c0 u t0 a0 H5) in (ex2_ind A -(\lambda (a4: A).(arity g c0 u a4)) (\lambda (_: A).(arity g (CHead c0 (Bind -b) u) t0 a0)) (leq g a3 a0) (\lambda (x: A).(\lambda (_: (arity g c0 u -x)).(\lambda (H8: (arity g (CHead c0 (Bind b) u) t0 a0)).(H4 a0 H8)))) -H6))))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a2: A).(\lambda -(_: (arity g c0 u (asucc g a2))).(\lambda (H1: ((\forall (a3: A).((arity g c0 -u a3) \to (leq g (asucc g a2) a3))))).(\lambda (t0: T).(\lambda (a3: -A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u) t0 a3)).(\lambda (H3: -((\forall (a4: A).((arity g (CHead c0 (Bind Abst) u) t0 a4) \to (leq g a3 -a4))))).(\lambda (a0: A).(\lambda (H4: (arity g c0 (THead (Bind Abst) u t0) -a0)).(let H5 \def (arity_gen_abst g c0 u t0 a0 H4) in (ex3_2_ind A A (\lambda -(a4: A).(\lambda (a5: A).(eq A a0 (AHead a4 a5)))) (\lambda (a4: A).(\lambda -(_: A).(arity g c0 u (asucc g a4)))) (\lambda (_: A).(\lambda (a5: A).(arity -g (CHead c0 (Bind Abst) u) t0 a5))) (leq g (AHead a2 a3) a0) (\lambda (x0: -A).(\lambda (x1: A).(\lambda (H6: (eq A a0 (AHead x0 x1))).(\lambda (H7: -(arity g c0 u (asucc g x0))).(\lambda (H8: (arity g (CHead c0 (Bind Abst) u) -t0 x1)).(eq_ind_r A (AHead x0 x1) (\lambda (a: A).(leq g (AHead a2 a3) a)) -(leq_head g a2 x0 (asucc_inj g a2 x0 (H1 (asucc g x0) H7)) a3 x1 (H3 x1 H8)) -a0 H6)))))) H5))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a2: -A).(\lambda (_: (arity g c0 u a2)).(\lambda (_: ((\forall (a3: A).((arity g -c0 u a3) \to (leq g a2 a3))))).(\lambda (t0: T).(\lambda (a3: A).(\lambda (_: -(arity g c0 t0 (AHead a2 a3))).(\lambda (H3: ((\forall (a4: A).((arity g c0 -t0 a4) \to (leq g (AHead a2 a3) a4))))).(\lambda (a0: A).(\lambda (H4: (arity -g c0 (THead (Flat Appl) u t0) a0)).(let H5 \def (arity_gen_appl g c0 u t0 a0 -H4) in (ex2_ind A (\lambda (a4: A).(arity g c0 u a4)) (\lambda (a4: A).(arity -g c0 t0 (AHead a4 a0))) (leq g a3 a0) (\lambda (x: A).(\lambda (_: (arity g -c0 u x)).(\lambda (H7: (arity g c0 t0 (AHead x a0))).(ahead_inj_snd g a2 a3 x -a0 (H3 (AHead x a0) H7))))) H5))))))))))))) (\lambda (c0: C).(\lambda (u: -T).(\lambda (a: A).(\lambda (_: (arity g c0 u (asucc g a))).(\lambda (_: -((\forall (a2: A).((arity g c0 u a2) \to (leq g (asucc g a) a2))))).(\lambda -(t0: T).(\lambda (_: (arity g c0 t0 a)).(\lambda (H3: ((\forall (a2: -A).((arity g c0 t0 a2) \to (leq g a a2))))).(\lambda (a2: A).(\lambda (H4: -(arity g c0 (THead (Flat Cast) u t0) a2)).(let H5 \def (arity_gen_cast g c0 u -t0 a2 H4) in (land_ind (arity g c0 u (asucc g a2)) (arity g c0 t0 a2) (leq g -a a2) (\lambda (_: (arity g c0 u (asucc g a2))).(\lambda (H7: (arity g c0 t0 -a2)).(H3 a2 H7))) H5)))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda -(a2: A).(\lambda (_: (arity g c0 t0 a2)).(\lambda (H1: ((\forall (a3: -A).((arity g c0 t0 a3) \to (leq g a2 a3))))).(\lambda (a3: A).(\lambda (H2: -(leq g a2 a3)).(\lambda (a0: A).(\lambda (H3: (arity g c0 t0 a0)).(leq_trans -g a3 a2 (leq_sym g a2 a3 H2) a0 (H1 a0 H3))))))))))) c t a1 H))))). -(* COMMENTS -Initial nodes: 2947 -END *) - -theorem arity_repellent: - \forall (g: G).(\forall (c: C).(\forall (w: T).(\forall (t: T).(\forall (a1: -A).((arity g (CHead c (Bind Abst) w) t a1) \to (\forall (a2: A).((arity g c -(THead (Bind Abst) w t) a2) \to ((leq g a1 a2) \to (\forall (P: -Prop).P))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (w: T).(\lambda (t: T).(\lambda (a1: -A).(\lambda (H: (arity g (CHead c (Bind Abst) w) t a1)).(\lambda (a2: -A).(\lambda (H0: (arity g c (THead (Bind Abst) w t) a2)).(\lambda (H1: (leq g -a1 a2)).(\lambda (P: Prop).(let H_y \def (arity_repl g (CHead c (Bind Abst) -w) t a1 H a2 H1) in (let H2 \def (arity_gen_abst g c w t a2 H0) in (ex3_2_ind -A A (\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: -A).(\lambda (_: A).(arity g c w (asucc g a3)))) (\lambda (_: A).(\lambda (a4: -A).(arity g (CHead c (Bind Abst) w) t a4))) P (\lambda (x0: A).(\lambda (x1: -A).(\lambda (H3: (eq A a2 (AHead x0 x1))).(\lambda (_: (arity g c w (asucc g -x0))).(\lambda (H5: (arity g (CHead c (Bind Abst) w) t x1)).(let H6 \def -(eq_ind A a2 (\lambda (a: A).(arity g (CHead c (Bind Abst) w) t a)) H_y -(AHead x0 x1) H3) in (leq_ahead_false_2 g x1 x0 (arity_mono g (CHead c (Bind -Abst) w) t (AHead x0 x1) H6 x1 H5) P))))))) H2)))))))))))). -(* COMMENTS -Initial nodes: 283 -END *) - -theorem arity_appls_cast: - \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (vs: -TList).(\forall (a: A).((arity g c (THeads (Flat Appl) vs u) (asucc g a)) \to -((arity g c (THeads (Flat Appl) vs t) a) \to (arity g c (THeads (Flat Appl) -vs (THead (Flat Cast) u t)) a)))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (vs: -TList).(TList_ind (\lambda (t0: TList).(\forall (a: A).((arity g c (THeads -(Flat Appl) t0 u) (asucc g a)) \to ((arity g c (THeads (Flat Appl) t0 t) a) -\to (arity g c (THeads (Flat Appl) t0 (THead (Flat Cast) u t)) a))))) -(\lambda (a: A).(\lambda (H: (arity g c u (asucc g a))).(\lambda (H0: (arity -g c t a)).(arity_cast g c u a H t H0)))) (\lambda (t0: T).(\lambda (t1: -TList).(\lambda (H: ((\forall (a: A).((arity g c (THeads (Flat Appl) t1 u) -(asucc g a)) \to ((arity g c (THeads (Flat Appl) t1 t) a) \to (arity g c -(THeads (Flat Appl) t1 (THead (Flat Cast) u t)) a)))))).(\lambda (a: -A).(\lambda (H0: (arity g c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 u)) -(asucc g a))).(\lambda (H1: (arity g c (THead (Flat Appl) t0 (THeads (Flat -Appl) t1 t)) a)).(let H2 \def (arity_gen_appl g c t0 (THeads (Flat Appl) t1 -t) a H1) in (ex2_ind A (\lambda (a1: A).(arity g c t0 a1)) (\lambda (a1: -A).(arity g c (THeads (Flat Appl) t1 t) (AHead a1 a))) (arity g c (THead -(Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Flat Cast) u t))) a) (\lambda -(x: A).(\lambda (H3: (arity g c t0 x)).(\lambda (H4: (arity g c (THeads (Flat -Appl) t1 t) (AHead x a))).(let H5 \def (arity_gen_appl g c t0 (THeads (Flat -Appl) t1 u) (asucc g a) H0) in (ex2_ind A (\lambda (a1: A).(arity g c t0 a1)) -(\lambda (a1: A).(arity g c (THeads (Flat Appl) t1 u) (AHead a1 (asucc g -a)))) (arity g c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Flat -Cast) u t))) a) (\lambda (x0: A).(\lambda (H6: (arity g c t0 x0)).(\lambda -(H7: (arity g c (THeads (Flat Appl) t1 u) (AHead x0 (asucc g -a)))).(arity_appl g c t0 x H3 (THeads (Flat Appl) t1 (THead (Flat Cast) u t)) -a (H (AHead x a) (arity_repl g c (THeads (Flat Appl) t1 u) (AHead x (asucc g -a)) (arity_repl g c (THeads (Flat Appl) t1 u) (AHead x0 (asucc g a)) H7 -(AHead x (asucc g a)) (leq_head g x0 x (arity_mono g c t0 x0 H6 x H3) (asucc -g a) (asucc g a) (leq_refl g (asucc g a)))) (asucc g (AHead x a)) (leq_refl g -(asucc g (AHead x a)))) H4))))) H5))))) H2)))))))) vs))))). -(* COMMENTS -Initial nodes: 707 -END *) - -theorem arity_appls_abbr: - \forall (g: G).(\forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: -nat).((getl i c (CHead d (Bind Abbr) v)) \to (\forall (vs: TList).(\forall -(a: A).((arity g c (THeads (Flat Appl) vs (lift (S i) O v)) a) \to (arity g c -(THeads (Flat Appl) vs (TLRef i)) a))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: -nat).(\lambda (H: (getl i c (CHead d (Bind Abbr) v))).(\lambda (vs: -TList).(TList_ind (\lambda (t: TList).(\forall (a: A).((arity g c (THeads -(Flat Appl) t (lift (S i) O v)) a) \to (arity g c (THeads (Flat Appl) t -(TLRef i)) a)))) (\lambda (a: A).(\lambda (H0: (arity g c (lift (S i) O v) -a)).(arity_abbr g c d v i H a (arity_gen_lift g c v a (S i) O H0 d (getl_drop -Abbr c d v i H))))) (\lambda (t: T).(\lambda (t0: TList).(\lambda (H0: -((\forall (a: A).((arity g c (THeads (Flat Appl) t0 (lift (S i) O v)) a) \to -(arity g c (THeads (Flat Appl) t0 (TLRef i)) a))))).(\lambda (a: A).(\lambda -(H1: (arity g c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O -v))) a)).(let H2 \def (arity_gen_appl g c t (THeads (Flat Appl) t0 (lift (S -i) O v)) a H1) in (ex2_ind A (\lambda (a1: A).(arity g c t a1)) (\lambda (a1: -A).(arity g c (THeads (Flat Appl) t0 (lift (S i) O v)) (AHead a1 a))) (arity -g c (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) a) (\lambda (x: -A).(\lambda (H3: (arity g c t x)).(\lambda (H4: (arity g c (THeads (Flat -Appl) t0 (lift (S i) O v)) (AHead x a))).(arity_appl g c t x H3 (THeads (Flat -Appl) t0 (TLRef i)) a (H0 (AHead x a) H4))))) H2))))))) vs))))))). -(* COMMENTS -Initial nodes: 425 -END *) - -theorem arity_appls_bind: - \forall (g: G).(\forall (b: B).((not (eq B b Abst)) \to (\forall (c: -C).(\forall (v: T).(\forall (a1: A).((arity g c v a1) \to (\forall (t: -T).(\forall (vs: TList).(\forall (a2: A).((arity g (CHead c (Bind b) v) -(THeads (Flat Appl) (lifts (S O) O vs) t) a2) \to (arity g c (THeads (Flat -Appl) vs (THead (Bind b) v t)) a2))))))))))) -\def - \lambda (g: G).(\lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda -(c: C).(\lambda (v: T).(\lambda (a1: A).(\lambda (H0: (arity g c v -a1)).(\lambda (t: T).(\lambda (vs: TList).(TList_ind (\lambda (t0: -TList).(\forall (a2: A).((arity g (CHead c (Bind b) v) (THeads (Flat Appl) -(lifts (S O) O t0) t) a2) \to (arity g c (THeads (Flat Appl) t0 (THead (Bind -b) v t)) a2)))) (\lambda (a2: A).(\lambda (H1: (arity g (CHead c (Bind b) v) -t a2)).(arity_bind g b H c v a1 H0 t a2 H1))) (\lambda (t0: T).(\lambda (t1: -TList).(\lambda (H1: ((\forall (a2: A).((arity g (CHead c (Bind b) v) (THeads -(Flat Appl) (lifts (S O) O t1) t) a2) \to (arity g c (THeads (Flat Appl) t1 -(THead (Bind b) v t)) a2))))).(\lambda (a2: A).(\lambda (H2: (arity g (CHead -c (Bind b) v) (THead (Flat Appl) (lift (S O) O t0) (THeads (Flat Appl) (lifts -(S O) O t1) t)) a2)).(let H3 \def (arity_gen_appl g (CHead c (Bind b) v) -(lift (S O) O t0) (THeads (Flat Appl) (lifts (S O) O t1) t) a2 H2) in -(ex2_ind A (\lambda (a3: A).(arity g (CHead c (Bind b) v) (lift (S O) O t0) -a3)) (\lambda (a3: A).(arity g (CHead c (Bind b) v) (THeads (Flat Appl) -(lifts (S O) O t1) t) (AHead a3 a2))) (arity g c (THead (Flat Appl) t0 -(THeads (Flat Appl) t1 (THead (Bind b) v t))) a2) (\lambda (x: A).(\lambda -(H4: (arity g (CHead c (Bind b) v) (lift (S O) O t0) x)).(\lambda (H5: (arity -g (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O t1) t) (AHead x -a2))).(arity_appl g c t0 x (arity_gen_lift g (CHead c (Bind b) v) t0 x (S O) -O H4 c (drop_drop (Bind b) O c c (drop_refl c) v)) (THeads (Flat Appl) t1 -(THead (Bind b) v t)) a2 (H1 (AHead x a2) H5))))) H3))))))) vs))))))))). -(* COMMENTS -Initial nodes: 567 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/arity/subst0.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/arity/subst0.ma deleted file mode 100644 index 16046993b..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/arity/subst0.ma +++ /dev/null @@ -1,1137 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/arity/props.ma". - -include "Basic-1/fsubst0/fwd.ma". - -include "Basic-1/csubst0/getl.ma". - -include "Basic-1/subst0/dec.ma". - -include "Basic-1/subst0/fwd.ma". - -include "Basic-1/getl/getl.ma". - -theorem arity_gen_cvoid_subst0: - \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t -a) \to (\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d -(Bind Void) u)) \to (\forall (w: T).(\forall (v: T).((subst0 i w t v) \to -(\forall (P: Prop).P)))))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: -(arity g c t a)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (_: -A).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c0 (CHead d -(Bind Void) u)) \to (\forall (w: T).(\forall (v: T).((subst0 i w t0 v) \to -(\forall (P: Prop).P))))))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda -(d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d -(Bind Void) u))).(\lambda (w: T).(\lambda (v: T).(\lambda (H1: (subst0 i w -(TSort n) v)).(\lambda (P: Prop).(subst0_gen_sort w v i n H1 P))))))))))) -(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H0: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (a0: A).(\lambda (_: -(arity g d u a0)).(\lambda (_: ((\forall (d0: C).(\forall (u0: T).(\forall -(i0: nat).((getl i0 d (CHead d0 (Bind Void) u0)) \to (\forall (w: T).(\forall -(v: T).((subst0 i0 w u v) \to (\forall (P: Prop).P)))))))))).(\lambda (d0: -C).(\lambda (u0: T).(\lambda (i0: nat).(\lambda (H3: (getl i0 c0 (CHead d0 -(Bind Void) u0))).(\lambda (w: T).(\lambda (v: T).(\lambda (H4: (subst0 i0 w -(TLRef i) v)).(\lambda (P: Prop).(land_ind (eq nat i i0) (eq T v (lift (S i) -O w)) P (\lambda (H5: (eq nat i i0)).(\lambda (_: (eq T v (lift (S i) O -w))).(let H7 \def (eq_ind_r nat i0 (\lambda (n: nat).(getl n c0 (CHead d0 -(Bind Void) u0))) H3 i H5) in (let H8 \def (eq_ind C (CHead d (Bind Abbr) u) -(\lambda (c1: C).(getl i c0 c1)) H0 (CHead d0 (Bind Void) u0) (getl_mono c0 -(CHead d (Bind Abbr) u) i H0 (CHead d0 (Bind Void) u0) H7)) in (let H9 \def -(eq_ind C (CHead d (Bind Abbr) u) (\lambda (ee: C).(match ee in C return -(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b) -\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow -True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _) -\Rightarrow False])])) I (CHead d0 (Bind Void) u0) (getl_mono c0 (CHead d -(Bind Abbr) u) i H0 (CHead d0 (Bind Void) u0) H7)) in (False_ind P H9)))))) -(subst0_gen_lref w v i0 i H4)))))))))))))))))) (\lambda (c0: C).(\lambda (d: -C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind -Abst) u))).(\lambda (a0: A).(\lambda (_: (arity g d u (asucc g a0))).(\lambda -(_: ((\forall (d0: C).(\forall (u0: T).(\forall (i0: nat).((getl i0 d (CHead -d0 (Bind Void) u0)) \to (\forall (w: T).(\forall (v: T).((subst0 i0 w u v) -\to (\forall (P: Prop).P)))))))))).(\lambda (d0: C).(\lambda (u0: T).(\lambda -(i0: nat).(\lambda (H3: (getl i0 c0 (CHead d0 (Bind Void) u0))).(\lambda (w: -T).(\lambda (v: T).(\lambda (H4: (subst0 i0 w (TLRef i) v)).(\lambda (P: -Prop).(land_ind (eq nat i i0) (eq T v (lift (S i) O w)) P (\lambda (H5: (eq -nat i i0)).(\lambda (_: (eq T v (lift (S i) O w))).(let H7 \def (eq_ind_r nat -i0 (\lambda (n: nat).(getl n c0 (CHead d0 (Bind Void) u0))) H3 i H5) in (let -H8 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (c1: C).(getl i c0 c1)) H0 -(CHead d0 (Bind Void) u0) (getl_mono c0 (CHead d (Bind Abst) u) i H0 (CHead -d0 (Bind Void) u0) H7)) in (let H9 \def (eq_ind C (CHead d (Bind Abst) u) -(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind b) \Rightarrow (match b in B return (\lambda (_: -B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow True | Void -\Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead d0 (Bind Void) -u0) (getl_mono c0 (CHead d (Bind Abst) u) i H0 (CHead d0 (Bind Void) u0) H7)) -in (False_ind P H9)))))) (subst0_gen_lref w v i0 i H4)))))))))))))))))) -(\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (c0: C).(\lambda -(u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u a1)).(\lambda (H2: -((\forall (d: C).(\forall (u0: T).(\forall (i: nat).((getl i c0 (CHead d -(Bind Void) u0)) \to (\forall (w: T).(\forall (v: T).((subst0 i w u v) \to -(\forall (P: Prop).P)))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: -(arity g (CHead c0 (Bind b) u) t0 a2)).(\lambda (H4: ((\forall (d: -C).(\forall (u0: T).(\forall (i: nat).((getl i (CHead c0 (Bind b) u) (CHead d -(Bind Void) u0)) \to (\forall (w: T).(\forall (v: T).((subst0 i w t0 v) \to -(\forall (P: Prop).P)))))))))).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: -nat).(\lambda (H5: (getl i c0 (CHead d (Bind Void) u0))).(\lambda (w: -T).(\lambda (v: T).(\lambda (H6: (subst0 i w (THead (Bind b) u t0) -v)).(\lambda (P: Prop).(or3_ind (ex2 T (\lambda (u2: T).(eq T v (THead (Bind -b) u2 t0))) (\lambda (u2: T).(subst0 i w u u2))) (ex2 T (\lambda (t2: T).(eq -T v (THead (Bind b) u t2))) (\lambda (t2: T).(subst0 (s (Bind b) i) w t0 -t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Bind b) u2 -t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: -T).(\lambda (t2: T).(subst0 (s (Bind b) i) w t0 t2)))) P (\lambda (H7: (ex2 T -(\lambda (u2: T).(eq T v (THead (Bind b) u2 t0))) (\lambda (u2: T).(subst0 i -w u u2)))).(ex2_ind T (\lambda (u2: T).(eq T v (THead (Bind b) u2 t0))) -(\lambda (u2: T).(subst0 i w u u2)) P (\lambda (x: T).(\lambda (_: (eq T v -(THead (Bind b) x t0))).(\lambda (H9: (subst0 i w u x)).(H2 d u0 i H5 w x H9 -P)))) H7)) (\lambda (H7: (ex2 T (\lambda (t2: T).(eq T v (THead (Bind b) u -t2))) (\lambda (t2: T).(subst0 (s (Bind b) i) w t0 t2)))).(ex2_ind T (\lambda -(t2: T).(eq T v (THead (Bind b) u t2))) (\lambda (t2: T).(subst0 (s (Bind b) -i) w t0 t2)) P (\lambda (x: T).(\lambda (_: (eq T v (THead (Bind b) u -x))).(\lambda (H9: (subst0 (s (Bind b) i) w t0 x)).(H4 d u0 (S i) -(getl_clear_bind b (CHead c0 (Bind b) u) c0 u (clear_bind b c0 u) (CHead d -(Bind Void) u0) i H5) w x H9 P)))) H7)) (\lambda (H7: (ex3_2 T T (\lambda -(u2: T).(\lambda (t2: T).(eq T v (THead (Bind b) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda (t2: -T).(subst0 (s (Bind b) i) w t0 t2))))).(ex3_2_ind T T (\lambda (u2: -T).(\lambda (t2: T).(eq T v (THead (Bind b) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda (t2: -T).(subst0 (s (Bind b) i) w t0 t2))) P (\lambda (x0: T).(\lambda (x1: -T).(\lambda (_: (eq T v (THead (Bind b) x0 x1))).(\lambda (H9: (subst0 i w u -x0)).(\lambda (_: (subst0 (s (Bind b) i) w t0 x1)).(H2 d u0 i H5 w x0 H9 -P)))))) H7)) (subst0_gen_head (Bind b) w u t0 v i H6))))))))))))))))))))) -(\lambda (c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u -(asucc g a1))).(\lambda (H1: ((\forall (d: C).(\forall (u0: T).(\forall (i: -nat).((getl i c0 (CHead d (Bind Void) u0)) \to (\forall (w: T).(\forall (v: -T).((subst0 i w u v) \to (\forall (P: Prop).P)))))))))).(\lambda (t0: -T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u) t0 -a2)).(\lambda (H3: ((\forall (d: C).(\forall (u0: T).(\forall (i: nat).((getl -i (CHead c0 (Bind Abst) u) (CHead d (Bind Void) u0)) \to (\forall (w: -T).(\forall (v: T).((subst0 i w t0 v) \to (\forall (P: -Prop).P)))))))))).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda -(H4: (getl i c0 (CHead d (Bind Void) u0))).(\lambda (w: T).(\lambda (v: -T).(\lambda (H5: (subst0 i w (THead (Bind Abst) u t0) v)).(\lambda (P: -Prop).(or3_ind (ex2 T (\lambda (u2: T).(eq T v (THead (Bind Abst) u2 t0))) -(\lambda (u2: T).(subst0 i w u u2))) (ex2 T (\lambda (t2: T).(eq T v (THead -(Bind Abst) u t2))) (\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0 t2))) -(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Bind Abst) u2 -t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: -T).(\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0 t2)))) P (\lambda (H6: -(ex2 T (\lambda (u2: T).(eq T v (THead (Bind Abst) u2 t0))) (\lambda (u2: -T).(subst0 i w u u2)))).(ex2_ind T (\lambda (u2: T).(eq T v (THead (Bind -Abst) u2 t0))) (\lambda (u2: T).(subst0 i w u u2)) P (\lambda (x: T).(\lambda -(_: (eq T v (THead (Bind Abst) x t0))).(\lambda (H8: (subst0 i w u x)).(H1 d -u0 i H4 w x H8 P)))) H6)) (\lambda (H6: (ex2 T (\lambda (t2: T).(eq T v -(THead (Bind Abst) u t2))) (\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0 -t2)))).(ex2_ind T (\lambda (t2: T).(eq T v (THead (Bind Abst) u t2))) -(\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0 t2)) P (\lambda (x: -T).(\lambda (_: (eq T v (THead (Bind Abst) u x))).(\lambda (H8: (subst0 (s -(Bind Abst) i) w t0 x)).(H3 d u0 (S i) (getl_clear_bind Abst (CHead c0 (Bind -Abst) u) c0 u (clear_bind Abst c0 u) (CHead d (Bind Void) u0) i H4) w x H8 -P)))) H6)) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v -(THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w u -u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0 -t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Bind -Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda -(_: T).(\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0 t2))) P (\lambda (x0: -T).(\lambda (x1: T).(\lambda (_: (eq T v (THead (Bind Abst) x0 x1))).(\lambda -(H8: (subst0 i w u x0)).(\lambda (_: (subst0 (s (Bind Abst) i) w t0 x1)).(H1 -d u0 i H4 w x0 H8 P)))))) H6)) (subst0_gen_head (Bind Abst) w u t0 v i -H5))))))))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a1: -A).(\lambda (_: (arity g c0 u a1)).(\lambda (H1: ((\forall (d: C).(\forall -(u0: T).(\forall (i: nat).((getl i c0 (CHead d (Bind Void) u0)) \to (\forall -(w: T).(\forall (v: T).((subst0 i w u v) \to (\forall (P: -Prop).P)))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g c0 -t0 (AHead a1 a2))).(\lambda (H3: ((\forall (d: C).(\forall (u0: T).(\forall -(i: nat).((getl i c0 (CHead d (Bind Void) u0)) \to (\forall (w: T).(\forall -(v: T).((subst0 i w t0 v) \to (\forall (P: Prop).P)))))))))).(\lambda (d: -C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H4: (getl i c0 (CHead d (Bind -Void) u0))).(\lambda (w: T).(\lambda (v: T).(\lambda (H5: (subst0 i w (THead -(Flat Appl) u t0) v)).(\lambda (P: Prop).(or3_ind (ex2 T (\lambda (u2: T).(eq -T v (THead (Flat Appl) u2 t0))) (\lambda (u2: T).(subst0 i w u u2))) (ex2 T -(\lambda (t2: T).(eq T v (THead (Flat Appl) u t2))) (\lambda (t2: T).(subst0 -(s (Flat Appl) i) w t0 t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq -T v (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w -u u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s (Flat Appl) i) w t0 -t2)))) P (\lambda (H6: (ex2 T (\lambda (u2: T).(eq T v (THead (Flat Appl) u2 -t0))) (\lambda (u2: T).(subst0 i w u u2)))).(ex2_ind T (\lambda (u2: T).(eq T -v (THead (Flat Appl) u2 t0))) (\lambda (u2: T).(subst0 i w u u2)) P (\lambda -(x: T).(\lambda (_: (eq T v (THead (Flat Appl) x t0))).(\lambda (H8: (subst0 -i w u x)).(H1 d u0 i H4 w x H8 P)))) H6)) (\lambda (H6: (ex2 T (\lambda (t2: -T).(eq T v (THead (Flat Appl) u t2))) (\lambda (t2: T).(subst0 (s (Flat Appl) -i) w t0 t2)))).(ex2_ind T (\lambda (t2: T).(eq T v (THead (Flat Appl) u t2))) -(\lambda (t2: T).(subst0 (s (Flat Appl) i) w t0 t2)) P (\lambda (x: -T).(\lambda (_: (eq T v (THead (Flat Appl) u x))).(\lambda (H8: (subst0 (s -(Flat Appl) i) w t0 x)).(H3 d u0 i H4 w x H8 P)))) H6)) (\lambda (H6: (ex3_2 -T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Flat Appl) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda -(t2: T).(subst0 (s (Flat Appl) i) w t0 t2))))).(ex3_2_ind T T (\lambda (u2: -T).(\lambda (t2: T).(eq T v (THead (Flat Appl) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda (t2: -T).(subst0 (s (Flat Appl) i) w t0 t2))) P (\lambda (x0: T).(\lambda (x1: -T).(\lambda (_: (eq T v (THead (Flat Appl) x0 x1))).(\lambda (H8: (subst0 i w -u x0)).(\lambda (_: (subst0 (s (Flat Appl) i) w t0 x1)).(H1 d u0 i H4 w x0 H8 -P)))))) H6)) (subst0_gen_head (Flat Appl) w u t0 v i H5))))))))))))))))))) -(\lambda (c0: C).(\lambda (u: T).(\lambda (a0: A).(\lambda (_: (arity g c0 u -(asucc g a0))).(\lambda (H1: ((\forall (d: C).(\forall (u0: T).(\forall (i: -nat).((getl i c0 (CHead d (Bind Void) u0)) \to (\forall (w: T).(\forall (v: -T).((subst0 i w u v) \to (\forall (P: Prop).P)))))))))).(\lambda (t0: -T).(\lambda (_: (arity g c0 t0 a0)).(\lambda (H3: ((\forall (d: C).(\forall -(u0: T).(\forall (i: nat).((getl i c0 (CHead d (Bind Void) u0)) \to (\forall -(w: T).(\forall (v: T).((subst0 i w t0 v) \to (\forall (P: -Prop).P)))))))))).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda -(H4: (getl i c0 (CHead d (Bind Void) u0))).(\lambda (w: T).(\lambda (v: -T).(\lambda (H5: (subst0 i w (THead (Flat Cast) u t0) v)).(\lambda (P: -Prop).(or3_ind (ex2 T (\lambda (u2: T).(eq T v (THead (Flat Cast) u2 t0))) -(\lambda (u2: T).(subst0 i w u u2))) (ex2 T (\lambda (t2: T).(eq T v (THead -(Flat Cast) u t2))) (\lambda (t2: T).(subst0 (s (Flat Cast) i) w t0 t2))) -(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Flat Cast) u2 -t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: -T).(\lambda (t2: T).(subst0 (s (Flat Cast) i) w t0 t2)))) P (\lambda (H6: -(ex2 T (\lambda (u2: T).(eq T v (THead (Flat Cast) u2 t0))) (\lambda (u2: -T).(subst0 i w u u2)))).(ex2_ind T (\lambda (u2: T).(eq T v (THead (Flat -Cast) u2 t0))) (\lambda (u2: T).(subst0 i w u u2)) P (\lambda (x: T).(\lambda -(_: (eq T v (THead (Flat Cast) x t0))).(\lambda (H8: (subst0 i w u x)).(H1 d -u0 i H4 w x H8 P)))) H6)) (\lambda (H6: (ex2 T (\lambda (t2: T).(eq T v -(THead (Flat Cast) u t2))) (\lambda (t2: T).(subst0 (s (Flat Cast) i) w t0 -t2)))).(ex2_ind T (\lambda (t2: T).(eq T v (THead (Flat Cast) u t2))) -(\lambda (t2: T).(subst0 (s (Flat Cast) i) w t0 t2)) P (\lambda (x: -T).(\lambda (_: (eq T v (THead (Flat Cast) u x))).(\lambda (H8: (subst0 (s -(Flat Cast) i) w t0 x)).(H3 d u0 i H4 w x H8 P)))) H6)) (\lambda (H6: (ex3_2 -T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Flat Cast) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda -(t2: T).(subst0 (s (Flat Cast) i) w t0 t2))))).(ex3_2_ind T T (\lambda (u2: -T).(\lambda (t2: T).(eq T v (THead (Flat Cast) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda (t2: -T).(subst0 (s (Flat Cast) i) w t0 t2))) P (\lambda (x0: T).(\lambda (x1: -T).(\lambda (_: (eq T v (THead (Flat Cast) x0 x1))).(\lambda (H8: (subst0 i w -u x0)).(\lambda (_: (subst0 (s (Flat Cast) i) w t0 x1)).(H1 d u0 i H4 w x0 H8 -P)))))) H6)) (subst0_gen_head (Flat Cast) w u t0 v i H5)))))))))))))))))) -(\lambda (c0: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda (_: (arity g c0 -t0 a1)).(\lambda (H1: ((\forall (d: C).(\forall (u: T).(\forall (i: -nat).((getl i c0 (CHead d (Bind Void) u)) \to (\forall (w: T).(\forall (v: -T).((subst0 i w t0 v) \to (\forall (P: Prop).P)))))))))).(\lambda (a2: -A).(\lambda (_: (leq g a1 a2)).(\lambda (d: C).(\lambda (u: T).(\lambda (i: -nat).(\lambda (H3: (getl i c0 (CHead d (Bind Void) u))).(\lambda (w: -T).(\lambda (v: T).(\lambda (H4: (subst0 i w t0 v)).(\lambda (P: Prop).(H1 d -u i H3 w v H4 P)))))))))))))))) c t a H))))). -(* COMMENTS -Initial nodes: 4131 -END *) - -theorem arity_gen_cvoid: - \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t -a) \to (\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d -(Bind Void) u)) \to (ex T (\lambda (v: T).(eq T t (lift (S O) i v)))))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: -(arity g c t a)).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H0: (getl i c (CHead d (Bind Void) u))).(let H_x \def (dnf_dec u t i) in -(let H1 \def H_x in (ex_ind T (\lambda (v: T).(or (subst0 i u t (lift (S O) i -v)) (eq T t (lift (S O) i v)))) (ex T (\lambda (v: T).(eq T t (lift (S O) i -v)))) (\lambda (x: T).(\lambda (H2: (or (subst0 i u t (lift (S O) i x)) (eq T -t (lift (S O) i x)))).(or_ind (subst0 i u t (lift (S O) i x)) (eq T t (lift -(S O) i x)) (ex T (\lambda (v: T).(eq T t (lift (S O) i v)))) (\lambda (H3: -(subst0 i u t (lift (S O) i x))).(arity_gen_cvoid_subst0 g c t a H d u i H0 u -(lift (S O) i x) H3 (ex T (\lambda (v: T).(eq T t (lift (S O) i v)))))) -(\lambda (H3: (eq T t (lift (S O) i x))).(let H4 \def (eq_ind T t (\lambda -(t0: T).(arity g c t0 a)) H (lift (S O) i x) H3) in (eq_ind_r T (lift (S O) i -x) (\lambda (t0: T).(ex T (\lambda (v: T).(eq T t0 (lift (S O) i v))))) -(ex_intro T (\lambda (v: T).(eq T (lift (S O) i x) (lift (S O) i v))) x -(refl_equal T (lift (S O) i x))) t H3))) H2))) H1))))))))))). -(* COMMENTS -Initial nodes: 423 -END *) - -theorem arity_fsubst0: - \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (a: A).((arity g -c1 t1 a) \to (\forall (d1: C).(\forall (u: T).(\forall (i: nat).((getl i c1 -(CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u -c1 t1 c2 t2) \to (arity g c2 t2 a)))))))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (a: A).(\lambda -(H: (arity g c1 t1 a)).(arity_ind g (\lambda (c: C).(\lambda (t: T).(\lambda -(a0: A).(\forall (d1: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead -d1 (Bind Abbr) u)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u c t c2 -t2) \to (arity g c2 t2 a0))))))))))) (\lambda (c: C).(\lambda (n: -nat).(\lambda (d1: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (_: (getl i -c (CHead d1 (Bind Abbr) u))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H1: -(fsubst0 i u c (TSort n) c2 t2)).(let H_x \def (fsubst0_gen_base c c2 (TSort -n) t2 u i H1) in (let H2 \def H_x in (or3_ind (land (eq C c c2) (subst0 i u -(TSort n) t2)) (land (eq T (TSort n) t2) (csubst0 i u c c2)) (land (subst0 i -u (TSort n) t2) (csubst0 i u c c2)) (arity g c2 t2 (ASort O n)) (\lambda (H3: -(land (eq C c c2) (subst0 i u (TSort n) t2))).(land_ind (eq C c c2) (subst0 i -u (TSort n) t2) (arity g c2 t2 (ASort O n)) (\lambda (H4: (eq C c -c2)).(\lambda (H5: (subst0 i u (TSort n) t2)).(eq_ind C c (\lambda (c0: -C).(arity g c0 t2 (ASort O n))) (subst0_gen_sort u t2 i n H5 (arity g c t2 -(ASort O n))) c2 H4))) H3)) (\lambda (H3: (land (eq T (TSort n) t2) (csubst0 -i u c c2))).(land_ind (eq T (TSort n) t2) (csubst0 i u c c2) (arity g c2 t2 -(ASort O n)) (\lambda (H4: (eq T (TSort n) t2)).(\lambda (_: (csubst0 i u c -c2)).(eq_ind T (TSort n) (\lambda (t: T).(arity g c2 t (ASort O n))) -(arity_sort g c2 n) t2 H4))) H3)) (\lambda (H3: (land (subst0 i u (TSort n) -t2) (csubst0 i u c c2))).(land_ind (subst0 i u (TSort n) t2) (csubst0 i u c -c2) (arity g c2 t2 (ASort O n)) (\lambda (H4: (subst0 i u (TSort n) -t2)).(\lambda (_: (csubst0 i u c c2)).(subst0_gen_sort u t2 i n H4 (arity g -c2 t2 (ASort O n))))) H3)) H2)))))))))))) (\lambda (c: C).(\lambda (d: -C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d (Bind -Abbr) u))).(\lambda (a0: A).(\lambda (H1: (arity g d u a0)).(\lambda (H2: -((\forall (d1: C).(\forall (u0: T).(\forall (i0: nat).((getl i0 d (CHead d1 -(Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i0 u0 d u c2 -t2) \to (arity g c2 t2 a0)))))))))).(\lambda (d1: C).(\lambda (u0: -T).(\lambda (i0: nat).(\lambda (H3: (getl i0 c (CHead d1 (Bind Abbr) -u0))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H4: (fsubst0 i0 u0 c (TLRef -i) c2 t2)).(let H_x \def (fsubst0_gen_base c c2 (TLRef i) t2 u0 i0 H4) in -(let H5 \def H_x in (or3_ind (land (eq C c c2) (subst0 i0 u0 (TLRef i) t2)) -(land (eq T (TLRef i) t2) (csubst0 i0 u0 c c2)) (land (subst0 i0 u0 (TLRef i) -t2) (csubst0 i0 u0 c c2)) (arity g c2 t2 a0) (\lambda (H6: (land (eq C c c2) -(subst0 i0 u0 (TLRef i) t2))).(land_ind (eq C c c2) (subst0 i0 u0 (TLRef i) -t2) (arity g c2 t2 a0) (\lambda (H7: (eq C c c2)).(\lambda (H8: (subst0 i0 u0 -(TLRef i) t2)).(eq_ind C c (\lambda (c0: C).(arity g c0 t2 a0)) (land_ind (eq -nat i i0) (eq T t2 (lift (S i) O u0)) (arity g c t2 a0) (\lambda (H9: (eq nat -i i0)).(\lambda (H10: (eq T t2 (lift (S i) O u0))).(eq_ind_r T (lift (S i) O -u0) (\lambda (t: T).(arity g c t a0)) (let H11 \def (eq_ind_r nat i0 (\lambda -(n: nat).(getl n c (CHead d1 (Bind Abbr) u0))) H3 i H9) in (let H12 \def -(eq_ind C (CHead d (Bind Abbr) u) (\lambda (c0: C).(getl i c c0)) H0 (CHead -d1 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) i H0 (CHead d1 (Bind -Abbr) u0) H11)) in (let H13 \def (f_equal C C (\lambda (e: C).(match e in C -return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) -\Rightarrow c0])) (CHead d (Bind Abbr) u) (CHead d1 (Bind Abbr) u0) -(getl_mono c (CHead d (Bind Abbr) u) i H0 (CHead d1 (Bind Abbr) u0) H11)) in -((let H14 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: -C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d -(Bind Abbr) u) (CHead d1 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) -i H0 (CHead d1 (Bind Abbr) u0) H11)) in (\lambda (H15: (eq C d d1)).(let H16 -\def (eq_ind_r T u0 (\lambda (t: T).(getl i c (CHead d1 (Bind Abbr) t))) H12 -u H14) in (eq_ind T u (\lambda (t: T).(arity g c (lift (S i) O t) a0)) (let -H17 \def (eq_ind_r C d1 (\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr) u))) -H16 d H15) in (arity_lift g d u a0 H1 c (S i) O (getl_drop Abbr c d u i -H17))) u0 H14)))) H13)))) t2 H10))) (subst0_gen_lref u0 t2 i0 i H8)) c2 H7))) -H6)) (\lambda (H6: (land (eq T (TLRef i) t2) (csubst0 i0 u0 c c2))).(land_ind -(eq T (TLRef i) t2) (csubst0 i0 u0 c c2) (arity g c2 t2 a0) (\lambda (H7: (eq -T (TLRef i) t2)).(\lambda (H8: (csubst0 i0 u0 c c2)).(eq_ind T (TLRef i) -(\lambda (t: T).(arity g c2 t a0)) (lt_le_e i i0 (arity g c2 (TLRef i) a0) -(\lambda (H9: (lt i i0)).(let H10 \def (csubst0_getl_lt i0 i H9 c c2 u0 H8 -(CHead d (Bind Abbr) u) H0) in (or4_ind (getl i c2 (CHead d (Bind Abbr) u)) -(ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: -B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl i c2 (CHead e0 -(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda -(w: T).(subst0 (minus i0 (S i)) u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind -Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u1: T).(getl i c2 (CHead e2 (Bind b) u1)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S -i)) u0 e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead -e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(getl i c2 (CHead e2 (Bind b) w))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: -T).(subst0 (minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) -u0 e1 e2))))))) (arity g c2 (TLRef i) a0) (\lambda (H11: (getl i c2 (CHead d -(Bind Abbr) u))).(let H12 \def (eq_ind nat (minus i0 i) (\lambda (n: -nat).(getl n (CHead d (Bind Abbr) u) (CHead d1 (Bind Abbr) u0))) -(getl_conf_le i0 (CHead d1 (Bind Abbr) u0) c H3 (CHead d (Bind Abbr) u) i H0 -(le_S_n i i0 (le_S (S i) i0 H9))) (S (minus i0 (S i))) (minus_x_Sy i0 i H9)) -in (arity_abbr g c2 d u i H11 a0 H1))) (\lambda (H11: (ex3_4 B C T T (\lambda -(b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind -Abbr) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(getl i c2 (CHead e0 (Bind b) w)))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 -(minus i0 (S i)) u0 u1 w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda -(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead -e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(getl i c2 (CHead e0 (Bind b) w)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) -u0 u1 w))))) (arity g c2 (TLRef i) a0) (\lambda (x0: B).(\lambda (x1: -C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (eq C (CHead d (Bind -Abbr) u) (CHead x1 (Bind x0) x2))).(\lambda (H13: (getl i c2 (CHead x1 (Bind -x0) x3))).(\lambda (H14: (subst0 (minus i0 (S i)) u0 x2 x3)).(let H15 \def -(eq_ind nat (minus i0 i) (\lambda (n: nat).(getl n (CHead d (Bind Abbr) u) -(CHead d1 (Bind Abbr) u0))) (getl_conf_le i0 (CHead d1 (Bind Abbr) u0) c H3 -(CHead d (Bind Abbr) u) i H0 (le_S_n i i0 (le_S (S i) i0 H9))) (S (minus i0 -(S i))) (minus_x_Sy i0 i H9)) in (let H16 \def (f_equal C C (\lambda (e: -C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | -(CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) -x2) H12) in ((let H17 \def (f_equal C B (\lambda (e: C).(match e in C return -(\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) -\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead -x1 (Bind x0) x2) H12) in ((let H18 \def (f_equal C T (\lambda (e: C).(match e -in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) -\Rightarrow t])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H12) in -(\lambda (H19: (eq B Abbr x0)).(\lambda (H20: (eq C d x1)).(let H21 \def -(eq_ind_r T x2 (\lambda (t: T).(subst0 (minus i0 (S i)) u0 t x3)) H14 u H18) -in (let H22 \def (eq_ind_r C x1 (\lambda (c0: C).(getl i c2 (CHead c0 (Bind -x0) x3))) H13 d H20) in (let H23 \def (eq_ind_r B x0 (\lambda (b: B).(getl i -c2 (CHead d (Bind b) x3))) H22 Abbr H19) in (arity_abbr g c2 d x3 i H23 a0 -(H2 d1 u0 (r (Bind Abbr) (minus i0 (S i))) (getl_gen_S (Bind Abbr) d (CHead -d1 (Bind Abbr) u0) u (minus i0 (S i)) H15) d x3 (fsubst0_snd (r (Bind Abbr) -(minus i0 (S i))) u0 d u x3 H21))))))))) H17)) H16)))))))))) H11)) (\lambda -(H11: (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c2 -(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))).(ex3_4_ind B C C -T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C -(CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda -(_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c2 (CHead e2 (Bind b) -u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus i0 (S i)) u0 e1 e2))))) (arity g c2 (TLRef i) a0) (\lambda -(x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H12: (eq -C (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3))).(\lambda (H13: (getl i c2 -(CHead x2 (Bind x0) x3))).(\lambda (H14: (csubst0 (minus i0 (S i)) u0 x1 -x2)).(let H15 \def (eq_ind nat (minus i0 i) (\lambda (n: nat).(getl n (CHead -d (Bind Abbr) u) (CHead d1 (Bind Abbr) u0))) (getl_conf_le i0 (CHead d1 (Bind -Abbr) u0) c H3 (CHead d (Bind Abbr) u) i H0 (le_S_n i i0 (le_S (S i) i0 H9))) -(S (minus i0 (S i))) (minus_x_Sy i0 i H9)) in (let H16 \def (f_equal C C -(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abbr) u) -(CHead x1 (Bind x0) x3) H12) in ((let H17 \def (f_equal C B (\lambda (e: -C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | -(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind -b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) -(CHead x1 (Bind x0) x3) H12) in ((let H18 \def (f_equal C T (\lambda (e: -C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | -(CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) -x3) H12) in (\lambda (H19: (eq B Abbr x0)).(\lambda (H20: (eq C d x1)).(let -H21 \def (eq_ind_r T x3 (\lambda (t: T).(getl i c2 (CHead x2 (Bind x0) t))) -H13 u H18) in (let H22 \def (eq_ind_r C x1 (\lambda (c0: C).(csubst0 (minus -i0 (S i)) u0 c0 x2)) H14 d H20) in (let H23 \def (eq_ind_r B x0 (\lambda (b: -B).(getl i c2 (CHead x2 (Bind b) u))) H21 Abbr H19) in (arity_abbr g c2 x2 u -i H23 a0 (H2 d1 u0 (r (Bind Abbr) (minus i0 (S i))) (getl_gen_S (Bind Abbr) d -(CHead d1 (Bind Abbr) u0) u (minus i0 (S i)) H15) x2 u (fsubst0_fst (r (Bind -Abbr) (minus i0 (S i))) u0 d u x2 H22))))))))) H17)) H16)))))))))) H11)) -(\lambda (H11: (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 -(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(getl i c2 (CHead e2 (Bind b) w))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 -(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 -e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 -(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(getl i c2 (CHead e2 (Bind b) w))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 -(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) -(arity g c2 (TLRef i) a0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: -C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H12: (eq C (CHead d (Bind -Abbr) u) (CHead x1 (Bind x0) x3))).(\lambda (H13: (getl i c2 (CHead x2 (Bind -x0) x4))).(\lambda (H14: (subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H15: -(csubst0 (minus i0 (S i)) u0 x1 x2)).(let H16 \def (eq_ind nat (minus i0 i) -(\lambda (n: nat).(getl n (CHead d (Bind Abbr) u) (CHead d1 (Bind Abbr) u0))) -(getl_conf_le i0 (CHead d1 (Bind Abbr) u0) c H3 (CHead d (Bind Abbr) u) i H0 -(le_S_n i i0 (le_S (S i) i0 H9))) (S (minus i0 (S i))) (minus_x_Sy i0 i H9)) -in (let H17 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda -(_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) -(CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H12) in ((let H18 \def -(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with -[(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k in K return -(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow -Abbr])])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H12) in ((let H19 -\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind -Abbr) u) (CHead x1 (Bind x0) x3) H12) in (\lambda (H20: (eq B Abbr -x0)).(\lambda (H21: (eq C d x1)).(let H22 \def (eq_ind_r T x3 (\lambda (t: -T).(subst0 (minus i0 (S i)) u0 t x4)) H14 u H19) in (let H23 \def (eq_ind_r C -x1 (\lambda (c0: C).(csubst0 (minus i0 (S i)) u0 c0 x2)) H15 d H21) in (let -H24 \def (eq_ind_r B x0 (\lambda (b: B).(getl i c2 (CHead x2 (Bind b) x4))) -H13 Abbr H20) in (arity_abbr g c2 x2 x4 i H24 a0 (H2 d1 u0 (r (Bind Abbr) -(minus i0 (S i))) (getl_gen_S (Bind Abbr) d (CHead d1 (Bind Abbr) u0) u -(minus i0 (S i)) H16) x2 x4 (fsubst0_both (r (Bind Abbr) (minus i0 (S i))) u0 -d u x4 H22 x2 H23))))))))) H18)) H17)))))))))))) H11)) H10))) (\lambda (H9: -(le i0 i)).(arity_abbr g c2 d u i (csubst0_getl_ge i0 i H9 c c2 u0 H8 (CHead -d (Bind Abbr) u) H0) a0 H1))) t2 H7))) H6)) (\lambda (H6: (land (subst0 i0 u0 -(TLRef i) t2) (csubst0 i0 u0 c c2))).(land_ind (subst0 i0 u0 (TLRef i) t2) -(csubst0 i0 u0 c c2) (arity g c2 t2 a0) (\lambda (H7: (subst0 i0 u0 (TLRef i) -t2)).(\lambda (H8: (csubst0 i0 u0 c c2)).(land_ind (eq nat i i0) (eq T t2 -(lift (S i) O u0)) (arity g c2 t2 a0) (\lambda (H9: (eq nat i i0)).(\lambda -(H10: (eq T t2 (lift (S i) O u0))).(eq_ind_r T (lift (S i) O u0) (\lambda (t: -T).(arity g c2 t a0)) (let H11 \def (eq_ind_r nat i0 (\lambda (n: -nat).(csubst0 n u0 c c2)) H8 i H9) in (let H12 \def (eq_ind_r nat i0 (\lambda -(n: nat).(getl n c (CHead d1 (Bind Abbr) u0))) H3 i H9) in (let H13 \def -(eq_ind C (CHead d (Bind Abbr) u) (\lambda (c0: C).(getl i c c0)) H0 (CHead -d1 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) i H0 (CHead d1 (Bind -Abbr) u0) H12)) in (let H14 \def (f_equal C C (\lambda (e: C).(match e in C -return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) -\Rightarrow c0])) (CHead d (Bind Abbr) u) (CHead d1 (Bind Abbr) u0) -(getl_mono c (CHead d (Bind Abbr) u) i H0 (CHead d1 (Bind Abbr) u0) H12)) in -((let H15 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: -C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d -(Bind Abbr) u) (CHead d1 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) -i H0 (CHead d1 (Bind Abbr) u0) H12)) in (\lambda (H16: (eq C d d1)).(let H17 -\def (eq_ind_r T u0 (\lambda (t: T).(getl i c (CHead d1 (Bind Abbr) t))) H13 -u H15) in (let H18 \def (eq_ind_r T u0 (\lambda (t: T).(csubst0 i t c c2)) -H11 u H15) in (eq_ind T u (\lambda (t: T).(arity g c2 (lift (S i) O t) a0)) -(let H19 \def (eq_ind_r C d1 (\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr) -u))) H17 d H16) in (arity_lift g d u a0 H1 c2 (S i) O (getl_drop Abbr c2 d u -i (csubst0_getl_ge i i (le_n i) c c2 u H18 (CHead d (Bind Abbr) u) H19)))) u0 -H15))))) H14))))) t2 H10))) (subst0_gen_lref u0 t2 i0 i H7)))) H6)) -H5)))))))))))))))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda -(i: nat).(\lambda (H0: (getl i c (CHead d (Bind Abst) u))).(\lambda (a0: -A).(\lambda (H1: (arity g d u (asucc g a0))).(\lambda (H2: ((\forall (d1: -C).(\forall (u0: T).(\forall (i0: nat).((getl i0 d (CHead d1 (Bind Abbr) u0)) -\to (\forall (c2: C).(\forall (t2: T).((fsubst0 i0 u0 d u c2 t2) \to (arity g -c2 t2 (asucc g a0))))))))))).(\lambda (d1: C).(\lambda (u0: T).(\lambda (i0: -nat).(\lambda (H3: (getl i0 c (CHead d1 (Bind Abbr) u0))).(\lambda (c2: -C).(\lambda (t2: T).(\lambda (H4: (fsubst0 i0 u0 c (TLRef i) c2 t2)).(let H_x -\def (fsubst0_gen_base c c2 (TLRef i) t2 u0 i0 H4) in (let H5 \def H_x in -(or3_ind (land (eq C c c2) (subst0 i0 u0 (TLRef i) t2)) (land (eq T (TLRef i) -t2) (csubst0 i0 u0 c c2)) (land (subst0 i0 u0 (TLRef i) t2) (csubst0 i0 u0 c -c2)) (arity g c2 t2 a0) (\lambda (H6: (land (eq C c c2) (subst0 i0 u0 (TLRef -i) t2))).(land_ind (eq C c c2) (subst0 i0 u0 (TLRef i) t2) (arity g c2 t2 a0) -(\lambda (H7: (eq C c c2)).(\lambda (H8: (subst0 i0 u0 (TLRef i) t2)).(eq_ind -C c (\lambda (c0: C).(arity g c0 t2 a0)) (land_ind (eq nat i i0) (eq T t2 -(lift (S i) O u0)) (arity g c t2 a0) (\lambda (H9: (eq nat i i0)).(\lambda -(H10: (eq T t2 (lift (S i) O u0))).(eq_ind_r T (lift (S i) O u0) (\lambda (t: -T).(arity g c t a0)) (let H11 \def (eq_ind_r nat i0 (\lambda (n: nat).(getl n -c (CHead d1 (Bind Abbr) u0))) H3 i H9) in (let H12 \def (eq_ind C (CHead d -(Bind Abst) u) (\lambda (c0: C).(getl i c c0)) H0 (CHead d1 (Bind Abbr) u0) -(getl_mono c (CHead d (Bind Abst) u) i H0 (CHead d1 (Bind Abbr) u0) H11)) in -(let H13 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (ee: C).(match ee in -C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k -_) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b) -\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow -False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) -\Rightarrow False])])) I (CHead d1 (Bind Abbr) u0) (getl_mono c (CHead d -(Bind Abst) u) i H0 (CHead d1 (Bind Abbr) u0) H11)) in (False_ind (arity g c -(lift (S i) O u0) a0) H13)))) t2 H10))) (subst0_gen_lref u0 t2 i0 i H8)) c2 -H7))) H6)) (\lambda (H6: (land (eq T (TLRef i) t2) (csubst0 i0 u0 c -c2))).(land_ind (eq T (TLRef i) t2) (csubst0 i0 u0 c c2) (arity g c2 t2 a0) -(\lambda (H7: (eq T (TLRef i) t2)).(\lambda (H8: (csubst0 i0 u0 c -c2)).(eq_ind T (TLRef i) (\lambda (t: T).(arity g c2 t a0)) (lt_le_e i i0 -(arity g c2 (TLRef i) a0) (\lambda (H9: (lt i i0)).(let H10 \def -(csubst0_getl_lt i0 i H9 c c2 u0 H8 (CHead d (Bind Abst) u) H0) in (or4_ind -(getl i c2 (CHead d (Bind Abst) u)) (ex3_4 B C T T (\lambda (b: B).(\lambda -(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead -e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(getl i c2 (CHead e0 (Bind b) w)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) -u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1)))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c2 -(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) (ex4_5 B C C T T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1))))))) (\lambda -(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl -i c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))) (arity g c2 (TLRef i) a0) -(\lambda (H11: (getl i c2 (CHead d (Bind Abst) u))).(let H12 \def (eq_ind nat -(minus i0 i) (\lambda (n: nat).(getl n (CHead d (Bind Abst) u) (CHead d1 -(Bind Abbr) u0))) (getl_conf_le i0 (CHead d1 (Bind Abbr) u0) c H3 (CHead d -(Bind Abst) u) i H0 (le_S_n i i0 (le_S (S i) i0 H9))) (S (minus i0 (S i))) -(minus_x_Sy i0 i H9)) in (arity_abst g c2 d u i H11 a0 H1))) (\lambda (H11: -(ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(eq C (CHead d (Bind Abst) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: -B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl i c2 (CHead e0 -(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda -(w: T).(subst0 (minus i0 (S i)) u0 u1 w))))))).(ex3_4_ind B C T T (\lambda -(b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind -Abst) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(getl i c2 (CHead e0 (Bind b) w)))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 -(minus i0 (S i)) u0 u1 w))))) (arity g c2 (TLRef i) a0) (\lambda (x0: -B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (eq C -(CHead d (Bind Abst) u) (CHead x1 (Bind x0) x2))).(\lambda (H13: (getl i c2 -(CHead x1 (Bind x0) x3))).(\lambda (H14: (subst0 (minus i0 (S i)) u0 x2 -x3)).(let H15 \def (eq_ind nat (minus i0 i) (\lambda (n: nat).(getl n (CHead -d (Bind Abst) u) (CHead d1 (Bind Abbr) u0))) (getl_conf_le i0 (CHead d1 (Bind -Abbr) u0) c H3 (CHead d (Bind Abst) u) i H0 (le_S_n i i0 (le_S (S i) i0 H9))) -(S (minus i0 (S i))) (minus_x_Sy i0 i H9)) in (let H16 \def (f_equal C C -(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abst) u) -(CHead x1 (Bind x0) x2) H12) in ((let H17 \def (f_equal C B (\lambda (e: -C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abst | -(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind -b) \Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead d (Bind Abst) u) -(CHead x1 (Bind x0) x2) H12) in ((let H18 \def (f_equal C T (\lambda (e: -C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | -(CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) -x2) H12) in (\lambda (H19: (eq B Abst x0)).(\lambda (H20: (eq C d x1)).(let -H21 \def (eq_ind_r T x2 (\lambda (t: T).(subst0 (minus i0 (S i)) u0 t x3)) -H14 u H18) in (let H22 \def (eq_ind_r C x1 (\lambda (c0: C).(getl i c2 (CHead -c0 (Bind x0) x3))) H13 d H20) in (let H23 \def (eq_ind_r B x0 (\lambda (b: -B).(getl i c2 (CHead d (Bind b) x3))) H22 Abst H19) in (arity_abst g c2 d x3 -i H23 a0 (H2 d1 u0 (r (Bind Abst) (minus i0 (S i))) (getl_gen_S (Bind Abst) d -(CHead d1 (Bind Abbr) u0) u (minus i0 (S i)) H15) d x3 (fsubst0_snd (r (Bind -Abst) (minus i0 (S i))) u0 d u x3 H21))))))))) H17)) H16)))))))))) H11)) -(\lambda (H11: (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1)))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c2 -(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))).(ex3_4_ind B C C -T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C -(CHead d (Bind Abst) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda -(_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c2 (CHead e2 (Bind b) -u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus i0 (S i)) u0 e1 e2))))) (arity g c2 (TLRef i) a0) (\lambda -(x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H12: (eq -C (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3))).(\lambda (H13: (getl i c2 -(CHead x2 (Bind x0) x3))).(\lambda (H14: (csubst0 (minus i0 (S i)) u0 x1 -x2)).(let H15 \def (eq_ind nat (minus i0 i) (\lambda (n: nat).(getl n (CHead -d (Bind Abst) u) (CHead d1 (Bind Abbr) u0))) (getl_conf_le i0 (CHead d1 (Bind -Abbr) u0) c H3 (CHead d (Bind Abst) u) i H0 (le_S_n i i0 (le_S (S i) i0 H9))) -(S (minus i0 (S i))) (minus_x_Sy i0 i H9)) in (let H16 \def (f_equal C C -(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abst) u) -(CHead x1 (Bind x0) x3) H12) in ((let H17 \def (f_equal C B (\lambda (e: -C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abst | -(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind -b) \Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead d (Bind Abst) u) -(CHead x1 (Bind x0) x3) H12) in ((let H18 \def (f_equal C T (\lambda (e: -C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | -(CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) -x3) H12) in (\lambda (H19: (eq B Abst x0)).(\lambda (H20: (eq C d x1)).(let -H21 \def (eq_ind_r T x3 (\lambda (t: T).(getl i c2 (CHead x2 (Bind x0) t))) -H13 u H18) in (let H22 \def (eq_ind_r C x1 (\lambda (c0: C).(csubst0 (minus -i0 (S i)) u0 c0 x2)) H14 d H20) in (let H23 \def (eq_ind_r B x0 (\lambda (b: -B).(getl i c2 (CHead x2 (Bind b) u))) H21 Abst H19) in (arity_abst g c2 x2 u -i H23 a0 (H2 d1 u0 (r (Bind Abst) (minus i0 (S i))) (getl_gen_S (Bind Abst) d -(CHead d1 (Bind Abbr) u0) u (minus i0 (S i)) H15) x2 u (fsubst0_fst (r (Bind -Abst) (minus i0 (S i))) u0 d u x2 H22))))))))) H17)) H16)))))))))) H11)) -(\lambda (H11: (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead e1 -(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(getl i c2 (CHead e2 (Bind b) w))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 -(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 -e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead e1 -(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(getl i c2 (CHead e2 (Bind b) w))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 -(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) -(arity g c2 (TLRef i) a0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: -C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H12: (eq C (CHead d (Bind -Abst) u) (CHead x1 (Bind x0) x3))).(\lambda (H13: (getl i c2 (CHead x2 (Bind -x0) x4))).(\lambda (H14: (subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H15: -(csubst0 (minus i0 (S i)) u0 x1 x2)).(let H16 \def (eq_ind nat (minus i0 i) -(\lambda (n: nat).(getl n (CHead d (Bind Abst) u) (CHead d1 (Bind Abbr) u0))) -(getl_conf_le i0 (CHead d1 (Bind Abbr) u0) c H3 (CHead d (Bind Abst) u) i H0 -(le_S_n i i0 (le_S (S i) i0 H9))) (S (minus i0 (S i))) (minus_x_Sy i0 i H9)) -in (let H17 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda -(_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) -(CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3) H12) in ((let H18 \def -(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with -[(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match k in K return -(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow -Abst])])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3) H12) in ((let H19 -\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind -Abst) u) (CHead x1 (Bind x0) x3) H12) in (\lambda (H20: (eq B Abst -x0)).(\lambda (H21: (eq C d x1)).(let H22 \def (eq_ind_r T x3 (\lambda (t: -T).(subst0 (minus i0 (S i)) u0 t x4)) H14 u H19) in (let H23 \def (eq_ind_r C -x1 (\lambda (c0: C).(csubst0 (minus i0 (S i)) u0 c0 x2)) H15 d H21) in (let -H24 \def (eq_ind_r B x0 (\lambda (b: B).(getl i c2 (CHead x2 (Bind b) x4))) -H13 Abst H20) in (arity_abst g c2 x2 x4 i H24 a0 (H2 d1 u0 (r (Bind Abst) -(minus i0 (S i))) (getl_gen_S (Bind Abst) d (CHead d1 (Bind Abbr) u0) u -(minus i0 (S i)) H16) x2 x4 (fsubst0_both (r (Bind Abst) (minus i0 (S i))) u0 -d u x4 H22 x2 H23))))))))) H18)) H17)))))))))))) H11)) H10))) (\lambda (H9: -(le i0 i)).(arity_abst g c2 d u i (csubst0_getl_ge i0 i H9 c c2 u0 H8 (CHead -d (Bind Abst) u) H0) a0 H1))) t2 H7))) H6)) (\lambda (H6: (land (subst0 i0 u0 -(TLRef i) t2) (csubst0 i0 u0 c c2))).(land_ind (subst0 i0 u0 (TLRef i) t2) -(csubst0 i0 u0 c c2) (arity g c2 t2 a0) (\lambda (H7: (subst0 i0 u0 (TLRef i) -t2)).(\lambda (H8: (csubst0 i0 u0 c c2)).(land_ind (eq nat i i0) (eq T t2 -(lift (S i) O u0)) (arity g c2 t2 a0) (\lambda (H9: (eq nat i i0)).(\lambda -(H10: (eq T t2 (lift (S i) O u0))).(eq_ind_r T (lift (S i) O u0) (\lambda (t: -T).(arity g c2 t a0)) (let H11 \def (eq_ind_r nat i0 (\lambda (n: -nat).(csubst0 n u0 c c2)) H8 i H9) in (let H12 \def (eq_ind_r nat i0 (\lambda -(n: nat).(getl n c (CHead d1 (Bind Abbr) u0))) H3 i H9) in (let H13 \def -(eq_ind C (CHead d (Bind Abst) u) (\lambda (c0: C).(getl i c c0)) H0 (CHead -d1 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abst) u) i H0 (CHead d1 (Bind -Abbr) u0) H12)) in (let H14 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda -(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind b) \Rightarrow (match b in B return (\lambda (_: -B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow True | Void -\Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead d1 (Bind Abbr) -u0) (getl_mono c (CHead d (Bind Abst) u) i H0 (CHead d1 (Bind Abbr) u0) H12)) -in (False_ind (arity g c2 (lift (S i) O u0) a0) H14))))) t2 H10))) -(subst0_gen_lref u0 t2 i0 i H7)))) H6)) H5)))))))))))))))))) (\lambda (b: -B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c: C).(\lambda (u: -T).(\lambda (a1: A).(\lambda (H1: (arity g c u a1)).(\lambda (H2: ((\forall -(d1: C).(\forall (u0: T).(\forall (i: nat).((getl i c (CHead d1 (Bind Abbr) -u0)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 c u c2 t2) \to -(arity g c2 t2 a1)))))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (_: -(arity g (CHead c (Bind b) u) t a2)).(\lambda (H4: ((\forall (d1: C).(\forall -(u0: T).(\forall (i: nat).((getl i (CHead c (Bind b) u) (CHead d1 (Bind Abbr) -u0)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 (CHead c (Bind b) -u) t c2 t2) \to (arity g c2 t2 a2)))))))))).(\lambda (d1: C).(\lambda (u0: -T).(\lambda (i: nat).(\lambda (H5: (getl i c (CHead d1 (Bind Abbr) -u0))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H6: (fsubst0 i u0 c (THead -(Bind b) u t) c2 t2)).(let H_x \def (fsubst0_gen_base c c2 (THead (Bind b) u -t) t2 u0 i H6) in (let H7 \def H_x in (or3_ind (land (eq C c c2) (subst0 i u0 -(THead (Bind b) u t) t2)) (land (eq T (THead (Bind b) u t) t2) (csubst0 i u0 -c c2)) (land (subst0 i u0 (THead (Bind b) u t) t2) (csubst0 i u0 c c2)) -(arity g c2 t2 a2) (\lambda (H8: (land (eq C c c2) (subst0 i u0 (THead (Bind -b) u t) t2))).(land_ind (eq C c c2) (subst0 i u0 (THead (Bind b) u t) t2) -(arity g c2 t2 a2) (\lambda (H9: (eq C c c2)).(\lambda (H10: (subst0 i u0 -(THead (Bind b) u t) t2)).(eq_ind C c (\lambda (c0: C).(arity g c0 t2 a2)) -(or3_ind (ex2 T (\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t))) (\lambda -(u2: T).(subst0 i u0 u u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead (Bind b) -u t3))) (\lambda (t3: T).(subst0 (s (Bind b) i) u0 t t3))) (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) u2 t3)))) (\lambda -(u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: -T).(subst0 (s (Bind b) i) u0 t t3)))) (arity g c t2 a2) (\lambda (H11: (ex2 T -(\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t))) (\lambda (u2: T).(subst0 i -u0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t))) -(\lambda (u2: T).(subst0 i u0 u u2)) (arity g c t2 a2) (\lambda (x: -T).(\lambda (H12: (eq T t2 (THead (Bind b) x t))).(\lambda (H13: (subst0 i u0 -u x)).(eq_ind_r T (THead (Bind b) x t) (\lambda (t0: T).(arity g c t0 a2)) -(arity_bind g b H0 c x a1 (H2 d1 u0 i H5 c x (fsubst0_snd i u0 c u x H13)) t -a2 (H4 d1 u0 (S i) (getl_clear_bind b (CHead c (Bind b) u) c u (clear_bind b -c u) (CHead d1 (Bind Abbr) u0) i H5) (CHead c (Bind b) x) t (fsubst0_fst (S -i) u0 (CHead c (Bind b) u) t (CHead c (Bind b) x) (csubst0_snd_bind b i u0 u -x H13 c)))) t2 H12)))) H11)) (\lambda (H11: (ex2 T (\lambda (t3: T).(eq T t2 -(THead (Bind b) u t3))) (\lambda (t3: T).(subst0 (s (Bind b) i) u0 t -t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead (Bind b) u t3))) (\lambda -(t3: T).(subst0 (s (Bind b) i) u0 t t3)) (arity g c t2 a2) (\lambda (x: -T).(\lambda (H12: (eq T t2 (THead (Bind b) u x))).(\lambda (H13: (subst0 (s -(Bind b) i) u0 t x)).(eq_ind_r T (THead (Bind b) u x) (\lambda (t0: T).(arity -g c t0 a2)) (arity_bind g b H0 c u a1 H1 x a2 (H4 d1 u0 (S i) -(getl_clear_bind b (CHead c (Bind b) u) c u (clear_bind b c u) (CHead d1 -(Bind Abbr) u0) i H5) (CHead c (Bind b) u) x (fsubst0_snd (S i) u0 (CHead c -(Bind b) u) t x H13))) t2 H12)))) H11)) (\lambda (H11: (ex3_2 T T (\lambda -(u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: -T).(subst0 (s (Bind b) i) u0 t t3))))).(ex3_2_ind T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Bind b) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: -T).(subst0 (s (Bind b) i) u0 t t3))) (arity g c t2 a2) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H12: (eq T t2 (THead (Bind b) x0 x1))).(\lambda -(H13: (subst0 i u0 u x0)).(\lambda (H14: (subst0 (s (Bind b) i) u0 t -x1)).(eq_ind_r T (THead (Bind b) x0 x1) (\lambda (t0: T).(arity g c t0 a2)) -(arity_bind g b H0 c x0 a1 (H2 d1 u0 i H5 c x0 (fsubst0_snd i u0 c u x0 H13)) -x1 a2 (H4 d1 u0 (S i) (getl_clear_bind b (CHead c (Bind b) u) c u (clear_bind -b c u) (CHead d1 (Bind Abbr) u0) i H5) (CHead c (Bind b) x0) x1 (fsubst0_both -(S i) u0 (CHead c (Bind b) u) t x1 H14 (CHead c (Bind b) x0) -(csubst0_snd_bind b i u0 u x0 H13 c)))) t2 H12)))))) H11)) (subst0_gen_head -(Bind b) u0 u t t2 i H10)) c2 H9))) H8)) (\lambda (H8: (land (eq T (THead -(Bind b) u t) t2) (csubst0 i u0 c c2))).(land_ind (eq T (THead (Bind b) u t) -t2) (csubst0 i u0 c c2) (arity g c2 t2 a2) (\lambda (H9: (eq T (THead (Bind -b) u t) t2)).(\lambda (H10: (csubst0 i u0 c c2)).(eq_ind T (THead (Bind b) u -t) (\lambda (t0: T).(arity g c2 t0 a2)) (arity_bind g b H0 c2 u a1 (H2 d1 u0 -i H5 c2 u (fsubst0_fst i u0 c u c2 H10)) t a2 (H4 d1 u0 (S i) -(getl_clear_bind b (CHead c (Bind b) u) c u (clear_bind b c u) (CHead d1 -(Bind Abbr) u0) i H5) (CHead c2 (Bind b) u) t (fsubst0_fst (S i) u0 (CHead c -(Bind b) u) t (CHead c2 (Bind b) u) (csubst0_fst_bind b i c c2 u0 H10 u)))) -t2 H9))) H8)) (\lambda (H8: (land (subst0 i u0 (THead (Bind b) u t) t2) -(csubst0 i u0 c c2))).(land_ind (subst0 i u0 (THead (Bind b) u t) t2) -(csubst0 i u0 c c2) (arity g c2 t2 a2) (\lambda (H9: (subst0 i u0 (THead -(Bind b) u t) t2)).(\lambda (H10: (csubst0 i u0 c c2)).(or3_ind (ex2 T -(\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t))) (\lambda (u2: T).(subst0 i -u0 u u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead (Bind b) u t3))) (\lambda -(t3: T).(subst0 (s (Bind b) i) u0 t t3))) (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Bind b) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: -T).(subst0 (s (Bind b) i) u0 t t3)))) (arity g c2 t2 a2) (\lambda (H11: (ex2 -T (\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t))) (\lambda (u2: T).(subst0 -i u0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t))) -(\lambda (u2: T).(subst0 i u0 u u2)) (arity g c2 t2 a2) (\lambda (x: -T).(\lambda (H12: (eq T t2 (THead (Bind b) x t))).(\lambda (H13: (subst0 i u0 -u x)).(eq_ind_r T (THead (Bind b) x t) (\lambda (t0: T).(arity g c2 t0 a2)) -(arity_bind g b H0 c2 x a1 (H2 d1 u0 i H5 c2 x (fsubst0_both i u0 c u x H13 -c2 H10)) t a2 (H4 d1 u0 (S i) (getl_clear_bind b (CHead c (Bind b) u) c u -(clear_bind b c u) (CHead d1 (Bind Abbr) u0) i H5) (CHead c2 (Bind b) x) t -(fsubst0_fst (S i) u0 (CHead c (Bind b) u) t (CHead c2 (Bind b) x) -(csubst0_both_bind b i u0 u x H13 c c2 H10)))) t2 H12)))) H11)) (\lambda -(H11: (ex2 T (\lambda (t3: T).(eq T t2 (THead (Bind b) u t3))) (\lambda (t3: -T).(subst0 (s (Bind b) i) u0 t t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 -(THead (Bind b) u t3))) (\lambda (t3: T).(subst0 (s (Bind b) i) u0 t t3)) -(arity g c2 t2 a2) (\lambda (x: T).(\lambda (H12: (eq T t2 (THead (Bind b) u -x))).(\lambda (H13: (subst0 (s (Bind b) i) u0 t x)).(eq_ind_r T (THead (Bind -b) u x) (\lambda (t0: T).(arity g c2 t0 a2)) (arity_bind g b H0 c2 u a1 (H2 -d1 u0 i H5 c2 u (fsubst0_fst i u0 c u c2 H10)) x a2 (H4 d1 u0 (S i) -(getl_clear_bind b (CHead c (Bind b) u) c u (clear_bind b c u) (CHead d1 -(Bind Abbr) u0) i H5) (CHead c2 (Bind b) u) x (fsubst0_both (S i) u0 (CHead c -(Bind b) u) t x H13 (CHead c2 (Bind b) u) (csubst0_fst_bind b i c c2 u0 H10 -u)))) t2 H12)))) H11)) (\lambda (H11: (ex3_2 T T (\lambda (u2: T).(\lambda -(t3: T).(eq T t2 (THead (Bind b) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind b) -i) u0 t t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 -(THead (Bind b) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u -u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind b) i) u0 t t3))) -(arity g c2 t2 a2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H12: (eq T t2 -(THead (Bind b) x0 x1))).(\lambda (H13: (subst0 i u0 u x0)).(\lambda (H14: -(subst0 (s (Bind b) i) u0 t x1)).(eq_ind_r T (THead (Bind b) x0 x1) (\lambda -(t0: T).(arity g c2 t0 a2)) (arity_bind g b H0 c2 x0 a1 (H2 d1 u0 i H5 c2 x0 -(fsubst0_both i u0 c u x0 H13 c2 H10)) x1 a2 (H4 d1 u0 (S i) (getl_clear_bind -b (CHead c (Bind b) u) c u (clear_bind b c u) (CHead d1 (Bind Abbr) u0) i H5) -(CHead c2 (Bind b) x0) x1 (fsubst0_both (S i) u0 (CHead c (Bind b) u) t x1 -H14 (CHead c2 (Bind b) x0) (csubst0_both_bind b i u0 u x0 H13 c c2 H10)))) t2 -H12)))))) H11)) (subst0_gen_head (Bind b) u0 u t t2 i H9)))) H8)) -H7))))))))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a1: -A).(\lambda (H0: (arity g c u (asucc g a1))).(\lambda (H1: ((\forall (d1: -C).(\forall (u0: T).(\forall (i: nat).((getl i c (CHead d1 (Bind Abbr) u0)) -\to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 c u c2 t2) \to (arity g -c2 t2 (asucc g a1))))))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (_: -(arity g (CHead c (Bind Abst) u) t a2)).(\lambda (H3: ((\forall (d1: -C).(\forall (u0: T).(\forall (i: nat).((getl i (CHead c (Bind Abst) u) (CHead -d1 (Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 -(CHead c (Bind Abst) u) t c2 t2) \to (arity g c2 t2 a2)))))))))).(\lambda -(d1: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H4: (getl i c (CHead d1 -(Bind Abbr) u0))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H5: (fsubst0 i -u0 c (THead (Bind Abst) u t) c2 t2)).(let H_x \def (fsubst0_gen_base c c2 -(THead (Bind Abst) u t) t2 u0 i H5) in (let H6 \def H_x in (or3_ind (land (eq -C c c2) (subst0 i u0 (THead (Bind Abst) u t) t2)) (land (eq T (THead (Bind -Abst) u t) t2) (csubst0 i u0 c c2)) (land (subst0 i u0 (THead (Bind Abst) u -t) t2) (csubst0 i u0 c c2)) (arity g c2 t2 (AHead a1 a2)) (\lambda (H7: (land -(eq C c c2) (subst0 i u0 (THead (Bind Abst) u t) t2))).(land_ind (eq C c c2) -(subst0 i u0 (THead (Bind Abst) u t) t2) (arity g c2 t2 (AHead a1 a2)) -(\lambda (H8: (eq C c c2)).(\lambda (H9: (subst0 i u0 (THead (Bind Abst) u t) -t2)).(eq_ind C c (\lambda (c0: C).(arity g c0 t2 (AHead a1 a2))) (or3_ind -(ex2 T (\lambda (u2: T).(eq T t2 (THead (Bind Abst) u2 t))) (\lambda (u2: -T).(subst0 i u0 u u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead (Bind Abst) u -t3))) (\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3))) (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: -T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3)))) (arity g c t2 -(AHead a1 a2)) (\lambda (H10: (ex2 T (\lambda (u2: T).(eq T t2 (THead (Bind -Abst) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T (\lambda (u2: -T).(eq T t2 (THead (Bind Abst) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)) -(arity g c t2 (AHead a1 a2)) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead -(Bind Abst) x t))).(\lambda (H12: (subst0 i u0 u x)).(eq_ind_r T (THead (Bind -Abst) x t) (\lambda (t0: T).(arity g c t0 (AHead a1 a2))) (arity_head g c x -a1 (H1 d1 u0 i H4 c x (fsubst0_snd i u0 c u x H12)) t a2 (H3 d1 u0 (S i) -(getl_clear_bind Abst (CHead c (Bind Abst) u) c u (clear_bind Abst c u) -(CHead d1 (Bind Abbr) u0) i H4) (CHead c (Bind Abst) x) t (fsubst0_fst (S i) -u0 (CHead c (Bind Abst) u) t (CHead c (Bind Abst) x) (csubst0_snd_bind Abst i -u0 u x H12 c)))) t2 H11)))) H10)) (\lambda (H10: (ex2 T (\lambda (t3: T).(eq -T t2 (THead (Bind Abst) u t3))) (\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 -t t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead (Bind Abst) u t3))) -(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3)) (arity g c t2 (AHead a1 -a2)) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Bind Abst) u -x))).(\lambda (H12: (subst0 (s (Bind Abst) i) u0 t x)).(eq_ind_r T (THead -(Bind Abst) u x) (\lambda (t0: T).(arity g c t0 (AHead a1 a2))) (arity_head g -c u a1 H0 x a2 (H3 d1 u0 (S i) (getl_clear_bind Abst (CHead c (Bind Abst) u) -c u (clear_bind Abst c u) (CHead d1 (Bind Abbr) u0) i H4) (CHead c (Bind -Abst) u) x (fsubst0_snd (S i) u0 (CHead c (Bind Abst) u) t x H12))) t2 -H11)))) H10)) (\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq -T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i -u0 u u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t -t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead -(Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) -(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3))) (arity -g c t2 (AHead a1 a2)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq T -t2 (THead (Bind Abst) x0 x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda -(H13: (subst0 (s (Bind Abst) i) u0 t x1)).(eq_ind_r T (THead (Bind Abst) x0 -x1) (\lambda (t0: T).(arity g c t0 (AHead a1 a2))) (arity_head g c x0 a1 (H1 -d1 u0 i H4 c x0 (fsubst0_snd i u0 c u x0 H12)) x1 a2 (H3 d1 u0 (S i) -(getl_clear_bind Abst (CHead c (Bind Abst) u) c u (clear_bind Abst c u) -(CHead d1 (Bind Abbr) u0) i H4) (CHead c (Bind Abst) x0) x1 (fsubst0_both (S -i) u0 (CHead c (Bind Abst) u) t x1 H13 (CHead c (Bind Abst) x0) -(csubst0_snd_bind Abst i u0 u x0 H12 c)))) t2 H11)))))) H10)) -(subst0_gen_head (Bind Abst) u0 u t t2 i H9)) c2 H8))) H7)) (\lambda (H7: -(land (eq T (THead (Bind Abst) u t) t2) (csubst0 i u0 c c2))).(land_ind (eq T -(THead (Bind Abst) u t) t2) (csubst0 i u0 c c2) (arity g c2 t2 (AHead a1 a2)) -(\lambda (H8: (eq T (THead (Bind Abst) u t) t2)).(\lambda (H9: (csubst0 i u0 -c c2)).(eq_ind T (THead (Bind Abst) u t) (\lambda (t0: T).(arity g c2 t0 -(AHead a1 a2))) (arity_head g c2 u a1 (H1 d1 u0 i H4 c2 u (fsubst0_fst i u0 c -u c2 H9)) t a2 (H3 d1 u0 (S i) (getl_clear_bind Abst (CHead c (Bind Abst) u) -c u (clear_bind Abst c u) (CHead d1 (Bind Abbr) u0) i H4) (CHead c2 (Bind -Abst) u) t (fsubst0_fst (S i) u0 (CHead c (Bind Abst) u) t (CHead c2 (Bind -Abst) u) (csubst0_fst_bind Abst i c c2 u0 H9 u)))) t2 H8))) H7)) (\lambda -(H7: (land (subst0 i u0 (THead (Bind Abst) u t) t2) (csubst0 i u0 c -c2))).(land_ind (subst0 i u0 (THead (Bind Abst) u t) t2) (csubst0 i u0 c c2) -(arity g c2 t2 (AHead a1 a2)) (\lambda (H8: (subst0 i u0 (THead (Bind Abst) u -t) t2)).(\lambda (H9: (csubst0 i u0 c c2)).(or3_ind (ex2 T (\lambda (u2: -T).(eq T t2 (THead (Bind Abst) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2))) -(ex2 T (\lambda (t3: T).(eq T t2 (THead (Bind Abst) u t3))) (\lambda (t3: -T).(subst0 (s (Bind Abst) i) u0 t t3))) (ex3_2 T T (\lambda (u2: T).(\lambda -(t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind -Abst) i) u0 t t3)))) (arity g c2 t2 (AHead a1 a2)) (\lambda (H10: (ex2 T -(\lambda (u2: T).(eq T t2 (THead (Bind Abst) u2 t))) (\lambda (u2: T).(subst0 -i u0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T t2 (THead (Bind Abst) u2 t))) -(\lambda (u2: T).(subst0 i u0 u u2)) (arity g c2 t2 (AHead a1 a2)) (\lambda -(x: T).(\lambda (H11: (eq T t2 (THead (Bind Abst) x t))).(\lambda (H12: -(subst0 i u0 u x)).(eq_ind_r T (THead (Bind Abst) x t) (\lambda (t0: -T).(arity g c2 t0 (AHead a1 a2))) (arity_head g c2 x a1 (H1 d1 u0 i H4 c2 x -(fsubst0_both i u0 c u x H12 c2 H9)) t a2 (H3 d1 u0 (S i) (getl_clear_bind -Abst (CHead c (Bind Abst) u) c u (clear_bind Abst c u) (CHead d1 (Bind Abbr) -u0) i H4) (CHead c2 (Bind Abst) x) t (fsubst0_fst (S i) u0 (CHead c (Bind -Abst) u) t (CHead c2 (Bind Abst) x) (csubst0_both_bind Abst i u0 u x H12 c c2 -H9)))) t2 H11)))) H10)) (\lambda (H10: (ex2 T (\lambda (t3: T).(eq T t2 -(THead (Bind Abst) u t3))) (\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t -t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead (Bind Abst) u t3))) -(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3)) (arity g c2 t2 (AHead a1 -a2)) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Bind Abst) u -x))).(\lambda (H12: (subst0 (s (Bind Abst) i) u0 t x)).(eq_ind_r T (THead -(Bind Abst) u x) (\lambda (t0: T).(arity g c2 t0 (AHead a1 a2))) (arity_head -g c2 u a1 (H1 d1 u0 i H4 c2 u (fsubst0_fst i u0 c u c2 H9)) x a2 (H3 d1 u0 (S -i) (getl_clear_bind Abst (CHead c (Bind Abst) u) c u (clear_bind Abst c u) -(CHead d1 (Bind Abbr) u0) i H4) (CHead c2 (Bind Abst) u) x (fsubst0_both (S -i) u0 (CHead c (Bind Abst) u) t x H12 (CHead c2 (Bind Abst) u) -(csubst0_fst_bind Abst i c c2 u0 H9 u)))) t2 H11)))) H10)) (\lambda (H10: -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: -T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3))))).(ex3_2_ind T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: -T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3))) (arity g c2 t2 -(AHead a1 a2)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq T t2 -(THead (Bind Abst) x0 x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda (H13: -(subst0 (s (Bind Abst) i) u0 t x1)).(eq_ind_r T (THead (Bind Abst) x0 x1) -(\lambda (t0: T).(arity g c2 t0 (AHead a1 a2))) (arity_head g c2 x0 a1 (H1 d1 -u0 i H4 c2 x0 (fsubst0_both i u0 c u x0 H12 c2 H9)) x1 a2 (H3 d1 u0 (S i) -(getl_clear_bind Abst (CHead c (Bind Abst) u) c u (clear_bind Abst c u) -(CHead d1 (Bind Abbr) u0) i H4) (CHead c2 (Bind Abst) x0) x1 (fsubst0_both (S -i) u0 (CHead c (Bind Abst) u) t x1 H13 (CHead c2 (Bind Abst) x0) -(csubst0_both_bind Abst i u0 u x0 H12 c c2 H9)))) t2 H11)))))) H10)) -(subst0_gen_head (Bind Abst) u0 u t t2 i H8)))) H7)) H6))))))))))))))))))) -(\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H0: (arity g c u -a1)).(\lambda (H1: ((\forall (d1: C).(\forall (u0: T).(\forall (i: -nat).((getl i c (CHead d1 (Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2: -T).((fsubst0 i u0 c u c2 t2) \to (arity g c2 t2 a1)))))))))).(\lambda (t: -T).(\lambda (a2: A).(\lambda (H2: (arity g c t (AHead a1 a2))).(\lambda (H3: -((\forall (d1: C).(\forall (u0: T).(\forall (i: nat).((getl i c (CHead d1 -(Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 c t c2 -t2) \to (arity g c2 t2 (AHead a1 a2))))))))))).(\lambda (d1: C).(\lambda (u0: -T).(\lambda (i: nat).(\lambda (H4: (getl i c (CHead d1 (Bind Abbr) -u0))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H5: (fsubst0 i u0 c (THead -(Flat Appl) u t) c2 t2)).(let H_x \def (fsubst0_gen_base c c2 (THead (Flat -Appl) u t) t2 u0 i H5) in (let H6 \def H_x in (or3_ind (land (eq C c c2) -(subst0 i u0 (THead (Flat Appl) u t) t2)) (land (eq T (THead (Flat Appl) u t) -t2) (csubst0 i u0 c c2)) (land (subst0 i u0 (THead (Flat Appl) u t) t2) -(csubst0 i u0 c c2)) (arity g c2 t2 a2) (\lambda (H7: (land (eq C c c2) -(subst0 i u0 (THead (Flat Appl) u t) t2))).(land_ind (eq C c c2) (subst0 i u0 -(THead (Flat Appl) u t) t2) (arity g c2 t2 a2) (\lambda (H8: (eq C c -c2)).(\lambda (H9: (subst0 i u0 (THead (Flat Appl) u t) t2)).(eq_ind C c -(\lambda (c0: C).(arity g c0 t2 a2)) (or3_ind (ex2 T (\lambda (u2: T).(eq T -t2 (THead (Flat Appl) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2))) (ex2 T -(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u t3))) (\lambda (t3: T).(subst0 -(s (Flat Appl) i) u0 t t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq -T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i -u0 u u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 t -t3)))) (arity g c t2 a2) (\lambda (H10: (ex2 T (\lambda (u2: T).(eq T t2 -(THead (Flat Appl) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T -(\lambda (u2: T).(eq T t2 (THead (Flat Appl) u2 t))) (\lambda (u2: T).(subst0 -i u0 u u2)) (arity g c t2 a2) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead -(Flat Appl) x t))).(\lambda (H12: (subst0 i u0 u x)).(eq_ind_r T (THead (Flat -Appl) x t) (\lambda (t0: T).(arity g c t0 a2)) (arity_appl g c x a1 (H1 d1 u0 -i H4 c x (fsubst0_snd i u0 c u x H12)) t a2 H2) t2 H11)))) H10)) (\lambda -(H10: (ex2 T (\lambda (t3: T).(eq T t2 (THead (Flat Appl) u t3))) (\lambda -(t3: T).(subst0 (s (Flat Appl) i) u0 t t3)))).(ex2_ind T (\lambda (t3: T).(eq -T t2 (THead (Flat Appl) u t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 -t t3)) (arity g c t2 a2) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat -Appl) u x))).(\lambda (H12: (subst0 (s (Flat Appl) i) u0 t x)).(eq_ind_r T -(THead (Flat Appl) u x) (\lambda (t0: T).(arity g c t0 a2)) (arity_appl g c u -a1 H0 x a2 (H3 d1 u0 i H4 c x (fsubst0_snd i u0 c t x H12))) t2 H11)))) H10)) -(\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead -(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) -(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 t -t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead -(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) -(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 t t3))) (arity -g c t2 a2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq T t2 (THead -(Flat Appl) x0 x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda (H13: -(subst0 (s (Flat Appl) i) u0 t x1)).(eq_ind_r T (THead (Flat Appl) x0 x1) -(\lambda (t0: T).(arity g c t0 a2)) (arity_appl g c x0 a1 (H1 d1 u0 i H4 c x0 -(fsubst0_snd i u0 c u x0 H12)) x1 a2 (H3 d1 u0 i H4 c x1 (fsubst0_snd i u0 c -t x1 H13))) t2 H11)))))) H10)) (subst0_gen_head (Flat Appl) u0 u t t2 i H9)) -c2 H8))) H7)) (\lambda (H7: (land (eq T (THead (Flat Appl) u t) t2) (csubst0 -i u0 c c2))).(land_ind (eq T (THead (Flat Appl) u t) t2) (csubst0 i u0 c c2) -(arity g c2 t2 a2) (\lambda (H8: (eq T (THead (Flat Appl) u t) t2)).(\lambda -(H9: (csubst0 i u0 c c2)).(eq_ind T (THead (Flat Appl) u t) (\lambda (t0: -T).(arity g c2 t0 a2)) (arity_appl g c2 u a1 (H1 d1 u0 i H4 c2 u (fsubst0_fst -i u0 c u c2 H9)) t a2 (H3 d1 u0 i H4 c2 t (fsubst0_fst i u0 c t c2 H9))) t2 -H8))) H7)) (\lambda (H7: (land (subst0 i u0 (THead (Flat Appl) u t) t2) -(csubst0 i u0 c c2))).(land_ind (subst0 i u0 (THead (Flat Appl) u t) t2) -(csubst0 i u0 c c2) (arity g c2 t2 a2) (\lambda (H8: (subst0 i u0 (THead -(Flat Appl) u t) t2)).(\lambda (H9: (csubst0 i u0 c c2)).(or3_ind (ex2 T -(\lambda (u2: T).(eq T t2 (THead (Flat Appl) u2 t))) (\lambda (u2: T).(subst0 -i u0 u u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead (Flat Appl) u t3))) -(\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 t t3))) (ex3_2 T T (\lambda -(u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: -T).(subst0 (s (Flat Appl) i) u0 t t3)))) (arity g c2 t2 a2) (\lambda (H10: -(ex2 T (\lambda (u2: T).(eq T t2 (THead (Flat Appl) u2 t))) (\lambda (u2: -T).(subst0 i u0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T t2 (THead (Flat -Appl) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)) (arity g c2 t2 a2) -(\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat Appl) x t))).(\lambda -(H12: (subst0 i u0 u x)).(eq_ind_r T (THead (Flat Appl) x t) (\lambda (t0: -T).(arity g c2 t0 a2)) (arity_appl g c2 x a1 (H1 d1 u0 i H4 c2 x -(fsubst0_both i u0 c u x H12 c2 H9)) t a2 (H3 d1 u0 i H4 c2 t (fsubst0_fst i -u0 c t c2 H9))) t2 H11)))) H10)) (\lambda (H10: (ex2 T (\lambda (t3: T).(eq T -t2 (THead (Flat Appl) u t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 t -t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead (Flat Appl) u t3))) -(\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 t t3)) (arity g c2 t2 a2) -(\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat Appl) u x))).(\lambda -(H12: (subst0 (s (Flat Appl) i) u0 t x)).(eq_ind_r T (THead (Flat Appl) u x) -(\lambda (t0: T).(arity g c2 t0 a2)) (arity_appl g c2 u a1 (H1 d1 u0 i H4 c2 -u (fsubst0_fst i u0 c u c2 H9)) x a2 (H3 d1 u0 i H4 c2 x (fsubst0_both i u0 c -t x H12 c2 H9))) t2 H11)))) H10)) (\lambda (H10: (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: -T).(subst0 (s (Flat Appl) i) u0 t t3))))).(ex3_2_ind T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: -T).(subst0 (s (Flat Appl) i) u0 t t3))) (arity g c2 t2 a2) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H11: (eq T t2 (THead (Flat Appl) x0 -x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda (H13: (subst0 (s (Flat -Appl) i) u0 t x1)).(eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda (t0: -T).(arity g c2 t0 a2)) (arity_appl g c2 x0 a1 (H1 d1 u0 i H4 c2 x0 -(fsubst0_both i u0 c u x0 H12 c2 H9)) x1 a2 (H3 d1 u0 i H4 c2 x1 -(fsubst0_both i u0 c t x1 H13 c2 H9))) t2 H11)))))) H10)) (subst0_gen_head -(Flat Appl) u0 u t t2 i H8)))) H7)) H6))))))))))))))))))) (\lambda (c: -C).(\lambda (u: T).(\lambda (a0: A).(\lambda (H0: (arity g c u (asucc g -a0))).(\lambda (H1: ((\forall (d1: C).(\forall (u0: T).(\forall (i: -nat).((getl i c (CHead d1 (Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2: -T).((fsubst0 i u0 c u c2 t2) \to (arity g c2 t2 (asucc g -a0))))))))))).(\lambda (t: T).(\lambda (H2: (arity g c t a0)).(\lambda (H3: -((\forall (d1: C).(\forall (u0: T).(\forall (i: nat).((getl i c (CHead d1 -(Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 c t c2 -t2) \to (arity g c2 t2 a0)))))))))).(\lambda (d1: C).(\lambda (u0: -T).(\lambda (i: nat).(\lambda (H4: (getl i c (CHead d1 (Bind Abbr) -u0))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H5: (fsubst0 i u0 c (THead -(Flat Cast) u t) c2 t2)).(let H_x \def (fsubst0_gen_base c c2 (THead (Flat -Cast) u t) t2 u0 i H5) in (let H6 \def H_x in (or3_ind (land (eq C c c2) -(subst0 i u0 (THead (Flat Cast) u t) t2)) (land (eq T (THead (Flat Cast) u t) -t2) (csubst0 i u0 c c2)) (land (subst0 i u0 (THead (Flat Cast) u t) t2) -(csubst0 i u0 c c2)) (arity g c2 t2 a0) (\lambda (H7: (land (eq C c c2) -(subst0 i u0 (THead (Flat Cast) u t) t2))).(land_ind (eq C c c2) (subst0 i u0 -(THead (Flat Cast) u t) t2) (arity g c2 t2 a0) (\lambda (H8: (eq C c -c2)).(\lambda (H9: (subst0 i u0 (THead (Flat Cast) u t) t2)).(eq_ind C c -(\lambda (c0: C).(arity g c0 t2 a0)) (or3_ind (ex2 T (\lambda (u2: T).(eq T -t2 (THead (Flat Cast) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2))) (ex2 T -(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u t3))) (\lambda (t3: T).(subst0 -(s (Flat Cast) i) u0 t t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq -T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i -u0 u u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t -t3)))) (arity g c t2 a0) (\lambda (H10: (ex2 T (\lambda (u2: T).(eq T t2 -(THead (Flat Cast) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T -(\lambda (u2: T).(eq T t2 (THead (Flat Cast) u2 t))) (\lambda (u2: T).(subst0 -i u0 u u2)) (arity g c t2 a0) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead -(Flat Cast) x t))).(\lambda (H12: (subst0 i u0 u x)).(eq_ind_r T (THead (Flat -Cast) x t) (\lambda (t0: T).(arity g c t0 a0)) (arity_cast g c x a0 (H1 d1 u0 -i H4 c x (fsubst0_snd i u0 c u x H12)) t H2) t2 H11)))) H10)) (\lambda (H10: -(ex2 T (\lambda (t3: T).(eq T t2 (THead (Flat Cast) u t3))) (\lambda (t3: -T).(subst0 (s (Flat Cast) i) u0 t t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 -(THead (Flat Cast) u t3))) (\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t -t3)) (arity g c t2 a0) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat -Cast) u x))).(\lambda (H12: (subst0 (s (Flat Cast) i) u0 t x)).(eq_ind_r T -(THead (Flat Cast) u x) (\lambda (t0: T).(arity g c t0 a0)) (arity_cast g c u -a0 H0 x (H3 d1 u0 i H4 c x (fsubst0_snd i u0 c t x H12))) t2 H11)))) H10)) -(\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead -(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) -(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t -t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead -(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) -(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t t3))) (arity -g c t2 a0) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq T t2 (THead -(Flat Cast) x0 x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda (H13: -(subst0 (s (Flat Cast) i) u0 t x1)).(eq_ind_r T (THead (Flat Cast) x0 x1) -(\lambda (t0: T).(arity g c t0 a0)) (arity_cast g c x0 a0 (H1 d1 u0 i H4 c x0 -(fsubst0_snd i u0 c u x0 H12)) x1 (H3 d1 u0 i H4 c x1 (fsubst0_snd i u0 c t -x1 H13))) t2 H11)))))) H10)) (subst0_gen_head (Flat Cast) u0 u t t2 i H9)) c2 -H8))) H7)) (\lambda (H7: (land (eq T (THead (Flat Cast) u t) t2) (csubst0 i -u0 c c2))).(land_ind (eq T (THead (Flat Cast) u t) t2) (csubst0 i u0 c c2) -(arity g c2 t2 a0) (\lambda (H8: (eq T (THead (Flat Cast) u t) t2)).(\lambda -(H9: (csubst0 i u0 c c2)).(eq_ind T (THead (Flat Cast) u t) (\lambda (t0: -T).(arity g c2 t0 a0)) (arity_cast g c2 u a0 (H1 d1 u0 i H4 c2 u (fsubst0_fst -i u0 c u c2 H9)) t (H3 d1 u0 i H4 c2 t (fsubst0_fst i u0 c t c2 H9))) t2 -H8))) H7)) (\lambda (H7: (land (subst0 i u0 (THead (Flat Cast) u t) t2) -(csubst0 i u0 c c2))).(land_ind (subst0 i u0 (THead (Flat Cast) u t) t2) -(csubst0 i u0 c c2) (arity g c2 t2 a0) (\lambda (H8: (subst0 i u0 (THead -(Flat Cast) u t) t2)).(\lambda (H9: (csubst0 i u0 c c2)).(or3_ind (ex2 T -(\lambda (u2: T).(eq T t2 (THead (Flat Cast) u2 t))) (\lambda (u2: T).(subst0 -i u0 u u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead (Flat Cast) u t3))) -(\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t t3))) (ex3_2 T T (\lambda -(u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: -T).(subst0 (s (Flat Cast) i) u0 t t3)))) (arity g c2 t2 a0) (\lambda (H10: -(ex2 T (\lambda (u2: T).(eq T t2 (THead (Flat Cast) u2 t))) (\lambda (u2: -T).(subst0 i u0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T t2 (THead (Flat -Cast) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)) (arity g c2 t2 a0) -(\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat Cast) x t))).(\lambda -(H12: (subst0 i u0 u x)).(eq_ind_r T (THead (Flat Cast) x t) (\lambda (t0: -T).(arity g c2 t0 a0)) (arity_cast g c2 x a0 (H1 d1 u0 i H4 c2 x -(fsubst0_both i u0 c u x H12 c2 H9)) t (H3 d1 u0 i H4 c2 t (fsubst0_fst i u0 -c t c2 H9))) t2 H11)))) H10)) (\lambda (H10: (ex2 T (\lambda (t3: T).(eq T t2 -(THead (Flat Cast) u t3))) (\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t -t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead (Flat Cast) u t3))) -(\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t t3)) (arity g c2 t2 a0) -(\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat Cast) u x))).(\lambda -(H12: (subst0 (s (Flat Cast) i) u0 t x)).(eq_ind_r T (THead (Flat Cast) u x) -(\lambda (t0: T).(arity g c2 t0 a0)) (arity_cast g c2 u a0 (H1 d1 u0 i H4 c2 -u (fsubst0_fst i u0 c u c2 H9)) x (H3 d1 u0 i H4 c2 x (fsubst0_both i u0 c t -x H12 c2 H9))) t2 H11)))) H10)) (\lambda (H10: (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: -T).(subst0 (s (Flat Cast) i) u0 t t3))))).(ex3_2_ind T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: -T).(subst0 (s (Flat Cast) i) u0 t t3))) (arity g c2 t2 a0) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H11: (eq T t2 (THead (Flat Cast) x0 -x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda (H13: (subst0 (s (Flat -Cast) i) u0 t x1)).(eq_ind_r T (THead (Flat Cast) x0 x1) (\lambda (t0: -T).(arity g c2 t0 a0)) (arity_cast g c2 x0 a0 (H1 d1 u0 i H4 c2 x0 -(fsubst0_both i u0 c u x0 H12 c2 H9)) x1 (H3 d1 u0 i H4 c2 x1 (fsubst0_both i -u0 c t x1 H13 c2 H9))) t2 H11)))))) H10)) (subst0_gen_head (Flat Cast) u0 u t -t2 i H8)))) H7)) H6)))))))))))))))))) (\lambda (c: C).(\lambda (t: -T).(\lambda (a1: A).(\lambda (_: (arity g c t a1)).(\lambda (H1: ((\forall -(d1: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d1 (Bind Abbr) -u)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u c t c2 t2) \to (arity -g c2 t2 a1)))))))))).(\lambda (a2: A).(\lambda (H2: (leq g a1 a2)).(\lambda -(d1: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H3: (getl i c (CHead d1 -(Bind Abbr) u))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H4: (fsubst0 i u -c t c2 t2)).(let H_x \def (fsubst0_gen_base c c2 t t2 u i H4) in (let H5 \def -H_x in (or3_ind (land (eq C c c2) (subst0 i u t t2)) (land (eq T t t2) -(csubst0 i u c c2)) (land (subst0 i u t t2) (csubst0 i u c c2)) (arity g c2 -t2 a2) (\lambda (H6: (land (eq C c c2) (subst0 i u t t2))).(land_ind (eq C c -c2) (subst0 i u t t2) (arity g c2 t2 a2) (\lambda (H7: (eq C c c2)).(\lambda -(H8: (subst0 i u t t2)).(eq_ind C c (\lambda (c0: C).(arity g c0 t2 a2)) -(arity_repl g c t2 a1 (H1 d1 u i H3 c t2 (fsubst0_snd i u c t t2 H8)) a2 H2) -c2 H7))) H6)) (\lambda (H6: (land (eq T t t2) (csubst0 i u c c2))).(land_ind -(eq T t t2) (csubst0 i u c c2) (arity g c2 t2 a2) (\lambda (H7: (eq T t -t2)).(\lambda (H8: (csubst0 i u c c2)).(eq_ind T t (\lambda (t0: T).(arity g -c2 t0 a2)) (arity_repl g c2 t a1 (H1 d1 u i H3 c2 t (fsubst0_fst i u c t c2 -H8)) a2 H2) t2 H7))) H6)) (\lambda (H6: (land (subst0 i u t t2) (csubst0 i u -c c2))).(land_ind (subst0 i u t t2) (csubst0 i u c c2) (arity g c2 t2 a2) -(\lambda (H7: (subst0 i u t t2)).(\lambda (H8: (csubst0 i u c -c2)).(arity_repl g c2 t2 a1 (H1 d1 u i H3 c2 t2 (fsubst0_both i u c t t2 H7 -c2 H8)) a2 H2))) H6)) H5))))))))))))))))) c1 t1 a H))))). -(* COMMENTS -Initial nodes: 20387 -END *) - -theorem arity_subst0: - \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (a: A).((arity g c -t1 a) \to (\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead -d (Bind Abbr) u)) \to (\forall (t2: T).((subst0 i u t1 t2) \to (arity g c t2 -a))))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (a: A).(\lambda (H: -(arity g c t1 a)).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (t2: T).(\lambda (H1: -(subst0 i u t1 t2)).(arity_fsubst0 g c t1 a H d u i H0 c t2 (fsubst0_snd i u -c t1 t2 H1)))))))))))). -(* COMMENTS -Initial nodes: 89 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/asucc/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/asucc/defs.ma deleted file mode 100644 index 2f1af0279..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/asucc/defs.ma +++ /dev/null @@ -1,28 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/A/defs.ma". - -include "Basic-1/G/defs.ma". - -definition asucc: - G \to (A \to A) -\def - let rec asucc (g: G) (l: A) on l: A \def (match l with [(ASort n0 n) -\Rightarrow (match n0 with [O \Rightarrow (ASort O (next g n)) | (S h) -\Rightarrow (ASort h n)]) | (AHead a1 a2) \Rightarrow (AHead a1 (asucc g -a2))]) in asucc. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/asucc/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/asucc/fwd.ma deleted file mode 100644 index 61fcb5799..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/asucc/fwd.ma +++ /dev/null @@ -1,99 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/asucc/defs.ma". - -theorem asucc_gen_sort: - \forall (g: G).(\forall (h: nat).(\forall (n: nat).(\forall (a: A).((eq A -(ASort h n) (asucc g a)) \to (ex_2 nat nat (\lambda (h0: nat).(\lambda (n0: -nat).(eq A a (ASort h0 n0))))))))) -\def - \lambda (g: G).(\lambda (h: nat).(\lambda (n: nat).(\lambda (a: A).(A_ind -(\lambda (a0: A).((eq A (ASort h n) (asucc g a0)) \to (ex_2 nat nat (\lambda -(h0: nat).(\lambda (n0: nat).(eq A a0 (ASort h0 n0))))))) (\lambda (n0: -nat).(\lambda (n1: nat).(\lambda (H: (eq A (ASort h n) (asucc g (ASort n0 -n1)))).(let H0 \def (f_equal A A (\lambda (e: A).e) (ASort h n) (match n0 -with [O \Rightarrow (ASort O (next g n1)) | (S h0) \Rightarrow (ASort h0 -n1)]) H) in (ex_2_intro nat nat (\lambda (h0: nat).(\lambda (n2: nat).(eq A -(ASort n0 n1) (ASort h0 n2)))) n0 n1 (refl_equal A (ASort n0 n1))))))) -(\lambda (a0: A).(\lambda (_: (((eq A (ASort h n) (asucc g a0)) \to (ex_2 nat -nat (\lambda (h0: nat).(\lambda (n0: nat).(eq A a0 (ASort h0 -n0)))))))).(\lambda (a1: A).(\lambda (_: (((eq A (ASort h n) (asucc g a1)) -\to (ex_2 nat nat (\lambda (h0: nat).(\lambda (n0: nat).(eq A a1 (ASort h0 -n0)))))))).(\lambda (H1: (eq A (ASort h n) (asucc g (AHead a0 a1)))).(let H2 -\def (eq_ind A (ASort h n) (\lambda (ee: A).(match ee in A return (\lambda -(_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow -False])) I (asucc g (AHead a0 a1)) H1) in (False_ind (ex_2 nat nat (\lambda -(h0: nat).(\lambda (n0: nat).(eq A (AHead a0 a1) (ASort h0 n0))))) H2))))))) -a)))). -(* COMMENTS -Initial nodes: 317 -END *) - -theorem asucc_gen_head: - \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a: A).((eq A -(AHead a1 a2) (asucc g a)) \to (ex2 A (\lambda (a0: A).(eq A a (AHead a1 -a0))) (\lambda (a0: A).(eq A a2 (asucc g a0)))))))) -\def - \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a: A).(A_ind -(\lambda (a0: A).((eq A (AHead a1 a2) (asucc g a0)) \to (ex2 A (\lambda (a3: -A).(eq A a0 (AHead a1 a3))) (\lambda (a3: A).(eq A a2 (asucc g a3)))))) -(\lambda (n: nat).(\lambda (n0: nat).(\lambda (H: (eq A (AHead a1 a2) (asucc -g (ASort n n0)))).(nat_ind (\lambda (n1: nat).((eq A (AHead a1 a2) (asucc g -(ASort n1 n0))) \to (ex2 A (\lambda (a0: A).(eq A (ASort n1 n0) (AHead a1 -a0))) (\lambda (a0: A).(eq A a2 (asucc g a0)))))) (\lambda (H0: (eq A (AHead -a1 a2) (asucc g (ASort O n0)))).(let H1 \def (eq_ind A (AHead a1 a2) (\lambda -(ee: A).(match ee in A return (\lambda (_: A).Prop) with [(ASort _ _) -\Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort O (next g n0)) -H0) in (False_ind (ex2 A (\lambda (a0: A).(eq A (ASort O n0) (AHead a1 a0))) -(\lambda (a0: A).(eq A a2 (asucc g a0)))) H1))) (\lambda (n1: nat).(\lambda -(_: (((eq A (AHead a1 a2) (asucc g (ASort n1 n0))) \to (ex2 A (\lambda (a0: -A).(eq A (ASort n1 n0) (AHead a1 a0))) (\lambda (a0: A).(eq A a2 (asucc g -a0))))))).(\lambda (H0: (eq A (AHead a1 a2) (asucc g (ASort (S n1) -n0)))).(let H1 \def (eq_ind A (AHead a1 a2) (\lambda (ee: A).(match ee in A -return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ -_) \Rightarrow True])) I (ASort n1 n0) H0) in (False_ind (ex2 A (\lambda (a0: -A).(eq A (ASort (S n1) n0) (AHead a1 a0))) (\lambda (a0: A).(eq A a2 (asucc g -a0)))) H1))))) n H)))) (\lambda (a0: A).(\lambda (H: (((eq A (AHead a1 a2) -(asucc g a0)) \to (ex2 A (\lambda (a3: A).(eq A a0 (AHead a1 a3))) (\lambda -(a3: A).(eq A a2 (asucc g a3))))))).(\lambda (a3: A).(\lambda (H0: (((eq A -(AHead a1 a2) (asucc g a3)) \to (ex2 A (\lambda (a4: A).(eq A a3 (AHead a1 -a4))) (\lambda (a4: A).(eq A a2 (asucc g a4))))))).(\lambda (H1: (eq A (AHead -a1 a2) (asucc g (AHead a0 a3)))).(let H2 \def (f_equal A A (\lambda (e: -A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a1 | -(AHead a4 _) \Rightarrow a4])) (AHead a1 a2) (AHead a0 (asucc g a3)) H1) in -((let H3 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: -A).A) with [(ASort _ _) \Rightarrow a2 | (AHead _ a4) \Rightarrow a4])) -(AHead a1 a2) (AHead a0 (asucc g a3)) H1) in (\lambda (H4: (eq A a1 a0)).(let -H5 \def (eq_ind_r A a0 (\lambda (a4: A).((eq A (AHead a1 a2) (asucc g a4)) -\to (ex2 A (\lambda (a5: A).(eq A a4 (AHead a1 a5))) (\lambda (a5: A).(eq A -a2 (asucc g a5)))))) H a1 H4) in (eq_ind A a1 (\lambda (a4: A).(ex2 A -(\lambda (a5: A).(eq A (AHead a4 a3) (AHead a1 a5))) (\lambda (a5: A).(eq A -a2 (asucc g a5))))) (let H6 \def (eq_ind A a2 (\lambda (a4: A).((eq A (AHead -a1 a4) (asucc g a3)) \to (ex2 A (\lambda (a5: A).(eq A a3 (AHead a1 a5))) -(\lambda (a5: A).(eq A a4 (asucc g a5)))))) H0 (asucc g a3) H3) in (let H7 -\def (eq_ind A a2 (\lambda (a4: A).((eq A (AHead a1 a4) (asucc g a1)) \to -(ex2 A (\lambda (a5: A).(eq A a1 (AHead a1 a5))) (\lambda (a5: A).(eq A a4 -(asucc g a5)))))) H5 (asucc g a3) H3) in (eq_ind_r A (asucc g a3) (\lambda -(a4: A).(ex2 A (\lambda (a5: A).(eq A (AHead a1 a3) (AHead a1 a5))) (\lambda -(a5: A).(eq A a4 (asucc g a5))))) (ex_intro2 A (\lambda (a4: A).(eq A (AHead -a1 a3) (AHead a1 a4))) (\lambda (a4: A).(eq A (asucc g a3) (asucc g a4))) a3 -(refl_equal A (AHead a1 a3)) (refl_equal A (asucc g a3))) a2 H3))) a0 H4)))) -H2))))))) a)))). -(* COMMENTS -Initial nodes: 957 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/cimp/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/cimp/defs.ma deleted file mode 100644 index b45b64b59..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/cimp/defs.ma +++ /dev/null @@ -1,25 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/getl/defs.ma". - -definition cimp: - C \to (C \to Prop) -\def - \lambda (c1: C).(\lambda (c2: C).(\forall (b: B).(\forall (d1: C).(\forall -(w: T).(\forall (h: nat).((getl h c1 (CHead d1 (Bind b) w)) \to (ex C -(\lambda (d2: C).(getl h c2 (CHead d2 (Bind b) w)))))))))). - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/cimp/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/cimp/props.ma deleted file mode 100644 index 0834a7afe..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/cimp/props.ma +++ /dev/null @@ -1,139 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/cimp/defs.ma". - -include "Basic-1/getl/getl.ma". - -theorem cimp_flat_sx: - \forall (f: F).(\forall (c: C).(\forall (v: T).(cimp (CHead c (Flat f) v) -c))) -\def - \lambda (f: F).(\lambda (c: C).(\lambda (v: T).(\lambda (b: B).(\lambda (d1: -C).(\lambda (w: T).(\lambda (h: nat).(\lambda (H: (getl h (CHead c (Flat f) -v) (CHead d1 (Bind b) w))).(nat_ind (\lambda (n: nat).((getl n (CHead c (Flat -f) v) (CHead d1 (Bind b) w)) \to (ex C (\lambda (d2: C).(getl n c (CHead d2 -(Bind b) w)))))) (\lambda (H0: (getl O (CHead c (Flat f) v) (CHead d1 (Bind -b) w))).(ex_intro C (\lambda (d2: C).(getl O c (CHead d2 (Bind b) w))) d1 -(getl_intro O c (CHead d1 (Bind b) w) c (drop_refl c) (clear_gen_flat f c -(CHead d1 (Bind b) w) v (getl_gen_O (CHead c (Flat f) v) (CHead d1 (Bind b) -w) H0))))) (\lambda (h0: nat).(\lambda (_: (((getl h0 (CHead c (Flat f) v) -(CHead d1 (Bind b) w)) \to (ex C (\lambda (d2: C).(getl h0 c (CHead d2 (Bind -b) w))))))).(\lambda (H0: (getl (S h0) (CHead c (Flat f) v) (CHead d1 (Bind -b) w))).(ex_intro C (\lambda (d2: C).(getl (S h0) c (CHead d2 (Bind b) w))) -d1 (getl_gen_S (Flat f) c (CHead d1 (Bind b) w) v h0 H0))))) h H)))))))). -(* COMMENTS -Initial nodes: 327 -END *) - -theorem cimp_flat_dx: - \forall (f: F).(\forall (c: C).(\forall (v: T).(cimp c (CHead c (Flat f) -v)))) -\def - \lambda (f: F).(\lambda (c: C).(\lambda (v: T).(\lambda (b: B).(\lambda (d1: -C).(\lambda (w: T).(\lambda (h: nat).(\lambda (H: (getl h c (CHead d1 (Bind -b) w))).(ex_intro C (\lambda (d2: C).(getl h (CHead c (Flat f) v) (CHead d2 -(Bind b) w))) d1 (getl_flat c (CHead d1 (Bind b) w) h H f v))))))))). -(* COMMENTS -Initial nodes: 83 -END *) - -theorem cimp_bind: - \forall (c1: C).(\forall (c2: C).((cimp c1 c2) \to (\forall (b: B).(\forall -(v: T).(cimp (CHead c1 (Bind b) v) (CHead c2 (Bind b) v)))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (H: ((\forall (b: B).(\forall (d1: -C).(\forall (w: T).(\forall (h: nat).((getl h c1 (CHead d1 (Bind b) w)) \to -(ex C (\lambda (d2: C).(getl h c2 (CHead d2 (Bind b) w))))))))))).(\lambda -(b: B).(\lambda (v: T).(\lambda (b0: B).(\lambda (d1: C).(\lambda (w: -T).(\lambda (h: nat).(\lambda (H0: (getl h (CHead c1 (Bind b) v) (CHead d1 -(Bind b0) w))).(nat_ind (\lambda (n: nat).((getl n (CHead c1 (Bind b) v) -(CHead d1 (Bind b0) w)) \to (ex C (\lambda (d2: C).(getl n (CHead c2 (Bind b) -v) (CHead d2 (Bind b0) w)))))) (\lambda (H1: (getl O (CHead c1 (Bind b) v) -(CHead d1 (Bind b0) w))).(let H2 \def (f_equal C C (\lambda (e: C).(match e -in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d1 | (CHead c _ _) -\Rightarrow c])) (CHead d1 (Bind b0) w) (CHead c1 (Bind b) v) (clear_gen_bind -b c1 (CHead d1 (Bind b0) w) v (getl_gen_O (CHead c1 (Bind b) v) (CHead d1 -(Bind b0) w) H1))) in ((let H3 \def (f_equal C B (\lambda (e: C).(match e in -C return (\lambda (_: C).B) with [(CSort _) \Rightarrow b0 | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b1) -\Rightarrow b1 | (Flat _) \Rightarrow b0])])) (CHead d1 (Bind b0) w) (CHead -c1 (Bind b) v) (clear_gen_bind b c1 (CHead d1 (Bind b0) w) v (getl_gen_O -(CHead c1 (Bind b) v) (CHead d1 (Bind b0) w) H1))) in ((let H4 \def (f_equal -C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) -\Rightarrow w | (CHead _ _ t) \Rightarrow t])) (CHead d1 (Bind b0) w) (CHead -c1 (Bind b) v) (clear_gen_bind b c1 (CHead d1 (Bind b0) w) v (getl_gen_O -(CHead c1 (Bind b) v) (CHead d1 (Bind b0) w) H1))) in (\lambda (H5: (eq B b0 -b)).(\lambda (_: (eq C d1 c1)).(eq_ind_r T v (\lambda (t: T).(ex C (\lambda -(d2: C).(getl O (CHead c2 (Bind b) v) (CHead d2 (Bind b0) t))))) (eq_ind_r B -b (\lambda (b1: B).(ex C (\lambda (d2: C).(getl O (CHead c2 (Bind b) v) -(CHead d2 (Bind b1) v))))) (ex_intro C (\lambda (d2: C).(getl O (CHead c2 -(Bind b) v) (CHead d2 (Bind b) v))) c2 (getl_refl b c2 v)) b0 H5) w H4)))) -H3)) H2))) (\lambda (h0: nat).(\lambda (_: (((getl h0 (CHead c1 (Bind b) v) -(CHead d1 (Bind b0) w)) \to (ex C (\lambda (d2: C).(getl h0 (CHead c2 (Bind -b) v) (CHead d2 (Bind b0) w))))))).(\lambda (H1: (getl (S h0) (CHead c1 (Bind -b) v) (CHead d1 (Bind b0) w))).(let H_x \def (H b0 d1 w (r (Bind b) h0) -(getl_gen_S (Bind b) c1 (CHead d1 (Bind b0) w) v h0 H1)) in (let H2 \def H_x -in (ex_ind C (\lambda (d2: C).(getl h0 c2 (CHead d2 (Bind b0) w))) (ex C -(\lambda (d2: C).(getl (S h0) (CHead c2 (Bind b) v) (CHead d2 (Bind b0) w)))) -(\lambda (x: C).(\lambda (H3: (getl h0 c2 (CHead x (Bind b0) w))).(ex_intro C -(\lambda (d2: C).(getl (S h0) (CHead c2 (Bind b) v) (CHead d2 (Bind b0) w))) -x (getl_head (Bind b) h0 c2 (CHead x (Bind b0) w) H3 v)))) H2)))))) h -H0)))))))))). -(* COMMENTS -Initial nodes: 817 -END *) - -theorem cimp_getl_conf: - \forall (c1: C).(\forall (c2: C).((cimp c1 c2) \to (\forall (b: B).(\forall -(d1: C).(\forall (w: T).(\forall (i: nat).((getl i c1 (CHead d1 (Bind b) w)) -\to (ex2 C (\lambda (d2: C).(cimp d1 d2)) (\lambda (d2: C).(getl i c2 (CHead -d2 (Bind b) w))))))))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (H: ((\forall (b: B).(\forall (d1: -C).(\forall (w: T).(\forall (h: nat).((getl h c1 (CHead d1 (Bind b) w)) \to -(ex C (\lambda (d2: C).(getl h c2 (CHead d2 (Bind b) w))))))))))).(\lambda -(b: B).(\lambda (d1: C).(\lambda (w: T).(\lambda (i: nat).(\lambda (H0: (getl -i c1 (CHead d1 (Bind b) w))).(let H_x \def (H b d1 w i H0) in (let H1 \def -H_x in (ex_ind C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind b) w))) (ex2 C -(\lambda (d2: C).(\forall (b0: B).(\forall (d3: C).(\forall (w0: T).(\forall -(h: nat).((getl h d1 (CHead d3 (Bind b0) w0)) \to (ex C (\lambda (d4: -C).(getl h d2 (CHead d4 (Bind b0) w0)))))))))) (\lambda (d2: C).(getl i c2 -(CHead d2 (Bind b) w)))) (\lambda (x: C).(\lambda (H2: (getl i c2 (CHead x -(Bind b) w))).(ex_intro2 C (\lambda (d2: C).(\forall (b0: B).(\forall (d3: -C).(\forall (w0: T).(\forall (h: nat).((getl h d1 (CHead d3 (Bind b0) w0)) -\to (ex C (\lambda (d4: C).(getl h d2 (CHead d4 (Bind b0) w0)))))))))) -(\lambda (d2: C).(getl i c2 (CHead d2 (Bind b) w))) x (\lambda (b0: -B).(\lambda (d0: C).(\lambda (w0: T).(\lambda (h: nat).(\lambda (H3: (getl h -d1 (CHead d0 (Bind b0) w0))).(let H_y \def (getl_trans (S h) c1 (CHead d1 -(Bind b) w) i H0) in (let H_x0 \def (H b0 d0 w0 (plus (S h) i) (H_y (CHead d0 -(Bind b0) w0) (getl_head (Bind b) h d1 (CHead d0 (Bind b0) w0) H3 w))) in -(let H4 \def H_x0 in (ex_ind C (\lambda (d2: C).(getl (S (plus h i)) c2 -(CHead d2 (Bind b0) w0))) (ex C (\lambda (d2: C).(getl h x (CHead d2 (Bind -b0) w0)))) (\lambda (x0: C).(\lambda (H5: (getl (S (plus h i)) c2 (CHead x0 -(Bind b0) w0))).(let H_y0 \def (getl_conf_le (S (plus h i)) (CHead x0 (Bind -b0) w0) c2 H5 (CHead x (Bind b) w) i H2) in (let H6 \def (refl_equal nat -(plus (S h) i)) in (let H7 \def (eq_ind nat (S (plus h i)) (\lambda (n: -nat).(getl (minus n i) (CHead x (Bind b) w) (CHead x0 (Bind b0) w0))) (H_y0 -(le_S i (plus h i) (le_plus_r h i))) (plus (S h) i) H6) in (let H8 \def -(eq_ind nat (minus (plus (S h) i) i) (\lambda (n: nat).(getl n (CHead x (Bind -b) w) (CHead x0 (Bind b0) w0))) H7 (S h) (minus_plus_r (S h) i)) in (ex_intro -C (\lambda (d2: C).(getl h x (CHead d2 (Bind b0) w0))) x0 (getl_gen_S (Bind -b) x (CHead x0 (Bind b0) w0) w h H8)))))))) H4))))))))) H2))) H1)))))))))). -(* COMMENTS -Initial nodes: 673 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/clear/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/clear/defs.ma deleted file mode 100644 index 30607a4fc..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/clear/defs.ma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/C/defs.ma". - -inductive clear: C \to (C \to Prop) \def -| clear_bind: \forall (b: B).(\forall (e: C).(\forall (u: T).(clear (CHead e -(Bind b) u) (CHead e (Bind b) u)))) -| clear_flat: \forall (e: C).(\forall (c: C).((clear e c) \to (\forall (f: -F).(\forall (u: T).(clear (CHead e (Flat f) u) c))))). - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/clear/drop.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/clear/drop.ma deleted file mode 100644 index aae3fedc0..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/clear/drop.ma +++ /dev/null @@ -1,181 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/clear/fwd.ma". - -include "Basic-1/drop/fwd.ma". - -theorem drop_clear: - \forall (c1: C).(\forall (c2: C).(\forall (i: nat).((drop (S i) O c1 c2) \to -(ex2_3 B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear c1 (CHead -e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e -c2)))))))) -\def - \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (i: -nat).((drop (S i) O c c2) \to (ex2_3 B C T (\lambda (b: B).(\lambda (e: -C).(\lambda (v: T).(clear c (CHead e (Bind b) v))))) (\lambda (_: B).(\lambda -(e: C).(\lambda (_: T).(drop i O e c2))))))))) (\lambda (n: nat).(\lambda -(c2: C).(\lambda (i: nat).(\lambda (H: (drop (S i) O (CSort n) c2)).(and3_ind -(eq C c2 (CSort n)) (eq nat (S i) O) (eq nat O O) (ex2_3 B C T (\lambda (b: -B).(\lambda (e: C).(\lambda (v: T).(clear (CSort n) (CHead e (Bind b) v))))) -(\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2))))) (\lambda -(_: (eq C c2 (CSort n))).(\lambda (H1: (eq nat (S i) O)).(\lambda (_: (eq nat -O O)).(let H3 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee in nat -return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow -True])) I O H1) in (False_ind (ex2_3 B C T (\lambda (b: B).(\lambda (e: -C).(\lambda (v: T).(clear (CSort n) (CHead e (Bind b) v))))) (\lambda (_: -B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2))))) H3))))) (drop_gen_sort -n (S i) O c2 H)))))) (\lambda (c: C).(\lambda (H: ((\forall (c2: C).(\forall -(i: nat).((drop (S i) O c c2) \to (ex2_3 B C T (\lambda (b: B).(\lambda (e: -C).(\lambda (v: T).(clear c (CHead e (Bind b) v))))) (\lambda (_: B).(\lambda -(e: C).(\lambda (_: T).(drop i O e c2)))))))))).(\lambda (k: K).(\lambda (t: -T).(\lambda (c2: C).(\lambda (i: nat).(\lambda (H0: (drop (S i) O (CHead c k -t) c2)).(K_ind (\lambda (k0: K).((drop (r k0 i) O c c2) \to (ex2_3 B C T -(\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear (CHead c k0 t) (CHead -e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e -c2))))))) (\lambda (b: B).(\lambda (H1: (drop (r (Bind b) i) O c -c2)).(ex2_3_intro B C T (\lambda (b0: B).(\lambda (e: C).(\lambda (v: -T).(clear (CHead c (Bind b) t) (CHead e (Bind b0) v))))) (\lambda (_: -B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2)))) b c t (clear_bind b c -t) H1))) (\lambda (f: F).(\lambda (H1: (drop (r (Flat f) i) O c c2)).(let H2 -\def (H c2 i H1) in (ex2_3_ind B C T (\lambda (b: B).(\lambda (e: C).(\lambda -(v: T).(clear c (CHead e (Bind b) v))))) (\lambda (_: B).(\lambda (e: -C).(\lambda (_: T).(drop i O e c2)))) (ex2_3 B C T (\lambda (b: B).(\lambda -(e: C).(\lambda (v: T).(clear (CHead c (Flat f) t) (CHead e (Bind b) v))))) -(\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2))))) (\lambda -(x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H3: (clear c (CHead x1 -(Bind x0) x2))).(\lambda (H4: (drop i O x1 c2)).(ex2_3_intro B C T (\lambda -(b: B).(\lambda (e: C).(\lambda (v: T).(clear (CHead c (Flat f) t) (CHead e -(Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e -c2)))) x0 x1 x2 (clear_flat c (CHead x1 (Bind x0) x2) H3 f t) H4)))))) H2)))) -k (drop_gen_drop k c c2 t i H0))))))))) c1). -(* COMMENTS -Initial nodes: 770 -END *) - -theorem drop_clear_O: - \forall (b: B).(\forall (c: C).(\forall (e1: C).(\forall (u: T).((clear c -(CHead e1 (Bind b) u)) \to (\forall (e2: C).(\forall (i: nat).((drop i O e1 -e2) \to (drop (S i) O c e2)))))))) -\def - \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e1: -C).(\forall (u: T).((clear c0 (CHead e1 (Bind b) u)) \to (\forall (e2: -C).(\forall (i: nat).((drop i O e1 e2) \to (drop (S i) O c0 e2)))))))) -(\lambda (n: nat).(\lambda (e1: C).(\lambda (u: T).(\lambda (H: (clear (CSort -n) (CHead e1 (Bind b) u))).(\lambda (e2: C).(\lambda (i: nat).(\lambda (_: -(drop i O e1 e2)).(clear_gen_sort (CHead e1 (Bind b) u) n H (drop (S i) O -(CSort n) e2))))))))) (\lambda (c0: C).(\lambda (H: ((\forall (e1: -C).(\forall (u: T).((clear c0 (CHead e1 (Bind b) u)) \to (\forall (e2: -C).(\forall (i: nat).((drop i O e1 e2) \to (drop (S i) O c0 -e2))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e1: C).(\lambda (u: -T).(\lambda (H0: (clear (CHead c0 k t) (CHead e1 (Bind b) u))).(\lambda (e2: -C).(\lambda (i: nat).(\lambda (H1: (drop i O e1 e2)).(K_ind (\lambda (k0: -K).((clear (CHead c0 k0 t) (CHead e1 (Bind b) u)) \to (drop (S i) O (CHead c0 -k0 t) e2))) (\lambda (b0: B).(\lambda (H2: (clear (CHead c0 (Bind b0) t) -(CHead e1 (Bind b) u))).(let H3 \def (f_equal C C (\lambda (e: C).(match e in -C return (\lambda (_: C).C) with [(CSort _) \Rightarrow e1 | (CHead c1 _ _) -\Rightarrow c1])) (CHead e1 (Bind b) u) (CHead c0 (Bind b0) t) -(clear_gen_bind b0 c0 (CHead e1 (Bind b) u) t H2)) in ((let H4 \def (f_equal -C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with [(CSort _) -\Rightarrow b | (CHead _ k0 _) \Rightarrow (match k0 in K return (\lambda (_: -K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow b])])) (CHead e1 -(Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e1 (Bind b) -u) t H2)) in ((let H5 \def (f_equal C T (\lambda (e: C).(match e in C return -(\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow -t0])) (CHead e1 (Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 -(CHead e1 (Bind b) u) t H2)) in (\lambda (H6: (eq B b b0)).(\lambda (H7: (eq -C e1 c0)).(let H8 \def (eq_ind C e1 (\lambda (c1: C).(drop i O c1 e2)) H1 c0 -H7) in (eq_ind B b (\lambda (b1: B).(drop (S i) O (CHead c0 (Bind b1) t) e2)) -(drop_drop (Bind b) i c0 e2 H8 t) b0 H6))))) H4)) H3)))) (\lambda (f: -F).(\lambda (H2: (clear (CHead c0 (Flat f) t) (CHead e1 (Bind b) -u))).(drop_drop (Flat f) i c0 e2 (H e1 u (clear_gen_flat f c0 (CHead e1 (Bind -b) u) t H2) e2 i H1) t))) k H0))))))))))) c)). -(* COMMENTS -Initial nodes: 619 -END *) - -theorem drop_clear_S: - \forall (x2: C).(\forall (x1: C).(\forall (h: nat).(\forall (d: nat).((drop -h (S d) x1 x2) \to (\forall (b: B).(\forall (c2: C).(\forall (u: T).((clear -x2 (CHead c2 (Bind b) u)) \to (ex2 C (\lambda (c1: C).(clear x1 (CHead c1 -(Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2))))))))))) -\def - \lambda (x2: C).(C_ind (\lambda (c: C).(\forall (x1: C).(\forall (h: -nat).(\forall (d: nat).((drop h (S d) x1 c) \to (\forall (b: B).(\forall (c2: -C).(\forall (u: T).((clear c (CHead c2 (Bind b) u)) \to (ex2 C (\lambda (c1: -C).(clear x1 (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 -c2)))))))))))) (\lambda (n: nat).(\lambda (x1: C).(\lambda (h: nat).(\lambda -(d: nat).(\lambda (_: (drop h (S d) x1 (CSort n))).(\lambda (b: B).(\lambda -(c2: C).(\lambda (u: T).(\lambda (H0: (clear (CSort n) (CHead c2 (Bind b) -u))).(clear_gen_sort (CHead c2 (Bind b) u) n H0 (ex2 C (\lambda (c1: -C).(clear x1 (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 -c2))))))))))))) (\lambda (c: C).(\lambda (H: ((\forall (x1: C).(\forall (h: -nat).(\forall (d: nat).((drop h (S d) x1 c) \to (\forall (b: B).(\forall (c2: -C).(\forall (u: T).((clear c (CHead c2 (Bind b) u)) \to (ex2 C (\lambda (c1: -C).(clear x1 (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 -c2))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (x1: C).(\lambda (h: -nat).(\lambda (d: nat).(\lambda (H0: (drop h (S d) x1 (CHead c k -t))).(\lambda (b: B).(\lambda (c2: C).(\lambda (u: T).(\lambda (H1: (clear -(CHead c k t) (CHead c2 (Bind b) u))).(ex2_ind C (\lambda (e: C).(eq C x1 -(CHead e k (lift h (r k d) t)))) (\lambda (e: C).(drop h (r k d) e c)) (ex2 C -(\lambda (c1: C).(clear x1 (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: -C).(drop h d c1 c2))) (\lambda (x: C).(\lambda (H2: (eq C x1 (CHead x k (lift -h (r k d) t)))).(\lambda (H3: (drop h (r k d) x c)).(eq_ind_r C (CHead x k -(lift h (r k d) t)) (\lambda (c0: C).(ex2 C (\lambda (c1: C).(clear c0 (CHead -c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2)))) (K_ind -(\lambda (k0: K).((clear (CHead c k0 t) (CHead c2 (Bind b) u)) \to ((drop h -(r k0 d) x c) \to (ex2 C (\lambda (c1: C).(clear (CHead x k0 (lift h (r k0 d) -t)) (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2)))))) -(\lambda (b0: B).(\lambda (H4: (clear (CHead c (Bind b0) t) (CHead c2 (Bind -b) u))).(\lambda (H5: (drop h (r (Bind b0) d) x c)).(let H6 \def (f_equal C C -(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) (CHead c2 (Bind b) u) -(CHead c (Bind b0) t) (clear_gen_bind b0 c (CHead c2 (Bind b) u) t H4)) in -((let H7 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: -C).B) with [(CSort _) \Rightarrow b | (CHead _ k0 _) \Rightarrow (match k0 in -K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) -\Rightarrow b])])) (CHead c2 (Bind b) u) (CHead c (Bind b0) t) -(clear_gen_bind b0 c (CHead c2 (Bind b) u) t H4)) in ((let H8 \def (f_equal C -T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) -\Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead c2 (Bind b) u) (CHead -c (Bind b0) t) (clear_gen_bind b0 c (CHead c2 (Bind b) u) t H4)) in (\lambda -(H9: (eq B b b0)).(\lambda (H10: (eq C c2 c)).(eq_ind_r T t (\lambda (t0: -T).(ex2 C (\lambda (c1: C).(clear (CHead x (Bind b0) (lift h (r (Bind b0) d) -t)) (CHead c1 (Bind b) (lift h d t0)))) (\lambda (c1: C).(drop h d c1 c2)))) -(eq_ind_r C c (\lambda (c0: C).(ex2 C (\lambda (c1: C).(clear (CHead x (Bind -b0) (lift h (r (Bind b0) d) t)) (CHead c1 (Bind b) (lift h d t)))) (\lambda -(c1: C).(drop h d c1 c0)))) (eq_ind_r B b0 (\lambda (b1: B).(ex2 C (\lambda -(c1: C).(clear (CHead x (Bind b0) (lift h (r (Bind b0) d) t)) (CHead c1 (Bind -b1) (lift h d t)))) (\lambda (c1: C).(drop h d c1 c)))) (ex_intro2 C (\lambda -(c1: C).(clear (CHead x (Bind b0) (lift h (r (Bind b0) d) t)) (CHead c1 (Bind -b0) (lift h d t)))) (\lambda (c1: C).(drop h d c1 c)) x (clear_bind b0 x -(lift h d t)) H5) b H9) c2 H10) u H8)))) H7)) H6))))) (\lambda (f: -F).(\lambda (H4: (clear (CHead c (Flat f) t) (CHead c2 (Bind b) u))).(\lambda -(H5: (drop h (r (Flat f) d) x c)).(let H6 \def (H x h d H5 b c2 u -(clear_gen_flat f c (CHead c2 (Bind b) u) t H4)) in (ex2_ind C (\lambda (c1: -C).(clear x (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 -c2)) (ex2 C (\lambda (c1: C).(clear (CHead x (Flat f) (lift h (r (Flat f) d) -t)) (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2))) -(\lambda (x0: C).(\lambda (H7: (clear x (CHead x0 (Bind b) (lift h d -u)))).(\lambda (H8: (drop h d x0 c2)).(ex_intro2 C (\lambda (c1: C).(clear -(CHead x (Flat f) (lift h (r (Flat f) d) t)) (CHead c1 (Bind b) (lift h d -u)))) (\lambda (c1: C).(drop h d c1 c2)) x0 (clear_flat x (CHead x0 (Bind b) -(lift h d u)) H7 f (lift h (r (Flat f) d) t)) H8)))) H6))))) k H1 H3) x1 -H2)))) (drop_gen_skip_r c x1 t h d k H0)))))))))))))) x2). -(* COMMENTS -Initial nodes: 1449 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/clear/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/clear/fwd.ma deleted file mode 100644 index d64ff77be..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/clear/fwd.ma +++ /dev/null @@ -1,164 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/clear/defs.ma". - -theorem clear_gen_sort: - \forall (x: C).(\forall (n: nat).((clear (CSort n) x) \to (\forall (P: -Prop).P))) -\def - \lambda (x: C).(\lambda (n: nat).(\lambda (H: (clear (CSort n) x)).(\lambda -(P: Prop).(insert_eq C (CSort n) (\lambda (c: C).(clear c x)) (\lambda (_: -C).P) (\lambda (y: C).(\lambda (H0: (clear y x)).(clear_ind (\lambda (c: -C).(\lambda (_: C).((eq C c (CSort n)) \to P))) (\lambda (b: B).(\lambda (e: -C).(\lambda (u: T).(\lambda (H1: (eq C (CHead e (Bind b) u) (CSort n))).(let -H2 \def (eq_ind C (CHead e (Bind b) u) (\lambda (ee: C).(match ee in C return -(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) -\Rightarrow True])) I (CSort n) H1) in (False_ind P H2)))))) (\lambda (e: -C).(\lambda (c: C).(\lambda (_: (clear e c)).(\lambda (_: (((eq C e (CSort -n)) \to P))).(\lambda (f: F).(\lambda (u: T).(\lambda (H3: (eq C (CHead e -(Flat f) u) (CSort n))).(let H4 \def (eq_ind C (CHead e (Flat f) u) (\lambda -(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H3) in -(False_ind P H4))))))))) y x H0))) H)))). -(* COMMENTS -Initial nodes: 215 -END *) - -theorem clear_gen_bind: - \forall (b: B).(\forall (e: C).(\forall (x: C).(\forall (u: T).((clear -(CHead e (Bind b) u) x) \to (eq C x (CHead e (Bind b) u)))))) -\def - \lambda (b: B).(\lambda (e: C).(\lambda (x: C).(\lambda (u: T).(\lambda (H: -(clear (CHead e (Bind b) u) x)).(insert_eq C (CHead e (Bind b) u) (\lambda -(c: C).(clear c x)) (\lambda (c: C).(eq C x c)) (\lambda (y: C).(\lambda (H0: -(clear y x)).(clear_ind (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead e -(Bind b) u)) \to (eq C c0 c)))) (\lambda (b0: B).(\lambda (e0: C).(\lambda -(u0: T).(\lambda (H1: (eq C (CHead e0 (Bind b0) u0) (CHead e (Bind b) -u))).(let H2 \def (f_equal C C (\lambda (e1: C).(match e1 in C return -(\lambda (_: C).C) with [(CSort _) \Rightarrow e0 | (CHead c _ _) \Rightarrow -c])) (CHead e0 (Bind b0) u0) (CHead e (Bind b) u) H1) in ((let H3 \def -(f_equal C B (\lambda (e1: C).(match e1 in C return (\lambda (_: C).B) with -[(CSort _) \Rightarrow b0 | (CHead _ k _) \Rightarrow (match k in K return -(\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow -b0])])) (CHead e0 (Bind b0) u0) (CHead e (Bind b) u) H1) in ((let H4 \def -(f_equal C T (\lambda (e1: C).(match e1 in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead e0 (Bind -b0) u0) (CHead e (Bind b) u) H1) in (\lambda (H5: (eq B b0 b)).(\lambda (H6: -(eq C e0 e)).(eq_ind_r T u (\lambda (t: T).(eq C (CHead e0 (Bind b0) t) -(CHead e0 (Bind b0) t))) (eq_ind_r C e (\lambda (c: C).(eq C (CHead c (Bind -b0) u) (CHead c (Bind b0) u))) (eq_ind_r B b (\lambda (b1: B).(eq C (CHead e -(Bind b1) u) (CHead e (Bind b1) u))) (refl_equal C (CHead e (Bind b) u)) b0 -H5) e0 H6) u0 H4)))) H3)) H2)))))) (\lambda (e0: C).(\lambda (c: C).(\lambda -(_: (clear e0 c)).(\lambda (_: (((eq C e0 (CHead e (Bind b) u)) \to (eq C c -e0)))).(\lambda (f: F).(\lambda (u0: T).(\lambda (H3: (eq C (CHead e0 (Flat -f) u0) (CHead e (Bind b) u))).(let H4 \def (eq_ind C (CHead e0 (Flat f) u0) -(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (CHead e (Bind b) u) H3) in (False_ind (eq C c (CHead e0 (Flat f) -u0)) H4))))))))) y x H0))) H))))). -(* COMMENTS -Initial nodes: 525 -END *) - -theorem clear_gen_flat: - \forall (f: F).(\forall (e: C).(\forall (x: C).(\forall (u: T).((clear -(CHead e (Flat f) u) x) \to (clear e x))))) -\def - \lambda (f: F).(\lambda (e: C).(\lambda (x: C).(\lambda (u: T).(\lambda (H: -(clear (CHead e (Flat f) u) x)).(insert_eq C (CHead e (Flat f) u) (\lambda -(c: C).(clear c x)) (\lambda (_: C).(clear e x)) (\lambda (y: C).(\lambda -(H0: (clear y x)).(clear_ind (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead -e (Flat f) u)) \to (clear e c0)))) (\lambda (b: B).(\lambda (e0: C).(\lambda -(u0: T).(\lambda (H1: (eq C (CHead e0 (Bind b) u0) (CHead e (Flat f) -u))).(let H2 \def (eq_ind C (CHead e0 (Bind b) u0) (\lambda (ee: C).(match ee -in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead -_ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) -\Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead e (Flat f) u) H1) -in (False_ind (clear e (CHead e0 (Bind b) u0)) H2)))))) (\lambda (e0: -C).(\lambda (c: C).(\lambda (H1: (clear e0 c)).(\lambda (H2: (((eq C e0 -(CHead e (Flat f) u)) \to (clear e c)))).(\lambda (f0: F).(\lambda (u0: -T).(\lambda (H3: (eq C (CHead e0 (Flat f0) u0) (CHead e (Flat f) u))).(let H4 -\def (f_equal C C (\lambda (e1: C).(match e1 in C return (\lambda (_: C).C) -with [(CSort _) \Rightarrow e0 | (CHead c0 _ _) \Rightarrow c0])) (CHead e0 -(Flat f0) u0) (CHead e (Flat f) u) H3) in ((let H5 \def (f_equal C F (\lambda -(e1: C).(match e1 in C return (\lambda (_: C).F) with [(CSort _) \Rightarrow -f0 | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).F) with -[(Bind _) \Rightarrow f0 | (Flat f1) \Rightarrow f1])])) (CHead e0 (Flat f0) -u0) (CHead e (Flat f) u) H3) in ((let H6 \def (f_equal C T (\lambda (e1: -C).(match e1 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | -(CHead _ _ t) \Rightarrow t])) (CHead e0 (Flat f0) u0) (CHead e (Flat f) u) -H3) in (\lambda (_: (eq F f0 f)).(\lambda (H8: (eq C e0 e)).(let H9 \def -(eq_ind C e0 (\lambda (c0: C).((eq C c0 (CHead e (Flat f) u)) \to (clear e -c))) H2 e H8) in (let H10 \def (eq_ind C e0 (\lambda (c0: C).(clear c0 c)) H1 -e H8) in H10))))) H5)) H4))))))))) y x H0))) H))))). -(* COMMENTS -Initial nodes: 453 -END *) - -theorem clear_gen_flat_r: - \forall (f: F).(\forall (x: C).(\forall (e: C).(\forall (u: T).((clear x -(CHead e (Flat f) u)) \to (\forall (P: Prop).P))))) -\def - \lambda (f: F).(\lambda (x: C).(\lambda (e: C).(\lambda (u: T).(\lambda (H: -(clear x (CHead e (Flat f) u))).(\lambda (P: Prop).(insert_eq C (CHead e -(Flat f) u) (\lambda (c: C).(clear x c)) (\lambda (_: C).P) (\lambda (y: -C).(\lambda (H0: (clear x y)).(clear_ind (\lambda (_: C).(\lambda (c0: -C).((eq C c0 (CHead e (Flat f) u)) \to P))) (\lambda (b: B).(\lambda (e0: -C).(\lambda (u0: T).(\lambda (H1: (eq C (CHead e0 (Bind b) u0) (CHead e (Flat -f) u))).(let H2 \def (eq_ind C (CHead e0 (Bind b) u0) (\lambda (ee: C).(match -ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | -(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead e (Flat -f) u) H1) in (False_ind P H2)))))) (\lambda (e0: C).(\lambda (c: C).(\lambda -(H1: (clear e0 c)).(\lambda (H2: (((eq C c (CHead e (Flat f) u)) \to -P))).(\lambda (_: F).(\lambda (_: T).(\lambda (H3: (eq C c (CHead e (Flat f) -u))).(let H4 \def (eq_ind C c (\lambda (c0: C).((eq C c0 (CHead e (Flat f) -u)) \to P)) H2 (CHead e (Flat f) u) H3) in (let H5 \def (eq_ind C c (\lambda -(c0: C).(clear e0 c0)) H1 (CHead e (Flat f) u) H3) in (H4 (refl_equal C -(CHead e (Flat f) u)))))))))))) x y H0))) H)))))). -(* COMMENTS -Initial nodes: 303 -END *) - -theorem clear_gen_all: - \forall (c1: C).(\forall (c2: C).((clear c1 c2) \to (ex_3 B C T (\lambda (b: -B).(\lambda (e: C).(\lambda (u: T).(eq C c2 (CHead e (Bind b) u)))))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (clear c1 c2)).(clear_ind -(\lambda (_: C).(\lambda (c0: C).(ex_3 B C T (\lambda (b: B).(\lambda (e: -C).(\lambda (u: T).(eq C c0 (CHead e (Bind b) u)))))))) (\lambda (b: -B).(\lambda (e: C).(\lambda (u: T).(ex_3_intro B C T (\lambda (b0: -B).(\lambda (e0: C).(\lambda (u0: T).(eq C (CHead e (Bind b) u) (CHead e0 -(Bind b0) u0))))) b e u (refl_equal C (CHead e (Bind b) u)))))) (\lambda (e: -C).(\lambda (c: C).(\lambda (H0: (clear e c)).(\lambda (H1: (ex_3 B C T -(\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(eq C c (CHead e0 (Bind b) -u))))))).(\lambda (_: F).(\lambda (_: T).(let H2 \def H1 in (ex_3_ind B C T -(\lambda (b: B).(\lambda (e0: C).(\lambda (u0: T).(eq C c (CHead e0 (Bind b) -u0))))) (ex_3 B C T (\lambda (b: B).(\lambda (e0: C).(\lambda (u0: T).(eq C c -(CHead e0 (Bind b) u0)))))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: -T).(\lambda (H3: (eq C c (CHead x1 (Bind x0) x2))).(let H4 \def (eq_ind C c -(\lambda (c0: C).(clear e c0)) H0 (CHead x1 (Bind x0) x2) H3) in (eq_ind_r C -(CHead x1 (Bind x0) x2) (\lambda (c0: C).(ex_3 B C T (\lambda (b: B).(\lambda -(e0: C).(\lambda (u0: T).(eq C c0 (CHead e0 (Bind b) u0))))))) (ex_3_intro B -C T (\lambda (b: B).(\lambda (e0: C).(\lambda (u0: T).(eq C (CHead x1 (Bind -x0) x2) (CHead e0 (Bind b) u0))))) x0 x1 x2 (refl_equal C (CHead x1 (Bind x0) -x2))) c H3)))))) H2)))))))) c1 c2 H))). -(* COMMENTS -Initial nodes: 381 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/clear/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/clear/props.ma deleted file mode 100644 index 68e250d76..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/clear/props.ma +++ /dev/null @@ -1,152 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/clear/fwd.ma". - -theorem clear_clear: - \forall (c1: C).(\forall (c2: C).((clear c1 c2) \to (clear c2 c2))) -\def - \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).((clear c c2) \to -(clear c2 c2)))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (H: (clear -(CSort n) c2)).(clear_gen_sort c2 n H (clear c2 c2))))) (\lambda (c: -C).(\lambda (H: ((\forall (c2: C).((clear c c2) \to (clear c2 -c2))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (H0: (clear -(CHead c k t) c2)).(K_ind (\lambda (k0: K).((clear (CHead c k0 t) c2) \to -(clear c2 c2))) (\lambda (b: B).(\lambda (H1: (clear (CHead c (Bind b) t) -c2)).(eq_ind_r C (CHead c (Bind b) t) (\lambda (c0: C).(clear c0 c0)) -(clear_bind b c t) c2 (clear_gen_bind b c c2 t H1)))) (\lambda (f: -F).(\lambda (H1: (clear (CHead c (Flat f) t) c2)).(H c2 (clear_gen_flat f c -c2 t H1)))) k H0))))))) c1). -(* COMMENTS -Initial nodes: 199 -END *) - -theorem clear_mono: - \forall (c: C).(\forall (c1: C).((clear c c1) \to (\forall (c2: C).((clear c -c2) \to (eq C c1 c2))))) -\def - \lambda (c: C).(C_ind (\lambda (c0: C).(\forall (c1: C).((clear c0 c1) \to -(\forall (c2: C).((clear c0 c2) \to (eq C c1 c2)))))) (\lambda (n: -nat).(\lambda (c1: C).(\lambda (_: (clear (CSort n) c1)).(\lambda (c2: -C).(\lambda (H0: (clear (CSort n) c2)).(clear_gen_sort c2 n H0 (eq C c1 -c2))))))) (\lambda (c0: C).(\lambda (H: ((\forall (c1: C).((clear c0 c1) \to -(\forall (c2: C).((clear c0 c2) \to (eq C c1 c2))))))).(\lambda (k: -K).(\lambda (t: T).(\lambda (c1: C).(\lambda (H0: (clear (CHead c0 k t) -c1)).(\lambda (c2: C).(\lambda (H1: (clear (CHead c0 k t) c2)).(K_ind -(\lambda (k0: K).((clear (CHead c0 k0 t) c1) \to ((clear (CHead c0 k0 t) c2) -\to (eq C c1 c2)))) (\lambda (b: B).(\lambda (H2: (clear (CHead c0 (Bind b) -t) c1)).(\lambda (H3: (clear (CHead c0 (Bind b) t) c2)).(eq_ind_r C (CHead c0 -(Bind b) t) (\lambda (c3: C).(eq C c1 c3)) (eq_ind_r C (CHead c0 (Bind b) t) -(\lambda (c3: C).(eq C c3 (CHead c0 (Bind b) t))) (refl_equal C (CHead c0 -(Bind b) t)) c1 (clear_gen_bind b c0 c1 t H2)) c2 (clear_gen_bind b c0 c2 t -H3))))) (\lambda (f: F).(\lambda (H2: (clear (CHead c0 (Flat f) t) -c1)).(\lambda (H3: (clear (CHead c0 (Flat f) t) c2)).(H c1 (clear_gen_flat f -c0 c1 t H2) c2 (clear_gen_flat f c0 c2 t H3))))) k H0 H1))))))))) c). -(* COMMENTS -Initial nodes: 357 -END *) - -theorem clear_trans: - \forall (c1: C).(\forall (c: C).((clear c1 c) \to (\forall (c2: C).((clear c -c2) \to (clear c1 c2))))) -\def - \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c0: C).((clear c c0) \to -(\forall (c2: C).((clear c0 c2) \to (clear c c2)))))) (\lambda (n: -nat).(\lambda (c: C).(\lambda (H: (clear (CSort n) c)).(\lambda (c2: -C).(\lambda (_: (clear c c2)).(clear_gen_sort c n H (clear (CSort n) -c2))))))) (\lambda (c: C).(\lambda (H: ((\forall (c0: C).((clear c c0) \to -(\forall (c2: C).((clear c0 c2) \to (clear c c2))))))).(\lambda (k: -K).(\lambda (t: T).(\lambda (c0: C).(\lambda (H0: (clear (CHead c k t) -c0)).(\lambda (c2: C).(\lambda (H1: (clear c0 c2)).(K_ind (\lambda (k0: -K).((clear (CHead c k0 t) c0) \to (clear (CHead c k0 t) c2))) (\lambda (b: -B).(\lambda (H2: (clear (CHead c (Bind b) t) c0)).(let H3 \def (eq_ind C c0 -(\lambda (c3: C).(clear c3 c2)) H1 (CHead c (Bind b) t) (clear_gen_bind b c -c0 t H2)) in (eq_ind_r C (CHead c (Bind b) t) (\lambda (c3: C).(clear (CHead -c (Bind b) t) c3)) (clear_bind b c t) c2 (clear_gen_bind b c c2 t H3))))) -(\lambda (f: F).(\lambda (H2: (clear (CHead c (Flat f) t) c0)).(clear_flat c -c2 (H c0 (clear_gen_flat f c c0 t H2) c2 H1) f t))) k H0))))))))) c1). -(* COMMENTS -Initial nodes: 299 -END *) - -theorem clear_ctail: - \forall (b: B).(\forall (c1: C).(\forall (c2: C).(\forall (u2: T).((clear c1 -(CHead c2 (Bind b) u2)) \to (\forall (k: K).(\forall (u1: T).(clear (CTail k -u1 c1) (CHead (CTail k u1 c2) (Bind b) u2)))))))) -\def - \lambda (b: B).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: -C).(\forall (u2: T).((clear c (CHead c2 (Bind b) u2)) \to (\forall (k: -K).(\forall (u1: T).(clear (CTail k u1 c) (CHead (CTail k u1 c2) (Bind b) -u2)))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (u2: T).(\lambda (H: -(clear (CSort n) (CHead c2 (Bind b) u2))).(\lambda (k: K).(\lambda (u1: -T).(K_ind (\lambda (k0: K).(clear (CHead (CSort n) k0 u1) (CHead (CTail k0 u1 -c2) (Bind b) u2))) (\lambda (b0: B).(clear_gen_sort (CHead c2 (Bind b) u2) n -H (clear (CHead (CSort n) (Bind b0) u1) (CHead (CTail (Bind b0) u1 c2) (Bind -b) u2)))) (\lambda (f: F).(clear_gen_sort (CHead c2 (Bind b) u2) n H (clear -(CHead (CSort n) (Flat f) u1) (CHead (CTail (Flat f) u1 c2) (Bind b) u2)))) -k))))))) (\lambda (c: C).(\lambda (H: ((\forall (c2: C).(\forall (u2: -T).((clear c (CHead c2 (Bind b) u2)) \to (\forall (k: K).(\forall (u1: -T).(clear (CTail k u1 c) (CHead (CTail k u1 c2) (Bind b) u2))))))))).(\lambda -(k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (u2: T).(\lambda (H0: (clear -(CHead c k t) (CHead c2 (Bind b) u2))).(\lambda (k0: K).(\lambda (u1: -T).(K_ind (\lambda (k1: K).((clear (CHead c k1 t) (CHead c2 (Bind b) u2)) \to -(clear (CHead (CTail k0 u1 c) k1 t) (CHead (CTail k0 u1 c2) (Bind b) u2)))) -(\lambda (b0: B).(\lambda (H1: (clear (CHead c (Bind b0) t) (CHead c2 (Bind -b) u2))).(let H2 \def (f_equal C C (\lambda (e: C).(match e in C return -(\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ _) -\Rightarrow c0])) (CHead c2 (Bind b) u2) (CHead c (Bind b0) t) -(clear_gen_bind b0 c (CHead c2 (Bind b) u2) t H1)) in ((let H3 \def (f_equal -C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with [(CSort _) -\Rightarrow b | (CHead _ k1 _) \Rightarrow (match k1 in K return (\lambda (_: -K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow b])])) (CHead c2 -(Bind b) u2) (CHead c (Bind b0) t) (clear_gen_bind b0 c (CHead c2 (Bind b) -u2) t H1)) in ((let H4 \def (f_equal C T (\lambda (e: C).(match e in C return -(\lambda (_: C).T) with [(CSort _) \Rightarrow u2 | (CHead _ _ t0) -\Rightarrow t0])) (CHead c2 (Bind b) u2) (CHead c (Bind b0) t) -(clear_gen_bind b0 c (CHead c2 (Bind b) u2) t H1)) in (\lambda (H5: (eq B b -b0)).(\lambda (H6: (eq C c2 c)).(eq_ind_r T t (\lambda (t0: T).(clear (CHead -(CTail k0 u1 c) (Bind b0) t) (CHead (CTail k0 u1 c2) (Bind b) t0))) (eq_ind_r -C c (\lambda (c0: C).(clear (CHead (CTail k0 u1 c) (Bind b0) t) (CHead (CTail -k0 u1 c0) (Bind b) t))) (eq_ind B b (\lambda (b1: B).(clear (CHead (CTail k0 -u1 c) (Bind b1) t) (CHead (CTail k0 u1 c) (Bind b) t))) (clear_bind b (CTail -k0 u1 c) t) b0 H5) c2 H6) u2 H4)))) H3)) H2)))) (\lambda (f: F).(\lambda (H1: -(clear (CHead c (Flat f) t) (CHead c2 (Bind b) u2))).(clear_flat (CTail k0 u1 -c) (CHead (CTail k0 u1 c2) (Bind b) u2) (H c2 u2 (clear_gen_flat f c (CHead -c2 (Bind b) u2) t H1) k0 u1) f t))) k H0)))))))))) c1)). -(* COMMENTS -Initial nodes: 819 -END *) - -theorem clear_cle: - \forall (c1: C).(\forall (c2: C).((clear c1 c2) \to (cle c2 c1))) -\def - \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).((clear c c2) \to -(le (cweight c2) (cweight c))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda -(H: (clear (CSort n) c2)).(clear_gen_sort c2 n H (le (cweight c2) O))))) -(\lambda (c: C).(\lambda (H: ((\forall (c2: C).((clear c c2) \to (le (cweight -c2) (cweight c)))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: -C).(\lambda (H0: (clear (CHead c k t) c2)).(K_ind (\lambda (k0: K).((clear -(CHead c k0 t) c2) \to (le (cweight c2) (plus (cweight c) (tweight t))))) -(\lambda (b: B).(\lambda (H1: (clear (CHead c (Bind b) t) c2)).(eq_ind_r C -(CHead c (Bind b) t) (\lambda (c0: C).(le (cweight c0) (plus (cweight c) -(tweight t)))) (le_n (plus (cweight c) (tweight t))) c2 (clear_gen_bind b c -c2 t H1)))) (\lambda (f: F).(\lambda (H1: (clear (CHead c (Flat f) t) -c2)).(le_plus_trans (cweight c2) (cweight c) (tweight t) (H c2 -(clear_gen_flat f c c2 t H1))))) k H0))))))) c1). -(* COMMENTS -Initial nodes: 247 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/clen/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/clen/defs.ma deleted file mode 100644 index 07539ffad..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/clen/defs.ma +++ /dev/null @@ -1,26 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/C/defs.ma". - -include "Basic-1/s/defs.ma". - -definition clen: - C \to nat -\def - let rec clen (c: C) on c: nat \def (match c with [(CSort _) \Rightarrow O | -(CHead c0 k _) \Rightarrow (s k (clen c0))]) in clen. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/clen/getl.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/clen/getl.ma deleted file mode 100644 index af8a96cf5..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/clen/getl.ma +++ /dev/null @@ -1,361 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/clen/defs.ma". - -include "Basic-1/getl/props.ma". - -theorem getl_ctail_clen: - \forall (b: B).(\forall (t: T).(\forall (c: C).(ex nat (\lambda (n: -nat).(getl (clen c) (CTail (Bind b) t c) (CHead (CSort n) (Bind b) t)))))) -\def - \lambda (b: B).(\lambda (t: T).(\lambda (c: C).(C_ind (\lambda (c0: C).(ex -nat (\lambda (n: nat).(getl (clen c0) (CTail (Bind b) t c0) (CHead (CSort n) -(Bind b) t))))) (\lambda (n: nat).(ex_intro nat (\lambda (n0: nat).(getl O -(CHead (CSort n) (Bind b) t) (CHead (CSort n0) (Bind b) t))) n (getl_refl b -(CSort n) t))) (\lambda (c0: C).(\lambda (H: (ex nat (\lambda (n: nat).(getl -(clen c0) (CTail (Bind b) t c0) (CHead (CSort n) (Bind b) t))))).(\lambda (k: -K).(\lambda (t0: T).(let H0 \def H in (ex_ind nat (\lambda (n: nat).(getl -(clen c0) (CTail (Bind b) t c0) (CHead (CSort n) (Bind b) t))) (ex nat -(\lambda (n: nat).(getl (s k (clen c0)) (CHead (CTail (Bind b) t c0) k t0) -(CHead (CSort n) (Bind b) t)))) (\lambda (x: nat).(\lambda (H1: (getl (clen -c0) (CTail (Bind b) t c0) (CHead (CSort x) (Bind b) t))).(K_ind (\lambda (k0: -K).(ex nat (\lambda (n: nat).(getl (s k0 (clen c0)) (CHead (CTail (Bind b) t -c0) k0 t0) (CHead (CSort n) (Bind b) t))))) (\lambda (b0: B).(ex_intro nat -(\lambda (n: nat).(getl (S (clen c0)) (CHead (CTail (Bind b) t c0) (Bind b0) -t0) (CHead (CSort n) (Bind b) t))) x (getl_head (Bind b0) (clen c0) (CTail -(Bind b) t c0) (CHead (CSort x) (Bind b) t) H1 t0))) (\lambda (f: -F).(ex_intro nat (\lambda (n: nat).(getl (clen c0) (CHead (CTail (Bind b) t -c0) (Flat f) t0) (CHead (CSort n) (Bind b) t))) x (getl_flat (CTail (Bind b) -t c0) (CHead (CSort x) (Bind b) t) (clen c0) H1 f t0))) k))) H0)))))) c))). -(* COMMENTS -Initial nodes: 459 -END *) - -theorem getl_gen_tail: - \forall (k: K).(\forall (b: B).(\forall (u1: T).(\forall (u2: T).(\forall -(c2: C).(\forall (c1: C).(\forall (i: nat).((getl i (CTail k u1 c1) (CHead c2 -(Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) -(\lambda (e: C).(getl i c1 (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: -nat).(eq nat i (clen c1))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: -nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n)))))))))))) -\def - \lambda (k: K).(\lambda (b: B).(\lambda (u1: T).(\lambda (u2: T).(\lambda -(c2: C).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (i: nat).((getl i -(CTail k u1 c) (CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C -c2 (CTail k u1 e))) (\lambda (e: C).(getl i c (CHead e (Bind b) u2)))) (ex4 -nat (\lambda (_: nat).(eq nat i (clen c))) (\lambda (_: nat).(eq K k (Bind -b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort -n)))))))) (\lambda (n: nat).(\lambda (i: nat).(nat_ind (\lambda (n0: -nat).((getl n0 (CTail k u1 (CSort n)) (CHead c2 (Bind b) u2)) \to (or (ex2 C -(\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl n0 (CSort n) -(CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat n0 (clen (CSort -n)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) -(\lambda (n1: nat).(eq C c2 (CSort n1))))))) (\lambda (H: (getl O (CHead -(CSort n) k u1) (CHead c2 (Bind b) u2))).(K_ind (\lambda (k0: K).((clear -(CHead (CSort n) k0 u1) (CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda (e: -C).(eq C c2 (CTail k0 u1 e))) (\lambda (e: C).(getl O (CSort n) (CHead e -(Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O O)) (\lambda (_: -nat).(eq K k0 (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: -nat).(eq C c2 (CSort n0))))))) (\lambda (b0: B).(\lambda (H0: (clear (CHead -(CSort n) (Bind b0) u1) (CHead c2 (Bind b) u2))).(let H1 \def (f_equal C C -(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow c2 | (CHead c _ _) \Rightarrow c])) (CHead c2 (Bind b) u2) (CHead -(CSort n) (Bind b0) u1) (clear_gen_bind b0 (CSort n) (CHead c2 (Bind b) u2) -u1 H0)) in ((let H2 \def (f_equal C B (\lambda (e: C).(match e in C return -(\lambda (_: C).B) with [(CSort _) \Rightarrow b | (CHead _ k0 _) \Rightarrow -(match k0 in K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | -(Flat _) \Rightarrow b])])) (CHead c2 (Bind b) u2) (CHead (CSort n) (Bind b0) -u1) (clear_gen_bind b0 (CSort n) (CHead c2 (Bind b) u2) u1 H0)) in ((let H3 -\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c2 -(Bind b) u2) (CHead (CSort n) (Bind b0) u1) (clear_gen_bind b0 (CSort n) -(CHead c2 (Bind b) u2) u1 H0)) in (\lambda (H4: (eq B b b0)).(\lambda (H5: -(eq C c2 (CSort n))).(eq_ind_r C (CSort n) (\lambda (c: C).(or (ex2 C -(\lambda (e: C).(eq C c (CTail (Bind b0) u1 e))) (\lambda (e: C).(getl O -(CSort n) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O O)) -(\lambda (_: nat).(eq K (Bind b0) (Bind b))) (\lambda (_: nat).(eq T u1 u2)) -(\lambda (n0: nat).(eq C c (CSort n0)))))) (eq_ind_r T u1 (\lambda (t: T).(or -(ex2 C (\lambda (e: C).(eq C (CSort n) (CTail (Bind b0) u1 e))) (\lambda (e: -C).(getl O (CSort n) (CHead e (Bind b) t)))) (ex4 nat (\lambda (_: nat).(eq -nat O O)) (\lambda (_: nat).(eq K (Bind b0) (Bind b))) (\lambda (_: nat).(eq -T u1 t)) (\lambda (n0: nat).(eq C (CSort n) (CSort n0)))))) (eq_ind_r B b0 -(\lambda (b1: B).(or (ex2 C (\lambda (e: C).(eq C (CSort n) (CTail (Bind b0) -u1 e))) (\lambda (e: C).(getl O (CSort n) (CHead e (Bind b1) u1)))) (ex4 nat -(\lambda (_: nat).(eq nat O O)) (\lambda (_: nat).(eq K (Bind b0) (Bind b1))) -(\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: nat).(eq C (CSort n) (CSort -n0)))))) (or_intror (ex2 C (\lambda (e: C).(eq C (CSort n) (CTail (Bind b0) -u1 e))) (\lambda (e: C).(getl O (CSort n) (CHead e (Bind b0) u1)))) (ex4 nat -(\lambda (_: nat).(eq nat O O)) (\lambda (_: nat).(eq K (Bind b0) (Bind b0))) -(\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: nat).(eq C (CSort n) (CSort -n0)))) (ex4_intro nat (\lambda (_: nat).(eq nat O O)) (\lambda (_: nat).(eq K -(Bind b0) (Bind b0))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: nat).(eq -C (CSort n) (CSort n0))) n (refl_equal nat O) (refl_equal K (Bind b0)) -(refl_equal T u1) (refl_equal C (CSort n)))) b H4) u2 H3) c2 H5)))) H2)) -H1)))) (\lambda (f: F).(\lambda (H0: (clear (CHead (CSort n) (Flat f) u1) -(CHead c2 (Bind b) u2))).(clear_gen_sort (CHead c2 (Bind b) u2) n -(clear_gen_flat f (CSort n) (CHead c2 (Bind b) u2) u1 H0) (or (ex2 C (\lambda -(e: C).(eq C c2 (CTail (Flat f) u1 e))) (\lambda (e: C).(getl O (CSort n) -(CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O O)) (\lambda -(_: nat).(eq K (Flat f) (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda -(n0: nat).(eq C c2 (CSort n0)))))))) k (getl_gen_O (CHead (CSort n) k u1) -(CHead c2 (Bind b) u2) H))) (\lambda (n0: nat).(\lambda (_: (((getl n0 (CHead -(CSort n) k u1) (CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C -c2 (CTail k u1 e))) (\lambda (e: C).(getl n0 (CSort n) (CHead e (Bind b) -u2)))) (ex4 nat (\lambda (_: nat).(eq nat n0 O)) (\lambda (_: nat).(eq K k -(Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n1: nat).(eq C c2 (CSort -n1)))))))).(\lambda (H0: (getl (S n0) (CHead (CSort n) k u1) (CHead c2 (Bind -b) u2))).(getl_gen_sort n (r k n0) (CHead c2 (Bind b) u2) (getl_gen_S k -(CSort n) (CHead c2 (Bind b) u2) u1 n0 H0) (or (ex2 C (\lambda (e: C).(eq C -c2 (CTail k u1 e))) (\lambda (e: C).(getl (S n0) (CSort n) (CHead e (Bind b) -u2)))) (ex4 nat (\lambda (_: nat).(eq nat (S n0) O)) (\lambda (_: nat).(eq K -k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n1: nat).(eq C c2 -(CSort n1))))))))) i))) (\lambda (c: C).(\lambda (H: ((\forall (i: -nat).((getl i (CTail k u1 c) (CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda -(e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl i c (CHead e (Bind b) -u2)))) (ex4 nat (\lambda (_: nat).(eq nat i (clen c))) (\lambda (_: nat).(eq -K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 -(CSort n))))))))).(\lambda (k0: K).(\lambda (t: T).(\lambda (i: nat).(nat_ind -(\lambda (n: nat).((getl n (CTail k u1 (CHead c k0 t)) (CHead c2 (Bind b) -u2)) \to (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: -C).(getl n (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: -nat).(eq nat n (clen (CHead c k0 t)))) (\lambda (_: nat).(eq K k (Bind b))) -(\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0))))))) -(\lambda (H0: (getl O (CHead (CTail k u1 c) k0 t) (CHead c2 (Bind b) -u2))).(K_ind (\lambda (k1: K).((clear (CHead (CTail k u1 c) k1 t) (CHead c2 -(Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) -(\lambda (e: C).(getl O (CHead c k1 t) (CHead e (Bind b) u2)))) (ex4 nat -(\lambda (_: nat).(eq nat O (s k1 (clen c)))) (\lambda (_: nat).(eq K k (Bind -b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort -n))))))) (\lambda (b0: B).(\lambda (H1: (clear (CHead (CTail k u1 c) (Bind -b0) t) (CHead c2 (Bind b) u2))).(let H2 \def (f_equal C C (\lambda (e: -C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | -(CHead c0 _ _) \Rightarrow c0])) (CHead c2 (Bind b) u2) (CHead (CTail k u1 c) -(Bind b0) t) (clear_gen_bind b0 (CTail k u1 c) (CHead c2 (Bind b) u2) t H1)) -in ((let H3 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda -(_: C).B) with [(CSort _) \Rightarrow b | (CHead _ k1 _) \Rightarrow (match -k1 in K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) -\Rightarrow b])])) (CHead c2 (Bind b) u2) (CHead (CTail k u1 c) (Bind b0) t) -(clear_gen_bind b0 (CTail k u1 c) (CHead c2 (Bind b) u2) t H1)) in ((let H4 -\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow u2 | (CHead _ _ t0) \Rightarrow t0])) (CHead c2 -(Bind b) u2) (CHead (CTail k u1 c) (Bind b0) t) (clear_gen_bind b0 (CTail k -u1 c) (CHead c2 (Bind b) u2) t H1)) in (\lambda (H5: (eq B b b0)).(\lambda -(H6: (eq C c2 (CTail k u1 c))).(eq_ind T u2 (\lambda (t0: T).(or (ex2 C -(\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c -(Bind b0) t0) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O -(s (Bind b0) (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: -nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n)))))) (eq_ind B b -(\lambda (b1: B).(or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) -(\lambda (e: C).(getl O (CHead c (Bind b1) u2) (CHead e (Bind b) u2)))) (ex4 -nat (\lambda (_: nat).(eq nat O (s (Bind b1) (clen c)))) (\lambda (_: -nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq -C c2 (CSort n)))))) (let H7 \def (eq_ind C c2 (\lambda (c0: C).(\forall (i0: -nat).((getl i0 (CTail k u1 c) (CHead c0 (Bind b) u2)) \to (or (ex2 C (\lambda -(e: C).(eq C c0 (CTail k u1 e))) (\lambda (e: C).(getl i0 c (CHead e (Bind b) -u2)))) (ex4 nat (\lambda (_: nat).(eq nat i0 (clen c))) (\lambda (_: nat).(eq -K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c0 -(CSort n)))))))) H (CTail k u1 c) H6) in (eq_ind_r C (CTail k u1 c) (\lambda -(c0: C).(or (ex2 C (\lambda (e: C).(eq C c0 (CTail k u1 e))) (\lambda (e: -C).(getl O (CHead c (Bind b) u2) (CHead e (Bind b) u2)))) (ex4 nat (\lambda -(_: nat).(eq nat O (s (Bind b) (clen c)))) (\lambda (_: nat).(eq K k (Bind -b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c0 (CSort -n)))))) (or_introl (ex2 C (\lambda (e: C).(eq C (CTail k u1 c) (CTail k u1 -e))) (\lambda (e: C).(getl O (CHead c (Bind b) u2) (CHead e (Bind b) u2)))) -(ex4 nat (\lambda (_: nat).(eq nat O (s (Bind b) (clen c)))) (\lambda (_: -nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq -C (CTail k u1 c) (CSort n)))) (ex_intro2 C (\lambda (e: C).(eq C (CTail k u1 -c) (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Bind b) u2) (CHead e -(Bind b) u2))) c (refl_equal C (CTail k u1 c)) (getl_refl b c u2))) c2 H6)) -b0 H5) t H4)))) H3)) H2)))) (\lambda (f: F).(\lambda (H1: (clear (CHead -(CTail k u1 c) (Flat f) t) (CHead c2 (Bind b) u2))).(let H2 \def (H O -(getl_intro O (CTail k u1 c) (CHead c2 (Bind b) u2) (CTail k u1 c) (drop_refl -(CTail k u1 c)) (clear_gen_flat f (CTail k u1 c) (CHead c2 (Bind b) u2) t -H1))) in (or_ind (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda -(e: C).(getl O c (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat -O (clen c))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 -u2)) (\lambda (n: nat).(eq C c2 (CSort n)))) (or (ex2 C (\lambda (e: C).(eq C -c2 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e -(Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) -(\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda -(n: nat).(eq C c2 (CSort n))))) (\lambda (H3: (ex2 C (\lambda (e: C).(eq C c2 -(CTail k u1 e))) (\lambda (e: C).(getl O c (CHead e (Bind b) u2))))).(ex2_ind -C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl O c (CHead -e (Bind b) u2))) (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) -(\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind b) u2)))) (ex4 -nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq -K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 -(CSort n))))) (\lambda (x: C).(\lambda (H4: (eq C c2 (CTail k u1 -x))).(\lambda (H5: (getl O c (CHead x (Bind b) u2))).(eq_ind_r C (CTail k u1 -x) (\lambda (c0: C).(or (ex2 C (\lambda (e: C).(eq C c0 (CTail k u1 e))) -(\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind b) u2)))) (ex4 -nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq -K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c0 -(CSort n)))))) (or_introl (ex2 C (\lambda (e: C).(eq C (CTail k u1 x) (CTail -k u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind b) -u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) (\lambda -(_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: -nat).(eq C (CTail k u1 x) (CSort n)))) (ex_intro2 C (\lambda (e: C).(eq C -(CTail k u1 x) (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) -(CHead e (Bind b) u2))) x (refl_equal C (CTail k u1 x)) (getl_flat c (CHead x -(Bind b) u2) O H5 f t))) c2 H4)))) H3)) (\lambda (H3: (ex4 nat (\lambda (_: -nat).(eq nat O (clen c))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: -nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n))))).(ex4_ind nat -(\lambda (_: nat).(eq nat O (clen c))) (\lambda (_: nat).(eq K k (Bind b))) -(\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n))) (or -(ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl O -(CHead c (Flat f) t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq -nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda -(_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n))))) (\lambda (x0: -nat).(\lambda (H4: (eq nat O (clen c))).(\lambda (H5: (eq K k (Bind -b))).(\lambda (H6: (eq T u1 u2)).(\lambda (H7: (eq C c2 (CSort -x0))).(eq_ind_r C (CSort x0) (\lambda (c0: C).(or (ex2 C (\lambda (e: C).(eq -C c0 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e -(Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) -(\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda -(n: nat).(eq C c0 (CSort n)))))) (eq_ind T u1 (\lambda (t0: T).(or (ex2 C -(\lambda (e: C).(eq C (CSort x0) (CTail k u1 e))) (\lambda (e: C).(getl O -(CHead c (Flat f) t) (CHead e (Bind b) t0)))) (ex4 nat (\lambda (_: nat).(eq -nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda -(_: nat).(eq T u1 t0)) (\lambda (n: nat).(eq C (CSort x0) (CSort n)))))) -(eq_ind_r K (Bind b) (\lambda (k1: K).(or (ex2 C (\lambda (e: C).(eq C (CSort -x0) (CTail k1 u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e -(Bind b) u1)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) -(\lambda (_: nat).(eq K k1 (Bind b))) (\lambda (_: nat).(eq T u1 u1)) -(\lambda (n: nat).(eq C (CSort x0) (CSort n)))))) (or_intror (ex2 C (\lambda -(e: C).(eq C (CSort x0) (CTail (Bind b) u1 e))) (\lambda (e: C).(getl O -(CHead c (Flat f) t) (CHead e (Bind b) u1)))) (ex4 nat (\lambda (_: nat).(eq -nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq K (Bind b) (Bind b))) -(\lambda (_: nat).(eq T u1 u1)) (\lambda (n: nat).(eq C (CSort x0) (CSort -n)))) (ex4_intro nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) -(\lambda (_: nat).(eq K (Bind b) (Bind b))) (\lambda (_: nat).(eq T u1 u1)) -(\lambda (n: nat).(eq C (CSort x0) (CSort n))) x0 H4 (refl_equal K (Bind b)) -(refl_equal T u1) (refl_equal C (CSort x0)))) k H5) u2 H6) c2 H7)))))) H3)) -H2)))) k0 (getl_gen_O (CHead (CTail k u1 c) k0 t) (CHead c2 (Bind b) u2) -H0))) (\lambda (n: nat).(\lambda (H0: (((getl n (CHead (CTail k u1 c) k0 t) -(CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 -e))) (\lambda (e: C).(getl n (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat -(\lambda (_: nat).(eq nat n (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind -b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort -n0)))))))).(\lambda (H1: (getl (S n) (CHead (CTail k u1 c) k0 t) (CHead c2 -(Bind b) u2))).(let H_x \def (H (r k0 n) (getl_gen_S k0 (CTail k u1 c) (CHead -c2 (Bind b) u2) t n H1)) in (let H2 \def H_x in (or_ind (ex2 C (\lambda (e: -C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl (r k0 n) c (CHead e (Bind -b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat (r k0 n) (clen c))) (\lambda (_: -nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: -nat).(eq C c2 (CSort n0)))) (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 -e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 -nat (\lambda (_: nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K -k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 -(CSort n0))))) (\lambda (H3: (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) -(\lambda (e: C).(getl (r k0 n) c (CHead e (Bind b) u2))))).(ex2_ind C -(\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl (r k0 n) c -(CHead e (Bind b) u2))) (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) -(\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat -(\lambda (_: nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k -(Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort -n0))))) (\lambda (x: C).(\lambda (H4: (eq C c2 (CTail k u1 x))).(\lambda (H5: -(getl (r k0 n) c (CHead x (Bind b) u2))).(let H6 \def (eq_ind C c2 (\lambda -(c0: C).(getl (r k0 n) (CTail k u1 c) (CHead c0 (Bind b) u2))) (getl_gen_S k0 -(CTail k u1 c) (CHead c2 (Bind b) u2) t n H1) (CTail k u1 x) H4) in (let H7 -\def (eq_ind C c2 (\lambda (c0: C).((getl n (CHead (CTail k u1 c) k0 t) -(CHead c0 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c0 (CTail k u1 -e))) (\lambda (e: C).(getl n (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat -(\lambda (_: nat).(eq nat n (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind -b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c0 (CSort -n0))))))) H0 (CTail k u1 x) H4) in (eq_ind_r C (CTail k u1 x) (\lambda (c0: -C).(or (ex2 C (\lambda (e: C).(eq C c0 (CTail k u1 e))) (\lambda (e: C).(getl -(S n) (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq -nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: -nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c0 (CSort n0)))))) (or_introl -(ex2 C (\lambda (e: C).(eq C (CTail k u1 x) (CTail k u1 e))) (\lambda (e: -C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: -nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) -(\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C (CTail k u1 x) -(CSort n0)))) (ex_intro2 C (\lambda (e: C).(eq C (CTail k u1 x) (CTail k u1 -e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u2))) x -(refl_equal C (CTail k u1 x)) (getl_head k0 n c (CHead x (Bind b) u2) H5 t))) -c2 H4)))))) H3)) (\lambda (H3: (ex4 nat (\lambda (_: nat).(eq nat (r k0 n) -(clen c))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 -u2)) (\lambda (n0: nat).(eq C c2 (CSort n0))))).(ex4_ind nat (\lambda (_: -nat).(eq nat (r k0 n) (clen c))) (\lambda (_: nat).(eq K k (Bind b))) -(\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0))) (or -(ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl (S n) -(CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat (S -n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: -nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0))))) (\lambda (x0: -nat).(\lambda (H4: (eq nat (r k0 n) (clen c))).(\lambda (H5: (eq K k (Bind -b))).(\lambda (H6: (eq T u1 u2)).(\lambda (H7: (eq C c2 (CSort x0))).(let H8 -\def (eq_ind C c2 (\lambda (c0: C).(getl (r k0 n) (CTail k u1 c) (CHead c0 -(Bind b) u2))) (getl_gen_S k0 (CTail k u1 c) (CHead c2 (Bind b) u2) t n H1) -(CSort x0) H7) in (let H9 \def (eq_ind C c2 (\lambda (c0: C).((getl n (CHead -(CTail k u1 c) k0 t) (CHead c0 (Bind b) u2)) \to (or (ex2 C (\lambda (e: -C).(eq C c0 (CTail k u1 e))) (\lambda (e: C).(getl n (CHead c k0 t) (CHead e -(Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat n (s k0 (clen c)))) -(\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda -(n0: nat).(eq C c0 (CSort n0))))))) H0 (CSort x0) H7) in (eq_ind_r C (CSort -x0) (\lambda (c0: C).(or (ex2 C (\lambda (e: C).(eq C c0 (CTail k u1 e))) -(\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat -(\lambda (_: nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k -(Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c0 (CSort -n0)))))) (let H10 \def (eq_ind_r T u2 (\lambda (t0: T).((getl n (CHead (CTail -k u1 c) k0 t) (CHead (CSort x0) (Bind b) t0)) \to (or (ex2 C (\lambda (e: -C).(eq C (CSort x0) (CTail k u1 e))) (\lambda (e: C).(getl n (CHead c k0 t) -(CHead e (Bind b) t0)))) (ex4 nat (\lambda (_: nat).(eq nat n (s k0 (clen -c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 t0)) -(\lambda (n0: nat).(eq C (CSort x0) (CSort n0))))))) H9 u1 H6) in (let H11 -\def (eq_ind_r T u2 (\lambda (t0: T).(getl (r k0 n) (CTail k u1 c) (CHead -(CSort x0) (Bind b) t0))) H8 u1 H6) in (eq_ind T u1 (\lambda (t0: T).(or (ex2 -C (\lambda (e: C).(eq C (CSort x0) (CTail k u1 e))) (\lambda (e: C).(getl (S -n) (CHead c k0 t) (CHead e (Bind b) t0)))) (ex4 nat (\lambda (_: nat).(eq nat -(S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: -nat).(eq T u1 t0)) (\lambda (n0: nat).(eq C (CSort x0) (CSort n0)))))) (let -H12 \def (eq_ind K k (\lambda (k1: K).((getl n (CHead (CTail k1 u1 c) k0 t) -(CHead (CSort x0) (Bind b) u1)) \to (or (ex2 C (\lambda (e: C).(eq C (CSort -x0) (CTail k1 u1 e))) (\lambda (e: C).(getl n (CHead c k0 t) (CHead e (Bind -b) u1)))) (ex4 nat (\lambda (_: nat).(eq nat n (s k0 (clen c)))) (\lambda (_: -nat).(eq K k1 (Bind b))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: -nat).(eq C (CSort x0) (CSort n0))))))) H10 (Bind b) H5) in (let H13 \def -(eq_ind K k (\lambda (k1: K).(getl (r k0 n) (CTail k1 u1 c) (CHead (CSort x0) -(Bind b) u1))) H11 (Bind b) H5) in (eq_ind_r K (Bind b) (\lambda (k1: K).(or -(ex2 C (\lambda (e: C).(eq C (CSort x0) (CTail k1 u1 e))) (\lambda (e: -C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u1)))) (ex4 nat (\lambda (_: -nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k1 (Bind b))) -(\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: nat).(eq C (CSort x0) (CSort -n0)))))) (eq_ind nat (r k0 n) (\lambda (n0: nat).(or (ex2 C (\lambda (e: -C).(eq C (CSort x0) (CTail (Bind b) u1 e))) (\lambda (e: C).(getl (S n) -(CHead c k0 t) (CHead e (Bind b) u1)))) (ex4 nat (\lambda (_: nat).(eq nat (S -n) (s k0 n0))) (\lambda (_: nat).(eq K (Bind b) (Bind b))) (\lambda (_: -nat).(eq T u1 u1)) (\lambda (n1: nat).(eq C (CSort x0) (CSort n1)))))) -(eq_ind_r nat (S n) (\lambda (n0: nat).(or (ex2 C (\lambda (e: C).(eq C -(CSort x0) (CTail (Bind b) u1 e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) -(CHead e (Bind b) u1)))) (ex4 nat (\lambda (_: nat).(eq nat (S n) n0)) -(\lambda (_: nat).(eq K (Bind b) (Bind b))) (\lambda (_: nat).(eq T u1 u1)) -(\lambda (n1: nat).(eq C (CSort x0) (CSort n1)))))) (or_intror (ex2 C -(\lambda (e: C).(eq C (CSort x0) (CTail (Bind b) u1 e))) (\lambda (e: -C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u1)))) (ex4 nat (\lambda (_: -nat).(eq nat (S n) (S n))) (\lambda (_: nat).(eq K (Bind b) (Bind b))) -(\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: nat).(eq C (CSort x0) (CSort -n0)))) (ex4_intro nat (\lambda (_: nat).(eq nat (S n) (S n))) (\lambda (_: -nat).(eq K (Bind b) (Bind b))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: -nat).(eq C (CSort x0) (CSort n0))) x0 (refl_equal nat (S n)) (refl_equal K -(Bind b)) (refl_equal T u1) (refl_equal C (CSort x0)))) (s k0 (r k0 n)) (s_r -k0 n)) (clen c) H4) k H5))) u2 H6))) c2 H7)))))))) H3)) H2)))))) i)))))) -c1)))))). -(* COMMENTS -Initial nodes: 7489 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/cnt/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/cnt/defs.ma deleted file mode 100644 index d6ab61d78..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/cnt/defs.ma +++ /dev/null @@ -1,23 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/T/defs.ma". - -inductive cnt: T \to Prop \def -| cnt_sort: \forall (n: nat).(cnt (TSort n)) -| cnt_head: \forall (t: T).((cnt t) \to (\forall (k: K).(\forall (v: T).(cnt -(THead k v t))))). - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/cnt/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/cnt/props.ma deleted file mode 100644 index 7fcd315eb..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/cnt/props.ma +++ /dev/null @@ -1,37 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/cnt/defs.ma". - -include "Basic-1/lift/fwd.ma". - -theorem cnt_lift: - \forall (t: T).((cnt t) \to (\forall (i: nat).(\forall (d: nat).(cnt (lift i -d t))))) -\def - \lambda (t: T).(\lambda (H: (cnt t)).(cnt_ind (\lambda (t0: T).(\forall (i: -nat).(\forall (d: nat).(cnt (lift i d t0))))) (\lambda (n: nat).(\lambda (i: -nat).(\lambda (d: nat).(eq_ind_r T (TSort n) (\lambda (t0: T).(cnt t0)) -(cnt_sort n) (lift i d (TSort n)) (lift_sort n i d))))) (\lambda (t0: -T).(\lambda (_: (cnt t0)).(\lambda (H1: ((\forall (i: nat).(\forall (d: -nat).(cnt (lift i d t0)))))).(\lambda (k: K).(\lambda (v: T).(\lambda (i: -nat).(\lambda (d: nat).(eq_ind_r T (THead k (lift i d v) (lift i (s k d) t0)) -(\lambda (t1: T).(cnt t1)) (cnt_head (lift i (s k d) t0) (H1 i (s k d)) k -(lift i d v)) (lift i d (THead k v t0)) (lift_head k v t0 i d))))))))) t H)). -(* COMMENTS -Initial nodes: 191 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/arity.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/arity.ma deleted file mode 100644 index fe021cd0f..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/arity.ma +++ /dev/null @@ -1,336 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csuba/getl.ma". - -include "Basic-1/csuba/props.ma". - -include "Basic-1/arity/props.ma". - -include "Basic-1/csubv/getl.ma". - -theorem csuba_arity: - \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1 -t a) \to (\forall (c2: C).((csuba g c1 c2) \to (arity g c2 t a))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: -(arity g c1 t a)).(arity_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda (a0: -A).(\forall (c2: C).((csuba g c c2) \to (arity g c2 t0 a0)))))) (\lambda (c: -C).(\lambda (n: nat).(\lambda (c2: C).(\lambda (_: (csuba g c -c2)).(arity_sort g c2 n))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: -T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d (Bind Abbr) -u))).(\lambda (a0: A).(\lambda (_: (arity g d u a0)).(\lambda (H2: ((\forall -(c2: C).((csuba g d c2) \to (arity g c2 u a0))))).(\lambda (c2: C).(\lambda -(H3: (csuba g c c2)).(let H4 \def (csuba_getl_abbr g c d u i H0 c2 H3) in -(ex2_ind C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda -(d2: C).(csuba g d d2)) (arity g c2 (TLRef i) a0) (\lambda (x: C).(\lambda -(H5: (getl i c2 (CHead x (Bind Abbr) u))).(\lambda (H6: (csuba g d -x)).(arity_abbr g c2 x u i H5 a0 (H2 x H6))))) H4)))))))))))) (\lambda (c: -C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c -(CHead d (Bind Abst) u))).(\lambda (a0: A).(\lambda (H1: (arity g d u (asucc -g a0))).(\lambda (H2: ((\forall (c2: C).((csuba g d c2) \to (arity g c2 u -(asucc g a0)))))).(\lambda (c2: C).(\lambda (H3: (csuba g c c2)).(let H4 \def -(csuba_getl_abst g c d u i H0 c2 H3) in (or_ind (ex2 C (\lambda (d2: C).(getl -i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d d2))) (ex4_3 C T -A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 -(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d u (asucc -g a1))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2 -a1))))) (arity g c2 (TLRef i) a0) (\lambda (H5: (ex2 C (\lambda (d2: C).(getl -i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d d2)))).(ex2_ind C -(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d d2)) (arity g c2 (TLRef i) a0) (\lambda (x: C).(\lambda (H6: -(getl i c2 (CHead x (Bind Abst) u))).(\lambda (H7: (csuba g d x)).(arity_abst -g c2 x u i H6 a0 (H2 x H7))))) H5)) (\lambda (H5: (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d d2)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d u (asucc g a1))))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2 -a1)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a1: A).(arity g d u (asucc g a1))))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a1: A).(arity g d2 u2 a1)))) (arity g c2 (TLRef i) a0) (\lambda -(x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H6: (getl i c2 (CHead x0 -(Bind Abbr) x1))).(\lambda (_: (csuba g d x0)).(\lambda (H8: (arity g d u -(asucc g x2))).(\lambda (H9: (arity g x0 x1 x2)).(arity_repl g c2 (TLRef i) -x2 (arity_abbr g c2 x0 x1 i H6 x2 H9) a0 (asucc_inj g x2 a0 (arity_mono g d u -(asucc g x2) H8 (asucc g a0) H1)))))))))) H5)) H4)))))))))))) (\lambda (b: -B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c: C).(\lambda (u: -T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H2: ((\forall -(c2: C).((csuba g c c2) \to (arity g c2 u a1))))).(\lambda (t0: T).(\lambda -(a2: A).(\lambda (_: (arity g (CHead c (Bind b) u) t0 a2)).(\lambda (H4: -((\forall (c2: C).((csuba g (CHead c (Bind b) u) c2) \to (arity g c2 t0 -a2))))).(\lambda (c2: C).(\lambda (H5: (csuba g c c2)).(arity_bind g b H0 c2 -u a1 (H2 c2 H5) t0 a2 (H4 (CHead c2 (Bind b) u) (csuba_head g c c2 H5 (Bind -b) u)))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a1: -A).(\lambda (_: (arity g c u (asucc g a1))).(\lambda (H1: ((\forall (c2: -C).((csuba g c c2) \to (arity g c2 u (asucc g a1)))))).(\lambda (t0: -T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c (Bind Abst) u) t0 -a2)).(\lambda (H3: ((\forall (c2: C).((csuba g (CHead c (Bind Abst) u) c2) -\to (arity g c2 t0 a2))))).(\lambda (c2: C).(\lambda (H4: (csuba g c -c2)).(arity_head g c2 u a1 (H1 c2 H4) t0 a2 (H3 (CHead c2 (Bind Abst) u) -(csuba_head g c c2 H4 (Bind Abst) u)))))))))))))) (\lambda (c: C).(\lambda -(u: T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H1: -((\forall (c2: C).((csuba g c c2) \to (arity g c2 u a1))))).(\lambda (t0: -T).(\lambda (a2: A).(\lambda (_: (arity g c t0 (AHead a1 a2))).(\lambda (H3: -((\forall (c2: C).((csuba g c c2) \to (arity g c2 t0 (AHead a1 -a2)))))).(\lambda (c2: C).(\lambda (H4: (csuba g c c2)).(arity_appl g c2 u a1 -(H1 c2 H4) t0 a2 (H3 c2 H4))))))))))))) (\lambda (c: C).(\lambda (u: -T).(\lambda (a0: A).(\lambda (_: (arity g c u (asucc g a0))).(\lambda (H1: -((\forall (c2: C).((csuba g c c2) \to (arity g c2 u (asucc g -a0)))))).(\lambda (t0: T).(\lambda (_: (arity g c t0 a0)).(\lambda (H3: -((\forall (c2: C).((csuba g c c2) \to (arity g c2 t0 a0))))).(\lambda (c2: -C).(\lambda (H4: (csuba g c c2)).(arity_cast g c2 u a0 (H1 c2 H4) t0 (H3 c2 -H4)))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda (_: -(arity g c t0 a1)).(\lambda (H1: ((\forall (c2: C).((csuba g c c2) \to (arity -g c2 t0 a1))))).(\lambda (a2: A).(\lambda (H2: (leq g a1 a2)).(\lambda (c2: -C).(\lambda (H3: (csuba g c c2)).(arity_repl g c2 t0 a1 (H1 c2 H3) a2 -H2)))))))))) c1 t a H))))). -(* COMMENTS -Initial nodes: 1505 -END *) - -theorem csuba_arity_rev: - \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1 -t a) \to (\forall (c2: C).((csuba g c2 c1) \to ((csubv c2 c1) \to (arity g c2 -t a)))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: -(arity g c1 t a)).(arity_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda (a0: -A).(\forall (c2: C).((csuba g c2 c) \to ((csubv c2 c) \to (arity g c2 t0 -a0))))))) (\lambda (c: C).(\lambda (n: nat).(\lambda (c2: C).(\lambda (_: -(csuba g c2 c)).(\lambda (_: (csubv c2 c)).(arity_sort g c2 n)))))) (\lambda -(c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl -i c (CHead d (Bind Abbr) u))).(\lambda (a0: A).(\lambda (H1: (arity g d u -a0)).(\lambda (H2: ((\forall (c2: C).((csuba g c2 d) \to ((csubv c2 d) \to -(arity g c2 u a0)))))).(\lambda (c2: C).(\lambda (H3: (csuba g c2 -c)).(\lambda (H4: (csubv c2 c)).(let H_x \def (csuba_getl_abbr_rev g c d u i -H0 c2 H3) in (let H5 \def H_x in (or3_ind (ex2 C (\lambda (d2: C).(getl i c2 -(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d2 d))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -d)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2 -(asucc g a1))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d -u a1))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d)))) (arity -g c2 (TLRef i) a0) (\lambda (H6: (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 -(Bind Abbr) u))) (\lambda (d2: C).(csuba g d2 d)))).(ex2_ind C (\lambda (d2: -C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d2 d)) -(arity g c2 (TLRef i) a0) (\lambda (x: C).(\lambda (H7: (getl i c2 (CHead x -(Bind Abbr) u))).(\lambda (H8: (csuba g x d)).(let H_x0 \def (csubv_getl_conf -c2 c H4 Abbr x u i H7) in (let H9 \def H_x0 in (ex2_3_ind B C T (\lambda (_: -B).(\lambda (d2: C).(\lambda (_: T).(csubv x d2)))) (\lambda (b2: B).(\lambda -(d2: C).(\lambda (v2: T).(getl i c (CHead d2 (Bind b2) v2))))) (arity g c2 -(TLRef i) a0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda -(H10: (csubv x x1)).(\lambda (H11: (getl i c (CHead x1 (Bind x0) x2))).(let -H12 \def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c0: C).(getl i c c0)) H0 -(CHead x1 (Bind x0) x2) (getl_mono c (CHead d (Bind Abbr) u) i H0 (CHead x1 -(Bind x0) x2) H11)) in (let H13 \def (f_equal C C (\lambda (e: C).(match e in -C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) -\Rightarrow c0])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) (getl_mono -c (CHead d (Bind Abbr) u) i H0 (CHead x1 (Bind x0) x2) H11)) in ((let H14 -\def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) -with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k in K -return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow -Abbr])])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) (getl_mono c (CHead -d (Bind Abbr) u) i H0 (CHead x1 (Bind x0) x2) H11)) in ((let H15 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind -Abbr) u) (CHead x1 (Bind x0) x2) (getl_mono c (CHead d (Bind Abbr) u) i H0 -(CHead x1 (Bind x0) x2) H11)) in (\lambda (H16: (eq B Abbr x0)).(\lambda -(H17: (eq C d x1)).(let H18 \def (eq_ind_r T x2 (\lambda (t0: T).(getl i c -(CHead x1 (Bind x0) t0))) H12 u H15) in (let H19 \def (eq_ind_r C x1 (\lambda -(c0: C).(getl i c (CHead c0 (Bind x0) u))) H18 d H17) in (let H20 \def -(eq_ind_r C x1 (\lambda (c0: C).(csubv x c0)) H10 d H17) in (let H21 \def -(eq_ind_r B x0 (\lambda (b: B).(getl i c (CHead d (Bind b) u))) H19 Abbr H16) -in (arity_abbr g c2 x u i H7 a0 (H2 x H8 H20))))))))) H14)) H13)))))))) -H9)))))) H6)) (\lambda (H6: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2 (asucc g a1))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d u a1)))))).(ex4_3_ind C T -A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 -(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d2 d)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2 -(asucc g a1))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d -u a1)))) (arity g c2 (TLRef i) a0) (\lambda (x0: C).(\lambda (x1: T).(\lambda -(x2: A).(\lambda (H7: (getl i c2 (CHead x0 (Bind Abst) x1))).(\lambda (_: -(csuba g x0 d)).(\lambda (H9: (arity g x0 x1 (asucc g x2))).(\lambda (H10: -(arity g d u x2)).(arity_repl g c2 (TLRef i) x2 (arity_abst g c2 x0 x1 i H7 -x2 H9) a0 (arity_mono g d u x2 H10 a0 H1))))))))) H6)) (\lambda (H6: (ex2_2 C -T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d))))).(ex2_2_ind C T (\lambda -(d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d))) (arity g c2 (TLRef i) a0) (\lambda -(x0: C).(\lambda (x1: T).(\lambda (H7: (getl i c2 (CHead x0 (Bind Void) -x1))).(\lambda (_: (csuba g x0 d)).(let H_x0 \def (csubv_getl_conf_void c2 c -H4 x0 x1 i H7) in (let H9 \def H_x0 in (ex2_2_ind C T (\lambda (d2: -C).(\lambda (_: T).(csubv x0 d2))) (\lambda (d2: C).(\lambda (v2: T).(getl i -c (CHead d2 (Bind Void) v2)))) (arity g c2 (TLRef i) a0) (\lambda (x2: -C).(\lambda (x3: T).(\lambda (_: (csubv x0 x2)).(\lambda (H11: (getl i c -(CHead x2 (Bind Void) x3))).(let H12 \def (eq_ind C (CHead d (Bind Abbr) u) -(\lambda (c0: C).(getl i c c0)) H0 (CHead x2 (Bind Void) x3) (getl_mono c -(CHead d (Bind Abbr) u) i H0 (CHead x2 (Bind Void) x3) H11)) in (let H13 \def -(eq_ind C (CHead d (Bind Abbr) u) (\lambda (ee: C).(match ee in C return -(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b) -\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow -True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _) -\Rightarrow False])])) I (CHead x2 (Bind Void) x3) (getl_mono c (CHead d -(Bind Abbr) u) i H0 (CHead x2 (Bind Void) x3) H11)) in (False_ind (arity g c2 -(TLRef i) a0) H13))))))) H9))))))) H6)) H5)))))))))))))) (\lambda (c: -C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c -(CHead d (Bind Abst) u))).(\lambda (a0: A).(\lambda (_: (arity g d u (asucc g -a0))).(\lambda (H2: ((\forall (c2: C).((csuba g c2 d) \to ((csubv c2 d) \to -(arity g c2 u (asucc g a0))))))).(\lambda (c2: C).(\lambda (H3: (csuba g c2 -c)).(\lambda (H4: (csubv c2 c)).(let H_x \def (csuba_getl_abst_rev g c d u i -H0 c2 H3) in (let H5 \def H_x in (or_ind (ex2 C (\lambda (d2: C).(getl i c2 -(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d)))) (arity g c2 (TLRef i) a0) -(\lambda (H6: (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d)))).(ex2_ind C (\lambda (d2: C).(getl i c2 -(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d)) (arity g c2 -(TLRef i) a0) (\lambda (x: C).(\lambda (H7: (getl i c2 (CHead x (Bind Abst) -u))).(\lambda (H8: (csuba g x d)).(let H_x0 \def (csubv_getl_conf c2 c H4 -Abst x u i H7) in (let H9 \def H_x0 in (ex2_3_ind B C T (\lambda (_: -B).(\lambda (d2: C).(\lambda (_: T).(csubv x d2)))) (\lambda (b2: B).(\lambda -(d2: C).(\lambda (v2: T).(getl i c (CHead d2 (Bind b2) v2))))) (arity g c2 -(TLRef i) a0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda -(H10: (csubv x x1)).(\lambda (H11: (getl i c (CHead x1 (Bind x0) x2))).(let -H12 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (c0: C).(getl i c c0)) H0 -(CHead x1 (Bind x0) x2) (getl_mono c (CHead d (Bind Abst) u) i H0 (CHead x1 -(Bind x0) x2) H11)) in (let H13 \def (f_equal C C (\lambda (e: C).(match e in -C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) -\Rightarrow c0])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x2) (getl_mono -c (CHead d (Bind Abst) u) i H0 (CHead x1 (Bind x0) x2) H11)) in ((let H14 -\def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) -with [(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match k in K -return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow -Abst])])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x2) (getl_mono c (CHead -d (Bind Abst) u) i H0 (CHead x1 (Bind x0) x2) H11)) in ((let H15 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind -Abst) u) (CHead x1 (Bind x0) x2) (getl_mono c (CHead d (Bind Abst) u) i H0 -(CHead x1 (Bind x0) x2) H11)) in (\lambda (H16: (eq B Abst x0)).(\lambda -(H17: (eq C d x1)).(let H18 \def (eq_ind_r T x2 (\lambda (t0: T).(getl i c -(CHead x1 (Bind x0) t0))) H12 u H15) in (let H19 \def (eq_ind_r C x1 (\lambda -(c0: C).(getl i c (CHead c0 (Bind x0) u))) H18 d H17) in (let H20 \def -(eq_ind_r C x1 (\lambda (c0: C).(csubv x c0)) H10 d H17) in (let H21 \def -(eq_ind_r B x0 (\lambda (b: B).(getl i c (CHead d (Bind b) u))) H19 Abst H16) -in (arity_abst g c2 x u i H7 a0 (H2 x H8 H20))))))))) H14)) H13)))))))) -H9)))))) H6)) (\lambda (H6: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(getl -i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g -d2 d))) (arity g c2 (TLRef i) a0) (\lambda (x0: C).(\lambda (x1: T).(\lambda -(H7: (getl i c2 (CHead x0 (Bind Void) x1))).(\lambda (_: (csuba g x0 d)).(let -H_x0 \def (csubv_getl_conf_void c2 c H4 x0 x1 i H7) in (let H9 \def H_x0 in -(ex2_2_ind C T (\lambda (d2: C).(\lambda (_: T).(csubv x0 d2))) (\lambda (d2: -C).(\lambda (v2: T).(getl i c (CHead d2 (Bind Void) v2)))) (arity g c2 (TLRef -i) a0) (\lambda (x2: C).(\lambda (x3: T).(\lambda (_: (csubv x0 x2)).(\lambda -(H11: (getl i c (CHead x2 (Bind Void) x3))).(let H12 \def (eq_ind C (CHead d -(Bind Abst) u) (\lambda (c0: C).(getl i c c0)) H0 (CHead x2 (Bind Void) x3) -(getl_mono c (CHead d (Bind Abst) u) i H0 (CHead x2 (Bind Void) x3) H11)) in -(let H13 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (ee: C).(match ee in -C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k -_) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b) -\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow -False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) -\Rightarrow False])])) I (CHead x2 (Bind Void) x3) (getl_mono c (CHead d -(Bind Abst) u) i H0 (CHead x2 (Bind Void) x3) H11)) in (False_ind (arity g c2 -(TLRef i) a0) H13))))))) H9))))))) H6)) H5)))))))))))))) (\lambda (b: -B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c: C).(\lambda (u: -T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H2: ((\forall -(c2: C).((csuba g c2 c) \to ((csubv c2 c) \to (arity g c2 u a1)))))).(\lambda -(t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c (Bind b) u) t0 -a2)).(\lambda (H4: ((\forall (c2: C).((csuba g c2 (CHead c (Bind b) u)) \to -((csubv c2 (CHead c (Bind b) u)) \to (arity g c2 t0 a2)))))).(\lambda (c2: -C).(\lambda (H5: (csuba g c2 c)).(\lambda (H6: (csubv c2 c)).(arity_bind g b -H0 c2 u a1 (H2 c2 H5 H6) t0 a2 (H4 (CHead c2 (Bind b) u) (csuba_head g c2 c -H5 (Bind b) u) (csubv_bind_same c2 c H6 b u u))))))))))))))))) (\lambda (c: -C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c u (asucc g -a1))).(\lambda (H1: ((\forall (c2: C).((csuba g c2 c) \to ((csubv c2 c) \to -(arity g c2 u (asucc g a1))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda -(_: (arity g (CHead c (Bind Abst) u) t0 a2)).(\lambda (H3: ((\forall (c2: -C).((csuba g c2 (CHead c (Bind Abst) u)) \to ((csubv c2 (CHead c (Bind Abst) -u)) \to (arity g c2 t0 a2)))))).(\lambda (c2: C).(\lambda (H4: (csuba g c2 -c)).(\lambda (H5: (csubv c2 c)).(arity_head g c2 u a1 (H1 c2 H4 H5) t0 a2 (H3 -(CHead c2 (Bind Abst) u) (csuba_head g c2 c H4 (Bind Abst) u) -(csubv_bind_same c2 c H5 Abst u u))))))))))))))) (\lambda (c: C).(\lambda (u: -T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H1: ((\forall -(c2: C).((csuba g c2 c) \to ((csubv c2 c) \to (arity g c2 u a1)))))).(\lambda -(t0: T).(\lambda (a2: A).(\lambda (_: (arity g c t0 (AHead a1 a2))).(\lambda -(H3: ((\forall (c2: C).((csuba g c2 c) \to ((csubv c2 c) \to (arity g c2 t0 -(AHead a1 a2))))))).(\lambda (c2: C).(\lambda (H4: (csuba g c2 c)).(\lambda -(H5: (csubv c2 c)).(arity_appl g c2 u a1 (H1 c2 H4 H5) t0 a2 (H3 c2 H4 -H5)))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a0: A).(\lambda -(_: (arity g c u (asucc g a0))).(\lambda (H1: ((\forall (c2: C).((csuba g c2 -c) \to ((csubv c2 c) \to (arity g c2 u (asucc g a0))))))).(\lambda (t0: -T).(\lambda (_: (arity g c t0 a0)).(\lambda (H3: ((\forall (c2: C).((csuba g -c2 c) \to ((csubv c2 c) \to (arity g c2 t0 a0)))))).(\lambda (c2: C).(\lambda -(H4: (csuba g c2 c)).(\lambda (H5: (csubv c2 c)).(arity_cast g c2 u a0 (H1 c2 -H4 H5) t0 (H3 c2 H4 H5))))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda -(a1: A).(\lambda (_: (arity g c t0 a1)).(\lambda (H1: ((\forall (c2: -C).((csuba g c2 c) \to ((csubv c2 c) \to (arity g c2 t0 a1)))))).(\lambda -(a2: A).(\lambda (H2: (leq g a1 a2)).(\lambda (c2: C).(\lambda (H3: (csuba g -c2 c)).(\lambda (H4: (csubv c2 c)).(arity_repl g c2 t0 a1 (H1 c2 H3 H4) a2 -H2))))))))))) c1 t a H))))). -(* COMMENTS -Initial nodes: 3597 -END *) - -theorem arity_appls_appl: - \forall (g: G).(\forall (c: C).(\forall (v: T).(\forall (a1: A).((arity g c -v a1) \to (\forall (u: T).((arity g c u (asucc g a1)) \to (\forall (t: -T).(\forall (vs: TList).(\forall (a2: A).((arity g c (THeads (Flat Appl) vs -(THead (Bind Abbr) v t)) a2) \to (arity g c (THeads (Flat Appl) vs (THead -(Flat Appl) v (THead (Bind Abst) u t))) a2))))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (v: T).(\lambda (a1: A).(\lambda (H: -(arity g c v a1)).(\lambda (u: T).(\lambda (H0: (arity g c u (asucc g -a1))).(\lambda (t: T).(\lambda (vs: TList).(TList_ind (\lambda (t0: -TList).(\forall (a2: A).((arity g c (THeads (Flat Appl) t0 (THead (Bind Abbr) -v t)) a2) \to (arity g c (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead -(Bind Abst) u t))) a2)))) (\lambda (a2: A).(\lambda (H1: (arity g c (THead -(Bind Abbr) v t) a2)).(let H_x \def (arity_gen_bind Abbr (\lambda (H2: (eq B -Abbr Abst)).(not_abbr_abst H2)) g c v t a2 H1) in (let H2 \def H_x in -(ex2_ind A (\lambda (a3: A).(arity g c v a3)) (\lambda (_: A).(arity g (CHead -c (Bind Abbr) v) t a2)) (arity g c (THead (Flat Appl) v (THead (Bind Abst) u -t)) a2) (\lambda (x: A).(\lambda (_: (arity g c v x)).(\lambda (H4: (arity g -(CHead c (Bind Abbr) v) t a2)).(arity_appl g c v a1 H (THead (Bind Abst) u t) -a2 (arity_head g c u a1 H0 t a2 (csuba_arity_rev g (CHead c (Bind Abbr) v) t -a2 H4 (CHead c (Bind Abst) u) (csuba_abst g c c (csuba_refl g c) u a1 H0 v H) -(csubv_bind c c (csubv_refl c) Abst (sym_not_eq B Void Abst not_void_abst) -Abbr u v))))))) H2))))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H1: -((\forall (a2: A).((arity g c (THeads (Flat Appl) t1 (THead (Bind Abbr) v t)) -a2) \to (arity g c (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind -Abst) u t))) a2))))).(\lambda (a2: A).(\lambda (H2: (arity g c (THead (Flat -Appl) t0 (THeads (Flat Appl) t1 (THead (Bind Abbr) v t))) a2)).(let H3 \def -(arity_gen_appl g c t0 (THeads (Flat Appl) t1 (THead (Bind Abbr) v t)) a2 H2) -in (ex2_ind A (\lambda (a3: A).(arity g c t0 a3)) (\lambda (a3: A).(arity g c -(THeads (Flat Appl) t1 (THead (Bind Abbr) v t)) (AHead a3 a2))) (arity g c -(THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead -(Bind Abst) u t)))) a2) (\lambda (x: A).(\lambda (H4: (arity g c t0 -x)).(\lambda (H5: (arity g c (THeads (Flat Appl) t1 (THead (Bind Abbr) v t)) -(AHead x a2))).(arity_appl g c t0 x H4 (THeads (Flat Appl) t1 (THead (Flat -Appl) v (THead (Bind Abst) u t))) a2 (H1 (AHead x a2) H5))))) H3))))))) -vs))))))))). -(* COMMENTS -Initial nodes: 687 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/clear.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/clear.ma deleted file mode 100644 index 2350fd1e8..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/clear.ma +++ /dev/null @@ -1,128 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csuba/defs.ma". - -include "Basic-1/clear/fwd.ma". - -theorem csuba_clear_conf: - \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csuba g c1 c2) \to -(\forall (e1: C).((clear c1 e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) -(\lambda (e2: C).(clear c2 e2)))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csuba g c1 -c2)).(csuba_ind g (\lambda (c: C).(\lambda (c0: C).(\forall (e1: C).((clear c -e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2: C).(clear c0 -e2))))))) (\lambda (n: nat).(\lambda (e1: C).(\lambda (H0: (clear (CSort n) -e1)).(clear_gen_sort e1 n H0 (ex2 C (\lambda (e2: C).(csuba g e1 e2)) -(\lambda (e2: C).(clear (CSort n) e2))))))) (\lambda (c3: C).(\lambda (c4: -C).(\lambda (H0: (csuba g c3 c4)).(\lambda (H1: ((\forall (e1: C).((clear c3 -e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2: C).(clear c4 -e2))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (e1: C).(\lambda (H2: -(clear (CHead c3 k u) e1)).(K_ind (\lambda (k0: K).((clear (CHead c3 k0 u) -e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2: C).(clear -(CHead c4 k0 u) e2))))) (\lambda (b: B).(\lambda (H3: (clear (CHead c3 (Bind -b) u) e1)).(eq_ind_r C (CHead c3 (Bind b) u) (\lambda (c: C).(ex2 C (\lambda -(e2: C).(csuba g c e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b) u) e2)))) -(ex_intro2 C (\lambda (e2: C).(csuba g (CHead c3 (Bind b) u) e2)) (\lambda -(e2: C).(clear (CHead c4 (Bind b) u) e2)) (CHead c4 (Bind b) u) (csuba_head g -c3 c4 H0 (Bind b) u) (clear_bind b c4 u)) e1 (clear_gen_bind b c3 e1 u H3)))) -(\lambda (f: F).(\lambda (H3: (clear (CHead c3 (Flat f) u) e1)).(let H4 \def -(H1 e1 (clear_gen_flat f c3 e1 u H3)) in (ex2_ind C (\lambda (e2: C).(csuba g -e1 e2)) (\lambda (e2: C).(clear c4 e2)) (ex2 C (\lambda (e2: C).(csuba g e1 -e2)) (\lambda (e2: C).(clear (CHead c4 (Flat f) u) e2))) (\lambda (x: -C).(\lambda (H5: (csuba g e1 x)).(\lambda (H6: (clear c4 x)).(ex_intro2 C -(\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2: C).(clear (CHead c4 (Flat f) -u) e2)) x H5 (clear_flat c4 x H6 f u))))) H4)))) k H2))))))))) (\lambda (c3: -C).(\lambda (c4: C).(\lambda (H0: (csuba g c3 c4)).(\lambda (_: ((\forall -(e1: C).((clear c3 e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) (\lambda -(e2: C).(clear c4 e2))))))).(\lambda (b: B).(\lambda (H2: (not (eq B b -Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (e1: C).(\lambda (H3: -(clear (CHead c3 (Bind Void) u1) e1)).(eq_ind_r C (CHead c3 (Bind Void) u1) -(\lambda (c: C).(ex2 C (\lambda (e2: C).(csuba g c e2)) (\lambda (e2: -C).(clear (CHead c4 (Bind b) u2) e2)))) (ex_intro2 C (\lambda (e2: C).(csuba -g (CHead c3 (Bind Void) u1) e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b) -u2) e2)) (CHead c4 (Bind b) u2) (csuba_void g c3 c4 H0 b H2 u1 u2) -(clear_bind b c4 u2)) e1 (clear_gen_bind Void c3 e1 u1 H3)))))))))))) -(\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csuba g c3 c4)).(\lambda (_: -((\forall (e1: C).((clear c3 e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) -(\lambda (e2: C).(clear c4 e2))))))).(\lambda (t: T).(\lambda (a: A).(\lambda -(H2: (arity g c3 t (asucc g a))).(\lambda (u: T).(\lambda (H3: (arity g c4 u -a)).(\lambda (e1: C).(\lambda (H4: (clear (CHead c3 (Bind Abst) t) -e1)).(eq_ind_r C (CHead c3 (Bind Abst) t) (\lambda (c: C).(ex2 C (\lambda -(e2: C).(csuba g c e2)) (\lambda (e2: C).(clear (CHead c4 (Bind Abbr) u) -e2)))) (ex_intro2 C (\lambda (e2: C).(csuba g (CHead c3 (Bind Abst) t) e2)) -(\lambda (e2: C).(clear (CHead c4 (Bind Abbr) u) e2)) (CHead c4 (Bind Abbr) -u) (csuba_abst g c3 c4 H0 t a H2 u H3) (clear_bind Abbr c4 u)) e1 -(clear_gen_bind Abst c3 e1 t H4))))))))))))) c1 c2 H)))). -(* COMMENTS -Initial nodes: 937 -END *) - -theorem csuba_clear_trans: - \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csuba g c2 c1) \to -(\forall (e1: C).((clear c1 e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) -(\lambda (e2: C).(clear c2 e2)))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csuba g c2 -c1)).(csuba_ind g (\lambda (c: C).(\lambda (c0: C).(\forall (e1: C).((clear -c0 e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) (\lambda (e2: C).(clear c -e2))))))) (\lambda (n: nat).(\lambda (e1: C).(\lambda (H0: (clear (CSort n) -e1)).(clear_gen_sort e1 n H0 (ex2 C (\lambda (e2: C).(csuba g e2 e1)) -(\lambda (e2: C).(clear (CSort n) e2))))))) (\lambda (c3: C).(\lambda (c4: -C).(\lambda (H0: (csuba g c3 c4)).(\lambda (H1: ((\forall (e1: C).((clear c4 -e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) (\lambda (e2: C).(clear c3 -e2))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (e1: C).(\lambda (H2: -(clear (CHead c4 k u) e1)).(K_ind (\lambda (k0: K).((clear (CHead c4 k0 u) -e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) (\lambda (e2: C).(clear -(CHead c3 k0 u) e2))))) (\lambda (b: B).(\lambda (H3: (clear (CHead c4 (Bind -b) u) e1)).(eq_ind_r C (CHead c4 (Bind b) u) (\lambda (c: C).(ex2 C (\lambda -(e2: C).(csuba g e2 c)) (\lambda (e2: C).(clear (CHead c3 (Bind b) u) e2)))) -(ex_intro2 C (\lambda (e2: C).(csuba g e2 (CHead c4 (Bind b) u))) (\lambda -(e2: C).(clear (CHead c3 (Bind b) u) e2)) (CHead c3 (Bind b) u) (csuba_head g -c3 c4 H0 (Bind b) u) (clear_bind b c3 u)) e1 (clear_gen_bind b c4 e1 u H3)))) -(\lambda (f: F).(\lambda (H3: (clear (CHead c4 (Flat f) u) e1)).(let H4 \def -(H1 e1 (clear_gen_flat f c4 e1 u H3)) in (ex2_ind C (\lambda (e2: C).(csuba g -e2 e1)) (\lambda (e2: C).(clear c3 e2)) (ex2 C (\lambda (e2: C).(csuba g e2 -e1)) (\lambda (e2: C).(clear (CHead c3 (Flat f) u) e2))) (\lambda (x: -C).(\lambda (H5: (csuba g x e1)).(\lambda (H6: (clear c3 x)).(ex_intro2 C -(\lambda (e2: C).(csuba g e2 e1)) (\lambda (e2: C).(clear (CHead c3 (Flat f) -u) e2)) x H5 (clear_flat c3 x H6 f u))))) H4)))) k H2))))))))) (\lambda (c3: -C).(\lambda (c4: C).(\lambda (H0: (csuba g c3 c4)).(\lambda (_: ((\forall -(e1: C).((clear c4 e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) (\lambda -(e2: C).(clear c3 e2))))))).(\lambda (b: B).(\lambda (H2: (not (eq B b -Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (e1: C).(\lambda (H3: -(clear (CHead c4 (Bind b) u2) e1)).(eq_ind_r C (CHead c4 (Bind b) u2) -(\lambda (c: C).(ex2 C (\lambda (e2: C).(csuba g e2 c)) (\lambda (e2: -C).(clear (CHead c3 (Bind Void) u1) e2)))) (ex_intro2 C (\lambda (e2: -C).(csuba g e2 (CHead c4 (Bind b) u2))) (\lambda (e2: C).(clear (CHead c3 -(Bind Void) u1) e2)) (CHead c3 (Bind Void) u1) (csuba_void g c3 c4 H0 b H2 u1 -u2) (clear_bind Void c3 u1)) e1 (clear_gen_bind b c4 e1 u2 H3)))))))))))) -(\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csuba g c3 c4)).(\lambda (_: -((\forall (e1: C).((clear c4 e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) -(\lambda (e2: C).(clear c3 e2))))))).(\lambda (t: T).(\lambda (a: A).(\lambda -(H2: (arity g c3 t (asucc g a))).(\lambda (u: T).(\lambda (H3: (arity g c4 u -a)).(\lambda (e1: C).(\lambda (H4: (clear (CHead c4 (Bind Abbr) u) -e1)).(eq_ind_r C (CHead c4 (Bind Abbr) u) (\lambda (c: C).(ex2 C (\lambda -(e2: C).(csuba g e2 c)) (\lambda (e2: C).(clear (CHead c3 (Bind Abst) t) -e2)))) (ex_intro2 C (\lambda (e2: C).(csuba g e2 (CHead c4 (Bind Abbr) u))) -(\lambda (e2: C).(clear (CHead c3 (Bind Abst) t) e2)) (CHead c3 (Bind Abst) -t) (csuba_abst g c3 c4 H0 t a H2 u H3) (clear_bind Abst c3 t)) e1 -(clear_gen_bind Abbr c4 e1 u H4))))))))))))) c2 c1 H)))). -(* COMMENTS -Initial nodes: 937 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/defs.ma deleted file mode 100644 index cc6a46cd8..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/defs.ma +++ /dev/null @@ -1,30 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/arity/defs.ma". - -inductive csuba (g: G): C \to (C \to Prop) \def -| csuba_sort: \forall (n: nat).(csuba g (CSort n) (CSort n)) -| csuba_head: \forall (c1: C).(\forall (c2: C).((csuba g c1 c2) \to (\forall -(k: K).(\forall (u: T).(csuba g (CHead c1 k u) (CHead c2 k u)))))) -| csuba_void: \forall (c1: C).(\forall (c2: C).((csuba g c1 c2) \to (\forall -(b: B).((not (eq B b Void)) \to (\forall (u1: T).(\forall (u2: T).(csuba g -(CHead c1 (Bind Void) u1) (CHead c2 (Bind b) u2)))))))) -| csuba_abst: \forall (c1: C).(\forall (c2: C).((csuba g c1 c2) \to (\forall -(t: T).(\forall (a: A).((arity g c1 t (asucc g a)) \to (\forall (u: -T).((arity g c2 u a) \to (csuba g (CHead c1 (Bind Abst) t) (CHead c2 (Bind -Abbr) u))))))))). - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/drop.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/drop.ma deleted file mode 100644 index 1047ac13d..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/drop.ma +++ /dev/null @@ -1,2468 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csuba/fwd.ma". - -include "Basic-1/drop/fwd.ma". - -theorem csuba_drop_abbr: - \forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).((drop i -O c1 (CHead d1 (Bind Abbr) u)) \to (\forall (g: G).(\forall (c2: C).((csuba g -c1 c2) \to (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr) u))) -(\lambda (d2: C).(csuba g d1 d2)))))))))) -\def - \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (d1: -C).(\forall (u: T).((drop n O c1 (CHead d1 (Bind Abbr) u)) \to (\forall (g: -G).(\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(drop n O c2 -(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))))))))) -(\lambda (c1: C).(\lambda (d1: C).(\lambda (u: T).(\lambda (H: (drop O O c1 -(CHead d1 (Bind Abbr) u))).(\lambda (g: G).(\lambda (c2: C).(\lambda (H0: -(csuba g c1 c2)).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csuba g c c2)) H0 -(CHead d1 (Bind Abbr) u) (drop_gen_refl c1 (CHead d1 (Bind Abbr) u) H)) in -(let H_x \def (csuba_gen_abbr g d1 c2 u H1) in (let H2 \def H_x in (ex2_ind C -(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba -g d1 d2)) (ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abbr) u))) -(\lambda (d2: C).(csuba g d1 d2))) (\lambda (x: C).(\lambda (H3: (eq C c2 -(CHead x (Bind Abbr) u))).(\lambda (H4: (csuba g d1 x)).(eq_ind_r C (CHead x -(Bind Abbr) u) (\lambda (c: C).(ex2 C (\lambda (d2: C).(drop O O c (CHead d2 -(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (ex_intro2 C (\lambda -(d2: C).(drop O O (CHead x (Bind Abbr) u) (CHead d2 (Bind Abbr) u))) (\lambda -(d2: C).(csuba g d1 d2)) x (drop_refl (CHead x (Bind Abbr) u)) H4) c2 H3)))) -H2))))))))))) (\lambda (n: nat).(\lambda (H: ((\forall (c1: C).(\forall (d1: -C).(\forall (u: T).((drop n O c1 (CHead d1 (Bind Abbr) u)) \to (\forall (g: -G).(\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(drop n O c2 -(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 -d2)))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (d1: -C).(\forall (u: T).((drop (S n) O c (CHead d1 (Bind Abbr) u)) \to (\forall -(g: G).(\forall (c2: C).((csuba g c c2) \to (ex2 C (\lambda (d2: C).(drop (S -n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))))))) -(\lambda (n0: nat).(\lambda (d1: C).(\lambda (u: T).(\lambda (H0: (drop (S n) -O (CSort n0) (CHead d1 (Bind Abbr) u))).(\lambda (g: G).(\lambda (c2: -C).(\lambda (_: (csuba g (CSort n0) c2)).(and3_ind (eq C (CHead d1 (Bind -Abbr) u) (CSort n0)) (eq nat (S n) O) (eq nat O O) (ex2 C (\lambda (d2: -C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 -d2))) (\lambda (_: (eq C (CHead d1 (Bind Abbr) u) (CSort n0))).(\lambda (H3: -(eq nat (S n) O)).(\lambda (_: (eq nat O O)).(let H5 \def (eq_ind nat (S n) -(\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O -\Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind (ex2 C -(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: -C).(csuba g d1 d2))) H5))))) (drop_gen_sort n0 (S n) O (CHead d1 (Bind Abbr) -u) H0))))))))) (\lambda (c: C).(\lambda (H0: ((\forall (d1: C).(\forall (u: -T).((drop (S n) O c (CHead d1 (Bind Abbr) u)) \to (\forall (g: G).(\forall -(c2: C).((csuba g c c2) \to (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead -d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))))))))).(\lambda (k: -K).(\lambda (t: T).(\lambda (d1: C).(\lambda (u: T).(\lambda (H1: (drop (S n) -O (CHead c k t) (CHead d1 (Bind Abbr) u))).(\lambda (g: G).(\lambda (c2: -C).(\lambda (H2: (csuba g (CHead c k t) c2)).(K_ind (\lambda (k0: K).((csuba -g (CHead c k0 t) c2) \to ((drop (r k0 n) O c (CHead d1 (Bind Abbr) u)) \to -(ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda -(d2: C).(csuba g d1 d2)))))) (\lambda (b: B).(\lambda (H3: (csuba g (CHead c -(Bind b) t) c2)).(\lambda (H4: (drop (r (Bind b) n) O c (CHead d1 (Bind Abbr) -u))).(B_ind (\lambda (b0: B).((csuba g (CHead c (Bind b0) t) c2) \to ((drop -(r (Bind b0) n) O c (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: -C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 -d2)))))) (\lambda (H5: (csuba g (CHead c (Bind Abbr) t) c2)).(\lambda (H6: -(drop (r (Bind Abbr) n) O c (CHead d1 (Bind Abbr) u))).(let H_x \def -(csuba_gen_abbr g c c2 t H5) in (let H7 \def H_x in (ex2_ind C (\lambda (d2: -C).(eq C c2 (CHead d2 (Bind Abbr) t))) (\lambda (d2: C).(csuba g c d2)) (ex2 -C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: -C).(csuba g d1 d2))) (\lambda (x: C).(\lambda (H8: (eq C c2 (CHead x (Bind -Abbr) t))).(\lambda (H9: (csuba g c x)).(eq_ind_r C (CHead x (Bind Abbr) t) -(\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind -Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H10 \def (H c d1 u H6 g x -H9) in (ex2_ind C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u))) -(\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(drop (S n) O -(CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g -d1 d2))) (\lambda (x0: C).(\lambda (H11: (drop n O x (CHead x0 (Bind Abbr) -u))).(\lambda (H12: (csuba g d1 x0)).(let H13 \def (refl_equal nat (r (Bind -Abbr) n)) in (let H14 \def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x -(CHead x0 (Bind Abbr) u))) H11 (r (Bind Abbr) n) H13) in (ex_intro2 C -(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) -u))) (\lambda (d2: C).(csuba g d1 d2)) x0 (drop_drop (Bind Abbr) n x (CHead -x0 (Bind Abbr) u) H14 t) H12)))))) H10)) c2 H8)))) H7))))) (\lambda (H5: -(csuba g (CHead c (Bind Abst) t) c2)).(\lambda (H6: (drop (r (Bind Abst) n) O -c (CHead d1 (Bind Abbr) u))).(let H_x \def (csuba_gen_abst g c c2 t H5) in -(let H7 \def H_x in (or_ind (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind -Abst) t))) (\lambda (d2: C).(csuba g c d2))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g c d2)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex2 C -(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: -C).(csuba g d1 d2))) (\lambda (H8: (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 -(Bind Abst) t))) (\lambda (d2: C).(csuba g c d2)))).(ex2_ind C (\lambda (d2: -C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba g c d2)) (ex2 -C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: -C).(csuba g d1 d2))) (\lambda (x: C).(\lambda (H9: (eq C c2 (CHead x (Bind -Abst) t))).(\lambda (H10: (csuba g c x)).(eq_ind_r C (CHead x (Bind Abst) t) -(\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind -Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H11 \def (H c d1 u H6 g x -H10) in (ex2_ind C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u))) -(\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(drop (S n) O -(CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g -d1 d2))) (\lambda (x0: C).(\lambda (H12: (drop n O x (CHead x0 (Bind Abbr) -u))).(\lambda (H13: (csuba g d1 x0)).(let H14 \def (refl_equal nat (r (Bind -Abbr) n)) in (let H15 \def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x -(CHead x0 (Bind Abbr) u))) H12 (r (Bind Abbr) n) H14) in (ex_intro2 C -(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) -u))) (\lambda (d2: C).(csuba g d1 d2)) x0 (drop_drop (Bind Abst) n x (CHead -x0 (Bind Abbr) u) H15 t) H13)))))) H11)) c2 H9)))) H8)) (\lambda (H8: (ex4_3 -C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 -(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -c d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t (asucc -g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g c d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g c t (asucc g a))))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 a)))) (ex2 C (\lambda (d2: C).(drop (S n) O -c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda -(x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H9: (eq C c2 (CHead x0 -(Bind Abbr) x1))).(\lambda (H10: (csuba g c x0)).(\lambda (_: (arity g c t -(asucc g x2))).(\lambda (_: (arity g x0 x1 x2)).(eq_ind_r C (CHead x0 (Bind -Abbr) x1) (\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 -(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H13 \def (H c d1 u -H6 g x0 H10) in (ex2_ind C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind -Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(drop (S -n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abbr) u))) (\lambda (d2: -C).(csuba g d1 d2))) (\lambda (x: C).(\lambda (H14: (drop n O x0 (CHead x -(Bind Abbr) u))).(\lambda (H15: (csuba g d1 x)).(let H16 \def (refl_equal nat -(r (Bind Abbr) n)) in (let H17 \def (eq_ind nat n (\lambda (n0: nat).(drop n0 -O x0 (CHead x (Bind Abbr) u))) H14 (r (Bind Abbr) n) H16) in (ex_intro2 C -(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind -Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x (drop_drop (Bind Abbr) n x0 -(CHead x (Bind Abbr) u) H17 x1) H15)))))) H13)) c2 H9)))))))) H8)) H7))))) -(\lambda (H5: (csuba g (CHead c (Bind Void) t) c2)).(\lambda (H6: (drop (r -(Bind Void) n) O c (CHead d1 (Bind Abbr) u))).(let H_x \def (csuba_gen_void g -c c2 t H5) in (let H7 \def H_x in (ex2_3_ind B C T (\lambda (b0: B).(\lambda -(d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind b0) u2))))) (\lambda (_: -B).(\lambda (d2: C).(\lambda (_: T).(csuba g c d2)))) (ex2 C (\lambda (d2: -C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 -d2))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H8: (eq C -c2 (CHead x1 (Bind x0) x2))).(\lambda (H9: (csuba g c x1)).(eq_ind_r C (CHead -x1 (Bind x0) x2) (\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0 -(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H10 \def -(H c d1 u H6 g x1 H9) in (ex2_ind C (\lambda (d2: C).(drop n O x1 (CHead d2 -(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abbr) u))) (\lambda -(d2: C).(csuba g d1 d2))) (\lambda (x: C).(\lambda (H11: (drop n O x1 (CHead -x (Bind Abbr) u))).(\lambda (H12: (csuba g d1 x)).(let H13 \def (refl_equal -nat (r (Bind Abbr) n)) in (let H14 \def (eq_ind nat n (\lambda (n0: -nat).(drop n0 O x1 (CHead x (Bind Abbr) u))) H11 (r (Bind Abbr) n) H13) in -(ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2 -(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x (drop_drop (Bind x0) n -x1 (CHead x (Bind Abbr) u) H14 x2) H12)))))) H10)) c2 H8)))))) H7))))) b H3 -H4)))) (\lambda (f: F).(\lambda (H3: (csuba g (CHead c (Flat f) t) -c2)).(\lambda (H4: (drop (r (Flat f) n) O c (CHead d1 (Bind Abbr) u))).(let -H_x \def (csuba_gen_flat g c c2 t f H3) in (let H5 \def H_x in (ex2_2_ind C T -(\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Flat f) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g c d2))) (ex2 C (\lambda (d2: C).(drop (S n) -O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda -(x0: C).(\lambda (x1: T).(\lambda (H6: (eq C c2 (CHead x0 (Flat f) -x1))).(\lambda (H7: (csuba g c x0)).(eq_ind_r C (CHead x0 (Flat f) x1) -(\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind -Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H8 \def (H0 d1 u H4 g x0 -H7) in (ex2_ind C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind Abbr) -u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(drop (S n) O -(CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g -d1 d2))) (\lambda (x: C).(\lambda (H9: (drop (S n) O x0 (CHead x (Bind Abbr) -u))).(\lambda (H10: (csuba g d1 x)).(ex_intro2 C (\lambda (d2: C).(drop (S n) -O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g -d1 d2)) x (drop_drop (Flat f) n x0 (CHead x (Bind Abbr) u) H9 x1) H10)))) -H8)) c2 H6))))) H5)))))) k H2 (drop_gen_drop k c (CHead d1 (Bind Abbr) u) t n -H1)))))))))))) c1)))) i). -(* COMMENTS -Initial nodes: 3648 -END *) - -theorem csuba_drop_abst: - \forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u1: T).((drop i -O c1 (CHead d1 (Bind Abst) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba -g c1 c2) \to (or (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst) -u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a))))))))))))) -\def - \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (d1: -C).(\forall (u1: T).((drop n O c1 (CHead d1 (Bind Abst) u1)) \to (\forall (g: -G).(\forall (c2: C).((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(drop n -O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C -T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O c2 (CHead d2 -(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 -(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 -u2 a)))))))))))))) (\lambda (c1: C).(\lambda (d1: C).(\lambda (u1: -T).(\lambda (H: (drop O O c1 (CHead d1 (Bind Abst) u1))).(\lambda (g: -G).(\lambda (c2: C).(\lambda (H0: (csuba g c1 c2)).(let H1 \def (eq_ind C c1 -(\lambda (c: C).(csuba g c c2)) H0 (CHead d1 (Bind Abst) u1) (drop_gen_refl -c1 (CHead d1 (Bind Abst) u1) H)) in (let H_x \def (csuba_gen_abst g d1 c2 u1 -H1) in (let H2 \def H_x in (or_ind (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 -(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a))))) (or (ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) u1))) -(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(drop O O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (H3: -(ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: -C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind -Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: -C).(drop O O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 -d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O -O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: -A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda -(a: A).(arity g d2 u2 a)))))) (\lambda (x: C).(\lambda (H4: (eq C c2 (CHead x -(Bind Abst) u1))).(\lambda (H5: (csuba g d1 x)).(eq_ind_r C (CHead x (Bind -Abst) u1) (\lambda (c: C).(or (ex2 C (\lambda (d2: C).(drop O O c (CHead d2 -(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O c (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a))))))) (or_introl (ex2 C (\lambda (d2: C).(drop O O (CHead x (Bind Abst) -u1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T -A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O (CHead x (Bind -Abst) u1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: -C).(drop O O (CHead x (Bind Abst) u1) (CHead d2 (Bind Abst) u1))) (\lambda -(d2: C).(csuba g d1 d2)) x (drop_refl (CHead x (Bind Abst) u1)) H5)) c2 -H4)))) H3)) (\lambda (H3: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))).(ex4_3_ind C -T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 -(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 -(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 -u2 a)))) (or (ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) u1))) -(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(drop O O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H4: (eq C c2 (CHead x0 (Bind -Abbr) x1))).(\lambda (H5: (csuba g d1 x0)).(\lambda (H6: (arity g d1 u1 -(asucc g x2))).(\lambda (H7: (arity g x0 x1 x2)).(eq_ind_r C (CHead x0 (Bind -Abbr) x1) (\lambda (c: C).(or (ex2 C (\lambda (d2: C).(drop O O c (CHead d2 -(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O c (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a))))))) (or_intror (ex2 C (\lambda (d2: C).(drop O O (CHead x0 (Bind Abbr) -x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T -A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O (CHead x0 (Bind -Abbr) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O (CHead x0 (Bind Abbr) x1) -(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity -g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 a)))) x0 x1 x2 (drop_refl (CHead x0 (Bind Abbr) x1)) H5 H6 -H7)) c2 H4)))))))) H3)) H2))))))))))) (\lambda (n: nat).(\lambda (H: -((\forall (c1: C).(\forall (d1: C).(\forall (u1: T).((drop n O c1 (CHead d1 -(Bind Abst) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba g c1 c2) \to -(or (ex2 C (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abst) u1))) (\lambda -(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop n O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a))))))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (d1: -C).(\forall (u1: T).((drop (S n) O c (CHead d1 (Bind Abst) u1)) \to (\forall -(g: G).(\forall (c2: C).((csuba g c c2) \to (or (ex2 C (\lambda (d2: C).(drop -(S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O -c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: -A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda -(a: A).(arity g d2 u2 a))))))))))))) (\lambda (n0: nat).(\lambda (d1: -C).(\lambda (u1: T).(\lambda (H0: (drop (S n) O (CSort n0) (CHead d1 (Bind -Abst) u1))).(\lambda (g: G).(\lambda (c2: C).(\lambda (_: (csuba g (CSort n0) -c2)).(and3_ind (eq C (CHead d1 (Bind Abst) u1) (CSort n0)) (eq nat (S n) O) -(eq nat O O) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind -Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))) (\lambda (_: (eq C (CHead d1 (Bind Abst) u1) (CSort n0))).(\lambda -(H3: (eq nat (S n) O)).(\lambda (_: (eq nat O O)).(let H5 \def (eq_ind nat (S -n) (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O -\Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind (or (ex2 -C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: -C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) H5))))) -(drop_gen_sort n0 (S n) O (CHead d1 (Bind Abst) u1) H0))))))))) (\lambda (c: -C).(\lambda (H0: ((\forall (d1: C).(\forall (u1: T).((drop (S n) O c (CHead -d1 (Bind Abst) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba g c c2) \to -(or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) -(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (d1: C).(\lambda -(u1: T).(\lambda (H1: (drop (S n) O (CHead c k t) (CHead d1 (Bind Abst) -u1))).(\lambda (g: G).(\lambda (c2: C).(\lambda (H2: (csuba g (CHead c k t) -c2)).(K_ind (\lambda (k0: K).((csuba g (CHead c k0 t) c2) \to ((drop (r k0 n) -O c (CHead d1 (Bind Abst) u1)) \to (or (ex2 C (\lambda (d2: C).(drop (S n) O -c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T -A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead -d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity -g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 a))))))))) (\lambda (b: B).(\lambda (H3: (csuba g (CHead c -(Bind b) t) c2)).(\lambda (H4: (drop (r (Bind b) n) O c (CHead d1 (Bind Abst) -u1))).(B_ind (\lambda (b0: B).((csuba g (CHead c (Bind b0) t) c2) \to ((drop -(r (Bind b0) n) O c (CHead d1 (Bind Abst) u1)) \to (or (ex2 C (\lambda (d2: -C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 -d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S -n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))) (\lambda (H5: (csuba g -(CHead c (Bind Abbr) t) c2)).(\lambda (H6: (drop (r (Bind Abbr) n) O c (CHead -d1 (Bind Abst) u1))).(let H_x \def (csuba_gen_abbr g c c2 t H5) in (let H7 -\def H_x in (ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) -(\lambda (d2: C).(csuba g c d2)) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 -(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 -(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 -(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 -u2 a)))))) (\lambda (x: C).(\lambda (H8: (eq C c2 (CHead x (Bind Abbr) -t))).(\lambda (H9: (csuba g c x)).(eq_ind_r C (CHead x (Bind Abbr) t) -(\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind -Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a))))))) (let H10 \def (H c d1 u1 H6 g x H9) in (or_ind (ex2 C (\lambda (d2: -C).(drop n O x (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x -(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity -g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x -(Bind Abbr) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O -(CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda -(H11: (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) u1))) -(\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(drop n O x -(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C -(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) -u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t) -(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity -g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 a)))))) (\lambda (x0: C).(\lambda (H12: (drop n O x (CHead -x0 (Bind Abst) u1))).(\lambda (H13: (csuba g d1 x0)).(let H14 \def -(refl_equal nat (r (Bind Abbr) n)) in (let H15 \def (eq_ind nat n (\lambda -(n0: nat).(drop n0 O x (CHead x0 (Bind Abst) u1))) H12 (r (Bind Abbr) n) H14) -in (or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) -(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x -(Bind Abbr) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: -C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u1))) (\lambda -(d2: C).(csuba g d1 d2)) x0 (drop_drop (Bind Abbr) n x (CHead x0 (Bind Abst) -u1) H15 t) H13))))))) H11)) (\lambda (H11: (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abbr) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop n O x (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u1))) (\lambda -(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind -Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 -d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc -g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H12: -(drop n O x (CHead x0 (Bind Abbr) x1))).(\lambda (H13: (csuba g d1 -x0)).(\lambda (H14: (arity g d1 u1 (asucc g x2))).(\lambda (H15: (arity g x0 -x1 x2)).(let H16 \def (refl_equal nat (r (Bind Abbr) n)) in (let H17 \def -(eq_ind nat n (\lambda (n0: nat).(drop n0 O x (CHead x0 (Bind Abbr) x1))) H12 -(r (Bind Abbr) n) H16) in (or_intror (ex2 C (\lambda (d2: C).(drop (S n) O -(CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g -d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop -(S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C -T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x -(Bind Abbr) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x0 x1 x2 (drop_drop (Bind Abbr) -n x (CHead x0 (Bind Abbr) x1) H17 t) H13 H14 H15))))))))))) H11)) H10)) c2 -H8)))) H7))))) (\lambda (H5: (csuba g (CHead c (Bind Abst) t) c2)).(\lambda -(H6: (drop (r (Bind Abst) n) O c (CHead d1 (Bind Abst) u1))).(let H_x \def -(csuba_gen_abst g c c2 t H5) in (let H7 \def H_x in (or_ind (ex2 C (\lambda -(d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba g c d2))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 -(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g c d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g -c t (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity -g d2 u2 a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind -Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))) (\lambda (H8: (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) -t))) (\lambda (d2: C).(csuba g c d2)))).(ex2_ind C (\lambda (d2: C).(eq C c2 -(CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba g c d2)) (or (ex2 C -(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: -C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x: -C).(\lambda (H9: (eq C c2 (CHead x (Bind Abst) t))).(\lambda (H10: (csuba g c -x)).(eq_ind_r C (CHead x (Bind Abst) t) (\lambda (c0: C).(or (ex2 C (\lambda -(d2: C).(drop (S n) O c0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba -g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop (S n) O c0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda -(_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a))))))) (let H11 \def (H c d1 u1 H6 g -x H10) in (or_ind (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) -u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abbr) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (or -(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind -Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) -(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity -g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 a)))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x -(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C -(\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) u1))) (\lambda (d2: -C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind -Abst) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O -(CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x0: -C).(\lambda (H13: (drop n O x (CHead x0 (Bind Abst) u1))).(\lambda (H14: -(csuba g d1 x0)).(let H15 \def (refl_equal nat (r (Bind Abbr) n)) in (let H16 -\def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x (CHead x0 (Bind Abst) -u1))) H13 (r (Bind Abbr) n) H15) in (or_introl (ex2 C (\lambda (d2: C).(drop -(S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: -C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) -(ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 -(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) x0 (drop_drop (Bind Abst) -n x (CHead x0 (Bind Abst) u1) H16 t) H14))))))) H12)) (\lambda (H12: (ex4_3 C -T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 -(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 -(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 -u2 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop n O x (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u1))) (\lambda -(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind -Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 -d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc -g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H13: -(drop n O x (CHead x0 (Bind Abbr) x1))).(\lambda (H14: (csuba g d1 -x0)).(\lambda (H15: (arity g d1 u1 (asucc g x2))).(\lambda (H16: (arity g x0 -x1 x2)).(let H17 \def (refl_equal nat (r (Bind Abbr) n)) in (let H18 \def -(eq_ind nat n (\lambda (n0: nat).(drop n0 O x (CHead x0 (Bind Abbr) x1))) H13 -(r (Bind Abbr) n) H17) in (or_intror (ex2 C (\lambda (d2: C).(drop (S n) O -(CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g -d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop -(S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C -T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x -(Bind Abst) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x0 x1 x2 (drop_drop (Bind Abst) -n x (CHead x0 (Bind Abbr) x1) H18 t) H14 H15 H16))))))))))) H12)) H11)) c2 -H9)))) H8)) (\lambda (H8: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g c d2)))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g c t (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))).(ex4_3_ind C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g c d2)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C -(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: -C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H9: (eq C c2 (CHead x0 (Bind -Abbr) x1))).(\lambda (H10: (csuba g c x0)).(\lambda (_: (arity g c t (asucc g -x2))).(\lambda (_: (arity g x0 x1 x2)).(eq_ind_r C (CHead x0 (Bind Abbr) x1) -(\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind -Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a))))))) (let H13 \def (H c d1 u1 H6 g x0 H10) in (or_ind (ex2 C (\lambda -(d2: C).(drop n O x0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 -d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n -O x0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: -A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda -(a: A).(arity g d2 u2 a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead -x0 (Bind Abbr) x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 -d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S -n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda -(H14: (ex2 C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind Abst) u1))) -(\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(drop n O x0 -(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C -(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind -Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abbr) x1) -(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity -g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 a)))))) (\lambda (x: C).(\lambda (H15: (drop n O x0 (CHead -x (Bind Abst) u1))).(\lambda (H16: (csuba g d1 x)).(let H17 \def (refl_equal -nat (r (Bind Abbr) n)) in (let H18 \def (eq_ind nat n (\lambda (n0: -nat).(drop n0 O x0 (CHead x (Bind Abst) u1))) H15 (r (Bind Abbr) n) H17) in -(or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abbr) x1) -(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 -(Bind Abbr) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abst) u1))) -(\lambda (d2: C).(csuba g d1 d2)) x (drop_drop (Bind Abbr) n x0 (CHead x -(Bind Abst) u1) H18 x1) H16))))))) H14)) (\lambda (H14: (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x0 (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop n O x0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abst) u1))) -(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 -(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 -(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 -u2 a)))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: A).(\lambda (H15: -(drop n O x0 (CHead x3 (Bind Abbr) x4))).(\lambda (H16: (csuba g d1 -x3)).(\lambda (H17: (arity g d1 u1 (asucc g x5))).(\lambda (H18: (arity g x3 -x4 x5)).(let H19 \def (refl_equal nat (r (Bind Abbr) n)) in (let H20 \def -(eq_ind nat n (\lambda (n0: nat).(drop n0 O x0 (CHead x3 (Bind Abbr) x4))) -H15 (r (Bind Abbr) n) H19) in (or_intror (ex2 C (\lambda (d2: C).(drop (S n) -O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: -C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abbr) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) -(ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S -n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x3 x4 x5 -(drop_drop (Bind Abbr) n x0 (CHead x3 (Bind Abbr) x4) H20 x1) H16 H17 -H18))))))))))) H14)) H13)) c2 H9)))))))) H8)) H7))))) (\lambda (H5: (csuba g -(CHead c (Bind Void) t) c2)).(\lambda (H6: (drop (r (Bind Void) n) O c (CHead -d1 (Bind Abst) u1))).(let H_x \def (csuba_gen_void g c c2 t H5) in (let H7 -\def H_x in (ex2_3_ind B C T (\lambda (b0: B).(\lambda (d2: C).(\lambda (u2: -T).(eq C c2 (CHead d2 (Bind b0) u2))))) (\lambda (_: B).(\lambda (d2: -C).(\lambda (_: T).(csuba g c d2)))) (or (ex2 C (\lambda (d2: C).(drop (S n) -O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C -T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead -d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity -g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 a)))))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: -T).(\lambda (H8: (eq C c2 (CHead x1 (Bind x0) x2))).(\lambda (H9: (csuba g c -x1)).(eq_ind_r C (CHead x1 (Bind x0) x2) (\lambda (c0: C).(or (ex2 C (\lambda -(d2: C).(drop (S n) O c0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba -g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop (S n) O c0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda -(_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a))))))) (let H10 \def (H c d1 u1 H6 g -x1 H9) in (or_ind (ex2 C (\lambda (d2: C).(drop n O x1 (CHead d2 (Bind Abst) -u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop n O x1 (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x1 (Bind x0) x2) -(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x1 -(Bind x0) x2) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (H11: (ex2 C (\lambda -(d2: C).(drop n O x1 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 -d2)))).(ex2_ind C (\lambda (d2: C).(drop n O x1 (CHead d2 (Bind Abst) u1))) -(\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: C).(drop (S n) O -(CHead x1 (Bind x0) x2) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g -d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop -(S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x: -C).(\lambda (H12: (drop n O x1 (CHead x (Bind Abst) u1))).(\lambda (H13: -(csuba g d1 x)).(let H14 \def (refl_equal nat (r (Bind Abbr) n)) in (let H15 -\def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x1 (CHead x (Bind Abst) -u1))) H12 (r (Bind Abbr) n) H14) in (or_introl (ex2 C (\lambda (d2: C).(drop -(S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abst) u1))) (\lambda (d2: -C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abbr) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) -(ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2 -(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) x (drop_drop (Bind x0) n -x1 (CHead x (Bind Abst) u1) H15 x2) H13))))))) H11)) (\lambda (H11: (ex4_3 C -T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x1 (CHead d2 -(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 -(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 -u2 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop n O x1 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abst) u1))) (\lambda -(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind -Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 -d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc -g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: A).(\lambda (H12: -(drop n O x1 (CHead x3 (Bind Abbr) x4))).(\lambda (H13: (csuba g d1 -x3)).(\lambda (H14: (arity g d1 u1 (asucc g x5))).(\lambda (H15: (arity g x3 -x4 x5)).(let H16 \def (refl_equal nat (r (Bind Abbr) n)) in (let H17 \def -(eq_ind nat n (\lambda (n0: nat).(drop n0 O x1 (CHead x3 (Bind Abbr) x4))) -H12 (r (Bind Abbr) n) H16) in (or_intror (ex2 C (\lambda (d2: C).(drop (S n) -O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba -g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abbr) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) -(ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S -n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x3 x4 x5 -(drop_drop (Bind x0) n x1 (CHead x3 (Bind Abbr) x4) H17 x2) H13 H14 -H15))))))))))) H11)) H10)) c2 H8)))))) H7))))) b H3 H4)))) (\lambda (f: -F).(\lambda (H3: (csuba g (CHead c (Flat f) t) c2)).(\lambda (H4: (drop (r -(Flat f) n) O c (CHead d1 (Bind Abst) u1))).(let H_x \def (csuba_gen_flat g c -c2 t f H3) in (let H5 \def H_x in (ex2_2_ind C T (\lambda (d2: C).(\lambda -(u2: T).(eq C c2 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g c d2))) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 -(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind -Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 -d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc -g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C c2 (CHead x0 -(Flat f) x1))).(\lambda (H7: (csuba g c x0)).(eq_ind_r C (CHead x0 (Flat f) -x1) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 -(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind -Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 -d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc -g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a))))))) (let H8 \def (H0 d1 u1 H4 g x0 H7) in (or_ind (ex2 C (\lambda (d2: -C).(drop (S n) O x0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 -d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S -n) O x0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (or (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u1))) (\lambda -(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))) (\lambda (H9: (ex2 C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 -(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda -(d2: C).(drop (S n) O x0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba -g d1 d2)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) -(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 -(Flat f) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x: C).(\lambda (H10: -(drop (S n) O x0 (CHead x (Bind Abst) u1))).(\lambda (H11: (csuba g d1 -x)).(or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) -(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 -(Flat f) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u1))) (\lambda -(d2: C).(csuba g d1 d2)) x (drop_drop (Flat f) n x0 (CHead x (Bind Abst) u1) -H10 x1) H11))))) H9)) (\lambda (H9: (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(drop (S n) O x0 (CHead d2 (Bind Abbr) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop (S n) O x0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda -(_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u1))) (\lambda -(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))) (\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: A).(\lambda (H10: -(drop (S n) O x0 (CHead x2 (Bind Abbr) x3))).(\lambda (H11: (csuba g d1 -x2)).(\lambda (H12: (arity g d1 u1 (asucc g x4))).(\lambda (H13: (arity g x2 -x3 x4)).(or_intror (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) -x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T -A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 -(Flat f) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) -(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity -g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 a)))) x2 x3 x4 (drop_drop (Flat f) n x0 (CHead x2 (Bind -Abbr) x3) H10 x1) H11 H12 H13))))))))) H9)) H8)) c2 H6))))) H5)))))) k H2 -(drop_gen_drop k c (CHead d1 (Bind Abst) u1) t n H1)))))))))))) c1)))) i). -(* COMMENTS -Initial nodes: 12528 -END *) - -theorem csuba_drop_abst_rev: - \forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).((drop i -O c1 (CHead d1 (Bind Abst) u)) \to (\forall (g: G).(\forall (c2: C).((csuba g -c2 c1) \to (or (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst) -u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda -(u2: T).(drop i O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda -(_: T).(csuba g d2 d1)))))))))))) -\def - \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (d1: -C).(\forall (u: T).((drop n O c1 (CHead d1 (Bind Abst) u)) \to (\forall (g: -G).(\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: C).(drop n -O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop n O c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))))) (\lambda (c1: -C).(\lambda (d1: C).(\lambda (u: T).(\lambda (H: (drop O O c1 (CHead d1 (Bind -Abst) u))).(\lambda (g: G).(\lambda (c2: C).(\lambda (H0: (csuba g c2 -c1)).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csuba g c2 c)) H0 (CHead d1 -(Bind Abst) u) (drop_gen_refl c1 (CHead d1 (Bind Abst) u) H)) in (let H_x -\def (csuba_gen_abst_rev g d1 c2 u H1) in (let H2 \def H_x in (or_ind (ex2 C -(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba -g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or -(ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O -c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))) (\lambda (H3: (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) -u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(eq C c2 -(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C -(\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O -c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))) (\lambda (x: C).(\lambda (H4: (eq C c2 (CHead x (Bind Abst) -u))).(\lambda (H5: (csuba g x d1)).(eq_ind_r C (CHead x (Bind Abst) u) -(\lambda (c: C).(or (ex2 C (\lambda (d2: C).(drop O O c (CHead d2 (Bind Abst) -u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda -(u2: T).(drop O O c (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda -(_: T).(csuba g d2 d1)))))) (or_introl (ex2 C (\lambda (d2: C).(drop O O -(CHead x (Bind Abst) u) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g -d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O (CHead x -(Bind Abst) u) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop O O (CHead x (Bind -Abst) u) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x -(drop_refl (CHead x (Bind Abst) u)) H5)) c2 H4)))) H3)) (\lambda (H3: (ex2_2 -C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda -(d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1))) (or (ex2 C (\lambda (d2: C).(drop O O c2 -(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop O O c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (H4: (eq C c2 (CHead x0 (Bind Void) -x1))).(\lambda (H5: (csuba g x0 d1)).(eq_ind_r C (CHead x0 (Bind Void) x1) -(\lambda (c: C).(or (ex2 C (\lambda (d2: C).(drop O O c (CHead d2 (Bind Abst) -u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda -(u2: T).(drop O O c (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda -(_: T).(csuba g d2 d1)))))) (or_intror (ex2 C (\lambda (d2: C).(drop O O -(CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba -g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O (CHead x0 -(Bind Void) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: -T).(drop O O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1 (drop_refl (CHead x0 (Bind -Void) x1)) H5)) c2 H4))))) H3)) H2))))))))))) (\lambda (n: nat).(\lambda (H: -((\forall (c1: C).(\forall (d1: C).(\forall (u: T).((drop n O c1 (CHead d1 -(Bind Abst) u)) \to (\forall (g: G).(\forall (c2: C).((csuba g c2 c1) \to (or -(ex2 C (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O -c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (d1: -C).(\forall (u: T).((drop (S n) O c (CHead d1 (Bind Abst) u)) \to (\forall -(g: G).(\forall (c2: C).((csuba g c2 c) \to (or (ex2 C (\lambda (d2: C).(drop -(S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))))))) -(\lambda (n0: nat).(\lambda (d1: C).(\lambda (u: T).(\lambda (H0: (drop (S n) -O (CSort n0) (CHead d1 (Bind Abst) u))).(\lambda (g: G).(\lambda (c2: -C).(\lambda (_: (csuba g c2 (CSort n0))).(and3_ind (eq C (CHead d1 (Bind -Abst) u) (CSort n0)) (eq nat (S n) O) (eq nat O O) (or (ex2 C (\lambda (d2: -C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 -d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) -(\lambda (_: (eq C (CHead d1 (Bind Abst) u) (CSort n0))).(\lambda (H3: (eq -nat (S n) O)).(\lambda (_: (eq nat O O)).(let H5 \def (eq_ind nat (S n) -(\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O -\Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind (or (ex2 -C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) -O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g -d2 d1))))) H5))))) (drop_gen_sort n0 (S n) O (CHead d1 (Bind Abst) u) -H0))))))))) (\lambda (c: C).(\lambda (H0: ((\forall (d1: C).(\forall (u: -T).((drop (S n) O c (CHead d1 (Bind Abst) u)) \to (\forall (g: G).(\forall -(c2: C).((csuba g c2 c) \to (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 -(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))))).(\lambda -(k: K).(\lambda (t: T).(\lambda (d1: C).(\lambda (u: T).(\lambda (H1: (drop -(S n) O (CHead c k t) (CHead d1 (Bind Abst) u))).(\lambda (g: G).(\lambda -(c2: C).(\lambda (H2: (csuba g c2 (CHead c k t))).(K_ind (\lambda (k0: -K).((csuba g c2 (CHead c k0 t)) \to ((drop (r k0 n) O c (CHead d1 (Bind Abst) -u)) \to (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) -u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda -(u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))))))) (\lambda (b: B).(\lambda (H3: -(csuba g c2 (CHead c (Bind b) t))).(\lambda (H4: (drop (r (Bind b) n) O c -(CHead d1 (Bind Abst) u))).(B_ind (\lambda (b0: B).((csuba g c2 (CHead c -(Bind b0) t)) \to ((drop (r (Bind b0) n) O c (CHead d1 (Bind Abst) u)) \to -(or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda -(_: T).(csuba g d2 d1)))))))) (\lambda (H5: (csuba g c2 (CHead c (Bind Abbr) -t))).(\lambda (H6: (drop (r (Bind Abbr) n) O c (CHead d1 (Bind Abst) -u))).(let H_x \def (csuba_gen_abbr_rev g c c2 t H5) in (let H7 \def H_x in -(or3_ind (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) (\lambda -(d2: C).(csuba g d2 c))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 c)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 c)))) (or (ex2 C (\lambda (d2: -C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 -d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) -(\lambda (H8: (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) -(\lambda (d2: C).(csuba g d2 c)))).(ex2_ind C (\lambda (d2: C).(eq C c2 -(CHead d2 (Bind Abbr) t))) (\lambda (d2: C).(csuba g d2 c)) (or (ex2 C -(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) -O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g -d2 d1))))) (\lambda (x: C).(\lambda (H9: (eq C c2 (CHead x (Bind Abbr) -t))).(\lambda (H10: (csuba g x c)).(eq_ind_r C (CHead x (Bind Abbr) t) -(\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind -Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 (Bind Void) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H11 \def (H c d1 u H6 g x -H10) in (or_ind (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) -u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda -(u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda -(_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x -(Bind Abbr) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind -Abbr) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba -g d2 d1))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 -(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: -C).(drop n O x (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) -(or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 -(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda -(x0: C).(\lambda (H13: (drop n O x (CHead x0 (Bind Abst) u))).(\lambda (H14: -(csuba g x0 d1)).(or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x -(Bind Abbr) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind -Abbr) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba -g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) -t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x0 (drop_drop -(Bind Abbr) n x (CHead x0 (Bind Abst) u) H13 t) H14))))) H12)) (\lambda (H12: -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind -C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or (ex2 C (\lambda -(d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (H13: (drop n O x (CHead x0 (Bind Void) -x1))).(\lambda (H14: (csuba g x0 d1)).(or_intror (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u))) (\lambda -(d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop -(S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1 -(drop_drop (Bind Abbr) n x (CHead x0 (Bind Void) x1) H13 t) H14)))))) H12)) -H11)) c2 H9)))) H8)) (\lambda (H8: (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 c)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t a)))))).(ex4_3_ind C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -c)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc -g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t a)))) -(or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda -(_: T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: -A).(\lambda (H9: (eq C c2 (CHead x0 (Bind Abst) x1))).(\lambda (H10: (csuba g -x0 c)).(\lambda (_: (arity g x0 x1 (asucc g x2))).(\lambda (_: (arity g c t -x2)).(eq_ind_r C (CHead x0 (Bind Abst) x1) (\lambda (c0: C).(or (ex2 C -(\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) -O c0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g -d2 d1)))))) (let H13 \def (H c d1 u H6 g x0 H10) in (or_ind (ex2 C (\lambda -(d2: C).(drop n O x0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 -d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or -(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 -(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda -(H14: (ex2 C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop n O x0 -(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C -(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind -Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda -(x: C).(\lambda (H15: (drop n O x0 (CHead x (Bind Abst) u))).(\lambda (H16: -(csuba g x d1)).(or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 -(Bind Abst) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind -Abst) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x0 -(Bind Abst) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) -x (drop_drop (Bind Abst) n x0 (CHead x (Bind Abst) u) H15 x1) H16))))) H14)) -(\lambda (H14: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead -d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or -(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 -(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda -(x3: C).(\lambda (x4: T).(\lambda (H15: (drop n O x0 (CHead x3 (Bind Void) -x4))).(\lambda (H16: (csuba g x3 d1)).(or_intror (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda -(d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x3 x4 -(drop_drop (Bind Abst) n x0 (CHead x3 (Bind Void) x4) H15 x1) H16)))))) H14)) -H13)) c2 H9)))))))) H8)) (\lambda (H8: (ex2_2 C T (\lambda (d2: C).(\lambda -(u2: T).(eq C c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 c))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(eq C -c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -c))) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda -(_: T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H9: -(eq C c2 (CHead x0 (Bind Void) x1))).(\lambda (H10: (csuba g x0 c)).(eq_ind_r -C (CHead x0 (Bind Void) x1) (\lambda (c0: C).(or (ex2 C (\lambda (d2: -C).(drop (S n) O c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 -d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let -H11 \def (H c d1 u H6 g x0 H10) in (or_ind (ex2 C (\lambda (d2: C).(drop n O -x0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H12: (ex2 C -(\lambda (d2: C).(drop n O x0 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop n O x0 (CHead d2 -(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda -(H13: (drop n O x0 (CHead x (Bind Abst) u))).(\lambda (H14: (csuba g x -d1)).(or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) -x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) -(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop -(Bind Void) n x0 (CHead x (Bind Abst) u) H13 x1) H14))))) H12)) (\lambda -(H12: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead -d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or -(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 -(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda -(x2: C).(\lambda (x3: T).(\lambda (H13: (drop n O x0 (CHead x2 (Bind Void) -x3))).(\lambda (H14: (csuba g x2 d1)).(or_intror (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda -(d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x2 x3 -(drop_drop (Bind Void) n x0 (CHead x2 (Bind Void) x3) H13 x1) H14)))))) H12)) -H11)) c2 H9))))) H8)) H7))))) (\lambda (H5: (csuba g c2 (CHead c (Bind Abst) -t))).(\lambda (H6: (drop (r (Bind Abst) n) O c (CHead d1 (Bind Abst) -u))).(let H_x \def (csuba_gen_abst_rev g c c2 t H5) in (let H7 \def H_x in -(or_ind (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda -(d2: C).(csuba g d2 c))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C -c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -c)))) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda -(_: T).(csuba g d2 d1))))) (\lambda (H8: (ex2 C (\lambda (d2: C).(eq C c2 -(CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba g d2 c)))).(ex2_ind C -(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba -g d2 c)) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) -u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda -(u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda (H9: (eq C c2 -(CHead x (Bind Abst) t))).(\lambda (H10: (csuba g x c)).(eq_ind_r C (CHead x -(Bind Abst) t) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 -(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H11 \def (H -c d1 u H6 g x H10) in (or_ind (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 -(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: C).(drop (S n) -O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba -g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead -x (Bind Abst) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x -(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C -(\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind -Abst) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 -C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abst) t) -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))) (\lambda (x0: C).(\lambda (H13: (drop n O x (CHead x0 (Bind Abst) -u))).(\lambda (H14: (csuba g x0 d1)).(or_introl (ex2 C (\lambda (d2: C).(drop -(S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) -O (CHead x (Bind Abst) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S -n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1)) x0 (drop_drop (Bind Abst) n x (CHead x0 (Bind Abst) u) -H13 t) H14))))) H12)) (\lambda (H12: (ex2_2 C T (\lambda (d2: C).(\lambda -(u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda -(_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: -T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind -Abst) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 -C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abst) t) -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H13: (drop n O x (CHead -x0 (Bind Void) x1))).(\lambda (H14: (csuba g x0 d1)).(or_intror (ex2 C -(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) -u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda -(u2: T).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda -(d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1 -(drop_drop (Bind Abst) n x (CHead x0 (Bind Void) x1) H13 t) H14)))))) H12)) -H11)) c2 H9)))) H8)) (\lambda (H8: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(eq C c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 c))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(eq C -c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -c))) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda -(_: T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H9: -(eq C c2 (CHead x0 (Bind Void) x1))).(\lambda (H10: (csuba g x0 c)).(eq_ind_r -C (CHead x0 (Bind Void) x1) (\lambda (c0: C).(or (ex2 C (\lambda (d2: -C).(drop (S n) O c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 -d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let -H11 \def (H c d1 u H6 g x0 H10) in (or_ind (ex2 C (\lambda (d2: C).(drop n O -x0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H12: (ex2 C -(\lambda (d2: C).(drop n O x0 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop n O x0 (CHead d2 -(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda -(H13: (drop n O x0 (CHead x (Bind Abst) u))).(\lambda (H14: (csuba g x -d1)).(or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) -x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) -(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop -(Bind Void) n x0 (CHead x (Bind Abst) u) H13 x1) H14))))) H12)) (\lambda -(H12: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead -d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or -(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 -(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda -(x2: C).(\lambda (x3: T).(\lambda (H13: (drop n O x0 (CHead x2 (Bind Void) -x3))).(\lambda (H14: (csuba g x2 d1)).(or_intror (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda -(d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x2 x3 -(drop_drop (Bind Void) n x0 (CHead x2 (Bind Void) x3) H13 x1) H14)))))) H12)) -H11)) c2 H9))))) H8)) H7))))) (\lambda (H5: (csuba g c2 (CHead c (Bind Void) -t))).(\lambda (H6: (drop (r (Bind Void) n) O c (CHead d1 (Bind Abst) -u))).(let H_x \def (csuba_gen_void_rev g c c2 t H5) in (let H7 \def H_x in -(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Void) t))) (\lambda (d2: -C).(csuba g d2 c)) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 -(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda (H8: (eq -C c2 (CHead x (Bind Void) t))).(\lambda (H9: (csuba g x c)).(eq_ind_r C -(CHead x (Bind Void) t) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S -n) O c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 -C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H10 \def (H -c d1 u H6 g x H9) in (or_ind (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 -(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: C).(drop (S n) -O (CHead x (Bind Void) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba -g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead -x (Bind Void) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))) (\lambda (H11: (ex2 C (\lambda (d2: C).(drop n O x -(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C -(\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind -Void) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 -C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Void) t) -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))) (\lambda (x0: C).(\lambda (H12: (drop n O x (CHead x0 (Bind Abst) -u))).(\lambda (H13: (csuba g x0 d1)).(or_introl (ex2 C (\lambda (d2: C).(drop -(S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) -O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S -n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1)) x0 (drop_drop (Bind Void) n x (CHead x0 (Bind Abst) u) -H12 t) H13))))) H11)) (\lambda (H11: (ex2_2 C T (\lambda (d2: C).(\lambda -(u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda -(_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: -T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind -Void) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 -C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Void) t) -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H12: (drop n O x (CHead -x0 (Bind Void) x1))).(\lambda (H13: (csuba g x0 d1)).(or_intror (ex2 C -(\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abst) -u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda -(u2: T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda -(d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1 -(drop_drop (Bind Void) n x (CHead x0 (Bind Void) x1) H12 t) H13)))))) H11)) -H10)) c2 H8)))) H7))))) b H3 H4)))) (\lambda (f: F).(\lambda (H3: (csuba g c2 -(CHead c (Flat f) t))).(\lambda (H4: (drop (r (Flat f) n) O c (CHead d1 (Bind -Abst) u))).(let H_x \def (csuba_gen_flat_rev g c c2 t f H3) in (let H5 \def -H_x in (ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 -(Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 c))) (or (ex2 C -(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) -O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g -d2 d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C c2 (CHead x0 -(Flat f) x1))).(\lambda (H7: (csuba g x0 c)).(eq_ind_r C (CHead x0 (Flat f) -x1) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 -(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 (Bind Void) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H8 \def (H0 d1 u H4 g x0 -H7) in (or_ind (ex2 C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind Abst) -u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda -(u2: T).(drop (S n) O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: C).(drop (S n) -O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g -d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 -(Flat f) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))) (\lambda (H9: (ex2 C (\lambda (d2: C).(drop (S n) O x0 -(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C -(\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat -f) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C -T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f) x1) -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))) (\lambda (x: C).(\lambda (H10: (drop (S n) O x0 (CHead x (Bind Abst) -u))).(\lambda (H11: (csuba g x d1)).(or_introl (ex2 C (\lambda (d2: C).(drop -(S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) -O (CHead x0 (Flat f) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S -n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1)) x (drop_drop (Flat f) n x0 (CHead x (Bind Abst) u) H10 -x1) H11))))) H9)) (\lambda (H9: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda -(_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda -(_: T).(csuba g d2 d1))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 -(Flat f) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f) -x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))) (\lambda (x2: C).(\lambda (x3: T).(\lambda (H10: (drop (S n) O x0 -(CHead x2 (Bind Void) x3))).(\lambda (H11: (csuba g x2 d1)).(or_intror (ex2 C -(\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) -u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda -(u2: T).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda -(d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x2 x3 -(drop_drop (Flat f) n x0 (CHead x2 (Bind Void) x3) H10 x1) H11)))))) H9)) -H8)) c2 H6))))) H5)))))) k H2 (drop_gen_drop k c (CHead d1 (Bind Abst) u) t n -H1)))))))))))) c1)))) i). -(* COMMENTS -Initial nodes: 11438 -END *) - -theorem csuba_drop_abbr_rev: - \forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u1: T).((drop i -O c1 (CHead d1 (Bind Abbr) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba -g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop i O c2 (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))))))) -\def - \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (d1: -C).(\forall (u1: T).((drop n O c1 (CHead d1 (Bind Abbr) u1)) \to (\forall (g: -G).(\forall (c2: C).((csuba g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(drop n -O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C -T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O c2 (CHead d2 -(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O c2 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))))))))))) (\lambda (c1: C).(\lambda (d1: C).(\lambda (u1: T).(\lambda -(H: (drop O O c1 (CHead d1 (Bind Abbr) u1))).(\lambda (g: G).(\lambda (c2: -C).(\lambda (H0: (csuba g c2 c1)).(let H1 \def (eq_ind C c1 (\lambda (c: -C).(csuba g c2 c)) H0 (CHead d1 (Bind Abbr) u1) (drop_gen_refl c1 (CHead d1 -(Bind Abbr) u1) H)) in (let H_x \def (csuba_gen_abbr_rev g d1 c2 u1 H1) in -(let H2 \def H_x in (or3_ind (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 -C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: -C).(drop O O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O -O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop O O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))) (\lambda (H3: (ex2 C (\lambda (d2: C).(eq C c2 (CHead -d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda -(d2: C).(eq C c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1)) (or3 (ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abbr) u1))) -(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(drop O O c2 (CHead d2 (Bind Abst) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop O O c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda -(H4: (eq C c2 (CHead x (Bind Abbr) u1))).(\lambda (H5: (csuba g x -d1)).(eq_ind_r C (CHead x (Bind Abbr) u1) (\lambda (c: C).(or3 (ex2 C -(\lambda (d2: C).(drop O O c (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(drop O O c (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda -(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop O O c (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))))) (or3_intro0 (ex2 C (\lambda (d2: -C).(drop O O (CHead x (Bind Abbr) u1) (CHead d2 (Bind Abbr) u1))) (\lambda -(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop O O (CHead x (Bind Abbr) u1) (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O (CHead x (Bind Abbr) -u1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))) (ex_intro2 C (\lambda (d2: C).(drop O O (CHead x (Bind Abbr) u1) -(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_refl -(CHead x (Bind Abbr) u1)) H5)) c2 H4)))) H3)) (\lambda (H3: (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(eq C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C (\lambda (d2: -C).(drop O O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O -O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop O O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: -A).(\lambda (H4: (eq C c2 (CHead x0 (Bind Abst) x1))).(\lambda (H5: (csuba g -x0 d1)).(\lambda (H6: (arity g x0 x1 (asucc g x2))).(\lambda (H7: (arity g d1 -u1 x2)).(eq_ind_r C (CHead x0 (Bind Abst) x1) (\lambda (c: C).(or3 (ex2 C -(\lambda (d2: C).(drop O O c (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(drop O O c (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda -(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop O O c (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))))) (or3_intro1 (ex2 C (\lambda (d2: -C).(drop O O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abbr) u1))) (\lambda -(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop O O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O (CHead x0 (Bind Abst) -x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop O O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x0 x1 x2 -(drop_refl (CHead x0 (Bind Abst) x1)) H5 H6 H7)) c2 H4)))))))) H3)) (\lambda -(H3: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind -C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C (\lambda (d2: -C).(drop O O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O -O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop O O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H4: (eq C -c2 (CHead x0 (Bind Void) x1))).(\lambda (H5: (csuba g x0 d1)).(eq_ind_r C -(CHead x0 (Bind Void) x1) (\lambda (c: C).(or3 (ex2 C (\lambda (d2: C).(drop -O O c (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C -T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O c (CHead d2 -(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O c (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) -(or3_intro2 (ex2 C (\lambda (d2: C).(drop O O (CHead x0 (Bind Void) x1) -(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O (CHead x0 (Bind -Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop O O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T -(\lambda (d2: C).(\lambda (u2: T).(drop O O (CHead x0 (Bind Void) x1) (CHead -d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 -x1 (drop_refl (CHead x0 (Bind Void) x1)) H5)) c2 H4))))) H3)) H2))))))))))) -(\lambda (n: nat).(\lambda (H: ((\forall (c1: C).(\forall (d1: C).(\forall -(u1: T).((drop n O c1 (CHead d1 (Bind Abbr) u1)) \to (\forall (g: G).(\forall -(c2: C).((csuba g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(drop n O c2 (CHead -d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O c2 (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O c2 (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (d1: -C).(\forall (u1: T).((drop (S n) O c (CHead d1 (Bind Abbr) u1)) \to (\forall -(g: G).(\forall (c2: C).((csuba g c2 c) \to (or3 (ex2 C (\lambda (d2: -C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S -n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1)))))))))))) (\lambda (n0: -nat).(\lambda (d1: C).(\lambda (u1: T).(\lambda (H0: (drop (S n) O (CSort n0) -(CHead d1 (Bind Abbr) u1))).(\lambda (g: G).(\lambda (c2: C).(\lambda (_: -(csuba g c2 (CSort n0))).(and3_ind (eq C (CHead d1 (Bind Abbr) u1) (CSort -n0)) (eq nat (S n) O) (eq nat O O) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O -c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T -A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead -d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda -(_: T).(csuba g d2 d1))))) (\lambda (_: (eq C (CHead d1 (Bind Abbr) u1) -(CSort n0))).(\lambda (H3: (eq nat (S n) O)).(\lambda (_: (eq nat O O)).(let -H5 \def (eq_ind nat (S n) (\lambda (ee: nat).(match ee in nat return (\lambda -(_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H3) -in (False_ind (or3 (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) H5))))) -(drop_gen_sort n0 (S n) O (CHead d1 (Bind Abbr) u1) H0))))))))) (\lambda (c: -C).(\lambda (H0: ((\forall (d1: C).(\forall (u1: T).((drop (S n) O c (CHead -d1 (Bind Abbr) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba g c2 c) \to -(or3 (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) -(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 -C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))))).(\lambda -(k: K).(\lambda (t: T).(\lambda (d1: C).(\lambda (u1: T).(\lambda (H1: (drop -(S n) O (CHead c k t) (CHead d1 (Bind Abbr) u1))).(\lambda (g: G).(\lambda -(c2: C).(\lambda (H2: (csuba g c2 (CHead c k t))).(K_ind (\lambda (k0: -K).((csuba g c2 (CHead c k0 t)) \to ((drop (r k0 n) O c (CHead d1 (Bind Abbr) -u1)) \to (or3 (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))) (\lambda -(b: B).(\lambda (H3: (csuba g c2 (CHead c (Bind b) t))).(\lambda (H4: (drop -(r (Bind b) n) O c (CHead d1 (Bind Abbr) u1))).(B_ind (\lambda (b0: -B).((csuba g c2 (CHead c (Bind b0) t)) \to ((drop (r (Bind b0) n) O c (CHead -d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead -d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))))))) (\lambda (H5: (csuba g c2 (CHead c (Bind Abbr) t))).(\lambda (H6: -(drop (r (Bind Abbr) n) O c (CHead d1 (Bind Abbr) u1))).(let H_x \def -(csuba_gen_abbr_rev g c c2 t H5) in (let H7 \def H_x in (or3_ind (ex2 C -(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) (\lambda (d2: C).(csuba -g d2 c))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq -C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 c)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g c t a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C -c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -c)))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda -(H8: (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) (\lambda -(d2: C).(csuba g d2 c)))).(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 -(Bind Abbr) t))) (\lambda (d2: C).(csuba g d2 c)) (or3 (ex2 C (\lambda (d2: -C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S -n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda (H9: (eq -C c2 (CHead x (Bind Abbr) t))).(\lambda (H10: (csuba g x c)).(eq_ind_r C -(CHead x (Bind Abbr) t) (\lambda (c0: C).(or3 (ex2 C (\lambda (d2: C).(drop -(S n) O c0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O -c0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O c0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda -(_: T).(csuba g d2 d1)))))) (let H11 \def (H c d1 u1 H6 g x H10) in (or3_ind -(ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(drop n O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda -(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(drop (S -n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 -C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abbr) t) -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: -C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) -(or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 -(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) -t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: -C).(\lambda (H13: (drop n O x (CHead x0 (Bind Abbr) u1))).(\lambda (H14: -(csuba g x0 d1)).(or3_intro0 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x -(Bind Abbr) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O -(CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abbr) t) -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) -(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x0 (drop_drop -(Bind Abbr) n x (CHead x0 (Bind Abbr) u1) H13 t) H14))))) H12)) (\lambda -(H12: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n -O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C -(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t) -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H13: (drop n O x (CHead x0 -(Bind Abst) x1))).(\lambda (H14: (csuba g x0 d1)).(\lambda (H15: (arity g x0 -x1 (asucc g x2))).(\lambda (H16: (arity g d1 u1 x2)).(or3_intro1 (ex2 C -(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t) -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex4_3_intro C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x -(Bind Abbr) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a)))) x0 x1 x2 (drop_drop (Bind Abbr) n -x (CHead x0 (Bind Abst) x1) H13 t) H14 H15 H16))))))))) H12)) (\lambda (H12: -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind -C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C -(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t) -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (H13: (drop n O x (CHead x0 (Bind Void) -x1))).(\lambda (H14: (csuba g x0 d1)).(or3_intro2 (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u1))) (\lambda -(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x -(Bind Abbr) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1 (drop_drop (Bind -Abbr) n x (CHead x0 (Bind Void) x1) H13 t) H14)))))) H12)) H11)) c2 H9)))) -H8)) (\lambda (H8: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(eq C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 c)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g c t a)))))).(ex4_3_ind C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 c)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t a)))) (or3 (ex2 -C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H9: (eq C c2 (CHead x0 (Bind -Abst) x1))).(\lambda (H10: (csuba g x0 c)).(\lambda (_: (arity g x0 x1 (asucc -g x2))).(\lambda (_: (arity g c t x2)).(eq_ind_r C (CHead x0 (Bind Abst) x1) -(\lambda (c0: C).(or3 (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 -(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0 -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))))) (let H13 \def (H c d1 u1 H6 g x0 H10) in (or3_ind (ex2 C (\lambda -(d2: C).(drop n O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n -O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 -(Bind Abst) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S -n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))) (\lambda (H14: (ex2 C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: -C).(drop n O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) -(or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead -d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abst) -x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda -(H15: (drop n O x0 (CHead x (Bind Abbr) u1))).(\lambda (H16: (csuba g x -d1)).(or3_intro0 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) -x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T -A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 -(Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C -(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop (Bind Abst) n x0 -(CHead x (Bind Abbr) u1) H15 x1) H16))))) H14)) (\lambda (H14: (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x0 (CHead d2 -(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop n O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abbr) u1))) -(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 -(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead -x0 (Bind Abst) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: -A).(\lambda (H15: (drop n O x0 (CHead x3 (Bind Abst) x4))).(\lambda (H16: -(csuba g x3 d1)).(\lambda (H17: (arity g x3 x4 (asucc g x5))).(\lambda (H18: -(arity g d1 u1 x5)).(or3_intro1 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead -x0 (Bind Abst) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S -n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x3 x4 x5 -(drop_drop (Bind Abst) n x0 (CHead x3 (Bind Abst) x4) H15 x1) H16 H17 -H18))))))))) H14)) (\lambda (H14: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(drop -n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g -d2 d1))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) -(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 -(Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda -(x3: C).(\lambda (x4: T).(\lambda (H15: (drop n O x0 (CHead x3 (Bind Void) -x4))).(\lambda (H16: (csuba g x3 d1)).(or3_intro2 (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abbr) u1))) -(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 -(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead -x0 (Bind Abst) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x3 x4 (drop_drop (Bind -Abst) n x0 (CHead x3 (Bind Void) x4) H15 x1) H16)))))) H14)) H13)) c2 -H9)))))))) H8)) (\lambda (H8: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(eq C c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 c))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(eq C -c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -c))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda -(x0: C).(\lambda (x1: T).(\lambda (H9: (eq C c2 (CHead x0 (Bind Void) -x1))).(\lambda (H10: (csuba g x0 c)).(eq_ind_r C (CHead x0 (Bind Void) x1) -(\lambda (c0: C).(or3 (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 -(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0 -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))))) (let H11 \def (H c d1 u1 H6 g x0 H10) in (or3_ind (ex2 C (\lambda -(d2: C).(drop n O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n -O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 -(Bind Void) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S -n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: -C).(drop n O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) -(or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead -d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Void) -x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda -(H13: (drop n O x0 (CHead x (Bind Abbr) u1))).(\lambda (H14: (csuba g x -d1)).(or3_intro0 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) -x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T -A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 -(Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C -(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop (Bind Void) n x0 -(CHead x (Bind Abbr) u1) H13 x1) H14))))) H12)) (\lambda (H12: (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x0 (CHead d2 -(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop n O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abbr) u1))) -(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 -(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead -x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))) (\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: -A).(\lambda (H13: (drop n O x0 (CHead x2 (Bind Abst) x3))).(\lambda (H14: -(csuba g x2 d1)).(\lambda (H15: (arity g x2 x3 (asucc g x4))).(\lambda (H16: -(arity g d1 u1 x4)).(or3_intro1 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead -x0 (Bind Void) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S -n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x2 x3 x4 -(drop_drop (Bind Void) n x0 (CHead x2 (Bind Abst) x3) H13 x1) H14 H15 -H16))))))))) H12)) (\lambda (H12: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(drop -n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g -d2 d1))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) -(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 -(Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda -(x2: C).(\lambda (x3: T).(\lambda (H13: (drop n O x0 (CHead x2 (Bind Void) -x3))).(\lambda (H14: (csuba g x2 d1)).(or3_intro2 (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abbr) u1))) -(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 -(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead -x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x2 x3 (drop_drop (Bind -Void) n x0 (CHead x2 (Bind Void) x3) H13 x1) H14)))))) H12)) H11)) c2 H9))))) -H8)) H7))))) (\lambda (H5: (csuba g c2 (CHead c (Bind Abst) t))).(\lambda -(H6: (drop (r (Bind Abst) n) O c (CHead d1 (Bind Abbr) u1))).(let H_x \def -(csuba_gen_abst_rev g c c2 t H5) in (let H7 \def H_x in (or_ind (ex2 C -(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba -g d2 c))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 c)))) (or3 -(ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda -(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H8: -(ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2: -C).(csuba g d2 c)))).(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind -Abst) t))) (\lambda (d2: C).(csuba g d2 c)) (or3 (ex2 C (\lambda (d2: -C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S -n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda (H9: (eq -C c2 (CHead x (Bind Abst) t))).(\lambda (H10: (csuba g x c)).(eq_ind_r C -(CHead x (Bind Abst) t) (\lambda (c0: C).(or3 (ex2 C (\lambda (d2: C).(drop -(S n) O c0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O -c0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O c0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda -(_: T).(csuba g d2 d1)))))) (let H11 \def (H c d1 u1 H6 g x H10) in (or3_ind -(ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(drop n O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda -(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(drop (S -n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 -C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abst) t) -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: -C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) -(or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 -(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) -t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: -C).(\lambda (H13: (drop n O x (CHead x0 (Bind Abbr) u1))).(\lambda (H14: -(csuba g x0 d1)).(or3_intro0 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x -(Bind Abst) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O -(CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abst) t) -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) -(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x0 (drop_drop -(Bind Abst) n x (CHead x0 (Bind Abbr) u1) H13 t) H14))))) H12)) (\lambda -(H12: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n -O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C -(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H13: (drop n O x (CHead x0 -(Bind Abst) x1))).(\lambda (H14: (csuba g x0 d1)).(\lambda (H15: (arity g x0 -x1 (asucc g x2))).(\lambda (H16: (arity g d1 u1 x2)).(or3_intro1 (ex2 C -(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex4_3_intro C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x -(Bind Abst) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a)))) x0 x1 x2 (drop_drop (Bind Abst) n -x (CHead x0 (Bind Abst) x1) H13 t) H14 H15 H16))))))))) H12)) (\lambda (H12: -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind -C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C -(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (H13: (drop n O x (CHead x0 (Bind Void) -x1))).(\lambda (H14: (csuba g x0 d1)).(or3_intro2 (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u1))) (\lambda -(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x -(Bind Abst) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1 (drop_drop (Bind -Abst) n x (CHead x0 (Bind Void) x1) H13 t) H14)))))) H12)) H11)) c2 H9)))) -H8)) (\lambda (H8: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -c))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 c))) (or3 -(ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda -(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (H9: (eq C c2 (CHead x0 (Bind Void) -x1))).(\lambda (H10: (csuba g x0 c)).(eq_ind_r C (CHead x0 (Bind Void) x1) -(\lambda (c0: C).(or3 (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 -(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0 -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))))) (let H11 \def (H c d1 u1 H6 g x0 H10) in (or3_ind (ex2 C (\lambda -(d2: C).(drop n O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n -O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 -(Bind Void) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S -n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: -C).(drop n O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) -(or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead -d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Void) -x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda -(H13: (drop n O x0 (CHead x (Bind Abbr) u1))).(\lambda (H14: (csuba g x -d1)).(or3_intro0 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) -x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T -A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 -(Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C -(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop (Bind Void) n x0 -(CHead x (Bind Abbr) u1) H13 x1) H14))))) H12)) (\lambda (H12: (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x0 (CHead d2 -(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop n O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abbr) u1))) -(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 -(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead -x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))) (\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: -A).(\lambda (H13: (drop n O x0 (CHead x2 (Bind Abst) x3))).(\lambda (H14: -(csuba g x2 d1)).(\lambda (H15: (arity g x2 x3 (asucc g x4))).(\lambda (H16: -(arity g d1 u1 x4)).(or3_intro1 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead -x0 (Bind Void) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S -n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x2 x3 x4 -(drop_drop (Bind Void) n x0 (CHead x2 (Bind Abst) x3) H13 x1) H14 H15 -H16))))))))) H12)) (\lambda (H12: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(drop -n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g -d2 d1))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) -(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 -(Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda -(x2: C).(\lambda (x3: T).(\lambda (H13: (drop n O x0 (CHead x2 (Bind Void) -x3))).(\lambda (H14: (csuba g x2 d1)).(or3_intro2 (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abbr) u1))) -(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 -(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead -x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x2 x3 (drop_drop (Bind -Void) n x0 (CHead x2 (Bind Void) x3) H13 x1) H14)))))) H12)) H11)) c2 H9))))) -H8)) H7))))) (\lambda (H5: (csuba g c2 (CHead c (Bind Void) t))).(\lambda -(H6: (drop (r (Bind Void) n) O c (CHead d1 (Bind Abbr) u1))).(let H_x \def -(csuba_gen_void_rev g c c2 t H5) in (let H7 \def H_x in (ex2_ind C (\lambda -(d2: C).(eq C c2 (CHead d2 (Bind Void) t))) (\lambda (d2: C).(csuba g d2 c)) -(or3 (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) -(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 -C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: -C).(\lambda (H8: (eq C c2 (CHead x (Bind Void) t))).(\lambda (H9: (csuba g x -c)).(eq_ind_r C (CHead x (Bind Void) t) (\lambda (c0: C).(or3 (ex2 C (\lambda -(d2: C).(drop (S n) O c0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba -g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop (S n) O c0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda -(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 (Bind Void) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H10 \def (H c d1 u1 H6 g x -H9) in (or3_ind (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 -C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C -(\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H11: (ex2 C -(\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop n O x (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) (or3 (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) u1))) (\lambda -(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x -(Bind Void) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (H12: (drop n O x (CHead x0 -(Bind Abbr) u1))).(\lambda (H13: (csuba g x0 d1)).(or3_intro0 (ex2 C (\lambda -(d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) u1))) -(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x -(Bind Void) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x -(Bind Void) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) -x0 (drop_drop (Bind Void) n x (CHead x0 (Bind Abbr) u1) H12 t) H13))))) H11)) -(\lambda (H11: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop n O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 -(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H12: (drop n O x (CHead x0 -(Bind Abst) x1))).(\lambda (H13: (csuba g x0 d1)).(\lambda (H14: (arity g x0 -x1 (asucc g x2))).(\lambda (H15: (arity g d1 u1 x2)).(or3_intro1 (ex2 C -(\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex4_3_intro C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x -(Bind Void) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a)))) x0 x1 x2 (drop_drop (Bind Void) n -x (CHead x0 (Bind Abst) x1) H12 t) H13 H14 H15))))))))) H11)) (\lambda (H11: -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind -C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C -(\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (H12: (drop n O x (CHead x0 (Bind Void) -x1))).(\lambda (H13: (csuba g x0 d1)).(or3_intro2 (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) u1))) (\lambda -(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x -(Bind Void) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1 (drop_drop (Bind -Void) n x (CHead x0 (Bind Void) x1) H12 t) H13)))))) H11)) H10)) c2 H8)))) -H7))))) b H3 H4)))) (\lambda (f: F).(\lambda (H3: (csuba g c2 (CHead c (Flat -f) t))).(\lambda (H4: (drop (r (Flat f) n) O c (CHead d1 (Bind Abbr) -u1))).(let H_x \def (csuba_gen_flat_rev g c c2 t f H3) in (let H5 \def H_x in -(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Flat f) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 c))) (or3 (ex2 C (\lambda -(d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba -g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda -(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (x1: -T).(\lambda (H6: (eq C c2 (CHead x0 (Flat f) x1))).(\lambda (H7: (csuba g x0 -c)).(eq_ind_r C (CHead x0 (Flat f) x1) (\lambda (c0: C).(or3 (ex2 C (\lambda -(d2: C).(drop (S n) O c0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba -g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop (S n) O c0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda -(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 (Bind Void) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H8 \def (H0 d1 u1 H4 g x0 -H7) in (or3_ind (ex2 C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O x0 (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O x0 (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C -(\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Void) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H9: (ex2 C (\lambda -(d2: C).(drop (S n) O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba -g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) (or3 (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u1))) (\lambda -(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f) -x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))) (\lambda (x: C).(\lambda (H10: (drop (S n) O x0 (CHead x (Bind Abbr) -u1))).(\lambda (H11: (csuba g x d1)).(or3_intro0 (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u1))) (\lambda -(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f) -x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) -(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop -(Flat f) n x0 (CHead x (Bind Abbr) u1) H10 x1) H11))))) H9)) (\lambda (H9: -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O -x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop (S n) O x0 (CHead d2 (Bind Abst) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C -(\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Void) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x2: C).(\lambda (x3: -T).(\lambda (x4: A).(\lambda (H10: (drop (S n) O x0 (CHead x2 (Bind Abst) -x3))).(\lambda (H11: (csuba g x2 d1)).(\lambda (H12: (arity g x2 x3 (asucc g -x4))).(\lambda (H13: (arity g d1 u1 x4)).(or3_intro1 (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u1))) (\lambda -(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f) -x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x2 x3 x4 -(drop_drop (Flat f) n x0 (CHead x2 (Bind Abst) x3) H10 x1) H11 H12 -H13))))))))) H9)) (\lambda (H9: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda -(_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda -(_: T).(csuba g d2 d1))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 -(Flat f) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O -(CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f) x1) (CHead -d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) -(\lambda (x2: C).(\lambda (x3: T).(\lambda (H10: (drop (S n) O x0 (CHead x2 -(Bind Void) x3))).(\lambda (H11: (csuba g x2 d1)).(or3_intro2 (ex2 C (\lambda -(d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u1))) -(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead -x0 (Flat f) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Void) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1))) x2 x3 (drop_drop (Flat f) n x0 -(CHead x2 (Bind Void) x3) H10 x1) H11)))))) H9)) H8)) c2 H6))))) H5)))))) k -H2 (drop_gen_drop k c (CHead d1 (Bind Abbr) u1) t n H1)))))))))))) c1)))) i). -(* COMMENTS -Initial nodes: 23852 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/fwd.ma deleted file mode 100644 index a618761fe..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/fwd.ma +++ /dev/null @@ -1,1083 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csuba/defs.ma". - -theorem csuba_gen_abbr: - \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g -(CHead d1 (Bind Abbr) u) c) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 -(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))))) -\def - \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u: T).(\lambda (H: -(csuba g (CHead d1 (Bind Abbr) u) c)).(insert_eq C (CHead d1 (Bind Abbr) u) -(\lambda (c0: C).(csuba g c0 c)) (\lambda (_: C).(ex2 C (\lambda (d2: C).(eq -C c (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (\lambda -(y: C).(\lambda (H0: (csuba g y c)).(csuba_ind g (\lambda (c0: C).(\lambda -(c1: C).((eq C c0 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(eq C -c1 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))) (\lambda -(n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Abbr) u))).(let H2 -\def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return (\lambda (_: -C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow -False])) I (CHead d1 (Bind Abbr) u) H1) in (False_ind (ex2 C (\lambda (d2: -C).(eq C (CSort n) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 -d2))) H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 -c2)).(\lambda (H2: (((eq C c1 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda -(d2: C).(eq C c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 -d2)))))).(\lambda (k: K).(\lambda (u0: T).(\lambda (H3: (eq C (CHead c1 k u0) -(CHead d1 (Bind Abbr) u))).(let H4 \def (f_equal C C (\lambda (e: C).(match e -in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _ -_) \Rightarrow c0])) (CHead c1 k u0) (CHead d1 (Bind Abbr) u) H3) in ((let H5 -\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) -with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c1 k -u0) (CHead d1 (Bind Abbr) u) H3) in ((let H6 \def (f_equal C T (\lambda (e: -C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | -(CHead _ _ t) \Rightarrow t])) (CHead c1 k u0) (CHead d1 (Bind Abbr) u) H3) -in (\lambda (H7: (eq K k (Bind Abbr))).(\lambda (H8: (eq C c1 d1)).(eq_ind_r -T u (\lambda (t: T).(ex2 C (\lambda (d2: C).(eq C (CHead c2 k t) (CHead d2 -(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (eq_ind_r K (Bind Abbr) -(\lambda (k0: K).(ex2 C (\lambda (d2: C).(eq C (CHead c2 k0 u) (CHead d2 -(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H9 \def (eq_ind C -c1 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda -(d2: C).(eq C c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 -d2))))) H2 d1 H8) in (let H10 \def (eq_ind C c1 (\lambda (c0: C).(csuba g c0 -c2)) H1 d1 H8) in (ex_intro2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Abbr) -u) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) c2 -(refl_equal C (CHead c2 (Bind Abbr) u)) H10))) k H7) u0 H6)))) H5)) -H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1 -c2)).(\lambda (_: (((eq C c1 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda -(d2: C).(eq C c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 -d2)))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: -T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u1) (CHead d1 -(Bind Abbr) u))).(let H5 \def (eq_ind C (CHead c1 (Bind Void) u1) (\lambda -(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind b0) \Rightarrow (match b0 in B return (\lambda (_: -B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False | Void -\Rightarrow True]) | (Flat _) \Rightarrow False])])) I (CHead d1 (Bind Abbr) -u) H4) in (False_ind (ex2 C (\lambda (d2: C).(eq C (CHead c2 (Bind b) u2) -(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) H5))))))))))) -(\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1 c2)).(\lambda (_: -(((eq C c1 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(eq C c2 -(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))).(\lambda (t: -T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g a))).(\lambda (u0: -T).(\lambda (_: (arity g c2 u0 a)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) -t) (CHead d1 (Bind Abbr) u))).(let H6 \def (eq_ind C (CHead c1 (Bind Abst) t) -(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind b) \Rightarrow (match b in B return (\lambda (_: -B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow True | Void -\Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead d1 (Bind Abbr) -u) H5) in (False_ind (ex2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Abbr) u0) -(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) H6)))))))))))) -y c H0))) H))))). -(* COMMENTS -Initial nodes: 1117 -END *) - -theorem csuba_gen_void: - \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).((csuba g -(CHead d1 (Bind Void) u1) c) \to (ex2_3 B C T (\lambda (b: B).(\lambda (d2: -C).(\lambda (u2: T).(eq C c (CHead d2 (Bind b) u2))))) (\lambda (_: -B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))))))))) -\def - \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda -(H: (csuba g (CHead d1 (Bind Void) u1) c)).(insert_eq C (CHead d1 (Bind Void) -u1) (\lambda (c0: C).(csuba g c0 c)) (\lambda (_: C).(ex2_3 B C T (\lambda -(b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Bind b) u2))))) -(\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))))) -(\lambda (y: C).(\lambda (H0: (csuba g y c)).(csuba_ind g (\lambda (c0: -C).(\lambda (c1: C).((eq C c0 (CHead d1 (Bind Void) u1)) \to (ex2_3 B C T -(\lambda (b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 (Bind b) -u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 -d2)))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind -Void) u1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C -return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) -\Rightarrow False])) I (CHead d1 (Bind Void) u1) H1) in (False_ind (ex2_3 B C -T (\lambda (b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C (CSort n) (CHead d2 -(Bind b) u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 -d2))))) H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 -c2)).(\lambda (H2: (((eq C c1 (CHead d1 (Bind Void) u1)) \to (ex2_3 B C T -(\lambda (b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind b) -u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 -d2)))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c1 k u) -(CHead d1 (Bind Void) u1))).(let H4 \def (f_equal C C (\lambda (e: C).(match -e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _ -_) \Rightarrow c0])) (CHead c1 k u) (CHead d1 (Bind Void) u1) H3) in ((let H5 -\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) -with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c1 k -u) (CHead d1 (Bind Void) u1) H3) in ((let H6 \def (f_equal C T (\lambda (e: -C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | -(CHead _ _ t) \Rightarrow t])) (CHead c1 k u) (CHead d1 (Bind Void) u1) H3) -in (\lambda (H7: (eq K k (Bind Void))).(\lambda (H8: (eq C c1 d1)).(eq_ind_r -T u1 (\lambda (t: T).(ex2_3 B C T (\lambda (b: B).(\lambda (d2: C).(\lambda -(u2: T).(eq C (CHead c2 k t) (CHead d2 (Bind b) u2))))) (\lambda (_: -B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))))) (eq_ind_r K (Bind -Void) (\lambda (k0: K).(ex2_3 B C T (\lambda (b: B).(\lambda (d2: C).(\lambda -(u2: T).(eq C (CHead c2 k0 u1) (CHead d2 (Bind b) u2))))) (\lambda (_: -B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))))) (let H9 \def (eq_ind -C c1 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Void) u1)) \to (ex2_3 B C T -(\lambda (b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind b) -u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 -d2))))))) H2 d1 H8) in (let H10 \def (eq_ind C c1 (\lambda (c0: C).(csuba g -c0 c2)) H1 d1 H8) in (ex2_3_intro B C T (\lambda (b: B).(\lambda (d2: -C).(\lambda (u2: T).(eq C (CHead c2 (Bind Void) u1) (CHead d2 (Bind b) -u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))) -Void c2 u1 (refl_equal C (CHead c2 (Bind Void) u1)) H10))) k H7) u H6)))) -H5)) H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 -c2)).(\lambda (H2: (((eq C c1 (CHead d1 (Bind Void) u1)) \to (ex2_3 B C T -(\lambda (b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind b) -u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 -d2)))))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u0: -T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u0) (CHead d1 -(Bind Void) u1))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C -return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _ _) -\Rightarrow c0])) (CHead c1 (Bind Void) u0) (CHead d1 (Bind Void) u1) H4) in -((let H6 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: -C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead -c1 (Bind Void) u0) (CHead d1 (Bind Void) u1) H4) in (\lambda (H7: (eq C c1 -d1)).(let H8 \def (eq_ind C c1 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind -Void) u1)) \to (ex2_3 B C T (\lambda (b0: B).(\lambda (d2: C).(\lambda (u3: -T).(eq C c2 (CHead d2 (Bind b0) u3))))) (\lambda (_: B).(\lambda (d2: -C).(\lambda (_: T).(csuba g d1 d2))))))) H2 d1 H7) in (let H9 \def (eq_ind C -c1 (\lambda (c0: C).(csuba g c0 c2)) H1 d1 H7) in (ex2_3_intro B C T (\lambda -(b0: B).(\lambda (d2: C).(\lambda (u3: T).(eq C (CHead c2 (Bind b) u2) (CHead -d2 (Bind b0) u3))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba -g d1 d2)))) b c2 u2 (refl_equal C (CHead c2 (Bind b) u2)) H9))))) -H5))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1 -c2)).(\lambda (_: (((eq C c1 (CHead d1 (Bind Void) u1)) \to (ex2_3 B C T -(\lambda (b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind b) -u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 -d2)))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc -g a))).(\lambda (u: T).(\lambda (_: (arity g c2 u a)).(\lambda (H5: (eq C -(CHead c1 (Bind Abst) t) (CHead d1 (Bind Void) u1))).(let H6 \def (eq_ind C -(CHead c1 (Bind Abst) t) (\lambda (ee: C).(match ee in C return (\lambda (_: -C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match -k in K return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B -return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow -True | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead d1 -(Bind Void) u1) H5) in (False_ind (ex2_3 B C T (\lambda (b: B).(\lambda (d2: -C).(\lambda (u2: T).(eq C (CHead c2 (Bind Abbr) u) (CHead d2 (Bind b) u2))))) -(\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))))) -H6)))))))))))) y c H0))) H))))). -(* COMMENTS -Initial nodes: 1418 -END *) - -theorem csuba_gen_abst: - \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).((csuba g -(CHead d1 (Bind Abst) u1) c) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead -d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))))))) -\def - \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda -(H: (csuba g (CHead d1 (Bind Abst) u1) c)).(insert_eq C (CHead d1 (Bind Abst) -u1) (\lambda (c0: C).(csuba g c0 c)) (\lambda (_: C).(or (ex2 C (\lambda (d2: -C).(eq C c (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead -d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity -g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 a))))))) (\lambda (y: C).(\lambda (H0: (csuba g y -c)).(csuba_ind g (\lambda (c0: C).(\lambda (c1: C).((eq C c0 (CHead d1 (Bind -Abst) u1)) \to (or (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abst) -u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind Abbr) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))) -(\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Abst) -u1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return -(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) -\Rightarrow False])) I (CHead d1 (Bind Abst) u1) H1) in (False_ind (or (ex2 C -(\lambda (d2: C).(eq C (CSort n) (CHead d2 (Bind Abst) u1))) (\lambda (d2: -C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(eq C (CSort n) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) H2)))) -(\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda -(H2: (((eq C c1 (CHead d1 (Bind Abst) u1)) \to (or (ex2 C (\lambda (d2: -C).(eq C c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 -(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity -g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 a))))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: -(eq C (CHead c1 k u) (CHead d1 (Bind Abst) u1))).(let H4 \def (f_equal C C -(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow c1 | (CHead c0 _ _) \Rightarrow c0])) (CHead c1 k u) (CHead d1 -(Bind Abst) u1) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e in -C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) -\Rightarrow k0])) (CHead c1 k u) (CHead d1 (Bind Abst) u1) H3) in ((let H6 -\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u) -(CHead d1 (Bind Abst) u1) H3) in (\lambda (H7: (eq K k (Bind Abst))).(\lambda -(H8: (eq C c1 d1)).(eq_ind_r T u1 (\lambda (t: T).(or (ex2 C (\lambda (d2: -C).(eq C (CHead c2 k t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g -d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C -(CHead c2 k t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a))))))) (eq_ind_r K (Bind Abst) -(\lambda (k0: K).(or (ex2 C (\lambda (d2: C).(eq C (CHead c2 k0 u1) (CHead d2 -(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c2 k0 u1) (CHead d2 -(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 -(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 -u2 a))))))) (let H9 \def (eq_ind C c1 (\lambda (c0: C).((eq C c0 (CHead d1 -(Bind Abst) u1)) \to (or (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind -Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))))) H2 -d1 H8) in (let H10 \def (eq_ind C c1 (\lambda (c0: C).(csuba g c0 c2)) H1 d1 -H8) in (or_introl (ex2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Abst) u1) -(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abst) -u1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: -A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda -(a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: C).(eq C (CHead c2 -(Bind Abst) u1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) -c2 (refl_equal C (CHead c2 (Bind Abst) u1)) H10)))) k H7) u H6)))) H5)) -H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1 -c2)).(\lambda (_: (((eq C c1 (CHead d1 (Bind Abst) u1)) \to (or (ex2 C -(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba -g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq -C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: -A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda -(a: A).(arity g d2 u2 a))))))))).(\lambda (b: B).(\lambda (_: (not (eq B b -Void))).(\lambda (u0: T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind -Void) u0) (CHead d1 (Bind Abst) u1))).(let H5 \def (eq_ind C (CHead c1 (Bind -Void) u0) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with -[(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return -(\lambda (_: K).Prop) with [(Bind b0) \Rightarrow (match b0 in B return -(\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False | -Void \Rightarrow True]) | (Flat _) \Rightarrow False])])) I (CHead d1 (Bind -Abst) u1) H4) in (False_ind (or (ex2 C (\lambda (d2: C).(eq C (CHead c2 (Bind -b) u2) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 -C T A (\lambda (d2: C).(\lambda (u3: T).(\lambda (_: A).(eq C (CHead c2 (Bind -b) u2) (CHead d2 (Bind Abbr) u3))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u3: T).(\lambda (a: A).(arity g d2 u3 a)))))) H5))))))))))) (\lambda (c1: -C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c1 -(CHead d1 (Bind Abst) u1)) \to (or (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 -(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a))))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (H3: (arity g c1 t (asucc -g a))).(\lambda (u: T).(\lambda (H4: (arity g c2 u a)).(\lambda (H5: (eq C -(CHead c1 (Bind Abst) t) (CHead d1 (Bind Abst) u1))).(let H6 \def (f_equal C -C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow c1 | (CHead c0 _ _) \Rightarrow c0])) (CHead c1 (Bind Abst) t) -(CHead d1 (Bind Abst) u1) H5) in ((let H7 \def (f_equal C T (\lambda (e: -C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow t | -(CHead _ _ t0) \Rightarrow t0])) (CHead c1 (Bind Abst) t) (CHead d1 (Bind -Abst) u1) H5) in (\lambda (H8: (eq C c1 d1)).(let H9 \def (eq_ind T t -(\lambda (t0: T).(arity g c1 t0 (asucc g a))) H3 u1 H7) in (let H10 \def -(eq_ind C c1 (\lambda (c0: C).(arity g c0 u1 (asucc g a))) H9 d1 H8) in (let -H11 \def (eq_ind C c1 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abst) u1)) -\to (or (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) u1))) (\lambda -(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 (asucc g a0))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 a0)))))))) H2 d1 H8) -in (let H12 \def (eq_ind C c1 (\lambda (c0: C).(csuba g c0 c2)) H1 d1 H8) in -(or_intror (ex2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Abbr) u) (CHead d2 -(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) u) -(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity -g d1 u1 (asucc g a0))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: -A).(arity g d2 u2 a0))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) u) (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 (asucc g -a0))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 -a0)))) c2 u a (refl_equal C (CHead c2 (Bind Abbr) u)) H12 H10 H4)))))))) -H6)))))))))))) y c H0))) H))))). -(* COMMENTS -Initial nodes: 2550 -END *) - -theorem csuba_gen_flat: - \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).(\forall -(f: F).((csuba g (CHead d1 (Flat f) u1) c) \to (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d1 d2))))))))) -\def - \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda -(f: F).(\lambda (H: (csuba g (CHead d1 (Flat f) u1) c)).(insert_eq C (CHead -d1 (Flat f) u1) (\lambda (c0: C).(csuba g c0 c)) (\lambda (_: C).(ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d1 d2))))) (\lambda (y: C).(\lambda (H0: -(csuba g y c)).(csuba_ind g (\lambda (c0: C).(\lambda (c1: C).((eq C c0 -(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq -C c1 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 -d2))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Flat f) -u1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return -(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) -\Rightarrow False])) I (CHead d1 (Flat f) u1) H1) in (False_ind (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(eq C (CSort n) (CHead d2 (Flat f) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))) H2)))) (\lambda (c1: -C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c1 -(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq -C c2 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 -d2))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c1 k u) -(CHead d1 (Flat f) u1))).(let H4 \def (f_equal C C (\lambda (e: C).(match e -in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _ -_) \Rightarrow c0])) (CHead c1 k u) (CHead d1 (Flat f) u1) H3) in ((let H5 -\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) -with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c1 k -u) (CHead d1 (Flat f) u1) H3) in ((let H6 \def (f_equal C T (\lambda (e: -C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | -(CHead _ _ t) \Rightarrow t])) (CHead c1 k u) (CHead d1 (Flat f) u1) H3) in -(\lambda (H7: (eq K k (Flat f))).(\lambda (H8: (eq C c1 d1)).(eq_ind_r T u1 -(\lambda (t: T).(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c2 -k t) (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 -d2))))) (eq_ind_r K (Flat f) (\lambda (k0: K).(ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(eq C (CHead c2 k0 u1) (CHead d2 (Flat f) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d1 d2))))) (let H9 \def (eq_ind C c1 -(\lambda (c0: C).((eq C c0 (CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda -(d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Flat f) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d1 d2)))))) H2 d1 H8) in (let H10 \def (eq_ind C -c1 (\lambda (c0: C).(csuba g c0 c2)) H1 d1 H8) in (ex2_2_intro C T (\lambda -(d2: C).(\lambda (u2: T).(eq C (CHead c2 (Flat f) u1) (CHead d2 (Flat f) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))) c2 u1 (refl_equal C -(CHead c2 (Flat f) u1)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: -C).(\lambda (c2: C).(\lambda (_: (csuba g c1 c2)).(\lambda (_: (((eq C c1 -(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq -C c2 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 -d2))))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u0: -T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u0) (CHead d1 -(Flat f) u1))).(let H5 \def (eq_ind C (CHead c1 (Bind Void) u0) (\lambda (ee: -C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow -False | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) -with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead d1 -(Flat f) u1) H4) in (False_ind (ex2_2 C T (\lambda (d2: C).(\lambda (u3: -T).(eq C (CHead c2 (Bind b) u2) (CHead d2 (Flat f) u3)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d1 d2)))) H5))))))))))) (\lambda (c1: C).(\lambda -(c2: C).(\lambda (_: (csuba g c1 c2)).(\lambda (_: (((eq C c1 (CHead d1 (Flat -f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 -(Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 -d2))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g -a))).(\lambda (u: T).(\lambda (_: (arity g c2 u a)).(\lambda (H5: (eq C -(CHead c1 (Bind Abst) t) (CHead d1 (Flat f) u1))).(let H6 \def (eq_ind C -(CHead c1 (Bind Abst) t) (\lambda (ee: C).(match ee in C return (\lambda (_: -C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match -k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat -_) \Rightarrow False])])) I (CHead d1 (Flat f) u1) H5) in (False_ind (ex2_2 C -T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c2 (Bind Abbr) u) (CHead d2 -(Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))) -H6)))))))))))) y c H0))) H)))))). -(* COMMENTS -Initial nodes: 1183 -END *) - -theorem csuba_gen_bind: - \forall (g: G).(\forall (b1: B).(\forall (e1: C).(\forall (c2: C).(\forall -(v1: T).((csuba g (CHead e1 (Bind b1) v1) c2) \to (ex2_3 B C T (\lambda (b2: -B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) -(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2)))))))))) -\def - \lambda (g: G).(\lambda (b1: B).(\lambda (e1: C).(\lambda (c2: C).(\lambda -(v1: T).(\lambda (H: (csuba g (CHead e1 (Bind b1) v1) c2)).(insert_eq C -(CHead e1 (Bind b1) v1) (\lambda (c: C).(csuba g c c2)) (\lambda (_: -C).(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 -(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: -T).(csuba g e1 e2)))))) (\lambda (y: C).(\lambda (H0: (csuba g y -c2)).(csuba_ind g (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead e1 (Bind -b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: -T).(eq C c0 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: -C).(\lambda (_: T).(csuba g e1 e2)))))))) (\lambda (n: nat).(\lambda (H1: (eq -C (CSort n) (CHead e1 (Bind b1) v1))).(let H2 \def (eq_ind C (CSort n) -(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 (Bind b1) -v1) H1) in (False_ind (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda -(v2: T).(eq C (CSort n) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda -(e2: C).(\lambda (_: T).(csuba g e1 e2))))) H2)))) (\lambda (c1: C).(\lambda -(c3: C).(\lambda (H1: (csuba g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 -(Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda -(v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: -C).(\lambda (_: T).(csuba g e1 e2)))))))).(\lambda (k: K).(\lambda (u: -T).(\lambda (H3: (eq C (CHead c1 k u) (CHead e1 (Bind b1) v1))).(let H4 \def -(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with -[(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u) -(CHead e1 (Bind b1) v1) H3) in ((let H5 \def (f_equal C K (\lambda (e: -C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | -(CHead _ k0 _) \Rightarrow k0])) (CHead c1 k u) (CHead e1 (Bind b1) v1) H3) -in ((let H6 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda -(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) -(CHead c1 k u) (CHead e1 (Bind b1) v1) H3) in (\lambda (H7: (eq K k (Bind -b1))).(\lambda (H8: (eq C c1 e1)).(eq_ind_r T v1 (\lambda (t: T).(ex2_3 B C T -(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 k t) -(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: -T).(csuba g e1 e2)))))) (eq_ind_r K (Bind b1) (\lambda (k0: K).(ex2_3 B C T -(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 k0 v1) -(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: -T).(csuba g e1 e2)))))) (let H9 \def (eq_ind C c1 (\lambda (c: C).((eq C c -(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: -C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: -B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2))))))) H2 e1 H8) in (let -H10 \def (eq_ind C c1 (\lambda (c: C).(csuba g c c3)) H1 e1 H8) in -(ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C -(CHead c3 (Bind b1) v1) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda -(e2: C).(\lambda (_: T).(csuba g e1 e2)))) b1 c3 v1 (refl_equal C (CHead c3 -(Bind b1) v1)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: -C).(\lambda (c3: C).(\lambda (H1: (csuba g c1 c3)).(\lambda (H2: (((eq C c1 -(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: -C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: -B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2)))))))).(\lambda (b: -B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: -T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) -v1))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda -(_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) -(CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) v1) H4) in ((let H6 \def -(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with -[(CSort _) \Rightarrow Void | (CHead _ k _) \Rightarrow (match k in K return -(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow -Void])])) (CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) v1) H4) in ((let H7 -\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead c1 -(Bind Void) u1) (CHead e1 (Bind b1) v1) H4) in (\lambda (H8: (eq B Void -b1)).(\lambda (H9: (eq C c1 e1)).(let H10 \def (eq_ind C c1 (\lambda (c: -C).((eq C c (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: -B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) -(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2))))))) H2 e1 -H9) in (let H11 \def (eq_ind C c1 (\lambda (c: C).(csuba g c c3)) H1 e1 H9) -in (let H12 \def (eq_ind_r B b1 (\lambda (b0: B).((eq C e1 (CHead e1 (Bind -b0) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: -T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: -C).(\lambda (_: T).(csuba g e1 e2))))))) H10 Void H8) in (ex2_3_intro B C T -(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind b) -u2) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: -T).(csuba g e1 e2)))) b c3 u2 (refl_equal C (CHead c3 (Bind b) u2)) -H11))))))) H6)) H5))))))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1: -(csuba g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind b1) v1)) \to (ex2_3 -B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 -(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g -e1 e2)))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (H3: (arity g c1 t -(asucc g a))).(\lambda (u: T).(\lambda (_: (arity g c3 u a)).(\lambda (H5: -(eq C (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1))).(let H6 \def -(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with -[(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 (Bind -Abst) t) (CHead e1 (Bind b1) v1) H5) in ((let H7 \def (f_equal C B (\lambda -(e: C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow -Abst | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with -[(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead c1 (Bind -Abst) t) (CHead e1 (Bind b1) v1) H5) in ((let H8 \def (f_equal C T (\lambda -(e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow t -| (CHead _ _ t0) \Rightarrow t0])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind -b1) v1) H5) in (\lambda (H9: (eq B Abst b1)).(\lambda (H10: (eq C c1 -e1)).(let H11 \def (eq_ind T t (\lambda (t0: T).(arity g c1 t0 (asucc g a))) -H3 v1 H8) in (let H12 \def (eq_ind C c1 (\lambda (c: C).(arity g c v1 (asucc -g a))) H11 e1 H10) in (let H13 \def (eq_ind C c1 (\lambda (c: C).((eq C c -(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: -C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: -B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2))))))) H2 e1 H10) in (let -H14 \def (eq_ind C c1 (\lambda (c: C).(csuba g c c3)) H1 e1 H10) in (let H15 -\def (eq_ind_r B b1 (\lambda (b: B).((eq C e1 (CHead e1 (Bind b) v1)) \to -(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 -(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: -T).(csuba g e1 e2))))))) H13 Abst H9) in (ex2_3_intro B C T (\lambda (b2: -B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 -(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g -e1 e2)))) Abbr c3 u (refl_equal C (CHead c3 (Bind Abbr) u)) H14))))))))) H7)) -H6)))))))))))) y c2 H0))) H)))))). -(* COMMENTS -Initial nodes: 1889 -END *) - -theorem csuba_gen_abst_rev: - \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g c -(CHead d1 (Bind Abst) u)) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead d2 -(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(eq C c (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1))))))))) -\def - \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u: T).(\lambda (H: -(csuba g c (CHead d1 (Bind Abst) u))).(insert_eq C (CHead d1 (Bind Abst) u) -(\lambda (c0: C).(csuba g c c0)) (\lambda (_: C).(or (ex2 C (\lambda (d2: -C).(eq C c (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (\lambda (y: -C).(\lambda (H0: (csuba g c y)).(csuba_ind g (\lambda (c0: C).(\lambda (c1: -C).((eq C c1 (CHead d1 (Bind Abst) u)) \to (or (ex2 C (\lambda (d2: C).(eq C -c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(eq C c0 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))) (\lambda (n: -nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Abst) u))).(let H2 \def -(eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return (\lambda (_: -C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow -False])) I (CHead d1 (Bind Abst) u) H1) in (False_ind (or (ex2 C (\lambda -(d2: C).(eq C (CSort n) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g -d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CSort n) (CHead -d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) -H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 -c2)).(\lambda (H2: (((eq C c2 (CHead d1 (Bind Abst) u)) \to (or (ex2 C -(\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba -g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))))))).(\lambda (k: K).(\lambda (u0: T).(\lambda (H3: (eq C (CHead c2 k -u0) (CHead d1 (Bind Abst) u))).(let H4 \def (f_equal C C (\lambda (e: -C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | -(CHead c0 _ _) \Rightarrow c0])) (CHead c2 k u0) (CHead d1 (Bind Abst) u) H3) -in ((let H5 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda -(_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) -(CHead c2 k u0) (CHead d1 (Bind Abst) u) H3) in ((let H6 \def (f_equal C T -(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) -\Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead c2 k u0) (CHead d1 -(Bind Abst) u) H3) in (\lambda (H7: (eq K k (Bind Abst))).(\lambda (H8: (eq C -c2 d1)).(eq_ind_r T u (\lambda (t: T).(or (ex2 C (\lambda (d2: C).(eq C -(CHead c1 k t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 k t) (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) -(eq_ind_r K (Bind Abst) (\lambda (k0: K).(or (ex2 C (\lambda (d2: C).(eq C -(CHead c1 k0 u) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 k0 u) (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let -H9 \def (eq_ind C c2 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abst) u)) \to -(or (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))))) H2 d1 H8) in (let H10 \def (eq_ind C c2 (\lambda (c0: C).(csuba g -c1 c0)) H1 d1 H8) in (or_introl (ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind -Abst) u) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 -C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 (Bind Abst) u) (CHead -d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) -(ex_intro2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abst) u) (CHead d2 (Bind -Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) c1 (refl_equal C (CHead c1 (Bind -Abst) u)) H10)))) k H7) u0 H6)))) H5)) H4))))))))) (\lambda (c1: C).(\lambda -(c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c2 (CHead d1 -(Bind Abst) u)) \to (or (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind -Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(eq C c1 (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))))))).(\lambda (b: B).(\lambda (H3: (not -(eq B b Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead -c2 (Bind b) u2) (CHead d1 (Bind Abst) u))).(let H5 \def (f_equal C C (\lambda -(e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 -| (CHead c0 _ _) \Rightarrow c0])) (CHead c2 (Bind b) u2) (CHead d1 (Bind -Abst) u) H4) in ((let H6 \def (f_equal C B (\lambda (e: C).(match e in C -return (\lambda (_: C).B) with [(CSort _) \Rightarrow b | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b0) -\Rightarrow b0 | (Flat _) \Rightarrow b])])) (CHead c2 (Bind b) u2) (CHead d1 -(Bind Abst) u) H4) in ((let H7 \def (f_equal C T (\lambda (e: C).(match e in -C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u2 | (CHead _ _ t) -\Rightarrow t])) (CHead c2 (Bind b) u2) (CHead d1 (Bind Abst) u) H4) in -(\lambda (H8: (eq B b Abst)).(\lambda (H9: (eq C c2 d1)).(let H10 \def -(eq_ind B b (\lambda (b0: B).(not (eq B b0 Void))) H3 Abst H8) in (let H11 -\def (eq_ind C c2 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abst) u)) \to -(or (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u3: T).(eq C c1 -(CHead d2 (Bind Void) u3)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))))) H2 d1 H9) in (let H12 \def (eq_ind C c2 (\lambda (c0: C).(csuba g -c1 c0)) H1 d1 H9) in (or_intror (ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind -Void) u1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u3: T).(eq C (CHead c1 (Bind Void) u1) -(CHead d2 (Bind Void) u3)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u3: T).(eq C (CHead c1 -(Bind Void) u1) (CHead d2 (Bind Void) u3)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))) c1 u1 (refl_equal C (CHead c1 (Bind Void) u1)) -H12)))))))) H6)) H5))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: -(csuba g c1 c2)).(\lambda (_: (((eq C c2 (CHead d1 (Bind Abst) u)) \to (or -(ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc -g a))).(\lambda (u0: T).(\lambda (_: (arity g c2 u0 a)).(\lambda (H5: (eq C -(CHead c2 (Bind Abbr) u0) (CHead d1 (Bind Abst) u))).(let H6 \def (eq_ind C -(CHead c2 (Bind Abbr) u0) (\lambda (ee: C).(match ee in C return (\lambda (_: -C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match -k in K return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B -return (\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow -False | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead -d1 (Bind Abst) u) H5) in (False_ind (or (ex2 C (\lambda (d2: C).(eq C (CHead -c1 (Bind Abst) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 -d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 (Bind -Abst) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba -g d2 d1))))) H6)))))))))))) c y H0))) H))))). -(* COMMENTS -Initial nodes: 1980 -END *) - -theorem csuba_gen_void_rev: - \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g c -(CHead d1 (Bind Void) u)) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind -Void) u))) (\lambda (d2: C).(csuba g d2 d1))))))) -\def - \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u: T).(\lambda (H: -(csuba g c (CHead d1 (Bind Void) u))).(insert_eq C (CHead d1 (Bind Void) u) -(\lambda (c0: C).(csuba g c c0)) (\lambda (_: C).(ex2 C (\lambda (d2: C).(eq -C c (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))) (\lambda -(y: C).(\lambda (H0: (csuba g c y)).(csuba_ind g (\lambda (c0: C).(\lambda -(c1: C).((eq C c1 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda (d2: C).(eq C -c0 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))))) (\lambda -(n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Void) u))).(let H2 -\def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return (\lambda (_: -C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow -False])) I (CHead d1 (Bind Void) u) H1) in (False_ind (ex2 C (\lambda (d2: -C).(eq C (CSort n) (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 -d1))) H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 -c2)).(\lambda (H2: (((eq C c2 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda -(d2: C).(eq C c1 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 -d1)))))).(\lambda (k: K).(\lambda (u0: T).(\lambda (H3: (eq C (CHead c2 k u0) -(CHead d1 (Bind Void) u))).(let H4 \def (f_equal C C (\lambda (e: C).(match e -in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ -_) \Rightarrow c0])) (CHead c2 k u0) (CHead d1 (Bind Void) u) H3) in ((let H5 -\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) -with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c2 k -u0) (CHead d1 (Bind Void) u) H3) in ((let H6 \def (f_equal C T (\lambda (e: -C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | -(CHead _ _ t) \Rightarrow t])) (CHead c2 k u0) (CHead d1 (Bind Void) u) H3) -in (\lambda (H7: (eq K k (Bind Void))).(\lambda (H8: (eq C c2 d1)).(eq_ind_r -T u (\lambda (t: T).(ex2 C (\lambda (d2: C).(eq C (CHead c1 k t) (CHead d2 -(Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))) (eq_ind_r K (Bind Void) -(\lambda (k0: K).(ex2 C (\lambda (d2: C).(eq C (CHead c1 k0 u) (CHead d2 -(Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))) (let H9 \def (eq_ind C -c2 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda -(d2: C).(eq C c1 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 -d1))))) H2 d1 H8) in (let H10 \def (eq_ind C c2 (\lambda (c0: C).(csuba g c1 -c0)) H1 d1 H8) in (ex_intro2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Void) -u) (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)) c1 -(refl_equal C (CHead c1 (Bind Void) u)) H10))) k H7) u0 H6)))) H5)) -H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 -c2)).(\lambda (H2: (((eq C c2 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda -(d2: C).(eq C c1 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 -d1)))))).(\lambda (b: B).(\lambda (H3: (not (eq B b Void))).(\lambda (u1: -T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c2 (Bind b) u2) (CHead d1 -(Bind Void) u))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C -return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ _) -\Rightarrow c0])) (CHead c2 (Bind b) u2) (CHead d1 (Bind Void) u) H4) in -((let H6 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: -C).B) with [(CSort _) \Rightarrow b | (CHead _ k _) \Rightarrow (match k in K -return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) -\Rightarrow b])])) (CHead c2 (Bind b) u2) (CHead d1 (Bind Void) u) H4) in -((let H7 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: -C).T) with [(CSort _) \Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead -c2 (Bind b) u2) (CHead d1 (Bind Void) u) H4) in (\lambda (H8: (eq B b -Void)).(\lambda (H9: (eq C c2 d1)).(let H10 \def (eq_ind B b (\lambda (b0: -B).(not (eq B b0 Void))) H3 Void H8) in (let H11 \def (eq_ind C c2 (\lambda -(c0: C).((eq C c0 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda (d2: C).(eq C -c1 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1))))) H2 d1 H9) -in (let H12 \def (eq_ind C c2 (\lambda (c0: C).(csuba g c1 c0)) H1 d1 H9) in -(let H13 \def (match (H10 (refl_equal B Void)) in False return (\lambda (_: -False).(ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Void) u1) (CHead d2 -(Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))) with []) in H13))))))) -H6)) H5))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1 -c2)).(\lambda (_: (((eq C c2 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda -(d2: C).(eq C c1 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 -d1)))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g -a))).(\lambda (u0: T).(\lambda (_: (arity g c2 u0 a)).(\lambda (H5: (eq C -(CHead c2 (Bind Abbr) u0) (CHead d1 (Bind Void) u))).(let H6 \def (eq_ind C -(CHead c2 (Bind Abbr) u0) (\lambda (ee: C).(match ee in C return (\lambda (_: -C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match -k in K return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B -return (\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow -False | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead -d1 (Bind Void) u) H5) in (False_ind (ex2 C (\lambda (d2: C).(eq C (CHead c1 -(Bind Abst) t) (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1))) -H6)))))))))))) c y H0))) H))))). -(* COMMENTS -Initial nodes: 1326 -END *) - -theorem csuba_gen_abbr_rev: - \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).((csuba g c -(CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(eq C c (CHead d2 -(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))) -\def - \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda -(H: (csuba g c (CHead d1 (Bind Abbr) u1))).(insert_eq C (CHead d1 (Bind Abbr) -u1) (\lambda (c0: C).(csuba g c c0)) (\lambda (_: C).(or3 (ex2 C (\lambda -(d2: C).(eq C c (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq -C c (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))))) (\lambda (y: C).(\lambda (H0: (csuba g c y)).(csuba_ind g (\lambda -(c0: C).(\lambda (c1: C).((eq C c1 (CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C -(\lambda (d2: C).(eq C c0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba -g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq -C c0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq -C c0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g -d2 d1)))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind -Abbr) u1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C -return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) -\Rightarrow False])) I (CHead d1 (Bind Abbr) u1) H1) in (False_ind (or3 (ex2 -C (\lambda (d2: C).(eq C (CSort n) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(eq C (CSort n) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(eq C (CSort n) (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) H2)))) (\lambda -(c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C -c2 (CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(eq C c1 -(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c2 k u) -(CHead d1 (Bind Abbr) u1))).(let H4 \def (f_equal C C (\lambda (e: C).(match -e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ -_) \Rightarrow c0])) (CHead c2 k u) (CHead d1 (Bind Abbr) u1) H3) in ((let H5 -\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) -with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c2 k -u) (CHead d1 (Bind Abbr) u1) H3) in ((let H6 \def (f_equal C T (\lambda (e: -C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | -(CHead _ _ t) \Rightarrow t])) (CHead c2 k u) (CHead d1 (Bind Abbr) u1) H3) -in (\lambda (H7: (eq K k (Bind Abbr))).(\lambda (H8: (eq C c2 d1)).(eq_ind_r -T u1 (\lambda (t: T).(or3 (ex2 C (\lambda (d2: C).(eq C (CHead c1 k t) (CHead -d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c1 k t) (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 k t) -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))))) (eq_ind_r K (Bind Abbr) (\lambda (k0: K).(or3 (ex2 C (\lambda (d2: -C).(eq C (CHead c1 k0 u1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba -g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq -C (CHead c1 k0 u1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda -(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(eq C (CHead c1 k0 u1) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H9 \def (eq_ind C -c2 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C -(\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba -g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq -C c1 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq -C c1 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g -d2 d1))))))) H2 d1 H8) in (let H10 \def (eq_ind C c2 (\lambda (c0: C).(csuba -g c1 c0)) H1 d1 H8) in (or3_intro0 (ex2 C (\lambda (d2: C).(eq C (CHead c1 -(Bind Abbr) u1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C -(CHead c1 (Bind Abbr) u1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 (Bind Abbr) u1) (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) -(ex_intro2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abbr) u1) (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) c1 (refl_equal C (CHead c1 -(Bind Abbr) u1)) H10)))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: -C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c2 -(CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(eq C c1 (CHead -d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))).(\lambda (b: -B).(\lambda (H3: (not (eq B b Void))).(\lambda (u0: T).(\lambda (u2: -T).(\lambda (H4: (eq C (CHead c2 (Bind b) u2) (CHead d1 (Bind Abbr) -u1))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda -(_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) -(CHead c2 (Bind b) u2) (CHead d1 (Bind Abbr) u1) H4) in ((let H6 \def -(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with -[(CSort _) \Rightarrow b | (CHead _ k _) \Rightarrow (match k in K return -(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow -b])])) (CHead c2 (Bind b) u2) (CHead d1 (Bind Abbr) u1) H4) in ((let H7 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c2 (Bind b) -u2) (CHead d1 (Bind Abbr) u1) H4) in (\lambda (H8: (eq B b Abbr)).(\lambda -(H9: (eq C c2 d1)).(let H10 \def (eq_ind B b (\lambda (b0: B).(not (eq B b0 -Void))) H3 Abbr H8) in (let H11 \def (eq_ind C c2 (\lambda (c0: C).((eq C c0 -(CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(eq C c1 (CHead -d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u3: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind Abst) -u3))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u3: T).(\lambda (a: A).(arity g d2 u3 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u3: T).(eq C c1 (CHead d2 (Bind Void) -u3)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))) H2 d1 H9) in -(let H12 \def (eq_ind C c2 (\lambda (c0: C).(csuba g c1 c0)) H1 d1 H9) in -(or3_intro2 (ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Void) u0) (CHead d2 -(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u3: T).(\lambda (_: A).(eq C (CHead c1 (Bind Void) u0) -(CHead d2 (Bind Abst) u3))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u3: T).(\lambda (a: -A).(arity g d2 u3 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u3: T).(eq -C (CHead c1 (Bind Void) u0) (CHead d2 (Bind Void) u3)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: -C).(\lambda (u3: T).(eq C (CHead c1 (Bind Void) u0) (CHead d2 (Bind Void) -u3)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) c1 u0 (refl_equal C -(CHead c1 (Bind Void) u0)) H12)))))))) H6)) H5))))))))))) (\lambda (c1: -C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c2 -(CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(eq C c1 (CHead -d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))).(\lambda (t: -T).(\lambda (a: A).(\lambda (H3: (arity g c1 t (asucc g a))).(\lambda (u: -T).(\lambda (H4: (arity g c2 u a)).(\lambda (H5: (eq C (CHead c2 (Bind Abbr) -u) (CHead d1 (Bind Abbr) u1))).(let H6 \def (f_equal C C (\lambda (e: -C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | -(CHead c0 _ _) \Rightarrow c0])) (CHead c2 (Bind Abbr) u) (CHead d1 (Bind -Abbr) u1) H5) in ((let H7 \def (f_equal C T (\lambda (e: C).(match e in C -return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) -\Rightarrow t0])) (CHead c2 (Bind Abbr) u) (CHead d1 (Bind Abbr) u1) H5) in -(\lambda (H8: (eq C c2 d1)).(let H9 \def (eq_ind T u (\lambda (t0: T).(arity -g c2 t0 a)) H4 u1 H7) in (let H10 \def (eq_ind C c2 (\lambda (c0: C).(arity g -c0 u1 a)) H9 d1 H8) in (let H11 \def (eq_ind C c2 (\lambda (c0: C).((eq C c0 -(CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(eq C c1 (CHead -d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 (asucc g -a0))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 -a0))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))) H2 d1 H8) -in (let H12 \def (eq_ind C c2 (\lambda (c0: C).(csuba g c1 c0)) H1 d1 H8) in -(or3_intro1 (ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abst) t) (CHead d2 -(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) t) -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: -A).(arity g d2 u2 (asucc g a0))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a0: A).(arity g d1 u1 a0))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(eq C (CHead c1 (Bind Abst) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))) (ex4_3_intro C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) t) (CHead d2 -(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 -(asucc g a0))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 -u1 a0)))) c1 t a (refl_equal C (CHead c1 (Bind Abst) t)) H12 H3 H10)))))))) -H6)))))))))))) c y H0))) H))))). -(* COMMENTS -Initial nodes: 3459 -END *) - -theorem csuba_gen_flat_rev: - \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).(\forall -(f: F).((csuba g c (CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1))))))))) -\def - \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda -(f: F).(\lambda (H: (csuba g c (CHead d1 (Flat f) u1))).(insert_eq C (CHead -d1 (Flat f) u1) (\lambda (c0: C).(csuba g c c0)) (\lambda (_: C).(ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (y: C).(\lambda (H0: -(csuba g c y)).(csuba_ind g (\lambda (c0: C).(\lambda (c1: C).((eq C c1 -(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq -C c0 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Flat f) -u1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return -(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) -\Rightarrow False])) I (CHead d1 (Flat f) u1) H1) in (False_ind (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(eq C (CSort n) (CHead d2 (Flat f) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) H2)))) (\lambda (c1: -C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c2 -(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq -C c1 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c2 k u) -(CHead d1 (Flat f) u1))).(let H4 \def (f_equal C C (\lambda (e: C).(match e -in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ -_) \Rightarrow c0])) (CHead c2 k u) (CHead d1 (Flat f) u1) H3) in ((let H5 -\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) -with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c2 k -u) (CHead d1 (Flat f) u1) H3) in ((let H6 \def (f_equal C T (\lambda (e: -C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | -(CHead _ _ t) \Rightarrow t])) (CHead c2 k u) (CHead d1 (Flat f) u1) H3) in -(\lambda (H7: (eq K k (Flat f))).(\lambda (H8: (eq C c2 d1)).(eq_ind_r T u1 -(\lambda (t: T).(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 -k t) (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))) (eq_ind_r K (Flat f) (\lambda (k0: K).(ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(eq C (CHead c1 k0 u1) (CHead d2 (Flat f) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (let H9 \def (eq_ind C c2 -(\lambda (c0: C).((eq C c0 (CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda -(d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 (Flat f) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))))) H2 d1 H8) in (let H10 \def (eq_ind C -c2 (\lambda (c0: C).(csuba g c1 c0)) H1 d1 H8) in (ex2_2_intro C T (\lambda -(d2: C).(\lambda (u2: T).(eq C (CHead c1 (Flat f) u1) (CHead d2 (Flat f) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) c1 u1 (refl_equal C -(CHead c1 (Flat f) u1)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: -C).(\lambda (c2: C).(\lambda (_: (csuba g c1 c2)).(\lambda (_: (((eq C c2 -(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq -C c1 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u0: -T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c2 (Bind b) u2) (CHead d1 -(Flat f) u1))).(let H5 \def (eq_ind C (CHead c2 (Bind b) u2) (\lambda (ee: -C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow -False | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) -with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead d1 -(Flat f) u1) H4) in (False_ind (ex2_2 C T (\lambda (d2: C).(\lambda (u3: -T).(eq C (CHead c1 (Bind Void) u0) (CHead d2 (Flat f) u3)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))) H5))))))))))) (\lambda (c1: C).(\lambda -(c2: C).(\lambda (_: (csuba g c1 c2)).(\lambda (_: (((eq C c2 (CHead d1 (Flat -f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 -(Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g -a))).(\lambda (u: T).(\lambda (_: (arity g c2 u a)).(\lambda (H5: (eq C -(CHead c2 (Bind Abbr) u) (CHead d1 (Flat f) u1))).(let H6 \def (eq_ind C -(CHead c2 (Bind Abbr) u) (\lambda (ee: C).(match ee in C return (\lambda (_: -C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match -k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat -_) \Rightarrow False])])) I (CHead d1 (Flat f) u1) H5) in (False_ind (ex2_2 C -T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 (Bind Abst) t) (CHead d2 -(Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) -H6)))))))))))) c y H0))) H)))))). -(* COMMENTS -Initial nodes: 1183 -END *) - -theorem csuba_gen_bind_rev: - \forall (g: G).(\forall (b1: B).(\forall (e1: C).(\forall (c2: C).(\forall -(v1: T).((csuba g c2 (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: -B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) -(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1)))))))))) -\def - \lambda (g: G).(\lambda (b1: B).(\lambda (e1: C).(\lambda (c2: C).(\lambda -(v1: T).(\lambda (H: (csuba g c2 (CHead e1 (Bind b1) v1))).(insert_eq C -(CHead e1 (Bind b1) v1) (\lambda (c: C).(csuba g c2 c)) (\lambda (_: -C).(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 -(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: -T).(csuba g e2 e1)))))) (\lambda (y: C).(\lambda (H0: (csuba g c2 -y)).(csuba_ind g (\lambda (c: C).(\lambda (c0: C).((eq C c0 (CHead e1 (Bind -b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: -T).(eq C c (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: -C).(\lambda (_: T).(csuba g e2 e1)))))))) (\lambda (n: nat).(\lambda (H1: (eq -C (CSort n) (CHead e1 (Bind b1) v1))).(let H2 \def (eq_ind C (CSort n) -(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 (Bind b1) -v1) H1) in (False_ind (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda -(v2: T).(eq C (CSort n) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda -(e2: C).(\lambda (_: T).(csuba g e2 e1))))) H2)))) (\lambda (c1: C).(\lambda -(c3: C).(\lambda (H1: (csuba g c1 c3)).(\lambda (H2: (((eq C c3 (CHead e1 -(Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda -(v2: T).(eq C c1 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: -C).(\lambda (_: T).(csuba g e2 e1)))))))).(\lambda (k: K).(\lambda (u: -T).(\lambda (H3: (eq C (CHead c3 k u) (CHead e1 (Bind b1) v1))).(let H4 \def -(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with -[(CSort _) \Rightarrow c3 | (CHead c _ _) \Rightarrow c])) (CHead c3 k u) -(CHead e1 (Bind b1) v1) H3) in ((let H5 \def (f_equal C K (\lambda (e: -C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | -(CHead _ k0 _) \Rightarrow k0])) (CHead c3 k u) (CHead e1 (Bind b1) v1) H3) -in ((let H6 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda -(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) -(CHead c3 k u) (CHead e1 (Bind b1) v1) H3) in (\lambda (H7: (eq K k (Bind -b1))).(\lambda (H8: (eq C c3 e1)).(eq_ind_r T v1 (\lambda (t: T).(ex2_3 B C T -(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c1 k t) -(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: -T).(csuba g e2 e1)))))) (eq_ind_r K (Bind b1) (\lambda (k0: K).(ex2_3 B C T -(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c1 k0 v1) -(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: -T).(csuba g e2 e1)))))) (let H9 \def (eq_ind C c3 (\lambda (c: C).((eq C c -(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: -C).(\lambda (v2: T).(eq C c1 (CHead e2 (Bind b2) v2))))) (\lambda (_: -B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1))))))) H2 e1 H8) in (let -H10 \def (eq_ind C c3 (\lambda (c: C).(csuba g c1 c)) H1 e1 H8) in -(ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C -(CHead c1 (Bind b1) v1) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda -(e2: C).(\lambda (_: T).(csuba g e2 e1)))) b1 c1 v1 (refl_equal C (CHead c1 -(Bind b1) v1)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: -C).(\lambda (c3: C).(\lambda (H1: (csuba g c1 c3)).(\lambda (H2: (((eq C c3 -(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: -C).(\lambda (v2: T).(eq C c1 (CHead e2 (Bind b2) v2))))) (\lambda (_: -B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1)))))))).(\lambda (b: -B).(\lambda (H3: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: -T).(\lambda (H4: (eq C (CHead c3 (Bind b) u2) (CHead e1 (Bind b1) v1))).(let -H5 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) -with [(CSort _) \Rightarrow c3 | (CHead c _ _) \Rightarrow c])) (CHead c3 -(Bind b) u2) (CHead e1 (Bind b1) v1) H4) in ((let H6 \def (f_equal C B -(\lambda (e: C).(match e in C return (\lambda (_: C).B) with [(CSort _) -\Rightarrow b | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: -K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow b])])) (CHead c3 -(Bind b) u2) (CHead e1 (Bind b1) v1) H4) in ((let H7 \def (f_equal C T -(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) -\Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c3 (Bind b) u2) (CHead -e1 (Bind b1) v1) H4) in (\lambda (H8: (eq B b b1)).(\lambda (H9: (eq C c3 -e1)).(let H10 \def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Void))) H3 b1 -H8) in (let H11 \def (eq_ind C c3 (\lambda (c: C).((eq C c (CHead e1 (Bind -b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: -T).(eq C c1 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: -C).(\lambda (_: T).(csuba g e2 e1))))))) H2 e1 H9) in (let H12 \def (eq_ind C -c3 (\lambda (c: C).(csuba g c1 c)) H1 e1 H9) in (ex2_3_intro B C T (\lambda -(b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c1 (Bind Void) u1) -(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: -T).(csuba g e2 e1)))) Void c1 u1 (refl_equal C (CHead c1 (Bind Void) u1)) -H12))))))) H6)) H5))))))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1: -(csuba g c1 c3)).(\lambda (H2: (((eq C c3 (CHead e1 (Bind b1) v1)) \to (ex2_3 -B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c1 (CHead e2 -(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g -e2 e1)))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t -(asucc g a))).(\lambda (u: T).(\lambda (H4: (arity g c3 u a)).(\lambda (H5: -(eq C (CHead c3 (Bind Abbr) u) (CHead e1 (Bind b1) v1))).(let H6 \def -(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with -[(CSort _) \Rightarrow c3 | (CHead c _ _) \Rightarrow c])) (CHead c3 (Bind -Abbr) u) (CHead e1 (Bind b1) v1) H5) in ((let H7 \def (f_equal C B (\lambda -(e: C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow -Abbr | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with -[(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead c3 (Bind -Abbr) u) (CHead e1 (Bind b1) v1) H5) in ((let H8 \def (f_equal C T (\lambda -(e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u -| (CHead _ _ t0) \Rightarrow t0])) (CHead c3 (Bind Abbr) u) (CHead e1 (Bind -b1) v1) H5) in (\lambda (H9: (eq B Abbr b1)).(\lambda (H10: (eq C c3 -e1)).(let H11 \def (eq_ind T u (\lambda (t0: T).(arity g c3 t0 a)) H4 v1 H8) -in (let H12 \def (eq_ind C c3 (\lambda (c: C).(arity g c v1 a)) H11 e1 H10) -in (let H13 \def (eq_ind C c3 (\lambda (c: C).((eq C c (CHead e1 (Bind b1) -v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq -C c1 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda -(_: T).(csuba g e2 e1))))))) H2 e1 H10) in (let H14 \def (eq_ind C c3 -(\lambda (c: C).(csuba g c1 c)) H1 e1 H10) in (let H15 \def (eq_ind_r B b1 -(\lambda (b: B).((eq C e1 (CHead e1 (Bind b) v1)) \to (ex2_3 B C T (\lambda -(b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c1 (CHead e2 (Bind b2) -v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 -e1))))))) H13 Abbr H9) in (ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: -C).(\lambda (v2: T).(eq C (CHead c1 (Bind Abst) t) (CHead e2 (Bind b2) -v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1)))) -Abst c1 t (refl_equal C (CHead c1 (Bind Abst) t)) H14))))))))) H7)) -H6)))))))))))) c2 y H0))) H)))))). -(* COMMENTS -Initial nodes: 1831 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/getl.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/getl.ma deleted file mode 100644 index b49d9b7c2..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/getl.ma +++ /dev/null @@ -1,1178 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csuba/drop.ma". - -include "Basic-1/csuba/clear.ma". - -include "Basic-1/getl/clear.ma". - -theorem csuba_getl_abbr: - \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).(\forall -(i: nat).((getl i c1 (CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csuba g -c1 c2) \to (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) -(\lambda (d2: C).(csuba g d1 d2)))))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u: T).(\lambda -(i: nat).(\lambda (H: (getl i c1 (CHead d1 (Bind Abbr) u))).(let H0 \def -(getl_gen_all c1 (CHead d1 (Bind Abbr) u) i H) in (ex2_ind C (\lambda (e: -C).(drop i O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abbr) u))) -(\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(getl i c2 -(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))) (\lambda (x: -C).(\lambda (H1: (drop i O c1 x)).(\lambda (H2: (clear x (CHead d1 (Bind -Abbr) u))).(C_ind (\lambda (c: C).((drop i O c1 c) \to ((clear c (CHead d1 -(Bind Abbr) u)) \to (\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda -(d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 -d2)))))))) (\lambda (n: nat).(\lambda (_: (drop i O c1 (CSort n))).(\lambda -(H4: (clear (CSort n) (CHead d1 (Bind Abbr) u))).(clear_gen_sort (CHead d1 -(Bind Abbr) u) n H4 (\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda -(d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 -d2))))))))) (\lambda (x0: C).(\lambda (_: (((drop i O c1 x0) \to ((clear x0 -(CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csuba g c1 c2) \to (ex2 C -(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: -C).(csuba g d1 d2))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (H3: -(drop i O c1 (CHead x0 k t))).(\lambda (H4: (clear (CHead x0 k t) (CHead d1 -(Bind Abbr) u))).(K_ind (\lambda (k0: K).((drop i O c1 (CHead x0 k0 t)) \to -((clear (CHead x0 k0 t) (CHead d1 (Bind Abbr) u)) \to (\forall (c2: -C).((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind -Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))))) (\lambda (b: B).(\lambda -(H5: (drop i O c1 (CHead x0 (Bind b) t))).(\lambda (H6: (clear (CHead x0 -(Bind b) t) (CHead d1 (Bind Abbr) u))).(let H7 \def (f_equal C C (\lambda (e: -C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d1 | -(CHead c _ _) \Rightarrow c])) (CHead d1 (Bind Abbr) u) (CHead x0 (Bind b) t) -(clear_gen_bind b x0 (CHead d1 (Bind Abbr) u) t H6)) in ((let H8 \def -(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with -[(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow (match k0 in K -return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) -\Rightarrow Abbr])])) (CHead d1 (Bind Abbr) u) (CHead x0 (Bind b) t) -(clear_gen_bind b x0 (CHead d1 (Bind Abbr) u) t H6)) in ((let H9 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d1 (Bind -Abbr) u) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abbr) u) -t H6)) in (\lambda (H10: (eq B Abbr b)).(\lambda (H11: (eq C d1 x0)).(\lambda -(c2: C).(\lambda (H12: (csuba g c1 c2)).(let H13 \def (eq_ind_r T t (\lambda -(t0: T).(drop i O c1 (CHead x0 (Bind b) t0))) H5 u H9) in (let H14 \def -(eq_ind_r B b (\lambda (b0: B).(drop i O c1 (CHead x0 (Bind b0) u))) H13 Abbr -H10) in (let H15 \def (eq_ind_r C x0 (\lambda (c: C).(drop i O c1 (CHead c -(Bind Abbr) u))) H14 d1 H11) in (let H16 \def (csuba_drop_abbr i c1 d1 u H15 -g c2 H12) in (ex2_ind C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr) -u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(getl i c2 -(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x1: -C).(\lambda (H17: (drop i O c2 (CHead x1 (Bind Abbr) u))).(\lambda (H18: -(csuba g d1 x1)).(ex_intro2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind -Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x1 (getl_intro i c2 (CHead x1 -(Bind Abbr) u) (CHead x1 (Bind Abbr) u) H17 (clear_bind Abbr x1 u)) H18)))) -H16)))))))))) H8)) H7))))) (\lambda (f: F).(\lambda (H5: (drop i O c1 (CHead -x0 (Flat f) t))).(\lambda (H6: (clear (CHead x0 (Flat f) t) (CHead d1 (Bind -Abbr) u))).(let H7 \def H5 in (unintro C c1 (\lambda (c: C).((drop i O c -(CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g c c2) \to (ex2 C -(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: -C).(csuba g d1 d2))))))) (nat_ind (\lambda (n: nat).(\forall (x1: C).((drop n -O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g x1 c2) \to (ex2 C -(\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: -C).(csuba g d1 d2)))))))) (\lambda (x1: C).(\lambda (H8: (drop O O x1 (CHead -x0 (Flat f) t))).(\lambda (c2: C).(\lambda (H9: (csuba g x1 c2)).(let H10 -\def (eq_ind C x1 (\lambda (c: C).(csuba g c c2)) H9 (CHead x0 (Flat f) t) -(drop_gen_refl x1 (CHead x0 (Flat f) t) H8)) in (let H_y \def (clear_flat x0 -(CHead d1 (Bind Abbr) u) (clear_gen_flat f x0 (CHead d1 (Bind Abbr) u) t H6) -f t) in (let H11 \def (csuba_clear_conf g (CHead x0 (Flat f) t) c2 H10 (CHead -d1 (Bind Abbr) u) H_y) in (ex2_ind C (\lambda (e2: C).(csuba g (CHead d1 -(Bind Abbr) u) e2)) (\lambda (e2: C).(clear c2 e2)) (ex2 C (\lambda (d2: -C).(getl O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) -(\lambda (x2: C).(\lambda (H12: (csuba g (CHead d1 (Bind Abbr) u) -x2)).(\lambda (H13: (clear c2 x2)).(let H_x \def (csuba_gen_abbr g d1 x2 u -H12) in (let H14 \def H_x in (ex2_ind C (\lambda (d2: C).(eq C x2 (CHead d2 -(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: -C).(getl O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) -(\lambda (x3: C).(\lambda (H15: (eq C x2 (CHead x3 (Bind Abbr) u))).(\lambda -(H16: (csuba g d1 x3)).(let H17 \def (eq_ind C x2 (\lambda (c: C).(clear c2 -c)) H13 (CHead x3 (Bind Abbr) u) H15) in (ex_intro2 C (\lambda (d2: C).(getl -O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x3 -(getl_intro O c2 (CHead x3 (Bind Abbr) u) c2 (drop_refl c2) H17) H16))))) -H14)))))) H11)))))))) (\lambda (n: nat).(\lambda (H8: ((\forall (x1: -C).((drop n O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g x1 c2) -\to (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))) (\lambda -(d2: C).(csuba g d1 d2))))))))).(\lambda (x1: C).(\lambda (H9: (drop (S n) O -x1 (CHead x0 (Flat f) t))).(\lambda (c2: C).(\lambda (H10: (csuba g x1 -c2)).(let H11 \def (drop_clear x1 (CHead x0 (Flat f) t) n H9) in (ex2_3_ind B -C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear x1 (CHead e (Bind -b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop n O e (CHead -x0 (Flat f) t))))) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind -Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x2: B).(\lambda (x3: -C).(\lambda (x4: T).(\lambda (H12: (clear x1 (CHead x3 (Bind x2) -x4))).(\lambda (H13: (drop n O x3 (CHead x0 (Flat f) t))).(let H14 \def -(csuba_clear_conf g x1 c2 H10 (CHead x3 (Bind x2) x4) H12) in (ex2_ind C -(\lambda (e2: C).(csuba g (CHead x3 (Bind x2) x4) e2)) (\lambda (e2: -C).(clear c2 e2)) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind -Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x5: C).(\lambda (H15: -(csuba g (CHead x3 (Bind x2) x4) x5)).(\lambda (H16: (clear c2 x5)).(let H_x -\def (csuba_gen_bind g x2 x3 x5 x4 H15) in (let H17 \def H_x in (ex2_3_ind B -C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C x5 (CHead e2 -(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g -x3 e2)))) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) u))) -(\lambda (d2: C).(csuba g d1 d2))) (\lambda (x6: B).(\lambda (x7: C).(\lambda -(x8: T).(\lambda (H18: (eq C x5 (CHead x7 (Bind x6) x8))).(\lambda (H19: -(csuba g x3 x7)).(let H20 \def (eq_ind C x5 (\lambda (c: C).(clear c2 c)) H16 -(CHead x7 (Bind x6) x8) H18) in (let H21 \def (H8 x3 H13 x7 H19) in (ex2_ind -C (\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abbr) u))) (\lambda (d2: -C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind -Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x9: C).(\lambda (H22: -(getl n x7 (CHead x9 (Bind Abbr) u))).(\lambda (H23: (csuba g d1 -x9)).(ex_intro2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) u))) -(\lambda (d2: C).(csuba g d1 d2)) x9 (getl_clear_bind x6 c2 x7 x8 H20 (CHead -x9 (Bind Abbr) u) n H22) H23)))) H21)))))))) H17)))))) H14))))))) H11)))))))) -i) H7))))) k H3 H4))))))) x H1 H2)))) H0))))))). -(* COMMENTS -Initial nodes: 2319 -END *) - -theorem csuba_getl_abst: - \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u1: T).(\forall -(i: nat).((getl i c1 (CHead d1 (Bind Abst) u1)) \to (\forall (c2: C).((csuba -g c1 c2) \to (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) -u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a))))))))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u1: T).(\lambda -(i: nat).(\lambda (H: (getl i c1 (CHead d1 (Bind Abst) u1))).(let H0 \def -(getl_gen_all c1 (CHead d1 (Bind Abst) u1) i H) in (ex2_ind C (\lambda (e: -C).(drop i O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abst) u1))) -(\forall (c2: C).((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(getl i c2 -(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind -Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 -d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc -g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))))) (\lambda (x: C).(\lambda (H1: (drop i O c1 x)).(\lambda (H2: (clear -x (CHead d1 (Bind Abst) u1))).(C_ind (\lambda (c: C).((drop i O c1 c) \to -((clear c (CHead d1 (Bind Abst) u1)) \to (\forall (c2: C).((csuba g c1 c2) -\to (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) -(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))))) (\lambda -(n: nat).(\lambda (_: (drop i O c1 (CSort n))).(\lambda (H4: (clear (CSort n) -(CHead d1 (Bind Abst) u1))).(clear_gen_sort (CHead d1 (Bind Abst) u1) n H4 -(\forall (c2: C).((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(getl i c2 -(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind -Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 -d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc -g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))))))))) (\lambda (x0: C).(\lambda (_: (((drop i O c1 x0) \to ((clear x0 -(CHead d1 (Bind Abst) u1)) \to (\forall (c2: C).((csuba g c1 c2) \to (or (ex2 -C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: -C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))))))))).(\lambda (k: K).(\lambda -(t: T).(\lambda (H3: (drop i O c1 (CHead x0 k t))).(\lambda (H4: (clear -(CHead x0 k t) (CHead d1 (Bind Abst) u1))).(K_ind (\lambda (k0: K).((drop i O -c1 (CHead x0 k0 t)) \to ((clear (CHead x0 k0 t) (CHead d1 (Bind Abst) u1)) -\to (\forall (c2: C).((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(getl i -c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T -A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 -(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 -(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 -u2 a))))))))))) (\lambda (b: B).(\lambda (H5: (drop i O c1 (CHead x0 (Bind b) -t))).(\lambda (H6: (clear (CHead x0 (Bind b) t) (CHead d1 (Bind Abst) -u1))).(let H7 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda -(_: C).C) with [(CSort _) \Rightarrow d1 | (CHead c _ _) \Rightarrow c])) -(CHead d1 (Bind Abst) u1) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead -d1 (Bind Abst) u1) t H6)) in ((let H8 \def (f_equal C B (\lambda (e: -C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abst | -(CHead _ k0 _) \Rightarrow (match k0 in K return (\lambda (_: K).B) with -[(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abst])])) (CHead d1 (Bind -Abst) u1) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abst) -u1) t H6)) in ((let H9 \def (f_equal C T (\lambda (e: C).(match e in C return -(\lambda (_: C).T) with [(CSort _) \Rightarrow u1 | (CHead _ _ t0) -\Rightarrow t0])) (CHead d1 (Bind Abst) u1) (CHead x0 (Bind b) t) -(clear_gen_bind b x0 (CHead d1 (Bind Abst) u1) t H6)) in (\lambda (H10: (eq B -Abst b)).(\lambda (H11: (eq C d1 x0)).(\lambda (c2: C).(\lambda (H12: (csuba -g c1 c2)).(let H13 \def (eq_ind_r T t (\lambda (t0: T).(drop i O c1 (CHead x0 -(Bind b) t0))) H5 u1 H9) in (let H14 \def (eq_ind_r B b (\lambda (b0: -B).(drop i O c1 (CHead x0 (Bind b0) u1))) H13 Abst H10) in (let H15 \def -(eq_ind_r C x0 (\lambda (c: C).(drop i O c1 (CHead c (Bind Abst) u1))) H14 d1 -H11) in (let H16 \def (csuba_drop_abst i c1 d1 u1 H15 g c2 H12) in (or_ind -(ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst) u1))) (\lambda -(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (or (ex2 C -(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: -C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (H17: (ex2 C (\lambda -(d2: C).(drop i O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 -d2)))).(ex2_ind C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst) u1))) -(\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: C).(getl i c2 -(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind -Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 -d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc -g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))) (\lambda (x1: C).(\lambda (H18: (drop i O c2 (CHead x1 (Bind Abst) -u1))).(\lambda (H19: (csuba g d1 x1)).(or_introl (ex2 C (\lambda (d2: -C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 -(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity -g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: C).(getl i c2 (CHead d2 -(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) x1 (getl_intro i c2 -(CHead x1 (Bind Abst) u1) (CHead x1 (Bind Abst) u1) H18 (clear_bind Abst x1 -u1)) H19))))) H17)) (\lambda (H17: (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))).(ex4_3_ind C -T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 -(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 -(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 -u2 a)))) (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) -(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x1: -C).(\lambda (x2: T).(\lambda (x3: A).(\lambda (H18: (drop i O c2 (CHead x1 -(Bind Abbr) x2))).(\lambda (H19: (csuba g d1 x1)).(\lambda (H20: (arity g d1 -u1 (asucc g x3))).(\lambda (H21: (arity g x1 x2 x3)).(or_intror (ex2 C -(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: -C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) -x1 x2 x3 (getl_intro i c2 (CHead x1 (Bind Abbr) x2) (CHead x1 (Bind Abbr) x2) -H18 (clear_bind Abbr x1 x2)) H19 H20 H21))))))))) H17)) H16)))))))))) H8)) -H7))))) (\lambda (f: F).(\lambda (H5: (drop i O c1 (CHead x0 (Flat f) -t))).(\lambda (H6: (clear (CHead x0 (Flat f) t) (CHead d1 (Bind Abst) -u1))).(let H7 \def H5 in (unintro C c1 (\lambda (c: C).((drop i O c (CHead x0 -(Flat f) t)) \to (\forall (c2: C).((csuba g c c2) \to (or (ex2 C (\lambda -(d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 -d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i -c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: -A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda -(a: A).(arity g d2 u2 a)))))))))) (nat_ind (\lambda (n: nat).(\forall (x1: -C).((drop n O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g x1 c2) -\to (or (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) u1))) -(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(getl n c2 (CHead d2 (Bind Abbr) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))))) (\lambda -(x1: C).(\lambda (H8: (drop O O x1 (CHead x0 (Flat f) t))).(\lambda (c2: -C).(\lambda (H9: (csuba g x1 c2)).(let H10 \def (eq_ind C x1 (\lambda (c: -C).(csuba g c c2)) H9 (CHead x0 (Flat f) t) (drop_gen_refl x1 (CHead x0 (Flat -f) t) H8)) in (let H_y \def (clear_flat x0 (CHead d1 (Bind Abst) u1) -(clear_gen_flat f x0 (CHead d1 (Bind Abst) u1) t H6) f t) in (let H11 \def -(csuba_clear_conf g (CHead x0 (Flat f) t) c2 H10 (CHead d1 (Bind Abst) u1) -H_y) in (ex2_ind C (\lambda (e2: C).(csuba g (CHead d1 (Bind Abst) u1) e2)) -(\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(getl O c2 (CHead -d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))) (\lambda (x2: C).(\lambda (H12: (csuba g (CHead d1 (Bind Abst) u1) -x2)).(\lambda (H13: (clear c2 x2)).(let H_x \def (csuba_gen_abst g d1 x2 u1 -H12) in (let H14 \def H_x in (or_ind (ex2 C (\lambda (d2: C).(eq C x2 (CHead -d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C x2 (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a))))) (or (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u1))) -(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda -(H15: (ex2 C (\lambda (d2: C).(eq C x2 (CHead d2 (Bind Abst) u1))) (\lambda -(d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(eq C x2 (CHead d2 -(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: -C).(getl O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 -(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity -g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 a)))))) (\lambda (x3: C).(\lambda (H16: (eq C x2 (CHead x3 -(Bind Abst) u1))).(\lambda (H17: (csuba g d1 x3)).(let H18 \def (eq_ind C x2 -(\lambda (c: C).(clear c2 c)) H13 (CHead x3 (Bind Abst) u1) H16) in -(or_introl (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u1))) -(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C -(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: -C).(csuba g d1 d2)) x3 (getl_intro O c2 (CHead x3 (Bind Abst) u1) c2 -(drop_refl c2) H18) H17)))))) H15)) (\lambda (H15: (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(eq C x2 (CHead d2 (Bind Abbr) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(eq C x2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2: -C).(getl O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 -(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity -g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 a)))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: -A).(\lambda (H16: (eq C x2 (CHead x3 (Bind Abbr) x4))).(\lambda (H17: (csuba -g d1 x3)).(\lambda (H18: (arity g d1 u1 (asucc g x5))).(\lambda (H19: (arity -g x3 x4 x5)).(let H20 \def (eq_ind C x2 (\lambda (c: C).(clear c2 c)) H13 -(CHead x3 (Bind Abbr) x4) H16) in (or_intror (ex2 C (\lambda (d2: C).(getl O -c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T -A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 -(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 -(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 -u2 a))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(getl O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x3 x4 x5 (getl_intro O c2 (CHead -x3 (Bind Abbr) x4) c2 (drop_refl c2) H20) H17 H18 H19)))))))))) H15)) -H14)))))) H11)))))))) (\lambda (n: nat).(\lambda (H8: ((\forall (x1: -C).((drop n O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g x1 c2) -\to (or (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) u1))) -(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(getl n c2 (CHead d2 (Bind Abbr) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))))))))).(\lambda (x1: C).(\lambda (H9: (drop (S n) O x1 (CHead x0 (Flat -f) t))).(\lambda (c2: C).(\lambda (H10: (csuba g x1 c2)).(let H11 \def -(drop_clear x1 (CHead x0 (Flat f) t) n H9) in (ex2_3_ind B C T (\lambda (b: -B).(\lambda (e: C).(\lambda (v: T).(clear x1 (CHead e (Bind b) v))))) -(\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop n O e (CHead x0 (Flat -f) t))))) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) -u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: T).(\lambda (H12: -(clear x1 (CHead x3 (Bind x2) x4))).(\lambda (H13: (drop n O x3 (CHead x0 -(Flat f) t))).(let H14 \def (csuba_clear_conf g x1 c2 H10 (CHead x3 (Bind x2) -x4) H12) in (ex2_ind C (\lambda (e2: C).(csuba g (CHead x3 (Bind x2) x4) e2)) -(\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 -(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 -(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 -(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 -u2 a)))))) (\lambda (x5: C).(\lambda (H15: (csuba g (CHead x3 (Bind x2) x4) -x5)).(\lambda (H16: (clear c2 x5)).(let H_x \def (csuba_gen_bind g x2 x3 x5 -x4 H15) in (let H17 \def H_x in (ex2_3_ind B C T (\lambda (b2: B).(\lambda -(e2: C).(\lambda (v2: T).(eq C x5 (CHead e2 (Bind b2) v2))))) (\lambda (_: -B).(\lambda (e2: C).(\lambda (_: T).(csuba g x3 e2)))) (or (ex2 C (\lambda -(d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g -d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl -(S n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x6: B).(\lambda (x7: -C).(\lambda (x8: T).(\lambda (H18: (eq C x5 (CHead x7 (Bind x6) -x8))).(\lambda (H19: (csuba g x3 x7)).(let H20 \def (eq_ind C x5 (\lambda (c: -C).(clear c2 c)) H16 (CHead x7 (Bind x6) x8) H18) in (let H21 \def (H8 x3 H13 -x7 H19) in (or_ind (ex2 C (\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abst) -u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(getl n x7 (CHead d2 (Bind Abbr) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (or -(ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u1))) (\lambda -(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda -(H22: (ex2 C (\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abst) u1))) (\lambda -(d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(getl n x7 (CHead d2 -(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: -C).(getl (S n) c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 -d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S -n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: -A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda -(a: A).(arity g d2 u2 a)))))) (\lambda (x9: C).(\lambda (H23: (getl n x7 -(CHead x9 (Bind Abst) u1))).(\lambda (H24: (csuba g d1 x9)).(or_introl (ex2 C -(\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: -C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(getl (S n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda -(_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: -C).(getl (S n) c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 -d2)) x9 (getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Abst) u1) n H23) -H24))))) H22)) (\lambda (H22: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(getl n x7 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))).(ex4_3_ind C -T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl n x7 (CHead d2 -(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 -(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 -u2 a)))) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) -u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))) (\lambda (x9: C).(\lambda (x10: T).(\lambda (x11: A).(\lambda (H23: -(getl n x7 (CHead x9 (Bind Abbr) x10))).(\lambda (H24: (csuba g d1 -x9)).(\lambda (H25: (arity g d1 u1 (asucc g x11))).(\lambda (H26: (arity g x9 -x10 x11)).(or_intror (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind -Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(getl (S n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x9 x10 x11 (getl_clear_bind x6 -c2 x7 x8 H20 (CHead x9 (Bind Abbr) x10) n H23) H24 H25 H26))))))))) H22)) -H21)))))))) H17)))))) H14))))))) H11)))))))) i) H7))))) k H3 H4))))))) x H1 -H2)))) H0))))))). -(* COMMENTS -Initial nodes: 6437 -END *) - -theorem csuba_getl_abst_rev: - \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).(\forall -(i: nat).((getl i c1 (CHead d1 (Bind Abst) u)) \to (\forall (c2: C).((csuba g -c2 c1) \to (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1)))))))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u: T).(\lambda -(i: nat).(\lambda (H: (getl i c1 (CHead d1 (Bind Abst) u))).(let H0 \def -(getl_gen_all c1 (CHead d1 (Bind Abst) u) i H) in (ex2_ind C (\lambda (e: -C).(drop i O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abst) u))) -(\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: C).(getl i c2 -(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))) (\lambda (x: -C).(\lambda (H1: (drop i O c1 x)).(\lambda (H2: (clear x (CHead d1 (Bind -Abst) u))).(C_ind (\lambda (c: C).((drop i O c1 c) \to ((clear c (CHead d1 -(Bind Abst) u)) \to (\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda -(d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 -d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))))) -(\lambda (n: nat).(\lambda (_: (drop i O c1 (CSort n))).(\lambda (H4: (clear -(CSort n) (CHead d1 (Bind Abst) u))).(clear_gen_sort (CHead d1 (Bind Abst) u) -n H4 (\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: C).(getl -i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))) (\lambda (x0: -C).(\lambda (_: (((drop i O c1 x0) \to ((clear x0 (CHead d1 (Bind Abst) u)) -\to (\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: C).(getl i -c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))).(\lambda (k: -K).(\lambda (t: T).(\lambda (H3: (drop i O c1 (CHead x0 k t))).(\lambda (H4: -(clear (CHead x0 k t) (CHead d1 (Bind Abst) u))).(K_ind (\lambda (k0: -K).((drop i O c1 (CHead x0 k0 t)) \to ((clear (CHead x0 k0 t) (CHead d1 (Bind -Abst) u)) \to (\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: -C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))))) (\lambda (b: -B).(\lambda (H5: (drop i O c1 (CHead x0 (Bind b) t))).(\lambda (H6: (clear -(CHead x0 (Bind b) t) (CHead d1 (Bind Abst) u))).(let H7 \def (f_equal C C -(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow d1 | (CHead c _ _) \Rightarrow c])) (CHead d1 (Bind Abst) u) -(CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abst) u) t H6)) in -((let H8 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: -C).B) with [(CSort _) \Rightarrow Abst | (CHead _ k0 _) \Rightarrow (match k0 -in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) -\Rightarrow Abst])])) (CHead d1 (Bind Abst) u) (CHead x0 (Bind b) t) -(clear_gen_bind b x0 (CHead d1 (Bind Abst) u) t H6)) in ((let H9 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d1 (Bind -Abst) u) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abst) u) -t H6)) in (\lambda (H10: (eq B Abst b)).(\lambda (H11: (eq C d1 x0)).(\lambda -(c2: C).(\lambda (H12: (csuba g c2 c1)).(let H13 \def (eq_ind_r T t (\lambda -(t0: T).(drop i O c1 (CHead x0 (Bind b) t0))) H5 u H9) in (let H14 \def -(eq_ind_r B b (\lambda (b0: B).(drop i O c1 (CHead x0 (Bind b0) u))) H13 Abst -H10) in (let H15 \def (eq_ind_r C x0 (\lambda (c: C).(drop i O c1 (CHead c -(Bind Abst) u))) H14 d1 H11) in (let H16 \def (csuba_drop_abst_rev i c1 d1 u -H15 g c2 H12) in (or_ind (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind -Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop i O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: C).(getl i c2 -(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H17: (ex2 C -(\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop i O c2 (CHead d2 -(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: -C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x1: -C).(\lambda (H18: (drop i O c2 (CHead x1 (Bind Abst) u))).(\lambda (H19: -(csuba g x1 d1)).(or_introl (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 -(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(getl i -c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x1 -(getl_intro i c2 (CHead x1 (Bind Abst) u) (CHead x1 (Bind Abst) u) H18 -(clear_bind Abst x1 u)) H19))))) H17)) (\lambda (H17: (ex2_2 C T (\lambda -(d2: C).(\lambda (u2: T).(drop i O c2 (CHead d2 (Bind Void) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: -C).(\lambda (u2: T).(drop i O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1))) (or (ex2 C (\lambda (d2: C).(getl i c2 -(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x1: -C).(\lambda (x2: T).(\lambda (H18: (drop i O c2 (CHead x1 (Bind Void) -x2))).(\lambda (H19: (csuba g x1 d1)).(or_intror (ex2 C (\lambda (d2: -C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T -(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x1 x2 (getl_intro i c2 -(CHead x1 (Bind Void) x2) (CHead x1 (Bind Void) x2) H18 (clear_bind Void x1 -x2)) H19)))))) H17)) H16)))))))))) H8)) H7))))) (\lambda (f: F).(\lambda (H5: -(drop i O c1 (CHead x0 (Flat f) t))).(\lambda (H6: (clear (CHead x0 (Flat f) -t) (CHead d1 (Bind Abst) u))).(let H7 \def H5 in (unintro C c1 (\lambda (c: -C).((drop i O c (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g c2 c) -\to (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))))))) (nat_ind (\lambda (n: nat).(\forall (x1: C).((drop -n O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g c2 x1) \to (or -(ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl n c2 -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))))))))) (\lambda (x1: C).(\lambda (H8: (drop O O x1 (CHead x0 (Flat f) -t))).(\lambda (c2: C).(\lambda (H9: (csuba g c2 x1)).(let H10 \def (eq_ind C -x1 (\lambda (c: C).(csuba g c2 c)) H9 (CHead x0 (Flat f) t) (drop_gen_refl x1 -(CHead x0 (Flat f) t) H8)) in (let H_y \def (clear_flat x0 (CHead d1 (Bind -Abst) u) (clear_gen_flat f x0 (CHead d1 (Bind Abst) u) t H6) f t) in (let H11 -\def (csuba_clear_trans g (CHead x0 (Flat f) t) c2 H10 (CHead d1 (Bind Abst) -u) H_y) in (ex2_ind C (\lambda (e2: C).(csuba g e2 (CHead d1 (Bind Abst) u))) -(\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(getl O c2 (CHead -d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda -(d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 (Bind Void) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x2: C).(\lambda (H12: -(csuba g x2 (CHead d1 (Bind Abst) u))).(\lambda (H13: (clear c2 x2)).(let H_x -\def (csuba_gen_abst_rev g d1 x2 u H12) in (let H14 \def H_x in (or_ind (ex2 -C (\lambda (d2: C).(eq C x2 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C x2 -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))) (or (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(getl O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))) (\lambda (H15: (ex2 C (\lambda (d2: C).(eq C x2 (CHead -d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda -(d2: C).(eq C x2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) -(or (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u))) (\lambda -(d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl -O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g -d2 d1))))) (\lambda (x3: C).(\lambda (H16: (eq C x2 (CHead x3 (Bind Abst) -u))).(\lambda (H17: (csuba g x3 d1)).(let H18 \def (eq_ind C x2 (\lambda (c: -C).(clear c2 c)) H13 (CHead x3 (Bind Abst) u) H16) in (or_introl (ex2 C -(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl O c2 -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))) (ex_intro2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1)) x3 (getl_intro O c2 (CHead x3 (Bind Abst) -u) c2 (drop_refl c2) H18) H17)))))) H15)) (\lambda (H15: (ex2_2 C T (\lambda -(d2: C).(\lambda (u2: T).(eq C x2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: -C).(\lambda (u2: T).(eq C x2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1))) (or (ex2 C (\lambda (d2: C).(getl O c2 -(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x3: -C).(\lambda (x4: T).(\lambda (H16: (eq C x2 (CHead x3 (Bind Void) -x4))).(\lambda (H17: (csuba g x3 d1)).(let H18 \def (eq_ind C x2 (\lambda (c: -C).(clear c2 c)) H13 (CHead x3 (Bind Void) x4) H16) in (or_intror (ex2 C -(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl O c2 -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead -d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x3 -x4 (getl_intro O c2 (CHead x3 (Bind Void) x4) c2 (drop_refl c2) H18) -H17))))))) H15)) H14)))))) H11)))))))) (\lambda (n: nat).(\lambda (H8: -((\forall (x1: C).((drop n O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: -C).((csuba g c2 x1) \to (or (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 -(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(getl n c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1))))))))))).(\lambda (x1: C).(\lambda (H9: -(drop (S n) O x1 (CHead x0 (Flat f) t))).(\lambda (c2: C).(\lambda (H10: -(csuba g c2 x1)).(let H11 \def (drop_clear x1 (CHead x0 (Flat f) t) n H9) in -(ex2_3_ind B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear x1 -(CHead e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: -T).(drop n O e (CHead x0 (Flat f) t))))) (or (ex2 C (\lambda (d2: C).(getl (S -n) c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C -T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x2: -B).(\lambda (x3: C).(\lambda (x4: T).(\lambda (H12: (clear x1 (CHead x3 (Bind -x2) x4))).(\lambda (H13: (drop n O x3 (CHead x0 (Flat f) t))).(let H14 \def -(csuba_clear_trans g x1 c2 H10 (CHead x3 (Bind x2) x4) H12) in (ex2_ind C -(\lambda (e2: C).(csuba g e2 (CHead x3 (Bind x2) x4))) (\lambda (e2: -C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind -Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x5: C).(\lambda (H15: (csuba -g x5 (CHead x3 (Bind x2) x4))).(\lambda (H16: (clear c2 x5)).(let H_x \def -(csuba_gen_bind_rev g x2 x3 x5 x4 H15) in (let H17 \def H_x in (ex2_3_ind B C -T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C x5 (CHead e2 (Bind -b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 -x3)))) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))) (\lambda (x6: B).(\lambda (x7: C).(\lambda (x8: -T).(\lambda (H18: (eq C x5 (CHead x7 (Bind x6) x8))).(\lambda (H19: (csuba g -x7 x3)).(let H20 \def (eq_ind C x5 (\lambda (c: C).(clear c2 c)) H16 (CHead -x7 (Bind x6) x8) H18) in (let H21 \def (H8 x3 H13 x7 H19) in (or_ind (ex2 C -(\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl n x7 -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))) (\lambda (H22: (ex2 C (\lambda (d2: C).(getl n x7 -(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C -(\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 -(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x9: C).(\lambda (H23: (getl -n x7 (CHead x9 (Bind Abst) u))).(\lambda (H24: (csuba g x9 d1)).(or_introl -(ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u))) (\lambda -(d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl -(S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba -g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind -Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x9 (getl_clear_bind x6 c2 x7 x8 -H20 (CHead x9 (Bind Abst) u) n H23) H24))))) H22)) (\lambda (H22: (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(getl n x7 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda -(d2: C).(\lambda (u2: T).(getl n x7 (CHead d2 (Bind Void) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1))) (or (ex2 C (\lambda (d2: C).(getl -(S n) c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 -C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x9: -C).(\lambda (x10: T).(\lambda (H23: (getl n x7 (CHead x9 (Bind Void) -x10))).(\lambda (H24: (csuba g x9 d1)).(or_intror (ex2 C (\lambda (d2: -C).(getl (S n) c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 -d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) -(ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x9 x10 -(getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Void) x10) n H23) H24)))))) -H22)) H21)))))))) H17)))))) H14))))))) H11)))))))) i) H7))))) k H3 H4))))))) -x H1 H2)))) H0))))))). -(* COMMENTS -Initial nodes: 4703 -END *) - -theorem csuba_getl_abbr_rev: - \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u1: T).(\forall -(i: nat).((getl i c1 (CHead d1 (Bind Abbr) u1)) \to (\forall (c2: C).((csuba -g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 -C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u1: T).(\lambda -(i: nat).(\lambda (H: (getl i c1 (CHead d1 (Bind Abbr) u1))).(let H0 \def -(getl_gen_all c1 (CHead d1 (Bind Abbr) u1) i H) in (ex2_ind C (\lambda (e: -C).(drop i O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abbr) u1))) -(\forall (c2: C).((csuba g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(getl i c2 -(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))) -(\lambda (x: C).(\lambda (H1: (drop i O c1 x)).(\lambda (H2: (clear x (CHead -d1 (Bind Abbr) u1))).(C_ind (\lambda (c: C).((drop i O c1 c) \to ((clear c -(CHead d1 (Bind Abbr) u1)) \to (\forall (c2: C).((csuba g c2 c1) \to (or3 -(ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))))))))) (\lambda (n: nat).(\lambda (_: -(drop i O c1 (CSort n))).(\lambda (H4: (clear (CSort n) (CHead d1 (Bind Abbr) -u1))).(clear_gen_sort (CHead d1 (Bind Abbr) u1) n H4 (\forall (c2: C).((csuba -g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 -C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))) (\lambda (x0: -C).(\lambda (_: (((drop i O c1 x0) \to ((clear x0 (CHead d1 (Bind Abbr) u1)) -\to (\forall (c2: C).((csuba g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(getl -i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C -T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 -(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (H3: (drop i O c1 -(CHead x0 k t))).(\lambda (H4: (clear (CHead x0 k t) (CHead d1 (Bind Abbr) -u1))).(K_ind (\lambda (k0: K).((drop i O c1 (CHead x0 k0 t)) \to ((clear -(CHead x0 k0 t) (CHead d1 (Bind Abbr) u1)) \to (\forall (c2: C).((csuba g c2 -c1) \to (or3 (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) -(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))))) (\lambda (b: -B).(\lambda (H5: (drop i O c1 (CHead x0 (Bind b) t))).(\lambda (H6: (clear -(CHead x0 (Bind b) t) (CHead d1 (Bind Abbr) u1))).(let H7 \def (f_equal C C -(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow d1 | (CHead c _ _) \Rightarrow c])) (CHead d1 (Bind Abbr) u1) -(CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abbr) u1) t H6)) -in ((let H8 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda -(_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow -(match k0 in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | -(Flat _) \Rightarrow Abbr])])) (CHead d1 (Bind Abbr) u1) (CHead x0 (Bind b) -t) (clear_gen_bind b x0 (CHead d1 (Bind Abbr) u1) t H6)) in ((let H9 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow u1 | (CHead _ _ t0) \Rightarrow t0])) (CHead d1 (Bind -Abbr) u1) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abbr) -u1) t H6)) in (\lambda (H10: (eq B Abbr b)).(\lambda (H11: (eq C d1 -x0)).(\lambda (c2: C).(\lambda (H12: (csuba g c2 c1)).(let H13 \def (eq_ind_r -T t (\lambda (t0: T).(drop i O c1 (CHead x0 (Bind b) t0))) H5 u1 H9) in (let -H14 \def (eq_ind_r B b (\lambda (b0: B).(drop i O c1 (CHead x0 (Bind b0) -u1))) H13 Abbr H10) in (let H15 \def (eq_ind_r C x0 (\lambda (c: C).(drop i O -c1 (CHead c (Bind Abbr) u1))) H14 d1 H11) in (let H16 \def -(csuba_drop_abbr_rev i c1 d1 u1 H15 g c2 H12) in (or3_ind (ex2 C (\lambda -(d2: C).(drop i O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop i -O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop i O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 -C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H17: (ex2 C -(\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop i O c2 (CHead d2 -(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) (or3 (ex2 C (\lambda (d2: -C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))) (\lambda (x1: C).(\lambda (H18: (drop i O c2 (CHead x1 -(Bind Abbr) u1))).(\lambda (H19: (csuba g x1 d1)).(or3_intro0 (ex2 C (\lambda -(d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i -c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(getl i c2 (CHead d2 -(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x1 (getl_intro i c2 -(CHead x1 (Bind Abbr) u1) (CHead x1 (Bind Abbr) u1) H18 (clear_bind Abbr x1 -u1)) H19))))) H17)) (\lambda (H17: (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abst) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T -A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 -(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a)))) (or3 (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) -(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x1: -C).(\lambda (x2: T).(\lambda (x3: A).(\lambda (H18: (drop i O c2 (CHead x1 -(Bind Abst) x2))).(\lambda (H19: (csuba g x1 d1)).(\lambda (H20: (arity g x1 -x2 (asucc g x3))).(\lambda (H21: (arity g d1 u1 x3)).(or3_intro1 (ex2 C -(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))) (ex4_3_intro C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x1 x2 x3 -(getl_intro i c2 (CHead x1 (Bind Abst) x2) (CHead x1 (Bind Abst) x2) H18 -(clear_bind Abst x1 x2)) H19 H20 H21))))))))) H17)) (\lambda (H17: (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop i O c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda -(d2: C).(\lambda (u2: T).(drop i O c2 (CHead d2 (Bind Void) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C (\lambda (d2: C).(getl -i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C -T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 -(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) -(\lambda (x1: C).(\lambda (x2: T).(\lambda (H18: (drop i O c2 (CHead x1 (Bind -Void) x2))).(\lambda (H19: (csuba g x1 d1)).(or3_intro2 (ex2 C (\lambda (d2: -C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: -T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))) x1 x2 (getl_intro i c2 (CHead x1 (Bind Void) x2) (CHead -x1 (Bind Void) x2) H18 (clear_bind Void x1 x2)) H19)))))) H17)) H16)))))))))) -H8)) H7))))) (\lambda (f: F).(\lambda (H5: (drop i O c1 (CHead x0 (Flat f) -t))).(\lambda (H6: (clear (CHead x0 (Flat f) t) (CHead d1 (Bind Abbr) -u1))).(let H7 \def H5 in (unintro C c1 (\lambda (c: C).((drop i O c (CHead x0 -(Flat f) t)) \to (\forall (c2: C).((csuba g c2 c) \to (or3 (ex2 C (\lambda -(d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i -c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))))))) (nat_ind (\lambda (n: nat).(\forall (x1: C).((drop -n O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g c2 x1) \to (or3 -(ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(getl n c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(getl n c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))))))))) (\lambda (x1: C).(\lambda (H8: -(drop O O x1 (CHead x0 (Flat f) t))).(\lambda (c2: C).(\lambda (H9: (csuba g -c2 x1)).(let H10 \def (eq_ind C x1 (\lambda (c: C).(csuba g c2 c)) H9 (CHead -x0 (Flat f) t) (drop_gen_refl x1 (CHead x0 (Flat f) t) H8)) in (let H_y \def -(clear_flat x0 (CHead d1 (Bind Abbr) u1) (clear_gen_flat f x0 (CHead d1 (Bind -Abbr) u1) t H6) f t) in (let H11 \def (csuba_clear_trans g (CHead x0 (Flat f) -t) c2 H10 (CHead d1 (Bind Abbr) u1) H_y) in (ex2_ind C (\lambda (e2: -C).(csuba g e2 (CHead d1 (Bind Abbr) u1))) (\lambda (e2: C).(clear c2 e2)) -(or3 (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u1))) (\lambda -(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x2: -C).(\lambda (H12: (csuba g x2 (CHead d1 (Bind Abbr) u1))).(\lambda (H13: -(clear c2 x2)).(let H_x \def (csuba_gen_abbr_rev g d1 x2 u1 H12) in (let H14 -\def H_x in (or3_ind (ex2 C (\lambda (d2: C).(eq C x2 (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(eq C x2 (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 -C T (\lambda (d2: C).(\lambda (u2: T).(eq C x2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: -C).(getl O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(getl O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))) (\lambda (H15: (ex2 C (\lambda (d2: C).(eq C x2 (CHead -d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda -(d2: C).(eq C x2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1)) (or3 (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u1))) -(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abst) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x3: -C).(\lambda (H16: (eq C x2 (CHead x3 (Bind Abbr) u1))).(\lambda (H17: (csuba -g x3 d1)).(let H18 \def (eq_ind C x2 (\lambda (c: C).(clear c2 c)) H13 (CHead -x3 (Bind Abbr) u1) H16) in (or3_intro0 (ex2 C (\lambda (d2: C).(getl O c2 -(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) -(ex_intro2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u1))) (\lambda -(d2: C).(csuba g d2 d1)) x3 (getl_intro O c2 (CHead x3 (Bind Abbr) u1) c2 -(drop_refl c2) H18) H17)))))) H15)) (\lambda (H15: (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(eq C x2 (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 -a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(eq C x2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C (\lambda (d2: -C).(getl O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(getl O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: -A).(\lambda (H16: (eq C x2 (CHead x3 (Bind Abst) x4))).(\lambda (H17: (csuba -g x3 d1)).(\lambda (H18: (arity g x3 x4 (asucc g x5))).(\lambda (H19: (arity -g d1 u1 x5)).(let H20 \def (eq_ind C x2 (\lambda (c: C).(clear c2 c)) H13 -(CHead x3 (Bind Abst) x4) H16) in (or3_intro1 (ex2 C (\lambda (d2: C).(getl O -c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T -A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 -(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) -(ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O -c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a)))) x3 x4 x5 (getl_intro O c2 (CHead x3 (Bind Abst) -x4) c2 (drop_refl c2) H20) H17 H18 H19)))))))))) H15)) (\lambda (H15: (ex2_2 -C T (\lambda (d2: C).(\lambda (u2: T).(eq C x2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda -(d2: C).(\lambda (u2: T).(eq C x2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C (\lambda (d2: C).(getl O c2 -(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) -(\lambda (x3: C).(\lambda (x4: T).(\lambda (H16: (eq C x2 (CHead x3 (Bind -Void) x4))).(\lambda (H17: (csuba g x3 d1)).(let H18 \def (eq_ind C x2 -(\lambda (c: C).(clear c2 c)) H13 (CHead x3 (Bind Void) x4) H16) in -(or3_intro2 (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u1))) -(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abst) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda -(d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 (Bind Void) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1))) x3 x4 (getl_intro O c2 (CHead x3 -(Bind Void) x4) c2 (drop_refl c2) H18) H17))))))) H15)) H14)))))) H11)))))))) -(\lambda (n: nat).(\lambda (H8: ((\forall (x1: C).((drop n O x1 (CHead x0 -(Flat f) t)) \to (\forall (c2: C).((csuba g c2 x1) \to (or3 (ex2 C (\lambda -(d2: C).(getl n c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl n -c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(getl n c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))))))))).(\lambda (x1: C).(\lambda (H9: (drop (S n) O x1 -(CHead x0 (Flat f) t))).(\lambda (c2: C).(\lambda (H10: (csuba g c2 x1)).(let -H11 \def (drop_clear x1 (CHead x0 (Flat f) t) n H9) in (ex2_3_ind B C T -(\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear x1 (CHead e (Bind b) -v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop n O e (CHead x0 -(Flat f) t))))) (or3 (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda -(x2: B).(\lambda (x3: C).(\lambda (x4: T).(\lambda (H12: (clear x1 (CHead x3 -(Bind x2) x4))).(\lambda (H13: (drop n O x3 (CHead x0 (Flat f) t))).(let H14 -\def (csuba_clear_trans g x1 c2 H10 (CHead x3 (Bind x2) x4) H12) in (ex2_ind -C (\lambda (e2: C).(csuba g e2 (CHead x3 (Bind x2) x4))) (\lambda (e2: -C).(clear c2 e2)) (or3 (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda -(x5: C).(\lambda (H15: (csuba g x5 (CHead x3 (Bind x2) x4))).(\lambda (H16: -(clear c2 x5)).(let H_x \def (csuba_gen_bind_rev g x2 x3 x5 x4 H15) in (let -H17 \def H_x in (ex2_3_ind B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda -(v2: T).(eq C x5 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: -C).(\lambda (_: T).(csuba g e2 x3)))) (or3 (ex2 C (\lambda (d2: C).(getl (S -n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C -T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead -d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))) (\lambda (x6: B).(\lambda (x7: C).(\lambda (x8: -T).(\lambda (H18: (eq C x5 (CHead x7 (Bind x6) x8))).(\lambda (H19: (csuba g -x7 x3)).(let H20 \def (eq_ind C x5 (\lambda (c: C).(clear c2 c)) H16 (CHead -x7 (Bind x6) x8) H18) in (let H21 \def (H8 x3 H13 x7 H19) in (or3_ind (ex2 C -(\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(getl n x7 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(getl n x7 (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(getl (S -n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C -T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead -d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))) (\lambda (H22: (ex2 C (\lambda (d2: C).(getl n x7 -(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C -(\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1)) (or3 (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 -(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda -(x9: C).(\lambda (H23: (getl n x7 (CHead x9 (Bind Abbr) u1))).(\lambda (H24: -(csuba g x9 d1)).(or3_intro0 (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 -(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C -(\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1)) x9 (getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Abbr) -u1) n H23) H24))))) H22)) (\lambda (H22: (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(getl n x7 (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 -a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(getl n x7 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C (\lambda (d2: -C).(getl (S n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S -n) c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))) (\lambda (x9: C).(\lambda (x10: T).(\lambda (x11: -A).(\lambda (H23: (getl n x7 (CHead x9 (Bind Abst) x10))).(\lambda (H24: -(csuba g x9 d1)).(\lambda (H25: (arity g x9 x10 (asucc g x11))).(\lambda -(H26: (arity g d1 u1 x11)).(or3_intro1 (ex2 C (\lambda (d2: C).(getl (S n) c2 -(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 -(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead -d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) -(ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S -n) c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a)))) x9 x10 x11 (getl_clear_bind x6 c2 x7 x8 H20 -(CHead x9 (Bind Abst) x10) n H23) H24 H25 H26))))))))) H22)) (\lambda (H22: -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl n x7 (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind C T -(\lambda (d2: C).(\lambda (u2: T).(getl n x7 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C (\lambda (d2: -C).(getl (S n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S -n) c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))) (\lambda (x9: C).(\lambda (x10: T).(\lambda (H23: -(getl n x7 (CHead x9 (Bind Void) x10))).(\lambda (H24: (csuba g x9 -d1)).(or3_intro2 (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro -C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x9 x10 -(getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Void) x10) n H23) H24)))))) -H22)) H21)))))))) H17)))))) H14))))))) H11)))))))) i) H7))))) k H3 H4))))))) -x H1 H2)))) H0))))))). -(* COMMENTS -Initial nodes: 9091 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/props.ma deleted file mode 100644 index 823e88825..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/props.ma +++ /dev/null @@ -1,28 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csuba/defs.ma". - -theorem csuba_refl: - \forall (g: G).(\forall (c: C).(csuba g c c)) -\def - \lambda (g: G).(\lambda (c: C).(C_ind (\lambda (c0: C).(csuba g c0 c0)) -(\lambda (n: nat).(csuba_sort g n)) (\lambda (c0: C).(\lambda (H: (csuba g c0 -c0)).(\lambda (k: K).(\lambda (t: T).(csuba_head g c0 c0 H k t))))) c)). -(* COMMENTS -Initial nodes: 53 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/arity.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/arity.ma deleted file mode 100644 index 62922d311..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/arity.ma +++ /dev/null @@ -1,42 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubc/csuba.ma". - -theorem csubc_arity_conf: - \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to -(\forall (t: T).(\forall (a: A).((arity g c1 t a) \to (arity g c2 t a))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubc g c1 -c2)).(\lambda (t: T).(\lambda (a: A).(\lambda (H0: (arity g c1 t -a)).(csuba_arity g c1 t a H0 c2 (csubc_csuba g c1 c2 H)))))))). -(* COMMENTS -Initial nodes: 51 -END *) - -theorem csubc_arity_trans: - \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to -((csubv c1 c2) \to (\forall (t: T).(\forall (a: A).((arity g c2 t a) \to -(arity g c1 t a)))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubc g c1 -c2)).(\lambda (H0: (csubv c1 c2)).(\lambda (t: T).(\lambda (a: A).(\lambda -(H1: (arity g c2 t a)).(csuba_arity_rev g c2 t a H1 c1 (csubc_csuba g c1 c2 -H) H0)))))))). -(* COMMENTS -Initial nodes: 59 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/clear.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/clear.ma deleted file mode 100644 index 185bdff75..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/clear.ma +++ /dev/null @@ -1,170 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubc/fwd.ma". - -theorem csubc_clear_conf: - \forall (g: G).(\forall (c1: C).(\forall (e1: C).((clear c1 e1) \to (\forall -(c2: C).((csubc g c1 c2) \to (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda -(e2: C).(csubc g e1 e2)))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (e1: C).(\lambda (H: (clear c1 -e1)).(clear_ind (\lambda (c: C).(\lambda (c0: C).(\forall (c2: C).((csubc g c -c2) \to (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2: C).(csubc g c0 -e2))))))) (\lambda (b: B).(\lambda (e: C).(\lambda (u: T).(\lambda (c2: -C).(\lambda (H0: (csubc g (CHead e (Bind b) u) c2)).(let H_x \def -(csubc_gen_head_l g e c2 u (Bind b) H0) in (let H1 \def H_x in (or3_ind (ex2 -C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind b) u))) (\lambda (c3: C).(csubc g -e c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K -(Bind b) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq -C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda -(_: A).(csubc g e c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 -g (asucc g a) e u)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g -a c3 w))))) (ex4_3 B C T (\lambda (b0: B).(\lambda (c3: C).(\lambda (v2: -T).(eq C c2 (CHead c3 (Bind b0) v2))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: T).(eq K (Bind b) (Bind Void))))) (\lambda (b0: B).(\lambda -(_: C).(\lambda (_: T).(not (eq B b0 Void))))) (\lambda (_: B).(\lambda (c3: -C).(\lambda (_: T).(csubc g e c3))))) (ex2 C (\lambda (e2: C).(clear c2 e2)) -(\lambda (e2: C).(csubc g (CHead e (Bind b) u) e2))) (\lambda (H2: (ex2 C -(\lambda (c3: C).(eq C c2 (CHead c3 (Bind b) u))) (\lambda (c3: C).(csubc g e -c3)))).(ex2_ind C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind b) u))) (\lambda -(c3: C).(csubc g e c3)) (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2: -C).(csubc g (CHead e (Bind b) u) e2))) (\lambda (x: C).(\lambda (H3: (eq C c2 -(CHead x (Bind b) u))).(\lambda (H4: (csubc g e x)).(eq_ind_r C (CHead x -(Bind b) u) (\lambda (c: C).(ex2 C (\lambda (e2: C).(clear c e2)) (\lambda -(e2: C).(csubc g (CHead e (Bind b) u) e2)))) (ex_intro2 C (\lambda (e2: -C).(clear (CHead x (Bind b) u) e2)) (\lambda (e2: C).(csubc g (CHead e (Bind -b) u) e2)) (CHead x (Bind b) u) (clear_bind b x u) (csubc_head g e x H4 (Bind -b) u)) c2 H3)))) H2)) (\lambda (H2: (ex5_3 C T A (\lambda (_: C).(\lambda (_: -T).(\lambda (_: A).(eq K (Bind b) (Bind Abst))))) (\lambda (c3: C).(\lambda -(w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: -C).(\lambda (_: T).(\lambda (_: A).(csubc g e c3)))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(sc3 g (asucc g a) e u)))) (\lambda (c3: C).(\lambda -(w: T).(\lambda (a: A).(sc3 g a c3 w)))))).(ex5_3_ind C T A (\lambda (_: -C).(\lambda (_: T).(\lambda (_: A).(eq K (Bind b) (Bind Abst))))) (\lambda -(c3: C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) -(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g e c3)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) e u)))) (\lambda -(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w)))) (ex2 C (\lambda -(e2: C).(clear c2 e2)) (\lambda (e2: C).(csubc g (CHead e (Bind b) u) e2))) -(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H3: (eq K (Bind -b) (Bind Abst))).(\lambda (H4: (eq C c2 (CHead x0 (Bind Abbr) x1))).(\lambda -(H5: (csubc g e x0)).(\lambda (H6: (sc3 g (asucc g x2) e u)).(\lambda (H7: -(sc3 g x2 x0 x1)).(eq_ind_r C (CHead x0 (Bind Abbr) x1) (\lambda (c: C).(ex2 -C (\lambda (e2: C).(clear c e2)) (\lambda (e2: C).(csubc g (CHead e (Bind b) -u) e2)))) (let H8 \def (f_equal K B (\lambda (e0: K).(match e0 in K return -(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow b])) -(Bind b) (Bind Abst) H3) in (eq_ind_r B Abst (\lambda (b0: B).(ex2 C (\lambda -(e2: C).(clear (CHead x0 (Bind Abbr) x1) e2)) (\lambda (e2: C).(csubc g -(CHead e (Bind b0) u) e2)))) (ex_intro2 C (\lambda (e2: C).(clear (CHead x0 -(Bind Abbr) x1) e2)) (\lambda (e2: C).(csubc g (CHead e (Bind Abst) u) e2)) -(CHead x0 (Bind Abbr) x1) (clear_bind Abbr x0 x1) (csubc_abst g e x0 H5 u x2 -H6 x1 H7)) b H8)) c2 H4))))))))) H2)) (\lambda (H2: (ex4_3 B C T (\lambda -(b0: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b0) -v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K (Bind b) (Bind -Void))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b0 -Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g e -c3)))))).(ex4_3_ind B C T (\lambda (b0: B).(\lambda (c3: C).(\lambda (v2: -T).(eq C c2 (CHead c3 (Bind b0) v2))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: T).(eq K (Bind b) (Bind Void))))) (\lambda (b0: B).(\lambda -(_: C).(\lambda (_: T).(not (eq B b0 Void))))) (\lambda (_: B).(\lambda (c3: -C).(\lambda (_: T).(csubc g e c3)))) (ex2 C (\lambda (e2: C).(clear c2 e2)) -(\lambda (e2: C).(csubc g (CHead e (Bind b) u) e2))) (\lambda (x0: -B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H3: (eq C c2 (CHead x1 (Bind -x0) x2))).(\lambda (H4: (eq K (Bind b) (Bind Void))).(\lambda (H5: (not (eq B -x0 Void))).(\lambda (H6: (csubc g e x1)).(eq_ind_r C (CHead x1 (Bind x0) x2) -(\lambda (c: C).(ex2 C (\lambda (e2: C).(clear c e2)) (\lambda (e2: C).(csubc -g (CHead e (Bind b) u) e2)))) (let H7 \def (f_equal K B (\lambda (e0: -K).(match e0 in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | -(Flat _) \Rightarrow b])) (Bind b) (Bind Void) H4) in (eq_ind_r B Void -(\lambda (b0: B).(ex2 C (\lambda (e2: C).(clear (CHead x1 (Bind x0) x2) e2)) -(\lambda (e2: C).(csubc g (CHead e (Bind b0) u) e2)))) (ex_intro2 C (\lambda -(e2: C).(clear (CHead x1 (Bind x0) x2) e2)) (\lambda (e2: C).(csubc g (CHead -e (Bind Void) u) e2)) (CHead x1 (Bind x0) x2) (clear_bind x0 x1 x2) -(csubc_void g e x1 H6 x0 H5 u x2)) b H7)) c2 H3)))))))) H2)) H1)))))))) -(\lambda (e: C).(\lambda (c: C).(\lambda (_: (clear e c)).(\lambda (H1: -((\forall (c2: C).((csubc g e c2) \to (ex2 C (\lambda (e2: C).(clear c2 e2)) -(\lambda (e2: C).(csubc g c e2))))))).(\lambda (f: F).(\lambda (u: -T).(\lambda (c2: C).(\lambda (H2: (csubc g (CHead e (Flat f) u) c2)).(let H_x -\def (csubc_gen_head_l g e c2 u (Flat f) H2) in (let H3 \def H_x in (or3_ind -(ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 (Flat f) u))) (\lambda (c3: -C).(csubc g e c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: -A).(eq K (Flat f) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda -(_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: -T).(\lambda (_: A).(csubc g e c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(sc3 g (asucc g a) e u)))) (\lambda (c3: C).(\lambda (w: T).(\lambda -(a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: -C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: T).(eq K (Flat f) (Bind Void))))) (\lambda -(b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: -B).(\lambda (c3: C).(\lambda (_: T).(csubc g e c3))))) (ex2 C (\lambda (e2: -C).(clear c2 e2)) (\lambda (e2: C).(csubc g c e2))) (\lambda (H4: (ex2 C -(\lambda (c3: C).(eq C c2 (CHead c3 (Flat f) u))) (\lambda (c3: C).(csubc g e -c3)))).(ex2_ind C (\lambda (c3: C).(eq C c2 (CHead c3 (Flat f) u))) (\lambda -(c3: C).(csubc g e c3)) (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2: -C).(csubc g c e2))) (\lambda (x: C).(\lambda (H5: (eq C c2 (CHead x (Flat f) -u))).(\lambda (H6: (csubc g e x)).(eq_ind_r C (CHead x (Flat f) u) (\lambda -(c0: C).(ex2 C (\lambda (e2: C).(clear c0 e2)) (\lambda (e2: C).(csubc g c -e2)))) (let H_x0 \def (H1 x H6) in (let H7 \def H_x0 in (ex2_ind C (\lambda -(e2: C).(clear x e2)) (\lambda (e2: C).(csubc g c e2)) (ex2 C (\lambda (e2: -C).(clear (CHead x (Flat f) u) e2)) (\lambda (e2: C).(csubc g c e2))) -(\lambda (x0: C).(\lambda (H8: (clear x x0)).(\lambda (H9: (csubc g c -x0)).(ex_intro2 C (\lambda (e2: C).(clear (CHead x (Flat f) u) e2)) (\lambda -(e2: C).(csubc g c e2)) x0 (clear_flat x x0 H8 f u) H9)))) H7))) c2 H5)))) -H4)) (\lambda (H4: (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: -A).(eq K (Flat f) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda -(_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: -T).(\lambda (_: A).(csubc g e c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(sc3 g (asucc g a) e u)))) (\lambda (c3: C).(\lambda (w: T).(\lambda -(a: A).(sc3 g a c3 w)))))).(ex5_3_ind C T A (\lambda (_: C).(\lambda (_: -T).(\lambda (_: A).(eq K (Flat f) (Bind Abst))))) (\lambda (c3: C).(\lambda -(w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: -C).(\lambda (_: T).(\lambda (_: A).(csubc g e c3)))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(sc3 g (asucc g a) e u)))) (\lambda (c3: C).(\lambda -(w: T).(\lambda (a: A).(sc3 g a c3 w)))) (ex2 C (\lambda (e2: C).(clear c2 -e2)) (\lambda (e2: C).(csubc g c e2))) (\lambda (x0: C).(\lambda (x1: -T).(\lambda (x2: A).(\lambda (H5: (eq K (Flat f) (Bind Abst))).(\lambda (H6: -(eq C c2 (CHead x0 (Bind Abbr) x1))).(\lambda (_: (csubc g e x0)).(\lambda -(_: (sc3 g (asucc g x2) e u)).(\lambda (_: (sc3 g x2 x0 x1)).(eq_ind_r C -(CHead x0 (Bind Abbr) x1) (\lambda (c0: C).(ex2 C (\lambda (e2: C).(clear c0 -e2)) (\lambda (e2: C).(csubc g c e2)))) (let H10 \def (eq_ind K (Flat f) -(\lambda (ee: K).(match ee in K return (\lambda (_: K).Prop) with [(Bind _) -\Rightarrow False | (Flat _) \Rightarrow True])) I (Bind Abst) H5) in -(False_ind (ex2 C (\lambda (e2: C).(clear (CHead x0 (Bind Abbr) x1) e2)) -(\lambda (e2: C).(csubc g c e2))) H10)) c2 H6))))))))) H4)) (\lambda (H4: -(ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 -(CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -T).(eq K (Flat f) (Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda -(_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: -T).(csubc g e c3)))))).(ex4_3_ind B C T (\lambda (b: B).(\lambda (c3: -C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: T).(eq K (Flat f) (Bind Void))))) (\lambda -(b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: -B).(\lambda (c3: C).(\lambda (_: T).(csubc g e c3)))) (ex2 C (\lambda (e2: -C).(clear c2 e2)) (\lambda (e2: C).(csubc g c e2))) (\lambda (x0: B).(\lambda -(x1: C).(\lambda (x2: T).(\lambda (H5: (eq C c2 (CHead x1 (Bind x0) -x2))).(\lambda (H6: (eq K (Flat f) (Bind Void))).(\lambda (_: (not (eq B x0 -Void))).(\lambda (_: (csubc g e x1)).(eq_ind_r C (CHead x1 (Bind x0) x2) -(\lambda (c0: C).(ex2 C (\lambda (e2: C).(clear c0 e2)) (\lambda (e2: -C).(csubc g c e2)))) (let H9 \def (eq_ind K (Flat f) (\lambda (ee: K).(match -ee in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat -_) \Rightarrow True])) I (Bind Void) H6) in (False_ind (ex2 C (\lambda (e2: -C).(clear (CHead x1 (Bind x0) x2) e2)) (\lambda (e2: C).(csubc g c e2))) H9)) -c2 H5)))))))) H4)) H3))))))))))) c1 e1 H)))). -(* COMMENTS -Initial nodes: 2837 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/csuba.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/csuba.ma deleted file mode 100644 index 2abfe2a1f..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/csuba.ma +++ /dev/null @@ -1,40 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubc/defs.ma". - -include "Basic-1/sc3/props.ma". - -theorem csubc_csuba: - \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to (csuba -g c1 c2)))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubc g c1 -c2)).(csubc_ind g (\lambda (c: C).(\lambda (c0: C).(csuba g c c0))) (\lambda -(n: nat).(csuba_refl g (CSort n))) (\lambda (c3: C).(\lambda (c4: C).(\lambda -(_: (csubc g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (k: K).(\lambda -(v: T).(csuba_head g c3 c4 H1 k v))))))) (\lambda (c3: C).(\lambda (c4: -C).(\lambda (_: (csubc g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (b: -B).(\lambda (H2: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: -T).(csuba_void g c3 c4 H1 b H2 u1 u2))))))))) (\lambda (c3: C).(\lambda (c4: -C).(\lambda (_: (csubc g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (v: -T).(\lambda (a: A).(\lambda (H2: (sc3 g (asucc g a) c3 v)).(\lambda (w: -T).(\lambda (H3: (sc3 g a c4 w)).(csuba_abst g c3 c4 H1 v a (sc3_arity_gen g -c3 v (asucc g a) H2) w (sc3_arity_gen g c4 w a H3))))))))))) c1 c2 H)))). -(* COMMENTS -Initial nodes: 231 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/defs.ma deleted file mode 100644 index 73a4c5625..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/defs.ma +++ /dev/null @@ -1,30 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/sc3/defs.ma". - -inductive csubc (g: G): C \to (C \to Prop) \def -| csubc_sort: \forall (n: nat).(csubc g (CSort n) (CSort n)) -| csubc_head: \forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to (\forall -(k: K).(\forall (v: T).(csubc g (CHead c1 k v) (CHead c2 k v)))))) -| csubc_void: \forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to (\forall -(b: B).((not (eq B b Void)) \to (\forall (u1: T).(\forall (u2: T).(csubc g -(CHead c1 (Bind Void) u1) (CHead c2 (Bind b) u2)))))))) -| csubc_abst: \forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to (\forall -(v: T).(\forall (a: A).((sc3 g (asucc g a) c1 v) \to (\forall (w: T).((sc3 g -a c2 w) \to (csubc g (CHead c1 (Bind Abst) v) (CHead c2 (Bind Abbr) -w))))))))). - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/drop.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/drop.ma deleted file mode 100644 index 195d36491..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/drop.ma +++ /dev/null @@ -1,475 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubc/fwd.ma". - -include "Basic-1/sc3/props.ma". - -theorem csubc_drop_conf_O: - \forall (g: G).(\forall (c1: C).(\forall (e1: C).(\forall (h: nat).((drop h -O c1 e1) \to (\forall (c2: C).((csubc g c1 c2) \to (ex2 C (\lambda (e2: -C).(drop h O c2 e2)) (\lambda (e2: C).(csubc g e1 e2))))))))) -\def - \lambda (g: G).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (e1: -C).(\forall (h: nat).((drop h O c e1) \to (\forall (c2: C).((csubc g c c2) -\to (ex2 C (\lambda (e2: C).(drop h O c2 e2)) (\lambda (e2: C).(csubc g e1 -e2))))))))) (\lambda (n: nat).(\lambda (e1: C).(\lambda (h: nat).(\lambda (H: -(drop h O (CSort n) e1)).(\lambda (c2: C).(\lambda (H0: (csubc g (CSort n) -c2)).(and3_ind (eq C e1 (CSort n)) (eq nat h O) (eq nat O O) (ex2 C (\lambda -(e2: C).(drop h O c2 e2)) (\lambda (e2: C).(csubc g e1 e2))) (\lambda (H1: -(eq C e1 (CSort n))).(\lambda (H2: (eq nat h O)).(\lambda (_: (eq nat O -O)).(eq_ind_r nat O (\lambda (n0: nat).(ex2 C (\lambda (e2: C).(drop n0 O c2 -e2)) (\lambda (e2: C).(csubc g e1 e2)))) (eq_ind_r C (CSort n) (\lambda (c: -C).(ex2 C (\lambda (e2: C).(drop O O c2 e2)) (\lambda (e2: C).(csubc g c -e2)))) (ex_intro2 C (\lambda (e2: C).(drop O O c2 e2)) (\lambda (e2: -C).(csubc g (CSort n) e2)) c2 (drop_refl c2) H0) e1 H1) h H2)))) -(drop_gen_sort n h O e1 H)))))))) (\lambda (c: C).(\lambda (H: ((\forall (e1: -C).(\forall (h: nat).((drop h O c e1) \to (\forall (c2: C).((csubc g c c2) -\to (ex2 C (\lambda (e2: C).(drop h O c2 e2)) (\lambda (e2: C).(csubc g e1 -e2)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e1: C).(\lambda (h: -nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c k t) e1) \to (\forall -(c2: C).((csubc g (CHead c k t) c2) \to (ex2 C (\lambda (e2: C).(drop n O c2 -e2)) (\lambda (e2: C).(csubc g e1 e2))))))) (\lambda (H0: (drop O O (CHead c -k t) e1)).(\lambda (c2: C).(\lambda (H1: (csubc g (CHead c k t) c2)).(eq_ind -C (CHead c k t) (\lambda (c0: C).(ex2 C (\lambda (e2: C).(drop O O c2 e2)) -(\lambda (e2: C).(csubc g c0 e2)))) (ex_intro2 C (\lambda (e2: C).(drop O O -c2 e2)) (\lambda (e2: C).(csubc g (CHead c k t) e2)) c2 (drop_refl c2) H1) e1 -(drop_gen_refl (CHead c k t) e1 H0))))) (\lambda (n: nat).(\lambda (H0: -(((drop n O (CHead c k t) e1) \to (\forall (c2: C).((csubc g (CHead c k t) -c2) \to (ex2 C (\lambda (e2: C).(drop n O c2 e2)) (\lambda (e2: C).(csubc g -e1 e2)))))))).(\lambda (H1: (drop (S n) O (CHead c k t) e1)).(\lambda (c2: -C).(\lambda (H2: (csubc g (CHead c k t) c2)).(let H_x \def (csubc_gen_head_l -g c c2 t k H2) in (let H3 \def H_x in (or3_ind (ex2 C (\lambda (c3: C).(eq C -c2 (CHead c3 k t))) (\lambda (c3: C).(csubc g c c3))) (ex5_3 C T A (\lambda -(_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3: -C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) -(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c c3)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c t)))) (\lambda -(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T -(\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b) -v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind -Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b -Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c c3))))) -(ex2 C (\lambda (e2: C).(drop (S n) O c2 e2)) (\lambda (e2: C).(csubc g e1 -e2))) (\lambda (H4: (ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 k t))) -(\lambda (c3: C).(csubc g c c3)))).(ex2_ind C (\lambda (c3: C).(eq C c2 -(CHead c3 k t))) (\lambda (c3: C).(csubc g c c3)) (ex2 C (\lambda (e2: -C).(drop (S n) O c2 e2)) (\lambda (e2: C).(csubc g e1 e2))) (\lambda (x: -C).(\lambda (H5: (eq C c2 (CHead x k t))).(\lambda (H6: (csubc g c -x)).(eq_ind_r C (CHead x k t) (\lambda (c0: C).(ex2 C (\lambda (e2: C).(drop -(S n) O c0 e2)) (\lambda (e2: C).(csubc g e1 e2)))) (let H_x0 \def (H e1 (r k -n) (drop_gen_drop k c e1 t n H1) x H6) in (let H7 \def H_x0 in (ex2_ind C -(\lambda (e2: C).(drop (r k n) O x e2)) (\lambda (e2: C).(csubc g e1 e2)) -(ex2 C (\lambda (e2: C).(drop (S n) O (CHead x k t) e2)) (\lambda (e2: -C).(csubc g e1 e2))) (\lambda (x0: C).(\lambda (H8: (drop (r k n) O x -x0)).(\lambda (H9: (csubc g e1 x0)).(ex_intro2 C (\lambda (e2: C).(drop (S n) -O (CHead x k t) e2)) (\lambda (e2: C).(csubc g e1 e2)) x0 (drop_drop k n x x0 -H8 t) H9)))) H7))) c2 H5)))) H4)) (\lambda (H4: (ex5_3 C T A (\lambda (_: -C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3: -C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) -(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c c3)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c t)))) (\lambda -(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w)))))).(ex5_3_ind C T A -(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) -(\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind -Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c -c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c -t)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w)))) (ex2 -C (\lambda (e2: C).(drop (S n) O c2 e2)) (\lambda (e2: C).(csubc g e1 e2))) -(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H5: (eq K k -(Bind Abst))).(\lambda (H6: (eq C c2 (CHead x0 (Bind Abbr) x1))).(\lambda -(H7: (csubc g c x0)).(\lambda (_: (sc3 g (asucc g x2) c t)).(\lambda (_: (sc3 -g x2 x0 x1)).(eq_ind_r C (CHead x0 (Bind Abbr) x1) (\lambda (c0: C).(ex2 C -(\lambda (e2: C).(drop (S n) O c0 e2)) (\lambda (e2: C).(csubc g e1 e2)))) -(let H10 \def (eq_ind K k (\lambda (k0: K).(drop (r k0 n) O c e1)) -(drop_gen_drop k c e1 t n H1) (Bind Abst) H5) in (let H11 \def (eq_ind K k -(\lambda (k0: K).((drop n O (CHead c k0 t) e1) \to (\forall (c3: C).((csubc g -(CHead c k0 t) c3) \to (ex2 C (\lambda (e2: C).(drop n O c3 e2)) (\lambda -(e2: C).(csubc g e1 e2))))))) H0 (Bind Abst) H5) in (let H_x0 \def (H e1 (r -(Bind Abst) n) H10 x0 H7) in (let H12 \def H_x0 in (ex2_ind C (\lambda (e2: -C).(drop n O x0 e2)) (\lambda (e2: C).(csubc g e1 e2)) (ex2 C (\lambda (e2: -C).(drop (S n) O (CHead x0 (Bind Abbr) x1) e2)) (\lambda (e2: C).(csubc g e1 -e2))) (\lambda (x: C).(\lambda (H13: (drop n O x0 x)).(\lambda (H14: (csubc g -e1 x)).(ex_intro2 C (\lambda (e2: C).(drop (S n) O (CHead x0 (Bind Abbr) x1) -e2)) (\lambda (e2: C).(csubc g e1 e2)) x (drop_drop (Bind Abbr) n x0 x H13 -x1) H14)))) H12))))) c2 H6))))))))) H4)) (\lambda (H4: (ex4_3 B C T (\lambda -(b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b) v2))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind Void))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) -(\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c -c3)))))).(ex4_3_ind B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: -T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: -C).(\lambda (_: T).(csubc g c c3)))) (ex2 C (\lambda (e2: C).(drop (S n) O c2 -e2)) (\lambda (e2: C).(csubc g e1 e2))) (\lambda (x0: B).(\lambda (x1: -C).(\lambda (x2: T).(\lambda (H5: (eq C c2 (CHead x1 (Bind x0) x2))).(\lambda -(H6: (eq K k (Bind Void))).(\lambda (_: (not (eq B x0 Void))).(\lambda (H8: -(csubc g c x1)).(eq_ind_r C (CHead x1 (Bind x0) x2) (\lambda (c0: C).(ex2 C -(\lambda (e2: C).(drop (S n) O c0 e2)) (\lambda (e2: C).(csubc g e1 e2)))) -(let H9 \def (eq_ind K k (\lambda (k0: K).(drop (r k0 n) O c e1)) -(drop_gen_drop k c e1 t n H1) (Bind Void) H6) in (let H10 \def (eq_ind K k -(\lambda (k0: K).((drop n O (CHead c k0 t) e1) \to (\forall (c3: C).((csubc g -(CHead c k0 t) c3) \to (ex2 C (\lambda (e2: C).(drop n O c3 e2)) (\lambda -(e2: C).(csubc g e1 e2))))))) H0 (Bind Void) H6) in (let H_x0 \def (H e1 (r -(Bind Void) n) H9 x1 H8) in (let H11 \def H_x0 in (ex2_ind C (\lambda (e2: -C).(drop n O x1 e2)) (\lambda (e2: C).(csubc g e1 e2)) (ex2 C (\lambda (e2: -C).(drop (S n) O (CHead x1 (Bind x0) x2) e2)) (\lambda (e2: C).(csubc g e1 -e2))) (\lambda (x: C).(\lambda (H12: (drop n O x1 x)).(\lambda (H13: (csubc g -e1 x)).(ex_intro2 C (\lambda (e2: C).(drop (S n) O (CHead x1 (Bind x0) x2) -e2)) (\lambda (e2: C).(csubc g e1 e2)) x (drop_drop (Bind x0) n x1 x H12 x2) -H13)))) H11))))) c2 H5)))))))) H4)) H3)))))))) h))))))) c1)). -(* COMMENTS -Initial nodes: 2389 -END *) - -theorem drop_csubc_trans: - \forall (g: G).(\forall (c2: C).(\forall (e2: C).(\forall (d: nat).(\forall -(h: nat).((drop h d c2 e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C -(\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c2 c1)))))))))) -\def - \lambda (g: G).(\lambda (c2: C).(C_ind (\lambda (c: C).(\forall (e2: -C).(\forall (d: nat).(\forall (h: nat).((drop h d c e2) \to (\forall (e1: -C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda -(c1: C).(csubc g c c1)))))))))) (\lambda (n: nat).(\lambda (e2: C).(\lambda -(d: nat).(\lambda (h: nat).(\lambda (H: (drop h d (CSort n) e2)).(\lambda -(e1: C).(\lambda (H0: (csubc g e2 e1)).(and3_ind (eq C e2 (CSort n)) (eq nat -h O) (eq nat d O) (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: -C).(csubc g (CSort n) c1))) (\lambda (H1: (eq C e2 (CSort n))).(\lambda (H2: -(eq nat h O)).(\lambda (H3: (eq nat d O)).(eq_ind_r nat O (\lambda (n0: -nat).(ex2 C (\lambda (c1: C).(drop n0 d c1 e1)) (\lambda (c1: C).(csubc g -(CSort n) c1)))) (eq_ind_r nat O (\lambda (n0: nat).(ex2 C (\lambda (c1: -C).(drop O n0 c1 e1)) (\lambda (c1: C).(csubc g (CSort n) c1)))) (let H4 \def -(eq_ind C e2 (\lambda (c: C).(csubc g c e1)) H0 (CSort n) H1) in (ex_intro2 C -(\lambda (c1: C).(drop O O c1 e1)) (\lambda (c1: C).(csubc g (CSort n) c1)) -e1 (drop_refl e1) H4)) d H3) h H2)))) (drop_gen_sort n h d e2 H))))))))) -(\lambda (c: C).(\lambda (H: ((\forall (e2: C).(\forall (d: nat).(\forall (h: -nat).((drop h d c e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C -(\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c -c1))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e2: C).(\lambda (d: -nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n (CHead c k t) -e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop h -n c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1)))))))) (\lambda (h: -nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c k t) e2) \to (\forall -(e1: C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop n O c1 e1)) -(\lambda (c1: C).(csubc g (CHead c k t) c1))))))) (\lambda (H0: (drop O O -(CHead c k t) e2)).(\lambda (e1: C).(\lambda (H1: (csubc g e2 e1)).(let H2 -\def (eq_ind_r C e2 (\lambda (c0: C).(csubc g c0 e1)) H1 (CHead c k t) -(drop_gen_refl (CHead c k t) e2 H0)) in (ex_intro2 C (\lambda (c1: C).(drop O -O c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1)) e1 (drop_refl e1) -H2))))) (\lambda (n: nat).(\lambda (_: (((drop n O (CHead c k t) e2) \to -(\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop n O c1 -e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1)))))))).(\lambda (H1: (drop -(S n) O (CHead c k t) e2)).(\lambda (e1: C).(\lambda (H2: (csubc g e2 -e1)).(let H_x \def (H e2 O (r k n) (drop_gen_drop k c e2 t n H1) e1 H2) in -(let H3 \def H_x in (ex2_ind C (\lambda (c1: C).(drop (r k n) O c1 e1)) -(\lambda (c1: C).(csubc g c c1)) (ex2 C (\lambda (c1: C).(drop (S n) O c1 -e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1))) (\lambda (x: C).(\lambda -(H4: (drop (r k n) O x e1)).(\lambda (H5: (csubc g c x)).(ex_intro2 C -(\lambda (c1: C).(drop (S n) O c1 e1)) (\lambda (c1: C).(csubc g (CHead c k -t) c1)) (CHead x k t) (drop_drop k n x e1 H4 t) (csubc_head g c x H5 k t))))) -H3)))))))) h)) (\lambda (n: nat).(\lambda (H0: ((\forall (h: nat).((drop h n -(CHead c k t) e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda -(c1: C).(drop h n c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t) -c1))))))))).(\lambda (h: nat).(\lambda (H1: (drop h (S n) (CHead c k t) -e2)).(\lambda (e1: C).(\lambda (H2: (csubc g e2 e1)).(ex3_2_ind C T (\lambda -(e: C).(\lambda (v: T).(eq C e2 (CHead e k v)))) (\lambda (_: C).(\lambda (v: -T).(eq T t (lift h (r k n) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r k -n) c e))) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1: -C).(csubc g (CHead c k t) c1))) (\lambda (x0: C).(\lambda (x1: T).(\lambda -(H3: (eq C e2 (CHead x0 k x1))).(\lambda (H4: (eq T t (lift h (r k n) -x1))).(\lambda (H5: (drop h (r k n) c x0)).(let H6 \def (eq_ind C e2 (\lambda -(c0: C).(csubc g c0 e1)) H2 (CHead x0 k x1) H3) in (let H7 \def (eq_ind C e2 -(\lambda (c0: C).(\forall (h0: nat).((drop h0 n (CHead c k t) c0) \to -(\forall (e3: C).((csubc g c0 e3) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 -e3)) (\lambda (c1: C).(csubc g (CHead c k t) c1)))))))) H0 (CHead x0 k x1) -H3) in (let H8 \def (eq_ind T t (\lambda (t0: T).(\forall (h0: nat).((drop h0 -n (CHead c k t0) (CHead x0 k x1)) \to (\forall (e3: C).((csubc g (CHead x0 k -x1) e3) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 e3)) (\lambda (c1: -C).(csubc g (CHead c k t0) c1)))))))) H7 (lift h (r k n) x1) H4) in (eq_ind_r -T (lift h (r k n) x1) (\lambda (t0: T).(ex2 C (\lambda (c1: C).(drop h (S n) -c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t0) c1)))) (let H_x \def -(csubc_gen_head_l g x0 e1 x1 k H6) in (let H9 \def H_x in (or3_ind (ex2 C -(\lambda (c3: C).(eq C e1 (CHead c3 k x1))) (\lambda (c3: C).(csubc g x0 -c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k -(Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C e1 -(CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: -A).(csubc g x0 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g -(asucc g a) x0 x1)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g -a c3 w))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: -T).(eq C e1 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: -C).(\lambda (_: T).(csubc g x0 c3))))) (ex2 C (\lambda (c1: C).(drop h (S n) -c1 e1)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1))) -(\lambda (H10: (ex2 C (\lambda (c3: C).(eq C e1 (CHead c3 k x1))) (\lambda -(c3: C).(csubc g x0 c3)))).(ex2_ind C (\lambda (c3: C).(eq C e1 (CHead c3 k -x1))) (\lambda (c3: C).(csubc g x0 c3)) (ex2 C (\lambda (c1: C).(drop h (S n) -c1 e1)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1))) -(\lambda (x: C).(\lambda (H11: (eq C e1 (CHead x k x1))).(\lambda (H12: -(csubc g x0 x)).(eq_ind_r C (CHead x k x1) (\lambda (c0: C).(ex2 C (\lambda -(c1: C).(drop h (S n) c1 c0)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r -k n) x1)) c1)))) (let H_x0 \def (H x0 (r k n) h H5 x H12) in (let H13 \def -H_x0 in (ex2_ind C (\lambda (c1: C).(drop h (r k n) c1 x)) (\lambda (c1: -C).(csubc g c c1)) (ex2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x k x1))) -(\lambda (c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1))) (\lambda (x2: -C).(\lambda (H14: (drop h (r k n) x2 x)).(\lambda (H15: (csubc g c -x2)).(ex_intro2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x k x1))) (\lambda -(c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1)) (CHead x2 k (lift h (r -k n) x1)) (drop_skip k h n x2 x H14 x1) (csubc_head g c x2 H15 k (lift h (r k -n) x1)))))) H13))) e1 H11)))) H10)) (\lambda (H10: (ex5_3 C T A (\lambda (_: -C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3: -C).(\lambda (w: T).(\lambda (_: A).(eq C e1 (CHead c3 (Bind Abbr) w))))) -(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g x0 c3)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) x0 x1)))) (\lambda -(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w)))))).(ex5_3_ind C T A -(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) -(\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C e1 (CHead c3 (Bind -Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g x0 -c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) x0 -x1)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w)))) -(ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1: C).(csubc g -(CHead c k (lift h (r k n) x1)) c1))) (\lambda (x2: C).(\lambda (x3: -T).(\lambda (x4: A).(\lambda (H11: (eq K k (Bind Abst))).(\lambda (H12: (eq C -e1 (CHead x2 (Bind Abbr) x3))).(\lambda (H13: (csubc g x0 x2)).(\lambda (H14: -(sc3 g (asucc g x4) x0 x1)).(\lambda (H15: (sc3 g x4 x2 x3)).(eq_ind_r C -(CHead x2 (Bind Abbr) x3) (\lambda (c0: C).(ex2 C (\lambda (c1: C).(drop h (S -n) c1 c0)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1)))) -(let H16 \def (eq_ind K k (\lambda (k0: K).(\forall (h0: nat).((drop h0 n -(CHead c k0 (lift h (r k0 n) x1)) (CHead x0 k0 x1)) \to (\forall (e3: -C).((csubc g (CHead x0 k0 x1) e3) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 -e3)) (\lambda (c1: C).(csubc g (CHead c k0 (lift h (r k0 n) x1)) c1)))))))) -H8 (Bind Abst) H11) in (let H17 \def (eq_ind K k (\lambda (k0: K).(drop h (r -k0 n) c x0)) H5 (Bind Abst) H11) in (eq_ind_r K (Bind Abst) (\lambda (k0: -K).(ex2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x2 (Bind Abbr) x3))) -(\lambda (c1: C).(csubc g (CHead c k0 (lift h (r k0 n) x1)) c1)))) (let H_x0 -\def (H x0 (r (Bind Abst) n) h H17 x2 H13) in (let H18 \def H_x0 in (ex2_ind -C (\lambda (c1: C).(drop h n c1 x2)) (\lambda (c1: C).(csubc g c c1)) (ex2 C -(\lambda (c1: C).(drop h (S n) c1 (CHead x2 (Bind Abbr) x3))) (\lambda (c1: -C).(csubc g (CHead c (Bind Abst) (lift h (r (Bind Abst) n) x1)) c1))) -(\lambda (x: C).(\lambda (H19: (drop h n x x2)).(\lambda (H20: (csubc g c -x)).(ex_intro2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x2 (Bind Abbr) -x3))) (\lambda (c1: C).(csubc g (CHead c (Bind Abst) (lift h (r (Bind Abst) -n) x1)) c1)) (CHead x (Bind Abbr) (lift h n x3)) (drop_skip_bind h n x x2 H19 -Abbr x3) (csubc_abst g c x H20 (lift h (r (Bind Abst) n) x1) x4 (sc3_lift g -(asucc g x4) x0 x1 H14 c h (r (Bind Abst) n) H17) (lift h n x3) (sc3_lift g -x4 x2 x3 H15 x h n H19)))))) H18))) k H11))) e1 H12))))))))) H10)) (\lambda -(H10: (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C e1 -(CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: -T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: -T).(csubc g x0 c3)))))).(ex4_3_ind B C T (\lambda (b: B).(\lambda (c3: -C).(\lambda (v2: T).(eq C e1 (CHead c3 (Bind b) v2))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: -B).(\lambda (c3: C).(\lambda (_: T).(csubc g x0 c3)))) (ex2 C (\lambda (c1: -C).(drop h (S n) c1 e1)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r k n) -x1)) c1))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: T).(\lambda (H11: -(eq C e1 (CHead x3 (Bind x2) x4))).(\lambda (H12: (eq K k (Bind -Void))).(\lambda (H13: (not (eq B x2 Void))).(\lambda (H14: (csubc g x0 -x3)).(eq_ind_r C (CHead x3 (Bind x2) x4) (\lambda (c0: C).(ex2 C (\lambda -(c1: C).(drop h (S n) c1 c0)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r -k n) x1)) c1)))) (let H15 \def (eq_ind K k (\lambda (k0: K).(\forall (h0: -nat).((drop h0 n (CHead c k0 (lift h (r k0 n) x1)) (CHead x0 k0 x1)) \to -(\forall (e3: C).((csubc g (CHead x0 k0 x1) e3) \to (ex2 C (\lambda (c1: -C).(drop h0 n c1 e3)) (\lambda (c1: C).(csubc g (CHead c k0 (lift h (r k0 n) -x1)) c1)))))))) H8 (Bind Void) H12) in (let H16 \def (eq_ind K k (\lambda -(k0: K).(drop h (r k0 n) c x0)) H5 (Bind Void) H12) in (eq_ind_r K (Bind -Void) (\lambda (k0: K).(ex2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x3 -(Bind x2) x4))) (\lambda (c1: C).(csubc g (CHead c k0 (lift h (r k0 n) x1)) -c1)))) (let H_x0 \def (H x0 (r (Bind Void) n) h H16 x3 H14) in (let H17 \def -H_x0 in (ex2_ind C (\lambda (c1: C).(drop h n c1 x3)) (\lambda (c1: C).(csubc -g c c1)) (ex2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x3 (Bind x2) x4))) -(\lambda (c1: C).(csubc g (CHead c (Bind Void) (lift h (r (Bind Void) n) x1)) -c1))) (\lambda (x: C).(\lambda (H18: (drop h n x x3)).(\lambda (H19: (csubc g -c x)).(ex_intro2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x3 (Bind x2) -x4))) (\lambda (c1: C).(csubc g (CHead c (Bind Void) (lift h (r (Bind Void) -n) x1)) c1)) (CHead x (Bind x2) (lift h n x4)) (drop_skip_bind h n x x3 H18 -x2 x4) (csubc_void g c x H19 x2 H13 (lift h (r (Bind Void) n) x1) (lift h n -x4)))))) H17))) k H12))) e1 H11)))))))) H10)) H9))) t H4))))))))) -(drop_gen_skip_l c e2 t h n k H1)))))))) d))))))) c2)). -(* COMMENTS -Initial nodes: 3747 -END *) - -theorem csubc_drop_conf_rev: - \forall (g: G).(\forall (c2: C).(\forall (e2: C).(\forall (d: nat).(\forall -(h: nat).((drop h d c2 e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C -(\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c1 c2)))))))))) -\def - \lambda (g: G).(\lambda (c2: C).(C_ind (\lambda (c: C).(\forall (e2: -C).(\forall (d: nat).(\forall (h: nat).((drop h d c e2) \to (\forall (e1: -C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda -(c1: C).(csubc g c1 c)))))))))) (\lambda (n: nat).(\lambda (e2: C).(\lambda -(d: nat).(\lambda (h: nat).(\lambda (H: (drop h d (CSort n) e2)).(\lambda -(e1: C).(\lambda (H0: (csubc g e1 e2)).(and3_ind (eq C e2 (CSort n)) (eq nat -h O) (eq nat d O) (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: -C).(csubc g c1 (CSort n)))) (\lambda (H1: (eq C e2 (CSort n))).(\lambda (H2: -(eq nat h O)).(\lambda (H3: (eq nat d O)).(eq_ind_r nat O (\lambda (n0: -nat).(ex2 C (\lambda (c1: C).(drop n0 d c1 e1)) (\lambda (c1: C).(csubc g c1 -(CSort n))))) (eq_ind_r nat O (\lambda (n0: nat).(ex2 C (\lambda (c1: -C).(drop O n0 c1 e1)) (\lambda (c1: C).(csubc g c1 (CSort n))))) (let H4 \def -(eq_ind C e2 (\lambda (c: C).(csubc g e1 c)) H0 (CSort n) H1) in (ex_intro2 C -(\lambda (c1: C).(drop O O c1 e1)) (\lambda (c1: C).(csubc g c1 (CSort n))) -e1 (drop_refl e1) H4)) d H3) h H2)))) (drop_gen_sort n h d e2 H))))))))) -(\lambda (c: C).(\lambda (H: ((\forall (e2: C).(\forall (d: nat).(\forall (h: -nat).((drop h d c e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C -(\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c1 -c))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e2: C).(\lambda (d: -nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n (CHead c k t) -e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop h -n c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t))))))))) (\lambda (h: -nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c k t) e2) \to (\forall -(e1: C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop n O c1 e1)) -(\lambda (c1: C).(csubc g c1 (CHead c k t)))))))) (\lambda (H0: (drop O O -(CHead c k t) e2)).(\lambda (e1: C).(\lambda (H1: (csubc g e1 e2)).(let H2 -\def (eq_ind_r C e2 (\lambda (c0: C).(csubc g e1 c0)) H1 (CHead c k t) -(drop_gen_refl (CHead c k t) e2 H0)) in (ex_intro2 C (\lambda (c1: C).(drop O -O c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t))) e1 (drop_refl e1) -H2))))) (\lambda (n: nat).(\lambda (_: (((drop n O (CHead c k t) e2) \to -(\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop n O c1 -e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t))))))))).(\lambda (H1: (drop -(S n) O (CHead c k t) e2)).(\lambda (e1: C).(\lambda (H2: (csubc g e1 -e2)).(let H_x \def (H e2 O (r k n) (drop_gen_drop k c e2 t n H1) e1 H2) in -(let H3 \def H_x in (ex2_ind C (\lambda (c1: C).(drop (r k n) O c1 e1)) -(\lambda (c1: C).(csubc g c1 c)) (ex2 C (\lambda (c1: C).(drop (S n) O c1 -e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t)))) (\lambda (x: C).(\lambda -(H4: (drop (r k n) O x e1)).(\lambda (H5: (csubc g x c)).(ex_intro2 C -(\lambda (c1: C).(drop (S n) O c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c -k t))) (CHead x k t) (drop_drop k n x e1 H4 t) (csubc_head g x c H5 k t))))) -H3)))))))) h)) (\lambda (n: nat).(\lambda (H0: ((\forall (h: nat).((drop h n -(CHead c k t) e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda -(c1: C).(drop h n c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k -t)))))))))).(\lambda (h: nat).(\lambda (H1: (drop h (S n) (CHead c k t) -e2)).(\lambda (e1: C).(\lambda (H2: (csubc g e1 e2)).(ex3_2_ind C T (\lambda -(e: C).(\lambda (v: T).(eq C e2 (CHead e k v)))) (\lambda (_: C).(\lambda (v: -T).(eq T t (lift h (r k n) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r k -n) c e))) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1: -C).(csubc g c1 (CHead c k t)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda -(H3: (eq C e2 (CHead x0 k x1))).(\lambda (H4: (eq T t (lift h (r k n) -x1))).(\lambda (H5: (drop h (r k n) c x0)).(let H6 \def (eq_ind C e2 (\lambda -(c0: C).(csubc g e1 c0)) H2 (CHead x0 k x1) H3) in (let H7 \def (eq_ind C e2 -(\lambda (c0: C).(\forall (h0: nat).((drop h0 n (CHead c k t) c0) \to -(\forall (e3: C).((csubc g e3 c0) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 -e3)) (\lambda (c1: C).(csubc g c1 (CHead c k t))))))))) H0 (CHead x0 k x1) -H3) in (let H8 \def (eq_ind T t (\lambda (t0: T).(\forall (h0: nat).((drop h0 -n (CHead c k t0) (CHead x0 k x1)) \to (\forall (e3: C).((csubc g e3 (CHead x0 -k x1)) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 e3)) (\lambda (c1: C).(csubc -g c1 (CHead c k t0))))))))) H7 (lift h (r k n) x1) H4) in (eq_ind_r T (lift h -(r k n) x1) (\lambda (t0: T).(ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) -(\lambda (c1: C).(csubc g c1 (CHead c k t0))))) (let H_x \def -(csubc_gen_head_r g x0 e1 x1 k H6) in (let H9 \def H_x in (or3_ind (ex2 C -(\lambda (c1: C).(eq C e1 (CHead c1 k x1))) (\lambda (c1: C).(csubc g c1 -x0))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k -(Bind Abbr))))) (\lambda (c1: C).(\lambda (v: T).(\lambda (_: A).(eq C e1 -(CHead c1 (Bind Abst) v))))) (\lambda (c1: C).(\lambda (_: T).(\lambda (_: -A).(csubc g c1 x0)))) (\lambda (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g -(asucc g a) c1 v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a -x0 x1))))) (ex4_3 B C T (\lambda (_: B).(\lambda (c1: C).(\lambda (v1: T).(eq -C e1 (CHead c1 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda -(_: T).(eq K k (Bind b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: -T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c1: C).(\lambda (_: -T).(csubc g c1 x0))))) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda -(c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1))))) (\lambda (H10: (ex2 C -(\lambda (c1: C).(eq C e1 (CHead c1 k x1))) (\lambda (c1: C).(csubc g c1 -x0)))).(ex2_ind C (\lambda (c1: C).(eq C e1 (CHead c1 k x1))) (\lambda (c1: -C).(csubc g c1 x0)) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda -(c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1))))) (\lambda (x: -C).(\lambda (H11: (eq C e1 (CHead x k x1))).(\lambda (H12: (csubc g x -x0)).(eq_ind_r C (CHead x k x1) (\lambda (c0: C).(ex2 C (\lambda (c1: -C).(drop h (S n) c1 c0)) (\lambda (c1: C).(csubc g c1 (CHead c k (lift h (r k -n) x1)))))) (let H_x0 \def (H x0 (r k n) h H5 x H12) in (let H13 \def H_x0 in -(ex2_ind C (\lambda (c1: C).(drop h (r k n) c1 x)) (\lambda (c1: C).(csubc g -c1 c)) (ex2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x k x1))) (\lambda -(c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1))))) (\lambda (x2: -C).(\lambda (H14: (drop h (r k n) x2 x)).(\lambda (H15: (csubc g x2 -c)).(ex_intro2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x k x1))) (\lambda -(c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1)))) (CHead x2 k (lift h (r -k n) x1)) (drop_skip k h n x2 x H14 x1) (csubc_head g x2 c H15 k (lift h (r k -n) x1)))))) H13))) e1 H11)))) H10)) (\lambda (H10: (ex5_3 C T A (\lambda (_: -C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c1: -C).(\lambda (v: T).(\lambda (_: A).(eq C e1 (CHead c1 (Bind Abst) v))))) -(\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 x0)))) (\lambda -(c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a x0 x1)))))).(ex5_3_ind C T A -(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) -(\lambda (c1: C).(\lambda (v: T).(\lambda (_: A).(eq C e1 (CHead c1 (Bind -Abst) v))))) (\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 -x0)))) (\lambda (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1 -v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a x0 x1)))) (ex2 -C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead -c k (lift h (r k n) x1))))) (\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: -A).(\lambda (H11: (eq K k (Bind Abbr))).(\lambda (H12: (eq C e1 (CHead x2 -(Bind Abst) x3))).(\lambda (H13: (csubc g x2 x0)).(\lambda (H14: (sc3 g -(asucc g x4) x2 x3)).(\lambda (H15: (sc3 g x4 x0 x1)).(eq_ind_r C (CHead x2 -(Bind Abst) x3) (\lambda (c0: C).(ex2 C (\lambda (c1: C).(drop h (S n) c1 -c0)) (\lambda (c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1)))))) (let -H16 \def (eq_ind K k (\lambda (k0: K).(\forall (h0: nat).((drop h0 n (CHead c -k0 (lift h (r k0 n) x1)) (CHead x0 k0 x1)) \to (\forall (e3: C).((csubc g e3 -(CHead x0 k0 x1)) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 e3)) (\lambda -(c1: C).(csubc g c1 (CHead c k0 (lift h (r k0 n) x1)))))))))) H8 (Bind Abbr) -H11) in (let H17 \def (eq_ind K k (\lambda (k0: K).(drop h (r k0 n) c x0)) H5 -(Bind Abbr) H11) in (eq_ind_r K (Bind Abbr) (\lambda (k0: K).(ex2 C (\lambda -(c1: C).(drop h (S n) c1 (CHead x2 (Bind Abst) x3))) (\lambda (c1: C).(csubc -g c1 (CHead c k0 (lift h (r k0 n) x1)))))) (let H_x0 \def (H x0 (r (Bind -Abbr) n) h H17 x2 H13) in (let H18 \def H_x0 in (ex2_ind C (\lambda (c1: -C).(drop h n c1 x2)) (\lambda (c1: C).(csubc g c1 c)) (ex2 C (\lambda (c1: -C).(drop h (S n) c1 (CHead x2 (Bind Abst) x3))) (\lambda (c1: C).(csubc g c1 -(CHead c (Bind Abbr) (lift h (r (Bind Abbr) n) x1))))) (\lambda (x: -C).(\lambda (H19: (drop h n x x2)).(\lambda (H20: (csubc g x c)).(ex_intro2 C -(\lambda (c1: C).(drop h (S n) c1 (CHead x2 (Bind Abst) x3))) (\lambda (c1: -C).(csubc g c1 (CHead c (Bind Abbr) (lift h (r (Bind Abbr) n) x1)))) (CHead x -(Bind Abst) (lift h n x3)) (drop_skip_bind h n x x2 H19 Abst x3) (csubc_abst -g x c H20 (lift h n x3) x4 (sc3_lift g (asucc g x4) x2 x3 H14 x h n H19) -(lift h (r (Bind Abbr) n) x1) (sc3_lift g x4 x0 x1 H15 c h (r (Bind Abbr) n) -H17)))))) H18))) k H11))) e1 H12))))))))) H10)) (\lambda (H10: (ex4_3 B C T -(\lambda (_: B).(\lambda (c1: C).(\lambda (v1: T).(eq C e1 (CHead c1 (Bind -Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind -b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) -(\lambda (_: B).(\lambda (c1: C).(\lambda (_: T).(csubc g c1 -x0)))))).(ex4_3_ind B C T (\lambda (_: B).(\lambda (c1: C).(\lambda (v1: -T).(eq C e1 (CHead c1 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c1: -C).(\lambda (_: T).(csubc g c1 x0)))) (ex2 C (\lambda (c1: C).(drop h (S n) -c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1))))) -(\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: T).(\lambda (H11: (eq C e1 -(CHead x3 (Bind Void) x4))).(\lambda (H12: (eq K k (Bind x2))).(\lambda (H13: -(not (eq B x2 Void))).(\lambda (H14: (csubc g x3 x0)).(eq_ind_r C (CHead x3 -(Bind Void) x4) (\lambda (c0: C).(ex2 C (\lambda (c1: C).(drop h (S n) c1 -c0)) (\lambda (c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1)))))) (let -H15 \def (eq_ind K k (\lambda (k0: K).(\forall (h0: nat).((drop h0 n (CHead c -k0 (lift h (r k0 n) x1)) (CHead x0 k0 x1)) \to (\forall (e3: C).((csubc g e3 -(CHead x0 k0 x1)) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 e3)) (\lambda -(c1: C).(csubc g c1 (CHead c k0 (lift h (r k0 n) x1)))))))))) H8 (Bind x2) -H12) in (let H16 \def (eq_ind K k (\lambda (k0: K).(drop h (r k0 n) c x0)) H5 -(Bind x2) H12) in (eq_ind_r K (Bind x2) (\lambda (k0: K).(ex2 C (\lambda (c1: -C).(drop h (S n) c1 (CHead x3 (Bind Void) x4))) (\lambda (c1: C).(csubc g c1 -(CHead c k0 (lift h (r k0 n) x1)))))) (let H_x0 \def (H x0 (r (Bind x2) n) h -H16 x3 H14) in (let H17 \def H_x0 in (ex2_ind C (\lambda (c1: C).(drop h n c1 -x3)) (\lambda (c1: C).(csubc g c1 c)) (ex2 C (\lambda (c1: C).(drop h (S n) -c1 (CHead x3 (Bind Void) x4))) (\lambda (c1: C).(csubc g c1 (CHead c (Bind -x2) (lift h (r (Bind x2) n) x1))))) (\lambda (x: C).(\lambda (H18: (drop h n -x x3)).(\lambda (H19: (csubc g x c)).(ex_intro2 C (\lambda (c1: C).(drop h (S -n) c1 (CHead x3 (Bind Void) x4))) (\lambda (c1: C).(csubc g c1 (CHead c (Bind -x2) (lift h (r (Bind x2) n) x1)))) (CHead x (Bind Void) (lift h n x4)) -(drop_skip_bind h n x x3 H18 Void x4) (csubc_void g x c H19 x2 H13 (lift h n -x4) (lift h (r (Bind x2) n) x1)))))) H17))) k H12))) e1 H11)))))))) H10)) -H9))) t H4))))))))) (drop_gen_skip_l c e2 t h n k H1)))))))) d))))))) c2)). -(* COMMENTS -Initial nodes: 3747 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/drop1.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/drop1.ma deleted file mode 100644 index 7c539cb4a..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/drop1.ma +++ /dev/null @@ -1,92 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubc/drop.ma". - -theorem drop1_csubc_trans: - \forall (g: G).(\forall (hds: PList).(\forall (c2: C).(\forall (e2: -C).((drop1 hds c2 e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C -(\lambda (c1: C).(drop1 hds c1 e1)) (\lambda (c1: C).(csubc g c2 c1))))))))) -\def - \lambda (g: G).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall -(c2: C).(\forall (e2: C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e2 -e1) \to (ex2 C (\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c2 -c1))))))))) (\lambda (c2: C).(\lambda (e2: C).(\lambda (H: (drop1 PNil c2 -e2)).(\lambda (e1: C).(\lambda (H0: (csubc g e2 e1)).(let H_y \def -(drop1_gen_pnil c2 e2 H) in (let H1 \def (eq_ind_r C e2 (\lambda (c: -C).(csubc g c e1)) H0 c2 H_y) in (ex_intro2 C (\lambda (c1: C).(drop1 PNil c1 -e1)) (\lambda (c1: C).(csubc g c2 c1)) e1 (drop1_nil e1) H1)))))))) (\lambda -(n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c2: -C).(\forall (e2: C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e2 e1) -\to (ex2 C (\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c2 -c1)))))))))).(\lambda (c2: C).(\lambda (e2: C).(\lambda (H0: (drop1 (PCons n -n0 p) c2 e2)).(\lambda (e1: C).(\lambda (H1: (csubc g e2 e1)).(let H_x \def -(drop1_gen_pcons c2 e2 p n n0 H0) in (let H2 \def H_x in (ex2_ind C (\lambda -(c3: C).(drop n n0 c2 c3)) (\lambda (c3: C).(drop1 p c3 e2)) (ex2 C (\lambda -(c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c2 c1))) -(\lambda (x: C).(\lambda (H3: (drop n n0 c2 x)).(\lambda (H4: (drop1 p x -e2)).(let H_x0 \def (H x e2 H4 e1 H1) in (let H5 \def H_x0 in (ex2_ind C -(\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g x c1)) (ex2 C -(\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c2 -c1))) (\lambda (x0: C).(\lambda (H6: (drop1 p x0 e1)).(\lambda (H7: (csubc g -x x0)).(let H_x1 \def (drop_csubc_trans g c2 x n0 n H3 x0 H7) in (let H8 \def -H_x1 in (ex2_ind C (\lambda (c1: C).(drop n n0 c1 x0)) (\lambda (c1: -C).(csubc g c2 c1)) (ex2 C (\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) -(\lambda (c1: C).(csubc g c2 c1))) (\lambda (x1: C).(\lambda (H9: (drop n n0 -x1 x0)).(\lambda (H10: (csubc g c2 x1)).(ex_intro2 C (\lambda (c1: C).(drop1 -(PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c2 c1)) x1 (drop1_cons x1 x0 -n n0 H9 e1 p H6) H10)))) H8)))))) H5)))))) H2)))))))))))) hds)). -(* COMMENTS -Initial nodes: 551 -END *) - -theorem csubc_drop1_conf_rev: - \forall (g: G).(\forall (hds: PList).(\forall (c2: C).(\forall (e2: -C).((drop1 hds c2 e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C -(\lambda (c1: C).(drop1 hds c1 e1)) (\lambda (c1: C).(csubc g c1 c2))))))))) -\def - \lambda (g: G).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall -(c2: C).(\forall (e2: C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e1 -e2) \to (ex2 C (\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c1 -c2))))))))) (\lambda (c2: C).(\lambda (e2: C).(\lambda (H: (drop1 PNil c2 -e2)).(\lambda (e1: C).(\lambda (H0: (csubc g e1 e2)).(let H_y \def -(drop1_gen_pnil c2 e2 H) in (let H1 \def (eq_ind_r C e2 (\lambda (c: -C).(csubc g e1 c)) H0 c2 H_y) in (ex_intro2 C (\lambda (c1: C).(drop1 PNil c1 -e1)) (\lambda (c1: C).(csubc g c1 c2)) e1 (drop1_nil e1) H1)))))))) (\lambda -(n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c2: -C).(\forall (e2: C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e1 e2) -\to (ex2 C (\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c1 -c2)))))))))).(\lambda (c2: C).(\lambda (e2: C).(\lambda (H0: (drop1 (PCons n -n0 p) c2 e2)).(\lambda (e1: C).(\lambda (H1: (csubc g e1 e2)).(let H_x \def -(drop1_gen_pcons c2 e2 p n n0 H0) in (let H2 \def H_x in (ex2_ind C (\lambda -(c3: C).(drop n n0 c2 c3)) (\lambda (c3: C).(drop1 p c3 e2)) (ex2 C (\lambda -(c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c1 c2))) -(\lambda (x: C).(\lambda (H3: (drop n n0 c2 x)).(\lambda (H4: (drop1 p x -e2)).(let H_x0 \def (H x e2 H4 e1 H1) in (let H5 \def H_x0 in (ex2_ind C -(\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c1 x)) (ex2 C -(\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c1 -c2))) (\lambda (x0: C).(\lambda (H6: (drop1 p x0 e1)).(\lambda (H7: (csubc g -x0 x)).(let H_x1 \def (csubc_drop_conf_rev g c2 x n0 n H3 x0 H7) in (let H8 -\def H_x1 in (ex2_ind C (\lambda (c1: C).(drop n n0 c1 x0)) (\lambda (c1: -C).(csubc g c1 c2)) (ex2 C (\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) -(\lambda (c1: C).(csubc g c1 c2))) (\lambda (x1: C).(\lambda (H9: (drop n n0 -x1 x0)).(\lambda (H10: (csubc g x1 c2)).(ex_intro2 C (\lambda (c1: C).(drop1 -(PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c1 c2)) x1 (drop1_cons x1 x0 -n n0 H9 e1 p H6) H10)))) H8)))))) H5)))))) H2)))))))))))) hds)). -(* COMMENTS -Initial nodes: 551 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/fwd.ma deleted file mode 100644 index fe04ddd45..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/fwd.ma +++ /dev/null @@ -1,673 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubc/defs.ma". - -theorem csubc_gen_sort_l: - \forall (g: G).(\forall (x: C).(\forall (n: nat).((csubc g (CSort n) x) \to -(eq C x (CSort n))))) -\def - \lambda (g: G).(\lambda (x: C).(\lambda (n: nat).(\lambda (H: (csubc g -(CSort n) x)).(insert_eq C (CSort n) (\lambda (c: C).(csubc g c x)) (\lambda -(c: C).(eq C x c)) (\lambda (y: C).(\lambda (H0: (csubc g y x)).(csubc_ind g -(\lambda (c: C).(\lambda (c0: C).((eq C c (CSort n)) \to (eq C c0 c)))) -(\lambda (n0: nat).(\lambda (H1: (eq C (CSort n0) (CSort n))).(let H2 \def -(f_equal C nat (\lambda (e: C).(match e in C return (\lambda (_: C).nat) with -[(CSort n1) \Rightarrow n1 | (CHead _ _ _) \Rightarrow n0])) (CSort n0) -(CSort n) H1) in (eq_ind_r nat n (\lambda (n1: nat).(eq C (CSort n1) (CSort -n1))) (refl_equal C (CSort n)) n0 H2)))) (\lambda (c1: C).(\lambda (c2: -C).(\lambda (_: (csubc g c1 c2)).(\lambda (_: (((eq C c1 (CSort n)) \to (eq C -c2 c1)))).(\lambda (k: K).(\lambda (v: T).(\lambda (H3: (eq C (CHead c1 k v) -(CSort n))).(let H4 \def (eq_ind C (CHead c1 k v) (\lambda (ee: C).(match ee -in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead -_ _ _) \Rightarrow True])) I (CSort n) H3) in (False_ind (eq C (CHead c2 k v) -(CHead c1 k v)) H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: -(csubc g c1 c2)).(\lambda (_: (((eq C c1 (CSort n)) \to (eq C c2 -c1)))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: -T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u1) (CSort -n))).(let H5 \def (eq_ind C (CHead c1 (Bind Void) u1) (\lambda (ee: C).(match -ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | -(CHead _ _ _) \Rightarrow True])) I (CSort n) H4) in (False_ind (eq C (CHead -c2 (Bind b) u2) (CHead c1 (Bind Void) u1)) H5))))))))))) (\lambda (c1: -C).(\lambda (c2: C).(\lambda (_: (csubc g c1 c2)).(\lambda (_: (((eq C c1 -(CSort n)) \to (eq C c2 c1)))).(\lambda (v: T).(\lambda (a: A).(\lambda (_: -(sc3 g (asucc g a) c1 v)).(\lambda (w: T).(\lambda (_: (sc3 g a c2 -w)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) v) (CSort n))).(let H6 \def -(eq_ind C (CHead c1 (Bind Abst) v) (\lambda (ee: C).(match ee in C return -(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) -\Rightarrow True])) I (CSort n) H5) in (False_ind (eq C (CHead c2 (Bind Abbr) -w) (CHead c1 (Bind Abst) v)) H6)))))))))))) y x H0))) H)))). -(* COMMENTS -Initial nodes: 533 -END *) - -theorem csubc_gen_head_l: - \forall (g: G).(\forall (c1: C).(\forall (x: C).(\forall (v: T).(\forall (k: -K).((csubc g (CHead c1 k v) x) \to (or3 (ex2 C (\lambda (c2: C).(eq C x -(CHead c2 k v))) (\lambda (c2: C).(csubc g c1 c2))) (ex5_3 C T A (\lambda (_: -C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c2: -C).(\lambda (w: T).(\lambda (_: A).(eq C x (CHead c2 (Bind Abbr) w))))) -(\lambda (c2: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c2)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda -(c2: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T -(\lambda (b: B).(\lambda (c2: C).(\lambda (v2: T).(eq C x (CHead c2 (Bind b) -v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind -Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b -Void))))) (\lambda (_: B).(\lambda (c2: C).(\lambda (_: T).(csubc g c1 -c2))))))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (x: C).(\lambda (v: T).(\lambda (k: -K).(\lambda (H: (csubc g (CHead c1 k v) x)).(insert_eq C (CHead c1 k v) -(\lambda (c: C).(csubc g c x)) (\lambda (_: C).(or3 (ex2 C (\lambda (c2: -C).(eq C x (CHead c2 k v))) (\lambda (c2: C).(csubc g c1 c2))) (ex5_3 C T A -(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) -(\lambda (c2: C).(\lambda (w: T).(\lambda (_: A).(eq C x (CHead c2 (Bind -Abbr) w))))) (\lambda (c2: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 -c2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 -v)))) (\lambda (c2: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c2 w))))) -(ex4_3 B C T (\lambda (b: B).(\lambda (c2: C).(\lambda (v2: T).(eq C x (CHead -c2 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k -(Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b -Void))))) (\lambda (_: B).(\lambda (c2: C).(\lambda (_: T).(csubc g c1 -c2))))))) (\lambda (y: C).(\lambda (H0: (csubc g y x)).(csubc_ind g (\lambda -(c: C).(\lambda (c0: C).((eq C c (CHead c1 k v)) \to (or3 (ex2 C (\lambda -(c2: C).(eq C c0 (CHead c2 k v))) (\lambda (c2: C).(csubc g c1 c2))) (ex5_3 C -T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) -(\lambda (c2: C).(\lambda (w: T).(\lambda (_: A).(eq C c0 (CHead c2 (Bind -Abbr) w))))) (\lambda (c2: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 -c2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 -v)))) (\lambda (c2: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c2 w))))) -(ex4_3 B C T (\lambda (b: B).(\lambda (c2: C).(\lambda (v2: T).(eq C c0 -(CHead c2 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: -T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c2: C).(\lambda (_: -T).(csubc g c1 c2))))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) -(CHead c1 k v))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee -in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ -_ _) \Rightarrow False])) I (CHead c1 k v) H1) in (False_ind (or3 (ex2 C -(\lambda (c2: C).(eq C (CSort n) (CHead c2 k v))) (\lambda (c2: C).(csubc g -c1 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k -(Bind Abst))))) (\lambda (c2: C).(\lambda (w: T).(\lambda (_: A).(eq C (CSort -n) (CHead c2 (Bind Abbr) w))))) (\lambda (c2: C).(\lambda (_: T).(\lambda (_: -A).(csubc g c1 c2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g -(asucc g a) c1 v)))) (\lambda (c2: C).(\lambda (w: T).(\lambda (a: A).(sc3 g -a c2 w))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c2: C).(\lambda (v2: -T).(eq C (CSort n) (CHead c2 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c2: -C).(\lambda (_: T).(csubc g c1 c2)))))) H2)))) (\lambda (c0: C).(\lambda (c2: -C).(\lambda (H1: (csubc g c0 c2)).(\lambda (H2: (((eq C c0 (CHead c1 k v)) -\to (or3 (ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 k v))) (\lambda (c3: -C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda -(_: A).(eq K k (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: -A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: -T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (c3: C).(\lambda (w: -T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda (b: B).(\lambda -(c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: -B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 c3))))))))).(\lambda (k0: -K).(\lambda (v0: T).(\lambda (H3: (eq C (CHead c0 k0 v0) (CHead c1 k -v))).(let H4 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda -(_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) -(CHead c0 k0 v0) (CHead c1 k v) H3) in ((let H5 \def (f_equal C K (\lambda -(e: C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k0 -| (CHead _ k1 _) \Rightarrow k1])) (CHead c0 k0 v0) (CHead c1 k v) H3) in -((let H6 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: -C).T) with [(CSort _) \Rightarrow v0 | (CHead _ _ t) \Rightarrow t])) (CHead -c0 k0 v0) (CHead c1 k v) H3) in (\lambda (H7: (eq K k0 k)).(\lambda (H8: (eq -C c0 c1)).(eq_ind_r T v (\lambda (t: T).(or3 (ex2 C (\lambda (c3: C).(eq C -(CHead c2 k0 t) (CHead c3 k v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C -T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) -(\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C (CHead c2 k0 t) (CHead -c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc -g c1 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g -a) c1 v)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 -w))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C -(CHead c2 k0 t) (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: -C).(\lambda (_: T).(csubc g c1 c3))))))) (eq_ind_r K k (\lambda (k1: K).(or3 -(ex2 C (\lambda (c3: C).(eq C (CHead c2 k1 v) (CHead c3 k v))) (\lambda (c3: -C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda -(_: A).(eq K k (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: -A).(eq C (CHead c2 k1 v) (CHead c3 (Bind Abbr) w))))) (\lambda (c3: -C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (c3: -C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda -(b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C (CHead c2 k1 v) (CHead c3 -(Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k -(Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b -Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 -c3))))))) (let H9 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 k v)) -\to (or3 (ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 k v))) (\lambda (c3: -C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda -(_: A).(eq K k (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: -A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: -T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (c3: C).(\lambda (w: -T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda (b: B).(\lambda -(c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: -B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 c3)))))))) H2 c1 H8) in (let -H10 \def (eq_ind C c0 (\lambda (c: C).(csubc g c c2)) H1 c1 H8) in -(or3_intro0 (ex2 C (\lambda (c3: C).(eq C (CHead c2 k v) (CHead c3 k v))) -(\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: -T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3: C).(\lambda (w: -T).(\lambda (_: A).(eq C (CHead c2 k v) (CHead c3 (Bind Abbr) w))))) (\lambda -(c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (c3: -C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda -(b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C (CHead c2 k v) (CHead c3 (Bind -b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind -Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b -Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 -c3))))) (ex_intro2 C (\lambda (c3: C).(eq C (CHead c2 k v) (CHead c3 k v))) -(\lambda (c3: C).(csubc g c1 c3)) c2 (refl_equal C (CHead c2 k v)) H10)))) k0 -H7) v0 H6)))) H5)) H4))))))))) (\lambda (c0: C).(\lambda (c2: C).(\lambda -(H1: (csubc g c0 c2)).(\lambda (H2: (((eq C c0 (CHead c1 k v)) \to (or3 (ex2 -C (\lambda (c3: C).(eq C c2 (CHead c3 k v))) (\lambda (c3: C).(csubc g c1 -c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k -(Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C c2 -(CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: -A).(csubc g c1 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g -(asucc g a) c1 v)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g -a c3 w))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: -T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: -C).(\lambda (_: T).(csubc g c1 c3))))))))).(\lambda (b: B).(\lambda (H3: (not -(eq B b Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead -c0 (Bind Void) u1) (CHead c1 k v))).(let H5 \def (f_equal C C (\lambda (e: -C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | -(CHead c _ _) \Rightarrow c])) (CHead c0 (Bind Void) u1) (CHead c1 k v) H4) -in ((let H6 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda -(_: C).K) with [(CSort _) \Rightarrow (Bind Void) | (CHead _ k0 _) -\Rightarrow k0])) (CHead c0 (Bind Void) u1) (CHead c1 k v) H4) in ((let H7 -\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead c0 -(Bind Void) u1) (CHead c1 k v) H4) in (\lambda (H8: (eq K (Bind Void) -k)).(\lambda (H9: (eq C c0 c1)).(let H10 \def (eq_ind C c0 (\lambda (c: -C).((eq C c (CHead c1 k v)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c2 (CHead -c3 k v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: -C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3: -C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) -(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda -(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T -(\lambda (b0: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind -b0) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind -Void))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b0 -Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 -c3)))))))) H2 c1 H9) in (let H11 \def (eq_ind C c0 (\lambda (c: C).(csubc g c -c2)) H1 c1 H9) in (let H12 \def (eq_ind_r K k (\lambda (k0: K).((eq C c1 -(CHead c1 k0 v)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 k0 v))) -(\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: -T).(\lambda (_: A).(eq K k0 (Bind Abst))))) (\lambda (c3: C).(\lambda (w: -T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: -C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (c3: -C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda -(b0: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b0) -v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k0 (Bind -Void))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b0 -Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 -c3)))))))) H10 (Bind Void) H8) in (eq_ind K (Bind Void) (\lambda (k0: K).(or3 -(ex2 C (\lambda (c3: C).(eq C (CHead c2 (Bind b) u2) (CHead c3 k0 v))) -(\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: -T).(\lambda (_: A).(eq K k0 (Bind Abst))))) (\lambda (c3: C).(\lambda (w: -T).(\lambda (_: A).(eq C (CHead c2 (Bind b) u2) (CHead c3 (Bind Abbr) w))))) -(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda -(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T -(\lambda (b0: B).(\lambda (c3: C).(\lambda (v2: T).(eq C (CHead c2 (Bind b) -u2) (CHead c3 (Bind b0) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -T).(eq K k0 (Bind Void))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: -T).(not (eq B b0 Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: -T).(csubc g c1 c3))))))) (or3_intro2 (ex2 C (\lambda (c3: C).(eq C (CHead c2 -(Bind b) u2) (CHead c3 (Bind Void) v))) (\lambda (c3: C).(csubc g c1 c3))) -(ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K (Bind -Void) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C -(CHead c2 (Bind b) u2) (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda -(_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (c3: C).(\lambda (w: -T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda (b0: B).(\lambda -(c3: C).(\lambda (v2: T).(eq C (CHead c2 (Bind b) u2) (CHead c3 (Bind b0) -v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K (Bind Void) -(Bind Void))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B -b0 Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 -c3))))) (ex4_3_intro B C T (\lambda (b0: B).(\lambda (c3: C).(\lambda (v2: -T).(eq C (CHead c2 (Bind b) u2) (CHead c3 (Bind b0) v2))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: T).(eq K (Bind Void) (Bind Void))))) (\lambda -(b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b0 Void))))) (\lambda (_: -B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 c3)))) b c2 u2 (refl_equal C -(CHead c2 (Bind b) u2)) (refl_equal K (Bind Void)) H3 H11)) k H8))))))) H6)) -H5))))))))))) (\lambda (c0: C).(\lambda (c2: C).(\lambda (H1: (csubc g c0 -c2)).(\lambda (H2: (((eq C c0 (CHead c1 k v)) \to (or3 (ex2 C (\lambda (c3: -C).(eq C c2 (CHead c3 k v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A -(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) -(\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind -Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 -c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 -v)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) -(ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 -(CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: -T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: -T).(csubc g c1 c3))))))))).(\lambda (v0: T).(\lambda (a: A).(\lambda (H3: -(sc3 g (asucc g a) c0 v0)).(\lambda (w: T).(\lambda (H4: (sc3 g a c2 -w)).(\lambda (H5: (eq C (CHead c0 (Bind Abst) v0) (CHead c1 k v))).(let H6 -\def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) -with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 -(Bind Abst) v0) (CHead c1 k v) H5) in ((let H7 \def (f_equal C K (\lambda (e: -C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow (Bind -Abst) | (CHead _ k0 _) \Rightarrow k0])) (CHead c0 (Bind Abst) v0) (CHead c1 -k v) H5) in ((let H8 \def (f_equal C T (\lambda (e: C).(match e in C return -(\lambda (_: C).T) with [(CSort _) \Rightarrow v0 | (CHead _ _ t) \Rightarrow -t])) (CHead c0 (Bind Abst) v0) (CHead c1 k v) H5) in (\lambda (H9: (eq K -(Bind Abst) k)).(\lambda (H10: (eq C c0 c1)).(let H11 \def (eq_ind T v0 -(\lambda (t: T).(sc3 g (asucc g a) c0 t)) H3 v H8) in (let H12 \def (eq_ind C -c0 (\lambda (c: C).(sc3 g (asucc g a) c v)) H11 c1 H10) in (let H13 \def -(eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 k v)) \to (or3 (ex2 C -(\lambda (c3: C).(eq C c2 (CHead c3 k v))) (\lambda (c3: C).(csubc g c1 c3))) -(ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind -Abst))))) (\lambda (c3: C).(\lambda (w0: T).(\lambda (_: A).(eq C c2 (CHead -c3 (Bind Abbr) w0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: -A).(csubc g c1 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g -(asucc g a0) c1 v)))) (\lambda (c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 -g a0 c3 w0))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: -T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: -C).(\lambda (_: T).(csubc g c1 c3)))))))) H2 c1 H10) in (let H14 \def (eq_ind -C c0 (\lambda (c: C).(csubc g c c2)) H1 c1 H10) in (let H15 \def (eq_ind_r K -k (\lambda (k0: K).((eq C c1 (CHead c1 k0 v)) \to (or3 (ex2 C (\lambda (c3: -C).(eq C c2 (CHead c3 k0 v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A -(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k0 (Bind Abst))))) -(\lambda (c3: C).(\lambda (w0: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind -Abbr) w0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 -c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g (asucc g a0) -c1 v)))) (\lambda (c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 g a0 c3 -w0))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C -c2 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -T).(eq K k0 (Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: -T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: -T).(csubc g c1 c3)))))))) H13 (Bind Abst) H9) in (eq_ind K (Bind Abst) -(\lambda (k0: K).(or3 (ex2 C (\lambda (c3: C).(eq C (CHead c2 (Bind Abbr) w) -(CHead c3 k0 v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda -(_: C).(\lambda (_: T).(\lambda (_: A).(eq K k0 (Bind Abst))))) (\lambda (c3: -C).(\lambda (w0: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) w) (CHead c3 -(Bind Abbr) w0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g -c1 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g (asucc g -a0) c1 v)))) (\lambda (c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 g a0 c3 -w0))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C -(CHead c2 (Bind Abbr) w) (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda -(_: C).(\lambda (_: T).(eq K k0 (Bind Void))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: -C).(\lambda (_: T).(csubc g c1 c3))))))) (or3_intro1 (ex2 C (\lambda (c3: -C).(eq C (CHead c2 (Bind Abbr) w) (CHead c3 (Bind Abst) v))) (\lambda (c3: -C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda -(_: A).(eq K (Bind Abst) (Bind Abst))))) (\lambda (c3: C).(\lambda (w0: -T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) w) (CHead c3 (Bind Abbr) -w0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g (asucc g a0) c1 v)))) -(\lambda (c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 g a0 c3 w0))))) -(ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C (CHead -c2 (Bind Abbr) w) (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: T).(eq K (Bind Abst) (Bind Void))))) (\lambda (b: B).(\lambda -(_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: -C).(\lambda (_: T).(csubc g c1 c3))))) (ex5_3_intro C T A (\lambda (_: -C).(\lambda (_: T).(\lambda (_: A).(eq K (Bind Abst) (Bind Abst))))) (\lambda -(c3: C).(\lambda (w0: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) w) -(CHead c3 (Bind Abbr) w0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: -A).(csubc g c1 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g -(asucc g a0) c1 v)))) (\lambda (c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 -g a0 c3 w0)))) c2 w a (refl_equal K (Bind Abst)) (refl_equal C (CHead c2 -(Bind Abbr) w)) H14 H12 H4)) k H9))))))))) H7)) H6)))))))))))) y x H0))) -H)))))). -(* COMMENTS -Initial nodes: 5205 -END *) - -theorem csubc_gen_sort_r: - \forall (g: G).(\forall (x: C).(\forall (n: nat).((csubc g x (CSort n)) \to -(eq C x (CSort n))))) -\def - \lambda (g: G).(\lambda (x: C).(\lambda (n: nat).(\lambda (H: (csubc g x -(CSort n))).(insert_eq C (CSort n) (\lambda (c: C).(csubc g x c)) (\lambda -(c: C).(eq C x c)) (\lambda (y: C).(\lambda (H0: (csubc g x y)).(csubc_ind g -(\lambda (c: C).(\lambda (c0: C).((eq C c0 (CSort n)) \to (eq C c c0)))) -(\lambda (n0: nat).(\lambda (H1: (eq C (CSort n0) (CSort n))).(let H2 \def -(f_equal C nat (\lambda (e: C).(match e in C return (\lambda (_: C).nat) with -[(CSort n1) \Rightarrow n1 | (CHead _ _ _) \Rightarrow n0])) (CSort n0) -(CSort n) H1) in (eq_ind_r nat n (\lambda (n1: nat).(eq C (CSort n1) (CSort -n1))) (refl_equal C (CSort n)) n0 H2)))) (\lambda (c1: C).(\lambda (c2: -C).(\lambda (_: (csubc g c1 c2)).(\lambda (_: (((eq C c2 (CSort n)) \to (eq C -c1 c2)))).(\lambda (k: K).(\lambda (v: T).(\lambda (H3: (eq C (CHead c2 k v) -(CSort n))).(let H4 \def (eq_ind C (CHead c2 k v) (\lambda (ee: C).(match ee -in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead -_ _ _) \Rightarrow True])) I (CSort n) H3) in (False_ind (eq C (CHead c1 k v) -(CHead c2 k v)) H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: -(csubc g c1 c2)).(\lambda (_: (((eq C c2 (CSort n)) \to (eq C c1 -c2)))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: -T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c2 (Bind b) u2) (CSort -n))).(let H5 \def (eq_ind C (CHead c2 (Bind b) u2) (\lambda (ee: C).(match ee -in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead -_ _ _) \Rightarrow True])) I (CSort n) H4) in (False_ind (eq C (CHead c1 -(Bind Void) u1) (CHead c2 (Bind b) u2)) H5))))))))))) (\lambda (c1: -C).(\lambda (c2: C).(\lambda (_: (csubc g c1 c2)).(\lambda (_: (((eq C c2 -(CSort n)) \to (eq C c1 c2)))).(\lambda (v: T).(\lambda (a: A).(\lambda (_: -(sc3 g (asucc g a) c1 v)).(\lambda (w: T).(\lambda (_: (sc3 g a c2 -w)).(\lambda (H5: (eq C (CHead c2 (Bind Abbr) w) (CSort n))).(let H6 \def -(eq_ind C (CHead c2 (Bind Abbr) w) (\lambda (ee: C).(match ee in C return -(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) -\Rightarrow True])) I (CSort n) H5) in (False_ind (eq C (CHead c1 (Bind Abst) -v) (CHead c2 (Bind Abbr) w)) H6)))))))))))) x y H0))) H)))). -(* COMMENTS -Initial nodes: 533 -END *) - -theorem csubc_gen_head_r: - \forall (g: G).(\forall (c2: C).(\forall (x: C).(\forall (w: T).(\forall (k: -K).((csubc g x (CHead c2 k w)) \to (or3 (ex2 C (\lambda (c1: C).(eq C x -(CHead c1 k w))) (\lambda (c1: C).(csubc g c1 c2))) (ex5_3 C T A (\lambda (_: -C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c1: -C).(\lambda (v: T).(\lambda (_: A).(eq C x (CHead c1 (Bind Abst) v))))) -(\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c2)))) (\lambda -(c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T -(\lambda (_: B).(\lambda (c1: C).(\lambda (v1: T).(eq C x (CHead c1 (Bind -Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind -b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) -(\lambda (_: B).(\lambda (c1: C).(\lambda (_: T).(csubc g c1 c2))))))))))) -\def - \lambda (g: G).(\lambda (c2: C).(\lambda (x: C).(\lambda (w: T).(\lambda (k: -K).(\lambda (H: (csubc g x (CHead c2 k w))).(insert_eq C (CHead c2 k w) -(\lambda (c: C).(csubc g x c)) (\lambda (_: C).(or3 (ex2 C (\lambda (c1: -C).(eq C x (CHead c1 k w))) (\lambda (c1: C).(csubc g c1 c2))) (ex5_3 C T A -(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) -(\lambda (c1: C).(\lambda (v: T).(\lambda (_: A).(eq C x (CHead c1 (Bind -Abst) v))))) (\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 -c2)))) (\lambda (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1 -v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) -(ex4_3 B C T (\lambda (_: B).(\lambda (c1: C).(\lambda (v1: T).(eq C x (CHead -c1 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(eq K -k (Bind b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b -Void))))) (\lambda (_: B).(\lambda (c1: C).(\lambda (_: T).(csubc g c1 -c2))))))) (\lambda (y: C).(\lambda (H0: (csubc g x y)).(csubc_ind g (\lambda -(c: C).(\lambda (c0: C).((eq C c0 (CHead c2 k w)) \to (or3 (ex2 C (\lambda -(c1: C).(eq C c (CHead c1 k w))) (\lambda (c1: C).(csubc g c1 c2))) (ex5_3 C -T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) -(\lambda (c1: C).(\lambda (v: T).(\lambda (_: A).(eq C c (CHead c1 (Bind -Abst) v))))) (\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 -c2)))) (\lambda (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1 -v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) -(ex4_3 B C T (\lambda (_: B).(\lambda (c1: C).(\lambda (v1: T).(eq C c (CHead -c1 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(eq K -k (Bind b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b -Void))))) (\lambda (_: B).(\lambda (c1: C).(\lambda (_: T).(csubc g c1 -c2))))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead c2 k -w))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return -(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) -\Rightarrow False])) I (CHead c2 k w) H1) in (False_ind (or3 (ex2 C (\lambda -(c1: C).(eq C (CSort n) (CHead c1 k w))) (\lambda (c1: C).(csubc g c1 c2))) -(ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind -Abbr))))) (\lambda (c1: C).(\lambda (v: T).(\lambda (_: A).(eq C (CSort n) -(CHead c1 (Bind Abst) v))))) (\lambda (c1: C).(\lambda (_: T).(\lambda (_: -A).(csubc g c1 c2)))) (\lambda (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g -(asucc g a) c1 v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a -c2 w))))) (ex4_3 B C T (\lambda (_: B).(\lambda (c1: C).(\lambda (v1: T).(eq -C (CSort n) (CHead c1 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c1: -C).(\lambda (_: T).(csubc g c1 c2)))))) H2)))) (\lambda (c1: C).(\lambda (c0: -C).(\lambda (H1: (csubc g c1 c0)).(\lambda (H2: (((eq C c0 (CHead c2 k w)) -\to (or3 (ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k w))) (\lambda (c3: -C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda -(_: A).(eq K k (Bind Abbr))))) (\lambda (c3: C).(\lambda (v: T).(\lambda (_: -A).(eq C c1 (CHead c3 (Bind Abst) v))))) (\lambda (c3: C).(\lambda (_: -T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v: -T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T (\lambda (_: B).(\lambda -(c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind Void) v1))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: -B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 c2))))))))).(\lambda (k0: -K).(\lambda (v: T).(\lambda (H3: (eq C (CHead c0 k0 v) (CHead c2 k w))).(let -H4 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) -with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k0 -v) (CHead c2 k w) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e -in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k1 -_) \Rightarrow k1])) (CHead c0 k0 v) (CHead c2 k w) H3) in ((let H6 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow v | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 v) -(CHead c2 k w) H3) in (\lambda (H7: (eq K k0 k)).(\lambda (H8: (eq C c0 -c2)).(eq_ind_r T w (\lambda (t: T).(or3 (ex2 C (\lambda (c3: C).(eq C (CHead -c1 k0 t) (CHead c3 k w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A -(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) -(\lambda (c3: C).(\lambda (v0: T).(\lambda (_: A).(eq C (CHead c1 k0 t) -(CHead c3 (Bind Abst) v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: -A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (a: A).(sc3 -g (asucc g a) c3 v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 -g a c2 w))))) (ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: -T).(eq C (CHead c1 k0 t) (CHead c3 (Bind Void) v1))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: -B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 c2))))))) (eq_ind_r K k -(\lambda (k1: K).(or3 (ex2 C (\lambda (c3: C).(eq C (CHead c1 k1 w) (CHead c3 -k w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: -C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c3: -C).(\lambda (v0: T).(\lambda (_: A).(eq C (CHead c1 k1 w) (CHead c3 (Bind -Abst) v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 -c2)))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (a: A).(sc3 g (asucc g a) -c3 v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) -(ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead -c1 k1 w) (CHead c3 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: -C).(\lambda (_: T).(csubc g c3 c2))))))) (let H9 \def (eq_ind C c0 (\lambda -(c: C).((eq C c (CHead c2 k w)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c1 -(CHead c3 k w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: -C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c3: -C).(\lambda (v0: T).(\lambda (_: A).(eq C c1 (CHead c3 (Bind Abst) v0))))) -(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda -(c3: C).(\lambda (v0: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v0)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C -T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind -Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind -b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) -(\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 c2)))))))) H2 c2 -H8) in (let H10 \def (eq_ind C c0 (\lambda (c: C).(csubc g c1 c)) H1 c2 H8) -in (or3_intro0 (ex2 C (\lambda (c3: C).(eq C (CHead c1 k w) (CHead c3 k w))) -(\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: -T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c3: C).(\lambda (v0: -T).(\lambda (_: A).(eq C (CHead c1 k w) (CHead c3 (Bind Abst) v0))))) -(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda -(c3: C).(\lambda (v0: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v0)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C -T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead c1 k w) -(CHead c3 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: -T).(eq K k (Bind b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not -(eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g -c3 c2))))) (ex_intro2 C (\lambda (c3: C).(eq C (CHead c1 k w) (CHead c3 k -w))) (\lambda (c3: C).(csubc g c3 c2)) c1 (refl_equal C (CHead c1 k w)) -H10)))) k0 H7) v H6)))) H5)) H4))))))))) (\lambda (c1: C).(\lambda (c0: -C).(\lambda (H1: (csubc g c1 c0)).(\lambda (H2: (((eq C c0 (CHead c2 k w)) -\to (or3 (ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k w))) (\lambda (c3: -C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda -(_: A).(eq K k (Bind Abbr))))) (\lambda (c3: C).(\lambda (v: T).(\lambda (_: -A).(eq C c1 (CHead c3 (Bind Abst) v))))) (\lambda (c3: C).(\lambda (_: -T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v: -T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T (\lambda (_: B).(\lambda -(c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind Void) v1))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: -B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 c2))))))))).(\lambda (b: -B).(\lambda (H3: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: -T).(\lambda (H4: (eq C (CHead c0 (Bind b) u2) (CHead c2 k w))).(let H5 \def -(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with -[(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 (Bind b) -u2) (CHead c2 k w) H4) in ((let H6 \def (f_equal C K (\lambda (e: C).(match e -in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow (Bind b) | (CHead -_ k0 _) \Rightarrow k0])) (CHead c0 (Bind b) u2) (CHead c2 k w) H4) in ((let -H7 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c0 -(Bind b) u2) (CHead c2 k w) H4) in (\lambda (H8: (eq K (Bind b) k)).(\lambda -(H9: (eq C c0 c2)).(let H10 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead -c2 k w)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k w))) (\lambda -(c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: -T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c3: C).(\lambda (v: -T).(\lambda (_: A).(eq C c1 (CHead c3 (Bind Abst) v))))) (\lambda (c3: -C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: -C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T (\lambda -(_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind Void) -v1))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind -b0))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b0 -Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 -c2)))))))) H2 c2 H9) in (let H11 \def (eq_ind C c0 (\lambda (c: C).(csubc g -c1 c)) H1 c2 H9) in (let H12 \def (eq_ind_r K k (\lambda (k0: K).((eq C c2 -(CHead c2 k0 w)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k0 w))) -(\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: -T).(\lambda (_: A).(eq K k0 (Bind Abbr))))) (\lambda (c3: C).(\lambda (v: -T).(\lambda (_: A).(eq C c1 (CHead c3 (Bind Abst) v))))) (\lambda (c3: -C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: -C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T (\lambda -(_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind Void) -v1))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(eq K k0 (Bind -b0))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b0 -Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 -c2)))))))) H10 (Bind b) H8) in (eq_ind K (Bind b) (\lambda (k0: K).(or3 (ex2 -C (\lambda (c3: C).(eq C (CHead c1 (Bind Void) u1) (CHead c3 k0 w))) (\lambda -(c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: -T).(\lambda (_: A).(eq K k0 (Bind Abbr))))) (\lambda (c3: C).(\lambda (v: -T).(\lambda (_: A).(eq C (CHead c1 (Bind Void) u1) (CHead c3 (Bind Abst) -v))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) -(\lambda (c3: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C -T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead c1 (Bind -Void) u1) (CHead c3 (Bind Void) v1))))) (\lambda (b0: B).(\lambda (_: -C).(\lambda (_: T).(eq K k0 (Bind b0))))) (\lambda (b0: B).(\lambda (_: -C).(\lambda (_: T).(not (eq B b0 Void))))) (\lambda (_: B).(\lambda (c3: -C).(\lambda (_: T).(csubc g c3 c2))))))) (or3_intro2 (ex2 C (\lambda (c3: -C).(eq C (CHead c1 (Bind Void) u1) (CHead c3 (Bind b) w))) (\lambda (c3: -C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda -(_: A).(eq K (Bind b) (Bind Abbr))))) (\lambda (c3: C).(\lambda (v: -T).(\lambda (_: A).(eq C (CHead c1 (Bind Void) u1) (CHead c3 (Bind Abst) -v))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) -(\lambda (c3: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C -T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead c1 (Bind -Void) u1) (CHead c3 (Bind Void) v1))))) (\lambda (b0: B).(\lambda (_: -C).(\lambda (_: T).(eq K (Bind b) (Bind b0))))) (\lambda (b0: B).(\lambda (_: -C).(\lambda (_: T).(not (eq B b0 Void))))) (\lambda (_: B).(\lambda (c3: -C).(\lambda (_: T).(csubc g c3 c2))))) (ex4_3_intro B C T (\lambda (_: -B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead c1 (Bind Void) u1) (CHead -c3 (Bind Void) v1))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(eq K -(Bind b) (Bind b0))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not -(eq B b0 Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g -c3 c2)))) b c1 u1 (refl_equal C (CHead c1 (Bind Void) u1)) (refl_equal K -(Bind b)) H3 H11)) k H8))))))) H6)) H5))))))))))) (\lambda (c1: C).(\lambda -(c0: C).(\lambda (H1: (csubc g c1 c0)).(\lambda (H2: (((eq C c0 (CHead c2 k -w)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k w))) (\lambda (c3: -C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda -(_: A).(eq K k (Bind Abbr))))) (\lambda (c3: C).(\lambda (v: T).(\lambda (_: -A).(eq C c1 (CHead c3 (Bind Abst) v))))) (\lambda (c3: C).(\lambda (_: -T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v: -T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T (\lambda (_: B).(\lambda -(c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind Void) v1))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: -B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 c2))))))))).(\lambda (v: -T).(\lambda (a: A).(\lambda (H3: (sc3 g (asucc g a) c1 v)).(\lambda (w0: -T).(\lambda (H4: (sc3 g a c0 w0)).(\lambda (H5: (eq C (CHead c0 (Bind Abbr) -w0) (CHead c2 k w))).(let H6 \def (f_equal C C (\lambda (e: C).(match e in C -return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) -\Rightarrow c])) (CHead c0 (Bind Abbr) w0) (CHead c2 k w) H5) in ((let H7 -\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) -with [(CSort _) \Rightarrow (Bind Abbr) | (CHead _ k0 _) \Rightarrow k0])) -(CHead c0 (Bind Abbr) w0) (CHead c2 k w) H5) in ((let H8 \def (f_equal C T -(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) -\Rightarrow w0 | (CHead _ _ t) \Rightarrow t])) (CHead c0 (Bind Abbr) w0) -(CHead c2 k w) H5) in (\lambda (H9: (eq K (Bind Abbr) k)).(\lambda (H10: (eq -C c0 c2)).(let H11 \def (eq_ind T w0 (\lambda (t: T).(sc3 g a c0 t)) H4 w H8) -in (let H12 \def (eq_ind C c0 (\lambda (c: C).(sc3 g a c w)) H11 c2 H10) in -(let H13 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c2 k w)) \to (or3 -(ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k w))) (\lambda (c3: C).(csubc g -c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k -(Bind Abbr))))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (_: A).(eq C c1 -(CHead c3 (Bind Abst) v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: -A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (a0: A).(sc3 -g (asucc g a0) c3 v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: -A).(sc3 g a0 c2 w))))) (ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda -(v1: T).(eq C c1 (CHead c3 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: -C).(\lambda (_: T).(csubc g c3 c2)))))))) H2 c2 H10) in (let H14 \def (eq_ind -C c0 (\lambda (c: C).(csubc g c1 c)) H1 c2 H10) in (let H15 \def (eq_ind_r K -k (\lambda (k0: K).((eq C c2 (CHead c2 k0 w)) \to (or3 (ex2 C (\lambda (c3: -C).(eq C c1 (CHead c3 k0 w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A -(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k0 (Bind Abbr))))) -(\lambda (c3: C).(\lambda (v0: T).(\lambda (_: A).(eq C c1 (CHead c3 (Bind -Abst) v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 -c2)))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (a0: A).(sc3 g (asucc g a0) -c3 v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g a0 c2 -w))))) (ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C -c1 (CHead c3 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: -T).(eq K k0 (Bind b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not -(eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g -c3 c2)))))))) H13 (Bind Abbr) H9) in (eq_ind K (Bind Abbr) (\lambda (k0: -K).(or3 (ex2 C (\lambda (c3: C).(eq C (CHead c1 (Bind Abst) v) (CHead c3 k0 -w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda -(_: T).(\lambda (_: A).(eq K k0 (Bind Abbr))))) (\lambda (c3: C).(\lambda -(v0: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) v) (CHead c3 (Bind Abst) -v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) -(\lambda (c3: C).(\lambda (v0: T).(\lambda (a0: A).(sc3 g (asucc g a0) c3 -v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g a0 c2 w))))) -(ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead -c1 (Bind Abst) v) (CHead c3 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (_: T).(eq K k0 (Bind b))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: -C).(\lambda (_: T).(csubc g c3 c2))))))) (or3_intro1 (ex2 C (\lambda (c3: -C).(eq C (CHead c1 (Bind Abst) v) (CHead c3 (Bind Abbr) w))) (\lambda (c3: -C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda -(_: A).(eq K (Bind Abbr) (Bind Abbr))))) (\lambda (c3: C).(\lambda (v0: -T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) v) (CHead c3 (Bind Abst) -v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) -(\lambda (c3: C).(\lambda (v0: T).(\lambda (a0: A).(sc3 g (asucc g a0) c3 -v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g a0 c2 w))))) -(ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead -c1 (Bind Abst) v) (CHead c3 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (_: T).(eq K (Bind Abbr) (Bind b))))) (\lambda (b: B).(\lambda -(_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: -C).(\lambda (_: T).(csubc g c3 c2))))) (ex5_3_intro C T A (\lambda (_: -C).(\lambda (_: T).(\lambda (_: A).(eq K (Bind Abbr) (Bind Abbr))))) (\lambda -(c3: C).(\lambda (v0: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) v) -(CHead c3 (Bind Abst) v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: -A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (a0: A).(sc3 -g (asucc g a0) c3 v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: -A).(sc3 g a0 c2 w)))) c1 v a (refl_equal K (Bind Abbr)) (refl_equal C (CHead -c1 (Bind Abst) v)) H14 H3 H12)) k H9))))))))) H7)) H6)))))))))))) x y H0))) -H)))))). -(* COMMENTS -Initial nodes: 5197 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/getl.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/getl.ma deleted file mode 100644 index 244c84927..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/getl.ma +++ /dev/null @@ -1,45 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubc/drop.ma". - -include "Basic-1/csubc/clear.ma". - -theorem csubc_getl_conf: - \forall (g: G).(\forall (c1: C).(\forall (e1: C).(\forall (i: nat).((getl i -c1 e1) \to (\forall (c2: C).((csubc g c1 c2) \to (ex2 C (\lambda (e2: -C).(getl i c2 e2)) (\lambda (e2: C).(csubc g e1 e2))))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (e1: C).(\lambda (i: nat).(\lambda -(H: (getl i c1 e1)).(\lambda (c2: C).(\lambda (H0: (csubc g c1 c2)).(let H1 -\def (getl_gen_all c1 e1 i H) in (ex2_ind C (\lambda (e: C).(drop i O c1 e)) -(\lambda (e: C).(clear e e1)) (ex2 C (\lambda (e2: C).(getl i c2 e2)) -(\lambda (e2: C).(csubc g e1 e2))) (\lambda (x: C).(\lambda (H2: (drop i O c1 -x)).(\lambda (H3: (clear x e1)).(let H_x \def (csubc_drop_conf_O g c1 x i H2 -c2 H0) in (let H4 \def H_x in (ex2_ind C (\lambda (e2: C).(drop i O c2 e2)) -(\lambda (e2: C).(csubc g x e2)) (ex2 C (\lambda (e2: C).(getl i c2 e2)) -(\lambda (e2: C).(csubc g e1 e2))) (\lambda (x0: C).(\lambda (H5: (drop i O -c2 x0)).(\lambda (H6: (csubc g x x0)).(let H_x0 \def (csubc_clear_conf g x e1 -H3 x0 H6) in (let H7 \def H_x0 in (ex2_ind C (\lambda (e2: C).(clear x0 e2)) -(\lambda (e2: C).(csubc g e1 e2)) (ex2 C (\lambda (e2: C).(getl i c2 e2)) -(\lambda (e2: C).(csubc g e1 e2))) (\lambda (x1: C).(\lambda (H8: (clear x0 -x1)).(\lambda (H9: (csubc g e1 x1)).(ex_intro2 C (\lambda (e2: C).(getl i c2 -e2)) (\lambda (e2: C).(csubc g e1 e2)) x1 (getl_intro i c2 x1 x0 H5 H8) -H9)))) H7)))))) H4)))))) H1)))))))). -(* COMMENTS -Initial nodes: 315 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/props.ma deleted file mode 100644 index d6399181c..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/props.ma +++ /dev/null @@ -1,30 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubc/defs.ma". - -include "Basic-1/sc3/props.ma". - -theorem csubc_refl: - \forall (g: G).(\forall (c: C).(csubc g c c)) -\def - \lambda (g: G).(\lambda (c: C).(C_ind (\lambda (c0: C).(csubc g c0 c0)) -(\lambda (n: nat).(csubc_sort g n)) (\lambda (c0: C).(\lambda (H: (csubc g c0 -c0)).(\lambda (k: K).(\lambda (t: T).(csubc_head g c0 c0 H k t))))) c)). -(* COMMENTS -Initial nodes: 53 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubst0/clear.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubst0/clear.ma deleted file mode 100644 index 0700d7d48..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubst0/clear.ma +++ /dev/null @@ -1,1139 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubst0/props.ma". - -include "Basic-1/csubst0/fwd.ma". - -include "Basic-1/clear/fwd.ma". - -theorem csubst0_clear_O: - \forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 O v c1 c2) \to -(\forall (c: C).((clear c1 c) \to (clear c2 c)))))) -\def - \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: -T).((csubst0 O v c c2) \to (\forall (c0: C).((clear c c0) \to (clear c2 -c0))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (H: -(csubst0 O v (CSort n) c2)).(\lambda (c: C).(\lambda (_: (clear (CSort n) -c)).(csubst0_gen_sort c2 v O n H (clear c2 c)))))))) (\lambda (c: C).(\lambda -(H: ((\forall (c2: C).(\forall (v: T).((csubst0 O v c c2) \to (\forall (c0: -C).((clear c c0) \to (clear c2 c0)))))))).(\lambda (k: K).(\lambda (t: -T).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 O v (CHead c k t) -c2)).(\lambda (c0: C).(\lambda (H1: (clear (CHead c k t) c0)).(or3_ind (ex3_2 -T nat (\lambda (_: T).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (u2: -T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: -nat).(subst0 j v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq -nat O (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k -t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (ex4_3 T C nat -(\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j))))) -(\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k -u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t -u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c -c3))))) (clear c2 c0) (\lambda (H2: (ex3_2 T nat (\lambda (_: T).(\lambda (j: -nat).(eq nat O (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead -c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t -u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: nat).(eq nat O (s k -j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda -(u2: T).(\lambda (j: nat).(subst0 j v t u2))) (clear c2 c0) (\lambda (x0: -T).(\lambda (x1: nat).(\lambda (H3: (eq nat O (s k x1))).(\lambda (H4: (eq C -c2 (CHead c k x0))).(\lambda (H5: (subst0 x1 v t x0)).(eq_ind_r C (CHead c k -x0) (\lambda (c3: C).(clear c3 c0)) (K_ind (\lambda (k0: K).((clear (CHead c -k0 t) c0) \to ((eq nat O (s k0 x1)) \to (clear (CHead c k0 x0) c0)))) -(\lambda (b: B).(\lambda (_: (clear (CHead c (Bind b) t) c0)).(\lambda (H7: -(eq nat O (s (Bind b) x1))).(let H8 \def (eq_ind nat O (\lambda (ee: -nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True -| (S _) \Rightarrow False])) I (S x1) H7) in (False_ind (clear (CHead c (Bind -b) x0) c0) H8))))) (\lambda (f: F).(\lambda (H6: (clear (CHead c (Flat f) t) -c0)).(\lambda (H7: (eq nat O (s (Flat f) x1))).(let H8 \def (eq_ind_r nat x1 -(\lambda (n: nat).(subst0 n v t x0)) H5 O H7) in (clear_flat c c0 -(clear_gen_flat f c c0 t H6) f x0))))) k H1 H3) c2 H4)))))) H2)) (\lambda -(H2: (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j)))) -(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: -C).(\lambda (j: nat).(csubst0 j v c c3))))).(ex3_2_ind C nat (\lambda (_: -C).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (c3: C).(\lambda (_: -nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j -v c c3))) (clear c2 c0) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H3: (eq -nat O (s k x1))).(\lambda (H4: (eq C c2 (CHead x0 k t))).(\lambda (H5: -(csubst0 x1 v c x0)).(eq_ind_r C (CHead x0 k t) (\lambda (c3: C).(clear c3 -c0)) (K_ind (\lambda (k0: K).((clear (CHead c k0 t) c0) \to ((eq nat O (s k0 -x1)) \to (clear (CHead x0 k0 t) c0)))) (\lambda (b: B).(\lambda (_: (clear -(CHead c (Bind b) t) c0)).(\lambda (H7: (eq nat O (s (Bind b) x1))).(let H8 -\def (eq_ind nat O (\lambda (ee: nat).(match ee in nat return (\lambda (_: -nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x1) H7) -in (False_ind (clear (CHead x0 (Bind b) t) c0) H8))))) (\lambda (f: -F).(\lambda (H6: (clear (CHead c (Flat f) t) c0)).(\lambda (H7: (eq nat O (s -(Flat f) x1))).(let H8 \def (eq_ind_r nat x1 (\lambda (n: nat).(csubst0 n v c -x0)) H5 O H7) in (clear_flat x0 c0 (H x0 v H8 c0 (clear_gen_flat f c c0 t -H6)) f t))))) k H1 H3) c2 H4)))))) H2)) (\lambda (H2: (ex4_3 T C nat (\lambda -(_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j))))) (\lambda (u2: -T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda -(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: -T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))))).(ex4_3_ind T C -nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j))))) -(\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k -u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t -u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c -c3)))) (clear c2 c0) (\lambda (x0: T).(\lambda (x1: C).(\lambda (x2: -nat).(\lambda (H3: (eq nat O (s k x2))).(\lambda (H4: (eq C c2 (CHead x1 k -x0))).(\lambda (H5: (subst0 x2 v t x0)).(\lambda (H6: (csubst0 x2 v c -x1)).(eq_ind_r C (CHead x1 k x0) (\lambda (c3: C).(clear c3 c0)) (K_ind -(\lambda (k0: K).((clear (CHead c k0 t) c0) \to ((eq nat O (s k0 x2)) \to -(clear (CHead x1 k0 x0) c0)))) (\lambda (b: B).(\lambda (_: (clear (CHead c -(Bind b) t) c0)).(\lambda (H8: (eq nat O (s (Bind b) x2))).(let H9 \def -(eq_ind nat O (\lambda (ee: nat).(match ee in nat return (\lambda (_: -nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x2) H8) -in (False_ind (clear (CHead x1 (Bind b) x0) c0) H9))))) (\lambda (f: -F).(\lambda (H7: (clear (CHead c (Flat f) t) c0)).(\lambda (H8: (eq nat O (s -(Flat f) x2))).(let H9 \def (eq_ind_r nat x2 (\lambda (n: nat).(csubst0 n v c -x1)) H6 O H8) in (let H10 \def (eq_ind_r nat x2 (\lambda (n: nat).(subst0 n v -t x0)) H5 O H8) in (clear_flat x1 c0 (H x1 v H9 c0 (clear_gen_flat f c c0 t -H7)) f x0)))))) k H1 H3) c2 H4)))))))) H2)) (csubst0_gen_head k c c2 t v O -H0))))))))))) c1). -(* COMMENTS -Initial nodes: 1582 -END *) - -theorem csubst0_clear_O_back: - \forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 O v c1 c2) \to -(\forall (c: C).((clear c2 c) \to (clear c1 c)))))) -\def - \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: -T).((csubst0 O v c c2) \to (\forall (c0: C).((clear c2 c0) \to (clear c -c0))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (H: -(csubst0 O v (CSort n) c2)).(\lambda (c: C).(\lambda (_: (clear c2 -c)).(csubst0_gen_sort c2 v O n H (clear (CSort n) c)))))))) (\lambda (c: -C).(\lambda (H: ((\forall (c2: C).(\forall (v: T).((csubst0 O v c c2) \to -(\forall (c0: C).((clear c2 c0) \to (clear c c0)))))))).(\lambda (k: -K).(\lambda (t: T).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 O -v (CHead c k t) c2)).(\lambda (c0: C).(\lambda (H1: (clear c2 c0)).(or3_ind -(ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda -(u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: -T).(\lambda (j: nat).(subst0 j v t u2)))) (ex3_2 C nat (\lambda (_: -C).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (c3: C).(\lambda (_: -nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j -v c c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: -nat).(eq nat O (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: -nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda -(j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: -nat).(csubst0 j v c c3))))) (clear (CHead c k t) c0) (\lambda (H2: (ex3_2 T -nat (\lambda (_: T).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (u2: -T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: -nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: -nat).(eq nat O (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead -c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))) (clear -(CHead c k t) c0) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H3: (eq nat O -(s k x1))).(\lambda (H4: (eq C c2 (CHead c k x0))).(\lambda (H5: (subst0 x1 v -t x0)).(let H6 \def (eq_ind C c2 (\lambda (c3: C).(clear c3 c0)) H1 (CHead c -k x0) H4) in (K_ind (\lambda (k0: K).((eq nat O (s k0 x1)) \to ((clear (CHead -c k0 x0) c0) \to (clear (CHead c k0 t) c0)))) (\lambda (b: B).(\lambda (H7: -(eq nat O (s (Bind b) x1))).(\lambda (_: (clear (CHead c (Bind b) x0) -c0)).(let H9 \def (eq_ind nat O (\lambda (ee: nat).(match ee in nat return -(\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) -I (S x1) H7) in (False_ind (clear (CHead c (Bind b) t) c0) H9))))) (\lambda -(f: F).(\lambda (H7: (eq nat O (s (Flat f) x1))).(\lambda (H8: (clear (CHead -c (Flat f) x0) c0)).(let H9 \def (eq_ind_r nat x1 (\lambda (n: nat).(subst0 n -v t x0)) H5 O H7) in (clear_flat c c0 (clear_gen_flat f c c0 x0 H8) f t))))) -k H3 H6))))))) H2)) (\lambda (H2: (ex3_2 C nat (\lambda (_: C).(\lambda (j: -nat).(eq nat O (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead -c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c -c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat O (s k -j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda -(c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (clear (CHead c k t) c0) -(\lambda (x0: C).(\lambda (x1: nat).(\lambda (H3: (eq nat O (s k -x1))).(\lambda (H4: (eq C c2 (CHead x0 k t))).(\lambda (H5: (csubst0 x1 v c -x0)).(let H6 \def (eq_ind C c2 (\lambda (c3: C).(clear c3 c0)) H1 (CHead x0 k -t) H4) in (K_ind (\lambda (k0: K).((eq nat O (s k0 x1)) \to ((clear (CHead x0 -k0 t) c0) \to (clear (CHead c k0 t) c0)))) (\lambda (b: B).(\lambda (H7: (eq -nat O (s (Bind b) x1))).(\lambda (_: (clear (CHead x0 (Bind b) t) c0)).(let -H9 \def (eq_ind nat O (\lambda (ee: nat).(match ee in nat return (\lambda (_: -nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x1) H7) -in (False_ind (clear (CHead c (Bind b) t) c0) H9))))) (\lambda (f: -F).(\lambda (H7: (eq nat O (s (Flat f) x1))).(\lambda (H8: (clear (CHead x0 -(Flat f) t) c0)).(let H9 \def (eq_ind_r nat x1 (\lambda (n: nat).(csubst0 n v -c x0)) H5 O H7) in (clear_flat c c0 (H x0 v H9 c0 (clear_gen_flat f x0 c0 t -H8)) f t))))) k H3 H6))))))) H2)) (\lambda (H2: (ex4_3 T C nat (\lambda (_: -T).(\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j))))) (\lambda (u2: -T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda -(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: -T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))))).(ex4_3_ind T C -nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j))))) -(\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k -u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t -u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c -c3)))) (clear (CHead c k t) c0) (\lambda (x0: T).(\lambda (x1: C).(\lambda -(x2: nat).(\lambda (H3: (eq nat O (s k x2))).(\lambda (H4: (eq C c2 (CHead x1 -k x0))).(\lambda (H5: (subst0 x2 v t x0)).(\lambda (H6: (csubst0 x2 v c -x1)).(let H7 \def (eq_ind C c2 (\lambda (c3: C).(clear c3 c0)) H1 (CHead x1 k -x0) H4) in (K_ind (\lambda (k0: K).((eq nat O (s k0 x2)) \to ((clear (CHead -x1 k0 x0) c0) \to (clear (CHead c k0 t) c0)))) (\lambda (b: B).(\lambda (H8: -(eq nat O (s (Bind b) x2))).(\lambda (_: (clear (CHead x1 (Bind b) x0) -c0)).(let H10 \def (eq_ind nat O (\lambda (ee: nat).(match ee in nat return -(\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) -I (S x2) H8) in (False_ind (clear (CHead c (Bind b) t) c0) H10))))) (\lambda -(f: F).(\lambda (H8: (eq nat O (s (Flat f) x2))).(\lambda (H9: (clear (CHead -x1 (Flat f) x0) c0)).(let H10 \def (eq_ind_r nat x2 (\lambda (n: -nat).(csubst0 n v c x1)) H6 O H8) in (let H11 \def (eq_ind_r nat x2 (\lambda -(n: nat).(subst0 n v t x0)) H5 O H8) in (clear_flat c c0 (H x1 v H10 c0 -(clear_gen_flat f x1 c0 x0 H9)) f t)))))) k H3 H7))))))))) H2)) -(csubst0_gen_head k c c2 t v O H0))))))))))) c1). -(* COMMENTS -Initial nodes: 1606 -END *) - -theorem csubst0_clear_S: - \forall (c1: C).(\forall (c2: C).(\forall (v: T).(\forall (i: nat).((csubst0 -(S i) v c1 c2) \to (\forall (c: C).((clear c1 c) \to (or4 (clear c2 c) (ex3_4 -B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq -C c (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: -T).(\lambda (u2: T).(clear c2 (CHead e (Bind b) u2)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) -(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(clear c2 (CHead e2 (Bind b) u)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b) u1))))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 i v e1 e2)))))))))))))) -\def - \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: -T).(\forall (i: nat).((csubst0 (S i) v c c2) \to (\forall (c0: C).((clear c -c0) \to (or4 (clear c2 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: -C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) -(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 -(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind -b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(clear c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e2 -(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 -e2))))))))))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda -(i: nat).(\lambda (H: (csubst0 (S i) v (CSort n) c2)).(\lambda (c: -C).(\lambda (_: (clear (CSort n) c)).(csubst0_gen_sort c2 v (S i) n H (or4 -(clear c2 c) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: -T).(\lambda (_: T).(eq C c (CHead e (Bind b) u1)))))) (\lambda (b: -B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e (Bind -b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind b) u)))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c2 -(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c -(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))))))))))) -(\lambda (c: C).(\lambda (H: ((\forall (c2: C).(\forall (v: T).(\forall (i: -nat).((csubst0 (S i) v c c2) \to (\forall (c0: C).((clear c c0) \to (or4 -(clear c2 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: -T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: -B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e (Bind -b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c2 -(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 -(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 -e2)))))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda -(v: T).(\lambda (i: nat).(\lambda (H0: (csubst0 (S i) v (CHead c k t) -c2)).(\lambda (c0: C).(\lambda (H1: (clear (CHead c k t) c0)).(or3_ind (ex3_2 -T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (S i) (s k j)))) (\lambda -(u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: -T).(\lambda (j: nat).(subst0 j v t u2)))) (ex3_2 C nat (\lambda (_: -C).(\lambda (j: nat).(eq nat (S i) (s k j)))) (\lambda (c3: C).(\lambda (_: -nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j -v c c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: -nat).(eq nat (S i) (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: -nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda -(j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: -nat).(csubst0 j v c c3))))) (or4 (clear c2 c0) (ex3_4 B C T T (\lambda (b: -B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind -b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: -T).(clear c2 (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 -(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(clear c2 (CHead e2 (Bind b) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) -(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear -c2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -i v e1 e2)))))))) (\lambda (H2: (ex3_2 T nat (\lambda (_: T).(\lambda (j: -nat).(eq nat (S i) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 -(CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t -u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (S i) (s k -j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda -(u2: T).(\lambda (j: nat).(subst0 j v t u2))) (or4 (clear c2 c0) (ex3_4 B C T -T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 -(CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: -T).(\lambda (u2: T).(clear c2 (CHead e (Bind b) u2)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) -(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(clear c2 (CHead e2 (Bind b) u)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 i v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: -nat).(\lambda (H3: (eq nat (S i) (s k x1))).(\lambda (H4: (eq C c2 (CHead c k -x0))).(\lambda (H5: (subst0 x1 v t x0)).(eq_ind_r C (CHead c k x0) (\lambda -(c3: C).(or4 (clear c3 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: -C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) -(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c3 -(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind -b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(clear c3 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear c3 (CHead e2 -(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 -e2))))))))) (K_ind (\lambda (k0: K).((clear (CHead c k0 t) c0) \to ((eq nat -(S i) (s k0 x1)) \to (or4 (clear (CHead c k0 x0) c0) (ex3_4 B C T T (\lambda -(b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e -(Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda -(u2: T).(clear (CHead c k0 x0) (CHead e (Bind b) u2)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) -(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead c k0 x0) (CHead e2 (Bind b) -u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind -b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (u2: T).(clear (CHead c k0 x0) (CHead e2 (Bind b) u2))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))))))) -(\lambda (b: B).(\lambda (H6: (clear (CHead c (Bind b) t) c0)).(\lambda (H7: -(eq nat (S i) (s (Bind b) x1))).(let H8 \def (f_equal nat nat (\lambda (e: -nat).(match e in nat return (\lambda (_: nat).nat) with [O \Rightarrow i | (S -n) \Rightarrow n])) (S i) (S x1) H7) in (let H9 \def (eq_ind_r nat x1 -(\lambda (n: nat).(subst0 n v t x0)) H5 i H8) in (eq_ind_r C (CHead c (Bind -b) t) (\lambda (c3: C).(or4 (clear (CHead c (Bind b) x0) c3) (ex3_4 B C T T -(\lambda (b0: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c3 -(CHead e (Bind b0) u1)))))) (\lambda (b0: B).(\lambda (e: C).(\lambda (_: -T).(\lambda (u2: T).(clear (CHead c (Bind b) x0) (CHead e (Bind b0) u2)))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v -u1 u2)))))) (ex3_4 B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C c3 (CHead e1 (Bind b0) u)))))) (\lambda (b0: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead c (Bind b) -x0) (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda -(b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq -C c3 (CHead e1 (Bind b0) u1))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead c (Bind b) x0) (CHead -e2 (Bind b0) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -i v e1 e2))))))))) (or4_intro1 (clear (CHead c (Bind b) x0) (CHead c (Bind b) -t)) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e: C).(\lambda (u1: T).(\lambda -(_: T).(eq C (CHead c (Bind b) t) (CHead e (Bind b0) u1)))))) (\lambda (b0: -B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead c (Bind b) -x0) (CHead e (Bind b0) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b0: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c (Bind b) -t) (CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(clear (CHead c (Bind b) x0) (CHead e2 (Bind b0) u)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e1 -(Bind b0) u1))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead c (Bind b) x0) (CHead e2 -(Bind b0) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 -e2))))))) (ex3_4_intro B C T T (\lambda (b0: B).(\lambda (e: C).(\lambda (u1: -T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e (Bind b0) u1)))))) -(\lambda (b0: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear -(CHead c (Bind b) x0) (CHead e (Bind b0) u2)))))) (\lambda (_: B).(\lambda -(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))) b c t x0 -(refl_equal C (CHead c (Bind b) t)) (clear_bind b c x0) H9)) c0 -(clear_gen_bind b c c0 t H6))))))) (\lambda (f: F).(\lambda (H6: (clear -(CHead c (Flat f) t) c0)).(\lambda (H7: (eq nat (S i) (s (Flat f) x1))).(let -H8 \def (f_equal nat nat (\lambda (e: nat).e) (S i) (s (Flat f) x1) H7) in -(let H9 \def (eq_ind_r nat x1 (\lambda (n: nat).(subst0 n v t x0)) H5 (S i) -H8) in (or4_intro0 (clear (CHead c (Flat f) x0) c0) (ex3_4 B C T T (\lambda -(b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e -(Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda -(u2: T).(clear (CHead c (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) -(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead c (Flat f) x0) (CHead e2 -(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 -(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (u2: T).(clear (CHead c (Flat f) x0) (CHead e2 (Bind b) -u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) -(clear_flat c c0 (clear_gen_flat f c c0 t H6) f x0))))))) k H1 H3) c2 -H4)))))) H2)) (\lambda (H2: (ex3_2 C nat (\lambda (_: C).(\lambda (j: -nat).(eq nat (S i) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 -(CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c -c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S i) (s k -j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda -(c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (or4 (clear c2 c0) (ex3_4 B C -T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 -(CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: -T).(\lambda (u2: T).(clear c2 (CHead e (Bind b) u2)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) -(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(clear c2 (CHead e2 (Bind b) u)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 i v e1 e2)))))))) (\lambda (x0: C).(\lambda (x1: -nat).(\lambda (H3: (eq nat (S i) (s k x1))).(\lambda (H4: (eq C c2 (CHead x0 -k t))).(\lambda (H5: (csubst0 x1 v c x0)).(eq_ind_r C (CHead x0 k t) (\lambda -(c3: C).(or4 (clear c3 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: -C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) -(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c3 -(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind -b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(clear c3 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear c3 (CHead e2 -(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 -e2))))))))) (K_ind (\lambda (k0: K).((clear (CHead c k0 t) c0) \to ((eq nat -(S i) (s k0 x1)) \to (or4 (clear (CHead x0 k0 t) c0) (ex3_4 B C T T (\lambda -(b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e -(Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda -(u2: T).(clear (CHead x0 k0 t) (CHead e (Bind b) u2)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) -(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 k0 t) (CHead e2 (Bind b) -u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind -b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (u2: T).(clear (CHead x0 k0 t) (CHead e2 (Bind b) u2))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))))))) -(\lambda (b: B).(\lambda (H6: (clear (CHead c (Bind b) t) c0)).(\lambda (H7: -(eq nat (S i) (s (Bind b) x1))).(let H8 \def (f_equal nat nat (\lambda (e: -nat).(match e in nat return (\lambda (_: nat).nat) with [O \Rightarrow i | (S -n) \Rightarrow n])) (S i) (S x1) H7) in (let H9 \def (eq_ind_r nat x1 -(\lambda (n: nat).(csubst0 n v c x0)) H5 i H8) in (eq_ind_r C (CHead c (Bind -b) t) (\lambda (c3: C).(or4 (clear (CHead x0 (Bind b) t) c3) (ex3_4 B C T T -(\lambda (b0: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c3 -(CHead e (Bind b0) u1)))))) (\lambda (b0: B).(\lambda (e: C).(\lambda (_: -T).(\lambda (u2: T).(clear (CHead x0 (Bind b) t) (CHead e (Bind b0) u2)))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v -u1 u2)))))) (ex3_4 B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C c3 (CHead e1 (Bind b0) u)))))) (\lambda (b0: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Bind b) -t) (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c3 -(CHead e1 (Bind b0) u1))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Bind b) t) (CHead e2 -(Bind b0) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 -e2))))))))) (or4_intro2 (clear (CHead x0 (Bind b) t) (CHead c (Bind b) t)) -(ex3_4 B C T T (\lambda (b0: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: -T).(eq C (CHead c (Bind b) t) (CHead e (Bind b0) u1)))))) (\lambda (b0: -B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Bind b) -t) (CHead e (Bind b0) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b0: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c (Bind b) -t) (CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(clear (CHead x0 (Bind b) t) (CHead e2 (Bind b0) u)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e1 -(Bind b0) u1))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Bind b) t) (CHead e2 -(Bind b0) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 -e2))))))) (ex3_4_intro B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C (CHead c (Bind b) t) (CHead e1 (Bind b0) u)))))) -(\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear -(CHead x0 (Bind b) t) (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2))))) b c x0 t -(refl_equal C (CHead c (Bind b) t)) (clear_bind b x0 t) H9)) c0 -(clear_gen_bind b c c0 t H6))))))) (\lambda (f: F).(\lambda (H6: (clear -(CHead c (Flat f) t) c0)).(\lambda (H7: (eq nat (S i) (s (Flat f) x1))).(let -H8 \def (f_equal nat nat (\lambda (e: nat).e) (S i) (s (Flat f) x1) H7) in -(let H9 \def (eq_ind_r nat x1 (\lambda (n: nat).(csubst0 n v c x0)) H5 (S i) -H8) in (let H10 \def (H x0 v i H9 c0 (clear_gen_flat f c c0 t H6)) in -(or4_ind (clear x0 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: -C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) -(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x0 -(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind -b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(clear x0 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear x0 (CHead e2 -(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 -e2))))))) (or4 (clear (CHead x0 (Flat f) t) c0) (ex3_4 B C T T (\lambda (b: -B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind -b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: -T).(clear (CHead x0 (Flat f) t) (CHead e (Bind b) u2)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) -(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Flat f) t) (CHead e2 -(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 -(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) -u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 -e2)))))))) (\lambda (H11: (clear x0 c0)).(or4_intro0 (clear (CHead x0 (Flat -f) t) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: -T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: -B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) -t) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind -b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) -(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear -(CHead x0 (Flat f) t) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda -(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 -u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (clear_flat x0 c0 H11 f t))) -(\lambda (H11: (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: -T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: -B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x0 (CHead e (Bind -b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 i v u1 u2))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e: -C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) -(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x0 -(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 i v u1 u2))))) (or4 (clear (CHead x0 (Flat f) t) -c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda -(_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: -C).(\lambda (_: T).(\lambda (u2: 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B).(\lambda (x3: -C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H12: (eq C c0 (CHead x3 (Bind -x2) x4))).(\lambda (H13: (clear x0 (CHead x3 (Bind x2) x5))).(\lambda (H14: -(subst0 i v x4 x5)).(or4_intro1 (clear (CHead x0 (Flat f) t) c0) (ex3_4 B C T -T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 -(CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: -T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e (Bind b) u2)))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v -u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Flat f) -t) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 -(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 -(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 -e2))))))) (ex3_4_intro B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: -T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: -B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) -t) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 i v u1 u2))))) x2 x3 x4 x5 H12 (clear_flat x0 -(CHead x3 (Bind x2) x5) H13 f t) H14))))))))) H11)) (\lambda (H11: (ex3_4 B C -C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 -(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 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C).(\lambda (u: T).(clear (CHead x0 (Flat f) -t) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 -(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 -(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 -e2)))))))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: -T).(\lambda (H12: (eq C c0 (CHead x3 (Bind x2) x5))).(\lambda (H13: (clear x0 -(CHead x4 (Bind x2) x5))).(\lambda (H14: (csubst0 i v x3 x4)).(or4_intro2 -(clear (CHead x0 (Flat f) t) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: -C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) -(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear -(CHead x0 (Flat f) t) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 -(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u2))))))) (\lambda -(_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (ex3_4_intro B C -C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 -(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v e1 e2))))) x2 x3 x4 x5 H12 (clear_flat x0 (CHead x4 (Bind x2) x5) H13 f t) -H14))))))))) H11)) (\lambda (H11: (ex4_5 B C C T T (\lambda (b: B).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 -(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (u2: T).(clear x0 (CHead e2 (Bind b) u2))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 -i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))).(ex4_5_ind B C C T T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear x0 (CHead e2 -(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 -e2)))))) (or4 (clear (CHead x0 (Flat f) t) c0) (ex3_4 B C T T (\lambda (b: -B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind -b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: -T).(clear (CHead x0 (Flat f) t) (CHead e (Bind b) u2)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) -(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Flat f) t) (CHead e2 -(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 -(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) -u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 -e2)))))))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: -T).(\lambda (x6: T).(\lambda (H12: (eq C c0 (CHead x3 (Bind x2) -x5))).(\lambda (H13: (clear x0 (CHead x4 (Bind x2) x6))).(\lambda (H14: -(subst0 i v x5 x6)).(\lambda (H15: (csubst0 i v x3 x4)).(or4_intro3 (clear -(CHead x0 (Flat f) t) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: -C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) -(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear -(CHead x0 (Flat f) t) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 -(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u2))))))) (\lambda -(_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (ex4_5_intro B C -C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear -(CHead x0 (Flat f) t) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda -(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 -u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 i v e1 e2)))))) x2 x3 x4 x5 x6 H12 (clear_flat x0 -(CHead x4 (Bind x2) x6) H13 f t) H14 H15))))))))))) H11)) H10))))))) k H1 H3) -c2 H4)))))) H2)) (\lambda (H2: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: -C).(\lambda (j: nat).(eq nat (S i) (s k j))))) (\lambda (u2: T).(\lambda (c3: -C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda -(_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: -C).(\lambda (j: nat).(csubst0 j v c c3)))))).(ex4_3_ind T C nat (\lambda (_: -T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S i) (s k j))))) (\lambda (u2: -T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda -(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: -T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (or4 (clear c2 -c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda -(_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: -C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e (Bind b) u2)))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v -u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c2 (CHead e2 (Bind -b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind -b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 -i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))) (\lambda (x0: T).(\lambda -(x1: C).(\lambda (x2: nat).(\lambda (H3: (eq nat (S i) (s k x2))).(\lambda -(H4: (eq C c2 (CHead x1 k x0))).(\lambda (H5: (subst0 x2 v t x0)).(\lambda -(H6: (csubst0 x2 v c x1)).(eq_ind_r C (CHead x1 k x0) (\lambda (c3: C).(or4 -(clear c3 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: -T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: -B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c3 (CHead e (Bind -b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c3 -(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 -(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(clear c3 (CHead e2 (Bind b) u2))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))))) (K_ind -(\lambda (k0: K).((clear (CHead c k0 t) c0) \to ((eq nat (S i) (s k0 x2)) \to -(or4 (clear (CHead x1 k0 x0) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: -C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) -(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear -(CHead x1 k0 x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 -(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(clear (CHead x1 k0 x0) (CHead e2 (Bind b) u)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(clear (CHead x1 k0 x0) (CHead e2 (Bind b) u2))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 -i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))))))) (\lambda (b: B).(\lambda -(H7: (clear (CHead c (Bind b) t) c0)).(\lambda (H8: (eq nat (S i) (s (Bind b) -x2))).(let H9 \def (f_equal nat nat (\lambda (e: nat).(match e in nat return -(\lambda (_: nat).nat) with [O \Rightarrow i | (S n) \Rightarrow n])) (S i) -(S x2) H8) in (let H10 \def (eq_ind_r nat x2 (\lambda (n: nat).(csubst0 n v c -x1)) H6 i H9) in (let H11 \def (eq_ind_r nat x2 (\lambda (n: nat).(subst0 n v -t x0)) H5 i H9) in (eq_ind_r C (CHead c (Bind b) t) (\lambda (c3: C).(or4 -(clear (CHead x1 (Bind b) x0) c3) (ex3_4 B C T T (\lambda (b0: B).(\lambda -(e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c3 (CHead e (Bind b0) u1)))))) -(\lambda (b0: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear -(CHead x1 (Bind b) x0) (CHead e (Bind b0) u2)))))) (\lambda (_: B).(\lambda -(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C -T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3 -(CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(clear (CHead x1 (Bind b) x0) (CHead e2 (Bind b0) u)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C c3 (CHead e1 (Bind b0) u1))))))) -(\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(clear (CHead x1 (Bind b) x0) (CHead e2 (Bind b0) u2))))))) (\lambda -(_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))))) (or4_intro3 -(clear (CHead x1 (Bind b) x0) (CHead c (Bind b) t)) (ex3_4 B C T T (\lambda -(b0: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead c (Bind -b) t) (CHead e (Bind b0) u1)))))) (\lambda (b0: B).(\lambda (e: C).(\lambda -(_: T).(\lambda (u2: T).(clear (CHead x1 (Bind b) x0) (CHead e (Bind b0) -u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b0: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c (Bind b) t) (CHead e1 (Bind -b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(clear (CHead x1 (Bind b) x0) (CHead e2 (Bind b0) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) -(ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e1 (Bind b0) -u1))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (u2: T).(clear (CHead x1 (Bind b) x0) (CHead e2 (Bind b0) -u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) -(ex4_5_intro B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e1 -(Bind b0) u1))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Bind b) x0) (CHead e2 -(Bind b0) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 -e2)))))) b c x1 t x0 (refl_equal C (CHead c (Bind b) t)) (clear_bind b x1 x0) -H11 H10)) c0 (clear_gen_bind b c c0 t H7)))))))) (\lambda (f: F).(\lambda -(H7: (clear (CHead c (Flat f) t) c0)).(\lambda (H8: (eq nat (S i) (s (Flat f) -x2))).(let H9 \def (f_equal nat nat (\lambda (e: nat).e) (S i) (s (Flat f) -x2) H8) in (let H10 \def (eq_ind_r nat x2 (\lambda (n: nat).(csubst0 n v c -x1)) H6 (S i) H9) in (let H11 \def (eq_ind_r nat x2 (\lambda (n: nat).(subst0 -n v t x0)) H5 (S i) H9) in (let H12 \def (H x1 v i H10 c0 (clear_gen_flat f c -c0 t H7)) in (or4_ind (clear x1 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda -(e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) -(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x1 -(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind -b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(clear x1 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear x1 (CHead e2 -(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 -e2))))))) (or4 (clear (CHead x1 (Flat f) x0) c0) (ex3_4 B C T T (\lambda (b: -B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind -b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: -T).(clear (CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) -(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2 -(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 -(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) -u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 -e2)))))))) (\lambda (H13: (clear x1 c0)).(or4_intro0 (clear (CHead x1 (Flat -f) x0) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: -T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: -B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) -x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind -b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) -(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear -(CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda -(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 -u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (clear_flat x1 c0 H13 f x0))) -(\lambda (H13: (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: -T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: -B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x1 (CHead e (Bind -b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 i v u1 u2))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e: -C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) -(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x1 -(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 i v u1 u2))))) (or4 (clear (CHead x1 (Flat f) x0) -c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda -(_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: -C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e -(Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear -(CHead x1 (Flat f) x0) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C -C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear -(CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda -(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 -u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 i v e1 e2)))))))) (\lambda (x3: B).(\lambda (x4: -C).(\lambda (x5: T).(\lambda (x6: T).(\lambda (H14: (eq C c0 (CHead x4 (Bind -x3) x5))).(\lambda (H15: (clear x1 (CHead x4 (Bind x3) x6))).(\lambda (H16: -(subst0 i v x5 x6)).(or4_intro1 (clear (CHead x1 (Flat f) x0) c0) (ex3_4 B C -T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 -(CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: -T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v -u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x1 (Flat f) -x0) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 -(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 -(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 -e2))))))) (ex3_4_intro B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: -T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: -B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) -x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 i v u1 u2))))) x3 x4 x5 x6 H14 (clear_flat x1 -(CHead x4 (Bind x3) x6) H15 f x0) H16))))))))) H13)) (\lambda (H13: (ex3_4 B -C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C -c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(clear x1 (CHead e2 (Bind b) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 -e2))))))).(ex3_4_ind B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x1 (CHead e2 (Bind -b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 i v e1 e2))))) (or4 (clear (CHead x1 (Flat f) x0) c0) (ex3_4 B C -T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 -(CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: -T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v -u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x1 (Flat f) -x0) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 -(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 -(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 -e2)))))))) (\lambda (x3: B).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: -T).(\lambda (H14: (eq C c0 (CHead x4 (Bind x3) x6))).(\lambda (H15: (clear x1 -(CHead x5 (Bind x3) x6))).(\lambda (H16: (csubst0 i v x4 x5)).(or4_intro2 -(clear (CHead x1 (Flat f) x0) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: -C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) -(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear -(CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda -(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C -T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 -(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda -(_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (ex3_4_intro B C -C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 -(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v e1 e2))))) x3 x4 x5 x6 H14 (clear_flat x1 (CHead x5 (Bind x3) x6) H15 f x0) -H16))))))))) H13)) (\lambda (H13: (ex4_5 B C C T T (\lambda (b: B).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 -(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (u2: T).(clear x1 (CHead e2 (Bind b) u2))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 -i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))).(ex4_5_ind B C C T T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear x1 (CHead e2 -(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 -e2)))))) (or4 (clear (CHead x1 (Flat f) x0) c0) (ex3_4 B C T T (\lambda (b: -B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind -b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: -T).(clear (CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) -(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2 -(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 -(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) -u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 -e2)))))))) (\lambda (x3: B).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: -T).(\lambda (x7: T).(\lambda (H14: (eq C c0 (CHead x4 (Bind x3) -x6))).(\lambda (H15: (clear x1 (CHead x5 (Bind x3) x7))).(\lambda (H16: -(subst0 i v x6 x7)).(\lambda (H17: (csubst0 i v x4 x5)).(or4_intro3 (clear -(CHead x1 (Flat f) x0) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: -C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) -(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear -(CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda -(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C -T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 -(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda -(_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (ex4_5_intro B C -C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear -(CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda -(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 -u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 i v e1 e2)))))) x3 x4 x5 x6 x7 H14 (clear_flat x1 -(CHead x5 (Bind x3) x7) H15 f x0) H16 H17))))))))))) H13)) H12)))))))) k H1 -H3) c2 H4)))))))) H2)) (csubst0_gen_head k c c2 t v (S i) H0)))))))))))) c1). -(* COMMENTS -Initial nodes: 14968 -END *) - -theorem csubst0_clear_trans: - \forall (c1: C).(\forall (c2: C).(\forall (v: T).(\forall (i: nat).((csubst0 -i v c1 c2) \to (\forall (e2: C).((clear c2 e2) \to (or (clear c1 e2) (ex2 C -(\lambda (e1: C).(csubst0 i v e1 e2)) (\lambda (e1: C).(clear c1 e1)))))))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda (i: nat).(\lambda -(H: (csubst0 i v c1 c2)).(csubst0_ind (\lambda (n: nat).(\lambda (t: -T).(\lambda (c: C).(\lambda (c0: C).(\forall (e2: C).((clear c0 e2) \to (or -(clear c e2) (ex2 C (\lambda (e1: C).(csubst0 n t e1 e2)) (\lambda (e1: -C).(clear c e1)))))))))) (\lambda (k: K).(\lambda (i0: nat).(\lambda (v0: -T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H0: (subst0 i0 v0 u1 -u2)).(\lambda (c: C).(\lambda (e2: C).(\lambda (H1: (clear (CHead c k u2) -e2)).(K_ind (\lambda (k0: K).((clear (CHead c k0 u2) e2) \to (or (clear -(CHead c k0 u1) e2) (ex2 C (\lambda (e1: C).(csubst0 (s k0 i0) v0 e1 e2)) -(\lambda (e1: C).(clear (CHead c k0 u1) e1)))))) (\lambda (b: B).(\lambda -(H2: (clear (CHead c (Bind b) u2) e2)).(eq_ind_r C (CHead c (Bind b) u2) -(\lambda (c0: C).(or (clear (CHead c (Bind b) u1) c0) (ex2 C (\lambda (e1: -C).(csubst0 (s (Bind b) i0) v0 e1 c0)) (\lambda (e1: C).(clear (CHead c (Bind -b) u1) e1))))) (or_intror (clear (CHead c (Bind b) u1) (CHead c (Bind b) u2)) -(ex2 C (\lambda (e1: C).(csubst0 (S i0) v0 e1 (CHead c (Bind b) u2))) -(\lambda (e1: C).(clear (CHead c (Bind b) u1) e1))) (ex_intro2 C (\lambda -(e1: C).(csubst0 (S i0) v0 e1 (CHead c (Bind b) u2))) (\lambda (e1: C).(clear -(CHead c (Bind b) u1) e1)) (CHead c (Bind b) u1) (csubst0_snd_bind b i0 v0 u1 -u2 H0 c) (clear_bind b c u1))) e2 (clear_gen_bind b c e2 u2 H2)))) (\lambda -(f: F).(\lambda (H2: (clear (CHead c (Flat f) u2) e2)).(or_introl (clear -(CHead c (Flat f) u1) e2) (ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2)) -(\lambda (e1: C).(clear (CHead c (Flat f) u1) e1))) (clear_flat c e2 -(clear_gen_flat f c e2 u2 H2) f u1)))) k H1)))))))))) (\lambda (k: -K).(\lambda (i0: nat).(\lambda (c3: C).(\lambda (c4: C).(\lambda (v0: -T).(\lambda (H0: (csubst0 i0 v0 c3 c4)).(\lambda (H1: ((\forall (e2: -C).((clear c4 e2) \to (or (clear c3 e2) (ex2 C (\lambda (e1: C).(csubst0 i0 -v0 e1 e2)) (\lambda (e1: C).(clear c3 e1)))))))).(\lambda (u: T).(\lambda -(e2: C).(\lambda (H2: (clear (CHead c4 k u) e2)).(K_ind (\lambda (k0: -K).((clear (CHead c4 k0 u) e2) \to (or (clear (CHead c3 k0 u) e2) (ex2 C -(\lambda (e1: C).(csubst0 (s k0 i0) v0 e1 e2)) (\lambda (e1: C).(clear (CHead -c3 k0 u) e1)))))) (\lambda (b: B).(\lambda (H3: (clear (CHead c4 (Bind b) u) -e2)).(eq_ind_r C (CHead c4 (Bind b) u) (\lambda (c: C).(or (clear (CHead c3 -(Bind b) u) c) (ex2 C (\lambda (e1: C).(csubst0 (s (Bind b) i0) v0 e1 c)) -(\lambda (e1: C).(clear (CHead c3 (Bind b) u) e1))))) (or_intror (clear -(CHead c3 (Bind b) u) (CHead c4 (Bind b) u)) (ex2 C (\lambda (e1: C).(csubst0 -(S i0) v0 e1 (CHead c4 (Bind b) u))) (\lambda (e1: C).(clear (CHead c3 (Bind -b) u) e1))) (ex_intro2 C (\lambda (e1: C).(csubst0 (S i0) v0 e1 (CHead c4 -(Bind b) u))) (\lambda (e1: C).(clear (CHead c3 (Bind b) u) e1)) (CHead c3 -(Bind b) u) (csubst0_fst_bind b i0 c3 c4 v0 H0 u) (clear_bind b c3 u))) e2 -(clear_gen_bind b c4 e2 u H3)))) (\lambda (f: F).(\lambda (H3: (clear (CHead -c4 (Flat f) u) e2)).(let H_x \def (H1 e2 (clear_gen_flat f c4 e2 u H3)) in -(let H4 \def H_x in (or_ind (clear c3 e2) (ex2 C (\lambda (e1: C).(csubst0 i0 -v0 e1 e2)) (\lambda (e1: C).(clear c3 e1))) (or (clear (CHead c3 (Flat f) u) -e2) (ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear -(CHead c3 (Flat f) u) e1)))) (\lambda (H5: (clear c3 e2)).(or_introl (clear -(CHead c3 (Flat f) u) e2) (ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2)) -(\lambda (e1: C).(clear (CHead c3 (Flat f) u) e1))) (clear_flat c3 e2 H5 f -u))) (\lambda (H5: (ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda -(e1: C).(clear c3 e1)))).(ex2_ind C (\lambda (e1: C).(csubst0 i0 v0 e1 e2)) -(\lambda (e1: C).(clear c3 e1)) (or (clear (CHead c3 (Flat f) u) e2) (ex2 C -(\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear (CHead c3 -(Flat f) u) e1)))) (\lambda (x: C).(\lambda (H6: (csubst0 i0 v0 x -e2)).(\lambda (H7: (clear c3 x)).(or_intror (clear (CHead c3 (Flat f) u) e2) -(ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear (CHead -c3 (Flat f) u) e1))) (ex_intro2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2)) -(\lambda (e1: C).(clear (CHead c3 (Flat f) u) e1)) x H6 (clear_flat c3 x H7 f -u)))))) H5)) H4))))) k H2))))))))))) (\lambda (k: K).(\lambda (i0: -nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H0: (subst0 -i0 v0 u1 u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (H1: (csubst0 i0 v0 -c3 c4)).(\lambda (H2: ((\forall (e2: C).((clear c4 e2) \to (or (clear c3 e2) -(ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear c3 -e1)))))))).(\lambda (e2: C).(\lambda (H3: (clear (CHead c4 k u2) e2)).(K_ind -(\lambda (k0: K).((clear (CHead c4 k0 u2) e2) \to (or (clear (CHead c3 k0 u1) -e2) (ex2 C (\lambda (e1: C).(csubst0 (s k0 i0) v0 e1 e2)) (\lambda (e1: -C).(clear (CHead c3 k0 u1) e1)))))) (\lambda (b: B).(\lambda (H4: (clear -(CHead c4 (Bind b) u2) e2)).(eq_ind_r C (CHead c4 (Bind b) u2) (\lambda (c: -C).(or (clear (CHead c3 (Bind b) u1) c) (ex2 C (\lambda (e1: C).(csubst0 (s -(Bind b) i0) v0 e1 c)) (\lambda (e1: C).(clear (CHead c3 (Bind b) u1) e1))))) -(or_intror (clear (CHead c3 (Bind b) u1) (CHead c4 (Bind b) u2)) (ex2 C -(\lambda (e1: C).(csubst0 (S i0) v0 e1 (CHead c4 (Bind b) u2))) (\lambda (e1: -C).(clear (CHead c3 (Bind b) u1) e1))) (ex_intro2 C (\lambda (e1: C).(csubst0 -(S i0) v0 e1 (CHead c4 (Bind b) u2))) (\lambda (e1: C).(clear (CHead c3 (Bind -b) u1) e1)) (CHead c3 (Bind b) u1) (csubst0_both_bind b i0 v0 u1 u2 H0 c3 c4 -H1) (clear_bind b c3 u1))) e2 (clear_gen_bind b c4 e2 u2 H4)))) (\lambda (f: -F).(\lambda (H4: (clear (CHead c4 (Flat f) u2) e2)).(let H_x \def (H2 e2 -(clear_gen_flat f c4 e2 u2 H4)) in (let H5 \def H_x in (or_ind (clear c3 e2) -(ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear c3 -e1))) (or (clear (CHead c3 (Flat f) u1) e2) (ex2 C (\lambda (e1: C).(csubst0 -i0 v0 e1 e2)) (\lambda (e1: C).(clear (CHead c3 (Flat f) u1) e1)))) (\lambda -(H6: (clear c3 e2)).(or_introl (clear (CHead c3 (Flat f) u1) e2) (ex2 C -(\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear (CHead c3 -(Flat f) u1) e1))) (clear_flat c3 e2 H6 f u1))) (\lambda (H6: (ex2 C (\lambda -(e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear c3 e1)))).(ex2_ind C -(\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear c3 e1)) (or -(clear (CHead c3 (Flat f) u1) e2) (ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 -e2)) (\lambda (e1: C).(clear (CHead c3 (Flat f) u1) e1)))) (\lambda (x: -C).(\lambda (H7: (csubst0 i0 v0 x e2)).(\lambda (H8: (clear c3 x)).(or_intror -(clear (CHead c3 (Flat f) u1) e2) (ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 -e2)) (\lambda (e1: C).(clear (CHead c3 (Flat f) u1) e1))) (ex_intro2 C -(\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear (CHead c3 -(Flat f) u1) e1)) x H7 (clear_flat c3 x H8 f u1)))))) H6)) H5))))) k -H3))))))))))))) i v c1 c2 H))))). -(* COMMENTS -Initial nodes: 2085 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubst0/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubst0/defs.ma deleted file mode 100644 index 0068e19f6..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubst0/defs.ma +++ /dev/null @@ -1,32 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/subst0/defs.ma". - -include "Basic-1/C/defs.ma". - -inductive csubst0: nat \to (T \to (C \to (C \to Prop))) \def -| csubst0_snd: \forall (k: K).(\forall (i: nat).(\forall (v: T).(\forall (u1: -T).(\forall (u2: T).((subst0 i v u1 u2) \to (\forall (c: C).(csubst0 (s k i) -v (CHead c k u1) (CHead c k u2)))))))) -| csubst0_fst: \forall (k: K).(\forall (i: nat).(\forall (c1: C).(\forall -(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (u: T).(csubst0 (s -k i) v (CHead c1 k u) (CHead c2 k u)))))))) -| csubst0_both: \forall (k: K).(\forall (i: nat).(\forall (v: T).(\forall -(u1: T).(\forall (u2: T).((subst0 i v u1 u2) \to (\forall (c1: C).(\forall -(c2: C).((csubst0 i v c1 c2) \to (csubst0 (s k i) v (CHead c1 k u1) (CHead c2 -k u2)))))))))). - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubst0/drop.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubst0/drop.ma deleted file mode 100644 index b6bc6cae7..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubst0/drop.ma +++ /dev/null @@ -1,6294 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubst0/fwd.ma". - -include "Basic-1/drop/fwd.ma". - -include "Basic-1/s/props.ma". - -theorem csubst0_drop_gt: - \forall (n: nat).(\forall (i: nat).((lt i n) \to (\forall (c1: C).(\forall -(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n O -c1 e) \to (drop n O c2 e))))))))) -\def - \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (i: nat).((lt i n0) -\to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) -\to (\forall (e: C).((drop n0 O c1 e) \to (drop n0 O c2 e)))))))))) (\lambda -(i: nat).(\lambda (H: (lt i O)).(\lambda (c1: C).(\lambda (c2: C).(\lambda -(v: T).(\lambda (_: (csubst0 i v c1 c2)).(\lambda (e: C).(\lambda (_: (drop O -O c1 e)).(lt_x_O i H (drop O O c2 e)))))))))) (\lambda (n0: nat).(\lambda (H: -((\forall (i: nat).((lt i n0) \to (\forall (c1: C).(\forall (c2: C).(\forall -(v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n0 O c1 e) \to (drop -n0 O c2 e))))))))))).(\lambda (i: nat).(\lambda (H0: (lt i (S n0))).(\lambda -(c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v -c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (drop (S n0) O c2 e))))))) -(\lambda (n1: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (_: (csubst0 i v -(CSort n1) c2)).(\lambda (e: C).(\lambda (H2: (drop (S n0) O (CSort n1) -e)).(and3_ind (eq C e (CSort n1)) (eq nat (S n0) O) (eq nat O O) (drop (S n0) -O c2 e) (\lambda (H3: (eq C e (CSort n1))).(\lambda (H4: (eq nat (S n0) -O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n1) (\lambda (c: C).(drop -(S n0) O c2 c)) (let H6 \def (eq_ind nat (S n0) (\lambda (ee: nat).(match ee -in nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) -\Rightarrow True])) I O H4) in (False_ind (drop (S n0) O c2 (CSort n1)) H6)) -e H3)))) (drop_gen_sort n1 (S n0) O e H2)))))))) (\lambda (c: C).(\lambda -(H1: ((\forall (c2: C).(\forall (v: T).((csubst0 i v c c2) \to (\forall (e: -C).((drop (S n0) O c e) \to (drop (S n0) O c2 e)))))))).(\lambda (k: -K).(\lambda (t: T).(\lambda (c2: C).(\lambda (v: T).(\lambda (H2: (csubst0 i -v (CHead c k t) c2)).(\lambda (e: C).(\lambda (H3: (drop (S n0) O (CHead c k -t) e)).(or3_ind (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k -j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda -(u2: T).(\lambda (j: nat).(subst0 j v t u2)))) (ex3_2 C nat (\lambda (_: -C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_: -nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j -v c c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: -nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: -nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda -(j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: -nat).(csubst0 j v c c3))))) (drop (S n0) O c2 e) (\lambda (H4: (ex3_2 T nat -(\lambda (_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: -T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: -nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: -nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead -c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))) (drop (S -n0) O c2 e) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H5: (eq nat i (s k -x1))).(\lambda (H6: (eq C c2 (CHead c k x0))).(\lambda (_: (subst0 x1 v t -x0)).(eq_ind_r C (CHead c k x0) (\lambda (c0: C).(drop (S n0) O c0 e)) (let -H8 \def (eq_ind nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0: -T).((csubst0 n1 v0 c c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (drop -(S n0) O c3 e0))))))) H1 (s k x1) H5) in (let H9 \def (eq_ind nat i (\lambda -(n1: nat).(lt n1 (S n0))) H0 (s k x1) H5) in (K_ind (\lambda (k0: K).((drop -(r k0 n0) O c e) \to (((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1) -v0 c c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3 -e0))))))) \to ((lt (s k0 x1) (S n0)) \to (drop (S n0) O (CHead c k0 x0) -e))))) (\lambda (b: B).(\lambda (H10: (drop (r (Bind b) n0) O c e)).(\lambda -(_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to -(\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3 -e0)))))))).(\lambda (_: (lt (s (Bind b) x1) (S n0))).(drop_drop (Bind b) n0 c -e H10 x0))))) (\lambda (f: F).(\lambda (H10: (drop (r (Flat f) n0) O c -e)).(\lambda (_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Flat f) x1) -v0 c c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3 -e0)))))))).(\lambda (H12: (lt (s (Flat f) x1) (S n0))).(or_ind (eq nat x1 O) -(ex2 nat (\lambda (m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0))) -(drop (S n0) O (CHead c (Flat f) x0) e) (\lambda (_: (eq nat x1 -O)).(drop_drop (Flat f) n0 c e H10 x0)) (\lambda (H13: (ex2 nat (\lambda (m: -nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda -(m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O -(CHead c (Flat f) x0) e) (\lambda (x: nat).(\lambda (_: (eq nat x1 (S -x))).(\lambda (_: (lt x n0)).(drop_drop (Flat f) n0 c e H10 x0)))) H13)) -(lt_gen_xS x1 n0 H12)))))) k (drop_gen_drop k c e t n0 H3) H8 H9))) c2 -H6)))))) H4)) (\lambda (H4: (ex3_2 C nat (\lambda (_: C).(\lambda (j: -nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead -c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c -c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k -j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda -(c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (drop (S n0) O c2 e) (\lambda -(x0: C).(\lambda (x1: nat).(\lambda (H5: (eq nat i (s k x1))).(\lambda (H6: -(eq C c2 (CHead x0 k t))).(\lambda (H7: (csubst0 x1 v c x0)).(eq_ind_r C -(CHead x0 k t) (\lambda (c0: C).(drop (S n0) O c0 e)) (let H8 \def (eq_ind -nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c -c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3 -e0))))))) H1 (s k x1) H5) in (let H9 \def (eq_ind nat i (\lambda (n1: -nat).(lt n1 (S n0))) H0 (s k x1) H5) in (K_ind (\lambda (k0: K).((drop (r k0 -n0) O c e) \to (((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c -c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3 -e0))))))) \to ((lt (s k0 x1) (S n0)) \to (drop (S n0) O (CHead x0 k0 t) -e))))) (\lambda (b: B).(\lambda (H10: (drop (r (Bind b) n0) O c e)).(\lambda -(_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to -(\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3 -e0)))))))).(\lambda (H12: (lt (s (Bind b) x1) (S n0))).(drop_drop (Bind b) n0 -x0 e (H x1 (lt_S_n x1 n0 H12) c x0 v H7 e H10) t))))) (\lambda (f: -F).(\lambda (H10: (drop (r (Flat f) n0) O c e)).(\lambda (H11: ((\forall (c3: -C).(\forall (v0: T).((csubst0 (s (Flat f) x1) v0 c c3) \to (\forall (e0: -C).((drop (S n0) O c e0) \to (drop (S n0) O c3 e0)))))))).(\lambda (H12: (lt -(s (Flat f) x1) (S n0))).(or_ind (eq nat x1 O) (ex2 nat (\lambda (m: nat).(eq -nat x1 (S m))) (\lambda (m: nat).(lt m n0))) (drop (S n0) O (CHead x0 (Flat -f) t) e) (\lambda (_: (eq nat x1 O)).(drop_drop (Flat f) n0 x0 e (H11 x0 v H7 -e H10) t)) (\lambda (H13: (ex2 nat (\lambda (m: nat).(eq nat x1 (S m))) -(\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda (m: nat).(eq nat x1 (S -m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O (CHead x0 (Flat f) t) e) -(\lambda (x: nat).(\lambda (_: (eq nat x1 (S x))).(\lambda (_: (lt x -n0)).(drop_drop (Flat f) n0 x0 e (H11 x0 v H7 e H10) t)))) H13)) (lt_gen_xS -x1 n0 H12)))))) k (drop_gen_drop k c e t n0 H3) H8 H9))) c2 H6)))))) H4)) -(\lambda (H4: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: -nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: -nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda -(j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: -nat).(csubst0 j v c c3)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_: -C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: -C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda -(_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: -C).(\lambda (j: nat).(csubst0 j v c c3)))) (drop (S n0) O c2 e) (\lambda (x0: -T).(\lambda (x1: C).(\lambda (x2: nat).(\lambda (H5: (eq nat i (s k -x2))).(\lambda (H6: (eq C c2 (CHead x1 k x0))).(\lambda (_: (subst0 x2 v t -x0)).(\lambda (H8: (csubst0 x2 v c x1)).(eq_ind_r C (CHead x1 k x0) (\lambda -(c0: C).(drop (S n0) O c0 e)) (let H9 \def (eq_ind nat i (\lambda (n1: -nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall -(e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3 e0))))))) H1 (s k x2) H5) -in (let H10 \def (eq_ind nat i (\lambda (n1: nat).(lt n1 (S n0))) H0 (s k x2) -H5) in (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to (((\forall (c3: -C).(\forall (v0: T).((csubst0 (s k0 x2) v0 c c3) \to (\forall (e0: C).((drop -(S n0) O c e0) \to (drop (S n0) O c3 e0))))))) \to ((lt (s k0 x2) (S n0)) \to -(drop (S n0) O (CHead x1 k0 x0) e))))) (\lambda (b: B).(\lambda (H11: (drop -(r (Bind b) n0) O c e)).(\lambda (_: ((\forall (c3: C).(\forall (v0: -T).((csubst0 (s (Bind b) x2) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c -e0) \to (drop (S n0) O c3 e0)))))))).(\lambda (H13: (lt (s (Bind b) x2) (S -n0))).(drop_drop (Bind b) n0 x1 e (H x2 (lt_S_n x2 n0 H13) c x1 v H8 e H11) -x0))))) (\lambda (f: F).(\lambda (H11: (drop (r (Flat f) n0) O c e)).(\lambda -(H12: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Flat f) x2) v0 c c3) -\to (\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3 -e0)))))))).(\lambda (H13: (lt (s (Flat f) x2) (S n0))).(or_ind (eq nat x2 O) -(ex2 nat (\lambda (m: nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m n0))) -(drop (S n0) O (CHead x1 (Flat f) x0) e) (\lambda (_: (eq nat x2 -O)).(drop_drop (Flat f) n0 x1 e (H12 x1 v H8 e H11) x0)) (\lambda (H14: (ex2 -nat (\lambda (m: nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m -n0)))).(ex2_ind nat (\lambda (m: nat).(eq nat x2 (S m))) (\lambda (m: -nat).(lt m n0)) (drop (S n0) O (CHead x1 (Flat f) x0) e) (\lambda (x: -nat).(\lambda (_: (eq nat x2 (S x))).(\lambda (_: (lt x n0)).(drop_drop (Flat -f) n0 x1 e (H12 x1 v H8 e H11) x0)))) H14)) (lt_gen_xS x2 n0 H13)))))) k -(drop_gen_drop k c e t n0 H3) H9 H10))) c2 H6)))))))) H4)) (csubst0_gen_head -k c c2 t v i H2))))))))))) c1)))))) n). -(* COMMENTS -Initial nodes: 3092 -END *) - -theorem csubst0_drop_gt_back: - \forall (n: nat).(\forall (i: nat).((lt i n) \to (\forall (c1: C).(\forall -(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n O -c2 e) \to (drop n O c1 e))))))))) -\def - \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (i: nat).((lt i n0) -\to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) -\to (\forall (e: C).((drop n0 O c2 e) \to (drop n0 O c1 e)))))))))) (\lambda -(i: nat).(\lambda (H: (lt i O)).(\lambda (c1: C).(\lambda (c2: C).(\lambda -(v: T).(\lambda (_: (csubst0 i v c1 c2)).(\lambda (e: C).(\lambda (_: (drop O -O c2 e)).(lt_x_O i H (drop O O c1 e)))))))))) (\lambda (n0: nat).(\lambda (H: -((\forall (i: nat).((lt i n0) \to (\forall (c1: C).(\forall (c2: C).(\forall -(v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n0 O c2 e) \to (drop -n0 O c1 e))))))))))).(\lambda (i: nat).(\lambda (H0: (lt i (S n0))).(\lambda -(c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v -c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c e))))))) -(\lambda (n1: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (H1: (csubst0 i -v (CSort n1) c2)).(\lambda (e: C).(\lambda (_: (drop (S n0) O c2 -e)).(csubst0_gen_sort c2 v i n1 H1 (drop (S n0) O (CSort n1) e)))))))) -(\lambda (c: C).(\lambda (H1: ((\forall (c2: C).(\forall (v: T).((csubst0 i v -c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c -e)))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (v: -T).(\lambda (H2: (csubst0 i v (CHead c k t) c2)).(\lambda (e: C).(\lambda -(H3: (drop (S n0) O c2 e)).(or3_ind (ex3_2 T nat (\lambda (_: T).(\lambda (j: -nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead -c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2)))) (ex3_2 C -nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: -C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: -nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: -C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: -C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda -(_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: -C).(\lambda (j: nat).(csubst0 j v c c3))))) (drop (S n0) O (CHead c k t) e) -(\lambda (H4: (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k -j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda -(u2: T).(\lambda (j: nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: -T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: -nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j -v t u2))) (drop (S n0) O (CHead c k t) e) (\lambda (x0: T).(\lambda (x1: -nat).(\lambda (H5: (eq nat i (s k x1))).(\lambda (H6: (eq C c2 (CHead c k -x0))).(\lambda (_: (subst0 x1 v t x0)).(let H8 \def (eq_ind C c2 (\lambda -(c0: C).(drop (S n0) O c0 e)) H3 (CHead c k x0) H6) in (let H9 \def (eq_ind -nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c -c3) \to (\forall (e0: C).((drop (S n0) O c3 e0) \to (drop (S n0) O c -e0))))))) H1 (s k x1) H5) in (let H10 \def (eq_ind nat i (\lambda (n1: -nat).(lt n1 (S n0))) H0 (s k x1) H5) in (K_ind (\lambda (k0: K).(((\forall -(c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c c3) \to (\forall (e0: -C).((drop (S n0) O c3 e0) \to (drop (S n0) O c e0))))))) \to ((lt (s k0 x1) -(S n0)) \to ((drop (r k0 n0) O c e) \to (drop (S n0) O (CHead c k0 t) e))))) -(\lambda (b: B).(\lambda (_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s -(Bind b) x1) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3 e0) \to (drop -(S n0) O c e0)))))))).(\lambda (_: (lt (s (Bind b) x1) (S n0))).(\lambda -(H13: (drop (r (Bind b) n0) O c e)).(drop_drop (Bind b) n0 c e H13 t))))) -(\lambda (f: F).(\lambda (_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s -(Flat f) x1) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3 e0) \to (drop -(S n0) O c e0)))))))).(\lambda (H12: (lt (s (Flat f) x1) (S n0))).(\lambda -(H13: (drop (r (Flat f) n0) O c e)).(or_ind (eq nat x1 O) (ex2 nat (\lambda -(m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0))) (drop (S n0) O -(CHead c (Flat f) t) e) (\lambda (_: (eq nat x1 O)).(drop_drop (Flat f) n0 c -e H13 t)) (\lambda (H14: (ex2 nat (\lambda (m: nat).(eq nat x1 (S m))) -(\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda (m: nat).(eq nat x1 (S -m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O (CHead c (Flat f) t) e) -(\lambda (x: nat).(\lambda (_: (eq nat x1 (S x))).(\lambda (_: (lt x -n0)).(drop_drop (Flat f) n0 c e H13 t)))) H14)) (lt_gen_xS x1 n0 H12)))))) k -H9 H10 (drop_gen_drop k c e x0 n0 H8)))))))))) H4)) (\lambda (H4: (ex3_2 C -nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: -C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: -nat).(csubst0 j v c c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: -nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead -c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (drop (S -n0) O (CHead c k t) e) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H5: (eq -nat i (s k x1))).(\lambda (H6: (eq C c2 (CHead x0 k t))).(\lambda (H7: -(csubst0 x1 v c x0)).(let H8 \def (eq_ind C c2 (\lambda (c0: C).(drop (S n0) -O c0 e)) H3 (CHead x0 k t) H6) in (let H9 \def (eq_ind nat i (\lambda (n1: -nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall -(e0: C).((drop (S n0) O c3 e0) \to (drop (S n0) O c e0))))))) H1 (s k x1) H5) -in (let H10 \def (eq_ind nat i (\lambda (n1: nat).(lt n1 (S n0))) H0 (s k x1) -H5) in (K_ind (\lambda (k0: K).(((\forall (c3: C).(\forall (v0: T).((csubst0 -(s k0 x1) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3 e0) \to (drop (S -n0) O c e0))))))) \to ((lt (s k0 x1) (S n0)) \to ((drop (r k0 n0) O x0 e) \to -(drop (S n0) O (CHead c k0 t) e))))) (\lambda (b: B).(\lambda (_: ((\forall -(c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to (\forall (e0: -C).((drop (S n0) O c3 e0) \to (drop (S n0) O c e0)))))))).(\lambda (H12: (lt -(s (Bind b) x1) (S n0))).(\lambda (H13: (drop (r (Bind b) n0) O x0 -e)).(drop_drop (Bind b) n0 c e (H x1 (lt_S_n x1 n0 H12) c x0 v H7 e H13) -t))))) (\lambda (f: F).(\lambda (H11: ((\forall (c3: C).(\forall (v0: -T).((csubst0 (s (Flat f) x1) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3 -e0) \to (drop (S n0) O c e0)))))))).(\lambda (H12: (lt (s (Flat f) x1) (S -n0))).(\lambda (H13: (drop (r (Flat f) n0) O x0 e)).(or_ind (eq nat x1 O) -(ex2 nat (\lambda (m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0))) -(drop (S n0) O (CHead c (Flat f) t) e) (\lambda (_: (eq nat x1 O)).(drop_drop -(Flat f) n0 c e (H11 x0 v H7 e H13) t)) (\lambda (H14: (ex2 nat (\lambda (m: -nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda -(m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O -(CHead c (Flat f) t) e) (\lambda (x: nat).(\lambda (_: (eq nat x1 (S -x))).(\lambda (_: (lt x n0)).(drop_drop (Flat f) n0 c e (H11 x0 v H7 e H13) -t)))) H14)) (lt_gen_xS x1 n0 H12)))))) k H9 H10 (drop_gen_drop k x0 e t n0 -H8)))))))))) H4)) (\lambda (H4: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: -C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: -C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda -(_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: -C).(\lambda (j: nat).(csubst0 j v c c3)))))).(ex4_3_ind T C nat (\lambda (_: -T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: -T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda -(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: -T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (drop (S n0) O -(CHead c k t) e) (\lambda (x0: T).(\lambda (x1: C).(\lambda (x2: -nat).(\lambda (H5: (eq nat i (s k x2))).(\lambda (H6: (eq C c2 (CHead x1 k -x0))).(\lambda (_: (subst0 x2 v t x0)).(\lambda (H8: (csubst0 x2 v c -x1)).(let H9 \def (eq_ind C c2 (\lambda (c0: C).(drop (S n0) O c0 e)) H3 -(CHead x1 k x0) H6) in (let H10 \def (eq_ind nat i (\lambda (n1: -nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall -(e0: C).((drop (S n0) O c3 e0) \to (drop (S n0) O c e0))))))) H1 (s k x2) H5) -in (let H11 \def (eq_ind nat i (\lambda (n1: nat).(lt n1 (S n0))) H0 (s k x2) -H5) in (K_ind (\lambda (k0: K).(((\forall (c3: C).(\forall (v0: T).((csubst0 -(s k0 x2) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3 e0) \to (drop (S -n0) O c e0))))))) \to ((lt (s k0 x2) (S n0)) \to ((drop (r k0 n0) O x1 e) \to -(drop (S n0) O (CHead c k0 t) e))))) (\lambda (b: B).(\lambda (_: ((\forall -(c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x2) v0 c c3) \to (\forall (e0: -C).((drop (S n0) O c3 e0) \to (drop (S n0) O c e0)))))))).(\lambda (H13: (lt -(s (Bind b) x2) (S n0))).(\lambda (H14: (drop (r (Bind b) n0) O x1 -e)).(drop_drop (Bind b) n0 c e (H x2 (lt_S_n x2 n0 H13) c x1 v H8 e H14) -t))))) (\lambda (f: F).(\lambda (H12: ((\forall (c3: C).(\forall (v0: -T).((csubst0 (s (Flat f) x2) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3 -e0) \to (drop (S n0) O c e0)))))))).(\lambda (H13: (lt (s (Flat f) x2) (S -n0))).(\lambda (H14: (drop (r (Flat f) n0) O x1 e)).(or_ind (eq nat x2 O) -(ex2 nat (\lambda (m: nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m n0))) -(drop (S n0) O (CHead c (Flat f) t) e) (\lambda (_: (eq nat x2 O)).(drop_drop -(Flat f) n0 c e (H12 x1 v H8 e H14) t)) (\lambda (H15: (ex2 nat (\lambda (m: -nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda -(m: nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O -(CHead c (Flat f) t) e) (\lambda (x: nat).(\lambda (_: (eq nat x2 (S -x))).(\lambda (_: (lt x n0)).(drop_drop (Flat f) n0 c e (H12 x1 v H8 e H14) -t)))) H15)) (lt_gen_xS x2 n0 H13)))))) k H10 H11 (drop_gen_drop k x1 e x0 n0 -H9)))))))))))) H4)) (csubst0_gen_head k c c2 t v i H2))))))))))) c1)))))) n). -(* COMMENTS -Initial nodes: 2989 -END *) - -theorem csubst0_drop_lt: - \forall (n: nat).(\forall (i: nat).((lt n i) \to (\forall (c1: C).(\forall -(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n O -c1 e) \to (or4 (drop n O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: -K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e0 k -w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (s k n)) v u w)))))) (ex3_4 K C C T (\lambda (k: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k -u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop n O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1 e2)))))) (ex4_5 K C C -T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda -(_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e2 k w))))))) -(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (s k n)) v u w)))))) (\lambda (k: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k -n)) v e1 e2)))))))))))))))) -\def - \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (i: nat).((lt n0 i) -\to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) -\to (\forall (e: C).((drop n0 O c1 e) \to (or4 (drop n0 O c2 e) (ex3_4 K C T -T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda -(w: T).(drop n0 O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n0)) v u w)))))) -(ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop n0 O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k n0)) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 -O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n0)) v u w)))))) -(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus i (s k n0)) v e1 e2))))))))))))))))) (\lambda (i: -nat).(\lambda (_: (lt O i)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (v: -T).(\lambda (H0: (csubst0 i v c1 c2)).(\lambda (e: C).(\lambda (H1: (drop O O -c1 e)).(eq_ind C c1 (\lambda (c: C).(or4 (drop O O c2 c) (ex3_4 K C T T -(\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c -(CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda -(w: T).(drop O O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k O)) v u w)))))) -(ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop O O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k O)) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 k u))))))) (\lambda (k: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O -c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k O)) v u w)))))) -(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus i (s k O)) v e1 e2))))))))) (csubst0_ind (\lambda (n0: -nat).(\lambda (t: T).(\lambda (c: C).(\lambda (c0: C).(or4 (drop O O c0 c) -(ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: -T).(eq C c (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop O O c0 (CHead e0 k w)))))) (\lambda (k: K).(\lambda -(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n0 (s k O)) t u w)))))) -(ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop O O c0 (CHead e2 k u)))))) (\lambda (k: K).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n0 (s k O)) t e1 -e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 k u))))))) (\lambda (k: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O -c0 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n0 (s k O)) t u w)))))) -(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus n0 (s k O)) t e1 e2)))))))))))) (\lambda (k: -K).(\lambda (i0: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: -T).(\lambda (H2: (subst0 i0 v0 u1 u2)).(\lambda (c: C).(let H3 \def (eq_ind_r -nat i0 (\lambda (n0: nat).(subst0 n0 v0 u1 u2)) H2 (minus (s k i0) (s k O)) -(s_arith0 k i0)) in (or4_intro1 (drop O O (CHead c k u2) (CHead c k u1)) -(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: -T).(eq C (CHead c k u1) (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c k u2) (CHead e0 k0 -w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus (s k i0) (s k0 O)) v0 u w)))))) (ex3_4 K C C T (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c k u1) -(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop O O (CHead c k u2) (CHead e2 k0 u)))))) (\lambda -(k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s -k i0) (s k0 O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c k u1) -(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c k u2) (CHead e2 k0 -w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 O)) v0 u w)))))) (\lambda -(k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: -T).(csubst0 (minus (s k i0) (s k0 O)) v0 e1 e2))))))) (ex3_4_intro K C T T -(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C -(CHead c k u1) (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c k u2) (CHead e0 k0 -w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus (s k i0) (s k0 O)) v0 u w))))) k c u1 u2 (refl_equal C -(CHead c k u1)) (drop_refl (CHead c k u2)) H3)))))))))) (\lambda (k: -K).(\lambda (i0: nat).(\lambda (c3: C).(\lambda (c4: C).(\lambda (v0: -T).(\lambda (H2: (csubst0 i0 v0 c3 c4)).(\lambda (H3: (or4 (drop O O c4 c3) -(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: -T).(eq C c3 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda -(_: T).(\lambda (w: T).(drop O O c4 (CHead e0 k0 w)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k0 -O)) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u: T).(eq C c3 (CHead e1 k0 u)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O c4 (CHead e2 k0 -u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus i0 (s k0 O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 -(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e2 k0 w))))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i0 (s k0 O)) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k0 -O)) v0 e1 e2))))))))).(\lambda (u: T).(let H4 \def (eq_ind_r nat i0 (\lambda -(n0: nat).(csubst0 n0 v0 c3 c4)) H2 (minus (s k i0) (s k O)) (s_arith0 k i0)) -in (let H5 \def (eq_ind_r nat i0 (\lambda (n0: nat).(or4 (drop O O c4 c3) -(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda -(_: T).(eq C c3 (CHead e0 k0 u0)))))) (\lambda (k0: K).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e0 k0 w)))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus n0 (s -k0 O)) v0 u0 w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u0: T).(eq C c3 (CHead e1 k0 u0)))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop O O c4 (CHead -e2 k0 u0)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 (minus n0 (s k0 O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq -C c3 (CHead e1 k0 u0))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e2 k0 w))))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda -(w: T).(subst0 (minus n0 (s k0 O)) v0 u0 w)))))) (\lambda (k0: K).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n0 -(s k0 O)) v0 e1 e2))))))))) H3 (minus (s k i0) (s k O)) (s_arith0 k i0)) in -(or4_intro2 (drop O O (CHead c4 k u) (CHead c3 k u)) (ex3_4 K C T T (\lambda -(k0: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 k -u) (CHead e0 k0 u0)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop O O (CHead c4 k u) (CHead e0 k0 w)))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k -i0) (s k0 O)) v0 u0 w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead c3 k u) (CHead e1 k0 -u0)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: -T).(drop O O (CHead c4 k u) (CHead e2 k0 u0)))))) (\lambda (k0: K).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 O)) -v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 k u) (CHead e1 k0 -u0))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop O O (CHead c4 k u) (CHead e2 k0 w))))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: -T).(subst0 (minus (s k i0) (s k0 O)) v0 u0 w)))))) (\lambda (k0: K).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k -i0) (s k0 O)) v0 e1 e2))))))) (ex3_4_intro K C C T (\lambda (k0: K).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead c3 k u) (CHead e1 k0 -u0)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: -T).(drop O O (CHead c4 k u) (CHead e2 k0 u0)))))) (\lambda (k0: K).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 O)) -v0 e1 e2))))) k c3 c4 u (refl_equal C (CHead c3 k u)) (drop_refl (CHead c4 k -u)) H4)))))))))))) (\lambda (k: K).(\lambda (i0: nat).(\lambda (v0: -T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 i0 v0 u1 -u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (H3: (csubst0 i0 v0 c3 -c4)).(\lambda (_: (or4 (drop O O c4 c3) (ex3_4 K C T T (\lambda (k0: -K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e0 k0 -u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(drop O O c4 (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k0 O)) v0 u w)))))) -(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c3 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u: T).(drop O O c4 (CHead e2 k0 u)))))) (\lambda (k0: -K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (s k0 -O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 k0 -u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop O O c4 (CHead e2 k0 w))))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus i0 (s k0 O)) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k0 O)) v0 e1 -e2))))))))).(let H5 \def (eq_ind_r nat i0 (\lambda (n0: nat).(subst0 n0 v0 u1 -u2)) H2 (minus (s k i0) (s k O)) (s_arith0 k i0)) in (let H6 \def (eq_ind_r -nat i0 (\lambda (n0: nat).(csubst0 n0 v0 c3 c4)) H3 (minus (s k i0) (s k O)) -(s_arith0 k i0)) in (or4_intro3 (drop O O (CHead c4 k u2) (CHead c3 k u1)) -(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: -T).(eq C (CHead c3 k u1) (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 k u2) (CHead e0 k0 -w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus (s k i0) (s k0 O)) v0 u w)))))) (ex3_4 K C C T (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c3 k -u1) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop O O (CHead c4 k u2) (CHead e2 k0 u)))))) (\lambda -(k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s -k i0) (s k0 O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 k u1) -(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 k u2) (CHead e2 k0 -w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 O)) v0 u w)))))) (\lambda -(k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: -T).(csubst0 (minus (s k i0) (s k0 O)) v0 e1 e2))))))) (ex4_5_intro K C C T T -(\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda -(_: T).(eq C (CHead c3 k u1) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 k -u2) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 O)) v0 u -w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 O)) v0 e1 e2)))))) k c3 c4 -u1 u2 (refl_equal C (CHead c3 k u1)) (drop_refl (CHead c4 k u2)) H5 -H6)))))))))))))) i v c1 c2 H0) e (drop_gen_refl c1 e H1)))))))))) (\lambda -(n0: nat).(\lambda (IHn: ((\forall (i: nat).((lt n0 i) \to (\forall (c1: -C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: -C).((drop n0 O c1 e) \to (or4 (drop n0 O c2 e) (ex3_4 K C T T (\lambda (k: -K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k -u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(drop n0 O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda -(u: T).(\lambda (w: T).(subst0 (minus i (s k n0)) v u w)))))) (ex3_4 K C C T -(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda -(u: T).(drop n0 O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k n0)) v e1 e2)))))) -(ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c2 (CHead -e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus i (s k n0)) v u w)))))) (\lambda (k: -K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -(minus i (s k n0)) v e1 e2)))))))))))))))))).(\lambda (i: nat).(\lambda (H: -(lt (S n0) i)).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: -C).(\forall (v: T).((csubst0 i v c c2) \to (\forall (e: C).((drop (S n0) O c -e) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: -K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead -e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k -u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w)))))) -(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus i (s k (S n0))) v e1 e2)))))))))))))) (\lambda (n1: -nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (_: (csubst0 i v (CSort n1) -c2)).(\lambda (e: C).(\lambda (H1: (drop (S n0) O (CSort n1) e)).(and3_ind -(eq C e (CSort n1)) (eq nat (S n0) O) (eq nat O O) (or4 (drop (S n0) O c2 e) -(ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: -T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: -K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S -n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda -(_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) -(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 -(minus i (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) -(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k (S -n0))) v e1 e2)))))))) (\lambda (H2: (eq C e (CSort n1))).(\lambda (H3: (eq -nat (S n0) O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n1) (\lambda (c: -C).(or4 (drop (S n0) O c2 c) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 k u)))))) (\lambda (k: -K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead -e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 k -u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 k u))))))) (\lambda (k: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w)))))) -(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus i (s k (S n0))) v e1 e2))))))))) (let H5 \def (eq_ind -nat (S n0) (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) -with [O \Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind -(or4 (drop (S n0) O c2 (CSort n1)) (ex3_4 K C T T (\lambda (k: K).(\lambda -(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CSort n1) (CHead e0 k u)))))) -(\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) -O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w)))))) (ex3_4 K C C T -(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CSort -n1) (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: -K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k -(S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CSort n1) (CHead e1 -k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k: -K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus i (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1 -e2)))))))) H5)) e H2)))) (drop_gen_sort n1 (S n0) O e H1)))))))) (\lambda (c: -C).(\lambda (H0: ((\forall (c2: C).(\forall (v: T).((csubst0 i v c c2) \to -(\forall (e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C -T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda -(w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w)))))) -(ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: -K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k -(S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k -u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k: -K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus i (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1 -e2))))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda -(v: T).(\lambda (H1: (csubst0 i v (CHead c k t) c2)).(\lambda (e: C).(\lambda -(H2: (drop (S n0) O (CHead c k t) e)).(or3_ind (ex3_2 T nat (\lambda (_: -T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: -nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j -v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k -j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda -(c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_: -T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: -T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda -(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: -T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))) (or4 (drop (S -n0) O c2 e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda -(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k0 -w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 -u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O c2 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k0 (S n0))) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O c2 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S -n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 (minus i (s k0 (S n0))) v e1 e2)))))))) -(\lambda (H3: (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k -j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda -(u2: T).(\lambda (j: nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: -T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: -nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j -v t u2))) (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k0: K).(\lambda -(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda -(k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 -(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (ex3_4 K C C T -(\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k0 u)))))) (\lambda (k0: -K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k0 -(S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 -u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k0 w))))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus i (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k0 -(S n0))) v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H4: -(eq nat i (s k x1))).(\lambda (H5: (eq C c2 (CHead c k x0))).(\lambda (_: -(subst0 x1 v t x0)).(eq_ind_r C (CHead c k x0) (\lambda (c0: C).(or4 (drop (S -n0) O c0 e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda -(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e0 k0 -w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 -u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O c0 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k0 (S n0))) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O c0 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S -n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 (minus i (s k0 (S n0))) v e1 e2))))))))) (let -H7 \def (eq_ind nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0: -T).((csubst0 n1 v0 c c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (or4 -(drop (S n0) O c3 e0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c3 -(CHead e1 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus n1 (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C -T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e0 -(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O c3 (CHead e2 k0 u)))))) (\lambda (k0: -K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n1 (s k0 -(S n0))) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 -u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus n1 (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 (s k0 -(S n0))) v0 e1 e2)))))))))))))) H0 (s k x1) H4) in (let H8 \def (eq_ind nat i -(\lambda (n1: nat).(lt (S n0) n1)) H (s k x1) H4) in (eq_ind_r nat (s k x1) -(\lambda (n1: nat).(or4 (drop (S n0) O (CHead c k x0) e) (ex3_4 K C T T -(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead c k x0) (CHead e0 k0 w)))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus n1 (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O -(CHead c k x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n1 (s k0 (S n0))) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead c k x0) (CHead e2 k0 w))))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus n1 (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 (s k0 -(S n0))) v e1 e2))))))))) (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to -(((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c c3) \to (\forall -(e0: C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T -(\lambda (k1: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 -(CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k1 w)))))) (\lambda (k1: -K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 x1) -(s k1 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k1 u)))))) (\lambda -(k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 -(CHead e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus (s k0 x1) (s k1 (S n0))) v0 e1 e2)))))) -(ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq C e0 (CHead e1 k1 u))))))) (\lambda (k1: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O c3 (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 x1) (s k1 (S n0))) v0 -u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (minus (s k0 x1) (s k1 (S n0))) v0 e1 -e2)))))))))))))) \to ((lt (S n0) (s k0 x1)) \to (or4 (drop (S n0) O (CHead c -k0 x0) e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda -(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c k0 x0) (CHead -e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus (s k0 x1) (s k1 (S n0))) v u w)))))) (ex3_4 K C C T -(\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O (CHead c k0 x0) (CHead e2 k1 u)))))) -(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 -(minus (s k0 x1) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c k0 x0) (CHead e2 -k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus (s k0 x1) (s k1 (S n0))) v u w)))))) -(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus (s k0 x1) (s k1 (S n0))) v e1 e2)))))))))))) (\lambda -(b: B).(\lambda (H9: (drop (r (Bind b) n0) O c e)).(\lambda (_: ((\forall -(c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to (\forall (e0: -C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T -(\lambda (k0: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 -(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) -x1) (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 -(CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v0 e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u))))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) -x1) (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 -(S n0))) v0 e1 e2))))))))))))))).(\lambda (_: (lt (S n0) (s (Bind b) -x1))).(or4_intro0 (drop (S n0) O (CHead c (Bind b) x0) e) (ex3_4 K C T T -(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead c (Bind b) x0) (CHead e0 k0 w)))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 -u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O (CHead c (Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0: -K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind -b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c (Bind b) x0) -(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S -n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1 -e2))))))) (drop_drop (Bind b) n0 c e H9 x0)))))) (\lambda (f: F).(\lambda -(H9: (drop (r (Flat f) n0) O c e)).(\lambda (_: ((\forall (c3: C).(\forall -(v0: T).((csubst0 (s (Flat f) x1) v0 c c3) \to (\forall (e0: C).((drop (S n0) -O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T (\lambda (k0: -K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 -u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S -n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 (CHead -e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v0 e1 e2)))))) (ex4_5 K C C -T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c3 -(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S -n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S -n0))) v0 e1 e2))))))))))))))).(\lambda (_: (lt (S n0) (s (Flat f) -x1))).(or4_intro0 (drop (S n0) O (CHead c (Flat f) x0) e) (ex3_4 K C T T -(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead c (Flat f) x0) (CHead e0 k0 w)))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 -u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O (CHead c (Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: -K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat -f) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c (Flat f) x0) -(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S -n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1 -e2))))))) (drop_drop (Flat f) n0 c e H9 x0)))))) k (drop_gen_drop k c e t n0 -H2) H7 H8) i H4))) c2 H5)))))) H3)) (\lambda (H3: (ex3_2 C nat (\lambda (_: -C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_: -nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j -v c c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k -j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda -(c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (or4 (drop (S n0) O c2 e) -(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: -T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda -(_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k0 w)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S -n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda -(_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k0 -u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus i (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq -C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k0 w))))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k0 -(S n0))) v e1 e2)))))))) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H4: -(eq nat i (s k x1))).(\lambda (H5: (eq C c2 (CHead x0 k t))).(\lambda (H6: -(csubst0 x1 v c x0)).(eq_ind_r C (CHead x0 k t) (\lambda (c0: C).(or4 (drop -(S n0) O c0 e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda -(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e0 k0 -w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 -u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O c0 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k0 (S n0))) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O c0 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S -n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 (minus i (s k0 (S n0))) v e1 e2))))))))) (let -H7 \def (eq_ind nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0: -T).((csubst0 n1 v0 c c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (or4 -(drop (S n0) O c3 e0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c3 -(CHead e1 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus n1 (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C -T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e0 -(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O c3 (CHead e2 k0 u)))))) (\lambda (k0: -K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n1 (s k0 -(S n0))) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 -u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus n1 (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 (s k0 -(S n0))) v0 e1 e2)))))))))))))) H0 (s k x1) H4) in (let H8 \def (eq_ind nat i -(\lambda (n1: nat).(lt (S n0) n1)) H (s k x1) H4) in (eq_ind_r nat (s k x1) -(\lambda (n1: nat).(or4 (drop (S n0) O (CHead x0 k t) e) (ex3_4 K C T T -(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x0 k t) (CHead e0 k0 w)))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus n1 (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O -(CHead x0 k t) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n1 (s k0 (S n0))) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x0 k t) (CHead e2 k0 w))))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus n1 (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 (s k0 -(S n0))) v e1 e2))))))))) (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to -(((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c c3) \to (\forall -(e0: C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T -(\lambda (k1: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 -(CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k1 w)))))) (\lambda (k1: -K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 x1) -(s k1 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k1 u)))))) (\lambda -(k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 -(CHead e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus (s k0 x1) (s k1 (S n0))) v0 e1 e2)))))) -(ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq C e0 (CHead e1 k1 u))))))) (\lambda (k1: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O c3 (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 x1) (s k1 (S n0))) v0 -u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (minus (s k0 x1) (s k1 (S n0))) v0 e1 -e2)))))))))))))) \to ((lt (S n0) (s k0 x1)) \to (or4 (drop (S n0) O (CHead x0 -k0 t) e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda -(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 k0 t) (CHead -e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus (s k0 x1) (s k1 (S n0))) v u w)))))) (ex3_4 K C C T -(\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O (CHead x0 k0 t) (CHead e2 k1 u)))))) -(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 -(minus (s k0 x1) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 k0 t) (CHead e2 -k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus (s k0 x1) (s k1 (S n0))) v u w)))))) -(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus (s k0 x1) (s k1 (S n0))) v e1 e2)))))))))))) (\lambda -(b: B).(\lambda (H9: (drop (r (Bind b) n0) O c e)).(\lambda (_: ((\forall -(c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to (\forall (e0: -C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T -(\lambda (k0: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 -(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) -x1) (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 -(CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v0 e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u))))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) -x1) (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 -(S n0))) v0 e1 e2))))))))))))))).(\lambda (H11: (lt (S n0) (s (Bind b) -x1))).(let H12 \def (IHn x1 (le_S_n (S n0) x1 H11) c x0 v H6 e H9) in -(or4_ind (drop n0 O x0 e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: -K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x0 (CHead e0 -k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus x1 (s k0 n0)) v u w)))))) (ex3_4 K C C T (\lambda (k0: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 -u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop n0 O x0 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus x1 (s k0 n0)) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop n0 O x0 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x1 (s k0 -n0)) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 (minus x1 (s k0 n0)) v e1 e2))))))) (or4 -(drop (S n0) O (CHead x0 (Bind b) t) e) (ex3_4 K C T T (\lambda (k0: -K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 -u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 k0 w)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) -x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 -(Bind b) t) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s -(Bind b) x1) (s k0 (S n0))) v e1 e2)))))))) (\lambda (H13: (drop n0 O x0 -e)).(or4_intro0 (drop (S n0) O (CHead x0 (Bind b) t) e) (ex3_4 K C T T -(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 k0 w)))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 -u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 u)))))) (\lambda (k0: -K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind -b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) -(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S -n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1 -e2))))))) (drop_drop (Bind b) n0 x0 e H13 t))) (\lambda (H13: (ex3_4 K C T T -(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop n0 O x0 (CHead e0 k0 w)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x1 (s k0 -n0)) v u w))))))).(ex3_4_ind K C T T (\lambda (k0: K).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: -K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x0 (CHead e0 -k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus x1 (s k0 n0)) v u w))))) (or4 (drop (S n0) O (CHead x0 -(Bind b) t) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda -(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) -(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) -(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 -u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C -T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead -x0 (Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) -x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 -(S n0))) v e1 e2)))))))) (\lambda (x2: K).(\lambda (x3: C).(\lambda (x4: -T).(\lambda (x5: T).(\lambda (H14: (eq C e (CHead x3 x2 x4))).(\lambda (H15: -(drop n0 O x0 (CHead x3 x2 x5))).(\lambda (H16: (subst0 (minus x1 (s x2 n0)) -v x4 x5)).(eq_ind_r C (CHead x3 x2 x4) (\lambda (c0: C).(or4 (drop (S n0) O -(CHead x0 (Bind b) t) c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 k0 u)))))) (\lambda -(k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O -(CHead x0 (Bind b) t) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S -n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u: T).(eq C c0 (CHead e1 k0 u)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 -(Bind b) t) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 k0 u))))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s -(Bind b) x1) (s k0 (S n0))) v e1 e2))))))))) (or4_intro1 (drop (S n0) O -(CHead x0 (Bind b) t) (CHead x3 x2 x4)) (ex3_4 K C T T (\lambda (k0: -K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x4) -(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 k0 w)))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 x2 -x4) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 u)))))) -(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 -(minus (s (Bind b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq -C (CHead x3 x2 x4) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 -(Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda -(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 -(S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S -n0))) v e1 e2))))))) (ex3_4_intro K C T T (\lambda (k0: K).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x4) (CHead e0 k0 -u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 k0 w)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) -x1) (s k0 (S n0))) v u w))))) x2 x3 x4 x5 (refl_equal C (CHead x3 x2 x4)) -(drop_drop (Bind b) n0 x0 (CHead x3 x2 x5) H15 t) (eq_ind_r nat (S (s x2 n0)) -(\lambda (n1: nat).(subst0 (minus (s 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(e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S -n0))) v e1 e2))))))) (ex3_4_intro K C C T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 x2 x5) (CHead e1 k0 -u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 u)))))) (\lambda (k0: -K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind -b) x1) (s k0 (S n0))) v e1 e2))))) x2 x3 x4 x5 (refl_equal C (CHead x3 x2 -x5)) (drop_drop (Bind b) n0 x0 (CHead x4 x2 x5) H15 t) (eq_ind_r nat (S (s x2 -n0)) (\lambda (n1: nat).(csubst0 (minus (s (Bind b) x1) n1) v x3 x4)) H16 (s -x2 (S n0)) (s_S x2 n0)))) e H14)))))))) H13)) (\lambda (H13: (ex4_5 K C C T T -(\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda -(_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x0 (CHead e2 -k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus x1 (s k0 n0)) v u w)))))) (\lambda (k0: -K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -(minus x1 (s k0 n0)) v e1 e2)))))))).(ex4_5_ind K C C T T (\lambda (k0: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x0 (CHead e2 k0 w))))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus x1 (s k0 n0)) v u w)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus x1 (s k0 -n0)) v e1 e2)))))) (or4 (drop (S n0) O (CHead x0 (Bind b) t) e) (ex3_4 K C T -T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O 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-(e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 -u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C -T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(eq C c0 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead -x0 (Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) -x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 -(S n0))) v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead x0 (Bind b) t) -(CHead x3 x2 x5)) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda -(u: T).(\lambda (_: T).(eq C (CHead x3 x2 x5) (CHead e0 k0 u)))))) (\lambda -(k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O -(CHead x0 (Bind b) t) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S -n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u: T).(eq C (CHead x3 x2 x5) (CHead e1 k0 u)))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O -(CHead x0 (Bind b) t) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S -n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x5) -(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) -(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind 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K).(\lambda (e0: C).(\lambda -(u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda -(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) -(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) -(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 -u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C -T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead -x0 (Flat f) t) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda 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(\lambda (k0: K).(\lambda (_: C).(\lambda -(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 -(S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S -n0))) v e1 e2))))))) (ex3_4_intro K C C T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 x2 x5) (CHead e1 k0 -u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 u)))))) (\lambda (k0: -K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat -f) x1) (s k0 (S n0))) v e1 e2))))) x2 x3 x4 x5 (refl_equal C (CHead x3 x2 -x5)) (drop_drop (Flat f) n0 x0 (CHead x4 x2 x5) H15 t) H16)) e H14)))))))) -H13)) (\lambda (H13: (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 -u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O x0 (CHead e2 k0 w))))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus x1 (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus x1 (s k0 -(S n0))) v e1 e2)))))))).(ex4_5_ind K C C T T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 -u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O x0 (CHead e2 k0 w))))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus x1 (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus x1 (s k0 -(S n0))) v e1 e2)))))) (or4 (drop (S n0) O (CHead x0 (Flat f) t) e) (ex3_4 K -C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C -e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 k0 w)))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 -u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 u)))))) (\lambda (k0: -K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat -f) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) -(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S -n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1 -e2)))))))) (\lambda (x2: K).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: -T).(\lambda (x6: T).(\lambda (H14: (eq C e (CHead x3 x2 x5))).(\lambda (H15: -(drop (S n0) O x0 (CHead x4 x2 x6))).(\lambda (H16: (subst0 (minus x1 (s x2 -(S n0))) v x5 x6)).(\lambda (H17: (csubst0 (minus x1 (s x2 (S n0))) v x3 -x4)).(eq_ind_r C (CHead x3 x2 x5) (\lambda (c0: C).(or4 (drop (S n0) O (CHead -x0 (Flat f) t) c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda -(u: T).(\lambda (_: T).(eq C c0 (CHead e0 k0 u)))))) (\lambda (k0: -K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 -(Flat f) t) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda -(u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S n0))) v u -w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C c0 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda -(_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) -(CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 k0 u))))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 w))))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s -(Flat f) x1) (s k0 (S n0))) v e1 e2))))))))) (or4_intro3 (drop (S n0) O -(CHead x0 (Flat f) t) (CHead x3 x2 x5)) (ex3_4 K C T T (\lambda (k0: -K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x5) -(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 k0 w)))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 x2 -x5) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 u)))))) -(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 -(minus (s (Flat f) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq -C (CHead x3 x2 x5) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 -(Flat f) t) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda -(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 -(S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S -n0))) v e1 e2))))))) (ex4_5_intro K C C T T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x5) -(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) -(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S -n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1 -e2)))))) x2 x3 x4 x5 x6 (refl_equal C (CHead x3 x2 x5)) (drop_drop (Flat f) -n0 x0 (CHead x4 x2 x6) H15 t) H16 H17)) e H14)))))))))) H13)) H12)))))) k -(drop_gen_drop k c e t n0 H2) H7 H8) i H4))) c2 H5)))))) H3)) (\lambda (H3: -(ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s -k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead -c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t -u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c -c3)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: -nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: -nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda -(j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: -nat).(csubst0 j v c c3)))) (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda -(k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 -u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O c2 (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S n0))) v u -w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k0 u)))))) -(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 -(minus i (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k0 w))))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k0 -(S n0))) v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: C).(\lambda (x2: -nat).(\lambda (H4: (eq nat i (s k x2))).(\lambda (H5: (eq C c2 (CHead x1 k -x0))).(\lambda (_: (subst0 x2 v t x0)).(\lambda (H7: (csubst0 x2 v c -x1)).(eq_ind_r C (CHead x1 k x0) (\lambda (c0: C).(or4 (drop (S n0) O c0 e) -(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: -T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda -(_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e0 k0 w)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S -n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda -(_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c0 (CHead e2 k0 -u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus i (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq -C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 k0 w))))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k0 -(S n0))) v e1 e2))))))))) (let H8 \def (eq_ind nat i (\lambda (n1: -nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall -(e0: C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T -(\lambda (k0: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 -(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n1 (s k0 (S -n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 (CHead -e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus n1 (s k0 (S n0))) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq -C e0 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c3 (CHead e2 k0 w))))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus n1 (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 -(s k0 (S n0))) v0 e1 e2)))))))))))))) H0 (s k x2) H4) in (let H9 \def (eq_ind -nat i (\lambda (n1: nat).(lt (S n0) n1)) H (s k x2) H4) in (eq_ind_r nat (s k -x2) (\lambda (n1: nat).(or4 (drop (S n0) O (CHead x1 k x0) e) (ex3_4 K C T T -(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 k x0) (CHead e0 k0 w)))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus n1 (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O -(CHead x1 k x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n1 (s k0 (S n0))) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x1 k x0) (CHead e2 k0 w))))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus n1 (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 (s k0 -(S n0))) v e1 e2))))))))) (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to -(((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x2) v0 c c3) \to (\forall -(e0: C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T -(\lambda (k1: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 -(CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k1 w)))))) (\lambda (k1: -K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 x2) -(s k1 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k1 u)))))) (\lambda -(k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 -(CHead e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus (s k0 x2) (s k1 (S n0))) v0 e1 e2)))))) -(ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq C e0 (CHead e1 k1 u))))))) (\lambda (k1: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O c3 (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 x2) (s k1 (S n0))) v0 -u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (minus (s k0 x2) (s k1 (S n0))) v0 e1 -e2)))))))))))))) \to ((lt (S n0) (s k0 x2)) \to (or4 (drop (S n0) O (CHead x1 -k0 x0) e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda -(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 k0 x0) -(CHead e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus (s k0 x2) (s k1 (S n0))) v u w)))))) (ex3_4 -K C C T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq -C e (CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O (CHead x1 k0 x0) (CHead e2 k1 u)))))) -(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 -(minus (s k0 x2) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 k0 x0) (CHead e2 -k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus (s k0 x2) (s k1 (S n0))) v u w)))))) -(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus (s k0 x2) (s k1 (S n0))) v e1 e2)))))))))))) (\lambda -(b: B).(\lambda (H10: (drop (r (Bind b) n0) O c e)).(\lambda (_: ((\forall -(c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x2) v0 c c3) \to (\forall (e0: -C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T -(\lambda (k0: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 -(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) -x2) (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 -(CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v0 e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u))))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda 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(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus x2 (s k0 n0)) v u w))))) (or4 (drop (S n0) O (CHead x1 -(Bind b) x0) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda -(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) -(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) -(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 -u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C -T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead -x1 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) -x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 -(S n0))) v e1 e2)))))))) (\lambda (x3: K).(\lambda (x4: C).(\lambda (x5: -T).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x4 x3 x5))).(\lambda (H16: -(drop n0 O x1 (CHead x4 x3 x6))).(\lambda (H17: (subst0 (minus x2 (s x3 n0)) -v x5 x6)).(eq_ind_r C (CHead x4 x3 x5) (\lambda (c0: C).(or4 (drop (S n0) O -(CHead x1 (Bind b) x0) c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 k0 u)))))) (\lambda -(k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O -(CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S -n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u: T).(eq C c0 (CHead e1 k0 u)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 -(Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 k0 u))))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: 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(n1: nat).(subst0 (minus (s (Bind b) x2) n1) v x5 x6)) H17 (s -x3 (S n0)) (s_S x3 n0)))) e H15)))))))) H14)) (\lambda (H14: (ex3_4 K C C T -(\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop n0 O x1 (CHead e2 k0 u)))))) (\lambda (k0: -K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus x2 (s k0 -n0)) v e1 e2))))))).(ex3_4_ind K C C T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O x1 (CHead e2 -k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus x2 (s k0 n0)) v e1 e2))))) (or4 (drop (S n0) O (CHead x1 -(Bind b) x0) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda -(e0: C).(\lambda (_: T).(\lambda 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(e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 -(S n0))) v e1 e2)))))))) (\lambda (x3: K).(\lambda (x4: C).(\lambda (x5: -C).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x4 x3 x6))).(\lambda (H16: -(drop n0 O x1 (CHead x5 x3 x6))).(\lambda (H17: (csubst0 (minus x2 (s x3 n0)) -v x4 x5)).(eq_ind_r C (CHead x4 x3 x6) (\lambda (c0: C).(or4 (drop (S n0) O -(CHead x1 (Bind b) x0) c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 k0 u)))))) (\lambda -(k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O -(CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S -n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u: T).(eq C c0 (CHead e1 k0 u)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 -(Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 k0 u))))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s -(Bind b) x2) (s k0 (S n0))) v e1 e2))))))))) (or4_intro2 (drop (S n0) O -(CHead x1 (Bind b) x0) (CHead x4 x3 x6)) (ex3_4 K C T T (\lambda (k0: -K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6) -(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop 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C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 -u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0: -K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind -b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) -(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S -n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 -e2)))))))) (\lambda (x3: K).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: -T).(\lambda (x7: T).(\lambda (H15: (eq C e (CHead x4 x3 x6))).(\lambda (H16: -(drop n0 O x1 (CHead x5 x3 x7))).(\lambda (H17: (subst0 (minus x2 (s x3 n0)) -v x6 x7)).(\lambda (H18: (csubst0 (minus x2 (s x3 n0)) v x4 x5)).(eq_ind_r C -(CHead x4 x3 x6) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Bind b) x0) -c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda -(_: T).(eq C c0 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) -(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) -(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c0 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 -u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C -T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(eq C c0 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead -x1 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) -x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 -(S n0))) v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead x1 (Bind b) x0) -(CHead x4 x3 x6)) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda -(u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6) (CHead e0 k0 u)))))) (\lambda -(k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O -(CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S -n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u: T).(eq C (CHead x4 x3 x6) (CHead e1 k0 u)))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O -(CHead x1 (Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S -n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6) -(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) -(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S -n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 -e2))))))) (ex4_5_intro K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6) (CHead e1 k0 -u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 -w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) -(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 e2)))))) x3 x4 x5 -x6 x7 (refl_equal C (CHead x4 x3 x6)) (drop_drop (Bind b) n0 x1 (CHead x5 x3 -x7) H16 x0) (eq_ind_r nat (S (s x3 n0)) (\lambda (n1: nat).(subst0 (minus (s -(Bind 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(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v0 e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u))))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) -x2) (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 -(S n0))) v0 e1 e2))))))))))))))).(\lambda (_: (lt (S n0) (s (Flat f) -x2))).(let H13 \def (H11 x1 v H7 e H10) in (or4_ind (drop (S n0) O x1 e) -(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: -T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda -(_: T).(\lambda (w: T).(drop (S n0) O x1 (CHead e0 k0 w)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x2 (s k0 (S -n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda -(_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O x1 (CHead e2 k0 -u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus x2 (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq -C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x1 (CHead e2 k0 w))))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus x2 (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus x2 -(s k0 (S n0))) v e1 e2))))))) (or4 (drop (S n0) O (CHead 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x0) -(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S -n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 -e2)))))))) (\lambda (H14: (drop (S n0) O x1 e)).(or4_intro0 (drop (S n0) O -(CHead x1 (Flat f) x0) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: -K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 -(Flat f) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda -(u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u -w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0) -(CHead e2 k0 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(_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus x2 (s k0 (S n0))) v u w))))))).(ex3_4_ind K C T T (\lambda -(k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 -u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O x1 (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x2 (s k0 (S n0))) v u -w))))) (or4 (drop (S n0) O (CHead x1 (Flat f) x0) e) (ex3_4 K C T T (\lambda -(k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 -u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 k0 w)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) -x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda 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C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 -(Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x5) (CHead e1 k0 -u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 -w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) -(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2))))))) -(ex3_4_intro K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C (CHead x4 x3 x5) (CHead e0 k0 u)))))) (\lambda (k0: -K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 -(Flat f) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda -(u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u -w))))) x3 x4 x5 x6 (refl_equal C (CHead x4 x3 x5)) (drop_drop (Flat f) n0 x1 -(CHead x4 x3 x6) H16 x0) H17)) e H15)))))))) H14)) (\lambda (H14: (ex3_4 K C -C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O x1 (CHead e2 k0 u)))))) (\lambda (k0: -K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus x2 (s k0 -(S n0))) v e1 e2))))))).(ex3_4_ind K C C T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O x1 (CHead -e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus x2 (s k0 (S n0))) v e1 e2))))) (or4 (drop (S n0) O (CHead -x1 (Flat f) x0) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda -(u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda -(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) -(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) -(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 -u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C -T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead -x1 (Flat f) x0) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) -x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 -(S n0))) v e1 e2)))))))) (\lambda (x3: K).(\lambda (x4: C).(\lambda (x5: -C).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x4 x3 x6))).(\lambda (H16: -(drop (S n0) O x1 (CHead x5 x3 x6))).(\lambda (H17: (csubst0 (minus x2 (s x3 -(S n0))) v x4 x5)).(eq_ind_r C (CHead x4 x3 x6) (\lambda (c0: C).(or4 (drop -(S n0) O (CHead x1 (Flat f) x0) c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda -(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 k0 u)))))) -(\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O (CHead x1 (Flat f) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda -(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 -(S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 k0 u)))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O -(CHead x1 (Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S -n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 k0 -u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 -w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) -(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2))))))))) -(or4_intro2 (drop (S n0) O (CHead x1 (Flat f) x0) (CHead x4 x3 x6)) (ex3_4 K -C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C -(CHead x4 x3 x6) (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) -(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) -(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C (CHead x4 x3 x6) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0) -(CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6) (CHead e1 k0 -u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 -w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) -(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2))))))) -(ex3_4_intro K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C (CHead x4 x3 x6) (CHead e1 k0 u)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 -(Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 -e2))))) x3 x4 x5 x6 (refl_equal C (CHead x4 x3 x6)) (drop_drop (Flat f) n0 x1 -(CHead x5 x3 x6) H16 x0) H17)) e H15)))))))) H14)) (\lambda (H14: (ex4_5 K C -C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x1 -(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x2 (s k0 (S n0))) v u -w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (minus x2 (s k0 (S n0))) v e1 -e2)))))))).(ex4_5_ind K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O x1 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x2 (s k0 (S -n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 (minus x2 (s k0 (S n0))) v e1 e2)))))) (or4 -(drop (S n0) O (CHead x1 (Flat f) x0) e) (ex3_4 K C T T (\lambda (k0: -K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 -u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 k0 w)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) -x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 -(Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 w))))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s -(Flat f) x2) (s k0 (S n0))) v e1 e2)))))))) (\lambda (x3: K).(\lambda (x4: -C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (x7: T).(\lambda (H15: (eq C e -(CHead x4 x3 x6))).(\lambda (H16: (drop (S n0) O x1 (CHead x5 x3 -x7))).(\lambda (H17: (subst0 (minus x2 (s x3 (S n0))) v x6 x7)).(\lambda -(H18: (csubst0 (minus x2 (s x3 (S n0))) v x4 x5)).(eq_ind_r C (CHead x4 x3 -x6) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Flat f) x0) c0) (ex3_4 K -C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C -c0 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 k0 w)))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 -k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: -K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat -f) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 -(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) -(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S -n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 -e2))))))))) (or4_intro3 (drop (S n0) O (CHead x1 (Flat f) x0) (CHead x4 x3 -x6)) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C (CHead x4 x3 x6) (CHead e0 k0 u)))))) (\lambda (k0: -K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 -(Flat f) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda -(u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u -w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C (CHead x4 x3 x6) (CHead e1 k0 u)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 -(Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6) (CHead e1 k0 -u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 -w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) -(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2))))))) -(ex4_5_intro K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6) (CHead e1 k0 -u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 -w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) -(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2)))))) x3 x4 x5 -x6 x7 (refl_equal C (CHead x4 x3 x6)) (drop_drop (Flat f) n0 x1 (CHead x5 x3 -x7) H16 x0) H17 H18)) e H15)))))))))) H14)) H13)))))) k (drop_gen_drop k c e -t n0 H2) H8 H9) i H4))) c2 H5)))))))) H3)) (csubst0_gen_head k c c2 t v i -H1))))))))))) c1)))))) n). -(* COMMENTS -Initial nodes: 39886 -END *) - -theorem csubst0_drop_eq: - \forall (n: nat).(\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 -n v c1 c2) \to (\forall (e: C).((drop n O c1 e) \to (or4 (drop n O c2 e) -(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: -T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e0 (Flat f) w)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u -w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n O c2 (CHead e2 -(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 -(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(drop n O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O -v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))) -\def - \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (c1: C).(\forall (c2: -C).(\forall (v: T).((csubst0 n0 v c1 c2) \to (\forall (e: C).((drop n0 O c1 -e) \to (or4 (drop n0 O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O -c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat -f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop n0 O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: -T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c2 (CHead e2 -(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2))))))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda -(H: (csubst0 O v c1 c2)).(\lambda (e: C).(\lambda (H0: (drop O O c1 -e)).(eq_ind C c1 (\lambda (c: C).(or4 (drop O O c2 c) (ex3_4 F C T T (\lambda -(f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 -(Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda -(w: T).(drop O O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c -(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop O O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) -(ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq C c (CHead e1 (Flat f) u))))))) (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O -c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2))))))))) (insert_eq nat O (\lambda (n0: nat).(csubst0 n0 v c1 c2)) -(\lambda (n0: nat).(or4 (drop n0 n0 c2 c1) (ex3_4 F C T T (\lambda (f: -F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c1 (CHead e0 (Flat -f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(drop n0 n0 c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v u w)))))) (ex3_4 F C C T -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c1 -(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop n0 n0 c2 (CHead e2 (Flat f) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 v e1 -e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C c1 (CHead e1 (Flat f) u))))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop n0 n0 c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v u w)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 n0 v e1 e2))))))))) (\lambda (y: nat).(\lambda (H1: (csubst0 -y v c1 c2)).(csubst0_ind (\lambda (n0: nat).(\lambda (t: T).(\lambda (c: -C).(\lambda (c0: C).((eq nat n0 O) \to (or4 (drop n0 n0 c0 c) (ex3_4 F C T T -(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c -(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop n0 n0 c0 (CHead e0 (Flat f) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 t u w)))))) -(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop n0 n0 c0 (CHead e2 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 -t e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Flat f) u))))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop n0 n0 c0 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 t u w)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 n0 t e1 e2))))))))))))) (\lambda (k: K).(K_ind (\lambda (k0: -K).(\forall (i: nat).(\forall (v0: T).(\forall (u1: T).(\forall (u2: -T).((subst0 i v0 u1 u2) \to (\forall (c: C).((eq nat (s k0 i) O) \to (or4 -(drop (s k0 i) (s k0 i) (CHead c k0 u2) (CHead c k0 u1)) (ex3_4 F C T T -(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead -c k0 u1) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda -(_: T).(\lambda (w: T).(drop (s k0 i) (s k0 i) (CHead c k0 u2) (CHead e0 -(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (s k0 i) v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c k0 u1) (CHead e1 (Flat f) -u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (s k0 i) (s k0 i) (CHead c k0 u2) (CHead e2 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (s -k0 i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c k0 u1) -(CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) (s k0 i) (CHead c k0 u2) -(CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (s k0 i) v0 u w)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -(s k0 i) v0 e1 e2)))))))))))))))) (\lambda (b: B).(\lambda (i: nat).(\lambda -(v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0 i v0 u1 -u2)).(\lambda (c: C).(\lambda (H3: (eq nat (S i) O)).(let H4 \def (eq_ind nat -(S i) (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with -[O \Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind (or4 -(drop (S i) (S i) (CHead c (Bind b) u2) (CHead c (Bind b) u1)) (ex3_4 F C T T -(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead -c (Bind b) u1) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(drop (S i) (S i) (CHead c (Bind b) u2) -(CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (S i) v0 u w)))))) (ex3_4 F C C T (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c (Bind b) -u1) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S i) (S i) (CHead c (Bind b) u2) (CHead e2 (Flat f) -u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (S i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c (Bind -b) u1) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S i) (S i) (CHead c (Bind b) -u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (S i) v0 u w)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -(S i) v0 e1 e2)))))))) H4)))))))))) (\lambda (f: F).(\lambda (i: -nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 -i v0 u1 u2)).(\lambda (c: C).(\lambda (H3: (eq nat i O)).(let H4 \def (eq_ind -nat i (\lambda (n0: nat).(subst0 n0 v0 u1 u2)) H2 O H3) in (eq_ind_r nat O -(\lambda (n0: nat).(or4 (drop n0 n0 (CHead c (Flat f) u2) (CHead c (Flat f) -u1)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C (CHead c (Flat f) u1) (CHead e0 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0 -(CHead c (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda -(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v0 u w)))))) (ex3_4 F C C T -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C -(CHead c (Flat f) u1) (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda -(_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 n0 (CHead c (Flat f) u2) -(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C -(CHead c (Flat f) u1) (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0 (CHead c -(Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v0 u w)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 n0 v0 e1 e2))))))))) (or4_intro1 (drop O O (CHead c (Flat f) -u2) (CHead c (Flat f) u1)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c (Flat f) u1) (CHead e0 -(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda -(w: T).(drop O O (CHead c (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) -(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C (CHead c (Flat f) u1) (CHead e1 (Flat f0) u)))))) (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O (CHead c (Flat -f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda -(f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq -C (CHead c (Flat f) u1) (CHead e1 (Flat f0) u))))))) (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O -(CHead c (Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda -(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v0 e1 e2))))))) (ex3_4_intro F C T T (\lambda (f0: -F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c (Flat f) -u1) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda -(_: T).(\lambda (w: T).(drop O O (CHead c (Flat f) u2) (CHead e0 (Flat f0) -w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v0 u w))))) f c u1 u2 (refl_equal C (CHead c (Flat f) u1)) -(drop_refl (CHead c (Flat f) u2)) H4)) i H3)))))))))) k)) (\lambda (k: -K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (c3: C).(\forall (c4: -C).(\forall (v0: T).((csubst0 i v0 c3 c4) \to ((((eq nat i O) \to (or4 (drop -i i c4 c3) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C c3 (CHead e0 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop i i c4 (CHead e0 -(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 i v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 (Flat f) u)))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop i i c4 -(CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 i v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 -(CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop i i c4 (CHead e2 (Flat f) w))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 i v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v0 e1 e2)))))))))) \to (\forall -(u: T).((eq nat (s k0 i) O) \to (or4 (drop (s k0 i) (s k0 i) (CHead c4 k0 u) -(CHead c3 k0 u)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda -(u0: T).(\lambda (_: T).(eq C (CHead c3 k0 u) (CHead e0 (Flat f) u0)))))) -(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 -i) (s k0 i) (CHead c4 k0 u) (CHead e0 (Flat f) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (s k0 i) v0 u0 -w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u0: T).(eq C (CHead c3 k0 u) (CHead e1 (Flat f) u0)))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s k0 -i) (s k0 i) (CHead c4 k0 u) (CHead e2 (Flat f) u0)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (s k0 i) v0 e1 -e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 k0 u) (CHead e1 (Flat f) -u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (s k0 i) (s k0 i) (CHead c4 k0 u) (CHead e2 (Flat f) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: -T).(\lambda (w: T).(subst0 (s k0 i) v0 u0 w)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (s k0 i) v0 -e1 e2))))))))))))))))) (\lambda (b: B).(\lambda (i: nat).(\lambda (c3: -C).(\lambda (c4: C).(\lambda (v0: T).(\lambda (_: (csubst0 i v0 c3 -c4)).(\lambda (_: (((eq nat i O) \to (or4 (drop i i c4 c3) (ex3_4 F C T T -(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 -(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop i i c4 (CHead e0 (Flat f) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w)))))) -(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c3 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop i i c4 (CHead e2 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f) u))))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop i i c4 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (u: T).(\lambda (H4: (eq nat -(S i) O)).(let H5 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee in nat -return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow -True])) I O H4) in (False_ind (or4 (drop (S i) (S i) (CHead c4 (Bind b) u) -(CHead c3 (Bind b) u)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: -C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 (Bind b) u) (CHead e0 -(Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda -(w: T).(drop (S i) (S i) (CHead c4 (Bind b) u) (CHead e0 (Flat f) w)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (S -i) v0 u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u0: T).(eq C (CHead c3 (Bind b) u) (CHead e1 (Flat f) u0)))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S i) -(S i) (CHead c4 (Bind b) u) (CHead e2 (Flat f) u0)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (S i) v0 e1 -e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 (Bind b) u) (CHead e1 -(Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(drop (S i) (S i) (CHead c4 (Bind b) u) (CHead e2 -(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda -(u0: T).(\lambda (w: T).(subst0 (S i) v0 u0 w)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (S i) v0 e1 -e2)))))))) H5))))))))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (c3: -C).(\lambda (c4: C).(\lambda (v0: T).(\lambda (H2: (csubst0 i v0 c3 -c4)).(\lambda (H3: (((eq nat i O) \to (or4 (drop i i c4 c3) (ex3_4 F C T T -(\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 -(CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop i i c4 (CHead e0 (Flat f0) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w)))))) -(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c3 (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop i i c4 (CHead e2 (Flat f0) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f0) u))))))) -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(w: T).(drop i i c4 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (u: T).(\lambda (H4: (eq nat i -O)).(let H5 \def (eq_ind nat i (\lambda (n0: nat).((eq nat n0 O) \to (or4 -(drop n0 n0 c4 c3) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda -(u0: T).(\lambda (_: T).(eq C c3 (CHead e0 (Flat f0) u0)))))) (\lambda (f0: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0 c4 (CHead e0 -(Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda -(w: T).(subst0 n0 v0 u0 w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u0: T).(eq C c3 (CHead e1 (Flat f0) u0)))))) -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop n0 -n0 c4 (CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda -(f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq -C c3 (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0 c4 (CHead e2 (Flat f0) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: -T).(\lambda (w: T).(subst0 n0 v0 u0 w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 n0 v0 e1 -e2)))))))))) H3 O H4) in (let H6 \def (eq_ind nat i (\lambda (n0: -nat).(csubst0 n0 v0 c3 c4)) H2 O H4) in (eq_ind_r nat O (\lambda (n0: -nat).(or4 (drop n0 n0 (CHead c4 (Flat f) u) (CHead c3 (Flat f) u)) (ex3_4 F C -T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C -(CHead c3 (Flat f) u) (CHead e0 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda -(e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0 (CHead c4 (Flat f) u) -(CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: -T).(\lambda (w: T).(subst0 n0 v0 u0 w)))))) (ex3_4 F C C T (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead c3 (Flat f) -u) (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u0: T).(drop n0 n0 (CHead c4 (Flat f) u) (CHead e2 (Flat -f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 (Flat f) -u) (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0 (CHead c4 (Flat f) u) -(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u0: T).(\lambda (w: T).(subst0 n0 v0 u0 w)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -n0 v0 e1 e2))))))))) (or4_intro2 (drop O O (CHead c4 (Flat f) u) (CHead c3 -(Flat f) u)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: -T).(\lambda (_: T).(eq C (CHead c3 (Flat f) u) (CHead e0 (Flat f0) u0)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O -(CHead c4 (Flat f) u) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda -(_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v0 u0 w)))))) (ex3_4 F C C -T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C -(CHead c3 (Flat f) u) (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda -(_: C).(\lambda (e2: C).(\lambda (u0: T).(drop O O (CHead c4 (Flat f) u) -(CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C -(CHead c3 (Flat f) u) (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 -(Flat f) u) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v0 u0 w)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v0 e1 e2))))))) (ex3_4_intro F C C T (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead c3 (Flat f) -u) (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u0: T).(drop O O (CHead c4 (Flat f) u) (CHead e2 (Flat f0) -u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v0 e1 e2))))) f c3 c4 u (refl_equal C (CHead c3 (Flat f) u)) -(drop_refl (CHead c4 (Flat f) u)) H6)) i H4)))))))))))) k)) (\lambda (k: -K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (v0: T).(\forall (u1: -T).(\forall (u2: T).((subst0 i v0 u1 u2) \to (\forall (c3: C).(\forall (c4: -C).((csubst0 i v0 c3 c4) \to ((((eq nat i O) \to (or4 (drop i i c4 c3) (ex3_4 -F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq -C c3 (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop i i c4 (CHead e0 (Flat f) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w)))))) -(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c3 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop i i c4 (CHead e2 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f) u))))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop i i c4 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 i v0 e1 e2)))))))))) \to ((eq nat (s k0 i) O) \to (or4 (drop -(s k0 i) (s k0 i) (CHead c4 k0 u2) (CHead c3 k0 u1)) (ex3_4 F C T T (\lambda -(f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 k0 -u1) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (s k0 i) (s k0 i) (CHead c4 k0 u2) (CHead e0 (Flat -f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (s k0 i) v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c3 k0 u1) (CHead e1 (Flat f) -u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (s k0 i) (s k0 i) (CHead c4 k0 u2) (CHead e2 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (s -k0 i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 k0 u1) -(CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) (s k0 i) (CHead c4 k0 u2) -(CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (s k0 i) v0 u w)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -(s k0 i) v0 e1 e2))))))))))))))))))) (\lambda (b: B).(\lambda (i: -nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0 -i v0 u1 u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (_: (csubst0 i v0 c3 -c4)).(\lambda (_: (((eq nat i O) \to (or4 (drop i i c4 c3) (ex3_4 F C T T -(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 -(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop i i c4 (CHead e0 (Flat f) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w)))))) -(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c3 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop i i c4 (CHead e2 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f) u))))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop i i c4 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (H5: (eq nat (S i) O)).(let H6 -\def (eq_ind nat (S i) (\lambda (ee: nat).(match ee in nat return (\lambda -(_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H5) -in (False_ind (or4 (drop (S i) (S i) (CHead c4 (Bind b) u2) (CHead c3 (Bind -b) u1)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C (CHead c3 (Bind b) u1) (CHead e0 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S i) -(S i) (CHead c4 (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (S i) v0 u w)))))) -(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C (CHead c3 (Bind b) u1) (CHead e1 (Flat f) u)))))) (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S i) (S i) (CHead -c4 (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (S i) v0 e1 e2)))))) (ex4_5 F C -C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(eq C (CHead c3 (Bind b) u1) (CHead e1 (Flat f) u))))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S i) (S i) (CHead c4 (Bind b) u2) (CHead e2 (Flat f) w))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (S i) v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (S i) v0 e1 e2)))))))) -H6))))))))))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (v0: T).(\lambda -(u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 i v0 u1 u2)).(\lambda (c3: -C).(\lambda (c4: C).(\lambda (H3: (csubst0 i v0 c3 c4)).(\lambda (H4: (((eq -nat i O) \to (or4 (drop i i c4 c3) (ex3_4 F C T T (\lambda (f0: F).(\lambda -(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e0 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop i i -c4 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 i v0 u w)))))) (ex3_4 F C C T (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 (Flat -f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop i i c4 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v0 e1 e2)))))) (ex4_5 F C C T -T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda -(_: T).(eq C c3 (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop i i c4 (CHead e2 -(Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (w: T).(subst0 i v0 u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v0 e1 -e2))))))))))).(\lambda (H5: (eq nat i O)).(let H6 \def (eq_ind nat i (\lambda -(n0: nat).((eq nat n0 O) \to (or4 (drop n0 n0 c4 c3) (ex3_4 F C T T (\lambda -(f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e0 -(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda -(w: T).(drop n0 n0 c4 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda -(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v0 u w)))))) (ex3_4 F C C T -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3 -(CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop n0 n0 c4 (CHead e2 (Flat f0) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 v0 e1 -e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f0) u))))))) -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(w: T).(drop n0 n0 c4 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda -(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v0 u -w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))))))) H4 O H5) in (let H7 \def -(eq_ind nat i (\lambda (n0: nat).(csubst0 n0 v0 c3 c4)) H3 O H5) in (let H8 -\def (eq_ind nat i (\lambda (n0: nat).(subst0 n0 v0 u1 u2)) H2 O H5) in -(eq_ind_r nat O (\lambda (n0: nat).(or4 (drop n0 n0 (CHead c4 (Flat f) u2) -(CHead c3 (Flat f) u1)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 (Flat f) u1) (CHead e0 -(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda -(w: T).(drop n0 n0 (CHead c4 (Flat f) u2) (CHead e0 (Flat f0) w)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v0 -u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 n0 -(CHead c4 (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F -C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(eq C (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u))))))) -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(w: T).(drop n0 n0 (CHead c4 (Flat f) u2) (CHead e2 (Flat f0) w))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 n0 v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 n0 v0 e1 e2))))))))) (or4_intro3 -(drop O O (CHead c4 (Flat f) u2) (CHead c3 (Flat f) u1)) (ex3_4 F C T T -(\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C -(CHead c3 (Flat f) u1) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda -(e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 (Flat f) u2) -(CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v0 u w)))))) (ex3_4 F C C T (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c3 (Flat f) -u1) (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u: T).(drop O O (CHead c4 (Flat f) u2) (CHead e2 (Flat f0) -u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 (Flat f) -u1) (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 (Flat f) u2) -(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v0 e1 e2))))))) (ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 (Flat f) -u1) (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 (Flat f) u2) -(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v0 e1 e2)))))) f c3 c4 u1 u2 (refl_equal C (CHead c3 (Flat f) u1)) -(drop_refl (CHead c4 (Flat f) u2)) H8 H7)) i H5))))))))))))))) k)) y v c1 c2 -H1))) H) e (drop_gen_refl c1 e H0)))))))) (\lambda (n0: nat).(\lambda (IHn: -((\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 n0 v c1 c2) \to -(\forall (e: C).((drop n0 O c1 e) \to (or4 (drop n0 O c2 e) (ex3_4 F C T T -(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop n0 O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O c2 (CHead e2 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O -v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop n0 O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2)))))))))))))))).(\lambda (c1: C).(C_ind (\lambda -(c: C).(\forall (c2: C).(\forall (v: T).((csubst0 (S n0) v c c2) \to (\forall -(e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 F C T T -(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 (Flat f) -u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) -u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O -v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))) (\lambda (n1: -nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (_: (csubst0 (S n0) v (CSort -n1) c2)).(\lambda (e: C).(\lambda (H0: (drop (S n0) O (CSort n1) -e)).(and3_ind (eq C e (CSort n1)) (eq nat (S n0) O) (eq nat O O) (or4 (drop -(S n0) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead -e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda -(w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) -O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda -(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C -e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2)))))))) (\lambda (H1: (eq C e (CSort n1))).(\lambda (H2: (eq nat (S n0) -O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n1) (\lambda (c: C).(or4 -(drop (S n0) O c2 c) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda -(u: T).(\lambda (_: T).(eq C c (CHead e0 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead -e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda -(w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Flat f) u)))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) -O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda -(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C -c (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2))))))))) (let H4 \def (eq_ind nat (S n0) (\lambda (ee: nat).(match ee in -nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) -\Rightarrow True])) I O H2) in (False_ind (or4 (drop (S n0) O c2 (CSort n1)) -(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: -T).(eq C (CSort n1) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 (Flat f) -w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C (CSort n1) (CHead e1 (Flat f) -u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: -T).(eq C (CSort n1) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead -e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2)))))))) H4)) e H1)))) (drop_gen_sort n1 (S n0) O e H0)))))))) (\lambda (c: -C).(\lambda (H: ((\forall (c2: C).(\forall (v: T).((csubst0 (S n0) v c c2) -\to (\forall (e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 -F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq -C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 (Flat f) -u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) -u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O -v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))))))).(\lambda (k: K).(\lambda -(t: T).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 (S n0) v -(CHead c k t) c2)).(\lambda (e: C).(\lambda (H1: (drop (S n0) O (CHead c k t) -e)).(or3_ind (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (S n0) (s -k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) -(\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2)))) (ex3_2 C nat (\lambda -(_: C).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (c3: C).(\lambda -(_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: -nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: -C).(\lambda (j: nat).(eq nat (S n0) (s k j))))) (\lambda (u2: T).(\lambda -(c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: -T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: -T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))) (or4 (drop (S -n0) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead -e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda -(w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) -O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda -(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C -e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2)))))))) (\lambda (H2: (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq -nat (S n0) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k -u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))))).(ex3_2_ind T -nat (\lambda (_: T).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (u2: -T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: -nat).(subst0 j v t u2))) (or4 (drop (S n0) O c2 e) (ex3_4 F C T T (\lambda -(f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 -(Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda -(w: T).(drop (S n0) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda -(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) -(ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda -(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H3: (eq nat -(S n0) (s k x1))).(\lambda (H4: (eq C c2 (CHead c k x0))).(\lambda (H5: -(subst0 x1 v t x0)).(eq_ind_r C (CHead c k x0) (\lambda (c0: C).(or4 (drop (S -n0) O c0 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead -e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda -(w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) -O c0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda -(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C -e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 (Flat f) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2))))))))) (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to ((eq nat (S -n0) (s k0 x1)) \to (or4 (drop (S n0) O (CHead c k0 x0) e) (ex3_4 F C T T -(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead c k0 x0) (CHead e0 (Flat f) w)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u -w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c -k0 x0) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda -(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C -e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c k0 x0) (CHead e2 -(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2))))))))))) (\lambda (b: B).(\lambda (H6: (drop (r (Bind b) n0) O c -e)).(\lambda (H7: (eq nat (S n0) (s (Bind b) x1))).(let H8 \def (f_equal nat -nat (\lambda (e0: nat).(match e0 in nat return (\lambda (_: nat).nat) with [O -\Rightarrow n0 | (S n1) \Rightarrow n1])) (S n0) (S x1) H7) in (let H9 \def -(eq_ind_r nat x1 (\lambda (n1: nat).(subst0 n1 v t x0)) H5 n0 H8) in -(or4_intro0 (drop (S n0) O (CHead c (Bind b) x0) e) (ex3_4 F C T T (\lambda -(f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 -(Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda -(w: T).(drop (S n0) O (CHead c (Bind b) x0) (CHead e0 (Flat f) w)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u -w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c -(Bind b) x0) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: -T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c -(Bind b) x0) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2))))))) (drop_drop (Bind b) n0 c e H6 x0))))))) -(\lambda (f: F).(\lambda (H6: (drop (r (Flat f) n0) O c e)).(\lambda (H7: (eq -nat (S n0) (s (Flat f) x1))).(let H8 \def (f_equal nat nat (\lambda (e0: -nat).e0) (S n0) (s (Flat f) x1) H7) in (let H9 \def (eq_ind_r nat x1 (\lambda -(n1: nat).(subst0 n1 v t x0)) H5 (S n0) H8) in (or4_intro0 (drop (S n0) O -(CHead c (Flat f) x0) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O (CHead c (Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) x0) -(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c (Flat f) x0) -(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2))))))) (drop_drop (Flat f) n0 c e H6 x0))))))) k (drop_gen_drop k c -e t n0 H1) H3) c2 H4)))))) H2)) (\lambda (H2: (ex3_2 C nat (\lambda (_: -C).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (c3: C).(\lambda (_: -nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j -v c c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) -(s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) -(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (or4 (drop (S n0) O -c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead -e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda -(w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) -O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda -(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C -e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2)))))))) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H3: (eq nat (S n0) -(s k x1))).(\lambda (H4: (eq C c2 (CHead x0 k t))).(\lambda (H5: (csubst0 x1 -v c x0)).(eq_ind_r C (CHead x0 k t) (\lambda (c0: C).(or4 (drop (S n0) O c0 -e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda -(_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e0 (Flat f) -w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) -O c0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda -(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C -e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 (Flat f) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2))))))))) (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to ((eq nat (S -n0) (s k0 x1)) \to (or4 (drop (S n0) O (CHead x0 k0 t) e) (ex3_4 F C T T -(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x0 k0 t) (CHead e0 (Flat f) w)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u -w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 -k0 t) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda -(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C -e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 k0 t) (CHead e2 -(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2))))))))))) (\lambda (b: B).(\lambda (H6: (drop (r (Bind b) n0) O c -e)).(\lambda (H7: (eq nat (S n0) (s (Bind b) x1))).(let H8 \def (f_equal nat -nat (\lambda (e0: nat).(match e0 in nat return (\lambda (_: nat).nat) with [O -\Rightarrow n0 | (S n1) \Rightarrow n1])) (S n0) (S x1) H7) in (let H9 \def -(eq_ind_r nat x1 (\lambda (n1: nat).(csubst0 n1 v c x0)) H5 n0 H8) in (let -H10 \def (IHn c x0 v H9 e H6) in (or4_ind (drop n0 O x0 e) (ex3_4 F C T T -(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop n0 O x0 (CHead e0 (Flat f) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O x0 (CHead e2 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O -v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop n0 O x0 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead x0 (Bind b) t) e) -(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: -T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) -(CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: 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C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 (Flat x2) -x5) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) -u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v e1 e2))))) x2 x3 x4 x5 (refl_equal C (CHead x3 (Flat x2) x5)) -(drop_drop (Bind b) n0 x0 (CHead x4 (Flat x2) x5) H13 t) H14)) e H12)))))))) -H11)) (\lambda (H11: (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) -u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop n0 O x0 (CHead e2 (Flat f) w))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O -v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda -(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C -e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x0 (CHead e2 (Flat f) w))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) -O (CHead x0 (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) -O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda -(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) 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c0) (ex3_4 F C T T -(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 -(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f) -w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Flat f) u)))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) -O (CHead x0 (Bind b) t) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C -C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat f) u))))))) (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O -v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O -(CHead x0 (Bind b) t) (CHead x3 (Flat x2) x5)) (ex3_4 F C T T (\lambda (f: -F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) -x5) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f) -w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1 -(Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O -v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1 -(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat -f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) -(ex4_5_intro F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1 -(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat -f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) -x2 x3 x4 x5 x6 (refl_equal C (CHead x3 (Flat x2) x5)) (drop_drop (Bind b) n0 -x0 (CHead x4 (Flat x2) x6) H13 t) H14 H15)) e H12)))))))))) H11)) H10))))))) -(\lambda (f: F).(\lambda (H6: (drop (r (Flat f) n0) O c e)).(\lambda (H7: (eq -nat (S n0) (s (Flat f) x1))).(let H8 \def (f_equal nat nat (\lambda (e0: -nat).e0) (S n0) (s (Flat f) x1) H7) in (let H9 \def (eq_ind_r nat x1 (\lambda -(n1: nat).(csubst0 n1 v c x0)) H5 (S n0) H8) in (let H10 \def (H x0 v H9 e -H6) in (or4_ind (drop (S n0) O x0 e) (ex3_4 F C T T (\lambda (f0: F).(\lambda -(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O x0 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda -(u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat -f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O x0 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C -C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O x0 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead x0 (Flat f) t) e) -(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: -T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) -(CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat -f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u)))))) (\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 -e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H11: -(drop (S n0) O x0 e)).(or4_intro0 (drop (S n0) O (CHead x0 (Flat f) t) e) -(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: -T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) -(CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat -f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u)))))) (\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 -e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: 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(CHead e0 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) -(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) -(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: T).(\lambda -(x5: T).(\lambda (H12: (eq C e (CHead x3 (Flat x2) x4))).(\lambda (H13: (drop -(S n0) O x0 (CHead x3 (Flat x2) x5))).(\lambda (H14: (subst0 O v x4 -x5)).(eq_ind_r C (CHead x3 (Flat x2) x4) (\lambda (c0: C).(or4 (drop (S n0) O -(CHead x0 (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c0 (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) -(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 -(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) -(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead x0 (Flat f) t) (CHead x3 -(Flat x2) x4)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x4) (CHead e0 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: 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T).(\lambda (_: T).(eq C -(CHead x3 (Flat x2) x4) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda -(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) -(CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w))))) x2 x3 x4 x5 (refl_equal C (CHead x3 -(Flat x2) x4)) (drop_drop (Flat f) n0 x0 (CHead x3 (Flat x2) x5) H13 t) H14)) -e H12)))))))) H11)) (\lambda (H11: (ex3_4 F C C T (\lambda (f0: F).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S -n0) O x0 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C -C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O x0 (CHead e2 (Flat f0) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) -(or4 (drop (S n0) O (CHead x0 (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: -F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat -f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) -(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) -(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda -(x5: T).(\lambda (H12: (eq C e (CHead x3 (Flat x2) x5))).(\lambda (H13: (drop -(S n0) O x0 (CHead x4 (Flat x2) x5))).(\lambda (H14: (csubst0 O v x3 -x4)).(eq_ind_r C (CHead x3 (Flat x2) x5) (\lambda (c0: C).(or4 (drop (S n0) O -(CHead x0 (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c0 (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) -(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 -(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) -(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead x0 (Flat f) t) (CHead x3 -(Flat x2) x5)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C (CHead x3 (Flat x2) x5) (CHead e1 (Flat f0) u)))))) (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 -(Flat f) t) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda -(_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1 (Flat f0) u))))))) (\lambda -(f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C -C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C -(CHead x3 (Flat x2) x5) (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda -(_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) -(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2))))) x2 x3 x4 x5 (refl_equal C (CHead -x3 (Flat x2) x5)) (drop_drop (Flat f) n0 x0 (CHead x4 (Flat x2) x5) H13 t) -H14)) e H12)))))))) H11)) (\lambda (H11: (ex4_5 F C C T T (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x0 (CHead e2 (Flat f0) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2)))))))).(ex4_5_ind F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(w: T).(drop (S n0) O x0 (CHead e2 (Flat f0) w))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O -v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) O (CHead x0 -(Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 -(Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) -u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat -f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: -T).(\lambda (x6: T).(\lambda (H12: (eq C e (CHead x3 (Flat x2) x5))).(\lambda -(H13: (drop (S n0) O x0 (CHead x4 (Flat x2) x6))).(\lambda (H14: (subst0 O v -x5 x6)).(\lambda (H15: (csubst0 O v x3 x4)).(eq_ind_r C (CHead x3 (Flat x2) -x5) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x0 (Flat f) t) c0) (ex3_4 F C -T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C -c0 (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) -w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S -n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) -(ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat f0) u))))))) (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O -v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O -(CHead x0 (Flat f) t) (CHead x3 (Flat x2) x5)) (ex3_4 F C T T (\lambda (f0: -F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) -x5) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda -(_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat -f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1 -(Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(u: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O -v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1 -(Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat -f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) -(ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1 -(Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat -f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) -x2 x3 x4 x5 x6 (refl_equal C (CHead x3 (Flat x2) x5)) (drop_drop (Flat f) n0 -x0 (CHead x4 (Flat x2) x6) H13 t) H14 H15)) e H12)))))))))) H11)) H10))))))) -k (drop_gen_drop k c e t n0 H1) H3) c2 H4)))))) H2)) (\lambda (H2: (ex4_3 T C -nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k -j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 -k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t -u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c -c3)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: -nat).(eq nat (S n0) (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda -(_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: -C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: -C).(\lambda (j: nat).(csubst0 j v c c3)))) (or4 (drop (S n0) O c2 e) (ex3_4 F -C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C -e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 (Flat f) -u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) -u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O -v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: -C).(\lambda (x2: nat).(\lambda (H3: (eq nat (S n0) (s k x2))).(\lambda (H4: -(eq C c2 (CHead x1 k x0))).(\lambda (H5: (subst0 x2 v t x0)).(\lambda (H6: -(csubst0 x2 v c x1)).(eq_ind_r C (CHead x1 k x0) (\lambda (c0: C).(or4 (drop -(S n0) O c0 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead -e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda -(w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) -O c0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda -(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C -e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 (Flat f) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2))))))))) (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to ((eq nat (S -n0) (s k0 x2)) \to (or4 (drop (S n0) O (CHead x1 k0 x0) e) (ex3_4 F C T T -(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 k0 x0) (CHead e0 (Flat f) w)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u -w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 -k0 x0) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda -(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C -e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 k0 x0) (CHead e2 -(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2))))))))))) (\lambda (b: B).(\lambda (H7: (drop (r (Bind b) n0) O c -e)).(\lambda (H8: (eq nat (S n0) (s (Bind b) x2))).(let H9 \def (f_equal nat -nat (\lambda (e0: nat).(match e0 in nat return (\lambda (_: nat).nat) with [O -\Rightarrow n0 | (S n1) \Rightarrow n1])) (S n0) (S x2) H8) in (let H10 \def -(eq_ind_r nat x2 (\lambda (n1: nat).(csubst0 n1 v c x1)) H6 n0 H9) in (let -H11 \def (eq_ind_r nat x2 (\lambda (n1: nat).(subst0 n1 v t x0)) H5 n0 H9) in -(let H12 \def (IHn c x1 v H10 e H7) in (or4_ind (drop n0 O x1 e) (ex3_4 F C T -T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop n0 O x1 (CHead e0 (Flat f) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O x1 (CHead e2 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O -v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop n0 O x1 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead x1 (Bind b) x0) -e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda -(_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) -(CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat -f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u)))))) (\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 -e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) w))))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H13: -(drop n0 O x1 e)).(or4_intro0 (drop (S n0) O (CHead x1 (Bind b) x0) e) (ex3_4 -F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq -C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) -w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) -O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C -C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) w))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O -v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Bind b) n0 x1 e H13 -x0))) (\lambda (H13: (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda -(u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x1 (CHead e0 -(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w))))))).(ex3_4_ind F C T T (\lambda (f: F).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O -x1 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w))))) (or4 (drop (S n0) O (CHead x1 (Bind -b) x0) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 -(Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) -u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) -u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: T).(\lambda (x6: -T).(\lambda (H14: (eq C e (CHead x4 (Flat x3) x5))).(\lambda (H15: (drop n0 O -x1 (CHead x4 (Flat x3) x6))).(\lambda (H16: (subst0 O v x5 x6)).(eq_ind_r C -(CHead x4 (Flat x3) x5) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Bind -b) x0) c0) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 -(Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 -(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) -u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat -f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2))))))))) (or4_intro1 (drop (S n0) O (CHead x1 (Bind b) x0) (CHead x4 (Flat -x3) x5)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x5) (CHead e0 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) -O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda -(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead -x4 (Flat x3) x5) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) -(CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C -(CHead x4 (Flat x3) x5) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead -x1 (Bind b) x0) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C T T (\lambda (f: -F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) -x5) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) -w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w))))) x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x3) x5)) -(drop_drop (Bind b) n0 x1 (CHead x4 (Flat x3) x6) H15 x0) H16)) e H14)))))))) -H13)) (\lambda (H13: (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O x1 (CHead e2 -(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O -x1 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2))))) (or4 (drop (S n0) O (CHead x1 -(Bind b) x0) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 -(Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) -u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) -u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: -T).(\lambda (H14: (eq C e (CHead x4 (Flat x3) x6))).(\lambda (H15: (drop n0 O -x1 (CHead x5 (Flat x3) x6))).(\lambda (H16: (csubst0 O v x4 x5)).(eq_ind_r C -(CHead x4 (Flat x3) x6) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Bind -b) x0) c0) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 -(Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 -(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) -u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat -f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2))))))))) (or4_intro2 (drop (S n0) O (CHead x1 (Bind b) x0) (CHead x4 (Flat -x3) x6)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) -O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda -(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead -x4 (Flat x3) x6) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) -(CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C -(CHead x4 (Flat x3) x6) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead -x1 (Bind b) x0) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C C T (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x4 (Flat x3) -x6) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) -u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v e1 e2))))) x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x3) x6)) -(drop_drop (Bind b) n0 x1 (CHead x5 (Flat x3) x6) H15 x0) H16)) e H14)))))))) -H13)) (\lambda (H13: (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) -u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop n0 O x1 (CHead e2 (Flat f) w))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O -v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda -(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C -e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x1 (CHead e2 (Flat f) w))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) -O (CHead x1 (Bind b) x0) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) -O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda -(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) -u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) -u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: -T).(\lambda (x7: T).(\lambda (H14: (eq C e (CHead x4 (Flat x3) x6))).(\lambda -(H15: (drop n0 O x1 (CHead x5 (Flat x3) x7))).(\lambda (H16: (subst0 O v x6 -x7)).(\lambda (H17: (csubst0 O v x4 x5)).(eq_ind_r C (CHead x4 (Flat x3) x6) -(\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Bind b) x0) c0) (ex3_4 F C T -T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 -(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) -w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Flat f) u)))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) -O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C -C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat f) u))))))) (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) w))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O -v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O -(CHead x1 (Bind b) x0) (CHead x4 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f: -F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) -x6) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) -w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 -(Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O -v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 -(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat -f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) -(ex4_5_intro F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 -(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat -f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) -x3 x4 x5 x6 x7 (refl_equal C (CHead x4 (Flat x3) x6)) (drop_drop (Bind b) n0 -x1 (CHead x5 (Flat x3) x7) H15 x0) H16 H17)) e H14)))))))))) H13)) -H12)))))))) (\lambda (f: F).(\lambda (H7: (drop (r (Flat f) n0) O c -e)).(\lambda (H8: (eq nat (S n0) (s (Flat f) x2))).(let H9 \def (f_equal nat -nat (\lambda (e0: nat).e0) (S n0) (s (Flat f) x2) H8) in (let H10 \def -(eq_ind_r nat x2 (\lambda (n1: nat).(csubst0 n1 v c x1)) H6 (S n0) H9) in -(let H11 \def (eq_ind_r nat x2 (\lambda (n1: nat).(subst0 n1 v t x0)) H5 (S -n0) H9) in (let H12 \def (H x1 v H10 e H7) in (or4_ind (drop (S n0) O x1 e) -(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: -T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x1 (CHead e0 (Flat f0) -w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S -n0) O x1 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda -(_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x1 (CHead -e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2))))))) (or4 (drop (S n0) O (CHead x1 (Flat f) x0) e) (ex3_4 F C T T -(\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) -w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S -n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) -(ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O -v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H13: (drop (S n0) O -x1 e)).(or4_intro0 (drop (S n0) O (CHead x1 (Flat f) x0) e) (ex3_4 F C T T -(\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) -w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S -n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) -(ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O -v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Flat f) n0 x1 e H13 -x0))) (\lambda (H13: (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O x1 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda -(u: T).(\lambda (w: T).(subst0 O v u w))))))).(ex3_4_ind F C T T (\lambda -(f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 -(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda -(w: T).(drop (S n0) O x1 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda -(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))) (or4 (drop (S n0) -O (CHead x1 (Flat f) x0) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0) -(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) -(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: T).(\lambda -(x6: T).(\lambda (H14: (eq C e (CHead x4 (Flat x3) x5))).(\lambda (H15: (drop -(S n0) O x1 (CHead x4 (Flat x3) x6))).(\lambda (H16: (subst0 O v x5 -x6)).(eq_ind_r C (CHead x4 (Flat x3) x5) (\lambda (c0: C).(or4 (drop (S n0) O -(CHead x1 (Flat f) x0) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c0 (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0) -(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 -(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) -(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead x1 (Flat f) x0) (CHead -x4 (Flat x3) x5)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda -(u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x5) (CHead e0 (Flat f0) -u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C (CHead x4 (Flat x3) x5) (CHead e1 (Flat f0) u)))))) (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 -(Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda -(_: T).(eq C (CHead x4 (Flat x3) x5) (CHead e1 (Flat f0) u))))))) (\lambda -(f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C -T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C -(CHead x4 (Flat x3) x5) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda -(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) -(CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w))))) x3 x4 x5 x6 (refl_equal C (CHead x4 -(Flat x3) x5)) (drop_drop (Flat f) n0 x1 (CHead x4 (Flat x3) x6) H15 x0) -H16)) e H14)))))))) H13)) (\lambda (H13: (ex3_4 F C C T (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat -f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O x1 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind -F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq -C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u: T).(drop (S n0) O x1 (CHead e2 (Flat f0) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O -v e1 e2))))) (or4 (drop (S n0) O (CHead x1 (Flat f) x0) e) (ex3_4 F C T T -(\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) -w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S -n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) -(ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O -v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: -C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (H14: (eq C e (CHead x4 (Flat -x3) x6))).(\lambda (H15: (drop (S n0) O x1 (CHead x5 (Flat x3) x6))).(\lambda -(H16: (csubst0 O v x4 x5)).(eq_ind_r C (CHead x4 (Flat x3) x6) (\lambda (c0: -C).(or4 (drop (S n0) O (CHead x1 (Flat f) x0) c0) (ex3_4 F C T T (\lambda -(f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 -(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda -(w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u -w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C c0 (CHead e1 (Flat f0) u)))))) (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 -(Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda -(_: T).(eq C c0 (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 -(Flat f) x0) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead x1 (Flat -f) x0) (CHead x4 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0 -(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda -(w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u -w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S -n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) -(ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) -u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) -(ex3_4_intro F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S -n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) -x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x3) x6)) (drop_drop (Flat f) n0 x1 -(CHead x5 (Flat x3) x6) H15 x0) H16)) e H14)))))))) H13)) (\lambda (H13: -(ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O x1 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x1 (CHead e2 (Flat f0) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) -(or4 (drop (S n0) O (CHead x1 (Flat f) x0) e) (ex3_4 F C T T (\lambda (f0: -F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat -f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0) -(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) -(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda -(x6: T).(\lambda (x7: T).(\lambda (H14: (eq C e (CHead x4 (Flat x3) -x6))).(\lambda (H15: (drop (S n0) O x1 (CHead x5 (Flat x3) x7))).(\lambda -(H16: (subst0 O v x6 x7)).(\lambda (H17: (csubst0 O v x4 x5)).(eq_ind_r C -(CHead x4 (Flat x3) x6) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Flat -f) x0) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f0) u)))))) (\lambda (f0: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 -(Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 -(CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) -u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat -f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2))))))))) (or4_intro3 (drop (S n0) O (CHead x1 (Flat f) x0) (CHead x4 (Flat -x3) x6)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) u)))))) (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 -(Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda -(_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) u))))))) (\lambda -(f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex4_5_intro F C -C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) u))))))) -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) x3 x4 x5 x6 x7 -(refl_equal C (CHead x4 (Flat x3) x6)) (drop_drop (Flat f) n0 x1 (CHead x5 -(Flat x3) x7) H15 x0) H16 H17)) e H14)))))))))) H13)) H12)))))))) k -(drop_gen_drop k c e t n0 H1) H3) c2 H4)))))))) H2)) (csubst0_gen_head k c c2 -t v (S n0) H0))))))))))) c1)))) n). -(* COMMENTS -Initial nodes: 36162 -END *) - -theorem csubst0_drop_eq_back: - \forall (n: nat).(\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 -n v c1 c2) \to (\forall (e: C).((drop n O c2 e) \to (or4 (drop n O c1 e) -(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: -T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(drop n O c1 (CHead e0 (Flat f) u1)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v -u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n O c1 (CHead e1 -(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat -f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (_: T).(drop n O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 -O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))) -\def - \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (c1: C).(\forall (c2: -C).(\forall (v: T).((csubst0 n0 v c1 c2) \to (\forall (e: C).((drop n0 O c2 -e) \to (or4 (drop n0 O c1 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) -(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O -c1 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat -f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(drop n0 O c1 (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c1 (CHead e1 -(Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2))))))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda -(H: (csubst0 O v c1 c2)).(\lambda (e: C).(\lambda (H0: (drop O O c2 -e)).(eq_ind C c2 (\lambda (c: C).(or4 (drop O O c1 c) (ex3_4 F C T T (\lambda -(f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c (CHead e0 -(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda -(_: T).(drop O O c1 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c -(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(drop O O c1 (CHead e1 (Flat f) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) -(ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (u2: T).(eq C c (CHead e2 (Flat f) u2))))))) (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop O -O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda -(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2))))))))) (insert_eq nat O (\lambda (n0: nat).(csubst0 n0 v c1 c2)) -(\lambda (n0: nat).(or4 (drop n0 n0 c1 c2) (ex3_4 F C T T (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c2 (CHead e0 (Flat -f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(drop n0 n0 c1 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 n0 v u1 u2)))))) (ex3_4 F C C T -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c2 -(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(drop n0 n0 c1 (CHead e1 (Flat f) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 v e1 -e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C c2 (CHead e2 (Flat f) u2))))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop n0 n0 c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda -(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 n0 v u1 -u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 n0 v e1 e2))))))))) (\lambda (y: nat).(\lambda -(H1: (csubst0 y v c1 c2)).(csubst0_ind (\lambda (n0: nat).(\lambda (t: -T).(\lambda (c: C).(\lambda (c0: C).((eq nat n0 O) \to (or4 (drop n0 n0 c c0) -(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: -T).(eq C c0 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(drop n0 n0 c (CHead e0 (Flat f) u1)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 n0 -t u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 n0 c (CHead e1 -(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 n0 t e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e2 -(Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (_: T).(drop n0 n0 c (CHead e1 (Flat f) u1))))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 n0 t u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 n0 t e1 e2))))))))))))) (\lambda -(k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (v0: T).(\forall -(u1: T).(\forall (u2: T).((subst0 i v0 u1 u2) \to (\forall (c: C).((eq nat (s -k0 i) O) \to (or4 (drop (s k0 i) (s k0 i) (CHead c k0 u1) (CHead c k0 u2)) -(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: -T).(eq C (CHead c k0 u2) (CHead e0 (Flat f) u4)))))) (\lambda (f: F).(\lambda -(e0: C).(\lambda (u3: T).(\lambda (_: T).(drop (s k0 i) (s k0 i) (CHead c k0 -u1) (CHead e0 (Flat f) u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: -T).(\lambda (u4: T).(subst0 (s k0 i) v0 u3 u4)))))) (ex3_4 F C C T (\lambda -(f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c k0 u2) -(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(drop (s k0 i) (s k0 i) (CHead c k0 u1) (CHead e1 (Flat f) -u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (s k0 i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c k0 -u2) (CHead e2 (Flat f) u4))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u3: T).(\lambda (_: T).(drop (s k0 i) (s k0 i) (CHead c k0 -u1) (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u3: T).(\lambda (u4: T).(subst0 (s k0 i) v0 u3 u4)))))) (\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: -T).(csubst0 (s k0 i) v0 e1 e2)))))))))))))))) (\lambda (b: B).(\lambda (i: -nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0 -i v0 u1 u2)).(\lambda (c: C).(\lambda (H3: (eq nat (S i) O)).(let H4 \def -(eq_ind nat (S i) (\lambda (ee: nat).(match ee in nat return (\lambda (_: -nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H3) in -(False_ind (or4 (drop (S i) (S i) (CHead c (Bind b) u1) (CHead c (Bind b) -u2)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda -(u4: T).(eq C (CHead c (Bind b) u2) (CHead e0 (Flat f) u4)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop (S i) (S i) (CHead -c (Bind b) u1) (CHead e0 (Flat f) u3)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u3: T).(\lambda (u4: T).(subst0 (S i) v0 u3 u4)))))) (ex3_4 F C -C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C -(CHead c (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S i) (S i) (CHead c (Bind b) -u1) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (S i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda -(f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq -C (CHead c (Bind b) u2) (CHead e2 (Flat f) u4))))))) (\lambda (f: F).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (_: T).(drop (S i) (S i) -(CHead c (Bind b) u1) (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda -(_: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 (S i) v0 u3 -u4)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (S i) v0 e1 e2)))))))) H4)))))))))) (\lambda (f: -F).(\lambda (i: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: -T).(\lambda (H2: (subst0 i v0 u1 u2)).(\lambda (c: C).(\lambda (H3: (eq nat i -O)).(let H4 \def (eq_ind nat i (\lambda (n0: nat).(subst0 n0 v0 u1 u2)) H2 O -H3) in (eq_ind_r nat O (\lambda (n0: nat).(or4 (drop n0 n0 (CHead c (Flat f) -u1) (CHead c (Flat f) u2)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c (Flat f) u2) (CHead e0 -(Flat f0) u4)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u3: -T).(\lambda (_: T).(drop n0 n0 (CHead c (Flat f) u1) (CHead e0 (Flat f0) -u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: -T).(subst0 n0 v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c (Flat f) u2) (CHead e2 -(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(drop n0 n0 (CHead c (Flat f) u1) (CHead e1 (Flat f0) u)))))) (\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 v0 e1 -e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c (Flat f) u2) (CHead e2 -(Flat f0) u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u3: T).(\lambda (_: T).(drop n0 n0 (CHead c (Flat f) u1) (CHead -e1 (Flat f0) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u3: T).(\lambda (u4: T).(subst0 n0 v0 u3 u4)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -n0 v0 e1 e2))))))))) (or4_intro1 (drop O O (CHead c (Flat f) u1) (CHead c -(Flat f) u2)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (u4: T).(eq C (CHead c (Flat f) u2) (CHead e0 (Flat f0) u4)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop O O -(CHead c (Flat f) u1) (CHead e0 (Flat f0) u3)))))) (\lambda (_: F).(\lambda -(_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (ex3_4 F C -C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C -(CHead c (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u: T).(drop O O (CHead c (Flat f) u1) -(CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C -(CHead c (Flat f) u2) (CHead e2 (Flat f0) u4))))))) (\lambda (f0: F).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c -(Flat f) u1) (CHead e1 (Flat f0) u3))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v0 e1 e2))))))) (ex3_4_intro F C T T (\lambda (f0: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c (Flat f) -u2) (CHead e0 (Flat f0) u4)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda -(u3: T).(\lambda (_: T).(drop O O (CHead c (Flat f) u1) (CHead e0 (Flat f0) -u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: -T).(subst0 O v0 u3 u4))))) f c u1 u2 (refl_equal C (CHead c (Flat f) u2)) -(drop_refl (CHead c (Flat f) u1)) H4)) i H3)))))))))) k)) (\lambda (k: -K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (c3: C).(\forall (c4: -C).(\forall (v0: T).((csubst0 i v0 c3 c4) \to ((((eq nat i O) \to (or4 (drop -i i c3 c4) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (u2: T).(eq C c4 (CHead e0 (Flat f) u2)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop i i c3 (CHead e0 -(Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 i v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(eq C c4 (CHead e2 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop i i c3 -(CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 i v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 -(CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(drop i i c3 (CHead e1 (Flat f) u1))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 i v0 u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v0 e1 e2)))))))))) \to -(\forall (u: T).((eq nat (s k0 i) O) \to (or4 (drop (s k0 i) (s k0 i) (CHead -c3 k0 u) (CHead c4 k0 u)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 k0 u) (CHead e0 (Flat f) -u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(drop (s k0 i) (s k0 i) (CHead c3 k0 u) (CHead e0 (Flat f) u1)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (s -k0 i) v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u0: T).(eq C (CHead c4 k0 u) (CHead e2 (Flat f) u0)))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(drop (s k0 -i) (s k0 i) (CHead c3 k0 u) (CHead e1 (Flat f) u0)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (s k0 i) v0 e1 -e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 k0 u) (CHead e2 (Flat f) -u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (_: T).(drop (s k0 i) (s k0 i) (CHead c3 k0 u) (CHead e1 (Flat f) -u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 (s k0 i) v0 u1 u2)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (s k0 i) v0 -e1 e2))))))))))))))))) (\lambda (b: B).(\lambda (i: nat).(\lambda (c3: -C).(\lambda (c4: C).(\lambda (v0: T).(\lambda (_: (csubst0 i v0 c3 -c4)).(\lambda (_: (((eq nat i O) \to (or4 (drop i i c3 c4) (ex3_4 F C T T -(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 -(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(drop i i c3 (CHead e0 (Flat f) u1)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v0 u1 u2)))))) -(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(eq C c4 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(drop i i c3 (CHead e1 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e2 (Flat f) u2))))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop i i c3 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v0 u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (u: T).(\lambda (H4: (eq nat -(S i) O)).(let H5 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee in nat -return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow -True])) I O H4) in (False_ind (or4 (drop (S i) (S i) (CHead c3 (Bind b) u) -(CHead c4 (Bind b) u)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 (Bind b) u) (CHead e0 -(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S i) (S i) (CHead c3 (Bind b) u) (CHead e0 (Flat f) u1)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (S -i) v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u0: T).(eq C (CHead c4 (Bind b) u) (CHead e2 (Flat f) -u0)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: -T).(drop (S i) (S i) (CHead c3 (Bind b) u) (CHead e1 (Flat f) u0)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (S -i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 (Bind b) u) (CHead -e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S i) (S i) (CHead c3 (Bind b) u) -(CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (S i) v0 u1 u2)))))) (\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: -T).(csubst0 (S i) v0 e1 e2)))))))) H5))))))))))) (\lambda (f: F).(\lambda (i: -nat).(\lambda (c3: C).(\lambda (c4: C).(\lambda (v0: T).(\lambda (H2: -(csubst0 i v0 c3 c4)).(\lambda (H3: (((eq nat i O) \to (or4 (drop i i c3 c4) -(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda -(u2: T).(eq C c4 (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(drop i i c3 (CHead e0 (Flat f0) u1)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i -v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C c4 (CHead e2 (Flat f0) u)))))) (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop i i c3 (CHead e1 -(Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 i v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e2 -(Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(drop i i c3 (CHead e1 (Flat f0) -u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 i v0 u1 u2)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v0 e1 -e2))))))))))).(\lambda (u: T).(\lambda (H4: (eq nat i O)).(let H5 \def -(eq_ind nat i (\lambda (n0: nat).((eq nat n0 O) \to (or4 (drop n0 n0 c3 c4) -(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda -(u2: T).(eq C c4 (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(drop n0 n0 c3 (CHead e0 (Flat f0) -u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 n0 v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u0: T).(eq C c4 (CHead e2 (Flat f0) u0)))))) -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(drop n0 -n0 c3 (CHead e1 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda -(f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq -C c4 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 n0 c3 (CHead e1 (Flat f0) -u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 n0 v0 u1 u2)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 n0 v0 e1 -e2)))))))))) H3 O H4) in (let H6 \def (eq_ind nat i (\lambda (n0: -nat).(csubst0 n0 v0 c3 c4)) H2 O H4) in (eq_ind_r nat O (\lambda (n0: -nat).(or4 (drop n0 n0 (CHead c3 (Flat f) u) (CHead c4 (Flat f) u)) (ex3_4 F C -T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C -(CHead c4 (Flat f) u) (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda -(e0: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 n0 (CHead c3 (Flat f) u) -(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 n0 v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(eq C (CHead c4 (Flat f) -u) (CHead e2 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u0: T).(drop n0 n0 (CHead c3 (Flat f) u) (CHead e1 (Flat f0) -u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 (Flat f) -u) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 n0 (CHead c3 (Flat f) u) -(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 n0 v0 u1 u2)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -n0 v0 e1 e2))))))))) (or4_intro2 (drop O O (CHead c3 (Flat f) u) (CHead c4 -(Flat f) u)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (u2: T).(eq C (CHead c4 (Flat f) u) (CHead e0 (Flat f0) u2)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop O O -(CHead c3 (Flat f) u) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda -(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (ex3_4 F C -C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(eq C -(CHead c4 (Flat f) u) (CHead e2 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u0: T).(drop O O (CHead c3 (Flat f) u) -(CHead e1 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C -(CHead c4 (Flat f) u) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop O O (CHead c3 -(Flat f) u) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v0 e1 e2))))))) (ex3_4_intro F C C T (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(eq C (CHead c4 (Flat f) -u) (CHead e2 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u0: T).(drop O O (CHead c3 (Flat f) u) (CHead e1 (Flat f0) -u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v0 e1 e2))))) f c3 c4 u (refl_equal C (CHead c4 (Flat f) u)) -(drop_refl (CHead c3 (Flat f) u)) H6)) i H4)))))))))))) k)) (\lambda (k: -K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (v0: T).(\forall (u1: -T).(\forall (u2: T).((subst0 i v0 u1 u2) \to (\forall (c3: C).(\forall (c4: -C).((csubst0 i v0 c3 c4) \to ((((eq nat i O) \to (or4 (drop i i c3 c4) (ex3_4 -F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq -C c4 (CHead e0 (Flat f) u4)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda -(u3: T).(\lambda (_: T).(drop i i c3 (CHead e0 (Flat f) u3)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 i v0 u3 u4)))))) -(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(eq C c4 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(drop i i c3 (CHead e1 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u4: T).(eq C c4 (CHead e2 (Flat f) u4))))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda -(_: T).(drop i i c3 (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 i v0 u3 u4)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 i v0 e1 e2)))))))))) \to ((eq nat (s k0 i) O) \to (or4 (drop -(s k0 i) (s k0 i) (CHead c3 k0 u1) (CHead c4 k0 u2)) (ex3_4 F C T T (\lambda -(f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 k0 -u2) (CHead e0 (Flat f) u4)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda -(u3: T).(\lambda (_: T).(drop (s k0 i) (s k0 i) (CHead c3 k0 u1) (CHead e0 -(Flat f) u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda -(u4: T).(subst0 (s k0 i) v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c4 k0 u2) -(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(drop (s k0 i) (s k0 i) (CHead c3 k0 u1) (CHead e1 (Flat -f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (s k0 i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 k0 -u2) (CHead e2 (Flat f) u4))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u3: T).(\lambda (_: T).(drop (s k0 i) (s k0 i) (CHead c3 k0 -u1) (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u3: T).(\lambda (u4: T).(subst0 (s k0 i) v0 u3 u4)))))) (\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: -T).(csubst0 (s k0 i) v0 e1 e2))))))))))))))))))) (\lambda (b: B).(\lambda (i: -nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0 -i v0 u1 u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (_: (csubst0 i v0 c3 -c4)).(\lambda (_: (((eq nat i O) \to (or4 (drop i i c3 c4) (ex3_4 F C T T -(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C c4 -(CHead e0 (Flat f) u4)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u3: -T).(\lambda (_: T).(drop i i c3 (CHead e0 (Flat f) u3)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 i v0 u3 u4)))))) -(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(eq C c4 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(drop i i c3 (CHead e1 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u4: T).(eq C c4 (CHead e2 (Flat f) u4))))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda -(_: T).(drop i i c3 (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 i v0 u3 u4)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (H5: (eq nat (S i) O)).(let H6 -\def (eq_ind nat (S i) (\lambda (ee: nat).(match ee in nat return (\lambda -(_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H5) -in (False_ind (or4 (drop (S i) (S i) (CHead c3 (Bind b) u1) (CHead c4 (Bind -b) u2)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (u4: T).(eq C (CHead c4 (Bind b) u2) (CHead e0 (Flat f) u4)))))) -(\lambda (f: F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop (S i) -(S i) (CHead c3 (Bind b) u1) (CHead e0 (Flat f) u3)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 (S i) v0 u3 -u4)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C (CHead c4 (Bind b) u2) (CHead e2 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S i) -(S i) (CHead c3 (Bind b) u1) (CHead e1 (Flat f) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (S i) v0 e1 -e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 (Bind b) u2) (CHead e2 -(Flat f) u4))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u3: T).(\lambda (_: T).(drop (S i) (S i) (CHead c3 (Bind b) u1) (CHead e1 -(Flat f) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda -(u3: T).(\lambda (u4: T).(subst0 (S i) v0 u3 u4)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -(S i) v0 e1 e2)))))))) H6))))))))))))) (\lambda (f: F).(\lambda (i: -nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 -i v0 u1 u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (H3: (csubst0 i v0 c3 -c4)).(\lambda (H4: (((eq nat i O) \to (or4 (drop i i c3 c4) (ex3_4 F C T T -(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C c4 -(CHead e0 (Flat f0) u4)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u3: -T).(\lambda (_: T).(drop i i c3 (CHead e0 (Flat f0) u3)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 i v0 u3 u4)))))) -(ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(eq C c4 (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(drop i i c3 (CHead e1 (Flat f0) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C c4 (CHead e2 (Flat f0) -u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: -T).(\lambda (_: T).(drop i i c3 (CHead e1 (Flat f0) u3))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 -i v0 u3 u4)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (H5: (eq nat i -O)).(let H6 \def (eq_ind nat i (\lambda (n0: nat).((eq nat n0 O) \to (or4 -(drop n0 n0 c3 c4) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda -(_: T).(\lambda (u4: T).(eq C c4 (CHead e0 (Flat f0) u4)))))) (\lambda (f0: -F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop n0 n0 c3 (CHead e0 -(Flat f0) u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda -(u4: T).(subst0 n0 v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda -(_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c4 (CHead e2 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 n0 -c3 (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C c4 -(CHead e2 (Flat f0) u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u3: T).(\lambda (_: T).(drop n0 n0 c3 (CHead e1 (Flat f0) -u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u3: -T).(\lambda (u4: T).(subst0 n0 v0 u3 u4)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 n0 v0 e1 -e2)))))))))) H4 O H5) in (let H7 \def (eq_ind nat i (\lambda (n0: -nat).(csubst0 n0 v0 c3 c4)) H3 O H5) in (let H8 \def (eq_ind nat i (\lambda -(n0: nat).(subst0 n0 v0 u1 u2)) H2 O H5) in (eq_ind_r nat O (\lambda (n0: -nat).(or4 (drop n0 n0 (CHead c3 (Flat f) u1) (CHead c4 (Flat f) u2)) (ex3_4 F -C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq -C (CHead c4 (Flat f) u2) (CHead e0 (Flat f0) u4)))))) (\lambda (f0: -F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop n0 n0 (CHead c3 -(Flat f) u1) (CHead e0 (Flat f0) u3)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u3: T).(\lambda (u4: T).(subst0 n0 v0 u3 u4)))))) (ex3_4 F C C T -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C -(CHead c4 (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 n0 (CHead c3 (Flat f) u1) -(CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C -(CHead c4 (Flat f) u2) (CHead e2 (Flat f0) u4))))))) (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (_: T).(drop n0 -n0 (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u3))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 -n0 v0 u3 u4)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 n0 v0 e1 e2))))))))) (or4_intro3 (drop O O -(CHead c3 (Flat f) u1) (CHead c4 (Flat f) u2)) (ex3_4 F C T T (\lambda (f0: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 (Flat f) -u2) (CHead e0 (Flat f0) u4)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda -(u3: T).(\lambda (_: T).(drop O O (CHead c3 (Flat f) u1) (CHead e0 (Flat f0) -u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: -T).(subst0 O v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c4 (Flat f) u2) (CHead e2 -(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(drop O O (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u)))))) (\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 -e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 (Flat f) u2) (CHead e2 -(Flat f0) u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c3 (Flat f) u1) (CHead -e1 (Flat f0) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v0 e1 e2))))))) (ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 (Flat f) -u2) (CHead e2 (Flat f0) u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c3 (Flat f) u1) -(CHead e1 (Flat f0) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v0 e1 e2)))))) f c3 c4 u1 u2 (refl_equal C (CHead c4 (Flat f) u2)) -(drop_refl (CHead c3 (Flat f) u1)) H8 H7)) i H5))))))))))))))) k)) y v c1 c2 -H1))) H) e (drop_gen_refl c2 e H0)))))))) (\lambda (n0: nat).(\lambda (IHn: -((\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 n0 v c1 c2) \to -(\forall (e: C).((drop n0 O c2 e) \to (or4 (drop n0 O c1 e) (ex3_4 F C T T -(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e -(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(drop n0 O c1 (CHead e0 (Flat f) u1)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c1 (CHead e1 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O -v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop n0 O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda -(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 -u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))))).(\lambda (c1: C).(C_ind -(\lambda (c: C).(\forall (c2: C).(\forall (v: T).((csubst0 (S n0) v c c2) \to -(\forall (e: C).((drop (S n0) O c2 e) \to (or4 (drop (S n0) O c e) (ex3_4 F C -T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e -(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f) u1)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O c (CHead e1 (Flat f) -u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat -f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f) u1))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 -O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))) (\lambda (n1: -nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (H: (csubst0 (S n0) v (CSort -n1) c2)).(\lambda (e: C).(\lambda (_: (drop (S n0) O c2 e)).(csubst0_gen_sort -c2 v (S n0) n1 H (or4 (drop (S n0) O (CSort n1) e) (ex3_4 F C T T (\lambda -(f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 -(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S n0) O (CSort n1) (CHead e0 (Flat f) u1)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CSort n1) (CHead e1 (Flat -f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat -f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O (CSort n1) (CHead e1 (Flat f) u1))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))))))) -(\lambda (c: C).(\lambda (H: ((\forall (c2: C).(\forall (v: T).((csubst0 (S -n0) v c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (or4 (drop (S n0) O -c e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda -(u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f) -u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) -O c (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e -(CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f) -u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2))))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda -(v: T).(\lambda (H0: (csubst0 (S n0) v (CHead c k t) c2)).(\lambda (e: -C).(\lambda (H1: (drop (S n0) O c2 e)).(or3_ind (ex3_2 T nat (\lambda (_: -T).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (u2: T).(\lambda (_: -nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j -v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k -j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda -(c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_: -T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k j))))) (\lambda (u2: -T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda -(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: -T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))) (or4 (drop (S -n0) O (CHead c k t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) -(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O (CHead c k t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e -(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O -v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u1))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 -O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H2: (ex3_2 T nat -(\lambda (_: T).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (u2: -T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: -nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: -nat).(eq nat (S n0) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 -(CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))) (or4 -(drop (S n0) O (CHead c k t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) -(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O (CHead c k t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e -(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O -v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u1))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 -O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: T).(\lambda -(x1: nat).(\lambda (H3: (eq nat (S n0) (s k x1))).(\lambda (H4: (eq C c2 -(CHead c k x0))).(\lambda (H5: (subst0 x1 v t x0)).(let H6 \def (eq_ind C c2 -(\lambda (c0: C).(drop (S n0) O c0 e)) H1 (CHead c k x0) H4) in (K_ind -(\lambda (k0: K).((eq nat (S n0) (s k0 x1)) \to ((drop (r k0 n0) O c e) \to -(or4 (drop (S n0) O (CHead c k0 t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda -(e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) -(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O (CHead c k0 t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e -(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(drop (S n0) O (CHead c k0 t) (CHead e1 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O -v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S n0) O (CHead c k0 t) (CHead e1 (Flat f) u1))))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))) (\lambda (b: -B).(\lambda (H7: (eq nat (S n0) (s (Bind b) x1))).(\lambda (H8: (drop (r -(Bind b) n0) O c e)).(let H9 \def (f_equal nat nat (\lambda (e0: nat).(match -e0 in nat return (\lambda (_: nat).nat) with [O \Rightarrow n0 | (S n1) -\Rightarrow n1])) (S n0) (S x1) H7) in (let H10 \def (eq_ind_r nat x1 -(\lambda (n1: nat).(subst0 n1 v t x0)) H5 n0 H9) in (or4_intro0 (drop (S n0) -O (CHead c (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) -(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead -e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e -(CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) -(CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2))))))) (drop_drop (Bind b) n0 c e H8 t))))))) (\lambda (f: -F).(\lambda (H7: (eq nat (S n0) (s (Flat f) x1))).(\lambda (H8: (drop (r -(Flat f) n0) O c e)).(let H9 \def (f_equal nat nat (\lambda (e0: nat).e0) (S -n0) x1 H7) in (let H10 \def (eq_ind_r nat x1 (\lambda (n1: nat).(subst0 n1 v -t x0)) H5 (S n0) H9) in (or4_intro0 (drop (S n0) O (CHead c (Flat f) t) e) -(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda -(u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) -(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat -f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 -e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop -(Flat f) n0 c e H8 t))))))) k H3 (drop_gen_drop k c e x0 n0 H6)))))))) H2)) -(\lambda (H2: (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) -(s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) -(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))).(ex3_2_ind C nat -(\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (c3: -C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: -nat).(csubst0 j v c c3))) (or4 (drop (S n0) O (CHead c k t) e) (ex3_4 F C T T -(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e -(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O (CHead c k t) (CHead e0 (Flat f) u1)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v -u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c k -t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e -(CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k t) (CHead e1 -(Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2)))))))) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H3: (eq nat (S n0) -(s k x1))).(\lambda (H4: (eq C c2 (CHead x0 k t))).(\lambda 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(u2: T).(eq -C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k0 t) (CHead -e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2))))))))))) (\lambda (b: B).(\lambda (H7: (eq nat (S n0) (s (Bind b) -x1))).(\lambda (H8: (drop (r (Bind b) n0) O x0 e)).(let H9 \def (f_equal nat -nat (\lambda (e0: nat).(match e0 in nat return (\lambda (_: nat).nat) with [O -\Rightarrow n0 | (S n1) \Rightarrow n1])) (S n0) (S x1) H7) in (let H10 \def -(eq_ind_r nat x1 (\lambda (n1: nat).(csubst0 n1 v c x0)) H5 n0 H9) in (let -H11 \def (IHn c x0 v H10 e H8) in (or4_ind (drop n0 O c e) (ex3_4 F C T T -(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e -(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda 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-(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: -T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) -(CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat -f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 -e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) -(\lambda (_: F).(\lambda (_: 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C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop -(Bind b) n0 c e H12 t))) (\lambda (H12: (ex3_4 F C T T (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat -f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))).(ex3_4_ind F C -T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e -(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: 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(u2: T).(subst0 O v u1 -u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c -(Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c -(Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead c (Bind -b) t) (CHead x3 (Flat x2) x5)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0 -(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v -u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C (CHead x3 (Flat x2) x5) (CHead e2 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) -O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C -C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e2 (Flat f) -u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) -u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) -(ex3_4_intro F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0 (Flat f) u2)))))) -(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))) -x2 x3 x4 x5 (refl_equal C (CHead x3 (Flat x2) x5)) (drop_drop (Bind b) n0 c -(CHead x3 (Flat x2) x4) H14 t) H15)) e H13)))))))) H12)) (\lambda (H12: -(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 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(\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 -e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda -(x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: T).(\lambda (H13: (eq -C e (CHead x4 (Flat x2) x5))).(\lambda (H14: (drop n0 O c (CHead x3 (Flat x2) -x5))).(\lambda (H15: (csubst0 O v x3 x4)).(eq_ind_r C (CHead x4 (Flat x2) x5) -(\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T -(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 -(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) -u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) -O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C -C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 -O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro2 (drop (S n0) O -(CHead c (Bind b) t) (CHead x4 (Flat x2) x5)) (ex3_4 F C T T (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat -x2) x5) (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda -(u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat -f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 -(Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O -v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 -(Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat -f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) -(ex3_4_intro F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) -O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) x2 x3 x4 x5 -(refl_equal C (CHead x4 (Flat x2) x5)) (drop_drop (Bind b) n0 c (CHead x3 -(Flat x2) x5) H14 t) H15)) e H13)))))))) H12)) (\lambda (H12: (ex4_5 F C C T -T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c (CHead e1 -(Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2)))))))).(ex4_5_ind F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop n0 O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e) -(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: -T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) -(CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat -f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 -e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda -(x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: T).(\lambda (x6: -T).(\lambda (H13: (eq C e (CHead x4 (Flat x2) x6))).(\lambda (H14: (drop n0 O -c (CHead x3 (Flat x2) x5))).(\lambda (H15: (subst0 O v x5 x6)).(\lambda (H16: -(csubst0 O v x3 x4)).(eq_ind_r C (CHead x4 (Flat x2) x6) (\lambda (c0: -C).(or4 (drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat -f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 -u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c -(Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c -(Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead c (Bind -b) t) (CHead x4 (Flat x2) x6)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e0 -(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v -u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) -O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C -C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 (Flat f) -u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) -u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) -(ex4_5_intro F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 -(Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat -f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) -x2 x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x2) x6)) (drop_drop (Bind b) n0 -c (CHead x3 (Flat x2) x5) H14 t) H15 H16)) e H13)))))))))) H12)) H11))))))) -(\lambda (f: F).(\lambda (H7: (eq nat (S n0) (s (Flat f) x1))).(\lambda (H8: -(drop (r (Flat f) n0) O x0 e)).(let H9 \def (f_equal nat nat (\lambda (e0: -nat).e0) (S n0) x1 H7) in (let H10 \def (eq_ind_r nat x1 (\lambda (n1: -nat).(csubst0 n1 v c x0)) H5 (S n0) H9) in (let H11 \def (H x0 v H10 e H8) in -(or4_ind (drop (S n0) O c e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O c (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat -f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(drop (S n0) O c (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead -e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: 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-(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 -O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H12: (drop (S n0) -O c e)).(or4_intro0 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T -(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e -(CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) -u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S -n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) -(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 -O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Flat f) n0 c e -H12 t))) (\lambda (H12: (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O c (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))).(ex3_4_ind F C T T (\lambda -(f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 -(Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f0) u1)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))) -(or4 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat -f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 -u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c -(Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c -(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda -(x4: T).(\lambda (x5: T).(\lambda (H13: (eq C e (CHead x3 (Flat x2) -x5))).(\lambda (H14: (drop (S n0) O c (CHead x3 (Flat x2) x4))).(\lambda -(H15: (subst0 O v x4 x5)).(eq_ind_r C (CHead x3 (Flat x2) x5) (\lambda (c0: -C).(or4 (drop (S n0) O (CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat -f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 -u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c -(Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(eq C c0 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c -(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead c (Flat -f) t) (CHead x3 (Flat x2) x5)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0 -(Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) -u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x3 (Flat x2) x5) (CHead e2 -(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O -v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e2 -(Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) -(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2))))))) (ex3_4_intro F C T T (\lambda (f0: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0 -(Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) -u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 O v u1 u2))))) x2 x3 x4 x5 (refl_equal C (CHead x3 (Flat x2) x5)) -(drop_drop (Flat f) n0 c (CHead x3 (Flat x2) x4) H14 t) H15)) e H13)))))))) -H12)) (\lambda (H12: (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O c (CHead -e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat -f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(drop (S n0) O c (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) (or4 (drop (S n0) -O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead -e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e -(CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) -(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda -(x5: T).(\lambda (H13: (eq C e (CHead x4 (Flat x2) x5))).(\lambda (H14: (drop -(S n0) O c (CHead x3 (Flat x2) x5))).(\lambda (H15: (csubst0 O v x3 -x4)).(eq_ind_r C (CHead x4 (Flat x2) x5) (\lambda (c0: C).(or4 (drop (S n0) O -(CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat f0) u2)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead -e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 -(CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) -(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead c (Flat f) t) (CHead x4 -(Flat x2) x5)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x5) (CHead e0 (Flat f0) -u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 -u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S -n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) -(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 (Flat f0) -u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) -u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) -(ex3_4_intro F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S -n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) -x2 x3 x4 x5 (refl_equal C (CHead x4 (Flat x2) x5)) (drop_drop (Flat f) n0 c -(CHead x3 (Flat x2) x5) H14 t) H15)) e H13)))))))) H12)) (\lambda (H12: -(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O c (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e -(CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f0) -u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) -(or4 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat -f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 -u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c -(Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c -(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda -(x4: C).(\lambda (x5: T).(\lambda (x6: T).(\lambda (H13: (eq C e (CHead x4 -(Flat x2) x6))).(\lambda (H14: (drop (S n0) O c (CHead x3 (Flat x2) -x5))).(\lambda (H15: (subst0 O v x5 x6)).(\lambda (H16: (csubst0 O v x3 -x4)).(eq_ind_r C (CHead x4 (Flat x2) x6) (\lambda (c0: C).(or4 (drop (S n0) O -(CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat f0) u2)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead -e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 -(CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) -(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead c (Flat f) t) (CHead x4 -(Flat x2) x6)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e0 (Flat f0) -u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 -u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S -n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) -(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 (Flat f0) -u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) -u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) -(ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 -(Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) -(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2)))))) x2 x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x2) x6)) -(drop_drop (Flat f) n0 c (CHead x3 (Flat x2) x5) H14 t) H15 H16)) e -H13)))))))))) H12)) H11))))))) k H3 (drop_gen_drop k x0 e t n0 H6)))))))) -H2)) (\lambda (H2: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda -(j: nat).(eq nat (S n0) (s k j))))) (\lambda (u2: T).(\lambda (c3: -C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda -(_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: -C).(\lambda (j: nat).(csubst0 j v c c3)))))).(ex4_3_ind T C nat (\lambda (_: -T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k j))))) (\lambda (u2: -T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda -(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: -T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (or4 (drop (S n0) -O (CHead c k t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda -(_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c -k t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat -f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) -(ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O (CHead c k t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: C).(\lambda -(x2: nat).(\lambda (H3: (eq nat (S n0) (s k x2))).(\lambda (H4: (eq C c2 -(CHead x1 k x0))).(\lambda (H5: (subst0 x2 v t x0)).(\lambda (H6: (csubst0 x2 -v c x1)).(let H7 \def (eq_ind C c2 (\lambda (c0: C).(drop (S n0) O c0 e)) H1 -(CHead x1 k x0) H4) in (K_ind (\lambda (k0: K).((eq nat (S n0) (s k0 x2)) \to -((drop (r k0 n0) O x1 e) \to (or4 (drop (S n0) O (CHead c k0 t) e) (ex3_4 F C -T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e -(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O (CHead c k0 t) (CHead e0 (Flat f) u1)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v -u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c -k0 t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda -(f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq -C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k0 t) (CHead -e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2))))))))))) (\lambda (b: B).(\lambda (H8: (eq nat (S n0) (s (Bind b) -x2))).(\lambda (H9: (drop (r (Bind b) n0) O x1 e)).(let H10 \def (f_equal nat -nat (\lambda (e0: nat).(match e0 in nat return (\lambda (_: nat).nat) with [O -\Rightarrow n0 | (S n1) \Rightarrow n1])) (S n0) (S x2) H8) in (let H11 \def -(eq_ind_r nat x2 (\lambda (n1: nat).(csubst0 n1 v c x1)) H6 n0 H10) in (let -H12 \def (eq_ind_r nat x2 (\lambda (n1: nat).(subst0 n1 v t x0)) H5 n0 H10) -in (let H13 \def (IHn c x1 v H11 e H9) in (or4_ind (drop n0 O c e) (ex3_4 F C -T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e -(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O -v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop n0 O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e) -(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: -T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) -(CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat -f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 -e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda -(H14: (drop n0 O c e)).(or4_intro0 (drop (S n0) O (CHead c (Bind b) t) e) -(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: -T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) -(CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat -f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 -e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop -(Bind b) n0 c e H14 t))) (\lambda (H14: (ex3_4 F C T T (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat -f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))).(ex3_4_ind F C -T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e -(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))) -(or4 (drop (S n0) O (CHead c (Bind b) t) e) (ex3_4 F C T T (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat -f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 -u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c -(Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c -(Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda -(x5: T).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x4 (Flat x3) -x6))).(\lambda (H16: (drop n0 O c (CHead x4 (Flat x3) x5))).(\lambda (H17: -(subst0 O v x5 x6)).(eq_ind_r C (CHead x4 (Flat x3) x6) (\lambda (c0: C).(or4 -(drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat -f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 -u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c -(Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c -(Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead c (Bind -b) t) (CHead x4 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0 -(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v -u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C (CHead x4 (Flat x3) x6) (CHead e2 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) -O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C -C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (u2: T).(eq C (CHead x4 (Flat x3) x6) (CHead e2 (Flat f) -u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) -u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) -(ex3_4_intro F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (u2: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0 (Flat f) u2)))))) -(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))) -x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x3) x6)) (drop_drop (Bind b) n0 c -(CHead x4 (Flat x3) x5) H16 t) H17)) e H15)))))))) H14)) (\lambda (H14: -(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O -v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1 -(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 O v e1 e2))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e) -(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: -T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) -(CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat -f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 -e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda -(x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (H15: (eq -C e (CHead x5 (Flat x3) x6))).(\lambda (H16: (drop n0 O c (CHead x4 (Flat x3) -x6))).(\lambda (H17: (csubst0 O v x4 x5)).(eq_ind_r C (CHead x5 (Flat x3) x6) -(\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T -(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 -(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) -u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) -O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C -C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 -O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro2 (drop (S n0) O -(CHead c (Bind b) t) (CHead x5 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat -x3) x6) (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda -(u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat -f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2 -(Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O -v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2 -(Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat -f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) -(ex3_4_intro F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) -O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) x3 x4 x5 x6 -(refl_equal C (CHead x5 (Flat x3) x6)) (drop_drop (Bind b) n0 c (CHead x4 -(Flat x3) x6) H16 t) H17)) e H15)))))))) H14)) (\lambda (H14: (ex4_5 F C C T -T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c (CHead e1 -(Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2)))))))).(ex4_5_ind F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop n0 O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e) -(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: -T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) -(CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat -f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 -e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda -(x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (x7: -T).(\lambda (H15: (eq C e (CHead x5 (Flat x3) x7))).(\lambda (H16: (drop n0 O -c (CHead x4 (Flat x3) x6))).(\lambda (H17: (subst0 O v x6 x7)).(\lambda (H18: -(csubst0 O v x4 x5)).(eq_ind_r C (CHead x5 (Flat x3) x7) (\lambda (c0: -C).(or4 (drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat -f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 -u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c -(Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c -(Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead c (Bind -b) t) (CHead x5 (Flat x3) x7)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e0 -(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v -u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) -O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C -C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2 (Flat f) -u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) -u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) -(ex4_5_intro F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2 -(Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat -f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) -x3 x4 x5 x6 x7 (refl_equal C (CHead x5 (Flat x3) x7)) (drop_drop (Bind b) n0 -c (CHead x4 (Flat x3) x6) H16 t) H17 H18)) e H15)))))))))) H14)) H13)))))))) -(\lambda (f: F).(\lambda (H8: (eq nat (S n0) (s (Flat f) x2))).(\lambda (H9: -(drop (r (Flat f) n0) O x1 e)).(let H10 \def (f_equal nat nat (\lambda (e0: -nat).e0) (S n0) x2 H8) in (let H11 \def (eq_ind_r nat x2 (\lambda (n1: -nat).(csubst0 n1 v c x1)) H6 (S n0) H10) in (let H12 \def (eq_ind_r nat x2 -(\lambda (n1: nat).(subst0 n1 v t x0)) H5 (S n0) H10) in (let H13 \def (H x1 -v H11 e H9) in (or4_ind (drop (S n0) O c e) (ex3_4 F C T T (\lambda (f0: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat -f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(drop (S n0) O c (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e -(CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(drop (S n0) O c (CHead e1 (Flat f0) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) -(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O c (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead c (Flat f) t) e) -(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda -(u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) -(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat -f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 -e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda -(H14: (drop (S n0) O c e)).(or4_intro0 (drop (S n0) O (CHead c (Flat f) t) e) -(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda -(u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) -(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat -f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 -e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop -(Flat f) n0 c e H14 t))) (\lambda (H14: (ex3_4 F C T T (\lambda (f0: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat -f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(drop (S n0) O c (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))).(ex3_4_ind F C -T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C -e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda -(u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f0) u1)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v -u1 u2))))) (or4 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T -(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e -(CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) -u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S -n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) -(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 -O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x3: F).(\lambda -(x4: C).(\lambda (x5: T).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x4 -(Flat x3) x6))).(\lambda (H16: (drop (S n0) O c (CHead x4 (Flat x3) -x5))).(\lambda (H17: (subst0 O v x5 x6)).(eq_ind_r C (CHead x4 (Flat x3) x6) -(\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Flat f) t) c0) (ex3_4 F C T T -(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 -(CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) -u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S -n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) -(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 -O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro1 (drop (S n0) O -(CHead c (Flat f) t) (CHead x4 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f0: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat -x3) x6) (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) -(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x4 (Flat x3) -x6) (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat -f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat -x3) x6) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c -(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C T T (\lambda (f0: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat -x3) x6) (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) -(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2))))) x3 x4 x5 x6 (refl_equal C (CHead -x4 (Flat x3) x6)) (drop_drop (Flat f) n0 c (CHead x4 (Flat x3) x5) H16 t) -H17)) e H15)))))))) H14)) (\lambda (H14: (ex3_4 F C C T (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat -f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(drop (S n0) O c (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C -C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e -(CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(drop (S n0) O c (CHead e1 (Flat f0) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) -(or4 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat -f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 -u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c -(Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c -(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda -(x5: C).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x5 (Flat x3) -x6))).(\lambda (H16: (drop (S n0) O c (CHead x4 (Flat x3) x6))).(\lambda -(H17: (csubst0 O v x4 x5)).(eq_ind_r C (CHead x5 (Flat x3) x6) (\lambda (c0: -C).(or4 (drop (S n0) O (CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat -f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 -u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c -(Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(eq C c0 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c -(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead c (Flat -f) t) (CHead x5 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x6) (CHead e0 -(Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) -u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2 -(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O -v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2 -(Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) -(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2))))))) (ex3_4_intro F C C T (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2 -(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O -v e1 e2))))) x3 x4 x5 x6 (refl_equal C (CHead x5 (Flat x3) x6)) (drop_drop -(Flat f) n0 c (CHead x4 (Flat x3) x6) H16 t) H17)) e H15)))))))) H14)) -(\lambda (H14: (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) -u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f0) u1))))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C -C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O c (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) O (CHead c (Flat f) t) e) -(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda -(u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) -(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat -f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 -e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda -(x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (x7: -T).(\lambda (H15: (eq C e (CHead x5 (Flat x3) x7))).(\lambda (H16: (drop (S -n0) O c (CHead x4 (Flat x3) x6))).(\lambda (H17: (subst0 O v x6 x7)).(\lambda -(H18: (csubst0 O v x4 x5)).(eq_ind_r C (CHead x5 (Flat x3) x7) (\lambda (c0: -C).(or4 (drop (S n0) O (CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat -f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 -u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c -(Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(eq C c0 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c -(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead c (Flat -f) t) (CHead x5 (Flat x3) x7)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e0 -(Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) -u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2 -(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O -v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2 -(Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) -(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2))))))) (ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat -x3) x7) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c -(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2)))))) x3 x4 x5 x6 x7 (refl_equal C (CHead x5 (Flat -x3) x7)) (drop_drop (Flat f) n0 c (CHead x4 (Flat x3) x6) H16 t) H17 H18)) e -H15)))))))))) H14)) H13)))))))) k H3 (drop_gen_drop k x1 e x0 n0 H7)))))))))) -H2)) (csubst0_gen_head k c c2 t v (S n0) H0))))))))))) c1)))) n). -(* COMMENTS -Initial nodes: 34765 -END *) - -theorem csubst0_drop_lt_back: - \forall (n: nat).(\forall (i: nat).((lt n i) \to (\forall (c1: C).(\forall -(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e2: C).((drop n O -c2 e2) \to (or (drop n O c1 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) -v e1 e2)) (\lambda (e1: C).(drop n O c1 e1)))))))))))) -\def - \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (i: nat).((lt n0 i) -\to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) -\to (\forall (e2: C).((drop n0 O c2 e2) \to (or (drop n0 O c1 e2) (ex2 C -(\lambda (e1: C).(csubst0 (minus i n0) v e1 e2)) (\lambda (e1: C).(drop n0 O -c1 e1))))))))))))) (\lambda (i: nat).(\lambda (_: (lt O i)).(\lambda (c1: -C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 i v c1 -c2)).(\lambda (e2: C).(\lambda (H1: (drop O O c2 e2)).(eq_ind C c2 (\lambda -(c: C).(or (drop O O c1 c) (ex2 C (\lambda (e1: C).(csubst0 (minus i O) v e1 -c)) (\lambda (e1: C).(drop O O c1 e1))))) (eq_ind nat i (\lambda (n0: -nat).(or (drop O O c1 c2) (ex2 C (\lambda (e1: C).(csubst0 n0 v e1 c2)) -(\lambda (e1: C).(drop O O c1 e1))))) (or_intror (drop O O c1 c2) (ex2 C -(\lambda (e1: C).(csubst0 i v e1 c2)) (\lambda (e1: C).(drop O O c1 e1))) -(ex_intro2 C (\lambda (e1: C).(csubst0 i v e1 c2)) (\lambda (e1: C).(drop O O -c1 e1)) c1 H0 (drop_refl c1))) (minus i O) (minus_n_O i)) e2 (drop_gen_refl -c2 e2 H1)))))))))) (\lambda (n0: nat).(\lambda (IHn: ((\forall (i: nat).((lt -n0 i) \to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 -c2) \to (\forall (e2: C).((drop n0 O c2 e2) \to (or (drop n0 O c1 e2) (ex2 C -(\lambda (e1: C).(csubst0 (minus i n0) v e1 e2)) (\lambda (e1: C).(drop n0 O -c1 e1)))))))))))))).(\lambda (i: nat).(\lambda (H: (lt (S n0) i)).(\lambda -(c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v -c c2) \to (\forall (e2: C).((drop (S n0) O c2 e2) \to (or (drop (S n0) O c -e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i (S n0)) v e1 e2)) (\lambda (e1: -C).(drop (S n0) O c e1)))))))))) (\lambda (n1: nat).(\lambda (c2: C).(\lambda -(v: T).(\lambda (H0: (csubst0 i v (CSort n1) c2)).(\lambda (e2: C).(\lambda -(_: (drop (S n0) O c2 e2)).(csubst0_gen_sort c2 v i n1 H0 (or (drop (S n0) O -(CSort n1) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i (S n0)) v e1 e2)) -(\lambda (e1: C).(drop (S n0) O (CSort n1) e1))))))))))) (\lambda (c: -C).(\lambda (H0: ((\forall (c2: C).(\forall (v: T).((csubst0 i v c c2) \to -(\forall (e2: C).((drop (S n0) O c2 e2) \to (or (drop (S n0) O c e2) (ex2 C -(\lambda (e1: C).(csubst0 (minus i (S n0)) v e1 e2)) (\lambda (e1: C).(drop -(S n0) O c e1))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: -C).(\lambda (v: T).(\lambda (H1: (csubst0 i v (CHead c k t) c2)).(\lambda -(e2: C).(\lambda (H2: (drop (S n0) O c2 e2)).(or3_ind (ex3_2 T nat (\lambda -(_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: -nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j -v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k -j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda -(c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_: -T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: -T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda -(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: -T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))) (or (drop (S n0) -O (CHead c k t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i (S n0)) v e1 -e2)) (\lambda (e1: C).(drop (S n0) O (CHead c k t) e1)))) (\lambda (H3: -(ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda -(u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: -T).(\lambda (j: nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: -T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: -nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j -v t u2))) (or (drop (S n0) O (CHead c k t) e2) (ex2 C (\lambda (e1: -C).(csubst0 (minus i (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead -c k t) e1)))) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H4: (eq nat i (s -k x1))).(\lambda (H5: (eq C c2 (CHead c k x0))).(\lambda (_: (subst0 x1 v t -x0)).(let H7 \def (eq_ind C c2 (\lambda (c0: C).(drop (S n0) O c0 e2)) H2 -(CHead c k x0) H5) in (let H8 \def (eq_ind nat i (\lambda (n1: nat).(\forall -(c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall (e3: C).((drop (S -n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: C).(csubst0 -(minus n1 (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop (S n0) O c e1)))))))))) -H0 (s k x1) H4) in (let H9 \def (eq_ind nat i (\lambda (n1: nat).(lt (S n0) -n1)) H (s k x1) H4) in (eq_ind_r nat (s k x1) (\lambda (n1: nat).(or (drop (S -n0) O (CHead c k t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus n1 (S n0)) v -e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c k t) e1))))) (K_ind (\lambda -(k0: K).(((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c c3) \to -(\forall (e3: C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C -(\lambda (e1: C).(csubst0 (minus (s k0 x1) (S n0)) v0 e1 e3)) (\lambda (e1: -C).(drop (S n0) O c e1)))))))))) \to ((lt (S n0) (s k0 x1)) \to ((drop (r k0 -n0) O c e2) \to (or (drop (S n0) O (CHead c k0 t) e2) (ex2 C (\lambda (e1: -C).(csubst0 (minus (s k0 x1) (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) -O (CHead c k0 t) e1)))))))) (\lambda (b: B).(\lambda (_: ((\forall (c3: -C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to (\forall (e3: -C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: -C).(csubst0 (minus (s (Bind b) x1) (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop -(S n0) O c e1))))))))))).(\lambda (_: (lt (S n0) (s (Bind b) x1))).(\lambda -(H12: (drop (r (Bind b) n0) O c e2)).(or_introl (drop (S n0) O (CHead c (Bind -b) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x1 n0) v e1 e2)) (\lambda -(e1: C).(drop (S n0) O (CHead c (Bind b) t) e1))) (drop_drop (Bind b) n0 c e2 -H12 t)))))) (\lambda (f: F).(\lambda (_: ((\forall (c3: C).(\forall (v0: -T).((csubst0 (s (Flat f) x1) v0 c c3) \to (\forall (e3: C).((drop (S n0) O c3 -e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: C).(csubst0 (minus (s -(Flat f) x1) (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop (S n0) O c -e1))))))))))).(\lambda (_: (lt (S n0) (s (Flat f) x1))).(\lambda (H12: (drop -(r (Flat f) n0) O c e2)).(or_introl (drop (S n0) O (CHead c (Flat f) t) e2) -(ex2 C (\lambda (e1: C).(csubst0 (minus x1 (S n0)) v e1 e2)) (\lambda (e1: -C).(drop (S n0) O (CHead c (Flat f) t) e1))) (drop_drop (Flat f) n0 c e2 H12 -t)))))) k H8 H9 (drop_gen_drop k c e2 x0 n0 H7)) i H4))))))))) H3)) (\lambda -(H3: (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j)))) -(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: -C).(\lambda (j: nat).(csubst0 j v c c3))))).(ex3_2_ind C nat (\lambda (_: -C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_: -nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j -v c c3))) (or (drop (S n0) O (CHead c k t) e2) (ex2 C (\lambda (e1: -C).(csubst0 (minus i (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead -c k t) e1)))) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H4: (eq nat i (s -k x1))).(\lambda (H5: (eq C c2 (CHead x0 k t))).(\lambda (H6: (csubst0 x1 v c -x0)).(let H7 \def (eq_ind C c2 (\lambda (c0: C).(drop (S n0) O c0 e2)) H2 -(CHead x0 k t) H5) in (let H8 \def (eq_ind nat i (\lambda (n1: nat).(\forall -(c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall (e3: C).((drop (S -n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: C).(csubst0 -(minus n1 (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop (S n0) O c e1)))))))))) -H0 (s k x1) H4) in (let H9 \def (eq_ind nat i (\lambda (n1: nat).(lt (S n0) -n1)) H (s k x1) H4) in (eq_ind_r nat (s k x1) (\lambda (n1: nat).(or (drop (S -n0) O (CHead c k t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus n1 (S n0)) v -e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c k t) e1))))) (K_ind (\lambda -(k0: K).(((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c c3) \to -(\forall (e3: C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C -(\lambda (e1: C).(csubst0 (minus (s k0 x1) (S n0)) v0 e1 e3)) (\lambda (e1: -C).(drop (S n0) O c e1)))))))))) \to ((lt (S n0) (s k0 x1)) \to ((drop (r k0 -n0) O x0 e2) \to (or (drop (S n0) O (CHead c k0 t) e2) (ex2 C (\lambda (e1: -C).(csubst0 (minus (s k0 x1) (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) -O (CHead c k0 t) e1)))))))) (\lambda (b: B).(\lambda (_: ((\forall (c3: -C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to (\forall (e3: -C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: -C).(csubst0 (minus (s (Bind b) x1) (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop -(S n0) O c e1))))))))))).(\lambda (H11: (lt (S n0) (s (Bind b) x1))).(\lambda -(H12: (drop (r (Bind b) n0) O x0 e2)).(let H_x \def (IHn x1 (lt_S_n n0 x1 -H11) c x0 v H6 e2 H12) in (let H13 \def H_x in (or_ind (drop n0 O c e2) (ex2 -C (\lambda (e1: C).(csubst0 (minus x1 n0) v e1 e2)) (\lambda (e1: C).(drop n0 -O c e1))) (or (drop (S n0) O (CHead c (Bind b) t) e2) (ex2 C (\lambda (e1: -C).(csubst0 (minus x1 n0) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c -(Bind b) t) e1)))) (\lambda (H14: (drop n0 O c e2)).(or_introl (drop (S n0) O -(CHead c (Bind b) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x1 n0) v e1 -e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Bind b) t) e1))) (drop_drop -(Bind b) n0 c e2 H14 t))) (\lambda (H14: (ex2 C (\lambda (e1: C).(csubst0 -(minus x1 n0) v e1 e2)) (\lambda (e1: C).(drop n0 O c e1)))).(ex2_ind C -(\lambda (e1: C).(csubst0 (minus x1 n0) v e1 e2)) (\lambda (e1: C).(drop n0 O -c e1)) (or (drop (S n0) O (CHead c (Bind b) t) e2) (ex2 C (\lambda (e1: -C).(csubst0 (minus x1 n0) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c -(Bind b) t) e1)))) (\lambda (x: C).(\lambda (H15: (csubst0 (minus x1 n0) v x -e2)).(\lambda (H16: (drop n0 O c x)).(or_intror (drop (S n0) O (CHead c (Bind -b) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x1 n0) v e1 e2)) (\lambda -(e1: C).(drop (S n0) O (CHead c (Bind b) t) e1))) (ex_intro2 C (\lambda (e1: -C).(csubst0 (minus x1 n0) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c -(Bind b) t) e1)) x H15 (drop_drop (Bind b) n0 c x H16 t)))))) H14)) -H13))))))) (\lambda (f: F).(\lambda (H10: ((\forall (c3: C).(\forall (v0: -T).((csubst0 (s (Flat f) x1) v0 c c3) \to (\forall (e3: C).((drop (S n0) O c3 -e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: C).(csubst0 (minus (s -(Flat f) x1) (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop (S n0) O c -e1))))))))))).(\lambda (_: (lt (S n0) (s (Flat f) x1))).(\lambda (H12: (drop -(r (Flat f) n0) O x0 e2)).(let H_x \def (H10 x0 v H6 e2 H12) in (let H13 \def -H_x in (or_ind (drop (S n0) O c e2) (ex2 C (\lambda (e1: C).(csubst0 (minus -x1 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O c e1))) (or (drop (S n0) -O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x1 (S n0)) -v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Flat f) t) e1)))) -(\lambda (H14: (drop (S n0) O c e2)).(or_introl (drop (S n0) O (CHead c (Flat -f) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x1 (S n0)) v e1 e2)) -(\lambda (e1: C).(drop (S n0) O (CHead c (Flat f) t) e1))) (drop_drop (Flat -f) n0 c e2 H14 t))) (\lambda (H14: (ex2 C (\lambda (e1: C).(csubst0 (minus x1 -(S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O c e1)))).(ex2_ind C -(\lambda (e1: C).(csubst0 (minus x1 (S n0)) v e1 e2)) (\lambda (e1: C).(drop -(S n0) O c e1)) (or (drop (S n0) O (CHead c (Flat f) t) e2) (ex2 C (\lambda -(e1: C).(csubst0 (minus x1 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O -(CHead c (Flat f) t) e1)))) (\lambda (x: C).(\lambda (H15: (csubst0 (minus x1 -(S n0)) v x e2)).(\lambda (H16: (drop (S n0) O c x)).(or_intror (drop (S n0) -O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x1 (S n0)) -v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Flat f) t) e1))) -(ex_intro2 C (\lambda (e1: C).(csubst0 (minus x1 (S n0)) v e1 e2)) (\lambda -(e1: C).(drop (S n0) O (CHead c (Flat f) t) e1)) x H15 (drop_drop (Flat f) n0 -c x H16 t)))))) H14)) H13))))))) k H8 H9 (drop_gen_drop k x0 e2 t n0 H7)) i -H4))))))))) H3)) (\lambda (H3: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: -C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: -C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda -(_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: -C).(\lambda (j: nat).(csubst0 j v c c3)))))).(ex4_3_ind T C nat (\lambda (_: -T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: -T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda -(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: -T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (or (drop (S n0) -O (CHead c k t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i (S n0)) v e1 -e2)) (\lambda (e1: C).(drop (S n0) O (CHead c k t) e1)))) (\lambda (x0: -T).(\lambda (x1: C).(\lambda (x2: nat).(\lambda (H4: (eq nat i (s k -x2))).(\lambda (H5: (eq C c2 (CHead x1 k x0))).(\lambda (_: (subst0 x2 v t -x0)).(\lambda (H7: (csubst0 x2 v c x1)).(let H8 \def (eq_ind C c2 (\lambda -(c0: C).(drop (S n0) O c0 e2)) H2 (CHead x1 k x0) H5) in (let H9 \def (eq_ind -nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c -c3) \to (\forall (e3: C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) -(ex2 C (\lambda (e1: C).(csubst0 (minus n1 (S n0)) v0 e1 e3)) (\lambda (e1: -C).(drop (S n0) O c e1)))))))))) H0 (s k x2) H4) in (let H10 \def (eq_ind nat -i (\lambda (n1: nat).(lt (S n0) n1)) H (s k x2) H4) in (eq_ind_r nat (s k x2) -(\lambda (n1: nat).(or (drop (S n0) O (CHead c k t) e2) (ex2 C (\lambda (e1: -C).(csubst0 (minus n1 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O -(CHead c k t) e1))))) (K_ind (\lambda (k0: K).(((\forall (c3: C).(\forall -(v0: T).((csubst0 (s k0 x2) v0 c c3) \to (\forall (e3: C).((drop (S n0) O c3 -e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: C).(csubst0 (minus (s -k0 x2) (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop (S n0) O c e1)))))))))) \to -((lt (S n0) (s k0 x2)) \to ((drop (r k0 n0) O x1 e2) \to (or (drop (S n0) O -(CHead c k0 t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus (s k0 x2) (S n0)) -v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c k0 t) e1)))))))) (\lambda -(b: B).(\lambda (_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Bind b) -x2) v0 c c3) \to (\forall (e3: C).((drop (S n0) O c3 e3) \to (or (drop (S n0) -O c e3) (ex2 C (\lambda (e1: C).(csubst0 (minus (s (Bind b) x2) (S n0)) v0 e1 -e3)) (\lambda (e1: C).(drop (S n0) O c e1))))))))))).(\lambda (H12: (lt (S -n0) (s (Bind b) x2))).(\lambda (H13: (drop (r (Bind b) n0) O x1 e2)).(let H_x -\def (IHn x2 (lt_S_n n0 x2 H12) c x1 v H7 e2 H13) in (let H14 \def H_x in -(or_ind (drop n0 O c e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x2 n0) v e1 -e2)) (\lambda (e1: C).(drop n0 O c e1))) (or (drop (S n0) O (CHead c (Bind b) -t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x2 n0) v e1 e2)) (\lambda (e1: -C).(drop (S n0) O (CHead c (Bind b) t) e1)))) (\lambda (H15: (drop n0 O c -e2)).(or_introl (drop (S n0) O (CHead c (Bind b) t) e2) (ex2 C (\lambda (e1: -C).(csubst0 (minus x2 n0) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c -(Bind b) t) e1))) (drop_drop (Bind b) n0 c e2 H15 t))) (\lambda (H15: (ex2 C -(\lambda (e1: C).(csubst0 (minus x2 n0) v e1 e2)) (\lambda (e1: C).(drop n0 O -c e1)))).(ex2_ind C (\lambda (e1: C).(csubst0 (minus x2 n0) v e1 e2)) -(\lambda (e1: C).(drop n0 O c e1)) (or (drop (S n0) O (CHead c (Bind b) t) -e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x2 n0) v e1 e2)) (\lambda (e1: -C).(drop (S n0) O (CHead c (Bind b) t) e1)))) (\lambda (x: C).(\lambda (H16: -(csubst0 (minus x2 n0) v x e2)).(\lambda (H17: (drop n0 O c x)).(or_intror -(drop (S n0) O (CHead c (Bind b) t) e2) (ex2 C (\lambda (e1: C).(csubst0 -(minus x2 n0) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Bind b) t) -e1))) (ex_intro2 C (\lambda (e1: C).(csubst0 (minus x2 n0) v e1 e2)) (\lambda -(e1: C).(drop (S n0) O (CHead c (Bind b) t) e1)) x H16 (drop_drop (Bind b) n0 -c x H17 t)))))) H15)) H14))))))) (\lambda (f: F).(\lambda (H11: ((\forall -(c3: C).(\forall (v0: T).((csubst0 (s (Flat f) x2) v0 c c3) \to (\forall (e3: -C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: -C).(csubst0 (minus (s (Flat f) x2) (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop -(S n0) O c e1))))))))))).(\lambda (_: (lt (S n0) (s (Flat f) x2))).(\lambda -(H13: (drop (r (Flat f) n0) O x1 e2)).(let H_x \def (H11 x1 v H7 e2 H13) in -(let H14 \def H_x in (or_ind (drop (S n0) O c e2) (ex2 C (\lambda (e1: -C).(csubst0 (minus x2 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O c -e1))) (or (drop (S n0) O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1: -C).(csubst0 (minus x2 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O -(CHead c (Flat f) t) e1)))) (\lambda (H15: (drop (S n0) O c e2)).(or_introl -(drop (S n0) O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1: C).(csubst0 -(minus x2 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Flat f) -t) e1))) (drop_drop (Flat f) n0 c e2 H15 t))) (\lambda (H15: (ex2 C (\lambda -(e1: C).(csubst0 (minus x2 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O -c e1)))).(ex2_ind C (\lambda (e1: C).(csubst0 (minus x2 (S n0)) v e1 e2)) -(\lambda (e1: C).(drop (S n0) O c e1)) (or (drop (S n0) O (CHead c (Flat f) -t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x2 (S n0)) v e1 e2)) (\lambda -(e1: C).(drop (S n0) O (CHead c (Flat f) t) e1)))) (\lambda (x: C).(\lambda -(H16: (csubst0 (minus x2 (S n0)) v x e2)).(\lambda (H17: (drop (S n0) O c -x)).(or_intror (drop (S n0) O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1: -C).(csubst0 (minus x2 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O -(CHead c (Flat f) t) e1))) (ex_intro2 C (\lambda (e1: C).(csubst0 (minus x2 -(S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Flat f) t) e1)) x -H16 (drop_drop (Flat f) n0 c x H17 t)))))) H15)) H14))))))) k H9 H10 -(drop_gen_drop k x1 e2 x0 n0 H8)) i H4))))))))))) H3)) (csubst0_gen_head k c -c2 t v i H1))))))))))) c1)))))) n). -(* COMMENTS -Initial nodes: 5939 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubst0/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubst0/fwd.ma deleted file mode 100644 index 9b3de2983..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubst0/fwd.ma +++ /dev/null @@ -1,462 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubst0/defs.ma". - -theorem csubst0_gen_sort: - \forall (x: C).(\forall (v: T).(\forall (i: nat).(\forall (n: nat).((csubst0 -i v (CSort n) x) \to (\forall (P: Prop).P))))) -\def - \lambda (x: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda -(H: (csubst0 i v (CSort n) x)).(\lambda (P: Prop).(insert_eq C (CSort n) -(\lambda (c: C).(csubst0 i v c x)) (\lambda (_: C).P) (\lambda (y: -C).(\lambda (H0: (csubst0 i v y x)).(csubst0_ind (\lambda (_: nat).(\lambda -(_: T).(\lambda (c: C).(\lambda (_: C).((eq C c (CSort n)) \to P))))) -(\lambda (k: K).(\lambda (i0: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda -(u2: T).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (c: C).(\lambda (H2: (eq -C (CHead c k u1) (CSort n))).(let H3 \def (eq_ind C (CHead c k u1) (\lambda -(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H2) in -(False_ind P H3)))))))))) (\lambda (k: K).(\lambda (i0: nat).(\lambda (c1: -C).(\lambda (c2: C).(\lambda (v0: T).(\lambda (_: (csubst0 i0 v0 c1 -c2)).(\lambda (_: (((eq C c1 (CSort n)) \to P))).(\lambda (u: T).(\lambda -(H3: (eq C (CHead c1 k u) (CSort n))).(let H4 \def (eq_ind C (CHead c1 k u) -(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H3) in -(False_ind P H4))))))))))) (\lambda (k: K).(\lambda (i0: nat).(\lambda (v0: -T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0 i0 v0 u1 -u2)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csubst0 i0 v0 c1 -c2)).(\lambda (_: (((eq C c1 (CSort n)) \to P))).(\lambda (H4: (eq C (CHead -c1 k u1) (CSort n))).(let H5 \def (eq_ind C (CHead c1 k u1) (\lambda (ee: -C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow -False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H4) in (False_ind P -H5))))))))))))) i v y x H0))) H)))))). -(* COMMENTS -Initial nodes: 355 -END *) - -theorem csubst0_gen_head: - \forall (k: K).(\forall (c1: C).(\forall (x: C).(\forall (u1: T).(\forall -(v: T).(\forall (i: nat).((csubst0 i v (CHead c1 k u1) x) \to (or3 (ex3_2 T -nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: -T).(\lambda (_: nat).(eq C x (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: -nat).(subst0 j v u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq -nat i (s k j)))) (\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2 k -u1)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 c2)))) (ex4_3 T C -nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) -(\lambda (u2: T).(\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2 k -u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 -u2)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 -c2)))))))))))) -\def - \lambda (k: K).(\lambda (c1: C).(\lambda (x: C).(\lambda (u1: T).(\lambda -(v: T).(\lambda (i: nat).(\lambda (H: (csubst0 i v (CHead c1 k u1) -x)).(insert_eq C (CHead c1 k u1) (\lambda (c: C).(csubst0 i v c x)) (\lambda -(_: C).(or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k -j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C x (CHead c1 k u2)))) (\lambda -(u2: T).(\lambda (j: nat).(subst0 j v u1 u2)))) (ex3_2 C nat (\lambda (_: -C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c2: C).(\lambda (_: -nat).(eq C x (CHead c2 k u1)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j -v c1 c2)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: -nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c2: C).(\lambda (_: -nat).(eq C x (CHead c2 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda -(j: nat).(subst0 j v u1 u2)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: -nat).(csubst0 j v c1 c2))))))) (\lambda (y: C).(\lambda (H0: (csubst0 i v y -x)).(csubst0_ind (\lambda (n: nat).(\lambda (t: T).(\lambda (c: C).(\lambda -(c0: C).((eq C c (CHead c1 k u1)) \to (or3 (ex3_2 T nat (\lambda (_: -T).(\lambda (j: nat).(eq nat n (s k j)))) (\lambda (u2: T).(\lambda (_: -nat).(eq C c0 (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j -t u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat n (s k -j)))) (\lambda (c2: C).(\lambda (_: nat).(eq C c0 (CHead c2 k u1)))) (\lambda -(c2: C).(\lambda (j: nat).(csubst0 j t c1 c2)))) (ex4_3 T C nat (\lambda (_: -T).(\lambda (_: C).(\lambda (j: nat).(eq nat n (s k j))))) (\lambda (u2: -T).(\lambda (c2: C).(\lambda (_: nat).(eq C c0 (CHead c2 k u2))))) (\lambda -(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j t u1 u2)))) (\lambda (_: -T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j t c1 c2))))))))))) (\lambda -(k0: K).(\lambda (i0: nat).(\lambda (v0: T).(\lambda (u0: T).(\lambda (u2: -T).(\lambda (H1: (subst0 i0 v0 u0 u2)).(\lambda (c: C).(\lambda (H2: (eq C -(CHead c k0 u0) (CHead c1 k u1))).(let H3 \def (f_equal C C (\lambda (e: -C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c | -(CHead c0 _ _) \Rightarrow c0])) (CHead c k0 u0) (CHead c1 k u1) H2) in ((let -H4 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) -with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead c k0 -u0) (CHead c1 k u1) H2) in ((let H5 \def (f_equal C T (\lambda (e: C).(match -e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ -t) \Rightarrow t])) (CHead c k0 u0) (CHead c1 k u1) H2) in (\lambda (H6: (eq -K k0 k)).(\lambda (H7: (eq C c c1)).(eq_ind_r C c1 (\lambda (c0: C).(or3 -(ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) -(\lambda (u3: T).(\lambda (_: nat).(eq C (CHead c0 k0 u2) (CHead c1 k u3)))) -(\lambda (u3: T).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (ex3_2 C nat -(\lambda (_: C).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) (\lambda (c2: -C).(\lambda (_: nat).(eq C (CHead c0 k0 u2) (CHead c2 k u1)))) (\lambda (c2: -C).(\lambda (j: nat).(csubst0 j v0 c1 c2)))) (ex4_3 T C nat (\lambda (_: -T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k0 i0) (s k j))))) (\lambda -(u3: T).(\lambda (c2: C).(\lambda (_: nat).(eq C (CHead c0 k0 u2) (CHead c2 k -u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 -u3)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v0 c1 -c2))))))) (let H8 \def (eq_ind T u0 (\lambda (t: T).(subst0 i0 v0 t u2)) H1 -u1 H5) in (eq_ind_r K k (\lambda (k1: K).(or3 (ex3_2 T nat (\lambda (_: -T).(\lambda (j: nat).(eq nat (s k1 i0) (s k j)))) (\lambda (u3: T).(\lambda -(_: nat).(eq C (CHead c1 k1 u2) (CHead c1 k u3)))) (\lambda (u3: T).(\lambda -(j: nat).(subst0 j v0 u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: -nat).(eq nat (s k1 i0) (s k j)))) (\lambda (c2: C).(\lambda (_: nat).(eq C -(CHead c1 k1 u2) (CHead c2 k u1)))) (\lambda (c2: C).(\lambda (j: -nat).(csubst0 j v0 c1 c2)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: -C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j))))) (\lambda (u3: T).(\lambda -(c2: C).(\lambda (_: nat).(eq C (CHead c1 k1 u2) (CHead c2 k u3))))) (\lambda -(u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (\lambda (_: -T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v0 c1 c2))))))) (or3_intro0 -(ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) -(\lambda (u3: T).(\lambda (_: nat).(eq C (CHead c1 k u2) (CHead c1 k u3)))) -(\lambda (u3: T).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (ex3_2 C nat -(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (c2: -C).(\lambda (_: nat).(eq C (CHead c1 k u2) (CHead c2 k u1)))) (\lambda (c2: -C).(\lambda (j: nat).(csubst0 j v0 c1 c2)))) (ex4_3 T C nat (\lambda (_: -T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))) (\lambda -(u3: T).(\lambda (c2: C).(\lambda (_: nat).(eq C (CHead c1 k u2) (CHead c2 k -u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 -u3)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v0 c1 -c2))))) (ex3_2_intro T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i0) -(s k j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C (CHead c1 k u2) (CHead c1 -k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j v0 u1 u3))) u2 i0 -(refl_equal nat (s k i0)) (refl_equal C (CHead c1 k u2)) H8)) k0 H6)) c -H7)))) H4)) H3)))))))))) (\lambda (k0: K).(\lambda (i0: nat).(\lambda (c0: -C).(\lambda (c2: C).(\lambda (v0: T).(\lambda (H1: (csubst0 i0 v0 c0 -c2)).(\lambda (H2: (((eq C c0 (CHead c1 k u1)) \to (or3 (ex3_2 T nat (\lambda -(_: T).(\lambda (j: nat).(eq nat i0 (s k j)))) (\lambda (u2: T).(\lambda (_: -nat).(eq C c2 (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j -v0 u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i0 (s k -j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u1)))) (\lambda -(c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: -T).(\lambda (_: C).(\lambda (j: nat).(eq nat i0 (s k j))))) (\lambda (u2: -T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda -(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 u2)))) (\lambda (_: -T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3))))))))).(\lambda -(u: T).(\lambda (H3: (eq C (CHead c0 k0 u) (CHead c1 k u1))).(let H4 \def -(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with -[(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k0 u) -(CHead c1 k u1) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e in -C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) -\Rightarrow k1])) (CHead c0 k0 u) (CHead c1 k u1) H3) in ((let H6 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 u) -(CHead c1 k u1) H3) in (\lambda (H7: (eq K k0 k)).(\lambda (H8: (eq C c0 -c1)).(eq_ind_r T u1 (\lambda (t: T).(or3 (ex3_2 T nat (\lambda (_: -T).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) (\lambda (u2: T).(\lambda -(_: nat).(eq C (CHead c2 k0 t) (CHead c1 k u2)))) (\lambda (u2: T).(\lambda -(j: nat).(subst0 j v0 u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: -nat).(eq nat (s k0 i0) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C -(CHead c2 k0 t) (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: -nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: -C).(\lambda (j: nat).(eq nat (s k0 i0) (s k j))))) (\lambda (u2: T).(\lambda -(c3: C).(\lambda (_: nat).(eq C (CHead c2 k0 t) (CHead c3 k u2))))) (\lambda -(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 u2)))) (\lambda (_: -T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3))))))) (let H9 \def -(eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 k u1)) \to (or3 (ex3_2 T nat -(\lambda (_: T).(\lambda (j: nat).(eq nat i0 (s k j)))) (\lambda (u2: -T).(\lambda (_: nat).(eq C c2 (CHead c1 k u2)))) (\lambda (u2: T).(\lambda -(j: nat).(subst0 j v0 u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: -nat).(eq nat i0 (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead -c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 -T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat i0 (s k -j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 -k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 -u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 -c3)))))))) H2 c1 H8) in (let H10 \def (eq_ind C c0 (\lambda (c: C).(csubst0 -i0 v0 c c2)) H1 c1 H8) in (eq_ind_r K k (\lambda (k1: K).(or3 (ex3_2 T nat -(\lambda (_: T).(\lambda (j: nat).(eq nat (s k1 i0) (s k j)))) (\lambda (u2: -T).(\lambda (_: nat).(eq C (CHead c2 k1 u1) (CHead c1 k u2)))) (\lambda (u2: -T).(\lambda (j: nat).(subst0 j v0 u1 u2)))) (ex3_2 C nat (\lambda (_: -C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j)))) (\lambda (c3: C).(\lambda -(_: nat).(eq C (CHead c2 k1 u1) (CHead c3 k u1)))) (\lambda (c3: C).(\lambda -(j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: -C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j))))) (\lambda (u2: T).(\lambda -(c3: C).(\lambda (_: nat).(eq C (CHead c2 k1 u1) (CHead c3 k u2))))) (\lambda -(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 u2)))) (\lambda (_: -T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3))))))) (or3_intro1 -(ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) -(\lambda (u2: T).(\lambda (_: nat).(eq C (CHead c2 k u1) (CHead c1 k u2)))) -(\lambda (u2: T).(\lambda (j: nat).(subst0 j v0 u1 u2)))) (ex3_2 C nat -(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (c3: -C).(\lambda (_: nat).(eq C (CHead c2 k u1) (CHead c3 k u1)))) (\lambda (c3: -C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: -T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))) (\lambda -(u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k u1) (CHead c3 k -u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 -u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 -c3))))) (ex3_2_intro C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) -(s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k u1) (CHead c3 -k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3))) c2 i0 -(refl_equal nat (s k i0)) (refl_equal C (CHead c2 k u1)) H10)) k0 H7))) u -H6)))) H5)) H4))))))))))) (\lambda (k0: K).(\lambda (i0: nat).(\lambda (v0: -T).(\lambda (u0: T).(\lambda (u2: T).(\lambda (H1: (subst0 i0 v0 u0 -u2)).(\lambda (c0: C).(\lambda (c2: C).(\lambda (H2: (csubst0 i0 v0 c0 -c2)).(\lambda (H3: (((eq C c0 (CHead c1 k u1)) \to (or3 (ex3_2 T nat (\lambda -(_: T).(\lambda (j: nat).(eq nat i0 (s k j)))) (\lambda (u3: T).(\lambda (_: -nat).(eq C c2 (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j -v0 u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i0 (s k -j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u1)))) (\lambda -(c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: -T).(\lambda (_: C).(\lambda (j: nat).(eq nat i0 (s k j))))) (\lambda (u3: -T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u3))))) (\lambda -(u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (\lambda (_: -T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3))))))))).(\lambda -(H4: (eq C (CHead c0 k0 u0) (CHead c1 k u1))).(let H5 \def (f_equal C C -(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k0 u0) (CHead c1 k -u1) H4) in ((let H6 \def (f_equal C K (\lambda (e: C).(match e in C return -(\lambda (_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) -\Rightarrow k1])) (CHead c0 k0 u0) (CHead c1 k u1) H4) in ((let H7 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 u0) -(CHead c1 k u1) H4) in (\lambda (H8: (eq K k0 k)).(\lambda (H9: (eq C c0 -c1)).(let H10 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 k u1)) \to -(or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i0 (s k j)))) -(\lambda (u3: T).(\lambda (_: nat).(eq C c2 (CHead c1 k u3)))) (\lambda (u3: -T).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (ex3_2 C nat (\lambda (_: -C).(\lambda (j: nat).(eq nat i0 (s k j)))) (\lambda (c3: C).(\lambda (_: -nat).(eq C c2 (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 -j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: -nat).(eq nat i0 (s k j))))) (\lambda (u3: T).(\lambda (c3: C).(\lambda (_: -nat).(eq C c2 (CHead c3 k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda -(j: nat).(subst0 j v0 u1 u3)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: -nat).(csubst0 j v0 c1 c3)))))))) H3 c1 H9) in (let H11 \def (eq_ind C c0 -(\lambda (c: C).(csubst0 i0 v0 c c2)) H2 c1 H9) in (let H12 \def (eq_ind T u0 -(\lambda (t: T).(subst0 i0 v0 t u2)) H1 u1 H7) in (eq_ind_r K k (\lambda (k1: -K).(or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k1 i0) (s k -j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C (CHead c2 k1 u2) (CHead c1 k -u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (ex3_2 C nat -(\lambda (_: C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j)))) (\lambda (c3: -C).(\lambda (_: nat).(eq C (CHead c2 k1 u2) (CHead c3 k u1)))) (\lambda (c3: -C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: -T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j))))) (\lambda -(u3: T).(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k1 u2) (CHead c3 k -u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 -u3)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 -c3))))))) (or3_intro2 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat -(s k i0) (s k j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C (CHead c2 k u2) -(CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j v0 u1 u3)))) -(ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) -(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k u2) (CHead c3 k u1)))) -(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat -(\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k -j))))) (\lambda (u3: T).(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k -u2) (CHead c3 k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: -nat).(subst0 j v0 u1 u3)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: -nat).(csubst0 j v0 c1 c3))))) (ex4_3_intro T C nat (\lambda (_: T).(\lambda -(_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))) (\lambda (u3: -T).(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k u2) (CHead c3 k -u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 -u3)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 -c3)))) u2 c2 i0 (refl_equal nat (s k i0)) (refl_equal C (CHead c2 k u2)) H12 -H11)) k0 H8))))))) H6)) H5))))))))))))) i v y x H0))) H))))))). -(* COMMENTS -Initial nodes: 4039 -END *) - -theorem csubst0_gen_S_bind_2: - \forall (b: B).(\forall (x: C).(\forall (c2: C).(\forall (v: T).(\forall -(v2: T).(\forall (i: nat).((csubst0 (S i) v x (CHead c2 (Bind b) v2)) \to -(or3 (ex2 T (\lambda (v1: T).(subst0 i v v1 v2)) (\lambda (v1: T).(eq C x -(CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i v c1 c2)) -(\lambda (c1: C).(eq C x (CHead c1 (Bind b) v2)))) (ex3_2 C T (\lambda (_: -C).(\lambda (v1: T).(subst0 i v v1 v2))) (\lambda (c1: C).(\lambda (_: -T).(csubst0 i v c1 c2))) (\lambda (c1: C).(\lambda (v1: T).(eq C x (CHead c1 -(Bind b) v1)))))))))))) -\def - \lambda (b: B).(\lambda (x: C).(\lambda (c2: C).(\lambda (v: T).(\lambda -(v2: T).(\lambda (i: nat).(\lambda (H: (csubst0 (S i) v x (CHead c2 (Bind b) -v2))).(insert_eq C (CHead c2 (Bind b) v2) (\lambda (c: C).(csubst0 (S i) v x -c)) (\lambda (_: C).(or3 (ex2 T (\lambda (v1: T).(subst0 i v v1 v2)) (\lambda -(v1: T).(eq C x (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i -v c1 c2)) (\lambda (c1: C).(eq C x (CHead c1 (Bind b) v2)))) (ex3_2 C T -(\lambda (_: C).(\lambda (v1: T).(subst0 i v v1 v2))) (\lambda (c1: -C).(\lambda (_: T).(csubst0 i v c1 c2))) (\lambda (c1: C).(\lambda (v1: -T).(eq C x (CHead c1 (Bind b) v1))))))) (\lambda (y: C).(\lambda (H0: -(csubst0 (S i) v x y)).(insert_eq nat (S i) (\lambda (n: nat).(csubst0 n v x -y)) (\lambda (_: nat).((eq C y (CHead c2 (Bind b) v2)) \to (or3 (ex2 T -(\lambda (v1: T).(subst0 i v v1 v2)) (\lambda (v1: T).(eq C x (CHead c2 (Bind -b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i v c1 c2)) (\lambda (c1: C).(eq C -x (CHead c1 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: -T).(subst0 i v v1 v2))) (\lambda (c1: C).(\lambda (_: T).(csubst0 i v c1 -c2))) (\lambda (c1: C).(\lambda (v1: T).(eq C x (CHead c1 (Bind b) v1)))))))) -(\lambda (y0: nat).(\lambda (H1: (csubst0 y0 v x y)).(csubst0_ind (\lambda -(n: nat).(\lambda (t: T).(\lambda (c: C).(\lambda (c0: C).((eq nat n (S i)) -\to ((eq C c0 (CHead c2 (Bind b) v2)) \to (or3 (ex2 T (\lambda (v1: -T).(subst0 i t v1 v2)) (\lambda (v1: T).(eq C c (CHead c2 (Bind b) v1)))) -(ex2 C (\lambda (c1: C).(csubst0 i t c1 c2)) (\lambda (c1: C).(eq C c (CHead -c1 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i t v1 -v2))) (\lambda (c1: C).(\lambda (_: T).(csubst0 i t c1 c2))) (\lambda (c1: -C).(\lambda (v1: T).(eq C c (CHead c1 (Bind b) v1)))))))))))) (\lambda (k: -K).(\lambda (i0: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: -T).(\lambda (H2: (subst0 i0 v0 u1 u2)).(\lambda (c: C).(\lambda (H3: (eq nat -(s k i0) (S i))).(\lambda (H4: (eq C (CHead c k u2) (CHead c2 (Bind b) -v2))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda -(_: C).C) with [(CSort _) \Rightarrow c | (CHead c0 _ _) \Rightarrow c0])) -(CHead c k u2) (CHead c2 (Bind b) v2) H4) in ((let H6 \def (f_equal C K -(\lambda (e: C).(match e in C return (\lambda (_: C).K) with [(CSort _) -\Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c k u2) (CHead c2 -(Bind b) v2) H4) in ((let H7 \def (f_equal C T (\lambda (e: C).(match e in C -return (\lambda (_: C).T) with [(CSort _) \Rightarrow u2 | (CHead _ _ t) -\Rightarrow t])) (CHead c k u2) (CHead c2 (Bind b) v2) H4) in (\lambda (H8: -(eq K k (Bind b))).(\lambda (H9: (eq C c c2)).(eq_ind_r C c2 (\lambda (c0: -C).(or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C -(CHead c0 k u1) (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i -v0 c1 c2)) (\lambda (c1: C).(eq C (CHead c0 k u1) (CHead c1 (Bind b) v2)))) -(ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda -(c1: C).(\lambda (_: T).(csubst0 i v0 c1 c2))) (\lambda (c1: C).(\lambda (v1: -T).(eq C (CHead c0 k u1) (CHead c1 (Bind b) v1))))))) (let H10 \def (eq_ind T -u2 (\lambda (t: T).(subst0 i0 v0 u1 t)) H2 v2 H7) in (let H11 \def (eq_ind K -k (\lambda (k0: K).(eq nat (s k0 i0) (S i))) H3 (Bind b) H8) in (eq_ind_r K -(Bind b) (\lambda (k0: K).(or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) -(\lambda (v1: T).(eq C (CHead c2 k0 u1) (CHead c2 (Bind b) v1)))) (ex2 C -(\lambda (c1: C).(csubst0 i v0 c1 c2)) (\lambda (c1: C).(eq C (CHead c2 k0 -u1) (CHead c1 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: -T).(subst0 i v0 v1 v2))) (\lambda (c1: C).(\lambda (_: T).(csubst0 i v0 c1 -c2))) (\lambda (c1: C).(\lambda (v1: T).(eq C (CHead c2 k0 u1) (CHead c1 -(Bind b) v1))))))) (let H12 \def (f_equal nat nat (\lambda (e: nat).(match e -in nat return (\lambda (_: nat).nat) with [O \Rightarrow i0 | (S n) -\Rightarrow n])) (S i0) (S i) H11) in (let H13 \def (eq_ind nat i0 (\lambda -(n: nat).(subst0 n v0 u1 v2)) H10 i H12) in (or3_intro0 (ex2 T (\lambda (v1: -T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C (CHead c2 (Bind b) u1) (CHead -c2 (Bind b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i v0 c1 c2)) (\lambda -(c1: C).(eq C (CHead c2 (Bind b) u1) (CHead c1 (Bind b) v2)))) (ex3_2 C T -(\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda (c1: -C).(\lambda (_: T).(csubst0 i v0 c1 c2))) (\lambda (c1: C).(\lambda (v1: -T).(eq C (CHead c2 (Bind b) u1) (CHead c1 (Bind b) v1))))) (ex_intro2 T -(\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C (CHead c2 (Bind -b) u1) (CHead c2 (Bind b) v1))) u1 H13 (refl_equal C (CHead c2 (Bind b) -u1)))))) k H8))) c H9)))) H6)) H5))))))))))) (\lambda (k: K).(\lambda (i0: -nat).(\lambda (c1: C).(\lambda (c0: C).(\lambda (v0: T).(\lambda (H2: -(csubst0 i0 v0 c1 c0)).(\lambda (H3: (((eq nat i0 (S i)) \to ((eq C c0 (CHead -c2 (Bind b) v2)) \to (or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) -(\lambda (v1: T).(eq C c1 (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3: -C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C c1 (CHead c3 (Bind b) v2)))) -(ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda -(c3: C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1: -T).(eq C c1 (CHead c3 (Bind b) v1)))))))))).(\lambda (u: T).(\lambda (H4: (eq -nat (s k i0) (S i))).(\lambda (H5: (eq C (CHead c0 k u) (CHead c2 (Bind b) -v2))).(let H6 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda -(_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) -(CHead c0 k u) (CHead c2 (Bind b) v2) H5) in ((let H7 \def (f_equal C K -(\lambda (e: C).(match e in C return (\lambda (_: C).K) with [(CSort _) -\Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c0 k u) (CHead c2 -(Bind b) v2) H5) in ((let H8 \def (f_equal C T (\lambda (e: C).(match e in C -return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) -\Rightarrow t])) (CHead c0 k u) (CHead c2 (Bind b) v2) H5) in (\lambda (H9: -(eq K k (Bind b))).(\lambda (H10: (eq C c0 c2)).(eq_ind_r T v2 (\lambda (t: -T).(or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C -(CHead c1 k t) (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3: C).(csubst0 i -v0 c3 c2)) (\lambda (c3: C).(eq C (CHead c1 k t) (CHead c3 (Bind b) v2)))) -(ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda -(c3: C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1: -T).(eq C (CHead c1 k t) (CHead c3 (Bind b) v1))))))) (let H11 \def (eq_ind C -c0 (\lambda (c: C).((eq nat i0 (S i)) \to ((eq C c (CHead c2 (Bind b) v2)) -\to (or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C -c1 (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3: C).(csubst0 i v0 c3 c2)) -(\lambda (c3: C).(eq C c1 (CHead c3 (Bind b) v2)))) (ex3_2 C T (\lambda (_: -C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda (c3: C).(\lambda (_: -T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1: T).(eq C c1 (CHead -c3 (Bind b) v1))))))))) H3 c2 H10) in (let H12 \def (eq_ind C c0 (\lambda (c: -C).(csubst0 i0 v0 c1 c)) H2 c2 H10) in (let H13 \def (eq_ind K k (\lambda -(k0: K).(eq nat (s k0 i0) (S i))) H4 (Bind b) H9) in (eq_ind_r K (Bind b) -(\lambda (k0: K).(or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda -(v1: T).(eq C (CHead c1 k0 v2) (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3: -C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C (CHead c1 k0 v2) (CHead c3 -(Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 -v2))) (\lambda (c3: C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3: -C).(\lambda (v1: T).(eq C (CHead c1 k0 v2) (CHead c3 (Bind b) v1))))))) (let -H14 \def (f_equal nat nat (\lambda (e: nat).(match e in nat return (\lambda -(_: nat).nat) with [O \Rightarrow i0 | (S n) \Rightarrow n])) (S i0) (S i) -H13) in (let H15 \def (eq_ind nat i0 (\lambda (n: nat).((eq nat n (S i)) \to -((eq C c2 (CHead c2 (Bind b) v2)) \to (or3 (ex2 T (\lambda (v1: T).(subst0 i -v0 v1 v2)) (\lambda (v1: T).(eq C c1 (CHead c2 (Bind b) v1)))) (ex2 C -(\lambda (c3: C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C c1 (CHead c3 -(Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 -v2))) (\lambda (c3: C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3: -C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind b) v1))))))))) H11 i H14) in -(let H16 \def (eq_ind nat i0 (\lambda (n: nat).(csubst0 n v0 c1 c2)) H12 i -H14) in (or3_intro1 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda -(v1: T).(eq C (CHead c1 (Bind b) v2) (CHead c2 (Bind b) v1)))) (ex2 C -(\lambda (c3: C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C (CHead c1 (Bind -b) v2) (CHead c3 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: -T).(subst0 i v0 v1 v2))) (\lambda (c3: C).(\lambda (_: T).(csubst0 i v0 c3 -c2))) (\lambda (c3: C).(\lambda (v1: T).(eq C (CHead c1 (Bind b) v2) (CHead -c3 (Bind b) v1))))) (ex_intro2 C (\lambda (c3: C).(csubst0 i v0 c3 c2)) -(\lambda (c3: C).(eq C (CHead c1 (Bind b) v2) (CHead c3 (Bind b) v2))) c1 H16 -(refl_equal C (CHead c1 (Bind b) v2))))))) k H9)))) u H8)))) H7)) -H6)))))))))))) (\lambda (k: K).(\lambda (i0: nat).(\lambda (v0: T).(\lambda -(u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 i0 v0 u1 u2)).(\lambda (c1: -C).(\lambda (c0: C).(\lambda (H3: (csubst0 i0 v0 c1 c0)).(\lambda (H4: (((eq -nat i0 (S i)) \to ((eq C c0 (CHead c2 (Bind b) v2)) \to (or3 (ex2 T (\lambda -(v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C c1 (CHead c2 (Bind b) -v1)))) (ex2 C (\lambda (c3: C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C -c1 (CHead c3 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: -T).(subst0 i v0 v1 v2))) (\lambda (c3: C).(\lambda (_: T).(csubst0 i v0 c3 -c2))) (\lambda (c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind b) -v1)))))))))).(\lambda (H5: (eq nat (s k i0) (S i))).(\lambda (H6: (eq C -(CHead c0 k u2) (CHead c2 (Bind b) v2))).(let H7 \def (f_equal C C (\lambda -(e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 -| (CHead c _ _) \Rightarrow c])) (CHead c0 k u2) (CHead c2 (Bind b) v2) H6) -in ((let H8 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda -(_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) -(CHead c0 k u2) (CHead c2 (Bind b) v2) H6) in ((let H9 \def (f_equal C T -(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) -\Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c0 k u2) (CHead c2 -(Bind b) v2) H6) in (\lambda (H10: (eq K k (Bind b))).(\lambda (H11: (eq C c0 -c2)).(let H12 \def (eq_ind C c0 (\lambda (c: C).((eq nat i0 (S i)) \to ((eq C -c (CHead c2 (Bind b) v2)) \to (or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 -v2)) (\lambda (v1: T).(eq C c1 (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3: -C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C c1 (CHead c3 (Bind b) v2)))) -(ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda -(c3: C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1: -T).(eq C c1 (CHead c3 (Bind b) v1))))))))) H4 c2 H11) in (let H13 \def -(eq_ind C c0 (\lambda (c: C).(csubst0 i0 v0 c1 c)) H3 c2 H11) in (let H14 -\def (eq_ind T u2 (\lambda (t: T).(subst0 i0 v0 u1 t)) H2 v2 H9) in (let H15 -\def (eq_ind K k (\lambda (k0: K).(eq nat (s k0 i0) (S i))) H5 (Bind b) H10) -in (eq_ind_r K (Bind b) (\lambda (k0: K).(or3 (ex2 T (\lambda (v1: T).(subst0 -i v0 v1 v2)) (\lambda (v1: T).(eq C (CHead c1 k0 u1) (CHead c2 (Bind b) -v1)))) (ex2 C (\lambda (c3: C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C -(CHead c1 k0 u1) (CHead c3 (Bind b) v2)))) (ex3_2 C T (\lambda (_: -C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda (c3: C).(\lambda (_: -T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1: T).(eq C (CHead c1 -k0 u1) (CHead c3 (Bind b) v1))))))) (let H16 \def (f_equal nat nat (\lambda -(e: nat).(match e in nat return (\lambda (_: nat).nat) with [O \Rightarrow i0 -| (S n) \Rightarrow n])) (S i0) (S i) H15) in (let H17 \def (eq_ind nat i0 -(\lambda (n: nat).((eq nat n (S i)) \to ((eq C c2 (CHead c2 (Bind b) v2)) \to -(or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C c1 -(CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3: C).(csubst0 i v0 c3 c2)) -(\lambda (c3: C).(eq C c1 (CHead c3 (Bind b) v2)))) (ex3_2 C T (\lambda (_: -C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda (c3: C).(\lambda (_: -T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1: T).(eq C c1 (CHead -c3 (Bind b) v1))))))))) H12 i H16) in (let H18 \def (eq_ind nat i0 (\lambda -(n: nat).(csubst0 n v0 c1 c2)) H13 i H16) in (let H19 \def (eq_ind nat i0 -(\lambda (n: nat).(subst0 n v0 u1 v2)) H14 i H16) in (or3_intro2 (ex2 T -(\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C (CHead c1 (Bind -b) u1) (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3: C).(csubst0 i v0 c3 -c2)) (\lambda (c3: C).(eq C (CHead c1 (Bind b) u1) (CHead c3 (Bind b) v2)))) -(ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda -(c3: C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1: -T).(eq C (CHead c1 (Bind b) u1) (CHead c3 (Bind b) v1))))) (ex3_2_intro C T -(\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda (c3: -C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1: -T).(eq C (CHead c1 (Bind b) u1) (CHead c3 (Bind b) v1)))) c1 u1 H19 H18 -(refl_equal C (CHead c1 (Bind b) u1)))))))) k H10)))))))) H8)) -H7)))))))))))))) y0 v x y H1))) H0))) H))))))). -(* COMMENTS -Initial nodes: 3878 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubst0/getl.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubst0/getl.ma deleted file mode 100644 index 2701af000..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubst0/getl.ma +++ /dev/null @@ -1,1157 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubst0/clear.ma". - -include "Basic-1/csubst0/drop.ma". - -include "Basic-1/getl/fwd.ma". - -theorem csubst0_getl_ge: - \forall (i: nat).(\forall (n: nat).((le i n) \to (\forall (c1: C).(\forall -(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((getl n c1 -e) \to (getl n c2 e))))))))) -\def - \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (le i n)).(\lambda (c1: -C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 i v c1 -c2)).(\lambda (e: C).(\lambda (H1: (getl n c1 e)).(let H2 \def (getl_gen_all -c1 e n H1) in (ex2_ind C (\lambda (e0: C).(drop n O c1 e0)) (\lambda (e0: -C).(clear e0 e)) (getl n c2 e) (\lambda (x: C).(\lambda (H3: (drop n O c1 -x)).(\lambda (H4: (clear x e)).(lt_eq_gt_e i n (getl n c2 e) (\lambda (H5: -(lt i n)).(getl_intro n c2 e x (csubst0_drop_gt n i H5 c1 c2 v H0 x H3) H4)) -(\lambda (H5: (eq nat i n)).(let H6 \def (eq_ind_r nat n (\lambda (n0: -nat).(drop n0 O c1 x)) H3 i H5) in (let H7 \def (eq_ind_r nat n (\lambda (n0: -nat).(le i n0)) H i H5) in (eq_ind nat i (\lambda (n0: nat).(getl n0 c2 e)) -(let H8 \def (csubst0_drop_eq i c1 c2 v H0 x H6) in (or4_ind (drop i O c2 x) -(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: -T).(eq C x (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(drop i O c2 (CHead e0 (Flat f) w)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u -w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C x (CHead e1 (Flat f) u)))))) (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop i O c2 (CHead e2 -(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C x (CHead e1 -(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(drop i O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O -v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (getl i c2 e) (\lambda (H9: -(drop i O c2 x)).(getl_intro i c2 e x H9 H4)) (\lambda (H9: (ex3_4 F C T T -(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C x -(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop i O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u -w))))))).(ex3_4_ind F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C x (CHead e0 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop i O c2 (CHead e0 -(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w))))) (getl i c2 e) (\lambda (x0: F).(\lambda (x1: -C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H10: (eq C x (CHead x1 (Flat -x0) x2))).(\lambda (H11: (drop i O c2 (CHead x1 (Flat x0) x3))).(\lambda (_: -(subst0 O v x2 x3)).(let H13 \def (eq_ind C x (\lambda (c: C).(clear c e)) H4 -(CHead x1 (Flat x0) x2) H10) in (getl_intro i c2 e (CHead x1 (Flat x0) x3) -H11 (clear_flat x1 e (clear_gen_flat x0 x1 e x2 H13) x0 x3)))))))))) H9)) -(\lambda (H9: (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C x (CHead e1 (Flat f) u)))))) (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop i O c2 (CHead e2 -(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C x (CHead e1 (Flat f) u)))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop i O c2 -(CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2))))) (getl i c2 e) (\lambda (x0: -F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H10: (eq C x -(CHead x1 (Flat x0) x3))).(\lambda (H11: (drop i O c2 (CHead x2 (Flat x0) -x3))).(\lambda (H12: (csubst0 O v x1 x2)).(let H13 \def (eq_ind C x (\lambda -(c: C).(clear c e)) H4 (CHead x1 (Flat x0) x3) H10) in (getl_intro i c2 e -(CHead x2 (Flat x0) x3) H11 (clear_flat x2 e (csubst0_clear_O x1 x2 v H12 e -(clear_gen_flat x0 x1 e x3 H13)) x0 x3)))))))))) H9)) (\lambda (H9: (ex4_5 F -C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(eq C x (CHead e1 (Flat f) u))))))) (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop i O -c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C x (CHead e1 (Flat f) -u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop i O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O -v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (getl i c2 e) (\lambda (x0: -F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: -T).(\lambda (H10: (eq C x (CHead x1 (Flat x0) x3))).(\lambda (H11: (drop i O -c2 (CHead x2 (Flat x0) x4))).(\lambda (_: (subst0 O v x3 x4)).(\lambda (H13: -(csubst0 O v x1 x2)).(let H14 \def (eq_ind C x (\lambda (c: C).(clear c e)) -H4 (CHead x1 (Flat x0) x3) H10) in (getl_intro i c2 e (CHead x2 (Flat x0) x4) -H11 (clear_flat x2 e (csubst0_clear_O x1 x2 v H13 e (clear_gen_flat x0 x1 e -x3 H14)) x0 x4)))))))))))) H9)) H8)) n H5)))) (\lambda (H5: (lt n -i)).(le_lt_false i n H H5 (getl n c2 e))))))) H2)))))))))). -(* COMMENTS -Initial nodes: 1525 -END *) - -theorem csubst0_getl_lt: - \forall (i: nat).(\forall (n: nat).((lt n i) \to (\forall (c1: C).(\forall -(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((getl n c1 -e) \to (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) -(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 -(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))))))))))) -\def - \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (lt n i)).(\lambda (c1: -C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 i v c1 -c2)).(\lambda (e: C).(\lambda (H1: (getl n c1 e)).(let H2 \def (getl_gen_all -c1 e n H1) in (ex2_ind C (\lambda (e0: C).(drop n O c1 e0)) (\lambda (e0: -C).(clear e0 e)) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda -(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) -(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 -(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x: -C).(\lambda (H3: (drop n O c1 x)).(\lambda (H4: (clear x e)).(let H5 \def -(csubst0_drop_lt n i H c1 c2 v H0 x H3) in (or4_ind (drop n O c2 x) (ex3_4 K -C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C -x (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop n O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda -(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n)) v u w)))))) -(ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C x (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop n O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C x (CHead e1 k u))))))) (\lambda (k: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n O -c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n)) v u w)))))) -(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus i (s k n)) v e1 e2))))))) (or4 (getl n c2 e) (ex3_4 B -C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C -e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 -(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))))) (\lambda (H6: (drop n O c2 x)).(or4_intro0 (getl n c2 e) -(ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: -T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 -(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: -T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 -(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -(minus i (S n)) v e1 e2))))))) (getl_intro n c2 e x H6 H4))) (\lambda (H6: -(ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: -T).(eq C x (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop n O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda -(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n)) v u -w))))))).(ex3_4_ind K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C x (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e0 k w)))))) (\lambda -(k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k -n)) v u w))))) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda -(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) -(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 -(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x0: -K).(\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq C x -(CHead x1 x0 x2))).(\lambda (H8: (drop n O c2 (CHead x1 x0 x3))).(\lambda -(H9: (subst0 (minus i (s x0 n)) v x2 x3)).(let H10 \def (eq_ind C x (\lambda -(c: C).(clear c e)) H4 (CHead x1 x0 x2) H7) in (K_ind (\lambda (k: K).((drop -n O c2 (CHead x1 k x3)) \to ((subst0 (minus i (s k n)) v x2 x3) \to ((clear -(CHead x1 k x2) e) \to (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: -B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind -b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 -B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq -C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))))))) (\lambda (b: -B).(\lambda (H11: (drop n O c2 (CHead x1 (Bind b) x3))).(\lambda (H12: -(subst0 (minus i (s (Bind b) n)) v x2 x3)).(\lambda (H13: (clear (CHead x1 -(Bind b) x2) e)).(eq_ind_r C (CHead x1 (Bind b) x2) (\lambda (c: C).(or4 -(getl n c2 c) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C c (CHead e0 (Bind b0) u)))))) (\lambda (b0: -B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 -(Bind b0) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda -(w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b0: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind -b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(getl n c2 (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) -(ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b0) u))))))) (\lambda (b0: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n -c2 (CHead e2 (Bind b0) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro1 (getl n c2 -(CHead x1 (Bind b) x2)) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Bind b) x2) (CHead e0 -(Bind b0) u)))))) (\lambda (b0: B).(\lambda (e0: C).(\lambda (_: T).(\lambda -(w: T).(getl n c2 (CHead e0 (Bind b0) w)))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 -B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq -C (CHead x1 (Bind b) x2) (CHead e1 (Bind b0) u)))))) (\lambda (b0: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 -(Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C -(CHead x1 (Bind b) x2) (CHead e1 (Bind b0) u))))))) (\lambda (b0: B).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 -(Bind b0) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -(minus i (S n)) v e1 e2))))))) (ex3_4_intro B C T T (\lambda (b0: B).(\lambda -(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Bind b) x2) (CHead -e0 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b0) w)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w))))) b x1 x2 x3 (refl_equal C (CHead x1 (Bind b) x2)) (getl_intro n c2 -(CHead x1 (Bind b) x3) (CHead x1 (Bind b) x3) H11 (clear_bind b x1 x3)) H12)) -e (clear_gen_bind b x1 e x2 H13)))))) (\lambda (f: F).(\lambda (H11: (drop n -O c2 (CHead x1 (Flat f) x3))).(\lambda (_: (subst0 (minus i (s (Flat f) n)) v -x2 x3)).(\lambda (H13: (clear (CHead x1 (Flat f) x2) e)).(or4_intro0 (getl n -c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: -B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 -(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind -b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) -(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n -c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (getl_intro n c2 e (CHead x1 -(Flat f) x3) H11 (clear_flat x1 e (clear_gen_flat f x1 e x2 H13) f x3))))))) -x0 H8 H9 H10))))))))) H6)) (\lambda (H6: (ex3_4 K C C T (\lambda (k: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C x (CHead e1 k -u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop n O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1 e2))))))).(ex3_4_ind -K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq -C x (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop n O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1 -e2))))) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) -(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 -(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x0: -K).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H7: (eq C x -(CHead x1 x0 x3))).(\lambda (H8: (drop n O c2 (CHead x2 x0 x3))).(\lambda -(H9: (csubst0 (minus i (s x0 n)) v x1 x2)).(let H10 \def (eq_ind C x (\lambda -(c: C).(clear c e)) H4 (CHead x1 x0 x3) H7) in (K_ind (\lambda (k: K).((drop -n O c2 (CHead x2 k x3)) \to ((csubst0 (minus i (s k n)) v x1 x2) \to ((clear -(CHead x1 k x3) e) \to (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: -B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind -b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 -B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq -C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))))))) (\lambda (b: -B).(\lambda (H11: (drop n O c2 (CHead x2 (Bind b) x3))).(\lambda (H12: -(csubst0 (minus i (s (Bind b) n)) v x1 x2)).(\lambda (H13: (clear (CHead x1 -(Bind b) x3) e)).(eq_ind_r C (CHead x1 (Bind b) x3) (\lambda (c: C).(or4 -(getl n c2 c) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C c (CHead e0 (Bind b0) u)))))) (\lambda (b0: -B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 -(Bind b0) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda -(w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b0: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind -b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(getl n c2 (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) -(ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b0) u))))))) (\lambda (b0: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n -c2 (CHead e2 (Bind b0) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro2 (getl n c2 -(CHead x1 (Bind b) x3)) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Bind b) x3) (CHead e0 -(Bind b0) u)))))) (\lambda (b0: B).(\lambda (e0: C).(\lambda (_: T).(\lambda -(w: T).(getl n c2 (CHead e0 (Bind b0) w)))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 -B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq -C (CHead x1 (Bind b) x3) (CHead e1 (Bind b0) u)))))) (\lambda (b0: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 -(Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C -(CHead x1 (Bind b) x3) (CHead e1 (Bind b0) u))))))) (\lambda (b0: B).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 -(Bind b0) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -(minus i (S n)) v e1 e2))))))) (ex3_4_intro B C C T (\lambda (b0: B).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x1 (Bind b) x3) (CHead -e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b0) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2))))) b x1 x2 x3 (refl_equal C (CHead x1 (Bind b) x3)) (getl_intro n -c2 (CHead x2 (Bind b) x3) (CHead x2 (Bind b) x3) H11 (clear_bind b x2 x3)) -H12)) e (clear_gen_bind b x1 e x3 H13)))))) (\lambda (f: F).(\lambda (H11: -(drop n O c2 (CHead x2 (Flat f) x3))).(\lambda (H12: (csubst0 (minus i (s -(Flat f) n)) v x1 x2)).(\lambda (H13: (clear (CHead x1 (Flat f) x3) e)).(let -H14 \def (eq_ind nat (minus i n) (\lambda (n0: nat).(csubst0 n0 v x1 x2)) H12 -(S (minus i (S n))) (minus_x_Sy i n H)) in (let H15 \def (csubst0_clear_S x1 -x2 v (minus i (S n)) H14 e (clear_gen_flat f x1 e x3 H13)) in (or4_ind (clear -x2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u1)))))) (\lambda (b: -B).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e0 -(Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 (minus i (S n)) v u1 u2)))))) (ex3_4 B C C T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind -b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(clear x2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) -(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u1))))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear -x2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (minus i (S n)) v u1 u2)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (or4 (getl n c2 e) (ex3_4 B C -T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 -(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))))) (\lambda (H16: (clear x2 e)).(or4_intro0 (getl n c2 e) (ex3_4 -B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq -C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 -(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2))))))) (getl_intro n c2 e (CHead x2 (Flat f) x3) H11 (clear_flat x2 e -H16 f x3)))) (\lambda (H16: (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u1)))))) -(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(clear x2 -(CHead e0 (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 (minus i (S n)) v u1 u2))))))).(ex3_4_ind B C T T -(\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C e -(CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: -T).(\lambda (u2: T).(clear x2 (CHead e0 (Bind b) u2)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (minus i (S n)) -v u1 u2))))) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) -(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 -(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x4: -B).(\lambda (x5: C).(\lambda (x6: T).(\lambda (x7: T).(\lambda (H17: (eq C e -(CHead x5 (Bind x4) x6))).(\lambda (H18: (clear x2 (CHead x5 (Bind x4) -x7))).(\lambda (H19: (subst0 (minus i (S n)) v x6 x7)).(eq_ind_r C (CHead x5 -(Bind x4) x6) (\lambda (c: C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda (b: -B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Bind -b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 -B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq -C c (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b) u))))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 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-(minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) -x6) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2))))))) (ex3_4_intro B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda -(u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) x6) (CHead e0 (Bind b) -u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w))))) x4 x5 -x6 x7 (refl_equal C (CHead x5 (Bind x4) x6)) (getl_intro n c2 (CHead x5 (Bind -x4) x7) (CHead x2 (Flat f) x3) H11 (clear_flat x2 (CHead x5 (Bind x4) x7) H18 -f x3)) H19)) e H17)))))))) H16)) (\lambda (H16: (ex3_4 B C C T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind -b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(clear x2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 -e2))))))).(ex3_4_ind B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x2 (CHead e2 (Bind -b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus i (S n)) v e1 e2))))) (or4 (getl n c2 e) (ex3_4 B C T T -(\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 -(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))))) (\lambda (x4: B).(\lambda (x5: C).(\lambda (x6: C).(\lambda -(x7: T).(\lambda (H17: (eq C e (CHead x5 (Bind x4) x7))).(\lambda (H18: -(clear x2 (CHead x6 (Bind x4) x7))).(\lambda (H19: (csubst0 (minus i (S n)) v -x5 x6)).(eq_ind_r C (CHead x5 (Bind x4) x7) (\lambda (c: C).(or4 (getl n c2 -c) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda -(_: T).(eq C c (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind b) u)))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 -(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: -T).(eq C c (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 -(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -(minus i (S n)) v e1 e2))))))))) (or4_intro2 (getl n c2 (CHead x5 (Bind x4) -x7)) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda -(_: T).(eq C (CHead x5 (Bind x4) x7) (CHead e0 (Bind b) u)))))) (\lambda (b: -B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 -(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x5 (Bind x4) -x7) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) x7) (CHead e1 -(Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))) -(ex3_4_intro B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C (CHead x5 (Bind x4) x7) (CHead e1 (Bind b) u)))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 -(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))) x4 x5 x6 x7 -(refl_equal C (CHead x5 (Bind x4) x7)) (getl_intro n c2 (CHead x6 (Bind x4) -x7) (CHead x2 (Flat f) x3) H11 (clear_flat x2 (CHead x6 (Bind x4) x7) H18 f -x3)) H19)) e H17)))))))) H16)) (\lambda (H16: (ex4_5 B C C T T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C e -(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e2 (Bind b) u2))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 (minus i (S n)) v u1 u2)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C e (CHead e1 (Bind -b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (u2: T).(clear x2 (CHead e2 (Bind b) u2))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 -(minus i (S n)) v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) -(or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda -(u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: -B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 -(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind -b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) -(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n -c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x4: B).(\lambda -(x5: C).(\lambda (x6: C).(\lambda (x7: T).(\lambda (x8: T).(\lambda (H17: (eq -C e (CHead x5 (Bind x4) x7))).(\lambda (H18: (clear x2 (CHead x6 (Bind x4) -x8))).(\lambda (H19: (subst0 (minus i (S n)) v x7 x8)).(\lambda (H20: -(csubst0 (minus i (S n)) v x5 x6)).(eq_ind_r C (CHead x5 (Bind x4) x7) -(\lambda (c: C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda (b: B).(\lambda -(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Bind b) u)))))) -(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 -(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c -(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b) u))))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro3 -(getl n c2 (CHead x5 (Bind x4) x7)) (ex3_4 B C T T (\lambda (b: B).(\lambda -(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) x7) (CHead -e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda -(w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 -B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq -C (CHead x5 (Bind x4) x7) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda -(_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 -(minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) -x7) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2))))))) (ex4_5_intro B C C T T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) -x7) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))) x4 x5 x6 x7 x8 (refl_equal C (CHead x5 (Bind x4) x7)) -(getl_intro n c2 (CHead x6 (Bind x4) x8) (CHead x2 (Flat f) x3) H11 -(clear_flat x2 (CHead x6 (Bind x4) x8) H18 f x3)) H19 H20)) e H17)))))))))) -H16)) H15))))))) x0 H8 H9 H10))))))))) H6)) (\lambda (H6: (ex4_5 K C C T T -(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: -T).(eq C x (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e2 k w))))))) (\lambda -(k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (s k n)) v u w)))))) (\lambda (k: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k -n)) v e1 e2)))))))).(ex4_5_ind K C C T T (\lambda (k: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C x (CHead e1 k -u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop n O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda -(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k -n)) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1 e2)))))) (or4 (getl n -c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: -B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 -(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind -b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) -(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n -c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x0: K).(\lambda -(x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H7: (eq -C x (CHead x1 x0 x3))).(\lambda (H8: (drop n O c2 (CHead x2 x0 x4))).(\lambda -(H9: (subst0 (minus i (s x0 n)) v x3 x4)).(\lambda (H10: (csubst0 (minus i (s -x0 n)) v x1 x2)).(let H11 \def (eq_ind C x (\lambda (c: C).(clear c e)) H4 -(CHead x1 x0 x3) H7) in (K_ind (\lambda (k: K).((drop n O c2 (CHead x2 k x4)) -\to ((subst0 (minus i (s k n)) v x3 x4) \to ((csubst0 (minus i (s k n)) v x1 -x2) \to ((clear (CHead x1 k x3) e) \to (or4 (getl n c2 e) (ex3_4 B C T T -(\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 -(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2))))))))))))) (\lambda (b: B).(\lambda (H12: (drop n O c2 (CHead x2 -(Bind b) x4))).(\lambda (H13: (subst0 (minus i (s (Bind b) n)) v x3 -x4)).(\lambda (H14: (csubst0 (minus i (s (Bind b) n)) v x1 x2)).(\lambda -(H15: (clear (CHead x1 (Bind b) x3) e)).(eq_ind_r C (CHead x1 (Bind b) x3) -(\lambda (c: C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda (b0: B).(\lambda -(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Bind b0) u)))))) -(\lambda (b0: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 -(CHead e0 (Bind b0) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T -(\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c -(CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b0) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b0) u))))))) -(\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(w: T).(getl n c2 (CHead e2 (Bind b0) w))))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro3 -(getl n c2 (CHead x1 (Bind b) x3)) (ex3_4 B C T T (\lambda (b0: B).(\lambda -(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Bind b) x3) (CHead -e0 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b0) w)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (ex3_4 B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C (CHead x1 (Bind b) x3) (CHead e1 (Bind b0) u)))))) -(\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 -(CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T -(\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda -(_: T).(eq C (CHead x1 (Bind b) x3) (CHead e1 (Bind b0) u))))))) (\lambda -(b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(getl n c2 (CHead e2 (Bind b0) w))))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (ex4_5_intro B C C -T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(eq C (CHead x1 (Bind b) x3) (CHead e1 (Bind b0) u))))))) -(\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(w: T).(getl n c2 (CHead e2 (Bind b0) w))))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) b x1 x2 x3 x4 -(refl_equal C (CHead x1 (Bind b) x3)) (getl_intro n c2 (CHead x2 (Bind b) x4) -(CHead x2 (Bind b) x4) H12 (clear_bind b x2 x4)) H13 H14)) e (clear_gen_bind -b x1 e x3 H15))))))) (\lambda (f: F).(\lambda (H12: (drop n O c2 (CHead x2 -(Flat f) x4))).(\lambda (_: (subst0 (minus i (s (Flat f) n)) v x3 -x4)).(\lambda (H14: (csubst0 (minus i (s (Flat f) n)) v x1 x2)).(\lambda -(H15: (clear (CHead x1 (Flat f) x3) e)).(let H16 \def (eq_ind nat (minus i n) -(\lambda (n0: nat).(csubst0 n0 v x1 x2)) H14 (S (minus i (S n))) (minus_x_Sy -i n H)) in (let H17 \def (csubst0_clear_S x1 x2 v (minus i (S n)) H16 e -(clear_gen_flat f x1 e x3 H15)) in (or4_ind (clear x2 e) (ex3_4 B C T T -(\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C e -(CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: -T).(\lambda (u2: T).(clear x2 (CHead e0 (Bind b) u2)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (minus i (S n)) -v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x2 (CHead e2 (Bind -b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C e -(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e2 (Bind b) u2))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 (minus i (S n)) v u1 u2)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2))))))) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda -(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) -(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 -(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (H18: -(clear x2 e)).(or4_intro0 (getl n c2 e) (ex3_4 B C T T (\lambda (b: -B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind -b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 -B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq -C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (getl_intro n c2 e -(CHead x2 (Flat f) x4) H12 (clear_flat x2 e H18 f x4)))) (\lambda (H18: -(ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(eq C e (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e0 (Bind b) u2)))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 -(minus i (S n)) v u1 u2))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda -(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u1)))))) -(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(clear x2 -(CHead e0 (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 (minus i (S n)) v u1 u2))))) (or4 (getl n c2 e) -(ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: -T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 -(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: -T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 -(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -(minus i (S n)) v e1 e2)))))))) (\lambda (x5: B).(\lambda (x6: C).(\lambda -(x7: T).(\lambda (x8: T).(\lambda (H19: (eq C e (CHead x6 (Bind x5) -x7))).(\lambda (H20: (clear x2 (CHead x6 (Bind x5) x8))).(\lambda (H21: -(subst0 (minus i (S n)) v x7 x8)).(eq_ind_r C (CHead x6 (Bind x5) x7) -(\lambda (c: C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda (b: B).(\lambda -(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Bind b) u)))))) -(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 -(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c -(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b) u))))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro1 -(getl n c2 (CHead x6 (Bind x5) x7)) (ex3_4 B C T T (\lambda (b: B).(\lambda -(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) x7) (CHead -e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda -(w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 -B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq -C (CHead x6 (Bind x5) x7) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda -(_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 -(minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) -x7) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2))))))) (ex3_4_intro B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda -(u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) x7) (CHead e0 (Bind b) -u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w))))) x5 x6 -x7 x8 (refl_equal C (CHead x6 (Bind x5) x7)) (getl_intro n c2 (CHead x6 (Bind -x5) x8) (CHead x2 (Flat f) x4) H12 (clear_flat x2 (CHead x6 (Bind x5) x8) H20 -f x4)) H21)) e H19)))))))) H18)) (\lambda (H18: (ex3_4 B C C T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind -b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(clear x2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 -e2))))))).(ex3_4_ind B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x2 (CHead e2 (Bind -b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus i (S n)) v e1 e2))))) (or4 (getl n c2 e) (ex3_4 B C T T -(\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 -(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))))) (\lambda (x5: B).(\lambda (x6: C).(\lambda (x7: C).(\lambda -(x8: T).(\lambda (H19: (eq C e (CHead x6 (Bind x5) x8))).(\lambda (H20: -(clear x2 (CHead x7 (Bind x5) x8))).(\lambda (H21: (csubst0 (minus i (S n)) v -x6 x7)).(eq_ind_r C (CHead x6 (Bind x5) x8) (\lambda (c: C).(or4 (getl n c2 -c) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda -(_: T).(eq C c (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind b) u)))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 -(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: -T).(eq C c (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 -(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -(minus i (S n)) v e1 e2))))))))) (or4_intro2 (getl n c2 (CHead x6 (Bind x5) -x8)) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda -(_: T).(eq C (CHead x6 (Bind x5) x8) (CHead e0 (Bind b) u)))))) (\lambda (b: -B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 -(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x6 (Bind x5) -x8) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) x8) (CHead e1 -(Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))) -(ex3_4_intro B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C (CHead x6 (Bind x5) x8) (CHead e1 (Bind b) u)))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 -(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))) x5 x6 x7 x8 -(refl_equal C (CHead x6 (Bind x5) x8)) (getl_intro n c2 (CHead x7 (Bind x5) -x8) (CHead x2 (Flat f) x4) H12 (clear_flat x2 (CHead x7 (Bind x5) x8) H20 f -x4)) H21)) e H19)))))))) H18)) (\lambda (H18: (ex4_5 B C C T T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C e -(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e2 (Bind b) u2))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 (minus i (S n)) v u1 u2)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C e (CHead e1 (Bind -b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (u2: T).(clear x2 (CHead e2 (Bind b) u2))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 -(minus i (S n)) v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) -(or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda -(u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: -B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 -(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind -b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) -(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n -c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x5: B).(\lambda -(x6: C).(\lambda (x7: C).(\lambda (x8: T).(\lambda (x9: T).(\lambda (H19: (eq -C e (CHead x6 (Bind x5) x8))).(\lambda (H20: (clear x2 (CHead x7 (Bind x5) -x9))).(\lambda (H21: (subst0 (minus i (S n)) v x8 x9)).(\lambda (H22: -(csubst0 (minus i (S n)) v x6 x7)).(eq_ind_r C (CHead x6 (Bind x5) x8) -(\lambda (c: C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda (b: B).(\lambda -(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Bind b) u)))))) -(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 -(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c -(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b) u))))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro3 -(getl n c2 (CHead x6 (Bind x5) x8)) (ex3_4 B C T T (\lambda (b: B).(\lambda -(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) x8) (CHead -e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda -(w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 -B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq -C (CHead x6 (Bind x5) x8) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda -(_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 -(minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) -x8) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2))))))) (ex4_5_intro B C C T T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) -x8) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))) x5 x6 x7 x8 x9 (refl_equal C (CHead x6 (Bind x5) x8)) -(getl_intro n c2 (CHead x7 (Bind x5) x9) (CHead x2 (Flat f) x4) H12 -(clear_flat x2 (CHead x7 (Bind x5) x9) H20 f x4)) H21 H22)) e H19)))))))))) -H18)) H17)))))))) x0 H8 H9 H10 H11))))))))))) H6)) H5))))) H2)))))))))). -(* COMMENTS -Initial nodes: 17179 -END *) - -theorem csubst0_getl_ge_back: - \forall (i: nat).(\forall (n: nat).((le i n) \to (\forall (c1: C).(\forall -(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((getl n c2 -e) \to (getl n c1 e))))))))) -\def - \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (le i n)).(\lambda (c1: -C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 i v c1 -c2)).(\lambda (e: C).(\lambda (H1: (getl n c2 e)).(let H2 \def (getl_gen_all -c2 e n H1) in (ex2_ind C (\lambda (e0: C).(drop n O c2 e0)) (\lambda (e0: -C).(clear e0 e)) (getl n c1 e) (\lambda (x: C).(\lambda (H3: (drop n O c2 -x)).(\lambda (H4: (clear x e)).(lt_eq_gt_e i n (getl n c1 e) (\lambda (H5: -(lt i n)).(getl_intro n c1 e x (csubst0_drop_gt_back n i H5 c1 c2 v H0 x H3) -H4)) (\lambda (H5: (eq nat i n)).(let H6 \def (eq_ind_r nat n (\lambda (n0: -nat).(drop n0 O c2 x)) H3 i H5) in (let H7 \def (eq_ind_r nat n (\lambda (n0: -nat).(le i n0)) H i H5) in (eq_ind nat i (\lambda (n0: nat).(getl n0 c1 e)) -(let H8 \def (csubst0_drop_eq_back i c1 c2 v H0 x H6) in (or4_ind (drop i O -c1 x) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (u2: T).(eq C x (CHead e0 (Flat f) u2)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop i O c1 (CHead e0 -(Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(eq C x (CHead e2 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop i O c1 -(CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C x -(CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(drop i O c1 (CHead e1 (Flat f) u1))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (getl i c1 -e) (\lambda (H9: (drop i O c1 x)).(getl_intro i c1 e x H9 H4)) (\lambda (H9: -(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: -T).(eq C x (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(drop i O c1 (CHead e0 (Flat f) u1)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v -u1 u2))))))).(ex3_4_ind F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (u2: T).(eq C x (CHead e0 (Flat f) u2)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop i O c1 (CHead e0 -(Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 O v u1 u2))))) (getl i c1 e) (\lambda (x0: F).(\lambda (x1: -C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H10: (eq C x (CHead x1 (Flat -x0) x3))).(\lambda (H11: (drop i O c1 (CHead x1 (Flat x0) x2))).(\lambda (_: -(subst0 O v x2 x3)).(let H13 \def (eq_ind C x (\lambda (c: C).(clear c e)) H4 -(CHead x1 (Flat x0) x3) H10) in (getl_intro i c1 e (CHead x1 (Flat x0) x2) -H11 (clear_flat x1 e (clear_gen_flat x0 x1 e x3 H13) x0 x2)))))))))) H9)) -(\lambda (H9: (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C x (CHead e2 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop i O c1 (CHead e1 -(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda -(_: C).(\lambda (e2: C).(\lambda (u: T).(eq C x (CHead e2 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop i O c1 -(CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2))))) (getl i c1 e) (\lambda (x0: -F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H10: (eq C x -(CHead x2 (Flat x0) x3))).(\lambda (H11: (drop i O c1 (CHead x1 (Flat x0) -x3))).(\lambda (H12: (csubst0 O v x1 x2)).(let H13 \def (eq_ind C x (\lambda -(c: C).(clear c e)) H4 (CHead x2 (Flat x0) x3) H10) in (getl_intro i c1 e -(CHead x1 (Flat x0) x3) H11 (clear_flat x1 e (csubst0_clear_O_back x1 x2 v -H12 e (clear_gen_flat x0 x2 e x3 H13)) x0 x3)))))))))) H9)) (\lambda (H9: -(ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (u2: T).(eq C x (CHead e2 (Flat f) u2))))))) (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop i -O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda -(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C x (CHead e2 (Flat -f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (_: T).(drop i O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 -O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (getl i c1 e) (\lambda (x0: -F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: -T).(\lambda (H10: (eq C x (CHead x2 (Flat x0) x4))).(\lambda (H11: (drop i O -c1 (CHead x1 (Flat x0) x3))).(\lambda (_: (subst0 O v x3 x4)).(\lambda (H13: -(csubst0 O v x1 x2)).(let H14 \def (eq_ind C x (\lambda (c: C).(clear c e)) -H4 (CHead x2 (Flat x0) x4) H10) in (getl_intro i c1 e (CHead x1 (Flat x0) x3) -H11 (clear_flat x1 e (csubst0_clear_O_back x1 x2 v H13 e (clear_gen_flat x0 -x2 e x4 H14)) x0 x3)))))))))))) H9)) H8)) n H5)))) (\lambda (H5: (lt n -i)).(le_lt_false i n H H5 (getl n c1 e))))))) H2)))))))))). -(* COMMENTS -Initial nodes: 1525 -END *) - -theorem csubst0_getl_lt_back: - \forall (n: nat).(\forall (i: nat).((lt n i) \to (\forall (c1: C).(\forall -(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e2: C).((getl n c2 -e2) \to (or (getl n c1 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) v e1 -e2)) (\lambda (e1: C).(getl n c1 e1)))))))))))) -\def - \lambda (n: nat).(\lambda (i: nat).(\lambda (H: (lt n i)).(\lambda (c1: -C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 i v c1 -c2)).(\lambda (e2: C).(\lambda (H1: (getl n c2 e2)).(let H2 \def -(getl_gen_all c2 e2 n H1) in (ex2_ind C (\lambda (e: C).(drop n O c2 e)) -(\lambda (e: C).(clear e e2)) (or (getl n c1 e2) (ex2 C (\lambda (e1: -C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(getl n c1 e1)))) (\lambda -(x: C).(\lambda (H3: (drop n O c2 x)).(\lambda (H4: (clear x e2)).(let H_x -\def (csubst0_drop_lt_back n i H c1 c2 v H0 x H3) in (let H5 \def H_x in -(or_ind (drop n O c1 x) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) v e1 x)) -(\lambda (e1: C).(drop n O c1 e1))) (or (getl n c1 e2) (ex2 C (\lambda (e1: -C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(getl n c1 e1)))) (\lambda -(H6: (drop n O c1 x)).(or_introl (getl n c1 e2) (ex2 C (\lambda (e1: -C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(getl n c1 e1))) -(getl_intro n c1 e2 x H6 H4))) (\lambda (H6: (ex2 C (\lambda (e1: C).(csubst0 -(minus i n) v e1 x)) (\lambda (e1: C).(drop n O c1 e1)))).(ex2_ind C (\lambda -(e1: C).(csubst0 (minus i n) v e1 x)) (\lambda (e1: C).(drop n O c1 e1)) (or -(getl n c1 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) v e1 e2)) -(\lambda (e1: C).(getl n c1 e1)))) (\lambda (x0: C).(\lambda (H7: (csubst0 -(minus i n) v x0 x)).(\lambda (H8: (drop n O c1 x0)).(let H_x0 \def -(csubst0_clear_trans x0 x v (minus i n) H7 e2 H4) in (let H9 \def H_x0 in -(or_ind (clear x0 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) v e1 e2)) -(\lambda (e1: C).(clear x0 e1))) (or (getl n c1 e2) (ex2 C (\lambda (e1: -C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(getl n c1 e1)))) (\lambda -(H10: (clear x0 e2)).(or_introl (getl n c1 e2) (ex2 C (\lambda (e1: -C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(getl n c1 e1))) -(getl_intro n c1 e2 x0 H8 H10))) (\lambda (H10: (ex2 C (\lambda (e1: -C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(clear x0 e1)))).(ex2_ind -C (\lambda (e1: C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(clear x0 -e1)) (or (getl n c1 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) v e1 -e2)) (\lambda (e1: C).(getl n c1 e1)))) (\lambda (x1: C).(\lambda (H11: -(csubst0 (minus i n) v x1 e2)).(\lambda (H12: (clear x0 x1)).(or_intror (getl -n c1 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: -C).(getl n c1 e1))) (ex_intro2 C (\lambda (e1: C).(csubst0 (minus i n) v e1 -e2)) (\lambda (e1: C).(getl n c1 e1)) x1 H11 (getl_intro n c1 x1 x0 H8 -H12)))))) H10)) H9)))))) H6)) H5)))))) H2)))))))))). -(* COMMENTS -Initial nodes: 801 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubst0/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubst0/props.ma deleted file mode 100644 index 28b75ec37..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubst0/props.ma +++ /dev/null @@ -1,61 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubst0/defs.ma". - -theorem csubst0_snd_bind: - \forall (b: B).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall -(u2: T).((subst0 i v u1 u2) \to (\forall (c: C).(csubst0 (S i) v (CHead c -(Bind b) u1) (CHead c (Bind b) u2)))))))) -\def - \lambda (b: B).(\lambda (i: nat).(\lambda (v: T).(\lambda (u1: T).(\lambda -(u2: T).(\lambda (H: (subst0 i v u1 u2)).(\lambda (c: C).(eq_ind nat (s (Bind -b) i) (\lambda (n: nat).(csubst0 n v (CHead c (Bind b) u1) (CHead c (Bind b) -u2))) (csubst0_snd (Bind b) i v u1 u2 H c) (S i) (refl_equal nat (S -i))))))))). -(* COMMENTS -Initial nodes: 91 -END *) - -theorem csubst0_fst_bind: - \forall (b: B).(\forall (i: nat).(\forall (c1: C).(\forall (c2: C).(\forall -(v: T).((csubst0 i v c1 c2) \to (\forall (u: T).(csubst0 (S i) v (CHead c1 -(Bind b) u) (CHead c2 (Bind b) u)))))))) -\def - \lambda (b: B).(\lambda (i: nat).(\lambda (c1: C).(\lambda (c2: C).(\lambda -(v: T).(\lambda (H: (csubst0 i v c1 c2)).(\lambda (u: T).(eq_ind nat (s (Bind -b) i) (\lambda (n: nat).(csubst0 n v (CHead c1 (Bind b) u) (CHead c2 (Bind b) -u))) (csubst0_fst (Bind b) i c1 c2 v H u) (S i) (refl_equal nat (S i))))))))). -(* COMMENTS -Initial nodes: 91 -END *) - -theorem csubst0_both_bind: - \forall (b: B).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall -(u2: T).((subst0 i v u1 u2) \to (\forall (c1: C).(\forall (c2: C).((csubst0 i -v c1 c2) \to (csubst0 (S i) v (CHead c1 (Bind b) u1) (CHead c2 (Bind b) -u2)))))))))) -\def - \lambda (b: B).(\lambda (i: nat).(\lambda (v: T).(\lambda (u1: T).(\lambda -(u2: T).(\lambda (H: (subst0 i v u1 u2)).(\lambda (c1: C).(\lambda (c2: -C).(\lambda (H0: (csubst0 i v c1 c2)).(eq_ind nat (s (Bind b) i) (\lambda (n: -nat).(csubst0 n v (CHead c1 (Bind b) u1) (CHead c2 (Bind b) u2))) -(csubst0_both (Bind b) i v u1 u2 H c1 c2 H0) (S i) (refl_equal nat (S -i))))))))))). -(* COMMENTS -Initial nodes: 107 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubst1/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubst1/defs.ma deleted file mode 100644 index d6ba0a942..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubst1/defs.ma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubst0/defs.ma". - -inductive csubst1 (i: nat) (v: T) (c1: C): C \to Prop \def -| csubst1_refl: csubst1 i v c1 c1 -| csubst1_sing: \forall (c2: C).((csubst0 i v c1 c2) \to (csubst1 i v c1 c2)). - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubst1/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubst1/fwd.ma deleted file mode 100644 index fe71a56fd..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubst1/fwd.ma +++ /dev/null @@ -1,117 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubst1/defs.ma". - -include "Basic-1/csubst0/fwd.ma". - -include "Basic-1/subst1/props.ma". - -theorem csubst1_gen_head: - \forall (k: K).(\forall (c1: C).(\forall (x: C).(\forall (u1: T).(\forall -(v: T).(\forall (i: nat).((csubst1 (s k i) v (CHead c1 k u1) x) \to (ex3_2 T -C (\lambda (u2: T).(\lambda (c2: C).(eq C x (CHead c2 k u2)))) (\lambda (u2: -T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c2: -C).(csubst1 i v c1 c2)))))))))) -\def - \lambda (k: K).(\lambda (c1: C).(\lambda (x: C).(\lambda (u1: T).(\lambda -(v: T).(\lambda (i: nat).(\lambda (H: (csubst1 (s k i) v (CHead c1 k u1) -x)).(csubst1_ind (s k i) v (CHead c1 k u1) (\lambda (c: C).(ex3_2 T C -(\lambda (u2: T).(\lambda (c2: C).(eq C c (CHead c2 k u2)))) (\lambda (u2: -T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c2: -C).(csubst1 i v c1 c2))))) (ex3_2_intro T C (\lambda (u2: T).(\lambda (c2: -C).(eq C (CHead c1 k u1) (CHead c2 k u2)))) (\lambda (u2: T).(\lambda (_: -C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c2: C).(csubst1 i v c1 -c2))) u1 c1 (refl_equal C (CHead c1 k u1)) (subst1_refl i v u1) (csubst1_refl -i v c1)) (\lambda (c2: C).(\lambda (H0: (csubst0 (s k i) v (CHead c1 k u1) -c2)).(or3_ind (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i) -(s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c1 k u2)))) -(\lambda (u2: T).(\lambda (j: nat).(subst0 j v u1 u2)))) (ex3_2 C nat -(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i) (s k j)))) (\lambda (c3: -C).(\lambda (_: nat).(eq C c2 (CHead c3 k u1)))) (\lambda (c3: C).(\lambda -(j: nat).(csubst0 j v c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: -C).(\lambda (j: nat).(eq nat (s k i) (s k j))))) (\lambda (u2: T).(\lambda -(c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: -T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u2)))) (\lambda (_: -T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3))))) (ex3_2 T C -(\lambda (u2: T).(\lambda (c3: C).(eq C c2 (CHead c3 k u2)))) (\lambda (u2: -T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3: -C).(csubst1 i v c1 c3)))) (\lambda (H1: (ex3_2 T nat (\lambda (_: T).(\lambda -(j: nat).(eq nat (s k i) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C -c2 (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v u1 -u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i) (s -k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c1 k u2)))) -(\lambda (u2: T).(\lambda (j: nat).(subst0 j v u1 u2))) (ex3_2 T C (\lambda -(u2: T).(\lambda (c3: C).(eq C c2 (CHead c3 k u2)))) (\lambda (u2: -T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3: -C).(csubst1 i v c1 c3)))) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H2: -(eq nat (s k i) (s k x1))).(\lambda (H3: (eq C c2 (CHead c1 k x0))).(\lambda -(H4: (subst0 x1 v u1 x0)).(eq_ind_r C (CHead c1 k x0) (\lambda (c: C).(ex3_2 -T C (\lambda (u2: T).(\lambda (c3: C).(eq C c (CHead c3 k u2)))) (\lambda -(u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3: -C).(csubst1 i v c1 c3))))) (let H5 \def (eq_ind_r nat x1 (\lambda (n: -nat).(subst0 n v u1 x0)) H4 i (s_inj k i x1 H2)) in (ex3_2_intro T C (\lambda -(u2: T).(\lambda (c3: C).(eq C (CHead c1 k x0) (CHead c3 k u2)))) (\lambda -(u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3: -C).(csubst1 i v c1 c3))) x0 c1 (refl_equal C (CHead c1 k x0)) (subst1_single -i v u1 x0 H5) (csubst1_refl i v c1))) c2 H3)))))) H1)) (\lambda (H1: (ex3_2 C -nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i) (s k j)))) (\lambda -(c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u1)))) (\lambda (c3: -C).(\lambda (j: nat).(csubst0 j v c1 c3))))).(ex3_2_ind C nat (\lambda (_: -C).(\lambda (j: nat).(eq nat (s k i) (s k j)))) (\lambda (c3: C).(\lambda (_: -nat).(eq C c2 (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 -j v c1 c3))) (ex3_2 T C (\lambda (u2: T).(\lambda (c3: C).(eq C c2 (CHead c3 -k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: -T).(\lambda (c3: C).(csubst1 i v c1 c3)))) (\lambda (x0: C).(\lambda (x1: -nat).(\lambda (H2: (eq nat (s k i) (s k x1))).(\lambda (H3: (eq C c2 (CHead -x0 k u1))).(\lambda (H4: (csubst0 x1 v c1 x0)).(eq_ind_r C (CHead x0 k u1) -(\lambda (c: C).(ex3_2 T C (\lambda (u2: T).(\lambda (c3: C).(eq C c (CHead -c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda -(_: T).(\lambda (c3: C).(csubst1 i v c1 c3))))) (let H5 \def (eq_ind_r nat x1 -(\lambda (n: nat).(csubst0 n v c1 x0)) H4 i (s_inj k i x1 H2)) in -(ex3_2_intro T C (\lambda (u2: T).(\lambda (c3: C).(eq C (CHead x0 k u1) -(CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) -(\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1 c3))) u1 x0 (refl_equal C -(CHead x0 k u1)) (subst1_refl i v u1) (csubst1_sing i v c1 x0 H5))) c2 -H3)))))) H1)) (\lambda (H1: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: -C).(\lambda (j: nat).(eq nat (s k i) (s k j))))) (\lambda (u2: T).(\lambda -(c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: -T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u2)))) (\lambda (_: -T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))))).(ex4_3_ind T C -nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i) (s k -j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 -k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 -u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 -c3)))) (ex3_2 T C (\lambda (u2: T).(\lambda (c3: C).(eq C c2 (CHead c3 k -u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: -T).(\lambda (c3: C).(csubst1 i v c1 c3)))) (\lambda (x0: T).(\lambda (x1: -C).(\lambda (x2: nat).(\lambda (H2: (eq nat (s k i) (s k x2))).(\lambda (H3: -(eq C c2 (CHead x1 k x0))).(\lambda (H4: (subst0 x2 v u1 x0)).(\lambda (H5: -(csubst0 x2 v c1 x1)).(eq_ind_r C (CHead x1 k x0) (\lambda (c: C).(ex3_2 T C -(\lambda (u2: T).(\lambda (c3: C).(eq C c (CHead c3 k u2)))) (\lambda (u2: -T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3: -C).(csubst1 i v c1 c3))))) (let H6 \def (eq_ind_r nat x2 (\lambda (n: -nat).(csubst0 n v c1 x1)) H5 i (s_inj k i x2 H2)) in (let H7 \def (eq_ind_r -nat x2 (\lambda (n: nat).(subst0 n v u1 x0)) H4 i (s_inj k i x2 H2)) in -(ex3_2_intro T C (\lambda (u2: T).(\lambda (c3: C).(eq C (CHead x1 k x0) -(CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) -(\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1 c3))) x0 x1 (refl_equal C -(CHead x1 k x0)) (subst1_single i v u1 x0 H7) (csubst1_sing i v c1 x1 H6)))) -c2 H3)))))))) H1)) (csubst0_gen_head k c1 c2 u1 v (s k i) H0)))) x H))))))). -(* COMMENTS -Initial nodes: 1817 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubst1/getl.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubst1/getl.ma deleted file mode 100644 index e24ba9533..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubst1/getl.ma +++ /dev/null @@ -1,283 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubst1/props.ma". - -include "Basic-1/csubst0/getl.ma". - -include "Basic-1/subst1/props.ma". - -include "Basic-1/drop/props.ma". - -theorem csubst1_getl_ge: - \forall (i: nat).(\forall (n: nat).((le i n) \to (\forall (c1: C).(\forall -(c2: C).(\forall (v: T).((csubst1 i v c1 c2) \to (\forall (e: C).((getl n c1 -e) \to (getl n c2 e))))))))) -\def - \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (le i n)).(\lambda (c1: -C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst1 i v c1 -c2)).(csubst1_ind i v c1 (\lambda (c: C).(\forall (e: C).((getl n c1 e) \to -(getl n c e)))) (\lambda (e: C).(\lambda (H1: (getl n c1 e)).H1)) (\lambda -(c3: C).(\lambda (H1: (csubst0 i v c1 c3)).(\lambda (e: C).(\lambda (H2: -(getl n c1 e)).(csubst0_getl_ge i n H c1 c3 v H1 e H2))))) c2 H0))))))). -(* COMMENTS -Initial nodes: 111 -END *) - -theorem csubst1_getl_lt: - \forall (i: nat).(\forall (n: nat).((lt n i) \to (\forall (c1: C).(\forall -(c2: C).(\forall (v: T).((csubst1 i v c1 c2) \to (\forall (e1: C).((getl n c1 -e1) \to (ex2 C (\lambda (e2: C).(csubst1 (minus i n) v e1 e2)) (\lambda (e2: -C).(getl n c2 e2))))))))))) -\def - \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (lt n i)).(\lambda (c1: -C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst1 i v c1 -c2)).(csubst1_ind i v c1 (\lambda (c: C).(\forall (e1: C).((getl n c1 e1) \to -(ex2 C (\lambda (e2: C).(csubst1 (minus i n) v e1 e2)) (\lambda (e2: C).(getl -n c e2)))))) (\lambda (e1: C).(\lambda (H1: (getl n c1 e1)).(eq_ind_r nat (S -(minus i (S n))) (\lambda (n0: nat).(ex2 C (\lambda (e2: C).(csubst1 n0 v e1 -e2)) (\lambda (e2: C).(getl n c1 e2)))) (ex_intro2 C (\lambda (e2: -C).(csubst1 (S (minus i (S n))) v e1 e2)) (\lambda (e2: C).(getl n c1 e2)) e1 -(csubst1_refl (S (minus i (S n))) v e1) H1) (minus i n) (minus_x_Sy i n H)))) -(\lambda (c3: C).(\lambda (H1: (csubst0 i v c1 c3)).(\lambda (e1: C).(\lambda -(H2: (getl n c1 e1)).(eq_ind_r nat (S (minus i (S n))) (\lambda (n0: -nat).(ex2 C (\lambda (e2: C).(csubst1 n0 v e1 e2)) (\lambda (e2: C).(getl n -c3 e2)))) (let H3 \def (csubst0_getl_lt i n H c1 c3 v H1 e1 H2) in (or4_ind -(getl n c3 e1) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e1 (CHead e0 (Bind b) u)))))) (\lambda (b: -B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e0 -(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: -B).(\lambda (e2: C).(\lambda (_: C).(\lambda (u: T).(eq C e1 (CHead e2 (Bind -b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e3: C).(\lambda (u: -T).(getl n c3 (CHead e3 (Bind b) u)))))) (\lambda (_: B).(\lambda (e2: -C).(\lambda (e3: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e2 e3)))))) -(ex4_5 B C C T T (\lambda (b: B).(\lambda (e2: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq C e1 (CHead e2 (Bind b) u))))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e3: C).(\lambda (_: T).(\lambda (w: T).(getl n -c3 (CHead e3 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) -(\lambda (_: B).(\lambda (e2: C).(\lambda (e3: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus i (S n)) v e2 e3))))))) (ex2 C (\lambda (e2: -C).(csubst1 (S (minus i (S n))) v e1 e2)) (\lambda (e2: C).(getl n c3 e2))) -(\lambda (H4: (getl n c3 e1)).(ex_intro2 C (\lambda (e2: C).(csubst1 (S -(minus i (S n))) v e1 e2)) (\lambda (e2: C).(getl n c3 e2)) e1 (csubst1_refl -(S (minus i (S n))) v e1) H4)) (\lambda (H4: (ex3_4 B C T T (\lambda (b: -B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e1 (CHead e0 (Bind -b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(getl n c3 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u -w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e1 (CHead e0 (Bind b) u)))))) (\lambda (b: -B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e0 -(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (S n)) v u w))))) (ex2 C (\lambda (e2: C).(csubst1 (S -(minus i (S n))) v e1 e2)) (\lambda (e2: C).(getl n c3 e2))) (\lambda (x0: -B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H5: (eq C e1 -(CHead x1 (Bind x0) x2))).(\lambda (H6: (getl n c3 (CHead x1 (Bind x0) -x3))).(\lambda (H7: (subst0 (minus i (S n)) v x2 x3)).(eq_ind_r C (CHead x1 -(Bind x0) x2) (\lambda (c: C).(ex2 C (\lambda (e2: C).(csubst1 (S (minus i (S -n))) v c e2)) (\lambda (e2: C).(getl n c3 e2)))) (ex_intro2 C (\lambda (e2: -C).(csubst1 (S (minus i (S n))) v (CHead x1 (Bind x0) x2) e2)) (\lambda (e2: -C).(getl n c3 e2)) (CHead x1 (Bind x0) x3) (csubst1_sing (S (minus i (S n))) -v (CHead x1 (Bind x0) x2) (CHead x1 (Bind x0) x3) (csubst0_snd_bind x0 (minus -i (S n)) v x2 x3 H7 x1)) H6) e1 H5)))))))) H4)) (\lambda (H4: (ex3_4 B C C T -(\lambda (b: B).(\lambda (e2: C).(\lambda (_: C).(\lambda (u: T).(eq C e1 -(CHead e2 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e3: -C).(\lambda (u: T).(getl n c3 (CHead e3 (Bind b) u)))))) (\lambda (_: -B).(\lambda (e2: C).(\lambda (e3: C).(\lambda (_: T).(csubst0 (minus i (S n)) -v e2 e3))))))).(ex3_4_ind B C C T (\lambda (b: B).(\lambda (e2: C).(\lambda -(_: C).(\lambda (u: T).(eq C e1 (CHead e2 (Bind b) u)))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e3: C).(\lambda (u: T).(getl n c3 (CHead e3 -(Bind b) u)))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (e3: C).(\lambda -(_: T).(csubst0 (minus i (S n)) v e2 e3))))) (ex2 C (\lambda (e2: C).(csubst1 -(S (minus i (S n))) v e1 e2)) (\lambda (e2: C).(getl n c3 e2))) (\lambda (x0: -B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H5: (eq C e1 -(CHead x1 (Bind x0) x3))).(\lambda (H6: (getl n c3 (CHead x2 (Bind x0) -x3))).(\lambda (H7: (csubst0 (minus i (S n)) v x1 x2)).(eq_ind_r C (CHead x1 -(Bind x0) x3) (\lambda (c: C).(ex2 C (\lambda (e2: C).(csubst1 (S (minus i (S -n))) v c e2)) (\lambda (e2: C).(getl n c3 e2)))) (ex_intro2 C (\lambda (e2: -C).(csubst1 (S (minus i (S n))) v (CHead x1 (Bind x0) x3) e2)) (\lambda (e2: -C).(getl n c3 e2)) (CHead x2 (Bind x0) x3) (csubst1_sing (S (minus i (S n))) -v (CHead x1 (Bind x0) x3) (CHead x2 (Bind x0) x3) (csubst0_fst_bind x0 (minus -i (S n)) x1 x2 v H7 x3)) H6) e1 H5)))))))) H4)) (\lambda (H4: (ex4_5 B C C T -T (\lambda (b: B).(\lambda (e2: C).(\lambda (_: C).(\lambda (u: T).(\lambda -(_: T).(eq C e1 (CHead e2 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e3: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e3 -(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: -B).(\lambda (e2: C).(\lambda (e3: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -(minus i (S n)) v e2 e3)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda -(e2: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e1 (CHead e2 -(Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e3: C).(\lambda -(_: T).(\lambda (w: T).(getl n c3 (CHead e3 (Bind b) w))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (e3: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e2 e3)))))) -(ex2 C (\lambda (e2: C).(csubst1 (S (minus i (S n))) v e1 e2)) (\lambda (e2: -C).(getl n c3 e2))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: -C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H5: (eq C e1 (CHead x1 (Bind -x0) x3))).(\lambda (H6: (getl n c3 (CHead x2 (Bind x0) x4))).(\lambda (H7: -(subst0 (minus i (S n)) v x3 x4)).(\lambda (H8: (csubst0 (minus i (S n)) v x1 -x2)).(eq_ind_r C (CHead x1 (Bind x0) x3) (\lambda (c: C).(ex2 C (\lambda (e2: -C).(csubst1 (S (minus i (S n))) v c e2)) (\lambda (e2: C).(getl n c3 e2)))) -(ex_intro2 C (\lambda (e2: C).(csubst1 (S (minus i (S n))) v (CHead x1 (Bind -x0) x3) e2)) (\lambda (e2: C).(getl n c3 e2)) (CHead x2 (Bind x0) x4) -(csubst1_sing (S (minus i (S n))) v (CHead x1 (Bind x0) x3) (CHead x2 (Bind -x0) x4) (csubst0_both_bind x0 (minus i (S n)) v x3 x4 H7 x1 x2 H8)) H6) e1 -H5)))))))))) H4)) H3)) (minus i n) (minus_x_Sy i n H)))))) c2 H0))))))). -(* COMMENTS -Initial nodes: 2035 -END *) - -theorem csubst1_getl_ge_back: - \forall (i: nat).(\forall (n: nat).((le i n) \to (\forall (c1: C).(\forall -(c2: C).(\forall (v: T).((csubst1 i v c1 c2) \to (\forall (e: C).((getl n c2 -e) \to (getl n c1 e))))))))) -\def - \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (le i n)).(\lambda (c1: -C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst1 i v c1 -c2)).(csubst1_ind i v c1 (\lambda (c: C).(\forall (e: C).((getl n c e) \to -(getl n c1 e)))) (\lambda (e: C).(\lambda (H1: (getl n c1 e)).H1)) (\lambda -(c3: C).(\lambda (H1: (csubst0 i v c1 c3)).(\lambda (e: C).(\lambda (H2: -(getl n c3 e)).(csubst0_getl_ge_back i n H c1 c3 v H1 e H2))))) c2 H0))))))). -(* COMMENTS -Initial nodes: 111 -END *) - -theorem getl_csubst1: - \forall (d: nat).(\forall (c: C).(\forall (e: C).(\forall (u: T).((getl d c -(CHead e (Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0: C).(\lambda (_: -C).(csubst1 d u c a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) d a0 -a)))))))) -\def - \lambda (d: nat).(nat_ind (\lambda (n: nat).(\forall (c: C).(\forall (e: -C).(\forall (u: T).((getl n c (CHead e (Bind Abbr) u)) \to (ex2_2 C C -(\lambda (a0: C).(\lambda (_: C).(csubst1 n u c a0))) (\lambda (a0: -C).(\lambda (a: C).(drop (S O) n a0 a))))))))) (\lambda (c: C).(C_ind -(\lambda (c0: C).(\forall (e: C).(\forall (u: T).((getl O c0 (CHead e (Bind -Abbr) u)) \to (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O u c0 -a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0 a)))))))) (\lambda -(n: nat).(\lambda (e: C).(\lambda (u: T).(\lambda (H: (getl O (CSort n) -(CHead e (Bind Abbr) u))).(getl_gen_sort n O (CHead e (Bind Abbr) u) H (ex2_2 -C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O u (CSort n) a0))) (\lambda -(a0: C).(\lambda (a: C).(drop (S O) O a0 a))))))))) (\lambda (c0: C).(\lambda -(H: ((\forall (e: C).(\forall (u: T).((getl O c0 (CHead e (Bind Abbr) u)) \to -(ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O u c0 a0))) (\lambda -(a0: C).(\lambda (a: C).(drop (S O) O a0 a))))))))).(\lambda (k: K).(K_ind -(\lambda (k0: K).(\forall (t: T).(\forall (e: C).(\forall (u: T).((getl O -(CHead c0 k0 t) (CHead e (Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0: -C).(\lambda (_: C).(csubst1 O u (CHead c0 k0 t) a0))) (\lambda (a0: -C).(\lambda (a: C).(drop (S O) O a0 a))))))))) (\lambda (b: B).(\lambda (t: -T).(\lambda (e: C).(\lambda (u: T).(\lambda (H0: (getl O (CHead c0 (Bind b) -t) (CHead e (Bind Abbr) u))).(let H1 \def (f_equal C C (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow e | -(CHead c1 _ _) \Rightarrow c1])) (CHead e (Bind Abbr) u) (CHead c0 (Bind b) -t) (clear_gen_bind b c0 (CHead e (Bind Abbr) u) t (getl_gen_O (CHead c0 (Bind -b) t) (CHead e (Bind Abbr) u) H0))) in ((let H2 \def (f_equal C B (\lambda -(e0: C).(match e0 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow -Abbr | (CHead _ k0 _) \Rightarrow (match k0 in K return (\lambda (_: K).B) -with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead e -(Bind Abbr) u) (CHead c0 (Bind b) t) (clear_gen_bind b c0 (CHead e (Bind -Abbr) u) t (getl_gen_O (CHead c0 (Bind b) t) (CHead e (Bind Abbr) u) H0))) in -((let H3 \def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda -(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) -(CHead e (Bind Abbr) u) (CHead c0 (Bind b) t) (clear_gen_bind b c0 (CHead e -(Bind Abbr) u) t (getl_gen_O (CHead c0 (Bind b) t) (CHead e (Bind Abbr) u) -H0))) in (\lambda (H4: (eq B Abbr b)).(\lambda (_: (eq C e c0)).(eq_ind_r T t -(\lambda (t0: T).(ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O t0 -(CHead c0 (Bind b) t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0 -a))))) (eq_ind B Abbr (\lambda (b0: B).(ex2_2 C C (\lambda (a0: C).(\lambda -(_: C).(csubst1 O t (CHead c0 (Bind b0) t) a0))) (\lambda (a0: C).(\lambda -(a: C).(drop (S O) O a0 a))))) (ex2_2_intro C C (\lambda (a0: C).(\lambda (_: -C).(csubst1 O t (CHead c0 (Bind Abbr) t) a0))) (\lambda (a0: C).(\lambda (a: -C).(drop (S O) O a0 a))) (CHead c0 (Bind Abbr) t) c0 (csubst1_refl O t (CHead -c0 (Bind Abbr) t)) (drop_drop (Bind Abbr) O c0 c0 (drop_refl c0) t)) b H4) u -H3)))) H2)) H1))))))) (\lambda (f: F).(\lambda (t: T).(\lambda (e: -C).(\lambda (u: T).(\lambda (H0: (getl O (CHead c0 (Flat f) t) (CHead e (Bind -Abbr) u))).(let H_x \def (subst1_ex u t O) in (let H1 \def H_x in (ex_ind T -(\lambda (t2: T).(subst1 O u t (lift (S O) O t2))) (ex2_2 C C (\lambda (a0: -C).(\lambda (_: C).(csubst1 O u (CHead c0 (Flat f) t) a0))) (\lambda (a0: -C).(\lambda (a: C).(drop (S O) O a0 a)))) (\lambda (x: T).(\lambda (H2: -(subst1 O u t (lift (S O) O x))).(let H3 \def (H e u (getl_intro O c0 (CHead -e (Bind Abbr) u) c0 (drop_refl c0) (clear_gen_flat f c0 (CHead e (Bind Abbr) -u) t (getl_gen_O (CHead c0 (Flat f) t) (CHead e (Bind Abbr) u) H0)))) in -(ex2_2_ind C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O u c0 a0))) -(\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0 a))) (ex2_2 C C (\lambda -(a0: C).(\lambda (_: C).(csubst1 O u (CHead c0 (Flat f) t) a0))) (\lambda -(a0: C).(\lambda (a: C).(drop (S O) O a0 a)))) (\lambda (x0: C).(\lambda (x1: -C).(\lambda (H4: (csubst1 O u c0 x0)).(\lambda (H5: (drop (S O) O x0 -x1)).(ex2_2_intro C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O u (CHead c0 -(Flat f) t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0 a))) -(CHead x0 (Flat f) (lift (S O) O x)) x1 (csubst1_flat f O u t (lift (S O) O -x) H2 c0 x0 H4) (drop_drop (Flat f) O x0 x1 H5 (lift (S O) O x))))))) H3)))) -H1)))))))) k)))) c)) (\lambda (n: nat).(\lambda (H: ((\forall (c: C).(\forall -(e: C).(\forall (u: T).((getl n c (CHead e (Bind Abbr) u)) \to (ex2_2 C C -(\lambda (a0: C).(\lambda (_: C).(csubst1 n u c a0))) (\lambda (a0: -C).(\lambda (a: C).(drop (S O) n a0 a)))))))))).(\lambda (c: C).(C_ind -(\lambda (c0: C).(\forall (e: C).(\forall (u: T).((getl (S n) c0 (CHead e -(Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S -n) u c0 a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a)))))))) -(\lambda (n0: nat).(\lambda (e: C).(\lambda (u: T).(\lambda (H0: (getl (S n) -(CSort n0) (CHead e (Bind Abbr) u))).(getl_gen_sort n0 (S n) (CHead e (Bind -Abbr) u) H0 (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u -(CSort n0) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 -a))))))))) (\lambda (c0: C).(\lambda (H0: ((\forall (e: C).(\forall (u: -T).((getl (S n) c0 (CHead e (Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0: -C).(\lambda (_: C).(csubst1 (S n) u c0 a0))) (\lambda (a0: C).(\lambda (a: -C).(drop (S O) (S n) a0 a))))))))).(\lambda (k: K).(K_ind (\lambda (k0: -K).(\forall (t: T).(\forall (e: C).(\forall (u: T).((getl (S n) (CHead c0 k0 -t) (CHead e (Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0: C).(\lambda (_: -C).(csubst1 (S n) u (CHead c0 k0 t) a0))) (\lambda (a0: C).(\lambda (a: -C).(drop (S O) (S n) a0 a))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda -(e: C).(\lambda (u: T).(\lambda (H1: (getl (S n) (CHead c0 (Bind b) t) (CHead -e (Bind Abbr) u))).(let H_x \def (subst1_ex u t n) in (let H2 \def H_x in -(ex_ind T (\lambda (t2: T).(subst1 n u t (lift (S O) n t2))) (ex2_2 C C -(\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u (CHead c0 (Bind b) t) a0))) -(\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a)))) (\lambda (x: -T).(\lambda (H3: (subst1 n u t (lift (S O) n x))).(let H4 \def (H c0 e u -(getl_gen_S (Bind b) c0 (CHead e (Bind Abbr) u) t n H1)) in (ex2_2_ind C C -(\lambda (a0: C).(\lambda (_: C).(csubst1 n u c0 a0))) (\lambda (a0: -C).(\lambda (a: C).(drop (S O) n a0 a))) (ex2_2 C C (\lambda (a0: C).(\lambda -(_: C).(csubst1 (S n) u (CHead c0 (Bind b) t) a0))) (\lambda (a0: C).(\lambda -(a: C).(drop (S O) (S n) a0 a)))) (\lambda (x0: C).(\lambda (x1: C).(\lambda -(H5: (csubst1 n u c0 x0)).(\lambda (H6: (drop (S O) n x0 x1)).(ex2_2_intro C -C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u (CHead c0 (Bind b) t) -a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a))) (CHead x0 -(Bind b) (lift (S O) n x)) (CHead x1 (Bind b) x) (csubst1_bind b n u t (lift -(S O) n x) H3 c0 x0 H5) (drop_skip_bind (S O) n x0 x1 H6 b x)))))) H4)))) -H2)))))))) (\lambda (f: F).(\lambda (t: T).(\lambda (e: C).(\lambda (u: -T).(\lambda (H1: (getl (S n) (CHead c0 (Flat f) t) (CHead e (Bind Abbr) -u))).(let H_x \def (subst1_ex u t (S n)) in (let H2 \def H_x in (ex_ind T -(\lambda (t2: T).(subst1 (S n) u t (lift (S O) (S n) t2))) (ex2_2 C C -(\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u (CHead c0 (Flat f) t) a0))) -(\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a)))) (\lambda (x: -T).(\lambda (H3: (subst1 (S n) u t (lift (S O) (S n) x))).(let H4 \def (H0 e -u (getl_gen_S (Flat f) c0 (CHead e (Bind Abbr) u) t n H1)) in (ex2_2_ind C C -(\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u c0 a0))) (\lambda (a0: -C).(\lambda (a: C).(drop (S O) (S n) a0 a))) (ex2_2 C C (\lambda (a0: -C).(\lambda (_: C).(csubst1 (S n) u (CHead c0 (Flat f) t) a0))) (\lambda (a0: -C).(\lambda (a: C).(drop (S O) (S n) a0 a)))) (\lambda (x0: C).(\lambda (x1: -C).(\lambda (H5: (csubst1 (S n) u c0 x0)).(\lambda (H6: (drop (S O) (S n) x0 -x1)).(ex2_2_intro C C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u -(CHead c0 (Flat f) t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S -n) a0 a))) (CHead x0 (Flat f) (lift (S O) (S n) x)) (CHead x1 (Flat f) x) -(csubst1_flat f (S n) u t (lift (S O) (S n) x) H3 c0 x0 H5) (drop_skip_flat -(S O) n x0 x1 H6 f x)))))) H4)))) H2)))))))) k)))) c)))) d). -(* COMMENTS -Initial nodes: 2467 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubst1/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubst1/props.ma deleted file mode 100644 index 518a86564..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubst1/props.ma +++ /dev/null @@ -1,75 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubst1/defs.ma". - -include "Basic-1/subst1/defs.ma". - -theorem csubst1_head: - \forall (k: K).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall -(u2: T).((subst1 i v u1 u2) \to (\forall (c1: C).(\forall (c2: C).((csubst1 i -v c1 c2) \to (csubst1 (s k i) v (CHead c1 k u1) (CHead c2 k u2)))))))))) -\def - \lambda (k: K).(\lambda (i: nat).(\lambda (v: T).(\lambda (u1: T).(\lambda -(u2: T).(\lambda (H: (subst1 i v u1 u2)).(subst1_ind i v u1 (\lambda (t: -T).(\forall (c1: C).(\forall (c2: C).((csubst1 i v c1 c2) \to (csubst1 (s k -i) v (CHead c1 k u1) (CHead c2 k t)))))) (\lambda (c1: C).(\lambda (c2: -C).(\lambda (H0: (csubst1 i v c1 c2)).(csubst1_ind i v c1 (\lambda (c: -C).(csubst1 (s k i) v (CHead c1 k u1) (CHead c k u1))) (csubst1_refl (s k i) -v (CHead c1 k u1)) (\lambda (c3: C).(\lambda (H1: (csubst0 i v c1 -c3)).(csubst1_sing (s k i) v (CHead c1 k u1) (CHead c3 k u1) (csubst0_fst k i -c1 c3 v H1 u1)))) c2 H0)))) (\lambda (t2: T).(\lambda (H0: (subst0 i v u1 -t2)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csubst1 i v c1 -c2)).(csubst1_ind i v c1 (\lambda (c: C).(csubst1 (s k i) v (CHead c1 k u1) -(CHead c k t2))) (csubst1_sing (s k i) v (CHead c1 k u1) (CHead c1 k t2) -(csubst0_snd k i v u1 t2 H0 c1)) (\lambda (c3: C).(\lambda (H2: (csubst0 i v -c1 c3)).(csubst1_sing (s k i) v (CHead c1 k u1) (CHead c3 k t2) (csubst0_both -k i v u1 t2 H0 c1 c3 H2)))) c2 H1)))))) u2 H)))))). -(* COMMENTS -Initial nodes: 365 -END *) - -theorem csubst1_bind: - \forall (b: B).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall -(u2: T).((subst1 i v u1 u2) \to (\forall (c1: C).(\forall (c2: C).((csubst1 i -v c1 c2) \to (csubst1 (S i) v (CHead c1 (Bind b) u1) (CHead c2 (Bind b) -u2)))))))))) -\def - \lambda (b: B).(\lambda (i: nat).(\lambda (v: T).(\lambda (u1: T).(\lambda -(u2: T).(\lambda (H: (subst1 i v u1 u2)).(\lambda (c1: C).(\lambda (c2: -C).(\lambda (H0: (csubst1 i v c1 c2)).(eq_ind nat (s (Bind b) i) (\lambda (n: -nat).(csubst1 n v (CHead c1 (Bind b) u1) (CHead c2 (Bind b) u2))) -(csubst1_head (Bind b) i v u1 u2 H c1 c2 H0) (S i) (refl_equal nat (S -i))))))))))). -(* COMMENTS -Initial nodes: 107 -END *) - -theorem csubst1_flat: - \forall (f: F).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall -(u2: T).((subst1 i v u1 u2) \to (\forall (c1: C).(\forall (c2: C).((csubst1 i -v c1 c2) \to (csubst1 i v (CHead c1 (Flat f) u1) (CHead c2 (Flat f) -u2)))))))))) -\def - \lambda (f: F).(\lambda (i: nat).(\lambda (v: T).(\lambda (u1: T).(\lambda -(u2: T).(\lambda (H: (subst1 i v u1 u2)).(\lambda (c1: C).(\lambda (c2: -C).(\lambda (H0: (csubst1 i v c1 c2)).(eq_ind nat (s (Flat f) i) (\lambda (n: -nat).(csubst1 n v (CHead c1 (Flat f) u1) (CHead c2 (Flat f) u2))) -(csubst1_head (Flat f) i v u1 u2 H c1 c2 H0) i (refl_equal nat i)))))))))). -(* COMMENTS -Initial nodes: 103 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/clear.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/clear.ma deleted file mode 100644 index c60700147..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/clear.ma +++ /dev/null @@ -1,74 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubt/defs.ma". - -include "Basic-1/clear/fwd.ma". - -theorem csubt_clear_conf: - \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to -(\forall (e1: C).((clear c1 e1) \to (ex2 C (\lambda (e2: C).(csubt g e1 e2)) -(\lambda (e2: C).(clear c2 e2)))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubt g c1 -c2)).(csubt_ind g (\lambda (c: C).(\lambda (c0: C).(\forall (e1: C).((clear c -e1) \to (ex2 C (\lambda (e2: C).(csubt g e1 e2)) (\lambda (e2: C).(clear c0 -e2))))))) (\lambda (n: nat).(\lambda (e1: C).(\lambda (H0: (clear (CSort n) -e1)).(clear_gen_sort e1 n H0 (ex2 C (\lambda (e2: C).(csubt g e1 e2)) -(\lambda (e2: C).(clear (CSort n) e2))))))) (\lambda (c3: C).(\lambda (c4: -C).(\lambda (H0: (csubt g c3 c4)).(\lambda (H1: ((\forall (e1: C).((clear c3 -e1) \to (ex2 C (\lambda (e2: C).(csubt g e1 e2)) (\lambda (e2: C).(clear c4 -e2))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (e1: C).(\lambda (H2: -(clear (CHead c3 k u) e1)).(K_ind (\lambda (k0: K).((clear (CHead c3 k0 u) -e1) \to (ex2 C (\lambda (e2: C).(csubt g e1 e2)) (\lambda (e2: C).(clear -(CHead c4 k0 u) e2))))) (\lambda (b: B).(\lambda (H3: (clear (CHead c3 (Bind -b) u) e1)).(eq_ind_r C (CHead c3 (Bind b) u) (\lambda (c: C).(ex2 C (\lambda -(e2: C).(csubt g c e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b) u) e2)))) -(ex_intro2 C (\lambda (e2: C).(csubt g (CHead c3 (Bind b) u) e2)) (\lambda -(e2: C).(clear (CHead c4 (Bind b) u) e2)) (CHead c4 (Bind b) u) (csubt_head g -c3 c4 H0 (Bind b) u) (clear_bind b c4 u)) e1 (clear_gen_bind b c3 e1 u H3)))) -(\lambda (f: F).(\lambda (H3: (clear (CHead c3 (Flat f) u) e1)).(let H4 \def -(H1 e1 (clear_gen_flat f c3 e1 u H3)) in (ex2_ind C (\lambda (e2: C).(csubt g -e1 e2)) (\lambda (e2: C).(clear c4 e2)) (ex2 C (\lambda (e2: C).(csubt g e1 -e2)) (\lambda (e2: C).(clear (CHead c4 (Flat f) u) e2))) (\lambda (x: -C).(\lambda (H5: (csubt g e1 x)).(\lambda (H6: (clear c4 x)).(ex_intro2 C -(\lambda (e2: C).(csubt g e1 e2)) (\lambda (e2: C).(clear (CHead c4 (Flat f) -u) e2)) x H5 (clear_flat c4 x H6 f u))))) H4)))) k H2))))))))) (\lambda (c3: -C).(\lambda (c4: C).(\lambda (H0: (csubt g c3 c4)).(\lambda (_: ((\forall -(e1: C).((clear c3 e1) \to (ex2 C (\lambda (e2: C).(csubt g e1 e2)) (\lambda -(e2: C).(clear c4 e2))))))).(\lambda (b: B).(\lambda (H2: (not (eq B b -Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (e1: C).(\lambda (H3: -(clear (CHead c3 (Bind Void) u1) e1)).(eq_ind_r C (CHead c3 (Bind Void) u1) -(\lambda (c: C).(ex2 C (\lambda (e2: C).(csubt g c e2)) (\lambda (e2: -C).(clear (CHead c4 (Bind b) u2) e2)))) (ex_intro2 C (\lambda (e2: C).(csubt -g (CHead c3 (Bind Void) u1) e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b) -u2) e2)) (CHead c4 (Bind b) u2) (csubt_void g c3 c4 H0 b H2 u1 u2) -(clear_bind b c4 u2)) e1 (clear_gen_bind Void c3 e1 u1 H3)))))))))))) -(\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csubt g c3 c4)).(\lambda (_: -((\forall (e1: C).((clear c3 e1) \to (ex2 C (\lambda (e2: C).(csubt g e1 e2)) -(\lambda (e2: C).(clear c4 e2))))))).(\lambda (u: T).(\lambda (t: T).(\lambda -(H2: (ty3 g c3 u t)).(\lambda (H3: (ty3 g c4 u t)).(\lambda (e1: C).(\lambda -(H4: (clear (CHead c3 (Bind Abst) t) e1)).(eq_ind_r C (CHead c3 (Bind Abst) -t) (\lambda (c: C).(ex2 C (\lambda (e2: C).(csubt g c e2)) (\lambda (e2: -C).(clear (CHead c4 (Bind Abbr) u) e2)))) (ex_intro2 C (\lambda (e2: -C).(csubt g (CHead c3 (Bind Abst) t) e2)) (\lambda (e2: C).(clear (CHead c4 -(Bind Abbr) u) e2)) (CHead c4 (Bind Abbr) u) (csubt_abst g c3 c4 H0 u t H2 -H3) (clear_bind Abbr c4 u)) e1 (clear_gen_bind Abst c3 e1 t H4)))))))))))) c1 -c2 H)))). -(* COMMENTS -Initial nodes: 929 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/csuba.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/csuba.ma deleted file mode 100644 index b2cf1183f..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/csuba.ma +++ /dev/null @@ -1,42 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/ty3/arity.ma". - -theorem csubt_csuba: - \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to (csuba -g c1 c2)))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubt g c1 -c2)).(csubt_ind g (\lambda (c: C).(\lambda (c0: C).(csuba g c c0))) (\lambda -(n: nat).(csuba_refl g (CSort n))) (\lambda (c3: C).(\lambda (c4: C).(\lambda -(_: (csubt g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (k: K).(\lambda -(u: T).(csuba_head g c3 c4 H1 k u))))))) (\lambda (c3: C).(\lambda (c4: -C).(\lambda (_: (csubt g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (b: -B).(\lambda (H2: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: -T).(csuba_void g c3 c4 H1 b H2 u1 u2))))))))) (\lambda (c3: C).(\lambda (c4: -C).(\lambda (_: (csubt g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (u: -T).(\lambda (t: T).(\lambda (H2: (ty3 g c3 u t)).(\lambda (_: (ty3 g c4 u -t)).(let H_x \def (ty3_arity g c3 u t H2) in (let H4 \def H_x in (ex2_ind A -(\lambda (a1: A).(arity g c3 u a1)) (\lambda (a1: A).(arity g c3 t (asucc g -a1))) (csuba g (CHead c3 (Bind Abst) t) (CHead c4 (Bind Abbr) u)) (\lambda -(x: A).(\lambda (H5: (arity g c3 u x)).(\lambda (H6: (arity g c3 t (asucc g -x))).(csuba_abst g c3 c4 H1 t x H6 u (csuba_arity g c3 u x H5 c4 H1))))) -H4))))))))))) c1 c2 H)))). -(* COMMENTS -Initial nodes: 313 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/defs.ma deleted file mode 100644 index a12c9f829..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/defs.ma +++ /dev/null @@ -1,29 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/ty3/defs.ma". - -inductive csubt (g: G): C \to (C \to Prop) \def -| csubt_sort: \forall (n: nat).(csubt g (CSort n) (CSort n)) -| csubt_head: \forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to (\forall -(k: K).(\forall (u: T).(csubt g (CHead c1 k u) (CHead c2 k u)))))) -| csubt_void: \forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to (\forall -(b: B).((not (eq B b Void)) \to (\forall (u1: T).(\forall (u2: T).(csubt g -(CHead c1 (Bind Void) u1) (CHead c2 (Bind b) u2)))))))) -| csubt_abst: \forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to (\forall -(u: T).(\forall (t: T).((ty3 g c1 u t) \to ((ty3 g c2 u t) \to (csubt g -(CHead c1 (Bind Abst) t) (CHead c2 (Bind Abbr) u)))))))). - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/drop.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/drop.ma deleted file mode 100644 index adaedcc9a..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/drop.ma +++ /dev/null @@ -1,590 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubt/fwd.ma". - -include "Basic-1/drop/fwd.ma". - -theorem csubt_drop_flat: - \forall (g: G).(\forall (f: F).(\forall (n: nat).(\forall (c1: C).(\forall -(c2: C).((csubt g c1 c2) \to (\forall (d1: C).(\forall (u: T).((drop n O c1 -(CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda -(d2: C).(drop n O c2 (CHead d2 (Flat f) u)))))))))))) -\def - \lambda (g: G).(\lambda (f: F).(\lambda (n: nat).(nat_ind (\lambda (n0: -nat).(\forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to (\forall (d1: -C).(\forall (u: T).((drop n0 O c1 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda -(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 (Flat f) -u))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubt g c1 -c2)).(\lambda (d1: C).(\lambda (u: T).(\lambda (H0: (drop O O c1 (CHead d1 -(Flat f) u))).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c2)) H -(CHead d1 (Flat f) u) (drop_gen_refl c1 (CHead d1 (Flat f) u) H0)) in (let -H_x \def (csubt_gen_flat g d1 c2 u f H1) in (let H2 \def H_x in (ex2_ind C -(\lambda (e2: C).(eq C c2 (CHead e2 (Flat f) u))) (\lambda (e2: C).(csubt g -d1 e2)) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O -c2 (CHead d2 (Flat f) u)))) (\lambda (x: C).(\lambda (H3: (eq C c2 (CHead x -(Flat f) u))).(\lambda (H4: (csubt g d1 x)).(eq_ind_r C (CHead x (Flat f) u) -(\lambda (c: C).(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(drop O O c (CHead d2 (Flat f) u))))) (ex_intro2 C (\lambda (d2: C).(csubt -g d1 d2)) (\lambda (d2: C).(drop O O (CHead x (Flat f) u) (CHead d2 (Flat f) -u))) x H4 (drop_refl (CHead x (Flat f) u))) c2 H3)))) H2)))))))))) (\lambda -(n0: nat).(\lambda (H: ((\forall (c1: C).(\forall (c2: C).((csubt g c1 c2) -\to (\forall (d1: C).(\forall (u: T).((drop n0 O c1 (CHead d1 (Flat f) u)) -\to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c2 -(CHead d2 (Flat f) u)))))))))))).(\lambda (c1: C).(\lambda (c2: C).(\lambda -(H0: (csubt g c1 c2)).(csubt_ind g (\lambda (c: C).(\lambda (c0: C).(\forall -(d1: C).(\forall (u: T).((drop (S n0) O c (CHead d1 (Flat f) u)) \to (ex2 C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O c0 (CHead -d2 (Flat f) u))))))))) (\lambda (n1: nat).(\lambda (d1: C).(\lambda (u: -T).(\lambda (H1: (drop (S n0) O (CSort n1) (CHead d1 (Flat f) u))).(and3_ind -(eq C (CHead d1 (Flat f) u) (CSort n1)) (eq nat (S n0) O) (eq nat O O) (ex2 C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) -(CHead d2 (Flat f) u)))) (\lambda (_: (eq C (CHead d1 (Flat f) u) (CSort -n1))).(\lambda (H3: (eq nat (S n0) O)).(\lambda (_: (eq nat O O)).(let H5 -\def (eq_ind nat (S n0) (\lambda (ee: nat).(match ee in nat return (\lambda -(_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H3) -in (False_ind (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop -(S n0) O (CSort n1) (CHead d2 (Flat f) u)))) H5))))) (drop_gen_sort n1 (S n0) -O (CHead d1 (Flat f) u) H1)))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda -(H1: (csubt g c0 c3)).(\lambda (H2: ((\forall (d1: C).(\forall (u: T).((drop -(S n0) O c0 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 -d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Flat f) -u))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (u: T).(\forall -(d1: C).(\forall (u0: T).((drop (S n0) O (CHead c0 k0 u) (CHead d1 (Flat f) -u0)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S -n0) O (CHead c3 k0 u) (CHead d2 (Flat f) u0))))))))) (\lambda (b: B).(\lambda -(u: T).(\lambda (d1: C).(\lambda (u0: T).(\lambda (H3: (drop (S n0) O (CHead -c0 (Bind b) u) (CHead d1 (Flat f) u0))).(ex2_ind C (\lambda (d2: C).(csubt g -d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Flat f) u0))) (ex2 C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 -(Bind b) u) (CHead d2 (Flat f) u0)))) (\lambda (x: C).(\lambda (H4: (csubt g -d1 x)).(\lambda (H5: (drop n0 O c3 (CHead x (Flat f) u0))).(ex_intro2 C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 -(Bind b) u) (CHead d2 (Flat f) u0))) x H4 (drop_drop (Bind b) n0 c3 (CHead x -(Flat f) u0) H5 u))))) (H c0 c3 H1 d1 u0 (drop_gen_drop (Bind b) c0 (CHead d1 -(Flat f) u0) u n0 H3)))))))) (\lambda (f0: F).(\lambda (u: T).(\lambda (d1: -C).(\lambda (u0: T).(\lambda (H3: (drop (S n0) O (CHead c0 (Flat f0) u) -(CHead d1 (Flat f) u0))).(ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Flat f) u0))) (ex2 C (\lambda -(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f0) -u) (CHead d2 (Flat f) u0)))) (\lambda (x: C).(\lambda (H4: (csubt g d1 -x)).(\lambda (H5: (drop (S n0) O c3 (CHead x (Flat f) u0))).(ex_intro2 C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 -(Flat f0) u) (CHead d2 (Flat f) u0))) x H4 (drop_drop (Flat f0) n0 c3 (CHead -x (Flat f) u0) H5 u))))) (H2 d1 u0 (drop_gen_drop (Flat f0) c0 (CHead d1 -(Flat f) u0) u n0 H3)))))))) k)))))) (\lambda (c0: C).(\lambda (c3: -C).(\lambda (H1: (csubt g c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (u: -T).((drop (S n0) O c0 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: -C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Flat f) -u))))))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: -T).(\lambda (u2: T).(\lambda (d1: C).(\lambda (u: T).(\lambda (H4: (drop (S -n0) O (CHead c0 (Bind Void) u1) (CHead d1 (Flat f) u))).(ex2_ind C (\lambda -(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Flat f) -u))) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O -(CHead c3 (Bind b) u2) (CHead d2 (Flat f) u)))) (\lambda (x: C).(\lambda (H5: -(csubt g d1 x)).(\lambda (H6: (drop n0 O c3 (CHead x (Flat f) u))).(ex_intro2 -C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 -(Bind b) u2) (CHead d2 (Flat f) u))) x H5 (drop_drop (Bind b) n0 c3 (CHead x -(Flat f) u) H6 u2))))) (H c0 c3 H1 d1 u (drop_gen_drop (Bind Void) c0 (CHead -d1 (Flat f) u) u1 n0 H4)))))))))))))) (\lambda (c0: C).(\lambda (c3: -C).(\lambda (H1: (csubt g c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (u: -T).((drop (S n0) O c0 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: -C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Flat f) -u))))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u -t)).(\lambda (_: (ty3 g c3 u t)).(\lambda (d1: C).(\lambda (u0: T).(\lambda -(H5: (drop (S n0) O (CHead c0 (Bind Abst) t) (CHead d1 (Flat f) -u0))).(ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 -O c3 (CHead d2 (Flat f) u0))) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Flat f) -u0)))) (\lambda (x: C).(\lambda (H6: (csubt g d1 x)).(\lambda (H7: (drop n0 O -c3 (CHead x (Flat f) u0))).(ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Flat f) -u0))) x H6 (drop_drop (Bind Abbr) n0 c3 (CHead x (Flat f) u0) H7 u))))) (H c0 -c3 H1 d1 u0 (drop_gen_drop (Bind Abst) c0 (CHead d1 (Flat f) u0) t n0 -H5)))))))))))))) c1 c2 H0)))))) n))). -(* COMMENTS -Initial nodes: 2090 -END *) - -theorem csubt_drop_abbr: - \forall (g: G).(\forall (n: nat).(\forall (c1: C).(\forall (c2: C).((csubt g -c1 c2) \to (\forall (d1: C).(\forall (u: T).((drop n O c1 (CHead d1 (Bind -Abbr) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop -n O c2 (CHead d2 (Bind Abbr) u))))))))))) -\def - \lambda (g: G).(\lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (c1: -C).(\forall (c2: C).((csubt g c1 c2) \to (\forall (d1: C).(\forall (u: -T).((drop n0 O c1 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: -C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 (Bind Abbr) -u))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubt g c1 -c2)).(\lambda (d1: C).(\lambda (u: T).(\lambda (H0: (drop O O c1 (CHead d1 -(Bind Abbr) u))).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c2)) H -(CHead d1 (Bind Abbr) u) (drop_gen_refl c1 (CHead d1 (Bind Abbr) u) H0)) in -(let H2 \def (csubt_gen_abbr g d1 c2 u H1) in (ex2_ind C (\lambda (e2: C).(eq -C c2 (CHead e2 (Bind Abbr) u))) (\lambda (e2: C).(csubt g d1 e2)) (ex2 C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 -(Bind Abbr) u)))) (\lambda (x: C).(\lambda (H3: (eq C c2 (CHead x (Bind Abbr) -u))).(\lambda (H4: (csubt g d1 x)).(eq_ind_r C (CHead x (Bind Abbr) u) -(\lambda (c: C).(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(drop O O c (CHead d2 (Bind Abbr) u))))) (ex_intro2 C (\lambda (d2: -C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O (CHead x (Bind Abbr) u) (CHead -d2 (Bind Abbr) u))) x H4 (drop_refl (CHead x (Bind Abbr) u))) c2 H3)))) -H2))))))))) (\lambda (n0: nat).(\lambda (H: ((\forall (c1: C).(\forall (c2: -C).((csubt g c1 c2) \to (\forall (d1: C).(\forall (u: T).((drop n0 O c1 -(CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(drop n0 O c2 (CHead d2 (Bind Abbr) u)))))))))))).(\lambda -(c1: C).(\lambda (c2: C).(\lambda (H0: (csubt g c1 c2)).(csubt_ind g (\lambda -(c: C).(\lambda (c0: C).(\forall (d1: C).(\forall (u: T).((drop (S n0) O c -(CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(drop (S n0) O c0 (CHead d2 (Bind Abbr) u))))))))) (\lambda -(n1: nat).(\lambda (d1: C).(\lambda (u: T).(\lambda (H1: (drop (S n0) O -(CSort n1) (CHead d1 (Bind Abbr) u))).(and3_ind (eq C (CHead d1 (Bind Abbr) -u) (CSort n1)) (eq nat (S n0) O) (eq nat O O) (ex2 C (\lambda (d2: C).(csubt -g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) -u)))) (\lambda (_: (eq C (CHead d1 (Bind Abbr) u) (CSort n1))).(\lambda (H3: -(eq nat (S n0) O)).(\lambda (_: (eq nat O O)).(let H5 \def (eq_ind nat (S n0) -(\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O -\Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind (ex2 C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) -(CHead d2 (Bind Abbr) u)))) H5))))) (drop_gen_sort n1 (S n0) O (CHead d1 -(Bind Abbr) u) H1)))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H1: -(csubt g c0 c3)).(\lambda (H2: ((\forall (d1: C).(\forall (u: T).((drop (S -n0) O c0 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 -d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abbr) -u))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (u: T).(\forall -(d1: C).(\forall (u0: T).((drop (S n0) O (CHead c0 k0 u) (CHead d1 (Bind -Abbr) u0)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(drop (S n0) O (CHead c3 k0 u) (CHead d2 (Bind Abbr) u0))))))))) (\lambda -(b: B).(\lambda (u: T).(\lambda (d1: C).(\lambda (u0: T).(\lambda (H3: (drop -(S n0) O (CHead c0 (Bind b) u) (CHead d1 (Bind Abbr) u0))).(ex2_ind C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 -(Bind Abbr) u0))) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abbr) u0)))) (\lambda -(x: C).(\lambda (H4: (csubt g d1 x)).(\lambda (H5: (drop n0 O c3 (CHead x -(Bind Abbr) u0))).(ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda -(d2: C).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abbr) u0))) x H4 -(drop_drop (Bind b) n0 c3 (CHead x (Bind Abbr) u0) H5 u))))) (H c0 c3 H1 d1 -u0 (drop_gen_drop (Bind b) c0 (CHead d1 (Bind Abbr) u0) u n0 H3)))))))) -(\lambda (f: F).(\lambda (u: T).(\lambda (d1: C).(\lambda (u0: T).(\lambda -(H3: (drop (S n0) O (CHead c0 (Flat f) u) (CHead d1 (Bind Abbr) -u0))).(ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S -n0) O c3 (CHead d2 (Bind Abbr) u0))) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind Abbr) -u0)))) (\lambda (x: C).(\lambda (H4: (csubt g d1 x)).(\lambda (H5: (drop (S -n0) O c3 (CHead x (Bind Abbr) u0))).(ex_intro2 C (\lambda (d2: C).(csubt g d1 -d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind -Abbr) u0))) x H4 (drop_drop (Flat f) n0 c3 (CHead x (Bind Abbr) u0) H5 u))))) -(H2 d1 u0 (drop_gen_drop (Flat f) c0 (CHead d1 (Bind Abbr) u0) u n0 -H3)))))))) k)))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H1: (csubt g -c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (u: T).((drop (S n0) O c0 -(CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u))))))))).(\lambda -(b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: -T).(\lambda (d1: C).(\lambda (u: T).(\lambda (H4: (drop (S n0) O (CHead c0 -(Bind Void) u1) (CHead d1 (Bind Abbr) u))).(ex2_ind C (\lambda (d2: C).(csubt -g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abbr) u))) (ex2 C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 -(Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda (x: C).(\lambda (H5: (csubt -g d1 x)).(\lambda (H6: (drop n0 O c3 (CHead x (Bind Abbr) u))).(ex_intro2 C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 -(Bind b) u2) (CHead d2 (Bind Abbr) u))) x H5 (drop_drop (Bind b) n0 c3 (CHead -x (Bind Abbr) u) H6 u2))))) (H c0 c3 H1 d1 u (drop_gen_drop (Bind Void) c0 -(CHead d1 (Bind Abbr) u) u1 n0 H4)))))))))))))) (\lambda (c0: C).(\lambda -(c3: C).(\lambda (H1: (csubt g c0 c3)).(\lambda (_: ((\forall (d1: -C).(\forall (u: T).((drop (S n0) O c0 (CHead d1 (Bind Abbr) u)) \to (ex2 C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead -d2 (Bind Abbr) u))))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (_: (ty3 g -c0 u t)).(\lambda (_: (ty3 g c3 u t)).(\lambda (d1: C).(\lambda (u0: -T).(\lambda (H5: (drop (S n0) O (CHead c0 (Bind Abst) t) (CHead d1 (Bind -Abbr) u0))).(ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(drop n0 O c3 (CHead d2 (Bind Abbr) u0))) (ex2 C (\lambda (d2: C).(csubt g -d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 -(Bind Abbr) u0)))) (\lambda (x: C).(\lambda (H6: (csubt g d1 x)).(\lambda -(H7: (drop n0 O c3 (CHead x (Bind Abbr) u0))).(ex_intro2 C (\lambda (d2: -C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) -(CHead d2 (Bind Abbr) u0))) x H6 (drop_drop (Bind Abbr) n0 c3 (CHead x (Bind -Abbr) u0) H7 u))))) (H c0 c3 H1 d1 u0 (drop_gen_drop (Bind Abst) c0 (CHead d1 -(Bind Abbr) u0) t n0 H5)))))))))))))) c1 c2 H0)))))) n)). -(* COMMENTS -Initial nodes: 2084 -END *) - -theorem csubt_drop_abst: - \forall (g: G).(\forall (n: nat).(\forall (c1: C).(\forall (c2: C).((csubt g -c1 c2) \to (\forall (d1: C).(\forall (t: T).((drop n O c1 (CHead d1 (Bind -Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(drop n O c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: -C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop n -O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u -t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))))) -\def - \lambda (g: G).(\lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (c1: -C).(\forall (c2: C).((csubt g c1 c2) \to (\forall (d1: C).(\forall (t: -T).((drop n0 O c1 (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: -C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 (Bind Abst) -t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda -(d2: C).(\lambda (u: T).(drop n0 O c2 (CHead d2 (Bind Abbr) u)))) (\lambda -(_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 -g d2 u t)))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubt g -c1 c2)).(\lambda (d1: C).(\lambda (t: T).(\lambda (H0: (drop O O c1 (CHead d1 -(Bind Abst) t))).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c2)) H -(CHead d1 (Bind Abst) t) (drop_gen_refl c1 (CHead d1 (Bind Abst) t) H0)) in -(let H2 \def (csubt_gen_abst g d1 c2 t H1) in (or_ind (ex2 C (\lambda (e2: -C).(eq C c2 (CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csubt g d1 e2))) -(ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) -v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g d1 e2))) (\lambda (_: -C).(\lambda (v2: T).(ty3 g d1 v2 t))) (\lambda (e2: C).(\lambda (v2: T).(ty3 -g e2 v2 t)))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(drop O O c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: -C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop O -O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u -t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (H3: (ex2 C -(\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csubt -g d1 e2)))).(ex2_ind C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abst) t))) -(\lambda (e2: C).(csubt g d1 e2)) (or (ex2 C (\lambda (d2: C).(csubt g d1 -d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T -(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda -(u: T).(drop O O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: -T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) -(\lambda (x: C).(\lambda (H4: (eq C c2 (CHead x (Bind Abst) t))).(\lambda -(H5: (csubt g d1 x)).(eq_ind_r C (CHead x (Bind Abst) t) (\lambda (c: C).(or -(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O c (CHead -d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 -d2))) (\lambda (d2: C).(\lambda (u: T).(drop O O c (CHead d2 (Bind Abbr) -u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: -C).(\lambda (u: T).(ty3 g d2 u t)))))) (or_introl (ex2 C (\lambda (d2: -C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O (CHead x (Bind Abst) t) (CHead -d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 -d2))) (\lambda (d2: C).(\lambda (u: T).(drop O O (CHead x (Bind Abst) t) -(CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) -(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: -C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O (CHead x (Bind Abst) t) (CHead -d2 (Bind Abst) t))) x H5 (drop_refl (CHead x (Bind Abst) t)))) c2 H4)))) H3)) -(\lambda (H3: (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 -(Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g d1 e2))) -(\lambda (_: C).(\lambda (v2: T).(ty3 g d1 v2 t))) (\lambda (e2: C).(\lambda -(v2: T).(ty3 g e2 v2 t))))).(ex4_2_ind C T (\lambda (e2: C).(\lambda (v2: -T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: -T).(csubt g d1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g d1 v2 t))) -(\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 t))) (or (ex2 C (\lambda (d2: -C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) t)))) -(ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: -C).(\lambda (u: T).(drop O O c2 (CHead d2 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-u) (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: -T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead -c3 (Flat f) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: -T).(ty3 g d1 u0 t))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))))) -(\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: (csubt g d1 x0)).(\lambda -(H6: (drop (S n0) O c3 (CHead x0 (Bind Abbr) x1))).(\lambda (H7: (ty3 g d1 x1 -t)).(\lambda (H8: (ty3 g x0 x1 t)).(or_intror (ex2 C (\lambda (d2: C).(csubt -g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 -(Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 -d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Flat f) u) -(CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g d1 u0 -t))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))) (ex4_2_intro C T -(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda -(u0: T).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind Abbr) u0)))) -(\lambda (_: C).(\lambda (u0: T).(ty3 g d1 u0 t))) (\lambda (d2: C).(\lambda -(u0: T).(ty3 g d2 u0 t))) x0 x1 H5 (drop_drop (Flat f) n0 c3 (CHead x0 (Bind -Abbr) x1) H6 u) H7 H8)))))))) H4)) (H2 d1 t (drop_gen_drop (Flat f) c0 (CHead -d1 (Bind Abst) t) u n0 H3)))))))) k)))))) (\lambda (c0: C).(\lambda (c3: -C).(\lambda (H1: (csubt g c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (t: -T).((drop (S n0) O c0 (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: -C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abst) -t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda -(d2: C).(\lambda (u: T).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u)))) -(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda -(u: T).(ty3 g d2 u t)))))))))).(\lambda (b: B).(\lambda (_: (not (eq B b -Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (d1: C).(\lambda (t: -T).(\lambda (H4: (drop (S n0) O (CHead c0 (Bind Void) u1) (CHead d1 (Bind -Abst) t))).(or_ind (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(drop n0 O c3 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: -C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop -n0 O c3 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 -u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (or (ex2 C (\lambda -(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) -u2) (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: -T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O (CHead -c3 (Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: -T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) -(\lambda (H5: (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop -n0 O c3 (CHead d2 (Bind Abst) t))))).(ex2_ind C (\lambda (d2: C).(csubt g d1 -d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abst) t))) (or (ex2 C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 -(Bind b) u2) (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda -(_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O -(CHead c3 (Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda -(u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) -(\lambda (x: C).(\lambda (H6: (csubt g d1 x)).(\lambda (H7: (drop n0 O c3 -(CHead x (Bind Abst) t))).(or_introl (ex2 C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abst) -t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda -(d2: C).(\lambda (u: T).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind -Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: -C).(\lambda (u: T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csubt g -d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 -(Bind Abst) t))) x H6 (drop_drop (Bind b) n0 c3 (CHead x (Bind Abst) t) H7 -u2)))))) H5)) (\lambda (H5: (ex4_2 C T (\lambda (d2: C).(\lambda (_: -T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop n0 O c3 (CHead d2 -(Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda -(d2: C).(\lambda (u: T).(ty3 g d2 u t))))).(ex4_2_ind C T (\lambda (d2: -C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop -n0 O c3 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 -u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))) (or (ex2 C (\lambda -(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) -u2) (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: -T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O (CHead -c3 (Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: -T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) -(\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (csubt g d1 x0)).(\lambda -(H7: (drop n0 O c3 (CHead x0 (Bind Abbr) x1))).(\lambda (H8: (ty3 g d1 x1 -t)).(\lambda (H9: (ty3 g x0 x1 t)).(or_intror (ex2 C (\lambda (d2: C).(csubt -g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 -(Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 -d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O (CHead c3 (Bind b) u2) -(CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) -(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (ex4_2_intro C T (\lambda -(d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: -T).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda -(_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 -g d2 u t))) x0 x1 H6 (drop_drop (Bind b) n0 c3 (CHead x0 (Bind Abbr) x1) H7 -u2) H8 H9)))))))) H5)) (H c0 c3 H1 d1 t (drop_gen_drop (Bind Void) c0 (CHead -d1 (Bind Abst) t) u1 n0 H4)))))))))))))) (\lambda (c0: C).(\lambda (c3: -C).(\lambda (H1: (csubt g c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (t: -T).((drop (S n0) O c0 (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: -C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abst) -t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda -(d2: C).(\lambda (u: T).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u)))) -(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda -(u: T).(ty3 g d2 u t)))))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (_: -(ty3 g c0 u t)).(\lambda (_: (ty3 g c3 u t)).(\lambda (d1: C).(\lambda (t0: -T).(\lambda (H5: (drop (S n0) O (CHead c0 (Bind Abst) t) (CHead d1 (Bind -Abst) t0))).(or_ind (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(drop n0 O c3 (CHead d2 (Bind Abst) t0)))) (ex4_2 C T (\lambda (d2: -C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop -n0 O c3 (CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g -d1 u0 t0))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0)))) (or (ex2 C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 -(Bind Abbr) u) (CHead d2 (Bind Abst) t0)))) (ex4_2 C T (\lambda (d2: -C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop -(S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (_: -C).(\lambda (u0: T).(ty3 g d1 u0 t0))) (\lambda (d2: C).(\lambda (u0: T).(ty3 -g d2 u0 t0))))) (\lambda (H6: (ex2 C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abst) t0))))).(ex2_ind C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 -(Bind Abst) t0))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abst) t0)))) -(ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: -C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind -Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g d1 u0 t0))) (\lambda -(d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0))))) (\lambda (x: C).(\lambda (H7: -(csubt g d1 x)).(\lambda (H8: (drop n0 O c3 (CHead x (Bind Abst) -t0))).(or_introl (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abst) t0)))) -(ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: -C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind -Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g d1 u0 t0))) (\lambda -(d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0)))) (ex_intro2 C (\lambda (d2: -C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) -(CHead d2 (Bind Abst) t0))) x H7 (drop_drop (Bind Abbr) n0 c3 (CHead x (Bind -Abst) t0) H8 u)))))) H6)) (\lambda (H6: (ex4_2 C T (\lambda (d2: C).(\lambda -(_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop n0 O c3 -(CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g d1 u0 -t0))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0))))).(ex4_2_ind C T -(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda -(u0: T).(drop n0 O c3 (CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda -(u0: T).(ty3 g d1 u0 t0))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 -t0))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S -n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abst) t0)))) (ex4_2 C T -(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda -(u0: T).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) -(\lambda (_: C).(\lambda (u0: T).(ty3 g d1 u0 t0))) (\lambda (d2: C).(\lambda -(u0: T).(ty3 g d2 u0 t0))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H7: -(csubt g d1 x0)).(\lambda (H8: (drop n0 O c3 (CHead x0 (Bind Abbr) -x1))).(\lambda (H9: (ty3 g d1 x1 t0)).(\lambda (H10: (ty3 g x0 x1 -t0)).(or_intror (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abst) t0)))) -(ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: -C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind -Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g d1 u0 t0))) (\lambda -(d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0)))) (ex4_2_intro C T (\lambda (d2: -C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop -(S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (_: -C).(\lambda (u0: T).(ty3 g d1 u0 t0))) (\lambda (d2: C).(\lambda (u0: T).(ty3 -g d2 u0 t0))) x0 x1 H7 (drop_drop (Bind Abbr) n0 c3 (CHead x0 (Bind Abbr) x1) -H8 u) H9 H10)))))))) H6)) (H c0 c3 H1 d1 t0 (drop_gen_drop (Bind Abst) c0 -(CHead d1 (Bind Abst) t0) t n0 H5)))))))))))))) c1 c2 H0)))))) n)). -(* COMMENTS -Initial nodes: 7940 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/fwd.ma deleted file mode 100644 index 63a3eca47..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/fwd.ma +++ /dev/null @@ -1,398 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubt/defs.ma". - -theorem csubt_gen_abbr: - \forall (g: G).(\forall (e1: C).(\forall (c2: C).(\forall (v: T).((csubt g -(CHead e1 (Bind Abbr) v) c2) \to (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 -(Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2))))))) -\def - \lambda (g: G).(\lambda (e1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda -(H: (csubt g (CHead e1 (Bind Abbr) v) c2)).(insert_eq C (CHead e1 (Bind Abbr) -v) (\lambda (c: C).(csubt g c c2)) (\lambda (_: C).(ex2 C (\lambda (e2: -C).(eq C c2 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))) -(\lambda (y: C).(\lambda (H0: (csubt g y c2)).(csubt_ind g (\lambda (c: -C).(\lambda (c0: C).((eq C c (CHead e1 (Bind Abbr) v)) \to (ex2 C (\lambda -(e2: C).(eq C c0 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 -e2)))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead e1 (Bind -Abbr) v))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C -return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) -\Rightarrow False])) I (CHead e1 (Bind Abbr) v) H1) in (False_ind (ex2 C -(\lambda (e2: C).(eq C (CSort n) (CHead e2 (Bind Abbr) v))) (\lambda (e2: -C).(csubt g e1 e2))) H2)))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1: -(csubt g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind Abbr) v)) \to (ex2 -C (\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abbr) v))) (\lambda (e2: -C).(csubt g e1 e2)))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C -(CHead c1 k u) (CHead e1 (Bind Abbr) v))).(let H4 \def (f_equal C C (\lambda -(e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 -| (CHead c _ _) \Rightarrow c])) (CHead c1 k u) (CHead e1 (Bind Abbr) v) H3) -in ((let H5 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda -(_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) -(CHead c1 k u) (CHead e1 (Bind Abbr) v) H3) in ((let H6 \def (f_equal C T -(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) -\Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u) (CHead e1 (Bind -Abbr) v) H3) in (\lambda (H7: (eq K k (Bind Abbr))).(\lambda (H8: (eq C c1 -e1)).(eq_ind_r T v (\lambda (t: T).(ex2 C (\lambda (e2: C).(eq C (CHead c3 k -t) (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))) (eq_ind_r K -(Bind Abbr) (\lambda (k0: K).(ex2 C (\lambda (e2: C).(eq C (CHead c3 k0 v) -(CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))) (let H9 \def -(eq_ind C c1 (\lambda (c: C).((eq C c (CHead e1 (Bind Abbr) v)) \to (ex2 C -(\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt -g e1 e2))))) H2 e1 H8) in (let H10 \def (eq_ind C c1 (\lambda (c: C).(csubt g -c c3)) H1 e1 H8) in (ex_intro2 C (\lambda (e2: C).(eq C (CHead c3 (Bind Abbr) -v) (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)) c3 -(refl_equal C (CHead c3 (Bind Abbr) v)) H10))) k H7) u H6)))) H5)) -H4))))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (_: (csubt g c1 -c3)).(\lambda (_: (((eq C c1 (CHead e1 (Bind Abbr) v)) \to (ex2 C (\lambda -(e2: C).(eq C c3 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 -e2)))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: -T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u1) (CHead e1 -(Bind Abbr) v))).(let H5 \def (eq_ind C (CHead c1 (Bind Void) u1) (\lambda -(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind b0) \Rightarrow (match b0 in B return (\lambda (_: -B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False | Void -\Rightarrow True]) | (Flat _) \Rightarrow False])])) I (CHead e1 (Bind Abbr) -v) H4) in (False_ind (ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind b) u2) -(CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2))) H5))))))))))) -(\lambda (c1: C).(\lambda (c3: C).(\lambda (_: (csubt g c1 c3)).(\lambda (_: -(((eq C c1 (CHead e1 (Bind Abbr) v)) \to (ex2 C (\lambda (e2: C).(eq C c3 -(CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))))).(\lambda (u: -T).(\lambda (t: T).(\lambda (_: (ty3 g c1 u t)).(\lambda (_: (ty3 g c3 u -t)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) t) (CHead e1 (Bind Abbr) -v))).(let H6 \def (eq_ind C (CHead c1 (Bind Abst) t) (\lambda (ee: C).(match -ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | -(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr -\Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat -_) \Rightarrow False])])) I (CHead e1 (Bind Abbr) v) H5) in (False_ind (ex2 C -(\lambda (e2: C).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 (Bind Abbr) v))) -(\lambda (e2: C).(csubt g e1 e2))) H6))))))))))) y c2 H0))) H))))). -(* COMMENTS -Initial nodes: 1111 -END *) - -theorem csubt_gen_abst: - \forall (g: G).(\forall (e1: C).(\forall (c2: C).(\forall (v1: T).((csubt g -(CHead e1 (Bind Abst) v1) c2) \to (or (ex2 C (\lambda (e2: C).(eq C c2 (CHead -e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda -(e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: -C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g -e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))))))))) -\def - \lambda (g: G).(\lambda (e1: C).(\lambda (c2: C).(\lambda (v1: T).(\lambda -(H: (csubt g (CHead e1 (Bind Abst) v1) c2)).(insert_eq C (CHead e1 (Bind -Abst) v1) (\lambda (c: C).(csubt g c c2)) (\lambda (_: C).(or (ex2 C (\lambda -(e2: C).(eq C c2 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 -e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind -Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: -C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 -g e2 v2 v1)))))) (\lambda (y: C).(\lambda (H0: (csubt g y c2)).(csubt_ind g -(\lambda (c: C).(\lambda (c0: C).((eq C c (CHead e1 (Bind Abst) v1)) \to (or -(ex2 C (\lambda (e2: C).(eq C c0 (CHead e2 (Bind Abst) v1))) (\lambda (e2: -C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c0 -(CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 -e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: -C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))))) (\lambda (n: nat).(\lambda (H1: -(eq C (CSort n) (CHead e1 (Bind Abst) v1))).(let H2 \def (eq_ind C (CSort n) -(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 (Bind Abst) -v1) H1) in (False_ind (or (ex2 C (\lambda (e2: C).(eq C (CSort n) (CHead e2 -(Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: -C).(\lambda (v2: T).(eq C (CSort n) (CHead e2 (Bind Abbr) v2)))) (\lambda -(e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: -T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))))) -H2)))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1: (csubt g c1 -c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind Abst) v1)) \to (or (ex2 C -(\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt -g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 -(Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) -(\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda -(v2: T).(ty3 g e2 v2 v1)))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: -(eq C (CHead c1 k u) (CHead e1 (Bind Abst) v1))).(let H4 \def (f_equal C C -(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u) (CHead e1 -(Bind Abst) v1) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e in -C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) -\Rightarrow k0])) (CHead c1 k u) (CHead e1 (Bind Abst) v1) H3) in ((let H6 -\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u) -(CHead e1 (Bind Abst) v1) H3) in (\lambda (H7: (eq K k (Bind Abst))).(\lambda -(H8: (eq C c1 e1)).(eq_ind_r T v1 (\lambda (t: T).(or (ex2 C (\lambda (e2: -C).(eq C (CHead c3 k t) (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g -e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 k t) -(CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 -e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: -C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))) (eq_ind_r K (Bind Abst) (\lambda -(k0: K).(or (ex2 C (\lambda (e2: C).(eq C (CHead c3 k0 v1) (CHead e2 (Bind -Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: -C).(\lambda (v2: T).(eq C (CHead c3 k0 v1) (CHead e2 (Bind Abbr) v2)))) -(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda -(v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 -v1)))))) (let H9 \def (eq_ind C c1 (\lambda (c: C).((eq C c (CHead e1 (Bind -Abst) v1)) \to (or (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abst) -v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda -(v2: T).(eq C c3 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: -T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) -(\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))))))) H2 e1 H8) in (let -H10 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H8) in (or_introl -(ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind Abst) v1) (CHead e2 (Bind Abst) -v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda -(v2: T).(eq C (CHead c3 (Bind Abst) v1) (CHead e2 (Bind Abbr) v2)))) (\lambda -(e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: -T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))) -(ex_intro2 C (\lambda (e2: C).(eq C (CHead c3 (Bind Abst) v1) (CHead e2 (Bind -Abst) v1))) (\lambda (e2: C).(csubt g e1 e2)) c3 (refl_equal C (CHead c3 -(Bind Abst) v1)) H10)))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: -C).(\lambda (c3: C).(\lambda (_: (csubt g c1 c3)).(\lambda (_: (((eq C c1 -(CHead e1 (Bind Abst) v1)) \to (or (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 -(Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: -C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: -C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g -e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 -v1)))))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: -T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u1) (CHead e1 -(Bind Abst) v1))).(let H5 \def (eq_ind C (CHead c1 (Bind Void) u1) (\lambda -(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind b0) \Rightarrow (match b0 in B return (\lambda (_: -B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False | Void -\Rightarrow True]) | (Flat _) \Rightarrow False])])) I (CHead e1 (Bind Abst) -v1) H4) in (False_ind (or (ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind b) -u2) (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T -(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind b) u2) (CHead e2 -(Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) -(\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda -(v2: T).(ty3 g e2 v2 v1))))) H5))))))))))) (\lambda (c1: C).(\lambda (c3: -C).(\lambda (H1: (csubt g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind -Abst) v1)) \to (or (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abst) -v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda -(v2: T).(eq C c3 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: -T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) -(\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))))).(\lambda (u: -T).(\lambda (t: T).(\lambda (H3: (ty3 g c1 u t)).(\lambda (H4: (ty3 g c3 u -t)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) t) (CHead e1 (Bind Abst) -v1))).(let H6 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda -(_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) -(CHead c1 (Bind Abst) t) (CHead e1 (Bind Abst) v1) H5) in ((let H7 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow t | (CHead _ _ t0) \Rightarrow t0])) (CHead c1 (Bind -Abst) t) (CHead e1 (Bind Abst) v1) H5) in (\lambda (H8: (eq C c1 e1)).(let H9 -\def (eq_ind T t (\lambda (t0: T).(ty3 g c3 u t0)) H4 v1 H7) in (let H10 \def -(eq_ind T t (\lambda (t0: T).(ty3 g c1 u t0)) H3 v1 H7) in (let H11 \def -(eq_ind C c1 (\lambda (c: C).(ty3 g c u v1)) H10 e1 H8) in (let H12 \def -(eq_ind C c1 (\lambda (c: C).((eq C c (CHead e1 (Bind Abst) v1)) \to (or (ex2 -C (\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abst) v1))) (\lambda (e2: -C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c3 -(CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 -e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: -C).(\lambda (v2: T).(ty3 g e2 v2 v1))))))) H2 e1 H8) in (let H13 \def (eq_ind -C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H8) in (or_intror (ex2 C (\lambda -(e2: C).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 (Bind Abst) v1))) (\lambda -(e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C -(CHead c3 (Bind Abbr) u) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: -C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g -e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))) -(ex4_2_intro C T (\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind -Abbr) u) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt -g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: -C).(\lambda (v2: T).(ty3 g e2 v2 v1))) c3 u (refl_equal C (CHead c3 (Bind -Abbr) u)) H13 H11 H9))))))))) H6))))))))))) y c2 H0))) H))))). -(* COMMENTS -Initial nodes: 2362 -END *) - -theorem csubt_gen_flat: - \forall (g: G).(\forall (e1: C).(\forall (c2: C).(\forall (v: T).(\forall -(f: F).((csubt g (CHead e1 (Flat f) v) c2) \to (ex2 C (\lambda (e2: C).(eq C -c2 (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g e1 e2)))))))) -\def - \lambda (g: G).(\lambda (e1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda -(f: F).(\lambda (H: (csubt g (CHead e1 (Flat f) v) c2)).(insert_eq C (CHead -e1 (Flat f) v) (\lambda (c: C).(csubt g c c2)) (\lambda (_: C).(ex2 C -(\lambda (e2: C).(eq C c2 (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g -e1 e2)))) (\lambda (y: C).(\lambda (H0: (csubt g y c2)).(csubt_ind g (\lambda -(c: C).(\lambda (c0: C).((eq C c (CHead e1 (Flat f) v)) \to (ex2 C (\lambda -(e2: C).(eq C c0 (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g e1 -e2)))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead e1 (Flat f) -v))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return -(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) -\Rightarrow False])) I (CHead e1 (Flat f) v) H1) in (False_ind (ex2 C -(\lambda (e2: C).(eq C (CSort n) (CHead e2 (Flat f) v))) (\lambda (e2: -C).(csubt g e1 e2))) H2)))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1: -(csubt g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Flat f) v)) \to (ex2 C -(\lambda (e2: C).(eq C c3 (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g -e1 e2)))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c1 k -u) (CHead e1 (Flat f) v))).(let H4 \def (f_equal C C (\lambda (e: C).(match e -in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) -\Rightarrow c])) (CHead c1 k u) (CHead e1 (Flat f) v) H3) in ((let H5 \def -(f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with -[(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c1 k u) -(CHead e1 (Flat f) v) H3) in ((let H6 \def (f_equal C T (\lambda (e: -C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | -(CHead _ _ t) \Rightarrow t])) (CHead c1 k u) (CHead e1 (Flat f) v) H3) in -(\lambda (H7: (eq K k (Flat f))).(\lambda (H8: (eq C c1 e1)).(eq_ind_r T v -(\lambda (t: T).(ex2 C (\lambda (e2: C).(eq C (CHead c3 k t) (CHead e2 (Flat -f) v))) (\lambda (e2: C).(csubt g e1 e2)))) (eq_ind_r K (Flat f) (\lambda -(k0: K).(ex2 C (\lambda (e2: C).(eq C (CHead c3 k0 v) (CHead e2 (Flat f) v))) -(\lambda (e2: C).(csubt g e1 e2)))) (let H9 \def (eq_ind C c1 (\lambda (c: -C).((eq C c (CHead e1 (Flat f) v)) \to (ex2 C (\lambda (e2: C).(eq C c3 -(CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g e1 e2))))) H2 e1 H8) in -(let H10 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H8) in -(ex_intro2 C (\lambda (e2: C).(eq C (CHead c3 (Flat f) v) (CHead e2 (Flat f) -v))) (\lambda (e2: C).(csubt g e1 e2)) c3 (refl_equal C (CHead c3 (Flat f) -v)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: C).(\lambda (c3: -C).(\lambda (_: (csubt g c1 c3)).(\lambda (_: (((eq C c1 (CHead e1 (Flat f) -v)) \to (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 (Flat f) v))) (\lambda -(e2: C).(csubt g e1 e2)))))).(\lambda (b: B).(\lambda (_: (not (eq B b -Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind -Void) u1) (CHead e1 (Flat f) v))).(let H5 \def (eq_ind C (CHead c1 (Bind -Void) u1) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with -[(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return -(\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow -False])])) I (CHead e1 (Flat f) v) H4) in (False_ind (ex2 C (\lambda (e2: -C).(eq C (CHead c3 (Bind b) u2) (CHead e2 (Flat f) v))) (\lambda (e2: -C).(csubt g e1 e2))) H5))))))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda -(_: (csubt g c1 c3)).(\lambda (_: (((eq C c1 (CHead e1 (Flat f) v)) \to (ex2 -C (\lambda (e2: C).(eq C c3 (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g -e1 e2)))))).(\lambda (u: T).(\lambda (t: T).(\lambda (_: (ty3 g c1 u -t)).(\lambda (_: (ty3 g c3 u t)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) t) -(CHead e1 (Flat f) v))).(let H6 \def (eq_ind C (CHead c1 (Bind Abst) t) -(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow -False])])) I (CHead e1 (Flat f) v) H5) in (False_ind (ex2 C (\lambda (e2: -C).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 (Flat f) v))) (\lambda (e2: -C).(csubt g e1 e2))) H6))))))))))) y c2 H0))) H)))))). -(* COMMENTS -Initial nodes: 1103 -END *) - -theorem csubt_gen_bind: - \forall (g: G).(\forall (b1: B).(\forall (e1: C).(\forall (c2: C).(\forall -(v1: T).((csubt g (CHead e1 (Bind b1) v1) c2) \to (ex2_3 B C T (\lambda (b2: -B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) -(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2)))))))))) -\def - \lambda (g: G).(\lambda (b1: B).(\lambda (e1: C).(\lambda (c2: C).(\lambda -(v1: T).(\lambda (H: (csubt g (CHead e1 (Bind b1) v1) c2)).(insert_eq C -(CHead e1 (Bind b1) v1) (\lambda (c: C).(csubt g c c2)) (\lambda (_: -C).(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 -(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: -T).(csubt g e1 e2)))))) (\lambda (y: C).(\lambda (H0: (csubt g y -c2)).(csubt_ind g (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead e1 (Bind -b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: -T).(eq C c0 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: -C).(\lambda (_: T).(csubt g e1 e2)))))))) (\lambda (n: nat).(\lambda (H1: (eq -C (CSort n) (CHead e1 (Bind b1) v1))).(let H2 \def (eq_ind C (CSort n) -(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 (Bind b1) -v1) H1) in (False_ind (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda -(v2: T).(eq C (CSort n) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda -(e2: C).(\lambda (_: T).(csubt g e1 e2))))) H2)))) (\lambda (c1: C).(\lambda -(c3: C).(\lambda (H1: (csubt g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 -(Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda -(v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: -C).(\lambda (_: T).(csubt g e1 e2)))))))).(\lambda (k: K).(\lambda (u: -T).(\lambda (H3: (eq C (CHead c1 k u) (CHead e1 (Bind b1) v1))).(let H4 \def -(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with -[(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u) -(CHead e1 (Bind b1) v1) H3) in ((let H5 \def (f_equal C K (\lambda (e: -C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | -(CHead _ k0 _) \Rightarrow k0])) (CHead c1 k u) (CHead e1 (Bind b1) v1) H3) -in ((let H6 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda -(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) -(CHead c1 k u) (CHead e1 (Bind b1) v1) H3) in (\lambda (H7: (eq K k (Bind -b1))).(\lambda (H8: (eq C c1 e1)).(eq_ind_r T v1 (\lambda (t: T).(ex2_3 B C T -(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 k t) -(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: -T).(csubt g e1 e2)))))) (eq_ind_r K (Bind b1) (\lambda (k0: K).(ex2_3 B C T -(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 k0 v1) -(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: -T).(csubt g e1 e2)))))) (let H9 \def (eq_ind C c1 (\lambda (c: C).((eq C c -(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: -C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: -B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))))))) H2 e1 H8) in (let -H10 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H8) in -(ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C -(CHead c3 (Bind b1) v1) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda -(e2: C).(\lambda (_: T).(csubt g e1 e2)))) b1 c3 v1 (refl_equal C (CHead c3 -(Bind b1) v1)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: -C).(\lambda (c3: C).(\lambda (H1: (csubt g c1 c3)).(\lambda (H2: (((eq C c1 -(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: -C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: -B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2)))))))).(\lambda (b: -B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: -T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) -v1))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda -(_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) -(CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) v1) H4) in ((let H6 \def -(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with -[(CSort _) \Rightarrow Void | (CHead _ k _) \Rightarrow (match k in K return -(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow -Void])])) (CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) v1) H4) in ((let H7 -\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead c1 -(Bind Void) u1) (CHead e1 (Bind b1) v1) H4) in (\lambda (H8: (eq B Void -b1)).(\lambda (H9: (eq C c1 e1)).(let H10 \def (eq_ind C c1 (\lambda (c: -C).((eq C c (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: -B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) -(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))))))) H2 e1 -H9) in (let H11 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H9) -in (let H12 \def (eq_ind_r B b1 (\lambda (b0: B).((eq C e1 (CHead e1 (Bind -b0) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: -T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: -C).(\lambda (_: T).(csubt g e1 e2))))))) H10 Void H8) in (ex2_3_intro B C T -(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind b) -u2) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: -T).(csubt g e1 e2)))) b c3 u2 (refl_equal C (CHead c3 (Bind b) u2)) -H11))))))) H6)) H5))))))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1: -(csubt g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind b1) v1)) \to (ex2_3 -B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 -(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g -e1 e2)))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H3: (ty3 g c1 u -t)).(\lambda (H4: (ty3 g c3 u t)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) -t) (CHead e1 (Bind b1) v1))).(let H6 \def (f_equal C C (\lambda (e: C).(match -e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ -_) \Rightarrow c])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H5) in -((let H7 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: -C).B) with [(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match k -in K return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) -\Rightarrow Abst])])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H5) in -((let H8 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: -C).T) with [(CSort _) \Rightarrow t | (CHead _ _ t0) \Rightarrow t0])) (CHead -c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H5) in (\lambda (H9: (eq B Abst -b1)).(\lambda (H10: (eq C c1 e1)).(let H11 \def (eq_ind T t (\lambda (t0: -T).(ty3 g c3 u t0)) H4 v1 H8) in (let H12 \def (eq_ind T t (\lambda (t0: -T).(ty3 g c1 u t0)) H3 v1 H8) in (let H13 \def (eq_ind C c1 (\lambda (c: -C).(ty3 g c u v1)) H12 e1 H10) in (let H14 \def (eq_ind C c1 (\lambda (c: -C).((eq C c (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: -B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) -(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))))))) H2 e1 -H10) in (let H15 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H10) -in (let H16 \def (eq_ind_r B b1 (\lambda (b: B).((eq C e1 (CHead e1 (Bind b) -v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq -C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda -(_: T).(csubt g e1 e2))))))) H14 Abst H9) in (ex2_3_intro B C T (\lambda (b2: -B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 -(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g -e1 e2)))) Abbr c3 u (refl_equal C (CHead c3 (Bind Abbr) u)) H15)))))))))) -H7)) H6))))))))))) y c2 H0))) H)))))). -(* COMMENTS -Initial nodes: 1899 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/getl.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/getl.ma deleted file mode 100644 index df14528ed..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/getl.ma +++ /dev/null @@ -1,426 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubt/clear.ma". - -include "Basic-1/csubt/drop.ma". - -include "Basic-1/getl/clear.ma". - -theorem csubt_getl_abbr: - \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).(\forall -(n: nat).((getl n c1 (CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csubt g -c1 c2) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n -c2 (CHead d2 (Bind Abbr) u))))))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u: T).(\lambda -(n: nat).(\lambda (H: (getl n c1 (CHead d1 (Bind Abbr) u))).(let H0 \def -(getl_gen_all c1 (CHead d1 (Bind Abbr) u) n H) in (ex2_ind C (\lambda (e: -C).(drop n O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abbr) u))) -(\forall (c2: C).((csubt g c1 c2) \to (ex2 C (\lambda (d2: C).(csubt g d1 -d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u)))))) (\lambda (x: -C).(\lambda (H1: (drop n O c1 x)).(\lambda (H2: (clear x (CHead d1 (Bind -Abbr) u))).(C_ind (\lambda (c: C).((drop n O c1 c) \to ((clear c (CHead d1 -(Bind Abbr) u)) \to (\forall (c2: C).((csubt g c1 c2) \to (ex2 C (\lambda -(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) -u))))))))) (\lambda (n0: nat).(\lambda (_: (drop n O c1 (CSort n0))).(\lambda -(H4: (clear (CSort n0) (CHead d1 (Bind Abbr) u))).(clear_gen_sort (CHead d1 -(Bind Abbr) u) n0 H4 (\forall (c2: C).((csubt g c1 c2) \to (ex2 C (\lambda -(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) -u)))))))))) (\lambda (x0: C).(\lambda (_: (((drop n O c1 x0) \to ((clear x0 -(CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csubt g c1 c2) \to (ex2 C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind -Abbr) u)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (H3: (drop n O c1 -(CHead x0 k t))).(\lambda (H4: (clear (CHead x0 k t) (CHead d1 (Bind Abbr) -u))).(K_ind (\lambda (k0: K).((drop n O c1 (CHead x0 k0 t)) \to ((clear -(CHead x0 k0 t) (CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csubt g c1 -c2) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 -(CHead d2 (Bind Abbr) u))))))))) (\lambda (b: B).(\lambda (H5: (drop n O c1 -(CHead x0 (Bind b) t))).(\lambda (H6: (clear (CHead x0 (Bind b) t) (CHead d1 -(Bind Abbr) u))).(let H7 \def (f_equal C C (\lambda (e: C).(match e in C -return (\lambda (_: C).C) with [(CSort _) \Rightarrow d1 | (CHead c _ _) -\Rightarrow c])) (CHead d1 (Bind Abbr) u) (CHead x0 (Bind b) t) -(clear_gen_bind b x0 (CHead d1 (Bind Abbr) u) t H6)) in ((let H8 \def -(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with -[(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow (match k0 in K -return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) -\Rightarrow Abbr])])) (CHead d1 (Bind Abbr) u) (CHead x0 (Bind b) t) -(clear_gen_bind b x0 (CHead d1 (Bind Abbr) u) t H6)) in ((let H9 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d1 (Bind -Abbr) u) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abbr) u) -t H6)) in (\lambda (H10: (eq B Abbr b)).(\lambda (H11: (eq C d1 x0)).(\lambda -(c2: C).(\lambda (H12: (csubt g c1 c2)).(let H13 \def (eq_ind_r T t (\lambda -(t0: T).(drop n O c1 (CHead x0 (Bind b) t0))) H5 u H9) in (let H14 \def -(eq_ind_r B b (\lambda (b0: B).(drop n O c1 (CHead x0 (Bind b0) u))) H13 Abbr -H10) in (let H15 \def (eq_ind_r C x0 (\lambda (c: C).(drop n O c1 (CHead c -(Bind Abbr) u))) H14 d1 H11) in (ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abbr) u))) (ex2 C (\lambda (d2: -C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u)))) -(\lambda (x1: C).(\lambda (H16: (csubt g d1 x1)).(\lambda (H17: (drop n O c2 -(CHead x1 (Bind Abbr) u))).(ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))) x1 H16 (getl_intro n -c2 (CHead x1 (Bind Abbr) u) (CHead x1 (Bind Abbr) u) H17 (clear_bind Abbr x1 -u)))))) (csubt_drop_abbr g n c1 c2 H12 d1 u H15)))))))))) H8)) H7))))) -(\lambda (f: F).(\lambda (H5: (drop n O c1 (CHead x0 (Flat f) t))).(\lambda -(H6: (clear (CHead x0 (Flat f) t) (CHead d1 (Bind Abbr) u))).(let H7 \def H5 -in (unintro C c1 (\lambda (c: C).((drop n O c (CHead x0 (Flat f) t)) \to -(\forall (c2: C).((csubt g c c2) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u)))))))) (nat_ind (\lambda -(n0: nat).(\forall (x1: C).((drop n0 O x1 (CHead x0 (Flat f) t)) \to (\forall -(c2: C).((csubt g x1 c2) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(getl n0 c2 (CHead d2 (Bind Abbr) u))))))))) (\lambda (x1: -C).(\lambda (H8: (drop O O x1 (CHead x0 (Flat f) t))).(\lambda (c2: -C).(\lambda (H9: (csubt g x1 c2)).(let H10 \def (eq_ind C x1 (\lambda (c: -C).(csubt g c c2)) H9 (CHead x0 (Flat f) t) (drop_gen_refl x1 (CHead x0 (Flat -f) t) H8)) in (let H_y \def (clear_flat x0 (CHead d1 (Bind Abbr) u) -(clear_gen_flat f x0 (CHead d1 (Bind Abbr) u) t H6) f t) in (let H11 \def -(csubt_clear_conf g (CHead x0 (Flat f) t) c2 H10 (CHead d1 (Bind Abbr) u) -H_y) in (ex2_ind C (\lambda (e2: C).(csubt g (CHead d1 (Bind Abbr) u) e2)) -(\lambda (e2: C).(clear c2 e2)) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (x2: -C).(\lambda (H12: (csubt g (CHead d1 (Bind Abbr) u) x2)).(\lambda (H13: -(clear c2 x2)).(let H14 \def (csubt_gen_abbr g d1 x2 u H12) in (ex2_ind C -(\lambda (e2: C).(eq C x2 (CHead e2 (Bind Abbr) u))) (\lambda (e2: C).(csubt -g d1 e2)) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl O -c2 (CHead d2 (Bind Abbr) u)))) (\lambda (x3: C).(\lambda (H15: (eq C x2 -(CHead x3 (Bind Abbr) u))).(\lambda (H16: (csubt g d1 x3)).(let H17 \def -(eq_ind C x2 (\lambda (c: C).(clear c2 c)) H13 (CHead x3 (Bind Abbr) u) H15) -in (ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl O c2 -(CHead d2 (Bind Abbr) u))) x3 H16 (getl_intro O c2 (CHead x3 (Bind Abbr) u) -c2 (drop_refl c2) H17)))))) H14))))) H11)))))))) (\lambda (n0: nat).(\lambda -(H8: ((\forall (x1: C).((drop n0 O x1 (CHead x0 (Flat f) t)) \to (\forall -(c2: C).((csubt g x1 c2) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(getl n0 c2 (CHead d2 (Bind Abbr) u)))))))))).(\lambda (x1: -C).(\lambda (H9: (drop (S n0) O x1 (CHead x0 (Flat f) t))).(\lambda (c2: -C).(\lambda (H10: (csubt g x1 c2)).(let H11 \def (drop_clear x1 (CHead x0 -(Flat f) t) n0 H9) in (ex2_3_ind B C T (\lambda (b: B).(\lambda (e: -C).(\lambda (v: T).(clear x1 (CHead e (Bind b) v))))) (\lambda (_: -B).(\lambda (e: C).(\lambda (_: T).(drop n0 O e (CHead x0 (Flat f) t))))) -(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 -(CHead d2 (Bind Abbr) u)))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: -T).(\lambda (H12: (clear x1 (CHead x3 (Bind x2) x4))).(\lambda (H13: (drop n0 -O x3 (CHead x0 (Flat f) t))).(let H14 \def (csubt_clear_conf g x1 c2 H10 -(CHead x3 (Bind x2) x4) H12) in (ex2_ind C (\lambda (e2: C).(csubt g (CHead -x3 (Bind x2) x4) e2)) (\lambda (e2: C).(clear c2 e2)) (ex2 C (\lambda (d2: -C).(csubt g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr) -u)))) (\lambda (x5: C).(\lambda (H15: (csubt g (CHead x3 (Bind x2) x4) -x5)).(\lambda (H16: (clear c2 x5)).(let H17 \def (csubt_gen_bind g x2 x3 x5 -x4 H15) in (ex2_3_ind B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: -T).(eq C x5 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: -C).(\lambda (_: T).(csubt g x3 e2)))) (ex2 C (\lambda (d2: C).(csubt g d1 -d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda -(x6: B).(\lambda (x7: C).(\lambda (x8: T).(\lambda (H18: (eq C x5 (CHead x7 -(Bind x6) x8))).(\lambda (H19: (csubt g x3 x7)).(let H20 \def (eq_ind C x5 -(\lambda (c: C).(clear c2 c)) H16 (CHead x7 (Bind x6) x8) H18) in (let H21 -\def (H8 x3 H13 x7 H19) in (ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(getl n0 x7 (CHead d2 (Bind Abbr) u))) (ex2 C (\lambda (d2: -C).(csubt g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr) -u)))) (\lambda (x9: C).(\lambda (H22: (csubt g d1 x9)).(\lambda (H23: (getl -n0 x7 (CHead x9 (Bind Abbr) u))).(ex_intro2 C (\lambda (d2: C).(csubt g d1 -d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr) u))) x9 H22 -(getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Abbr) u) n0 H23))))) -H21)))))))) H17))))) H14))))))) H11)))))))) n) H7))))) k H3 H4))))))) x H1 -H2)))) H0))))))). -(* COMMENTS -Initial nodes: 2313 -END *) - -theorem csubt_getl_abst: - \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (t: T).(\forall -(n: nat).((getl n c1 (CHead d1 (Bind Abst) t)) \to (\forall (c2: C).((csubt g -c1 c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: -C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n -c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u -t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (t: T).(\lambda -(n: nat).(\lambda (H: (getl n c1 (CHead d1 (Bind Abst) t))).(let H0 \def -(getl_gen_all c1 (CHead d1 (Bind Abst) t) n H) in (ex2_ind C (\lambda (e: -C).(drop n O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abst) t))) -(\forall (c2: C).((csubt g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 -d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T -(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda -(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: -T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))))) -(\lambda (x: C).(\lambda (H1: (drop n O c1 x)).(\lambda (H2: (clear x (CHead -d1 (Bind Abst) t))).(C_ind (\lambda (c: C).((drop n O c1 c) \to ((clear c -(CHead d1 (Bind Abst) t)) \to (\forall (c2: C).((csubt g c1 c2) \to (or (ex2 -C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 -(Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 -d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) -(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda -(u: T).(ty3 g d2 u t)))))))))) (\lambda (n0: nat).(\lambda (_: (drop n O c1 -(CSort n0))).(\lambda (H4: (clear (CSort n0) (CHead d1 (Bind Abst) -t))).(clear_gen_sort (CHead d1 (Bind Abst) t) n0 H4 (\forall (c2: C).((csubt -g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: -C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n -c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u -t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))))))))) (\lambda (x0: -C).(\lambda (_: (((drop n O c1 x0) \to ((clear x0 (CHead d1 (Bind Abst) t)) -\to (\forall (c2: C).((csubt g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csubt -g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T -(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda -(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: -T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u -t))))))))))).(\lambda (k: K).(\lambda (t0: T).(\lambda (H3: (drop n O c1 -(CHead x0 k t0))).(\lambda (H4: (clear (CHead x0 k t0) (CHead d1 (Bind Abst) -t))).(K_ind (\lambda (k0: K).((drop n O c1 (CHead x0 k0 t0)) \to ((clear -(CHead x0 k0 t0) (CHead d1 (Bind Abst) t)) \to (\forall (c2: C).((csubt g c1 -c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n -c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: -T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 -(Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda -(d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))) (\lambda (b: B).(\lambda (H5: -(drop n O c1 (CHead x0 (Bind b) t0))).(\lambda (H6: (clear (CHead x0 (Bind b) -t0) (CHead d1 (Bind Abst) t))).(let H7 \def (f_equal C C (\lambda (e: -C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d1 | -(CHead c _ _) \Rightarrow c])) (CHead d1 (Bind Abst) t) (CHead x0 (Bind b) -t0) (clear_gen_bind b x0 (CHead d1 (Bind Abst) t) t0 H6)) in ((let H8 \def -(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with -[(CSort _) \Rightarrow Abst | (CHead _ k0 _) \Rightarrow (match k0 in K -return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) -\Rightarrow Abst])])) (CHead d1 (Bind Abst) t) (CHead x0 (Bind b) t0) -(clear_gen_bind b x0 (CHead d1 (Bind Abst) t) t0 H6)) in ((let H9 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow t | (CHead _ _ t1) \Rightarrow t1])) (CHead d1 (Bind -Abst) t) (CHead x0 (Bind b) t0) (clear_gen_bind b x0 (CHead d1 (Bind Abst) t) -t0 H6)) in (\lambda (H10: (eq B Abst b)).(\lambda (H11: (eq C d1 -x0)).(\lambda (c2: C).(\lambda (H12: (csubt g c1 c2)).(let H13 \def (eq_ind_r -T t0 (\lambda (t1: T).(drop n O c1 (CHead x0 (Bind b) t1))) H5 t H9) in (let -H14 \def (eq_ind_r B b (\lambda (b0: B).(drop n O c1 (CHead x0 (Bind b0) t))) -H13 Abst H10) in (let H15 \def (eq_ind_r C x0 (\lambda (c: C).(drop n O c1 -(CHead c (Bind Abst) t))) H14 d1 H11) in (or_ind (ex2 C (\lambda (d2: -C).(csubt g d1 d2)) (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abst) t)))) -(ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: -C).(\lambda (u: T).(drop n O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: -C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g -d2 u t)))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: -C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n -c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u -t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (H16: (ex2 -C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n O c2 (CHead d2 -(Bind Abst) t))))).(ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(drop n O c2 (CHead d2 (Bind Abst) t))) (or (ex2 C (\lambda (d2: C).(csubt -g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T -(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda -(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: -T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) -(\lambda (x1: C).(\lambda (H17: (csubt g d1 x1)).(\lambda (H18: (drop n O c2 -(CHead x1 (Bind Abst) t))).(or_introl (ex2 C (\lambda (d2: C).(csubt g d1 -d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T -(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda -(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: -T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) -(ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 -(CHead d2 (Bind Abst) t))) x1 H17 (getl_intro n c2 (CHead x1 (Bind Abst) t) -(CHead x1 (Bind Abst) t) H18 (clear_bind Abst x1 t))))))) H16)) (\lambda -(H16: (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda -(d2: C).(\lambda (u: T).(drop n O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: -C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g -d2 u t))))).(ex4_2_ind C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) -(\lambda (d2: C).(\lambda (u: T).(drop n O c2 (CHead d2 (Bind Abbr) u)))) -(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda -(u: T).(ty3 g d2 u t))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda -(d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: -C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n -c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u -t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x1: -C).(\lambda (x2: T).(\lambda (H17: (csubt g d1 x1)).(\lambda (H18: (drop n O -c2 (CHead x1 (Bind Abbr) x2))).(\lambda (H19: (ty3 g d1 x2 t)).(\lambda (H20: -(ty3 g x1 x2 t)).(or_intror (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda -(d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: -C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n -c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u -t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (ex4_2_intro C T -(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda -(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: -T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))) x1 x2 -H17 (getl_intro n c2 (CHead x1 (Bind Abbr) x2) (CHead x1 (Bind Abbr) x2) H18 -(clear_bind Abbr x1 x2)) H19 H20)))))))) H16)) (csubt_drop_abst g n c1 c2 H12 -d1 t H15)))))))))) H8)) H7))))) (\lambda (f: F).(\lambda (H5: (drop n O c1 -(CHead x0 (Flat f) t0))).(\lambda (H6: (clear (CHead x0 (Flat f) t0) (CHead -d1 (Bind Abst) t))).(let H7 \def H5 in (unintro C c1 (\lambda (c: C).((drop n -O c (CHead x0 (Flat f) t0)) \to (\forall (c2: C).((csubt g c c2) \to (or (ex2 -C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 -(Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 -d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) -(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda -(u: T).(ty3 g d2 u t))))))))) (nat_ind (\lambda (n0: nat).(\forall (x1: -C).((drop n0 O x1 (CHead x0 (Flat f) t0)) \to (\forall (c2: C).((csubt g x1 -c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl -n0 c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: -T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n0 c2 (CHead d2 -(Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda -(d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))) (\lambda (x1: C).(\lambda -(H8: (drop O O x1 (CHead x0 (Flat f) t0))).(\lambda (c2: C).(\lambda (H9: -(csubt g x1 c2)).(let H10 \def (eq_ind C x1 (\lambda (c: C).(csubt g c c2)) -H9 (CHead x0 (Flat f) t0) (drop_gen_refl x1 (CHead x0 (Flat f) t0) H8)) in -(let H_y \def (clear_flat x0 (CHead d1 (Bind Abst) t) (clear_gen_flat f x0 -(CHead d1 (Bind Abst) t) t0 H6) f t0) in (let H11 \def (csubt_clear_conf g -(CHead x0 (Flat f) t0) c2 H10 (CHead d1 (Bind Abst) t) H_y) in (ex2_ind C -(\lambda (e2: C).(csubt g (CHead d1 (Bind Abst) t) e2)) (\lambda (e2: -C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: -C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl O -c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u -t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x2: -C).(\lambda (H12: (csubt g (CHead d1 (Bind Abst) t) x2)).(\lambda (H13: -(clear c2 x2)).(let H14 \def (csubt_gen_abst g d1 x2 t H12) in (or_ind (ex2 C -(\lambda (e2: C).(eq C x2 (CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csubt -g d1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C x2 (CHead e2 -(Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g d1 e2))) -(\lambda (_: C).(\lambda (v2: T).(ty3 g d1 v2 t))) (\lambda (e2: C).(\lambda -(v2: T).(ty3 g e2 v2 t)))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda -(d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: -T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: -T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) -(\lambda (H15: (ex2 C (\lambda (e2: C).(eq C x2 (CHead e2 (Bind Abst) t))) -(\lambda (e2: C).(csubt g d1 e2)))).(ex2_ind C (\lambda (e2: C).(eq C x2 -(CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csubt g d1 e2)) (or (ex2 C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind -Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) -(\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr) u)))) -(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda -(u: T).(ty3 g d2 u t))))) (\lambda (x3: C).(\lambda (H16: (eq C x2 (CHead x3 -(Bind Abst) t))).(\lambda (H17: (csubt g d1 x3)).(let H18 \def (eq_ind C x2 -(\lambda (c: C).(clear c2 c)) H13 (CHead x3 (Bind Abst) t) H16) in (or_introl -(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead -d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 -d2))) (\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr) u)))) -(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda -(u: T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t))) x3 H17 (getl_intro O -c2 (CHead x3 (Bind Abst) t) c2 (drop_refl c2) H18))))))) H15)) (\lambda (H15: -(ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C x2 (CHead e2 (Bind Abbr) -v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g d1 e2))) (\lambda (_: -C).(\lambda (v2: T).(ty3 g d1 v2 t))) (\lambda (e2: C).(\lambda (v2: T).(ty3 -g e2 v2 t))))).(ex4_2_ind C T (\lambda (e2: C).(\lambda (v2: T).(eq C x2 -(CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g d1 -e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g d1 v2 t))) (\lambda (e2: -C).(\lambda (v2: T).(ty3 g e2 v2 t))) (or (ex2 C (\lambda (d2: C).(csubt g d1 -d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T -(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda -(u: T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: -T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) -(\lambda (x3: C).(\lambda (x4: T).(\lambda (H16: (eq C x2 (CHead x3 (Bind -Abbr) x4))).(\lambda (H17: (csubt g d1 x3)).(\lambda (H18: (ty3 g d1 x4 -t)).(\lambda (H19: (ty3 g x3 x4 t)).(let H20 \def (eq_ind C x2 (\lambda (c: -C).(clear c2 c)) H13 (CHead x3 (Bind Abbr) x4) H16) in (or_intror (ex2 C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind -Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) -(\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr) u)))) -(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda -(u: T).(ty3 g d2 u t)))) (ex4_2_intro C T (\lambda (d2: C).(\lambda (_: -T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 -(Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda -(d2: C).(\lambda (u: T).(ty3 g d2 u t))) x3 x4 H17 (getl_intro O c2 (CHead x3 -(Bind Abbr) x4) c2 (drop_refl c2) H20) H18 H19))))))))) H15)) H14))))) -H11)))))))) (\lambda (n0: nat).(\lambda (H8: ((\forall (x1: C).((drop n0 O x1 -(CHead x0 (Flat f) t0)) \to (\forall (c2: C).((csubt g x1 c2) \to (or (ex2 C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n0 c2 (CHead d2 -(Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 -d2))) (\lambda (d2: C).(\lambda (u: T).(getl n0 c2 (CHead d2 (Bind Abbr) -u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: -C).(\lambda (u: T).(ty3 g d2 u t))))))))))).(\lambda (x1: C).(\lambda (H9: -(drop (S n0) O x1 (CHead x0 (Flat f) t0))).(\lambda (c2: C).(\lambda (H10: -(csubt g x1 c2)).(let H11 \def (drop_clear x1 (CHead x0 (Flat f) t0) n0 H9) -in (ex2_3_ind B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear x1 -(CHead e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: -T).(drop n0 O e (CHead x0 (Flat f) t0))))) (or (ex2 C (\lambda (d2: C).(csubt -g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex4_2 -C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: -C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: -C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g -d2 u t))))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: T).(\lambda (H12: -(clear x1 (CHead x3 (Bind x2) x4))).(\lambda (H13: (drop n0 O x3 (CHead x0 -(Flat f) t0))).(let H14 \def (csubt_clear_conf g x1 c2 H10 (CHead x3 (Bind -x2) x4) H12) in (ex2_ind C (\lambda (e2: C).(csubt g (CHead x3 (Bind x2) x4) -e2)) (\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(csubt g d1 -d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T -(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda -(u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda -(u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) -(\lambda (x5: C).(\lambda (H15: (csubt g (CHead x3 (Bind x2) x4) -x5)).(\lambda (H16: (clear c2 x5)).(let H17 \def (csubt_gen_bind g x2 x3 x5 -x4 H15) in (ex2_3_ind B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: -T).(eq C x5 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: -C).(\lambda (_: T).(csubt g x3 e2)))) (or (ex2 C (\lambda (d2: C).(csubt g d1 -d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T -(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda -(u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda -(u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) -(\lambda (x6: B).(\lambda (x7: C).(\lambda (x8: T).(\lambda (H18: (eq C x5 -(CHead x7 (Bind x6) x8))).(\lambda (H19: (csubt g x3 x7)).(let H20 \def -(eq_ind C x5 (\lambda (c: C).(clear c2 c)) H16 (CHead x7 (Bind x6) x8) H18) -in (let H21 \def (H8 x3 H13 x7 H19) in (or_ind (ex2 C (\lambda (d2: C).(csubt -g d1 d2)) (\lambda (d2: C).(getl n0 x7 (CHead d2 (Bind Abst) t)))) (ex4_2 C T -(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda -(u: T).(getl n0 x7 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: -T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (or -(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 -(CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: -T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead -d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) -(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (H22: (ex2 C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n0 x7 (CHead d2 -(Bind Abst) t))))).(ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(getl n0 x7 (CHead d2 (Bind Abst) t))) (or (ex2 C (\lambda (d2: C).(csubt -g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex4_2 -C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: -C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: -C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g -d2 u t))))) (\lambda (x9: C).(\lambda (H23: (csubt g d1 x9)).(\lambda (H24: -(getl n0 x7 (CHead x9 (Bind Abst) t))).(or_introl (ex2 C (\lambda (d2: -C).(csubt g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) -t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda -(d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda -(_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 -g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(getl (S n0) c2 (CHead d2 (Bind Abst) t))) x9 H23 (getl_clear_bind x6 c2 -x7 x8 H20 (CHead x9 (Bind Abst) t) n0 H24)))))) H22)) (\lambda (H22: (ex4_2 C -T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: -C).(\lambda (u: T).(getl n0 x7 (CHead d2 (Bind Abbr) u)))) (\lambda (_: -C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g -d2 u t))))).(ex4_2_ind C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) -(\lambda (d2: C).(\lambda (u: T).(getl n0 x7 (CHead d2 (Bind Abbr) u)))) -(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda -(u: T).(ty3 g d2 u t))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda -(d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: -C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl -(S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g -d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x9: -C).(\lambda (x10: T).(\lambda (H23: (csubt g d1 x9)).(\lambda (H24: (getl n0 -x7 (CHead x9 (Bind Abbr) x10))).(\lambda (H25: (ty3 g d1 x10 t)).(\lambda -(H26: (ty3 g x9 x10 t)).(or_intror (ex2 C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T -(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda -(u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda -(u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) -(ex4_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda -(d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda -(_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 -g d2 u t))) x9 x10 H23 (getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Abbr) -x10) n0 H24) H25 H26)))))))) H22)) H21)))))))) H17))))) H14))))))) -H11)))))))) n) H7))))) k H3 H4))))))) x H1 H2)))) H0))))))). -(* COMMENTS -Initial nodes: 5861 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/pc3.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/pc3.ma deleted file mode 100644 index 900ba49e9..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/pc3.ma +++ /dev/null @@ -1,62 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubt/getl.ma". - -include "Basic-1/pc3/left.ma". - -theorem csubt_pr2: - \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pr2 c1 -t1 t2) \to (\forall (c2: C).((csubt g c1 c2) \to (pr2 c2 t1 t2))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H: (pr2 c1 t1 t2)).(pr2_ind (\lambda (c: C).(\lambda (t: T).(\lambda (t0: -T).(\forall (c2: C).((csubt g c c2) \to (pr2 c2 t t0)))))) (\lambda (c: -C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(\lambda (c2: -C).(\lambda (_: (csubt g c c2)).(pr2_free c2 t3 t4 H0))))))) (\lambda (c: -C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c -(CHead d (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: -(pr0 t3 t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (c2: -C).(\lambda (H3: (csubt g c c2)).(let H4 \def (csubt_getl_abbr g c d u i H0 -c2 H3) in (ex2_ind C (\lambda (d2: C).(csubt g d d2)) (\lambda (d2: C).(getl -i c2 (CHead d2 (Bind Abbr) u))) (pr2 c2 t3 t) (\lambda (x: C).(\lambda (_: -(csubt g d x)).(\lambda (H6: (getl i c2 (CHead x (Bind Abbr) u))).(pr2_delta -c2 x u i H6 t3 t4 H1 t H2)))) H4)))))))))))))) c1 t1 t2 H))))). -(* COMMENTS -Initial nodes: 245 -END *) - -theorem csubt_pc3: - \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pc3 c1 -t1 t2) \to (\forall (c2: C).((csubt g c1 c2) \to (pc3 c2 t1 t2))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H: (pc3 c1 t1 t2)).(pc3_ind_left c1 (\lambda (t: T).(\lambda (t0: -T).(\forall (c2: C).((csubt g c1 c2) \to (pc3 c2 t t0))))) (\lambda (t: -T).(\lambda (c2: C).(\lambda (_: (csubt g c1 c2)).(pc3_refl c2 t)))) (\lambda -(t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c1 t0 t3)).(\lambda (t4: -T).(\lambda (_: (pc3 c1 t3 t4)).(\lambda (H2: ((\forall (c2: C).((csubt g c1 -c2) \to (pc3 c2 t3 t4))))).(\lambda (c2: C).(\lambda (H3: (csubt g c1 -c2)).(pc3_t t3 c2 t0 (pc3_pr2_r c2 t0 t3 (csubt_pr2 g c1 t0 t3 H0 c2 H3)) t4 -(H2 c2 H3)))))))))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c1 -t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3 c1 t0 t4)).(\lambda (H2: ((\forall -(c2: C).((csubt g c1 c2) \to (pc3 c2 t0 t4))))).(\lambda (c2: C).(\lambda -(H3: (csubt g c1 c2)).(pc3_t t0 c2 t3 (pc3_pr2_x c2 t3 t0 (csubt_pr2 g c1 t0 -t3 H0 c2 H3)) t4 (H2 c2 H3)))))))))) t1 t2 H))))). -(* COMMENTS -Initial nodes: 245 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/props.ma deleted file mode 100644 index 2efc87ccb..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/props.ma +++ /dev/null @@ -1,28 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubt/defs.ma". - -theorem csubt_refl: - \forall (g: G).(\forall (c: C).(csubt g c c)) -\def - \lambda (g: G).(\lambda (c: C).(C_ind (\lambda (c0: C).(csubt g c0 c0)) -(\lambda (n: nat).(csubt_sort g n)) (\lambda (c0: C).(\lambda (H: (csubt g c0 -c0)).(\lambda (k: K).(\lambda (t: T).(csubt_head g c0 c0 H k t))))) c)). -(* COMMENTS -Initial nodes: 53 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/ty3.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/ty3.ma deleted file mode 100644 index e6199ac15..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/ty3.ma +++ /dev/null @@ -1,102 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubt/pc3.ma". - -include "Basic-1/csubt/props.ma". - -theorem csubt_ty3: - \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c1 -t1 t2) \to (\forall (c2: C).((csubt g c1 c2) \to (ty3 g c2 t1 t2))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H: (ty3 g c1 t1 t2)).(ty3_ind g (\lambda (c: C).(\lambda (t: T).(\lambda -(t0: T).(\forall (c2: C).((csubt g c c2) \to (ty3 g c2 t t0)))))) (\lambda -(c: C).(\lambda (t0: T).(\lambda (t: T).(\lambda (_: (ty3 g c t0 t)).(\lambda -(H1: ((\forall (c2: C).((csubt g c c2) \to (ty3 g c2 t0 t))))).(\lambda (u: -T).(\lambda (t3: T).(\lambda (_: (ty3 g c u t3)).(\lambda (H3: ((\forall (c2: -C).((csubt g c c2) \to (ty3 g c2 u t3))))).(\lambda (H4: (pc3 c t3 -t0)).(\lambda (c2: C).(\lambda (H5: (csubt g c c2)).(ty3_conv g c2 t0 t (H1 -c2 H5) u t3 (H3 c2 H5) (csubt_pc3 g c t3 t0 H4 c2 H5)))))))))))))) (\lambda -(c: C).(\lambda (m: nat).(\lambda (c2: C).(\lambda (_: (csubt g c -c2)).(ty3_sort g c2 m))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d: -C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind Abbr) u))).(\lambda -(t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2: ((\forall (c2: C).((csubt g -d c2) \to (ty3 g c2 u t))))).(\lambda (c2: C).(\lambda (H3: (csubt g c -c2)).(let H4 \def (csubt_getl_abbr g c d u n H0 c2 H3) in (ex2_ind C (\lambda -(d2: C).(csubt g d d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) -u))) (ty3 g c2 (TLRef n) (lift (S n) O t)) (\lambda (x: C).(\lambda (H5: -(csubt g d x)).(\lambda (H6: (getl n c2 (CHead x (Bind Abbr) u))).(ty3_abbr g -n c2 x u H6 t (H2 x H5))))) H4)))))))))))) (\lambda (n: nat).(\lambda (c: -C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind -Abst) u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2: -((\forall (c2: C).((csubt g d c2) \to (ty3 g c2 u t))))).(\lambda (c2: -C).(\lambda (H3: (csubt g c c2)).(let H4 \def (csubt_getl_abst g c d u n H0 -c2 H3) in (or_ind (ex2 C (\lambda (d2: C).(csubt g d d2)) (\lambda (d2: -C).(getl n c2 (CHead d2 (Bind Abst) u)))) (ex4_2 C T (\lambda (d2: -C).(\lambda (_: T).(csubt g d d2))) (\lambda (d2: C).(\lambda (u0: T).(getl n -c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g d u0 -u))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 u)))) (ty3 g c2 (TLRef n) -(lift (S n) O u)) (\lambda (H5: (ex2 C (\lambda (d2: C).(csubt g d d2)) -(\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) u))))).(ex2_ind C (\lambda -(d2: C).(csubt g d d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) -u))) (ty3 g c2 (TLRef n) (lift (S n) O u)) (\lambda (x: C).(\lambda (H6: -(csubt g d x)).(\lambda (H7: (getl n c2 (CHead x (Bind Abst) u))).(ty3_abst g -n c2 x u H7 t (H2 x H6))))) H5)) (\lambda (H5: (ex4_2 C T (\lambda (d2: -C).(\lambda (_: T).(csubt g d d2))) (\lambda (d2: C).(\lambda (u0: T).(getl n -c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g d u0 -u))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 u))))).(ex4_2_ind C T -(\lambda (d2: C).(\lambda (_: T).(csubt g d d2))) (\lambda (d2: C).(\lambda -(u0: T).(getl n c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: -T).(ty3 g d u0 u))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 u))) (ty3 -g c2 (TLRef n) (lift (S n) O u)) (\lambda (x0: C).(\lambda (x1: T).(\lambda -(_: (csubt g d x0)).(\lambda (H7: (getl n c2 (CHead x0 (Bind Abbr) -x1))).(\lambda (_: (ty3 g d x1 u)).(\lambda (H9: (ty3 g x0 x1 u)).(ty3_abbr g -n c2 x0 x1 H7 u H9))))))) H5)) H4)))))))))))) (\lambda (c: C).(\lambda (u: -T).(\lambda (t: T).(\lambda (_: (ty3 g c u t)).(\lambda (H1: ((\forall (c2: -C).((csubt g c c2) \to (ty3 g c2 u t))))).(\lambda (b: B).(\lambda (t0: -T).(\lambda (t3: T).(\lambda (_: (ty3 g (CHead c (Bind b) u) t0 t3)).(\lambda -(H3: ((\forall (c2: C).((csubt g (CHead c (Bind b) u) c2) \to (ty3 g c2 t0 -t3))))).(\lambda (c2: C).(\lambda (H4: (csubt g c c2)).(ty3_bind g c2 u t (H1 -c2 H4) b t0 t3 (H3 (CHead c2 (Bind b) u) (csubt_head g c c2 H4 (Bind b) -u))))))))))))))) (\lambda (c: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: -(ty3 g c w u)).(\lambda (H1: ((\forall (c2: C).((csubt g c c2) \to (ty3 g c2 -w u))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c v (THead (Bind -Abst) u t))).(\lambda (H3: ((\forall (c2: C).((csubt g c c2) \to (ty3 g c2 v -(THead (Bind Abst) u t)))))).(\lambda (c2: C).(\lambda (H4: (csubt g c -c2)).(ty3_appl g c2 w u (H1 c2 H4) v t (H3 c2 H4))))))))))))) (\lambda (c: -C).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (ty3 g c t0 t3)).(\lambda -(H1: ((\forall (c2: C).((csubt g c c2) \to (ty3 g c2 t0 t3))))).(\lambda (t4: -T).(\lambda (_: (ty3 g c t3 t4)).(\lambda (H3: ((\forall (c2: C).((csubt g c -c2) \to (ty3 g c2 t3 t4))))).(\lambda (c2: C).(\lambda (H4: (csubt g c -c2)).(ty3_cast g c2 t0 t3 (H1 c2 H4) t4 (H3 c2 H4)))))))))))) c1 t1 t2 H))))). -(* COMMENTS -Initial nodes: 1325 -END *) - -theorem csubt_ty3_ld: - \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (v: T).((ty3 g c u -v) \to (\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c (Bind Abst) v) t1 -t2) \to (ty3 g (CHead c (Bind Abbr) u) t1 t2)))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (v: T).(\lambda (H: -(ty3 g c u v)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (ty3 g (CHead -c (Bind Abst) v) t1 t2)).(csubt_ty3 g (CHead c (Bind Abst) v) t1 t2 H0 (CHead -c (Bind Abbr) u) (csubt_abst g c c (csubt_refl g c) u v H H))))))))). -(* COMMENTS -Initial nodes: 91 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubv/clear.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubv/clear.ma deleted file mode 100644 index 5c54f5b9a..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubv/clear.ma +++ /dev/null @@ -1,194 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubv/defs.ma". - -include "Basic-1/clear/fwd.ma". - -theorem csubv_clear_conf: - \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (b1: -B).(\forall (d1: C).(\forall (v1: T).((clear c1 (CHead d1 (Bind b1) v1)) \to -(ex2_3 B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 -d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda (v2: T).(clear c2 (CHead d2 -(Bind b2) v2)))))))))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubv c1 c2)).(csubv_ind -(\lambda (c: C).(\lambda (c0: C).(\forall (b1: B).(\forall (d1: C).(\forall -(v1: T).((clear c (CHead d1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (_: -B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2: -B).(\lambda (d2: C).(\lambda (v2: T).(clear c0 (CHead d2 (Bind b2) -v2)))))))))))) (\lambda (n: nat).(\lambda (b1: B).(\lambda (d1: C).(\lambda -(v1: T).(\lambda (H0: (clear (CSort n) (CHead d1 (Bind b1) -v1))).(clear_gen_sort (CHead d1 (Bind b1) v1) n H0 (ex2_3 B C T (\lambda (_: -B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2: -B).(\lambda (d2: C).(\lambda (v2: T).(clear (CSort n) (CHead d2 (Bind b2) -v2)))))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csubv c3 -c4)).(\lambda (_: ((\forall (b1: B).(\forall (d1: C).(\forall (v1: T).((clear -c3 (CHead d1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (_: B).(\lambda (d2: -C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2: B).(\lambda (d2: -C).(\lambda (v2: T).(clear c4 (CHead d2 (Bind b2) v2)))))))))))).(\lambda -(v1: T).(\lambda (v2: T).(\lambda (b1: B).(\lambda (d1: C).(\lambda (v0: -T).(\lambda (H2: (clear (CHead c3 (Bind Void) v1) (CHead d1 (Bind b1) -v0))).(let H3 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda -(_: C).C) with [(CSort _) \Rightarrow d1 | (CHead c _ _) \Rightarrow c])) -(CHead d1 (Bind b1) v0) (CHead c3 (Bind Void) v1) (clear_gen_bind Void c3 -(CHead d1 (Bind b1) v0) v1 H2)) in ((let H4 \def (f_equal C B (\lambda (e: -C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow b1 | -(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind -b) \Rightarrow b | (Flat _) \Rightarrow b1])])) (CHead d1 (Bind b1) v0) -(CHead c3 (Bind Void) v1) (clear_gen_bind Void c3 (CHead d1 (Bind b1) v0) v1 -H2)) in ((let H5 \def (f_equal C T (\lambda (e: C).(match e in C return -(\lambda (_: C).T) with [(CSort _) \Rightarrow v0 | (CHead _ _ t) \Rightarrow -t])) (CHead d1 (Bind b1) v0) (CHead c3 (Bind Void) v1) (clear_gen_bind Void -c3 (CHead d1 (Bind b1) v0) v1 H2)) in (\lambda (_: (eq B b1 Void)).(\lambda -(H7: (eq C d1 c3)).(eq_ind_r C c3 (\lambda (c: C).(ex2_3 B C T (\lambda (_: -B).(\lambda (d2: C).(\lambda (_: T).(csubv c d2)))) (\lambda (b2: B).(\lambda -(d2: C).(\lambda (v3: T).(clear (CHead c4 (Bind Void) v2) (CHead d2 (Bind b2) -v3))))))) (ex2_3_intro B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: -T).(csubv c3 d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda (v3: T).(clear -(CHead c4 (Bind Void) v2) (CHead d2 (Bind b2) v3))))) Void c4 v2 H0 -(clear_bind Void c4 v2)) d1 H7)))) H4)) H3)))))))))))) (\lambda (c3: -C).(\lambda (c4: C).(\lambda (H0: (csubv c3 c4)).(\lambda (_: ((\forall (b1: -B).(\forall (d1: C).(\forall (v1: T).((clear c3 (CHead d1 (Bind b1) v1)) \to -(ex2_3 B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 -d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda (v2: T).(clear c4 (CHead d2 -(Bind b2) v2)))))))))))).(\lambda (b1: B).(\lambda (_: (not (eq B b1 -Void))).(\lambda (b2: B).(\lambda (v1: T).(\lambda (v2: T).(\lambda (b0: -B).(\lambda (d1: C).(\lambda (v0: T).(\lambda (H3: (clear (CHead c3 (Bind b1) -v1) (CHead d1 (Bind b0) v0))).(let H4 \def (f_equal C C (\lambda (e: -C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d1 | -(CHead c _ _) \Rightarrow c])) (CHead d1 (Bind b0) v0) (CHead c3 (Bind b1) -v1) (clear_gen_bind b1 c3 (CHead d1 (Bind b0) v0) v1 H3)) in ((let H5 \def -(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with -[(CSort _) \Rightarrow b0 | (CHead _ k _) \Rightarrow (match k in K return -(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow -b0])])) (CHead d1 (Bind b0) v0) (CHead c3 (Bind b1) v1) (clear_gen_bind b1 c3 -(CHead d1 (Bind b0) v0) v1 H3)) in ((let H6 \def (f_equal C T (\lambda (e: -C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow v0 | -(CHead _ _ t) \Rightarrow t])) (CHead d1 (Bind b0) v0) (CHead c3 (Bind b1) -v1) (clear_gen_bind b1 c3 (CHead d1 (Bind b0) v0) v1 H3)) in (\lambda (_: (eq -B b0 b1)).(\lambda (H8: (eq C d1 c3)).(eq_ind_r C c3 (\lambda (c: C).(ex2_3 B -C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv c d2)))) (\lambda -(b3: B).(\lambda (d2: C).(\lambda (v3: T).(clear (CHead c4 (Bind b2) v2) -(CHead d2 (Bind b3) v3))))))) (ex2_3_intro B C T (\lambda (_: B).(\lambda -(d2: C).(\lambda (_: T).(csubv c3 d2)))) (\lambda (b3: B).(\lambda (d2: -C).(\lambda (v3: T).(clear (CHead c4 (Bind b2) v2) (CHead d2 (Bind b3) -v3))))) b2 c4 v2 H0 (clear_bind b2 c4 v2)) d1 H8)))) H5)) H4))))))))))))))) -(\lambda (c3: C).(\lambda (c4: C).(\lambda (_: (csubv c3 c4)).(\lambda (H1: -((\forall (b1: B).(\forall (d1: C).(\forall (v1: T).((clear c3 (CHead d1 -(Bind b1) v1)) \to (ex2_3 B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: -T).(csubv d1 d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda (v2: T).(clear -c4 (CHead d2 (Bind b2) v2)))))))))))).(\lambda (f1: F).(\lambda (f2: -F).(\lambda (v1: T).(\lambda (v2: T).(\lambda (b1: B).(\lambda (d1: -C).(\lambda (v0: T).(\lambda (H2: (clear (CHead c3 (Flat f1) v1) (CHead d1 -(Bind b1) v0))).(let H_x \def (H1 b1 d1 v0 (clear_gen_flat f1 c3 (CHead d1 -(Bind b1) v0) v1 H2)) in (let H3 \def H_x in (ex2_3_ind B C T (\lambda (_: -B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2: -B).(\lambda (d2: C).(\lambda (v3: T).(clear c4 (CHead d2 (Bind b2) v3))))) -(ex2_3 B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 -d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda (v3: T).(clear (CHead c4 -(Flat f2) v2) (CHead d2 (Bind b2) v3)))))) (\lambda (x0: B).(\lambda (x1: -C).(\lambda (x2: T).(\lambda (H4: (csubv d1 x1)).(\lambda (H5: (clear c4 -(CHead x1 (Bind x0) x2))).(ex2_3_intro B C T (\lambda (_: B).(\lambda (d2: -C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2: B).(\lambda (d2: -C).(\lambda (v3: T).(clear (CHead c4 (Flat f2) v2) (CHead d2 (Bind b2) -v3))))) x0 x1 x2 H4 (clear_flat c4 (CHead x1 (Bind x0) x2) H5 f2 v2))))))) -H3))))))))))))))) c1 c2 H))). -(* COMMENTS -Initial nodes: 1357 -END *) - -theorem csubv_clear_conf_void: - \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (d1: -C).(\forall (v1: T).((clear c1 (CHead d1 (Bind Void) v1)) \to (ex2_2 C T -(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2))) (\lambda (d2: C).(\lambda -(v2: T).(clear c2 (CHead d2 (Bind Void) v2)))))))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubv c1 c2)).(csubv_ind -(\lambda (c: C).(\lambda (c0: C).(\forall (d1: C).(\forall (v1: T).((clear c -(CHead d1 (Bind Void) v1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (_: -T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(clear c0 (CHead d2 -(Bind Void) v2)))))))))) (\lambda (n: nat).(\lambda (d1: C).(\lambda (v1: -T).(\lambda (H0: (clear (CSort n) (CHead d1 (Bind Void) v1))).(clear_gen_sort -(CHead d1 (Bind Void) v1) n H0 (ex2_2 C T (\lambda (d2: C).(\lambda (_: -T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(clear (CSort n) (CHead -d2 (Bind Void) v2)))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: -(csubv c3 c4)).(\lambda (_: ((\forall (d1: C).(\forall (v1: T).((clear c3 -(CHead d1 (Bind Void) v1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (_: -T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(clear c4 (CHead d2 -(Bind Void) v2)))))))))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (d1: -C).(\lambda (v0: T).(\lambda (H2: (clear (CHead c3 (Bind Void) v1) (CHead d1 -(Bind Void) v0))).(let H3 \def (f_equal C C (\lambda (e: C).(match e in C -return (\lambda (_: C).C) with [(CSort _) \Rightarrow d1 | (CHead c _ _) -\Rightarrow c])) (CHead d1 (Bind Void) v0) (CHead c3 (Bind Void) v1) -(clear_gen_bind Void c3 (CHead d1 (Bind Void) v0) v1 H2)) in ((let H4 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow v0 | (CHead _ _ t) \Rightarrow t])) (CHead d1 (Bind -Void) v0) (CHead c3 (Bind Void) v1) (clear_gen_bind Void c3 (CHead d1 (Bind -Void) v0) v1 H2)) in (\lambda (H5: (eq C d1 c3)).(eq_ind_r C c3 (\lambda (c: -C).(ex2_2 C T (\lambda (d2: C).(\lambda (_: T).(csubv c d2))) (\lambda (d2: -C).(\lambda (v3: T).(clear (CHead c4 (Bind Void) v2) (CHead d2 (Bind Void) -v3)))))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csubv c3 d2))) -(\lambda (d2: C).(\lambda (v3: T).(clear (CHead c4 (Bind Void) v2) (CHead d2 -(Bind Void) v3)))) c4 v2 H0 (clear_bind Void c4 v2)) d1 H5))) H3))))))))))) -(\lambda (c3: C).(\lambda (c4: C).(\lambda (_: (csubv c3 c4)).(\lambda (_: -((\forall (d1: C).(\forall (v1: T).((clear c3 (CHead d1 (Bind Void) v1)) \to -(ex2_2 C T (\lambda (d2: C).(\lambda (_: T).(csubv d1 d2))) (\lambda (d2: -C).(\lambda (v2: T).(clear c4 (CHead d2 (Bind Void) v2)))))))))).(\lambda -(b1: B).(\lambda (H2: (not (eq B b1 Void))).(\lambda (b2: B).(\lambda (v1: -T).(\lambda (v2: T).(\lambda (d1: C).(\lambda (v0: T).(\lambda (H3: (clear -(CHead c3 (Bind b1) v1) (CHead d1 (Bind Void) v0))).(let H4 \def (f_equal C C -(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow d1 | (CHead c _ _) \Rightarrow c])) (CHead d1 (Bind Void) v0) -(CHead c3 (Bind b1) v1) (clear_gen_bind b1 c3 (CHead d1 (Bind Void) v0) v1 -H3)) in ((let H5 \def (f_equal C B (\lambda (e: C).(match e in C return -(\lambda (_: C).B) with [(CSort _) \Rightarrow Void | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) -\Rightarrow b | (Flat _) \Rightarrow Void])])) (CHead d1 (Bind Void) v0) -(CHead c3 (Bind b1) v1) (clear_gen_bind b1 c3 (CHead d1 (Bind Void) v0) v1 -H3)) in ((let H6 \def (f_equal C T (\lambda (e: C).(match e in C return -(\lambda (_: C).T) with [(CSort _) \Rightarrow v0 | (CHead _ _ t) \Rightarrow -t])) (CHead d1 (Bind Void) v0) (CHead c3 (Bind b1) v1) (clear_gen_bind b1 c3 -(CHead d1 (Bind Void) v0) v1 H3)) in (\lambda (H7: (eq B Void b1)).(\lambda -(H8: (eq C d1 c3)).(eq_ind_r C c3 (\lambda (c: C).(ex2_2 C T (\lambda (d2: -C).(\lambda (_: T).(csubv c d2))) (\lambda (d2: C).(\lambda (v3: T).(clear -(CHead c4 (Bind b2) v2) (CHead d2 (Bind Void) v3)))))) (let H9 \def (eq_ind_r -B b1 (\lambda (b: B).(not (eq B b Void))) H2 Void H7) in (let H10 \def (match -(H9 (refl_equal B Void)) in False return (\lambda (_: False).(ex2_2 C T -(\lambda (d2: C).(\lambda (_: T).(csubv c3 d2))) (\lambda (d2: C).(\lambda -(v3: T).(clear (CHead c4 (Bind b2) v2) (CHead d2 (Bind Void) v3)))))) with -[]) in H10)) d1 H8)))) H5)) H4)))))))))))))) (\lambda (c3: C).(\lambda (c4: -C).(\lambda (_: (csubv c3 c4)).(\lambda (H1: ((\forall (d1: C).(\forall (v1: -T).((clear c3 (CHead d1 (Bind Void) v1)) \to (ex2_2 C T (\lambda (d2: -C).(\lambda (_: T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(clear -c4 (CHead d2 (Bind Void) v2)))))))))).(\lambda (f1: F).(\lambda (f2: -F).(\lambda (v1: T).(\lambda (v2: T).(\lambda (d1: C).(\lambda (v0: -T).(\lambda (H2: (clear (CHead c3 (Flat f1) v1) (CHead d1 (Bind Void) -v0))).(let H_x \def (H1 d1 v0 (clear_gen_flat f1 c3 (CHead d1 (Bind Void) v0) -v1 H2)) in (let H3 \def H_x in (ex2_2_ind C T (\lambda (d2: C).(\lambda (_: -T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v3: T).(clear c4 (CHead d2 -(Bind Void) v3)))) (ex2_2 C T (\lambda (d2: C).(\lambda (_: T).(csubv d1 -d2))) (\lambda (d2: C).(\lambda (v3: T).(clear (CHead c4 (Flat f2) v2) (CHead -d2 (Bind Void) v3))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H4: (csubv -d1 x0)).(\lambda (H5: (clear c4 (CHead x0 (Bind Void) x1))).(ex2_2_intro C T -(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2))) (\lambda (d2: C).(\lambda -(v3: T).(clear (CHead c4 (Flat f2) v2) (CHead d2 (Bind Void) v3)))) x0 x1 H4 -(clear_flat c4 (CHead x0 (Bind Void) x1) H5 f2 v2)))))) H3)))))))))))))) c1 -c2 H))). -(* COMMENTS -Initial nodes: 1205 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubv/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubv/defs.ma deleted file mode 100644 index 6ffb63608..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubv/defs.ma +++ /dev/null @@ -1,30 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/C/defs.ma". - -inductive csubv: C \to (C \to Prop) \def -| csubv_sort: \forall (n: nat).(csubv (CSort n) (CSort n)) -| csubv_void: \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall -(v1: T).(\forall (v2: T).(csubv (CHead c1 (Bind Void) v1) (CHead c2 (Bind -Void) v2)))))) -| csubv_bind: \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall -(b1: B).((not (eq B b1 Void)) \to (\forall (b2: B).(\forall (v1: T).(\forall -(v2: T).(csubv (CHead c1 (Bind b1) v1) (CHead c2 (Bind b2) v2))))))))) -| csubv_flat: \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall -(f1: F).(\forall (f2: F).(\forall (v1: T).(\forall (v2: T).(csubv (CHead c1 -(Flat f1) v1) (CHead c2 (Flat f2) v2)))))))). - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubv/drop.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubv/drop.ma deleted file mode 100644 index 007670dbb..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubv/drop.ma +++ /dev/null @@ -1,115 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubv/props.ma". - -include "Basic-1/drop/fwd.ma". - -theorem csubv_drop_conf: - \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (e1: -C).(\forall (h: nat).((drop h O c1 e1) \to (ex2 C (\lambda (e2: C).(csubv e1 -e2)) (\lambda (e2: C).(drop h O c2 e2)))))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubv c1 c2)).(csubv_ind -(\lambda (c: C).(\lambda (c0: C).(\forall (e1: C).(\forall (h: nat).((drop h -O c e1) \to (ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop h O -c0 e2)))))))) (\lambda (n: nat).(\lambda (e1: C).(\lambda (h: nat).(\lambda -(H0: (drop h O (CSort n) e1)).(and3_ind (eq C e1 (CSort n)) (eq nat h O) (eq -nat O O) (ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop h O -(CSort n) e2))) (\lambda (H1: (eq C e1 (CSort n))).(\lambda (H2: (eq nat h -O)).(\lambda (_: (eq nat O O)).(eq_ind_r nat O (\lambda (n0: nat).(ex2 C -(\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop n0 O (CSort n) e2)))) -(eq_ind_r C (CSort n) (\lambda (c: C).(ex2 C (\lambda (e2: C).(csubv c e2)) -(\lambda (e2: C).(drop O O (CSort n) e2)))) (ex_intro2 C (\lambda (e2: -C).(csubv (CSort n) e2)) (\lambda (e2: C).(drop O O (CSort n) e2)) (CSort n) -(csubv_refl (CSort n)) (drop_refl (CSort n))) e1 H1) h H2)))) (drop_gen_sort -n h O e1 H0)))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csubv c3 -c4)).(\lambda (H1: ((\forall (e1: C).(\forall (h: nat).((drop h O c3 e1) \to -(ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop h O c4 -e2)))))))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (e1: C).(\lambda (h: -nat).(\lambda (H2: (drop h O (CHead c3 (Bind Void) v1) e1)).(nat_ind (\lambda -(n: nat).((drop n O (CHead c3 (Bind Void) v1) e1) \to (ex2 C (\lambda (e2: -C).(csubv e1 e2)) (\lambda (e2: C).(drop n O (CHead c4 (Bind Void) v2) -e2))))) (\lambda (H3: (drop O O (CHead c3 (Bind Void) v1) e1)).(eq_ind C -(CHead c3 (Bind Void) v1) (\lambda (c: C).(ex2 C (\lambda (e2: C).(csubv c -e2)) (\lambda (e2: C).(drop O O (CHead c4 (Bind Void) v2) e2)))) (ex_intro2 C -(\lambda (e2: C).(csubv (CHead c3 (Bind Void) v1) e2)) (\lambda (e2: C).(drop -O O (CHead c4 (Bind Void) v2) e2)) (CHead c4 (Bind Void) v2) (csubv_bind_same -c3 c4 H0 Void v1 v2) (drop_refl (CHead c4 (Bind Void) v2))) e1 (drop_gen_refl -(CHead c3 (Bind Void) v1) e1 H3))) (\lambda (h0: nat).(\lambda (_: (((drop h0 -O (CHead c3 (Bind Void) v1) e1) \to (ex2 C (\lambda (e2: C).(csubv e1 e2)) -(\lambda (e2: C).(drop h0 O (CHead c4 (Bind Void) v2) e2)))))).(\lambda (H3: -(drop (S h0) O (CHead c3 (Bind Void) v1) e1)).(let H_x \def (H1 e1 (r (Bind -Void) h0) (drop_gen_drop (Bind Void) c3 e1 v1 h0 H3)) in (let H4 \def H_x in -(ex2_ind C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop h0 O c4 -e2)) (ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop (S h0) O -(CHead c4 (Bind Void) v2) e2))) (\lambda (x: C).(\lambda (H5: (csubv e1 -x)).(\lambda (H6: (drop h0 O c4 x)).(ex_intro2 C (\lambda (e2: C).(csubv e1 -e2)) (\lambda (e2: C).(drop (S h0) O (CHead c4 (Bind Void) v2) e2)) x H5 -(drop_drop (Bind Void) h0 c4 x H6 v2))))) H4)))))) h H2)))))))))) (\lambda -(c3: C).(\lambda (c4: C).(\lambda (H0: (csubv c3 c4)).(\lambda (H1: ((\forall -(e1: C).(\forall (h: nat).((drop h O c3 e1) \to (ex2 C (\lambda (e2: -C).(csubv e1 e2)) (\lambda (e2: C).(drop h O c4 e2)))))))).(\lambda (b1: -B).(\lambda (H2: (not (eq B b1 Void))).(\lambda (b2: B).(\lambda (v1: -T).(\lambda (v2: T).(\lambda (e1: C).(\lambda (h: nat).(\lambda (H3: (drop h -O (CHead c3 (Bind b1) v1) e1)).(nat_ind (\lambda (n: nat).((drop n O (CHead -c3 (Bind b1) v1) e1) \to (ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: -C).(drop n O (CHead c4 (Bind b2) v2) e2))))) (\lambda (H4: (drop O O (CHead -c3 (Bind b1) v1) e1)).(eq_ind C (CHead c3 (Bind b1) v1) (\lambda (c: C).(ex2 -C (\lambda (e2: C).(csubv c e2)) (\lambda (e2: C).(drop O O (CHead c4 (Bind -b2) v2) e2)))) (ex_intro2 C (\lambda (e2: C).(csubv (CHead c3 (Bind b1) v1) -e2)) (\lambda (e2: C).(drop O O (CHead c4 (Bind b2) v2) e2)) (CHead c4 (Bind -b2) v2) (csubv_bind c3 c4 H0 b1 H2 b2 v1 v2) (drop_refl (CHead c4 (Bind b2) -v2))) e1 (drop_gen_refl (CHead c3 (Bind b1) v1) e1 H4))) (\lambda (h0: -nat).(\lambda (_: (((drop h0 O (CHead c3 (Bind b1) v1) e1) \to (ex2 C -(\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop h0 O (CHead c4 (Bind -b2) v2) e2)))))).(\lambda (H4: (drop (S h0) O (CHead c3 (Bind b1) v1) -e1)).(let H_x \def (H1 e1 (r (Bind b1) h0) (drop_gen_drop (Bind b1) c3 e1 v1 -h0 H4)) in (let H5 \def H_x in (ex2_ind C (\lambda (e2: C).(csubv e1 e2)) -(\lambda (e2: C).(drop h0 O c4 e2)) (ex2 C (\lambda (e2: C).(csubv e1 e2)) -(\lambda (e2: C).(drop (S h0) O (CHead c4 (Bind b2) v2) e2))) (\lambda (x: -C).(\lambda (H6: (csubv e1 x)).(\lambda (H7: (drop h0 O c4 x)).(ex_intro2 C -(\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop (S h0) O (CHead c4 -(Bind b2) v2) e2)) x H6 (drop_drop (Bind b2) h0 c4 x H7 v2))))) H5)))))) h -H3))))))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csubv c3 -c4)).(\lambda (H1: ((\forall (e1: C).(\forall (h: nat).((drop h O c3 e1) \to -(ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop h O c4 -e2)))))))).(\lambda (f1: F).(\lambda (f2: F).(\lambda (v1: T).(\lambda (v2: -T).(\lambda (e1: C).(\lambda (h: nat).(\lambda (H2: (drop h O (CHead c3 (Flat -f1) v1) e1)).(nat_ind (\lambda (n: nat).((drop n O (CHead c3 (Flat f1) v1) -e1) \to (ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop n O -(CHead c4 (Flat f2) v2) e2))))) (\lambda (H3: (drop O O (CHead c3 (Flat f1) -v1) e1)).(eq_ind C (CHead c3 (Flat f1) v1) (\lambda (c: C).(ex2 C (\lambda -(e2: C).(csubv c e2)) (\lambda (e2: C).(drop O O (CHead c4 (Flat f2) v2) -e2)))) (ex_intro2 C (\lambda (e2: C).(csubv (CHead c3 (Flat f1) v1) e2)) -(\lambda (e2: C).(drop O O (CHead c4 (Flat f2) v2) e2)) (CHead c4 (Flat f2) -v2) (csubv_flat c3 c4 H0 f1 f2 v1 v2) (drop_refl (CHead c4 (Flat f2) v2))) e1 -(drop_gen_refl (CHead c3 (Flat f1) v1) e1 H3))) (\lambda (h0: nat).(\lambda -(_: (((drop h0 O (CHead c3 (Flat f1) v1) e1) \to (ex2 C (\lambda (e2: -C).(csubv e1 e2)) (\lambda (e2: C).(drop h0 O (CHead c4 (Flat f2) v2) -e2)))))).(\lambda (H3: (drop (S h0) O (CHead c3 (Flat f1) v1) e1)).(let H_x -\def (H1 e1 (r (Flat f1) h0) (drop_gen_drop (Flat f1) c3 e1 v1 h0 H3)) in -(let H4 \def H_x in (ex2_ind C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: -C).(drop (S h0) O c4 e2)) (ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda -(e2: C).(drop (S h0) O (CHead c4 (Flat f2) v2) e2))) (\lambda (x: C).(\lambda -(H5: (csubv e1 x)).(\lambda (H6: (drop (S h0) O c4 x)).(ex_intro2 C (\lambda -(e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop (S h0) O (CHead c4 (Flat f2) -v2) e2)) x H5 (drop_drop (Flat f2) h0 c4 x H6 v2))))) H4)))))) h -H2)))))))))))) c1 c2 H))). -(* COMMENTS -Initial nodes: 1897 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubv/getl.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubv/getl.ma deleted file mode 100644 index bbba95084..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubv/getl.ma +++ /dev/null @@ -1,90 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubv/clear.ma". - -include "Basic-1/csubv/drop.ma". - -include "Basic-1/getl/fwd.ma". - -theorem csubv_getl_conf: - \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (b1: -B).(\forall (d1: C).(\forall (v1: T).(\forall (i: nat).((getl i c1 (CHead d1 -(Bind b1) v1)) \to (ex2_3 B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: -T).(csubv d1 d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda (v2: T).(getl -i c2 (CHead d2 (Bind b2) v2))))))))))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubv c1 c2)).(\lambda (b1: -B).(\lambda (d1: C).(\lambda (v1: T).(\lambda (i: nat).(\lambda (H0: (getl i -c1 (CHead d1 (Bind b1) v1))).(let H1 \def (getl_gen_all c1 (CHead d1 (Bind -b1) v1) i H0) in (ex2_ind C (\lambda (e: C).(drop i O c1 e)) (\lambda (e: -C).(clear e (CHead d1 (Bind b1) v1))) (ex2_3 B C T (\lambda (_: B).(\lambda -(d2: C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2: B).(\lambda (d2: -C).(\lambda (v2: T).(getl i c2 (CHead d2 (Bind b2) v2)))))) (\lambda (x: -C).(\lambda (H2: (drop i O c1 x)).(\lambda (H3: (clear x (CHead d1 (Bind b1) -v1))).(let H_x \def (csubv_drop_conf c1 c2 H x i H2) in (let H4 \def H_x in -(ex2_ind C (\lambda (e2: C).(csubv x e2)) (\lambda (e2: C).(drop i O c2 e2)) -(ex2_3 B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 -d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda (v2: T).(getl i c2 (CHead -d2 (Bind b2) v2)))))) (\lambda (x0: C).(\lambda (H5: (csubv x x0)).(\lambda -(H6: (drop i O c2 x0)).(let H_x0 \def (csubv_clear_conf x x0 H5 b1 d1 v1 H3) -in (let H7 \def H_x0 in (ex2_3_ind B C T (\lambda (_: B).(\lambda (d2: -C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2: B).(\lambda (d2: -C).(\lambda (v2: T).(clear x0 (CHead d2 (Bind b2) v2))))) (ex2_3 B C T -(\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2)))) (\lambda -(b2: B).(\lambda (d2: C).(\lambda (v2: T).(getl i c2 (CHead d2 (Bind b2) -v2)))))) (\lambda (x1: B).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H8: -(csubv d1 x2)).(\lambda (H9: (clear x0 (CHead x2 (Bind x1) x3))).(ex2_3_intro -B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2)))) -(\lambda (b2: B).(\lambda (d2: C).(\lambda (v2: T).(getl i c2 (CHead d2 (Bind -b2) v2))))) x1 x2 x3 H8 (getl_intro i c2 (CHead x2 (Bind x1) x3) x0 H6 -H9))))))) H7)))))) H4)))))) H1))))))))). -(* COMMENTS -Initial nodes: 455 -END *) - -theorem csubv_getl_conf_void: - \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (d1: -C).(\forall (v1: T).(\forall (i: nat).((getl i c1 (CHead d1 (Bind Void) v1)) -\to (ex2_2 C T (\lambda (d2: C).(\lambda (_: T).(csubv d1 d2))) (\lambda (d2: -C).(\lambda (v2: T).(getl i c2 (CHead d2 (Bind Void) v2))))))))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubv c1 c2)).(\lambda (d1: -C).(\lambda (v1: T).(\lambda (i: nat).(\lambda (H0: (getl i c1 (CHead d1 -(Bind Void) v1))).(let H1 \def (getl_gen_all c1 (CHead d1 (Bind Void) v1) i -H0) in (ex2_ind C (\lambda (e: C).(drop i O c1 e)) (\lambda (e: C).(clear e -(CHead d1 (Bind Void) v1))) (ex2_2 C T (\lambda (d2: C).(\lambda (_: -T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(getl i c2 (CHead d2 -(Bind Void) v2))))) (\lambda (x: C).(\lambda (H2: (drop i O c1 x)).(\lambda -(H3: (clear x (CHead d1 (Bind Void) v1))).(let H_x \def (csubv_drop_conf c1 -c2 H x i H2) in (let H4 \def H_x in (ex2_ind C (\lambda (e2: C).(csubv x e2)) -(\lambda (e2: C).(drop i O c2 e2)) (ex2_2 C T (\lambda (d2: C).(\lambda (_: -T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(getl i c2 (CHead d2 -(Bind Void) v2))))) (\lambda (x0: C).(\lambda (H5: (csubv x x0)).(\lambda -(H6: (drop i O c2 x0)).(let H_x0 \def (csubv_clear_conf_void x x0 H5 d1 v1 -H3) in (let H7 \def H_x0 in (ex2_2_ind C T (\lambda (d2: C).(\lambda (_: -T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(clear x0 (CHead d2 -(Bind Void) v2)))) (ex2_2 C T (\lambda (d2: C).(\lambda (_: T).(csubv d1 -d2))) (\lambda (d2: C).(\lambda (v2: T).(getl i c2 (CHead d2 (Bind Void) -v2))))) (\lambda (x1: C).(\lambda (x2: T).(\lambda (H8: (csubv d1 -x1)).(\lambda (H9: (clear x0 (CHead x1 (Bind Void) x2))).(ex2_2_intro C T -(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2))) (\lambda (d2: C).(\lambda -(v2: T).(getl i c2 (CHead d2 (Bind Void) v2)))) x1 x2 H8 (getl_intro i c2 -(CHead x1 (Bind Void) x2) x0 H6 H9)))))) H7)))))) H4)))))) H1)))))))). -(* COMMENTS -Initial nodes: 417 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubv/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubv/props.ma deleted file mode 100644 index 716922824..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/csubv/props.ma +++ /dev/null @@ -1,48 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubv/defs.ma". - -include "Basic-1/T/props.ma". - -theorem csubv_bind_same: - \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (b: B).(\forall -(v1: T).(\forall (v2: T).(csubv (CHead c1 (Bind b) v1) (CHead c2 (Bind b) -v2))))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubv c1 c2)).(\lambda (b: -B).(B_ind (\lambda (b0: B).(\forall (v1: T).(\forall (v2: T).(csubv (CHead c1 -(Bind b0) v1) (CHead c2 (Bind b0) v2))))) (\lambda (v1: T).(\lambda (v2: -T).(csubv_bind c1 c2 H Abbr (\lambda (H0: (eq B Abbr Void)).(not_abbr_void -H0)) Abbr v1 v2))) (\lambda (v1: T).(\lambda (v2: T).(csubv_bind c1 c2 H Abst -(sym_not_eq B Void Abst not_void_abst) Abst v1 v2))) (\lambda (v1: -T).(\lambda (v2: T).(csubv_void c1 c2 H v1 v2))) b)))). -(* COMMENTS -Initial nodes: 121 -END *) - -theorem csubv_refl: - \forall (c: C).(csubv c c) -\def - \lambda (c: C).(C_ind (\lambda (c0: C).(csubv c0 c0)) (\lambda (n: -nat).(csubv_sort n)) (\lambda (c0: C).(\lambda (H: (csubv c0 c0)).(\lambda -(k: K).(K_ind (\lambda (k0: K).(\forall (t: T).(csubv (CHead c0 k0 t) (CHead -c0 k0 t)))) (\lambda (b: B).(\lambda (t: T).(csubv_bind_same c0 c0 H b t t))) -(\lambda (f: F).(\lambda (t: T).(csubv_flat c0 c0 H f f t t))) k)))) c). -(* COMMENTS -Initial nodes: 93 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/definitions.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/definitions.ma deleted file mode 100644 index 5863b4243..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/definitions.ma +++ /dev/null @@ -1,68 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/tlt/defs.ma". - -include "Basic-1/iso/defs.ma". - -include "Basic-1/clen/defs.ma". - -include "Basic-1/flt/defs.ma". - -include "Basic-1/app/defs.ma". - -include "Basic-1/cnt/defs.ma". - -include "Basic-1/cimp/defs.ma". - -include "Basic-1/csubv/defs.ma". - -include "Basic-1/subst/defs.ma". - -include "Basic-1/subst1/defs.ma". - -include "Basic-1/csubst1/defs.ma". - -include "Basic-1/fsubst0/defs.ma". - -include "Basic-1/next_plus/defs.ma". - -include "Basic-1/sty1/defs.ma". - -include "Basic-1/llt/defs.ma". - -include "Basic-1/aprem/defs.ma". - -include "Basic-1/ex0/defs.ma". - -include "Basic-1/wcpr0/defs.ma". - -include "Basic-1/csuba/defs.ma". - -include "Basic-1/nf2/defs.ma". - -include "Basic-1/ex2/defs.ma". - -include "Basic-1/csubc/defs.ma". - -include "Basic-1/pc1/defs.ma". - -include "Basic-1/ex1/defs.ma". - -include "Basic-1/csubt/defs.ma". - -include "Basic-1/wf3/defs.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/drop/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/drop/defs.ma deleted file mode 100644 index 987ae0ceb..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/drop/defs.ma +++ /dev/null @@ -1,31 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/C/defs.ma". - -include "Basic-1/lift/defs.ma". - -include "Basic-1/r/defs.ma". - -inductive drop: nat \to (nat \to (C \to (C \to Prop))) \def -| drop_refl: \forall (c: C).(drop O O c c) -| drop_drop: \forall (k: K).(\forall (h: nat).(\forall (c: C).(\forall (e: -C).((drop (r k h) O c e) \to (\forall (u: T).(drop (S h) O (CHead c k u) -e)))))) -| drop_skip: \forall (k: K).(\forall (h: nat).(\forall (d: nat).(\forall (c: -C).(\forall (e: C).((drop h (r k d) c e) \to (\forall (u: T).(drop h (S d) -(CHead c k (lift h (r k d) u)) (CHead e k u)))))))). - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/drop/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/drop/fwd.ma deleted file mode 100644 index 48495c148..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/drop/fwd.ma +++ /dev/null @@ -1,384 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/drop/defs.ma". - -theorem drop_gen_sort: - \forall (n: nat).(\forall (h: nat).(\forall (d: nat).(\forall (x: C).((drop -h d (CSort n) x) \to (and3 (eq C x (CSort n)) (eq nat h O) (eq nat d O)))))) -\def - \lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (x: -C).(\lambda (H: (drop h d (CSort n) x)).(insert_eq C (CSort n) (\lambda (c: -C).(drop h d c x)) (\lambda (c: C).(and3 (eq C x c) (eq nat h O) (eq nat d -O))) (\lambda (y: C).(\lambda (H0: (drop h d y x)).(drop_ind (\lambda (n0: -nat).(\lambda (n1: nat).(\lambda (c: C).(\lambda (c0: C).((eq C c (CSort n)) -\to (and3 (eq C c0 c) (eq nat n0 O) (eq nat n1 O))))))) (\lambda (c: -C).(\lambda (H1: (eq C c (CSort n))).(let H2 \def (f_equal C C (\lambda (e: -C).e) c (CSort n) H1) in (eq_ind_r C (CSort n) (\lambda (c0: C).(and3 (eq C -c0 c0) (eq nat O O) (eq nat O O))) (and3_intro (eq C (CSort n) (CSort n)) (eq -nat O O) (eq nat O O) (refl_equal C (CSort n)) (refl_equal nat O) (refl_equal -nat O)) c H2)))) (\lambda (k: K).(\lambda (h0: nat).(\lambda (c: C).(\lambda -(e: C).(\lambda (_: (drop (r k h0) O c e)).(\lambda (_: (((eq C c (CSort n)) -\to (and3 (eq C e c) (eq nat (r k h0) O) (eq nat O O))))).(\lambda (u: -T).(\lambda (H3: (eq C (CHead c k u) (CSort n))).(let H4 \def (eq_ind C -(CHead c k u) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) -with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I -(CSort n) H3) in (False_ind (and3 (eq C e (CHead c k u)) (eq nat (S h0) O) -(eq nat O O)) H4)))))))))) (\lambda (k: K).(\lambda (h0: nat).(\lambda (d0: -nat).(\lambda (c: C).(\lambda (e: C).(\lambda (_: (drop h0 (r k d0) c -e)).(\lambda (_: (((eq C c (CSort n)) \to (and3 (eq C e c) (eq nat h0 O) (eq -nat (r k d0) O))))).(\lambda (u: T).(\lambda (H3: (eq C (CHead c k (lift h0 -(r k d0) u)) (CSort n))).(let H4 \def (eq_ind C (CHead c k (lift h0 (r k d0) -u)) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort -_) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H3) in -(False_ind (and3 (eq C (CHead e k u) (CHead c k (lift h0 (r k d0) u))) (eq -nat h0 O) (eq nat (S d0) O)) H4))))))))))) h d y x H0))) H))))). -(* COMMENTS -Initial nodes: 595 -END *) - -theorem drop_gen_refl: - \forall (x: C).(\forall (e: C).((drop O O x e) \to (eq C x e))) -\def - \lambda (x: C).(\lambda (e: C).(\lambda (H: (drop O O x e)).(insert_eq nat O -(\lambda (n: nat).(drop n O x e)) (\lambda (_: nat).(eq C x e)) (\lambda (y: -nat).(\lambda (H0: (drop y O x e)).(insert_eq nat O (\lambda (n: nat).(drop y -n x e)) (\lambda (n: nat).((eq nat y n) \to (eq C x e))) (\lambda (y0: -nat).(\lambda (H1: (drop y y0 x e)).(drop_ind (\lambda (n: nat).(\lambda (n0: -nat).(\lambda (c: C).(\lambda (c0: C).((eq nat n0 O) \to ((eq nat n n0) \to -(eq C c c0))))))) (\lambda (c: C).(\lambda (_: (eq nat O O)).(\lambda (_: (eq -nat O O)).(refl_equal C c)))) (\lambda (k: K).(\lambda (h: nat).(\lambda (c: -C).(\lambda (e0: C).(\lambda (_: (drop (r k h) O c e0)).(\lambda (_: (((eq -nat O O) \to ((eq nat (r k h) O) \to (eq C c e0))))).(\lambda (u: T).(\lambda -(_: (eq nat O O)).(\lambda (H5: (eq nat (S h) O)).(let H6 \def (eq_ind nat (S -h) (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O -\Rightarrow False | (S _) \Rightarrow True])) I O H5) in (False_ind (eq C -(CHead c k u) e0) H6))))))))))) (\lambda (k: K).(\lambda (h: nat).(\lambda -(d: nat).(\lambda (c: C).(\lambda (e0: C).(\lambda (H2: (drop h (r k d) c -e0)).(\lambda (H3: (((eq nat (r k d) O) \to ((eq nat h (r k d)) \to (eq C c -e0))))).(\lambda (u: T).(\lambda (H4: (eq nat (S d) O)).(\lambda (H5: (eq nat -h (S d))).(let H6 \def (f_equal nat nat (\lambda (e1: nat).e1) h (S d) H5) in -(let H7 \def (eq_ind nat h (\lambda (n: nat).((eq nat (r k d) O) \to ((eq nat -n (r k d)) \to (eq C c e0)))) H3 (S d) H6) in (let H8 \def (eq_ind nat h -(\lambda (n: nat).(drop n (r k d) c e0)) H2 (S d) H6) in (eq_ind_r nat (S d) -(\lambda (n: nat).(eq C (CHead c k (lift n (r k d) u)) (CHead e0 k u))) (let -H9 \def (eq_ind nat (S d) (\lambda (ee: nat).(match ee in nat return (\lambda -(_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H4) -in (False_ind (eq C (CHead c k (lift (S d) (r k d) u)) (CHead e0 k u)) H9)) h -H6)))))))))))))) y y0 x e H1))) H0))) H))). -(* COMMENTS -Initial nodes: 561 -END *) - -theorem drop_gen_drop: - \forall (k: K).(\forall (c: C).(\forall (x: C).(\forall (u: T).(\forall (h: -nat).((drop (S h) O (CHead c k u) x) \to (drop (r k h) O c x)))))) -\def - \lambda (k: K).(\lambda (c: C).(\lambda (x: C).(\lambda (u: T).(\lambda (h: -nat).(\lambda (H: (drop (S h) O (CHead c k u) x)).(insert_eq C (CHead c k u) -(\lambda (c0: C).(drop (S h) O c0 x)) (\lambda (_: C).(drop (r k h) O c x)) -(\lambda (y: C).(\lambda (H0: (drop (S h) O y x)).(insert_eq nat O (\lambda -(n: nat).(drop (S h) n y x)) (\lambda (n: nat).((eq C y (CHead c k u)) \to -(drop (r k h) n c x))) (\lambda (y0: nat).(\lambda (H1: (drop (S h) y0 y -x)).(insert_eq nat (S h) (\lambda (n: nat).(drop n y0 y x)) (\lambda (_: -nat).((eq nat y0 O) \to ((eq C y (CHead c k u)) \to (drop (r k h) y0 c x)))) -(\lambda (y1: nat).(\lambda (H2: (drop y1 y0 y x)).(drop_ind (\lambda (n: -nat).(\lambda (n0: nat).(\lambda (c0: C).(\lambda (c1: C).((eq nat n (S h)) -\to ((eq nat n0 O) \to ((eq C c0 (CHead c k u)) \to (drop (r k h) n0 c -c1)))))))) (\lambda (c0: C).(\lambda (H3: (eq nat O (S h))).(\lambda (_: (eq -nat O O)).(\lambda (H5: (eq C c0 (CHead c k u))).(eq_ind_r C (CHead c k u) -(\lambda (c1: C).(drop (r k h) O c c1)) (let H6 \def (eq_ind nat O (\lambda -(ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow -True | (S _) \Rightarrow False])) I (S h) H3) in (False_ind (drop (r k h) O c -(CHead c k u)) H6)) c0 H5))))) (\lambda (k0: K).(\lambda (h0: nat).(\lambda -(c0: C).(\lambda (e: C).(\lambda (H3: (drop (r k0 h0) O c0 e)).(\lambda (H4: -(((eq nat (r k0 h0) (S h)) \to ((eq nat O O) \to ((eq C c0 (CHead c k u)) \to -(drop (r k h) O c e)))))).(\lambda (u0: T).(\lambda (H5: (eq nat (S h0) (S -h))).(\lambda (_: (eq nat O O)).(\lambda (H7: (eq C (CHead c0 k0 u0) (CHead c -k u))).(let H8 \def (f_equal C C (\lambda (e0: C).(match e0 in C return -(\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c1 _ _) -\Rightarrow c1])) (CHead c0 k0 u0) (CHead c k u) H7) in ((let H9 \def -(f_equal C K (\lambda (e0: C).(match e0 in C return (\lambda (_: C).K) with -[(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead c0 k0 u0) -(CHead c k u) H7) in ((let H10 \def (f_equal C T (\lambda (e0: C).(match e0 -in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) -\Rightarrow t])) (CHead c0 k0 u0) (CHead c k u) H7) in (\lambda (H11: (eq K -k0 k)).(\lambda (H12: (eq C c0 c)).(let H13 \def (eq_ind C c0 (\lambda (c1: -C).((eq nat (r k0 h0) (S h)) \to ((eq nat O O) \to ((eq C c1 (CHead c k u)) -\to (drop (r k h) O c e))))) H4 c H12) in (let H14 \def (eq_ind C c0 (\lambda -(c1: C).(drop (r k0 h0) O c1 e)) H3 c H12) in (let H15 \def (eq_ind K k0 -(\lambda (k1: K).((eq nat (r k1 h0) (S h)) \to ((eq nat O O) \to ((eq C c -(CHead c k u)) \to (drop (r k h) O c e))))) H13 k H11) in (let H16 \def -(eq_ind K k0 (\lambda (k1: K).(drop (r k1 h0) O c e)) H14 k H11) in (let H17 -\def (f_equal nat nat (\lambda (e0: nat).(match e0 in nat return (\lambda (_: -nat).nat) with [O \Rightarrow h0 | (S n) \Rightarrow n])) (S h0) (S h) H5) in -(let H18 \def (eq_ind nat h0 (\lambda (n: nat).((eq nat (r k n) (S h)) \to -((eq nat O O) \to ((eq C c (CHead c k u)) \to (drop (r k h) O c e))))) H15 h -H17) in (let H19 \def (eq_ind nat h0 (\lambda (n: nat).(drop (r k n) O c e)) -H16 h H17) in H19)))))))))) H9)) H8)))))))))))) (\lambda (k0: K).(\lambda -(h0: nat).(\lambda (d: nat).(\lambda (c0: C).(\lambda (e: C).(\lambda (H3: -(drop h0 (r k0 d) c0 e)).(\lambda (H4: (((eq nat h0 (S h)) \to ((eq nat (r k0 -d) O) \to ((eq C c0 (CHead c k u)) \to (drop (r k h) (r k0 d) c -e)))))).(\lambda (u0: T).(\lambda (H5: (eq nat h0 (S h))).(\lambda (H6: (eq -nat (S d) O)).(\lambda (H7: (eq C (CHead c0 k0 (lift h0 (r k0 d) u0)) (CHead -c k u))).(let H8 \def (eq_ind nat h0 (\lambda (n: nat).(eq C (CHead c0 k0 -(lift n (r k0 d) u0)) (CHead c k u))) H7 (S h) H5) in (let H9 \def (eq_ind -nat h0 (\lambda (n: nat).((eq nat n (S h)) \to ((eq nat (r k0 d) O) \to ((eq -C c0 (CHead c k u)) \to (drop (r k h) (r k0 d) c e))))) H4 (S h) H5) in (let -H10 \def (eq_ind nat h0 (\lambda (n: nat).(drop n (r k0 d) c0 e)) H3 (S h) -H5) in (let H11 \def (f_equal C C (\lambda (e0: C).(match e0 in C return -(\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c1 _ _) -\Rightarrow c1])) (CHead c0 k0 (lift (S h) (r k0 d) u0)) (CHead c k u) H8) in -((let H12 \def (f_equal C K (\lambda (e0: C).(match e0 in C return (\lambda -(_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) -(CHead c0 k0 (lift (S h) (r k0 d) u0)) (CHead c k u) H8) in ((let H13 \def -(f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d0: nat) (t: -T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) -\Rightarrow (TLRef (match (blt i d0) with [true \Rightarrow i | false -\Rightarrow (f i)])) | (THead k1 u1 t0) \Rightarrow (THead k1 (lref_map f d0 -u1) (lref_map f (s k1 d0) t0))]) in lref_map) (\lambda (x0: nat).(plus x0 (S -h))) (r k0 d) u0) | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 (lift (S h) -(r k0 d) u0)) (CHead c k u) H8) in (\lambda (H14: (eq K k0 k)).(\lambda (H15: -(eq C c0 c)).(let H16 \def (eq_ind C c0 (\lambda (c1: C).((eq nat (S h) (S -h)) \to ((eq nat (r k0 d) O) \to ((eq C c1 (CHead c k u)) \to (drop (r k h) -(r k0 d) c e))))) H9 c H15) in (let H17 \def (eq_ind C c0 (\lambda (c1: -C).(drop (S h) (r k0 d) c1 e)) H10 c H15) in (let H18 \def (eq_ind K k0 -(\lambda (k1: K).(eq T (lift (S h) (r k1 d) u0) u)) H13 k H14) in (let H19 -\def (eq_ind K k0 (\lambda (k1: K).((eq nat (S h) (S h)) \to ((eq nat (r k1 -d) O) \to ((eq C c (CHead c k u)) \to (drop (r k h) (r k1 d) c e))))) H16 k -H14) in (let H20 \def (eq_ind K k0 (\lambda (k1: K).(drop (S h) (r k1 d) c -e)) H17 k H14) in (eq_ind_r K k (\lambda (k1: K).(drop (r k h) (S d) c (CHead -e k1 u0))) (let H21 \def (eq_ind_r T u (\lambda (t: T).((eq nat (S h) (S h)) -\to ((eq nat (r k d) O) \to ((eq C c (CHead c k t)) \to (drop (r k h) (r k d) -c e))))) H19 (lift (S h) (r k d) u0) H18) in (let H22 \def (eq_ind nat (S d) -(\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O -\Rightarrow False | (S _) \Rightarrow True])) I O H6) in (False_ind (drop (r -k h) (S d) c (CHead e k u0)) H22))) k0 H14))))))))) H12)) H11)))))))))))))))) -y1 y0 y x H2))) H1))) H0))) H)))))). -(* COMMENTS -Initial nodes: 1856 -END *) - -theorem drop_gen_skip_r: - \forall (c: C).(\forall (x: C).(\forall (u: T).(\forall (h: nat).(\forall -(d: nat).(\forall (k: K).((drop h (S d) x (CHead c k u)) \to (ex2 C (\lambda -(e: C).(eq C x (CHead e k (lift h (r k d) u)))) (\lambda (e: C).(drop h (r k -d) e c))))))))) -\def - \lambda (c: C).(\lambda (x: C).(\lambda (u: T).(\lambda (h: nat).(\lambda -(d: nat).(\lambda (k: K).(\lambda (H: (drop h (S d) x (CHead c k -u))).(insert_eq C (CHead c k u) (\lambda (c0: C).(drop h (S d) x c0)) -(\lambda (_: C).(ex2 C (\lambda (e: C).(eq C x (CHead e k (lift h (r k d) -u)))) (\lambda (e: C).(drop h (r k d) e c)))) (\lambda (y: C).(\lambda (H0: -(drop h (S d) x y)).(insert_eq nat (S d) (\lambda (n: nat).(drop h n x y)) -(\lambda (_: nat).((eq C y (CHead c k u)) \to (ex2 C (\lambda (e: C).(eq C x -(CHead e k (lift h (r k d) u)))) (\lambda (e: C).(drop h (r k d) e c))))) -(\lambda (y0: nat).(\lambda (H1: (drop h y0 x y)).(drop_ind (\lambda (n: -nat).(\lambda (n0: nat).(\lambda (c0: C).(\lambda (c1: C).((eq nat n0 (S d)) -\to ((eq C c1 (CHead c k u)) \to (ex2 C (\lambda (e: C).(eq C c0 (CHead e k -(lift n (r k d) u)))) (\lambda (e: C).(drop n (r k d) e c))))))))) (\lambda -(c0: C).(\lambda (H2: (eq nat O (S d))).(\lambda (H3: (eq C c0 (CHead c k -u))).(eq_ind_r C (CHead c k u) (\lambda (c1: C).(ex2 C (\lambda (e: C).(eq C -c1 (CHead e k (lift O (r k d) u)))) (\lambda (e: C).(drop O (r k d) e c)))) -(let H4 \def (eq_ind nat O (\lambda (ee: nat).(match ee in nat return -(\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) -I (S d) H2) in (False_ind (ex2 C (\lambda (e: C).(eq C (CHead c k u) (CHead e -k (lift O (r k d) u)))) (\lambda (e: C).(drop O (r k d) e c))) H4)) c0 H3)))) -(\lambda (k0: K).(\lambda (h0: nat).(\lambda (c0: C).(\lambda (e: C).(\lambda -(H2: (drop (r k0 h0) O c0 e)).(\lambda (H3: (((eq nat O (S d)) \to ((eq C e -(CHead c k u)) \to (ex2 C (\lambda (e0: C).(eq C c0 (CHead e0 k (lift (r k0 -h0) (r k d) u)))) (\lambda (e0: C).(drop (r k0 h0) (r k d) e0 -c))))))).(\lambda (u0: T).(\lambda (H4: (eq nat O (S d))).(\lambda (H5: (eq C -e (CHead c k u))).(let H6 \def (eq_ind C e (\lambda (c1: C).((eq nat O (S d)) -\to ((eq C c1 (CHead c k u)) \to (ex2 C (\lambda (e0: C).(eq C c0 (CHead e0 k -(lift (r k0 h0) (r k d) u)))) (\lambda (e0: C).(drop (r k0 h0) (r k d) e0 -c)))))) H3 (CHead c k u) H5) in (let H7 \def (eq_ind C e (\lambda (c1: -C).(drop (r k0 h0) O c0 c1)) H2 (CHead c k u) H5) in (let H8 \def (eq_ind nat -O (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O -\Rightarrow True | (S _) \Rightarrow False])) I (S d) H4) in (False_ind (ex2 -C (\lambda (e0: C).(eq C (CHead c0 k0 u0) (CHead e0 k (lift (S h0) (r k d) -u)))) (\lambda (e0: C).(drop (S h0) (r k d) e0 c))) H8))))))))))))) (\lambda -(k0: K).(\lambda (h0: nat).(\lambda (d0: nat).(\lambda (c0: C).(\lambda (e: -C).(\lambda (H2: (drop h0 (r k0 d0) c0 e)).(\lambda (H3: (((eq nat (r k0 d0) -(S d)) \to ((eq C e (CHead c k u)) \to (ex2 C (\lambda (e0: C).(eq C c0 -(CHead e0 k (lift h0 (r k d) u)))) (\lambda (e0: C).(drop h0 (r k d) e0 -c))))))).(\lambda (u0: T).(\lambda (H4: (eq nat (S d0) (S d))).(\lambda (H5: -(eq C (CHead e k0 u0) (CHead c k u))).(let H6 \def (f_equal C C (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow e | -(CHead c1 _ _) \Rightarrow c1])) (CHead e k0 u0) (CHead c k u) H5) in ((let -H7 \def (f_equal C K (\lambda (e0: C).(match e0 in C return (\lambda (_: -C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) -(CHead e k0 u0) (CHead c k u) H5) in ((let H8 \def (f_equal C T (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | -(CHead _ _ t) \Rightarrow t])) (CHead e k0 u0) (CHead c k u) H5) in (\lambda -(H9: (eq K k0 k)).(\lambda (H10: (eq C e c)).(eq_ind_r T u (\lambda (t: -T).(ex2 C (\lambda (e0: C).(eq C (CHead c0 k0 (lift h0 (r k0 d0) t)) (CHead -e0 k (lift h0 (r k d) u)))) (\lambda (e0: C).(drop h0 (r k d) e0 c)))) (let -H11 \def (eq_ind C e (\lambda (c1: C).((eq nat (r k0 d0) (S d)) \to ((eq C c1 -(CHead c k u)) \to (ex2 C (\lambda (e0: C).(eq C c0 (CHead e0 k (lift h0 (r k -d) u)))) (\lambda (e0: C).(drop h0 (r k d) e0 c)))))) H3 c H10) in (let H12 -\def (eq_ind C e (\lambda (c1: C).(drop h0 (r k0 d0) c0 c1)) H2 c H10) in -(let H13 \def (eq_ind K k0 (\lambda (k1: K).((eq nat (r k1 d0) (S d)) \to -((eq C c (CHead c k u)) \to (ex2 C (\lambda (e0: C).(eq C c0 (CHead e0 k -(lift h0 (r k d) u)))) (\lambda (e0: C).(drop h0 (r k d) e0 c)))))) H11 k H9) -in (let H14 \def (eq_ind K k0 (\lambda (k1: K).(drop h0 (r k1 d0) c0 c)) H12 -k H9) in (eq_ind_r K k (\lambda (k1: K).(ex2 C (\lambda (e0: C).(eq C (CHead -c0 k1 (lift h0 (r k1 d0) u)) (CHead e0 k (lift h0 (r k d) u)))) (\lambda (e0: -C).(drop h0 (r k d) e0 c)))) (let H15 \def (f_equal nat nat (\lambda (e0: -nat).(match e0 in nat return (\lambda (_: nat).nat) with [O \Rightarrow d0 | -(S n) \Rightarrow n])) (S d0) (S d) H4) in (let H16 \def (eq_ind nat d0 -(\lambda (n: nat).((eq nat (r k n) (S d)) \to ((eq C c (CHead c k u)) \to -(ex2 C (\lambda (e0: C).(eq C c0 (CHead e0 k (lift h0 (r k d) u)))) (\lambda -(e0: C).(drop h0 (r k d) e0 c)))))) H13 d H15) in (let H17 \def (eq_ind nat -d0 (\lambda (n: nat).(drop h0 (r k n) c0 c)) H14 d H15) in (eq_ind_r nat d -(\lambda (n: nat).(ex2 C (\lambda (e0: C).(eq C (CHead c0 k (lift h0 (r k n) -u)) (CHead e0 k (lift h0 (r k d) u)))) (\lambda (e0: C).(drop h0 (r k d) e0 -c)))) (ex_intro2 C (\lambda (e0: C).(eq C (CHead c0 k (lift h0 (r k d) u)) -(CHead e0 k (lift h0 (r k d) u)))) (\lambda (e0: C).(drop h0 (r k d) e0 c)) -c0 (refl_equal C (CHead c0 k (lift h0 (r k d) u))) H17) d0 H15)))) k0 H9))))) -u0 H8)))) H7)) H6)))))))))))) h y0 x y H1))) H0))) H))))))). -(* COMMENTS -Initial nodes: 1758 -END *) - -theorem drop_gen_skip_l: - \forall (c: C).(\forall (x: C).(\forall (u: T).(\forall (h: nat).(\forall -(d: nat).(\forall (k: K).((drop h (S d) (CHead c k u) x) \to (ex3_2 C T -(\lambda (e: C).(\lambda (v: T).(eq C x (CHead e k v)))) (\lambda (_: -C).(\lambda (v: T).(eq T u (lift h (r k d) v)))) (\lambda (e: C).(\lambda (_: -T).(drop h (r k d) c e)))))))))) -\def - \lambda (c: C).(\lambda (x: C).(\lambda (u: T).(\lambda (h: nat).(\lambda -(d: nat).(\lambda (k: K).(\lambda (H: (drop h (S d) (CHead c k u) -x)).(insert_eq C (CHead c k u) (\lambda (c0: C).(drop h (S d) c0 x)) (\lambda -(_: C).(ex3_2 C T (\lambda (e: C).(\lambda (v: T).(eq C x (CHead e k v)))) -(\lambda (_: C).(\lambda (v: T).(eq T u (lift h (r k d) v)))) (\lambda (e: -C).(\lambda (_: T).(drop h (r k d) c e))))) (\lambda (y: C).(\lambda (H0: -(drop h (S d) y x)).(insert_eq nat (S d) (\lambda (n: nat).(drop h n y x)) -(\lambda (_: nat).((eq C y (CHead c k u)) \to (ex3_2 C T (\lambda (e: -C).(\lambda (v: T).(eq C x (CHead e k v)))) (\lambda (_: C).(\lambda (v: -T).(eq T u (lift h (r k d) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r k -d) c e)))))) (\lambda (y0: nat).(\lambda (H1: (drop h y0 y x)).(drop_ind -(\lambda (n: nat).(\lambda (n0: nat).(\lambda (c0: C).(\lambda (c1: C).((eq -nat n0 (S d)) \to ((eq C c0 (CHead c k u)) \to (ex3_2 C T (\lambda (e: -C).(\lambda (v: T).(eq C c1 (CHead e k v)))) (\lambda (_: C).(\lambda (v: -T).(eq T u (lift n (r k d) v)))) (\lambda (e: C).(\lambda (_: T).(drop n (r k -d) c e)))))))))) (\lambda (c0: C).(\lambda (H2: (eq nat O (S d))).(\lambda -(H3: (eq C c0 (CHead c k u))).(eq_ind_r C (CHead c k u) (\lambda (c1: -C).(ex3_2 C T (\lambda (e: C).(\lambda (v: T).(eq C c1 (CHead e k v)))) -(\lambda (_: C).(\lambda (v: T).(eq T u (lift O (r k d) v)))) (\lambda (e: -C).(\lambda (_: T).(drop O (r k d) c e))))) (let H4 \def (eq_ind nat O -(\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O -\Rightarrow True | (S _) \Rightarrow False])) I (S d) H2) in (False_ind -(ex3_2 C T (\lambda (e: C).(\lambda (v: T).(eq C (CHead c k u) (CHead e k -v)))) (\lambda (_: C).(\lambda (v: T).(eq T u (lift O (r k d) v)))) (\lambda -(e: C).(\lambda (_: T).(drop O (r k d) c e)))) H4)) c0 H3)))) (\lambda (k0: -K).(\lambda (h0: nat).(\lambda (c0: C).(\lambda (e: C).(\lambda (H2: (drop (r -k0 h0) O c0 e)).(\lambda (H3: (((eq nat O (S d)) \to ((eq C c0 (CHead c k u)) -\to (ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 k v)))) -(\lambda (_: C).(\lambda (v: T).(eq T u (lift (r k0 h0) (r k d) v)))) -(\lambda (e0: C).(\lambda (_: T).(drop (r k0 h0) (r k d) c -e0)))))))).(\lambda (u0: T).(\lambda (H4: (eq nat O (S d))).(\lambda (H5: (eq -C (CHead c0 k0 u0) (CHead c k u))).(let H6 \def (f_equal C C (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | -(CHead c1 _ _) \Rightarrow c1])) (CHead c0 k0 u0) (CHead c k u) H5) in ((let -H7 \def (f_equal C K (\lambda (e0: C).(match e0 in C return (\lambda (_: -C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) -(CHead c0 k0 u0) (CHead c k u) H5) in ((let H8 \def (f_equal C T (\lambda -(e0: C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow -u0 | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 u0) (CHead c k u) H5) in -(\lambda (H9: (eq K k0 k)).(\lambda (H10: (eq C c0 c)).(let H11 \def (eq_ind -C c0 (\lambda (c1: C).((eq nat O (S d)) \to ((eq C c1 (CHead c k u)) \to -(ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 k v)))) -(\lambda (_: C).(\lambda (v: T).(eq T u (lift (r k0 h0) (r k d) v)))) -(\lambda (e0: C).(\lambda (_: T).(drop (r k0 h0) (r k d) c e0))))))) H3 c -H10) in (let H12 \def (eq_ind C c0 (\lambda (c1: C).(drop (r k0 h0) O c1 e)) -H2 c H10) in (let H13 \def (eq_ind K k0 (\lambda (k1: K).((eq nat O (S d)) -\to ((eq C c (CHead c k u)) \to (ex3_2 C T (\lambda (e0: C).(\lambda (v: -T).(eq C e (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T u (lift (r -k1 h0) (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop (r k1 h0) (r k d) -c e0))))))) H11 k H9) in (let H14 \def (eq_ind K k0 (\lambda (k1: K).(drop (r -k1 h0) O c e)) H12 k H9) in (let H15 \def (eq_ind nat O (\lambda (ee: -nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True -| (S _) \Rightarrow False])) I (S d) H4) in (False_ind (ex3_2 C T (\lambda -(e0: C).(\lambda (v: T).(eq C e (CHead e0 k v)))) (\lambda (_: C).(\lambda -(v: T).(eq T u (lift (S h0) (r k d) v)))) (\lambda (e0: C).(\lambda (_: -T).(drop (S h0) (r k d) c e0)))) H15))))))))) H7)) H6))))))))))) (\lambda -(k0: K).(\lambda (h0: nat).(\lambda (d0: nat).(\lambda (c0: C).(\lambda (e: -C).(\lambda (H2: (drop h0 (r k0 d0) c0 e)).(\lambda (H3: (((eq nat (r k0 d0) -(S d)) \to ((eq C c0 (CHead c k u)) \to (ex3_2 C T (\lambda (e0: C).(\lambda -(v: T).(eq C e (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T u -(lift h0 (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h0 (r k d) c -e0)))))))).(\lambda (u0: T).(\lambda (H4: (eq nat (S d0) (S d))).(\lambda -(H5: (eq C (CHead c0 k0 (lift h0 (r k0 d0) u0)) (CHead c k u))).(let H6 \def -(f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with -[(CSort _) \Rightarrow c0 | (CHead c1 _ _) \Rightarrow c1])) (CHead c0 k0 -(lift h0 (r k0 d0) u0)) (CHead c k u) H5) in ((let H7 \def (f_equal C K -(\lambda (e0: C).(match e0 in C return (\lambda (_: C).K) with [(CSort _) -\Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead c0 k0 (lift h0 (r k0 -d0) u0)) (CHead c k u) H5) in ((let H8 \def (f_equal C T (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow ((let -rec lref_map (f: ((nat \to nat))) (d1: nat) (t: T) on t: T \def (match t with -[(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i -d1) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k1 u1 t0) -\Rightarrow (THead k1 (lref_map f d1 u1) (lref_map f (s k1 d1) t0))]) in -lref_map) (\lambda (x0: nat).(plus x0 h0)) (r k0 d0) u0) | (CHead _ _ t) -\Rightarrow t])) (CHead c0 k0 (lift h0 (r k0 d0) u0)) (CHead c k u) H5) in -(\lambda (H9: (eq K k0 k)).(\lambda (H10: (eq C c0 c)).(let H11 \def (eq_ind -C c0 (\lambda (c1: C).((eq nat (r k0 d0) (S d)) \to ((eq C c1 (CHead c k u)) -\to (ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 k v)))) -(\lambda (_: C).(\lambda (v: T).(eq T u (lift h0 (r k d) v)))) (\lambda (e0: -C).(\lambda (_: T).(drop h0 (r k d) c e0))))))) H3 c H10) in (let H12 \def -(eq_ind C c0 (\lambda (c1: C).(drop h0 (r k0 d0) c1 e)) H2 c H10) in (let H13 -\def (eq_ind K k0 (\lambda (k1: K).(eq T (lift h0 (r k1 d0) u0) u)) H8 k H9) -in (let H14 \def (eq_ind K k0 (\lambda (k1: K).((eq nat (r k1 d0) (S d)) \to -((eq C c (CHead c k u)) \to (ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C -e (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T u (lift h0 (r k d) -v)))) (\lambda (e0: C).(\lambda (_: T).(drop h0 (r k d) c e0))))))) H11 k H9) -in (let H15 \def (eq_ind K k0 (\lambda (k1: K).(drop h0 (r k1 d0) c e)) H12 k -H9) in (eq_ind_r K k (\lambda (k1: K).(ex3_2 C T (\lambda (e0: C).(\lambda -(v: T).(eq C (CHead e k1 u0) (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: -T).(eq T u (lift h0 (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h0 -(r k d) c e0))))) (let H16 \def (eq_ind_r T u (\lambda (t: T).((eq nat (r k -d0) (S d)) \to ((eq C c (CHead c k t)) \to (ex3_2 C T (\lambda (e0: -C).(\lambda (v: T).(eq C e (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: -T).(eq T t (lift h0 (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h0 -(r k d) c e0))))))) H14 (lift h0 (r k d0) u0) H13) in (eq_ind T (lift h0 (r k -d0) u0) (\lambda (t: T).(ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C -(CHead e k u0) (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T t -(lift h0 (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h0 (r k d) c -e0))))) (let H17 \def (f_equal nat nat (\lambda (e0: nat).(match e0 in nat -return (\lambda (_: nat).nat) with [O \Rightarrow d0 | (S n) \Rightarrow n])) -(S d0) (S d) H4) in (let H18 \def (eq_ind nat d0 (\lambda (n: nat).((eq nat -(r k n) (S d)) \to ((eq C c (CHead c k (lift h0 (r k n) u0))) \to (ex3_2 C T -(\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 k v)))) (\lambda (_: -C).(\lambda (v: T).(eq T (lift h0 (r k n) u0) (lift h0 (r k d) v)))) (\lambda -(e0: C).(\lambda (_: T).(drop h0 (r k d) c e0))))))) H16 d H17) in (let H19 -\def (eq_ind nat d0 (\lambda (n: nat).(drop h0 (r k n) c e)) H15 d H17) in -(eq_ind_r nat d (\lambda (n: nat).(ex3_2 C T (\lambda (e0: C).(\lambda (v: -T).(eq C (CHead e k u0) (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq -T (lift h0 (r k n) u0) (lift h0 (r k d) v)))) (\lambda (e0: C).(\lambda (_: -T).(drop h0 (r k d) c e0))))) (ex3_2_intro C T (\lambda (e0: C).(\lambda (v: -T).(eq C (CHead e k u0) (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq -T (lift h0 (r k d) u0) (lift h0 (r k d) v)))) (\lambda (e0: C).(\lambda (_: -T).(drop h0 (r k d) c e0))) e u0 (refl_equal C (CHead e k u0)) (refl_equal T -(lift h0 (r k d) u0)) H19) d0 H17)))) u H13)) k0 H9))))))))) H7)) -H6)))))))))))) h y0 y x H1))) H0))) H))))))). -(* COMMENTS -Initial nodes: 2574 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/drop/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/drop/props.ma deleted file mode 100644 index ac802b105..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/drop/props.ma +++ /dev/null @@ -1,737 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/drop/fwd.ma". - -include "Basic-1/lift/props.ma". - -include "Basic-1/r/props.ma". - -theorem drop_skip_bind: - \forall (h: nat).(\forall (d: nat).(\forall (c: C).(\forall (e: C).((drop h -d c e) \to (\forall (b: B).(\forall (u: T).(drop h (S d) (CHead c (Bind b) -(lift h d u)) (CHead e (Bind b) u)))))))) -\def - \lambda (h: nat).(\lambda (d: nat).(\lambda (c: C).(\lambda (e: C).(\lambda -(H: (drop h d c e)).(\lambda (b: B).(\lambda (u: T).(eq_ind nat (r (Bind b) -d) (\lambda (n: nat).(drop h (S d) (CHead c (Bind b) (lift h n u)) (CHead e -(Bind b) u))) (drop_skip (Bind b) h d c e H u) d (refl_equal nat d)))))))). -(* COMMENTS -Initial nodes: 95 -END *) - -theorem drop_skip_flat: - \forall (h: nat).(\forall (d: nat).(\forall (c: C).(\forall (e: C).((drop h -(S d) c e) \to (\forall (f: F).(\forall (u: T).(drop h (S d) (CHead c (Flat -f) (lift h (S d) u)) (CHead e (Flat f) u)))))))) -\def - \lambda (h: nat).(\lambda (d: nat).(\lambda (c: C).(\lambda (e: C).(\lambda -(H: (drop h (S d) c e)).(\lambda (f: F).(\lambda (u: T).(eq_ind nat (r (Flat -f) d) (\lambda (n: nat).(drop h (S d) (CHead c (Flat f) (lift h n u)) (CHead -e (Flat f) u))) (drop_skip (Flat f) h d c e H u) (S d) (refl_equal nat (S -d))))))))). -(* COMMENTS -Initial nodes: 101 -END *) - -theorem drop_S: - \forall (b: B).(\forall (c: C).(\forall (e: C).(\forall (u: T).(\forall (h: -nat).((drop h O c (CHead e (Bind b) u)) \to (drop (S h) O c e)))))) -\def - \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e: -C).(\forall (u: T).(\forall (h: nat).((drop h O c0 (CHead e (Bind b) u)) \to -(drop (S h) O c0 e)))))) (\lambda (n: nat).(\lambda (e: C).(\lambda (u: -T).(\lambda (h: nat).(\lambda (H: (drop h O (CSort n) (CHead e (Bind b) -u))).(and3_ind (eq C (CHead e (Bind b) u) (CSort n)) (eq nat h O) (eq nat O -O) (drop (S h) O (CSort n) e) (\lambda (H0: (eq C (CHead e (Bind b) u) (CSort -n))).(\lambda (H1: (eq nat h O)).(\lambda (_: (eq nat O O)).(eq_ind_r nat O -(\lambda (n0: nat).(drop (S n0) O (CSort n) e)) (let H3 \def (eq_ind C (CHead -e (Bind b) u) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) -with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I -(CSort n) H0) in (False_ind (drop (S O) O (CSort n) e) H3)) h H1)))) -(drop_gen_sort n h O (CHead e (Bind b) u) H))))))) (\lambda (c0: C).(\lambda -(H: ((\forall (e: C).(\forall (u: T).(\forall (h: nat).((drop h O c0 (CHead e -(Bind b) u)) \to (drop (S h) O c0 e))))))).(\lambda (k: K).(\lambda (t: -T).(\lambda (e: C).(\lambda (u: T).(\lambda (h: nat).(nat_ind (\lambda (n: -nat).((drop n O (CHead c0 k t) (CHead e (Bind b) u)) \to (drop (S n) O (CHead -c0 k t) e))) (\lambda (H0: (drop O O (CHead c0 k t) (CHead e (Bind b) -u))).(let H1 \def (f_equal C C (\lambda (e0: C).(match e0 in C return -(\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c1 _ _) -\Rightarrow c1])) (CHead c0 k t) (CHead e (Bind b) u) (drop_gen_refl (CHead -c0 k t) (CHead e (Bind b) u) H0)) in ((let H2 \def (f_equal C K (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | -(CHead _ k0 _) \Rightarrow k0])) (CHead c0 k t) (CHead e (Bind b) u) -(drop_gen_refl (CHead c0 k t) (CHead e (Bind b) u) H0)) in ((let H3 \def -(f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow t | (CHead _ _ t0) \Rightarrow t0])) (CHead c0 k t) -(CHead e (Bind b) u) (drop_gen_refl (CHead c0 k t) (CHead e (Bind b) u) H0)) -in (\lambda (H4: (eq K k (Bind b))).(\lambda (H5: (eq C c0 e)).(eq_ind C c0 -(\lambda (c1: C).(drop (S O) O (CHead c0 k t) c1)) (eq_ind_r K (Bind b) -(\lambda (k0: K).(drop (S O) O (CHead c0 k0 t) c0)) (drop_drop (Bind b) O c0 -c0 (drop_refl c0) t) k H4) e H5)))) H2)) H1))) (\lambda (n: nat).(\lambda (_: -(((drop n O (CHead c0 k t) (CHead e (Bind b) u)) \to (drop (S n) O (CHead c0 -k t) e)))).(\lambda (H1: (drop (S n) O (CHead c0 k t) (CHead e (Bind b) -u))).(drop_drop k (S n) c0 e (eq_ind_r nat (S (r k n)) (\lambda (n0: -nat).(drop n0 O c0 e)) (H e u (r k n) (drop_gen_drop k c0 (CHead e (Bind b) -u) t n H1)) (r k (S n)) (r_S k n)) t)))) h)))))))) c)). -(* COMMENTS -Initial nodes: 807 -END *) - -theorem drop_ctail: - \forall (c1: C).(\forall (c2: C).(\forall (d: nat).(\forall (h: nat).((drop -h d c1 c2) \to (\forall (k: K).(\forall (u: T).(drop h d (CTail k u c1) -(CTail k u c2)))))))) -\def - \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (d: -nat).(\forall (h: nat).((drop h d c c2) \to (\forall (k: K).(\forall (u: -T).(drop h d (CTail k u c) (CTail k u c2))))))))) (\lambda (n: nat).(\lambda -(c2: C).(\lambda (d: nat).(\lambda (h: nat).(\lambda (H: (drop h d (CSort n) -c2)).(\lambda (k: K).(\lambda (u: T).(and3_ind (eq C c2 (CSort n)) (eq nat h -O) (eq nat d O) (drop h d (CTail k u (CSort n)) (CTail k u c2)) (\lambda (H0: -(eq C c2 (CSort n))).(\lambda (H1: (eq nat h O)).(\lambda (H2: (eq nat d -O)).(eq_ind_r nat O (\lambda (n0: nat).(drop n0 d (CTail k u (CSort n)) -(CTail k u c2))) (eq_ind_r nat O (\lambda (n0: nat).(drop O n0 (CTail k u -(CSort n)) (CTail k u c2))) (eq_ind_r C (CSort n) (\lambda (c: C).(drop O O -(CTail k u (CSort n)) (CTail k u c))) (drop_refl (CTail k u (CSort n))) c2 -H0) d H2) h H1)))) (drop_gen_sort n h d c2 H))))))))) (\lambda (c2: -C).(\lambda (IHc: ((\forall (c3: C).(\forall (d: nat).(\forall (h: -nat).((drop h d c2 c3) \to (\forall (k: K).(\forall (u: T).(drop h d (CTail k -u c2) (CTail k u c3)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c3: -C).(\lambda (d: nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n -(CHead c2 k t) c3) \to (\forall (k0: K).(\forall (u: T).(drop h n (CTail k0 u -(CHead c2 k t)) (CTail k0 u c3))))))) (\lambda (h: nat).(nat_ind (\lambda (n: -nat).((drop n O (CHead c2 k t) c3) \to (\forall (k0: K).(\forall (u: T).(drop -n O (CTail k0 u (CHead c2 k t)) (CTail k0 u c3)))))) (\lambda (H: (drop O O -(CHead c2 k t) c3)).(\lambda (k0: K).(\lambda (u: T).(eq_ind C (CHead c2 k t) -(\lambda (c: C).(drop O O (CTail k0 u (CHead c2 k t)) (CTail k0 u c))) -(drop_refl (CTail k0 u (CHead c2 k t))) c3 (drop_gen_refl (CHead c2 k t) c3 -H))))) (\lambda (n: nat).(\lambda (_: (((drop n O (CHead c2 k t) c3) \to -(\forall (k0: K).(\forall (u: T).(drop n O (CTail k0 u (CHead c2 k t)) (CTail -k0 u c3))))))).(\lambda (H0: (drop (S n) O (CHead c2 k t) c3)).(\lambda (k0: -K).(\lambda (u: T).(drop_drop k n (CTail k0 u c2) (CTail k0 u c3) (IHc c3 O -(r k n) (drop_gen_drop k c2 c3 t n H0) k0 u) t)))))) h)) (\lambda (n: -nat).(\lambda (H: ((\forall (h: nat).((drop h n (CHead c2 k t) c3) \to -(\forall (k0: K).(\forall (u: T).(drop h n (CTail k0 u (CHead c2 k t)) (CTail -k0 u c3)))))))).(\lambda (h: nat).(\lambda (H0: (drop h (S n) (CHead c2 k t) -c3)).(\lambda (k0: K).(\lambda (u: T).(ex3_2_ind C T (\lambda (e: C).(\lambda -(v: T).(eq C c3 (CHead e k v)))) (\lambda (_: C).(\lambda (v: T).(eq T t -(lift h (r k n) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r k n) c2 e))) -(drop h (S n) (CTail k0 u (CHead c2 k t)) (CTail k0 u c3)) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (H1: (eq C c3 (CHead x0 k x1))).(\lambda (H2: -(eq T t (lift h (r k n) x1))).(\lambda (H3: (drop h (r k n) c2 x0)).(let H4 -\def (eq_ind C c3 (\lambda (c: C).(\forall (h0: nat).((drop h0 n (CHead c2 k -t) c) \to (\forall (k1: K).(\forall (u0: T).(drop h0 n (CTail k1 u0 (CHead c2 -k t)) (CTail k1 u0 c))))))) H (CHead x0 k x1) H1) in (eq_ind_r C (CHead x0 k -x1) (\lambda (c: C).(drop h (S n) (CTail k0 u (CHead c2 k t)) (CTail k0 u -c))) (let H5 \def (eq_ind T t (\lambda (t0: T).(\forall (h0: nat).((drop h0 n -(CHead c2 k t0) (CHead x0 k x1)) \to (\forall (k1: K).(\forall (u0: T).(drop -h0 n (CTail k1 u0 (CHead c2 k t0)) (CTail k1 u0 (CHead x0 k x1)))))))) H4 -(lift h (r k n) x1) H2) in (eq_ind_r T (lift h (r k n) x1) (\lambda (t0: -T).(drop h (S n) (CTail k0 u (CHead c2 k t0)) (CTail k0 u (CHead x0 k x1)))) -(drop_skip k h n (CTail k0 u c2) (CTail k0 u x0) (IHc x0 (r k n) h H3 k0 u) -x1) t H2)) c3 H1))))))) (drop_gen_skip_l c2 c3 t h n k H0)))))))) d))))))) -c1). -(* COMMENTS -Initial nodes: 1211 -END *) - -theorem drop_mono: - \forall (c: C).(\forall (x1: C).(\forall (d: nat).(\forall (h: nat).((drop h -d c x1) \to (\forall (x2: C).((drop h d c x2) \to (eq C x1 x2))))))) -\def - \lambda (c: C).(C_ind (\lambda (c0: C).(\forall (x1: C).(\forall (d: -nat).(\forall (h: nat).((drop h d c0 x1) \to (\forall (x2: C).((drop h d c0 -x2) \to (eq C x1 x2)))))))) (\lambda (n: nat).(\lambda (x1: C).(\lambda (d: -nat).(\lambda (h: nat).(\lambda (H: (drop h d (CSort n) x1)).(\lambda (x2: -C).(\lambda (H0: (drop h d (CSort n) x2)).(and3_ind (eq C x2 (CSort n)) (eq -nat h O) (eq nat d O) (eq C x1 x2) (\lambda (H1: (eq C x2 (CSort -n))).(\lambda (H2: (eq nat h O)).(\lambda (H3: (eq nat d O)).(and3_ind (eq C -x1 (CSort n)) (eq nat h O) (eq nat d O) (eq C x1 x2) (\lambda (H4: (eq C x1 -(CSort n))).(\lambda (H5: (eq nat h O)).(\lambda (H6: (eq nat d O)).(eq_ind_r -C (CSort n) (\lambda (c0: C).(eq C x1 c0)) (let H7 \def (eq_ind nat h -(\lambda (n0: nat).(eq nat n0 O)) H2 O H5) in (let H8 \def (eq_ind nat d -(\lambda (n0: nat).(eq nat n0 O)) H3 O H6) in (eq_ind_r C (CSort n) (\lambda -(c0: C).(eq C c0 (CSort n))) (refl_equal C (CSort n)) x1 H4))) x2 H1)))) -(drop_gen_sort n h d x1 H))))) (drop_gen_sort n h d x2 H0))))))))) (\lambda -(c0: C).(\lambda (H: ((\forall (x1: C).(\forall (d: nat).(\forall (h: -nat).((drop h d c0 x1) \to (\forall (x2: C).((drop h d c0 x2) \to (eq C x1 -x2))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (x1: C).(\lambda (d: -nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n (CHead c0 k t) -x1) \to (\forall (x2: C).((drop h n (CHead c0 k t) x2) \to (eq C x1 x2)))))) -(\lambda (h: nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c0 k t) x1) -\to (\forall (x2: C).((drop n O (CHead c0 k t) x2) \to (eq C x1 x2))))) -(\lambda (H0: (drop O O (CHead c0 k t) x1)).(\lambda (x2: C).(\lambda (H1: -(drop O O (CHead c0 k t) x2)).(eq_ind C (CHead c0 k t) (\lambda (c1: C).(eq C -x1 c1)) (eq_ind C (CHead c0 k t) (\lambda (c1: C).(eq C c1 (CHead c0 k t))) -(refl_equal C (CHead c0 k t)) x1 (drop_gen_refl (CHead c0 k t) x1 H0)) x2 -(drop_gen_refl (CHead c0 k t) x2 H1))))) (\lambda (n: nat).(\lambda (_: -(((drop n O (CHead c0 k t) x1) \to (\forall (x2: C).((drop n O (CHead c0 k t) -x2) \to (eq C x1 x2)))))).(\lambda (H1: (drop (S n) O (CHead c0 k t) -x1)).(\lambda (x2: C).(\lambda (H2: (drop (S n) O (CHead c0 k t) x2)).(H x1 O -(r k n) (drop_gen_drop k c0 x1 t n H1) x2 (drop_gen_drop k c0 x2 t n -H2))))))) h)) (\lambda (n: nat).(\lambda (H0: ((\forall (h: nat).((drop h n -(CHead c0 k t) x1) \to (\forall (x2: C).((drop h n (CHead c0 k t) x2) \to (eq -C x1 x2))))))).(\lambda (h: nat).(\lambda (H1: (drop h (S n) (CHead c0 k t) -x1)).(\lambda (x2: C).(\lambda (H2: (drop h (S n) (CHead c0 k t) -x2)).(ex3_2_ind C T (\lambda (e: C).(\lambda (v: T).(eq C x2 (CHead e k v)))) -(\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r k n) v)))) (\lambda (e: -C).(\lambda (_: T).(drop h (r k n) c0 e))) (eq C x1 x2) (\lambda (x0: -C).(\lambda (x3: T).(\lambda (H3: (eq C x2 (CHead x0 k x3))).(\lambda (H4: -(eq T t (lift h (r k n) x3))).(\lambda (H5: (drop h (r k n) c0 -x0)).(ex3_2_ind C T (\lambda (e: C).(\lambda (v: T).(eq C x1 (CHead e k v)))) -(\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r k n) v)))) (\lambda (e: -C).(\lambda (_: T).(drop h (r k n) c0 e))) (eq C x1 x2) (\lambda (x4: -C).(\lambda (x5: T).(\lambda (H6: (eq C x1 (CHead x4 k x5))).(\lambda (H7: -(eq T t (lift h (r k n) x5))).(\lambda (H8: (drop h (r k n) c0 x4)).(eq_ind_r -C (CHead x0 k x3) (\lambda (c1: C).(eq C x1 c1)) (let H9 \def (eq_ind C x1 -(\lambda (c1: C).(\forall (h0: nat).((drop h0 n (CHead c0 k t) c1) \to -(\forall (x6: C).((drop h0 n (CHead c0 k t) x6) \to (eq C c1 x6)))))) H0 -(CHead x4 k x5) H6) in (eq_ind_r C (CHead x4 k x5) (\lambda (c1: C).(eq C c1 -(CHead x0 k x3))) (let H10 \def (eq_ind T t (\lambda (t0: T).(\forall (h0: -nat).((drop h0 n (CHead c0 k t0) (CHead x4 k x5)) \to (\forall (x6: C).((drop -h0 n (CHead c0 k t0) x6) \to (eq C (CHead x4 k x5) x6)))))) H9 (lift h (r k -n) x5) H7) in (let H11 \def (eq_ind T t (\lambda (t0: T).(eq T t0 (lift h (r -k n) x3))) H4 (lift h (r k n) x5) H7) in (let H12 \def (eq_ind T x5 (\lambda -(t0: T).(\forall (h0: nat).((drop h0 n (CHead c0 k (lift h (r k n) t0)) -(CHead x4 k t0)) \to (\forall (x6: C).((drop h0 n (CHead c0 k (lift h (r k n) -t0)) x6) \to (eq C (CHead x4 k t0) x6)))))) H10 x3 (lift_inj x5 x3 h (r k n) -H11)) in (eq_ind_r T x3 (\lambda (t0: T).(eq C (CHead x4 k t0) (CHead x0 k -x3))) (f_equal3 C K T C CHead x4 x0 k k x3 x3 (sym_eq C x0 x4 (H x0 (r k n) h -H5 x4 H8)) (refl_equal K k) (refl_equal T x3)) x5 (lift_inj x5 x3 h (r k n) -H11))))) x1 H6)) x2 H3)))))) (drop_gen_skip_l c0 x1 t h n k H1))))))) -(drop_gen_skip_l c0 x2 t h n k H2)))))))) d))))))) c). -(* COMMENTS -Initial nodes: 1539 -END *) - -theorem drop_conf_lt: - \forall (k: K).(\forall (i: nat).(\forall (u: T).(\forall (c0: C).(\forall -(c: C).((drop i O c (CHead c0 k u)) \to (\forall (e: C).(\forall (h: -nat).(\forall (d: nat).((drop h (S (plus i d)) c e) \to (ex3_2 T C (\lambda -(v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda -(e0: C).(drop i O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop -h (r k d) c0 e0))))))))))))) -\def - \lambda (k: K).(\lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (u: -T).(\forall (c0: C).(\forall (c: C).((drop n O c (CHead c0 k u)) \to (\forall -(e: C).(\forall (h: nat).(\forall (d: nat).((drop h (S (plus n d)) c e) \to -(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) -(\lambda (v: T).(\lambda (e0: C).(drop n O e (CHead e0 k v)))) (\lambda (_: -T).(\lambda (e0: C).(drop h (r k d) c0 e0))))))))))))) (\lambda (u: -T).(\lambda (c0: C).(\lambda (c: C).(\lambda (H: (drop O O c (CHead c0 k -u))).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H0: (drop -h (S (plus O d)) c e)).(let H1 \def (eq_ind C c (\lambda (c1: C).(drop h (S -(plus O d)) c1 e)) H0 (CHead c0 k u) (drop_gen_refl c (CHead c0 k u) H)) in -(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 k v)))) -(\lambda (_: C).(\lambda (v: T).(eq T u (lift h (r k (plus O d)) v)))) -(\lambda (e0: C).(\lambda (_: T).(drop h (r k (plus O d)) c0 e0))) (ex3_2 T C -(\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: -T).(\lambda (e0: C).(drop O O e (CHead e0 k v)))) (\lambda (_: T).(\lambda -(e0: C).(drop h (r k d) c0 e0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda -(H2: (eq C e (CHead x0 k x1))).(\lambda (H3: (eq T u (lift h (r k (plus O d)) -x1))).(\lambda (H4: (drop h (r k (plus O d)) c0 x0)).(eq_ind_r C (CHead x0 k -x1) (\lambda (c1: C).(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift -h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop O O c1 (CHead e0 k -v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0))))) (eq_ind_r T -(lift h (r k (plus O d)) x1) (\lambda (t: T).(ex3_2 T C (\lambda (v: -T).(\lambda (_: C).(eq T t (lift h (r k d) v)))) (\lambda (v: T).(\lambda -(e0: C).(drop O O (CHead x0 k x1) (CHead e0 k v)))) (\lambda (_: T).(\lambda -(e0: C).(drop h (r k d) c0 e0))))) (ex3_2_intro T C (\lambda (v: T).(\lambda -(_: C).(eq T (lift h (r k (plus O d)) x1) (lift h (r k d) v)))) (\lambda (v: -T).(\lambda (e0: C).(drop O O (CHead x0 k x1) (CHead e0 k v)))) (\lambda (_: -T).(\lambda (e0: C).(drop h (r k d) c0 e0))) x1 x0 (refl_equal T (lift h (r k -d) x1)) (drop_refl (CHead x0 k x1)) H4) u H3) e H2)))))) (drop_gen_skip_l c0 -e u h (plus O d) k H1))))))))))) (\lambda (i0: nat).(\lambda (H: ((\forall -(u: T).(\forall (c0: C).(\forall (c: C).((drop i0 O c (CHead c0 k u)) \to -(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h (S (plus i0 d)) -c e) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) -v)))) (\lambda (v: T).(\lambda (e0: C).(drop i0 O e (CHead e0 k v)))) -(\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0)))))))))))))).(\lambda -(u: T).(\lambda (c0: C).(\lambda (c: C).(C_ind (\lambda (c1: C).((drop (S i0) -O c1 (CHead c0 k u)) \to (\forall (e: C).(\forall (h: nat).(\forall (d: -nat).((drop h (S (plus (S i0) d)) c1 e) \to (ex3_2 T C (\lambda (v: -T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda -(e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: -C).(drop h (r k d) c0 e0)))))))))) (\lambda (n: nat).(\lambda (_: (drop (S -i0) O (CSort n) (CHead c0 k u))).(\lambda (e: C).(\lambda (h: nat).(\lambda -(d: nat).(\lambda (H1: (drop h (S (plus (S i0) d)) (CSort n) e)).(and3_ind -(eq C e (CSort n)) (eq nat h O) (eq nat (S (plus (S i0) d)) O) (ex3_2 T C -(\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: -T).(\lambda (e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda (_: -T).(\lambda (e0: C).(drop h (r k d) c0 e0)))) (\lambda (_: (eq C e (CSort -n))).(\lambda (_: (eq nat h O)).(\lambda (H4: (eq nat (S (plus (S i0) d)) -O)).(let H5 \def (eq_ind nat (S (plus (S i0) d)) (\lambda (ee: nat).(match ee -in nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) -\Rightarrow True])) I O H4) in (False_ind (ex3_2 T C (\lambda (v: T).(\lambda -(_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop -(S i0) O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) -c0 e0)))) H5))))) (drop_gen_sort n h (S (plus (S i0) d)) e H1)))))))) -(\lambda (c1: C).(\lambda (H0: (((drop (S i0) O c1 (CHead c0 k u)) \to -(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h (S (plus (S i0) -d)) c1 e) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k -d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S i0) O e (CHead e0 k v)))) -(\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0))))))))))).(\lambda -(k0: K).(K_ind (\lambda (k1: K).(\forall (t: T).((drop (S i0) O (CHead c1 k1 -t) (CHead c0 k u)) \to (\forall (e: C).(\forall (h: nat).(\forall (d: -nat).((drop h (S (plus (S i0) d)) (CHead c1 k1 t) e) \to (ex3_2 T C (\lambda -(v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda -(e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: -C).(drop h (r k d) c0 e0))))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda -(H1: (drop (S i0) O (CHead c1 (Bind b) t) (CHead c0 k u))).(\lambda (e: -C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2: (drop h (S (plus (S i0) -d)) (CHead c1 (Bind b) t) e)).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: -T).(eq C e (CHead e0 (Bind b) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t -(lift h (r (Bind b) (plus (S i0) d)) v)))) (\lambda (e0: C).(\lambda (_: -T).(drop h (r (Bind b) (plus (S i0) d)) c1 e0))) (ex3_2 T C (\lambda (v: -T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda -(e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: -C).(drop h (r k d) c0 e0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H3: -(eq C e (CHead x0 (Bind b) x1))).(\lambda (_: (eq T t (lift h (r (Bind b) -(plus (S i0) d)) x1))).(\lambda (H5: (drop h (r (Bind b) (plus (S i0) d)) c1 -x0)).(eq_ind_r C (CHead x0 (Bind b) x1) (\lambda (c2: C).(ex3_2 T C (\lambda -(v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda -(e0: C).(drop (S i0) O c2 (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: -C).(drop h (r k d) c0 e0))))) (let H6 \def (H u c0 c1 (drop_gen_drop (Bind b) -c1 (CHead c0 k u) t i0 H1) x0 h d H5) in (ex3_2_ind T C (\lambda (v: -T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda -(e0: C).(drop i0 O x0 (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: -C).(drop h (r k d) c0 e0))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T -u (lift h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S i0) O -(CHead x0 (Bind b) x1) (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: -C).(drop h (r k d) c0 e0)))) (\lambda (x2: T).(\lambda (x3: C).(\lambda (H7: -(eq T u (lift h (r k d) x2))).(\lambda (H8: (drop i0 O x0 (CHead x3 k -x2))).(\lambda (H9: (drop h (r k d) c0 x3)).(ex3_2_intro T C (\lambda (v: -T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda -(e0: C).(drop (S i0) O (CHead x0 (Bind b) x1) (CHead e0 k v)))) (\lambda (_: -T).(\lambda (e0: C).(drop h (r k d) c0 e0))) x2 x3 H7 (drop_drop (Bind b) i0 -x0 (CHead x3 k x2) H8 x1) H9)))))) H6)) e H3)))))) (drop_gen_skip_l c1 e t h -(plus (S i0) d) (Bind b) H2))))))))) (\lambda (f: F).(\lambda (t: T).(\lambda -(H1: (drop (S i0) O (CHead c1 (Flat f) t) (CHead c0 k u))).(\lambda (e: -C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2: (drop h (S (plus (S i0) -d)) (CHead c1 (Flat f) t) e)).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: -T).(eq C e (CHead e0 (Flat f) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t -(lift h (r (Flat f) (plus (S i0) d)) v)))) (\lambda (e0: C).(\lambda (_: -T).(drop h (r (Flat f) (plus (S i0) d)) c1 e0))) (ex3_2 T C (\lambda (v: -T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda -(e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: -C).(drop h (r k d) c0 e0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H3: -(eq C e (CHead x0 (Flat f) x1))).(\lambda (_: (eq T t (lift h (r (Flat f) -(plus (S i0) d)) x1))).(\lambda (H5: (drop h (r (Flat f) (plus (S i0) d)) c1 -x0)).(eq_ind_r C (CHead x0 (Flat f) x1) (\lambda (c2: C).(ex3_2 T C (\lambda -(v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda -(e0: C).(drop (S i0) O c2 (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: -C).(drop h (r k d) c0 e0))))) (ex3_2_ind T C (\lambda (v: T).(\lambda (_: -C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S -i0) O x0 (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) -c0 e0))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) -v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S i0) O (CHead x0 (Flat f) x1) -(CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0)))) -(\lambda (x2: T).(\lambda (x3: C).(\lambda (H6: (eq T u (lift h (r k d) -x2))).(\lambda (H7: (drop (S i0) O x0 (CHead x3 k x2))).(\lambda (H8: (drop h -(r k d) c0 x3)).(ex3_2_intro T C (\lambda (v: T).(\lambda (_: C).(eq T u -(lift h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S i0) O (CHead -x0 (Flat f) x1) (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r -k d) c0 e0))) x2 x3 H6 (drop_drop (Flat f) i0 x0 (CHead x3 k x2) H7 x1) -H8)))))) (H0 (drop_gen_drop (Flat f) c1 (CHead c0 k u) t i0 H1) x0 h d H5)) e -H3)))))) (drop_gen_skip_l c1 e t h (plus (S i0) d) (Flat f) H2))))))))) -k0)))) c)))))) i)). -(* COMMENTS -Initial nodes: 2972 -END *) - -theorem drop_conf_ge: - \forall (i: nat).(\forall (a: C).(\forall (c: C).((drop i O c a) \to -(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to ((le -(plus d h) i) \to (drop (minus i h) O e a))))))))) -\def - \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (a: C).(\forall (c: -C).((drop n O c a) \to (\forall (e: C).(\forall (h: nat).(\forall (d: -nat).((drop h d c e) \to ((le (plus d h) n) \to (drop (minus n h) O e -a)))))))))) (\lambda (a: C).(\lambda (c: C).(\lambda (H: (drop O O c -a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H0: (drop h -d c e)).(\lambda (H1: (le (plus d h) O)).(let H2 \def (eq_ind C c (\lambda -(c0: C).(drop h d c0 e)) H0 a (drop_gen_refl c a H)) in (let H_y \def -(le_n_O_eq (plus d h) H1) in (land_ind (eq nat d O) (eq nat h O) (drop (minus -O h) O e a) (\lambda (H3: (eq nat d O)).(\lambda (H4: (eq nat h O)).(let H5 -\def (eq_ind nat d (\lambda (n: nat).(drop h n a e)) H2 O H3) in (let H6 \def -(eq_ind nat h (\lambda (n: nat).(drop n O a e)) H5 O H4) in (eq_ind_r nat O -(\lambda (n: nat).(drop (minus O n) O e a)) (eq_ind C a (\lambda (c0: -C).(drop (minus O O) O c0 a)) (drop_refl a) e (drop_gen_refl a e H6)) h -H4))))) (plus_O d h (sym_eq nat O (plus d h) H_y))))))))))))) (\lambda (i0: -nat).(\lambda (H: ((\forall (a: C).(\forall (c: C).((drop i0 O c a) \to -(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to ((le -(plus d h) i0) \to (drop (minus i0 h) O e a))))))))))).(\lambda (a: -C).(\lambda (c: C).(C_ind (\lambda (c0: C).((drop (S i0) O c0 a) \to (\forall -(e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c0 e) \to ((le (plus d -h) (S i0)) \to (drop (minus (S i0) h) O e a)))))))) (\lambda (n: -nat).(\lambda (H0: (drop (S i0) O (CSort n) a)).(\lambda (e: C).(\lambda (h: -nat).(\lambda (d: nat).(\lambda (H1: (drop h d (CSort n) e)).(\lambda (H2: -(le (plus d h) (S i0))).(and3_ind (eq C e (CSort n)) (eq nat h O) (eq nat d -O) (drop (minus (S i0) h) O e a) (\lambda (H3: (eq C e (CSort n))).(\lambda -(H4: (eq nat h O)).(\lambda (H5: (eq nat d O)).(and3_ind (eq C a (CSort n)) -(eq nat (S i0) O) (eq nat O O) (drop (minus (S i0) h) O e a) (\lambda (H6: -(eq C a (CSort n))).(\lambda (H7: (eq nat (S i0) O)).(\lambda (_: (eq nat O -O)).(let H9 \def (eq_ind nat d (\lambda (n0: nat).(le (plus n0 h) (S i0))) H2 -O H5) in (let H10 \def (eq_ind nat h (\lambda (n0: nat).(le (plus O n0) (S -i0))) H9 O H4) in (eq_ind_r nat O (\lambda (n0: nat).(drop (minus (S i0) n0) -O e a)) (eq_ind_r C (CSort n) (\lambda (c0: C).(drop (minus (S i0) O) O c0 -a)) (eq_ind_r C (CSort n) (\lambda (c0: C).(drop (minus (S i0) O) O (CSort n) -c0)) (let H11 \def (eq_ind nat (S i0) (\lambda (ee: nat).(match ee in nat -return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow -True])) I O H7) in (False_ind (drop (minus (S i0) O) O (CSort n) (CSort n)) -H11)) a H6) e H3) h H4)))))) (drop_gen_sort n (S i0) O a H0))))) -(drop_gen_sort n h d e H1))))))))) (\lambda (c0: C).(\lambda (H0: (((drop (S -i0) O c0 a) \to (\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h -d c0 e) \to ((le (plus d h) (S i0)) \to (drop (minus (S i0) h) O e -a))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t: T).((drop (S -i0) O (CHead c0 k0 t) a) \to (\forall (e: C).(\forall (h: nat).(\forall (d: -nat).((drop h d (CHead c0 k0 t) e) \to ((le (plus d h) (S i0)) \to (drop -(minus (S i0) h) O e a))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (H1: -(drop (S i0) O (CHead c0 (Bind b) t) a)).(\lambda (e: C).(\lambda (h: -nat).(\lambda (d: nat).(\lambda (H2: (drop h d (CHead c0 (Bind b) t) -e)).(\lambda (H3: (le (plus d h) (S i0))).(nat_ind (\lambda (n: nat).((drop h -n (CHead c0 (Bind b) t) e) \to ((le (plus n h) (S i0)) \to (drop (minus (S -i0) h) O e a)))) (\lambda (H4: (drop h O (CHead c0 (Bind b) t) e)).(\lambda -(H5: (le (plus O h) (S i0))).(nat_ind (\lambda (n: nat).((drop n O (CHead c0 -(Bind b) t) e) \to ((le (plus O n) (S i0)) \to (drop (minus (S i0) n) O e -a)))) (\lambda (H6: (drop O O (CHead c0 (Bind b) t) e)).(\lambda (_: (le -(plus O O) (S i0))).(eq_ind C (CHead c0 (Bind b) t) (\lambda (c1: C).(drop -(minus (S i0) O) O c1 a)) (drop_drop (Bind b) i0 c0 a (drop_gen_drop (Bind b) -c0 a t i0 H1) t) e (drop_gen_refl (CHead c0 (Bind b) t) e H6)))) (\lambda -(h0: nat).(\lambda (_: (((drop h0 O (CHead c0 (Bind b) t) e) \to ((le (plus O -h0) (S i0)) \to (drop (minus (S i0) h0) O e a))))).(\lambda (H6: (drop (S h0) -O (CHead c0 (Bind b) t) e)).(\lambda (H7: (le (plus O (S h0)) (S i0))).(H a -c0 (drop_gen_drop (Bind b) c0 a t i0 H1) e h0 O (drop_gen_drop (Bind b) c0 e -t h0 H6) (le_S_n (plus O h0) i0 H7)))))) h H4 H5))) (\lambda (d0: -nat).(\lambda (_: (((drop h d0 (CHead c0 (Bind b) t) e) \to ((le (plus d0 h) -(S i0)) \to (drop (minus (S i0) h) O e a))))).(\lambda (H4: (drop h (S d0) -(CHead c0 (Bind b) t) e)).(\lambda (H5: (le (plus (S d0) h) (S -i0))).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 (Bind -b) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r (Bind b) d0) -v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r (Bind b) d0) c0 e0))) (drop -(minus (S i0) h) O e a) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C -e (CHead x0 (Bind b) x1))).(\lambda (_: (eq T t (lift h (r (Bind b) d0) -x1))).(\lambda (H8: (drop h (r (Bind b) d0) c0 x0)).(eq_ind_r C (CHead x0 -(Bind b) x1) (\lambda (c1: C).(drop (minus (S i0) h) O c1 a)) (eq_ind nat (S -(minus i0 h)) (\lambda (n: nat).(drop n O (CHead x0 (Bind b) x1) a)) -(drop_drop (Bind b) (minus i0 h) x0 a (H a c0 (drop_gen_drop (Bind b) c0 a t -i0 H1) x0 h d0 H8 (le_S_n (plus d0 h) i0 H5)) x1) (minus (S i0) h) -(minus_Sn_m i0 h (le_trans_plus_r d0 h i0 (le_S_n (plus d0 h) i0 H5)))) e -H6)))))) (drop_gen_skip_l c0 e t h d0 (Bind b) H4)))))) d H2 H3))))))))) -(\lambda (f: F).(\lambda (t: T).(\lambda (H1: (drop (S i0) O (CHead c0 (Flat -f) t) a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2: -(drop h d (CHead c0 (Flat f) t) e)).(\lambda (H3: (le (plus d h) (S -i0))).(nat_ind (\lambda (n: nat).((drop h n (CHead c0 (Flat f) t) e) \to ((le -(plus n h) (S i0)) \to (drop (minus (S i0) h) O e a)))) (\lambda (H4: (drop h -O (CHead c0 (Flat f) t) e)).(\lambda (H5: (le (plus O h) (S i0))).(nat_ind -(\lambda (n: nat).((drop n O (CHead c0 (Flat f) t) e) \to ((le (plus O n) (S -i0)) \to (drop (minus (S i0) n) O e a)))) (\lambda (H6: (drop O O (CHead c0 -(Flat f) t) e)).(\lambda (_: (le (plus O O) (S i0))).(eq_ind C (CHead c0 -(Flat f) t) (\lambda (c1: C).(drop (minus (S i0) O) O c1 a)) (drop_drop (Flat -f) i0 c0 a (drop_gen_drop (Flat f) c0 a t i0 H1) t) e (drop_gen_refl (CHead -c0 (Flat f) t) e H6)))) (\lambda (h0: nat).(\lambda (_: (((drop h0 O (CHead -c0 (Flat f) t) e) \to ((le (plus O h0) (S i0)) \to (drop (minus (S i0) h0) O -e a))))).(\lambda (H6: (drop (S h0) O (CHead c0 (Flat f) t) e)).(\lambda (H7: -(le (plus O (S h0)) (S i0))).(H0 (drop_gen_drop (Flat f) c0 a t i0 H1) e (S -h0) O (drop_gen_drop (Flat f) c0 e t h0 H6) H7))))) h H4 H5))) (\lambda (d0: -nat).(\lambda (_: (((drop h d0 (CHead c0 (Flat f) t) e) \to ((le (plus d0 h) -(S i0)) \to (drop (minus (S i0) h) O e a))))).(\lambda (H4: (drop h (S d0) -(CHead c0 (Flat f) t) e)).(\lambda (H5: (le (plus (S d0) h) (S -i0))).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 (Flat -f) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r (Flat f) d0) -v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r (Flat f) d0) c0 e0))) (drop -(minus (S i0) h) O e a) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C -e (CHead x0 (Flat f) x1))).(\lambda (_: (eq T t (lift h (r (Flat f) d0) -x1))).(\lambda (H8: (drop h (r (Flat f) d0) c0 x0)).(eq_ind_r C (CHead x0 -(Flat f) x1) (\lambda (c1: C).(drop (minus (S i0) h) O c1 a)) (let H9 \def -(eq_ind_r nat (minus (S i0) h) (\lambda (n: nat).(drop n O x0 a)) (H0 -(drop_gen_drop (Flat f) c0 a t i0 H1) x0 h (S d0) H8 H5) (S (minus i0 h)) -(minus_Sn_m i0 h (le_trans_plus_r d0 h i0 (le_S_n (plus d0 h) i0 H5)))) in -(eq_ind nat (S (minus i0 h)) (\lambda (n: nat).(drop n O (CHead x0 (Flat f) -x1) a)) (drop_drop (Flat f) (minus i0 h) x0 a H9 x1) (minus (S i0) h) -(minus_Sn_m i0 h (le_trans_plus_r d0 h i0 (le_S_n (plus d0 h) i0 H5))))) e -H6)))))) (drop_gen_skip_l c0 e t h d0 (Flat f) H4)))))) d H2 H3))))))))) -k)))) c))))) i). -(* COMMENTS -Initial nodes: 2726 -END *) - -theorem drop_conf_rev: - \forall (j: nat).(\forall (e1: C).(\forall (e2: C).((drop j O e1 e2) \to -(\forall (c2: C).(\forall (i: nat).((drop i O c2 e2) \to (ex2 C (\lambda (c1: -C).(drop j O c1 c2)) (\lambda (c1: C).(drop i j c1 e1))))))))) -\def - \lambda (j: nat).(nat_ind (\lambda (n: nat).(\forall (e1: C).(\forall (e2: -C).((drop n O e1 e2) \to (\forall (c2: C).(\forall (i: nat).((drop i O c2 e2) -\to (ex2 C (\lambda (c1: C).(drop n O c1 c2)) (\lambda (c1: C).(drop i n c1 -e1)))))))))) (\lambda (e1: C).(\lambda (e2: C).(\lambda (H: (drop O O e1 -e2)).(\lambda (c2: C).(\lambda (i: nat).(\lambda (H0: (drop i O c2 e2)).(let -H1 \def (eq_ind_r C e2 (\lambda (c: C).(drop i O c2 c)) H0 e1 (drop_gen_refl -e1 e2 H)) in (ex_intro2 C (\lambda (c1: C).(drop O O c1 c2)) (\lambda (c1: -C).(drop i O c1 e1)) c2 (drop_refl c2) H1)))))))) (\lambda (j0: nat).(\lambda -(IHj: ((\forall (e1: C).(\forall (e2: C).((drop j0 O e1 e2) \to (\forall (c2: -C).(\forall (i: nat).((drop i O c2 e2) \to (ex2 C (\lambda (c1: C).(drop j0 O -c1 c2)) (\lambda (c1: C).(drop i j0 c1 e1))))))))))).(\lambda (e1: C).(C_ind -(\lambda (c: C).(\forall (e2: C).((drop (S j0) O c e2) \to (\forall (c2: -C).(\forall (i: nat).((drop i O c2 e2) \to (ex2 C (\lambda (c1: C).(drop (S -j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 c))))))))) (\lambda (n: -nat).(\lambda (e2: C).(\lambda (H: (drop (S j0) O (CSort n) e2)).(\lambda -(c2: C).(\lambda (i: nat).(\lambda (H0: (drop i O c2 e2)).(and3_ind (eq C e2 -(CSort n)) (eq nat (S j0) O) (eq nat O O) (ex2 C (\lambda (c1: C).(drop (S -j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 (CSort n)))) (\lambda (H1: -(eq C e2 (CSort n))).(\lambda (H2: (eq nat (S j0) O)).(\lambda (_: (eq nat O -O)).(let H4 \def (eq_ind C e2 (\lambda (c: C).(drop i O c2 c)) H0 (CSort n) -H1) in (let H5 \def (eq_ind nat (S j0) (\lambda (ee: nat).(match ee in nat -return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow -True])) I O H2) in (False_ind (ex2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) -(\lambda (c1: C).(drop i (S j0) c1 (CSort n)))) H5)))))) (drop_gen_sort n (S -j0) O e2 H)))))))) (\lambda (e2: C).(\lambda (IHe1: ((\forall (e3: C).((drop -(S j0) O e2 e3) \to (\forall (c2: C).(\forall (i: nat).((drop i O c2 e3) \to -(ex2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S -j0) c1 e2)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e3: C).(\lambda -(H: (drop (S j0) O (CHead e2 k t) e3)).(\lambda (c2: C).(\lambda (i: -nat).(\lambda (H0: (drop i O c2 e3)).(K_ind (\lambda (k0: K).((drop (r k0 j0) -O e2 e3) \to (ex2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: -C).(drop i (S j0) c1 (CHead e2 k0 t)))))) (\lambda (b: B).(\lambda (H1: (drop -(r (Bind b) j0) O e2 e3)).(let H_x \def (IHj e2 e3 H1 c2 i H0) in (let H2 -\def H_x in (ex2_ind C (\lambda (c1: C).(drop j0 O c1 c2)) (\lambda (c1: -C).(drop i j0 c1 e2)) (ex2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda -(c1: C).(drop i (S j0) c1 (CHead e2 (Bind b) t)))) (\lambda (x: C).(\lambda -(H3: (drop j0 O x c2)).(\lambda (H4: (drop i j0 x e2)).(ex_intro2 C (\lambda -(c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 (CHead e2 -(Bind b) t))) (CHead x (Bind b) (lift i (r (Bind b) j0) t)) (drop_drop (Bind -b) j0 x c2 H3 (lift i (r (Bind b) j0) t)) (drop_skip (Bind b) i j0 x e2 H4 -t))))) H2))))) (\lambda (f: F).(\lambda (H1: (drop (r (Flat f) j0) O e2 -e3)).(let H_x \def (IHe1 e3 H1 c2 i H0) in (let H2 \def H_x in (ex2_ind C -(\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 -e2)) (ex2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i -(S j0) c1 (CHead e2 (Flat f) t)))) (\lambda (x: C).(\lambda (H3: (drop (S j0) -O x c2)).(\lambda (H4: (drop i (S j0) x e2)).(ex_intro2 C (\lambda (c1: -C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 (CHead e2 (Flat -f) t))) (CHead x (Flat f) (lift i (r (Flat f) j0) t)) (drop_drop (Flat f) j0 -x c2 H3 (lift i (r (Flat f) j0) t)) (drop_skip (Flat f) i j0 x e2 H4 t))))) -H2))))) k (drop_gen_drop k e2 e3 t j0 H))))))))))) e1)))) j). -(* COMMENTS -Initial nodes: 1154 -END *) - -theorem drop_trans_le: - \forall (i: nat).(\forall (d: nat).((le i d) \to (\forall (c1: C).(\forall -(c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((drop i O -c2 e2) \to (ex2 C (\lambda (e1: C).(drop i O c1 e1)) (\lambda (e1: C).(drop h -(minus d i) e1 e2))))))))))) -\def - \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (d: nat).((le n d) \to -(\forall (c1: C).(\forall (c2: C).(\forall (h: nat).((drop h d c1 c2) \to -(\forall (e2: C).((drop n O c2 e2) \to (ex2 C (\lambda (e1: C).(drop n O c1 -e1)) (\lambda (e1: C).(drop h (minus d n) e1 e2)))))))))))) (\lambda (d: -nat).(\lambda (_: (le O d)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (h: -nat).(\lambda (H0: (drop h d c1 c2)).(\lambda (e2: C).(\lambda (H1: (drop O O -c2 e2)).(let H2 \def (eq_ind C c2 (\lambda (c: C).(drop h d c1 c)) H0 e2 -(drop_gen_refl c2 e2 H1)) in (eq_ind nat d (\lambda (n: nat).(ex2 C (\lambda -(e1: C).(drop O O c1 e1)) (\lambda (e1: C).(drop h n e1 e2)))) (ex_intro2 C -(\lambda (e1: C).(drop O O c1 e1)) (\lambda (e1: C).(drop h d e1 e2)) c1 -(drop_refl c1) H2) (minus d O) (minus_n_O d))))))))))) (\lambda (i0: -nat).(\lambda (IHi: ((\forall (d: nat).((le i0 d) \to (\forall (c1: -C).(\forall (c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: -C).((drop i0 O c2 e2) \to (ex2 C (\lambda (e1: C).(drop i0 O c1 e1)) (\lambda -(e1: C).(drop h (minus d i0) e1 e2))))))))))))).(\lambda (d: nat).(nat_ind -(\lambda (n: nat).((le (S i0) n) \to (\forall (c1: C).(\forall (c2: -C).(\forall (h: nat).((drop h n c1 c2) \to (\forall (e2: C).((drop (S i0) O -c2 e2) \to (ex2 C (\lambda (e1: C).(drop (S i0) O c1 e1)) (\lambda (e1: -C).(drop h (minus n (S i0)) e1 e2))))))))))) (\lambda (H: (le (S i0) -O)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (h: nat).(\lambda (_: (drop h -O c1 c2)).(\lambda (e2: C).(\lambda (_: (drop (S i0) O c2 e2)).(ex2_ind nat -(\lambda (n: nat).(eq nat O (S n))) (\lambda (n: nat).(le i0 n)) (ex2 C -(\lambda (e1: C).(drop (S i0) O c1 e1)) (\lambda (e1: C).(drop h (minus O (S -i0)) e1 e2))) (\lambda (x: nat).(\lambda (H2: (eq nat O (S x))).(\lambda (_: -(le i0 x)).(let H4 \def (eq_ind nat O (\lambda (ee: nat).(match ee in nat -return (\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow -False])) I (S x) H2) in (False_ind (ex2 C (\lambda (e1: C).(drop (S i0) O c1 -e1)) (\lambda (e1: C).(drop h (minus O (S i0)) e1 e2))) H4))))) (le_gen_S i0 -O H))))))))) (\lambda (d0: nat).(\lambda (_: (((le (S i0) d0) \to (\forall -(c1: C).(\forall (c2: C).(\forall (h: nat).((drop h d0 c1 c2) \to (\forall -(e2: C).((drop (S i0) O c2 e2) \to (ex2 C (\lambda (e1: C).(drop (S i0) O c1 -e1)) (\lambda (e1: C).(drop h (minus d0 (S i0)) e1 e2)))))))))))).(\lambda -(H: (le (S i0) (S d0))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: -C).(\forall (h: nat).((drop h (S d0) c c2) \to (\forall (e2: C).((drop (S i0) -O c2 e2) \to (ex2 C (\lambda (e1: C).(drop (S i0) O c e1)) (\lambda (e1: -C).(drop h (minus (S d0) (S i0)) e1 e2))))))))) (\lambda (n: nat).(\lambda -(c2: C).(\lambda (h: nat).(\lambda (H0: (drop h (S d0) (CSort n) -c2)).(\lambda (e2: C).(\lambda (H1: (drop (S i0) O c2 e2)).(and3_ind (eq C c2 -(CSort n)) (eq nat h O) (eq nat (S d0) O) (ex2 C (\lambda (e1: C).(drop (S -i0) O (CSort n) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2))) -(\lambda (H2: (eq C c2 (CSort n))).(\lambda (_: (eq nat h O)).(\lambda (_: -(eq nat (S d0) O)).(let H5 \def (eq_ind C c2 (\lambda (c: C).(drop (S i0) O c -e2)) H1 (CSort n) H2) in (and3_ind (eq C e2 (CSort n)) (eq nat (S i0) O) (eq -nat O O) (ex2 C (\lambda (e1: C).(drop (S i0) O (CSort n) e1)) (\lambda (e1: -C).(drop h (minus (S d0) (S i0)) e1 e2))) (\lambda (H6: (eq C e2 (CSort -n))).(\lambda (H7: (eq nat (S i0) O)).(\lambda (_: (eq nat O O)).(eq_ind_r C -(CSort n) (\lambda (c: C).(ex2 C (\lambda (e1: C).(drop (S i0) O (CSort n) -e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 c)))) (let H9 \def -(eq_ind nat (S i0) (\lambda (ee: nat).(match ee in nat return (\lambda (_: -nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H7) in -(False_ind (ex2 C (\lambda (e1: C).(drop (S i0) O (CSort n) e1)) (\lambda -(e1: C).(drop h (minus (S d0) (S i0)) e1 (CSort n)))) H9)) e2 H6)))) -(drop_gen_sort n (S i0) O e2 H5)))))) (drop_gen_sort n h (S d0) c2 H0)))))))) -(\lambda (c2: C).(\lambda (IHc: ((\forall (c3: C).(\forall (h: nat).((drop h -(S d0) c2 c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to (ex2 C (\lambda -(e1: C).(drop (S i0) O c2 e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) -e1 e2)))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t: -T).(\forall (c3: C).(\forall (h: nat).((drop h (S d0) (CHead c2 k0 t) c3) \to -(\forall (e2: C).((drop (S i0) O c3 e2) \to (ex2 C (\lambda (e1: C).(drop (S -i0) O (CHead c2 k0 t) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 -e2)))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (c3: C).(\lambda (h: -nat).(\lambda (H0: (drop h (S d0) (CHead c2 (Bind b) t) c3)).(\lambda (e2: -C).(\lambda (H1: (drop (S i0) O c3 e2)).(ex3_2_ind C T (\lambda (e: -C).(\lambda (v: T).(eq C c3 (CHead e (Bind b) v)))) (\lambda (_: C).(\lambda -(v: T).(eq T t (lift h (r (Bind b) d0) v)))) (\lambda (e: C).(\lambda (_: -T).(drop h (r (Bind b) d0) c2 e))) (ex2 C (\lambda (e1: C).(drop (S i0) O -(CHead c2 (Bind b) t) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 -e2))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H2: (eq C c3 (CHead x0 -(Bind b) x1))).(\lambda (H3: (eq T t (lift h (r (Bind b) d0) x1))).(\lambda -(H4: (drop h (r (Bind b) d0) c2 x0)).(let H5 \def (eq_ind C c3 (\lambda (c: -C).(drop (S i0) O c e2)) H1 (CHead x0 (Bind b) x1) H2) in (eq_ind_r T (lift h -(r (Bind b) d0) x1) (\lambda (t0: T).(ex2 C (\lambda (e1: C).(drop (S i0) O -(CHead c2 (Bind b) t0) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 -e2)))) (ex2_ind C (\lambda (e1: C).(drop i0 O c2 e1)) (\lambda (e1: C).(drop -h (minus d0 i0) e1 e2)) (ex2 C (\lambda (e1: C).(drop (S i0) O (CHead c2 -(Bind b) (lift h (r (Bind b) d0) x1)) e1)) (\lambda (e1: C).(drop h (minus (S -d0) (S i0)) e1 e2))) (\lambda (x: C).(\lambda (H6: (drop i0 O c2 x)).(\lambda -(H7: (drop h (minus d0 i0) x e2)).(ex_intro2 C (\lambda (e1: C).(drop (S i0) -O (CHead c2 (Bind b) (lift h (r (Bind b) d0) x1)) e1)) (\lambda (e1: C).(drop -h (minus (S d0) (S i0)) e1 e2)) x (drop_drop (Bind b) i0 c2 x H6 (lift h (r -(Bind b) d0) x1)) H7)))) (IHi d0 (le_S_n i0 d0 H) c2 x0 h H4 e2 -(drop_gen_drop (Bind b) x0 e2 x1 i0 H5))) t H3))))))) (drop_gen_skip_l c2 c3 -t h d0 (Bind b) H0))))))))) (\lambda (f: F).(\lambda (t: T).(\lambda (c3: -C).(\lambda (h: nat).(\lambda (H0: (drop h (S d0) (CHead c2 (Flat f) t) -c3)).(\lambda (e2: C).(\lambda (H1: (drop (S i0) O c3 e2)).(ex3_2_ind C T -(\lambda (e: C).(\lambda (v: T).(eq C c3 (CHead e (Flat f) v)))) (\lambda (_: -C).(\lambda (v: T).(eq T t (lift h (r (Flat f) d0) v)))) (\lambda (e: -C).(\lambda (_: T).(drop h (r (Flat f) d0) c2 e))) (ex2 C (\lambda (e1: -C).(drop (S i0) O (CHead c2 (Flat f) t) e1)) (\lambda (e1: C).(drop h (minus -(S d0) (S i0)) e1 e2))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H2: (eq C -c3 (CHead x0 (Flat f) x1))).(\lambda (H3: (eq T t (lift h (r (Flat f) d0) -x1))).(\lambda (H4: (drop h (r (Flat f) d0) c2 x0)).(let H5 \def (eq_ind C c3 -(\lambda (c: C).(drop (S i0) O c e2)) H1 (CHead x0 (Flat f) x1) H2) in -(eq_ind_r T (lift h (r (Flat f) d0) x1) (\lambda (t0: T).(ex2 C (\lambda (e1: -C).(drop (S i0) O (CHead c2 (Flat f) t0) e1)) (\lambda (e1: C).(drop h (minus -(S d0) (S i0)) e1 e2)))) (ex2_ind C (\lambda (e1: C).(drop (S i0) O c2 e1)) -(\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2)) (ex2 C (\lambda (e1: -C).(drop (S i0) O (CHead c2 (Flat f) (lift h (r (Flat f) d0) x1)) e1)) -(\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2))) (\lambda (x: -C).(\lambda (H6: (drop (S i0) O c2 x)).(\lambda (H7: (drop h (minus (S d0) (S -i0)) x e2)).(ex_intro2 C (\lambda (e1: C).(drop (S i0) O (CHead c2 (Flat f) -(lift h (r (Flat f) d0) x1)) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S -i0)) e1 e2)) x (drop_drop (Flat f) i0 c2 x H6 (lift h (r (Flat f) d0) x1)) -H7)))) (IHc x0 h H4 e2 (drop_gen_drop (Flat f) x0 e2 x1 i0 H5))) t H3))))))) -(drop_gen_skip_l c2 c3 t h d0 (Flat f) H0))))))))) k)))) c1))))) d)))) i). -(* COMMENTS -Initial nodes: 2453 -END *) - -theorem drop_trans_ge: - \forall (i: nat).(\forall (c1: C).(\forall (c2: C).(\forall (d: -nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((drop i O c2 -e2) \to ((le d i) \to (drop (plus i h) O c1 e2))))))))) -\def - \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (c2: -C).(\forall (d: nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: -C).((drop n O c2 e2) \to ((le d n) \to (drop (plus n h) O c1 e2)))))))))) -(\lambda (c1: C).(\lambda (c2: C).(\lambda (d: nat).(\lambda (h: -nat).(\lambda (H: (drop h d c1 c2)).(\lambda (e2: C).(\lambda (H0: (drop O O -c2 e2)).(\lambda (H1: (le d O)).(eq_ind C c2 (\lambda (c: C).(drop (plus O h) -O c1 c)) (let H_y \def (le_n_O_eq d H1) in (let H2 \def (eq_ind_r nat d -(\lambda (n: nat).(drop h n c1 c2)) H O H_y) in H2)) e2 (drop_gen_refl c2 e2 -H0)))))))))) (\lambda (i0: nat).(\lambda (IHi: ((\forall (c1: C).(\forall -(c2: C).(\forall (d: nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall -(e2: C).((drop i0 O c2 e2) \to ((le d i0) \to (drop (plus i0 h) O c1 -e2))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: -C).(\forall (d: nat).(\forall (h: nat).((drop h d c c2) \to (\forall (e2: -C).((drop (S i0) O c2 e2) \to ((le d (S i0)) \to (drop (plus (S i0) h) O c -e2))))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (d: nat).(\lambda (h: -nat).(\lambda (H: (drop h d (CSort n) c2)).(\lambda (e2: C).(\lambda (H0: -(drop (S i0) O c2 e2)).(\lambda (H1: (le d (S i0))).(and3_ind (eq C c2 (CSort -n)) (eq nat h O) (eq nat d O) (drop (S (plus i0 h)) O (CSort n) e2) (\lambda -(H2: (eq C c2 (CSort n))).(\lambda (H3: (eq nat h O)).(\lambda (H4: (eq nat d -O)).(eq_ind_r nat O (\lambda (n0: nat).(drop (S (plus i0 n0)) O (CSort n) -e2)) (let H5 \def (eq_ind nat d (\lambda (n0: nat).(le n0 (S i0))) H1 O H4) -in (let H6 \def (eq_ind C c2 (\lambda (c: C).(drop (S i0) O c e2)) H0 (CSort -n) H2) in (and3_ind (eq C e2 (CSort n)) (eq nat (S i0) O) (eq nat O O) (drop -(S (plus i0 O)) O (CSort n) e2) (\lambda (H7: (eq C e2 (CSort n))).(\lambda -(H8: (eq nat (S i0) O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n) -(\lambda (c: C).(drop (S (plus i0 O)) O (CSort n) c)) (let H10 \def (eq_ind -nat (S i0) (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) -with [O \Rightarrow False | (S _) \Rightarrow True])) I O H8) in (False_ind -(drop (S (plus i0 O)) O (CSort n) (CSort n)) H10)) e2 H7)))) (drop_gen_sort n -(S i0) O e2 H6)))) h H3)))) (drop_gen_sort n h d c2 H)))))))))) (\lambda (c2: -C).(\lambda (IHc: ((\forall (c3: C).(\forall (d: nat).(\forall (h: -nat).((drop h d c2 c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to ((le d -(S i0)) \to (drop (S (plus i0 h)) O c2 e2)))))))))).(\lambda (k: K).(\lambda -(t: T).(\lambda (c3: C).(\lambda (d: nat).(nat_ind (\lambda (n: nat).(\forall -(h: nat).((drop h n (CHead c2 k t) c3) \to (\forall (e2: C).((drop (S i0) O -c3 e2) \to ((le n (S i0)) \to (drop (S (plus i0 h)) O (CHead c2 k t) -e2))))))) (\lambda (h: nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c2 k -t) c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to ((le O (S i0)) \to -(drop (S (plus i0 n)) O (CHead c2 k t) e2)))))) (\lambda (H: (drop O O (CHead -c2 k t) c3)).(\lambda (e2: C).(\lambda (H0: (drop (S i0) O c3 e2)).(\lambda -(_: (le O (S i0))).(let H2 \def (eq_ind_r C c3 (\lambda (c: C).(drop (S i0) O -c e2)) H0 (CHead c2 k t) (drop_gen_refl (CHead c2 k t) c3 H)) in (eq_ind nat -i0 (\lambda (n: nat).(drop (S n) O (CHead c2 k t) e2)) (drop_drop k i0 c2 e2 -(drop_gen_drop k c2 e2 t i0 H2) t) (plus i0 O) (plus_n_O i0))))))) (\lambda -(n: nat).(\lambda (_: (((drop n O (CHead c2 k t) c3) \to (\forall (e2: -C).((drop (S i0) O c3 e2) \to ((le O (S i0)) \to (drop (S (plus i0 n)) O -(CHead c2 k t) e2))))))).(\lambda (H0: (drop (S n) O (CHead c2 k t) -c3)).(\lambda (e2: C).(\lambda (H1: (drop (S i0) O c3 e2)).(\lambda (H2: (le -O (S i0))).(eq_ind nat (S (plus i0 n)) (\lambda (n0: nat).(drop (S n0) O -(CHead c2 k t) e2)) (drop_drop k (S (plus i0 n)) c2 e2 (eq_ind_r nat (S (r k -(plus i0 n))) (\lambda (n0: nat).(drop n0 O c2 e2)) (eq_ind_r nat (plus i0 (r -k n)) (\lambda (n0: nat).(drop (S n0) O c2 e2)) (IHc c3 O (r k n) -(drop_gen_drop k c2 c3 t n H0) e2 H1 H2) (r k (plus i0 n)) (r_plus_sym k i0 -n)) (r k (S (plus i0 n))) (r_S k (plus i0 n))) t) (plus i0 (S n)) (plus_n_Sm -i0 n)))))))) h)) (\lambda (d0: nat).(\lambda (IHd: ((\forall (h: nat).((drop -h d0 (CHead c2 k t) c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to ((le -d0 (S i0)) \to (drop (S (plus i0 h)) O (CHead c2 k t) e2)))))))).(\lambda (h: -nat).(\lambda (H: (drop h (S d0) (CHead c2 k t) c3)).(\lambda (e2: -C).(\lambda (H0: (drop (S i0) O c3 e2)).(\lambda (H1: (le (S d0) (S -i0))).(ex3_2_ind C T (\lambda (e: C).(\lambda (v: T).(eq C c3 (CHead e k -v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r k d0) v)))) (\lambda -(e: C).(\lambda (_: T).(drop h (r k d0) c2 e))) (drop (S (plus i0 h)) O -(CHead c2 k t) e2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H2: (eq C c3 -(CHead x0 k x1))).(\lambda (H3: (eq T t (lift h (r k d0) x1))).(\lambda (H4: -(drop h (r k d0) c2 x0)).(let H5 \def (eq_ind C c3 (\lambda (c: C).(\forall -(h0: nat).((drop h0 d0 (CHead c2 k t) c) \to (\forall (e3: C).((drop (S i0) O -c e3) \to ((le d0 (S i0)) \to (drop (S (plus i0 h0)) O (CHead c2 k t) -e3))))))) IHd (CHead x0 k x1) H2) in (let H6 \def (eq_ind C c3 (\lambda (c: -C).(drop (S i0) O c e2)) H0 (CHead x0 k x1) H2) in (let H7 \def (eq_ind T t -(\lambda (t0: T).(\forall (h0: nat).((drop h0 d0 (CHead c2 k t0) (CHead x0 k -x1)) \to (\forall (e3: C).((drop (S i0) O (CHead x0 k x1) e3) \to ((le d0 (S -i0)) \to (drop (S (plus i0 h0)) O (CHead c2 k t0) e3))))))) H5 (lift h (r k -d0) x1) H3) in (eq_ind_r T (lift h (r k d0) x1) (\lambda (t0: T).(drop (S -(plus i0 h)) O (CHead c2 k t0) e2)) (drop_drop k (plus i0 h) c2 e2 (K_ind -(\lambda (k0: K).((drop h (r k0 d0) c2 x0) \to ((drop (r k0 i0) O x0 e2) \to -(drop (r k0 (plus i0 h)) O c2 e2)))) (\lambda (b: B).(\lambda (H8: (drop h (r -(Bind b) d0) c2 x0)).(\lambda (H9: (drop (r (Bind b) i0) O x0 e2)).(IHi c2 x0 -(r (Bind b) d0) h H8 e2 H9 (le_S_n (r (Bind b) d0) i0 H1))))) (\lambda (f: -F).(\lambda (H8: (drop h (r (Flat f) d0) c2 x0)).(\lambda (H9: (drop (r (Flat -f) i0) O x0 e2)).(IHc x0 (r (Flat f) d0) h H8 e2 H9 H1)))) k H4 -(drop_gen_drop k x0 e2 x1 i0 H6)) (lift h (r k d0) x1)) t H3))))))))) -(drop_gen_skip_l c2 c3 t h d0 k H))))))))) d))))))) c1)))) i). -(* COMMENTS -Initial nodes: 2020 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/drop1/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/drop1/defs.ma deleted file mode 100644 index c0e14f438..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/drop1/defs.ma +++ /dev/null @@ -1,35 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/drop/defs.ma". - -include "Basic-1/lift1/defs.ma". - -inductive drop1: PList \to (C \to (C \to Prop)) \def -| drop1_nil: \forall (c: C).(drop1 PNil c c) -| drop1_cons: \forall (c1: C).(\forall (c2: C).(\forall (h: nat).(\forall (d: -nat).((drop h d c1 c2) \to (\forall (c3: C).(\forall (hds: PList).((drop1 hds -c2 c3) \to (drop1 (PCons h d hds) c1 c3)))))))). - -definition ptrans: - PList \to (nat \to PList) -\def - let rec ptrans (hds: PList) on hds: (nat \to PList) \def (\lambda (i: -nat).(match hds with [PNil \Rightarrow PNil | (PCons h d hds0) \Rightarrow -(let j \def (trans hds0 i) in (let q \def (ptrans hds0 i) in (match (blt j d) -with [true \Rightarrow (PCons h (minus d (S j)) q) | false \Rightarrow -q])))])) in ptrans. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/drop1/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/drop1/fwd.ma deleted file mode 100644 index 6e4d1789e..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/drop1/fwd.ma +++ /dev/null @@ -1,81 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/drop1/defs.ma". - -theorem drop1_gen_pnil: - \forall (c1: C).(\forall (c2: C).((drop1 PNil c1 c2) \to (eq C c1 c2))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (drop1 PNil c1 c2)).(insert_eq -PList PNil (\lambda (p: PList).(drop1 p c1 c2)) (\lambda (_: PList).(eq C c1 -c2)) (\lambda (y: PList).(\lambda (H0: (drop1 y c1 c2)).(drop1_ind (\lambda -(p: PList).(\lambda (c: C).(\lambda (c0: C).((eq PList p PNil) \to (eq C c -c0))))) (\lambda (c: C).(\lambda (_: (eq PList PNil PNil)).(refl_equal C c))) -(\lambda (c3: C).(\lambda (c4: C).(\lambda (h: nat).(\lambda (d: -nat).(\lambda (_: (drop h d c3 c4)).(\lambda (c5: C).(\lambda (hds: -PList).(\lambda (_: (drop1 hds c4 c5)).(\lambda (_: (((eq PList hds PNil) \to -(eq C c4 c5)))).(\lambda (H4: (eq PList (PCons h d hds) PNil)).(let H5 \def -(eq_ind PList (PCons h d hds) (\lambda (ee: PList).(match ee in PList return -(\lambda (_: PList).Prop) with [PNil \Rightarrow False | (PCons _ _ _) -\Rightarrow True])) I PNil H4) in (False_ind (eq C c3 c5) H5)))))))))))) y c1 -c2 H0))) H))). -(* COMMENTS -Initial nodes: 198 -END *) - -theorem drop1_gen_pcons: - \forall (c1: C).(\forall (c3: C).(\forall (hds: PList).(\forall (h: -nat).(\forall (d: nat).((drop1 (PCons h d hds) c1 c3) \to (ex2 C (\lambda -(c2: C).(drop h d c1 c2)) (\lambda (c2: C).(drop1 hds c2 c3)))))))) -\def - \lambda (c1: C).(\lambda (c3: C).(\lambda (hds: PList).(\lambda (h: -nat).(\lambda (d: nat).(\lambda (H: (drop1 (PCons h d hds) c1 c3)).(insert_eq -PList (PCons h d hds) (\lambda (p: PList).(drop1 p c1 c3)) (\lambda (_: -PList).(ex2 C (\lambda (c2: C).(drop h d c1 c2)) (\lambda (c2: C).(drop1 hds -c2 c3)))) (\lambda (y: PList).(\lambda (H0: (drop1 y c1 c3)).(drop1_ind -(\lambda (p: PList).(\lambda (c: C).(\lambda (c0: C).((eq PList p (PCons h d -hds)) \to (ex2 C (\lambda (c2: C).(drop h d c c2)) (\lambda (c2: C).(drop1 -hds c2 c0))))))) (\lambda (c: C).(\lambda (H1: (eq PList PNil (PCons h d -hds))).(let H2 \def (eq_ind PList PNil (\lambda (ee: PList).(match ee in -PList return (\lambda (_: PList).Prop) with [PNil \Rightarrow True | (PCons _ -_ _) \Rightarrow False])) I (PCons h d hds) H1) in (False_ind (ex2 C (\lambda -(c2: C).(drop h d c c2)) (\lambda (c2: C).(drop1 hds c2 c))) H2)))) (\lambda -(c2: C).(\lambda (c4: C).(\lambda (h0: nat).(\lambda (d0: nat).(\lambda (H1: -(drop h0 d0 c2 c4)).(\lambda (c5: C).(\lambda (hds0: PList).(\lambda (H2: -(drop1 hds0 c4 c5)).(\lambda (H3: (((eq PList hds0 (PCons h d hds)) \to (ex2 -C (\lambda (c6: C).(drop h d c4 c6)) (\lambda (c6: C).(drop1 hds c6 -c5)))))).(\lambda (H4: (eq PList (PCons h0 d0 hds0) (PCons h d hds))).(let H5 -\def (f_equal PList nat (\lambda (e: PList).(match e in PList return (\lambda -(_: PList).nat) with [PNil \Rightarrow h0 | (PCons n _ _) \Rightarrow n])) -(PCons h0 d0 hds0) (PCons h d hds) H4) in ((let H6 \def (f_equal PList nat -(\lambda (e: PList).(match e in PList return (\lambda (_: PList).nat) with -[PNil \Rightarrow d0 | (PCons _ n _) \Rightarrow n])) (PCons h0 d0 hds0) -(PCons h d hds) H4) in ((let H7 \def (f_equal PList PList (\lambda (e: -PList).(match e in PList return (\lambda (_: PList).PList) with [PNil -\Rightarrow hds0 | (PCons _ _ p) \Rightarrow p])) (PCons h0 d0 hds0) (PCons h -d hds) H4) in (\lambda (H8: (eq nat d0 d)).(\lambda (H9: (eq nat h0 h)).(let -H10 \def (eq_ind PList hds0 (\lambda (p: PList).((eq PList p (PCons h d hds)) -\to (ex2 C (\lambda (c6: C).(drop h d c4 c6)) (\lambda (c6: C).(drop1 hds c6 -c5))))) H3 hds H7) in (let H11 \def (eq_ind PList hds0 (\lambda (p: -PList).(drop1 p c4 c5)) H2 hds H7) in (let H12 \def (eq_ind nat d0 (\lambda -(n: nat).(drop h0 n c2 c4)) H1 d H8) in (let H13 \def (eq_ind nat h0 (\lambda -(n: nat).(drop n d c2 c4)) H12 h H9) in (ex_intro2 C (\lambda (c6: C).(drop h -d c2 c6)) (\lambda (c6: C).(drop1 hds c6 c5)) c4 H13 H11)))))))) H6)) -H5)))))))))))) y c1 c3 H0))) H)))))). -(* COMMENTS -Initial nodes: 587 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/drop1/getl.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/drop1/getl.ma deleted file mode 100644 index dbee79850..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/drop1/getl.ma +++ /dev/null @@ -1,110 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/drop1/fwd.ma". - -include "Basic-1/getl/drop.ma". - -theorem drop1_getl_trans: - \forall (hds: PList).(\forall (c1: C).(\forall (c2: C).((drop1 hds c2 c1) -\to (\forall (b: B).(\forall (e1: C).(\forall (v: T).(\forall (i: nat).((getl -i c1 (CHead e1 (Bind b) v)) \to (ex2 C (\lambda (e2: C).(drop1 (ptrans hds i) -e2 e1)) (\lambda (e2: C).(getl (trans hds i) c2 (CHead e2 (Bind b) (lift1 -(ptrans hds i) v))))))))))))) -\def - \lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (c1: -C).(\forall (c2: C).((drop1 p c2 c1) \to (\forall (b: B).(\forall (e1: -C).(\forall (v: T).(\forall (i: nat).((getl i c1 (CHead e1 (Bind b) v)) \to -(ex2 C (\lambda (e2: C).(drop1 (ptrans p i) e2 e1)) (\lambda (e2: C).(getl -(trans p i) c2 (CHead e2 (Bind b) (lift1 (ptrans p i) v)))))))))))))) -(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (drop1 PNil c2 c1)).(\lambda -(b: B).(\lambda (e1: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H0: (getl -i c1 (CHead e1 (Bind b) v))).(let H_y \def (drop1_gen_pnil c2 c1 H) in -(eq_ind_r C c1 (\lambda (c: C).(ex2 C (\lambda (e2: C).(drop1 PNil e2 e1)) -(\lambda (e2: C).(getl i c (CHead e2 (Bind b) v))))) (ex_intro2 C (\lambda -(e2: C).(drop1 PNil e2 e1)) (\lambda (e2: C).(getl i c1 (CHead e2 (Bind b) -v))) e1 (drop1_nil e1) H0) c2 H_y)))))))))) (\lambda (h: nat).(\lambda (d: -nat).(\lambda (hds0: PList).(\lambda (H: ((\forall (c1: C).(\forall (c2: -C).((drop1 hds0 c2 c1) \to (\forall (b: B).(\forall (e1: C).(\forall (v: -T).(\forall (i: nat).((getl i c1 (CHead e1 (Bind b) v)) \to (ex2 C (\lambda -(e2: C).(drop1 (ptrans hds0 i) e2 e1)) (\lambda (e2: C).(getl (trans hds0 i) -c2 (CHead e2 (Bind b) (lift1 (ptrans hds0 i) v))))))))))))))).(\lambda (c1: -C).(\lambda (c2: C).(\lambda (H0: (drop1 (PCons h d hds0) c2 c1)).(\lambda -(b: B).(\lambda (e1: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H1: (getl -i c1 (CHead e1 (Bind b) v))).(let H_x \def (drop1_gen_pcons c2 c1 hds0 h d -H0) in (let H2 \def H_x in (ex2_ind C (\lambda (c3: C).(drop h d c2 c3)) -(\lambda (c3: C).(drop1 hds0 c3 c1)) (ex2 C (\lambda (e2: C).(drop1 (match -(blt (trans hds0 i) d) with [true \Rightarrow (PCons h (minus d (S (trans -hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)]) e2 e1)) -(\lambda (e2: C).(getl (match (blt (trans hds0 i) d) with [true \Rightarrow -(trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 -(Bind b) (lift1 (match (blt (trans hds0 i) d) with [true \Rightarrow (PCons h -(minus d (S (trans hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans -hds0 i)]) v))))) (\lambda (x: C).(\lambda (H3: (drop h d c2 x)).(\lambda (H4: -(drop1 hds0 x c1)).(xinduction bool (blt (trans hds0 i) d) (\lambda (b0: -bool).(ex2 C (\lambda (e2: C).(drop1 (match b0 with [true \Rightarrow (PCons -h (minus d (S (trans hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans -hds0 i)]) e2 e1)) (\lambda (e2: C).(getl (match b0 with [true \Rightarrow -(trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 -(Bind b) (lift1 (match b0 with [true \Rightarrow (PCons h (minus d (S (trans -hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)]) v)))))) -(\lambda (x_x: bool).(bool_ind (\lambda (b0: bool).((eq bool (blt (trans hds0 -i) d) b0) \to (ex2 C (\lambda (e2: C).(drop1 (match b0 with [true \Rightarrow -(PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) | false \Rightarrow -(ptrans hds0 i)]) e2 e1)) (\lambda (e2: C).(getl (match b0 with [true -\Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2 -(CHead e2 (Bind b) (lift1 (match b0 with [true \Rightarrow (PCons h (minus d -(S (trans hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)]) -v))))))) (\lambda (H5: (eq bool (blt (trans hds0 i) d) true)).(let H_x0 \def -(H c1 x H4 b e1 v i H1) in (let H6 \def H_x0 in (ex2_ind C (\lambda (e2: -C).(drop1 (ptrans hds0 i) e2 e1)) (\lambda (e2: C).(getl (trans hds0 i) x -(CHead e2 (Bind b) (lift1 (ptrans hds0 i) v)))) (ex2 C (\lambda (e2: -C).(drop1 (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) e2 e1)) -(\lambda (e2: C).(getl (trans hds0 i) c2 (CHead e2 (Bind b) (lift1 (PCons h -(minus d (S (trans hds0 i))) (ptrans hds0 i)) v))))) (\lambda (x0: -C).(\lambda (H7: (drop1 (ptrans hds0 i) x0 e1)).(\lambda (H8: (getl (trans -hds0 i) x (CHead x0 (Bind b) (lift1 (ptrans hds0 i) v)))).(let H_x1 \def -(drop_getl_trans_lt (trans hds0 i) d (blt_lt d (trans hds0 i) H5) c2 x h H3 b -x0 (lift1 (ptrans hds0 i) v) H8) in (let H9 \def H_x1 in (ex2_ind C (\lambda -(e2: C).(getl (trans hds0 i) c2 (CHead e2 (Bind b) (lift h (minus d (S (trans -hds0 i))) (lift1 (ptrans hds0 i) v))))) (\lambda (e2: C).(drop h (minus d (S -(trans hds0 i))) e2 x0)) (ex2 C (\lambda (e2: C).(drop1 (PCons h (minus d (S -(trans hds0 i))) (ptrans hds0 i)) e2 e1)) (\lambda (e2: C).(getl (trans hds0 -i) c2 (CHead e2 (Bind b) (lift1 (PCons h (minus d (S (trans hds0 i))) (ptrans -hds0 i)) v))))) (\lambda (x1: C).(\lambda (H10: (getl (trans hds0 i) c2 -(CHead x1 (Bind b) (lift h (minus d (S (trans hds0 i))) (lift1 (ptrans hds0 -i) v))))).(\lambda (H11: (drop h (minus d (S (trans hds0 i))) x1 -x0)).(ex_intro2 C (\lambda (e2: C).(drop1 (PCons h (minus d (S (trans hds0 -i))) (ptrans hds0 i)) e2 e1)) (\lambda (e2: C).(getl (trans hds0 i) c2 (CHead -e2 (Bind b) (lift1 (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) -v)))) x1 (drop1_cons x1 x0 h (minus d (S (trans hds0 i))) H11 e1 (ptrans hds0 -i) H7) H10)))) H9)))))) H6)))) (\lambda (H5: (eq bool (blt (trans hds0 i) d) -false)).(let H_x0 \def (H c1 x H4 b e1 v i H1) in (let H6 \def H_x0 in -(ex2_ind C (\lambda (e2: C).(drop1 (ptrans hds0 i) e2 e1)) (\lambda (e2: -C).(getl (trans hds0 i) x (CHead e2 (Bind b) (lift1 (ptrans hds0 i) v)))) -(ex2 C (\lambda (e2: C).(drop1 (ptrans hds0 i) e2 e1)) (\lambda (e2: C).(getl -(plus (trans hds0 i) h) c2 (CHead e2 (Bind b) (lift1 (ptrans hds0 i) v))))) -(\lambda (x0: C).(\lambda (H7: (drop1 (ptrans hds0 i) x0 e1)).(\lambda (H8: -(getl (trans hds0 i) x (CHead x0 (Bind b) (lift1 (ptrans hds0 i) v)))).(let -H9 \def (drop_getl_trans_ge (trans hds0 i) c2 x d h H3 (CHead x0 (Bind b) -(lift1 (ptrans hds0 i) v)) H8) in (ex_intro2 C (\lambda (e2: C).(drop1 -(ptrans hds0 i) e2 e1)) (\lambda (e2: C).(getl (plus (trans hds0 i) h) c2 -(CHead e2 (Bind b) (lift1 (ptrans hds0 i) v)))) x0 H7 (H9 (bge_le d (trans -hds0 i) H5))))))) H6)))) x_x)))))) H2))))))))))))))) hds). -(* COMMENTS -Initial nodes: 1674 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/drop1/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/drop1/props.ma deleted file mode 100644 index 240219b95..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/drop1/props.ma +++ /dev/null @@ -1,97 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/drop1/fwd.ma". - -include "Basic-1/drop/props.ma". - -include "Basic-1/getl/defs.ma". - -theorem drop1_skip_bind: - \forall (b: B).(\forall (e: C).(\forall (hds: PList).(\forall (c: -C).(\forall (u: T).((drop1 hds c e) \to (drop1 (Ss hds) (CHead c (Bind b) -(lift1 hds u)) (CHead e (Bind b) u))))))) -\def - \lambda (b: B).(\lambda (e: C).(\lambda (hds: PList).(PList_ind (\lambda (p: -PList).(\forall (c: C).(\forall (u: T).((drop1 p c e) \to (drop1 (Ss p) -(CHead c (Bind b) (lift1 p u)) (CHead e (Bind b) u)))))) (\lambda (c: -C).(\lambda (u: T).(\lambda (H: (drop1 PNil c e)).(let H_y \def -(drop1_gen_pnil c e H) in (eq_ind_r C e (\lambda (c0: C).(drop1 PNil (CHead -c0 (Bind b) u) (CHead e (Bind b) u))) (drop1_nil (CHead e (Bind b) u)) c -H_y))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda -(H: ((\forall (c: C).(\forall (u: T).((drop1 p c e) \to (drop1 (Ss p) (CHead -c (Bind b) (lift1 p u)) (CHead e (Bind b) u))))))).(\lambda (c: C).(\lambda -(u: T).(\lambda (H0: (drop1 (PCons n n0 p) c e)).(let H_x \def -(drop1_gen_pcons c e p n n0 H0) in (let H1 \def H_x in (ex2_ind C (\lambda -(c2: C).(drop n n0 c c2)) (\lambda (c2: C).(drop1 p c2 e)) (drop1 (PCons n (S -n0) (Ss p)) (CHead c (Bind b) (lift n n0 (lift1 p u))) (CHead e (Bind b) u)) -(\lambda (x: C).(\lambda (H2: (drop n n0 c x)).(\lambda (H3: (drop1 p x -e)).(drop1_cons (CHead c (Bind b) (lift n n0 (lift1 p u))) (CHead x (Bind b) -(lift1 p u)) n (S n0) (drop_skip_bind n n0 c x H2 b (lift1 p u)) (CHead e -(Bind b) u) (Ss p) (H x u H3))))) H1)))))))))) hds))). -(* COMMENTS -Initial nodes: 379 -END *) - -theorem drop1_cons_tail: - \forall (c2: C).(\forall (c3: C).(\forall (h: nat).(\forall (d: nat).((drop -h d c2 c3) \to (\forall (hds: PList).(\forall (c1: C).((drop1 hds c1 c2) \to -(drop1 (PConsTail hds h d) c1 c3)))))))) -\def - \lambda (c2: C).(\lambda (c3: C).(\lambda (h: nat).(\lambda (d: -nat).(\lambda (H: (drop h d c2 c3)).(\lambda (hds: PList).(PList_ind (\lambda -(p: PList).(\forall (c1: C).((drop1 p c1 c2) \to (drop1 (PConsTail p h d) c1 -c3)))) (\lambda (c1: C).(\lambda (H0: (drop1 PNil c1 c2)).(let H_y \def -(drop1_gen_pnil c1 c2 H0) in (eq_ind_r C c2 (\lambda (c: C).(drop1 (PCons h d -PNil) c c3)) (drop1_cons c2 c3 h d H c3 PNil (drop1_nil c3)) c1 H_y)))) -(\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H0: -((\forall (c1: C).((drop1 p c1 c2) \to (drop1 (PConsTail p h d) c1 -c3))))).(\lambda (c1: C).(\lambda (H1: (drop1 (PCons n n0 p) c1 c2)).(let H_x -\def (drop1_gen_pcons c1 c2 p n n0 H1) in (let H2 \def H_x in (ex2_ind C -(\lambda (c4: C).(drop n n0 c1 c4)) (\lambda (c4: C).(drop1 p c4 c2)) (drop1 -(PCons n n0 (PConsTail p h d)) c1 c3) (\lambda (x: C).(\lambda (H3: (drop n -n0 c1 x)).(\lambda (H4: (drop1 p x c2)).(drop1_cons c1 x n n0 H3 c3 -(PConsTail p h d) (H0 x H4))))) H2))))))))) hds)))))). -(* COMMENTS -Initial nodes: 271 -END *) - -theorem drop1_trans: - \forall (is1: PList).(\forall (c1: C).(\forall (c0: C).((drop1 is1 c1 c0) -\to (\forall (is2: PList).(\forall (c2: C).((drop1 is2 c0 c2) \to (drop1 -(papp is1 is2) c1 c2))))))) -\def - \lambda (is1: PList).(PList_ind (\lambda (p: PList).(\forall (c1: -C).(\forall (c0: C).((drop1 p c1 c0) \to (\forall (is2: PList).(\forall (c2: -C).((drop1 is2 c0 c2) \to (drop1 (papp p is2) c1 c2)))))))) (\lambda (c1: -C).(\lambda (c0: C).(\lambda (H: (drop1 PNil c1 c0)).(\lambda (is2: -PList).(\lambda (c2: C).(\lambda (H0: (drop1 is2 c0 c2)).(let H_y \def -(drop1_gen_pnil c1 c0 H) in (let H1 \def (eq_ind_r C c0 (\lambda (c: -C).(drop1 is2 c c2)) H0 c1 H_y) in H1)))))))) (\lambda (n: nat).(\lambda (n0: -nat).(\lambda (p: PList).(\lambda (H: ((\forall (c1: C).(\forall (c0: -C).((drop1 p c1 c0) \to (\forall (is2: PList).(\forall (c2: C).((drop1 is2 c0 -c2) \to (drop1 (papp p is2) c1 c2))))))))).(\lambda (c1: C).(\lambda (c0: -C).(\lambda (H0: (drop1 (PCons n n0 p) c1 c0)).(\lambda (is2: PList).(\lambda -(c2: C).(\lambda (H1: (drop1 is2 c0 c2)).(let H_x \def (drop1_gen_pcons c1 c0 -p n n0 H0) in (let H2 \def H_x in (ex2_ind C (\lambda (c3: C).(drop n n0 c1 -c3)) (\lambda (c3: C).(drop1 p c3 c0)) (drop1 (PCons n n0 (papp p is2)) c1 -c2) (\lambda (x: C).(\lambda (H3: (drop n n0 c1 x)).(\lambda (H4: (drop1 p x -c0)).(drop1_cons c1 x n n0 H3 c2 (papp p is2) (H x c0 H4 is2 c2 H1))))) -H2))))))))))))) is1). -(* COMMENTS -Initial nodes: 287 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/ex0/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/ex0/defs.ma deleted file mode 100644 index 6ab98b1af..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/ex0/defs.ma +++ /dev/null @@ -1,32 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/A/defs.ma". - -include "Basic-1/G/defs.ma". - -definition gz: - G -\def - mk_G S lt_n_Sn. - -inductive leqz: A \to (A \to Prop) \def -| leqz_sort: \forall (h1: nat).(\forall (h2: nat).(\forall (n1: nat).(\forall -(n2: nat).((eq nat (plus h1 n2) (plus h2 n1)) \to (leqz (ASort h1 n1) (ASort -h2 n2)))))) -| leqz_head: \forall (a1: A).(\forall (a2: A).((leqz a1 a2) \to (\forall (a3: -A).(\forall (a4: A).((leqz a3 a4) \to (leqz (AHead a1 a3) (AHead a2 a4))))))). - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/ex0/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/ex0/props.ma deleted file mode 100644 index 96dd77da3..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/ex0/props.ma +++ /dev/null @@ -1,207 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/ex0/defs.ma". - -include "Basic-1/leq/defs.ma". - -include "Basic-1/aplus/props.ma". - -theorem aplus_gz_le: - \forall (k: nat).(\forall (h: nat).(\forall (n: nat).((le h k) \to (eq A -(aplus gz (ASort h n) k) (ASort O (plus (minus k h) n)))))) -\def - \lambda (k: nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).(\forall (n0: -nat).((le h n) \to (eq A (aplus gz (ASort h n0) n) (ASort O (plus (minus n h) -n0))))))) (\lambda (h: nat).(\lambda (n: nat).(\lambda (H: (le h O)).(let H_y -\def (le_n_O_eq h H) in (eq_ind nat O (\lambda (n0: nat).(eq A (ASort n0 n) -(ASort O n))) (refl_equal A (ASort O n)) h H_y))))) (\lambda (k0: -nat).(\lambda (IH: ((\forall (h: nat).(\forall (n: nat).((le h k0) \to (eq A -(aplus gz (ASort h n) k0) (ASort O (plus (minus k0 h) n)))))))).(\lambda (h: -nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).((le n (S k0)) \to (eq A -(asucc gz (aplus gz (ASort n n0) k0)) (ASort O (plus (match n with [O -\Rightarrow (S k0) | (S l) \Rightarrow (minus k0 l)]) n0)))))) (\lambda (n: -nat).(\lambda (_: (le O (S k0))).(eq_ind A (aplus gz (asucc gz (ASort O n)) -k0) (\lambda (a: A).(eq A a (ASort O (S (plus k0 n))))) (eq_ind_r A (ASort O -(plus (minus k0 O) (S n))) (\lambda (a: A).(eq A a (ASort O (S (plus k0 -n))))) (eq_ind nat k0 (\lambda (n0: nat).(eq A (ASort O (plus n0 (S n))) -(ASort O (S (plus k0 n))))) (eq_ind nat (S (plus k0 n)) (\lambda (n0: -nat).(eq A (ASort O n0) (ASort O (S (plus k0 n))))) (refl_equal A (ASort O (S -(plus k0 n)))) (plus k0 (S n)) (plus_n_Sm k0 n)) (minus k0 O) (minus_n_O k0)) -(aplus gz (ASort O (S n)) k0) (IH O (S n) (le_O_n k0))) (asucc gz (aplus gz -(ASort O n) k0)) (aplus_asucc gz k0 (ASort O n))))) (\lambda (n: -nat).(\lambda (_: ((\forall (n0: nat).((le n (S k0)) \to (eq A (asucc gz -(aplus gz (ASort n n0) k0)) (ASort O (plus (match n with [O \Rightarrow (S -k0) | (S l) \Rightarrow (minus k0 l)]) n0))))))).(\lambda (n0: nat).(\lambda -(H0: (le (S n) (S k0))).(let H_y \def (le_S_n n k0 H0) in (eq_ind A (aplus gz -(ASort n n0) k0) (\lambda (a: A).(eq A (asucc gz (aplus gz (ASort (S n) n0) -k0)) a)) (eq_ind A (aplus gz (asucc gz (ASort (S n) n0)) k0) (\lambda (a: -A).(eq A a (aplus gz (ASort n n0) k0))) (refl_equal A (aplus gz (ASort n n0) -k0)) (asucc gz (aplus gz (ASort (S n) n0) k0)) (aplus_asucc gz k0 (ASort (S -n) n0))) (ASort O (plus (minus k0 n) n0)) (IH n n0 H_y))))))) h)))) k). -(* COMMENTS -Initial nodes: 683 -END *) - -theorem aplus_gz_ge: - \forall (n: nat).(\forall (k: nat).(\forall (h: nat).((le k h) \to (eq A -(aplus gz (ASort h n) k) (ASort (minus h k) n))))) -\def - \lambda (n: nat).(\lambda (k: nat).(nat_ind (\lambda (n0: nat).(\forall (h: -nat).((le n0 h) \to (eq A (aplus gz (ASort h n) n0) (ASort (minus h n0) -n))))) (\lambda (h: nat).(\lambda (_: (le O h)).(eq_ind nat h (\lambda (n0: -nat).(eq A (ASort h n) (ASort n0 n))) (refl_equal A (ASort h n)) (minus h O) -(minus_n_O h)))) (\lambda (k0: nat).(\lambda (IH: ((\forall (h: nat).((le k0 -h) \to (eq A (aplus gz (ASort h n) k0) (ASort (minus h k0) n)))))).(\lambda -(h: nat).(nat_ind (\lambda (n0: nat).((le (S k0) n0) \to (eq A (asucc gz -(aplus gz (ASort n0 n) k0)) (ASort (minus n0 (S k0)) n)))) (\lambda (H: (le -(S k0) O)).(ex2_ind nat (\lambda (n0: nat).(eq nat O (S n0))) (\lambda (n0: -nat).(le k0 n0)) (eq A (asucc gz (aplus gz (ASort O n) k0)) (ASort O n)) -(\lambda (x: nat).(\lambda (H0: (eq nat O (S x))).(\lambda (_: (le k0 -x)).(let H2 \def (eq_ind nat O (\lambda (ee: nat).(match ee in nat return -(\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) -I (S x) H0) in (False_ind (eq A (asucc gz (aplus gz (ASort O n) k0)) (ASort O -n)) H2))))) (le_gen_S k0 O H))) (\lambda (n0: nat).(\lambda (_: (((le (S k0) -n0) \to (eq A (asucc gz (aplus gz (ASort n0 n) k0)) (ASort (minus n0 (S k0)) -n))))).(\lambda (H0: (le (S k0) (S n0))).(let H_y \def (le_S_n k0 n0 H0) in -(eq_ind A (aplus gz (ASort n0 n) k0) (\lambda (a: A).(eq A (asucc gz (aplus -gz (ASort (S n0) n) k0)) a)) (eq_ind A (aplus gz (asucc gz (ASort (S n0) n)) -k0) (\lambda (a: A).(eq A a (aplus gz (ASort n0 n) k0))) (refl_equal A (aplus -gz (ASort n0 n) k0)) (asucc gz (aplus gz (ASort (S n0) n) k0)) (aplus_asucc -gz k0 (ASort (S n0) n))) (ASort (minus n0 k0) n) (IH n0 H_y)))))) h)))) k)). -(* COMMENTS -Initial nodes: 524 -END *) - -theorem next_plus_gz: - \forall (n: nat).(\forall (h: nat).(eq nat (next_plus gz n h) (plus h n))) -\def - \lambda (n: nat).(\lambda (h: nat).(nat_ind (\lambda (n0: nat).(eq nat -(next_plus gz n n0) (plus n0 n))) (refl_equal nat n) (\lambda (n0: -nat).(\lambda (H: (eq nat (next_plus gz n n0) (plus n0 n))).(f_equal nat nat -S (next_plus gz n n0) (plus n0 n) H))) h)). -(* COMMENTS -Initial nodes: 77 -END *) - -theorem leqz_leq: - \forall (a1: A).(\forall (a2: A).((leq gz a1 a2) \to (leqz a1 a2))) -\def - \lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq gz a1 a2)).(leq_ind gz -(\lambda (a: A).(\lambda (a0: A).(leqz a a0))) (\lambda (h1: nat).(\lambda -(h2: nat).(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda -(H0: (eq A (aplus gz (ASort h1 n1) k) (aplus gz (ASort h2 n2) k))).(lt_le_e k -h1 (leqz (ASort h1 n1) (ASort h2 n2)) (\lambda (H1: (lt k h1)).(lt_le_e k h2 -(leqz (ASort h1 n1) (ASort h2 n2)) (\lambda (H2: (lt k h2)).(let H3 \def -(eq_ind A (aplus gz (ASort h1 n1) k) (\lambda (a: A).(eq A a (aplus gz (ASort -h2 n2) k))) H0 (ASort (minus h1 k) n1) (aplus_gz_ge n1 k h1 (le_S_n k h1 -(le_S (S k) h1 H1)))) in (let H4 \def (eq_ind A (aplus gz (ASort h2 n2) k) -(\lambda (a: A).(eq A (ASort (minus h1 k) n1) a)) H3 (ASort (minus h2 k) n2) -(aplus_gz_ge n2 k h2 (le_S_n k h2 (le_S (S k) h2 H2)))) in (let H5 \def -(f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with -[(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow ((let rec minus (n: nat) -on n: (nat \to nat) \def (\lambda (m: nat).(match n with [O \Rightarrow O | -(S k0) \Rightarrow (match m with [O \Rightarrow (S k0) | (S l) \Rightarrow -(minus k0 l)])])) in minus) h1 k)])) (ASort (minus h1 k) n1) (ASort (minus h2 -k) n2) H4) in ((let H6 \def (f_equal A nat (\lambda (e: A).(match e in A -return (\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) -\Rightarrow n1])) (ASort (minus h1 k) n1) (ASort (minus h2 k) n2) H4) in -(\lambda (H7: (eq nat (minus h1 k) (minus h2 k))).(eq_ind nat n1 (\lambda (n: -nat).(leqz (ASort h1 n1) (ASort h2 n))) (eq_ind nat h1 (\lambda (n: -nat).(leqz (ASort h1 n1) (ASort n n1))) (leqz_sort h1 h1 n1 n1 (refl_equal -nat (plus h1 n1))) h2 (minus_minus k h1 h2 (le_S_n k h1 (le_S (S k) h1 H1)) -(le_S_n k h2 (le_S (S k) h2 H2)) H7)) n2 H6))) H5))))) (\lambda (H2: (le h2 -k)).(let H3 \def (eq_ind A (aplus gz (ASort h1 n1) k) (\lambda (a: A).(eq A a -(aplus gz (ASort h2 n2) k))) H0 (ASort (minus h1 k) n1) (aplus_gz_ge n1 k h1 -(le_S_n k h1 (le_S (S k) h1 H1)))) in (let H4 \def (eq_ind A (aplus gz (ASort -h2 n2) k) (\lambda (a: A).(eq A (ASort (minus h1 k) n1) a)) H3 (ASort O (plus -(minus k h2) n2)) (aplus_gz_le k h2 n2 H2)) in (let H5 \def (eq_ind nat -(minus h1 k) (\lambda (n: nat).(eq A (ASort n n1) (ASort O (plus (minus k h2) -n2)))) H4 (S (minus h1 (S k))) (minus_x_Sy h1 k H1)) in (let H6 \def (eq_ind -A (ASort (S (minus h1 (S k))) n1) (\lambda (ee: A).(match ee in A return -(\lambda (_: A).Prop) with [(ASort n _) \Rightarrow (match n in nat return -(\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True]) -| (AHead _ _) \Rightarrow False])) I (ASort O (plus (minus k h2) n2)) H5) in -(False_ind (leqz (ASort h1 n1) (ASort h2 n2)) H6)))))))) (\lambda (H1: (le h1 -k)).(lt_le_e k h2 (leqz (ASort h1 n1) (ASort h2 n2)) (\lambda (H2: (lt k -h2)).(let H3 \def (eq_ind A (aplus gz (ASort h1 n1) k) (\lambda (a: A).(eq A -a (aplus gz (ASort h2 n2) k))) H0 (ASort O (plus (minus k h1) n1)) -(aplus_gz_le k h1 n1 H1)) in (let H4 \def (eq_ind A (aplus gz (ASort h2 n2) -k) (\lambda (a: A).(eq A (ASort O (plus (minus k h1) n1)) a)) H3 (ASort -(minus h2 k) n2) (aplus_gz_ge n2 k h2 (le_S_n k h2 (le_S (S k) h2 H2)))) in -(let H5 \def (sym_eq A (ASort O (plus (minus k h1) n1)) (ASort (minus h2 k) -n2) H4) in (let H6 \def (eq_ind nat (minus h2 k) (\lambda (n: nat).(eq A -(ASort n n2) (ASort O (plus (minus k h1) n1)))) H5 (S (minus h2 (S k))) -(minus_x_Sy h2 k H2)) in (let H7 \def (eq_ind A (ASort (S (minus h2 (S k))) -n2) (\lambda (ee: A).(match ee in A return (\lambda (_: A).Prop) with [(ASort -n _) \Rightarrow (match n in nat return (\lambda (_: nat).Prop) with [O -\Rightarrow False | (S _) \Rightarrow True]) | (AHead _ _) \Rightarrow -False])) I (ASort O (plus (minus k h1) n1)) H6) in (False_ind (leqz (ASort h1 -n1) (ASort h2 n2)) H7))))))) (\lambda (H2: (le h2 k)).(let H3 \def (eq_ind A -(aplus gz (ASort h1 n1) k) (\lambda (a: A).(eq A a (aplus gz (ASort h2 n2) -k))) H0 (ASort O (plus (minus k h1) n1)) (aplus_gz_le k h1 n1 H1)) in (let H4 -\def (eq_ind A (aplus gz (ASort h2 n2) k) (\lambda (a: A).(eq A (ASort O -(plus (minus k h1) n1)) a)) H3 (ASort O (plus (minus k h2) n2)) (aplus_gz_le -k h2 n2 H2)) in (let H5 \def (f_equal A nat (\lambda (e: A).(match e in A -return (\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) -\Rightarrow ((let rec plus (n: nat) on n: (nat \to nat) \def (\lambda (m: -nat).(match n with [O \Rightarrow m | (S p) \Rightarrow (S (plus p m))])) in -plus) (minus k h1) n1)])) (ASort O (plus (minus k h1) n1)) (ASort O (plus -(minus k h2) n2)) H4) in (let H_y \def (plus_plus k h1 h2 n1 n2 H1 H2 H5) in -(leqz_sort h1 h2 n1 n2 H_y))))))))))))))) (\lambda (a0: A).(\lambda (a3: -A).(\lambda (_: (leq gz a0 a3)).(\lambda (H1: (leqz a0 a3)).(\lambda (a4: -A).(\lambda (a5: A).(\lambda (_: (leq gz a4 a5)).(\lambda (H3: (leqz a4 -a5)).(leqz_head a0 a3 H1 a4 a5 H3))))))))) a1 a2 H))). -(* COMMENTS -Initial nodes: 1375 -END *) - -theorem leq_leqz: - \forall (a1: A).(\forall (a2: A).((leqz a1 a2) \to (leq gz a1 a2))) -\def - \lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leqz a1 a2)).(leqz_ind -(\lambda (a: A).(\lambda (a0: A).(leq gz a a0))) (\lambda (h1: nat).(\lambda -(h2: nat).(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (H0: (eq nat (plus -h1 n2) (plus h2 n1))).(leq_sort gz h1 h2 n1 n2 (plus h1 h2) (eq_ind_r A -(ASort (minus h1 (plus h1 h2)) (next_plus gz n1 (minus (plus h1 h2) h1))) -(\lambda (a: A).(eq A a (aplus gz (ASort h2 n2) (plus h1 h2)))) (eq_ind_r A -(ASort (minus h2 (plus h1 h2)) (next_plus gz n2 (minus (plus h1 h2) h2))) -(\lambda (a: A).(eq A (ASort (minus h1 (plus h1 h2)) (next_plus gz n1 (minus -(plus h1 h2) h1))) a)) (eq_ind_r nat h2 (\lambda (n: nat).(eq A (ASort (minus -h1 (plus h1 h2)) (next_plus gz n1 n)) (ASort (minus h2 (plus h1 h2)) -(next_plus gz n2 (minus (plus h1 h2) h2))))) (eq_ind_r nat h1 (\lambda (n: -nat).(eq A (ASort (minus h1 (plus h1 h2)) (next_plus gz n1 h2)) (ASort (minus -h2 (plus h1 h2)) (next_plus gz n2 n)))) (eq_ind_r nat O (\lambda (n: nat).(eq -A (ASort n (next_plus gz n1 h2)) (ASort (minus h2 (plus h1 h2)) (next_plus gz -n2 h1)))) (eq_ind_r nat O (\lambda (n: nat).(eq A (ASort O (next_plus gz n1 -h2)) (ASort n (next_plus gz n2 h1)))) (eq_ind_r nat (plus h2 n1) (\lambda (n: -nat).(eq A (ASort O n) (ASort O (next_plus gz n2 h1)))) (eq_ind_r nat (plus -h1 n2) (\lambda (n: nat).(eq A (ASort O (plus h2 n1)) (ASort O n))) (f_equal -nat A (ASort O) (plus h2 n1) (plus h1 n2) (sym_eq nat (plus h1 n2) (plus h2 -n1) H0)) (next_plus gz n2 h1) (next_plus_gz n2 h1)) (next_plus gz n1 h2) -(next_plus_gz n1 h2)) (minus h2 (plus h1 h2)) (O_minus h2 (plus h1 h2) -(le_plus_r h1 h2))) (minus h1 (plus h1 h2)) (O_minus h1 (plus h1 h2) -(le_plus_l h1 h2))) (minus (plus h1 h2) h2) (minus_plus_r h1 h2)) (minus -(plus h1 h2) h1) (minus_plus h1 h2)) (aplus gz (ASort h2 n2) (plus h1 h2)) -(aplus_asort_simpl gz (plus h1 h2) h2 n2)) (aplus gz (ASort h1 n1) (plus h1 -h2)) (aplus_asort_simpl gz (plus h1 h2) h1 n1)))))))) (\lambda (a0: -A).(\lambda (a3: A).(\lambda (_: (leqz a0 a3)).(\lambda (H1: (leq gz a0 -a3)).(\lambda (a4: A).(\lambda (a5: A).(\lambda (_: (leqz a4 a5)).(\lambda -(H3: (leq gz a4 a5)).(leq_head gz a0 a3 H1 a4 a5 H3))))))))) a1 a2 H))). -(* COMMENTS -Initial nodes: 717 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/ex1/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/ex1/defs.ma deleted file mode 100644 index dc18f6fc2..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/ex1/defs.ma +++ /dev/null @@ -1,29 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/C/defs.ma". - -definition ex1_c: - C -\def - CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) -(Bind Abst) (TLRef O). - -definition ex1_t: - T -\def - THead (Flat Appl) (TLRef O) (THead (Bind Abst) (TLRef (S (S O))) (TSort O)). - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/ex1/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/ex1/props.ma deleted file mode 100644 index 5c442f5bb..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/ex1/props.ma +++ /dev/null @@ -1,536 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/ex1/defs.ma". - -include "Basic-1/ty3/fwd.ma". - -include "Basic-1/pc3/fwd.ma". - -include "Basic-1/nf2/pr3.ma". - -include "Basic-1/nf2/props.ma". - -include "Basic-1/arity/defs.ma". - -include "Basic-1/leq/props.ma". - -theorem ex1__leq_sort_SS: - \forall (g: G).(\forall (k: nat).(\forall (n: nat).(leq g (ASort k n) (asucc -g (asucc g (ASort (S (S k)) n)))))) -\def - \lambda (g: G).(\lambda (k: nat).(\lambda (n: nat).(leq_refl g (asucc g -(asucc g (ASort (S (S k)) n)))))). -(* COMMENTS -Initial nodes: 27 -END *) - -theorem ex1_arity: - \forall (g: G).(arity g ex1_c ex1_t (ASort O O)) -\def - \lambda (g: G).(arity_appl g (CHead (CHead (CHead (CSort O) (Bind Abst) -(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef O) (ASort (S -(S O)) O) (arity_abst g (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) -(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) (TLRef O) O (getl_refl Abst (CHead -(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (TLRef O)) -(ASort (S (S O)) O) (arity_abst g (CHead (CHead (CSort O) (Bind Abst) (TSort -O)) (Bind Abst) (TSort O)) (CHead (CSort O) (Bind Abst) (TSort O)) (TSort O) -O (getl_refl Abst (CHead (CSort O) (Bind Abst) (TSort O)) (TSort O)) (asucc g -(ASort (S (S O)) O)) (arity_repl g (CHead (CSort O) (Bind Abst) (TSort O)) -(TSort O) (ASort O O) (arity_sort g (CHead (CSort O) (Bind Abst) (TSort O)) -O) (asucc g (asucc g (ASort (S (S O)) O))) (ex1__leq_sort_SS g O O)))) (THead -(Bind Abst) (TLRef (S (S O))) (TSort O)) (ASort O O) (arity_head g (CHead -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind -Abst) (TLRef O)) (TLRef (S (S O))) (ASort (S (S O)) O) (arity_abst g (CHead -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind -Abst) (TLRef O)) (CSort O) (TSort O) (S (S O)) (getl_clear_bind Abst (CHead -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind -Abst) (TLRef O)) (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) -(TSort O)) (TLRef O) (clear_bind Abst (CHead (CHead (CSort O) (Bind Abst) -(TSort O)) (Bind Abst) (TSort O)) (TLRef O)) (CHead (CSort O) (Bind Abst) -(TSort O)) (S O) (getl_head (Bind Abst) O (CHead (CSort O) (Bind Abst) (TSort -O)) (CHead (CSort O) (Bind Abst) (TSort O)) (getl_refl Abst (CSort O) (TSort -O)) (TSort O))) (asucc g (ASort (S (S O)) O)) (arity_repl g (CSort O) (TSort -O) (ASort O O) (arity_sort g (CSort O) O) (asucc g (asucc g (ASort (S (S O)) -O))) (ex1__leq_sort_SS g O O))) (TSort O) (ASort O O) (arity_sort g (CHead -(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) -(Bind Abst) (TLRef O)) (Bind Abst) (TLRef (S (S O)))) O))). -(* COMMENTS -Initial nodes: 753 -END *) - -theorem ex1_ty3: - \forall (g: G).(\forall (u: T).((ty3 g ex1_c ex1_t u) \to (\forall (P: -Prop).P))) -\def - \lambda (g: G).(\lambda (u: T).(\lambda (H: (ty3 g (CHead (CHead (CHead -(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef -O)) (THead (Flat Appl) (TLRef O) (THead (Bind Abst) (TLRef (S (S O))) (TSort -O))) u)).(\lambda (P: Prop).(ex3_2_ind T T (\lambda (u0: T).(\lambda (t: -T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) -(TSort O)) (Bind Abst) (TLRef O)) (THead (Flat Appl) (TLRef O) (THead (Bind -Abst) u0 t)) u))) (\lambda (u0: T).(\lambda (t: T).(ty3 g (CHead (CHead -(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) -(TLRef O)) (THead (Bind Abst) (TLRef (S (S O))) (TSort O)) (THead (Bind Abst) -u0 t)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g (CHead (CHead (CHead (CSort -O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) -(TLRef O) u0))) P (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 (CHead -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind -Abst) (TLRef O)) (THead (Flat Appl) (TLRef O) (THead (Bind Abst) x0 x1)) -u)).(\lambda (H1: (ty3 g (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort -O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (THead (Bind Abst) (TLRef -(S (S O))) (TSort O)) (THead (Bind Abst) x0 x1))).(\lambda (H2: (ty3 g (CHead -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind -Abst) (TLRef O)) (TLRef O) x0)).(or_ind (ex3_3 C T T (\lambda (_: C).(\lambda -(_: T).(\lambda (t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort -O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S O) O t) x0)))) -(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl O (CHead (CHead (CHead -(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef -O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(t: T).(ty3 g e u0 t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: -T).(\lambda (_: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) -(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S O) O u0) x0)))) -(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl O (CHead (CHead (CHead -(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef -O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(t: T).(ty3 g e u0 t))))) P (\lambda (H3: (ex3_3 C T T (\lambda (_: -C).(\lambda (_: T).(\lambda (t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S O) O -t) x0)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl O (CHead -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind -Abst) (TLRef O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (t: T).(ty3 g e u0 t)))))).(ex3_3_ind C T T (\lambda (_: -C).(\lambda (_: T).(\lambda (t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S O) O -t) x0)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl O (CHead -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind -Abst) (TLRef O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (t: T).(ty3 g e u0 t)))) P (\lambda (x2: C).(\lambda (x3: -T).(\lambda (x4: T).(\lambda (_: (pc3 (CHead (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S O) O -x4) x0)).(\lambda (H5: (getl O (CHead (CHead (CHead (CSort O) (Bind Abst) -(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead x2 (Bind -Abbr) x3))).(\lambda (_: (ty3 g x2 x3 x4)).(ex3_2_ind T T (\lambda (t2: -T).(\lambda (_: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) -(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (THead (Bind Abst) (TLRef (S (S -O))) t2) (THead (Bind Abst) x0 x1)))) (\lambda (_: T).(\lambda (t: T).(ty3 g -(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) -(Bind Abst) (TLRef O)) (TLRef (S (S O))) t))) (\lambda (t2: T).(\lambda (_: -T).(ty3 g (CHead (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind -Abst) (TSort O)) (Bind Abst) (TLRef O)) (Bind Abst) (TLRef (S (S O)))) (TSort -O) t2))) P (\lambda (x5: T).(\lambda (x6: T).(\lambda (_: (pc3 (CHead (CHead -(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) -(TLRef O)) (THead (Bind Abst) (TLRef (S (S O))) x5) (THead (Bind Abst) x0 -x1))).(\lambda (H8: (ty3 g (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort -O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) -x6)).(\lambda (_: (ty3 g (CHead (CHead (CHead (CHead (CSort O) (Bind Abst) -(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (Bind Abst) (TLRef -(S (S O)))) (TSort O) x5)).(or_ind (ex3_3 C T T (\lambda (_: C).(\lambda (_: -T).(\lambda (t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) -(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O t) x6)))) -(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) (CHead -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind -Abst) (TLRef O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (t: T).(ty3 g e u0 t))))) (ex3_3 C T T (\lambda (_: C).(\lambda -(u0: T).(\lambda (_: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) -(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) -O u0) x6)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) -(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) -(Bind Abst) (TLRef O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda -(u0: T).(\lambda (t: T).(ty3 g e u0 t))))) P (\lambda (H10: (ex3_3 C T T -(\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 (CHead (CHead (CHead -(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef -O)) (lift (S (S (S O))) O t) x6)))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(_: T).(getl (S (S O)) (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) -(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead e (Bind Abbr) u0))))) -(\lambda (e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 -t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 -(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) -(Bind Abst) (TLRef O)) (lift (S (S (S O))) O t) x6)))) (\lambda (e: -C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) (CHead (CHead (CHead -(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef -O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(t: T).(ty3 g e u0 t)))) P (\lambda (x7: C).(\lambda (x8: T).(\lambda (x9: -T).(\lambda (_: (pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) -(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O x9) -x6)).(\lambda (H12: (getl (S (S O)) (CHead (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead x7 -(Bind Abbr) x8))).(\lambda (_: (ty3 g x7 x8 x9)).(let H14 \def (getl_gen_all -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead -x7 (Bind Abbr) x8) (r (Bind Abst) (S O)) (getl_gen_S (Bind Abst) (CHead -(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead x7 -(Bind Abbr) x8) (TLRef O) (S O) H12)) in (ex2_ind C (\lambda (e: C).(drop (S -O) O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) -e)) (\lambda (e: C).(clear e (CHead x7 (Bind Abbr) x8))) P (\lambda (x: -C).(\lambda (_: (drop (S O) O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) -(Bind Abst) (TSort O)) x)).(\lambda (_: (clear x (CHead x7 (Bind Abbr) -x8))).(let H17 \def (eq_ind C (CHead x2 (Bind Abbr) x3) (\lambda (ee: -C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow -False | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) -with [(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with -[Abbr \Rightarrow True | Abst \Rightarrow False | Void \Rightarrow False]) | -(Flat _) \Rightarrow False])])) I (CHead (CHead (CHead (CSort O) (Bind Abst) -(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (clear_gen_bind Abst -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead -x2 (Bind Abbr) x3) (TLRef O) (getl_gen_O (CHead (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead x2 -(Bind Abbr) x3) H5))) in (False_ind P H17))))) H14)))))))) H10)) (\lambda -(H10: (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 -(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) -(Bind Abst) (TLRef O)) (lift (S (S (S O))) O u0) x6)))) (\lambda (e: -C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) (CHead (CHead (CHead -(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef -O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(t: T).(ty3 g e u0 t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (u0: -T).(\lambda (_: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) -(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O u0) -x6)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) -(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) -(Bind Abst) (TLRef O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda -(u0: T).(\lambda (t: T).(ty3 g e u0 t)))) P (\lambda (x7: C).(\lambda (x8: -T).(\lambda (x9: T).(\lambda (_: (pc3 (CHead (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S -O))) O x8) x6)).(\lambda (H12: (getl (S (S O)) (CHead (CHead (CHead (CSort O) -(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead -x7 (Bind Abst) x8))).(\lambda (_: (ty3 g x7 x8 x9)).(let H14 \def -(getl_gen_all (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) -(TSort O)) (CHead x7 (Bind Abst) x8) (r (Bind Abst) (S O)) (getl_gen_S (Bind -Abst) (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) -(CHead x7 (Bind Abst) x8) (TLRef O) (S O) H12)) in (ex2_ind C (\lambda (e: -C).(drop (S O) O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) -(TSort O)) e)) (\lambda (e: C).(clear e (CHead x7 (Bind Abst) x8))) P -(\lambda (x: C).(\lambda (_: (drop (S O) O (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) x)).(\lambda (_: (clear x (CHead x7 -(Bind Abst) x8))).(let H17 \def (eq_ind C (CHead x2 (Bind Abbr) x3) (\lambda -(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind b) \Rightarrow (match b in B return (\lambda (_: -B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | Void -\Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead (CHead (CHead -(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef -O)) (clear_gen_bind Abst (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind -Abst) (TSort O)) (CHead x2 (Bind Abbr) x3) (TLRef O) (getl_gen_O (CHead -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind -Abst) (TLRef O)) (CHead x2 (Bind Abbr) x3) H5))) in (False_ind P H17))))) -H14)))))))) H10)) (ty3_gen_lref g (CHead (CHead (CHead (CSort O) (Bind Abst) -(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) x6 (S (S O)) -H8))))))) (ty3_gen_bind g Abst (CHead (CHead (CHead (CSort O) (Bind Abst) -(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) -(TSort O) (THead (Bind Abst) x0 x1) H1)))))))) H3)) (\lambda (H3: (ex3_3 C T -T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 (CHead (CHead (CHead -(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef -O)) (lift (S O) O u0) x0)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: -T).(getl O (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) -(TSort O)) (Bind Abst) (TLRef O)) (CHead e (Bind Abst) u0))))) (\lambda (e: -C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t)))))).(ex3_3_ind C T T -(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 (CHead (CHead (CHead -(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef -O)) (lift (S O) O u0) x0)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: -T).(getl O (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) -(TSort O)) (Bind Abst) (TLRef O)) (CHead e (Bind Abst) u0))))) (\lambda (e: -C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t)))) P (\lambda (x2: -C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H4: (pc3 (CHead (CHead (CHead -(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef -O)) (lift (S O) O x3) x0)).(\lambda (H5: (getl O (CHead (CHead (CHead (CSort -O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) -(CHead x2 (Bind Abst) x3))).(\lambda (H6: (ty3 g x2 x3 x4)).(ex3_2_ind T T -(\lambda (t2: T).(\lambda (_: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (THead (Bind -Abst) (TLRef (S (S O))) t2) (THead (Bind Abst) x0 x1)))) (\lambda (_: -T).(\lambda (t: T).(ty3 g (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort -O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) t))) -(\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead (CHead (CHead (CHead (CSort O) -(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (Bind -Abst) (TLRef (S (S O)))) (TSort O) t2))) P (\lambda (x5: T).(\lambda (x6: -T).(\lambda (H7: (pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) -(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (THead (Bind Abst) (TLRef (S (S -O))) x5) (THead (Bind Abst) x0 x1))).(\lambda (H8: (ty3 g (CHead (CHead -(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) -(TLRef O)) (TLRef (S (S O))) x6)).(\lambda (_: (ty3 g (CHead (CHead (CHead -(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) -(TLRef O)) (Bind Abst) (TLRef (S (S O)))) (TSort O) x5)).(or_ind (ex3_3 C T T -(\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 (CHead (CHead (CHead -(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef -O)) (lift (S (S (S O))) O t) x6)))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(_: T).(getl (S (S O)) (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) -(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead e (Bind Abbr) u0))))) -(\lambda (e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t))))) (ex3_3 C -T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 (CHead (CHead -(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) -(TLRef O)) (lift (S (S (S O))) O u0) x6)))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (_: T).(getl (S (S O)) (CHead (CHead (CHead (CSort O) (Bind Abst) -(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead e (Bind Abst) -u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t))))) P -(\lambda (H10: (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: -T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) -(TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O t) x6)))) (\lambda -(e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) (CHead (CHead (CHead -(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef -O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(t: T).(ty3 g e u0 t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (_: -T).(\lambda (t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) -(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O t) x6)))) -(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) (CHead -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind -Abst) (TLRef O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (t: T).(ty3 g e u0 t)))) P (\lambda (x7: C).(\lambda (x8: -T).(\lambda (x9: T).(\lambda (_: (pc3 (CHead (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S -O))) O x9) x6)).(\lambda (H12: (getl (S (S O)) (CHead (CHead (CHead (CSort O) -(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead -x7 (Bind Abbr) x8))).(\lambda (_: (ty3 g x7 x8 x9)).(let H14 \def -(getl_gen_all (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) -(TSort O)) (CHead x7 (Bind Abbr) x8) (r (Bind Abst) (S O)) (getl_gen_S (Bind -Abst) (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) -(CHead x7 (Bind Abbr) x8) (TLRef O) (S O) H12)) in (ex2_ind C (\lambda (e: -C).(drop (S O) O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) -(TSort O)) e)) (\lambda (e: C).(clear e (CHead x7 (Bind Abbr) x8))) P -(\lambda (x: C).(\lambda (H15: (drop (S O) O (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) x)).(\lambda (H16: (clear x (CHead x7 -(Bind Abbr) x8))).(let H17 \def (f_equal C C (\lambda (e: C).(match e in C -return (\lambda (_: C).C) with [(CSort _) \Rightarrow x2 | (CHead c _ _) -\Rightarrow c])) (CHead x2 (Bind Abst) x3) (CHead (CHead (CHead (CSort O) -(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) -(clear_gen_bind Abst (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind -Abst) (TSort O)) (CHead x2 (Bind Abst) x3) (TLRef O) (getl_gen_O (CHead -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind -Abst) (TLRef O)) (CHead x2 (Bind Abst) x3) H5))) in ((let H18 \def (f_equal C -T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) -\Rightarrow x3 | (CHead _ _ t) \Rightarrow t])) (CHead x2 (Bind Abst) x3) -(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) -(Bind Abst) (TLRef O)) (clear_gen_bind Abst (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead x2 (Bind Abst) x3) (TLRef O) -(getl_gen_O (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) -(TSort O)) (Bind Abst) (TLRef O)) (CHead x2 (Bind Abst) x3) H5))) in (\lambda -(H19: (eq C x2 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) -(TSort O)))).(let H20 \def (eq_ind T x3 (\lambda (t: T).(ty3 g x2 t x4)) H6 -(TLRef O) H18) in (let H21 \def (eq_ind T x3 (\lambda (t: T).(pc3 (CHead -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind -Abst) (TLRef O)) (lift (S O) O t) x0)) H4 (TLRef O) H18) in (let H22 \def -(eq_ind C x2 (\lambda (c: C).(ty3 g c (TLRef O) x4)) H20 (CHead (CHead (CSort -O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) H19) in (let H23 \def -(eq_ind_r C x (\lambda (c: C).(clear c (CHead x7 (Bind Abbr) x8))) H16 (CHead -(CSort O) (Bind Abst) (TSort O)) (drop_gen_refl (CHead (CSort O) (Bind Abst) -(TSort O)) x (drop_gen_drop (Bind Abst) (CHead (CSort O) (Bind Abst) (TSort -O)) x (TSort O) O H15))) in (let H24 \def (eq_ind C (CHead x7 (Bind Abbr) x8) -(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind b) \Rightarrow (match b in B return (\lambda (_: -B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | Void -\Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead (CSort O) -(Bind Abst) (TSort O)) (clear_gen_bind Abst (CSort O) (CHead x7 (Bind Abbr) -x8) (TSort O) H23)) in (False_ind P H24)))))))) H17))))) H14)))))))) H10)) -(\lambda (H10: (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: -T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) -(TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O u0) x6)))) (\lambda -(e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) (CHead (CHead (CHead -(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef -O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(t: T).(ty3 g e u0 t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (u0: -T).(\lambda (_: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) -(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O u0) -x6)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) -(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) -(Bind Abst) (TLRef O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda -(u0: T).(\lambda (t: T).(ty3 g e u0 t)))) P (\lambda (x7: C).(\lambda (x8: -T).(\lambda (x9: T).(\lambda (H11: (pc3 (CHead (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S -O))) O x8) x6)).(\lambda (H12: (getl (S (S O)) (CHead (CHead (CHead (CSort O) -(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead -x7 (Bind Abst) x8))).(\lambda (H13: (ty3 g x7 x8 x9)).(let H14 \def -(getl_gen_all (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) -(TSort O)) (CHead x7 (Bind Abst) x8) (r (Bind Abst) (S O)) (getl_gen_S (Bind -Abst) (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) -(CHead x7 (Bind Abst) x8) (TLRef O) (S O) H12)) in (ex2_ind C (\lambda (e: -C).(drop (S O) O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) -(TSort O)) e)) (\lambda (e: C).(clear e (CHead x7 (Bind Abst) x8))) P -(\lambda (x: C).(\lambda (H15: (drop (S O) O (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) x)).(\lambda (H16: (clear x (CHead x7 -(Bind Abst) x8))).(let H17 \def (f_equal C C (\lambda (e: C).(match e in C -return (\lambda (_: C).C) with [(CSort _) \Rightarrow x2 | (CHead c _ _) -\Rightarrow c])) (CHead x2 (Bind Abst) x3) (CHead (CHead (CHead (CSort O) -(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) -(clear_gen_bind Abst (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind -Abst) (TSort O)) (CHead x2 (Bind Abst) x3) (TLRef O) (getl_gen_O (CHead -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind -Abst) (TLRef O)) (CHead x2 (Bind Abst) x3) H5))) in ((let H18 \def (f_equal C -T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) -\Rightarrow x3 | (CHead _ _ t) \Rightarrow t])) (CHead x2 (Bind Abst) x3) -(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) -(Bind Abst) (TLRef O)) (clear_gen_bind Abst (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead x2 (Bind Abst) x3) (TLRef O) -(getl_gen_O (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) -(TSort O)) (Bind Abst) (TLRef O)) (CHead x2 (Bind Abst) x3) H5))) in (\lambda -(H19: (eq C x2 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) -(TSort O)))).(let H20 \def (eq_ind T x3 (\lambda (t: T).(ty3 g x2 t x4)) H6 -(TLRef O) H18) in (let H21 \def (eq_ind T x3 (\lambda (t: T).(pc3 (CHead -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind -Abst) (TLRef O)) (lift (S O) O t) x0)) H4 (TLRef O) H18) in (let H22 \def -(eq_ind C x2 (\lambda (c: C).(ty3 g c (TLRef O) x4)) H20 (CHead (CHead (CSort -O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) H19) in (let H23 \def -(eq_ind_r C x (\lambda (c: C).(clear c (CHead x7 (Bind Abst) x8))) H16 (CHead -(CSort O) (Bind Abst) (TSort O)) (drop_gen_refl (CHead (CSort O) (Bind Abst) -(TSort O)) x (drop_gen_drop (Bind Abst) (CHead (CSort O) (Bind Abst) (TSort -O)) x (TSort O) O H15))) in (let H24 \def (f_equal C C (\lambda (e: C).(match -e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow x7 | (CHead c _ -_) \Rightarrow c])) (CHead x7 (Bind Abst) x8) (CHead (CSort O) (Bind Abst) -(TSort O)) (clear_gen_bind Abst (CSort O) (CHead x7 (Bind Abst) x8) (TSort O) -H23)) in ((let H25 \def (f_equal C T (\lambda (e: C).(match e in C return -(\lambda (_: C).T) with [(CSort _) \Rightarrow x8 | (CHead _ _ t) \Rightarrow -t])) (CHead x7 (Bind Abst) x8) (CHead (CSort O) (Bind Abst) (TSort O)) -(clear_gen_bind Abst (CSort O) (CHead x7 (Bind Abst) x8) (TSort O) H23)) in -(\lambda (H26: (eq C x7 (CSort O))).(let H27 \def (eq_ind T x8 (\lambda (t: -T).(ty3 g x7 t x9)) H13 (TSort O) H25) in (let H28 \def (eq_ind T x8 (\lambda -(t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) -(TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O t) x6)) H11 (TSort O) -H25) in (let H29 \def (eq_ind C x7 (\lambda (c: C).(ty3 g c (TSort O) x9)) -H27 (CSort O) H26) in (or_ind (ex3_3 C T T (\lambda (_: C).(\lambda (_: -T).(\lambda (t: T).(pc3 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind -Abst) (TSort O)) (lift (S O) O t) x4)))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (_: T).(getl O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) -(Bind Abst) (TSort O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda -(u0: T).(\lambda (t: T).(ty3 g e u0 t))))) (ex3_3 C T T (\lambda (_: -C).(\lambda (u0: T).(\lambda (_: T).(pc3 (CHead (CHead (CSort O) (Bind Abst) -(TSort O)) (Bind Abst) (TSort O)) (lift (S O) O u0) x4)))) (\lambda (e: -C).(\lambda (u0: T).(\lambda (_: T).(getl O (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead e (Bind Abst) u0))))) (\lambda -(e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t))))) P (\lambda (H30: -(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 (CHead -(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (lift (S O) O -t) x4)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl O (CHead -(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead e (Bind -Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 -t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (lift -(S O) O t) x4)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl O -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead -e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g -e u0 t)))) P (\lambda (x10: C).(\lambda (x11: T).(\lambda (x12: T).(\lambda -(_: (pc3 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort -O)) (lift (S O) O x12) x4)).(\lambda (H32: (getl O (CHead (CHead (CSort O) -(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead x10 (Bind Abbr) -x11))).(\lambda (_: (ty3 g x10 x11 x12)).(let H34 \def (eq_ind C (CHead x10 -(Bind Abbr) x11) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) -with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K -return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B return -(\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | -Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead (CHead -(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (clear_gen_bind Abst -(CHead (CSort O) (Bind Abst) (TSort O)) (CHead x10 (Bind Abbr) x11) (TSort O) -(getl_gen_O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort -O)) (CHead x10 (Bind Abbr) x11) H32))) in (False_ind P H34)))))))) H30)) -(\lambda (H30: (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: -T).(pc3 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) -(lift (S O) O u0) x4)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: -T).(getl O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort -O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(t: T).(ty3 g e u0 t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (u0: -T).(\lambda (_: T).(pc3 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind -Abst) (TSort O)) (lift (S O) O u0) x4)))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (_: T).(getl O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) -(Bind Abst) (TSort O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda -(u0: T).(\lambda (t: T).(ty3 g e u0 t)))) P (\lambda (x10: C).(\lambda (x11: -T).(\lambda (x12: T).(\lambda (H31: (pc3 (CHead (CHead (CSort O) (Bind Abst) -(TSort O)) (Bind Abst) (TSort O)) (lift (S O) O x11) x4)).(\lambda (H32: -(getl O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) -(CHead x10 (Bind Abst) x11))).(\lambda (H33: (ty3 g x10 x11 x12)).(let H34 -\def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) -with [(CSort _) \Rightarrow x10 | (CHead c _ _) \Rightarrow c])) (CHead x10 -(Bind Abst) x11) (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) -(TSort O)) (clear_gen_bind Abst (CHead (CSort O) (Bind Abst) (TSort O)) -(CHead x10 (Bind Abst) x11) (TSort O) (getl_gen_O (CHead (CHead (CSort O) -(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead x10 (Bind Abst) x11) -H32))) in ((let H35 \def (f_equal C T (\lambda (e: C).(match e in C return -(\lambda (_: C).T) with [(CSort _) \Rightarrow x11 | (CHead _ _ t) -\Rightarrow t])) (CHead x10 (Bind Abst) x11) (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) (clear_gen_bind Abst (CHead (CSort O) -(Bind Abst) (TSort O)) (CHead x10 (Bind Abst) x11) (TSort O) (getl_gen_O -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead -x10 (Bind Abst) x11) H32))) in (\lambda (H36: (eq C x10 (CHead (CSort O) -(Bind Abst) (TSort O)))).(let H37 \def (eq_ind T x11 (\lambda (t: T).(ty3 g -x10 t x12)) H33 (TSort O) H35) in (let H38 \def (eq_ind T x11 (\lambda (t: -T).(pc3 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) -(lift (S O) O t) x4)) H31 (TSort O) H35) in (let H39 \def (eq_ind C x10 -(\lambda (c: C).(ty3 g c (TSort O) x12)) H37 (CHead (CSort O) (Bind Abst) -(TSort O)) H36) in (land_ind (pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) -(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) -x0) (\forall (b: B).(\forall (u0: T).(pc3 (CHead (CHead (CHead (CHead (CSort -O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (Bind -b) u0) x5 x1))) P (\lambda (H40: (pc3 (CHead (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S -O))) x0)).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pc3 (CHead (CHead -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind -Abst) (TLRef O)) (Bind b) u0) x5 x1))))).(let H42 \def (eq_ind T (lift (S O) -O (TLRef O)) (\lambda (t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) -(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) -t)) (pc3_t x0 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind -Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) H40 (lift (S O) O -(TLRef O)) (ex2_sym T (pr3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort -O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S O) O (TLRef O))) -(pr3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort -O)) (Bind Abst) (TLRef O)) x0) H21)) (TLRef (plus O (S O))) (lift_lref_ge O -(S O) O (le_n O))) in (let H43 \def H42 in (ex2_ind T (\lambda (t: T).(pr3 -(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) -(Bind Abst) (TLRef O)) (TLRef (S (S O))) t)) (\lambda (t: T).(pr3 (CHead -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind -Abst) (TLRef O)) (TLRef (S O)) t)) P (\lambda (x13: T).(\lambda (H44: (pr3 -(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) -(Bind Abst) (TLRef O)) (TLRef (S (S O))) x13)).(\lambda (H45: (pr3 (CHead -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind -Abst) (TLRef O)) (TLRef (S O)) x13)).(let H46 \def (eq_ind_r T x13 (\lambda -(t: T).(eq T (TLRef (S (S O))) t)) (nf2_pr3_unfold (CHead (CHead (CHead -(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef -O)) (TLRef (S (S O))) x13 H44 (nf2_lref_abst (CHead (CHead (CHead (CSort O) -(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CSort -O) (TSort O) (S (S O)) (getl_clear_bind Abst (CHead (CHead (CHead (CSort O) -(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead -(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (TLRef O) -(clear_bind Abst (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) -(TSort O)) (TLRef O)) (CHead (CSort O) (Bind Abst) (TSort O)) (S O) -(getl_head (Bind Abst) O (CHead (CSort O) (Bind Abst) (TSort O)) (CHead -(CSort O) (Bind Abst) (TSort O)) (getl_refl Abst (CSort O) (TSort O)) (TSort -O))))) (TLRef (S O)) (nf2_pr3_unfold (CHead (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S O)) -x13 H45 (nf2_lref_abst (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) -(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead (CSort O) (Bind Abst) -(TSort O)) (TSort O) (S O) (getl_head (Bind Abst) O (CHead (CHead (CSort O) -(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) (getl_refl Abst (CHead (CSort O) -(Bind Abst) (TSort O)) (TSort O)) (TLRef O))))) in (let H47 \def (eq_ind T -(TLRef (S (S O))) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef n) \Rightarrow (match n -in nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S n0) -\Rightarrow (match n0 in nat return (\lambda (_: nat).Prop) with [O -\Rightarrow False | (S _) \Rightarrow True])]) | (THead _ _ _) \Rightarrow -False])) I (TLRef (S O)) H46) in (False_ind P H47)))))) H43))))) -(pc3_gen_abst (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind -Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) x0 x5 x1 H7))))))) -H34)))))))) H30)) (ty3_gen_lref g (CHead (CHead (CSort O) (Bind Abst) (TSort -O)) (Bind Abst) (TSort O)) x4 O H22))))))) H24)))))))) H17))))) H14)))))))) -H10)) (ty3_gen_lref g (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) -(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) x6 (S (S O)) H8))))))) -(ty3_gen_bind g Abst (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) -(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) (TSort O) -(THead (Bind Abst) x0 x1) H1)))))))) H3)) (ty3_gen_lref g (CHead (CHead -(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) -(TLRef O)) x0 O H2))))))) (ty3_gen_appl g (CHead (CHead (CHead (CSort O) -(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef -O) (THead (Bind Abst) (TLRef (S (S O))) (TSort O)) u H))))). -(* COMMENTS -Initial nodes: 9973 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/ex2/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/ex2/defs.ma deleted file mode 100644 index 35b5df73e..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/ex2/defs.ma +++ /dev/null @@ -1,28 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/C/defs.ma". - -definition ex2_c: - C -\def - CSort O. - -definition ex2_t: - T -\def - THead (Flat Appl) (TSort O) (TSort O). - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/ex2/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/ex2/props.ma deleted file mode 100644 index 4dfa6a582..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/ex2/props.ma +++ /dev/null @@ -1,159 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/ex2/defs.ma". - -include "Basic-1/nf2/defs.ma". - -include "Basic-1/pr2/fwd.ma". - -include "Basic-1/arity/fwd.ma". - -theorem ex2_nf2: - nf2 ex2_c ex2_t -\def - \lambda (t2: T).(\lambda (H: (pr2 (CSort O) (THead (Flat Appl) (TSort O) -(TSort O)) t2)).(let H0 \def (pr2_gen_appl (CSort O) (TSort O) (TSort O) t2 -H) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead -(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 (CSort O) (TSort -O) u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 (CSort O) (TSort O) t3)))) -(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(eq T (TSort O) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda -(_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 -t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: -T).(pr2 (CSort O) (TSort O) u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead (CSort O) -(Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B -b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (TSort O) (THead (Bind b) y1 -z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: -T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat -Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 (CSort O) (TSort -O) u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CSort O) y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead (CSort O) (Bind b) y2) z1 z2)))))))) (eq T (THead (Flat -Appl) (TSort O) (TSort O)) t2) (\lambda (H1: (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 (CSort O) (TSort O) u2))) (\lambda (_: T).(\lambda -(t3: T).(pr2 (CSort O) (TSort O) t3))))).(ex3_2_ind T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 (CSort O) (TSort O) u2))) (\lambda (_: T).(\lambda -(t3: T).(pr2 (CSort O) (TSort O) t3))) (eq T (THead (Flat Appl) (TSort O) -(TSort O)) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H2: (eq T t2 -(THead (Flat Appl) x0 x1))).(\lambda (H3: (pr2 (CSort O) (TSort O) -x0)).(\lambda (H4: (pr2 (CSort O) (TSort O) x1)).(let H5 \def (eq_ind T x1 -(\lambda (t: T).(eq T t2 (THead (Flat Appl) x0 t))) H2 (TSort O) -(pr2_gen_sort (CSort O) x1 O H4)) in (let H6 \def (eq_ind T x0 (\lambda (t: -T).(eq T t2 (THead (Flat Appl) t (TSort O)))) H5 (TSort O) (pr2_gen_sort -(CSort O) x0 O H3)) in (eq_ind_r T (THead (Flat Appl) (TSort O) (TSort O)) -(\lambda (t: T).(eq T (THead (Flat Appl) (TSort O) (TSort O)) t)) (refl_equal -T (THead (Flat Appl) (TSort O) (TSort O))) t2 H6)))))))) H1)) (\lambda (H1: -(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(eq T (TSort O) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda -(_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 -t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: -T).(pr2 (CSort O) (TSort O) u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead (CSort O) -(Bind b) u) z1 t3))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T (TSort O) (THead (Bind Abst) y1 -z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: -T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 (CSort O) (TSort O) u2))))) (\lambda -(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: -B).(\forall (u: T).(pr2 (CHead (CSort O) (Bind b) u) z1 t3))))))) (eq T -(THead (Flat Appl) (TSort O) (TSort O)) t2) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H2: (eq T (TSort O) (THead -(Bind Abst) x0 x1))).(\lambda (H3: (eq T t2 (THead (Bind Abbr) x2 -x3))).(\lambda (H4: (pr2 (CSort O) (TSort O) x2)).(\lambda (_: ((\forall (b: -B).(\forall (u: T).(pr2 (CHead (CSort O) (Bind b) u) x1 x3))))).(let H6 \def -(eq_ind T x2 (\lambda (t: T).(eq T t2 (THead (Bind Abbr) t x3))) H3 (TSort O) -(pr2_gen_sort (CSort O) x2 O H4)) in (eq_ind_r T (THead (Bind Abbr) (TSort O) -x3) (\lambda (t: T).(eq T (THead (Flat Appl) (TSort O) (TSort O)) t)) (let H7 -\def (eq_ind T (TSort O) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | -(THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) x0 x1) H2) in -(False_ind (eq T (THead (Flat Appl) (TSort O) (TSort O)) (THead (Bind Abbr) -(TSort O) x3)) H7)) t2 H6)))))))))) H1)) (\lambda (H1: (ex6_6 B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T -(TSort O) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T -t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 (CSort O) (TSort O) u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CSort O) y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda -(z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead (CSort -O) (Bind b) y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (TSort O) -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 (CSort O) (TSort O) u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 -(CSort O) y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead (CSort O) -(Bind b) y2) z1 z2))))))) (eq T (THead (Flat Appl) (TSort O) (TSort O)) t2) -(\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda -(x4: T).(\lambda (x5: T).(\lambda (_: (not (eq B x0 Abst))).(\lambda (H3: (eq -T (TSort O) (THead (Bind x0) x1 x2))).(\lambda (H4: (eq T t2 (THead (Bind x0) -x5 (THead (Flat Appl) (lift (S O) O x4) x3)))).(\lambda (H5: (pr2 (CSort O) -(TSort O) x4)).(\lambda (H6: (pr2 (CSort O) x1 x5)).(\lambda (_: (pr2 (CHead -(CSort O) (Bind x0) x5) x2 x3)).(let H_y \def (pr2_gen_csort x1 x5 O H6) in -(let H8 \def (eq_ind T x4 (\lambda (t: T).(eq T t2 (THead (Bind x0) x5 (THead -(Flat Appl) (lift (S O) O t) x3)))) H4 (TSort O) (pr2_gen_sort (CSort O) x4 O -H5)) in (eq_ind_r T (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O -(TSort O)) x3)) (\lambda (t: T).(eq T (THead (Flat Appl) (TSort O) (TSort O)) -t)) (let H9 \def (eq_ind T (TSort O) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Bind x0) x1 -x2) H3) in (False_ind (eq T (THead (Flat Appl) (TSort O) (TSort O)) (THead -(Bind x0) x5 (THead (Flat Appl) (lift (S O) O (TSort O)) x3))) H9)) t2 -H8))))))))))))))) H1)) H0))). -(* COMMENTS -Initial nodes: 1939 -END *) - -theorem ex2_arity: - \forall (g: G).(\forall (a: A).((arity g ex2_c ex2_t a) \to (\forall (P: -Prop).P))) -\def - \lambda (g: G).(\lambda (a: A).(\lambda (H: (arity g (CSort O) (THead (Flat -Appl) (TSort O) (TSort O)) a)).(\lambda (P: Prop).(let H0 \def -(arity_gen_appl g (CSort O) (TSort O) (TSort O) a H) in (ex2_ind A (\lambda -(a1: A).(arity g (CSort O) (TSort O) a1)) (\lambda (a1: A).(arity g (CSort O) -(TSort O) (AHead a1 a))) P (\lambda (x: A).(\lambda (_: (arity g (CSort O) -(TSort O) x)).(\lambda (H2: (arity g (CSort O) (TSort O) (AHead x a))).(let -H_x \def (leq_gen_head1 g x a (ASort O O) (arity_gen_sort g (CSort O) O -(AHead x a) H2)) in (let H3 \def H_x in (ex3_2_ind A A (\lambda (a3: -A).(\lambda (_: A).(leq g x a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a -a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort O O) (AHead a3 a4)))) P -(\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g x x0)).(\lambda (_: -(leq g a x1)).(\lambda (H6: (eq A (ASort O O) (AHead x0 x1))).(let H7 \def -(eq_ind A (ASort O O) (\lambda (ee: A).(match ee in A return (\lambda (_: -A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow -False])) I (AHead x0 x1) H6) in (False_ind P H7))))))) H3)))))) H0))))). -(* COMMENTS -Initial nodes: 289 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/flt/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/flt/defs.ma deleted file mode 100644 index 3191e1e38..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/flt/defs.ma +++ /dev/null @@ -1,29 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/C/defs.ma". - -definition fweight: - C \to (T \to nat) -\def - \lambda (c: C).(\lambda (t: T).(plus (cweight c) (tweight t))). - -definition flt: - C \to (T \to (C \to (T \to Prop))) -\def - \lambda (c1: C).(\lambda (t1: T).(\lambda (c2: C).(\lambda (t2: T).(lt -(fweight c1 t1) (fweight c2 t2))))). - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/flt/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/flt/props.ma deleted file mode 100644 index 57df41528..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/flt/props.ma +++ /dev/null @@ -1,154 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/flt/defs.ma". - -include "Basic-1/C/props.ma". - -theorem flt_thead_sx: - \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).(flt c u c -(THead k u t))))) -\def - \lambda (_: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t: -T).(le_lt_plus_plus (cweight c) (cweight c) (tweight u) (S (plus (tweight u) -(tweight t))) (le_n (cweight c)) (le_n_S (tweight u) (plus (tweight u) -(tweight t)) (le_plus_l (tweight u) (tweight t))))))). -(* COMMENTS -Initial nodes: 65 -END *) - -theorem flt_thead_dx: - \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).(flt c t c -(THead k u t))))) -\def - \lambda (_: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t: -T).(le_lt_plus_plus (cweight c) (cweight c) (tweight t) (S (plus (tweight u) -(tweight t))) (le_n (cweight c)) (le_n_S (tweight t) (plus (tweight u) -(tweight t)) (le_plus_r (tweight u) (tweight t))))))). -(* COMMENTS -Initial nodes: 65 -END *) - -theorem flt_shift: - \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).(flt (CHead c -k u) t c (THead k u t))))) -\def - \lambda (_: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(eq_ind nat -(S (plus (cweight c) (plus (tweight u) (tweight t)))) (\lambda (n: nat).(lt -(plus (plus (cweight c) (tweight u)) (tweight t)) n)) (eq_ind_r nat (plus -(plus (cweight c) (tweight u)) (tweight t)) (\lambda (n: nat).(lt (plus (plus -(cweight c) (tweight u)) (tweight t)) (S n))) (le_n (S (plus (plus (cweight -c) (tweight u)) (tweight t)))) (plus (cweight c) (plus (tweight u) (tweight -t))) (plus_assoc_l (cweight c) (tweight u) (tweight t))) (plus (cweight c) (S -(plus (tweight u) (tweight t)))) (plus_n_Sm (cweight c) (plus (tweight u) -(tweight t))))))). -(* COMMENTS -Initial nodes: 179 -END *) - -theorem flt_arith0: - \forall (k: K).(\forall (c: C).(\forall (t: T).(\forall (i: nat).(flt c t -(CHead c k t) (TLRef i))))) -\def - \lambda (_: K).(\lambda (c: C).(\lambda (t: T).(\lambda (_: -nat).(lt_x_plus_x_Sy (plus (cweight c) (tweight t)) O)))). -(* COMMENTS -Initial nodes: 21 -END *) - -theorem flt_arith1: - \forall (k1: K).(\forall (c1: C).(\forall (c2: C).(\forall (t1: T).((cle -(CHead c1 k1 t1) c2) \to (\forall (k2: K).(\forall (t2: T).(\forall (i: -nat).(flt c1 t1 (CHead c2 k2 t2) (TLRef i))))))))) -\def - \lambda (_: K).(\lambda (c1: C).(\lambda (c2: C).(\lambda (t1: T).(\lambda -(H: (le (plus (cweight c1) (tweight t1)) (cweight c2))).(\lambda (_: -K).(\lambda (t2: T).(\lambda (_: nat).(le_lt_trans (plus (cweight c1) -(tweight t1)) (cweight c2) (plus (plus (cweight c2) (tweight t2)) (S O)) H -(eq_ind_r nat (plus (S O) (plus (cweight c2) (tweight t2))) (\lambda (n: -nat).(lt (cweight c2) n)) (le_lt_n_Sm (cweight c2) (plus (cweight c2) -(tweight t2)) (le_plus_l (cweight c2) (tweight t2))) (plus (plus (cweight c2) -(tweight t2)) (S O)) (plus_sym (plus (cweight c2) (tweight t2)) (S -O))))))))))). -(* COMMENTS -Initial nodes: 151 -END *) - -theorem flt_arith2: - \forall (c1: C).(\forall (c2: C).(\forall (t1: T).(\forall (i: nat).((flt c1 -t1 c2 (TLRef i)) \to (\forall (k2: K).(\forall (t2: T).(\forall (j: nat).(flt -c1 t1 (CHead c2 k2 t2) (TLRef j))))))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (t1: T).(\lambda (_: nat).(\lambda -(H: (lt (plus (cweight c1) (tweight t1)) (plus (cweight c2) (S O)))).(\lambda -(_: K).(\lambda (t2: T).(\lambda (_: nat).(lt_le_trans (plus (cweight c1) -(tweight t1)) (plus (cweight c2) (S O)) (plus (plus (cweight c2) (tweight -t2)) (S O)) H (le_plus_plus (cweight c2) (plus (cweight c2) (tweight t2)) (S -O) (S O) (le_plus_l (cweight c2) (tweight t2)) (le_n (S O))))))))))). -(* COMMENTS -Initial nodes: 115 -END *) - -theorem flt_trans: - \forall (c1: C).(\forall (c2: C).(\forall (t1: T).(\forall (t2: T).((flt c1 -t1 c2 t2) \to (\forall (c3: C).(\forall (t3: T).((flt c2 t2 c3 t3) \to (flt -c1 t1 c3 t3)))))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H: (lt (fweight c1 t1) (fweight c2 t2))).(\lambda (c3: C).(\lambda (t3: -T).(\lambda (H0: (lt (fweight c2 t2) (fweight c3 t3))).(lt_trans (fweight c1 -t1) (fweight c2 t2) (fweight c3 t3) H H0)))))))). -(* COMMENTS -Initial nodes: 63 -END *) - -theorem flt_wf__q_ind: - \forall (P: ((C \to (T \to Prop)))).(((\forall (n: nat).((\lambda (P0: ((C -\to (T \to Prop)))).(\lambda (n0: nat).(\forall (c: C).(\forall (t: T).((eq -nat (fweight c t) n0) \to (P0 c t)))))) P n))) \to (\forall (c: C).(\forall -(t: T).(P c t)))) -\def - let Q \def (\lambda (P: ((C \to (T \to Prop)))).(\lambda (n: nat).(\forall -(c: C).(\forall (t: T).((eq nat (fweight c t) n) \to (P c t)))))) in (\lambda -(P: ((C \to (T \to Prop)))).(\lambda (H: ((\forall (n: nat).(\forall (c: -C).(\forall (t: T).((eq nat (fweight c t) n) \to (P c t))))))).(\lambda (c: -C).(\lambda (t: T).(H (fweight c t) c t (refl_equal nat (fweight c t))))))). -(* COMMENTS -Initial nodes: 85 -END *) - -theorem flt_wf_ind: - \forall (P: ((C \to (T \to Prop)))).(((\forall (c2: C).(\forall (t2: -T).(((\forall (c1: C).(\forall (t1: T).((flt c1 t1 c2 t2) \to (P c1 t1))))) -\to (P c2 t2))))) \to (\forall (c: C).(\forall (t: T).(P c t)))) -\def - let Q \def (\lambda (P: ((C \to (T \to Prop)))).(\lambda (n: nat).(\forall -(c: C).(\forall (t: T).((eq nat (fweight c t) n) \to (P c t)))))) in (\lambda -(P: ((C \to (T \to Prop)))).(\lambda (H: ((\forall (c2: C).(\forall (t2: -T).(((\forall (c1: C).(\forall (t1: T).((flt c1 t1 c2 t2) \to (P c1 t1))))) -\to (P c2 t2)))))).(\lambda (c: C).(\lambda (t: T).(flt_wf__q_ind P (\lambda -(n: nat).(lt_wf_ind n (Q P) (\lambda (n0: nat).(\lambda (H0: ((\forall (m: -nat).((lt m n0) \to (Q P m))))).(\lambda (c0: C).(\lambda (t0: T).(\lambda -(H1: (eq nat (fweight c0 t0) n0)).(let H2 \def (eq_ind_r nat n0 (\lambda (n1: -nat).(\forall (m: nat).((lt m n1) \to (\forall (c1: C).(\forall (t1: T).((eq -nat (fweight c1 t1) m) \to (P c1 t1))))))) H0 (fweight c0 t0) H1) in (H c0 t0 -(\lambda (c1: C).(\lambda (t1: T).(\lambda (H3: (flt c1 t1 c0 t0)).(H2 -(fweight c1 t1) H3 c1 t1 (refl_equal nat (fweight c1 t1))))))))))))))) c -t))))). -(* COMMENTS -Initial nodes: 211 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/fsubst0/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/fsubst0/defs.ma deleted file mode 100644 index f39baebf7..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/fsubst0/defs.ma +++ /dev/null @@ -1,26 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubst0/defs.ma". - -inductive fsubst0 (i: nat) (v: T) (c1: C) (t1: T): C \to (T \to Prop) \def -| fsubst0_snd: \forall (t2: T).((subst0 i v t1 t2) \to (fsubst0 i v c1 t1 c1 -t2)) -| fsubst0_fst: \forall (c2: C).((csubst0 i v c1 c2) \to (fsubst0 i v c1 t1 c2 -t1)) -| fsubst0_both: \forall (t2: T).((subst0 i v t1 t2) \to (\forall (c2: -C).((csubst0 i v c1 c2) \to (fsubst0 i v c1 t1 c2 t2)))). - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/fsubst0/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/fsubst0/fwd.ma deleted file mode 100644 index 3c1212510..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/fsubst0/fwd.ma +++ /dev/null @@ -1,43 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/fsubst0/defs.ma". - -theorem fsubst0_gen_base: - \forall (c1: C).(\forall (c2: C).(\forall (t1: T).(\forall (t2: T).(\forall -(v: T).(\forall (i: nat).((fsubst0 i v c1 t1 c2 t2) \to (or3 (land (eq C c1 -c2) (subst0 i v t1 t2)) (land (eq T t1 t2) (csubst0 i v c1 c2)) (land (subst0 -i v t1 t2) (csubst0 i v c1 c2))))))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(v: T).(\lambda (i: nat).(\lambda (H: (fsubst0 i v c1 t1 c2 t2)).(fsubst0_ind -i v c1 t1 (\lambda (c: C).(\lambda (t: T).(or3 (land (eq C c1 c) (subst0 i v -t1 t)) (land (eq T t1 t) (csubst0 i v c1 c)) (land (subst0 i v t1 t) (csubst0 -i v c1 c))))) (\lambda (t0: T).(\lambda (H0: (subst0 i v t1 t0)).(or3_intro0 -(land (eq C c1 c1) (subst0 i v t1 t0)) (land (eq T t1 t0) (csubst0 i v c1 -c1)) (land (subst0 i v t1 t0) (csubst0 i v c1 c1)) (conj (eq C c1 c1) (subst0 -i v t1 t0) (refl_equal C c1) H0)))) (\lambda (c0: C).(\lambda (H0: (csubst0 i -v c1 c0)).(or3_intro1 (land (eq C c1 c0) (subst0 i v t1 t1)) (land (eq T t1 -t1) (csubst0 i v c1 c0)) (land (subst0 i v t1 t1) (csubst0 i v c1 c0)) (conj -(eq T t1 t1) (csubst0 i v c1 c0) (refl_equal T t1) H0)))) (\lambda (t0: -T).(\lambda (H0: (subst0 i v t1 t0)).(\lambda (c0: C).(\lambda (H1: (csubst0 -i v c1 c0)).(or3_intro2 (land (eq C c1 c0) (subst0 i v t1 t0)) (land (eq T t1 -t0) (csubst0 i v c1 c0)) (land (subst0 i v t1 t0) (csubst0 i v c1 c0)) (conj -(subst0 i v t1 t0) (csubst0 i v c1 c0) H0 H1)))))) c2 t2 H))))))). -(* COMMENTS -Initial nodes: 431 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/getl/clear.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/getl/clear.ma deleted file mode 100644 index de0a85bd3..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/getl/clear.ma +++ /dev/null @@ -1,153 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/getl/props.ma". - -include "Basic-1/clear/drop.ma". - -theorem clear_getl_trans: - \forall (i: nat).(\forall (c2: C).(\forall (c3: C).((getl i c2 c3) \to -(\forall (c1: C).((clear c1 c2) \to (getl i c1 c3)))))) -\def - \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c2: C).(\forall (c3: -C).((getl n c2 c3) \to (\forall (c1: C).((clear c1 c2) \to (getl n c1 -c3))))))) (\lambda (c2: C).(\lambda (c3: C).(\lambda (H: (getl O c2 -c3)).(\lambda (c1: C).(\lambda (H0: (clear c1 c2)).(getl_intro O c1 c3 c1 -(drop_refl c1) (clear_trans c1 c2 H0 c3 (getl_gen_O c2 c3 H)))))))) (\lambda -(n: nat).(\lambda (_: ((\forall (c2: C).(\forall (c3: C).((getl n c2 c3) \to -(\forall (c1: C).((clear c1 c2) \to (getl n c1 c3)))))))).(\lambda (c2: -C).(C_ind (\lambda (c: C).(\forall (c3: C).((getl (S n) c c3) \to (\forall -(c1: C).((clear c1 c) \to (getl (S n) c1 c3)))))) (\lambda (n0: nat).(\lambda -(c3: C).(\lambda (H0: (getl (S n) (CSort n0) c3)).(\lambda (c1: C).(\lambda -(_: (clear c1 (CSort n0))).(getl_gen_sort n0 (S n) c3 H0 (getl (S n) c1 -c3))))))) (\lambda (c: C).(\lambda (_: ((\forall (c3: C).((getl (S n) c c3) -\to (\forall (c1: C).((clear c1 c) \to (getl (S n) c1 c3))))))).(\lambda (k: -K).(\lambda (t: T).(\lambda (c3: C).(\lambda (H1: (getl (S n) (CHead c k t) -c3)).(\lambda (c1: C).(\lambda (H2: (clear c1 (CHead c k t))).(K_ind (\lambda -(k0: K).((getl (S n) (CHead c k0 t) c3) \to ((clear c1 (CHead c k0 t)) \to -(getl (S n) c1 c3)))) (\lambda (b: B).(\lambda (H3: (getl (S n) (CHead c -(Bind b) t) c3)).(\lambda (H4: (clear c1 (CHead c (Bind b) t))).(let H5 \def -(getl_gen_all c c3 (r (Bind b) n) (getl_gen_S (Bind b) c c3 t n H3)) in -(ex2_ind C (\lambda (e: C).(drop n O c e)) (\lambda (e: C).(clear e c3)) -(getl (S n) c1 c3) (\lambda (x: C).(\lambda (H6: (drop n O c x)).(\lambda -(H7: (clear x c3)).(getl_intro (S n) c1 c3 x (drop_clear_O b c1 c t H4 x n -H6) H7)))) H5))))) (\lambda (f: F).(\lambda (_: (getl (S n) (CHead c (Flat f) -t) c3)).(\lambda (H4: (clear c1 (CHead c (Flat f) t))).(clear_gen_flat_r f c1 -c t H4 (getl (S n) c1 c3))))) k H1 H2))))))))) c2)))) i). -(* COMMENTS -Initial nodes: 525 -END *) - -theorem getl_clear_trans: - \forall (i: nat).(\forall (c1: C).(\forall (c2: C).((getl i c1 c2) \to -(\forall (c3: C).((clear c2 c3) \to (getl i c1 c3)))))) -\def - \lambda (i: nat).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (getl i c1 -c2)).(\lambda (c3: C).(\lambda (H0: (clear c2 c3)).(let H1 \def (getl_gen_all -c1 c2 i H) in (ex2_ind C (\lambda (e: C).(drop i O c1 e)) (\lambda (e: -C).(clear e c2)) (getl i c1 c3) (\lambda (x: C).(\lambda (H2: (drop i O c1 -x)).(\lambda (H3: (clear x c2)).(let H4 \def (clear_gen_all x c2 H3) in -(ex_3_ind B C T (\lambda (b: B).(\lambda (e: C).(\lambda (u: T).(eq C c2 -(CHead e (Bind b) u))))) (getl i c1 c3) (\lambda (x0: B).(\lambda (x1: -C).(\lambda (x2: T).(\lambda (H5: (eq C c2 (CHead x1 (Bind x0) x2))).(let H6 -\def (eq_ind C c2 (\lambda (c: C).(clear x c)) H3 (CHead x1 (Bind x0) x2) H5) -in (let H7 \def (eq_ind C c2 (\lambda (c: C).(clear c c3)) H0 (CHead x1 (Bind -x0) x2) H5) in (eq_ind_r C (CHead x1 (Bind x0) x2) (\lambda (c: C).(getl i c1 -c)) (getl_intro i c1 (CHead x1 (Bind x0) x2) x H2 H6) c3 (clear_gen_bind x0 -x1 c3 x2 H7)))))))) H4))))) H1))))))). -(* COMMENTS -Initial nodes: 269 -END *) - -theorem getl_clear_bind: - \forall (b: B).(\forall (c: C).(\forall (e1: C).(\forall (v: T).((clear c -(CHead e1 (Bind b) v)) \to (\forall (e2: C).(\forall (n: nat).((getl n e1 e2) -\to (getl (S n) c e2)))))))) -\def - \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e1: -C).(\forall (v: T).((clear c0 (CHead e1 (Bind b) v)) \to (\forall (e2: -C).(\forall (n: nat).((getl n e1 e2) \to (getl (S n) c0 e2)))))))) (\lambda -(n: nat).(\lambda (e1: C).(\lambda (v: T).(\lambda (H: (clear (CSort n) -(CHead e1 (Bind b) v))).(\lambda (e2: C).(\lambda (n0: nat).(\lambda (_: -(getl n0 e1 e2)).(clear_gen_sort (CHead e1 (Bind b) v) n H (getl (S n0) -(CSort n) e2))))))))) (\lambda (c0: C).(\lambda (H: ((\forall (e1: -C).(\forall (v: T).((clear c0 (CHead e1 (Bind b) v)) \to (\forall (e2: -C).(\forall (n: nat).((getl n e1 e2) \to (getl (S n) c0 e2))))))))).(\lambda -(k: K).(\lambda (t: T).(\lambda (e1: C).(\lambda (v: T).(\lambda (H0: (clear -(CHead c0 k t) (CHead e1 (Bind b) v))).(\lambda (e2: C).(\lambda (n: -nat).(\lambda (H1: (getl n e1 e2)).(K_ind (\lambda (k0: K).((clear (CHead c0 -k0 t) (CHead e1 (Bind b) v)) \to (getl (S n) (CHead c0 k0 t) e2))) (\lambda -(b0: B).(\lambda (H2: (clear (CHead c0 (Bind b0) t) (CHead e1 (Bind b) -v))).(let H3 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda -(_: C).C) with [(CSort _) \Rightarrow e1 | (CHead c1 _ _) \Rightarrow c1])) -(CHead e1 (Bind b) v) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e1 -(Bind b) v) t H2)) in ((let H4 \def (f_equal C B (\lambda (e: C).(match e in -C return (\lambda (_: C).B) with [(CSort _) \Rightarrow b | (CHead _ k0 _) -\Rightarrow (match k0 in K return (\lambda (_: K).B) with [(Bind b1) -\Rightarrow b1 | (Flat _) \Rightarrow b])])) (CHead e1 (Bind b) v) (CHead c0 -(Bind b0) t) (clear_gen_bind b0 c0 (CHead e1 (Bind b) v) t H2)) in ((let H5 -\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow v | (CHead _ _ t0) \Rightarrow t0])) (CHead e1 -(Bind b) v) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e1 (Bind b) -v) t H2)) in (\lambda (H6: (eq B b b0)).(\lambda (H7: (eq C e1 c0)).(let H8 -\def (eq_ind C e1 (\lambda (c1: C).(getl n c1 e2)) H1 c0 H7) in (eq_ind B b -(\lambda (b1: B).(getl (S n) (CHead c0 (Bind b1) t) e2)) (getl_head (Bind b) -n c0 e2 H8 t) b0 H6))))) H4)) H3)))) (\lambda (f: F).(\lambda (H2: (clear -(CHead c0 (Flat f) t) (CHead e1 (Bind b) v))).(getl_flat c0 e2 (S n) (H e1 v -(clear_gen_flat f c0 (CHead e1 (Bind b) v) t H2) e2 n H1) f t))) k -H0))))))))))) c)). -(* COMMENTS -Initial nodes: 599 -END *) - -theorem getl_clear_conf: - \forall (i: nat).(\forall (c1: C).(\forall (c3: C).((getl i c1 c3) \to -(\forall (c2: C).((clear c1 c2) \to (getl i c2 c3)))))) -\def - \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (c3: -C).((getl n c1 c3) \to (\forall (c2: C).((clear c1 c2) \to (getl n c2 -c3))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H: (getl O c1 -c3)).(\lambda (c2: C).(\lambda (H0: (clear c1 c2)).(eq_ind C c3 (\lambda (c: -C).(getl O c c3)) (let H1 \def (clear_gen_all c1 c3 (getl_gen_O c1 c3 H)) in -(ex_3_ind B C T (\lambda (b: B).(\lambda (e: C).(\lambda (u: T).(eq C c3 -(CHead e (Bind b) u))))) (getl O c3 c3) (\lambda (x0: B).(\lambda (x1: -C).(\lambda (x2: T).(\lambda (H2: (eq C c3 (CHead x1 (Bind x0) x2))).(let H3 -\def (eq_ind C c3 (\lambda (c: C).(clear c1 c)) (getl_gen_O c1 c3 H) (CHead -x1 (Bind x0) x2) H2) in (eq_ind_r C (CHead x1 (Bind x0) x2) (\lambda (c: -C).(getl O c c)) (getl_refl x0 x1 x2) c3 H2)))))) H1)) c2 (clear_mono c1 c3 -(getl_gen_O c1 c3 H) c2 H0))))))) (\lambda (n: nat).(\lambda (_: ((\forall -(c1: C).(\forall (c3: C).((getl n c1 c3) \to (\forall (c2: C).((clear c1 c2) -\to (getl n c2 c3)))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall -(c3: C).((getl (S n) c c3) \to (\forall (c2: C).((clear c c2) \to (getl (S n) -c2 c3)))))) (\lambda (n0: nat).(\lambda (c3: C).(\lambda (H0: (getl (S n) -(CSort n0) c3)).(\lambda (c2: C).(\lambda (_: (clear (CSort n0) -c2)).(getl_gen_sort n0 (S n) c3 H0 (getl (S n) c2 c3))))))) (\lambda (c: -C).(\lambda (H0: ((\forall (c3: C).((getl (S n) c c3) \to (\forall (c2: -C).((clear c c2) \to (getl (S n) c2 c3))))))).(\lambda (k: K).(\lambda (t: -T).(\lambda (c3: C).(\lambda (H1: (getl (S n) (CHead c k t) c3)).(\lambda -(c2: C).(\lambda (H2: (clear (CHead c k t) c2)).(K_ind (\lambda (k0: -K).((getl (S n) (CHead c k0 t) c3) \to ((clear (CHead c k0 t) c2) \to (getl -(S n) c2 c3)))) (\lambda (b: B).(\lambda (H3: (getl (S n) (CHead c (Bind b) -t) c3)).(\lambda (H4: (clear (CHead c (Bind b) t) c2)).(eq_ind_r C (CHead c -(Bind b) t) (\lambda (c0: C).(getl (S n) c0 c3)) (getl_head (Bind b) n c c3 -(getl_gen_S (Bind b) c c3 t n H3) t) c2 (clear_gen_bind b c c2 t H4))))) -(\lambda (f: F).(\lambda (H3: (getl (S n) (CHead c (Flat f) t) c3)).(\lambda -(H4: (clear (CHead c (Flat f) t) c2)).(H0 c3 (getl_gen_S (Flat f) c c3 t n -H3) c2 (clear_gen_flat f c c2 t H4))))) k H1 H2))))))))) c1)))) i). -(* COMMENTS -Initial nodes: 641 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/getl/dec.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/getl/dec.ma deleted file mode 100644 index 278362590..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/getl/dec.ma +++ /dev/null @@ -1,100 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/getl/props.ma". - -theorem getl_dec: - \forall (c: C).(\forall (i: nat).(or (ex_3 C B T (\lambda (e: C).(\lambda -(b: B).(\lambda (v: T).(getl i c (CHead e (Bind b) v)))))) (\forall (d: -C).((getl i c d) \to (\forall (P: Prop).P))))) -\def - \lambda (c: C).(C_ind (\lambda (c0: C).(\forall (i: nat).(or (ex_3 C B T -(\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl i c0 (CHead e (Bind b) -v)))))) (\forall (d: C).((getl i c0 d) \to (\forall (P: Prop).P)))))) -(\lambda (n: nat).(\lambda (i: nat).(or_intror (ex_3 C B T (\lambda (e: -C).(\lambda (b: B).(\lambda (v: T).(getl i (CSort n) (CHead e (Bind b) -v)))))) (\forall (d: C).((getl i (CSort n) d) \to (\forall (P: Prop).P))) -(\lambda (d: C).(\lambda (H: (getl i (CSort n) d)).(\lambda (P: -Prop).(getl_gen_sort n i d H P))))))) (\lambda (c0: C).(\lambda (H: ((\forall -(i: nat).(or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: -T).(getl i c0 (CHead e (Bind b) v)))))) (\forall (d: C).((getl i c0 d) \to -(\forall (P: Prop).P))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (i: -nat).(nat_ind (\lambda (n: nat).(or (ex_3 C B T (\lambda (e: C).(\lambda (b: -B).(\lambda (v: T).(getl n (CHead c0 k t) (CHead e (Bind b) v)))))) (\forall -(d: C).((getl n (CHead c0 k t) d) \to (\forall (P: Prop).P))))) (K_ind -(\lambda (k0: K).(or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: -T).(getl O (CHead c0 k0 t) (CHead e (Bind b) v)))))) (\forall (d: C).((getl O -(CHead c0 k0 t) d) \to (\forall (P: Prop).P))))) (\lambda (b: B).(or_introl -(ex_3 C B T (\lambda (e: C).(\lambda (b0: B).(\lambda (v: T).(getl O (CHead -c0 (Bind b) t) (CHead e (Bind b0) v)))))) (\forall (d: C).((getl O (CHead c0 -(Bind b) t) d) \to (\forall (P: Prop).P))) (ex_3_intro C B T (\lambda (e: -C).(\lambda (b0: B).(\lambda (v: T).(getl O (CHead c0 (Bind b) t) (CHead e -(Bind b0) v))))) c0 b t (getl_refl b c0 t)))) (\lambda (f: F).(let H_x \def -(H O) in (let H0 \def H_x in (or_ind (ex_3 C B T (\lambda (e: C).(\lambda (b: -B).(\lambda (v: T).(getl O c0 (CHead e (Bind b) v)))))) (\forall (d: -C).((getl O c0 d) \to (\forall (P: Prop).P))) (or (ex_3 C B T (\lambda (e: -C).(\lambda (b: B).(\lambda (v: T).(getl O (CHead c0 (Flat f) t) (CHead e -(Bind b) v)))))) (\forall (d: C).((getl O (CHead c0 (Flat f) t) d) \to -(\forall (P: Prop).P)))) (\lambda (H1: (ex_3 C B T (\lambda (e: C).(\lambda -(b: B).(\lambda (v: T).(getl O c0 (CHead e (Bind b) v))))))).(ex_3_ind C B T -(\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl O c0 (CHead e (Bind b) -v))))) (or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl -O (CHead c0 (Flat f) t) (CHead e (Bind b) v)))))) (\forall (d: C).((getl O -(CHead c0 (Flat f) t) d) \to (\forall (P: Prop).P)))) (\lambda (x0: -C).(\lambda (x1: B).(\lambda (x2: T).(\lambda (H2: (getl O c0 (CHead x0 (Bind -x1) x2))).(or_introl (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: -T).(getl O (CHead c0 (Flat f) t) (CHead e (Bind b) v)))))) (\forall (d: -C).((getl O (CHead c0 (Flat f) t) d) \to (\forall (P: Prop).P))) (ex_3_intro -C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl O (CHead c0 (Flat -f) t) (CHead e (Bind b) v))))) x0 x1 x2 (getl_flat c0 (CHead x0 (Bind x1) x2) -O H2 f t))))))) H1)) (\lambda (H1: ((\forall (d: C).((getl O c0 d) \to -(\forall (P: Prop).P))))).(or_intror (ex_3 C B T (\lambda (e: C).(\lambda (b: -B).(\lambda (v: T).(getl O (CHead c0 (Flat f) t) (CHead e (Bind b) v)))))) -(\forall (d: C).((getl O (CHead c0 (Flat f) t) d) \to (\forall (P: Prop).P))) -(\lambda (d: C).(\lambda (H2: (getl O (CHead c0 (Flat f) t) d)).(\lambda (P: -Prop).(H1 d (getl_intro O c0 d c0 (drop_refl c0) (clear_gen_flat f c0 d t -(getl_gen_O (CHead c0 (Flat f) t) d H2))) P)))))) H0)))) k) (\lambda (n: -nat).(\lambda (_: (or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda -(v: T).(getl n (CHead c0 k t) (CHead e (Bind b) v)))))) (\forall (d: -C).((getl n (CHead c0 k t) d) \to (\forall (P: Prop).P))))).(let H_x \def (H -(r k n)) in (let H1 \def H_x in (or_ind (ex_3 C B T (\lambda (e: C).(\lambda -(b: B).(\lambda (v: T).(getl (r k n) c0 (CHead e (Bind b) v)))))) (\forall -(d: C).((getl (r k n) c0 d) \to (\forall (P: Prop).P))) (or (ex_3 C B T -(\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl (S n) (CHead c0 k t) -(CHead e (Bind b) v)))))) (\forall (d: C).((getl (S n) (CHead c0 k t) d) \to -(\forall (P: Prop).P)))) (\lambda (H2: (ex_3 C B T (\lambda (e: C).(\lambda -(b: B).(\lambda (v: T).(getl (r k n) c0 (CHead e (Bind b) v))))))).(ex_3_ind -C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl (r k n) c0 (CHead -e (Bind b) v))))) (or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda -(v: T).(getl (S n) (CHead c0 k t) (CHead e (Bind b) v)))))) (\forall (d: -C).((getl (S n) (CHead c0 k t) d) \to (\forall (P: Prop).P)))) (\lambda (x0: -C).(\lambda (x1: B).(\lambda (x2: T).(\lambda (H3: (getl (r k n) c0 (CHead x0 -(Bind x1) x2))).(or_introl (ex_3 C B T (\lambda (e: C).(\lambda (b: -B).(\lambda (v: T).(getl (S n) (CHead c0 k t) (CHead e (Bind b) v)))))) -(\forall (d: C).((getl (S n) (CHead c0 k t) d) \to (\forall (P: Prop).P))) -(ex_3_intro C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl (S n) -(CHead c0 k t) (CHead e (Bind b) v))))) x0 x1 x2 (getl_head k n c0 (CHead x0 -(Bind x1) x2) H3 t))))))) H2)) (\lambda (H2: ((\forall (d: C).((getl (r k n) -c0 d) \to (\forall (P: Prop).P))))).(or_intror (ex_3 C B T (\lambda (e: -C).(\lambda (b: B).(\lambda (v: T).(getl (S n) (CHead c0 k t) (CHead e (Bind -b) v)))))) (\forall (d: C).((getl (S n) (CHead c0 k t) d) \to (\forall (P: -Prop).P))) (\lambda (d: C).(\lambda (H3: (getl (S n) (CHead c0 k t) -d)).(\lambda (P: Prop).(H2 d (getl_gen_S k c0 d t n H3) P)))))) H1))))) -i)))))) c). -(* COMMENTS -Initial nodes: 1563 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/getl/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/getl/defs.ma deleted file mode 100644 index a6efd2eee..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/getl/defs.ma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/drop/defs.ma". - -include "Basic-1/clear/defs.ma". - -inductive getl (h: nat) (c1: C) (c2: C): Prop \def -| getl_intro: \forall (e: C).((drop h O c1 e) \to ((clear e c2) \to (getl h -c1 c2))). - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/getl/drop.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/getl/drop.ma deleted file mode 100644 index e4404e137..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/getl/drop.ma +++ /dev/null @@ -1,514 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/getl/props.ma". - -include "Basic-1/clear/drop.ma". - -theorem getl_drop: - \forall (b: B).(\forall (c: C).(\forall (e: C).(\forall (u: T).(\forall (h: -nat).((getl h c (CHead e (Bind b) u)) \to (drop (S h) O c e)))))) -\def - \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e: -C).(\forall (u: T).(\forall (h: nat).((getl h c0 (CHead e (Bind b) u)) \to -(drop (S h) O c0 e)))))) (\lambda (n: nat).(\lambda (e: C).(\lambda (u: -T).(\lambda (h: nat).(\lambda (H: (getl h (CSort n) (CHead e (Bind b) -u))).(getl_gen_sort n h (CHead e (Bind b) u) H (drop (S h) O (CSort n) -e))))))) (\lambda (c0: C).(\lambda (H: ((\forall (e: C).(\forall (u: -T).(\forall (h: nat).((getl h c0 (CHead e (Bind b) u)) \to (drop (S h) O c0 -e))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e: C).(\lambda (u: -T).(\lambda (h: nat).(nat_ind (\lambda (n: nat).((getl n (CHead c0 k t) -(CHead e (Bind b) u)) \to (drop (S n) O (CHead c0 k t) e))) (\lambda (H0: -(getl O (CHead c0 k t) (CHead e (Bind b) u))).(K_ind (\lambda (k0: K).((clear -(CHead c0 k0 t) (CHead e (Bind b) u)) \to (drop (S O) O (CHead c0 k0 t) e))) -(\lambda (b0: B).(\lambda (H1: (clear (CHead c0 (Bind b0) t) (CHead e (Bind -b) u))).(let H2 \def (f_equal C C (\lambda (e0: C).(match e0 in C return -(\lambda (_: C).C) with [(CSort _) \Rightarrow e | (CHead c1 _ _) \Rightarrow -c1])) (CHead e (Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 -(CHead e (Bind b) u) t H1)) in ((let H3 \def (f_equal C B (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow b | -(CHead _ k0 _) \Rightarrow (match k0 in K return (\lambda (_: K).B) with -[(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow b])])) (CHead e (Bind b) u) -(CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e (Bind b) u) t H1)) in -((let H4 \def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda -(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) -(CHead e (Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e -(Bind b) u) t H1)) in (\lambda (H5: (eq B b b0)).(\lambda (H6: (eq C e -c0)).(eq_ind_r C c0 (\lambda (c1: C).(drop (S O) O (CHead c0 (Bind b0) t) -c1)) (eq_ind B b (\lambda (b1: B).(drop (S O) O (CHead c0 (Bind b1) t) c0)) -(drop_drop (Bind b) O c0 c0 (drop_refl c0) t) b0 H5) e H6)))) H3)) H2)))) -(\lambda (f: F).(\lambda (H1: (clear (CHead c0 (Flat f) t) (CHead e (Bind b) -u))).(drop_clear_O b (CHead c0 (Flat f) t) e u (clear_flat c0 (CHead e (Bind -b) u) (clear_gen_flat f c0 (CHead e (Bind b) u) t H1) f t) e O (drop_refl -e)))) k (getl_gen_O (CHead c0 k t) (CHead e (Bind b) u) H0))) (\lambda (n: -nat).(\lambda (_: (((getl n (CHead c0 k t) (CHead e (Bind b) u)) \to (drop (S -n) O (CHead c0 k t) e)))).(\lambda (H1: (getl (S n) (CHead c0 k t) (CHead e -(Bind b) u))).(drop_drop k (S n) c0 e (eq_ind_r nat (S (r k n)) (\lambda (n0: -nat).(drop n0 O c0 e)) (H e u (r k n) (getl_gen_S k c0 (CHead e (Bind b) u) t -n H1)) (r k (S n)) (r_S k n)) t)))) h)))))))) c)). -(* COMMENTS -Initial nodes: 827 -END *) - -theorem getl_drop_conf_lt: - \forall (b: B).(\forall (c: C).(\forall (c0: C).(\forall (u: T).(\forall (i: -nat).((getl i c (CHead c0 (Bind b) u)) \to (\forall (e: C).(\forall (h: -nat).(\forall (d: nat).((drop h (S (plus i d)) c e) \to (ex3_2 T C (\lambda -(v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: -C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop -h d c0 e0))))))))))))) -\def - \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (c1: -C).(\forall (u: T).(\forall (i: nat).((getl i c0 (CHead c1 (Bind b) u)) \to -(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h (S (plus i d)) -c0 e) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) -(\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda -(_: T).(\lambda (e0: C).(drop h d c1 e0))))))))))))) (\lambda (n: -nat).(\lambda (c0: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H: (getl i -(CSort n) (CHead c0 (Bind b) u))).(\lambda (e: C).(\lambda (h: nat).(\lambda -(d: nat).(\lambda (_: (drop h (S (plus i d)) (CSort n) e)).(getl_gen_sort n i -(CHead c0 (Bind b) u) H (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u -(lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) -v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c0 e0)))))))))))))) (\lambda -(c0: C).(\lambda (H: ((\forall (c1: C).(\forall (u: T).(\forall (i: -nat).((getl i c0 (CHead c1 (Bind b) u)) \to (\forall (e: C).(\forall (h: -nat).(\forall (d: nat).((drop h (S (plus i d)) c0 e) \to (ex3_2 T C (\lambda -(v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: -C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop -h d c1 e0)))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c1: -C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i (CHead c0 k t) -(CHead c1 (Bind b) u))).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: -nat).(\lambda (H1: (drop h (S (plus i d)) (CHead c0 k t) e)).(let H2 \def -(getl_gen_all (CHead c0 k t) (CHead c1 (Bind b) u) i H0) in (ex2_ind C -(\lambda (e0: C).(drop i O (CHead c0 k t) e0)) (\lambda (e0: C).(clear e0 -(CHead c1 (Bind b) u))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u -(lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) -v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x: -C).(\lambda (H3: (drop i O (CHead c0 k t) x)).(\lambda (H4: (clear x (CHead -c1 (Bind b) u))).(C_ind (\lambda (c2: C).((drop i O (CHead c0 k t) c2) \to -((clear c2 (CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: -C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead -e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) -(\lambda (n: nat).(\lambda (_: (drop i O (CHead c0 k t) (CSort n))).(\lambda -(H6: (clear (CSort n) (CHead c1 (Bind b) u))).(clear_gen_sort (CHead c1 (Bind -b) u) n H6 (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) -(\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda -(_: T).(\lambda (e0: C).(drop h d c1 e0)))))))) (\lambda (x0: C).(\lambda -(IHx: (((drop i O (CHead c0 k t) x0) \to ((clear x0 (CHead c1 (Bind b) u)) -\to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) -(\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda -(_: T).(\lambda (e0: C).(drop h d c1 e0)))))))).(\lambda (k0: K).(\lambda -(t0: T).(\lambda (H5: (drop i O (CHead c0 k t) (CHead x0 k0 t0))).(\lambda -(H6: (clear (CHead x0 k0 t0) (CHead c1 (Bind b) u))).(K_ind (\lambda (k1: -K).((drop i O (CHead c0 k t) (CHead x0 k1 t0)) \to ((clear (CHead x0 k1 t0) -(CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u -(lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) -v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) (\lambda (b0: -B).(\lambda (H7: (drop i O (CHead c0 k t) (CHead x0 (Bind b0) t0))).(\lambda -(H8: (clear (CHead x0 (Bind b0) t0) (CHead c1 (Bind b) u))).(let H9 \def -(f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with -[(CSort _) \Rightarrow c1 | (CHead c2 _ _) \Rightarrow c2])) (CHead c1 (Bind -b) u) (CHead x0 (Bind b0) t0) (clear_gen_bind b0 x0 (CHead c1 (Bind b) u) t0 -H8)) in ((let H10 \def (f_equal C B (\lambda (e0: C).(match e0 in C return -(\lambda (_: C).B) with [(CSort _) \Rightarrow b | (CHead _ k1 _) \Rightarrow -(match k1 in K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | -(Flat _) \Rightarrow b])])) (CHead c1 (Bind b) u) (CHead x0 (Bind b0) t0) -(clear_gen_bind b0 x0 (CHead c1 (Bind b) u) t0 H8)) in ((let H11 \def -(f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow u | (CHead _ _ t1) \Rightarrow t1])) (CHead c1 (Bind -b) u) (CHead x0 (Bind b0) t0) (clear_gen_bind b0 x0 (CHead c1 (Bind b) u) t0 -H8)) in (\lambda (H12: (eq B b b0)).(\lambda (H13: (eq C c1 x0)).(let H14 -\def (eq_ind_r T t0 (\lambda (t1: T).(drop i O (CHead c0 k t) (CHead x0 (Bind -b0) t1))) H7 u H11) in (let H15 \def (eq_ind_r B b0 (\lambda (b1: B).(drop i -O (CHead c0 k t) (CHead x0 (Bind b1) u))) H14 b H12) in (let H16 \def -(eq_ind_r C x0 (\lambda (c2: C).((drop i O (CHead c0 k t) c2) \to ((clear c2 -(CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u -(lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) -v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) IHx c1 H13) in -(let H17 \def (eq_ind_r C x0 (\lambda (c2: C).(drop i O (CHead c0 k t) (CHead -c2 (Bind b) u))) H15 c1 H13) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: -C).(eq T u (lift h (r (Bind b) d) v)))) (\lambda (v: T).(\lambda (e0: -C).(drop i O e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: -C).(drop h (r (Bind b) d) c1 e0))) (ex3_2 T C (\lambda (v: T).(\lambda (_: -C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead -e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))) -(\lambda (x1: T).(\lambda (x2: C).(\lambda (H18: (eq T u (lift h (r (Bind b) -d) x1))).(\lambda (H19: (drop i O e (CHead x2 (Bind b) x1))).(\lambda (H20: -(drop h (r (Bind b) d) c1 x2)).(let H21 \def (eq_ind T u (\lambda (t1: -T).((drop i O (CHead c0 k t) c1) \to ((clear c1 (CHead c1 (Bind b) t1)) \to -(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T t1 (lift h d v)))) (\lambda -(v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: -T).(\lambda (e0: C).(drop h d c1 e0))))))) H16 (lift h (r (Bind b) d) x1) -H18) in (eq_ind_r T (lift h (r (Bind b) d) x1) (\lambda (t1: T).(ex3_2 T C -(\lambda (v: T).(\lambda (_: C).(eq T t1 (lift h d v)))) (\lambda (v: -T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: -T).(\lambda (e0: C).(drop h d c1 e0))))) (ex3_2_intro T C (\lambda (v: -T).(\lambda (_: C).(eq T (lift h (r (Bind b) d) x1) (lift h d v)))) (\lambda -(v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: -T).(\lambda (e0: C).(drop h d c1 e0))) x1 x2 (refl_equal T (lift h d x1)) -(getl_intro i e (CHead x2 (Bind b) x1) (CHead x2 (Bind b) x1) H19 (clear_bind -b x2 x1)) H20) u H18))))))) (drop_conf_lt (Bind b) i u c1 (CHead c0 k t) H17 -e h d H1))))))))) H10)) H9))))) (\lambda (f: F).(\lambda (H7: (drop i O -(CHead c0 k t) (CHead x0 (Flat f) t0))).(\lambda (H8: (clear (CHead x0 (Flat -f) t0) (CHead c1 (Bind b) u))).(nat_ind (\lambda (n: nat).((drop h (S (plus n -d)) (CHead c0 k t) e) \to ((drop n O (CHead c0 k t) (CHead x0 (Flat f) t0)) -\to ((((drop n O (CHead c0 k t) x0) \to ((clear x0 (CHead c1 (Bind b) u)) \to -(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda -(v: T).(\lambda (e0: C).(getl n e (CHead e0 (Bind b) v)))) (\lambda (_: -T).(\lambda (e0: C).(drop h d c1 e0))))))) \to (ex3_2 T C (\lambda (v: -T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: -C).(getl n e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop -h d c1 e0)))))))) (\lambda (H9: (drop h (S (plus O d)) (CHead c0 k t) -e)).(\lambda (H10: (drop O O (CHead c0 k t) (CHead x0 (Flat f) t0))).(\lambda -(IHx0: (((drop O O (CHead c0 k t) x0) \to ((clear x0 (CHead c1 (Bind b) u)) -\to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) -(\lambda (v: T).(\lambda (e0: C).(getl O e (CHead e0 (Bind b) v)))) (\lambda -(_: T).(\lambda (e0: C).(drop h d c1 e0)))))))).(let H11 \def (f_equal C C -(\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow c0 | (CHead c2 _ _) \Rightarrow c2])) (CHead c0 k t) (CHead x0 -(Flat f) t0) (drop_gen_refl (CHead c0 k t) (CHead x0 (Flat f) t0) H10)) in -((let H12 \def (f_equal C K (\lambda (e0: C).(match e0 in C return (\lambda -(_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k1 _) \Rightarrow k1])) -(CHead c0 k t) (CHead x0 (Flat f) t0) (drop_gen_refl (CHead c0 k t) (CHead x0 -(Flat f) t0) H10)) in ((let H13 \def (f_equal C T (\lambda (e0: C).(match e0 -in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow t | (CHead _ _ t1) -\Rightarrow t1])) (CHead c0 k t) (CHead x0 (Flat f) t0) (drop_gen_refl (CHead -c0 k t) (CHead x0 (Flat f) t0) H10)) in (\lambda (H14: (eq K k (Flat -f))).(\lambda (H15: (eq C c0 x0)).(let H16 \def (eq_ind_r C x0 (\lambda (c2: -C).(clear c2 (CHead c1 (Bind b) u))) (clear_gen_flat f x0 (CHead c1 (Bind b) -u) t0 H8) c0 H15) in (let H17 \def (eq_ind_r C x0 (\lambda (c2: C).((drop O O -(CHead c0 k t) c2) \to ((clear c2 (CHead c1 (Bind b) u)) \to (ex3_2 T C -(\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: -T).(\lambda (e0: C).(getl O e (CHead e0 (Bind b) v)))) (\lambda (_: -T).(\lambda (e0: C).(drop h d c1 e0))))))) IHx0 c0 H15) in (let H18 \def -(eq_ind K k (\lambda (k1: K).((drop O O (CHead c0 k1 t) c0) \to ((clear c0 -(CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u -(lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl O e (CHead e0 (Bind b) -v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) H17 (Flat f) -H14) in (let H19 \def (eq_ind K k (\lambda (k1: K).(drop h (S (plus O d)) -(CHead c0 k1 t) e)) H9 (Flat f) H14) in (ex3_2_ind C T (\lambda (e0: -C).(\lambda (v: T).(eq C e (CHead e0 (Flat f) v)))) (\lambda (_: C).(\lambda -(v: T).(eq T t (lift h (r (Flat f) (plus O d)) v)))) (\lambda (e0: -C).(\lambda (_: T).(drop h (r (Flat f) (plus O d)) c0 e0))) (ex3_2 T C -(\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: -T).(\lambda (e0: C).(getl O e (CHead e0 (Bind b) v)))) (\lambda (_: -T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x1: C).(\lambda (x2: -T).(\lambda (H20: (eq C e (CHead x1 (Flat f) x2))).(\lambda (H21: (eq T t -(lift h (r (Flat f) (plus O d)) x2))).(\lambda (H22: (drop h (r (Flat f) -(plus O d)) c0 x1)).(let H23 \def (f_equal T T (\lambda (e0: T).e0) t (lift h -(r (Flat f) (plus O d)) x2) H21) in (let H24 \def (eq_ind C e (\lambda (c2: -C).((drop O O (CHead c0 (Flat f) t) c0) \to ((clear c0 (CHead c1 (Bind b) u)) -\to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) -(\lambda (v: T).(\lambda (e0: C).(getl O c2 (CHead e0 (Bind b) v)))) (\lambda -(_: T).(\lambda (e0: C).(drop h d c1 e0))))))) H18 (CHead x1 (Flat f) x2) -H20) in (eq_ind_r C (CHead x1 (Flat f) x2) (\lambda (c2: C).(ex3_2 T C -(\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: -T).(\lambda (e0: C).(getl O c2 (CHead e0 (Bind b) v)))) (\lambda (_: -T).(\lambda (e0: C).(drop h d c1 e0))))) (let H25 \def (eq_ind T t (\lambda -(t1: T).((drop O O (CHead c0 (Flat f) t1) c0) \to ((clear c0 (CHead c1 (Bind -b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) -(\lambda (v: T).(\lambda (e0: C).(getl O (CHead x1 (Flat f) x2) (CHead e0 -(Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) H24 -(lift h (S d) x2) H23) in (let H26 \def (H c1 u O (getl_intro O c0 (CHead c1 -(Bind b) u) c0 (drop_refl c0) H16) x1 h d H22) in (ex3_2_ind T C (\lambda (v: -T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: -C).(getl O x1 (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop -h d c1 e0))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d -v)))) (\lambda (v: T).(\lambda (e0: C).(getl O (CHead x1 (Flat f) x2) (CHead -e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))) -(\lambda (x3: T).(\lambda (x4: C).(\lambda (H27: (eq T u (lift h d -x3))).(\lambda (H28: (getl O x1 (CHead x4 (Bind b) x3))).(\lambda (H29: (drop -h d c1 x4)).(let H30 \def (eq_ind T u (\lambda (t1: T).((drop O O (CHead c0 -(Flat f) (lift h (S d) x2)) c0) \to ((clear c0 (CHead c1 (Bind b) t1)) \to -(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T t1 (lift h d v)))) (\lambda -(v: T).(\lambda (e0: C).(getl O (CHead x1 (Flat f) x2) (CHead e0 (Bind b) -v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) H25 (lift h d -x3) H27) in (let H31 \def (eq_ind T u (\lambda (t1: T).(clear c0 (CHead c1 -(Bind b) t1))) H16 (lift h d x3) H27) in (eq_ind_r T (lift h d x3) (\lambda -(t1: T).(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T t1 (lift h d v)))) -(\lambda (v: T).(\lambda (e0: C).(getl O (CHead x1 (Flat f) x2) (CHead e0 -(Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))) -(ex3_2_intro T C (\lambda (v: T).(\lambda (_: C).(eq T (lift h d x3) (lift h -d v)))) (\lambda (v: T).(\lambda (e0: C).(getl O (CHead x1 (Flat f) x2) -(CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))) -x3 x4 (refl_equal T (lift h d x3)) (getl_flat x1 (CHead x4 (Bind b) x3) O H28 -f x2) H29) u H27)))))))) H26))) e H20)))))))) (drop_gen_skip_l c0 e t h (plus -O d) (Flat f) H19))))))))) H12)) H11))))) (\lambda (i0: nat).(\lambda (IHi: -(((drop h (S (plus i0 d)) (CHead c0 k t) e) \to ((drop i0 O (CHead c0 k t) -(CHead x0 (Flat f) t0)) \to ((((drop i0 O (CHead c0 k t) x0) \to ((clear x0 -(CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u -(lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i0 e (CHead e0 (Bind -b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) \to (ex3_2 T -C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: -T).(\lambda (e0: C).(getl i0 e (CHead e0 (Bind b) v)))) (\lambda (_: -T).(\lambda (e0: C).(drop h d c1 e0))))))))).(\lambda (H9: (drop h (S (plus -(S i0) d)) (CHead c0 k t) e)).(\lambda (H10: (drop (S i0) O (CHead c0 k t) -(CHead x0 (Flat f) t0))).(\lambda (IHx0: (((drop (S i0) O (CHead c0 k t) x0) -\to ((clear x0 (CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda -(_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl (S i0) -e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 -e0)))))))).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 -k v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r k (plus (S i0) d)) -v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r k (plus (S i0) d)) c0 e0))) -(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda -(v: T).(\lambda (e0: C).(getl (S i0) e (CHead e0 (Bind b) v)))) (\lambda (_: -T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x1: C).(\lambda (x2: -T).(\lambda (H11: (eq C e (CHead x1 k x2))).(\lambda (H12: (eq T t (lift h (r -k (plus (S i0) d)) x2))).(\lambda (H13: (drop h (r k (plus (S i0) d)) c0 -x1)).(let H14 \def (f_equal T T (\lambda (e0: T).e0) t (lift h (r k (plus (S -i0) d)) x2) H12) in (let H15 \def (eq_ind C e (\lambda (c2: C).((drop h (S -(plus i0 d)) (CHead c0 k t) c2) \to ((drop i0 O (CHead c0 k t) (CHead x0 -(Flat f) t0)) \to ((((drop i0 O (CHead c0 k t) x0) \to ((clear x0 (CHead c1 -(Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d -v)))) (\lambda (v: T).(\lambda (e0: C).(getl i0 c2 (CHead e0 (Bind b) v)))) -(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) \to (ex3_2 T C -(\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: -T).(\lambda (e0: C).(getl i0 c2 (CHead e0 (Bind b) v)))) (\lambda (_: -T).(\lambda (e0: C).(drop h d c1 e0)))))))) IHi (CHead x1 k x2) H11) in (let -H16 \def (eq_ind C e (\lambda (c2: C).((drop (S i0) O (CHead c0 k t) x0) \to -((clear x0 (CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: -C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl (S i0) c2 -(CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 -e0))))))) IHx0 (CHead x1 k x2) H11) in (eq_ind_r C (CHead x1 k x2) (\lambda -(c2: C).(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) -(\lambda (v: T).(\lambda (e0: C).(getl (S i0) c2 (CHead e0 (Bind b) v)))) -(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))) (let H17 \def (eq_ind T -t (\lambda (t1: T).((drop (S i0) O (CHead c0 k t1) x0) \to ((clear x0 (CHead -c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift -h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl (S i0) (CHead x1 k x2) -(CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 -e0))))))) H16 (lift h (r k (S (plus i0 d))) x2) H14) in (let H18 \def (eq_ind -T t (\lambda (t1: T).((drop h (S (plus i0 d)) (CHead c0 k t1) (CHead x1 k -x2)) \to ((drop i0 O (CHead c0 k t1) (CHead x0 (Flat f) t0)) \to ((((drop i0 -O (CHead c0 k t1) x0) \to ((clear x0 (CHead c1 (Bind b) u)) \to (ex3_2 T C -(\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: -T).(\lambda (e0: C).(getl i0 (CHead x1 k x2) (CHead e0 (Bind b) v)))) -(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) \to (ex3_2 T C -(\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: -T).(\lambda (e0: C).(getl i0 (CHead x1 k x2) (CHead e0 (Bind b) v)))) -(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))))))) H15 (lift h (r k (S -(plus i0 d))) x2) H14) in (let H19 \def (eq_ind nat (r k (plus (S i0) d)) -(\lambda (n: nat).(drop h n c0 x1)) H13 (plus (r k (S i0)) d) (r_plus k (S -i0) d)) in (let H20 \def (eq_ind nat (r k (S i0)) (\lambda (n: nat).(drop h -(plus n d) c0 x1)) H19 (S (r k i0)) (r_S k i0)) in (let H21 \def (H c1 u (r k -i0) (getl_intro (r k i0) c0 (CHead c1 (Bind b) u) (CHead x0 (Flat f) t0) -(drop_gen_drop k c0 (CHead x0 (Flat f) t0) t i0 H10) (clear_flat x0 (CHead c1 -(Bind b) u) (clear_gen_flat f x0 (CHead c1 (Bind b) u) t0 H8) f t0)) x1 h d -H20) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d -v)))) (\lambda (v: T).(\lambda (e0: C).(getl (r k i0) x1 (CHead e0 (Bind b) -v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))) (ex3_2 T C (\lambda -(v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: -C).(getl (S i0) (CHead x1 k x2) (CHead e0 (Bind b) v)))) (\lambda (_: -T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x3: T).(\lambda (x4: -C).(\lambda (H22: (eq T u (lift h d x3))).(\lambda (H23: (getl (r k i0) x1 -(CHead x4 (Bind b) x3))).(\lambda (H24: (drop h d c1 x4)).(let H25 \def -(eq_ind T u (\lambda (t1: T).((drop (S i0) O (CHead c0 k (lift h (r k (S -(plus i0 d))) x2)) x0) \to ((clear x0 (CHead c1 (Bind b) t1)) \to (ex3_2 T C -(\lambda (v: T).(\lambda (_: C).(eq T t1 (lift h d v)))) (\lambda (v: -T).(\lambda (e0: C).(getl (S i0) (CHead x1 k x2) (CHead e0 (Bind b) v)))) -(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) H17 (lift h d x3) -H22) in (let H26 \def (eq_ind T u (\lambda (t1: T).(clear x0 (CHead c1 (Bind -b) t1))) (clear_gen_flat f x0 (CHead c1 (Bind b) u) t0 H8) (lift h d x3) H22) -in (eq_ind_r T (lift h d x3) (\lambda (t1: T).(ex3_2 T C (\lambda (v: -T).(\lambda (_: C).(eq T t1 (lift h d v)))) (\lambda (v: T).(\lambda (e0: -C).(getl (S i0) (CHead x1 k x2) (CHead e0 (Bind b) v)))) (\lambda (_: -T).(\lambda (e0: C).(drop h d c1 e0))))) (ex3_2_intro T C (\lambda (v: -T).(\lambda (_: C).(eq T (lift h d x3) (lift h d v)))) (\lambda (v: -T).(\lambda (e0: C).(getl (S i0) (CHead x1 k x2) (CHead e0 (Bind b) v)))) -(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))) x3 x4 (refl_equal T (lift -h d x3)) (getl_head k i0 x1 (CHead x4 (Bind b) x3) H23 x2) H24) u H22)))))))) -H21)))))) e H11))))))))) (drop_gen_skip_l c0 e t h (plus (S i0) d) k -H9))))))) i H1 H7 IHx)))) k0 H5 H6))))))) x H3 H4)))) H2)))))))))))))) c)). -(* COMMENTS -Initial nodes: 6137 -END *) - -theorem getl_drop_conf_ge: - \forall (i: nat).(\forall (a: C).(\forall (c: C).((getl i c a) \to (\forall -(e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to ((le (plus d -h) i) \to (getl (minus i h) e a))))))))) -\def - \lambda (i: nat).(\lambda (a: C).(\lambda (c: C).(\lambda (H: (getl i c -a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H0: (drop h -d c e)).(\lambda (H1: (le (plus d h) i)).(let H2 \def (getl_gen_all c a i H) -in (ex2_ind C (\lambda (e0: C).(drop i O c e0)) (\lambda (e0: C).(clear e0 -a)) (getl (minus i h) e a) (\lambda (x: C).(\lambda (H3: (drop i O c -x)).(\lambda (H4: (clear x a)).(getl_intro (minus i h) e a x (drop_conf_ge i -x c H3 e h d H0 H1) H4)))) H2)))))))))). -(* COMMENTS -Initial nodes: 141 -END *) - -theorem getl_conf_ge_drop: - \forall (b: B).(\forall (c1: C).(\forall (e: C).(\forall (u: T).(\forall (i: -nat).((getl i c1 (CHead e (Bind b) u)) \to (\forall (c2: C).((drop (S O) i c1 -c2) \to (drop i O c2 e)))))))) -\def - \lambda (b: B).(\lambda (c1: C).(\lambda (e: C).(\lambda (u: T).(\lambda (i: -nat).(\lambda (H: (getl i c1 (CHead e (Bind b) u))).(\lambda (c2: C).(\lambda -(H0: (drop (S O) i c1 c2)).(let H3 \def (eq_ind nat (minus (S i) (S O)) -(\lambda (n: nat).(drop n O c2 e)) (drop_conf_ge (S i) e c1 (getl_drop b c1 e -u i H) c2 (S O) i H0 (eq_ind_r nat (plus (S O) i) (\lambda (n: nat).(le n (S -i))) (le_n (S i)) (plus i (S O)) (plus_sym i (S O)))) i (minus_Sx_SO i)) in -H3)))))))). -(* COMMENTS -Initial nodes: 151 -END *) - -theorem getl_drop_conf_rev: - \forall (j: nat).(\forall (e1: C).(\forall (e2: C).((drop j O e1 e2) \to -(\forall (b: B).(\forall (c2: C).(\forall (v2: T).(\forall (i: nat).((getl i -c2 (CHead e2 (Bind b) v2)) \to (ex2 C (\lambda (c1: C).(drop j O c1 c2)) -(\lambda (c1: C).(drop (S i) j c1 e1))))))))))) -\def - \lambda (j: nat).(\lambda (e1: C).(\lambda (e2: C).(\lambda (H: (drop j O e1 -e2)).(\lambda (b: B).(\lambda (c2: C).(\lambda (v2: T).(\lambda (i: -nat).(\lambda (H0: (getl i c2 (CHead e2 (Bind b) v2))).(drop_conf_rev j e1 e2 -H c2 (S i) (getl_drop b c2 e2 v2 i H0)))))))))). -(* COMMENTS -Initial nodes: 69 -END *) - -theorem drop_getl_trans_lt: - \forall (i: nat).(\forall (d: nat).((lt i d) \to (\forall (c1: C).(\forall -(c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (b: B).(\forall (e2: -C).(\forall (v: T).((getl i c2 (CHead e2 (Bind b) v)) \to (ex2 C (\lambda -(e1: C).(getl i c1 (CHead e1 (Bind b) (lift h (minus d (S i)) v)))) (\lambda -(e1: C).(drop h (minus d (S i)) e1 e2))))))))))))) -\def - \lambda (i: nat).(\lambda (d: nat).(\lambda (H: (lt i d)).(\lambda (c1: -C).(\lambda (c2: C).(\lambda (h: nat).(\lambda (H0: (drop h d c1 -c2)).(\lambda (b: B).(\lambda (e2: C).(\lambda (v: T).(\lambda (H1: (getl i -c2 (CHead e2 (Bind b) v))).(let H2 \def (getl_gen_all c2 (CHead e2 (Bind b) -v) i H1) in (ex2_ind C (\lambda (e: C).(drop i O c2 e)) (\lambda (e: -C).(clear e (CHead e2 (Bind b) v))) (ex2 C (\lambda (e1: C).(getl i c1 (CHead -e1 (Bind b) (lift h (minus d (S i)) v)))) (\lambda (e1: C).(drop h (minus d -(S i)) e1 e2))) (\lambda (x: C).(\lambda (H3: (drop i O c2 x)).(\lambda (H4: -(clear x (CHead e2 (Bind b) v))).(ex2_ind C (\lambda (e1: C).(drop i O c1 -e1)) (\lambda (e1: C).(drop h (minus d i) e1 x)) (ex2 C (\lambda (e1: -C).(getl i c1 (CHead e1 (Bind b) (lift h (minus d (S i)) v)))) (\lambda (e1: -C).(drop h (minus d (S i)) e1 e2))) (\lambda (x0: C).(\lambda (H5: (drop i O -c1 x0)).(\lambda (H6: (drop h (minus d i) x0 x)).(let H7 \def (eq_ind nat -(minus d i) (\lambda (n: nat).(drop h n x0 x)) H6 (S (minus d (S i))) -(minus_x_Sy d i H)) in (let H8 \def (drop_clear_S x x0 h (minus d (S i)) H7 b -e2 v H4) in (ex2_ind C (\lambda (c3: C).(clear x0 (CHead c3 (Bind b) (lift h -(minus d (S i)) v)))) (\lambda (c3: C).(drop h (minus d (S i)) c3 e2)) (ex2 C -(\lambda (e1: C).(getl i c1 (CHead e1 (Bind b) (lift h (minus d (S i)) v)))) -(\lambda (e1: C).(drop h (minus d (S i)) e1 e2))) (\lambda (x1: C).(\lambda -(H9: (clear x0 (CHead x1 (Bind b) (lift h (minus d (S i)) v)))).(\lambda -(H10: (drop h (minus d (S i)) x1 e2)).(ex_intro2 C (\lambda (e1: C).(getl i -c1 (CHead e1 (Bind b) (lift h (minus d (S i)) v)))) (\lambda (e1: C).(drop h -(minus d (S i)) e1 e2)) x1 (getl_intro i c1 (CHead x1 (Bind b) (lift h (minus -d (S i)) v)) x0 H5 H9) H10)))) H8)))))) (drop_trans_le i d (le_S_n i d (le_S -(S i) d H)) c1 c2 h H0 x H3))))) H2)))))))))))). -(* COMMENTS -Initial nodes: 627 -END *) - -theorem drop_getl_trans_le: - \forall (i: nat).(\forall (d: nat).((le i d) \to (\forall (c1: C).(\forall -(c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((getl i c2 -e2) \to (ex3_2 C C (\lambda (e0: C).(\lambda (_: C).(drop i O c1 e0))) -(\lambda (e0: C).(\lambda (e1: C).(drop h (minus d i) e0 e1))) (\lambda (_: -C).(\lambda (e1: C).(clear e1 e2)))))))))))) -\def - \lambda (i: nat).(\lambda (d: nat).(\lambda (H: (le i d)).(\lambda (c1: -C).(\lambda (c2: C).(\lambda (h: nat).(\lambda (H0: (drop h d c1 -c2)).(\lambda (e2: C).(\lambda (H1: (getl i c2 e2)).(let H2 \def -(getl_gen_all c2 e2 i H1) in (ex2_ind C (\lambda (e: C).(drop i O c2 e)) -(\lambda (e: C).(clear e e2)) (ex3_2 C C (\lambda (e0: C).(\lambda (_: -C).(drop i O c1 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d i) -e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear e1 e2)))) (\lambda (x: -C).(\lambda (H3: (drop i O c2 x)).(\lambda (H4: (clear x e2)).(let H5 \def -(drop_trans_le i d H c1 c2 h H0 x H3) in (ex2_ind C (\lambda (e1: C).(drop i -O c1 e1)) (\lambda (e1: C).(drop h (minus d i) e1 x)) (ex3_2 C C (\lambda -(e0: C).(\lambda (_: C).(drop i O c1 e0))) (\lambda (e0: C).(\lambda (e1: -C).(drop h (minus d i) e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear e1 -e2)))) (\lambda (x0: C).(\lambda (H6: (drop i O c1 x0)).(\lambda (H7: (drop h -(minus d i) x0 x)).(ex3_2_intro C C (\lambda (e0: C).(\lambda (_: C).(drop i -O c1 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d i) e0 e1))) -(\lambda (_: C).(\lambda (e1: C).(clear e1 e2))) x0 x H6 H7 H4)))) H5))))) -H2)))))))))). -(* COMMENTS -Initial nodes: 323 -END *) - -theorem drop_getl_trans_ge: - \forall (i: nat).(\forall (c1: C).(\forall (c2: C).(\forall (d: -nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((getl i c2 e2) -\to ((le d i) \to (getl (plus i h) c1 e2))))))))) -\def - \lambda (i: nat).(\lambda (c1: C).(\lambda (c2: C).(\lambda (d: -nat).(\lambda (h: nat).(\lambda (H: (drop h d c1 c2)).(\lambda (e2: -C).(\lambda (H0: (getl i c2 e2)).(\lambda (H1: (le d i)).(let H2 \def -(getl_gen_all c2 e2 i H0) in (ex2_ind C (\lambda (e: C).(drop i O c2 e)) -(\lambda (e: C).(clear e e2)) (getl (plus i h) c1 e2) (\lambda (x: -C).(\lambda (H3: (drop i O c2 x)).(\lambda (H4: (clear x e2)).(getl_intro -(plus i h) c1 e2 x (drop_trans_ge i c1 c2 d h H x H3 H1) H4)))) H2)))))))))). -(* COMMENTS -Initial nodes: 137 -END *) - -theorem getl_drop_trans: - \forall (c1: C).(\forall (c2: C).(\forall (h: nat).((getl h c1 c2) \to -(\forall (e2: C).(\forall (i: nat).((drop (S i) O c2 e2) \to (drop (S (plus i -h)) O c1 e2))))))) -\def - \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (h: -nat).((getl h c c2) \to (\forall (e2: C).(\forall (i: nat).((drop (S i) O c2 -e2) \to (drop (S (plus i h)) O c e2)))))))) (\lambda (n: nat).(\lambda (c2: -C).(\lambda (h: nat).(\lambda (H: (getl h (CSort n) c2)).(\lambda (e2: -C).(\lambda (i: nat).(\lambda (_: (drop (S i) O c2 e2)).(getl_gen_sort n h c2 -H (drop (S (plus i h)) O (CSort n) e2))))))))) (\lambda (c2: C).(\lambda -(IHc: ((\forall (c3: C).(\forall (h: nat).((getl h c2 c3) \to (\forall (e2: -C).(\forall (i: nat).((drop (S i) O c3 e2) \to (drop (S (plus i h)) O c2 -e2))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t: T).(\forall -(c3: C).(\forall (h: nat).((getl h (CHead c2 k0 t) c3) \to (\forall (e2: -C).(\forall (i: nat).((drop (S i) O c3 e2) \to (drop (S (plus i h)) O (CHead -c2 k0 t) e2))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (c3: -C).(\lambda (h: nat).(nat_ind (\lambda (n: nat).((getl n (CHead c2 (Bind b) -t) c3) \to (\forall (e2: C).(\forall (i: nat).((drop (S i) O c3 e2) \to (drop -(S (plus i n)) O (CHead c2 (Bind b) t) e2)))))) (\lambda (H: (getl O (CHead -c2 (Bind b) t) c3)).(\lambda (e2: C).(\lambda (i: nat).(\lambda (H0: (drop (S -i) O c3 e2)).(let H1 \def (eq_ind C c3 (\lambda (c: C).(drop (S i) O c e2)) -H0 (CHead c2 (Bind b) t) (clear_gen_bind b c2 c3 t (getl_gen_O (CHead c2 -(Bind b) t) c3 H))) in (eq_ind nat i (\lambda (n: nat).(drop (S n) O (CHead -c2 (Bind b) t) e2)) (drop_drop (Bind b) i c2 e2 (drop_gen_drop (Bind b) c2 e2 -t i H1) t) (plus i O) (plus_n_O i))))))) (\lambda (n: nat).(\lambda (_: -(((getl n (CHead c2 (Bind b) t) c3) \to (\forall (e2: C).(\forall (i: -nat).((drop (S i) O c3 e2) \to (drop (S (plus i n)) O (CHead c2 (Bind b) t) -e2))))))).(\lambda (H0: (getl (S n) (CHead c2 (Bind b) t) c3)).(\lambda (e2: -C).(\lambda (i: nat).(\lambda (H1: (drop (S i) O c3 e2)).(eq_ind nat (plus (S -i) n) (\lambda (n0: nat).(drop (S n0) O (CHead c2 (Bind b) t) e2)) (drop_drop -(Bind b) (plus (S i) n) c2 e2 (IHc c3 n (getl_gen_S (Bind b) c2 c3 t n H0) e2 -i H1) t) (plus i (S n)) (plus_Snm_nSm i n)))))))) h))))) (\lambda (f: -F).(\lambda (t: T).(\lambda (c3: C).(\lambda (h: nat).(nat_ind (\lambda (n: -nat).((getl n (CHead c2 (Flat f) t) c3) \to (\forall (e2: C).(\forall (i: -nat).((drop (S i) O c3 e2) \to (drop (S (plus i n)) O (CHead c2 (Flat f) t) -e2)))))) (\lambda (H: (getl O (CHead c2 (Flat f) t) c3)).(\lambda (e2: -C).(\lambda (i: nat).(\lambda (H0: (drop (S i) O c3 e2)).(drop_drop (Flat f) -(plus i O) c2 e2 (IHc c3 O (getl_intro O c2 c3 c2 (drop_refl c2) -(clear_gen_flat f c2 c3 t (getl_gen_O (CHead c2 (Flat f) t) c3 H))) e2 i H0) -t))))) (\lambda (n: nat).(\lambda (_: (((getl n (CHead c2 (Flat f) t) c3) \to -(\forall (e2: C).(\forall (i: nat).((drop (S i) O c3 e2) \to (drop (S (plus i -n)) O (CHead c2 (Flat f) t) e2))))))).(\lambda (H0: (getl (S n) (CHead c2 -(Flat f) t) c3)).(\lambda (e2: C).(\lambda (i: nat).(\lambda (H1: (drop (S i) -O c3 e2)).(drop_drop (Flat f) (plus i (S n)) c2 e2 (IHc c3 (S n) (getl_gen_S -(Flat f) c2 c3 t n H0) e2 i H1) t))))))) h))))) k)))) c1). -(* COMMENTS -Initial nodes: 953 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/getl/flt.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/getl/flt.ma deleted file mode 100644 index 3d6c90f73..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/getl/flt.ma +++ /dev/null @@ -1,67 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/getl/fwd.ma". - -include "Basic-1/clear/props.ma". - -include "Basic-1/flt/props.ma". - -theorem getl_flt: - \forall (b: B).(\forall (c: C).(\forall (e: C).(\forall (u: T).(\forall (i: -nat).((getl i c (CHead e (Bind b) u)) \to (flt e u c (TLRef i))))))) -\def - \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e: -C).(\forall (u: T).(\forall (i: nat).((getl i c0 (CHead e (Bind b) u)) \to -(flt e u c0 (TLRef i))))))) (\lambda (n: nat).(\lambda (e: C).(\lambda (u: -T).(\lambda (i: nat).(\lambda (H: (getl i (CSort n) (CHead e (Bind b) -u))).(getl_gen_sort n i (CHead e (Bind b) u) H (flt e u (CSort n) (TLRef -i)))))))) (\lambda (c0: C).(\lambda (H: ((\forall (e: C).(\forall (u: -T).(\forall (i: nat).((getl i c0 (CHead e (Bind b) u)) \to (flt e u c0 (TLRef -i)))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e: C).(\lambda (u: -T).(\lambda (i: nat).(nat_ind (\lambda (n: nat).((getl n (CHead c0 k t) -(CHead e (Bind b) u)) \to (flt e u (CHead c0 k t) (TLRef n)))) (\lambda (H0: -(getl O (CHead c0 k t) (CHead e (Bind b) u))).(K_ind (\lambda (k0: K).((clear -(CHead c0 k0 t) (CHead e (Bind b) u)) \to (flt e u (CHead c0 k0 t) (TLRef -O)))) (\lambda (b0: B).(\lambda (H1: (clear (CHead c0 (Bind b0) t) (CHead e -(Bind b) u))).(let H2 \def (f_equal C C (\lambda (e0: C).(match e0 in C -return (\lambda (_: C).C) with [(CSort _) \Rightarrow e | (CHead c1 _ _) -\Rightarrow c1])) (CHead e (Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind -b0 c0 (CHead e (Bind b) u) t H1)) in ((let H3 \def (f_equal C B (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow b | -(CHead _ k0 _) \Rightarrow (match k0 in K return (\lambda (_: K).B) with -[(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow b])])) (CHead e (Bind b) u) -(CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e (Bind b) u) t H1)) in -((let H4 \def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda -(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) -(CHead e (Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e -(Bind b) u) t H1)) in (\lambda (H5: (eq B b b0)).(\lambda (H6: (eq C e -c0)).(eq_ind_r T t (\lambda (t0: T).(flt e t0 (CHead c0 (Bind b0) t) (TLRef -O))) (eq_ind_r C c0 (\lambda (c1: C).(flt c1 t (CHead c0 (Bind b0) t) (TLRef -O))) (eq_ind B b (\lambda (b1: B).(flt c0 t (CHead c0 (Bind b1) t) (TLRef -O))) (flt_arith0 (Bind b) c0 t O) b0 H5) e H6) u H4)))) H3)) H2)))) (\lambda -(f: F).(\lambda (H1: (clear (CHead c0 (Flat f) t) (CHead e (Bind b) -u))).(flt_arith1 (Bind b) e c0 u (clear_cle c0 (CHead e (Bind b) u) -(clear_gen_flat f c0 (CHead e (Bind b) u) t H1)) (Flat f) t O))) k -(getl_gen_O (CHead c0 k t) (CHead e (Bind b) u) H0))) (\lambda (n: -nat).(\lambda (_: (((getl n (CHead c0 k t) (CHead e (Bind b) u)) \to (flt e u -(CHead c0 k t) (TLRef n))))).(\lambda (H1: (getl (S n) (CHead c0 k t) (CHead -e (Bind b) u))).(let H_y \def (H e u (r k n) (getl_gen_S k c0 (CHead e (Bind -b) u) t n H1)) in (flt_arith2 e c0 u (r k n) H_y k t (S n)))))) i)))))))) c)). -(* COMMENTS -Initial nodes: 815 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/getl/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/getl/fwd.ma deleted file mode 100644 index b378f61fc..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/getl/fwd.ma +++ /dev/null @@ -1,148 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/getl/defs.ma". - -include "Basic-1/drop/fwd.ma". - -include "Basic-1/clear/fwd.ma". - -theorem getl_gen_all: - \forall (c1: C).(\forall (c2: C).(\forall (i: nat).((getl i c1 c2) \to (ex2 -C (\lambda (e: C).(drop i O c1 e)) (\lambda (e: C).(clear e c2)))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (i: nat).(\lambda (H: (getl i c1 -c2)).(getl_ind i c1 c2 (ex2 C (\lambda (e: C).(drop i O c1 e)) (\lambda (e: -C).(clear e c2))) (\lambda (e: C).(\lambda (H0: (drop i O c1 e)).(\lambda -(H1: (clear e c2)).(ex_intro2 C (\lambda (e0: C).(drop i O c1 e0)) (\lambda -(e0: C).(clear e0 c2)) e H0 H1)))) H)))). -(* COMMENTS -Initial nodes: 95 -END *) - -theorem getl_gen_sort: - \forall (n: nat).(\forall (h: nat).(\forall (x: C).((getl h (CSort n) x) \to -(\forall (P: Prop).P)))) -\def - \lambda (n: nat).(\lambda (h: nat).(\lambda (x: C).(\lambda (H: (getl h -(CSort n) x)).(\lambda (P: Prop).(let H0 \def (getl_gen_all (CSort n) x h H) -in (ex2_ind C (\lambda (e: C).(drop h O (CSort n) e)) (\lambda (e: C).(clear -e x)) P (\lambda (x0: C).(\lambda (H1: (drop h O (CSort n) x0)).(\lambda (H2: -(clear x0 x)).(and3_ind (eq C x0 (CSort n)) (eq nat h O) (eq nat O O) P -(\lambda (H3: (eq C x0 (CSort n))).(\lambda (_: (eq nat h O)).(\lambda (_: -(eq nat O O)).(let H6 \def (eq_ind C x0 (\lambda (c: C).(clear c x)) H2 -(CSort n) H3) in (clear_gen_sort x n H6 P))))) (drop_gen_sort n h O x0 -H1))))) H0)))))). -(* COMMENTS -Initial nodes: 179 -END *) - -theorem getl_gen_O: - \forall (e: C).(\forall (x: C).((getl O e x) \to (clear e x))) -\def - \lambda (e: C).(\lambda (x: C).(\lambda (H: (getl O e x)).(let H0 \def -(getl_gen_all e x O H) in (ex2_ind C (\lambda (e0: C).(drop O O e e0)) -(\lambda (e0: C).(clear e0 x)) (clear e x) (\lambda (x0: C).(\lambda (H1: -(drop O O e x0)).(\lambda (H2: (clear x0 x)).(let H3 \def (eq_ind_r C x0 -(\lambda (c: C).(clear c x)) H2 e (drop_gen_refl e x0 H1)) in H3)))) H0)))). -(* COMMENTS -Initial nodes: 99 -END *) - -theorem getl_gen_S: - \forall (k: K).(\forall (c: C).(\forall (x: C).(\forall (u: T).(\forall (h: -nat).((getl (S h) (CHead c k u) x) \to (getl (r k h) c x)))))) -\def - \lambda (k: K).(\lambda (c: C).(\lambda (x: C).(\lambda (u: T).(\lambda (h: -nat).(\lambda (H: (getl (S h) (CHead c k u) x)).(let H0 \def (getl_gen_all -(CHead c k u) x (S h) H) in (ex2_ind C (\lambda (e: C).(drop (S h) O (CHead c -k u) e)) (\lambda (e: C).(clear e x)) (getl (r k h) c x) (\lambda (x0: -C).(\lambda (H1: (drop (S h) O (CHead c k u) x0)).(\lambda (H2: (clear x0 -x)).(getl_intro (r k h) c x x0 (drop_gen_drop k c x0 u h H1) H2)))) H0))))))). -(* COMMENTS -Initial nodes: 145 -END *) - -theorem getl_gen_2: - \forall (c1: C).(\forall (c2: C).(\forall (i: nat).((getl i c1 c2) \to (ex_3 -B C T (\lambda (b: B).(\lambda (c: C).(\lambda (v: T).(eq C c2 (CHead c (Bind -b) v))))))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (i: nat).(\lambda (H: (getl i c1 -c2)).(let H0 \def (getl_gen_all c1 c2 i H) in (ex2_ind C (\lambda (e: -C).(drop i O c1 e)) (\lambda (e: C).(clear e c2)) (ex_3 B C T (\lambda (b: -B).(\lambda (c: C).(\lambda (v: T).(eq C c2 (CHead c (Bind b) v)))))) -(\lambda (x: C).(\lambda (_: (drop i O c1 x)).(\lambda (H2: (clear x -c2)).(let H3 \def (clear_gen_all x c2 H2) in (ex_3_ind B C T (\lambda (b: -B).(\lambda (e: C).(\lambda (u: T).(eq C c2 (CHead e (Bind b) u))))) (ex_3 B -C T (\lambda (b: B).(\lambda (c: C).(\lambda (v: T).(eq C c2 (CHead c (Bind -b) v)))))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H4: -(eq C c2 (CHead x1 (Bind x0) x2))).(let H5 \def (eq_ind C c2 (\lambda (c: -C).(clear x c)) H2 (CHead x1 (Bind x0) x2) H4) in (eq_ind_r C (CHead x1 (Bind -x0) x2) (\lambda (c: C).(ex_3 B C T (\lambda (b: B).(\lambda (c0: C).(\lambda -(v: T).(eq C c (CHead c0 (Bind b) v))))))) (ex_3_intro B C T (\lambda (b: -B).(\lambda (c: C).(\lambda (v: T).(eq C (CHead x1 (Bind x0) x2) (CHead c -(Bind b) v))))) x0 x1 x2 (refl_equal C (CHead x1 (Bind x0) x2))) c2 H4)))))) -H3))))) H0))))). -(* COMMENTS -Initial nodes: 325 -END *) - -theorem getl_gen_flat: - \forall (f: F).(\forall (e: C).(\forall (d: C).(\forall (v: T).(\forall (i: -nat).((getl i (CHead e (Flat f) v) d) \to (getl i e d)))))) -\def - \lambda (f: F).(\lambda (e: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: -nat).(nat_ind (\lambda (n: nat).((getl n (CHead e (Flat f) v) d) \to (getl n -e d))) (\lambda (H: (getl O (CHead e (Flat f) v) d)).(getl_intro O e d e -(drop_refl e) (clear_gen_flat f e d v (getl_gen_O (CHead e (Flat f) v) d -H)))) (\lambda (n: nat).(\lambda (_: (((getl n (CHead e (Flat f) v) d) \to -(getl n e d)))).(\lambda (H0: (getl (S n) (CHead e (Flat f) v) -d)).(getl_gen_S (Flat f) e d v n H0)))) i))))). -(* COMMENTS -Initial nodes: 155 -END *) - -theorem getl_gen_bind: - \forall (b: B).(\forall (e: C).(\forall (d: C).(\forall (v: T).(\forall (i: -nat).((getl i (CHead e (Bind b) v) d) \to (or (land (eq nat i O) (eq C d -(CHead e (Bind b) v))) (ex2 nat (\lambda (j: nat).(eq nat i (S j))) (\lambda -(j: nat).(getl j e d))))))))) -\def - \lambda (b: B).(\lambda (e: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: -nat).(nat_ind (\lambda (n: nat).((getl n (CHead e (Bind b) v) d) \to (or -(land (eq nat n O) (eq C d (CHead e (Bind b) v))) (ex2 nat (\lambda (j: -nat).(eq nat n (S j))) (\lambda (j: nat).(getl j e d)))))) (\lambda (H: (getl -O (CHead e (Bind b) v) d)).(eq_ind_r C (CHead e (Bind b) v) (\lambda (c: -C).(or (land (eq nat O O) (eq C c (CHead e (Bind b) v))) (ex2 nat (\lambda -(j: nat).(eq nat O (S j))) (\lambda (j: nat).(getl j e c))))) (or_introl -(land (eq nat O O) (eq C (CHead e (Bind b) v) (CHead e (Bind b) v))) (ex2 nat -(\lambda (j: nat).(eq nat O (S j))) (\lambda (j: nat).(getl j e (CHead e -(Bind b) v)))) (conj (eq nat O O) (eq C (CHead e (Bind b) v) (CHead e (Bind -b) v)) (refl_equal nat O) (refl_equal C (CHead e (Bind b) v)))) d -(clear_gen_bind b e d v (getl_gen_O (CHead e (Bind b) v) d H)))) (\lambda (n: -nat).(\lambda (_: (((getl n (CHead e (Bind b) v) d) \to (or (land (eq nat n -O) (eq C d (CHead e (Bind b) v))) (ex2 nat (\lambda (j: nat).(eq nat n (S -j))) (\lambda (j: nat).(getl j e d))))))).(\lambda (H0: (getl (S n) (CHead e -(Bind b) v) d)).(or_intror (land (eq nat (S n) O) (eq C d (CHead e (Bind b) -v))) (ex2 nat (\lambda (j: nat).(eq nat (S n) (S j))) (\lambda (j: nat).(getl -j e d))) (ex_intro2 nat (\lambda (j: nat).(eq nat (S n) (S j))) (\lambda (j: -nat).(getl j e d)) n (refl_equal nat (S n)) (getl_gen_S (Bind b) e d v n -H0)))))) i))))). -(* COMMENTS -Initial nodes: 525 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/getl/getl.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/getl/getl.ma deleted file mode 100644 index 71327becf..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/getl/getl.ma +++ /dev/null @@ -1,57 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/getl/drop.ma". - -include "Basic-1/getl/clear.ma". - -theorem getl_conf_le: - \forall (i: nat).(\forall (a: C).(\forall (c: C).((getl i c a) \to (\forall -(e: C).(\forall (h: nat).((getl h c e) \to ((le h i) \to (getl (minus i h) e -a)))))))) -\def - \lambda (i: nat).(\lambda (a: C).(\lambda (c: C).(\lambda (H: (getl i c -a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (H0: (getl h c e)).(\lambda -(H1: (le h i)).(let H2 \def (getl_gen_all c e h H0) in (ex2_ind C (\lambda -(e0: C).(drop h O c e0)) (\lambda (e0: C).(clear e0 e)) (getl (minus i h) e -a) (\lambda (x: C).(\lambda (H3: (drop h O c x)).(\lambda (H4: (clear x -e)).(getl_clear_conf (minus i h) x a (getl_drop_conf_ge i a c H x h O H3 H1) -e H4)))) H2))))))))). -(* COMMENTS -Initial nodes: 133 -END *) - -theorem getl_trans: - \forall (i: nat).(\forall (c1: C).(\forall (c2: C).(\forall (h: nat).((getl -h c1 c2) \to (\forall (e2: C).((getl i c2 e2) \to (getl (plus i h) c1 -e2))))))) -\def - \lambda (i: nat).(\lambda (c1: C).(\lambda (c2: C).(\lambda (h: -nat).(\lambda (H: (getl h c1 c2)).(\lambda (e2: C).(\lambda (H0: (getl i c2 -e2)).(let H1 \def (getl_gen_all c2 e2 i H0) in (ex2_ind C (\lambda (e: -C).(drop i O c2 e)) (\lambda (e: C).(clear e e2)) (getl (plus i h) c1 e2) -(\lambda (x: C).(\lambda (H2: (drop i O c2 x)).(\lambda (H3: (clear x -e2)).(nat_ind (\lambda (n: nat).((drop n O c2 x) \to (getl (plus n h) c1 -e2))) (\lambda (H4: (drop O O c2 x)).(let H5 \def (eq_ind_r C x (\lambda (c: -C).(clear c e2)) H3 c2 (drop_gen_refl c2 x H4)) in (getl_clear_trans (plus O -h) c1 c2 H e2 H5))) (\lambda (i0: nat).(\lambda (_: (((drop i0 O c2 x) \to -(getl (plus i0 h) c1 e2)))).(\lambda (H4: (drop (S i0) O c2 x)).(let H_y \def -(getl_drop_trans c1 c2 h H x i0 H4) in (getl_intro (plus (S i0) h) c1 e2 x -H_y H3))))) i H2)))) H1)))))))). -(* COMMENTS -Initial nodes: 247 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/getl/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/getl/props.ma deleted file mode 100644 index 9399ad4ab..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/getl/props.ma +++ /dev/null @@ -1,104 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/getl/fwd.ma". - -include "Basic-1/drop/props.ma". - -include "Basic-1/clear/props.ma". - -theorem getl_refl: - \forall (b: B).(\forall (c: C).(\forall (u: T).(getl O (CHead c (Bind b) u) -(CHead c (Bind b) u)))) -\def - \lambda (b: B).(\lambda (c: C).(\lambda (u: T).(getl_intro O (CHead c (Bind -b) u) (CHead c (Bind b) u) (CHead c (Bind b) u) (drop_refl (CHead c (Bind b) -u)) (clear_bind b c u)))). -(* COMMENTS -Initial nodes: 59 -END *) - -theorem getl_head: - \forall (k: K).(\forall (h: nat).(\forall (c: C).(\forall (e: C).((getl (r k -h) c e) \to (\forall (u: T).(getl (S h) (CHead c k u) e)))))) -\def - \lambda (k: K).(\lambda (h: nat).(\lambda (c: C).(\lambda (e: C).(\lambda -(H: (getl (r k h) c e)).(\lambda (u: T).(let H0 \def (getl_gen_all c e (r k -h) H) in (ex2_ind C (\lambda (e0: C).(drop (r k h) O c e0)) (\lambda (e0: -C).(clear e0 e)) (getl (S h) (CHead c k u) e) (\lambda (x: C).(\lambda (H1: -(drop (r k h) O c x)).(\lambda (H2: (clear x e)).(getl_intro (S h) (CHead c k -u) e x (drop_drop k h c x H1 u) H2)))) H0))))))). -(* COMMENTS -Initial nodes: 137 -END *) - -theorem getl_flat: - \forall (c: C).(\forall (e: C).(\forall (h: nat).((getl h c e) \to (\forall -(f: F).(\forall (u: T).(getl h (CHead c (Flat f) u) e)))))) -\def - \lambda (c: C).(\lambda (e: C).(\lambda (h: nat).(\lambda (H: (getl h c -e)).(\lambda (f: F).(\lambda (u: T).(let H0 \def (getl_gen_all c e h H) in -(ex2_ind C (\lambda (e0: C).(drop h O c e0)) (\lambda (e0: C).(clear e0 e)) -(getl h (CHead c (Flat f) u) e) (\lambda (x: C).(\lambda (H1: (drop h O c -x)).(\lambda (H2: (clear x e)).(nat_ind (\lambda (n: nat).((drop n O c x) \to -(getl n (CHead c (Flat f) u) e))) (\lambda (H3: (drop O O c x)).(let H4 \def -(eq_ind_r C x (\lambda (c0: C).(clear c0 e)) H2 c (drop_gen_refl c x H3)) in -(getl_intro O (CHead c (Flat f) u) e (CHead c (Flat f) u) (drop_refl (CHead c -(Flat f) u)) (clear_flat c e H4 f u)))) (\lambda (h0: nat).(\lambda (_: -(((drop h0 O c x) \to (getl h0 (CHead c (Flat f) u) e)))).(\lambda (H3: (drop -(S h0) O c x)).(getl_intro (S h0) (CHead c (Flat f) u) e x (drop_drop (Flat -f) h0 c x H3 u) H2)))) h H1)))) H0))))))). -(* COMMENTS -Initial nodes: 285 -END *) - -theorem getl_ctail: - \forall (b: B).(\forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: -nat).((getl i c (CHead d (Bind b) u)) \to (\forall (k: K).(\forall (v: -T).(getl i (CTail k v c) (CHead (CTail k v d) (Bind b) u))))))))) -\def - \lambda (b: B).(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: -nat).(\lambda (H: (getl i c (CHead d (Bind b) u))).(\lambda (k: K).(\lambda -(v: T).(let H0 \def (getl_gen_all c (CHead d (Bind b) u) i H) in (ex2_ind C -(\lambda (e: C).(drop i O c e)) (\lambda (e: C).(clear e (CHead d (Bind b) -u))) (getl i (CTail k v c) (CHead (CTail k v d) (Bind b) u)) (\lambda (x: -C).(\lambda (H1: (drop i O c x)).(\lambda (H2: (clear x (CHead d (Bind b) -u))).(getl_intro i (CTail k v c) (CHead (CTail k v d) (Bind b) u) (CTail k v -x) (drop_ctail c x O i H1 k v) (clear_ctail b x d u H2 k v))))) H0))))))))). -(* COMMENTS -Initial nodes: 203 -END *) - -theorem getl_mono: - \forall (c: C).(\forall (x1: C).(\forall (h: nat).((getl h c x1) \to -(\forall (x2: C).((getl h c x2) \to (eq C x1 x2)))))) -\def - \lambda (c: C).(\lambda (x1: C).(\lambda (h: nat).(\lambda (H: (getl h c -x1)).(\lambda (x2: C).(\lambda (H0: (getl h c x2)).(let H1 \def (getl_gen_all -c x2 h H0) in (ex2_ind C (\lambda (e: C).(drop h O c e)) (\lambda (e: -C).(clear e x2)) (eq C x1 x2) (\lambda (x: C).(\lambda (H2: (drop h O c -x)).(\lambda (H3: (clear x x2)).(let H4 \def (getl_gen_all c x1 h H) in -(ex2_ind C (\lambda (e: C).(drop h O c e)) (\lambda (e: C).(clear e x1)) (eq -C x1 x2) (\lambda (x0: C).(\lambda (H5: (drop h O c x0)).(\lambda (H6: (clear -x0 x1)).(let H7 \def (eq_ind C x (\lambda (c0: C).(drop h O c c0)) H2 x0 -(drop_mono c x O h H2 x0 H5)) in (let H8 \def (eq_ind_r C x0 (\lambda (c0: -C).(drop h O c c0)) H7 x (drop_mono c x O h H2 x0 H5)) in (let H9 \def -(eq_ind_r C x0 (\lambda (c0: C).(clear c0 x1)) H6 x (drop_mono c x O h H2 x0 -H5)) in (clear_mono x x1 H9 x2 H3))))))) H4))))) H1))))))). -(* COMMENTS -Initial nodes: 269 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/iso/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/iso/defs.ma deleted file mode 100644 index 202b8a300..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/iso/defs.ma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/T/defs.ma". - -inductive iso: T \to (T \to Prop) \def -| iso_sort: \forall (n1: nat).(\forall (n2: nat).(iso (TSort n1) (TSort n2))) -| iso_lref: \forall (i1: nat).(\forall (i2: nat).(iso (TLRef i1) (TLRef i2))) -| iso_head: \forall (v1: T).(\forall (v2: T).(\forall (t1: T).(\forall (t2: -T).(\forall (k: K).(iso (THead k v1 t1) (THead k v2 t2)))))). - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/iso/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/iso/fwd.ma deleted file mode 100644 index 761e982f7..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/iso/fwd.ma +++ /dev/null @@ -1,191 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/iso/defs.ma". - -include "Basic-1/tlist/defs.ma". - -theorem iso_gen_sort: - \forall (u2: T).(\forall (n1: nat).((iso (TSort n1) u2) \to (ex nat (\lambda -(n2: nat).(eq T u2 (TSort n2)))))) -\def - \lambda (u2: T).(\lambda (n1: nat).(\lambda (H: (iso (TSort n1) -u2)).(insert_eq T (TSort n1) (\lambda (t: T).(iso t u2)) (\lambda (_: T).(ex -nat (\lambda (n2: nat).(eq T u2 (TSort n2))))) (\lambda (y: T).(\lambda (H0: -(iso y u2)).(iso_ind (\lambda (t: T).(\lambda (t0: T).((eq T t (TSort n1)) -\to (ex nat (\lambda (n2: nat).(eq T t0 (TSort n2))))))) (\lambda (n0: -nat).(\lambda (n2: nat).(\lambda (H1: (eq T (TSort n0) (TSort n1))).(let H2 -\def (f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) -with [(TSort n) \Rightarrow n | (TLRef _) \Rightarrow n0 | (THead _ _ _) -\Rightarrow n0])) (TSort n0) (TSort n1) H1) in (ex_intro nat (\lambda (n3: -nat).(eq T (TSort n2) (TSort n3))) n2 (refl_equal T (TSort n2))))))) (\lambda -(i1: nat).(\lambda (i2: nat).(\lambda (H1: (eq T (TLRef i1) (TSort n1))).(let -H2 \def (eq_ind T (TLRef i1) (\lambda (ee: T).(match ee in T return (\lambda -(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | -(THead _ _ _) \Rightarrow False])) I (TSort n1) H1) in (False_ind (ex nat -(\lambda (n2: nat).(eq T (TLRef i2) (TSort n2)))) H2))))) (\lambda (v1: -T).(\lambda (v2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k: -K).(\lambda (H1: (eq T (THead k v1 t1) (TSort n1))).(let H2 \def (eq_ind T -(THead k v1 t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) -with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ -_) \Rightarrow True])) I (TSort n1) H1) in (False_ind (ex nat (\lambda (n2: -nat).(eq T (THead k v2 t2) (TSort n2)))) H2)))))))) y u2 H0))) H))). -(* COMMENTS -Initial nodes: 321 -END *) - -theorem iso_gen_lref: - \forall (u2: T).(\forall (n1: nat).((iso (TLRef n1) u2) \to (ex nat (\lambda -(n2: nat).(eq T u2 (TLRef n2)))))) -\def - \lambda (u2: T).(\lambda (n1: nat).(\lambda (H: (iso (TLRef n1) -u2)).(insert_eq T (TLRef n1) (\lambda (t: T).(iso t u2)) (\lambda (_: T).(ex -nat (\lambda (n2: nat).(eq T u2 (TLRef n2))))) (\lambda (y: T).(\lambda (H0: -(iso y u2)).(iso_ind (\lambda (t: T).(\lambda (t0: T).((eq T t (TLRef n1)) -\to (ex nat (\lambda (n2: nat).(eq T t0 (TLRef n2))))))) (\lambda (n0: -nat).(\lambda (n2: nat).(\lambda (H1: (eq T (TSort n0) (TLRef n1))).(let H2 -\def (eq_ind T (TSort n0) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | -(THead _ _ _) \Rightarrow False])) I (TLRef n1) H1) in (False_ind (ex nat -(\lambda (n3: nat).(eq T (TSort n2) (TLRef n3)))) H2))))) (\lambda (i1: -nat).(\lambda (i2: nat).(\lambda (H1: (eq T (TLRef i1) (TLRef n1))).(let H2 -\def (f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) -with [(TSort _) \Rightarrow i1 | (TLRef n) \Rightarrow n | (THead _ _ _) -\Rightarrow i1])) (TLRef i1) (TLRef n1) H1) in (ex_intro nat (\lambda (n2: -nat).(eq T (TLRef i2) (TLRef n2))) i2 (refl_equal T (TLRef i2))))))) (\lambda -(v1: T).(\lambda (v2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k: -K).(\lambda (H1: (eq T (THead k v1 t1) (TLRef n1))).(let H2 \def (eq_ind T -(THead k v1 t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) -with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ -_) \Rightarrow True])) I (TLRef n1) H1) in (False_ind (ex nat (\lambda (n2: -nat).(eq T (THead k v2 t2) (TLRef n2)))) H2)))))))) y u2 H0))) H))). -(* COMMENTS -Initial nodes: 321 -END *) - -theorem iso_gen_head: - \forall (k: K).(\forall (v1: T).(\forall (t1: T).(\forall (u2: T).((iso -(THead k v1 t1) u2) \to (ex_2 T T (\lambda (v2: T).(\lambda (t2: T).(eq T u2 -(THead k v2 t2))))))))) -\def - \lambda (k: K).(\lambda (v1: T).(\lambda (t1: T).(\lambda (u2: T).(\lambda -(H: (iso (THead k v1 t1) u2)).(insert_eq T (THead k v1 t1) (\lambda (t: -T).(iso t u2)) (\lambda (_: T).(ex_2 T T (\lambda (v2: T).(\lambda (t2: -T).(eq T u2 (THead k v2 t2)))))) (\lambda (y: T).(\lambda (H0: (iso y -u2)).(iso_ind (\lambda (t: T).(\lambda (t0: T).((eq T t (THead k v1 t1)) \to -(ex_2 T T (\lambda (v2: T).(\lambda (t2: T).(eq T t0 (THead k v2 t2)))))))) -(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (H1: (eq T (TSort n1) (THead k -v1 t1))).(let H2 \def (eq_ind T (TSort n1) (\lambda (ee: T).(match ee in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead k v1 t1) H1) -in (False_ind (ex_2 T T (\lambda (v2: T).(\lambda (t2: T).(eq T (TSort n2) -(THead k v2 t2))))) H2))))) (\lambda (i1: nat).(\lambda (i2: nat).(\lambda -(H1: (eq T (TLRef i1) (THead k v1 t1))).(let H2 \def (eq_ind T (TLRef i1) -(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow -False])) I (THead k v1 t1) H1) in (False_ind (ex_2 T T (\lambda (v2: -T).(\lambda (t2: T).(eq T (TLRef i2) (THead k v2 t2))))) H2))))) (\lambda -(v0: T).(\lambda (v2: T).(\lambda (t0: T).(\lambda (t2: T).(\lambda (k0: -K).(\lambda (H1: (eq T (THead k0 v0 t0) (THead k v1 t1))).(let H2 \def -(f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with -[(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _ _) -\Rightarrow k1])) (THead k0 v0 t0) (THead k v1 t1) H1) in ((let H3 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow v0 | (TLRef _) \Rightarrow v0 | (THead _ t _) -\Rightarrow t])) (THead k0 v0 t0) (THead k v1 t1) H1) in ((let H4 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) -\Rightarrow t])) (THead k0 v0 t0) (THead k v1 t1) H1) in (\lambda (_: (eq T -v0 v1)).(\lambda (H6: (eq K k0 k)).(eq_ind_r K k (\lambda (k1: K).(ex_2 T T -(\lambda (v3: T).(\lambda (t3: T).(eq T (THead k1 v2 t2) (THead k v3 t3)))))) -(ex_2_intro T T (\lambda (v3: T).(\lambda (t3: T).(eq T (THead k v2 t2) -(THead k v3 t3)))) v2 t2 (refl_equal T (THead k v2 t2))) k0 H6)))) H3)) -H2)))))))) y u2 H0))) H))))). -(* COMMENTS -Initial nodes: 545 -END *) - -theorem iso_flats_lref_bind_false: - \forall (f: F).(\forall (b: B).(\forall (i: nat).(\forall (v: T).(\forall -(t: T).(\forall (vs: TList).((iso (THeads (Flat f) vs (TLRef i)) (THead (Bind -b) v t)) \to (\forall (P: Prop).P))))))) -\def - \lambda (f: F).(\lambda (b: B).(\lambda (i: nat).(\lambda (v: T).(\lambda -(t: T).(\lambda (vs: TList).(TList_ind (\lambda (t0: TList).((iso (THeads -(Flat f) t0 (TLRef i)) (THead (Bind b) v t)) \to (\forall (P: Prop).P))) -(\lambda (H: (iso (TLRef i) (THead (Bind b) v t))).(\lambda (P: Prop).(let -H_x \def (iso_gen_lref (THead (Bind b) v t) i H) in (let H0 \def H_x in -(ex_ind nat (\lambda (n2: nat).(eq T (THead (Bind b) v t) (TLRef n2))) P -(\lambda (x: nat).(\lambda (H1: (eq T (THead (Bind b) v t) (TLRef x))).(let -H2 \def (eq_ind T (THead (Bind b) v t) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef x) H1) in -(False_ind P H2)))) H0))))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda -(_: (((iso (THeads (Flat f) t1 (TLRef i)) (THead (Bind b) v t)) \to (\forall -(P: Prop).P)))).(\lambda (H0: (iso (THead (Flat f) t0 (THeads (Flat f) t1 -(TLRef i))) (THead (Bind b) v t))).(\lambda (P: Prop).(let H_x \def -(iso_gen_head (Flat f) t0 (THeads (Flat f) t1 (TLRef i)) (THead (Bind b) v t) -H0) in (let H1 \def H_x in (ex_2_ind T T (\lambda (v2: T).(\lambda (t2: -T).(eq T (THead (Bind b) v t) (THead (Flat f) v2 t2)))) P (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H2: (eq T (THead (Bind b) v t) (THead (Flat f) -x0 x1))).(let H3 \def (eq_ind T (THead (Bind b) v t) (\lambda (ee: T).(match -ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | -(TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return -(\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow -False])])) I (THead (Flat f) x0 x1) H2) in (False_ind P H3))))) H1)))))))) -vs)))))). -(* COMMENTS -Initial nodes: 391 -END *) - -theorem iso_flats_flat_bind_false: - \forall (f1: F).(\forall (f2: F).(\forall (b: B).(\forall (v: T).(\forall -(v2: T).(\forall (t: T).(\forall (t2: T).(\forall (vs: TList).((iso (THeads -(Flat f1) vs (THead (Flat f2) v2 t2)) (THead (Bind b) v t)) \to (\forall (P: -Prop).P))))))))) -\def - \lambda (f1: F).(\lambda (f2: F).(\lambda (b: B).(\lambda (v: T).(\lambda -(v2: T).(\lambda (t: T).(\lambda (t2: T).(\lambda (vs: TList).(TList_ind -(\lambda (t0: TList).((iso (THeads (Flat f1) t0 (THead (Flat f2) v2 t2)) -(THead (Bind b) v t)) \to (\forall (P: Prop).P))) (\lambda (H: (iso (THead -(Flat f2) v2 t2) (THead (Bind b) v t))).(\lambda (P: Prop).(let H_x \def -(iso_gen_head (Flat f2) v2 t2 (THead (Bind b) v t) H) in (let H0 \def H_x in -(ex_2_ind T T (\lambda (v3: T).(\lambda (t3: T).(eq T (THead (Bind b) v t) -(THead (Flat f2) v3 t3)))) P (\lambda (x0: T).(\lambda (x1: T).(\lambda (H1: -(eq T (THead (Bind b) v t) (THead (Flat f2) x0 x1))).(let H2 \def (eq_ind T -(THead (Bind b) v t) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat -f2) x0 x1) H1) in (False_ind P H2))))) H0))))) (\lambda (t0: T).(\lambda (t1: -TList).(\lambda (_: (((iso (THeads (Flat f1) t1 (THead (Flat f2) v2 t2)) -(THead (Bind b) v t)) \to (\forall (P: Prop).P)))).(\lambda (H0: (iso (THead -(Flat f1) t0 (THeads (Flat f1) t1 (THead (Flat f2) v2 t2))) (THead (Bind b) v -t))).(\lambda (P: Prop).(let H_x \def (iso_gen_head (Flat f1) t0 (THeads -(Flat f1) t1 (THead (Flat f2) v2 t2)) (THead (Bind b) v t) H0) in (let H1 -\def H_x in (ex_2_ind T T (\lambda (v3: T).(\lambda (t3: T).(eq T (THead -(Bind b) v t) (THead (Flat f1) v3 t3)))) P (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H2: (eq T (THead (Bind b) v t) (THead (Flat f1) x0 x1))).(let H3 -\def (eq_ind T (THead (Bind b) v t) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow -False])])) I (THead (Flat f1) x0 x1) H2) in (False_ind P H3))))) H1)))))))) -vs)))))))). -(* COMMENTS -Initial nodes: 461 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/iso/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/iso/props.ma deleted file mode 100644 index af521d073..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/iso/props.ma +++ /dev/null @@ -1,56 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/iso/fwd.ma". - -theorem iso_refl: - \forall (t: T).(iso t t) -\def - \lambda (t: T).(T_ind (\lambda (t0: T).(iso t0 t0)) (\lambda (n: -nat).(iso_sort n n)) (\lambda (n: nat).(iso_lref n n)) (\lambda (k: -K).(\lambda (t0: T).(\lambda (_: (iso t0 t0)).(\lambda (t1: T).(\lambda (_: -(iso t1 t1)).(iso_head t0 t0 t1 t1 k)))))) t). -(* COMMENTS -Initial nodes: 59 -END *) - -theorem iso_trans: - \forall (t1: T).(\forall (t2: T).((iso t1 t2) \to (\forall (t3: T).((iso t2 -t3) \to (iso t1 t3))))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (iso t1 t2)).(iso_ind (\lambda -(t: T).(\lambda (t0: T).(\forall (t3: T).((iso t0 t3) \to (iso t t3))))) -(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (t3: T).(\lambda (H0: (iso -(TSort n2) t3)).(let H_x \def (iso_gen_sort t3 n2 H0) in (let H1 \def H_x in -(ex_ind nat (\lambda (n3: nat).(eq T t3 (TSort n3))) (iso (TSort n1) t3) -(\lambda (x: nat).(\lambda (H2: (eq T t3 (TSort x))).(eq_ind_r T (TSort x) -(\lambda (t: T).(iso (TSort n1) t)) (iso_sort n1 x) t3 H2))) H1))))))) -(\lambda (i1: nat).(\lambda (i2: nat).(\lambda (t3: T).(\lambda (H0: (iso -(TLRef i2) t3)).(let H_x \def (iso_gen_lref t3 i2 H0) in (let H1 \def H_x in -(ex_ind nat (\lambda (n2: nat).(eq T t3 (TLRef n2))) (iso (TLRef i1) t3) -(\lambda (x: nat).(\lambda (H2: (eq T t3 (TLRef x))).(eq_ind_r T (TLRef x) -(\lambda (t: T).(iso (TLRef i1) t)) (iso_lref i1 x) t3 H2))) H1))))))) -(\lambda (v1: T).(\lambda (v2: T).(\lambda (t3: T).(\lambda (t4: T).(\lambda -(k: K).(\lambda (t5: T).(\lambda (H0: (iso (THead k v2 t4) t5)).(let H_x \def -(iso_gen_head k v2 t4 t5 H0) in (let H1 \def H_x in (ex_2_ind T T (\lambda -(v3: T).(\lambda (t6: T).(eq T t5 (THead k v3 t6)))) (iso (THead k v1 t3) t5) -(\lambda (x0: T).(\lambda (x1: T).(\lambda (H2: (eq T t5 (THead k x0 -x1))).(eq_ind_r T (THead k x0 x1) (\lambda (t: T).(iso (THead k v1 t3) t)) -(iso_head v1 x0 t3 x1 k) t5 H2)))) H1)))))))))) t1 t2 H))). -(* COMMENTS -Initial nodes: 351 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/leq/asucc.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/leq/asucc.ma deleted file mode 100644 index fd9e7c1d3..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/leq/asucc.ma +++ /dev/null @@ -1,479 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/leq/props.ma". - -theorem asucc_repl: - \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (leq g -(asucc g a1) (asucc g a2))))) -\def - \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1 -a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(leq g (asucc g a) (asucc g -a0)))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: nat).(\lambda (n2: -nat).(\lambda (k: nat).(\lambda (H0: (eq A (aplus g (ASort h1 n1) k) (aplus g -(ASort h2 n2) k))).(nat_ind (\lambda (n: nat).((eq A (aplus g (ASort n n1) k) -(aplus g (ASort h2 n2) k)) \to (leq g (match n with [O \Rightarrow (ASort O -(next g n1)) | (S h) \Rightarrow (ASort h n1)]) (match h2 with [O \Rightarrow -(ASort O (next g n2)) | (S h) \Rightarrow (ASort h n2)])))) (\lambda (H1: (eq -A (aplus g (ASort O n1) k) (aplus g (ASort h2 n2) k))).(nat_ind (\lambda (n: -nat).((eq A (aplus g (ASort O n1) k) (aplus g (ASort n n2) k)) \to (leq g -(ASort O (next g n1)) (match n with [O \Rightarrow (ASort O (next g n2)) | (S -h) \Rightarrow (ASort h n2)])))) (\lambda (H2: (eq A (aplus g (ASort O n1) k) -(aplus g (ASort O n2) k))).(leq_sort g O O (next g n1) (next g n2) k (eq_ind -A (aplus g (ASort O n1) (S k)) (\lambda (a: A).(eq A a (aplus g (ASort O -(next g n2)) k))) (eq_ind A (aplus g (ASort O n2) (S k)) (\lambda (a: A).(eq -A (aplus g (ASort O n1) (S k)) a)) (eq_ind_r A (aplus g (ASort O n2) k) -(\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g (ASort O n2) k)))) -(refl_equal A (asucc g (aplus g (ASort O n2) k))) (aplus g (ASort O n1) k) -H2) (aplus g (ASort O (next g n2)) k) (aplus_sort_O_S_simpl g n2 k)) (aplus g -(ASort O (next g n1)) k) (aplus_sort_O_S_simpl g n1 k)))) (\lambda (h3: -nat).(\lambda (_: (((eq A (aplus g (ASort O n1) k) (aplus g (ASort h3 n2) k)) -\to (leq g (ASort O (next g n1)) (match h3 with [O \Rightarrow (ASort O (next -g n2)) | (S h) \Rightarrow (ASort h n2)]))))).(\lambda (H2: (eq A (aplus g -(ASort O n1) k) (aplus g (ASort (S h3) n2) k))).(leq_sort g O h3 (next g n1) -n2 k (eq_ind A (aplus g (ASort O n1) (S k)) (\lambda (a: A).(eq A a (aplus g -(ASort h3 n2) k))) (eq_ind A (aplus g (ASort (S h3) n2) (S k)) (\lambda (a: -A).(eq A (aplus g (ASort O n1) (S k)) a)) (eq_ind_r A (aplus g (ASort (S h3) -n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g (ASort (S h3) n2) -k)))) (refl_equal A (asucc g (aplus g (ASort (S h3) n2) k))) (aplus g (ASort -O n1) k) H2) (aplus g (ASort h3 n2) k) (aplus_sort_S_S_simpl g n2 h3 k)) -(aplus g (ASort O (next g n1)) k) (aplus_sort_O_S_simpl g n1 k)))))) h2 H1)) -(\lambda (h3: nat).(\lambda (IHh1: (((eq A (aplus g (ASort h3 n1) k) (aplus g -(ASort h2 n2) k)) \to (leq g (match h3 with [O \Rightarrow (ASort O (next g -n1)) | (S h) \Rightarrow (ASort h n1)]) (match h2 with [O \Rightarrow (ASort -O (next g n2)) | (S h) \Rightarrow (ASort h n2)]))))).(\lambda (H1: (eq A -(aplus g (ASort (S h3) n1) k) (aplus g (ASort h2 n2) k))).(nat_ind (\lambda -(n: nat).((eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort n n2) k)) \to -((((eq A (aplus g (ASort h3 n1) k) (aplus g (ASort n n2) k)) \to (leq g -(match h3 with [O \Rightarrow (ASort O (next g n1)) | (S h) \Rightarrow -(ASort h n1)]) (match n with [O \Rightarrow (ASort O (next g n2)) | (S h) -\Rightarrow (ASort h n2)])))) \to (leq g (ASort h3 n1) (match n with [O -\Rightarrow (ASort O (next g n2)) | (S h) \Rightarrow (ASort h n2)]))))) -(\lambda (H2: (eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort O n2) -k))).(\lambda (_: (((eq A (aplus g (ASort h3 n1) k) (aplus g (ASort O n2) k)) -\to (leq g (match h3 with [O \Rightarrow (ASort O (next g n1)) | (S h) -\Rightarrow (ASort h n1)]) (ASort O (next g n2)))))).(leq_sort g h3 O n1 -(next g n2) k (eq_ind A (aplus g (ASort O n2) (S k)) (\lambda (a: A).(eq A -(aplus g (ASort h3 n1) k) a)) (eq_ind A (aplus g (ASort (S h3) n1) (S k)) -(\lambda (a: A).(eq A a (aplus g (ASort O n2) (S k)))) (eq_ind_r A (aplus g -(ASort O n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g (ASort O -n2) k)))) (refl_equal A (asucc g (aplus g (ASort O n2) k))) (aplus g (ASort -(S h3) n1) k) H2) (aplus g (ASort h3 n1) k) (aplus_sort_S_S_simpl g n1 h3 k)) -(aplus g (ASort O (next g n2)) k) (aplus_sort_O_S_simpl g n2 k))))) (\lambda -(h4: nat).(\lambda (_: (((eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort -h4 n2) k)) \to ((((eq A (aplus g (ASort h3 n1) k) (aplus g (ASort h4 n2) k)) -\to (leq g (match h3 with [O \Rightarrow (ASort O (next g n1)) | (S h) -\Rightarrow (ASort h n1)]) (match h4 with [O \Rightarrow (ASort O (next g -n2)) | (S h) \Rightarrow (ASort h n2)])))) \to (leq g (ASort h3 n1) (match h4 -with [O \Rightarrow (ASort O (next g n2)) | (S h) \Rightarrow (ASort h -n2)])))))).(\lambda (H2: (eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort -(S h4) n2) k))).(\lambda (_: (((eq A (aplus g (ASort h3 n1) k) (aplus g -(ASort (S h4) n2) k)) \to (leq g (match h3 with [O \Rightarrow (ASort O (next -g n1)) | (S h) \Rightarrow (ASort h n1)]) (ASort h4 n2))))).(leq_sort g h3 h4 -n1 n2 k (eq_ind A (aplus g (ASort (S h3) n1) (S k)) (\lambda (a: A).(eq A a -(aplus g (ASort h4 n2) k))) (eq_ind A (aplus g (ASort (S h4) n2) (S k)) -(\lambda (a: A).(eq A (aplus g (ASort (S h3) n1) (S k)) a)) (eq_ind_r A -(aplus g (ASort (S h4) n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g -(aplus g (ASort (S h4) n2) k)))) (refl_equal A (asucc g (aplus g (ASort (S -h4) n2) k))) (aplus g (ASort (S h3) n1) k) H2) (aplus g (ASort h4 n2) k) -(aplus_sort_S_S_simpl g n2 h4 k)) (aplus g (ASort h3 n1) k) -(aplus_sort_S_S_simpl g n1 h3 k))))))) h2 H1 IHh1)))) h1 H0))))))) (\lambda -(a3: A).(\lambda (a4: A).(\lambda (H0: (leq g a3 a4)).(\lambda (_: (leq g -(asucc g a3) (asucc g a4))).(\lambda (a5: A).(\lambda (a6: A).(\lambda (_: -(leq g a5 a6)).(\lambda (H3: (leq g (asucc g a5) (asucc g a6))).(leq_head g -a3 a4 H0 (asucc g a5) (asucc g a6) H3))))))))) a1 a2 H)))). -(* COMMENTS -Initial nodes: 1907 -END *) - -theorem asucc_inj: - \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (asucc g a1) (asucc -g a2)) \to (leq g a1 a2)))) -\def - \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2: -A).((leq g (asucc g a) (asucc g a2)) \to (leq g a a2)))) (\lambda (n: -nat).(\lambda (n0: nat).(\lambda (a2: A).(A_ind (\lambda (a: A).((leq g -(asucc g (ASort n n0)) (asucc g a)) \to (leq g (ASort n n0) a))) (\lambda -(n1: nat).(\lambda (n2: nat).(\lambda (H: (leq g (asucc g (ASort n n0)) -(asucc g (ASort n1 n2)))).(nat_ind (\lambda (n3: nat).((leq g (asucc g (ASort -n3 n0)) (asucc g (ASort n1 n2))) \to (leq g (ASort n3 n0) (ASort n1 n2)))) -(\lambda (H0: (leq g (asucc g (ASort O n0)) (asucc g (ASort n1 -n2)))).(nat_ind (\lambda (n3: nat).((leq g (asucc g (ASort O n0)) (asucc g -(ASort n3 n2))) \to (leq g (ASort O n0) (ASort n3 n2)))) (\lambda (H1: (leq g -(asucc g (ASort O n0)) (asucc g (ASort O n2)))).(let H_x \def (leq_gen_sort1 -g O (next g n0) (ASort O (next g n2)) H1) in (let H2 \def H_x in (ex2_3_ind -nat nat nat (\lambda (n3: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A -(aplus g (ASort O (next g n0)) k) (aplus g (ASort h2 n3) k))))) (\lambda (n3: -nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A (ASort O (next g n2)) (ASort -h2 n3))))) (leq g (ASort O n0) (ASort O n2)) (\lambda (x0: nat).(\lambda (x1: -nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort O (next g n0)) -x2) (aplus g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort O (next g n2)) -(ASort x1 x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e in A -return (\lambda (_: A).nat) with [(ASort n3 _) \Rightarrow n3 | (AHead _ _) -\Rightarrow O])) (ASort O (next g n2)) (ASort x1 x0) H4) in ((let H6 \def -(f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with -[(ASort _ n3) \Rightarrow n3 | (AHead _ _) \Rightarrow ((match g with [(mk_G -next _) \Rightarrow next]) n2)])) (ASort O (next g n2)) (ASort x1 x0) H4) in -(\lambda (H7: (eq nat O x1)).(let H8 \def (eq_ind_r nat x1 (\lambda (n3: -nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort n3 x0) x2))) H3 -O H7) in (let H9 \def (eq_ind_r nat x0 (\lambda (n3: nat).(eq A (aplus g -(ASort O (next g n0)) x2) (aplus g (ASort O n3) x2))) H8 (next g n2) H6) in -(let H10 \def (eq_ind_r A (aplus g (ASort O (next g n0)) x2) (\lambda (a: -A).(eq A a (aplus g (ASort O (next g n2)) x2))) H9 (aplus g (ASort O n0) (S -x2)) (aplus_sort_O_S_simpl g n0 x2)) in (let H11 \def (eq_ind_r A (aplus g -(ASort O (next g n2)) x2) (\lambda (a: A).(eq A (aplus g (ASort O n0) (S x2)) -a)) H10 (aplus g (ASort O n2) (S x2)) (aplus_sort_O_S_simpl g n2 x2)) in -(leq_sort g O O n0 n2 (S x2) H11))))))) H5))))))) H2)))) (\lambda (n3: -nat).(\lambda (_: (((leq g (asucc g (ASort O n0)) (asucc g (ASort n3 n2))) -\to (leq g (ASort O n0) (ASort n3 n2))))).(\lambda (H1: (leq g (asucc g -(ASort O n0)) (asucc g (ASort (S n3) n2)))).(let H_x \def (leq_gen_sort1 g O -(next g n0) (ASort n3 n2) H1) in (let H2 \def H_x in (ex2_3_ind nat nat nat -(\lambda (n4: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort -O (next g n0)) k) (aplus g (ASort h2 n4) k))))) (\lambda (n4: nat).(\lambda -(h2: nat).(\lambda (_: nat).(eq A (ASort n3 n2) (ASort h2 n4))))) (leq g -(ASort O n0) (ASort (S n3) n2)) (\lambda (x0: nat).(\lambda (x1: -nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort O (next g n0)) -x2) (aplus g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort n3 n2) (ASort x1 -x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e in A return -(\lambda (_: A).nat) with [(ASort n4 _) \Rightarrow n4 | (AHead _ _) -\Rightarrow n3])) (ASort n3 n2) (ASort x1 x0) H4) in ((let H6 \def (f_equal A -nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort _ -n4) \Rightarrow n4 | (AHead _ _) \Rightarrow n2])) (ASort n3 n2) (ASort x1 -x0) H4) in (\lambda (H7: (eq nat n3 x1)).(let H8 \def (eq_ind_r nat x1 -(\lambda (n4: nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort -n4 x0) x2))) H3 n3 H7) in (let H9 \def (eq_ind_r nat x0 (\lambda (n4: -nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort n3 n4) x2))) H8 -n2 H6) in (let H10 \def (eq_ind_r A (aplus g (ASort O (next g n0)) x2) -(\lambda (a: A).(eq A a (aplus g (ASort n3 n2) x2))) H9 (aplus g (ASort O n0) -(S x2)) (aplus_sort_O_S_simpl g n0 x2)) in (let H11 \def (eq_ind_r A (aplus g -(ASort n3 n2) x2) (\lambda (a: A).(eq A (aplus g (ASort O n0) (S x2)) a)) H10 -(aplus g (ASort (S n3) n2) (S x2)) (aplus_sort_S_S_simpl g n2 n3 x2)) in -(leq_sort g O (S n3) n0 n2 (S x2) H11))))))) H5))))))) H2)))))) n1 H0)) -(\lambda (n3: nat).(\lambda (IHn: (((leq g (asucc g (ASort n3 n0)) (asucc g -(ASort n1 n2))) \to (leq g (ASort n3 n0) (ASort n1 n2))))).(\lambda (H0: (leq -g (asucc g (ASort (S n3) n0)) (asucc g (ASort n1 n2)))).(nat_ind (\lambda -(n4: nat).((leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort n4 n2))) \to -((((leq g (asucc g (ASort n3 n0)) (asucc g (ASort n4 n2))) \to (leq g (ASort -n3 n0) (ASort n4 n2)))) \to (leq g (ASort (S n3) n0) (ASort n4 n2))))) -(\lambda (H1: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort O -n2)))).(\lambda (_: (((leq g (asucc g (ASort n3 n0)) (asucc g (ASort O n2))) -\to (leq g (ASort n3 n0) (ASort O n2))))).(let H_x \def (leq_gen_sort1 g n3 -n0 (ASort O (next g n2)) H1) in (let H2 \def H_x in (ex2_3_ind nat nat nat -(\lambda (n4: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort -n3 n0) k) (aplus g (ASort h2 n4) k))))) (\lambda (n4: nat).(\lambda (h2: -nat).(\lambda (_: nat).(eq A (ASort O (next g n2)) (ASort h2 n4))))) (leq g -(ASort (S n3) n0) (ASort O n2)) (\lambda (x0: nat).(\lambda (x1: -nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort n3 n0) x2) (aplus -g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort O (next g n2)) (ASort x1 -x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e in A return -(\lambda (_: A).nat) with [(ASort n4 _) \Rightarrow n4 | (AHead _ _) -\Rightarrow O])) (ASort O (next g n2)) (ASort x1 x0) H4) in ((let H6 \def -(f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with -[(ASort _ n4) \Rightarrow n4 | (AHead _ _) \Rightarrow ((match g with [(mk_G -next _) \Rightarrow next]) n2)])) (ASort O (next g n2)) (ASort x1 x0) H4) in -(\lambda (H7: (eq nat O x1)).(let H8 \def (eq_ind_r nat x1 (\lambda (n4: -nat).(eq A (aplus g (ASort n3 n0) x2) (aplus g (ASort n4 x0) x2))) H3 O H7) -in (let H9 \def (eq_ind_r nat x0 (\lambda (n4: nat).(eq A (aplus g (ASort n3 -n0) x2) (aplus g (ASort O n4) x2))) H8 (next g n2) H6) in (let H10 \def -(eq_ind_r A (aplus g (ASort n3 n0) x2) (\lambda (a: A).(eq A a (aplus g -(ASort O (next g n2)) x2))) H9 (aplus g (ASort (S n3) n0) (S x2)) -(aplus_sort_S_S_simpl g n0 n3 x2)) in (let H11 \def (eq_ind_r A (aplus g -(ASort O (next g n2)) x2) (\lambda (a: A).(eq A (aplus g (ASort (S n3) n0) (S -x2)) a)) H10 (aplus g (ASort O n2) (S x2)) (aplus_sort_O_S_simpl g n2 x2)) in -(leq_sort g (S n3) O n0 n2 (S x2) H11))))))) H5))))))) H2))))) (\lambda (n4: -nat).(\lambda (_: (((leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort n4 -n2))) \to ((((leq g (asucc g (ASort n3 n0)) (asucc g (ASort n4 n2))) \to (leq -g (ASort n3 n0) (ASort n4 n2)))) \to (leq g (ASort (S n3) n0) (ASort n4 -n2)))))).(\lambda (H1: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort (S -n4) n2)))).(\lambda (_: (((leq g (asucc g (ASort n3 n0)) (asucc g (ASort (S -n4) n2))) \to (leq g (ASort n3 n0) (ASort (S n4) n2))))).(let H_x \def -(leq_gen_sort1 g n3 n0 (ASort n4 n2) H1) in (let H2 \def H_x in (ex2_3_ind -nat nat nat (\lambda (n5: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A -(aplus g (ASort n3 n0) k) (aplus g (ASort h2 n5) k))))) (\lambda (n5: -nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A (ASort n4 n2) (ASort h2 -n5))))) (leq g (ASort (S n3) n0) (ASort (S n4) n2)) (\lambda (x0: -nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g -(ASort n3 n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort n4 -n2) (ASort x1 x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e in A -return (\lambda (_: A).nat) with [(ASort n5 _) \Rightarrow n5 | (AHead _ _) -\Rightarrow n4])) (ASort n4 n2) (ASort x1 x0) H4) in ((let H6 \def (f_equal A -nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort _ -n5) \Rightarrow n5 | (AHead _ _) \Rightarrow n2])) (ASort n4 n2) (ASort x1 -x0) H4) in (\lambda (H7: (eq nat n4 x1)).(let H8 \def (eq_ind_r nat x1 -(\lambda (n5: nat).(eq A (aplus g (ASort n3 n0) x2) (aplus g (ASort n5 x0) -x2))) H3 n4 H7) in (let H9 \def (eq_ind_r nat x0 (\lambda (n5: nat).(eq A -(aplus g (ASort n3 n0) x2) (aplus g (ASort n4 n5) x2))) H8 n2 H6) in (let H10 -\def (eq_ind_r A (aplus g (ASort n3 n0) x2) (\lambda (a: A).(eq A a (aplus g -(ASort n4 n2) x2))) H9 (aplus g (ASort (S n3) n0) (S x2)) -(aplus_sort_S_S_simpl g n0 n3 x2)) in (let H11 \def (eq_ind_r A (aplus g -(ASort n4 n2) x2) (\lambda (a: A).(eq A (aplus g (ASort (S n3) n0) (S x2)) -a)) H10 (aplus g (ASort (S n4) n2) (S x2)) (aplus_sort_S_S_simpl g n2 n4 x2)) -in (leq_sort g (S n3) (S n4) n0 n2 (S x2) H11))))))) H5))))))) H2))))))) n1 -H0 IHn)))) n H)))) (\lambda (a: A).(\lambda (H: (((leq g (asucc g (ASort n -n0)) (asucc g a)) \to (leq g (ASort n n0) a)))).(\lambda (a0: A).(\lambda -(H0: (((leq g (asucc g (ASort n n0)) (asucc g a0)) \to (leq g (ASort n n0) -a0)))).(\lambda (H1: (leq g (asucc g (ASort n n0)) (asucc g (AHead a -a0)))).(nat_ind (\lambda (n1: nat).((((leq g (asucc g (ASort n1 n0)) (asucc g -a)) \to (leq g (ASort n1 n0) a))) \to ((((leq g (asucc g (ASort n1 n0)) -(asucc g a0)) \to (leq g (ASort n1 n0) a0))) \to ((leq g (asucc g (ASort n1 -n0)) (asucc g (AHead a a0))) \to (leq g (ASort n1 n0) (AHead a a0)))))) -(\lambda (_: (((leq g (asucc g (ASort O n0)) (asucc g a)) \to (leq g (ASort O -n0) a)))).(\lambda (_: (((leq g (asucc g (ASort O n0)) (asucc g a0)) \to (leq -g (ASort O n0) a0)))).(\lambda (H4: (leq g (asucc g (ASort O n0)) (asucc g -(AHead a a0)))).(let H_x \def (leq_gen_sort1 g O (next g n0) (AHead a (asucc -g a0)) H4) in (let H5 \def H_x in (ex2_3_ind nat nat nat (\lambda (n2: -nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort O (next g -n0)) k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: -nat).(\lambda (_: nat).(eq A (AHead a (asucc g a0)) (ASort h2 n2))))) (leq g -(ASort O n0) (AHead a a0)) (\lambda (x0: nat).(\lambda (x1: nat).(\lambda -(x2: nat).(\lambda (_: (eq A (aplus g (ASort O (next g n0)) x2) (aplus g -(ASort x1 x0) x2))).(\lambda (H7: (eq A (AHead a (asucc g a0)) (ASort x1 -x0))).(let H8 \def (eq_ind A (AHead a (asucc g a0)) (\lambda (ee: A).(match -ee in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | -(AHead _ _) \Rightarrow True])) I (ASort x1 x0) H7) in (False_ind (leq g -(ASort O n0) (AHead a a0)) H8))))))) H5)))))) (\lambda (n1: nat).(\lambda (_: -(((((leq g (asucc g (ASort n1 n0)) (asucc g a)) \to (leq g (ASort n1 n0) a))) -\to ((((leq g (asucc g (ASort n1 n0)) (asucc g a0)) \to (leq g (ASort n1 n0) -a0))) \to ((leq g (asucc g (ASort n1 n0)) (asucc g (AHead a a0))) \to (leq g -(ASort n1 n0) (AHead a a0))))))).(\lambda (_: (((leq g (asucc g (ASort (S n1) -n0)) (asucc g a)) \to (leq g (ASort (S n1) n0) a)))).(\lambda (_: (((leq g -(asucc g (ASort (S n1) n0)) (asucc g a0)) \to (leq g (ASort (S n1) n0) -a0)))).(\lambda (H4: (leq g (asucc g (ASort (S n1) n0)) (asucc g (AHead a -a0)))).(let H_x \def (leq_gen_sort1 g n1 n0 (AHead a (asucc g a0)) H4) in -(let H5 \def H_x in (ex2_3_ind nat nat nat (\lambda (n2: nat).(\lambda (h2: -nat).(\lambda (k: nat).(eq A (aplus g (ASort n1 n0) k) (aplus g (ASort h2 n2) -k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A (AHead a -(asucc g a0)) (ASort h2 n2))))) (leq g (ASort (S n1) n0) (AHead a a0)) -(\lambda (x0: nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (_: (eq A -(aplus g (ASort n1 n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H7: (eq A -(AHead a (asucc g a0)) (ASort x1 x0))).(let H8 \def (eq_ind A (AHead a (asucc -g a0)) (\lambda (ee: A).(match ee in A return (\lambda (_: A).Prop) with -[(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort x1 -x0) H7) in (False_ind (leq g (ASort (S n1) n0) (AHead a a0)) H8))))))) -H5)))))))) n H H0 H1)))))) a2)))) (\lambda (a: A).(\lambda (_: ((\forall (a2: -A).((leq g (asucc g a) (asucc g a2)) \to (leq g a a2))))).(\lambda (a0: -A).(\lambda (H0: ((\forall (a2: A).((leq g (asucc g a0) (asucc g a2)) \to -(leq g a0 a2))))).(\lambda (a2: A).(A_ind (\lambda (a3: A).((leq g (asucc g -(AHead a a0)) (asucc g a3)) \to (leq g (AHead a a0) a3))) (\lambda (n: -nat).(\lambda (n0: nat).(\lambda (H1: (leq g (asucc g (AHead a a0)) (asucc g -(ASort n n0)))).(nat_ind (\lambda (n1: nat).((leq g (asucc g (AHead a a0)) -(asucc g (ASort n1 n0))) \to (leq g (AHead a a0) (ASort n1 n0)))) (\lambda -(H2: (leq g (asucc g (AHead a a0)) (asucc g (ASort O n0)))).(let H_x \def -(leq_gen_head1 g a (asucc g a0) (ASort O (next g n0)) H2) in (let H3 \def H_x -in (ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq g a a3))) (\lambda -(_: A).(\lambda (a4: A).(leq g (asucc g a0) a4))) (\lambda (a3: A).(\lambda -(a4: A).(eq A (ASort O (next g n0)) (AHead a3 a4)))) (leq g (AHead a a0) -(ASort O n0)) (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g a -x0)).(\lambda (_: (leq g (asucc g a0) x1)).(\lambda (H6: (eq A (ASort O (next -g n0)) (AHead x0 x1))).(let H7 \def (eq_ind A (ASort O (next g n0)) (\lambda -(ee: A).(match ee in A return (\lambda (_: A).Prop) with [(ASort _ _) -\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H6) in -(False_ind (leq g (AHead a a0) (ASort O n0)) H7))))))) H3)))) (\lambda (n1: -nat).(\lambda (_: (((leq g (asucc g (AHead a a0)) (asucc g (ASort n1 n0))) -\to (leq g (AHead a a0) (ASort n1 n0))))).(\lambda (H2: (leq g (asucc g -(AHead a a0)) (asucc g (ASort (S n1) n0)))).(let H_x \def (leq_gen_head1 g a -(asucc g a0) (ASort n1 n0) H2) in (let H3 \def H_x in (ex3_2_ind A A (\lambda -(a3: A).(\lambda (_: A).(leq g a a3))) (\lambda (_: A).(\lambda (a4: A).(leq -g (asucc g a0) a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort n1 n0) -(AHead a3 a4)))) (leq g (AHead a a0) (ASort (S n1) n0)) (\lambda (x0: -A).(\lambda (x1: A).(\lambda (_: (leq g a x0)).(\lambda (_: (leq g (asucc g -a0) x1)).(\lambda (H6: (eq A (ASort n1 n0) (AHead x0 x1))).(let H7 \def -(eq_ind A (ASort n1 n0) (\lambda (ee: A).(match ee in A return (\lambda (_: -A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow -False])) I (AHead x0 x1) H6) in (False_ind (leq g (AHead a a0) (ASort (S n1) -n0)) H7))))))) H3)))))) n H1)))) (\lambda (a3: A).(\lambda (_: (((leq g -(asucc g (AHead a a0)) (asucc g a3)) \to (leq g (AHead a a0) a3)))).(\lambda -(a4: A).(\lambda (_: (((leq g (asucc g (AHead a a0)) (asucc g a4)) \to (leq g -(AHead a a0) a4)))).(\lambda (H3: (leq g (asucc g (AHead a a0)) (asucc g -(AHead a3 a4)))).(let H_x \def (leq_gen_head1 g a (asucc g a0) (AHead a3 -(asucc g a4)) H3) in (let H4 \def H_x in (ex3_2_ind A A (\lambda (a5: -A).(\lambda (_: A).(leq g a a5))) (\lambda (_: A).(\lambda (a6: A).(leq g -(asucc g a0) a6))) (\lambda (a5: A).(\lambda (a6: A).(eq A (AHead a3 (asucc g -a4)) (AHead a5 a6)))) (leq g (AHead a a0) (AHead a3 a4)) (\lambda (x0: -A).(\lambda (x1: A).(\lambda (H5: (leq g a x0)).(\lambda (H6: (leq g (asucc g -a0) x1)).(\lambda (H7: (eq A (AHead a3 (asucc g a4)) (AHead x0 x1))).(let H8 -\def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) -with [(ASort _ _) \Rightarrow a3 | (AHead a5 _) \Rightarrow a5])) (AHead a3 -(asucc g a4)) (AHead x0 x1) H7) in ((let H9 \def (f_equal A A (\lambda (e: -A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow -((let rec asucc (g0: G) (l: A) on l: A \def (match l with [(ASort n0 n) -\Rightarrow (match n0 with [O \Rightarrow (ASort O (next g0 n)) | (S h) -\Rightarrow (ASort h n)]) | (AHead a5 a6) \Rightarrow (AHead a5 (asucc g0 -a6))]) in asucc) g a4) | (AHead _ a5) \Rightarrow a5])) (AHead a3 (asucc g -a4)) (AHead x0 x1) H7) in (\lambda (H10: (eq A a3 x0)).(let H11 \def -(eq_ind_r A x1 (\lambda (a5: A).(leq g (asucc g a0) a5)) H6 (asucc g a4) H9) -in (let H12 \def (eq_ind_r A x0 (\lambda (a5: A).(leq g a a5)) H5 a3 H10) in -(leq_head g a a3 H12 a0 a4 (H0 a4 H11)))))) H8))))))) H4)))))))) a2)))))) -a1)). -(* COMMENTS -Initial nodes: 4697 -END *) - -theorem leq_asucc: - \forall (g: G).(\forall (a: A).(ex A (\lambda (a0: A).(leq g a (asucc g -a0))))) -\def - \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(ex A (\lambda (a1: -A).(leq g a0 (asucc g a1))))) (\lambda (n: nat).(\lambda (n0: nat).(ex_intro -A (\lambda (a0: A).(leq g (ASort n n0) (asucc g a0))) (ASort (S n) n0) -(leq_refl g (ASort n n0))))) (\lambda (a0: A).(\lambda (_: (ex A (\lambda -(a1: A).(leq g a0 (asucc g a1))))).(\lambda (a1: A).(\lambda (H0: (ex A -(\lambda (a2: A).(leq g a1 (asucc g a2))))).(let H1 \def H0 in (ex_ind A -(\lambda (a2: A).(leq g a1 (asucc g a2))) (ex A (\lambda (a2: A).(leq g -(AHead a0 a1) (asucc g a2)))) (\lambda (x: A).(\lambda (H2: (leq g a1 (asucc -g x))).(ex_intro A (\lambda (a2: A).(leq g (AHead a0 a1) (asucc g a2))) -(AHead a0 x) (leq_head g a0 a0 (leq_refl g a0) a1 (asucc g x) H2)))) H1)))))) -a)). -(* COMMENTS -Initial nodes: 221 -END *) - -theorem leq_ahead_asucc_false: - \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (AHead a1 a2) -(asucc g a1)) \to (\forall (P: Prop).P)))) -\def - \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2: -A).((leq g (AHead a a2) (asucc g a)) \to (\forall (P: Prop).P)))) (\lambda -(n: nat).(\lambda (n0: nat).(\lambda (a2: A).(\lambda (H: (leq g (AHead -(ASort n n0) a2) (match n with [O \Rightarrow (ASort O (next g n0)) | (S h) -\Rightarrow (ASort h n0)]))).(\lambda (P: Prop).(nat_ind (\lambda (n1: -nat).((leq g (AHead (ASort n1 n0) a2) (match n1 with [O \Rightarrow (ASort O -(next g n0)) | (S h) \Rightarrow (ASort h n0)])) \to P)) (\lambda (H0: (leq g -(AHead (ASort O n0) a2) (ASort O (next g n0)))).(let H_x \def (leq_gen_head1 -g (ASort O n0) a2 (ASort O (next g n0)) H0) in (let H1 \def H_x in (ex3_2_ind -A A (\lambda (a3: A).(\lambda (_: A).(leq g (ASort O n0) a3))) (\lambda (_: -A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A -(ASort O (next g n0)) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: -A).(\lambda (_: (leq g (ASort O n0) x0)).(\lambda (_: (leq g a2 x1)).(\lambda -(H4: (eq A (ASort O (next g n0)) (AHead x0 x1))).(let H5 \def (eq_ind A -(ASort O (next g n0)) (\lambda (ee: A).(match ee in A return (\lambda (_: -A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow -False])) I (AHead x0 x1) H4) in (False_ind P H5))))))) H1)))) (\lambda (n1: -nat).(\lambda (_: (((leq g (AHead (ASort n1 n0) a2) (match n1 with [O -\Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)])) \to -P))).(\lambda (H0: (leq g (AHead (ASort (S n1) n0) a2) (ASort n1 n0))).(let -H_x \def (leq_gen_head1 g (ASort (S n1) n0) a2 (ASort n1 n0) H0) in (let H1 -\def H_x in (ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq g (ASort (S -n1) n0) a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: -A).(\lambda (a4: A).(eq A (ASort n1 n0) (AHead a3 a4)))) P (\lambda (x0: -A).(\lambda (x1: A).(\lambda (_: (leq g (ASort (S n1) n0) x0)).(\lambda (_: -(leq g a2 x1)).(\lambda (H4: (eq A (ASort n1 n0) (AHead x0 x1))).(let H5 \def -(eq_ind A (ASort n1 n0) (\lambda (ee: A).(match ee in A return (\lambda (_: -A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow -False])) I (AHead x0 x1) H4) in (False_ind P H5))))))) H1)))))) n H)))))) -(\lambda (a: A).(\lambda (_: ((\forall (a2: A).((leq g (AHead a a2) (asucc g -a)) \to (\forall (P: Prop).P))))).(\lambda (a0: A).(\lambda (_: ((\forall -(a2: A).((leq g (AHead a0 a2) (asucc g a0)) \to (\forall (P: -Prop).P))))).(\lambda (a2: A).(\lambda (H1: (leq g (AHead (AHead a a0) a2) -(AHead a (asucc g a0)))).(\lambda (P: Prop).(let H_x \def (leq_gen_head1 g -(AHead a a0) a2 (AHead a (asucc g a0)) H1) in (let H2 \def H_x in (ex3_2_ind -A A (\lambda (a3: A).(\lambda (_: A).(leq g (AHead a a0) a3))) (\lambda (_: -A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A -(AHead a (asucc g a0)) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: -A).(\lambda (H3: (leq g (AHead a a0) x0)).(\lambda (H4: (leq g a2 -x1)).(\lambda (H5: (eq A (AHead a (asucc g a0)) (AHead x0 x1))).(let H6 \def -(f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with -[(ASort _ _) \Rightarrow a | (AHead a3 _) \Rightarrow a3])) (AHead a (asucc g -a0)) (AHead x0 x1) H5) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e -in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow ((let rec asucc -(g0: G) (l: A) on l: A \def (match l with [(ASort n0 n) \Rightarrow (match n0 -with [O \Rightarrow (ASort O (next g0 n)) | (S h) \Rightarrow (ASort h n)]) | -(AHead a3 a4) \Rightarrow (AHead a3 (asucc g0 a4))]) in asucc) g a0) | (AHead -_ a3) \Rightarrow a3])) (AHead a (asucc g a0)) (AHead x0 x1) H5) in (\lambda -(H8: (eq A a x0)).(let H9 \def (eq_ind_r A x1 (\lambda (a3: A).(leq g a2 a3)) -H4 (asucc g a0) H7) in (let H10 \def (eq_ind_r A x0 (\lambda (a3: A).(leq g -(AHead a a0) a3)) H3 a H8) in (leq_ahead_false_1 g a a0 H10 P))))) H6))))))) -H2)))))))))) a1)). -(* COMMENTS -Initial nodes: 927 -END *) - -theorem leq_asucc_false: - \forall (g: G).(\forall (a: A).((leq g (asucc g a) a) \to (\forall (P: -Prop).P))) -\def - \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).((leq g (asucc g a0) -a0) \to (\forall (P: Prop).P))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda -(H: (leq g (match n with [O \Rightarrow (ASort O (next g n0)) | (S h) -\Rightarrow (ASort h n0)]) (ASort n n0))).(\lambda (P: Prop).(nat_ind -(\lambda (n1: nat).((leq g (match n1 with [O \Rightarrow (ASort O (next g -n0)) | (S h) \Rightarrow (ASort h n0)]) (ASort n1 n0)) \to P)) (\lambda (H0: -(leq g (ASort O (next g n0)) (ASort O n0))).(let H_x \def (leq_gen_sort1 g O -(next g n0) (ASort O n0) H0) in (let H1 \def H_x in (ex2_3_ind nat nat nat -(\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort -O (next g n0)) k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda -(h2: nat).(\lambda (_: nat).(eq A (ASort O n0) (ASort h2 n2))))) P (\lambda -(x0: nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (H2: (eq A (aplus g -(ASort O (next g n0)) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H3: (eq A -(ASort O n0) (ASort x1 x0))).(let H4 \def (f_equal A nat (\lambda (e: -A).(match e in A return (\lambda (_: A).nat) with [(ASort n1 _) \Rightarrow -n1 | (AHead _ _) \Rightarrow O])) (ASort O n0) (ASort x1 x0) H3) in ((let H5 -\def (f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) -with [(ASort _ n1) \Rightarrow n1 | (AHead _ _) \Rightarrow n0])) (ASort O -n0) (ASort x1 x0) H3) in (\lambda (H6: (eq nat O x1)).(let H7 \def (eq_ind_r -nat x1 (\lambda (n1: nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g -(ASort n1 x0) x2))) H2 O H6) in (let H8 \def (eq_ind_r nat x0 (\lambda (n1: -nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort O n1) x2))) H7 -n0 H5) in (let H9 \def (eq_ind_r A (aplus g (ASort O (next g n0)) x2) -(\lambda (a0: A).(eq A a0 (aplus g (ASort O n0) x2))) H8 (aplus g (ASort O -n0) (S x2)) (aplus_sort_O_S_simpl g n0 x2)) in (let H_y \def (aplus_inj g (S -x2) x2 (ASort O n0) H9) in (le_Sx_x x2 (eq_ind_r nat x2 (\lambda (n1: -nat).(le n1 x2)) (le_n x2) (S x2) H_y) P))))))) H4))))))) H1)))) (\lambda -(n1: nat).(\lambda (_: (((leq g (match n1 with [O \Rightarrow (ASort O (next -g n0)) | (S h) \Rightarrow (ASort h n0)]) (ASort n1 n0)) \to P))).(\lambda -(H0: (leq g (ASort n1 n0) (ASort (S n1) n0))).(let H_x \def (leq_gen_sort1 g -n1 n0 (ASort (S n1) n0) H0) in (let H1 \def H_x in (ex2_3_ind nat nat nat -(\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort -n1 n0) k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: -nat).(\lambda (_: nat).(eq A (ASort (S n1) n0) (ASort h2 n2))))) P (\lambda -(x0: nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (H2: (eq A (aplus g -(ASort n1 n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H3: (eq A (ASort (S -n1) n0) (ASort x1 x0))).(let H4 \def (f_equal A nat (\lambda (e: A).(match e -in A return (\lambda (_: A).nat) with [(ASort n2 _) \Rightarrow n2 | (AHead _ -_) \Rightarrow (S n1)])) (ASort (S n1) n0) (ASort x1 x0) H3) in ((let H5 \def -(f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with -[(ASort _ n2) \Rightarrow n2 | (AHead _ _) \Rightarrow n0])) (ASort (S n1) -n0) (ASort x1 x0) H3) in (\lambda (H6: (eq nat (S n1) x1)).(let H7 \def -(eq_ind_r nat x1 (\lambda (n2: nat).(eq A (aplus g (ASort n1 n0) x2) (aplus g -(ASort n2 x0) x2))) H2 (S n1) H6) in (let H8 \def (eq_ind_r nat x0 (\lambda -(n2: nat).(eq A (aplus g (ASort n1 n0) x2) (aplus g (ASort (S n1) n2) x2))) -H7 n0 H5) in (let H9 \def (eq_ind_r A (aplus g (ASort n1 n0) x2) (\lambda -(a0: A).(eq A a0 (aplus g (ASort (S n1) n0) x2))) H8 (aplus g (ASort (S n1) -n0) (S x2)) (aplus_sort_S_S_simpl g n0 n1 x2)) in (let H_y \def (aplus_inj g -(S x2) x2 (ASort (S n1) n0) H9) in (le_Sx_x x2 (eq_ind_r nat x2 (\lambda (n2: -nat).(le n2 x2)) (le_n x2) (S x2) H_y) P))))))) H4))))))) H1)))))) n H))))) -(\lambda (a0: A).(\lambda (_: (((leq g (asucc g a0) a0) \to (\forall (P: -Prop).P)))).(\lambda (a1: A).(\lambda (H0: (((leq g (asucc g a1) a1) \to -(\forall (P: Prop).P)))).(\lambda (H1: (leq g (AHead a0 (asucc g a1)) (AHead -a0 a1))).(\lambda (P: Prop).(let H_x \def (leq_gen_head1 g a0 (asucc g a1) -(AHead a0 a1) H1) in (let H2 \def H_x in (ex3_2_ind A A (\lambda (a3: -A).(\lambda (_: A).(leq g a0 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g -(asucc g a1) a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (AHead a0 a1) -(AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (H3: (leq g a0 -x0)).(\lambda (H4: (leq g (asucc g a1) x1)).(\lambda (H5: (eq A (AHead a0 a1) -(AHead x0 x1))).(let H6 \def (f_equal A A (\lambda (e: A).(match e in A -return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead a2 _) -\Rightarrow a2])) (AHead a0 a1) (AHead x0 x1) H5) in ((let H7 \def (f_equal A -A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) -\Rightarrow a1 | (AHead _ a2) \Rightarrow a2])) (AHead a0 a1) (AHead x0 x1) -H5) in (\lambda (H8: (eq A a0 x0)).(let H9 \def (eq_ind_r A x1 (\lambda (a2: -A).(leq g (asucc g a1) a2)) H4 a1 H7) in (let H10 \def (eq_ind_r A x0 -(\lambda (a2: A).(leq g a0 a2)) H3 a0 H8) in (H0 H9 P))))) H6))))))) -H2))))))))) a)). -(* COMMENTS -Initial nodes: 1327 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/leq/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/leq/defs.ma deleted file mode 100644 index 5a5308280..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/leq/defs.ma +++ /dev/null @@ -1,26 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/aplus/defs.ma". - -inductive leq (g: G): A \to (A \to Prop) \def -| leq_sort: \forall (h1: nat).(\forall (h2: nat).(\forall (n1: nat).(\forall -(n2: nat).(\forall (k: nat).((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort -h2 n2) k)) \to (leq g (ASort h1 n1) (ASort h2 n2))))))) -| leq_head: \forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall (a3: -A).(\forall (a4: A).((leq g a3 a4) \to (leq g (AHead a1 a3) (AHead a2 -a4))))))). - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/leq/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/leq/fwd.ma deleted file mode 100644 index e259fa19e..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/leq/fwd.ma +++ /dev/null @@ -1,244 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/leq/defs.ma". - -theorem leq_gen_sort1: - \forall (g: G).(\forall (h1: nat).(\forall (n1: nat).(\forall (a2: A).((leq -g (ASort h1 n1) a2) \to (ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: -nat).(\lambda (k: nat).(eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) -k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a2 -(ASort h2 n2)))))))))) -\def - \lambda (g: G).(\lambda (h1: nat).(\lambda (n1: nat).(\lambda (a2: -A).(\lambda (H: (leq g (ASort h1 n1) a2)).(insert_eq A (ASort h1 n1) (\lambda -(a: A).(leq g a a2)) (\lambda (a: A).(ex2_3 nat nat nat (\lambda (n2: -nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g a k) (aplus g (ASort -h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A -a2 (ASort h2 n2))))))) (\lambda (y: A).(\lambda (H0: (leq g y a2)).(leq_ind g -(\lambda (a: A).(\lambda (a0: A).((eq A a (ASort h1 n1)) \to (ex2_3 nat nat -nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g a -k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: -nat).(\lambda (_: nat).(eq A a0 (ASort h2 n2))))))))) (\lambda (h0: -nat).(\lambda (h2: nat).(\lambda (n0: nat).(\lambda (n2: nat).(\lambda (k: -nat).(\lambda (H1: (eq A (aplus g (ASort h0 n0) k) (aplus g (ASort h2 n2) -k))).(\lambda (H2: (eq A (ASort h0 n0) (ASort h1 n1))).(let H3 \def (f_equal -A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort -n _) \Rightarrow n | (AHead _ _) \Rightarrow h0])) (ASort h0 n0) (ASort h1 -n1) H2) in ((let H4 \def (f_equal A nat (\lambda (e: A).(match e in A return -(\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) -\Rightarrow n0])) (ASort h0 n0) (ASort h1 n1) H2) in (\lambda (H5: (eq nat h0 -h1)).(let H6 \def (eq_ind nat n0 (\lambda (n: nat).(eq A (aplus g (ASort h0 -n) k) (aplus g (ASort h2 n2) k))) H1 n1 H4) in (eq_ind_r nat n1 (\lambda (n: -nat).(ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (k0: -nat).(eq A (aplus g (ASort h0 n) k0) (aplus g (ASort h3 n3) k0))))) (\lambda -(n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A (ASort h2 n2) (ASort h3 -n3))))))) (let H7 \def (eq_ind nat h0 (\lambda (n: nat).(eq A (aplus g (ASort -n n1) k) (aplus g (ASort h2 n2) k))) H6 h1 H5) in (eq_ind_r nat h1 (\lambda -(n: nat).(ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda -(k0: nat).(eq A (aplus g (ASort n n1) k0) (aplus g (ASort h3 n3) k0))))) -(\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A (ASort h2 n2) -(ASort h3 n3))))))) (ex2_3_intro nat nat nat (\lambda (n3: nat).(\lambda (h3: -nat).(\lambda (k0: nat).(eq A (aplus g (ASort h1 n1) k0) (aplus g (ASort h3 -n3) k0))))) (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A -(ASort h2 n2) (ASort h3 n3))))) n2 h2 k H7 (refl_equal A (ASort h2 n2))) h0 -H5)) n0 H4)))) H3))))))))) (\lambda (a1: A).(\lambda (a3: A).(\lambda (_: -(leq g a1 a3)).(\lambda (_: (((eq A a1 (ASort h1 n1)) \to (ex2_3 nat nat nat -(\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g a1 k) -(aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda -(_: nat).(eq A a3 (ASort h2 n2))))))))).(\lambda (a4: A).(\lambda (a5: -A).(\lambda (_: (leq g a4 a5)).(\lambda (_: (((eq A a4 (ASort h1 n1)) \to -(ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: -nat).(eq A (aplus g a4 k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: -nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a5 (ASort h2 -n2))))))))).(\lambda (H5: (eq A (AHead a1 a4) (ASort h1 n1))).(let H6 \def -(eq_ind A (AHead a1 a4) (\lambda (ee: A).(match ee in A return (\lambda (_: -A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow -True])) I (ASort h1 n1) H5) in (False_ind (ex2_3 nat nat nat (\lambda (n2: -nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (AHead a1 a4) k) -(aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda -(_: nat).(eq A (AHead a3 a5) (ASort h2 n2)))))) H6))))))))))) y a2 H0))) -H))))). -(* COMMENTS -Initial nodes: 913 -END *) - -theorem leq_gen_head1: - \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a: A).((leq g -(AHead a1 a2) a) \to (ex3_2 A A (\lambda (a3: A).(\lambda (_: A).(leq g a1 -a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: -A).(\lambda (a4: A).(eq A a (AHead a3 a4))))))))) -\def - \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a: A).(\lambda -(H: (leq g (AHead a1 a2) a)).(insert_eq A (AHead a1 a2) (\lambda (a0: A).(leq -g a0 a)) (\lambda (_: A).(ex3_2 A A (\lambda (a3: A).(\lambda (_: A).(leq g -a1 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: -A).(\lambda (a4: A).(eq A a (AHead a3 a4)))))) (\lambda (y: A).(\lambda (H0: -(leq g y a)).(leq_ind g (\lambda (a0: A).(\lambda (a3: A).((eq A a0 (AHead a1 -a2)) \to (ex3_2 A A (\lambda (a4: A).(\lambda (_: A).(leq g a1 a4))) (\lambda -(_: A).(\lambda (a5: A).(leq g a2 a5))) (\lambda (a4: A).(\lambda (a5: A).(eq -A a3 (AHead a4 a5)))))))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: -nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (_: (eq A (aplus g (ASort -h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (H2: (eq A (ASort h1 n1) -(AHead a1 a2))).(let H3 \def (eq_ind A (ASort h1 n1) (\lambda (ee: A).(match -ee in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | -(AHead _ _) \Rightarrow False])) I (AHead a1 a2) H2) in (False_ind (ex3_2 A A -(\lambda (a3: A).(\lambda (_: A).(leq g a1 a3))) (\lambda (_: A).(\lambda -(a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort h2 n2) -(AHead a3 a4))))) H3))))))))) (\lambda (a0: A).(\lambda (a3: A).(\lambda (H1: -(leq g a0 a3)).(\lambda (H2: (((eq A a0 (AHead a1 a2)) \to (ex3_2 A A -(\lambda (a4: A).(\lambda (_: A).(leq g a1 a4))) (\lambda (_: A).(\lambda -(a5: A).(leq g a2 a5))) (\lambda (a4: A).(\lambda (a5: A).(eq A a3 (AHead a4 -a5)))))))).(\lambda (a4: A).(\lambda (a5: A).(\lambda (H3: (leq g a4 -a5)).(\lambda (H4: (((eq A a4 (AHead a1 a2)) \to (ex3_2 A A (\lambda (a6: -A).(\lambda (_: A).(leq g a1 a6))) (\lambda (_: A).(\lambda (a7: A).(leq g a2 -a7))) (\lambda (a6: A).(\lambda (a7: A).(eq A a5 (AHead a6 -a7)))))))).(\lambda (H5: (eq A (AHead a0 a4) (AHead a1 a2))).(let H6 \def -(f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with -[(ASort _ _) \Rightarrow a0 | (AHead a6 _) \Rightarrow a6])) (AHead a0 a4) -(AHead a1 a2) H5) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e in A -return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a4 | (AHead _ a6) -\Rightarrow a6])) (AHead a0 a4) (AHead a1 a2) H5) in (\lambda (H8: (eq A a0 -a1)).(let H9 \def (eq_ind A a4 (\lambda (a6: A).((eq A a6 (AHead a1 a2)) \to -(ex3_2 A A (\lambda (a7: A).(\lambda (_: A).(leq g a1 a7))) (\lambda (_: -A).(\lambda (a8: A).(leq g a2 a8))) (\lambda (a7: A).(\lambda (a8: A).(eq A -a5 (AHead a7 a8))))))) H4 a2 H7) in (let H10 \def (eq_ind A a4 (\lambda (a6: -A).(leq g a6 a5)) H3 a2 H7) in (let H11 \def (eq_ind A a0 (\lambda (a6: -A).((eq A a6 (AHead a1 a2)) \to (ex3_2 A A (\lambda (a7: A).(\lambda (_: -A).(leq g a1 a7))) (\lambda (_: A).(\lambda (a8: A).(leq g a2 a8))) (\lambda -(a7: A).(\lambda (a8: A).(eq A a3 (AHead a7 a8))))))) H2 a1 H8) in (let H12 -\def (eq_ind A a0 (\lambda (a6: A).(leq g a6 a3)) H1 a1 H8) in (ex3_2_intro A -A (\lambda (a6: A).(\lambda (_: A).(leq g a1 a6))) (\lambda (_: A).(\lambda -(a7: A).(leq g a2 a7))) (\lambda (a6: A).(\lambda (a7: A).(eq A (AHead a3 a5) -(AHead a6 a7)))) a3 a5 H12 H10 (refl_equal A (AHead a3 a5))))))))) -H6))))))))))) y a H0))) H))))). -(* COMMENTS -Initial nodes: 797 -END *) - -theorem leq_gen_sort2: - \forall (g: G).(\forall (h1: nat).(\forall (n1: nat).(\forall (a2: A).((leq -g a2 (ASort h1 n1)) \to (ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: -nat).(\lambda (k: nat).(eq A (aplus g (ASort h2 n2) k) (aplus g (ASort h1 n1) -k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a2 -(ASort h2 n2)))))))))) -\def - \lambda (g: G).(\lambda (h1: nat).(\lambda (n1: nat).(\lambda (a2: -A).(\lambda (H: (leq g a2 (ASort h1 n1))).(insert_eq A (ASort h1 n1) (\lambda -(a: A).(leq g a2 a)) (\lambda (a: A).(ex2_3 nat nat nat (\lambda (n2: -nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort h2 n2) k) -(aplus g a k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq -A a2 (ASort h2 n2))))))) (\lambda (y: A).(\lambda (H0: (leq g a2 y)).(leq_ind -g (\lambda (a: A).(\lambda (a0: A).((eq A a0 (ASort h1 n1)) \to (ex2_3 nat -nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus -g (ASort h2 n2) k) (aplus g a0 k))))) (\lambda (n2: nat).(\lambda (h2: -nat).(\lambda (_: nat).(eq A a (ASort h2 n2))))))))) (\lambda (h0: -nat).(\lambda (h2: nat).(\lambda (n0: nat).(\lambda (n2: nat).(\lambda (k: -nat).(\lambda (H1: (eq A (aplus g (ASort h0 n0) k) (aplus g (ASort h2 n2) -k))).(\lambda (H2: (eq A (ASort h2 n2) (ASort h1 n1))).(let H3 \def (f_equal -A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort -n _) \Rightarrow n | (AHead _ _) \Rightarrow h2])) (ASort h2 n2) (ASort h1 -n1) H2) in ((let H4 \def (f_equal A nat (\lambda (e: A).(match e in A return -(\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) -\Rightarrow n2])) (ASort h2 n2) (ASort h1 n1) H2) in (\lambda (H5: (eq nat h2 -h1)).(let H6 \def (eq_ind nat n2 (\lambda (n: nat).(eq A (aplus g (ASort h0 -n0) k) (aplus g (ASort h2 n) k))) H1 n1 H4) in (eq_ind_r nat n1 (\lambda (n: -nat).(ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (k0: -nat).(eq A (aplus g (ASort h3 n3) k0) (aplus g (ASort h2 n) k0))))) (\lambda -(n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A (ASort h0 n0) (ASort h3 -n3))))))) (let H7 \def (eq_ind nat h2 (\lambda (n: nat).(eq A (aplus g (ASort -h0 n0) k) (aplus g (ASort n n1) k))) H6 h1 H5) in (eq_ind_r nat h1 (\lambda -(n: nat).(ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda -(k0: nat).(eq A (aplus g (ASort h3 n3) k0) (aplus g (ASort n n1) k0))))) -(\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A (ASort h0 n0) -(ASort h3 n3))))))) (ex2_3_intro nat nat nat (\lambda (n3: nat).(\lambda (h3: -nat).(\lambda (k0: nat).(eq A (aplus g (ASort h3 n3) k0) (aplus g (ASort h1 -n1) k0))))) (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A -(ASort h0 n0) (ASort h3 n3))))) n0 h0 k H7 (refl_equal A (ASort h0 n0))) h2 -H5)) n2 H4)))) H3))))))))) (\lambda (a1: A).(\lambda (a3: A).(\lambda (_: -(leq g a1 a3)).(\lambda (_: (((eq A a3 (ASort h1 n1)) \to (ex2_3 nat nat nat -(\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort -h2 n2) k) (aplus g a3 k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda -(_: nat).(eq A a1 (ASort h2 n2))))))))).(\lambda (a4: A).(\lambda (a5: -A).(\lambda (_: (leq g a4 a5)).(\lambda (_: (((eq A a5 (ASort h1 n1)) \to -(ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: -nat).(eq A (aplus g (ASort h2 n2) k) (aplus g a5 k))))) (\lambda (n2: -nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a4 (ASort h2 -n2))))))))).(\lambda (H5: (eq A (AHead a3 a5) (ASort h1 n1))).(let H6 \def -(eq_ind A (AHead a3 a5) (\lambda (ee: A).(match ee in A return (\lambda (_: -A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow -True])) I (ASort h1 n1) H5) in (False_ind (ex2_3 nat nat nat (\lambda (n2: -nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort h2 n2) k) -(aplus g (AHead a3 a5) k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda -(_: nat).(eq A (AHead a1 a4) (ASort h2 n2)))))) H6))))))))))) a2 y H0))) -H))))). -(* COMMENTS -Initial nodes: 913 -END *) - -theorem leq_gen_head2: - \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a: A).((leq g a -(AHead a1 a2)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (_: A).(leq g a3 -a1))) (\lambda (_: A).(\lambda (a4: A).(leq g a4 a2))) (\lambda (a3: -A).(\lambda (a4: A).(eq A a (AHead a3 a4))))))))) -\def - \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a: A).(\lambda -(H: (leq g a (AHead a1 a2))).(insert_eq A (AHead a1 a2) (\lambda (a0: A).(leq -g a a0)) (\lambda (_: A).(ex3_2 A A (\lambda (a3: A).(\lambda (_: A).(leq g -a3 a1))) (\lambda (_: A).(\lambda (a4: A).(leq g a4 a2))) (\lambda (a3: -A).(\lambda (a4: A).(eq A a (AHead a3 a4)))))) (\lambda (y: A).(\lambda (H0: -(leq g a y)).(leq_ind g (\lambda (a0: A).(\lambda (a3: A).((eq A a3 (AHead a1 -a2)) \to (ex3_2 A A (\lambda (a4: A).(\lambda (_: A).(leq g a4 a1))) (\lambda -(_: A).(\lambda (a5: A).(leq g a5 a2))) (\lambda (a4: A).(\lambda (a5: A).(eq -A a0 (AHead a4 a5)))))))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: -nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (_: (eq A (aplus g (ASort -h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (H2: (eq A (ASort h2 n2) -(AHead a1 a2))).(let H3 \def (eq_ind A (ASort h2 n2) (\lambda (ee: A).(match -ee in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | -(AHead _ _) \Rightarrow False])) I (AHead a1 a2) H2) in (False_ind (ex3_2 A A -(\lambda (a3: A).(\lambda (_: A).(leq g a3 a1))) (\lambda (_: A).(\lambda -(a4: A).(leq g a4 a2))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort h1 n1) -(AHead a3 a4))))) H3))))))))) (\lambda (a0: A).(\lambda (a3: A).(\lambda (H1: -(leq g a0 a3)).(\lambda (H2: (((eq A a3 (AHead a1 a2)) \to (ex3_2 A A -(\lambda (a4: A).(\lambda (_: A).(leq g a4 a1))) (\lambda (_: A).(\lambda -(a5: A).(leq g a5 a2))) (\lambda (a4: A).(\lambda (a5: A).(eq A a0 (AHead a4 -a5)))))))).(\lambda (a4: A).(\lambda (a5: A).(\lambda (H3: (leq g a4 -a5)).(\lambda (H4: (((eq A a5 (AHead a1 a2)) \to (ex3_2 A A (\lambda (a6: -A).(\lambda (_: A).(leq g a6 a1))) (\lambda (_: A).(\lambda (a7: A).(leq g a7 -a2))) (\lambda (a6: A).(\lambda (a7: A).(eq A a4 (AHead a6 -a7)))))))).(\lambda (H5: (eq A (AHead a3 a5) (AHead a1 a2))).(let H6 \def -(f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with -[(ASort _ _) \Rightarrow a3 | (AHead a6 _) \Rightarrow a6])) (AHead a3 a5) -(AHead a1 a2) H5) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e in A -return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a5 | (AHead _ a6) -\Rightarrow a6])) (AHead a3 a5) (AHead a1 a2) H5) in (\lambda (H8: (eq A a3 -a1)).(let H9 \def (eq_ind A a5 (\lambda (a6: A).((eq A a6 (AHead a1 a2)) \to -(ex3_2 A A (\lambda (a7: A).(\lambda (_: A).(leq g a7 a1))) (\lambda (_: -A).(\lambda (a8: A).(leq g a8 a2))) (\lambda (a7: A).(\lambda (a8: A).(eq A -a4 (AHead a7 a8))))))) H4 a2 H7) in (let H10 \def (eq_ind A a5 (\lambda (a6: -A).(leq g a4 a6)) H3 a2 H7) in (let H11 \def (eq_ind A a3 (\lambda (a6: -A).((eq A a6 (AHead a1 a2)) \to (ex3_2 A A (\lambda (a7: A).(\lambda (_: -A).(leq g a7 a1))) (\lambda (_: A).(\lambda (a8: A).(leq g a8 a2))) (\lambda -(a7: A).(\lambda (a8: A).(eq A a0 (AHead a7 a8))))))) H2 a1 H8) in (let H12 -\def (eq_ind A a3 (\lambda (a6: A).(leq g a0 a6)) H1 a1 H8) in (ex3_2_intro A -A (\lambda (a6: A).(\lambda (_: A).(leq g a6 a1))) (\lambda (_: A).(\lambda -(a7: A).(leq g a7 a2))) (\lambda (a6: A).(\lambda (a7: A).(eq A (AHead a0 a4) -(AHead a6 a7)))) a0 a4 H12 H10 (refl_equal A (AHead a0 a4))))))))) -H6))))))))))) a y H0))) H))))). -(* COMMENTS -Initial nodes: 797 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/leq/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/leq/props.ma deleted file mode 100644 index b83fc503e..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/leq/props.ma +++ /dev/null @@ -1,233 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/leq/fwd.ma". - -include "Basic-1/aplus/props.ma". - -theorem ahead_inj_snd: - \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a3: A).(\forall -(a4: A).((leq g (AHead a1 a2) (AHead a3 a4)) \to (leq g a2 a4)))))) -\def - \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a3: A).(\lambda -(a4: A).(\lambda (H: (leq g (AHead a1 a2) (AHead a3 a4))).(let H_x \def -(leq_gen_head1 g a1 a2 (AHead a3 a4) H) in (let H0 \def H_x in (ex3_2_ind A A -(\lambda (a5: A).(\lambda (_: A).(leq g a1 a5))) (\lambda (_: A).(\lambda -(a6: A).(leq g a2 a6))) (\lambda (a5: A).(\lambda (a6: A).(eq A (AHead a3 a4) -(AHead a5 a6)))) (leq g a2 a4) (\lambda (x0: A).(\lambda (x1: A).(\lambda -(H1: (leq g a1 x0)).(\lambda (H2: (leq g a2 x1)).(\lambda (H3: (eq A (AHead -a3 a4) (AHead x0 x1))).(let H4 \def (f_equal A A (\lambda (e: A).(match e in -A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a3 | (AHead a _) -\Rightarrow a])) (AHead a3 a4) (AHead x0 x1) H3) in ((let H5 \def (f_equal A -A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) -\Rightarrow a4 | (AHead _ a) \Rightarrow a])) (AHead a3 a4) (AHead x0 x1) H3) -in (\lambda (H6: (eq A a3 x0)).(let H7 \def (eq_ind_r A x1 (\lambda (a: -A).(leq g a2 a)) H2 a4 H5) in (let H8 \def (eq_ind_r A x0 (\lambda (a: -A).(leq g a1 a)) H1 a3 H6) in H7)))) H4))))))) H0)))))))). -(* COMMENTS -Initial nodes: 259 -END *) - -theorem leq_refl: - \forall (g: G).(\forall (a: A).(leq g a a)) -\def - \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(leq g a0 a0)) -(\lambda (n: nat).(\lambda (n0: nat).(leq_sort g n n n0 n0 O (refl_equal A -(aplus g (ASort n n0) O))))) (\lambda (a0: A).(\lambda (H: (leq g a0 -a0)).(\lambda (a1: A).(\lambda (H0: (leq g a1 a1)).(leq_head g a0 a0 H a1 a1 -H0))))) a)). -(* COMMENTS -Initial nodes: 87 -END *) - -theorem leq_eq: - \forall (g: G).(\forall (a1: A).(\forall (a2: A).((eq A a1 a2) \to (leq g a1 -a2)))) -\def - \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (eq A a1 -a2)).(eq_ind A a1 (\lambda (a: A).(leq g a1 a)) (leq_refl g a1) a2 H)))). -(* COMMENTS -Initial nodes: 39 -END *) - -theorem leq_sym: - \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (leq g -a2 a1)))) -\def - \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1 -a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(leq g a0 a))) (\lambda (h1: -nat).(\lambda (h2: nat).(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (k: -nat).(\lambda (H0: (eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) -k))).(leq_sort g h2 h1 n2 n1 k (sym_eq A (aplus g (ASort h1 n1) k) (aplus g -(ASort h2 n2) k) H0)))))))) (\lambda (a3: A).(\lambda (a4: A).(\lambda (_: -(leq g a3 a4)).(\lambda (H1: (leq g a4 a3)).(\lambda (a5: A).(\lambda (a6: -A).(\lambda (_: (leq g a5 a6)).(\lambda (H3: (leq g a6 a5)).(leq_head g a4 a3 -H1 a6 a5 H3))))))))) a1 a2 H)))). -(* COMMENTS -Initial nodes: 173 -END *) - -theorem leq_trans: - \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall -(a3: A).((leq g a2 a3) \to (leq g a1 a3)))))) -\def - \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1 -a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(\forall (a3: A).((leq g a0 -a3) \to (leq g a a3))))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: -nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (H0: (eq A (aplus g (ASort -h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (a3: A).(\lambda (H1: (leq g -(ASort h2 n2) a3)).(let H_x \def (leq_gen_sort1 g h2 n2 a3 H1) in (let H2 -\def H_x in (ex2_3_ind nat nat nat (\lambda (n3: nat).(\lambda (h3: -nat).(\lambda (k0: nat).(eq A (aplus g (ASort h2 n2) k0) (aplus g (ASort h3 -n3) k0))))) (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A a3 -(ASort h3 n3))))) (leq g (ASort h1 n1) a3) (\lambda (x0: nat).(\lambda (x1: -nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort h2 n2) x2) (aplus -g (ASort x1 x0) x2))).(\lambda (H4: (eq A a3 (ASort x1 x0))).(let H5 \def -(f_equal A A (\lambda (e: A).e) a3 (ASort x1 x0) H4) in (eq_ind_r A (ASort x1 -x0) (\lambda (a: A).(leq g (ASort h1 n1) a)) (lt_le_e k x2 (leq g (ASort h1 -n1) (ASort x1 x0)) (\lambda (H6: (lt k x2)).(let H_y \def (aplus_reg_r g -(ASort h1 n1) (ASort h2 n2) k k H0 (minus x2 k)) in (let H7 \def (eq_ind_r -nat (plus (minus x2 k) k) (\lambda (n: nat).(eq A (aplus g (ASort h1 n1) n) -(aplus g (ASort h2 n2) n))) H_y x2 (le_plus_minus_sym k x2 (le_trans k (S k) -x2 (le_S k k (le_n k)) H6))) in (leq_sort g h1 x1 n1 x0 x2 (trans_eq A (aplus -g (ASort h1 n1) x2) (aplus g (ASort h2 n2) x2) (aplus g (ASort x1 x0) x2) H7 -H3))))) (\lambda (H6: (le x2 k)).(let H_y \def (aplus_reg_r g (ASort h2 n2) -(ASort x1 x0) x2 x2 H3 (minus k x2)) in (let H7 \def (eq_ind_r nat (plus -(minus k x2) x2) (\lambda (n: nat).(eq A (aplus g (ASort h2 n2) n) (aplus g -(ASort x1 x0) n))) H_y k (le_plus_minus_sym x2 k H6)) in (leq_sort g h1 x1 n1 -x0 k (trans_eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k) (aplus g -(ASort x1 x0) k) H0 H7)))))) a3 H5))))))) H2))))))))))) (\lambda (a3: -A).(\lambda (a4: A).(\lambda (_: (leq g a3 a4)).(\lambda (H1: ((\forall (a5: -A).((leq g a4 a5) \to (leq g a3 a5))))).(\lambda (a5: A).(\lambda (a6: -A).(\lambda (_: (leq g a5 a6)).(\lambda (H3: ((\forall (a7: A).((leq g a6 a7) -\to (leq g a5 a7))))).(\lambda (a0: A).(\lambda (H4: (leq g (AHead a4 a6) -a0)).(let H_x \def (leq_gen_head1 g a4 a6 a0 H4) in (let H5 \def H_x in -(ex3_2_ind A A (\lambda (a7: A).(\lambda (_: A).(leq g a4 a7))) (\lambda (_: -A).(\lambda (a8: A).(leq g a6 a8))) (\lambda (a7: A).(\lambda (a8: A).(eq A -a0 (AHead a7 a8)))) (leq g (AHead a3 a5) a0) (\lambda (x0: A).(\lambda (x1: -A).(\lambda (H6: (leq g a4 x0)).(\lambda (H7: (leq g a6 x1)).(\lambda (H8: -(eq A a0 (AHead x0 x1))).(let H9 \def (f_equal A A (\lambda (e: A).e) a0 -(AHead x0 x1) H8) in (eq_ind_r A (AHead x0 x1) (\lambda (a: A).(leq g (AHead -a3 a5) a)) (leq_head g a3 x0 (H1 x0 H6) a5 x1 (H3 x1 H7)) a0 H9))))))) -H5))))))))))))) a1 a2 H)))). -(* COMMENTS -Initial nodes: 869 -END *) - -theorem leq_ahead_false_1: - \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (AHead a1 a2) a1) -\to (\forall (P: Prop).P)))) -\def - \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2: -A).((leq g (AHead a a2) a) \to (\forall (P: Prop).P)))) (\lambda (n: -nat).(\lambda (n0: nat).(\lambda (a2: A).(\lambda (H: (leq g (AHead (ASort n -n0) a2) (ASort n n0))).(\lambda (P: Prop).(nat_ind (\lambda (n1: nat).((leq g -(AHead (ASort n1 n0) a2) (ASort n1 n0)) \to P)) (\lambda (H0: (leq g (AHead -(ASort O n0) a2) (ASort O n0))).(let H_x \def (leq_gen_head1 g (ASort O n0) -a2 (ASort O n0) H0) in (let H1 \def H_x in (ex3_2_ind A A (\lambda (a3: -A).(\lambda (_: A).(leq g (ASort O n0) a3))) (\lambda (_: A).(\lambda (a4: -A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort O n0) -(AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g -(ASort O n0) x0)).(\lambda (_: (leq g a2 x1)).(\lambda (H4: (eq A (ASort O -n0) (AHead x0 x1))).(let H5 \def (eq_ind A (ASort O n0) (\lambda (ee: -A).(match ee in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow -True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H4) in (False_ind P -H5))))))) H1)))) (\lambda (n1: nat).(\lambda (_: (((leq g (AHead (ASort n1 -n0) a2) (ASort n1 n0)) \to P))).(\lambda (H0: (leq g (AHead (ASort (S n1) n0) -a2) (ASort (S n1) n0))).(let H_x \def (leq_gen_head1 g (ASort (S n1) n0) a2 -(ASort (S n1) n0) H0) in (let H1 \def H_x in (ex3_2_ind A A (\lambda (a3: -A).(\lambda (_: A).(leq g (ASort (S n1) n0) a3))) (\lambda (_: A).(\lambda -(a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort (S n1) -n0) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g -(ASort (S n1) n0) x0)).(\lambda (_: (leq g a2 x1)).(\lambda (H4: (eq A (ASort -(S n1) n0) (AHead x0 x1))).(let H5 \def (eq_ind A (ASort (S n1) n0) (\lambda -(ee: A).(match ee in A return (\lambda (_: A).Prop) with [(ASort _ _) -\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H4) in -(False_ind P H5))))))) H1)))))) n H)))))) (\lambda (a: A).(\lambda (H: -((\forall (a2: A).((leq g (AHead a a2) a) \to (\forall (P: -Prop).P))))).(\lambda (a0: A).(\lambda (_: ((\forall (a2: A).((leq g (AHead -a0 a2) a0) \to (\forall (P: Prop).P))))).(\lambda (a2: A).(\lambda (H1: (leq -g (AHead (AHead a a0) a2) (AHead a a0))).(\lambda (P: Prop).(let H_x \def -(leq_gen_head1 g (AHead a a0) a2 (AHead a a0) H1) in (let H2 \def H_x in -(ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq g (AHead a a0) a3))) -(\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda -(a4: A).(eq A (AHead a a0) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: -A).(\lambda (H3: (leq g (AHead a a0) x0)).(\lambda (H4: (leq g a2 -x1)).(\lambda (H5: (eq A (AHead a a0) (AHead x0 x1))).(let H6 \def (f_equal A -A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) -\Rightarrow a | (AHead a3 _) \Rightarrow a3])) (AHead a a0) (AHead x0 x1) H5) -in ((let H7 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda -(_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead _ a3) \Rightarrow a3])) -(AHead a a0) (AHead x0 x1) H5) in (\lambda (H8: (eq A a x0)).(let H9 \def -(eq_ind_r A x1 (\lambda (a3: A).(leq g a2 a3)) H4 a0 H7) in (let H10 \def -(eq_ind_r A x0 (\lambda (a3: A).(leq g (AHead a a0) a3)) H3 a H8) in (H a0 -H10 P))))) H6))))))) H2)))))))))) a1)). -(* COMMENTS -Initial nodes: 797 -END *) - -theorem leq_ahead_false_2: - \forall (g: G).(\forall (a2: A).(\forall (a1: A).((leq g (AHead a1 a2) a2) -\to (\forall (P: Prop).P)))) -\def - \lambda (g: G).(\lambda (a2: A).(A_ind (\lambda (a: A).(\forall (a1: -A).((leq g (AHead a1 a) a) \to (\forall (P: Prop).P)))) (\lambda (n: -nat).(\lambda (n0: nat).(\lambda (a1: A).(\lambda (H: (leq g (AHead a1 (ASort -n n0)) (ASort n n0))).(\lambda (P: Prop).(nat_ind (\lambda (n1: nat).((leq g -(AHead a1 (ASort n1 n0)) (ASort n1 n0)) \to P)) (\lambda (H0: (leq g (AHead -a1 (ASort O n0)) (ASort O n0))).(let H_x \def (leq_gen_head1 g a1 (ASort O -n0) (ASort O n0) H0) in (let H1 \def H_x in (ex3_2_ind A A (\lambda (a3: -A).(\lambda (_: A).(leq g a1 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g -(ASort O n0) a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort O n0) -(AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g a1 -x0)).(\lambda (_: (leq g (ASort O n0) x1)).(\lambda (H4: (eq A (ASort O n0) -(AHead x0 x1))).(let H5 \def (eq_ind A (ASort O n0) (\lambda (ee: A).(match -ee in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | -(AHead _ _) \Rightarrow False])) I (AHead x0 x1) H4) in (False_ind P -H5))))))) H1)))) (\lambda (n1: nat).(\lambda (_: (((leq g (AHead a1 (ASort n1 -n0)) (ASort n1 n0)) \to P))).(\lambda (H0: (leq g (AHead a1 (ASort (S n1) -n0)) (ASort (S n1) n0))).(let H_x \def (leq_gen_head1 g a1 (ASort (S n1) n0) -(ASort (S n1) n0) H0) in (let H1 \def H_x in (ex3_2_ind A A (\lambda (a3: -A).(\lambda (_: A).(leq g a1 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g -(ASort (S n1) n0) a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort (S n1) -n0) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g -a1 x0)).(\lambda (_: (leq g (ASort (S n1) n0) x1)).(\lambda (H4: (eq A (ASort -(S n1) n0) (AHead x0 x1))).(let H5 \def (eq_ind A (ASort (S n1) n0) (\lambda -(ee: A).(match ee in A return (\lambda (_: A).Prop) with [(ASort _ _) -\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H4) in -(False_ind P H5))))))) H1)))))) n H)))))) (\lambda (a: A).(\lambda (_: -((\forall (a1: A).((leq g (AHead a1 a) a) \to (\forall (P: -Prop).P))))).(\lambda (a0: A).(\lambda (H0: ((\forall (a1: A).((leq g (AHead -a1 a0) a0) \to (\forall (P: Prop).P))))).(\lambda (a1: A).(\lambda (H1: (leq -g (AHead a1 (AHead a a0)) (AHead a a0))).(\lambda (P: Prop).(let H_x \def -(leq_gen_head1 g a1 (AHead a a0) (AHead a a0) H1) in (let H2 \def H_x in -(ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq g a1 a3))) (\lambda (_: -A).(\lambda (a4: A).(leq g (AHead a a0) a4))) (\lambda (a3: A).(\lambda (a4: -A).(eq A (AHead a a0) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: -A).(\lambda (H3: (leq g a1 x0)).(\lambda (H4: (leq g (AHead a a0) -x1)).(\lambda (H5: (eq A (AHead a a0) (AHead x0 x1))).(let H6 \def (f_equal A -A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) -\Rightarrow a | (AHead a3 _) \Rightarrow a3])) (AHead a a0) (AHead x0 x1) H5) -in ((let H7 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda -(_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead _ a3) \Rightarrow a3])) -(AHead a a0) (AHead x0 x1) H5) in (\lambda (H8: (eq A a x0)).(let H9 \def -(eq_ind_r A x1 (\lambda (a3: A).(leq g (AHead a a0) a3)) H4 a0 H7) in (let -H10 \def (eq_ind_r A x0 (\lambda (a3: A).(leq g a1 a3)) H3 a H8) in (H0 a H9 -P))))) H6))))))) H2)))))))))) a2)). -(* COMMENTS -Initial nodes: 797 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/lift/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/lift/defs.ma deleted file mode 100644 index 046506672..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/lift/defs.ma +++ /dev/null @@ -1,42 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/tlist/defs.ma". - -include "Basic-1/s/defs.ma". - -definition lref_map: - ((nat \to nat)) \to (nat \to (T \to T)) -\def - let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t -with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match -(blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u -t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in -lref_map. - -definition lift: - nat \to (nat \to (T \to T)) -\def - \lambda (h: nat).(\lambda (i: nat).(\lambda (t: T).(lref_map (\lambda (x: -nat).(plus x h)) i t))). - -definition lifts: - nat \to (nat \to (TList \to TList)) -\def - let rec lifts (h: nat) (d: nat) (ts: TList) on ts: TList \def (match ts with -[TNil \Rightarrow TNil | (TCons t ts0) \Rightarrow (TCons (lift h d t) (lifts -h d ts0))]) in lifts. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/lift/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/lift/fwd.ma deleted file mode 100644 index 324fed2fb..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/lift/fwd.ma +++ /dev/null @@ -1,434 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/lift/defs.ma". - -theorem lift_sort: - \forall (n: nat).(\forall (h: nat).(\forall (d: nat).(eq T (lift h d (TSort -n)) (TSort n)))) -\def - \lambda (n: nat).(\lambda (_: nat).(\lambda (_: nat).(refl_equal T (TSort -n)))). -(* COMMENTS -Initial nodes: 13 -END *) - -theorem lift_lref_lt: - \forall (n: nat).(\forall (h: nat).(\forall (d: nat).((lt n d) \to (eq T -(lift h d (TLRef n)) (TLRef n))))) -\def - \lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (lt n -d)).(eq_ind bool true (\lambda (b: bool).(eq T (TLRef (match b with [true -\Rightarrow n | false \Rightarrow (plus n h)])) (TLRef n))) (refl_equal T -(TLRef n)) (blt n d) (sym_eq bool (blt n d) true (lt_blt d n H)))))). -(* COMMENTS -Initial nodes: 72 -END *) - -theorem lift_lref_ge: - \forall (n: nat).(\forall (h: nat).(\forall (d: nat).((le d n) \to (eq T -(lift h d (TLRef n)) (TLRef (plus n h)))))) -\def - \lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (le d -n)).(eq_ind bool false (\lambda (b: bool).(eq T (TLRef (match b with [true -\Rightarrow n | false \Rightarrow (plus n h)])) (TLRef (plus n h)))) -(refl_equal T (TLRef (plus n h))) (blt n d) (sym_eq bool (blt n d) false -(le_bge d n H)))))). -(* COMMENTS -Initial nodes: 80 -END *) - -theorem lift_head: - \forall (k: K).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall -(d: nat).(eq T (lift h d (THead k u t)) (THead k (lift h d u) (lift h (s k d) -t))))))) -\def - \lambda (k: K).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda -(d: nat).(refl_equal T (THead k (lift h d u) (lift h (s k d) t))))))). -(* COMMENTS -Initial nodes: 37 -END *) - -theorem lift_bind: - \forall (b: B).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall -(d: nat).(eq T (lift h d (THead (Bind b) u t)) (THead (Bind b) (lift h d u) -(lift h (S d) t))))))) -\def - \lambda (b: B).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda -(d: nat).(refl_equal T (THead (Bind b) (lift h d u) (lift h (S d) t))))))). -(* COMMENTS -Initial nodes: 37 -END *) - -theorem lift_flat: - \forall (f: F).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall -(d: nat).(eq T (lift h d (THead (Flat f) u t)) (THead (Flat f) (lift h d u) -(lift h d t))))))) -\def - \lambda (f: F).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda -(d: nat).(refl_equal T (THead (Flat f) (lift h d u) (lift h d t))))))). -(* COMMENTS -Initial nodes: 35 -END *) - -theorem lift_gen_sort: - \forall (h: nat).(\forall (d: nat).(\forall (n: nat).(\forall (t: T).((eq T -(TSort n) (lift h d t)) \to (eq T t (TSort n)))))) -\def - \lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (t: T).(T_ind -(\lambda (t0: T).((eq T (TSort n) (lift h d t0)) \to (eq T t0 (TSort n)))) -(\lambda (n0: nat).(\lambda (H: (eq T (TSort n) (lift h d (TSort -n0)))).(sym_eq T (TSort n) (TSort n0) H))) (\lambda (n0: nat).(\lambda (H: -(eq T (TSort n) (lift h d (TLRef n0)))).(lt_le_e n0 d (eq T (TLRef n0) (TSort -n)) (\lambda (_: (lt n0 d)).(let H1 \def (eq_ind T (lift h d (TLRef n0)) -(\lambda (t0: T).(eq T (TSort n) t0)) H (TLRef n0) (lift_lref_lt n0 h d (let -H1 \def (eq_ind T (TSort n) (\lambda (ee: T).(match ee in T return (\lambda -(_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | -(THead _ _ _) \Rightarrow False])) I (lift h d (TLRef n0)) H) in (False_ind -(lt n0 d) H1)))) in (let H2 \def (eq_ind T (TSort n) (\lambda (ee: T).(match -ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | -(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef n0) -H1) in (False_ind (eq T (TLRef n0) (TSort n)) H2)))) (\lambda (_: (le d -n0)).(let H1 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t0: T).(eq T -(TSort n) t0)) H (TLRef (plus n0 h)) (lift_lref_ge n0 h d (let H1 \def -(eq_ind T (TSort n) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | -(THead _ _ _) \Rightarrow False])) I (lift h d (TLRef n0)) H) in (False_ind -(le d n0) H1)))) in (let H2 \def (eq_ind T (TSort n) (\lambda (ee: T).(match -ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | -(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef -(plus n0 h)) H1) in (False_ind (eq T (TLRef n0) (TSort n)) H2))))))) (\lambda -(k: K).(\lambda (t0: T).(\lambda (_: (((eq T (TSort n) (lift h d t0)) \to (eq -T t0 (TSort n))))).(\lambda (t1: T).(\lambda (_: (((eq T (TSort n) (lift h d -t1)) \to (eq T t1 (TSort n))))).(\lambda (H1: (eq T (TSort n) (lift h d -(THead k t0 t1)))).(let H2 \def (eq_ind T (lift h d (THead k t0 t1)) (\lambda -(t2: T).(eq T (TSort n) t2)) H1 (THead k (lift h d t0) (lift h (s k d) t1)) -(lift_head k t0 t1 h d)) in (let H3 \def (eq_ind T (TSort n) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I -(THead k (lift h d t0) (lift h (s k d) t1)) H2) in (False_ind (eq T (THead k -t0 t1) (TSort n)) H3))))))))) t)))). -(* COMMENTS -Initial nodes: 613 -END *) - -theorem lift_gen_lref: - \forall (t: T).(\forall (d: nat).(\forall (h: nat).(\forall (i: nat).((eq T -(TLRef i) (lift h d t)) \to (or (land (lt i d) (eq T t (TLRef i))) (land (le -(plus d h) i) (eq T t (TLRef (minus i h))))))))) -\def - \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d: nat).(\forall (h: -nat).(\forall (i: nat).((eq T (TLRef i) (lift h d t0)) \to (or (land (lt i d) -(eq T t0 (TLRef i))) (land (le (plus d h) i) (eq T t0 (TLRef (minus i -h)))))))))) (\lambda (n: nat).(\lambda (d: nat).(\lambda (h: nat).(\lambda -(i: nat).(\lambda (H: (eq T (TLRef i) (lift h d (TSort n)))).(let H0 \def -(eq_ind T (lift h d (TSort n)) (\lambda (t0: T).(eq T (TLRef i) t0)) H (TSort -n) (lift_sort n h d)) in (let H1 \def (eq_ind T (TLRef i) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I -(TSort n) H0) in (False_ind (or (land (lt i d) (eq T (TSort n) (TLRef i))) -(land (le (plus d h) i) (eq T (TSort n) (TLRef (minus i h))))) H1)))))))) -(\lambda (n: nat).(\lambda (d: nat).(\lambda (h: nat).(\lambda (i: -nat).(\lambda (H: (eq T (TLRef i) (lift h d (TLRef n)))).(lt_le_e n d (or -(land (lt i d) (eq T (TLRef n) (TLRef i))) (land (le (plus d h) i) (eq T -(TLRef n) (TLRef (minus i h))))) (\lambda (H0: (lt n d)).(let H1 \def (eq_ind -T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (TLRef i) t0)) H (TLRef n) -(lift_lref_lt n h d H0)) in (let H2 \def (f_equal T nat (\lambda (e: -T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow i | -(TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow i])) (TLRef i) (TLRef -n) H1) in (eq_ind_r nat n (\lambda (n0: nat).(or (land (lt n0 d) (eq T (TLRef -n) (TLRef n0))) (land (le (plus d h) n0) (eq T (TLRef n) (TLRef (minus n0 -h)))))) (or_introl (land (lt n d) (eq T (TLRef n) (TLRef n))) (land (le (plus -d h) n) (eq T (TLRef n) (TLRef (minus n h)))) (conj (lt n d) (eq T (TLRef n) -(TLRef n)) H0 (refl_equal T (TLRef n)))) i H2)))) (\lambda (H0: (le d -n)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (TLRef -i) t0)) H (TLRef (plus n h)) (lift_lref_ge n h d H0)) in (let H2 \def -(f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with -[(TSort _) \Rightarrow i | (TLRef n0) \Rightarrow n0 | (THead _ _ _) -\Rightarrow i])) (TLRef i) (TLRef (plus n h)) H1) in (eq_ind_r nat (plus n h) -(\lambda (n0: nat).(or (land (lt n0 d) (eq T (TLRef n) (TLRef n0))) (land (le -(plus d h) n0) (eq T (TLRef n) (TLRef (minus n0 h)))))) (eq_ind_r nat n -(\lambda (n0: nat).(or (land (lt (plus n h) d) (eq T (TLRef n) (TLRef (plus n -h)))) (land (le (plus d h) (plus n h)) (eq T (TLRef n) (TLRef n0))))) -(or_intror (land (lt (plus n h) d) (eq T (TLRef n) (TLRef (plus n h)))) (land -(le (plus d h) (plus n h)) (eq T (TLRef n) (TLRef n))) (conj (le (plus d h) -(plus n h)) (eq T (TLRef n) (TLRef n)) (le_plus_plus d n h h H0 (le_n h)) -(refl_equal T (TLRef n)))) (minus (plus n h) h) (minus_plus_r n h)) i -H2)))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (_: ((\forall (d: -nat).(\forall (h: nat).(\forall (i: nat).((eq T (TLRef i) (lift h d t0)) \to -(or (land (lt i d) (eq T t0 (TLRef i))) (land (le (plus d h) i) (eq T t0 -(TLRef (minus i h))))))))))).(\lambda (t1: T).(\lambda (_: ((\forall (d: -nat).(\forall (h: nat).(\forall (i: nat).((eq T (TLRef i) (lift h d t1)) \to -(or (land (lt i d) (eq T t1 (TLRef i))) (land (le (plus d h) i) (eq T t1 -(TLRef (minus i h))))))))))).(\lambda (d: nat).(\lambda (h: nat).(\lambda (i: -nat).(\lambda (H1: (eq T (TLRef i) (lift h d (THead k t0 t1)))).(let H2 \def -(eq_ind T (lift h d (THead k t0 t1)) (\lambda (t2: T).(eq T (TLRef i) t2)) H1 -(THead k (lift h d t0) (lift h (s k d) t1)) (lift_head k t0 t1 h d)) in (let -H3 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda -(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | -(THead _ _ _) \Rightarrow False])) I (THead k (lift h d t0) (lift h (s k d) -t1)) H2) in (False_ind (or (land (lt i d) (eq T (THead k t0 t1) (TLRef i))) -(land (le (plus d h) i) (eq T (THead k t0 t1) (TLRef (minus i h))))) -H3)))))))))))) t). -(* COMMENTS -Initial nodes: 1221 -END *) - -theorem lift_gen_lref_lt: - \forall (h: nat).(\forall (d: nat).(\forall (n: nat).((lt n d) \to (\forall -(t: T).((eq T (TLRef n) (lift h d t)) \to (eq T t (TLRef n))))))) -\def - \lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (H: (lt n -d)).(\lambda (t: T).(\lambda (H0: (eq T (TLRef n) (lift h d t))).(let H_x -\def (lift_gen_lref t d h n H0) in (let H1 \def H_x in (or_ind (land (lt n d) -(eq T t (TLRef n))) (land (le (plus d h) n) (eq T t (TLRef (minus n h)))) (eq -T t (TLRef n)) (\lambda (H2: (land (lt n d) (eq T t (TLRef n)))).(land_ind -(lt n d) (eq T t (TLRef n)) (eq T t (TLRef n)) (\lambda (_: (lt n -d)).(\lambda (H4: (eq T t (TLRef n))).(eq_ind_r T (TLRef n) (\lambda (t0: -T).(eq T t0 (TLRef n))) (refl_equal T (TLRef n)) t H4))) H2)) (\lambda (H2: -(land (le (plus d h) n) (eq T t (TLRef (minus n h))))).(land_ind (le (plus d -h) n) (eq T t (TLRef (minus n h))) (eq T t (TLRef n)) (\lambda (H3: (le (plus -d h) n)).(\lambda (H4: (eq T t (TLRef (minus n h)))).(eq_ind_r T (TLRef -(minus n h)) (\lambda (t0: T).(eq T t0 (TLRef n))) (le_false (plus d h) n (eq -T (TLRef (minus n h)) (TLRef n)) H3 (lt_le_S n (plus d h) (le_plus_trans (S -n) d h H))) t H4))) H2)) H1)))))))). -(* COMMENTS -Initial nodes: 363 -END *) - -theorem lift_gen_lref_false: - \forall (h: nat).(\forall (d: nat).(\forall (n: nat).((le d n) \to ((lt n -(plus d h)) \to (\forall (t: T).((eq T (TLRef n) (lift h d t)) \to (\forall -(P: Prop).P))))))) -\def - \lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (H: (le d -n)).(\lambda (H0: (lt n (plus d h))).(\lambda (t: T).(\lambda (H1: (eq T -(TLRef n) (lift h d t))).(\lambda (P: Prop).(let H_x \def (lift_gen_lref t d -h n H1) in (let H2 \def H_x in (or_ind (land (lt n d) (eq T t (TLRef n))) -(land (le (plus d h) n) (eq T t (TLRef (minus n h)))) P (\lambda (H3: (land -(lt n d) (eq T t (TLRef n)))).(land_ind (lt n d) (eq T t (TLRef n)) P -(\lambda (H4: (lt n d)).(\lambda (_: (eq T t (TLRef n))).(le_false d n P H -H4))) H3)) (\lambda (H3: (land (le (plus d h) n) (eq T t (TLRef (minus n -h))))).(land_ind (le (plus d h) n) (eq T t (TLRef (minus n h))) P (\lambda -(H4: (le (plus d h) n)).(\lambda (_: (eq T t (TLRef (minus n h)))).(le_false -(plus d h) n P H4 H0))) H3)) H2)))))))))). -(* COMMENTS -Initial nodes: 269 -END *) - -theorem lift_gen_lref_ge: - \forall (h: nat).(\forall (d: nat).(\forall (n: nat).((le d n) \to (\forall -(t: T).((eq T (TLRef (plus n h)) (lift h d t)) \to (eq T t (TLRef n))))))) -\def - \lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (H: (le d -n)).(\lambda (t: T).(\lambda (H0: (eq T (TLRef (plus n h)) (lift h d -t))).(let H_x \def (lift_gen_lref t d h (plus n h) H0) in (let H1 \def H_x in -(or_ind (land (lt (plus n h) d) (eq T t (TLRef (plus n h)))) (land (le (plus -d h) (plus n h)) (eq T t (TLRef (minus (plus n h) h)))) (eq T t (TLRef n)) -(\lambda (H2: (land (lt (plus n h) d) (eq T t (TLRef (plus n h))))).(land_ind -(lt (plus n h) d) (eq T t (TLRef (plus n h))) (eq T t (TLRef n)) (\lambda -(H3: (lt (plus n h) d)).(\lambda (H4: (eq T t (TLRef (plus n h)))).(eq_ind_r -T (TLRef (plus n h)) (\lambda (t0: T).(eq T t0 (TLRef n))) (le_false d n (eq -T (TLRef (plus n h)) (TLRef n)) H (lt_le_S n d (simpl_lt_plus_r h n d -(lt_le_trans (plus n h) d (plus d h) H3 (le_plus_l d h))))) t H4))) H2)) -(\lambda (H2: (land (le (plus d h) (plus n h)) (eq T t (TLRef (minus (plus n -h) h))))).(land_ind (le (plus d h) (plus n h)) (eq T t (TLRef (minus (plus n -h) h))) (eq T t (TLRef n)) (\lambda (_: (le (plus d h) (plus n h))).(\lambda -(H4: (eq T t (TLRef (minus (plus n h) h)))).(eq_ind_r T (TLRef (minus (plus n -h) h)) (\lambda (t0: T).(eq T t0 (TLRef n))) (f_equal nat T TLRef (minus -(plus n h) h) n (minus_plus_r n h)) t H4))) H2)) H1)))))))). -(* COMMENTS -Initial nodes: 473 -END *) - -theorem lift_gen_head: - \forall (k: K).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h: -nat).(\forall (d: nat).((eq T (THead k u t) (lift h d x)) \to (ex3_2 T T -(\lambda (y: T).(\lambda (z: T).(eq T x (THead k y z)))) (\lambda (y: -T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: -T).(eq T t (lift h (s k d) z))))))))))) -\def - \lambda (k: K).(\lambda (u: T).(\lambda (t: T).(\lambda (x: T).(T_ind -(\lambda (t0: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k u t) -(lift h d t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead -k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda -(_: T).(\lambda (z: T).(eq T t (lift h (s k d) z))))))))) (\lambda (n: -nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (eq T (THead k u t) -(lift h d (TSort n)))).(let H0 \def (eq_ind T (lift h d (TSort n)) (\lambda -(t0: T).(eq T (THead k u t) t0)) H (TSort n) (lift_sort n h d)) in (let H1 -\def (eq_ind T (THead k u t) (\lambda (ee: T).(match ee in T return (\lambda -(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False -| (THead _ _ _) \Rightarrow True])) I (TSort n) H0) in (False_ind (ex3_2 T T -(\lambda (y: T).(\lambda (z: T).(eq T (TSort n) (THead k y z)))) (\lambda (y: -T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: -T).(eq T t (lift h (s k d) z))))) H1))))))) (\lambda (n: nat).(\lambda (h: -nat).(\lambda (d: nat).(\lambda (H: (eq T (THead k u t) (lift h d (TLRef -n)))).(lt_le_e n d (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) -(THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) -(\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k d) z))))) (\lambda (H0: -(lt n d)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T -(THead k u t) t0)) H (TLRef n) (lift_lref_lt n h d H0)) in (let H2 \def -(eq_ind T (THead k u t) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead _ _ _) \Rightarrow True])) I (TLRef n) H1) in (False_ind (ex3_2 T T -(\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead k y z)))) (\lambda (y: -T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: -T).(eq T t (lift h (s k d) z))))) H2)))) (\lambda (H0: (le d n)).(let H1 \def -(eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (THead k u t) t0)) H -(TLRef (plus n h)) (lift_lref_ge n h d H0)) in (let H2 \def (eq_ind T (THead -k u t) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) -\Rightarrow True])) I (TLRef (plus n h)) H1) in (False_ind (ex3_2 T T -(\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead k y z)))) (\lambda (y: -T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: -T).(eq T t (lift h (s k d) z))))) H2))))))))) (\lambda (k0: K).(\lambda (t0: -T).(\lambda (H: ((\forall (h: nat).(\forall (d: nat).((eq T (THead k u t) -(lift h d t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead -k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda -(_: T).(\lambda (z: T).(eq T t (lift h (s k d) z)))))))))).(\lambda (t1: -T).(\lambda (H0: ((\forall (h: nat).(\forall (d: nat).((eq T (THead k u t) -(lift h d t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t1 (THead -k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda -(_: T).(\lambda (z: T).(eq T t (lift h (s k d) z)))))))))).(\lambda (h: -nat).(\lambda (d: nat).(\lambda (H1: (eq T (THead k u t) (lift h d (THead k0 -t0 t1)))).(let H2 \def (eq_ind T (lift h d (THead k0 t0 t1)) (\lambda (t2: -T).(eq T (THead k u t) t2)) H1 (THead k0 (lift h d t0) (lift h (s k0 d) t1)) -(lift_head k0 t0 t1 h d)) in (let H3 \def (f_equal T K (\lambda (e: T).(match -e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) -\Rightarrow k | (THead k1 _ _) \Rightarrow k1])) (THead k u t) (THead k0 -(lift h d t0) (lift h (s k0 d) t1)) H2) in ((let H4 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t2 _) \Rightarrow t2])) -(THead k u t) (THead k0 (lift h d t0) (lift h (s k0 d) t1)) H2) in ((let H5 -\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t2) -\Rightarrow t2])) (THead k u t) (THead k0 (lift h d t0) (lift h (s k0 d) t1)) -H2) in (\lambda (H6: (eq T u (lift h d t0))).(\lambda (H7: (eq K k k0)).(let -H8 \def (eq_ind_r K k0 (\lambda (k1: K).(eq T t (lift h (s k1 d) t1))) H5 k -H7) in (eq_ind K k (\lambda (k1: K).(ex3_2 T T (\lambda (y: T).(\lambda (z: -T).(eq T (THead k1 t0 t1) (THead k y z)))) (\lambda (y: T).(\lambda (_: -T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s -k d) z)))))) (let H9 \def (eq_ind T t (\lambda (t2: T).(\forall (h0: -nat).(\forall (d0: nat).((eq T (THead k u t2) (lift h0 d0 t1)) \to (ex3_2 T T -(\lambda (y: T).(\lambda (z: T).(eq T t1 (THead k y z)))) (\lambda (y: -T).(\lambda (_: T).(eq T u (lift h0 d0 y)))) (\lambda (_: T).(\lambda (z: -T).(eq T t2 (lift h0 (s k d0) z))))))))) H0 (lift h (s k d) t1) H8) in (let -H10 \def (eq_ind T t (\lambda (t2: T).(\forall (h0: nat).(\forall (d0: -nat).((eq T (THead k u t2) (lift h0 d0 t0)) \to (ex3_2 T T (\lambda (y: -T).(\lambda (z: T).(eq T t0 (THead k y z)))) (\lambda (y: T).(\lambda (_: -T).(eq T u (lift h0 d0 y)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift -h0 (s k d0) z))))))))) H (lift h (s k d) t1) H8) in (eq_ind_r T (lift h (s k -d) t1) (\lambda (t2: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T -(THead k t0 t1) (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u -(lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h (s k d) -z)))))) (let H11 \def (eq_ind T u (\lambda (t2: T).(\forall (h0: -nat).(\forall (d0: nat).((eq T (THead k t2 (lift h (s k d) t1)) (lift h0 d0 -t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead k y z)))) -(\lambda (y: T).(\lambda (_: T).(eq T t2 (lift h0 d0 y)))) (\lambda (_: -T).(\lambda (z: T).(eq T (lift h (s k d) t1) (lift h0 (s k d0) z))))))))) H10 -(lift h d t0) H6) in (let H12 \def (eq_ind T u (\lambda (t2: T).(\forall (h0: -nat).(\forall (d0: nat).((eq T (THead k t2 (lift h (s k d) t1)) (lift h0 d0 -t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t1 (THead k y z)))) -(\lambda (y: T).(\lambda (_: T).(eq T t2 (lift h0 d0 y)))) (\lambda (_: -T).(\lambda (z: T).(eq T (lift h (s k d) t1) (lift h0 (s k d0) z))))))))) H9 -(lift h d t0) H6) in (eq_ind_r T (lift h d t0) (\lambda (t2: T).(ex3_2 T T -(\lambda (y: T).(\lambda (z: T).(eq T (THead k t0 t1) (THead k y z)))) -(\lambda (y: T).(\lambda (_: T).(eq T t2 (lift h d y)))) (\lambda (_: -T).(\lambda (z: T).(eq T (lift h (s k d) t1) (lift h (s k d) z)))))) -(ex3_2_intro T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k t0 t1) (THead -k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t0) (lift h d y)))) -(\lambda (_: T).(\lambda (z: T).(eq T (lift h (s k d) t1) (lift h (s k d) -z)))) t0 t1 (refl_equal T (THead k t0 t1)) (refl_equal T (lift h d t0)) -(refl_equal T (lift h (s k d) t1))) u H6))) t H8))) k0 H7))))) H4)) -H3))))))))))) x)))). -(* COMMENTS -Initial nodes: 2083 -END *) - -theorem lift_gen_bind: - \forall (b: B).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h: -nat).(\forall (d: nat).((eq T (THead (Bind b) u t) (lift h d x)) \to (ex3_2 T -T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Bind b) y z)))) (\lambda -(y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: -T).(eq T t (lift h (S d) z))))))))))) -\def - \lambda (b: B).(\lambda (u: T).(\lambda (t: T).(\lambda (x: T).(\lambda (h: -nat).(\lambda (d: nat).(\lambda (H: (eq T (THead (Bind b) u t) (lift h d -x))).(let H_x \def (lift_gen_head (Bind b) u t x h d H) in (let H0 \def H_x -in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Bind b) y -z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: -T).(\lambda (z: T).(eq T t (lift h (S d) z)))) (ex3_2 T T (\lambda (y: -T).(\lambda (z: T).(eq T x (THead (Bind b) y z)))) (\lambda (y: T).(\lambda -(_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift -h (S d) z))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H1: (eq T x (THead -(Bind b) x0 x1))).(\lambda (H2: (eq T u (lift h d x0))).(\lambda (H3: (eq T t -(lift h (S d) x1))).(eq_ind_r T (THead (Bind b) x0 x1) (\lambda (t0: -T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead (Bind b) y -z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: -T).(\lambda (z: T).(eq T t (lift h (S d) z)))))) (eq_ind_r T (lift h (S d) -x1) (\lambda (t0: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead -(Bind b) x0 x1) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T -u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t0 (lift h (S d) -z)))))) (eq_ind_r T (lift h d x0) (\lambda (t0: T).(ex3_2 T T (\lambda (y: -T).(\lambda (z: T).(eq T (THead (Bind b) x0 x1) (THead (Bind b) y z)))) -(\lambda (y: T).(\lambda (_: T).(eq T t0 (lift h d y)))) (\lambda (_: -T).(\lambda (z: T).(eq T (lift h (S d) x1) (lift h (S d) z)))))) (ex3_2_intro -T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Bind b) x0 x1) (THead (Bind -b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d x0) (lift h d -y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h (S d) x1) (lift h (S d) -z)))) x0 x1 (refl_equal T (THead (Bind b) x0 x1)) (refl_equal T (lift h d -x0)) (refl_equal T (lift h (S d) x1))) u H2) t H3) x H1)))))) H0))))))))). -(* COMMENTS -Initial nodes: 637 -END *) - -theorem lift_gen_flat: - \forall (f: F).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h: -nat).(\forall (d: nat).((eq T (THead (Flat f) u t) (lift h d x)) \to (ex3_2 T -T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Flat f) y z)))) (\lambda -(y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: -T).(eq T t (lift h d z))))))))))) -\def - \lambda (f: F).(\lambda (u: T).(\lambda (t: T).(\lambda (x: T).(\lambda (h: -nat).(\lambda (d: nat).(\lambda (H: (eq T (THead (Flat f) u t) (lift h d -x))).(let H_x \def (lift_gen_head (Flat f) u t x h d H) in (let H0 \def H_x -in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Flat f) y -z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: -T).(\lambda (z: T).(eq T t (lift h d z)))) (ex3_2 T T (\lambda (y: -T).(\lambda (z: T).(eq T x (THead (Flat f) y z)))) (\lambda (y: T).(\lambda -(_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift -h d z))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H1: (eq T x (THead -(Flat f) x0 x1))).(\lambda (H2: (eq T u (lift h d x0))).(\lambda (H3: (eq T t -(lift h d x1))).(eq_ind_r T (THead (Flat f) x0 x1) (\lambda (t0: T).(ex3_2 T -T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead (Flat f) y z)))) (\lambda -(y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: -T).(eq T t (lift h d z)))))) (eq_ind_r T (lift h d x1) (\lambda (t0: -T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Flat f) x0 x1) -(THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d -y)))) (\lambda (_: T).(\lambda (z: T).(eq T t0 (lift h d z)))))) (eq_ind_r T -(lift h d x0) (\lambda (t0: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq -T (THead (Flat f) x0 x1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: -T).(eq T t0 (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h d -x1) (lift h d z)))))) (ex3_2_intro T T (\lambda (y: T).(\lambda (z: T).(eq T -(THead (Flat f) x0 x1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: -T).(eq T (lift h d x0) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T -(lift h d x1) (lift h d z)))) x0 x1 (refl_equal T (THead (Flat f) x0 x1)) -(refl_equal T (lift h d x0)) (refl_equal T (lift h d x1))) u H2) t H3) x -H1)))))) H0))))))))). -(* COMMENTS -Initial nodes: 615 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/lift/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/lift/props.ma deleted file mode 100644 index f0ed22451..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/lift/props.ma +++ /dev/null @@ -1,592 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/lift/fwd.ma". - -include "Basic-1/s/props.ma". - -theorem thead_x_lift_y_y: - \forall (k: K).(\forall (t: T).(\forall (v: T).(\forall (h: nat).(\forall -(d: nat).((eq T (THead k v (lift h d t)) t) \to (\forall (P: Prop).P)))))) -\def - \lambda (k: K).(\lambda (t: T).(T_ind (\lambda (t0: T).(\forall (v: -T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k v (lift h d t0)) t0) -\to (\forall (P: Prop).P)))))) (\lambda (n: nat).(\lambda (v: T).(\lambda (h: -nat).(\lambda (d: nat).(\lambda (H: (eq T (THead k v (lift h d (TSort n))) -(TSort n))).(\lambda (P: Prop).(let H0 \def (eq_ind T (THead k v (lift h d -(TSort n))) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) -\Rightarrow True])) I (TSort n) H) in (False_ind P H0)))))))) (\lambda (n: -nat).(\lambda (v: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (eq T -(THead k v (lift h d (TLRef n))) (TLRef n))).(\lambda (P: Prop).(let H0 \def -(eq_ind T (THead k v (lift h d (TLRef n))) (\lambda (ee: T).(match ee in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H) in -(False_ind P H0)))))))) (\lambda (k0: K).(\lambda (t0: T).(\lambda (_: -((\forall (v: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k v (lift -h d t0)) t0) \to (\forall (P: Prop).P))))))).(\lambda (t1: T).(\lambda (H0: -((\forall (v: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k v (lift -h d t1)) t1) \to (\forall (P: Prop).P))))))).(\lambda (v: T).(\lambda (h: -nat).(\lambda (d: nat).(\lambda (H1: (eq T (THead k v (lift h d (THead k0 t0 -t1))) (THead k0 t0 t1))).(\lambda (P: Prop).(let H2 \def (f_equal T K -(\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) -\Rightarrow k | (TLRef _) \Rightarrow k | (THead k1 _ _) \Rightarrow k1])) -(THead k v (lift h d (THead k0 t0 t1))) (THead k0 t0 t1) H1) in ((let H3 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow v | (TLRef _) \Rightarrow v | (THead _ t2 _) -\Rightarrow t2])) (THead k v (lift h d (THead k0 t0 t1))) (THead k0 t0 t1) -H1) in ((let H4 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow (THead k0 ((let rec lref_map -(f: ((nat \to nat))) (d0: nat) (t2: T) on t2: T \def (match t2 with [(TSort -n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) -with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k1 u t3) -\Rightarrow (THead k1 (lref_map f d0 u) (lref_map f (s k1 d0) t3))]) in -lref_map) (\lambda (x: nat).(plus x h)) d t0) ((let rec lref_map (f: ((nat -\to nat))) (d0: nat) (t2: T) on t2: T \def (match t2 with [(TSort n) -\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) with -[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k1 u t3) -\Rightarrow (THead k1 (lref_map f d0 u) (lref_map f (s k1 d0) t3))]) in -lref_map) (\lambda (x: nat).(plus x h)) (s k0 d) t1)) | (TLRef _) \Rightarrow -(THead k0 ((let rec lref_map (f: ((nat \to nat))) (d0: nat) (t2: T) on t2: T -\def (match t2 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow -(TLRef (match (blt i d0) with [true \Rightarrow i | false \Rightarrow (f -i)])) | (THead k1 u t3) \Rightarrow (THead k1 (lref_map f d0 u) (lref_map f -(s k1 d0) t3))]) in lref_map) (\lambda (x: nat).(plus x h)) d t0) ((let rec -lref_map (f: ((nat \to nat))) (d0: nat) (t2: T) on t2: T \def (match t2 with -[(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i -d0) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k1 u t3) -\Rightarrow (THead k1 (lref_map f d0 u) (lref_map f (s k1 d0) t3))]) in -lref_map) (\lambda (x: nat).(plus x h)) (s k0 d) t1)) | (THead _ _ t2) -\Rightarrow t2])) (THead k v (lift h d (THead k0 t0 t1))) (THead k0 t0 t1) -H1) in (\lambda (_: (eq T v t0)).(\lambda (H6: (eq K k k0)).(let H7 \def -(eq_ind K k (\lambda (k1: K).(\forall (v0: T).(\forall (h0: nat).(\forall -(d0: nat).((eq T (THead k1 v0 (lift h0 d0 t1)) t1) \to (\forall (P0: -Prop).P0)))))) H0 k0 H6) in (let H8 \def (eq_ind T (lift h d (THead k0 t0 -t1)) (\lambda (t2: T).(eq T t2 t1)) H4 (THead k0 (lift h d t0) (lift h (s k0 -d) t1)) (lift_head k0 t0 t1 h d)) in (H7 (lift h d t0) h (s k0 d) H8 P)))))) -H3)) H2)))))))))))) t)). -(* COMMENTS -Initial nodes: 887 -END *) - -theorem lift_r: - \forall (t: T).(\forall (d: nat).(eq T (lift O d t) t)) -\def - \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d: nat).(eq T (lift O d t0) -t0))) (\lambda (n: nat).(\lambda (_: nat).(refl_equal T (TSort n)))) (\lambda -(n: nat).(\lambda (d: nat).(lt_le_e n d (eq T (lift O d (TLRef n)) (TLRef n)) -(\lambda (H: (lt n d)).(eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 (TLRef -n))) (refl_equal T (TLRef n)) (lift O d (TLRef n)) (lift_lref_lt n O d H))) -(\lambda (H: (le d n)).(eq_ind_r T (TLRef (plus n O)) (\lambda (t0: T).(eq T -t0 (TLRef n))) (f_equal nat T TLRef (plus n O) n (sym_eq nat n (plus n O) -(plus_n_O n))) (lift O d (TLRef n)) (lift_lref_ge n O d H)))))) (\lambda (k: -K).(\lambda (t0: T).(\lambda (H: ((\forall (d: nat).(eq T (lift O d t0) -t0)))).(\lambda (t1: T).(\lambda (H0: ((\forall (d: nat).(eq T (lift O d t1) -t1)))).(\lambda (d: nat).(eq_ind_r T (THead k (lift O d t0) (lift O (s k d) -t1)) (\lambda (t2: T).(eq T t2 (THead k t0 t1))) (f_equal3 K T T T THead k k -(lift O d t0) t0 (lift O (s k d) t1) t1 (refl_equal K k) (H d) (H0 (s k d))) -(lift O d (THead k t0 t1)) (lift_head k t0 t1 O d)))))))) t). -(* COMMENTS -Initial nodes: 367 -END *) - -theorem lift_lref_gt: - \forall (d: nat).(\forall (n: nat).((lt d n) \to (eq T (lift (S O) d (TLRef -(pred n))) (TLRef n)))) -\def - \lambda (d: nat).(\lambda (n: nat).(\lambda (H: (lt d n)).(eq_ind_r T (TLRef -(plus (pred n) (S O))) (\lambda (t: T).(eq T t (TLRef n))) (eq_ind nat (plus -(S O) (pred n)) (\lambda (n0: nat).(eq T (TLRef n0) (TLRef n))) (eq_ind nat n -(\lambda (n0: nat).(eq T (TLRef n0) (TLRef n))) (refl_equal T (TLRef n)) (S -(pred n)) (S_pred n d H)) (plus (pred n) (S O)) (plus_sym (S O) (pred n))) -(lift (S O) d (TLRef (pred n))) (lift_lref_ge (pred n) (S O) d (le_S_n d -(pred n) (eq_ind nat n (\lambda (n0: nat).(le (S d) n0)) H (S (pred n)) -(S_pred n d H))))))). -(* COMMENTS -Initial nodes: 193 -END *) - -theorem lifts_tapp: - \forall (h: nat).(\forall (d: nat).(\forall (v: T).(\forall (vs: TList).(eq -TList (lifts h d (TApp vs v)) (TApp (lifts h d vs) (lift h d v)))))) -\def - \lambda (h: nat).(\lambda (d: nat).(\lambda (v: T).(\lambda (vs: -TList).(TList_ind (\lambda (t: TList).(eq TList (lifts h d (TApp t v)) (TApp -(lifts h d t) (lift h d v)))) (refl_equal TList (TCons (lift h d v) TNil)) -(\lambda (t: T).(\lambda (t0: TList).(\lambda (H: (eq TList (lifts h d (TApp -t0 v)) (TApp (lifts h d t0) (lift h d v)))).(eq_ind_r TList (TApp (lifts h d -t0) (lift h d v)) (\lambda (t1: TList).(eq TList (TCons (lift h d t) t1) -(TCons (lift h d t) (TApp (lifts h d t0) (lift h d v))))) (refl_equal TList -(TCons (lift h d t) (TApp (lifts h d t0) (lift h d v)))) (lifts h d (TApp t0 -v)) H)))) vs)))). -(* COMMENTS -Initial nodes: 215 -END *) - -theorem lift_inj: - \forall (x: T).(\forall (t: T).(\forall (h: nat).(\forall (d: nat).((eq T -(lift h d x) (lift h d t)) \to (eq T x t))))) -\def - \lambda (x: T).(T_ind (\lambda (t: T).(\forall (t0: T).(\forall (h: -nat).(\forall (d: nat).((eq T (lift h d t) (lift h d t0)) \to (eq T t -t0)))))) (\lambda (n: nat).(\lambda (t: T).(\lambda (h: nat).(\lambda (d: -nat).(\lambda (H: (eq T (lift h d (TSort n)) (lift h d t))).(let H0 \def -(eq_ind T (lift h d (TSort n)) (\lambda (t0: T).(eq T t0 (lift h d t))) H -(TSort n) (lift_sort n h d)) in (sym_eq T t (TSort n) (lift_gen_sort h d n t -H0)))))))) (\lambda (n: nat).(\lambda (t: T).(\lambda (h: nat).(\lambda (d: -nat).(\lambda (H: (eq T (lift h d (TLRef n)) (lift h d t))).(lt_le_e n d (eq -T (TLRef n) t) (\lambda (H0: (lt n d)).(let H1 \def (eq_ind T (lift h d -(TLRef n)) (\lambda (t0: T).(eq T t0 (lift h d t))) H (TLRef n) (lift_lref_lt -n h d H0)) in (sym_eq T t (TLRef n) (lift_gen_lref_lt h d n (lt_le_trans n d -d H0 (le_n d)) t H1)))) (\lambda (H0: (le d n)).(let H1 \def (eq_ind T (lift -h d (TLRef n)) (\lambda (t0: T).(eq T t0 (lift h d t))) H (TLRef (plus n h)) -(lift_lref_ge n h d H0)) in (sym_eq T t (TLRef n) (lift_gen_lref_ge h d n H0 -t H1)))))))))) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t: -T).(((\forall (t0: T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t) -(lift h d t0)) \to (eq T t t0)))))) \to (\forall (t0: T).(((\forall (t1: -T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t0) (lift h d t1)) -\to (eq T t0 t1)))))) \to (\forall (t1: T).(\forall (h: nat).(\forall (d: -nat).((eq T (lift h d (THead k0 t t0)) (lift h d t1)) \to (eq T (THead k0 t -t0) t1)))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (H: ((\forall (t0: -T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t) (lift h d t0)) \to -(eq T t t0))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (t1: T).(\forall -(h: nat).(\forall (d: nat).((eq T (lift h d t0) (lift h d t1)) \to (eq T t0 -t1))))))).(\lambda (t1: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: -(eq T (lift h d (THead (Bind b) t t0)) (lift h d t1))).(let H2 \def (eq_ind T -(lift h d (THead (Bind b) t t0)) (\lambda (t2: T).(eq T t2 (lift h d t1))) H1 -(THead (Bind b) (lift h d t) (lift h (S d) t0)) (lift_bind b t t0 h d)) in -(ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T t1 (THead (Bind b) y -z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t) (lift h d y)))) -(\lambda (_: T).(\lambda (z: T).(eq T (lift h (S d) t0) (lift h (S d) z)))) -(eq T (THead (Bind b) t t0) t1) (\lambda (x0: T).(\lambda (x1: T).(\lambda -(H3: (eq T t1 (THead (Bind b) x0 x1))).(\lambda (H4: (eq T (lift h d t) (lift -h d x0))).(\lambda (H5: (eq T (lift h (S d) t0) (lift h (S d) x1))).(eq_ind_r -T (THead (Bind b) x0 x1) (\lambda (t2: T).(eq T (THead (Bind b) t t0) t2)) -(f_equal3 K T T T THead (Bind b) (Bind b) t x0 t0 x1 (refl_equal K (Bind b)) -(H x0 h d H4) (H0 x1 h (S d) H5)) t1 H3)))))) (lift_gen_bind b (lift h d t) -(lift h (S d) t0) t1 h d H2)))))))))))) (\lambda (f: F).(\lambda (t: -T).(\lambda (H: ((\forall (t0: T).(\forall (h: nat).(\forall (d: nat).((eq T -(lift h d t) (lift h d t0)) \to (eq T t t0))))))).(\lambda (t0: T).(\lambda -(H0: ((\forall (t1: T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d -t0) (lift h d t1)) \to (eq T t0 t1))))))).(\lambda (t1: T).(\lambda (h: -nat).(\lambda (d: nat).(\lambda (H1: (eq T (lift h d (THead (Flat f) t t0)) -(lift h d t1))).(let H2 \def (eq_ind T (lift h d (THead (Flat f) t t0)) -(\lambda (t2: T).(eq T t2 (lift h d t1))) H1 (THead (Flat f) (lift h d t) -(lift h d t0)) (lift_flat f t t0 h d)) in (ex3_2_ind T T (\lambda (y: -T).(\lambda (z: T).(eq T t1 (THead (Flat f) y z)))) (\lambda (y: T).(\lambda -(_: T).(eq T (lift h d t) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq -T (lift h d t0) (lift h d z)))) (eq T (THead (Flat f) t t0) t1) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H3: (eq T t1 (THead (Flat f) x0 x1))).(\lambda -(H4: (eq T (lift h d t) (lift h d x0))).(\lambda (H5: (eq T (lift h d t0) -(lift h d x1))).(eq_ind_r T (THead (Flat f) x0 x1) (\lambda (t2: T).(eq T -(THead (Flat f) t t0) t2)) (f_equal3 K T T T THead (Flat f) (Flat f) t x0 t0 -x1 (refl_equal K (Flat f)) (H x0 h d H4) (H0 x1 h d H5)) t1 H3)))))) -(lift_gen_flat f (lift h d t) (lift h d t0) t1 h d H2)))))))))))) k)) x). -(* COMMENTS -Initial nodes: 1391 -END *) - -theorem lift_gen_lift: - \forall (t1: T).(\forall (x: T).(\forall (h1: nat).(\forall (h2: -nat).(\forall (d1: nat).(\forall (d2: nat).((le d1 d2) \to ((eq T (lift h1 d1 -t1) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1 -t2))) (\lambda (t2: T).(eq T t1 (lift h2 d2 t2))))))))))) -\def - \lambda (t1: T).(T_ind (\lambda (t: T).(\forall (x: T).(\forall (h1: -nat).(\forall (h2: nat).(\forall (d1: nat).(\forall (d2: nat).((le d1 d2) \to -((eq T (lift h1 d1 t) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2: -T).(eq T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T t (lift h2 d2 -t2)))))))))))) (\lambda (n: nat).(\lambda (x: T).(\lambda (h1: nat).(\lambda -(h2: nat).(\lambda (d1: nat).(\lambda (d2: nat).(\lambda (_: (le d1 -d2)).(\lambda (H0: (eq T (lift h1 d1 (TSort n)) (lift h2 (plus d2 h1) -x))).(let H1 \def (eq_ind T (lift h1 d1 (TSort n)) (\lambda (t: T).(eq T t -(lift h2 (plus d2 h1) x))) H0 (TSort n) (lift_sort n h1 d1)) in (eq_ind_r T -(TSort n) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h1 d1 t2))) -(\lambda (t2: T).(eq T (TSort n) (lift h2 d2 t2))))) (ex_intro2 T (\lambda -(t2: T).(eq T (TSort n) (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TSort n) -(lift h2 d2 t2))) (TSort n) (eq_ind_r T (TSort n) (\lambda (t: T).(eq T -(TSort n) t)) (refl_equal T (TSort n)) (lift h1 d1 (TSort n)) (lift_sort n h1 -d1)) (eq_ind_r T (TSort n) (\lambda (t: T).(eq T (TSort n) t)) (refl_equal T -(TSort n)) (lift h2 d2 (TSort n)) (lift_sort n h2 d2))) x (lift_gen_sort h2 -(plus d2 h1) n x H1))))))))))) (\lambda (n: nat).(\lambda (x: T).(\lambda -(h1: nat).(\lambda (h2: nat).(\lambda (d1: nat).(\lambda (d2: nat).(\lambda -(H: (le d1 d2)).(\lambda (H0: (eq T (lift h1 d1 (TLRef n)) (lift h2 (plus d2 -h1) x))).(lt_le_e n d1 (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1 t2))) -(\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2)))) (\lambda (H1: (lt n -d1)).(let H2 \def (eq_ind T (lift h1 d1 (TLRef n)) (\lambda (t: T).(eq T t -(lift h2 (plus d2 h1) x))) H0 (TLRef n) (lift_lref_lt n h1 d1 H1)) in -(eq_ind_r T (TLRef n) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift -h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2))))) (ex_intro2 T -(\lambda (t2: T).(eq T (TLRef n) (lift h1 d1 t2))) (\lambda (t2: T).(eq T -(TLRef n) (lift h2 d2 t2))) (TLRef n) (eq_ind_r T (TLRef n) (\lambda (t: -T).(eq T (TLRef n) t)) (refl_equal T (TLRef n)) (lift h1 d1 (TLRef n)) -(lift_lref_lt n h1 d1 H1)) (eq_ind_r T (TLRef n) (\lambda (t: T).(eq T (TLRef -n) t)) (refl_equal T (TLRef n)) (lift h2 d2 (TLRef n)) (lift_lref_lt n h2 d2 -(lt_le_trans n d1 d2 H1 H)))) x (lift_gen_lref_lt h2 (plus d2 h1) n -(lt_le_trans n d1 (plus d2 h1) H1 (le_plus_trans d1 d2 h1 H)) x H2)))) -(\lambda (H1: (le d1 n)).(let H2 \def (eq_ind T (lift h1 d1 (TLRef n)) -(\lambda (t: T).(eq T t (lift h2 (plus d2 h1) x))) H0 (TLRef (plus n h1)) -(lift_lref_ge n h1 d1 H1)) in (lt_le_e n d2 (ex2 T (\lambda (t2: T).(eq T x -(lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2)))) -(\lambda (H3: (lt n d2)).(eq_ind_r T (TLRef (plus n h1)) (\lambda (t: T).(ex2 -T (\lambda (t2: T).(eq T t (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) -(lift h2 d2 t2))))) (ex_intro2 T (\lambda (t2: T).(eq T (TLRef (plus n h1)) -(lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2))) (TLRef -n) (eq_ind_r T (TLRef (plus n h1)) (\lambda (t: T).(eq T (TLRef (plus n h1)) -t)) (refl_equal T (TLRef (plus n h1))) (lift h1 d1 (TLRef n)) (lift_lref_ge n -h1 d1 H1)) (eq_ind_r T (TLRef n) (\lambda (t: T).(eq T (TLRef n) t)) -(refl_equal T (TLRef n)) (lift h2 d2 (TLRef n)) (lift_lref_lt n h2 d2 H3))) x -(lift_gen_lref_lt h2 (plus d2 h1) (plus n h1) (lt_reg_r n d2 h1 H3) x H2))) -(\lambda (H3: (le d2 n)).(lt_le_e n (plus d2 h2) (ex2 T (\lambda (t2: T).(eq -T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2)))) -(\lambda (H4: (lt n (plus d2 h2))).(lift_gen_lref_false h2 (plus d2 h1) (plus -n h1) (le_plus_plus d2 n h1 h1 H3 (le_n h1)) (eq_ind_r nat (plus (plus d2 h2) -h1) (\lambda (n0: nat).(lt (plus n h1) n0)) (lt_reg_r n (plus d2 h2) h1 H4) -(plus (plus d2 h1) h2) (plus_permute_2_in_3 d2 h1 h2)) x H2 (ex2 T (\lambda -(t2: T).(eq T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 -d2 t2)))))) (\lambda (H4: (le (plus d2 h2) n)).(let H5 \def (eq_ind nat (plus -n h1) (\lambda (n0: nat).(eq T (TLRef n0) (lift h2 (plus d2 h1) x))) H2 (plus -(minus (plus n h1) h2) h2) (le_plus_minus_sym h2 (plus n h1) (le_plus_trans -h2 n h1 (le_trans h2 (plus d2 h2) n (le_plus_r d2 h2) H4)))) in (eq_ind_r T -(TLRef (minus (plus n h1) h2)) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T -t (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2))))) -(ex_intro2 T (\lambda (t2: T).(eq T (TLRef (minus (plus n h1) h2)) (lift h1 -d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2))) (TLRef (minus n -h2)) (eq_ind_r nat (plus (minus n h2) h1) (\lambda (n0: nat).(eq T (TLRef n0) -(lift h1 d1 (TLRef (minus n h2))))) (eq_ind_r T (TLRef (plus (minus n h2) -h1)) (\lambda (t: T).(eq T (TLRef (plus (minus n h2) h1)) t)) (refl_equal T -(TLRef (plus (minus n h2) h1))) (lift h1 d1 (TLRef (minus n h2))) -(lift_lref_ge (minus n h2) h1 d1 (le_trans d1 d2 (minus n h2) H (le_minus d2 -n h2 H4)))) (minus (plus n h1) h2) (le_minus_plus h2 n (le_trans h2 (plus d2 -h2) n (le_plus_r d2 h2) H4) h1)) (eq_ind_r nat (plus (minus n h2) h2) -(\lambda (n0: nat).(eq T (TLRef n0) (lift h2 d2 (TLRef (minus n0 h2))))) -(eq_ind_r T (TLRef (plus (minus (plus (minus n h2) h2) h2) h2)) (\lambda (t: -T).(eq T (TLRef (plus (minus n h2) h2)) t)) (f_equal nat T TLRef (plus (minus -n h2) h2) (plus (minus (plus (minus n h2) h2) h2) h2) (f_equal2 nat nat nat -plus (minus n h2) (minus (plus (minus n h2) h2) h2) h2 h2 (sym_eq nat (minus -(plus (minus n h2) h2) h2) (minus n h2) (minus_plus_r (minus n h2) h2)) -(refl_equal nat h2))) (lift h2 d2 (TLRef (minus (plus (minus n h2) h2) h2))) -(lift_lref_ge (minus (plus (minus n h2) h2) h2) h2 d2 (le_minus d2 (plus -(minus n h2) h2) h2 (le_plus_plus d2 (minus n h2) h2 h2 (le_minus d2 n h2 H4) -(le_n h2))))) n (le_plus_minus_sym h2 n (le_trans h2 (plus d2 h2) n -(le_plus_r d2 h2) H4)))) x (lift_gen_lref_ge h2 (plus d2 h1) (minus (plus n -h1) h2) (arith0 h2 d2 n H4 h1) x H5)))))))))))))))))) (\lambda (k: -K).(\lambda (t: T).(\lambda (H: ((\forall (x: T).(\forall (h1: nat).(\forall -(h2: nat).(\forall (d1: nat).(\forall (d2: nat).((le d1 d2) \to ((eq T (lift -h1 d1 t) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2: T).(eq T x (lift -h1 d1 t2))) (\lambda (t2: T).(eq T t (lift h2 d2 t2))))))))))))).(\lambda -(t0: T).(\lambda (H0: ((\forall (x: T).(\forall (h1: nat).(\forall (h2: -nat).(\forall (d1: nat).(\forall (d2: nat).((le d1 d2) \to ((eq T (lift h1 d1 -t0) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1 -t2))) (\lambda (t2: T).(eq T t0 (lift h2 d2 t2))))))))))))).(\lambda (x: -T).(\lambda (h1: nat).(\lambda (h2: nat).(\lambda (d1: nat).(\lambda (d2: -nat).(\lambda (H1: (le d1 d2)).(\lambda (H2: (eq T (lift h1 d1 (THead k t -t0)) (lift h2 (plus d2 h1) x))).(K_ind (\lambda (k0: K).((eq T (lift h1 d1 -(THead k0 t t0)) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2: T).(eq T -x (lift h1 d1 t2))) (\lambda (t2: T).(eq T (THead k0 t t0) (lift h2 d2 -t2)))))) (\lambda (b: B).(\lambda (H3: (eq T (lift h1 d1 (THead (Bind b) t -t0)) (lift h2 (plus d2 h1) x))).(let H4 \def (eq_ind T (lift h1 d1 (THead -(Bind b) t t0)) (\lambda (t2: T).(eq T t2 (lift h2 (plus d2 h1) x))) H3 -(THead (Bind b) (lift h1 d1 t) (lift h1 (S d1) t0)) (lift_bind b t t0 h1 d1)) -in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Bind b) y -z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h1 d1 t) (lift h2 (plus d2 -h1) y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h1 (S d1) t0) (lift h2 -(S (plus d2 h1)) z)))) (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1 t2))) -(\lambda (t2: T).(eq T (THead (Bind b) t t0) (lift h2 d2 t2)))) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H5: (eq T x (THead (Bind b) x0 x1))).(\lambda -(H6: (eq T (lift h1 d1 t) (lift h2 (plus d2 h1) x0))).(\lambda (H7: (eq T -(lift h1 (S d1) t0) (lift h2 (S (plus d2 h1)) x1))).(eq_ind_r T (THead (Bind -b) x0 x1) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h1 d1 t3))) -(\lambda (t3: T).(eq T (THead (Bind b) t t0) (lift h2 d2 t3))))) (ex2_ind T -(\lambda (t2: T).(eq T x0 (lift h1 d1 t2))) (\lambda (t2: T).(eq T t (lift h2 -d2 t2))) (ex2 T (\lambda (t2: T).(eq T (THead (Bind b) x0 x1) (lift h1 d1 -t2))) (\lambda (t2: T).(eq T (THead (Bind b) t t0) (lift h2 d2 t2)))) -(\lambda (x2: T).(\lambda (H8: (eq T x0 (lift h1 d1 x2))).(\lambda (H9: (eq T -t (lift h2 d2 x2))).(eq_ind_r T (lift h1 d1 x2) (\lambda (t2: T).(ex2 T -(\lambda (t3: T).(eq T (THead (Bind b) t2 x1) (lift h1 d1 t3))) (\lambda (t3: -T).(eq T (THead (Bind b) t t0) (lift h2 d2 t3))))) (eq_ind_r T (lift h2 d2 -x2) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead (Bind b) (lift h1 -d1 x2) x1) (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead (Bind b) t2 t0) -(lift h2 d2 t3))))) (let H10 \def (refl_equal nat (plus (S d2) h1)) in (let -H11 \def (eq_ind nat (S (plus d2 h1)) (\lambda (n: nat).(eq T (lift h1 (S d1) -t0) (lift h2 n x1))) H7 (plus (S d2) h1) H10) in (ex2_ind T (\lambda (t2: -T).(eq T x1 (lift h1 (S d1) t2))) (\lambda (t2: T).(eq T t0 (lift h2 (S d2) -t2))) (ex2 T (\lambda (t2: T).(eq T (THead (Bind b) (lift h1 d1 x2) x1) (lift -h1 d1 t2))) (\lambda (t2: T).(eq T (THead (Bind b) (lift h2 d2 x2) t0) (lift -h2 d2 t2)))) (\lambda (x3: T).(\lambda (H12: (eq T x1 (lift h1 (S d1) -x3))).(\lambda (H13: (eq T t0 (lift h2 (S d2) x3))).(eq_ind_r T (lift h1 (S -d1) x3) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead (Bind b) (lift -h1 d1 x2) t2) (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead (Bind b) (lift -h2 d2 x2) t0) (lift h2 d2 t3))))) (eq_ind_r T (lift h2 (S d2) x3) (\lambda -(t2: T).(ex2 T (\lambda (t3: T).(eq T (THead (Bind b) (lift h1 d1 x2) (lift -h1 (S d1) x3)) (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead (Bind b) (lift -h2 d2 x2) t2) (lift h2 d2 t3))))) (ex_intro2 T (\lambda (t2: T).(eq T (THead -(Bind b) (lift h1 d1 x2) (lift h1 (S d1) x3)) (lift h1 d1 t2))) (\lambda (t2: -T).(eq T (THead (Bind b) (lift h2 d2 x2) (lift h2 (S d2) x3)) (lift h2 d2 -t2))) (THead (Bind b) x2 x3) (eq_ind_r T (THead (Bind b) (lift h1 d1 x2) -(lift h1 (S d1) x3)) (\lambda (t2: T).(eq T (THead (Bind b) (lift h1 d1 x2) -(lift h1 (S d1) x3)) t2)) (refl_equal T (THead (Bind b) (lift h1 d1 x2) (lift -h1 (S d1) x3))) (lift h1 d1 (THead (Bind b) x2 x3)) (lift_bind b x2 x3 h1 -d1)) (eq_ind_r T (THead (Bind b) (lift h2 d2 x2) (lift h2 (S d2) x3)) -(\lambda (t2: T).(eq T (THead (Bind b) (lift h2 d2 x2) (lift h2 (S d2) x3)) -t2)) (refl_equal T (THead (Bind b) (lift h2 d2 x2) (lift h2 (S d2) x3))) -(lift h2 d2 (THead (Bind b) x2 x3)) (lift_bind b x2 x3 h2 d2))) t0 H13) x1 -H12)))) (H0 x1 h1 h2 (S d1) (S d2) (le_n_S d1 d2 H1) H11)))) t H9) x0 H8)))) -(H x0 h1 h2 d1 d2 H1 H6)) x H5)))))) (lift_gen_bind b (lift h1 d1 t) (lift h1 -(S d1) t0) x h2 (plus d2 h1) H4))))) (\lambda (f: F).(\lambda (H3: (eq T -(lift h1 d1 (THead (Flat f) t t0)) (lift h2 (plus d2 h1) x))).(let H4 \def -(eq_ind T (lift h1 d1 (THead (Flat f) t t0)) (\lambda (t2: T).(eq T t2 (lift -h2 (plus d2 h1) x))) H3 (THead (Flat f) (lift h1 d1 t) (lift h1 d1 t0)) -(lift_flat f t t0 h1 d1)) in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: -T).(eq T x (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T -(lift h1 d1 t) (lift h2 (plus d2 h1) y)))) (\lambda (_: T).(\lambda (z: -T).(eq T (lift h1 d1 t0) (lift h2 (plus d2 h1) z)))) (ex2 T (\lambda (t2: -T).(eq T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T (THead (Flat f) t t0) -(lift h2 d2 t2)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T x -(THead (Flat f) x0 x1))).(\lambda (H6: (eq T (lift h1 d1 t) (lift h2 (plus d2 -h1) x0))).(\lambda (H7: (eq T (lift h1 d1 t0) (lift h2 (plus d2 h1) -x1))).(eq_ind_r T (THead (Flat f) x0 x1) (\lambda (t2: T).(ex2 T (\lambda -(t3: T).(eq T t2 (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead (Flat f) t -t0) (lift h2 d2 t3))))) (ex2_ind T (\lambda (t2: T).(eq T x0 (lift h1 d1 -t2))) (\lambda (t2: T).(eq T t (lift h2 d2 t2))) (ex2 T (\lambda (t2: T).(eq -T (THead (Flat f) x0 x1) (lift h1 d1 t2))) (\lambda (t2: T).(eq T (THead -(Flat f) t t0) (lift h2 d2 t2)))) (\lambda (x2: T).(\lambda (H8: (eq T x0 -(lift h1 d1 x2))).(\lambda (H9: (eq T t (lift h2 d2 x2))).(eq_ind_r T (lift -h1 d1 x2) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead (Flat f) t2 -x1) (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead (Flat f) t t0) (lift h2 -d2 t3))))) (eq_ind_r T (lift h2 d2 x2) (\lambda (t2: T).(ex2 T (\lambda (t3: -T).(eq T (THead (Flat f) (lift h1 d1 x2) x1) (lift h1 d1 t3))) (\lambda (t3: -T).(eq T (THead (Flat f) t2 t0) (lift h2 d2 t3))))) (ex2_ind T (\lambda (t2: -T).(eq T x1 (lift h1 d1 t2))) (\lambda (t2: T).(eq T t0 (lift h2 d2 t2))) -(ex2 T (\lambda (t2: T).(eq T (THead (Flat f) (lift h1 d1 x2) x1) (lift h1 d1 -t2))) (\lambda (t2: T).(eq T (THead (Flat f) (lift h2 d2 x2) t0) (lift h2 d2 -t2)))) (\lambda (x3: T).(\lambda (H10: (eq T x1 (lift h1 d1 x3))).(\lambda -(H11: (eq T t0 (lift h2 d2 x3))).(eq_ind_r T (lift h1 d1 x3) (\lambda (t2: -T).(ex2 T (\lambda (t3: T).(eq T (THead (Flat f) (lift h1 d1 x2) t2) (lift h1 -d1 t3))) (\lambda (t3: T).(eq T (THead (Flat f) (lift h2 d2 x2) t0) (lift h2 -d2 t3))))) (eq_ind_r T (lift h2 d2 x3) (\lambda (t2: T).(ex2 T (\lambda (t3: -T).(eq T (THead (Flat f) (lift h1 d1 x2) (lift h1 d1 x3)) (lift h1 d1 t3))) -(\lambda (t3: T).(eq T (THead (Flat f) (lift h2 d2 x2) t2) (lift h2 d2 -t3))))) (ex_intro2 T (\lambda (t2: T).(eq T (THead (Flat f) (lift h1 d1 x2) -(lift h1 d1 x3)) (lift h1 d1 t2))) (\lambda (t2: T).(eq T (THead (Flat f) -(lift h2 d2 x2) (lift h2 d2 x3)) (lift h2 d2 t2))) (THead (Flat f) x2 x3) -(eq_ind_r T (THead (Flat f) (lift h1 d1 x2) (lift h1 d1 x3)) (\lambda (t2: -T).(eq T (THead (Flat f) (lift h1 d1 x2) (lift h1 d1 x3)) t2)) (refl_equal T -(THead (Flat f) (lift h1 d1 x2) (lift h1 d1 x3))) (lift h1 d1 (THead (Flat f) -x2 x3)) (lift_flat f x2 x3 h1 d1)) (eq_ind_r T (THead (Flat f) (lift h2 d2 -x2) (lift h2 d2 x3)) (\lambda (t2: T).(eq T (THead (Flat f) (lift h2 d2 x2) -(lift h2 d2 x3)) t2)) (refl_equal T (THead (Flat f) (lift h2 d2 x2) (lift h2 -d2 x3))) (lift h2 d2 (THead (Flat f) x2 x3)) (lift_flat f x2 x3 h2 d2))) t0 -H11) x1 H10)))) (H0 x1 h1 h2 d1 d2 H1 H7)) t H9) x0 H8)))) (H x0 h1 h2 d1 d2 -H1 H6)) x H5)))))) (lift_gen_flat f (lift h1 d1 t) (lift h1 d1 t0) x h2 (plus -d2 h1) H4))))) k H2))))))))))))) t1). -(* COMMENTS -Initial nodes: 5037 -END *) - -theorem lifts_inj: - \forall (xs: TList).(\forall (ts: TList).(\forall (h: nat).(\forall (d: -nat).((eq TList (lifts h d xs) (lifts h d ts)) \to (eq TList xs ts))))) -\def - \lambda (xs: TList).(TList_ind (\lambda (t: TList).(\forall (ts: -TList).(\forall (h: nat).(\forall (d: nat).((eq TList (lifts h d t) (lifts h -d ts)) \to (eq TList t ts)))))) (\lambda (ts: TList).(TList_ind (\lambda (t: -TList).(\forall (h: nat).(\forall (d: nat).((eq TList (lifts h d TNil) (lifts -h d t)) \to (eq TList TNil t))))) (\lambda (_: nat).(\lambda (_: -nat).(\lambda (_: (eq TList TNil TNil)).(refl_equal TList TNil)))) (\lambda -(t: T).(\lambda (t0: TList).(\lambda (_: ((\forall (h: nat).(\forall (d: -nat).((eq TList TNil (lifts h d t0)) \to (eq TList TNil t0)))))).(\lambda (h: -nat).(\lambda (d: nat).(\lambda (H0: (eq TList TNil (TCons (lift h d t) -(lifts h d t0)))).(let H1 \def (eq_ind TList TNil (\lambda (ee: TList).(match -ee in TList return (\lambda (_: TList).Prop) with [TNil \Rightarrow True | -(TCons _ _) \Rightarrow False])) I (TCons (lift h d t) (lifts h d t0)) H0) in -(False_ind (eq TList TNil (TCons t t0)) H1)))))))) ts)) (\lambda (t: -T).(\lambda (t0: TList).(\lambda (H: ((\forall (ts: TList).(\forall (h: -nat).(\forall (d: nat).((eq TList (lifts h d t0) (lifts h d ts)) \to (eq -TList t0 ts))))))).(\lambda (ts: TList).(TList_ind (\lambda (t1: -TList).(\forall (h: nat).(\forall (d: nat).((eq TList (lifts h d (TCons t -t0)) (lifts h d t1)) \to (eq TList (TCons t t0) t1))))) (\lambda (h: -nat).(\lambda (d: nat).(\lambda (H0: (eq TList (TCons (lift h d t) (lifts h d -t0)) TNil)).(let H1 \def (eq_ind TList (TCons (lift h d t) (lifts h d t0)) -(\lambda (ee: TList).(match ee in TList return (\lambda (_: TList).Prop) with -[TNil \Rightarrow False | (TCons _ _) \Rightarrow True])) I TNil H0) in -(False_ind (eq TList (TCons t t0) TNil) H1))))) (\lambda (t1: T).(\lambda -(t2: TList).(\lambda (_: ((\forall (h: nat).(\forall (d: nat).((eq TList -(TCons (lift h d t) (lifts h d t0)) (lifts h d t2)) \to (eq TList (TCons t -t0) t2)))))).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: (eq TList -(TCons (lift h d t) (lifts h d t0)) (TCons (lift h d t1) (lifts h d -t2)))).(let H2 \def (f_equal TList T (\lambda (e: TList).(match e in TList -return (\lambda (_: TList).T) with [TNil \Rightarrow ((let rec lref_map (f: -((nat \to nat))) (d0: nat) (t3: T) on t3: T \def (match t3 with [(TSort n) -\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) with -[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t4) \Rightarrow -(THead k (lref_map f d0 u) (lref_map f (s k d0) t4))]) in lref_map) (\lambda -(x: nat).(plus x h)) d t) | (TCons t3 _) \Rightarrow t3])) (TCons (lift h d -t) (lifts h d t0)) (TCons (lift h d t1) (lifts h d t2)) H1) in ((let H3 \def -(f_equal TList TList (\lambda (e: TList).(match e in TList return (\lambda -(_: TList).TList) with [TNil \Rightarrow ((let rec lifts (h0: nat) (d0: nat) -(ts0: TList) on ts0: TList \def (match ts0 with [TNil \Rightarrow TNil | -(TCons t3 ts1) \Rightarrow (TCons (lift h0 d0 t3) (lifts h0 d0 ts1))]) in -lifts) h d t0) | (TCons _ t3) \Rightarrow t3])) (TCons (lift h d t) (lifts h -d t0)) (TCons (lift h d t1) (lifts h d t2)) H1) in (\lambda (H4: (eq T (lift -h d t) (lift h d t1))).(eq_ind T t (\lambda (t3: T).(eq TList (TCons t t0) -(TCons t3 t2))) (f_equal2 T TList TList TCons t t t0 t2 (refl_equal T t) (H -t2 h d H3)) t1 (lift_inj t t1 h d H4)))) H2)))))))) ts))))) xs). -(* COMMENTS -Initial nodes: 772 -END *) - -theorem lift_free: - \forall (t: T).(\forall (h: nat).(\forall (k: nat).(\forall (d: -nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to (eq T (lift k e -(lift h d t)) (lift (plus k h) d t)))))))) -\def - \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (h: nat).(\forall (k: -nat).(\forall (d: nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to -(eq T (lift k e (lift h d t0)) (lift (plus k h) d t0))))))))) (\lambda (n: -nat).(\lambda (h: nat).(\lambda (k: nat).(\lambda (d: nat).(\lambda (e: -nat).(\lambda (_: (le e (plus d h))).(\lambda (_: (le d e)).(eq_ind_r T -(TSort n) (\lambda (t0: T).(eq T (lift k e t0) (lift (plus k h) d (TSort -n)))) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq T t0 (lift (plus k h) d -(TSort n)))) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq T (TSort n) t0)) -(refl_equal T (TSort n)) (lift (plus k h) d (TSort n)) (lift_sort n (plus k -h) d)) (lift k e (TSort n)) (lift_sort n k e)) (lift h d (TSort n)) -(lift_sort n h d))))))))) (\lambda (n: nat).(\lambda (h: nat).(\lambda (k: -nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (H: (le e (plus d -h))).(\lambda (H0: (le d e)).(lt_le_e n d (eq T (lift k e (lift h d (TLRef -n))) (lift (plus k h) d (TLRef n))) (\lambda (H1: (lt n d)).(eq_ind_r T -(TLRef n) (\lambda (t0: T).(eq T (lift k e t0) (lift (plus k h) d (TLRef -n)))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 (lift (plus k h) d -(TLRef n)))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (TLRef n) t0)) -(refl_equal T (TLRef n)) (lift (plus k h) d (TLRef n)) (lift_lref_lt n (plus -k h) d H1)) (lift k e (TLRef n)) (lift_lref_lt n k e (lt_le_trans n d e H1 -H0))) (lift h d (TLRef n)) (lift_lref_lt n h d H1))) (\lambda (H1: (le d -n)).(eq_ind_r T (TLRef (plus n h)) (\lambda (t0: T).(eq T (lift k e t0) (lift -(plus k h) d (TLRef n)))) (eq_ind_r T (TLRef (plus (plus n h) k)) (\lambda -(t0: T).(eq T t0 (lift (plus k h) d (TLRef n)))) (eq_ind_r T (TLRef (plus n -(plus k h))) (\lambda (t0: T).(eq T (TLRef (plus (plus n h) k)) t0)) (f_equal -nat T TLRef (plus (plus n h) k) (plus n (plus k h)) -(plus_permute_2_in_3_assoc n h k)) (lift (plus k h) d (TLRef n)) -(lift_lref_ge n (plus k h) d H1)) (lift k e (TLRef (plus n h))) (lift_lref_ge -(plus n h) k e (le_trans e (plus d h) (plus n h) H (le_plus_plus d n h h H1 -(le_n h))))) (lift h d (TLRef n)) (lift_lref_ge n h d H1))))))))))) (\lambda -(k: K).(\lambda (t0: T).(\lambda (H: ((\forall (h: nat).(\forall (k0: -nat).(\forall (d: nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to -(eq T (lift k0 e (lift h d t0)) (lift (plus k0 h) d t0)))))))))).(\lambda -(t1: T).(\lambda (H0: ((\forall (h: nat).(\forall (k0: nat).(\forall (d: -nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to (eq T (lift k0 e -(lift h d t1)) (lift (plus k0 h) d t1)))))))))).(\lambda (h: nat).(\lambda -(k0: nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (H1: (le e (plus d -h))).(\lambda (H2: (le d e)).(eq_ind_r T (THead k (lift h d t0) (lift h (s k -d) t1)) (\lambda (t2: T).(eq T (lift k0 e t2) (lift (plus k0 h) d (THead k t0 -t1)))) (eq_ind_r T (THead k (lift k0 e (lift h d t0)) (lift k0 (s k e) (lift -h (s k d) t1))) (\lambda (t2: T).(eq T t2 (lift (plus k0 h) d (THead k t0 -t1)))) (eq_ind_r T (THead k (lift (plus k0 h) d t0) (lift (plus k0 h) (s k d) -t1)) (\lambda (t2: T).(eq T (THead k (lift k0 e (lift h d t0)) (lift k0 (s k -e) (lift h (s k d) t1))) t2)) (f_equal3 K T T T THead k k (lift k0 e (lift h -d t0)) (lift (plus k0 h) d t0) (lift k0 (s k e) (lift h (s k d) t1)) (lift -(plus k0 h) (s k d) t1) (refl_equal K k) (H h k0 d e H1 H2) (H0 h k0 (s k d) -(s k e) (eq_ind nat (s k (plus d h)) (\lambda (n: nat).(le (s k e) n)) (s_le -k e (plus d h) H1) (plus (s k d) h) (s_plus k d h)) (s_le k d e H2))) (lift -(plus k0 h) d (THead k t0 t1)) (lift_head k t0 t1 (plus k0 h) d)) (lift k0 e -(THead k (lift h d t0) (lift h (s k d) t1))) (lift_head k (lift h d t0) (lift -h (s k d) t1) k0 e)) (lift h d (THead k t0 t1)) (lift_head k t0 t1 h -d))))))))))))) t). -(* COMMENTS -Initial nodes: 1407 -END *) - -theorem lift_d: - \forall (t: T).(\forall (h: nat).(\forall (k: nat).(\forall (d: -nat).(\forall (e: nat).((le e d) \to (eq T (lift h (plus k d) (lift k e t)) -(lift k e (lift h d t)))))))) -\def - \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (h: nat).(\forall (k: -nat).(\forall (d: nat).(\forall (e: nat).((le e d) \to (eq T (lift h (plus k -d) (lift k e t0)) (lift k e (lift h d t0))))))))) (\lambda (n: nat).(\lambda -(h: nat).(\lambda (k: nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (_: -(le e d)).(eq_ind_r T (TSort n) (\lambda (t0: T).(eq T (lift h (plus k d) t0) -(lift k e (lift h d (TSort n))))) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq -T t0 (lift k e (lift h d (TSort n))))) (eq_ind_r T (TSort n) (\lambda (t0: -T).(eq T (TSort n) (lift k e t0))) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq -T (TSort n) t0)) (refl_equal T (TSort n)) (lift k e (TSort n)) (lift_sort n k -e)) (lift h d (TSort n)) (lift_sort n h d)) (lift h (plus k d) (TSort n)) -(lift_sort n h (plus k d))) (lift k e (TSort n)) (lift_sort n k e)))))))) -(\lambda (n: nat).(\lambda (h: nat).(\lambda (k: nat).(\lambda (d: -nat).(\lambda (e: nat).(\lambda (H: (le e d)).(lt_le_e n e (eq T (lift h -(plus k d) (lift k e (TLRef n))) (lift k e (lift h d (TLRef n)))) (\lambda -(H0: (lt n e)).(let H1 \def (lt_le_trans n e d H0 H) in (eq_ind_r T (TLRef n) -(\lambda (t0: T).(eq T (lift h (plus k d) t0) (lift k e (lift h d (TLRef -n))))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 (lift k e (lift h d -(TLRef n))))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (TLRef n) (lift k -e t0))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (TLRef n) t0)) -(refl_equal T (TLRef n)) (lift k e (TLRef n)) (lift_lref_lt n k e H0)) (lift -h d (TLRef n)) (lift_lref_lt n h d H1)) (lift h (plus k d) (TLRef n)) -(lift_lref_lt n h (plus k d) (lt_le_trans n d (plus k d) H1 (le_plus_r k -d)))) (lift k e (TLRef n)) (lift_lref_lt n k e H0)))) (\lambda (H0: (le e -n)).(eq_ind_r T (TLRef (plus n k)) (\lambda (t0: T).(eq T (lift h (plus k d) -t0) (lift k e (lift h d (TLRef n))))) (eq_ind_r nat (plus d k) (\lambda (n0: -nat).(eq T (lift h n0 (TLRef (plus n k))) (lift k e (lift h d (TLRef n))))) -(lt_le_e n d (eq T (lift h (plus d k) (TLRef (plus n k))) (lift k e (lift h d -(TLRef n)))) (\lambda (H1: (lt n d)).(eq_ind_r T (TLRef (plus n k)) (\lambda -(t0: T).(eq T t0 (lift k e (lift h d (TLRef n))))) (eq_ind_r T (TLRef n) -(\lambda (t0: T).(eq T (TLRef (plus n k)) (lift k e t0))) (eq_ind_r T (TLRef -(plus n k)) (\lambda (t0: T).(eq T (TLRef (plus n k)) t0)) (refl_equal T -(TLRef (plus n k))) (lift k e (TLRef n)) (lift_lref_ge n k e H0)) (lift h d -(TLRef n)) (lift_lref_lt n h d H1)) (lift h (plus d k) (TLRef (plus n k))) -(lift_lref_lt (plus n k) h (plus d k) (lt_reg_r n d k H1)))) (\lambda (H1: -(le d n)).(eq_ind_r T (TLRef (plus (plus n k) h)) (\lambda (t0: T).(eq T t0 -(lift k e (lift h d (TLRef n))))) (eq_ind_r T (TLRef (plus n h)) (\lambda -(t0: T).(eq T (TLRef (plus (plus n k) h)) (lift k e t0))) (eq_ind_r T (TLRef -(plus (plus n h) k)) (\lambda (t0: T).(eq T (TLRef (plus (plus n k) h)) t0)) -(f_equal nat T TLRef (plus (plus n k) h) (plus (plus n h) k) (sym_eq nat -(plus (plus n h) k) (plus (plus n k) h) (plus_permute_2_in_3 n h k))) (lift k -e (TLRef (plus n h))) (lift_lref_ge (plus n h) k e (le_plus_trans e n h H0))) -(lift h d (TLRef n)) (lift_lref_ge n h d H1)) (lift h (plus d k) (TLRef (plus -n k))) (lift_lref_ge (plus n k) h (plus d k) (le_plus_plus d n k k H1 (le_n -k)))))) (plus k d) (plus_sym k d)) (lift k e (TLRef n)) (lift_lref_ge n k e -H0)))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (H: ((\forall (h: -nat).(\forall (k0: nat).(\forall (d: nat).(\forall (e: nat).((le e d) \to (eq -T (lift h (plus k0 d) (lift k0 e t0)) (lift k0 e (lift h d -t0)))))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (h: nat).(\forall (k0: -nat).(\forall (d: nat).(\forall (e: nat).((le e d) \to (eq T (lift h (plus k0 -d) (lift k0 e t1)) (lift k0 e (lift h d t1)))))))))).(\lambda (h: -nat).(\lambda (k0: nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (H1: (le -e d)).(eq_ind_r T (THead k (lift k0 e t0) (lift k0 (s k e) t1)) (\lambda (t2: -T).(eq T (lift h (plus k0 d) t2) (lift k0 e (lift h d (THead k t0 t1))))) -(eq_ind_r T (THead k (lift h (plus k0 d) (lift k0 e t0)) (lift h (s k (plus -k0 d)) (lift k0 (s k e) t1))) (\lambda (t2: T).(eq T t2 (lift k0 e (lift h d -(THead k t0 t1))))) (eq_ind_r T (THead k (lift h d t0) (lift h (s k d) t1)) -(\lambda (t2: T).(eq T (THead k (lift h (plus k0 d) (lift k0 e t0)) (lift h -(s k (plus k0 d)) (lift k0 (s k e) t1))) (lift k0 e t2))) (eq_ind_r T (THead -k (lift k0 e (lift h d t0)) (lift k0 (s k e) (lift h (s k d) t1))) (\lambda -(t2: T).(eq T (THead k (lift h (plus k0 d) (lift k0 e t0)) (lift h (s k (plus -k0 d)) (lift k0 (s k e) t1))) t2)) (eq_ind_r nat (plus k0 (s k d)) (\lambda -(n: nat).(eq T (THead k (lift h (plus k0 d) (lift k0 e t0)) (lift h n (lift -k0 (s k e) t1))) (THead k (lift k0 e (lift h d t0)) (lift k0 (s k e) (lift h -(s k d) t1))))) (f_equal3 K T T T THead k k (lift h (plus k0 d) (lift k0 e -t0)) (lift k0 e (lift h d t0)) (lift h (plus k0 (s k d)) (lift k0 (s k e) -t1)) (lift k0 (s k e) (lift h (s k d) t1)) (refl_equal K k) (H h k0 d e H1) -(H0 h k0 (s k d) (s k e) (s_le k e d H1))) (s k (plus k0 d)) (s_plus_sym k k0 -d)) (lift k0 e (THead k (lift h d t0) (lift h (s k d) t1))) (lift_head k -(lift h d t0) (lift h (s k d) t1) k0 e)) (lift h d (THead k t0 t1)) -(lift_head k t0 t1 h d)) (lift h (plus k0 d) (THead k (lift k0 e t0) (lift k0 -(s k e) t1))) (lift_head k (lift k0 e t0) (lift k0 (s k e) t1) h (plus k0 -d))) (lift k0 e (THead k t0 t1)) (lift_head k t0 t1 k0 e)))))))))))) t). -(* COMMENTS -Initial nodes: 2143 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/lift/tlt.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/lift/tlt.ma deleted file mode 100644 index 1d8edc7df..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/lift/tlt.ma +++ /dev/null @@ -1,299 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/lift/fwd.ma". - -include "Basic-1/tlt/props.ma". - -theorem lift_weight_map: - \forall (t: T).(\forall (h: nat).(\forall (d: nat).(\forall (f: ((nat \to -nat))).(((\forall (m: nat).((le d m) \to (eq nat (f m) O)))) \to (eq nat -(weight_map f (lift h d t)) (weight_map f t)))))) -\def - \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (h: nat).(\forall (d: -nat).(\forall (f: ((nat \to nat))).(((\forall (m: nat).((le d m) \to (eq nat -(f m) O)))) \to (eq nat (weight_map f (lift h d t0)) (weight_map f t0))))))) -(\lambda (n: nat).(\lambda (_: nat).(\lambda (d: nat).(\lambda (f: ((nat \to -nat))).(\lambda (_: ((\forall (m: nat).((le d m) \to (eq nat (f m) -O))))).(refl_equal nat (weight_map f (TSort n)))))))) (\lambda (n: -nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (f: ((nat \to -nat))).(\lambda (H: ((\forall (m: nat).((le d m) \to (eq nat (f m) -O))))).(lt_le_e n d (eq nat (weight_map f (lift h d (TLRef n))) (weight_map f -(TLRef n))) (\lambda (H0: (lt n d)).(eq_ind_r T (TLRef n) (\lambda (t0: -T).(eq nat (weight_map f t0) (weight_map f (TLRef n)))) (refl_equal nat -(weight_map f (TLRef n))) (lift h d (TLRef n)) (lift_lref_lt n h d H0))) -(\lambda (H0: (le d n)).(eq_ind_r T (TLRef (plus n h)) (\lambda (t0: T).(eq -nat (weight_map f t0) (weight_map f (TLRef n)))) (eq_ind_r nat O (\lambda -(n0: nat).(eq nat (f (plus n h)) n0)) (H (plus n h) (le_plus_trans d n h H0)) -(f n) (H n H0)) (lift h d (TLRef n)) (lift_lref_ge n h d H0))))))))) (\lambda -(k: K).(\lambda (t0: T).(\lambda (H: ((\forall (h: nat).(\forall (d: -nat).(\forall (f: ((nat \to nat))).(((\forall (m: nat).((le d m) \to (eq nat -(f m) O)))) \to (eq nat (weight_map f (lift h d t0)) (weight_map f -t0)))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (h: nat).(\forall (d: -nat).(\forall (f: ((nat \to nat))).(((\forall (m: nat).((le d m) \to (eq nat -(f m) O)))) \to (eq nat (weight_map f (lift h d t1)) (weight_map f -t1)))))))).(\lambda (h: nat).(\lambda (d: nat).(\lambda (f: ((nat \to -nat))).(\lambda (H1: ((\forall (m: nat).((le d m) \to (eq nat (f m) -O))))).(K_ind (\lambda (k0: K).(eq nat (weight_map f (lift h d (THead k0 t0 -t1))) (weight_map f (THead k0 t0 t1)))) (\lambda (b: B).(eq_ind_r T (THead -(Bind b) (lift h d t0) (lift h (s (Bind b) d) t1)) (\lambda (t2: T).(eq nat -(weight_map f t2) (weight_map f (THead (Bind b) t0 t1)))) (B_ind (\lambda -(b0: B).(eq nat (match b0 with [Abbr \Rightarrow (S (plus (weight_map f (lift -h d t0)) (weight_map (wadd f (S (weight_map f (lift h d t0)))) (lift h (S d) -t1)))) | Abst \Rightarrow (S (plus (weight_map f (lift h d t0)) (weight_map -(wadd f O) (lift h (S d) t1)))) | Void \Rightarrow (S (plus (weight_map f -(lift h d t0)) (weight_map (wadd f O) (lift h (S d) t1))))]) (match b0 with -[Abbr \Rightarrow (S (plus (weight_map f t0) (weight_map (wadd f (S -(weight_map f t0))) t1))) | Abst \Rightarrow (S (plus (weight_map f t0) -(weight_map (wadd f O) t1))) | Void \Rightarrow (S (plus (weight_map f t0) -(weight_map (wadd f O) t1)))]))) (eq_ind_r nat (weight_map f t0) (\lambda (n: -nat).(eq nat (S (plus n (weight_map (wadd f (S n)) (lift h (S d) t1)))) (S -(plus (weight_map f t0) (weight_map (wadd f (S (weight_map f t0))) t1))))) -(eq_ind_r nat (weight_map (wadd f (S (weight_map f t0))) t1) (\lambda (n: -nat).(eq nat (S (plus (weight_map f t0) n)) (S (plus (weight_map f t0) -(weight_map (wadd f (S (weight_map f t0))) t1))))) (refl_equal nat (S (plus -(weight_map f t0) (weight_map (wadd f (S (weight_map f t0))) t1)))) -(weight_map (wadd f (S (weight_map f t0))) (lift h (S d) t1)) (H0 h (S d) -(wadd f (S (weight_map f t0))) (\lambda (m: nat).(\lambda (H2: (le (S d) -m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S n))) (\lambda (n: nat).(le d -n)) (eq nat (wadd f (S (weight_map f t0)) m) O) (\lambda (x: nat).(\lambda -(H3: (eq nat m (S x))).(\lambda (H4: (le d x)).(eq_ind_r nat (S x) (\lambda -(n: nat).(eq nat (wadd f (S (weight_map f t0)) n) O)) (H1 x H4) m H3)))) -(le_gen_S d m H2)))))) (weight_map f (lift h d t0)) (H h d f H1)) (eq_ind_r -nat (weight_map (wadd f O) t1) (\lambda (n: nat).(eq nat (S (plus (weight_map -f (lift h d t0)) n)) (S (plus (weight_map f t0) (weight_map (wadd f O) -t1))))) (f_equal nat nat S (plus (weight_map f (lift h d t0)) (weight_map -(wadd f O) t1)) (plus (weight_map f t0) (weight_map (wadd f O) t1)) (f_equal2 -nat nat nat plus (weight_map f (lift h d t0)) (weight_map f t0) (weight_map -(wadd f O) t1) (weight_map (wadd f O) t1) (H h d f H1) (refl_equal nat -(weight_map (wadd f O) t1)))) (weight_map (wadd f O) (lift h (S d) t1)) (H0 h -(S d) (wadd f O) (\lambda (m: nat).(\lambda (H2: (le (S d) m)).(ex2_ind nat -(\lambda (n: nat).(eq nat m (S n))) (\lambda (n: nat).(le d n)) (eq nat (wadd -f O m) O) (\lambda (x: nat).(\lambda (H3: (eq nat m (S x))).(\lambda (H4: (le -d x)).(eq_ind_r nat (S x) (\lambda (n: nat).(eq nat (wadd f O n) O)) (H1 x -H4) m H3)))) (le_gen_S d m H2)))))) (eq_ind_r nat (weight_map (wadd f O) t1) -(\lambda (n: nat).(eq nat (S (plus (weight_map f (lift h d t0)) n)) (S (plus -(weight_map f t0) (weight_map (wadd f O) t1))))) (f_equal nat nat S (plus -(weight_map f (lift h d t0)) (weight_map (wadd f O) t1)) (plus (weight_map f -t0) (weight_map (wadd f O) t1)) (f_equal2 nat nat nat plus (weight_map f -(lift h d t0)) (weight_map f t0) (weight_map (wadd f O) t1) (weight_map (wadd -f O) t1) (H h d f H1) (refl_equal nat (weight_map (wadd f O) t1)))) -(weight_map (wadd f O) (lift h (S d) t1)) (H0 h (S d) (wadd f O) (\lambda (m: -nat).(\lambda (H2: (le (S d) m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S -n))) (\lambda (n: nat).(le d n)) (eq nat (wadd f O m) O) (\lambda (x: -nat).(\lambda (H3: (eq nat m (S x))).(\lambda (H4: (le d x)).(eq_ind_r nat (S -x) (\lambda (n: nat).(eq nat (wadd f O n) O)) (H1 x H4) m H3)))) (le_gen_S d -m H2)))))) b) (lift h d (THead (Bind b) t0 t1)) (lift_head (Bind b) t0 t1 h -d))) (\lambda (f0: F).(eq_ind_r T (THead (Flat f0) (lift h d t0) (lift h (s -(Flat f0) d) t1)) (\lambda (t2: T).(eq nat (weight_map f t2) (weight_map f -(THead (Flat f0) t0 t1)))) (f_equal nat nat S (plus (weight_map f (lift h d -t0)) (weight_map f (lift h d t1))) (plus (weight_map f t0) (weight_map f t1)) -(f_equal2 nat nat nat plus (weight_map f (lift h d t0)) (weight_map f t0) -(weight_map f (lift h d t1)) (weight_map f t1) (H h d f H1) (H0 h d f H1))) -(lift h d (THead (Flat f0) t0 t1)) (lift_head (Flat f0) t0 t1 h d))) -k)))))))))) t). -(* COMMENTS -Initial nodes: 1969 -END *) - -theorem lift_weight: - \forall (t: T).(\forall (h: nat).(\forall (d: nat).(eq nat (weight (lift h d -t)) (weight t)))) -\def - \lambda (t: T).(\lambda (h: nat).(\lambda (d: nat).(lift_weight_map t h d -(\lambda (_: nat).O) (\lambda (m: nat).(\lambda (_: (le d m)).(refl_equal nat -O)))))). -(* COMMENTS -Initial nodes: 31 -END *) - -theorem lift_weight_add: - \forall (w: nat).(\forall (t: T).(\forall (h: nat).(\forall (d: -nat).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall -(m: nat).((lt m d) \to (eq nat (g m) (f m))))) \to ((eq nat (g d) w) \to -(((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f m))))) \to (eq nat -(weight_map f (lift h d t)) (weight_map g (lift (S h) d t))))))))))) -\def - \lambda (w: nat).(\lambda (t: T).(T_ind (\lambda (t0: T).(\forall (h: -nat).(\forall (d: nat).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (m: nat).((lt m d) \to (eq nat (g m) (f m))))) \to ((eq nat -(g d) w) \to (((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f m))))) -\to (eq nat (weight_map f (lift h d t0)) (weight_map g (lift (S h) d -t0))))))))))) (\lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda -(f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (_: ((\forall (m: -nat).((lt m d) \to (eq nat (g m) (f m)))))).(\lambda (_: (eq nat (g d) -w)).(\lambda (_: ((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f -m)))))).(refl_equal nat (weight_map g (lift (S h) d (TSort n)))))))))))) -(\lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (f: ((nat \to -nat))).(\lambda (g: ((nat \to nat))).(\lambda (H: ((\forall (m: nat).((lt m -d) \to (eq nat (g m) (f m)))))).(\lambda (_: (eq nat (g d) w)).(\lambda (H1: -((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f m)))))).(lt_le_e n d -(eq nat (weight_map f (lift h d (TLRef n))) (weight_map g (lift (S h) d -(TLRef n)))) (\lambda (H2: (lt n d)).(eq_ind_r T (TLRef n) (\lambda (t0: -T).(eq nat (weight_map f t0) (weight_map g (lift (S h) d (TLRef n))))) -(eq_ind_r T (TLRef n) (\lambda (t0: T).(eq nat (weight_map f (TLRef n)) -(weight_map g t0))) (sym_eq nat (g n) (f n) (H n H2)) (lift (S h) d (TLRef -n)) (lift_lref_lt n (S h) d H2)) (lift h d (TLRef n)) (lift_lref_lt n h d -H2))) (\lambda (H2: (le d n)).(eq_ind_r T (TLRef (plus n h)) (\lambda (t0: -T).(eq nat (weight_map f t0) (weight_map g (lift (S h) d (TLRef n))))) -(eq_ind_r T (TLRef (plus n (S h))) (\lambda (t0: T).(eq nat (weight_map f -(TLRef (plus n h))) (weight_map g t0))) (eq_ind nat (S (plus n h)) (\lambda -(n0: nat).(eq nat (f (plus n h)) (g n0))) (sym_eq nat (g (S (plus n h))) (f -(plus n h)) (H1 (plus n h) (le_plus_trans d n h H2))) (plus n (S h)) -(plus_n_Sm n h)) (lift (S h) d (TLRef n)) (lift_lref_ge n (S h) d H2)) (lift -h d (TLRef n)) (lift_lref_ge n h d H2)))))))))))) (\lambda (k: K).(\lambda -(t0: T).(\lambda (H: ((\forall (h: nat).(\forall (d: nat).(\forall (f: ((nat -\to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).((lt m d) \to -(eq nat (g m) (f m))))) \to ((eq nat (g d) w) \to (((\forall (m: nat).((le d -m) \to (eq nat (g (S m)) (f m))))) \to (eq nat (weight_map f (lift h d t0)) -(weight_map g (lift (S h) d t0)))))))))))).(\lambda (t1: T).(\lambda (H0: -((\forall (h: nat).(\forall (d: nat).(\forall (f: ((nat \to nat))).(\forall -(g: ((nat \to nat))).(((\forall (m: nat).((lt m d) \to (eq nat (g m) (f -m))))) \to ((eq nat (g d) w) \to (((\forall (m: nat).((le d m) \to (eq nat (g -(S m)) (f m))))) \to (eq nat (weight_map f (lift h d t1)) (weight_map g (lift -(S h) d t1)))))))))))).(\lambda (h: nat).(\lambda (d: nat).(\lambda (f: ((nat -\to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H1: ((\forall (m: -nat).((lt m d) \to (eq nat (g m) (f m)))))).(\lambda (H2: (eq nat (g d) -w)).(\lambda (H3: ((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f -m)))))).(K_ind (\lambda (k0: K).(eq nat (weight_map f (lift h d (THead k0 t0 -t1))) (weight_map g (lift (S h) d (THead k0 t0 t1))))) (\lambda (b: -B).(eq_ind_r T (THead (Bind b) (lift h d t0) (lift h (s (Bind b) d) t1)) -(\lambda (t2: T).(eq nat (weight_map f t2) (weight_map g (lift (S h) d (THead -(Bind b) t0 t1))))) (eq_ind_r T (THead (Bind b) (lift (S h) d t0) (lift (S h) -(s (Bind b) d) t1)) (\lambda (t2: T).(eq nat (weight_map f (THead (Bind b) -(lift h d t0) (lift h (s (Bind b) d) t1))) (weight_map g t2))) (B_ind -(\lambda (b0: B).(eq nat (match b0 with [Abbr \Rightarrow (S (plus -(weight_map f (lift h d t0)) (weight_map (wadd f (S (weight_map f (lift h d -t0)))) (lift h (S d) t1)))) | Abst \Rightarrow (S (plus (weight_map f (lift h -d t0)) (weight_map (wadd f O) (lift h (S d) t1)))) | Void \Rightarrow (S -(plus (weight_map f (lift h d t0)) (weight_map (wadd f O) (lift h (S d) -t1))))]) (match b0 with [Abbr \Rightarrow (S (plus (weight_map g (lift (S h) -d t0)) (weight_map (wadd g (S (weight_map g (lift (S h) d t0)))) (lift (S h) -(S d) t1)))) | Abst \Rightarrow (S (plus (weight_map g (lift (S h) d t0)) -(weight_map (wadd g O) (lift (S h) (S d) t1)))) | Void \Rightarrow (S (plus -(weight_map g (lift (S h) d t0)) (weight_map (wadd g O) (lift (S h) (S d) -t1))))]))) (f_equal nat nat S (plus (weight_map f (lift h d t0)) (weight_map -(wadd f (S (weight_map f (lift h d t0)))) (lift h (S d) t1))) (plus -(weight_map g (lift (S h) d t0)) (weight_map (wadd g (S (weight_map g (lift -(S h) d t0)))) (lift (S h) (S d) t1))) (f_equal2 nat nat nat plus (weight_map -f (lift h d t0)) (weight_map g (lift (S h) d t0)) (weight_map (wadd f (S -(weight_map f (lift h d t0)))) (lift h (S d) t1)) (weight_map (wadd g (S -(weight_map g (lift (S h) d t0)))) (lift (S h) (S d) t1)) (H h d f g H1 H2 -H3) (H0 h (S d) (wadd f (S (weight_map f (lift h d t0)))) (wadd g (S -(weight_map g (lift (S h) d t0)))) (\lambda (m: nat).(\lambda (H4: (lt m (S -d))).(or_ind (eq nat m O) (ex2 nat (\lambda (m0: nat).(eq nat m (S m0))) -(\lambda (m0: nat).(lt m0 d))) (eq nat (wadd g (S (weight_map g (lift (S h) d -t0))) m) (wadd f (S (weight_map f (lift h d t0))) m)) (\lambda (H5: (eq nat m -O)).(eq_ind_r nat O (\lambda (n: nat).(eq nat (wadd g (S (weight_map g (lift -(S h) d t0))) n) (wadd f (S (weight_map f (lift h d t0))) n))) (f_equal nat -nat S (weight_map g (lift (S h) d t0)) (weight_map f (lift h d t0)) (sym_eq -nat (weight_map f (lift h d t0)) (weight_map g (lift (S h) d t0)) (H h d f g -H1 H2 H3))) m H5)) (\lambda (H5: (ex2 nat (\lambda (m0: nat).(eq nat m (S -m0))) (\lambda (m0: nat).(lt m0 d)))).(ex2_ind nat (\lambda (m0: nat).(eq nat -m (S m0))) (\lambda (m0: nat).(lt m0 d)) (eq nat (wadd g (S (weight_map g -(lift (S h) d t0))) m) (wadd f (S (weight_map f (lift h d t0))) m)) (\lambda -(x: nat).(\lambda (H6: (eq nat m (S x))).(\lambda (H7: (lt x d)).(eq_ind_r -nat (S x) (\lambda (n: nat).(eq nat (wadd g (S (weight_map g (lift (S h) d -t0))) n) (wadd f (S (weight_map f (lift h d t0))) n))) (H1 x H7) m H6)))) -H5)) (lt_gen_xS m d H4)))) H2 (\lambda (m: nat).(\lambda (H4: (le (S d) -m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S n))) (\lambda (n: nat).(le d -n)) (eq nat (g m) (wadd f (S (weight_map f (lift h d t0))) m)) (\lambda (x: -nat).(\lambda (H5: (eq nat m (S x))).(\lambda (H6: (le d x)).(eq_ind_r nat (S -x) (\lambda (n: nat).(eq nat (g n) (wadd f (S (weight_map f (lift h d t0))) -n))) (H3 x H6) m H5)))) (le_gen_S d m H4))))))) (f_equal nat nat S (plus -(weight_map f (lift h d t0)) (weight_map (wadd f O) (lift h (S d) t1))) (plus -(weight_map g (lift (S h) d t0)) (weight_map (wadd g O) (lift (S h) (S d) -t1))) (f_equal2 nat nat nat plus (weight_map f (lift h d t0)) (weight_map g -(lift (S h) d t0)) (weight_map (wadd f O) (lift h (S d) t1)) (weight_map -(wadd g O) (lift (S h) (S d) t1)) (H h d f g H1 H2 H3) (H0 h (S d) (wadd f O) -(wadd g O) (\lambda (m: nat).(\lambda (H4: (lt m (S d))).(or_ind (eq nat m O) -(ex2 nat (\lambda (m0: nat).(eq nat m (S m0))) (\lambda (m0: nat).(lt m0 d))) -(eq nat (wadd g O m) (wadd f O m)) (\lambda (H5: (eq nat m O)).(eq_ind_r nat -O (\lambda (n: nat).(eq nat (wadd g O n) (wadd f O n))) (refl_equal nat O) m -H5)) (\lambda (H5: (ex2 nat (\lambda (m0: nat).(eq nat m (S m0))) (\lambda -(m0: nat).(lt m0 d)))).(ex2_ind nat (\lambda (m0: nat).(eq nat m (S m0))) -(\lambda (m0: nat).(lt m0 d)) (eq nat (wadd g O m) (wadd f O m)) (\lambda (x: -nat).(\lambda (H6: (eq nat m (S x))).(\lambda (H7: (lt x d)).(eq_ind_r nat (S -x) (\lambda (n: nat).(eq nat (wadd g O n) (wadd f O n))) (H1 x H7) m H6)))) -H5)) (lt_gen_xS m d H4)))) H2 (\lambda (m: nat).(\lambda (H4: (le (S d) -m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S n))) (\lambda (n: nat).(le d -n)) (eq nat (g m) (wadd f O m)) (\lambda (x: nat).(\lambda (H5: (eq nat m (S -x))).(\lambda (H6: (le d x)).(eq_ind_r nat (S x) (\lambda (n: nat).(eq nat (g -n) (wadd f O n))) (H3 x H6) m H5)))) (le_gen_S d m H4))))))) (f_equal nat nat -S (plus (weight_map f (lift h d t0)) (weight_map (wadd f O) (lift h (S d) -t1))) (plus (weight_map g (lift (S h) d t0)) (weight_map (wadd g O) (lift (S -h) (S d) t1))) (f_equal2 nat nat nat plus (weight_map f (lift h d t0)) -(weight_map g (lift (S h) d t0)) (weight_map (wadd f O) (lift h (S d) t1)) -(weight_map (wadd g O) (lift (S h) (S d) t1)) (H h d f g H1 H2 H3) (H0 h (S -d) (wadd f O) (wadd g O) (\lambda (m: nat).(\lambda (H4: (lt m (S -d))).(or_ind (eq nat m O) (ex2 nat (\lambda (m0: nat).(eq nat m (S m0))) -(\lambda (m0: nat).(lt m0 d))) (eq nat (wadd g O m) (wadd f O m)) (\lambda -(H5: (eq nat m O)).(eq_ind_r nat O (\lambda (n: nat).(eq nat (wadd g O n) -(wadd f O n))) (refl_equal nat O) m H5)) (\lambda (H5: (ex2 nat (\lambda (m0: -nat).(eq nat m (S m0))) (\lambda (m0: nat).(lt m0 d)))).(ex2_ind nat (\lambda -(m0: nat).(eq nat m (S m0))) (\lambda (m0: nat).(lt m0 d)) (eq nat (wadd g O -m) (wadd f O m)) (\lambda (x: nat).(\lambda (H6: (eq nat m (S x))).(\lambda -(H7: (lt x d)).(eq_ind_r nat (S x) (\lambda (n: nat).(eq nat (wadd g O n) -(wadd f O n))) (H1 x H7) m H6)))) H5)) (lt_gen_xS m d H4)))) H2 (\lambda (m: -nat).(\lambda (H4: (le (S d) m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S -n))) (\lambda (n: nat).(le d n)) (eq nat (g m) (wadd f O m)) (\lambda (x: -nat).(\lambda (H5: (eq nat m (S x))).(\lambda (H6: (le d x)).(eq_ind_r nat (S -x) (\lambda (n: nat).(eq nat (g n) (wadd f O n))) (H3 x H6) m H5)))) -(le_gen_S d m H4))))))) b) (lift (S h) d (THead (Bind b) t0 t1)) (lift_head -(Bind b) t0 t1 (S h) d)) (lift h d (THead (Bind b) t0 t1)) (lift_head (Bind -b) t0 t1 h d))) (\lambda (f0: F).(eq_ind_r T (THead (Flat f0) (lift h d t0) -(lift h (s (Flat f0) d) t1)) (\lambda (t2: T).(eq nat (weight_map f t2) -(weight_map g (lift (S h) d (THead (Flat f0) t0 t1))))) (eq_ind_r T (THead -(Flat f0) (lift (S h) d t0) (lift (S h) (s (Flat f0) d) t1)) (\lambda (t2: -T).(eq nat (weight_map f (THead (Flat f0) (lift h d t0) (lift h (s (Flat f0) -d) t1))) (weight_map g t2))) (f_equal nat nat S (plus (weight_map f (lift h d -t0)) (weight_map f (lift h d t1))) (plus (weight_map g (lift (S h) d t0)) -(weight_map g (lift (S h) d t1))) (f_equal2 nat nat nat plus (weight_map f -(lift h d t0)) (weight_map g (lift (S h) d t0)) (weight_map f (lift h d t1)) -(weight_map g (lift (S h) d t1)) (H h d f g H1 H2 H3) (H0 h d f g H1 H2 H3))) -(lift (S h) d (THead (Flat f0) t0 t1)) (lift_head (Flat f0) t0 t1 (S h) d)) -(lift h d (THead (Flat f0) t0 t1)) (lift_head (Flat f0) t0 t1 h d))) -k))))))))))))) t)). -(* COMMENTS -Initial nodes: 3697 -END *) - -theorem lift_weight_add_O: - \forall (w: nat).(\forall (t: T).(\forall (h: nat).(\forall (f: ((nat \to -nat))).(eq nat (weight_map f (lift h O t)) (weight_map (wadd f w) (lift (S h) -O t)))))) -\def - \lambda (w: nat).(\lambda (t: T).(\lambda (h: nat).(\lambda (f: ((nat \to -nat))).(lift_weight_add (plus (wadd f w O) O) t h O f (wadd f w) (\lambda (m: -nat).(\lambda (H: (lt m O)).(lt_x_O m H (eq nat (wadd f w m) (f m))))) -(plus_n_O (wadd f w O)) (\lambda (m: nat).(\lambda (_: (le O m)).(refl_equal -nat (f m)))))))). -(* COMMENTS -Initial nodes: 93 -END *) - -theorem lift_tlt_dx: - \forall (k: K).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall -(d: nat).(tlt t (THead k u (lift h d t))))))) -\def - \lambda (k: K).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda -(d: nat).(eq_ind nat (weight (lift h d t)) (\lambda (n: nat).(lt n (weight -(THead k u (lift h d t))))) (tlt_head_dx k u (lift h d t)) (weight t) -(lift_weight t h d)))))). -(* COMMENTS -Initial nodes: 71 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/lift1/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/lift1/defs.ma deleted file mode 100644 index 1f473cbce..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/lift1/defs.ma +++ /dev/null @@ -1,40 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/lift/defs.ma". - -definition trans: - PList \to (nat \to nat) -\def - let rec trans (hds: PList) on hds: (nat \to nat) \def (\lambda (i: -nat).(match hds with [PNil \Rightarrow i | (PCons h d hds0) \Rightarrow (let -j \def (trans hds0 i) in (match (blt j d) with [true \Rightarrow j | false -\Rightarrow (plus j h)]))])) in trans. - -definition lift1: - PList \to (T \to T) -\def - let rec lift1 (hds: PList) on hds: (T \to T) \def (\lambda (t: T).(match hds -with [PNil \Rightarrow t | (PCons h d hds0) \Rightarrow (lift h d (lift1 hds0 -t))])) in lift1. - -definition lifts1: - PList \to (TList \to TList) -\def - let rec lifts1 (hds: PList) (ts: TList) on ts: TList \def (match ts with -[TNil \Rightarrow TNil | (TCons t ts0) \Rightarrow (TCons (lift1 hds t) -(lifts1 hds ts0))]) in lifts1. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/lift1/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/lift1/fwd.ma deleted file mode 100644 index e9ae2d11c..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/lift1/fwd.ma +++ /dev/null @@ -1,164 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/lift1/defs.ma". - -include "Basic-1/lift/fwd.ma". - -theorem lift1_sort: - \forall (n: nat).(\forall (is: PList).(eq T (lift1 is (TSort n)) (TSort n))) -\def - \lambda (n: nat).(\lambda (is: PList).(PList_ind (\lambda (p: PList).(eq T -(lift1 p (TSort n)) (TSort n))) (refl_equal T (TSort n)) (\lambda (n0: -nat).(\lambda (n1: nat).(\lambda (p: PList).(\lambda (H: (eq T (lift1 p -(TSort n)) (TSort n))).(eq_ind_r T (TSort n) (\lambda (t: T).(eq T (lift n0 -n1 t) (TSort n))) (refl_equal T (TSort n)) (lift1 p (TSort n)) H))))) is)). -(* COMMENTS -Initial nodes: 99 -END *) - -theorem lift1_lref: - \forall (hds: PList).(\forall (i: nat).(eq T (lift1 hds (TLRef i)) (TLRef -(trans hds i)))) -\def - \lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (i: nat).(eq T -(lift1 p (TLRef i)) (TLRef (trans p i))))) (\lambda (i: nat).(refl_equal T -(TLRef i))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda -(H: ((\forall (i: nat).(eq T (lift1 p (TLRef i)) (TLRef (trans p -i)))))).(\lambda (i: nat).(eq_ind_r T (TLRef (trans p i)) (\lambda (t: T).(eq -T (lift n n0 t) (TLRef (match (blt (trans p i) n0) with [true \Rightarrow -(trans p i) | false \Rightarrow (plus (trans p i) n)])))) (refl_equal T -(TLRef (match (blt (trans p i) n0) with [true \Rightarrow (trans p i) | false -\Rightarrow (plus (trans p i) n)]))) (lift1 p (TLRef i)) (H i))))))) hds). -(* COMMENTS -Initial nodes: 165 -END *) - -theorem lift1_bind: - \forall (b: B).(\forall (hds: PList).(\forall (u: T).(\forall (t: T).(eq T -(lift1 hds (THead (Bind b) u t)) (THead (Bind b) (lift1 hds u) (lift1 (Ss -hds) t)))))) -\def - \lambda (b: B).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall -(u: T).(\forall (t: T).(eq T (lift1 p (THead (Bind b) u t)) (THead (Bind b) -(lift1 p u) (lift1 (Ss p) t)))))) (\lambda (u: T).(\lambda (t: T).(refl_equal -T (THead (Bind b) u t)))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: -PList).(\lambda (H: ((\forall (u: T).(\forall (t: T).(eq T (lift1 p (THead -(Bind b) u t)) (THead (Bind b) (lift1 p u) (lift1 (Ss p) t))))))).(\lambda -(u: T).(\lambda (t: T).(eq_ind_r T (THead (Bind b) (lift1 p u) (lift1 (Ss p) -t)) (\lambda (t0: T).(eq T (lift n n0 t0) (THead (Bind b) (lift n n0 (lift1 p -u)) (lift n (S n0) (lift1 (Ss p) t))))) (eq_ind_r T (THead (Bind b) (lift n -n0 (lift1 p u)) (lift n (S n0) (lift1 (Ss p) t))) (\lambda (t0: T).(eq T t0 -(THead (Bind b) (lift n n0 (lift1 p u)) (lift n (S n0) (lift1 (Ss p) t))))) -(refl_equal T (THead (Bind b) (lift n n0 (lift1 p u)) (lift n (S n0) (lift1 -(Ss p) t)))) (lift n n0 (THead (Bind b) (lift1 p u) (lift1 (Ss p) t))) -(lift_bind b (lift1 p u) (lift1 (Ss p) t) n n0)) (lift1 p (THead (Bind b) u -t)) (H u t)))))))) hds)). -(* COMMENTS -Initial nodes: 379 -END *) - -theorem lift1_flat: - \forall (f: F).(\forall (hds: PList).(\forall (u: T).(\forall (t: T).(eq T -(lift1 hds (THead (Flat f) u t)) (THead (Flat f) (lift1 hds u) (lift1 hds -t)))))) -\def - \lambda (f: F).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall -(u: T).(\forall (t: T).(eq T (lift1 p (THead (Flat f) u t)) (THead (Flat f) -(lift1 p u) (lift1 p t)))))) (\lambda (u: T).(\lambda (t: T).(refl_equal T -(THead (Flat f) u t)))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: -PList).(\lambda (H: ((\forall (u: T).(\forall (t: T).(eq T (lift1 p (THead -(Flat f) u t)) (THead (Flat f) (lift1 p u) (lift1 p t))))))).(\lambda (u: -T).(\lambda (t: T).(eq_ind_r T (THead (Flat f) (lift1 p u) (lift1 p t)) -(\lambda (t0: T).(eq T (lift n n0 t0) (THead (Flat f) (lift n n0 (lift1 p u)) -(lift n n0 (lift1 p t))))) (eq_ind_r T (THead (Flat f) (lift n n0 (lift1 p -u)) (lift n n0 (lift1 p t))) (\lambda (t0: T).(eq T t0 (THead (Flat f) (lift -n n0 (lift1 p u)) (lift n n0 (lift1 p t))))) (refl_equal T (THead (Flat f) -(lift n n0 (lift1 p u)) (lift n n0 (lift1 p t)))) (lift n n0 (THead (Flat f) -(lift1 p u) (lift1 p t))) (lift_flat f (lift1 p u) (lift1 p t) n n0)) (lift1 -p (THead (Flat f) u t)) (H u t)))))))) hds)). -(* COMMENTS -Initial nodes: 353 -END *) - -theorem lift1_cons_tail: - \forall (t: T).(\forall (h: nat).(\forall (d: nat).(\forall (hds: PList).(eq -T (lift1 (PConsTail hds h d) t) (lift1 hds (lift h d t)))))) -\def - \lambda (t: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (hds: -PList).(PList_ind (\lambda (p: PList).(eq T (lift1 (PConsTail p h d) t) -(lift1 p (lift h d t)))) (refl_equal T (lift h d t)) (\lambda (n: -nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: (eq T (lift1 -(PConsTail p h d) t) (lift1 p (lift h d t)))).(eq_ind_r T (lift1 p (lift h d -t)) (\lambda (t0: T).(eq T (lift n n0 t0) (lift n n0 (lift1 p (lift h d -t))))) (refl_equal T (lift n n0 (lift1 p (lift h d t)))) (lift1 (PConsTail p -h d) t) H))))) hds)))). -(* COMMENTS -Initial nodes: 171 -END *) - -theorem lifts1_flat: - \forall (f: F).(\forall (hds: PList).(\forall (t: T).(\forall (ts: -TList).(eq T (lift1 hds (THeads (Flat f) ts t)) (THeads (Flat f) (lifts1 hds -ts) (lift1 hds t)))))) -\def - \lambda (f: F).(\lambda (hds: PList).(\lambda (t: T).(\lambda (ts: -TList).(TList_ind (\lambda (t0: TList).(eq T (lift1 hds (THeads (Flat f) t0 -t)) (THeads (Flat f) (lifts1 hds t0) (lift1 hds t)))) (refl_equal T (lift1 -hds t)) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H: (eq T (lift1 hds -(THeads (Flat f) t1 t)) (THeads (Flat f) (lifts1 hds t1) (lift1 hds -t)))).(eq_ind_r T (THead (Flat f) (lift1 hds t0) (lift1 hds (THeads (Flat f) -t1 t))) (\lambda (t2: T).(eq T t2 (THead (Flat f) (lift1 hds t0) (THeads -(Flat f) (lifts1 hds t1) (lift1 hds t))))) (eq_ind_r T (THeads (Flat f) -(lifts1 hds t1) (lift1 hds t)) (\lambda (t2: T).(eq T (THead (Flat f) (lift1 -hds t0) t2) (THead (Flat f) (lift1 hds t0) (THeads (Flat f) (lifts1 hds t1) -(lift1 hds t))))) (refl_equal T (THead (Flat f) (lift1 hds t0) (THeads (Flat -f) (lifts1 hds t1) (lift1 hds t)))) (lift1 hds (THeads (Flat f) t1 t)) H) -(lift1 hds (THead (Flat f) t0 (THeads (Flat f) t1 t))) (lift1_flat f hds t0 -(THeads (Flat f) t1 t)))))) ts)))). -(* COMMENTS -Initial nodes: 329 -END *) - -theorem lifts1_nil: - \forall (ts: TList).(eq TList (lifts1 PNil ts) ts) -\def - \lambda (ts: TList).(TList_ind (\lambda (t: TList).(eq TList (lifts1 PNil t) -t)) (refl_equal TList TNil) (\lambda (t: T).(\lambda (t0: TList).(\lambda (H: -(eq TList (lifts1 PNil t0) t0)).(eq_ind_r TList t0 (\lambda (t1: TList).(eq -TList (TCons t t1) (TCons t t0))) (refl_equal TList (TCons t t0)) (lifts1 -PNil t0) H)))) ts). -(* COMMENTS -Initial nodes: 83 -END *) - -theorem lifts1_cons: - \forall (h: nat).(\forall (d: nat).(\forall (hds: PList).(\forall (ts: -TList).(eq TList (lifts1 (PCons h d hds) ts) (lifts h d (lifts1 hds ts)))))) -\def - \lambda (h: nat).(\lambda (d: nat).(\lambda (hds: PList).(\lambda (ts: -TList).(TList_ind (\lambda (t: TList).(eq TList (lifts1 (PCons h d hds) t) -(lifts h d (lifts1 hds t)))) (refl_equal TList TNil) (\lambda (t: T).(\lambda -(t0: TList).(\lambda (H: (eq TList (lifts1 (PCons h d hds) t0) (lifts h d -(lifts1 hds t0)))).(eq_ind_r TList (lifts h d (lifts1 hds t0)) (\lambda (t1: -TList).(eq TList (TCons (lift h d (lift1 hds t)) t1) (TCons (lift h d (lift1 -hds t)) (lifts h d (lifts1 hds t0))))) (refl_equal TList (TCons (lift h d -(lift1 hds t)) (lifts h d (lifts1 hds t0)))) (lifts1 (PCons h d hds) t0) -H)))) ts)))). -(* COMMENTS -Initial nodes: 187 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/lift1/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/lift1/props.ma deleted file mode 100644 index ebda0267b..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/lift1/props.ma +++ /dev/null @@ -1,139 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/lift/props.ma". - -include "Basic-1/drop1/defs.ma". - -theorem lift1_lift1: - \forall (is1: PList).(\forall (is2: PList).(\forall (t: T).(eq T (lift1 is1 -(lift1 is2 t)) (lift1 (papp is1 is2) t)))) -\def - \lambda (is1: PList).(PList_ind (\lambda (p: PList).(\forall (is2: -PList).(\forall (t: T).(eq T (lift1 p (lift1 is2 t)) (lift1 (papp p is2) -t))))) (\lambda (is2: PList).(\lambda (t: T).(refl_equal T (lift1 is2 t)))) -(\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: -((\forall (is2: PList).(\forall (t: T).(eq T (lift1 p (lift1 is2 t)) (lift1 -(papp p is2) t)))))).(\lambda (is2: PList).(\lambda (t: T).(f_equal3 nat nat -T T lift n n n0 n0 (lift1 p (lift1 is2 t)) (lift1 (papp p is2) t) (refl_equal -nat n) (refl_equal nat n0) (H is2 t)))))))) is1). -(* COMMENTS -Initial nodes: 145 -END *) - -theorem lift1_xhg: - \forall (hds: PList).(\forall (t: T).(eq T (lift1 (Ss hds) (lift (S O) O t)) -(lift (S O) O (lift1 hds t)))) -\def - \lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (t: T).(eq T -(lift1 (Ss p) (lift (S O) O t)) (lift (S O) O (lift1 p t))))) (\lambda (t: -T).(refl_equal T (lift (S O) O t))) (\lambda (h: nat).(\lambda (d: -nat).(\lambda (p: PList).(\lambda (H: ((\forall (t: T).(eq T (lift1 (Ss p) -(lift (S O) O t)) (lift (S O) O (lift1 p t)))))).(\lambda (t: T).(eq_ind_r T -(lift (S O) O (lift1 p t)) (\lambda (t0: T).(eq T (lift h (S d) t0) (lift (S -O) O (lift h d (lift1 p t))))) (eq_ind nat (plus (S O) d) (\lambda (n: -nat).(eq T (lift h n (lift (S O) O (lift1 p t))) (lift (S O) O (lift h d -(lift1 p t))))) (eq_ind_r T (lift (S O) O (lift h d (lift1 p t))) (\lambda -(t0: T).(eq T t0 (lift (S O) O (lift h d (lift1 p t))))) (refl_equal T (lift -(S O) O (lift h d (lift1 p t)))) (lift h (plus (S O) d) (lift (S O) O (lift1 -p t))) (lift_d (lift1 p t) h (S O) d O (le_O_n d))) (S d) (refl_equal nat (S -d))) (lift1 (Ss p) (lift (S O) O t)) (H t))))))) hds). -(* COMMENTS -Initial nodes: 371 -END *) - -theorem lifts1_xhg: - \forall (hds: PList).(\forall (ts: TList).(eq TList (lifts1 (Ss hds) (lifts -(S O) O ts)) (lifts (S O) O (lifts1 hds ts)))) -\def - \lambda (hds: PList).(\lambda (ts: TList).(TList_ind (\lambda (t: TList).(eq -TList (lifts1 (Ss hds) (lifts (S O) O t)) (lifts (S O) O (lifts1 hds t)))) -(refl_equal TList TNil) (\lambda (t: T).(\lambda (t0: TList).(\lambda (H: (eq -TList (lifts1 (Ss hds) (lifts (S O) O t0)) (lifts (S O) O (lifts1 hds -t0)))).(eq_ind_r T (lift (S O) O (lift1 hds t)) (\lambda (t1: T).(eq TList -(TCons t1 (lifts1 (Ss hds) (lifts (S O) O t0))) (TCons (lift (S O) O (lift1 -hds t)) (lifts (S O) O (lifts1 hds t0))))) (eq_ind_r TList (lifts (S O) O -(lifts1 hds t0)) (\lambda (t1: TList).(eq TList (TCons (lift (S O) O (lift1 -hds t)) t1) (TCons (lift (S O) O (lift1 hds t)) (lifts (S O) O (lifts1 hds -t0))))) (refl_equal TList (TCons (lift (S O) O (lift1 hds t)) (lifts (S O) O -(lifts1 hds t0)))) (lifts1 (Ss hds) (lifts (S O) O t0)) H) (lift1 (Ss hds) -(lift (S O) O t)) (lift1_xhg hds t))))) ts)). -(* COMMENTS -Initial nodes: 307 -END *) - -theorem lift1_free: - \forall (hds: PList).(\forall (i: nat).(\forall (t: T).(eq T (lift1 hds -(lift (S i) O t)) (lift (S (trans hds i)) O (lift1 (ptrans hds i) t))))) -\def - \lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (i: -nat).(\forall (t: T).(eq T (lift1 p (lift (S i) O t)) (lift (S (trans p i)) O -(lift1 (ptrans p i) t)))))) (\lambda (i: nat).(\lambda (t: T).(refl_equal T -(lift (S i) O t)))) (\lambda (h: nat).(\lambda (d: nat).(\lambda (hds0: -PList).(\lambda (H: ((\forall (i: nat).(\forall (t: T).(eq T (lift1 hds0 -(lift (S i) O t)) (lift (S (trans hds0 i)) O (lift1 (ptrans hds0 i) -t))))))).(\lambda (i: nat).(\lambda (t: T).(eq_ind_r T (lift (S (trans hds0 -i)) O (lift1 (ptrans hds0 i) t)) (\lambda (t0: T).(eq T (lift h d t0) (lift -(S (match (blt (trans hds0 i) d) with [true \Rightarrow (trans hds0 i) | -false \Rightarrow (plus (trans hds0 i) h)])) O (lift1 (match (blt (trans hds0 -i) d) with [true \Rightarrow (PCons h (minus d (S (trans hds0 i))) (ptrans -hds0 i)) | false \Rightarrow (ptrans hds0 i)]) t)))) (xinduction bool (blt -(trans hds0 i) d) (\lambda (b: bool).(eq T (lift h d (lift (S (trans hds0 i)) -O (lift1 (ptrans hds0 i) t))) (lift (S (match b with [true \Rightarrow (trans -hds0 i) | false \Rightarrow (plus (trans hds0 i) h)])) O (lift1 (match b with -[true \Rightarrow (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) | -false \Rightarrow (ptrans hds0 i)]) t)))) (\lambda (x_x: bool).(bool_ind -(\lambda (b: bool).((eq bool (blt (trans hds0 i) d) b) \to (eq T (lift h d -(lift (S (trans hds0 i)) O (lift1 (ptrans hds0 i) t))) (lift (S (match b with -[true \Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i) -h)])) O (lift1 (match b with [true \Rightarrow (PCons h (minus d (S (trans -hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)]) t))))) -(\lambda (H0: (eq bool (blt (trans hds0 i) d) true)).(eq_ind_r nat (plus (S -(trans hds0 i)) (minus d (S (trans hds0 i)))) (\lambda (n: nat).(eq T (lift h -n (lift (S (trans hds0 i)) O (lift1 (ptrans hds0 i) t))) (lift (S (trans hds0 -i)) O (lift1 (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) t)))) -(eq_ind_r T (lift (S (trans hds0 i)) O (lift h (minus d (S (trans hds0 i))) -(lift1 (ptrans hds0 i) t))) (\lambda (t0: T).(eq T t0 (lift (S (trans hds0 -i)) O (lift1 (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) t)))) -(refl_equal T (lift (S (trans hds0 i)) O (lift1 (PCons h (minus d (S (trans -hds0 i))) (ptrans hds0 i)) t))) (lift h (plus (S (trans hds0 i)) (minus d (S -(trans hds0 i)))) (lift (S (trans hds0 i)) O (lift1 (ptrans hds0 i) t))) -(lift_d (lift1 (ptrans hds0 i) t) h (S (trans hds0 i)) (minus d (S (trans -hds0 i))) O (le_O_n (minus d (S (trans hds0 i)))))) d (le_plus_minus (S -(trans hds0 i)) d (bge_le (S (trans hds0 i)) d (le_bge (S (trans hds0 i)) d -(lt_le_S (trans hds0 i) d (blt_lt d (trans hds0 i) H0))))))) (\lambda (H0: -(eq bool (blt (trans hds0 i) d) false)).(eq_ind_r T (lift (plus h (S (trans -hds0 i))) O (lift1 (ptrans hds0 i) t)) (\lambda (t0: T).(eq T t0 (lift (S -(plus (trans hds0 i) h)) O (lift1 (ptrans hds0 i) t)))) (eq_ind nat (S (plus -h (trans hds0 i))) (\lambda (n: nat).(eq T (lift n O (lift1 (ptrans hds0 i) -t)) (lift (S (plus (trans hds0 i) h)) O (lift1 (ptrans hds0 i) t)))) -(eq_ind_r nat (plus (trans hds0 i) h) (\lambda (n: nat).(eq T (lift (S n) O -(lift1 (ptrans hds0 i) t)) (lift (S (plus (trans hds0 i) h)) O (lift1 (ptrans -hds0 i) t)))) (refl_equal T (lift (S (plus (trans hds0 i) h)) O (lift1 -(ptrans hds0 i) t))) (plus h (trans hds0 i)) (plus_sym h (trans hds0 i))) -(plus h (S (trans hds0 i))) (plus_n_Sm h (trans hds0 i))) (lift h d (lift (S -(trans hds0 i)) O (lift1 (ptrans hds0 i) t))) (lift_free (lift1 (ptrans hds0 -i) t) (S (trans hds0 i)) h O d (eq_ind nat (S (plus O (trans hds0 i))) -(\lambda (n: nat).(le d n)) (eq_ind_r nat (plus (trans hds0 i) O) (\lambda -(n: nat).(le d (S n))) (le_S d (plus (trans hds0 i) O) (le_plus_trans d -(trans hds0 i) O (bge_le d (trans hds0 i) H0))) (plus O (trans hds0 i)) -(plus_sym O (trans hds0 i))) (plus O (S (trans hds0 i))) (plus_n_Sm O (trans -hds0 i))) (le_O_n d)))) x_x))) (lift1 hds0 (lift (S i) O t)) (H i t)))))))) -hds). -(* COMMENTS -Initial nodes: 1339 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/llt/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/llt/defs.ma deleted file mode 100644 index 96d869935..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/llt/defs.ma +++ /dev/null @@ -1,30 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/A/defs.ma". - -definition lweight: - A \to nat -\def - let rec lweight (a: A) on a: nat \def (match a with [(ASort _ _) \Rightarrow -O | (AHead a1 a2) \Rightarrow (S (plus (lweight a1) (lweight a2)))]) in -lweight. - -definition llt: - A \to (A \to Prop) -\def - \lambda (a1: A).(\lambda (a2: A).(lt (lweight a1) (lweight a2))). - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/llt/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/llt/props.ma deleted file mode 100644 index ef1f15a96..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/llt/props.ma +++ /dev/null @@ -1,114 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/llt/defs.ma". - -include "Basic-1/leq/defs.ma". - -theorem lweight_repl: - \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (eq nat -(lweight a1) (lweight a2))))) -\def - \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1 -a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(eq nat (lweight a) (lweight -a0)))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: nat).(\lambda (n2: -nat).(\lambda (k: nat).(\lambda (_: (eq A (aplus g (ASort h1 n1) k) (aplus g -(ASort h2 n2) k))).(refl_equal nat O))))))) (\lambda (a0: A).(\lambda (a3: -A).(\lambda (_: (leq g a0 a3)).(\lambda (H1: (eq nat (lweight a0) (lweight -a3))).(\lambda (a4: A).(\lambda (a5: A).(\lambda (_: (leq g a4 a5)).(\lambda -(H3: (eq nat (lweight a4) (lweight a5))).(f_equal nat nat S (plus (lweight -a0) (lweight a4)) (plus (lweight a3) (lweight a5)) (f_equal2 nat nat nat plus -(lweight a0) (lweight a3) (lweight a4) (lweight a5) H1 H3)))))))))) a1 a2 -H)))). -(* COMMENTS -Initial nodes: 189 -END *) - -theorem llt_repl: - \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall -(a3: A).((llt a1 a3) \to (llt a2 a3)))))) -\def - \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1 -a2)).(\lambda (a3: A).(\lambda (H0: (lt (lweight a1) (lweight a3))).(let H1 -\def (eq_ind nat (lweight a1) (\lambda (n: nat).(lt n (lweight a3))) H0 -(lweight a2) (lweight_repl g a1 a2 H)) in H1)))))). -(* COMMENTS -Initial nodes: 61 -END *) - -theorem llt_trans: - \forall (a1: A).(\forall (a2: A).(\forall (a3: A).((llt a1 a2) \to ((llt a2 -a3) \to (llt a1 a3))))) -\def - \lambda (a1: A).(\lambda (a2: A).(\lambda (a3: A).(\lambda (H: (lt (lweight -a1) (lweight a2))).(\lambda (H0: (lt (lweight a2) (lweight a3))).(lt_trans -(lweight a1) (lweight a2) (lweight a3) H H0))))). -(* COMMENTS -Initial nodes: 43 -END *) - -theorem llt_head_sx: - \forall (a1: A).(\forall (a2: A).(llt a1 (AHead a1 a2))) -\def - \lambda (a1: A).(\lambda (a2: A).(le_n_S (lweight a1) (plus (lweight a1) -(lweight a2)) (le_plus_l (lweight a1) (lweight a2)))). -(* COMMENTS -Initial nodes: 29 -END *) - -theorem llt_head_dx: - \forall (a1: A).(\forall (a2: A).(llt a2 (AHead a1 a2))) -\def - \lambda (a1: A).(\lambda (a2: A).(le_n_S (lweight a2) (plus (lweight a1) -(lweight a2)) (le_plus_r (lweight a1) (lweight a2)))). -(* COMMENTS -Initial nodes: 29 -END *) - -theorem llt_wf__q_ind: - \forall (P: ((A \to Prop))).(((\forall (n: nat).((\lambda (P0: ((A \to -Prop))).(\lambda (n0: nat).(\forall (a: A).((eq nat (lweight a) n0) \to (P0 -a))))) P n))) \to (\forall (a: A).(P a))) -\def - let Q \def (\lambda (P: ((A \to Prop))).(\lambda (n: nat).(\forall (a: -A).((eq nat (lweight a) n) \to (P a))))) in (\lambda (P: ((A \to -Prop))).(\lambda (H: ((\forall (n: nat).(\forall (a: A).((eq nat (lweight a) -n) \to (P a)))))).(\lambda (a: A).(H (lweight a) a (refl_equal nat (lweight -a)))))). -(* COMMENTS -Initial nodes: 61 -END *) - -theorem llt_wf_ind: - \forall (P: ((A \to Prop))).(((\forall (a2: A).(((\forall (a1: A).((llt a1 -a2) \to (P a1)))) \to (P a2)))) \to (\forall (a: A).(P a))) -\def - let Q \def (\lambda (P: ((A \to Prop))).(\lambda (n: nat).(\forall (a: -A).((eq nat (lweight a) n) \to (P a))))) in (\lambda (P: ((A \to -Prop))).(\lambda (H: ((\forall (a2: A).(((\forall (a1: A).((lt (lweight a1) -(lweight a2)) \to (P a1)))) \to (P a2))))).(\lambda (a: A).(llt_wf__q_ind -(\lambda (a0: A).(P a0)) (\lambda (n: nat).(lt_wf_ind n (Q (\lambda (a0: -A).(P a0))) (\lambda (n0: nat).(\lambda (H0: ((\forall (m: nat).((lt m n0) -\to (Q (\lambda (a0: A).(P a0)) m))))).(\lambda (a0: A).(\lambda (H1: (eq nat -(lweight a0) n0)).(let H2 \def (eq_ind_r nat n0 (\lambda (n1: nat).(\forall -(m: nat).((lt m n1) \to (\forall (a1: A).((eq nat (lweight a1) m) \to (P -a1)))))) H0 (lweight a0) H1) in (H a0 (\lambda (a1: A).(\lambda (H3: (lt -(lweight a1) (lweight a0))).(H2 (lweight a1) H3 a1 (refl_equal nat (lweight -a1))))))))))))) a)))). -(* COMMENTS -Initial nodes: 179 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/next_plus/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/next_plus/defs.ma deleted file mode 100644 index 1ff7ecdff..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/next_plus/defs.ma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/G/defs.ma". - -definition next_plus: - G \to (nat \to (nat \to nat)) -\def - let rec next_plus (g: G) (n: nat) (i: nat) on i: nat \def (match i with [O -\Rightarrow n | (S i0) \Rightarrow (next g (next_plus g n i0))]) in next_plus. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/next_plus/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/next_plus/props.ma deleted file mode 100644 index 258c3a711..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/next_plus/props.ma +++ /dev/null @@ -1,68 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/next_plus/defs.ma". - -theorem next_plus_assoc: - \forall (g: G).(\forall (n: nat).(\forall (h1: nat).(\forall (h2: nat).(eq -nat (next_plus g (next_plus g n h1) h2) (next_plus g n (plus h1 h2)))))) -\def - \lambda (g: G).(\lambda (n: nat).(\lambda (h1: nat).(nat_ind (\lambda (n0: -nat).(\forall (h2: nat).(eq nat (next_plus g (next_plus g n n0) h2) -(next_plus g n (plus n0 h2))))) (\lambda (h2: nat).(refl_equal nat (next_plus -g n h2))) (\lambda (n0: nat).(\lambda (_: ((\forall (h2: nat).(eq nat -(next_plus g (next_plus g n n0) h2) (next_plus g n (plus n0 h2)))))).(\lambda -(h2: nat).(nat_ind (\lambda (n1: nat).(eq nat (next_plus g (next g (next_plus -g n n0)) n1) (next g (next_plus g n (plus n0 n1))))) (eq_ind nat n0 (\lambda -(n1: nat).(eq nat (next g (next_plus g n n0)) (next g (next_plus g n n1)))) -(refl_equal nat (next g (next_plus g n n0))) (plus n0 O) (plus_n_O n0)) -(\lambda (n1: nat).(\lambda (H0: (eq nat (next_plus g (next g (next_plus g n -n0)) n1) (next g (next_plus g n (plus n0 n1))))).(eq_ind nat (S (plus n0 n1)) -(\lambda (n2: nat).(eq nat (next g (next_plus g (next g (next_plus g n n0)) -n1)) (next g (next_plus g n n2)))) (f_equal nat nat (next g) (next_plus g -(next g (next_plus g n n0)) n1) (next g (next_plus g n (plus n0 n1))) H0) -(plus n0 (S n1)) (plus_n_Sm n0 n1)))) h2)))) h1))). -(* COMMENTS -Initial nodes: 351 -END *) - -theorem next_plus_next: - \forall (g: G).(\forall (n: nat).(\forall (h: nat).(eq nat (next_plus g -(next g n) h) (next g (next_plus g n h))))) -\def - \lambda (g: G).(\lambda (n: nat).(\lambda (h: nat).(eq_ind_r nat (next_plus -g n (plus (S O) h)) (\lambda (n0: nat).(eq nat n0 (next g (next_plus g n -h)))) (refl_equal nat (next g (next_plus g n h))) (next_plus g (next_plus g n -(S O)) h) (next_plus_assoc g n (S O) h)))). -(* COMMENTS -Initial nodes: 87 -END *) - -theorem next_plus_lt: - \forall (g: G).(\forall (h: nat).(\forall (n: nat).(lt n (next_plus g (next -g n) h)))) -\def - \lambda (g: G).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall (n0: -nat).(lt n0 (next_plus g (next g n0) n)))) (\lambda (n: nat).(next_lt g n)) -(\lambda (n: nat).(\lambda (H: ((\forall (n0: nat).(lt n0 (next_plus g (next -g n0) n))))).(\lambda (n0: nat).(eq_ind nat (next_plus g (next g (next g n0)) -n) (\lambda (n1: nat).(lt n0 n1)) (lt_trans n0 (next g n0) (next_plus g (next -g (next g n0)) n) (next_lt g n0) (H (next g n0))) (next g (next_plus g (next -g n0) n)) (next_plus_next g (next g n0) n))))) h)). -(* COMMENTS -Initial nodes: 153 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/arity.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/arity.ma deleted file mode 100644 index 98770d9e9..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/arity.ma +++ /dev/null @@ -1,496 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/nf2/fwd.ma". - -include "Basic-1/arity/subst0.ma". - -theorem arity_nf2_inv_all: - \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t -a) \to ((nf2 c t) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t -(THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c w))) -(\lambda (w: T).(\lambda (u: T).(nf2 (CHead c (Bind Abst) w) u)))) (ex nat -(\lambda (n: nat).(eq T t (TSort n)))) (ex3_2 TList nat (\lambda (ws: -TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i))))) -(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c (TLRef i))))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: -(arity g c t a)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (_: -A).((nf2 c0 t0) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t0 -(THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) -(\lambda (w: T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat -(\lambda (n: nat).(eq T t0 (TSort n)))) (ex3_2 TList nat (\lambda (ws: -TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i))))) -(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))))))) (\lambda (c0: C).(\lambda -(n: nat).(\lambda (_: (nf2 c0 (TSort n))).(or3_intro1 (ex3_2 T T (\lambda (w: -T).(\lambda (u: T).(eq T (TSort n) (THead (Bind Abst) w u)))) (\lambda (w: -T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead -c0 (Bind Abst) w) u)))) (ex nat (\lambda (n0: nat).(eq T (TSort n) (TSort -n0)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (TSort -n) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef -i))))) (ex_intro nat (\lambda (n0: nat).(eq T (TSort n) (TSort n0))) n -(refl_equal T (TSort n))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: -T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr) -u))).(\lambda (a0: A).(\lambda (_: (arity g d u a0)).(\lambda (_: (((nf2 d u) -\to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u (THead (Bind -Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 d w))) (\lambda (w: -T).(\lambda (u0: T).(nf2 (CHead d (Bind Abst) w) u0)))) (ex nat (\lambda (n: -nat).(eq T u (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i0: -nat).(eq T u (THeads (Flat Appl) ws (TLRef i0))))) (\lambda (ws: -TList).(\lambda (_: nat).(nfs2 d ws))) (\lambda (_: TList).(\lambda (i0: -nat).(nf2 d (TLRef i0))))))))).(\lambda (H3: (nf2 c0 (TLRef -i))).(nf2_gen_lref c0 d u i H0 H3 (or3 (ex3_2 T T (\lambda (w: T).(\lambda -(u0: T).(eq T (TLRef i) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda -(_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind -Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (TLRef i) (TSort n)))) (ex3_2 -TList nat (\lambda (ws: TList).(\lambda (i0: nat).(eq T (TLRef i) (THeads -(Flat Appl) ws (TLRef i0))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 -ws))) (\lambda (_: TList).(\lambda (i0: nat).(nf2 c0 (TLRef -i0)))))))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda -(i: nat).(\lambda (_: (getl i c0 (CHead d (Bind Abst) u))).(\lambda (a0: -A).(\lambda (_: (arity g d u (asucc g a0))).(\lambda (_: (((nf2 d u) \to (or3 -(ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u (THead (Bind Abst) w -u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 d w))) (\lambda (w: T).(\lambda -(u0: T).(nf2 (CHead d (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T u -(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i0: nat).(eq T u -(THeads (Flat Appl) ws (TLRef i0))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 d ws))) (\lambda (_: TList).(\lambda (i0: nat).(nf2 d (TLRef -i0))))))))).(\lambda (H3: (nf2 c0 (TLRef i))).(or3_intro2 (ex3_2 T T (\lambda -(w: T).(\lambda (u0: T).(eq T (TLRef i) (THead (Bind Abst) w u0)))) (\lambda -(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 -(CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (TLRef i) -(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i0: nat).(eq T -(TLRef i) (THeads (Flat Appl) ws (TLRef i0))))) (\lambda (ws: TList).(\lambda -(_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i0: nat).(nf2 c0 (TLRef -i0))))) (ex3_2_intro TList nat (\lambda (ws: TList).(\lambda (i0: nat).(eq T -(TLRef i) (THeads (Flat Appl) ws (TLRef i0))))) (\lambda (ws: TList).(\lambda -(_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i0: nat).(nf2 c0 (TLRef -i0)))) TNil i (refl_equal T (TLRef i)) I H3))))))))))) (\lambda (b: -B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c0: C).(\lambda (u: -T).(\lambda (a1: A).(\lambda (_: (arity g c0 u a1)).(\lambda (_: (((nf2 c0 u) -\to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u (THead (Bind -Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: -T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: -nat).(eq T u (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: -nat).(eq T u (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: -TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: -nat).(nf2 c0 (TLRef i))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda -(H3: (arity g (CHead c0 (Bind b) u) t0 a2)).(\lambda (_: (((nf2 (CHead c0 -(Bind b) u) t0) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T t0 -(THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 (CHead c0 -(Bind b) u) w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead (CHead c0 (Bind -b) u) (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort n)))) -(ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads -(Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 -(CHead c0 (Bind b) u) ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 (CHead -c0 (Bind b) u) (TLRef i))))))))).(\lambda (H5: (nf2 c0 (THead (Bind b) u -t0))).(B_ind (\lambda (b0: B).((not (eq B b0 Abst)) \to ((arity g (CHead c0 -(Bind b0) u) t0 a2) \to ((nf2 c0 (THead (Bind b0) u t0)) \to (or3 (ex3_2 T T -(\lambda (w: T).(\lambda (u0: T).(eq T (THead (Bind b0) u t0) (THead (Bind -Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: -T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: -nat).(eq T (THead (Bind b0) u t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws: -TList).(\lambda (i: nat).(eq T (THead (Bind b0) u t0) (THeads (Flat Appl) ws -(TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda -(_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))))))) (\lambda (_: (not (eq -B Abbr Abst))).(\lambda (_: (arity g (CHead c0 (Bind Abbr) u) t0 -a2)).(\lambda (H8: (nf2 c0 (THead (Bind Abbr) u t0))).(nf2_gen_abbr c0 u t0 -H8 (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Bind Abbr) -u t0) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 -w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) -(ex nat (\lambda (n: nat).(eq T (THead (Bind Abbr) u t0) (TSort n)))) (ex3_2 -TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Bind Abbr) u -t0) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef -i)))))))))) (\lambda (H6: (not (eq B Abst Abst))).(\lambda (_: (arity g -(CHead c0 (Bind Abst) u) t0 a2)).(\lambda (_: (nf2 c0 (THead (Bind Abst) u -t0))).(let H9 \def (match (H6 (refl_equal B Abst)) in False return (\lambda -(_: False).(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead -(Bind Abst) u t0) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: -T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) -w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind Abst) u t0) (TSort -n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead -(Bind Abst) u t0) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: -TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: -nat).(nf2 c0 (TLRef i))))))) with []) in H9)))) (\lambda (_: (not (eq B Void -Abst))).(\lambda (H7: (arity g (CHead c0 (Bind Void) u) t0 a2)).(\lambda (H8: -(nf2 c0 (THead (Bind Void) u t0))).(let H9 \def (arity_gen_cvoid g (CHead c0 -(Bind Void) u) t0 a2 H7 c0 u O (getl_refl Void c0 u)) in (ex_ind T (\lambda -(v: T).(eq T t0 (lift (S O) O v))) (or3 (ex3_2 T T (\lambda (w: T).(\lambda -(u0: T).(eq T (THead (Bind Void) u t0) (THead (Bind Abst) w u0)))) (\lambda -(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 -(CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind -Void) u t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: -nat).(eq T (THead (Bind Void) u t0) (THeads (Flat Appl) ws (TLRef i))))) -(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))) (\lambda (x: T).(\lambda -(H10: (eq T t0 (lift (S O) O x))).(let H11 \def (eq_ind T t0 (\lambda (t1: -T).(nf2 c0 (THead (Bind Void) u t1))) H8 (lift (S O) O x) H10) in (eq_ind_r T -(lift (S O) O x) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (w: T).(\lambda -(u0: T).(eq T (THead (Bind Void) u t1) (THead (Bind Abst) w u0)))) (\lambda -(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 -(CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind -Void) u t1) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: -nat).(eq T (THead (Bind Void) u t1) (THeads (Flat Appl) ws (TLRef i))))) -(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))) (nf2_gen_void c0 u x H11 -(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Bind Void) u -(lift (S O) O x)) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: -T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) -w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind Void) u (lift (S O) O -x)) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq -T (THead (Bind Void) u (lift (S O) O x)) (THeads (Flat Appl) ws (TLRef i))))) -(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))) t0 H10)))) H9))))) b H0 H3 -H5))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda -(_: (arity g c0 u (asucc g a1))).(\lambda (_: (((nf2 c0 u) \to (or3 (ex3_2 T -T (\lambda (w: T).(\lambda (u0: T).(eq T u (THead (Bind Abst) w u0)))) -(\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: -T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T u -(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T u -(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef -i))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c0 -(Bind Abst) u) t0 a2)).(\lambda (_: (((nf2 (CHead c0 (Bind Abst) u) t0) \to -(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) -w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 (CHead c0 (Bind Abst) u) w))) -(\lambda (w: T).(\lambda (u0: T).(nf2 (CHead (CHead c0 (Bind Abst) u) (Bind -Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort n)))) (ex3_2 TList -nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws -(TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 (CHead c0 (Bind -Abst) u) ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 (CHead c0 (Bind -Abst) u) (TLRef i))))))))).(\lambda (H4: (nf2 c0 (THead (Bind Abst) u -t0))).(let H5 \def (nf2_gen_abst c0 u t0 H4) in (land_ind (nf2 c0 u) (nf2 -(CHead c0 (Bind Abst) u) t0) (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: -T).(eq T (THead (Bind Abst) u t0) (THead (Bind Abst) w u0)))) (\lambda (w: -T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead -c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind Abst) u -t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq -T (THead (Bind Abst) u t0) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: -TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: -nat).(nf2 c0 (TLRef i)))))) (\lambda (H6: (nf2 c0 u)).(\lambda (H7: (nf2 -(CHead c0 (Bind Abst) u) t0)).(or3_intro0 (ex3_2 T T (\lambda (w: T).(\lambda -(u0: T).(eq T (THead (Bind Abst) u t0) (THead (Bind Abst) w u0)))) (\lambda -(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 -(CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind -Abst) u t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: -nat).(eq T (THead (Bind Abst) u t0) (THeads (Flat Appl) ws (TLRef i))))) -(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))) (ex3_2_intro T T (\lambda (w: -T).(\lambda (u0: T).(eq T (THead (Bind Abst) u t0) (THead (Bind Abst) w -u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda -(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0))) u t0 (refl_equal T (THead (Bind -Abst) u t0)) H6 H7)))) H5)))))))))))) (\lambda (c0: C).(\lambda (u: -T).(\lambda (a1: A).(\lambda (_: (arity g c0 u a1)).(\lambda (_: (((nf2 c0 u) -\to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u (THead (Bind -Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: -T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: -nat).(eq T u (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: -nat).(eq T u (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: -TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: -nat).(nf2 c0 (TLRef i))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda -(H2: (arity g c0 t0 (AHead a1 a2))).(\lambda (H3: (((nf2 c0 t0) \to (or3 -(ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w -u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda -(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T -t0 (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T -t0 (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef -i))))))))).(\lambda (H4: (nf2 c0 (THead (Flat Appl) u t0))).(let H5 \def -(nf2_gen_flat Appl c0 u t0 H4) in (land_ind (nf2 c0 u) (nf2 c0 t0) (or3 -(ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t0) -(THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) -(\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat -(\lambda (n: nat).(eq T (THead (Flat Appl) u t0) (TSort n)))) (ex3_2 TList -nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u t0) -(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef -i)))))) (\lambda (H6: (nf2 c0 u)).(\lambda (H7: (nf2 c0 t0)).(let H_x \def -(H3 H7) in (let H8 \def H_x in (or3_ind (ex3_2 T T (\lambda (w: T).(\lambda -(u0: T).(eq T t0 (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: -T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) -w) u0)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort n)))) (ex3_2 TList nat -(\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef -i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))) (or3 (ex3_2 T T (\lambda (w: -T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t0) (THead (Bind Abst) w -u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda -(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T -(THead (Flat Appl) u t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws: -TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u t0) (THeads (Flat Appl) -ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) -(\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))) (\lambda (H9: -(ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w -u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda -(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0))))).(ex3_2_ind T T (\lambda (w: -T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w u0)))) (\lambda (w: -T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead -c0 (Bind Abst) w) u0))) (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq -T (THead (Flat Appl) u t0) (THead (Bind Abst) w u0)))) (\lambda (w: -T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead -c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u -t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq -T (THead (Flat Appl) u t0) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: -TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: -nat).(nf2 c0 (TLRef i)))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H10: -(eq T t0 (THead (Bind Abst) x0 x1))).(\lambda (_: (nf2 c0 x0)).(\lambda (_: -(nf2 (CHead c0 (Bind Abst) x0) x1)).(let H13 \def (eq_ind T t0 (\lambda (t1: -T).(nf2 c0 (THead (Flat Appl) u t1))) H4 (THead (Bind Abst) x0 x1) H10) in -(let H14 \def (eq_ind T t0 (\lambda (t1: T).(arity g c0 t1 (AHead a1 a2))) H2 -(THead (Bind Abst) x0 x1) H10) in (eq_ind_r T (THead (Bind Abst) x0 x1) -(\lambda (t1: T).(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T -(THead (Flat Appl) u t1) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda -(_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind -Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u t1) -(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T -(THead (Flat Appl) u t1) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: -TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: -nat).(nf2 c0 (TLRef i))))))) (nf2_gen_beta c0 u x0 x1 H13 (or3 (ex3_2 T T -(\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl) u (THead (Bind -Abst) x0 x1)) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: -T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) -w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u (THead (Bind -Abst) x0 x1)) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: -nat).(eq T (THead (Flat Appl) u (THead (Bind Abst) x0 x1)) (THeads (Flat -Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) -(\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))) t0 H10)))))))) -H9)) (\lambda (H9: (ex nat (\lambda (n: nat).(eq T t0 (TSort n))))).(ex_ind -nat (\lambda (n: nat).(eq T t0 (TSort n))) (or3 (ex3_2 T T (\lambda (w: -T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t0) (THead (Bind Abst) w -u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda -(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T -(THead (Flat Appl) u t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws: -TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u t0) (THeads (Flat Appl) -ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) -(\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))) (\lambda (x: -nat).(\lambda (H10: (eq T t0 (TSort x))).(let H11 \def (eq_ind T t0 (\lambda -(t1: T).(nf2 c0 (THead (Flat Appl) u t1))) H4 (TSort x) H10) in (let H12 \def -(eq_ind T t0 (\lambda (t1: T).(arity g c0 t1 (AHead a1 a2))) H2 (TSort x) -H10) in (eq_ind_r T (TSort x) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (w: -T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t1) (THead (Bind Abst) w -u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda -(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T -(THead (Flat Appl) u t1) (TSort n)))) (ex3_2 TList nat (\lambda (ws: -TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u t1) (THeads (Flat Appl) -ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) -(\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))) (let H_x0 \def -(leq_gen_head1 g a1 a2 (ASort O x) (arity_gen_sort g c0 x (AHead a1 a2) H12)) -in (let H13 \def H_x0 in (ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq -g a1 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: -A).(\lambda (a4: A).(eq A (ASort O x) (AHead a3 a4)))) (or3 (ex3_2 T T -(\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl) u (TSort x)) (THead -(Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda -(w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda -(n: nat).(eq T (THead (Flat Appl) u (TSort x)) (TSort n)))) (ex3_2 TList nat -(\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u (TSort x)) -(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef -i)))))) (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g a1 -x0)).(\lambda (_: (leq g a2 x1)).(\lambda (H16: (eq A (ASort O x) (AHead x0 -x1))).(let H17 \def (eq_ind A (ASort O x) (\lambda (ee: A).(match ee in A -return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) -\Rightarrow False])) I (AHead x0 x1) H16) in (False_ind (or3 (ex3_2 T T -(\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl) u (TSort x)) (THead -(Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda -(w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda -(n: nat).(eq T (THead (Flat Appl) u (TSort x)) (TSort n)))) (ex3_2 TList nat -(\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u (TSort x)) -(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef -i)))))) H17))))))) H13))) t0 H10))))) H9)) (\lambda (H9: (ex3_2 TList nat -(\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef -i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))).(ex3_2_ind TList nat (\lambda -(ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i))))) -(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))) (or3 (ex3_2 T T (\lambda (w: -T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t0) (THead (Bind Abst) w -u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda -(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T -(THead (Flat Appl) u t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws: -TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u t0) (THeads (Flat Appl) -ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) -(\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))) (\lambda (x0: -TList).(\lambda (x1: nat).(\lambda (H10: (eq T t0 (THeads (Flat Appl) x0 -(TLRef x1)))).(\lambda (H11: (nfs2 c0 x0)).(\lambda (H12: (nf2 c0 (TLRef -x1))).(let H13 \def (eq_ind T t0 (\lambda (t1: T).(nf2 c0 (THead (Flat Appl) -u t1))) H4 (THeads (Flat Appl) x0 (TLRef x1)) H10) in (let H14 \def (eq_ind T -t0 (\lambda (t1: T).(arity g c0 t1 (AHead a1 a2))) H2 (THeads (Flat Appl) x0 -(TLRef x1)) H10) in (eq_ind_r T (THeads (Flat Appl) x0 (TLRef x1)) (\lambda -(t1: T).(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat -Appl) u t1) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 -c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) -(ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u t1) (TSort n)))) (ex3_2 -TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u -t1) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef -i))))))) (or3_intro2 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead -(Flat Appl) u (THeads (Flat Appl) x0 (TLRef x1))) (THead (Bind Abst) w u0)))) -(\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: -T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T -(THead (Flat Appl) u (THeads (Flat Appl) x0 (TLRef x1))) (TSort n)))) (ex3_2 -TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u -(THeads (Flat Appl) x0 (TLRef x1))) (THeads (Flat Appl) ws (TLRef i))))) -(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))) (ex3_2_intro TList nat -(\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u (THeads -(Flat Appl) x0 (TLRef x1))) (THeads (Flat Appl) ws (TLRef i))))) (\lambda -(ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: -nat).(nf2 c0 (TLRef i)))) (TCons u x0) x1 (refl_equal T (THead (Flat Appl) u -(THeads (Flat Appl) x0 (TLRef x1)))) (conj (nf2 c0 u) (nfs2 c0 x0) H6 H11) -H12)) t0 H10)))))))) H9)) H8))))) H5)))))))))))) (\lambda (c0: C).(\lambda -(u: T).(\lambda (a0: A).(\lambda (_: (arity g c0 u (asucc g a0))).(\lambda -(_: (((nf2 c0 u) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u -(THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) -(\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat -(\lambda (n: nat).(eq T u (TSort n)))) (ex3_2 TList nat (\lambda (ws: -TList).(\lambda (i: nat).(eq T u (THeads (Flat Appl) ws (TLRef i))))) -(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))))).(\lambda (t0: T).(\lambda -(_: (arity g c0 t0 a0)).(\lambda (_: (((nf2 c0 t0) \to (or3 (ex3_2 T T -(\lambda (w: T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w u0)))) -(\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: -T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T t0 -(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 -(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef -i))))))))).(\lambda (H4: (nf2 c0 (THead (Flat Cast) u t0))).(nf2_gen_cast c0 -u t0 H4 (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat -Cast) u t0) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 -c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) -(ex nat (\lambda (n: nat).(eq T (THead (Flat Cast) u t0) (TSort n)))) (ex3_2 -TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Cast) u -t0) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef -i)))))))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda -(_: (arity g c0 t0 a1)).(\lambda (H1: (((nf2 c0 t0) \to (or3 (ex3_2 T T -(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda -(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 -(CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort -n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 -(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef -i))))))))).(\lambda (a2: A).(\lambda (_: (leq g a1 a2)).(\lambda (H3: (nf2 c0 -t0)).(let H_x \def (H1 H3) in (let H4 \def H_x in (or3_ind (ex3_2 T T -(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda -(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 -(CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort -n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 -(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef -i))))) (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind -Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: -T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: -nat).(eq T t0 (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: -nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: -TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: -nat).(nf2 c0 (TLRef i)))))) (\lambda (H5: (ex3_2 T T (\lambda (w: T).(\lambda -(u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: -T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) -u))))).(ex3_2_ind T T (\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind -Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: -T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u))) (or3 (ex3_2 T T -(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda -(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 -(CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort -n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 -(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef -i)))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H6: (eq T t0 (THead (Bind -Abst) x0 x1))).(\lambda (H7: (nf2 c0 x0)).(\lambda (H8: (nf2 (CHead c0 (Bind -Abst) x0) x1)).(eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t1: T).(or3 -(ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t1 (THead (Bind Abst) w -u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda -(u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t1 -(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t1 -(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef -i))))))) (or3_intro0 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T (THead -(Bind Abst) x0 x1) (THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: -T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) -u)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind Abst) x0 x1) (TSort n)))) -(ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Bind -Abst) x0 x1) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: -TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: -nat).(nf2 c0 (TLRef i))))) (ex3_2_intro T T (\lambda (w: T).(\lambda (u: -T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) w u)))) (\lambda (w: -T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead -c0 (Bind Abst) w) u))) x0 x1 (refl_equal T (THead (Bind Abst) x0 x1)) H7 H8)) -t0 H6)))))) H5)) (\lambda (H5: (ex nat (\lambda (n: nat).(eq T t0 (TSort -n))))).(ex_ind nat (\lambda (n: nat).(eq T t0 (TSort n))) (or3 (ex3_2 T T -(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda -(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 -(CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort -n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 -(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef -i)))))) (\lambda (x: nat).(\lambda (H6: (eq T t0 (TSort x))).(eq_ind_r T -(TSort x) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u: -T).(eq T t1 (THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 -c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) -(ex nat (\lambda (n: nat).(eq T t1 (TSort n)))) (ex3_2 TList nat (\lambda -(ws: TList).(\lambda (i: nat).(eq T t1 (THeads (Flat Appl) ws (TLRef i))))) -(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))) (or3_intro1 (ex3_2 T T -(\lambda (w: T).(\lambda (u: T).(eq T (TSort x) (THead (Bind Abst) w u)))) -(\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: -T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T (TSort -x) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T -(TSort x) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda -(_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef -i))))) (ex_intro nat (\lambda (n: nat).(eq T (TSort x) (TSort n))) x -(refl_equal T (TSort x)))) t0 H6))) H5)) (\lambda (H5: (ex3_2 TList nat -(\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef -i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))).(ex3_2_ind TList nat (\lambda -(ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i))))) -(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))) (or3 (ex3_2 T T (\lambda (w: -T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda (w: -T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead -c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort n)))) -(ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads -(Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 -ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))) (\lambda -(x0: TList).(\lambda (x1: nat).(\lambda (H6: (eq T t0 (THeads (Flat Appl) x0 -(TLRef x1)))).(\lambda (H7: (nfs2 c0 x0)).(\lambda (H8: (nf2 c0 (TLRef -x1))).(eq_ind_r T (THeads (Flat Appl) x0 (TLRef x1)) (\lambda (t1: T).(or3 -(ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t1 (THead (Bind Abst) w -u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda -(u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t1 -(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t1 -(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef -i))))))) (or3_intro2 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T (THeads -(Flat Appl) x0 (TLRef x1)) (THead (Bind Abst) w u)))) (\lambda (w: -T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead -c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T (THeads (Flat Appl) -x0 (TLRef x1)) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda -(i: nat).(eq T (THeads (Flat Appl) x0 (TLRef x1)) (THeads (Flat Appl) ws -(TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda -(_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))) (ex3_2_intro TList nat -(\lambda (ws: TList).(\lambda (i: nat).(eq T (THeads (Flat Appl) x0 (TLRef -x1)) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef -i)))) x0 x1 (refl_equal T (THeads (Flat Appl) x0 (TLRef x1))) H7 H8)) t0 -H6)))))) H5)) H4))))))))))) c t a H))))). -(* COMMENTS -Initial nodes: 9193 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/dec.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/dec.ma deleted file mode 100644 index 33b652baf..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/dec.ma +++ /dev/null @@ -1,200 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/nf2/defs.ma". - -include "Basic-1/pr2/clen.ma". - -include "Basic-1/pr2/fwd.ma". - -include "Basic-1/pr0/dec.ma". - -include "Basic-1/C/props.ma". - -theorem nf2_dec: - \forall (c: C).(\forall (t1: T).(or (nf2 c t1) (ex2 T (\lambda (t2: T).((eq -T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c t1 t2))))) -\def - \lambda (c: C).(c_tail_ind (\lambda (c0: C).(\forall (t1: T).(or (\forall -(t2: T).((pr2 c0 t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 -t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c0 t1 t2)))))) (\lambda -(n: nat).(\lambda (t1: T).(let H_x \def (nf0_dec t1) in (let H \def H_x in -(or_ind (\forall (t2: T).((pr0 t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: -T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t1 t2))) -(or (\forall (t2: T).((pr2 (CSort n) t1 t2) \to (eq T t1 t2))) (ex2 T -(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: -T).(pr2 (CSort n) t1 t2)))) (\lambda (H0: ((\forall (t2: T).((pr0 t1 t2) \to -(eq T t1 t2))))).(or_introl (\forall (t2: T).((pr2 (CSort n) t1 t2) \to (eq T -t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) -(\lambda (t2: T).(pr2 (CSort n) t1 t2))) (\lambda (t2: T).(\lambda (H1: (pr2 -(CSort n) t1 t2)).(let H_y \def (pr2_gen_csort t1 t2 n H1) in (H0 t2 -H_y)))))) (\lambda (H0: (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall -(P: Prop).P))) (\lambda (t2: T).(pr0 t1 t2)))).(ex2_ind T (\lambda (t2: -T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t1 t2)) -(or (\forall (t2: T).((pr2 (CSort n) t1 t2) \to (eq T t1 t2))) (ex2 T -(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: -T).(pr2 (CSort n) t1 t2)))) (\lambda (x: T).(\lambda (H1: (((eq T t1 x) \to -(\forall (P: Prop).P)))).(\lambda (H2: (pr0 t1 x)).(or_intror (\forall (t2: -T).((pr2 (CSort n) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T -t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CSort n) t1 t2))) -(ex_intro2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) -(\lambda (t2: T).(pr2 (CSort n) t1 t2)) x H1 (pr2_free (CSort n) t1 x -H2)))))) H0)) H))))) (\lambda (c0: C).(\lambda (H: ((\forall (t1: T).(or -(\forall (t2: T).((pr2 c0 t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: -T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c0 t1 -t2))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (t1: T).(let H_x \def (H -t1) in (let H0 \def H_x in (or_ind (\forall (t2: T).((pr2 c0 t1 t2) \to (eq T -t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) -(\lambda (t2: T).(pr2 c0 t1 t2))) (or (\forall (t2: T).((pr2 (CTail k t c0) -t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall -(P: Prop).P))) (\lambda (t2: T).(pr2 (CTail k t c0) t1 t2)))) (\lambda (H1: -((\forall (t2: T).((pr2 c0 t1 t2) \to (eq T t1 t2))))).(K_ind (\lambda (k0: -K).(or (\forall (t2: T).((pr2 (CTail k0 t c0) t1 t2) \to (eq T t1 t2))) (ex2 -T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: -T).(pr2 (CTail k0 t c0) t1 t2))))) (\lambda (b: B).(B_ind (\lambda (b0: -B).(or (\forall (t2: T).((pr2 (CTail (Bind b0) t c0) t1 t2) \to (eq T t1 -t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) -(\lambda (t2: T).(pr2 (CTail (Bind b0) t c0) t1 t2))))) (let H_x0 \def -(dnf_dec t t1 (clen c0)) in (let H2 \def H_x0 in (ex_ind T (\lambda (v: -T).(or (subst0 (clen c0) t t1 (lift (S O) (clen c0) v)) (eq T t1 (lift (S O) -(clen c0) v)))) (or (\forall (t2: T).((pr2 (CTail (Bind Abbr) t c0) t1 t2) -\to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) t1 t2)))) (\lambda -(x: T).(\lambda (H3: (or (subst0 (clen c0) t t1 (lift (S O) (clen c0) x)) (eq -T t1 (lift (S O) (clen c0) x)))).(or_ind (subst0 (clen c0) t t1 (lift (S O) -(clen c0) x)) (eq T t1 (lift (S O) (clen c0) x)) (or (\forall (t2: T).((pr2 -(CTail (Bind Abbr) t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: -T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail -(Bind Abbr) t c0) t1 t2)))) (\lambda (H4: (subst0 (clen c0) t t1 (lift (S O) -(clen c0) x))).(let H_x1 \def (getl_ctail_clen Abbr t c0) in (let H5 \def -H_x1 in (ex_ind nat (\lambda (n: nat).(getl (clen c0) (CTail (Bind Abbr) t -c0) (CHead (CSort n) (Bind Abbr) t))) (or (\forall (t2: T).((pr2 (CTail (Bind -Abbr) t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) -\to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) t1 -t2)))) (\lambda (x0: nat).(\lambda (H6: (getl (clen c0) (CTail (Bind Abbr) t -c0) (CHead (CSort x0) (Bind Abbr) t))).(or_intror (\forall (t2: T).((pr2 -(CTail (Bind Abbr) t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: -T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail -(Bind Abbr) t c0) t1 t2))) (ex_intro2 T (\lambda (t2: T).((eq T t1 t2) \to -(\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) t1 -t2)) (lift (S O) (clen c0) x) (\lambda (H7: (eq T t1 (lift (S O) (clen c0) -x))).(\lambda (P: Prop).(let H8 \def (eq_ind T t1 (\lambda (t0: T).(subst0 -(clen c0) t t0 (lift (S O) (clen c0) x))) H4 (lift (S O) (clen c0) x) H7) in -(subst0_gen_lift_false x t (lift (S O) (clen c0) x) (S O) (clen c0) (clen c0) -(le_n (clen c0)) (eq_ind_r nat (plus (S O) (clen c0)) (\lambda (n: nat).(lt -(clen c0) n)) (le_n (plus (S O) (clen c0))) (plus (clen c0) (S O)) (plus_sym -(clen c0) (S O))) H8 P)))) (pr2_delta (CTail (Bind Abbr) t c0) (CSort x0) t -(clen c0) H6 t1 t1 (pr0_refl t1) (lift (S O) (clen c0) x) H4))))) H5)))) -(\lambda (H4: (eq T t1 (lift (S O) (clen c0) x))).(let H5 \def (eq_ind T t1 -(\lambda (t0: T).(\forall (t2: T).((pr2 c0 t0 t2) \to (eq T t0 t2)))) H1 -(lift (S O) (clen c0) x) H4) in (eq_ind_r T (lift (S O) (clen c0) x) (\lambda -(t0: T).(or (\forall (t2: T).((pr2 (CTail (Bind Abbr) t c0) t0 t2) \to (eq T -t0 t2))) (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: Prop).P))) -(\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) t0 t2))))) (or_introl (\forall -(t2: T).((pr2 (CTail (Bind Abbr) t c0) (lift (S O) (clen c0) x) t2) \to (eq T -(lift (S O) (clen c0) x) t2))) (ex2 T (\lambda (t2: T).((eq T (lift (S O) -(clen c0) x) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail -(Bind Abbr) t c0) (lift (S O) (clen c0) x) t2))) (\lambda (t2: T).(\lambda -(H6: (pr2 (CTail (Bind Abbr) t c0) (lift (S O) (clen c0) x) t2)).(let H_x1 -\def (pr2_gen_ctail (Bind Abbr) c0 t (lift (S O) (clen c0) x) t2 H6) in (let -H7 \def H_x1 in (or_ind (pr2 c0 (lift (S O) (clen c0) x) t2) (ex3 T (\lambda -(_: T).(eq K (Bind Abbr) (Bind Abbr))) (\lambda (t0: T).(pr0 (lift (S O) -(clen c0) x) t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2))) (eq T (lift -(S O) (clen c0) x) t2) (\lambda (H8: (pr2 c0 (lift (S O) (clen c0) x) -t2)).(H5 t2 H8)) (\lambda (H8: (ex3 T (\lambda (_: T).(eq K (Bind Abbr) (Bind -Abbr))) (\lambda (t0: T).(pr0 (lift (S O) (clen c0) x) t0)) (\lambda (t0: -T).(subst0 (clen c0) t t0 t2)))).(ex3_ind T (\lambda (_: T).(eq K (Bind Abbr) -(Bind Abbr))) (\lambda (t0: T).(pr0 (lift (S O) (clen c0) x) t0)) (\lambda -(t0: T).(subst0 (clen c0) t t0 t2)) (eq T (lift (S O) (clen c0) x) t2) -(\lambda (x0: T).(\lambda (_: (eq K (Bind Abbr) (Bind Abbr))).(\lambda (H10: -(pr0 (lift (S O) (clen c0) x) x0)).(\lambda (H11: (subst0 (clen c0) t x0 -t2)).(ex2_ind T (\lambda (t3: T).(eq T x0 (lift (S O) (clen c0) t3))) -(\lambda (t3: T).(pr0 x t3)) (eq T (lift (S O) (clen c0) x) t2) (\lambda (x1: -T).(\lambda (H12: (eq T x0 (lift (S O) (clen c0) x1))).(\lambda (_: (pr0 x -x1)).(let H14 \def (eq_ind T x0 (\lambda (t0: T).(subst0 (clen c0) t t0 t2)) -H11 (lift (S O) (clen c0) x1) H12) in (subst0_gen_lift_false x1 t t2 (S O) -(clen c0) (clen c0) (le_n (clen c0)) (eq_ind_r nat (plus (S O) (clen c0)) -(\lambda (n: nat).(lt (clen c0) n)) (le_n (plus (S O) (clen c0))) (plus (clen -c0) (S O)) (plus_sym (clen c0) (S O))) H14 (eq T (lift (S O) (clen c0) x) -t2)))))) (pr0_gen_lift x x0 (S O) (clen c0) H10)))))) H8)) H7)))))) t1 H4))) -H3))) H2))) (or_introl (\forall (t2: T).((pr2 (CTail (Bind Abst) t c0) t1 t2) -\to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abst) t c0) t1 t2))) (\lambda -(t2: T).(\lambda (H2: (pr2 (CTail (Bind Abst) t c0) t1 t2)).(let H_x0 \def -(pr2_gen_ctail (Bind Abst) c0 t t1 t2 H2) in (let H3 \def H_x0 in (or_ind -(pr2 c0 t1 t2) (ex3 T (\lambda (_: T).(eq K (Bind Abst) (Bind Abbr))) -(\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2))) -(eq T t1 t2) (\lambda (H4: (pr2 c0 t1 t2)).(H1 t2 H4)) (\lambda (H4: (ex3 T -(\lambda (_: T).(eq K (Bind Abst) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) -(\lambda (t0: T).(subst0 (clen c0) t t0 t2)))).(ex3_ind T (\lambda (_: T).(eq -K (Bind Abst) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: -T).(subst0 (clen c0) t t0 t2)) (eq T t1 t2) (\lambda (x0: T).(\lambda (H5: -(eq K (Bind Abst) (Bind Abbr))).(\lambda (_: (pr0 t1 x0)).(\lambda (_: -(subst0 (clen c0) t x0 t2)).(let H8 \def (eq_ind K (Bind Abst) (\lambda (ee: -K).(match ee in K return (\lambda (_: K).Prop) with [(Bind b0) \Rightarrow -(match b0 in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | -Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) \Rightarrow -False])) I (Bind Abbr) H5) in (False_ind (eq T t1 t2) H8)))))) H4)) H3)))))) -(or_introl (\forall (t2: T).((pr2 (CTail (Bind Void) t c0) t1 t2) \to (eq T -t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) -(\lambda (t2: T).(pr2 (CTail (Bind Void) t c0) t1 t2))) (\lambda (t2: -T).(\lambda (H2: (pr2 (CTail (Bind Void) t c0) t1 t2)).(let H_x0 \def -(pr2_gen_ctail (Bind Void) c0 t t1 t2 H2) in (let H3 \def H_x0 in (or_ind -(pr2 c0 t1 t2) (ex3 T (\lambda (_: T).(eq K (Bind Void) (Bind Abbr))) -(\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2))) -(eq T t1 t2) (\lambda (H4: (pr2 c0 t1 t2)).(H1 t2 H4)) (\lambda (H4: (ex3 T -(\lambda (_: T).(eq K (Bind Void) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) -(\lambda (t0: T).(subst0 (clen c0) t t0 t2)))).(ex3_ind T (\lambda (_: T).(eq -K (Bind Void) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: -T).(subst0 (clen c0) t t0 t2)) (eq T t1 t2) (\lambda (x0: T).(\lambda (H5: -(eq K (Bind Void) (Bind Abbr))).(\lambda (_: (pr0 t1 x0)).(\lambda (_: -(subst0 (clen c0) t x0 t2)).(let H8 \def (eq_ind K (Bind Void) (\lambda (ee: -K).(match ee in K return (\lambda (_: K).Prop) with [(Bind b0) \Rightarrow -(match b0 in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | -Abst \Rightarrow False | Void \Rightarrow True]) | (Flat _) \Rightarrow -False])) I (Bind Abbr) H5) in (False_ind (eq T t1 t2) H8)))))) H4)) H3)))))) -b)) (\lambda (f: F).(or_introl (\forall (t2: T).((pr2 (CTail (Flat f) t c0) -t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall -(P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Flat f) t c0) t1 t2))) (\lambda -(t2: T).(\lambda (H2: (pr2 (CTail (Flat f) t c0) t1 t2)).(let H_x0 \def -(pr2_gen_ctail (Flat f) c0 t t1 t2 H2) in (let H3 \def H_x0 in (or_ind (pr2 -c0 t1 t2) (ex3 T (\lambda (_: T).(eq K (Flat f) (Bind Abbr))) (\lambda (t0: -T).(pr0 t1 t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2))) (eq T t1 t2) -(\lambda (H4: (pr2 c0 t1 t2)).(H1 t2 H4)) (\lambda (H4: (ex3 T (\lambda (_: -T).(eq K (Flat f) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: -T).(subst0 (clen c0) t t0 t2)))).(ex3_ind T (\lambda (_: T).(eq K (Flat f) -(Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(subst0 (clen -c0) t t0 t2)) (eq T t1 t2) (\lambda (x0: T).(\lambda (H5: (eq K (Flat f) -(Bind Abbr))).(\lambda (_: (pr0 t1 x0)).(\lambda (_: (subst0 (clen c0) t x0 -t2)).(let H8 \def (eq_ind K (Flat f) (\lambda (ee: K).(match ee in K return -(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])) I (Bind Abbr) H5) in (False_ind (eq T t1 t2) H8)))))) H4)) H3))))))) -k)) (\lambda (H1: (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr2 c0 t1 t2)))).(ex2_ind T (\lambda (t2: -T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c0 t1 t2)) -(or (\forall (t2: T).((pr2 (CTail k t c0) t1 t2) \to (eq T t1 t2))) (ex2 T -(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: -T).(pr2 (CTail k t c0) t1 t2)))) (\lambda (x: T).(\lambda (H2: (((eq T t1 x) -\to (\forall (P: Prop).P)))).(\lambda (H3: (pr2 c0 t1 x)).(or_intror (\forall -(t2: T).((pr2 (CTail k t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: -T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail k t -c0) t1 t2))) (ex_intro2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr2 (CTail k t c0) t1 t2)) x H2 (pr2_ctail c0 t1 -x H3 k t)))))) H1)) H0)))))))) c). -(* COMMENTS -Initial nodes: 3653 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/defs.ma deleted file mode 100644 index 98e931c0b..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/defs.ma +++ /dev/null @@ -1,31 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr2/defs.ma". - -definition nf2: - C \to (T \to Prop) -\def - \lambda (c: C).(\lambda (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (eq T t1 -t2)))). - -definition nfs2: - C \to (TList \to Prop) -\def - let rec nfs2 (c: C) (ts: TList) on ts: Prop \def (match ts with [TNil -\Rightarrow True | (TCons t ts0) \Rightarrow (land (nf2 c t) (nfs2 c ts0))]) -in nfs2. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/fwd.ma deleted file mode 100644 index 9138ff2fa..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/fwd.ma +++ /dev/null @@ -1,220 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/nf2/defs.ma". - -include "Basic-1/pr2/clen.ma". - -include "Basic-1/subst0/dec.ma". - -include "Basic-1/T/props.ma". - -theorem nf2_gen_lref: - \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c -(CHead d (Bind Abbr) u)) \to ((nf2 c (TLRef i)) \to (\forall (P: Prop).P)))))) -\def - \lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H0: ((\forall (t2: T).((pr2 -c (TLRef i) t2) \to (eq T (TLRef i) t2))))).(\lambda (P: -Prop).(lift_gen_lref_false (S i) O i (le_O_n i) (le_n (plus O (S i))) u (H0 -(lift (S i) O u) (pr2_delta c d u i H (TLRef i) (TLRef i) (pr0_refl (TLRef -i)) (lift (S i) O u) (subst0_lref u i))) P))))))). -(* COMMENTS -Initial nodes: 129 -END *) - -theorem nf2_gen_abst: - \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Bind Abst) u -t)) \to (land (nf2 c u) (nf2 (CHead c (Bind Abst) u) t))))) -\def - \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: ((\forall (t2: -T).((pr2 c (THead (Bind Abst) u t) t2) \to (eq T (THead (Bind Abst) u t) -t2))))).(conj (\forall (t2: T).((pr2 c u t2) \to (eq T u t2))) (\forall (t2: -T).((pr2 (CHead c (Bind Abst) u) t t2) \to (eq T t t2))) (\lambda (t2: -T).(\lambda (H0: (pr2 c u t2)).(let H1 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | -(TLRef _) \Rightarrow u | (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abst) -u t) (THead (Bind Abst) t2 t) (H (THead (Bind Abst) t2 t) (pr2_head_1 c u t2 -H0 (Bind Abst) t))) in (let H2 \def (eq_ind_r T t2 (\lambda (t0: T).(pr2 c u -t0)) H0 u H1) in (eq_ind T u (\lambda (t0: T).(eq T u t0)) (refl_equal T u) -t2 H1))))) (\lambda (t2: T).(\lambda (H0: (pr2 (CHead c (Bind Abst) u) t -t2)).(let H1 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda -(_: T).T) with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ -_ t0) \Rightarrow t0])) (THead (Bind Abst) u t) (THead (Bind Abst) u t2) (H -(THead (Bind Abst) u t2) (let H_y \def (pr2_gen_cbind Abst c u t t2 H0) in -H_y))) in (let H2 \def (eq_ind_r T t2 (\lambda (t0: T).(pr2 (CHead c (Bind -Abst) u) t t0)) H0 t H1) in (eq_ind T t (\lambda (t0: T).(eq T t t0)) -(refl_equal T t) t2 H1))))))))). -(* COMMENTS -Initial nodes: 353 -END *) - -theorem nf2_gen_cast: - \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Flat Cast) u -t)) \to (\forall (P: Prop).P)))) -\def - \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (nf2 c (THead -(Flat Cast) u t))).(\lambda (P: Prop).(thead_x_y_y (Flat Cast) u t (H t -(pr2_free c (THead (Flat Cast) u t) t (pr0_tau t t (pr0_refl t) u))) P))))). -(* COMMENTS -Initial nodes: 65 -END *) - -theorem nf2_gen_beta: - \forall (c: C).(\forall (u: T).(\forall (v: T).(\forall (t: T).((nf2 c -(THead (Flat Appl) u (THead (Bind Abst) v t))) \to (\forall (P: Prop).P))))) -\def - \lambda (c: C).(\lambda (u: T).(\lambda (v: T).(\lambda (t: T).(\lambda (H: -((\forall (t2: T).((pr2 c (THead (Flat Appl) u (THead (Bind Abst) v t)) t2) -\to (eq T (THead (Flat Appl) u (THead (Bind Abst) v t)) t2))))).(\lambda (P: -Prop).(let H0 \def (eq_ind T (THead (Flat Appl) u (THead (Bind Abst) v t)) -(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow -(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | -(Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u t) (H (THead (Bind -Abbr) u t) (pr2_free c (THead (Flat Appl) u (THead (Bind Abst) v t)) (THead -(Bind Abbr) u t) (pr0_beta v u u (pr0_refl u) t t (pr0_refl t))))) in -(False_ind P H0))))))). -(* COMMENTS -Initial nodes: 183 -END *) - -theorem nf2_gen_flat: - \forall (f: F).(\forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c -(THead (Flat f) u t)) \to (land (nf2 c u) (nf2 c t)))))) -\def - \lambda (f: F).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: -((\forall (t2: T).((pr2 c (THead (Flat f) u t) t2) \to (eq T (THead (Flat f) -u t) t2))))).(conj (\forall (t2: T).((pr2 c u t2) \to (eq T u t2))) (\forall -(t2: T).((pr2 c t t2) \to (eq T t t2))) (\lambda (t2: T).(\lambda (H0: (pr2 c -u t2)).(let H1 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | -(THead _ t0 _) \Rightarrow t0])) (THead (Flat f) u t) (THead (Flat f) t2 t) -(H (THead (Flat f) t2 t) (pr2_head_1 c u t2 H0 (Flat f) t))) in H1))) -(\lambda (t2: T).(\lambda (H0: (pr2 c t t2)).(let H1 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t0) \Rightarrow t0])) -(THead (Flat f) u t) (THead (Flat f) u t2) (H (THead (Flat f) u t2) -(pr2_head_2 c u t t2 (Flat f) (pr2_cflat c t t2 H0 f u)))) in H1)))))))). -(* COMMENTS -Initial nodes: 251 -END *) - -theorem nf2_gen__nf2_gen_aux: - \forall (b: B).(\forall (x: T).(\forall (u: T).(\forall (d: nat).((eq T -(THead (Bind b) u (lift (S O) d x)) x) \to (\forall (P: Prop).P))))) -\def - \lambda (b: B).(\lambda (x: T).(T_ind (\lambda (t: T).(\forall (u: -T).(\forall (d: nat).((eq T (THead (Bind b) u (lift (S O) d t)) t) \to -(\forall (P: Prop).P))))) (\lambda (n: nat).(\lambda (u: T).(\lambda (d: -nat).(\lambda (H: (eq T (THead (Bind b) u (lift (S O) d (TSort n))) (TSort -n))).(\lambda (P: Prop).(let H0 \def (eq_ind T (THead (Bind b) u (lift (S O) -d (TSort n))) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) -with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ -_) \Rightarrow True])) I (TSort n) H) in (False_ind P H0))))))) (\lambda (n: -nat).(\lambda (u: T).(\lambda (d: nat).(\lambda (H: (eq T (THead (Bind b) u -(lift (S O) d (TLRef n))) (TLRef n))).(\lambda (P: Prop).(let H0 \def (eq_ind -T (THead (Bind b) u (lift (S O) d (TLRef n))) (\lambda (ee: T).(match ee in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H) in -(False_ind P H0))))))) (\lambda (k: K).(\lambda (t: T).(\lambda (_: ((\forall -(u: T).(\forall (d: nat).((eq T (THead (Bind b) u (lift (S O) d t)) t) \to -(\forall (P: Prop).P)))))).(\lambda (t0: T).(\lambda (H0: ((\forall (u: -T).(\forall (d: nat).((eq T (THead (Bind b) u (lift (S O) d t0)) t0) \to -(\forall (P: Prop).P)))))).(\lambda (u: T).(\lambda (d: nat).(\lambda (H1: -(eq T (THead (Bind b) u (lift (S O) d (THead k t t0))) (THead k t -t0))).(\lambda (P: Prop).(let H2 \def (f_equal T K (\lambda (e: T).(match e -in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow (Bind b) | (TLRef -_) \Rightarrow (Bind b) | (THead k0 _ _) \Rightarrow k0])) (THead (Bind b) u -(lift (S O) d (THead k t t0))) (THead k t t0) H1) in ((let H3 \def (f_equal T -T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t1 _) \Rightarrow t1])) -(THead (Bind b) u (lift (S O) d (THead k t t0))) (THead k t t0) H1) in ((let -H4 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow (THead k ((let rec lref_map (f: ((nat \to nat))) -(d0: nat) (t1: T) on t1: T \def (match t1 with [(TSort n) \Rightarrow (TSort -n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) with [true \Rightarrow i -| false \Rightarrow (f i)])) | (THead k0 u0 t2) \Rightarrow (THead k0 -(lref_map f d0 u0) (lref_map f (s k0 d0) t2))]) in lref_map) (\lambda (x0: -nat).(plus x0 (S O))) d t) ((let rec lref_map (f: ((nat \to nat))) (d0: nat) -(t1: T) on t1: T \def (match t1 with [(TSort n) \Rightarrow (TSort n) | -(TLRef i) \Rightarrow (TLRef (match (blt i d0) with [true \Rightarrow i | -false \Rightarrow (f i)])) | (THead k0 u0 t2) \Rightarrow (THead k0 (lref_map -f d0 u0) (lref_map f (s k0 d0) t2))]) in lref_map) (\lambda (x0: nat).(plus -x0 (S O))) (s k d) t0)) | (TLRef _) \Rightarrow (THead k ((let rec lref_map -(f: ((nat \to nat))) (d0: nat) (t1: T) on t1: T \def (match t1 with [(TSort -n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) -with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k0 u0 t2) -\Rightarrow (THead k0 (lref_map f d0 u0) (lref_map f (s k0 d0) t2))]) in -lref_map) (\lambda (x0: nat).(plus x0 (S O))) d t) ((let rec lref_map (f: -((nat \to nat))) (d0: nat) (t1: T) on t1: T \def (match t1 with [(TSort n) -\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) with -[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k0 u0 t2) -\Rightarrow (THead k0 (lref_map f d0 u0) (lref_map f (s k0 d0) t2))]) in -lref_map) (\lambda (x0: nat).(plus x0 (S O))) (s k d) t0)) | (THead _ _ t1) -\Rightarrow t1])) (THead (Bind b) u (lift (S O) d (THead k t t0))) (THead k t -t0) H1) in (\lambda (_: (eq T u t)).(\lambda (H6: (eq K (Bind b) k)).(let H7 -\def (eq_ind_r K k (\lambda (k0: K).(eq T (lift (S O) d (THead k0 t t0)) t0)) -H4 (Bind b) H6) in (let H8 \def (eq_ind T (lift (S O) d (THead (Bind b) t -t0)) (\lambda (t1: T).(eq T t1 t0)) H7 (THead (Bind b) (lift (S O) d t) (lift -(S O) (S d) t0)) (lift_bind b t t0 (S O) d)) in (H0 (lift (S O) d t) (S d) H8 -P)))))) H3)) H2))))))))))) x)). -(* COMMENTS -Initial nodes: 935 -END *) - -theorem nf2_gen_abbr: - \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Bind Abbr) u -t)) \to (\forall (P: Prop).P)))) -\def - \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: ((\forall (t2: -T).((pr2 c (THead (Bind Abbr) u t) t2) \to (eq T (THead (Bind Abbr) u t) -t2))))).(\lambda (P: Prop).(let H_x \def (dnf_dec u t O) in (let H0 \def H_x -in (ex_ind T (\lambda (v: T).(or (subst0 O u t (lift (S O) O v)) (eq T t -(lift (S O) O v)))) P (\lambda (x: T).(\lambda (H1: (or (subst0 O u t (lift -(S O) O x)) (eq T t (lift (S O) O x)))).(or_ind (subst0 O u t (lift (S O) O -x)) (eq T t (lift (S O) O x)) P (\lambda (H2: (subst0 O u t (lift (S O) O -x))).(let H3 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda -(_: T).T) with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ -_ t0) \Rightarrow t0])) (THead (Bind Abbr) u t) (THead (Bind Abbr) u (lift (S -O) O x)) (H (THead (Bind Abbr) u (lift (S O) O x)) (pr2_free c (THead (Bind -Abbr) u t) (THead (Bind Abbr) u (lift (S O) O x)) (pr0_delta u u (pr0_refl u) -t t (pr0_refl t) (lift (S O) O x) H2)))) in (let H4 \def (eq_ind T t (\lambda -(t0: T).(subst0 O u t0 (lift (S O) O x))) H2 (lift (S O) O x) H3) in -(subst0_refl u (lift (S O) O x) O H4 P)))) (\lambda (H2: (eq T t (lift (S O) -O x))).(let H3 \def (eq_ind T t (\lambda (t0: T).(\forall (t2: T).((pr2 c -(THead (Bind Abbr) u t0) t2) \to (eq T (THead (Bind Abbr) u t0) t2)))) H -(lift (S O) O x) H2) in (nf2_gen__nf2_gen_aux Abbr x u O (H3 x (pr2_free c -(THead (Bind Abbr) u (lift (S O) O x)) x (pr0_zeta Abbr not_abbr_abst x x -(pr0_refl x) u))) P))) H1))) H0))))))). -(* COMMENTS -Initial nodes: 511 -END *) - -theorem nf2_gen_void: - \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Bind Void) u -(lift (S O) O t))) \to (\forall (P: Prop).P)))) -\def - \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: ((\forall (t2: -T).((pr2 c (THead (Bind Void) u (lift (S O) O t)) t2) \to (eq T (THead (Bind -Void) u (lift (S O) O t)) t2))))).(\lambda (P: Prop).(nf2_gen__nf2_gen_aux -Void t u O (H t (pr2_free c (THead (Bind Void) u (lift (S O) O t)) t -(pr0_zeta Void (sym_not_eq B Abst Void not_abst_void) t t (pr0_refl t) u))) -P))))). -(* COMMENTS -Initial nodes: 121 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/iso.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/iso.ma deleted file mode 100644 index 6a2ce00f8..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/iso.ma +++ /dev/null @@ -1,130 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/nf2/pr3.ma". - -include "Basic-1/pr3/fwd.ma". - -include "Basic-1/iso/props.ma". - -theorem nf2_iso_appls_lref: - \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (vs: -TList).(\forall (u: T).((pr3 c (THeads (Flat Appl) vs (TLRef i)) u) \to (iso -(THeads (Flat Appl) vs (TLRef i)) u)))))) -\def - \lambda (c: C).(\lambda (i: nat).(\lambda (H: (nf2 c (TLRef i))).(\lambda -(vs: TList).(TList_ind (\lambda (t: TList).(\forall (u: T).((pr3 c (THeads -(Flat Appl) t (TLRef i)) u) \to (iso (THeads (Flat Appl) t (TLRef i)) u)))) -(\lambda (u: T).(\lambda (H0: (pr3 c (TLRef i) u)).(let H_y \def -(nf2_pr3_unfold c (TLRef i) u H0 H) in (let H1 \def (eq_ind_r T u (\lambda -(t: T).(pr3 c (TLRef i) t)) H0 (TLRef i) H_y) in (eq_ind T (TLRef i) (\lambda -(t: T).(iso (TLRef i) t)) (iso_refl (TLRef i)) u H_y))))) (\lambda (t: -T).(\lambda (t0: TList).(\lambda (H0: ((\forall (u: T).((pr3 c (THeads (Flat -Appl) t0 (TLRef i)) u) \to (iso (THeads (Flat Appl) t0 (TLRef i)) -u))))).(\lambda (u: T).(\lambda (H1: (pr3 c (THead (Flat Appl) t (THeads -(Flat Appl) t0 (TLRef i))) u)).(let H2 \def (pr3_gen_appl c t (THeads (Flat -Appl) t0 (TLRef i)) u H1) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda -(t2: T).(eq T u (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c t u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) -t0 (TLRef i)) t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) u))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))) (\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat -Appl) t0 (TLRef i)) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u0: -T).(pr3 (CHead c (Bind b) u0) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat -Appl) t0 (TLRef i)) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda -(_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: -T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) -u))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) -y2) z1 z2)))))))) (iso (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef -i))) u) (\lambda (H3: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T u -(THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) -(\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) -t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T u (THead (Flat -Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_: -T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) t2))) (iso -(THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) u) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H4: (eq T u (THead (Flat Appl) x0 -x1))).(\lambda (_: (pr3 c t x0)).(\lambda (_: (pr3 c (THeads (Flat Appl) t0 -(TLRef i)) x1)).(eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda (t1: T).(iso -(THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) t1)) (iso_head t x0 -(THeads (Flat Appl) t0 (TLRef i)) x1 (Flat Appl)) u H4)))))) H3)) (\lambda -(H3: (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t2: T).(pr3 c (THead (Bind Abbr) u2 t2) u))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))) (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat -Appl) t0 (TLRef i)) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u0: -T).(pr3 (CHead c (Bind b) u0) z1 t2))))))))).(ex4_4_ind T T T T (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind -Abbr) u2 t2) u))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c t u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(t2: T).(\forall (b: B).(\forall (u0: T).(pr3 (CHead c (Bind b) u0) z1 -t2))))))) (iso (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) u) -(\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda -(_: (pr3 c (THead (Bind Abbr) x2 x3) u)).(\lambda (_: (pr3 c t x2)).(\lambda -(H6: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind Abst) x0 -x1))).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pr3 (CHead c (Bind b) -u0) x1 x3))))).(let H_y \def (H0 (THead (Bind Abst) x0 x1) H6) in -(iso_flats_lref_bind_false Appl Abst i x0 x1 t0 H_y (iso (THead (Flat Appl) t -(THeads (Flat Appl) t0 (TLRef i))) u))))))))))) H3)) (\lambda (H3: (ex6_6 B T -T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind b) y1 z1)))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda -(u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift -(S O) O u2) z2)) u))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 -(CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat -Appl) t0 (TLRef i)) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda -(_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: -T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) -u))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) -y2) z1 z2))))))) (iso (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) -u) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: -T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (not (eq B x0 -Abst))).(\lambda (H5: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind -x0) x1 x2))).(\lambda (_: (pr3 c (THead (Bind x0) x5 (THead (Flat Appl) (lift -(S O) O x4) x3)) u)).(\lambda (_: (pr3 c t x4)).(\lambda (_: (pr3 c x1 -x5)).(\lambda (_: (pr3 (CHead c (Bind x0) x5) x2 x3)).(let H_y \def (H0 -(THead (Bind x0) x1 x2) H5) in (iso_flats_lref_bind_false Appl x0 i x1 x2 t0 -H_y (iso (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) -u))))))))))))))) H3)) H2))))))) vs)))). -(* COMMENTS -Initial nodes: 1817 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/lift1.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/lift1.ma deleted file mode 100644 index d50790336..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/lift1.ma +++ /dev/null @@ -1,41 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/nf2/props.ma". - -include "Basic-1/drop1/fwd.ma". - -theorem nf2_lift1: - \forall (e: C).(\forall (hds: PList).(\forall (c: C).(\forall (t: T).((drop1 -hds c e) \to ((nf2 e t) \to (nf2 c (lift1 hds t))))))) -\def - \lambda (e: C).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall -(c: C).(\forall (t: T).((drop1 p c e) \to ((nf2 e t) \to (nf2 c (lift1 p -t))))))) (\lambda (c: C).(\lambda (t: T).(\lambda (H: (drop1 PNil c -e)).(\lambda (H0: (nf2 e t)).(let H_y \def (drop1_gen_pnil c e H) in -(eq_ind_r C e (\lambda (c0: C).(nf2 c0 t)) H0 c H_y)))))) (\lambda (n: -nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c: -C).(\forall (t: T).((drop1 p c e) \to ((nf2 e t) \to (nf2 c (lift1 p -t)))))))).(\lambda (c: C).(\lambda (t: T).(\lambda (H0: (drop1 (PCons n n0 p) -c e)).(\lambda (H1: (nf2 e t)).(let H_x \def (drop1_gen_pcons c e p n n0 H0) -in (let H2 \def H_x in (ex2_ind C (\lambda (c2: C).(drop n n0 c c2)) (\lambda -(c2: C).(drop1 p c2 e)) (nf2 c (lift n n0 (lift1 p t))) (\lambda (x: -C).(\lambda (H3: (drop n n0 c x)).(\lambda (H4: (drop1 p x e)).(nf2_lift x -(lift1 p t) (H x t H4 H1) c n n0 H3)))) H2))))))))))) hds)). -(* COMMENTS -Initial nodes: 249 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/pr3.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/pr3.ma deleted file mode 100644 index 3db24223e..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/pr3.ma +++ /dev/null @@ -1,56 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/nf2/defs.ma". - -include "Basic-1/pr3/pr3.ma". - -theorem nf2_pr3_unfold: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to ((nf2 c -t1) \to (eq T t1 t2))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 -t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).((nf2 c t) \to (eq T t -t0)))) (\lambda (t: T).(\lambda (H0: (nf2 c t)).(H0 t (pr2_free c t t -(pr0_refl t))))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c t3 -t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2: (((nf2 c t0) -\to (eq T t0 t4)))).(\lambda (H3: (nf2 c t3)).(let H4 \def H3 in (let H5 \def -(eq_ind T t3 (\lambda (t: T).(nf2 c t)) H3 t0 (H4 t0 H0)) in (let H6 \def -(eq_ind T t3 (\lambda (t: T).(pr2 c t t0)) H0 t0 (H4 t0 H0)) in (eq_ind_r T -t0 (\lambda (t: T).(eq T t t4)) (H2 H5) t3 (H4 t0 H0)))))))))))) t1 t2 H)))). -(* COMMENTS -Initial nodes: 187 -END *) - -theorem nf2_pr3_confluence: - \forall (c: C).(\forall (t1: T).((nf2 c t1) \to (\forall (t2: T).((nf2 c t2) -\to (\forall (t: T).((pr3 c t t1) \to ((pr3 c t t2) \to (eq T t1 t2)))))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (H: (nf2 c t1)).(\lambda (t2: -T).(\lambda (H0: (nf2 c t2)).(\lambda (t: T).(\lambda (H1: (pr3 c t -t1)).(\lambda (H2: (pr3 c t t2)).(ex2_ind T (\lambda (t0: T).(pr3 c t2 t0)) -(\lambda (t0: T).(pr3 c t1 t0)) (eq T t1 t2) (\lambda (x: T).(\lambda (H3: -(pr3 c t2 x)).(\lambda (H4: (pr3 c t1 x)).(let H_y \def (nf2_pr3_unfold c t1 -x H4 H) in (let H5 \def (eq_ind_r T x (\lambda (t0: T).(pr3 c t1 t0)) H4 t1 -H_y) in (let H6 \def (eq_ind_r T x (\lambda (t0: T).(pr3 c t2 t0)) H3 t1 H_y) -in (let H_y0 \def (nf2_pr3_unfold c t2 t1 H6 H0) in (let H7 \def (eq_ind T t2 -(\lambda (t0: T).(pr3 c t0 t1)) H6 t1 H_y0) in (eq_ind_r T t1 (\lambda (t0: -T).(eq T t1 t0)) (refl_equal T t1) t2 H_y0))))))))) (pr3_confluence c t t2 H2 -t1 H1))))))))). -(* COMMENTS -Initial nodes: 215 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/props.ma deleted file mode 100644 index 2f0f092f2..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/props.ma +++ /dev/null @@ -1,341 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/nf2/defs.ma". - -include "Basic-1/pr2/fwd.ma". - -theorem nf2_sort: - \forall (c: C).(\forall (n: nat).(nf2 c (TSort n))) -\def - \lambda (c: C).(\lambda (n: nat).(\lambda (t2: T).(\lambda (H: (pr2 c (TSort -n) t2)).(eq_ind_r T (TSort n) (\lambda (t: T).(eq T (TSort n) t)) (refl_equal -T (TSort n)) t2 (pr2_gen_sort c t2 n H))))). -(* COMMENTS -Initial nodes: 55 -END *) - -theorem nf2_csort_lref: - \forall (n: nat).(\forall (i: nat).(nf2 (CSort n) (TLRef i))) -\def - \lambda (n: nat).(\lambda (i: nat).(\lambda (t2: T).(\lambda (H: (pr2 (CSort -n) (TLRef i) t2)).(let H0 \def (pr2_gen_lref (CSort n) t2 i H) in (or_ind (eq -T t2 (TLRef i)) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i (CSort n) -(CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T t2 (lift (S -i) O u))))) (eq T (TLRef i) t2) (\lambda (H1: (eq T t2 (TLRef i))).(eq_ind_r -T (TLRef i) (\lambda (t: T).(eq T (TLRef i) t)) (refl_equal T (TLRef i)) t2 -H1)) (\lambda (H1: (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i (CSort -n) (CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T t2 (lift -(S i) O u)))))).(ex2_2_ind C T (\lambda (d: C).(\lambda (u: T).(getl i (CSort -n) (CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T t2 (lift -(S i) O u)))) (eq T (TLRef i) t2) (\lambda (x0: C).(\lambda (x1: T).(\lambda -(H2: (getl i (CSort n) (CHead x0 (Bind Abbr) x1))).(\lambda (H3: (eq T t2 -(lift (S i) O x1))).(eq_ind_r T (lift (S i) O x1) (\lambda (t: T).(eq T -(TLRef i) t)) (getl_gen_sort n i (CHead x0 (Bind Abbr) x1) H2 (eq T (TLRef i) -(lift (S i) O x1))) t2 H3))))) H1)) H0))))). -(* COMMENTS -Initial nodes: 355 -END *) - -theorem nf2_abst: - \forall (c: C).(\forall (u: T).((nf2 c u) \to (\forall (b: B).(\forall (v: -T).(\forall (t: T).((nf2 (CHead c (Bind b) v) t) \to (nf2 c (THead (Bind -Abst) u t)))))))) -\def - \lambda (c: C).(\lambda (u: T).(\lambda (H: ((\forall (t2: T).((pr2 c u t2) -\to (eq T u t2))))).(\lambda (b: B).(\lambda (v: T).(\lambda (t: T).(\lambda -(H0: ((\forall (t2: T).((pr2 (CHead c (Bind b) v) t t2) \to (eq T t -t2))))).(\lambda (t2: T).(\lambda (H1: (pr2 c (THead (Bind Abst) u t) -t2)).(let H2 \def (pr2_gen_abst c u t t2 H1) in (ex3_2_ind T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c u u2))) (\lambda (_: T).(\lambda (t3: T).(\forall -(b0: B).(\forall (u0: T).(pr2 (CHead c (Bind b0) u0) t t3))))) (eq T (THead -(Bind Abst) u t) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H3: (eq T t2 -(THead (Bind Abst) x0 x1))).(\lambda (H4: (pr2 c u x0)).(\lambda (H5: -((\forall (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind b0) u0) t -x1))))).(eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t0: T).(eq T (THead -(Bind Abst) u t) t0)) (f_equal3 K T T T THead (Bind Abst) (Bind Abst) u x0 t -x1 (refl_equal K (Bind Abst)) (H x0 H4) (H0 x1 (H5 b v))) t2 H3)))))) -H2)))))))))). -(* COMMENTS -Initial nodes: 299 -END *) - -theorem nf2_abst_shift: - \forall (c: C).(\forall (u: T).((nf2 c u) \to (\forall (t: T).((nf2 (CHead c -(Bind Abst) u) t) \to (nf2 c (THead (Bind Abst) u t)))))) -\def - \lambda (c: C).(\lambda (u: T).(\lambda (H: ((\forall (t2: T).((pr2 c u t2) -\to (eq T u t2))))).(\lambda (t: T).(\lambda (H0: ((\forall (t2: T).((pr2 -(CHead c (Bind Abst) u) t t2) \to (eq T t t2))))).(\lambda (t2: T).(\lambda -(H1: (pr2 c (THead (Bind Abst) u t) t2)).(let H2 \def (pr2_gen_abst c u t t2 -H1) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u u2))) (\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) -u0) t t3))))) (eq T (THead (Bind Abst) u t) t2) (\lambda (x0: T).(\lambda -(x1: T).(\lambda (H3: (eq T t2 (THead (Bind Abst) x0 x1))).(\lambda (H4: (pr2 -c u x0)).(\lambda (H5: ((\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind -b) u0) t x1))))).(eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t0: T).(eq T -(THead (Bind Abst) u t) t0)) (f_equal3 K T T T THead (Bind Abst) (Bind Abst) -u x0 t x1 (refl_equal K (Bind Abst)) (H x0 H4) (H0 x1 (H5 Abst u))) t2 -H3)))))) H2)))))))). -(* COMMENTS -Initial nodes: 295 -END *) - -theorem nfs2_tapp: - \forall (c: C).(\forall (t: T).(\forall (ts: TList).((nfs2 c (TApp ts t)) -\to (land (nfs2 c ts) (nf2 c t))))) -\def - \lambda (c: C).(\lambda (t: T).(\lambda (ts: TList).(TList_ind (\lambda (t0: -TList).((nfs2 c (TApp t0 t)) \to (land (nfs2 c t0) (nf2 c t)))) (\lambda (H: -(land (nf2 c t) True)).(let H0 \def H in (land_ind (nf2 c t) True (land True -(nf2 c t)) (\lambda (H1: (nf2 c t)).(\lambda (_: True).(conj True (nf2 c t) I -H1))) H0))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H: (((nfs2 c -(TApp t1 t)) \to (land (nfs2 c t1) (nf2 c t))))).(\lambda (H0: (land (nf2 c -t0) (nfs2 c (TApp t1 t)))).(let H1 \def H0 in (land_ind (nf2 c t0) (nfs2 c -(TApp t1 t)) (land (land (nf2 c t0) (nfs2 c t1)) (nf2 c t)) (\lambda (H2: -(nf2 c t0)).(\lambda (H3: (nfs2 c (TApp t1 t))).(let H_x \def (H H3) in (let -H4 \def H_x in (land_ind (nfs2 c t1) (nf2 c t) (land (land (nf2 c t0) (nfs2 c -t1)) (nf2 c t)) (\lambda (H5: (nfs2 c t1)).(\lambda (H6: (nf2 c t)).(conj -(land (nf2 c t0) (nfs2 c t1)) (nf2 c t) (conj (nf2 c t0) (nfs2 c t1) H2 H5) -H6))) H4))))) H1)))))) ts))). -(* COMMENTS -Initial nodes: 295 -END *) - -theorem nf2_appls_lref: - \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (vs: -TList).((nfs2 c vs) \to (nf2 c (THeads (Flat Appl) vs (TLRef i))))))) -\def - \lambda (c: C).(\lambda (i: nat).(\lambda (H: (nf2 c (TLRef i))).(\lambda -(vs: TList).(TList_ind (\lambda (t: TList).((nfs2 c t) \to (nf2 c (THeads -(Flat Appl) t (TLRef i))))) (\lambda (_: True).H) (\lambda (t: T).(\lambda -(t0: TList).(\lambda (H0: (((nfs2 c t0) \to (nf2 c (THeads (Flat Appl) t0 -(TLRef i)))))).(\lambda (H1: (land (nf2 c t) (nfs2 c t0))).(let H2 \def H1 in -(land_ind (nf2 c t) (nfs2 c t0) (nf2 c (THead (Flat Appl) t (THeads (Flat -Appl) t0 (TLRef i)))) (\lambda (H3: (nf2 c t)).(\lambda (H4: (nfs2 c -t0)).(let H_y \def (H0 H4) in (\lambda (t2: T).(\lambda (H5: (pr2 c (THead -(Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) t2)).(let H6 \def -(pr2_gen_appl c t (THeads (Flat Appl) t0 (TLRef i)) t2 H5) in (or3_ind (ex3_2 -T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c t u2))) (\lambda (_: T).(\lambda (t3: -T).(pr2 c (THeads (Flat Appl) t0 (TLRef i)) t3)))) (ex4_4 T T T T (\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THeads (Flat -Appl) t0 (TLRef i)) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda -(_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 -t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: -T).(pr2 c t u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 -t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) -(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(eq T (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead -(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t -u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (eq T (THead (Flat Appl) t -(THeads (Flat Appl) t0 (TLRef i))) t2) (\lambda (H7: (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c t u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c -(THeads (Flat Appl) t0 (TLRef i)) t3))))).(ex3_2_ind T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c t u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c -(THeads (Flat Appl) t0 (TLRef i)) t3))) (eq T (THead (Flat Appl) t (THeads -(Flat Appl) t0 (TLRef i))) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda -(H8: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H9: (pr2 c t -x0)).(\lambda (H10: (pr2 c (THeads (Flat Appl) t0 (TLRef i)) x1)).(eq_ind_r T -(THead (Flat Appl) x0 x1) (\lambda (t1: T).(eq T (THead (Flat Appl) t (THeads -(Flat Appl) t0 (TLRef i))) t1)) (let H11 \def (eq_ind_r T x1 (\lambda (t1: -T).(pr2 c (THeads (Flat Appl) t0 (TLRef i)) t1)) H10 (THeads (Flat Appl) t0 -(TLRef i)) (H_y x1 H10)) in (eq_ind T (THeads (Flat Appl) t0 (TLRef i)) -(\lambda (t1: T).(eq T (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef -i))) (THead (Flat Appl) x0 t1))) (let H12 \def (eq_ind_r T x0 (\lambda (t1: -T).(pr2 c t t1)) H9 t (H3 x0 H9)) in (eq_ind T t (\lambda (t1: T).(eq T -(THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) (THead (Flat Appl) t1 -(THeads (Flat Appl) t0 (TLRef i))))) (refl_equal T (THead (Flat Appl) t -(THeads (Flat Appl) t0 (TLRef i)))) x0 (H3 x0 H9))) x1 (H_y x1 H10))) t2 -H8)))))) H7)) (\lambda (H7: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THeads (Flat Appl) t0 (TLRef i)) -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t u2))))) (\lambda -(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: -B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3))))))))).(ex4_4_ind T T T -T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THeads (Flat Appl) t0 (TLRef i)) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c t u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) -z1 t3))))))) (eq T (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) -t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: -T).(\lambda (H8: (eq T (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind Abst) -x0 x1))).(\lambda (H9: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (_: (pr2 -c t x2)).(\lambda (_: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) -u) x1 x3))))).(eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda (t1: T).(eq T -(THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) t1)) (TList_ind -(\lambda (t1: TList).((nf2 c (THeads (Flat Appl) t1 (TLRef i))) \to ((eq T -(THeads (Flat Appl) t1 (TLRef i)) (THead (Bind Abst) x0 x1)) \to (eq T (THead -(Flat Appl) t (THeads (Flat Appl) t1 (TLRef i))) (THead (Bind Abbr) x2 -x3))))) (\lambda (_: (nf2 c (THeads (Flat Appl) TNil (TLRef i)))).(\lambda -(H13: (eq T (THeads (Flat Appl) TNil (TLRef i)) (THead (Bind Abst) x0 -x1))).(let H14 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) x0 -x1) H13) in (False_ind (eq T (THead (Flat Appl) t (THeads (Flat Appl) TNil -(TLRef i))) (THead (Bind Abbr) x2 x3)) H14)))) (\lambda (t1: T).(\lambda (t3: -TList).(\lambda (_: (((nf2 c (THeads (Flat Appl) t3 (TLRef i))) \to ((eq T -(THeads (Flat Appl) t3 (TLRef i)) (THead (Bind Abst) x0 x1)) \to (eq T (THead -(Flat Appl) t (THeads (Flat Appl) t3 (TLRef i))) (THead (Bind Abbr) x2 -x3)))))).(\lambda (_: (nf2 c (THeads (Flat Appl) (TCons t1 t3) (TLRef -i)))).(\lambda (H13: (eq T (THeads (Flat Appl) (TCons t1 t3) (TLRef i)) -(THead (Bind Abst) x0 x1))).(let H14 \def (eq_ind T (THead (Flat Appl) t1 -(THeads (Flat Appl) t3 (TLRef i))) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind Abst) x0 x1) H13) in (False_ind (eq T (THead (Flat -Appl) t (THeads (Flat Appl) (TCons t1 t3) (TLRef i))) (THead (Bind Abbr) x2 -x3)) H14))))))) t0 H_y H8) t2 H9))))))))) H7)) (\lambda (H7: (ex6_6 B T T T T -T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq -T (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind b) y1 z1)))))))) (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: -T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S -O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) -y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B -b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THeads (Flat Appl) t0 (TLRef i)) -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c t u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) (eq T (THead -(Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) t2) (\lambda (x0: -B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: -T).(\lambda (x5: T).(\lambda (_: (not (eq B x0 Abst))).(\lambda (H9: (eq T -(THeads (Flat Appl) t0 (TLRef i)) (THead (Bind x0) x1 x2))).(\lambda (H10: -(eq T t2 (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) -x3)))).(\lambda (_: (pr2 c t x4)).(\lambda (_: (pr2 c x1 x5)).(\lambda (_: -(pr2 (CHead c (Bind x0) x5) x2 x3)).(eq_ind_r T (THead (Bind x0) x5 (THead -(Flat Appl) (lift (S O) O x4) x3)) (\lambda (t1: T).(eq T (THead (Flat Appl) -t (THeads (Flat Appl) t0 (TLRef i))) t1)) (TList_ind (\lambda (t1: -TList).((nf2 c (THeads (Flat Appl) t1 (TLRef i))) \to ((eq T (THeads (Flat -Appl) t1 (TLRef i)) (THead (Bind x0) x1 x2)) \to (eq T (THead (Flat Appl) t -(THeads (Flat Appl) t1 (TLRef i))) (THead (Bind x0) x5 (THead (Flat Appl) -(lift (S O) O x4) x3)))))) (\lambda (_: (nf2 c (THeads (Flat Appl) TNil -(TLRef i)))).(\lambda (H15: (eq T (THeads (Flat Appl) TNil (TLRef i)) (THead -(Bind x0) x1 x2))).(let H16 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match -ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | -(TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead -(Bind x0) x1 x2) H15) in (False_ind (eq T (THead (Flat Appl) t (THeads (Flat -Appl) TNil (TLRef i))) (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O -x4) x3))) H16)))) (\lambda (t1: T).(\lambda (t3: TList).(\lambda (_: (((nf2 c -(THeads (Flat Appl) t3 (TLRef i))) \to ((eq T (THeads (Flat Appl) t3 (TLRef -i)) (THead (Bind x0) x1 x2)) \to (eq T (THead (Flat Appl) t (THeads (Flat -Appl) t3 (TLRef i))) (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) -x3))))))).(\lambda (_: (nf2 c (THeads (Flat Appl) (TCons t1 t3) (TLRef -i)))).(\lambda (H15: (eq T (THeads (Flat Appl) (TCons t1 t3) (TLRef i)) -(THead (Bind x0) x1 x2))).(let H16 \def (eq_ind T (THead (Flat Appl) t1 -(THeads (Flat Appl) t3 (TLRef i))) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind x0) x1 x2) H15) in (False_ind (eq T (THead (Flat -Appl) t (THeads (Flat Appl) (TCons t1 t3) (TLRef i))) (THead (Bind x0) x5 -(THead (Flat Appl) (lift (S O) O x4) x3))) H16))))))) t0 H_y H9) t2 -H10))))))))))))) H7)) H6))))))) H2)))))) vs)))). -(* COMMENTS -Initial nodes: 2915 -END *) - -theorem nf2_appl_lref: - \forall (c: C).(\forall (u: T).((nf2 c u) \to (\forall (i: nat).((nf2 c -(TLRef i)) \to (nf2 c (THead (Flat Appl) u (TLRef i))))))) -\def - \lambda (c: C).(\lambda (u: T).(\lambda (H: (nf2 c u)).(\lambda (i: -nat).(\lambda (H0: (nf2 c (TLRef i))).(let H_y \def (nf2_appls_lref c i H0 -(TCons u TNil)) in (H_y (conj (nf2 c u) True H I))))))). -(* COMMENTS -Initial nodes: 49 -END *) - -theorem nf2_lref_abst: - \forall (c: C).(\forall (e: C).(\forall (u: T).(\forall (i: nat).((getl i c -(CHead e (Bind Abst) u)) \to (nf2 c (TLRef i)))))) -\def - \lambda (c: C).(\lambda (e: C).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H: (getl i c (CHead e (Bind Abst) u))).(\lambda (t2: T).(\lambda (H0: (pr2 c -(TLRef i) t2)).(let H1 \def (pr2_gen_lref c t2 i H0) in (or_ind (eq T t2 -(TLRef i)) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c (CHead d -(Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T t2 (lift (S i) O -u0))))) (eq T (TLRef i) t2) (\lambda (H2: (eq T t2 (TLRef i))).(eq_ind_r T -(TLRef i) (\lambda (t: T).(eq T (TLRef i) t)) (refl_equal T (TLRef i)) t2 -H2)) (\lambda (H2: (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c -(CHead d (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T t2 (lift -(S i) O u0)))))).(ex2_2_ind C T (\lambda (d: C).(\lambda (u0: T).(getl i c -(CHead d (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T t2 (lift -(S i) O u0)))) (eq T (TLRef i) t2) (\lambda (x0: C).(\lambda (x1: T).(\lambda -(H3: (getl i c (CHead x0 (Bind Abbr) x1))).(\lambda (H4: (eq T t2 (lift (S i) -O x1))).(eq_ind_r T (lift (S i) O x1) (\lambda (t: T).(eq T (TLRef i) t)) -(let H5 \def (eq_ind C (CHead e (Bind Abst) u) (\lambda (c0: C).(getl i c -c0)) H (CHead x0 (Bind Abbr) x1) (getl_mono c (CHead e (Bind Abst) u) i H -(CHead x0 (Bind Abbr) x1) H3)) in (let H6 \def (eq_ind C (CHead e (Bind Abst) -u) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort -_) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return -(\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B return -(\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow True | -Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead x0 (Bind -Abbr) x1) (getl_mono c (CHead e (Bind Abst) u) i H (CHead x0 (Bind Abbr) x1) -H3)) in (False_ind (eq T (TLRef i) (lift (S i) O x1)) H6))) t2 H4))))) H2)) -H1)))))))). -(* COMMENTS -Initial nodes: 494 -END *) - -theorem nf2_lift: - \forall (d: C).(\forall (t: T).((nf2 d t) \to (\forall (c: C).(\forall (h: -nat).(\forall (i: nat).((drop h i c d) \to (nf2 c (lift h i t)))))))) -\def - \lambda (d: C).(\lambda (t: T).(\lambda (H: ((\forall (t2: T).((pr2 d t t2) -\to (eq T t t2))))).(\lambda (c: C).(\lambda (h: nat).(\lambda (i: -nat).(\lambda (H0: (drop h i c d)).(\lambda (t2: T).(\lambda (H1: (pr2 c -(lift h i t) t2)).(let H2 \def (pr2_gen_lift c t t2 h i H1 d H0) in (ex2_ind -T (\lambda (t3: T).(eq T t2 (lift h i t3))) (\lambda (t3: T).(pr2 d t t3)) -(eq T (lift h i t) t2) (\lambda (x: T).(\lambda (H3: (eq T t2 (lift h i -x))).(\lambda (H4: (pr2 d t x)).(eq_ind_r T (lift h i x) (\lambda (t0: T).(eq -T (lift h i t) t0)) (let H_y \def (H x H4) in (let H5 \def (eq_ind_r T x -(\lambda (t0: T).(pr2 d t t0)) H4 t H_y) in (eq_ind T t (\lambda (t0: T).(eq -T (lift h i t) (lift h i t0))) (refl_equal T (lift h i t)) x H_y))) t2 H3)))) -H2)))))))))). -(* COMMENTS -Initial nodes: 245 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pc1/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pc1/defs.ma deleted file mode 100644 index 80726a188..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pc1/defs.ma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr1/defs.ma". - -definition pc1: - T \to (T \to Prop) -\def - \lambda (t1: T).(\lambda (t2: T).(ex2 T (\lambda (t: T).(pr1 t1 t)) (\lambda -(t: T).(pr1 t2 t)))). - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pc1/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pc1/props.ma deleted file mode 100644 index 0da1a5aca..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pc1/props.ma +++ /dev/null @@ -1,146 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pc1/defs.ma". - -include "Basic-1/pr1/pr1.ma". - -theorem pc1_pr0_r: - \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (pc1 t1 t2))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(ex_intro2 T -(\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)) t2 (pr1_pr0 t1 t2 H) -(pr1_refl t2)))). -(* COMMENTS -Initial nodes: 43 -END *) - -theorem pc1_pr0_x: - \forall (t1: T).(\forall (t2: T).((pr0 t2 t1) \to (pc1 t1 t2))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t2 t1)).(ex_intro2 T -(\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)) t1 (pr1_refl t1) -(pr1_pr0 t2 t1 H)))). -(* COMMENTS -Initial nodes: 43 -END *) - -theorem pc1_refl: - \forall (t: T).(pc1 t t) -\def - \lambda (t: T).(ex_intro2 T (\lambda (t0: T).(pr1 t t0)) (\lambda (t0: -T).(pr1 t t0)) t (pr1_refl t) (pr1_refl t)). -(* COMMENTS -Initial nodes: 31 -END *) - -theorem pc1_pr0_u: - \forall (t2: T).(\forall (t1: T).((pr0 t1 t2) \to (\forall (t3: T).((pc1 t2 -t3) \to (pc1 t1 t3))))) -\def - \lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pr0 t1 t2)).(\lambda (t3: -T).(\lambda (H0: (pc1 t2 t3)).(let H1 \def H0 in (ex2_ind T (\lambda (t: -T).(pr1 t2 t)) (\lambda (t: T).(pr1 t3 t)) (pc1 t1 t3) (\lambda (x: -T).(\lambda (H2: (pr1 t2 x)).(\lambda (H3: (pr1 t3 x)).(ex_intro2 T (\lambda -(t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t3 t)) x (pr1_sing t2 t1 H x H2) -H3)))) H1)))))). -(* COMMENTS -Initial nodes: 97 -END *) - -theorem pc1_s: - \forall (t2: T).(\forall (t1: T).((pc1 t1 t2) \to (pc1 t2 t1))) -\def - \lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pc1 t1 t2)).(let H0 \def H in -(ex2_ind T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)) (pc1 t2 -t1) (\lambda (x: T).(\lambda (H1: (pr1 t1 x)).(\lambda (H2: (pr1 t2 -x)).(ex_intro2 T (\lambda (t: T).(pr1 t2 t)) (\lambda (t: T).(pr1 t1 t)) x H2 -H1)))) H0)))). -(* COMMENTS -Initial nodes: 79 -END *) - -theorem pc1_head_1: - \forall (u1: T).(\forall (u2: T).((pc1 u1 u2) \to (\forall (t: T).(\forall -(k: K).(pc1 (THead k u1 t) (THead k u2 t)))))) -\def - \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pc1 u1 u2)).(\lambda (t: -T).(\lambda (k: K).(let H0 \def H in (ex2_ind T (\lambda (t0: T).(pr1 u1 t0)) -(\lambda (t0: T).(pr1 u2 t0)) (pc1 (THead k u1 t) (THead k u2 t)) (\lambda -(x: T).(\lambda (H1: (pr1 u1 x)).(\lambda (H2: (pr1 u2 x)).(ex_intro2 T -(\lambda (t0: T).(pr1 (THead k u1 t) t0)) (\lambda (t0: T).(pr1 (THead k u2 -t) t0)) (THead k x t) (pr1_head_1 u1 x H1 t k) (pr1_head_1 u2 x H2 t k))))) -H0)))))). -(* COMMENTS -Initial nodes: 133 -END *) - -theorem pc1_head_2: - \forall (t1: T).(\forall (t2: T).((pc1 t1 t2) \to (\forall (u: T).(\forall -(k: K).(pc1 (THead k u t1) (THead k u t2)))))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc1 t1 t2)).(\lambda (u: -T).(\lambda (k: K).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr1 t1 t)) -(\lambda (t: T).(pr1 t2 t)) (pc1 (THead k u t1) (THead k u t2)) (\lambda (x: -T).(\lambda (H1: (pr1 t1 x)).(\lambda (H2: (pr1 t2 x)).(ex_intro2 T (\lambda -(t: T).(pr1 (THead k u t1) t)) (\lambda (t: T).(pr1 (THead k u t2) t)) (THead -k u x) (pr1_head_2 t1 x H1 u k) (pr1_head_2 t2 x H2 u k))))) H0)))))). -(* COMMENTS -Initial nodes: 133 -END *) - -theorem pc1_t: - \forall (t2: T).(\forall (t1: T).((pc1 t1 t2) \to (\forall (t3: T).((pc1 t2 -t3) \to (pc1 t1 t3))))) -\def - \lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pc1 t1 t2)).(\lambda (t3: -T).(\lambda (H0: (pc1 t2 t3)).(let H1 \def H0 in (ex2_ind T (\lambda (t: -T).(pr1 t2 t)) (\lambda (t: T).(pr1 t3 t)) (pc1 t1 t3) (\lambda (x: -T).(\lambda (H2: (pr1 t2 x)).(\lambda (H3: (pr1 t3 x)).(let H4 \def H in -(ex2_ind T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)) (pc1 t1 -t3) (\lambda (x0: T).(\lambda (H5: (pr1 t1 x0)).(\lambda (H6: (pr1 t2 -x0)).(ex2_ind T (\lambda (t: T).(pr1 x0 t)) (\lambda (t: T).(pr1 x t)) (pc1 -t1 t3) (\lambda (x1: T).(\lambda (H7: (pr1 x0 x1)).(\lambda (H8: (pr1 x -x1)).(ex_intro2 T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t3 t)) x1 -(pr1_t x0 t1 H5 x1 H7) (pr1_t x t3 H3 x1 H8))))) (pr1_confluence t2 x0 H6 x -H2))))) H4))))) H1)))))). -(* COMMENTS -Initial nodes: 203 -END *) - -theorem pc1_pr0_u2: - \forall (t0: T).(\forall (t1: T).((pr0 t0 t1) \to (\forall (t2: T).((pc1 t0 -t2) \to (pc1 t1 t2))))) -\def - \lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr0 t0 t1)).(\lambda (t2: -T).(\lambda (H0: (pc1 t0 t2)).(pc1_t t0 t1 (pc1_pr0_x t1 t0 H) t2 H0))))). -(* COMMENTS -Initial nodes: 35 -END *) - -theorem pc1_head: - \forall (u1: T).(\forall (u2: T).((pc1 u1 u2) \to (\forall (t1: T).(\forall -(t2: T).((pc1 t1 t2) \to (\forall (k: K).(pc1 (THead k u1 t1) (THead k u2 -t2)))))))) -\def - \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pc1 u1 u2)).(\lambda (t1: -T).(\lambda (t2: T).(\lambda (H0: (pc1 t1 t2)).(\lambda (k: K).(pc1_t (THead -k u2 t1) (THead k u1 t1) (pc1_head_1 u1 u2 H t1 k) (THead k u2 t2) -(pc1_head_2 t1 t2 H0 u2 k)))))))). -(* COMMENTS -Initial nodes: 71 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/dec.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/dec.ma deleted file mode 100644 index aa7a4d89e..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/dec.ma +++ /dev/null @@ -1,152 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/ty3/arity_props.ma". - -include "Basic-1/nf2/fwd.ma". - -theorem pc3_dec: - \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (t1: T).((ty3 g c -u1 t1) \to (\forall (u2: T).(\forall (t2: T).((ty3 g c u2 t2) \to (or (pc3 c -u1 u2) ((pc3 c u1 u2) \to False))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda -(H: (ty3 g c u1 t1)).(\lambda (u2: T).(\lambda (t2: T).(\lambda (H0: (ty3 g c -u2 t2)).(let H_y \def (ty3_sn3 g c u1 t1 H) in (let H_y0 \def (ty3_sn3 g c u2 -t2 H0) in (let H_x \def (nf2_sn3 c u1 H_y) in (let H1 \def H_x in (ex2_ind T -(\lambda (u: T).(pr3 c u1 u)) (\lambda (u: T).(nf2 c u)) (or (pc3 c u1 u2) -((pc3 c u1 u2) \to False)) (\lambda (x: T).(\lambda (H2: (pr3 c u1 -x)).(\lambda (H3: (nf2 c x)).(let H_x0 \def (nf2_sn3 c u2 H_y0) in (let H4 -\def H_x0 in (ex2_ind T (\lambda (u: T).(pr3 c u2 u)) (\lambda (u: T).(nf2 c -u)) (or (pc3 c u1 u2) ((pc3 c u1 u2) \to False)) (\lambda (x0: T).(\lambda -(H5: (pr3 c u2 x0)).(\lambda (H6: (nf2 c x0)).(let H_x1 \def (term_dec x x0) -in (let H7 \def H_x1 in (or_ind (eq T x x0) ((eq T x x0) \to (\forall (P: -Prop).P)) (or (pc3 c u1 u2) ((pc3 c u1 u2) \to False)) (\lambda (H8: (eq T x -x0)).(let H9 \def (eq_ind_r T x0 (\lambda (t: T).(nf2 c t)) H6 x H8) in (let -H10 \def (eq_ind_r T x0 (\lambda (t: T).(pr3 c u2 t)) H5 x H8) in (or_introl -(pc3 c u1 u2) ((pc3 c u1 u2) \to False) (pc3_pr3_t c u1 x H2 u2 H10))))) -(\lambda (H8: (((eq T x x0) \to (\forall (P: Prop).P)))).(or_intror (pc3 c u1 -u2) ((pc3 c u1 u2) \to False) (\lambda (H9: (pc3 c u1 u2)).(let H10 \def H9 -in (ex2_ind T (\lambda (t: T).(pr3 c u1 t)) (\lambda (t: T).(pr3 c u2 t)) -False (\lambda (x1: T).(\lambda (H11: (pr3 c u1 x1)).(\lambda (H12: (pr3 c u2 -x1)).(let H_x2 \def (pr3_confluence c u2 x0 H5 x1 H12) in (let H13 \def H_x2 -in (ex2_ind T (\lambda (t: T).(pr3 c x0 t)) (\lambda (t: T).(pr3 c x1 t)) -False (\lambda (x2: T).(\lambda (H14: (pr3 c x0 x2)).(\lambda (H15: (pr3 c x1 -x2)).(let H_y1 \def (nf2_pr3_unfold c x0 x2 H14 H6) in (let H16 \def -(eq_ind_r T x2 (\lambda (t: T).(pr3 c x1 t)) H15 x0 H_y1) in (let H17 \def -(nf2_pr3_confluence c x H3 x0 H6 u1 H2) in (H8 (H17 (pr3_t x1 u1 c H11 x0 -H16)) False))))))) H13)))))) H10))))) H7)))))) H4)))))) H1)))))))))))). -(* COMMENTS -Initial nodes: 551 -END *) - -theorem pc3_abst_dec: - \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (t1: T).((ty3 g c -u1 t1) \to (\forall (u2: T).(\forall (t2: T).((ty3 g c u2 t2) \to (or (ex4_2 -T T (\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst) u2 u)))) -(\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u) t1))) -(\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_: T).(\lambda -(v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind Abst) u2 u)) -\to False)))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda -(H: (ty3 g c u1 t1)).(\lambda (u2: T).(\lambda (t2: T).(\lambda (H0: (ty3 g c -u2 t2)).(let H1 \def (ty3_sn3 g c u1 t1 H) in (let H2 \def (ty3_sn3 g c u2 t2 -H0) in (let H_x \def (nf2_sn3 c u1 H1) in (let H3 \def H_x in (ex2_ind T -(\lambda (u: T).(pr3 c u1 u)) (\lambda (u: T).(nf2 c u)) (or (ex4_2 T T -(\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst) u2 u)))) -(\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u) t1))) -(\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_: T).(\lambda -(v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind Abst) u2 u)) -\to False))) (\lambda (x: T).(\lambda (H4: (pr3 c u1 x)).(\lambda (H5: (nf2 c -x)).(let H_x0 \def (nf2_sn3 c u2 H2) in (let H6 \def H_x0 in (ex2_ind T -(\lambda (u: T).(pr3 c u2 u)) (\lambda (u: T).(nf2 c u)) (or (ex4_2 T T -(\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst) u2 u)))) -(\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u) t1))) -(\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_: T).(\lambda -(v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind Abst) u2 u)) -\to False))) (\lambda (x0: T).(\lambda (H7: (pr3 c u2 x0)).(\lambda (H8: (nf2 -c x0)).(let H_x1 \def (abst_dec x x0) in (let H9 \def H_x1 in (or_ind (ex T -(\lambda (t: T).(eq T x (THead (Bind Abst) x0 t)))) (\forall (t: T).((eq T x -(THead (Bind Abst) x0 t)) \to (\forall (P: Prop).P))) (or (ex4_2 T T (\lambda -(u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst) u2 u)))) (\lambda (u: -T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u) t1))) (\lambda (_: -T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_: T).(\lambda (v2: T).(nf2 c -v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind Abst) u2 u)) \to False))) -(\lambda (H10: (ex T (\lambda (t: T).(eq T x (THead (Bind Abst) x0 -t))))).(ex_ind T (\lambda (t: T).(eq T x (THead (Bind Abst) x0 t))) (or -(ex4_2 T T (\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst) u2 -u)))) (\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u) -t1))) (\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_: -T).(\lambda (v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind -Abst) u2 u)) \to False))) (\lambda (x1: T).(\lambda (H11: (eq T x (THead -(Bind Abst) x0 x1))).(let H12 \def (eq_ind T x (\lambda (t: T).(nf2 c t)) H5 -(THead (Bind Abst) x0 x1) H11) in (let H13 \def (eq_ind T x (\lambda (t: -T).(pr3 c u1 t)) H4 (THead (Bind Abst) x0 x1) H11) in (let H_y \def -(ty3_sred_pr3 c u1 (THead (Bind Abst) x0 x1) H13 g t1 H) in (or_introl (ex4_2 -T T (\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst) u2 u)))) -(\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u) t1))) -(\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_: T).(\lambda -(v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind Abst) u2 u)) -\to False)) (ex4_2_intro T T (\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead -(Bind Abst) u2 u)))) (\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind -Abst) v2 u) t1))) (\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda -(_: T).(\lambda (v2: T).(nf2 c v2))) x1 x0 (pc3_pr3_t c u1 (THead (Bind Abst) -x0 x1) H13 (THead (Bind Abst) u2 x1) (pr3_head_12 c u2 x0 H7 (Bind Abst) x1 -x1 (pr3_refl (CHead c (Bind Abst) x0) x1))) H_y H7 H8))))))) H10)) (\lambda -(H10: ((\forall (t: T).((eq T x (THead (Bind Abst) x0 t)) \to (\forall (P: -Prop).P))))).(or_intror (ex4_2 T T (\lambda (u: T).(\lambda (_: T).(pc3 c u1 -(THead (Bind Abst) u2 u)))) (\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead -(Bind Abst) v2 u) t1))) (\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) -(\lambda (_: T).(\lambda (v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1 -(THead (Bind Abst) u2 u)) \to False)) (\lambda (u: T).(\lambda (H11: (pc3 c -u1 (THead (Bind Abst) u2 u))).(let H12 \def H11 in (ex2_ind T (\lambda (t: -T).(pr3 c u1 t)) (\lambda (t: T).(pr3 c (THead (Bind Abst) u2 u) t)) False -(\lambda (x1: T).(\lambda (H13: (pr3 c u1 x1)).(\lambda (H14: (pr3 c (THead -(Bind Abst) u2 u) x1)).(ex2_ind T (\lambda (t: T).(pr3 c x1 t)) (\lambda (t: -T).(pr3 c x t)) False (\lambda (x2: T).(\lambda (H15: (pr3 c x1 x2)).(\lambda -(H16: (pr3 c x x2)).(let H_y \def (nf2_pr3_unfold c x x2 H16 H5) in (let H17 -\def (eq_ind_r T x2 (\lambda (t: T).(pr3 c x1 t)) H15 x H_y) in (let H18 \def -(pr3_gen_abst c u2 u x1 H14) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: -T).(eq T x1 (THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: -T).(pr3 c u2 u3))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall -(u0: T).(pr3 (CHead c (Bind b) u0) u t3))))) False (\lambda (x3: T).(\lambda -(x4: T).(\lambda (H19: (eq T x1 (THead (Bind Abst) x3 x4))).(\lambda (H20: -(pr3 c u2 x3)).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pr3 (CHead c -(Bind b) u0) u x4))))).(let H22 \def (eq_ind T x1 (\lambda (t: T).(pr3 c t -x)) H17 (THead (Bind Abst) x3 x4) H19) in (let H23 \def (pr3_gen_abst c x3 x4 -x H22) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead -(Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c x3 u3))) -(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr3 (CHead -c (Bind b) u0) x4 t3))))) False (\lambda (x5: T).(\lambda (x6: T).(\lambda -(H24: (eq T x (THead (Bind Abst) x5 x6))).(\lambda (H25: (pr3 c x3 -x5)).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pr3 (CHead c (Bind b) -u0) x4 x6))))).(let H27 \def (eq_ind T x (\lambda (t: T).(\forall (t0: -T).((eq T t (THead (Bind Abst) x0 t0)) \to (\forall (P: Prop).P)))) H10 -(THead (Bind Abst) x5 x6) H24) in (let H28 \def (eq_ind T x (\lambda (t: -T).(nf2 c t)) H5 (THead (Bind Abst) x5 x6) H24) in (let H29 \def -(nf2_gen_abst c x5 x6 H28) in (land_ind (nf2 c x5) (nf2 (CHead c (Bind Abst) -x5) x6) False (\lambda (H30: (nf2 c x5)).(\lambda (_: (nf2 (CHead c (Bind -Abst) x5) x6)).(let H32 \def (nf2_pr3_confluence c x0 H8 x5 H30 u2 H7) in -(H27 x6 (sym_eq T (THead (Bind Abst) x0 x6) (THead (Bind Abst) x5 x6) -(f_equal3 K T T T THead (Bind Abst) (Bind Abst) x0 x5 x6 x6 (refl_equal K -(Bind Abst)) (H32 (pr3_t x3 u2 c H20 x5 H25)) (refl_equal T x6))) False)))) -H29))))))))) H23)))))))) H18))))))) (pr3_confluence c u1 x1 H13 x H4))))) -H12)))))) H9)))))) H6)))))) H3)))))))))))). -(* COMMENTS -Initial nodes: 1759 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/defs.ma deleted file mode 100644 index e7ea2b24f..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/defs.ma +++ /dev/null @@ -1,31 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr3/defs.ma". - -definition pc3: - C \to (T \to (T \to Prop)) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(ex2 T (\lambda (t: T).(pr3 -c t1 t)) (\lambda (t: T).(pr3 c t2 t))))). - -inductive pc3_left (c: C): T \to (T \to Prop) \def -| pc3_left_r: \forall (t: T).(pc3_left c t t) -| pc3_left_ur: \forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall -(t3: T).((pc3_left c t2 t3) \to (pc3_left c t1 t3))))) -| pc3_left_ux: \forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall -(t3: T).((pc3_left c t1 t3) \to (pc3_left c t2 t3))))). - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/fsubst0.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/fsubst0.ma deleted file mode 100644 index c563ca397..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/fsubst0.ma +++ /dev/null @@ -1,726 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pc3/left.ma". - -include "Basic-1/fsubst0/defs.ma". - -include "Basic-1/csubst0/getl.ma". - -theorem pc3_pr2_fsubst0: - \forall (c1: C).(\forall (t1: T).(\forall (t: T).((pr2 c1 t1 t) \to (\forall -(i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t2: T).((fsubst0 i u c1 -t1 c2 t2) \to (\forall (e: C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3 -c2 t2 t))))))))))) -\def - \lambda (c1: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H: (pr2 c1 t1 -t)).(pr2_ind (\lambda (c: C).(\lambda (t0: T).(\lambda (t2: T).(\forall (i: -nat).(\forall (u: T).(\forall (c2: C).(\forall (t3: T).((fsubst0 i u c t0 c2 -t3) \to (\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) \to (pc3 c2 t3 -t2))))))))))) (\lambda (c: C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H0: -(pr0 t2 t3)).(\lambda (i: nat).(\lambda (u: T).(\lambda (c2: C).(\lambda (t0: -T).(\lambda (H1: (fsubst0 i u c t2 c2 t0)).(fsubst0_ind i u c t2 (\lambda -(c0: C).(\lambda (t4: T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) -\to (pc3 c0 t4 t3))))) (\lambda (t4: T).(\lambda (H2: (subst0 i u t2 -t4)).(\lambda (e: C).(\lambda (H3: (getl i c (CHead e (Bind Abbr) -u))).(or_ind (pr0 t4 t3) (ex2 T (\lambda (w2: T).(pr0 t4 w2)) (\lambda (w2: -T).(subst0 i u t3 w2))) (pc3 c t4 t3) (\lambda (H4: (pr0 t4 t3)).(pc3_pr2_r c -t4 t3 (pr2_free c t4 t3 H4))) (\lambda (H4: (ex2 T (\lambda (w2: T).(pr0 t4 -w2)) (\lambda (w2: T).(subst0 i u t3 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 -t4 w2)) (\lambda (w2: T).(subst0 i u t3 w2)) (pc3 c t4 t3) (\lambda (x: -T).(\lambda (H5: (pr0 t4 x)).(\lambda (H6: (subst0 i u t3 x)).(pc3_pr2_u c x -t4 (pr2_free c t4 x H5) t3 (pc3_pr2_x c x t3 (pr2_delta c e u i H3 t3 t3 -(pr0_refl t3) x H6)))))) H4)) (pr0_subst0 t2 t3 H0 u t4 i H2 u (pr0_refl -u))))))) (\lambda (c0: C).(\lambda (_: (csubst0 i u c c0)).(\lambda (e: -C).(\lambda (_: (getl i c (CHead e (Bind Abbr) u))).(pc3_pr2_r c0 t2 t3 -(pr2_free c0 t2 t3 H0)))))) (\lambda (t4: T).(\lambda (H2: (subst0 i u t2 -t4)).(\lambda (c0: C).(\lambda (H3: (csubst0 i u c c0)).(\lambda (e: -C).(\lambda (H4: (getl i c (CHead e (Bind Abbr) u))).(or_ind (pr0 t4 t3) (ex2 -T (\lambda (w2: T).(pr0 t4 w2)) (\lambda (w2: T).(subst0 i u t3 w2))) (pc3 c0 -t4 t3) (\lambda (H5: (pr0 t4 t3)).(pc3_pr2_r c0 t4 t3 (pr2_free c0 t4 t3 -H5))) (\lambda (H5: (ex2 T (\lambda (w2: T).(pr0 t4 w2)) (\lambda (w2: -T).(subst0 i u t3 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t4 w2)) (\lambda -(w2: T).(subst0 i u t3 w2)) (pc3 c0 t4 t3) (\lambda (x: T).(\lambda (H6: (pr0 -t4 x)).(\lambda (H7: (subst0 i u t3 x)).(pc3_pr2_u c0 x t4 (pr2_free c0 t4 x -H6) t3 (pc3_pr2_x c0 x t3 (pr2_delta c0 e u i (csubst0_getl_ge i i (le_n i) c -c0 u H3 (CHead e (Bind Abbr) u) H4) t3 t3 (pr0_refl t3) x H7)))))) H5)) -(pr0_subst0 t2 t3 H0 u t4 i H2 u (pr0_refl u))))))))) c2 t0 H1)))))))))) -(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (t2: T).(\lambda (t3: -T).(\lambda (H1: (pr0 t2 t3)).(\lambda (t0: T).(\lambda (H2: (subst0 i u t3 -t0)).(\lambda (i0: nat).(\lambda (u0: T).(\lambda (c2: C).(\lambda (t4: -T).(\lambda (H3: (fsubst0 i0 u0 c t2 c2 t4)).(fsubst0_ind i0 u0 c t2 (\lambda -(c0: C).(\lambda (t5: T).(\forall (e: C).((getl i0 c (CHead e (Bind Abbr) -u0)) \to (pc3 c0 t5 t0))))) (\lambda (t5: T).(\lambda (H4: (subst0 i0 u0 t2 -t5)).(\lambda (e: C).(\lambda (H5: (getl i0 c (CHead e (Bind Abbr) -u0))).(pc3_t t2 c t5 (pc3_s c t5 t2 (pc3_pr2_r c t2 t5 (pr2_delta c e u0 i0 -H5 t2 t2 (pr0_refl t2) t5 H4))) t0 (pc3_pr2_r c t2 t0 (pr2_delta c d u i H0 -t2 t3 H1 t0 H2))))))) (\lambda (c0: C).(\lambda (H4: (csubst0 i0 u0 c -c0)).(\lambda (e: C).(\lambda (H5: (getl i0 c (CHead e (Bind Abbr) -u0))).(lt_le_e i i0 (pc3 c0 t2 t0) (\lambda (H6: (lt i i0)).(let H7 \def -(csubst0_getl_lt i0 i H6 c c0 u0 H4 (CHead d (Bind Abbr) u) H0) in (or4_ind -(getl i c0 (CHead d (Bind Abbr) u)) (ex3_4 B C T T (\lambda (b: B).(\lambda -(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead -e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) -u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c0 -(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) (ex4_5 B C C T T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda -(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl -i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))) (pc3 c0 t2 t0) (\lambda (H8: -(getl i c0 (CHead d (Bind Abbr) u))).(pc3_pr2_r c0 t2 t0 (pr2_delta c0 d u i -H8 t2 t3 H1 t0 H2))) (\lambda (H8: (ex3_4 B C T T (\lambda (b: B).(\lambda -(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead -e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) -u0 u1 w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda -(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b) -u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w))))) -(pc3 c0 t2 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda -(x3: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) -x2))).(\lambda (H10: (getl i c0 (CHead x1 (Bind x0) x3))).(\lambda (H11: -(subst0 (minus i0 (S i)) u0 x2 x3)).(let H12 \def (f_equal C C (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | -(CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) -x2) H9) in ((let H13 \def (f_equal C B (\lambda (e0: C).(match e0 in C return -(\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) -\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead -x1 (Bind x0) x2) H9) in ((let H14 \def (f_equal C T (\lambda (e0: C).(match -e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ -t5) \Rightarrow t5])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H9) in -(\lambda (H15: (eq B Abbr x0)).(\lambda (H16: (eq C d x1)).(let H17 \def -(eq_ind_r T x2 (\lambda (t5: T).(subst0 (minus i0 (S i)) u0 t5 x3)) H11 u -H14) in (let H18 \def (eq_ind_r C x1 (\lambda (c3: C).(getl i c0 (CHead c3 -(Bind x0) x3))) H10 d H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b: -B).(getl i c0 (CHead d (Bind b) x3))) H18 Abbr H15) in (ex2_ind T (\lambda -(t5: T).(subst0 i x3 t3 t5)) (\lambda (t5: T).(subst0 (S (plus (minus i0 (S -i)) i)) u0 t0 t5)) (pc3 c0 t2 t0) (\lambda (x: T).(\lambda (H20: (subst0 i x3 -t3 x)).(\lambda (H21: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let -H22 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n: -nat).(subst0 n u0 t0 x)) H21 i0 (lt_plus_minus_r i i0 H6)) in (pc3_pr2_u c0 x -t2 (pr2_delta c0 d x3 i H19 t2 t3 H1 x H20) t0 (pc3_pr2_x c0 x t0 (pr2_delta -c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H4 (CHead e (Bind Abbr) -u0) H5) t0 t0 (pr0_refl t0) x H22))))))) (subst0_subst0_back t3 t0 u i H2 x3 -u0 (minus i0 (S i)) H17)))))))) H13)) H12))))))))) H8)) (\lambda (H8: (ex3_4 -B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq -C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 -(Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))).(ex3_4_ind B C C T (\lambda -(b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind -Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 (Bind b) u1)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S -i)) u0 e1 e2))))) (pc3 c0 t2 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda -(x2: C).(\lambda (x3: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead -x1 (Bind x0) x3))).(\lambda (H10: (getl i c0 (CHead x2 (Bind x0) -x3))).(\lambda (H11: (csubst0 (minus i0 (S i)) u0 x1 x2)).(let H12 \def -(f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with -[(CSort _) \Rightarrow d | (CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind -Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H13 \def (f_equal C B (\lambda -(e0: C).(match e0 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow -Abbr | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with -[(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) -u) (CHead x1 (Bind x0) x3) H9) in ((let H14 \def (f_equal C T (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | -(CHead _ _ t5) \Rightarrow t5])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) -x3) H9) in (\lambda (H15: (eq B Abbr x0)).(\lambda (H16: (eq C d x1)).(let -H17 \def (eq_ind_r T x3 (\lambda (t5: T).(getl i c0 (CHead x2 (Bind x0) t5))) -H10 u H14) in (let H18 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus -i0 (S i)) u0 c3 x2)) H11 d H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b: -B).(getl i c0 (CHead x2 (Bind b) u))) H17 Abbr H15) in (pc3_pr2_r c0 t2 t0 -(pr2_delta c0 x2 u i H19 t2 t3 H1 t0 H2)))))))) H13)) H12))))))))) H8)) -(\lambda (H8: (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 -(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 -(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 -e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 -(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 -(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) -(pc3 c0 t2 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda -(x3: T).(\lambda (x4: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead -x1 (Bind x0) x3))).(\lambda (H10: (getl i c0 (CHead x2 (Bind x0) -x4))).(\lambda (H11: (subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H12: -(csubst0 (minus i0 (S i)) u0 x1 x2)).(let H13 \def (f_equal C C (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | -(CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) -x3) H9) in ((let H14 \def (f_equal C B (\lambda (e0: C).(match e0 in C return -(\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) -\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead -x1 (Bind x0) x3) H9) in ((let H15 \def (f_equal C T (\lambda (e0: C).(match -e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ -t5) \Rightarrow t5])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in -(\lambda (H16: (eq B Abbr x0)).(\lambda (H17: (eq C d x1)).(let H18 \def -(eq_ind_r T x3 (\lambda (t5: T).(subst0 (minus i0 (S i)) u0 t5 x4)) H11 u -H15) in (let H19 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus i0 (S -i)) u0 c3 x2)) H12 d H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b: -B).(getl i c0 (CHead x2 (Bind b) x4))) H10 Abbr H16) in (ex2_ind T (\lambda -(t5: T).(subst0 i x4 t3 t5)) (\lambda (t5: T).(subst0 (S (plus (minus i0 (S -i)) i)) u0 t0 t5)) (pc3 c0 t2 t0) (\lambda (x: T).(\lambda (H21: (subst0 i x4 -t3 x)).(\lambda (H22: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let -H23 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n: -nat).(subst0 n u0 t0 x)) H22 i0 (lt_plus_minus_r i i0 H6)) in (pc3_pr2_u c0 x -t2 (pr2_delta c0 x2 x4 i H20 t2 t3 H1 x H21) t0 (pc3_pr2_x c0 x t0 (pr2_delta -c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H4 (CHead e (Bind Abbr) -u0) H5) t0 t0 (pr0_refl t0) x H23))))))) (subst0_subst0_back t3 t0 u i H2 x4 -u0 (minus i0 (S i)) H18)))))))) H14)) H13))))))))))) H8)) H7))) (\lambda (H6: -(le i0 i)).(pc3_pr2_r c0 t2 t0 (pr2_delta c0 d u i (csubst0_getl_ge i0 i H6 c -c0 u0 H4 (CHead d (Bind Abbr) u) H0) t2 t3 H1 t0 H2)))))))) (\lambda (t5: -T).(\lambda (H4: (subst0 i0 u0 t2 t5)).(\lambda (c0: C).(\lambda (H5: -(csubst0 i0 u0 c c0)).(\lambda (e: C).(\lambda (H6: (getl i0 c (CHead e (Bind -Abbr) u0))).(lt_le_e i i0 (pc3 c0 t5 t0) (\lambda (H7: (lt i i0)).(let H8 -\def (csubst0_getl_lt i0 i H7 c c0 u0 H5 (CHead d (Bind Abbr) u) H0) in -(or4_ind (getl i c0 (CHead d (Bind Abbr) u)) (ex3_4 B C T T (\lambda (b: -B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind -Abbr) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 -(minus i0 (S i)) u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 -(Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(u1: T).(getl i c0 (CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) -(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) -u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 -(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))) -(pc3 c0 t5 t0) (\lambda (H9: (getl i c0 (CHead d (Bind Abbr) u))).(pc3_pr2_u2 -c0 t2 t5 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 -(CHead e (Bind Abbr) u0) H6) t2 t2 (pr0_refl t2) t5 H4) t0 (pc3_pr2_r c0 t2 -t0 (pr2_delta c0 d u i H9 t2 t3 H1 t0 H2)))) (\lambda (H9: (ex3_4 B C T T -(\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C -(CHead d (Bind Abbr) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda -(e0: C).(\lambda (_: T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 -(minus i0 (S i)) u0 u1 w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda -(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead -e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) -u0 u1 w))))) (pc3 c0 t5 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: -T).(\lambda (x3: T).(\lambda (H10: (eq C (CHead d (Bind Abbr) u) (CHead x1 -(Bind x0) x2))).(\lambda (H11: (getl i c0 (CHead x1 (Bind x0) x3))).(\lambda -(H12: (subst0 (minus i0 (S i)) u0 x2 x3)).(let H13 \def (f_equal C C (\lambda -(e0: C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow -d | (CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind -x0) x2) H10) in ((let H14 \def (f_equal C B (\lambda (e0: C).(match e0 in C -return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) -\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead -x1 (Bind x0) x2) H10) in ((let H15 \def (f_equal C T (\lambda (e0: C).(match -e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ -t6) \Rightarrow t6])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H10) in -(\lambda (H16: (eq B Abbr x0)).(\lambda (H17: (eq C d x1)).(let H18 \def -(eq_ind_r T x2 (\lambda (t6: T).(subst0 (minus i0 (S i)) u0 t6 x3)) H12 u -H15) in (let H19 \def (eq_ind_r C x1 (\lambda (c3: C).(getl i c0 (CHead c3 -(Bind x0) x3))) H11 d H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b: -B).(getl i c0 (CHead d (Bind b) x3))) H19 Abbr H16) in (ex2_ind T (\lambda -(t6: T).(subst0 i x3 t3 t6)) (\lambda (t6: T).(subst0 (S (plus (minus i0 (S -i)) i)) u0 t0 t6)) (pc3 c0 t5 t0) (\lambda (x: T).(\lambda (H21: (subst0 i x3 -t3 x)).(\lambda (H22: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let -H23 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n: -nat).(subst0 n u0 t0 x)) H22 i0 (lt_plus_minus_r i i0 H7)) in (pc3_pr2_u2 c0 -t2 t5 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 -(CHead e (Bind Abbr) u0) H6) t2 t2 (pr0_refl t2) t5 H4) t0 (pc3_pr2_u c0 x t2 -(pr2_delta c0 d x3 i H20 t2 t3 H1 x H21) t0 (pc3_pr2_x c0 x t0 (pr2_delta c0 -e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e (Bind Abbr) u0) -H6) t0 t0 (pr0_refl t0) x H23)))))))) (subst0_subst0_back t3 t0 u i H2 x3 u0 -(minus i0 (S i)) H18)))))))) H14)) H13))))))))) H9)) (\lambda (H9: (ex3_4 B C -C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C -(CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda -(_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 (Bind b) -u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))).(ex3_4_ind B C C T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind -Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 (Bind b) u1)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S -i)) u0 e1 e2))))) (pc3 c0 t5 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda -(x2: C).(\lambda (x3: T).(\lambda (H10: (eq C (CHead d (Bind Abbr) u) (CHead -x1 (Bind x0) x3))).(\lambda (H11: (getl i c0 (CHead x2 (Bind x0) -x3))).(\lambda (H12: (csubst0 (minus i0 (S i)) u0 x1 x2)).(let H13 \def -(f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with -[(CSort _) \Rightarrow d | (CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind -Abbr) u) (CHead x1 (Bind x0) x3) H10) in ((let H14 \def (f_equal C B (\lambda -(e0: C).(match e0 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow -Abbr | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with -[(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) -u) (CHead x1 (Bind x0) x3) H10) in ((let H15 \def (f_equal C T (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | -(CHead _ _ t6) \Rightarrow t6])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) -x3) H10) in (\lambda (H16: (eq B Abbr x0)).(\lambda (H17: (eq C d x1)).(let -H18 \def (eq_ind_r T x3 (\lambda (t6: T).(getl i c0 (CHead x2 (Bind x0) t6))) -H11 u H15) in (let H19 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus -i0 (S i)) u0 c3 x2)) H12 d H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b: -B).(getl i c0 (CHead x2 (Bind b) u))) H18 Abbr H16) in (pc3_pr2_u2 c0 t2 t5 -(pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e -(Bind Abbr) u0) H6) t2 t2 (pr0_refl t2) t5 H4) t0 (pc3_pr2_r c0 t2 t0 -(pr2_delta c0 x2 u i H20 t2 t3 H1 t0 H2))))))))) H14)) H13))))))))) H9)) -(\lambda (H9: (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 -(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 -(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 -e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 -(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 -(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) -(pc3 c0 t5 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda -(x3: T).(\lambda (x4: T).(\lambda (H10: (eq C (CHead d (Bind Abbr) u) (CHead -x1 (Bind x0) x3))).(\lambda (H11: (getl i c0 (CHead x2 (Bind x0) -x4))).(\lambda (H12: (subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H13: -(csubst0 (minus i0 (S i)) u0 x1 x2)).(let H14 \def (f_equal C C (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | -(CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) -x3) H10) in ((let H15 \def (f_equal C B (\lambda (e0: C).(match e0 in C -return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) -\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead -x1 (Bind x0) x3) H10) in ((let H16 \def (f_equal C T (\lambda (e0: C).(match -e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ -t6) \Rightarrow t6])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H10) in -(\lambda (H17: (eq B Abbr x0)).(\lambda (H18: (eq C d x1)).(let H19 \def -(eq_ind_r T x3 (\lambda (t6: T).(subst0 (minus i0 (S i)) u0 t6 x4)) H12 u -H16) in (let H20 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus i0 (S -i)) u0 c3 x2)) H13 d H18) in (let H21 \def (eq_ind_r B x0 (\lambda (b: -B).(getl i c0 (CHead x2 (Bind b) x4))) H11 Abbr H17) in (ex2_ind T (\lambda -(t6: T).(subst0 i x4 t3 t6)) (\lambda (t6: T).(subst0 (S (plus (minus i0 (S -i)) i)) u0 t0 t6)) (pc3 c0 t5 t0) (\lambda (x: T).(\lambda (H22: (subst0 i x4 -t3 x)).(\lambda (H23: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let -H24 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n: -nat).(subst0 n u0 t0 x)) H23 i0 (lt_plus_minus_r i i0 H7)) in (pc3_pr2_u2 c0 -t2 t5 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 -(CHead e (Bind Abbr) u0) H6) t2 t2 (pr0_refl t2) t5 H4) t0 (pc3_pr2_u c0 x t2 -(pr2_delta c0 x2 x4 i H21 t2 t3 H1 x H22) t0 (pc3_pr2_x c0 x t0 (pr2_delta c0 -e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e (Bind Abbr) u0) -H6) t0 t0 (pr0_refl t0) x H24)))))))) (subst0_subst0_back t3 t0 u i H2 x4 u0 -(minus i0 (S i)) H19)))))))) H15)) H14))))))))))) H9)) H8))) (\lambda (H7: -(le i0 i)).(pc3_pr2_u2 c0 t2 t5 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 -(le_n i0) c c0 u0 H5 (CHead e (Bind Abbr) u0) H6) t2 t2 (pr0_refl t2) t5 H4) -t0 (pc3_pr2_r c0 t2 t0 (pr2_delta c0 d u i (csubst0_getl_ge i0 i H7 c c0 u0 -H5 (CHead d (Bind Abbr) u) H0) t2 t3 H1 t0 H2))))))))))) c2 t4 -H3)))))))))))))))) c1 t1 t H)))). -(* COMMENTS -Initial nodes: 6455 -END *) - -theorem pc3_pr2_fsubst0_back: - \forall (c1: C).(\forall (t: T).(\forall (t1: T).((pr2 c1 t t1) \to (\forall -(i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t2: T).((fsubst0 i u c1 -t1 c2 t2) \to (\forall (e: C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3 -c2 t t2))))))))))) -\def - \lambda (c1: C).(\lambda (t: T).(\lambda (t1: T).(\lambda (H: (pr2 c1 t -t1)).(pr2_ind (\lambda (c: C).(\lambda (t0: T).(\lambda (t2: T).(\forall (i: -nat).(\forall (u: T).(\forall (c2: C).(\forall (t3: T).((fsubst0 i u c t2 c2 -t3) \to (\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) \to (pc3 c2 t0 -t3))))))))))) (\lambda (c: C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H0: -(pr0 t2 t3)).(\lambda (i: nat).(\lambda (u: T).(\lambda (c2: C).(\lambda (t0: -T).(\lambda (H1: (fsubst0 i u c t3 c2 t0)).(fsubst0_ind i u c t3 (\lambda -(c0: C).(\lambda (t4: T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) -\to (pc3 c0 t2 t4))))) (\lambda (t4: T).(\lambda (H2: (subst0 i u t3 -t4)).(\lambda (e: C).(\lambda (H3: (getl i c (CHead e (Bind Abbr) -u))).(pc3_pr2_u c t3 t2 (pr2_free c t2 t3 H0) t4 (pc3_pr2_r c t3 t4 -(pr2_delta c e u i H3 t3 t3 (pr0_refl t3) t4 H2))))))) (\lambda (c0: -C).(\lambda (_: (csubst0 i u c c0)).(\lambda (e: C).(\lambda (_: (getl i c -(CHead e (Bind Abbr) u))).(pc3_pr2_r c0 t2 t3 (pr2_free c0 t2 t3 H0)))))) -(\lambda (t4: T).(\lambda (H2: (subst0 i u t3 t4)).(\lambda (c0: C).(\lambda -(H3: (csubst0 i u c c0)).(\lambda (e: C).(\lambda (H4: (getl i c (CHead e -(Bind Abbr) u))).(pc3_pr2_u c0 t3 t2 (pr2_free c0 t2 t3 H0) t4 (pc3_pr2_r c0 -t3 t4 (pr2_delta c0 e u i (csubst0_getl_ge i i (le_n i) c c0 u H3 (CHead e -(Bind Abbr) u) H4) t3 t3 (pr0_refl t3) t4 H2))))))))) c2 t0 H1)))))))))) -(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (t2: T).(\lambda (t3: -T).(\lambda (H1: (pr0 t2 t3)).(\lambda (t0: T).(\lambda (H2: (subst0 i u t3 -t0)).(\lambda (i0: nat).(\lambda (u0: T).(\lambda (c2: C).(\lambda (t4: -T).(\lambda (H3: (fsubst0 i0 u0 c t0 c2 t4)).(fsubst0_ind i0 u0 c t0 (\lambda -(c0: C).(\lambda (t5: T).(\forall (e: C).((getl i0 c (CHead e (Bind Abbr) -u0)) \to (pc3 c0 t2 t5))))) (\lambda (t5: T).(\lambda (H4: (subst0 i0 u0 t0 -t5)).(\lambda (e: C).(\lambda (H5: (getl i0 c (CHead e (Bind Abbr) -u0))).(pc3_t t3 c t2 (pc3_pr3_r c t2 t3 (pr3_pr2 c t2 t3 (pr2_free c t2 t3 -H1))) t5 (pc3_pr3_r c t3 t5 (pr3_sing c t0 t3 (pr2_delta c d u i H0 t3 t3 -(pr0_refl t3) t0 H2) t5 (pr3_pr2 c t0 t5 (pr2_delta c e u0 i0 H5 t0 t0 -(pr0_refl t0) t5 H4))))))))) (\lambda (c0: C).(\lambda (H4: (csubst0 i0 u0 c -c0)).(\lambda (e: C).(\lambda (H5: (getl i0 c (CHead e (Bind Abbr) -u0))).(lt_le_e i i0 (pc3 c0 t2 t0) (\lambda (H6: (lt i i0)).(let H7 \def -(csubst0_getl_lt i0 i H6 c c0 u0 H4 (CHead d (Bind Abbr) u) H0) in (or4_ind -(getl i c0 (CHead d (Bind Abbr) u)) (ex3_4 B C T T (\lambda (b: B).(\lambda -(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead -e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) -u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c0 -(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) (ex4_5 B C C T T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda -(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl -i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))) (pc3 c0 t2 t0) (\lambda (H8: -(getl i c0 (CHead d (Bind Abbr) u))).(pc3_pr2_r c0 t2 t0 (pr2_delta c0 d u i -H8 t2 t3 H1 t0 H2))) (\lambda (H8: (ex3_4 B C T T (\lambda (b: B).(\lambda -(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead -e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) -u0 u1 w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda -(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b) -u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w))))) -(pc3 c0 t2 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda -(x3: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) -x2))).(\lambda (H10: (getl i c0 (CHead x1 (Bind x0) x3))).(\lambda (H11: -(subst0 (minus i0 (S i)) u0 x2 x3)).(let H12 \def (f_equal C C (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | -(CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) -x2) H9) in ((let H13 \def (f_equal C B (\lambda (e0: C).(match e0 in C return -(\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) -\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead -x1 (Bind x0) x2) H9) in ((let H14 \def (f_equal C T (\lambda (e0: C).(match -e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ -t5) \Rightarrow t5])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H9) in -(\lambda (H15: (eq B Abbr x0)).(\lambda (H16: (eq C d x1)).(let H17 \def -(eq_ind_r T x2 (\lambda (t5: T).(subst0 (minus i0 (S i)) u0 t5 x3)) H11 u -H14) in (let H18 \def (eq_ind_r C x1 (\lambda (c3: C).(getl i c0 (CHead c3 -(Bind x0) x3))) H10 d H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b: -B).(getl i c0 (CHead d (Bind b) x3))) H18 Abbr H15) in (ex2_ind T (\lambda -(t5: T).(subst0 i x3 t3 t5)) (\lambda (t5: T).(subst0 (S (plus (minus i0 (S -i)) i)) u0 t0 t5)) (pc3 c0 t2 t0) (\lambda (x: T).(\lambda (H20: (subst0 i x3 -t3 x)).(\lambda (H21: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let -H22 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n: -nat).(subst0 n u0 t0 x)) H21 i0 (lt_plus_minus_r i i0 H6)) in (pc3_pr2_u c0 x -t2 (pr2_delta c0 d x3 i H19 t2 t3 H1 x H20) t0 (pc3_pr2_x c0 x t0 (pr2_delta -c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H4 (CHead e (Bind Abbr) -u0) H5) t0 t0 (pr0_refl t0) x H22))))))) (subst0_subst0_back t3 t0 u i H2 x3 -u0 (minus i0 (S i)) H17)))))))) H13)) H12))))))))) H8)) (\lambda (H8: (ex3_4 -B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq -C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 -(Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))).(ex3_4_ind B C C T (\lambda -(b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind -Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 (Bind b) u1)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S -i)) u0 e1 e2))))) (pc3 c0 t2 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda -(x2: C).(\lambda (x3: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead -x1 (Bind x0) x3))).(\lambda (H10: (getl i c0 (CHead x2 (Bind x0) -x3))).(\lambda (H11: (csubst0 (minus i0 (S i)) u0 x1 x2)).(let H12 \def -(f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with -[(CSort _) \Rightarrow d | (CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind -Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H13 \def (f_equal C B (\lambda -(e0: C).(match e0 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow -Abbr | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with -[(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) -u) (CHead x1 (Bind x0) x3) H9) in ((let H14 \def (f_equal C T (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | -(CHead _ _ t5) \Rightarrow t5])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) -x3) H9) in (\lambda (H15: (eq B Abbr x0)).(\lambda (H16: (eq C d x1)).(let -H17 \def (eq_ind_r T x3 (\lambda (t5: T).(getl i c0 (CHead x2 (Bind x0) t5))) -H10 u H14) in (let H18 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus -i0 (S i)) u0 c3 x2)) H11 d H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b: -B).(getl i c0 (CHead x2 (Bind b) u))) H17 Abbr H15) in (pc3_pr2_r c0 t2 t0 -(pr2_delta c0 x2 u i H19 t2 t3 H1 t0 H2)))))))) H13)) H12))))))))) H8)) -(\lambda (H8: (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 -(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 -(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 -e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 -(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 -(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) -(pc3 c0 t2 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda -(x3: T).(\lambda (x4: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead -x1 (Bind x0) x3))).(\lambda (H10: (getl i c0 (CHead x2 (Bind x0) -x4))).(\lambda (H11: (subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H12: -(csubst0 (minus i0 (S i)) u0 x1 x2)).(let H13 \def (f_equal C C (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | -(CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) -x3) H9) in ((let H14 \def (f_equal C B (\lambda (e0: C).(match e0 in C return -(\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) -\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead -x1 (Bind x0) x3) H9) in ((let H15 \def (f_equal C T (\lambda (e0: C).(match -e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ -t5) \Rightarrow t5])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in -(\lambda (H16: (eq B Abbr x0)).(\lambda (H17: (eq C d x1)).(let H18 \def -(eq_ind_r T x3 (\lambda (t5: T).(subst0 (minus i0 (S i)) u0 t5 x4)) H11 u -H15) in (let H19 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus i0 (S -i)) u0 c3 x2)) H12 d H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b: -B).(getl i c0 (CHead x2 (Bind b) x4))) H10 Abbr H16) in (ex2_ind T (\lambda -(t5: T).(subst0 i x4 t3 t5)) (\lambda (t5: T).(subst0 (S (plus (minus i0 (S -i)) i)) u0 t0 t5)) (pc3 c0 t2 t0) (\lambda (x: T).(\lambda (H21: (subst0 i x4 -t3 x)).(\lambda (H22: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let -H23 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n: -nat).(subst0 n u0 t0 x)) H22 i0 (lt_plus_minus_r i i0 H6)) in (pc3_pr2_u c0 x -t2 (pr2_delta c0 x2 x4 i H20 t2 t3 H1 x H21) t0 (pc3_pr2_x c0 x t0 (pr2_delta -c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H4 (CHead e (Bind Abbr) -u0) H5) t0 t0 (pr0_refl t0) x H23))))))) (subst0_subst0_back t3 t0 u i H2 x4 -u0 (minus i0 (S i)) H18)))))))) H14)) H13))))))))))) H8)) H7))) (\lambda (H6: -(le i0 i)).(pc3_pr2_r c0 t2 t0 (pr2_delta c0 d u i (csubst0_getl_ge i0 i H6 c -c0 u0 H4 (CHead d (Bind Abbr) u) H0) t2 t3 H1 t0 H2)))))))) (\lambda (t5: -T).(\lambda (H4: (subst0 i0 u0 t0 t5)).(\lambda (c0: C).(\lambda (H5: -(csubst0 i0 u0 c c0)).(\lambda (e: C).(\lambda (H6: (getl i0 c (CHead e (Bind -Abbr) u0))).(lt_le_e i i0 (pc3 c0 t2 t5) (\lambda (H7: (lt i i0)).(let H8 -\def (csubst0_getl_lt i0 i H7 c c0 u0 H5 (CHead d (Bind Abbr) u) H0) in -(or4_ind (getl i c0 (CHead d (Bind Abbr) u)) (ex3_4 B C T T (\lambda (b: -B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind -Abbr) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 -(minus i0 (S i)) u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 -(Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(u1: T).(getl i c0 (CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) -(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) -u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 -(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))) -(pc3 c0 t2 t5) (\lambda (H9: (getl i c0 (CHead d (Bind Abbr) u))).(pc3_pr2_u -c0 t3 t2 (pr2_free c0 t2 t3 H1) t5 (pc3_pr3_r c0 t3 t5 (pr3_sing c0 t0 t3 -(pr2_delta c0 d u i H9 t3 t3 (pr0_refl t3) t0 H2) t5 (pr3_pr2 c0 t0 t5 -(pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e -(Bind Abbr) u0) H6) t0 t0 (pr0_refl t0) t5 H4)))))) (\lambda (H9: (ex3_4 B C -T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C -(CHead d (Bind Abbr) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda -(e0: C).(\lambda (_: T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 -(minus i0 (S i)) u0 u1 w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda -(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead -e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) -u0 u1 w))))) (pc3 c0 t2 t5) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: -T).(\lambda (x3: T).(\lambda (H10: (eq C (CHead d (Bind Abbr) u) (CHead x1 -(Bind x0) x2))).(\lambda (H11: (getl i c0 (CHead x1 (Bind x0) x3))).(\lambda -(H12: (subst0 (minus i0 (S i)) u0 x2 x3)).(let H13 \def (f_equal C C (\lambda -(e0: C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow -d | (CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind -x0) x2) H10) in ((let H14 \def (f_equal C B (\lambda (e0: C).(match e0 in C -return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) -\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead -x1 (Bind x0) x2) H10) in ((let H15 \def (f_equal C T (\lambda (e0: C).(match -e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ -t6) \Rightarrow t6])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H10) in -(\lambda (H16: (eq B Abbr x0)).(\lambda (H17: (eq C d x1)).(let H18 \def -(eq_ind_r T x2 (\lambda (t6: T).(subst0 (minus i0 (S i)) u0 t6 x3)) H12 u -H15) in (let H19 \def (eq_ind_r C x1 (\lambda (c3: C).(getl i c0 (CHead c3 -(Bind x0) x3))) H11 d H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b: -B).(getl i c0 (CHead d (Bind b) x3))) H19 Abbr H16) in (ex2_ind T (\lambda -(t6: T).(subst0 i x3 t3 t6)) (\lambda (t6: T).(subst0 (S (plus (minus i0 (S -i)) i)) u0 t0 t6)) (pc3 c0 t2 t5) (\lambda (x: T).(\lambda (H21: (subst0 i x3 -t3 x)).(\lambda (H22: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let -H23 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n: -nat).(subst0 n u0 t0 x)) H22 i0 (lt_plus_minus_r i i0 H7)) in (pc3_pr2_u c0 x -t2 (pr2_delta c0 d x3 i H20 t2 t3 H1 x H21) t5 (pc3_pr2_u2 c0 t0 x (pr2_delta -c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e (Bind Abbr) -u0) H6) t0 t0 (pr0_refl t0) x H23) t5 (pc3_pr2_r c0 t0 t5 (pr2_delta c0 e u0 -i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e (Bind Abbr) u0) H6) -t0 t0 (pr0_refl t0) t5 H4)))))))) (subst0_subst0_back t3 t0 u i H2 x3 u0 -(minus i0 (S i)) H18)))))))) H14)) H13))))))))) H9)) (\lambda (H9: (ex3_4 B C -C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C -(CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda -(_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 (Bind b) -u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))).(ex3_4_ind B C C T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind -Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 (Bind b) u1)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S -i)) u0 e1 e2))))) (pc3 c0 t2 t5) (\lambda (x0: B).(\lambda (x1: C).(\lambda -(x2: C).(\lambda (x3: T).(\lambda (H10: (eq C (CHead d (Bind Abbr) u) (CHead -x1 (Bind x0) x3))).(\lambda (H11: (getl i c0 (CHead x2 (Bind x0) -x3))).(\lambda (H12: (csubst0 (minus i0 (S i)) u0 x1 x2)).(let H13 \def -(f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with -[(CSort _) \Rightarrow d | (CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind -Abbr) u) (CHead x1 (Bind x0) x3) H10) in ((let H14 \def (f_equal C B (\lambda -(e0: C).(match e0 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow -Abbr | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with -[(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) -u) (CHead x1 (Bind x0) x3) H10) in ((let H15 \def (f_equal C T (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | -(CHead _ _ t6) \Rightarrow t6])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) -x3) H10) in (\lambda (H16: (eq B Abbr x0)).(\lambda (H17: (eq C d x1)).(let -H18 \def (eq_ind_r T x3 (\lambda (t6: T).(getl i c0 (CHead x2 (Bind x0) t6))) -H11 u H15) in (let H19 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus -i0 (S i)) u0 c3 x2)) H12 d H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b: -B).(getl i c0 (CHead x2 (Bind b) u))) H18 Abbr H16) in (pc3_pr2_u c0 t0 t2 -(pr2_delta c0 x2 u i H20 t2 t3 H1 t0 H2) t5 (pc3_pr2_r c0 t0 t5 (pr2_delta c0 -e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e (Bind Abbr) u0) -H6) t0 t0 (pr0_refl t0) t5 H4))))))))) H14)) H13))))))))) H9)) (\lambda (H9: -(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) -u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 -(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 -e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 -(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 -(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) -(pc3 c0 t2 t5) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda -(x3: T).(\lambda (x4: T).(\lambda (H10: (eq C (CHead d (Bind Abbr) u) (CHead -x1 (Bind x0) x3))).(\lambda (H11: (getl i c0 (CHead x2 (Bind x0) -x4))).(\lambda (H12: (subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H13: -(csubst0 (minus i0 (S i)) u0 x1 x2)).(let H14 \def (f_equal C C (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | -(CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) -x3) H10) in ((let H15 \def (f_equal C B (\lambda (e0: C).(match e0 in C -return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) -\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead -x1 (Bind x0) x3) H10) in ((let H16 \def (f_equal C T (\lambda (e0: C).(match -e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ -t6) \Rightarrow t6])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H10) in -(\lambda (H17: (eq B Abbr x0)).(\lambda (H18: (eq C d x1)).(let H19 \def -(eq_ind_r T x3 (\lambda (t6: T).(subst0 (minus i0 (S i)) u0 t6 x4)) H12 u -H16) in (let H20 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus i0 (S -i)) u0 c3 x2)) H13 d H18) in (let H21 \def (eq_ind_r B x0 (\lambda (b: -B).(getl i c0 (CHead x2 (Bind b) x4))) H11 Abbr H17) in (ex2_ind T (\lambda -(t6: T).(subst0 i x4 t3 t6)) (\lambda (t6: T).(subst0 (S (plus (minus i0 (S -i)) i)) u0 t0 t6)) (pc3 c0 t2 t5) (\lambda (x: T).(\lambda (H22: (subst0 i x4 -t3 x)).(\lambda (H23: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let -H24 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n: -nat).(subst0 n u0 t0 x)) H23 i0 (lt_plus_minus_r i i0 H7)) in (pc3_pr2_u c0 x -t2 (pr2_delta c0 x2 x4 i H21 t2 t3 H1 x H22) t5 (pc3_pr2_u2 c0 t0 x -(pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e -(Bind Abbr) u0) H6) t0 t0 (pr0_refl t0) x H24) t5 (pc3_pr2_r c0 t0 t5 -(pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e -(Bind Abbr) u0) H6) t0 t0 (pr0_refl t0) t5 H4)))))))) (subst0_subst0_back t3 -t0 u i H2 x4 u0 (minus i0 (S i)) H19)))))))) H15)) H14))))))))))) H9)) H8))) -(\lambda (H7: (le i0 i)).(pc3_pr2_u c0 t0 t2 (pr2_delta c0 d u i -(csubst0_getl_ge i0 i H7 c c0 u0 H5 (CHead d (Bind Abbr) u) H0) t2 t3 H1 t0 -H2) t5 (pc3_pr2_r c0 t0 t5 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n -i0) c c0 u0 H5 (CHead e (Bind Abbr) u0) H6) t0 t0 (pr0_refl t0) t5 -H4))))))))))) c2 t4 H3)))))))))))))))) c1 t t1 H)))). -(* COMMENTS -Initial nodes: 6191 -END *) - -theorem pc3_fsubst0: - \forall (c1: C).(\forall (t1: T).(\forall (t: T).((pc3 c1 t1 t) \to (\forall -(i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t2: T).((fsubst0 i u c1 -t1 c2 t2) \to (\forall (e: C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3 -c2 t2 t))))))))))) -\def - \lambda (c1: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H: (pc3 c1 t1 -t)).(pc3_ind_left c1 (\lambda (t0: T).(\lambda (t2: T).(\forall (i: -nat).(\forall (u: T).(\forall (c2: C).(\forall (t3: T).((fsubst0 i u c1 t0 c2 -t3) \to (\forall (e: C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3 c2 t3 -t2)))))))))) (\lambda (t0: T).(\lambda (i: nat).(\lambda (u: T).(\lambda (c2: -C).(\lambda (t2: T).(\lambda (H0: (fsubst0 i u c1 t0 c2 t2)).(fsubst0_ind i u -c1 t0 (\lambda (c: C).(\lambda (t3: T).(\forall (e: C).((getl i c1 (CHead e -(Bind Abbr) u)) \to (pc3 c t3 t0))))) (\lambda (t3: T).(\lambda (H1: (subst0 -i u t0 t3)).(\lambda (e: C).(\lambda (H2: (getl i c1 (CHead e (Bind Abbr) -u))).(pc3_pr2_x c1 t3 t0 (pr2_delta c1 e u i H2 t0 t0 (pr0_refl t0) t3 -H1)))))) (\lambda (c0: C).(\lambda (_: (csubst0 i u c1 c0)).(\lambda (e: -C).(\lambda (_: (getl i c1 (CHead e (Bind Abbr) u))).(pc3_refl c0 t0))))) -(\lambda (t3: T).(\lambda (H1: (subst0 i u t0 t3)).(\lambda (c0: C).(\lambda -(H2: (csubst0 i u c1 c0)).(\lambda (e: C).(\lambda (H3: (getl i c1 (CHead e -(Bind Abbr) u))).(pc3_pr2_x c0 t3 t0 (pr2_delta c0 e u i (csubst0_getl_ge i i -(le_n i) c1 c0 u H2 (CHead e (Bind Abbr) u) H3) t0 t0 (pr0_refl t0) t3 -H1)))))))) c2 t2 H0))))))) (\lambda (t0: T).(\lambda (t2: T).(\lambda (H0: -(pr2 c1 t0 t2)).(\lambda (t3: T).(\lambda (H1: (pc3 c1 t2 t3)).(\lambda (H2: -((\forall (i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t4: -T).((fsubst0 i u c1 t2 c2 t4) \to (\forall (e: C).((getl i c1 (CHead e (Bind -Abbr) u)) \to (pc3 c2 t4 t3)))))))))).(\lambda (i: nat).(\lambda (u: -T).(\lambda (c2: C).(\lambda (t4: T).(\lambda (H3: (fsubst0 i u c1 t0 c2 -t4)).(fsubst0_ind i u c1 t0 (\lambda (c: C).(\lambda (t5: T).(\forall (e: -C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3 c t5 t3))))) (\lambda (t5: -T).(\lambda (H4: (subst0 i u t0 t5)).(\lambda (e: C).(\lambda (H5: (getl i c1 -(CHead e (Bind Abbr) u))).(pc3_t t2 c1 t5 (pc3_pr2_fsubst0 c1 t0 t2 H0 i u c1 -t5 (fsubst0_snd i u c1 t0 t5 H4) e H5) t3 H1))))) (\lambda (c0: C).(\lambda -(H4: (csubst0 i u c1 c0)).(\lambda (e: C).(\lambda (H5: (getl i c1 (CHead e -(Bind Abbr) u))).(pc3_t t2 c0 t0 (pc3_pr2_fsubst0 c1 t0 t2 H0 i u c0 t0 -(fsubst0_fst i u c1 t0 c0 H4) e H5) t3 (H2 i u c0 t2 (fsubst0_fst i u c1 t2 -c0 H4) e H5)))))) (\lambda (t5: T).(\lambda (H4: (subst0 i u t0 t5)).(\lambda -(c0: C).(\lambda (H5: (csubst0 i u c1 c0)).(\lambda (e: C).(\lambda (H6: -(getl i c1 (CHead e (Bind Abbr) u))).(pc3_t t2 c0 t5 (pc3_pr2_fsubst0 c1 t0 -t2 H0 i u c0 t5 (fsubst0_both i u c1 t0 t5 H4 c0 H5) e H6) t3 (H2 i u c0 t2 -(fsubst0_fst i u c1 t2 c0 H5) e H6)))))))) c2 t4 H3)))))))))))) (\lambda (t0: -T).(\lambda (t2: T).(\lambda (H0: (pr2 c1 t0 t2)).(\lambda (t3: T).(\lambda -(H1: (pc3 c1 t0 t3)).(\lambda (H2: ((\forall (i: nat).(\forall (u: -T).(\forall (c2: C).(\forall (t4: T).((fsubst0 i u c1 t0 c2 t4) \to (\forall -(e: C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3 c2 t4 -t3)))))))))).(\lambda (i: nat).(\lambda (u: T).(\lambda (c2: C).(\lambda (t4: -T).(\lambda (H3: (fsubst0 i u c1 t2 c2 t4)).(fsubst0_ind i u c1 t2 (\lambda -(c: C).(\lambda (t5: T).(\forall (e: C).((getl i c1 (CHead e (Bind Abbr) u)) -\to (pc3 c t5 t3))))) (\lambda (t5: T).(\lambda (H4: (subst0 i u t2 -t5)).(\lambda (e: C).(\lambda (H5: (getl i c1 (CHead e (Bind Abbr) -u))).(pc3_t t0 c1 t5 (pc3_s c1 t5 t0 (pc3_pr2_fsubst0_back c1 t0 t2 H0 i u c1 -t5 (fsubst0_snd i u c1 t2 t5 H4) e H5)) t3 H1))))) (\lambda (c0: C).(\lambda -(H4: (csubst0 i u c1 c0)).(\lambda (e: C).(\lambda (H5: (getl i c1 (CHead e -(Bind Abbr) u))).(pc3_t t0 c0 t2 (pc3_s c0 t2 t0 (pc3_pr2_fsubst0_back c1 t0 -t2 H0 i u c0 t2 (fsubst0_fst i u c1 t2 c0 H4) e H5)) t3 (H2 i u c0 t0 -(fsubst0_fst i u c1 t0 c0 H4) e H5)))))) (\lambda (t5: T).(\lambda (H4: -(subst0 i u t2 t5)).(\lambda (c0: C).(\lambda (H5: (csubst0 i u c1 -c0)).(\lambda (e: C).(\lambda (H6: (getl i c1 (CHead e (Bind Abbr) -u))).(pc3_t t0 c0 t5 (pc3_s c0 t5 t0 (pc3_pr2_fsubst0_back c1 t0 t2 H0 i u c0 -t5 (fsubst0_both i u c1 t2 t5 H4 c0 H5) e H6)) t3 (H2 i u c0 t0 (fsubst0_fst -i u c1 t0 c0 H5) e H6)))))))) c2 t4 H3)))))))))))) t1 t H)))). -(* COMMENTS -Initial nodes: 1249 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/fwd.ma deleted file mode 100644 index c89e4563c..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/fwd.ma +++ /dev/null @@ -1,333 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pc3/props.ma". - -include "Basic-1/pr3/fwd.ma". - -theorem pc3_gen_sort: - \forall (c: C).(\forall (m: nat).(\forall (n: nat).((pc3 c (TSort m) (TSort -n)) \to (eq nat m n)))) -\def - \lambda (c: C).(\lambda (m: nat).(\lambda (n: nat).(\lambda (H: (pc3 c -(TSort m) (TSort n))).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr3 c -(TSort m) t)) (\lambda (t: T).(pr3 c (TSort n) t)) (eq nat m n) (\lambda (x: -T).(\lambda (H1: (pr3 c (TSort m) x)).(\lambda (H2: (pr3 c (TSort n) x)).(let -H3 \def (eq_ind T x (\lambda (t: T).(eq T t (TSort n))) (pr3_gen_sort c x n -H2) (TSort m) (pr3_gen_sort c x m H1)) in (let H4 \def (f_equal T nat -(\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort n0) -\Rightarrow n0 | (TLRef _) \Rightarrow m | (THead _ _ _) \Rightarrow m])) -(TSort m) (TSort n) H3) in H4))))) H0))))). -(* COMMENTS -Initial nodes: 153 -END *) - -theorem pc3_gen_abst: - \forall (c: C).(\forall (u1: T).(\forall (u2: T).(\forall (t1: T).(\forall -(t2: T).((pc3 c (THead (Bind Abst) u1 t1) (THead (Bind Abst) u2 t2)) \to -(land (pc3 c u1 u2) (\forall (b: B).(\forall (u: T).(pc3 (CHead c (Bind b) u) -t1 t2))))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (t1: T).(\lambda -(t2: T).(\lambda (H: (pc3 c (THead (Bind Abst) u1 t1) (THead (Bind Abst) u2 -t2))).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr3 c (THead (Bind Abst) -u1 t1) t)) (\lambda (t: T).(pr3 c (THead (Bind Abst) u2 t2) t)) (land (pc3 c -u1 u2) (\forall (b: B).(\forall (u: T).(pc3 (CHead c (Bind b) u) t1 t2)))) -(\lambda (x: T).(\lambda (H1: (pr3 c (THead (Bind Abst) u1 t1) x)).(\lambda -(H2: (pr3 c (THead (Bind Abst) u2 t2) x)).(let H3 \def (pr3_gen_abst c u2 t2 -x H2) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead -(Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c u2 u3))) -(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead -c (Bind b) u) t2 t3))))) (land (pc3 c u1 u2) (\forall (b: B).(\forall (u: -T).(pc3 (CHead c (Bind b) u) t1 t2)))) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H4: (eq T x (THead (Bind Abst) x0 x1))).(\lambda (H5: (pr3 c u2 -x0)).(\lambda (H6: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -t2 x1))))).(let H7 \def (pr3_gen_abst c u1 t1 x H1) in (ex3_2_ind T T -(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u3 t3)))) -(\lambda (u3: T).(\lambda (_: T).(pr3 c u1 u3))) (\lambda (_: T).(\lambda -(t3: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t1 t3))))) -(land (pc3 c u1 u2) (\forall (b: B).(\forall (u: T).(pc3 (CHead c (Bind b) u) -t1 t2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H8: (eq T x (THead -(Bind Abst) x2 x3))).(\lambda (H9: (pr3 c u1 x2)).(\lambda (H10: ((\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t1 x3))))).(let H11 \def -(eq_ind T x (\lambda (t: T).(eq T t (THead (Bind Abst) x0 x1))) H4 (THead -(Bind Abst) x2 x3) H8) in (let H12 \def (f_equal T T (\lambda (e: T).(match e -in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x2 | (TLRef _) -\Rightarrow x2 | (THead _ t _) \Rightarrow t])) (THead (Bind Abst) x2 x3) -(THead (Bind Abst) x0 x1) H11) in ((let H13 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x3 | -(TLRef _) \Rightarrow x3 | (THead _ _ t) \Rightarrow t])) (THead (Bind Abst) -x2 x3) (THead (Bind Abst) x0 x1) H11) in (\lambda (H14: (eq T x2 x0)).(let -H15 \def (eq_ind T x3 (\lambda (t: T).(\forall (b: B).(\forall (u: T).(pr3 -(CHead c (Bind b) u) t1 t)))) H10 x1 H13) in (let H16 \def (eq_ind T x2 -(\lambda (t: T).(pr3 c u1 t)) H9 x0 H14) in (conj (pc3 c u1 u2) (\forall (b: -B).(\forall (u: T).(pc3 (CHead c (Bind b) u) t1 t2))) (pc3_pr3_t c u1 x0 H16 -u2 H5) (\lambda (b: B).(\lambda (u: T).(pc3_pr3_t (CHead c (Bind b) u) t1 x1 -(H15 b u) t2 (H6 b u))))))))) H12)))))))) H7))))))) H3))))) H0))))))). -(* COMMENTS -Initial nodes: 715 -END *) - -theorem pc3_gen_abst_shift: - \forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).((pc3 c -(THead (Bind Abst) u t1) (THead (Bind Abst) u t2)) \to (pc3 (CHead c (Bind -Abst) u) t1 t2))))) -\def - \lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H: (pc3 c (THead (Bind Abst) u t1) (THead (Bind Abst) u t2))).(let H_x \def -(pc3_gen_abst c u u t1 t2 H) in (let H0 \def H_x in (land_ind (pc3 c u u) -(\forall (b: B).(\forall (u0: T).(pc3 (CHead c (Bind b) u0) t1 t2))) (pc3 -(CHead c (Bind Abst) u) t1 t2) (\lambda (_: (pc3 c u u)).(\lambda (H2: -((\forall (b: B).(\forall (u0: T).(pc3 (CHead c (Bind b) u0) t1 t2))))).(H2 -Abst u))) H0))))))). -(* COMMENTS -Initial nodes: 129 -END *) - -theorem pc3_gen_lift: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).(\forall (h: nat).(\forall -(d: nat).((pc3 c (lift h d t1) (lift h d t2)) \to (\forall (e: C).((drop h d -c e) \to (pc3 e t1 t2)))))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (h: nat).(\lambda -(d: nat).(\lambda (H: (pc3 c (lift h d t1) (lift h d t2))).(\lambda (e: -C).(\lambda (H0: (drop h d c e)).(let H1 \def H in (ex2_ind T (\lambda (t: -T).(pr3 c (lift h d t1) t)) (\lambda (t: T).(pr3 c (lift h d t2) t)) (pc3 e -t1 t2) (\lambda (x: T).(\lambda (H2: (pr3 c (lift h d t1) x)).(\lambda (H3: -(pr3 c (lift h d t2) x)).(let H4 \def (pr3_gen_lift c t2 x h d H3 e H0) in -(ex2_ind T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr3 e -t2 t3)) (pc3 e t1 t2) (\lambda (x0: T).(\lambda (H5: (eq T x (lift h d -x0))).(\lambda (H6: (pr3 e t2 x0)).(let H7 \def (pr3_gen_lift c t1 x h d H2 e -H0) in (ex2_ind T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: -T).(pr3 e t1 t3)) (pc3 e t1 t2) (\lambda (x1: T).(\lambda (H8: (eq T x (lift -h d x1))).(\lambda (H9: (pr3 e t1 x1)).(let H10 \def (eq_ind T x (\lambda (t: -T).(eq T t (lift h d x0))) H5 (lift h d x1) H8) in (let H11 \def (eq_ind T x1 -(\lambda (t: T).(pr3 e t1 t)) H9 x0 (lift_inj x1 x0 h d H10)) in (pc3_pr3_t e -t1 x0 H11 t2 H6)))))) H7))))) H4))))) H1))))))))). -(* COMMENTS -Initial nodes: 363 -END *) - -theorem pc3_gen_not_abst: - \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (t1: -T).(\forall (t2: T).(\forall (u1: T).(\forall (u2: T).((pc3 c (THead (Bind b) -u1 t1) (THead (Bind Abst) u2 t2)) \to (pc3 (CHead c (Bind b) u1) t1 (lift (S -O) O (THead (Bind Abst) u2 t2)))))))))) -\def - \lambda (b: B).(B_ind (\lambda (b0: B).((not (eq B b0 Abst)) \to (\forall -(c: C).(\forall (t1: T).(\forall (t2: T).(\forall (u1: T).(\forall (u2: -T).((pc3 c (THead (Bind b0) u1 t1) (THead (Bind Abst) u2 t2)) \to (pc3 (CHead -c (Bind b0) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2))))))))))) (\lambda -(_: (not (eq B Abbr Abst))).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H0: (pc3 c (THead (Bind Abbr) -u1 t1) (THead (Bind Abst) u2 t2))).(let H1 \def H0 in (ex2_ind T (\lambda (t: -T).(pr3 c (THead (Bind Abbr) u1 t1) t)) (\lambda (t: T).(pr3 c (THead (Bind -Abst) u2 t2) t)) (pc3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O (THead (Bind -Abst) u2 t2))) (\lambda (x: T).(\lambda (H2: (pr3 c (THead (Bind Abbr) u1 t1) -x)).(\lambda (H3: (pr3 c (THead (Bind Abst) u2 t2) x)).(let H4 \def -(pr3_gen_abbr c u1 t1 x H2) in (or_ind (ex3_2 T T (\lambda (u3: T).(\lambda -(t3: T).(eq T x (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: -T).(pr3 c u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr3 (CHead c (Bind Abbr) -u1) t1 t3)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O x)) (pc3 (CHead -c (Bind Abbr) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2))) (\lambda (H5: -(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3 -t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c u1 u3))) (\lambda (_: -T).(\lambda (t3: T).(pr3 (CHead c (Bind Abbr) u1) t1 t3))))).(ex3_2_ind T T -(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3 t3)))) -(\lambda (u3: T).(\lambda (_: T).(pr3 c u1 u3))) (\lambda (_: T).(\lambda -(t3: T).(pr3 (CHead c (Bind Abbr) u1) t1 t3))) (pc3 (CHead c (Bind Abbr) u1) -t1 (lift (S O) O (THead (Bind Abst) u2 t2))) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H6: (eq T x (THead (Bind Abbr) x0 x1))).(\lambda (_: (pr3 c u1 -x0)).(\lambda (_: (pr3 (CHead c (Bind Abbr) u1) t1 x1)).(let H9 \def -(pr3_gen_abst c u2 t2 x H3) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: -T).(eq T x (THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 -c u2 u3))) (\lambda (_: T).(\lambda (t3: T).(\forall (b0: B).(\forall (u: -T).(pr3 (CHead c (Bind b0) u) t2 t3))))) (pc3 (CHead c (Bind Abbr) u1) t1 -(lift (S O) O (THead (Bind Abst) u2 t2))) (\lambda (x2: T).(\lambda (x3: -T).(\lambda (H10: (eq T x (THead (Bind Abst) x2 x3))).(\lambda (_: (pr3 c u2 -x2)).(\lambda (_: ((\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) -u) t2 x3))))).(let H13 \def (eq_ind T x (\lambda (t: T).(eq T t (THead (Bind -Abbr) x0 x1))) H6 (THead (Bind Abst) x2 x3) H10) in (let H14 \def (eq_ind T -(THead (Bind Abst) x2 x3) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind b0) \Rightarrow (match b0 in B return (\lambda (_: B).Prop) with [Abbr -\Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat -_) \Rightarrow False])])) I (THead (Bind Abbr) x0 x1) H13) in (False_ind (pc3 -(CHead c (Bind Abbr) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2))) -H14)))))))) H9))))))) H5)) (\lambda (H5: (pr3 (CHead c (Bind Abbr) u1) t1 -(lift (S O) O x))).(let H6 \def (pr3_gen_abst c u2 t2 x H3) in (ex3_2_ind T T -(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u3 t3)))) -(\lambda (u3: T).(\lambda (_: T).(pr3 c u2 u3))) (\lambda (_: T).(\lambda -(t3: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t2 -t3))))) (pc3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O (THead (Bind Abst) u2 -t2))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: (eq T x (THead (Bind -Abst) x0 x1))).(\lambda (H8: (pr3 c u2 x0)).(\lambda (H9: ((\forall (b0: -B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t2 x1))))).(let H10 \def -(eq_ind T x (\lambda (t: T).(pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O -t))) H5 (THead (Bind Abst) x0 x1) H7) in (pc3_pr3_t (CHead c (Bind Abbr) u1) -t1 (lift (S O) O (THead (Bind Abst) x0 x1)) H10 (lift (S O) O (THead (Bind -Abst) u2 t2)) (pr3_lift (CHead c (Bind Abbr) u1) c (S O) O (drop_drop (Bind -Abbr) O c c (drop_refl c) u1) (THead (Bind Abst) u2 t2) (THead (Bind Abst) x0 -x1) (pr3_head_12 c u2 x0 H8 (Bind Abst) t2 x1 (H9 Abst x0)))))))))) H6))) -H4))))) H1))))))))) (\lambda (H: (not (eq B Abst Abst))).(\lambda (c: -C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (u1: T).(\lambda (u2: -T).(\lambda (_: (pc3 c (THead (Bind Abst) u1 t1) (THead (Bind Abst) u2 -t2))).(let H1 \def (match (H (refl_equal B Abst)) in False return (\lambda -(_: False).(pc3 (CHead c (Bind Abst) u1) t1 (lift (S O) O (THead (Bind Abst) -u2 t2)))) with []) in H1)))))))) (\lambda (_: (not (eq B Void -Abst))).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (u1: -T).(\lambda (u2: T).(\lambda (H0: (pc3 c (THead (Bind Void) u1 t1) (THead -(Bind Abst) u2 t2))).(let H1 \def H0 in (ex2_ind T (\lambda (t: T).(pr3 c -(THead (Bind Void) u1 t1) t)) (\lambda (t: T).(pr3 c (THead (Bind Abst) u2 -t2) t)) (pc3 (CHead c (Bind Void) u1) t1 (lift (S O) O (THead (Bind Abst) u2 -t2))) (\lambda (x: T).(\lambda (H2: (pr3 c (THead (Bind Void) u1 t1) -x)).(\lambda (H3: (pr3 c (THead (Bind Abst) u2 t2) x)).(let H4 \def -(pr3_gen_void c u1 t1 x H2) in (or_ind (ex3_2 T T (\lambda (u3: T).(\lambda -(t3: T).(eq T x (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: -T).(pr3 c u1 u3))) (\lambda (_: T).(\lambda (t3: T).(\forall (b0: B).(\forall -(u: T).(pr3 (CHead c (Bind b0) u) t1 t3)))))) (pr3 (CHead c (Bind Void) u1) -t1 (lift (S O) O x)) (pc3 (CHead c (Bind Void) u1) t1 (lift (S O) O (THead -(Bind Abst) u2 t2))) (\lambda (H5: (ex3_2 T T (\lambda (u3: T).(\lambda (t3: -T).(eq T x (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 -c u1 u3))) (\lambda (_: T).(\lambda (t3: T).(\forall (b0: B).(\forall (u: -T).(pr3 (CHead c (Bind b0) u) t1 t3))))))).(ex3_2_ind T T (\lambda (u3: -T).(\lambda (t3: T).(eq T x (THead (Bind Void) u3 t3)))) (\lambda (u3: -T).(\lambda (_: T).(pr3 c u1 u3))) (\lambda (_: T).(\lambda (t3: T).(\forall -(b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t1 t3))))) (pc3 (CHead c -(Bind Void) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2))) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H6: (eq T x (THead (Bind Void) x0 -x1))).(\lambda (_: (pr3 c u1 x0)).(\lambda (_: ((\forall (b0: B).(\forall (u: -T).(pr3 (CHead c (Bind b0) u) t1 x1))))).(let H9 \def (pr3_gen_abst c u2 t2 x -H3) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind -Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c u2 u3))) (\lambda (_: -T).(\lambda (t3: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) -u) t2 t3))))) (pc3 (CHead c (Bind Void) u1) t1 (lift (S O) O (THead (Bind -Abst) u2 t2))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H10: (eq T x -(THead (Bind Abst) x2 x3))).(\lambda (_: (pr3 c u2 x2)).(\lambda (_: -((\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t2 x3))))).(let -H13 \def (eq_ind T x (\lambda (t: T).(eq T t (THead (Bind Void) x0 x1))) H6 -(THead (Bind Abst) x2 x3) H10) in (let H14 \def (eq_ind T (THead (Bind Abst) -x2 x3) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b0) -\Rightarrow (match b0 in B return (\lambda (_: B).Prop) with [Abbr -\Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat -_) \Rightarrow False])])) I (THead (Bind Void) x0 x1) H13) in (False_ind (pc3 -(CHead c (Bind Void) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2))) -H14)))))))) H9))))))) H5)) (\lambda (H5: (pr3 (CHead c (Bind Void) u1) t1 -(lift (S O) O x))).(let H6 \def (pr3_gen_abst c u2 t2 x H3) in (ex3_2_ind T T -(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u3 t3)))) -(\lambda (u3: T).(\lambda (_: T).(pr3 c u2 u3))) (\lambda (_: T).(\lambda -(t3: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t2 -t3))))) (pc3 (CHead c (Bind Void) u1) t1 (lift (S O) O (THead (Bind Abst) u2 -t2))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: (eq T x (THead (Bind -Abst) x0 x1))).(\lambda (H8: (pr3 c u2 x0)).(\lambda (H9: ((\forall (b0: -B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t2 x1))))).(let H10 \def -(eq_ind T x (\lambda (t: T).(pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O -t))) H5 (THead (Bind Abst) x0 x1) H7) in (pc3_pr3_t (CHead c (Bind Void) u1) -t1 (lift (S O) O (THead (Bind Abst) x0 x1)) H10 (lift (S O) O (THead (Bind -Abst) u2 t2)) (pr3_lift (CHead c (Bind Void) u1) c (S O) O (drop_drop (Bind -Void) O c c (drop_refl c) u1) (THead (Bind Abst) u2 t2) (THead (Bind Abst) x0 -x1) (pr3_head_12 c u2 x0 H8 (Bind Abst) t2 x1 (H9 Abst x0)))))))))) H6))) -H4))))) H1))))))))) b). -(* COMMENTS -Initial nodes: 2427 -END *) - -theorem pc3_gen_lift_abst: - \forall (c: C).(\forall (t: T).(\forall (t2: T).(\forall (u2: T).(\forall -(h: nat).(\forall (d: nat).((pc3 c (lift h d t) (THead (Bind Abst) u2 t2)) -\to (\forall (e: C).((drop h d c e) \to (ex3_2 T T (\lambda (u1: T).(\lambda -(t1: T).(pr3 e t (THead (Bind Abst) u1 t1)))) (\lambda (u1: T).(\lambda (_: -T).(pr3 c u2 (lift h d u1)))) (\lambda (_: T).(\lambda (t1: T).(\forall (b: -B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t2 (lift h (S d) -t1))))))))))))))) -\def - \lambda (c: C).(\lambda (t: T).(\lambda (t2: T).(\lambda (u2: T).(\lambda -(h: nat).(\lambda (d: nat).(\lambda (H: (pc3 c (lift h d t) (THead (Bind -Abst) u2 t2))).(\lambda (e: C).(\lambda (H0: (drop h d c e)).(let H1 \def H -in (ex2_ind T (\lambda (t0: T).(pr3 c (lift h d t) t0)) (\lambda (t0: T).(pr3 -c (THead (Bind Abst) u2 t2) t0)) (ex3_2 T T (\lambda (u1: T).(\lambda (t1: -T).(pr3 e t (THead (Bind Abst) u1 t1)))) (\lambda (u1: T).(\lambda (_: -T).(pr3 c u2 (lift h d u1)))) (\lambda (_: T).(\lambda (t1: T).(\forall (b: -B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t2 (lift h (S d) t1))))))) -(\lambda (x: T).(\lambda (H2: (pr3 c (lift h d t) x)).(\lambda (H3: (pr3 c -(THead (Bind Abst) u2 t2) x)).(let H4 \def (pr3_gen_lift c t x h d H2 e H0) -in (ex2_ind T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr3 -e t t3)) (ex3_2 T T (\lambda (u1: T).(\lambda (t1: T).(pr3 e t (THead (Bind -Abst) u1 t1)))) (\lambda (u1: T).(\lambda (_: T).(pr3 c u2 (lift h d u1)))) -(\lambda (_: T).(\lambda (t1: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead -c (Bind b) u) t2 (lift h (S d) t1))))))) (\lambda (x0: T).(\lambda (H5: (eq T -x (lift h d x0))).(\lambda (H6: (pr3 e t x0)).(let H7 \def (pr3_gen_abst c u2 -t2 x H3) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead -(Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c u2 u3))) -(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead -c (Bind b) u) t2 t3))))) (ex3_2 T T (\lambda (u1: T).(\lambda (t1: T).(pr3 e -t (THead (Bind Abst) u1 t1)))) (\lambda (u1: T).(\lambda (_: T).(pr3 c u2 -(lift h d u1)))) (\lambda (_: T).(\lambda (t1: T).(\forall (b: B).(\forall -(u: T).(pr3 (CHead c (Bind b) u) t2 (lift h (S d) t1))))))) (\lambda (x1: -T).(\lambda (x2: T).(\lambda (H8: (eq T x (THead (Bind Abst) x1 -x2))).(\lambda (H9: (pr3 c u2 x1)).(\lambda (H10: ((\forall (b: B).(\forall -(u: T).(pr3 (CHead c (Bind b) u) t2 x2))))).(let H11 \def (eq_ind T x -(\lambda (t0: T).(eq T t0 (lift h d x0))) H5 (THead (Bind Abst) x1 x2) H8) in -(ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x0 (THead (Bind Abst) y -z)))) (\lambda (y: T).(\lambda (_: T).(eq T x1 (lift h d y)))) (\lambda (_: -T).(\lambda (z: T).(eq T x2 (lift h (S d) z)))) (ex3_2 T T (\lambda (u1: -T).(\lambda (t1: T).(pr3 e t (THead (Bind Abst) u1 t1)))) (\lambda (u1: -T).(\lambda (_: T).(pr3 c u2 (lift h d u1)))) (\lambda (_: T).(\lambda (t1: -T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t2 (lift h (S d) -t1))))))) (\lambda (x3: T).(\lambda (x4: T).(\lambda (H12: (eq T x0 (THead -(Bind Abst) x3 x4))).(\lambda (H13: (eq T x1 (lift h d x3))).(\lambda (H14: -(eq T x2 (lift h (S d) x4))).(let H15 \def (eq_ind T x2 (\lambda (t0: -T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t2 t0)))) H10 -(lift h (S d) x4) H14) in (let H16 \def (eq_ind T x1 (\lambda (t0: T).(pr3 c -u2 t0)) H9 (lift h d x3) H13) in (let H17 \def (eq_ind T x0 (\lambda (t0: -T).(pr3 e t t0)) H6 (THead (Bind Abst) x3 x4) H12) in (ex3_2_intro T T -(\lambda (u1: T).(\lambda (t1: T).(pr3 e t (THead (Bind Abst) u1 t1)))) -(\lambda (u1: T).(\lambda (_: T).(pr3 c u2 (lift h d u1)))) (\lambda (_: -T).(\lambda (t1: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -t2 (lift h (S d) t1)))))) x3 x4 H17 H16 H15))))))))) (lift_gen_bind Abst x1 -x2 x0 h d H11)))))))) H7))))) H4))))) H1)))))))))). -(* COMMENTS -Initial nodes: 973 -END *) - -theorem pc3_gen_sort_abst: - \forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (n: nat).((pc3 c -(TSort n) (THead (Bind Abst) u t)) \to (\forall (P: Prop).P))))) -\def - \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (n: nat).(\lambda -(H: (pc3 c (TSort n) (THead (Bind Abst) u t))).(\lambda (P: Prop).(let H0 -\def H in (ex2_ind T (\lambda (t0: T).(pr3 c (TSort n) t0)) (\lambda (t0: -T).(pr3 c (THead (Bind Abst) u t) t0)) P (\lambda (x: T).(\lambda (H1: (pr3 c -(TSort n) x)).(\lambda (H2: (pr3 c (THead (Bind Abst) u t) x)).(let H3 \def -(pr3_gen_abst c u t x H2) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: -T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 -c u u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u0: -T).(pr3 (CHead c (Bind b) u0) t t2))))) P (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H4: (eq T x (THead (Bind Abst) x0 x1))).(\lambda (_: (pr3 c u -x0)).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pr3 (CHead c (Bind b) -u0) t x1))))).(let H7 \def (eq_ind T x (\lambda (t0: T).(eq T t0 (TSort n))) -(pr3_gen_sort c x n H1) (THead (Bind Abst) x0 x1) H4) in (let H8 \def (eq_ind -T (THead (Bind Abst) x0 x1) (\lambda (ee: T).(match ee in T return (\lambda -(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False -| (THead _ _ _) \Rightarrow True])) I (TSort n) H7) in (False_ind P -H8)))))))) H3))))) H0))))))). -(* COMMENTS -Initial nodes: 303 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/left.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/left.ma deleted file mode 100644 index 12d22d10c..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/left.ma +++ /dev/null @@ -1,125 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pc3/props.ma". - -theorem pc3_ind_left__pc3_left_pr3: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to -(pc3_left c t1 t2)))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 -t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(pc3_left c t t0))) (\lambda -(t: T).(pc3_left_r c t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 -c t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2: -(pc3_left c t0 t4)).(pc3_left_ur c t3 t0 H0 t4 H2))))))) t1 t2 H)))). -(* COMMENTS -Initial nodes: 87 -END *) - -theorem pc3_ind_left__pc3_left_trans: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3_left c t1 t2) \to -(\forall (t3: T).((pc3_left c t2 t3) \to (pc3_left c t1 t3)))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3_left c t1 -t2)).(pc3_left_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (t3: -T).((pc3_left c t0 t3) \to (pc3_left c t t3))))) (\lambda (t: T).(\lambda -(t3: T).(\lambda (H0: (pc3_left c t t3)).H0))) (\lambda (t0: T).(\lambda (t3: -T).(\lambda (H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t3 -t4)).(\lambda (H2: ((\forall (t5: T).((pc3_left c t4 t5) \to (pc3_left c t3 -t5))))).(\lambda (t5: T).(\lambda (H3: (pc3_left c t4 t5)).(pc3_left_ur c t0 -t3 H0 t5 (H2 t5 H3)))))))))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: -(pr2 c t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t0 t4)).(\lambda -(H2: ((\forall (t5: T).((pc3_left c t4 t5) \to (pc3_left c t0 -t5))))).(\lambda (t5: T).(\lambda (H3: (pc3_left c t4 t5)).(pc3_left_ux c t0 -t3 H0 t5 (H2 t5 H3)))))))))) t1 t2 H)))). -(* COMMENTS -Initial nodes: 195 -END *) - -theorem pc3_ind_left__pc3_left_sym: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3_left c t1 t2) \to -(pc3_left c t2 t1)))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3_left c t1 -t2)).(pc3_left_ind c (\lambda (t: T).(\lambda (t0: T).(pc3_left c t0 t))) -(\lambda (t: T).(pc3_left_r c t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda -(H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t3 -t4)).(\lambda (H2: (pc3_left c t4 t3)).(pc3_ind_left__pc3_left_trans c t4 t3 -H2 t0 (pc3_left_ux c t0 t3 H0 t0 (pc3_left_r c t0))))))))) (\lambda (t0: -T).(\lambda (t3: T).(\lambda (H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda -(_: (pc3_left c t0 t4)).(\lambda (H2: (pc3_left c t4 -t0)).(pc3_ind_left__pc3_left_trans c t4 t0 H2 t3 (pc3_left_ur c t0 t3 H0 t3 -(pc3_left_r c t3))))))))) t1 t2 H)))). -(* COMMENTS -Initial nodes: 163 -END *) - -theorem pc3_ind_left__pc3_left_pc3: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to -(pc3_left c t1 t2)))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c t1 -t2)).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: -T).(pr3 c t2 t)) (pc3_left c t1 t2) (\lambda (x: T).(\lambda (H1: (pr3 c t1 -x)).(\lambda (H2: (pr3 c t2 x)).(pc3_ind_left__pc3_left_trans c t1 x -(pc3_ind_left__pc3_left_pr3 c t1 x H1) t2 (pc3_ind_left__pc3_left_sym c t2 x -(pc3_ind_left__pc3_left_pr3 c t2 x H2)))))) H0))))). -(* COMMENTS -Initial nodes: 105 -END *) - -theorem pc3_ind_left__pc3_pc3_left: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3_left c t1 t2) \to -(pc3 c t1 t2)))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3_left c t1 -t2)).(pc3_left_ind c (\lambda (t: T).(\lambda (t0: T).(pc3 c t t0))) (\lambda -(t: T).(pc3_refl c t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c -t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t3 t4)).(\lambda (H2: (pc3 -c t3 t4)).(pc3_t t3 c t0 (pc3_pr2_r c t0 t3 H0) t4 H2))))))) (\lambda (t0: -T).(\lambda (t3: T).(\lambda (H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda -(_: (pc3_left c t0 t4)).(\lambda (H2: (pc3 c t0 t4)).(pc3_t t0 c t3 -(pc3_pr2_x c t3 t0 H0) t4 H2))))))) t1 t2 H)))). -(* COMMENTS -Initial nodes: 147 -END *) - -theorem pc3_ind_left: - \forall (c: C).(\forall (P: ((T \to (T \to Prop)))).(((\forall (t: T).(P t -t))) \to (((\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall (t3: -T).((pc3 c t2 t3) \to ((P t2 t3) \to (P t1 t3)))))))) \to (((\forall (t1: -T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall (t3: T).((pc3 c t1 t3) \to -((P t1 t3) \to (P t2 t3)))))))) \to (\forall (t: T).(\forall (t0: T).((pc3 c -t t0) \to (P t t0)))))))) -\def - \lambda (c: C).(\lambda (P: ((T \to (T \to Prop)))).(\lambda (H: ((\forall -(t: T).(P t t)))).(\lambda (H0: ((\forall (t1: T).(\forall (t2: T).((pr2 c t1 -t2) \to (\forall (t3: T).((pc3 c t2 t3) \to ((P t2 t3) \to (P t1 -t3))))))))).(\lambda (H1: ((\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) -\to (\forall (t3: T).((pc3 c t1 t3) \to ((P t1 t3) \to (P t2 -t3))))))))).(\lambda (t: T).(\lambda (t0: T).(\lambda (H2: (pc3 c t -t0)).(pc3_left_ind c (\lambda (t1: T).(\lambda (t2: T).(P t1 t2))) H (\lambda -(t1: T).(\lambda (t2: T).(\lambda (H3: (pr2 c t1 t2)).(\lambda (t3: -T).(\lambda (H4: (pc3_left c t2 t3)).(\lambda (H5: (P t2 t3)).(H0 t1 t2 H3 t3 -(pc3_ind_left__pc3_pc3_left c t2 t3 H4) H5))))))) (\lambda (t1: T).(\lambda -(t2: T).(\lambda (H3: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda (H4: (pc3_left -c t1 t3)).(\lambda (H5: (P t1 t3)).(H1 t1 t2 H3 t3 -(pc3_ind_left__pc3_pc3_left c t1 t3 H4) H5))))))) t t0 -(pc3_ind_left__pc3_left_pc3 c t t0 H2))))))))). -(* COMMENTS -Initial nodes: 225 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/nf2.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/nf2.ma deleted file mode 100644 index ed930b85e..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/nf2.ma +++ /dev/null @@ -1,52 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pc3/defs.ma". - -include "Basic-1/nf2/pr3.ma". - -theorem pc3_nf2: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to ((nf2 c -t1) \to ((nf2 c t2) \to (eq T t1 t2)))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c t1 -t2)).(\lambda (H0: (nf2 c t1)).(\lambda (H1: (nf2 c t2)).(let H2 \def H in -(ex2_ind T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) (eq T -t1 t2) (\lambda (x: T).(\lambda (H3: (pr3 c t1 x)).(\lambda (H4: (pr3 c t2 -x)).(let H_y \def (nf2_pr3_unfold c t1 x H3 H0) in (let H5 \def (eq_ind_r T x -(\lambda (t: T).(pr3 c t2 t)) H4 t1 H_y) in (let H6 \def (eq_ind_r T x -(\lambda (t: T).(pr3 c t1 t)) H3 t1 H_y) in (let H_y0 \def (nf2_pr3_unfold c -t2 t1 H5 H1) in (let H7 \def (eq_ind T t2 (\lambda (t: T).(pr3 c t t1)) H5 t1 -H_y0) in (eq_ind_r T t1 (\lambda (t: T).(eq T t1 t)) (refl_equal T t1) t2 -H_y0))))))))) H2))))))). -(* COMMENTS -Initial nodes: 195 -END *) - -theorem pc3_nf2_unfold: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to ((nf2 c -t2) \to (pr3 c t1 t2))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c t1 -t2)).(\lambda (H0: (nf2 c t2)).(let H1 \def H in (ex2_ind T (\lambda (t: -T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) (pr3 c t1 t2) (\lambda (x: -T).(\lambda (H2: (pr3 c t1 x)).(\lambda (H3: (pr3 c t2 x)).(let H_y \def -(nf2_pr3_unfold c t2 x H3 H0) in (let H4 \def (eq_ind_r T x (\lambda (t: -T).(pr3 c t1 t)) H2 t2 H_y) in H4))))) H1)))))). -(* COMMENTS -Initial nodes: 109 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/pc1.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/pc1.ma deleted file mode 100644 index dd89f951d..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/pc1.ma +++ /dev/null @@ -1,36 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pc3/defs.ma". - -include "Basic-1/pc1/defs.ma". - -include "Basic-1/pr3/pr1.ma". - -theorem pc3_pc1: - \forall (t1: T).(\forall (t2: T).((pc1 t1 t2) \to (\forall (c: C).(pc3 c t1 -t2)))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc1 t1 t2)).(\lambda (c: -C).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: -T).(pr1 t2 t)) (pc3 c t1 t2) (\lambda (x: T).(\lambda (H1: (pr1 t1 -x)).(\lambda (H2: (pr1 t2 x)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) -(\lambda (t: T).(pr3 c t2 t)) x (pr3_pr1 t1 x H1 c) (pr3_pr1 t2 x H2 c))))) -H0))))). -(* COMMENTS -Initial nodes: 103 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/props.ma deleted file mode 100644 index b0a9a2f50..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/props.ma +++ /dev/null @@ -1,483 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pc3/defs.ma". - -include "Basic-1/pr3/pr3.ma". - -theorem clear_pc3_trans: - \forall (c2: C).(\forall (t1: T).(\forall (t2: T).((pc3 c2 t1 t2) \to -(\forall (c1: C).((clear c1 c2) \to (pc3 c1 t1 t2)))))) -\def - \lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c2 t1 -t2)).(\lambda (c1: C).(\lambda (H0: (clear c1 c2)).(let H1 \def H in (ex2_ind -T (\lambda (t: T).(pr3 c2 t1 t)) (\lambda (t: T).(pr3 c2 t2 t)) (pc3 c1 t1 -t2) (\lambda (x: T).(\lambda (H2: (pr3 c2 t1 x)).(\lambda (H3: (pr3 c2 t2 -x)).(ex_intro2 T (\lambda (t: T).(pr3 c1 t1 t)) (\lambda (t: T).(pr3 c1 t2 -t)) x (clear_pr3_trans c2 t1 x H2 c1 H0) (clear_pr3_trans c2 t2 x H3 c1 -H0))))) H1))))))). -(* COMMENTS -Initial nodes: 129 -END *) - -theorem pc3_pr2_r: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (pc3 c -t1 t2)))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 -t2)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) -t2 (pr3_pr2 c t1 t2 H) (pr3_refl c t2))))). -(* COMMENTS -Initial nodes: 55 -END *) - -theorem pc3_pr2_x: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t2 t1) \to (pc3 c -t1 t2)))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t2 -t1)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) -t1 (pr3_refl c t1) (pr3_pr2 c t2 t1 H))))). -(* COMMENTS -Initial nodes: 55 -END *) - -theorem pc3_pr3_r: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (pc3 c -t1 t2)))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 -t2)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) -t2 H (pr3_refl c t2))))). -(* COMMENTS -Initial nodes: 47 -END *) - -theorem pc3_pr3_x: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t2 t1) \to (pc3 c -t1 t2)))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t2 -t1)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) -t1 (pr3_refl c t1) H)))). -(* COMMENTS -Initial nodes: 47 -END *) - -theorem pc3_pr3_t: - \forall (c: C).(\forall (t1: T).(\forall (t0: T).((pr3 c t1 t0) \to (\forall -(t2: T).((pr3 c t2 t0) \to (pc3 c t1 t2)))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t0: T).(\lambda (H: (pr3 c t1 -t0)).(\lambda (t2: T).(\lambda (H0: (pr3 c t2 t0)).(ex_intro2 T (\lambda (t: -T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) t0 H H0)))))). -(* COMMENTS -Initial nodes: 53 -END *) - -theorem pc3_refl: - \forall (c: C).(\forall (t: T).(pc3 c t t)) -\def - \lambda (c: C).(\lambda (t: T).(ex_intro2 T (\lambda (t0: T).(pr3 c t t0)) -(\lambda (t0: T).(pr3 c t t0)) t (pr3_refl c t) (pr3_refl c t))). -(* COMMENTS -Initial nodes: 41 -END *) - -theorem pc3_s: - \forall (c: C).(\forall (t2: T).(\forall (t1: T).((pc3 c t1 t2) \to (pc3 c -t2 t1)))) -\def - \lambda (c: C).(\lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pc3 c t1 -t2)).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: -T).(pr3 c t2 t)) (pc3 c t2 t1) (\lambda (x: T).(\lambda (H1: (pr3 c t1 -x)).(\lambda (H2: (pr3 c t2 x)).(ex_intro2 T (\lambda (t: T).(pr3 c t2 t)) -(\lambda (t: T).(pr3 c t1 t)) x H2 H1)))) H0))))). -(* COMMENTS -Initial nodes: 97 -END *) - -theorem pc3_thin_dx: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to (\forall -(u: T).(\forall (f: F).(pc3 c (THead (Flat f) u t1) (THead (Flat f) u -t2))))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c t1 -t2)).(\lambda (u: T).(\lambda (f: F).(let H0 \def H in (ex2_ind T (\lambda -(t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) (pc3 c (THead (Flat f) u -t1) (THead (Flat f) u t2)) (\lambda (x: T).(\lambda (H1: (pr3 c t1 -x)).(\lambda (H2: (pr3 c t2 x)).(ex_intro2 T (\lambda (t: T).(pr3 c (THead -(Flat f) u t1) t)) (\lambda (t: T).(pr3 c (THead (Flat f) u t2) t)) (THead -(Flat f) u x) (pr3_thin_dx c t1 x H1 u f) (pr3_thin_dx c t2 x H2 u f))))) -H0))))))). -(* COMMENTS -Initial nodes: 165 -END *) - -theorem pc3_head_1: - \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pc3 c u1 u2) \to (\forall -(k: K).(\forall (t: T).(pc3 c (THead k u1 t) (THead k u2 t))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pc3 c u1 -u2)).(\lambda (k: K).(\lambda (t: T).(let H0 \def H in (ex2_ind T (\lambda -(t0: T).(pr3 c u1 t0)) (\lambda (t0: T).(pr3 c u2 t0)) (pc3 c (THead k u1 t) -(THead k u2 t)) (\lambda (x: T).(\lambda (H1: (pr3 c u1 x)).(\lambda (H2: -(pr3 c u2 x)).(ex_intro2 T (\lambda (t0: T).(pr3 c (THead k u1 t) t0)) -(\lambda (t0: T).(pr3 c (THead k u2 t) t0)) (THead k x t) (pr3_head_12 c u1 x -H1 k t t (pr3_refl (CHead c k x) t)) (pr3_head_12 c u2 x H2 k t t (pr3_refl -(CHead c k x) t)))))) H0))))))). -(* COMMENTS -Initial nodes: 183 -END *) - -theorem pc3_head_2: - \forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).(\forall -(k: K).((pc3 (CHead c k u) t1 t2) \to (pc3 c (THead k u t1) (THead k u -t2))))))) -\def - \lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(k: K).(\lambda (H: (pc3 (CHead c k u) t1 t2)).(let H0 \def H in (ex2_ind T -(\lambda (t: T).(pr3 (CHead c k u) t1 t)) (\lambda (t: T).(pr3 (CHead c k u) -t2 t)) (pc3 c (THead k u t1) (THead k u t2)) (\lambda (x: T).(\lambda (H1: -(pr3 (CHead c k u) t1 x)).(\lambda (H2: (pr3 (CHead c k u) t2 x)).(ex_intro2 -T (\lambda (t: T).(pr3 c (THead k u t1) t)) (\lambda (t: T).(pr3 c (THead k u -t2) t)) (THead k u x) (pr3_head_12 c u u (pr3_refl c u) k t1 x H1) -(pr3_head_12 c u u (pr3_refl c u) k t2 x H2))))) H0))))))). -(* COMMENTS -Initial nodes: 201 -END *) - -theorem pc3_pr2_u: - \forall (c: C).(\forall (t2: T).(\forall (t1: T).((pr2 c t1 t2) \to (\forall -(t3: T).((pc3 c t2 t3) \to (pc3 c t1 t3)))))) -\def - \lambda (c: C).(\lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pr2 c t1 -t2)).(\lambda (t3: T).(\lambda (H0: (pc3 c t2 t3)).(let H1 \def H0 in -(ex2_ind T (\lambda (t: T).(pr3 c t2 t)) (\lambda (t: T).(pr3 c t3 t)) (pc3 c -t1 t3) (\lambda (x: T).(\lambda (H2: (pr3 c t2 x)).(\lambda (H3: (pr3 c t3 -x)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t3 t)) -x (pr3_sing c t2 t1 H x H2) H3)))) H1))))))). -(* COMMENTS -Initial nodes: 119 -END *) - -theorem pc3_t: - \forall (t2: T).(\forall (c: C).(\forall (t1: T).((pc3 c t1 t2) \to (\forall -(t3: T).((pc3 c t2 t3) \to (pc3 c t1 t3)))))) -\def - \lambda (t2: T).(\lambda (c: C).(\lambda (t1: T).(\lambda (H: (pc3 c t1 -t2)).(\lambda (t3: T).(\lambda (H0: (pc3 c t2 t3)).(let H1 \def H0 in -(ex2_ind T (\lambda (t: T).(pr3 c t2 t)) (\lambda (t: T).(pr3 c t3 t)) (pc3 c -t1 t3) (\lambda (x: T).(\lambda (H2: (pr3 c t2 x)).(\lambda (H3: (pr3 c t3 -x)).(let H4 \def H in (ex2_ind T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: -T).(pr3 c t2 t)) (pc3 c t1 t3) (\lambda (x0: T).(\lambda (H5: (pr3 c t1 -x0)).(\lambda (H6: (pr3 c t2 x0)).(ex2_ind T (\lambda (t: T).(pr3 c x0 t)) -(\lambda (t: T).(pr3 c x t)) (pc3 c t1 t3) (\lambda (x1: T).(\lambda (H7: -(pr3 c x0 x1)).(\lambda (H8: (pr3 c x x1)).(pc3_pr3_t c t1 x1 (pr3_t x0 t1 c -H5 x1 H7) t3 (pr3_t x t3 c H3 x1 H8))))) (pr3_confluence c t2 x0 H6 x H2))))) -H4))))) H1))))))). -(* COMMENTS -Initial nodes: 233 -END *) - -theorem pc3_pr2_u2: - \forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr2 c t0 t1) \to (\forall -(t2: T).((pc3 c t0 t2) \to (pc3 c t1 t2)))))) -\def - \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr2 c t0 -t1)).(\lambda (t2: T).(\lambda (H0: (pc3 c t0 t2)).(pc3_t t0 c t1 (pc3_pr2_x -c t1 t0 H) t2 H0)))))). -(* COMMENTS -Initial nodes: 45 -END *) - -theorem pc3_pr3_conf: - \forall (c: C).(\forall (t: T).(\forall (t1: T).((pc3 c t t1) \to (\forall -(t2: T).((pr3 c t t2) \to (pc3 c t2 t1)))))) -\def - \lambda (c: C).(\lambda (t: T).(\lambda (t1: T).(\lambda (H: (pc3 c t -t1)).(\lambda (t2: T).(\lambda (H0: (pr3 c t t2)).(pc3_t t c t2 (pc3_pr3_x c -t2 t H0) t1 H)))))). -(* COMMENTS -Initial nodes: 45 -END *) - -theorem pc3_head_12: - \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pc3 c u1 u2) \to (\forall -(k: K).(\forall (t1: T).(\forall (t2: T).((pc3 (CHead c k u2) t1 t2) \to (pc3 -c (THead k u1 t1) (THead k u2 t2))))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pc3 c u1 -u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pc3 -(CHead c k u2) t1 t2)).(pc3_t (THead k u2 t1) c (THead k u1 t1) (pc3_head_1 c -u1 u2 H k t1) (THead k u2 t2) (pc3_head_2 c u2 t1 t2 k H0))))))))). -(* COMMENTS -Initial nodes: 89 -END *) - -theorem pc3_head_21: - \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pc3 c u1 u2) \to (\forall -(k: K).(\forall (t1: T).(\forall (t2: T).((pc3 (CHead c k u1) t1 t2) \to (pc3 -c (THead k u1 t1) (THead k u2 t2))))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pc3 c u1 -u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pc3 -(CHead c k u1) t1 t2)).(pc3_t (THead k u1 t2) c (THead k u1 t1) (pc3_head_2 c -u1 t1 t2 k H0) (THead k u2 t2) (pc3_head_1 c u1 u2 H k t2))))))))). -(* COMMENTS -Initial nodes: 89 -END *) - -theorem pc3_pr0_pr2_t: - \forall (u1: T).(\forall (u2: T).((pr0 u2 u1) \to (\forall (c: C).(\forall -(t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pc3 -(CHead c k u1) t1 t2)))))))) -\def - \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr0 u2 u1)).(\lambda (c: -C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k: K).(\lambda (H0: (pr2 -(CHead c k u2) t1 t2)).(insert_eq C (CHead c k u2) (\lambda (c0: C).(pr2 c0 -t1 t2)) (\lambda (_: C).(pc3 (CHead c k u1) t1 t2)) (\lambda (y: C).(\lambda -(H1: (pr2 y t1 t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: -T).((eq C c0 (CHead c k u2)) \to (pc3 (CHead c k u1) t t0))))) (\lambda (c0: -C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (H3: -(eq C c0 (CHead c k u2))).(let H4 \def (f_equal C C (\lambda (e: C).e) c0 -(CHead c k u2) H3) in (pc3_pr2_r (CHead c k u1) t3 t4 (pr2_free (CHead c k -u1) t3 t4 H2)))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: -T).(\lambda (i: nat).(\lambda (H2: (getl i c0 (CHead d (Bind Abbr) -u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda -(t: T).(\lambda (H4: (subst0 i u t4 t)).(\lambda (H5: (eq C c0 (CHead c k -u2))).(let H6 \def (f_equal C C (\lambda (e: C).e) c0 (CHead c k u2) H5) in -(let H7 \def (eq_ind C c0 (\lambda (c1: C).(getl i c1 (CHead d (Bind Abbr) -u))) H2 (CHead c k u2) H6) in (nat_ind (\lambda (n: nat).((getl n (CHead c k -u2) (CHead d (Bind Abbr) u)) \to ((subst0 n u t4 t) \to (pc3 (CHead c k u1) -t3 t)))) (\lambda (H8: (getl O (CHead c k u2) (CHead d (Bind Abbr) -u))).(\lambda (H9: (subst0 O u t4 t)).(K_ind (\lambda (k0: K).((clear (CHead -c k0 u2) (CHead d (Bind Abbr) u)) \to (pc3 (CHead c k0 u1) t3 t))) (\lambda -(b: B).(\lambda (H10: (clear (CHead c (Bind b) u2) (CHead d (Bind Abbr) -u))).(let H11 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda -(_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) -(CHead d (Bind Abbr) u) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d -(Bind Abbr) u) u2 H10)) in ((let H12 \def (f_equal C B (\lambda (e: C).(match -e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ -k0 _) \Rightarrow (match k0 in K return (\lambda (_: K).B) with [(Bind b0) -\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) -(CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 H10)) in -((let H13 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: -C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead -d (Bind Abbr) u) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind -Abbr) u) u2 H10)) in (\lambda (H14: (eq B Abbr b)).(\lambda (_: (eq C d -c)).(let H16 \def (eq_ind T u (\lambda (t0: T).(subst0 O t0 t4 t)) H9 u2 H13) -in (eq_ind B Abbr (\lambda (b0: B).(pc3 (CHead c (Bind b0) u1) t3 t)) -(ex2_ind T (\lambda (t0: T).(subst0 O u1 t4 t0)) (\lambda (t0: T).(pr0 t t0)) -(pc3 (CHead c (Bind Abbr) u1) t3 t) (\lambda (x: T).(\lambda (H17: (subst0 O -u1 t4 x)).(\lambda (H18: (pr0 t x)).(pc3_pr3_t (CHead c (Bind Abbr) u1) t3 x -(pr3_pr2 (CHead c (Bind Abbr) u1) t3 x (pr2_delta (CHead c (Bind Abbr) u1) c -u1 O (getl_refl Abbr c u1) t3 t4 H3 x H17)) t (pr3_pr2 (CHead c (Bind Abbr) -u1) t x (pr2_free (CHead c (Bind Abbr) u1) t x H18)))))) (pr0_subst0_fwd u2 -t4 t O H16 u1 H)) b H14))))) H12)) H11)))) (\lambda (f: F).(\lambda (H10: -(clear (CHead c (Flat f) u2) (CHead d (Bind Abbr) u))).(clear_pc3_trans -(CHead d (Bind Abbr) u) t3 t (pc3_pr2_r (CHead d (Bind Abbr) u) t3 t -(pr2_delta (CHead d (Bind Abbr) u) d u O (getl_refl Abbr d u) t3 t4 H3 t H9)) -(CHead c (Flat f) u1) (clear_flat c (CHead d (Bind Abbr) u) (clear_gen_flat f -c (CHead d (Bind Abbr) u) u2 H10) f u1)))) k (getl_gen_O (CHead c k u2) -(CHead d (Bind Abbr) u) H8)))) (\lambda (i0: nat).(\lambda (IHi: (((getl i0 -(CHead c k u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t4 t) \to (pc3 -(CHead c k u1) t3 t))))).(\lambda (H8: (getl (S i0) (CHead c k u2) (CHead d -(Bind Abbr) u))).(\lambda (H9: (subst0 (S i0) u t4 t)).(K_ind (\lambda (k0: -K).((((getl i0 (CHead c k0 u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t4 -t) \to (pc3 (CHead c k0 u1) t3 t)))) \to ((getl (r k0 i0) c (CHead d (Bind -Abbr) u)) \to (pc3 (CHead c k0 u1) t3 t)))) (\lambda (b: B).(\lambda (_: -(((getl i0 (CHead c (Bind b) u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u -t4 t) \to (pc3 (CHead c (Bind b) u1) t3 t))))).(\lambda (H10: (getl (r (Bind -b) i0) c (CHead d (Bind Abbr) u))).(pc3_pr2_r (CHead c (Bind b) u1) t3 t -(pr2_delta (CHead c (Bind b) u1) d u (S i0) (getl_head (Bind b) i0 c (CHead d -(Bind Abbr) u) H10 u1) t3 t4 H3 t H9))))) (\lambda (f: F).(\lambda (_: -(((getl i0 (CHead c (Flat f) u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u -t4 t) \to (pc3 (CHead c (Flat f) u1) t3 t))))).(\lambda (H10: (getl (r (Flat -f) i0) c (CHead d (Bind Abbr) u))).(pc3_pr2_r (CHead c (Flat f) u1) t3 t -(pr2_cflat c t3 t (pr2_delta c d u (r (Flat f) i0) H10 t3 t4 H3 t H9) f -u1))))) k IHi (getl_gen_S k c (CHead d (Bind Abbr) u) u2 i0 H8)))))) i H7 -H4)))))))))))))) y t1 t2 H1))) H0)))))))). -(* COMMENTS -Initial nodes: 1533 -END *) - -theorem pc3_pr2_pr2_t: - \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr2 c u2 u1) \to (\forall -(t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pc3 -(CHead c k u1) t1 t2)))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr2 c u2 -u1)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (t1: -T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c0 k t) t1 t2) \to (pc3 -(CHead c0 k t0) t1 t2)))))))) (\lambda (c0: C).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (H0: (pr0 t1 t2)).(\lambda (t0: T).(\lambda (t3: T).(\lambda (k: -K).(\lambda (H1: (pr2 (CHead c0 k t1) t0 t3)).(pc3_pr0_pr2_t t2 t1 H0 c0 t0 -t3 k H1))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda -(i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t1: -T).(\lambda (t2: T).(\lambda (H1: (pr0 t1 t2)).(\lambda (t: T).(\lambda (H2: -(subst0 i u t2 t)).(\lambda (t0: T).(\lambda (t3: T).(\lambda (k: K).(\lambda -(H3: (pr2 (CHead c0 k t1) t0 t3)).(insert_eq C (CHead c0 k t1) (\lambda (c1: -C).(pr2 c1 t0 t3)) (\lambda (_: C).(pc3 (CHead c0 k t) t0 t3)) (\lambda (y: -C).(\lambda (H4: (pr2 y t0 t3)).(pr2_ind (\lambda (c1: C).(\lambda (t4: -T).(\lambda (t5: T).((eq C c1 (CHead c0 k t1)) \to (pc3 (CHead c0 k t) t4 -t5))))) (\lambda (c1: C).(\lambda (t4: T).(\lambda (t5: T).(\lambda (H5: (pr0 -t4 t5)).(\lambda (_: (eq C c1 (CHead c0 k t1))).(pc3_pr2_r (CHead c0 k t) t4 -t5 (pr2_free (CHead c0 k t) t4 t5 H5))))))) (\lambda (c1: C).(\lambda (d0: -C).(\lambda (u0: T).(\lambda (i0: nat).(\lambda (H5: (getl i0 c1 (CHead d0 -(Bind Abbr) u0))).(\lambda (t4: T).(\lambda (t5: T).(\lambda (H6: (pr0 t4 -t5)).(\lambda (t6: T).(\lambda (H7: (subst0 i0 u0 t5 t6)).(\lambda (H8: (eq C -c1 (CHead c0 k t1))).(let H9 \def (eq_ind C c1 (\lambda (c2: C).(getl i0 c2 -(CHead d0 (Bind Abbr) u0))) H5 (CHead c0 k t1) H8) in (nat_ind (\lambda (n: -nat).((getl n (CHead c0 k t1) (CHead d0 (Bind Abbr) u0)) \to ((subst0 n u0 t5 -t6) \to (pc3 (CHead c0 k t) t4 t6)))) (\lambda (H10: (getl O (CHead c0 k t1) -(CHead d0 (Bind Abbr) u0))).(\lambda (H11: (subst0 O u0 t5 t6)).(K_ind -(\lambda (k0: K).((clear (CHead c0 k0 t1) (CHead d0 (Bind Abbr) u0)) \to (pc3 -(CHead c0 k0 t) t4 t6))) (\lambda (b: B).(\lambda (H12: (clear (CHead c0 -(Bind b) t1) (CHead d0 (Bind Abbr) u0))).(let H13 \def (f_equal C C (\lambda -(e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d0 -| (CHead c2 _ _) \Rightarrow c2])) (CHead d0 (Bind Abbr) u0) (CHead c0 (Bind -b) t1) (clear_gen_bind b c0 (CHead d0 (Bind Abbr) u0) t1 H12)) in ((let H14 -\def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) -with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow (match k0 in K -return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) -\Rightarrow Abbr])])) (CHead d0 (Bind Abbr) u0) (CHead c0 (Bind b) t1) -(clear_gen_bind b c0 (CHead d0 (Bind Abbr) u0) t1 H12)) in ((let H15 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow u0 | (CHead _ _ t7) \Rightarrow t7])) (CHead d0 (Bind -Abbr) u0) (CHead c0 (Bind b) t1) (clear_gen_bind b c0 (CHead d0 (Bind Abbr) -u0) t1 H12)) in (\lambda (H16: (eq B Abbr b)).(\lambda (_: (eq C d0 c0)).(let -H18 \def (eq_ind T u0 (\lambda (t7: T).(subst0 O t7 t5 t6)) H11 t1 H15) in -(eq_ind B Abbr (\lambda (b0: B).(pc3 (CHead c0 (Bind b0) t) t4 t6)) (ex2_ind -T (\lambda (t7: T).(subst0 O t2 t5 t7)) (\lambda (t7: T).(pr0 t6 t7)) (pc3 -(CHead c0 (Bind Abbr) t) t4 t6) (\lambda (x: T).(\lambda (H19: (subst0 O t2 -t5 x)).(\lambda (H20: (pr0 t6 x)).(ex2_ind T (\lambda (t7: T).(subst0 O t t5 -t7)) (\lambda (t7: T).(subst0 (S (plus i O)) u x t7)) (pc3 (CHead c0 (Bind -Abbr) t) t4 t6) (\lambda (x0: T).(\lambda (H21: (subst0 O t t5 x0)).(\lambda -(H22: (subst0 (S (plus i O)) u x x0)).(let H23 \def (f_equal nat nat S (plus -i O) i (sym_eq nat i (plus i O) (plus_n_O i))) in (let H24 \def (eq_ind nat -(S (plus i O)) (\lambda (n: nat).(subst0 n u x x0)) H22 (S i) H23) in -(pc3_pr2_u (CHead c0 (Bind Abbr) t) x0 t4 (pr2_delta (CHead c0 (Bind Abbr) t) -c0 t O (getl_refl Abbr c0 t) t4 t5 H6 x0 H21) t6 (pc3_pr2_x (CHead c0 (Bind -Abbr) t) x0 t6 (pr2_delta (CHead c0 (Bind Abbr) t) d u (S i) (getl_head (Bind -Abbr) i c0 (CHead d (Bind Abbr) u) H0 t) t6 x H20 x0 H24)))))))) -(subst0_subst0_back t5 x t2 O H19 t u i H2))))) (pr0_subst0_fwd t1 t5 t6 O -H18 t2 H1)) b H16))))) H14)) H13)))) (\lambda (f: F).(\lambda (H12: (clear -(CHead c0 (Flat f) t1) (CHead d0 (Bind Abbr) u0))).(clear_pc3_trans (CHead d0 -(Bind Abbr) u0) t4 t6 (pc3_pr2_r (CHead d0 (Bind Abbr) u0) t4 t6 (pr2_delta -(CHead d0 (Bind Abbr) u0) d0 u0 O (getl_refl Abbr d0 u0) t4 t5 H6 t6 H11)) -(CHead c0 (Flat f) t) (clear_flat c0 (CHead d0 (Bind Abbr) u0) -(clear_gen_flat f c0 (CHead d0 (Bind Abbr) u0) t1 H12) f t)))) k (getl_gen_O -(CHead c0 k t1) (CHead d0 (Bind Abbr) u0) H10)))) (\lambda (i1: nat).(\lambda -(_: (((getl i1 (CHead c0 k t1) (CHead d0 (Bind Abbr) u0)) \to ((subst0 i1 u0 -t5 t6) \to (pc3 (CHead c0 k t) t4 t6))))).(\lambda (H10: (getl (S i1) (CHead -c0 k t1) (CHead d0 (Bind Abbr) u0))).(\lambda (H11: (subst0 (S i1) u0 t5 -t6)).(K_ind (\lambda (k0: K).((getl (r k0 i1) c0 (CHead d0 (Bind Abbr) u0)) -\to (pc3 (CHead c0 k0 t) t4 t6))) (\lambda (b: B).(\lambda (H12: (getl (r -(Bind b) i1) c0 (CHead d0 (Bind Abbr) u0))).(pc3_pr2_r (CHead c0 (Bind b) t) -t4 t6 (pr2_delta (CHead c0 (Bind b) t) d0 u0 (S i1) (getl_head (Bind b) i1 c0 -(CHead d0 (Bind Abbr) u0) H12 t) t4 t5 H6 t6 H11)))) (\lambda (f: F).(\lambda -(H12: (getl (r (Flat f) i1) c0 (CHead d0 (Bind Abbr) u0))).(pc3_pr2_r (CHead -c0 (Flat f) t) t4 t6 (pr2_cflat c0 t4 t6 (pr2_delta c0 d0 u0 (r (Flat f) i1) -H12 t4 t5 H6 t6 H11) f t)))) k (getl_gen_S k c0 (CHead d0 (Bind Abbr) u0) t1 -i1 H10)))))) i0 H9 H7))))))))))))) y t0 t3 H4))) H3))))))))))))))) c u2 u1 -H)))). -(* COMMENTS -Initial nodes: 1671 -END *) - -theorem pc3_pr2_pr3_t: - \forall (c: C).(\forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall -(k: K).((pr3 (CHead c k u2) t1 t2) \to (\forall (u1: T).((pr2 c u2 u1) \to -(pc3 (CHead c k u1) t1 t2)))))))) -\def - \lambda (c: C).(\lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(k: K).(\lambda (H: (pr3 (CHead c k u2) t1 t2)).(pr3_ind (CHead c k u2) -(\lambda (t: T).(\lambda (t0: T).(\forall (u1: T).((pr2 c u2 u1) \to (pc3 -(CHead c k u1) t t0))))) (\lambda (t: T).(\lambda (u1: T).(\lambda (_: (pr2 c -u2 u1)).(pc3_refl (CHead c k u1) t)))) (\lambda (t0: T).(\lambda (t3: -T).(\lambda (H0: (pr2 (CHead c k u2) t3 t0)).(\lambda (t4: T).(\lambda (_: -(pr3 (CHead c k u2) t0 t4)).(\lambda (H2: ((\forall (u1: T).((pr2 c u2 u1) -\to (pc3 (CHead c k u1) t0 t4))))).(\lambda (u1: T).(\lambda (H3: (pr2 c u2 -u1)).(pc3_t t0 (CHead c k u1) t3 (pc3_pr2_pr2_t c u1 u2 H3 t3 t0 k H0) t4 (H2 -u1 H3)))))))))) t1 t2 H)))))). -(* COMMENTS -Initial nodes: 199 -END *) - -theorem pc3_pr3_pc3_t: - \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u2 u1) \to (\forall -(t1: T).(\forall (t2: T).(\forall (k: K).((pc3 (CHead c k u2) t1 t2) \to (pc3 -(CHead c k u1) t1 t2)))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u2 -u1)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (t1: T).(\forall -(t2: T).(\forall (k: K).((pc3 (CHead c k t) t1 t2) \to (pc3 (CHead c k t0) t1 -t2))))))) (\lambda (t: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k: -K).(\lambda (H0: (pc3 (CHead c k t) t1 t2)).H0))))) (\lambda (t2: T).(\lambda -(t1: T).(\lambda (H0: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda (_: (pr3 c t2 -t3)).(\lambda (H2: ((\forall (t4: T).(\forall (t5: T).(\forall (k: K).((pc3 -(CHead c k t2) t4 t5) \to (pc3 (CHead c k t3) t4 t5))))))).(\lambda (t0: -T).(\lambda (t4: T).(\lambda (k: K).(\lambda (H3: (pc3 (CHead c k t1) t0 -t4)).(H2 t0 t4 k (let H4 \def H3 in (ex2_ind T (\lambda (t: T).(pr3 (CHead c -k t1) t0 t)) (\lambda (t: T).(pr3 (CHead c k t1) t4 t)) (pc3 (CHead c k t2) -t0 t4) (\lambda (x: T).(\lambda (H5: (pr3 (CHead c k t1) t0 x)).(\lambda (H6: -(pr3 (CHead c k t1) t4 x)).(pc3_t x (CHead c k t2) t0 (pc3_pr2_pr3_t c t1 t0 -x k H5 t2 H0) t4 (pc3_s (CHead c k t2) x t4 (pc3_pr2_pr3_t c t1 t4 x k H6 t2 -H0)))))) H4))))))))))))) u2 u1 H)))). -(* COMMENTS -Initial nodes: 319 -END *) - -theorem pc3_lift: - \forall (c: C).(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h -d c e) \to (\forall (t1: T).(\forall (t2: T).((pc3 e t1 t2) \to (pc3 c (lift -h d t1) (lift h d t2))))))))) -\def - \lambda (c: C).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda -(H: (drop h d c e)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pc3 e t1 -t2)).(let H1 \def H0 in (ex2_ind T (\lambda (t: T).(pr3 e t1 t)) (\lambda (t: -T).(pr3 e t2 t)) (pc3 c (lift h d t1) (lift h d t2)) (\lambda (x: T).(\lambda -(H2: (pr3 e t1 x)).(\lambda (H3: (pr3 e t2 x)).(pc3_pr3_t c (lift h d t1) -(lift h d x) (pr3_lift c e h d H t1 x H2) (lift h d t2) (pr3_lift c e h d H -t2 x H3))))) H1))))))))). -(* COMMENTS -Initial nodes: 159 -END *) - -theorem pc3_eta: - \forall (c: C).(\forall (t: T).(\forall (w: T).(\forall (u: T).((pc3 c t -(THead (Bind Abst) w u)) \to (\forall (v: T).((pc3 c v w) \to (pc3 c (THead -(Bind Abst) v (THead (Flat Appl) (TLRef O) (lift (S O) O t))) t))))))) -\def - \lambda (c: C).(\lambda (t: T).(\lambda (w: T).(\lambda (u: T).(\lambda (H: -(pc3 c t (THead (Bind Abst) w u))).(\lambda (v: T).(\lambda (H0: (pc3 c v -w)).(pc3_t (THead (Bind Abst) w (THead (Flat Appl) (TLRef O) (lift (S O) O -(THead (Bind Abst) w u)))) c (THead (Bind Abst) v (THead (Flat Appl) (TLRef -O) (lift (S O) O t))) (pc3_head_21 c v w H0 (Bind Abst) (THead (Flat Appl) -(TLRef O) (lift (S O) O t)) (THead (Flat Appl) (TLRef O) (lift (S O) O (THead -(Bind Abst) w u))) (pc3_thin_dx (CHead c (Bind Abst) v) (lift (S O) O t) -(lift (S O) O (THead (Bind Abst) w u)) (pc3_lift (CHead c (Bind Abst) v) c (S -O) O (drop_drop (Bind Abst) O c c (drop_refl c) v) t (THead (Bind Abst) w u) -H) (TLRef O) Appl)) t (pc3_t (THead (Bind Abst) w u) c (THead (Bind Abst) w -(THead (Flat Appl) (TLRef O) (lift (S O) O (THead (Bind Abst) w u)))) -(pc3_pr3_r c (THead (Bind Abst) w (THead (Flat Appl) (TLRef O) (lift (S O) O -(THead (Bind Abst) w u)))) (THead (Bind Abst) w u) (pr3_eta c w u w (pr3_refl -c w))) t (pc3_s c (THead (Bind Abst) w u) t H))))))))). -(* COMMENTS -Initial nodes: 399 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/subst1.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/subst1.ma deleted file mode 100644 index 3af245b3a..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/subst1.ma +++ /dev/null @@ -1,48 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pc3/props.ma". - -include "Basic-1/pr3/subst1.ma". - -theorem pc3_gen_cabbr: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to (\forall -(e: C).(\forall (u: T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) -\to (\forall (a0: C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d -a0 a) \to (\forall (x1: T).((subst1 d u t1 (lift (S O) d x1)) \to (\forall -(x2: T).((subst1 d u t2 (lift (S O) d x2)) \to (pc3 a x1 x2)))))))))))))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c t1 -t2)).(\lambda (e: C).(\lambda (u: T).(\lambda (d: nat).(\lambda (H0: (getl d -c (CHead e (Bind Abbr) u))).(\lambda (a0: C).(\lambda (H1: (csubst1 d u c -a0)).(\lambda (a: C).(\lambda (H2: (drop (S O) d a0 a)).(\lambda (x1: -T).(\lambda (H3: (subst1 d u t1 (lift (S O) d x1))).(\lambda (x2: T).(\lambda -(H4: (subst1 d u t2 (lift (S O) d x2))).(let H5 \def H in (ex2_ind T (\lambda -(t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) (pc3 a x1 x2) (\lambda (x: -T).(\lambda (H6: (pr3 c t1 x)).(\lambda (H7: (pr3 c t2 x)).(ex2_ind T -(\lambda (x3: T).(subst1 d u x (lift (S O) d x3))) (\lambda (x3: T).(pr3 a x2 -x3)) (pc3 a x1 x2) (\lambda (x0: T).(\lambda (H8: (subst1 d u x (lift (S O) d -x0))).(\lambda (H9: (pr3 a x2 x0)).(ex2_ind T (\lambda (x3: T).(subst1 d u x -(lift (S O) d x3))) (\lambda (x3: T).(pr3 a x1 x3)) (pc3 a x1 x2) (\lambda -(x3: T).(\lambda (H10: (subst1 d u x (lift (S O) d x3))).(\lambda (H11: (pr3 -a x1 x3)).(let H12 \def (eq_ind T x3 (\lambda (t: T).(pr3 a x1 t)) H11 x0 -(subst1_confluence_lift x x3 u d H10 x0 H8)) in (pc3_pr3_t a x1 x0 H12 x2 -H9))))) (pr3_gen_cabbr c t1 x H6 e u d H0 a0 H1 a H2 x1 H3))))) -(pr3_gen_cabbr c t2 x H7 e u d H0 a0 H1 a H2 x2 H4))))) H5))))))))))))))))). -(* COMMENTS -Initial nodes: 405 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/wcpr0.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/wcpr0.ma deleted file mode 100644 index 2b51edf62..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/wcpr0.ma +++ /dev/null @@ -1,96 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pc3/props.ma". - -include "Basic-1/wcpr0/getl.ma". - -theorem pc3_wcpr0__pc3_wcpr0_t_aux: - \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (k: K).(\forall -(u: T).(\forall (t1: T).(\forall (t2: T).((pr3 (CHead c1 k u) t1 t2) \to (pc3 -(CHead c2 k u) t1 t2)))))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(\lambda (k: -K).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 -(CHead c1 k u) t1 t2)).(pr3_ind (CHead c1 k u) (\lambda (t: T).(\lambda (t0: -T).(pc3 (CHead c2 k u) t t0))) (\lambda (t: T).(pc3_refl (CHead c2 k u) t)) -(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr2 (CHead c1 k u) t4 -t3)).(\lambda (t5: T).(\lambda (_: (pr3 (CHead c1 k u) t3 t5)).(\lambda (H3: -(pc3 (CHead c2 k u) t3 t5)).(pc3_t t3 (CHead c2 k u) t4 (insert_eq C (CHead -c1 k u) (\lambda (c: C).(pr2 c t4 t3)) (\lambda (_: C).(pc3 (CHead c2 k u) t4 -t3)) (\lambda (y: C).(\lambda (H4: (pr2 y t4 t3)).(pr2_ind (\lambda (c: -C).(\lambda (t: T).(\lambda (t0: T).((eq C c (CHead c1 k u)) \to (pc3 (CHead -c2 k u) t t0))))) (\lambda (c: C).(\lambda (t6: T).(\lambda (t0: T).(\lambda -(H5: (pr0 t6 t0)).(\lambda (_: (eq C c (CHead c1 k u))).(pc3_pr2_r (CHead c2 -k u) t6 t0 (pr2_free (CHead c2 k u) t6 t0 H5))))))) (\lambda (c: C).(\lambda -(d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H5: (getl i c (CHead d -(Bind Abbr) u0))).(\lambda (t6: T).(\lambda (t0: T).(\lambda (H6: (pr0 t6 -t0)).(\lambda (t: T).(\lambda (H7: (subst0 i u0 t0 t)).(\lambda (H8: (eq C c -(CHead c1 k u))).(let H9 \def (eq_ind C c (\lambda (c0: C).(getl i c0 (CHead -d (Bind Abbr) u0))) H5 (CHead c1 k u) H8) in (ex3_2_ind C T (\lambda (e2: -C).(\lambda (u2: T).(getl i (CHead c2 k u) (CHead e2 (Bind Abbr) u2)))) -(\lambda (e2: C).(\lambda (_: T).(wcpr0 d e2))) (\lambda (_: C).(\lambda (u2: -T).(pr0 u0 u2))) (pc3 (CHead c2 k u) t6 t) (\lambda (x0: C).(\lambda (x1: -T).(\lambda (H10: (getl i (CHead c2 k u) (CHead x0 (Bind Abbr) x1))).(\lambda -(_: (wcpr0 d x0)).(\lambda (H12: (pr0 u0 x1)).(ex2_ind T (\lambda (t7: -T).(subst0 i x1 t0 t7)) (\lambda (t7: T).(pr0 t t7)) (pc3 (CHead c2 k u) t6 -t) (\lambda (x: T).(\lambda (H13: (subst0 i x1 t0 x)).(\lambda (H14: (pr0 t -x)).(pc3_pr2_u (CHead c2 k u) x t6 (pr2_delta (CHead c2 k u) x0 x1 i H10 t6 -t0 H6 x H13) t (pc3_pr2_x (CHead c2 k u) x t (pr2_free (CHead c2 k u) t x -H14)))))) (pr0_subst0_fwd u0 t0 t i H7 x1 H12))))))) (wcpr0_getl (CHead c1 k -u) (CHead c2 k u) (wcpr0_comp c1 c2 H u u (pr0_refl u) k) i d u0 (Bind Abbr) -H9)))))))))))))) y t4 t3 H4))) H1) t5 H3))))))) t1 t2 H0)))))))). -(* COMMENTS -Initial nodes: 689 -END *) - -theorem pc3_wcpr0_t: - \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (t1: -T).(\forall (t2: T).((pr3 c1 t1 t2) \to (pc3 c2 t1 t2)))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(wcpr0_ind -(\lambda (c: C).(\lambda (c0: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 -t2) \to (pc3 c0 t1 t2)))))) (\lambda (c: C).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (H0: (pr3 c t1 t2)).(pc3_pr3_r c t1 t2 H0))))) (\lambda (c0: -C).(\lambda (c3: C).(\lambda (H0: (wcpr0 c0 c3)).(\lambda (_: ((\forall (t1: -T).(\forall (t2: T).((pr3 c0 t1 t2) \to (pc3 c3 t1 t2)))))).(\lambda (u1: -T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 u2)).(\lambda (k: K).(\lambda (t1: -T).(\lambda (t2: T).(\lambda (H3: (pr3 (CHead c0 k u1) t1 t2)).(let H4 \def -(pc3_pr2_pr3_t c0 u1 t1 t2 k H3 u2 (pr2_free c0 u1 u2 H2)) in (ex2_ind T -(\lambda (t: T).(pr3 (CHead c0 k u2) t1 t)) (\lambda (t: T).(pr3 (CHead c0 k -u2) t2 t)) (pc3 (CHead c3 k u2) t1 t2) (\lambda (x: T).(\lambda (H5: (pr3 -(CHead c0 k u2) t1 x)).(\lambda (H6: (pr3 (CHead c0 k u2) t2 x)).(pc3_t x -(CHead c3 k u2) t1 (pc3_wcpr0__pc3_wcpr0_t_aux c0 c3 H0 k u2 t1 x H5) t2 -(pc3_s (CHead c3 k u2) x t2 (pc3_wcpr0__pc3_wcpr0_t_aux c0 c3 H0 k u2 t2 x -H6)))))) H4))))))))))))) c1 c2 H))). -(* COMMENTS -Initial nodes: 299 -END *) - -theorem pc3_wcpr0: - \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (t1: -T).(\forall (t2: T).((pc3 c1 t1 t2) \to (pc3 c2 t1 t2)))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(\lambda (t1: -T).(\lambda (t2: T).(\lambda (H0: (pc3 c1 t1 t2)).(let H1 \def H0 in (ex2_ind -T (\lambda (t: T).(pr3 c1 t1 t)) (\lambda (t: T).(pr3 c1 t2 t)) (pc3 c2 t1 -t2) (\lambda (x: T).(\lambda (H2: (pr3 c1 t1 x)).(\lambda (H3: (pr3 c1 t2 -x)).(pc3_t x c2 t1 (pc3_wcpr0_t c1 c2 H t1 x H2) t2 (pc3_s c2 x t2 -(pc3_wcpr0_t c1 c2 H t2 x H3)))))) H1))))))). -(* COMMENTS -Initial nodes: 121 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr0/dec.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr0/dec.ma deleted file mode 100644 index c28504bee..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr0/dec.ma +++ /dev/null @@ -1,529 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr0/fwd.ma". - -include "Basic-1/subst0/dec.ma". - -include "Basic-1/T/dec.ma". - -include "Basic-1/T/props.ma". - -theorem nf0_dec: - \forall (t1: T).(or (\forall (t2: T).((pr0 t1 t2) \to (eq T t1 t2))) (ex2 T -(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: -T).(pr0 t1 t2)))) -\def - \lambda (t1: T).(T_ind (\lambda (t: T).(or (\forall (t2: T).((pr0 t t2) \to -(eq T t t2))) (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 t t2))))) (\lambda (n: nat).(or_introl -(\forall (t2: T).((pr0 (TSort n) t2) \to (eq T (TSort n) t2))) (ex2 T -(\lambda (t2: T).((eq T (TSort n) t2) \to (\forall (P: Prop).P))) (\lambda -(t2: T).(pr0 (TSort n) t2))) (\lambda (t2: T).(\lambda (H: (pr0 (TSort n) -t2)).(eq_ind_r T (TSort n) (\lambda (t: T).(eq T (TSort n) t)) (refl_equal T -(TSort n)) t2 (pr0_gen_sort t2 n H)))))) (\lambda (n: nat).(or_introl -(\forall (t2: T).((pr0 (TLRef n) t2) \to (eq T (TLRef n) t2))) (ex2 T -(\lambda (t2: T).((eq T (TLRef n) t2) \to (\forall (P: Prop).P))) (\lambda -(t2: T).(pr0 (TLRef n) t2))) (\lambda (t2: T).(\lambda (H: (pr0 (TLRef n) -t2)).(eq_ind_r T (TLRef n) (\lambda (t: T).(eq T (TLRef n) t)) (refl_equal T -(TLRef n)) t2 (pr0_gen_lref t2 n H)))))) (\lambda (k: K).(\lambda (t: -T).(\lambda (H: (or (\forall (t2: T).((pr0 t t2) \to (eq T t t2))) (ex2 T -(\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: -T).(pr0 t t2))))).(\lambda (t0: T).(\lambda (H0: (or (\forall (t2: T).((pr0 -t0 t2) \to (eq T t0 t2))) (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall -(P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2))))).(K_ind (\lambda (k0: K).(or -(\forall (t2: T).((pr0 (THead k0 t t0) t2) \to (eq T (THead k0 t t0) t2))) -(ex2 T (\lambda (t2: T).((eq T (THead k0 t t0) t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 (THead k0 t t0) t2))))) (\lambda (b: -B).(B_ind (\lambda (b0: B).(or (\forall (t2: T).((pr0 (THead (Bind b0) t t0) -t2) \to (eq T (THead (Bind b0) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T -(THead (Bind b0) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 -(THead (Bind b0) t t0) t2))))) (or_intror (\forall (t2: T).((pr0 (THead (Bind -Abbr) t t0) t2) \to (eq T (THead (Bind Abbr) t t0) t2))) (ex2 T (\lambda (t2: -T).((eq T (THead (Bind Abbr) t t0) t2) \to (\forall (P: Prop).P))) (\lambda -(t2: T).(pr0 (THead (Bind Abbr) t t0) t2))) (let H_x \def (dnf_dec t t0 O) in -(let H1 \def H_x in (ex_ind T (\lambda (v: T).(or (subst0 O t t0 (lift (S O) -O v)) (eq T t0 (lift (S O) O v)))) (ex2 T (\lambda (t2: T).((eq T (THead -(Bind Abbr) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 -(THead (Bind Abbr) t t0) t2))) (\lambda (x: T).(\lambda (H2: (or (subst0 O t -t0 (lift (S O) O x)) (eq T t0 (lift (S O) O x)))).(or_ind (subst0 O t t0 -(lift (S O) O x)) (eq T t0 (lift (S O) O x)) (ex2 T (\lambda (t2: T).((eq T -(THead (Bind Abbr) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: -T).(pr0 (THead (Bind Abbr) t t0) t2))) (\lambda (H3: (subst0 O t t0 (lift (S -O) O x))).(ex_intro2 T (\lambda (t2: T).((eq T (THead (Bind Abbr) t t0) t2) -\to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abbr) t t0) -t2)) (THead (Bind Abbr) t (lift (S O) O x)) (\lambda (H4: (eq T (THead (Bind -Abbr) t t0) (THead (Bind Abbr) t (lift (S O) O x)))).(\lambda (P: Prop).(let -H5 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t2) -\Rightarrow t2])) (THead (Bind Abbr) t t0) (THead (Bind Abbr) t (lift (S O) O -x)) H4) in (let H6 \def (eq_ind T t0 (\lambda (t2: T).(subst0 O t t2 (lift (S -O) O x))) H3 (lift (S O) O x) H5) in (subst0_refl t (lift (S O) O x) O H6 -P))))) (pr0_delta t t (pr0_refl t) t0 t0 (pr0_refl t0) (lift (S O) O x) H3))) -(\lambda (H3: (eq T t0 (lift (S O) O x))).(eq_ind_r T (lift (S O) O x) -(\lambda (t2: T).(ex2 T (\lambda (t3: T).((eq T (THead (Bind Abbr) t t2) t3) -\to (\forall (P: Prop).P))) (\lambda (t3: T).(pr0 (THead (Bind Abbr) t t2) -t3)))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Bind Abbr) t (lift (S O) -O x)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind -Abbr) t (lift (S O) O x)) t2)) x (\lambda (H4: (eq T (THead (Bind Abbr) t -(lift (S O) O x)) x)).(\lambda (P: Prop).(thead_x_lift_y_y (Bind Abbr) x t (S -O) O H4 P))) (pr0_zeta Abbr not_abbr_abst x x (pr0_refl x) t)) t0 H3)) H2))) -H1)))) (let H1 \def H in (or_ind (\forall (t2: T).((pr0 t t2) \to (eq T t -t2))) (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) -(\lambda (t2: T).(pr0 t t2))) (or (\forall (t2: T).((pr0 (THead (Bind Abst) t -t0) t2) \to (eq T (THead (Bind Abst) t t0) t2))) (ex2 T (\lambda (t2: T).((eq -T (THead (Bind Abst) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: -T).(pr0 (THead (Bind Abst) t t0) t2)))) (\lambda (H2: ((\forall (t2: T).((pr0 -t t2) \to (eq T t t2))))).(let H3 \def H0 in (or_ind (\forall (t2: T).((pr0 -t0 t2) \to (eq T t0 t2))) (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall -(P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2))) (or (\forall (t2: T).((pr0 -(THead (Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0) t2))) (ex2 T -(\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2)))) (\lambda -(H4: ((\forall (t2: T).((pr0 t0 t2) \to (eq T t0 t2))))).(or_introl (\forall -(t2: T).((pr0 (THead (Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0) -t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to -(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2))) -(\lambda (t2: T).(\lambda (H5: (pr0 (THead (Bind Abst) t t0) t2)).(ex3_2_ind -T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t0 t3))) (eq T (THead (Bind Abst) t t0) t2) (\lambda (x0: T).(\lambda -(x1: T).(\lambda (H6: (eq T t2 (THead (Bind Abst) x0 x1))).(\lambda (H7: (pr0 -t x0)).(\lambda (H8: (pr0 t0 x1)).(let H_y \def (H4 x1 H8) in (let H_y0 \def -(H2 x0 H7) in (let H9 \def (eq_ind_r T x1 (\lambda (t3: T).(pr0 t0 t3)) H8 t0 -H_y) in (let H10 \def (eq_ind_r T x1 (\lambda (t3: T).(eq T t2 (THead (Bind -Abst) x0 t3))) H6 t0 H_y) in (let H11 \def (eq_ind_r T x0 (\lambda (t3: -T).(pr0 t t3)) H7 t H_y0) in (let H12 \def (eq_ind_r T x0 (\lambda (t3: -T).(eq T t2 (THead (Bind Abst) t3 t0))) H10 t H_y0) in (eq_ind_r T (THead -(Bind Abst) t t0) (\lambda (t3: T).(eq T (THead (Bind Abst) t t0) t3)) -(refl_equal T (THead (Bind Abst) t t0)) t2 H12)))))))))))) (pr0_gen_abst t t0 -t2 H5)))))) (\lambda (H4: (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall -(P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2)))).(ex2_ind T (\lambda (t2: -T).((eq T t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2)) -(or (\forall (t2: T).((pr0 (THead (Bind Abst) t t0) t2) \to (eq T (THead -(Bind Abst) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abst) t -t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) -t t0) t2)))) (\lambda (x: T).(\lambda (H5: (((eq T t0 x) \to (\forall (P: -Prop).P)))).(\lambda (H6: (pr0 t0 x)).(or_intror (\forall (t2: T).((pr0 -(THead (Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0) t2))) (ex2 T -(\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2))) (ex_intro2 T -(\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2)) (THead (Bind -Abst) t x) (\lambda (H7: (eq T (THead (Bind Abst) t t0) (THead (Bind Abst) t -x))).(\lambda (P: Prop).(let H8 \def (f_equal T T (\lambda (e: T).(match e in -T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) -\Rightarrow t0 | (THead _ _ t2) \Rightarrow t2])) (THead (Bind Abst) t t0) -(THead (Bind Abst) t x) H7) in (let H9 \def (eq_ind_r T x (\lambda (t2: -T).(pr0 t0 t2)) H6 t0 H8) in (let H10 \def (eq_ind_r T x (\lambda (t2: -T).((eq T t0 t2) \to (\forall (P0: Prop).P0))) H5 t0 H8) in (H10 (refl_equal -T t0) P)))))) (pr0_comp t t (pr0_refl t) t0 x H6 (Bind Abst))))))) H4)) H3))) -(\lambda (H2: (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 t t2)))).(ex2_ind T (\lambda (t2: T).((eq T -t t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2)) (or (\forall -(t2: T).((pr0 (THead (Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0) -t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to -(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2)))) -(\lambda (x: T).(\lambda (H3: (((eq T t x) \to (\forall (P: -Prop).P)))).(\lambda (H4: (pr0 t x)).(or_intror (\forall (t2: T).((pr0 (THead -(Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0) t2))) (ex2 T -(\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2))) (ex_intro2 T -(\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2)) (THead (Bind -Abst) x t0) (\lambda (H5: (eq T (THead (Bind Abst) t t0) (THead (Bind Abst) x -t0))).(\lambda (P: Prop).(let H6 \def (f_equal T T (\lambda (e: T).(match e -in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t | (TLRef _) -\Rightarrow t | (THead _ t2 _) \Rightarrow t2])) (THead (Bind Abst) t t0) -(THead (Bind Abst) x t0) H5) in (let H7 \def (eq_ind_r T x (\lambda (t2: -T).(pr0 t t2)) H4 t H6) in (let H8 \def (eq_ind_r T x (\lambda (t2: T).((eq T -t t2) \to (\forall (P0: Prop).P0))) H3 t H6) in (H8 (refl_equal T t) P)))))) -(pr0_comp t x H4 t0 t0 (pr0_refl t0) (Bind Abst))))))) H2)) H1)) (let H_x -\def (dnf_dec t t0 O) in (let H1 \def H_x in (ex_ind T (\lambda (v: T).(or -(subst0 O t t0 (lift (S O) O v)) (eq T t0 (lift (S O) O v)))) (or (\forall -(t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) -t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to -(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)))) -(\lambda (x: T).(\lambda (H2: (or (subst0 O t t0 (lift (S O) O x)) (eq T t0 -(lift (S O) O x)))).(or_ind (subst0 O t t0 (lift (S O) O x)) (eq T t0 (lift -(S O) O x)) (or (\forall (t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T -(THead (Bind Void) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind -Void) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead -(Bind Void) t t0) t2)))) (\lambda (H3: (subst0 O t t0 (lift (S O) O x))).(let -H4 \def H in (or_ind (\forall (t2: T).((pr0 t t2) \to (eq T t t2))) (ex2 T -(\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: -T).(pr0 t t2))) (or (\forall (t2: T).((pr0 (THead (Bind Void) t t0) t2) \to -(eq T (THead (Bind Void) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead -(Bind Void) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 -(THead (Bind Void) t t0) t2)))) (\lambda (H5: ((\forall (t2: T).((pr0 t t2) -\to (eq T t t2))))).(let H6 \def H0 in (or_ind (\forall (t2: T).((pr0 t0 t2) -\to (eq T t0 t2))) (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 t0 t2))) (or (\forall (t2: T).((pr0 (THead -(Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) t2))) (ex2 T -(\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)))) (\lambda -(H7: ((\forall (t2: T).((pr0 t0 t2) \to (eq T t0 t2))))).(or_introl (\forall -(t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) -t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to -(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2))) -(\lambda (t2: T).(\lambda (H8: (pr0 (THead (Bind Void) t t0) t2)).(or_ind -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda -(t3: T).(pr0 t0 t3)))) (pr0 t0 (lift (S O) O t2)) (eq T (THead (Bind Void) t -t0) t2) (\lambda (H9: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 -(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) -(\lambda (_: T).(\lambda (t3: T).(pr0 t0 t3))))).(ex3_2_ind T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t0 -t3))) (eq T (THead (Bind Void) t t0) t2) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H10: (eq T t2 (THead (Bind Void) x0 x1))).(\lambda (H11: (pr0 t -x0)).(\lambda (H12: (pr0 t0 x1)).(let H_y \def (H7 x1 H12) in (let H_y0 \def -(H5 x0 H11) in (let H13 \def (eq_ind_r T x1 (\lambda (t3: T).(pr0 t0 t3)) H12 -t0 H_y) in (let H14 \def (eq_ind_r T x1 (\lambda (t3: T).(eq T t2 (THead -(Bind Void) x0 t3))) H10 t0 H_y) in (let H15 \def (eq_ind_r T x0 (\lambda -(t3: T).(pr0 t t3)) H11 t H_y0) in (let H16 \def (eq_ind_r T x0 (\lambda (t3: -T).(eq T t2 (THead (Bind Void) t3 t0))) H14 t H_y0) in (eq_ind_r T (THead -(Bind Void) t t0) (\lambda (t3: T).(eq T (THead (Bind Void) t t0) t3)) -(refl_equal T (THead (Bind Void) t t0)) t2 H16)))))))))))) H9)) (\lambda (H9: -(pr0 t0 (lift (S O) O t2))).(let H_y \def (H7 (lift (S O) O t2) H9) in (let -H10 \def (eq_ind T t0 (\lambda (t3: T).(subst0 O t t3 (lift (S O) O x))) H3 -(lift (S O) O t2) H_y) in (eq_ind_r T (lift (S O) O t2) (\lambda (t3: T).(eq -T (THead (Bind Void) t t3) t2)) (subst0_gen_lift_false t2 t (lift (S O) O x) -(S O) O O (le_n O) (eq_ind_r nat (plus (S O) O) (\lambda (n: nat).(lt O n)) -(le_n (plus (S O) O)) (plus O (S O)) (plus_sym O (S O))) H10 (eq T (THead -(Bind Void) t (lift (S O) O t2)) t2)) t0 H_y)))) (pr0_gen_void t t0 t2 -H8)))))) (\lambda (H7: (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 t0 t2)))).(ex2_ind T (\lambda (t2: T).((eq T -t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2)) (or (\forall -(t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) -t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to -(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)))) -(\lambda (x0: T).(\lambda (H8: (((eq T t0 x0) \to (\forall (P: -Prop).P)))).(\lambda (H9: (pr0 t0 x0)).(or_intror (\forall (t2: T).((pr0 -(THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) t2))) (ex2 T -(\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2))) (ex_intro2 T -(\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)) (THead (Bind -Void) t x0) (\lambda (H10: (eq T (THead (Bind Void) t t0) (THead (Bind Void) -t x0))).(\lambda (P: Prop).(let H11 \def (f_equal T T (\lambda (e: T).(match -e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) -\Rightarrow t0 | (THead _ _ t2) \Rightarrow t2])) (THead (Bind Void) t t0) -(THead (Bind Void) t x0) H10) in (let H12 \def (eq_ind_r T x0 (\lambda (t2: -T).(pr0 t0 t2)) H9 t0 H11) in (let H13 \def (eq_ind_r T x0 (\lambda (t2: -T).((eq T t0 t2) \to (\forall (P0: Prop).P0))) H8 t0 H11) in (H13 (refl_equal -T t0) P)))))) (pr0_comp t t (pr0_refl t) t0 x0 H9 (Bind Void))))))) H7)) -H6))) (\lambda (H5: (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 t t2)))).(ex2_ind T (\lambda (t2: T).((eq T -t t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2)) (or (\forall -(t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) -t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to -(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)))) -(\lambda (x0: T).(\lambda (H6: (((eq T t x0) \to (\forall (P: -Prop).P)))).(\lambda (H7: (pr0 t x0)).(or_intror (\forall (t2: T).((pr0 -(THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) t2))) (ex2 T -(\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2))) (ex_intro2 T -(\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)) (THead (Bind -Void) x0 t0) (\lambda (H8: (eq T (THead (Bind Void) t t0) (THead (Bind Void) -x0 t0))).(\lambda (P: Prop).(let H9 \def (f_equal T T (\lambda (e: T).(match -e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t | (TLRef _) -\Rightarrow t | (THead _ t2 _) \Rightarrow t2])) (THead (Bind Void) t t0) -(THead (Bind Void) x0 t0) H8) in (let H10 \def (eq_ind_r T x0 (\lambda (t2: -T).(pr0 t t2)) H7 t H9) in (let H11 \def (eq_ind_r T x0 (\lambda (t2: T).((eq -T t t2) \to (\forall (P0: Prop).P0))) H6 t H9) in (H11 (refl_equal T t) -P)))))) (pr0_comp t x0 H7 t0 t0 (pr0_refl t0) (Bind Void))))))) H5)) H4))) -(\lambda (H3: (eq T t0 (lift (S O) O x))).(let H4 \def (eq_ind T t0 (\lambda -(t2: T).(or (\forall (t3: T).((pr0 t2 t3) \to (eq T t2 t3))) (ex2 T (\lambda -(t3: T).((eq T t2 t3) \to (\forall (P: Prop).P))) (\lambda (t3: T).(pr0 t2 -t3))))) H0 (lift (S O) O x) H3) in (eq_ind_r T (lift (S O) O x) (\lambda (t2: -T).(or (\forall (t3: T).((pr0 (THead (Bind Void) t t2) t3) \to (eq T (THead -(Bind Void) t t2) t3))) (ex2 T (\lambda (t3: T).((eq T (THead (Bind Void) t -t2) t3) \to (\forall (P: Prop).P))) (\lambda (t3: T).(pr0 (THead (Bind Void) -t t2) t3))))) (or_intror (\forall (t2: T).((pr0 (THead (Bind Void) t (lift (S -O) O x)) t2) \to (eq T (THead (Bind Void) t (lift (S O) O x)) t2))) (ex2 T -(\lambda (t2: T).((eq T (THead (Bind Void) t (lift (S O) O x)) t2) \to -(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t (lift (S -O) O x)) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Bind Void) t -(lift (S O) O x)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 -(THead (Bind Void) t (lift (S O) O x)) t2)) x (\lambda (H5: (eq T (THead -(Bind Void) t (lift (S O) O x)) x)).(\lambda (P: Prop).(thead_x_lift_y_y -(Bind Void) x t (S O) O H5 P))) (pr0_zeta Void (sym_not_eq B Abst Void -not_abst_void) x x (pr0_refl x) t))) t0 H3))) H2))) H1))) b)) (\lambda (f: -F).(F_ind (\lambda (f0: F).(or (\forall (t2: T).((pr0 (THead (Flat f0) t t0) -t2) \to (eq T (THead (Flat f0) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T -(THead (Flat f0) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 -(THead (Flat f0) t t0) t2))))) (let H_x \def (binder_dec t0) in (let H1 \def -H_x in (or_ind (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: -T).(eq T t0 (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: -T).(\forall (u: T).((eq T t0 (THead (Bind b) w u)) \to (\forall (P: -Prop).P))))) (or (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq -T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat -Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead -(Flat Appl) t t0) t2)))) (\lambda (H2: (ex_3 B T T (\lambda (b: B).(\lambda -(w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w u))))))).(ex_3_ind B T T -(\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w -u))))) (or (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T -(THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat -Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead -(Flat Appl) t t0) t2)))) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: -T).(\lambda (H3: (eq T t0 (THead (Bind x0) x1 x2))).(let H4 \def (eq_ind T t0 -(\lambda (t2: T).(or (\forall (t3: T).((pr0 t2 t3) \to (eq T t2 t3))) (ex2 T -(\lambda (t3: T).((eq T t2 t3) \to (\forall (P: Prop).P))) (\lambda (t3: -T).(pr0 t2 t3))))) H0 (THead (Bind x0) x1 x2) H3) in (eq_ind_r T (THead (Bind -x0) x1 x2) (\lambda (t2: T).(or (\forall (t3: T).((pr0 (THead (Flat Appl) t -t2) t3) \to (eq T (THead (Flat Appl) t t2) t3))) (ex2 T (\lambda (t3: T).((eq -T (THead (Flat Appl) t t2) t3) \to (\forall (P: Prop).P))) (\lambda (t3: -T).(pr0 (THead (Flat Appl) t t2) t3))))) (B_ind (\lambda (b: B).((or (\forall -(t2: T).((pr0 (THead (Bind b) x1 x2) t2) \to (eq T (THead (Bind b) x1 x2) -t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind b) x1 x2) t2) \to (\forall -(P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind b) x1 x2) t2)))) \to (or -(\forall (t2: T).((pr0 (THead (Flat Appl) t (THead (Bind b) x1 x2)) t2) \to -(eq T (THead (Flat Appl) t (THead (Bind b) x1 x2)) t2))) (ex2 T (\lambda (t2: -T).((eq T (THead (Flat Appl) t (THead (Bind b) x1 x2)) t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead (Bind b) x1 x2)) -t2)))))) (\lambda (_: (or (\forall (t2: T).((pr0 (THead (Bind Abbr) x1 x2) -t2) \to (eq T (THead (Bind Abbr) x1 x2) t2))) (ex2 T (\lambda (t2: T).((eq T -(THead (Bind Abbr) x1 x2) t2) \to (\forall (P: Prop).P))) (\lambda (t2: -T).(pr0 (THead (Bind Abbr) x1 x2) t2))))).(or_intror (\forall (t2: T).((pr0 -(THead (Flat Appl) t (THead (Bind Abbr) x1 x2)) t2) \to (eq T (THead (Flat -Appl) t (THead (Bind Abbr) x1 x2)) t2))) (ex2 T (\lambda (t2: T).((eq T -(THead (Flat Appl) t (THead (Bind Abbr) x1 x2)) t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead (Bind Abbr) x1 -x2)) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t (THead -(Bind Abbr) x1 x2)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 -(THead (Flat Appl) t (THead (Bind Abbr) x1 x2)) t2)) (THead (Bind Abbr) x1 -(THead (Flat Appl) (lift (S O) O t) x2)) (\lambda (H6: (eq T (THead (Flat -Appl) t (THead (Bind Abbr) x1 x2)) (THead (Bind Abbr) x1 (THead (Flat Appl) -(lift (S O) O t) x2)))).(\lambda (P: Prop).(let H7 \def (eq_ind T (THead -(Flat Appl) t (THead (Bind Abbr) x1 x2)) (\lambda (ee: T).(match ee in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead _ _ t2) \Rightarrow (match t2 in T return (\lambda -(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False -| (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])])) I (THead (Bind -Abbr) x1 (THead (Flat Appl) (lift (S O) O t) x2)) H6) in (False_ind P H7)))) -(pr0_upsilon Abbr not_abbr_abst t t (pr0_refl t) x1 x1 (pr0_refl x1) x2 x2 -(pr0_refl x2))))) (\lambda (_: (or (\forall (t2: T).((pr0 (THead (Bind Abst) -x1 x2) t2) \to (eq T (THead (Bind Abst) x1 x2) t2))) (ex2 T (\lambda (t2: -T).((eq T (THead (Bind Abst) x1 x2) t2) \to (\forall (P: Prop).P))) (\lambda -(t2: T).(pr0 (THead (Bind Abst) x1 x2) t2))))).(or_intror (\forall (t2: -T).((pr0 (THead (Flat Appl) t (THead (Bind Abst) x1 x2)) t2) \to (eq T (THead -(Flat Appl) t (THead (Bind Abst) x1 x2)) t2))) (ex2 T (\lambda (t2: T).((eq T -(THead (Flat Appl) t (THead (Bind Abst) x1 x2)) t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead (Bind Abst) x1 -x2)) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t (THead -(Bind Abst) x1 x2)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 -(THead (Flat Appl) t (THead (Bind Abst) x1 x2)) t2)) (THead (Bind Abbr) t x2) -(\lambda (H6: (eq T (THead (Flat Appl) t (THead (Bind Abst) x1 x2)) (THead -(Bind Abbr) t x2))).(\lambda (P: Prop).(let H7 \def (eq_ind T (THead (Flat -Appl) t (THead (Bind Abst) x1 x2)) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind Abbr) t x2) H6) in (False_ind P H7)))) (pr0_beta x1 -t t (pr0_refl t) x2 x2 (pr0_refl x2))))) (\lambda (_: (or (\forall (t2: -T).((pr0 (THead (Bind Void) x1 x2) t2) \to (eq T (THead (Bind Void) x1 x2) -t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) x1 x2) t2) \to -(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) x1 x2) -t2))))).(or_intror (\forall (t2: T).((pr0 (THead (Flat Appl) t (THead (Bind -Void) x1 x2)) t2) \to (eq T (THead (Flat Appl) t (THead (Bind Void) x1 x2)) -t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t (THead (Bind Void) -x1 x2)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat -Appl) t (THead (Bind Void) x1 x2)) t2))) (ex_intro2 T (\lambda (t2: T).((eq T -(THead (Flat Appl) t (THead (Bind Void) x1 x2)) t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead (Bind Void) x1 -x2)) t2)) (THead (Bind Void) x1 (THead (Flat Appl) (lift (S O) O t) x2)) -(\lambda (H6: (eq T (THead (Flat Appl) t (THead (Bind Void) x1 x2)) (THead -(Bind Void) x1 (THead (Flat Appl) (lift (S O) O t) x2)))).(\lambda (P: -Prop).(let H7 \def (eq_ind T (THead (Flat Appl) t (THead (Bind Void) x1 x2)) -(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ t2) \Rightarrow -(match t2 in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False -| (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K -return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) -\Rightarrow False])])])) I (THead (Bind Void) x1 (THead (Flat Appl) (lift (S -O) O t) x2)) H6) in (False_ind P H7)))) (pr0_upsilon Void (sym_not_eq B Abst -Void not_abst_void) t t (pr0_refl t) x1 x1 (pr0_refl x1) x2 x2 (pr0_refl -x2))))) x0 H4) t0 H3)))))) H2)) (\lambda (H2: ((\forall (b: B).(\forall (w: -T).(\forall (u: T).((eq T t0 (THead (Bind b) w u)) \to (\forall (P: -Prop).P))))))).(let H3 \def H in (or_ind (\forall (t2: T).((pr0 t t2) \to (eq -T t t2))) (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) -(\lambda (t2: T).(pr0 t t2))) (or (\forall (t2: T).((pr0 (THead (Flat Appl) t -t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq -T (THead (Flat Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: -T).(pr0 (THead (Flat Appl) t t0) t2)))) (\lambda (H4: ((\forall (t2: T).((pr0 -t t2) \to (eq T t t2))))).(let H5 \def H0 in (or_ind (\forall (t2: T).((pr0 -t0 t2) \to (eq T t0 t2))) (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall -(P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2))) (or (\forall (t2: T).((pr0 -(THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T -(\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)))) (\lambda -(H6: ((\forall (t2: T).((pr0 t0 t2) \to (eq T t0 t2))))).(or_introl (\forall -(t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) -t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to -(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2))) -(\lambda (t2: T).(\lambda (H7: (pr0 (THead (Flat Appl) t t0) t2)).(or3_ind -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda -(t3: T).(pr0 t0 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T t0 (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 -(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr0 t u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t0 (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead -(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 t -u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(t3: T).(pr0 z1 t3)))))))) (eq T (THead (Flat Appl) t t0) t2) (\lambda (H8: -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda -(t3: T).(pr0 t0 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq -T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t -u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t0 t3))) (eq T (THead (Flat Appl) -t t0) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H9: (eq T t2 (THead -(Flat Appl) x0 x1))).(\lambda (H10: (pr0 t x0)).(\lambda (H11: (pr0 t0 -x1)).(let H_y \def (H6 x1 H11) in (let H_y0 \def (H4 x0 H10) in (let H12 \def -(eq_ind_r T x1 (\lambda (t3: T).(pr0 t0 t3)) H11 t0 H_y) in (let H13 \def -(eq_ind_r T x1 (\lambda (t3: T).(eq T t2 (THead (Flat Appl) x0 t3))) H9 t0 -H_y) in (let H14 \def (eq_ind_r T x0 (\lambda (t3: T).(pr0 t t3)) H10 t H_y0) -in (let H15 \def (eq_ind_r T x0 (\lambda (t3: T).(eq T t2 (THead (Flat Appl) -t3 t0))) H13 t H_y0) in (eq_ind_r T (THead (Flat Appl) t t0) (\lambda (t3: -T).(eq T (THead (Flat Appl) t t0) t3)) (refl_equal T (THead (Flat Appl) t -t0)) t2 H15)))))))))))) H8)) (\lambda (H8: (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t0 (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr0 t u2))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))).(ex4_4_ind T T T T -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t0 -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 t u2))))) (\lambda -(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))) (eq -T (THead (Flat Appl) t t0) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda -(x2: T).(\lambda (x3: T).(\lambda (H9: (eq T t0 (THead (Bind Abst) x0 -x1))).(\lambda (H10: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (_: (pr0 t -x2)).(\lambda (_: (pr0 x1 x3)).(eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda -(t3: T).(eq T (THead (Flat Appl) t t0) t3)) (let H13 \def (eq_ind T t0 -(\lambda (t3: T).(\forall (t4: T).((pr0 t3 t4) \to (eq T t3 t4)))) H6 (THead -(Bind Abst) x0 x1) H9) in (let H14 \def (eq_ind T t0 (\lambda (t3: -T).(\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t3 (THead (Bind b) -w u)) \to (\forall (P: Prop).P)))))) H2 (THead (Bind Abst) x0 x1) H9) in -(eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t3: T).(eq T (THead (Flat -Appl) t t3) (THead (Bind Abbr) x2 x3))) (H14 Abst x0 x1 (H13 (THead (Bind -Abst) x0 x1) (pr0_refl (THead (Bind Abst) x0 x1))) (eq T (THead (Flat Appl) t -(THead (Bind Abst) x0 x1)) (THead (Bind Abbr) x2 x3))) t0 H9))) t2 -H10))))))))) H8)) (\lambda (H8: (ex6_6 B T T T T T (\lambda (b: B).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not -(eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t0 (THead (Bind b) -y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead (Flat -Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 t u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 -t3))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T t0 (THead (Bind b) y1 z1)))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead (Flat Appl) (lift -(S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 t u2))))))) (\lambda -(_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: -T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))) -(eq T (THead (Flat Appl) t t0) t2) (\lambda (x0: B).(\lambda (x1: T).(\lambda -(x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (not -(eq B x0 Abst))).(\lambda (H10: (eq T t0 (THead (Bind x0) x1 x2))).(\lambda -(H11: (eq T t2 (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) -x5)))).(\lambda (_: (pr0 t x3)).(\lambda (_: (pr0 x1 x4)).(\lambda (_: (pr0 -x2 x5)).(eq_ind_r T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) -x5)) (\lambda (t3: T).(eq T (THead (Flat Appl) t t0) t3)) (let H15 \def -(eq_ind T t0 (\lambda (t3: T).(\forall (t4: T).((pr0 t3 t4) \to (eq T t3 -t4)))) H6 (THead (Bind x0) x1 x2) H10) in (let H16 \def (eq_ind T t0 (\lambda -(t3: T).(\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t3 (THead -(Bind b) w u)) \to (\forall (P: Prop).P)))))) H2 (THead (Bind x0) x1 x2) H10) -in (eq_ind_r T (THead (Bind x0) x1 x2) (\lambda (t3: T).(eq T (THead (Flat -Appl) t t3) (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)))) -(H16 x0 x1 x2 (H15 (THead (Bind x0) x1 x2) (pr0_refl (THead (Bind x0) x1 -x2))) (eq T (THead (Flat Appl) t (THead (Bind x0) x1 x2)) (THead (Bind x0) x4 -(THead (Flat Appl) (lift (S O) O x3) x5)))) t0 H10))) t2 H11))))))))))))) -H8)) (pr0_gen_appl t t0 t2 H7)))))) (\lambda (H6: (ex2 T (\lambda (t2: -T).((eq T t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t0 -t2)))).(ex2_ind T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: Prop).P))) -(\lambda (t2: T).(pr0 t0 t2)) (or (\forall (t2: T).((pr0 (THead (Flat Appl) t -t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq -T (THead (Flat Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: -T).(pr0 (THead (Flat Appl) t t0) t2)))) (\lambda (x: T).(\lambda (H7: (((eq T -t0 x) \to (\forall (P: Prop).P)))).(\lambda (H8: (pr0 t0 x)).(or_intror -(\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat -Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) -\to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) -t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to -(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)) -(THead (Flat Appl) t x) (\lambda (H9: (eq T (THead (Flat Appl) t t0) (THead -(Flat Appl) t x))).(\lambda (P: Prop).(let H10 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | -(TLRef _) \Rightarrow t0 | (THead _ _ t2) \Rightarrow t2])) (THead (Flat -Appl) t t0) (THead (Flat Appl) t x) H9) in (let H11 \def (eq_ind_r T x -(\lambda (t2: T).(pr0 t0 t2)) H8 t0 H10) in (let H12 \def (eq_ind_r T x -(\lambda (t2: T).((eq T t0 t2) \to (\forall (P0: Prop).P0))) H7 t0 H10) in -(H12 (refl_equal T t0) P)))))) (pr0_comp t t (pr0_refl t) t0 x H8 (Flat -Appl))))))) H6)) H5))) (\lambda (H4: (ex2 T (\lambda (t2: T).((eq T t t2) \to -(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2)))).(ex2_ind T (\lambda -(t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2)) -(or (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead -(Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t -t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) -t t0) t2)))) (\lambda (x: T).(\lambda (H5: (((eq T t x) \to (\forall (P: -Prop).P)))).(\lambda (H6: (pr0 t x)).(or_intror (\forall (t2: T).((pr0 (THead -(Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T -(\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2))) (ex_intro2 T -(\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)) (THead (Flat -Appl) x t0) (\lambda (H7: (eq T (THead (Flat Appl) t t0) (THead (Flat Appl) x -t0))).(\lambda (P: Prop).(let H8 \def (f_equal T T (\lambda (e: T).(match e -in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t | (TLRef _) -\Rightarrow t | (THead _ t2 _) \Rightarrow t2])) (THead (Flat Appl) t t0) -(THead (Flat Appl) x t0) H7) in (let H9 \def (eq_ind_r T x (\lambda (t2: -T).(pr0 t t2)) H6 t H8) in (let H10 \def (eq_ind_r T x (\lambda (t2: T).((eq -T t t2) \to (\forall (P0: Prop).P0))) H5 t H8) in (H10 (refl_equal T t) -P)))))) (pr0_comp t x H6 t0 t0 (pr0_refl t0) (Flat Appl))))))) H4)) H3))) -H1))) (or_intror (\forall (t2: T).((pr0 (THead (Flat Cast) t t0) t2) \to (eq -T (THead (Flat Cast) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat -Cast) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead -(Flat Cast) t t0) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Flat -Cast) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead -(Flat Cast) t t0) t2)) t0 (\lambda (H1: (eq T (THead (Flat Cast) t t0) -t0)).(\lambda (P: Prop).(thead_x_y_y (Flat Cast) t t0 H1 P))) (pr0_tau t0 t0 -(pr0_refl t0) t))) f)) k)))))) t1). -(* COMMENTS -Initial nodes: 10459 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr0/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr0/defs.ma deleted file mode 100644 index 0568e070c..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr0/defs.ma +++ /dev/null @@ -1,40 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/subst0/defs.ma". - -inductive pr0: T \to (T \to Prop) \def -| pr0_refl: \forall (t: T).(pr0 t t) -| pr0_comp: \forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (t1: -T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (k: K).(pr0 (THead k u1 t1) -(THead k u2 t2)))))))) -| pr0_beta: \forall (u: T).(\forall (v1: T).(\forall (v2: T).((pr0 v1 v2) \to -(\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (pr0 (THead (Flat Appl) v1 -(THead (Bind Abst) u t1)) (THead (Bind Abbr) v2 t2)))))))) -| pr0_upsilon: \forall (b: B).((not (eq B b Abst)) \to (\forall (v1: -T).(\forall (v2: T).((pr0 v1 v2) \to (\forall (u1: T).(\forall (u2: T).((pr0 -u1 u2) \to (\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (pr0 (THead -(Flat Appl) v1 (THead (Bind b) u1 t1)) (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t2))))))))))))) -| pr0_delta: \forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (t1: -T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (w: T).((subst0 O u2 t2 w) \to -(pr0 (THead (Bind Abbr) u1 t1) (THead (Bind Abbr) u2 w))))))))) -| pr0_zeta: \forall (b: B).((not (eq B b Abst)) \to (\forall (t1: T).(\forall -(t2: T).((pr0 t1 t2) \to (\forall (u: T).(pr0 (THead (Bind b) u (lift (S O) O -t1)) t2)))))) -| pr0_tau: \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (u: -T).(pr0 (THead (Flat Cast) u t1) t2)))). - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr0/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr0/fwd.ma deleted file mode 100644 index 46caceab4..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr0/fwd.ma +++ /dev/null @@ -1,2018 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr0/props.ma". - -theorem pr0_gen_sort: - \forall (x: T).(\forall (n: nat).((pr0 (TSort n) x) \to (eq T x (TSort n)))) -\def - \lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr0 (TSort n) x)).(insert_eq -T (TSort n) (\lambda (t: T).(pr0 t x)) (\lambda (t: T).(eq T x t)) (\lambda -(y: T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: T).(\lambda (t0: -T).((eq T t (TSort n)) \to (eq T t0 t)))) (\lambda (t: T).(\lambda (H1: (eq T -t (TSort n))).(let H2 \def (f_equal T T (\lambda (e: T).e) t (TSort n) H1) in -(eq_ind_r T (TSort n) (\lambda (t0: T).(eq T t0 t0)) (refl_equal T (TSort n)) -t H2)))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda -(_: (((eq T u1 (TSort n)) \to (eq T u2 u1)))).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2 -t1)))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u1 t1) (TSort n))).(let -H6 \def (eq_ind T (THead k u1 t1) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H5) in -(False_ind (eq T (THead k u2 t2) (THead k u1 t1)) H6)))))))))))) (\lambda (u: -T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: -(((eq T v1 (TSort n)) \to (eq T v2 v1)))).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2 -t1)))).(\lambda (H5: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t1)) -(TSort n))).(let H6 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u -t1)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) -\Rightarrow True])) I (TSort n) H5) in (False_ind (eq T (THead (Bind Abbr) v2 -t2) (THead (Flat Appl) v1 (THead (Bind Abst) u t1))) H6)))))))))))) (\lambda -(b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: -T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (TSort n)) \to (eq T v2 -v1)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda -(_: (((eq T u1 (TSort n)) \to (eq T u2 u1)))).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2 -t1)))).(\lambda (H8: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t1)) -(TSort n))).(let H9 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 -t1)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) -\Rightarrow True])) I (TSort n) H8) in (False_ind (eq T (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Flat Appl) v1 (THead (Bind -b) u1 t1))) H9))))))))))))))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda -(_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 (TSort n)) \to (eq T u2 -u1)))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda -(_: (((eq T t1 (TSort n)) \to (eq T t2 t1)))).(\lambda (w: T).(\lambda (_: -(subst0 O u2 t2 w)).(\lambda (H6: (eq T (THead (Bind Abbr) u1 t1) (TSort -n))).(let H7 \def (eq_ind T (THead (Bind Abbr) u1 t1) (\lambda (ee: T).(match -ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | -(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) -H6) in (False_ind (eq T (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u1 t1)) -H7))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda -(t1: T).(\lambda (t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 -(TSort n)) \to (eq T t2 t1)))).(\lambda (u: T).(\lambda (H4: (eq T (THead -(Bind b) u (lift (S O) O t1)) (TSort n))).(let H5 \def (eq_ind T (THead (Bind -b) u (lift (S O) O t1)) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead _ _ _) \Rightarrow True])) I (TSort n) H4) in (False_ind (eq T t2 -(THead (Bind b) u (lift (S O) O t1))) H5)))))))))) (\lambda (t1: T).(\lambda -(t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq -T t2 t1)))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u t1) -(TSort n))).(let H4 \def (eq_ind T (THead (Flat Cast) u t1) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I -(TSort n) H3) in (False_ind (eq T t2 (THead (Flat Cast) u t1)) H4)))))))) y x -H0))) H))). -(* COMMENTS -Initial nodes: 1045 -END *) - -theorem pr0_gen_lref: - \forall (x: T).(\forall (n: nat).((pr0 (TLRef n) x) \to (eq T x (TLRef n)))) -\def - \lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr0 (TLRef n) x)).(insert_eq -T (TLRef n) (\lambda (t: T).(pr0 t x)) (\lambda (t: T).(eq T x t)) (\lambda -(y: T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: T).(\lambda (t0: -T).((eq T t (TLRef n)) \to (eq T t0 t)))) (\lambda (t: T).(\lambda (H1: (eq T -t (TLRef n))).(let H2 \def (f_equal T T (\lambda (e: T).e) t (TLRef n) H1) in -(eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 t0)) (refl_equal T (TLRef n)) -t H2)))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda -(_: (((eq T u1 (TLRef n)) \to (eq T u2 u1)))).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (eq T t2 -t1)))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u1 t1) (TLRef n))).(let -H6 \def (eq_ind T (THead k u1 t1) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H5) in -(False_ind (eq T (THead k u2 t2) (THead k u1 t1)) H6)))))))))))) (\lambda (u: -T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: -(((eq T v1 (TLRef n)) \to (eq T v2 v1)))).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (eq T t2 -t1)))).(\lambda (H5: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t1)) -(TLRef n))).(let H6 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u -t1)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) -\Rightarrow True])) I (TLRef n) H5) in (False_ind (eq T (THead (Bind Abbr) v2 -t2) (THead (Flat Appl) v1 (THead (Bind Abst) u t1))) H6)))))))))))) (\lambda -(b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: -T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (TLRef n)) \to (eq T v2 -v1)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda -(_: (((eq T u1 (TLRef n)) \to (eq T u2 u1)))).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (eq T t2 -t1)))).(\lambda (H8: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t1)) -(TLRef n))).(let H9 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 -t1)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) -\Rightarrow True])) I (TLRef n) H8) in (False_ind (eq T (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Flat Appl) v1 (THead (Bind -b) u1 t1))) H9))))))))))))))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda -(_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 (TLRef n)) \to (eq T u2 -u1)))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda -(_: (((eq T t1 (TLRef n)) \to (eq T t2 t1)))).(\lambda (w: T).(\lambda (_: -(subst0 O u2 t2 w)).(\lambda (H6: (eq T (THead (Bind Abbr) u1 t1) (TLRef -n))).(let H7 \def (eq_ind T (THead (Bind Abbr) u1 t1) (\lambda (ee: T).(match -ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | -(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) -H6) in (False_ind (eq T (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u1 t1)) -H7))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda -(t1: T).(\lambda (t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 -(TLRef n)) \to (eq T t2 t1)))).(\lambda (u: T).(\lambda (H4: (eq T (THead -(Bind b) u (lift (S O) O t1)) (TLRef n))).(let H5 \def (eq_ind T (THead (Bind -b) u (lift (S O) O t1)) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead _ _ _) \Rightarrow True])) I (TLRef n) H4) in (False_ind (eq T t2 -(THead (Bind b) u (lift (S O) O t1))) H5)))))))))) (\lambda (t1: T).(\lambda -(t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (eq -T t2 t1)))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u t1) -(TLRef n))).(let H4 \def (eq_ind T (THead (Flat Cast) u t1) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I -(TLRef n) H3) in (False_ind (eq T t2 (THead (Flat Cast) u t1)) H4)))))))) y x -H0))) H))). -(* COMMENTS -Initial nodes: 1045 -END *) - -theorem pr0_gen_abst: - \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Bind Abst) u1 -t1) x) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind -Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: -T).(\lambda (t2: T).(pr0 t1 t2))))))) -\def - \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead -(Bind Abst) u1 t1) x)).(insert_eq T (THead (Bind Abst) u1 t1) (\lambda (t: -T).(pr0 t x)) (\lambda (_: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 -u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))))) (\lambda (y: -T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: T).(\lambda (t0: T).((eq T -t (THead (Bind Abst) u1 t1)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T t0 (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))))))) (\lambda -(t: T).(\lambda (H1: (eq T t (THead (Bind Abst) u1 t1))).(let H2 \def -(f_equal T T (\lambda (e: T).e) t (THead (Bind Abst) u1 t1) H1) in (eq_ind_r -T (THead (Bind Abst) u1 t1) (\lambda (t0: T).(ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T t0 (THead (Bind Abst) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 -t2))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Bind -Abst) u1 t1) (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))) u1 t1 -(refl_equal T (THead (Bind Abst) u1 t1)) (pr0_refl u1) (pr0_refl t1)) t -H2)))) (\lambda (u0: T).(\lambda (u2: T).(\lambda (H1: (pr0 u0 u2)).(\lambda -(H2: (((eq T u0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T (\lambda (u3: -T).(\lambda (t2: T).(eq T u2 (THead (Bind Abst) u3 t2)))) (\lambda (u3: -T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 -t2))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H3: (pr0 t0 -t2)).(\lambda (H4: (((eq T t0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T -(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u3 t3)))) -(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3))))))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u0 t0) -(THead (Bind Abst) u1 t1))).(let H6 \def (f_equal T K (\lambda (e: T).(match -e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) -\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u0 t0) (THead (Bind -Abst) u1 t1) H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) -\Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead k u0 t0) (THead (Bind -Abst) u1 t1) H5) in ((let H8 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) -\Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k u0 t0) (THead (Bind -Abst) u1 t1) H5) in (\lambda (H9: (eq T u0 u1)).(\lambda (H10: (eq K k (Bind -Abst))).(eq_ind_r K (Bind Abst) (\lambda (k0: K).(ex3_2 T T (\lambda (u3: -T).(\lambda (t3: T).(eq T (THead k0 u2 t2) (THead (Bind Abst) u3 t3)))) -(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3))))) (let H11 \def (eq_ind T t0 (\lambda (t: T).((eq T t (THead -(Bind Abst) u1 t1)) \to (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 -(THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) -(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))))) H4 t1 H8) in (let H12 \def -(eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H3 t1 H8) in (let H13 \def (eq_ind T -u0 (\lambda (t: T).((eq T t (THead (Bind Abst) u1 t1)) \to (ex3_2 T T -(\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead (Bind Abst) u3 t3)))) -(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3)))))) H2 u1 H9) in (let H14 \def (eq_ind T u0 (\lambda (t: -T).(pr0 t u2)) H1 u1 H9) in (ex3_2_intro T T (\lambda (u3: T).(\lambda (t3: -T).(eq T (THead (Bind Abst) u2 t2) (THead (Bind Abst) u3 t3)))) (\lambda (u3: -T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 -t3))) u2 t2 (refl_equal T (THead (Bind Abst) u2 t2)) H14 H12))))) k H10)))) -H7)) H6)))))))))))) (\lambda (u: T).(\lambda (v1: T).(\lambda (v2: -T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind Abst) u1 -t1)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind -Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: -T).(\lambda (t2: T).(pr0 t1 t2))))))).(\lambda (t0: T).(\lambda (t2: -T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Abst) u1 -t1)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t1 t3))))))).(\lambda (H5: (eq T (THead (Flat Appl) -v1 (THead (Bind Abst) u t0)) (THead (Bind Abst) u1 t1))).(let H6 \def (eq_ind -T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (ee: T).(match ee -in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef -_) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return -(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind Abst) u1 t1) H5) in (False_ind (ex3_2 T T (\lambda -(u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) (THead (Bind Abst) -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t1 t3)))) H6)))))))))))) (\lambda (b: B).(\lambda -(_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 -v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T -(\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind Abst) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: -T).(pr0 t1 t2))))))).(\lambda (u0: T).(\lambda (u2: T).(\lambda (_: (pr0 u0 -u2)).(\lambda (_: (((eq T u0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T -(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Bind Abst) u3 t2)))) -(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: -T).(pr0 t1 t2))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 -t2)).(\lambda (_: (((eq T t0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T -(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u3 t3)))) -(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3))))))).(\lambda (H8: (eq T (THead (Flat Appl) v1 (THead (Bind -b) u0 t0)) (THead (Bind Abst) u1 t1))).(let H9 \def (eq_ind T (THead (Flat -Appl) v1 (THead (Bind b) u0 t0)) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind Abst) u1 t1) H8) in (False_ind (ex3_2 T T (\lambda -(u3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t2)) (THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: -T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) -H9))))))))))))))))) (\lambda (u0: T).(\lambda (u2: T).(\lambda (_: (pr0 u0 -u2)).(\lambda (_: (((eq T u0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T -(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Bind Abst) u3 t2)))) -(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: -T).(pr0 t1 t2))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 -t2)).(\lambda (_: (((eq T t0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T -(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u3 t3)))) -(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3))))))).(\lambda (w: T).(\lambda (_: (subst0 O u2 t2 -w)).(\lambda (H6: (eq T (THead (Bind Abbr) u0 t0) (THead (Bind Abst) u1 -t1))).(let H7 \def (eq_ind T (THead (Bind Abbr) u0 t0) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K -return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B return -(\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | -Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (THead (Bind -Abst) u1 t1) H6) in (False_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t3: -T).(eq T (THead (Bind Abbr) u2 w) (THead (Bind Abst) u3 t3)))) (\lambda (u3: -T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 -t3)))) H7))))))))))))) (\lambda (b: B).(\lambda (H1: (not (eq B b -Abst))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda -(H3: (((eq T t0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 -t3))))))).(\lambda (u: T).(\lambda (H4: (eq T (THead (Bind b) u (lift (S O) O -t0)) (THead (Bind Abst) u1 t1))).(let H5 \def (f_equal T B (\lambda (e: -T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | -(TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k in K return -(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow -b])])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abst) u1 t1) H4) in -((let H6 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: -T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) -\Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abst) u1 -t1) H4) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat -\to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow -(TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true -\Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t3) \Rightarrow -(THead k (lref_map f d u0) (lref_map f (s k d) t3))]) in lref_map) (\lambda -(x0: nat).(plus x0 (S O))) O t0) | (TLRef _) \Rightarrow ((let rec lref_map -(f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) -\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with -[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t3) -\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t3))]) in -lref_map) (\lambda (x0: nat).(plus x0 (S O))) O t0) | (THead _ _ t) -\Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abst) u1 -t1) H4) in (\lambda (_: (eq T u u1)).(\lambda (H9: (eq B b Abst)).(let H10 -\def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H1 Abst H9) in (let -H11 \def (eq_ind_r T t1 (\lambda (t: T).((eq T t0 (THead (Bind Abst) u1 t)) -\to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t t3)))))) H3 (lift (S O) O t0) H7) in (eq_ind T -(lift (S O) O t0) (\lambda (t: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t3: -T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t t3))))) (let H12 -\def (match (H10 (refl_equal B Abst)) in False return (\lambda (_: -False).(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr0 (lift (S O) O t0) t3))))) with []) in H12) t1 -H7)))))) H6)) H5)))))))))) (\lambda (t0: T).(\lambda (t2: T).(\lambda (_: -(pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T -T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3))))))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u -t0) (THead (Bind Abst) u1 t1))).(let H4 \def (eq_ind T (THead (Flat Cast) u -t0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort -_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) -\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u1 -t1) H3) in (False_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 -(THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) -(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) H4)))))))) y x H0))) H)))). -(* COMMENTS -Initial nodes: 2838 -END *) - -theorem pr0_gen_appl: - \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Flat Appl) u1 -t1) x) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead -(Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda -(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T x (THead (Bind b) -v2 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: B).(\lambda -(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 -u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(t2: T).(pr0 z1 t2)))))))))))) -\def - \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead -(Flat Appl) u1 t1) x)).(insert_eq T (THead (Flat Appl) u1 t1) (\lambda (t: -T).(pr0 t x)) (\lambda (_: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 -u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda -(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) -(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T x -(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda -(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: -T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 -v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))))))) (\lambda (y: -T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: T).(\lambda (t0: T).((eq T -t (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda -(t2: T).(eq T t0 (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T -T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda -(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) -(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T -t0 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: -T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 -v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))))))))) (\lambda (t: -T).(\lambda (H1: (eq T t (THead (Flat Appl) u1 t1))).(let H2 \def (f_equal T -T (\lambda (e: T).e) t (THead (Flat Appl) u1 t1) H1) in (eq_ind_r T (THead -(Flat Appl) u1 t1) (\lambda (t0: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda -(t2: T).(eq T t0 (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T -T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda -(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) -(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T -t0 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: -T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 -v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))))))) (or3_intro0 (ex3_2 T -T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Appl) u1 t1) (THead -(Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda -(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t2: T).(eq T (THead (Flat Appl) u1 t1) (THead (Bind Abbr) u2 t2)))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 -u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: -T).(pr0 z1 t2)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B -b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(v2: T).(\lambda (t2: T).(eq T (THead (Flat Appl) u1 t1) (THead (Bind b) v2 -(THead (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 -u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(t2: T).(pr0 z1 t2)))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: -T).(eq T (THead (Flat Appl) u1 t1) (THead (Flat Appl) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 -t2))) u1 t1 (refl_equal T (THead (Flat Appl) u1 t1)) (pr0_refl u1) (pr0_refl -t1))) t H2)))) (\lambda (u0: T).(\lambda (u2: T).(\lambda (H1: (pr0 u0 -u2)).(\lambda (H2: (((eq T u0 (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T -(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) -(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: -T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Bind -Abbr) u3 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda -(_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(u3: T).(\lambda (v2: T).(\lambda (t2: T).(eq T u2 (THead (Bind b) v2 (THead -(Flat Appl) (lift (S O) O u3) t2))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 -u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(t2: T).(pr0 z1 t2)))))))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda -(H3: (pr0 t0 t2)).(\lambda (H4: (((eq T t0 (THead (Flat Appl) u1 t1)) \to -(or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) -u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 -z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: -T).(eq T t2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -b) v2 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda -(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3)))))))))))).(\lambda (k: K).(\lambda (H5: (eq -T (THead k u0 t0) (THead (Flat Appl) u1 t1))).(let H6 \def (f_equal T K -(\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) -\Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) -(THead k u0 t0) (THead (Flat Appl) u1 t1) H5) in ((let H7 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) -(THead k u0 t0) (THead (Flat Appl) u1 t1) H5) in ((let H8 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) -(THead k u0 t0) (THead (Flat Appl) u1 t1) H5) in (\lambda (H9: (eq T u0 -u1)).(\lambda (H10: (eq K k (Flat Appl))).(eq_ind_r K (Flat Appl) (\lambda -(k0: K).(or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead k0 u2 -t2) (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 -u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead -(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: -T).(\lambda (t3: T).(eq T (THead k0 u2 t2) (THead (Bind Abbr) u3 t3)))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 -u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: -T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B -b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda -(v2: T).(\lambda (t3: T).(eq T (THead k0 u2 t2) (THead (Bind b) v2 (THead -(Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 -u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(t3: T).(pr0 z1 t3)))))))))) (let H11 \def (eq_ind T t0 (\lambda (t: T).((eq -T t (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda -(t3: T).(eq T t2 (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: -T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T -T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: -T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda -(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) -(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T -t2 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: -T).(\lambda (_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 -v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))))))) H4 t1 H8) in (let -H12 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H3 t1 H8) in (let H13 \def -(eq_ind T u0 (\lambda (t: T).((eq T t (THead (Flat Appl) u1 t1)) \to (or3 -(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead (Flat Appl) u3 -t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 -z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: -T).(eq T u2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T u2 (THead (Bind -b) v2 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda -(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3))))))))))) H2 u1 H9) in (let H14 \def (eq_ind -T u0 (\lambda (t: T).(pr0 t u2)) H1 u1 H9) in (or3_intro0 (ex3_2 T T (\lambda -(u3: T).(\lambda (t3: T).(eq T (THead (Flat Appl) u2 t2) (THead (Flat Appl) -u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 -z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: -T).(eq T (THead (Flat Appl) u2 t2) (THead (Bind Abbr) u3 t3)))))) (\lambda -(_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 -t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) -(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (v2: T).(\lambda -(t3: T).(eq T (THead (Flat Appl) u2 t2) (THead (Bind b) v2 (THead (Flat Appl) -(lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 -t3)))))))) (ex3_2_intro T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead -(Flat Appl) u2 t2) (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: -T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) u2 t2 -(refl_equal T (THead (Flat Appl) u2 t2)) H14 H12)))))) k H10)))) H7)) -H6)))))))))))) (\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda -(H1: (pr0 v1 v2)).(\lambda (H2: (((eq T v1 (THead (Flat Appl) u1 t1)) \to -(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Flat Appl) -u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: -T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 -z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: -T).(eq T v2 (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (v3: T).(\lambda (t2: T).(eq T v2 (THead (Bind -b) v3 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda -(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (t2: T).(pr0 z1 t2)))))))))))).(\lambda (t0: T).(\lambda (t2: -T).(\lambda (H3: (pr0 t0 t2)).(\lambda (H4: (((eq T t0 (THead (Flat Appl) u1 -t1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead -(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda -(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T t2 (THead (Bind -b) v3 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda -(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3)))))))))))).(\lambda (H5: (eq T (THead (Flat -Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat Appl) u1 t1))).(let H6 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t _) -\Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat -Appl) u1 t1) H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow (THead (Bind Abst) u -t0) | (TLRef _) \Rightarrow (THead (Bind Abst) u t0) | (THead _ _ t) -\Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat -Appl) u1 t1) H5) in (\lambda (H8: (eq T v1 u1)).(let H9 \def (eq_ind T v1 -(\lambda (t: T).((eq T t (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T v2 (THead (Flat Appl) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T v2 (THead (Bind -Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T v2 (THead (Bind b) v3 (THead -(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 -u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(t3: T).(pr0 z1 t3))))))))))) H2 u1 H8) in (let H10 \def (eq_ind T v1 -(\lambda (t: T).(pr0 t v2)) H1 u1 H8) in (let H11 \def (eq_ind_r T t1 -(\lambda (t: T).((eq T t0 (THead (Flat Appl) u1 t)) \to (or3 (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v3 (THead -(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 -u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(t3: T).(pr0 z1 t3))))))))))) H4 (THead (Bind Abst) u t0) H7) in (let H12 -\def (eq_ind_r T t1 (\lambda (t: T).((eq T u1 (THead (Flat Appl) u1 t)) \to -(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T v2 (THead (Flat Appl) -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 -z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: -T).(eq T v2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T v2 (THead (Bind -b) v3 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda -(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3))))))))))) H9 (THead (Bind Abst) u t0) H7) in -(eq_ind T (THead (Bind Abst) u t0) (\lambda (t: T).(or3 (ex3_2 T T (\lambda -(u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) (THead (Flat Appl) -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 -z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: -T).(eq T (THead (Bind Abbr) v2 t2) (THead (Bind Abbr) u2 t3)))))) (\lambda -(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 -t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) -(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(eq T t (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v3: T).(\lambda -(t3: T).(eq T (THead (Bind Abbr) v2 t2) (THead (Bind b) v3 (THead (Flat Appl) -(lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 -t3)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T -(THead (Bind Abbr) v2 t2) (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 (THead -(Bind Abst) u t0) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) u t0) (THead -(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) (THead (Bind Abbr) u2 -t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: -T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B -b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) u t0) (THead -(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T (THead (Bind -Abbr) v2 t2) (THead (Bind b) v3 (THead (Flat Appl) (lift (S O) O u2) -t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda -(y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 -y1 v3))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))))) (ex4_4_intro T T T T -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind Abst) u t0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) -v2 t2) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))) u t0 v2 t2 -(refl_equal T (THead (Bind Abst) u t0)) (refl_equal T (THead (Bind Abbr) v2 -t2)) H10 H3)) t1 H7))))))) H6)))))))))))) (\lambda (b: B).(\lambda (H1: (not -(eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H2: (pr0 v1 -v2)).(\lambda (H3: (((eq T v1 (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T -(\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Flat Appl) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: -T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind -Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T (\lambda (b0: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (v3: T).(\lambda (t2: T).(eq T v2 (THead (Bind b0) v3 (THead -(Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 -u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(t2: T).(pr0 z1 t2)))))))))))).(\lambda (u0: T).(\lambda (u2: T).(\lambda -(H4: (pr0 u0 u2)).(\lambda (H5: (((eq T u0 (THead (Flat Appl) u1 t1)) \to -(or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) -u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: -T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 -z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: -T).(eq T u2 (THead (Bind Abbr) u3 t2)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T -(\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t2: T).(eq T u2 (THead (Bind -b0) v3 (THead (Flat Appl) (lift (S O) O u3) t2))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda -(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (t2: T).(pr0 z1 t2)))))))))))).(\lambda (t0: T).(\lambda (t2: -T).(\lambda (H6: (pr0 t0 t2)).(\lambda (H7: (((eq T t0 (THead (Flat Appl) u1 -t1)) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead -(Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda -(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda -(t3: T).(eq T t2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T -(\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T t2 (THead (Bind -b0) v3 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda -(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3)))))))))))).(\lambda (H8: (eq T (THead (Flat -Appl) v1 (THead (Bind b) u0 t0)) (THead (Flat Appl) u1 t1))).(let H9 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t _) -\Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead (Flat -Appl) u1 t1) H8) in ((let H10 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow (THead (Bind b) u0 t0) -| (TLRef _) \Rightarrow (THead (Bind b) u0 t0) | (THead _ _ t) \Rightarrow -t])) (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead (Flat Appl) u1 t1) -H8) in (\lambda (H11: (eq T v1 u1)).(let H12 \def (eq_ind T v1 (\lambda (t: -T).((eq T t (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T (\lambda (u3: -T).(\lambda (t3: T).(eq T v2 (THead (Flat Appl) u3 t3)))) (\lambda (u3: -T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 -t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T v2 (THead (Bind -Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda -(_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b0: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T v2 (THead (Bind b0) v3 (THead -(Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 -u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(t3: T).(pr0 z1 t3))))))))))) H3 u1 H11) in (let H13 \def (eq_ind T v1 -(\lambda (t: T).(pr0 t v2)) H2 u1 H11) in (let H14 \def (eq_ind_r T t1 -(\lambda (t: T).((eq T t0 (THead (Flat Appl) u1 t)) \to (or3 (ex3_2 T T -(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u3 t3)))) -(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind -Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda -(_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b0: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind -b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T t2 (THead (Bind b0) v3 (THead -(Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 -u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(t3: T).(pr0 z1 t3))))))))))) H7 (THead (Bind b) u0 t0) H10) in (let H15 \def -(eq_ind_r T t1 (\lambda (t: T).((eq T u0 (THead (Flat Appl) u1 t)) \to (or3 -(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead (Flat Appl) u3 -t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 -z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: -T).(eq T u2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T -(\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t -(THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T u2 (THead (Bind -b0) v3 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda -(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3))))))))))) H5 (THead (Bind b) u0 t0) H10) in -(let H16 \def (eq_ind_r T t1 (\lambda (t: T).((eq T u1 (THead (Flat Appl) u1 -t)) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T v2 (THead -(Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda -(_: T).(\lambda (t3: T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda -(t3: T).(eq T v2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T -(\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t -(THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T v2 (THead (Bind -b0) v3 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda -(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3))))))))))) H12 (THead (Bind b) u0 t0) H10) in -(eq_ind T (THead (Bind b) u0 t0) (\lambda (t: T).(or3 (ex3_2 T T (\lambda -(u3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t2)) (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: -T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t t3)))) (ex4_4 T T T -T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: -T).(\lambda (t3: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O -v2) t2)) (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T -(\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t -(THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Bind b0) v3 (THead (Flat -Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 -t3)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T -(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Flat -Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: -T).(\lambda (t3: T).(pr0 (THead (Bind b) u0 t0) t3)))) (ex4_4 T T T T -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind b) u0 t0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Bind Abbr) u3 t3)))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 -u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: -T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B -b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) u0 t0) (THead -(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Bind b0) v3 (THead (Flat -Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 -t3)))))))) (ex6_6_intro B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 -Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) u0 t0) (THead (Bind -b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t2)) (THead (Bind b0) v3 (THead (Flat Appl) -(lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 -t3))))))) b u0 t0 v2 u2 t2 H1 (refl_equal T (THead (Bind b) u0 t0)) -(refl_equal T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2))) -H13 H4 H6)) t1 H10)))))))) H9))))))))))))))))) (\lambda (u0: T).(\lambda (u2: -T).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (((eq T u0 (THead (Flat Appl) u1 -t1)) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead -(Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda -(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda -(t2: T).(eq T u2 (THead (Bind Abbr) u3 t2)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t2: T).(eq T u2 (THead (Bind -b) v2 (THead (Flat Appl) (lift (S O) O u3) t2))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda -(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (t2: T).(pr0 z1 t2)))))))))))).(\lambda (t0: T).(\lambda (t2: -T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u1 -t1)) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead -(Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda -(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda -(t3: T).(eq T t2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -b) v2 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda -(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3)))))))))))).(\lambda (w: T).(\lambda (_: -(subst0 O u2 t2 w)).(\lambda (H6: (eq T (THead (Bind Abbr) u0 t0) (THead -(Flat Appl) u1 t1))).(let H7 \def (eq_ind T (THead (Bind Abbr) u0 t0) -(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow -(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | -(Flat _) \Rightarrow False])])) I (THead (Flat Appl) u1 t1) H6) in (False_ind -(or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 -w) (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 -u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead -(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: -T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u3 -t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: -T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B -b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda -(v2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 w) (THead (Bind b) v2 -(THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 -u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(t3: T).(pr0 z1 t3))))))))) H7))))))))))))) (\lambda (b: B).(\lambda (_: (not -(eq B b Abst))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 -t2)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b0: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b0) v2 (THead -(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 -u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(t3: T).(pr0 z1 t3)))))))))))).(\lambda (u: T).(\lambda (H4: (eq T (THead -(Bind b) u (lift (S O) O t0)) (THead (Flat Appl) u1 t1))).(let H5 \def -(eq_ind T (THead (Bind b) u (lift (S O) O t0)) (\lambda (ee: T).(match ee in -T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow -False])])) I (THead (Flat Appl) u1 t1) H4) in (False_ind (or3 (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b0: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b0) v2 (THead -(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 -u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(t3: T).(pr0 z1 t3))))))))) H5)))))))))) (\lambda (t0: T).(\lambda (t2: -T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u1 -t1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead -(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda -(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda -(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3)))))))))))).(\lambda (u: T).(\lambda (H3: (eq -T (THead (Flat Cast) u t0) (THead (Flat Appl) u1 t1))).(let H4 \def (eq_ind T -(THead (Flat Cast) u t0) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f in F return -(\lambda (_: F).Prop) with [Appl \Rightarrow False | Cast \Rightarrow -True])])])) I (THead (Flat Appl) u1 t1) H3) in (False_ind (or3 (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead -(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 -u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(t3: T).(pr0 z1 t3))))))))) H4)))))))) y x H0))) H)))). -(* COMMENTS -Initial nodes: 12299 -END *) - -theorem pr0_gen_cast: - \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Flat Cast) u1 -t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead -(Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda -(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 x))))) -\def - \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead -(Flat Cast) u1 t1) x)).(insert_eq T (THead (Flat Cast) u1 t1) (\lambda (t: -T).(pr0 t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 -u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 x))) -(\lambda (y: T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: T).(\lambda -(t0: T).((eq T t (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T t0 (THead (Flat Cast) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 -t2)))) (pr0 t1 t0))))) (\lambda (t: T).(\lambda (H1: (eq T t (THead (Flat -Cast) u1 t1))).(let H2 \def (f_equal T T (\lambda (e: T).e) t (THead (Flat -Cast) u1 t1) H1) in (eq_ind_r T (THead (Flat Cast) u1 t1) (\lambda (t0: -T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Flat -Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: -T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 t0))) (or_introl (ex3_2 T T -(\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Cast) u1 t1) (THead -(Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda -(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (THead (Flat Cast) u1 t1)) -(ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Cast) -u1 t1) (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 -u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))) u1 t1 (refl_equal T -(THead (Flat Cast) u1 t1)) (pr0_refl u1) (pr0_refl t1))) t H2)))) (\lambda -(u0: T).(\lambda (u2: T).(\lambda (H1: (pr0 u0 u2)).(\lambda (H2: (((eq T u0 -(THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: -T).(eq T u2 (THead (Flat Cast) u3 t2)))) (\lambda (u3: T).(\lambda (_: -T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 -u2))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H3: (pr0 t0 t2)).(\lambda -(H4: (((eq T t0 (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: -T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u3 t3)))) (\lambda (u3: -T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 -t3)))) (pr0 t1 t2))))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u0 t0) -(THead (Flat Cast) u1 t1))).(let H6 \def (f_equal T K (\lambda (e: T).(match -e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) -\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u0 t0) (THead (Flat -Cast) u1 t1) H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) -\Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead k u0 t0) (THead (Flat -Cast) u1 t1) H5) in ((let H8 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) -\Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k u0 t0) (THead (Flat -Cast) u1 t1) H5) in (\lambda (H9: (eq T u0 u1)).(\lambda (H10: (eq K k (Flat -Cast))).(eq_ind_r K (Flat Cast) (\lambda (k0: K).(or (ex3_2 T T (\lambda (u3: -T).(\lambda (t3: T).(eq T (THead k0 u2 t2) (THead (Flat Cast) u3 t3)))) -(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3)))) (pr0 t1 (THead k0 u2 t2)))) (let H11 \def (eq_ind T t0 -(\lambda (t: T).((eq T t (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T -(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u3 t3)))) -(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3)))) (pr0 t1 t2)))) H4 t1 H8) in (let H12 \def (eq_ind T t0 -(\lambda (t: T).(pr0 t t2)) H3 t1 H8) in (let H13 \def (eq_ind T u0 (\lambda -(t: T).((eq T t (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: -T).(\lambda (t3: T).(eq T u2 (THead (Flat Cast) u3 t3)))) (\lambda (u3: -T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 -t3)))) (pr0 t1 u2)))) H2 u1 H9) in (let H14 \def (eq_ind T u0 (\lambda (t: -T).(pr0 t u2)) H1 u1 H9) in (or_introl (ex3_2 T T (\lambda (u3: T).(\lambda -(t3: T).(eq T (THead (Flat Cast) u2 t2) (THead (Flat Cast) u3 t3)))) (\lambda -(u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 -t1 t3)))) (pr0 t1 (THead (Flat Cast) u2 t2)) (ex3_2_intro T T (\lambda (u3: -T).(\lambda (t3: T).(eq T (THead (Flat Cast) u2 t2) (THead (Flat Cast) u3 -t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t1 t3))) u2 t2 (refl_equal T (THead (Flat Cast) u2 -t2)) H14 H12)))))) k H10)))) H7)) H6)))))))))))) (\lambda (u: T).(\lambda -(v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 -(THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T v2 (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 -v2))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda -(_: (((eq T t0 (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 -t3)))) (pr0 t1 t2))))).(\lambda (H5: (eq T (THead (Flat Appl) v1 (THead (Bind -Abst) u t0)) (THead (Flat Cast) u1 t1))).(let H6 \def (eq_ind T (THead (Flat -Appl) v1 (THead (Bind Abst) u t0)) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f -in F return (\lambda (_: F).Prop) with [Appl \Rightarrow True | Cast -\Rightarrow False])])])) I (THead (Flat Cast) u1 t1) H5) in (False_ind (or -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) -(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) -(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (THead (Bind Abbr) v2 -t2))) H6)))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b -Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda -(_: (((eq T v1 (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T v2 (THead (Flat Cast) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 -t2)))) (pr0 t1 v2))))).(\lambda (u0: T).(\lambda (u2: T).(\lambda (_: (pr0 u0 -u2)).(\lambda (_: (((eq T u0 (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T -(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Cast) u3 t2)))) -(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: -T).(pr0 t1 t2)))) (pr0 t1 u2))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda -(_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Flat Cast) u1 t1)) \to (or -(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u3 -t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2))))).(\lambda (H8: (eq T (THead -(Flat Appl) v1 (THead (Bind b) u0 t0)) (THead (Flat Cast) u1 t1))).(let H9 -\def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K -return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) -\Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl \Rightarrow -True | Cast \Rightarrow False])])])) I (THead (Flat Cast) u1 t1) H8) in -(False_ind (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Flat Cast) u3 -t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t2)))) H9))))))))))))))))) (\lambda (u0: T).(\lambda -(u2: T).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (((eq T u0 (THead (Flat Cast) -u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead -(Flat Cast) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda -(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 u2))))).(\lambda (t0: -T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead -(Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq -T t2 (THead (Flat Cast) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 -u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 -t2))))).(\lambda (w: T).(\lambda (_: (subst0 O u2 t2 w)).(\lambda (H6: (eq T -(THead (Bind Abbr) u0 t0) (THead (Flat Cast) u1 t1))).(let H7 \def (eq_ind T -(THead (Bind Abbr) u0 t0) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat -Cast) u1 t1) H6) in (False_ind (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: -T).(eq T (THead (Bind Abbr) u2 w) (THead (Flat Cast) u3 t3)))) (\lambda (u3: -T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 -t3)))) (pr0 t1 (THead (Bind Abbr) u2 w))) H7))))))))))))) (\lambda (b: -B).(\lambda (_: (not (eq B b Abst))).(\lambda (t0: T).(\lambda (t2: -T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Flat Cast) u1 -t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead -(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda -(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2))))).(\lambda (u: -T).(\lambda (H4: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead (Flat -Cast) u1 t1))).(let H5 \def (eq_ind T (THead (Bind b) u (lift (S O) O t0)) -(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow -(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | -(Flat _) \Rightarrow False])])) I (THead (Flat Cast) u1 t1) H4) in (False_ind -(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2)) H5)))))))))) (\lambda (t0: -T).(\lambda (t2: T).(\lambda (H1: (pr0 t0 t2)).(\lambda (H2: (((eq T t0 -(THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: -T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 -t2))))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u t0) (THead -(Flat Cast) u1 t1))).(let H4 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) -\Rightarrow u | (THead _ t _) \Rightarrow t])) (THead (Flat Cast) u t0) -(THead (Flat Cast) u1 t1) H3) in ((let H5 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | -(TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Flat Cast) -u t0) (THead (Flat Cast) u1 t1) H3) in (\lambda (_: (eq T u u1)).(let H7 \def -(eq_ind T t0 (\lambda (t: T).((eq T t (THead (Flat Cast) u1 t1)) \to (or -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2)))) H2 t1 H5) in (let H8 \def -(eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H1 t1 H5) in (or_intror (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3)))) (pr0 t1 t2) H8))))) H4)))))))) y x H0))) H)))). -(* COMMENTS -Initial nodes: 2911 -END *) - -theorem pr0_gen_abbr: - \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Bind Abbr) u1 -t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead -(Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda -(u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y: T).(pr0 t1 y)) -(\lambda (y: T).(subst0 O u2 y t2))))))) (pr0 t1 (lift (S O) O x)))))) -\def - \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead -(Bind Abbr) u1 t1) x)).(insert_eq T (THead (Bind Abbr) u1 t1) (\lambda (t: -T).(pr0 t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 -u1 u2))) (\lambda (u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda -(y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y t2))))))) (pr0 t1 (lift (S -O) O x)))) (\lambda (y: T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: -T).(\lambda (t0: T).((eq T t (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T -(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: -T).(or (pr0 t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: -T).(subst0 O u2 y0 t2))))))) (pr0 t1 (lift (S O) O t0)))))) (\lambda (t: -T).(\lambda (H1: (eq T t (THead (Bind Abbr) u1 t1))).(let H2 \def (f_equal T -T (\lambda (e: T).e) t (THead (Bind Abbr) u1 t1) H1) in (eq_ind_r T (THead -(Bind Abbr) u1 t1) (\lambda (t0: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda -(t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T -(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 t2))))))) (pr0 -t1 (lift (S O) O t0)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T (THead (Bind Abbr) u1 t1) (THead (Bind Abbr) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: T).(or (pr0 -t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 -t2))))))) (pr0 t1 (lift (S O) O (THead (Bind Abbr) u1 t1))) (ex3_2_intro T T -(\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Bind Abbr) u1 t1) (THead -(Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda -(u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) -(\lambda (y0: T).(subst0 O u2 y0 t2)))))) u1 t1 (refl_equal T (THead (Bind -Abbr) u1 t1)) (pr0_refl u1) (or_introl (pr0 t1 t1) (ex2 T (\lambda (y0: -T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u1 y0 t1))) (pr0_refl t1)))) t -H2)))) (\lambda (u0: T).(\lambda (u2: T).(\lambda (H1: (pr0 u0 u2)).(\lambda -(H2: (((eq T u0 (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: -T).(\lambda (t2: T).(eq T u2 (THead (Bind Abbr) u3 t2)))) (\lambda (u3: -T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t2: T).(or (pr0 -t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 -t2))))))) (pr0 t1 (lift (S O) O u2)))))).(\lambda (t0: T).(\lambda (t2: -T).(\lambda (H3: (pr0 t0 t2)).(\lambda (H4: (((eq T t0 (THead (Bind Abbr) u1 -t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead -(Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda -(u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) -(\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O -t2)))))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u0 t0) (THead (Bind -Abbr) u1 t1))).(let H6 \def (f_equal T K (\lambda (e: T).(match e in T return -(\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | -(THead k0 _ _) \Rightarrow k0])) (THead k u0 t0) (THead (Bind Abbr) u1 t1) -H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 -| (THead _ t _) \Rightarrow t])) (THead k u0 t0) (THead (Bind Abbr) u1 t1) -H5) in ((let H8 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 -| (THead _ _ t) \Rightarrow t])) (THead k u0 t0) (THead (Bind Abbr) u1 t1) -H5) in (\lambda (H9: (eq T u0 u1)).(\lambda (H10: (eq K k (Bind -Abbr))).(eq_ind_r K (Bind Abbr) (\lambda (k0: K).(or (ex3_2 T T (\lambda (u3: -T).(\lambda (t3: T).(eq T (THead k0 u2 t2) (THead (Bind Abbr) u3 t3)))) -(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: -T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: -T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O (THead k0 u2 t2))))) (let -H11 \def (eq_ind T t0 (\lambda (t: T).((eq T t (THead (Bind Abbr) u1 t1)) \to -(or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) -u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: -T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) -(\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O t2))))) H4 -t1 H8) in (let H12 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H3 t1 H8) in -(let H13 \def (eq_ind T u0 (\lambda (t: T).((eq T t (THead (Bind Abbr) u1 -t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead -(Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda -(u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) -(\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O u2))))) H2 -u1 H9) in (let H14 \def (eq_ind T u0 (\lambda (t: T).(pr0 t u2)) H1 u1 H9) in -(or_introl (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind -Abbr) u2 t2) (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: -T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T -(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 -t1 (lift (S O) O (THead (Bind Abbr) u2 t2))) (ex3_2_intro T T (\lambda (u3: -T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 t2) (THead (Bind Abbr) u3 -t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: -T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) -(\lambda (y0: T).(subst0 O u3 y0 t3)))))) u2 t2 (refl_equal T (THead (Bind -Abbr) u2 t2)) H14 (or_introl (pr0 t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) -(\lambda (y0: T).(subst0 O u2 y0 t2))) H12))))))) k H10)))) H7)) -H6)))))))))))) (\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: -(pr0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind Abbr) u1 t1)) \to (or -(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind Abbr) u2 -t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: -T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) -(\lambda (y0: T).(subst0 O u2 y0 t2))))))) (pr0 t1 (lift (S O) O -v2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda -(_: (((eq T t0 (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0 -t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 -t3))))))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (H5: (eq T (THead (Flat -Appl) v1 (THead (Bind Abst) u t0)) (THead (Bind Abbr) u1 t1))).(let H6 \def -(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K -return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) -\Rightarrow True])])) I (THead (Bind Abbr) u1 t1) H5) in (False_ind (or -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) -(THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) -(\lambda (u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: -T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S -O) O (THead (Bind Abbr) v2 t2)))) H6)))))))))))) (\lambda (b: B).(\lambda (_: -(not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 -v2)).(\lambda (_: (((eq T v1 (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T -(\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind Abbr) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: -T).(or (pr0 t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: -T).(subst0 O u2 y0 t2))))))) (pr0 t1 (lift (S O) O v2)))))).(\lambda (u0: -T).(\lambda (u2: T).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (((eq T u0 (THead -(Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq -T u2 (THead (Bind Abbr) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 -u3))) (\lambda (u3: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y0: -T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 t2))))))) (pr0 t1 (lift (S -O) O u2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 -t2)).(\lambda (_: (((eq T t0 (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T -(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u3 t3)))) -(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: -T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: -T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (H8: (eq -T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead (Bind Abbr) u1 -t1))).(let H9 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) -(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow -(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | -(Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u1 t1) H8) in (False_ind -(or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Bind Abbr) u3 t3)))) -(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: -T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: -T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t2))))) H9))))))))))))))))) (\lambda (u0: -T).(\lambda (u2: T).(\lambda (H1: (pr0 u0 u2)).(\lambda (H2: (((eq T u0 -(THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: -T).(eq T u2 (THead (Bind Abbr) u3 t2)))) (\lambda (u3: T).(\lambda (_: -T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T -(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 t2))))))) (pr0 -t1 (lift (S O) O u2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H3: -(pr0 t0 t2)).(\lambda (H4: (((eq T t0 (THead (Bind Abbr) u1 t1)) \to (or -(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u3 -t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: -T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) -(\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O -t2)))))).(\lambda (w: T).(\lambda (H5: (subst0 O u2 t2 w)).(\lambda (H6: (eq -T (THead (Bind Abbr) u0 t0) (THead (Bind Abbr) u1 t1))).(let H7 \def (f_equal -T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) -(THead (Bind Abbr) u0 t0) (THead (Bind Abbr) u1 t1) H6) in ((let H8 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) -\Rightarrow t])) (THead (Bind Abbr) u0 t0) (THead (Bind Abbr) u1 t1) H6) in -(\lambda (H9: (eq T u0 u1)).(let H10 \def (eq_ind T t0 (\lambda (t: T).((eq T -t (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda -(t3: T).(eq T t2 (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: -T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T -(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 -t1 (lift (S O) O t2))))) H4 t1 H8) in (let H11 \def (eq_ind T t0 (\lambda (t: -T).(pr0 t t2)) H3 t1 H8) in (let H12 \def (eq_ind T u0 (\lambda (t: T).((eq T -t (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda -(t3: T).(eq T u2 (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: -T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T -(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 -t1 (lift (S O) O u2))))) H2 u1 H9) in (let H13 \def (eq_ind T u0 (\lambda (t: -T).(pr0 t u2)) H1 u1 H9) in (or_introl (ex3_2 T T (\lambda (u3: T).(\lambda -(t3: T).(eq T (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u3 t3)))) (\lambda -(u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or -(pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O -u3 y0 t3))))))) (pr0 t1 (lift (S O) O (THead (Bind Abbr) u2 w))) (ex3_2_intro -T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 w) (THead -(Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda -(u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) -(\lambda (y0: T).(subst0 O u3 y0 t3)))))) u2 w (refl_equal T (THead (Bind -Abbr) u2 w)) H13 (or_intror (pr0 t1 w) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) -(\lambda (y0: T).(subst0 O u2 y0 w))) (ex_intro2 T (\lambda (y0: T).(pr0 t1 -y0)) (\lambda (y0: T).(subst0 O u2 y0 w)) t2 H11 H5)))))))))) H7))))))))))))) -(\lambda (b: B).(\lambda (H1: (not (eq B b Abst))).(\lambda (t0: T).(\lambda -(t2: T).(\lambda (H2: (pr0 t0 t2)).(\lambda (H3: (((eq T t0 (THead (Bind -Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 -(THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) -(\lambda (u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: -T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S -O) O t2)))))).(\lambda (u: T).(\lambda (H4: (eq T (THead (Bind b) u (lift (S -O) O t0)) (THead (Bind Abbr) u1 t1))).(let H5 \def (f_equal T B (\lambda (e: -T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | -(TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k in K return -(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow -b])])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abbr) u1 t1) H4) in -((let H6 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: -T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) -\Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abbr) u1 -t1) H4) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat -\to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow -(TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true -\Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t3) \Rightarrow -(THead k (lref_map f d u0) (lref_map f (s k d) t3))]) in lref_map) (\lambda -(x0: nat).(plus x0 (S O))) O t0) | (TLRef _) \Rightarrow ((let rec lref_map -(f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) -\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with -[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t3) -\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t3))]) in -lref_map) (\lambda (x0: nat).(plus x0 (S O))) O t0) | (THead _ _ t) -\Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abbr) u1 -t1) H4) in (\lambda (_: (eq T u u1)).(\lambda (H9: (eq B b Abbr)).(let H10 -\def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H1 Abbr H9) in (let -H11 \def (eq_ind_r T t1 (\lambda (t: T).((eq T t0 (THead (Bind Abbr) u1 t)) -\to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: -T).(\lambda (t3: T).(or (pr0 t t3) (ex2 T (\lambda (y0: T).(pr0 t y0)) -(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t (lift (S O) O t2))))) H3 -(lift (S O) O t0) H7) in (eq_ind T (lift (S O) O t0) (\lambda (t: T).(or -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: -T).(\lambda (t3: T).(or (pr0 t t3) (ex2 T (\lambda (y0: T).(pr0 t y0)) -(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t (lift (S O) O t2)))) -(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: -T).(\lambda (t3: T).(or (pr0 (lift (S O) O t0) t3) (ex2 T (\lambda (y0: -T).(pr0 (lift (S O) O t0) y0)) (\lambda (y0: T).(subst0 O u2 y0 t3))))))) -(pr0 (lift (S O) O t0) (lift (S O) O t2)) (pr0_lift t0 t2 H2 (S O) O)) t1 -H7)))))) H6)) H5)))))))))) (\lambda (t0: T).(\lambda (t2: T).(\lambda (_: -(pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Abbr) u1 t1)) \to (or -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: -T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) -(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S O) O -t2)))))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u t0) (THead -(Bind Abbr) u1 t1))).(let H4 \def (eq_ind T (THead (Flat Cast) u t0) (\lambda -(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow -(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | -(Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u1 t1) H3) in (False_ind -(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: -T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) -(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S O) O t2))) -H4)))))))) y x H0))) H)))). -(* COMMENTS -Initial nodes: 4711 -END *) - -theorem pr0_gen_void: - \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Bind Void) u1 -t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead -(Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda -(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O x)))))) -\def - \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead -(Bind Void) u1 t1) x)).(insert_eq T (THead (Bind Void) u1 t1) (\lambda (t: -T).(pr0 t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 -u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) -O x)))) (\lambda (y: T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: -T).(\lambda (t0: T).((eq T t (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T -(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Void) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: -T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O t0)))))) (\lambda (t: T).(\lambda -(H1: (eq T t (THead (Bind Void) u1 t1))).(let H2 \def (f_equal T T (\lambda -(e: T).e) t (THead (Bind Void) u1 t1) H1) in (eq_ind_r T (THead (Bind Void) -u1 t1) (\lambda (t0: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq -T t0 (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 -u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O -t0)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead -(Bind Void) u1 t1) (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 -(lift (S O) O (THead (Bind Void) u1 t1))) (ex3_2_intro T T (\lambda (u2: -T).(\lambda (t2: T).(eq T (THead (Bind Void) u1 t1) (THead (Bind Void) u2 -t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: -T).(\lambda (t2: T).(pr0 t1 t2))) u1 t1 (refl_equal T (THead (Bind Void) u1 -t1)) (pr0_refl u1) (pr0_refl t1))) t H2)))) (\lambda (u0: T).(\lambda (u2: -T).(\lambda (H1: (pr0 u0 u2)).(\lambda (H2: (((eq T u0 (THead (Bind Void) u1 -t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead -(Bind Void) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda -(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O -u2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H3: (pr0 t0 -t2)).(\lambda (H4: (((eq T t0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T -(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u3 t3)))) -(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (k: K).(\lambda -(H5: (eq T (THead k u0 t0) (THead (Bind Void) u1 t1))).(let H6 \def (f_equal -T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) -\Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) -(THead k u0 t0) (THead (Bind Void) u1 t1) H5) in ((let H7 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) -(THead k u0 t0) (THead (Bind Void) u1 t1) H5) in ((let H8 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) -(THead k u0 t0) (THead (Bind Void) u1 t1) H5) in (\lambda (H9: (eq T u0 -u1)).(\lambda (H10: (eq K k (Bind Void))).(eq_ind_r K (Bind Void) (\lambda -(k0: K).(or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead k0 u2 -t2) (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 -u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O -(THead k0 u2 t2))))) (let H11 \def (eq_ind T t0 (\lambda (t: T).((eq T t -(THead (Bind Void) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: -T).(eq T t2 (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: -T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 -(lift (S O) O t2))))) H4 t1 H8) in (let H12 \def (eq_ind T t0 (\lambda (t: -T).(pr0 t t2)) H3 t1 H8) in (let H13 \def (eq_ind T u0 (\lambda (t: T).((eq T -t (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda -(t3: T).(eq T u2 (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: -T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 -(lift (S O) O u2))))) H2 u1 H9) in (let H14 \def (eq_ind T u0 (\lambda (t: -T).(pr0 t u2)) H1 u1 H9) in (or_introl (ex3_2 T T (\lambda (u3: T).(\lambda -(t3: T).(eq T (THead (Bind Void) u2 t2) (THead (Bind Void) u3 t3)))) (\lambda -(u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 -t1 t3)))) (pr0 t1 (lift (S O) O (THead (Bind Void) u2 t2))) (ex3_2_intro T T -(\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Void) u2 t2) (THead -(Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda -(_: T).(\lambda (t3: T).(pr0 t1 t3))) u2 t2 (refl_equal T (THead (Bind Void) -u2 t2)) H14 H12)))))) k H10)))) H7)) H6)))))))))))) (\lambda (u: T).(\lambda -(v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 -(THead (Bind Void) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T v2 (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 -(lift (S O) O v2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 -t2)).(\lambda (_: (((eq T t0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (H5: (eq T (THead -(Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Bind Void) u1 t1))).(let H6 -\def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K -return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) -\Rightarrow True])])) I (THead (Bind Void) u1 t1) H5) in (False_ind (or -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) -(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) -(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O (THead -(Bind Abbr) v2 t2)))) H6)))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B -b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 -v2)).(\lambda (_: (((eq T v1 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T -(\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind Void) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: -T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O v2)))))).(\lambda (u0: T).(\lambda -(u2: T).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (((eq T u0 (THead (Bind Void) -u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead -(Bind Void) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda -(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O -u2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda -(_: (((eq T t0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: -T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u3 t3)))) (\lambda (u3: -T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 -t3)))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (H8: (eq T (THead (Flat Appl) -v1 (THead (Bind b) u0 t0)) (THead (Bind Void) u1 t1))).(let H9 \def (eq_ind T -(THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (\lambda (ee: T).(match ee in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind Void) u1 t1) H8) in (False_ind (or (ex3_2 T T -(\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t2)) (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda -(_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 -(lift (S O) O (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2))))) -H9))))))))))))))))) (\lambda (u0: T).(\lambda (u2: T).(\lambda (_: (pr0 u0 -u2)).(\lambda (_: (((eq T u0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T -(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Bind Void) u3 t2)))) -(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: -T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O u2)))))).(\lambda (t0: T).(\lambda -(t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Void) -u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead -(Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda -(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O -t2)))))).(\lambda (w: T).(\lambda (_: (subst0 O u2 t2 w)).(\lambda (H6: (eq T -(THead (Bind Abbr) u0 t0) (THead (Bind Void) u1 t1))).(let H7 \def (eq_ind T -(THead (Bind Abbr) u0 t0) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr -\Rightarrow True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat -_) \Rightarrow False])])) I (THead (Bind Void) u1 t1) H6) in (False_ind (or -(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 w) -(THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) -(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O (THead -(Bind Abbr) u2 w)))) H7))))))))))))) (\lambda (b: B).(\lambda (H1: (not (eq B -b Abst))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H2: (pr0 t0 -t2)).(\lambda (H3: (((eq T t0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (u: T).(\lambda -(H4: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Void) u1 -t1))).(let H5 \def (f_equal T B (\lambda (e: T).(match e in T return (\lambda -(_: T).B) with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead k -_ _) \Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b0) -\Rightarrow b0 | (Flat _) \Rightarrow b])])) (THead (Bind b) u (lift (S O) O -t0)) (THead (Bind Void) u1 t1) H4) in ((let H6 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | -(TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t])) (THead (Bind b) u -(lift (S O) O t0)) (THead (Bind Void) u1 t1) H4) in ((let H7 \def (f_equal T -T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T -\def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow -(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) -| (THead k u0 t3) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) -t3))]) in lref_map) (\lambda (x0: nat).(plus x0 (S O))) O t0) | (TLRef _) -\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T -\def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow -(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) -| (THead k u0 t3) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) -t3))]) in lref_map) (\lambda (x0: nat).(plus x0 (S O))) O t0) | (THead _ _ t) -\Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Void) u1 -t1) H4) in (\lambda (_: (eq T u u1)).(\lambda (H9: (eq B b Void)).(let H10 -\def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H1 Void H9) in (let -H11 \def (eq_ind_r T t1 (\lambda (t: T).((eq T t0 (THead (Bind Void) u1 t)) -\to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t t3)))) (pr0 t (lift (S O) O t2))))) H3 (lift (S O) -O t0) H7) in (eq_ind T (lift (S O) O t0) (\lambda (t: T).(or (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t t3)))) (pr0 t (lift (S O) O t2)))) (or_intror (ex3_2 T T (\lambda -(u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 (lift -(S O) O t0) t3)))) (pr0 (lift (S O) O t0) (lift (S O) O t2)) (pr0_lift t0 t2 -H2 (S O) O)) t1 H7)))))) H6)) H5)))))))))) (\lambda (t0: T).(\lambda (t2: -T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Void) u1 -t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead -(Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda -(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O -t2)))))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u t0) (THead -(Bind Void) u1 t1))).(let H4 \def (eq_ind T (THead (Flat Cast) u t0) (\lambda -(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow -(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | -(Flat _) \Rightarrow True])])) I (THead (Bind Void) u1 t1) H3) in (False_ind -(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2))) H4)))))))) y x -H0))) H)))). -(* COMMENTS -Initial nodes: 3436 -END *) - -theorem pr0_gen_lift: - \forall (t1: T).(\forall (x: T).(\forall (h: nat).(\forall (d: nat).((pr0 -(lift h d t1) x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda -(t2: T).(pr0 t1 t2))))))) -\def - \lambda (t1: T).(\lambda (x: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda -(H: (pr0 (lift h d t1) x)).(insert_eq T (lift h d t1) (\lambda (t: T).(pr0 t -x)) (\lambda (_: T).(ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda -(t2: T).(pr0 t1 t2)))) (\lambda (y: T).(\lambda (H0: (pr0 y x)).(unintro nat -d (\lambda (n: nat).((eq T y (lift h n t1)) \to (ex2 T (\lambda (t2: T).(eq T -x (lift h n t2))) (\lambda (t2: T).(pr0 t1 t2))))) (unintro T t1 (\lambda (t: -T).(\forall (x0: nat).((eq T y (lift h x0 t)) \to (ex2 T (\lambda (t2: T).(eq -T x (lift h x0 t2))) (\lambda (t2: T).(pr0 t t2)))))) (pr0_ind (\lambda (t: -T).(\lambda (t0: T).(\forall (x0: T).(\forall (x1: nat).((eq T t (lift h x1 -x0)) \to (ex2 T (\lambda (t2: T).(eq T t0 (lift h x1 t2))) (\lambda (t2: -T).(pr0 x0 t2)))))))) (\lambda (t: T).(\lambda (x0: T).(\lambda (x1: -nat).(\lambda (H1: (eq T t (lift h x1 x0))).(ex_intro2 T (\lambda (t2: T).(eq -T t (lift h x1 t2))) (\lambda (t2: T).(pr0 x0 t2)) x0 H1 (pr0_refl x0)))))) -(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda (H2: -((\forall (x0: T).(\forall (x1: nat).((eq T u1 (lift h x1 x0)) \to (ex2 T -(\lambda (t2: T).(eq T u2 (lift h x1 t2))) (\lambda (t2: T).(pr0 x0 -t2)))))))).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2 -t3)).(\lambda (H4: ((\forall (x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 -x0)) \to (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: -T).(pr0 x0 t4)))))))).(\lambda (k: K).(\lambda (x0: T).(\lambda (x1: -nat).(\lambda (H5: (eq T (THead k u1 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(Bind Abbr) u2 w) (lift h x1 -t4))) (\lambda (t4: T).(pr0 t t4)))) (ex2_ind T (\lambda (t4: T).(eq T t3 -(lift h (S x1) t4))) (\lambda (t4: T).(pr0 x3 t4)) (ex2 T (\lambda (t4: -T).(eq T (THead (Bind Abbr) u2 w) (lift h x1 t4))) (\lambda (t4: T).(pr0 -(THead (Bind Abbr) x2 x3) t4))) (\lambda (x4: T).(\lambda (H_x: (eq T t3 -(lift h (S x1) x4))).(\lambda (H10: (pr0 x3 x4)).(let H11 \def (eq_ind T t3 -(\lambda (t: T).(subst0 O u2 t w)) H5 (lift h (S x1) x4) H_x) in (ex2_ind T -(\lambda (t4: T).(eq T u2 (lift h x1 t4))) (\lambda (t4: T).(pr0 x2 t4)) (ex2 -T (\lambda (t4: T).(eq T (THead (Bind Abbr) u2 w) (lift h x1 t4))) (\lambda -(t4: T).(pr0 (THead (Bind Abbr) x2 x3) t4))) (\lambda (x5: T).(\lambda (H_x0: -(eq T u2 (lift h x1 x5))).(\lambda (H12: (pr0 x2 x5)).(eq_ind_r T (lift h x1 -x5) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) t w) -(lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind Abbr) x2 x3) t4)))) (let -H13 \def (eq_ind T u2 (\lambda (t: T).(subst0 O t (lift h (S x1) x4) w)) H11 -(lift h x1 x5) H_x0) in (let H14 \def (refl_equal nat (S (plus O x1))) in -(let H15 \def (eq_ind nat (S x1) (\lambda (n: nat).(subst0 O (lift h x1 x5) -(lift h n x4) w)) H13 (S (plus O x1)) H14) in (ex2_ind T (\lambda (t4: T).(eq -T w (lift h (S (plus O x1)) t4))) (\lambda (t4: T).(subst0 O x5 x4 t4)) (ex2 -T (\lambda (t4: T).(eq T (THead (Bind Abbr) (lift h x1 x5) w) (lift h x1 -t4))) (\lambda (t4: T).(pr0 (THead (Bind Abbr) x2 x3) t4))) (\lambda (x6: -T).(\lambda (H16: (eq T w (lift h (S (plus O x1)) x6))).(\lambda (H17: -(subst0 O x5 x4 x6)).(eq_ind_r T (lift h (S (plus O x1)) x6) (\lambda (t: -T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) (lift h x1 x5) t) (lift h -x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind Abbr) x2 x3) t4)))) (ex_intro2 T -(\lambda (t4: T).(eq T (THead (Bind Abbr) (lift h x1 x5) (lift h (S (plus O -x1)) x6)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind Abbr) x2 x3) -t4)) (THead (Bind Abbr) x5 x6) (sym_eq T (lift h x1 (THead (Bind Abbr) x5 -x6)) (THead (Bind Abbr) (lift h x1 x5) (lift h (S (plus O x1)) x6)) -(lift_bind Abbr x5 x6 h (plus O x1))) (pr0_delta x2 x5 H12 x3 x4 H10 x6 H17)) -w H16)))) (subst0_gen_lift_lt x5 x4 w O h x1 H15))))) u2 H_x0)))) (H2 x2 x1 -H8)))))) (H4 x3 (S x1) H9)) x0 H7)))))) (lift_gen_bind Abbr u1 t2 x0 h x1 -H6))))))))))))))) (\lambda (b: B).(\lambda (H1: (not (eq B b Abst))).(\lambda -(t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2 t3)).(\lambda (H3: ((\forall -(x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to (ex2 T (\lambda (t4: -T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4)))))))).(\lambda (u: -T).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H4: (eq T (THead (Bind b) u -(lift (S O) O t2)) (lift h x1 x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda -(z: T).(eq T x0 (THead (Bind b) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq -T u (lift h x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift (S O) O t2) -(lift h (S x1) z)))) (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) -(\lambda (t4: T).(pr0 x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda -(H5: (eq T x0 (THead (Bind b) x2 x3))).(\lambda (_: (eq T u (lift h x1 -x2))).(\lambda (H7: (eq T (lift (S O) O t2) (lift h (S x1) x3))).(eq_ind_r T -(THead (Bind b) x2 x3) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T t3 (lift -h x1 t4))) (\lambda (t4: T).(pr0 t t4)))) (let H8 \def (eq_ind_r nat (plus (S -O) x1) (\lambda (n: nat).(eq nat (S x1) n)) (refl_equal nat (plus (S O) x1)) -(plus x1 (S O)) (plus_sym x1 (S O))) in (let H9 \def (eq_ind nat (S x1) -(\lambda (n: nat).(eq T (lift (S O) O t2) (lift h n x3))) H7 (plus x1 (S O)) -H8) in (ex2_ind T (\lambda (t4: T).(eq T x3 (lift (S O) O t4))) (\lambda (t4: -T).(eq T t2 (lift h x1 t4))) (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 -t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 x3) t4))) (\lambda (x4: -T).(\lambda (H10: (eq T x3 (lift (S O) O x4))).(\lambda (H11: (eq T t2 (lift -h x1 x4))).(eq_ind_r T (lift (S O) O x4) (\lambda (t: T).(ex2 T (\lambda (t4: -T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 t) -t4)))) (ex2_ind T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: -T).(pr0 x4 t4)) (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda -(t4: T).(pr0 (THead (Bind b) x2 (lift (S O) O x4)) t4))) (\lambda (x5: -T).(\lambda (H_x: (eq T t3 (lift h x1 x5))).(\lambda (H12: (pr0 x4 -x5)).(eq_ind_r T (lift h x1 x5) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T -t (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 (lift (S O) O -x4)) t4)))) (ex_intro2 T (\lambda (t4: T).(eq T (lift h x1 x5) (lift h x1 -t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 (lift (S O) O x4)) t4)) x5 -(refl_equal T (lift h x1 x5)) (pr0_zeta b H1 x4 x5 H12 x2)) t3 H_x)))) (H3 x4 -x1 H11)) x3 H10)))) (lift_gen_lift t2 x3 (S O) h O x1 (le_O_n x1) H9)))) x0 -H5)))))) (lift_gen_bind b u (lift (S O) O t2) x0 h x1 H4)))))))))))) (\lambda -(t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2 t3)).(\lambda (H2: ((\forall -(x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to (ex2 T (\lambda (t4: -T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4)))))))).(\lambda (u: -T).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H3: (eq T (THead (Flat Cast) -u t2) (lift h x1 x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T -x0 (THead (Flat Cast) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u (lift -h x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h x1 z)))) (ex2 T -(\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4))) -(\lambda (x2: T).(\lambda (x3: T).(\lambda (H4: (eq T x0 (THead (Flat Cast) -x2 x3))).(\lambda (_: (eq T u (lift h x1 x2))).(\lambda (H6: (eq T t2 (lift h -x1 x3))).(eq_ind_r T (THead (Flat Cast) x2 x3) (\lambda (t: T).(ex2 T -(\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 t t4)))) -(ex2_ind T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 -x3 t4)) (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: -T).(pr0 (THead (Flat Cast) x2 x3) t4))) (\lambda (x4: T).(\lambda (H_x: (eq T -t3 (lift h x1 x4))).(\lambda (H7: (pr0 x3 x4)).(eq_ind_r T (lift h x1 x4) -(\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T t (lift h x1 t4))) (\lambda -(t4: T).(pr0 (THead (Flat Cast) x2 x3) t4)))) (ex_intro2 T (\lambda (t4: -T).(eq T (lift h x1 x4) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat -Cast) x2 x3) t4)) x4 (refl_equal T (lift h x1 x4)) (pr0_tau x3 x4 H7 x2)) t3 -H_x)))) (H2 x3 x1 H6)) x0 H4)))))) (lift_gen_flat Cast u t2 x0 h x1 -H3)))))))))) y x H0))))) H))))). -(* COMMENTS -Initial nodes: 7569 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr0/pr0.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr0/pr0.ma deleted file mode 100644 index 9a3b397fe..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr0/pr0.ma +++ /dev/null @@ -1,2507 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr0/fwd.ma". - -include "Basic-1/lift/tlt.ma". - -theorem pr0_confluence__pr0_cong_upsilon_refl: - \forall (b: B).((not (eq B b Abst)) \to (\forall (u0: T).(\forall (u3: -T).((pr0 u0 u3) \to (\forall (t4: T).(\forall (t5: T).((pr0 t4 t5) \to -(\forall (u2: T).(\forall (v2: T).(\forall (x: T).((pr0 u2 x) \to ((pr0 v2 x) -\to (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u0 t4)) -t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O -v2) t5)) t))))))))))))))) -\def - \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (u0: T).(\lambda -(u3: T).(\lambda (H0: (pr0 u0 u3)).(\lambda (t4: T).(\lambda (t5: T).(\lambda -(H1: (pr0 t4 t5)).(\lambda (u2: T).(\lambda (v2: T).(\lambda (x: T).(\lambda -(H2: (pr0 u2 x)).(\lambda (H3: (pr0 v2 x)).(ex_intro2 T (\lambda (t: T).(pr0 -(THead (Flat Appl) u2 (THead (Bind b) u0 t4)) t)) (\lambda (t: T).(pr0 (THead -(Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t)) (THead (Bind b) u3 -(THead (Flat Appl) (lift (S O) O x) t5)) (pr0_upsilon b H u2 x H2 u0 u3 H0 t4 -t5 H1) (pr0_comp u3 u3 (pr0_refl u3) (THead (Flat Appl) (lift (S O) O v2) t5) -(THead (Flat Appl) (lift (S O) O x) t5) (pr0_comp (lift (S O) O v2) (lift (S -O) O x) (pr0_lift v2 x H3 (S O) O) t5 t5 (pr0_refl t5) (Flat Appl)) (Bind -b))))))))))))))). -(* COMMENTS -Initial nodes: 257 -END *) - -theorem pr0_confluence__pr0_cong_upsilon_cong: - \forall (b: B).((not (eq B b Abst)) \to (\forall (u2: T).(\forall (v2: -T).(\forall (x: T).((pr0 u2 x) \to ((pr0 v2 x) \to (\forall (t2: T).(\forall -(t5: T).(\forall (x0: T).((pr0 t2 x0) \to ((pr0 t5 x0) \to (\forall (u5: -T).(\forall (u3: T).(\forall (x1: T).((pr0 u5 x1) \to ((pr0 u3 x1) \to (ex2 T -(\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 t2)) t)) -(\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) -t5)) t))))))))))))))))))) -\def - \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (u2: T).(\lambda -(v2: T).(\lambda (x: T).(\lambda (H0: (pr0 u2 x)).(\lambda (H1: (pr0 v2 -x)).(\lambda (t2: T).(\lambda (t5: T).(\lambda (x0: T).(\lambda (H2: (pr0 t2 -x0)).(\lambda (H3: (pr0 t5 x0)).(\lambda (u5: T).(\lambda (u3: T).(\lambda -(x1: T).(\lambda (H4: (pr0 u5 x1)).(\lambda (H5: (pr0 u3 x1)).(ex_intro2 T -(\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 t2)) t)) -(\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) -t5)) t)) (THead (Bind b) x1 (THead (Flat Appl) (lift (S O) O x) x0)) -(pr0_upsilon b H u2 x H0 u5 x1 H4 t2 x0 H2) (pr0_comp u3 x1 H5 (THead (Flat -Appl) (lift (S O) O v2) t5) (THead (Flat Appl) (lift (S O) O x) x0) (pr0_comp -(lift (S O) O v2) (lift (S O) O x) (pr0_lift v2 x H1 (S O) O) t5 x0 H3 (Flat -Appl)) (Bind b))))))))))))))))))). -(* COMMENTS -Initial nodes: 269 -END *) - -theorem pr0_confluence__pr0_cong_upsilon_delta: - (not (eq B Abbr Abst)) \to (\forall (u5: T).(\forall (t2: T).(\forall (w: -T).((subst0 O u5 t2 w) \to (\forall (u2: T).(\forall (v2: T).(\forall (x: -T).((pr0 u2 x) \to ((pr0 v2 x) \to (\forall (t5: T).(\forall (x0: T).((pr0 t2 -x0) \to ((pr0 t5 x0) \to (\forall (u3: T).(\forall (x1: T).((pr0 u5 x1) \to -((pr0 u3 x1) \to (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead -(Bind Abbr) u5 w)) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 (THead -(Flat Appl) (lift (S O) O v2) t5)) t)))))))))))))))))))) -\def - \lambda (H: (not (eq B Abbr Abst))).(\lambda (u5: T).(\lambda (t2: -T).(\lambda (w: T).(\lambda (H0: (subst0 O u5 t2 w)).(\lambda (u2: -T).(\lambda (v2: T).(\lambda (x: T).(\lambda (H1: (pr0 u2 x)).(\lambda (H2: -(pr0 v2 x)).(\lambda (t5: T).(\lambda (x0: T).(\lambda (H3: (pr0 t2 -x0)).(\lambda (H4: (pr0 t5 x0)).(\lambda (u3: T).(\lambda (x1: T).(\lambda -(H5: (pr0 u5 x1)).(\lambda (H6: (pr0 u3 x1)).(or_ind (pr0 w x0) (ex2 T -(\lambda (w2: T).(pr0 w w2)) (\lambda (w2: T).(subst0 O x1 x0 w2))) (ex2 T -(\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abbr) u5 w)) t)) -(\lambda (t: T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O -v2) t5)) t))) (\lambda (H7: (pr0 w x0)).(ex_intro2 T (\lambda (t: T).(pr0 -(THead (Flat Appl) u2 (THead (Bind Abbr) u5 w)) t)) (\lambda (t: T).(pr0 -(THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t)) (THead -(Bind Abbr) x1 (THead (Flat Appl) (lift (S O) O x) x0)) (pr0_upsilon Abbr H -u2 x H1 u5 x1 H5 w x0 H7) (pr0_comp u3 x1 H6 (THead (Flat Appl) (lift (S O) O -v2) t5) (THead (Flat Appl) (lift (S O) O x) x0) (pr0_comp (lift (S O) O v2) -(lift (S O) O x) (pr0_lift v2 x H2 (S O) O) t5 x0 H4 (Flat Appl)) (Bind -Abbr)))) (\lambda (H7: (ex2 T (\lambda (w2: T).(pr0 w w2)) (\lambda (w2: -T).(subst0 O x1 x0 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 w w2)) (\lambda -(w2: T).(subst0 O x1 x0 w2)) (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) -u2 (THead (Bind Abbr) u5 w)) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 -(THead (Flat Appl) (lift (S O) O v2) t5)) t))) (\lambda (x2: T).(\lambda (H8: -(pr0 w x2)).(\lambda (H9: (subst0 O x1 x0 x2)).(ex_intro2 T (\lambda (t: -T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abbr) u5 w)) t)) (\lambda (t: -T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t)) -(THead (Bind Abbr) x1 (THead (Flat Appl) (lift (S O) O x) x2)) (pr0_upsilon -Abbr H u2 x H1 u5 x1 H5 w x2 H8) (pr0_delta u3 x1 H6 (THead (Flat Appl) (lift -(S O) O v2) t5) (THead (Flat Appl) (lift (S O) O x) x0) (pr0_comp (lift (S O) -O v2) (lift (S O) O x) (pr0_lift v2 x H2 (S O) O) t5 x0 H4 (Flat Appl)) -(THead (Flat Appl) (lift (S O) O x) x2) (subst0_snd (Flat Appl) x1 x2 x0 O H9 -(lift (S O) O x))))))) H7)) (pr0_subst0 t2 x0 H3 u5 w O H0 x1 -H5))))))))))))))))))). -(* COMMENTS -Initial nodes: 769 -END *) - -theorem pr0_confluence__pr0_cong_upsilon_zeta: - \forall (b: B).((not (eq B b Abst)) \to (\forall (u0: T).(\forall (u3: -T).((pr0 u0 u3) \to (\forall (u2: T).(\forall (v2: T).(\forall (x0: T).((pr0 -u2 x0) \to ((pr0 v2 x0) \to (\forall (x: T).(\forall (t3: T).(\forall (x1: -T).((pr0 x x1) \to ((pr0 t3 x1) \to (ex2 T (\lambda (t: T).(pr0 (THead (Flat -Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) -(lift (S O) O v2) (lift (S O) O x))) t))))))))))))))))) -\def - \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (u0: T).(\lambda -(u3: T).(\lambda (_: (pr0 u0 u3)).(\lambda (u2: T).(\lambda (v2: T).(\lambda -(x0: T).(\lambda (H1: (pr0 u2 x0)).(\lambda (H2: (pr0 v2 x0)).(\lambda (x: -T).(\lambda (t3: T).(\lambda (x1: T).(\lambda (H3: (pr0 x x1)).(\lambda (H4: -(pr0 t3 x1)).(eq_ind T (lift (S O) O (THead (Flat Appl) v2 x)) (\lambda (t: -T).(ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3) t0)) (\lambda (t0: -T).(pr0 (THead (Bind b) u3 t) t0)))) (ex_intro2 T (\lambda (t: T).(pr0 (THead -(Flat Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (lift (S O) O -(THead (Flat Appl) v2 x))) t)) (THead (Flat Appl) x0 x1) (pr0_comp u2 x0 H1 -t3 x1 H4 (Flat Appl)) (pr0_zeta b H (THead (Flat Appl) v2 x) (THead (Flat -Appl) x0 x1) (pr0_comp v2 x0 H2 x x1 H3 (Flat Appl)) u3)) (THead (Flat Appl) -(lift (S O) O v2) (lift (S O) O x)) (lift_flat Appl v2 x (S O) -O)))))))))))))))). -(* COMMENTS -Initial nodes: 283 -END *) - -theorem pr0_confluence__pr0_cong_delta: - \forall (u3: T).(\forall (t5: T).(\forall (w: T).((subst0 O u3 t5 w) \to -(\forall (u2: T).(\forall (x: T).((pr0 u2 x) \to ((pr0 u3 x) \to (\forall -(t3: T).(\forall (x0: T).((pr0 t3 x0) \to ((pr0 t5 x0) \to (ex2 T (\lambda -(t: T).(pr0 (THead (Bind Abbr) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind -Abbr) u3 w) t)))))))))))))) -\def - \lambda (u3: T).(\lambda (t5: T).(\lambda (w: T).(\lambda (H: (subst0 O u3 -t5 w)).(\lambda (u2: T).(\lambda (x: T).(\lambda (H0: (pr0 u2 x)).(\lambda -(H1: (pr0 u3 x)).(\lambda (t3: T).(\lambda (x0: T).(\lambda (H2: (pr0 t3 -x0)).(\lambda (H3: (pr0 t5 x0)).(or_ind (pr0 w x0) (ex2 T (\lambda (w2: -T).(pr0 w w2)) (\lambda (w2: T).(subst0 O x x0 w2))) (ex2 T (\lambda (t: -T).(pr0 (THead (Bind Abbr) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) -u3 w) t))) (\lambda (H4: (pr0 w x0)).(ex_intro2 T (\lambda (t: T).(pr0 (THead -(Bind Abbr) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w) t)) -(THead (Bind Abbr) x x0) (pr0_comp u2 x H0 t3 x0 H2 (Bind Abbr)) (pr0_comp u3 -x H1 w x0 H4 (Bind Abbr)))) (\lambda (H4: (ex2 T (\lambda (w2: T).(pr0 w w2)) -(\lambda (w2: T).(subst0 O x x0 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 w -w2)) (\lambda (w2: T).(subst0 O x x0 w2)) (ex2 T (\lambda (t: T).(pr0 (THead -(Bind Abbr) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w) t))) -(\lambda (x1: T).(\lambda (H5: (pr0 w x1)).(\lambda (H6: (subst0 O x x0 -x1)).(ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 t3) t)) (\lambda -(t: T).(pr0 (THead (Bind Abbr) u3 w) t)) (THead (Bind Abbr) x x1) (pr0_delta -u2 x H0 t3 x0 H2 x1 H6) (pr0_comp u3 x H1 w x1 H5 (Bind Abbr)))))) H4)) -(pr0_subst0 t5 x0 H3 u3 w O H x H1))))))))))))). -(* COMMENTS -Initial nodes: 409 -END *) - -theorem pr0_confluence__pr0_upsilon_upsilon: - \forall (b: B).((not (eq B b Abst)) \to (\forall (v1: T).(\forall (v2: -T).(\forall (x0: T).((pr0 v1 x0) \to ((pr0 v2 x0) \to (\forall (u1: -T).(\forall (u2: T).(\forall (x1: T).((pr0 u1 x1) \to ((pr0 u2 x1) \to -(\forall (t1: T).(\forall (t2: T).(\forall (x2: T).((pr0 t1 x2) \to ((pr0 t2 -x2) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u1 (THead (Flat Appl) -(lift (S O) O v1) t1)) t)) (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t2)) t))))))))))))))))))) -\def - \lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda -(v2: T).(\lambda (x0: T).(\lambda (H0: (pr0 v1 x0)).(\lambda (H1: (pr0 v2 -x0)).(\lambda (u1: T).(\lambda (u2: T).(\lambda (x1: T).(\lambda (H2: (pr0 u1 -x1)).(\lambda (H3: (pr0 u2 x1)).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(x2: T).(\lambda (H4: (pr0 t1 x2)).(\lambda (H5: (pr0 t2 x2)).(ex_intro2 T -(\lambda (t: T).(pr0 (THead (Bind b) u1 (THead (Flat Appl) (lift (S O) O v1) -t1)) t)) (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t2)) t)) (THead (Bind b) x1 (THead (Flat Appl) (lift (S O) O x0) -x2)) (pr0_comp u1 x1 H2 (THead (Flat Appl) (lift (S O) O v1) t1) (THead (Flat -Appl) (lift (S O) O x0) x2) (pr0_comp (lift (S O) O v1) (lift (S O) O x0) -(pr0_lift v1 x0 H0 (S O) O) t1 x2 H4 (Flat Appl)) (Bind b)) (pr0_comp u2 x1 -H3 (THead (Flat Appl) (lift (S O) O v2) t2) (THead (Flat Appl) (lift (S O) O -x0) x2) (pr0_comp (lift (S O) O v2) (lift (S O) O x0) (pr0_lift v2 x0 H1 (S -O) O) t2 x2 H5 (Flat Appl)) (Bind b))))))))))))))))))). -(* COMMENTS -Initial nodes: 347 -END *) - -theorem pr0_confluence__pr0_delta_delta: - \forall (u2: T).(\forall (t3: T).(\forall (w: T).((subst0 O u2 t3 w) \to -(\forall (u3: T).(\forall (t5: T).(\forall (w0: T).((subst0 O u3 t5 w0) \to -(\forall (x: T).((pr0 u2 x) \to ((pr0 u3 x) \to (\forall (x0: T).((pr0 t3 x0) -\to ((pr0 t5 x0) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) -(\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t)))))))))))))))) -\def - \lambda (u2: T).(\lambda (t3: T).(\lambda (w: T).(\lambda (H: (subst0 O u2 -t3 w)).(\lambda (u3: T).(\lambda (t5: T).(\lambda (w0: T).(\lambda (H0: -(subst0 O u3 t5 w0)).(\lambda (x: T).(\lambda (H1: (pr0 u2 x)).(\lambda (H2: -(pr0 u3 x)).(\lambda (x0: T).(\lambda (H3: (pr0 t3 x0)).(\lambda (H4: (pr0 t5 -x0)).(or_ind (pr0 w0 x0) (ex2 T (\lambda (w2: T).(pr0 w0 w2)) (\lambda (w2: -T).(subst0 O x x0 w2))) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) -t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (H5: (pr0 w0 -x0)).(or_ind (pr0 w x0) (ex2 T (\lambda (w2: T).(pr0 w w2)) (\lambda (w2: -T).(subst0 O x x0 w2))) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) -t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (H6: (pr0 w -x0)).(ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda -(t: T).(pr0 (THead (Bind Abbr) u3 w0) t)) (THead (Bind Abbr) x x0) (pr0_comp -u2 x H1 w x0 H6 (Bind Abbr)) (pr0_comp u3 x H2 w0 x0 H5 (Bind Abbr)))) -(\lambda (H6: (ex2 T (\lambda (w2: T).(pr0 w w2)) (\lambda (w2: T).(subst0 O -x x0 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 w w2)) (\lambda (w2: T).(subst0 -O x x0 w2)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda -(t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (x1: T).(\lambda (H7: -(pr0 w x1)).(\lambda (H8: (subst0 O x x0 x1)).(ex_intro2 T (\lambda (t: -T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) -u3 w0) t)) (THead (Bind Abbr) x x1) (pr0_comp u2 x H1 w x1 H7 (Bind Abbr)) -(pr0_delta u3 x H2 w0 x0 H5 x1 H8))))) H6)) (pr0_subst0 t3 x0 H3 u2 w O H x -H1))) (\lambda (H5: (ex2 T (\lambda (w2: T).(pr0 w0 w2)) (\lambda (w2: -T).(subst0 O x x0 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 w0 w2)) (\lambda -(w2: T).(subst0 O x x0 w2)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 -w) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (x1: -T).(\lambda (H6: (pr0 w0 x1)).(\lambda (H7: (subst0 O x x0 x1)).(or_ind (pr0 -w x0) (ex2 T (\lambda (w2: T).(pr0 w w2)) (\lambda (w2: T).(subst0 O x x0 -w2))) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: -T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (H8: (pr0 w x0)).(ex_intro2 T -(\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead -(Bind Abbr) u3 w0) t)) (THead (Bind Abbr) x x1) (pr0_delta u2 x H1 w x0 H8 x1 -H7) (pr0_comp u3 x H2 w0 x1 H6 (Bind Abbr)))) (\lambda (H8: (ex2 T (\lambda -(w2: T).(pr0 w w2)) (\lambda (w2: T).(subst0 O x x0 w2)))).(ex2_ind T -(\lambda (w2: T).(pr0 w w2)) (\lambda (w2: T).(subst0 O x x0 w2)) (ex2 T -(\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead -(Bind Abbr) u3 w0) t))) (\lambda (x2: T).(\lambda (H9: (pr0 w x2)).(\lambda -(H10: (subst0 O x x0 x2)).(or4_ind (eq T x2 x1) (ex2 T (\lambda (t: -T).(subst0 O x x2 t)) (\lambda (t: T).(subst0 O x x1 t))) (subst0 O x x2 x1) -(subst0 O x x1 x2) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) -(\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (H11: (eq T x2 -x1)).(let H12 \def (eq_ind T x2 (\lambda (t: T).(pr0 w t)) H9 x1 H11) in -(ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: -T).(pr0 (THead (Bind Abbr) u3 w0) t)) (THead (Bind Abbr) x x1) (pr0_comp u2 x -H1 w x1 H12 (Bind Abbr)) (pr0_comp u3 x H2 w0 x1 H6 (Bind Abbr))))) (\lambda -(H11: (ex2 T (\lambda (t: T).(subst0 O x x2 t)) (\lambda (t: T).(subst0 O x -x1 t)))).(ex2_ind T (\lambda (t: T).(subst0 O x x2 t)) (\lambda (t: -T).(subst0 O x x1 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) -t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (x3: -T).(\lambda (H12: (subst0 O x x2 x3)).(\lambda (H13: (subst0 O x x1 -x3)).(ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda -(t: T).(pr0 (THead (Bind Abbr) u3 w0) t)) (THead (Bind Abbr) x x3) (pr0_delta -u2 x H1 w x2 H9 x3 H12) (pr0_delta u3 x H2 w0 x1 H6 x3 H13))))) H11)) -(\lambda (H11: (subst0 O x x2 x1)).(ex_intro2 T (\lambda (t: T).(pr0 (THead -(Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t)) -(THead (Bind Abbr) x x1) (pr0_delta u2 x H1 w x2 H9 x1 H11) (pr0_comp u3 x H2 -w0 x1 H6 (Bind Abbr)))) (\lambda (H11: (subst0 O x x1 x2)).(ex_intro2 T -(\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead -(Bind Abbr) u3 w0) t)) (THead (Bind Abbr) x x2) (pr0_comp u2 x H1 w x2 H9 -(Bind Abbr)) (pr0_delta u3 x H2 w0 x1 H6 x2 H11))) (subst0_confluence_eq x0 -x2 x O H10 x1 H7))))) H8)) (pr0_subst0 t3 x0 H3 u2 w O H x H1))))) H5)) -(pr0_subst0 t5 x0 H4 u3 w0 O H0 x H2))))))))))))))). -(* COMMENTS -Initial nodes: 1501 -END *) - -theorem pr0_confluence__pr0_delta_tau: - \forall (u2: T).(\forall (t3: T).(\forall (w: T).((subst0 O u2 t3 w) \to -(\forall (t4: T).((pr0 (lift (S O) O t4) t3) \to (\forall (t2: T).(ex2 T -(\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 t2 -t))))))))) -\def - \lambda (u2: T).(\lambda (t3: T).(\lambda (w: T).(\lambda (H: (subst0 O u2 -t3 w)).(\lambda (t4: T).(\lambda (H0: (pr0 (lift (S O) O t4) t3)).(\lambda -(t2: T).(ex2_ind T (\lambda (t5: T).(eq T t3 (lift (S O) O t5))) (\lambda -(t5: T).(pr0 t4 t5)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) -(\lambda (t: T).(pr0 t2 t))) (\lambda (x: T).(\lambda (H1: (eq T t3 (lift (S -O) O x))).(\lambda (_: (pr0 t4 x)).(let H3 \def (eq_ind T t3 (\lambda (t: -T).(subst0 O u2 t w)) H (lift (S O) O x) H1) in (subst0_gen_lift_false x u2 w -(S O) O O (le_n O) (eq_ind_r nat (plus (S O) O) (\lambda (n: nat).(lt O n)) -(le_n (plus (S O) O)) (plus O (S O)) (plus_sym O (S O))) H3 (ex2 T (\lambda -(t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 t2 t)))))))) -(pr0_gen_lift t4 t3 (S O) O H0)))))))). -(* COMMENTS -Initial nodes: 257 -END *) - -theorem pr0_confluence: - \forall (t0: T).(\forall (t1: T).((pr0 t0 t1) \to (\forall (t2: T).((pr0 t0 -t2) \to (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t))))))) -\def - \lambda (t0: T).(tlt_wf_ind (\lambda (t: T).(\forall (t1: T).((pr0 t t1) \to -(\forall (t2: T).((pr0 t t2) \to (ex2 T (\lambda (t3: T).(pr0 t1 t3)) -(\lambda (t3: T).(pr0 t2 t3)))))))) (\lambda (t: T).(\lambda (H: ((\forall -(v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 -v t2) \to (ex2 T (\lambda (t3: T).(pr0 t1 t3)) (\lambda (t3: T).(pr0 t2 -t3))))))))))).(\lambda (t1: T).(\lambda (H0: (pr0 t t1)).(\lambda (t2: -T).(\lambda (H1: (pr0 t t2)).(let H2 \def (match H0 in pr0 return (\lambda -(t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).((eq T t3 t) \to ((eq T t4 -t1) \to (ex2 T (\lambda (t5: T).(pr0 t1 t5)) (\lambda (t5: T).(pr0 t2 -t5)))))))) with [(pr0_refl t3) \Rightarrow (\lambda (H2: (eq T t3 -t)).(\lambda (H3: (eq T t3 t1)).(eq_ind T t (\lambda (t4: T).((eq T t4 t1) -\to (ex2 T (\lambda (t5: T).(pr0 t1 t5)) (\lambda (t5: T).(pr0 t2 t5))))) -(\lambda (H4: (eq T t t1)).(eq_ind T t1 (\lambda (_: T).(ex2 T (\lambda (t5: -T).(pr0 t1 t5)) (\lambda (t5: T).(pr0 t2 t5)))) (let H5 \def (match H1 in pr0 -return (\lambda (t4: T).(\lambda (t5: T).(\lambda (_: (pr0 t4 t5)).((eq T t4 -t) \to ((eq T t5 t2) \to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: -T).(pr0 t2 t6)))))))) with [(pr0_refl t4) \Rightarrow (\lambda (H5: (eq T t4 -t)).(\lambda (H6: (eq T t4 t2)).(eq_ind T t (\lambda (t5: T).((eq T t5 t2) -\to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6))))) -(\lambda (H7: (eq T t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t6: -T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6)))) (let H8 \def (eq_ind T t -(\lambda (t5: T).(eq T t4 t5)) H5 t2 H7) in (let H9 \def (eq_ind T t (\lambda -(t5: T).(eq T t5 t1)) H4 t2 H7) in (let H10 \def (eq_ind T t (\lambda (t5: -T).(eq T t3 t5)) H2 t2 H7) in (let H11 \def (eq_ind T t (\lambda (t5: -T).(\forall (v: T).((tlt v t5) \to (\forall (t6: T).((pr0 v t6) \to (\forall -(t7: T).((pr0 v t7) \to (ex2 T (\lambda (t8: T).(pr0 t6 t8)) (\lambda (t8: -T).(pr0 t7 t8)))))))))) H t2 H7) in (let H12 \def (eq_ind T t2 (\lambda (t5: -T).(\forall (v: T).((tlt v t5) \to (\forall (t6: T).((pr0 v t6) \to (\forall -(t7: T).((pr0 v t7) \to (ex2 T (\lambda (t8: T).(pr0 t6 t8)) (\lambda (t8: -T).(pr0 t7 t8)))))))))) H11 t1 H9) in (eq_ind_r T t1 (\lambda (t5: T).(ex2 T -(\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t5 t6)))) (let H13 \def -(eq_ind T t2 (\lambda (t5: T).(eq T t3 t5)) H10 t1 H9) in (ex_intro2 T -(\lambda (t5: T).(pr0 t1 t5)) (\lambda (t5: T).(pr0 t1 t5)) t1 (pr0_refl t1) -(pr0_refl t1))) t2 H9)))))) t (sym_eq T t t2 H7))) t4 (sym_eq T t4 t H5) -H6))) | (pr0_comp u1 u2 H5 t4 t5 H6 k) \Rightarrow (\lambda (H7: (eq T (THead -k u1 t4) t)).(\lambda (H8: (eq T (THead k u2 t5) t2)).(eq_ind T (THead k u1 -t4) (\lambda (_: T).((eq T (THead k u2 t5) t2) \to ((pr0 u1 u2) \to ((pr0 t4 -t5) \to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 -t7))))))) (\lambda (H9: (eq T (THead k u2 t5) t2)).(eq_ind T (THead k u2 t5) -(\lambda (t6: T).((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t7: -T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t6 t7)))))) (\lambda (H10: (pr0 u1 -u2)).(\lambda (H11: (pr0 t4 t5)).(let H12 \def (eq_ind_r T t (\lambda (t6: -T).(eq T t6 t1)) H4 (THead k u1 t4) H7) in (eq_ind T (THead k u1 t4) (\lambda -(t6: T).(ex2 T (\lambda (t7: T).(pr0 t6 t7)) (\lambda (t7: T).(pr0 (THead k -u2 t5) t7)))) (let H13 \def (eq_ind_r T t (\lambda (t6: T).(eq T t3 t6)) H2 -(THead k u1 t4) H7) in (let H14 \def (eq_ind_r T t (\lambda (t6: T).(\forall -(v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall (t8: -T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 -t8 t9)))))))))) H (THead k u1 t4) H7) in (ex_intro2 T (\lambda (t6: T).(pr0 -(THead k u1 t4) t6)) (\lambda (t6: T).(pr0 (THead k u2 t5) t6)) (THead k u2 -t5) (pr0_comp u1 u2 H10 t4 t5 H11 k) (pr0_refl (THead k u2 t5))))) t1 H12)))) -t2 H9)) t H7 H8 H5 H6))) | (pr0_beta u v1 v2 H5 t4 t5 H6) \Rightarrow -(\lambda (H7: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t4)) -t)).(\lambda (H8: (eq T (THead (Bind Abbr) v2 t5) t2)).(eq_ind T (THead (Flat -Appl) v1 (THead (Bind Abst) u t4)) (\lambda (_: T).((eq T (THead (Bind Abbr) -v2 t5) t2) \to ((pr0 v1 v2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t7: T).(pr0 -t1 t7)) (\lambda (t7: T).(pr0 t2 t7))))))) (\lambda (H9: (eq T (THead (Bind -Abbr) v2 t5) t2)).(eq_ind T (THead (Bind Abbr) v2 t5) (\lambda (t6: T).((pr0 -v1 v2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda -(t7: T).(pr0 t6 t7)))))) (\lambda (H10: (pr0 v1 v2)).(\lambda (H11: (pr0 t4 -t5)).(let H12 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 t1)) H4 (THead -(Flat Appl) v1 (THead (Bind Abst) u t4)) H7) in (eq_ind T (THead (Flat Appl) -v1 (THead (Bind Abst) u t4)) (\lambda (t6: T).(ex2 T (\lambda (t7: T).(pr0 t6 -t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t5) t7)))) (let H13 \def -(eq_ind_r T t (\lambda (t6: T).(eq T t3 t6)) H2 (THead (Flat Appl) v1 (THead -(Bind Abst) u t4)) H7) in (let H14 \def (eq_ind_r T t (\lambda (t6: -T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall -(t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: -T).(pr0 t8 t9)))))))))) H (THead (Flat Appl) v1 (THead (Bind Abst) u t4)) H7) -in (ex_intro2 T (\lambda (t6: T).(pr0 (THead (Flat Appl) v1 (THead (Bind -Abst) u t4)) t6)) (\lambda (t6: T).(pr0 (THead (Bind Abbr) v2 t5) t6)) (THead -(Bind Abbr) v2 t5) (pr0_beta u v1 v2 H10 t4 t5 H11) (pr0_refl (THead (Bind -Abbr) v2 t5))))) t1 H12)))) t2 H9)) t H7 H8 H5 H6))) | (pr0_upsilon b H5 v1 -v2 H6 u1 u2 H7 t4 t5 H8) \Rightarrow (\lambda (H9: (eq T (THead (Flat Appl) -v1 (THead (Bind b) u1 t4)) t)).(\lambda (H10: (eq T (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t5)) t2)).(eq_ind T (THead (Flat Appl) v1 -(THead (Bind b) u1 t4)) (\lambda (_: T).((eq T (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t5)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1 -v2) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t7: T).(pr0 t1 -t7)) (\lambda (t7: T).(pr0 t2 t7))))))))) (\lambda (H11: (eq T (THead (Bind -b) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) t2)).(eq_ind T (THead (Bind -b) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) (\lambda (t6: T).((not (eq B -b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T -(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t6 t7)))))))) (\lambda -(H12: (not (eq B b Abst))).(\lambda (H13: (pr0 v1 v2)).(\lambda (H14: (pr0 u1 -u2)).(\lambda (H15: (pr0 t4 t5)).(let H16 \def (eq_ind_r T t (\lambda (t6: -T).(eq T t6 t1)) H4 (THead (Flat Appl) v1 (THead (Bind b) u1 t4)) H9) in -(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t4)) (\lambda (t6: T).(ex2 -T (\lambda (t7: T).(pr0 t6 t7)) (\lambda (t7: T).(pr0 (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t5)) t7)))) (let H17 \def (eq_ind_r T t -(\lambda (t6: T).(eq T t3 t6)) H2 (THead (Flat Appl) v1 (THead (Bind b) u1 -t4)) H9) in (let H18 \def (eq_ind_r T t (\lambda (t6: T).(\forall (v: -T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall (t8: T).((pr0 v -t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8 -t9)))))))))) H (THead (Flat Appl) v1 (THead (Bind b) u1 t4)) H9) in -(pr0_confluence__pr0_cong_upsilon_refl b H12 u1 u2 H14 t4 t5 H15 v1 v2 v2 H13 -(pr0_refl v2)))) t1 H16)))))) t2 H11)) t H9 H10 H5 H6 H7 H8))) | (pr0_delta -u1 u2 H5 t4 t5 H6 w H7) \Rightarrow (\lambda (H8: (eq T (THead (Bind Abbr) u1 -t4) t)).(\lambda (H9: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead -(Bind Abbr) u1 t4) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t2) \to -((pr0 u1 u2) \to ((pr0 t4 t5) \to ((subst0 O u2 t5 w) \to (ex2 T (\lambda -(t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)))))))) (\lambda (H10: (eq T -(THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda -(t6: T).((pr0 u1 u2) \to ((pr0 t4 t5) \to ((subst0 O u2 t5 w) \to (ex2 T -(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t6 t7))))))) (\lambda -(H11: (pr0 u1 u2)).(\lambda (H12: (pr0 t4 t5)).(\lambda (H13: (subst0 O u2 t5 -w)).(let H14 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 t1)) H4 (THead -(Bind Abbr) u1 t4) H8) in (eq_ind T (THead (Bind Abbr) u1 t4) (\lambda (t6: -T).(ex2 T (\lambda (t7: T).(pr0 t6 t7)) (\lambda (t7: T).(pr0 (THead (Bind -Abbr) u2 w) t7)))) (let H15 \def (eq_ind_r T t (\lambda (t6: T).(eq T t3 t6)) -H2 (THead (Bind Abbr) u1 t4) H8) in (let H16 \def (eq_ind_r T t (\lambda (t6: -T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall -(t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: -T).(pr0 t8 t9)))))))))) H (THead (Bind Abbr) u1 t4) H8) in (ex_intro2 T -(\lambda (t6: T).(pr0 (THead (Bind Abbr) u1 t4) t6)) (\lambda (t6: T).(pr0 -(THead (Bind Abbr) u2 w) t6)) (THead (Bind Abbr) u2 w) (pr0_delta u1 u2 H11 -t4 t5 H12 w H13) (pr0_refl (THead (Bind Abbr) u2 w))))) t1 H14))))) t2 H10)) -t H8 H9 H5 H6 H7))) | (pr0_zeta b H5 t4 t5 H6 u) \Rightarrow (\lambda (H7: -(eq T (THead (Bind b) u (lift (S O) O t4)) t)).(\lambda (H8: (eq T t5 -t2)).(eq_ind T (THead (Bind b) u (lift (S O) O t4)) (\lambda (_: T).((eq T t5 -t2) \to ((not (eq B b Abst)) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t7: -T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))))))) (\lambda (H9: (eq T t5 -t2)).(eq_ind T t2 (\lambda (t6: T).((not (eq B b Abst)) \to ((pr0 t4 t6) \to -(ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)))))) -(\lambda (H10: (not (eq B b Abst))).(\lambda (H11: (pr0 t4 t2)).(let H12 \def -(eq_ind_r T t (\lambda (t6: T).(eq T t6 t1)) H4 (THead (Bind b) u (lift (S O) -O t4)) H7) in (eq_ind T (THead (Bind b) u (lift (S O) O t4)) (\lambda (t6: -T).(ex2 T (\lambda (t7: T).(pr0 t6 t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let -H13 \def (eq_ind_r T t (\lambda (t6: T).(eq T t3 t6)) H2 (THead (Bind b) u -(lift (S O) O t4)) H7) in (let H14 \def (eq_ind_r T t (\lambda (t6: -T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall -(t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: -T).(pr0 t8 t9)))))))))) H (THead (Bind b) u (lift (S O) O t4)) H7) in -(ex_intro2 T (\lambda (t6: T).(pr0 (THead (Bind b) u (lift (S O) O t4)) t6)) -(\lambda (t6: T).(pr0 t2 t6)) t2 (pr0_zeta b H10 t4 t2 H11 u) (pr0_refl -t2)))) t1 H12)))) t5 (sym_eq T t5 t2 H9))) t H7 H8 H5 H6))) | (pr0_tau t4 t5 -H5 u) \Rightarrow (\lambda (H6: (eq T (THead (Flat Cast) u t4) t)).(\lambda -(H7: (eq T t5 t2)).(eq_ind T (THead (Flat Cast) u t4) (\lambda (_: T).((eq T -t5 t2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda -(t7: T).(pr0 t2 t7)))))) (\lambda (H8: (eq T t5 t2)).(eq_ind T t2 (\lambda -(t6: T).((pr0 t4 t6) \to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: -T).(pr0 t2 t7))))) (\lambda (H9: (pr0 t4 t2)).(let H10 \def (eq_ind_r T t -(\lambda (t6: T).(eq T t6 t1)) H4 (THead (Flat Cast) u t4) H6) in (eq_ind T -(THead (Flat Cast) u t4) (\lambda (t6: T).(ex2 T (\lambda (t7: T).(pr0 t6 -t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let H11 \def (eq_ind_r T t (\lambda -(t6: T).(eq T t3 t6)) H2 (THead (Flat Cast) u t4) H6) in (let H12 \def -(eq_ind_r T t (\lambda (t6: T).(\forall (v: T).((tlt v t6) \to (\forall (t7: -T).((pr0 v t7) \to (\forall (t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: -T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead (Flat Cast) u -t4) H6) in (ex_intro2 T (\lambda (t6: T).(pr0 (THead (Flat Cast) u t4) t6)) -(\lambda (t6: T).(pr0 t2 t6)) t2 (pr0_tau t4 t2 H9 u) (pr0_refl t2)))) t1 -H10))) t5 (sym_eq T t5 t2 H8))) t H6 H7 H5)))]) in (H5 (refl_equal T t) -(refl_equal T t2))) t (sym_eq T t t1 H4))) t3 (sym_eq T t3 t H2) H3))) | -(pr0_comp u1 u2 H2 t3 t4 H3 k) \Rightarrow (\lambda (H4: (eq T (THead k u1 -t3) t)).(\lambda (H5: (eq T (THead k u2 t4) t1)).(eq_ind T (THead k u1 t3) -(\lambda (_: T).((eq T (THead k u2 t4) t1) \to ((pr0 u1 u2) \to ((pr0 t3 t4) -\to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6))))))) -(\lambda (H6: (eq T (THead k u2 t4) t1)).(eq_ind T (THead k u2 t4) (\lambda -(t5: T).((pr0 u1 u2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t5 -t6)) (\lambda (t6: T).(pr0 t2 t6)))))) (\lambda (H7: (pr0 u1 u2)).(\lambda -(H8: (pr0 t3 t4)).(let H9 \def (match H1 in pr0 return (\lambda (t5: -T).(\lambda (t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 t) \to ((eq T t6 t2) -\to (ex2 T (\lambda (t7: T).(pr0 (THead k u2 t4) t7)) (\lambda (t7: T).(pr0 -t2 t7)))))))) with [(pr0_refl t5) \Rightarrow (\lambda (H9: (eq T t5 -t)).(\lambda (H10: (eq T t5 t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2) -\to (ex2 T (\lambda (t7: T).(pr0 (THead k u2 t4) t7)) (\lambda (t7: T).(pr0 -t2 t7))))) (\lambda (H11: (eq T t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T -(\lambda (t7: T).(pr0 (THead k u2 t4) t7)) (\lambda (t7: T).(pr0 t2 t7)))) -(let H12 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 t2)) H11 (THead k u1 -t3) H4) in (eq_ind T (THead k u1 t3) (\lambda (t6: T).(ex2 T (\lambda (t7: -T).(pr0 (THead k u2 t4) t7)) (\lambda (t7: T).(pr0 t6 t7)))) (let H13 \def -(eq_ind_r T t (\lambda (t6: T).(eq T t5 t6)) H9 (THead k u1 t3) H4) in (let -H14 \def (eq_ind_r T t (\lambda (t6: T).(\forall (v: T).((tlt v t6) \to -(\forall (t7: T).((pr0 v t7) \to (\forall (t8: T).((pr0 v t8) \to (ex2 T -(\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead -k u1 t3) H4) in (ex_intro2 T (\lambda (t6: T).(pr0 (THead k u2 t4) t6)) -(\lambda (t6: T).(pr0 (THead k u1 t3) t6)) (THead k u2 t4) (pr0_refl (THead k -u2 t4)) (pr0_comp u1 u2 H7 t3 t4 H8 k)))) t2 H12)) t (sym_eq T t t2 H11))) t5 -(sym_eq T t5 t H9) H10))) | (pr0_comp u0 u3 H9 t5 t6 H10 k0) \Rightarrow -(\lambda (H11: (eq T (THead k0 u0 t5) t)).(\lambda (H12: (eq T (THead k0 u3 -t6) t2)).(eq_ind T (THead k0 u0 t5) (\lambda (_: T).((eq T (THead k0 u3 t6) -t2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead -k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H13: (eq T (THead -k0 u3 t6) t2)).(eq_ind T (THead k0 u3 t6) (\lambda (t7: T).((pr0 u0 u3) \to -((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda -(t8: T).(pr0 t7 t8)))))) (\lambda (H14: (pr0 u0 u3)).(\lambda (H15: (pr0 t5 -t6)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead k u1 t3) t7)) -H4 (THead k0 u0 t5) H11) in (let H17 \def (f_equal T K (\lambda (e: T).(match -e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) -\Rightarrow k | (THead k1 _ _) \Rightarrow k1])) (THead k u1 t3) (THead k0 u0 -t5) H16) in ((let H18 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 -| (THead _ t7 _) \Rightarrow t7])) (THead k u1 t3) (THead k0 u0 t5) H16) in -((let H19 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: -T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ -t7) \Rightarrow t7])) (THead k u1 t3) (THead k0 u0 t5) H16) in (\lambda (H20: -(eq T u1 u0)).(\lambda (H21: (eq K k k0)).(let H22 \def (eq_ind_r T t -(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) -\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) -(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead k0 u0 t5) H11) in (eq_ind_r -K k0 (\lambda (k1: K).(ex2 T (\lambda (t7: T).(pr0 (THead k1 u2 t4) t7)) -(\lambda (t7: T).(pr0 (THead k0 u3 t6) t7)))) (let H23 \def (eq_ind T u1 -(\lambda (t7: T).(pr0 t7 u2)) H7 u0 H20) in (let H24 \def (eq_ind T t3 -(\lambda (t7: T).(pr0 t7 t4)) H8 t5 H19) in (ex2_ind T (\lambda (t7: T).(pr0 -t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead k0 -u2 t4) t7)) (\lambda (t7: T).(pr0 (THead k0 u3 t6) t7))) (\lambda (x: -T).(\lambda (H25: (pr0 t4 x)).(\lambda (H26: (pr0 t6 x)).(ex2_ind T (\lambda -(t7: T).(pr0 u2 t7)) (\lambda (t7: T).(pr0 u3 t7)) (ex2 T (\lambda (t7: -T).(pr0 (THead k0 u2 t4) t7)) (\lambda (t7: T).(pr0 (THead k0 u3 t6) t7))) -(\lambda (x0: T).(\lambda (H27: (pr0 u2 x0)).(\lambda (H28: (pr0 u3 -x0)).(ex_intro2 T (\lambda (t7: T).(pr0 (THead k0 u2 t4) t7)) (\lambda (t7: -T).(pr0 (THead k0 u3 t6) t7)) (THead k0 x0 x) (pr0_comp u2 x0 H27 t4 x H25 -k0) (pr0_comp u3 x0 H28 t6 x H26 k0))))) (H22 u0 (tlt_head_sx k0 u0 t5) u2 -H23 u3 H14))))) (H22 t5 (tlt_head_dx k0 u0 t5) t4 H24 t6 H15)))) k H21))))) -H18)) H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_beta u v1 v2 H9 t5 t6 H10) -\Rightarrow (\lambda (H11: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u -t5)) t)).(\lambda (H12: (eq T (THead (Bind Abbr) v2 t6) t2)).(eq_ind T (THead -(Flat Appl) v1 (THead (Bind Abst) u t5)) (\lambda (_: T).((eq T (THead (Bind -Abbr) v2 t6) t2) \to ((pr0 v1 v2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: -T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda -(H13: (eq T (THead (Bind Abbr) v2 t6) t2)).(eq_ind T (THead (Bind Abbr) v2 -t6) (\lambda (t7: T).((pr0 v1 v2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: -T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (H14: -(pr0 v1 v2)).(\lambda (H15: (pr0 t5 t6)).(let H16 \def (eq_ind_r T t (\lambda -(t7: T).(eq T (THead k u1 t3) t7)) H4 (THead (Flat Appl) v1 (THead (Bind -Abst) u t5)) H11) in (let H17 \def (f_equal T K (\lambda (e: T).(match e in T -return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) -\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) (THead (Flat -Appl) v1 (THead (Bind Abst) u t5)) H16) in ((let H18 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) -(THead k u1 t3) (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) H16) in ((let -H19 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7) -\Rightarrow t7])) (THead k u1 t3) (THead (Flat Appl) v1 (THead (Bind Abst) u -t5)) H16) in (\lambda (H20: (eq T u1 v1)).(\lambda (H21: (eq K k (Flat -Appl))).(let H22 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v -t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to -(ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 -t10)))))))))) H (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) H11) in -(eq_ind_r K (Flat Appl) (\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 (THead -k0 u2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t6) t7)))) (let -H23 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H7 v1 H20) in (let H24 -\def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H8 (THead (Bind Abst) u t5) -H19) in (let H25 \def (match H24 in pr0 return (\lambda (t7: T).(\lambda (t8: -T).(\lambda (_: (pr0 t7 t8)).((eq T t7 (THead (Bind Abst) u t5)) \to ((eq T -t8 t4) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) -(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9)))))))) with [(pr0_refl -t7) \Rightarrow (\lambda (H25: (eq T t7 (THead (Bind Abst) u t5))).(\lambda -(H26: (eq T t7 t4)).(eq_ind T (THead (Bind Abst) u t5) (\lambda (t8: T).((eq -T t8 t4) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) -(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9))))) (\lambda (H27: (eq T -(THead (Bind Abst) u t5) t4)).(eq_ind T (THead (Bind Abst) u t5) (\lambda -(t8: T).(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t8) t9)) (\lambda -(t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9)))) (ex2_ind T (\lambda (t8: -T).(pr0 u2 t8)) (\lambda (t8: T).(pr0 v2 t8)) (ex2 T (\lambda (t8: T).(pr0 -(THead (Flat Appl) u2 (THead (Bind Abst) u t5)) t8)) (\lambda (t8: T).(pr0 -(THead (Bind Abbr) v2 t6) t8))) (\lambda (x: T).(\lambda (H28: (pr0 u2 -x)).(\lambda (H29: (pr0 v2 x)).(ex_intro2 T (\lambda (t8: T).(pr0 (THead -(Flat Appl) u2 (THead (Bind Abst) u t5)) t8)) (\lambda (t8: T).(pr0 (THead -(Bind Abbr) v2 t6) t8)) (THead (Bind Abbr) x t6) (pr0_beta u u2 x H28 t5 t6 -H15) (pr0_comp v2 x H29 t6 t6 (pr0_refl t6) (Bind Abbr)))))) (H22 v1 -(tlt_head_sx (Flat Appl) v1 (THead (Bind Abst) u t5)) u2 H23 v2 H14)) t4 -H27)) t7 (sym_eq T t7 (THead (Bind Abst) u t5) H25) H26))) | (pr0_comp u0 u3 -H25 t7 t8 H26 k0) \Rightarrow (\lambda (H27: (eq T (THead k0 u0 t7) (THead -(Bind Abst) u t5))).(\lambda (H28: (eq T (THead k0 u3 t8) t4)).((let H29 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow t7 | (TLRef _) \Rightarrow t7 | (THead _ _ t9) -\Rightarrow t9])) (THead k0 u0 t7) (THead (Bind Abst) u t5) H27) in ((let H30 -\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t9 _) -\Rightarrow t9])) (THead k0 u0 t7) (THead (Bind Abst) u t5) H27) in ((let H31 -\def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) -with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _ _) -\Rightarrow k1])) (THead k0 u0 t7) (THead (Bind Abst) u t5) H27) in (eq_ind K -(Bind Abst) (\lambda (k1: K).((eq T u0 u) \to ((eq T t7 t5) \to ((eq T (THead -k1 u3 t8) t4) \to ((pr0 u0 u3) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: -T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind -Abbr) v2 t6) t9))))))))) (\lambda (H32: (eq T u0 u)).(eq_ind T u (\lambda -(t9: T).((eq T t7 t5) \to ((eq T (THead (Bind Abst) u3 t8) t4) \to ((pr0 t9 -u3) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 -t4) t10)) (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t6) t10)))))))) -(\lambda (H33: (eq T t7 t5)).(eq_ind T t5 (\lambda (t9: T).((eq T (THead -(Bind Abst) u3 t8) t4) \to ((pr0 u u3) \to ((pr0 t9 t8) \to (ex2 T (\lambda -(t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda (t10: T).(pr0 (THead -(Bind Abbr) v2 t6) t10))))))) (\lambda (H34: (eq T (THead (Bind Abst) u3 t8) -t4)).(eq_ind T (THead (Bind Abst) u3 t8) (\lambda (t9: T).((pr0 u u3) \to -((pr0 t5 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t9) -t10)) (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t6) t10)))))) (\lambda (_: -(pr0 u u3)).(\lambda (H36: (pr0 t5 t8)).(ex2_ind T (\lambda (t9: T).(pr0 t8 -t9)) (\lambda (t9: T).(pr0 t6 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat -Appl) u2 (THead (Bind Abst) u3 t8)) t9)) (\lambda (t9: T).(pr0 (THead (Bind -Abbr) v2 t6) t9))) (\lambda (x: T).(\lambda (H37: (pr0 t8 x)).(\lambda (H38: -(pr0 t6 x)).(ex2_ind T (\lambda (t9: T).(pr0 u2 t9)) (\lambda (t9: T).(pr0 v2 -t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u3 -t8)) t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9))) (\lambda (x0: -T).(\lambda (H39: (pr0 u2 x0)).(\lambda (H40: (pr0 v2 x0)).(ex_intro2 T -(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u3 t8)) t9)) -(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9)) (THead (Bind Abbr) x0 x) -(pr0_beta u3 u2 x0 H39 t8 x H37) (pr0_comp v2 x0 H40 t6 x H38 (Bind -Abbr)))))) (H22 v1 (tlt_head_sx (Flat Appl) v1 (THead (Bind Abst) u t5)) u2 -H23 v2 H14))))) (H22 t5 (tlt_trans (THead (Bind Abst) u t5) t5 (THead (Flat -Appl) v1 (THead (Bind Abst) u t5)) (tlt_head_dx (Bind Abst) u t5) -(tlt_head_dx (Flat Appl) v1 (THead (Bind Abst) u t5))) t8 H36 t6 H15)))) t4 -H34)) t7 (sym_eq T t7 t5 H33))) u0 (sym_eq T u0 u H32))) k0 (sym_eq K k0 -(Bind Abst) H31))) H30)) H29)) H28 H25 H26))) | (pr0_beta u0 v0 v3 H25 t7 t8 -H26) \Rightarrow (\lambda (H27: (eq T (THead (Flat Appl) v0 (THead (Bind -Abst) u0 t7)) (THead (Bind Abst) u t5))).(\lambda (H28: (eq T (THead (Bind -Abbr) v3 t8) t4)).((let H29 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind -Abst) u0 t7)) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _) -\Rightarrow (match k0 in K return (\lambda (_: K).Prop) with [(Bind _) -\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u t5) -H27) in (False_ind ((eq T (THead (Bind Abbr) v3 t8) t4) \to ((pr0 v0 v3) \to -((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) -(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9)))))) H29)) H28 H25 H26))) -| (pr0_upsilon b H25 v0 v3 H26 u0 u3 H27 t7 t8 H28) \Rightarrow (\lambda -(H29: (eq T (THead (Flat Appl) v0 (THead (Bind b) u0 t7)) (THead (Bind Abst) -u t5))).(\lambda (H30: (eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S -O) O v3) t8)) t4)).((let H31 \def (eq_ind T (THead (Flat Appl) v0 (THead -(Bind b) u0 t7)) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) -with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ -_) \Rightarrow (match k0 in K return (\lambda (_: K).Prop) with [(Bind _) -\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u t5) -H29) in (False_ind ((eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O -v3) t8)) t4) \to ((not (eq B b Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3) \to -((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) -(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9)))))))) H31)) H30 H25 H26 -H27 H28))) | (pr0_delta u0 u3 H25 t7 t8 H26 w H27) \Rightarrow (\lambda (H28: -(eq T (THead (Bind Abbr) u0 t7) (THead (Bind Abst) u t5))).(\lambda (H29: (eq -T (THead (Bind Abbr) u3 w) t4)).((let H30 \def (eq_ind T (THead (Bind Abbr) -u0 t7) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _) -\Rightarrow (match k0 in K return (\lambda (_: K).Prop) with [(Bind b) -\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow -True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _) -\Rightarrow False])])) I (THead (Bind Abst) u t5) H28) in (False_ind ((eq T -(THead (Bind Abbr) u3 w) t4) \to ((pr0 u0 u3) \to ((pr0 t7 t8) \to ((subst0 O -u3 t8 w) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) -(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9))))))) H30)) H29 H25 H26 -H27))) | (pr0_zeta b H25 t7 t8 H26 u0) \Rightarrow (\lambda (H27: (eq T -(THead (Bind b) u0 (lift (S O) O t7)) (THead (Bind Abst) u t5))).(\lambda -(H28: (eq T t8 t4)).((let H29 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: -((nat \to nat))) (d: nat) (t9: T) on t9: T \def (match t9 with [(TSort n) -\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with -[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k0 u3 t10) -\Rightarrow (THead k0 (lref_map f d u3) (lref_map f (s k0 d) t10))]) in -lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (TLRef _) \Rightarrow -((let rec lref_map (f: ((nat \to nat))) (d: nat) (t9: T) on t9: T \def (match -t9 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef -(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | -(THead k0 u3 t10) \Rightarrow (THead k0 (lref_map f d u3) (lref_map f (s k0 -d) t10))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (THead _ _ -t9) \Rightarrow t9])) (THead (Bind b) u0 (lift (S O) O t7)) (THead (Bind -Abst) u t5) H27) in ((let H30 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) -\Rightarrow u0 | (THead _ t9 _) \Rightarrow t9])) (THead (Bind b) u0 (lift (S -O) O t7)) (THead (Bind Abst) u t5) H27) in ((let H31 \def (f_equal T B -(\lambda (e: T).(match e in T return (\lambda (_: T).B) with [(TSort _) -\Rightarrow b | (TLRef _) \Rightarrow b | (THead k0 _ _) \Rightarrow (match -k0 in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) -\Rightarrow b])])) (THead (Bind b) u0 (lift (S O) O t7)) (THead (Bind Abst) u -t5) H27) in (eq_ind B Abst (\lambda (b0: B).((eq T u0 u) \to ((eq T (lift (S -O) O t7) t5) \to ((eq T t8 t4) \to ((not (eq B b0 Abst)) \to ((pr0 t7 t8) \to -(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: -T).(pr0 (THead (Bind Abbr) v2 t6) t9))))))))) (\lambda (H32: (eq T u0 -u)).(eq_ind T u (\lambda (_: T).((eq T (lift (S O) O t7) t5) \to ((eq T t8 -t4) \to ((not (eq B Abst Abst)) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: -T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda (t10: T).(pr0 (THead (Bind -Abbr) v2 t6) t10)))))))) (\lambda (H33: (eq T (lift (S O) O t7) t5)).(eq_ind -T (lift (S O) O t7) (\lambda (_: T).((eq T t8 t4) \to ((not (eq B Abst Abst)) -\to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) -t10)) (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t6) t10))))))) (\lambda -(H34: (eq T t8 t4)).(eq_ind T t4 (\lambda (t9: T).((not (eq B Abst Abst)) \to -((pr0 t7 t9) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) -t10)) (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t6) t10)))))) (\lambda -(H35: (not (eq B Abst Abst))).(\lambda (_: (pr0 t7 t4)).(let H37 \def (match -(H35 (refl_equal B Abst)) in False return (\lambda (_: False).(ex2 T (\lambda -(t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead -(Bind Abbr) v2 t6) t9)))) with []) in H37))) t8 (sym_eq T t8 t4 H34))) t5 -H33)) u0 (sym_eq T u0 u H32))) b (sym_eq B b Abst H31))) H30)) H29)) H28 H25 -H26))) | (pr0_tau t7 t8 H25 u0) \Rightarrow (\lambda (H26: (eq T (THead (Flat -Cast) u0 t7) (THead (Bind Abst) u t5))).(\lambda (H27: (eq T t8 t4)).((let -H28 \def (eq_ind T (THead (Flat Cast) u0 t7) (\lambda (e: T).(match e in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind Abst) u t5) H26) in (False_ind ((eq T t8 t4) \to -((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) -(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9))))) H28)) H27 H25)))]) in -(H25 (refl_equal T (THead (Bind Abst) u t5)) (refl_equal T t4))))) k H21))))) -H18)) H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_upsilon b H9 v1 v2 H10 u0 -u3 H11 t5 t6 H12) \Rightarrow (\lambda (H13: (eq T (THead (Flat Appl) v1 -(THead (Bind b) u0 t5)) t)).(\lambda (H14: (eq T (THead (Bind b) u3 (THead -(Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Flat Appl) v1 -(THead (Bind b) u0 t5)) (\lambda (_: T).((eq T (THead (Bind b) u3 (THead -(Flat Appl) (lift (S O) O v2) t6)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1 -v2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead -k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H15: (eq T -(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T -(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) (\lambda (t7: -T).((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) -\to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 -t7 t8)))))))) (\lambda (H16: (not (eq B b Abst))).(\lambda (H17: (pr0 v1 -v2)).(\lambda (H18: (pr0 u0 u3)).(\lambda (H19: (pr0 t5 t6)).(let H20 \def -(eq_ind_r T t (\lambda (t7: T).(eq T (THead k u1 t3) t7)) H4 (THead (Flat -Appl) v1 (THead (Bind b) u0 t5)) H13) in (let H21 \def (f_equal T K (\lambda -(e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k -| (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) -(THead (Flat Appl) v1 (THead (Bind b) u0 t5)) H20) in ((let H22 \def (f_equal -T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) -(THead k u1 t3) (THead (Flat Appl) v1 (THead (Bind b) u0 t5)) H20) in ((let -H23 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7) -\Rightarrow t7])) (THead k u1 t3) (THead (Flat Appl) v1 (THead (Bind b) u0 -t5)) H20) in (\lambda (H24: (eq T u1 v1)).(\lambda (H25: (eq K k (Flat -Appl))).(let H26 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v -t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to -(ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 -t10)))))))))) H (THead (Flat Appl) v1 (THead (Bind b) u0 t5)) H13) in -(eq_ind_r K (Flat Appl) (\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 (THead -k0 u2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) -(lift (S O) O v2) t6)) t7)))) (let H27 \def (eq_ind T u1 (\lambda (t7: -T).(pr0 t7 u2)) H7 v1 H24) in (let H28 \def (eq_ind T t3 (\lambda (t7: -T).(pr0 t7 t4)) H8 (THead (Bind b) u0 t5) H23) in (let H29 \def (match H28 in -pr0 return (\lambda (t7: T).(\lambda (t8: T).(\lambda (_: (pr0 t7 t8)).((eq T -t7 (THead (Bind b) u0 t5)) \to ((eq T t8 t4) \to (ex2 T (\lambda (t9: T).(pr0 -(THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 -(THead (Flat Appl) (lift (S O) O v2) t6)) t9)))))))) with [(pr0_refl t7) -\Rightarrow (\lambda (H29: (eq T t7 (THead (Bind b) u0 t5))).(\lambda (H30: -(eq T t7 t4)).(eq_ind T (THead (Bind b) u0 t5) (\lambda (t8: T).((eq T t8 t4) -\to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: -T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9))))) -(\lambda (H31: (eq T (THead (Bind b) u0 t5) t4)).(eq_ind T (THead (Bind b) u0 -t5) (\lambda (t8: T).(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t8) -t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) -O v2) t6)) t9)))) (ex2_ind T (\lambda (t8: T).(pr0 u2 t8)) (\lambda (t8: -T).(pr0 v2 t8)) (ex2 T (\lambda (t8: T).(pr0 (THead (Flat Appl) u2 (THead -(Bind b) u0 t5)) t8)) (\lambda (t8: T).(pr0 (THead (Bind b) u3 (THead (Flat -Appl) (lift (S O) O v2) t6)) t8))) (\lambda (x: T).(\lambda (H32: (pr0 u2 -x)).(\lambda (H33: (pr0 v2 x)).(pr0_confluence__pr0_cong_upsilon_refl b H16 -u0 u3 H18 t5 t6 H19 u2 v2 x H32 H33)))) (H26 v1 (tlt_head_sx (Flat Appl) v1 -(THead (Bind b) u0 t5)) u2 H27 v2 H17)) t4 H31)) t7 (sym_eq T t7 (THead (Bind -b) u0 t5) H29) H30))) | (pr0_comp u4 u5 H29 t7 t8 H30 k0) \Rightarrow -(\lambda (H31: (eq T (THead k0 u4 t7) (THead (Bind b) u0 t5))).(\lambda (H32: -(eq T (THead k0 u5 t8) t4)).((let H33 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t7 | -(TLRef _) \Rightarrow t7 | (THead _ _ t9) \Rightarrow t9])) (THead k0 u4 t7) -(THead (Bind b) u0 t5) H31) in ((let H34 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u4 | -(TLRef _) \Rightarrow u4 | (THead _ t9 _) \Rightarrow t9])) (THead k0 u4 t7) -(THead (Bind b) u0 t5) H31) in ((let H35 \def (f_equal T K (\lambda (e: -T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k0 | -(TLRef _) \Rightarrow k0 | (THead k1 _ _) \Rightarrow k1])) (THead k0 u4 t7) -(THead (Bind b) u0 t5) H31) in (eq_ind K (Bind b) (\lambda (k1: K).((eq T u4 -u0) \to ((eq T t7 t5) \to ((eq T (THead k1 u5 t8) t4) \to ((pr0 u4 u5) \to -((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) -(\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) -t6)) t9))))))))) (\lambda (H36: (eq T u4 u0)).(eq_ind T u0 (\lambda (t9: -T).((eq T t7 t5) \to ((eq T (THead (Bind b) u5 t8) t4) \to ((pr0 t9 u5) \to -((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) -t10)) (\lambda (t10: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S -O) O v2) t6)) t10)))))))) (\lambda (H37: (eq T t7 t5)).(eq_ind T t5 (\lambda -(t9: T).((eq T (THead (Bind b) u5 t8) t4) \to ((pr0 u0 u5) \to ((pr0 t9 t8) -\to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda -(t10: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) -t10))))))) (\lambda (H38: (eq T (THead (Bind b) u5 t8) t4)).(eq_ind T (THead -(Bind b) u5 t8) (\lambda (t9: T).((pr0 u0 u5) \to ((pr0 t5 t8) \to (ex2 T -(\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t9) t10)) (\lambda (t10: T).(pr0 -(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t10)))))) -(\lambda (H39: (pr0 u0 u5)).(\lambda (H40: (pr0 t5 t8)).(ex2_ind T (\lambda -(t9: T).(pr0 t8 t9)) (\lambda (t9: T).(pr0 t6 t9)) (ex2 T (\lambda (t9: -T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 t8)) t9)) (\lambda (t9: -T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9))) -(\lambda (x: T).(\lambda (H41: (pr0 t8 x)).(\lambda (H42: (pr0 t6 -x)).(ex2_ind T (\lambda (t9: T).(pr0 u5 t9)) (\lambda (t9: T).(pr0 u3 t9)) -(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 t8)) -t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) -O v2) t6)) t9))) (\lambda (x0: T).(\lambda (H43: (pr0 u5 x0)).(\lambda (H44: -(pr0 u3 x0)).(ex2_ind T (\lambda (t9: T).(pr0 u2 t9)) (\lambda (t9: T).(pr0 -v2 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 -t8)) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift -(S O) O v2) t6)) t9))) (\lambda (x1: T).(\lambda (H45: (pr0 u2 x1)).(\lambda -(H46: (pr0 v2 x1)).(pr0_confluence__pr0_cong_upsilon_cong b H16 u2 v2 x1 H45 -H46 t8 t6 x H41 H42 u5 u3 x0 H43 H44)))) (H26 v1 (tlt_head_sx (Flat Appl) v1 -(THead (Bind b) u0 t5)) u2 H27 v2 H17))))) (H26 u0 (tlt_trans (THead (Bind b) -u0 t5) u0 (THead (Flat Appl) v1 (THead (Bind b) u0 t5)) (tlt_head_sx (Bind b) -u0 t5) (tlt_head_dx (Flat Appl) v1 (THead (Bind b) u0 t5))) u5 H39 u3 -H18))))) (H26 t5 (tlt_trans (THead (Bind b) u0 t5) t5 (THead (Flat Appl) v1 -(THead (Bind b) u0 t5)) (tlt_head_dx (Bind b) u0 t5) (tlt_head_dx (Flat Appl) -v1 (THead (Bind b) u0 t5))) t8 H40 t6 H19)))) t4 H38)) t7 (sym_eq T t7 t5 -H37))) u4 (sym_eq T u4 u0 H36))) k0 (sym_eq K k0 (Bind b) H35))) H34)) H33)) -H32 H29 H30))) | (pr0_beta u v0 v3 H29 t7 t8 H30) \Rightarrow (\lambda (H31: -(eq T (THead (Flat Appl) v0 (THead (Bind Abst) u t7)) (THead (Bind b) u0 -t5))).(\lambda (H32: (eq T (THead (Bind Abbr) v3 t8) t4)).((let H33 \def -(eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u t7)) (\lambda (e: -T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in -K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) -\Rightarrow True])])) I (THead (Bind b) u0 t5) H31) in (False_ind ((eq T -(THead (Bind Abbr) v3 t8) t4) \to ((pr0 v0 v3) \to ((pr0 t7 t8) \to (ex2 T -(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 -(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9)))))) H33)) -H32 H29 H30))) | (pr0_upsilon b0 H29 v0 v3 H30 u4 u5 H31 t7 t8 H32) -\Rightarrow (\lambda (H33: (eq T (THead (Flat Appl) v0 (THead (Bind b0) u4 -t7)) (THead (Bind b) u0 t5))).(\lambda (H34: (eq T (THead (Bind b0) u5 (THead -(Flat Appl) (lift (S O) O v3) t8)) t4)).((let H35 \def (eq_ind T (THead (Flat -Appl) v0 (THead (Bind b0) u4 t7)) (\lambda (e: T).(match e in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind b) u0 t5) H33) in (False_ind ((eq T (THead (Bind b0) -u5 (THead (Flat Appl) (lift (S O) O v3) t8)) t4) \to ((not (eq B b0 Abst)) -\to ((pr0 v0 v3) \to ((pr0 u4 u5) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: -T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) -u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9)))))))) H35)) H34 H29 H30 H31 -H32))) | (pr0_delta u4 u5 H29 t7 t8 H30 w H31) \Rightarrow (\lambda (H32: (eq -T (THead (Bind Abbr) u4 t7) (THead (Bind b) u0 t5))).(\lambda (H33: (eq T -(THead (Bind Abbr) u5 w) t4)).((let H34 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t7 | -(TLRef _) \Rightarrow t7 | (THead _ _ t9) \Rightarrow t9])) (THead (Bind -Abbr) u4 t7) (THead (Bind b) u0 t5) H32) in ((let H35 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow u4 | (TLRef _) \Rightarrow u4 | (THead _ t9 _) \Rightarrow t9])) -(THead (Bind Abbr) u4 t7) (THead (Bind b) u0 t5) H32) in ((let H36 \def -(f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with -[(TSort _) \Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | (THead k0 _ _) -\Rightarrow (match k0 in K return (\lambda (_: K).B) with [(Bind b0) -\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (THead (Bind Abbr) u4 t7) -(THead (Bind b) u0 t5) H32) in (eq_ind B Abbr (\lambda (b0: B).((eq T u4 u0) -\to ((eq T t7 t5) \to ((eq T (THead (Bind Abbr) u5 w) t4) \to ((pr0 u4 u5) -\to ((pr0 t7 t8) \to ((subst0 O u5 t8 w) \to (ex2 T (\lambda (t9: T).(pr0 -(THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind b0) u3 -(THead (Flat Appl) (lift (S O) O v2) t6)) t9)))))))))) (\lambda (H37: (eq T -u4 u0)).(eq_ind T u0 (\lambda (t9: T).((eq T t7 t5) \to ((eq T (THead (Bind -Abbr) u5 w) t4) \to ((pr0 t9 u5) \to ((pr0 t7 t8) \to ((subst0 O u5 t8 w) \to -(ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda (t10: -T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) -t10))))))))) (\lambda (H38: (eq T t7 t5)).(eq_ind T t5 (\lambda (t9: T).((eq -T (THead (Bind Abbr) u5 w) t4) \to ((pr0 u0 u5) \to ((pr0 t9 t8) \to ((subst0 -O u5 t8 w) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) -(\lambda (t10: T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O -v2) t6)) t10)))))))) (\lambda (H39: (eq T (THead (Bind Abbr) u5 w) -t4)).(eq_ind T (THead (Bind Abbr) u5 w) (\lambda (t9: T).((pr0 u0 u5) \to -((pr0 t5 t8) \to ((subst0 O u5 t8 w) \to (ex2 T (\lambda (t10: T).(pr0 (THead -(Flat Appl) u2 t9) t10)) (\lambda (t10: T).(pr0 (THead (Bind Abbr) u3 (THead -(Flat Appl) (lift (S O) O v2) t6)) t10))))))) (\lambda (H40: (pr0 u0 -u5)).(\lambda (H41: (pr0 t5 t8)).(\lambda (H42: (subst0 O u5 t8 w)).(let H43 -\def (eq_ind_r B b (\lambda (b0: B).(\forall (v: T).((tlt v (THead (Flat -Appl) v1 (THead (Bind b0) u0 t5))) \to (\forall (t9: T).((pr0 v t9) \to -(\forall (t10: T).((pr0 v t10) \to (ex2 T (\lambda (t11: T).(pr0 t9 t11)) -(\lambda (t11: T).(pr0 t10 t11)))))))))) H26 Abbr H36) in (let H44 \def -(eq_ind_r B b (\lambda (b0: B).(eq T t3 (THead (Bind b0) u0 t5))) H23 Abbr -H36) in (let H45 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 Abst))) -H16 Abbr H36) in (ex2_ind T (\lambda (t9: T).(pr0 t8 t9)) (\lambda (t9: -T).(pr0 t6 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead -(Bind Abbr) u5 w)) t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) u3 (THead -(Flat Appl) (lift (S O) O v2) t6)) t9))) (\lambda (x: T).(\lambda (H46: (pr0 -t8 x)).(\lambda (H47: (pr0 t6 x)).(ex2_ind T (\lambda (t9: T).(pr0 u5 t9)) -(\lambda (t9: T).(pr0 u3 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) -u2 (THead (Bind Abbr) u5 w)) t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) u3 -(THead (Flat Appl) (lift (S O) O v2) t6)) t9))) (\lambda (x0: T).(\lambda -(H48: (pr0 u5 x0)).(\lambda (H49: (pr0 u3 x0)).(ex2_ind T (\lambda (t9: -T).(pr0 u2 t9)) (\lambda (t9: T).(pr0 v2 t9)) (ex2 T (\lambda (t9: T).(pr0 -(THead (Flat Appl) u2 (THead (Bind Abbr) u5 w)) t9)) (\lambda (t9: T).(pr0 -(THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9))) -(\lambda (x1: T).(\lambda (H50: (pr0 u2 x1)).(\lambda (H51: (pr0 v2 -x1)).(pr0_confluence__pr0_cong_upsilon_delta H45 u5 t8 w H42 u2 v2 x1 H50 H51 -t6 x H46 H47 u3 x0 H48 H49)))) (H43 v1 (tlt_head_sx (Flat Appl) v1 (THead -(Bind Abbr) u0 t5)) u2 H27 v2 H17))))) (H43 u0 (tlt_trans (THead (Bind Abbr) -u0 t5) u0 (THead (Flat Appl) v1 (THead (Bind Abbr) u0 t5)) (tlt_head_sx (Bind -Abbr) u0 t5) (tlt_head_dx (Flat Appl) v1 (THead (Bind Abbr) u0 t5))) u5 H40 -u3 H18))))) (H43 t5 (tlt_trans (THead (Bind Abbr) u0 t5) t5 (THead (Flat -Appl) v1 (THead (Bind Abbr) u0 t5)) (tlt_head_dx (Bind Abbr) u0 t5) -(tlt_head_dx (Flat Appl) v1 (THead (Bind Abbr) u0 t5))) t8 H41 t6 H19)))))))) -t4 H39)) t7 (sym_eq T t7 t5 H38))) u4 (sym_eq T u4 u0 H37))) b H36)) H35)) -H34)) H33 H29 H30 H31))) | (pr0_zeta b0 H29 t7 t8 H30 u) \Rightarrow (\lambda -(H31: (eq T (THead (Bind b0) u (lift (S O) O t7)) (THead (Bind b) u0 -t5))).(\lambda (H32: (eq T t8 t4)).((let H33 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let -rec lref_map (f: ((nat \to nat))) (d: nat) (t9: T) on t9: T \def (match t9 -with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match -(blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k0 -u4 t10) \Rightarrow (THead k0 (lref_map f d u4) (lref_map f (s k0 d) t10))]) -in lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (TLRef _) \Rightarrow -((let rec lref_map (f: ((nat \to nat))) (d: nat) (t9: T) on t9: T \def (match -t9 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef -(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | -(THead k0 u4 t10) \Rightarrow (THead k0 (lref_map f d u4) (lref_map f (s k0 -d) t10))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (THead _ _ -t9) \Rightarrow t9])) (THead (Bind b0) u (lift (S O) O t7)) (THead (Bind b) -u0 t5) H31) in ((let H34 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) -\Rightarrow u | (THead _ t9 _) \Rightarrow t9])) (THead (Bind b0) u (lift (S -O) O t7)) (THead (Bind b) u0 t5) H31) in ((let H35 \def (f_equal T B (\lambda -(e: T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b0 -| (TLRef _) \Rightarrow b0 | (THead k0 _ _) \Rightarrow (match k0 in K return -(\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow -b0])])) (THead (Bind b0) u (lift (S O) O t7)) (THead (Bind b) u0 t5) H31) in -(eq_ind B b (\lambda (b1: B).((eq T u u0) \to ((eq T (lift (S O) O t7) t5) -\to ((eq T t8 t4) \to ((not (eq B b1 Abst)) \to ((pr0 t7 t8) \to (ex2 T -(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 -(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9))))))))) -(\lambda (H36: (eq T u u0)).(eq_ind T u0 (\lambda (_: T).((eq T (lift (S O) O -t7) t5) \to ((eq T t8 t4) \to ((not (eq B b Abst)) \to ((pr0 t7 t8) \to (ex2 -T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda (t10: -T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) -t10)))))))) (\lambda (H37: (eq T (lift (S O) O t7) t5)).(eq_ind T (lift (S O) -O t7) (\lambda (_: T).((eq T t8 t4) \to ((not (eq B b Abst)) \to ((pr0 t7 t8) -\to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda -(t10: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) -t10))))))) (\lambda (H38: (eq T t8 t4)).(eq_ind T t4 (\lambda (t9: T).((not -(eq B b Abst)) \to ((pr0 t7 t9) \to (ex2 T (\lambda (t10: T).(pr0 (THead -(Flat Appl) u2 t4) t10)) (\lambda (t10: T).(pr0 (THead (Bind b) u3 (THead -(Flat Appl) (lift (S O) O v2) t6)) t10)))))) (\lambda (H39: (not (eq B b -Abst))).(\lambda (H40: (pr0 t7 t4)).(let H41 \def (eq_ind_r T t5 (\lambda -(t9: T).(\forall (v: T).((tlt v (THead (Flat Appl) v1 (THead (Bind b) u0 -t9))) \to (\forall (t10: T).((pr0 v t10) \to (\forall (t11: T).((pr0 v t11) -\to (ex2 T (\lambda (t12: T).(pr0 t10 t12)) (\lambda (t12: T).(pr0 t11 -t12)))))))))) H26 (lift (S O) O t7) H37) in (let H42 \def (eq_ind_r T t5 -(\lambda (t9: T).(eq T t3 (THead (Bind b) u0 t9))) H23 (lift (S O) O t7) H37) -in (let H43 \def (eq_ind_r T t5 (\lambda (t9: T).(pr0 t9 t6)) H19 (lift (S O) -O t7) H37) in (ex2_ind T (\lambda (t9: T).(eq T t6 (lift (S O) O t9))) -(\lambda (t9: T).(pr0 t7 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) -u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift -(S O) O v2) t6)) t9))) (\lambda (x: T).(\lambda (H44: (eq T t6 (lift (S O) O -x))).(\lambda (H45: (pr0 t7 x)).(eq_ind_r T (lift (S O) O x) (\lambda (t9: -T).(ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda -(t10: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t9)) -t10)))) (ex2_ind T (\lambda (t9: T).(pr0 x t9)) (\lambda (t9: T).(pr0 t4 t9)) -(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: -T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) (lift (S O) O -x))) t9))) (\lambda (x0: T).(\lambda (H46: (pr0 x x0)).(\lambda (H47: (pr0 t4 -x0)).(ex2_ind T (\lambda (t9: T).(pr0 u2 t9)) (\lambda (t9: T).(pr0 v2 t9)) -(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: -T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) (lift (S O) O -x))) t9))) (\lambda (x1: T).(\lambda (H48: (pr0 u2 x1)).(\lambda (H49: (pr0 -v2 x1)).(pr0_confluence__pr0_cong_upsilon_zeta b H39 u0 u3 H18 u2 v2 x1 H48 -H49 x t4 x0 H46 H47)))) (H41 v1 (tlt_head_sx (Flat Appl) v1 (THead (Bind b) -u0 (lift (S O) O t7))) u2 H27 v2 H17))))) (H41 t7 (tlt_trans (THead (Bind b) -u0 (lift (S O) O t7)) t7 (THead (Flat Appl) v1 (THead (Bind b) u0 (lift (S O) -O t7))) (lift_tlt_dx (Bind b) u0 t7 (S O) O) (tlt_head_dx (Flat Appl) v1 -(THead (Bind b) u0 (lift (S O) O t7)))) x H45 t4 H40)) t6 H44)))) -(pr0_gen_lift t7 t6 (S O) O H43))))))) t8 (sym_eq T t8 t4 H38))) t5 H37)) u -(sym_eq T u u0 H36))) b0 (sym_eq B b0 b H35))) H34)) H33)) H32 H29 H30))) | -(pr0_tau t7 t8 H29 u) \Rightarrow (\lambda (H30: (eq T (THead (Flat Cast) u -t7) (THead (Bind b) u0 t5))).(\lambda (H31: (eq T t8 t4)).((let H32 \def -(eq_ind T (THead (Flat Cast) u t7) (\lambda (e: T).(match e in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind b) u0 t5) H30) in (False_ind ((eq T t8 t4) \to ((pr0 -t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) -(\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) -t6)) t9))))) H32)) H31 H29)))]) in (H29 (refl_equal T (THead (Bind b) u0 t5)) -(refl_equal T t4))))) k H25))))) H22)) H21))))))) t2 H15)) t H13 H14 H9 H10 -H11 H12))) | (pr0_delta u0 u3 H9 t5 t6 H10 w H11) \Rightarrow (\lambda (H12: -(eq T (THead (Bind Abbr) u0 t5) t)).(\lambda (H13: (eq T (THead (Bind Abbr) -u3 w) t2)).(eq_ind T (THead (Bind Abbr) u0 t5) (\lambda (_: T).((eq T (THead -(Bind Abbr) u3 w) t2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to ((subst0 O u3 t6 -w) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: -T).(pr0 t2 t8)))))))) (\lambda (H14: (eq T (THead (Bind Abbr) u3 w) -t2)).(eq_ind T (THead (Bind Abbr) u3 w) (\lambda (t7: T).((pr0 u0 u3) \to -((pr0 t5 t6) \to ((subst0 O u3 t6 w) \to (ex2 T (\lambda (t8: T).(pr0 (THead -k u2 t4) t8)) (\lambda (t8: T).(pr0 t7 t8))))))) (\lambda (H15: (pr0 u0 -u3)).(\lambda (H16: (pr0 t5 t6)).(\lambda (H17: (subst0 O u3 t6 w)).(let H18 -\def (eq_ind_r T t (\lambda (t7: T).(eq T (THead k u1 t3) t7)) H4 (THead -(Bind Abbr) u0 t5) H12) in (let H19 \def (f_equal T K (\lambda (e: T).(match -e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) -\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) (THead (Bind -Abbr) u0 t5) H18) in ((let H20 \def (f_equal T T (\lambda (e: T).(match e in -T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) -\Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t3) (THead -(Bind Abbr) u0 t5) H18) in ((let H21 \def (f_equal T T (\lambda (e: T).(match -e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) -\Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t3) (THead -(Bind Abbr) u0 t5) H18) in (\lambda (H22: (eq T u1 u0)).(\lambda (H23: (eq K -k (Bind Abbr))).(let H24 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: -T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v -t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 -t10)))))))))) H (THead (Bind Abbr) u0 t5) H12) in (eq_ind_r K (Bind Abbr) -(\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 (THead k0 u2 t4) t7)) (\lambda -(t7: T).(pr0 (THead (Bind Abbr) u3 w) t7)))) (let H25 \def (eq_ind T u1 -(\lambda (t7: T).(pr0 t7 u2)) H7 u0 H22) in (let H26 \def (eq_ind T t3 -(\lambda (t7: T).(pr0 t7 t4)) H8 t5 H21) in (ex2_ind T (\lambda (t7: T).(pr0 -t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead -(Bind Abbr) u2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u3 w) t7))) -(\lambda (x: T).(\lambda (H27: (pr0 t4 x)).(\lambda (H28: (pr0 t6 -x)).(ex2_ind T (\lambda (t7: T).(pr0 u2 t7)) (\lambda (t7: T).(pr0 u3 t7)) -(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 t4) t7)) (\lambda (t7: -T).(pr0 (THead (Bind Abbr) u3 w) t7))) (\lambda (x0: T).(\lambda (H29: (pr0 -u2 x0)).(\lambda (H30: (pr0 u3 x0)).(pr0_confluence__pr0_cong_delta u3 t6 w -H17 u2 x0 H29 H30 t4 x H27 H28)))) (H24 u0 (tlt_head_sx (Bind Abbr) u0 t5) u2 -H25 u3 H15))))) (H24 t5 (tlt_head_dx (Bind Abbr) u0 t5) t4 H26 t6 H16)))) k -H23))))) H20)) H19)))))) t2 H14)) t H12 H13 H9 H10 H11))) | (pr0_zeta b H9 t5 -t6 H10 u) \Rightarrow (\lambda (H11: (eq T (THead (Bind b) u (lift (S O) O -t5)) t)).(\lambda (H12: (eq T t6 t2)).(eq_ind T (THead (Bind b) u (lift (S O) -O t5)) (\lambda (_: T).((eq T t6 t2) \to ((not (eq B b Abst)) \to ((pr0 t5 -t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: -T).(pr0 t2 t8))))))) (\lambda (H13: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: -T).((not (eq B b Abst)) \to ((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 -(THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H14: (not -(eq B b Abst))).(\lambda (H15: (pr0 t5 t2)).(let H16 \def (eq_ind_r T t -(\lambda (t7: T).(eq T (THead k u1 t3) t7)) H4 (THead (Bind b) u (lift (S O) -O t5)) H11) in (let H17 \def (f_equal T K (\lambda (e: T).(match e in T -return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) -\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) (THead (Bind -b) u (lift (S O) O t5)) H16) in ((let H18 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | -(TLRef _) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t3) -(THead (Bind b) u (lift (S O) O t5)) H16) in ((let H19 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) -(THead k u1 t3) (THead (Bind b) u (lift (S O) O t5)) H16) in (\lambda (H20: -(eq T u1 u)).(\lambda (H21: (eq K k (Bind b))).(let H22 \def (eq_ind_r T t -(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) -\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) -(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Bind b) u (lift (S O) O -t5)) H11) in (eq_ind_r K (Bind b) (\lambda (k0: K).(ex2 T (\lambda (t7: -T).(pr0 (THead k0 u2 t4) t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let H23 \def -(eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H7 u H20) in (let H24 \def (eq_ind -T t3 (\lambda (t7: T).(pr0 t7 t4)) H8 (lift (S O) O t5) H19) in (ex2_ind T -(\lambda (t7: T).(eq T t4 (lift (S O) O t7))) (\lambda (t7: T).(pr0 t5 t7)) -(ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 t4) t7)) (\lambda (t7: -T).(pr0 t2 t7))) (\lambda (x: T).(\lambda (H25: (eq T t4 (lift (S O) O -x))).(\lambda (H26: (pr0 t5 x)).(eq_ind_r T (lift (S O) O x) (\lambda (t7: -T).(ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 t7) t8)) (\lambda (t8: -T).(pr0 t2 t8)))) (ex2_ind T (\lambda (t7: T).(pr0 x t7)) (\lambda (t7: -T).(pr0 t2 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (lift (S O) O -x)) t7)) (\lambda (t7: T).(pr0 t2 t7))) (\lambda (x0: T).(\lambda (H27: (pr0 -x x0)).(\lambda (H28: (pr0 t2 x0)).(ex_intro2 T (\lambda (t7: T).(pr0 (THead -(Bind b) u2 (lift (S O) O x)) t7)) (\lambda (t7: T).(pr0 t2 t7)) x0 (pr0_zeta -b H14 x x0 H27 u2) H28)))) (H22 t5 (lift_tlt_dx (Bind b) u t5 (S O) O) x H26 -t2 H15)) t4 H25)))) (pr0_gen_lift t5 t4 (S O) O H24)))) k H21))))) H18)) -H17))))) t6 (sym_eq T t6 t2 H13))) t H11 H12 H9 H10))) | (pr0_tau t5 t6 H9 u) -\Rightarrow (\lambda (H10: (eq T (THead (Flat Cast) u t5) t)).(\lambda (H11: -(eq T t6 t2)).(eq_ind T (THead (Flat Cast) u t5) (\lambda (_: T).((eq T t6 -t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) -(\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H12: (eq T t6 t2)).(eq_ind T t2 -(\lambda (t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 -t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))) (\lambda (H13: (pr0 t5 t2)).(let -H14 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead k u1 t3) t7)) H4 (THead -(Flat Cast) u t5) H10) in (let H15 \def (f_equal T K (\lambda (e: T).(match e -in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) -\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) (THead (Flat -Cast) u t5) H14) in ((let H16 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) -\Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t3) (THead -(Flat Cast) u t5) H14) in ((let H17 \def (f_equal T T (\lambda (e: T).(match -e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) -\Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t3) (THead -(Flat Cast) u t5) H14) in (\lambda (H18: (eq T u1 u)).(\lambda (H19: (eq K k -(Flat Cast))).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: -T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v -t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 -t10)))))))))) H (THead (Flat Cast) u t5) H10) in (eq_ind_r K (Flat Cast) -(\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 (THead k0 u2 t4) t7)) (\lambda -(t7: T).(pr0 t2 t7)))) (let H21 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 -u2)) H7 u H18) in (let H22 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H8 -t5 H17) in (ex2_ind T (\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t2 -t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Flat Cast) u2 t4) t7)) (\lambda -(t7: T).(pr0 t2 t7))) (\lambda (x: T).(\lambda (H23: (pr0 t4 x)).(\lambda -(H24: (pr0 t2 x)).(ex_intro2 T (\lambda (t7: T).(pr0 (THead (Flat Cast) u2 -t4) t7)) (\lambda (t7: T).(pr0 t2 t7)) x (pr0_tau t4 x H23 u2) H24)))) (H20 -t5 (tlt_head_dx (Flat Cast) u t5) t4 H22 t2 H13)))) k H19))))) H16)) H15)))) -t6 (sym_eq T t6 t2 H12))) t H10 H11 H9)))]) in (H9 (refl_equal T t) -(refl_equal T t2))))) t1 H6)) t H4 H5 H2 H3))) | (pr0_beta u v1 v2 H2 t3 t4 -H3) \Rightarrow (\lambda (H4: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) -u t3)) t)).(\lambda (H5: (eq T (THead (Bind Abbr) v2 t4) t1)).(eq_ind T -(THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (\lambda (_: T).((eq T (THead -(Bind Abbr) v2 t4) t1) \to ((pr0 v1 v2) \to ((pr0 t3 t4) \to (ex2 T (\lambda -(t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6))))))) (\lambda (H6: (eq T -(THead (Bind Abbr) v2 t4) t1)).(eq_ind T (THead (Bind Abbr) v2 t4) (\lambda -(t5: T).((pr0 v1 v2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t5 -t6)) (\lambda (t6: T).(pr0 t2 t6)))))) (\lambda (H7: (pr0 v1 v2)).(\lambda -(H8: (pr0 t3 t4)).(let H9 \def (match H1 in pr0 return (\lambda (t5: -T).(\lambda (t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 t) \to ((eq T t6 t2) -\to (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: -T).(pr0 t2 t7)))))))) with [(pr0_refl t5) \Rightarrow (\lambda (H9: (eq T t5 -t)).(\lambda (H10: (eq T t5 t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2) -\to (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: -T).(pr0 t2 t7))))) (\lambda (H11: (eq T t t2)).(eq_ind T t2 (\lambda (_: -T).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: -T).(pr0 t2 t7)))) (let H12 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 t2)) -H11 (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) H4) in (eq_ind T (THead -(Flat Appl) v1 (THead (Bind Abst) u t3)) (\lambda (t6: T).(ex2 T (\lambda -(t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 t6 t7)))) -(let H13 \def (eq_ind_r T t (\lambda (t6: T).(eq T t5 t6)) H9 (THead (Flat -Appl) v1 (THead (Bind Abst) u t3)) H4) in (let H14 \def (eq_ind_r T t -(\lambda (t6: T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) -\to (\forall (t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) -(\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead (Flat Appl) v1 (THead (Bind -Abst) u t3)) H4) in (ex_intro2 T (\lambda (t6: T).(pr0 (THead (Bind Abbr) v2 -t4) t6)) (\lambda (t6: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u -t3)) t6)) (THead (Bind Abbr) v2 t4) (pr0_refl (THead (Bind Abbr) v2 t4)) -(pr0_beta u v1 v2 H7 t3 t4 H8)))) t2 H12)) t (sym_eq T t t2 H11))) t5 (sym_eq -T t5 t H9) H10))) | (pr0_comp u1 u2 H9 t5 t6 H10 k) \Rightarrow (\lambda -(H11: (eq T (THead k u1 t5) t)).(\lambda (H12: (eq T (THead k u2 t6) -t2)).(eq_ind T (THead k u1 t5) (\lambda (_: T).((eq T (THead k u2 t6) t2) \to -((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind -Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H13: (eq T -(THead k u2 t6) t2)).(eq_ind T (THead k u2 t6) (\lambda (t7: T).((pr0 u1 u2) -\to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) -t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (H14: (pr0 u1 u2)).(\lambda -(H15: (pr0 t5 t6)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead -(Flat Appl) v1 (THead (Bind Abst) u t3)) t7)) H4 (THead k u1 t5) H11) in (let -H17 \def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) -with [(TSort _) \Rightarrow (Flat Appl) | (TLRef _) \Rightarrow (Flat Appl) | -(THead k0 _ _) \Rightarrow k0])) (THead (Flat Appl) v1 (THead (Bind Abst) u -t3)) (THead k u1 t5) H16) in ((let H18 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | -(TLRef _) \Rightarrow v1 | (THead _ t7 _) \Rightarrow t7])) (THead (Flat -Appl) v1 (THead (Bind Abst) u t3)) (THead k u1 t5) H16) in ((let H19 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow (THead (Bind Abst) u t3) | (TLRef _) \Rightarrow -(THead (Bind Abst) u t3) | (THead _ _ t7) \Rightarrow t7])) (THead (Flat -Appl) v1 (THead (Bind Abst) u t3)) (THead k u1 t5) H16) in (\lambda (H20: (eq -T v1 u1)).(\lambda (H21: (eq K (Flat Appl) k)).(eq_ind K (Flat Appl) (\lambda -(k0: K).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda -(t7: T).(pr0 (THead k0 u2 t6) t7)))) (let H22 \def (eq_ind_r K k (\lambda -(k0: K).(eq T (THead k0 u1 t5) t)) H11 (Flat Appl) H21) in (let H23 \def -(eq_ind_r T t5 (\lambda (t7: T).(pr0 t7 t6)) H15 (THead (Bind Abst) u t3) -H19) in (let H24 \def (match H23 in pr0 return (\lambda (t7: T).(\lambda (t8: -T).(\lambda (_: (pr0 t7 t8)).((eq T t7 (THead (Bind Abst) u t3)) \to ((eq T -t8 t6) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) -(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t6) t9)))))))) with [(pr0_refl -t7) \Rightarrow (\lambda (H24: (eq T t7 (THead (Bind Abst) u t3))).(\lambda -(H25: (eq T t7 t6)).(eq_ind T (THead (Bind Abst) u t3) (\lambda (t8: T).((eq -T t8 t6) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) -(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t6) t9))))) (\lambda (H26: (eq T -(THead (Bind Abst) u t3) t6)).(eq_ind T (THead (Bind Abst) u t3) (\lambda -(t8: T).(ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda -(t9: T).(pr0 (THead (Flat Appl) u2 t8) t9)))) (let H27 \def (eq_ind_r T t5 -(\lambda (t8: T).(eq T (THead (Flat Appl) u1 t8) t)) H22 (THead (Bind Abst) u -t3) H19) in (let H28 \def (eq_ind_r T t (\lambda (t8: T).(\forall (v: -T).((tlt v t8) \to (\forall (t9: T).((pr0 v t9) \to (\forall (t10: T).((pr0 v -t10) \to (ex2 T (\lambda (t11: T).(pr0 t9 t11)) (\lambda (t11: T).(pr0 t10 -t11)))))))))) H (THead (Flat Appl) u1 (THead (Bind Abst) u t3)) H27) in (let -H29 \def (eq_ind T v1 (\lambda (t8: T).(pr0 t8 v2)) H7 u1 H20) in (ex2_ind T -(\lambda (t8: T).(pr0 v2 t8)) (\lambda (t8: T).(pr0 u2 t8)) (ex2 T (\lambda -(t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 (THead -(Flat Appl) u2 (THead (Bind Abst) u t3)) t8))) (\lambda (x: T).(\lambda (H30: -(pr0 v2 x)).(\lambda (H31: (pr0 u2 x)).(ex_intro2 T (\lambda (t8: T).(pr0 -(THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 (THead (Flat Appl) u2 -(THead (Bind Abst) u t3)) t8)) (THead (Bind Abbr) x t4) (pr0_comp v2 x H30 t4 -t4 (pr0_refl t4) (Bind Abbr)) (pr0_beta u u2 x H31 t3 t4 H8))))) (H28 u1 -(tlt_head_sx (Flat Appl) u1 (THead (Bind Abst) u t3)) v2 H29 u2 H14))))) t6 -H26)) t7 (sym_eq T t7 (THead (Bind Abst) u t3) H24) H25))) | (pr0_comp u0 u3 -H24 t7 t8 H25 k0) \Rightarrow (\lambda (H26: (eq T (THead k0 u0 t7) (THead -(Bind Abst) u t3))).(\lambda (H27: (eq T (THead k0 u3 t8) t6)).((let H28 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow t7 | (TLRef _) \Rightarrow t7 | (THead _ _ t9) -\Rightarrow t9])) (THead k0 u0 t7) (THead (Bind Abst) u t3) H26) in ((let H29 -\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t9 _) -\Rightarrow t9])) (THead k0 u0 t7) (THead (Bind Abst) u t3) H26) in ((let H30 -\def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) -with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _ _) -\Rightarrow k1])) (THead k0 u0 t7) (THead (Bind Abst) u t3) H26) in (eq_ind K -(Bind Abst) (\lambda (k1: K).((eq T u0 u) \to ((eq T t7 t3) \to ((eq T (THead -k1 u3 t8) t6) \to ((pr0 u0 u3) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: -T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Flat -Appl) u2 t6) t9))))))))) (\lambda (H31: (eq T u0 u)).(eq_ind T u (\lambda -(t9: T).((eq T t7 t3) \to ((eq T (THead (Bind Abst) u3 t8) t6) \to ((pr0 t9 -u3) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 -t4) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t6) t10)))))))) -(\lambda (H32: (eq T t7 t3)).(eq_ind T t3 (\lambda (t9: T).((eq T (THead -(Bind Abst) u3 t8) t6) \to ((pr0 u u3) \to ((pr0 t9 t8) \to (ex2 T (\lambda -(t10: T).(pr0 (THead (Bind Abbr) v2 t4) t10)) (\lambda (t10: T).(pr0 (THead -(Flat Appl) u2 t6) t10))))))) (\lambda (H33: (eq T (THead (Bind Abst) u3 t8) -t6)).(eq_ind T (THead (Bind Abst) u3 t8) (\lambda (t9: T).((pr0 u u3) \to -((pr0 t3 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t4) -t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t9) t10)))))) (\lambda (_: -(pr0 u u3)).(\lambda (H35: (pr0 t3 t8)).(let H36 \def (eq_ind_r T t5 (\lambda -(t9: T).(eq T (THead (Flat Appl) u1 t9) t)) H22 (THead (Bind Abst) u t3) H19) -in (let H37 \def (eq_ind_r T t (\lambda (t9: T).(\forall (v: T).((tlt v t9) -\to (\forall (t10: T).((pr0 v t10) \to (\forall (t11: T).((pr0 v t11) \to -(ex2 T (\lambda (t12: T).(pr0 t10 t12)) (\lambda (t12: T).(pr0 t11 -t12)))))))))) H (THead (Flat Appl) u1 (THead (Bind Abst) u t3)) H36) in (let -H38 \def (eq_ind T v1 (\lambda (t9: T).(pr0 t9 v2)) H7 u1 H20) in (ex2_ind T -(\lambda (t9: T).(pr0 v2 t9)) (\lambda (t9: T).(pr0 u2 t9)) (ex2 T (\lambda -(t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead -(Flat Appl) u2 (THead (Bind Abst) u3 t8)) t9))) (\lambda (x: T).(\lambda -(H39: (pr0 v2 x)).(\lambda (H40: (pr0 u2 x)).(ex2_ind T (\lambda (t9: T).(pr0 -t8 t9)) (\lambda (t9: T).(pr0 t4 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead -(Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead -(Bind Abst) u3 t8)) t9))) (\lambda (x0: T).(\lambda (H41: (pr0 t8 -x0)).(\lambda (H42: (pr0 t4 x0)).(ex_intro2 T (\lambda (t9: T).(pr0 (THead -(Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead -(Bind Abst) u3 t8)) t9)) (THead (Bind Abbr) x x0) (pr0_comp v2 x H39 t4 x0 -H42 (Bind Abbr)) (pr0_beta u3 u2 x H40 t8 x0 H41))))) (H37 t3 (tlt_trans -(THead (Bind Abst) u t3) t3 (THead (Flat Appl) u1 (THead (Bind Abst) u t3)) -(tlt_head_dx (Bind Abst) u t3) (tlt_head_dx (Flat Appl) u1 (THead (Bind Abst) -u t3))) t8 H35 t4 H8))))) (H37 u1 (tlt_head_sx (Flat Appl) u1 (THead (Bind -Abst) u t3)) v2 H38 u2 H14))))))) t6 H33)) t7 (sym_eq T t7 t3 H32))) u0 -(sym_eq T u0 u H31))) k0 (sym_eq K k0 (Bind Abst) H30))) H29)) H28)) H27 H24 -H25))) | (pr0_beta u0 v0 v3 H24 t7 t8 H25) \Rightarrow (\lambda (H26: (eq T -(THead (Flat Appl) v0 (THead (Bind Abst) u0 t7)) (THead (Bind Abst) u -t3))).(\lambda (H27: (eq T (THead (Bind Abbr) v3 t8) t6)).((let H28 \def -(eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u0 t7)) (\lambda (e: -T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in -K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) -\Rightarrow True])])) I (THead (Bind Abst) u t3) H26) in (False_ind ((eq T -(THead (Bind Abbr) v3 t8) t6) \to ((pr0 v0 v3) \to ((pr0 t7 t8) \to (ex2 T -(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 -(THead (Flat Appl) u2 t6) t9)))))) H28)) H27 H24 H25))) | (pr0_upsilon b H24 -v0 v3 H25 u0 u3 H26 t7 t8 H27) \Rightarrow (\lambda (H28: (eq T (THead (Flat -Appl) v0 (THead (Bind b) u0 t7)) (THead (Bind Abst) u t3))).(\lambda (H29: -(eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v3) t8)) t6)).((let -H30 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind b) u0 t7)) (\lambda (e: -T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in -K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) -\Rightarrow True])])) I (THead (Bind Abst) u t3) H28) in (False_ind ((eq T -(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v3) t8)) t6) \to ((not -(eq B b Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3) \to ((pr0 t7 t8) \to (ex2 T -(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 -(THead (Flat Appl) u2 t6) t9)))))))) H30)) H29 H24 H25 H26 H27))) | -(pr0_delta u0 u3 H24 t7 t8 H25 w H26) \Rightarrow (\lambda (H27: (eq T (THead -(Bind Abbr) u0 t7) (THead (Bind Abst) u t3))).(\lambda (H28: (eq T (THead -(Bind Abbr) u3 w) t6)).((let H29 \def (eq_ind T (THead (Bind Abbr) u0 t7) -(\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow -(match k0 in K return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match -b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst -\Rightarrow False | Void \Rightarrow False]) | (Flat _) \Rightarrow -False])])) I (THead (Bind Abst) u t3) H27) in (False_ind ((eq T (THead (Bind -Abbr) u3 w) t6) \to ((pr0 u0 u3) \to ((pr0 t7 t8) \to ((subst0 O u3 t8 w) \to -(ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: -T).(pr0 (THead (Flat Appl) u2 t6) t9))))))) H29)) H28 H24 H25 H26))) | -(pr0_zeta b H24 t7 t8 H25 u0) \Rightarrow (\lambda (H26: (eq T (THead (Bind -b) u0 (lift (S O) O t7)) (THead (Bind Abst) u t3))).(\lambda (H27: (eq T t8 -t6)).((let H28 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat -\to nat))) (d: nat) (t9: T) on t9: T \def (match t9 with [(TSort n) -\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with -[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k0 u3 t10) -\Rightarrow (THead k0 (lref_map f d u3) (lref_map f (s k0 d) t10))]) in -lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (TLRef _) \Rightarrow -((let rec lref_map (f: ((nat \to nat))) (d: nat) (t9: T) on t9: T \def (match -t9 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef -(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | -(THead k0 u3 t10) \Rightarrow (THead k0 (lref_map f d u3) (lref_map f (s k0 -d) t10))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (THead _ _ -t9) \Rightarrow t9])) (THead (Bind b) u0 (lift (S O) O t7)) (THead (Bind -Abst) u t3) H26) in ((let H29 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) -\Rightarrow u0 | (THead _ t9 _) \Rightarrow t9])) (THead (Bind b) u0 (lift (S -O) O t7)) (THead (Bind Abst) u t3) H26) in ((let H30 \def (f_equal T B -(\lambda (e: T).(match e in T return (\lambda (_: T).B) with [(TSort _) -\Rightarrow b | (TLRef _) \Rightarrow b | (THead k0 _ _) \Rightarrow (match -k0 in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) -\Rightarrow b])])) (THead (Bind b) u0 (lift (S O) O t7)) (THead (Bind Abst) u -t3) H26) in (eq_ind B Abst (\lambda (b0: B).((eq T u0 u) \to ((eq T (lift (S -O) O t7) t3) \to ((eq T t8 t6) \to ((not (eq B b0 Abst)) \to ((pr0 t7 t8) \to -(ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: -T).(pr0 (THead (Flat Appl) u2 t6) t9))))))))) (\lambda (H31: (eq T u0 -u)).(eq_ind T u (\lambda (_: T).((eq T (lift (S O) O t7) t3) \to ((eq T t8 -t6) \to ((not (eq B Abst Abst)) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: -T).(pr0 (THead (Bind Abbr) v2 t4) t10)) (\lambda (t10: T).(pr0 (THead (Flat -Appl) u2 t6) t10)))))))) (\lambda (H32: (eq T (lift (S O) O t7) t3)).(eq_ind -T (lift (S O) O t7) (\lambda (_: T).((eq T t8 t6) \to ((not (eq B Abst Abst)) -\to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t4) -t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t6) t10))))))) (\lambda -(H33: (eq T t8 t6)).(eq_ind T t6 (\lambda (t9: T).((not (eq B Abst Abst)) \to -((pr0 t7 t9) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t4) -t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t6) t10)))))) (\lambda -(H34: (not (eq B Abst Abst))).(\lambda (_: (pr0 t7 t6)).(let H36 \def (match -(H34 (refl_equal B Abst)) in False return (\lambda (_: False).(ex2 T (\lambda -(t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead -(Flat Appl) u2 t6) t9)))) with []) in H36))) t8 (sym_eq T t8 t6 H33))) t3 -H32)) u0 (sym_eq T u0 u H31))) b (sym_eq B b Abst H30))) H29)) H28)) H27 H24 -H25))) | (pr0_tau t7 t8 H24 u0) \Rightarrow (\lambda (H25: (eq T (THead (Flat -Cast) u0 t7) (THead (Bind Abst) u t3))).(\lambda (H26: (eq T t8 t6)).((let -H27 \def (eq_ind T (THead (Flat Cast) u0 t7) (\lambda (e: T).(match e in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind Abst) u t3) H25) in (False_ind ((eq T t8 t6) \to -((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) -(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t6) t9))))) H27)) H26 H24)))]) in -(H24 (refl_equal T (THead (Bind Abst) u t3)) (refl_equal T t6))))) k H21)))) -H18)) H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_beta u0 v0 v3 H9 t5 t6 -H10) \Rightarrow (\lambda (H11: (eq T (THead (Flat Appl) v0 (THead (Bind -Abst) u0 t5)) t)).(\lambda (H12: (eq T (THead (Bind Abbr) v3 t6) t2)).(eq_ind -T (THead (Flat Appl) v0 (THead (Bind Abst) u0 t5)) (\lambda (_: T).((eq T -(THead (Bind Abbr) v3 t6) t2) \to ((pr0 v0 v3) \to ((pr0 t5 t6) \to (ex2 T -(\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 -t8))))))) (\lambda (H13: (eq T (THead (Bind Abbr) v3 t6) t2)).(eq_ind T -(THead (Bind Abbr) v3 t6) (\lambda (t7: T).((pr0 v0 v3) \to ((pr0 t5 t6) \to -(ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: -T).(pr0 t7 t8)))))) (\lambda (H14: (pr0 v0 v3)).(\lambda (H15: (pr0 t5 -t6)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1 -(THead (Bind Abst) u t3)) t7)) H4 (THead (Flat Appl) v0 (THead (Bind Abst) u0 -t5)) H11) in (let H17 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 -| (THead _ t7 _) \Rightarrow t7])) (THead (Flat Appl) v1 (THead (Bind Abst) u -t3)) (THead (Flat Appl) v0 (THead (Bind Abst) u0 t5)) H16) in ((let H18 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ _ t7) -\Rightarrow (match t7 in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t8 _) \Rightarrow t8])])) -(THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (THead (Flat Appl) v0 (THead -(Bind Abst) u0 t5)) H16) in ((let H19 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | -(TLRef _) \Rightarrow t3 | (THead _ _ t7) \Rightarrow (match t7 in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 -| (THead _ _ t8) \Rightarrow t8])])) (THead (Flat Appl) v1 (THead (Bind Abst) -u t3)) (THead (Flat Appl) v0 (THead (Bind Abst) u0 t5)) H16) in (\lambda (_: -(eq T u u0)).(\lambda (H21: (eq T v1 v0)).(let H22 \def (eq_ind_r T t -(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) -\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) -(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Appl) v0 (THead (Bind -Abst) u0 t5)) H11) in (let H23 \def (eq_ind T v1 (\lambda (t7: T).(pr0 t7 -v2)) H7 v0 H21) in (let H24 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) -H8 t5 H19) in (ex2_ind T (\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 -t6 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda -(t7: T).(pr0 (THead (Bind Abbr) v3 t6) t7))) (\lambda (x: T).(\lambda (H25: -(pr0 t4 x)).(\lambda (H26: (pr0 t6 x)).(ex2_ind T (\lambda (t7: T).(pr0 v2 -t7)) (\lambda (t7: T).(pr0 v3 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind -Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v3 t6) t7))) -(\lambda (x0: T).(\lambda (H27: (pr0 v2 x0)).(\lambda (H28: (pr0 v3 -x0)).(ex_intro2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) -(\lambda (t7: T).(pr0 (THead (Bind Abbr) v3 t6) t7)) (THead (Bind Abbr) x0 x) -(pr0_comp v2 x0 H27 t4 x H25 (Bind Abbr)) (pr0_comp v3 x0 H28 t6 x H26 (Bind -Abbr)))))) (H22 v0 (tlt_head_sx (Flat Appl) v0 (THead (Bind Abst) u0 t5)) v2 -H23 v3 H14))))) (H22 t5 (tlt_trans (THead (Bind Abst) u0 t5) t5 (THead (Flat -Appl) v0 (THead (Bind Abst) u0 t5)) (tlt_head_dx (Bind Abst) u0 t5) -(tlt_head_dx (Flat Appl) v0 (THead (Bind Abst) u0 t5))) t4 H24 t6 H15)))))))) -H18)) H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_upsilon b H9 v0 v3 H10 u1 -u2 H11 t5 t6 H12) \Rightarrow (\lambda (H13: (eq T (THead (Flat Appl) v0 -(THead (Bind b) u1 t5)) t)).(\lambda (H14: (eq T (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v3) t6)) t2)).(eq_ind T (THead (Flat Appl) v0 -(THead (Bind b) u1 t5)) (\lambda (_: T).((eq T (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v3) t6)) t2) \to ((not (eq B b Abst)) \to ((pr0 v0 -v3) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead -(Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H15: -(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v3) t6)) -t2)).(eq_ind T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v3) t6)) -(\lambda (t7: T).((not (eq B b Abst)) \to ((pr0 v0 v3) \to ((pr0 u1 u2) \to -((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) -(\lambda (t8: T).(pr0 t7 t8)))))))) (\lambda (H16: (not (eq B b -Abst))).(\lambda (_: (pr0 v0 v3)).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (pr0 -t5 t6)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) -v1 (THead (Bind Abst) u t3)) t7)) H4 (THead (Flat Appl) v0 (THead (Bind b) u1 -t5)) H13) in (let H21 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 -| (THead _ t7 _) \Rightarrow t7])) (THead (Flat Appl) v1 (THead (Bind Abst) u -t3)) (THead (Flat Appl) v0 (THead (Bind b) u1 t5)) H20) in ((let H22 \def -(f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with -[(TSort _) \Rightarrow Abst | (TLRef _) \Rightarrow Abst | (THead _ _ t7) -\Rightarrow (match t7 in T return (\lambda (_: T).B) with [(TSort _) -\Rightarrow Abst | (TLRef _) \Rightarrow Abst | (THead k _ _) \Rightarrow -(match k in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | -(Flat _) \Rightarrow Abst])])])) (THead (Flat Appl) v1 (THead (Bind Abst) u -t3)) (THead (Flat Appl) v0 (THead (Bind b) u1 t5)) H20) in ((let H23 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ _ t7) -\Rightarrow (match t7 in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t8 _) \Rightarrow t8])])) -(THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (THead (Flat Appl) v0 (THead -(Bind b) u1 t5)) H20) in ((let H24 \def (f_equal T T (\lambda (e: T).(match e -in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) -\Rightarrow t3 | (THead _ _ t7) \Rightarrow (match t7 in T return (\lambda -(_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead -_ _ t8) \Rightarrow t8])])) (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) -(THead (Flat Appl) v0 (THead (Bind b) u1 t5)) H20) in (\lambda (_: (eq T u -u1)).(\lambda (H26: (eq B Abst b)).(\lambda (_: (eq T v1 v0)).(eq_ind B Abst -(\lambda (b0: B).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) -(\lambda (t7: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O -v3) t6)) t7)))) (let H28 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 -Abst))) H16 Abst H26) in (let H29 \def (match (H28 (refl_equal B Abst)) in -False return (\lambda (_: False).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind -Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abst) u2 (THead (Flat -Appl) (lift (S O) O v3) t6)) t7)))) with []) in H29)) b H26))))) H23)) H22)) -H21))))))) t2 H15)) t H13 H14 H9 H10 H11 H12))) | (pr0_delta u1 u2 H9 t5 t6 -H10 w H11) \Rightarrow (\lambda (H12: (eq T (THead (Bind Abbr) u1 t5) -t)).(\lambda (H13: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind -Abbr) u1 t5) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t2) \to ((pr0 u1 -u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T (\lambda (t8: T).(pr0 -(THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8)))))))) (\lambda -(H14: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u2 w) -(\lambda (t7: T).((pr0 u1 u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to -(ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: -T).(pr0 t7 t8))))))) (\lambda (_: (pr0 u1 u2)).(\lambda (_: (pr0 t5 -t6)).(\lambda (_: (subst0 O u2 t6 w)).(let H18 \def (eq_ind_r T t (\lambda -(t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) t7)) H4 (THead -(Bind Abbr) u1 t5) H12) in (let H19 \def (eq_ind T (THead (Flat Appl) v1 -(THead (Bind Abst) u t3)) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind -Abbr) u1 t5) H18) in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind -Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7))) -H19)))))) t2 H14)) t H12 H13 H9 H10 H11))) | (pr0_zeta b H9 t5 t6 H10 u0) -\Rightarrow (\lambda (H11: (eq T (THead (Bind b) u0 (lift (S O) O t5)) -t)).(\lambda (H12: (eq T t6 t2)).(eq_ind T (THead (Bind b) u0 (lift (S O) O -t5)) (\lambda (_: T).((eq T t6 t2) \to ((not (eq B b Abst)) \to ((pr0 t5 t6) -\to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: -T).(pr0 t2 t8))))))) (\lambda (H13: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: -T).((not (eq B b Abst)) \to ((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 -(THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (_: -(not (eq B b Abst))).(\lambda (_: (pr0 t5 t2)).(let H16 \def (eq_ind_r T t -(\lambda (t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) t7)) -H4 (THead (Bind b) u0 (lift (S O) O t5)) H11) in (let H17 \def (eq_ind T -(THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (\lambda (ee: T).(match ee in -T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind b) u0 (lift (S O) O t5)) H16) in (False_ind (ex2 T -(\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 t2 -t7))) H17))))) t6 (sym_eq T t6 t2 H13))) t H11 H12 H9 H10))) | (pr0_tau t5 t6 -H9 u0) \Rightarrow (\lambda (H10: (eq T (THead (Flat Cast) u0 t5) -t)).(\lambda (H11: (eq T t6 t2)).(eq_ind T (THead (Flat Cast) u0 t5) (\lambda -(_: T).((eq T t6 t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead -(Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H12: (eq -T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8: -T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))) -(\lambda (_: (pr0 t5 t2)).(let H14 \def (eq_ind_r T t (\lambda (t7: T).(eq T -(THead (Flat Appl) v1 (THead (Bind Abst) u t3)) t7)) H4 (THead (Flat Cast) u0 -t5) H10) in (let H15 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) -u t3)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) -\Rightarrow False | (Flat f) \Rightarrow (match f in F return (\lambda (_: -F).Prop) with [Appl \Rightarrow True | Cast \Rightarrow False])])])) I (THead -(Flat Cast) u0 t5) H14) in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead -(Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 t2 t7))) H15)))) t6 (sym_eq T -t6 t2 H12))) t H10 H11 H9)))]) in (H9 (refl_equal T t) (refl_equal T t2))))) -t1 H6)) t H4 H5 H2 H3))) | (pr0_upsilon b H2 v1 v2 H3 u1 u2 H4 t3 t4 H5) -\Rightarrow (\lambda (H6: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) -t)).(\lambda (H7: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O -v2) t4)) t1)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) -(\lambda (_: T).((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O -v2) t4)) t1) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to -((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 -t2 t6))))))))) (\lambda (H8: (eq T (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4)) t1)).(eq_ind T (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4)) (\lambda (t5: T).((not (eq B b Abst)) \to ((pr0 v1 v2) -\to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t5 t6)) -(\lambda (t6: T).(pr0 t2 t6)))))))) (\lambda (H9: (not (eq B b -Abst))).(\lambda (H10: (pr0 v1 v2)).(\lambda (H11: (pr0 u1 u2)).(\lambda -(H12: (pr0 t3 t4)).(let H13 \def (match H1 in pr0 return (\lambda (t5: -T).(\lambda (t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 t) \to ((eq T t6 t2) -\to (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift -(S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 t2 t7)))))))) with [(pr0_refl t5) -\Rightarrow (\lambda (H13: (eq T t5 t)).(\lambda (H14: (eq T t5 t2)).(eq_ind -T t (\lambda (t6: T).((eq T t6 t2) \to (ex2 T (\lambda (t7: T).(pr0 (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: -T).(pr0 t2 t7))))) (\lambda (H15: (eq T t t2)).(eq_ind T t2 (\lambda (_: -T).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift -(S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let H16 \def (eq_ind_r -T t (\lambda (t6: T).(eq T t6 t2)) H15 (THead (Flat Appl) v1 (THead (Bind b) -u1 t3)) H6) in (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) -(\lambda (t6: T).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 t6 t7)))) (let H17 -\def (eq_ind_r T t (\lambda (t6: T).(eq T t5 t6)) H13 (THead (Flat Appl) v1 -(THead (Bind b) u1 t3)) H6) in (let H18 \def (eq_ind_r T t (\lambda (t6: -T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall -(t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: -T).(pr0 t8 t9)))))))))) H (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) H6) -in (ex2_sym T (pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 t3))) (pr0 (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) -(pr0_confluence__pr0_cong_upsilon_refl b H9 u1 u2 H11 t3 t4 H12 v1 v2 v2 H10 -(pr0_refl v2))))) t2 H16)) t (sym_eq T t t2 H15))) t5 (sym_eq T t5 t H13) -H14))) | (pr0_comp u0 u3 H13 t5 t6 H14 k) \Rightarrow (\lambda (H15: (eq T -(THead k u0 t5) t)).(\lambda (H16: (eq T (THead k u3 t6) t2)).(eq_ind T -(THead k u0 t5) (\lambda (_: T).((eq T (THead k u3 t6) t2) \to ((pr0 u0 u3) -\to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) -(\lambda (H17: (eq T (THead k u3 t6) t2)).(eq_ind T (THead k u3 t6) (\lambda -(t7: T).((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: -T).(pr0 t7 t8)))))) (\lambda (H18: (pr0 u0 u3)).(\lambda (H19: (pr0 t5 -t6)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1 -(THead (Bind b) u1 t3)) t7)) H6 (THead k u0 t5) H15) in (let H21 \def -(f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with -[(TSort _) \Rightarrow (Flat Appl) | (TLRef _) \Rightarrow (Flat Appl) | -(THead k0 _ _) \Rightarrow k0])) (THead (Flat Appl) v1 (THead (Bind b) u1 -t3)) (THead k u0 t5) H20) in ((let H22 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | -(TLRef _) \Rightarrow v1 | (THead _ t7 _) \Rightarrow t7])) (THead (Flat -Appl) v1 (THead (Bind b) u1 t3)) (THead k u0 t5) H20) in ((let H23 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow (THead (Bind b) u1 t3) | (TLRef _) \Rightarrow (THead -(Bind b) u1 t3) | (THead _ _ t7) \Rightarrow t7])) (THead (Flat Appl) v1 -(THead (Bind b) u1 t3)) (THead k u0 t5) H20) in (\lambda (H24: (eq T v1 -u0)).(\lambda (H25: (eq K (Flat Appl) k)).(eq_ind K (Flat Appl) (\lambda (k0: -K).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift -(S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 (THead k0 u3 t6) t7)))) (let H26 -\def (eq_ind_r K k (\lambda (k0: K).(eq T (THead k0 u0 t5) t)) H15 (Flat -Appl) H25) in (let H27 \def (eq_ind_r T t5 (\lambda (t7: T).(pr0 t7 t6)) H19 -(THead (Bind b) u1 t3) H23) in (let H28 \def (match H27 in pr0 return -(\lambda (t7: T).(\lambda (t8: T).(\lambda (_: (pr0 t7 t8)).((eq T t7 (THead -(Bind b) u1 t3)) \to ((eq T t8 t6) \to (ex2 T (\lambda (t9: T).(pr0 (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: -T).(pr0 (THead (Flat Appl) u3 t6) t9)))))))) with [(pr0_refl t7) \Rightarrow -(\lambda (H28: (eq T t7 (THead (Bind b) u1 t3))).(\lambda (H29: (eq T t7 -t6)).(eq_ind T (THead (Bind b) u1 t3) (\lambda (t8: T).((eq T t8 t6) \to (ex2 -T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O -v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9))))) -(\lambda (H30: (eq T (THead (Bind b) u1 t3) t6)).(eq_ind T (THead (Bind b) u1 -t3) (\lambda (t8: T).(ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat -Appl) u3 t8) t9)))) (let H31 \def (eq_ind_r T t5 (\lambda (t8: T).(eq T -(THead (Flat Appl) u0 t8) t)) H26 (THead (Bind b) u1 t3) H23) in (let H32 -\def (eq_ind_r T t (\lambda (t8: T).(\forall (v: T).((tlt v t8) \to (\forall -(t9: T).((pr0 v t9) \to (\forall (t10: T).((pr0 v t10) \to (ex2 T (\lambda -(t11: T).(pr0 t9 t11)) (\lambda (t11: T).(pr0 t10 t11)))))))))) H (THead -(Flat Appl) u0 (THead (Bind b) u1 t3)) H31) in (let H33 \def (eq_ind T v1 -(\lambda (t8: T).(pr0 t8 v2)) H10 u0 H24) in (ex2_ind T (\lambda (t8: T).(pr0 -v2 t8)) (\lambda (t8: T).(pr0 u3 t8)) (ex2 T (\lambda (t8: T).(pr0 (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: -T).(pr0 (THead (Flat Appl) u3 (THead (Bind b) u1 t3)) t8))) (\lambda (x: -T).(\lambda (H34: (pr0 v2 x)).(\lambda (H35: (pr0 u3 x)).(ex2_sym T (pr0 -(THead (Flat Appl) u3 (THead (Bind b) u1 t3))) (pr0 (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t4))) (pr0_confluence__pr0_cong_upsilon_refl b -H9 u1 u2 H11 t3 t4 H12 u3 v2 x H35 H34))))) (H32 u0 (tlt_head_sx (Flat Appl) -u0 (THead (Bind b) u1 t3)) v2 H33 u3 H18))))) t6 H30)) t7 (sym_eq T t7 (THead -(Bind b) u1 t3) H28) H29))) | (pr0_comp u4 u5 H28 t7 t8 H29 k0) \Rightarrow -(\lambda (H30: (eq T (THead k0 u4 t7) (THead (Bind b) u1 t3))).(\lambda (H31: -(eq T (THead k0 u5 t8) t6)).((let H32 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t7 | -(TLRef _) \Rightarrow t7 | (THead _ _ t9) \Rightarrow t9])) (THead k0 u4 t7) -(THead (Bind b) u1 t3) H30) in ((let H33 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u4 | -(TLRef _) \Rightarrow u4 | (THead _ t9 _) \Rightarrow t9])) (THead k0 u4 t7) -(THead (Bind b) u1 t3) H30) in ((let H34 \def (f_equal T K (\lambda (e: -T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k0 | -(TLRef _) \Rightarrow k0 | (THead k1 _ _) \Rightarrow k1])) (THead k0 u4 t7) -(THead (Bind b) u1 t3) H30) in (eq_ind K (Bind b) (\lambda (k1: K).((eq T u4 -u1) \to ((eq T t7 t3) \to ((eq T (THead k1 u5 t8) t6) \to ((pr0 u4 u5) \to -((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 -t6) t9))))))))) (\lambda (H35: (eq T u4 u1)).(eq_ind T u1 (\lambda (t9: -T).((eq T t7 t3) \to ((eq T (THead (Bind b) u5 t8) t6) \to ((pr0 t9 u5) \to -((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat -Appl) u3 t6) t10)))))))) (\lambda (H36: (eq T t7 t3)).(eq_ind T t3 (\lambda -(t9: T).((eq T (THead (Bind b) u5 t8) t6) \to ((pr0 u1 u5) \to ((pr0 t9 t8) -\to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift -(S O) O v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6) -t10))))))) (\lambda (H37: (eq T (THead (Bind b) u5 t8) t6)).(eq_ind T (THead -(Bind b) u5 t8) (\lambda (t9: T).((pr0 u1 u5) \to ((pr0 t3 t8) \to (ex2 T -(\lambda (t10: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O -v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t9) t10)))))) -(\lambda (H38: (pr0 u1 u5)).(\lambda (H39: (pr0 t3 t8)).(let H40 \def -(eq_ind_r T t5 (\lambda (t9: T).(eq T (THead (Flat Appl) u0 t9) t)) H26 -(THead (Bind b) u1 t3) H23) in (let H41 \def (eq_ind_r T t (\lambda (t9: -T).(\forall (v: T).((tlt v t9) \to (\forall (t10: T).((pr0 v t10) \to -(\forall (t11: T).((pr0 v t11) \to (ex2 T (\lambda (t12: T).(pr0 t10 t12)) -(\lambda (t12: T).(pr0 t11 t12)))))))))) H (THead (Flat Appl) u0 (THead (Bind -b) u1 t3)) H40) in (let H42 \def (eq_ind T v1 (\lambda (t9: T).(pr0 t9 v2)) -H10 u0 H24) in (ex2_ind T (\lambda (t9: T).(pr0 v2 t9)) (\lambda (t9: T).(pr0 -u3 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 -(THead (Bind b) u5 t8)) t9))) (\lambda (x: T).(\lambda (H43: (pr0 v2 -x)).(\lambda (H44: (pr0 u3 x)).(ex2_ind T (\lambda (t9: T).(pr0 t8 t9)) -(\lambda (t9: T).(pr0 t4 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead -(Flat Appl) u3 (THead (Bind b) u5 t8)) t9))) (\lambda (x0: T).(\lambda (H45: -(pr0 t8 x0)).(\lambda (H46: (pr0 t4 x0)).(ex2_ind T (\lambda (t9: T).(pr0 u5 -t9)) (\lambda (t9: T).(pr0 u2 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Bind -b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 -(THead (Flat Appl) u3 (THead (Bind b) u5 t8)) t9))) (\lambda (x1: T).(\lambda -(H47: (pr0 u5 x1)).(\lambda (H48: (pr0 u2 x1)).(ex2_sym T (pr0 (THead (Flat -Appl) u3 (THead (Bind b) u5 t8))) (pr0 (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4))) (pr0_confluence__pr0_cong_upsilon_cong b H9 u3 v2 x -H44 H43 t8 t4 x0 H45 H46 u5 u2 x1 H47 H48))))) (H41 u1 (tlt_trans (THead -(Bind b) u1 t3) u1 (THead (Flat Appl) u0 (THead (Bind b) u1 t3)) (tlt_head_sx -(Bind b) u1 t3) (tlt_head_dx (Flat Appl) u0 (THead (Bind b) u1 t3))) u5 H38 -u2 H11))))) (H41 t3 (tlt_trans (THead (Bind b) u1 t3) t3 (THead (Flat Appl) -u0 (THead (Bind b) u1 t3)) (tlt_head_dx (Bind b) u1 t3) (tlt_head_dx (Flat -Appl) u0 (THead (Bind b) u1 t3))) t8 H39 t4 H12))))) (H41 u0 (tlt_head_sx -(Flat Appl) u0 (THead (Bind b) u1 t3)) v2 H42 u3 H18))))))) t6 H37)) t7 -(sym_eq T t7 t3 H36))) u4 (sym_eq T u4 u1 H35))) k0 (sym_eq K k0 (Bind b) -H34))) H33)) H32)) H31 H28 H29))) | (pr0_beta u v0 v3 H28 t7 t8 H29) -\Rightarrow (\lambda (H30: (eq T (THead (Flat Appl) v0 (THead (Bind Abst) u -t7)) (THead (Bind b) u1 t3))).(\lambda (H31: (eq T (THead (Bind Abbr) v3 t8) -t6)).((let H32 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u t7)) -(\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow -(match k0 in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False -| (Flat _) \Rightarrow True])])) I (THead (Bind b) u1 t3) H30) in (False_ind -((eq T (THead (Bind Abbr) v3 t8) t6) \to ((pr0 v0 v3) \to ((pr0 t7 t8) \to -(ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9)))))) -H32)) H31 H28 H29))) | (pr0_upsilon b0 H28 v0 v3 H29 u4 u5 H30 t7 t8 H31) -\Rightarrow (\lambda (H32: (eq T (THead (Flat Appl) v0 (THead (Bind b0) u4 -t7)) (THead (Bind b) u1 t3))).(\lambda (H33: (eq T (THead (Bind b0) u5 (THead -(Flat Appl) (lift (S O) O v3) t8)) t6)).((let H34 \def (eq_ind T (THead (Flat -Appl) v0 (THead (Bind b0) u4 t7)) (\lambda (e: T).(match e in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind b) u1 t3) H32) in (False_ind ((eq T (THead (Bind b0) -u5 (THead (Flat Appl) (lift (S O) O v3) t8)) t6) \to ((not (eq B b0 Abst)) -\to ((pr0 v0 v3) \to ((pr0 u4 u5) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: -T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) -(\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9)))))))) H34)) H33 H28 H29 -H30 H31))) | (pr0_delta u4 u5 H28 t7 t8 H29 w H30) \Rightarrow (\lambda (H31: -(eq T (THead (Bind Abbr) u4 t7) (THead (Bind b) u1 t3))).(\lambda (H32: (eq T -(THead (Bind Abbr) u5 w) t6)).((let H33 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t7 | -(TLRef _) \Rightarrow t7 | (THead _ _ t9) \Rightarrow t9])) (THead (Bind -Abbr) u4 t7) (THead (Bind b) u1 t3) H31) in ((let H34 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow u4 | (TLRef _) \Rightarrow u4 | (THead _ t9 _) \Rightarrow t9])) -(THead (Bind Abbr) u4 t7) (THead (Bind b) u1 t3) H31) in ((let H35 \def -(f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with -[(TSort _) \Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | (THead k0 _ _) -\Rightarrow (match k0 in K return (\lambda (_: K).B) with [(Bind b0) -\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (THead (Bind Abbr) u4 t7) -(THead (Bind b) u1 t3) H31) in (eq_ind B Abbr (\lambda (b0: B).((eq T u4 u1) -\to ((eq T t7 t3) \to ((eq T (THead (Bind Abbr) u5 w) t6) \to ((pr0 u4 u5) -\to ((pr0 t7 t8) \to ((subst0 O u5 t8 w) \to (ex2 T (\lambda (t9: T).(pr0 -(THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda -(t9: T).(pr0 (THead (Flat Appl) u3 t6) t9)))))))))) (\lambda (H36: (eq T u4 -u1)).(eq_ind T u1 (\lambda (t9: T).((eq T t7 t3) \to ((eq T (THead (Bind -Abbr) u5 w) t6) \to ((pr0 t9 u5) \to ((pr0 t7 t8) \to ((subst0 O u5 t8 w) \to -(ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift -(S O) O v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6) -t10))))))))) (\lambda (H37: (eq T t7 t3)).(eq_ind T t3 (\lambda (t9: T).((eq -T (THead (Bind Abbr) u5 w) t6) \to ((pr0 u1 u5) \to ((pr0 t9 t8) \to ((subst0 -O u5 t8 w) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) u2 (THead -(Flat Appl) (lift (S O) O v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat -Appl) u3 t6) t10)))))))) (\lambda (H38: (eq T (THead (Bind Abbr) u5 w) -t6)).(eq_ind T (THead (Bind Abbr) u5 w) (\lambda (t9: T).((pr0 u1 u5) \to -((pr0 t3 t8) \to ((subst0 O u5 t8 w) \to (ex2 T (\lambda (t10: T).(pr0 (THead -(Bind Abbr) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t10)) (\lambda (t10: -T).(pr0 (THead (Flat Appl) u3 t9) t10))))))) (\lambda (H39: (pr0 u1 -u5)).(\lambda (H40: (pr0 t3 t8)).(\lambda (H41: (subst0 O u5 t8 w)).(let H42 -\def (eq_ind_r B b (\lambda (b0: B).(eq T (THead (Bind b0) u1 t3) t5)) H23 -Abbr H35) in (let H43 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 -Abst))) H9 Abbr H35) in (let H44 \def (eq_ind_r B b (\lambda (b0: B).(eq T -(THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t1)) H8 Abbr -H35) in (let H45 \def (eq_ind_r T t5 (\lambda (t9: T).(eq T (THead (Flat -Appl) u0 t9) t)) H26 (THead (Bind Abbr) u1 t3) H42) in (let H46 \def -(eq_ind_r T t (\lambda (t9: T).(\forall (v: T).((tlt v t9) \to (\forall (t10: -T).((pr0 v t10) \to (\forall (t11: T).((pr0 v t11) \to (ex2 T (\lambda (t12: -T).(pr0 t10 t12)) (\lambda (t12: T).(pr0 t11 t12)))))))))) H (THead (Flat -Appl) u0 (THead (Bind Abbr) u1 t3)) H45) in (let H47 \def (eq_ind T v1 -(\lambda (t9: T).(pr0 t9 v2)) H10 u0 H24) in (ex2_ind T (\lambda (t9: T).(pr0 -v2 t9)) (\lambda (t9: T).(pr0 u3 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead -(Bind Abbr) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: -T).(pr0 (THead (Flat Appl) u3 (THead (Bind Abbr) u5 w)) t9))) (\lambda (x: -T).(\lambda (H48: (pr0 v2 x)).(\lambda (H49: (pr0 u3 x)).(ex2_ind T (\lambda -(t9: T).(pr0 t8 t9)) (\lambda (t9: T).(pr0 t4 t9)) (ex2 T (\lambda (t9: -T).(pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) -(\lambda (t9: T).(pr0 (THead (Flat Appl) u3 (THead (Bind Abbr) u5 w)) t9))) -(\lambda (x0: T).(\lambda (H50: (pr0 t8 x0)).(\lambda (H51: (pr0 t4 -x0)).(ex2_ind T (\lambda (t9: T).(pr0 u5 t9)) (\lambda (t9: T).(pr0 u2 t9)) -(ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift -(S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 (THead -(Bind Abbr) u5 w)) t9))) (\lambda (x1: T).(\lambda (H52: (pr0 u5 -x1)).(\lambda (H53: (pr0 u2 x1)).(ex2_sym T (pr0 (THead (Flat Appl) u3 (THead -(Bind Abbr) u5 w))) (pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift (S O) -O v2) t4))) (pr0_confluence__pr0_cong_upsilon_delta H43 u5 t8 w H41 u3 v2 x -H49 H48 t4 x0 H50 H51 u2 x1 H52 H53))))) (H46 u1 (tlt_trans (THead (Bind -Abbr) u1 t3) u1 (THead (Flat Appl) u0 (THead (Bind Abbr) u1 t3)) (tlt_head_sx -(Bind Abbr) u1 t3) (tlt_head_dx (Flat Appl) u0 (THead (Bind Abbr) u1 t3))) u5 -H39 u2 H11))))) (H46 t3 (tlt_trans (THead (Bind Abbr) u1 t3) t3 (THead (Flat -Appl) u0 (THead (Bind Abbr) u1 t3)) (tlt_head_dx (Bind Abbr) u1 t3) -(tlt_head_dx (Flat Appl) u0 (THead (Bind Abbr) u1 t3))) t8 H40 t4 H12))))) -(H46 u0 (tlt_head_sx (Flat Appl) u0 (THead (Bind Abbr) u1 t3)) v2 H47 u3 -H18))))))))))) t6 H38)) t7 (sym_eq T t7 t3 H37))) u4 (sym_eq T u4 u1 H36))) b -H35)) H34)) H33)) H32 H28 H29 H30))) | (pr0_zeta b0 H28 t7 t8 H29 u) -\Rightarrow (\lambda (H30: (eq T (THead (Bind b0) u (lift (S O) O t7)) (THead -(Bind b) u1 t3))).(\lambda (H31: (eq T t8 t6)).((let H32 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t9: T) on t9: T -\def (match t9 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow -(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) -| (THead k0 u4 t10) \Rightarrow (THead k0 (lref_map f d u4) (lref_map f (s k0 -d) t10))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (TLRef _) -\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t9: T) on t9: T -\def (match t9 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow -(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) -| (THead k0 u4 t10) \Rightarrow (THead k0 (lref_map f d u4) (lref_map f (s k0 -d) t10))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (THead _ _ -t9) \Rightarrow t9])) (THead (Bind b0) u (lift (S O) O t7)) (THead (Bind b) -u1 t3) H30) in ((let H33 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) -\Rightarrow u | (THead _ t9 _) \Rightarrow t9])) (THead (Bind b0) u (lift (S -O) O t7)) (THead (Bind b) u1 t3) H30) in ((let H34 \def (f_equal T B (\lambda -(e: T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b0 -| (TLRef _) \Rightarrow b0 | (THead k0 _ _) \Rightarrow (match k0 in K return -(\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow -b0])])) (THead (Bind b0) u (lift (S O) O t7)) (THead (Bind b) u1 t3) H30) in -(eq_ind B b (\lambda (b1: B).((eq T u u1) \to ((eq T (lift (S O) O t7) t3) -\to ((eq T t8 t6) \to ((not (eq B b1 Abst)) \to ((pr0 t7 t8) \to (ex2 T -(\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) -t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9))))))))) -(\lambda (H35: (eq T u u1)).(eq_ind T u1 (\lambda (_: T).((eq T (lift (S O) O -t7) t3) \to ((eq T t8 t6) \to ((not (eq B b Abst)) \to ((pr0 t7 t8) \to (ex2 -T (\lambda (t10: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O -v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6) t10)))))))) -(\lambda (H36: (eq T (lift (S O) O t7) t3)).(eq_ind T (lift (S O) O t7) -(\lambda (_: T).((eq T t8 t6) \to ((not (eq B b Abst)) \to ((pr0 t7 t8) \to -(ex2 T (\lambda (t10: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6) -t10))))))) (\lambda (H37: (eq T t8 t6)).(eq_ind T t6 (\lambda (t9: T).((not -(eq B b Abst)) \to ((pr0 t7 t9) \to (ex2 T (\lambda (t10: T).(pr0 (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t10)) (\lambda (t10: -T).(pr0 (THead (Flat Appl) u3 t6) t10)))))) (\lambda (H38: (not (eq B b -Abst))).(\lambda (H39: (pr0 t7 t6)).(let H40 \def (eq_ind_r T t3 (\lambda -(t9: T).(eq T (THead (Bind b) u1 t9) t5)) H23 (lift (S O) O t7) H36) in (let -H41 \def (eq_ind_r T t5 (\lambda (t9: T).(eq T (THead (Flat Appl) u0 t9) t)) -H26 (THead (Bind b) u1 (lift (S O) O t7)) H40) in (let H42 \def (eq_ind_r T t -(\lambda (t9: T).(\forall (v: T).((tlt v t9) \to (\forall (t10: T).((pr0 v -t10) \to (\forall (t11: T).((pr0 v t11) \to (ex2 T (\lambda (t12: T).(pr0 t10 -t12)) (\lambda (t12: T).(pr0 t11 t12)))))))))) H (THead (Flat Appl) u0 (THead -(Bind b) u1 (lift (S O) O t7))) H41) in (let H43 \def (eq_ind_r T t3 (\lambda -(t9: T).(pr0 t9 t4)) H12 (lift (S O) O t7) H36) in (ex2_ind T (\lambda (t9: -T).(eq T t4 (lift (S O) O t9))) (\lambda (t9: T).(pr0 t7 t9)) (ex2 T (\lambda -(t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9))) (\lambda (x: -T).(\lambda (H44: (eq T t4 (lift (S O) O x))).(\lambda (H45: (pr0 t7 -x)).(eq_ind_r T (lift (S O) O x) (\lambda (t9: T).(ex2 T (\lambda (t10: -T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t9)) t10)) -(\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6) t10)))) (let H46 \def -(eq_ind T v1 (\lambda (t9: T).(pr0 t9 v2)) H10 u0 H24) in (ex2_ind T (\lambda -(t9: T).(pr0 v2 t9)) (\lambda (t9: T).(pr0 u3 t9)) (ex2 T (\lambda (t9: -T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (lift (S O) O -x))) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9))) (\lambda (x0: -T).(\lambda (H47: (pr0 v2 x0)).(\lambda (H48: (pr0 u3 x0)).(ex2_ind T -(\lambda (t9: T).(pr0 x t9)) (\lambda (t9: T).(pr0 t6 t9)) (ex2 T (\lambda -(t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (lift (S -O) O x))) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9))) (\lambda -(x1: T).(\lambda (H49: (pr0 x x1)).(\lambda (H50: (pr0 t6 x1)).(ex2_sym T -(pr0 (THead (Flat Appl) u3 t6)) (pr0 (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) (lift (S O) O x)))) (pr0_confluence__pr0_cong_upsilon_zeta -b H38 u1 u2 H11 u3 v2 x0 H48 H47 x t6 x1 H49 H50))))) (H42 t7 (tlt_trans -(THead (Bind b) u1 (lift (S O) O t7)) t7 (THead (Flat Appl) u0 (THead (Bind -b) u1 (lift (S O) O t7))) (lift_tlt_dx (Bind b) u1 t7 (S O) O) (tlt_head_dx -(Flat Appl) u0 (THead (Bind b) u1 (lift (S O) O t7)))) x H45 t6 H39))))) (H42 -u0 (tlt_head_sx (Flat Appl) u0 (THead (Bind b) u1 (lift (S O) O t7))) v2 H46 -u3 H18))) t4 H44)))) (pr0_gen_lift t7 t4 (S O) O H43)))))))) t8 (sym_eq T t8 -t6 H37))) t3 H36)) u (sym_eq T u u1 H35))) b0 (sym_eq B b0 b H34))) H33)) -H32)) H31 H28 H29))) | (pr0_tau t7 t8 H28 u) \Rightarrow (\lambda (H29: (eq T -(THead (Flat Cast) u t7) (THead (Bind b) u1 t3))).(\lambda (H30: (eq T t8 -t6)).((let H31 \def (eq_ind T (THead (Flat Cast) u t7) (\lambda (e: T).(match -e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | -(TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K -return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) -\Rightarrow True])])) I (THead (Bind b) u1 t3) H29) in (False_ind ((eq T t8 -t6) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead -(Flat Appl) u3 t6) t9))))) H31)) H30 H28)))]) in (H28 (refl_equal T (THead -(Bind b) u1 t3)) (refl_equal T t6))))) k H25)))) H22)) H21))))) t2 H17)) t -H15 H16 H13 H14))) | (pr0_beta u v0 v3 H13 t5 t6 H14) \Rightarrow (\lambda -(H15: (eq T (THead (Flat Appl) v0 (THead (Bind Abst) u t5)) t)).(\lambda -(H16: (eq T (THead (Bind Abbr) v3 t6) t2)).(eq_ind T (THead (Flat Appl) v0 -(THead (Bind Abst) u t5)) (\lambda (_: T).((eq T (THead (Bind Abbr) v3 t6) -t2) \to ((pr0 v0 v3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: -T).(pr0 t2 t8))))))) (\lambda (H17: (eq T (THead (Bind Abbr) v3 t6) -t2)).(eq_ind T (THead (Bind Abbr) v3 t6) (\lambda (t7: T).((pr0 v0 v3) \to -((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda -(_: (pr0 v0 v3)).(\lambda (_: (pr0 t5 t6)).(let H20 \def (eq_ind_r T t -(\lambda (t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) t7)) H6 -(THead (Flat Appl) v0 (THead (Bind Abst) u t5)) H15) in (let H21 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t7 _) -\Rightarrow t7])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat -Appl) v0 (THead (Bind Abst) u t5)) H20) in ((let H22 \def (f_equal T B -(\lambda (e: T).(match e in T return (\lambda (_: T).B) with [(TSort _) -\Rightarrow b | (TLRef _) \Rightarrow b | (THead _ _ t7) \Rightarrow (match -t7 in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | (TLRef _) -\Rightarrow b | (THead k _ _) \Rightarrow (match k in K return (\lambda (_: -K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow b])])])) (THead -(Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) v0 (THead (Bind -Abst) u t5)) H20) in ((let H23 \def (f_equal T T (\lambda (e: T).(match e in -T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) -\Rightarrow u1 | (THead _ _ t7) \Rightarrow (match t7 in T return (\lambda -(_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead -_ t8 _) \Rightarrow t8])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) -(THead (Flat Appl) v0 (THead (Bind Abst) u t5)) H20) in ((let H24 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7) -\Rightarrow (match t7 in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t8) \Rightarrow -t8])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) v0 -(THead (Bind Abst) u t5)) H20) in (\lambda (_: (eq T u1 u)).(\lambda (H26: -(eq B b Abst)).(\lambda (H27: (eq T v1 v0)).(let H28 \def (eq_ind_r T t -(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) -\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) -(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Appl) v0 (THead (Bind -Abst) u t5)) H15) in (let H29 \def (eq_ind T v1 (\lambda (t7: T).(pr0 t7 v2)) -H10 v0 H27) in (eq_ind_r B Abst (\lambda (b0: B).(ex2 T (\lambda (t7: T).(pr0 -(THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda -(t7: T).(pr0 (THead (Bind Abbr) v3 t6) t7)))) (let H30 \def (eq_ind B b -(\lambda (b0: B).(not (eq B b0 Abst))) H9 Abst H26) in (let H31 \def (match -(H30 (refl_equal B Abst)) in False return (\lambda (_: False).(ex2 T (\lambda -(t7: T).(pr0 (THead (Bind Abst) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v3 t6) t7)))) with []) in H31)) -b H26))))))) H23)) H22)) H21))))) t2 H17)) t H15 H16 H13 H14))) | -(pr0_upsilon b0 H13 v0 v3 H14 u0 u3 H15 t5 t6 H16) \Rightarrow (\lambda (H17: -(eq T (THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) t)).(\lambda (H18: (eq T -(THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) t2)).(eq_ind T -(THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) (\lambda (_: T).((eq T (THead -(Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) t2) \to ((not (eq B b0 -Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda -(t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -t8)) (\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H19: (eq T (THead (Bind -b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) t2)).(eq_ind T (THead (Bind -b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) (\lambda (t7: T).((not (eq B -b0 Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T -(\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) -t4)) t8)) (\lambda (t8: T).(pr0 t7 t8)))))))) (\lambda (_: (not (eq B b0 -Abst))).(\lambda (H21: (pr0 v0 v3)).(\lambda (H22: (pr0 u0 u3)).(\lambda -(H23: (pr0 t5 t6)).(let H24 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead -(Flat Appl) v1 (THead (Bind b) u1 t3)) t7)) H6 (THead (Flat Appl) v0 (THead -(Bind b0) u0 t5)) H17) in (let H25 \def (f_equal T T (\lambda (e: T).(match e -in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) -\Rightarrow v1 | (THead _ t7 _) \Rightarrow t7])) (THead (Flat Appl) v1 -(THead (Bind b) u1 t3)) (THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) H24) -in ((let H26 \def (f_equal T B (\lambda (e: T).(match e in T return (\lambda -(_: T).B) with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead _ -_ t7) \Rightarrow (match t7 in T return (\lambda (_: T).B) with [(TSort _) -\Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k -in K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) -\Rightarrow b])])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead -(Flat Appl) v0 (THead (Bind b0) u0 t5)) H24) in ((let H27 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ _ t7) \Rightarrow (match -t7 in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) -\Rightarrow u1 | (THead _ t8 _) \Rightarrow t8])])) (THead (Flat Appl) v1 -(THead (Bind b) u1 t3)) (THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) H24) -in ((let H28 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda -(_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead -_ _ t7) \Rightarrow (match t7 in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t8) \Rightarrow -t8])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) v0 -(THead (Bind b0) u0 t5)) H24) in (\lambda (H29: (eq T u1 u0)).(\lambda (H30: -(eq B b b0)).(\lambda (H31: (eq T v1 v0)).(let H32 \def (eq_ind_r T t -(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) -\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) -(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Appl) v0 (THead (Bind -b0) u0 t5)) H17) in (let H33 \def (eq_ind T v1 (\lambda (t7: T).(pr0 t7 v2)) -H10 v0 H31) in (eq_ind_r B b0 (\lambda (b1: B).(ex2 T (\lambda (t7: T).(pr0 -(THead (Bind b1) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda -(t7: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) -t7)))) (let H34 \def (eq_ind B b (\lambda (b1: B).(not (eq B b1 Abst))) H9 b0 -H30) in (let H35 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H11 u0 H29) -in (let H36 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H12 t5 H28) in -(ex2_ind T (\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T -(\lambda (t7: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O -v2) t4)) t7)) (\lambda (t7: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) -(lift (S O) O v3) t6)) t7))) (\lambda (x: T).(\lambda (H37: (pr0 t4 -x)).(\lambda (H38: (pr0 t6 x)).(ex2_ind T (\lambda (t7: T).(pr0 u2 t7)) -(\lambda (t7: T).(pr0 u3 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b0) -u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 -(THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) t7))) (\lambda -(x0: T).(\lambda (H39: (pr0 u2 x0)).(\lambda (H40: (pr0 u3 x0)).(ex2_ind T -(\lambda (t7: T).(pr0 v2 t7)) (\lambda (t7: T).(pr0 v3 t7)) (ex2 T (\lambda -(t7: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -t7)) (\lambda (t7: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) -O v3) t6)) t7))) (\lambda (x1: T).(\lambda (H41: (pr0 v2 x1)).(\lambda (H42: -(pr0 v3 x1)).(pr0_confluence__pr0_upsilon_upsilon b0 H34 v2 v3 x1 H41 H42 u2 -u3 x0 H39 H40 t4 t6 x H37 H38)))) (H32 v0 (tlt_head_sx (Flat Appl) v0 (THead -(Bind b0) u0 t5)) v2 H33 v3 H21))))) (H32 u0 (tlt_trans (THead (Bind b0) u0 -t5) u0 (THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) (tlt_head_sx (Bind b0) -u0 t5) (tlt_head_dx (Flat Appl) v0 (THead (Bind b0) u0 t5))) u2 H35 u3 -H22))))) (H32 t5 (tlt_trans (THead (Bind b0) u0 t5) t5 (THead (Flat Appl) v0 -(THead (Bind b0) u0 t5)) (tlt_head_dx (Bind b0) u0 t5) (tlt_head_dx (Flat -Appl) v0 (THead (Bind b0) u0 t5))) t4 H36 t6 H23))))) b H30))))))) H27)) -H26)) H25))))))) t2 H19)) t H17 H18 H13 H14 H15 H16))) | (pr0_delta u0 u3 H13 -t5 t6 H14 w H15) \Rightarrow (\lambda (H16: (eq T (THead (Bind Abbr) u0 t5) -t)).(\lambda (H17: (eq T (THead (Bind Abbr) u3 w) t2)).(eq_ind T (THead (Bind -Abbr) u0 t5) (\lambda (_: T).((eq T (THead (Bind Abbr) u3 w) t2) \to ((pr0 u0 -u3) \to ((pr0 t5 t6) \to ((subst0 O u3 t6 w) \to (ex2 T (\lambda (t8: T).(pr0 -(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda -(t8: T).(pr0 t2 t8)))))))) (\lambda (H18: (eq T (THead (Bind Abbr) u3 w) -t2)).(eq_ind T (THead (Bind Abbr) u3 w) (\lambda (t7: T).((pr0 u0 u3) \to -((pr0 t5 t6) \to ((subst0 O u3 t6 w) \to (ex2 T (\lambda (t8: T).(pr0 (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: -T).(pr0 t7 t8))))))) (\lambda (_: (pr0 u0 u3)).(\lambda (_: (pr0 t5 -t6)).(\lambda (_: (subst0 O u3 t6 w)).(let H22 \def (eq_ind_r T t (\lambda -(t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) t7)) H6 (THead -(Bind Abbr) u0 t5) H16) in (let H23 \def (eq_ind T (THead (Flat Appl) v1 -(THead (Bind b) u1 t3)) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind -Abbr) u0 t5) H22) in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 -(THead (Bind Abbr) u3 w) t7))) H23)))))) t2 H18)) t H16 H17 H13 H14 H15))) | -(pr0_zeta b0 H13 t5 t6 H14 u) \Rightarrow (\lambda (H15: (eq T (THead (Bind -b0) u (lift (S O) O t5)) t)).(\lambda (H16: (eq T t6 t2)).(eq_ind T (THead -(Bind b0) u (lift (S O) O t5)) (\lambda (_: T).((eq T t6 t2) \to ((not (eq B -b0 Abst)) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t2 -t8))))))) (\lambda (H17: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((not -(eq B b0 Abst)) \to ((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: -T).(pr0 t2 t8)))))) (\lambda (_: (not (eq B b0 Abst))).(\lambda (_: (pr0 t5 -t2)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1 -(THead (Bind b) u1 t3)) t7)) H6 (THead (Bind b0) u (lift (S O) O t5)) H15) in -(let H21 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) -(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow -(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | -(Flat _) \Rightarrow True])])) I (THead (Bind b0) u (lift (S O) O t5)) H20) -in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 t2 t7))) H21))))) t6 -(sym_eq T t6 t2 H17))) t H15 H16 H13 H14))) | (pr0_tau t5 t6 H13 u) -\Rightarrow (\lambda (H14: (eq T (THead (Flat Cast) u t5) t)).(\lambda (H15: -(eq T t6 t2)).(eq_ind T (THead (Flat Cast) u t5) (\lambda (_: T).((eq T t6 -t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t2 -t8)))))) (\lambda (H16: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5 -t7) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t2 t8))))) (\lambda (_: -(pr0 t5 t2)).(let H18 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat -Appl) v1 (THead (Bind b) u1 t3)) t7)) H6 (THead (Flat Cast) u t5) H14) in -(let H19 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) -(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow -(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | -(Flat f) \Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl -\Rightarrow True | Cast \Rightarrow False])])])) I (THead (Flat Cast) u t5) -H18) in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 t2 t7))) -H19)))) t6 (sym_eq T t6 t2 H16))) t H14 H15 H13)))]) in (H13 (refl_equal T t) -(refl_equal T t2))))))) t1 H8)) t H6 H7 H2 H3 H4 H5))) | (pr0_delta u1 u2 H2 -t3 t4 H3 w H4) \Rightarrow (\lambda (H5: (eq T (THead (Bind Abbr) u1 t3) -t)).(\lambda (H6: (eq T (THead (Bind Abbr) u2 w) t1)).(eq_ind T (THead (Bind -Abbr) u1 t3) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t1) \to ((pr0 u1 -u2) \to ((pr0 t3 t4) \to ((subst0 O u2 t4 w) \to (ex2 T (\lambda (t6: T).(pr0 -t1 t6)) (\lambda (t6: T).(pr0 t2 t6)))))))) (\lambda (H7: (eq T (THead (Bind -Abbr) u2 w) t1)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t5: T).((pr0 u1 -u2) \to ((pr0 t3 t4) \to ((subst0 O u2 t4 w) \to (ex2 T (\lambda (t6: T).(pr0 -t5 t6)) (\lambda (t6: T).(pr0 t2 t6))))))) (\lambda (H8: (pr0 u1 -u2)).(\lambda (H9: (pr0 t3 t4)).(\lambda (H10: (subst0 O u2 t4 w)).(let H11 -\def (match H1 in pr0 return (\lambda (t5: T).(\lambda (t6: T).(\lambda (_: -(pr0 t5 t6)).((eq T t5 t) \to ((eq T t6 t2) \to (ex2 T (\lambda (t7: T).(pr0 -(THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7)))))))) with -[(pr0_refl t5) \Rightarrow (\lambda (H11: (eq T t5 t)).(\lambda (H12: (eq T -t5 t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2) \to (ex2 T (\lambda (t7: -T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7))))) -(\lambda (H13: (eq T t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t7: -T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let -H14 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 t2)) H13 (THead (Bind Abbr) -u1 t3) H5) in (eq_ind T (THead (Bind Abbr) u1 t3) (\lambda (t6: T).(ex2 T -(\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t6 -t7)))) (let H15 \def (eq_ind_r T t (\lambda (t6: T).(eq T t5 t6)) H11 (THead -(Bind Abbr) u1 t3) H5) in (let H16 \def (eq_ind_r T t (\lambda (t6: -T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall -(t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: -T).(pr0 t8 t9)))))))))) H (THead (Bind Abbr) u1 t3) H5) in (ex_intro2 T -(\lambda (t6: T).(pr0 (THead (Bind Abbr) u2 w) t6)) (\lambda (t6: T).(pr0 -(THead (Bind Abbr) u1 t3) t6)) (THead (Bind Abbr) u2 w) (pr0_refl (THead -(Bind Abbr) u2 w)) (pr0_delta u1 u2 H8 t3 t4 H9 w H10)))) t2 H14)) t (sym_eq -T t t2 H13))) t5 (sym_eq T t5 t H11) H12))) | (pr0_comp u0 u3 H11 t5 t6 H12 -k) \Rightarrow (\lambda (H13: (eq T (THead k u0 t5) t)).(\lambda (H14: (eq T -(THead k u3 t6) t2)).(eq_ind T (THead k u0 t5) (\lambda (_: T).((eq T (THead -k u3 t6) t2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: -T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) -(\lambda (H15: (eq T (THead k u3 t6) t2)).(eq_ind T (THead k u3 t6) (\lambda -(t7: T).((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead -(Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (H16: (pr0 -u0 u3)).(\lambda (H17: (pr0 t5 t6)).(let H18 \def (eq_ind_r T t (\lambda (t7: -T).(eq T (THead (Bind Abbr) u1 t3) t7)) H5 (THead k u0 t5) H13) in (let H19 -\def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) -with [(TSort _) \Rightarrow (Bind Abbr) | (TLRef _) \Rightarrow (Bind Abbr) | -(THead k0 _ _) \Rightarrow k0])) (THead (Bind Abbr) u1 t3) (THead k u0 t5) -H18) in ((let H20 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 -| (THead _ t7 _) \Rightarrow t7])) (THead (Bind Abbr) u1 t3) (THead k u0 t5) -H18) in ((let H21 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 -| (THead _ _ t7) \Rightarrow t7])) (THead (Bind Abbr) u1 t3) (THead k u0 t5) -H18) in (\lambda (H22: (eq T u1 u0)).(\lambda (H23: (eq K (Bind Abbr) -k)).(eq_ind K (Bind Abbr) (\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 -(THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 (THead k0 u3 t6) t7)))) -(let H24 \def (eq_ind_r K k (\lambda (k0: K).(eq T (THead k0 u0 t5) t)) H13 -(Bind Abbr) H23) in (let H25 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: -T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v -t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 -t10)))))))))) H (THead (Bind Abbr) u0 t5) H24) in (let H26 \def (eq_ind T u1 -(\lambda (t7: T).(pr0 t7 u2)) H8 u0 H22) in (let H27 \def (eq_ind T t3 -(\lambda (t7: T).(pr0 t7 t4)) H9 t5 H21) in (ex2_ind T (\lambda (t7: T).(pr0 -t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead -(Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u3 t6) t7))) -(\lambda (x: T).(\lambda (H28: (pr0 t4 x)).(\lambda (H29: (pr0 t6 -x)).(ex2_ind T (\lambda (t7: T).(pr0 u2 t7)) (\lambda (t7: T).(pr0 u3 t7)) -(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: -T).(pr0 (THead (Bind Abbr) u3 t6) t7))) (\lambda (x0: T).(\lambda (H30: (pr0 -u2 x0)).(\lambda (H31: (pr0 u3 x0)).(ex2_sym T (pr0 (THead (Bind Abbr) u3 -t6)) (pr0 (THead (Bind Abbr) u2 w)) (pr0_confluence__pr0_cong_delta u2 t4 w -H10 u3 x0 H31 H30 t6 x H29 H28))))) (H25 u0 (tlt_head_sx (Bind Abbr) u0 t5) -u2 H26 u3 H16))))) (H25 t5 (tlt_head_dx (Bind Abbr) u0 t5) t4 H27 t6 -H17)))))) k H23)))) H20)) H19))))) t2 H15)) t H13 H14 H11 H12))) | (pr0_beta -u v1 v2 H11 t5 t6 H12) \Rightarrow (\lambda (H13: (eq T (THead (Flat Appl) v1 -(THead (Bind Abst) u t5)) t)).(\lambda (H14: (eq T (THead (Bind Abbr) v2 t6) -t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) (\lambda (_: -T).((eq T (THead (Bind Abbr) v2 t6) t2) \to ((pr0 v1 v2) \to ((pr0 t5 t6) \to -(ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: -T).(pr0 t2 t8))))))) (\lambda (H15: (eq T (THead (Bind Abbr) v2 t6) -t2)).(eq_ind T (THead (Bind Abbr) v2 t6) (\lambda (t7: T).((pr0 v1 v2) \to -((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) -(\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 -t5 t6)).(let H18 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind Abbr) -u1 t3) t7)) H5 (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) H13) in (let -H19 \def (eq_ind T (THead (Bind Abbr) u1 t3) (\lambda (ee: T).(match ee in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow -False])])) I (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) H18) in -(False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) -(\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t6) t7))) H19))))) t2 H15)) t H13 -H14 H11 H12))) | (pr0_upsilon b H11 v1 v2 H12 u0 u3 H13 t5 t6 H14) -\Rightarrow (\lambda (H15: (eq T (THead (Flat Appl) v1 (THead (Bind b) u0 -t5)) t)).(\lambda (H16: (eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S -O) O v2) t6)) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u0 t5)) -(\lambda (_: T).((eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O -v2) t6)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u3) \to -((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) -(\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H17: (eq T (THead (Bind b) u3 -(THead (Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Bind b) u3 -(THead (Flat Appl) (lift (S O) O v2) t6)) (\lambda (t7: T).((not (eq B b -Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda -(t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t7 -t8)))))))) (\lambda (_: (not (eq B b Abst))).(\lambda (_: (pr0 v1 -v2)).(\lambda (_: (pr0 u0 u3)).(\lambda (_: (pr0 t5 t6)).(let H22 \def -(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind Abbr) u1 t3) t7)) H5 (THead -(Flat Appl) v1 (THead (Bind b) u0 t5)) H15) in (let H23 \def (eq_ind T (THead -(Bind Abbr) u1 t3) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat -Appl) v1 (THead (Bind b) u0 t5)) H22) in (False_ind (ex2 T (\lambda (t7: -T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 (THead (Bind b) -u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t7))) H23))))))) t2 H17)) t H15 -H16 H11 H12 H13 H14))) | (pr0_delta u0 u3 H11 t5 t6 H12 w0 H13) \Rightarrow -(\lambda (H14: (eq T (THead (Bind Abbr) u0 t5) t)).(\lambda (H15: (eq T -(THead (Bind Abbr) u3 w0) t2)).(eq_ind T (THead (Bind Abbr) u0 t5) (\lambda -(_: T).((eq T (THead (Bind Abbr) u3 w0) t2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) -\to ((subst0 O u3 t6 w0) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) -u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8)))))))) (\lambda (H16: (eq T (THead -(Bind Abbr) u3 w0) t2)).(eq_ind T (THead (Bind Abbr) u3 w0) (\lambda (t7: -T).((pr0 u0 u3) \to ((pr0 t5 t6) \to ((subst0 O u3 t6 w0) \to (ex2 T (\lambda -(t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t7 t8))))))) -(\lambda (H17: (pr0 u0 u3)).(\lambda (H18: (pr0 t5 t6)).(\lambda (H19: -(subst0 O u3 t6 w0)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T -(THead (Bind Abbr) u1 t3) t7)) H5 (THead (Bind Abbr) u0 t5) H14) in (let H21 -\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t7 _) -\Rightarrow t7])) (THead (Bind Abbr) u1 t3) (THead (Bind Abbr) u0 t5) H20) in -((let H22 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: -T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ -t7) \Rightarrow t7])) (THead (Bind Abbr) u1 t3) (THead (Bind Abbr) u0 t5) -H20) in (\lambda (H23: (eq T u1 u0)).(let H24 \def (eq_ind_r T t (\lambda -(t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to -(\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) -(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Bind Abbr) u0 t5) H14) in -(let H25 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H8 u0 H23) in (let -H26 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H9 t5 H22) in (ex2_ind T -(\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda -(t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 (THead (Bind -Abbr) u3 w0) t7))) (\lambda (x: T).(\lambda (H27: (pr0 t4 x)).(\lambda (H28: -(pr0 t6 x)).(ex2_ind T (\lambda (t7: T).(pr0 u2 t7)) (\lambda (t7: T).(pr0 u3 -t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: -T).(pr0 (THead (Bind Abbr) u3 w0) t7))) (\lambda (x0: T).(\lambda (H29: (pr0 -u2 x0)).(\lambda (H30: (pr0 u3 x0)).(pr0_confluence__pr0_delta_delta u2 t4 w -H10 u3 t6 w0 H19 x0 H29 H30 x H27 H28)))) (H24 u0 (tlt_head_sx (Bind Abbr) u0 -t5) u2 H25 u3 H17))))) (H24 t5 (tlt_head_dx (Bind Abbr) u0 t5) t4 H26 t6 -H18))))))) H21)))))) t2 H16)) t H14 H15 H11 H12 H13))) | (pr0_zeta b H11 t5 -t6 H12 u) \Rightarrow (\lambda (H13: (eq T (THead (Bind b) u (lift (S O) O -t5)) t)).(\lambda (H14: (eq T t6 t2)).(eq_ind T (THead (Bind b) u (lift (S O) -O t5)) (\lambda (_: T).((eq T t6 t2) \to ((not (eq B b Abst)) \to ((pr0 t5 -t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda -(t8: T).(pr0 t2 t8))))))) (\lambda (H15: (eq T t6 t2)).(eq_ind T t2 (\lambda -(t7: T).((not (eq B b Abst)) \to ((pr0 t5 t7) \to (ex2 T (\lambda (t8: -T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) -(\lambda (H16: (not (eq B b Abst))).(\lambda (H17: (pr0 t5 t2)).(let H18 \def -(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind Abbr) u1 t3) t7)) H5 (THead -(Bind b) u (lift (S O) O t5)) H13) in (let H19 \def (f_equal T B (\lambda (e: -T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow Abbr | -(TLRef _) \Rightarrow Abbr | (THead k _ _) \Rightarrow (match k in K return -(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow -Abbr])])) (THead (Bind Abbr) u1 t3) (THead (Bind b) u (lift (S O) O t5)) H18) -in ((let H20 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda -(_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead -_ t7 _) \Rightarrow t7])) (THead (Bind Abbr) u1 t3) (THead (Bind b) u (lift -(S O) O t5)) H18) in ((let H21 \def (f_equal T T (\lambda (e: T).(match e in -T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) -\Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead (Bind Abbr) u1 t3) -(THead (Bind b) u (lift (S O) O t5)) H18) in (\lambda (H22: (eq T u1 -u)).(\lambda (H23: (eq B Abbr b)).(let H24 \def (eq_ind_r B b (\lambda (b0: -B).(not (eq B b0 Abst))) H16 Abbr H23) in (let H25 \def (eq_ind_r B b -(\lambda (b0: B).(eq T (THead (Bind b0) u (lift (S O) O t5)) t)) H13 Abbr -H23) in (let H26 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v -t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to -(ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 -t10)))))))))) H (THead (Bind Abbr) u (lift (S O) O t5)) H25) in (let H27 \def -(eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H8 u H22) in (let H28 \def (eq_ind -T t3 (\lambda (t7: T).(pr0 t7 t4)) H9 (lift (S O) O t5) H21) in (ex2_ind T -(\lambda (t7: T).(eq T t4 (lift (S O) O t7))) (\lambda (t7: T).(pr0 t5 t7)) -(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: -T).(pr0 t2 t7))) (\lambda (x: T).(\lambda (H29: (eq T t4 (lift (S O) O -x))).(\lambda (H30: (pr0 t5 x)).(let H31 \def (eq_ind T t4 (\lambda (t7: -T).(subst0 O u2 t7 w)) H10 (lift (S O) O x) H29) in (ex2_ind T (\lambda (t7: -T).(pr0 x t7)) (\lambda (t7: T).(pr0 t2 t7)) (ex2 T (\lambda (t7: T).(pr0 -(THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7))) (\lambda (x0: -T).(\lambda (_: (pr0 x x0)).(\lambda (_: (pr0 t2 -x0)).(pr0_confluence__pr0_delta_tau u2 (lift (S O) O x) w H31 x (pr0_refl -(lift (S O) O x)) t2)))) (H26 t5 (lift_tlt_dx (Bind Abbr) u t5 (S O) O) x H30 -t2 H17)))))) (pr0_gen_lift t5 t4 (S O) O H28)))))))))) H20)) H19))))) t6 -(sym_eq T t6 t2 H15))) t H13 H14 H11 H12))) | (pr0_tau t5 t6 H11 u) -\Rightarrow (\lambda (H12: (eq T (THead (Flat Cast) u t5) t)).(\lambda (H13: -(eq T t6 t2)).(eq_ind T (THead (Flat Cast) u t5) (\lambda (_: T).((eq T t6 -t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 -w) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H14: (eq T t6 -t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8: -T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8))))) -(\lambda (_: (pr0 t5 t2)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T -(THead (Bind Abbr) u1 t3) t7)) H5 (THead (Flat Cast) u t5) H12) in (let H17 -\def (eq_ind T (THead (Bind Abbr) u1 t3) (\lambda (ee: T).(match ee in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow -False])])) I (THead (Flat Cast) u t5) H16) in (False_ind (ex2 T (\lambda (t7: -T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7))) H17)))) -t6 (sym_eq T t6 t2 H14))) t H12 H13 H11)))]) in (H11 (refl_equal T t) -(refl_equal T t2)))))) t1 H7)) t H5 H6 H2 H3 H4))) | (pr0_zeta b H2 t3 t4 H3 -u) \Rightarrow (\lambda (H4: (eq T (THead (Bind b) u (lift (S O) O t3)) -t)).(\lambda (H5: (eq T t4 t1)).(eq_ind T (THead (Bind b) u (lift (S O) O -t3)) (\lambda (_: T).((eq T t4 t1) \to ((not (eq B b Abst)) \to ((pr0 t3 t4) -\to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6))))))) -(\lambda (H6: (eq T t4 t1)).(eq_ind T t1 (\lambda (t5: T).((not (eq B b -Abst)) \to ((pr0 t3 t5) \to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda -(t6: T).(pr0 t2 t6)))))) (\lambda (H7: (not (eq B b Abst))).(\lambda (H8: -(pr0 t3 t1)).(let H9 \def (match H1 in pr0 return (\lambda (t5: T).(\lambda -(t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 t) \to ((eq T t6 t2) \to (ex2 T -(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)))))))) with -[(pr0_refl t5) \Rightarrow (\lambda (H9: (eq T t5 t)).(\lambda (H10: (eq T t5 -t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2) \to (ex2 T (\lambda (t7: -T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))))) (\lambda (H11: (eq T t -t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t7: T).(pr0 t1 t7)) -(\lambda (t7: T).(pr0 t2 t7)))) (let H12 \def (eq_ind_r T t (\lambda (t6: -T).(eq T t6 t2)) H11 (THead (Bind b) u (lift (S O) O t3)) H4) in (eq_ind T -(THead (Bind b) u (lift (S O) O t3)) (\lambda (t6: T).(ex2 T (\lambda (t7: -T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t6 t7)))) (let H13 \def (eq_ind_r T t -(\lambda (t6: T).(eq T t5 t6)) H9 (THead (Bind b) u (lift (S O) O t3)) H4) in -(let H14 \def (eq_ind_r T t (\lambda (t6: T).(\forall (v: T).((tlt v t6) \to -(\forall (t7: T).((pr0 v t7) \to (\forall (t8: T).((pr0 v t8) \to (ex2 T -(\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead -(Bind b) u (lift (S O) O t3)) H4) in (ex_intro2 T (\lambda (t6: T).(pr0 t1 -t6)) (\lambda (t6: T).(pr0 (THead (Bind b) u (lift (S O) O t3)) t6)) t1 -(pr0_refl t1) (pr0_zeta b H7 t3 t1 H8 u)))) t2 H12)) t (sym_eq T t t2 H11))) -t5 (sym_eq T t5 t H9) H10))) | (pr0_comp u1 u2 H9 t5 t6 H10 k) \Rightarrow -(\lambda (H11: (eq T (THead k u1 t5) t)).(\lambda (H12: (eq T (THead k u2 t6) -t2)).(eq_ind T (THead k u1 t5) (\lambda (_: T).((eq T (THead k u2 t6) t2) \to -((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) -(\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H13: (eq T (THead k u2 t6) -t2)).(eq_ind T (THead k u2 t6) (\lambda (t7: T).((pr0 u1 u2) \to ((pr0 t5 t6) -\to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 t8)))))) -(\lambda (_: (pr0 u1 u2)).(\lambda (H15: (pr0 t5 t6)).(let H16 \def (eq_ind_r -T t (\lambda (t7: T).(eq T (THead (Bind b) u (lift (S O) O t3)) t7)) H4 -(THead k u1 t5) H11) in (let H17 \def (f_equal T K (\lambda (e: T).(match e -in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow (Bind b) | (TLRef -_) \Rightarrow (Bind b) | (THead k0 _ _) \Rightarrow k0])) (THead (Bind b) u -(lift (S O) O t3)) (THead k u1 t5) H16) in ((let H18 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t7 _) \Rightarrow t7])) -(THead (Bind b) u (lift (S O) O t3)) (THead k u1 t5) H16) in ((let H19 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t7: -T) on t7: T \def (match t7 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) -\Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false -\Rightarrow (f i)])) | (THead k0 u0 t8) \Rightarrow (THead k0 (lref_map f d -u0) (lref_map f (s k0 d) t8))]) in lref_map) (\lambda (x: nat).(plus x (S -O))) O t3) | (TLRef _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) -(d: nat) (t7: T) on t7: T \def (match t7 with [(TSort n) \Rightarrow (TSort -n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | -false \Rightarrow (f i)])) | (THead k0 u0 t8) \Rightarrow (THead k0 (lref_map -f d u0) (lref_map f (s k0 d) t8))]) in lref_map) (\lambda (x: nat).(plus x (S -O))) O t3) | (THead _ _ t7) \Rightarrow t7])) (THead (Bind b) u (lift (S O) O -t3)) (THead k u1 t5) H16) in (\lambda (_: (eq T u u1)).(\lambda (H21: (eq K -(Bind b) k)).(eq_ind K (Bind b) (\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 -t1 t7)) (\lambda (t7: T).(pr0 (THead k0 u2 t6) t7)))) (let H22 \def (eq_ind_r -K k (\lambda (k0: K).(eq T (THead k0 u1 t5) t)) H11 (Bind b) H21) in (let H23 -\def (eq_ind_r T t5 (\lambda (t7: T).(pr0 t7 t6)) H15 (lift (S O) O t3) H19) -in (ex2_ind T (\lambda (t7: T).(eq T t6 (lift (S O) O t7))) (\lambda (t7: -T).(pr0 t3 t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 -(THead (Bind b) u2 t6) t7))) (\lambda (x: T).(\lambda (H24: (eq T t6 (lift (S -O) O x))).(\lambda (H25: (pr0 t3 x)).(let H26 \def (eq_ind_r T t5 (\lambda -(t7: T).(eq T (THead (Bind b) u1 t7) t)) H22 (lift (S O) O t3) H19) in (let -H27 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to -(\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T -(\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H -(THead (Bind b) u1 (lift (S O) O t3)) H26) in (eq_ind_r T (lift (S O) O x) -(\lambda (t7: T).(ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 -(THead (Bind b) u2 t7) t8)))) (ex2_ind T (\lambda (t7: T).(pr0 x t7)) -(\lambda (t7: T).(pr0 t1 t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda -(t7: T).(pr0 (THead (Bind b) u2 (lift (S O) O x)) t7))) (\lambda (x0: -T).(\lambda (H28: (pr0 x x0)).(\lambda (H29: (pr0 t1 x0)).(ex_intro2 T -(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Bind b) u2 (lift -(S O) O x)) t7)) x0 H29 (pr0_zeta b H7 x x0 H28 u2))))) (H27 t3 (lift_tlt_dx -(Bind b) u1 t3 (S O) O) x H25 t1 H8)) t6 H24)))))) (pr0_gen_lift t3 t6 (S O) -O H23)))) k H21)))) H18)) H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_beta -u0 v1 v2 H9 t5 t6 H10) \Rightarrow (\lambda (H11: (eq T (THead (Flat Appl) v1 -(THead (Bind Abst) u0 t5)) t)).(\lambda (H12: (eq T (THead (Bind Abbr) v2 t6) -t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t5)) (\lambda (_: -T).((eq T (THead (Bind Abbr) v2 t6) t2) \to ((pr0 v1 v2) \to ((pr0 t5 t6) \to -(ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))))) -(\lambda (H13: (eq T (THead (Bind Abbr) v2 t6) t2)).(eq_ind T (THead (Bind -Abbr) v2 t6) (\lambda (t7: T).((pr0 v1 v2) \to ((pr0 t5 t6) \to (ex2 T -(\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (_: -(pr0 v1 v2)).(\lambda (_: (pr0 t5 t6)).(let H16 \def (eq_ind_r T t (\lambda -(t7: T).(eq T (THead (Bind b) u (lift (S O) O t3)) t7)) H4 (THead (Flat Appl) -v1 (THead (Bind Abst) u0 t5)) H11) in (let H17 \def (eq_ind T (THead (Bind b) -u (lift (S O) O t3)) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat -Appl) v1 (THead (Bind Abst) u0 t5)) H16) in (False_ind (ex2 T (\lambda (t7: -T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t6) t7))) -H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_upsilon b0 H9 v1 v2 H10 u1 u2 -H11 t5 t6 H12) \Rightarrow (\lambda (H13: (eq T (THead (Flat Appl) v1 (THead -(Bind b0) u1 t5)) t)).(\lambda (H14: (eq T (THead (Bind b0) u2 (THead (Flat -Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Flat Appl) v1 (THead -(Bind b0) u1 t5)) (\lambda (_: T).((eq T (THead (Bind b0) u2 (THead (Flat -Appl) (lift (S O) O v2) t6)) t2) \to ((not (eq B b0 Abst)) \to ((pr0 v1 v2) -\to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) -(\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H15: (eq T (THead (Bind b0) u2 -(THead (Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Bind b0) u2 -(THead (Flat Appl) (lift (S O) O v2) t6)) (\lambda (t7: T).((not (eq B b0 -Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda -(t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 t8)))))))) (\lambda (_: (not -(eq B b0 Abst))).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 u1 u2)).(\lambda -(_: (pr0 t5 t6)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead -(Bind b) u (lift (S O) O t3)) t7)) H4 (THead (Flat Appl) v1 (THead (Bind b0) -u1 t5)) H13) in (let H21 \def (eq_ind T (THead (Bind b) u (lift (S O) O t3)) -(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow -(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | -(Flat _) \Rightarrow False])])) I (THead (Flat Appl) v1 (THead (Bind b0) u1 -t5)) H20) in (False_ind (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: -T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t6)) t7))) -H21))))))) t2 H15)) t H13 H14 H9 H10 H11 H12))) | (pr0_delta u1 u2 H9 t5 t6 -H10 w H11) \Rightarrow (\lambda (H12: (eq T (THead (Bind Abbr) u1 t5) -t)).(\lambda (H13: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind -Abbr) u1 t5) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t2) \to ((pr0 u1 -u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T (\lambda (t8: T).(pr0 -t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))))) (\lambda (H14: (eq T (THead (Bind -Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t7: T).((pr0 u1 -u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T (\lambda (t8: T).(pr0 -t1 t8)) (\lambda (t8: T).(pr0 t7 t8))))))) (\lambda (_: (pr0 u1 u2)).(\lambda -(H16: (pr0 t5 t6)).(\lambda (H17: (subst0 O u2 t6 w)).(let H18 \def (eq_ind_r -T t (\lambda (t7: T).(eq T (THead (Bind b) u (lift (S O) O t3)) t7)) H4 -(THead (Bind Abbr) u1 t5) H12) in (let H19 \def (f_equal T B (\lambda (e: -T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | -(TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k in K return -(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow -b])])) (THead (Bind b) u (lift (S O) O t3)) (THead (Bind Abbr) u1 t5) H18) in -((let H20 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: -T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t7 -_) \Rightarrow t7])) (THead (Bind b) u (lift (S O) O t3)) (THead (Bind Abbr) -u1 t5) H18) in ((let H21 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: -((nat \to nat))) (d: nat) (t7: T) on t7: T \def (match t7 with [(TSort n) -\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with -[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t8) -\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t8))]) in -lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (TLRef _) \Rightarrow -((let rec lref_map (f: ((nat \to nat))) (d: nat) (t7: T) on t7: T \def (match -t7 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef -(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | -(THead k u0 t8) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) -t8))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (THead _ _ t7) -\Rightarrow t7])) (THead (Bind b) u (lift (S O) O t3)) (THead (Bind Abbr) u1 -t5) H18) in (\lambda (_: (eq T u u1)).(\lambda (H23: (eq B b Abbr)).(let H24 -\def (eq_ind_r T t5 (\lambda (t7: T).(pr0 t7 t6)) H16 (lift (S O) O t3) H21) -in (ex2_ind T (\lambda (t7: T).(eq T t6 (lift (S O) O t7))) (\lambda (t7: -T).(pr0 t3 t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 -(THead (Bind Abbr) u2 w) t7))) (\lambda (x: T).(\lambda (H25: (eq T t6 (lift -(S O) O x))).(\lambda (H26: (pr0 t3 x)).(let H27 \def (eq_ind_r T t5 (\lambda -(t7: T).(eq T (THead (Bind Abbr) u1 t7) t)) H12 (lift (S O) O t3) H21) in -(let H28 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to -(\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T -(\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H -(THead (Bind Abbr) u1 (lift (S O) O t3)) H27) in (let H29 \def (eq_ind T t6 -(\lambda (t7: T).(subst0 O u2 t7 w)) H17 (lift (S O) O x) H25) in (let H30 -\def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H7 Abbr H23) in -(ex2_ind T (\lambda (t7: T).(pr0 x t7)) (\lambda (t7: T).(pr0 t1 t7)) (ex2 T -(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) -t7))) (\lambda (x0: T).(\lambda (_: (pr0 x x0)).(\lambda (_: (pr0 t1 -x0)).(ex2_sym T (pr0 (THead (Bind Abbr) u2 w)) (pr0 t1) -(pr0_confluence__pr0_delta_tau u2 (lift (S O) O x) w H29 x (pr0_refl (lift (S -O) O x)) t1))))) (H28 t3 (lift_tlt_dx (Bind Abbr) u1 t3 (S O) O) x H26 t1 -H8))))))))) (pr0_gen_lift t3 t6 (S O) O H24)))))) H20)) H19)))))) t2 H14)) t -H12 H13 H9 H10 H11))) | (pr0_zeta b0 H9 t5 t6 H10 u0) \Rightarrow (\lambda -(H11: (eq T (THead (Bind b0) u0 (lift (S O) O t5)) t)).(\lambda (H12: (eq T -t6 t2)).(eq_ind T (THead (Bind b0) u0 (lift (S O) O t5)) (\lambda (_: T).((eq -T t6 t2) \to ((not (eq B b0 Abst)) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: -T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H13: (eq T t6 -t2)).(eq_ind T t2 (\lambda (t7: T).((not (eq B b0 Abst)) \to ((pr0 t5 t7) \to -(ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))) -(\lambda (_: (not (eq B b0 Abst))).(\lambda (H15: (pr0 t5 t2)).(let H16 \def -(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind b) u (lift (S O) O t3)) -t7)) H4 (THead (Bind b0) u0 (lift (S O) O t5)) H11) in (let H17 \def (f_equal -T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with [(TSort _) -\Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k -in K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) -\Rightarrow b])])) (THead (Bind b) u (lift (S O) O t3)) (THead (Bind b0) u0 -(lift (S O) O t5)) H16) in ((let H18 \def (f_equal T T (\lambda (e: T).(match -e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) -\Rightarrow u | (THead _ t7 _) \Rightarrow t7])) (THead (Bind b) u (lift (S -O) O t3)) (THead (Bind b0) u0 (lift (S O) O t5)) H16) in ((let H19 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t7: -T) on t7: T \def (match t7 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) -\Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false -\Rightarrow (f i)])) | (THead k u1 t8) \Rightarrow (THead k (lref_map f d u1) -(lref_map f (s k d) t8))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O -t3) | (TLRef _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) -(t7: T) on t7: T \def (match t7 with [(TSort n) \Rightarrow (TSort n) | -(TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | -false \Rightarrow (f i)])) | (THead k u1 t8) \Rightarrow (THead k (lref_map f -d u1) (lref_map f (s k d) t8))]) in lref_map) (\lambda (x: nat).(plus x (S -O))) O t3) | (THead _ _ t7) \Rightarrow t7])) (THead (Bind b) u (lift (S O) O -t3)) (THead (Bind b0) u0 (lift (S O) O t5)) H16) in (\lambda (_: (eq T u -u0)).(\lambda (H21: (eq B b b0)).(let H22 \def (eq_ind_r T t (\lambda (t7: -T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall -(t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: -T).(pr0 t9 t10)))))))))) H (THead (Bind b0) u0 (lift (S O) O t5)) H11) in -(let H23 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t1)) H8 t5 (lift_inj t3 -t5 (S O) O H19)) in (let H24 \def (eq_ind B b (\lambda (b1: B).(not (eq B b1 -Abst))) H7 b0 H21) in (ex2_ind T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: -T).(pr0 t2 t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 -t7))) (\lambda (x: T).(\lambda (H25: (pr0 t1 x)).(\lambda (H26: (pr0 t2 -x)).(ex_intro2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)) -x H25 H26)))) (H22 t5 (lift_tlt_dx (Bind b0) u0 t5 (S O) O) t1 H23 t2 -H15)))))))) H18)) H17))))) t6 (sym_eq T t6 t2 H13))) t H11 H12 H9 H10))) | -(pr0_tau t5 t6 H9 u0) \Rightarrow (\lambda (H10: (eq T (THead (Flat Cast) u0 -t5) t)).(\lambda (H11: (eq T t6 t2)).(eq_ind T (THead (Flat Cast) u0 t5) -(\lambda (_: T).((eq T t6 t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: -T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H12: (eq T t6 -t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8: -T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))) (\lambda (_: (pr0 t5 -t2)).(let H14 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind b) u -(lift (S O) O t3)) t7)) H4 (THead (Flat Cast) u0 t5) H10) in (let H15 \def -(eq_ind T (THead (Bind b) u (lift (S O) O t3)) (\lambda (ee: T).(match ee in -T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow -False])])) I (THead (Flat Cast) u0 t5) H14) in (False_ind (ex2 T (\lambda -(t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))) H15)))) t6 (sym_eq T t6 -t2 H12))) t H10 H11 H9)))]) in (H9 (refl_equal T t) (refl_equal T t2))))) t4 -(sym_eq T t4 t1 H6))) t H4 H5 H2 H3))) | (pr0_tau t3 t4 H2 u) \Rightarrow -(\lambda (H3: (eq T (THead (Flat Cast) u t3) t)).(\lambda (H4: (eq T t4 -t1)).(eq_ind T (THead (Flat Cast) u t3) (\lambda (_: T).((eq T t4 t1) \to -((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 -t2 t6)))))) (\lambda (H5: (eq T t4 t1)).(eq_ind T t1 (\lambda (t5: T).((pr0 -t3 t5) \to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 -t6))))) (\lambda (H6: (pr0 t3 t1)).(let H7 \def (match H1 in pr0 return -(\lambda (t5: T).(\lambda (t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 t) \to -((eq T t6 t2) \to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 -t2 t7)))))))) with [(pr0_refl t5) \Rightarrow (\lambda (H7: (eq T t5 -t)).(\lambda (H8: (eq T t5 t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2) -\to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))))) -(\lambda (H9: (eq T t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t7: -T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let H10 \def (eq_ind_r T t -(\lambda (t6: T).(eq T t6 t2)) H9 (THead (Flat Cast) u t3) H3) in (eq_ind T -(THead (Flat Cast) u t3) (\lambda (t6: T).(ex2 T (\lambda (t7: T).(pr0 t1 -t7)) (\lambda (t7: T).(pr0 t6 t7)))) (let H11 \def (eq_ind_r T t (\lambda -(t6: T).(eq T t5 t6)) H7 (THead (Flat Cast) u t3) H3) in (let H12 \def -(eq_ind_r T t (\lambda (t6: T).(\forall (v: T).((tlt v t6) \to (\forall (t7: -T).((pr0 v t7) \to (\forall (t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: -T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead (Flat Cast) u -t3) H3) in (ex_intro2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 -(THead (Flat Cast) u t3) t6)) t1 (pr0_refl t1) (pr0_tau t3 t1 H6 u)))) t2 -H10)) t (sym_eq T t t2 H9))) t5 (sym_eq T t5 t H7) H8))) | (pr0_comp u1 u2 H7 -t5 t6 H8 k) \Rightarrow (\lambda (H9: (eq T (THead k u1 t5) t)).(\lambda -(H10: (eq T (THead k u2 t6) t2)).(eq_ind T (THead k u1 t5) (\lambda (_: -T).((eq T (THead k u2 t6) t2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T -(\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda -(H11: (eq T (THead k u2 t6) t2)).(eq_ind T (THead k u2 t6) (\lambda (t7: -T).((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) -(\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (_: (pr0 u1 u2)).(\lambda (H13: -(pr0 t5 t6)).(let H14 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat -Cast) u t3) t7)) H3 (THead k u1 t5) H9) in (let H15 \def (f_equal T K -(\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) -\Rightarrow (Flat Cast) | (TLRef _) \Rightarrow (Flat Cast) | (THead k0 _ _) -\Rightarrow k0])) (THead (Flat Cast) u t3) (THead k u1 t5) H14) in ((let H16 -\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t7 _) -\Rightarrow t7])) (THead (Flat Cast) u t3) (THead k u1 t5) H14) in ((let H17 -\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7) -\Rightarrow t7])) (THead (Flat Cast) u t3) (THead k u1 t5) H14) in (\lambda -(_: (eq T u u1)).(\lambda (H19: (eq K (Flat Cast) k)).(eq_ind K (Flat Cast) -(\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 -(THead k0 u2 t6) t7)))) (let H20 \def (eq_ind_r K k (\lambda (k0: K).(eq T -(THead k0 u1 t5) t)) H9 (Flat Cast) H19) in (let H21 \def (eq_ind_r T t -(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) -\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) -(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Cast) u1 t5) H20) in -(let H22 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t1)) H6 t5 H17) in -(ex2_ind T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T -(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Flat Cast) u2 t6) -t7))) (\lambda (x: T).(\lambda (H23: (pr0 t1 x)).(\lambda (H24: (pr0 t6 -x)).(ex_intro2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead -(Flat Cast) u2 t6) t7)) x H23 (pr0_tau t6 x H24 u2))))) (H21 t5 (tlt_head_dx -(Flat Cast) u1 t5) t1 H22 t6 H13))))) k H19)))) H16)) H15))))) t2 H11)) t H9 -H10 H7 H8))) | (pr0_beta u0 v1 v2 H7 t5 t6 H8) \Rightarrow (\lambda (H9: (eq -T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t5)) t)).(\lambda (H10: (eq T -(THead (Bind Abbr) v2 t6) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind -Abst) u0 t5)) (\lambda (_: T).((eq T (THead (Bind Abbr) v2 t6) t2) \to ((pr0 -v1 v2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda -(t8: T).(pr0 t2 t8))))))) (\lambda (H11: (eq T (THead (Bind Abbr) v2 t6) -t2)).(eq_ind T (THead (Bind Abbr) v2 t6) (\lambda (t7: T).((pr0 v1 v2) \to -((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 -t7 t8)))))) (\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 t5 t6)).(let H14 \def -(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Cast) u t3) t7)) H3 (THead -(Flat Appl) v1 (THead (Bind Abst) u0 t5)) H9) in (let H15 \def (eq_ind T -(THead (Flat Cast) u t3) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f in F return -(\lambda (_: F).Prop) with [Appl \Rightarrow False | Cast \Rightarrow -True])])])) I (THead (Flat Appl) v1 (THead (Bind Abst) u0 t5)) H14) in -(False_ind (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead -(Bind Abbr) v2 t6) t7))) H15))))) t2 H11)) t H9 H10 H7 H8))) | (pr0_upsilon b -H7 v1 v2 H8 u1 u2 H9 t5 t6 H10) \Rightarrow (\lambda (H11: (eq T (THead (Flat -Appl) v1 (THead (Bind b) u1 t5)) t)).(\lambda (H12: (eq T (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Flat Appl) -v1 (THead (Bind b) u1 t5)) (\lambda (_: T).((eq T (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t6)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1 -v2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 -t8)) (\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H13: (eq T (THead (Bind -b) u2 (THead (Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Bind -b) u2 (THead (Flat Appl) (lift (S O) O v2) t6)) (\lambda (t7: T).((not (eq B -b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T -(\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 t8)))))))) (\lambda -(_: (not (eq B b Abst))).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 u1 -u2)).(\lambda (_: (pr0 t5 t6)).(let H18 \def (eq_ind_r T t (\lambda (t7: -T).(eq T (THead (Flat Cast) u t3) t7)) H3 (THead (Flat Appl) v1 (THead (Bind -b) u1 t5)) H11) in (let H19 \def (eq_ind T (THead (Flat Cast) u t3) (\lambda -(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow -(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | -(Flat f) \Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl -\Rightarrow False | Cast \Rightarrow True])])])) I (THead (Flat Appl) v1 -(THead (Bind b) u1 t5)) H18) in (False_ind (ex2 T (\lambda (t7: T).(pr0 t1 -t7)) (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) -O v2) t6)) t7))) H19))))))) t2 H13)) t H11 H12 H7 H8 H9 H10))) | (pr0_delta -u1 u2 H7 t5 t6 H8 w H9) \Rightarrow (\lambda (H10: (eq T (THead (Bind Abbr) -u1 t5) t)).(\lambda (H11: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T -(THead (Bind Abbr) u1 t5) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t2) -\to ((pr0 u1 u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T (\lambda -(t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))))) (\lambda (H12: (eq T -(THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda -(t7: T).((pr0 u1 u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T -(\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 t8))))))) (\lambda (_: -(pr0 u1 u2)).(\lambda (_: (pr0 t5 t6)).(\lambda (_: (subst0 O u2 t6 w)).(let -H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Cast) u t3) t7)) -H3 (THead (Bind Abbr) u1 t5) H10) in (let H17 \def (eq_ind T (THead (Flat -Cast) u t3) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) -\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u1 -t5) H16) in (False_ind (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: -T).(pr0 (THead (Bind Abbr) u2 w) t7))) H17)))))) t2 H12)) t H10 H11 H7 H8 -H9))) | (pr0_zeta b H7 t5 t6 H8 u0) \Rightarrow (\lambda (H9: (eq T (THead -(Bind b) u0 (lift (S O) O t5)) t)).(\lambda (H10: (eq T t6 t2)).(eq_ind T -(THead (Bind b) u0 (lift (S O) O t5)) (\lambda (_: T).((eq T t6 t2) \to ((not -(eq B b Abst)) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) -(\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H11: (eq T t6 t2)).(eq_ind T t2 -(\lambda (t7: T).((not (eq B b Abst)) \to ((pr0 t5 t7) \to (ex2 T (\lambda -(t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (_: (not (eq -B b Abst))).(\lambda (_: (pr0 t5 t2)).(let H14 \def (eq_ind_r T t (\lambda -(t7: T).(eq T (THead (Flat Cast) u t3) t7)) H3 (THead (Bind b) u0 (lift (S O) -O t5)) H9) in (let H15 \def (eq_ind T (THead (Flat Cast) u t3) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K -return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) -\Rightarrow True])])) I (THead (Bind b) u0 (lift (S O) O t5)) H14) in -(False_ind (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 -t7))) H15))))) t6 (sym_eq T t6 t2 H11))) t H9 H10 H7 H8))) | (pr0_tau t5 t6 -H7 u0) \Rightarrow (\lambda (H8: (eq T (THead (Flat Cast) u0 t5) t)).(\lambda -(H9: (eq T t6 t2)).(eq_ind T (THead (Flat Cast) u0 t5) (\lambda (_: T).((eq T -t6 t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda -(t8: T).(pr0 t2 t8)))))) (\lambda (H10: (eq T t6 t2)).(eq_ind T t2 (\lambda -(t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: -T).(pr0 t2 t8))))) (\lambda (H11: (pr0 t5 t2)).(let H12 \def (eq_ind_r T t -(\lambda (t7: T).(eq T (THead (Flat Cast) u t3) t7)) H3 (THead (Flat Cast) u0 -t5) H8) in (let H13 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | -(THead _ t7 _) \Rightarrow t7])) (THead (Flat Cast) u t3) (THead (Flat Cast) -u0 t5) H12) in ((let H14 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) -\Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead (Flat Cast) u t3) -(THead (Flat Cast) u0 t5) H12) in (\lambda (_: (eq T u u0)).(let H16 \def -(eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: -T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: -T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Cast) -u0 t5) H8) in (let H17 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t1)) H6 t5 -H14) in (ex2_ind T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 -t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))) -(\lambda (x: T).(\lambda (H18: (pr0 t1 x)).(\lambda (H19: (pr0 t2 -x)).(ex_intro2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)) -x H18 H19)))) (H16 t5 (tlt_head_dx (Flat Cast) u0 t5) t1 H17 t2 H11)))))) -H13)))) t6 (sym_eq T t6 t2 H10))) t H8 H9 H7)))]) in (H7 (refl_equal T t) -(refl_equal T t2)))) t4 (sym_eq T t4 t1 H5))) t H3 H4 H2)))]) in (H2 -(refl_equal T t) (refl_equal T t1))))))))) t0). -(* COMMENTS -Initial nodes: 46103 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr0/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr0/props.ma deleted file mode 100644 index d7f69d691..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr0/props.ma +++ /dev/null @@ -1,1758 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr0/defs.ma". - -include "Basic-1/subst0/subst0.ma". - -theorem pr0_lift: - \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (h: nat).(\forall -(d: nat).(pr0 (lift h d t1) (lift h d t2)))))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(pr0_ind (\lambda -(t: T).(\lambda (t0: T).(\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t) -(lift h d t0)))))) (\lambda (t: T).(\lambda (h: nat).(\lambda (d: -nat).(pr0_refl (lift h d t))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda -(_: (pr0 u1 u2)).(\lambda (H1: ((\forall (h: nat).(\forall (d: nat).(pr0 -(lift h d u1) (lift h d u2)))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda -(_: (pr0 t3 t4)).(\lambda (H3: ((\forall (h: nat).(\forall (d: nat).(pr0 -(lift h d t3) (lift h d t4)))))).(\lambda (k: K).(\lambda (h: nat).(\lambda -(d: nat).(eq_ind_r T (THead k (lift h d u1) (lift h (s k d) t3)) (\lambda (t: -T).(pr0 t (lift h d (THead k u2 t4)))) (eq_ind_r T (THead k (lift h d u2) -(lift h (s k d) t4)) (\lambda (t: T).(pr0 (THead k (lift h d u1) (lift h (s k -d) t3)) t)) (pr0_comp (lift h d u1) (lift h d u2) (H1 h d) (lift h (s k d) -t3) (lift h (s k d) t4) (H3 h (s k d)) k) (lift h d (THead k u2 t4)) -(lift_head k u2 t4 h d)) (lift h d (THead k u1 t3)) (lift_head k u1 t3 h -d))))))))))))) (\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: -(pr0 v1 v2)).(\lambda (H1: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h -d v1) (lift h d v2)))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 -t3 t4)).(\lambda (H3: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t3) -(lift h d t4)))))).(\lambda (h: nat).(\lambda (d: nat).(eq_ind_r T (THead -(Flat Appl) (lift h d v1) (lift h (s (Flat Appl) d) (THead (Bind Abst) u -t3))) (\lambda (t: T).(pr0 t (lift h d (THead (Bind Abbr) v2 t4)))) (eq_ind_r -T (THead (Bind Abst) (lift h (s (Flat Appl) d) u) (lift h (s (Bind Abst) (s -(Flat Appl) d)) t3)) (\lambda (t: T).(pr0 (THead (Flat Appl) (lift h d v1) t) -(lift h d (THead (Bind Abbr) v2 t4)))) (eq_ind_r T (THead (Bind Abbr) (lift h -d v2) (lift h (s (Bind Abbr) d) t4)) (\lambda (t: T).(pr0 (THead (Flat Appl) -(lift h d v1) (THead (Bind Abst) (lift h (s (Flat Appl) d) u) (lift h (s -(Bind Abst) (s (Flat Appl) d)) t3))) t)) (pr0_beta (lift h (s (Flat Appl) d) -u) (lift h d v1) (lift h d v2) (H1 h d) (lift h (s (Bind Abst) (s (Flat Appl) -d)) t3) (lift h (s (Bind Abbr) d) t4) (H3 h (s (Bind Abbr) d))) (lift h d -(THead (Bind Abbr) v2 t4)) (lift_head (Bind Abbr) v2 t4 h d)) (lift h (s -(Flat Appl) d) (THead (Bind Abst) u t3)) (lift_head (Bind Abst) u t3 h (s -(Flat Appl) d))) (lift h d (THead (Flat Appl) v1 (THead (Bind Abst) u t3))) -(lift_head (Flat Appl) v1 (THead (Bind Abst) u t3) h d))))))))))))) (\lambda -(b: B).(\lambda (H0: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: -T).(\lambda (_: (pr0 v1 v2)).(\lambda (H2: ((\forall (h: nat).(\forall (d: -nat).(pr0 (lift h d v1) (lift h d v2)))))).(\lambda (u1: T).(\lambda (u2: -T).(\lambda (_: (pr0 u1 u2)).(\lambda (H4: ((\forall (h: nat).(\forall (d: -nat).(pr0 (lift h d u1) (lift h d u2)))))).(\lambda (t3: T).(\lambda (t4: -T).(\lambda (_: (pr0 t3 t4)).(\lambda (H6: ((\forall (h: nat).(\forall (d: -nat).(pr0 (lift h d t3) (lift h d t4)))))).(\lambda (h: nat).(\lambda (d: -nat).(eq_ind_r T (THead (Flat Appl) (lift h d v1) (lift h (s (Flat Appl) d) -(THead (Bind b) u1 t3))) (\lambda (t: T).(pr0 t (lift h d (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t4))))) (eq_ind_r T (THead (Bind b) -(lift h (s (Flat Appl) d) u1) (lift h (s (Bind b) (s (Flat Appl) d)) t3)) -(\lambda (t: T).(pr0 (THead (Flat Appl) (lift h d v1) t) (lift h d (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))))) (eq_ind_r T (THead -(Bind b) (lift h d u2) (lift h (s (Bind b) d) (THead (Flat Appl) (lift (S O) -O v2) t4))) (\lambda (t: T).(pr0 (THead (Flat Appl) (lift h d v1) (THead -(Bind b) (lift h (s (Flat Appl) d) u1) (lift h (s (Bind b) (s (Flat Appl) d)) -t3))) t)) (eq_ind_r T (THead (Flat Appl) (lift h (s (Bind b) d) (lift (S O) O -v2)) (lift h (s (Flat Appl) (s (Bind b) d)) t4)) (\lambda (t: T).(pr0 (THead -(Flat Appl) (lift h d v1) (THead (Bind b) (lift h (s (Flat Appl) d) u1) (lift -h (s (Bind b) (s (Flat Appl) d)) t3))) (THead (Bind b) (lift h d u2) t))) -(eq_ind nat (plus (S O) d) (\lambda (n: nat).(pr0 (THead (Flat Appl) (lift h -d v1) (THead (Bind b) (lift h d u1) (lift h n t3))) (THead (Bind b) (lift h d -u2) (THead (Flat Appl) (lift h n (lift (S O) O v2)) (lift h n t4))))) -(eq_ind_r T (lift (S O) O (lift h d v2)) (\lambda (t: T).(pr0 (THead (Flat -Appl) (lift h d v1) (THead (Bind b) (lift h d u1) (lift h (plus (S O) d) -t3))) (THead (Bind b) (lift h d u2) (THead (Flat Appl) t (lift h (plus (S O) -d) t4))))) (pr0_upsilon b H0 (lift h d v1) (lift h d v2) (H2 h d) (lift h d -u1) (lift h d u2) (H4 h d) (lift h (plus (S O) d) t3) (lift h (plus (S O) d) -t4) (H6 h (plus (S O) d))) (lift h (plus (S O) d) (lift (S O) O v2)) (lift_d -v2 h (S O) d O (le_O_n d))) (S d) (refl_equal nat (S d))) (lift h (s (Bind b) -d) (THead (Flat Appl) (lift (S O) O v2) t4)) (lift_head (Flat Appl) (lift (S -O) O v2) t4 h (s (Bind b) d))) (lift h d (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4))) (lift_head (Bind b) u2 (THead (Flat Appl) (lift -(S O) O v2) t4) h d)) (lift h (s (Flat Appl) d) (THead (Bind b) u1 t3)) -(lift_head (Bind b) u1 t3 h (s (Flat Appl) d))) (lift h d (THead (Flat Appl) -v1 (THead (Bind b) u1 t3))) (lift_head (Flat Appl) v1 (THead (Bind b) u1 t3) -h d)))))))))))))))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 -u2)).(\lambda (H1: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d u1) -(lift h d u2)))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 -t4)).(\lambda (H3: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t3) -(lift h d t4)))))).(\lambda (w: T).(\lambda (H4: (subst0 O u2 t4 w)).(\lambda -(h: nat).(\lambda (d: nat).(eq_ind_r T (THead (Bind Abbr) (lift h d u1) (lift -h (s (Bind Abbr) d) t3)) (\lambda (t: T).(pr0 t (lift h d (THead (Bind Abbr) -u2 w)))) (eq_ind_r T (THead (Bind Abbr) (lift h d u2) (lift h (s (Bind Abbr) -d) w)) (\lambda (t: T).(pr0 (THead (Bind Abbr) (lift h d u1) (lift h (s (Bind -Abbr) d) t3)) t)) (pr0_delta (lift h d u1) (lift h d u2) (H1 h d) (lift h (S -d) t3) (lift h (S d) t4) (H3 h (S d)) (lift h (S d) w) (let d' \def (S d) in -(eq_ind nat (minus (S d) (S O)) (\lambda (n: nat).(subst0 O (lift h n u2) -(lift h d' t4) (lift h d' w))) (subst0_lift_lt t4 w u2 O H4 (S d) (le_n_S O d -(le_O_n d)) h) d (eq_ind nat d (\lambda (n: nat).(eq nat n d)) (refl_equal -nat d) (minus d O) (minus_n_O d))))) (lift h d (THead (Bind Abbr) u2 w)) -(lift_head (Bind Abbr) u2 w h d)) (lift h d (THead (Bind Abbr) u1 t3)) -(lift_head (Bind Abbr) u1 t3 h d)))))))))))))) (\lambda (b: B).(\lambda (H0: -(not (eq B b Abst))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 -t4)).(\lambda (H2: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t3) -(lift h d t4)))))).(\lambda (u: T).(\lambda (h: nat).(\lambda (d: -nat).(eq_ind_r T (THead (Bind b) (lift h d u) (lift h (s (Bind b) d) (lift (S -O) O t3))) (\lambda (t: T).(pr0 t (lift h d t4))) (eq_ind nat (plus (S O) d) -(\lambda (n: nat).(pr0 (THead (Bind b) (lift h d u) (lift h n (lift (S O) O -t3))) (lift h d t4))) (eq_ind_r T (lift (S O) O (lift h d t3)) (\lambda (t: -T).(pr0 (THead (Bind b) (lift h d u) t) (lift h d t4))) (pr0_zeta b H0 (lift -h d t3) (lift h d t4) (H2 h d) (lift h d u)) (lift h (plus (S O) d) (lift (S -O) O t3)) (lift_d t3 h (S O) d O (le_O_n d))) (S d) (refl_equal nat (S d))) -(lift h d (THead (Bind b) u (lift (S O) O t3))) (lift_head (Bind b) u (lift -(S O) O t3) h d))))))))))) (\lambda (t3: T).(\lambda (t4: T).(\lambda (_: -(pr0 t3 t4)).(\lambda (H1: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h -d t3) (lift h d t4)))))).(\lambda (u: T).(\lambda (h: nat).(\lambda (d: -nat).(eq_ind_r T (THead (Flat Cast) (lift h d u) (lift h (s (Flat Cast) d) -t3)) (\lambda (t: T).(pr0 t (lift h d t4))) (pr0_tau (lift h (s (Flat Cast) -d) t3) (lift h d t4) (H1 h d) (lift h d u)) (lift h d (THead (Flat Cast) u -t3)) (lift_head (Flat Cast) u t3 h d))))))))) t1 t2 H))). -(* COMMENTS -Initial nodes: 2845 -END *) - -theorem pr0_subst0_back: - \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst0 -i u2 t1 t2) \to (\forall (u1: T).((pr0 u1 u2) \to (ex2 T (\lambda (t: -T).(subst0 i u1 t1 t)) (\lambda (t: T).(pr0 t t2))))))))) -\def - \lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (i: nat).(\lambda -(H: (subst0 i u2 t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t: -T).(\lambda (t0: T).(\lambda (t3: T).(\forall (u1: T).((pr0 u1 t) \to (ex2 T -(\lambda (t4: T).(subst0 n u1 t0 t4)) (\lambda (t4: T).(pr0 t4 t3))))))))) -(\lambda (v: T).(\lambda (i0: nat).(\lambda (u1: T).(\lambda (H0: (pr0 u1 -v)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1 (TLRef i0) t)) (\lambda (t: -T).(pr0 t (lift (S i0) O v))) (lift (S i0) O u1) (subst0_lref u1 i0) -(pr0_lift u1 v H0 (S i0) O)))))) (\lambda (v: T).(\lambda (u3: T).(\lambda -(u1: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v u1 u3)).(\lambda (H1: -((\forall (u4: T).((pr0 u4 v) \to (ex2 T (\lambda (t: T).(subst0 i0 u4 u1 t)) -(\lambda (t: T).(pr0 t u3))))))).(\lambda (t: T).(\lambda (k: K).(\lambda -(u0: T).(\lambda (H2: (pr0 u0 v)).(ex2_ind T (\lambda (t0: T).(subst0 i0 u0 -u1 t0)) (\lambda (t0: T).(pr0 t0 u3)) (ex2 T (\lambda (t0: T).(subst0 i0 u0 -(THead k u1 t) t0)) (\lambda (t0: T).(pr0 t0 (THead k u3 t)))) (\lambda (x: -T).(\lambda (H3: (subst0 i0 u0 u1 x)).(\lambda (H4: (pr0 x u3)).(ex_intro2 T -(\lambda (t0: T).(subst0 i0 u0 (THead k u1 t) t0)) (\lambda (t0: T).(pr0 t0 -(THead k u3 t))) (THead k x t) (subst0_fst u0 x u1 i0 H3 t k) (pr0_comp x u3 -H4 t t (pr0_refl t) k))))) (H1 u0 H2)))))))))))) (\lambda (k: K).(\lambda (v: -T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (i0: nat).(\lambda (_: (subst0 -(s k i0) v t4 t3)).(\lambda (H1: ((\forall (u1: T).((pr0 u1 v) \to (ex2 T -(\lambda (t: T).(subst0 (s k i0) u1 t4 t)) (\lambda (t: T).(pr0 t -t3))))))).(\lambda (u: T).(\lambda (u1: T).(\lambda (H2: (pr0 u1 v)).(ex2_ind -T (\lambda (t: T).(subst0 (s k i0) u1 t4 t)) (\lambda (t: T).(pr0 t t3)) (ex2 -T (\lambda (t: T).(subst0 i0 u1 (THead k u t4) t)) (\lambda (t: T).(pr0 t -(THead k u t3)))) (\lambda (x: T).(\lambda (H3: (subst0 (s k i0) u1 t4 -x)).(\lambda (H4: (pr0 x t3)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1 -(THead k u t4) t)) (\lambda (t: T).(pr0 t (THead k u t3))) (THead k u x) -(subst0_snd k u1 x t4 i0 H3 u) (pr0_comp u u (pr0_refl u) x t3 H4 k))))) (H1 -u1 H2)))))))))))) (\lambda (v: T).(\lambda (u1: T).(\lambda (u3: T).(\lambda -(i0: nat).(\lambda (_: (subst0 i0 v u1 u3)).(\lambda (H1: ((\forall (u4: -T).((pr0 u4 v) \to (ex2 T (\lambda (t: T).(subst0 i0 u4 u1 t)) (\lambda (t: -T).(pr0 t u3))))))).(\lambda (k: K).(\lambda (t3: T).(\lambda (t4: -T).(\lambda (_: (subst0 (s k i0) v t3 t4)).(\lambda (H3: ((\forall (u4: -T).((pr0 u4 v) \to (ex2 T (\lambda (t: T).(subst0 (s k i0) u4 t3 t)) (\lambda -(t: T).(pr0 t t4))))))).(\lambda (u0: T).(\lambda (H4: (pr0 u0 v)).(ex2_ind T -(\lambda (t: T).(subst0 (s k i0) u0 t3 t)) (\lambda (t: T).(pr0 t t4)) (ex2 T -(\lambda (t: T).(subst0 i0 u0 (THead k u1 t3) t)) (\lambda (t: T).(pr0 t -(THead k u3 t4)))) (\lambda (x: T).(\lambda (H5: (subst0 (s k i0) u0 t3 -x)).(\lambda (H6: (pr0 x t4)).(ex2_ind T (\lambda (t: T).(subst0 i0 u0 u1 t)) -(\lambda (t: T).(pr0 t u3)) (ex2 T (\lambda (t: T).(subst0 i0 u0 (THead k u1 -t3) t)) (\lambda (t: T).(pr0 t (THead k u3 t4)))) (\lambda (x0: T).(\lambda -(H7: (subst0 i0 u0 u1 x0)).(\lambda (H8: (pr0 x0 u3)).(ex_intro2 T (\lambda -(t: T).(subst0 i0 u0 (THead k u1 t3) t)) (\lambda (t: T).(pr0 t (THead k u3 -t4))) (THead k x0 x) (subst0_both u0 u1 x0 i0 H7 k t3 x H5) (pr0_comp x0 u3 -H8 x t4 H6 k))))) (H1 u0 H4))))) (H3 u0 H4))))))))))))))) i u2 t1 t2 H))))). -(* COMMENTS -Initial nodes: 979 -END *) - -theorem pr0_subst0_fwd: - \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst0 -i u2 t1 t2) \to (\forall (u1: T).((pr0 u2 u1) \to (ex2 T (\lambda (t: -T).(subst0 i u1 t1 t)) (\lambda (t: T).(pr0 t2 t))))))))) -\def - \lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (i: nat).(\lambda -(H: (subst0 i u2 t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t: -T).(\lambda (t0: T).(\lambda (t3: T).(\forall (u1: T).((pr0 t u1) \to (ex2 T -(\lambda (t4: T).(subst0 n u1 t0 t4)) (\lambda (t4: T).(pr0 t3 t4))))))))) -(\lambda (v: T).(\lambda (i0: nat).(\lambda (u1: T).(\lambda (H0: (pr0 v -u1)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1 (TLRef i0) t)) (\lambda (t: -T).(pr0 (lift (S i0) O v) t)) (lift (S i0) O u1) (subst0_lref u1 i0) -(pr0_lift v u1 H0 (S i0) O)))))) (\lambda (v: T).(\lambda (u3: T).(\lambda -(u1: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v u1 u3)).(\lambda (H1: -((\forall (u4: T).((pr0 v u4) \to (ex2 T (\lambda (t: T).(subst0 i0 u4 u1 t)) -(\lambda (t: T).(pr0 u3 t))))))).(\lambda (t: T).(\lambda (k: K).(\lambda -(u0: T).(\lambda (H2: (pr0 v u0)).(ex2_ind T (\lambda (t0: T).(subst0 i0 u0 -u1 t0)) (\lambda (t0: T).(pr0 u3 t0)) (ex2 T (\lambda (t0: T).(subst0 i0 u0 -(THead k u1 t) t0)) (\lambda (t0: T).(pr0 (THead k u3 t) t0))) (\lambda (x: -T).(\lambda (H3: (subst0 i0 u0 u1 x)).(\lambda (H4: (pr0 u3 x)).(ex_intro2 T -(\lambda (t0: T).(subst0 i0 u0 (THead k u1 t) t0)) (\lambda (t0: T).(pr0 -(THead k u3 t) t0)) (THead k x t) (subst0_fst u0 x u1 i0 H3 t k) (pr0_comp u3 -x H4 t t (pr0_refl t) k))))) (H1 u0 H2)))))))))))) (\lambda (k: K).(\lambda -(v: T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (i0: nat).(\lambda (_: -(subst0 (s k i0) v t4 t3)).(\lambda (H1: ((\forall (u1: T).((pr0 v u1) \to -(ex2 T (\lambda (t: T).(subst0 (s k i0) u1 t4 t)) (\lambda (t: T).(pr0 t3 -t))))))).(\lambda (u: T).(\lambda (u1: T).(\lambda (H2: (pr0 v u1)).(ex2_ind -T (\lambda (t: T).(subst0 (s k i0) u1 t4 t)) (\lambda (t: T).(pr0 t3 t)) (ex2 -T (\lambda (t: T).(subst0 i0 u1 (THead k u t4) t)) (\lambda (t: T).(pr0 -(THead k u t3) t))) (\lambda (x: T).(\lambda (H3: (subst0 (s k i0) u1 t4 -x)).(\lambda (H4: (pr0 t3 x)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1 -(THead k u t4) t)) (\lambda (t: T).(pr0 (THead k u t3) t)) (THead k u x) -(subst0_snd k u1 x t4 i0 H3 u) (pr0_comp u u (pr0_refl u) t3 x H4 k))))) (H1 -u1 H2)))))))))))) (\lambda (v: T).(\lambda (u1: T).(\lambda (u3: T).(\lambda -(i0: nat).(\lambda (_: (subst0 i0 v u1 u3)).(\lambda (H1: ((\forall (u4: -T).((pr0 v u4) \to (ex2 T (\lambda (t: T).(subst0 i0 u4 u1 t)) (\lambda (t: -T).(pr0 u3 t))))))).(\lambda (k: K).(\lambda (t3: T).(\lambda (t4: -T).(\lambda (_: (subst0 (s k i0) v t3 t4)).(\lambda (H3: ((\forall (u4: -T).((pr0 v u4) \to (ex2 T (\lambda (t: T).(subst0 (s k i0) u4 t3 t)) (\lambda -(t: T).(pr0 t4 t))))))).(\lambda (u0: T).(\lambda (H4: (pr0 v u0)).(ex2_ind T -(\lambda (t: T).(subst0 (s k i0) u0 t3 t)) (\lambda (t: T).(pr0 t4 t)) (ex2 T -(\lambda (t: T).(subst0 i0 u0 (THead k u1 t3) t)) (\lambda (t: T).(pr0 (THead -k u3 t4) t))) (\lambda (x: T).(\lambda (H5: (subst0 (s k i0) u0 t3 -x)).(\lambda (H6: (pr0 t4 x)).(ex2_ind T (\lambda (t: T).(subst0 i0 u0 u1 t)) -(\lambda (t: T).(pr0 u3 t)) (ex2 T (\lambda (t: T).(subst0 i0 u0 (THead k u1 -t3) t)) (\lambda (t: T).(pr0 (THead k u3 t4) t))) (\lambda (x0: T).(\lambda -(H7: (subst0 i0 u0 u1 x0)).(\lambda (H8: (pr0 u3 x0)).(ex_intro2 T (\lambda -(t: T).(subst0 i0 u0 (THead k u1 t3) t)) (\lambda (t: T).(pr0 (THead k u3 t4) -t)) (THead k x0 x) (subst0_both u0 u1 x0 i0 H7 k t3 x H5) (pr0_comp u3 x0 H8 -t4 x H6 k))))) (H1 u0 H4))))) (H3 u0 H4))))))))))))))) i u2 t1 t2 H))))). -(* COMMENTS -Initial nodes: 979 -END *) - -theorem pr0_subst0: - \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (v1: T).(\forall -(w1: T).(\forall (i: nat).((subst0 i v1 t1 w1) \to (\forall (v2: T).((pr0 v1 -v2) \to (or (pr0 w1 t2) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: -T).(subst0 i v2 t2 w2)))))))))))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(pr0_ind (\lambda -(t: T).(\lambda (t0: T).(\forall (v1: T).(\forall (w1: T).(\forall (i: -nat).((subst0 i v1 t w1) \to (\forall (v2: T).((pr0 v1 v2) \to (or (pr0 w1 -t0) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t0 -w2)))))))))))) (\lambda (t: T).(\lambda (v1: T).(\lambda (w1: T).(\lambda (i: -nat).(\lambda (H0: (subst0 i v1 t w1)).(\lambda (v2: T).(\lambda (H1: (pr0 v1 -v2)).(or_intror (pr0 w1 t) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: -T).(subst0 i v2 t w2))) (ex2_sym T (subst0 i v2 t) (pr0 w1) (pr0_subst0_fwd -v1 t w1 i H0 v2 H1)))))))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (H0: -(pr0 u1 u2)).(\lambda (H1: ((\forall (v1: T).(\forall (w1: T).(\forall (i: -nat).((subst0 i v1 u1 w1) \to (\forall (v2: T).((pr0 v1 v2) \to (or (pr0 w1 -u2) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 u2 -w2)))))))))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 -t4)).(\lambda (H3: ((\forall (v1: T).(\forall (w1: T).(\forall (i: -nat).((subst0 i v1 t3 w1) \to (\forall (v2: T).((pr0 v1 v2) \to (or (pr0 w1 -t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 -w2)))))))))))).(\lambda (k: K).(\lambda (v1: T).(\lambda (w1: T).(\lambda (i: -nat).(\lambda (H4: (subst0 i v1 (THead k u1 t3) w1)).(\lambda (v2: -T).(\lambda (H5: (pr0 v1 v2)).(or3_ind (ex2 T (\lambda (u3: T).(eq T w1 -(THead k u3 t3))) (\lambda (u3: T).(subst0 i v1 u1 u3))) (ex2 T (\lambda (t5: -T).(eq T w1 (THead k u1 t5))) (\lambda (t5: T).(subst0 (s k i) v1 t3 t5))) -(ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T w1 (THead k u3 t5)))) -(\lambda (u3: T).(\lambda (_: T).(subst0 i v1 u1 u3))) (\lambda (_: -T).(\lambda (t5: T).(subst0 (s k i) v1 t3 t5)))) (or (pr0 w1 (THead k u2 t4)) -(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 (THead k -u2 t4) w2)))) (\lambda (H6: (ex2 T (\lambda (u3: T).(eq T w1 (THead k u3 -t3))) (\lambda (u3: T).(subst0 i v1 u1 u3)))).(ex2_ind T (\lambda (u3: T).(eq -T w1 (THead k u3 t3))) (\lambda (u3: T).(subst0 i v1 u1 u3)) (or (pr0 w1 -(THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: -T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x: T).(\lambda (H7: (eq T w1 -(THead k x t3))).(\lambda (H8: (subst0 i v1 u1 x)).(eq_ind_r T (THead k x t3) -(\lambda (t: T).(or (pr0 t (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 t -w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))))) (or_ind (pr0 x u2) -(ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 u2 w2))) -(or (pr0 (THead k x t3) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead -k x t3) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda -(H9: (pr0 x u2)).(or_introl (pr0 (THead k x t3) (THead k u2 t4)) (ex2 T -(\lambda (w2: T).(pr0 (THead k x t3) w2)) (\lambda (w2: T).(subst0 i v2 -(THead k u2 t4) w2))) (pr0_comp x u2 H9 t3 t4 H2 k))) (\lambda (H9: (ex2 T -(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 u2 w2)))).(ex2_ind -T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 u2 w2)) (or (pr0 -(THead k x t3) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x t3) -w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x0: -T).(\lambda (H10: (pr0 x x0)).(\lambda (H11: (subst0 i v2 u2 x0)).(or_intror -(pr0 (THead k x t3) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x -t3) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))) (ex_intro2 T -(\lambda (w2: T).(pr0 (THead k x t3) w2)) (\lambda (w2: T).(subst0 i v2 -(THead k u2 t4) w2)) (THead k x0 t4) (pr0_comp x x0 H10 t3 t4 H2 k) -(subst0_fst v2 x0 u2 i H11 t4 k)))))) H9)) (H1 v1 x i H8 v2 H5)) w1 H7)))) -H6)) (\lambda (H6: (ex2 T (\lambda (t5: T).(eq T w1 (THead k u1 t5))) -(\lambda (t5: T).(subst0 (s k i) v1 t3 t5)))).(ex2_ind T (\lambda (t5: T).(eq -T w1 (THead k u1 t5))) (\lambda (t5: T).(subst0 (s k i) v1 t3 t5)) (or (pr0 -w1 (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: -T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x: T).(\lambda (H7: (eq T w1 -(THead k u1 x))).(\lambda (H8: (subst0 (s k i) v1 t3 x)).(eq_ind_r T (THead k -u1 x) (\lambda (t: T).(or (pr0 t (THead k u2 t4)) (ex2 T (\lambda (w2: -T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))))) (or_ind -(pr0 x t4) (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s k -i) v2 t4 w2))) (or (pr0 (THead k u1 x) (THead k u2 t4)) (ex2 T (\lambda (w2: -T).(pr0 (THead k u1 x) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) -w2)))) (\lambda (H9: (pr0 x t4)).(or_introl (pr0 (THead k u1 x) (THead k u2 -t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k u1 x) w2)) (\lambda (w2: -T).(subst0 i v2 (THead k u2 t4) w2))) (pr0_comp u1 u2 H0 x t4 H9 k))) -(\lambda (H9: (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s -k i) v2 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: -T).(subst0 (s k i) v2 t4 w2)) (or (pr0 (THead k u1 x) (THead k u2 t4)) (ex2 T -(\lambda (w2: T).(pr0 (THead k u1 x) w2)) (\lambda (w2: T).(subst0 i v2 -(THead k u2 t4) w2)))) (\lambda (x0: T).(\lambda (H10: (pr0 x x0)).(\lambda -(H11: (subst0 (s k i) v2 t4 x0)).(or_intror (pr0 (THead k u1 x) (THead k u2 -t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k u1 x) w2)) (\lambda (w2: -T).(subst0 i v2 (THead k u2 t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 -(THead k u1 x) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)) (THead -k u2 x0) (pr0_comp u1 u2 H0 x x0 H10 k) (subst0_snd k v2 x0 t4 i H11 u2)))))) -H9)) (H3 v1 x (s k i) H8 v2 H5)) w1 H7)))) H6)) (\lambda (H6: (ex3_2 T T -(\lambda (u3: T).(\lambda (t5: T).(eq T w1 (THead k u3 t5)))) (\lambda (u3: -T).(\lambda (_: T).(subst0 i v1 u1 u3))) (\lambda (_: T).(\lambda (t5: -T).(subst0 (s k i) v1 t3 t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda -(t5: T).(eq T w1 (THead k u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 -i v1 u1 u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s k i) v1 t3 t5))) -(or (pr0 w1 (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda -(w2: T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H7: (eq T w1 (THead k x0 x1))).(\lambda (H8: (subst0 i v1 u1 -x0)).(\lambda (H9: (subst0 (s k i) v1 t3 x1)).(eq_ind_r T (THead k x0 x1) -(\lambda (t: T).(or (pr0 t (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 t -w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))))) (or_ind (pr0 x1 -t4) (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s k i) v2 -t4 w2))) (or (pr0 (THead k x0 x1) (THead k u2 t4)) (ex2 T (\lambda (w2: -T).(pr0 (THead k x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) -w2)))) (\lambda (H10: (pr0 x1 t4)).(or_ind (pr0 x0 u2) (ex2 T (\lambda (w2: -T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 u2 w2))) (or (pr0 (THead k x0 -x1) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x0 x1) w2)) -(\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (H11: (pr0 x0 -u2)).(or_introl (pr0 (THead k x0 x1) (THead k u2 t4)) (ex2 T (\lambda (w2: -T).(pr0 (THead k x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) -w2))) (pr0_comp x0 u2 H11 x1 t4 H10 k))) (\lambda (H11: (ex2 T (\lambda (w2: -T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 u2 w2)))).(ex2_ind T (\lambda -(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 u2 w2)) (or (pr0 (THead k -x0 x1) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x0 x1) w2)) -(\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x: T).(\lambda -(H12: (pr0 x0 x)).(\lambda (H13: (subst0 i v2 u2 x)).(or_intror (pr0 (THead k -x0 x1) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x0 x1) w2)) -(\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))) (ex_intro2 T (\lambda -(w2: T).(pr0 (THead k x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 -t4) w2)) (THead k x t4) (pr0_comp x0 x H12 x1 t4 H10 k) (subst0_fst v2 x u2 i -H13 t4 k)))))) H11)) (H1 v1 x0 i H8 v2 H5))) (\lambda (H10: (ex2 T (\lambda -(w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s k i) v2 t4 w2)))).(ex2_ind T -(\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s k i) v2 t4 w2)) (or -(pr0 (THead k x0 x1) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k -x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x: -T).(\lambda (H11: (pr0 x1 x)).(\lambda (H12: (subst0 (s k i) v2 t4 -x)).(or_ind (pr0 x0 u2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: -T).(subst0 i v2 u2 w2))) (or (pr0 (THead k x0 x1) (THead k u2 t4)) (ex2 T -(\lambda (w2: T).(pr0 (THead k x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 -(THead k u2 t4) w2)))) (\lambda (H13: (pr0 x0 u2)).(or_intror (pr0 (THead k -x0 x1) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x0 x1) w2)) -(\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))) (ex_intro2 T (\lambda -(w2: T).(pr0 (THead k x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 -t4) w2)) (THead k u2 x) (pr0_comp x0 u2 H13 x1 x H11 k) (subst0_snd k v2 x t4 -i H12 u2)))) (\lambda (H13: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda -(w2: T).(subst0 i v2 u2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) -(\lambda (w2: T).(subst0 i v2 u2 w2)) (or (pr0 (THead k x0 x1) (THead k u2 -t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x0 x1) w2)) (\lambda (w2: -T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x2: T).(\lambda (H14: (pr0 -x0 x2)).(\lambda (H15: (subst0 i v2 u2 x2)).(or_intror (pr0 (THead k x0 x1) -(THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x0 x1) w2)) (\lambda -(w2: T).(subst0 i v2 (THead k u2 t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 -(THead k x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)) -(THead k x2 x) (pr0_comp x0 x2 H14 x1 x H11 k) (subst0_both v2 u2 x2 i H15 k -t4 x H12)))))) H13)) (H1 v1 x0 i H8 v2 H5))))) H10)) (H3 v1 x1 (s k i) H9 v2 -H5)) w1 H7)))))) H6)) (subst0_gen_head k v1 u1 t3 w1 i H4))))))))))))))))) -(\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H0: (pr0 v1 -v2)).(\lambda (H1: ((\forall (v3: T).(\forall (w1: T).(\forall (i: -nat).((subst0 i v3 v1 w1) \to (\forall (v4: T).((pr0 v3 v4) \to (or (pr0 w1 -v2) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v4 v2 -w2)))))))))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 -t4)).(\lambda (H3: ((\forall (v3: T).(\forall (w1: T).(\forall (i: -nat).((subst0 i v3 t3 w1) \to (\forall (v4: T).((pr0 v3 v4) \to (or (pr0 w1 -t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v4 t4 -w2)))))))))))).(\lambda (v0: T).(\lambda (w1: T).(\lambda (i: nat).(\lambda -(H4: (subst0 i v0 (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) -w1)).(\lambda (v3: T).(\lambda (H5: (pr0 v0 v3)).(or3_ind (ex2 T (\lambda -(u2: T).(eq T w1 (THead (Flat Appl) u2 (THead (Bind Abst) u t3)))) (\lambda -(u2: T).(subst0 i v0 v1 u2))) (ex2 T (\lambda (t5: T).(eq T w1 (THead (Flat -Appl) v1 t5))) (\lambda (t5: T).(subst0 (s (Flat Appl) i) v0 (THead (Bind -Abst) u t3) t5))) (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T w1 -(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 v1 -u2))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Flat Appl) i) v0 (THead -(Bind Abst) u t3) t5)))) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T -(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind -Abbr) v2 t4) w2)))) (\lambda (H6: (ex2 T (\lambda (u2: T).(eq T w1 (THead -(Flat Appl) u2 (THead (Bind Abst) u t3)))) (\lambda (u2: T).(subst0 i v0 v1 -u2)))).(ex2_ind T (\lambda (u2: T).(eq T w1 (THead (Flat Appl) u2 (THead -(Bind Abst) u t3)))) (\lambda (u2: T).(subst0 i v0 v1 u2)) (or (pr0 w1 (THead -(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (x: T).(\lambda -(H7: (eq T w1 (THead (Flat Appl) x (THead (Bind Abst) u t3)))).(\lambda (H8: -(subst0 i v0 v1 x)).(eq_ind_r T (THead (Flat Appl) x (THead (Bind Abst) u -t3)) (\lambda (t: T).(or (pr0 t (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda -(w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) -w2))))) (or_ind (pr0 x v2) (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: -T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat Appl) x (THead (Bind Abst) u -t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat -Appl) x (THead (Bind Abst) u t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead -(Bind Abbr) v2 t4) w2)))) (\lambda (H9: (pr0 x v2)).(or_introl (pr0 (THead -(Flat Appl) x (THead (Bind Abst) u t3)) (THead (Bind Abbr) v2 t4)) (ex2 T -(\lambda (w2: T).(pr0 (THead (Flat Appl) x (THead (Bind Abst) u t3)) w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (pr0_beta u x -v2 H9 t3 t4 H2))) (\lambda (H9: (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda -(w2: T).(subst0 i v3 v2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) -(\lambda (w2: T).(subst0 i v3 v2 w2)) (or (pr0 (THead (Flat Appl) x (THead -(Bind Abst) u t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 -(THead (Flat Appl) x (THead (Bind Abst) u t3)) w2)) (\lambda (w2: T).(subst0 -i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (x0: T).(\lambda (H10: (pr0 x -x0)).(\lambda (H11: (subst0 i v3 v2 x0)).(or_intror (pr0 (THead (Flat Appl) x -(THead (Bind Abst) u t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x (THead (Bind Abst) u t3)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x (THead (Bind Abst) u t3)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)) (THead (Bind Abbr) x0 t4) -(pr0_beta u x x0 H10 t3 t4 H2) (subst0_fst v3 x0 v2 i H11 t4 (Bind -Abbr))))))) H9)) (H1 v0 x i H8 v3 H5)) w1 H7)))) H6)) (\lambda (H6: (ex2 T -(\lambda (t5: T).(eq T w1 (THead (Flat Appl) v1 t5))) (\lambda (t5: -T).(subst0 (s (Flat Appl) i) v0 (THead (Bind Abst) u t3) t5)))).(ex2_ind T -(\lambda (t5: T).(eq T w1 (THead (Flat Appl) v1 t5))) (\lambda (t5: -T).(subst0 (s (Flat Appl) i) v0 (THead (Bind Abst) u t3) t5)) (or (pr0 w1 -(THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (x: T).(\lambda -(H7: (eq T w1 (THead (Flat Appl) v1 x))).(\lambda (H8: (subst0 (s (Flat Appl) -i) v0 (THead (Bind Abst) u t3) x)).(or3_ind (ex2 T (\lambda (u2: T).(eq T x -(THead (Bind Abst) u2 t3))) (\lambda (u2: T).(subst0 (s (Flat Appl) i) v0 u -u2))) (ex2 T (\lambda (t5: T).(eq T x (THead (Bind Abst) u t5))) (\lambda -(t5: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5))) (ex3_2 T T -(\lambda (u2: T).(\lambda (t5: T).(eq T x (THead (Bind Abst) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u u2))) -(\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 -t3 t5)))) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 -w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) -(\lambda (H9: (ex2 T (\lambda (u2: T).(eq T x (THead (Bind Abst) u2 t3))) -(\lambda (u2: T).(subst0 (s (Flat Appl) i) v0 u u2)))).(ex2_ind T (\lambda -(u2: T).(eq T x (THead (Bind Abst) u2 t3))) (\lambda (u2: T).(subst0 (s (Flat -Appl) i) v0 u u2)) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda -(w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) -w2)))) (\lambda (x0: T).(\lambda (H10: (eq T x (THead (Bind Abst) x0 -t3))).(\lambda (_: (subst0 (s (Flat Appl) i) v0 u x0)).(let H12 \def (eq_ind -T x (\lambda (t: T).(eq T w1 (THead (Flat Appl) v1 t))) H7 (THead (Bind Abst) -x0 t3) H10) in (eq_ind_r T (THead (Flat Appl) v1 (THead (Bind Abst) x0 t3)) -(\lambda (t: T).(or (pr0 t (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: -T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) -w2))))) (or_introl (pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 t3)) -(THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 -(THead (Bind Abst) x0 t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind -Abbr) v2 t4) w2))) (pr0_beta x0 v1 v2 H0 t3 t4 H2)) w1 H12))))) H9)) (\lambda -(H9: (ex2 T (\lambda (t5: T).(eq T x (THead (Bind Abst) u t5))) (\lambda (t5: -T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5)))).(ex2_ind T (\lambda -(t5: T).(eq T x (THead (Bind Abst) u t5))) (\lambda (t5: T).(subst0 (s (Bind -Abst) (s (Flat Appl) i)) v0 t3 t5)) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) -(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead -(Bind Abbr) v2 t4) w2)))) (\lambda (x0: T).(\lambda (H10: (eq T x (THead -(Bind Abst) u x0))).(\lambda (H11: (subst0 (s (Bind Abst) (s (Flat Appl) i)) -v0 t3 x0)).(let H12 \def (eq_ind T x (\lambda (t: T).(eq T w1 (THead (Flat -Appl) v1 t))) H7 (THead (Bind Abst) u x0) H10) in (eq_ind_r T (THead (Flat -Appl) v1 (THead (Bind Abst) u x0)) (\lambda (t: T).(or (pr0 t (THead (Bind -Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i -v3 (THead (Bind Abbr) v2 t4) w2))))) (or_ind (pr0 x0 t4) (ex2 T (\lambda (w2: -T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 -t4 w2))) (or (pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u x0)) (THead -(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead -(Bind Abst) u x0)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 -t4) w2)))) (\lambda (H13: (pr0 x0 t4)).(or_introl (pr0 (THead (Flat Appl) v1 -(THead (Bind Abst) u x0)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u x0)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (pr0_beta u v1 v2 H0 x0 t4 -H13))) (\lambda (H13: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: -T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda -(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) -i)) v3 t4 w2)) (or (pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u x0)) -(THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 -(THead (Bind Abst) u x0)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind -Abbr) v2 t4) w2)))) (\lambda (x1: T).(\lambda (H14: (pr0 x0 x1)).(\lambda -(H15: (subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 x1)).(or_intror (pr0 -(THead (Flat Appl) v1 (THead (Bind Abst) u x0)) (THead (Bind Abbr) v2 t4)) -(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u x0)) -w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (ex_intro2 -T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u x0)) w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)) (THead (Bind -Abbr) v2 x1) (pr0_beta u v1 v2 H0 x0 x1 H14) (subst0_snd (Bind Abbr) v3 x1 t4 -i H15 v2)))))) H13)) (H3 v0 x0 (s (Bind Abst) (s (Flat Appl) i)) H11 v3 H5)) -w1 H12))))) H9)) (\lambda (H9: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: -T).(eq T x (THead (Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 (s (Flat Appl) i) v0 u u2))) (\lambda (_: T).(\lambda (t5: -T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5))))).(ex3_2_ind T T -(\lambda (u2: T).(\lambda (t5: T).(eq T x (THead (Bind Abst) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u u2))) -(\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 -t3 t5))) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 -w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) -(\lambda (x0: T).(\lambda (x1: T).(\lambda (H10: (eq T x (THead (Bind Abst) -x0 x1))).(\lambda (_: (subst0 (s (Flat Appl) i) v0 u x0)).(\lambda (H12: -(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 x1)).(let H13 \def (eq_ind T -x (\lambda (t: T).(eq T w1 (THead (Flat Appl) v1 t))) H7 (THead (Bind Abst) -x0 x1) H10) in (eq_ind_r T (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) -(\lambda (t: T).(or (pr0 t (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: -T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) -w2))))) (or_ind (pr0 x1 t4) (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda -(w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2))) (or (pr0 (THead -(Flat Appl) v1 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) v2 t4)) (ex2 T -(\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H14: -(pr0 x1 t4)).(or_introl (pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) -(THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 -(THead (Bind Abst) x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind -Abbr) v2 t4) w2))) (pr0_beta x0 v1 v2 H0 x1 t4 H14))) (\lambda (H14: (ex2 T -(\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s -(Flat Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x1 w2)) -(\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2)) (or -(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) v2 -t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 -x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) -(\lambda (x2: T).(\lambda (H15: (pr0 x1 x2)).(\lambda (H16: (subst0 (s (Bind -Abst) (s (Flat Appl) i)) v3 t4 x2)).(or_intror (pr0 (THead (Flat Appl) v1 -(THead (Bind Abst) x0 x1)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)) (THead (Bind Abbr) v2 x2) -(pr0_beta x0 v1 v2 H0 x1 x2 H15) (subst0_snd (Bind Abbr) v3 x2 t4 i H16 -v2)))))) H14)) (H3 v0 x1 (s (Bind Abst) (s (Flat Appl) i)) H12 v3 H5)) w1 -H13))))))) H9)) (subst0_gen_head (Bind Abst) v0 u t3 x (s (Flat Appl) i) -H8))))) H6)) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T -w1 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 -v1 u2))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Flat Appl) i) v0 (THead -(Bind Abst) u t3) t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: -T).(eq T w1 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 i v0 v1 u2))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Flat -Appl) i) v0 (THead (Bind Abst) u t3) t5))) (or (pr0 w1 (THead (Bind Abbr) v2 -t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 -(THead (Bind Abbr) v2 t4) w2)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda -(H7: (eq T w1 (THead (Flat Appl) x0 x1))).(\lambda (H8: (subst0 i v0 v1 -x0)).(\lambda (H9: (subst0 (s (Flat Appl) i) v0 (THead (Bind Abst) u t3) -x1)).(or3_ind (ex2 T (\lambda (u2: T).(eq T x1 (THead (Bind Abst) u2 t3))) -(\lambda (u2: T).(subst0 (s (Flat Appl) i) v0 u u2))) (ex2 T (\lambda (t5: -T).(eq T x1 (THead (Bind Abst) u t5))) (\lambda (t5: T).(subst0 (s (Bind -Abst) (s (Flat Appl) i)) v0 t3 t5))) (ex3_2 T T (\lambda (u2: T).(\lambda -(t5: T).(eq T x1 (THead (Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 (s (Flat Appl) i) v0 u u2))) (\lambda (_: T).(\lambda (t5: -T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5)))) (or (pr0 w1 (THead -(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H10: (ex2 T -(\lambda (u2: T).(eq T x1 (THead (Bind Abst) u2 t3))) (\lambda (u2: -T).(subst0 (s (Flat Appl) i) v0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T x1 -(THead (Bind Abst) u2 t3))) (\lambda (u2: T).(subst0 (s (Flat Appl) i) v0 u -u2)) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 -w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda -(x: T).(\lambda (H11: (eq T x1 (THead (Bind Abst) x t3))).(\lambda (_: -(subst0 (s (Flat Appl) i) v0 u x)).(let H13 \def (eq_ind T x1 (\lambda (t: -T).(eq T w1 (THead (Flat Appl) x0 t))) H7 (THead (Bind Abst) x t3) H11) in -(eq_ind_r T (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) (\lambda (t: -T).(or (pr0 t (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 t w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))))) (or_ind (pr0 -x0 v2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 -w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) (THead (Bind -Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind -Abst) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) -w2)))) (\lambda (H14: (pr0 x0 v2)).(or_introl (pr0 (THead (Flat Appl) x0 -(THead (Bind Abst) x t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (pr0_beta x x0 v2 H14 t3 t4 -H2))) (\lambda (H14: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: -T).(subst0 i v3 v2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda -(w2: T).(subst0 i v3 v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind -Abst) x t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead -(Flat Appl) x0 (THead (Bind Abst) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 -(THead (Bind Abbr) v2 t4) w2)))) (\lambda (x2: T).(\lambda (H15: (pr0 x0 -x2)).(\lambda (H16: (subst0 i v3 v2 x2)).(or_intror (pr0 (THead (Flat Appl) -x0 (THead (Bind Abst) x t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)) (THead (Bind Abbr) x2 t4) -(pr0_beta x x0 x2 H15 t3 t4 H2) (subst0_fst v3 x2 v2 i H16 t4 (Bind -Abbr))))))) H14)) (H1 v0 x0 i H8 v3 H5)) w1 H13))))) H10)) (\lambda (H10: -(ex2 T (\lambda (t5: T).(eq T x1 (THead (Bind Abst) u t5))) (\lambda (t5: -T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5)))).(ex2_ind T (\lambda -(t5: T).(eq T x1 (THead (Bind Abst) u t5))) (\lambda (t5: T).(subst0 (s (Bind -Abst) (s (Flat Appl) i)) v0 t3 t5)) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) -(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead -(Bind Abbr) v2 t4) w2)))) (\lambda (x: T).(\lambda (H11: (eq T x1 (THead -(Bind Abst) u x))).(\lambda (H12: (subst0 (s (Bind Abst) (s (Flat Appl) i)) -v0 t3 x)).(let H13 \def (eq_ind T x1 (\lambda (t: T).(eq T w1 (THead (Flat -Appl) x0 t))) H7 (THead (Bind Abst) u x) H11) in (eq_ind_r T (THead (Flat -Appl) x0 (THead (Bind Abst) u x)) (\lambda (t: T).(or (pr0 t (THead (Bind -Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i -v3 (THead (Bind Abbr) v2 t4) w2))))) (or_ind (pr0 x t4) (ex2 T (\lambda (w2: -T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 -t4 w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind -Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind -Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) -w2)))) (\lambda (H14: (pr0 x t4)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2: -T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat -Appl) x0 (THead (Bind Abst) u x)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda -(w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) w2)) (\lambda -(w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H15: (pr0 x0 -v2)).(or_introl (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead -(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead -(Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 -t4) w2))) (pr0_beta u x0 v2 H15 x t4 H14))) (\lambda (H15: (ex2 T (\lambda -(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)))).(ex2_ind T -(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)) (or (pr0 -(THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind Abbr) v2 t4)) -(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) -w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda -(x2: T).(\lambda (H16: (pr0 x0 x2)).(\lambda (H17: (subst0 i v3 v2 -x2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead -(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead -(Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 -t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead -(Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 -t4) w2)) (THead (Bind Abbr) x2 t4) (pr0_beta u x0 x2 H16 x t4 H14) -(subst0_fst v3 x2 v2 i H17 t4 (Bind Abbr))))))) H15)) (H1 v0 x0 i H8 v3 H5))) -(\lambda (H14: (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 -(s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda (w2: -T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 -t4 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind -Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind -Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) -w2)))) (\lambda (x2: T).(\lambda (H15: (pr0 x x2)).(\lambda (H16: (subst0 (s -(Bind Abst) (s (Flat Appl) i)) v3 t4 x2)).(or_ind (pr0 x0 v2) (ex2 T (\lambda -(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead -(Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind Abbr) v2 t4)) (ex2 T -(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H17: -(pr0 x0 v2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) -(THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 -(THead (Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind -Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 -(THead (Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind -Abbr) v2 t4) w2)) (THead (Bind Abbr) v2 x2) (pr0_beta u x0 v2 H17 x x2 H15) -(subst0_snd (Bind Abbr) v3 x2 t4 i H16 v2)))) (\lambda (H17: (ex2 T (\lambda -(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)))).(ex2_ind T -(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)) (or (pr0 -(THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind Abbr) v2 t4)) -(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) -w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda -(x3: T).(\lambda (H18: (pr0 x0 x3)).(\lambda (H19: (subst0 i v3 v2 -x3)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead -(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead -(Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 -t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead -(Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 -t4) w2)) (THead (Bind Abbr) x3 x2) (pr0_beta u x0 x3 H18 x x2 H15) -(subst0_both v3 v2 x3 i H19 (Bind Abbr) t4 x2 H16)))))) H17)) (H1 v0 x0 i H8 -v3 H5))))) H14)) (H3 v0 x (s (Bind Abst) (s (Flat Appl) i)) H12 v3 H5)) w1 -H13))))) H10)) (\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: -T).(eq T x1 (THead (Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 (s (Flat Appl) i) v0 u u2))) (\lambda (_: T).(\lambda (t5: -T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5))))).(ex3_2_ind T T -(\lambda (u2: T).(\lambda (t5: T).(eq T x1 (THead (Bind Abst) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u u2))) -(\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 -t3 t5))) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 -w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) -(\lambda (x2: T).(\lambda (x3: T).(\lambda (H11: (eq T x1 (THead (Bind Abst) -x2 x3))).(\lambda (_: (subst0 (s (Flat Appl) i) v0 u x2)).(\lambda (H13: -(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 x3)).(let H14 \def (eq_ind T -x1 (\lambda (t: T).(eq T w1 (THead (Flat Appl) x0 t))) H7 (THead (Bind Abst) -x2 x3) H11) in (eq_ind_r T (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) -(\lambda (t: T).(or (pr0 t (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: -T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) -w2))))) (or_ind (pr0 x3 t4) (ex2 T (\lambda (w2: T).(pr0 x3 w2)) (\lambda -(w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2))) (or (pr0 (THead -(Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead (Bind Abbr) v2 t4)) (ex2 T -(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H15: -(pr0 x3 t4)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) -(\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat Appl) x0 (THead -(Bind Abst) x2 x3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 -(THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H16: (pr0 x0 -v2)).(or_introl (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead -(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead -(Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 -t4) w2))) (pr0_beta x2 x0 v2 H16 x3 t4 H15))) (\lambda (H16: (ex2 T (\lambda -(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)))).(ex2_ind T -(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)) (or (pr0 -(THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead (Bind Abbr) v2 t4)) -(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) -w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda -(x: T).(\lambda (H17: (pr0 x0 x)).(\lambda (H18: (subst0 i v3 v2 -x)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead -(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead -(Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 -t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead -(Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 -t4) w2)) (THead (Bind Abbr) x t4) (pr0_beta x2 x0 x H17 x3 t4 H15) -(subst0_fst v3 x v2 i H18 t4 (Bind Abbr))))))) H16)) (H1 v0 x0 i H8 v3 H5))) -(\lambda (H15: (ex2 T (\lambda (w2: T).(pr0 x3 w2)) (\lambda (w2: T).(subst0 -(s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda (w2: -T).(pr0 x3 w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 -t4 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead -(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead -(Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 -t4) w2)))) (\lambda (x: T).(\lambda (H16: (pr0 x3 x)).(\lambda (H17: (subst0 -(s (Bind Abst) (s (Flat Appl) i)) v3 t4 x)).(or_ind (pr0 x0 v2) (ex2 T -(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 -(THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead (Bind Abbr) v2 t4)) -(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) -w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda -(H18: (pr0 x0 v2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) -x2 x3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat -Appl) x0 (THead (Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead -(Bind Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat -Appl) x0 (THead (Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead -(Bind Abbr) v2 t4) w2)) (THead (Bind Abbr) v2 x) (pr0_beta x2 x0 v2 H18 x3 x -H16) (subst0_snd (Bind Abbr) v3 x t4 i H17 v2)))) (\lambda (H18: (ex2 T -(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 -w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 -v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead -(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead -(Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 -t4) w2)))) (\lambda (x4: T).(\lambda (H19: (pr0 x0 x4)).(\lambda (H20: -(subst0 i v3 v2 x4)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) -x2 x3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat -Appl) x0 (THead (Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead -(Bind Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat -Appl) x0 (THead (Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead -(Bind Abbr) v2 t4) w2)) (THead (Bind Abbr) x4 x) (pr0_beta x2 x0 x4 H19 x3 x -H16) (subst0_both v3 v2 x4 i H20 (Bind Abbr) t4 x H17)))))) H18)) (H1 v0 x0 i -H8 v3 H5))))) H15)) (H3 v0 x3 (s (Bind Abst) (s (Flat Appl) i)) H13 v3 H5)) -w1 H14))))))) H10)) (subst0_gen_head (Bind Abst) v0 u t3 x1 (s (Flat Appl) i) -H9))))))) H6)) (subst0_gen_head (Flat Appl) v0 v1 (THead (Bind Abst) u t3) w1 -i H4))))))))))))))))) (\lambda (b: B).(\lambda (H0: (not (eq B b -Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H1: (pr0 v1 v2)).(\lambda -(H2: ((\forall (v3: T).(\forall (w1: T).(\forall (i: nat).((subst0 i v3 v1 -w1) \to (\forall (v4: T).((pr0 v3 v4) \to (or (pr0 w1 v2) (ex2 T (\lambda -(w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v4 v2 w2)))))))))))).(\lambda -(u1: T).(\lambda (u2: T).(\lambda (H3: (pr0 u1 u2)).(\lambda (H4: ((\forall -(v3: T).(\forall (w1: T).(\forall (i: nat).((subst0 i v3 u1 w1) \to (\forall -(v4: T).((pr0 v3 v4) \to (or (pr0 w1 u2) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) -(\lambda (w2: T).(subst0 i v4 u2 w2)))))))))))).(\lambda (t3: T).(\lambda -(t4: T).(\lambda (H5: (pr0 t3 t4)).(\lambda (H6: ((\forall (v3: T).(\forall -(w1: T).(\forall (i: nat).((subst0 i v3 t3 w1) \to (\forall (v4: T).((pr0 v3 -v4) \to (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: -T).(subst0 i v4 t4 w2)))))))))))).(\lambda (v0: T).(\lambda (w1: T).(\lambda -(i: nat).(\lambda (H7: (subst0 i v0 (THead (Flat Appl) v1 (THead (Bind b) u1 -t3)) w1)).(\lambda (v3: T).(\lambda (H8: (pr0 v0 v3)).(or3_ind (ex2 T -(\lambda (u3: T).(eq T w1 (THead (Flat Appl) u3 (THead (Bind b) u1 t3)))) -(\lambda (u3: T).(subst0 i v0 v1 u3))) (ex2 T (\lambda (t5: T).(eq T w1 -(THead (Flat Appl) v1 t5))) (\lambda (t5: T).(subst0 (s (Flat Appl) i) v0 -(THead (Bind b) u1 t3) t5))) (ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq -T w1 (THead (Flat Appl) u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i -v0 v1 u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Flat Appl) i) v0 -(THead (Bind b) u1 t3) t5)))) (or (pr0 w1 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda -(w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) -t4)) w2)))) (\lambda (H9: (ex2 T (\lambda (u3: T).(eq T w1 (THead (Flat Appl) -u3 (THead (Bind b) u1 t3)))) (\lambda (u3: T).(subst0 i v0 v1 u3)))).(ex2_ind -T (\lambda (u3: T).(eq T w1 (THead (Flat Appl) u3 (THead (Bind b) u1 t3)))) -(\lambda (u3: T).(subst0 i v0 v1 u3)) (or (pr0 w1 (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t4)) w2)))) (\lambda (x: T).(\lambda (H10: (eq T w1 (THead (Flat -Appl) x (THead (Bind b) u1 t3)))).(\lambda (H11: (subst0 i v0 v1 -x)).(eq_ind_r T (THead (Flat Appl) x (THead (Bind b) u1 t3)) (\lambda (t: -T).(or (pr0 t (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) -(ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2))))) (or_ind (pr0 x -v2) (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v3 v2 -w2))) (or (pr0 (THead (Flat Appl) x (THead (Bind b) u1 t3)) (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 -(THead (Flat Appl) x (THead (Bind b) u1 t3)) w2)) (\lambda (w2: T).(subst0 i -v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) -(\lambda (H12: (pr0 x v2)).(or_introl (pr0 (THead (Flat Appl) x (THead (Bind -b) u1 t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 -T (\lambda (w2: T).(pr0 (THead (Flat Appl) x (THead (Bind b) u1 t3)) w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t4)) w2))) (pr0_upsilon b H0 x v2 H12 u1 u2 H3 t3 t4 H5))) (\lambda -(H12: (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v3 v2 -w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v3 -v2 w2)) (or (pr0 (THead (Flat Appl) x (THead (Bind b) u1 t3)) (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 -(THead (Flat Appl) x (THead (Bind b) u1 t3)) w2)) (\lambda (w2: T).(subst0 i -v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) -(\lambda (x0: T).(\lambda (H13: (pr0 x x0)).(\lambda (H14: (subst0 i v3 v2 -x0)).(or_intror (pr0 (THead (Flat Appl) x (THead (Bind b) u1 t3)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x (THead (Bind b) u1 t3)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x (THead (Bind b) -u1 t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 (THead (Flat Appl) (lift -(S O) O x0) t4)) (pr0_upsilon b H0 x x0 H13 u1 u2 H3 t3 t4 H5) (subst0_snd -(Bind b) v3 (THead (Flat Appl) (lift (S O) O x0) t4) (THead (Flat Appl) (lift -(S O) O v2) t4) i (subst0_fst v3 (lift (S O) O x0) (lift (S O) O v2) (s (Bind -b) i) (subst0_lift_ge_s v2 x0 v3 i H14 O (le_O_n i) b) t4 (Flat Appl)) -u2)))))) H12)) (H2 v0 x i H11 v3 H8)) w1 H10)))) H9)) (\lambda (H9: (ex2 T -(\lambda (t5: T).(eq T w1 (THead (Flat Appl) v1 t5))) (\lambda (t5: -T).(subst0 (s (Flat Appl) i) v0 (THead (Bind b) u1 t3) t5)))).(ex2_ind T -(\lambda (t5: T).(eq T w1 (THead (Flat Appl) v1 t5))) (\lambda (t5: -T).(subst0 (s (Flat Appl) i) v0 (THead (Bind b) u1 t3) t5)) (or (pr0 w1 -(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda -(w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x: T).(\lambda (H10: (eq -T w1 (THead (Flat Appl) v1 x))).(\lambda (H11: (subst0 (s (Flat Appl) i) v0 -(THead (Bind b) u1 t3) x)).(or3_ind (ex2 T (\lambda (u3: T).(eq T x (THead -(Bind b) u3 t3))) (\lambda (u3: T).(subst0 (s (Flat Appl) i) v0 u1 u3))) (ex2 -T (\lambda (t5: T).(eq T x (THead (Bind b) u1 t5))) (\lambda (t5: T).(subst0 -(s (Bind b) (s (Flat Appl) i)) v0 t3 t5))) (ex3_2 T T (\lambda (u3: -T).(\lambda (t5: T).(eq T x (THead (Bind b) u3 t5)))) (\lambda (u3: -T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u1 u3))) (\lambda (_: -T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 t5)))) (or -(pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T -(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H12: (ex2 T -(\lambda (u3: T).(eq T x (THead (Bind b) u3 t3))) (\lambda (u3: T).(subst0 (s -(Flat Appl) i) v0 u1 u3)))).(ex2_ind T (\lambda (u3: T).(eq T x (THead (Bind -b) u3 t3))) (\lambda (u3: T).(subst0 (s (Flat Appl) i) v0 u1 u3)) (or (pr0 w1 -(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda -(w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x0: T).(\lambda (H13: (eq -T x (THead (Bind b) x0 t3))).(\lambda (H14: (subst0 (s (Flat Appl) i) v0 u1 -x0)).(let H15 \def (eq_ind T x (\lambda (t: T).(eq T w1 (THead (Flat Appl) v1 -t))) H10 (THead (Bind b) x0 t3) H13) in (eq_ind_r T (THead (Flat Appl) v1 -(THead (Bind b) x0 t3)) (\lambda (t: T).(or (pr0 t (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 t w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t4)) w2))))) (or_ind (pr0 x0 u2) (ex2 T (\lambda (w2: T).(pr0 x0 -w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 u2 w2))) (or (pr0 (THead -(Flat Appl) v1 (THead (Bind b) x0 t3)) (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 -(THead (Bind b) x0 t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H16: (pr0 x0 -u2)).(or_introl (pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 t3)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 t3)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2))) (pr0_upsilon b H0 v1 v2 H1 x0 u2 H16 t3 t4 H5))) (\lambda (H16: (ex2 T -(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 -u2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 -(s (Flat Appl) i) v3 u2 w2)) (or (pr0 (THead (Flat Appl) v1 (THead (Bind b) -x0 t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T -(\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 t3)) w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t4)) w2)))) (\lambda (x1: T).(\lambda (H17: (pr0 x0 x1)).(\lambda -(H18: (subst0 (s (Flat Appl) i) v3 u2 x1)).(or_intror (pr0 (THead (Flat Appl) -v1 (THead (Bind b) x0 t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) -O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind -b) x0 t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead -(Flat Appl) v1 (THead (Bind b) x0 t3)) w2)) (\lambda (w2: T).(subst0 i v3 -(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)) (THead -(Bind b) x1 (THead (Flat Appl) (lift (S O) O v2) t4)) (pr0_upsilon b H0 v1 v2 -H1 x0 x1 H17 t3 t4 H5) (subst0_fst v3 x1 u2 i H18 (THead (Flat Appl) (lift (S -O) O v2) t4) (Bind b))))))) H16)) (H4 v0 x0 (s (Flat Appl) i) H14 v3 H8)) w1 -H15))))) H12)) (\lambda (H12: (ex2 T (\lambda (t5: T).(eq T x (THead (Bind b) -u1 t5))) (\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 -t5)))).(ex2_ind T (\lambda (t5: T).(eq T x (THead (Bind b) u1 t5))) (\lambda -(t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 t5)) (or (pr0 w1 (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x0: T).(\lambda (H13: (eq T x -(THead (Bind b) u1 x0))).(\lambda (H14: (subst0 (s (Bind b) (s (Flat Appl) -i)) v0 t3 x0)).(let H15 \def (eq_ind T x (\lambda (t: T).(eq T w1 (THead -(Flat Appl) v1 t))) H10 (THead (Bind b) u1 x0) H13) in (eq_ind_r T (THead -(Flat Appl) v1 (THead (Bind b) u1 x0)) (\lambda (t: T).(or (pr0 t (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) w2))))) (or_ind (pr0 x0 t4) (ex2 T (\lambda (w2: -T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 -w2))) (or (pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 x0)) (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 -(THead (Flat Appl) v1 (THead (Bind b) u1 x0)) w2)) (\lambda (w2: T).(subst0 i -v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) -(\lambda (H16: (pr0 x0 t4)).(or_introl (pr0 (THead (Flat Appl) v1 (THead -(Bind b) u1 x0)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) -t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 -x0)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4)) w2))) (pr0_upsilon b H0 v1 v2 H1 u1 u2 H3 x0 t4 H16))) -(\lambda (H16: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 -(s (Bind b) (s (Flat Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 -x0 w2)) (\lambda (w2: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 w2)) -(or (pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 x0)) (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 -(THead (Flat Appl) v1 (THead (Bind b) u1 x0)) w2)) (\lambda (w2: T).(subst0 i -v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) -(\lambda (x1: T).(\lambda (H17: (pr0 x0 x1)).(\lambda (H18: (subst0 (s (Bind -b) (s (Flat Appl) i)) v3 t4 x1)).(or_intror (pr0 (THead (Flat Appl) v1 (THead -(Bind b) u1 x0)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) -t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 -x0)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat -Appl) v1 (THead (Bind b) u1 x0)) w2)) (\lambda (w2: T).(subst0 i v3 (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) x1)) (pr0_upsilon b H0 v1 v2 H1 u1 u2 H3 -x0 x1 H17) (subst0_snd (Bind b) v3 (THead (Flat Appl) (lift (S O) O v2) x1) -(THead (Flat Appl) (lift (S O) O v2) t4) i (subst0_snd (Flat Appl) v3 x1 t4 -(s (Bind b) i) H18 (lift (S O) O v2)) u2)))))) H16)) (H6 v0 x0 (s (Bind b) (s -(Flat Appl) i)) H14 v3 H8)) w1 H15))))) H12)) (\lambda (H12: (ex3_2 T T -(\lambda (u3: T).(\lambda (t5: T).(eq T x (THead (Bind b) u3 t5)))) (\lambda -(u3: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u1 u3))) (\lambda (_: -T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 -t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t5: T).(eq T x (THead (Bind -b) u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u1 -u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) -v0 t3 t5))) (or (pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O -v2) t4))) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 -(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda -(x0: T).(\lambda (x1: T).(\lambda (H13: (eq T x (THead (Bind b) x0 -x1))).(\lambda (H14: (subst0 (s (Flat Appl) i) v0 u1 x0)).(\lambda (H15: -(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 x1)).(let H16 \def (eq_ind T x -(\lambda (t: T).(eq T w1 (THead (Flat Appl) v1 t))) H10 (THead (Bind b) x0 -x1) H13) in (eq_ind_r T (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) -(\lambda (t: T).(or (pr0 t (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) -O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 -(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2))))) (or_ind -(pr0 x1 t4) (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s -(Bind b) (s (Flat Appl) i)) v3 t4 w2))) (or (pr0 (THead (Flat Appl) v1 (THead -(Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) -t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 -x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4)) w2)))) (\lambda (H17: (pr0 x1 t4)).(or_ind (pr0 x0 u2) -(ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) -i) v3 u2 w2))) (or (pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2)))) (\lambda (H18: (pr0 x0 u2)).(or_introl (pr0 (THead (Flat Appl) v1 -(THead (Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O -v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) -x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) w2))) (pr0_upsilon b H0 v1 v2 H1 x0 u2 H18 x1 t4 -H17))) (\lambda (H18: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: -T).(subst0 (s (Flat Appl) i) v3 u2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 -w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 u2 w2)) (or (pr0 (THead -(Flat Appl) v1 (THead (Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 -(THead (Bind b) x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x2: T).(\lambda -(H19: (pr0 x0 x2)).(\lambda (H20: (subst0 (s (Flat Appl) i) v3 u2 -x2)).(or_intror (pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind -b) x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) x2 (THead (Flat Appl) (lift -(S O) O v2) t4)) (pr0_upsilon b H0 v1 v2 H1 x0 x2 H19 x1 t4 H17) (subst0_fst -v3 x2 u2 i H20 (THead (Flat Appl) (lift (S O) O v2) t4) (Bind b))))))) H18)) -(H4 v0 x0 (s (Flat Appl) i) H14 v3 H8))) (\lambda (H17: (ex2 T (\lambda (w2: -T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 -w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s -(Bind b) (s (Flat Appl) i)) v3 t4 w2)) (or (pr0 (THead (Flat Appl) v1 (THead -(Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) -t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 -x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4)) w2)))) (\lambda (x2: T).(\lambda (H18: (pr0 x1 -x2)).(\lambda (H19: (subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 x2)).(or_ind -(pr0 x0 u2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s -(Flat Appl) i) v3 u2 w2))) (or (pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 -x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T -(\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t4)) w2)))) (\lambda (H20: (pr0 x0 u2)).(or_intror (pr0 (THead (Flat -Appl) v1 (THead (Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift -(S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead -(Bind b) x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 -(THead (Flat Appl) v1 (THead (Bind b) x0 x1)) w2)) (\lambda (w2: T).(subst0 i -v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) x2)) (pr0_upsilon b H0 v1 v2 -H1 x0 u2 H20 x1 x2 H18) (subst0_snd (Bind b) v3 (THead (Flat Appl) (lift (S -O) O v2) x2) (THead (Flat Appl) (lift (S O) O v2) t4) i (subst0_snd (Flat -Appl) v3 x2 t4 (s (Bind b) i) H19 (lift (S O) O v2)) u2)))) (\lambda (H20: -(ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) -i) v3 u2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: -T).(subst0 (s (Flat Appl) i) v3 u2 w2)) (or (pr0 (THead (Flat Appl) v1 (THead -(Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) -t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 -x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4)) w2)))) (\lambda (x3: T).(\lambda (H21: (pr0 x0 -x3)).(\lambda (H22: (subst0 (s (Flat Appl) i) v3 u2 x3)).(or_intror (pr0 -(THead (Flat Appl) v1 (THead (Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) -v1 (THead (Bind b) x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2)) (THead (Bind b) x3 (THead (Flat Appl) (lift (S O) O v2) x2)) -(pr0_upsilon b H0 v1 v2 H1 x0 x3 H21 x1 x2 H18) (subst0_both v3 u2 x3 i H22 -(Bind b) (THead (Flat Appl) (lift (S O) O v2) t4) (THead (Flat Appl) (lift (S -O) O v2) x2) (subst0_snd (Flat Appl) v3 x2 t4 (s (Bind b) i) H19 (lift (S O) -O v2)))))))) H20)) (H4 v0 x0 (s (Flat Appl) i) H14 v3 H8))))) H17)) (H6 v0 x1 -(s (Bind b) (s (Flat Appl) i)) H15 v3 H8)) w1 H16))))))) H12)) -(subst0_gen_head (Bind b) v0 u1 t3 x (s (Flat Appl) i) H11))))) H9)) (\lambda -(H9: (ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T w1 (THead (Flat Appl) -u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v0 v1 u3))) (\lambda (_: -T).(\lambda (t5: T).(subst0 (s (Flat Appl) i) v0 (THead (Bind b) u1 t3) -t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t5: T).(eq T w1 (THead -(Flat Appl) u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v0 v1 u3))) -(\lambda (_: T).(\lambda (t5: T).(subst0 (s (Flat Appl) i) v0 (THead (Bind b) -u1 t3) t5))) (or (pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O -v2) t4))) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 -(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda -(x0: T).(\lambda (x1: T).(\lambda (H10: (eq T w1 (THead (Flat Appl) x0 -x1))).(\lambda (H11: (subst0 i v0 v1 x0)).(\lambda (H12: (subst0 (s (Flat -Appl) i) v0 (THead (Bind b) u1 t3) x1)).(or3_ind (ex2 T (\lambda (u3: T).(eq -T x1 (THead (Bind b) u3 t3))) (\lambda (u3: T).(subst0 (s (Flat Appl) i) v0 -u1 u3))) (ex2 T (\lambda (t5: T).(eq T x1 (THead (Bind b) u1 t5))) (\lambda -(t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 t5))) (ex3_2 T T -(\lambda (u3: T).(\lambda (t5: T).(eq T x1 (THead (Bind b) u3 t5)))) (\lambda -(u3: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u1 u3))) (\lambda (_: -T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 t5)))) (or -(pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T -(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H13: (ex2 T -(\lambda (u3: T).(eq T x1 (THead (Bind b) u3 t3))) (\lambda (u3: T).(subst0 -(s (Flat Appl) i) v0 u1 u3)))).(ex2_ind T (\lambda (u3: T).(eq T x1 (THead -(Bind b) u3 t3))) (\lambda (u3: T).(subst0 (s (Flat Appl) i) v0 u1 u3)) (or -(pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T -(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x: T).(\lambda -(H14: (eq T x1 (THead (Bind b) x t3))).(\lambda (H15: (subst0 (s (Flat Appl) -i) v0 u1 x)).(let H16 \def (eq_ind T x1 (\lambda (t: T).(eq T w1 (THead (Flat -Appl) x0 t))) H10 (THead (Bind b) x t3) H14) in (eq_ind_r T (THead (Flat -Appl) x0 (THead (Bind b) x t3)) (\lambda (t: T).(or (pr0 t (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 t -w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4)) w2))))) (or_ind (pr0 x u2) (ex2 T (\lambda (w2: -T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 u2 w2))) (or -(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat -Appl) x0 (THead (Bind b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H17: -(pr0 x u2)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda -(w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) -x t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T -(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t4)) w2)))) (\lambda (H18: (pr0 x0 v2)).(or_introl (pr0 (THead (Flat -Appl) x0 (THead (Bind b) x t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift -(S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead -(Bind b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t4)) w2))) (pr0_upsilon b H0 x0 v2 H18 x u2 H17 -t3 t4 H5))) (\lambda (H18: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: -T).(subst0 i v3 v2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda -(w2: T).(subst0 i v3 v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x -t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T -(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t4)) w2)))) (\lambda (x2: T).(\lambda (H19: (pr0 x0 x2)).(\lambda -(H20: (subst0 i v3 v2 x2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind -b) x t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 -T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 -(THead (Bind b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O x2) t4)) (pr0_upsilon b H0 x0 x2 H19 x u2 H17 t3 t4 -H5) (subst0_snd (Bind b) v3 (THead (Flat Appl) (lift (S O) O x2) t4) (THead -(Flat Appl) (lift (S O) O v2) t4) i (subst0_fst v3 (lift (S O) O x2) (lift (S -O) O v2) (s (Bind b) i) (subst0_lift_ge_s v2 x2 v3 i H20 O (le_O_n i) b) t4 -(Flat Appl)) u2)))))) H18)) (H2 v0 x0 i H11 v3 H8))) (\lambda (H17: (ex2 T -(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 u2 -w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s -(Flat Appl) i) v3 u2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x -t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T -(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t4)) w2)))) (\lambda (x2: T).(\lambda (H18: (pr0 x x2)).(\lambda -(H19: (subst0 (s (Flat Appl) i) v3 u2 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H20 x x2 H18 t3 t4 H5) (subst0_fst -v3 x2 u2 i H19 (THead (Flat Appl) (lift (S O) O v2) t4) (Bind b))))) (\lambda -(H20: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 -w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 -v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 -(THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) (\lambda (w2: T).(subst0 i -v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) -(\lambda (x3: T).(\lambda (H21: (pr0 x0 x3)).(\lambda (H22: (subst0 i v3 v2 -x3)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind -b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) x2 (THead (Flat Appl) (lift -(S O) O x3) t4)) (pr0_upsilon b H0 x0 x3 H21 x x2 H18 t3 t4 H5) (subst0_both -v3 u2 x2 i H19 (Bind b) (THead (Flat Appl) (lift (S O) O v2) t4) (THead (Flat -Appl) (lift (S O) O x3) t4) (subst0_fst v3 (lift (S O) O x3) (lift (S O) O -v2) (s (Bind b) i) (subst0_lift_ge_s v2 x3 v3 i H22 O (le_O_n i) b) t4 (Flat -Appl)))))))) H20)) (H2 v0 x0 i H11 v3 H8))))) H17)) (H4 v0 x (s (Flat Appl) -i) H15 v3 H8)) w1 H16))))) H13)) (\lambda (H13: (ex2 T (\lambda (t5: T).(eq T -x1 (THead (Bind b) u1 t5))) (\lambda (t5: T).(subst0 (s (Bind b) (s (Flat -Appl) i)) v0 t3 t5)))).(ex2_ind T (\lambda (t5: T).(eq T x1 (THead (Bind b) -u1 t5))) (\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 t5)) -(or (pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) -(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x: -T).(\lambda (H14: (eq T x1 (THead (Bind b) u1 x))).(\lambda (H15: (subst0 (s -(Bind b) (s (Flat Appl) i)) v0 t3 x)).(let H16 \def (eq_ind T x1 (\lambda (t: -T).(eq T w1 (THead (Flat Appl) x0 t))) H10 (THead (Bind b) u1 x) H14) in -(eq_ind_r T (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (\lambda (t: T).(or -(pr0 t (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T -(\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t4)) w2))))) (or_ind (pr0 x t4) (ex2 T -(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind b) (s (Flat -Appl) i)) v3 t4 w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) -(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda -(w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2)))) (\lambda (H17: (pr0 x t4)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2: -T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat -Appl) x0 (THead (Bind b) u1 x)) (THead (Bind b) u2 (THead (Flat Appl) (lift -(S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead -(Bind b) u1 x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H18: (pr0 x0 -v2)).(or_introl (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2))) (pr0_upsilon b H0 x0 v2 H18 u1 u2 H3 x t4 H17))) (\lambda (H18: (ex2 T -(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 -w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 -v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 -(THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: T).(subst0 i -v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) -(\lambda (x2: T).(\lambda (H19: (pr0 x0 x2)).(\lambda (H20: (subst0 i v3 v2 -x2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind -b) u1 x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 (THead (Flat Appl) (lift -(S O) O x2) t4)) (pr0_upsilon b H0 x0 x2 H19 u1 u2 H3 x t4 H17) (subst0_snd -(Bind b) v3 (THead (Flat Appl) (lift (S O) O x2) t4) (THead (Flat Appl) (lift -(S O) O v2) t4) i (subst0_fst v3 (lift (S O) O x2) (lift (S O) O v2) (s (Bind -b) i) (subst0_lift_ge_s v2 x2 v3 i H20 O (le_O_n i) b) t4 (Flat Appl)) -u2)))))) H18)) (H2 v0 x0 i H11 v3 H8))) (\lambda (H17: (ex2 T (\lambda (w2: -T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 -w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s -(Bind b) (s (Flat Appl) i)) v3 t4 w2)) (or (pr0 (THead (Flat Appl) x0 (THead -(Bind b) u1 x)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) -(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) -w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4)) w2)))) (\lambda (x2: T).(\lambda (H18: (pr0 x -x2)).(\lambda (H19: (subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 x2)).(or_ind -(pr0 x0 v2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i -v3 v2 w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2)))) (\lambda (H20: (pr0 x0 v2)).(or_intror (pr0 (THead (Flat Appl) x0 -(THead (Bind b) u1 x)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O -v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) -u1 x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead -(Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: T).(subst0 i v3 -(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) x2)) (pr0_upsilon b H0 x0 v2 -H20 u1 u2 H3 x x2 H18) (subst0_snd (Bind b) v3 (THead (Flat Appl) (lift (S O) -O v2) x2) (THead (Flat Appl) (lift (S O) O v2) t4) i (subst0_snd (Flat Appl) -v3 x2 t4 (s (Bind b) i) H19 (lift (S O) O v2)) u2)))) (\lambda (H20: (ex2 T -(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 -w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 -v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 -(THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: T).(subst0 i -v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) -(\lambda (x3: T).(\lambda (H21: (pr0 x0 x3)).(\lambda (H22: (subst0 i v3 v2 -x3)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind -b) u1 x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 (THead (Flat Appl) (lift -(S O) O x3) x2)) (pr0_upsilon b H0 x0 x3 H21 u1 u2 H3 x x2 H18) (subst0_snd -(Bind b) v3 (THead (Flat Appl) (lift (S O) O x3) x2) (THead (Flat Appl) (lift -(S O) O v2) t4) i (subst0_both v3 (lift (S O) O v2) (lift (S O) O x3) (s -(Bind b) i) (subst0_lift_ge_s v2 x3 v3 i H22 O (le_O_n i) b) (Flat Appl) t4 -x2 H19) u2)))))) H20)) (H2 v0 x0 i H11 v3 H8))))) H17)) (H6 v0 x (s (Bind b) -(s (Flat Appl) i)) H15 v3 H8)) w1 H16))))) H13)) (\lambda (H13: (ex3_2 T T -(\lambda (u3: T).(\lambda (t5: T).(eq T x1 (THead (Bind b) u3 t5)))) (\lambda -(u3: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u1 u3))) (\lambda (_: -T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 -t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t5: T).(eq T x1 (THead -(Bind b) u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) -v0 u1 u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat -Appl) i)) v0 t3 t5))) (or (pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift -(S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H14: (eq T x1 (THead (Bind -b) x2 x3))).(\lambda (H15: (subst0 (s (Flat Appl) i) v0 u1 x2)).(\lambda -(H16: (subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 x3)).(let H17 \def (eq_ind -T x1 (\lambda (t: T).(eq T w1 (THead (Flat Appl) x0 t))) H10 (THead (Bind b) -x2 x3) H14) in (eq_ind_r T (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) -(\lambda (t: T).(or (pr0 t (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) -O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 -(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2))))) (or_ind -(pr0 x3 t4) (ex2 T (\lambda (w2: T).(pr0 x3 w2)) (\lambda (w2: T).(subst0 (s -(Bind b) (s (Flat Appl) i)) v3 t4 w2))) (or (pr0 (THead (Flat Appl) x0 (THead -(Bind b) x2 x3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) -t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 -x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4)) w2)))) (\lambda (H18: (pr0 x3 t4)).(or_ind (pr0 x2 u2) -(ex2 T (\lambda (w2: T).(pr0 x2 w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) -i) v3 u2 w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2)))) (\lambda (H19: (pr0 x2 u2)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2: -T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat -Appl) x0 (THead (Bind b) x2 x3)) (THead (Bind b) u2 (THead (Flat Appl) (lift -(S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead -(Bind b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H20: (pr0 x0 -v2)).(or_introl (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2))) (pr0_upsilon b H0 x0 v2 H20 x2 u2 H19 x3 t4 H18))) (\lambda (H20: (ex2 -T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 -w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 -v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead (Bind -b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 -(THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i -v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) -(\lambda (x: T).(\lambda (H21: (pr0 x0 x)).(\lambda (H22: (subst0 i v3 v2 -x)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind -b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 (THead (Flat Appl) (lift -(S O) O x) t4)) (pr0_upsilon b H0 x0 x H21 x2 u2 H19 x3 t4 H18) (subst0_snd -(Bind b) v3 (THead (Flat Appl) (lift (S O) O x) t4) (THead (Flat Appl) (lift -(S O) O v2) t4) i (subst0_fst v3 (lift (S O) O x) (lift (S O) O v2) (s (Bind -b) i) (subst0_lift_ge_s v2 x v3 i H22 O (le_O_n i) b) t4 (Flat Appl)) -u2)))))) H20)) (H2 v0 x0 i H11 v3 H8))) (\lambda (H19: (ex2 T (\lambda (w2: -T).(pr0 x2 w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 u2 -w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x2 w2)) (\lambda (w2: T).(subst0 (s -(Flat Appl) i) v3 u2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 -x3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T -(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t4)) w2)))) (\lambda (x: T).(\lambda (H20: (pr0 x2 x)).(\lambda -(H21: (subst0 (s (Flat Appl) i) v3 u2 x)).(or_ind (pr0 x0 v2) (ex2 T (\lambda -(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead -(Flat Appl) x0 (THead (Bind b) x2 x3)) (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 -(THead (Bind b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H22: (pr0 x0 -v2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind -b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) x (THead (Flat Appl) (lift -(S O) O v2) t4)) (pr0_upsilon b H0 x0 v2 H22 x2 x H20 x3 t4 H18) (subst0_fst -v3 x u2 i H21 (THead (Flat Appl) (lift (S O) O v2) t4) (Bind b))))) (\lambda -(H22: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 -w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 -v2 w2)) (or (pr0 (THead (Flat Appl) x0 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(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: -T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x: T).(\lambda (H8: -(eq T w1 (THead (Bind Abbr) u1 x))).(\lambda (H9: (subst0 (s (Bind Abbr) i) -v1 t3 x)).(eq_ind_r T (THead (Bind Abbr) u1 x) (\lambda (t: T).(or (pr0 t -(THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: -T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))))) (or_ind (pr0 x t4) (ex2 T -(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind Abbr) i) v2 t4 -w2))) (or (pr0 (THead (Bind Abbr) u1 x) (THead (Bind Abbr) u2 w)) (ex2 T -(\lambda (w2: T).(pr0 (THead (Bind Abbr) u1 x) w2)) (\lambda (w2: T).(subst0 -i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (H10: (pr0 x t4)).(or_introl -(pr0 (THead (Bind Abbr) u1 x) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: -T).(pr0 (THead (Bind Abbr) u1 x) w2)) (\lambda (w2: T).(subst0 i v2 (THead -(Bind Abbr) u2 w) w2))) (pr0_delta u1 u2 H0 x t4 H10 w H4))) (\lambda (H10: -(ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind Abbr) -i) v2 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: -T).(subst0 (s (Bind Abbr) i) v2 t4 w2)) (or (pr0 (THead (Bind Abbr) u1 x) -(THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) u1 -x) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) -(\lambda (x0: T).(\lambda (H11: (pr0 x x0)).(\lambda (H12: (subst0 (s (Bind -Abbr) i) v2 t4 x0)).(ex2_ind T (\lambda (t: T).(subst0 O u2 x0 t)) (\lambda -(t: T).(subst0 (s (Bind Abbr) i) v2 w t)) (or (pr0 (THead (Bind Abbr) u1 x) -(THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) u1 -x) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) -(\lambda (x1: T).(\lambda (H13: (subst0 O u2 x0 x1)).(\lambda (H14: (subst0 -(s (Bind Abbr) i) v2 w x1)).(or_intror (pr0 (THead (Bind Abbr) u1 x) (THead -(Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) u1 x) w2)) -(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))) (ex_intro2 T -(\lambda (w2: T).(pr0 (THead (Bind Abbr) u1 x) w2)) (\lambda (w2: T).(subst0 -i v2 (THead (Bind Abbr) u2 w) w2)) (THead (Bind Abbr) u2 x1) (pr0_delta u1 u2 -H0 x x0 H11 x1 H13) (subst0_snd (Bind Abbr) v2 x1 w i H14 u2)))))) -(subst0_confluence_neq t4 x0 v2 (s (Bind Abbr) i) H12 w u2 O H4 (sym_not_eq -nat O (S i) (O_S i))))))) H10)) (H3 v1 x (s (Bind Abbr) i) H9 v2 H6)) w1 -H8)))) H7)) (\lambda (H7: (ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T -w1 (THead (Bind Abbr) u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v1 -u1 u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind Abbr) i) v1 t3 -t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t5: T).(eq T w1 (THead -(Bind Abbr) u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v1 u1 u3))) -(\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind Abbr) i) v1 t3 t5))) (or -(pr0 w1 (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) -(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H8: (eq T w1 (THead (Bind Abbr) x0 -x1))).(\lambda (H9: (subst0 i v1 u1 x0)).(\lambda (H10: (subst0 (s (Bind -Abbr) i) v1 t3 x1)).(eq_ind_r T (THead (Bind Abbr) x0 x1) (\lambda (t: T).(or -(pr0 t (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda -(w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))))) (or_ind (pr0 x1 t4) -(ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s (Bind Abbr) -i) v2 t4 w2))) (or (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) -(ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: -T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (H11: (pr0 x1 -t4)).(or_ind (pr0 x0 u2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: -T).(subst0 i v2 u2 w2))) (or (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind -Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) -(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (H12: -(pr0 x0 u2)).(or_introl (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 -w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: -T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))) (pr0_delta x0 u2 H12 x1 t4 H11 -w H4))) (\lambda (H12: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: -T).(subst0 i v2 u2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda -(w2: T).(subst0 i v2 u2 w2)) (or (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind -Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) -(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x: -T).(\lambda (H13: (pr0 x0 x)).(\lambda (H14: (subst0 i v2 u2 x)).(ex2_ind T -(\lambda (t: T).(subst0 O x t4 t)) (\lambda (t: T).(subst0 (S (plus i O)) v2 -w t)) (or (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T -(\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 -i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x2: T).(\lambda (H15: (subst0 -O x t4 x2)).(\lambda (H16: (subst0 (S (plus i O)) v2 w x2)).(let H17 \def -(f_equal nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O i))) in -(let H18 \def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n v2 w -x2)) H16 (S i) H17) in (or_intror (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind -Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) -(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))) (ex_intro2 T -(\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 -i v2 (THead (Bind Abbr) u2 w) w2)) (THead (Bind Abbr) x x2) (pr0_delta x0 x -H13 x1 t4 H11 x2 H15) (subst0_both v2 u2 x i H14 (Bind Abbr) w x2 H18)))))))) -(subst0_subst0_back t4 w u2 O H4 x v2 i H14))))) H12)) (H1 v1 x0 i H9 v2 -H6))) (\lambda (H11: (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: -T).(subst0 (s (Bind Abbr) i) v2 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x1 -w2)) (\lambda (w2: T).(subst0 (s (Bind Abbr) i) v2 t4 w2)) (or (pr0 (THead -(Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 -(THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind -Abbr) u2 w) w2)))) (\lambda (x: T).(\lambda (H12: (pr0 x1 x)).(\lambda (H13: -(subst0 (s (Bind Abbr) i) v2 t4 x)).(or_ind (pr0 x0 u2) (ex2 T (\lambda (w2: -T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 u2 w2))) (or (pr0 (THead (Bind -Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead -(Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 -w) w2)))) (\lambda (H14: (pr0 x0 u2)).(ex2_ind T (\lambda (t: T).(subst0 O u2 -x t)) (\lambda (t: T).(subst0 (s (Bind Abbr) i) v2 w t)) (or (pr0 (THead -(Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 -(THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind -Abbr) u2 w) w2)))) (\lambda (x2: T).(\lambda (H15: (subst0 O u2 x -x2)).(\lambda (H16: (subst0 (s (Bind Abbr) i) v2 w x2)).(or_intror (pr0 -(THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: -T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead -(Bind Abbr) u2 w) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) -x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)) -(THead (Bind Abbr) u2 x2) (pr0_delta x0 u2 H14 x1 x H12 x2 H15) (subst0_snd -(Bind Abbr) v2 x2 w i H16 u2)))))) (subst0_confluence_neq t4 x v2 (s (Bind -Abbr) i) H13 w u2 O H4 (sym_not_eq nat O (S i) (O_S i))))) (\lambda (H14: -(ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 u2 -w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 -u2 w2)) (or (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T -(\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 -i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x2: T).(\lambda (H15: (pr0 x0 -x2)).(\lambda (H16: (subst0 i v2 u2 x2)).(ex2_ind T (\lambda (t: T).(subst0 O -x2 t4 t)) (\lambda (t: T).(subst0 (S (plus i O)) v2 w t)) (or (pr0 (THead -(Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 -(THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind -Abbr) u2 w) w2)))) (\lambda (x3: T).(\lambda (H17: (subst0 O x2 t4 -x3)).(\lambda (H18: (subst0 (S (plus i O)) v2 w x3)).(let H19 \def (f_equal -nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O i))) in (let H20 -\def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n v2 w x3)) H18 (S -i) H19) in (ex2_ind T (\lambda (t: T).(subst0 (s (Bind Abbr) i) v2 x3 t)) -(\lambda (t: T).(subst0 O x2 x t)) (or (pr0 (THead (Bind Abbr) x0 x1) (THead -(Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) -w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda -(x4: T).(\lambda (H21: (subst0 (s (Bind Abbr) i) v2 x3 x4)).(\lambda (H22: -(subst0 O x2 x x4)).(or_intror (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind -Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) -(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))) (ex_intro2 T -(\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 -i v2 (THead (Bind Abbr) u2 w) w2)) (THead (Bind Abbr) x2 x4) (pr0_delta x0 x2 -H15 x1 x H12 x4 H22) (subst0_both v2 u2 x2 i H16 (Bind Abbr) w x4 -(subst0_trans x3 w v2 (s (Bind Abbr) i) H20 x4 H21))))))) -(subst0_confluence_neq t4 x3 x2 O H17 x v2 (s (Bind Abbr) i) H13 (O_S -i)))))))) (subst0_subst0_back t4 w u2 O H4 x2 v2 i H16))))) H14)) (H1 v1 x0 i -H9 v2 H6))))) H11)) (H3 v1 x1 (s (Bind Abbr) i) H10 v2 H6)) w1 H8)))))) H7)) -(subst0_gen_head (Bind Abbr) v1 u1 t3 w1 i H5)))))))))))))))))) (\lambda (b: -B).(\lambda (H0: (not (eq B b Abst))).(\lambda (t3: T).(\lambda (t4: -T).(\lambda (H1: (pr0 t3 t4)).(\lambda (H2: ((\forall (v1: T).(\forall (w1: -T).(\forall (i: nat).((subst0 i v1 t3 w1) \to (\forall (v2: T).((pr0 v1 v2) -\to (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: -T).(subst0 i v2 t4 w2)))))))))))).(\lambda (u: T).(\lambda (v1: T).(\lambda -(w1: T).(\lambda (i: nat).(\lambda (H3: (subst0 i v1 (THead (Bind b) u (lift -(S O) O t3)) w1)).(\lambda (v2: T).(\lambda (H4: (pr0 v1 v2)).(or3_ind (ex2 T -(\lambda (u2: T).(eq T w1 (THead (Bind b) u2 (lift (S O) O t3)))) (\lambda -(u2: T).(subst0 i v1 u u2))) (ex2 T (\lambda (t5: T).(eq T w1 (THead (Bind b) -u t5))) (\lambda (t5: T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5))) -(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T w1 (THead (Bind b) u2 -t5)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: -T).(\lambda (t5: T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5)))) (or -(pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i -v2 t4 w2)))) (\lambda (H5: (ex2 T (\lambda (u2: T).(eq T w1 (THead (Bind b) -u2 (lift (S O) O t3)))) (\lambda (u2: T).(subst0 i v1 u u2)))).(ex2_ind T -(\lambda (u2: T).(eq T w1 (THead (Bind b) u2 (lift (S O) O t3)))) (\lambda -(u2: T).(subst0 i v1 u u2)) (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 -w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H6: -(eq T w1 (THead (Bind b) x (lift (S O) O t3)))).(\lambda (_: (subst0 i v1 u -x)).(eq_ind_r T (THead (Bind b) x (lift (S O) O t3)) (\lambda (t: T).(or (pr0 -t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4 -w2))))) (or_introl (pr0 (THead (Bind b) x (lift (S O) O t3)) t4) (ex2 T -(\lambda (w2: T).(pr0 (THead (Bind b) x (lift (S O) O t3)) w2)) (\lambda (w2: -T).(subst0 i v2 t4 w2))) (pr0_zeta b H0 t3 t4 H1 x)) w1 H6)))) H5)) (\lambda -(H5: (ex2 T (\lambda (t5: T).(eq T w1 (THead (Bind b) u t5))) (\lambda (t5: -T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5)))).(ex2_ind T (\lambda -(t5: T).(eq T w1 (THead (Bind b) u t5))) (\lambda (t5: T).(subst0 (s (Bind b) -i) v1 (lift (S O) O t3) t5)) (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 -w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H6: -(eq T w1 (THead (Bind b) u x))).(\lambda (H7: (subst0 (s (Bind b) i) v1 (lift -(S O) O t3) x)).(ex2_ind T (\lambda (t5: T).(eq T x (lift (S O) O t5))) -(\lambda (t5: T).(subst0 (minus (s (Bind b) i) (S O)) v1 t3 t5)) (or (pr0 w1 -t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 -w2)))) (\lambda (x0: T).(\lambda (H8: (eq T x (lift (S O) O x0))).(\lambda -(H9: (subst0 (minus (s (Bind b) i) (S O)) v1 t3 x0)).(let H10 \def (eq_ind T -x (\lambda (t: T).(eq T w1 (THead (Bind b) u t))) H6 (lift (S O) O x0) H8) in -(eq_ind_r T (THead (Bind b) u (lift (S O) O x0)) (\lambda (t: T).(or (pr0 t -t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4 -w2))))) (let H11 \def (eq_ind_r nat (minus i O) (\lambda (n: nat).(subst0 n -v1 t3 x0)) H9 i (minus_n_O i)) in (or_ind (pr0 x0 t4) (ex2 T (\lambda (w2: -T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (or (pr0 (THead (Bind -b) u (lift (S O) O x0)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) u -(lift (S O) O x0)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda -(H12: (pr0 x0 t4)).(or_introl (pr0 (THead (Bind b) u (lift (S O) O x0)) t4) -(ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) u (lift (S O) O x0)) w2)) -(\lambda (w2: T).(subst0 i v2 t4 w2))) (pr0_zeta b H0 x0 t4 H12 u))) (\lambda -(H12: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 t4 -w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 -t4 w2)) (or (pr0 (THead (Bind b) u (lift (S O) O x0)) t4) (ex2 T (\lambda -(w2: T).(pr0 (THead (Bind b) u (lift (S O) O x0)) w2)) (\lambda (w2: -T).(subst0 i v2 t4 w2)))) (\lambda (x1: T).(\lambda (H13: (pr0 x0 -x1)).(\lambda (H14: (subst0 i v2 t4 x1)).(or_intror (pr0 (THead (Bind b) u -(lift (S O) O x0)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) u (lift -(S O) O x0)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (ex_intro2 T -(\lambda (w2: T).(pr0 (THead (Bind b) u (lift (S O) O x0)) w2)) (\lambda (w2: -T).(subst0 i v2 t4 w2)) x1 (pr0_zeta b H0 x0 x1 H13 u) H14))))) H12)) (H2 v1 -x0 i H11 v2 H4))) w1 H10))))) (subst0_gen_lift_ge v1 t3 x (s (Bind b) i) (S -O) O H7 (le_n_S O i (le_O_n i))))))) H5)) (\lambda (H5: (ex3_2 T T (\lambda -(u2: T).(\lambda (t5: T).(eq T w1 (THead (Bind b) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: -T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5))))).(ex3_2_ind T T -(\lambda (u2: T).(\lambda (t5: T).(eq T w1 (THead (Bind b) u2 t5)))) (\lambda -(u2: T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: -T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5))) (or (pr0 w1 t4) (ex2 T -(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) -(\lambda (x0: T).(\lambda (x1: T).(\lambda (H6: (eq T w1 (THead (Bind b) x0 -x1))).(\lambda (_: (subst0 i v1 u x0)).(\lambda (H8: (subst0 (s (Bind b) i) -v1 (lift (S O) O t3) x1)).(ex2_ind T (\lambda (t5: T).(eq T x1 (lift (S O) O -t5))) (\lambda (t5: T).(subst0 (minus (s (Bind b) i) (S O)) v1 t3 t5)) (or -(pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i -v2 t4 w2)))) (\lambda (x: T).(\lambda (H9: (eq T x1 (lift (S O) O -x))).(\lambda (H10: (subst0 (minus (s (Bind b) i) (S O)) v1 t3 x)).(let H11 -\def (eq_ind T x1 (\lambda (t: T).(eq T w1 (THead (Bind b) x0 t))) H6 (lift -(S O) O x) H9) in (eq_ind_r T (THead (Bind b) x0 (lift (S O) O x)) (\lambda -(t: T).(or (pr0 t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: -T).(subst0 i v2 t4 w2))))) (let H12 \def (eq_ind_r nat (minus i O) (\lambda -(n: nat).(subst0 n v1 t3 x)) H10 i (minus_n_O i)) in (or_ind (pr0 x t4) (ex2 -T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (or -(pr0 (THead (Bind b) x0 (lift (S O) O x)) t4) (ex2 T (\lambda (w2: T).(pr0 -(THead (Bind b) x0 (lift (S O) O x)) w2)) (\lambda (w2: T).(subst0 i v2 t4 -w2)))) (\lambda (H13: (pr0 x t4)).(or_introl (pr0 (THead (Bind b) x0 (lift (S -O) O x)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) x0 (lift (S O) O -x)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (pr0_zeta b H0 x t4 H13 x0))) -(\lambda (H13: (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i -v2 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 -i v2 t4 w2)) (or (pr0 (THead (Bind b) x0 (lift (S O) O x)) t4) (ex2 T -(\lambda (w2: T).(pr0 (THead (Bind b) x0 (lift (S O) O x)) w2)) (\lambda (w2: -T).(subst0 i v2 t4 w2)))) (\lambda (x2: T).(\lambda (H14: (pr0 x -x2)).(\lambda (H15: (subst0 i v2 t4 x2)).(or_intror (pr0 (THead (Bind b) x0 -(lift (S O) O x)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) x0 (lift -(S O) O x)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (ex_intro2 T (\lambda -(w2: T).(pr0 (THead (Bind b) x0 (lift (S O) O x)) w2)) (\lambda (w2: -T).(subst0 i v2 t4 w2)) x2 (pr0_zeta b H0 x x2 H14 x0) H15))))) H13)) (H2 v1 -x i H12 v2 H4))) w1 H11))))) (subst0_gen_lift_ge v1 t3 x1 (s (Bind b) i) (S -O) O H8 (le_n_S O i (le_O_n i))))))))) H5)) (subst0_gen_head (Bind b) v1 u -(lift (S O) O t3) w1 i H3))))))))))))))) (\lambda (t3: T).(\lambda (t4: -T).(\lambda (H0: (pr0 t3 t4)).(\lambda (H1: ((\forall (v1: T).(\forall (w1: -T).(\forall (i: nat).((subst0 i v1 t3 w1) \to (\forall (v2: T).((pr0 v1 v2) -\to (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: -T).(subst0 i v2 t4 w2)))))))))))).(\lambda (u: T).(\lambda (v1: T).(\lambda -(w1: T).(\lambda (i: nat).(\lambda (H2: (subst0 i v1 (THead (Flat Cast) u t3) -w1)).(\lambda (v2: T).(\lambda (H3: (pr0 v1 v2)).(or3_ind (ex2 T (\lambda -(u2: T).(eq T w1 (THead (Flat Cast) u2 t3))) (\lambda (u2: T).(subst0 i v1 u -u2))) (ex2 T (\lambda (t5: T).(eq T w1 (THead (Flat Cast) u t5))) (\lambda -(t5: T).(subst0 (s (Flat Cast) i) v1 t3 t5))) (ex3_2 T T (\lambda (u2: -T).(\lambda (t5: T).(eq T w1 (THead (Flat Cast) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: -T).(subst0 (s (Flat Cast) i) v1 t3 t5)))) (or (pr0 w1 t4) (ex2 T (\lambda -(w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (H4: -(ex2 T (\lambda (u2: T).(eq T w1 (THead (Flat Cast) u2 t3))) (\lambda (u2: -T).(subst0 i v1 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T w1 (THead (Flat -Cast) u2 t3))) (\lambda (u2: T).(subst0 i v1 u u2)) (or (pr0 w1 t4) (ex2 T -(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) -(\lambda (x: T).(\lambda (H5: (eq T w1 (THead (Flat Cast) x t3))).(\lambda -(_: (subst0 i v1 u x)).(eq_ind_r T (THead (Flat Cast) x t3) (\lambda (t: -T).(or (pr0 t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: -T).(subst0 i v2 t4 w2))))) (or_introl (pr0 (THead (Flat Cast) x t3) t4) (ex2 -T (\lambda (w2: T).(pr0 (THead (Flat Cast) x t3) w2)) (\lambda (w2: -T).(subst0 i v2 t4 w2))) (pr0_tau t3 t4 H0 x)) w1 H5)))) H4)) (\lambda (H4: -(ex2 T (\lambda (t5: T).(eq T w1 (THead (Flat Cast) u t5))) (\lambda (t5: -T).(subst0 (s (Flat Cast) i) v1 t3 t5)))).(ex2_ind T (\lambda (t5: T).(eq T -w1 (THead (Flat Cast) u t5))) (\lambda (t5: T).(subst0 (s (Flat Cast) i) v1 -t3 t5)) (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: -T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H5: (eq T w1 (THead (Flat -Cast) u x))).(\lambda (H6: (subst0 (s (Flat Cast) i) v1 t3 x)).(eq_ind_r T -(THead (Flat Cast) u x) (\lambda (t: T).(or (pr0 t t4) (ex2 T (\lambda (w2: -T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))))) (or_ind (pr0 x t4) -(ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Flat Cast) -i) v2 t4 w2))) (or (pr0 (THead (Flat Cast) u x) t4) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) -(\lambda (H7: (pr0 x t4)).(or_introl (pr0 (THead (Flat Cast) u x) t4) (ex2 T -(\lambda (w2: T).(pr0 (THead (Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i -v2 t4 w2))) (pr0_tau x t4 H7 u))) (\lambda (H7: (ex2 T (\lambda (w2: T).(pr0 -x w2)) (\lambda (w2: T).(subst0 (s (Flat Cast) i) v2 t4 w2)))).(ex2_ind T -(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Flat Cast) i) v2 t4 -w2)) (or (pr0 (THead (Flat Cast) u x) t4) (ex2 T (\lambda (w2: T).(pr0 (THead -(Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x0: -T).(\lambda (H8: (pr0 x x0)).(\lambda (H9: (subst0 (s (Flat Cast) i) v2 t4 -x0)).(or_intror (pr0 (THead (Flat Cast) u x) t4) (ex2 T (\lambda (w2: T).(pr0 -(THead (Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) -(ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) u x) w2)) (\lambda (w2: -T).(subst0 i v2 t4 w2)) x0 (pr0_tau x x0 H8 u) H9))))) H7)) (H1 v1 x (s (Flat -Cast) i) H6 v2 H3)) w1 H5)))) H4)) (\lambda (H4: (ex3_2 T T (\lambda (u2: -T).(\lambda (t5: T).(eq T w1 (THead (Flat Cast) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: -T).(subst0 (s (Flat Cast) i) v1 t3 t5))))).(ex3_2_ind T T (\lambda (u2: -T).(\lambda (t5: T).(eq T w1 (THead (Flat Cast) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: -T).(subst0 (s (Flat Cast) i) v1 t3 t5))) (or (pr0 w1 t4) (ex2 T (\lambda (w2: -T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H5: (eq T w1 (THead (Flat Cast) x0 -x1))).(\lambda (_: (subst0 i v1 u x0)).(\lambda (H7: (subst0 (s (Flat Cast) -i) v1 t3 x1)).(eq_ind_r T (THead (Flat Cast) x0 x1) (\lambda (t: T).(or (pr0 -t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4 -w2))))) (or_ind (pr0 x1 t4) (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda -(w2: T).(subst0 (s (Flat Cast) i) v2 t4 w2))) (or (pr0 (THead (Flat Cast) x0 -x1) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) x0 x1) w2)) (\lambda -(w2: T).(subst0 i v2 t4 w2)))) (\lambda (H8: (pr0 x1 t4)).(or_introl (pr0 -(THead (Flat Cast) x0 x1) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) -x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (pr0_tau x1 t4 H8 x0))) -(\lambda (H8: (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 -(s (Flat Cast) i) v2 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x1 w2)) -(\lambda (w2: T).(subst0 (s (Flat Cast) i) v2 t4 w2)) (or (pr0 (THead (Flat -Cast) x0 x1) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) x0 x1) w2)) -(\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H9: (pr0 x1 -x)).(\lambda (H10: (subst0 (s (Flat Cast) i) v2 t4 x)).(or_intror (pr0 (THead -(Flat Cast) x0 x1) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) x0 x1) -w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (ex_intro2 T (\lambda (w2: -T).(pr0 (THead (Flat Cast) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)) -x (pr0_tau x1 x H9 x0) H10))))) H8)) (H1 v1 x1 (s (Flat Cast) i) H7 v2 H3)) -w1 H5)))))) H4)) (subst0_gen_head (Flat Cast) v1 u t3 w1 i H2))))))))))))) t1 -t2 H))). -(* COMMENTS -Initial nodes: 38857 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr0/subst1.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr0/subst1.ma deleted file mode 100644 index 877f87f01..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr0/subst1.ma +++ /dev/null @@ -1,105 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr0/props.ma". - -include "Basic-1/subst1/defs.ma". - -theorem pr0_delta1: - \forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (t1: T).(\forall -(t2: T).((pr0 t1 t2) \to (\forall (w: T).((subst1 O u2 t2 w) \to (pr0 (THead -(Bind Abbr) u1 t1) (THead (Bind Abbr) u2 w))))))))) -\def - \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr0 u1 u2)).(\lambda (t1: -T).(\lambda (t2: T).(\lambda (H0: (pr0 t1 t2)).(\lambda (w: T).(\lambda (H1: -(subst1 O u2 t2 w)).(subst1_ind O u2 t2 (\lambda (t: T).(pr0 (THead (Bind -Abbr) u1 t1) (THead (Bind Abbr) u2 t))) (pr0_comp u1 u2 H t1 t2 H0 (Bind -Abbr)) (\lambda (t0: T).(\lambda (H2: (subst0 O u2 t2 t0)).(pr0_delta u1 u2 H -t1 t2 H0 t0 H2))) w H1)))))))). -(* COMMENTS -Initial nodes: 115 -END *) - -theorem pr0_subst1_back: - \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst1 -i u2 t1 t2) \to (\forall (u1: T).((pr0 u1 u2) \to (ex2 T (\lambda (t: -T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t t2))))))))) -\def - \lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (i: nat).(\lambda -(H: (subst1 i u2 t1 t2)).(subst1_ind i u2 t1 (\lambda (t: T).(\forall (u1: -T).((pr0 u1 u2) \to (ex2 T (\lambda (t0: T).(subst1 i u1 t1 t0)) (\lambda -(t0: T).(pr0 t0 t)))))) (\lambda (u1: T).(\lambda (_: (pr0 u1 u2)).(ex_intro2 -T (\lambda (t: T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t t1)) t1 -(subst1_refl i u1 t1) (pr0_refl t1)))) (\lambda (t0: T).(\lambda (H0: (subst0 -i u2 t1 t0)).(\lambda (u1: T).(\lambda (H1: (pr0 u1 u2)).(ex2_ind T (\lambda -(t: T).(subst0 i u1 t1 t)) (\lambda (t: T).(pr0 t t0)) (ex2 T (\lambda (t: -T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t t0))) (\lambda (x: T).(\lambda -(H2: (subst0 i u1 t1 x)).(\lambda (H3: (pr0 x t0)).(ex_intro2 T (\lambda (t: -T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t t0)) x (subst1_single i u1 t1 x -H2) H3)))) (pr0_subst0_back u2 t1 t0 i H0 u1 H1)))))) t2 H))))). -(* COMMENTS -Initial nodes: 251 -END *) - -theorem pr0_subst1_fwd: - \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst1 -i u2 t1 t2) \to (\forall (u1: T).((pr0 u2 u1) \to (ex2 T (\lambda (t: -T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t2 t))))))))) -\def - \lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (i: nat).(\lambda -(H: (subst1 i u2 t1 t2)).(subst1_ind i u2 t1 (\lambda (t: T).(\forall (u1: -T).((pr0 u2 u1) \to (ex2 T (\lambda (t0: T).(subst1 i u1 t1 t0)) (\lambda -(t0: T).(pr0 t t0)))))) (\lambda (u1: T).(\lambda (_: (pr0 u2 u1)).(ex_intro2 -T (\lambda (t: T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t1 t)) t1 -(subst1_refl i u1 t1) (pr0_refl t1)))) (\lambda (t0: T).(\lambda (H0: (subst0 -i u2 t1 t0)).(\lambda (u1: T).(\lambda (H1: (pr0 u2 u1)).(ex2_ind T (\lambda -(t: T).(subst0 i u1 t1 t)) (\lambda (t: T).(pr0 t0 t)) (ex2 T (\lambda (t: -T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t0 t))) (\lambda (x: T).(\lambda -(H2: (subst0 i u1 t1 x)).(\lambda (H3: (pr0 t0 x)).(ex_intro2 T (\lambda (t: -T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t0 t)) x (subst1_single i u1 t1 x -H2) H3)))) (pr0_subst0_fwd u2 t1 t0 i H0 u1 H1)))))) t2 H))))). -(* COMMENTS -Initial nodes: 251 -END *) - -theorem pr0_subst1: - \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (v1: T).(\forall -(w1: T).(\forall (i: nat).((subst1 i v1 t1 w1) \to (\forall (v2: T).((pr0 v1 -v2) \to (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst1 i v2 t2 -w2))))))))))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(\lambda (v1: -T).(\lambda (w1: T).(\lambda (i: nat).(\lambda (H0: (subst1 i v1 t1 -w1)).(subst1_ind i v1 t1 (\lambda (t: T).(\forall (v2: T).((pr0 v1 v2) \to -(ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst1 i v2 t2 w2)))))) -(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(ex_intro2 T (\lambda (w2: T).(pr0 -t1 w2)) (\lambda (w2: T).(subst1 i v2 t2 w2)) t2 H (subst1_refl i v2 t2)))) -(\lambda (t0: T).(\lambda (H1: (subst0 i v1 t1 t0)).(\lambda (v2: T).(\lambda -(H2: (pr0 v1 v2)).(or_ind (pr0 t0 t2) (ex2 T (\lambda (w2: T).(pr0 t0 w2)) -(\lambda (w2: T).(subst0 i v2 t2 w2))) (ex2 T (\lambda (w2: T).(pr0 t0 w2)) -(\lambda (w2: T).(subst1 i v2 t2 w2))) (\lambda (H3: (pr0 t0 t2)).(ex_intro2 -T (\lambda (w2: T).(pr0 t0 w2)) (\lambda (w2: T).(subst1 i v2 t2 w2)) t2 H3 -(subst1_refl i v2 t2))) (\lambda (H3: (ex2 T (\lambda (w2: T).(pr0 t0 w2)) -(\lambda (w2: T).(subst0 i v2 t2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t0 -w2)) (\lambda (w2: T).(subst0 i v2 t2 w2)) (ex2 T (\lambda (w2: T).(pr0 t0 -w2)) (\lambda (w2: T).(subst1 i v2 t2 w2))) (\lambda (x: T).(\lambda (H4: -(pr0 t0 x)).(\lambda (H5: (subst0 i v2 t2 x)).(ex_intro2 T (\lambda (w2: -T).(pr0 t0 w2)) (\lambda (w2: T).(subst1 i v2 t2 w2)) x H4 (subst1_single i -v2 t2 x H5))))) H3)) (pr0_subst0 t1 t2 H v1 t0 i H1 v2 H2)))))) w1 H0))))))). -(* COMMENTS -Initial nodes: 385 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr1/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr1/defs.ma deleted file mode 100644 index 84db71acb..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr1/defs.ma +++ /dev/null @@ -1,23 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr0/defs.ma". - -inductive pr1: T \to (T \to Prop) \def -| pr1_refl: \forall (t: T).(pr1 t t) -| pr1_sing: \forall (t2: T).(\forall (t1: T).((pr0 t1 t2) \to (\forall (t3: -T).((pr1 t2 t3) \to (pr1 t1 t3))))). - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr1/pr1.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr1/pr1.ma deleted file mode 100644 index ec469da50..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr1/pr1.ma +++ /dev/null @@ -1,70 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr1/props.ma". - -include "Basic-1/pr0/pr0.ma". - -theorem pr1_strip: - \forall (t0: T).(\forall (t1: T).((pr1 t0 t1) \to (\forall (t2: T).((pr0 t0 -t2) \to (ex2 T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t))))))) -\def - \lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr1 t0 t1)).(pr1_ind (\lambda -(t: T).(\lambda (t2: T).(\forall (t3: T).((pr0 t t3) \to (ex2 T (\lambda (t4: -T).(pr1 t2 t4)) (\lambda (t4: T).(pr1 t3 t4))))))) (\lambda (t: T).(\lambda -(t2: T).(\lambda (H0: (pr0 t t2)).(ex_intro2 T (\lambda (t3: T).(pr1 t t3)) -(\lambda (t3: T).(pr1 t2 t3)) t2 (pr1_pr0 t t2 H0) (pr1_refl t2))))) (\lambda -(t2: T).(\lambda (t3: T).(\lambda (H0: (pr0 t3 t2)).(\lambda (t4: T).(\lambda -(_: (pr1 t2 t4)).(\lambda (H2: ((\forall (t5: T).((pr0 t2 t5) \to (ex2 T -(\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t))))))).(\lambda (t5: -T).(\lambda (H3: (pr0 t3 t5)).(ex2_ind T (\lambda (t: T).(pr0 t5 t)) (\lambda -(t: T).(pr0 t2 t)) (ex2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 -t))) (\lambda (x: T).(\lambda (H4: (pr0 t5 x)).(\lambda (H5: (pr0 t2 x)).(let -H6 \def (H2 x H5) in (ex2_ind T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: -T).(pr1 x t)) (ex2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t))) -(\lambda (x0: T).(\lambda (H7: (pr1 t4 x0)).(\lambda (H8: (pr1 x -x0)).(ex_intro2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t)) x0 -H7 (pr1_t x t5 (pr1_pr0 t5 x H4) x0 H8))))) H6))))) (pr0_confluence t3 t5 H3 -t2 H0)))))))))) t0 t1 H))). -(* COMMENTS -Initial nodes: 317 -END *) - -theorem pr1_confluence: - \forall (t0: T).(\forall (t1: T).((pr1 t0 t1) \to (\forall (t2: T).((pr1 t0 -t2) \to (ex2 T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t))))))) -\def - \lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr1 t0 t1)).(pr1_ind (\lambda -(t: T).(\lambda (t2: T).(\forall (t3: T).((pr1 t t3) \to (ex2 T (\lambda (t4: -T).(pr1 t2 t4)) (\lambda (t4: T).(pr1 t3 t4))))))) (\lambda (t: T).(\lambda -(t2: T).(\lambda (H0: (pr1 t t2)).(ex_intro2 T (\lambda (t3: T).(pr1 t t3)) -(\lambda (t3: T).(pr1 t2 t3)) t2 H0 (pr1_refl t2))))) (\lambda (t2: -T).(\lambda (t3: T).(\lambda (H0: (pr0 t3 t2)).(\lambda (t4: T).(\lambda (_: -(pr1 t2 t4)).(\lambda (H2: ((\forall (t5: T).((pr1 t2 t5) \to (ex2 T (\lambda -(t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t))))))).(\lambda (t5: T).(\lambda -(H3: (pr1 t3 t5)).(let H_x \def (pr1_strip t3 t5 H3 t2 H0) in (let H4 \def -H_x in (ex2_ind T (\lambda (t: T).(pr1 t5 t)) (\lambda (t: T).(pr1 t2 t)) -(ex2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t))) (\lambda (x: -T).(\lambda (H5: (pr1 t5 x)).(\lambda (H6: (pr1 t2 x)).(let H_x0 \def (H2 x -H6) in (let H7 \def H_x0 in (ex2_ind T (\lambda (t: T).(pr1 t4 t)) (\lambda -(t: T).(pr1 x t)) (ex2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 -t))) (\lambda (x0: T).(\lambda (H8: (pr1 t4 x0)).(\lambda (H9: (pr1 x -x0)).(ex_intro2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t)) x0 -H8 (pr1_t x t5 H5 x0 H9))))) H7)))))) H4))))))))))) t0 t1 H))). -(* COMMENTS -Initial nodes: 311 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr1/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr1/props.ma deleted file mode 100644 index 0615bed96..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr1/props.ma +++ /dev/null @@ -1,126 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr1/defs.ma". - -include "Basic-1/pr0/subst1.ma". - -include "Basic-1/subst1/props.ma". - -include "Basic-1/T/props.ma". - -theorem pr1_pr0: - \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (pr1 t1 t2))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(pr1_sing t2 t1 H -t2 (pr1_refl t2)))). -(* COMMENTS -Initial nodes: 23 -END *) - -theorem pr1_t: - \forall (t2: T).(\forall (t1: T).((pr1 t1 t2) \to (\forall (t3: T).((pr1 t2 -t3) \to (pr1 t1 t3))))) -\def - \lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pr1 t1 t2)).(pr1_ind (\lambda -(t: T).(\lambda (t0: T).(\forall (t3: T).((pr1 t0 t3) \to (pr1 t t3))))) -(\lambda (t: T).(\lambda (t3: T).(\lambda (H0: (pr1 t t3)).H0))) (\lambda -(t0: T).(\lambda (t3: T).(\lambda (H0: (pr0 t3 t0)).(\lambda (t4: T).(\lambda -(_: (pr1 t0 t4)).(\lambda (H2: ((\forall (t5: T).((pr1 t4 t5) \to (pr1 t0 -t5))))).(\lambda (t5: T).(\lambda (H3: (pr1 t4 t5)).(pr1_sing t0 t3 H0 t5 (H2 -t5 H3)))))))))) t1 t2 H))). -(* COMMENTS -Initial nodes: 103 -END *) - -theorem pr1_head_1: - \forall (u1: T).(\forall (u2: T).((pr1 u1 u2) \to (\forall (t: T).(\forall -(k: K).(pr1 (THead k u1 t) (THead k u2 t)))))) -\def - \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr1 u1 u2)).(\lambda (t: -T).(\lambda (k: K).(pr1_ind (\lambda (t0: T).(\lambda (t1: T).(pr1 (THead k -t0 t) (THead k t1 t)))) (\lambda (t0: T).(pr1_refl (THead k t0 t))) (\lambda -(t2: T).(\lambda (t1: T).(\lambda (H0: (pr0 t1 t2)).(\lambda (t3: T).(\lambda -(_: (pr1 t2 t3)).(\lambda (H2: (pr1 (THead k t2 t) (THead k t3 t))).(pr1_sing -(THead k t2 t) (THead k t1 t) (pr0_comp t1 t2 H0 t t (pr0_refl t) k) (THead k -t3 t) H2))))))) u1 u2 H))))). -(* COMMENTS -Initial nodes: 137 -END *) - -theorem pr1_head_2: - \forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall (u: T).(\forall -(k: K).(pr1 (THead k u t1) (THead k u t2)))))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr1 t1 t2)).(\lambda (u: -T).(\lambda (k: K).(pr1_ind (\lambda (t: T).(\lambda (t0: T).(pr1 (THead k u -t) (THead k u t0)))) (\lambda (t: T).(pr1_refl (THead k u t))) (\lambda (t0: -T).(\lambda (t3: T).(\lambda (H0: (pr0 t3 t0)).(\lambda (t4: T).(\lambda (_: -(pr1 t0 t4)).(\lambda (H2: (pr1 (THead k u t0) (THead k u t4))).(pr1_sing -(THead k u t0) (THead k u t3) (pr0_comp u u (pr0_refl u) t3 t0 H0 k) (THead k -u t4) H2))))))) t1 t2 H))))). -(* COMMENTS -Initial nodes: 137 -END *) - -theorem pr1_comp: - \forall (v: T).(\forall (w: T).((pr1 v w) \to (\forall (t: T).(\forall (u: -T).((pr1 t u) \to (\forall (k: K).(pr1 (THead k v t) (THead k w u)))))))) -\def - \lambda (v: T).(\lambda (w: T).(\lambda (H: (pr1 v w)).(pr1_ind (\lambda (t: -T).(\lambda (t0: T).(\forall (t1: T).(\forall (u: T).((pr1 t1 u) \to (\forall -(k: K).(pr1 (THead k t t1) (THead k t0 u)))))))) (\lambda (t: T).(\lambda -(t0: T).(\lambda (u: T).(\lambda (H0: (pr1 t0 u)).(\lambda (k: K).(pr1_head_2 -t0 u H0 t k)))))) (\lambda (t2: T).(\lambda (t1: T).(\lambda (H0: (pr0 t1 -t2)).(\lambda (t3: T).(\lambda (H1: (pr1 t2 t3)).(\lambda (_: ((\forall (t: -T).(\forall (u: T).((pr1 t u) \to (\forall (k: K).(pr1 (THead k t2 t) (THead -k t3 u)))))))).(\lambda (t: T).(\lambda (u: T).(\lambda (H3: (pr1 t -u)).(\lambda (k: K).(pr1_ind (\lambda (t0: T).(\lambda (t4: T).(pr1 (THead k -t1 t0) (THead k t3 t4)))) (\lambda (t0: T).(pr1_head_1 t1 t3 (pr1_sing t2 t1 -H0 t3 H1) t0 k)) (\lambda (t0: T).(\lambda (t4: T).(\lambda (H4: (pr0 t4 -t0)).(\lambda (t5: T).(\lambda (_: (pr1 t0 t5)).(\lambda (H6: (pr1 (THead k -t1 t0) (THead k t3 t5))).(pr1_sing (THead k t1 t0) (THead k t1 t4) (pr0_comp -t1 t1 (pr0_refl t1) t4 t0 H4 k) (THead k t3 t5) H6))))))) t u H3))))))))))) v -w H))). -(* COMMENTS -Initial nodes: 273 -END *) - -theorem pr1_eta: - \forall (w: T).(\forall (u: T).(let t \def (THead (Bind Abst) w u) in -(\forall (v: T).((pr1 v w) \to (pr1 (THead (Bind Abst) v (THead (Flat Appl) -(TLRef O) (lift (S O) O t))) t))))) -\def - \lambda (w: T).(\lambda (u: T).(let t \def (THead (Bind Abst) w u) in -(\lambda (v: T).(\lambda (H: (pr1 v w)).(eq_ind_r T (THead (Bind Abst) (lift -(S O) O w) (lift (S O) (S O) u)) (\lambda (t0: T).(pr1 (THead (Bind Abst) v -(THead (Flat Appl) (TLRef O) t0)) (THead (Bind Abst) w u))) (pr1_comp v w H -(THead (Flat Appl) (TLRef O) (THead (Bind Abst) (lift (S O) O w) (lift (S O) -(S O) u))) u (pr1_sing (THead (Bind Abbr) (TLRef O) (lift (S O) (S O) u)) -(THead (Flat Appl) (TLRef O) (THead (Bind Abst) (lift (S O) O w) (lift (S O) -(S O) u))) (pr0_beta (lift (S O) O w) (TLRef O) (TLRef O) (pr0_refl (TLRef -O)) (lift (S O) (S O) u) (lift (S O) (S O) u) (pr0_refl (lift (S O) (S O) -u))) u (pr1_sing (THead (Bind Abbr) (TLRef O) (lift (S O) O u)) (THead (Bind -Abbr) (TLRef O) (lift (S O) (S O) u)) (pr0_delta1 (TLRef O) (TLRef O) -(pr0_refl (TLRef O)) (lift (S O) (S O) u) (lift (S O) (S O) u) (pr0_refl -(lift (S O) (S O) u)) (lift (S O) O u) (subst1_lift_S u O O (le_n O))) u -(pr1_pr0 (THead (Bind Abbr) (TLRef O) (lift (S O) O u)) u (pr0_zeta Abbr -not_abbr_abst u u (pr0_refl u) (TLRef O))))) (Bind Abst)) (lift (S O) O -(THead (Bind Abst) w u)) (lift_bind Abst w u (S O) O)))))). -(* COMMENTS -Initial nodes: 463 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr2/clen.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr2/clen.ma deleted file mode 100644 index df9ea1fd9..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr2/clen.ma +++ /dev/null @@ -1,161 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr2/props.ma". - -include "Basic-1/clen/getl.ma". - -theorem pr2_gen_ctail: - \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall -(t2: T).((pr2 (CTail k u c) t1 t2) \to (or (pr2 c t1 t2) (ex3 T (\lambda (_: -T).(eq K k (Bind Abbr))) (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(subst0 -(clen c) u t t2))))))))) -\def - \lambda (k: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda -(t2: T).(\lambda (H: (pr2 (CTail k u c) t1 t2)).(insert_eq C (CTail k u c) -(\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(or (pr2 c t1 t2) (ex3 T -(\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t: T).(pr0 t1 t)) (\lambda -(t: T).(subst0 (clen c) u t t2))))) (\lambda (y: C).(\lambda (H0: (pr2 y t1 -t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 -(CTail k u c)) \to (or (pr2 c t t0) (ex3 T (\lambda (_: T).(eq K k (Bind -Abbr))) (\lambda (t3: T).(pr0 t t3)) (\lambda (t3: T).(subst0 (clen c) u t3 -t0)))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: -(pr0 t3 t4)).(\lambda (_: (eq C c0 (CTail k u c))).(or_introl (pr2 c t3 t4) -(ex3 T (\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t: T).(pr0 t3 t)) -(\lambda (t: T).(subst0 (clen c) u t t4))) (pr2_free c t3 t4 H1))))))) -(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda -(H1: (getl i c0 (CHead d (Bind Abbr) u0))).(\lambda (t3: T).(\lambda (t4: -T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: (subst0 i u0 t4 -t)).(\lambda (H4: (eq C c0 (CTail k u c))).(let H5 \def (eq_ind C c0 (\lambda -(c1: C).(getl i c1 (CHead d (Bind Abbr) u0))) H1 (CTail k u c) H4) in (let -H_x \def (getl_gen_tail k Abbr u u0 d c i H5) in (let H6 \def H_x in (or_ind -(ex2 C (\lambda (e: C).(eq C d (CTail k u e))) (\lambda (e: C).(getl i c -(CHead e (Bind Abbr) u0)))) (ex4 nat (\lambda (_: nat).(eq nat i (clen c))) -(\lambda (_: nat).(eq K k (Bind Abbr))) (\lambda (_: nat).(eq T u u0)) -(\lambda (n: nat).(eq C d (CSort n)))) (or (pr2 c t3 t) (ex3 T (\lambda (_: -T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0: -T).(subst0 (clen c) u t0 t)))) (\lambda (H7: (ex2 C (\lambda (e: C).(eq C d -(CTail k u e))) (\lambda (e: C).(getl i c (CHead e (Bind Abbr) -u0))))).(ex2_ind C (\lambda (e: C).(eq C d (CTail k u e))) (\lambda (e: -C).(getl i c (CHead e (Bind Abbr) u0))) (or (pr2 c t3 t) (ex3 T (\lambda (_: -T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0: -T).(subst0 (clen c) u t0 t)))) (\lambda (x: C).(\lambda (_: (eq C d (CTail k -u x))).(\lambda (H9: (getl i c (CHead x (Bind Abbr) u0))).(or_introl (pr2 c -t3 t) (ex3 T (\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3 -t0)) (\lambda (t0: T).(subst0 (clen c) u t0 t))) (pr2_delta c x u0 i H9 t3 t4 -H2 t H3))))) H7)) (\lambda (H7: (ex4 nat (\lambda (_: nat).(eq nat i (clen -c))) (\lambda (_: nat).(eq K k (Bind Abbr))) (\lambda (_: nat).(eq T u u0)) -(\lambda (n: nat).(eq C d (CSort n))))).(ex4_ind nat (\lambda (_: nat).(eq -nat i (clen c))) (\lambda (_: nat).(eq K k (Bind Abbr))) (\lambda (_: -nat).(eq T u u0)) (\lambda (n: nat).(eq C d (CSort n))) (or (pr2 c t3 t) (ex3 -T (\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3 t0)) -(\lambda (t0: T).(subst0 (clen c) u t0 t)))) (\lambda (x0: nat).(\lambda (H8: -(eq nat i (clen c))).(\lambda (H9: (eq K k (Bind Abbr))).(\lambda (H10: (eq T -u u0)).(\lambda (_: (eq C d (CSort x0))).(let H12 \def (eq_ind nat i (\lambda -(n: nat).(subst0 n u0 t4 t)) H3 (clen c) H8) in (let H13 \def (eq_ind_r T u0 -(\lambda (t0: T).(subst0 (clen c) t0 t4 t)) H12 u H10) in (eq_ind_r K (Bind -Abbr) (\lambda (k0: K).(or (pr2 c t3 t) (ex3 T (\lambda (_: T).(eq K k0 (Bind -Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0 -t))))) (or_intror (pr2 c t3 t) (ex3 T (\lambda (_: T).(eq K (Bind Abbr) (Bind -Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0 -t))) (ex3_intro T (\lambda (_: T).(eq K (Bind Abbr) (Bind Abbr))) (\lambda -(t0: T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0 t)) t4 -(refl_equal K (Bind Abbr)) H2 H13)) k H9)))))))) H7)) H6))))))))))))))) y t1 -t2 H0))) H)))))). -(* COMMENTS -Initial nodes: 1161 -END *) - -theorem pr2_gen_cbind: - \forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall -(t2: T).((pr2 (CHead c (Bind b) v) t1 t2) \to (pr2 c (THead (Bind b) v t1) -(THead (Bind b) v t2))))))) -\def - \lambda (b: B).(\lambda (c: C).(\lambda (v: T).(\lambda (t1: T).(\lambda -(t2: T).(\lambda (H: (pr2 (CHead c (Bind b) v) t1 t2)).(insert_eq C (CHead c -(Bind b) v) (\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 c (THead -(Bind b) v t1) (THead (Bind b) v t2))) (\lambda (y: C).(\lambda (H0: (pr2 y -t1 t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 -(CHead c (Bind b) v)) \to (pr2 c (THead (Bind b) v t) (THead (Bind b) v -t0)))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: -(pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c (Bind b) v))).(pr2_free c (THead -(Bind b) v t3) (THead (Bind b) v t4) (pr0_comp v v (pr0_refl v) t3 t4 H1 -(Bind b)))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: -nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t3: -T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: -(subst0 i u t4 t)).(\lambda (H4: (eq C c0 (CHead c (Bind b) v))).(let H5 \def -(eq_ind C c0 (\lambda (c1: C).(getl i c1 (CHead d (Bind Abbr) u))) H1 (CHead -c (Bind b) v) H4) in (let H_x \def (getl_gen_bind b c (CHead d (Bind Abbr) u) -v i H5) in (let H6 \def H_x in (or_ind (land (eq nat i O) (eq C (CHead d -(Bind Abbr) u) (CHead c (Bind b) v))) (ex2 nat (\lambda (j: nat).(eq nat i (S -j))) (\lambda (j: nat).(getl j c (CHead d (Bind Abbr) u)))) (pr2 c (THead -(Bind b) v t3) (THead (Bind b) v t)) (\lambda (H7: (land (eq nat i O) (eq C -(CHead d (Bind Abbr) u) (CHead c (Bind b) v)))).(land_ind (eq nat i O) (eq C -(CHead d (Bind Abbr) u) (CHead c (Bind b) v)) (pr2 c (THead (Bind b) v t3) -(THead (Bind b) v t)) (\lambda (H8: (eq nat i O)).(\lambda (H9: (eq C (CHead -d (Bind Abbr) u) (CHead c (Bind b) v))).(let H10 \def (f_equal C C (\lambda -(e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d -| (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u) (CHead c (Bind b) -v) H9) in ((let H11 \def (f_equal C B (\lambda (e: C).(match e in C return -(\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b0) -\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) -(CHead c (Bind b) v) H9) in ((let H12 \def (f_equal C T (\lambda (e: -C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | -(CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind Abbr) u) (CHead c (Bind b) v) -H9) in (\lambda (H13: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H15 \def -(eq_ind nat i (\lambda (n: nat).(subst0 n u t4 t)) H3 O H8) in (let H16 \def -(eq_ind T u (\lambda (t0: T).(subst0 O t0 t4 t)) H15 v H12) in (eq_ind B Abbr -(\lambda (b0: B).(pr2 c (THead (Bind b0) v t3) (THead (Bind b0) v t))) -(pr2_free c (THead (Bind Abbr) v t3) (THead (Bind Abbr) v t) (pr0_delta v v -(pr0_refl v) t3 t4 H2 t H16)) b H13)))))) H11)) H10)))) H7)) (\lambda (H7: -(ex2 nat (\lambda (j: nat).(eq nat i (S j))) (\lambda (j: nat).(getl j c -(CHead d (Bind Abbr) u))))).(ex2_ind nat (\lambda (j: nat).(eq nat i (S j))) -(\lambda (j: nat).(getl j c (CHead d (Bind Abbr) u))) (pr2 c (THead (Bind b) -v t3) (THead (Bind b) v t)) (\lambda (x: nat).(\lambda (H8: (eq nat i (S -x))).(\lambda (H9: (getl x c (CHead d (Bind Abbr) u))).(let H10 \def (f_equal -nat nat (\lambda (e: nat).e) i (S x) H8) in (let H11 \def (eq_ind nat i -(\lambda (n: nat).(subst0 n u t4 t)) H3 (S x) H10) in (pr2_head_2 c v t3 t -(Bind b) (pr2_delta (CHead c (Bind b) v) d u (S x) (getl_clear_bind b (CHead -c (Bind b) v) c v (clear_bind b c v) (CHead d (Bind Abbr) u) x H9) t3 t4 H2 t -H11))))))) H7)) H6))))))))))))))) y t1 t2 H0))) H)))))). -(* COMMENTS -Initial nodes: 1085 -END *) - -theorem pr2_gen_cflat: - \forall (f: F).(\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall -(t2: T).((pr2 (CHead c (Flat f) v) t1 t2) \to (pr2 c t1 t2)))))) -\def - \lambda (f: F).(\lambda (c: C).(\lambda (v: T).(\lambda (t1: T).(\lambda -(t2: T).(\lambda (H: (pr2 (CHead c (Flat f) v) t1 t2)).(insert_eq C (CHead c -(Flat f) v) (\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 c t1 t2)) -(\lambda (y: C).(\lambda (H0: (pr2 y t1 t2)).(pr2_ind (\lambda (c0: -C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CHead c (Flat f) v)) \to (pr2 -c t t0))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: -(pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c (Flat f) v))).(pr2_free c t3 t4 -H1)))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: -nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t3: -T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: -(subst0 i u t4 t)).(\lambda (H4: (eq C c0 (CHead c (Flat f) v))).(let H5 \def -(eq_ind C c0 (\lambda (c1: C).(getl i c1 (CHead d (Bind Abbr) u))) H1 (CHead -c (Flat f) v) H4) in (let H_y \def (getl_gen_flat f c (CHead d (Bind Abbr) u) -v i H5) in (pr2_delta c d u i H_y t3 t4 H2 t H3)))))))))))))) y t1 t2 H0))) -H)))))). -(* COMMENTS -Initial nodes: 293 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr2/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr2/defs.ma deleted file mode 100644 index e21568232..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr2/defs.ma +++ /dev/null @@ -1,28 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr0/defs.ma". - -include "Basic-1/getl/defs.ma". - -inductive pr2: C \to (T \to (T \to Prop)) \def -| pr2_free: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to -(pr2 c t1 t2)))) -| pr2_delta: \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: -nat).((getl i c (CHead d (Bind Abbr) u)) \to (\forall (t1: T).(\forall (t2: -T).((pr0 t1 t2) \to (\forall (t: T).((subst0 i u t2 t) \to (pr2 c t1 -t)))))))))). - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr2/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr2/fwd.ma deleted file mode 100644 index 1a18f7dbc..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr2/fwd.ma +++ /dev/null @@ -1,3343 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr2/defs.ma". - -include "Basic-1/pr0/fwd.ma". - -include "Basic-1/getl/drop.ma". - -include "Basic-1/getl/clear.ma". - -theorem pr2_gen_sort: - \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr2 c (TSort n) x) \to -(eq T x (TSort n))))) -\def - \lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr2 c (TSort -n) x)).(insert_eq T (TSort n) (\lambda (t: T).(pr2 c t x)) (\lambda (t: -T).(eq T x t)) (\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda -(_: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (TSort n)) \to (eq T t0 -t))))) (\lambda (_: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H1: (pr0 -t1 t2)).(\lambda (H2: (eq T t1 (TSort n))).(let H3 \def (eq_ind T t1 (\lambda -(t: T).(pr0 t t2)) H1 (TSort n) H2) in (eq_ind_r T (TSort n) (\lambda (t: -T).(eq T t2 t)) (eq_ind_r T (TSort n) (\lambda (t: T).(eq T t (TSort n))) -(refl_equal T (TSort n)) t2 (pr0_gen_sort t2 n H3)) t1 H2))))))) (\lambda -(c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (_: (getl -i c0 (CHead d (Bind Abbr) u))).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H2: (pr0 t1 t2)).(\lambda (t: T).(\lambda (H3: (subst0 i u t2 t)).(\lambda -(H4: (eq T t1 (TSort n))).(let H5 \def (eq_ind T t1 (\lambda (t0: T).(pr0 t0 -t2)) H2 (TSort n) H4) in (eq_ind_r T (TSort n) (\lambda (t0: T).(eq T t t0)) -(let H6 \def (eq_ind T t2 (\lambda (t0: T).(subst0 i u t0 t)) H3 (TSort n) -(pr0_gen_sort t2 n H5)) in (subst0_gen_sort u t i n H6 (eq T t (TSort n)))) -t1 H4))))))))))))) c y x H0))) H)))). -(* COMMENTS -Initial nodes: 347 -END *) - -theorem pr2_gen_lref: - \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr2 c (TLRef n) x) \to -(or (eq T x (TLRef n)) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c -(CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T x (lift (S -n) O u))))))))) -\def - \lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr2 c (TLRef -n) x)).(insert_eq T (TLRef n) (\lambda (t: T).(pr2 c t x)) (\lambda (t: -T).(or (eq T x t) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c (CHead -d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T x (lift (S n) O -u))))))) (\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0: -C).(\lambda (t: T).(\lambda (t0: T).((eq T t (TLRef n)) \to (or (eq T t0 t) -(ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c0 (CHead d (Bind Abbr) -u)))) (\lambda (_: C).(\lambda (u: T).(eq T t0 (lift (S n) O u)))))))))) -(\lambda (c0: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H1: (pr0 t1 -t2)).(\lambda (H2: (eq T t1 (TLRef n))).(let H3 \def (eq_ind T t1 (\lambda -(t: T).(pr0 t t2)) H1 (TLRef n) H2) in (eq_ind_r T (TLRef n) (\lambda (t: -T).(or (eq T t2 t) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c0 -(CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T t2 (lift (S -n) O u))))))) (eq_ind_r T (TLRef n) (\lambda (t: T).(or (eq T t (TLRef n)) -(ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c0 (CHead d (Bind Abbr) -u)))) (\lambda (_: C).(\lambda (u: T).(eq T t (lift (S n) O u))))))) -(or_introl (eq T (TLRef n) (TLRef n)) (ex2_2 C T (\lambda (d: C).(\lambda (u: -T).(getl n c0 (CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq -T (TLRef n) (lift (S n) O u))))) (refl_equal T (TLRef n))) t2 (pr0_gen_lref -t2 n H3)) t1 H2))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: -T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr) -u))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H2: (pr0 t1 t2)).(\lambda -(t: T).(\lambda (H3: (subst0 i u t2 t)).(\lambda (H4: (eq T t1 (TLRef -n))).(let H5 \def (eq_ind T t1 (\lambda (t0: T).(pr0 t0 t2)) H2 (TLRef n) H4) -in (eq_ind_r T (TLRef n) (\lambda (t0: T).(or (eq T t t0) (ex2_2 C T (\lambda -(d0: C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind Abbr) u0)))) (\lambda (_: -C).(\lambda (u0: T).(eq T t (lift (S n) O u0))))))) (let H6 \def (eq_ind T t2 -(\lambda (t0: T).(subst0 i u t0 t)) H3 (TLRef n) (pr0_gen_lref t2 n H5)) in -(land_ind (eq nat n i) (eq T t (lift (S n) O u)) (or (eq T t (TLRef n)) -(ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind Abbr) -u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T t (lift (S n) O u0)))))) -(\lambda (H7: (eq nat n i)).(\lambda (H8: (eq T t (lift (S n) O -u))).(eq_ind_r T (lift (S n) O u) (\lambda (t0: T).(or (eq T t0 (TLRef n)) -(ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind Abbr) -u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T t0 (lift (S n) O u0))))))) (let -H9 \def (eq_ind_r nat i (\lambda (n0: nat).(getl n0 c0 (CHead d (Bind Abbr) -u))) H1 n H7) in (or_intror (eq T (lift (S n) O u) (TLRef n)) (ex2_2 C T -(\lambda (d0: C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind Abbr) u0)))) -(\lambda (_: C).(\lambda (u0: T).(eq T (lift (S n) O u) (lift (S n) O u0))))) -(ex2_2_intro C T (\lambda (d0: C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind -Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T (lift (S n) O u) (lift (S -n) O u0)))) d u H9 (refl_equal T (lift (S n) O u))))) t H8))) -(subst0_gen_lref u t i n H6))) t1 H4))))))))))))) c y x H0))) H)))). -(* COMMENTS -Initial nodes: 1003 -END *) - -theorem pr2_gen_abst: - \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c -(THead (Bind Abst) u1 t1) x) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 -c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: -T).(pr2 (CHead c (Bind b) u) t1 t2)))))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda -(H: (pr2 c (THead (Bind Abst) u1 t1) x)).(insert_eq T (THead (Bind Abst) u1 -t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall -(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2))))))) (\lambda (y: -T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0: C).(\lambda (t: -T).(\lambda (t0: T).((eq T t (THead (Bind Abst) u1 t1)) \to (ex3_2 T T -(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abst) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 -t2)))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H1: -(pr0 t0 t2)).(\lambda (H2: (eq T t0 (THead (Bind Abst) u1 t1))).(let H3 \def -(eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Bind Abst) u1 t1) H2) in -(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: -T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall -(u: T).(pr2 (CHead c0 (Bind b) u) t1 t3)))))) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H4: (eq T t2 (THead (Bind Abst) x0 x1))).(\lambda (H5: (pr0 u1 -x0)).(\lambda (H6: (pr0 t1 x1)).(eq_ind_r T (THead (Bind Abst) x0 x1) -(\lambda (t: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead -(Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) -(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead -c0 (Bind b) u) t1 t3))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: -T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall -(b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 t3))))) x0 x1 -(refl_equal T (THead (Bind Abst) x0 x1)) (pr2_free c0 u1 x0 H5) (\lambda (b: -B).(\lambda (u: T).(pr2_free (CHead c0 (Bind b) u) t1 x1 H6)))) t2 H4)))))) -(pr0_gen_abst u1 t1 t2 H3)))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda -(u: T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr) -u))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H2: (pr0 t0 t2)).(\lambda -(t: T).(\lambda (H3: (subst0 i u t2 t)).(\lambda (H4: (eq T t0 (THead (Bind -Abst) u1 t1))).(let H5 \def (eq_ind T t0 (\lambda (t3: T).(pr0 t3 t2)) H2 -(THead (Bind Abst) u1 t1) H4) in (ex3_2_ind T T (\lambda (u2: T).(\lambda -(t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 -t3)))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H6: (eq T t2 (THead -(Bind Abst) x0 x1))).(\lambda (H7: (pr0 u1 x0)).(\lambda (H8: (pr0 t1 -x1)).(let H9 \def (eq_ind T t2 (\lambda (t3: T).(subst0 i u t3 t)) H3 (THead -(Bind Abst) x0 x1) H6) in (or3_ind (ex2 T (\lambda (u2: T).(eq T t (THead -(Bind Abst) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2))) (ex2 T (\lambda -(t3: T).(eq T t (THead (Bind Abst) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind -Abst) i) u x1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t -(THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 -u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u x1 t3)))) -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) t1 t3)))))) (\lambda (H10: (ex2 T (\lambda (u2: T).(eq T t (THead (Bind -Abst) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2)))).(ex2_ind T (\lambda -(u2: T).(eq T t (THead (Bind Abst) u2 x1))) (\lambda (u2: T).(subst0 i u x0 -u2)) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst) -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) t1 t3)))))) (\lambda (x2: T).(\lambda (H11: (eq T t (THead (Bind Abst) x2 -x1))).(\lambda (H12: (subst0 i u x0 x2)).(eq_ind_r T (THead (Bind Abst) x2 -x1) (\lambda (t3: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 -(THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 -(CHead c0 (Bind b) u0) t1 t4))))))) (ex3_2_intro T T (\lambda (u2: -T).(\lambda (t3: T).(eq T (THead (Bind Abst) x2 x1) (THead (Bind Abst) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) t1 t3))))) x2 x1 (refl_equal T (THead (Bind Abst) x2 x1)) (pr2_delta c0 d -u i H1 u1 x0 H7 x2 H12) (\lambda (b: B).(\lambda (u0: T).(pr2_free (CHead c0 -(Bind b) u0) t1 x1 H8)))) t H11)))) H10)) (\lambda (H10: (ex2 T (\lambda (t3: -T).(eq T t (THead (Bind Abst) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind -Abst) i) u x1 t3)))).(ex2_ind T (\lambda (t3: T).(eq T t (THead (Bind Abst) -x0 t3))) (\lambda (t3: T).(subst0 (s (Bind Abst) i) u x1 t3)) (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 -t3)))))) (\lambda (x2: T).(\lambda (H11: (eq T t (THead (Bind Abst) x0 -x2))).(\lambda (H12: (subst0 (s (Bind Abst) i) u x1 x2)).(eq_ind_r T (THead -(Bind Abst) x0 x2) (\lambda (t3: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t4: -T).(eq T t3 (THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall -(u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t4))))))) (ex3_2_intro T T (\lambda -(u2: T).(\lambda (t3: T).(eq T (THead (Bind Abst) x0 x2) (THead (Bind Abst) -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) t1 t3))))) x0 x2 (refl_equal T (THead (Bind Abst) x0 x2)) (pr2_free c0 u1 -x0 H7) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u -(S i) (getl_head (Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 x1 H8 x2 -H12)))) t H11)))) H10)) (\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda -(t3: T).(eq T t (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind -Abst) i) u x1 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T -t (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u -x0 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u x1 -t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst) -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) t1 t3)))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H11: (eq T t -(THead (Bind Abst) x2 x3))).(\lambda (H12: (subst0 i u x0 x2)).(\lambda (H13: -(subst0 (s (Bind Abst) i) u x1 x3)).(eq_ind_r T (THead (Bind Abst) x2 x3) -(\lambda (t3: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead -(Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) -(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead -c0 (Bind b) u0) t1 t4))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: -T).(eq T (THead (Bind Abst) x2 x3) (THead (Bind Abst) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall -(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))))) x2 x3 -(refl_equal T (THead (Bind Abst) x2 x3)) (pr2_delta c0 d u i H1 u1 x0 H7 x2 -H12) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u -(S i) (getl_head (Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 x1 H8 x3 -H13)))) t H11)))))) H10)) (subst0_gen_head (Bind Abst) u x0 x1 t i H9)))))))) -(pr0_gen_abst u1 t1 t2 H5)))))))))))))) c y x H0))) H))))). -(* COMMENTS -Initial nodes: 2383 -END *) - -theorem pr2_gen_cast: - \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c -(THead (Flat Cast) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda -(t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c t1 t2)))) (pr2 c -t1 x)))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda -(H: (pr2 c (THead (Flat Cast) u1 t1) x)).(insert_eq T (THead (Flat Cast) u1 -t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c t1 -t2)))) (pr2 c t1 x))) (\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind -(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Flat Cast) -u1 t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead -(Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) -(\lambda (_: T).(\lambda (t2: T).(pr2 c0 t1 t2)))) (pr2 c0 t1 t0)))))) -(\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H1: (pr0 t0 -t2)).(\lambda (H2: (eq T t0 (THead (Flat Cast) u1 t1))).(let H3 \def (eq_ind -T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Flat Cast) u1 t1) H2) in (or_ind -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2) (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -t1 t3)))) (pr2 c0 t1 t2)) (\lambda (H4: (ex3_2 T T (\lambda (u2: T).(\lambda -(t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind -T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 -(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t2)) -(\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T t2 (THead (Flat Cast) -x0 x1))).(\lambda (H6: (pr0 u1 x0)).(\lambda (H7: (pr0 t1 x1)).(eq_ind_r T -(THead (Flat Cast) x0 x1) (\lambda (t: T).(or (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -t1 t3)))) (pr2 c0 t1 t))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda -(t3: T).(eq T (THead (Flat Cast) x0 x1) (THead (Flat Cast) u2 t3)))) (\lambda -(u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: -T).(pr2 c0 t1 t3)))) (pr2 c0 t1 (THead (Flat Cast) x0 x1)) (ex3_2_intro T T -(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Cast) x0 x1) (THead -(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) -(\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3))) x0 x1 (refl_equal T (THead -(Flat Cast) x0 x1)) (pr2_free c0 u1 x0 H6) (pr2_free c0 t1 x1 H7))) t2 -H5)))))) H4)) (\lambda (H4: (pr0 t1 t2)).(or_intror (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -t1 t3)))) (pr2 c0 t1 t2) (pr2_free c0 t1 t2 H4))) (pr0_gen_cast u1 t1 t2 -H3)))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: -nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t0: -T).(\lambda (t2: T).(\lambda (H2: (pr0 t0 t2)).(\lambda (t: T).(\lambda (H3: -(subst0 i u t2 t)).(\lambda (H4: (eq T t0 (THead (Flat Cast) u1 t1))).(let H5 -\def (eq_ind T t0 (\lambda (t3: T).(pr0 t3 t2)) H2 (THead (Flat Cast) u1 t1) -H4) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead -(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda -(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2) (or (ex3_2 T T (\lambda -(u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -t1 t3)))) (pr2 c0 t1 t)) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda -(t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind -T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t -(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t)) -(\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: (eq T t2 (THead (Flat Cast) -x0 x1))).(\lambda (H8: (pr0 u1 x0)).(\lambda (H9: (pr0 t1 x1)).(let H10 \def -(eq_ind T t2 (\lambda (t3: T).(subst0 i u t3 t)) H3 (THead (Flat Cast) x0 x1) -H7) in (or3_ind (ex2 T (\lambda (u2: T).(eq T t (THead (Flat Cast) u2 x1))) -(\lambda (u2: T).(subst0 i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T t (THead -(Flat Cast) x0 t3))) (\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3))) -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: -T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3)))) (or (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t)) (\lambda (H11: (ex2 T (\lambda (u2: -T).(eq T t (THead (Flat Cast) u2 x1))) (\lambda (u2: T).(subst0 i u x0 -u2)))).(ex2_ind T (\lambda (u2: T).(eq T t (THead (Flat Cast) u2 x1))) -(\lambda (u2: T).(subst0 i u x0 u2)) (or (ex3_2 T T (\lambda (u2: T).(\lambda -(t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 -c0 t1 t)) (\lambda (x2: T).(\lambda (H12: (eq T t (THead (Flat Cast) x2 -x1))).(\lambda (H13: (subst0 i u x0 x2)).(or_introl (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -t1 t3)))) (pr2 c0 t1 t) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: -T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 -c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3))) x2 x1 H12 -(pr2_delta c0 d u i H1 u1 x0 H8 x2 H13) (pr2_free c0 t1 x1 H9)))))) H11)) -(\lambda (H11: (ex2 T (\lambda (t3: T).(eq T t (THead (Flat Cast) x0 t3))) -(\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3)))).(ex2_ind T (\lambda -(t3: T).(eq T t (THead (Flat Cast) x0 t3))) (\lambda (t3: T).(subst0 (s (Flat -Cast) i) u x1 t3)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t -(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t)) -(\lambda (x2: T).(\lambda (H12: (eq T t (THead (Flat Cast) x0 x2))).(\lambda -(H13: (subst0 (s (Flat Cast) i) u x1 x2)).(or_introl (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -t1 t3)))) (pr2 c0 t1 t) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: -T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 -c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3))) x0 x2 H12 -(pr2_free c0 u1 x0 H8) (pr2_delta c0 d u i H1 t1 x1 H9 x2 H13)))))) H11)) -(\lambda (H11: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead -(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) -(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 -t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat -Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) -(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3))) (or -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t)) (\lambda (x2: -T).(\lambda (x3: T).(\lambda (H12: (eq T t (THead (Flat Cast) x2 -x3))).(\lambda (H13: (subst0 i u x0 x2)).(\lambda (H14: (subst0 (s (Flat -Cast) i) u x1 x3)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: -T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 -c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t) -(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast) -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr2 c0 t1 t3))) x2 x3 H12 (pr2_delta c0 d u i H1 u1 x0 -H8 x2 H13) (pr2_delta c0 d u i H1 t1 x1 H9 x3 H14)))))))) H11)) -(subst0_gen_head (Flat Cast) u x0 x1 t i H10)))))))) H6)) (\lambda (H6: (pr0 -t1 t2)).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t -(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t) -(pr2_delta c0 d u i H1 t1 t2 H6 t H3))) (pr0_gen_cast u1 t1 t2 -H5)))))))))))))) c y x H0))) H))))). -(* COMMENTS -Initial nodes: 2659 -END *) - -theorem pr2_gen_csort: - \forall (t1: T).(\forall (t2: T).(\forall (n: nat).((pr2 (CSort n) t1 t2) -\to (pr0 t1 t2)))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (n: nat).(\lambda (H: (pr2 (CSort -n) t1 t2)).(insert_eq C (CSort n) (\lambda (c: C).(pr2 c t1 t2)) (\lambda (_: -C).(pr0 t1 t2)) (\lambda (y: C).(\lambda (H0: (pr2 y t1 t2)).(pr2_ind -(\lambda (c: C).(\lambda (t: T).(\lambda (t0: T).((eq C c (CSort n)) \to (pr0 -t t0))))) (\lambda (c: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: -(pr0 t3 t4)).(\lambda (_: (eq C c (CSort n))).H1))))) (\lambda (c: -C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H1: (getl i c -(CHead d (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 -t3 t4)).(\lambda (t: T).(\lambda (_: (subst0 i u t4 t)).(\lambda (H4: (eq C c -(CSort n))).(let H5 \def (eq_ind C c (\lambda (c0: C).(getl i c0 (CHead d -(Bind Abbr) u))) H1 (CSort n) H4) in (getl_gen_sort n i (CHead d (Bind Abbr) -u) H5 (pr0 t3 t)))))))))))))) y t1 t2 H0))) H)))). -(* COMMENTS -Initial nodes: 221 -END *) - -theorem pr2_gen_appl: - \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c -(THead (Flat Appl) u1 t1) x) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda -(t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c t1 t2)))) (ex4_4 T -T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T -t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall -(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T -T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq -T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead -(Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda -(H: (pr2 c (THead (Flat Appl) u1 t1) x)).(insert_eq T (THead (Flat Appl) u1 -t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(or3 (ex3_2 T T (\lambda -(u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c t1 -t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind -Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) -z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) -(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) -z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) -y2) z1 z2)))))))))) (\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind -(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Flat Appl) -u1 t1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead -(Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) -(\lambda (_: T).(\lambda (t2: T).(pr2 c0 t1 t2)))) (ex4_4 T T T T (\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead -(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall -(b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) z1 t2)))))))) (ex6_6 B T T -T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T -t0 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 -y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 -z2))))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: T).(\lambda -(H1: (pr0 t0 t2)).(\lambda (H2: (eq T t0 (THead (Flat Appl) u1 t1))).(let H3 -\def (eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Flat Appl) u1 t1) -H2) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead -(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda -(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda -(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3)))))))) (or3 (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) -u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda -(u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift -(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 -(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (H4: (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 -t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead -(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda -(_: T).(\lambda (t3: T).(pr0 t1 t3))) (or3 (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) -u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda -(u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift -(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 -(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H5: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H6: (pr0 u1 -x0)).(\lambda (H7: (pr0 t1 x1)).(eq_ind_r T (THead (Flat Appl) x0 x1) -(\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t -(THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall -(b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T -T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t -(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda -(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 -y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 -z2)))))))))) (or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T -(THead (Flat Appl) x0 x1) (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) -x0 x1) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall -(u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Flat -Appl) x0 x1) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) -z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2)))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T -(THead (Flat Appl) x0 x1) (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -t1 t3))) x0 x1 (refl_equal T (THead (Flat Appl) x0 x1)) (pr2_free c0 u1 x0 -H6) (pr2_free c0 t1 x1 H7))) t2 H5)))))) H4)) (\lambda (H4: (ex4_4 T T T T -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda -(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 -t3))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda -(t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 -T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq -T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall -(b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T -T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T -t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 -y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 -z2))))))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: -T).(\lambda (H5: (eq T t1 (THead (Bind Abst) x0 x1))).(\lambda (H6: (eq T t2 -(THead (Bind Abbr) x2 x3))).(\lambda (H7: (pr0 u1 x2)).(\lambda (H8: (pr0 x1 -x3)).(eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda (t: T).(or3 (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t -(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 -(CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead -(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 -u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (eq_ind_r T (THead (Bind -Abst) x0 x1) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: -T).(eq T (THead (Bind Abbr) x2 x3) (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) -x2 x3) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall -(u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind -Abbr) x2 x3) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) -z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: -T).(eq T (THead (Bind Abbr) x2 x3) (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) u2 -t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: -T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) -u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead -(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind -Abbr) x2 x3) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) -z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead -(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) u2 -t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: -T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) -u) z1 t3))))))) x0 x1 x2 x3 (refl_equal T (THead (Bind Abst) x0 x1)) -(refl_equal T (THead (Bind Abbr) x2 x3)) (pr2_free c0 u1 x2 H7) (\lambda (b: -B).(\lambda (u: T).(pr2_free (CHead c0 (Bind b) u) x1 x3 H8))))) t1 H5) t2 -H6))))))))) H4)) (\lambda (H4: (ex6_6 B T T T T T (\lambda (b: B).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not -(eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) -y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead (Flat -Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 -t3))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead (Flat Appl) (lift -(S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 -t3))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead -(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) -(\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead -(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall -(b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T -T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T -t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 -y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 -z2))))))))) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: -T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H5: (not (eq B x0 -Abst))).(\lambda (H6: (eq T t1 (THead (Bind x0) x1 x2))).(\lambda (H7: (eq T -t2 (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)))).(\lambda -(H8: (pr0 u1 x3)).(\lambda (H9: (pr0 x1 x4)).(\lambda (H10: (pr0 x2 -x5)).(eq_ind_r T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) -x5)) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T -t (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall -(b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T -T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t -(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda -(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 -y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 -z2)))))))))) (eq_ind_r T (THead (Bind x0) x1 x2) (\lambda (t: T).(or3 (ex3_2 -T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat -Appl) (lift (S O) O x3) x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) -x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind Abbr) u2 t3)))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: -T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) -(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T t (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T -(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind b) -y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 -c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat -Appl) (lift (S O) O x3) x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) -(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 -(CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind -x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T -(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind b) -y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 -c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) -(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda -(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 -y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))) -x0 x1 x2 x5 x3 x4 H5 (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T -(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5))) (pr2_free c0 -u1 x3 H8) (pr2_free c0 x1 x4 H9) (pr2_free (CHead c0 (Bind x0) x4) x2 x5 -H10))) t1 H6) t2 H7))))))))))))) H4)) (pr0_gen_appl u1 t1 t2 H3)))))))) -(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t0: T).(\lambda (t2: -T).(\lambda (H2: (pr0 t0 t2)).(\lambda (t: T).(\lambda (H3: (subst0 i u t2 -t)).(\lambda (H4: (eq T t0 (THead (Flat Appl) u1 t1))).(let H5 \def (eq_ind T -t0 (\lambda (t3: T).(pr0 t3 t2)) H2 (THead (Flat Appl) u1 t1) H4) in (or3_ind -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 -z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: -T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda -(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3)))))))) (or3 (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind -Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda -(u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift -(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 -(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (H6: (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 -t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead -(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda -(_: T).(\lambda (t3: T).(pr0 t1 t3))) (or3 (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind -Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda -(u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift -(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 -(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H7: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H8: (pr0 u1 -x0)).(\lambda (H9: (pr0 t1 x1)).(let H10 \def (eq_ind T t2 (\lambda (t3: -T).(subst0 i u t3 t)) H3 (THead (Flat Appl) x0 x1) H7) in (or3_ind (ex2 T -(\lambda (u2: T).(eq T t (THead (Flat Appl) u2 x1))) (\lambda (u2: T).(subst0 -i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T t (THead (Flat Appl) x0 t3))) -(\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 t3))) (ex3_2 T T (\lambda -(u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3: -T).(subst0 (s (Flat Appl) i) u x1 t3)))) (or3 (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind -Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda -(u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift -(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 -(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (H11: (ex2 T (\lambda (u2: -T).(eq T t (THead (Flat Appl) u2 x1))) (\lambda (u2: T).(subst0 i u x0 -u2)))).(ex2_ind T (\lambda (u2: T).(eq T t (THead (Flat Appl) u2 x1))) -(\lambda (u2: T).(subst0 i u x0 u2)) (or3 (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind -Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda -(u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift -(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 -(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x2: T).(\lambda (H12: (eq T t -(THead (Flat Appl) x2 x1))).(\lambda (H13: (subst0 i u x0 x2)).(eq_ind_r T -(THead (Flat Appl) x2 x1) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: -T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 -t1 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind -Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda -(u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift -(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 -(CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro0 (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x1) (THead (Flat Appl) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t3: T).(eq T (THead (Flat Appl) x2 x1) (THead (Bind Abbr) u2 t3)))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: -T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) -(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T -(THead (Flat Appl) x2 x1) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O -u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2)))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T -(THead (Flat Appl) x2 x1) (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -t1 t3))) x2 x1 (refl_equal T (THead (Flat Appl) x2 x1)) (pr2_delta c0 d u i -H1 u1 x0 H8 x2 H13) (pr2_free c0 t1 x1 H9))) t H12)))) H11)) (\lambda (H11: -(ex2 T (\lambda (t3: T).(eq T t (THead (Flat Appl) x0 t3))) (\lambda (t3: -T).(subst0 (s (Flat Appl) i) u x1 t3)))).(ex2_ind T (\lambda (t3: T).(eq T t -(THead (Flat Appl) x0 t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 -t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat -Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda -(_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall -(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) -y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 -c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x2: T).(\lambda -(H12: (eq T t (THead (Flat Appl) x0 x2))).(\lambda (H13: (subst0 (s (Flat -Appl) i) u x1 x2)).(eq_ind_r T (THead (Flat Appl) x0 x2) (\lambda (t3: -T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat -Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda -(_: T).(\lambda (t4: T).(pr2 c0 t1 t4)))) (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall -(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) -(or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat -Appl) x0 x2) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 -T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq -T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x0 x2) (THead (Bind Abbr) -u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: -T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda -(u2: T).(\lambda (y2: T).(eq T (THead (Flat Appl) x0 x2) (THead (Bind b) y2 -(THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 -u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) (ex3_2_intro T T (\lambda -(u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x0 x2) (THead (Flat Appl) -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr2 c0 t1 t3))) x0 x2 (refl_equal T (THead (Flat Appl) -x0 x2)) (pr2_free c0 u1 x0 H8) (pr2_delta c0 d u i H1 t1 x1 H9 x2 H13))) t -H12)))) H11)) (\lambda (H11: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq -T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u -x0 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 -t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat -Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) -(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 t3))) (or3 -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall -(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) -y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 -c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x2: T).(\lambda -(x3: T).(\lambda (H12: (eq T t (THead (Flat Appl) x2 x3))).(\lambda (H13: -(subst0 i u x0 x2)).(\lambda (H14: (subst0 (s (Flat Appl) i) u x1 -x3)).(eq_ind_r T (THead (Flat Appl) x2 x3) (\lambda (t3: T).(or3 (ex3_2 T T -(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t4: T).(pr2 c0 t1 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 -(THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 -(CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead -(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 -u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro0 (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x3) (THead -(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) -(\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead -(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x3) (THead (Bind Abbr) u2 -t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: -T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda -(u2: T).(\lambda (y2: T).(eq T (THead (Flat Appl) x2 x3) (THead (Bind b) y2 -(THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 -u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) (ex3_2_intro T T (\lambda -(u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x3) (THead (Flat Appl) -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr2 c0 t1 t3))) x2 x3 (refl_equal T (THead (Flat Appl) -x2 x3)) (pr2_delta c0 d u i H1 u1 x0 H8 x2 H13) (pr2_delta c0 d u i H1 t1 x1 -H9 x3 H14))) t H12)))))) H11)) (subst0_gen_head (Flat Appl) u x0 x1 t i -H10)))))))) H6)) (\lambda (H6: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 -(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (t3: T).(pr0 z1 t3))))))).(ex4_4_ind T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))) (or3 (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t -(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 -(CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead -(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 -u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x0: T).(\lambda -(x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq T t1 (THead (Bind -Abst) x0 x1))).(\lambda (H8: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda -(H9: (pr0 u1 x2)).(\lambda (H10: (pr0 x1 x3)).(let H11 \def (eq_ind T t2 -(\lambda (t3: T).(subst0 i u t3 t)) H3 (THead (Bind Abbr) x2 x3) H8) in -(eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t3: T).(or3 (ex3_2 T T -(\lambda (u2: T).(\lambda (t4: T).(eq T t (THead (Flat Appl) u2 t4)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t4: T).(pr2 c0 t3 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T t3 (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t -(THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 -(CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t3 (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead -(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 -u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_ind (ex2 T (\lambda -(u2: T).(eq T t (THead (Bind Abbr) u2 x3))) (\lambda (u2: T).(subst0 i u x2 -u2))) (ex2 T (\lambda (t3: T).(eq T t (THead (Bind Abbr) x2 t3))) (\lambda -(t3: T).(subst0 (s (Bind Abbr) i) u x3 t3))) (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i u x2 u2))) (\lambda (_: T).(\lambda (t3: -T).(subst0 (s (Bind Abbr) i) u x3 t3)))) (or3 (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 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z2))))))))) (\lambda (H12: (ex2 T (\lambda (u2: T).(eq T t (THead -(Bind Abbr) u2 x3))) (\lambda (u2: T).(subst0 i u x2 u2)))).(ex2_ind T -(\lambda (u2: T).(eq T t (THead (Bind Abbr) u2 x3))) (\lambda (u2: T).(subst0 -i u x2 u2)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead -(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) -(\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind Abst) x0 x1) t3)))) -(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda -(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind -Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead -(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) -y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 -c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x4: T).(\lambda -(H13: (eq T t (THead (Bind Abbr) x4 x3))).(\lambda (H14: (subst0 i u x2 -x4)).(eq_ind_r T (THead (Bind Abbr) x4 x3) (\lambda (t3: T).(or3 (ex3_2 T T -(\lambda (u2: T).(\lambda (t4: 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T).(\lambda (z2: T).(\lambda (u2: -T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S -O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: -T).(eq T (THead (Bind Abbr) x4 x3) (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x3) (THead (Bind Abbr) u2 -t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: -T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead -(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind -Abbr) x4 x3) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) -z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead -(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x3) (THead (Bind Abbr) u2 -t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: -T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) z1 t3))))))) x0 x1 x4 x3 (refl_equal T (THead (Bind Abst) x0 x1)) -(refl_equal T (THead (Bind Abbr) x4 x3)) (pr2_delta c0 d u i H1 u1 x2 H9 x4 -H14) (\lambda (b: B).(\lambda (u0: T).(pr2_free (CHead c0 (Bind b) u0) x1 x3 -H10))))) t H13)))) H12)) (\lambda (H12: (ex2 T (\lambda (t3: T).(eq T t -(THead (Bind Abbr) x2 t3))) (\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x3 -t3)))).(ex2_ind T (\lambda (t3: T).(eq T t (THead (Bind Abbr) x2 t3))) -(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x3 t3)) (or3 (ex3_2 T T (\lambda -(u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall -(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T -T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: -T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) -O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2))))))))) (\lambda (x4: T).(\lambda (H13: (eq T t (THead (Bind Abbr) -x2 x4))).(\lambda (H14: (subst0 (s (Bind Abbr) i) u x3 x4)).(eq_ind_r T -(THead (Bind Abbr) x2 x4) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: -T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 -(THead (Bind Abst) x0 x1) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall -(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T -T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: -T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S -O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: -T).(eq T (THead (Bind Abbr) x2 x4) (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x4) (THead (Bind Abbr) u2 -t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: -T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead -(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind -Abbr) x2 x4) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) -z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead -(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x4) (THead (Bind Abbr) u2 -t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: -T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) z1 t3))))))) x0 x1 x2 x4 (refl_equal T (THead (Bind Abst) x0 x1)) -(refl_equal T (THead (Bind Abbr) x2 x4)) (pr2_free c0 u1 x2 H9) (\lambda (b: -B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) -(getl_clear_bind b (CHead c0 (Bind b) u0) c0 u0 (clear_bind b c0 u0) (CHead d -(Bind Abbr) u) i H1) x1 x3 H10 x4 H14))))) t H13)))) H12)) (\lambda (H12: -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x2 u2))) (\lambda (_: -T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x3 t3))))).(ex3_2_ind T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 i u x2 u2))) (\lambda (_: -T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x3 t3))) (or3 (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t3: T).(pr2 c0 (THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) -x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 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T).(\lambda (H13: (eq T t -(THead (Bind Abbr) x4 x5))).(\lambda (H14: (subst0 i u x2 x4)).(\lambda (H15: -(subst0 (s (Bind Abbr) i) u x3 x5)).(eq_ind_r T (THead (Bind Abbr) x4 x5) -(\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 -(THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind Abst) x0 x1) -t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1 -z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: -T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall -(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) -z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: -T).(eq T (THead (Bind Abbr) x4 x5) (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2 -t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: -T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead -(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind -Abbr) x4 x5) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) -z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead -(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2 -t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: -T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) z1 t3))))))) x0 x1 x4 x5 (refl_equal T (THead (Bind Abst) x0 x1)) -(refl_equal T (THead (Bind Abbr) x4 x5)) (pr2_delta c0 d u i H1 u1 x2 H9 x4 -H14) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u -(S i) (getl_clear_bind b (CHead c0 (Bind b) u0) c0 u0 (clear_bind b c0 u0) -(CHead d (Bind Abbr) u) i H1) x1 x3 H10 x5 H15))))) t H13)))))) H12)) -(subst0_gen_head (Bind Abbr) u x2 x3 t i H11)) t1 H7)))))))))) H6)) (\lambda -(H6: (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) -(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda -(t3: T).(eq T t2 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) -t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda -(y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 -y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))))).(ex6_6_ind B T T T T -T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq -T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead -(Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda -(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3))))))) (or3 (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind -Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda -(u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift -(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 -(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x0: B).(\lambda (x1: -T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: -T).(\lambda (H7: (not (eq B x0 Abst))).(\lambda (H8: (eq T t1 (THead (Bind -x0) x1 x2))).(\lambda (H9: (eq T t2 (THead (Bind x0) x4 (THead (Flat Appl) -(lift (S O) O x3) x5)))).(\lambda (H10: (pr0 u1 x3)).(\lambda (H11: (pr0 x1 -x4)).(\lambda (H12: (pr0 x2 x5)).(let H13 \def (eq_ind T t2 (\lambda (t3: -T).(subst0 i u t3 t)) H3 (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O -x3) x5)) H9) in (eq_ind_r T (THead (Bind x0) x1 x2) (\lambda (t3: T).(or3 -(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t (THead (Flat Appl) u2 -t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t4: T).(pr2 c0 t3 t4)))) (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t3 (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t4: T).(eq T t (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall -(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t3 -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) -y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 -c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_ind (ex2 T (\lambda -(u2: T).(eq T t (THead (Bind x0) u2 (THead (Flat Appl) (lift (S O) O x3) -x5)))) (\lambda (u2: T).(subst0 i u x4 u2))) (ex2 T (\lambda (t3: T).(eq T t -(THead (Bind x0) x4 t3))) (\lambda (t3: T).(subst0 (s (Bind x0) i) u (THead -(Flat Appl) (lift (S O) O x3) x5) t3))) (ex3_2 T T (\lambda (u2: T).(\lambda -(t3: T).(eq T t (THead (Bind x0) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 i u x4 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind x0) -i) u (THead (Flat Appl) (lift (S O) O x3) x5) t3)))) (or3 (ex3_2 T T (\lambda -(u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall -(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 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-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) -(\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) -(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind -Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead -(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 -u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x6: T).(\lambda -(H15: (eq T t (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) -x5)))).(\lambda (H16: (subst0 i u x4 x6)).(eq_ind_r T (THead (Bind x0) x6 -(THead (Flat Appl) (lift (S O) O x3) x5)) (\lambda (t3: T).(or3 (ex3_2 T T -(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 -x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall -(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T -T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: -T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S -O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (u2: 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T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B -b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead -(Flat Appl) (lift (S O) O x3) x5)) (THead (Bind b) y2 (THead (Flat Appl) -(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 -(CHead c0 (Bind b) y2) z1 z2))))))) x0 x1 x2 x5 x3 x6 H7 (refl_equal T (THead -(Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x6 (THead (Flat Appl) (lift -(S O) O x3) x5))) (pr2_free c0 u1 x3 H10) (pr2_delta c0 d u i H1 x1 x4 H11 x6 -H16) (pr2_free (CHead c0 (Bind x0) x6) x2 x5 H12))) t H15)))) H14)) (\lambda -(H14: (ex2 T (\lambda (t3: T).(eq T t (THead (Bind x0) x4 t3))) (\lambda (t3: -T).(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) O x3) x5) -t3)))).(ex2_ind T (\lambda (t3: T).(eq T t (THead (Bind x0) x4 t3))) (\lambda -(t3: T).(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) O x3) x5) -t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat -Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda -(_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: 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(CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) -z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2)))))))))) (or3_ind (ex2 T (\lambda (u2: T).(eq T x6 (THead (Flat -Appl) u2 x5))) (\lambda (u2: T).(subst0 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(THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) -x4 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall -(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(eq T (THead (Bind x0) x4 t3) (THead (Bind b) y2 (THead (Flat 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T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))) -x0 x1 x2 x7 x3 x4 H7 (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T -(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x7))) (pr2_free c0 -u1 x3 H10) (pr2_free c0 x1 x4 H11) (pr2_delta (CHead c0 (Bind x0) x4) d u (S -i) (getl_clear_bind x0 (CHead c0 (Bind x0) x4) c0 x4 (clear_bind x0 c0 x4) -(CHead d (Bind Abbr) u) i H1) x2 x5 H12 x7 H19))) x6 H18)))) H17)) (\lambda -(H17: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x6 (THead (Flat -Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0) i) u -(lift (S O) O x3) u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat -Appl) (s (Bind x0) i)) u x5 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda -(t3: T).(eq T x6 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) u2))) (\lambda (_: T).(\lambda -(t3: T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3))) (or3 (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 x6) (THead (Flat -Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda -(_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) -x4 x6) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall -(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind b) y2 (THead (Flat Appl) -(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 -(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x7: T).(\lambda (x8: -T).(\lambda (H18: (eq T x6 (THead (Flat Appl) x7 x8))).(\lambda (H19: (subst0 -(s (Bind x0) i) u (lift (S O) O x3) x7)).(\lambda (H20: (subst0 (s (Flat -Appl) (s (Bind x0) i)) u x5 x8)).(eq_ind_r T (THead (Flat Appl) x7 x8) -(\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T -(THead (Bind x0) x4 t3) (THead (Flat Appl) u2 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(Bind x0) x1 -x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) x7 x8)) -(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 -(CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind -x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T -(THead (Bind x0) x4 (THead (Flat Appl) x7 x8)) (THead (Bind b) y2 (THead -(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 -u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x9: T).(\lambda -(H21: (eq T x7 (lift (S O) O x9))).(\lambda (H22: (subst0 (minus (s (Bind x0) -i) (S O)) u x3 x9)).(let H23 \def (eq_ind nat (minus (s (Bind x0) i) (S O)) -(\lambda (n: nat).(subst0 n u x3 x9)) H22 i (s_arith1 x0 i)) in (eq_ind_r T -(lift (S O) O x9) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda -(t4: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) t3 x8)) (THead (Flat -Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda -(_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) -x4 (THead (Flat Appl) t3 x8)) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall -(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T -T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: -T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) t3 x8)) -(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda -(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda 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T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B -b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead -(Flat Appl) (lift (S O) O x9) x8)) (THead (Bind b) y2 (THead (Flat Appl) -(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 -(CHead c0 (Bind b) y2) z1 z2))))))) x0 x1 x2 x8 x9 x4 H7 (refl_equal T (THead -(Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x4 (THead (Flat Appl) (lift -(S O) O x9) x8))) (pr2_delta c0 d u i H1 u1 x3 H10 x9 H23) (pr2_free c0 x1 x4 -H11) (pr2_delta (CHead c0 (Bind x0) x4) d u (S i) (getl_clear_bind x0 (CHead -c0 (Bind x0) x4) c0 x4 (clear_bind x0 c0 x4) (CHead d (Bind Abbr) u) i H1) x2 -x5 H12 x8 H20))) x7 H21))))) (subst0_gen_lift_ge u x3 x7 (s (Bind x0) i) (S -O) O H19 (le_n_S O i (le_O_n i)))) x6 H18)))))) H17)) (subst0_gen_head (Flat -Appl) u (lift (S O) O x3) x5 x6 (s (Bind x0) i) H16)) t H15)))) H14)) -(\lambda (H14: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead -(Bind x0) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x4 u2))) -(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind x0) i) u (THead (Flat Appl) -(lift (S O) O x3) x5) t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: -T).(eq T t (THead (Bind x0) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 i u x4 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind x0) -i) u (THead (Flat Appl) (lift (S O) O x3) x5) t3))) (or3 (ex3_2 T T (\lambda -(u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall -(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda 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(Bind -x0) i)) u x5 t3)))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T -(THead (Bind x0) x6 x7) (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind Abbr) u2 t3)))))) (\lambda -(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall -(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T -T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: 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x6 x7) (THead (Flat Appl) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) -x6 x7) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall -(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: 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(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) -x6 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall -(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(eq T (THead (Bind x0) x6 t3) (THead (Bind b) y2 (THead (Flat Appl) -(lift (S O) O u2) z2))))))))) (\lambda (_: 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x5)) (THead (Bind -Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead -(Flat Appl) t3 x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) -z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 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u i H1 u1 x3 H10 x9 H23) (pr2_delta c0 d u -i H1 x1 x4 H11 x6 H16) (pr2_free (CHead c0 (Bind x0) x6) x2 x5 H12))) x8 -H21))))) (subst0_gen_lift_ge u x3 x8 (s (Bind x0) i) (S O) O H20 (le_n_S O i -(le_O_n i)))) x7 H19)))) H18)) (\lambda (H18: (ex2 T (\lambda (t3: T).(eq T -x7 (THead (Flat Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s -(Flat Appl) (s (Bind x0) i)) u x5 t3)))).(ex2_ind T (\lambda (t3: T).(eq T x7 -(THead (Flat Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat -Appl) (s (Bind x0) i)) u x5 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda -(t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Flat Appl) u2 t3)))) (\lambda -(u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: -T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 -x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (t3: T).(eq T (THead 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T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall -(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T -T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: -T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 t3) (THead (Bind b) y2 (THead -(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 -u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat -Appl) (lift (S O) O x3) x8)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8)) -(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 -(CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind -x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T -(THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8)) (THead (Bind b) -y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 -c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8)) -(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda -(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 -y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))) -x0 x1 x2 x8 x3 x6 H7 (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T -(THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8))) (pr2_free c0 -u1 x3 H10) (pr2_delta c0 d u i H1 x1 x4 H11 x6 H16) (pr2_delta (CHead c0 -(Bind x0) x6) d u (S i) (getl_clear_bind x0 (CHead c0 (Bind x0) x6) c0 x6 -(clear_bind x0 c0 x6) (CHead d (Bind Abbr) u) i H1) x2 x5 H12 x8 H20))) x7 -H19)))) H18)) (\lambda (H18: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq -T x7 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s -(Bind x0) i) u (lift (S O) O x3) u2))) (\lambda (_: T).(\lambda (t3: -T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3))))).(ex3_2_ind T T -(\lambda (u2: T).(\lambda (t3: T).(eq T x7 (THead (Flat Appl) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) -u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind x0) -i)) u x5 t3))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead -(Bind x0) x6 x7) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) -x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 -z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: -T).(eq T (THead (Bind x0) x6 x7) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall -(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T -T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: -T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind b) y2 (THead -(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 -u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x8: T).(\lambda -(x9: T).(\lambda (H19: (eq T x7 (THead (Flat Appl) x8 x9))).(\lambda (H20: -(subst0 (s (Bind x0) i) u (lift (S O) O x3) x8)).(\lambda (H21: (subst0 (s -(Flat Appl) (s (Bind x0) i)) u x5 x9)).(eq_ind_r T (THead (Flat Appl) x8 x9) -(\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T -(THead (Bind x0) x6 t3) (THead (Flat Appl) u2 t4)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 -(THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t4: T).(eq T (THead (Bind x0) x6 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda -(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall -(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T -T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: -T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 t3) (THead (Bind b) y2 (THead -(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 -u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (ex2_ind T (\lambda (t3: -T).(eq T x8 (lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus (s (Bind x0) -i) (S O)) u x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T -(THead (Bind x0) x6 (THead (Flat Appl) x8 x9)) (THead (Flat Appl) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 -x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x8 x9)) -(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 -(CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind -x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T -(THead (Bind x0) x6 (THead (Flat Appl) x8 x9)) (THead (Bind b) y2 (THead -(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 -u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x10: T).(\lambda -(H22: (eq T x8 (lift (S O) O x10))).(\lambda (H23: (subst0 (minus (s (Bind -x0) i) (S O)) u x3 x10)).(let H24 \def (eq_ind nat (minus (s (Bind x0) i) (S -O)) (\lambda (n: nat).(subst0 n u x3 x10)) H23 i (s_arith1 x0 i)) in -(eq_ind_r T (lift (S O) O x10) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: -T).(\lambda (t4: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t3 x9)) -(THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) -(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) -x6 (THead (Flat Appl) t3 x9)) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall -(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T -T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: -T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t3 x9)) -(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda -(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 -y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 -z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T -(THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x10) x9)) (THead (Flat -Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda -(_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) -x6 (THead (Flat Appl) (lift (S O) O x10) x9)) (THead (Bind Abbr) u2 t3)))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: -T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) -(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: -T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) -O x10) x9)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) -z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B -b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead -(Flat Appl) (lift (S O) O x10) x9)) (THead (Bind b) y2 (THead (Flat Appl) -(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 -(CHead c0 (Bind b) y2) z1 z2))))))) x0 x1 x2 x9 x10 x6 H7 (refl_equal T -(THead (Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x6 (THead (Flat Appl) -(lift (S O) O x10) x9))) (pr2_delta c0 d u i H1 u1 x3 H10 x10 H24) (pr2_delta -c0 d u i H1 x1 x4 H11 x6 H16) (pr2_delta (CHead c0 (Bind x0) x6) d u (S i) -(getl_clear_bind x0 (CHead c0 (Bind x0) x6) c0 x6 (clear_bind x0 c0 x6) -(CHead d (Bind Abbr) u) i H1) x2 x5 H12 x9 H21))) x8 H22))))) -(subst0_gen_lift_ge u x3 x8 (s (Bind x0) i) (S O) O H20 (le_n_S O i (le_O_n -i)))) x7 H19)))))) H18)) (subst0_gen_head (Flat Appl) u (lift (S O) O x3) x5 -x7 (s (Bind x0) i) H17)) t H15)))))) H14)) (subst0_gen_head (Bind x0) u x4 -(THead (Flat Appl) (lift (S O) O x3) x5) t i H13)) t1 H8)))))))))))))) H6)) -(pr0_gen_appl u1 t1 t2 H5)))))))))))))) c y x H0))) H))))). -(* COMMENTS -Initial nodes: 38859 -END *) - -theorem pr2_gen_abbr: - \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c -(THead (Bind Abbr) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda -(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(or3 (\forall (b: -B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2))) (ex2 T (\lambda (u: -T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t2))) (ex3_2 T -T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) -(\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: -T).(pr2 (CHead c (Bind Abbr) u1) z t2)))))))) (\forall (b: B).(\forall (u: -T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda -(H: (pr2 c (THead (Bind Abbr) u1 t1) x)).(insert_eq T (THead (Bind Abbr) u1 -t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(or3 -(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2))) (ex2 T -(\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 -t2))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) -u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: -T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t2)))))))) (\forall (b: -B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))) -(\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0: C).(\lambda -(t: T).(\lambda (t0: T).((eq T t (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T -T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t2: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 -t2))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind -Abbr) u) t1 t2))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 -(Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) -(\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t2)))))))) -(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O -t0))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H1: -(pr0 t0 t2)).(\lambda (H2: (eq T t0 (THead (Bind Abbr) u1 t1))).(let H3 \def 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(\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 -(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T -(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) -t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind -Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: -T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: -B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))) -(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 -(Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: -T).(pr2 (CHead c0 (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y0: -T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: -T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead -c0 (Bind Abbr) u1) z t3))))))) x3 x1 H13 (pr2_delta c0 d u i H1 u1 x0 H8 x3 -H14) (or3_intro1 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) -t1 x1))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 -(Bind Abbr) u0) t1 x1))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 -(CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 -z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z x1)))) -(ex_intro2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 -(Bind Abbr) u0) t1 x1)) x0 H8 (pr2_delta (CHead c0 (Bind Abbr) x0) c0 x0 O -(getl_refl Abbr c0 x0) t1 x2 H9 x1 H10)))))))) (pr0_subst0_back x0 x2 x1 O -H10 u1 H8))))) H12)) (\lambda (H12: (ex2 T (\lambda (t3: T).(eq T t (THead -(Bind Abbr) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x1 -t3)))).(ex2_ind T (\lambda (t3: T).(eq T t (THead (Bind Abbr) x0 t3))) -(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x1 t3)) (or (ex3_2 T T (\lambda -(u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 -(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T -(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) -t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind -Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: -T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: -B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))) -(\lambda (x3: T).(\lambda (H13: (eq T t (THead (Bind Abbr) x0 x3))).(\lambda -(H14: (subst0 (s (Bind Abbr) i) u x1 x3)).(ex2_ind T (\lambda (t3: T).(subst0 -O u1 x2 t3)) (\lambda (t3: T).(pr0 t3 x1)) (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 -(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T -(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) -t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind -Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: -T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: -B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))) -(\lambda (x4: T).(\lambda (H15: (subst0 O u1 x2 x4)).(\lambda (H16: (pr0 x4 -x1)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead -(Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) -(\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 -(CHead c0 (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda -(u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y0: -T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: -T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead -c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead -c0 (Bind b) u0) t1 (lift (S O) O t)))) (ex3_2_intro T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 -(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T -(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) -t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind -Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: -T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3))))))) x0 x3 H13 -(pr2_free c0 u1 x0 H8) (or3_intro2 (\forall (b: B).(\forall (u0: T).(pr2 -(CHead c0 (Bind b) u0) t1 x3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda -(u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 x3))) (ex3_2 T T (\lambda (y0: -T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: -T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead -c0 (Bind Abbr) u1) z x3)))) (ex3_2_intro T T (\lambda (y0: T).(\lambda (_: -T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: -T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) -u1) z x3))) x4 x1 (pr2_delta (CHead c0 (Bind Abbr) u1) c0 u1 O (getl_refl -Abbr c0 u1) t1 x2 H9 x4 H15) H16 (pr2_delta (CHead c0 (Bind Abbr) u1) d u (S -i) (getl_head (Bind Abbr) i c0 (CHead d (Bind Abbr) u) H1 u1) x1 x1 (pr0_refl -x1) x3 H14)))))))) (pr0_subst0_back x0 x2 x1 O H10 u1 H8))))) H12)) (\lambda -(H12: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: -T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x1 t3))))).(ex3_2_ind T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: -T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x1 t3))) (or (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 -t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 -(Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 -(CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 -z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z -t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 -(lift (S O) O t))))) (\lambda (x3: T).(\lambda (x4: T).(\lambda (H13: (eq T t -(THead (Bind Abbr) x3 x4))).(\lambda (H14: (subst0 i u x0 x3)).(\lambda (H15: -(subst0 (s (Bind Abbr) i) u x1 x4)).(ex2_ind T (\lambda (t3: T).(subst0 O u1 -x2 t3)) (\lambda (t3: T).(pr0 t3 x1)) (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 -(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T -(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) -t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind -Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: -T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: -B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))) -(\lambda (x5: T).(\lambda (H16: (subst0 O u1 x2 x5)).(\lambda (H17: (pr0 x5 -x1)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead -(Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) -(\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 -(CHead c0 (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda -(u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y0: -T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: -T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead -c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead -c0 (Bind b) u0) t1 (lift (S O) O t)))) (ex3_2_intro T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 -(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T -(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) -t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind -Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: -T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3))))))) x3 x4 H13 -(pr2_delta c0 d u i H1 u1 x0 H8 x3 H14) (or3_intro2 (\forall (b: B).(\forall -(u0: T).(pr2 (CHead c0 (Bind b) u0) t1 x4))) (ex2 T (\lambda (u0: T).(pr0 u1 -u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 x4))) (ex3_2 T T -(\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) -(\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: -T).(pr2 (CHead c0 (Bind Abbr) u1) z x4)))) (ex3_2_intro T T (\lambda (y0: -T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: -T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead -c0 (Bind Abbr) u1) z x4))) x5 x1 (pr2_delta (CHead c0 (Bind Abbr) u1) c0 u1 O -(getl_refl Abbr c0 u1) t1 x2 H9 x5 H16) H17 (pr2_delta (CHead c0 (Bind Abbr) -u1) d u (S i) (getl_head (Bind Abbr) i c0 (CHead d (Bind Abbr) u) H1 u1) x1 -x1 (pr0_refl x1) x4 H15)))))))) (pr0_subst0_back x0 x2 x1 O H10 u1 H8))))))) -H12)) (subst0_gen_head (Bind Abbr) u x0 x1 t i H11)))))) H_x0)) H_x)))))) -H6)) (\lambda (H6: (pr0 t1 (lift (S O) O t2))).(or_intror (ex3_2 T T (\lambda -(u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 -(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T -(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) -t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind -Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: -T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: -B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))) -(\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) -(getl_head (Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 (lift (S O) O t2) -H6 (lift (S O) O t) (subst0_lift_ge_S t2 t u i H3 O (le_O_n i))))))) -(pr0_gen_abbr u1 t1 t2 H5)))))))))))))) c y x H0))) H))))). -(* COMMENTS -Initial nodes: 11671 -END *) - -theorem pr2_gen_void: - \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c -(THead (Bind Void) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda -(t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall -(u: T).(pr2 (CHead c (Bind b) u) t1 t2)))))) (\forall (b: B).(\forall (u: -T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda -(H: (pr2 c (THead (Bind Void) u1 t1) x)).(insert_eq T (THead (Bind Void) u1 -t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall -(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2)))))) (\forall (b: -B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))) -(\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0: C).(\lambda -(t: T).(\lambda (t0: T).((eq T t (THead (Bind Void) u1 t1)) \to (or (ex3_2 T -T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Void) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 -t2)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 (lift -(S O) O t0))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: -T).(\lambda (H1: (pr0 t0 t2)).(\lambda (H2: (eq T t0 (THead (Bind Void) u1 -t1))).(let H3 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Bind -Void) u1 t1) H2) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq -T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 -u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O -t2)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda -(_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind -b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) -t1 (lift (S O) O t2))))) (\lambda (H4: (ex3_2 T T (\lambda (u2: T).(\lambda -(t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind -T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 -(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 -(CHead c0 (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead -c0 (Bind b) u) t1 (lift (S O) O t2))))) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H5: (eq T t2 (THead (Bind Void) x0 x1))).(\lambda (H6: (pr0 u1 -x0)).(\lambda (H7: (pr0 t1 x1)).(eq_ind_r T (THead (Bind Void) x0 x1) -(\lambda (t: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t -(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 -(CHead c0 (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead -c0 (Bind b) u) t1 (lift (S O) O t)))))) (or_introl (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T (THead (Bind Void) x0 x1) (THead (Bind Void) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) -u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 -(lift (S O) O (THead (Bind Void) x0 x1))))) (ex3_2_intro T T (\lambda (u2: -T).(\lambda (t3: T).(eq T (THead (Bind Void) x0 x1) (THead (Bind Void) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) -u) t1 t3))))) x0 x1 (refl_equal T (THead (Bind Void) x0 x1)) (pr2_free c0 u1 -x0 H6) (\lambda (b: B).(\lambda (u: T).(pr2_free (CHead c0 (Bind b) u) t1 x1 -H7))))) t2 H5)))))) H4)) (\lambda (H4: (pr0 t1 (lift (S O) O t2))).(or_intror -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) -u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 -(lift (S O) O t2)))) (\lambda (b: B).(\lambda (u: T).(pr2_free (CHead c0 -(Bind b) u) t1 (lift (S O) O t2) H4))))) (pr0_gen_void u1 t1 t2 H3)))))))) -(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t0: T).(\lambda (t2: -T).(\lambda (H2: (pr0 t0 t2)).(\lambda (t: T).(\lambda (H3: (subst0 i u t2 -t)).(\lambda (H4: (eq T t0 (THead (Bind Void) u1 t1))).(let H5 \def (eq_ind T -t0 (\lambda (t3: T).(pr0 t3 t2)) H2 (THead (Bind Void) u1 t1) H4) in (or_ind -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2)) (or (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 -t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 -(lift (S O) O t))))) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: -T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind -T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t -(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 -(CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 -(CHead c0 (Bind b) u0) t1 (lift (S O) O t))))) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H7: (eq T t2 (THead (Bind Void) x0 x1))).(\lambda (H8: (pr0 u1 -x0)).(\lambda (H9: (pr0 t1 x1)).(let H10 \def (eq_ind T t2 (\lambda (t3: -T).(subst0 i u t3 t)) H3 (THead (Bind Void) x0 x1) H7) in (or3_ind (ex2 T -(\lambda (u2: T).(eq T t (THead (Bind Void) u2 x1))) (\lambda (u2: T).(subst0 -i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T t (THead (Bind Void) x0 t3))) -(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3))) (ex3_2 T T (\lambda -(u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3: -T).(subst0 (s (Bind Void) i) u x1 t3)))) (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall -(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: -B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))) -(\lambda (H11: (ex2 T (\lambda (u2: T).(eq T t (THead (Bind Void) u2 x1))) -(\lambda (u2: T).(subst0 i u x0 u2)))).(ex2_ind T (\lambda (u2: T).(eq T t -(THead (Bind Void) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2)) (or (ex3_2 T -T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 -t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 -(lift (S O) O t))))) (\lambda (x2: T).(\lambda (H12: (eq T t (THead (Bind -Void) x2 x1))).(\lambda (H13: (subst0 i u x0 x2)).(or_introl (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 -t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 -(lift (S O) O t)))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T -t (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 -(CHead c0 (Bind b) u0) t1 t3))))) x2 x1 H12 (pr2_delta c0 d u i H1 u1 x0 H8 -x2 H13) (\lambda (b: B).(\lambda (u0: T).(pr2_free (CHead c0 (Bind b) u0) t1 -x1 H9)))))))) H11)) (\lambda (H11: (ex2 T (\lambda (t3: T).(eq T t (THead -(Bind Void) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 -t3)))).(ex2_ind T (\lambda (t3: T).(eq T t (THead (Bind Void) x0 t3))) -(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3)) (or (ex3_2 T T (\lambda -(u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall -(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: -B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))) -(\lambda (x2: T).(\lambda (H12: (eq T t (THead (Bind Void) x0 x2))).(\lambda -(H13: (subst0 (s (Bind Void) i) u x1 x2)).(or_introl (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall -(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: -B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))) -(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) t1 t3))))) x0 x2 H12 (pr2_free c0 u1 x0 H8) (\lambda (b: B).(\lambda (u0: -T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) (getl_head (Bind b) i c0 -(CHead d (Bind Abbr) u) H1 u0) t1 x1 H9 x2 H13)))))))) H11)) (\lambda (H11: -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: -T).(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3))))).(ex3_2_ind T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: -T).(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3))) (or (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 -t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 -(lift (S O) O t))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (eq T t -(THead (Bind Void) x2 x3))).(\lambda (H13: (subst0 i u x0 x2)).(\lambda (H14: -(subst0 (s (Bind Void) i) u x1 x3)).(or_introl (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall -(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: -B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))) -(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) t1 t3))))) x2 x3 H12 (pr2_delta c0 d u i H1 u1 x0 H8 x2 H13) (\lambda (b: -B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) (getl_head -(Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 x1 H9 x3 H14)))))))))) H11)) -(subst0_gen_head (Bind Void) u x0 x1 t i H10)))))))) H6)) (\lambda (H6: (pr0 -t1 (lift (S O) O t2))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: -T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 -c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: -T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0: -T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))) (\lambda (b: -B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) (getl_head -(Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 (lift (S O) O t2) H6 (lift (S -O) O t) (subst0_lift_ge_S t2 t u i H3 O (le_O_n i))))))) (pr0_gen_void u1 t1 -t2 H5)))))))))))))) c y x H0))) H))))). -(* COMMENTS -Initial nodes: 3467 -END *) - -theorem pr2_gen_lift: - \forall (c: C).(\forall (t1: T).(\forall (x: T).(\forall (h: nat).(\forall -(d: nat).((pr2 c (lift h d t1) x) \to (\forall (e: C).((drop h d c e) \to -(ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr2 e t1 -t2)))))))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (x: T).(\lambda (h: nat).(\lambda -(d: nat).(\lambda (H: (pr2 c (lift h d t1) x)).(insert_eq T (lift h d t1) -(\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(\forall (e: C).((drop h d c e) -\to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr2 e -t1 t2)))))) (\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0: -C).(\lambda (t: T).(\lambda (t0: T).((eq T t (lift h d t1)) \to (\forall (e: -C).((drop h d c0 e) \to (ex2 T (\lambda (t2: T).(eq T t0 (lift h d t2))) -(\lambda (t2: T).(pr2 e t1 t2))))))))) (\lambda (c0: C).(\lambda (t0: -T).(\lambda (t2: T).(\lambda (H1: (pr0 t0 t2)).(\lambda (H2: (eq T t0 (lift h -d t1))).(\lambda (e: C).(\lambda (_: (drop h d c0 e)).(let H4 \def (eq_ind T -t0 (\lambda (t: T).(pr0 t t2)) H1 (lift h d t1) H2) in (ex2_ind T (\lambda -(t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(pr0 t1 t3)) (ex2 T -(\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))) -(\lambda (x0: T).(\lambda (H5: (eq T t2 (lift h d x0))).(\lambda (H6: (pr0 t1 -x0)).(eq_ind_r T (lift h d x0) (\lambda (t: T).(ex2 T (\lambda (t3: T).(eq T -t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)))) (ex_intro2 T (\lambda -(t3: T).(eq T (lift h d x0) (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)) -x0 (refl_equal T (lift h d x0)) (pr2_free e t1 x0 H6)) t2 H5)))) -(pr0_gen_lift t1 t2 h d H4)))))))))) (\lambda (c0: C).(\lambda (d0: -C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d0 (Bind -Abbr) u))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H2: (pr0 t0 -t2)).(\lambda (t: T).(\lambda (H3: (subst0 i u t2 t)).(\lambda (H4: (eq T t0 -(lift h d t1))).(\lambda (e: C).(\lambda (H5: (drop h d c0 e)).(let H6 \def -(eq_ind T t0 (\lambda (t3: T).(pr0 t3 t2)) H2 (lift h d t1) H4) in (ex2_ind T -(\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(pr0 t1 t3)) (ex2 -T (\lambda (t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))) -(\lambda (x0: T).(\lambda (H7: (eq T t2 (lift h d x0))).(\lambda (H8: (pr0 t1 -x0)).(let H9 \def (eq_ind T t2 (\lambda (t3: T).(subst0 i u t3 t)) H3 (lift h -d x0) H7) in (lt_le_e i d (ex2 T (\lambda (t3: T).(eq T t (lift h d t3))) -(\lambda (t3: T).(pr2 e t1 t3))) (\lambda (H10: (lt i d)).(let H11 \def -(eq_ind nat d (\lambda (n: nat).(subst0 i u (lift h n x0) t)) H9 (S (plus i -(minus d (S i)))) (lt_plus_minus i d H10)) in (let H12 \def (eq_ind nat d -(\lambda (n: nat).(drop h n c0 e)) H5 (S (plus i (minus d (S i)))) -(lt_plus_minus i d H10)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: -C).(eq T u (lift h (minus d (S i)) v)))) (\lambda (v: T).(\lambda (e0: -C).(getl i e (CHead e0 (Bind Abbr) v)))) (\lambda (_: T).(\lambda (e0: -C).(drop h (minus d (S i)) d0 e0))) (ex2 T (\lambda (t3: T).(eq T t (lift h d -t3))) (\lambda (t3: T).(pr2 e t1 t3))) (\lambda (x1: T).(\lambda (x2: -C).(\lambda (H13: (eq T u (lift h (minus d (S i)) x1))).(\lambda (H14: (getl -i e (CHead x2 (Bind Abbr) x1))).(\lambda (_: (drop h (minus d (S i)) d0 -x2)).(let H16 \def (eq_ind T u (\lambda (t3: T).(subst0 i t3 (lift h (S (plus -i (minus d (S i)))) x0) t)) H11 (lift h (minus d (S i)) x1) H13) in (ex2_ind -T (\lambda (t3: T).(eq T t (lift h (S (plus i (minus d (S i)))) t3))) -(\lambda (t3: T).(subst0 i x1 x0 t3)) (ex2 T (\lambda (t3: T).(eq T t (lift h -d t3))) (\lambda (t3: T).(pr2 e t1 t3))) (\lambda (x3: T).(\lambda (H17: (eq -T t (lift h (S (plus i (minus d (S i)))) x3))).(\lambda (H18: (subst0 i x1 x0 -x3)).(let H19 \def (eq_ind_r nat (S (plus i (minus d (S i)))) (\lambda (n: -nat).(eq T t (lift h n x3))) H17 d (lt_plus_minus i d H10)) in (ex_intro2 T -(\lambda (t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)) x3 -H19 (pr2_delta e x2 x1 i H14 t1 x0 H8 x3 H18)))))) (subst0_gen_lift_lt x1 x0 -t i h (minus d (S i)) H16)))))))) (getl_drop_conf_lt Abbr c0 d0 u i H1 e h -(minus d (S i)) H12))))) (\lambda (H10: (le d i)).(lt_le_e i (plus d h) (ex2 -T (\lambda (t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))) -(\lambda (H11: (lt i (plus d h))).(subst0_gen_lift_false x0 u t h d i H10 H11 -H9 (ex2 T (\lambda (t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 -t3))))) (\lambda (H11: (le (plus d h) i)).(ex2_ind T (\lambda (t3: T).(eq T t -(lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u x0 t3)) (ex2 T -(\lambda (t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))) -(\lambda (x1: T).(\lambda (H12: (eq T t (lift h d x1))).(\lambda (H13: -(subst0 (minus i h) u x0 x1)).(ex_intro2 T (\lambda (t3: T).(eq T t (lift h d -t3))) (\lambda (t3: T).(pr2 e t1 t3)) x1 H12 (pr2_delta e d0 u (minus i h) -(getl_drop_conf_ge i (CHead d0 (Bind Abbr) u) c0 H1 e h d H5 H11) t1 x0 H8 x1 -H13))))) (subst0_gen_lift_ge u x0 t i h d H9 H11)))))))))) (pr0_gen_lift t1 -t2 h d H6)))))))))))))))) c y x H0))) H)))))). -(* COMMENTS -Initial nodes: 1579 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr2/pr2.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr2/pr2.ma deleted file mode 100644 index f8df4e9e2..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr2/pr2.ma +++ /dev/null @@ -1,258 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr2/defs.ma". - -include "Basic-1/pr0/pr0.ma". - -include "Basic-1/getl/props.ma". - -theorem pr2_confluence__pr2_free_free: - \forall (c: C).(\forall (t0: T).(\forall (t1: T).(\forall (t2: T).((pr0 t0 -t1) \to ((pr0 t0 t2) \to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: -T).(pr2 c t2 t)))))))) -\def - \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H: (pr0 t0 t1)).(\lambda (H0: (pr0 t0 t2)).(ex2_ind T (\lambda (t: T).(pr0 -t2 t)) (\lambda (t: T).(pr0 t1 t)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) -(\lambda (t: T).(pr2 c t2 t))) (\lambda (x: T).(\lambda (H1: (pr0 t2 -x)).(\lambda (H2: (pr0 t1 x)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) -(\lambda (t: T).(pr2 c t2 t)) x (pr2_free c t1 x H2) (pr2_free c t2 x H1))))) -(pr0_confluence t0 t2 H0 t1 H))))))). -(* COMMENTS -Initial nodes: 135 -END *) - -theorem pr2_confluence__pr2_free_delta: - \forall (c: C).(\forall (d: C).(\forall (t0: T).(\forall (t1: T).(\forall -(t2: T).(\forall (t4: T).(\forall (u: T).(\forall (i: nat).((pr0 t0 t1) \to -((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t4) \to ((subst0 i u t4 t2) -\to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 -t)))))))))))))) -\def - \lambda (c: C).(\lambda (d: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda -(t2: T).(\lambda (t4: T).(\lambda (u: T).(\lambda (i: nat).(\lambda (H: (pr0 -t0 t1)).(\lambda (H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H1: (pr0 -t0 t4)).(\lambda (H2: (subst0 i u t4 t2)).(ex2_ind T (\lambda (t: T).(pr0 t4 -t)) (\lambda (t: T).(pr0 t1 t)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda -(t: T).(pr2 c t2 t))) (\lambda (x: T).(\lambda (H3: (pr0 t4 x)).(\lambda (H4: -(pr0 t1 x)).(or_ind (pr0 t2 x) (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda -(w2: T).(subst0 i u x w2))) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: -T).(pr2 c t2 t))) (\lambda (H5: (pr0 t2 x)).(ex_intro2 T (\lambda (t: T).(pr2 -c t1 t)) (\lambda (t: T).(pr2 c t2 t)) x (pr2_free c t1 x H4) (pr2_free c t2 -x H5))) (\lambda (H5: (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: -T).(subst0 i u x w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t2 w2)) (\lambda -(w2: T).(subst0 i u x w2)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: -T).(pr2 c t2 t))) (\lambda (x0: T).(\lambda (H6: (pr0 t2 x0)).(\lambda (H7: -(subst0 i u x x0)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: -T).(pr2 c t2 t)) x0 (pr2_delta c d u i H0 t1 x H4 x0 H7) (pr2_free c t2 x0 -H6))))) H5)) (pr0_subst0 t4 x H3 u t2 i H2 u (pr0_refl u)))))) -(pr0_confluence t0 t4 H1 t1 H))))))))))))). -(* COMMENTS -Initial nodes: 403 -END *) - -theorem pr2_confluence__pr2_delta_delta: - \forall (c: C).(\forall (d: C).(\forall (d0: C).(\forall (t0: T).(\forall -(t1: T).(\forall (t2: T).(\forall (t3: T).(\forall (t4: T).(\forall (u: -T).(\forall (u0: T).(\forall (i: nat).(\forall (i0: nat).((getl i c (CHead d -(Bind Abbr) u)) \to ((pr0 t0 t3) \to ((subst0 i u t3 t1) \to ((getl i0 c -(CHead d0 (Bind Abbr) u0)) \to ((pr0 t0 t4) \to ((subst0 i0 u0 t4 t2) \to -(ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 -t)))))))))))))))))))) -\def - \lambda (c: C).(\lambda (d: C).(\lambda (d0: C).(\lambda (t0: T).(\lambda -(t1: T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (u: -T).(\lambda (u0: T).(\lambda (i: nat).(\lambda (i0: nat).(\lambda (H: (getl i -c (CHead d (Bind Abbr) u))).(\lambda (H0: (pr0 t0 t3)).(\lambda (H1: (subst0 -i u t3 t1)).(\lambda (H2: (getl i0 c (CHead d0 (Bind Abbr) u0))).(\lambda -(H3: (pr0 t0 t4)).(\lambda (H4: (subst0 i0 u0 t4 t2)).(ex2_ind T (\lambda (t: -T).(pr0 t4 t)) (\lambda (t: T).(pr0 t3 t)) (ex2 T (\lambda (t: T).(pr2 c t1 -t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (x: T).(\lambda (H5: (pr0 t4 -x)).(\lambda (H6: (pr0 t3 x)).(or_ind (pr0 t1 x) (ex2 T (\lambda (w2: T).(pr0 -t1 w2)) (\lambda (w2: T).(subst0 i u x w2))) (ex2 T (\lambda (t: T).(pr2 c t1 -t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (H7: (pr0 t1 x)).(or_ind (pr0 t2 -x) (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: T).(subst0 i0 u0 x -w2))) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) -(\lambda (H8: (pr0 t2 x)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda -(t: T).(pr2 c t2 t)) x (pr2_free c t1 x H7) (pr2_free c t2 x H8))) (\lambda -(H8: (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: T).(subst0 i0 u0 x -w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: T).(subst0 i0 -u0 x w2)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) -(\lambda (x0: T).(\lambda (H9: (pr0 t2 x0)).(\lambda (H10: (subst0 i0 u0 x -x0)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t)) -x0 (pr2_delta c d0 u0 i0 H2 t1 x H7 x0 H10) (pr2_free c t2 x0 H9))))) H8)) -(pr0_subst0 t4 x H5 u0 t2 i0 H4 u0 (pr0_refl u0)))) (\lambda (H7: (ex2 T -(\lambda (w2: T).(pr0 t1 w2)) (\lambda (w2: T).(subst0 i u x w2)))).(ex2_ind -T (\lambda (w2: T).(pr0 t1 w2)) (\lambda (w2: T).(subst0 i u x w2)) (ex2 T -(\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (x0: -T).(\lambda (H8: (pr0 t1 x0)).(\lambda (H9: (subst0 i u x x0)).(or_ind (pr0 -t2 x) (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: T).(subst0 i0 u0 x -w2))) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) -(\lambda (H10: (pr0 t2 x)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) -(\lambda (t: T).(pr2 c t2 t)) x0 (pr2_free c t1 x0 H8) (pr2_delta c d u i H -t2 x H10 x0 H9))) (\lambda (H10: (ex2 T (\lambda (w2: T).(pr0 t2 w2)) -(\lambda (w2: T).(subst0 i0 u0 x w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t2 -w2)) (\lambda (w2: T).(subst0 i0 u0 x w2)) (ex2 T (\lambda (t: T).(pr2 c t1 -t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (x1: T).(\lambda (H11: (pr0 t2 -x1)).(\lambda (H12: (subst0 i0 u0 x x1)).(neq_eq_e i i0 (ex2 T (\lambda (t: -T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (H13: (not (eq nat i -i0))).(ex2_ind T (\lambda (t: T).(subst0 i u x1 t)) (\lambda (t: T).(subst0 -i0 u0 x0 t)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 -t))) (\lambda (x2: T).(\lambda (H14: (subst0 i u x1 x2)).(\lambda (H15: -(subst0 i0 u0 x0 x2)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: -T).(pr2 c t2 t)) x2 (pr2_delta c d0 u0 i0 H2 t1 x0 H8 x2 H15) (pr2_delta c d -u i H t2 x1 H11 x2 H14))))) (subst0_confluence_neq x x1 u0 i0 H12 x0 u i H9 -(sym_not_eq nat i i0 H13)))) (\lambda (H13: (eq nat i i0)).(let H14 \def -(eq_ind_r nat i0 (\lambda (n: nat).(subst0 n u0 x x1)) H12 i H13) in (let H15 -\def (eq_ind_r nat i0 (\lambda (n: nat).(getl n c (CHead d0 (Bind Abbr) u0))) -H2 i H13) in (let H16 \def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c0: -C).(getl i c c0)) H (CHead d0 (Bind Abbr) u0) (getl_mono c (CHead d (Bind -Abbr) u) i H (CHead d0 (Bind Abbr) u0) H15)) in (let H17 \def (f_equal C C -(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abbr) u) -(CHead d0 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) i H (CHead d0 -(Bind Abbr) u0) H15)) in ((let H18 \def (f_equal C T (\lambda (e: C).(match e -in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) -\Rightarrow t])) (CHead d (Bind Abbr) u) (CHead d0 (Bind Abbr) u0) (getl_mono -c (CHead d (Bind Abbr) u) i H (CHead d0 (Bind Abbr) u0) H15)) in (\lambda -(H19: (eq C d d0)).(let H20 \def (eq_ind_r T u0 (\lambda (t: T).(subst0 i t x -x1)) H14 u H18) in (let H21 \def (eq_ind_r T u0 (\lambda (t: T).(getl i c -(CHead d0 (Bind Abbr) t))) H16 u H18) in (let H22 \def (eq_ind_r C d0 -(\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr) u))) H21 d H19) in (or4_ind -(eq T x1 x0) (ex2 T (\lambda (t: T).(subst0 i u x1 t)) (\lambda (t: -T).(subst0 i u x0 t))) (subst0 i u x1 x0) (subst0 i u x0 x1) (ex2 T (\lambda -(t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (H23: (eq T x1 -x0)).(let H24 \def (eq_ind T x1 (\lambda (t: T).(pr0 t2 t)) H11 x0 H23) in -(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t)) x0 -(pr2_free c t1 x0 H8) (pr2_free c t2 x0 H24)))) (\lambda (H23: (ex2 T -(\lambda (t: T).(subst0 i u x1 t)) (\lambda (t: T).(subst0 i u x0 -t)))).(ex2_ind T (\lambda (t: T).(subst0 i u x1 t)) (\lambda (t: T).(subst0 i -u x0 t)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) -(\lambda (x2: T).(\lambda (H24: (subst0 i u x1 x2)).(\lambda (H25: (subst0 i -u x0 x2)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c -t2 t)) x2 (pr2_delta c d u i H22 t1 x0 H8 x2 H25) (pr2_delta c d u i H22 t2 -x1 H11 x2 H24))))) H23)) (\lambda (H23: (subst0 i u x1 x0)).(ex_intro2 T -(\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t)) x0 (pr2_free c t1 -x0 H8) (pr2_delta c d u i H22 t2 x1 H11 x0 H23))) (\lambda (H23: (subst0 i u -x0 x1)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 -t)) x1 (pr2_delta c d u i H22 t1 x0 H8 x1 H23) (pr2_free c t2 x1 H11))) -(subst0_confluence_eq x x1 u i H20 x0 H9))))))) H17)))))))))) H10)) -(pr0_subst0 t4 x H5 u0 t2 i0 H4 u0 (pr0_refl u0)))))) H7)) (pr0_subst0 t3 x -H6 u t1 i H1 u (pr0_refl u)))))) (pr0_confluence t0 t4 H3 t3 -H0))))))))))))))))))). -(* COMMENTS -Initial nodes: 1901 -END *) - -theorem pr2_confluence: - \forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr2 c t0 t1) \to (\forall -(t2: T).((pr2 c t0 t2) \to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: -T).(pr2 c t2 t)))))))) -\def - \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr2 c t0 -t1)).(\lambda (t2: T).(\lambda (H0: (pr2 c t0 t2)).(let H1 \def (match H in -pr2 return (\lambda (c0: C).(\lambda (t: T).(\lambda (t3: T).(\lambda (_: -(pr2 c0 t t3)).((eq C c0 c) \to ((eq T t t0) \to ((eq T t3 t1) \to (ex2 T -(\lambda (t4: T).(pr2 c t1 t4)) (\lambda (t4: T).(pr2 c t2 t4)))))))))) with -[(pr2_free c0 t3 t4 H1) \Rightarrow (\lambda (H2: (eq C c0 c)).(\lambda (H3: -(eq T t3 t0)).(\lambda (H4: (eq T t4 t1)).(eq_ind C c (\lambda (_: C).((eq T -t3 t0) \to ((eq T t4 t1) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t: T).(pr2 c -t1 t)) (\lambda (t: T).(pr2 c t2 t))))))) (\lambda (H5: (eq T t3 t0)).(eq_ind -T t0 (\lambda (t: T).((eq T t4 t1) \to ((pr0 t t4) \to (ex2 T (\lambda (t5: -T).(pr2 c t1 t5)) (\lambda (t5: T).(pr2 c t2 t5)))))) (\lambda (H6: (eq T t4 -t1)).(eq_ind T t1 (\lambda (t: T).((pr0 t0 t) \to (ex2 T (\lambda (t5: -T).(pr2 c t1 t5)) (\lambda (t5: T).(pr2 c t2 t5))))) (\lambda (H7: (pr0 t0 -t1)).(let H8 \def (match H0 in pr2 return (\lambda (c1: C).(\lambda (t: -T).(\lambda (t5: T).(\lambda (_: (pr2 c1 t t5)).((eq C c1 c) \to ((eq T t t0) -\to ((eq T t5 t2) \to (ex2 T (\lambda (t6: T).(pr2 c t1 t6)) (\lambda (t6: -T).(pr2 c t2 t6)))))))))) with [(pr2_free c1 t5 t6 H8) \Rightarrow (\lambda -(H9: (eq C c1 c)).(\lambda (H10: (eq T t5 t0)).(\lambda (H11: (eq T t6 -t2)).(eq_ind C c (\lambda (_: C).((eq T t5 t0) \to ((eq T t6 t2) \to ((pr0 t5 -t6) \to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 -t))))))) (\lambda (H12: (eq T t5 t0)).(eq_ind T t0 (\lambda (t: T).((eq T t6 -t2) \to ((pr0 t t6) \to (ex2 T (\lambda (t7: T).(pr2 c t1 t7)) (\lambda (t7: -T).(pr2 c t2 t7)))))) (\lambda (H13: (eq T t6 t2)).(eq_ind T t2 (\lambda (t: -T).((pr0 t0 t) \to (ex2 T (\lambda (t7: T).(pr2 c t1 t7)) (\lambda (t7: -T).(pr2 c t2 t7))))) (\lambda (H14: (pr0 t0 -t2)).(pr2_confluence__pr2_free_free c t0 t1 t2 H7 H14)) t6 (sym_eq T t6 t2 -H13))) t5 (sym_eq T t5 t0 H12))) c1 (sym_eq C c1 c H9) H10 H11 H8)))) | -(pr2_delta c1 d u i H8 t5 t6 H9 t H10) \Rightarrow (\lambda (H11: (eq C c1 -c)).(\lambda (H12: (eq T t5 t0)).(\lambda (H13: (eq T t t2)).(eq_ind C c -(\lambda (c2: C).((eq T t5 t0) \to ((eq T t t2) \to ((getl i c2 (CHead d -(Bind Abbr) u)) \to ((pr0 t5 t6) \to ((subst0 i u t6 t) \to (ex2 T (\lambda -(t7: T).(pr2 c t1 t7)) (\lambda (t7: T).(pr2 c t2 t7))))))))) (\lambda (H14: -(eq T t5 t0)).(eq_ind T t0 (\lambda (t7: T).((eq T t t2) \to ((getl i c -(CHead d (Bind Abbr) u)) \to ((pr0 t7 t6) \to ((subst0 i u t6 t) \to (ex2 T -(\lambda (t8: T).(pr2 c t1 t8)) (\lambda (t8: T).(pr2 c t2 t8)))))))) -(\lambda (H15: (eq T t t2)).(eq_ind T t2 (\lambda (t7: T).((getl i c (CHead d -(Bind Abbr) u)) \to ((pr0 t0 t6) \to ((subst0 i u t6 t7) \to (ex2 T (\lambda -(t8: T).(pr2 c t1 t8)) (\lambda (t8: T).(pr2 c t2 t8))))))) (\lambda (H16: -(getl i c (CHead d (Bind Abbr) u))).(\lambda (H17: (pr0 t0 t6)).(\lambda -(H18: (subst0 i u t6 t2)).(pr2_confluence__pr2_free_delta c d t0 t1 t2 t6 u i -H7 H16 H17 H18)))) t (sym_eq T t t2 H15))) t5 (sym_eq T t5 t0 H14))) c1 -(sym_eq C c1 c H11) H12 H13 H8 H9 H10))))]) in (H8 (refl_equal C c) -(refl_equal T t0) (refl_equal T t2)))) t4 (sym_eq T t4 t1 H6))) t3 (sym_eq T -t3 t0 H5))) c0 (sym_eq C c0 c H2) H3 H4 H1)))) | (pr2_delta c0 d u i H1 t3 t4 -H2 t H3) \Rightarrow (\lambda (H4: (eq C c0 c)).(\lambda (H5: (eq T t3 -t0)).(\lambda (H6: (eq T t t1)).(eq_ind C c (\lambda (c1: C).((eq T t3 t0) -\to ((eq T t t1) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t3 t4) -\to ((subst0 i u t4 t) \to (ex2 T (\lambda (t5: T).(pr2 c t1 t5)) (\lambda -(t5: T).(pr2 c t2 t5))))))))) (\lambda (H7: (eq T t3 t0)).(eq_ind T t0 -(\lambda (t5: T).((eq T t t1) \to ((getl i c (CHead d (Bind Abbr) u)) \to -((pr0 t5 t4) \to ((subst0 i u t4 t) \to (ex2 T (\lambda (t6: T).(pr2 c t1 -t6)) (\lambda (t6: T).(pr2 c t2 t6)))))))) (\lambda (H8: (eq T t t1)).(eq_ind -T t1 (\lambda (t5: T).((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t4) -\to ((subst0 i u t4 t5) \to (ex2 T (\lambda (t6: T).(pr2 c t1 t6)) (\lambda -(t6: T).(pr2 c t2 t6))))))) (\lambda (H9: (getl i c (CHead d (Bind Abbr) -u))).(\lambda (H10: (pr0 t0 t4)).(\lambda (H11: (subst0 i u t4 t1)).(let H12 -\def (match H0 in pr2 return (\lambda (c1: C).(\lambda (t5: T).(\lambda (t6: -T).(\lambda (_: (pr2 c1 t5 t6)).((eq C c1 c) \to ((eq T t5 t0) \to ((eq T t6 -t2) \to (ex2 T (\lambda (t7: T).(pr2 c t1 t7)) (\lambda (t7: T).(pr2 c t2 -t7)))))))))) with [(pr2_free c1 t5 t6 H12) \Rightarrow (\lambda (H13: (eq C -c1 c)).(\lambda (H14: (eq T t5 t0)).(\lambda (H15: (eq T t6 t2)).(eq_ind C c -(\lambda (_: C).((eq T t5 t0) \to ((eq T t6 t2) \to ((pr0 t5 t6) \to (ex2 T -(\lambda (t7: T).(pr2 c t1 t7)) (\lambda (t7: T).(pr2 c t2 t7))))))) (\lambda -(H16: (eq T t5 t0)).(eq_ind T t0 (\lambda (t7: T).((eq T t6 t2) \to ((pr0 t7 -t6) \to (ex2 T (\lambda (t8: T).(pr2 c t1 t8)) (\lambda (t8: T).(pr2 c t2 -t8)))))) (\lambda (H17: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t0 -t7) \to (ex2 T (\lambda (t8: T).(pr2 c t1 t8)) (\lambda (t8: T).(pr2 c t2 -t8))))) (\lambda (H18: (pr0 t0 t2)).(ex2_sym T (pr2 c t2) (pr2 c t1) -(pr2_confluence__pr2_free_delta c d t0 t2 t1 t4 u i H18 H9 H10 H11))) t6 -(sym_eq T t6 t2 H17))) t5 (sym_eq T t5 t0 H16))) c1 (sym_eq C c1 c H13) H14 -H15 H12)))) | (pr2_delta c1 d0 u0 i0 H12 t5 t6 H13 t7 H14) \Rightarrow -(\lambda (H15: (eq C c1 c)).(\lambda (H16: (eq T t5 t0)).(\lambda (H17: (eq T -t7 t2)).(eq_ind C c (\lambda (c2: C).((eq T t5 t0) \to ((eq T t7 t2) \to -((getl i0 c2 (CHead d0 (Bind Abbr) u0)) \to ((pr0 t5 t6) \to ((subst0 i0 u0 -t6 t7) \to (ex2 T (\lambda (t8: T).(pr2 c t1 t8)) (\lambda (t8: T).(pr2 c t2 -t8))))))))) (\lambda (H18: (eq T t5 t0)).(eq_ind T t0 (\lambda (t8: T).((eq T -t7 t2) \to ((getl i0 c (CHead d0 (Bind Abbr) u0)) \to ((pr0 t8 t6) \to -((subst0 i0 u0 t6 t7) \to (ex2 T (\lambda (t9: T).(pr2 c t1 t9)) (\lambda -(t9: T).(pr2 c t2 t9)))))))) (\lambda (H19: (eq T t7 t2)).(eq_ind T t2 -(\lambda (t8: T).((getl i0 c (CHead d0 (Bind Abbr) u0)) \to ((pr0 t0 t6) \to -((subst0 i0 u0 t6 t8) \to (ex2 T (\lambda (t9: T).(pr2 c t1 t9)) (\lambda -(t9: T).(pr2 c t2 t9))))))) (\lambda (H20: (getl i0 c (CHead d0 (Bind Abbr) -u0))).(\lambda (H21: (pr0 t0 t6)).(\lambda (H22: (subst0 i0 u0 t6 -t2)).(pr2_confluence__pr2_delta_delta c d d0 t0 t1 t2 t4 t6 u u0 i i0 H9 H10 -H11 H20 H21 H22)))) t7 (sym_eq T t7 t2 H19))) t5 (sym_eq T t5 t0 H18))) c1 -(sym_eq C c1 c H15) H16 H17 H12 H13 H14))))]) in (H12 (refl_equal C c) -(refl_equal T t0) (refl_equal T t2)))))) t (sym_eq T t t1 H8))) t3 (sym_eq T -t3 t0 H7))) c0 (sym_eq C c0 c H4) H5 H6 H1 H2 H3))))]) in (H1 (refl_equal C -c) (refl_equal T t0) (refl_equal T t1)))))))). -(* COMMENTS -Initial nodes: 2087 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr2/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr2/props.ma deleted file mode 100644 index 2faeb6ebd..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr2/props.ma +++ /dev/null @@ -1,307 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr2/defs.ma". - -include "Basic-1/pr0/props.ma". - -include "Basic-1/getl/drop.ma". - -include "Basic-1/getl/clear.ma". - -theorem pr2_thin_dx: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall -(u: T).(\forall (f: F).(pr2 c (THead (Flat f) u t1) (THead (Flat f) u -t2))))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 -t2)).(\lambda (u: T).(\lambda (f: F).(pr2_ind (\lambda (c0: C).(\lambda (t: -T).(\lambda (t0: T).(pr2 c0 (THead (Flat f) u t) (THead (Flat f) u t0))))) -(\lambda (c0: C).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr0 t0 -t3)).(pr2_free c0 (THead (Flat f) u t0) (THead (Flat f) u t3) (pr0_comp u u -(pr0_refl u) t0 t3 H0 (Flat f))))))) (\lambda (c0: C).(\lambda (d: -C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind -Abbr) u0))).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H1: (pr0 t0 -t3)).(\lambda (t: T).(\lambda (H2: (subst0 i u0 t3 t)).(pr2_delta c0 d u0 i -H0 (THead (Flat f) u t0) (THead (Flat f) u t3) (pr0_comp u u (pr0_refl u) t0 -t3 H1 (Flat f)) (THead (Flat f) u t) (subst0_snd (Flat f) u0 t t3 i H2 -u)))))))))))) c t1 t2 H)))))). -(* COMMENTS -Initial nodes: 239 -END *) - -theorem pr2_head_1: - \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr2 c u1 u2) \to (\forall -(k: K).(\forall (t: T).(pr2 c (THead k u1 t) (THead k u2 t))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr2 c u1 -u2)).(\lambda (k: K).(\lambda (t: T).(pr2_ind (\lambda (c0: C).(\lambda (t0: -T).(\lambda (t1: T).(pr2 c0 (THead k t0 t) (THead k t1 t))))) (\lambda (c0: -C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr0 t1 t2)).(pr2_free c0 -(THead k t1 t) (THead k t2 t) (pr0_comp t1 t2 H0 t t (pr0_refl t) k)))))) -(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H0: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (H1: (pr0 t1 t2)).(\lambda (t0: T).(\lambda (H2: (subst0 i u t2 -t0)).(pr2_delta c0 d u i H0 (THead k t1 t) (THead k t2 t) (pr0_comp t1 t2 H1 -t t (pr0_refl t) k) (THead k t0 t) (subst0_fst u t0 t2 i H2 t k)))))))))))) c -u1 u2 H)))))). -(* COMMENTS -Initial nodes: 219 -END *) - -theorem pr2_head_2: - \forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).(\forall -(k: K).((pr2 (CHead c k u) t1 t2) \to (pr2 c (THead k u t1) (THead k u -t2))))))) -\def - \lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(k: K).(\lambda (H: (pr2 (CHead c k u) t1 t2)).(insert_eq C (CHead c k u) -(\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 c (THead k u t1) (THead -k u t2))) (\lambda (y: C).(\lambda (H0: (pr2 y t1 t2)).(pr2_ind (\lambda (c0: -C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CHead c k u)) \to (pr2 c -(THead k u t) (THead k u t0)))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda -(t4: T).(\lambda (H1: (pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c k -u))).(pr2_free c (THead k u t3) (THead k u t4) (pr0_comp u u (pr0_refl u) t3 -t4 H1 k))))))) (K_ind (\lambda (k0: K).(\forall (c0: C).(\forall (d: -C).(\forall (u0: T).(\forall (i: nat).((getl i c0 (CHead d (Bind Abbr) u0)) -\to (\forall (t3: T).(\forall (t4: T).((pr0 t3 t4) \to (\forall (t: -T).((subst0 i u0 t4 t) \to ((eq C c0 (CHead c k0 u)) \to (pr2 c (THead k0 u -t3) (THead k0 u t)))))))))))))) (\lambda (b: B).(\lambda (c0: C).(\lambda (d: -C).(\lambda (u0: T).(\lambda (i: nat).(nat_ind (\lambda (n: nat).((getl n c0 -(CHead d (Bind Abbr) u0)) \to (\forall (t3: T).(\forall (t4: T).((pr0 t3 t4) -\to (\forall (t: T).((subst0 n u0 t4 t) \to ((eq C c0 (CHead c (Bind b) u)) -\to (pr2 c (THead (Bind b) u t3) (THead (Bind b) u t)))))))))) (\lambda (H1: -(getl O c0 (CHead d (Bind Abbr) u0))).(\lambda (t3: T).(\lambda (t4: -T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: (subst0 O u0 t4 -t)).(\lambda (H4: (eq C c0 (CHead c (Bind b) u))).(let H5 \def (eq_ind C c0 -(\lambda (c1: C).(getl O c1 (CHead d (Bind Abbr) u0))) H1 (CHead c (Bind b) -u) H4) in (let H6 \def (f_equal C C (\lambda (e: C).(match e in C return -(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow -c1])) (CHead d (Bind Abbr) u0) (CHead c (Bind b) u) (clear_gen_bind b c -(CHead d (Bind Abbr) u0) u (getl_gen_O (CHead c (Bind b) u) (CHead d (Bind -Abbr) u0) H5))) in ((let H7 \def (f_equal C B (\lambda (e: C).(match e in C -return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _) -\Rightarrow (match k0 in K return (\lambda (_: K).B) with [(Bind b0) -\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u0) -(CHead c (Bind b) u) (clear_gen_bind b c (CHead d (Bind Abbr) u0) u -(getl_gen_O (CHead c (Bind b) u) (CHead d (Bind Abbr) u0) H5))) in ((let H8 -\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow u0 | (CHead _ _ t0) \Rightarrow t0])) (CHead d -(Bind Abbr) u0) (CHead c (Bind b) u) (clear_gen_bind b c (CHead d (Bind Abbr) -u0) u (getl_gen_O (CHead c (Bind b) u) (CHead d (Bind Abbr) u0) H5))) in -(\lambda (H9: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H11 \def (eq_ind T -u0 (\lambda (t0: T).(subst0 O t0 t4 t)) H3 u H8) in (eq_ind B Abbr (\lambda -(b0: B).(pr2 c (THead (Bind b0) u t3) (THead (Bind b0) u t))) (pr2_free c -(THead (Bind Abbr) u t3) (THead (Bind Abbr) u t) (pr0_delta u u (pr0_refl u) -t3 t4 H2 t H11)) b H9))))) H7)) H6)))))))))) (\lambda (n: nat).(\lambda (H1: -(((getl n c0 (CHead d (Bind Abbr) u0)) \to (\forall (t3: T).(\forall (t4: -T).((pr0 t3 t4) \to (\forall (t: T).((subst0 n u0 t4 t) \to ((eq C c0 (CHead -c (Bind b) u)) \to (pr2 c (THead (Bind b) u t3) (THead (Bind b) u -t))))))))))).(\lambda (H2: (getl (S n) c0 (CHead d (Bind Abbr) u0))).(\lambda -(t3: T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda (t: T).(\lambda -(H4: (subst0 (S n) u0 t4 t)).(\lambda (H5: (eq C c0 (CHead c (Bind b) -u))).(let H6 \def (eq_ind C c0 (\lambda (c1: C).(getl (S n) c1 (CHead d (Bind -Abbr) u0))) H2 (CHead c (Bind b) u) H5) in (let H7 \def (eq_ind C c0 (\lambda -(c1: C).((getl n c1 (CHead d (Bind Abbr) u0)) \to (\forall (t5: T).(\forall -(t6: T).((pr0 t5 t6) \to (\forall (t0: T).((subst0 n u0 t6 t0) \to ((eq C c1 -(CHead c (Bind b) u)) \to (pr2 c (THead (Bind b) u t5) (THead (Bind b) u -t0)))))))))) H1 (CHead c (Bind b) u) H5) in (pr2_delta c d u0 (r (Bind b) n) -(getl_gen_S (Bind b) c (CHead d (Bind Abbr) u0) u n H6) (THead (Bind b) u t3) -(THead (Bind b) u t4) (pr0_comp u u (pr0_refl u) t3 t4 H3 (Bind b)) (THead -(Bind b) u t) (subst0_snd (Bind b) u0 t t4 (r (Bind b) n) H4 u))))))))))))) -i)))))) (\lambda (f: F).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: -T).(\lambda (i: nat).(nat_ind (\lambda (n: nat).((getl n c0 (CHead d (Bind -Abbr) u0)) \to (\forall (t3: T).(\forall (t4: T).((pr0 t3 t4) \to (\forall -(t: T).((subst0 n u0 t4 t) \to ((eq C c0 (CHead c (Flat f) u)) \to (pr2 c -(THead (Flat f) u t3) (THead (Flat f) u t)))))))))) (\lambda (H1: (getl O c0 -(CHead d (Bind Abbr) u0))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: -(pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: (subst0 O u0 t4 t)).(\lambda (H4: -(eq C c0 (CHead c (Flat f) u))).(let H5 \def (eq_ind C c0 (\lambda (c1: -C).(getl O c1 (CHead d (Bind Abbr) u0))) H1 (CHead c (Flat f) u) H4) in -(pr2_delta c d u0 O (getl_intro O c (CHead d (Bind Abbr) u0) c (drop_refl c) -(clear_gen_flat f c (CHead d (Bind Abbr) u0) u (getl_gen_O (CHead c (Flat f) -u) (CHead d (Bind Abbr) u0) H5))) (THead (Flat f) u t3) (THead (Flat f) u t4) -(pr0_comp u u (pr0_refl u) t3 t4 H2 (Flat f)) (THead (Flat f) u t) -(subst0_snd (Flat f) u0 t t4 O H3 u)))))))))) (\lambda (n: nat).(\lambda (H1: -(((getl n c0 (CHead d (Bind Abbr) u0)) \to (\forall (t3: T).(\forall (t4: -T).((pr0 t3 t4) \to (\forall (t: T).((subst0 n u0 t4 t) \to ((eq C c0 (CHead -c (Flat f) u)) \to (pr2 c (THead (Flat f) u t3) (THead (Flat f) u -t))))))))))).(\lambda (H2: (getl (S n) c0 (CHead d (Bind Abbr) u0))).(\lambda -(t3: T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda (t: T).(\lambda -(H4: (subst0 (S n) u0 t4 t)).(\lambda (H5: (eq C c0 (CHead c (Flat f) -u))).(let H6 \def (eq_ind C c0 (\lambda (c1: C).(getl (S n) c1 (CHead d (Bind -Abbr) u0))) H2 (CHead c (Flat f) u) H5) in (let H7 \def (eq_ind C c0 (\lambda -(c1: C).((getl n c1 (CHead d (Bind Abbr) u0)) \to (\forall (t5: T).(\forall -(t6: T).((pr0 t5 t6) \to (\forall (t0: T).((subst0 n u0 t6 t0) \to ((eq C c1 -(CHead c (Flat f) u)) \to (pr2 c (THead (Flat f) u t5) (THead (Flat f) u -t0)))))))))) H1 (CHead c (Flat f) u) H5) in (pr2_delta c d u0 (r (Flat f) n) -(getl_gen_S (Flat f) c (CHead d (Bind Abbr) u0) u n H6) (THead (Flat f) u t3) -(THead (Flat f) u t4) (pr0_comp u u (pr0_refl u) t3 t4 H3 (Flat f)) (THead -(Flat f) u t) (subst0_snd (Flat f) u0 t t4 (r (Flat f) n) H4 u))))))))))))) -i)))))) k) y t1 t2 H0))) H)))))). -(* COMMENTS -Initial nodes: 1947 -END *) - -theorem clear_pr2_trans: - \forall (c2: C).(\forall (t1: T).(\forall (t2: T).((pr2 c2 t1 t2) \to -(\forall (c1: C).((clear c1 c2) \to (pr2 c1 t1 t2)))))) -\def - \lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c2 t1 -t2)).(pr2_ind (\lambda (c: C).(\lambda (t: T).(\lambda (t0: T).(\forall (c1: -C).((clear c1 c) \to (pr2 c1 t t0)))))) (\lambda (c: C).(\lambda (t3: -T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(\lambda (c1: C).(\lambda (_: -(clear c1 c)).(pr2_free c1 t3 t4 H0))))))) (\lambda (c: C).(\lambda (d: -C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d (Bind -Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 -t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (c1: -C).(\lambda (H3: (clear c1 c)).(pr2_delta c1 d u i (clear_getl_trans i c -(CHead d (Bind Abbr) u) H0 c1 H3) t3 t4 H1 t H2))))))))))))) c2 t1 t2 H)))). -(* COMMENTS -Initial nodes: 171 -END *) - -theorem pr2_cflat: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall -(f: F).(\forall (v: T).(pr2 (CHead c (Flat f) v) t1 t2)))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 -t2)).(\lambda (f: F).(\lambda (v: T).(pr2_ind (\lambda (c0: C).(\lambda (t: -T).(\lambda (t0: T).(pr2 (CHead c0 (Flat f) v) t t0)))) (\lambda (c0: -C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(pr2_free -(CHead c0 (Flat f) v) t3 t4 H0))))) (\lambda (c0: C).(\lambda (d: C).(\lambda -(u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr) -u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 t4)).(\lambda -(t: T).(\lambda (H2: (subst0 i u t4 t)).(pr2_delta (CHead c0 (Flat f) v) d u -i (getl_flat c0 (CHead d (Bind Abbr) u) i H0 f v) t3 t4 H1 t H2))))))))))) c -t1 t2 H)))))). -(* COMMENTS -Initial nodes: 175 -END *) - -theorem pr2_ctail: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall -(k: K).(\forall (u: T).(pr2 (CTail k u c) t1 t2)))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 -t2)).(\lambda (k: K).(\lambda (u: T).(pr2_ind (\lambda (c0: C).(\lambda (t: -T).(\lambda (t0: T).(pr2 (CTail k u c0) t t0)))) (\lambda (c0: C).(\lambda -(t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(pr2_free (CTail k u c0) -t3 t4 H0))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: -nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr) u0))).(\lambda (t3: -T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H2: -(subst0 i u0 t4 t)).(pr2_delta (CTail k u c0) (CTail k u d) u0 i (getl_ctail -Abbr c0 d u0 i H0 k u) t3 t4 H1 t H2))))))))))) c t1 t2 H)))))). -(* COMMENTS -Initial nodes: 171 -END *) - -theorem pr2_change: - \forall (b: B).((not (eq B b Abbr)) \to (\forall (c: C).(\forall (v1: -T).(\forall (t1: T).(\forall (t2: T).((pr2 (CHead c (Bind b) v1) t1 t2) \to -(\forall (v2: T).(pr2 (CHead c (Bind b) v2) t1 t2)))))))) -\def - \lambda (b: B).(\lambda (H: (not (eq B b Abbr))).(\lambda (c: C).(\lambda -(v1: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr2 (CHead c (Bind -b) v1) t1 t2)).(\lambda (v2: T).(insert_eq C (CHead c (Bind b) v1) (\lambda -(c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 (CHead c (Bind b) v2) t1 t2)) -(\lambda (y: C).(\lambda (H1: (pr2 y t1 t2)).(pr2_ind (\lambda (c0: -C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CHead c (Bind b) v1)) \to (pr2 -(CHead c (Bind b) v2) t t0))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda -(t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c (Bind b) -v1))).(pr2_free (CHead c (Bind b) v2) t3 t4 H2)))))) (\lambda (c0: -C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H2: (getl i c0 -(CHead d (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H3: -(pr0 t3 t4)).(\lambda (t: T).(\lambda (H4: (subst0 i u t4 t)).(\lambda (H5: -(eq C c0 (CHead c (Bind b) v1))).(let H6 \def (eq_ind C c0 (\lambda (c1: -C).(getl i c1 (CHead d (Bind Abbr) u))) H2 (CHead c (Bind b) v1) H5) in -(nat_ind (\lambda (n: nat).((getl n (CHead c (Bind b) v1) (CHead d (Bind -Abbr) u)) \to ((subst0 n u t4 t) \to (pr2 (CHead c (Bind b) v2) t3 t)))) -(\lambda (H7: (getl O (CHead c (Bind b) v1) (CHead d (Bind Abbr) -u))).(\lambda (H8: (subst0 O u t4 t)).(let H9 \def (f_equal C C (\lambda (e: -C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | -(CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u) (CHead c (Bind b) -v1) (clear_gen_bind b c (CHead d (Bind Abbr) u) v1 (getl_gen_O (CHead c (Bind -b) v1) (CHead d (Bind Abbr) u) H7))) in ((let H10 \def (f_equal C B (\lambda -(e: C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow -Abbr | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with -[(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind -Abbr) u) (CHead c (Bind b) v1) (clear_gen_bind b c (CHead d (Bind Abbr) u) v1 -(getl_gen_O (CHead c (Bind b) v1) (CHead d (Bind Abbr) u) H7))) in ((let H11 -\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d -(Bind Abbr) u) (CHead c (Bind b) v1) (clear_gen_bind b c (CHead d (Bind Abbr) -u) v1 (getl_gen_O (CHead c (Bind b) v1) (CHead d (Bind Abbr) u) H7))) in -(\lambda (H12: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H14 \def (eq_ind -T u (\lambda (t0: T).(subst0 O t0 t4 t)) H8 v1 H11) in (let H15 \def -(eq_ind_r B b (\lambda (b0: B).(not (eq B b0 Abbr))) H Abbr H12) in (eq_ind B -Abbr (\lambda (b0: B).(pr2 (CHead c (Bind b0) v2) t3 t)) (let H16 \def (match -(H15 (refl_equal B Abbr)) in False return (\lambda (_: False).(pr2 (CHead c -(Bind Abbr) v2) t3 t)) with []) in H16) b H12)))))) H10)) H9)))) (\lambda -(i0: nat).(\lambda (_: (((getl i0 (CHead c (Bind b) v1) (CHead d (Bind Abbr) -u)) \to ((subst0 i0 u t4 t) \to (pr2 (CHead c (Bind b) v2) t3 t))))).(\lambda -(H7: (getl (S i0) (CHead c (Bind b) v1) (CHead d (Bind Abbr) u))).(\lambda -(H8: (subst0 (S i0) u t4 t)).(pr2_delta (CHead c (Bind b) v2) d u (S i0) -(getl_head (Bind b) i0 c (CHead d (Bind Abbr) u) (getl_gen_S (Bind b) c -(CHead d (Bind Abbr) u) v1 i0 H7) v2) t3 t4 H3 t H8))))) i H6 H4))))))))))))) -y t1 t2 H1))) H0)))))))). -(* COMMENTS -Initial nodes: 913 -END *) - -theorem pr2_lift: - \forall (c: C).(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h -d c e) \to (\forall (t1: T).(\forall (t2: T).((pr2 e t1 t2) \to (pr2 c (lift -h d t1) (lift h d t2))))))))) -\def - \lambda (c: C).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda -(H: (drop h d c e)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr2 e t1 -t2)).(insert_eq C e (\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 c -(lift h d t1) (lift h d t2))) (\lambda (y: C).(\lambda (H1: (pr2 y t1 -t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 e) -\to (pr2 c (lift h d t) (lift h d t0)))))) (\lambda (c0: C).(\lambda (t3: -T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (_: (eq C c0 -e)).(pr2_free c (lift h d t3) (lift h d t4) (pr0_lift t3 t4 H2 h d))))))) -(\lambda (c0: C).(\lambda (d0: C).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H2: (getl i c0 (CHead d0 (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: -T).(\lambda (H3: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H4: (subst0 i u t4 -t)).(\lambda (H5: (eq C c0 e)).(let H6 \def (eq_ind C c0 (\lambda (c1: -C).(getl i c1 (CHead d0 (Bind Abbr) u))) H2 e H5) in (lt_le_e i d (pr2 c -(lift h d t3) (lift h d t)) (\lambda (H7: (lt i d)).(let H8 \def -(drop_getl_trans_le i d (le_S_n i d (le_S (S i) d H7)) c e h H (CHead d0 -(Bind Abbr) u) H6) in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop i -O c e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d i) e0 e1))) -(\lambda (_: C).(\lambda (e1: C).(clear e1 (CHead d0 (Bind Abbr) u)))) (pr2 c -(lift h d t3) (lift h d t)) (\lambda (x0: C).(\lambda (x1: C).(\lambda (H9: -(drop i O c x0)).(\lambda (H10: (drop h (minus d i) x0 x1)).(\lambda (H11: -(clear x1 (CHead d0 (Bind Abbr) u))).(let H12 \def (eq_ind nat (minus d i) -(\lambda (n: nat).(drop h n x0 x1)) H10 (S (minus d (S i))) (minus_x_Sy d i -H7)) in (let H13 \def (drop_clear_S x1 x0 h (minus d (S i)) H12 Abbr d0 u -H11) in (ex2_ind C (\lambda (c1: C).(clear x0 (CHead c1 (Bind Abbr) (lift h -(minus d (S i)) u)))) (\lambda (c1: C).(drop h (minus d (S i)) c1 d0)) (pr2 c -(lift h d t3) (lift h d t)) (\lambda (x: C).(\lambda (H14: (clear x0 (CHead x -(Bind Abbr) (lift h (minus d (S i)) u)))).(\lambda (_: (drop h (minus d (S -i)) x d0)).(pr2_delta c x (lift h (minus d (S i)) u) i (getl_intro i c (CHead -x (Bind Abbr) (lift h (minus d (S i)) u)) x0 H9 H14) (lift h d t3) (lift h d -t4) (pr0_lift t3 t4 H3 h d) (lift h d t) (subst0_lift_lt t4 t u i H4 d H7 -h))))) H13)))))))) H8))) (\lambda (H7: (le d i)).(pr2_delta c d0 u (plus i h) -(drop_getl_trans_ge i c e d h H (CHead d0 (Bind Abbr) u) H6 H7) (lift h d t3) -(lift h d t4) (pr0_lift t3 t4 H3 h d) (lift h d t) (subst0_lift_ge t4 t u i h -H4 d H7)))))))))))))))) y t1 t2 H1))) H0)))))))). -(* COMMENTS -Initial nodes: 849 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr2/subst1.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr2/subst1.ma deleted file mode 100644 index 47704dd52..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr2/subst1.ma +++ /dev/null @@ -1,281 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr2/defs.ma". - -include "Basic-1/pr0/subst1.ma". - -include "Basic-1/pr0/fwd.ma". - -include "Basic-1/csubst1/getl.ma". - -include "Basic-1/csubst1/fwd.ma". - -include "Basic-1/subst1/subst1.ma". - -include "Basic-1/getl/drop.ma". - -theorem pr2_delta1: - \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c -(CHead d (Bind Abbr) u)) \to (\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) -\to (\forall (t: T).((subst1 i u t2 t) \to (pr2 c t1 t)))))))))) -\def - \lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H: (getl i c (CHead d (Bind Abbr) u))).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (H0: (pr0 t1 t2)).(\lambda (t: T).(\lambda (H1: (subst1 i u t2 -t)).(subst1_ind i u t2 (\lambda (t0: T).(pr2 c t1 t0)) (pr2_free c t1 t2 H0) -(\lambda (t0: T).(\lambda (H2: (subst0 i u t2 t0)).(pr2_delta c d u i H t1 t2 -H0 t0 H2))) t H1)))))))))). -(* COMMENTS -Initial nodes: 111 -END *) - -theorem pr2_subst1: - \forall (c: C).(\forall (e: C).(\forall (v: T).(\forall (i: nat).((getl i c -(CHead e (Bind Abbr) v)) \to (\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) -\to (\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T (\lambda (w2: T).(pr2 c -w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2)))))))))))) -\def - \lambda (c: C).(\lambda (e: C).(\lambda (v: T).(\lambda (i: nat).(\lambda -(H: (getl i c (CHead e (Bind Abbr) v))).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (H0: (pr2 c t1 t2)).(insert_eq C c (\lambda (c0: C).(pr2 c0 t1 -t2)) (\lambda (c0: C).(\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T -(\lambda (w2: T).(pr2 c0 w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2)))))) -(\lambda (y: C).(\lambda (H1: (pr2 y t1 t2)).(pr2_ind (\lambda (c0: -C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 c) \to (\forall (w1: -T).((subst1 i v t w1) \to (ex2 T (\lambda (w2: T).(pr2 c0 w1 w2)) (\lambda -(w2: T).(subst1 i v t0 w2))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda -(t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (H3: (eq C c0 c)).(\lambda (w1: -T).(\lambda (H4: (subst1 i v t3 w1)).(eq_ind_r C c (\lambda (c1: C).(ex2 T -(\lambda (w2: T).(pr2 c1 w1 w2)) (\lambda (w2: T).(subst1 i v t4 w2)))) -(ex2_ind T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst1 i v t4 w2)) -(ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t4 w2))) -(\lambda (x: T).(\lambda (H5: (pr0 w1 x)).(\lambda (H6: (subst1 i v t4 -x)).(ex_intro2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v -t4 w2)) x (pr2_free c w1 x H5) H6)))) (pr0_subst1 t3 t4 H2 v w1 i H4 v -(pr0_refl v))) c0 H3)))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: -T).(\lambda (i0: nat).(\lambda (H2: (getl i0 c0 (CHead d (Bind Abbr) -u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda -(t: T).(\lambda (H4: (subst0 i0 u t4 t)).(\lambda (H5: (eq C c0 c)).(\lambda -(w1: T).(\lambda (H6: (subst1 i v t3 w1)).(let H7 \def (eq_ind C c0 (\lambda -(c1: C).(getl i0 c1 (CHead d (Bind Abbr) u))) H2 c H5) in (eq_ind_r C c -(\lambda (c1: C).(ex2 T (\lambda (w2: T).(pr2 c1 w1 w2)) (\lambda (w2: -T).(subst1 i v t w2)))) (ex2_ind T (\lambda (w2: T).(pr0 w1 w2)) (\lambda -(w2: T).(subst1 i v t4 w2)) (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda -(w2: T).(subst1 i v t w2))) (\lambda (x: T).(\lambda (H8: (pr0 w1 -x)).(\lambda (H9: (subst1 i v t4 x)).(neq_eq_e i i0 (ex2 T (\lambda (w2: -T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t w2))) (\lambda (H10: (not -(eq nat i i0))).(ex2_ind T (\lambda (t0: T).(subst1 i v t t0)) (\lambda (t0: -T).(subst1 i0 u x t0)) (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: -T).(subst1 i v t w2))) (\lambda (x0: T).(\lambda (H11: (subst1 i v t -x0)).(\lambda (H12: (subst1 i0 u x x0)).(ex_intro2 T (\lambda (w2: T).(pr2 c -w1 w2)) (\lambda (w2: T).(subst1 i v t w2)) x0 (pr2_delta1 c d u i0 H7 w1 x -H8 x0 H12) H11)))) (subst1_confluence_neq t4 t u i0 (subst1_single i0 u t4 t -H4) x v i H9 (sym_not_eq nat i i0 H10)))) (\lambda (H10: (eq nat i i0)).(let -H11 \def (eq_ind_r nat i0 (\lambda (n: nat).(subst0 n u t4 t)) H4 i H10) in -(let H12 \def (eq_ind_r nat i0 (\lambda (n: nat).(getl n c (CHead d (Bind -Abbr) u))) H7 i H10) in (let H13 \def (eq_ind C (CHead e (Bind Abbr) v) -(\lambda (c1: C).(getl i c c1)) H (CHead d (Bind Abbr) u) (getl_mono c (CHead -e (Bind Abbr) v) i H (CHead d (Bind Abbr) u) H12)) in (let H14 \def (f_equal -C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow e | (CHead c1 _ _) \Rightarrow c1])) (CHead e (Bind Abbr) v) -(CHead d (Bind Abbr) u) (getl_mono c (CHead e (Bind Abbr) v) i H (CHead d -(Bind Abbr) u) H12)) in ((let H15 \def (f_equal C T (\lambda (e0: C).(match -e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow v | (CHead _ _ -t0) \Rightarrow t0])) (CHead e (Bind Abbr) v) (CHead d (Bind Abbr) u) -(getl_mono c (CHead e (Bind Abbr) v) i H (CHead d (Bind Abbr) u) H12)) in -(\lambda (H16: (eq C e d)).(let H17 \def (eq_ind_r T u (\lambda (t0: T).(getl -i c (CHead d (Bind Abbr) t0))) H13 v H15) in (let H18 \def (eq_ind_r T u -(\lambda (t0: T).(subst0 i t0 t4 t)) H11 v H15) in (let H19 \def (eq_ind_r C -d (\lambda (c1: C).(getl i c (CHead c1 (Bind Abbr) v))) H17 e H16) in -(ex2_ind T (\lambda (t0: T).(subst1 i v t t0)) (\lambda (t0: T).(subst1 i v x -t0)) (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t -w2))) (\lambda (x0: T).(\lambda (H20: (subst1 i v t x0)).(\lambda (H21: -(subst1 i v x x0)).(ex_intro2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: -T).(subst1 i v t w2)) x0 (pr2_delta1 c e v i H19 w1 x H8 x0 H21) H20)))) -(subst1_confluence_eq t4 t v i (subst1_single i v t4 t H18) x H9))))))) -H14)))))))))) (pr0_subst1 t3 t4 H3 v w1 i H6 v (pr0_refl v))) c0 -H5))))))))))))))) y t1 t2 H1))) H0)))))))). -(* COMMENTS -Initial nodes: 1311 -END *) - -theorem pr2_gen_cabbr: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall -(e: C).(\forall (u: T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) -\to (\forall (a0: C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d -a0 a) \to (\forall (x1: T).((subst1 d u t1 (lift (S O) d x1)) \to (ex2 T -(\lambda (x2: T).(subst1 d u t2 (lift (S O) d x2))) (\lambda (x2: T).(pr2 a -x1 x2)))))))))))))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 -t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (e: -C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u)) \to -(\forall (a0: C).((csubst1 d u c0 a0) \to (\forall (a: C).((drop (S O) d a0 -a) \to (\forall (x1: T).((subst1 d u t (lift (S O) d x1)) \to (ex2 T (\lambda -(x2: T).(subst1 d u t0 (lift (S O) d x2))) (\lambda (x2: T).(pr2 a x1 -x2)))))))))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: -T).(\lambda (H0: (pr0 t3 t4)).(\lambda (e: C).(\lambda (u: T).(\lambda (d: -nat).(\lambda (_: (getl d c0 (CHead e (Bind Abbr) u))).(\lambda (a0: -C).(\lambda (_: (csubst1 d u c0 a0)).(\lambda (a: C).(\lambda (_: (drop (S O) -d a0 a)).(\lambda (x1: T).(\lambda (H4: (subst1 d u t3 (lift (S O) d -x1))).(ex2_ind T (\lambda (w2: T).(pr0 (lift (S O) d x1) w2)) (\lambda (w2: -T).(subst1 d u t4 w2)) (ex2 T (\lambda (x2: T).(subst1 d u t4 (lift (S O) d -x2))) (\lambda (x2: T).(pr2 a x1 x2))) (\lambda (x: T).(\lambda (H5: (pr0 -(lift (S O) d x1) x)).(\lambda (H6: (subst1 d u t4 x)).(ex2_ind T (\lambda -(t5: T).(eq T x (lift (S O) d t5))) (\lambda (t5: T).(pr0 x1 t5)) (ex2 T -(\lambda (x2: T).(subst1 d u t4 (lift (S O) d x2))) (\lambda (x2: T).(pr2 a -x1 x2))) (\lambda (x0: T).(\lambda (H7: (eq T x (lift (S O) d x0))).(\lambda -(H8: (pr0 x1 x0)).(let H9 \def (eq_ind T x (\lambda (t: T).(subst1 d u t4 t)) -H6 (lift (S O) d x0) H7) in (ex_intro2 T (\lambda (x2: T).(subst1 d u t4 -(lift (S O) d x2))) (\lambda (x2: T).(pr2 a x1 x2)) x0 H9 (pr2_free a x1 x0 -H8)))))) (pr0_gen_lift x1 x (S O) d H5))))) (pr0_subst1 t3 t4 H0 u (lift (S -O) d x1) d H4 u (pr0_refl u))))))))))))))))) (\lambda (c0: C).(\lambda (d: -C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind -Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 -t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (e: -C).(\lambda (u0: T).(\lambda (d0: nat).(\lambda (H3: (getl d0 c0 (CHead e -(Bind Abbr) u0))).(\lambda (a0: C).(\lambda (H4: (csubst1 d0 u0 c0 -a0)).(\lambda (a: C).(\lambda (H5: (drop (S O) d0 a0 a)).(\lambda (x1: -T).(\lambda (H6: (subst1 d0 u0 t3 (lift (S O) d0 x1))).(ex2_ind T (\lambda -(w2: T).(pr0 (lift (S O) d0 x1) w2)) (\lambda (w2: T).(subst1 d0 u0 t4 w2)) -(ex2 T (\lambda (x2: T).(subst1 d0 u0 t (lift (S O) d0 x2))) (\lambda (x2: -T).(pr2 a x1 x2))) (\lambda (x: T).(\lambda (H7: (pr0 (lift (S O) d0 x1) -x)).(\lambda (H8: (subst1 d0 u0 t4 x)).(ex2_ind T (\lambda (t5: T).(eq T x -(lift (S O) d0 t5))) (\lambda (t5: T).(pr0 x1 t5)) (ex2 T (\lambda (x2: -T).(subst1 d0 u0 t (lift (S O) d0 x2))) (\lambda (x2: T).(pr2 a x1 x2))) -(\lambda (x0: T).(\lambda (H9: (eq T x (lift (S O) d0 x0))).(\lambda (H10: -(pr0 x1 x0)).(let H11 \def (eq_ind T x (\lambda (t0: T).(subst1 d0 u0 t4 t0)) -H8 (lift (S O) d0 x0) H9) in (lt_eq_gt_e i d0 (ex2 T (\lambda (x2: T).(subst1 -d0 u0 t (lift (S O) d0 x2))) (\lambda (x2: T).(pr2 a x1 x2))) (\lambda (H12: -(lt i d0)).(ex2_ind T (\lambda (t0: T).(subst1 d0 u0 t t0)) (\lambda (t0: -T).(subst1 i u (lift (S O) d0 x0) t0)) (ex2 T (\lambda (x2: T).(subst1 d0 u0 -t (lift (S O) d0 x2))) (\lambda (x2: T).(pr2 a x1 x2))) (\lambda (x2: -T).(\lambda (H13: (subst1 d0 u0 t x2)).(\lambda (H14: (subst1 i u (lift (S O) -d0 x0) x2)).(ex2_ind C (\lambda (e2: C).(csubst1 (minus d0 i) u0 (CHead d -(Bind Abbr) u) e2)) (\lambda (e2: C).(getl i a0 e2)) (ex2 T (\lambda (x3: -T).(subst1 d0 u0 t (lift (S O) d0 x3))) (\lambda (x3: T).(pr2 a x1 x3))) -(\lambda (x3: C).(\lambda (H15: (csubst1 (minus d0 i) u0 (CHead d (Bind Abbr) -u) x3)).(\lambda (H16: (getl i a0 x3)).(let H17 \def (eq_ind nat (minus d0 i) -(\lambda (n: nat).(csubst1 n u0 (CHead d (Bind Abbr) u) x3)) H15 (S (minus d0 -(S i))) (minus_x_Sy d0 i H12)) in (let H18 \def (csubst1_gen_head (Bind Abbr) -d x3 u u0 (minus d0 (S i)) H17) in (ex3_2_ind T C (\lambda (u2: T).(\lambda -(c2: C).(eq C x3 (CHead c2 (Bind Abbr) u2)))) (\lambda (u2: T).(\lambda (_: -C).(subst1 (minus d0 (S i)) u0 u u2))) (\lambda (_: T).(\lambda (c2: -C).(csubst1 (minus d0 (S i)) u0 d c2))) (ex2 T (\lambda (x4: T).(subst1 d0 u0 -t (lift (S O) d0 x4))) (\lambda (x4: T).(pr2 a x1 x4))) (\lambda (x4: -T).(\lambda (x5: C).(\lambda (H19: (eq C x3 (CHead x5 (Bind Abbr) -x4))).(\lambda (H20: (subst1 (minus d0 (S i)) u0 u x4)).(\lambda (_: (csubst1 -(minus d0 (S i)) u0 d x5)).(let H22 \def (eq_ind C x3 (\lambda (c1: C).(getl -i a0 c1)) H16 (CHead x5 (Bind Abbr) x4) H19) in (let H23 \def (eq_ind nat d0 -(\lambda (n: nat).(drop (S O) n a0 a)) H5 (S (plus i (minus d0 (S i)))) -(lt_plus_minus i d0 H12)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: -C).(eq T x4 (lift (S O) (minus d0 (S i)) v)))) (\lambda (v: T).(\lambda (e0: -C).(getl i a (CHead e0 (Bind Abbr) v)))) (\lambda (_: T).(\lambda (e0: -C).(drop (S O) (minus d0 (S i)) x5 e0))) (ex2 T (\lambda (x6: T).(subst1 d0 -u0 t (lift (S O) d0 x6))) (\lambda (x6: T).(pr2 a x1 x6))) (\lambda (x6: -T).(\lambda (x7: C).(\lambda (H24: (eq T x4 (lift (S O) (minus d0 (S i)) -x6))).(\lambda (H25: (getl i a (CHead x7 (Bind Abbr) x6))).(\lambda (_: (drop -(S O) (minus d0 (S i)) x5 x7)).(let H27 \def (eq_ind T x4 (\lambda (t0: -T).(subst1 (minus d0 (S i)) u0 u t0)) H20 (lift (S O) (minus d0 (S i)) x6) -H24) in (ex2_ind T (\lambda (t0: T).(subst1 i (lift (S O) (minus d0 (S i)) -x6) (lift (S O) d0 x0) t0)) (\lambda (t0: T).(subst1 (S (plus (minus d0 (S -i)) i)) u0 x2 t0)) (ex2 T (\lambda (x8: T).(subst1 d0 u0 t (lift (S O) d0 -x8))) (\lambda (x8: T).(pr2 a x1 x8))) (\lambda (x8: T).(\lambda (H28: -(subst1 i (lift (S O) (minus d0 (S i)) x6) (lift (S O) d0 x0) x8)).(\lambda -(H29: (subst1 (S (plus (minus d0 (S i)) i)) u0 x2 x8)).(let H30 \def (eq_ind -nat d0 (\lambda (n: nat).(subst1 i (lift (S O) (minus d0 (S i)) x6) (lift (S -O) n x0) x8)) H28 (S (plus i (minus d0 (S i)))) (lt_plus_minus i d0 H12)) in -(ex2_ind T (\lambda (t5: T).(eq T x8 (lift (S O) (S (plus i (minus d0 (S -i)))) t5))) (\lambda (t5: T).(subst1 i x6 x0 t5)) (ex2 T (\lambda (x9: -T).(subst1 d0 u0 t (lift (S O) d0 x9))) (\lambda (x9: T).(pr2 a x1 x9))) -(\lambda (x9: T).(\lambda (H31: (eq T x8 (lift (S O) (S (plus i (minus d0 (S -i)))) x9))).(\lambda (H32: (subst1 i x6 x0 x9)).(let H33 \def (eq_ind T x8 -(\lambda (t0: T).(subst1 (S (plus (minus d0 (S i)) i)) u0 x2 t0)) H29 (lift -(S O) (S (plus i (minus d0 (S i)))) x9) H31) in (let H34 \def (eq_ind_r nat -(S (plus i (minus d0 (S i)))) (\lambda (n: nat).(subst1 (S (plus (minus d0 (S -i)) i)) u0 x2 (lift (S O) n x9))) H33 d0 (lt_plus_minus i d0 H12)) in (let -H35 \def (eq_ind_r nat (S (plus (minus d0 (S i)) i)) (\lambda (n: -nat).(subst1 n u0 x2 (lift (S O) d0 x9))) H34 d0 (lt_plus_minus_r i d0 H12)) -in (ex_intro2 T (\lambda (x10: T).(subst1 d0 u0 t (lift (S O) d0 x10))) -(\lambda (x10: T).(pr2 a x1 x10)) x9 (subst1_trans x2 t u0 d0 H13 (lift (S O) -d0 x9) H35) (pr2_delta1 a x7 x6 i H25 x1 x0 H10 x9 H32)))))))) -(subst1_gen_lift_lt x6 x0 x8 i (S O) (minus d0 (S i)) H30)))))) -(subst1_subst1_back (lift (S O) d0 x0) x2 u i H14 (lift (S O) (minus d0 (S -i)) x6) u0 (minus d0 (S i)) H27)))))))) (getl_drop_conf_lt Abbr a0 x5 x4 i -H22 a (S O) (minus d0 (S i)) H23))))))))) H18)))))) (csubst1_getl_lt d0 i H12 -c0 a0 u0 H4 (CHead d (Bind Abbr) u) H0))))) (subst1_confluence_neq t4 t u i -(subst1_single i u t4 t H2) (lift (S O) d0 x0) u0 d0 H11 (lt_neq i d0 H12)))) -(\lambda (H12: (eq nat i d0)).(let H13 \def (eq_ind_r nat d0 (\lambda (n: -nat).(subst1 n u0 t4 (lift (S O) n x0))) H11 i H12) in (let H14 \def -(eq_ind_r nat d0 (\lambda (n: nat).(drop (S O) n a0 a)) H5 i H12) in (let H15 -\def (eq_ind_r nat d0 (\lambda (n: nat).(csubst1 n u0 c0 a0)) H4 i H12) in -(let H16 \def (eq_ind_r nat d0 (\lambda (n: nat).(getl n c0 (CHead e (Bind -Abbr) u0))) H3 i H12) in (eq_ind nat i (\lambda (n: nat).(ex2 T (\lambda (x2: -T).(subst1 n u0 t (lift (S O) n x2))) (\lambda (x2: T).(pr2 a x1 x2)))) (let -H17 \def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c1: C).(getl i c0 c1)) -H0 (CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead -e (Bind Abbr) u0) H16)) in (let H18 \def (f_equal C C (\lambda (e0: C).(match -e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ -_) \Rightarrow c1])) (CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0) -(getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead e (Bind Abbr) u0) H16)) in -((let H19 \def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda -(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) -(CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d (Bind -Abbr) u) i H0 (CHead e (Bind Abbr) u0) H16)) in (\lambda (H20: (eq C d -e)).(let H21 \def (eq_ind_r T u0 (\lambda (t0: T).(getl i c0 (CHead e (Bind -Abbr) t0))) H17 u H19) in (let H22 \def (eq_ind_r T u0 (\lambda (t0: -T).(subst1 i t0 t4 (lift (S O) i x0))) H13 u H19) in (let H23 \def (eq_ind_r -T u0 (\lambda (t0: T).(csubst1 i t0 c0 a0)) H15 u H19) in (eq_ind T u -(\lambda (t0: T).(ex2 T (\lambda (x2: T).(subst1 i t0 t (lift (S O) i x2))) -(\lambda (x2: T).(pr2 a x1 x2)))) (let H24 \def (eq_ind_r C e (\lambda (c1: -C).(getl i c0 (CHead c1 (Bind Abbr) u))) H21 d H20) in (ex2_ind T (\lambda -(t0: T).(subst1 i u t t0)) (\lambda (t0: T).(subst1 i u (lift (S O) i x0) -t0)) (ex2 T (\lambda (x2: T).(subst1 i u t (lift (S O) i x2))) (\lambda (x2: -T).(pr2 a x1 x2))) (\lambda (x2: T).(\lambda (H25: (subst1 i u t -x2)).(\lambda (H26: (subst1 i u (lift (S O) i x0) x2)).(let H27 \def (eq_ind -T x2 (\lambda (t0: T).(subst1 i u t t0)) H25 (lift (S O) i x0) -(subst1_gen_lift_eq x0 u x2 (S O) i i (le_n i) (eq_ind_r nat (plus (S O) i) -(\lambda (n: nat).(lt i n)) (le_n (plus (S O) i)) (plus i (S O)) (plus_sym i -(S O))) H26)) in (ex_intro2 T (\lambda (x3: T).(subst1 i u t (lift (S O) i -x3))) (\lambda (x3: T).(pr2 a x1 x3)) x0 H27 (pr2_free a x1 x0 H10)))))) -(subst1_confluence_eq t4 t u i (subst1_single i u t4 t H2) (lift (S O) i x0) -H22))) u0 H19)))))) H18))) d0 H12)))))) (\lambda (H12: (lt d0 i)).(ex2_ind T -(\lambda (t0: T).(subst1 d0 u0 t t0)) (\lambda (t0: T).(subst1 i u (lift (S -O) d0 x0) t0)) (ex2 T (\lambda (x2: T).(subst1 d0 u0 t (lift (S O) d0 x2))) -(\lambda (x2: T).(pr2 a x1 x2))) (\lambda (x2: T).(\lambda (H13: (subst1 d0 -u0 t x2)).(\lambda (H14: (subst1 i u (lift (S O) d0 x0) x2)).(ex2_ind T -(\lambda (t5: T).(eq T x2 (lift (S O) d0 t5))) (\lambda (t5: T).(subst1 -(minus i (S O)) u x0 t5)) (ex2 T (\lambda (x3: T).(subst1 d0 u0 t (lift (S O) -d0 x3))) (\lambda (x3: T).(pr2 a x1 x3))) (\lambda (x3: T).(\lambda (H15: (eq -T x2 (lift (S O) d0 x3))).(\lambda (H16: (subst1 (minus i (S O)) u x0 -x3)).(let H17 \def (eq_ind T x2 (\lambda (t0: T).(subst1 d0 u0 t t0)) H13 -(lift (S O) d0 x3) H15) in (ex_intro2 T (\lambda (x4: T).(subst1 d0 u0 t -(lift (S O) d0 x4))) (\lambda (x4: T).(pr2 a x1 x4)) x3 H17 (pr2_delta1 a d u -(minus i (S O)) (getl_drop_conf_ge i (CHead d (Bind Abbr) u) a0 -(csubst1_getl_ge d0 i (le_S_n d0 i (le_S (S d0) i H12)) c0 a0 u0 H4 (CHead d -(Bind Abbr) u) H0) a (S O) d0 H5 (eq_ind_r nat (plus (S O) d0) (\lambda (n: -nat).(le n i)) H12 (plus d0 (S O)) (plus_sym d0 (S O)))) x1 x0 H10 x3 -H16)))))) (subst1_gen_lift_ge u x0 x2 i (S O) d0 H14 (eq_ind_r nat (plus (S -O) d0) (\lambda (n: nat).(le n i)) H12 (plus d0 (S O)) (plus_sym d0 (S -O)))))))) (subst1_confluence_neq t4 t u i (subst1_single i u t4 t H2) (lift -(S O) d0 x0) u0 d0 H11 (sym_not_eq nat d0 i (lt_neq d0 i H12)))))))))) -(pr0_gen_lift x1 x (S O) d0 H7))))) (pr0_subst1 t3 t4 H1 u0 (lift (S O) d0 -x1) d0 H6 u0 (pr0_refl u0))))))))))))))))))))))) c t1 t2 H)))). -(* COMMENTS -Initial nodes: 3757 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/defs.ma deleted file mode 100644 index 9619ee1d7..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/defs.ma +++ /dev/null @@ -1,23 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr2/defs.ma". - -inductive pr3 (c: C): T \to (T \to Prop) \def -| pr3_refl: \forall (t: T).(pr3 c t t) -| pr3_sing: \forall (t2: T).(\forall (t1: T).((pr2 c t1 t2) \to (\forall (t3: -T).((pr3 c t2 t3) \to (pr3 c t1 t3))))). - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/fwd.ma deleted file mode 100644 index 726c420f0..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/fwd.ma +++ /dev/null @@ -1,1604 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr3/props.ma". - -include "Basic-1/pr2/fwd.ma". - -theorem pr3_gen_sort: - \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr3 c (TSort n) x) \to -(eq T x (TSort n))))) -\def - \lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr3 c (TSort -n) x)).(insert_eq T (TSort n) (\lambda (t: T).(pr3 c t x)) (\lambda (t: -T).(eq T x t)) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(pr3_ind c (\lambda -(t: T).(\lambda (t0: T).((eq T t (TSort n)) \to (eq T t0 t)))) (\lambda (t: -T).(\lambda (_: (eq T t (TSort n))).(refl_equal T t))) (\lambda (t2: -T).(\lambda (t1: T).(\lambda (H1: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda -(_: (pr3 c t2 t3)).(\lambda (H3: (((eq T t2 (TSort n)) \to (eq T t3 -t2)))).(\lambda (H4: (eq T t1 (TSort n))).(let H5 \def (eq_ind T t1 (\lambda -(t: T).(pr2 c t t2)) H1 (TSort n) H4) in (eq_ind_r T (TSort n) (\lambda (t: -T).(eq T t3 t)) (let H6 \def (eq_ind T t2 (\lambda (t: T).((eq T t (TSort n)) -\to (eq T t3 t))) H3 (TSort n) (pr2_gen_sort c t2 n H5)) in (H6 (refl_equal T -(TSort n)))) t1 H4))))))))) y x H0))) H)))). -(* COMMENTS -Initial nodes: 253 -END *) - -theorem pr3_gen_abst: - \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c -(THead (Bind Abst) u1 t1) x) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 -c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: -T).(pr3 (CHead c (Bind b) u) t1 t2)))))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda -(H: (pr3 c (THead (Bind Abst) u1 t1) x)).(insert_eq T (THead (Bind Abst) u1 -t1) (\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t1 t2))))))) (\lambda (y: -T).(\lambda (H0: (pr3 c y x)).(unintro T t1 (\lambda (t: T).((eq T y (THead -(Bind Abst) u1 t)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x -(THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) -(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead -c (Bind b) u) t t2)))))))) (unintro T u1 (\lambda (t: T).(\forall (x0: -T).((eq T y (THead (Bind Abst) t x0)) \to (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_: T).(\lambda (t2: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x0 t2))))))))) (pr3_ind c -(\lambda (t: T).(\lambda (t0: T).(\forall (x0: T).(\forall (x1: T).((eq T t -(THead (Bind Abst) x0 x1)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T t0 (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall -(u: T).(pr3 (CHead c (Bind b) u) x1 t2))))))))))) (\lambda (t: T).(\lambda -(x0: T).(\lambda (x1: T).(\lambda (H1: (eq T t (THead (Bind Abst) x0 -x1))).(ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T t (THead (Bind -Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: -T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -x1 t2))))) x0 x1 H1 (pr3_refl c x0) (\lambda (b: B).(\lambda (u: T).(pr3_refl -(CHead c (Bind b) u) x1)))))))) (\lambda (t2: T).(\lambda (t3: T).(\lambda -(H1: (pr2 c t3 t2)).(\lambda (t4: T).(\lambda (_: (pr3 c t2 t4)).(\lambda -(H3: ((\forall (x0: T).(\forall (x1: T).((eq T t2 (THead (Bind Abst) x0 x1)) -\to (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst) -u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: -T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -x1 t5))))))))))).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t3 -(THead (Bind Abst) x0 x1))).(let H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c -t t2)) H1 (THead (Bind Abst) x0 x1) H4) in (let H6 \def (pr2_gen_abst c x0 x1 -t2 H5) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead -(Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead -c (Bind b) u) x1 t5))))) (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T -t4 (THead (Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 -u2))) (\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 -(CHead c (Bind b) u) x1 t5)))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda -(H7: (eq T t2 (THead (Bind Abst) x2 x3))).(\lambda (H8: (pr2 c x0 -x2)).(\lambda (H9: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) -x1 x3))))).(let H10 \def (eq_ind T t2 (\lambda (t: T).(\forall (x4: -T).(\forall (x5: T).((eq T t (THead (Bind Abst) x4 x5)) \to (ex3_2 T T -(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x4 u2))) (\lambda (_: T).(\lambda -(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x5 -t5)))))))))) H3 (THead (Bind Abst) x2 x3) H7) in (let H11 \def (H10 x2 x3 -(refl_equal T (THead (Bind Abst) x2 x3))) in (ex3_2_ind T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 t5))))) (ex3_2 T T -(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) -(\lambda (x4: T).(\lambda (x5: T).(\lambda (H12: (eq T t4 (THead (Bind Abst) -x4 x5))).(\lambda (H13: (pr3 c x2 x4)).(\lambda (H14: ((\forall (b: -B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 x5))))).(ex3_2_intro T T -(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5))))) -x4 x5 H12 (pr3_sing c x2 x0 H8 x4 H13) (\lambda (b: B).(\lambda (u: -T).(pr3_sing (CHead c (Bind b) u) x3 x1 (H9 b u) x5 (H14 b u)))))))))) -H11)))))))) H6)))))))))))) y x H0))))) H))))). -(* COMMENTS -Initial nodes: 1261 -END *) - -theorem pr3_gen_cast: - \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c -(THead (Flat Cast) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda -(t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t1 t2)))) (pr3 c -t1 x)))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda -(H: (pr3 c (THead (Flat Cast) u1 t1) x)).(insert_eq T (THead (Flat Cast) u1 -t1) (\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t1 -t2)))) (pr3 c t1 x))) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(unintro T -t1 (\lambda (t: T).((eq T y (THead (Flat Cast) u1 t)) \to (or (ex3_2 T T -(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda -(t2: T).(pr3 c t t2)))) (pr3 c t x)))) (unintro T u1 (\lambda (t: T).(\forall -(x0: T).((eq T y (THead (Flat Cast) t x0)) \to (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c x0 -t2)))) (pr3 c x0 x))))) (pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall -(x0: T).(\forall (x1: T).((eq T t (THead (Flat Cast) x0 x1)) \to (or (ex3_2 T -T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Flat Cast) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t2: T).(pr3 c x1 t2)))) (pr3 c x1 t0))))))) (\lambda (t: T).(\lambda (x0: -T).(\lambda (x1: T).(\lambda (H1: (eq T t (THead (Flat Cast) x0 -x1))).(eq_ind_r T (THead (Flat Cast) x0 x1) (\lambda (t0: T).(or (ex3_2 T T -(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Flat Cast) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t2: T).(pr3 c x1 t2)))) (pr3 c x1 t0))) (or_introl (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T (THead (Flat Cast) x0 x1) (THead (Flat Cast) u2 -t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: -T).(\lambda (t2: T).(pr3 c x1 t2)))) (pr3 c x1 (THead (Flat Cast) x0 x1)) -(ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Cast) -x0 x1) (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c -x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c x1 t2))) x0 x1 (refl_equal T -(THead (Flat Cast) x0 x1)) (pr3_refl c x0) (pr3_refl c x1))) t H1))))) -(\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c t3 t2)).(\lambda (t4: -T).(\lambda (H2: (pr3 c t2 t4)).(\lambda (H3: ((\forall (x0: T).(\forall (x1: -T).((eq T t2 (THead (Flat Cast) x0 x1)) \to (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 -t5)))) (pr3 c x1 t4))))))).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: -(eq T t3 (THead (Flat Cast) x0 x1))).(let H5 \def (eq_ind T t3 (\lambda (t: -T).(pr2 c t t2)) H1 (THead (Flat Cast) x0 x1) H4) in (let H6 \def -(pr2_gen_cast c x0 x1 t2 H5) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda -(t5: T).(eq T t2 (THead (Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr2 c x1 t5)))) (pr2 c -x1 t2) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat -Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: -T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4)) (\lambda (H7: (ex3_2 T T -(\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Flat Cast) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda -(t5: T).(pr2 c x1 t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: -T).(eq T t2 (THead (Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr2 c x1 t5))) (or -(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 -t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: -T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4)) (\lambda (x2: T).(\lambda -(x3: T).(\lambda (H8: (eq T t2 (THead (Flat Cast) x2 x3))).(\lambda (H9: (pr2 -c x0 x2)).(\lambda (H10: (pr2 c x1 x3)).(let H11 \def (eq_ind T t2 (\lambda -(t: T).(\forall (x4: T).(\forall (x5: T).((eq T t (THead (Flat Cast) x4 x5)) -\to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat -Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x4 u2))) (\lambda (_: -T).(\lambda (t5: T).(pr3 c x5 t5)))) (pr3 c x5 t4)))))) H3 (THead (Flat Cast) -x2 x3) H8) in (let H12 \def (H11 x2 x3 (refl_equal T (THead (Flat Cast) x2 -x3))) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead -(Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr3 c x3 t5)))) (pr3 c x3 t4) (or (ex3_2 T -T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4)) (\lambda (H13: (ex3_2 T T (\lambda -(u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x3 -t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead -(Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr3 c x3 t5))) (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 -t5)))) (pr3 c x1 t4)) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H14: (eq T -t4 (THead (Flat Cast) x4 x5))).(\lambda (H15: (pr3 c x2 x4)).(\lambda (H16: -(pr3 c x3 x5)).(eq_ind_r T (THead (Flat Cast) x4 x5) (\lambda (t: T).(or -(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t (THead (Flat Cast) u2 -t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: -T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t))) (or_introl (ex3_2 T T -(\lambda (u2: T).(\lambda (t5: T).(eq T (THead (Flat Cast) x4 x5) (THead -(Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 (THead (Flat -Cast) x4 x5)) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t5: T).(eq T (THead -(Flat Cast) x4 x5) (THead (Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5))) x4 x5 -(refl_equal T (THead (Flat Cast) x4 x5)) (pr3_sing c x2 x0 H9 x4 H15) -(pr3_sing c x3 x1 H10 x5 H16))) t4 H14)))))) H13)) (\lambda (H13: (pr3 c x3 -t4)).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead -(Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4) (pr3_sing c -x3 x1 H10 t4 H13))) H12)))))))) H7)) (\lambda (H7: (pr2 c x1 t2)).(or_intror -(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 -t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: -T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4) (pr3_sing c t2 x1 H7 t4 -H2))) H6)))))))))))) y x H0))))) H))))). -(* COMMENTS -Initial nodes: 2001 -END *) - -theorem pr3_gen_lift: - \forall (c: C).(\forall (t1: T).(\forall (x: T).(\forall (h: nat).(\forall -(d: nat).((pr3 c (lift h d t1) x) \to (\forall (e: C).((drop h d c e) \to -(ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr3 e t1 -t2)))))))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (x: T).(\lambda (h: nat).(\lambda -(d: nat).(\lambda (H: (pr3 c (lift h d t1) x)).(insert_eq T (lift h d t1) -(\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(\forall (e: C).((drop h d c e) -\to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr3 e -t1 t2)))))) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(unintro T t1 (\lambda -(t: T).((eq T y (lift h d t)) \to (\forall (e: C).((drop h d c e) \to (ex2 T -(\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr3 e t t2))))))) -(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (x0: T).((eq T t (lift h -d x0)) \to (\forall (e: C).((drop h d c e) \to (ex2 T (\lambda (t2: T).(eq T -t0 (lift h d t2))) (\lambda (t2: T).(pr3 e x0 t2))))))))) (\lambda (t: -T).(\lambda (x0: T).(\lambda (H1: (eq T t (lift h d x0))).(\lambda (e: -C).(\lambda (_: (drop h d c e)).(ex_intro2 T (\lambda (t2: T).(eq T t (lift h -d t2))) (\lambda (t2: T).(pr3 e x0 t2)) x0 H1 (pr3_refl e x0))))))) (\lambda -(t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c t3 t2)).(\lambda (t4: -T).(\lambda (_: (pr3 c t2 t4)).(\lambda (H3: ((\forall (x0: T).((eq T t2 -(lift h d x0)) \to (\forall (e: C).((drop h d c e) \to (ex2 T (\lambda (t5: -T).(eq T t4 (lift h d t5))) (\lambda (t5: T).(pr3 e x0 t5))))))))).(\lambda -(x0: T).(\lambda (H4: (eq T t3 (lift h d x0))).(\lambda (e: C).(\lambda (H5: -(drop h d c e)).(let H6 \def (eq_ind T t3 (\lambda (t: T).(pr2 c t t2)) H1 -(lift h d x0) H4) in (let H7 \def (pr2_gen_lift c x0 t2 h d H6 e H5) in -(ex2_ind T (\lambda (t5: T).(eq T t2 (lift h d t5))) (\lambda (t5: T).(pr2 e -x0 t5)) (ex2 T (\lambda (t5: T).(eq T t4 (lift h d t5))) (\lambda (t5: -T).(pr3 e x0 t5))) (\lambda (x1: T).(\lambda (H8: (eq T t2 (lift h d -x1))).(\lambda (H9: (pr2 e x0 x1)).(ex2_ind T (\lambda (t5: T).(eq T t4 (lift -h d t5))) (\lambda (t5: T).(pr3 e x1 t5)) (ex2 T (\lambda (t5: T).(eq T t4 -(lift h d t5))) (\lambda (t5: T).(pr3 e x0 t5))) (\lambda (x2: T).(\lambda -(H10: (eq T t4 (lift h d x2))).(\lambda (H11: (pr3 e x1 x2)).(ex_intro2 T -(\lambda (t5: T).(eq T t4 (lift h d t5))) (\lambda (t5: T).(pr3 e x0 t5)) x2 -H10 (pr3_sing e x1 x0 H9 x2 H11))))) (H3 x1 H8 e H5))))) H7))))))))))))) y x -H0)))) H)))))). -(* COMMENTS -Initial nodes: 689 -END *) - -theorem pr3_gen_lref: - \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr3 c (TLRef n) x) \to -(or (eq T x (TLRef n)) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda -(_: T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: -T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(v: T).(eq T x (lift (S n) O v)))))))))) -\def - \lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr3 c (TLRef -n) x)).(insert_eq T (TLRef n) (\lambda (t: T).(pr3 c t x)) (\lambda (t: -T).(or (eq T x t) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: -T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: -T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(v: T).(eq T x (lift (S n) O v)))))))) (\lambda (y: T).(\lambda (H0: (pr3 c y -x)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).((eq T t (TLRef n)) \to (or -(eq T t0 t) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: -T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: -T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(v: T).(eq T t0 (lift (S n) O v)))))))))) (\lambda (t: T).(\lambda (_: (eq T -t (TLRef n))).(or_introl (eq T t t) (ex3_3 C T T (\lambda (d: C).(\lambda (u: -T).(\lambda (_: T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: -C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda -(_: T).(\lambda (v: T).(eq T t (lift (S n) O v)))))) (refl_equal T t)))) -(\lambda (t2: T).(\lambda (t1: T).(\lambda (H1: (pr2 c t1 t2)).(\lambda (t3: -T).(\lambda (H2: (pr3 c t2 t3)).(\lambda (H3: (((eq T t2 (TLRef n)) \to (or -(eq T t3 t2) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: -T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: -T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(v: T).(eq T t3 (lift (S n) O v)))))))))).(\lambda (H4: (eq T t1 (TLRef -n))).(let H5 \def (eq_ind T t1 (\lambda (t: T).(pr2 c t t2)) H1 (TLRef n) H4) -in (eq_ind_r T (TLRef n) (\lambda (t: T).(or (eq T t3 t) (ex3_3 C T T -(\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead d (Bind -Abbr) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T t3 (lift (S n) O -v)))))))) (let H6 \def (pr2_gen_lref c t2 n H5) in (or_ind (eq T t2 (TLRef -n)) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) -u)))) (\lambda (_: C).(\lambda (u: T).(eq T t2 (lift (S n) O u))))) (or (eq T -t3 (TLRef n)) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: -T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: -T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(v: T).(eq T t3 (lift (S n) O v))))))) (\lambda (H7: (eq T t2 (TLRef -n))).(let H8 \def (eq_ind T t2 (\lambda (t: T).((eq T t (TLRef n)) \to (or -(eq T t3 t) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: -T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: -T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(v: T).(eq T t3 (lift (S n) O v))))))))) H3 (TLRef n) H7) in (let H9 \def -(eq_ind T t2 (\lambda (t: T).(pr3 c t t3)) H2 (TLRef n) H7) in (H8 -(refl_equal T (TLRef n)))))) (\lambda (H7: (ex2_2 C T (\lambda (d: -C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda (_: -C).(\lambda (u: T).(eq T t2 (lift (S n) O u)))))).(ex2_2_ind C T (\lambda (d: -C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda (_: -C).(\lambda (u: T).(eq T t2 (lift (S n) O u)))) (or (eq T t3 (TLRef n)) -(ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead -d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (v: T).(pr3 d u -v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T t3 (lift (S n) O -v))))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H8: (getl n c (CHead x0 -(Bind Abbr) x1))).(\lambda (H9: (eq T t2 (lift (S n) O x1))).(let H10 \def -(eq_ind T t2 (\lambda (t: T).((eq T t (TLRef n)) \to (or (eq T t3 t) (ex3_3 C -T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead d (Bind -Abbr) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T t3 (lift (S n) O -v))))))))) H3 (lift (S n) O x1) H9) in (let H11 \def (eq_ind T t2 (\lambda -(t: T).(pr3 c t t3)) H2 (lift (S n) O x1) H9) in (let H12 \def (pr3_gen_lift -c x1 t3 (S n) O H11 x0 (getl_drop Abbr c x0 x1 n H8)) in (ex2_ind T (\lambda -(t4: T).(eq T t3 (lift (S n) O t4))) (\lambda (t4: T).(pr3 x0 x1 t4)) (or (eq -T t3 (TLRef n)) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: -T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: -T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(v: T).(eq T t3 (lift (S n) O v))))))) (\lambda (x2: T).(\lambda (H13: (eq T -t3 (lift (S n) O x2))).(\lambda (H14: (pr3 x0 x1 x2)).(or_intror (eq T t3 -(TLRef n)) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(getl -n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (v: -T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T t3 -(lift (S n) O v)))))) (ex3_3_intro C T T (\lambda (d: C).(\lambda (u: -T).(\lambda (_: T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: -C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda -(_: T).(\lambda (v: T).(eq T t3 (lift (S n) O v))))) x0 x1 x2 H8 H14 H13))))) -H12)))))))) H7)) H6)) t1 H4))))))))) y x H0))) H)))). -(* COMMENTS -Initial nodes: 1515 -END *) - -theorem pr3_gen_void: - \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c -(THead (Bind Void) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda -(t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall -(u: T).(pr3 (CHead c (Bind b) u) t1 t2)))))) (pr3 (CHead c (Bind Void) u1) t1 -(lift (S O) O x))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda -(H: (pr3 c (THead (Bind Void) u1 t1) x)).(insert_eq T (THead (Bind Void) u1 -t1) (\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t1 t2)))))) (pr3 (CHead c -(Bind Void) u1) t1 (lift (S O) O x)))) (\lambda (y: T).(\lambda (H0: (pr3 c y -x)).(unintro T t1 (\lambda (t: T).((eq T y (THead (Bind Void) u1 t)) \to (or -(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 -t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: -T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -t t2)))))) (pr3 (CHead c (Bind Void) u1) t (lift (S O) O x))))) (unintro T u1 -(\lambda (t: T).(\forall (x0: T).((eq T y (THead (Bind Void) t x0)) \to (or -(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 -t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_: -T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -x0 t2)))))) (pr3 (CHead c (Bind Void) t) x0 (lift (S O) O x)))))) (pr3_ind c -(\lambda (t: T).(\lambda (t0: T).(\forall (x0: T).(\forall (x1: T).((eq T t -(THead (Bind Void) x0 x1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T t0 (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall -(u: T).(pr3 (CHead c (Bind b) u) x1 t2)))))) (pr3 (CHead c (Bind Void) x0) x1 -(lift (S O) O t0)))))))) (\lambda (t: T).(\lambda (x0: T).(\lambda (x1: -T).(\lambda (H1: (eq T t (THead (Bind Void) x0 x1))).(eq_ind_r T (THead (Bind -Void) x0 x1) (\lambda (t0: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T t0 (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall -(u: T).(pr3 (CHead c (Bind b) u) x1 t2)))))) (pr3 (CHead c (Bind Void) x0) x1 -(lift (S O) O t0)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T (THead (Bind Void) x0 x1) (THead (Bind Void) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t2)))))) (pr3 (CHead c -(Bind Void) x0) x1 (lift (S O) O (THead (Bind Void) x0 x1))) (ex3_2_intro T T -(\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Bind Void) x0 x1) (THead -(Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) -(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead -c (Bind b) u) x1 t2))))) x0 x1 (refl_equal T (THead (Bind Void) x0 x1)) -(pr3_refl c x0) (\lambda (b: B).(\lambda (u: T).(pr3_refl (CHead c (Bind b) -u) x1))))) t H1))))) (\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c -t3 t2)).(\lambda (t4: T).(\lambda (H2: (pr3 c t2 t4)).(\lambda (H3: ((\forall -(x0: T).(\forall (x1: T).((eq T t2 (THead (Bind Void) x0 x1)) \to (or (ex3_2 -T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) -(pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4)))))))).(\lambda (x0: -T).(\lambda (x1: T).(\lambda (H4: (eq T t3 (THead (Bind Void) x0 x1))).(let -H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c t t2)) H1 (THead (Bind Void) x0 -x1) H4) in (let H6 \def (pr2_gen_void c x0 x1 t2 H5) in (or_ind (ex3_2 T T -(\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind Void) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda -(t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 t5)))))) -(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 (lift (S O) O -t2)))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind -Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: -T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4))) (\lambda -(H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind Void) -u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: -T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) -x1 t5))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead -(Bind Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead -c (Bind b) u) x1 t5))))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq -T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 -u2))) (\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 -(CHead c (Bind b) u) x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) -O t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H8: (eq T t2 (THead (Bind -Void) x2 x3))).(\lambda (H9: (pr2 c x0 x2)).(\lambda (H10: ((\forall (b: -B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 x3))))).(let H11 \def (eq_ind -T t2 (\lambda (t: T).(\forall (x4: T).(\forall (x5: T).((eq T t (THead (Bind -Void) x4 x5)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 -(THead (Bind Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x4 u2))) -(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead -c (Bind b) u) x5 t5)))))) (pr3 (CHead c (Bind Void) x4) x5 (lift (S O) O -t4))))))) H3 (THead (Bind Void) x2 x3) H8) in (let H12 \def (H11 x2 x3 -(refl_equal T (THead (Bind Void) x2 x3))) in (or_ind (ex3_2 T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 t5)))))) (pr3 (CHead c -(Bind Void) x2) x3 (lift (S O) O t4)) (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) (pr3 (CHead c -(Bind Void) x0) x1 (lift (S O) O t4))) (\lambda (H13: (ex3_2 T T (\lambda -(u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 t5))))))).(ex3_2_ind T T -(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda -(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 t5))))) -(or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) -u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: -T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4))) (\lambda -(x4: T).(\lambda (x5: T).(\lambda (H14: (eq T t4 (THead (Bind Void) x4 -x5))).(\lambda (H15: (pr3 c x2 x4)).(\lambda (H16: ((\forall (b: B).(\forall -(u: T).(pr3 (CHead c (Bind b) u) x3 x5))))).(or_introl (ex3_2 T T (\lambda -(u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) (pr3 (CHead c -(Bind Void) x0) x1 (lift (S O) O t4)) (ex3_2_intro T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5))))) x4 x5 H14 -(pr3_sing c x2 x0 H9 x4 H15) (\lambda (b: B).(\lambda (u: T).(pr3_sing (CHead -c (Bind b) u) x3 x1 (H10 b u) x5 (H16 b u))))))))))) H13)) (\lambda (H13: -(pr3 (CHead c (Bind Void) x2) x3 (lift (S O) O t4))).(or_intror (ex3_2 T T -(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) -(pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4)) (pr3_sing (CHead c (Bind -Void) x0) x3 x1 (H10 Void x0) (lift (S O) O t4) (pr3_pr2_pr3_t c x2 x3 (lift -(S O) O t4) (Bind Void) H13 x0 H9)))) H12)))))))) H7)) (\lambda (H7: -((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 (lift (S O) O -t2)))))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 -(THead (Bind Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead -c (Bind b) u) x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4)) -(pr3_sing (CHead c (Bind Void) x0) (lift (S O) O t2) x1 (H7 Void x0) (lift (S -O) O t4) (pr3_lift (CHead c (Bind Void) x0) c (S O) O (drop_drop (Bind Void) -O c c (drop_refl c) x0) t2 t4 H2)))) H6)))))))))))) y x H0))))) H))))). -(* COMMENTS -Initial nodes: 2645 -END *) - -theorem pr3_gen_abbr: - \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c -(THead (Bind Abbr) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda -(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) -u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O x))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda -(H: (pr3 c (THead (Bind Abbr) u1 t1) x)).(insert_eq T (THead (Bind Abbr) u1 -t1) (\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 -(CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S -O) O x)))) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(unintro T t1 (\lambda -(t: T).((eq T y (THead (Bind Abbr) u1 t)) \to (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 -(CHead c (Bind Abbr) u1) t t2)))) (pr3 (CHead c (Bind Abbr) u1) t (lift (S O) -O x))))) (unintro T u1 (\lambda (t: T).(\forall (x0: T).((eq T y (THead (Bind -Abbr) t x0)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x -(THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) -(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) t) x0 t2)))) (pr3 -(CHead c (Bind Abbr) t) x0 (lift (S O) O x)))))) (pr3_ind c (\lambda (t: -T).(\lambda (t0: T).(\forall (x0: T).(\forall (x1: T).((eq T t (THead (Bind -Abbr) x0 x1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 -(THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) -(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) x0) x1 t2)))) (pr3 -(CHead c (Bind Abbr) x0) x1 (lift (S O) O t0)))))))) (\lambda (t: T).(\lambda -(x0: T).(\lambda (x1: T).(\lambda (H1: (eq T t (THead (Bind Abbr) x0 -x1))).(eq_ind_r T (THead (Bind Abbr) x0 x1) (\lambda (t0: T).(or (ex3_2 T T -(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t2: T).(pr3 (CHead c (Bind Abbr) x0) x1 t2)))) (pr3 (CHead c (Bind Abbr) x0) -x1 (lift (S O) O t0)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 -(CHead c (Bind Abbr) x0) x1 t2)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S -O) O (THead (Bind Abbr) x0 x1))) (ex3_2_intro T T (\lambda (u2: T).(\lambda -(t2: T).(eq T (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 t2)))) (\lambda -(u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 -(CHead c (Bind Abbr) x0) x1 t2))) x0 x1 (refl_equal T (THead (Bind Abbr) x0 -x1)) (pr3_refl c x0) (pr3_refl (CHead c (Bind Abbr) x0) x1))) t H1))))) -(\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c t3 t2)).(\lambda (t4: -T).(\lambda (H2: (pr3 c t2 t4)).(\lambda (H3: ((\forall (x0: T).(\forall (x1: -T).((eq T t2 (THead (Bind Abbr) x0 x1)) \to (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S -O) O t4)))))))).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t3 -(THead (Bind Abbr) x0 x1))).(let H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c -t t2)) H1 (THead (Bind Abbr) x0 x1) H4) in (let H6 \def (pr2_gen_abbr c x0 x1 -t2 H5) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 -(THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 -(CHead c (Bind b) u) x1 t5))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: -T).(pr2 (CHead c (Bind Abbr) u) x1 t5))) (ex3_2 T T (\lambda (y0: T).(\lambda -(_: T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: -T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) -z t5)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 -(lift (S O) O t2)))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T -t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 -u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 -t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (H7: -(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind Abbr) u2 -t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: -T).(\lambda (t5: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind -b) u) x1 t5))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead -c (Bind Abbr) u) x1 t5))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 -(CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 -z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) z -t5))))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead -(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 -(CHead c (Bind b) u) x1 t5))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: -T).(pr2 (CHead c (Bind Abbr) u) x1 t5))) (ex3_2 T T (\lambda (y0: T).(\lambda -(_: T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: -T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) -z t5))))))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead -(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 -(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (x2: T).(\lambda -(x3: T).(\lambda (H8: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (H9: (pr2 -c x0 x2)).(\lambda (H10: (or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c -(Bind b) u) x1 x3))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 -(CHead c (Bind Abbr) u) x1 x3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: -T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: -T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) -z x3)))))).(or3_ind (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) -x1 x3))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead c -(Bind Abbr) u) x1 x3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 -(CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 -z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) z x3)))) -(or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) -u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: -T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c -(Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (H11: ((\forall (b: -B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 x3))))).(let H12 \def (eq_ind -T t2 (\lambda (t: T).(\forall (x4: T).(\forall (x5: T).((eq T t (THead (Bind -Abbr) x4 x5)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 -(THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x4 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x4) x5 t5)))) (pr3 -(CHead c (Bind Abbr) x4) x5 (lift (S O) O t4))))))) H3 (THead (Bind Abbr) x2 -x3) H8) in (let H13 \def (H12 x2 x3 (refl_equal T (THead (Bind Abbr) x2 x3))) -in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind -Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: -T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 t5)))) (pr3 (CHead c -(Bind Abbr) x2) x3 (lift (S O) O t4)) (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S -O) O t4))) (\lambda (H14: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T -t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 -u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 -t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead -(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 t5))) (or -(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 -t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: -T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c -(Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (x4: T).(\lambda (x5: -T).(\lambda (H15: (eq T t4 (THead (Bind Abbr) x4 x5))).(\lambda (H16: (pr3 c -x2 x4)).(\lambda (H17: (pr3 (CHead c (Bind Abbr) x2) x3 x5)).(eq_ind_r T -(THead (Bind Abbr) x4 x5) (\lambda (t: T).(or (ex3_2 T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t (THead (Bind Abbr) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S -O) O t)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T -(THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S -O) O (THead (Bind Abbr) x4 x5))) (ex3_2_intro T T (\lambda (u2: T).(\lambda -(t5: T).(eq T (THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2 t5)))) (\lambda -(u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x0) x1 t5))) x4 x5 (refl_equal T (THead (Bind Abbr) x4 -x5)) (pr3_sing c x2 x0 H9 x4 H16) (pr3_sing (CHead c (Bind Abbr) x0) x3 x1 -(H11 Abbr x0) x5 (pr3_pr2_pr3_t c x2 x3 x5 (Bind Abbr) H17 x0 H9)))) t4 -H15)))))) H14)) (\lambda (H14: (pr3 (CHead c (Bind Abbr) x2) x3 (lift (S O) O -t4))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead -(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 -(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4)) (pr3_sing (CHead c (Bind Abbr) -x0) x3 x1 (H11 Abbr x0) (lift (S O) O t4) (pr3_pr2_pr3_t c x2 x3 (lift (S O) -O t4) (Bind Abbr) H14 x0 H9)))) H13)))) (\lambda (H11: (ex2 T (\lambda (u: -T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) x1 -x3)))).(ex2_ind T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead c -(Bind Abbr) u) x1 x3)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T -t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 -u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 -t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (x4: -T).(\lambda (H12: (pr0 x0 x4)).(\lambda (H13: (pr2 (CHead c (Bind Abbr) x4) -x1 x3)).(let H14 \def (eq_ind T t2 (\lambda (t: T).(\forall (x5: T).(\forall -(x6: T).((eq T t (THead (Bind Abbr) x5 x6)) \to (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x5 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x5) x6 t5)))) (pr3 (CHead c (Bind Abbr) x5) x6 (lift (S -O) O t4))))))) H3 (THead (Bind Abbr) x2 x3) H8) in (let H15 \def (H14 x2 x3 -(refl_equal T (THead (Bind Abbr) x2 x3))) in (or_ind (ex3_2 T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x2) x3 t5)))) (pr3 (CHead c (Bind Abbr) x2) x3 (lift (S -O) O t4)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead -(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 -(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (H16: (ex3_2 T T -(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda -(t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 t5))))).(ex3_2_ind T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x2) x3 t5))) (or (ex3_2 T T (\lambda (u2: T).(\lambda -(t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) -x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda -(x5: T).(\lambda (x6: T).(\lambda (H17: (eq T t4 (THead (Bind Abbr) x5 -x6))).(\lambda (H18: (pr3 c x2 x5)).(\lambda (H19: (pr3 (CHead c (Bind Abbr) -x2) x3 x6)).(eq_ind_r T (THead (Bind Abbr) x5 x6) (\lambda (t: T).(or (ex3_2 -T T (\lambda (u2: T).(\lambda (t5: T).(eq T t (THead (Bind Abbr) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) -x1 (lift (S O) O t)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t5: -T).(eq T (THead (Bind Abbr) x5 x6) (THead (Bind Abbr) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S -O) O (THead (Bind Abbr) x5 x6))) (ex3_2_intro T T (\lambda (u2: T).(\lambda -(t5: T).(eq T (THead (Bind Abbr) x5 x6) (THead (Bind Abbr) u2 t5)))) (\lambda -(u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x0) x1 t5))) x5 x6 (refl_equal T (THead (Bind Abbr) x5 -x6)) (pr3_sing c x2 x0 H9 x5 H18) (pr3_t x3 x1 (CHead c (Bind Abbr) x0) -(pr3_pr0_pr2_t x0 x4 H12 c x1 x3 (Bind Abbr) H13) x6 (pr3_pr2_pr3_t c x2 x3 -x6 (Bind Abbr) H19 x0 H9)))) t4 H17)))))) H16)) (\lambda (H16: (pr3 (CHead c -(Bind Abbr) x2) x3 (lift (S O) O t4))).(or_intror (ex3_2 T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S -O) O t4)) (pr3_t x3 x1 (CHead c (Bind Abbr) x0) (pr3_pr0_pr2_t x0 x4 H12 c x1 -x3 (Bind Abbr) H13) (lift (S O) O t4) (pr3_pr2_pr3_t c x2 x3 (lift (S O) O -t4) (Bind Abbr) H16 x0 H9)))) H15)))))) H11)) (\lambda (H11: (ex3_2 T T -(\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) -(\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: -T).(pr2 (CHead c (Bind Abbr) x0) z x3))))).(ex3_2_ind T T (\lambda (y0: -T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: -T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c -(Bind Abbr) x0) z x3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq -T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 -u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 -t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (x4: -T).(\lambda (x5: T).(\lambda (H12: (pr2 (CHead c (Bind Abbr) x0) x1 -x4)).(\lambda (H13: (pr0 x4 x5)).(\lambda (H14: (pr2 (CHead c (Bind Abbr) x0) -x5 x3)).(let H15 \def (eq_ind T t2 (\lambda (t: T).(\forall (x6: T).(\forall -(x7: T).((eq T t (THead (Bind Abbr) x6 x7)) \to (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x6 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x6) x7 t5)))) (pr3 (CHead c (Bind Abbr) x6) x7 (lift (S -O) O t4))))))) H3 (THead (Bind Abbr) x2 x3) H8) in (let H16 \def (H15 x2 x3 -(refl_equal T (THead (Bind Abbr) x2 x3))) in (or_ind (ex3_2 T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x2) x3 t5)))) (pr3 (CHead c (Bind Abbr) x2) x3 (lift (S -O) O t4)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead -(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 -(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (H17: (ex3_2 T T -(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda -(t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 t5))))).(ex3_2_ind T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x2) x3 t5))) (or (ex3_2 T T (\lambda (u2: T).(\lambda -(t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) -x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda -(x6: T).(\lambda (x7: T).(\lambda (H18: (eq T t4 (THead (Bind Abbr) x6 -x7))).(\lambda (H19: (pr3 c x2 x6)).(\lambda (H20: (pr3 (CHead c (Bind Abbr) -x2) x3 x7)).(eq_ind_r T (THead (Bind Abbr) x6 x7) (\lambda (t: T).(or (ex3_2 -T T (\lambda (u2: T).(\lambda (t5: T).(eq T t (THead (Bind Abbr) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) -x1 (lift (S O) O t)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t5: -T).(eq T (THead (Bind Abbr) x6 x7) (THead (Bind Abbr) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S -O) O (THead (Bind Abbr) x6 x7))) (ex3_2_intro T T (\lambda (u2: T).(\lambda -(t5: T).(eq T (THead (Bind Abbr) x6 x7) (THead (Bind Abbr) u2 t5)))) (\lambda -(u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x0) x1 t5))) x6 x7 (refl_equal T (THead (Bind Abbr) x6 -x7)) (pr3_sing c x2 x0 H9 x6 H19) (pr3_sing (CHead c (Bind Abbr) x0) x4 x1 -H12 x7 (pr3_sing (CHead c (Bind Abbr) x0) x5 x4 (pr2_free (CHead c (Bind -Abbr) x0) x4 x5 H13) x7 (pr3_sing (CHead c (Bind Abbr) x0) x3 x5 H14 x7 -(pr3_pr2_pr3_t c x2 x3 x7 (Bind Abbr) H20 x0 H9)))))) t4 H18)))))) H17)) -(\lambda (H17: (pr3 (CHead c (Bind Abbr) x2) x3 (lift (S O) O -t4))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead -(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 -(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4)) (pr3_sing (CHead c (Bind Abbr) -x0) x4 x1 H12 (lift (S O) O t4) (pr3_sing (CHead c (Bind Abbr) x0) x5 x4 -(pr2_free (CHead c (Bind Abbr) x0) x4 x5 H13) (lift (S O) O t4) (pr3_sing -(CHead c (Bind Abbr) x0) x3 x5 H14 (lift (S O) O t4) (pr3_pr2_pr3_t c x2 x3 -(lift (S O) O t4) (Bind Abbr) H17 x0 H9)))))) H16)))))))) H11)) H10)))))) -H7)) (\lambda (H7: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) -x1 (lift (S O) O t2)))))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda -(t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) -x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4)) (pr3_sing -(CHead c (Bind Abbr) x0) (lift (S O) O t2) x1 (H7 Abbr x0) (lift (S O) O t4) -(pr3_lift (CHead c (Bind Abbr) x0) c (S O) O (drop_drop (Bind Abbr) O c c -(drop_refl c) x0) t2 t4 H2)))) H6)))))))))))) y x H0))))) H))))). -(* COMMENTS -Initial nodes: 5983 -END *) - -theorem pr3_gen_appl: - \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c -(THead (Flat Appl) u1 t1) x) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda -(t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t1 t2)))) (ex4_4 T -T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 -c (THead (Bind Abbr) u2 t2) x))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (_: T).(pr3 c u1 u2))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c t1 (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T -T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 -c t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) x))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda -(H: (pr3 c (THead (Flat Appl) u1 t1) x)).(insert_eq T (THead (Flat Appl) u1 -t1) (\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(or3 (ex3_2 T T (\lambda -(u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t1 -t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) x))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))))) -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c t1 -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) -(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(pr3 c t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) -x))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) -y2) z1 z2)))))))))) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(unintro T t1 -(\lambda (t: T).((eq T y (THead (Flat Appl) u1 t)) \to (or3 (ex3_2 T T -(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda -(t2: T).(pr3 c t t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) x))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 -u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(pr3 c t (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 -(CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c t (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead -(Flat Appl) (lift (S O) O u2) z2)) x))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 -u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))) (unintro T u1 (\lambda -(t: T).(\forall (x0: T).((eq T y (THead (Flat Appl) t x0)) \to (or3 (ex3_2 T -T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_: T).(\lambda (t2: -T).(pr3 c x0 t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) x))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))) (\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x0 (THead -(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) -(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(pr3 c x0 (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) -x))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) -y2) z1 z2)))))))))))) (pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall -(x0: T).(\forall (x1: T).((eq T t (THead (Flat Appl) x0 x1)) \to (or3 (ex3_2 -T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Flat Appl) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t2: T).(pr3 c x1 t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) t0))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 -u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 -(CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead -(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) -y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t0))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))))))) -(\lambda (t: T).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H1: (eq T t -(THead (Flat Appl) x0 x1))).(eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda -(t0: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead -(Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) -(\lambda (_: T).(\lambda (t2: T).(pr3 c x1 t2)))) (ex4_4 T T T T (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind -Abbr) u2 t2) t0))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T -T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 -c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t0))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))) -(or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat -Appl) x0 x1) (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c x1 t2)))) (ex4_4 T -T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 -c (THead (Bind Abbr) u2 t2) (THead (Flat Appl) x0 x1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))) -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) -(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) -(THead (Flat Appl) x0 x1)))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 -(CHead c (Bind b) y2) z1 z2)))))))) (ex3_2_intro T T (\lambda (u2: -T).(\lambda (t2: T).(eq T (THead (Flat Appl) x0 x1) (THead (Flat Appl) u2 -t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: -T).(\lambda (t2: T).(pr3 c x1 t2))) x0 x1 (refl_equal T (THead (Flat Appl) x0 -x1)) (pr3_refl c x0) (pr3_refl c x1))) t H1))))) (\lambda (t2: T).(\lambda -(t3: T).(\lambda (H1: (pr2 c t3 t2)).(\lambda (t4: T).(\lambda (H2: (pr3 c t2 -t4)).(\lambda (H3: ((\forall (x0: T).(\forall (x1: T).((eq T t2 (THead (Flat -Appl) x0 x1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 -(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind -Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T -T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 -c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 -z2)))))))))))))).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t3 -(THead (Flat Appl) x0 x1))).(let H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c -t t2)) H1 (THead (Flat Appl) x0 x1) H4) in (let H6 \def (pr2_gen_appl c x0 x1 -t2 H5) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 -(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr2 c x1 t5)))) (ex4_4 T T T T (\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x1 (THead -(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t5: T).(eq T t2 (THead (Bind Abbr) u2 t5)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T -T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq -T x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead -(Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (or3 (ex3_2 -T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 -u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 -(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead -(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) -y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda -(H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Flat Appl) -u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: -T).(\lambda (t5: T).(pr2 c x1 t5))))).(ex3_2_ind T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t2 (THead (Flat Appl) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr2 c x1 -t5))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat -Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: -T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda -(_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) -t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 -c x0 u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 -(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead -(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) -y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda -(x2: T).(\lambda (x3: T).(\lambda (H8: (eq T t2 (THead (Flat Appl) x2 -x3))).(\lambda (H9: (pr2 c x0 x2)).(\lambda (H10: (pr2 c x1 x3)).(let H11 -\def (eq_ind T t2 (\lambda (t: T).(\forall (x4: T).(\forall (x5: T).((eq T t -(THead (Flat Appl) x4 x5)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: -T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c x4 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x5 t5)))) (ex4_4 T -T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 -c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (_: T).(pr3 c x4 u2))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c x5 (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T -T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 -c x5 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c x4 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))))) H3 -(THead (Flat Appl) x2 x3) H8) in (let H12 \def (eq_ind T t2 (\lambda (t: -T).(pr3 c t t4)) H2 (THead (Flat Appl) x2 x3) H8) in (let H13 \def (H11 x2 x3 -(refl_equal T (THead (Flat Appl) x2 x3))) in (or3_ind (ex3_2 T T (\lambda -(u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x3 -t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))))) -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x3 -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) -(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(pr3 c x3 (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) -t4))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) -y2) z1 z2)))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 -(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind -Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T -T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 -c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda -(H14: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat -Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: -T).(\lambda (t5: T).(pr3 c x3 t5))))).(ex3_2_ind T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x3 -t5))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat -Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: -T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda -(_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) -t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 -c x0 u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall 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-y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) -(ex3_2_intro T T (\lambda (u2: T).(\lambda (t5: T).(eq T (THead (Flat Appl) -x4 x5) (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c -x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5))) x4 x5 (refl_equal T -(THead (Flat Appl) x4 x5)) (pr3_sing c x2 x0 H9 x4 H16) (pr3_sing c x3 x1 H10 -x5 H17))) t4 H15)))))) H14)) (\lambda (H14: (ex4_4 T T T T (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind -Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr3 c x2 u2))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c x3 (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5))))))))).(ex4_4_ind T -T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 -c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (_: T).(pr3 c x2 u2))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c x3 (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5))))))) (or3 (ex3_2 T T -(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 -u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (t5: 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Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead -(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) -y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (ex4_4_intro -T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: -T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 -t5))))))) x4 x5 x6 x7 H15 (pr3_sing c x2 x0 H9 x6 H16) (pr3_sing c x3 x1 H10 -(THead (Bind Abst) x4 x5) H17) H18)))))))))) H14)) (\lambda (H14: (ex6_6 B T -T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(pr3 c x3 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c -(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr3 c x2 u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 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(or3 (ex3_2 T T (\lambda (u2: T).(\lambda -(t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T -T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 -c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T -T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 -c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (z2: 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t4))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (ex6_6_intro -B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(pr3 c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c -(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 -y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))) -x4 x5 x6 x7 x8 x9 H15 (pr3_sing c x3 x1 H10 (THead (Bind x4) x5 x6) H16) H17 -(pr3_sing c x2 x0 H9 x8 H18) H19 H20)))))))))))))) H14)) H13))))))))) H7)) -(\lambda (H7: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind -Abbr) u2 t5)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c x0 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) -z1 t5))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (_: T).(eq T x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind -Abbr) u2 t5)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c x0 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) -z1 t5))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 -(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind -Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T -T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 -c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda -(x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H8: (eq -T x1 (THead (Bind Abst) x2 x3))).(\lambda (H9: (eq T t2 (THead (Bind Abbr) x4 -x5))).(\lambda (H10: (pr2 c x0 x4)).(\lambda (H11: ((\forall (b: B).(\forall -(u: T).(pr2 (CHead c (Bind b) u) x3 x5))))).(eq_ind_r T (THead (Bind Abst) x2 -x3) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T -t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 -u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c t t5)))) (ex4_4 T T T T -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c -(THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c t (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T -T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 -c t (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))) (let H12 -\def (eq_ind T t2 (\lambda (t: T).(\forall (x6: T).(\forall (x7: T).((eq T t -(THead (Flat Appl) x6 x7)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: -T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c x6 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x7 t5)))) (ex4_4 T -T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 -c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (_: T).(pr3 c x6 u2))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c x7 (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T -T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 -c x7 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c x6 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))))) H3 -(THead (Bind Abbr) x4 x5) H9) in (let H13 \def (eq_ind T t2 (\lambda (t: -T).(pr3 c t t4)) H2 (THead (Bind Abbr) x4 x5) H9) in (or3_intro1 (ex3_2 T T -(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t5: T).(pr3 c (THead (Bind Abst) x2 x3) t5)))) (ex4_4 T T T T (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind -Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) x2 x3) (THead -(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) -(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) x2 x3) (THead (Bind b) y1 -z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: -T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat -Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 -u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (ex4_4_intro T T T T -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c -(THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) x2 x3) (THead -(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -z1 t5))))))) x2 x3 x4 x5 H13 (pr3_pr2 c x0 x4 H10) (pr3_refl c (THead (Bind -Abst) x2 x3)) (\lambda (b: B).(\lambda (u: T).(pr3_pr2 (CHead c (Bind b) u) -x3 x5 (H11 b u)))))))) x1 H8))))))))) H7)) (\lambda (H7: (ex6_6 B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T x1 -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind -B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T -t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 -y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) -(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) -u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: -T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda -(_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) -t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 -c x0 u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 -(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead -(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) -y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda -(x2: B).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (x6: -T).(\lambda (x7: T).(\lambda (H8: (not (eq B x2 Abst))).(\lambda (H9: (eq T -x1 (THead (Bind x2) x3 x4))).(\lambda (H10: (eq T t2 (THead (Bind x2) x7 -(THead (Flat Appl) (lift (S O) O x6) x5)))).(\lambda (H11: (pr2 c x0 -x6)).(\lambda (H12: (pr2 c x3 x7)).(\lambda (H13: (pr2 (CHead c (Bind x2) x7) -x4 x5)).(eq_ind_r T (THead (Bind x2) x3 x4) (\lambda (t: T).(or3 (ex3_2 T T -(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t5: T).(pr3 c t t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 -u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(pr3 c t (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 -(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c t (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead -(Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 -u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))) (let H14 \def (eq_ind T t2 -(\lambda (t: T).(\forall (x8: T).(\forall (x9: T).((eq T t (THead (Flat Appl) -x8 x9)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead -(Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x8 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr3 c x9 t5)))) (ex4_4 T T T T (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind -Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr3 c x8 u2))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c x9 (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T -T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 -c x9 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c x8 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))))) H3 -(THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) O x6) x5)) H10) in (let -H15 \def (eq_ind T t2 (\lambda (t: T).(pr3 c t t4)) H2 (THead (Bind x2) x7 -(THead (Flat Appl) (lift (S O) O x6) x5)) H10) in (or3_intro2 (ex3_2 T T -(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t5: T).(pr3 c (THead (Bind x2) x3 x4) t5)))) (ex4_4 T T T T (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind -Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind x2) x3 x4) (THead -(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) -(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(pr3 c (THead (Bind x2) x3 x4) (THead (Bind b) y1 -z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: -T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat -Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 -u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c -(THead (Bind x2) x3 x4) (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) -t4))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) -y2) z1 z2))))))) x2 x3 x4 x5 x6 x7 H8 (pr3_refl c (THead (Bind x2) x3 x4)) -H15 (pr3_pr2 c x0 x6 H11) (pr3_pr2 c x3 x7 H12) (pr3_pr2 (CHead c (Bind x2) -x7) x4 x5 H13))))) x1 H9))))))))))))) H7)) H6)))))))))))) y x H0))))) H))))). -(* COMMENTS -Initial nodes: 12691 -END *) - -theorem pr3_gen_bind: - \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (u1: -T).(\forall (t1: T).(\forall (x: T).((pr3 c (THead (Bind b) u1 t1) x) \to (or -(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind b) u2 -t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: -T).(\lambda (t2: T).(pr3 (CHead c (Bind b) u1) t1 t2)))) (pr3 (CHead c (Bind -b) u1) t1 (lift (S O) O x))))))))) -\def - \lambda (b: B).(B_ind (\lambda (b0: B).((not (eq B b0 Abst)) \to (\forall -(c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c (THead (Bind -b0) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x -(THead (Bind b0) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) -(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind b0) u1) t1 t2)))) (pr3 -(CHead c (Bind b0) u1) t1 (lift (S O) O x)))))))))) (\lambda (_: (not (eq B -Abbr Abst))).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: -T).(\lambda (H0: (pr3 c (THead (Bind Abbr) u1 t1) x)).(let H1 \def -(pr3_gen_abbr c u1 t1 x H0) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda -(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) -u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O x)) (or (ex3_2 T -T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda -(t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) -t1 (lift (S O) O x))) (\lambda (H2: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 -c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 -t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind -Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: -T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2))) (or (ex3_2 T T -(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda -(t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) -t1 (lift (S O) O x))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H3: (eq T x -(THead (Bind Abbr) x0 x1))).(\lambda (H4: (pr3 c u1 x0)).(\lambda (H5: (pr3 -(CHead c (Bind Abbr) u1) t1 x1)).(or_introl (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 -(CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S -O) O x)) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead -(Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) -(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2))) x0 x1 -H3 H4 H5))))))) H2)) (\lambda (H2: (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S -O) O x))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x -(THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) -(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2)))) (pr3 -(CHead c (Bind Abbr) u1) t1 (lift (S O) O x)) H2)) H1)))))))) (\lambda (H: -(not (eq B Abst Abst))).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: -T).(\lambda (x: T).(\lambda (_: (pr3 c (THead (Bind Abst) u1 t1) x)).(let H1 -\def (match (H (refl_equal B Abst)) in False return (\lambda (_: False).(or -(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 -t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: -T).(\lambda (t2: T).(pr3 (CHead c (Bind Abst) u1) t1 t2)))) (pr3 (CHead c -(Bind Abst) u1) t1 (lift (S O) O x)))) with []) in H1))))))) (\lambda (_: -(not (eq B Void Abst))).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: -T).(\lambda (x: T).(\lambda (H0: (pr3 c (THead (Bind Void) u1 t1) x)).(let H1 -\def (pr3_gen_void c u1 t1 x H0) in (or_ind (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall -(b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t1 t2)))))) (pr3 (CHead c -(Bind Void) u1) t1 (lift (S O) O x)) (or (ex3_2 T T (\lambda (u2: T).(\lambda -(t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) -u1) t1 t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O x))) (\lambda -(H2: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) -u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: -T).(\lambda (t2: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) -u) t1 t2))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T x -(THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) -(\lambda (_: T).(\lambda (t2: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead -c (Bind b0) u) t1 t2))))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 -c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) u1) t1 -t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O x))) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H3: (eq T x (THead (Bind Void) x0 -x1))).(\lambda (H4: (pr3 c u1 x0)).(\lambda (H5: ((\forall (b0: B).(\forall -(u: T).(pr3 (CHead c (Bind b0) u) t1 x1))))).(or_introl (ex3_2 T T (\lambda -(u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 -(CHead c (Bind Void) u1) t1 t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S -O) O x)) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead -(Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) -(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) u1) t1 t2))) x0 x1 -H3 H4 (H5 Void u1)))))))) H2)) (\lambda (H2: (pr3 (CHead c (Bind Void) u1) t1 -(lift (S O) O x))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 -c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) u1) t1 -t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O x)) H2)) H1)))))))) b). -(* COMMENTS -Initial nodes: 1721 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/iso.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/iso.ma deleted file mode 100644 index c7b2fb7aa..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/iso.ma +++ /dev/null @@ -1,1155 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr3/fwd.ma". - -include "Basic-1/iso/props.ma". - -include "Basic-1/tlist/props.ma". - -theorem pr3_iso_appls_abbr: - \forall (c: C).(\forall (d: C).(\forall (w: T).(\forall (i: nat).((getl i c -(CHead d (Bind Abbr) w)) \to (\forall (vs: TList).(let u1 \def (THeads (Flat -Appl) vs (TLRef i)) in (\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to -(\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) vs (lift (S i) O w)) -u2)))))))))) -\def - \lambda (c: C).(\lambda (d: C).(\lambda (w: T).(\lambda (i: nat).(\lambda -(H: (getl i c (CHead d (Bind Abbr) w))).(\lambda (vs: TList).(TList_ind -(\lambda (t: TList).(let u1 \def (THeads (Flat Appl) t (TLRef i)) in (\forall -(u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to -(pr3 c (THeads (Flat Appl) t (lift (S i) O w)) u2)))))) (\lambda (u2: -T).(\lambda (H0: (pr3 c (TLRef i) u2)).(\lambda (H1: (((iso (TLRef i) u2) \to -(\forall (P: Prop).P)))).(let H2 \def (pr3_gen_lref c u2 i H0) in (or_ind (eq -T u2 (TLRef i)) (ex3_3 C T T (\lambda (d0: C).(\lambda (u: T).(\lambda (_: -T).(getl i c (CHead d0 (Bind Abbr) u))))) (\lambda (d0: C).(\lambda (u: -T).(\lambda (v: T).(pr3 d0 u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(v: T).(eq T u2 (lift (S i) O v)))))) (pr3 c (lift (S i) O w) u2) (\lambda -(H3: (eq T u2 (TLRef i))).(let H4 \def (eq_ind T u2 (\lambda (t: T).((iso -(TLRef i) t) \to (\forall (P: Prop).P))) H1 (TLRef i) H3) in (eq_ind_r T -(TLRef i) (\lambda (t: T).(pr3 c (lift (S i) O w) t)) (H4 (iso_refl (TLRef -i)) (pr3 c (lift (S i) O w) (TLRef i))) u2 H3))) (\lambda (H3: (ex3_3 C T T -(\lambda (d0: C).(\lambda (u: T).(\lambda (_: T).(getl i c (CHead d0 (Bind -Abbr) u))))) (\lambda (d0: C).(\lambda (u: T).(\lambda (v: T).(pr3 d0 u v)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T u2 (lift (S i) O -v))))))).(ex3_3_ind C T T (\lambda (d0: C).(\lambda (u: T).(\lambda (_: -T).(getl i c (CHead d0 (Bind Abbr) u))))) (\lambda (d0: C).(\lambda (u: -T).(\lambda (v: T).(pr3 d0 u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(v: T).(eq T u2 (lift (S i) O v))))) (pr3 c (lift (S i) O w) u2) (\lambda -(x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H4: (getl i c (CHead x0 -(Bind Abbr) x1))).(\lambda (H5: (pr3 x0 x1 x2)).(\lambda (H6: (eq T u2 (lift -(S i) O x2))).(let H7 \def (eq_ind T u2 (\lambda (t: T).((iso (TLRef i) t) -\to (\forall (P: Prop).P))) H1 (lift (S i) O x2) H6) in (eq_ind_r T (lift (S -i) O x2) (\lambda (t: T).(pr3 c (lift (S i) O w) t)) (let H8 \def (eq_ind C -(CHead d (Bind Abbr) w) (\lambda (c0: C).(getl i c c0)) H (CHead x0 (Bind -Abbr) x1) (getl_mono c (CHead d (Bind Abbr) w) i H (CHead x0 (Bind Abbr) x1) -H4)) in (let H9 \def (f_equal C C (\lambda (e: C).(match e in C return -(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow -c0])) (CHead d (Bind Abbr) w) (CHead x0 (Bind Abbr) x1) (getl_mono c (CHead d -(Bind Abbr) w) i H (CHead x0 (Bind Abbr) x1) H4)) in ((let H10 \def (f_equal -C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) -\Rightarrow w | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abbr) w) (CHead -x0 (Bind Abbr) x1) (getl_mono c (CHead d (Bind Abbr) w) i H (CHead x0 (Bind -Abbr) x1) H4)) in (\lambda (H11: (eq C d x0)).(let H12 \def (eq_ind_r T x1 -(\lambda (t: T).(getl i c (CHead x0 (Bind Abbr) t))) H8 w H10) in (let H13 -\def (eq_ind_r T x1 (\lambda (t: T).(pr3 x0 t x2)) H5 w H10) in (let H14 \def -(eq_ind_r C x0 (\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr) w))) H12 d -H11) in (let H15 \def (eq_ind_r C x0 (\lambda (c0: C).(pr3 c0 w x2)) H13 d -H11) in (pr3_lift c d (S i) O (getl_drop Abbr c d w i H14) w x2 H15))))))) -H9))) u2 H6)))))))) H3)) H2))))) (\lambda (t: T).(\lambda (t0: -TList).(\lambda (H0: ((\forall (u2: T).((pr3 c (THeads (Flat Appl) t0 (TLRef -i)) u2) \to ((((iso (THeads (Flat Appl) t0 (TLRef i)) u2) \to (\forall (P: -Prop).P))) \to (pr3 c (THeads (Flat Appl) t0 (lift (S i) O w)) -u2)))))).(\lambda (u2: T).(\lambda (H1: (pr3 c (THead (Flat Appl) t (THeads -(Flat Appl) t0 (TLRef i))) u2)).(\lambda (H2: (((iso (THead (Flat Appl) t -(THeads (Flat Appl) t0 (TLRef i))) u2) \to (\forall (P: Prop).P)))).(let H3 -\def (pr3_gen_appl c t (THeads (Flat Appl) t0 (TLRef i)) u2 H1) in (or3_ind -(ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 -t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t u3))) (\lambda (_: -T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) t2)))) (ex4_4 T -T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 -c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u3: T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) -(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead -(Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t -u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (pr3 c (THead (Flat Appl) t -(THeads (Flat Appl) t0 (lift (S i) O w))) u2) (\lambda (H4: (ex3_2 T T -(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) -(\lambda (u3: T).(\lambda (_: T).(pr3 c t u3))) (\lambda (_: T).(\lambda (t2: -T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) t2))))).(ex3_2_ind T T (\lambda -(u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: -T).(\lambda (_: T).(pr3 c t u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c -(THeads (Flat Appl) t0 (TLRef i)) t2))) (pr3 c (THead (Flat Appl) t (THeads -(Flat Appl) t0 (lift (S i) O w))) u2) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H5: (eq T u2 (THead (Flat Appl) x0 x1))).(\lambda (_: (pr3 c t -x0)).(\lambda (_: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) x1)).(let H8 \def -(eq_ind T u2 (\lambda (t1: T).((iso (THead (Flat Appl) t (THeads (Flat Appl) -t0 (TLRef i))) t1) \to (\forall (P: Prop).P))) H2 (THead (Flat Appl) x0 x1) -H5) in (eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda (t1: T).(pr3 c (THead -(Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O w))) t1)) (H8 (iso_head t -x0 (THeads (Flat Appl) t0 (TLRef i)) x1 (Flat Appl)) (pr3 c (THead (Flat -Appl) t (THeads (Flat Appl) t0 (lift (S i) O w))) (THead (Flat Appl) x0 x1))) -u2 H5))))))) H4)) (\lambda (H4: (ex4_4 T T T T (\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t -u3))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2))))))))).(ex4_4_ind T -T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 -c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u3: T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -z1 t2))))))) (pr3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O -w))) u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: -T).(\lambda (H5: (pr3 c (THead (Bind Abbr) x2 x3) u2)).(\lambda (H6: (pr3 c t -x2)).(\lambda (H7: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind -Abst) x0 x1))).(\lambda (H8: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c -(Bind b) u) x1 x3))))).(pr3_t (THead (Bind Abbr) t x1) (THead (Flat Appl) t -(THeads (Flat Appl) t0 (lift (S i) O w))) c (pr3_t (THead (Flat Appl) t -(THead (Bind Abst) x0 x1)) (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift -(S i) O w))) c (pr3_thin_dx c (THeads (Flat Appl) t0 (lift (S i) O w)) (THead -(Bind Abst) x0 x1) (H0 (THead (Bind Abst) x0 x1) H7 (\lambda (H9: (iso -(THeads (Flat Appl) t0 (TLRef i)) (THead (Bind Abst) x0 x1))).(\lambda (P: -Prop).(iso_flats_lref_bind_false Appl Abst i x0 x1 t0 H9 P)))) t Appl) (THead -(Bind Abbr) t x1) (pr3_pr2 c (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) -(THead (Bind Abbr) t x1) (pr2_free c (THead (Flat Appl) t (THead (Bind Abst) -x0 x1)) (THead (Bind Abbr) t x1) (pr0_beta x0 t t (pr0_refl t) x1 x1 -(pr0_refl x1))))) u2 (pr3_t (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) t -x1) c (pr3_head_12 c t x2 H6 (Bind Abbr) x1 x3 (H8 Abbr x2)) u2 H5)))))))))) -H4)) (\lambda (H4: (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B -b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) -y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda -(_: T).(pr3 c t u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind -B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind b) y1 z1)))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda -(u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift -(S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t u3))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 -(CHead c (Bind b) y2) z1 z2))))))) (pr3 c (THead (Flat Appl) t (THeads (Flat -Appl) t0 (lift (S i) O w))) u2) (\lambda (x0: B).(\lambda (x1: T).(\lambda -(x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H5: (not -(eq B x0 Abst))).(\lambda (H6: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) -(THead (Bind x0) x1 x2))).(\lambda (H7: (pr3 c (THead (Bind x0) x5 (THead -(Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda (H8: (pr3 c t x4)).(\lambda -(H9: (pr3 c x1 x5)).(\lambda (H10: (pr3 (CHead c (Bind x0) x5) x2 x3)).(pr3_t -(THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) (THead (Flat -Appl) t (THeads (Flat Appl) t0 (lift (S i) O w))) c (pr3_t (THead (Bind x0) -x1 (THead (Flat Appl) (lift (S O) O t) x2)) (THead (Flat Appl) t (THeads -(Flat Appl) t0 (lift (S i) O w))) c (pr3_t (THead (Flat Appl) t (THead (Bind -x0) x1 x2)) (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O w))) c -(pr3_thin_dx c (THeads (Flat Appl) t0 (lift (S i) O w)) (THead (Bind x0) x1 -x2) (H0 (THead (Bind x0) x1 x2) H6 (\lambda (H11: (iso (THeads (Flat Appl) t0 -(TLRef i)) (THead (Bind x0) x1 x2))).(\lambda (P: -Prop).(iso_flats_lref_bind_false Appl x0 i x1 x2 t0 H11 P)))) t Appl) (THead -(Bind x0) x1 (THead (Flat Appl) (lift (S O) O t) x2)) (pr3_pr2 c (THead (Flat -Appl) t (THead (Bind x0) x1 x2)) (THead (Bind x0) x1 (THead (Flat Appl) (lift -(S O) O t) x2)) (pr2_free c (THead (Flat Appl) t (THead (Bind x0) x1 x2)) -(THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O t) x2)) (pr0_upsilon x0 -H5 t t (pr0_refl t) x1 x1 (pr0_refl x1) x2 x2 (pr0_refl x2))))) (THead (Bind -x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) (pr3_head_12 c x1 x1 -(pr3_refl c x1) (Bind x0) (THead (Flat Appl) (lift (S O) O t) x2) (THead -(Flat Appl) (lift (S O) O x4) x2) (pr3_head_12 (CHead c (Bind x0) x1) (lift -(S O) O t) (lift (S O) O x4) (pr3_lift (CHead c (Bind x0) x1) c (S O) O -(drop_drop (Bind x0) O c c (drop_refl c) x1) t x4 H8) (Flat Appl) x2 x2 -(pr3_refl (CHead (CHead c (Bind x0) x1) (Flat Appl) (lift (S O) O x4)) x2)))) -u2 (pr3_t (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) -(THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) c (pr3_head_12 -c x1 x5 H9 (Bind x0) (THead (Flat Appl) (lift (S O) O x4) x2) (THead (Flat -Appl) (lift (S O) O x4) x3) (pr3_thin_dx (CHead c (Bind x0) x5) x2 x3 H10 -(lift (S O) O x4) Appl)) u2 H7)))))))))))))) H4)) H3)))))))) vs)))))). -(* COMMENTS -Initial nodes: 3759 -END *) - -theorem pr3_iso_appls_cast: - \forall (c: C).(\forall (v: T).(\forall (t: T).(\forall (vs: TList).(let u1 -\def (THeads (Flat Appl) vs (THead (Flat Cast) v t)) in (\forall (u2: -T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to (pr3 c -(THeads (Flat Appl) vs t) u2)))))))) -\def - \lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (vs: -TList).(TList_ind (\lambda (t0: TList).(let u1 \def (THeads (Flat Appl) t0 -(THead (Flat Cast) v t)) in (\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 -u2) \to (\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) t0 t) u2)))))) -(\lambda (u2: T).(\lambda (H: (pr3 c (THead (Flat Cast) v t) u2)).(\lambda -(H0: (((iso (THead (Flat Cast) v t) u2) \to (\forall (P: Prop).P)))).(let H1 -\def (pr3_gen_cast c v t u2 H) in (or_ind (ex3_2 T T (\lambda (u3: -T).(\lambda (t2: T).(eq T u2 (THead (Flat Cast) u3 t2)))) (\lambda (u3: -T).(\lambda (_: T).(pr3 c v u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t -t2)))) (pr3 c t u2) (pr3 c t u2) (\lambda (H2: (ex3_2 T T (\lambda (u3: -T).(\lambda (t2: T).(eq T u2 (THead (Flat Cast) u3 t2)))) (\lambda (u3: -T).(\lambda (_: T).(pr3 c v u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t -t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead -(Flat Cast) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c v u3))) -(\lambda (_: T).(\lambda (t2: T).(pr3 c t t2))) (pr3 c t u2) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H3: (eq T u2 (THead (Flat Cast) x0 -x1))).(\lambda (_: (pr3 c v x0)).(\lambda (_: (pr3 c t x1)).(let H6 \def -(eq_ind T u2 (\lambda (t0: T).((iso (THead (Flat Cast) v t) t0) \to (\forall -(P: Prop).P))) H0 (THead (Flat Cast) x0 x1) H3) in (eq_ind_r T (THead (Flat -Cast) x0 x1) (\lambda (t0: T).(pr3 c t t0)) (H6 (iso_head v x0 t x1 (Flat -Cast)) (pr3 c t (THead (Flat Cast) x0 x1))) u2 H3))))))) H2)) (\lambda (H2: -(pr3 c t u2)).H2) H1))))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H: -((\forall (u2: T).((pr3 c (THeads (Flat Appl) t1 (THead (Flat Cast) v t)) u2) -\to ((((iso (THeads (Flat Appl) t1 (THead (Flat Cast) v t)) u2) \to (\forall -(P: Prop).P))) \to (pr3 c (THeads (Flat Appl) t1 t) u2)))))).(\lambda (u2: -T).(\lambda (H0: (pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead -(Flat Cast) v t))) u2)).(\lambda (H1: (((iso (THead (Flat Appl) t0 (THeads -(Flat Appl) t1 (THead (Flat Cast) v t))) u2) \to (\forall (P: -Prop).P)))).(let H2 \def (pr3_gen_appl c t0 (THeads (Flat Appl) t1 (THead -(Flat Cast) v t)) u2 H0) in (or3_ind (ex3_2 T T (\lambda (u3: T).(\lambda -(t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: -T).(pr3 c t0 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat -Appl) t1 (THead (Flat Cast) v t)) t2)))) (ex4_4 T T T T (\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind -Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: -T).(\lambda (_: T).(pr3 c t0 u3))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat -Cast) v t)) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 -(CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat -Appl) t1 (THead (Flat Cast) v t)) (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u3: T).(\lambda -(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u3) z2)) -u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) -y2) z1 z2)))))))) (pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 t)) u2) -(\lambda (H3: (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead -(Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3))) -(\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat -Cast) v t)) t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 -(THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3))) -(\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat -Cast) v t)) t2))) (pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 t)) u2) -(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T u2 (THead (Flat Appl) -x0 x1))).(\lambda (_: (pr3 c t0 x0)).(\lambda (_: (pr3 c (THeads (Flat Appl) -t1 (THead (Flat Cast) v t)) x1)).(let H7 \def (eq_ind T u2 (\lambda (t2: -T).((iso (THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Flat Cast) v -t))) t2) \to (\forall (P: Prop).P))) H1 (THead (Flat Appl) x0 x1) H4) in -(eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda (t2: T).(pr3 c (THead (Flat -Appl) t0 (THeads (Flat Appl) t1 t)) t2)) (H7 (iso_head t0 x0 (THeads (Flat -Appl) t1 (THead (Flat Cast) v t)) x1 (Flat Appl)) (pr3 c (THead (Flat Appl) -t0 (THeads (Flat Appl) t1 t)) (THead (Flat Appl) x0 x1))) u2 H4))))))) H3)) -(\lambda (H3: (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: -T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3))))) -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c -(THeads (Flat Appl) t1 (THead (Flat Cast) v t)) (THead (Bind Abst) y1 -z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: -T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 -t2))))))))).(ex4_4_ind T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: -T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3))))) -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c -(THeads (Flat Appl) t1 (THead (Flat Cast) v t)) (THead (Bind Abst) y1 -z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: -T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2))))))) -(pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 t)) u2) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H4: (pr3 c -(THead (Bind Abbr) x2 x3) u2)).(\lambda (H5: (pr3 c t0 x2)).(\lambda (H6: -(pr3 c (THeads (Flat Appl) t1 (THead (Flat Cast) v t)) (THead (Bind Abst) x0 -x1))).(\lambda (H7: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) -u) x1 x3))))).(pr3_t (THead (Bind Abbr) t0 x1) (THead (Flat Appl) t0 (THeads -(Flat Appl) t1 t)) c (pr3_t (THead (Flat Appl) t0 (THead (Bind Abst) x0 x1)) -(THead (Flat Appl) t0 (THeads (Flat Appl) t1 t)) c (pr3_thin_dx c (THeads -(Flat Appl) t1 t) (THead (Bind Abst) x0 x1) (H (THead (Bind Abst) x0 x1) H6 -(\lambda (H8: (iso (THeads (Flat Appl) t1 (THead (Flat Cast) v t)) (THead -(Bind Abst) x0 x1))).(\lambda (P: Prop).(iso_flats_flat_bind_false Appl Cast -Abst x0 v x1 t t1 H8 P)))) t0 Appl) (THead (Bind Abbr) t0 x1) (pr3_pr2 c -(THead (Flat Appl) t0 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) t0 x1) -(pr2_free c (THead (Flat Appl) t0 (THead (Bind Abst) x0 x1)) (THead (Bind -Abbr) t0 x1) (pr0_beta x0 t0 t0 (pr0_refl t0) x1 x1 (pr0_refl x1))))) u2 -(pr3_t (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) t0 x1) c (pr3_head_12 c -t0 x2 H5 (Bind Abbr) x1 x3 (H7 Abbr x2)) u2 H4)))))))))) H3)) (\lambda (H3: -(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat Cast) v t)) (THead (Bind b) -y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: -T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat -Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t0 -u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c -(THeads (Flat Appl) t1 (THead (Flat Cast) v t)) (THead (Bind b) y1 z1)))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda -(u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift -(S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 -(CHead c (Bind b) y2) z1 z2))))))) (pr3 c (THead (Flat Appl) t0 (THeads (Flat -Appl) t1 t)) u2) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda -(x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H4: (not (eq B x0 -Abst))).(\lambda (H5: (pr3 c (THeads (Flat Appl) t1 (THead (Flat Cast) v t)) -(THead (Bind x0) x1 x2))).(\lambda (H6: (pr3 c (THead (Bind x0) x5 (THead -(Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda (H7: (pr3 c t0 x4)).(\lambda -(H8: (pr3 c x1 x5)).(\lambda (H9: (pr3 (CHead c (Bind x0) x5) x2 x3)).(pr3_t -(THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) (THead (Flat -Appl) t0 (THeads (Flat Appl) t1 t)) c (pr3_t (THead (Bind x0) x1 (THead (Flat -Appl) (lift (S O) O t0) x2)) (THead (Flat Appl) t0 (THeads (Flat Appl) t1 t)) -c (pr3_t (THead (Flat Appl) t0 (THead (Bind x0) x1 x2)) (THead (Flat Appl) t0 -(THeads (Flat Appl) t1 t)) c (pr3_thin_dx c (THeads (Flat Appl) t1 t) (THead -(Bind x0) x1 x2) (H (THead (Bind x0) x1 x2) H5 (\lambda (H10: (iso (THeads -(Flat Appl) t1 (THead (Flat Cast) v t)) (THead (Bind x0) x1 x2))).(\lambda -(P: Prop).(iso_flats_flat_bind_false Appl Cast x0 x1 v x2 t t1 H10 P)))) t0 -Appl) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O t0) x2)) (pr3_pr2 -c (THead (Flat Appl) t0 (THead (Bind x0) x1 x2)) (THead (Bind x0) x1 (THead -(Flat Appl) (lift (S O) O t0) x2)) (pr2_free c (THead (Flat Appl) t0 (THead -(Bind x0) x1 x2)) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O t0) -x2)) (pr0_upsilon x0 H4 t0 t0 (pr0_refl t0) x1 x1 (pr0_refl x1) x2 x2 -(pr0_refl x2))))) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) -x2)) (pr3_head_12 c x1 x1 (pr3_refl c x1) (Bind x0) (THead (Flat Appl) (lift -(S O) O t0) x2) (THead (Flat Appl) (lift (S O) O x4) x2) (pr3_head_12 (CHead -c (Bind x0) x1) (lift (S O) O t0) (lift (S O) O x4) (pr3_lift (CHead c (Bind -x0) x1) c (S O) O (drop_drop (Bind x0) O c c (drop_refl c) x1) t0 x4 H7) -(Flat Appl) x2 x2 (pr3_refl (CHead (CHead c (Bind x0) x1) (Flat Appl) (lift -(S O) O x4)) x2)))) u2 (pr3_t (THead (Bind x0) x5 (THead (Flat Appl) (lift (S -O) O x4) x3)) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) c -(pr3_head_12 c x1 x5 H8 (Bind x0) (THead (Flat Appl) (lift (S O) O x4) x2) -(THead (Flat Appl) (lift (S O) O x4) x3) (pr3_thin_dx (CHead c (Bind x0) x5) -x2 x3 H9 (lift (S O) O x4) Appl)) u2 H6)))))))))))))) H3)) H2)))))))) vs)))). -(* COMMENTS -Initial nodes: 3297 -END *) - -theorem pr3_iso_appl_bind: - \forall (b: B).((not (eq B b Abst)) \to (\forall (v1: T).(\forall (v2: -T).(\forall (t: T).(let u1 \def (THead (Flat Appl) v1 (THead (Bind b) v2 t)) -in (\forall (c: C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to -(\forall (P: Prop).P))) \to (pr3 c (THead (Bind b) v2 (THead (Flat Appl) -(lift (S O) O v1) t)) u2)))))))))) -\def - \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (v1: T).(\lambda -(v2: T).(\lambda (t: T).(\lambda (c: C).(\lambda (u2: T).(\lambda (H0: (pr3 c -(THead (Flat Appl) v1 (THead (Bind b) v2 t)) u2)).(\lambda (H1: (((iso (THead -(Flat Appl) v1 (THead (Bind b) v2 t)) u2) \to (\forall (P: Prop).P)))).(let -H2 \def (pr3_gen_appl c v1 (THead (Bind b) v2 t) u2 H0) in (or3_ind (ex3_2 T -T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) -(\lambda (u3: T).(\lambda (_: T).(pr3 c v1 u3))) (\lambda (_: T).(\lambda -(t2: T).(pr3 c (THead (Bind b) v2 t) t2)))) (ex4_4 T T T T (\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind -Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: -T).(\lambda (_: T).(pr3 c v1 u3))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind b) v2 t) (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(t2: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) z1 -t2)))))))) (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) -(\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(pr3 c (THead (Bind b) v2 t) (THead (Bind b0) y1 -z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: -T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b0) y2 (THead (Flat -Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v1 -u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b0: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2)))))))) (pr3 c (THead (Bind b) v2 -(THead (Flat Appl) (lift (S O) O v1) t)) u2) (\lambda (H3: (ex3_2 T T -(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) -(\lambda (u3: T).(\lambda (_: T).(pr3 c v1 u3))) (\lambda (_: T).(\lambda -(t2: T).(pr3 c (THead (Bind b) v2 t) t2))))).(ex3_2_ind T T (\lambda (u3: -T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: -T).(\lambda (_: T).(pr3 c v1 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c -(THead (Bind b) v2 t) t2))) (pr3 c (THead (Bind b) v2 (THead (Flat Appl) -(lift (S O) O v1) t)) u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq -T u2 (THead (Flat Appl) x0 x1))).(\lambda (_: (pr3 c v1 x0)).(\lambda (_: -(pr3 c (THead (Bind b) v2 t) x1)).(let H7 \def (eq_ind T u2 (\lambda (t0: -T).((iso (THead (Flat Appl) v1 (THead (Bind b) v2 t)) t0) \to (\forall (P: -Prop).P))) H1 (THead (Flat Appl) x0 x1) H4) in (eq_ind_r T (THead (Flat Appl) -x0 x1) (\lambda (t0: T).(pr3 c (THead (Bind b) v2 (THead (Flat Appl) (lift (S -O) O v1) t)) t0)) (H7 (iso_head v1 x0 (THead (Bind b) v2 t) x1 (Flat Appl)) -(pr3 c (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) t)) (THead -(Flat Appl) x0 x1))) u2 H4))))))) H3)) (\lambda (H3: (ex4_4 T T T T (\lambda -(_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind -Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: -T).(\lambda (_: T).(pr3 c v1 u3))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind b) v2 t) (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(t2: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) z1 -t2))))))))).(ex4_4_ind T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: -T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v1 u3))))) -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c -(THead (Bind b) v2 t) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b0: B).(\forall (u: -T).(pr3 (CHead c (Bind b0) u) z1 t2))))))) (pr3 c (THead (Bind b) v2 (THead -(Flat Appl) (lift (S O) O v1) t)) u2) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H4: (pr3 c (THead (Bind Abbr) -x2 x3) u2)).(\lambda (H5: (pr3 c v1 x2)).(\lambda (H6: (pr3 c (THead (Bind b) -v2 t) (THead (Bind Abst) x0 x1))).(\lambda (H7: ((\forall (b0: B).(\forall -(u: T).(pr3 (CHead c (Bind b0) u) x1 x3))))).(pr3_t (THead (Bind Abbr) x2 x3) -(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) t)) c (let H_x \def -(pr3_gen_bind b H c v2 t (THead (Bind Abst) x0 x1) H6) in (let H8 \def H_x in -(or_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind Abst) -x0 x1) (THead (Bind b) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c v2 -u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind b) v2) t t2)))) -(pr3 (CHead c (Bind b) v2) t (lift (S O) O (THead (Bind Abst) x0 x1))) (pr3 c -(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) t)) (THead (Bind -Abbr) x2 x3)) (\lambda (H9: (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq -T (THead (Bind Abst) x0 x1) (THead (Bind b) u3 t2)))) (\lambda (u3: -T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 -(CHead c (Bind b) v2) t t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: -T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) u3 t2)))) (\lambda (u3: -T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 -(CHead c (Bind b) v2) t t2))) (pr3 c (THead (Bind b) v2 (THead (Flat Appl) -(lift (S O) O v1) t)) (THead (Bind Abbr) x2 x3)) (\lambda (x4: T).(\lambda -(x5: T).(\lambda (H10: (eq T (THead (Bind Abst) x0 x1) (THead (Bind b) x4 -x5))).(\lambda (H11: (pr3 c v2 x4)).(\lambda (H12: (pr3 (CHead c (Bind b) v2) -t x5)).(let H13 \def (f_equal T B (\lambda (e: T).(match e in T return -(\lambda (_: T).B) with [(TSort _) \Rightarrow Abst | (TLRef _) \Rightarrow -Abst | (THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).B) with -[(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abst])])) (THead (Bind Abst) -x0 x1) (THead (Bind b) x4 x5) H10) in ((let H14 \def (f_equal T T (\lambda -(e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 -| (TLRef _) \Rightarrow x0 | (THead _ t0 _) \Rightarrow t0])) (THead (Bind -Abst) x0 x1) (THead (Bind b) x4 x5) H10) in ((let H15 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow x1 | (TLRef _) \Rightarrow x1 | (THead _ _ t0) \Rightarrow t0])) -(THead (Bind Abst) x0 x1) (THead (Bind b) x4 x5) H10) in (\lambda (H16: (eq T -x0 x4)).(\lambda (H17: (eq B Abst b)).(let H18 \def (eq_ind_r T x5 (\lambda -(t0: T).(pr3 (CHead c (Bind b) v2) t t0)) H12 x1 H15) in (let H19 \def -(eq_ind_r T x4 (\lambda (t0: T).(pr3 c v2 t0)) H11 x0 H16) in (let H20 \def -(eq_ind_r B b (\lambda (b0: B).(pr3 (CHead c (Bind b0) v2) t x1)) H18 Abst -H17) in (let H21 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 Abst))) H -Abst H17) in (eq_ind B Abst (\lambda (b0: B).(pr3 c (THead (Bind b0) v2 -(THead (Flat Appl) (lift (S O) O v1) t)) (THead (Bind Abbr) x2 x3))) (let H22 -\def (match (H21 (refl_equal B Abst)) in False return (\lambda (_: -False).(pr3 c (THead (Bind Abst) v2 (THead (Flat Appl) (lift (S O) O v1) t)) -(THead (Bind Abbr) x2 x3))) with []) in H22) b H17)))))))) H14)) H13))))))) -H9)) (\lambda (H9: (pr3 (CHead c (Bind b) v2) t (lift (S O) O (THead (Bind -Abst) x0 x1)))).(pr3_t (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O -x2) (lift (S O) O (THead (Bind Abst) x0 x1)))) (THead (Bind b) v2 (THead -(Flat Appl) (lift (S O) O v1) t)) c (pr3_head_2 c v2 (THead (Flat Appl) (lift -(S O) O v1) t) (THead (Flat Appl) (lift (S O) O x2) (lift (S O) O (THead -(Bind Abst) x0 x1))) (Bind b) (pr3_flat (CHead c (Bind b) v2) (lift (S O) O -v1) (lift (S O) O x2) (pr3_lift (CHead c (Bind b) v2) c (S O) O (drop_drop -(Bind b) O c c (drop_refl c) v2) v1 x2 H5) t (lift (S O) O (THead (Bind Abst) -x0 x1)) H9 Appl)) (THead (Bind Abbr) x2 x3) (eq_ind T (lift (S O) O (THead -(Flat Appl) x2 (THead (Bind Abst) x0 x1))) (\lambda (t0: T).(pr3 c (THead -(Bind b) v2 t0) (THead (Bind Abbr) x2 x3))) (pr3_sing c (THead (Bind Abbr) x2 -x1) (THead (Bind b) v2 (lift (S O) O (THead (Flat Appl) x2 (THead (Bind Abst) -x0 x1)))) (pr2_free c (THead (Bind b) v2 (lift (S O) O (THead (Flat Appl) x2 -(THead (Bind Abst) x0 x1)))) (THead (Bind Abbr) x2 x1) (pr0_zeta b H (THead -(Flat Appl) x2 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) x2 x1) (pr0_beta -x0 x2 x2 (pr0_refl x2) x1 x1 (pr0_refl x1)) v2)) (THead (Bind Abbr) x2 x3) -(pr3_head_12 c x2 x2 (pr3_refl c x2) (Bind Abbr) x1 x3 (H7 Abbr x2))) (THead -(Flat Appl) (lift (S O) O x2) (lift (S O) O (THead (Bind Abst) x0 x1))) -(lift_flat Appl x2 (THead (Bind Abst) x0 x1) (S O) O)))) H8))) u2 H4))))))))) -H3)) (\lambda (H3: (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B -b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind b) v2 t) (THead -(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b0) -y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda -(_: T).(pr3 c v1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2))))))))).(ex6_6_ind -B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(pr3 c (THead (Bind b) v2 t) (THead (Bind b0) y1 z1)))))))) (\lambda -(b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u3: -T).(\lambda (y2: T).(pr3 c (THead (Bind b0) y2 (THead (Flat Appl) (lift (S O) -O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v1 u3))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 c y1 y2))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) -y2) z1 z2))))))) (pr3 c (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O -v1) t)) u2) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: -T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H4: (not (eq B x0 -Abst))).(\lambda (H5: (pr3 c (THead (Bind b) v2 t) (THead (Bind x0) x1 -x2))).(\lambda (H6: (pr3 c (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) -O x4) x3)) u2)).(\lambda (H7: (pr3 c v1 x4)).(\lambda (H8: (pr3 c x1 -x5)).(\lambda (H9: (pr3 (CHead c (Bind x0) x5) x2 x3)).(pr3_t (THead (Bind -x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) (THead (Bind b) v2 (THead -(Flat Appl) (lift (S O) O v1) t)) c (let H_x \def (pr3_gen_bind b H c v2 t -(THead (Bind x0) x1 x2) H5) in (let H10 \def H_x in (or_ind (ex3_2 T T -(\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind -b) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: -T).(\lambda (t2: T).(pr3 (CHead c (Bind b) v2) t t2)))) (pr3 (CHead c (Bind -b) v2) t (lift (S O) O (THead (Bind x0) x1 x2))) (pr3 c (THead (Bind b) v2 -(THead (Flat Appl) (lift (S O) O v1) t)) (THead (Bind x0) x5 (THead (Flat -Appl) (lift (S O) O x4) x3))) (\lambda (H11: (ex3_2 T T (\lambda (u3: -T).(\lambda (t2: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) u3 t2)))) -(\lambda (u3: T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: T).(\lambda -(t2: T).(pr3 (CHead c (Bind b) v2) t t2))))).(ex3_2_ind T T (\lambda (u3: -T).(\lambda (t2: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) u3 t2)))) -(\lambda (u3: T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: T).(\lambda -(t2: T).(pr3 (CHead c (Bind b) v2) t t2))) (pr3 c (THead (Bind b) v2 (THead -(Flat Appl) (lift (S O) O v1) t)) (THead (Bind x0) x5 (THead (Flat Appl) -(lift (S O) O x4) x3))) (\lambda (x6: T).(\lambda (x7: T).(\lambda (H12: (eq -T (THead (Bind x0) x1 x2) (THead (Bind b) x6 x7))).(\lambda (H13: (pr3 c v2 -x6)).(\lambda (H14: (pr3 (CHead c (Bind b) v2) t x7)).(let H15 \def (f_equal -T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with [(TSort _) -\Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead k _ _) \Rightarrow (match -k in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) -\Rightarrow x0])])) (THead (Bind x0) x1 x2) (THead (Bind b) x6 x7) H12) in -((let H16 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: -T).T) with [(TSort _) \Rightarrow x1 | (TLRef _) \Rightarrow x1 | (THead _ t0 -_) \Rightarrow t0])) (THead (Bind x0) x1 x2) (THead (Bind b) x6 x7) H12) in -((let H17 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: -T).T) with [(TSort _) \Rightarrow x2 | (TLRef _) \Rightarrow x2 | (THead _ _ -t0) \Rightarrow t0])) (THead (Bind x0) x1 x2) (THead (Bind b) x6 x7) H12) in -(\lambda (H18: (eq T x1 x6)).(\lambda (H19: (eq B x0 b)).(let H20 \def -(eq_ind_r T x7 (\lambda (t0: T).(pr3 (CHead c (Bind b) v2) t t0)) H14 x2 H17) -in (let H21 \def (eq_ind_r T x6 (\lambda (t0: T).(pr3 c v2 t0)) H13 x1 H18) -in (let H22 \def (eq_ind B x0 (\lambda (b0: B).(pr3 (CHead c (Bind b0) x5) x2 -x3)) H9 b H19) in (let H23 \def (eq_ind B x0 (\lambda (b0: B).(not (eq B b0 -Abst))) H4 b H19) in (eq_ind_r B b (\lambda (b0: B).(pr3 c (THead (Bind b) v2 -(THead (Flat Appl) (lift (S O) O v1) t)) (THead (Bind b0) x5 (THead (Flat -Appl) (lift (S O) O x4) x3)))) (pr3_head_21 c v2 x5 (pr3_t x1 v2 c H21 x5 H8) -(Bind b) (THead (Flat Appl) (lift (S O) O v1) t) (THead (Flat Appl) (lift (S -O) O x4) x3) (pr3_flat (CHead c (Bind b) v2) (lift (S O) O v1) (lift (S O) O -x4) (pr3_lift (CHead c (Bind b) v2) c (S O) O (drop_drop (Bind b) O c c -(drop_refl c) v2) v1 x4 H7) t x3 (pr3_t x2 t (CHead c (Bind b) v2) H20 x3 -(pr3_pr3_pr3_t c v2 x1 H21 x2 x3 (Bind b) (pr3_pr3_pr3_t c x1 x5 H8 x2 x3 -(Bind b) H22))) Appl)) x0 H19)))))))) H16)) H15))))))) H11)) (\lambda (H11: -(pr3 (CHead c (Bind b) v2) t (lift (S O) O (THead (Bind x0) x1 x2)))).(pr3_t -(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O x4) (lift (S O) O (THead -(Bind x0) x1 x2)))) (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) -t)) c (pr3_head_2 c v2 (THead (Flat Appl) (lift (S O) O v1) t) (THead (Flat -Appl) (lift (S O) O x4) (lift (S O) O (THead (Bind x0) x1 x2))) (Bind b) -(pr3_flat (CHead c (Bind b) v2) (lift (S O) O v1) (lift (S O) O x4) (pr3_lift -(CHead c (Bind b) v2) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) v2) -v1 x4 H7) t (lift (S O) O (THead (Bind x0) x1 x2)) H11 Appl)) (THead (Bind -x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) (eq_ind T (lift (S O) O -(THead (Flat Appl) x4 (THead (Bind x0) x1 x2))) (\lambda (t0: T).(pr3 c -(THead (Bind b) v2 t0) (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O -x4) x3)))) (pr3_sing c (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O -x4) x2)) (THead (Bind b) v2 (lift (S O) O (THead (Flat Appl) x4 (THead (Bind -x0) x1 x2)))) (pr2_free c (THead (Bind b) v2 (lift (S O) O (THead (Flat Appl) -x4 (THead (Bind x0) x1 x2)))) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S -O) O x4) x2)) (pr0_zeta b H (THead (Flat Appl) x4 (THead (Bind x0) x1 x2)) -(THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) (pr0_upsilon x0 -H4 x4 x4 (pr0_refl x4) x1 x1 (pr0_refl x1) x2 x2 (pr0_refl x2)) v2)) (THead -(Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) (pr3_head_12 c x1 x5 -H8 (Bind x0) (THead (Flat Appl) (lift (S O) O x4) x2) (THead (Flat Appl) -(lift (S O) O x4) x3) (pr3_thin_dx (CHead c (Bind x0) x5) x2 x3 H9 (lift (S -O) O x4) Appl))) (THead (Flat Appl) (lift (S O) O x4) (lift (S O) O (THead -(Bind x0) x1 x2))) (lift_flat Appl x4 (THead (Bind x0) x1 x2) (S O) O)))) -H10))) u2 H6))))))))))))) H3)) H2)))))))))). -(* COMMENTS -Initial nodes: 4805 -END *) - -theorem pr3_iso_appls_appl_bind: - \forall (b: B).((not (eq B b Abst)) \to (\forall (v: T).(\forall (u: -T).(\forall (t: T).(\forall (vs: TList).(let u1 \def (THeads (Flat Appl) vs -(THead (Flat Appl) v (THead (Bind b) u t))) in (\forall (c: C).(\forall (u2: -T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to (pr3 c -(THeads (Flat Appl) vs (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) -t))) u2))))))))))) -\def - \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (v: T).(\lambda -(u: T).(\lambda (t: T).(\lambda (vs: TList).(TList_ind (\lambda (t0: -TList).(let u1 \def (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind -b) u t))) in (\forall (c: C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 -u2) \to (\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) t0 (THead -(Bind b) u (THead (Flat Appl) (lift (S O) O v) t))) u2))))))) (\lambda (c: -C).(\lambda (u2: T).(\lambda (H0: (pr3 c (THead (Flat Appl) v (THead (Bind b) -u t)) u2)).(\lambda (H1: (((iso (THead (Flat Appl) v (THead (Bind b) u t)) -u2) \to (\forall (P: Prop).P)))).(pr3_iso_appl_bind b H v u t c u2 H0 H1))))) -(\lambda (t0: T).(\lambda (t1: TList).(\lambda (H0: ((\forall (c: C).(\forall -(u2: T).((pr3 c (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u -t))) u2) \to ((((iso (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind -b) u t))) u2) \to (\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) t1 -(THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t))) u2))))))).(\lambda -(c: C).(\lambda (u2: T).(\lambda (H1: (pr3 c (THead (Flat Appl) t0 (THeads -(Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t)))) u2)).(\lambda -(H2: (((iso (THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Flat Appl) v -(THead (Bind b) u t)))) u2) \to (\forall (P: Prop).P)))).(let H3 \def -(pr3_gen_appl c t0 (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind -b) u t))) u2 H1) in (or3_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq -T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t0 -u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t1 (THead -(Flat Appl) v (THead (Bind b) u t))) t2)))) (ex4_4 T T T T (\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind -Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: -T).(\lambda (_: T).(pr3 c t0 u3))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat -Appl) v (THead (Bind b) u t))) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b0: -B).(\forall (u0: T).(pr3 (CHead c (Bind b0) u0) z1 t2)))))))) (ex6_6 B T T T -T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 -c (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t))) (THead -(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b0) -y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda -(_: T).(pr3 c t0 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2)))))))) (pr3 c -(THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat -Appl) (lift (S O) O v) t)))) u2) (\lambda (H4: (ex3_2 T T (\lambda (u3: -T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: -T).(\lambda (_: T).(pr3 c t0 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c -(THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t))) -t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead -(Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3))) -(\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat -Appl) v (THead (Bind b) u t))) t2))) (pr3 c (THead (Flat Appl) t0 (THeads -(Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) -u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T u2 (THead (Flat -Appl) x0 x1))).(\lambda (_: (pr3 c t0 x0)).(\lambda (_: (pr3 c (THeads (Flat -Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t))) x1)).(let H8 \def -(eq_ind T u2 (\lambda (t2: T).((iso (THead (Flat Appl) t0 (THeads (Flat Appl) -t1 (THead (Flat Appl) v (THead (Bind b) u t)))) t2) \to (\forall (P: -Prop).P))) H2 (THead (Flat Appl) x0 x1) H5) in (eq_ind_r T (THead (Flat Appl) -x0 x1) (\lambda (t2: T).(pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 -(THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) t2)) (H8 -(iso_head t0 x0 (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u -t))) x1 (Flat Appl)) (pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 -(THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) (THead (Flat -Appl) x0 x1))) u2 H5))))))) H4)) (\lambda (H4: (ex4_4 T T T T (\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind -Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: -T).(\lambda (_: T).(pr3 c t0 u3))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat -Appl) v (THead (Bind b) u t))) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b0: -B).(\forall (u0: T).(pr3 (CHead c (Bind b0) u0) z1 t2))))))))).(ex4_4_ind T T -T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c -(THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u3: T).(\lambda (_: T).(pr3 c t0 u3))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat -Appl) v (THead (Bind b) u t))) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b0: -B).(\forall (u0: T).(pr3 (CHead c (Bind b0) u0) z1 t2))))))) (pr3 c (THead -(Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) -(lift (S O) O v) t)))) u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: -T).(\lambda (x3: T).(\lambda (H5: (pr3 c (THead (Bind Abbr) x2 x3) -u2)).(\lambda (H6: (pr3 c t0 x2)).(\lambda (H7: (pr3 c (THeads (Flat Appl) t1 -(THead (Flat Appl) v (THead (Bind b) u t))) (THead (Bind Abst) x0 -x1))).(\lambda (H8: ((\forall (b0: B).(\forall (u0: T).(pr3 (CHead c (Bind -b0) u0) x1 x3))))).(pr3_t (THead (Bind Abbr) t0 x1) (THead (Flat Appl) t0 -(THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) -t)))) c (pr3_t (THead (Flat Appl) t0 (THead (Bind Abst) x0 x1)) (THead (Flat -Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) (lift (S -O) O v) t)))) c (pr3_thin_dx c (THeads (Flat Appl) t1 (THead (Bind b) u -(THead (Flat Appl) (lift (S O) O v) t))) (THead (Bind Abst) x0 x1) (H0 c -(THead (Bind Abst) x0 x1) H7 (\lambda (H9: (iso (THeads (Flat Appl) t1 (THead -(Flat Appl) v (THead (Bind b) u t))) (THead (Bind Abst) x0 x1))).(\lambda (P: -Prop).(iso_flats_flat_bind_false Appl Appl Abst x0 v x1 (THead (Bind b) u t) -t1 H9 P)))) t0 Appl) (THead (Bind Abbr) t0 x1) (pr3_pr2 c (THead (Flat Appl) -t0 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) t0 x1) (pr2_free c (THead -(Flat Appl) t0 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) t0 x1) (pr0_beta -x0 t0 t0 (pr0_refl t0) x1 x1 (pr0_refl x1))))) u2 (pr3_t (THead (Bind Abbr) -x2 x3) (THead (Bind Abbr) t0 x1) c (pr3_head_12 c t0 x2 H6 (Bind Abbr) x1 x3 -(H8 Abbr x2)) u2 H5)))))))))) H4)) (\lambda (H4: (ex6_6 B T T T T T (\lambda -(b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c -(THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t))) (THead -(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b0) -y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda -(_: T).(pr3 c t0 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2))))))))).(ex6_6_ind -B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u -t))) (THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind -b0) y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda -(_: T).(pr3 c t0 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2))))))) (pr3 c -(THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat -Appl) (lift (S O) O v) t)))) u2) (\lambda (x0: B).(\lambda (x1: T).(\lambda -(x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H5: (not -(eq B x0 Abst))).(\lambda (H6: (pr3 c (THeads (Flat Appl) t1 (THead (Flat -Appl) v (THead (Bind b) u t))) (THead (Bind x0) x1 x2))).(\lambda (H7: (pr3 c -(THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda -(H8: (pr3 c t0 x4)).(\lambda (H9: (pr3 c x1 x5)).(\lambda (H10: (pr3 (CHead c -(Bind x0) x5) x2 x3)).(pr3_t (THead (Bind x0) x1 (THead (Flat Appl) (lift (S -O) O x4) x2)) (THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u -(THead (Flat Appl) (lift (S O) O v) t)))) c (pr3_t (THead (Bind x0) x1 (THead -(Flat Appl) (lift (S O) O t0) x2)) (THead (Flat Appl) t0 (THeads (Flat Appl) -t1 (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) c (pr3_t -(THead (Flat Appl) t0 (THead (Bind x0) x1 x2)) (THead (Flat Appl) t0 (THeads -(Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) c -(pr3_thin_dx c (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) -(lift (S O) O v) t))) (THead (Bind x0) x1 x2) (H0 c (THead (Bind x0) x1 x2) -H6 (\lambda (H11: (iso (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead -(Bind b) u t))) (THead (Bind x0) x1 x2))).(\lambda (P: -Prop).(iso_flats_flat_bind_false Appl Appl x0 x1 v x2 (THead (Bind b) u t) t1 -H11 P)))) t0 Appl) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O t0) -x2)) (pr3_pr2 c (THead (Flat Appl) t0 (THead (Bind x0) x1 x2)) (THead (Bind -x0) x1 (THead (Flat Appl) (lift (S O) O t0) x2)) (pr2_free c (THead (Flat -Appl) t0 (THead (Bind x0) x1 x2)) (THead (Bind x0) x1 (THead (Flat Appl) -(lift (S O) O t0) x2)) (pr0_upsilon x0 H5 t0 t0 (pr0_refl t0) x1 x1 (pr0_refl -x1) x2 x2 (pr0_refl x2))))) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S -O) O x4) x2)) (pr3_head_12 c x1 x1 (pr3_refl c x1) (Bind x0) (THead (Flat -Appl) (lift (S O) O t0) x2) (THead (Flat Appl) (lift (S O) O x4) x2) -(pr3_head_12 (CHead c (Bind x0) x1) (lift (S O) O t0) (lift (S O) O x4) -(pr3_lift (CHead c (Bind x0) x1) c (S O) O (drop_drop (Bind x0) O c c -(drop_refl c) x1) t0 x4 H8) (Flat Appl) x2 x2 (pr3_refl (CHead (CHead c (Bind -x0) x1) (Flat Appl) (lift (S O) O x4)) x2)))) u2 (pr3_t (THead (Bind x0) x5 -(THead (Flat Appl) (lift (S O) O x4) x3)) (THead (Bind x0) x1 (THead (Flat -Appl) (lift (S O) O x4) x2)) c (pr3_head_12 c x1 x5 H9 (Bind x0) (THead (Flat -Appl) (lift (S O) O x4) x2) (THead (Flat Appl) (lift (S O) O x4) x3) -(pr3_thin_dx (CHead c (Bind x0) x5) x2 x3 H10 (lift (S O) O x4) Appl)) u2 -H7)))))))))))))) H4)) H3))))))))) vs)))))). -(* COMMENTS -Initial nodes: 3571 -END *) - -theorem pr3_iso_appls_bind: - \forall (b: B).((not (eq B b Abst)) \to (\forall (vs: TList).(\forall (u: -T).(\forall (t: T).(let u1 \def (THeads (Flat Appl) vs (THead (Bind b) u t)) -in (\forall (c: C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to -(\forall (P: Prop).P))) \to (pr3 c (THead (Bind b) u (THeads (Flat Appl) -(lifts (S O) O vs) t)) u2)))))))))) -\def - \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (vs: -TList).(tlist_ind_rev (\lambda (t: TList).(\forall (u: T).(\forall (t0: -T).(let u1 \def (THeads (Flat Appl) t (THead (Bind b) u t0)) in (\forall (c: -C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: -Prop).P))) \to (pr3 c (THead (Bind b) u (THeads (Flat Appl) (lifts (S O) O t) -t0)) u2))))))))) (\lambda (u: T).(\lambda (t: T).(\lambda (c: C).(\lambda -(u2: T).(\lambda (H0: (pr3 c (THead (Bind b) u t) u2)).(\lambda (_: (((iso -(THead (Bind b) u t) u2) \to (\forall (P: Prop).P)))).H0)))))) (\lambda (ts: -TList).(\lambda (t: T).(\lambda (H0: ((\forall (u: T).(\forall (t0: -T).(\forall (c: C).(\forall (u2: T).((pr3 c (THeads (Flat Appl) ts (THead -(Bind b) u t0)) u2) \to ((((iso (THeads (Flat Appl) ts (THead (Bind b) u t0)) -u2) \to (\forall (P: Prop).P))) \to (pr3 c (THead (Bind b) u (THeads (Flat -Appl) (lifts (S O) O ts) t0)) u2))))))))).(\lambda (u: T).(\lambda (t0: -T).(\lambda (c: C).(\lambda (u2: T).(\lambda (H1: (pr3 c (THeads (Flat Appl) -(TApp ts t) (THead (Bind b) u t0)) u2)).(\lambda (H2: (((iso (THeads (Flat -Appl) (TApp ts t) (THead (Bind b) u t0)) u2) \to (\forall (P: -Prop).P)))).(eq_ind_r TList (TApp (lifts (S O) O ts) (lift (S O) O t)) -(\lambda (t1: TList).(pr3 c (THead (Bind b) u (THeads (Flat Appl) t1 t0)) -u2)) (eq_ind_r T (THeads (Flat Appl) (lifts (S O) O ts) (THead (Flat Appl) -(lift (S O) O t) t0)) (\lambda (t1: T).(pr3 c (THead (Bind b) u t1) u2)) (let -H3 \def (eq_ind T (THeads (Flat Appl) (TApp ts t) (THead (Bind b) u t0)) -(\lambda (t1: T).(pr3 c t1 u2)) H1 (THeads (Flat Appl) ts (THead (Flat Appl) -t (THead (Bind b) u t0))) (theads_tapp (Flat Appl) t (THead (Bind b) u t0) -ts)) in (let H4 \def (eq_ind T (THeads (Flat Appl) (TApp ts t) (THead (Bind -b) u t0)) (\lambda (t1: T).((iso t1 u2) \to (\forall (P: Prop).P))) H2 -(THeads (Flat Appl) ts (THead (Flat Appl) t (THead (Bind b) u t0))) -(theads_tapp (Flat Appl) t (THead (Bind b) u t0) ts)) in (TList_ind (\lambda -(t1: TList).(((\forall (u0: T).(\forall (t2: T).(\forall (c0: C).(\forall -(u3: T).((pr3 c0 (THeads (Flat Appl) t1 (THead (Bind b) u0 t2)) u3) \to -((((iso (THeads (Flat Appl) t1 (THead (Bind b) u0 t2)) u3) \to (\forall (P: -Prop).P))) \to (pr3 c0 (THead (Bind b) u0 (THeads (Flat Appl) (lifts (S O) O -t1) t2)) u3)))))))) \to ((pr3 c (THeads (Flat Appl) t1 (THead (Flat Appl) t -(THead (Bind b) u t0))) u2) \to ((((iso (THeads (Flat Appl) t1 (THead (Flat -Appl) t (THead (Bind b) u t0))) u2) \to (\forall (P: Prop).P))) \to (pr3 c -(THead (Bind b) u (THeads (Flat Appl) (lifts (S O) O t1) (THead (Flat Appl) -(lift (S O) O t) t0))) u2))))) (\lambda (_: ((\forall (u0: T).(\forall (t1: -T).(\forall (c0: C).(\forall (u3: T).((pr3 c0 (THeads (Flat Appl) TNil (THead -(Bind b) u0 t1)) u3) \to ((((iso (THeads (Flat Appl) TNil (THead (Bind b) u0 -t1)) u3) \to (\forall (P: Prop).P))) \to (pr3 c0 (THead (Bind b) u0 (THeads -(Flat Appl) (lifts (S O) O TNil) t1)) u3))))))))).(\lambda (H6: (pr3 c -(THeads (Flat Appl) TNil (THead (Flat Appl) t (THead (Bind b) u t0))) -u2)).(\lambda (H7: (((iso (THeads (Flat Appl) TNil (THead (Flat Appl) t -(THead (Bind b) u t0))) u2) \to (\forall (P: Prop).P)))).(pr3_iso_appl_bind b -H t u t0 c u2 H6 H7)))) (\lambda (t1: T).(\lambda (ts0: TList).(\lambda (_: -((((\forall (u0: T).(\forall (t2: T).(\forall (c0: C).(\forall (u3: T).((pr3 -c0 (THeads (Flat Appl) ts0 (THead (Bind b) u0 t2)) u3) \to ((((iso (THeads -(Flat Appl) ts0 (THead (Bind b) u0 t2)) u3) \to (\forall (P: Prop).P))) \to -(pr3 c0 (THead (Bind b) u0 (THeads (Flat Appl) (lifts (S O) O ts0) t2)) -u3)))))))) \to ((pr3 c (THeads (Flat Appl) ts0 (THead (Flat Appl) t (THead -(Bind b) u t0))) u2) \to ((((iso (THeads (Flat Appl) ts0 (THead (Flat Appl) t -(THead (Bind b) u t0))) u2) \to (\forall (P: Prop).P))) \to (pr3 c (THead -(Bind b) u (THeads (Flat Appl) (lifts (S O) O ts0) (THead (Flat Appl) (lift -(S O) O t) t0))) u2)))))).(\lambda (H5: ((\forall (u0: T).(\forall (t2: -T).(\forall (c0: C).(\forall (u3: T).((pr3 c0 (THeads (Flat Appl) (TCons t1 -ts0) (THead (Bind b) u0 t2)) u3) \to ((((iso (THeads (Flat Appl) (TCons t1 -ts0) (THead (Bind b) u0 t2)) u3) \to (\forall (P: Prop).P))) \to (pr3 c0 -(THead (Bind b) u0 (THeads (Flat Appl) (lifts (S O) O (TCons t1 ts0)) t2)) -u3))))))))).(\lambda (H6: (pr3 c (THeads (Flat Appl) (TCons t1 ts0) (THead -(Flat Appl) t (THead (Bind b) u t0))) u2)).(\lambda (H7: (((iso (THeads (Flat -Appl) (TCons t1 ts0) (THead (Flat Appl) t (THead (Bind b) u t0))) u2) \to -(\forall (P: Prop).P)))).(H5 u (THead (Flat Appl) (lift (S O) O t) t0) c u2 -(pr3_iso_appls_appl_bind b H t u t0 (TCons t1 ts0) c u2 H6 H7) (\lambda (H8: -(iso (THeads (Flat Appl) (TCons t1 ts0) (THead (Bind b) u (THead (Flat Appl) -(lift (S O) O t) t0))) u2)).(\lambda (P: Prop).(H7 (iso_trans (THeads (Flat -Appl) (TCons t1 ts0) (THead (Flat Appl) t (THead (Bind b) u t0))) (THeads -(Flat Appl) (TCons t1 ts0) (THead (Bind b) u (THead (Flat Appl) (lift (S O) O -t) t0))) (iso_head t1 t1 (THeads (Flat Appl) ts0 (THead (Flat Appl) t (THead -(Bind b) u t0))) (THeads (Flat Appl) ts0 (THead (Bind b) u (THead (Flat Appl) -(lift (S O) O t) t0))) (Flat Appl)) u2 H8) P)))))))))) ts H0 H3 H4))) (THeads -(Flat Appl) (TApp (lifts (S O) O ts) (lift (S O) O t)) t0) (theads_tapp (Flat -Appl) (lift (S O) O t) t0 (lifts (S O) O ts))) (lifts (S O) O (TApp ts t)) -(lifts_tapp (S O) O t ts))))))))))) vs))). -(* COMMENTS -Initial nodes: 1681 -END *) - -theorem pr3_iso_beta: - \forall (v: T).(\forall (w: T).(\forall (t: T).(let u1 \def (THead (Flat -Appl) v (THead (Bind Abst) w t)) in (\forall (c: C).(\forall (u2: T).((pr3 c -u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to (pr3 c (THead (Bind -Abbr) v t) u2)))))))) -\def - \lambda (v: T).(\lambda (w: T).(\lambda (t: T).(\lambda (c: C).(\lambda (u2: -T).(\lambda (H: (pr3 c (THead (Flat Appl) v (THead (Bind Abst) w t)) -u2)).(\lambda (H0: (((iso (THead (Flat Appl) v (THead (Bind Abst) w t)) u2) -\to (\forall (P: Prop).P)))).(let H1 \def (pr3_gen_appl c v (THead (Bind -Abst) w t) u2 H) in (or3_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq -T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c v -u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THead (Bind Abst) w t) t2)))) -(ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: -T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v u3))))) (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) -w t) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind -b) u) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B -b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) w t) (THead -(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) -y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda -(_: T).(pr3 c v u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (pr3 c -(THead (Bind Abbr) v t) u2) (\lambda (H2: (ex3_2 T T (\lambda (u3: -T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: -T).(\lambda (_: T).(pr3 c v u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c -(THead (Bind Abst) w t) t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: -T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: -T).(pr3 c v u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THead (Bind Abst) -w t) t2))) (pr3 c (THead (Bind Abbr) v t) u2) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H3: (eq T u2 (THead (Flat Appl) x0 x1))).(\lambda (_: (pr3 c v -x0)).(\lambda (_: (pr3 c (THead (Bind Abst) w t) x1)).(let H6 \def (eq_ind T -u2 (\lambda (t0: T).((iso (THead (Flat Appl) v (THead (Bind Abst) w t)) t0) -\to (\forall (P: Prop).P))) H0 (THead (Flat Appl) x0 x1) H3) in (eq_ind_r T -(THead (Flat Appl) x0 x1) (\lambda (t0: T).(pr3 c (THead (Bind Abbr) v t) -t0)) (H6 (iso_head v x0 (THead (Bind Abst) w t) x1 (Flat Appl)) (pr3 c (THead -(Bind Abbr) v t) (THead (Flat Appl) x0 x1))) u2 H3))))))) H2)) (\lambda (H2: -(ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: -T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v u3))))) (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) -w t) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind -b) u) z1 t2))))))))).(ex4_4_ind T T T T (\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v -u3))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(pr3 c (THead (Bind Abst) w t) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall -(u: T).(pr3 (CHead c (Bind b) u) z1 t2))))))) (pr3 c (THead (Bind Abbr) v t) -u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: -T).(\lambda (H3: (pr3 c (THead (Bind Abbr) x2 x3) u2)).(\lambda (H4: (pr3 c v -x2)).(\lambda (H5: (pr3 c (THead (Bind Abst) w t) (THead (Bind Abst) x0 -x1))).(\lambda (H6: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) -u) x1 x3))))).(let H7 \def (pr3_gen_abst c w t (THead (Bind Abst) x0 x1) H5) -in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind Abst) -x0 x1) (THead (Bind Abst) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c w -u3))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 -(CHead c (Bind b) u) t t2))))) (pr3 c (THead (Bind Abbr) v t) u2) (\lambda -(x4: T).(\lambda (x5: T).(\lambda (H8: (eq T (THead (Bind Abst) x0 x1) (THead -(Bind Abst) x4 x5))).(\lambda (H9: (pr3 c w x4)).(\lambda (H10: ((\forall (b: -B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t x5))))).(let H11 \def (f_equal -T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ t0 _) \Rightarrow t0])) -(THead (Bind Abst) x0 x1) (THead (Bind Abst) x4 x5) H8) in ((let H12 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow x1 | (TLRef _) \Rightarrow x1 | (THead _ _ t0) -\Rightarrow t0])) (THead (Bind Abst) x0 x1) (THead (Bind Abst) x4 x5) H8) in -(\lambda (H13: (eq T x0 x4)).(let H14 \def (eq_ind_r T x5 (\lambda (t0: -T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t t0)))) H10 x1 -H12) in (let H15 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 c w t0)) H9 x0 -H13) in (pr3_t (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) v t) c -(pr3_head_12 c v x2 H4 (Bind Abbr) t x3 (pr3_t x1 t (CHead c (Bind Abbr) x2) -(H14 Abbr x2) x3 (H6 Abbr x2))) u2 H3))))) H11))))))) H7)))))))))) H2)) -(\lambda (H2: (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) -(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) w t) (THead (Bind b) y1 -z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: -T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat -Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v -u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c -(THead (Bind Abst) w t) (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u3: T).(\lambda -(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u3) z2)) -u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v u3))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) -y2) z1 z2))))))) (pr3 c (THead (Bind Abbr) v t) u2) (\lambda (x0: B).(\lambda -(x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: -T).(\lambda (H3: (not (eq B x0 Abst))).(\lambda (H4: (pr3 c (THead (Bind -Abst) w t) (THead (Bind x0) x1 x2))).(\lambda (H5: (pr3 c (THead (Bind x0) x5 -(THead (Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda (_: (pr3 c v -x4)).(\lambda (_: (pr3 c x1 x5)).(\lambda (H8: (pr3 (CHead c (Bind x0) x5) x2 -x3)).(let H9 \def (pr3_gen_abst c w t (THead (Bind x0) x1 x2) H4) in -(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind x0) x1 -x2) (THead (Bind Abst) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c w -u3))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 -(CHead c (Bind b) u) t t2))))) (pr3 c (THead (Bind Abbr) v t) u2) (\lambda -(x6: T).(\lambda (x7: T).(\lambda (H10: (eq T (THead (Bind x0) x1 x2) (THead -(Bind Abst) x6 x7))).(\lambda (H11: (pr3 c w x6)).(\lambda (H12: ((\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t x7))))).(let H13 \def -(f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with -[(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead k _ _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) -\Rightarrow b | (Flat _) \Rightarrow x0])])) (THead (Bind x0) x1 x2) (THead -(Bind Abst) x6 x7) H10) in ((let H14 \def (f_equal T T (\lambda (e: T).(match -e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x1 | (TLRef _) -\Rightarrow x1 | (THead _ t0 _) \Rightarrow t0])) (THead (Bind x0) x1 x2) -(THead (Bind Abst) x6 x7) H10) in ((let H15 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x2 | -(TLRef _) \Rightarrow x2 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind x0) -x1 x2) (THead (Bind Abst) x6 x7) H10) in (\lambda (H16: (eq T x1 -x6)).(\lambda (H17: (eq B x0 Abst)).(let H18 \def (eq_ind_r T x7 (\lambda -(t0: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t t0)))) -H12 x2 H15) in (let H19 \def (eq_ind_r T x6 (\lambda (t0: T).(pr3 c w t0)) -H11 x1 H16) in (let H20 \def (eq_ind B x0 (\lambda (b: B).(pr3 (CHead c (Bind -b) x5) x2 x3)) H8 Abst H17) in (let H21 \def (eq_ind B x0 (\lambda (b: -B).(pr3 c (THead (Bind b) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) u2)) -H5 Abst H17) in (let H22 \def (eq_ind B x0 (\lambda (b: B).(not (eq B b -Abst))) H3 Abst H17) in (let H23 \def (match (H22 (refl_equal B Abst)) in -False return (\lambda (_: False).(pr3 c (THead (Bind Abbr) v t) u2)) with []) -in H23))))))))) H14)) H13))))))) H9)))))))))))))) H2)) H1)))))))). -(* COMMENTS -Initial nodes: 2459 -END *) - -theorem pr3_iso_appls_beta: - \forall (us: TList).(\forall (v: T).(\forall (w: T).(\forall (t: T).(let u1 -\def (THeads (Flat Appl) us (THead (Flat Appl) v (THead (Bind Abst) w t))) in -(\forall (c: C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to -(\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) us (THead (Bind Abbr) -v t)) u2))))))))) -\def - \lambda (us: TList).(TList_ind (\lambda (t: TList).(\forall (v: T).(\forall -(w: T).(\forall (t0: T).(let u1 \def (THeads (Flat Appl) t (THead (Flat Appl) -v (THead (Bind Abst) w t0))) in (\forall (c: C).(\forall (u2: T).((pr3 c u1 -u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to (pr3 c (THeads (Flat -Appl) t (THead (Bind Abbr) v t0)) u2)))))))))) (\lambda (v: T).(\lambda (w: -T).(\lambda (t: T).(\lambda (c: C).(\lambda (u2: T).(\lambda (H: (pr3 c -(THead (Flat Appl) v (THead (Bind Abst) w t)) u2)).(\lambda (H0: (((iso -(THead (Flat Appl) v (THead (Bind Abst) w t)) u2) \to (\forall (P: -Prop).P)))).(pr3_iso_beta v w t c u2 H H0)))))))) (\lambda (t: T).(\lambda -(t0: TList).(\lambda (H: ((\forall (v: T).(\forall (w: T).(\forall (t1: -T).(\forall (c: C).(\forall (u2: T).((pr3 c (THeads (Flat Appl) t0 (THead -(Flat Appl) v (THead (Bind Abst) w t1))) u2) \to ((((iso (THeads (Flat Appl) -t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) u2) \to (\forall (P: -Prop).P))) \to (pr3 c (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1)) -u2)))))))))).(\lambda (v: T).(\lambda (w: T).(\lambda (t1: T).(\lambda (c: -C).(\lambda (u2: T).(\lambda (H0: (pr3 c (THead (Flat Appl) t (THeads (Flat -Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1)))) u2)).(\lambda (H1: -(((iso (THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Flat Appl) v -(THead (Bind Abst) w t1)))) u2) \to (\forall (P: Prop).P)))).(let H2 \def -(pr3_gen_appl c t (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind -Abst) w t1))) u2 H0) in (or3_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t2: -T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: -T).(pr3 c t u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) -t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) t2)))) (ex4_4 T T T T -(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c -(THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u3: T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat -Appl) v (THead (Bind Abst) w t1))) (THead (Bind Abst) y1 z1)))))) (\lambda -(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: -B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c -(THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) (THead -(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) -y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda -(_: T).(pr3 c t u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (pr3 c -(THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) u2) -(\lambda (H3: (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead -(Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t u3))) -(\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat -Appl) v (THead (Bind Abst) w t1))) t2))))).(ex3_2_ind T T (\lambda (u3: -T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: -T).(\lambda (_: T).(pr3 c t u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c -(THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) t2))) -(pr3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) -u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T u2 (THead (Flat -Appl) x0 x1))).(\lambda (_: (pr3 c t x0)).(\lambda (_: (pr3 c (THeads (Flat -Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) x1)).(let H7 \def -(eq_ind T u2 (\lambda (t2: T).((iso (THead (Flat Appl) t (THeads (Flat Appl) -t0 (THead (Flat Appl) v (THead (Bind Abst) w t1)))) t2) \to (\forall (P: -Prop).P))) H1 (THead (Flat Appl) x0 x1) H4) in (eq_ind_r T (THead (Flat Appl) -x0 x1) (\lambda (t2: T).(pr3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 -(THead (Bind Abbr) v t1))) t2)) (H7 (iso_head t x0 (THeads (Flat Appl) t0 -(THead (Flat Appl) v (THead (Bind Abst) w t1))) x1 (Flat Appl)) (pr3 c (THead -(Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) (THead (Flat -Appl) x0 x1))) u2 H4))))))) H3)) (\lambda (H3: (ex4_4 T T T T (\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind -Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: -T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat -Appl) v (THead (Bind Abst) w t1))) (THead (Bind Abst) y1 z1)))))) (\lambda -(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: -B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2))))))))).(ex4_4_ind T T T -T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c -(THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u3: T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat -Appl) v (THead (Bind Abst) w t1))) (THead (Bind Abst) y1 z1)))))) (\lambda -(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: -B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2))))))) (pr3 c (THead (Flat -Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) u2) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H4: (pr3 c -(THead (Bind Abbr) x2 x3) u2)).(\lambda (H5: (pr3 c t x2)).(\lambda (H6: (pr3 -c (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) -(THead (Bind Abst) x0 x1))).(\lambda (H7: ((\forall (b: B).(\forall (u: -T).(pr3 (CHead c (Bind b) u) x1 x3))))).(pr3_t (THead (Bind Abbr) t x1) -(THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) c -(pr3_t (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) (THead (Flat Appl) t -(THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) c (pr3_thin_dx c (THeads -(Flat Appl) t0 (THead (Bind Abbr) v t1)) (THead (Bind Abst) x0 x1) (H v w t1 -c (THead (Bind Abst) x0 x1) H6 (\lambda (H8: (iso (THeads (Flat Appl) t0 -(THead (Flat Appl) v (THead (Bind Abst) w t1))) (THead (Bind Abst) x0 -x1))).(\lambda (P: Prop).(iso_flats_flat_bind_false Appl Appl Abst x0 v x1 -(THead (Bind Abst) w t1) t0 H8 P)))) t Appl) (THead (Bind Abbr) t x1) -(pr3_pr2 c (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) -t x1) (pr2_free c (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) (THead -(Bind Abbr) t x1) (pr0_beta x0 t t (pr0_refl t) x1 x1 (pr0_refl x1))))) u2 -(pr3_t (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) t x1) c (pr3_head_12 c t -x2 H5 (Bind Abbr) x1 x3 (H7 Abbr x2)) u2 H4)))))))))) H3)) (\lambda (H3: -(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) -w t1))) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c -(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: -T).(\lambda (_: T).(pr3 c t u3))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 -y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 -z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat -Appl) v (THead (Bind Abst) w t1))) (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u3: T).(\lambda -(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u3) z2)) -u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t u3))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) -y2) z1 z2))))))) (pr3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (THead -(Bind Abbr) v t1))) u2) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: -T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H4: (not (eq -B x0 Abst))).(\lambda (H5: (pr3 c (THeads (Flat Appl) t0 (THead (Flat Appl) v -(THead (Bind Abst) w t1))) (THead (Bind x0) x1 x2))).(\lambda (H6: (pr3 c -(THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda -(H7: (pr3 c t x4)).(\lambda (H8: (pr3 c x1 x5)).(\lambda (H9: (pr3 (CHead c -(Bind x0) x5) x2 x3)).(pr3_t (THead (Bind x0) x1 (THead (Flat Appl) (lift (S -O) O x4) x2)) (THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) -v t1))) c (pr3_t (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O t) x2)) -(THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) c -(pr3_t (THead (Flat Appl) t (THead (Bind x0) x1 x2)) (THead (Flat Appl) t -(THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) c (pr3_thin_dx c (THeads -(Flat Appl) t0 (THead (Bind Abbr) v t1)) (THead (Bind x0) x1 x2) (H v w t1 c -(THead (Bind x0) x1 x2) H5 (\lambda (H10: (iso (THeads (Flat Appl) t0 (THead -(Flat Appl) v (THead (Bind Abst) w t1))) (THead (Bind x0) x1 x2))).(\lambda -(P: Prop).(iso_flats_flat_bind_false Appl Appl x0 x1 v x2 (THead (Bind Abst) -w t1) t0 H10 P)))) t Appl) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) -O t) x2)) (pr3_pr2 c (THead (Flat Appl) t (THead (Bind x0) x1 x2)) (THead -(Bind x0) x1 (THead (Flat Appl) (lift (S O) O t) x2)) (pr2_free c (THead -(Flat Appl) t (THead (Bind x0) x1 x2)) (THead (Bind x0) x1 (THead (Flat Appl) -(lift (S O) O t) x2)) (pr0_upsilon x0 H4 t t (pr0_refl t) x1 x1 (pr0_refl x1) -x2 x2 (pr0_refl x2))))) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O -x4) x2)) (pr3_head_12 c x1 x1 (pr3_refl c x1) (Bind x0) (THead (Flat Appl) -(lift (S O) O t) x2) (THead (Flat Appl) (lift (S O) O x4) x2) (pr3_head_12 -(CHead c (Bind x0) x1) (lift (S O) O t) (lift (S O) O x4) (pr3_lift (CHead c -(Bind x0) x1) c (S O) O (drop_drop (Bind x0) O c c (drop_refl c) x1) t x4 H7) -(Flat Appl) x2 x2 (pr3_refl (CHead (CHead c (Bind x0) x1) (Flat Appl) (lift -(S O) O x4)) x2)))) u2 (pr3_t (THead (Bind x0) x5 (THead (Flat Appl) (lift (S -O) O x4) x3)) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) c -(pr3_head_12 c x1 x5 H8 (Bind x0) (THead (Flat Appl) (lift (S O) O x4) x2) -(THead (Flat Appl) (lift (S O) O x4) x3) (pr3_thin_dx (CHead c (Bind x0) x5) -x2 x3 H9 (lift (S O) O x4) Appl)) u2 H6)))))))))))))) H3)) H2)))))))))))) us). -(* COMMENTS -Initial nodes: 3345 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/pr1.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/pr1.ma deleted file mode 100644 index e428daf37..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/pr1.ma +++ /dev/null @@ -1,34 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr3/defs.ma". - -include "Basic-1/pr1/defs.ma". - -theorem pr3_pr1: - \forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall (c: C).(pr3 c t1 -t2)))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr1 t1 t2)).(pr1_ind (\lambda -(t: T).(\lambda (t0: T).(\forall (c: C).(pr3 c t t0)))) (\lambda (t: -T).(\lambda (c: C).(pr3_refl c t))) (\lambda (t0: T).(\lambda (t3: -T).(\lambda (H0: (pr0 t3 t0)).(\lambda (t4: T).(\lambda (_: (pr1 t0 -t4)).(\lambda (H2: ((\forall (c: C).(pr3 c t0 t4)))).(\lambda (c: -C).(pr3_sing c t0 t3 (pr2_free c t3 t0 H0) t4 (H2 c))))))))) t1 t2 H))). -(* COMMENTS -Initial nodes: 95 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/pr3.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/pr3.ma deleted file mode 100644 index 935850b1c..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/pr3.ma +++ /dev/null @@ -1,74 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr3/props.ma". - -include "Basic-1/pr2/pr2.ma". - -theorem pr3_strip: - \forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr3 c t0 t1) \to (\forall -(t2: T).((pr2 c t0 t2) \to (ex2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: -T).(pr3 c t2 t)))))))) -\def - \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr3 c t0 -t1)).(pr3_ind c (\lambda (t: T).(\lambda (t2: T).(\forall (t3: T).((pr2 c t -t3) \to (ex2 T (\lambda (t4: T).(pr3 c t2 t4)) (\lambda (t4: T).(pr3 c t3 -t4))))))) (\lambda (t: T).(\lambda (t2: T).(\lambda (H0: (pr2 c t -t2)).(ex_intro2 T (\lambda (t3: T).(pr3 c t t3)) (\lambda (t3: T).(pr3 c t2 -t3)) t2 (pr3_pr2 c t t2 H0) (pr3_refl c t2))))) (\lambda (t2: T).(\lambda -(t3: T).(\lambda (H0: (pr2 c t3 t2)).(\lambda (t4: T).(\lambda (_: (pr3 c t2 -t4)).(\lambda (H2: ((\forall (t5: T).((pr2 c t2 t5) \to (ex2 T (\lambda (t: -T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t))))))).(\lambda (t5: T).(\lambda -(H3: (pr2 c t3 t5)).(ex2_ind T (\lambda (t: T).(pr2 c t5 t)) (\lambda (t: -T).(pr2 c t2 t)) (ex2 T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c -t5 t))) (\lambda (x: T).(\lambda (H4: (pr2 c t5 x)).(\lambda (H5: (pr2 c t2 -x)).(ex2_ind T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c x t)) -(ex2 T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t))) (\lambda -(x0: T).(\lambda (H6: (pr3 c t4 x0)).(\lambda (H7: (pr3 c x x0)).(ex_intro2 T -(\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t)) x0 H6 (pr3_sing c -x t5 H4 x0 H7))))) (H2 x H5))))) (pr2_confluence c t3 t5 H3 t2 H0)))))))))) -t0 t1 H)))). -(* COMMENTS -Initial nodes: 375 -END *) - -theorem pr3_confluence: - \forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr3 c t0 t1) \to (\forall -(t2: T).((pr3 c t0 t2) \to (ex2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: -T).(pr3 c t2 t)))))))) -\def - \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr3 c t0 -t1)).(pr3_ind c (\lambda (t: T).(\lambda (t2: T).(\forall (t3: T).((pr3 c t -t3) \to (ex2 T (\lambda (t4: T).(pr3 c t2 t4)) (\lambda (t4: T).(pr3 c t3 -t4))))))) (\lambda (t: T).(\lambda (t2: T).(\lambda (H0: (pr3 c t -t2)).(ex_intro2 T (\lambda (t3: T).(pr3 c t t3)) (\lambda (t3: T).(pr3 c t2 -t3)) t2 H0 (pr3_refl c t2))))) (\lambda (t2: T).(\lambda (t3: T).(\lambda -(H0: (pr2 c t3 t2)).(\lambda (t4: T).(\lambda (_: (pr3 c t2 t4)).(\lambda -(H2: ((\forall (t5: T).((pr3 c t2 t5) \to (ex2 T (\lambda (t: T).(pr3 c t4 -t)) (\lambda (t: T).(pr3 c t5 t))))))).(\lambda (t5: T).(\lambda (H3: (pr3 c -t3 t5)).(ex2_ind T (\lambda (t: T).(pr3 c t5 t)) (\lambda (t: T).(pr3 c t2 -t)) (ex2 T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t))) -(\lambda (x: T).(\lambda (H4: (pr3 c t5 x)).(\lambda (H5: (pr3 c t2 -x)).(ex2_ind T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c x t)) -(ex2 T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t))) (\lambda -(x0: T).(\lambda (H6: (pr3 c t4 x0)).(\lambda (H7: (pr3 c x x0)).(ex_intro2 T -(\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t)) x0 H6 (pr3_t x t5 -c H4 x0 H7))))) (H2 x H5))))) (pr3_strip c t3 t5 H3 t2 H0)))))))))) t0 t1 -H)))). -(* COMMENTS -Initial nodes: 367 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/props.ma deleted file mode 100644 index 777f5d85d..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/props.ma +++ /dev/null @@ -1,407 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr3/pr1.ma". - -include "Basic-1/pr2/props.ma". - -include "Basic-1/pr1/props.ma". - -theorem clear_pr3_trans: - \forall (c2: C).(\forall (t1: T).(\forall (t2: T).((pr3 c2 t1 t2) \to -(\forall (c1: C).((clear c1 c2) \to (pr3 c1 t1 t2)))))) -\def - \lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c2 t1 -t2)).(\lambda (c1: C).(\lambda (H0: (clear c1 c2)).(pr3_ind c2 (\lambda (t: -T).(\lambda (t0: T).(pr3 c1 t t0))) (\lambda (t: T).(pr3_refl c1 t)) (\lambda -(t3: T).(\lambda (t4: T).(\lambda (H1: (pr2 c2 t4 t3)).(\lambda (t5: -T).(\lambda (_: (pr3 c2 t3 t5)).(\lambda (H3: (pr3 c1 t3 t5)).(pr3_sing c1 t3 -t4 (clear_pr2_trans c2 t4 t3 H1 c1 H0) t5 H3))))))) t1 t2 H)))))). -(* COMMENTS -Initial nodes: 107 -END *) - -theorem pr3_pr2: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (pr3 c -t1 t2)))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 -t2)).(pr3_sing c t2 t1 H t2 (pr3_refl c t2))))). -(* COMMENTS -Initial nodes: 31 -END *) - -theorem pr3_t: - \forall (t2: T).(\forall (t1: T).(\forall (c: C).((pr3 c t1 t2) \to (\forall -(t3: T).((pr3 c t2 t3) \to (pr3 c t1 t3)))))) -\def - \lambda (t2: T).(\lambda (t1: T).(\lambda (c: C).(\lambda (H: (pr3 c t1 -t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (t3: T).((pr3 c t0 -t3) \to (pr3 c t t3))))) (\lambda (t: T).(\lambda (t3: T).(\lambda (H0: (pr3 -c t t3)).H0))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c t3 -t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2: ((\forall -(t5: T).((pr3 c t4 t5) \to (pr3 c t0 t5))))).(\lambda (t5: T).(\lambda (H3: -(pr3 c t4 t5)).(pr3_sing c t0 t3 H0 t5 (H2 t5 H3)))))))))) t1 t2 H)))). -(* COMMENTS -Initial nodes: 127 -END *) - -theorem pr3_thin_dx: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall -(u: T).(\forall (f: F).(pr3 c (THead (Flat f) u t1) (THead (Flat f) u -t2))))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 -t2)).(\lambda (u: T).(\lambda (f: F).(pr3_ind c (\lambda (t: T).(\lambda (t0: -T).(pr3 c (THead (Flat f) u t) (THead (Flat f) u t0)))) (\lambda (t: -T).(pr3_refl c (THead (Flat f) u t))) (\lambda (t0: T).(\lambda (t3: -T).(\lambda (H0: (pr2 c t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0 -t4)).(\lambda (H2: (pr3 c (THead (Flat f) u t0) (THead (Flat f) u -t4))).(pr3_sing c (THead (Flat f) u t0) (THead (Flat f) u t3) (pr2_thin_dx c -t3 t0 H0 u f) (THead (Flat f) u t4) H2))))))) t1 t2 H)))))). -(* COMMENTS -Initial nodes: 167 -END *) - -theorem pr3_head_1: - \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall -(k: K).(\forall (t: T).(pr3 c (THead k u1 t) (THead k u2 t))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1 -u2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (k: K).(\forall -(t1: T).(pr3 c (THead k t t1) (THead k t0 t1)))))) (\lambda (t: T).(\lambda -(k: K).(\lambda (t0: T).(pr3_refl c (THead k t t0))))) (\lambda (t2: -T).(\lambda (t1: T).(\lambda (H0: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda -(_: (pr3 c t2 t3)).(\lambda (H2: ((\forall (k: K).(\forall (t: T).(pr3 c -(THead k t2 t) (THead k t3 t)))))).(\lambda (k: K).(\lambda (t: T).(pr3_sing -c (THead k t2 t) (THead k t1 t) (pr2_head_1 c t1 t2 H0 k t) (THead k t3 t) -(H2 k t)))))))))) u1 u2 H)))). -(* COMMENTS -Initial nodes: 167 -END *) - -theorem pr3_head_2: - \forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).(\forall -(k: K).((pr3 (CHead c k u) t1 t2) \to (pr3 c (THead k u t1) (THead k u -t2))))))) -\def - \lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(k: K).(\lambda (H: (pr3 (CHead c k u) t1 t2)).(pr3_ind (CHead c k u) -(\lambda (t: T).(\lambda (t0: T).(pr3 c (THead k u t) (THead k u t0)))) -(\lambda (t: T).(pr3_refl c (THead k u t))) (\lambda (t0: T).(\lambda (t3: -T).(\lambda (H0: (pr2 (CHead c k u) t3 t0)).(\lambda (t4: T).(\lambda (_: -(pr3 (CHead c k u) t0 t4)).(\lambda (H2: (pr3 c (THead k u t0) (THead k u -t4))).(pr3_sing c (THead k u t0) (THead k u t3) (pr2_head_2 c u t3 t0 k H0) -(THead k u t4) H2))))))) t1 t2 H)))))). -(* COMMENTS -Initial nodes: 175 -END *) - -theorem pr3_head_21: - \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall -(k: K).(\forall (t1: T).(\forall (t2: T).((pr3 (CHead c k u1) t1 t2) \to (pr3 -c (THead k u1 t1) (THead k u2 t2))))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1 -u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 -(CHead c k u1) t1 t2)).(pr3_t (THead k u1 t2) (THead k u1 t1) c (pr3_head_2 c -u1 t1 t2 k H0) (THead k u2 t2) (pr3_head_1 c u1 u2 H k t2))))))))). -(* COMMENTS -Initial nodes: 89 -END *) - -theorem pr3_head_12: - \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall -(k: K).(\forall (t1: T).(\forall (t2: T).((pr3 (CHead c k u2) t1 t2) \to (pr3 -c (THead k u1 t1) (THead k u2 t2))))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1 -u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 -(CHead c k u2) t1 t2)).(pr3_t (THead k u2 t1) (THead k u1 t1) c (pr3_head_1 c -u1 u2 H k t1) (THead k u2 t2) (pr3_head_2 c u2 t1 t2 k H0))))))))). -(* COMMENTS -Initial nodes: 89 -END *) - -theorem pr3_cflat: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall -(f: F).(\forall (v: T).(pr3 (CHead c (Flat f) v) t1 t2)))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 -t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (f: F).(\forall (v: -T).(pr3 (CHead c (Flat f) v) t t0))))) (\lambda (t: T).(\lambda (f: -F).(\lambda (v: T).(pr3_refl (CHead c (Flat f) v) t)))) (\lambda (t3: -T).(\lambda (t4: T).(\lambda (H0: (pr2 c t4 t3)).(\lambda (t5: T).(\lambda -(_: (pr3 c t3 t5)).(\lambda (H2: ((\forall (f: F).(\forall (v: T).(pr3 (CHead -c (Flat f) v) t3 t5))))).(\lambda (f: F).(\lambda (v: T).(pr3_sing (CHead c -(Flat f) v) t3 t4 (pr2_cflat c t4 t3 H0 f v) t5 (H2 f v)))))))))) t1 t2 H)))). -(* COMMENTS -Initial nodes: 151 -END *) - -theorem pr3_flat: - \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall -(t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall (f: F).(pr3 c (THead -(Flat f) u1 t1) (THead (Flat f) u2 t2))))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1 -u2)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 c t1 t2)).(\lambda -(f: F).(pr3_head_12 c u1 u2 H (Flat f) t1 t2 (pr3_cflat c t1 t2 H0 f -u2))))))))). -(* COMMENTS -Initial nodes: 59 -END *) - -theorem pr3_pr0_pr2_t: - \forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (c: C).(\forall -(t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pr3 -(CHead c k u1) t1 t2)))))))) -\def - \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr0 u1 u2)).(\lambda (c: -C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k: K).(\lambda (H0: (pr2 -(CHead c k u2) t1 t2)).(insert_eq C (CHead c k u2) (\lambda (c0: C).(pr2 c0 -t1 t2)) (\lambda (_: C).(pr3 (CHead c k u1) t1 t2)) (\lambda (y: C).(\lambda -(H1: (pr2 y t1 t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: -T).((eq C c0 (CHead c k u2)) \to (pr3 (CHead c k u1) t t0))))) (\lambda (c0: -C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (_: -(eq C c0 (CHead c k u2))).(pr3_pr2 (CHead c k u1) t3 t4 (pr2_free (CHead c k -u1) t3 t4 H2))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda -(i: nat).(\lambda (H2: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t3: -T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H4: -(subst0 i u t4 t)).(\lambda (H5: (eq C c0 (CHead c k u2))).(let H6 \def -(eq_ind C c0 (\lambda (c1: C).(getl i c1 (CHead d (Bind Abbr) u))) H2 (CHead -c k u2) H5) in (nat_ind (\lambda (n: nat).((getl n (CHead c k u2) (CHead d -(Bind Abbr) u)) \to ((subst0 n u t4 t) \to (pr3 (CHead c k u1) t3 t)))) -(\lambda (H7: (getl O (CHead c k u2) (CHead d (Bind Abbr) u))).(\lambda (H8: -(subst0 O u t4 t)).(K_ind (\lambda (k0: K).((getl O (CHead c k0 u2) (CHead d -(Bind Abbr) u)) \to (pr3 (CHead c k0 u1) t3 t))) (\lambda (b: B).(\lambda -(H9: (getl O (CHead c (Bind b) u2) (CHead d (Bind Abbr) u))).(let H10 \def -(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with -[(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind -Abbr) u) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 -(getl_gen_O (CHead c (Bind b) u2) (CHead d (Bind Abbr) u) H9))) in ((let H11 -\def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) -with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow (match k0 in K -return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) -\Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead c (Bind b) u2) -(clear_gen_bind b c (CHead d (Bind Abbr) u) u2 (getl_gen_O (CHead c (Bind b) -u2) (CHead d (Bind Abbr) u) H9))) in ((let H12 \def (f_equal C T (\lambda (e: -C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | -(CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind Abbr) u) (CHead c (Bind b) -u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 (getl_gen_O (CHead c (Bind -b) u2) (CHead d (Bind Abbr) u) H9))) in (\lambda (H13: (eq B Abbr -b)).(\lambda (_: (eq C d c)).(let H15 \def (eq_ind T u (\lambda (t0: -T).(subst0 O t0 t4 t)) H8 u2 H12) in (eq_ind B Abbr (\lambda (b0: B).(pr3 -(CHead c (Bind b0) u1) t3 t)) (ex2_ind T (\lambda (t0: T).(subst0 O u1 t4 -t0)) (\lambda (t0: T).(pr0 t0 t)) (pr3 (CHead c (Bind Abbr) u1) t3 t) -(\lambda (x: T).(\lambda (H16: (subst0 O u1 t4 x)).(\lambda (H17: (pr0 x -t)).(pr3_sing (CHead c (Bind Abbr) u1) x t3 (pr2_delta (CHead c (Bind Abbr) -u1) c u1 O (getl_refl Abbr c u1) t3 t4 H3 x H16) t (pr3_pr2 (CHead c (Bind -Abbr) u1) x t (pr2_free (CHead c (Bind Abbr) u1) x t H17)))))) -(pr0_subst0_back u2 t4 t O H15 u1 H)) b H13))))) H11)) H10)))) (\lambda (f: -F).(\lambda (H9: (getl O (CHead c (Flat f) u2) (CHead d (Bind Abbr) -u))).(pr3_pr2 (CHead c (Flat f) u1) t3 t (pr2_cflat c t3 t (pr2_delta c d u O -(getl_intro O c (CHead d (Bind Abbr) u) c (drop_refl c) (clear_gen_flat f c -(CHead d (Bind Abbr) u) u2 (getl_gen_O (CHead c (Flat f) u2) (CHead d (Bind -Abbr) u) H9))) t3 t4 H3 t H8) f u1)))) k H7))) (\lambda (i0: nat).(\lambda -(IHi: (((getl i0 (CHead c k u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t4 -t) \to (pr3 (CHead c k u1) t3 t))))).(\lambda (H7: (getl (S i0) (CHead c k -u2) (CHead d (Bind Abbr) u))).(\lambda (H8: (subst0 (S i0) u t4 t)).(K_ind -(\lambda (k0: K).((getl (S i0) (CHead c k0 u2) (CHead d (Bind Abbr) u)) \to -((((getl i0 (CHead c k0 u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t4 t) -\to (pr3 (CHead c k0 u1) t3 t)))) \to (pr3 (CHead c k0 u1) t3 t)))) (\lambda -(b: B).(\lambda (H9: (getl (S i0) (CHead c (Bind b) u2) (CHead d (Bind Abbr) -u))).(\lambda (_: (((getl i0 (CHead c (Bind b) u2) (CHead d (Bind Abbr) u)) -\to ((subst0 i0 u t4 t) \to (pr3 (CHead c (Bind b) u1) t3 t))))).(pr3_pr2 -(CHead c (Bind b) u1) t3 t (pr2_delta (CHead c (Bind b) u1) d u (S i0) -(getl_head (Bind b) i0 c (CHead d (Bind Abbr) u) (getl_gen_S (Bind b) c -(CHead d (Bind Abbr) u) u2 i0 H9) u1) t3 t4 H3 t H8))))) (\lambda (f: -F).(\lambda (H9: (getl (S i0) (CHead c (Flat f) u2) (CHead d (Bind Abbr) -u))).(\lambda (_: (((getl i0 (CHead c (Flat f) u2) (CHead d (Bind Abbr) u)) -\to ((subst0 i0 u t4 t) \to (pr3 (CHead c (Flat f) u1) t3 t))))).(pr3_pr2 -(CHead c (Flat f) u1) t3 t (pr2_cflat c t3 t (pr2_delta c d u (r (Flat f) i0) -(getl_gen_S (Flat f) c (CHead d (Bind Abbr) u) u2 i0 H9) t3 t4 H3 t H8) f -u1))))) k H7 IHi))))) i H6 H4))))))))))))) y t1 t2 H1))) H0)))))))). -(* COMMENTS -Initial nodes: 1557 -END *) - -theorem pr3_pr2_pr2_t: - \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr2 c u1 u2) \to (\forall -(t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pr3 -(CHead c k u1) t1 t2)))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr2 c u1 -u2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (t1: -T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c0 k t0) t1 t2) \to (pr3 -(CHead c0 k t) t1 t2)))))))) (\lambda (c0: C).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (H0: (pr0 t1 t2)).(\lambda (t0: T).(\lambda (t3: T).(\lambda (k: -K).(\lambda (H1: (pr2 (CHead c0 k t2) t0 t3)).(pr3_pr0_pr2_t t1 t2 H0 c0 t0 -t3 k H1))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda -(i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t1: -T).(\lambda (t2: T).(\lambda (H1: (pr0 t1 t2)).(\lambda (t: T).(\lambda (H2: -(subst0 i u t2 t)).(\lambda (t0: T).(\lambda (t3: T).(\lambda (k: K).(\lambda -(H3: (pr2 (CHead c0 k t) t0 t3)).(insert_eq C (CHead c0 k t) (\lambda (c1: -C).(pr2 c1 t0 t3)) (\lambda (_: C).(pr3 (CHead c0 k t1) t0 t3)) (\lambda (y: -C).(\lambda (H4: (pr2 y t0 t3)).(pr2_ind (\lambda (c1: C).(\lambda (t4: -T).(\lambda (t5: T).((eq C c1 (CHead c0 k t)) \to (pr3 (CHead c0 k t1) t4 -t5))))) (\lambda (c1: C).(\lambda (t4: T).(\lambda (t5: T).(\lambda (H5: (pr0 -t4 t5)).(\lambda (_: (eq C c1 (CHead c0 k t))).(pr3_pr2 (CHead c0 k t1) t4 t5 -(pr2_free (CHead c0 k t1) t4 t5 H5))))))) (\lambda (c1: C).(\lambda (d0: -C).(\lambda (u0: T).(\lambda (i0: nat).(\lambda (H5: (getl i0 c1 (CHead d0 -(Bind Abbr) u0))).(\lambda (t4: T).(\lambda (t5: T).(\lambda (H6: (pr0 t4 -t5)).(\lambda (t6: T).(\lambda (H7: (subst0 i0 u0 t5 t6)).(\lambda (H8: (eq C -c1 (CHead c0 k t))).(let H9 \def (eq_ind C c1 (\lambda (c2: C).(getl i0 c2 -(CHead d0 (Bind Abbr) u0))) H5 (CHead c0 k t) H8) in (nat_ind (\lambda (n: -nat).((getl n (CHead c0 k t) (CHead d0 (Bind Abbr) u0)) \to ((subst0 n u0 t5 -t6) \to (pr3 (CHead c0 k t1) t4 t6)))) (\lambda (H10: (getl O (CHead c0 k t) -(CHead d0 (Bind Abbr) u0))).(\lambda (H11: (subst0 O u0 t5 t6)).(K_ind -(\lambda (k0: K).((clear (CHead c0 k0 t) (CHead d0 (Bind Abbr) u0)) \to (pr3 -(CHead c0 k0 t1) t4 t6))) (\lambda (b: B).(\lambda (H12: (clear (CHead c0 -(Bind b) t) (CHead d0 (Bind Abbr) u0))).(let H13 \def (f_equal C C (\lambda -(e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d0 -| (CHead c2 _ _) \Rightarrow c2])) (CHead d0 (Bind Abbr) u0) (CHead c0 (Bind -b) t) (clear_gen_bind b c0 (CHead d0 (Bind Abbr) u0) t H12)) in ((let H14 -\def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) -with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow (match k0 in K -return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) -\Rightarrow Abbr])])) (CHead d0 (Bind Abbr) u0) (CHead c0 (Bind b) t) -(clear_gen_bind b c0 (CHead d0 (Bind Abbr) u0) t H12)) in ((let H15 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow u0 | (CHead _ _ t7) \Rightarrow t7])) (CHead d0 (Bind -Abbr) u0) (CHead c0 (Bind b) t) (clear_gen_bind b c0 (CHead d0 (Bind Abbr) -u0) t H12)) in (\lambda (H16: (eq B Abbr b)).(\lambda (_: (eq C d0 c0)).(let -H18 \def (eq_ind T u0 (\lambda (t7: T).(subst0 O t7 t5 t6)) H11 t H15) in -(eq_ind B Abbr (\lambda (b0: B).(pr3 (CHead c0 (Bind b0) t1) t4 t6)) (ex2_ind -T (\lambda (t7: T).(subst0 O t2 t5 t7)) (\lambda (t7: T).(subst0 (S (plus i -O)) u t7 t6)) (pr3 (CHead c0 (Bind Abbr) t1) t4 t6) (\lambda (x: T).(\lambda -(H19: (subst0 O t2 t5 x)).(\lambda (H20: (subst0 (S (plus i O)) u x t6)).(let -H21 \def (f_equal nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O -i))) in (let H22 \def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n -u x t6)) H20 (S i) H21) in (ex2_ind T (\lambda (t7: T).(subst0 O t1 t5 t7)) -(\lambda (t7: T).(pr0 t7 x)) (pr3 (CHead c0 (Bind Abbr) t1) t4 t6) (\lambda -(x0: T).(\lambda (H23: (subst0 O t1 t5 x0)).(\lambda (H24: (pr0 x0 -x)).(pr3_sing (CHead c0 (Bind Abbr) t1) x0 t4 (pr2_delta (CHead c0 (Bind -Abbr) t1) c0 t1 O (getl_refl Abbr c0 t1) t4 t5 H6 x0 H23) t6 (pr3_pr2 (CHead -c0 (Bind Abbr) t1) x0 t6 (pr2_delta (CHead c0 (Bind Abbr) t1) d u (S i) -(getl_clear_bind Abbr (CHead c0 (Bind Abbr) t1) c0 t1 (clear_bind Abbr c0 t1) -(CHead d (Bind Abbr) u) i H0) x0 x H24 t6 H22)))))) (pr0_subst0_back t2 t5 x -O H19 t1 H1))))))) (subst0_subst0 t5 t6 t O H18 t2 u i H2)) b H16))))) H14)) -H13)))) (\lambda (f: F).(\lambda (H12: (clear (CHead c0 (Flat f) t) (CHead d0 -(Bind Abbr) u0))).(pr3_pr2 (CHead c0 (Flat f) t1) t4 t6 (pr2_cflat c0 t4 t6 -(pr2_delta c0 d0 u0 O (getl_intro O c0 (CHead d0 (Bind Abbr) u0) c0 -(drop_refl c0) (clear_gen_flat f c0 (CHead d0 (Bind Abbr) u0) t H12)) t4 t5 -H6 t6 H11) f t1)))) k (getl_gen_O (CHead c0 k t) (CHead d0 (Bind Abbr) u0) -H10)))) (\lambda (i1: nat).(\lambda (_: (((getl i1 (CHead c0 k t) (CHead d0 -(Bind Abbr) u0)) \to ((subst0 i1 u0 t5 t6) \to (pr3 (CHead c0 k t1) t4 -t6))))).(\lambda (H10: (getl (S i1) (CHead c0 k t) (CHead d0 (Bind Abbr) -u0))).(\lambda (H11: (subst0 (S i1) u0 t5 t6)).(K_ind (\lambda (k0: K).((getl -(S i1) (CHead c0 k0 t) (CHead d0 (Bind Abbr) u0)) \to (pr3 (CHead c0 k0 t1) -t4 t6))) (\lambda (b: B).(\lambda (H12: (getl (S i1) (CHead c0 (Bind b) t) -(CHead d0 (Bind Abbr) u0))).(pr3_pr2 (CHead c0 (Bind b) t1) t4 t6 (pr2_delta -(CHead c0 (Bind b) t1) d0 u0 (S i1) (getl_head (Bind b) i1 c0 (CHead d0 (Bind -Abbr) u0) (getl_gen_S (Bind b) c0 (CHead d0 (Bind Abbr) u0) t i1 H12) t1) t4 -t5 H6 t6 H11)))) (\lambda (f: F).(\lambda (H12: (getl (S i1) (CHead c0 (Flat -f) t) (CHead d0 (Bind Abbr) u0))).(pr3_pr2 (CHead c0 (Flat f) t1) t4 t6 -(pr2_cflat c0 t4 t6 (pr2_delta c0 d0 u0 (r (Flat f) i1) (getl_gen_S (Flat f) -c0 (CHead d0 (Bind Abbr) u0) t i1 H12) t4 t5 H6 t6 H11) f t1)))) k H10))))) -i0 H9 H7))))))))))))) y t0 t3 H4))) H3))))))))))))))) c u1 u2 H)))). -(* COMMENTS -Initial nodes: 1697 -END *) - -theorem pr3_pr2_pr3_t: - \forall (c: C).(\forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall -(k: K).((pr3 (CHead c k u2) t1 t2) \to (\forall (u1: T).((pr2 c u1 u2) \to -(pr3 (CHead c k u1) t1 t2)))))))) -\def - \lambda (c: C).(\lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(k: K).(\lambda (H: (pr3 (CHead c k u2) t1 t2)).(pr3_ind (CHead c k u2) -(\lambda (t: T).(\lambda (t0: T).(\forall (u1: T).((pr2 c u1 u2) \to (pr3 -(CHead c k u1) t t0))))) (\lambda (t: T).(\lambda (u1: T).(\lambda (_: (pr2 c -u1 u2)).(pr3_refl (CHead c k u1) t)))) (\lambda (t0: T).(\lambda (t3: -T).(\lambda (H0: (pr2 (CHead c k u2) t3 t0)).(\lambda (t4: T).(\lambda (_: -(pr3 (CHead c k u2) t0 t4)).(\lambda (H2: ((\forall (u1: T).((pr2 c u1 u2) -\to (pr3 (CHead c k u1) t0 t4))))).(\lambda (u1: T).(\lambda (H3: (pr2 c u1 -u2)).(pr3_t t0 t3 (CHead c k u1) (pr3_pr2_pr2_t c u1 u2 H3 t3 t0 k H0) t4 (H2 -u1 H3)))))))))) t1 t2 H)))))). -(* COMMENTS -Initial nodes: 199 -END *) - -theorem pr3_pr3_pr3_t: - \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall -(t1: T).(\forall (t2: T).(\forall (k: K).((pr3 (CHead c k u2) t1 t2) \to (pr3 -(CHead c k u1) t1 t2)))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1 -u2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (t1: T).(\forall -(t2: T).(\forall (k: K).((pr3 (CHead c k t0) t1 t2) \to (pr3 (CHead c k t) t1 -t2))))))) (\lambda (t: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k: -K).(\lambda (H0: (pr3 (CHead c k t) t1 t2)).H0))))) (\lambda (t2: T).(\lambda -(t1: T).(\lambda (H0: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda (_: (pr3 c t2 -t3)).(\lambda (H2: ((\forall (t4: T).(\forall (t5: T).(\forall (k: K).((pr3 -(CHead c k t3) t4 t5) \to (pr3 (CHead c k t2) t4 t5))))))).(\lambda (t0: -T).(\lambda (t4: T).(\lambda (k: K).(\lambda (H3: (pr3 (CHead c k t3) t0 -t4)).(pr3_pr2_pr3_t c t2 t0 t4 k (H2 t0 t4 k H3) t1 H0))))))))))) u1 u2 H)))). -(* COMMENTS -Initial nodes: 187 -END *) - -theorem pr3_lift: - \forall (c: C).(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h -d c e) \to (\forall (t1: T).(\forall (t2: T).((pr3 e t1 t2) \to (pr3 c (lift -h d t1) (lift h d t2))))))))) -\def - \lambda (c: C).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda -(H: (drop h d c e)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 e t1 -t2)).(pr3_ind e (\lambda (t: T).(\lambda (t0: T).(pr3 c (lift h d t) (lift h -d t0)))) (\lambda (t: T).(pr3_refl c (lift h d t))) (\lambda (t0: T).(\lambda -(t3: T).(\lambda (H1: (pr2 e t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 e t0 -t4)).(\lambda (H3: (pr3 c (lift h d t0) (lift h d t4))).(pr3_sing c (lift h d -t0) (lift h d t3) (pr2_lift c e h d H t3 t0 H1) (lift h d t4) H3))))))) t1 t2 -H0)))))))). -(* COMMENTS -Initial nodes: 167 -END *) - -theorem pr3_eta: - \forall (c: C).(\forall (w: T).(\forall (u: T).(let t \def (THead (Bind -Abst) w u) in (\forall (v: T).((pr3 c v w) \to (pr3 c (THead (Bind Abst) v -(THead (Flat Appl) (TLRef O) (lift (S O) O t))) t)))))) -\def - \lambda (c: C).(\lambda (w: T).(\lambda (u: T).(let t \def (THead (Bind -Abst) w u) in (\lambda (v: T).(\lambda (H: (pr3 c v w)).(eq_ind_r T (THead -(Bind Abst) (lift (S O) O w) (lift (S O) (S O) u)) (\lambda (t0: T).(pr3 c -(THead (Bind Abst) v (THead (Flat Appl) (TLRef O) t0)) (THead (Bind Abst) w -u))) (pr3_head_12 c v w H (Bind Abst) (THead (Flat Appl) (TLRef O) (THead -(Bind Abst) (lift (S O) O w) (lift (S O) (S O) u))) u (pr3_pr1 (THead (Flat -Appl) (TLRef O) (THead (Bind Abst) (lift (S O) O w) (lift (S O) (S O) u))) u -(pr1_sing (THead (Bind Abbr) (TLRef O) (lift (S O) (S O) u)) (THead (Flat -Appl) (TLRef O) (THead (Bind Abst) (lift (S O) O w) (lift (S O) (S O) u))) -(pr0_beta (lift (S O) O w) (TLRef O) (TLRef O) (pr0_refl (TLRef O)) (lift (S -O) (S O) u) (lift (S O) (S O) u) (pr0_refl (lift (S O) (S O) u))) u (pr1_sing -(THead (Bind Abbr) (TLRef O) (lift (S O) O u)) (THead (Bind Abbr) (TLRef O) -(lift (S O) (S O) u)) (pr0_delta1 (TLRef O) (TLRef O) (pr0_refl (TLRef O)) -(lift (S O) (S O) u) (lift (S O) (S O) u) (pr0_refl (lift (S O) (S O) u)) -(lift (S O) O u) (subst1_lift_S u O O (le_n O))) u (pr1_pr0 (THead (Bind -Abbr) (TLRef O) (lift (S O) O u)) u (pr0_zeta Abbr not_abbr_abst u u -(pr0_refl u) (TLRef O))))) (CHead c (Bind Abst) w))) (lift (S O) O (THead -(Bind Abst) w u)) (lift_bind Abst w u (S O) O))))))). -(* COMMENTS -Initial nodes: 523 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/subst1.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/subst1.ma deleted file mode 100644 index 3bcbd4965..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/subst1.ma +++ /dev/null @@ -1,95 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr3/defs.ma". - -include "Basic-1/pr2/subst1.ma". - -theorem pr3_subst1: - \forall (c: C).(\forall (e: C).(\forall (v: T).(\forall (i: nat).((getl i c -(CHead e (Bind Abbr) v)) \to (\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) -\to (\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T (\lambda (w2: T).(pr3 c -w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2)))))))))))) -\def - \lambda (c: C).(\lambda (e: C).(\lambda (v: T).(\lambda (i: nat).(\lambda -(H: (getl i c (CHead e (Bind Abbr) v))).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (H0: (pr3 c t1 t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: -T).(\forall (w1: T).((subst1 i v t w1) \to (ex2 T (\lambda (w2: T).(pr3 c w1 -w2)) (\lambda (w2: T).(subst1 i v t0 w2))))))) (\lambda (t: T).(\lambda (w1: -T).(\lambda (H1: (subst1 i v t w1)).(ex_intro2 T (\lambda (w2: T).(pr3 c w1 -w2)) (\lambda (w2: T).(subst1 i v t w2)) w1 (pr3_refl c w1) H1)))) (\lambda -(t3: T).(\lambda (t4: T).(\lambda (H1: (pr2 c t4 t3)).(\lambda (t5: -T).(\lambda (_: (pr3 c t3 t5)).(\lambda (H3: ((\forall (w1: T).((subst1 i v -t3 w1) \to (ex2 T (\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1 i -v t5 w2))))))).(\lambda (w1: T).(\lambda (H4: (subst1 i v t4 w1)).(ex2_ind T -(\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t3 w2)) (ex2 T -(\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1 i v t5 w2))) -(\lambda (x: T).(\lambda (H5: (pr2 c w1 x)).(\lambda (H6: (subst1 i v t3 -x)).(ex2_ind T (\lambda (w2: T).(pr3 c x w2)) (\lambda (w2: T).(subst1 i v t5 -w2)) (ex2 T (\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1 i v t5 -w2))) (\lambda (x0: T).(\lambda (H7: (pr3 c x x0)).(\lambda (H8: (subst1 i v -t5 x0)).(ex_intro2 T (\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1 -i v t5 w2)) x0 (pr3_sing c x w1 H5 x0 H7) H8)))) (H3 x H6))))) (pr2_subst1 c -e v i H t4 t3 H1 w1 H4)))))))))) t1 t2 H0)))))))). -(* COMMENTS -Initial nodes: 425 -END *) - -theorem pr3_gen_cabbr: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall -(e: C).(\forall (u: T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) -\to (\forall (a0: C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d -a0 a) \to (\forall (x1: T).((subst1 d u t1 (lift (S O) d x1)) \to (ex2 T -(\lambda (x2: T).(subst1 d u t2 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a -x1 x2)))))))))))))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 -t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (e: C).(\forall (u: -T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) \to (\forall (a0: -C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (\forall -(x1: T).((subst1 d u t (lift (S O) d x1)) \to (ex2 T (\lambda (x2: T).(subst1 -d u t0 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x1 x2))))))))))))))) -(\lambda (t: T).(\lambda (e: C).(\lambda (u: T).(\lambda (d: nat).(\lambda -(_: (getl d c (CHead e (Bind Abbr) u))).(\lambda (a0: C).(\lambda (_: -(csubst1 d u c a0)).(\lambda (a: C).(\lambda (_: (drop (S O) d a0 -a)).(\lambda (x1: T).(\lambda (H3: (subst1 d u t (lift (S O) d -x1))).(ex_intro2 T (\lambda (x2: T).(subst1 d u t (lift (S O) d x2))) -(\lambda (x2: T).(pr3 a x1 x2)) x1 H3 (pr3_refl a x1))))))))))))) (\lambda -(t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c t3 t0)).(\lambda (t4: -T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2: ((\forall (e: C).(\forall (u: -T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) \to (\forall (a0: -C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (\forall -(x1: T).((subst1 d u t0 (lift (S O) d x1)) \to (ex2 T (\lambda (x2: -T).(subst1 d u t4 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x1 -x2))))))))))))))).(\lambda (e: C).(\lambda (u: T).(\lambda (d: nat).(\lambda -(H3: (getl d c (CHead e (Bind Abbr) u))).(\lambda (a0: C).(\lambda (H4: -(csubst1 d u c a0)).(\lambda (a: C).(\lambda (H5: (drop (S O) d a0 -a)).(\lambda (x1: T).(\lambda (H6: (subst1 d u t3 (lift (S O) d -x1))).(ex2_ind T (\lambda (x2: T).(subst1 d u t0 (lift (S O) d x2))) (\lambda -(x2: T).(pr2 a x1 x2)) (ex2 T (\lambda (x2: T).(subst1 d u t4 (lift (S O) d -x2))) (\lambda (x2: T).(pr3 a x1 x2))) (\lambda (x: T).(\lambda (H7: (subst1 -d u t0 (lift (S O) d x))).(\lambda (H8: (pr2 a x1 x)).(ex2_ind T (\lambda -(x2: T).(subst1 d u t4 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x x2)) -(ex2 T (\lambda (x2: T).(subst1 d u t4 (lift (S O) d x2))) (\lambda (x2: -T).(pr3 a x1 x2))) (\lambda (x0: T).(\lambda (H9: (subst1 d u t4 (lift (S O) -d x0))).(\lambda (H10: (pr3 a x x0)).(ex_intro2 T (\lambda (x2: T).(subst1 d -u t4 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x1 x2)) x0 H9 (pr3_sing a x -x1 H8 x0 H10))))) (H2 e u d H3 a0 H4 a H5 x H7))))) (pr2_gen_cabbr c t3 t0 H0 -e u d H3 a0 H4 a H5 x1 H6)))))))))))))))))) t1 t2 H)))). -(* COMMENTS -Initial nodes: 731 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/wcpr0.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/wcpr0.ma deleted file mode 100644 index 052f4e5b2..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/wcpr0.ma +++ /dev/null @@ -1,66 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr3/props.ma". - -include "Basic-1/wcpr0/getl.ma". - -theorem pr3_wcpr0_t: - \forall (c1: C).(\forall (c2: C).((wcpr0 c2 c1) \to (\forall (t1: -T).(\forall (t2: T).((pr3 c1 t1 t2) \to (pr3 c2 t1 t2)))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c2 c1)).(wcpr0_ind -(\lambda (c: C).(\lambda (c0: C).(\forall (t1: T).(\forall (t2: T).((pr3 c0 -t1 t2) \to (pr3 c t1 t2)))))) (\lambda (c: C).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (H0: (pr3 c t1 t2)).H0)))) (\lambda (c0: C).(\lambda (c3: -C).(\lambda (H0: (wcpr0 c0 c3)).(\lambda (_: ((\forall (t1: T).(\forall (t2: -T).((pr3 c3 t1 t2) \to (pr3 c0 t1 t2)))))).(\lambda (u1: T).(\lambda (u2: -T).(\lambda (H2: (pr0 u1 u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (H3: (pr3 (CHead c3 k u2) t1 t2)).(pr3_ind (CHead c3 k u1) -(\lambda (t: T).(\lambda (t0: T).(pr3 (CHead c0 k u1) t t0))) (\lambda (t: -T).(pr3_refl (CHead c0 k u1) t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda -(H4: (pr2 (CHead c3 k u1) t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 (CHead -c3 k u1) t0 t4)).(\lambda (H6: (pr3 (CHead c0 k u1) t0 t4)).(pr3_t t0 t3 -(CHead c0 k u1) (insert_eq C (CHead c3 k u1) (\lambda (c: C).(pr2 c t3 t0)) -(\lambda (_: C).(pr3 (CHead c0 k u1) t3 t0)) (\lambda (y: C).(\lambda (H7: -(pr2 y t3 t0)).(pr2_ind (\lambda (c: C).(\lambda (t: T).(\lambda (t5: T).((eq -C c (CHead c3 k u1)) \to (pr3 (CHead c0 k u1) t t5))))) (\lambda (c: -C).(\lambda (t5: T).(\lambda (t6: T).(\lambda (H8: (pr0 t5 t6)).(\lambda (_: -(eq C c (CHead c3 k u1))).(pr3_pr2 (CHead c0 k u1) t5 t6 (pr2_free (CHead c0 -k u1) t5 t6 H8))))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: -T).(\lambda (i: nat).(\lambda (H8: (getl i c (CHead d (Bind Abbr) -u))).(\lambda (t5: T).(\lambda (t6: T).(\lambda (H9: (pr0 t5 t6)).(\lambda -(t: T).(\lambda (H10: (subst0 i u t6 t)).(\lambda (H11: (eq C c (CHead c3 k -u1))).(let H12 \def (eq_ind C c (\lambda (c4: C).(getl i c4 (CHead d (Bind -Abbr) u))) H8 (CHead c3 k u1) H11) in (ex3_2_ind C T (\lambda (e2: -C).(\lambda (u3: T).(getl i (CHead c0 k u1) (CHead e2 (Bind Abbr) u3)))) -(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 d))) (\lambda (_: C).(\lambda (u3: -T).(pr0 u3 u))) (pr3 (CHead c0 k u1) t5 t) (\lambda (x0: C).(\lambda (x1: -T).(\lambda (H13: (getl i (CHead c0 k u1) (CHead x0 (Bind Abbr) -x1))).(\lambda (_: (wcpr0 x0 d)).(\lambda (H15: (pr0 x1 u)).(ex2_ind T -(\lambda (t7: T).(subst0 i x1 t6 t7)) (\lambda (t7: T).(pr0 t7 t)) (pr3 -(CHead c0 k u1) t5 t) (\lambda (x: T).(\lambda (H16: (subst0 i x1 t6 -x)).(\lambda (H17: (pr0 x t)).(pr3_sing (CHead c0 k u1) x t5 (pr2_delta -(CHead c0 k u1) x0 x1 i H13 t5 t6 H9 x H16) t (pr3_pr2 (CHead c0 k u1) x t -(pr2_free (CHead c0 k u1) x t H17)))))) (pr0_subst0_back u t6 t i H10 x1 -H15))))))) (wcpr0_getl_back (CHead c3 k u1) (CHead c0 k u1) (wcpr0_comp c0 c3 -H0 u1 u1 (pr0_refl u1) k) i d u (Bind Abbr) H12)))))))))))))) y t3 t0 H7))) -H4) t4 H6))))))) t1 t2 (pr3_pr2_pr3_t c3 u2 t1 t2 k H3 u1 (pr2_free c3 u1 u2 -H2)))))))))))))) c2 c1 H))). -(* COMMENTS -Initial nodes: 799 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/preamble.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/preamble.ma deleted file mode 100644 index c9d6a172a..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/preamble.ma +++ /dev/null @@ -1,15 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "Ground-1/theory.ma". diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/r/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/r/defs.ma deleted file mode 100644 index cde40fdf8..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/r/defs.ma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/T/defs.ma". - -definition r: - K \to (nat \to nat) -\def - \lambda (k: K).(\lambda (i: nat).(match k with [(Bind _) \Rightarrow i | -(Flat _) \Rightarrow (S i)])). - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/r/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/r/props.ma deleted file mode 100644 index 0815aaf5e..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/r/props.ma +++ /dev/null @@ -1,117 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/r/defs.ma". - -include "Basic-1/s/defs.ma". - -theorem r_S: - \forall (k: K).(\forall (i: nat).(eq nat (r k (S i)) (S (r k i)))) -\def - \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(eq nat (r k0 (S -i)) (S (r k0 i))))) (\lambda (b: B).(\lambda (i: nat).(refl_equal nat (S (r -(Bind b) i))))) (\lambda (f: F).(\lambda (i: nat).(refl_equal nat (S (r (Flat -f) i))))) k). -(* COMMENTS -Initial nodes: 65 -END *) - -theorem r_plus: - \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (r k (plus i j)) -(plus (r k i) j)))) -\def - \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: -nat).(eq nat (r k0 (plus i j)) (plus (r k0 i) j))))) (\lambda (b: B).(\lambda -(i: nat).(\lambda (j: nat).(refl_equal nat (plus (r (Bind b) i) j))))) -(\lambda (f: F).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (plus (r -(Flat f) i) j))))) k). -(* COMMENTS -Initial nodes: 79 -END *) - -theorem r_plus_sym: - \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (r k (plus i j)) -(plus i (r k j))))) -\def - \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: -nat).(eq nat (r k0 (plus i j)) (plus i (r k0 j)))))) (\lambda (_: B).(\lambda -(i: nat).(\lambda (j: nat).(refl_equal nat (plus i j))))) (\lambda (_: -F).(\lambda (i: nat).(\lambda (j: nat).(plus_n_Sm i j)))) k). -(* COMMENTS -Initial nodes: 63 -END *) - -theorem r_minus: - \forall (i: nat).(\forall (n: nat).((lt n i) \to (\forall (k: K).(eq nat -(minus (r k i) (S n)) (r k (minus i (S n))))))) -\def - \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (lt n i)).(\lambda (k: -K).(K_ind (\lambda (k0: K).(eq nat (minus (r k0 i) (S n)) (r k0 (minus i (S -n))))) (\lambda (_: B).(refl_equal nat (minus i (S n)))) (\lambda (_: -F).(minus_x_Sy i n H)) k)))). -(* COMMENTS -Initial nodes: 69 -END *) - -theorem r_dis: - \forall (k: K).(\forall (P: Prop).(((((\forall (i: nat).(eq nat (r k i) i))) -\to P)) \to (((((\forall (i: nat).(eq nat (r k i) (S i)))) \to P)) \to P))) -\def - \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (P: Prop).(((((\forall (i: -nat).(eq nat (r k0 i) i))) \to P)) \to (((((\forall (i: nat).(eq nat (r k0 i) -(S i)))) \to P)) \to P)))) (\lambda (b: B).(\lambda (P: Prop).(\lambda (H: -((((\forall (i: nat).(eq nat (r (Bind b) i) i))) \to P))).(\lambda (_: -((((\forall (i: nat).(eq nat (r (Bind b) i) (S i)))) \to P))).(H (\lambda (i: -nat).(refl_equal nat i))))))) (\lambda (f: F).(\lambda (P: Prop).(\lambda (_: -((((\forall (i: nat).(eq nat (r (Flat f) i) i))) \to P))).(\lambda (H0: -((((\forall (i: nat).(eq nat (r (Flat f) i) (S i)))) \to P))).(H0 (\lambda -(i: nat).(refl_equal nat (S i)))))))) k). -(* COMMENTS -Initial nodes: 151 -END *) - -theorem s_r: - \forall (k: K).(\forall (i: nat).(eq nat (s k (r k i)) (S i))) -\def - \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(eq nat (s k0 (r k0 -i)) (S i)))) (\lambda (_: B).(\lambda (i: nat).(refl_equal nat (S i)))) -(\lambda (_: F).(\lambda (i: nat).(refl_equal nat (S i)))) k). -(* COMMENTS -Initial nodes: 51 -END *) - -theorem r_arith0: - \forall (k: K).(\forall (i: nat).(eq nat (minus (r k (S i)) (S O)) (r k i))) -\def - \lambda (k: K).(\lambda (i: nat).(eq_ind_r nat (S (r k i)) (\lambda (n: -nat).(eq nat (minus n (S O)) (r k i))) (eq_ind_r nat (r k i) (\lambda (n: -nat).(eq nat n (r k i))) (refl_equal nat (r k i)) (minus (S (r k i)) (S O)) -(minus_Sx_SO (r k i))) (r k (S i)) (r_S k i))). -(* COMMENTS -Initial nodes: 105 -END *) - -theorem r_arith1: - \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (minus (r k (S -i)) (S j)) (minus (r k i) j)))) -\def - \lambda (k: K).(\lambda (i: nat).(\lambda (j: nat).(eq_ind_r nat (S (r k i)) -(\lambda (n: nat).(eq nat (minus n (S j)) (minus (r k i) j))) (refl_equal nat -(minus (r k i) j)) (r k (S i)) (r_S k i)))). -(* COMMENTS -Initial nodes: 69 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/s/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/s/defs.ma deleted file mode 100644 index 0d1fb7914..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/s/defs.ma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/T/defs.ma". - -definition s: - K \to (nat \to nat) -\def - \lambda (k: K).(\lambda (i: nat).(match k with [(Bind _) \Rightarrow (S i) | -(Flat _) \Rightarrow i])). - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/s/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/s/props.ma deleted file mode 100644 index 3cb4fbd74..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/s/props.ma +++ /dev/null @@ -1,151 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/s/defs.ma". - -theorem s_S: - \forall (k: K).(\forall (i: nat).(eq nat (s k (S i)) (S (s k i)))) -\def - \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(eq nat (s k0 (S -i)) (S (s k0 i))))) (\lambda (b: B).(\lambda (i: nat).(refl_equal nat (S (s -(Bind b) i))))) (\lambda (f: F).(\lambda (i: nat).(refl_equal nat (S (s (Flat -f) i))))) k). -(* COMMENTS -Initial nodes: 65 -END *) - -theorem s_plus: - \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (s k (plus i j)) -(plus (s k i) j)))) -\def - \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: -nat).(eq nat (s k0 (plus i j)) (plus (s k0 i) j))))) (\lambda (b: B).(\lambda -(i: nat).(\lambda (j: nat).(refl_equal nat (plus (s (Bind b) i) j))))) -(\lambda (f: F).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (plus (s -(Flat f) i) j))))) k). -(* COMMENTS -Initial nodes: 79 -END *) - -theorem s_plus_sym: - \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (s k (plus i j)) -(plus i (s k j))))) -\def - \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: -nat).(eq nat (s k0 (plus i j)) (plus i (s k0 j)))))) (\lambda (_: B).(\lambda -(i: nat).(\lambda (j: nat).(eq_ind_r nat (plus i (S j)) (\lambda (n: nat).(eq -nat n (plus i (S j)))) (refl_equal nat (plus i (S j))) (S (plus i j)) -(plus_n_Sm i j))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (j: -nat).(refl_equal nat (plus i (s (Flat f) j)))))) k). -(* COMMENTS -Initial nodes: 117 -END *) - -theorem s_minus: - \forall (k: K).(\forall (i: nat).(\forall (j: nat).((le j i) \to (eq nat (s -k (minus i j)) (minus (s k i) j))))) -\def - \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: -nat).((le j i) \to (eq nat (s k0 (minus i j)) (minus (s k0 i) j)))))) -(\lambda (_: B).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (le j -i)).(eq_ind_r nat (minus (S i) j) (\lambda (n: nat).(eq nat n (minus (S i) -j))) (refl_equal nat (minus (S i) j)) (S (minus i j)) (minus_Sn_m i j H)))))) -(\lambda (f: F).(\lambda (i: nat).(\lambda (j: nat).(\lambda (_: (le j -i)).(refl_equal nat (minus (s (Flat f) i) j)))))) k). -(* COMMENTS -Initial nodes: 137 -END *) - -theorem minus_s_s: - \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (minus (s k i) (s -k j)) (minus i j)))) -\def - \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: -nat).(eq nat (minus (s k0 i) (s k0 j)) (minus i j))))) (\lambda (_: -B).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (minus i j))))) -(\lambda (_: F).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (minus i -j))))) k). -(* COMMENTS -Initial nodes: 67 -END *) - -theorem s_le: - \forall (k: K).(\forall (i: nat).(\forall (j: nat).((le i j) \to (le (s k i) -(s k j))))) -\def - \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: -nat).((le i j) \to (le (s k0 i) (s k0 j)))))) (\lambda (_: B).(\lambda (i: -nat).(\lambda (j: nat).(\lambda (H: (le i j)).(le_n_S i j H))))) (\lambda (_: -F).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (le i j)).H)))) k). -(* COMMENTS -Initial nodes: 65 -END *) - -theorem s_lt: - \forall (k: K).(\forall (i: nat).(\forall (j: nat).((lt i j) \to (lt (s k i) -(s k j))))) -\def - \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: -nat).((lt i j) \to (lt (s k0 i) (s k0 j)))))) (\lambda (_: B).(\lambda (i: -nat).(\lambda (j: nat).(\lambda (H: (lt i j)).(le_n_S (S i) j H))))) (\lambda -(_: F).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (lt i j)).H)))) k). -(* COMMENTS -Initial nodes: 67 -END *) - -theorem s_inj: - \forall (k: K).(\forall (i: nat).(\forall (j: nat).((eq nat (s k i) (s k j)) -\to (eq nat i j)))) -\def - \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: -nat).((eq nat (s k0 i) (s k0 j)) \to (eq nat i j))))) (\lambda (b: -B).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (eq nat (s (Bind b) i) (s -(Bind b) j))).(eq_add_S i j H))))) (\lambda (f: F).(\lambda (i: nat).(\lambda -(j: nat).(\lambda (H: (eq nat (s (Flat f) i) (s (Flat f) j))).H)))) k). -(* COMMENTS -Initial nodes: 97 -END *) - -theorem s_inc: - \forall (k: K).(\forall (i: nat).(le i (s k i))) -\def - \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(le i (s k0 i)))) -(\lambda (b: B).(\lambda (i: nat).(le_S_n i (s (Bind b) i) (le_S (S i) (s -(Bind b) i) (le_n (s (Bind b) i)))))) (\lambda (f: F).(\lambda (i: nat).(le_n -(s (Flat f) i)))) k). -(* COMMENTS -Initial nodes: 73 -END *) - -theorem s_arith0: - \forall (k: K).(\forall (i: nat).(eq nat (minus (s k i) (s k O)) i)) -\def - \lambda (k: K).(\lambda (i: nat).(eq_ind_r nat (minus i O) (\lambda (n: -nat).(eq nat n i)) (eq_ind nat i (\lambda (n: nat).(eq nat n i)) (refl_equal -nat i) (minus i O) (minus_n_O i)) (minus (s k i) (s k O)) (minus_s_s k i O))). -(* COMMENTS -Initial nodes: 77 -END *) - -theorem s_arith1: - \forall (b: B).(\forall (i: nat).(eq nat (minus (s (Bind b) i) (S O)) i)) -\def - \lambda (_: B).(\lambda (i: nat).(eq_ind nat i (\lambda (n: nat).(eq nat n -i)) (refl_equal nat i) (minus i O) (minus_n_O i))). -(* COMMENTS -Initial nodes: 35 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/sc3/arity.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/sc3/arity.ma deleted file mode 100644 index 651321580..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/sc3/arity.ma +++ /dev/null @@ -1,322 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubc/arity.ma". - -include "Basic-1/csubc/getl.ma". - -include "Basic-1/csubc/drop1.ma". - -include "Basic-1/csubc/props.ma". - -theorem sc3_arity_csubc: - \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1 -t a) \to (\forall (d1: C).(\forall (is: PList).((drop1 is d1 c1) \to (\forall -(c2: C).((csubc g d1 c2) \to (sc3 g a c2 (lift1 is t))))))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: -(arity g c1 t a)).(arity_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda (a0: -A).(\forall (d1: C).(\forall (is: PList).((drop1 is d1 c) \to (\forall (c2: -C).((csubc g d1 c2) \to (sc3 g a0 c2 (lift1 is t0)))))))))) (\lambda (c: -C).(\lambda (n: nat).(\lambda (d1: C).(\lambda (is: PList).(\lambda (_: -(drop1 is d1 c)).(\lambda (c2: C).(\lambda (_: (csubc g d1 c2)).(eq_ind_r T -(TSort n) (\lambda (t0: T).(land (arity g c2 t0 (ASort O n)) (sn3 c2 t0))) -(conj (arity g c2 (TSort n) (ASort O n)) (sn3 c2 (TSort n)) (arity_sort g c2 -n) (sn3_nf2 c2 (TSort n) (nf2_sort c2 n))) (lift1 is (TSort n)) (lift1_sort n -is))))))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: -nat).(\lambda (H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (a0: -A).(\lambda (_: (arity g d u a0)).(\lambda (H2: ((\forall (d1: C).(\forall -(is: PList).((drop1 is d1 d) \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g -a0 c2 (lift1 is u))))))))).(\lambda (d1: C).(\lambda (is: PList).(\lambda -(H3: (drop1 is d1 c)).(\lambda (c2: C).(\lambda (H4: (csubc g d1 c2)).(let -H_x \def (drop1_getl_trans is c d1 H3 Abbr d u i H0) in (let H5 \def H_x in -(ex2_ind C (\lambda (e2: C).(drop1 (ptrans is i) e2 d)) (\lambda (e2: -C).(getl (trans is i) d1 (CHead e2 (Bind Abbr) (lift1 (ptrans is i) u)))) -(sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x: C).(\lambda (_: (drop1 -(ptrans is i) x d)).(\lambda (H7: (getl (trans is i) d1 (CHead x (Bind Abbr) -(lift1 (ptrans is i) u)))).(let H_x0 \def (csubc_getl_conf g d1 (CHead x -(Bind Abbr) (lift1 (ptrans is i) u)) (trans is i) H7 c2 H4) in (let H8 \def -H_x0 in (ex2_ind C (\lambda (e2: C).(getl (trans is i) c2 e2)) (\lambda (e2: -C).(csubc g (CHead x (Bind Abbr) (lift1 (ptrans is i) u)) e2)) (sc3 g a0 c2 -(lift1 is (TLRef i))) (\lambda (x0: C).(\lambda (H9: (getl (trans is i) c2 -x0)).(\lambda (H10: (csubc g (CHead x (Bind Abbr) (lift1 (ptrans is i) u)) -x0)).(let H_x1 \def (csubc_gen_head_l g x x0 (lift1 (ptrans is i) u) (Bind -Abbr) H10) in (let H11 \def H_x1 in (or3_ind (ex2 C (\lambda (c3: C).(eq C x0 -(CHead c3 (Bind Abbr) (lift1 (ptrans is i) u)))) (\lambda (c3: C).(csubc g x -c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K -(Bind Abbr) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: -A).(eq C x0 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: -T).(\lambda (_: A).(csubc g x c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans is i) u))))) (\lambda (c3: -C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3 w))))) (ex4_3 B C T (\lambda -(b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C x0 (CHead c3 (Bind b) v2))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K (Bind Abbr) (Bind -Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b -Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g x c3))))) -(sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (H12: (ex2 C (\lambda (c3: C).(eq -C x0 (CHead c3 (Bind Abbr) (lift1 (ptrans is i) u)))) (\lambda (c3: C).(csubc -g x c3)))).(ex2_ind C (\lambda (c3: C).(eq C x0 (CHead c3 (Bind Abbr) (lift1 -(ptrans is i) u)))) (\lambda (c3: C).(csubc g x c3)) (sc3 g a0 c2 (lift1 is -(TLRef i))) (\lambda (x1: C).(\lambda (H13: (eq C x0 (CHead x1 (Bind Abbr) -(lift1 (ptrans is i) u)))).(\lambda (_: (csubc g x x1)).(let H15 \def (eq_ind -C x0 (\lambda (c0: C).(getl (trans is i) c2 c0)) H9 (CHead x1 (Bind Abbr) -(lift1 (ptrans is i) u)) H13) in (let H_y \def (sc3_abbr g a0 TNil) in -(eq_ind_r T (TLRef (trans is i)) (\lambda (t0: T).(sc3 g a0 c2 t0)) (H_y -(trans is i) x1 (lift1 (ptrans is i) u) c2 (eq_ind T (lift1 is (lift (S i) O -u)) (\lambda (t0: T).(sc3 g a0 c2 t0)) (eq_ind T (lift1 (PConsTail is (S i) -O) u) (\lambda (t0: T).(sc3 g a0 c2 t0)) (H2 d1 (PConsTail is (S i) O) -(drop1_cons_tail c d (S i) O (getl_drop Abbr c d u i H0) is d1 H3) c2 H4) -(lift1 is (lift (S i) O u)) (lift1_cons_tail u (S i) O is)) (lift (S (trans -is i)) O (lift1 (ptrans is i) u)) (lift1_free is i u)) H15) (lift1 is (TLRef -i)) (lift1_lref is i))))))) H12)) (\lambda (H12: (ex5_3 C T A (\lambda (_: -C).(\lambda (_: T).(\lambda (_: A).(eq K (Bind Abbr) (Bind Abst))))) (\lambda -(c3: C).(\lambda (w: T).(\lambda (_: A).(eq C x0 (CHead c3 (Bind Abbr) w))))) -(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g x c3)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans -is i) u))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3 -w)))))).(ex5_3_ind C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq -K (Bind Abbr) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: -A).(eq C x0 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: -T).(\lambda (_: A).(csubc g x c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans is i) u))))) (\lambda (c3: -C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3 w)))) (sc3 g a0 c2 (lift1 is -(TLRef i))) (\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: A).(\lambda (H13: -(eq K (Bind Abbr) (Bind Abst))).(\lambda (H14: (eq C x0 (CHead x1 (Bind Abbr) -x2))).(\lambda (_: (csubc g x x1)).(\lambda (_: (sc3 g (asucc g x3) x (lift1 -(ptrans is i) u))).(\lambda (_: (sc3 g x3 x1 x2)).(let H18 \def (eq_ind C x0 -(\lambda (c0: C).(getl (trans is i) c2 c0)) H9 (CHead x1 (Bind Abbr) x2) H14) -in (let H19 \def (eq_ind K (Bind Abbr) (\lambda (ee: K).(match ee in K return -(\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B return -(\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | -Void \Rightarrow False]) | (Flat _) \Rightarrow False])) I (Bind Abst) H13) -in (False_ind (sc3 g a0 c2 (lift1 is (TLRef i))) H19))))))))))) H12)) -(\lambda (H12: (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: -T).(eq C x0 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: T).(eq K (Bind Abbr) (Bind Void))))) (\lambda (b: B).(\lambda -(_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: -C).(\lambda (_: T).(csubc g x c3)))))).(ex4_3_ind B C T (\lambda (b: -B).(\lambda (c3: C).(\lambda (v2: T).(eq C x0 (CHead c3 (Bind b) v2))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K (Bind Abbr) (Bind -Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b -Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g x c3)))) -(sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x1: B).(\lambda (x2: C).(\lambda -(x3: T).(\lambda (H13: (eq C x0 (CHead x2 (Bind x1) x3))).(\lambda (H14: (eq -K (Bind Abbr) (Bind Void))).(\lambda (_: (not (eq B x1 Void))).(\lambda (_: -(csubc g x x2)).(let H17 \def (eq_ind C x0 (\lambda (c0: C).(getl (trans is -i) c2 c0)) H9 (CHead x2 (Bind x1) x3) H13) in (let H18 \def (eq_ind K (Bind -Abbr) (\lambda (ee: K).(match ee in K return (\lambda (_: K).Prop) with -[(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr -\Rightarrow True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat -_) \Rightarrow False])) I (Bind Void) H14) in (False_ind (sc3 g a0 c2 (lift1 -is (TLRef i))) H18)))))))))) H12)) H11)))))) H8)))))) H5)))))))))))))))) -(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H0: (getl i c (CHead d (Bind Abst) u))).(\lambda (a0: A).(\lambda (H1: -(arity g d u (asucc g a0))).(\lambda (_: ((\forall (d1: C).(\forall (is: -PList).((drop1 is d1 d) \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g -(asucc g a0) c2 (lift1 is u))))))))).(\lambda (d1: C).(\lambda (is: -PList).(\lambda (H3: (drop1 is d1 c)).(\lambda (c2: C).(\lambda (H4: (csubc g -d1 c2)).(let H5 \def H0 in (let H_x \def (drop1_getl_trans is c d1 H3 Abst d -u i H5) in (let H6 \def H_x in (ex2_ind C (\lambda (e2: C).(drop1 (ptrans is -i) e2 d)) (\lambda (e2: C).(getl (trans is i) d1 (CHead e2 (Bind Abst) (lift1 -(ptrans is i) u)))) (sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x: -C).(\lambda (H7: (drop1 (ptrans is i) x d)).(\lambda (H8: (getl (trans is i) -d1 (CHead x (Bind Abst) (lift1 (ptrans is i) u)))).(let H_x0 \def -(csubc_getl_conf g d1 (CHead x (Bind Abst) (lift1 (ptrans is i) u)) (trans is -i) H8 c2 H4) in (let H9 \def H_x0 in (ex2_ind C (\lambda (e2: C).(getl (trans -is i) c2 e2)) (\lambda (e2: C).(csubc g (CHead x (Bind Abst) (lift1 (ptrans -is i) u)) e2)) (sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x0: C).(\lambda -(H10: (getl (trans is i) c2 x0)).(\lambda (H11: (csubc g (CHead x (Bind Abst) -(lift1 (ptrans is i) u)) x0)).(let H_x1 \def (csubc_gen_head_l g x x0 (lift1 -(ptrans is i) u) (Bind Abst) H11) in (let H12 \def H_x1 in (or3_ind (ex2 C -(\lambda (c3: C).(eq C x0 (CHead c3 (Bind Abst) (lift1 (ptrans is i) u)))) -(\lambda (c3: C).(csubc g x c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: -T).(\lambda (_: A).(eq K (Bind Abst) (Bind Abst))))) (\lambda (c3: -C).(\lambda (w: T).(\lambda (_: A).(eq C x0 (CHead c3 (Bind Abbr) w))))) -(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g x c3)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans -is i) u))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3 -w))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C -x0 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -T).(eq K (Bind Abst) (Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda -(_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: -T).(csubc g x c3))))) (sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (H13: (ex2 -C (\lambda (c3: C).(eq C x0 (CHead c3 (Bind Abst) (lift1 (ptrans is i) u)))) -(\lambda (c3: C).(csubc g x c3)))).(ex2_ind C (\lambda (c3: C).(eq C x0 -(CHead c3 (Bind Abst) (lift1 (ptrans is i) u)))) (\lambda (c3: C).(csubc g x -c3)) (sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x1: C).(\lambda (H14: (eq C -x0 (CHead x1 (Bind Abst) (lift1 (ptrans is i) u)))).(\lambda (_: (csubc g x -x1)).(let H16 \def (eq_ind C x0 (\lambda (c0: C).(getl (trans is i) c2 c0)) -H10 (CHead x1 (Bind Abst) (lift1 (ptrans is i) u)) H14) in (let H_y \def -(sc3_abst g a0 TNil) in (eq_ind_r T (TLRef (trans is i)) (\lambda (t0: -T).(sc3 g a0 c2 t0)) (H_y c2 (trans is i) (csubc_arity_conf g d1 c2 H4 (TLRef -(trans is i)) a0 (eq_ind T (lift1 is (TLRef i)) (\lambda (t0: T).(arity g d1 -t0 a0)) (arity_lift1 g a0 c is d1 (TLRef i) H3 (arity_abst g c d u i H0 a0 -H1)) (TLRef (trans is i)) (lift1_lref is i))) (nf2_lref_abst c2 x1 (lift1 -(ptrans is i) u) (trans is i) H16) I) (lift1 is (TLRef i)) (lift1_lref is -i))))))) H13)) (\lambda (H13: (ex5_3 C T A (\lambda (_: C).(\lambda (_: -T).(\lambda (_: A).(eq K (Bind Abst) (Bind Abst))))) (\lambda (c3: -C).(\lambda (w: T).(\lambda (_: A).(eq C x0 (CHead c3 (Bind Abbr) w))))) -(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g x c3)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans -is i) u))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3 -w)))))).(ex5_3_ind C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq -K (Bind Abst) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: -A).(eq C x0 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: -T).(\lambda (_: A).(csubc g x c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans is i) u))))) (\lambda (c3: -C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3 w)))) (sc3 g a0 c2 (lift1 is -(TLRef i))) (\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: A).(\lambda (_: -(eq K (Bind Abst) (Bind Abst))).(\lambda (H15: (eq C x0 (CHead x1 (Bind Abbr) -x2))).(\lambda (_: (csubc g x x1)).(\lambda (H17: (sc3 g (asucc g x3) x -(lift1 (ptrans is i) u))).(\lambda (H18: (sc3 g x3 x1 x2)).(let H19 \def -(eq_ind C x0 (\lambda (c0: C).(getl (trans is i) c2 c0)) H10 (CHead x1 (Bind -Abbr) x2) H15) in (let H_y \def (sc3_abbr g a0 TNil) in (eq_ind_r T (TLRef -(trans is i)) (\lambda (t0: T).(sc3 g a0 c2 t0)) (H_y (trans is i) x1 x2 c2 -(let H_y0 \def (arity_lift1 g (asucc g a0) d (ptrans is i) x u H7 H1) in (let -H_y1 \def (sc3_arity_gen g x (lift1 (ptrans is i) u) (asucc g x3) H17) in -(sc3_repl g x3 c2 (lift (S (trans is i)) O x2) (sc3_lift g x3 x1 x2 H18 c2 (S -(trans is i)) O (getl_drop Abbr c2 x1 x2 (trans is i) H19)) a0 (asucc_inj g -x3 a0 (arity_mono g x (lift1 (ptrans is i) u) (asucc g x3) H_y1 (asucc g a0) -H_y0))))) H19) (lift1 is (TLRef i)) (lift1_lref is i)))))))))))) H13)) -(\lambda (H13: (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: -T).(eq C x0 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: T).(eq K (Bind Abst) (Bind Void))))) (\lambda (b: B).(\lambda -(_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: -C).(\lambda (_: T).(csubc g x c3)))))).(ex4_3_ind B C T (\lambda (b: -B).(\lambda (c3: C).(\lambda (v2: T).(eq C x0 (CHead c3 (Bind b) v2))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K (Bind Abst) (Bind -Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b -Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g x c3)))) -(sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x1: B).(\lambda (x2: C).(\lambda -(x3: T).(\lambda (H14: (eq C x0 (CHead x2 (Bind x1) x3))).(\lambda (H15: (eq -K (Bind Abst) (Bind Void))).(\lambda (_: (not (eq B x1 Void))).(\lambda (_: -(csubc g x x2)).(let H18 \def (eq_ind C x0 (\lambda (c0: C).(getl (trans is -i) c2 c0)) H10 (CHead x2 (Bind x1) x3) H14) in (let H19 \def (eq_ind K (Bind -Abst) (\lambda (ee: K).(match ee in K return (\lambda (_: K).Prop) with -[(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr -\Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat -_) \Rightarrow False])) I (Bind Void) H15) in (False_ind (sc3 g a0 c2 (lift1 -is (TLRef i))) H19)))))))))) H13)) H12)))))) H9)))))) H6))))))))))))))))) -(\lambda (b: B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c: C).(\lambda -(u: T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H2: -((\forall (d1: C).(\forall (is: PList).((drop1 is d1 c) \to (\forall (c2: -C).((csubc g d1 c2) \to (sc3 g a1 c2 (lift1 is u))))))))).(\lambda (t0: -T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c (Bind b) u) t0 -a2)).(\lambda (H4: ((\forall (d1: C).(\forall (is: PList).((drop1 is d1 -(CHead c (Bind b) u)) \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g a2 c2 -(lift1 is t0))))))))).(\lambda (d1: C).(\lambda (is: PList).(\lambda (H5: -(drop1 is d1 c)).(\lambda (c2: C).(\lambda (H6: (csubc g d1 c2)).(let H_y -\def (sc3_bind g b H0 a1 a2 TNil) in (eq_ind_r T (THead (Bind b) (lift1 is u) -(lift1 (Ss is) t0)) (\lambda (t1: T).(sc3 g a2 c2 t1)) (H_y c2 (lift1 is u) -(lift1 (Ss is) t0) (H4 (CHead d1 (Bind b) (lift1 is u)) (Ss is) -(drop1_skip_bind b c is d1 u H5) (CHead c2 (Bind b) (lift1 is u)) (csubc_head -g d1 c2 H6 (Bind b) (lift1 is u))) (H2 d1 is H5 c2 H6)) (lift1 is (THead -(Bind b) u t0)) (lift1_bind b is u t0))))))))))))))))))) (\lambda (c: -C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H0: (arity g c u (asucc g -a1))).(\lambda (H1: ((\forall (d1: C).(\forall (is: PList).((drop1 is d1 c) -\to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g (asucc g a1) c2 (lift1 is -u))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (H2: (arity g (CHead c -(Bind Abst) u) t0 a2)).(\lambda (H3: ((\forall (d1: C).(\forall (is: -PList).((drop1 is d1 (CHead c (Bind Abst) u)) \to (\forall (c2: C).((csubc g -d1 c2) \to (sc3 g a2 c2 (lift1 is t0))))))))).(\lambda (d1: C).(\lambda (is: -PList).(\lambda (H4: (drop1 is d1 c)).(\lambda (c2: C).(\lambda (H5: (csubc g -d1 c2)).(eq_ind_r T (THead (Bind Abst) (lift1 is u) (lift1 (Ss is) t0)) -(\lambda (t1: T).(land (arity g c2 t1 (AHead a1 a2)) (\forall (d: C).(\forall -(w: T).((sc3 g a1 d w) \to (\forall (is0: PList).((drop1 is0 d c2) \to (sc3 g -a2 d (THead (Flat Appl) w (lift1 is0 t1)))))))))) (conj (arity g c2 (THead -(Bind Abst) (lift1 is u) (lift1 (Ss is) t0)) (AHead a1 a2)) (\forall (d: -C).(\forall (w: T).((sc3 g a1 d w) \to (\forall (is0: PList).((drop1 is0 d -c2) \to (sc3 g a2 d (THead (Flat Appl) w (lift1 is0 (THead (Bind Abst) (lift1 -is u) (lift1 (Ss is) t0)))))))))) (csubc_arity_conf g d1 c2 H5 (THead (Bind -Abst) (lift1 is u) (lift1 (Ss is) t0)) (AHead a1 a2) (arity_head g d1 (lift1 -is u) a1 (arity_lift1 g (asucc g a1) c is d1 u H4 H0) (lift1 (Ss is) t0) a2 -(arity_lift1 g a2 (CHead c (Bind Abst) u) (Ss is) (CHead d1 (Bind Abst) -(lift1 is u)) t0 (drop1_skip_bind Abst c is d1 u H4) H2))) (\lambda (d: -C).(\lambda (w: T).(\lambda (H6: (sc3 g a1 d w)).(\lambda (is0: -PList).(\lambda (H7: (drop1 is0 d c2)).(eq_ind_r T (THead (Bind Abst) (lift1 -is0 (lift1 is u)) (lift1 (Ss is0) (lift1 (Ss is) t0))) (\lambda (t1: T).(sc3 -g a2 d (THead (Flat Appl) w t1))) (let H8 \def (sc3_appl g a1 a2 TNil) in (H8 -d w (lift1 (Ss is0) (lift1 (Ss is) t0)) (let H_y \def (sc3_bind g Abbr -(\lambda (H9: (eq B Abbr Abst)).(not_abbr_abst H9)) a1 a2 TNil) in (H_y d w -(lift1 (Ss is0) (lift1 (Ss is) t0)) (let H_x \def (csubc_drop1_conf_rev g is0 -d c2 H7 d1 H5) in (let H9 \def H_x in (ex2_ind C (\lambda (c3: C).(drop1 is0 -c3 d1)) (\lambda (c3: C).(csubc g c3 d)) (sc3 g a2 (CHead d (Bind Abbr) w) -(lift1 (Ss is0) (lift1 (Ss is) t0))) (\lambda (x: C).(\lambda (H10: (drop1 -is0 x d1)).(\lambda (H11: (csubc g x d)).(eq_ind_r T (lift1 (papp (Ss is0) -(Ss is)) t0) (\lambda (t1: T).(sc3 g a2 (CHead d (Bind Abbr) w) t1)) -(eq_ind_r PList (Ss (papp is0 is)) (\lambda (p: PList).(sc3 g a2 (CHead d -(Bind Abbr) w) (lift1 p t0))) (H3 (CHead x (Bind Abst) (lift1 (papp is0 is) -u)) (Ss (papp is0 is)) (drop1_skip_bind Abst c (papp is0 is) x u (drop1_trans -is0 x d1 H10 is c H4)) (CHead d (Bind Abbr) w) (csubc_abst g x d H11 (lift1 -(papp is0 is) u) a1 (H1 x (papp is0 is) (drop1_trans is0 x d1 H10 is c H4) x -(csubc_refl g x)) w H6)) (papp (Ss is0) (Ss is)) (papp_ss is0 is)) (lift1 (Ss -is0) (lift1 (Ss is) t0)) (lift1_lift1 (Ss is0) (Ss is) t0))))) H9))) H6)) H6 -(lift1 is0 (lift1 is u)) (sc3_lift1 g c2 (asucc g a1) is0 d (lift1 is u) (H1 -d1 is H4 c2 H5) H7))) (lift1 is0 (THead (Bind Abst) (lift1 is u) (lift1 (Ss -is) t0))) (lift1_bind Abst is0 (lift1 is u) (lift1 (Ss is) t0))))))))) (lift1 -is (THead (Bind Abst) u t0)) (lift1_bind Abst is u t0)))))))))))))))) -(\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c u -a1)).(\lambda (H1: ((\forall (d1: C).(\forall (is: PList).((drop1 is d1 c) -\to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g a1 c2 (lift1 is -u))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g c t0 -(AHead a1 a2))).(\lambda (H3: ((\forall (d1: C).(\forall (is: PList).((drop1 -is d1 c) \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g (AHead a1 a2) c2 -(lift1 is t0))))))))).(\lambda (d1: C).(\lambda (is: PList).(\lambda (H4: -(drop1 is d1 c)).(\lambda (c2: C).(\lambda (H5: (csubc g d1 c2)).(let H_y -\def (H1 d1 is H4 c2 H5) in (let H_y0 \def (H3 d1 is H4 c2 H5) in (let H6 -\def H_y0 in (land_ind (arity g c2 (lift1 is t0) (AHead a1 a2)) (\forall (d: -C).(\forall (w: T).((sc3 g a1 d w) \to (\forall (is0: PList).((drop1 is0 d -c2) \to (sc3 g a2 d (THead (Flat Appl) w (lift1 is0 (lift1 is t0))))))))) -(sc3 g a2 c2 (lift1 is (THead (Flat Appl) u t0))) (\lambda (_: (arity g c2 -(lift1 is t0) (AHead a1 a2))).(\lambda (H8: ((\forall (d: C).(\forall (w: -T).((sc3 g a1 d w) \to (\forall (is0: PList).((drop1 is0 d c2) \to (sc3 g a2 -d (THead (Flat Appl) w (lift1 is0 (lift1 is t0))))))))))).(let H_y1 \def (H8 -c2 (lift1 is u) H_y PNil) in (eq_ind_r T (THead (Flat Appl) (lift1 is u) -(lift1 is t0)) (\lambda (t1: T).(sc3 g a2 c2 t1)) (H_y1 (drop1_nil c2)) -(lift1 is (THead (Flat Appl) u t0)) (lift1_flat Appl is u t0))))) -H6)))))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a0: -A).(\lambda (_: (arity g c u (asucc g a0))).(\lambda (H1: ((\forall (d1: -C).(\forall (is: PList).((drop1 is d1 c) \to (\forall (c2: C).((csubc g d1 -c2) \to (sc3 g (asucc g a0) c2 (lift1 is u))))))))).(\lambda (t0: T).(\lambda -(_: (arity g c t0 a0)).(\lambda (H3: ((\forall (d1: C).(\forall (is: -PList).((drop1 is d1 c) \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g a0 -c2 (lift1 is t0))))))))).(\lambda (d1: C).(\lambda (is: PList).(\lambda (H4: -(drop1 is d1 c)).(\lambda (c2: C).(\lambda (H5: (csubc g d1 c2)).(let H_y -\def (sc3_cast g a0 TNil) in (eq_ind_r T (THead (Flat Cast) (lift1 is u) -(lift1 is t0)) (\lambda (t1: T).(sc3 g a0 c2 t1)) (H_y c2 (lift1 is u) (H1 d1 -is H4 c2 H5) (lift1 is t0) (H3 d1 is H4 c2 H5)) (lift1 is (THead (Flat Cast) -u t0)) (lift1_flat Cast is u t0)))))))))))))))) (\lambda (c: C).(\lambda (t0: -T).(\lambda (a1: A).(\lambda (_: (arity g c t0 a1)).(\lambda (H1: ((\forall -(d1: C).(\forall (is: PList).((drop1 is d1 c) \to (\forall (c2: C).((csubc g -d1 c2) \to (sc3 g a1 c2 (lift1 is t0))))))))).(\lambda (a2: A).(\lambda (H2: -(leq g a1 a2)).(\lambda (d1: C).(\lambda (is: PList).(\lambda (H3: (drop1 is -d1 c)).(\lambda (c2: C).(\lambda (H4: (csubc g d1 c2)).(sc3_repl g a1 c2 -(lift1 is t0) (H1 d1 is H3 c2 H4) a2 H2))))))))))))) c1 t a H))))). -(* COMMENTS -Initial nodes: 5940 -END *) - -theorem sc3_arity: - \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t -a) \to (sc3 g a c t))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: -(arity g c t a)).(let H_y \def (sc3_arity_csubc g c t a H c PNil) in (H_y -(drop1_nil c) c (csubc_refl g c))))))). -(* COMMENTS -Initial nodes: 47 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/sc3/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/sc3/defs.ma deleted file mode 100644 index 32fdfda84..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/sc3/defs.ma +++ /dev/null @@ -1,32 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/sn3/defs.ma". - -include "Basic-1/arity/defs.ma". - -include "Basic-1/drop1/defs.ma". - -definition sc3: - G \to (A \to (C \to (T \to Prop))) -\def - let rec sc3 (g: G) (a: A) on a: (C \to (T \to Prop)) \def (\lambda (c: -C).(\lambda (t: T).(match a with [(ASort h n) \Rightarrow (land (arity g c t -(ASort h n)) (sn3 c t)) | (AHead a1 a2) \Rightarrow (land (arity g c t (AHead -a1 a2)) (\forall (d: C).(\forall (w: T).((sc3 g a1 d w) \to (\forall (is: -PList).((drop1 is d c) \to (sc3 g a2 d (THead (Flat Appl) w (lift1 is -t)))))))))]))) in sc3. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/sc3/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/sc3/props.ma deleted file mode 100644 index e1d909251..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/sc3/props.ma +++ /dev/null @@ -1,728 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/sc3/defs.ma". - -include "Basic-1/sn3/lift1.ma". - -include "Basic-1/nf2/lift1.ma". - -include "Basic-1/csuba/arity.ma". - -include "Basic-1/arity/lift1.ma". - -include "Basic-1/arity/aprem.ma". - -include "Basic-1/llt/props.ma". - -include "Basic-1/drop1/getl.ma". - -include "Basic-1/drop1/props.ma". - -include "Basic-1/lift1/props.ma". - -theorem sc3_arity_gen: - \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((sc3 g a c -t) \to (arity g c t a))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(A_ind -(\lambda (a0: A).((sc3 g a0 c t) \to (arity g c t a0))) (\lambda (n: -nat).(\lambda (n0: nat).(\lambda (H: (land (arity g c t (ASort n n0)) (sn3 c -t))).(let H0 \def H in (land_ind (arity g c t (ASort n n0)) (sn3 c t) (arity -g c t (ASort n n0)) (\lambda (H1: (arity g c t (ASort n n0))).(\lambda (_: -(sn3 c t)).H1)) H0))))) (\lambda (a0: A).(\lambda (_: (((sc3 g a0 c t) \to -(arity g c t a0)))).(\lambda (a1: A).(\lambda (_: (((sc3 g a1 c t) \to (arity -g c t a1)))).(\lambda (H1: (land (arity g c t (AHead a0 a1)) (\forall (d: -C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) -\to (sc3 g a1 d (THead (Flat Appl) w (lift1 is t)))))))))).(let H2 \def H1 in -(land_ind (arity g c t (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g -a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat -Appl) w (lift1 is t)))))))) (arity g c t (AHead a0 a1)) (\lambda (H3: (arity -g c t (AHead a0 a1))).(\lambda (_: ((\forall (d: C).(\forall (w: T).((sc3 g -a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat -Appl) w (lift1 is t)))))))))).H3)) H2))))))) a)))). -(* COMMENTS -Initial nodes: 369 -END *) - -theorem sc3_repl: - \forall (g: G).(\forall (a1: A).(\forall (c: C).(\forall (t: T).((sc3 g a1 c -t) \to (\forall (a2: A).((leq g a1 a2) \to (sc3 g a2 c t))))))) -\def - \lambda (g: G).(\lambda (a1: A).(llt_wf_ind (\lambda (a: A).(\forall (c: -C).(\forall (t: T).((sc3 g a c t) \to (\forall (a2: A).((leq g a a2) \to (sc3 -g a2 c t))))))) (\lambda (a2: A).(A_ind (\lambda (a: A).(((\forall (a3: -A).((llt a3 a) \to (\forall (c: C).(\forall (t: T).((sc3 g a3 c t) \to -(\forall (a4: A).((leq g a3 a4) \to (sc3 g a4 c t))))))))) \to (\forall (c: -C).(\forall (t: T).((sc3 g a c t) \to (\forall (a3: A).((leq g a a3) \to (sc3 -g a3 c t)))))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (_: ((\forall -(a3: A).((llt a3 (ASort n n0)) \to (\forall (c: C).(\forall (t: T).((sc3 g a3 -c t) \to (\forall (a4: A).((leq g a3 a4) \to (sc3 g a4 c t)))))))))).(\lambda -(c: C).(\lambda (t: T).(\lambda (H0: (land (arity g c t (ASort n n0)) (sn3 c -t))).(\lambda (a3: A).(\lambda (H1: (leq g (ASort n n0) a3)).(let H2 \def H0 -in (land_ind (arity g c t (ASort n n0)) (sn3 c t) (sc3 g a3 c t) (\lambda -(H3: (arity g c t (ASort n n0))).(\lambda (H4: (sn3 c t)).(let H_y \def -(arity_repl g c t (ASort n n0) H3 a3 H1) in (let H_x \def (leq_gen_sort1 g n -n0 a3 H1) in (let H5 \def H_x in (ex2_3_ind nat nat nat (\lambda (n2: -nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort n n0) k) -(aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda -(_: nat).(eq A a3 (ASort h2 n2))))) (sc3 g a3 c t) (\lambda (x0: -nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (_: (eq A (aplus g (ASort -n n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H7: (eq A a3 (ASort x1 -x0))).(let H8 \def (f_equal A A (\lambda (e: A).e) a3 (ASort x1 x0) H7) in -(let H9 \def (eq_ind A a3 (\lambda (a: A).(arity g c t a)) H_y (ASort x1 x0) -H8) in (eq_ind_r A (ASort x1 x0) (\lambda (a: A).(sc3 g a c t)) (conj (arity -g c t (ASort x1 x0)) (sn3 c t) H9 H4) a3 H8)))))))) H5)))))) H2)))))))))) -(\lambda (a: A).(\lambda (_: ((((\forall (a3: A).((llt a3 a) \to (\forall (c: -C).(\forall (t: T).((sc3 g a3 c t) \to (\forall (a4: A).((leq g a3 a4) \to -(sc3 g a4 c t))))))))) \to (\forall (c: C).(\forall (t: T).((sc3 g a c t) \to -(\forall (a3: A).((leq g a a3) \to (sc3 g a3 c t))))))))).(\lambda (a0: -A).(\lambda (H0: ((((\forall (a3: A).((llt a3 a0) \to (\forall (c: -C).(\forall (t: T).((sc3 g a3 c t) \to (\forall (a4: A).((leq g a3 a4) \to -(sc3 g a4 c t))))))))) \to (\forall (c: C).(\forall (t: T).((sc3 g a0 c t) -\to (\forall (a3: A).((leq g a0 a3) \to (sc3 g a3 c t))))))))).(\lambda (H1: -((\forall (a3: A).((llt a3 (AHead a a0)) \to (\forall (c: C).(\forall (t: -T).((sc3 g a3 c t) \to (\forall (a4: A).((leq g a3 a4) \to (sc3 g a4 c -t)))))))))).(\lambda (c: C).(\lambda (t: T).(\lambda (H2: (land (arity g c t -(AHead a a0)) (\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall -(is: PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w (lift1 is -t)))))))))).(\lambda (a3: A).(\lambda (H3: (leq g (AHead a a0) a3)).(let H4 -\def H2 in (land_ind (arity g c t (AHead a a0)) (\forall (d: C).(\forall (w: -T).((sc3 g a d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a0 d -(THead (Flat Appl) w (lift1 is t)))))))) (sc3 g a3 c t) (\lambda (H5: (arity -g c t (AHead a a0))).(\lambda (H6: ((\forall (d: C).(\forall (w: T).((sc3 g a -d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat -Appl) w (lift1 is t)))))))))).(let H_x \def (leq_gen_head1 g a a0 a3 H3) in -(let H7 \def H_x in (ex3_2_ind A A (\lambda (a4: A).(\lambda (_: A).(leq g a -a4))) (\lambda (_: A).(\lambda (a5: A).(leq g a0 a5))) (\lambda (a4: -A).(\lambda (a5: A).(eq A a3 (AHead a4 a5)))) (sc3 g a3 c t) (\lambda (x0: -A).(\lambda (x1: A).(\lambda (H8: (leq g a x0)).(\lambda (H9: (leq g a0 -x1)).(\lambda (H10: (eq A a3 (AHead x0 x1))).(let H11 \def (f_equal A A -(\lambda (e: A).e) a3 (AHead x0 x1) H10) in (eq_ind_r A (AHead x0 x1) -(\lambda (a4: A).(sc3 g a4 c t)) (conj (arity g c t (AHead x0 x1)) (\forall -(d: C).(\forall (w: T).((sc3 g x0 d w) \to (\forall (is: PList).((drop1 is d -c) \to (sc3 g x1 d (THead (Flat Appl) w (lift1 is t)))))))) (arity_repl g c t -(AHead a a0) H5 (AHead x0 x1) (leq_head g a x0 H8 a0 x1 H9)) (\lambda (d: -C).(\lambda (w: T).(\lambda (H12: (sc3 g x0 d w)).(\lambda (is: -PList).(\lambda (H13: (drop1 is d c)).(H0 (\lambda (a4: A).(\lambda (H14: -(llt a4 a0)).(\lambda (c0: C).(\lambda (t0: T).(\lambda (H15: (sc3 g a4 c0 -t0)).(\lambda (a5: A).(\lambda (H16: (leq g a4 a5)).(H1 a4 (llt_trans a4 a0 -(AHead a a0) H14 (llt_head_dx a a0)) c0 t0 H15 a5 H16)))))))) d (THead (Flat -Appl) w (lift1 is t)) (H6 d w (H1 x0 (llt_repl g a x0 H8 (AHead a a0) -(llt_head_sx a a0)) d w H12 a (leq_sym g a x0 H8)) is H13) x1 H9))))))) a3 -H11))))))) H7))))) H4)))))))))))) a2)) a1)). -(* COMMENTS -Initial nodes: 1359 -END *) - -theorem sc3_lift: - \forall (g: G).(\forall (a: A).(\forall (e: C).(\forall (t: T).((sc3 g a e -t) \to (\forall (c: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) -\to (sc3 g a c (lift h d t)))))))))) -\def - \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(\forall (e: -C).(\forall (t: T).((sc3 g a0 e t) \to (\forall (c: C).(\forall (h: -nat).(\forall (d: nat).((drop h d c e) \to (sc3 g a0 c (lift h d t)))))))))) -(\lambda (n: nat).(\lambda (n0: nat).(\lambda (e: C).(\lambda (t: T).(\lambda -(H: (land (arity g e t (ASort n n0)) (sn3 e t))).(\lambda (c: C).(\lambda (h: -nat).(\lambda (d: nat).(\lambda (H0: (drop h d c e)).(let H1 \def H in -(land_ind (arity g e t (ASort n n0)) (sn3 e t) (land (arity g c (lift h d t) -(ASort n n0)) (sn3 c (lift h d t))) (\lambda (H2: (arity g e t (ASort n -n0))).(\lambda (H3: (sn3 e t)).(conj (arity g c (lift h d t) (ASort n n0)) -(sn3 c (lift h d t)) (arity_lift g e t (ASort n n0) H2 c h d H0) (sn3_lift e -t H3 c h d H0)))) H1))))))))))) (\lambda (a0: A).(\lambda (_: ((\forall (e: -C).(\forall (t: T).((sc3 g a0 e t) \to (\forall (c: C).(\forall (h: -nat).(\forall (d: nat).((drop h d c e) \to (sc3 g a0 c (lift h d -t))))))))))).(\lambda (a1: A).(\lambda (_: ((\forall (e: C).(\forall (t: -T).((sc3 g a1 e t) \to (\forall (c: C).(\forall (h: nat).(\forall (d: -nat).((drop h d c e) \to (sc3 g a1 c (lift h d t))))))))))).(\lambda (e: -C).(\lambda (t: T).(\lambda (H1: (land (arity g e t (AHead a0 a1)) (\forall -(d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d -e) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is t)))))))))).(\lambda (c: -C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2: (drop h d c e)).(let H3 -\def H1 in (land_ind (arity g e t (AHead a0 a1)) (\forall (d0: C).(\forall -(w: T).((sc3 g a0 d0 w) \to (\forall (is: PList).((drop1 is d0 e) \to (sc3 g -a1 d0 (THead (Flat Appl) w (lift1 is t)))))))) (land (arity g c (lift h d t) -(AHead a0 a1)) (\forall (d0: C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall -(is: PList).((drop1 is d0 c) \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is -(lift h d t)))))))))) (\lambda (H4: (arity g e t (AHead a0 a1))).(\lambda -(H5: ((\forall (d0: C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: -PList).((drop1 is d0 e) \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is -t)))))))))).(conj (arity g c (lift h d t) (AHead a0 a1)) (\forall (d0: -C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: PList).((drop1 is d0 c) -\to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is (lift h d t))))))))) -(arity_lift g e t (AHead a0 a1) H4 c h d H2) (\lambda (d0: C).(\lambda (w: -T).(\lambda (H6: (sc3 g a0 d0 w)).(\lambda (is: PList).(\lambda (H7: (drop1 -is d0 c)).(let H_y \def (H5 d0 w H6 (PConsTail is h d)) in (eq_ind T (lift1 -(PConsTail is h d) t) (\lambda (t0: T).(sc3 g a1 d0 (THead (Flat Appl) w -t0))) (H_y (drop1_cons_tail c e h d H2 is d0 H7)) (lift1 is (lift h d t)) -(lift1_cons_tail t h d is))))))))))) H3))))))))))))) a)). -(* COMMENTS -Initial nodes: 849 -END *) - -theorem sc3_lift1: - \forall (g: G).(\forall (e: C).(\forall (a: A).(\forall (hds: -PList).(\forall (c: C).(\forall (t: T).((sc3 g a e t) \to ((drop1 hds c e) -\to (sc3 g a c (lift1 hds t))))))))) -\def - \lambda (g: G).(\lambda (e: C).(\lambda (a: A).(\lambda (hds: -PList).(PList_ind (\lambda (p: PList).(\forall (c: C).(\forall (t: T).((sc3 g -a e t) \to ((drop1 p c e) \to (sc3 g a c (lift1 p t))))))) (\lambda (c: -C).(\lambda (t: T).(\lambda (H: (sc3 g a e t)).(\lambda (H0: (drop1 PNil c -e)).(let H_y \def (drop1_gen_pnil c e H0) in (eq_ind_r C e (\lambda (c0: -C).(sc3 g a c0 t)) H c H_y)))))) (\lambda (n: nat).(\lambda (n0: -nat).(\lambda (p: PList).(\lambda (H: ((\forall (c: C).(\forall (t: T).((sc3 -g a e t) \to ((drop1 p c e) \to (sc3 g a c (lift1 p t)))))))).(\lambda (c: -C).(\lambda (t: T).(\lambda (H0: (sc3 g a e t)).(\lambda (H1: (drop1 (PCons n -n0 p) c e)).(let H_x \def (drop1_gen_pcons c e p n n0 H1) in (let H2 \def H_x -in (ex2_ind C (\lambda (c2: C).(drop n n0 c c2)) (\lambda (c2: C).(drop1 p c2 -e)) (sc3 g a c (lift n n0 (lift1 p t))) (\lambda (x: C).(\lambda (H3: (drop n -n0 c x)).(\lambda (H4: (drop1 p x e)).(sc3_lift g a x (lift1 p t) (H x t H0 -H4) c n n0 H3)))) H2))))))))))) hds)))). -(* COMMENTS -Initial nodes: 289 -END *) - -theorem sc3_abbr: - \forall (g: G).(\forall (a: A).(\forall (vs: TList).(\forall (i: -nat).(\forall (d: C).(\forall (v: T).(\forall (c: C).((sc3 g a c (THeads -(Flat Appl) vs (lift (S i) O v))) \to ((getl i c (CHead d (Bind Abbr) v)) \to -(sc3 g a c (THeads (Flat Appl) vs (TLRef i))))))))))) -\def - \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(\forall (vs: -TList).(\forall (i: nat).(\forall (d: C).(\forall (v: T).(\forall (c: -C).((sc3 g a0 c (THeads (Flat Appl) vs (lift (S i) O v))) \to ((getl i c -(CHead d (Bind Abbr) v)) \to (sc3 g a0 c (THeads (Flat Appl) vs (TLRef -i))))))))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (vs: -TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(\lambda (c: -C).(\lambda (H: (land (arity g c (THeads (Flat Appl) vs (lift (S i) O v)) -(ASort n n0)) (sn3 c (THeads (Flat Appl) vs (lift (S i) O v))))).(\lambda -(H0: (getl i c (CHead d (Bind Abbr) v))).(let H1 \def H in (land_ind (arity g -c (THeads (Flat Appl) vs (lift (S i) O v)) (ASort n n0)) (sn3 c (THeads (Flat -Appl) vs (lift (S i) O v))) (land (arity g c (THeads (Flat Appl) vs (TLRef -i)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (TLRef i)))) (\lambda (H2: -(arity g c (THeads (Flat Appl) vs (lift (S i) O v)) (ASort n n0))).(\lambda -(H3: (sn3 c (THeads (Flat Appl) vs (lift (S i) O v)))).(conj (arity g c -(THeads (Flat Appl) vs (TLRef i)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs -(TLRef i))) (arity_appls_abbr g c d v i H0 vs (ASort n n0) H2) -(sn3_appls_abbr c d v i H0 vs H3)))) H1))))))))))) (\lambda (a0: A).(\lambda -(_: ((\forall (vs: TList).(\forall (i: nat).(\forall (d: C).(\forall (v: -T).(\forall (c: C).((sc3 g a0 c (THeads (Flat Appl) vs (lift (S i) O v))) \to -((getl i c (CHead d (Bind Abbr) v)) \to (sc3 g a0 c (THeads (Flat Appl) vs -(TLRef i)))))))))))).(\lambda (a1: A).(\lambda (H0: ((\forall (vs: -TList).(\forall (i: nat).(\forall (d: C).(\forall (v: T).(\forall (c: -C).((sc3 g a1 c (THeads (Flat Appl) vs (lift (S i) O v))) \to ((getl i c -(CHead d (Bind Abbr) v)) \to (sc3 g a1 c (THeads (Flat Appl) vs (TLRef -i)))))))))))).(\lambda (vs: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda -(v: T).(\lambda (c: C).(\lambda (H1: (land (arity g c (THeads (Flat Appl) vs -(lift (S i) O v)) (AHead a0 a1)) (\forall (d0: C).(\forall (w: T).((sc3 g a0 -d0 w) \to (\forall (is: PList).((drop1 is d0 c) \to (sc3 g a1 d0 (THead (Flat -Appl) w (lift1 is (THeads (Flat Appl) vs (lift (S i) O v)))))))))))).(\lambda -(H2: (getl i c (CHead d (Bind Abbr) v))).(let H3 \def H1 in (land_ind (arity -g c (THeads (Flat Appl) vs (lift (S i) O v)) (AHead a0 a1)) (\forall (d0: -C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: PList).((drop1 is d0 c) -\to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs (lift -(S i) O v)))))))))) (land (arity g c (THeads (Flat Appl) vs (TLRef i)) (AHead -a0 a1)) (\forall (d0: C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: -PList).((drop1 is d0 c) \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is -(THeads (Flat Appl) vs (TLRef i))))))))))) (\lambda (H4: (arity g c (THeads -(Flat Appl) vs (lift (S i) O v)) (AHead a0 a1))).(\lambda (H5: ((\forall (d0: -C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: PList).((drop1 is d0 c) -\to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs (lift -(S i) O v)))))))))))).(conj (arity g c (THeads (Flat Appl) vs (TLRef i)) -(AHead a0 a1)) (\forall (d0: C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall -(is: PList).((drop1 is d0 c) \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is -(THeads (Flat Appl) vs (TLRef i)))))))))) (arity_appls_abbr g c d v i H2 vs -(AHead a0 a1) H4) (\lambda (d0: C).(\lambda (w: T).(\lambda (H6: (sc3 g a0 d0 -w)).(\lambda (is: PList).(\lambda (H7: (drop1 is d0 c)).(let H_x \def -(drop1_getl_trans is c d0 H7 Abbr d v i H2) in (let H8 \def H_x in (ex2_ind C -(\lambda (e2: C).(drop1 (ptrans is i) e2 d)) (\lambda (e2: C).(getl (trans is -i) d0 (CHead e2 (Bind Abbr) (lift1 (ptrans is i) v)))) (sc3 g a1 d0 (THead -(Flat Appl) w (lift1 is (THeads (Flat Appl) vs (TLRef i))))) (\lambda (x: -C).(\lambda (_: (drop1 (ptrans is i) x d)).(\lambda (H10: (getl (trans is i) -d0 (CHead x (Bind Abbr) (lift1 (ptrans is i) v)))).(let H_y \def (H0 (TCons w -(lifts1 is vs))) in (eq_ind_r T (THeads (Flat Appl) (lifts1 is vs) (lift1 is -(TLRef i))) (\lambda (t: T).(sc3 g a1 d0 (THead (Flat Appl) w t))) (eq_ind_r -T (TLRef (trans is i)) (\lambda (t: T).(sc3 g a1 d0 (THead (Flat Appl) w -(THeads (Flat Appl) (lifts1 is vs) t)))) (H_y (trans is i) x (lift1 (ptrans -is i) v) d0 (eq_ind T (lift1 is (lift (S i) O v)) (\lambda (t: T).(sc3 g a1 -d0 (THead (Flat Appl) w (THeads (Flat Appl) (lifts1 is vs) t)))) (eq_ind T -(lift1 is (THeads (Flat Appl) vs (lift (S i) O v))) (\lambda (t: T).(sc3 g a1 -d0 (THead (Flat Appl) w t))) (H5 d0 w H6 is H7) (THeads (Flat Appl) (lifts1 -is vs) (lift1 is (lift (S i) O v))) (lifts1_flat Appl is (lift (S i) O v) -vs)) (lift (S (trans is i)) O (lift1 (ptrans is i) v)) (lift1_free is i v)) -H10) (lift1 is (TLRef i)) (lift1_lref is i)) (lift1 is (THeads (Flat Appl) vs -(TLRef i))) (lifts1_flat Appl is (TLRef i) vs)))))) H8))))))))))) -H3))))))))))))) a)). -(* COMMENTS -Initial nodes: 1563 -END *) - -theorem sc3_cast: - \forall (g: G).(\forall (a: A).(\forall (vs: TList).(\forall (c: C).(\forall -(u: T).((sc3 g (asucc g a) c (THeads (Flat Appl) vs u)) \to (\forall (t: -T).((sc3 g a c (THeads (Flat Appl) vs t)) \to (sc3 g a c (THeads (Flat Appl) -vs (THead (Flat Cast) u t)))))))))) -\def - \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(\forall (vs: -TList).(\forall (c: C).(\forall (u: T).((sc3 g (asucc g a0) c (THeads (Flat -Appl) vs u)) \to (\forall (t: T).((sc3 g a0 c (THeads (Flat Appl) vs t)) \to -(sc3 g a0 c (THeads (Flat Appl) vs (THead (Flat Cast) u t)))))))))) (\lambda -(n: nat).(\lambda (n0: nat).(\lambda (vs: TList).(\lambda (c: C).(\lambda (u: -T).(\lambda (H: (sc3 g (match n with [O \Rightarrow (ASort O (next g n0)) | -(S h) \Rightarrow (ASort h n0)]) c (THeads (Flat Appl) vs u))).(\lambda (t: -T).(\lambda (H0: (land (arity g c (THeads (Flat Appl) vs t) (ASort n n0)) -(sn3 c (THeads (Flat Appl) vs t)))).(nat_ind (\lambda (n1: nat).((sc3 g -(match n1 with [O \Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow -(ASort h n0)]) c (THeads (Flat Appl) vs u)) \to ((land (arity g c (THeads -(Flat Appl) vs t) (ASort n1 n0)) (sn3 c (THeads (Flat Appl) vs t))) \to (land -(arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort n1 n0)) -(sn3 c (THeads (Flat Appl) vs (THead (Flat Cast) u t))))))) (\lambda (H1: -(sc3 g (ASort O (next g n0)) c (THeads (Flat Appl) vs u))).(\lambda (H2: -(land (arity g c (THeads (Flat Appl) vs t) (ASort O n0)) (sn3 c (THeads (Flat -Appl) vs t)))).(let H3 \def H1 in (land_ind (arity g c (THeads (Flat Appl) vs -u) (ASort O (next g n0))) (sn3 c (THeads (Flat Appl) vs u)) (land (arity g c -(THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort O n0)) (sn3 c (THeads -(Flat Appl) vs (THead (Flat Cast) u t)))) (\lambda (H4: (arity g c (THeads -(Flat Appl) vs u) (ASort O (next g n0)))).(\lambda (H5: (sn3 c (THeads (Flat -Appl) vs u))).(let H6 \def H2 in (land_ind (arity g c (THeads (Flat Appl) vs -t) (ASort O n0)) (sn3 c (THeads (Flat Appl) vs t)) (land (arity g c (THeads -(Flat Appl) vs (THead (Flat Cast) u t)) (ASort O n0)) (sn3 c (THeads (Flat -Appl) vs (THead (Flat Cast) u t)))) (\lambda (H7: (arity g c (THeads (Flat -Appl) vs t) (ASort O n0))).(\lambda (H8: (sn3 c (THeads (Flat Appl) vs -t))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort -O n0)) (sn3 c (THeads (Flat Appl) vs (THead (Flat Cast) u t))) -(arity_appls_cast g c u t vs (ASort O n0) H4 H7) (sn3_appls_cast c vs u H5 t -H8)))) H6)))) H3)))) (\lambda (n1: nat).(\lambda (_: (((sc3 g (match n1 with -[O \Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)]) c -(THeads (Flat Appl) vs u)) \to ((land (arity g c (THeads (Flat Appl) vs t) -(ASort n1 n0)) (sn3 c (THeads (Flat Appl) vs t))) \to (land (arity g c -(THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort n1 n0)) (sn3 c (THeads -(Flat Appl) vs (THead (Flat Cast) u t)))))))).(\lambda (H1: (sc3 g (ASort n1 -n0) c (THeads (Flat Appl) vs u))).(\lambda (H2: (land (arity g c (THeads -(Flat Appl) vs t) (ASort (S n1) n0)) (sn3 c (THeads (Flat Appl) vs t)))).(let -H3 \def H1 in (land_ind (arity g c (THeads (Flat Appl) vs u) (ASort n1 n0)) -(sn3 c (THeads (Flat Appl) vs u)) (land (arity g c (THeads (Flat Appl) vs -(THead (Flat Cast) u t)) (ASort (S n1) n0)) (sn3 c (THeads (Flat Appl) vs -(THead (Flat Cast) u t)))) (\lambda (H4: (arity g c (THeads (Flat Appl) vs u) -(ASort n1 n0))).(\lambda (H5: (sn3 c (THeads (Flat Appl) vs u))).(let H6 \def -H2 in (land_ind (arity g c (THeads (Flat Appl) vs t) (ASort (S n1) n0)) (sn3 -c (THeads (Flat Appl) vs t)) (land (arity g c (THeads (Flat Appl) vs (THead -(Flat Cast) u t)) (ASort (S n1) n0)) (sn3 c (THeads (Flat Appl) vs (THead -(Flat Cast) u t)))) (\lambda (H7: (arity g c (THeads (Flat Appl) vs t) (ASort -(S n1) n0))).(\lambda (H8: (sn3 c (THeads (Flat Appl) vs t))).(conj (arity g -c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort (S n1) n0)) (sn3 c -(THeads (Flat Appl) vs (THead (Flat Cast) u t))) (arity_appls_cast g c u t vs -(ASort (S n1) n0) H4 H7) (sn3_appls_cast c vs u H5 t H8)))) H6)))) H3)))))) n -H H0))))))))) (\lambda (a0: A).(\lambda (_: ((\forall (vs: TList).(\forall -(c: C).(\forall (u: T).((sc3 g (asucc g a0) c (THeads (Flat Appl) vs u)) \to -(\forall (t: T).((sc3 g a0 c (THeads (Flat Appl) vs t)) \to (sc3 g a0 c -(THeads (Flat Appl) vs (THead (Flat Cast) u t))))))))))).(\lambda (a1: -A).(\lambda (H0: ((\forall (vs: TList).(\forall (c: C).(\forall (u: T).((sc3 -g (asucc g a1) c (THeads (Flat Appl) vs u)) \to (\forall (t: T).((sc3 g a1 c -(THeads (Flat Appl) vs t)) \to (sc3 g a1 c (THeads (Flat Appl) vs (THead -(Flat Cast) u t))))))))))).(\lambda (vs: TList).(\lambda (c: C).(\lambda (u: -T).(\lambda (H1: (land (arity g c (THeads (Flat Appl) vs u) (AHead a0 (asucc -g a1))) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: -PList).((drop1 is d c) \to (sc3 g (asucc g a1) d (THead (Flat Appl) w (lift1 -is (THeads (Flat Appl) vs u))))))))))).(\lambda (t: T).(\lambda (H2: (land -(arity g c (THeads (Flat Appl) vs t) (AHead a0 a1)) (\forall (d: C).(\forall -(w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 -d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs t))))))))))).(let H3 -\def H1 in (land_ind (arity g c (THeads (Flat Appl) vs u) (AHead a0 (asucc g -a1))) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: -PList).((drop1 is d c) \to (sc3 g (asucc g a1) d (THead (Flat Appl) w (lift1 -is (THeads (Flat Appl) vs u))))))))) (land (arity g c (THeads (Flat Appl) vs -(THead (Flat Cast) u t)) (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 -g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead -(Flat Appl) w (lift1 is (THeads (Flat Appl) vs (THead (Flat Cast) u -t))))))))))) (\lambda (H4: (arity g c (THeads (Flat Appl) vs u) (AHead a0 -(asucc g a1)))).(\lambda (H5: ((\forall (d: C).(\forall (w: T).((sc3 g a0 d -w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g (asucc g a1) d (THead -(Flat Appl) w (lift1 is (THeads (Flat Appl) vs u))))))))))).(let H6 \def H2 -in (land_ind (arity g c (THeads (Flat Appl) vs t) (AHead a0 a1)) (\forall (d: -C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) -\to (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs -t))))))))) (land (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) -(AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall -(is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is -(THeads (Flat Appl) vs (THead (Flat Cast) u t))))))))))) (\lambda (H7: (arity -g c (THeads (Flat Appl) vs t) (AHead a0 a1))).(\lambda (H8: ((\forall (d: -C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) -\to (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs -t))))))))))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) -(AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall -(is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is -(THeads (Flat Appl) vs (THead (Flat Cast) u t)))))))))) (arity_appls_cast g c -u t vs (AHead a0 a1) H4 H7) (\lambda (d: C).(\lambda (w: T).(\lambda (H9: -(sc3 g a0 d w)).(\lambda (is: PList).(\lambda (H10: (drop1 is d c)).(let H_y -\def (H0 (TCons w (lifts1 is vs))) in (eq_ind_r T (THeads (Flat Appl) (lifts1 -is vs) (lift1 is (THead (Flat Cast) u t))) (\lambda (t0: T).(sc3 g a1 d -(THead (Flat Appl) w t0))) (eq_ind_r T (THead (Flat Cast) (lift1 is u) (lift1 -is t)) (\lambda (t0: T).(sc3 g a1 d (THead (Flat Appl) w (THeads (Flat Appl) -(lifts1 is vs) t0)))) (H_y d (lift1 is u) (eq_ind T (lift1 is (THeads (Flat -Appl) vs u)) (\lambda (t0: T).(sc3 g (asucc g a1) d (THead (Flat Appl) w -t0))) (H5 d w H9 is H10) (THeads (Flat Appl) (lifts1 is vs) (lift1 is u)) -(lifts1_flat Appl is u vs)) (lift1 is t) (eq_ind T (lift1 is (THeads (Flat -Appl) vs t)) (\lambda (t0: T).(sc3 g a1 d (THead (Flat Appl) w t0))) (H8 d w -H9 is H10) (THeads (Flat Appl) (lifts1 is vs) (lift1 is t)) (lifts1_flat Appl -is t vs))) (lift1 is (THead (Flat Cast) u t)) (lift1_flat Cast is u t)) -(lift1 is (THeads (Flat Appl) vs (THead (Flat Cast) u t))) (lifts1_flat Appl -is (THead (Flat Cast) u t) vs))))))))))) H6)))) H3)))))))))))) a)). -(* COMMENTS -Initial nodes: 2625 -END *) - -theorem sc3_props__sc3_sn3_abst: - \forall (g: G).(\forall (a: A).(land (\forall (c: C).(\forall (t: T).((sc3 g -a c t) \to (sn3 c t)))) (\forall (vs: TList).(\forall (i: nat).(let t \def -(THeads (Flat Appl) vs (TLRef i)) in (\forall (c: C).((arity g c t a) \to -((nf2 c (TLRef i)) \to ((sns3 c vs) \to (sc3 g a c t)))))))))) -\def - \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(land (\forall (c: -C).(\forall (t: T).((sc3 g a0 c t) \to (sn3 c t)))) (\forall (vs: -TList).(\forall (i: nat).(let t \def (THeads (Flat Appl) vs (TLRef i)) in -(\forall (c: C).((arity g c t a0) \to ((nf2 c (TLRef i)) \to ((sns3 c vs) \to -(sc3 g a0 c t)))))))))) (\lambda (n: nat).(\lambda (n0: nat).(conj (\forall -(c: C).(\forall (t: T).((land (arity g c t (ASort n n0)) (sn3 c t)) \to (sn3 -c t)))) (\forall (vs: TList).(\forall (i: nat).(\forall (c: C).((arity g c -(THeads (Flat Appl) vs (TLRef i)) (ASort n n0)) \to ((nf2 c (TLRef i)) \to -((sns3 c vs) \to (land (arity g c (THeads (Flat Appl) vs (TLRef i)) (ASort n -n0)) (sn3 c (THeads (Flat Appl) vs (TLRef i)))))))))) (\lambda (c: -C).(\lambda (t: T).(\lambda (H: (land (arity g c t (ASort n n0)) (sn3 c -t))).(let H0 \def H in (land_ind (arity g c t (ASort n n0)) (sn3 c t) (sn3 c -t) (\lambda (_: (arity g c t (ASort n n0))).(\lambda (H2: (sn3 c t)).H2)) -H0))))) (\lambda (vs: TList).(\lambda (i: nat).(\lambda (c: C).(\lambda (H: -(arity g c (THeads (Flat Appl) vs (TLRef i)) (ASort n n0))).(\lambda (H0: -(nf2 c (TLRef i))).(\lambda (H1: (sns3 c vs)).(conj (arity g c (THeads (Flat -Appl) vs (TLRef i)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (TLRef i))) H -(sn3_appls_lref c i H0 vs H1))))))))))) (\lambda (a0: A).(\lambda (H: (land -(\forall (c: C).(\forall (t: T).((sc3 g a0 c t) \to (sn3 c t)))) (\forall -(vs: TList).(\forall (i: nat).(\forall (c: C).((arity g c (THeads (Flat Appl) -vs (TLRef i)) a0) \to ((nf2 c (TLRef i)) \to ((sns3 c vs) \to (sc3 g a0 c -(THeads (Flat Appl) vs (TLRef i))))))))))).(\lambda (a1: A).(\lambda (H0: -(land (\forall (c: C).(\forall (t: T).((sc3 g a1 c t) \to (sn3 c t)))) -(\forall (vs: TList).(\forall (i: nat).(\forall (c: C).((arity g c (THeads -(Flat Appl) vs (TLRef i)) a1) \to ((nf2 c (TLRef i)) \to ((sns3 c vs) \to -(sc3 g a1 c (THeads (Flat Appl) vs (TLRef i))))))))))).(conj (\forall (c: -C).(\forall (t: T).((land (arity g c t (AHead a0 a1)) (\forall (d: -C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) -\to (sc3 g a1 d (THead (Flat Appl) w (lift1 is t))))))))) \to (sn3 c t)))) -(\forall (vs: TList).(\forall (i: nat).(\forall (c: C).((arity g c (THeads -(Flat Appl) vs (TLRef i)) (AHead a0 a1)) \to ((nf2 c (TLRef i)) \to ((sns3 c -vs) \to (land (arity g c (THeads (Flat Appl) vs (TLRef i)) (AHead a0 a1)) -(\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: -PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads -(Flat Appl) vs (TLRef i))))))))))))))))) (\lambda (c: C).(\lambda (t: -T).(\lambda (H1: (land (arity g c t (AHead a0 a1)) (\forall (d: C).(\forall -(w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 -d (THead (Flat Appl) w (lift1 is t)))))))))).(let H2 \def H in (land_ind -(\forall (c0: C).(\forall (t0: T).((sc3 g a0 c0 t0) \to (sn3 c0 t0)))) -(\forall (vs: TList).(\forall (i: nat).(\forall (c0: C).((arity g c0 (THeads -(Flat Appl) vs (TLRef i)) a0) \to ((nf2 c0 (TLRef i)) \to ((sns3 c0 vs) \to -(sc3 g a0 c0 (THeads (Flat Appl) vs (TLRef i))))))))) (sn3 c t) (\lambda (_: -((\forall (c0: C).(\forall (t0: T).((sc3 g a0 c0 t0) \to (sn3 c0 -t0)))))).(\lambda (H4: ((\forall (vs: TList).(\forall (i: nat).(\forall (c0: -C).((arity g c0 (THeads (Flat Appl) vs (TLRef i)) a0) \to ((nf2 c0 (TLRef i)) -\to ((sns3 c0 vs) \to (sc3 g a0 c0 (THeads (Flat Appl) vs (TLRef -i))))))))))).(let H5 \def H0 in (land_ind (\forall (c0: C).(\forall (t0: -T).((sc3 g a1 c0 t0) \to (sn3 c0 t0)))) (\forall (vs: TList).(\forall (i: -nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) vs (TLRef i)) a1) \to -((nf2 c0 (TLRef i)) \to ((sns3 c0 vs) \to (sc3 g a1 c0 (THeads (Flat Appl) vs -(TLRef i))))))))) (sn3 c t) (\lambda (H6: ((\forall (c0: C).(\forall (t0: -T).((sc3 g a1 c0 t0) \to (sn3 c0 t0)))))).(\lambda (_: ((\forall (vs: -TList).(\forall (i: nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) vs -(TLRef i)) a1) \to ((nf2 c0 (TLRef i)) \to ((sns3 c0 vs) \to (sc3 g a1 c0 -(THeads (Flat Appl) vs (TLRef i))))))))))).(let H8 \def H1 in (land_ind -(arity g c t (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) -\to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w -(lift1 is t)))))))) (sn3 c t) (\lambda (H9: (arity g c t (AHead a0 -a1))).(\lambda (H10: ((\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to -(\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w -(lift1 is t)))))))))).(let H_y \def (arity_aprem g c t (AHead a0 a1) H9 O a0) -in (let H11 \def (H_y (aprem_zero a0 a1)) in (ex2_3_ind C T nat (\lambda (d: -C).(\lambda (_: T).(\lambda (j: nat).(drop j O d c)))) (\lambda (d: -C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g a0))))) (sn3 c t) -(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H12: (drop x2 -O x0 c)).(\lambda (H13: (arity g x0 x1 (asucc g a0))).(let H_y0 \def (H10 -(CHead x0 (Bind Abst) x1) (TLRef O) (H4 TNil O (CHead x0 (Bind Abst) x1) -(arity_abst g (CHead x0 (Bind Abst) x1) x0 x1 O (getl_refl Abst x0 x1) a0 -H13) (nf2_lref_abst (CHead x0 (Bind Abst) x1) x0 x1 O (getl_refl Abst x0 x1)) -I) (PCons (S x2) O PNil)) in (let H_y1 \def (H6 (CHead x0 (Bind Abst) x1) -(THead (Flat Appl) (TLRef O) (lift (S x2) O t)) (H_y0 (drop1_cons (CHead x0 -(Bind Abst) x1) c (S x2) O (drop_drop (Bind Abst) x2 x0 c H12 x1) c PNil -(drop1_nil c)))) in (let H_x \def (sn3_gen_flat Appl (CHead x0 (Bind Abst) -x1) (TLRef O) (lift (S x2) O t) H_y1) in (let H14 \def H_x in (land_ind (sn3 -(CHead x0 (Bind Abst) x1) (TLRef O)) (sn3 (CHead x0 (Bind Abst) x1) (lift (S -x2) O t)) (sn3 c t) (\lambda (_: (sn3 (CHead x0 (Bind Abst) x1) (TLRef -O))).(\lambda (H16: (sn3 (CHead x0 (Bind Abst) x1) (lift (S x2) O -t))).(sn3_gen_lift (CHead x0 (Bind Abst) x1) t (S x2) O H16 c (drop_drop -(Bind Abst) x2 x0 c H12 x1)))) H14)))))))))) H11))))) H8)))) H5)))) H2))))) -(\lambda (vs: TList).(\lambda (i: nat).(\lambda (c: C).(\lambda (H1: (arity g -c (THeads (Flat Appl) vs (TLRef i)) (AHead a0 a1))).(\lambda (H2: (nf2 c -(TLRef i))).(\lambda (H3: (sns3 c vs)).(conj (arity g c (THeads (Flat Appl) -vs (TLRef i)) (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) -\to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w -(lift1 is (THeads (Flat Appl) vs (TLRef i)))))))))) H1 (\lambda (d: -C).(\lambda (w: T).(\lambda (H4: (sc3 g a0 d w)).(\lambda (is: -PList).(\lambda (H5: (drop1 is d c)).(let H6 \def H in (land_ind (\forall -(c0: C).(\forall (t: T).((sc3 g a0 c0 t) \to (sn3 c0 t)))) (\forall (vs0: -TList).(\forall (i0: nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) -vs0 (TLRef i0)) a0) \to ((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a0 -c0 (THeads (Flat Appl) vs0 (TLRef i0))))))))) (sc3 g a1 d (THead (Flat Appl) -w (lift1 is (THeads (Flat Appl) vs (TLRef i))))) (\lambda (H7: ((\forall (c0: -C).(\forall (t: T).((sc3 g a0 c0 t) \to (sn3 c0 t)))))).(\lambda (_: -((\forall (vs0: TList).(\forall (i0: nat).(\forall (c0: C).((arity g c0 -(THeads (Flat Appl) vs0 (TLRef i0)) a0) \to ((nf2 c0 (TLRef i0)) \to ((sns3 -c0 vs0) \to (sc3 g a0 c0 (THeads (Flat Appl) vs0 (TLRef i0))))))))))).(let H9 -\def H0 in (land_ind (\forall (c0: C).(\forall (t: T).((sc3 g a1 c0 t) \to -(sn3 c0 t)))) (\forall (vs0: TList).(\forall (i0: nat).(\forall (c0: -C).((arity g c0 (THeads (Flat Appl) vs0 (TLRef i0)) a1) \to ((nf2 c0 (TLRef -i0)) \to ((sns3 c0 vs0) \to (sc3 g a1 c0 (THeads (Flat Appl) vs0 (TLRef -i0))))))))) (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs -(TLRef i))))) (\lambda (_: ((\forall (c0: C).(\forall (t: T).((sc3 g a1 c0 t) -\to (sn3 c0 t)))))).(\lambda (H11: ((\forall (vs0: TList).(\forall (i0: -nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) vs0 (TLRef i0)) a1) -\to ((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a1 c0 (THeads (Flat -Appl) vs0 (TLRef i0))))))))))).(let H_y \def (H11 (TCons w (lifts1 is vs))) -in (eq_ind_r T (THeads (Flat Appl) (lifts1 is vs) (lift1 is (TLRef i))) -(\lambda (t: T).(sc3 g a1 d (THead (Flat Appl) w t))) (eq_ind_r T (TLRef -(trans is i)) (\lambda (t: T).(sc3 g a1 d (THead (Flat Appl) w (THeads (Flat -Appl) (lifts1 is vs) t)))) (H_y (trans is i) d (eq_ind T (lift1 is (TLRef i)) -(\lambda (t: T).(arity g d (THead (Flat Appl) w (THeads (Flat Appl) (lifts1 -is vs) t)) a1)) (eq_ind T (lift1 is (THeads (Flat Appl) vs (TLRef i))) -(\lambda (t: T).(arity g d (THead (Flat Appl) w t) a1)) (arity_appl g d w a0 -(sc3_arity_gen g d w a0 H4) (lift1 is (THeads (Flat Appl) vs (TLRef i))) a1 -(arity_lift1 g (AHead a0 a1) c is d (THeads (Flat Appl) vs (TLRef i)) H5 H1)) -(THeads (Flat Appl) (lifts1 is vs) (lift1 is (TLRef i))) (lifts1_flat Appl is -(TLRef i) vs)) (TLRef (trans is i)) (lift1_lref is i)) (eq_ind T (lift1 is -(TLRef i)) (\lambda (t: T).(nf2 d t)) (nf2_lift1 c is d (TLRef i) H5 H2) -(TLRef (trans is i)) (lift1_lref is i)) (conj (sn3 d w) (sns3 d (lifts1 is -vs)) (H7 d w H4) (sns3_lifts1 c is d H5 vs H3))) (lift1 is (TLRef i)) -(lift1_lref is i)) (lift1 is (THeads (Flat Appl) vs (TLRef i))) (lifts1_flat -Appl is (TLRef i) vs))))) H9)))) H6))))))))))))))))))) a)). -(* COMMENTS -Initial nodes: 2737 -END *) - -theorem sc3_sn3: - \forall (g: G).(\forall (a: A).(\forall (c: C).(\forall (t: T).((sc3 g a c -t) \to (sn3 c t))))) -\def - \lambda (g: G).(\lambda (a: A).(\lambda (c: C).(\lambda (t: T).(\lambda (H: -(sc3 g a c t)).(let H_x \def (sc3_props__sc3_sn3_abst g a) in (let H0 \def -H_x in (land_ind (\forall (c0: C).(\forall (t0: T).((sc3 g a c0 t0) \to (sn3 -c0 t0)))) (\forall (vs: TList).(\forall (i: nat).(\forall (c0: C).((arity g -c0 (THeads (Flat Appl) vs (TLRef i)) a) \to ((nf2 c0 (TLRef i)) \to ((sns3 c0 -vs) \to (sc3 g a c0 (THeads (Flat Appl) vs (TLRef i))))))))) (sn3 c t) -(\lambda (H1: ((\forall (c0: C).(\forall (t0: T).((sc3 g a c0 t0) \to (sn3 c0 -t0)))))).(\lambda (_: ((\forall (vs: TList).(\forall (i: nat).(\forall (c0: -C).((arity g c0 (THeads (Flat Appl) vs (TLRef i)) a) \to ((nf2 c0 (TLRef i)) -\to ((sns3 c0 vs) \to (sc3 g a c0 (THeads (Flat Appl) vs (TLRef -i))))))))))).(H1 c t H))) H0))))))). -(* COMMENTS -Initial nodes: 203 -END *) - -theorem sc3_abst: - \forall (g: G).(\forall (a: A).(\forall (vs: TList).(\forall (c: C).(\forall -(i: nat).((arity g c (THeads (Flat Appl) vs (TLRef i)) a) \to ((nf2 c (TLRef -i)) \to ((sns3 c vs) \to (sc3 g a c (THeads (Flat Appl) vs (TLRef i)))))))))) -\def - \lambda (g: G).(\lambda (a: A).(\lambda (vs: TList).(\lambda (c: C).(\lambda -(i: nat).(\lambda (H: (arity g c (THeads (Flat Appl) vs (TLRef i)) -a)).(\lambda (H0: (nf2 c (TLRef i))).(\lambda (H1: (sns3 c vs)).(let H_x \def -(sc3_props__sc3_sn3_abst g a) in (let H2 \def H_x in (land_ind (\forall (c0: -C).(\forall (t: T).((sc3 g a c0 t) \to (sn3 c0 t)))) (\forall (vs0: -TList).(\forall (i0: nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) -vs0 (TLRef i0)) a) \to ((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a -c0 (THeads (Flat Appl) vs0 (TLRef i0))))))))) (sc3 g a c (THeads (Flat Appl) -vs (TLRef i))) (\lambda (_: ((\forall (c0: C).(\forall (t: T).((sc3 g a c0 t) -\to (sn3 c0 t)))))).(\lambda (H4: ((\forall (vs0: TList).(\forall (i0: -nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) vs0 (TLRef i0)) a) \to -((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a c0 (THeads (Flat Appl) -vs0 (TLRef i0))))))))))).(H4 vs i c H H0 H1))) H2)))))))))). -(* COMMENTS -Initial nodes: 249 -END *) - -theorem sc3_bind: - \forall (g: G).(\forall (b: B).((not (eq B b Abst)) \to (\forall (a1: -A).(\forall (a2: A).(\forall (vs: TList).(\forall (c: C).(\forall (v: -T).(\forall (t: T).((sc3 g a2 (CHead c (Bind b) v) (THeads (Flat Appl) (lifts -(S O) O vs) t)) \to ((sc3 g a1 c v) \to (sc3 g a2 c (THeads (Flat Appl) vs -(THead (Bind b) v t))))))))))))) -\def - \lambda (g: G).(\lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda -(a1: A).(\lambda (a2: A).(A_ind (\lambda (a: A).(\forall (vs: TList).(\forall -(c: C).(\forall (v: T).(\forall (t: T).((sc3 g a (CHead c (Bind b) v) (THeads -(Flat Appl) (lifts (S O) O vs) t)) \to ((sc3 g a1 c v) \to (sc3 g a c (THeads -(Flat Appl) vs (THead (Bind b) v t)))))))))) (\lambda (n: nat).(\lambda (n0: -nat).(\lambda (vs: TList).(\lambda (c: C).(\lambda (v: T).(\lambda (t: -T).(\lambda (H0: (land (arity g (CHead c (Bind b) v) (THeads (Flat Appl) -(lifts (S O) O vs) t) (ASort n n0)) (sn3 (CHead c (Bind b) v) (THeads (Flat -Appl) (lifts (S O) O vs) t)))).(\lambda (H1: (sc3 g a1 c v)).(let H2 \def H0 -in (land_ind (arity g (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O -vs) t) (ASort n n0)) (sn3 (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S -O) O vs) t)) (land (arity g c (THeads (Flat Appl) vs (THead (Bind b) v t)) -(ASort n n0)) (sn3 c (THeads (Flat Appl) vs (THead (Bind b) v t)))) (\lambda -(H3: (arity g (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O vs) t) -(ASort n n0))).(\lambda (H4: (sn3 (CHead c (Bind b) v) (THeads (Flat Appl) -(lifts (S O) O vs) t))).(conj (arity g c (THeads (Flat Appl) vs (THead (Bind -b) v t)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (THead (Bind b) v t))) -(arity_appls_bind g b H c v a1 (sc3_arity_gen g c v a1 H1) t vs (ASort n n0) -H3) (sn3_appls_bind b H c v (sc3_sn3 g a1 c v H1) vs t H4)))) H2)))))))))) -(\lambda (a: A).(\lambda (_: ((\forall (vs: TList).(\forall (c: C).(\forall -(v: T).(\forall (t: T).((sc3 g a (CHead c (Bind b) v) (THeads (Flat Appl) -(lifts (S O) O vs) t)) \to ((sc3 g a1 c v) \to (sc3 g a c (THeads (Flat Appl) -vs (THead (Bind b) v t))))))))))).(\lambda (a0: A).(\lambda (H1: ((\forall -(vs: TList).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sc3 g a0 (CHead -c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O vs) t)) \to ((sc3 g a1 c v) -\to (sc3 g a0 c (THeads (Flat Appl) vs (THead (Bind b) v -t))))))))))).(\lambda (vs: TList).(\lambda (c: C).(\lambda (v: T).(\lambda -(t: T).(\lambda (H2: (land (arity g (CHead c (Bind b) v) (THeads (Flat Appl) -(lifts (S O) O vs) t) (AHead a a0)) (\forall (d: C).(\forall (w: T).((sc3 g a -d w) \to (\forall (is: PList).((drop1 is d (CHead c (Bind b) v)) \to (sc3 g -a0 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) (lifts (S O) O vs) -t))))))))))).(\lambda (H3: (sc3 g a1 c v)).(let H4 \def H2 in (land_ind -(arity g (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O vs) t) -(AHead a a0)) (\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall -(is: PList).((drop1 is d (CHead c (Bind b) v)) \to (sc3 g a0 d (THead (Flat -Appl) w (lift1 is (THeads (Flat Appl) (lifts (S O) O vs) t))))))))) (land -(arity g c (THeads (Flat Appl) vs (THead (Bind b) v t)) (AHead a a0)) -(\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall (is: -PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w (lift1 is (THeads -(Flat Appl) vs (THead (Bind b) v t))))))))))) (\lambda (H5: (arity g (CHead c -(Bind b) v) (THeads (Flat Appl) (lifts (S O) O vs) t) (AHead a a0))).(\lambda -(H6: ((\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall (is: -PList).((drop1 is d (CHead c (Bind b) v)) \to (sc3 g a0 d (THead (Flat Appl) -w (lift1 is (THeads (Flat Appl) (lifts (S O) O vs) t))))))))))).(conj (arity -g c (THeads (Flat Appl) vs (THead (Bind b) v t)) (AHead a a0)) (\forall (d: -C).(\forall (w: T).((sc3 g a d w) \to (\forall (is: PList).((drop1 is d c) -\to (sc3 g a0 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs (THead -(Bind b) v t)))))))))) (arity_appls_bind g b H c v a1 (sc3_arity_gen g c v a1 -H3) t vs (AHead a a0) H5) (\lambda (d: C).(\lambda (w: T).(\lambda (H7: (sc3 -g a d w)).(\lambda (is: PList).(\lambda (H8: (drop1 is d c)).(let H_y \def -(H1 (TCons w (lifts1 is vs))) in (eq_ind_r T (THeads (Flat Appl) (lifts1 is -vs) (lift1 is (THead (Bind b) v t))) (\lambda (t0: T).(sc3 g a0 d (THead -(Flat Appl) w t0))) (eq_ind_r T (THead (Bind b) (lift1 is v) (lift1 (Ss is) -t)) (\lambda (t0: T).(sc3 g a0 d (THead (Flat Appl) w (THeads (Flat Appl) -(lifts1 is vs) t0)))) (H_y d (lift1 is v) (lift1 (Ss is) t) (eq_ind TList -(lifts1 (Ss is) (lifts (S O) O vs)) (\lambda (t0: TList).(sc3 g a0 (CHead d -(Bind b) (lift1 is v)) (THead (Flat Appl) (lift (S O) O w) (THeads (Flat -Appl) t0 (lift1 (Ss is) t))))) (eq_ind T (lift1 (Ss is) (THeads (Flat Appl) -(lifts (S O) O vs) t)) (\lambda (t0: T).(sc3 g a0 (CHead d (Bind b) (lift1 is -v)) (THead (Flat Appl) (lift (S O) O w) t0))) (H6 (CHead d (Bind b) (lift1 is -v)) (lift (S O) O w) (sc3_lift g a d w H7 (CHead d (Bind b) (lift1 is v)) (S -O) O (drop_drop (Bind b) O d d (drop_refl d) (lift1 is v))) (Ss is) -(drop1_skip_bind b c is d v H8)) (THeads (Flat Appl) (lifts1 (Ss is) (lifts -(S O) O vs)) (lift1 (Ss is) t)) (lifts1_flat Appl (Ss is) t (lifts (S O) O -vs))) (lifts (S O) O (lifts1 is vs)) (lifts1_xhg is vs)) (sc3_lift1 g c a1 is -d v H3 H8)) (lift1 is (THead (Bind b) v t)) (lift1_bind b is v t)) (lift1 is -(THeads (Flat Appl) vs (THead (Bind b) v t))) (lifts1_flat Appl is (THead -(Bind b) v t) vs))))))))))) H4)))))))))))) a2))))). -(* COMMENTS -Initial nodes: 1797 -END *) - -theorem sc3_appl: - \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (vs: -TList).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sc3 g a2 c (THeads -(Flat Appl) vs (THead (Bind Abbr) v t))) \to ((sc3 g a1 c v) \to (\forall (w: -T).((sc3 g (asucc g a1) c w) \to (sc3 g a2 c (THeads (Flat Appl) vs (THead -(Flat Appl) v (THead (Bind Abst) w t)))))))))))))) -\def - \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(A_ind (\lambda (a: -A).(\forall (vs: TList).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sc3 -g a c (THeads (Flat Appl) vs (THead (Bind Abbr) v t))) \to ((sc3 g a1 c v) -\to (\forall (w: T).((sc3 g (asucc g a1) c w) \to (sc3 g a c (THeads (Flat -Appl) vs (THead (Flat Appl) v (THead (Bind Abst) w t))))))))))))) (\lambda -(n: nat).(\lambda (n0: nat).(\lambda (vs: TList).(\lambda (c: C).(\lambda (v: -T).(\lambda (t: T).(\lambda (H: (land (arity g c (THeads (Flat Appl) vs -(THead (Bind Abbr) v t)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (THead -(Bind Abbr) v t))))).(\lambda (H0: (sc3 g a1 c v)).(\lambda (w: T).(\lambda -(H1: (sc3 g (asucc g a1) c w)).(let H2 \def H in (land_ind (arity g c (THeads -(Flat Appl) vs (THead (Bind Abbr) v t)) (ASort n n0)) (sn3 c (THeads (Flat -Appl) vs (THead (Bind Abbr) v t))) (land (arity g c (THeads (Flat Appl) vs -(THead (Flat Appl) v (THead (Bind Abst) w t))) (ASort n n0)) (sn3 c (THeads -(Flat Appl) vs (THead (Flat Appl) v (THead (Bind Abst) w t))))) (\lambda (H3: -(arity g c (THeads (Flat Appl) vs (THead (Bind Abbr) v t)) (ASort n -n0))).(\lambda (H4: (sn3 c (THeads (Flat Appl) vs (THead (Bind Abbr) v -t)))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat Appl) v (THead -(Bind Abst) w t))) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (THead (Flat -Appl) v (THead (Bind Abst) w t)))) (arity_appls_appl g c v a1 (sc3_arity_gen -g c v a1 H0) w (sc3_arity_gen g c w (asucc g a1) H1) t vs (ASort n n0) H3) -(sn3_appls_beta c v t vs H4 w (sc3_sn3 g (asucc g a1) c w H1))))) -H2)))))))))))) (\lambda (a: A).(\lambda (_: ((\forall (vs: TList).(\forall -(c: C).(\forall (v: T).(\forall (t: T).((sc3 g a c (THeads (Flat Appl) vs -(THead (Bind Abbr) v t))) \to ((sc3 g a1 c v) \to (\forall (w: T).((sc3 g -(asucc g a1) c w) \to (sc3 g a c (THeads (Flat Appl) vs (THead (Flat Appl) v -(THead (Bind Abst) w t)))))))))))))).(\lambda (a0: A).(\lambda (H0: ((\forall -(vs: TList).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sc3 g a0 c -(THeads (Flat Appl) vs (THead (Bind Abbr) v t))) \to ((sc3 g a1 c v) \to -(\forall (w: T).((sc3 g (asucc g a1) c w) \to (sc3 g a0 c (THeads (Flat Appl) -vs (THead (Flat Appl) v (THead (Bind Abst) w t)))))))))))))).(\lambda (vs: -TList).(\lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (H1: (land -(arity g c (THeads (Flat Appl) vs (THead (Bind Abbr) v t)) (AHead a a0)) -(\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall (is: -PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w (lift1 is (THeads -(Flat Appl) vs (THead (Bind Abbr) v t)))))))))))).(\lambda (H2: (sc3 g a1 c -v)).(\lambda (w: T).(\lambda (H3: (sc3 g (asucc g a1) c w)).(let H4 \def H1 -in (land_ind (arity g c (THeads (Flat Appl) vs (THead (Bind Abbr) v t)) -(AHead a a0)) (\forall (d: C).(\forall (w0: T).((sc3 g a d w0) \to (\forall -(is: PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w0 (lift1 is -(THeads (Flat Appl) vs (THead (Bind Abbr) v t)))))))))) (land (arity g c -(THeads (Flat Appl) vs (THead (Flat Appl) v (THead (Bind Abst) w t))) (AHead -a a0)) (\forall (d: C).(\forall (w0: T).((sc3 g a d w0) \to (\forall (is: -PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w0 (lift1 is -(THeads (Flat Appl) vs (THead (Flat Appl) v (THead (Bind Abst) w -t)))))))))))) (\lambda (H5: (arity g c (THeads (Flat Appl) vs (THead (Bind -Abbr) v t)) (AHead a a0))).(\lambda (H6: ((\forall (d: C).(\forall (w0: -T).((sc3 g a d w0) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a0 d -(THead (Flat Appl) w0 (lift1 is (THeads (Flat Appl) vs (THead (Bind Abbr) v -t)))))))))))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat Appl) v -(THead (Bind Abst) w t))) (AHead a a0)) (\forall (d: C).(\forall (w0: -T).((sc3 g a d w0) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a0 d -(THead (Flat Appl) w0 (lift1 is (THeads (Flat Appl) vs (THead (Flat Appl) v -(THead (Bind Abst) w t))))))))))) (arity_appls_appl g c v a1 (sc3_arity_gen g -c v a1 H2) w (sc3_arity_gen g c w (asucc g a1) H3) t vs (AHead a a0) H5) -(\lambda (d: C).(\lambda (w0: T).(\lambda (H7: (sc3 g a d w0)).(\lambda (is: -PList).(\lambda (H8: (drop1 is d c)).(eq_ind_r T (THeads (Flat Appl) (lifts1 -is vs) (lift1 is (THead (Flat Appl) v (THead (Bind Abst) w t)))) (\lambda -(t0: T).(sc3 g a0 d (THead (Flat Appl) w0 t0))) (eq_ind_r T (THead (Flat -Appl) (lift1 is v) (lift1 is (THead (Bind Abst) w t))) (\lambda (t0: T).(sc3 -g a0 d (THead (Flat Appl) w0 (THeads (Flat Appl) (lifts1 is vs) t0)))) -(eq_ind_r T (THead (Bind Abst) (lift1 is w) (lift1 (Ss is) t)) (\lambda (t0: -T).(sc3 g a0 d (THead (Flat Appl) w0 (THeads (Flat Appl) (lifts1 is vs) -(THead (Flat Appl) (lift1 is v) t0))))) (let H_y \def (H0 (TCons w0 (lifts1 -is vs))) in (H_y d (lift1 is v) (lift1 (Ss is) t) (eq_ind T (lift1 is (THead -(Bind Abbr) v t)) (\lambda (t0: T).(sc3 g a0 d (THead (Flat Appl) w0 (THeads -(Flat Appl) (lifts1 is vs) t0)))) (eq_ind T (lift1 is (THeads (Flat Appl) vs -(THead (Bind Abbr) v t))) (\lambda (t0: T).(sc3 g a0 d (THead (Flat Appl) w0 -t0))) (H6 d w0 H7 is H8) (THeads (Flat Appl) (lifts1 is vs) (lift1 is (THead -(Bind Abbr) v t))) (lifts1_flat Appl is (THead (Bind Abbr) v t) vs)) (THead -(Bind Abbr) (lift1 is v) (lift1 (Ss is) t)) (lift1_bind Abbr is v t)) -(sc3_lift1 g c a1 is d v H2 H8) (lift1 is w) (sc3_lift1 g c (asucc g a1) is d -w H3 H8))) (lift1 is (THead (Bind Abst) w t)) (lift1_bind Abst is w t)) -(lift1 is (THead (Flat Appl) v (THead (Bind Abst) w t))) (lift1_flat Appl is -v (THead (Bind Abst) w t))) (lift1 is (THeads (Flat Appl) vs (THead (Flat -Appl) v (THead (Bind Abst) w t)))) (lifts1_flat Appl is (THead (Flat Appl) v -(THead (Bind Abst) w t)) vs)))))))))) H4)))))))))))))) a2))). -(* COMMENTS -Initial nodes: 1901 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/sn3/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/sn3/defs.ma deleted file mode 100644 index 686e6c673..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/sn3/defs.ma +++ /dev/null @@ -1,29 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr3/defs.ma". - -inductive sn3 (c: C): T \to Prop \def -| sn3_sing: \forall (t1: T).(((\forall (t2: T).((((eq T t1 t2) \to (\forall -(P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2))))) \to (sn3 c t1)). - -definition sns3: - C \to (TList \to Prop) -\def - let rec sns3 (c: C) (ts: TList) on ts: Prop \def (match ts with [TNil -\Rightarrow True | (TCons t ts0) \Rightarrow (land (sn3 c t) (sns3 c ts0))]) -in sns3. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/sn3/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/sn3/fwd.ma deleted file mode 100644 index 68276fe9f..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/sn3/fwd.ma +++ /dev/null @@ -1,197 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/sn3/defs.ma". - -include "Basic-1/pr3/props.ma". - -theorem sn3_gen_bind: - \forall (b: B).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 c -(THead (Bind b) u t)) \to (land (sn3 c u) (sn3 (CHead c (Bind b) u) t)))))) -\def - \lambda (b: B).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: -(sn3 c (THead (Bind b) u t))).(insert_eq T (THead (Bind b) u t) (\lambda (t0: -T).(sn3 c t0)) (\lambda (_: T).(land (sn3 c u) (sn3 (CHead c (Bind b) u) t))) -(\lambda (y: T).(\lambda (H0: (sn3 c y)).(unintro T t (\lambda (t0: T).((eq T -y (THead (Bind b) u t0)) \to (land (sn3 c u) (sn3 (CHead c (Bind b) u) t0)))) -(unintro T u (\lambda (t0: T).(\forall (x: T).((eq T y (THead (Bind b) t0 x)) -\to (land (sn3 c t0) (sn3 (CHead c (Bind b) t0) x))))) (sn3_ind c (\lambda -(t0: T).(\forall (x: T).(\forall (x0: T).((eq T t0 (THead (Bind b) x x0)) \to -(land (sn3 c x) (sn3 (CHead c (Bind b) x) x0)))))) (\lambda (t1: T).(\lambda -(H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 -c t1 t2) \to (sn3 c t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2) -\to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (\forall (x: T).(\forall -(x0: T).((eq T t2 (THead (Bind b) x x0)) \to (land (sn3 c x) (sn3 (CHead c -(Bind b) x) x0)))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: (eq T -t1 (THead (Bind b) x x0))).(let H4 \def (eq_ind T t1 (\lambda (t0: -T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c -t0 t2) \to (\forall (x1: T).(\forall (x2: T).((eq T t2 (THead (Bind b) x1 -x2)) \to (land (sn3 c x1) (sn3 (CHead c (Bind b) x1) x2))))))))) H2 (THead -(Bind b) x x0) H3) in (let H5 \def (eq_ind T t1 (\lambda (t0: T).(\forall -(t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to -(sn3 c t2))))) H1 (THead (Bind b) x x0) H3) in (conj (sn3 c x) (sn3 (CHead c -(Bind b) x) x0) (sn3_sing c x (\lambda (t2: T).(\lambda (H6: (((eq T x t2) -\to (\forall (P: Prop).P)))).(\lambda (H7: (pr3 c x t2)).(let H8 \def (H4 -(THead (Bind b) t2 x0) (\lambda (H8: (eq T (THead (Bind b) x x0) (THead (Bind -b) t2 x0))).(\lambda (P: Prop).(let H9 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x | -(TLRef _) \Rightarrow x | (THead _ t0 _) \Rightarrow t0])) (THead (Bind b) x -x0) (THead (Bind b) t2 x0) H8) in (let H10 \def (eq_ind_r T t2 (\lambda (t0: -T).(pr3 c x t0)) H7 x H9) in (let H11 \def (eq_ind_r T t2 (\lambda (t0: -T).((eq T x t0) \to (\forall (P0: Prop).P0))) H6 x H9) in (H11 (refl_equal T -x) P)))))) (pr3_head_12 c x t2 H7 (Bind b) x0 x0 (pr3_refl (CHead c (Bind b) -t2) x0)) t2 x0 (refl_equal T (THead (Bind b) t2 x0))) in (land_ind (sn3 c t2) -(sn3 (CHead c (Bind b) t2) x0) (sn3 c t2) (\lambda (H9: (sn3 c t2)).(\lambda -(_: (sn3 (CHead c (Bind b) t2) x0)).H9)) H8)))))) (sn3_sing (CHead c (Bind b) -x) x0 (\lambda (t2: T).(\lambda (H6: (((eq T x0 t2) \to (\forall (P: -Prop).P)))).(\lambda (H7: (pr3 (CHead c (Bind b) x) x0 t2)).(let H8 \def (H4 -(THead (Bind b) x t2) (\lambda (H8: (eq T (THead (Bind b) x x0) (THead (Bind -b) x t2))).(\lambda (P: Prop).(let H9 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | -(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind b) x -x0) (THead (Bind b) x t2) H8) in (let H10 \def (eq_ind_r T t2 (\lambda (t0: -T).(pr3 (CHead c (Bind b) x) x0 t0)) H7 x0 H9) in (let H11 \def (eq_ind_r T -t2 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H6 x0 H9) in -(H11 (refl_equal T x0) P)))))) (pr3_head_12 c x x (pr3_refl c x) (Bind b) x0 -t2 H7) x t2 (refl_equal T (THead (Bind b) x t2))) in (land_ind (sn3 c x) (sn3 -(CHead c (Bind b) x) t2) (sn3 (CHead c (Bind b) x) t2) (\lambda (_: (sn3 c -x)).(\lambda (H10: (sn3 (CHead c (Bind b) x) t2)).H10)) H8))))))))))))))) y -H0))))) H))))). -(* COMMENTS -Initial nodes: 1055 -END *) - -theorem sn3_gen_flat: - \forall (f: F).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 c -(THead (Flat f) u t)) \to (land (sn3 c u) (sn3 c t)))))) -\def - \lambda (f: F).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: -(sn3 c (THead (Flat f) u t))).(insert_eq T (THead (Flat f) u t) (\lambda (t0: -T).(sn3 c t0)) (\lambda (_: T).(land (sn3 c u) (sn3 c t))) (\lambda (y: -T).(\lambda (H0: (sn3 c y)).(unintro T t (\lambda (t0: T).((eq T y (THead -(Flat f) u t0)) \to (land (sn3 c u) (sn3 c t0)))) (unintro T u (\lambda (t0: -T).(\forall (x: T).((eq T y (THead (Flat f) t0 x)) \to (land (sn3 c t0) (sn3 -c x))))) (sn3_ind c (\lambda (t0: T).(\forall (x: T).(\forall (x0: T).((eq T -t0 (THead (Flat f) x x0)) \to (land (sn3 c x) (sn3 c x0)))))) (\lambda (t1: -T).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: -Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H2: ((\forall -(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to -(\forall (x: T).(\forall (x0: T).((eq T t2 (THead (Flat f) x x0)) \to (land -(sn3 c x) (sn3 c x0)))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: -(eq T t1 (THead (Flat f) x x0))).(let H4 \def (eq_ind T t1 (\lambda (t0: -T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c -t0 t2) \to (\forall (x1: T).(\forall (x2: T).((eq T t2 (THead (Flat f) x1 -x2)) \to (land (sn3 c x1) (sn3 c x2))))))))) H2 (THead (Flat f) x x0) H3) in -(let H5 \def (eq_ind T t1 (\lambda (t0: T).(\forall (t2: T).((((eq T t0 t2) -\to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (sn3 c t2))))) H1 (THead -(Flat f) x x0) H3) in (conj (sn3 c x) (sn3 c x0) (sn3_sing c x (\lambda (t2: -T).(\lambda (H6: (((eq T x t2) \to (\forall (P: Prop).P)))).(\lambda (H7: -(pr3 c x t2)).(let H8 \def (H4 (THead (Flat f) t2 x0) (\lambda (H8: (eq T -(THead (Flat f) x x0) (THead (Flat f) t2 x0))).(\lambda (P: Prop).(let H9 -\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t0 _) -\Rightarrow t0])) (THead (Flat f) x x0) (THead (Flat f) t2 x0) H8) in (let -H10 \def (eq_ind_r T t2 (\lambda (t0: T).(pr3 c x t0)) H7 x H9) in (let H11 -\def (eq_ind_r T t2 (\lambda (t0: T).((eq T x t0) \to (\forall (P0: -Prop).P0))) H6 x H9) in (H11 (refl_equal T x) P)))))) (pr3_head_12 c x t2 H7 -(Flat f) x0 x0 (pr3_refl (CHead c (Flat f) t2) x0)) t2 x0 (refl_equal T -(THead (Flat f) t2 x0))) in (land_ind (sn3 c t2) (sn3 c x0) (sn3 c t2) -(\lambda (H9: (sn3 c t2)).(\lambda (_: (sn3 c x0)).H9)) H8)))))) (sn3_sing c -x0 (\lambda (t2: T).(\lambda (H6: (((eq T x0 t2) \to (\forall (P: -Prop).P)))).(\lambda (H7: (pr3 c x0 t2)).(let H8 \def (H4 (THead (Flat f) x -t2) (\lambda (H8: (eq T (THead (Flat f) x x0) (THead (Flat f) x -t2))).(\lambda (P: Prop).(let H9 \def (f_equal T T (\lambda (e: T).(match e -in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) -\Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat f) x x0) -(THead (Flat f) x t2) H8) in (let H10 \def (eq_ind_r T t2 (\lambda (t0: -T).(pr3 c x0 t0)) H7 x0 H9) in (let H11 \def (eq_ind_r T t2 (\lambda (t0: -T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H6 x0 H9) in (H11 (refl_equal -T x0) P)))))) (pr3_thin_dx c x0 t2 H7 x f) x t2 (refl_equal T (THead (Flat f) -x t2))) in (land_ind (sn3 c x) (sn3 c t2) (sn3 c t2) (\lambda (_: (sn3 c -x)).(\lambda (H10: (sn3 c t2)).H10)) H8))))))))))))))) y H0))))) H))))). -(* COMMENTS -Initial nodes: 925 -END *) - -theorem sn3_gen_head: - \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 c -(THead k u t)) \to (sn3 c u))))) -\def - \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (c: C).(\forall (u: -T).(\forall (t: T).((sn3 c (THead k0 u t)) \to (sn3 c u)))))) (\lambda (b: -B).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (sn3 c (THead -(Bind b) u t))).(let H_x \def (sn3_gen_bind b c u t H) in (let H0 \def H_x in -(land_ind (sn3 c u) (sn3 (CHead c (Bind b) u) t) (sn3 c u) (\lambda (H1: (sn3 -c u)).(\lambda (_: (sn3 (CHead c (Bind b) u) t)).H1)) H0)))))))) (\lambda (f: -F).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (sn3 c (THead -(Flat f) u t))).(let H_x \def (sn3_gen_flat f c u t H) in (let H0 \def H_x in -(land_ind (sn3 c u) (sn3 c t) (sn3 c u) (\lambda (H1: (sn3 c u)).(\lambda (_: -(sn3 c t)).H1)) H0)))))))) k). -(* COMMENTS -Initial nodes: 191 -END *) - -theorem sn3_gen_cflat: - \forall (f: F).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 (CHead -c (Flat f) u) t) \to (sn3 c t))))) -\def - \lambda (f: F).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: -(sn3 (CHead c (Flat f) u) t)).(sn3_ind (CHead c (Flat f) u) (\lambda (t0: -T).(sn3 c t0)) (\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 -t2) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Flat f) u) t1 t2) \to -(sn3 (CHead c (Flat f) u) t2)))))).(\lambda (H1: ((\forall (t2: T).((((eq T -t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Flat f) u) t1 t2) \to -(sn3 c t2)))))).(sn3_sing c t1 (\lambda (t2: T).(\lambda (H2: (((eq T t1 t2) -\to (\forall (P: Prop).P)))).(\lambda (H3: (pr3 c t1 t2)).(H1 t2 H2 -(pr3_cflat c t1 t2 H3 f u))))))))) t H))))). -(* COMMENTS -Initial nodes: 175 -END *) - -theorem sn3_gen_lift: - \forall (c1: C).(\forall (t: T).(\forall (h: nat).(\forall (d: nat).((sn3 c1 -(lift h d t)) \to (\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 t))))))) -\def - \lambda (c1: C).(\lambda (t: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda -(H: (sn3 c1 (lift h d t))).(insert_eq T (lift h d t) (\lambda (t0: T).(sn3 c1 -t0)) (\lambda (_: T).(\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 t)))) -(\lambda (y: T).(\lambda (H0: (sn3 c1 y)).(unintro T t (\lambda (t0: T).((eq -T y (lift h d t0)) \to (\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 t0))))) -(sn3_ind c1 (\lambda (t0: T).(\forall (x: T).((eq T t0 (lift h d x)) \to -(\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 x)))))) (\lambda (t1: -T).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: -Prop).P))) \to ((pr3 c1 t1 t2) \to (sn3 c1 t2)))))).(\lambda (H2: ((\forall -(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c1 t1 t2) \to -(\forall (x: T).((eq T t2 (lift h d x)) \to (\forall (c2: C).((drop h d c1 -c2) \to (sn3 c2 x)))))))))).(\lambda (x: T).(\lambda (H3: (eq T t1 (lift h d -x))).(\lambda (c2: C).(\lambda (H4: (drop h d c1 c2)).(let H5 \def (eq_ind T -t1 (\lambda (t0: T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P: -Prop).P))) \to ((pr3 c1 t0 t2) \to (\forall (x0: T).((eq T t2 (lift h d x0)) -\to (\forall (c3: C).((drop h d c1 c3) \to (sn3 c3 x0))))))))) H2 (lift h d -x) H3) in (let H6 \def (eq_ind T t1 (\lambda (t0: T).(\forall (t2: T).((((eq -T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c1 t0 t2) \to (sn3 c1 t2))))) -H1 (lift h d x) H3) in (sn3_sing c2 x (\lambda (t2: T).(\lambda (H7: (((eq T -x t2) \to (\forall (P: Prop).P)))).(\lambda (H8: (pr3 c2 x t2)).(H5 (lift h d -t2) (\lambda (H9: (eq T (lift h d x) (lift h d t2))).(\lambda (P: Prop).(let -H10 \def (eq_ind_r T t2 (\lambda (t0: T).(pr3 c2 x t0)) H8 x (lift_inj x t2 h -d H9)) in (let H11 \def (eq_ind_r T t2 (\lambda (t0: T).((eq T x t0) \to -(\forall (P0: Prop).P0))) H7 x (lift_inj x t2 h d H9)) in (H11 (refl_equal T -x) P))))) (pr3_lift c1 c2 h d H4 x t2 H8) t2 (refl_equal T (lift h d t2)) c2 -H4)))))))))))))) y H0)))) H))))). -(* COMMENTS -Initial nodes: 565 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/sn3/lift1.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/sn3/lift1.ma deleted file mode 100644 index 1b64c22bb..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/sn3/lift1.ma +++ /dev/null @@ -1,46 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/sn3/props.ma". - -include "Basic-1/drop1/fwd.ma". - -include "Basic-1/lift1/fwd.ma". - -theorem sns3_lifts1: - \forall (e: C).(\forall (hds: PList).(\forall (c: C).((drop1 hds c e) \to -(\forall (ts: TList).((sns3 e ts) \to (sns3 c (lifts1 hds ts))))))) -\def - \lambda (e: C).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall -(c: C).((drop1 p c e) \to (\forall (ts: TList).((sns3 e ts) \to (sns3 c -(lifts1 p ts))))))) (\lambda (c: C).(\lambda (H: (drop1 PNil c e)).(\lambda -(ts: TList).(\lambda (H0: (sns3 e ts)).(let H_y \def (drop1_gen_pnil c e H) -in (eq_ind_r C e (\lambda (c0: C).(sns3 c0 (lifts1 PNil ts))) (eq_ind_r TList -ts (\lambda (t: TList).(sns3 e t)) H0 (lifts1 PNil ts) (lifts1_nil ts)) c -H_y)))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda -(H: ((\forall (c: C).((drop1 p c e) \to (\forall (ts: TList).((sns3 e ts) \to -(sns3 c (lifts1 p ts)))))))).(\lambda (c: C).(\lambda (H0: (drop1 (PCons n n0 -p) c e)).(\lambda (ts: TList).(\lambda (H1: (sns3 e ts)).(let H_x \def -(drop1_gen_pcons c e p n n0 H0) in (let H2 \def H_x in (ex2_ind C (\lambda -(c2: C).(drop n n0 c c2)) (\lambda (c2: C).(drop1 p c2 e)) (sns3 c (lifts1 -(PCons n n0 p) ts)) (\lambda (x: C).(\lambda (H3: (drop n n0 c x)).(\lambda -(H4: (drop1 p x e)).(eq_ind_r TList (lifts n n0 (lifts1 p ts)) (\lambda (t: -TList).(sns3 c t)) (sns3_lifts c x n n0 H3 (lifts1 p ts) (H x H4 ts H1)) -(lifts1 (PCons n n0 p) ts) (lifts1_cons n n0 p ts))))) H2))))))))))) hds)). -(* COMMENTS -Initial nodes: 323 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/sn3/nf2.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/sn3/nf2.ma deleted file mode 100644 index 824428084..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/sn3/nf2.ma +++ /dev/null @@ -1,66 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/sn3/defs.ma". - -include "Basic-1/nf2/dec.ma". - -include "Basic-1/nf2/pr3.ma". - -theorem sn3_nf2: - \forall (c: C).(\forall (t: T).((nf2 c t) \to (sn3 c t))) -\def - \lambda (c: C).(\lambda (t: T).(\lambda (H: (nf2 c t)).(sn3_sing c t -(\lambda (t2: T).(\lambda (H0: (((eq T t t2) \to (\forall (P: -Prop).P)))).(\lambda (H1: (pr3 c t t2)).(let H_y \def (nf2_pr3_unfold c t t2 -H1 H) in (let H2 \def (eq_ind_r T t2 (\lambda (t0: T).(pr3 c t t0)) H1 t H_y) -in (let H3 \def (eq_ind_r T t2 (\lambda (t0: T).((eq T t t0) \to (\forall (P: -Prop).P))) H0 t H_y) in (eq_ind T t (\lambda (t0: T).(sn3 c t0)) (H3 -(refl_equal T t) (sn3 c t)) t2 H_y)))))))))). -(* COMMENTS -Initial nodes: 129 -END *) - -theorem nf2_sn3: - \forall (c: C).(\forall (t: T).((sn3 c t) \to (ex2 T (\lambda (u: T).(pr3 c -t u)) (\lambda (u: T).(nf2 c u))))) -\def - \lambda (c: C).(\lambda (t: T).(\lambda (H: (sn3 c t)).(sn3_ind c (\lambda -(t0: T).(ex2 T (\lambda (u: T).(pr3 c t0 u)) (\lambda (u: T).(nf2 c u)))) -(\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) \to (\forall -(P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H1: ((\forall -(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to -(ex2 T (\lambda (u: T).(pr3 c t2 u)) (\lambda (u: T).(nf2 c u)))))))).(let -H_x \def (nf2_dec c t1) in (let H2 \def H_x in (or_ind (nf2 c t1) (ex2 T -(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: -T).(pr2 c t1 t2))) (ex2 T (\lambda (u: T).(pr3 c t1 u)) (\lambda (u: T).(nf2 -c u))) (\lambda (H3: (nf2 c t1)).(ex_intro2 T (\lambda (u: T).(pr3 c t1 u)) -(\lambda (u: T).(nf2 c u)) t1 (pr3_refl c t1) H3)) (\lambda (H3: (ex2 T -(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: -T).(pr2 c t1 t2)))).(ex2_ind T (\lambda (t2: T).((eq T t1 t2) \to (\forall -(P: Prop).P))) (\lambda (t2: T).(pr2 c t1 t2)) (ex2 T (\lambda (u: T).(pr3 c -t1 u)) (\lambda (u: T).(nf2 c u))) (\lambda (x: T).(\lambda (H4: (((eq T t1 -x) \to (\forall (P: Prop).P)))).(\lambda (H5: (pr2 c t1 x)).(let H_y \def (H1 -x H4) in (let H6 \def (H_y (pr3_pr2 c t1 x H5)) in (ex2_ind T (\lambda (u: -T).(pr3 c x u)) (\lambda (u: T).(nf2 c u)) (ex2 T (\lambda (u: T).(pr3 c t1 -u)) (\lambda (u: T).(nf2 c u))) (\lambda (x0: T).(\lambda (H7: (pr3 c x -x0)).(\lambda (H8: (nf2 c x0)).(ex_intro2 T (\lambda (u: T).(pr3 c t1 u)) -(\lambda (u: T).(nf2 c u)) x0 (pr3_sing c x t1 H5 x0 H7) H8)))) H6)))))) H3)) -H2)))))) t H))). -(* COMMENTS -Initial nodes: 443 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/sn3/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/sn3/props.ma deleted file mode 100644 index ea72c8869..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/sn3/props.ma +++ /dev/null @@ -1,2575 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/sn3/nf2.ma". - -include "Basic-1/sn3/fwd.ma". - -include "Basic-1/nf2/iso.ma". - -include "Basic-1/pr3/iso.ma". - -theorem sn3_pr3_trans: - \forall (c: C).(\forall (t1: T).((sn3 c t1) \to (\forall (t2: T).((pr3 c t1 -t2) \to (sn3 c t2))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (H: (sn3 c t1)).(sn3_ind c (\lambda -(t: T).(\forall (t2: T).((pr3 c t t2) \to (sn3 c t2)))) (\lambda (t2: -T).(\lambda (H0: ((\forall (t3: T).((((eq T t2 t3) \to (\forall (P: -Prop).P))) \to ((pr3 c t2 t3) \to (sn3 c t3)))))).(\lambda (H1: ((\forall -(t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to -(\forall (t4: T).((pr3 c t3 t4) \to (sn3 c t4)))))))).(\lambda (t3: -T).(\lambda (H2: (pr3 c t2 t3)).(sn3_sing c t3 (\lambda (t0: T).(\lambda (H3: -(((eq T t3 t0) \to (\forall (P: Prop).P)))).(\lambda (H4: (pr3 c t3 t0)).(let -H_x \def (term_dec t2 t3) in (let H5 \def H_x in (or_ind (eq T t2 t3) ((eq T -t2 t3) \to (\forall (P: Prop).P)) (sn3 c t0) (\lambda (H6: (eq T t2 t3)).(let -H7 \def (eq_ind_r T t3 (\lambda (t: T).(pr3 c t t0)) H4 t2 H6) in (let H8 -\def (eq_ind_r T t3 (\lambda (t: T).((eq T t t0) \to (\forall (P: Prop).P))) -H3 t2 H6) in (let H9 \def (eq_ind_r T t3 (\lambda (t: T).(pr3 c t2 t)) H2 t2 -H6) in (H0 t0 H8 H7))))) (\lambda (H6: (((eq T t2 t3) \to (\forall (P: -Prop).P)))).(H1 t3 H6 H2 t0 H4)) H5)))))))))))) t1 H))). -(* COMMENTS -Initial nodes: 289 -END *) - -theorem sn3_pr2_intro: - \forall (c: C).(\forall (t1: T).(((\forall (t2: T).((((eq T t1 t2) \to -(\forall (P: Prop).P))) \to ((pr2 c t1 t2) \to (sn3 c t2))))) \to (sn3 c t1))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (H: ((\forall (t2: T).((((eq T t1 -t2) \to (\forall (P: Prop).P))) \to ((pr2 c t1 t2) \to (sn3 c -t2)))))).(sn3_sing c t1 (\lambda (t2: T).(\lambda (H0: (((eq T t1 t2) \to -(\forall (P: Prop).P)))).(\lambda (H1: (pr3 c t1 t2)).(let H2 \def H0 in -((let H3 \def H in (pr3_ind c (\lambda (t: T).(\lambda (t0: T).(((\forall -(t3: T).((((eq T t t3) \to (\forall (P: Prop).P))) \to ((pr2 c t t3) \to (sn3 -c t3))))) \to ((((eq T t t0) \to (\forall (P: Prop).P))) \to (sn3 c t0))))) -(\lambda (t: T).(\lambda (H4: ((\forall (t3: T).((((eq T t t3) \to (\forall -(P: Prop).P))) \to ((pr2 c t t3) \to (sn3 c t3)))))).(\lambda (H5: (((eq T t -t) \to (\forall (P: Prop).P)))).(H4 t H5 (pr2_free c t t (pr0_refl t)))))) -(\lambda (t3: T).(\lambda (t4: T).(\lambda (H4: (pr2 c t4 t3)).(\lambda (t5: -T).(\lambda (H5: (pr3 c t3 t5)).(\lambda (H6: ((((\forall (t6: T).((((eq T t3 -t6) \to (\forall (P: Prop).P))) \to ((pr2 c t3 t6) \to (sn3 c t6))))) \to -((((eq T t3 t5) \to (\forall (P: Prop).P))) \to (sn3 c t5))))).(\lambda (H7: -((\forall (t6: T).((((eq T t4 t6) \to (\forall (P: Prop).P))) \to ((pr2 c t4 -t6) \to (sn3 c t6)))))).(\lambda (H8: (((eq T t4 t5) \to (\forall (P: -Prop).P)))).(let H_x \def (term_dec t4 t3) in (let H9 \def H_x in (or_ind (eq -T t4 t3) ((eq T t4 t3) \to (\forall (P: Prop).P)) (sn3 c t5) (\lambda (H10: -(eq T t4 t3)).(let H11 \def (eq_ind T t4 (\lambda (t: T).((eq T t t5) \to -(\forall (P: Prop).P))) H8 t3 H10) in (let H12 \def (eq_ind T t4 (\lambda (t: -T).(\forall (t6: T).((((eq T t t6) \to (\forall (P: Prop).P))) \to ((pr2 c t -t6) \to (sn3 c t6))))) H7 t3 H10) in (let H13 \def (eq_ind T t4 (\lambda (t: -T).(pr2 c t t3)) H4 t3 H10) in (H6 H12 H11))))) (\lambda (H10: (((eq T t4 t3) -\to (\forall (P: Prop).P)))).(sn3_pr3_trans c t3 (H7 t3 H10 H4) t5 H5)) -H9))))))))))) t1 t2 H1 H3)) H2)))))))). -(* COMMENTS -Initial nodes: 467 -END *) - -theorem sn3_cast: - \forall (c: C).(\forall (u: T).((sn3 c u) \to (\forall (t: T).((sn3 c t) \to -(sn3 c (THead (Flat Cast) u t)))))) -\def - \lambda (c: C).(\lambda (u: T).(\lambda (H: (sn3 c u)).(sn3_ind c (\lambda -(t: T).(\forall (t0: T).((sn3 c t0) \to (sn3 c (THead (Flat Cast) t t0))))) -(\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) \to (\forall -(P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H1: ((\forall -(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to -(\forall (t: T).((sn3 c t) \to (sn3 c (THead (Flat Cast) t2 -t))))))))).(\lambda (t: T).(\lambda (H2: (sn3 c t)).(sn3_ind c (\lambda (t0: -T).(sn3 c (THead (Flat Cast) t1 t0))) (\lambda (t0: T).(\lambda (H3: -((\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t0 -t2) \to (sn3 c t2)))))).(\lambda (H4: ((\forall (t2: T).((((eq T t0 t2) \to -(\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (sn3 c (THead (Flat Cast) t1 -t2))))))).(sn3_pr2_intro c (THead (Flat Cast) t1 t0) (\lambda (t2: -T).(\lambda (H5: (((eq T (THead (Flat Cast) t1 t0) t2) \to (\forall (P: -Prop).P)))).(\lambda (H6: (pr2 c (THead (Flat Cast) t1 t0) t2)).(let H7 \def -(pr2_gen_cast c t1 t0 t2 H6) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda -(t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c t1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t0 t3)))) (pr2 c -t0 t2) (sn3 c t2) (\lambda (H8: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: -T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c t1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t0 -t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead -(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2))) -(\lambda (_: T).(\lambda (t3: T).(pr2 c t0 t3))) (sn3 c t2) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H9: (eq T t2 (THead (Flat Cast) x0 -x1))).(\lambda (H10: (pr2 c t1 x0)).(\lambda (H11: (pr2 c t0 x1)).(let H12 -\def (eq_ind T t2 (\lambda (t3: T).((eq T (THead (Flat Cast) t1 t0) t3) \to -(\forall (P: Prop).P))) H5 (THead (Flat Cast) x0 x1) H9) in (eq_ind_r T -(THead (Flat Cast) x0 x1) (\lambda (t3: T).(sn3 c t3)) (let H_x \def -(term_dec x0 t1) in (let H13 \def H_x in (or_ind (eq T x0 t1) ((eq T x0 t1) -\to (\forall (P: Prop).P)) (sn3 c (THead (Flat Cast) x0 x1)) (\lambda (H14: -(eq T x0 t1)).(let H15 \def (eq_ind T x0 (\lambda (t3: T).((eq T (THead (Flat -Cast) t1 t0) (THead (Flat Cast) t3 x1)) \to (\forall (P: Prop).P))) H12 t1 -H14) in (let H16 \def (eq_ind T x0 (\lambda (t3: T).(pr2 c t1 t3)) H10 t1 -H14) in (eq_ind_r T t1 (\lambda (t3: T).(sn3 c (THead (Flat Cast) t3 x1))) -(let H_x0 \def (term_dec t0 x1) in (let H17 \def H_x0 in (or_ind (eq T t0 x1) -((eq T t0 x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Cast) t1 x1)) -(\lambda (H18: (eq T t0 x1)).(let H19 \def (eq_ind_r T x1 (\lambda (t3: -T).((eq T (THead (Flat Cast) t1 t0) (THead (Flat Cast) t1 t3)) \to (\forall -(P: Prop).P))) H15 t0 H18) in (let H20 \def (eq_ind_r T x1 (\lambda (t3: -T).(pr2 c t0 t3)) H11 t0 H18) in (eq_ind T t0 (\lambda (t3: T).(sn3 c (THead -(Flat Cast) t1 t3))) (H19 (refl_equal T (THead (Flat Cast) t1 t0)) (sn3 c -(THead (Flat Cast) t1 t0))) x1 H18)))) (\lambda (H18: (((eq T t0 x1) \to -(\forall (P: Prop).P)))).(H4 x1 H18 (pr3_pr2 c t0 x1 H11))) H17))) x0 H14)))) -(\lambda (H14: (((eq T x0 t1) \to (\forall (P: Prop).P)))).(H1 x0 (\lambda -(H15: (eq T t1 x0)).(\lambda (P: Prop).(let H16 \def (eq_ind_r T x0 (\lambda -(t3: T).((eq T t3 t1) \to (\forall (P0: Prop).P0))) H14 t1 H15) in (let H17 -\def (eq_ind_r T x0 (\lambda (t3: T).((eq T (THead (Flat Cast) t1 t0) (THead -(Flat Cast) t3 x1)) \to (\forall (P0: Prop).P0))) H12 t1 H15) in (let H18 -\def (eq_ind_r T x0 (\lambda (t3: T).(pr2 c t1 t3)) H10 t1 H15) in (H16 -(refl_equal T t1) P)))))) (pr3_pr2 c t1 x0 H10) x1 (let H_x0 \def (term_dec -t0 x1) in (let H15 \def H_x0 in (or_ind (eq T t0 x1) ((eq T t0 x1) \to -(\forall (P: Prop).P)) (sn3 c x1) (\lambda (H16: (eq T t0 x1)).(let H17 \def -(eq_ind_r T x1 (\lambda (t3: T).((eq T (THead (Flat Cast) t1 t0) (THead (Flat -Cast) x0 t3)) \to (\forall (P: Prop).P))) H12 t0 H16) in (let H18 \def -(eq_ind_r T x1 (\lambda (t3: T).(pr2 c t0 t3)) H11 t0 H16) in (eq_ind T t0 -(\lambda (t3: T).(sn3 c t3)) (sn3_sing c t0 H3) x1 H16)))) (\lambda (H16: -(((eq T t0 x1) \to (\forall (P: Prop).P)))).(H3 x1 H16 (pr3_pr2 c t0 x1 -H11))) H15))))) H13))) t2 H9))))))) H8)) (\lambda (H8: (pr2 c t0 -t2)).(sn3_pr3_trans c t0 (sn3_sing c t0 H3) t2 (pr3_pr2 c t0 t2 H8))) -H7))))))))) t H2)))))) u H))). -(* COMMENTS -Initial nodes: 1239 -END *) - -theorem sn3_cflat: - \forall (c: C).(\forall (t: T).((sn3 c t) \to (\forall (f: F).(\forall (u: -T).(sn3 (CHead c (Flat f) u) t))))) -\def - \lambda (c: C).(\lambda (t: T).(\lambda (H: (sn3 c t)).(\lambda (f: -F).(\lambda (u: T).(sn3_ind c (\lambda (t0: T).(sn3 (CHead c (Flat f) u) t0)) -(\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) \to (\forall -(P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H1: ((\forall -(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to -(sn3 (CHead c (Flat f) u) t2)))))).(sn3_pr2_intro (CHead c (Flat f) u) t1 -(\lambda (t2: T).(\lambda (H2: (((eq T t1 t2) \to (\forall (P: -Prop).P)))).(\lambda (H3: (pr2 (CHead c (Flat f) u) t1 t2)).(H1 t2 H2 -(pr3_pr2 c t1 t2 (pr2_gen_cflat f c u t1 t2 H3)))))))))) t H))))). -(* COMMENTS -Initial nodes: 175 -END *) - -theorem sn3_shift: - \forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sn3 c -(THead (Bind b) v t)) \to (sn3 (CHead c (Bind b) v) t))))) -\def - \lambda (b: B).(\lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (H: -(sn3 c (THead (Bind b) v t))).(let H_x \def (sn3_gen_bind b c v t H) in (let -H0 \def H_x in (land_ind (sn3 c v) (sn3 (CHead c (Bind b) v) t) (sn3 (CHead c -(Bind b) v) t) (\lambda (_: (sn3 c v)).(\lambda (H2: (sn3 (CHead c (Bind b) -v) t)).H2)) H0))))))). -(* COMMENTS -Initial nodes: 95 -END *) - -theorem sn3_change: - \forall (b: B).((not (eq B b Abbr)) \to (\forall (c: C).(\forall (v1: -T).(\forall (t: T).((sn3 (CHead c (Bind b) v1) t) \to (\forall (v2: T).(sn3 -(CHead c (Bind b) v2) t))))))) -\def - \lambda (b: B).(\lambda (H: (not (eq B b Abbr))).(\lambda (c: C).(\lambda -(v1: T).(\lambda (t: T).(\lambda (H0: (sn3 (CHead c (Bind b) v1) t)).(\lambda -(v2: T).(sn3_ind (CHead c (Bind b) v1) (\lambda (t0: T).(sn3 (CHead c (Bind -b) v2) t0)) (\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) -\to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) v1) t1 t2) \to (sn3 -(CHead c (Bind b) v1) t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 -t2) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) v1) t1 t2) \to -(sn3 (CHead c (Bind b) v2) t2)))))).(sn3_pr2_intro (CHead c (Bind b) v2) t1 -(\lambda (t2: T).(\lambda (H3: (((eq T t1 t2) \to (\forall (P: -Prop).P)))).(\lambda (H4: (pr2 (CHead c (Bind b) v2) t1 t2)).(H2 t2 H3 -(pr3_pr2 (CHead c (Bind b) v1) t1 t2 (pr2_change b H c v2 t1 t2 H4 -v1)))))))))) t H0))))))). -(* COMMENTS -Initial nodes: 239 -END *) - -theorem sn3_gen_def: - \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c -(CHead d (Bind Abbr) v)) \to ((sn3 c (TLRef i)) \to (sn3 d v)))))) -\def - \lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda -(H: (getl i c (CHead d (Bind Abbr) v))).(\lambda (H0: (sn3 c (TLRef -i))).(sn3_gen_lift c v (S i) O (sn3_pr3_trans c (TLRef i) H0 (lift (S i) O v) -(pr3_pr2 c (TLRef i) (lift (S i) O v) (pr2_delta c d v i H (TLRef i) (TLRef -i) (pr0_refl (TLRef i)) (lift (S i) O v) (subst0_lref v i)))) d (getl_drop -Abbr c d v i H))))))). -(* COMMENTS -Initial nodes: 139 -END *) - -theorem sn3_cdelta: - \forall (v: T).(\forall (t: T).(\forall (i: nat).(((\forall (w: T).(ex T -(\lambda (u: T).(subst0 i w t u))))) \to (\forall (c: C).(\forall (d: -C).((getl i c (CHead d (Bind Abbr) v)) \to ((sn3 c t) \to (sn3 d v)))))))) -\def - \lambda (v: T).(\lambda (t: T).(\lambda (i: nat).(\lambda (H: ((\forall (w: -T).(ex T (\lambda (u: T).(subst0 i w t u)))))).(let H_x \def (H v) in (let H0 -\def H_x in (ex_ind T (\lambda (u: T).(subst0 i v t u)) (\forall (c: -C).(\forall (d: C).((getl i c (CHead d (Bind Abbr) v)) \to ((sn3 c t) \to -(sn3 d v))))) (\lambda (x: T).(\lambda (H1: (subst0 i v t x)).(subst0_ind -(\lambda (n: nat).(\lambda (t0: T).(\lambda (t1: T).(\lambda (_: T).(\forall -(c: C).(\forall (d: C).((getl n c (CHead d (Bind Abbr) t0)) \to ((sn3 c t1) -\to (sn3 d t0))))))))) (\lambda (v0: T).(\lambda (i0: nat).(\lambda (c: -C).(\lambda (d: C).(\lambda (H2: (getl i0 c (CHead d (Bind Abbr) -v0))).(\lambda (H3: (sn3 c (TLRef i0))).(sn3_gen_def c d v0 i0 H2 H3))))))) -(\lambda (v0: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i0: -nat).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (H3: ((\forall (c: -C).(\forall (d: C).((getl i0 c (CHead d (Bind Abbr) v0)) \to ((sn3 c u1) \to -(sn3 d v0))))))).(\lambda (t0: T).(\lambda (k: K).(\lambda (c: C).(\lambda -(d: C).(\lambda (H4: (getl i0 c (CHead d (Bind Abbr) v0))).(\lambda (H5: (sn3 -c (THead k u1 t0))).(let H_y \def (sn3_gen_head k c u1 t0 H5) in (H3 c d H4 -H_y)))))))))))))) (\lambda (k: K).(\lambda (v0: T).(\lambda (t2: T).(\lambda -(t1: T).(\lambda (i0: nat).(\lambda (H2: (subst0 (s k i0) v0 t1 t2)).(\lambda -(H3: ((\forall (c: C).(\forall (d: C).((getl (s k i0) c (CHead d (Bind Abbr) -v0)) \to ((sn3 c t1) \to (sn3 d v0))))))).(\lambda (u: T).(\lambda (c: -C).(\lambda (d: C).(\lambda (H4: (getl i0 c (CHead d (Bind Abbr) -v0))).(\lambda (H5: (sn3 c (THead k u t1))).(K_ind (\lambda (k0: K).((subst0 -(s k0 i0) v0 t1 t2) \to (((\forall (c0: C).(\forall (d0: C).((getl (s k0 i0) -c0 (CHead d0 (Bind Abbr) v0)) \to ((sn3 c0 t1) \to (sn3 d0 v0)))))) \to ((sn3 -c (THead k0 u t1)) \to (sn3 d v0))))) (\lambda (b: B).(\lambda (_: (subst0 (s -(Bind b) i0) v0 t1 t2)).(\lambda (H7: ((\forall (c0: C).(\forall (d0: -C).((getl (s (Bind b) i0) c0 (CHead d0 (Bind Abbr) v0)) \to ((sn3 c0 t1) \to -(sn3 d0 v0))))))).(\lambda (H8: (sn3 c (THead (Bind b) u t1))).(let H_x0 \def -(sn3_gen_bind b c u t1 H8) in (let H9 \def H_x0 in (land_ind (sn3 c u) (sn3 -(CHead c (Bind b) u) t1) (sn3 d v0) (\lambda (_: (sn3 c u)).(\lambda (H11: -(sn3 (CHead c (Bind b) u) t1)).(H7 (CHead c (Bind b) u) d (getl_clear_bind b -(CHead c (Bind b) u) c u (clear_bind b c u) (CHead d (Bind Abbr) v0) i0 H4) -H11))) H9))))))) (\lambda (f: F).(\lambda (_: (subst0 (s (Flat f) i0) v0 t1 -t2)).(\lambda (H7: ((\forall (c0: C).(\forall (d0: C).((getl (s (Flat f) i0) -c0 (CHead d0 (Bind Abbr) v0)) \to ((sn3 c0 t1) \to (sn3 d0 v0))))))).(\lambda -(H8: (sn3 c (THead (Flat f) u t1))).(let H_x0 \def (sn3_gen_flat f c u t1 H8) -in (let H9 \def H_x0 in (land_ind (sn3 c u) (sn3 c t1) (sn3 d v0) (\lambda -(_: (sn3 c u)).(\lambda (H11: (sn3 c t1)).(H7 c d H4 H11))) H9))))))) k H2 H3 -H5))))))))))))) (\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda -(i0: nat).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (H3: ((\forall (c: -C).(\forall (d: C).((getl i0 c (CHead d (Bind Abbr) v0)) \to ((sn3 c u1) \to -(sn3 d v0))))))).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(_: (subst0 (s k i0) v0 t1 t2)).(\lambda (_: ((\forall (c: C).(\forall (d: -C).((getl (s k i0) c (CHead d (Bind Abbr) v0)) \to ((sn3 c t1) \to (sn3 d -v0))))))).(\lambda (c: C).(\lambda (d: C).(\lambda (H6: (getl i0 c (CHead d -(Bind Abbr) v0))).(\lambda (H7: (sn3 c (THead k u1 t1))).(let H_y \def -(sn3_gen_head k c u1 t1 H7) in (H3 c d H6 H_y))))))))))))))))) i v t x H1))) -H0)))))). -(* COMMENTS -Initial nodes: 949 -END *) - -theorem sn3_cpr3_trans: - \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall -(k: K).(\forall (t: T).((sn3 (CHead c k u1) t) \to (sn3 (CHead c k u2) -t))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1 -u2)).(\lambda (k: K).(\lambda (t: T).(\lambda (H0: (sn3 (CHead c k u1) -t)).(sn3_ind (CHead c k u1) (\lambda (t0: T).(sn3 (CHead c k u2) t0)) -(\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) \to (\forall -(P: Prop).P))) \to ((pr3 (CHead c k u1) t1 t2) \to (sn3 (CHead c k u1) -t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: -Prop).P))) \to ((pr3 (CHead c k u1) t1 t2) \to (sn3 (CHead c k u2) -t2)))))).(sn3_sing (CHead c k u2) t1 (\lambda (t2: T).(\lambda (H3: (((eq T -t1 t2) \to (\forall (P: Prop).P)))).(\lambda (H4: (pr3 (CHead c k u2) t1 -t2)).(H2 t2 H3 (pr3_pr3_pr3_t c u1 u2 H t1 t2 k H4))))))))) t H0))))))). -(* COMMENTS -Initial nodes: 203 -END *) - -theorem sn3_bind: - \forall (b: B).(\forall (c: C).(\forall (u: T).((sn3 c u) \to (\forall (t: -T).((sn3 (CHead c (Bind b) u) t) \to (sn3 c (THead (Bind b) u t))))))) -\def - \lambda (b: B).(\lambda (c: C).(\lambda (u: T).(\lambda (H: (sn3 c -u)).(sn3_ind c (\lambda (t: T).(\forall (t0: T).((sn3 (CHead c (Bind b) t) -t0) \to (sn3 c (THead (Bind b) t t0))))) (\lambda (t1: T).(\lambda (_: -((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 -t2) \to (sn3 c t2)))))).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to -(\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (\forall (t: T).((sn3 (CHead c -(Bind b) t2) t) \to (sn3 c (THead (Bind b) t2 t))))))))).(\lambda (t: -T).(\lambda (H2: (sn3 (CHead c (Bind b) t1) t)).(sn3_ind (CHead c (Bind b) -t1) (\lambda (t0: T).(sn3 c (THead (Bind b) t1 t0))) (\lambda (t2: -T).(\lambda (H3: ((\forall (t3: T).((((eq T t2 t3) \to (\forall (P: -Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to (sn3 (CHead c (Bind b) -t1) t3)))))).(\lambda (H4: ((\forall (t3: T).((((eq T t2 t3) \to (\forall (P: -Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to (sn3 c (THead (Bind b) -t1 t3))))))).(sn3_sing c (THead (Bind b) t1 t2) (\lambda (t3: T).(\lambda -(H5: (((eq T (THead (Bind b) t1 t2) t3) \to (\forall (P: Prop).P)))).(\lambda -(H6: (pr3 c (THead (Bind b) t1 t2) t3)).(let H_x \def (bind_dec_not b Abst) -in (let H7 \def H_x in (or_ind (eq B b Abst) (not (eq B b Abst)) (sn3 c t3) -(\lambda (H8: (eq B b Abst)).(let H9 \def (eq_ind B b (\lambda (b0: B).(pr3 c -(THead (Bind b0) t1 t2) t3)) H6 Abst H8) in (let H10 \def (eq_ind B b -(\lambda (b0: B).((eq T (THead (Bind b0) t1 t2) t3) \to (\forall (P: -Prop).P))) H5 Abst H8) in (let H11 \def (eq_ind B b (\lambda (b0: B).(\forall -(t4: T).((((eq T t2 t4) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind -b0) t1) t2 t4) \to (sn3 c (THead (Bind b0) t1 t4)))))) H4 Abst H8) in (let -H12 \def (eq_ind B b (\lambda (b0: B).(\forall (t4: T).((((eq T t2 t4) \to -(\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b0) t1) t2 t4) \to (sn3 -(CHead c (Bind b0) t1) t4))))) H3 Abst H8) in (let H13 \def (eq_ind B b -(\lambda (b0: B).(\forall (t4: T).((((eq T t1 t4) \to (\forall (P: Prop).P))) -\to ((pr3 c t1 t4) \to (\forall (t0: T).((sn3 (CHead c (Bind b0) t4) t0) \to -(sn3 c (THead (Bind b0) t4 t0)))))))) H1 Abst H8) in (let H14 \def -(pr3_gen_abst c t1 t2 t3 H9) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: -T).(eq T t3 (THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b0: B).(\forall -(u0: T).(pr3 (CHead c (Bind b0) u0) t2 t4))))) (sn3 c t3) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H15: (eq T t3 (THead (Bind Abst) x0 -x1))).(\lambda (H16: (pr3 c t1 x0)).(\lambda (H17: ((\forall (b0: B).(\forall -(u0: T).(pr3 (CHead c (Bind b0) u0) t2 x1))))).(let H18 \def (eq_ind T t3 -(\lambda (t0: T).((eq T (THead (Bind Abst) t1 t2) t0) \to (\forall (P: -Prop).P))) H10 (THead (Bind Abst) x0 x1) H15) in (eq_ind_r T (THead (Bind -Abst) x0 x1) (\lambda (t0: T).(sn3 c t0)) (let H_x0 \def (term_dec t1 x0) in -(let H19 \def H_x0 in (or_ind (eq T t1 x0) ((eq T t1 x0) \to (\forall (P: -Prop).P)) (sn3 c (THead (Bind Abst) x0 x1)) (\lambda (H20: (eq T t1 x0)).(let -H21 \def (eq_ind_r T x0 (\lambda (t0: T).((eq T (THead (Bind Abst) t1 t2) -(THead (Bind Abst) t0 x1)) \to (\forall (P: Prop).P))) H18 t1 H20) in (let -H22 \def (eq_ind_r T x0 (\lambda (t0: T).(pr3 c t1 t0)) H16 t1 H20) in -(eq_ind T t1 (\lambda (t0: T).(sn3 c (THead (Bind Abst) t0 x1))) (let H_x1 -\def (term_dec t2 x1) in (let H23 \def H_x1 in (or_ind (eq T t2 x1) ((eq T t2 -x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Bind Abst) t1 x1)) (\lambda -(H24: (eq T t2 x1)).(let H25 \def (eq_ind_r T x1 (\lambda (t0: T).((eq T -(THead (Bind Abst) t1 t2) (THead (Bind Abst) t1 t0)) \to (\forall (P: -Prop).P))) H21 t2 H24) in (let H26 \def (eq_ind_r T x1 (\lambda (t0: -T).(\forall (b0: B).(\forall (u0: T).(pr3 (CHead c (Bind b0) u0) t2 t0)))) -H17 t2 H24) in (eq_ind T t2 (\lambda (t0: T).(sn3 c (THead (Bind Abst) t1 -t0))) (H25 (refl_equal T (THead (Bind Abst) t1 t2)) (sn3 c (THead (Bind Abst) -t1 t2))) x1 H24)))) (\lambda (H24: (((eq T t2 x1) \to (\forall (P: -Prop).P)))).(H11 x1 H24 (H17 Abst t1))) H23))) x0 H20)))) (\lambda (H20: -(((eq T t1 x0) \to (\forall (P: Prop).P)))).(let H_x1 \def (term_dec t2 x1) -in (let H21 \def H_x1 in (or_ind (eq T t2 x1) ((eq T t2 x1) \to (\forall (P: -Prop).P)) (sn3 c (THead (Bind Abst) x0 x1)) (\lambda (H22: (eq T t2 x1)).(let -H23 \def (eq_ind_r T x1 (\lambda (t0: T).(\forall (b0: B).(\forall (u0: -T).(pr3 (CHead c (Bind b0) u0) t2 t0)))) H17 t2 H22) in (eq_ind T t2 (\lambda -(t0: T).(sn3 c (THead (Bind Abst) x0 t0))) (H13 x0 H20 H16 t2 (sn3_cpr3_trans -c t1 x0 H16 (Bind Abst) t2 (sn3_sing (CHead c (Bind Abst) t1) t2 H12))) x1 -H22))) (\lambda (H22: (((eq T t2 x1) \to (\forall (P: Prop).P)))).(H13 x0 H20 -H16 x1 (sn3_cpr3_trans c t1 x0 H16 (Bind Abst) x1 (H12 x1 H22 (H17 Abst -t1))))) H21)))) H19))) t3 H15))))))) H14)))))))) (\lambda (H8: (not (eq B b -Abst))).(let H_x0 \def (pr3_gen_bind b H8 c t1 t2 t3 H6) in (let H9 \def H_x0 -in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind -b) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_: -T).(\lambda (t4: T).(pr3 (CHead c (Bind b) t1) t2 t4)))) (pr3 (CHead c (Bind -b) t1) t2 (lift (S O) O t3)) (sn3 c t3) (\lambda (H10: (ex3_2 T T (\lambda -(u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr3 -(CHead c (Bind b) t1) t2 t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda -(t4: T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr3 (CHead c (Bind b) -t1) t2 t4))) (sn3 c t3) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq -T t3 (THead (Bind b) x0 x1))).(\lambda (H12: (pr3 c t1 x0)).(\lambda (H13: -(pr3 (CHead c (Bind b) t1) t2 x1)).(let H14 \def (eq_ind T t3 (\lambda (t0: -T).((eq T (THead (Bind b) t1 t2) t0) \to (\forall (P: Prop).P))) H5 (THead -(Bind b) x0 x1) H11) in (eq_ind_r T (THead (Bind b) x0 x1) (\lambda (t0: -T).(sn3 c t0)) (let H_x1 \def (term_dec t1 x0) in (let H15 \def H_x1 in -(or_ind (eq T t1 x0) ((eq T t1 x0) \to (\forall (P: Prop).P)) (sn3 c (THead -(Bind b) x0 x1)) (\lambda (H16: (eq T t1 x0)).(let H17 \def (eq_ind_r T x0 -(\lambda (t0: T).((eq T (THead (Bind b) t1 t2) (THead (Bind b) t0 x1)) \to -(\forall (P: Prop).P))) H14 t1 H16) in (let H18 \def (eq_ind_r T x0 (\lambda -(t0: T).(pr3 c t1 t0)) H12 t1 H16) in (eq_ind T t1 (\lambda (t0: T).(sn3 c -(THead (Bind b) t0 x1))) (let H_x2 \def (term_dec t2 x1) in (let H19 \def -H_x2 in (or_ind (eq T t2 x1) ((eq T t2 x1) \to (\forall (P: Prop).P)) (sn3 c -(THead (Bind b) t1 x1)) (\lambda (H20: (eq T t2 x1)).(let H21 \def (eq_ind_r -T x1 (\lambda (t0: T).((eq T (THead (Bind b) t1 t2) (THead (Bind b) t1 t0)) -\to (\forall (P: Prop).P))) H17 t2 H20) in (let H22 \def (eq_ind_r T x1 -(\lambda (t0: T).(pr3 (CHead c (Bind b) t1) t2 t0)) H13 t2 H20) in (eq_ind T -t2 (\lambda (t0: T).(sn3 c (THead (Bind b) t1 t0))) (H21 (refl_equal T (THead -(Bind b) t1 t2)) (sn3 c (THead (Bind b) t1 t2))) x1 H20)))) (\lambda (H20: -(((eq T t2 x1) \to (\forall (P: Prop).P)))).(H4 x1 H20 H13)) H19))) x0 -H16)))) (\lambda (H16: (((eq T t1 x0) \to (\forall (P: Prop).P)))).(let H_x2 -\def (term_dec t2 x1) in (let H17 \def H_x2 in (or_ind (eq T t2 x1) ((eq T t2 -x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Bind b) x0 x1)) (\lambda (H18: -(eq T t2 x1)).(let H19 \def (eq_ind_r T x1 (\lambda (t0: T).(pr3 (CHead c -(Bind b) t1) t2 t0)) H13 t2 H18) in (eq_ind T t2 (\lambda (t0: T).(sn3 c -(THead (Bind b) x0 t0))) (H1 x0 H16 H12 t2 (sn3_cpr3_trans c t1 x0 H12 (Bind -b) t2 (sn3_sing (CHead c (Bind b) t1) t2 H3))) x1 H18))) (\lambda (H18: (((eq -T t2 x1) \to (\forall (P: Prop).P)))).(H1 x0 H16 H12 x1 (sn3_cpr3_trans c t1 -x0 H12 (Bind b) x1 (H3 x1 H18 H13)))) H17)))) H15))) t3 H11))))))) H10)) -(\lambda (H10: (pr3 (CHead c (Bind b) t1) t2 (lift (S O) O -t3))).(sn3_gen_lift (CHead c (Bind b) t1) t3 (S O) O (sn3_pr3_trans (CHead c -(Bind b) t1) t2 (sn3_sing (CHead c (Bind b) t1) t2 H3) (lift (S O) O t3) H10) -c (drop_drop (Bind b) O c c (drop_refl c) t1))) H9)))) H7)))))))))) t -H2)))))) u H)))). -(* COMMENTS -Initial nodes: 2401 -END *) - -theorem sn3_beta: - \forall (c: C).(\forall (v: T).(\forall (t: T).((sn3 c (THead (Bind Abbr) v -t)) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) v (THead -(Bind Abst) w t)))))))) -\def - \lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (H: (sn3 c (THead -(Bind Abbr) v t))).(insert_eq T (THead (Bind Abbr) v t) (\lambda (t0: T).(sn3 -c t0)) (\lambda (_: T).(\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat -Appl) v (THead (Bind Abst) w t)))))) (\lambda (y: T).(\lambda (H0: (sn3 c -y)).(unintro T t (\lambda (t0: T).((eq T y (THead (Bind Abbr) v t0)) \to -(\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) v (THead (Bind Abst) -w t0))))))) (unintro T v (\lambda (t0: T).(\forall (x: T).((eq T y (THead -(Bind Abbr) t0 x)) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat -Appl) t0 (THead (Bind Abst) w x)))))))) (sn3_ind c (\lambda (t0: T).(\forall -(x: T).(\forall (x0: T).((eq T t0 (THead (Bind Abbr) x x0)) \to (\forall (w: -T).((sn3 c w) \to (sn3 c (THead (Flat Appl) x (THead (Bind Abst) w -x0))))))))) (\lambda (t1: T).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) -\to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda -(H2: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 -c t1 t2) \to (\forall (x: T).(\forall (x0: T).((eq T t2 (THead (Bind Abbr) x -x0)) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) x (THead -(Bind Abst) w x0))))))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: -(eq T t1 (THead (Bind Abbr) x x0))).(\lambda (w: T).(\lambda (H4: (sn3 c -w)).(let H5 \def (eq_ind T t1 (\lambda (t0: T).(\forall (t2: T).((((eq T t0 -t2) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (\forall (x1: -T).(\forall (x2: T).((eq T t2 (THead (Bind Abbr) x1 x2)) \to (\forall (w0: -T).((sn3 c w0) \to (sn3 c (THead (Flat Appl) x1 (THead (Bind Abst) w0 -x2)))))))))))) H2 (THead (Bind Abbr) x x0) H3) in (let H6 \def (eq_ind T t1 -(\lambda (t0: T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) -\to ((pr3 c t0 t2) \to (sn3 c t2))))) H1 (THead (Bind Abbr) x x0) H3) in -(sn3_ind c (\lambda (t0: T).(sn3 c (THead (Flat Appl) x (THead (Bind Abst) t0 -x0)))) (\lambda (t2: T).(\lambda (H7: ((\forall (t3: T).((((eq T t2 t3) \to -(\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to (sn3 c t3)))))).(\lambda (H8: -((\forall (t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t2 -t3) \to (sn3 c (THead (Flat Appl) x (THead (Bind Abst) t3 -x0)))))))).(sn3_pr2_intro c (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) -(\lambda (t3: T).(\lambda (H9: (((eq T (THead (Flat Appl) x (THead (Bind -Abst) t2 x0)) t3) \to (\forall (P: Prop).P)))).(\lambda (H10: (pr2 c (THead -(Flat Appl) x (THead (Bind Abst) t2 x0)) t3)).(let H11 \def (pr2_gen_appl c x -(THead (Bind Abst) t2 x0) t3 H10) in (or3_ind (ex3_2 T T (\lambda (u2: -T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c -(THead (Bind Abst) t2 x0) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) t2 x0) -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda -(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: -B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4)))))))) (ex6_6 B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind Abst) t2 x0) (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) -z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) -y2) z1 z2)))))))) (sn3 c t3) (\lambda (H12: (ex3_2 T T (\lambda (u2: -T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c -(THead (Bind Abst) t2 x0) t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda -(t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind Abst) -t2 x0) t4))) (sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (H13: (eq -T t3 (THead (Flat Appl) x1 x2))).(\lambda (H14: (pr2 c x x1)).(\lambda (H15: -(pr2 c (THead (Bind Abst) t2 x0) x2)).(let H16 \def (eq_ind T t3 (\lambda -(t0: T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) t0) \to -(\forall (P: Prop).P))) H9 (THead (Flat Appl) x1 x2) H13) in (eq_ind_r T -(THead (Flat Appl) x1 x2) (\lambda (t0: T).(sn3 c t0)) (let H17 \def -(pr2_gen_abst c t2 x0 x2 H15) in (ex3_2_ind T T (\lambda (u2: T).(\lambda -(t4: T).(eq T x2 (THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c t2 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall -(u: T).(pr2 (CHead c (Bind b) u) x0 t4))))) (sn3 c (THead (Flat Appl) x1 x2)) -(\lambda (x3: T).(\lambda (x4: T).(\lambda (H18: (eq T x2 (THead (Bind Abst) -x3 x4))).(\lambda (H19: (pr2 c t2 x3)).(\lambda (H20: ((\forall (b: -B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x0 x4))))).(let H21 \def (eq_ind -T x2 (\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) -(THead (Flat Appl) x1 t0)) \to (\forall (P: Prop).P))) H16 (THead (Bind Abst) -x3 x4) H18) in (eq_ind_r T (THead (Bind Abst) x3 x4) (\lambda (t0: T).(sn3 c -(THead (Flat Appl) x1 t0))) (let H_x \def (term_dec t2 x3) in (let H22 \def -H_x in (or_ind (eq T t2 x3) ((eq T t2 x3) \to (\forall (P: Prop).P)) (sn3 c -(THead (Flat Appl) x1 (THead (Bind Abst) x3 x4))) (\lambda (H23: (eq T t2 -x3)).(let H24 \def (eq_ind_r T x3 (\lambda (t0: T).((eq T (THead (Flat Appl) -x (THead (Bind Abst) t2 x0)) (THead (Flat Appl) x1 (THead (Bind Abst) t0 -x4))) \to (\forall (P: Prop).P))) H21 t2 H23) in (let H25 \def (eq_ind_r T x3 -(\lambda (t0: T).(pr2 c t2 t0)) H19 t2 H23) in (eq_ind T t2 (\lambda (t0: -T).(sn3 c (THead (Flat Appl) x1 (THead (Bind Abst) t0 x4)))) (let H_x0 \def -(term_dec x x1) in (let H26 \def H_x0 in (or_ind (eq T x x1) ((eq T x x1) \to -(\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Bind Abst) t2 -x4))) (\lambda (H27: (eq T x x1)).(let H28 \def (eq_ind_r T x1 (\lambda (t0: -T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) (THead (Flat Appl) -t0 (THead (Bind Abst) t2 x4))) \to (\forall (P: Prop).P))) H24 x H27) in (let -H29 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14 x H27) in (eq_ind -T x (\lambda (t0: T).(sn3 c (THead (Flat Appl) t0 (THead (Bind Abst) t2 -x4)))) (let H_x1 \def (term_dec x0 x4) in (let H30 \def H_x1 in (or_ind (eq T -x0 x4) ((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x -(THead (Bind Abst) t2 x4))) (\lambda (H31: (eq T x0 x4)).(let H32 \def -(eq_ind_r T x4 (\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind -Abst) t2 x0)) (THead (Flat Appl) x (THead (Bind Abst) t2 t0))) \to (\forall -(P: Prop).P))) H28 x0 H31) in (let H33 \def (eq_ind_r T x4 (\lambda (t0: -T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 -H31) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Flat Appl) x (THead -(Bind Abst) t2 t0)))) (H32 (refl_equal T (THead (Flat Appl) x (THead (Bind -Abst) t2 x0))) (sn3 c (THead (Flat Appl) x (THead (Bind Abst) t2 x0)))) x4 -H31)))) (\lambda (H31: (((eq T x0 x4) \to (\forall (P: Prop).P)))).(H5 (THead -(Bind Abbr) x x4) (\lambda (H32: (eq T (THead (Bind Abbr) x x0) (THead (Bind -Abbr) x x4))).(\lambda (P: Prop).(let H33 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | -(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind -Abbr) x x0) (THead (Bind Abbr) x x4) H32) in (let H34 \def (eq_ind_r T x4 -(\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H31 x0 H33) in -(let H35 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: -T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H33) in (H34 (refl_equal T x0) -P)))))) (pr3_pr2 c (THead (Bind Abbr) x x0) (THead (Bind Abbr) x x4) -(pr2_head_2 c x x0 x4 (Bind Abbr) (H20 Abbr x))) x x4 (refl_equal T (THead -(Bind Abbr) x x4)) t2 (sn3_sing c t2 H7))) H30))) x1 H27)))) (\lambda (H27: -(((eq T x x1) \to (\forall (P: Prop).P)))).(H5 (THead (Bind Abbr) x1 x4) -(\lambda (H28: (eq T (THead (Bind Abbr) x x0) (THead (Bind Abbr) x1 -x4))).(\lambda (P: Prop).(let H29 \def (f_equal T T (\lambda (e: T).(match e -in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) -\Rightarrow x | (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abbr) x x0) -(THead (Bind Abbr) x1 x4) H28) in ((let H30 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | -(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind -Abbr) x x0) (THead (Bind Abbr) x1 x4) H28) in (\lambda (H31: (eq T x -x1)).(let H32 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall -(u: T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H30) in (let H33 \def -(eq_ind_r T x1 (\lambda (t0: T).((eq T x t0) \to (\forall (P0: Prop).P0))) -H27 x H31) in (let H34 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14 -x H31) in (H33 (refl_equal T x) P)))))) H29)))) (pr3_head_12 c x x1 (pr3_pr2 -c x x1 H14) (Bind Abbr) x0 x4 (pr3_pr2 (CHead c (Bind Abbr) x1) x0 x4 (H20 -Abbr x1))) x1 x4 (refl_equal T (THead (Bind Abbr) x1 x4)) t2 (sn3_sing c t2 -H7))) H26))) x3 H23)))) (\lambda (H23: (((eq T t2 x3) \to (\forall (P: -Prop).P)))).(let H_x0 \def (term_dec x x1) in (let H24 \def H_x0 in (or_ind -(eq T x x1) ((eq T x x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) -x1 (THead (Bind Abst) x3 x4))) (\lambda (H25: (eq T x x1)).(let H26 \def -(eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14 x H25) in (eq_ind T x -(\lambda (t0: T).(sn3 c (THead (Flat Appl) t0 (THead (Bind Abst) x3 x4)))) -(let H_x1 \def (term_dec x0 x4) in (let H27 \def H_x1 in (or_ind (eq T x0 x4) -((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x (THead -(Bind Abst) x3 x4))) (\lambda (H28: (eq T x0 x4)).(let H29 \def (eq_ind_r T -x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) -x0 t0)))) H20 x0 H28) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Flat -Appl) x (THead (Bind Abst) x3 t0)))) (H8 x3 H23 (pr3_pr2 c t2 x3 H19)) x4 -H28))) (\lambda (H28: (((eq T x0 x4) \to (\forall (P: Prop).P)))).(H5 (THead -(Bind Abbr) x x4) (\lambda (H29: (eq T (THead (Bind Abbr) x x0) (THead (Bind -Abbr) x x4))).(\lambda (P: Prop).(let H30 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | -(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind -Abbr) x x0) (THead (Bind Abbr) x x4) H29) in (let H31 \def (eq_ind_r T x4 -(\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H28 x0 H30) in -(let H32 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: -T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H30) in (H31 (refl_equal T x0) -P)))))) (pr3_pr2 c (THead (Bind Abbr) x x0) (THead (Bind Abbr) x x4) -(pr2_head_2 c x x0 x4 (Bind Abbr) (H20 Abbr x))) x x4 (refl_equal T (THead -(Bind Abbr) x x4)) x3 (H7 x3 H23 (pr3_pr2 c t2 x3 H19)))) H27))) x1 H25))) -(\lambda (H25: (((eq T x x1) \to (\forall (P: Prop).P)))).(H5 (THead (Bind -Abbr) x1 x4) (\lambda (H26: (eq T (THead (Bind Abbr) x x0) (THead (Bind Abbr) -x1 x4))).(\lambda (P: Prop).(let H27 \def (f_equal T T (\lambda (e: T).(match -e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) -\Rightarrow x | (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abbr) x x0) -(THead (Bind Abbr) x1 x4) H26) in ((let H28 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | -(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind -Abbr) x x0) (THead (Bind Abbr) x1 x4) H26) in (\lambda (H29: (eq T x -x1)).(let H30 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall -(u: T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H28) in (let H31 \def -(eq_ind_r T x1 (\lambda (t0: T).((eq T x t0) \to (\forall (P0: Prop).P0))) -H25 x H29) in (let H32 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14 -x H29) in (H31 (refl_equal T x) P)))))) H27)))) (pr3_head_12 c x x1 (pr3_pr2 -c x x1 H14) (Bind Abbr) x0 x4 (pr3_pr2 (CHead c (Bind Abbr) x1) x0 x4 (H20 -Abbr x1))) x1 x4 (refl_equal T (THead (Bind Abbr) x1 x4)) x3 (H7 x3 H23 -(pr3_pr2 c t2 x3 H19)))) H24)))) H22))) x2 H18))))))) H17)) t3 H13))))))) -H12)) (\lambda (H12: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) t2 x0) (THead -(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda -(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: -B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4))))))))).(ex4_4_ind T T T -T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind Abst) t2 x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind -Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t4: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) -z1 t4))))))) (sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: -T).(\lambda (x4: T).(\lambda (H13: (eq T (THead (Bind Abst) t2 x0) (THead -(Bind Abst) x1 x2))).(\lambda (H14: (eq T t3 (THead (Bind Abbr) x3 -x4))).(\lambda (H15: (pr2 c x x3)).(\lambda (H16: ((\forall (b: B).(\forall -(u: T).(pr2 (CHead c (Bind b) u) x2 x4))))).(let H17 \def (eq_ind T t3 -(\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) t0) -\to (\forall (P: Prop).P))) H9 (THead (Bind Abbr) x3 x4) H14) in (eq_ind_r T -(THead (Bind Abbr) x3 x4) (\lambda (t0: T).(sn3 c t0)) (let H18 \def (f_equal -T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow t2 | (TLRef _) \Rightarrow t2 | (THead _ t0 _) \Rightarrow t0])) -(THead (Bind Abst) t2 x0) (THead (Bind Abst) x1 x2) H13) in ((let H19 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ t0) -\Rightarrow t0])) (THead (Bind Abst) t2 x0) (THead (Bind Abst) x1 x2) H13) in -(\lambda (_: (eq T t2 x1)).(let H21 \def (eq_ind_r T x2 (\lambda (t0: -T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t0 x4)))) H16 x0 -H19) in (let H_x \def (term_dec x x3) in (let H22 \def H_x in (or_ind (eq T x -x3) ((eq T x x3) \to (\forall (P: Prop).P)) (sn3 c (THead (Bind Abbr) x3 x4)) -(\lambda (H23: (eq T x x3)).(let H24 \def (eq_ind_r T x3 (\lambda (t0: -T).(pr2 c x t0)) H15 x H23) in (eq_ind T x (\lambda (t0: T).(sn3 c (THead -(Bind Abbr) t0 x4))) (let H_x0 \def (term_dec x0 x4) in (let H25 \def H_x0 in -(or_ind (eq T x0 x4) ((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead -(Bind Abbr) x x4)) (\lambda (H26: (eq T x0 x4)).(let H27 \def (eq_ind_r T x4 -(\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x0 -t0)))) H21 x0 H26) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Bind Abbr) -x t0))) (sn3_sing c (THead (Bind Abbr) x x0) H6) x4 H26))) (\lambda (H26: -(((eq T x0 x4) \to (\forall (P: Prop).P)))).(H6 (THead (Bind Abbr) x x4) -(\lambda (H27: (eq T (THead (Bind Abbr) x x0) (THead (Bind Abbr) x -x4))).(\lambda (P: Prop).(let H28 \def (f_equal T T (\lambda (e: T).(match e -in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) -\Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind Abbr) x x0) -(THead (Bind Abbr) x x4) H27) in (let H29 \def (eq_ind_r T x4 (\lambda (t0: -T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H26 x0 H28) in (let H30 \def -(eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c -(Bind b) u) x0 t0)))) H21 x0 H28) in (H29 (refl_equal T x0) P)))))) (pr3_pr2 -c (THead (Bind Abbr) x x0) (THead (Bind Abbr) x x4) (pr2_head_2 c x x0 x4 -(Bind Abbr) (H21 Abbr x))))) H25))) x3 H23))) (\lambda (H23: (((eq T x x3) -\to (\forall (P: Prop).P)))).(H6 (THead (Bind Abbr) x3 x4) (\lambda (H24: (eq -T (THead (Bind Abbr) x x0) (THead (Bind Abbr) x3 x4))).(\lambda (P: -Prop).(let H25 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | -(THead _ t0 _) \Rightarrow t0])) (THead (Bind Abbr) x x0) (THead (Bind Abbr) -x3 x4) H24) in ((let H26 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) -\Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind Abbr) x x0) -(THead (Bind Abbr) x3 x4) H24) in (\lambda (H27: (eq T x x3)).(let H28 \def -(eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c -(Bind b) u) x0 t0)))) H21 x0 H26) in (let H29 \def (eq_ind_r T x3 (\lambda -(t0: T).((eq T x t0) \to (\forall (P0: Prop).P0))) H23 x H27) in (let H30 -\def (eq_ind_r T x3 (\lambda (t0: T).(pr2 c x t0)) H15 x H27) in (H29 -(refl_equal T x) P)))))) H25)))) (pr3_head_12 c x x3 (pr3_pr2 c x x3 H15) -(Bind Abbr) x0 x4 (pr3_pr2 (CHead c (Bind Abbr) x3) x0 x4 (H21 Abbr x3))))) -H22)))))) H18)) t3 H14)))))))))) H12)) (\lambda (H12: (ex6_6 B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind Abst) t2 x0) (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) -z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) -y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B -b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) t2 x0) (THead -(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) (sn3 c t3) -(\lambda (x1: B).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda -(x5: T).(\lambda (x6: T).(\lambda (H13: (not (eq B x1 Abst))).(\lambda (H14: -(eq T (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3))).(\lambda (H15: (eq -T t3 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))).(\lambda -(_: (pr2 c x x5)).(\lambda (H17: (pr2 c x2 x6)).(\lambda (H18: (pr2 (CHead c -(Bind x1) x6) x3 x4)).(let H19 \def (eq_ind T t3 (\lambda (t0: T).((eq T -(THead (Flat Appl) x (THead (Bind Abst) t2 x0)) t0) \to (\forall (P: -Prop).P))) H9 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)) -H15) in (eq_ind_r T (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) -x4)) (\lambda (t0: T).(sn3 c t0)) (let H20 \def (f_equal T B (\lambda (e: -T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow Abst | -(TLRef _) \Rightarrow Abst | (THead k _ _) \Rightarrow (match k in K return -(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow -Abst])])) (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3) H14) in ((let H21 -\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow t2 | (TLRef _) \Rightarrow t2 | (THead _ t0 _) -\Rightarrow t0])) (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3) H14) in -((let H22 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: -T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ -t0) \Rightarrow t0])) (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3) H14) -in (\lambda (H23: (eq T t2 x2)).(\lambda (H24: (eq B Abst x1)).(let H25 \def -(eq_ind_r T x3 (\lambda (t0: T).(pr2 (CHead c (Bind x1) x6) t0 x4)) H18 x0 -H22) in (let H26 \def (eq_ind_r T x2 (\lambda (t0: T).(pr2 c t0 x6)) H17 t2 -H23) in (let H27 \def (eq_ind_r B x1 (\lambda (b: B).(pr2 (CHead c (Bind b) -x6) x0 x4)) H25 Abst H24) in (let H28 \def (eq_ind_r B x1 (\lambda (b: -B).(not (eq B b Abst))) H13 Abst H24) in (eq_ind B Abst (\lambda (b: B).(sn3 -c (THead (Bind b) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))) (let H29 -\def (match (H28 (refl_equal B Abst)) in False return (\lambda (_: -False).(sn3 c (THead (Bind Abst) x6 (THead (Flat Appl) (lift (S O) O x5) -x4)))) with []) in H29) x1 H24)))))))) H21)) H20)) t3 H15)))))))))))))) H12)) -H11))))))))) w H4))))))))))) y H0))))) H)))). -(* COMMENTS -Initial nodes: 5699 -END *) - -theorem sn3_appl_lref: - \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (v: -T).((sn3 c v) \to (sn3 c (THead (Flat Appl) v (TLRef i))))))) -\def - \lambda (c: C).(\lambda (i: nat).(\lambda (H: (nf2 c (TLRef i))).(\lambda -(v: T).(\lambda (H0: (sn3 c v)).(sn3_ind c (\lambda (t: T).(sn3 c (THead -(Flat Appl) t (TLRef i)))) (\lambda (t1: T).(\lambda (_: ((\forall (t2: -T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c -t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: -Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c (THead (Flat Appl) t2 (TLRef -i)))))))).(sn3_pr2_intro c (THead (Flat Appl) t1 (TLRef i)) (\lambda (t2: -T).(\lambda (H3: (((eq T (THead (Flat Appl) t1 (TLRef i)) t2) \to (\forall -(P: Prop).P)))).(\lambda (H4: (pr2 c (THead (Flat Appl) t1 (TLRef i)) -t2)).(let H5 \def (pr2_gen_appl c t1 (TLRef i) t2 H4) in (or3_ind (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2))) (\lambda (_: T).(\lambda -(t3: T).(pr2 c (TLRef i) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i) (THead (Bind Abst) y1 -z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: -T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2))))) (\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall -(u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T -(TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T -t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c t1 u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 -y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) -(sn3 c t2) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T -t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c t1 -u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (TLRef i) t3))))).(ex3_2_ind T -T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2))) (\lambda (_: T).(\lambda -(t3: T).(pr2 c (TLRef i) t3))) (sn3 c t2) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H7: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H8: (pr2 c t1 -x0)).(\lambda (H9: (pr2 c (TLRef i) x1)).(let H10 \def (eq_ind T t2 (\lambda -(t: T).((eq T (THead (Flat Appl) t1 (TLRef i)) t) \to (\forall (P: Prop).P))) -H3 (THead (Flat Appl) x0 x1) H7) in (eq_ind_r T (THead (Flat Appl) x0 x1) -(\lambda (t: T).(sn3 c t)) (let H11 \def (eq_ind_r T x1 (\lambda (t: T).((eq -T (THead (Flat Appl) t1 (TLRef i)) (THead (Flat Appl) x0 t)) \to (\forall (P: -Prop).P))) H10 (TLRef i) (H x1 H9)) in (let H12 \def (eq_ind_r T x1 (\lambda -(t: T).(pr2 c (TLRef i) t)) H9 (TLRef i) (H x1 H9)) in (eq_ind T (TLRef i) -(\lambda (t: T).(sn3 c (THead (Flat Appl) x0 t))) (let H_x \def (term_dec t1 -x0) in (let H13 \def H_x in (or_ind (eq T t1 x0) ((eq T t1 x0) \to (\forall -(P: Prop).P)) (sn3 c (THead (Flat Appl) x0 (TLRef i))) (\lambda (H14: (eq T -t1 x0)).(let H15 \def (eq_ind_r T x0 (\lambda (t: T).((eq T (THead (Flat -Appl) t1 (TLRef i)) (THead (Flat Appl) t (TLRef i))) \to (\forall (P: -Prop).P))) H11 t1 H14) in (let H16 \def (eq_ind_r T x0 (\lambda (t: T).(pr2 c -t1 t)) H8 t1 H14) in (eq_ind T t1 (\lambda (t: T).(sn3 c (THead (Flat Appl) t -(TLRef i)))) (H15 (refl_equal T (THead (Flat Appl) t1 (TLRef i))) (sn3 c -(THead (Flat Appl) t1 (TLRef i)))) x0 H14)))) (\lambda (H14: (((eq T t1 x0) -\to (\forall (P: Prop).P)))).(H2 x0 H14 (pr3_pr2 c t1 x0 H8))) H13))) x1 (H -x1 H9)))) t2 H7))))))) H6)) (\lambda (H6: (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i) (THead -(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall -(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3))))))))).(ex4_4_ind T -T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T -(TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 -u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: -T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3))))))) -(sn3 c t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: -T).(\lambda (H7: (eq T (TLRef i) (THead (Bind Abst) x0 x1))).(\lambda (H8: -(eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (_: (pr2 c t1 x2)).(\lambda (_: -((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 x3))))).(let -H11 \def (eq_ind T t2 (\lambda (t: T).((eq T (THead (Flat Appl) t1 (TLRef i)) -t) \to (\forall (P: Prop).P))) H3 (THead (Bind Abbr) x2 x3) H8) in (eq_ind_r -T (THead (Bind Abbr) x2 x3) (\lambda (t: T).(sn3 c t)) (let H12 \def (eq_ind -T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) -\Rightarrow False])) I (THead (Bind Abst) x0 x1) H7) in (False_ind (sn3 c -(THead (Bind Abbr) x2 x3)) H12)) t2 H8)))))))))) H6)) (\lambda (H6: (ex6_6 B -T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T (TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda -(_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq -T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c t1 u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 -y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 -z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i) (THead (Bind b) y1 -z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: -T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat -Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 -(CHead c (Bind b) y2) z1 z2))))))) (sn3 c t2) (\lambda (x0: B).(\lambda (x1: -T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: -T).(\lambda (_: (not (eq B x0 Abst))).(\lambda (H8: (eq T (TLRef i) (THead -(Bind x0) x1 x2))).(\lambda (H9: (eq T t2 (THead (Bind x0) x5 (THead (Flat -Appl) (lift (S O) O x4) x3)))).(\lambda (_: (pr2 c t1 x4)).(\lambda (_: (pr2 -c x1 x5)).(\lambda (_: (pr2 (CHead c (Bind x0) x5) x2 x3)).(let H13 \def -(eq_ind T t2 (\lambda (t: T).((eq T (THead (Flat Appl) t1 (TLRef i)) t) \to -(\forall (P: Prop).P))) H3 (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) -O x4) x3)) H9) in (eq_ind_r T (THead (Bind x0) x5 (THead (Flat Appl) (lift (S -O) O x4) x3)) (\lambda (t: T).(sn3 c t)) (let H14 \def (eq_ind T (TLRef i) -(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow -False])) I (THead (Bind x0) x1 x2) H8) in (False_ind (sn3 c (THead (Bind x0) -x5 (THead (Flat Appl) (lift (S O) O x4) x3))) H14)) t2 H9)))))))))))))) H6)) -H5))))))))) v H0))))). -(* COMMENTS -Initial nodes: 2125 -END *) - -theorem sn3_appl_abbr: - \forall (c: C).(\forall (d: C).(\forall (w: T).(\forall (i: nat).((getl i c -(CHead d (Bind Abbr) w)) \to (\forall (v: T).((sn3 c (THead (Flat Appl) v -(lift (S i) O w))) \to (sn3 c (THead (Flat Appl) v (TLRef i))))))))) -\def - \lambda (c: C).(\lambda (d: C).(\lambda (w: T).(\lambda (i: nat).(\lambda -(H: (getl i c (CHead d (Bind Abbr) w))).(\lambda (v: T).(\lambda (H0: (sn3 c -(THead (Flat Appl) v (lift (S i) O w)))).(insert_eq T (THead (Flat Appl) v -(lift (S i) O w)) (\lambda (t: T).(sn3 c t)) (\lambda (_: T).(sn3 c (THead -(Flat Appl) v (TLRef i)))) (\lambda (y: T).(\lambda (H1: (sn3 c y)).(unintro -T v (\lambda (t: T).((eq T y (THead (Flat Appl) t (lift (S i) O w))) \to (sn3 -c (THead (Flat Appl) t (TLRef i))))) (sn3_ind c (\lambda (t: T).(\forall (x: -T).((eq T t (THead (Flat Appl) x (lift (S i) O w))) \to (sn3 c (THead (Flat -Appl) x (TLRef i)))))) (\lambda (t1: T).(\lambda (H2: ((\forall (t2: -T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c -t2)))))).(\lambda (H3: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: -Prop).P))) \to ((pr3 c t1 t2) \to (\forall (x: T).((eq T t2 (THead (Flat -Appl) x (lift (S i) O w))) \to (sn3 c (THead (Flat Appl) x (TLRef -i)))))))))).(\lambda (x: T).(\lambda (H4: (eq T t1 (THead (Flat Appl) x (lift -(S i) O w)))).(let H5 \def (eq_ind T t1 (\lambda (t: T).(\forall (t2: -T).((((eq T t t2) \to (\forall (P: Prop).P))) \to ((pr3 c t t2) \to (\forall -(x0: T).((eq T t2 (THead (Flat Appl) x0 (lift (S i) O w))) \to (sn3 c (THead -(Flat Appl) x0 (TLRef i))))))))) H3 (THead (Flat Appl) x (lift (S i) O w)) -H4) in (let H6 \def (eq_ind T t1 (\lambda (t: T).(\forall (t2: T).((((eq T t -t2) \to (\forall (P: Prop).P))) \to ((pr3 c t t2) \to (sn3 c t2))))) H2 -(THead (Flat Appl) x (lift (S i) O w)) H4) in (sn3_pr2_intro c (THead (Flat -Appl) x (TLRef i)) (\lambda (t2: T).(\lambda (H7: (((eq T (THead (Flat Appl) -x (TLRef i)) t2) \to (\forall (P: Prop).P)))).(\lambda (H8: (pr2 c (THead -(Flat Appl) x (TLRef i)) t2)).(let H9 \def (pr2_gen_appl c x (TLRef i) t2 H8) -in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead -(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) -(\lambda (_: T).(\lambda (t3: T).(pr2 c (TLRef i) t3)))) (ex4_4 T T T T -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T -(TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x -u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: -T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) -(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T (TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda -(_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq -T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 -y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) -(sn3 c t2) (\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T -t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x -u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (TLRef i) t3))))).(ex3_2_ind T -T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t3: -T).(pr2 c (TLRef i) t3))) (sn3 c t2) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H11: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H12: (pr2 c -x x0)).(\lambda (H13: (pr2 c (TLRef i) x1)).(let H14 \def (eq_ind T t2 -(\lambda (t: T).((eq T (THead (Flat Appl) x (TLRef i)) t) \to (\forall (P: -Prop).P))) H7 (THead (Flat Appl) x0 x1) H11) in (eq_ind_r T (THead (Flat -Appl) x0 x1) (\lambda (t: T).(sn3 c t)) (let H15 \def (pr2_gen_lref c x1 i -H13) in (or_ind (eq T x1 (TLRef i)) (ex2_2 C T (\lambda (d0: C).(\lambda (u: -T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq -T x1 (lift (S i) O u))))) (sn3 c (THead (Flat Appl) x0 x1)) (\lambda (H16: -(eq T x1 (TLRef i))).(let H17 \def (eq_ind T x1 (\lambda (t: T).((eq T (THead -(Flat Appl) x (TLRef i)) (THead (Flat Appl) x0 t)) \to (\forall (P: -Prop).P))) H14 (TLRef i) H16) in (eq_ind_r T (TLRef i) (\lambda (t: T).(sn3 c -(THead (Flat Appl) x0 t))) (let H_x \def (term_dec x x0) in (let H18 \def H_x -in (or_ind (eq T x x0) ((eq T x x0) \to (\forall (P: Prop).P)) (sn3 c (THead -(Flat Appl) x0 (TLRef i))) (\lambda (H19: (eq T x x0)).(let H20 \def -(eq_ind_r T x0 (\lambda (t: T).((eq T (THead (Flat Appl) x (TLRef i)) (THead -(Flat Appl) t (TLRef i))) \to (\forall (P: Prop).P))) H17 x H19) in (let H21 -\def (eq_ind_r T x0 (\lambda (t: T).(pr2 c x t)) H12 x H19) in (eq_ind T x -(\lambda (t: T).(sn3 c (THead (Flat Appl) t (TLRef i)))) (H20 (refl_equal T -(THead (Flat Appl) x (TLRef i))) (sn3 c (THead (Flat Appl) x (TLRef i)))) x0 -H19)))) (\lambda (H19: (((eq T x x0) \to (\forall (P: Prop).P)))).(H5 (THead -(Flat Appl) x0 (lift (S i) O w)) (\lambda (H20: (eq T (THead (Flat Appl) x -(lift (S i) O w)) (THead (Flat Appl) x0 (lift (S i) O w)))).(\lambda (P: -Prop).(let H21 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | -(THead _ t _) \Rightarrow t])) (THead (Flat Appl) x (lift (S i) O w)) (THead -(Flat Appl) x0 (lift (S i) O w)) H20) in (let H22 \def (eq_ind_r T x0 -(\lambda (t: T).((eq T x t) \to (\forall (P0: Prop).P0))) H19 x H21) in (let -H23 \def (eq_ind_r T x0 (\lambda (t: T).(pr2 c x t)) H12 x H21) in (H22 -(refl_equal T x) P)))))) (pr3_pr2 c (THead (Flat Appl) x (lift (S i) O w)) -(THead (Flat Appl) x0 (lift (S i) O w)) (pr2_head_1 c x x0 H12 (Flat Appl) -(lift (S i) O w))) x0 (refl_equal T (THead (Flat Appl) x0 (lift (S i) O -w))))) H18))) x1 H16))) (\lambda (H16: (ex2_2 C T (\lambda (d0: C).(\lambda -(u: T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: -T).(eq T x1 (lift (S i) O u)))))).(ex2_2_ind C T (\lambda (d0: C).(\lambda -(u: T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: -T).(eq T x1 (lift (S i) O u)))) (sn3 c (THead (Flat Appl) x0 x1)) (\lambda -(x2: C).(\lambda (x3: T).(\lambda (H17: (getl i c (CHead x2 (Bind Abbr) -x3))).(\lambda (H18: (eq T x1 (lift (S i) O x3))).(let H19 \def (eq_ind T x1 -(\lambda (t: T).((eq T (THead (Flat Appl) x (TLRef i)) (THead (Flat Appl) x0 -t)) \to (\forall (P: Prop).P))) H14 (lift (S i) O x3) H18) in (eq_ind_r T -(lift (S i) O x3) (\lambda (t: T).(sn3 c (THead (Flat Appl) x0 t))) (let H20 -\def (eq_ind C (CHead d (Bind Abbr) w) (\lambda (c0: C).(getl i c c0)) H -(CHead x2 (Bind Abbr) x3) (getl_mono c (CHead d (Bind Abbr) w) i H (CHead x2 -(Bind Abbr) x3) H17)) in (let H21 \def (f_equal C C (\lambda (e: C).(match e -in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) -\Rightarrow c0])) (CHead d (Bind Abbr) w) (CHead x2 (Bind Abbr) x3) -(getl_mono c (CHead d (Bind Abbr) w) i H (CHead x2 (Bind Abbr) x3) H17)) in -((let H22 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: -C).T) with [(CSort _) \Rightarrow w | (CHead _ _ t) \Rightarrow t])) (CHead d -(Bind Abbr) w) (CHead x2 (Bind Abbr) x3) (getl_mono c (CHead d (Bind Abbr) w) -i H (CHead x2 (Bind Abbr) x3) H17)) in (\lambda (H23: (eq C d x2)).(let H24 -\def (eq_ind_r T x3 (\lambda (t: T).(getl i c (CHead x2 (Bind Abbr) t))) H20 -w H22) in (eq_ind T w (\lambda (t: T).(sn3 c (THead (Flat Appl) x0 (lift (S -i) O t)))) (let H25 \def (eq_ind_r C x2 (\lambda (c0: C).(getl i c (CHead c0 -(Bind Abbr) w))) H24 d H23) in (let H_x \def (term_dec x x0) in (let H26 \def -H_x in (or_ind (eq T x x0) ((eq T x x0) \to (\forall (P: Prop).P)) (sn3 c -(THead (Flat Appl) x0 (lift (S i) O w))) (\lambda (H27: (eq T x x0)).(let H28 -\def (eq_ind_r T x0 (\lambda (t: T).(pr2 c x t)) H12 x H27) in (eq_ind T x -(\lambda (t: T).(sn3 c (THead (Flat Appl) t (lift (S i) O w)))) (sn3_sing c -(THead (Flat Appl) x (lift (S i) O w)) H6) x0 H27))) (\lambda (H27: (((eq T x -x0) \to (\forall (P: Prop).P)))).(H6 (THead (Flat Appl) x0 (lift (S i) O w)) -(\lambda (H28: (eq T (THead (Flat Appl) x (lift (S i) O w)) (THead (Flat -Appl) x0 (lift (S i) O w)))).(\lambda (P: Prop).(let H29 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t _) \Rightarrow t])) -(THead (Flat Appl) x (lift (S i) O w)) (THead (Flat Appl) x0 (lift (S i) O -w)) H28) in (let H30 \def (eq_ind_r T x0 (\lambda (t: T).((eq T x t) \to -(\forall (P0: Prop).P0))) H27 x H29) in (let H31 \def (eq_ind_r T x0 (\lambda -(t: T).(pr2 c x t)) H12 x H29) in (H30 (refl_equal T x) P)))))) (pr3_pr2 c -(THead (Flat Appl) x (lift (S i) O w)) (THead (Flat Appl) x0 (lift (S i) O -w)) (pr2_head_1 c x x0 H12 (Flat Appl) (lift (S i) O w))))) H26)))) x3 -H22)))) H21))) x1 H18)))))) H16)) H15)) t2 H11))))))) H10)) (\lambda (H10: -(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(eq T (TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda -(_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 -t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: -T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 -t3))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T (TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda -(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead -(Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind -b) u) z1 t3))))))) (sn3 c t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: -T).(\lambda (x3: T).(\lambda (H11: (eq T (TLRef i) (THead (Bind Abst) x0 -x1))).(\lambda (H12: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (_: (pr2 c -x x2)).(\lambda (_: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) -u) x1 x3))))).(let H15 \def (eq_ind T t2 (\lambda (t: T).((eq T (THead (Flat -Appl) x (TLRef i)) t) \to (\forall (P: Prop).P))) H7 (THead (Bind Abbr) x2 -x3) H12) in (eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda (t: T).(sn3 c t)) -(let H16 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) x0 -x1) H11) in (False_ind (sn3 c (THead (Bind Abbr) x2 x3)) H16)) t2 -H12)))))))))) H10)) (\lambda (H10: (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i) -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind -B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T (TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda -(_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq -T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 -y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) -(sn3 c t2) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: -T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (not (eq B x0 -Abst))).(\lambda (H12: (eq T (TLRef i) (THead (Bind x0) x1 x2))).(\lambda -(H13: (eq T t2 (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) -x3)))).(\lambda (_: (pr2 c x x4)).(\lambda (_: (pr2 c x1 x5)).(\lambda (_: -(pr2 (CHead c (Bind x0) x5) x2 x3)).(let H17 \def (eq_ind T t2 (\lambda (t: -T).((eq T (THead (Flat Appl) x (TLRef i)) t) \to (\forall (P: Prop).P))) H7 -(THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) H13) in -(eq_ind_r T (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) -(\lambda (t: T).(sn3 c t)) (let H18 \def (eq_ind T (TLRef i) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I -(THead (Bind x0) x1 x2) H12) in (False_ind (sn3 c (THead (Bind x0) x5 (THead -(Flat Appl) (lift (S O) O x4) x3))) H18)) t2 H13)))))))))))))) H10)) -H9))))))))))))) y H1)))) H0))))))). -(* COMMENTS -Initial nodes: 3727 -END *) - -theorem sn3_appl_cast: - \forall (c: C).(\forall (v: T).(\forall (u: T).((sn3 c (THead (Flat Appl) v -u)) \to (\forall (t: T).((sn3 c (THead (Flat Appl) v t)) \to (sn3 c (THead -(Flat Appl) v (THead (Flat Cast) u t)))))))) -\def - \lambda (c: C).(\lambda (v: T).(\lambda (u: T).(\lambda (H: (sn3 c (THead -(Flat Appl) v u))).(insert_eq T (THead (Flat Appl) v u) (\lambda (t: T).(sn3 -c t)) (\lambda (_: T).(\forall (t0: T).((sn3 c (THead (Flat Appl) v t0)) \to -(sn3 c (THead (Flat Appl) v (THead (Flat Cast) u t0)))))) (\lambda (y: -T).(\lambda (H0: (sn3 c y)).(unintro T u (\lambda (t: T).((eq T y (THead -(Flat Appl) v t)) \to (\forall (t0: T).((sn3 c (THead (Flat Appl) v t0)) \to -(sn3 c (THead (Flat Appl) v (THead (Flat Cast) t t0))))))) (unintro T v -(\lambda (t: T).(\forall (x: T).((eq T y (THead (Flat Appl) t x)) \to -(\forall (t0: T).((sn3 c (THead (Flat Appl) t t0)) \to (sn3 c (THead (Flat -Appl) t (THead (Flat Cast) x t0)))))))) (sn3_ind c (\lambda (t: T).(\forall -(x: T).(\forall (x0: T).((eq T t (THead (Flat Appl) x x0)) \to (\forall (t0: -T).((sn3 c (THead (Flat Appl) x t0)) \to (sn3 c (THead (Flat Appl) x (THead -(Flat Cast) x0 t0))))))))) (\lambda (t1: T).(\lambda (H1: ((\forall (t2: -T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c -t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: -Prop).P))) \to ((pr3 c t1 t2) \to (\forall (x: T).(\forall (x0: T).((eq T t2 -(THead (Flat Appl) x x0)) \to (\forall (t: T).((sn3 c (THead (Flat Appl) x -t)) \to (sn3 c (THead (Flat Appl) x (THead (Flat Cast) x0 -t))))))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: (eq T t1 (THead -(Flat Appl) x x0))).(\lambda (t: T).(\lambda (H4: (sn3 c (THead (Flat Appl) x -t))).(insert_eq T (THead (Flat Appl) x t) (\lambda (t0: T).(sn3 c t0)) -(\lambda (_: T).(sn3 c (THead (Flat Appl) x (THead (Flat Cast) x0 t)))) -(\lambda (y0: T).(\lambda (H5: (sn3 c y0)).(unintro T t (\lambda (t0: T).((eq -T y0 (THead (Flat Appl) x t0)) \to (sn3 c (THead (Flat Appl) x (THead (Flat -Cast) x0 t0))))) (sn3_ind c (\lambda (t0: T).(\forall (x1: T).((eq T t0 -(THead (Flat Appl) x x1)) \to (sn3 c (THead (Flat Appl) x (THead (Flat Cast) -x0 x1)))))) (\lambda (t0: T).(\lambda (H6: ((\forall (t2: T).((((eq T t0 t2) -\to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (sn3 c t2)))))).(\lambda -(H7: ((\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 -c t0 t2) \to (\forall (x1: T).((eq T t2 (THead (Flat Appl) x x1)) \to (sn3 c -(THead (Flat Appl) x (THead (Flat Cast) x0 x1)))))))))).(\lambda (x1: -T).(\lambda (H8: (eq T t0 (THead (Flat Appl) x x1))).(let H9 \def (eq_ind T -t0 (\lambda (t2: T).(\forall (t3: T).((((eq T t2 t3) \to (\forall (P: -Prop).P))) \to ((pr3 c t2 t3) \to (\forall (x2: T).((eq T t3 (THead (Flat -Appl) x x2)) \to (sn3 c (THead (Flat Appl) x (THead (Flat Cast) x0 -x2))))))))) H7 (THead (Flat Appl) x x1) H8) in (let H10 \def (eq_ind T t0 -(\lambda (t2: T).(\forall (t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) -\to ((pr3 c t2 t3) \to (sn3 c t3))))) H6 (THead (Flat Appl) x x1) H8) in (let -H11 \def (eq_ind T t1 (\lambda (t2: T).(\forall (t3: T).((((eq T t2 t3) \to -(\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to (\forall (x2: T).(\forall (x3: -T).((eq T t3 (THead (Flat Appl) x2 x3)) \to (\forall (t4: T).((sn3 c (THead -(Flat Appl) x2 t4)) \to (sn3 c (THead (Flat Appl) x2 (THead (Flat Cast) x3 -t4)))))))))))) H2 (THead (Flat Appl) x x0) H3) in (let H12 \def (eq_ind T t1 -(\lambda (t2: T).(\forall (t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) -\to ((pr3 c t2 t3) \to (sn3 c t3))))) H1 (THead (Flat Appl) x x0) H3) in -(sn3_pr2_intro c (THead (Flat Appl) x (THead (Flat Cast) x0 x1)) (\lambda -(t2: T).(\lambda (H13: (((eq T (THead (Flat Appl) x (THead (Flat Cast) x0 -x1)) t2) \to (\forall (P: Prop).P)))).(\lambda (H14: (pr2 c (THead (Flat -Appl) x (THead (Flat Cast) x0 x1)) t2)).(let H15 \def (pr2_gen_appl c x -(THead (Flat Cast) x0 x1) t2 H14) in (or3_ind (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c -(THead (Flat Cast) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Cast) x0 x1) -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda -(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: -B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T -T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq -T (THead (Flat Cast) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) -z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) -y2) z1 z2)))))))) (sn3 c t2) (\lambda (H16: (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c -(THead (Flat Cast) x0 x1) t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda -(t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Flat Cast) -x0 x1) t3))) (sn3 c t2) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H17: (eq -T t2 (THead (Flat Appl) x2 x3))).(\lambda (H18: (pr2 c x x2)).(\lambda (H19: -(pr2 c (THead (Flat Cast) x0 x1) x3)).(let H20 \def (eq_ind T t2 (\lambda -(t3: T).((eq T (THead (Flat Appl) x (THead (Flat Cast) x0 x1)) t3) \to -(\forall (P: Prop).P))) H13 (THead (Flat Appl) x2 x3) H17) in (eq_ind_r T -(THead (Flat Appl) x2 x3) (\lambda (t3: T).(sn3 c t3)) (let H21 \def -(pr2_gen_cast c x0 x1 x3 H19) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda -(t3: T).(eq T x3 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c x1 t3)))) (pr2 c -x1 x3) (sn3 c (THead (Flat Appl) x2 x3)) (\lambda (H22: (ex3_2 T T (\lambda -(u2: T).(\lambda (t3: T).(eq T x3 (THead (Flat Cast) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c x1 -t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x3 (THead -(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) -(\lambda (_: T).(\lambda (t3: T).(pr2 c x1 t3))) (sn3 c (THead (Flat Appl) x2 -x3)) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H23: (eq T x3 (THead (Flat -Cast) x4 x5))).(\lambda (H24: (pr2 c x0 x4)).(\lambda (H25: (pr2 c x1 -x5)).(let H26 \def (eq_ind T x3 (\lambda (t3: T).((eq T (THead (Flat Appl) x -(THead (Flat Cast) x0 x1)) (THead (Flat Appl) x2 t3)) \to (\forall (P: -Prop).P))) H20 (THead (Flat Cast) x4 x5) H23) in (eq_ind_r T (THead (Flat -Cast) x4 x5) (\lambda (t3: T).(sn3 c (THead (Flat Appl) x2 t3))) (let H_x -\def (term_dec (THead (Flat Appl) x x0) (THead (Flat Appl) x2 x4)) in (let -H27 \def H_x in (or_ind (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x2 -x4)) ((eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x2 x4)) \to (\forall -(P: Prop).P)) (sn3 c (THead (Flat Appl) x2 (THead (Flat Cast) x4 x5))) -(\lambda (H28: (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x2 -x4))).(let H29 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | -(THead _ t3 _) \Rightarrow t3])) (THead (Flat Appl) x x0) (THead (Flat Appl) -x2 x4) H28) in ((let H30 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) -\Rightarrow x0 | (THead _ _ t3) \Rightarrow t3])) (THead (Flat Appl) x x0) -(THead (Flat Appl) x2 x4) H28) in (\lambda (H31: (eq T x x2)).(let H32 \def -(eq_ind_r T x4 (\lambda (t3: T).((eq T (THead (Flat Appl) x (THead (Flat -Cast) x0 x1)) (THead (Flat Appl) x2 (THead (Flat Cast) t3 x5))) \to (\forall -(P: Prop).P))) H26 x0 H30) in (let H33 \def (eq_ind_r T x4 (\lambda (t3: -T).(pr2 c x0 t3)) H24 x0 H30) in (eq_ind T x0 (\lambda (t3: T).(sn3 c (THead -(Flat Appl) x2 (THead (Flat Cast) t3 x5)))) (let H34 \def (eq_ind_r T x2 -(\lambda (t3: T).((eq T (THead (Flat Appl) x (THead (Flat Cast) x0 x1)) -(THead (Flat Appl) t3 (THead (Flat Cast) x0 x5))) \to (\forall (P: Prop).P))) -H32 x H31) in (let H35 \def (eq_ind_r T x2 (\lambda (t3: T).(pr2 c x t3)) H18 -x H31) in (eq_ind T x (\lambda (t3: T).(sn3 c (THead (Flat Appl) t3 (THead -(Flat Cast) x0 x5)))) (let H_x0 \def (term_dec (THead (Flat Appl) x x1) -(THead (Flat Appl) x x5)) in (let H36 \def H_x0 in (or_ind (eq T (THead (Flat -Appl) x x1) (THead (Flat Appl) x x5)) ((eq T (THead (Flat Appl) x x1) (THead -(Flat Appl) x x5)) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x -(THead (Flat Cast) x0 x5))) (\lambda (H37: (eq T (THead (Flat Appl) x x1) -(THead (Flat Appl) x x5))).(let H38 \def (f_equal T T (\lambda (e: T).(match -e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x1 | (TLRef _) -\Rightarrow x1 | (THead _ _ t3) \Rightarrow t3])) (THead (Flat Appl) x x1) -(THead (Flat Appl) x x5) H37) in (let H39 \def (eq_ind_r T x5 (\lambda (t3: -T).((eq T (THead (Flat Appl) x (THead (Flat Cast) x0 x1)) (THead (Flat Appl) -x (THead (Flat Cast) x0 t3))) \to (\forall (P: Prop).P))) H34 x1 H38) in (let -H40 \def (eq_ind_r T x5 (\lambda (t3: T).(pr2 c x1 t3)) H25 x1 H38) in -(eq_ind T x1 (\lambda (t3: T).(sn3 c (THead (Flat Appl) x (THead (Flat Cast) -x0 t3)))) (H39 (refl_equal T (THead (Flat Appl) x (THead (Flat Cast) x0 x1))) -(sn3 c (THead (Flat Appl) x (THead (Flat Cast) x0 x1)))) x5 H38))))) (\lambda -(H37: (((eq T (THead (Flat Appl) x x1) (THead (Flat Appl) x x5)) \to (\forall -(P: Prop).P)))).(H9 (THead (Flat Appl) x x5) H37 (pr3_pr2 c (THead (Flat -Appl) x x1) (THead (Flat Appl) x x5) (pr2_thin_dx c x1 x5 H25 x Appl)) x5 -(refl_equal T (THead (Flat Appl) x x5)))) H36))) x2 H31))) x4 H30))))) H29))) -(\lambda (H28: (((eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x2 x4)) -\to (\forall (P: Prop).P)))).(let H_x0 \def (term_dec (THead (Flat Appl) x -x1) (THead (Flat Appl) x2 x5)) in (let H29 \def H_x0 in (or_ind (eq T (THead -(Flat Appl) x x1) (THead (Flat Appl) x2 x5)) ((eq T (THead (Flat Appl) x x1) -(THead (Flat Appl) x2 x5)) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat -Appl) x2 (THead (Flat Cast) x4 x5))) (\lambda (H30: (eq T (THead (Flat Appl) -x x1) (THead (Flat Appl) x2 x5))).(let H31 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x | -(TLRef _) \Rightarrow x | (THead _ t3 _) \Rightarrow t3])) (THead (Flat Appl) -x x1) (THead (Flat Appl) x2 x5) H30) in ((let H32 \def (f_equal T T (\lambda -(e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x1 -| (TLRef _) \Rightarrow x1 | (THead _ _ t3) \Rightarrow t3])) (THead (Flat -Appl) x x1) (THead (Flat Appl) x2 x5) H30) in (\lambda (H33: (eq T x -x2)).(let H34 \def (eq_ind_r T x5 (\lambda (t3: T).(pr2 c x1 t3)) H25 x1 H32) -in (eq_ind T x1 (\lambda (t3: T).(sn3 c (THead (Flat Appl) x2 (THead (Flat -Cast) x4 t3)))) (let H35 \def (eq_ind_r T x2 (\lambda (t3: T).((eq T (THead -(Flat Appl) x x0) (THead (Flat Appl) t3 x4)) \to (\forall (P: Prop).P))) H28 -x H33) in (let H36 \def (eq_ind_r T x2 (\lambda (t3: T).(pr2 c x t3)) H18 x -H33) in (eq_ind T x (\lambda (t3: T).(sn3 c (THead (Flat Appl) t3 (THead -(Flat Cast) x4 x1)))) (H11 (THead (Flat Appl) x x4) H35 (pr3_pr2 c (THead -(Flat Appl) x x0) (THead (Flat Appl) x x4) (pr2_thin_dx c x0 x4 H24 x Appl)) -x x4 (refl_equal T (THead (Flat Appl) x x4)) x1 (sn3_sing c (THead (Flat -Appl) x x1) H10)) x2 H33))) x5 H32)))) H31))) (\lambda (H30: (((eq T (THead -(Flat Appl) x x1) (THead (Flat Appl) x2 x5)) \to (\forall (P: -Prop).P)))).(H11 (THead (Flat Appl) x2 x4) H28 (pr3_flat c x x2 (pr3_pr2 c x -x2 H18) x0 x4 (pr3_pr2 c x0 x4 H24) Appl) x2 x4 (refl_equal T (THead (Flat -Appl) x2 x4)) x5 (H10 (THead (Flat Appl) x2 x5) H30 (pr3_flat c x x2 (pr3_pr2 -c x x2 H18) x1 x5 (pr3_pr2 c x1 x5 H25) Appl)))) H29)))) H27))) x3 H23))))))) -H22)) (\lambda (H22: (pr2 c x1 x3)).(let H_x \def (term_dec (THead (Flat -Appl) x x1) (THead (Flat Appl) x2 x3)) in (let H23 \def H_x in (or_ind (eq T -(THead (Flat Appl) x x1) (THead (Flat Appl) x2 x3)) ((eq T (THead (Flat Appl) -x x1) (THead (Flat Appl) x2 x3)) \to (\forall (P: Prop).P)) (sn3 c (THead -(Flat Appl) x2 x3)) (\lambda (H24: (eq T (THead (Flat Appl) x x1) (THead -(Flat Appl) x2 x3))).(let H25 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) -\Rightarrow x | (THead _ t3 _) \Rightarrow t3])) (THead (Flat Appl) x x1) -(THead (Flat Appl) x2 x3) H24) in ((let H26 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x1 | -(TLRef _) \Rightarrow x1 | (THead _ _ t3) \Rightarrow t3])) (THead (Flat -Appl) x x1) (THead (Flat Appl) x2 x3) H24) in (\lambda (H27: (eq T x -x2)).(let H28 \def (eq_ind_r T x3 (\lambda (t3: T).(pr2 c x1 t3)) H22 x1 H26) -in (let H29 \def (eq_ind_r T x3 (\lambda (t3: T).((eq T (THead (Flat Appl) x -(THead (Flat Cast) x0 x1)) (THead (Flat Appl) x2 t3)) \to (\forall (P: -Prop).P))) H20 x1 H26) in (eq_ind T x1 (\lambda (t3: T).(sn3 c (THead (Flat -Appl) x2 t3))) (let H30 \def (eq_ind_r T x2 (\lambda (t3: T).((eq T (THead -(Flat Appl) x (THead (Flat Cast) x0 x1)) (THead (Flat Appl) t3 x1)) \to -(\forall (P: Prop).P))) H29 x H27) in (let H31 \def (eq_ind_r T x2 (\lambda -(t3: T).(pr2 c x t3)) H18 x H27) in (eq_ind T x (\lambda (t3: T).(sn3 c -(THead (Flat Appl) t3 x1))) (sn3_sing c (THead (Flat Appl) x x1) H10) x2 -H27))) x3 H26))))) H25))) (\lambda (H24: (((eq T (THead (Flat Appl) x x1) -(THead (Flat Appl) x2 x3)) \to (\forall (P: Prop).P)))).(H10 (THead (Flat -Appl) x2 x3) H24 (pr3_flat c x x2 (pr3_pr2 c x x2 H18) x1 x3 (pr3_pr2 c x1 x3 -H22) Appl))) H23)))) H21)) t2 H17))))))) H16)) (\lambda (H16: (ex4_4 T T T T -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Flat Cast) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) -u0) z1 t3))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Cast) x0 x1) (THead -(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda -(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: -B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3))))))) (sn3 c t2) -(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda -(H17: (eq T (THead (Flat Cast) x0 x1) (THead (Bind Abst) x2 x3))).(\lambda -(H18: (eq T t2 (THead (Bind Abbr) x4 x5))).(\lambda (_: (pr2 c x -x4)).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) -u0) x3 x5))))).(let H21 \def (eq_ind T t2 (\lambda (t3: T).((eq T (THead -(Flat Appl) x (THead (Flat Cast) x0 x1)) t3) \to (\forall (P: Prop).P))) H13 -(THead (Bind Abbr) x4 x5) H18) in (eq_ind_r T (THead (Bind Abbr) x4 x5) -(\lambda (t3: T).(sn3 c t3)) (let H22 \def (eq_ind T (THead (Flat Cast) x0 -x1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort -_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) -\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) x2 -x3) H17) in (False_ind (sn3 c (THead (Bind Abbr) x4 x5)) H22)) t2 -H18)))))))))) H16)) (\lambda (H16: (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat -Cast) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T -t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 -y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 -z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Cast) x0 x1) (THead -(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) (sn3 c t2) -(\lambda (x2: B).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda -(x6: T).(\lambda (x7: T).(\lambda (_: (not (eq B x2 Abst))).(\lambda (H18: -(eq T (THead (Flat Cast) x0 x1) (THead (Bind x2) x3 x4))).(\lambda (H19: (eq -T t2 (THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) O x6) x5)))).(\lambda -(_: (pr2 c x x6)).(\lambda (_: (pr2 c x3 x7)).(\lambda (_: (pr2 (CHead c -(Bind x2) x7) x4 x5)).(let H23 \def (eq_ind T t2 (\lambda (t3: T).((eq T -(THead (Flat Appl) x (THead (Flat Cast) x0 x1)) t3) \to (\forall (P: -Prop).P))) H13 (THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) O x6) x5)) -H19) in (eq_ind_r T (THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) O x6) -x5)) (\lambda (t3: T).(sn3 c t3)) (let H24 \def (eq_ind T (THead (Flat Cast) -x0 x1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) -\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind x2) x3 x4) -H18) in (False_ind (sn3 c (THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) -O x6) x5))) H24)) t2 H19)))))))))))))) H16)) H15))))))))))))))) y0 H5)))) -H4))))))))) y H0))))) H)))). -(* COMMENTS -Initial nodes: 5149 -END *) - -theorem sn3_appl_bind: - \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (u: -T).((sn3 c u) \to (\forall (t: T).(\forall (v: T).((sn3 (CHead c (Bind b) u) -(THead (Flat Appl) (lift (S O) O v) t)) \to (sn3 c (THead (Flat Appl) v -(THead (Bind b) u t)))))))))) -\def - \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (c: C).(\lambda -(u: T).(\lambda (H0: (sn3 c u)).(sn3_ind c (\lambda (t: T).(\forall (t0: -T).(\forall (v: T).((sn3 (CHead c (Bind b) t) (THead (Flat Appl) (lift (S O) -O v) t0)) \to (sn3 c (THead (Flat Appl) v (THead (Bind b) t t0))))))) -(\lambda (t1: T).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall -(P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H2: ((\forall -(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to -(\forall (t: T).(\forall (v: T).((sn3 (CHead c (Bind b) t2) (THead (Flat -Appl) (lift (S O) O v) t)) \to (sn3 c (THead (Flat Appl) v (THead (Bind b) t2 -t))))))))))).(\lambda (t: T).(\lambda (v: T).(\lambda (H3: (sn3 (CHead c -(Bind b) t1) (THead (Flat Appl) (lift (S O) O v) t))).(insert_eq T (THead -(Flat Appl) (lift (S O) O v) t) (\lambda (t0: T).(sn3 (CHead c (Bind b) t1) -t0)) (\lambda (_: T).(sn3 c (THead (Flat Appl) v (THead (Bind b) t1 t)))) -(\lambda (y: T).(\lambda (H4: (sn3 (CHead c (Bind b) t1) y)).(unintro T t -(\lambda (t0: T).((eq T y (THead (Flat Appl) (lift (S O) O v) t0)) \to (sn3 c -(THead (Flat Appl) v (THead (Bind b) t1 t0))))) (unintro T v (\lambda (t0: -T).(\forall (x: T).((eq T y (THead (Flat Appl) (lift (S O) O t0) x)) \to (sn3 -c (THead (Flat Appl) t0 (THead (Bind b) t1 x)))))) (sn3_ind (CHead c (Bind b) -t1) (\lambda (t0: T).(\forall (x: T).(\forall (x0: T).((eq T t0 (THead (Flat -Appl) (lift (S O) O x) x0)) \to (sn3 c (THead (Flat Appl) x (THead (Bind b) -t1 x0))))))) (\lambda (t2: T).(\lambda (H5: ((\forall (t3: T).((((eq T t2 t3) -\to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to (sn3 -(CHead c (Bind b) t1) t3)))))).(\lambda (H6: ((\forall (t3: T).((((eq T t2 -t3) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to -(\forall (x: T).(\forall (x0: T).((eq T t3 (THead (Flat Appl) (lift (S O) O -x) x0)) \to (sn3 c (THead (Flat Appl) x (THead (Bind b) t1 -x0))))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H7: (eq T t2 (THead -(Flat Appl) (lift (S O) O x) x0))).(let H8 \def (eq_ind T t2 (\lambda (t0: -T).(\forall (t3: T).((((eq T t0 t3) \to (\forall (P: Prop).P))) \to ((pr3 -(CHead c (Bind b) t1) t0 t3) \to (\forall (x1: T).(\forall (x2: T).((eq T t3 -(THead (Flat Appl) (lift (S O) O x1) x2)) \to (sn3 c (THead (Flat Appl) x1 -(THead (Bind b) t1 x2)))))))))) H6 (THead (Flat Appl) (lift (S O) O x) x0) -H7) in (let H9 \def (eq_ind T t2 (\lambda (t0: T).(\forall (t3: T).((((eq T -t0 t3) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t0 t3) \to -(sn3 (CHead c (Bind b) t1) t3))))) H5 (THead (Flat Appl) (lift (S O) O x) x0) -H7) in (sn3_pr2_intro c (THead (Flat Appl) x (THead (Bind b) t1 x0)) (\lambda -(t3: T).(\lambda (H10: (((eq T (THead (Flat Appl) x (THead (Bind b) t1 x0)) -t3) \to (\forall (P: Prop).P)))).(\lambda (H11: (pr2 c (THead (Flat Appl) x -(THead (Bind b) t1 x0)) t3)).(let H12 \def (pr2_gen_appl c x (THead (Bind b) -t1 x0) t3 H11) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T -t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x -u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind b) t1 x0) t4)))) -(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(eq T (THead (Bind b) t1 x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind -Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t4: T).(\forall (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind b0) -u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B -b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) t1 x0) (THead -(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind -b0) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) -(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b0) y2) z1 z2)))))))) (sn3 c t3) -(\lambda (H13: (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead -(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) -(\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind b) t1 x0) -t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead -(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) -(\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind b) t1 x0) t4))) (sn3 c -t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (H14: (eq T t3 (THead (Flat -Appl) x1 x2))).(\lambda (H15: (pr2 c x x1)).(\lambda (H16: (pr2 c (THead -(Bind b) t1 x0) x2)).(let H17 \def (eq_ind T t3 (\lambda (t0: T).((eq T -(THead (Flat Appl) x (THead (Bind b) t1 x0)) t0) \to (\forall (P: Prop).P))) -H10 (THead (Flat Appl) x1 x2) H14) in (eq_ind_r T (THead (Flat Appl) x1 x2) -(\lambda (t0: T).(sn3 c t0)) (let H_x \def (pr3_gen_bind b H c t1 x0 x2) in -(let H18 \def (H_x (pr3_pr2 c (THead (Bind b) t1 x0) x2 H16)) in (or_ind -(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead (Bind b) u2 -t4)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_: -T).(\lambda (t4: T).(pr3 (CHead c (Bind b) t1) x0 t4)))) (pr3 (CHead c (Bind -b) t1) x0 (lift (S O) O x2)) (sn3 c (THead (Flat Appl) x1 x2)) (\lambda (H19: -(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead (Bind b) u2 -t4)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_: -T).(\lambda (t4: T).(pr3 (CHead c (Bind b) t1) x0 t4))))).(ex3_2_ind T T -(\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead (Bind b) u2 t4)))) (\lambda -(u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr3 -(CHead c (Bind b) t1) x0 t4))) (sn3 c (THead (Flat Appl) x1 x2)) (\lambda -(x3: T).(\lambda (x4: T).(\lambda (H20: (eq T x2 (THead (Bind b) x3 -x4))).(\lambda (H21: (pr3 c t1 x3)).(\lambda (H22: (pr3 (CHead c (Bind b) t1) -x0 x4)).(let H23 \def (eq_ind T x2 (\lambda (t0: T).((eq T (THead (Flat Appl) -x (THead (Bind b) t1 x0)) (THead (Flat Appl) x1 t0)) \to (\forall (P: -Prop).P))) H17 (THead (Bind b) x3 x4) H20) in (eq_ind_r T (THead (Bind b) x3 -x4) (\lambda (t0: T).(sn3 c (THead (Flat Appl) x1 t0))) (let H_x0 \def -(term_dec t1 x3) in (let H24 \def H_x0 in (or_ind (eq T t1 x3) ((eq T t1 x3) -\to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Bind b) x3 -x4))) (\lambda (H25: (eq T t1 x3)).(let H26 \def (eq_ind_r T x3 (\lambda (t0: -T).((eq T (THead (Flat Appl) x (THead (Bind b) t1 x0)) (THead (Flat Appl) x1 -(THead (Bind b) t0 x4))) \to (\forall (P: Prop).P))) H23 t1 H25) in (let H27 -\def (eq_ind_r T x3 (\lambda (t0: T).(pr3 c t1 t0)) H21 t1 H25) in (eq_ind T -t1 (\lambda (t0: T).(sn3 c (THead (Flat Appl) x1 (THead (Bind b) t0 x4)))) -(let H_x1 \def (term_dec x0 x4) in (let H28 \def H_x1 in (or_ind (eq T x0 x4) -((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead -(Bind b) t1 x4))) (\lambda (H29: (eq T x0 x4)).(let H30 \def (eq_ind_r T x4 -(\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind b) t1 x0)) (THead -(Flat Appl) x1 (THead (Bind b) t1 t0))) \to (\forall (P: Prop).P))) H26 x0 -H29) in (let H31 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 (CHead c (Bind b) -t1) x0 t0)) H22 x0 H29) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Flat -Appl) x1 (THead (Bind b) t1 t0)))) (let H_x2 \def (term_dec x x1) in (let H32 -\def H_x2 in (or_ind (eq T x x1) ((eq T x x1) \to (\forall (P: Prop).P)) (sn3 -c (THead (Flat Appl) x1 (THead (Bind b) t1 x0))) (\lambda (H33: (eq T x -x1)).(let H34 \def (eq_ind_r T x1 (\lambda (t0: T).((eq T (THead (Flat Appl) -x (THead (Bind b) t1 x0)) (THead (Flat Appl) t0 (THead (Bind b) t1 x0))) \to -(\forall (P: Prop).P))) H30 x H33) in (let H35 \def (eq_ind_r T x1 (\lambda -(t0: T).(pr2 c x t0)) H15 x H33) in (eq_ind T x (\lambda (t0: T).(sn3 c -(THead (Flat Appl) t0 (THead (Bind b) t1 x0)))) (H34 (refl_equal T (THead -(Flat Appl) x (THead (Bind b) t1 x0))) (sn3 c (THead (Flat Appl) x (THead -(Bind b) t1 x0)))) x1 H33)))) (\lambda (H33: (((eq T x x1) \to (\forall (P: -Prop).P)))).(H8 (THead (Flat Appl) (lift (S O) O x1) x0) (\lambda (H34: (eq T -(THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) -x0))).(\lambda (P: Prop).(let H35 \def (f_equal T T (\lambda (e: T).(match e -in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map -(f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n) -\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with -[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4) -\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in -lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow -((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match -t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef -(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | -(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) -t4))]) in lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _) -\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) -(lift (S O) O x1) x0) H34) in (let H36 \def (eq_ind_r T x1 (\lambda (t0: -T).((eq T x t0) \to (\forall (P0: Prop).P0))) H33 x (lift_inj x x1 (S O) O -H35)) in (let H37 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x -(lift_inj x x1 (S O) O H35)) in (H36 (refl_equal T x) P)))))) (pr3_flat -(CHead c (Bind b) t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c -(Bind b) t1) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1 -(pr3_pr2 c x x1 H15)) x0 x0 (pr3_refl (CHead c (Bind b) t1) x0) Appl) x1 x0 -(refl_equal T (THead (Flat Appl) (lift (S O) O x1) x0)))) H32))) x4 H29)))) -(\lambda (H29: (((eq T x0 x4) \to (\forall (P: Prop).P)))).(H8 (THead (Flat -Appl) (lift (S O) O x1) x4) (\lambda (H30: (eq T (THead (Flat Appl) (lift (S -O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) x4))).(\lambda (P: -Prop).(let H31 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat -\to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n) -\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with -[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4) -\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in -lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow -((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match -t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef -(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | -(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) -t4))]) in lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _) -\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) -(lift (S O) O x1) x4) H30) in ((let H32 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | -(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat -Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) x4) H30) in -(\lambda (H33: (eq T (lift (S O) O x) (lift (S O) O x1))).(let H34 \def -(eq_ind_r T x4 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) -H29 x0 H32) in (let H35 \def (eq_ind_r T x4 (\lambda (t0: T).((eq T (THead -(Flat Appl) x (THead (Bind b) t1 x0)) (THead (Flat Appl) x1 (THead (Bind b) -t1 t0))) \to (\forall (P0: Prop).P0))) H26 x0 H32) in (let H36 \def (eq_ind_r -T x4 (\lambda (t0: T).(pr3 (CHead c (Bind b) t1) x0 t0)) H22 x0 H32) in (let -H37 \def (eq_ind_r T x1 (\lambda (t0: T).((eq T (THead (Flat Appl) x (THead -(Bind b) t1 x0)) (THead (Flat Appl) t0 (THead (Bind b) t1 x0))) \to (\forall -(P0: Prop).P0))) H35 x (lift_inj x x1 (S O) O H33)) in (let H38 \def -(eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x (lift_inj x x1 (S O) O -H33)) in (H34 (refl_equal T x0) P)))))))) H31)))) (pr3_flat (CHead c (Bind b) -t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c (Bind b) t1) c (S -O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1 (pr3_pr2 c x x1 H15)) -x0 x4 H22 Appl) x1 x4 (refl_equal T (THead (Flat Appl) (lift (S O) O x1) -x4)))) H28))) x3 H25)))) (\lambda (H25: (((eq T t1 x3) \to (\forall (P: -Prop).P)))).(H2 x3 H25 H21 x4 x1 (sn3_cpr3_trans c t1 x3 H21 (Bind b) (THead -(Flat Appl) (lift (S O) O x1) x4) (let H_x1 \def (term_dec x0 x4) in (let H26 -\def H_x1 in (or_ind (eq T x0 x4) ((eq T x0 x4) \to (\forall (P: Prop).P)) -(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x1) x4)) (\lambda -(H27: (eq T x0 x4)).(let H28 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 (CHead -c (Bind b) t1) x0 t0)) H22 x0 H27) in (eq_ind T x0 (\lambda (t0: T).(sn3 -(CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x1) t0))) (let H_x2 -\def (term_dec x x1) in (let H29 \def H_x2 in (or_ind (eq T x x1) ((eq T x -x1) \to (\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat Appl) -(lift (S O) O x1) x0)) (\lambda (H30: (eq T x x1)).(let H31 \def (eq_ind_r T -x1 (\lambda (t0: T).(pr2 c x t0)) H15 x H30) in (eq_ind T x (\lambda (t0: -T).(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O t0) x0))) -(sn3_sing (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x) x0) H9) -x1 H30))) (\lambda (H30: (((eq T x x1) \to (\forall (P: Prop).P)))).(H9 -(THead (Flat Appl) (lift (S O) O x1) x0) (\lambda (H31: (eq T (THead (Flat -Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) -x0))).(\lambda (P: Prop).(let H32 \def (f_equal T T (\lambda (e: T).(match e -in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map -(f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n) -\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with -[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4) -\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in -lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow -((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match -t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef -(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | -(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) -t4))]) in lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _) -\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) -(lift (S O) O x1) x0) H31) in (let H33 \def (eq_ind_r T x1 (\lambda (t0: -T).((eq T x t0) \to (\forall (P0: Prop).P0))) H30 x (lift_inj x x1 (S O) O -H32)) in (let H34 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x -(lift_inj x x1 (S O) O H32)) in (H33 (refl_equal T x) P)))))) (pr3_flat -(CHead c (Bind b) t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c -(Bind b) t1) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1 -(pr3_pr2 c x x1 H15)) x0 x0 (pr3_refl (CHead c (Bind b) t1) x0) Appl))) -H29))) x4 H27))) (\lambda (H27: (((eq T x0 x4) \to (\forall (P: -Prop).P)))).(H9 (THead (Flat Appl) (lift (S O) O x1) x4) (\lambda (H28: (eq T -(THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) -x4))).(\lambda (P: Prop).(let H29 \def (f_equal T T (\lambda (e: T).(match e -in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map -(f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n) -\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with -[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4) -\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in -lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow -((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match -t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef -(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | -(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) -t4))]) in lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _) -\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) -(lift (S O) O x1) x4) H28) in ((let H30 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | -(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat -Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) x4) H28) in -(\lambda (H31: (eq T (lift (S O) O x) (lift (S O) O x1))).(let H32 \def -(eq_ind_r T x4 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) -H27 x0 H30) in (let H33 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 (CHead c -(Bind b) t1) x0 t0)) H22 x0 H30) in (let H34 \def (eq_ind_r T x1 (\lambda -(t0: T).(pr2 c x t0)) H15 x (lift_inj x x1 (S O) O H31)) in (H32 (refl_equal -T x0) P)))))) H29)))) (pr3_flat (CHead c (Bind b) t1) (lift (S O) O x) (lift -(S O) O x1) (pr3_lift (CHead c (Bind b) t1) c (S O) O (drop_drop (Bind b) O c -c (drop_refl c) t1) x x1 (pr3_pr2 c x x1 H15)) x0 x4 H22 Appl))) H26)))))) -H24))) x2 H20))))))) H19)) (\lambda (H19: (pr3 (CHead c (Bind b) t1) x0 (lift -(S O) O x2))).(sn3_gen_lift (CHead c (Bind b) t1) (THead (Flat Appl) x1 x2) -(S O) O (eq_ind_r T (THead (Flat Appl) (lift (S O) O x1) (lift (S O) (s (Flat -Appl) O) x2)) (\lambda (t0: T).(sn3 (CHead c (Bind b) t1) t0)) (sn3_pr3_trans -(CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x1) x0) (let H_x0 \def -(term_dec x x1) in (let H20 \def H_x0 in (or_ind (eq T x x1) ((eq T x x1) \to -(\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S -O) O x1) x0)) (\lambda (H21: (eq T x x1)).(let H22 \def (eq_ind_r T x1 -(\lambda (t0: T).(pr2 c x t0)) H15 x H21) in (eq_ind T x (\lambda (t0: -T).(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O t0) x0))) -(sn3_sing (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x) x0) H9) -x1 H21))) (\lambda (H21: (((eq T x x1) \to (\forall (P: Prop).P)))).(H9 -(THead (Flat Appl) (lift (S O) O x1) x0) (\lambda (H22: (eq T (THead (Flat -Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) -x0))).(\lambda (P: Prop).(let H23 \def (f_equal T T (\lambda (e: T).(match e -in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map -(f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n) -\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with -[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4) -\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in -lref_map) (\lambda (x3: nat).(plus x3 (S O))) O x) | (TLRef _) \Rightarrow -((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match -t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef -(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | -(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) -t4))]) in lref_map) (\lambda (x3: nat).(plus x3 (S O))) O x) | (THead _ t0 _) -\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) -(lift (S O) O x1) x0) H22) in (let H24 \def (eq_ind_r T x1 (\lambda (t0: -T).((eq T x t0) \to (\forall (P0: Prop).P0))) H21 x (lift_inj x x1 (S O) O -H23)) in (let H25 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x -(lift_inj x x1 (S O) O H23)) in (H24 (refl_equal T x) P)))))) (pr3_flat -(CHead c (Bind b) t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c -(Bind b) t1) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1 -(pr3_pr2 c x x1 H15)) x0 x0 (pr3_refl (CHead c (Bind b) t1) x0) Appl))) -H20))) (THead (Flat Appl) (lift (S O) O x1) (lift (S O) O x2)) (pr3_thin_dx -(CHead c (Bind b) t1) x0 (lift (S O) O x2) H19 (lift (S O) O x1) Appl)) (lift -(S O) O (THead (Flat Appl) x1 x2)) (lift_head (Flat Appl) x1 x2 (S O) O)) c -(drop_drop (Bind b) O c c (drop_refl c) t1))) H18))) t3 H14))))))) H13)) -(\lambda (H13: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T (THead (Bind b) t1 x0) (THead (Bind Abst) y1 -z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: -T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b0: B).(\forall (u0: -T).(pr2 (CHead c (Bind b0) u0) z1 t4))))))))).(ex4_4_ind T T T T (\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind -b) t1 x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x -u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: -T).(\forall (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind b0) u0) z1 t4))))))) -(sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: -T).(\lambda (H14: (eq T (THead (Bind b) t1 x0) (THead (Bind Abst) x1 -x2))).(\lambda (H15: (eq T t3 (THead (Bind Abbr) x3 x4))).(\lambda (_: (pr2 c -x x3)).(\lambda (H17: ((\forall (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind -b0) u0) x2 x4))))).(let H18 \def (eq_ind T t3 (\lambda (t0: T).((eq T (THead -(Flat Appl) x (THead (Bind b) t1 x0)) t0) \to (\forall (P: Prop).P))) H10 -(THead (Bind Abbr) x3 x4) H15) in (eq_ind_r T (THead (Bind Abbr) x3 x4) -(\lambda (t0: T).(sn3 c t0)) (let H19 \def (f_equal T B (\lambda (e: -T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | -(TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k in K return -(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow -b])])) (THead (Bind b) t1 x0) (THead (Bind Abst) x1 x2) H14) in ((let H20 -\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ t0 _) -\Rightarrow t0])) (THead (Bind b) t1 x0) (THead (Bind Abst) x1 x2) H14) in -((let H21 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: -T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ -t0) \Rightarrow t0])) (THead (Bind b) t1 x0) (THead (Bind Abst) x1 x2) H14) -in (\lambda (_: (eq T t1 x1)).(\lambda (H23: (eq B b Abst)).(let H24 \def -(eq_ind_r T x2 (\lambda (t0: T).(\forall (b0: B).(\forall (u0: T).(pr2 (CHead -c (Bind b0) u0) t0 x4)))) H17 x0 H21) in (let H25 \def (eq_ind B b (\lambda -(b0: B).((eq T (THead (Flat Appl) x (THead (Bind b0) t1 x0)) (THead (Bind -Abbr) x3 x4)) \to (\forall (P: Prop).P))) H18 Abst H23) in (let H26 \def -(eq_ind B b (\lambda (b0: B).(\forall (t4: T).((((eq T (THead (Flat Appl) -(lift (S O) O x) x0) t4) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind -b0) t1) (THead (Flat Appl) (lift (S O) O x) x0) t4) \to (sn3 (CHead c (Bind -b0) t1) t4))))) H9 Abst H23) in (let H27 \def (eq_ind B b (\lambda (b0: -B).(\forall (t4: T).((((eq T (THead (Flat Appl) (lift (S O) O x) x0) t4) \to -(\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b0) t1) (THead (Flat Appl) -(lift (S O) O x) x0) t4) \to (\forall (x5: T).(\forall (x6: T).((eq T t4 -(THead (Flat Appl) (lift (S O) O x5) x6)) \to (sn3 c (THead (Flat Appl) x5 -(THead (Bind b0) t1 x6)))))))))) H8 Abst H23) in (let H28 \def (eq_ind B b -(\lambda (b0: B).(\forall (t4: T).((((eq T t1 t4) \to (\forall (P: Prop).P))) -\to ((pr3 c t1 t4) \to (\forall (t0: T).(\forall (v0: T).((sn3 (CHead c (Bind -b0) t4) (THead (Flat Appl) (lift (S O) O v0) t0)) \to (sn3 c (THead (Flat -Appl) v0 (THead (Bind b0) t4 t0)))))))))) H2 Abst H23) in (let H29 \def -(eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H Abst H23) in (let H30 -\def (match (H29 (refl_equal B Abst)) in False return (\lambda (_: -False).(sn3 c (THead (Bind Abbr) x3 x4))) with []) in H30)))))))))) H20)) -H19)) t3 H15)))))))))) H13)) (\lambda (H13: (ex6_6 B T T T T T (\lambda (b0: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) -t1 x0) (THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T -t3 (THead (Bind b0) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 -y2))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b0) y2) z1 -z2))))))))).(ex6_6_ind B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 -Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) t1 x0) (THead (Bind -b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind b0) y2 (THead -(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x -u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b0: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c (Bind b0) y2) z1 z2))))))) (sn3 c t3) (\lambda (x1: -B).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: -T).(\lambda (x6: T).(\lambda (_: (not (eq B x1 Abst))).(\lambda (H15: (eq T -(THead (Bind b) t1 x0) (THead (Bind x1) x2 x3))).(\lambda (H16: (eq T t3 -(THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))).(\lambda -(H17: (pr2 c x x5)).(\lambda (H18: (pr2 c x2 x6)).(\lambda (H19: (pr2 (CHead -c (Bind x1) x6) x3 x4)).(let H20 \def (eq_ind T t3 (\lambda (t0: T).((eq T -(THead (Flat Appl) x (THead (Bind b) t1 x0)) t0) \to (\forall (P: Prop).P))) -H10 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)) H16) in -(eq_ind_r T (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)) -(\lambda (t0: T).(sn3 c t0)) (let H21 \def (f_equal T B (\lambda (e: -T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | -(TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k in K return -(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow -b])])) (THead (Bind b) t1 x0) (THead (Bind x1) x2 x3) H15) in ((let H22 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ t0 _) -\Rightarrow t0])) (THead (Bind b) t1 x0) (THead (Bind x1) x2 x3) H15) in -((let H23 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: -T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ -t0) \Rightarrow t0])) (THead (Bind b) t1 x0) (THead (Bind x1) x2 x3) H15) in -(\lambda (H24: (eq T t1 x2)).(\lambda (H25: (eq B b x1)).(let H26 \def -(eq_ind_r T x3 (\lambda (t0: T).(pr2 (CHead c (Bind x1) x6) t0 x4)) H19 x0 -H23) in (let H27 \def (eq_ind_r T x2 (\lambda (t0: T).(pr2 c t0 x6)) H18 t1 -H24) in (let H28 \def (eq_ind_r B x1 (\lambda (b0: B).(pr2 (CHead c (Bind b0) -x6) x0 x4)) H26 b H25) in (eq_ind B b (\lambda (b0: B).(sn3 c (THead (Bind -b0) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))) (sn3_pr3_trans c (THead -(Bind b) t1 (THead (Flat Appl) (lift (S O) O x5) x4)) (sn3_bind b c t1 -(sn3_sing c t1 H1) (THead (Flat Appl) (lift (S O) O x5) x4) (let H_x \def -(term_dec x x5) in (let H29 \def H_x in (or_ind (eq T x x5) ((eq T x x5) \to -(\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S -O) O x5) x4)) (\lambda (H30: (eq T x x5)).(let H31 \def (eq_ind_r T x5 -(\lambda (t0: T).(pr2 c x t0)) H17 x H30) in (eq_ind T x (\lambda (t0: -T).(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O t0) x4))) (let -H_x0 \def (term_dec x0 x4) in (let H32 \def H_x0 in (or_ind (eq T x0 x4) ((eq -T x0 x4) \to (\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat -Appl) (lift (S O) O x) x4)) (\lambda (H33: (eq T x0 x4)).(let H34 \def -(eq_ind_r T x4 (\lambda (t0: T).(pr2 (CHead c (Bind b) x6) x0 t0)) H28 x0 -H33) in (eq_ind T x0 (\lambda (t0: T).(sn3 (CHead c (Bind b) t1) (THead (Flat -Appl) (lift (S O) O x) t0))) (sn3_sing (CHead c (Bind b) t1) (THead (Flat -Appl) (lift (S O) O x) x0) H9) x4 H33))) (\lambda (H33: (((eq T x0 x4) \to -(\forall (P: Prop).P)))).(H9 (THead (Flat Appl) (lift (S O) O x) x4) (\lambda -(H34: (eq T (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift -(S O) O x) x4))).(\lambda (P: Prop).(let H35 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | -(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat -Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x) x4) H34) in -(let H36 \def (eq_ind_r T x4 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: -Prop).P0))) H33 x0 H35) in (let H37 \def (eq_ind_r T x4 (\lambda (t0: T).(pr2 -(CHead c (Bind b) x6) x0 t0)) H28 x0 H35) in (H36 (refl_equal T x0) P)))))) -(pr3_pr3_pr3_t c t1 x6 (pr3_pr2 c t1 x6 H27) (THead (Flat Appl) (lift (S O) O -x) x0) (THead (Flat Appl) (lift (S O) O x) x4) (Bind b) (pr3_pr2 (CHead c -(Bind b) x6) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift -(S O) O x) x4) (pr2_thin_dx (CHead c (Bind b) x6) x0 x4 H28 (lift (S O) O x) -Appl))))) H32))) x5 H30))) (\lambda (H30: (((eq T x x5) \to (\forall (P: -Prop).P)))).(H9 (THead (Flat Appl) (lift (S O) O x5) x4) (\lambda (H31: (eq T -(THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x5) -x4))).(\lambda (P: Prop).(let H32 \def (f_equal T T (\lambda (e: T).(match e -in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map -(f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n) -\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with -[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4) -\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in -lref_map) (\lambda (x7: nat).(plus x7 (S O))) O x) | (TLRef _) \Rightarrow -((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match -t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef -(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | -(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) -t4))]) in lref_map) (\lambda (x7: nat).(plus x7 (S O))) O x) | (THead _ t0 _) -\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) -(lift (S O) O x5) x4) H31) in ((let H33 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | -(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat -Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x5) x4) H31) in -(\lambda (H34: (eq T (lift (S O) O x) (lift (S O) O x5))).(let H35 \def -(eq_ind_r T x5 (\lambda (t0: T).((eq T x t0) \to (\forall (P0: Prop).P0))) -H30 x (lift_inj x x5 (S O) O H34)) in (let H36 \def (eq_ind_r T x5 (\lambda -(t0: T).(pr2 c x t0)) H17 x (lift_inj x x5 (S O) O H34)) in (let H37 \def -(eq_ind_r T x4 (\lambda (t0: T).(pr2 (CHead c (Bind b) x6) x0 t0)) H28 x0 -H33) in (H35 (refl_equal T x) P)))))) H32)))) (pr3_pr3_pr3_t c t1 x6 (pr3_pr2 -c t1 x6 H27) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift -(S O) O x5) x4) (Bind b) (pr3_flat (CHead c (Bind b) x6) (lift (S O) O x) -(lift (S O) O x5) (pr3_lift (CHead c (Bind b) x6) c (S O) O (drop_drop (Bind -b) O c c (drop_refl c) x6) x x5 (pr3_pr2 c x x5 H17)) x0 x4 (pr3_pr2 (CHead c -(Bind b) x6) x0 x4 H28) Appl)))) H29)))) (THead (Bind b) x6 (THead (Flat -Appl) (lift (S O) O x5) x4)) (pr3_pr2 c (THead (Bind b) t1 (THead (Flat Appl) -(lift (S O) O x5) x4)) (THead (Bind b) x6 (THead (Flat Appl) (lift (S O) O -x5) x4)) (pr2_head_1 c t1 x6 H27 (Bind b) (THead (Flat Appl) (lift (S O) O -x5) x4)))) x1 H25))))))) H22)) H21)) t3 H16)))))))))))))) H13)) -H12)))))))))))))) y H4))))) H3))))))) u H0))))). -(* COMMENTS -Initial nodes: 9191 -END *) - -theorem sn3_appl_appl: - \forall (v1: T).(\forall (t1: T).(let u1 \def (THead (Flat Appl) v1 t1) in -(\forall (c: C).((sn3 c u1) \to (\forall (v2: T).((sn3 c v2) \to (((\forall -(u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to -(sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c (THead (Flat Appl) v2 -u1))))))))) -\def - \lambda (v1: T).(\lambda (t1: T).(let u1 \def (THead (Flat Appl) v1 t1) in -(\lambda (c: C).(\lambda (H: (sn3 c (THead (Flat Appl) v1 t1))).(insert_eq T -(THead (Flat Appl) v1 t1) (\lambda (t: T).(sn3 c t)) (\lambda (t: T).(\forall -(v2: T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c t u2) \to ((((iso t u2) -\to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to -(sn3 c (THead (Flat Appl) v2 t)))))) (\lambda (y: T).(\lambda (H0: (sn3 c -y)).(unintro T t1 (\lambda (t: T).((eq T y (THead (Flat Appl) v1 t)) \to -(\forall (v2: T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c y u2) \to ((((iso -y u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) -\to (sn3 c (THead (Flat Appl) v2 y))))))) (unintro T v1 (\lambda (t: -T).(\forall (x: T).((eq T y (THead (Flat Appl) t x)) \to (\forall (v2: -T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c y u2) \to ((((iso y u2) \to -(\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c -(THead (Flat Appl) v2 y)))))))) (sn3_ind c (\lambda (t: T).(\forall (x: -T).(\forall (x0: T).((eq T t (THead (Flat Appl) x x0)) \to (\forall (v2: -T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c t u2) \to ((((iso t u2) \to -(\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c -(THead (Flat Appl) v2 t))))))))) (\lambda (t2: T).(\lambda (H1: ((\forall -(t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to -(sn3 c t3)))))).(\lambda (H2: ((\forall (t3: T).((((eq T t2 t3) \to (\forall -(P: Prop).P))) \to ((pr3 c t2 t3) \to (\forall (x: T).(\forall (x0: T).((eq T -t3 (THead (Flat Appl) x x0)) \to (\forall (v2: T).((sn3 c v2) \to (((\forall -(u2: T).((pr3 c t3 u2) \to ((((iso t3 u2) \to (\forall (P: Prop).P))) \to -(sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c (THead (Flat Appl) v2 -t3))))))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: (eq T t2 -(THead (Flat Appl) x x0))).(\lambda (v2: T).(\lambda (H4: (sn3 c -v2)).(sn3_ind c (\lambda (t: T).(((\forall (u2: T).((pr3 c t2 u2) \to ((((iso -t2 u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) t u2)))))) -\to (sn3 c (THead (Flat Appl) t t2)))) (\lambda (t0: T).(\lambda (H5: -((\forall (t3: T).((((eq T t0 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t0 -t3) \to (sn3 c t3)))))).(\lambda (H6: ((\forall (t3: T).((((eq T t0 t3) \to -(\forall (P: Prop).P))) \to ((pr3 c t0 t3) \to (((\forall (u2: T).((pr3 c t2 -u2) \to ((((iso t2 u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat -Appl) t3 u2)))))) \to (sn3 c (THead (Flat Appl) t3 t2)))))))).(\lambda (H7: -((\forall (u2: T).((pr3 c t2 u2) \to ((((iso t2 u2) \to (\forall (P: -Prop).P))) \to (sn3 c (THead (Flat Appl) t0 u2))))))).(let H8 \def (eq_ind T -t2 (\lambda (t: T).(\forall (u2: T).((pr3 c t u2) \to ((((iso t u2) \to -(\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) t0 u2)))))) H7 (THead -(Flat Appl) x x0) H3) in (let H9 \def (eq_ind T t2 (\lambda (t: T).(\forall -(t3: T).((((eq T t0 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t3) \to -(((\forall (u2: T).((pr3 c t u2) \to ((((iso t u2) \to (\forall (P: -Prop).P))) \to (sn3 c (THead (Flat Appl) t3 u2)))))) \to (sn3 c (THead (Flat -Appl) t3 t))))))) H6 (THead (Flat Appl) x x0) H3) in (let H10 \def (eq_ind T -t2 (\lambda (t: T).(\forall (t3: T).((((eq T t t3) \to (\forall (P: -Prop).P))) \to ((pr3 c t t3) \to (\forall (x1: T).(\forall (x2: T).((eq T t3 -(THead (Flat Appl) x1 x2)) \to (\forall (v3: T).((sn3 c v3) \to (((\forall -(u2: T).((pr3 c t3 u2) \to ((((iso t3 u2) \to (\forall (P: Prop).P))) \to -(sn3 c (THead (Flat Appl) v3 u2)))))) \to (sn3 c (THead (Flat Appl) v3 -t3)))))))))))) H2 (THead (Flat Appl) x x0) H3) in (let H11 \def (eq_ind T t2 -(\lambda (t: T).(\forall (t3: T).((((eq T t t3) \to (\forall (P: Prop).P))) -\to ((pr3 c t t3) \to (sn3 c t3))))) H1 (THead (Flat Appl) x x0) H3) in -(eq_ind_r T (THead (Flat Appl) x x0) (\lambda (t: T).(sn3 c (THead (Flat -Appl) t0 t))) (sn3_pr2_intro c (THead (Flat Appl) t0 (THead (Flat Appl) x -x0)) (\lambda (t3: T).(\lambda (H12: (((eq T (THead (Flat Appl) t0 (THead -(Flat Appl) x x0)) t3) \to (\forall (P: Prop).P)))).(\lambda (H13: (pr2 c -(THead (Flat Appl) t0 (THead (Flat Appl) x x0)) t3)).(let H14 \def -(pr2_gen_appl c t0 (THead (Flat Appl) x x0) t3 H13) in (or3_ind (ex3_2 T T -(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))) (\lambda (_: T).(\lambda -(t4: T).(pr2 c (THead (Flat Appl) x x0) t4)))) (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Appl) -x x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall -(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4)))))))) (ex6_6 B T T T -T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq -T (THead (Flat Appl) x x0) (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) -z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) -y2) z1 z2)))))))) (sn3 c t3) (\lambda (H15: (ex3_2 T T (\lambda (u2: -T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c t0 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c -(THead (Flat Appl) x x0) t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda -(t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c t0 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Flat Appl) -x x0) t4))) (sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (H16: (eq T -t3 (THead (Flat Appl) x1 x2))).(\lambda (H17: (pr2 c t0 x1)).(\lambda (H18: -(pr2 c (THead (Flat Appl) x x0) x2)).(let H19 \def (eq_ind T t3 (\lambda (t: -T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) t) \to (\forall (P: -Prop).P))) H12 (THead (Flat Appl) x1 x2) H16) in (eq_ind_r T (THead (Flat -Appl) x1 x2) (\lambda (t: T).(sn3 c t)) (let H20 \def (pr2_gen_appl c x x0 x2 -H18) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead -(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) -(\lambda (_: T).(\lambda (t4: T).(pr2 c x0 t4)))) (ex4_4 T T T T (\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x0 (THead -(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t4: T).(eq T x2 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda -(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: -B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4)))))))) (ex6_6 B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T x0 -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x2 (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (sn3 c -(THead (Flat Appl) x1 x2)) (\lambda (H21: (ex3_2 T T (\lambda (u2: -T).(\lambda (t4: T).(eq T x2 (THead (Flat Appl) u2 t4)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c x0 -t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead -(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) -(\lambda (_: T).(\lambda (t4: T).(pr2 c x0 t4))) (sn3 c (THead (Flat Appl) x1 -x2)) (\lambda (x3: T).(\lambda (x4: T).(\lambda (H22: (eq T x2 (THead (Flat -Appl) x3 x4))).(\lambda (H23: (pr2 c x x3)).(\lambda (H24: (pr2 c x0 -x4)).(let H25 \def (eq_ind T x2 (\lambda (t: T).((eq T (THead (Flat Appl) t0 -(THead (Flat Appl) x x0)) (THead (Flat Appl) x1 t)) \to (\forall (P: -Prop).P))) H19 (THead (Flat Appl) x3 x4) H22) in (eq_ind_r T (THead (Flat -Appl) x3 x4) (\lambda (t: T).(sn3 c (THead (Flat Appl) x1 t))) (let H_x \def -(term_dec (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4)) in (let H26 -\def H_x in (or_ind (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4)) -((eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4)) \to (\forall (P: -Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Flat Appl) x3 x4))) (\lambda -(H27: (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4))).(let H28 -\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t _) -\Rightarrow t])) (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4) H27) in -((let H29 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: -T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ -t) \Rightarrow t])) (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4) H27) -in (\lambda (H30: (eq T x x3)).(let H31 \def (eq_ind_r T x4 (\lambda (t: -T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) (THead (Flat Appl) -x1 (THead (Flat Appl) x3 t))) \to (\forall (P: Prop).P))) H25 x0 H29) in (let -H32 \def (eq_ind_r T x4 (\lambda (t: T).(pr2 c x0 t)) H24 x0 H29) in (eq_ind -T x0 (\lambda (t: T).(sn3 c (THead (Flat Appl) x1 (THead (Flat Appl) x3 t)))) -(let H33 \def (eq_ind_r T x3 (\lambda (t: T).((eq T (THead (Flat Appl) t0 -(THead (Flat Appl) x x0)) (THead (Flat Appl) x1 (THead (Flat Appl) t x0))) -\to (\forall (P: Prop).P))) H31 x H30) in (let H34 \def (eq_ind_r T x3 -(\lambda (t: T).(pr2 c x t)) H23 x H30) in (eq_ind T x (\lambda (t: T).(sn3 c -(THead (Flat Appl) x1 (THead (Flat Appl) t x0)))) (let H_x0 \def (term_dec t0 -x1) in (let H35 \def H_x0 in (or_ind (eq T t0 x1) ((eq T t0 x1) \to (\forall -(P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Flat Appl) x x0))) -(\lambda (H36: (eq T t0 x1)).(let H37 \def (eq_ind_r T x1 (\lambda (t: -T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) (THead (Flat Appl) -t (THead (Flat Appl) x x0))) \to (\forall (P: Prop).P))) H33 t0 H36) in (let -H38 \def (eq_ind_r T x1 (\lambda (t: T).(pr2 c t0 t)) H17 t0 H36) in (eq_ind -T t0 (\lambda (t: T).(sn3 c (THead (Flat Appl) t (THead (Flat Appl) x x0)))) -(H37 (refl_equal T (THead (Flat Appl) t0 (THead (Flat Appl) x x0))) (sn3 c -(THead (Flat Appl) t0 (THead (Flat Appl) x x0)))) x1 H36)))) (\lambda (H36: -(((eq T t0 x1) \to (\forall (P: Prop).P)))).(H9 x1 H36 (pr3_pr2 c t0 x1 H17) -(\lambda (u2: T).(\lambda (H37: (pr3 c (THead (Flat Appl) x x0) u2)).(\lambda -(H38: (((iso (THead (Flat Appl) x x0) u2) \to (\forall (P: -Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) t0 u2) (H8 u2 H37 H38) (THead -(Flat Appl) x1 u2) (pr3_pr2 c (THead (Flat Appl) t0 u2) (THead (Flat Appl) x1 -u2) (pr2_head_1 c t0 x1 H17 (Flat Appl) u2)))))))) H35))) x3 H30))) x4 -H29))))) H28))) (\lambda (H27: (((eq T (THead (Flat Appl) x x0) (THead (Flat -Appl) x3 x4)) \to (\forall (P: Prop).P)))).(H10 (THead (Flat Appl) x3 x4) H27 -(pr3_flat c x x3 (pr3_pr2 c x x3 H23) x0 x4 (pr3_pr2 c x0 x4 H24) Appl) x3 x4 -(refl_equal T (THead (Flat Appl) x3 x4)) x1 (sn3_pr3_trans c t0 (sn3_sing c -t0 H5) x1 (pr3_pr2 c t0 x1 H17)) (\lambda (u2: T).(\lambda (H28: (pr3 c -(THead (Flat Appl) x3 x4) u2)).(\lambda (H29: (((iso (THead (Flat Appl) x3 -x4) u2) \to (\forall (P: Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) t0 -u2) (H8 u2 (pr3_sing c (THead (Flat Appl) x x4) (THead (Flat Appl) x x0) -(pr2_thin_dx c x0 x4 H24 x Appl) u2 (pr3_sing c (THead (Flat Appl) x3 x4) -(THead (Flat Appl) x x4) (pr2_head_1 c x x3 H23 (Flat Appl) x4) u2 H28)) -(\lambda (H30: (iso (THead (Flat Appl) x x0) u2)).(\lambda (P: Prop).(H29 -(iso_trans (THead (Flat Appl) x3 x4) (THead (Flat Appl) x x0) (iso_head x3 x -x4 x0 (Flat Appl)) u2 H30) P)))) (THead (Flat Appl) x1 u2) (pr3_pr2 c (THead -(Flat Appl) t0 u2) (THead (Flat Appl) x1 u2) (pr2_head_1 c t0 x1 H17 (Flat -Appl) u2)))))))) H26))) x2 H22))))))) H21)) (\lambda (H21: (ex4_4 T T T T -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x0 -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t4: T).(eq T x2 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda -(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: -B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4))))))))).(ex4_4_ind T T T -T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x0 -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t4: T).(eq T x2 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda -(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: -B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4))))))) (sn3 c (THead (Flat -Appl) x1 x2)) (\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda -(x6: T).(\lambda (H22: (eq T x0 (THead (Bind Abst) x3 x4))).(\lambda (H23: -(eq T x2 (THead (Bind Abbr) x5 x6))).(\lambda (H24: (pr2 c x x5)).(\lambda -(H25: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x4 -x6))))).(let H26 \def (eq_ind T x2 (\lambda (t: T).((eq T (THead (Flat Appl) -t0 (THead (Flat Appl) x x0)) (THead (Flat Appl) x1 t)) \to (\forall (P: -Prop).P))) H19 (THead (Bind Abbr) x5 x6) H23) in (eq_ind_r T (THead (Bind -Abbr) x5 x6) (\lambda (t: T).(sn3 c (THead (Flat Appl) x1 t))) (let H27 \def -(eq_ind T x0 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) -x t)) (THead (Flat Appl) x1 (THead (Bind Abbr) x5 x6))) \to (\forall (P: -Prop).P))) H26 (THead (Bind Abst) x3 x4) H22) in (let H28 \def (eq_ind T x0 -(\lambda (t: T).(\forall (t4: T).((((eq T (THead (Flat Appl) x t) t4) \to -(\forall (P: Prop).P))) \to ((pr3 c (THead (Flat Appl) x t) t4) \to (sn3 c -t4))))) H11 (THead (Bind Abst) x3 x4) H22) in (let H29 \def (eq_ind T x0 -(\lambda (t: T).(\forall (t4: T).((((eq T (THead (Flat Appl) x t) t4) \to -(\forall (P: Prop).P))) \to ((pr3 c (THead (Flat Appl) x t) t4) \to (\forall -(x7: T).(\forall (x8: T).((eq T t4 (THead (Flat Appl) x7 x8)) \to (\forall -(v3: T).((sn3 c v3) \to (((\forall (u2: T).((pr3 c t4 u2) \to ((((iso t4 u2) -\to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v3 u2)))))) \to -(sn3 c (THead (Flat Appl) v3 t4)))))))))))) H10 (THead (Bind Abst) x3 x4) -H22) in (let H30 \def (eq_ind T x0 (\lambda (t: T).(\forall (u2: T).((pr3 c -(THead (Flat Appl) x t) u2) \to ((((iso (THead (Flat Appl) x t) u2) \to -(\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) t0 u2)))))) H8 (THead -(Bind Abst) x3 x4) H22) in (let H31 \def (eq_ind T x0 (\lambda (t: -T).(\forall (t4: T).((((eq T t0 t4) \to (\forall (P: Prop).P))) \to ((pr3 c -t0 t4) \to (((\forall (u2: T).((pr3 c (THead (Flat Appl) x t) u2) \to ((((iso -(THead (Flat Appl) x t) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead -(Flat Appl) t4 u2)))))) \to (sn3 c (THead (Flat Appl) t4 (THead (Flat Appl) x -t)))))))) H9 (THead (Bind Abst) x3 x4) H22) in (sn3_pr3_trans c (THead (Flat -Appl) t0 (THead (Bind Abbr) x5 x6)) (H30 (THead (Bind Abbr) x5 x6) (pr3_sing -c (THead (Bind Abbr) x x4) (THead (Flat Appl) x (THead (Bind Abst) x3 x4)) -(pr2_free c (THead (Flat Appl) x (THead (Bind Abst) x3 x4)) (THead (Bind -Abbr) x x4) (pr0_beta x3 x x (pr0_refl x) x4 x4 (pr0_refl x4))) (THead (Bind -Abbr) x5 x6) (pr3_head_12 c x x5 (pr3_pr2 c x x5 H24) (Bind Abbr) x4 x6 -(pr3_pr2 (CHead c (Bind Abbr) x5) x4 x6 (H25 Abbr x5)))) (\lambda (H32: (iso -(THead (Flat Appl) x (THead (Bind Abst) x3 x4)) (THead (Bind Abbr) x5 -x6))).(\lambda (P: Prop).(let H33 \def (match H32 in iso return (\lambda (t: -T).(\lambda (t4: T).(\lambda (_: (iso t t4)).((eq T t (THead (Flat Appl) x -(THead (Bind Abst) x3 x4))) \to ((eq T t4 (THead (Bind Abbr) x5 x6)) \to -P))))) with [(iso_sort n1 n2) \Rightarrow (\lambda (H33: (eq T (TSort n1) -(THead (Flat Appl) x (THead (Bind Abst) x3 x4)))).(\lambda (H34: (eq T (TSort -n2) (THead (Bind Abbr) x5 x6))).((let H35 \def (eq_ind T (TSort n1) (\lambda -(e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I -(THead (Flat Appl) x (THead (Bind Abst) x3 x4)) H33) in (False_ind ((eq T -(TSort n2) (THead (Bind Abbr) x5 x6)) \to P) H35)) H34))) | (iso_lref i1 i2) -\Rightarrow (\lambda (H33: (eq T (TLRef i1) (THead (Flat Appl) x (THead (Bind -Abst) x3 x4)))).(\lambda (H34: (eq T (TLRef i2) (THead (Bind Abbr) x5 -x6))).((let H35 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) x -(THead (Bind Abst) x3 x4)) H33) in (False_ind ((eq T (TLRef i2) (THead (Bind -Abbr) x5 x6)) \to P) H35)) H34))) | (iso_head v4 v5 t4 t5 k) \Rightarrow -(\lambda (H33: (eq T (THead k v4 t4) (THead (Flat Appl) x (THead (Bind Abst) -x3 x4)))).(\lambda (H34: (eq T (THead k v5 t5) (THead (Bind Abbr) x5 -x6))).((let H35 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow t4 | (TLRef _) \Rightarrow t4 -| (THead _ _ t) \Rightarrow t])) (THead k v4 t4) (THead (Flat Appl) x (THead -(Bind Abst) x3 x4)) H33) in ((let H36 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v4 | -(TLRef _) \Rightarrow v4 | (THead _ t _) \Rightarrow t])) (THead k v4 t4) -(THead (Flat Appl) x (THead (Bind Abst) x3 x4)) H33) in ((let H37 \def -(f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with -[(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) -\Rightarrow k0])) (THead k v4 t4) (THead (Flat Appl) x (THead (Bind Abst) x3 -x4)) H33) in (eq_ind K (Flat Appl) (\lambda (k0: K).((eq T v4 x) \to ((eq T -t4 (THead (Bind Abst) x3 x4)) \to ((eq T (THead k0 v5 t5) (THead (Bind Abbr) -x5 x6)) \to P)))) (\lambda (H38: (eq T v4 x)).(eq_ind T x (\lambda (_: -T).((eq T t4 (THead (Bind Abst) x3 x4)) \to ((eq T (THead (Flat Appl) v5 t5) -(THead (Bind Abbr) x5 x6)) \to P))) (\lambda (H39: (eq T t4 (THead (Bind -Abst) x3 x4))).(eq_ind T (THead (Bind Abst) x3 x4) (\lambda (_: T).((eq T -(THead (Flat Appl) v5 t5) (THead (Bind Abbr) x5 x6)) \to P)) (\lambda (H40: -(eq T (THead (Flat Appl) v5 t5) (THead (Bind Abbr) x5 x6))).(let H41 \def -(eq_ind T (THead (Flat Appl) v5 t5) (\lambda (e: T).(match e in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind Abbr) x5 x6) H40) in (False_ind P H41))) t4 (sym_eq -T t4 (THead (Bind Abst) x3 x4) H39))) v4 (sym_eq T v4 x H38))) k (sym_eq K k -(Flat Appl) H37))) H36)) H35)) H34)))]) in (H33 (refl_equal T (THead (Flat -Appl) x (THead (Bind Abst) x3 x4))) (refl_equal T (THead (Bind Abbr) x5 -x6))))))) (THead (Flat Appl) x1 (THead (Bind Abbr) x5 x6)) (pr3_pr2 c (THead -(Flat Appl) t0 (THead (Bind Abbr) x5 x6)) (THead (Flat Appl) x1 (THead (Bind -Abbr) x5 x6)) (pr2_head_1 c t0 x1 H17 (Flat Appl) (THead (Bind Abbr) x5 -x6))))))))) x2 H23)))))))))) H21)) (\lambda (H21: (ex6_6 B T T T T T (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T x0 -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x2 (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind -B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T x0 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T -x2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 -y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) -(sn3 c (THead (Flat Appl) x1 x2)) (\lambda (x3: B).(\lambda (x4: T).(\lambda -(x5: T).(\lambda (x6: T).(\lambda (x7: T).(\lambda (x8: T).(\lambda (H22: -(not (eq B x3 Abst))).(\lambda (H23: (eq T x0 (THead (Bind x3) x4 -x5))).(\lambda (H24: (eq T x2 (THead (Bind x3) x8 (THead (Flat Appl) (lift (S -O) O x7) x6)))).(\lambda (H25: (pr2 c x x7)).(\lambda (H26: (pr2 c x4 -x8)).(\lambda (H27: (pr2 (CHead c (Bind x3) x8) x5 x6)).(let H28 \def (eq_ind -T x2 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) -(THead (Flat Appl) x1 t)) \to (\forall (P: Prop).P))) H19 (THead (Bind x3) x8 -(THead (Flat Appl) (lift (S O) O x7) x6)) H24) in (eq_ind_r T (THead (Bind -x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)) (\lambda (t: T).(sn3 c -(THead (Flat Appl) x1 t))) (let H29 \def (eq_ind T x0 (\lambda (t: T).((eq T -(THead (Flat Appl) t0 (THead (Flat Appl) x t)) (THead (Flat Appl) x1 (THead -(Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))) \to (\forall (P: -Prop).P))) H28 (THead (Bind x3) x4 x5) H23) in (let H30 \def (eq_ind T x0 -(\lambda (t: T).(\forall (t4: T).((((eq T (THead (Flat Appl) x t) t4) \to -(\forall (P: Prop).P))) \to ((pr3 c (THead (Flat Appl) x t) t4) \to (sn3 c -t4))))) H11 (THead (Bind x3) x4 x5) H23) in (let H31 \def (eq_ind T x0 -(\lambda (t: T).(\forall (t4: T).((((eq T (THead (Flat Appl) x t) t4) \to -(\forall (P: Prop).P))) \to ((pr3 c (THead (Flat Appl) x t) t4) \to (\forall -(x9: T).(\forall (x10: T).((eq T t4 (THead (Flat Appl) x9 x10)) \to (\forall -(v3: T).((sn3 c v3) \to (((\forall (u2: T).((pr3 c t4 u2) \to ((((iso t4 u2) -\to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v3 u2)))))) \to -(sn3 c (THead (Flat Appl) v3 t4)))))))))))) H10 (THead (Bind x3) x4 x5) H23) -in (let H32 \def (eq_ind T x0 (\lambda (t: T).(\forall (u2: T).((pr3 c (THead -(Flat Appl) x t) u2) \to ((((iso (THead (Flat Appl) x t) u2) \to (\forall (P: -Prop).P))) \to (sn3 c (THead (Flat Appl) t0 u2)))))) H8 (THead (Bind x3) x4 -x5) H23) in (let H33 \def (eq_ind T x0 (\lambda (t: T).(\forall (t4: -T).((((eq T t0 t4) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t4) \to -(((\forall (u2: T).((pr3 c (THead (Flat Appl) x t) u2) \to ((((iso (THead -(Flat Appl) x t) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat -Appl) t4 u2)))))) \to (sn3 c (THead (Flat Appl) t4 (THead (Flat Appl) x -t)))))))) H9 (THead (Bind x3) x4 x5) H23) in (sn3_pr3_trans c (THead (Flat -Appl) t0 (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) (H32 -(THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)) (pr3_sing c -(THead (Bind x3) x4 (THead (Flat Appl) (lift (S O) O x) x5)) (THead (Flat -Appl) x (THead (Bind x3) x4 x5)) (pr2_free c (THead (Flat Appl) x (THead -(Bind x3) x4 x5)) (THead (Bind x3) x4 (THead (Flat Appl) (lift (S O) O x) -x5)) (pr0_upsilon x3 H22 x x (pr0_refl x) x4 x4 (pr0_refl x4) x5 x5 (pr0_refl -x5))) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)) -(pr3_head_12 c x4 x8 (pr3_pr2 c x4 x8 H26) (Bind x3) (THead (Flat Appl) (lift -(S O) O x) x5) (THead (Flat Appl) (lift (S O) O x7) x6) (pr3_head_12 (CHead c -(Bind x3) x8) (lift (S O) O x) (lift (S O) O x7) (pr3_lift (CHead c (Bind x3) -x8) c (S O) O (drop_drop (Bind x3) O c c (drop_refl c) x8) x x7 (pr3_pr2 c x -x7 H25)) (Flat Appl) x5 x6 (pr3_pr2 (CHead (CHead c (Bind x3) x8) (Flat Appl) -(lift (S O) O x7)) x5 x6 (pr2_cflat (CHead c (Bind x3) x8) x5 x6 H27 Appl -(lift (S O) O x7)))))) (\lambda (H34: (iso (THead (Flat Appl) x (THead (Bind -x3) x4 x5)) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) -x6)))).(\lambda (P: Prop).(let H35 \def (match H34 in iso return (\lambda (t: -T).(\lambda (t4: T).(\lambda (_: (iso t t4)).((eq T t (THead (Flat Appl) x -(THead (Bind x3) x4 x5))) \to ((eq T t4 (THead (Bind x3) x8 (THead (Flat -Appl) (lift (S O) O x7) x6))) \to P))))) with [(iso_sort n1 n2) \Rightarrow -(\lambda (H35: (eq T (TSort n1) (THead (Flat Appl) x (THead (Bind x3) x4 -x5)))).(\lambda (H36: (eq T (TSort n2) (THead (Bind x3) x8 (THead (Flat Appl) -(lift (S O) O x7) x6)))).((let H37 \def (eq_ind T (TSort n1) (\lambda (e: -T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I -(THead (Flat Appl) x (THead (Bind x3) x4 x5)) H35) in (False_ind ((eq T -(TSort n2) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to -P) H37)) H36))) | (iso_lref i1 i2) \Rightarrow (\lambda (H35: (eq T (TLRef -i1) (THead (Flat Appl) x (THead (Bind x3) x4 x5)))).(\lambda (H36: (eq T -(TLRef i2) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) -x6)))).((let H37 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) x -(THead (Bind x3) x4 x5)) H35) in (False_ind ((eq T (TLRef i2) (THead (Bind -x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to P) H37)) H36))) | -(iso_head v4 v5 t4 t5 k) \Rightarrow (\lambda (H35: (eq T (THead k v4 t4) -(THead (Flat Appl) x (THead (Bind x3) x4 x5)))).(\lambda (H36: (eq T (THead k -v5 t5) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))).((let -H37 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow t4 | (TLRef _) \Rightarrow t4 | (THead _ _ t) -\Rightarrow t])) (THead k v4 t4) (THead (Flat Appl) x (THead (Bind x3) x4 -x5)) H35) in ((let H38 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow v4 | (TLRef _) \Rightarrow v4 -| (THead _ t _) \Rightarrow t])) (THead k v4 t4) (THead (Flat Appl) x (THead -(Bind x3) x4 x5)) H35) in ((let H39 \def (f_equal T K (\lambda (e: T).(match -e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) -\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k v4 t4) (THead (Flat -Appl) x (THead (Bind x3) x4 x5)) H35) in (eq_ind K (Flat Appl) (\lambda (k0: -K).((eq T v4 x) \to ((eq T t4 (THead (Bind x3) x4 x5)) \to ((eq T (THead k0 -v5 t5) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to -P)))) (\lambda (H40: (eq T v4 x)).(eq_ind T x (\lambda (_: T).((eq T t4 -(THead (Bind x3) x4 x5)) \to ((eq T (THead (Flat Appl) v5 t5) (THead (Bind -x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to P))) (\lambda (H41: (eq -T t4 (THead (Bind x3) x4 x5))).(eq_ind T (THead (Bind x3) x4 x5) (\lambda (_: -T).((eq T (THead (Flat Appl) v5 t5) (THead (Bind x3) x8 (THead (Flat Appl) -(lift (S O) O x7) x6))) \to P)) (\lambda (H42: (eq T (THead (Flat Appl) v5 -t5) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))).(let H43 -\def (eq_ind T (THead (Flat Appl) v5 t5) (\lambda (e: T).(match e in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)) -H42) in (False_ind P H43))) t4 (sym_eq T t4 (THead (Bind x3) x4 x5) H41))) v4 -(sym_eq T v4 x H40))) k (sym_eq K k (Flat Appl) H39))) H38)) H37)) H36)))]) -in (H35 (refl_equal T (THead (Flat Appl) x (THead (Bind x3) x4 x5))) -(refl_equal T (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) -x6)))))))) (THead (Flat Appl) x1 (THead (Bind x3) x8 (THead (Flat Appl) (lift -(S O) O x7) x6))) (pr3_pr2 c (THead (Flat Appl) t0 (THead (Bind x3) x8 (THead -(Flat Appl) (lift (S O) O x7) x6))) (THead (Flat Appl) x1 (THead (Bind x3) x8 -(THead (Flat Appl) (lift (S O) O x7) x6))) (pr2_head_1 c t0 x1 H17 (Flat -Appl) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))))))))) -x2 H24)))))))))))))) H21)) H20)) t3 H16))))))) H15)) (\lambda (H15: (ex4_4 T -T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Flat Appl) x x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind -Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c t0 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t4: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) -z1 t4))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (_: T).(eq T (THead (Flat Appl) x x0) (THead (Bind Abst) y1 -z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: -T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))))) (\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall -(u: T).(pr2 (CHead c (Bind b) u) z1 t4))))))) (sn3 c t3) (\lambda (x1: -T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H16: (eq T -(THead (Flat Appl) x x0) (THead (Bind Abst) x1 x2))).(\lambda (H17: (eq T t3 -(THead (Bind Abbr) x3 x4))).(\lambda (_: (pr2 c t0 x3)).(\lambda (_: -((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x2 x4))))).(let -H20 \def (eq_ind T t3 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead -(Flat Appl) x x0)) t) \to (\forall (P: Prop).P))) H12 (THead (Bind Abbr) x3 -x4) H17) in (eq_ind_r T (THead (Bind Abbr) x3 x4) (\lambda (t: T).(sn3 c t)) -(let H21 \def (eq_ind T (THead (Flat Appl) x x0) (\lambda (ee: T).(match ee -in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef -_) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return -(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind Abst) x1 x2) H16) in (False_ind (sn3 c (THead (Bind -Abbr) x3 x4)) H21)) t3 H17)))))))))) H15)) (\lambda (H15: (ex6_6 B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Flat Appl) x x0) (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) -z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) -y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B -b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Appl) x x0) (THead -(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c t0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) (sn3 c t3) -(\lambda (x1: B).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda -(x5: T).(\lambda (x6: T).(\lambda (_: (not (eq B x1 Abst))).(\lambda (H17: -(eq T (THead (Flat Appl) x x0) (THead (Bind x1) x2 x3))).(\lambda (H18: (eq T -t3 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))).(\lambda -(_: (pr2 c t0 x5)).(\lambda (_: (pr2 c x2 x6)).(\lambda (_: (pr2 (CHead c -(Bind x1) x6) x3 x4)).(let H22 \def (eq_ind T t3 (\lambda (t: T).((eq T -(THead (Flat Appl) t0 (THead (Flat Appl) x x0)) t) \to (\forall (P: -Prop).P))) H12 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)) -H18) in (eq_ind_r T (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) -x4)) (\lambda (t: T).(sn3 c t)) (let H23 \def (eq_ind T (THead (Flat Appl) x -x0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort -_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) -\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind x1) x2 x3) -H17) in (False_ind (sn3 c (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) -O x5) x4))) H23)) t3 H18)))))))))))))) H15)) H14)))))) t2 H3))))))))) v2 -H4))))))))) y H0))))) H))))). -(* COMMENTS -Initial nodes: 9317 -END *) - -theorem sn3_appl_beta: - \forall (c: C).(\forall (u: T).(\forall (v: T).(\forall (t: T).((sn3 c -(THead (Flat Appl) u (THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w) -\to (sn3 c (THead (Flat Appl) u (THead (Flat Appl) v (THead (Bind Abst) w -t)))))))))) -\def - \lambda (c: C).(\lambda (u: T).(\lambda (v: T).(\lambda (t: T).(\lambda (H: -(sn3 c (THead (Flat Appl) u (THead (Bind Abbr) v t)))).(\lambda (w: -T).(\lambda (H0: (sn3 c w)).(let H_x \def (sn3_gen_flat Appl c u (THead (Bind -Abbr) v t) H) in (let H1 \def H_x in (land_ind (sn3 c u) (sn3 c (THead (Bind -Abbr) v t)) (sn3 c (THead (Flat Appl) u (THead (Flat Appl) v (THead (Bind -Abst) w t)))) (\lambda (H2: (sn3 c u)).(\lambda (H3: (sn3 c (THead (Bind -Abbr) v t))).(sn3_appl_appl v (THead (Bind Abst) w t) c (sn3_beta c v t H3 w -H0) u H2 (\lambda (u2: T).(\lambda (H4: (pr3 c (THead (Flat Appl) v (THead -(Bind Abst) w t)) u2)).(\lambda (H5: (((iso (THead (Flat Appl) v (THead (Bind -Abst) w t)) u2) \to (\forall (P: Prop).P)))).(sn3_pr3_trans c (THead (Flat -Appl) u (THead (Bind Abbr) v t)) H (THead (Flat Appl) u u2) (pr3_thin_dx c -(THead (Bind Abbr) v t) u2 (pr3_iso_beta v w t c u2 H4 H5) u Appl)))))))) -H1))))))))). -(* COMMENTS -Initial nodes: 289 -END *) - -theorem sn3_appl_appls: - \forall (v1: T).(\forall (t1: T).(\forall (vs: TList).(let u1 \def (THeads -(Flat Appl) (TCons v1 vs) t1) in (\forall (c: C).((sn3 c u1) \to (\forall -(v2: T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) -\to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to -(sn3 c (THead (Flat Appl) v2 u1)))))))))) -\def - \lambda (v1: T).(\lambda (t1: T).(\lambda (vs: TList).(let u1 \def (THeads -(Flat Appl) (TCons v1 vs) t1) in (\lambda (c: C).(\lambda (H: (sn3 c (THead -(Flat Appl) v1 (THeads (Flat Appl) vs t1)))).(\lambda (v2: T).(\lambda (H0: -(sn3 c v2)).(\lambda (H1: ((\forall (u2: T).((pr3 c (THead (Flat Appl) v1 -(THeads (Flat Appl) vs t1)) u2) \to ((((iso (THead (Flat Appl) v1 (THeads -(Flat Appl) vs t1)) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat -Appl) v2 u2))))))).(sn3_appl_appl v1 (THeads (Flat Appl) vs t1) c H v2 H0 -H1))))))))). -(* COMMENTS -Initial nodes: 141 -END *) - -theorem sn3_appls_lref: - \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (us: -TList).((sns3 c us) \to (sn3 c (THeads (Flat Appl) us (TLRef i))))))) -\def - \lambda (c: C).(\lambda (i: nat).(\lambda (H: (nf2 c (TLRef i))).(\lambda -(us: TList).(TList_ind (\lambda (t: TList).((sns3 c t) \to (sn3 c (THeads -(Flat Appl) t (TLRef i))))) (\lambda (_: True).(sn3_nf2 c (TLRef i) H)) -(\lambda (t: T).(\lambda (t0: TList).(TList_ind (\lambda (t1: TList).((((sns3 -c t1) \to (sn3 c (THeads (Flat Appl) t1 (TLRef i))))) \to ((land (sn3 c t) -(sns3 c t1)) \to (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t1 (TLRef -i))))))) (\lambda (_: (((sns3 c TNil) \to (sn3 c (THeads (Flat Appl) TNil -(TLRef i)))))).(\lambda (H1: (land (sn3 c t) (sns3 c TNil))).(let H2 \def H1 -in (land_ind (sn3 c t) True (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) -TNil (TLRef i)))) (\lambda (H3: (sn3 c t)).(\lambda (_: True).(sn3_appl_lref -c i H t H3))) H2)))) (\lambda (t1: T).(\lambda (t2: TList).(\lambda (_: -(((((sns3 c t2) \to (sn3 c (THeads (Flat Appl) t2 (TLRef i))))) \to ((land -(sn3 c t) (sns3 c t2)) \to (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t2 -(TLRef i)))))))).(\lambda (H1: (((sns3 c (TCons t1 t2)) \to (sn3 c (THeads -(Flat Appl) (TCons t1 t2) (TLRef i)))))).(\lambda (H2: (land (sn3 c t) (sns3 -c (TCons t1 t2)))).(let H3 \def H2 in (land_ind (sn3 c t) (land (sn3 c t1) -(sns3 c t2)) (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) (TCons t1 t2) -(TLRef i)))) (\lambda (H4: (sn3 c t)).(\lambda (H5: (land (sn3 c t1) (sns3 c -t2))).(land_ind (sn3 c t1) (sns3 c t2) (sn3 c (THead (Flat Appl) t (THeads -(Flat Appl) (TCons t1 t2) (TLRef i)))) (\lambda (H6: (sn3 c t1)).(\lambda -(H7: (sns3 c t2)).(sn3_appl_appls t1 (TLRef i) t2 c (H1 (conj (sn3 c t1) -(sns3 c t2) H6 H7)) t H4 (\lambda (u2: T).(\lambda (H8: (pr3 c (THeads (Flat -Appl) (TCons t1 t2) (TLRef i)) u2)).(\lambda (H9: (((iso (THeads (Flat Appl) -(TCons t1 t2) (TLRef i)) u2) \to (\forall (P: Prop).P)))).(H9 -(nf2_iso_appls_lref c i H (TCons t1 t2) u2 H8) (sn3 c (THead (Flat Appl) t -u2))))))))) H5))) H3))))))) t0))) us)))). -(* COMMENTS -Initial nodes: 577 -END *) - -theorem sn3_appls_cast: - \forall (c: C).(\forall (vs: TList).(\forall (u: T).((sn3 c (THeads (Flat -Appl) vs u)) \to (\forall (t: T).((sn3 c (THeads (Flat Appl) vs t)) \to (sn3 -c (THeads (Flat Appl) vs (THead (Flat Cast) u t)))))))) -\def - \lambda (c: C).(\lambda (vs: TList).(TList_ind (\lambda (t: TList).(\forall -(u: T).((sn3 c (THeads (Flat Appl) t u)) \to (\forall (t0: T).((sn3 c (THeads -(Flat Appl) t t0)) \to (sn3 c (THeads (Flat Appl) t (THead (Flat Cast) u -t0)))))))) (\lambda (u: T).(\lambda (H: (sn3 c u)).(\lambda (t: T).(\lambda -(H0: (sn3 c t)).(sn3_cast c u H t H0))))) (\lambda (t: T).(\lambda (t0: -TList).(TList_ind (\lambda (t1: TList).(((\forall (u: T).((sn3 c (THeads -(Flat Appl) t1 u)) \to (\forall (t2: T).((sn3 c (THeads (Flat Appl) t1 t2)) -\to (sn3 c (THeads (Flat Appl) t1 (THead (Flat Cast) u t2)))))))) \to -(\forall (u: T).((sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t1 u))) \to -(\forall (t2: T).((sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t1 t2))) -\to (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t1 (THead (Flat Cast) u -t2)))))))))) (\lambda (_: ((\forall (u: T).((sn3 c (THeads (Flat Appl) TNil -u)) \to (\forall (t1: T).((sn3 c (THeads (Flat Appl) TNil t1)) \to (sn3 c -(THeads (Flat Appl) TNil (THead (Flat Cast) u t1))))))))).(\lambda (u: -T).(\lambda (H0: (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) TNil -u)))).(\lambda (t1: T).(\lambda (H1: (sn3 c (THead (Flat Appl) t (THeads -(Flat Appl) TNil t1)))).(sn3_appl_cast c t u H0 t1 H1)))))) (\lambda (t1: -T).(\lambda (t2: TList).(\lambda (_: ((((\forall (u: T).((sn3 c (THeads (Flat -Appl) t2 u)) \to (\forall (t3: T).((sn3 c (THeads (Flat Appl) t2 t3)) \to -(sn3 c (THeads (Flat Appl) t2 (THead (Flat Cast) u t3)))))))) \to (\forall -(u: T).((sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t2 u))) \to (\forall -(t3: T).((sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t2 t3))) \to (sn3 c -(THead (Flat Appl) t (THeads (Flat Appl) t2 (THead (Flat Cast) u -t3))))))))))).(\lambda (H0: ((\forall (u: T).((sn3 c (THeads (Flat Appl) -(TCons t1 t2) u)) \to (\forall (t3: T).((sn3 c (THeads (Flat Appl) (TCons t1 -t2) t3)) \to (sn3 c (THeads (Flat Appl) (TCons t1 t2) (THead (Flat Cast) u -t3))))))))).(\lambda (u: T).(\lambda (H1: (sn3 c (THead (Flat Appl) t (THeads -(Flat Appl) (TCons t1 t2) u)))).(\lambda (t3: T).(\lambda (H2: (sn3 c (THead -(Flat Appl) t (THeads (Flat Appl) (TCons t1 t2) t3)))).(let H_x \def -(sn3_gen_flat Appl c t (THeads (Flat Appl) (TCons t1 t2) t3) H2) in (let H3 -\def H_x in (land_ind (sn3 c t) (sn3 c (THead (Flat Appl) t1 (THeads (Flat -Appl) t2 t3))) (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) (TCons t1 t2) -(THead (Flat Cast) u t3)))) (\lambda (_: (sn3 c t)).(\lambda (H5: (sn3 c -(THead (Flat Appl) t1 (THeads (Flat Appl) t2 t3)))).(let H6 \def H5 in (let -H_x0 \def (sn3_gen_flat Appl c t (THeads (Flat Appl) (TCons t1 t2) u) H1) in -(let H7 \def H_x0 in (land_ind (sn3 c t) (sn3 c (THead (Flat Appl) t1 (THeads -(Flat Appl) t2 u))) (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) (TCons t1 -t2) (THead (Flat Cast) u t3)))) (\lambda (H8: (sn3 c t)).(\lambda (H9: (sn3 c -(THead (Flat Appl) t1 (THeads (Flat Appl) t2 u)))).(let H10 \def H9 in -(sn3_appl_appls t1 (THead (Flat Cast) u t3) t2 c (H0 u H10 t3 H6) t H8 -(\lambda (u2: T).(\lambda (H11: (pr3 c (THeads (Flat Appl) (TCons t1 t2) -(THead (Flat Cast) u t3)) u2)).(\lambda (H12: (((iso (THeads (Flat Appl) -(TCons t1 t2) (THead (Flat Cast) u t3)) u2) \to (\forall (P: -Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) t (THeads (Flat Appl) (TCons -t1 t2) t3)) H2 (THead (Flat Appl) t u2) (pr3_thin_dx c (THeads (Flat Appl) -(TCons t1 t2) t3) u2 (pr3_iso_appls_cast c u t3 (TCons t1 t2) u2 H11 H12) t -Appl))))))))) H7)))))) H3))))))))))) t0))) vs)). -(* COMMENTS -Initial nodes: 1025 -END *) - -theorem sn3_appls_bind: - \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (u: -T).((sn3 c u) \to (\forall (vs: TList).(\forall (t: T).((sn3 (CHead c (Bind -b) u) (THeads (Flat Appl) (lifts (S O) O vs) t)) \to (sn3 c (THeads (Flat -Appl) vs (THead (Bind b) u t)))))))))) -\def - \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (c: C).(\lambda -(u: T).(\lambda (H0: (sn3 c u)).(\lambda (vs: TList).(TList_ind (\lambda (t: -TList).(\forall (t0: T).((sn3 (CHead c (Bind b) u) (THeads (Flat Appl) (lifts -(S O) O t) t0)) \to (sn3 c (THeads (Flat Appl) t (THead (Bind b) u t0)))))) -(\lambda (t: T).(\lambda (H1: (sn3 (CHead c (Bind b) u) t)).(sn3_bind b c u -H0 t H1))) (\lambda (v: T).(\lambda (vs0: TList).(TList_ind (\lambda (t: -TList).(((\forall (t0: T).((sn3 (CHead c (Bind b) u) (THeads (Flat Appl) -(lifts (S O) O t) t0)) \to (sn3 c (THeads (Flat Appl) t (THead (Bind b) u -t0)))))) \to (\forall (t0: T).((sn3 (CHead c (Bind b) u) (THead (Flat Appl) -(lift (S O) O v) (THeads (Flat Appl) (lifts (S O) O t) t0))) \to (sn3 c -(THead (Flat Appl) v (THeads (Flat Appl) t (THead (Bind b) u t0)))))))) -(\lambda (_: ((\forall (t: T).((sn3 (CHead c (Bind b) u) (THeads (Flat Appl) -(lifts (S O) O TNil) t)) \to (sn3 c (THeads (Flat Appl) TNil (THead (Bind b) -u t))))))).(\lambda (t: T).(\lambda (H2: (sn3 (CHead c (Bind b) u) (THead -(Flat Appl) (lift (S O) O v) (THeads (Flat Appl) (lifts (S O) O TNil) -t)))).(sn3_appl_bind b H c u H0 t v H2)))) (\lambda (t: T).(\lambda (t0: -TList).(\lambda (_: ((((\forall (t1: T).((sn3 (CHead c (Bind b) u) (THeads -(Flat Appl) (lifts (S O) O t0) t1)) \to (sn3 c (THeads (Flat Appl) t0 (THead -(Bind b) u t1)))))) \to (\forall (t1: T).((sn3 (CHead c (Bind b) u) (THead -(Flat Appl) (lift (S O) O v) (THeads (Flat Appl) (lifts (S O) O t0) t1))) \to -(sn3 c (THead (Flat Appl) v (THeads (Flat Appl) t0 (THead (Bind b) u -t1))))))))).(\lambda (H2: ((\forall (t1: T).((sn3 (CHead c (Bind b) u) -(THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1)) \to (sn3 c (THeads -(Flat Appl) (TCons t t0) (THead (Bind b) u t1))))))).(\lambda (t1: -T).(\lambda (H3: (sn3 (CHead c (Bind b) u) (THead (Flat Appl) (lift (S O) O -v) (THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1)))).(let H_x \def -(sn3_gen_flat Appl (CHead c (Bind b) u) (lift (S O) O v) (THeads (Flat Appl) -(lifts (S O) O (TCons t t0)) t1) H3) in (let H4 \def H_x in (land_ind (sn3 -(CHead c (Bind b) u) (lift (S O) O v)) (sn3 (CHead c (Bind b) u) (THead (Flat -Appl) (lift (S O) O t) (THeads (Flat Appl) (lifts (S O) O t0) t1))) (sn3 c -(THead (Flat Appl) v (THeads (Flat Appl) (TCons t t0) (THead (Bind b) u -t1)))) (\lambda (H5: (sn3 (CHead c (Bind b) u) (lift (S O) O v))).(\lambda -(H6: (sn3 (CHead c (Bind b) u) (THead (Flat Appl) (lift (S O) O t) (THeads -(Flat Appl) (lifts (S O) O t0) t1)))).(let H_y \def (sn3_gen_lift (CHead c -(Bind b) u) v (S O) O H5 c) in (sn3_appl_appls t (THead (Bind b) u t1) t0 c -(H2 t1 H6) v (H_y (drop_drop (Bind b) O c c (drop_refl c) u)) (\lambda (u2: -T).(\lambda (H7: (pr3 c (THeads (Flat Appl) (TCons t t0) (THead (Bind b) u -t1)) u2)).(\lambda (H8: (((iso (THeads (Flat Appl) (TCons t t0) (THead (Bind -b) u t1)) u2) \to (\forall (P: Prop).P)))).(let H9 \def (pr3_iso_appls_bind b -H (TCons t t0) u t1 c u2 H7 H8) in (sn3_pr3_trans c (THead (Flat Appl) v -(THead (Bind b) u (THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1))) -(sn3_appl_bind b H c u H0 (THeads (Flat Appl) (lifts (S O) O (TCons t t0)) -t1) v H3) (THead (Flat Appl) v u2) (pr3_flat c v v (pr3_refl c v) (THead -(Bind b) u (THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1)) u2 H9 -Appl)))))))))) H4))))))))) vs0))) vs)))))). -(* COMMENTS -Initial nodes: 1143 -END *) - -theorem sn3_appls_beta: - \forall (c: C).(\forall (v: T).(\forall (t: T).(\forall (us: TList).((sn3 c -(THeads (Flat Appl) us (THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c -w) \to (sn3 c (THeads (Flat Appl) us (THead (Flat Appl) v (THead (Bind Abst) -w t)))))))))) -\def - \lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (us: -TList).(TList_ind (\lambda (t0: TList).((sn3 c (THeads (Flat Appl) t0 (THead -(Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THeads (Flat -Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t)))))))) (\lambda (H: -(sn3 c (THead (Bind Abbr) v t))).(\lambda (w: T).(\lambda (H0: (sn3 c -w)).(sn3_beta c v t H w H0)))) (\lambda (u: T).(\lambda (us0: -TList).(TList_ind (\lambda (t0: TList).((((sn3 c (THeads (Flat Appl) t0 -(THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THeads -(Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t)))))))) \to ((sn3 -c (THead (Flat Appl) u (THeads (Flat Appl) t0 (THead (Bind Abbr) v t)))) \to -(\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) u (THeads (Flat -Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t)))))))))) (\lambda (_: -(((sn3 c (THeads (Flat Appl) TNil (THead (Bind Abbr) v t))) \to (\forall (w: -T).((sn3 c w) \to (sn3 c (THeads (Flat Appl) TNil (THead (Flat Appl) v (THead -(Bind Abst) w t))))))))).(\lambda (H0: (sn3 c (THead (Flat Appl) u (THeads -(Flat Appl) TNil (THead (Bind Abbr) v t))))).(\lambda (w: T).(\lambda (H1: -(sn3 c w)).(sn3_appl_beta c u v t H0 w H1))))) (\lambda (t0: T).(\lambda (t1: -TList).(\lambda (_: (((((sn3 c (THeads (Flat Appl) t1 (THead (Bind Abbr) v -t))) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THeads (Flat Appl) t1 (THead -(Flat Appl) v (THead (Bind Abst) w t)))))))) \to ((sn3 c (THead (Flat Appl) u -(THeads (Flat Appl) t1 (THead (Bind Abbr) v t)))) \to (\forall (w: T).((sn3 c -w) \to (sn3 c (THead (Flat Appl) u (THeads (Flat Appl) t1 (THead (Flat Appl) -v (THead (Bind Abst) w t))))))))))).(\lambda (H0: (((sn3 c (THeads (Flat -Appl) (TCons t0 t1) (THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w) -\to (sn3 c (THeads (Flat Appl) (TCons t0 t1) (THead (Flat Appl) v (THead -(Bind Abst) w t))))))))).(\lambda (H1: (sn3 c (THead (Flat Appl) u (THeads -(Flat Appl) (TCons t0 t1) (THead (Bind Abbr) v t))))).(\lambda (w: -T).(\lambda (H2: (sn3 c w)).(let H_x \def (sn3_gen_flat Appl c u (THeads -(Flat Appl) (TCons t0 t1) (THead (Bind Abbr) v t)) H1) in (let H3 \def H_x in -(land_ind (sn3 c u) (sn3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 -(THead (Bind Abbr) v t)))) (sn3 c (THead (Flat Appl) u (THeads (Flat Appl) -(TCons t0 t1) (THead (Flat Appl) v (THead (Bind Abst) w t))))) (\lambda (H4: -(sn3 c u)).(\lambda (H5: (sn3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 -(THead (Bind Abbr) v t))))).(sn3_appl_appls t0 (THead (Flat Appl) v (THead -(Bind Abst) w t)) t1 c (H0 H5 w H2) u H4 (\lambda (u2: T).(\lambda (H6: (pr3 -c (THeads (Flat Appl) (TCons t0 t1) (THead (Flat Appl) v (THead (Bind Abst) w -t))) u2)).(\lambda (H7: (((iso (THeads (Flat Appl) (TCons t0 t1) (THead (Flat -Appl) v (THead (Bind Abst) w t))) u2) \to (\forall (P: Prop).P)))).(let H8 -\def (pr3_iso_appls_beta (TCons t0 t1) v w t c u2 H6 H7) in (sn3_pr3_trans c -(THead (Flat Appl) u (THeads (Flat Appl) (TCons t0 t1) (THead (Bind Abbr) v -t))) H1 (THead (Flat Appl) u u2) (pr3_thin_dx c (THeads (Flat Appl) (TCons t0 -t1) (THead (Bind Abbr) v t)) u2 H8 u Appl))))))))) H3)))))))))) us0))) us)))). -(* COMMENTS -Initial nodes: 987 -END *) - -theorem sn3_lift: - \forall (d: C).(\forall (t: T).((sn3 d t) \to (\forall (c: C).(\forall (h: -nat).(\forall (i: nat).((drop h i c d) \to (sn3 c (lift h i t)))))))) -\def - \lambda (d: C).(\lambda (t: T).(\lambda (H: (sn3 d t)).(sn3_ind d (\lambda -(t0: T).(\forall (c: C).(\forall (h: nat).(\forall (i: nat).((drop h i c d) -\to (sn3 c (lift h i t0))))))) (\lambda (t1: T).(\lambda (_: ((\forall (t2: -T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 d t1 t2) \to (sn3 d -t2)))))).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: -Prop).P))) \to ((pr3 d t1 t2) \to (\forall (c: C).(\forall (h: nat).(\forall -(i: nat).((drop h i c d) \to (sn3 c (lift h i t2))))))))))).(\lambda (c: -C).(\lambda (h: nat).(\lambda (i: nat).(\lambda (H2: (drop h i c -d)).(sn3_pr2_intro c (lift h i t1) (\lambda (t2: T).(\lambda (H3: (((eq T -(lift h i t1) t2) \to (\forall (P: Prop).P)))).(\lambda (H4: (pr2 c (lift h i -t1) t2)).(let H5 \def (pr2_gen_lift c t1 t2 h i H4 d H2) in (ex2_ind T -(\lambda (t3: T).(eq T t2 (lift h i t3))) (\lambda (t3: T).(pr2 d t1 t3)) -(sn3 c t2) (\lambda (x: T).(\lambda (H6: (eq T t2 (lift h i x))).(\lambda -(H7: (pr2 d t1 x)).(let H8 \def (eq_ind T t2 (\lambda (t0: T).((eq T (lift h -i t1) t0) \to (\forall (P: Prop).P))) H3 (lift h i x) H6) in (eq_ind_r T -(lift h i x) (\lambda (t0: T).(sn3 c t0)) (H1 x (\lambda (H9: (eq T t1 -x)).(\lambda (P: Prop).(let H10 \def (eq_ind_r T x (\lambda (t0: T).((eq T -(lift h i t1) (lift h i t0)) \to (\forall (P0: Prop).P0))) H8 t1 H9) in (let -H11 \def (eq_ind_r T x (\lambda (t0: T).(pr2 d t1 t0)) H7 t1 H9) in (H10 -(refl_equal T (lift h i t1)) P))))) (pr3_pr2 d t1 x H7) c h i H2) t2 H6))))) -H5))))))))))))) t H))). -(* COMMENTS -Initial nodes: 439 -END *) - -theorem sn3_abbr: - \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c -(CHead d (Bind Abbr) v)) \to ((sn3 d v) \to (sn3 c (TLRef i))))))) -\def - \lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda -(H: (getl i c (CHead d (Bind Abbr) v))).(\lambda (H0: (sn3 d -v)).(sn3_pr2_intro c (TLRef i) (\lambda (t2: T).(\lambda (H1: (((eq T (TLRef -i) t2) \to (\forall (P: Prop).P)))).(\lambda (H2: (pr2 c (TLRef i) t2)).(let -H3 \def (pr2_gen_lref c t2 i H2) in (or_ind (eq T t2 (TLRef i)) (ex2_2 C T -(\lambda (d0: C).(\lambda (u: T).(getl i c (CHead d0 (Bind Abbr) u)))) -(\lambda (_: C).(\lambda (u: T).(eq T t2 (lift (S i) O u))))) (sn3 c t2) -(\lambda (H4: (eq T t2 (TLRef i))).(let H5 \def (eq_ind T t2 (\lambda (t: -T).((eq T (TLRef i) t) \to (\forall (P: Prop).P))) H1 (TLRef i) H4) in -(eq_ind_r T (TLRef i) (\lambda (t: T).(sn3 c t)) (H5 (refl_equal T (TLRef i)) -(sn3 c (TLRef i))) t2 H4))) (\lambda (H4: (ex2_2 C T (\lambda (d0: -C).(\lambda (u: T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_: -C).(\lambda (u: T).(eq T t2 (lift (S i) O u)))))).(ex2_2_ind C T (\lambda -(d0: C).(\lambda (u: T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_: -C).(\lambda (u: T).(eq T t2 (lift (S i) O u)))) (sn3 c t2) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (H5: (getl i c (CHead x0 (Bind Abbr) -x1))).(\lambda (H6: (eq T t2 (lift (S i) O x1))).(let H7 \def (eq_ind T t2 -(\lambda (t: T).((eq T (TLRef i) t) \to (\forall (P: Prop).P))) H1 (lift (S -i) O x1) H6) in (eq_ind_r T (lift (S i) O x1) (\lambda (t: T).(sn3 c t)) (let -H8 \def (eq_ind C (CHead d (Bind Abbr) v) (\lambda (c0: C).(getl i c c0)) H -(CHead x0 (Bind Abbr) x1) (getl_mono c (CHead d (Bind Abbr) v) i H (CHead x0 -(Bind Abbr) x1) H5)) in (let H9 \def (f_equal C C (\lambda (e: C).(match e in -C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) -\Rightarrow c0])) (CHead d (Bind Abbr) v) (CHead x0 (Bind Abbr) x1) -(getl_mono c (CHead d (Bind Abbr) v) i H (CHead x0 (Bind Abbr) x1) H5)) in -((let H10 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: -C).T) with [(CSort _) \Rightarrow v | (CHead _ _ t) \Rightarrow t])) (CHead d -(Bind Abbr) v) (CHead x0 (Bind Abbr) x1) (getl_mono c (CHead d (Bind Abbr) v) -i H (CHead x0 (Bind Abbr) x1) H5)) in (\lambda (H11: (eq C d x0)).(let H12 -\def (eq_ind_r T x1 (\lambda (t: T).(getl i c (CHead x0 (Bind Abbr) t))) H8 v -H10) in (eq_ind T v (\lambda (t: T).(sn3 c (lift (S i) O t))) (let H13 \def -(eq_ind_r C x0 (\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr) v))) H12 d -H11) in (sn3_lift d v H0 c (S i) O (getl_drop Abbr c d v i H13))) x1 H10)))) -H9))) t2 H6)))))) H4)) H3))))))))))). -(* COMMENTS -Initial nodes: 743 -END *) - -theorem sn3_appls_abbr: - \forall (c: C).(\forall (d: C).(\forall (w: T).(\forall (i: nat).((getl i c -(CHead d (Bind Abbr) w)) \to (\forall (vs: TList).((sn3 c (THeads (Flat Appl) -vs (lift (S i) O w))) \to (sn3 c (THeads (Flat Appl) vs (TLRef i))))))))) -\def - \lambda (c: C).(\lambda (d: C).(\lambda (w: T).(\lambda (i: nat).(\lambda -(H: (getl i c (CHead d (Bind Abbr) w))).(\lambda (vs: TList).(TList_ind -(\lambda (t: TList).((sn3 c (THeads (Flat Appl) t (lift (S i) O w))) \to (sn3 -c (THeads (Flat Appl) t (TLRef i))))) (\lambda (H0: (sn3 c (lift (S i) O -w))).(let H_y \def (sn3_gen_lift c w (S i) O H0 d (getl_drop Abbr c d w i H)) -in (sn3_abbr c d w i H H_y))) (\lambda (v: T).(\lambda (vs0: -TList).(TList_ind (\lambda (t: TList).((((sn3 c (THeads (Flat Appl) t (lift -(S i) O w))) \to (sn3 c (THeads (Flat Appl) t (TLRef i))))) \to ((sn3 c -(THead (Flat Appl) v (THeads (Flat Appl) t (lift (S i) O w)))) \to (sn3 c -(THead (Flat Appl) v (THeads (Flat Appl) t (TLRef i))))))) (\lambda (_: -(((sn3 c (THeads (Flat Appl) TNil (lift (S i) O w))) \to (sn3 c (THeads (Flat -Appl) TNil (TLRef i)))))).(\lambda (H1: (sn3 c (THead (Flat Appl) v (THeads -(Flat Appl) TNil (lift (S i) O w))))).(sn3_appl_abbr c d w i H v H1))) -(\lambda (t: T).(\lambda (t0: TList).(\lambda (_: (((((sn3 c (THeads (Flat -Appl) t0 (lift (S i) O w))) \to (sn3 c (THeads (Flat Appl) t0 (TLRef i))))) -\to ((sn3 c (THead (Flat Appl) v (THeads (Flat Appl) t0 (lift (S i) O w)))) -\to (sn3 c (THead (Flat Appl) v (THeads (Flat Appl) t0 (TLRef -i)))))))).(\lambda (H1: (((sn3 c (THeads (Flat Appl) (TCons t t0) (lift (S i) -O w))) \to (sn3 c (THeads (Flat Appl) (TCons t t0) (TLRef i)))))).(\lambda -(H2: (sn3 c (THead (Flat Appl) v (THeads (Flat Appl) (TCons t t0) (lift (S i) -O w))))).(let H_x \def (sn3_gen_flat Appl c v (THeads (Flat Appl) (TCons t -t0) (lift (S i) O w)) H2) in (let H3 \def H_x in (land_ind (sn3 c v) (sn3 c -(THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O w)))) (sn3 c (THead -(Flat Appl) v (THeads (Flat Appl) (TCons t t0) (TLRef i)))) (\lambda (H4: -(sn3 c v)).(\lambda (H5: (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 -(lift (S i) O w))))).(sn3_appl_appls t (TLRef i) t0 c (H1 H5) v H4 (\lambda -(u2: T).(\lambda (H6: (pr3 c (THeads (Flat Appl) (TCons t t0) (TLRef i)) -u2)).(\lambda (H7: (((iso (THeads (Flat Appl) (TCons t t0) (TLRef i)) u2) \to -(\forall (P: Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) v (THeads (Flat -Appl) (TCons t t0) (lift (S i) O w))) H2 (THead (Flat Appl) v u2) -(pr3_thin_dx c (THeads (Flat Appl) (TCons t t0) (lift (S i) O w)) u2 -(pr3_iso_appls_abbr c d w i H (TCons t t0) u2 H6 H7) v Appl)))))))) -H3)))))))) vs0))) vs)))))). -(* COMMENTS -Initial nodes: 797 -END *) - -theorem sns3_lifts: - \forall (c: C).(\forall (d: C).(\forall (h: nat).(\forall (i: nat).((drop h -i c d) \to (\forall (ts: TList).((sns3 d ts) \to (sns3 c (lifts h i ts)))))))) -\def - \lambda (c: C).(\lambda (d: C).(\lambda (h: nat).(\lambda (i: nat).(\lambda -(H: (drop h i c d)).(\lambda (ts: TList).(TList_ind (\lambda (t: -TList).((sns3 d t) \to (sns3 c (lifts h i t)))) (\lambda (H0: True).H0) -(\lambda (t: T).(\lambda (t0: TList).(\lambda (H0: (((sns3 d t0) \to (sns3 c -(lifts h i t0))))).(\lambda (H1: (land (sn3 d t) (sns3 d t0))).(let H2 \def -H1 in (land_ind (sn3 d t) (sns3 d t0) (land (sn3 c (lift h i t)) (sns3 c -(lifts h i t0))) (\lambda (H3: (sn3 d t)).(\lambda (H4: (sns3 d t0)).(conj -(sn3 c (lift h i t)) (sns3 c (lifts h i t0)) (sn3_lift d t H3 c h i H) (H0 -H4)))) H2)))))) ts)))))). -(* COMMENTS -Initial nodes: 185 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/spare.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/spare.ma deleted file mode 100644 index 95d2c24ee..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/spare.ma +++ /dev/null @@ -1,38 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/theory.ma". - -axiom pc3_gen_appls_sort_abst: - \forall (c: C).(\forall (vs: TList).(\forall (w: T).(\forall (u: T).(\forall -(n: nat).((pc3 c (THeads (Flat Appl) vs (TSort n)) (THead (Bind Abst) w u)) -\to False))))) -. - -axiom pc3_gen_appls_lref_abst: - \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c -(CHead d (Bind Abst) v)) \to (\forall (vs: TList).(\forall (w: T).(\forall -(u: T).((pc3 c (THeads (Flat Appl) vs (TLRef i)) (THead (Bind Abst) w u)) \to -False)))))))) -. - -axiom pc3_gen_appls_lref_sort: - \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c -(CHead d (Bind Abst) v)) \to (\forall (vs: TList).(\forall (ws: -TList).(\forall (n: nat).((pc3 c (THeads (Flat Appl) vs (TLRef i)) (THeads -(Flat Appl) ws (TSort n))) \to False)))))))) -. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/sty0/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/sty0/defs.ma deleted file mode 100644 index 6468d9dbb..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/sty0/defs.ma +++ /dev/null @@ -1,39 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/G/defs.ma". - -include "Basic-1/getl/defs.ma". - -inductive sty0 (g: G): C \to (T \to (T \to Prop)) \def -| sty0_sort: \forall (c: C).(\forall (n: nat).(sty0 g c (TSort n) (TSort -(next g n)))) -| sty0_abbr: \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: -nat).((getl i c (CHead d (Bind Abbr) v)) \to (\forall (w: T).((sty0 g d v w) -\to (sty0 g c (TLRef i) (lift (S i) O w)))))))) -| sty0_abst: \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: -nat).((getl i c (CHead d (Bind Abst) v)) \to (\forall (w: T).((sty0 g d v w) -\to (sty0 g c (TLRef i) (lift (S i) O v)))))))) -| sty0_bind: \forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t1: -T).(\forall (t2: T).((sty0 g (CHead c (Bind b) v) t1 t2) \to (sty0 g c (THead -(Bind b) v t1) (THead (Bind b) v t2))))))) -| sty0_appl: \forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall (t2: -T).((sty0 g c t1 t2) \to (sty0 g c (THead (Flat Appl) v t1) (THead (Flat -Appl) v t2)))))) -| sty0_cast: \forall (c: C).(\forall (v1: T).(\forall (v2: T).((sty0 g c v1 -v2) \to (\forall (t1: T).(\forall (t2: T).((sty0 g c t1 t2) \to (sty0 g c -(THead (Flat Cast) v1 t1) (THead (Flat Cast) v2 t2)))))))). - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/sty0/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/sty0/fwd.ma deleted file mode 100644 index 134ec3c10..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/sty0/fwd.ma +++ /dev/null @@ -1,562 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/sty0/defs.ma". - -theorem sty0_gen_sort: - \forall (g: G).(\forall (c: C).(\forall (x: T).(\forall (n: nat).((sty0 g c -(TSort n) x) \to (eq T x (TSort (next g n))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda -(H: (sty0 g c (TSort n) x)).(insert_eq T (TSort n) (\lambda (t: T).(sty0 g c -t x)) (\lambda (_: T).(eq T x (TSort (next g n)))) (\lambda (y: T).(\lambda -(H0: (sty0 g c y x)).(sty0_ind g (\lambda (_: C).(\lambda (t: T).(\lambda -(t0: T).((eq T t (TSort n)) \to (eq T t0 (TSort (next g n))))))) (\lambda (_: -C).(\lambda (n0: nat).(\lambda (H1: (eq T (TSort n0) (TSort n))).(let H2 \def -(f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with -[(TSort n1) \Rightarrow n1 | (TLRef _) \Rightarrow n0 | (THead _ _ _) -\Rightarrow n0])) (TSort n0) (TSort n) H1) in (eq_ind_r nat n (\lambda (n1: -nat).(eq T (TSort (next g n1)) (TSort (next g n)))) (refl_equal T (TSort -(next g n))) n0 H2))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (v: -T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind Abbr) -v))).(\lambda (w: T).(\lambda (_: (sty0 g d v w)).(\lambda (_: (((eq T v -(TSort n)) \to (eq T w (TSort (next g n)))))).(\lambda (H4: (eq T (TLRef i) -(TSort n))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (TSort n) H4) in -(False_ind (eq T (lift (S i) O w) (TSort (next g n))) H5))))))))))) (\lambda -(c0: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (_: (getl -i c0 (CHead d (Bind Abst) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v -w)).(\lambda (_: (((eq T v (TSort n)) \to (eq T w (TSort (next g -n)))))).(\lambda (H4: (eq T (TLRef i) (TSort n))).(let H5 \def (eq_ind T -(TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) -\Rightarrow False])) I (TSort n) H4) in (False_ind (eq T (lift (S i) O v) -(TSort (next g n))) H5))))))))))) (\lambda (b: B).(\lambda (c0: C).(\lambda -(v: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (sty0 g (CHead c0 (Bind -b) v) t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2 (TSort (next g -n)))))).(\lambda (H3: (eq T (THead (Bind b) v t1) (TSort n))).(let H4 \def -(eq_ind T (THead (Bind b) v t1) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H3) in -(False_ind (eq T (THead (Bind b) v t2) (TSort (next g n))) H4)))))))))) -(\lambda (c0: C).(\lambda (v: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(_: (sty0 g c0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2 (TSort -(next g n)))))).(\lambda (H3: (eq T (THead (Flat Appl) v t1) (TSort n))).(let -H4 \def (eq_ind T (THead (Flat Appl) v t1) (\lambda (ee: T).(match ee in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H3) in -(False_ind (eq T (THead (Flat Appl) v t2) (TSort (next g n))) H4))))))))) -(\lambda (c0: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (sty0 g c0 v1 -v2)).(\lambda (_: (((eq T v1 (TSort n)) \to (eq T v2 (TSort (next g -n)))))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (sty0 g c0 t1 -t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2 (TSort (next g -n)))))).(\lambda (H5: (eq T (THead (Flat Cast) v1 t1) (TSort n))).(let H6 -\def (eq_ind T (THead (Flat Cast) v1 t1) (\lambda (ee: T).(match ee in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H5) in -(False_ind (eq T (THead (Flat Cast) v2 t2) (TSort (next g n))) H6)))))))))))) -c y x H0))) H))))). -(* COMMENTS -Initial nodes: 869 -END *) - -theorem sty0_gen_lref: - \forall (g: G).(\forall (c: C).(\forall (x: T).(\forall (n: nat).((sty0 g c -(TLRef n) x) \to (or (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda -(_: T).(getl n c (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: -T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(t: T).(eq T x (lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda -(u: T).(\lambda (_: T).(getl n c (CHead e (Bind Abst) u))))) (\lambda (e: -C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq T x (lift (S n) O u))))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda -(H: (sty0 g c (TLRef n) x)).(insert_eq T (TLRef n) (\lambda (t: T).(sty0 g c -t x)) (\lambda (_: T).(or (ex3_3 C T T (\lambda (e: C).(\lambda (u: -T).(\lambda (_: T).(getl n c (CHead e (Bind Abbr) u))))) (\lambda (e: -C).(\lambda (u: T).(\lambda (t0: T).(sty0 g e u t0)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (t0: T).(eq T x (lift (S n) O t0)))))) (ex3_3 C T -T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead e (Bind -Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t0: T).(sty0 g e u -t0)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T x (lift (S n) O -u)))))))) (\lambda (y: T).(\lambda (H0: (sty0 g c y x)).(sty0_ind g (\lambda -(c0: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (TLRef n)) \to (or (ex3_3 C -T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind -Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t1: T).(sty0 g e u -t1)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t1: T).(eq T t0 (lift (S n) -O t1)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl -n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda -(t1: T).(sty0 g e u t1)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: -T).(eq T t0 (lift (S n) O u))))))))))) (\lambda (c0: C).(\lambda (n0: -nat).(\lambda (H1: (eq T (TSort n0) (TLRef n))).(let H2 \def (eq_ind T (TSort -n0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort -_) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow -False])) I (TLRef n) H1) in (False_ind (or (ex3_3 C T T (\lambda (e: -C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u))))) -(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (t: T).(eq T (TSort (next g n0)) (lift (S n) -O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl -n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: -T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T -(TSort (next g n0)) (lift (S n) O u))))))) H2))))) (\lambda (c0: C).(\lambda -(d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d -(Bind Abbr) v))).(\lambda (w: T).(\lambda (H2: (sty0 g d v w)).(\lambda (_: -(((eq T v (TLRef n)) \to (or (ex3_3 C T T (\lambda (e: C).(\lambda (u: -T).(\lambda (_: T).(getl n d (CHead e (Bind Abbr) u))))) (\lambda (e: -C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda -(_: T).(\lambda (t: T).(eq T w (lift (S n) O t)))))) (ex3_3 C T T (\lambda -(e: C).(\lambda (u: T).(\lambda (_: T).(getl n d (CHead e (Bind Abst) u))))) -(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda -(_: C).(\lambda (u: T).(\lambda (_: T).(eq T w (lift (S n) O -u)))))))))).(\lambda (H4: (eq T (TLRef i) (TLRef n))).(let H5 \def (f_equal T -nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort _) -\Rightarrow i | (TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow i])) -(TLRef i) (TLRef n) H4) in (let H6 \def (eq_ind nat i (\lambda (n0: -nat).(getl n0 c0 (CHead d (Bind Abbr) v))) H1 n H5) in (eq_ind_r nat n -(\lambda (n0: nat).(or (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda -(_: T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: -T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(t: T).(eq T (lift (S n0) O w) (lift (S n) O t)))))) (ex3_3 C T T (\lambda -(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) -(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda -(_: C).(\lambda (u: T).(\lambda (_: T).(eq T (lift (S n0) O w) (lift (S n) O -u)))))))) (or_introl (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda -(_: T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: -T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(t: T).(eq T (lift (S n) O w) (lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: -C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) -(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda -(_: C).(\lambda (u: T).(\lambda (_: T).(eq T (lift (S n) O w) (lift (S n) O -u)))))) (ex3_3_intro C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: -T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: -T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(t: T).(eq T (lift (S n) O w) (lift (S n) O t))))) d v w H6 H2 (refl_equal T -(lift (S n) O w)))) i H5)))))))))))) (\lambda (c0: C).(\lambda (d: -C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d (Bind -Abst) v))).(\lambda (w: T).(\lambda (H2: (sty0 g d v w)).(\lambda (_: (((eq T -v (TLRef n)) \to (or (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda -(_: T).(getl n d (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: -T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(t: T).(eq T w (lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda -(u: T).(\lambda (_: T).(getl n d (CHead e (Bind Abst) u))))) (\lambda (e: -C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq T w (lift (S n) O u)))))))))).(\lambda (H4: (eq T -(TLRef i) (TLRef n))).(let H5 \def (f_equal T nat (\lambda (e: T).(match e in -T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow i | (TLRef n0) -\Rightarrow n0 | (THead _ _ _) \Rightarrow i])) (TLRef i) (TLRef n) H4) in -(let H6 \def (eq_ind nat i (\lambda (n0: nat).(getl n0 c0 (CHead d (Bind -Abst) v))) H1 n H5) in (eq_ind_r nat n (\lambda (n0: nat).(or (ex3_3 C T T -(\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind -Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u -t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T (lift (S n0) O v) -(lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda -(_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: -T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda -(_: T).(eq T (lift (S n0) O v) (lift (S n) O u)))))))) (or_intror (ex3_3 C T -T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind -Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u -t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T (lift (S n) O v) -(lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda -(_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: -T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda -(_: T).(eq T (lift (S n) O v) (lift (S n) O u)))))) (ex3_3_intro C T T -(\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind -Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u -t)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T (lift (S n) O v) -(lift (S n) O u))))) d v w H6 H2 (refl_equal T (lift (S n) O v)))) i -H5)))))))))))) (\lambda (b: B).(\lambda (c0: C).(\lambda (v: T).(\lambda (t1: -T).(\lambda (t2: T).(\lambda (_: (sty0 g (CHead c0 (Bind b) v) t1 -t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (or (ex3_3 C T T (\lambda (e: -C).(\lambda (u: T).(\lambda (_: T).(getl n (CHead c0 (Bind b) v) (CHead e -(Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e -u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T t2 (lift (S n) -O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl -n (CHead c0 (Bind b) v) (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda -(u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(eq T t2 (lift (S n) O u)))))))))).(\lambda (H3: (eq T -(THead (Bind b) v t1) (TLRef n))).(let H4 \def (eq_ind T (THead (Bind b) v -t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort -_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) -\Rightarrow True])) I (TLRef n) H3) in (False_ind (or (ex3_3 C T T (\lambda -(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u))))) -(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (t: T).(eq T (THead (Bind b) v t2) (lift (S -n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: -T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: -T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda -(_: T).(eq T (THead (Bind b) v t2) (lift (S n) O u))))))) H4)))))))))) -(\lambda (c0: C).(\lambda (v: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(_: (sty0 g c0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (or (ex3_3 C T -T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind -Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u -t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T t2 (lift (S n) O -t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n -c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: -T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T t2 -(lift (S n) O u)))))))))).(\lambda (H3: (eq T (THead (Flat Appl) v t1) (TLRef -n))).(let H4 \def (eq_ind T (THead (Flat Appl) v t1) (\lambda (ee: T).(match -ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | -(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) -H3) in (False_ind (or (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda -(_: T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: -T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(t: T).(eq T (THead (Flat Appl) v t2) (lift (S n) O t)))))) (ex3_3 C T T -(\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind -Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u -t)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T (THead (Flat -Appl) v t2) (lift (S n) O u))))))) H4))))))))) (\lambda (c0: C).(\lambda (v1: -T).(\lambda (v2: T).(\lambda (_: (sty0 g c0 v1 v2)).(\lambda (_: (((eq T v1 -(TLRef n)) \to (or (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: -T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: -T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(t: T).(eq T v2 (lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda -(u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: -C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq T v2 (lift (S n) O u)))))))))).(\lambda (t1: -T).(\lambda (t2: T).(\lambda (_: (sty0 g c0 t1 t2)).(\lambda (_: (((eq T t1 -(TLRef n)) \to (or (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: -T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: -T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(t: T).(eq T t2 (lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda -(u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: -C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq T t2 (lift (S n) O u)))))))))).(\lambda (H5: (eq T -(THead (Flat Cast) v1 t1) (TLRef n))).(let H6 \def (eq_ind T (THead (Flat -Cast) v1 t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) -with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ -_) \Rightarrow True])) I (TLRef n) H5) in (False_ind (or (ex3_3 C T T -(\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind -Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u -t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T (THead (Flat -Cast) v2 t2) (lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: -T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: -C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq T (THead (Flat Cast) v2 t2) (lift (S n) O u))))))) -H6)))))))))))) c y x H0))) H))))). -(* COMMENTS -Initial nodes: 3231 -END *) - -theorem sty0_gen_bind: - \forall (g: G).(\forall (b: B).(\forall (c: C).(\forall (u: T).(\forall (t1: -T).(\forall (x: T).((sty0 g c (THead (Bind b) u t1) x) \to (ex2 T (\lambda -(t2: T).(sty0 g (CHead c (Bind b) u) t1 t2)) (\lambda (t2: T).(eq T x (THead -(Bind b) u t2)))))))))) -\def - \lambda (g: G).(\lambda (b: B).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: -T).(\lambda (x: T).(\lambda (H: (sty0 g c (THead (Bind b) u t1) -x)).(insert_eq T (THead (Bind b) u t1) (\lambda (t: T).(sty0 g c t x)) -(\lambda (_: T).(ex2 T (\lambda (t2: T).(sty0 g (CHead c (Bind b) u) t1 t2)) -(\lambda (t2: T).(eq T x (THead (Bind b) u t2))))) (\lambda (y: T).(\lambda -(H0: (sty0 g c y x)).(sty0_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda -(t0: T).((eq T t (THead (Bind b) u t1)) \to (ex2 T (\lambda (t2: T).(sty0 g -(CHead c0 (Bind b) u) t1 t2)) (\lambda (t2: T).(eq T t0 (THead (Bind b) u -t2)))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (H1: (eq T (TSort n) -(THead (Bind b) u t1))).(let H2 \def (eq_ind T (TSort n) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I -(THead (Bind b) u t1) H1) in (False_ind (ex2 T (\lambda (t2: T).(sty0 g -(CHead c0 (Bind b) u) t1 t2)) (\lambda (t2: T).(eq T (TSort (next g n)) -(THead (Bind b) u t2)))) H2))))) (\lambda (c0: C).(\lambda (d: C).(\lambda -(v: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind Abbr) -v))).(\lambda (w: T).(\lambda (_: (sty0 g d v w)).(\lambda (_: (((eq T v -(THead (Bind b) u t1)) \to (ex2 T (\lambda (t2: T).(sty0 g (CHead d (Bind b) -u) t1 t2)) (\lambda (t2: T).(eq T w (THead (Bind b) u t2))))))).(\lambda (H4: -(eq T (TLRef i) (THead (Bind b) u t1))).(let H5 \def (eq_ind T (TLRef i) -(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow -False])) I (THead (Bind b) u t1) H4) in (False_ind (ex2 T (\lambda (t2: -T).(sty0 g (CHead c0 (Bind b) u) t1 t2)) (\lambda (t2: T).(eq T (lift (S i) O -w) (THead (Bind b) u t2)))) H5))))))))))) (\lambda (c0: C).(\lambda (d: -C).(\lambda (v: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind -Abst) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v w)).(\lambda (_: (((eq T -v (THead (Bind b) u t1)) \to (ex2 T (\lambda (t2: T).(sty0 g (CHead d (Bind -b) u) t1 t2)) (\lambda (t2: T).(eq T w (THead (Bind b) u t2))))))).(\lambda -(H4: (eq T (TLRef i) (THead (Bind b) u t1))).(let H5 \def (eq_ind T (TLRef i) -(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow -False])) I (THead (Bind b) u t1) H4) in (False_ind (ex2 T (\lambda (t2: -T).(sty0 g (CHead c0 (Bind b) u) t1 t2)) (\lambda (t2: T).(eq T (lift (S i) O -v) (THead (Bind b) u t2)))) H5))))))))))) (\lambda (b0: B).(\lambda (c0: -C).(\lambda (v: T).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H1: (sty0 g -(CHead c0 (Bind b0) v) t0 t2)).(\lambda (H2: (((eq T t0 (THead (Bind b) u -t1)) \to (ex2 T (\lambda (t3: T).(sty0 g (CHead (CHead c0 (Bind b0) v) (Bind -b) u) t1 t3)) (\lambda (t3: T).(eq T t2 (THead (Bind b) u t3))))))).(\lambda -(H3: (eq T (THead (Bind b0) v t0) (THead (Bind b) u t1))).(let H4 \def -(f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with -[(TSort _) \Rightarrow b0 | (TLRef _) \Rightarrow b0 | (THead k _ _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b1) -\Rightarrow b1 | (Flat _) \Rightarrow b0])])) (THead (Bind b0) v t0) (THead -(Bind b) u t1) H3) in ((let H5 \def (f_equal T T (\lambda (e: T).(match e in -T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v | (TLRef _) -\Rightarrow v | (THead _ t _) \Rightarrow t])) (THead (Bind b0) v t0) (THead -(Bind b) u t1) H3) in ((let H6 \def (f_equal T T (\lambda (e: T).(match e in -T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) -\Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Bind b0) v t0) (THead -(Bind b) u t1) H3) in (\lambda (H7: (eq T v u)).(\lambda (H8: (eq B b0 -b)).(let H9 \def (eq_ind T t0 (\lambda (t: T).((eq T t (THead (Bind b) u t1)) -\to (ex2 T (\lambda (t3: T).(sty0 g (CHead (CHead c0 (Bind b0) v) (Bind b) u) -t1 t3)) (\lambda (t3: T).(eq T t2 (THead (Bind b) u t3)))))) H2 t1 H6) in -(let H10 \def (eq_ind T t0 (\lambda (t: T).(sty0 g (CHead c0 (Bind b0) v) t -t2)) H1 t1 H6) in (let H11 \def (eq_ind T v (\lambda (t: T).((eq T t1 (THead -(Bind b) u t1)) \to (ex2 T (\lambda (t3: T).(sty0 g (CHead (CHead c0 (Bind -b0) t) (Bind b) u) t1 t3)) (\lambda (t3: T).(eq T t2 (THead (Bind b) u -t3)))))) H9 u H7) in (let H12 \def (eq_ind T v (\lambda (t: T).(sty0 g (CHead -c0 (Bind b0) t) t1 t2)) H10 u H7) in (eq_ind_r T u (\lambda (t: T).(ex2 T -(\lambda (t3: T).(sty0 g (CHead c0 (Bind b) u) t1 t3)) (\lambda (t3: T).(eq T -(THead (Bind b0) t t2) (THead (Bind b) u t3))))) (let H13 \def (eq_ind B b0 -(\lambda (b1: B).((eq T t1 (THead (Bind b) u t1)) \to (ex2 T (\lambda (t3: -T).(sty0 g (CHead (CHead c0 (Bind b1) u) (Bind b) u) t1 t3)) (\lambda (t3: -T).(eq T t2 (THead (Bind b) u t3)))))) H11 b H8) in (let H14 \def (eq_ind B -b0 (\lambda (b1: B).(sty0 g (CHead c0 (Bind b1) u) t1 t2)) H12 b H8) in -(eq_ind_r B b (\lambda (b1: B).(ex2 T (\lambda (t3: T).(sty0 g (CHead c0 -(Bind b) u) t1 t3)) (\lambda (t3: T).(eq T (THead (Bind b1) u t2) (THead -(Bind b) u t3))))) (ex_intro2 T (\lambda (t3: T).(sty0 g (CHead c0 (Bind b) -u) t1 t3)) (\lambda (t3: T).(eq T (THead (Bind b) u t2) (THead (Bind b) u -t3))) t2 H14 (refl_equal T (THead (Bind b) u t2))) b0 H8))) v H7)))))))) H5)) -H4)))))))))) (\lambda (c0: C).(\lambda (v: T).(\lambda (t0: T).(\lambda (t2: -T).(\lambda (_: (sty0 g c0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind b) u -t1)) \to (ex2 T (\lambda (t3: T).(sty0 g (CHead c0 (Bind b) u) t1 t3)) -(\lambda (t3: T).(eq T t2 (THead (Bind b) u t3))))))).(\lambda (H3: (eq T -(THead (Flat Appl) v t0) (THead (Bind b) u t1))).(let H4 \def (eq_ind T -(THead (Flat Appl) v t0) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind -b) u t1) H3) in (False_ind (ex2 T (\lambda (t3: T).(sty0 g (CHead c0 (Bind b) -u) t1 t3)) (\lambda (t3: T).(eq T (THead (Flat Appl) v t2) (THead (Bind b) u -t3)))) H4))))))))) (\lambda (c0: C).(\lambda (v1: T).(\lambda (v2: -T).(\lambda (_: (sty0 g c0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind b) u -t1)) \to (ex2 T (\lambda (t2: T).(sty0 g (CHead c0 (Bind b) u) t1 t2)) -(\lambda (t2: T).(eq T v2 (THead (Bind b) u t2))))))).(\lambda (t0: -T).(\lambda (t2: T).(\lambda (_: (sty0 g c0 t0 t2)).(\lambda (_: (((eq T t0 -(THead (Bind b) u t1)) \to (ex2 T (\lambda (t3: T).(sty0 g (CHead c0 (Bind b) -u) t1 t3)) (\lambda (t3: T).(eq T t2 (THead (Bind b) u t3))))))).(\lambda -(H5: (eq T (THead (Flat Cast) v1 t0) (THead (Bind b) u t1))).(let H6 \def -(eq_ind T (THead (Flat Cast) v1 t0) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind b) u t1) H5) in (False_ind (ex2 T (\lambda (t3: -T).(sty0 g (CHead c0 (Bind b) u) t1 t3)) (\lambda (t3: T).(eq T (THead (Flat -Cast) v2 t2) (THead (Bind b) u t3)))) H6)))))))))))) c y x H0))) H))))))). -(* COMMENTS -Initial nodes: 1975 -END *) - -theorem sty0_gen_appl: - \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (x: -T).((sty0 g c (THead (Flat Appl) u t1) x) \to (ex2 T (\lambda (t2: T).(sty0 g -c t1 t2)) (\lambda (t2: T).(eq T x (THead (Flat Appl) u t2))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (x: -T).(\lambda (H: (sty0 g c (THead (Flat Appl) u t1) x)).(insert_eq T (THead -(Flat Appl) u t1) (\lambda (t: T).(sty0 g c t x)) (\lambda (_: T).(ex2 T -(\lambda (t2: T).(sty0 g c t1 t2)) (\lambda (t2: T).(eq T x (THead (Flat -Appl) u t2))))) (\lambda (y: T).(\lambda (H0: (sty0 g c y x)).(sty0_ind g -(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Flat Appl) -u t1)) \to (ex2 T (\lambda (t2: T).(sty0 g c0 t1 t2)) (\lambda (t2: T).(eq T -t0 (THead (Flat Appl) u t2)))))))) (\lambda (c0: C).(\lambda (n: -nat).(\lambda (H1: (eq T (TSort n) (THead (Flat Appl) u t1))).(let H2 \def -(eq_ind T (TSort n) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | -(THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) u t1) H1) in -(False_ind (ex2 T (\lambda (t2: T).(sty0 g c0 t1 t2)) (\lambda (t2: T).(eq T -(TSort (next g n)) (THead (Flat Appl) u t2)))) H2))))) (\lambda (c0: -C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (_: (getl i c0 -(CHead d (Bind Abbr) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v -w)).(\lambda (_: (((eq T v (THead (Flat Appl) u t1)) \to (ex2 T (\lambda (t2: -T).(sty0 g d t1 t2)) (\lambda (t2: T).(eq T w (THead (Flat Appl) u -t2))))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat Appl) u t1))).(let H5 -\def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | -(THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) u t1) H4) in -(False_ind (ex2 T (\lambda (t2: T).(sty0 g c0 t1 t2)) (\lambda (t2: T).(eq T -(lift (S i) O w) (THead (Flat Appl) u t2)))) H5))))))))))) (\lambda (c0: -C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (_: (getl i c0 -(CHead d (Bind Abst) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v -w)).(\lambda (_: (((eq T v (THead (Flat Appl) u t1)) \to (ex2 T (\lambda (t2: -T).(sty0 g d t1 t2)) (\lambda (t2: T).(eq T w (THead (Flat Appl) u -t2))))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat Appl) u t1))).(let H5 -\def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | -(THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) u t1) H4) in -(False_ind (ex2 T (\lambda (t2: T).(sty0 g c0 t1 t2)) (\lambda (t2: T).(eq T -(lift (S i) O v) (THead (Flat Appl) u t2)))) H5))))))))))) (\lambda (b: -B).(\lambda (c0: C).(\lambda (v: T).(\lambda (t0: T).(\lambda (t2: -T).(\lambda (_: (sty0 g (CHead c0 (Bind b) v) t0 t2)).(\lambda (_: (((eq T t0 -(THead (Flat Appl) u t1)) \to (ex2 T (\lambda (t3: T).(sty0 g (CHead c0 (Bind -b) v) t1 t3)) (\lambda (t3: T).(eq T t2 (THead (Flat Appl) u -t3))))))).(\lambda (H3: (eq T (THead (Bind b) v t0) (THead (Flat Appl) u -t1))).(let H4 \def (eq_ind T (THead (Bind b) v t0) (\lambda (ee: T).(match ee -in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef -_) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return -(\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow -False])])) I (THead (Flat Appl) u t1) H3) in (False_ind (ex2 T (\lambda (t3: -T).(sty0 g c0 t1 t3)) (\lambda (t3: T).(eq T (THead (Bind b) v t2) (THead -(Flat Appl) u t3)))) H4)))))))))) (\lambda (c0: C).(\lambda (v: T).(\lambda -(t0: T).(\lambda (t2: T).(\lambda (H1: (sty0 g c0 t0 t2)).(\lambda (H2: (((eq -T t0 (THead (Flat Appl) u t1)) \to (ex2 T (\lambda (t3: T).(sty0 g c0 t1 t3)) -(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u t3))))))).(\lambda (H3: (eq T -(THead (Flat Appl) v t0) (THead (Flat Appl) u t1))).(let H4 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow v | (TLRef _) \Rightarrow v | (THead _ t _) \Rightarrow t])) -(THead (Flat Appl) v t0) (THead (Flat Appl) u t1) H3) in ((let H5 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) -\Rightarrow t])) (THead (Flat Appl) v t0) (THead (Flat Appl) u t1) H3) in -(\lambda (H6: (eq T v u)).(let H7 \def (eq_ind T t0 (\lambda (t: T).((eq T t -(THead (Flat Appl) u t1)) \to (ex2 T (\lambda (t3: T).(sty0 g c0 t1 t3)) -(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u t3)))))) H2 t1 H5) in (let H8 -\def (eq_ind T t0 (\lambda (t: T).(sty0 g c0 t t2)) H1 t1 H5) in (eq_ind_r T -u (\lambda (t: T).(ex2 T (\lambda (t3: T).(sty0 g c0 t1 t3)) (\lambda (t3: -T).(eq T (THead (Flat Appl) t t2) (THead (Flat Appl) u t3))))) (ex_intro2 T -(\lambda (t3: T).(sty0 g c0 t1 t3)) (\lambda (t3: T).(eq T (THead (Flat Appl) -u t2) (THead (Flat Appl) u t3))) t2 H8 (refl_equal T (THead (Flat Appl) u -t2))) v H6))))) H4))))))))) (\lambda (c0: C).(\lambda (v1: T).(\lambda (v2: -T).(\lambda (_: (sty0 g c0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Flat Appl) -u t1)) \to (ex2 T (\lambda (t2: T).(sty0 g c0 t1 t2)) (\lambda (t2: T).(eq T -v2 (THead (Flat Appl) u t2))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda -(_: (sty0 g c0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u t1)) \to -(ex2 T (\lambda (t3: T).(sty0 g c0 t1 t3)) (\lambda (t3: T).(eq T t2 (THead -(Flat Appl) u t3))))))).(\lambda (H5: (eq T (THead (Flat Cast) v1 t0) (THead -(Flat Appl) u t1))).(let H6 \def (eq_ind T (THead (Flat Cast) v1 t0) (\lambda -(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow -(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | -(Flat f) \Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl -\Rightarrow False | Cast \Rightarrow True])])])) I (THead (Flat Appl) u t1) -H5) in (False_ind (ex2 T (\lambda (t3: T).(sty0 g c0 t1 t3)) (\lambda (t3: -T).(eq T (THead (Flat Cast) v2 t2) (THead (Flat Appl) u t3)))) H6)))))))))))) -c y x H0))) H)))))). -(* COMMENTS -Initial nodes: 1489 -END *) - -theorem sty0_gen_cast: - \forall (g: G).(\forall (c: C).(\forall (v1: T).(\forall (t1: T).(\forall -(x: T).((sty0 g c (THead (Flat Cast) v1 t1) x) \to (ex3_2 T T (\lambda (v2: -T).(\lambda (_: T).(sty0 g c v1 v2))) (\lambda (_: T).(\lambda (t2: T).(sty0 -g c t1 t2))) (\lambda (v2: T).(\lambda (t2: T).(eq T x (THead (Flat Cast) v2 -t2)))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (v1: T).(\lambda (t1: T).(\lambda -(x: T).(\lambda (H: (sty0 g c (THead (Flat Cast) v1 t1) x)).(insert_eq T -(THead (Flat Cast) v1 t1) (\lambda (t: T).(sty0 g c t x)) (\lambda (_: -T).(ex3_2 T T (\lambda (v2: T).(\lambda (_: T).(sty0 g c v1 v2))) (\lambda -(_: T).(\lambda (t2: T).(sty0 g c t1 t2))) (\lambda (v2: T).(\lambda (t2: -T).(eq T x (THead (Flat Cast) v2 t2)))))) (\lambda (y: T).(\lambda (H0: (sty0 -g c y x)).(sty0_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq -T t (THead (Flat Cast) v1 t1)) \to (ex3_2 T T (\lambda (v2: T).(\lambda (_: -T).(sty0 g c0 v1 v2))) (\lambda (_: T).(\lambda (t2: T).(sty0 g c0 t1 t2))) -(\lambda (v2: T).(\lambda (t2: T).(eq T t0 (THead (Flat Cast) v2 t2))))))))) -(\lambda (c0: C).(\lambda (n: nat).(\lambda (H1: (eq T (TSort n) (THead (Flat -Cast) v1 t1))).(let H2 \def (eq_ind T (TSort n) (\lambda (ee: T).(match ee in -T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) -v1 t1) H1) in (False_ind (ex3_2 T T (\lambda (v2: T).(\lambda (_: T).(sty0 g -c0 v1 v2))) (\lambda (_: T).(\lambda (t2: T).(sty0 g c0 t1 t2))) (\lambda -(v2: T).(\lambda (t2: T).(eq T (TSort (next g n)) (THead (Flat Cast) v2 -t2))))) H2))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: -nat).(\lambda (_: (getl i c0 (CHead d (Bind Abbr) v))).(\lambda (w: -T).(\lambda (_: (sty0 g d v w)).(\lambda (_: (((eq T v (THead (Flat Cast) v1 -t1)) \to (ex3_2 T T (\lambda (v2: T).(\lambda (_: T).(sty0 g d v1 v2))) -(\lambda (_: T).(\lambda (t2: T).(sty0 g d t1 t2))) (\lambda (v2: T).(\lambda -(t2: T).(eq T w (THead (Flat Cast) v2 t2)))))))).(\lambda (H4: (eq T (TLRef -i) (THead (Flat Cast) v1 t1))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I -(THead (Flat Cast) v1 t1) H4) in (False_ind (ex3_2 T T (\lambda (v2: -T).(\lambda (_: T).(sty0 g c0 v1 v2))) (\lambda (_: T).(\lambda (t2: T).(sty0 -g c0 t1 t2))) (\lambda (v2: T).(\lambda (t2: T).(eq T (lift (S i) O w) (THead -(Flat Cast) v2 t2))))) H5))))))))))) (\lambda (c0: C).(\lambda (d: -C).(\lambda (v: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind -Abst) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v w)).(\lambda (_: (((eq T -v (THead (Flat Cast) v1 t1)) \to (ex3_2 T T (\lambda (v2: T).(\lambda (_: -T).(sty0 g d v1 v2))) (\lambda (_: T).(\lambda (t2: T).(sty0 g d t1 t2))) -(\lambda (v2: T).(\lambda (t2: T).(eq T w (THead (Flat Cast) v2 -t2)))))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat Cast) v1 t1))).(let H5 -\def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | -(THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) v1 t1) H4) in -(False_ind (ex3_2 T T (\lambda (v2: T).(\lambda (_: T).(sty0 g c0 v1 v2))) -(\lambda (_: T).(\lambda (t2: T).(sty0 g c0 t1 t2))) (\lambda (v2: -T).(\lambda (t2: T).(eq T (lift (S i) O v) (THead (Flat Cast) v2 t2))))) -H5))))))))))) (\lambda (b: B).(\lambda (c0: C).(\lambda (v: T).(\lambda (t0: -T).(\lambda (t2: T).(\lambda (_: (sty0 g (CHead c0 (Bind b) v) t0 -t2)).(\lambda (_: (((eq T t0 (THead (Flat Cast) v1 t1)) \to (ex3_2 T T -(\lambda (v2: T).(\lambda (_: T).(sty0 g (CHead c0 (Bind b) v) v1 v2))) -(\lambda (_: T).(\lambda (t3: T).(sty0 g (CHead c0 (Bind b) v) t1 t3))) -(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) v2 -t3)))))))).(\lambda (H3: (eq T (THead (Bind b) v t0) (THead (Flat Cast) v1 -t1))).(let H4 \def (eq_ind T (THead (Bind b) v t0) (\lambda (ee: T).(match ee -in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef -_) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return -(\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow -False])])) I (THead (Flat Cast) v1 t1) H3) in (False_ind (ex3_2 T T (\lambda -(v2: T).(\lambda (_: T).(sty0 g c0 v1 v2))) (\lambda (_: T).(\lambda (t3: -T).(sty0 g c0 t1 t3))) (\lambda (v2: T).(\lambda (t3: T).(eq T (THead (Bind -b) v t2) (THead (Flat Cast) v2 t3))))) H4)))))))))) (\lambda (c0: C).(\lambda -(v: T).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (sty0 g c0 t0 -t2)).(\lambda (_: (((eq T t0 (THead (Flat Cast) v1 t1)) \to (ex3_2 T T -(\lambda (v2: T).(\lambda (_: T).(sty0 g c0 v1 v2))) (\lambda (_: T).(\lambda -(t3: T).(sty0 g c0 t1 t3))) (\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead -(Flat Cast) v2 t3)))))))).(\lambda (H3: (eq T (THead (Flat Appl) v t0) (THead -(Flat Cast) v1 t1))).(let H4 \def (eq_ind T (THead (Flat Appl) v t0) (\lambda -(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow -(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | -(Flat f) \Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl -\Rightarrow True | Cast \Rightarrow False])])])) I (THead (Flat Cast) v1 t1) -H3) in (False_ind (ex3_2 T T (\lambda (v2: T).(\lambda (_: T).(sty0 g c0 v1 -v2))) (\lambda (_: T).(\lambda (t3: T).(sty0 g c0 t1 t3))) (\lambda (v2: -T).(\lambda (t3: T).(eq T (THead (Flat Appl) v t2) (THead (Flat Cast) v2 -t3))))) H4))))))))) (\lambda (c0: C).(\lambda (v0: T).(\lambda (v2: -T).(\lambda (H1: (sty0 g c0 v0 v2)).(\lambda (H2: (((eq T v0 (THead (Flat -Cast) v1 t1)) \to (ex3_2 T T (\lambda (v3: T).(\lambda (_: T).(sty0 g c0 v1 -v3))) (\lambda (_: T).(\lambda (t2: T).(sty0 g c0 t1 t2))) (\lambda (v3: -T).(\lambda (t2: T).(eq T v2 (THead (Flat Cast) v3 t2)))))))).(\lambda (t0: -T).(\lambda (t2: T).(\lambda (H3: (sty0 g c0 t0 t2)).(\lambda (H4: (((eq T t0 -(THead (Flat Cast) v1 t1)) \to (ex3_2 T T (\lambda (v3: T).(\lambda (_: -T).(sty0 g c0 v1 v3))) (\lambda (_: T).(\lambda (t3: T).(sty0 g c0 t1 t3))) -(\lambda (v3: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) v3 -t3)))))))).(\lambda (H5: (eq T (THead (Flat Cast) v0 t0) (THead (Flat Cast) -v1 t1))).(let H6 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow v0 | (TLRef _) \Rightarrow v0 -| (THead _ t _) \Rightarrow t])) (THead (Flat Cast) v0 t0) (THead (Flat Cast) -v1 t1) H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 -| (THead _ _ t) \Rightarrow t])) (THead (Flat Cast) v0 t0) (THead (Flat Cast) -v1 t1) H5) in (\lambda (H8: (eq T v0 v1)).(let H9 \def (eq_ind T t0 (\lambda -(t: T).((eq T t (THead (Flat Cast) v1 t1)) \to (ex3_2 T T (\lambda (v3: -T).(\lambda (_: T).(sty0 g c0 v1 v3))) (\lambda (_: T).(\lambda (t3: T).(sty0 -g c0 t1 t3))) (\lambda (v3: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) -v3 t3))))))) H4 t1 H7) in (let H10 \def (eq_ind T t0 (\lambda (t: T).(sty0 g -c0 t t2)) H3 t1 H7) in (let H11 \def (eq_ind T v0 (\lambda (t: T).((eq T t -(THead (Flat Cast) v1 t1)) \to (ex3_2 T T (\lambda (v3: T).(\lambda (_: -T).(sty0 g c0 v1 v3))) (\lambda (_: T).(\lambda (t3: T).(sty0 g c0 t1 t3))) -(\lambda (v3: T).(\lambda (t3: T).(eq T v2 (THead (Flat Cast) v3 t3))))))) H2 -v1 H8) in (let H12 \def (eq_ind T v0 (\lambda (t: T).(sty0 g c0 t v2)) H1 v1 -H8) in (ex3_2_intro T T (\lambda (v3: T).(\lambda (_: T).(sty0 g c0 v1 v3))) -(\lambda (_: T).(\lambda (t3: T).(sty0 g c0 t1 t3))) (\lambda (v3: -T).(\lambda (t3: T).(eq T (THead (Flat Cast) v2 t2) (THead (Flat Cast) v3 -t3)))) v2 t2 H12 H10 (refl_equal T (THead (Flat Cast) v2 t2))))))))) -H6)))))))))))) c y x H0))) H)))))). -(* COMMENTS -Initial nodes: 1855 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/sty0/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/sty0/props.ma deleted file mode 100644 index 00c9f6818..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/sty0/props.ma +++ /dev/null @@ -1,217 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/sty0/defs.ma". - -include "Basic-1/getl/drop.ma". - -theorem sty0_lift: - \forall (g: G).(\forall (e: C).(\forall (t1: T).(\forall (t2: T).((sty0 g e -t1 t2) \to (\forall (c: C).(\forall (h: nat).(\forall (d: nat).((drop h d c -e) \to (sty0 g c (lift h d t1) (lift h d t2)))))))))) -\def - \lambda (g: G).(\lambda (e: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H: (sty0 g e t1 t2)).(sty0_ind g (\lambda (c: C).(\lambda (t: T).(\lambda -(t0: T).(\forall (c0: C).(\forall (h: nat).(\forall (d: nat).((drop h d c0 c) -\to (sty0 g c0 (lift h d t) (lift h d t0))))))))) (\lambda (c: C).(\lambda -(n: nat).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (_: -(drop h d c0 c)).(eq_ind_r T (TSort n) (\lambda (t: T).(sty0 g c0 t (lift h d -(TSort (next g n))))) (eq_ind_r T (TSort (next g n)) (\lambda (t: T).(sty0 g -c0 (TSort n) t)) (sty0_sort g c0 n) (lift h d (TSort (next g n))) (lift_sort -(next g n) h d)) (lift h d (TSort n)) (lift_sort n h d)))))))) (\lambda (c: -C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H0: (getl i c -(CHead d (Bind Abbr) v))).(\lambda (w: T).(\lambda (H1: (sty0 g d v -w)).(\lambda (H2: ((\forall (c0: C).(\forall (h: nat).(\forall (d0: -nat).((drop h d0 c0 d) \to (sty0 g c0 (lift h d0 v) (lift h d0 -w)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d0: nat).(\lambda (H3: -(drop h d0 c0 c)).(lt_le_e i d0 (sty0 g c0 (lift h d0 (TLRef i)) (lift h d0 -(lift (S i) O w))) (\lambda (H4: (lt i d0)).(let H5 \def (drop_getl_trans_le -i d0 (le_S_n i d0 (le_S (S i) d0 H4)) c0 c h H3 (CHead d (Bind Abbr) v) H0) -in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop i O c0 e0))) -(\lambda (e0: C).(\lambda (e1: C).(drop h (minus d0 i) e0 e1))) (\lambda (_: -C).(\lambda (e1: C).(clear e1 (CHead d (Bind Abbr) v)))) (sty0 g c0 (lift h -d0 (TLRef i)) (lift h d0 (lift (S i) O w))) (\lambda (x0: C).(\lambda (x1: -C).(\lambda (H6: (drop i O c0 x0)).(\lambda (H7: (drop h (minus d0 i) x0 -x1)).(\lambda (H8: (clear x1 (CHead d (Bind Abbr) v))).(let H9 \def (eq_ind -nat (minus d0 i) (\lambda (n: nat).(drop h n x0 x1)) H7 (S (minus d0 (S i))) -(minus_x_Sy d0 i H4)) in (let H10 \def (drop_clear_S x1 x0 h (minus d0 (S i)) -H9 Abbr d v H8) in (ex2_ind C (\lambda (c1: C).(clear x0 (CHead c1 (Bind -Abbr) (lift h (minus d0 (S i)) v)))) (\lambda (c1: C).(drop h (minus d0 (S -i)) c1 d)) (sty0 g c0 (lift h d0 (TLRef i)) (lift h d0 (lift (S i) O w))) -(\lambda (x: C).(\lambda (H11: (clear x0 (CHead x (Bind Abbr) (lift h (minus -d0 (S i)) v)))).(\lambda (H12: (drop h (minus d0 (S i)) x d)).(eq_ind_r T -(TLRef i) (\lambda (t: T).(sty0 g c0 t (lift h d0 (lift (S i) O w)))) (eq_ind -nat (plus (S i) (minus d0 (S i))) (\lambda (n: nat).(sty0 g c0 (TLRef i) -(lift h n (lift (S i) O w)))) (eq_ind_r T (lift (S i) O (lift h (minus d0 (S -i)) w)) (\lambda (t: T).(sty0 g c0 (TLRef i) t)) (eq_ind nat d0 (\lambda (_: -nat).(sty0 g c0 (TLRef i) (lift (S i) O (lift h (minus d0 (S i)) w)))) -(sty0_abbr g c0 x (lift h (minus d0 (S i)) v) i (getl_intro i c0 (CHead x -(Bind Abbr) (lift h (minus d0 (S i)) v)) x0 H6 H11) (lift h (minus d0 (S i)) -w) (H2 x h (minus d0 (S i)) H12)) (plus (S i) (minus d0 (S i))) -(le_plus_minus (S i) d0 H4)) (lift h (plus (S i) (minus d0 (S i))) (lift (S -i) O w)) (lift_d w h (S i) (minus d0 (S i)) O (le_O_n (minus d0 (S i))))) d0 -(le_plus_minus_r (S i) d0 H4)) (lift h d0 (TLRef i)) (lift_lref_lt i h d0 -H4))))) H10)))))))) H5))) (\lambda (H4: (le d0 i)).(eq_ind_r T (TLRef (plus i -h)) (\lambda (t: T).(sty0 g c0 t (lift h d0 (lift (S i) O w)))) (eq_ind nat -(S i) (\lambda (_: nat).(sty0 g c0 (TLRef (plus i h)) (lift h d0 (lift (S i) -O w)))) (eq_ind_r T (lift (plus h (S i)) O w) (\lambda (t: T).(sty0 g c0 -(TLRef (plus i h)) t)) (eq_ind_r nat (plus (S i) h) (\lambda (n: nat).(sty0 g -c0 (TLRef (plus i h)) (lift n O w))) (sty0_abbr g c0 d v (plus i h) -(drop_getl_trans_ge i c0 c d0 h H3 (CHead d (Bind Abbr) v) H0 H4) w H1) (plus -h (S i)) (plus_sym h (S i))) (lift h d0 (lift (S i) O w)) (lift_free w (S i) -h O d0 (le_S d0 i H4) (le_O_n d0))) (plus i (S O)) (eq_ind_r nat (plus (S O) -i) (\lambda (n: nat).(eq nat (S i) n)) (refl_equal nat (plus (S O) i)) (plus -i (S O)) (plus_sym i (S O)))) (lift h d0 (TLRef i)) (lift_lref_ge i h d0 -H4)))))))))))))))) (\lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda -(i: nat).(\lambda (H0: (getl i c (CHead d (Bind Abst) v))).(\lambda (w: -T).(\lambda (H1: (sty0 g d v w)).(\lambda (H2: ((\forall (c0: C).(\forall (h: -nat).(\forall (d0: nat).((drop h d0 c0 d) \to (sty0 g c0 (lift h d0 v) (lift -h d0 w)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d0: nat).(\lambda -(H3: (drop h d0 c0 c)).(lt_le_e i d0 (sty0 g c0 (lift h d0 (TLRef i)) (lift h -d0 (lift (S i) O v))) (\lambda (H4: (lt i d0)).(let H5 \def -(drop_getl_trans_le i d0 (le_S_n i d0 (le_S (S i) d0 H4)) c0 c h H3 (CHead d -(Bind Abst) v) H0) in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop i -O c0 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d0 i) e0 e1))) -(\lambda (_: C).(\lambda (e1: C).(clear e1 (CHead d (Bind Abst) v)))) (sty0 g -c0 (lift h d0 (TLRef i)) (lift h d0 (lift (S i) O v))) (\lambda (x0: -C).(\lambda (x1: C).(\lambda (H6: (drop i O c0 x0)).(\lambda (H7: (drop h -(minus d0 i) x0 x1)).(\lambda (H8: (clear x1 (CHead d (Bind Abst) v))).(let -H9 \def (eq_ind nat (minus d0 i) (\lambda (n: nat).(drop h n x0 x1)) H7 (S -(minus d0 (S i))) (minus_x_Sy d0 i H4)) in (let H10 \def (drop_clear_S x1 x0 -h (minus d0 (S i)) H9 Abst d v H8) in (ex2_ind C (\lambda (c1: C).(clear x0 -(CHead c1 (Bind Abst) (lift h (minus d0 (S i)) v)))) (\lambda (c1: C).(drop h -(minus d0 (S i)) c1 d)) (sty0 g c0 (lift h d0 (TLRef i)) (lift h d0 (lift (S -i) O v))) (\lambda (x: C).(\lambda (H11: (clear x0 (CHead x (Bind Abst) (lift -h (minus d0 (S i)) v)))).(\lambda (H12: (drop h (minus d0 (S i)) x -d)).(eq_ind_r T (TLRef i) (\lambda (t: T).(sty0 g c0 t (lift h d0 (lift (S i) -O v)))) (eq_ind nat (plus (S i) (minus d0 (S i))) (\lambda (n: nat).(sty0 g -c0 (TLRef i) (lift h n (lift (S i) O v)))) (eq_ind_r T (lift (S i) O (lift h -(minus d0 (S i)) v)) (\lambda (t: T).(sty0 g c0 (TLRef i) t)) (eq_ind nat d0 -(\lambda (_: nat).(sty0 g c0 (TLRef i) (lift (S i) O (lift h (minus d0 (S i)) -v)))) (sty0_abst g c0 x (lift h (minus d0 (S i)) v) i (getl_intro i c0 (CHead -x (Bind Abst) (lift h (minus d0 (S i)) v)) x0 H6 H11) (lift h (minus d0 (S -i)) w) (H2 x h (minus d0 (S i)) H12)) (plus (S i) (minus d0 (S i))) -(le_plus_minus (S i) d0 H4)) (lift h (plus (S i) (minus d0 (S i))) (lift (S -i) O v)) (lift_d v h (S i) (minus d0 (S i)) O (le_O_n (minus d0 (S i))))) d0 -(le_plus_minus_r (S i) d0 H4)) (lift h d0 (TLRef i)) (lift_lref_lt i h d0 -H4))))) H10)))))))) H5))) (\lambda (H4: (le d0 i)).(eq_ind_r T (TLRef (plus i -h)) (\lambda (t: T).(sty0 g c0 t (lift h d0 (lift (S i) O v)))) (eq_ind nat -(S i) (\lambda (_: nat).(sty0 g c0 (TLRef (plus i h)) (lift h d0 (lift (S i) -O v)))) (eq_ind_r T (lift (plus h (S i)) O v) (\lambda (t: T).(sty0 g c0 -(TLRef (plus i h)) t)) (eq_ind_r nat (plus (S i) h) (\lambda (n: nat).(sty0 g -c0 (TLRef (plus i h)) (lift n O v))) (sty0_abst g c0 d v (plus i h) -(drop_getl_trans_ge i c0 c d0 h H3 (CHead d (Bind Abst) v) H0 H4) w H1) (plus -h (S i)) (plus_sym h (S i))) (lift h d0 (lift (S i) O v)) (lift_free v (S i) -h O d0 (le_S d0 i H4) (le_O_n d0))) (plus i (S O)) (eq_ind_r nat (plus (S O) -i) (\lambda (n: nat).(eq nat (S i) n)) (refl_equal nat (plus (S O) i)) (plus -i (S O)) (plus_sym i (S O)))) (lift h d0 (TLRef i)) (lift_lref_ge i h d0 -H4)))))))))))))))) (\lambda (b: B).(\lambda (c: C).(\lambda (v: T).(\lambda -(t3: T).(\lambda (t4: T).(\lambda (_: (sty0 g (CHead c (Bind b) v) t3 -t4)).(\lambda (H1: ((\forall (c0: C).(\forall (h: nat).(\forall (d: -nat).((drop h d c0 (CHead c (Bind b) v)) \to (sty0 g c0 (lift h d t3) (lift h -d t4)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda -(H2: (drop h d c0 c)).(eq_ind_r T (THead (Bind b) (lift h d v) (lift h (s -(Bind b) d) t3)) (\lambda (t: T).(sty0 g c0 t (lift h d (THead (Bind b) v -t4)))) (eq_ind_r T (THead (Bind b) (lift h d v) (lift h (s (Bind b) d) t4)) -(\lambda (t: T).(sty0 g c0 (THead (Bind b) (lift h d v) (lift h (s (Bind b) -d) t3)) t)) (sty0_bind g b c0 (lift h d v) (lift h (S d) t3) (lift h (S d) -t4) (H1 (CHead c0 (Bind b) (lift h d v)) h (S d) (drop_skip_bind h d c0 c H2 -b v))) (lift h d (THead (Bind b) v t4)) (lift_head (Bind b) v t4 h d)) (lift -h d (THead (Bind b) v t3)) (lift_head (Bind b) v t3 h d))))))))))))) (\lambda -(c: C).(\lambda (v: T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (sty0 g -c t3 t4)).(\lambda (H1: ((\forall (c0: C).(\forall (h: nat).(\forall (d: -nat).((drop h d c0 c) \to (sty0 g c0 (lift h d t3) (lift h d -t4)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2: -(drop h d c0 c)).(eq_ind_r T (THead (Flat Appl) (lift h d v) (lift h (s (Flat -Appl) d) t3)) (\lambda (t: T).(sty0 g c0 t (lift h d (THead (Flat Appl) v -t4)))) (eq_ind_r T (THead (Flat Appl) (lift h d v) (lift h (s (Flat Appl) d) -t4)) (\lambda (t: T).(sty0 g c0 (THead (Flat Appl) (lift h d v) (lift h (s -(Flat Appl) d) t3)) t)) (sty0_appl g c0 (lift h d v) (lift h (s (Flat Appl) -d) t3) (lift h (s (Flat Appl) d) t4) (H1 c0 h (s (Flat Appl) d) H2)) (lift h -d (THead (Flat Appl) v t4)) (lift_head (Flat Appl) v t4 h d)) (lift h d -(THead (Flat Appl) v t3)) (lift_head (Flat Appl) v t3 h d)))))))))))) -(\lambda (c: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (sty0 g c v1 -v2)).(\lambda (H1: ((\forall (c0: C).(\forall (h: nat).(\forall (d: -nat).((drop h d c0 c) \to (sty0 g c0 (lift h d v1) (lift h d -v2)))))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (sty0 g c t3 -t4)).(\lambda (H3: ((\forall (c0: C).(\forall (h: nat).(\forall (d: -nat).((drop h d c0 c) \to (sty0 g c0 (lift h d t3) (lift h d -t4)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H4: -(drop h d c0 c)).(eq_ind_r T (THead (Flat Cast) (lift h d v1) (lift h (s -(Flat Cast) d) t3)) (\lambda (t: T).(sty0 g c0 t (lift h d (THead (Flat Cast) -v2 t4)))) (eq_ind_r T (THead (Flat Cast) (lift h d v2) (lift h (s (Flat Cast) -d) t4)) (\lambda (t: T).(sty0 g c0 (THead (Flat Cast) (lift h d v1) (lift h -(s (Flat Cast) d) t3)) t)) (sty0_cast g c0 (lift h d v1) (lift h d v2) (H1 c0 -h d H4) (lift h (s (Flat Cast) d) t3) (lift h (s (Flat Cast) d) t4) (H3 c0 h -(s (Flat Cast) d) H4)) (lift h d (THead (Flat Cast) v2 t4)) (lift_head (Flat -Cast) v2 t4 h d)) (lift h d (THead (Flat Cast) v1 t3)) (lift_head (Flat Cast) -v1 t3 h d))))))))))))))) e t1 t2 H))))). -(* COMMENTS -Initial nodes: 3677 -END *) - -theorem sty0_correct: - \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((sty0 g c -t1 t) \to (ex T (\lambda (t2: T).(sty0 g c t t2))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H: -(sty0 g c t1 t)).(sty0_ind g (\lambda (c0: C).(\lambda (_: T).(\lambda (t2: -T).(ex T (\lambda (t3: T).(sty0 g c0 t2 t3)))))) (\lambda (c0: C).(\lambda -(n: nat).(ex_intro T (\lambda (t2: T).(sty0 g c0 (TSort (next g n)) t2)) -(TSort (next g (next g n))) (sty0_sort g c0 (next g n))))) (\lambda (c0: -C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 -(CHead d (Bind Abbr) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v -w)).(\lambda (H2: (ex T (\lambda (t2: T).(sty0 g d w t2)))).(let H3 \def H2 -in (ex_ind T (\lambda (t2: T).(sty0 g d w t2)) (ex T (\lambda (t2: T).(sty0 g -c0 (lift (S i) O w) t2))) (\lambda (x: T).(\lambda (H4: (sty0 g d w -x)).(ex_intro T (\lambda (t2: T).(sty0 g c0 (lift (S i) O w) t2)) (lift (S i) -O x) (sty0_lift g d w x H4 c0 (S i) O (getl_drop Abbr c0 d v i H0))))) -H3)))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: -nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abst) v))).(\lambda (w: -T).(\lambda (H1: (sty0 g d v w)).(\lambda (H2: (ex T (\lambda (t2: T).(sty0 g -d w t2)))).(let H3 \def H2 in (ex_ind T (\lambda (t2: T).(sty0 g d w t2)) (ex -T (\lambda (t2: T).(sty0 g c0 (lift (S i) O v) t2))) (\lambda (x: T).(\lambda -(_: (sty0 g d w x)).(ex_intro T (\lambda (t2: T).(sty0 g c0 (lift (S i) O v) -t2)) (lift (S i) O w) (sty0_lift g d v w H1 c0 (S i) O (getl_drop Abst c0 d v -i H0))))) H3)))))))))) (\lambda (b: B).(\lambda (c0: C).(\lambda (v: -T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (sty0 g (CHead c0 (Bind b) -v) t2 t3)).(\lambda (H1: (ex T (\lambda (t4: T).(sty0 g (CHead c0 (Bind b) v) -t3 t4)))).(let H2 \def H1 in (ex_ind T (\lambda (t4: T).(sty0 g (CHead c0 -(Bind b) v) t3 t4)) (ex T (\lambda (t4: T).(sty0 g c0 (THead (Bind b) v t3) -t4))) (\lambda (x: T).(\lambda (H3: (sty0 g (CHead c0 (Bind b) v) t3 -x)).(ex_intro T (\lambda (t4: T).(sty0 g c0 (THead (Bind b) v t3) t4)) (THead -(Bind b) v x) (sty0_bind g b c0 v t3 x H3)))) H2))))))))) (\lambda (c0: -C).(\lambda (v: T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (sty0 g c0 -t2 t3)).(\lambda (H1: (ex T (\lambda (t4: T).(sty0 g c0 t3 t4)))).(let H2 -\def H1 in (ex_ind T (\lambda (t4: T).(sty0 g c0 t3 t4)) (ex T (\lambda (t4: -T).(sty0 g c0 (THead (Flat Appl) v t3) t4))) (\lambda (x: T).(\lambda (H3: -(sty0 g c0 t3 x)).(ex_intro T (\lambda (t4: T).(sty0 g c0 (THead (Flat Appl) -v t3) t4)) (THead (Flat Appl) v x) (sty0_appl g c0 v t3 x H3)))) H2)))))))) -(\lambda (c0: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (sty0 g c0 v1 -v2)).(\lambda (H1: (ex T (\lambda (t2: T).(sty0 g c0 v2 t2)))).(\lambda (t2: -T).(\lambda (t3: T).(\lambda (_: (sty0 g c0 t2 t3)).(\lambda (H3: (ex T -(\lambda (t4: T).(sty0 g c0 t3 t4)))).(let H4 \def H1 in (ex_ind T (\lambda -(t4: T).(sty0 g c0 v2 t4)) (ex T (\lambda (t4: T).(sty0 g c0 (THead (Flat -Cast) v2 t3) t4))) (\lambda (x: T).(\lambda (H5: (sty0 g c0 v2 x)).(let H6 -\def H3 in (ex_ind T (\lambda (t4: T).(sty0 g c0 t3 t4)) (ex T (\lambda (t4: -T).(sty0 g c0 (THead (Flat Cast) v2 t3) t4))) (\lambda (x0: T).(\lambda (H7: -(sty0 g c0 t3 x0)).(ex_intro T (\lambda (t4: T).(sty0 g c0 (THead (Flat Cast) -v2 t3) t4)) (THead (Flat Cast) x x0) (sty0_cast g c0 v2 x H5 t3 x0 H7)))) -H6)))) H4))))))))))) c t1 t H))))). -(* COMMENTS -Initial nodes: 991 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/sty1/cnt.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/sty1/cnt.ma deleted file mode 100644 index 0f39bdfd9..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/sty1/cnt.ma +++ /dev/null @@ -1,89 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/sty1/props.ma". - -include "Basic-1/cnt/props.ma". - -theorem sty1_cnt: - \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((sty0 g c -t1 t) \to (ex2 T (\lambda (t2: T).(sty1 g c t1 t2)) (\lambda (t2: T).(cnt -t2))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H: -(sty0 g c t1 t)).(sty0_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (_: -T).(ex2 T (\lambda (t3: T).(sty1 g c0 t0 t3)) (\lambda (t3: T).(cnt t3)))))) -(\lambda (c0: C).(\lambda (n: nat).(ex_intro2 T (\lambda (t2: T).(sty1 g c0 -(TSort n) t2)) (\lambda (t2: T).(cnt t2)) (TSort (next g n)) (sty1_sty0 g c0 -(TSort n) (TSort (next g n)) (sty0_sort g c0 n)) (cnt_sort (next g n))))) -(\lambda (c0: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda -(H0: (getl i c0 (CHead d (Bind Abbr) v))).(\lambda (w: T).(\lambda (_: (sty0 -g d v w)).(\lambda (H2: (ex2 T (\lambda (t2: T).(sty1 g d v t2)) (\lambda -(t2: T).(cnt t2)))).(let H3 \def H2 in (ex2_ind T (\lambda (t2: T).(sty1 g d -v t2)) (\lambda (t2: T).(cnt t2)) (ex2 T (\lambda (t2: T).(sty1 g c0 (TLRef -i) t2)) (\lambda (t2: T).(cnt t2))) (\lambda (x: T).(\lambda (H4: (sty1 g d v -x)).(\lambda (H5: (cnt x)).(ex_intro2 T (\lambda (t2: T).(sty1 g c0 (TLRef i) -t2)) (\lambda (t2: T).(cnt t2)) (lift (S i) O x) (sty1_abbr g c0 d v i H0 x -H4) (cnt_lift x H5 (S i) O))))) H3)))))))))) (\lambda (c0: C).(\lambda (d: -C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind -Abst) v))).(\lambda (w: T).(\lambda (H1: (sty0 g d v w)).(\lambda (H2: (ex2 T -(\lambda (t2: T).(sty1 g d v t2)) (\lambda (t2: T).(cnt t2)))).(let H3 \def -H2 in (ex2_ind T (\lambda (t2: T).(sty1 g d v t2)) (\lambda (t2: T).(cnt t2)) -(ex2 T (\lambda (t2: T).(sty1 g c0 (TLRef i) t2)) (\lambda (t2: T).(cnt t2))) -(\lambda (x: T).(\lambda (H4: (sty1 g d v x)).(\lambda (H5: (cnt -x)).(ex_intro2 T (\lambda (t2: T).(sty1 g c0 (TLRef i) t2)) (\lambda (t2: -T).(cnt t2)) (lift (S i) O x) (sty1_trans g c0 (TLRef i) (lift (S i) O v) -(sty1_sty0 g c0 (TLRef i) (lift (S i) O v) (sty0_abst g c0 d v i H0 w H1)) -(lift (S i) O x) (sty1_lift g d v x H4 c0 (S i) O (getl_drop Abst c0 d v i -H0))) (cnt_lift x H5 (S i) O))))) H3)))))))))) (\lambda (b: B).(\lambda (c0: -C).(\lambda (v: T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (sty0 g -(CHead c0 (Bind b) v) t2 t3)).(\lambda (H1: (ex2 T (\lambda (t4: T).(sty1 g -(CHead c0 (Bind b) v) t2 t4)) (\lambda (t4: T).(cnt t4)))).(let H2 \def H1 in -(ex2_ind T (\lambda (t4: T).(sty1 g (CHead c0 (Bind b) v) t2 t4)) (\lambda -(t4: T).(cnt t4)) (ex2 T (\lambda (t4: T).(sty1 g c0 (THead (Bind b) v t2) -t4)) (\lambda (t4: T).(cnt t4))) (\lambda (x: T).(\lambda (H3: (sty1 g (CHead -c0 (Bind b) v) t2 x)).(\lambda (H4: (cnt x)).(ex_intro2 T (\lambda (t4: -T).(sty1 g c0 (THead (Bind b) v t2) t4)) (\lambda (t4: T).(cnt t4)) (THead -(Bind b) v x) (sty1_bind g b c0 v t2 x H3) (cnt_head x H4 (Bind b) v))))) -H2))))))))) (\lambda (c0: C).(\lambda (v: T).(\lambda (t2: T).(\lambda (t3: -T).(\lambda (_: (sty0 g c0 t2 t3)).(\lambda (H1: (ex2 T (\lambda (t4: -T).(sty1 g c0 t2 t4)) (\lambda (t4: T).(cnt t4)))).(let H2 \def H1 in -(ex2_ind T (\lambda (t4: T).(sty1 g c0 t2 t4)) (\lambda (t4: T).(cnt t4)) -(ex2 T (\lambda (t4: T).(sty1 g c0 (THead (Flat Appl) v t2) t4)) (\lambda -(t4: T).(cnt t4))) (\lambda (x: T).(\lambda (H3: (sty1 g c0 t2 x)).(\lambda -(H4: (cnt x)).(ex_intro2 T (\lambda (t4: T).(sty1 g c0 (THead (Flat Appl) v -t2) t4)) (\lambda (t4: T).(cnt t4)) (THead (Flat Appl) v x) (sty1_appl g c0 v -t2 x H3) (cnt_head x H4 (Flat Appl) v))))) H2)))))))) (\lambda (c0: -C).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H0: (sty0 g c0 v1 -v2)).(\lambda (_: (ex2 T (\lambda (t2: T).(sty1 g c0 v1 t2)) (\lambda (t2: -T).(cnt t2)))).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (sty0 g c0 t2 -t3)).(\lambda (H3: (ex2 T (\lambda (t4: T).(sty1 g c0 t2 t4)) (\lambda (t4: -T).(cnt t4)))).(let H4 \def H3 in (ex2_ind T (\lambda (t4: T).(sty1 g c0 t2 -t4)) (\lambda (t4: T).(cnt t4)) (ex2 T (\lambda (t4: T).(sty1 g c0 (THead -(Flat Cast) v1 t2) t4)) (\lambda (t4: T).(cnt t4))) (\lambda (x: T).(\lambda -(H5: (sty1 g c0 t2 x)).(\lambda (H6: (cnt x)).(let H_x \def (sty1_cast2 g c0 -t2 x H5 v1 v2 H0) in (let H7 \def H_x in (ex2_ind T (\lambda (v3: T).(sty1 g -c0 v1 v3)) (\lambda (v3: T).(sty1 g c0 (THead (Flat Cast) v1 t2) (THead (Flat -Cast) v3 x))) (ex2 T (\lambda (t4: T).(sty1 g c0 (THead (Flat Cast) v1 t2) -t4)) (\lambda (t4: T).(cnt t4))) (\lambda (x0: T).(\lambda (_: (sty1 g c0 v1 -x0)).(\lambda (H9: (sty1 g c0 (THead (Flat Cast) v1 t2) (THead (Flat Cast) x0 -x))).(ex_intro2 T (\lambda (t4: T).(sty1 g c0 (THead (Flat Cast) v1 t2) t4)) -(\lambda (t4: T).(cnt t4)) (THead (Flat Cast) x0 x) H9 (cnt_head x H6 (Flat -Cast) x0))))) H7)))))) H4))))))))))) c t1 t H))))). -(* COMMENTS -Initial nodes: 1313 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/sty1/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/sty1/defs.ma deleted file mode 100644 index 8fd219524..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/sty1/defs.ma +++ /dev/null @@ -1,23 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/sty0/defs.ma". - -inductive sty1 (g: G) (c: C) (t1: T): T \to Prop \def -| sty1_sty0: \forall (t2: T).((sty0 g c t1 t2) \to (sty1 g c t1 t2)) -| sty1_sing: \forall (t: T).((sty1 g c t1 t) \to (\forall (t2: T).((sty0 g c -t t2) \to (sty1 g c t1 t2)))). - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/sty1/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/sty1/props.ma deleted file mode 100644 index 53061952e..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/sty1/props.ma +++ /dev/null @@ -1,163 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/sty1/defs.ma". - -include "Basic-1/sty0/props.ma". - -theorem sty1_trans: - \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((sty1 g c -t1 t) \to (\forall (t2: T).((sty1 g c t t2) \to (sty1 g c t1 t2))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H: -(sty1 g c t1 t)).(\lambda (t2: T).(\lambda (H0: (sty1 g c t t2)).(sty1_ind g -c t (\lambda (t0: T).(sty1 g c t1 t0)) (\lambda (t3: T).(\lambda (H1: (sty0 g -c t t3)).(sty1_sing g c t1 t H t3 H1))) (\lambda (t0: T).(\lambda (_: (sty1 g -c t t0)).(\lambda (H2: (sty1 g c t1 t0)).(\lambda (t3: T).(\lambda (H3: (sty0 -g c t0 t3)).(sty1_sing g c t1 t0 H2 t3 H3)))))) t2 H0))))))). -(* COMMENTS -Initial nodes: 131 -END *) - -theorem sty1_bind: - \forall (g: G).(\forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t1: -T).(\forall (t2: T).((sty1 g (CHead c (Bind b) v) t1 t2) \to (sty1 g c (THead -(Bind b) v t1) (THead (Bind b) v t2)))))))) -\def - \lambda (g: G).(\lambda (b: B).(\lambda (c: C).(\lambda (v: T).(\lambda (t1: -T).(\lambda (t2: T).(\lambda (H: (sty1 g (CHead c (Bind b) v) t1 -t2)).(sty1_ind g (CHead c (Bind b) v) t1 (\lambda (t: T).(sty1 g c (THead -(Bind b) v t1) (THead (Bind b) v t))) (\lambda (t3: T).(\lambda (H0: (sty0 g -(CHead c (Bind b) v) t1 t3)).(sty1_sty0 g c (THead (Bind b) v t1) (THead -(Bind b) v t3) (sty0_bind g b c v t1 t3 H0)))) (\lambda (t: T).(\lambda (_: -(sty1 g (CHead c (Bind b) v) t1 t)).(\lambda (H1: (sty1 g c (THead (Bind b) v -t1) (THead (Bind b) v t))).(\lambda (t3: T).(\lambda (H2: (sty0 g (CHead c -(Bind b) v) t t3)).(sty1_sing g c (THead (Bind b) v t1) (THead (Bind b) v t) -H1 (THead (Bind b) v t3) (sty0_bind g b c v t t3 H2))))))) t2 H))))))). -(* COMMENTS -Initial nodes: 259 -END *) - -theorem sty1_appl: - \forall (g: G).(\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall -(t2: T).((sty1 g c t1 t2) \to (sty1 g c (THead (Flat Appl) v t1) (THead (Flat -Appl) v t2))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (v: T).(\lambda (t1: T).(\lambda -(t2: T).(\lambda (H: (sty1 g c t1 t2)).(sty1_ind g c t1 (\lambda (t: T).(sty1 -g c (THead (Flat Appl) v t1) (THead (Flat Appl) v t))) (\lambda (t3: -T).(\lambda (H0: (sty0 g c t1 t3)).(sty1_sty0 g c (THead (Flat Appl) v t1) -(THead (Flat Appl) v t3) (sty0_appl g c v t1 t3 H0)))) (\lambda (t: -T).(\lambda (_: (sty1 g c t1 t)).(\lambda (H1: (sty1 g c (THead (Flat Appl) v -t1) (THead (Flat Appl) v t))).(\lambda (t3: T).(\lambda (H2: (sty0 g c t -t3)).(sty1_sing g c (THead (Flat Appl) v t1) (THead (Flat Appl) v t) H1 -(THead (Flat Appl) v t3) (sty0_appl g c v t t3 H2))))))) t2 H)))))). -(* COMMENTS -Initial nodes: 213 -END *) - -theorem sty1_lift: - \forall (g: G).(\forall (e: C).(\forall (t1: T).(\forall (t2: T).((sty1 g e -t1 t2) \to (\forall (c: C).(\forall (h: nat).(\forall (d: nat).((drop h d c -e) \to (sty1 g c (lift h d t1) (lift h d t2)))))))))) -\def - \lambda (g: G).(\lambda (e: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H: (sty1 g e t1 t2)).(sty1_ind g e t1 (\lambda (t: T).(\forall (c: -C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to (sty1 g c (lift h -d t1) (lift h d t))))))) (\lambda (t3: T).(\lambda (H0: (sty0 g e t1 -t3)).(\lambda (c: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: (drop -h d c e)).(sty1_sty0 g c (lift h d t1) (lift h d t3) (sty0_lift g e t1 t3 H0 -c h d H1)))))))) (\lambda (t: T).(\lambda (_: (sty1 g e t1 t)).(\lambda (H1: -((\forall (c: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to -(sty1 g c (lift h d t1) (lift h d t)))))))).(\lambda (t3: T).(\lambda (H2: -(sty0 g e t t3)).(\lambda (c: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda -(H3: (drop h d c e)).(sty1_sing g c (lift h d t1) (lift h d t) (H1 c h d H3) -(lift h d t3) (sty0_lift g e t t3 H2 c h d H3))))))))))) t2 H))))). -(* COMMENTS -Initial nodes: 277 -END *) - -theorem sty1_correct: - \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((sty1 g c -t1 t) \to (ex T (\lambda (t2: T).(sty0 g c t t2))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H: -(sty1 g c t1 t)).(sty1_ind g c t1 (\lambda (t0: T).(ex T (\lambda (t2: -T).(sty0 g c t0 t2)))) (\lambda (t2: T).(\lambda (H0: (sty0 g c t1 -t2)).(sty0_correct g c t1 t2 H0))) (\lambda (t0: T).(\lambda (_: (sty1 g c t1 -t0)).(\lambda (_: (ex T (\lambda (t2: T).(sty0 g c t0 t2)))).(\lambda (t2: -T).(\lambda (H2: (sty0 g c t0 t2)).(sty0_correct g c t0 t2 H2)))))) t H))))). -(* COMMENTS -Initial nodes: 123 -END *) - -theorem sty1_abbr: - \forall (g: G).(\forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: -nat).((getl i c (CHead d (Bind Abbr) v)) \to (\forall (w: T).((sty1 g d v w) -\to (sty1 g c (TLRef i) (lift (S i) O w))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: -nat).(\lambda (H: (getl i c (CHead d (Bind Abbr) v))).(\lambda (w: -T).(\lambda (H0: (sty1 g d v w)).(sty1_ind g d v (\lambda (t: T).(sty1 g c -(TLRef i) (lift (S i) O t))) (\lambda (t2: T).(\lambda (H1: (sty0 g d v -t2)).(sty1_sty0 g c (TLRef i) (lift (S i) O t2) (sty0_abbr g c d v i H t2 -H1)))) (\lambda (t: T).(\lambda (_: (sty1 g d v t)).(\lambda (H2: (sty1 g c -(TLRef i) (lift (S i) O t))).(\lambda (t2: T).(\lambda (H3: (sty0 g d t -t2)).(sty1_sing g c (TLRef i) (lift (S i) O t) H2 (lift (S i) O t2) -(sty0_lift g d t t2 H3 c (S i) O (getl_drop Abbr c d v i H)))))))) w -H0)))))))). -(* COMMENTS -Initial nodes: 231 -END *) - -theorem sty1_cast2: - \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((sty1 g c -t1 t2) \to (\forall (v1: T).(\forall (v2: T).((sty0 g c v1 v2) \to (ex2 T -(\lambda (v3: T).(sty1 g c v1 v3)) (\lambda (v3: T).(sty1 g c (THead (Flat -Cast) v1 t1) (THead (Flat Cast) v3 t2))))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H: (sty1 g c t1 t2)).(sty1_ind g c t1 (\lambda (t: T).(\forall (v1: -T).(\forall (v2: T).((sty0 g c v1 v2) \to (ex2 T (\lambda (v3: T).(sty1 g c -v1 v3)) (\lambda (v3: T).(sty1 g c (THead (Flat Cast) v1 t1) (THead (Flat -Cast) v3 t)))))))) (\lambda (t3: T).(\lambda (H0: (sty0 g c t1 t3)).(\lambda -(v1: T).(\lambda (v2: T).(\lambda (H1: (sty0 g c v1 v2)).(ex_intro2 T -(\lambda (v3: T).(sty1 g c v1 v3)) (\lambda (v3: T).(sty1 g c (THead (Flat -Cast) v1 t1) (THead (Flat Cast) v3 t3))) v2 (sty1_sty0 g c v1 v2 H1) -(sty1_sty0 g c (THead (Flat Cast) v1 t1) (THead (Flat Cast) v2 t3) (sty0_cast -g c v1 v2 H1 t1 t3 H0)))))))) (\lambda (t: T).(\lambda (_: (sty1 g c t1 -t)).(\lambda (H1: ((\forall (v1: T).(\forall (v2: T).((sty0 g c v1 v2) \to -(ex2 T (\lambda (v3: T).(sty1 g c v1 v3)) (\lambda (v3: T).(sty1 g c (THead -(Flat Cast) v1 t1) (THead (Flat Cast) v3 t))))))))).(\lambda (t3: T).(\lambda -(H2: (sty0 g c t t3)).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H3: (sty0 g -c v1 v2)).(let H_x \def (H1 v1 v2 H3) in (let H4 \def H_x in (ex2_ind T -(\lambda (v3: T).(sty1 g c v1 v3)) (\lambda (v3: T).(sty1 g c (THead (Flat -Cast) v1 t1) (THead (Flat Cast) v3 t))) (ex2 T (\lambda (v3: T).(sty1 g c v1 -v3)) (\lambda (v3: T).(sty1 g c (THead (Flat Cast) v1 t1) (THead (Flat Cast) -v3 t3)))) (\lambda (x: T).(\lambda (H5: (sty1 g c v1 x)).(\lambda (H6: (sty1 -g c (THead (Flat Cast) v1 t1) (THead (Flat Cast) x t))).(let H_x0 \def -(sty1_correct g c v1 x H5) in (let H7 \def H_x0 in (ex_ind T (\lambda (t4: -T).(sty0 g c x t4)) (ex2 T (\lambda (v3: T).(sty1 g c v1 v3)) (\lambda (v3: -T).(sty1 g c (THead (Flat Cast) v1 t1) (THead (Flat Cast) v3 t3)))) (\lambda -(x0: T).(\lambda (H8: (sty0 g c x x0)).(ex_intro2 T (\lambda (v3: T).(sty1 g -c v1 v3)) (\lambda (v3: T).(sty1 g c (THead (Flat Cast) v1 t1) (THead (Flat -Cast) v3 t3))) x0 (sty1_sing g c v1 x H5 x0 H8) (sty1_sing g c (THead (Flat -Cast) v1 t1) (THead (Flat Cast) x t) H6 (THead (Flat Cast) x0 t3) (sty0_cast -g c x x0 H8 t t3 H2))))) H7)))))) H4))))))))))) t2 H))))). -(* COMMENTS -Initial nodes: 657 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/subst/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/subst/defs.ma deleted file mode 100644 index e0ed86b52..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/subst/defs.ma +++ /dev/null @@ -1,27 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/lift/defs.ma". - -definition subst: - nat \to (T \to (T \to T)) -\def - let rec subst (d: nat) (v: T) (t: T) on t: T \def (match t with [(TSort n) -\Rightarrow (TSort n) | (TLRef i) \Rightarrow (match (blt i d) with [true -\Rightarrow (TLRef i) | false \Rightarrow (match (blt d i) with [true -\Rightarrow (TLRef (pred i)) | false \Rightarrow (lift d O v)])]) | (THead k -u t0) \Rightarrow (THead k (subst d v u) (subst (s k d) v t0))]) in subst. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/subst/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/subst/fwd.ma deleted file mode 100644 index a0678e5fb..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/subst/fwd.ma +++ /dev/null @@ -1,79 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/subst/defs.ma". - -theorem subst_sort: - \forall (v: T).(\forall (d: nat).(\forall (k: nat).(eq T (subst d v (TSort -k)) (TSort k)))) -\def - \lambda (_: T).(\lambda (_: nat).(\lambda (k: nat).(refl_equal T (TSort -k)))). -(* COMMENTS -Initial nodes: 13 -END *) - -theorem subst_lref_lt: - \forall (v: T).(\forall (d: nat).(\forall (i: nat).((lt i d) \to (eq T -(subst d v (TLRef i)) (TLRef i))))) -\def - \lambda (v: T).(\lambda (d: nat).(\lambda (i: nat).(\lambda (H: (lt i -d)).(eq_ind_r bool true (\lambda (b: bool).(eq T (match b with [true -\Rightarrow (TLRef i) | false \Rightarrow (match (blt d i) with [true -\Rightarrow (TLRef (pred i)) | false \Rightarrow (lift d O v)])]) (TLRef i))) -(refl_equal T (TLRef i)) (blt i d) (lt_blt d i H))))). -(* COMMENTS -Initial nodes: 73 -END *) - -theorem subst_lref_eq: - \forall (v: T).(\forall (i: nat).(eq T (subst i v (TLRef i)) (lift i O v))) -\def - \lambda (v: T).(\lambda (i: nat).(eq_ind_r bool false (\lambda (b: bool).(eq -T (match b with [true \Rightarrow (TLRef i) | false \Rightarrow (match b with -[true \Rightarrow (TLRef (pred i)) | false \Rightarrow (lift i O v)])]) (lift -i O v))) (refl_equal T (lift i O v)) (blt i i) (le_bge i i (le_n i)))). -(* COMMENTS -Initial nodes: 71 -END *) - -theorem subst_lref_gt: - \forall (v: T).(\forall (d: nat).(\forall (i: nat).((lt d i) \to (eq T -(subst d v (TLRef i)) (TLRef (pred i)))))) -\def - \lambda (v: T).(\lambda (d: nat).(\lambda (i: nat).(\lambda (H: (lt d -i)).(eq_ind_r bool false (\lambda (b: bool).(eq T (match b with [true -\Rightarrow (TLRef i) | false \Rightarrow (match (blt d i) with [true -\Rightarrow (TLRef (pred i)) | false \Rightarrow (lift d O v)])]) (TLRef -(pred i)))) (eq_ind_r bool true (\lambda (b: bool).(eq T (match b with [true -\Rightarrow (TLRef (pred i)) | false \Rightarrow (lift d O v)]) (TLRef (pred -i)))) (refl_equal T (TLRef (pred i))) (blt d i) (lt_blt i d H)) (blt i d) -(le_bge d i (lt_le_weak d i H)))))). -(* COMMENTS -Initial nodes: 130 -END *) - -theorem subst_head: - \forall (k: K).(\forall (w: T).(\forall (u: T).(\forall (t: T).(\forall (d: -nat).(eq T (subst d w (THead k u t)) (THead k (subst d w u) (subst (s k d) w -t))))))) -\def - \lambda (k: K).(\lambda (w: T).(\lambda (u: T).(\lambda (t: T).(\lambda (d: -nat).(refl_equal T (THead k (subst d w u) (subst (s k d) w t))))))). -(* COMMENTS -Initial nodes: 37 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/subst/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/subst/props.ma deleted file mode 100644 index 3bad044e3..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/subst/props.ma +++ /dev/null @@ -1,116 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/subst/fwd.ma". - -include "Basic-1/subst0/defs.ma". - -include "Basic-1/lift/props.ma". - -theorem subst_lift_SO: - \forall (v: T).(\forall (t: T).(\forall (d: nat).(eq T (subst d v (lift (S -O) d t)) t))) -\def - \lambda (v: T).(\lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d: nat).(eq -T (subst d v (lift (S O) d t0)) t0))) (\lambda (n: nat).(\lambda (d: -nat).(eq_ind_r T (TSort n) (\lambda (t0: T).(eq T (subst d v t0) (TSort n))) -(eq_ind_r T (TSort n) (\lambda (t0: T).(eq T t0 (TSort n))) (refl_equal T -(TSort n)) (subst d v (TSort n)) (subst_sort v d n)) (lift (S O) d (TSort n)) -(lift_sort n (S O) d)))) (\lambda (n: nat).(\lambda (d: nat).(lt_le_e n d (eq -T (subst d v (lift (S O) d (TLRef n))) (TLRef n)) (\lambda (H: (lt n -d)).(eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (subst d v t0) (TLRef n))) -(eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 (TLRef n))) (refl_equal T -(TLRef n)) (subst d v (TLRef n)) (subst_lref_lt v d n H)) (lift (S O) d -(TLRef n)) (lift_lref_lt n (S O) d H))) (\lambda (H: (le d n)).(eq_ind_r T -(TLRef (plus n (S O))) (\lambda (t0: T).(eq T (subst d v t0) (TLRef n))) -(eq_ind nat (S (plus n O)) (\lambda (n0: nat).(eq T (subst d v (TLRef n0)) -(TLRef n))) (eq_ind_r T (TLRef (pred (S (plus n O)))) (\lambda (t0: T).(eq T -t0 (TLRef n))) (eq_ind nat (plus n O) (\lambda (n0: nat).(eq T (TLRef n0) -(TLRef n))) (f_equal nat T TLRef (plus n O) n (sym_eq nat n (plus n O) -(plus_n_O n))) (pred (S (plus n O))) (pred_Sn (plus n O))) (subst d v (TLRef -(S (plus n O)))) (subst_lref_gt v d (S (plus n O)) (le_n_S d (plus n O) -(le_plus_trans d n O H)))) (plus n (S O)) (plus_n_Sm n O)) (lift (S O) d -(TLRef n)) (lift_lref_ge n (S O) d H)))))) (\lambda (k: K).(\lambda (t0: -T).(\lambda (H: ((\forall (d: nat).(eq T (subst d v (lift (S O) d t0)) -t0)))).(\lambda (t1: T).(\lambda (H0: ((\forall (d: nat).(eq T (subst d v -(lift (S O) d t1)) t1)))).(\lambda (d: nat).(eq_ind_r T (THead k (lift (S O) -d t0) (lift (S O) (s k d) t1)) (\lambda (t2: T).(eq T (subst d v t2) (THead k -t0 t1))) (eq_ind_r T (THead k (subst d v (lift (S O) d t0)) (subst (s k d) v -(lift (S O) (s k d) t1))) (\lambda (t2: T).(eq T t2 (THead k t0 t1))) -(f_equal3 K T T T THead k k (subst d v (lift (S O) d t0)) t0 (subst (s k d) v -(lift (S O) (s k d) t1)) t1 (refl_equal K k) (H d) (H0 (s k d))) (subst d v -(THead k (lift (S O) d t0) (lift (S O) (s k d) t1))) (subst_head k v (lift (S -O) d t0) (lift (S O) (s k d) t1) d)) (lift (S O) d (THead k t0 t1)) -(lift_head k t0 t1 (S O) d)))))))) t)). -(* COMMENTS -Initial nodes: 879 -END *) - -theorem subst_subst0: - \forall (v: T).(\forall (t1: T).(\forall (t2: T).(\forall (d: nat).((subst0 -d v t1 t2) \to (eq T (subst d v t1) (subst d v t2)))))) -\def - \lambda (v: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (d: nat).(\lambda -(H: (subst0 d v t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t: -T).(\lambda (t0: T).(\lambda (t3: T).(eq T (subst n t t0) (subst n t t3)))))) -(\lambda (v0: T).(\lambda (i: nat).(eq_ind_r T (lift i O v0) (\lambda (t: -T).(eq T t (subst i v0 (lift (S i) O v0)))) (eq_ind nat (plus (S O) i) -(\lambda (n: nat).(eq T (lift i O v0) (subst i v0 (lift n O v0)))) (eq_ind T -(lift (S O) i (lift i O v0)) (\lambda (t: T).(eq T (lift i O v0) (subst i v0 -t))) (eq_ind_r T (lift i O v0) (\lambda (t: T).(eq T (lift i O v0) t)) -(refl_equal T (lift i O v0)) (subst i v0 (lift (S O) i (lift i O v0))) -(subst_lift_SO v0 (lift i O v0) i)) (lift (plus (S O) i) O v0) (lift_free v0 -i (S O) O i (le_n (plus O i)) (le_O_n i))) (S i) (refl_equal nat (S i))) -(subst i v0 (TLRef i)) (subst_lref_eq v0 i)))) (\lambda (v0: T).(\lambda (u2: -T).(\lambda (u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v0 u1 -u2)).(\lambda (H1: (eq T (subst i v0 u1) (subst i v0 u2))).(\lambda (t: -T).(\lambda (k: K).(eq_ind_r T (THead k (subst i v0 u1) (subst (s k i) v0 t)) -(\lambda (t0: T).(eq T t0 (subst i v0 (THead k u2 t)))) (eq_ind_r T (THead k -(subst i v0 u2) (subst (s k i) v0 t)) (\lambda (t0: T).(eq T (THead k (subst -i v0 u1) (subst (s k i) v0 t)) t0)) (eq_ind_r T (subst i v0 u2) (\lambda (t0: -T).(eq T (THead k t0 (subst (s k i) v0 t)) (THead k (subst i v0 u2) (subst (s -k i) v0 t)))) (refl_equal T (THead k (subst i v0 u2) (subst (s k i) v0 t))) -(subst i v0 u1) H1) (subst i v0 (THead k u2 t)) (subst_head k v0 u2 t i)) -(subst i v0 (THead k u1 t)) (subst_head k v0 u1 t i)))))))))) (\lambda (k: -K).(\lambda (v0: T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (i: -nat).(\lambda (_: (subst0 (s k i) v0 t4 t3)).(\lambda (H1: (eq T (subst (s k -i) v0 t4) (subst (s k i) v0 t3))).(\lambda (u: T).(eq_ind_r T (THead k (subst -i v0 u) (subst (s k i) v0 t4)) (\lambda (t: T).(eq T t (subst i v0 (THead k u -t3)))) (eq_ind_r T (THead k (subst i v0 u) (subst (s k i) v0 t3)) (\lambda -(t: T).(eq T (THead k (subst i v0 u) (subst (s k i) v0 t4)) t)) (eq_ind_r T -(subst (s k i) v0 t3) (\lambda (t: T).(eq T (THead k (subst i v0 u) t) (THead -k (subst i v0 u) (subst (s k i) v0 t3)))) (refl_equal T (THead k (subst i v0 -u) (subst (s k i) v0 t3))) (subst (s k i) v0 t4) H1) (subst i v0 (THead k u -t3)) (subst_head k v0 u t3 i)) (subst i v0 (THead k u t4)) (subst_head k v0 u -t4 i)))))))))) (\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda -(i: nat).(\lambda (_: (subst0 i v0 u1 u2)).(\lambda (H1: (eq T (subst i v0 -u1) (subst i v0 u2))).(\lambda (k: K).(\lambda (t3: T).(\lambda (t4: -T).(\lambda (_: (subst0 (s k i) v0 t3 t4)).(\lambda (H3: (eq T (subst (s k i) -v0 t3) (subst (s k i) v0 t4))).(eq_ind_r T (THead k (subst i v0 u1) (subst (s -k i) v0 t3)) (\lambda (t: T).(eq T t (subst i v0 (THead k u2 t4)))) (eq_ind_r -T (THead k (subst i v0 u2) (subst (s k i) v0 t4)) (\lambda (t: T).(eq T -(THead k (subst i v0 u1) (subst (s k i) v0 t3)) t)) (eq_ind_r T (subst i v0 -u2) (\lambda (t: T).(eq T (THead k t (subst (s k i) v0 t3)) (THead k (subst i -v0 u2) (subst (s k i) v0 t4)))) (eq_ind_r T (subst (s k i) v0 t4) (\lambda -(t: T).(eq T (THead k (subst i v0 u2) t) (THead k (subst i v0 u2) (subst (s k -i) v0 t4)))) (refl_equal T (THead k (subst i v0 u2) (subst (s k i) v0 t4))) -(subst (s k i) v0 t3) H3) (subst i v0 u1) H1) (subst i v0 (THead k u2 t4)) -(subst_head k v0 u2 t4 i)) (subst i v0 (THead k u1 t3)) (subst_head k v0 u1 -t3 i))))))))))))) d v t1 t2 H))))). -(* COMMENTS -Initial nodes: 1363 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/subst0/dec.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/subst0/dec.ma deleted file mode 100644 index 0234ff06c..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/subst0/dec.ma +++ /dev/null @@ -1,182 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/subst0/defs.ma". - -include "Basic-1/lift/props.ma". - -theorem dnf_dec2: - \forall (t: T).(\forall (d: nat).(or (\forall (w: T).(ex T (\lambda (v: -T).(subst0 d w t (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T t (lift (S -O) d v)))))) -\def - \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d: nat).(or (\forall (w: -T).(ex T (\lambda (v: T).(subst0 d w t0 (lift (S O) d v))))) (ex T (\lambda -(v: T).(eq T t0 (lift (S O) d v))))))) (\lambda (n: nat).(\lambda (d: -nat).(or_intror (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (TSort n) -(lift (S O) d v))))) (ex T (\lambda (v: T).(eq T (TSort n) (lift (S O) d -v)))) (ex_intro T (\lambda (v: T).(eq T (TSort n) (lift (S O) d v))) (TSort -n) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq T (TSort n) t0)) (refl_equal T -(TSort n)) (lift (S O) d (TSort n)) (lift_sort n (S O) d)))))) (\lambda (n: -nat).(\lambda (d: nat).(lt_eq_gt_e n d (or (\forall (w: T).(ex T (\lambda (v: -T).(subst0 d w (TLRef n) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T -(TLRef n) (lift (S O) d v))))) (\lambda (H: (lt n d)).(or_intror (\forall (w: -T).(ex T (\lambda (v: T).(subst0 d w (TLRef n) (lift (S O) d v))))) (ex T -(\lambda (v: T).(eq T (TLRef n) (lift (S O) d v)))) (ex_intro T (\lambda (v: -T).(eq T (TLRef n) (lift (S O) d v))) (TLRef n) (eq_ind_r T (TLRef n) -(\lambda (t0: T).(eq T (TLRef n) t0)) (refl_equal T (TLRef n)) (lift (S O) d -(TLRef n)) (lift_lref_lt n (S O) d H))))) (\lambda (H: (eq nat n d)).(eq_ind -nat n (\lambda (n0: nat).(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 n0 -w (TLRef n) (lift (S O) n0 v))))) (ex T (\lambda (v: T).(eq T (TLRef n) (lift -(S O) n0 v)))))) (or_introl (\forall (w: T).(ex T (\lambda (v: T).(subst0 n w -(TLRef n) (lift (S O) n v))))) (ex T (\lambda (v: T).(eq T (TLRef n) (lift (S -O) n v)))) (\lambda (w: T).(ex_intro T (\lambda (v: T).(subst0 n w (TLRef n) -(lift (S O) n v))) (lift n O w) (eq_ind_r T (lift (plus (S O) n) O w) -(\lambda (t0: T).(subst0 n w (TLRef n) t0)) (subst0_lref w n) (lift (S O) n -(lift n O w)) (lift_free w n (S O) O n (le_n (plus O n)) (le_O_n n)))))) d -H)) (\lambda (H: (lt d n)).(or_intror (\forall (w: T).(ex T (\lambda (v: -T).(subst0 d w (TLRef n) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T -(TLRef n) (lift (S O) d v)))) (ex_intro T (\lambda (v: T).(eq T (TLRef n) -(lift (S O) d v))) (TLRef (pred n)) (eq_ind_r T (TLRef n) (\lambda (t0: -T).(eq T (TLRef n) t0)) (refl_equal T (TLRef n)) (lift (S O) d (TLRef (pred -n))) (lift_lref_gt d n H)))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda -(H: ((\forall (d: nat).(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w -t0 (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T t0 (lift (S O) d -v)))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (d: nat).(or (\forall (w: -T).(ex T (\lambda (v: T).(subst0 d w t1 (lift (S O) d v))))) (ex T (\lambda -(v: T).(eq T t1 (lift (S O) d v)))))))).(\lambda (d: nat).(let H_x \def (H d) -in (let H1 \def H_x in (or_ind (\forall (w: T).(ex T (\lambda (v: T).(subst0 -d w t0 (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T t0 (lift (S O) d -v)))) (or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) -(lift (S O) d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 t1) (lift (S O) -d v))))) (\lambda (H2: ((\forall (w: T).(ex T (\lambda (v: T).(subst0 d w t0 -(lift (S O) d v))))))).(let H_x0 \def (H0 (s k d)) in (let H3 \def H_x0 in -(or_ind (\forall (w: T).(ex T (\lambda (v: T).(subst0 (s k d) w t1 (lift (S -O) (s k d) v))))) (ex T (\lambda (v: T).(eq T t1 (lift (S O) (s k d) v)))) -(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) (lift -(S O) d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 t1) (lift (S O) d -v))))) (\lambda (H4: ((\forall (w: T).(ex T (\lambda (v: T).(subst0 (s k d) w -t1 (lift (S O) (s k d) v))))))).(or_introl (\forall (w: T).(ex T (\lambda (v: -T).(subst0 d w (THead k t0 t1) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq -T (THead k t0 t1) (lift (S O) d v)))) (\lambda (w: T).(let H_x1 \def (H4 w) -in (let H5 \def H_x1 in (ex_ind T (\lambda (v: T).(subst0 (s k d) w t1 (lift -(S O) (s k d) v))) (ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) (lift (S -O) d v)))) (\lambda (x: T).(\lambda (H6: (subst0 (s k d) w t1 (lift (S O) (s -k d) x))).(let H_x2 \def (H2 w) in (let H7 \def H_x2 in (ex_ind T (\lambda -(v: T).(subst0 d w t0 (lift (S O) d v))) (ex T (\lambda (v: T).(subst0 d w -(THead k t0 t1) (lift (S O) d v)))) (\lambda (x0: T).(\lambda (H8: (subst0 d -w t0 (lift (S O) d x0))).(ex_intro T (\lambda (v: T).(subst0 d w (THead k t0 -t1) (lift (S O) d v))) (THead k x0 x) (eq_ind_r T (THead k (lift (S O) d x0) -(lift (S O) (s k d) x)) (\lambda (t2: T).(subst0 d w (THead k t0 t1) t2)) -(subst0_both w t0 (lift (S O) d x0) d H8 k t1 (lift (S O) (s k d) x) H6) -(lift (S O) d (THead k x0 x)) (lift_head k x0 x (S O) d))))) H7))))) H5)))))) -(\lambda (H4: (ex T (\lambda (v: T).(eq T t1 (lift (S O) (s k d) -v))))).(ex_ind T (\lambda (v: T).(eq T t1 (lift (S O) (s k d) v))) (or -(\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) (lift (S O) -d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 t1) (lift (S O) d v))))) -(\lambda (x: T).(\lambda (H5: (eq T t1 (lift (S O) (s k d) x))).(eq_ind_r T -(lift (S O) (s k d) x) (\lambda (t2: T).(or (\forall (w: T).(ex T (\lambda -(v: T).(subst0 d w (THead k t0 t2) (lift (S O) d v))))) (ex T (\lambda (v: -T).(eq T (THead k t0 t2) (lift (S O) d v)))))) (or_introl (\forall (w: T).(ex -T (\lambda (v: T).(subst0 d w (THead k t0 (lift (S O) (s k d) x)) (lift (S O) -d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 (lift (S O) (s k d) x)) -(lift (S O) d v)))) (\lambda (w: T).(let H_x1 \def (H2 w) in (let H6 \def -H_x1 in (ex_ind T (\lambda (v: T).(subst0 d w t0 (lift (S O) d v))) (ex T -(\lambda (v: T).(subst0 d w (THead k t0 (lift (S O) (s k d) x)) (lift (S O) d -v)))) (\lambda (x0: T).(\lambda (H7: (subst0 d w t0 (lift (S O) d -x0))).(ex_intro T (\lambda (v: T).(subst0 d w (THead k t0 (lift (S O) (s k d) -x)) (lift (S O) d v))) (THead k x0 x) (eq_ind_r T (THead k (lift (S O) d x0) -(lift (S O) (s k d) x)) (\lambda (t2: T).(subst0 d w (THead k t0 (lift (S O) -(s k d) x)) t2)) (subst0_fst w (lift (S O) d x0) t0 d H7 (lift (S O) (s k d) -x) k) (lift (S O) d (THead k x0 x)) (lift_head k x0 x (S O) d))))) H6))))) t1 -H5))) H4)) H3)))) (\lambda (H2: (ex T (\lambda (v: T).(eq T t0 (lift (S O) d -v))))).(ex_ind T (\lambda (v: T).(eq T t0 (lift (S O) d v))) (or (\forall (w: -T).(ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) (lift (S O) d v))))) (ex -T (\lambda (v: T).(eq T (THead k t0 t1) (lift (S O) d v))))) (\lambda (x: -T).(\lambda (H3: (eq T t0 (lift (S O) d x))).(let H_x0 \def (H0 (s k d)) in -(let H4 \def H_x0 in (or_ind (\forall (w: T).(ex T (\lambda (v: T).(subst0 (s -k d) w t1 (lift (S O) (s k d) v))))) (ex T (\lambda (v: T).(eq T t1 (lift (S -O) (s k d) v)))) (or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead -k t0 t1) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 t1) -(lift (S O) d v))))) (\lambda (H5: ((\forall (w: T).(ex T (\lambda (v: -T).(subst0 (s k d) w t1 (lift (S O) (s k d) v))))))).(eq_ind_r T (lift (S O) -d x) (\lambda (t2: T).(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w -(THead k t2 t1) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T (THead k t2 -t1) (lift (S O) d v)))))) (or_introl (\forall (w: T).(ex T (\lambda (v: -T).(subst0 d w (THead k (lift (S O) d x) t1) (lift (S O) d v))))) (ex T -(\lambda (v: T).(eq T (THead k (lift (S O) d x) t1) (lift (S O) d v)))) -(\lambda (w: T).(let H_x1 \def (H5 w) in (let H6 \def H_x1 in (ex_ind T -(\lambda (v: T).(subst0 (s k d) w t1 (lift (S O) (s k d) v))) (ex T (\lambda -(v: T).(subst0 d w (THead k (lift (S O) d x) t1) (lift (S O) d v)))) (\lambda -(x0: T).(\lambda (H7: (subst0 (s k d) w t1 (lift (S O) (s k d) -x0))).(ex_intro T (\lambda (v: T).(subst0 d w (THead k (lift (S O) d x) t1) -(lift (S O) d v))) (THead k x x0) (eq_ind_r T (THead k (lift (S O) d x) (lift -(S O) (s k d) x0)) (\lambda (t2: T).(subst0 d w (THead k (lift (S O) d x) t1) -t2)) (subst0_snd k w (lift (S O) (s k d) x0) t1 d H7 (lift (S O) d x)) (lift -(S O) d (THead k x x0)) (lift_head k x x0 (S O) d))))) H6))))) t0 H3)) -(\lambda (H5: (ex T (\lambda (v: T).(eq T t1 (lift (S O) (s k d) -v))))).(ex_ind T (\lambda (v: T).(eq T t1 (lift (S O) (s k d) v))) (or -(\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) (lift (S O) -d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 t1) (lift (S O) d v))))) -(\lambda (x0: T).(\lambda (H6: (eq T t1 (lift (S O) (s k d) x0))).(eq_ind_r T -(lift (S O) (s k d) x0) (\lambda (t2: T).(or (\forall (w: T).(ex T (\lambda -(v: T).(subst0 d w (THead k t0 t2) (lift (S O) d v))))) (ex T (\lambda (v: -T).(eq T (THead k t0 t2) (lift (S O) d v)))))) (eq_ind_r T (lift (S O) d x) -(\lambda (t2: T).(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead -k t2 (lift (S O) (s k d) x0)) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq -T (THead k t2 (lift (S O) (s k d) x0)) (lift (S O) d v)))))) (or_intror -(\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead k (lift (S O) d x) -(lift (S O) (s k d) x0)) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T -(THead k (lift (S O) d x) (lift (S O) (s k d) x0)) (lift (S O) d v)))) -(ex_intro T (\lambda (v: T).(eq T (THead k (lift (S O) d x) (lift (S O) (s k -d) x0)) (lift (S O) d v))) (THead k x x0) (eq_ind_r T (THead k (lift (S O) d -x) (lift (S O) (s k d) x0)) (\lambda (t2: T).(eq T (THead k (lift (S O) d x) -(lift (S O) (s k d) x0)) t2)) (refl_equal T (THead k (lift (S O) d x) (lift -(S O) (s k d) x0))) (lift (S O) d (THead k x x0)) (lift_head k x x0 (S O) -d)))) t0 H3) t1 H6))) H5)) H4))))) H2)) H1))))))))) t). -(* COMMENTS -Initial nodes: 3549 -END *) - -theorem dnf_dec: - \forall (w: T).(\forall (t: T).(\forall (d: nat).(ex T (\lambda (v: T).(or -(subst0 d w t (lift (S O) d v)) (eq T t (lift (S O) d v))))))) -\def - \lambda (w: T).(\lambda (t: T).(\lambda (d: nat).(let H_x \def (dnf_dec2 t -d) in (let H \def H_x in (or_ind (\forall (w0: T).(ex T (\lambda (v: -T).(subst0 d w0 t (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T t (lift (S -O) d v)))) (ex T (\lambda (v: T).(or (subst0 d w t (lift (S O) d v)) (eq T t -(lift (S O) d v))))) (\lambda (H0: ((\forall (w0: T).(ex T (\lambda (v: -T).(subst0 d w0 t (lift (S O) d v))))))).(let H_x0 \def (H0 w) in (let H1 -\def H_x0 in (ex_ind T (\lambda (v: T).(subst0 d w t (lift (S O) d v))) (ex T -(\lambda (v: T).(or (subst0 d w t (lift (S O) d v)) (eq T t (lift (S O) d -v))))) (\lambda (x: T).(\lambda (H2: (subst0 d w t (lift (S O) d -x))).(ex_intro T (\lambda (v: T).(or (subst0 d w t (lift (S O) d v)) (eq T t -(lift (S O) d v)))) x (or_introl (subst0 d w t (lift (S O) d x)) (eq T t -(lift (S O) d x)) H2)))) H1)))) (\lambda (H0: (ex T (\lambda (v: T).(eq T t -(lift (S O) d v))))).(ex_ind T (\lambda (v: T).(eq T t (lift (S O) d v))) (ex -T (\lambda (v: T).(or (subst0 d w t (lift (S O) d v)) (eq T t (lift (S O) d -v))))) (\lambda (x: T).(\lambda (H1: (eq T t (lift (S O) d x))).(eq_ind_r T -(lift (S O) d x) (\lambda (t0: T).(ex T (\lambda (v: T).(or (subst0 d w t0 -(lift (S O) d v)) (eq T t0 (lift (S O) d v)))))) (ex_intro T (\lambda (v: -T).(or (subst0 d w (lift (S O) d x) (lift (S O) d v)) (eq T (lift (S O) d x) -(lift (S O) d v)))) x (or_intror (subst0 d w (lift (S O) d x) (lift (S O) d -x)) (eq T (lift (S O) d x) (lift (S O) d x)) (refl_equal T (lift (S O) d -x)))) t H1))) H0)) H))))). -(* COMMENTS -Initial nodes: 603 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/subst0/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/subst0/defs.ma deleted file mode 100644 index a493a7ac2..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/subst0/defs.ma +++ /dev/null @@ -1,32 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/lift/defs.ma". - -inductive subst0: nat \to (T \to (T \to (T \to Prop))) \def -| subst0_lref: \forall (v: T).(\forall (i: nat).(subst0 i v (TLRef i) (lift -(S i) O v))) -| subst0_fst: \forall (v: T).(\forall (u2: T).(\forall (u1: T).(\forall (i: -nat).((subst0 i v u1 u2) \to (\forall (t: T).(\forall (k: K).(subst0 i v -(THead k u1 t) (THead k u2 t)))))))) -| subst0_snd: \forall (k: K).(\forall (v: T).(\forall (t2: T).(\forall (t1: -T).(\forall (i: nat).((subst0 (s k i) v t1 t2) \to (\forall (u: T).(subst0 i -v (THead k u t1) (THead k u t2)))))))) -| subst0_both: \forall (v: T).(\forall (u1: T).(\forall (u2: T).(\forall (i: -nat).((subst0 i v u1 u2) \to (\forall (k: K).(\forall (t1: T).(\forall (t2: -T).((subst0 (s k i) v t1 t2) \to (subst0 i v (THead k u1 t1) (THead k u2 -t2)))))))))). - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/subst0/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/subst0/fwd.ma deleted file mode 100644 index 165555fe2..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/subst0/fwd.ma +++ /dev/null @@ -1,728 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/subst0/defs.ma". - -include "Basic-1/lift/props.ma". - -theorem subst0_gen_sort: - \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst0 -i v (TSort n) x) \to (\forall (P: Prop).P))))) -\def - \lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda -(H: (subst0 i v (TSort n) x)).(\lambda (P: Prop).(insert_eq T (TSort n) -(\lambda (t: T).(subst0 i v t x)) (\lambda (_: T).P) (\lambda (y: T).(\lambda -(H0: (subst0 i v y x)).(subst0_ind (\lambda (_: nat).(\lambda (_: T).(\lambda -(t0: T).(\lambda (_: T).((eq T t0 (TSort n)) \to P))))) (\lambda (_: -T).(\lambda (i0: nat).(\lambda (H1: (eq T (TLRef i0) (TSort n))).(let H2 \def -(eq_ind T (TLRef i0) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | -(THead _ _ _) \Rightarrow False])) I (TSort n) H1) in (False_ind P H2))))) -(\lambda (v0: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i0: -nat).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (_: (((eq T u1 (TSort n)) -\to P))).(\lambda (t: T).(\lambda (k: K).(\lambda (H3: (eq T (THead k u1 t) -(TSort n))).(let H4 \def (eq_ind T (THead k u1 t) (\lambda (ee: T).(match ee -in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef -_) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H3) in -(False_ind P H4))))))))))) (\lambda (k: K).(\lambda (v0: T).(\lambda (t2: -T).(\lambda (t1: T).(\lambda (i0: nat).(\lambda (_: (subst0 (s k i0) v0 t1 -t2)).(\lambda (_: (((eq T t1 (TSort n)) \to P))).(\lambda (u: T).(\lambda -(H3: (eq T (THead k u t1) (TSort n))).(let H4 \def (eq_ind T (THead k u t1) -(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow -True])) I (TSort n) H3) in (False_ind P H4))))))))))) (\lambda (v0: -T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i0: nat).(\lambda (_: (subst0 -i0 v0 u1 u2)).(\lambda (_: (((eq T u1 (TSort n)) \to P))).(\lambda (k: -K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (subst0 (s k i0) v0 t1 -t2)).(\lambda (_: (((eq T t1 (TSort n)) \to P))).(\lambda (H5: (eq T (THead k -u1 t1) (TSort n))).(let H6 \def (eq_ind T (THead k u1 t1) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I -(TSort n) H5) in (False_ind P H6)))))))))))))) i v y x H0))) H)))))). -(* COMMENTS -Initial nodes: 445 -END *) - -theorem subst0_gen_lref: - \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst0 -i v (TLRef n) x) \to (land (eq nat n i) (eq T x (lift (S n) O v))))))) -\def - \lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda -(H: (subst0 i v (TLRef n) x)).(insert_eq T (TLRef n) (\lambda (t: T).(subst0 -i v t x)) (\lambda (_: T).(land (eq nat n i) (eq T x (lift (S n) O v)))) -(\lambda (y: T).(\lambda (H0: (subst0 i v y x)).(subst0_ind (\lambda (n0: -nat).(\lambda (t: T).(\lambda (t0: T).(\lambda (t1: T).((eq T t0 (TLRef n)) -\to (land (eq nat n n0) (eq T t1 (lift (S n) O t)))))))) (\lambda (v0: -T).(\lambda (i0: nat).(\lambda (H1: (eq T (TLRef i0) (TLRef n))).(let H2 \def -(f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with -[(TSort _) \Rightarrow i0 | (TLRef n0) \Rightarrow n0 | (THead _ _ _) -\Rightarrow i0])) (TLRef i0) (TLRef n) H1) in (eq_ind_r nat n (\lambda (n0: -nat).(land (eq nat n n0) (eq T (lift (S n0) O v0) (lift (S n) O v0)))) (conj -(eq nat n n) (eq T (lift (S n) O v0) (lift (S n) O v0)) (refl_equal nat n) -(refl_equal T (lift (S n) O v0))) i0 H2))))) (\lambda (v0: T).(\lambda (u2: -T).(\lambda (u1: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v0 u1 -u2)).(\lambda (_: (((eq T u1 (TLRef n)) \to (land (eq nat n i0) (eq T u2 -(lift (S n) O v0)))))).(\lambda (t: T).(\lambda (k: K).(\lambda (H3: (eq T -(THead k u1 t) (TLRef n))).(let H4 \def (eq_ind T (THead k u1 t) (\lambda -(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow -True])) I (TLRef n) H3) in (False_ind (land (eq nat n i0) (eq T (THead k u2 -t) (lift (S n) O v0))) H4))))))))))) (\lambda (k: K).(\lambda (v0: -T).(\lambda (t2: T).(\lambda (t1: T).(\lambda (i0: nat).(\lambda (_: (subst0 -(s k i0) v0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (land (eq nat n (s -k i0)) (eq T t2 (lift (S n) O v0)))))).(\lambda (u: T).(\lambda (H3: (eq T -(THead k u t1) (TLRef n))).(let H4 \def (eq_ind T (THead k u t1) (\lambda -(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow -True])) I (TLRef n) H3) in (False_ind (land (eq nat n i0) (eq T (THead k u -t2) (lift (S n) O v0))) H4))))))))))) (\lambda (v0: T).(\lambda (u1: -T).(\lambda (u2: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v0 u1 -u2)).(\lambda (_: (((eq T u1 (TLRef n)) \to (land (eq nat n i0) (eq T u2 -(lift (S n) O v0)))))).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (_: (subst0 (s k i0) v0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef -n)) \to (land (eq nat n (s k i0)) (eq T t2 (lift (S n) O v0)))))).(\lambda -(H5: (eq T (THead k u1 t1) (TLRef n))).(let H6 \def (eq_ind T (THead k u1 t1) -(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow -True])) I (TLRef n) H5) in (False_ind (land (eq nat n i0) (eq T (THead k u2 -t2) (lift (S n) O v0))) H6)))))))))))))) i v y x H0))) H))))). -(* COMMENTS -Initial nodes: 779 -END *) - -theorem subst0_gen_head: - \forall (k: K).(\forall (v: T).(\forall (u1: T).(\forall (t1: T).(\forall -(x: T).(\forall (i: nat).((subst0 i v (THead k u1 t1) x) \to (or3 (ex2 T -(\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 -u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k u1 t2))) (\lambda (t2: -T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1 -u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1 t2))))))))))) -\def - \lambda (k: K).(\lambda (v: T).(\lambda (u1: T).(\lambda (t1: T).(\lambda -(x: T).(\lambda (i: nat).(\lambda (H: (subst0 i v (THead k u1 t1) -x)).(insert_eq T (THead k u1 t1) (\lambda (t: T).(subst0 i v t x)) (\lambda -(_: T).(or3 (ex2 T (\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2: -T).(subst0 i v u1 u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k u1 t2))) -(\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1 -t2)))))) (\lambda (y: T).(\lambda (H0: (subst0 i v y x)).(subst0_ind (\lambda -(n: nat).(\lambda (t: T).(\lambda (t0: T).(\lambda (t2: T).((eq T t0 (THead k -u1 t1)) \to (or3 (ex2 T (\lambda (u2: T).(eq T t2 (THead k u2 t1))) (\lambda -(u2: T).(subst0 n t u1 u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead k u1 -t3))) (\lambda (t3: T).(subst0 (s k n) t t1 t3))) (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 n t u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k n) t t1 -t3)))))))))) (\lambda (v0: T).(\lambda (i0: nat).(\lambda (H1: (eq T (TLRef -i0) (THead k u1 t1))).(let H2 \def (eq_ind T (TLRef i0) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I -(THead k u1 t1) H1) in (False_ind (or3 (ex2 T (\lambda (u2: T).(eq T (lift (S -i0) O v0) (THead k u2 t1))) (\lambda (u2: T).(subst0 i0 v0 u1 u2))) (ex2 T -(\lambda (t2: T).(eq T (lift (S i0) O v0) (THead k u1 t2))) (\lambda (t2: -T).(subst0 (s k i0) v0 t1 t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T (lift (S i0) O v0) (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 i0 v0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i0) -v0 t1 t2))))) H2))))) (\lambda (v0: T).(\lambda (u2: T).(\lambda (u0: -T).(\lambda (i0: nat).(\lambda (H1: (subst0 i0 v0 u0 u2)).(\lambda (H2: (((eq -T u0 (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T u2 (THead k u3 -t1))) (\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t2: T).(eq T -u2 (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))) (ex3_2 T T -(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead k u3 t2)))) (\lambda (u3: -T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t2: -T).(subst0 (s k i0) v0 t1 t2)))))))).(\lambda (t: T).(\lambda (k0: -K).(\lambda (H3: (eq T (THead k0 u0 t) (THead k u1 t1))).(let H4 \def -(f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with -[(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _ _) -\Rightarrow k1])) (THead k0 u0 t) (THead k u1 t1) H3) in ((let H5 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t0 _) -\Rightarrow t0])) (THead k0 u0 t) (THead k u1 t1) H3) in ((let H6 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t0) -\Rightarrow t0])) (THead k0 u0 t) (THead k u1 t1) H3) in (\lambda (H7: (eq T -u0 u1)).(\lambda (H8: (eq K k0 k)).(eq_ind_r K k (\lambda (k1: K).(or3 (ex2 T -(\lambda (u3: T).(eq T (THead k1 u2 t) (THead k u3 t1))) (\lambda (u3: -T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t2: T).(eq T (THead k1 u2 t) -(THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))) (ex3_2 T T -(\lambda (u3: T).(\lambda (t2: T).(eq T (THead k1 u2 t) (THead k u3 t2)))) -(\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: -T).(\lambda (t2: T).(subst0 (s k i0) v0 t1 t2)))))) (eq_ind_r T t1 (\lambda -(t0: T).(or3 (ex2 T (\lambda (u3: T).(eq T (THead k u2 t0) (THead k u3 t1))) -(\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t2: T).(eq T (THead -k u2 t0) (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))) -(ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead k u2 t0) (THead k -u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda -(_: T).(\lambda (t2: T).(subst0 (s k i0) v0 t1 t2)))))) (let H9 \def (eq_ind -T u0 (\lambda (t0: T).((eq T t0 (THead k u1 t1)) \to (or3 (ex2 T (\lambda -(u3: T).(eq T u2 (THead k u3 t1))) (\lambda (u3: T).(subst0 i0 v0 u1 u3))) -(ex2 T (\lambda (t2: T).(eq T u2 (THead k u1 t2))) (\lambda (t2: T).(subst0 -(s k i0) v0 t1 t2))) (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 -(THead k u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) -(\lambda (_: T).(\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))))))) H2 u1 H7) -in (let H10 \def (eq_ind T u0 (\lambda (t0: T).(subst0 i0 v0 t0 u2)) H1 u1 -H7) in (or3_intro0 (ex2 T (\lambda (u3: T).(eq T (THead k u2 t1) (THead k u3 -t1))) (\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t2: T).(eq T -(THead k u2 t1) (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1 -t2))) (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead k u2 t1) -(THead k u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) -(\lambda (_: T).(\lambda (t2: T).(subst0 (s k i0) v0 t1 t2)))) (ex_intro2 T -(\lambda (u3: T).(eq T (THead k u2 t1) (THead k u3 t1))) (\lambda (u3: -T).(subst0 i0 v0 u1 u3)) u2 (refl_equal T (THead k u2 t1)) H10)))) t H6) k0 -H8)))) H5)) H4))))))))))) (\lambda (k0: K).(\lambda (v0: T).(\lambda (t2: -T).(\lambda (t0: T).(\lambda (i0: nat).(\lambda (H1: (subst0 (s k0 i0) v0 t0 -t2)).(\lambda (H2: (((eq T t0 (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u2: -T).(eq T t2 (THead k u2 t1))) (\lambda (u2: T).(subst0 (s k0 i0) v0 u1 u2))) -(ex2 T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 -(s k (s k0 i0)) v0 t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq -T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s k0 i0) v0 -u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k0 i0)) v0 t1 -t3)))))))).(\lambda (u: T).(\lambda (H3: (eq T (THead k0 u t0) (THead k u1 -t1))).(let H4 \def (f_equal T K (\lambda (e: T).(match e in T return (\lambda -(_: T).K) with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead -k1 _ _) \Rightarrow k1])) (THead k0 u t0) (THead k u1 t1) H3) in ((let H5 -\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) -\Rightarrow t])) (THead k0 u t0) (THead k u1 t1) H3) in ((let H6 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) -\Rightarrow t])) (THead k0 u t0) (THead k u1 t1) H3) in (\lambda (H7: (eq T u -u1)).(\lambda (H8: (eq K k0 k)).(eq_ind_r T u1 (\lambda (t: T).(or3 (ex2 T -(\lambda (u2: T).(eq T (THead k0 t t2) (THead k u2 t1))) (\lambda (u2: -T).(subst0 i0 v0 u1 u2))) (ex2 T (\lambda (t3: T).(eq T (THead k0 t t2) -(THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T (THead k0 t t2) (THead k u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(subst0 (s k i0) v0 t1 t3)))))) (let H9 \def (eq_ind T t0 -(\lambda (t: T).((eq T t (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u2: -T).(eq T t2 (THead k u2 t1))) (\lambda (u2: T).(subst0 (s k0 i0) v0 u1 u2))) -(ex2 T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 -(s k (s k0 i0)) v0 t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq -T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s k0 i0) v0 -u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k0 i0)) v0 t1 -t3))))))) H2 t1 H6) in (let H10 \def (eq_ind T t0 (\lambda (t: T).(subst0 (s -k0 i0) v0 t t2)) H1 t1 H6) in (let H11 \def (eq_ind K k0 (\lambda (k1: -K).((eq T t1 (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u2: T).(eq T t2 -(THead k u2 t1))) (\lambda (u2: T).(subst0 (s k1 i0) v0 u1 u2))) (ex2 T -(\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k (s -k1 i0)) v0 t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 -(THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s k1 i0) v0 u1 -u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k1 i0)) v0 t1 -t3))))))) H9 k H8) in (let H12 \def (eq_ind K k0 (\lambda (k1: K).(subst0 (s -k1 i0) v0 t1 t2)) H10 k H8) in (eq_ind_r K k (\lambda (k1: K).(or3 (ex2 T -(\lambda (u2: T).(eq T (THead k1 u1 t2) (THead k u2 t1))) (\lambda (u2: -T).(subst0 i0 v0 u1 u2))) (ex2 T (\lambda (t3: T).(eq T (THead k1 u1 t2) -(THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T (THead k1 u1 t2) (THead k u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(subst0 (s k i0) v0 t1 t3)))))) (or3_intro1 (ex2 T -(\lambda (u2: T).(eq T (THead k u1 t2) (THead k u2 t1))) (\lambda (u2: -T).(subst0 i0 v0 u1 u2))) (ex2 T (\lambda (t3: T).(eq T (THead k u1 t2) -(THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T (THead k u1 t2) (THead k u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(subst0 (s k i0) v0 t1 t3)))) (ex_intro2 T (\lambda (t3: -T).(eq T (THead k u1 t2) (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) -v0 t1 t3)) t2 (refl_equal T (THead k u1 t2)) H12)) k0 H8))))) u H7)))) H5)) -H4))))))))))) (\lambda (v0: T).(\lambda (u0: T).(\lambda (u2: T).(\lambda -(i0: nat).(\lambda (H1: (subst0 i0 v0 u0 u2)).(\lambda (H2: (((eq T u0 (THead -k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T u2 (THead k u3 t1))) -(\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t2: T).(eq T u2 -(THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))) (ex3_2 T T -(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead k u3 t2)))) (\lambda (u3: -T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t2: -T).(subst0 (s k i0) v0 t1 t2)))))))).(\lambda (k0: K).(\lambda (t0: -T).(\lambda (t2: T).(\lambda (H3: (subst0 (s k0 i0) v0 t0 t2)).(\lambda (H4: -(((eq T t0 (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T t2 (THead -k u3 t1))) (\lambda (u3: T).(subst0 (s k0 i0) v0 u1 u3))) (ex2 T (\lambda -(t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k (s k0 i0)) -v0 t1 t3))) (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead k u3 -t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 (s k0 i0) v0 u1 u3))) -(\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k0 i0)) v0 t1 -t3)))))))).(\lambda (H5: (eq T (THead k0 u0 t0) (THead k u1 t1))).(let H6 -\def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) -with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _ _) -\Rightarrow k1])) (THead k0 u0 t0) (THead k u1 t1) H5) in ((let H7 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) -\Rightarrow t])) (THead k0 u0 t0) (THead k u1 t1) H5) in ((let H8 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) -\Rightarrow t])) (THead k0 u0 t0) (THead k u1 t1) H5) in (\lambda (H9: (eq T -u0 u1)).(\lambda (H10: (eq K k0 k)).(let H11 \def (eq_ind T t0 (\lambda (t: -T).((eq T t (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T t2 (THead -k u3 t1))) (\lambda (u3: T).(subst0 (s k0 i0) v0 u1 u3))) (ex2 T (\lambda -(t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k (s k0 i0)) -v0 t1 t3))) (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead k u3 -t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 (s k0 i0) v0 u1 u3))) -(\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k0 i0)) v0 t1 t3))))))) H4 -t1 H8) in (let H12 \def (eq_ind T t0 (\lambda (t: T).(subst0 (s k0 i0) v0 t -t2)) H3 t1 H8) in (let H13 \def (eq_ind K k0 (\lambda (k1: K).((eq T t1 -(THead k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T t2 (THead k u3 t1))) -(\lambda (u3: T).(subst0 (s k1 i0) v0 u1 u3))) (ex2 T (\lambda (t3: T).(eq T -t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k (s k1 i0)) v0 t1 t3))) -(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead k u3 t3)))) -(\lambda (u3: T).(\lambda (_: T).(subst0 (s k1 i0) v0 u1 u3))) (\lambda (_: -T).(\lambda (t3: T).(subst0 (s k (s k1 i0)) v0 t1 t3))))))) H11 k H10) in -(let H14 \def (eq_ind K k0 (\lambda (k1: K).(subst0 (s k1 i0) v0 t1 t2)) H12 -k H10) in (eq_ind_r K k (\lambda (k1: K).(or3 (ex2 T (\lambda (u3: T).(eq T -(THead k1 u2 t2) (THead k u3 t1))) (\lambda (u3: T).(subst0 i0 v0 u1 u3))) -(ex2 T (\lambda (t3: T).(eq T (THead k1 u2 t2) (THead k u1 t3))) (\lambda -(t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T (\lambda (u3: T).(\lambda -(t3: T).(eq T (THead k1 u2 t2) (THead k u3 t3)))) (\lambda (u3: T).(\lambda -(_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k -i0) v0 t1 t3)))))) (let H15 \def (eq_ind T u0 (\lambda (t: T).((eq T t (THead -k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T u2 (THead k u3 t1))) -(\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t3: T).(eq T u2 -(THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T -(\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead k u3 t3)))) (\lambda (u3: -T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t3: -T).(subst0 (s k i0) v0 t1 t3))))))) H2 u1 H9) in (let H16 \def (eq_ind T u0 -(\lambda (t: T).(subst0 i0 v0 t u2)) H1 u1 H9) in (or3_intro2 (ex2 T (\lambda -(u3: T).(eq T (THead k u2 t2) (THead k u3 t1))) (\lambda (u3: T).(subst0 i0 -v0 u1 u3))) (ex2 T (\lambda (t3: T).(eq T (THead k u2 t2) (THead k u1 t3))) -(\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T (\lambda (u3: -T).(\lambda (t3: T).(eq T (THead k u2 t2) (THead k u3 t3)))) (\lambda (u3: -T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t3: -T).(subst0 (s k i0) v0 t1 t3)))) (ex3_2_intro T T (\lambda (u3: T).(\lambda -(t3: T).(eq T (THead k u2 t2) (THead k u3 t3)))) (\lambda (u3: T).(\lambda -(_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k -i0) v0 t1 t3))) u2 t2 (refl_equal T (THead k u2 t2)) H16 H14)))) k0 -H10)))))))) H7)) H6)))))))))))))) i v y x H0))) H))))))). -(* COMMENTS -Initial nodes: 4255 -END *) - -theorem subst0_gen_lift_lt: - \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall -(h: nat).(\forall (d: nat).((subst0 i (lift h d u) (lift h (S (plus i d)) t1) -x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda -(t2: T).(subst0 i u t1 t2))))))))) -\def - \lambda (u: T).(\lambda (t1: T).(T_ind (\lambda (t: T).(\forall (x: -T).(\forall (i: nat).(\forall (h: nat).(\forall (d: nat).((subst0 i (lift h d -u) (lift h (S (plus i d)) t) x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h -(S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u t t2))))))))) (\lambda (n: -nat).(\lambda (x: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda (d: -nat).(\lambda (H: (subst0 i (lift h d u) (lift h (S (plus i d)) (TSort n)) -x)).(let H0 \def (eq_ind T (lift h (S (plus i d)) (TSort n)) (\lambda (t: -T).(subst0 i (lift h d u) t x)) H (TSort n) (lift_sort n h (S (plus i d)))) -in (subst0_gen_sort (lift h d u) x i n H0 (ex2 T (\lambda (t2: T).(eq T x -(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TSort n) -t2))))))))))) (\lambda (n: nat).(\lambda (x: T).(\lambda (i: nat).(\lambda -(h: nat).(\lambda (d: nat).(\lambda (H: (subst0 i (lift h d u) (lift h (S -(plus i d)) (TLRef n)) x)).(lt_le_e n (S (plus i d)) (ex2 T (\lambda (t2: -T).(eq T x (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef -n) t2))) (\lambda (H0: (lt n (S (plus i d)))).(let H1 \def (eq_ind T (lift h -(S (plus i d)) (TLRef n)) (\lambda (t: T).(subst0 i (lift h d u) t x)) H -(TLRef n) (lift_lref_lt n h (S (plus i d)) H0)) in (land_ind (eq nat n i) (eq -T x (lift (S n) O (lift h d u))) (ex2 T (\lambda (t2: T).(eq T x (lift h (S -(plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef n) t2))) (\lambda (H2: -(eq nat n i)).(\lambda (H3: (eq T x (lift (S n) O (lift h d u)))).(eq_ind_r T -(lift (S n) O (lift h d u)) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t -(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef n) t2)))) -(eq_ind_r nat i (\lambda (n0: nat).(ex2 T (\lambda (t2: T).(eq T (lift (S n0) -O (lift h d u)) (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u -(TLRef n0) t2)))) (eq_ind T (lift h (plus (S i) d) (lift (S i) O u)) (\lambda -(t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h (S (plus i d)) t2))) (\lambda -(t2: T).(subst0 i u (TLRef i) t2)))) (ex_intro2 T (\lambda (t2: T).(eq T -(lift h (S (plus i d)) (lift (S i) O u)) (lift h (S (plus i d)) t2))) -(\lambda (t2: T).(subst0 i u (TLRef i) t2)) (lift (S i) O u) (refl_equal T -(lift h (S (plus i d)) (lift (S i) O u))) (subst0_lref u i)) (lift (S i) O -(lift h d u)) (lift_d u h (S i) d O (le_O_n d))) n H2) x H3))) -(subst0_gen_lref (lift h d u) x i n H1)))) (\lambda (H0: (le (S (plus i d)) -n)).(let H1 \def (eq_ind T (lift h (S (plus i d)) (TLRef n)) (\lambda (t: -T).(subst0 i (lift h d u) t x)) H (TLRef (plus n h)) (lift_lref_ge n h (S -(plus i d)) H0)) in (land_ind (eq nat (plus n h) i) (eq T x (lift (S (plus n -h)) O (lift h d u))) (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) -t2))) (\lambda (t2: T).(subst0 i u (TLRef n) t2))) (\lambda (H2: (eq nat -(plus n h) i)).(\lambda (_: (eq T x (lift (S (plus n h)) O (lift h d -u)))).(let H4 \def (eq_ind_r nat i (\lambda (n0: nat).(le (S (plus n0 d)) n)) -H0 (plus n h) H2) in (le_false n (plus (plus n h) d) (ex2 T (\lambda (t2: -T).(eq T x (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef -n) t2))) (le_plus_trans n (plus n h) d (le_plus_l n h)) H4)))) -(subst0_gen_lref (lift h d u) x i (plus n h) H1))))))))))) (\lambda (k: -K).(\lambda (t: T).(\lambda (H: ((\forall (x: T).(\forall (i: nat).(\forall -(h: nat).(\forall (d: nat).((subst0 i (lift h d u) (lift h (S (plus i d)) t) -x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda -(t2: T).(subst0 i u t t2)))))))))).(\lambda (t0: T).(\lambda (H0: ((\forall -(x: T).(\forall (i: nat).(\forall (h: nat).(\forall (d: nat).((subst0 i (lift -h d u) (lift h (S (plus i d)) t0) x) \to (ex2 T (\lambda (t2: T).(eq T x -(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u t0 -t2)))))))))).(\lambda (x: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda (d: -nat).(\lambda (H1: (subst0 i (lift h d u) (lift h (S (plus i d)) (THead k t -t0)) x)).(let H2 \def (eq_ind T (lift h (S (plus i d)) (THead k t t0)) -(\lambda (t2: T).(subst0 i (lift h d u) t2 x)) H1 (THead k (lift h (S (plus i -d)) t) (lift h (s k (S (plus i d))) t0)) (lift_head k t t0 h (S (plus i d)))) -in (or3_ind (ex2 T (\lambda (u2: T).(eq T x (THead k u2 (lift h (s k (S (plus -i d))) t0)))) (\lambda (u2: T).(subst0 i (lift h d u) (lift h (S (plus i d)) -t) u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h (S (plus i d)) t) -t2))) (\lambda (t2: T).(subst0 (s k i) (lift h d u) (lift h (s k (S (plus i -d))) t0) t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k -u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i (lift h d u) (lift h (S -(plus i d)) t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) (lift h -d u) (lift h (s k (S (plus i d))) t0) t2)))) (ex2 T (\lambda (t2: T).(eq T x -(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) -t2))) (\lambda (H3: (ex2 T (\lambda (u2: T).(eq T x (THead k u2 (lift h (s k -(S (plus i d))) t0)))) (\lambda (u2: T).(subst0 i (lift h d u) (lift h (S -(plus i d)) t) u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead k u2 (lift h -(s k (S (plus i d))) t0)))) (\lambda (u2: T).(subst0 i (lift h d u) (lift h -(S (plus i d)) t) u2)) (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) -t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2))) (\lambda (x0: -T).(\lambda (H4: (eq T x (THead k x0 (lift h (s k (S (plus i d))) -t0)))).(\lambda (H5: (subst0 i (lift h d u) (lift h (S (plus i d)) t) -x0)).(eq_ind_r T (THead k x0 (lift h (s k (S (plus i d))) t0)) (\lambda (t2: -T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda -(t3: T).(subst0 i u (THead k t t0) t3)))) (ex2_ind T (\lambda (t2: T).(eq T -x0 (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u t t2)) (ex2 T -(\lambda (t2: T).(eq T (THead k x0 (lift h (s k (S (plus i d))) t0)) (lift h -(S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2))) -(\lambda (x1: T).(\lambda (H6: (eq T x0 (lift h (S (plus i d)) x1))).(\lambda -(H7: (subst0 i u t x1)).(eq_ind_r T (lift h (S (plus i d)) x1) (\lambda (t2: -T).(ex2 T (\lambda (t3: T).(eq T (THead k t2 (lift h (s k (S (plus i d))) -t0)) (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u (THead k t t0) -t3)))) (eq_ind T (lift h (S (plus i d)) (THead k x1 t0)) (\lambda (t2: -T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda -(t3: T).(subst0 i u (THead k t t0) t3)))) (ex_intro2 T (\lambda (t2: T).(eq T -(lift h (S (plus i d)) (THead k x1 t0)) (lift h (S (plus i d)) t2))) (\lambda -(t2: T).(subst0 i u (THead k t t0) t2)) (THead k x1 t0) (refl_equal T (lift h -(S (plus i d)) (THead k x1 t0))) (subst0_fst u x1 t i H7 t0 k)) (THead k -(lift h (S (plus i d)) x1) (lift h (s k (S (plus i d))) t0)) (lift_head k x1 -t0 h (S (plus i d)))) x0 H6)))) (H x0 i h d H5)) x H4)))) H3)) (\lambda (H3: -(ex2 T (\lambda (t2: T).(eq T x (THead k (lift h (S (plus i d)) t) t2))) -(\lambda (t2: T).(subst0 (s k i) (lift h d u) (lift h (s k (S (plus i d))) -t0) t2)))).(ex2_ind T (\lambda (t2: T).(eq T x (THead k (lift h (S (plus i -d)) t) t2))) (\lambda (t2: T).(subst0 (s k i) (lift h d u) (lift h (s k (S -(plus i d))) t0) t2)) (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) -t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2))) (\lambda (x0: -T).(\lambda (H4: (eq T x (THead k (lift h (S (plus i d)) t) x0))).(\lambda -(H5: (subst0 (s k i) (lift h d u) (lift h (s k (S (plus i d))) t0) -x0)).(eq_ind_r T (THead k (lift h (S (plus i d)) t) x0) (\lambda (t2: T).(ex2 -T (\lambda (t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda (t3: -T).(subst0 i u (THead k t t0) t3)))) (let H6 \def (eq_ind nat (s k (S (plus i -d))) (\lambda (n: nat).(subst0 (s k i) (lift h d u) (lift h n t0) x0)) H5 (S -(s k (plus i d))) (s_S k (plus i d))) in (let H7 \def (eq_ind nat (s k (plus -i d)) (\lambda (n: nat).(subst0 (s k i) (lift h d u) (lift h (S n) t0) x0)) -H6 (plus (s k i) d) (s_plus k i d)) in (ex2_ind T (\lambda (t2: T).(eq T x0 -(lift h (S (plus (s k i) d)) t2))) (\lambda (t2: T).(subst0 (s k i) u t0 t2)) -(ex2 T (\lambda (t2: T).(eq T (THead k (lift h (S (plus i d)) t) x0) (lift h -(S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2))) -(\lambda (x1: T).(\lambda (H8: (eq T x0 (lift h (S (plus (s k i) d)) -x1))).(\lambda (H9: (subst0 (s k i) u t0 x1)).(eq_ind_r T (lift h (S (plus (s -k i) d)) x1) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k (lift h -(S (plus i d)) t) t2) (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i -u (THead k t t0) t3)))) (eq_ind nat (s k (plus i d)) (\lambda (n: nat).(ex2 T -(\lambda (t2: T).(eq T (THead k (lift h (S (plus i d)) t) (lift h (S n) x1)) -(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) -t2)))) (eq_ind nat (s k (S (plus i d))) (\lambda (n: nat).(ex2 T (\lambda -(t2: T).(eq T (THead k (lift h (S (plus i d)) t) (lift h n x1)) (lift h (S -(plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2)))) (eq_ind -T (lift h (S (plus i d)) (THead k t x1)) (\lambda (t2: T).(ex2 T (\lambda -(t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u -(THead k t t0) t3)))) (ex_intro2 T (\lambda (t2: T).(eq T (lift h (S (plus i -d)) (THead k t x1)) (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u -(THead k t t0) t2)) (THead k t x1) (refl_equal T (lift h (S (plus i d)) -(THead k t x1))) (subst0_snd k u x1 t0 i H9 t)) (THead k (lift h (S (plus i -d)) t) (lift h (s k (S (plus i d))) x1)) (lift_head k t x1 h (S (plus i d)))) -(S (s k (plus i d))) (s_S k (plus i d))) (plus (s k i) d) (s_plus k i d)) x0 -H8)))) (H0 x0 (s k i) h d H7)))) x H4)))) H3)) (\lambda (H3: (ex3_2 T T -(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i (lift h d u) (lift h (S (plus i d)) t) u2))) -(\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) (lift h d u) (lift h (s k (S -(plus i d))) t0) t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq -T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i (lift h d -u) (lift h (S (plus i d)) t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 -(s k i) (lift h d u) (lift h (s k (S (plus i d))) t0) t2))) (ex2 T (\lambda -(t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u -(THead k t t0) t2))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T x -(THead k x0 x1))).(\lambda (H5: (subst0 i (lift h d u) (lift h (S (plus i d)) -t) x0)).(\lambda (H6: (subst0 (s k i) (lift h d u) (lift h (s k (S (plus i -d))) t0) x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t2: T).(ex2 T (\lambda -(t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u -(THead k t t0) t3)))) (let H7 \def (eq_ind nat (s k (S (plus i d))) (\lambda -(n: nat).(subst0 (s k i) (lift h d u) (lift h n t0) x1)) H6 (S (s k (plus i -d))) (s_S k (plus i d))) in (let H8 \def (eq_ind nat (s k (plus i d)) -(\lambda (n: nat).(subst0 (s k i) (lift h d u) (lift h (S n) t0) x1)) H7 -(plus (s k i) d) (s_plus k i d)) in (ex2_ind T (\lambda (t2: T).(eq T x1 -(lift h (S (plus (s k i) d)) t2))) (\lambda (t2: T).(subst0 (s k i) u t0 t2)) -(ex2 T (\lambda (t2: T).(eq T (THead k x0 x1) (lift h (S (plus i d)) t2))) -(\lambda (t2: T).(subst0 i u (THead k t t0) t2))) (\lambda (x2: T).(\lambda -(H9: (eq T x1 (lift h (S (plus (s k i) d)) x2))).(\lambda (H10: (subst0 (s k -i) u t0 x2)).(ex2_ind T (\lambda (t2: T).(eq T x0 (lift h (S (plus i d)) -t2))) (\lambda (t2: T).(subst0 i u t t2)) (ex2 T (\lambda (t2: T).(eq T -(THead k x0 x1) (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u -(THead k t t0) t2))) (\lambda (x3: T).(\lambda (H11: (eq T x0 (lift h (S -(plus i d)) x3))).(\lambda (H12: (subst0 i u t x3)).(eq_ind_r T (lift h (S -(plus i d)) x3) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k t2 -x1) (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u (THead k t t0) -t3)))) (eq_ind_r T (lift h (S (plus (s k i) d)) x2) (\lambda (t2: T).(ex2 T -(\lambda (t3: T).(eq T (THead k (lift h (S (plus i d)) x3) t2) (lift h (S -(plus i d)) t3))) (\lambda (t3: T).(subst0 i u (THead k t t0) t3)))) (eq_ind -nat (s k (plus i d)) (\lambda (n: nat).(ex2 T (\lambda (t2: T).(eq T (THead k -(lift h (S (plus i d)) x3) (lift h (S n) x2)) (lift h (S (plus i d)) t2))) -(\lambda (t2: T).(subst0 i u (THead k t t0) t2)))) (eq_ind nat (s k (S (plus -i d))) (\lambda (n: nat).(ex2 T (\lambda (t2: T).(eq T (THead k (lift h (S -(plus i d)) x3) (lift h n x2)) (lift h (S (plus i d)) t2))) (\lambda (t2: -T).(subst0 i u (THead k t t0) t2)))) (eq_ind T (lift h (S (plus i d)) (THead -k x3 x2)) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h (S (plus -i d)) t3))) (\lambda (t3: T).(subst0 i u (THead k t t0) t3)))) (ex_intro2 T -(\lambda (t2: T).(eq T (lift h (S (plus i d)) (THead k x3 x2)) (lift h (S -(plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2)) (THead k -x3 x2) (refl_equal T (lift h (S (plus i d)) (THead k x3 x2))) (subst0_both u -t x3 i H12 k t0 x2 H10)) (THead k (lift h (S (plus i d)) x3) (lift h (s k (S -(plus i d))) x2)) (lift_head k x3 x2 h (S (plus i d)))) (S (s k (plus i d))) -(s_S k (plus i d))) (plus (s k i) d) (s_plus k i d)) x1 H9) x0 H11)))) (H x0 -i h d H5))))) (H0 x1 (s k i) h d H8)))) x H4)))))) H3)) (subst0_gen_head k -(lift h d u) (lift h (S (plus i d)) t) (lift h (s k (S (plus i d))) t0) x i -H2))))))))))))) t1)). -(* COMMENTS -Initial nodes: 5157 -END *) - -theorem subst0_gen_lift_false: - \forall (t: T).(\forall (u: T).(\forall (x: T).(\forall (h: nat).(\forall -(d: nat).(\forall (i: nat).((le d i) \to ((lt i (plus d h)) \to ((subst0 i u -(lift h d t) x) \to (\forall (P: Prop).P))))))))) -\def - \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (u: T).(\forall (x: -T).(\forall (h: nat).(\forall (d: nat).(\forall (i: nat).((le d i) \to ((lt i -(plus d h)) \to ((subst0 i u (lift h d t0) x) \to (\forall (P: -Prop).P)))))))))) (\lambda (n: nat).(\lambda (u: T).(\lambda (x: T).(\lambda -(h: nat).(\lambda (d: nat).(\lambda (i: nat).(\lambda (_: (le d i)).(\lambda -(_: (lt i (plus d h))).(\lambda (H1: (subst0 i u (lift h d (TSort n)) -x)).(\lambda (P: Prop).(let H2 \def (eq_ind T (lift h d (TSort n)) (\lambda -(t0: T).(subst0 i u t0 x)) H1 (TSort n) (lift_sort n h d)) in -(subst0_gen_sort u x i n H2 P)))))))))))) (\lambda (n: nat).(\lambda (u: -T).(\lambda (x: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (i: -nat).(\lambda (H: (le d i)).(\lambda (H0: (lt i (plus d h))).(\lambda (H1: -(subst0 i u (lift h d (TLRef n)) x)).(\lambda (P: Prop).(lt_le_e n d P -(\lambda (H2: (lt n d)).(let H3 \def (eq_ind T (lift h d (TLRef n)) (\lambda -(t0: T).(subst0 i u t0 x)) H1 (TLRef n) (lift_lref_lt n h d H2)) in (land_ind -(eq nat n i) (eq T x (lift (S n) O u)) P (\lambda (H4: (eq nat n i)).(\lambda -(_: (eq T x (lift (S n) O u))).(let H6 \def (eq_ind nat n (\lambda (n0: -nat).(lt n0 d)) H2 i H4) in (le_false d i P H H6)))) (subst0_gen_lref u x i n -H3)))) (\lambda (H2: (le d n)).(let H3 \def (eq_ind T (lift h d (TLRef n)) -(\lambda (t0: T).(subst0 i u t0 x)) H1 (TLRef (plus n h)) (lift_lref_ge n h d -H2)) in (land_ind (eq nat (plus n h) i) (eq T x (lift (S (plus n h)) O u)) P -(\lambda (H4: (eq nat (plus n h) i)).(\lambda (_: (eq T x (lift (S (plus n -h)) O u))).(let H6 \def (eq_ind_r nat i (\lambda (n0: nat).(lt n0 (plus d -h))) H0 (plus n h) H4) in (le_false d n P H2 (lt_le_S n d (simpl_lt_plus_r h -n d H6)))))) (subst0_gen_lref u x i (plus n h) H3))))))))))))))) (\lambda (k: -K).(\lambda (t0: T).(\lambda (H: ((\forall (u: T).(\forall (x: T).(\forall -(h: nat).(\forall (d: nat).(\forall (i: nat).((le d i) \to ((lt i (plus d h)) -\to ((subst0 i u (lift h d t0) x) \to (\forall (P: -Prop).P))))))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (u: T).(\forall -(x: T).(\forall (h: nat).(\forall (d: nat).(\forall (i: nat).((le d i) \to -((lt i (plus d h)) \to ((subst0 i u (lift h d t1) x) \to (\forall (P: -Prop).P))))))))))).(\lambda (u: T).(\lambda (x: T).(\lambda (h: nat).(\lambda -(d: nat).(\lambda (i: nat).(\lambda (H1: (le d i)).(\lambda (H2: (lt i (plus -d h))).(\lambda (H3: (subst0 i u (lift h d (THead k t0 t1)) x)).(\lambda (P: -Prop).(let H4 \def (eq_ind T (lift h d (THead k t0 t1)) (\lambda (t2: -T).(subst0 i u t2 x)) H3 (THead k (lift h d t0) (lift h (s k d) t1)) -(lift_head k t0 t1 h d)) in (or3_ind (ex2 T (\lambda (u2: T).(eq T x (THead k -u2 (lift h (s k d) t1)))) (\lambda (u2: T).(subst0 i u (lift h d t0) u2))) -(ex2 T (\lambda (t2: T).(eq T x (THead k (lift h d t0) t2))) (\lambda (t2: -T).(subst0 (s k i) u (lift h (s k d) t1) t2))) (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 i u (lift h d t0) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 -(s k i) u (lift h (s k d) t1) t2)))) P (\lambda (H5: (ex2 T (\lambda (u2: -T).(eq T x (THead k u2 (lift h (s k d) t1)))) (\lambda (u2: T).(subst0 i u -(lift h d t0) u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead k u2 (lift h -(s k d) t1)))) (\lambda (u2: T).(subst0 i u (lift h d t0) u2)) P (\lambda -(x0: T).(\lambda (_: (eq T x (THead k x0 (lift h (s k d) t1)))).(\lambda (H7: -(subst0 i u (lift h d t0) x0)).(H u x0 h d i H1 H2 H7 P)))) H5)) (\lambda -(H5: (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h d t0) t2))) (\lambda -(t2: T).(subst0 (s k i) u (lift h (s k d) t1) t2)))).(ex2_ind T (\lambda (t2: -T).(eq T x (THead k (lift h d t0) t2))) (\lambda (t2: T).(subst0 (s k i) u -(lift h (s k d) t1) t2)) P (\lambda (x0: T).(\lambda (_: (eq T x (THead k -(lift h d t0) x0))).(\lambda (H7: (subst0 (s k i) u (lift h (s k d) t1) -x0)).(H0 u x0 h (s k d) (s k i) (s_le k d i H1) (eq_ind nat (s k (plus d h)) -(\lambda (n: nat).(lt (s k i) n)) (lt_le_S (s k i) (s k (plus d h)) (s_lt k i -(plus d h) H2)) (plus (s k d) h) (s_plus k d h)) H7 P)))) H5)) (\lambda (H5: -(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 i u (lift h d t0) u2))) (\lambda (_: -T).(\lambda (t2: T).(subst0 (s k i) u (lift h (s k d) t1) t2))))).(ex3_2_ind -T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda -(u2: T).(\lambda (_: T).(subst0 i u (lift h d t0) u2))) (\lambda (_: -T).(\lambda (t2: T).(subst0 (s k i) u (lift h (s k d) t1) t2))) P (\lambda -(x0: T).(\lambda (x1: T).(\lambda (_: (eq T x (THead k x0 x1))).(\lambda (H7: -(subst0 i u (lift h d t0) x0)).(\lambda (_: (subst0 (s k i) u (lift h (s k d) -t1) x1)).(H u x0 h d i H1 H2 H7 P)))))) H5)) (subst0_gen_head k u (lift h d -t0) (lift h (s k d) t1) x i H4))))))))))))))))) t). -(* COMMENTS -Initial nodes: 1621 -END *) - -theorem subst0_gen_lift_ge: - \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall -(h: nat).(\forall (d: nat).((subst0 i u (lift h d t1) x) \to ((le (plus d h) -i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: -T).(subst0 (minus i h) u t1 t2)))))))))) -\def - \lambda (u: T).(\lambda (t1: T).(T_ind (\lambda (t: T).(\forall (x: -T).(\forall (i: nat).(\forall (h: nat).(\forall (d: nat).((subst0 i u (lift h -d t) x) \to ((le (plus d h) i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d -t2))) (\lambda (t2: T).(subst0 (minus i h) u t t2)))))))))) (\lambda (n: -nat).(\lambda (x: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda (d: -nat).(\lambda (H: (subst0 i u (lift h d (TSort n)) x)).(\lambda (_: (le (plus -d h) i)).(let H1 \def (eq_ind T (lift h d (TSort n)) (\lambda (t: T).(subst0 -i u t x)) H (TSort n) (lift_sort n h d)) in (subst0_gen_sort u x i n H1 (ex2 -T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i -h) u (TSort n) t2)))))))))))) (\lambda (n: nat).(\lambda (x: T).(\lambda (i: -nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (subst0 i u (lift h d -(TLRef n)) x)).(\lambda (H0: (le (plus d h) i)).(lt_le_e n d (ex2 T (\lambda -(t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (TLRef -n) t2))) (\lambda (H1: (lt n d)).(let H2 \def (eq_ind T (lift h d (TLRef n)) -(\lambda (t: T).(subst0 i u t x)) H (TLRef n) (lift_lref_lt n h d H1)) in -(land_ind (eq nat n i) (eq T x (lift (S n) O u)) (ex2 T (\lambda (t2: T).(eq -T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (TLRef n) t2))) -(\lambda (H3: (eq nat n i)).(\lambda (_: (eq T x (lift (S n) O u))).(let H5 -\def (eq_ind nat n (\lambda (n0: nat).(lt n0 d)) H1 i H3) in (le_false (plus -d h) i (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: -T).(subst0 (minus i h) u (TLRef n) t2))) H0 (le_plus_trans (S i) d h H5))))) -(subst0_gen_lref u x i n H2)))) (\lambda (H1: (le d n)).(let H2 \def (eq_ind -T (lift h d (TLRef n)) (\lambda (t: T).(subst0 i u t x)) H (TLRef (plus n h)) -(lift_lref_ge n h d H1)) in (land_ind (eq nat (plus n h) i) (eq T x (lift (S -(plus n h)) O u)) (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda -(t2: T).(subst0 (minus i h) u (TLRef n) t2))) (\lambda (H3: (eq nat (plus n -h) i)).(\lambda (H4: (eq T x (lift (S (plus n h)) O u))).(eq_ind nat (plus n -h) (\lambda (n0: nat).(ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) -(\lambda (t2: T).(subst0 (minus n0 h) u (TLRef n) t2)))) (eq_ind_r T (lift (S -(plus n h)) O u) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h d -t2))) (\lambda (t2: T).(subst0 (minus (plus n h) h) u (TLRef n) t2)))) -(eq_ind_r nat n (\lambda (n0: nat).(ex2 T (\lambda (t2: T).(eq T (lift (S -(plus n h)) O u) (lift h d t2))) (\lambda (t2: T).(subst0 n0 u (TLRef n) -t2)))) (ex_intro2 T (\lambda (t2: T).(eq T (lift (S (plus n h)) O u) (lift h -d t2))) (\lambda (t2: T).(subst0 n u (TLRef n) t2)) (lift (S n) O u) -(eq_ind_r T (lift (plus h (S n)) O u) (\lambda (t: T).(eq T (lift (S (plus n -h)) O u) t)) (eq_ind_r nat (plus h n) (\lambda (n0: nat).(eq T (lift (S n0) O -u) (lift (plus h (S n)) O u))) (eq_ind_r nat (plus h (S n)) (\lambda (n0: -nat).(eq T (lift n0 O u) (lift (plus h (S n)) O u))) (refl_equal T (lift -(plus h (S n)) O u)) (S (plus h n)) (plus_n_Sm h n)) (plus n h) (plus_sym n -h)) (lift h d (lift (S n) O u)) (lift_free u (S n) h O d (le_trans_plus_r O d -(plus O (S n)) (le_plus_plus O O d (S n) (le_n O) (le_S d n H1))) (le_O_n -d))) (subst0_lref u n)) (minus (plus n h) h) (minus_plus_r n h)) x H4) i -H3))) (subst0_gen_lref u x i (plus n h) H2)))))))))))) (\lambda (k: -K).(\lambda (t: T).(\lambda (H: ((\forall (x: T).(\forall (i: nat).(\forall -(h: nat).(\forall (d: nat).((subst0 i u (lift h d t) x) \to ((le (plus d h) -i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: -T).(subst0 (minus i h) u t t2))))))))))).(\lambda (t0: T).(\lambda (H0: -((\forall (x: T).(\forall (i: nat).(\forall (h: nat).(\forall (d: -nat).((subst0 i u (lift h d t0) x) \to ((le (plus d h) i) \to (ex2 T (\lambda -(t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u t0 -t2))))))))))).(\lambda (x: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda -(d: nat).(\lambda (H1: (subst0 i u (lift h d (THead k t t0)) x)).(\lambda -(H2: (le (plus d h) i)).(let H3 \def (eq_ind T (lift h d (THead k t t0)) -(\lambda (t2: T).(subst0 i u t2 x)) H1 (THead k (lift h d t) (lift h (s k d) -t0)) (lift_head k t t0 h d)) in (or3_ind (ex2 T (\lambda (u2: T).(eq T x -(THead k u2 (lift h (s k d) t0)))) (\lambda (u2: T).(subst0 i u (lift h d t) -u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h d t) t2))) (\lambda -(t2: T).(subst0 (s k i) u (lift h (s k d) t0) t2))) (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 i u (lift h d t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s -k i) u (lift h (s k d) t0) t2)))) (ex2 T (\lambda (t2: T).(eq T x (lift h d -t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda -(H4: (ex2 T (\lambda (u2: T).(eq T x (THead k u2 (lift h (s k d) t0)))) -(\lambda (u2: T).(subst0 i u (lift h d t) u2)))).(ex2_ind T (\lambda (u2: -T).(eq T x (THead k u2 (lift h (s k d) t0)))) (\lambda (u2: T).(subst0 i u -(lift h d t) u2)) (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda -(t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (x0: T).(\lambda -(H5: (eq T x (THead k x0 (lift h (s k d) t0)))).(\lambda (H6: (subst0 i u -(lift h d t) x0)).(eq_ind_r T (THead k x0 (lift h (s k d) t0)) (\lambda (t2: -T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst0 -(minus i h) u (THead k t t0) t3)))) (ex2_ind T (\lambda (t2: T).(eq T x0 -(lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u t t2)) (ex2 T (\lambda -(t2: T).(eq T (THead k x0 (lift h (s k d) t0)) (lift h d t2))) (\lambda (t2: -T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (x1: T).(\lambda (H7: -(eq T x0 (lift h d x1))).(\lambda (H8: (subst0 (minus i h) u t x1)).(eq_ind_r -T (lift h d x1) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k t2 -(lift h (s k d) t0)) (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u -(THead k t t0) t3)))) (eq_ind T (lift h d (THead k x1 t0)) (\lambda (t2: -T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst0 -(minus i h) u (THead k t t0) t3)))) (ex_intro2 T (\lambda (t2: T).(eq T (lift -h d (THead k x1 t0)) (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u -(THead k t t0) t2)) (THead k x1 t0) (refl_equal T (lift h d (THead k x1 t0))) -(subst0_fst u x1 t (minus i h) H8 t0 k)) (THead k (lift h d x1) (lift h (s k -d) t0)) (lift_head k x1 t0 h d)) x0 H7)))) (H x0 i h d H6 H2)) x H5)))) H4)) -(\lambda (H4: (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h d t) t2))) -(\lambda (t2: T).(subst0 (s k i) u (lift h (s k d) t0) t2)))).(ex2_ind T -(\lambda (t2: T).(eq T x (THead k (lift h d t) t2))) (\lambda (t2: T).(subst0 -(s k i) u (lift h (s k d) t0) t2)) (ex2 T (\lambda (t2: T).(eq T x (lift h d -t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda -(x0: T).(\lambda (H5: (eq T x (THead k (lift h d t) x0))).(\lambda (H6: -(subst0 (s k i) u (lift h (s k d) t0) x0)).(eq_ind_r T (THead k (lift h d t) -x0) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3))) -(\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) (ex2_ind T -(\lambda (t2: T).(eq T x0 (lift h (s k d) t2))) (\lambda (t2: T).(subst0 -(minus (s k i) h) u t0 t2)) (ex2 T (\lambda (t2: T).(eq T (THead k (lift h d -t) x0) (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead k t t0) -t2))) (\lambda (x1: T).(\lambda (H7: (eq T x0 (lift h (s k d) x1))).(\lambda -(H8: (subst0 (minus (s k i) h) u t0 x1)).(eq_ind_r T (lift h (s k d) x1) -(\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k (lift h d t) t2) -(lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) -(eq_ind T (lift h d (THead k t x1)) (\lambda (t2: T).(ex2 T (\lambda (t3: -T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t -t0) t3)))) (let H9 \def (eq_ind_r nat (minus (s k i) h) (\lambda (n: -nat).(subst0 n u t0 x1)) H8 (s k (minus i h)) (s_minus k i h (le_trans_plus_r -d h i H2))) in (ex_intro2 T (\lambda (t2: T).(eq T (lift h d (THead k t x1)) -(lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead k t t0) t2)) -(THead k t x1) (refl_equal T (lift h d (THead k t x1))) (subst0_snd k u x1 t0 -(minus i h) H9 t))) (THead k (lift h d t) (lift h (s k d) x1)) (lift_head k t -x1 h d)) x0 H7)))) (H0 x0 (s k i) h (s k d) H6 (eq_ind nat (s k (plus d h)) -(\lambda (n: nat).(le n (s k i))) (s_le k (plus d h) i H2) (plus (s k d) h) -(s_plus k d h)))) x H5)))) H4)) (\lambda (H4: (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 i u (lift h d t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s -k i) u (lift h (s k d) t0) t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda -(t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i -u (lift h d t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) u (lift -h (s k d) t0) t2))) (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda -(t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (x0: T).(\lambda -(x1: T).(\lambda (H5: (eq T x (THead k x0 x1))).(\lambda (H6: (subst0 i u -(lift h d t) x0)).(\lambda (H7: (subst0 (s k i) u (lift h (s k d) t0) -x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq -T t2 (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) -t3)))) (ex2_ind T (\lambda (t2: T).(eq T x1 (lift h (s k d) t2))) (\lambda -(t2: T).(subst0 (minus (s k i) h) u t0 t2)) (ex2 T (\lambda (t2: T).(eq T -(THead k x0 x1) (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead -k t t0) t2))) (\lambda (x2: T).(\lambda (H8: (eq T x1 (lift h (s k d) -x2))).(\lambda (H9: (subst0 (minus (s k i) h) u t0 x2)).(ex2_ind T (\lambda -(t2: T).(eq T x0 (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u t -t2)) (ex2 T (\lambda (t2: T).(eq T (THead k x0 x1) (lift h d t2))) (\lambda -(t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (x3: T).(\lambda -(H10: (eq T x0 (lift h d x3))).(\lambda (H11: (subst0 (minus i h) u t -x3)).(eq_ind_r T (lift h d x3) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T -(THead k t2 x1) (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead -k t t0) t3)))) (eq_ind_r T (lift h (s k d) x2) (\lambda (t2: T).(ex2 T -(\lambda (t3: T).(eq T (THead k (lift h d x3) t2) (lift h d t3))) (\lambda -(t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) (eq_ind T (lift h d -(THead k x3 x2)) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d -t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) (let H12 -\def (eq_ind_r nat (minus (s k i) h) (\lambda (n: nat).(subst0 n u t0 x2)) H9 -(s k (minus i h)) (s_minus k i h (le_trans_plus_r d h i H2))) in (ex_intro2 T -(\lambda (t2: T).(eq T (lift h d (THead k x3 x2)) (lift h d t2))) (\lambda -(t2: T).(subst0 (minus i h) u (THead k t t0) t2)) (THead k x3 x2) (refl_equal -T (lift h d (THead k x3 x2))) (subst0_both u t x3 (minus i h) H11 k t0 x2 -H12))) (THead k (lift h d x3) (lift h (s k d) x2)) (lift_head k x3 x2 h d)) -x1 H8) x0 H10)))) (H x0 i h d H6 H2))))) (H0 x1 (s k i) h (s k d) H7 (eq_ind -nat (s k (plus d h)) (\lambda (n: nat).(le n (s k i))) (s_le k (plus d h) i -H2) (plus (s k d) h) (s_plus k d h)))) x H5)))))) H4)) (subst0_gen_head k u -(lift h d t) (lift h (s k d) t0) x i H3)))))))))))))) t1)). -(* COMMENTS -Initial nodes: 4191 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/subst0/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/subst0/props.ma deleted file mode 100644 index 5da05fa2a..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/subst0/props.ma +++ /dev/null @@ -1,241 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/subst0/fwd.ma". - -theorem subst0_refl: - \forall (u: T).(\forall (t: T).(\forall (d: nat).((subst0 d u t t) \to -(\forall (P: Prop).P)))) -\def - \lambda (u: T).(\lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d: -nat).((subst0 d u t0 t0) \to (\forall (P: Prop).P)))) (\lambda (n: -nat).(\lambda (d: nat).(\lambda (H: (subst0 d u (TSort n) (TSort -n))).(\lambda (P: Prop).(subst0_gen_sort u (TSort n) d n H P))))) (\lambda -(n: nat).(\lambda (d: nat).(\lambda (H: (subst0 d u (TLRef n) (TLRef -n))).(\lambda (P: Prop).(land_ind (eq nat n d) (eq T (TLRef n) (lift (S n) O -u)) P (\lambda (_: (eq nat n d)).(\lambda (H1: (eq T (TLRef n) (lift (S n) O -u))).(lift_gen_lref_false (S n) O n (le_O_n n) (le_n (plus O (S n))) u H1 -P))) (subst0_gen_lref u (TLRef n) d n H)))))) (\lambda (k: K).(\lambda (t0: -T).(\lambda (H: ((\forall (d: nat).((subst0 d u t0 t0) \to (\forall (P: -Prop).P))))).(\lambda (t1: T).(\lambda (H0: ((\forall (d: nat).((subst0 d u -t1 t1) \to (\forall (P: Prop).P))))).(\lambda (d: nat).(\lambda (H1: (subst0 -d u (THead k t0 t1) (THead k t0 t1))).(\lambda (P: Prop).(or3_ind (ex2 T -(\lambda (u2: T).(eq T (THead k t0 t1) (THead k u2 t1))) (\lambda (u2: -T).(subst0 d u t0 u2))) (ex2 T (\lambda (t2: T).(eq T (THead k t0 t1) (THead -k t0 t2))) (\lambda (t2: T).(subst0 (s k d) u t1 t2))) (ex3_2 T T (\lambda -(u2: T).(\lambda (t2: T).(eq T (THead k t0 t1) (THead k u2 t2)))) (\lambda -(u2: T).(\lambda (_: T).(subst0 d u t0 u2))) (\lambda (_: T).(\lambda (t2: -T).(subst0 (s k d) u t1 t2)))) P (\lambda (H2: (ex2 T (\lambda (u2: T).(eq T -(THead k t0 t1) (THead k u2 t1))) (\lambda (u2: T).(subst0 d u t0 -u2)))).(ex2_ind T (\lambda (u2: T).(eq T (THead k t0 t1) (THead k u2 t1))) -(\lambda (u2: T).(subst0 d u t0 u2)) P (\lambda (x: T).(\lambda (H3: (eq T -(THead k t0 t1) (THead k x t1))).(\lambda (H4: (subst0 d u t0 x)).(let H5 -\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ t2 _) -\Rightarrow t2])) (THead k t0 t1) (THead k x t1) H3) in (let H6 \def -(eq_ind_r T x (\lambda (t2: T).(subst0 d u t0 t2)) H4 t0 H5) in (H d H6 -P)))))) H2)) (\lambda (H2: (ex2 T (\lambda (t2: T).(eq T (THead k t0 t1) -(THead k t0 t2))) (\lambda (t2: T).(subst0 (s k d) u t1 t2)))).(ex2_ind T -(\lambda (t2: T).(eq T (THead k t0 t1) (THead k t0 t2))) (\lambda (t2: -T).(subst0 (s k d) u t1 t2)) P (\lambda (x: T).(\lambda (H3: (eq T (THead k -t0 t1) (THead k t0 x))).(\lambda (H4: (subst0 (s k d) u t1 x)).(let H5 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ _ t2) -\Rightarrow t2])) (THead k t0 t1) (THead k t0 x) H3) in (let H6 \def -(eq_ind_r T x (\lambda (t2: T).(subst0 (s k d) u t1 t2)) H4 t1 H5) in (H0 (s -k d) H6 P)))))) H2)) (\lambda (H2: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T (THead k t0 t1) (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 d u t0 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k d) u t1 -t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead k t0 -t1) (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 d u t0 u2))) -(\lambda (_: T).(\lambda (t2: T).(subst0 (s k d) u t1 t2))) P (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H3: (eq T (THead k t0 t1) (THead k x0 -x1))).(\lambda (H4: (subst0 d u t0 x0)).(\lambda (H5: (subst0 (s k d) u t1 -x1)).(let H6 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda -(_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead -_ t2 _) \Rightarrow t2])) (THead k t0 t1) (THead k x0 x1) H3) in ((let H7 -\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ _ t2) -\Rightarrow t2])) (THead k t0 t1) (THead k x0 x1) H3) in (\lambda (H8: (eq T -t0 x0)).(let H9 \def (eq_ind_r T x1 (\lambda (t2: T).(subst0 (s k d) u t1 -t2)) H5 t1 H7) in (let H10 \def (eq_ind_r T x0 (\lambda (t2: T).(subst0 d u -t0 t2)) H4 t0 H8) in (H d H10 P))))) H6))))))) H2)) (subst0_gen_head k u t0 -t1 (THead k t0 t1) d H1)))))))))) t)). -(* COMMENTS -Initial nodes: 1119 -END *) - -theorem subst0_lift_lt: - \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).((subst0 -i u t1 t2) \to (\forall (d: nat).((lt i d) \to (\forall (h: nat).(subst0 i -(lift h (minus d (S i)) u) (lift h d t1) (lift h d t2))))))))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H: (subst0 i u t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t: -T).(\lambda (t0: T).(\lambda (t3: T).(\forall (d: nat).((lt n d) \to (\forall -(h: nat).(subst0 n (lift h (minus d (S n)) t) (lift h d t0) (lift h d -t3))))))))) (\lambda (v: T).(\lambda (i0: nat).(\lambda (d: nat).(\lambda -(H0: (lt i0 d)).(\lambda (h: nat).(eq_ind_r T (TLRef i0) (\lambda (t: -T).(subst0 i0 (lift h (minus d (S i0)) v) t (lift h d (lift (S i0) O v)))) -(let w \def (minus d (S i0)) in (eq_ind nat (plus (S i0) (minus d (S i0))) -(\lambda (n: nat).(subst0 i0 (lift h w v) (TLRef i0) (lift h n (lift (S i0) O -v)))) (eq_ind_r T (lift (S i0) O (lift h (minus d (S i0)) v)) (\lambda (t: -T).(subst0 i0 (lift h w v) (TLRef i0) t)) (subst0_lref (lift h (minus d (S -i0)) v) i0) (lift h (plus (S i0) (minus d (S i0))) (lift (S i0) O v)) (lift_d -v h (S i0) (minus d (S i0)) O (le_O_n (minus d (S i0))))) d (le_plus_minus_r -(S i0) d H0))) (lift h d (TLRef i0)) (lift_lref_lt i0 h d H0))))))) (\lambda -(v: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i0: nat).(\lambda (_: -(subst0 i0 v u1 u2)).(\lambda (H1: ((\forall (d: nat).((lt i0 d) \to (\forall -(h: nat).(subst0 i0 (lift h (minus d (S i0)) v) (lift h d u1) (lift h d -u2))))))).(\lambda (t: T).(\lambda (k: K).(\lambda (d: nat).(\lambda (H2: (lt -i0 d)).(\lambda (h: nat).(eq_ind_r T (THead k (lift h d u1) (lift h (s k d) -t)) (\lambda (t0: T).(subst0 i0 (lift h (minus d (S i0)) v) t0 (lift h d -(THead k u2 t)))) (eq_ind_r T (THead k (lift h d u2) (lift h (s k d) t)) -(\lambda (t0: T).(subst0 i0 (lift h (minus d (S i0)) v) (THead k (lift h d -u1) (lift h (s k d) t)) t0)) (subst0_fst (lift h (minus d (S i0)) v) (lift h -d u2) (lift h d u1) i0 (H1 d H2 h) (lift h (s k d) t) k) (lift h d (THead k -u2 t)) (lift_head k u2 t h d)) (lift h d (THead k u1 t)) (lift_head k u1 t h -d))))))))))))) (\lambda (k: K).(\lambda (v: T).(\lambda (t0: T).(\lambda (t3: -T).(\lambda (i0: nat).(\lambda (_: (subst0 (s k i0) v t3 t0)).(\lambda (H1: -((\forall (d: nat).((lt (s k i0) d) \to (\forall (h: nat).(subst0 (s k i0) -(lift h (minus d (S (s k i0))) v) (lift h d t3) (lift h d t0))))))).(\lambda -(u0: T).(\lambda (d: nat).(\lambda (H2: (lt i0 d)).(\lambda (h: nat).(let H3 -\def (eq_ind_r nat (S (s k i0)) (\lambda (n: nat).(\forall (d0: nat).((lt (s -k i0) d0) \to (\forall (h0: nat).(subst0 (s k i0) (lift h0 (minus d0 n) v) -(lift h0 d0 t3) (lift h0 d0 t0)))))) H1 (s k (S i0)) (s_S k i0)) in (eq_ind_r -T (THead k (lift h d u0) (lift h (s k d) t3)) (\lambda (t: T).(subst0 i0 -(lift h (minus d (S i0)) v) t (lift h d (THead k u0 t0)))) (eq_ind_r T (THead -k (lift h d u0) (lift h (s k d) t0)) (\lambda (t: T).(subst0 i0 (lift h -(minus d (S i0)) v) (THead k (lift h d u0) (lift h (s k d) t3)) t)) (eq_ind -nat (minus (s k d) (s k (S i0))) (\lambda (n: nat).(subst0 i0 (lift h n v) -(THead k (lift h d u0) (lift h (s k d) t3)) (THead k (lift h d u0) (lift h (s -k d) t0)))) (subst0_snd k (lift h (minus (s k d) (s k (S i0))) v) (lift h (s -k d) t0) (lift h (s k d) t3) i0 (H3 (s k d) (s_lt k i0 d H2) h) (lift h d -u0)) (minus d (S i0)) (minus_s_s k d (S i0))) (lift h d (THead k u0 t0)) -(lift_head k u0 t0 h d)) (lift h d (THead k u0 t3)) (lift_head k u0 t3 h -d)))))))))))))) (\lambda (v: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda -(i0: nat).(\lambda (_: (subst0 i0 v u1 u2)).(\lambda (H1: ((\forall (d: -nat).((lt i0 d) \to (\forall (h: nat).(subst0 i0 (lift h (minus d (S i0)) v) -(lift h d u1) (lift h d u2))))))).(\lambda (k: K).(\lambda (t0: T).(\lambda -(t3: T).(\lambda (_: (subst0 (s k i0) v t0 t3)).(\lambda (H3: ((\forall (d: -nat).((lt (s k i0) d) \to (\forall (h: nat).(subst0 (s k i0) (lift h (minus d -(S (s k i0))) v) (lift h d t0) (lift h d t3))))))).(\lambda (d: nat).(\lambda -(H4: (lt i0 d)).(\lambda (h: nat).(let H5 \def (eq_ind_r nat (S (s k i0)) -(\lambda (n: nat).(\forall (d0: nat).((lt (s k i0) d0) \to (\forall (h0: -nat).(subst0 (s k i0) (lift h0 (minus d0 n) v) (lift h0 d0 t0) (lift h0 d0 -t3)))))) H3 (s k (S i0)) (s_S k i0)) in (eq_ind_r T (THead k (lift h d u1) -(lift h (s k d) t0)) (\lambda (t: T).(subst0 i0 (lift h (minus d (S i0)) v) t -(lift h d (THead k u2 t3)))) (eq_ind_r T (THead k (lift h d u2) (lift h (s k -d) t3)) (\lambda (t: T).(subst0 i0 (lift h (minus d (S i0)) v) (THead k (lift -h d u1) (lift h (s k d) t0)) t)) (subst0_both (lift h (minus d (S i0)) v) -(lift h d u1) (lift h d u2) i0 (H1 d H4 h) k (lift h (s k d) t0) (lift h (s k -d) t3) (eq_ind nat (minus (s k d) (s k (S i0))) (\lambda (n: nat).(subst0 (s -k i0) (lift h n v) (lift h (s k d) t0) (lift h (s k d) t3))) (H5 (s k d) -(s_lt k i0 d H4) h) (minus d (S i0)) (minus_s_s k d (S i0)))) (lift h d -(THead k u2 t3)) (lift_head k u2 t3 h d)) (lift h d (THead k u1 t0)) -(lift_head k u1 t0 h d))))))))))))))))) i u t1 t2 H))))). -(* COMMENTS -Initial nodes: 1805 -END *) - -theorem subst0_lift_ge: - \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).(\forall -(h: nat).((subst0 i u t1 t2) \to (\forall (d: nat).((le d i) \to (subst0 -(plus i h) u (lift h d t1) (lift h d t2))))))))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda -(h: nat).(\lambda (H: (subst0 i u t1 t2)).(subst0_ind (\lambda (n: -nat).(\lambda (t: T).(\lambda (t0: T).(\lambda (t3: T).(\forall (d: nat).((le -d n) \to (subst0 (plus n h) t (lift h d t0) (lift h d t3)))))))) (\lambda (v: -T).(\lambda (i0: nat).(\lambda (d: nat).(\lambda (H0: (le d i0)).(eq_ind_r T -(TLRef (plus i0 h)) (\lambda (t: T).(subst0 (plus i0 h) v t (lift h d (lift -(S i0) O v)))) (eq_ind_r T (lift (plus h (S i0)) O v) (\lambda (t: T).(subst0 -(plus i0 h) v (TLRef (plus i0 h)) t)) (eq_ind nat (S (plus h i0)) (\lambda -(n: nat).(subst0 (plus i0 h) v (TLRef (plus i0 h)) (lift n O v))) (eq_ind_r -nat (plus h i0) (\lambda (n: nat).(subst0 n v (TLRef n) (lift (S (plus h i0)) -O v))) (subst0_lref v (plus h i0)) (plus i0 h) (plus_sym i0 h)) (plus h (S -i0)) (plus_n_Sm h i0)) (lift h d (lift (S i0) O v)) (lift_free v (S i0) h O d -(le_S d i0 H0) (le_O_n d))) (lift h d (TLRef i0)) (lift_lref_ge i0 h d -H0)))))) (\lambda (v: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i0: -nat).(\lambda (_: (subst0 i0 v u1 u2)).(\lambda (H1: ((\forall (d: nat).((le -d i0) \to (subst0 (plus i0 h) v (lift h d u1) (lift h d u2)))))).(\lambda (t: -T).(\lambda (k: K).(\lambda (d: nat).(\lambda (H2: (le d i0)).(eq_ind_r T -(THead k (lift h d u1) (lift h (s k d) t)) (\lambda (t0: T).(subst0 (plus i0 -h) v t0 (lift h d (THead k u2 t)))) (eq_ind_r T (THead k (lift h d u2) (lift -h (s k d) t)) (\lambda (t0: T).(subst0 (plus i0 h) v (THead k (lift h d u1) -(lift h (s k d) t)) t0)) (subst0_fst v (lift h d u2) (lift h d u1) (plus i0 -h) (H1 d H2) (lift h (s k d) t) k) (lift h d (THead k u2 t)) (lift_head k u2 -t h d)) (lift h d (THead k u1 t)) (lift_head k u1 t h d)))))))))))) (\lambda -(k: K).(\lambda (v: T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (i0: -nat).(\lambda (_: (subst0 (s k i0) v t3 t0)).(\lambda (H1: ((\forall (d: -nat).((le d (s k i0)) \to (subst0 (plus (s k i0) h) v (lift h d t3) (lift h d -t0)))))).(\lambda (u0: T).(\lambda (d: nat).(\lambda (H2: (le d i0)).(let H3 -\def (eq_ind_r nat (plus (s k i0) h) (\lambda (n: nat).(\forall (d0: -nat).((le d0 (s k i0)) \to (subst0 n v (lift h d0 t3) (lift h d0 t0))))) H1 -(s k (plus i0 h)) (s_plus k i0 h)) in (eq_ind_r T (THead k (lift h d u0) -(lift h (s k d) t3)) (\lambda (t: T).(subst0 (plus i0 h) v t (lift h d (THead -k u0 t0)))) (eq_ind_r T (THead k (lift h d u0) (lift h (s k d) t0)) (\lambda -(t: T).(subst0 (plus i0 h) v (THead k (lift h d u0) (lift h (s k d) t3)) t)) -(subst0_snd k v (lift h (s k d) t0) (lift h (s k d) t3) (plus i0 h) (H3 (s k -d) (s_le k d i0 H2)) (lift h d u0)) (lift h d (THead k u0 t0)) (lift_head k -u0 t0 h d)) (lift h d (THead k u0 t3)) (lift_head k u0 t3 h d))))))))))))) -(\lambda (v: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i0: nat).(\lambda -(_: (subst0 i0 v u1 u2)).(\lambda (H1: ((\forall (d: nat).((le d i0) \to -(subst0 (plus i0 h) v (lift h d u1) (lift h d u2)))))).(\lambda (k: -K).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (subst0 (s k i0) v t0 -t3)).(\lambda (H3: ((\forall (d: nat).((le d (s k i0)) \to (subst0 (plus (s k -i0) h) v (lift h d t0) (lift h d t3)))))).(\lambda (d: nat).(\lambda (H4: (le -d i0)).(let H5 \def (eq_ind_r nat (plus (s k i0) h) (\lambda (n: -nat).(\forall (d0: nat).((le d0 (s k i0)) \to (subst0 n v (lift h d0 t0) -(lift h d0 t3))))) H3 (s k (plus i0 h)) (s_plus k i0 h)) in (eq_ind_r T -(THead k (lift h d u1) (lift h (s k d) t0)) (\lambda (t: T).(subst0 (plus i0 -h) v t (lift h d (THead k u2 t3)))) (eq_ind_r T (THead k (lift h d u2) (lift -h (s k d) t3)) (\lambda (t: T).(subst0 (plus i0 h) v (THead k (lift h d u1) -(lift h (s k d) t0)) t)) (subst0_both v (lift h d u1) (lift h d u2) (plus i0 -h) (H1 d H4) k (lift h (s k d) t0) (lift h (s k d) t3) (H5 (s k d) (s_le k d -i0 H4))) (lift h d (THead k u2 t3)) (lift_head k u2 t3 h d)) (lift h d (THead -k u1 t0)) (lift_head k u1 t0 h d)))))))))))))))) i u t1 t2 H)))))). -(* COMMENTS -Initial nodes: 1449 -END *) - -theorem subst0_lift_ge_S: - \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).((subst0 -i u t1 t2) \to (\forall (d: nat).((le d i) \to (subst0 (S i) u (lift (S O) d -t1) (lift (S O) d t2)))))))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H: (subst0 i u t1 t2)).(\lambda (d: nat).(\lambda (H0: (le d i)).(eq_ind nat -(plus i (S O)) (\lambda (n: nat).(subst0 n u (lift (S O) d t1) (lift (S O) d -t2))) (subst0_lift_ge t1 t2 u i (S O) H d H0) (S i) (eq_ind_r nat (plus (S O) -i) (\lambda (n: nat).(eq nat n (S i))) (refl_equal nat (S i)) (plus i (S O)) -(plus_sym i (S O)))))))))). -(* COMMENTS -Initial nodes: 137 -END *) - -theorem subst0_lift_ge_s: - \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).((subst0 -i u t1 t2) \to (\forall (d: nat).((le d i) \to (\forall (b: B).(subst0 (s -(Bind b) i) u (lift (S O) d t1) (lift (S O) d t2))))))))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H: (subst0 i u t1 t2)).(\lambda (d: nat).(\lambda (H0: (le d i)).(\lambda -(_: B).(subst0_lift_ge_S t1 t2 u i H d H0)))))))). -(* COMMENTS -Initial nodes: 43 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/subst0/subst0.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/subst0/subst0.ma deleted file mode 100644 index 66c167d2b..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/subst0/subst0.ma +++ /dev/null @@ -1,1407 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/subst0/props.ma". - -theorem subst0_subst0: - \forall (t1: T).(\forall (t2: T).(\forall (u2: T).(\forall (j: nat).((subst0 -j u2 t1 t2) \to (\forall (u1: T).(\forall (u: T).(\forall (i: nat).((subst0 i -u u1 u2) \to (ex2 T (\lambda (t: T).(subst0 j u1 t1 t)) (\lambda (t: -T).(subst0 (S (plus i j)) u t t2))))))))))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (u2: T).(\lambda (j: nat).(\lambda -(H: (subst0 j u2 t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t: -T).(\lambda (t0: T).(\lambda (t3: T).(\forall (u1: T).(\forall (u: -T).(\forall (i: nat).((subst0 i u u1 t) \to (ex2 T (\lambda (t4: T).(subst0 n -u1 t0 t4)) (\lambda (t4: T).(subst0 (S (plus i n)) u t4 t3))))))))))) -(\lambda (v: T).(\lambda (i: nat).(\lambda (u1: T).(\lambda (u: T).(\lambda -(i0: nat).(\lambda (H0: (subst0 i0 u u1 v)).(eq_ind nat (plus i0 (S i)) -(\lambda (n: nat).(ex2 T (\lambda (t: T).(subst0 i u1 (TLRef i) t)) (\lambda -(t: T).(subst0 n u t (lift (S i) O v))))) (ex_intro2 T (\lambda (t: -T).(subst0 i u1 (TLRef i) t)) (\lambda (t: T).(subst0 (plus i0 (S i)) u t -(lift (S i) O v))) (lift (S i) O u1) (subst0_lref u1 i) (subst0_lift_ge u1 v -u i0 (S i) H0 O (le_O_n i0))) (S (plus i0 i)) (sym_eq nat (S (plus i0 i)) -(plus i0 (S i)) (plus_n_Sm i0 i))))))))) (\lambda (v: T).(\lambda (u0: -T).(\lambda (u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1 -u0)).(\lambda (H1: ((\forall (u3: T).(\forall (u: T).(\forall (i0: -nat).((subst0 i0 u u3 v) \to (ex2 T (\lambda (t: T).(subst0 i u3 u1 t)) -(\lambda (t: T).(subst0 (S (plus i0 i)) u t u0))))))))).(\lambda (t: -T).(\lambda (k: K).(\lambda (u3: T).(\lambda (u: T).(\lambda (i0: -nat).(\lambda (H2: (subst0 i0 u u3 v)).(ex2_ind T (\lambda (t0: T).(subst0 i -u3 u1 t0)) (\lambda (t0: T).(subst0 (S (plus i0 i)) u t0 u0)) (ex2 T (\lambda -(t0: T).(subst0 i u3 (THead k u1 t) t0)) (\lambda (t0: T).(subst0 (S (plus i0 -i)) u t0 (THead k u0 t)))) (\lambda (x: T).(\lambda (H3: (subst0 i u3 u1 -x)).(\lambda (H4: (subst0 (S (plus i0 i)) u x u0)).(ex_intro2 T (\lambda (t0: -T).(subst0 i u3 (THead k u1 t) t0)) (\lambda (t0: T).(subst0 (S (plus i0 i)) -u t0 (THead k u0 t))) (THead k x t) (subst0_fst u3 x u1 i H3 t k) (subst0_fst -u u0 x (S (plus i0 i)) H4 t k))))) (H1 u3 u i0 H2)))))))))))))) (\lambda (k: -K).(\lambda (v: T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (i: -nat).(\lambda (_: (subst0 (s k i) v t3 t0)).(\lambda (H1: ((\forall (u1: -T).(\forall (u: T).(\forall (i0: nat).((subst0 i0 u u1 v) \to (ex2 T (\lambda -(t: T).(subst0 (s k i) u1 t3 t)) (\lambda (t: T).(subst0 (S (plus i0 (s k -i))) u t t0))))))))).(\lambda (u: T).(\lambda (u1: T).(\lambda (u0: -T).(\lambda (i0: nat).(\lambda (H2: (subst0 i0 u0 u1 v)).(ex2_ind T (\lambda -(t: T).(subst0 (s k i) u1 t3 t)) (\lambda (t: T).(subst0 (S (plus i0 (s k -i))) u0 t t0)) (ex2 T (\lambda (t: T).(subst0 i u1 (THead k u t3) t)) -(\lambda (t: T).(subst0 (S (plus i0 i)) u0 t (THead k u t0)))) (\lambda (x: -T).(\lambda (H3: (subst0 (s k i) u1 t3 x)).(\lambda (H4: (subst0 (S (plus i0 -(s k i))) u0 x t0)).(let H5 \def (eq_ind_r nat (plus i0 (s k i)) (\lambda (n: -nat).(subst0 (S n) u0 x t0)) H4 (s k (plus i0 i)) (s_plus_sym k i0 i)) in -(let H6 \def (eq_ind_r nat (S (s k (plus i0 i))) (\lambda (n: nat).(subst0 n -u0 x t0)) H5 (s k (S (plus i0 i))) (s_S k (plus i0 i))) in (ex_intro2 T -(\lambda (t: T).(subst0 i u1 (THead k u t3) t)) (\lambda (t: T).(subst0 (S -(plus i0 i)) u0 t (THead k u t0))) (THead k u x) (subst0_snd k u1 x t3 i H3 -u) (subst0_snd k u0 t0 x (S (plus i0 i)) H6 u))))))) (H1 u1 u0 i0 -H2)))))))))))))) (\lambda (v: T).(\lambda (u1: T).(\lambda (u0: T).(\lambda -(i: nat).(\lambda (_: (subst0 i v u1 u0)).(\lambda (H1: ((\forall (u3: -T).(\forall (u: T).(\forall (i0: nat).((subst0 i0 u u3 v) \to (ex2 T (\lambda -(t: T).(subst0 i u3 u1 t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u t -u0))))))))).(\lambda (k: K).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: -(subst0 (s k i) v t0 t3)).(\lambda (H3: ((\forall (u3: T).(\forall (u: -T).(\forall (i0: nat).((subst0 i0 u u3 v) \to (ex2 T (\lambda (t: T).(subst0 -(s k i) u3 t0 t)) (\lambda (t: T).(subst0 (S (plus i0 (s k i))) u t -t3))))))))).(\lambda (u3: T).(\lambda (u: T).(\lambda (i0: nat).(\lambda (H4: -(subst0 i0 u u3 v)).(ex2_ind T (\lambda (t: T).(subst0 (s k i) u3 t0 t)) -(\lambda (t: T).(subst0 (S (plus i0 (s k i))) u t t3)) (ex2 T (\lambda (t: -T).(subst0 i u3 (THead k u1 t0) t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u -t (THead k u0 t3)))) (\lambda (x: T).(\lambda (H5: (subst0 (s k i) u3 t0 -x)).(\lambda (H6: (subst0 (S (plus i0 (s k i))) u x t3)).(ex2_ind T (\lambda -(t: T).(subst0 i u3 u1 t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u t u0)) -(ex2 T (\lambda (t: T).(subst0 i u3 (THead k u1 t0) t)) (\lambda (t: -T).(subst0 (S (plus i0 i)) u t (THead k u0 t3)))) (\lambda (x0: T).(\lambda -(H7: (subst0 i u3 u1 x0)).(\lambda (H8: (subst0 (S (plus i0 i)) u x0 -u0)).(let H9 \def (eq_ind_r nat (plus i0 (s k i)) (\lambda (n: nat).(subst0 -(S n) u x t3)) H6 (s k (plus i0 i)) (s_plus_sym k i0 i)) in (let H10 \def -(eq_ind_r nat (S (s k (plus i0 i))) (\lambda (n: nat).(subst0 n u x t3)) H9 -(s k (S (plus i0 i))) (s_S k (plus i0 i))) in (ex_intro2 T (\lambda (t: -T).(subst0 i u3 (THead k u1 t0) t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u -t (THead k u0 t3))) (THead k x0 x) (subst0_both u3 u1 x0 i H7 k t0 x H5) -(subst0_both u x0 u0 (S (plus i0 i)) H8 k x t3 H10))))))) (H1 u3 u i0 H4))))) -(H3 u3 u i0 H4))))))))))))))))) j u2 t1 t2 H))))). -(* COMMENTS -Initial nodes: 1613 -END *) - -theorem subst0_subst0_back: - \forall (t1: T).(\forall (t2: T).(\forall (u2: T).(\forall (j: nat).((subst0 -j u2 t1 t2) \to (\forall (u1: T).(\forall (u: T).(\forall (i: nat).((subst0 i -u u2 u1) \to (ex2 T (\lambda (t: T).(subst0 j u1 t1 t)) (\lambda (t: -T).(subst0 (S (plus i j)) u t2 t))))))))))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (u2: T).(\lambda (j: nat).(\lambda -(H: (subst0 j u2 t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t: -T).(\lambda (t0: T).(\lambda (t3: T).(\forall (u1: T).(\forall (u: -T).(\forall (i: nat).((subst0 i u t u1) \to (ex2 T (\lambda (t4: T).(subst0 n -u1 t0 t4)) (\lambda (t4: T).(subst0 (S (plus i n)) u t3 t4))))))))))) -(\lambda (v: T).(\lambda (i: nat).(\lambda (u1: T).(\lambda (u: T).(\lambda -(i0: nat).(\lambda (H0: (subst0 i0 u v u1)).(eq_ind nat (plus i0 (S i)) -(\lambda (n: nat).(ex2 T (\lambda (t: T).(subst0 i u1 (TLRef i) t)) (\lambda -(t: T).(subst0 n u (lift (S i) O v) t)))) (ex_intro2 T (\lambda (t: -T).(subst0 i u1 (TLRef i) t)) (\lambda (t: T).(subst0 (plus i0 (S i)) u (lift -(S i) O v) t)) (lift (S i) O u1) (subst0_lref u1 i) (subst0_lift_ge v u1 u i0 -(S i) H0 O (le_O_n i0))) (S (plus i0 i)) (sym_eq nat (S (plus i0 i)) (plus i0 -(S i)) (plus_n_Sm i0 i))))))))) (\lambda (v: T).(\lambda (u0: T).(\lambda -(u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1 u0)).(\lambda (H1: -((\forall (u3: T).(\forall (u: T).(\forall (i0: nat).((subst0 i0 u v u3) \to -(ex2 T (\lambda (t: T).(subst0 i u3 u1 t)) (\lambda (t: T).(subst0 (S (plus -i0 i)) u u0 t))))))))).(\lambda (t: T).(\lambda (k: K).(\lambda (u3: -T).(\lambda (u: T).(\lambda (i0: nat).(\lambda (H2: (subst0 i0 u v -u3)).(ex2_ind T (\lambda (t0: T).(subst0 i u3 u1 t0)) (\lambda (t0: -T).(subst0 (S (plus i0 i)) u u0 t0)) (ex2 T (\lambda (t0: T).(subst0 i u3 -(THead k u1 t) t0)) (\lambda (t0: T).(subst0 (S (plus i0 i)) u (THead k u0 t) -t0))) (\lambda (x: T).(\lambda (H3: (subst0 i u3 u1 x)).(\lambda (H4: (subst0 -(S (plus i0 i)) u u0 x)).(ex_intro2 T (\lambda (t0: T).(subst0 i u3 (THead k -u1 t) t0)) (\lambda (t0: T).(subst0 (S (plus i0 i)) u (THead k u0 t) t0)) -(THead k x t) (subst0_fst u3 x u1 i H3 t k) (subst0_fst u x u0 (S (plus i0 -i)) H4 t k))))) (H1 u3 u i0 H2)))))))))))))) (\lambda (k: K).(\lambda (v: -T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (i: nat).(\lambda (_: (subst0 -(s k i) v t3 t0)).(\lambda (H1: ((\forall (u1: T).(\forall (u: T).(\forall -(i0: nat).((subst0 i0 u v u1) \to (ex2 T (\lambda (t: T).(subst0 (s k i) u1 -t3 t)) (\lambda (t: T).(subst0 (S (plus i0 (s k i))) u t0 t))))))))).(\lambda -(u: T).(\lambda (u1: T).(\lambda (u0: T).(\lambda (i0: nat).(\lambda (H2: -(subst0 i0 u0 v u1)).(ex2_ind T (\lambda (t: T).(subst0 (s k i) u1 t3 t)) -(\lambda (t: T).(subst0 (S (plus i0 (s k i))) u0 t0 t)) (ex2 T (\lambda (t: -T).(subst0 i u1 (THead k u t3) t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u0 -(THead k u t0) t))) (\lambda (x: T).(\lambda (H3: (subst0 (s k i) u1 t3 -x)).(\lambda (H4: (subst0 (S (plus i0 (s k i))) u0 t0 x)).(let H5 \def -(eq_ind_r nat (plus i0 (s k i)) (\lambda (n: nat).(subst0 (S n) u0 t0 x)) H4 -(s k (plus i0 i)) (s_plus_sym k i0 i)) in (let H6 \def (eq_ind_r nat (S (s k -(plus i0 i))) (\lambda (n: nat).(subst0 n u0 t0 x)) H5 (s k (S (plus i0 i))) -(s_S k (plus i0 i))) in (ex_intro2 T (\lambda (t: T).(subst0 i u1 (THead k u -t3) t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u0 (THead k u t0) t)) (THead -k u x) (subst0_snd k u1 x t3 i H3 u) (subst0_snd k u0 x t0 (S (plus i0 i)) H6 -u))))))) (H1 u1 u0 i0 H2)))))))))))))) (\lambda (v: T).(\lambda (u1: -T).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1 -u0)).(\lambda (H1: ((\forall (u3: T).(\forall (u: T).(\forall (i0: -nat).((subst0 i0 u v u3) \to (ex2 T (\lambda (t: T).(subst0 i u3 u1 t)) -(\lambda (t: T).(subst0 (S (plus i0 i)) u u0 t))))))))).(\lambda (k: -K).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (subst0 (s k i) v t0 -t3)).(\lambda (H3: ((\forall (u3: T).(\forall (u: T).(\forall (i0: -nat).((subst0 i0 u v u3) \to (ex2 T (\lambda (t: T).(subst0 (s k i) u3 t0 t)) -(\lambda (t: T).(subst0 (S (plus i0 (s k i))) u t3 t))))))))).(\lambda (u3: -T).(\lambda (u: T).(\lambda (i0: nat).(\lambda (H4: (subst0 i0 u v -u3)).(ex2_ind T (\lambda (t: T).(subst0 (s k i) u3 t0 t)) (\lambda (t: -T).(subst0 (S (plus i0 (s k i))) u t3 t)) (ex2 T (\lambda (t: T).(subst0 i u3 -(THead k u1 t0) t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u (THead k u0 t3) -t))) (\lambda (x: T).(\lambda (H5: (subst0 (s k i) u3 t0 x)).(\lambda (H6: -(subst0 (S (plus i0 (s k i))) u t3 x)).(ex2_ind T (\lambda (t: T).(subst0 i -u3 u1 t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u u0 t)) (ex2 T (\lambda -(t: T).(subst0 i u3 (THead k u1 t0) t)) (\lambda (t: T).(subst0 (S (plus i0 -i)) u (THead k u0 t3) t))) (\lambda (x0: T).(\lambda (H7: (subst0 i u3 u1 -x0)).(\lambda (H8: (subst0 (S (plus i0 i)) u u0 x0)).(let H9 \def (eq_ind_r -nat (plus i0 (s k i)) (\lambda (n: nat).(subst0 (S n) u t3 x)) H6 (s k (plus -i0 i)) (s_plus_sym k i0 i)) in (let H10 \def (eq_ind_r nat (S (s k (plus i0 -i))) (\lambda (n: nat).(subst0 n u t3 x)) H9 (s k (S (plus i0 i))) (s_S k -(plus i0 i))) in (ex_intro2 T (\lambda (t: T).(subst0 i u3 (THead k u1 t0) -t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u (THead k u0 t3) t)) (THead k x0 -x) (subst0_both u3 u1 x0 i H7 k t0 x H5) (subst0_both u u0 x0 (S (plus i0 i)) -H8 k t3 x H10))))))) (H1 u3 u i0 H4))))) (H3 u3 u i0 H4))))))))))))))))) j u2 -t1 t2 H))))). -(* COMMENTS -Initial nodes: 1613 -END *) - -theorem subst0_trans: - \forall (t2: T).(\forall (t1: T).(\forall (v: T).(\forall (i: nat).((subst0 -i v t1 t2) \to (\forall (t3: T).((subst0 i v t2 t3) \to (subst0 i v t1 -t3))))))) -\def - \lambda (t2: T).(\lambda (t1: T).(\lambda (v: T).(\lambda (i: nat).(\lambda -(H: (subst0 i v t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t: -T).(\lambda (t0: T).(\lambda (t3: T).(\forall (t4: T).((subst0 n t t3 t4) \to -(subst0 n t t0 t4))))))) (\lambda (v0: T).(\lambda (i0: nat).(\lambda (t3: -T).(\lambda (H0: (subst0 i0 v0 (lift (S i0) O v0) t3)).(subst0_gen_lift_false -v0 v0 t3 (S i0) O i0 (le_O_n i0) (le_n (plus O (S i0))) H0 (subst0 i0 v0 -(TLRef i0) t3)))))) (\lambda (v0: T).(\lambda (u2: T).(\lambda (u1: -T).(\lambda (i0: nat).(\lambda (H0: (subst0 i0 v0 u1 u2)).(\lambda (H1: -((\forall (t3: T).((subst0 i0 v0 u2 t3) \to (subst0 i0 v0 u1 t3))))).(\lambda -(t: T).(\lambda (k: K).(\lambda (t3: T).(\lambda (H2: (subst0 i0 v0 (THead k -u2 t) t3)).(or3_ind (ex2 T (\lambda (u3: T).(eq T t3 (THead k u3 t))) -(\lambda (u3: T).(subst0 i0 v0 u2 u3))) (ex2 T (\lambda (t4: T).(eq T t3 -(THead k u2 t4))) (\lambda (t4: T).(subst0 (s k i0) v0 t t4))) (ex3_2 T T -(\lambda (u3: T).(\lambda (t4: T).(eq T t3 (THead k u3 t4)))) (\lambda (u3: -T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) (\lambda (_: T).(\lambda (t4: -T).(subst0 (s k i0) v0 t t4)))) (subst0 i0 v0 (THead k u1 t) t3) (\lambda -(H3: (ex2 T (\lambda (u3: T).(eq T t3 (THead k u3 t))) (\lambda (u3: -T).(subst0 i0 v0 u2 u3)))).(ex2_ind T (\lambda (u3: T).(eq T t3 (THead k u3 -t))) (\lambda (u3: T).(subst0 i0 v0 u2 u3)) (subst0 i0 v0 (THead k u1 t) t3) -(\lambda (x: T).(\lambda (H4: (eq T t3 (THead k x t))).(\lambda (H5: (subst0 -i0 v0 u2 x)).(eq_ind_r T (THead k x t) (\lambda (t0: T).(subst0 i0 v0 (THead -k u1 t) t0)) (subst0_fst v0 x u1 i0 (H1 x H5) t k) t3 H4)))) H3)) (\lambda -(H3: (ex2 T (\lambda (t4: T).(eq T t3 (THead k u2 t4))) (\lambda (t4: -T).(subst0 (s k i0) v0 t t4)))).(ex2_ind T (\lambda (t4: T).(eq T t3 (THead k -u2 t4))) (\lambda (t4: T).(subst0 (s k i0) v0 t t4)) (subst0 i0 v0 (THead k -u1 t) t3) (\lambda (x: T).(\lambda (H4: (eq T t3 (THead k u2 x))).(\lambda -(H5: (subst0 (s k i0) v0 t x)).(eq_ind_r T (THead k u2 x) (\lambda (t0: -T).(subst0 i0 v0 (THead k u1 t) t0)) (subst0_both v0 u1 u2 i0 H0 k t x H5) t3 -H4)))) H3)) (\lambda (H3: (ex3_2 T T (\lambda (u3: T).(\lambda (t4: T).(eq T -t3 (THead k u3 t4)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) -(\lambda (_: T).(\lambda (t4: T).(subst0 (s k i0) v0 t t4))))).(ex3_2_ind T T -(\lambda (u3: T).(\lambda (t4: T).(eq T t3 (THead k u3 t4)))) (\lambda (u3: -T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) (\lambda (_: T).(\lambda (t4: -T).(subst0 (s k i0) v0 t t4))) (subst0 i0 v0 (THead k u1 t) t3) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H4: (eq T t3 (THead k x0 x1))).(\lambda (H5: -(subst0 i0 v0 u2 x0)).(\lambda (H6: (subst0 (s k i0) v0 t x1)).(eq_ind_r T -(THead k x0 x1) (\lambda (t0: T).(subst0 i0 v0 (THead k u1 t) t0)) -(subst0_both v0 u1 x0 i0 (H1 x0 H5) k t x1 H6) t3 H4)))))) H3)) -(subst0_gen_head k v0 u2 t t3 i0 H2)))))))))))) (\lambda (k: K).(\lambda (v0: -T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (i0: nat).(\lambda (H0: (subst0 -(s k i0) v0 t3 t0)).(\lambda (H1: ((\forall (t4: T).((subst0 (s k i0) v0 t0 -t4) \to (subst0 (s k i0) v0 t3 t4))))).(\lambda (u: T).(\lambda (t4: -T).(\lambda (H2: (subst0 i0 v0 (THead k u t0) t4)).(or3_ind (ex2 T (\lambda -(u2: T).(eq T t4 (THead k u2 t0))) (\lambda (u2: T).(subst0 i0 v0 u u2))) -(ex2 T (\lambda (t5: T).(eq T t4 (THead k u t5))) (\lambda (t5: T).(subst0 (s -k i0) v0 t0 t5))) (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 -(THead k u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u u2))) -(\lambda (_: T).(\lambda (t5: T).(subst0 (s k i0) v0 t0 t5)))) (subst0 i0 v0 -(THead k u t3) t4) (\lambda (H3: (ex2 T (\lambda (u2: T).(eq T t4 (THead k u2 -t0))) (\lambda (u2: T).(subst0 i0 v0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq -T t4 (THead k u2 t0))) (\lambda (u2: T).(subst0 i0 v0 u u2)) (subst0 i0 v0 -(THead k u t3) t4) (\lambda (x: T).(\lambda (H4: (eq T t4 (THead k x -t0))).(\lambda (H5: (subst0 i0 v0 u x)).(eq_ind_r T (THead k x t0) (\lambda -(t: T).(subst0 i0 v0 (THead k u t3) t)) (subst0_both v0 u x i0 H5 k t3 t0 H0) -t4 H4)))) H3)) (\lambda (H3: (ex2 T (\lambda (t5: T).(eq T t4 (THead k u -t5))) (\lambda (t5: T).(subst0 (s k i0) v0 t0 t5)))).(ex2_ind T (\lambda (t5: -T).(eq T t4 (THead k u t5))) (\lambda (t5: T).(subst0 (s k i0) v0 t0 t5)) -(subst0 i0 v0 (THead k u t3) t4) (\lambda (x: T).(\lambda (H4: (eq T t4 -(THead k u x))).(\lambda (H5: (subst0 (s k i0) v0 t0 x)).(eq_ind_r T (THead k -u x) (\lambda (t: T).(subst0 i0 v0 (THead k u t3) t)) (subst0_snd k v0 x t3 -i0 (H1 x H5) u) t4 H4)))) H3)) (\lambda (H3: (ex3_2 T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead k u2 t5)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 i0 v0 u u2))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s k i0) v0 -t0 t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead k -u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u u2))) (\lambda (_: -T).(\lambda (t5: T).(subst0 (s k i0) v0 t0 t5))) (subst0 i0 v0 (THead k u t3) -t4) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t4 (THead k x0 -x1))).(\lambda (H5: (subst0 i0 v0 u x0)).(\lambda (H6: (subst0 (s k i0) v0 t0 -x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t: T).(subst0 i0 v0 (THead k u t3) -t)) (subst0_both v0 u x0 i0 H5 k t3 x1 (H1 x1 H6)) t4 H4)))))) H3)) -(subst0_gen_head k v0 u t0 t4 i0 H2)))))))))))) (\lambda (v0: T).(\lambda -(u1: T).(\lambda (u2: T).(\lambda (i0: nat).(\lambda (H0: (subst0 i0 v0 u1 -u2)).(\lambda (H1: ((\forall (t3: T).((subst0 i0 v0 u2 t3) \to (subst0 i0 v0 -u1 t3))))).(\lambda (k: K).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H2: -(subst0 (s k i0) v0 t0 t3)).(\lambda (H3: ((\forall (t4: T).((subst0 (s k i0) -v0 t3 t4) \to (subst0 (s k i0) v0 t0 t4))))).(\lambda (t4: T).(\lambda (H4: -(subst0 i0 v0 (THead k u2 t3) t4)).(or3_ind (ex2 T (\lambda (u3: T).(eq T t4 -(THead k u3 t3))) (\lambda (u3: T).(subst0 i0 v0 u2 u3))) (ex2 T (\lambda -(t5: T).(eq T t4 (THead k u2 t5))) (\lambda (t5: T).(subst0 (s k i0) v0 t3 -t5))) (ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T t4 (THead k u3 -t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) (\lambda (_: -T).(\lambda (t5: T).(subst0 (s k i0) v0 t3 t5)))) (subst0 i0 v0 (THead k u1 -t0) t4) (\lambda (H5: (ex2 T (\lambda (u3: T).(eq T t4 (THead k u3 t3))) -(\lambda (u3: T).(subst0 i0 v0 u2 u3)))).(ex2_ind T (\lambda (u3: T).(eq T t4 -(THead k u3 t3))) (\lambda (u3: T).(subst0 i0 v0 u2 u3)) (subst0 i0 v0 (THead -k u1 t0) t4) (\lambda (x: T).(\lambda (H6: (eq T t4 (THead k x t3))).(\lambda -(H7: (subst0 i0 v0 u2 x)).(eq_ind_r T (THead k x t3) (\lambda (t: T).(subst0 -i0 v0 (THead k u1 t0) t)) (subst0_both v0 u1 x i0 (H1 x H7) k t0 t3 H2) t4 -H6)))) H5)) (\lambda (H5: (ex2 T (\lambda (t5: T).(eq T t4 (THead k u2 t5))) -(\lambda (t5: T).(subst0 (s k i0) v0 t3 t5)))).(ex2_ind T (\lambda (t5: -T).(eq T t4 (THead k u2 t5))) (\lambda (t5: T).(subst0 (s k i0) v0 t3 t5)) -(subst0 i0 v0 (THead k u1 t0) t4) (\lambda (x: T).(\lambda (H6: (eq T t4 -(THead k u2 x))).(\lambda (H7: (subst0 (s k i0) v0 t3 x)).(eq_ind_r T (THead -k u2 x) (\lambda (t: T).(subst0 i0 v0 (THead k u1 t0) t)) (subst0_both v0 u1 -u2 i0 H0 k t0 x (H3 x H7)) t4 H6)))) H5)) (\lambda (H5: (ex3_2 T T (\lambda -(u3: T).(\lambda (t5: T).(eq T t4 (THead k u3 t5)))) (\lambda (u3: -T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) (\lambda (_: T).(\lambda (t5: -T).(subst0 (s k i0) v0 t3 t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda -(t5: T).(eq T t4 (THead k u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 -i0 v0 u2 u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s k i0) v0 t3 t5))) -(subst0 i0 v0 (THead k u1 t0) t4) (\lambda (x0: T).(\lambda (x1: T).(\lambda -(H6: (eq T t4 (THead k x0 x1))).(\lambda (H7: (subst0 i0 v0 u2 x0)).(\lambda -(H8: (subst0 (s k i0) v0 t3 x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t: -T).(subst0 i0 v0 (THead k u1 t0) t)) (subst0_both v0 u1 x0 i0 (H1 x0 H7) k t0 -x1 (H3 x1 H8)) t4 H6)))))) H5)) (subst0_gen_head k v0 u2 t3 t4 i0 -H4))))))))))))))) i v t1 t2 H))))). -(* COMMENTS -Initial nodes: 2555 -END *) - -theorem subst0_confluence_neq: - \forall (t0: T).(\forall (t1: T).(\forall (u1: T).(\forall (i1: -nat).((subst0 i1 u1 t0 t1) \to (\forall (t2: T).(\forall (u2: T).(\forall -(i2: nat).((subst0 i2 u2 t0 t2) \to ((not (eq nat i1 i2)) \to (ex2 T (\lambda -(t: T).(subst0 i2 u2 t1 t)) (\lambda (t: T).(subst0 i1 u1 t2 t)))))))))))) -\def - \lambda (t0: T).(\lambda (t1: T).(\lambda (u1: T).(\lambda (i1: -nat).(\lambda (H: (subst0 i1 u1 t0 t1)).(subst0_ind (\lambda (n: -nat).(\lambda (t: T).(\lambda (t2: T).(\lambda (t3: T).(\forall (t4: -T).(\forall (u2: T).(\forall (i2: nat).((subst0 i2 u2 t2 t4) \to ((not (eq -nat n i2)) \to (ex2 T (\lambda (t5: T).(subst0 i2 u2 t3 t5)) (\lambda (t5: -T).(subst0 n t t4 t5)))))))))))) (\lambda (v: T).(\lambda (i: nat).(\lambda -(t2: T).(\lambda (u2: T).(\lambda (i2: nat).(\lambda (H0: (subst0 i2 u2 -(TLRef i) t2)).(\lambda (H1: (not (eq nat i i2))).(land_ind (eq nat i i2) (eq -T t2 (lift (S i) O u2)) (ex2 T (\lambda (t: T).(subst0 i2 u2 (lift (S i) O v) -t)) (\lambda (t: T).(subst0 i v t2 t))) (\lambda (H2: (eq nat i i2)).(\lambda -(H3: (eq T t2 (lift (S i) O u2))).(let H4 \def (eq_ind nat i (\lambda (n: -nat).(not (eq nat n i2))) H1 i2 H2) in (eq_ind_r T (lift (S i) O u2) (\lambda -(t: T).(ex2 T (\lambda (t3: T).(subst0 i2 u2 (lift (S i) O v) t3)) (\lambda -(t3: T).(subst0 i v t t3)))) (let H5 \def (match (H4 (refl_equal nat i2)) in -False return (\lambda (_: False).(ex2 T (\lambda (t: T).(subst0 i2 u2 (lift -(S i) O v) t)) (\lambda (t: T).(subst0 i v (lift (S i) O u2) t)))) with []) -in H5) t2 H3)))) (subst0_gen_lref u2 t2 i2 i H0))))))))) (\lambda (v: -T).(\lambda (u2: T).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H0: (subst0 -i v u0 u2)).(\lambda (H1: ((\forall (t2: T).(\forall (u3: T).(\forall (i2: -nat).((subst0 i2 u3 u0 t2) \to ((not (eq nat i i2)) \to (ex2 T (\lambda (t: -T).(subst0 i2 u3 u2 t)) (\lambda (t: T).(subst0 i v t2 t)))))))))).(\lambda -(t: T).(\lambda (k: K).(\lambda (t2: T).(\lambda (u3: T).(\lambda (i2: -nat).(\lambda (H2: (subst0 i2 u3 (THead k u0 t) t2)).(\lambda (H3: (not (eq -nat i i2))).(or3_ind (ex2 T (\lambda (u4: T).(eq T t2 (THead k u4 t))) -(\lambda (u4: T).(subst0 i2 u3 u0 u4))) (ex2 T (\lambda (t3: T).(eq T t2 -(THead k u0 t3))) (\lambda (t3: T).(subst0 (s k i2) u3 t t3))) (ex3_2 T T -(\lambda (u4: T).(\lambda (t3: T).(eq T t2 (THead k u4 t3)))) (\lambda (u4: -T).(\lambda (_: T).(subst0 i2 u3 u0 u4))) (\lambda (_: T).(\lambda (t3: -T).(subst0 (s k i2) u3 t t3)))) (ex2 T (\lambda (t3: T).(subst0 i2 u3 (THead -k u2 t) t3)) (\lambda (t3: T).(subst0 i v t2 t3))) (\lambda (H4: (ex2 T -(\lambda (u4: T).(eq T t2 (THead k u4 t))) (\lambda (u4: T).(subst0 i2 u3 u0 -u4)))).(ex2_ind T (\lambda (u4: T).(eq T t2 (THead k u4 t))) (\lambda (u4: -T).(subst0 i2 u3 u0 u4)) (ex2 T (\lambda (t3: T).(subst0 i2 u3 (THead k u2 t) -t3)) (\lambda (t3: T).(subst0 i v t2 t3))) (\lambda (x: T).(\lambda (H5: (eq -T t2 (THead k x t))).(\lambda (H6: (subst0 i2 u3 u0 x)).(eq_ind_r T (THead k -x t) (\lambda (t3: T).(ex2 T (\lambda (t4: T).(subst0 i2 u3 (THead k u2 t) -t4)) (\lambda (t4: T).(subst0 i v t3 t4)))) (ex2_ind T (\lambda (t3: -T).(subst0 i2 u3 u2 t3)) (\lambda (t3: T).(subst0 i v x t3)) (ex2 T (\lambda -(t3: T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i v (THead -k x t) t3))) (\lambda (x0: T).(\lambda (H7: (subst0 i2 u3 u2 x0)).(\lambda -(H8: (subst0 i v x x0)).(ex_intro2 T (\lambda (t3: T).(subst0 i2 u3 (THead k -u2 t) t3)) (\lambda (t3: T).(subst0 i v (THead k x t) t3)) (THead k x0 t) -(subst0_fst u3 x0 u2 i2 H7 t k) (subst0_fst v x0 x i H8 t k))))) (H1 x u3 i2 -H6 H3)) t2 H5)))) H4)) (\lambda (H4: (ex2 T (\lambda (t3: T).(eq T t2 (THead -k u0 t3))) (\lambda (t3: T).(subst0 (s k i2) u3 t t3)))).(ex2_ind T (\lambda -(t3: T).(eq T t2 (THead k u0 t3))) (\lambda (t3: T).(subst0 (s k i2) u3 t -t3)) (ex2 T (\lambda (t3: T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda (t3: -T).(subst0 i v t2 t3))) (\lambda (x: T).(\lambda (H5: (eq T t2 (THead k u0 -x))).(\lambda (H6: (subst0 (s k i2) u3 t x)).(eq_ind_r T (THead k u0 x) -(\lambda (t3: T).(ex2 T (\lambda (t4: T).(subst0 i2 u3 (THead k u2 t) t4)) -(\lambda (t4: T).(subst0 i v t3 t4)))) (ex_intro2 T (\lambda (t3: T).(subst0 -i2 u3 (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i v (THead k u0 x) t3)) -(THead k u2 x) (subst0_snd k u3 x t i2 H6 u2) (subst0_fst v u2 u0 i H0 x k)) -t2 H5)))) H4)) (\lambda (H4: (ex3_2 T T (\lambda (u4: T).(\lambda (t3: T).(eq -T t2 (THead k u4 t3)))) (\lambda (u4: T).(\lambda (_: T).(subst0 i2 u3 u0 -u4))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i2) u3 t -t3))))).(ex3_2_ind T T (\lambda (u4: T).(\lambda (t3: T).(eq T t2 (THead k u4 -t3)))) (\lambda (u4: T).(\lambda (_: T).(subst0 i2 u3 u0 u4))) (\lambda (_: -T).(\lambda (t3: T).(subst0 (s k i2) u3 t t3))) (ex2 T (\lambda (t3: -T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i v t2 t3))) -(\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T t2 (THead k x0 -x1))).(\lambda (H6: (subst0 i2 u3 u0 x0)).(\lambda (H7: (subst0 (s k i2) u3 t -x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t3: T).(ex2 T (\lambda (t4: -T).(subst0 i2 u3 (THead k u2 t) t4)) (\lambda (t4: T).(subst0 i v t3 t4)))) -(ex2_ind T (\lambda (t3: T).(subst0 i2 u3 u2 t3)) (\lambda (t3: T).(subst0 i -v x0 t3)) (ex2 T (\lambda (t3: T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda -(t3: T).(subst0 i v (THead k x0 x1) t3))) (\lambda (x: T).(\lambda (H8: -(subst0 i2 u3 u2 x)).(\lambda (H9: (subst0 i v x0 x)).(ex_intro2 T (\lambda -(t3: T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i v (THead -k x0 x1) t3)) (THead k x x1) (subst0_both u3 u2 x i2 H8 k t x1 H7) -(subst0_fst v x x0 i H9 x1 k))))) (H1 x0 u3 i2 H6 H3)) t2 H5)))))) H4)) -(subst0_gen_head k u3 u0 t t2 i2 H2))))))))))))))) (\lambda (k: K).(\lambda -(v: T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (i: nat).(\lambda (H0: -(subst0 (s k i) v t3 t2)).(\lambda (H1: ((\forall (t4: T).(\forall (u2: -T).(\forall (i2: nat).((subst0 i2 u2 t3 t4) \to ((not (eq nat (s k i) i2)) -\to (ex2 T (\lambda (t: T).(subst0 i2 u2 t2 t)) (\lambda (t: T).(subst0 (s k -i) v t4 t)))))))))).(\lambda (u: T).(\lambda (t4: T).(\lambda (u2: -T).(\lambda (i2: nat).(\lambda (H2: (subst0 i2 u2 (THead k u t3) -t4)).(\lambda (H3: (not (eq nat i i2))).(or3_ind (ex2 T (\lambda (u3: T).(eq -T t4 (THead k u3 t3))) (\lambda (u3: T).(subst0 i2 u2 u u3))) (ex2 T (\lambda -(t5: T).(eq T t4 (THead k u t5))) (\lambda (t5: T).(subst0 (s k i2) u2 t3 -t5))) (ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T t4 (THead k u3 -t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i2 u2 u u3))) (\lambda (_: -T).(\lambda (t5: T).(subst0 (s k i2) u2 t3 t5)))) (ex2 T (\lambda (t: -T).(subst0 i2 u2 (THead k u t2) t)) (\lambda (t: T).(subst0 i v t4 t))) -(\lambda (H4: (ex2 T (\lambda (u3: T).(eq T t4 (THead k u3 t3))) (\lambda -(u3: T).(subst0 i2 u2 u u3)))).(ex2_ind T (\lambda (u3: T).(eq T t4 (THead k -u3 t3))) (\lambda (u3: T).(subst0 i2 u2 u u3)) (ex2 T (\lambda (t: T).(subst0 -i2 u2 (THead k u t2) t)) (\lambda (t: T).(subst0 i v t4 t))) (\lambda (x: -T).(\lambda (H5: (eq T t4 (THead k x t3))).(\lambda (H6: (subst0 i2 u2 u -x)).(eq_ind_r T (THead k x t3) (\lambda (t: T).(ex2 T (\lambda (t5: -T).(subst0 i2 u2 (THead k u t2) t5)) (\lambda (t5: T).(subst0 i v t t5)))) -(ex_intro2 T (\lambda (t: T).(subst0 i2 u2 (THead k u t2) t)) (\lambda (t: -T).(subst0 i v (THead k x t3) t)) (THead k x t2) (subst0_fst u2 x u i2 H6 t2 -k) (subst0_snd k v t2 t3 i H0 x)) t4 H5)))) H4)) (\lambda (H4: (ex2 T -(\lambda (t5: T).(eq T t4 (THead k u t5))) (\lambda (t5: T).(subst0 (s k i2) -u2 t3 t5)))).(ex2_ind T (\lambda (t5: T).(eq T t4 (THead k u t5))) (\lambda -(t5: T).(subst0 (s k i2) u2 t3 t5)) (ex2 T (\lambda (t: T).(subst0 i2 u2 -(THead k u t2) t)) (\lambda (t: T).(subst0 i v t4 t))) (\lambda (x: -T).(\lambda (H5: (eq T t4 (THead k u x))).(\lambda (H6: (subst0 (s k i2) u2 -t3 x)).(eq_ind_r T (THead k u x) (\lambda (t: T).(ex2 T (\lambda (t5: -T).(subst0 i2 u2 (THead k u t2) t5)) (\lambda (t5: T).(subst0 i v t t5)))) -(ex2_ind T (\lambda (t: T).(subst0 (s k i2) u2 t2 t)) (\lambda (t: T).(subst0 -(s k i) v x t)) (ex2 T (\lambda (t: T).(subst0 i2 u2 (THead k u t2) t)) -(\lambda (t: T).(subst0 i v (THead k u x) t))) (\lambda (x0: T).(\lambda (H7: -(subst0 (s k i2) u2 t2 x0)).(\lambda (H8: (subst0 (s k i) v x x0)).(ex_intro2 -T (\lambda (t: T).(subst0 i2 u2 (THead k u t2) t)) (\lambda (t: T).(subst0 i -v (THead k u x) t)) (THead k u x0) (subst0_snd k u2 x0 t2 i2 H7 u) -(subst0_snd k v x0 x i H8 u))))) (H1 x u2 (s k i2) H6 (ex2_ind T (\lambda (t: -T).(subst0 (s k i2) u2 t2 t)) (\lambda (t: T).(subst0 (s k i) v x t)) ((eq -nat (s k i) (s k i2)) \to False) (\lambda (x0: T).(\lambda (_: (subst0 (s k -i2) u2 t2 x0)).(\lambda (_: (subst0 (s k i) v x x0)).(\lambda (H9: (eq nat (s -k i) (s k i2))).(H3 (s_inj k i i2 H9)))))) (H1 x u2 (s k i2) H6 (\lambda (H7: -(eq nat (s k i) (s k i2))).(H3 (s_inj k i i2 H7))))))) t4 H5)))) H4)) -(\lambda (H4: (ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T t4 (THead k -u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i2 u2 u u3))) (\lambda (_: -T).(\lambda (t5: T).(subst0 (s k i2) u2 t3 t5))))).(ex3_2_ind T T (\lambda -(u3: T).(\lambda (t5: T).(eq T t4 (THead k u3 t5)))) (\lambda (u3: -T).(\lambda (_: T).(subst0 i2 u2 u u3))) (\lambda (_: T).(\lambda (t5: -T).(subst0 (s k i2) u2 t3 t5))) (ex2 T (\lambda (t: T).(subst0 i2 u2 (THead k -u t2) t)) (\lambda (t: T).(subst0 i v t4 t))) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H5: (eq T t4 (THead k x0 x1))).(\lambda (H6: (subst0 i2 u2 u -x0)).(\lambda (H7: (subst0 (s k i2) u2 t3 x1)).(eq_ind_r T (THead k x0 x1) -(\lambda (t: T).(ex2 T (\lambda (t5: T).(subst0 i2 u2 (THead k u t2) t5)) -(\lambda (t5: T).(subst0 i v t t5)))) (ex2_ind T (\lambda (t: T).(subst0 (s k -i2) u2 t2 t)) (\lambda (t: T).(subst0 (s k i) v x1 t)) (ex2 T (\lambda (t: -T).(subst0 i2 u2 (THead k u t2) t)) (\lambda (t: T).(subst0 i v (THead k x0 -x1) t))) (\lambda (x: T).(\lambda (H8: (subst0 (s k i2) u2 t2 x)).(\lambda -(H9: (subst0 (s k i) v x1 x)).(ex_intro2 T (\lambda (t: T).(subst0 i2 u2 -(THead k u t2) t)) (\lambda (t: T).(subst0 i v (THead k x0 x1) t)) (THead k -x0 x) (subst0_both u2 u x0 i2 H6 k t2 x H8) (subst0_snd k v x x1 i H9 x0))))) -(H1 x1 u2 (s k i2) H7 (ex2_ind T (\lambda (t: T).(subst0 (s k i2) u2 t2 t)) -(\lambda (t: T).(subst0 (s k i) v x1 t)) ((eq nat (s k i) (s k i2)) \to -False) (\lambda (x: T).(\lambda (_: (subst0 (s k i2) u2 t2 x)).(\lambda (_: -(subst0 (s k i) v x1 x)).(\lambda (H10: (eq nat (s k i) (s k i2))).(H3 (s_inj -k i i2 H10)))))) (H1 x1 u2 (s k i2) H7 (\lambda (H8: (eq nat (s k i) (s k -i2))).(H3 (s_inj k i i2 H8))))))) t4 H5)))))) H4)) (subst0_gen_head k u2 u t3 -t4 i2 H2))))))))))))))) (\lambda (v: T).(\lambda (u0: T).(\lambda (u2: -T).(\lambda (i: nat).(\lambda (H0: (subst0 i v u0 u2)).(\lambda (H1: -((\forall (t2: T).(\forall (u3: T).(\forall (i2: nat).((subst0 i2 u3 u0 t2) -\to ((not (eq nat i i2)) \to (ex2 T (\lambda (t: T).(subst0 i2 u3 u2 t)) -(\lambda (t: T).(subst0 i v t2 t)))))))))).(\lambda (k: K).(\lambda (t2: -T).(\lambda (t3: T).(\lambda (H2: (subst0 (s k i) v t2 t3)).(\lambda (H3: -((\forall (t4: T).(\forall (u3: T).(\forall (i2: nat).((subst0 i2 u3 t2 t4) -\to ((not (eq nat (s k i) i2)) \to (ex2 T (\lambda (t: T).(subst0 i2 u3 t3 -t)) (\lambda (t: T).(subst0 (s k i) v t4 t)))))))))).(\lambda (t4: -T).(\lambda (u3: T).(\lambda (i2: nat).(\lambda (H4: (subst0 i2 u3 (THead k -u0 t2) t4)).(\lambda (H5: (not (eq nat i i2))).(or3_ind (ex2 T (\lambda (u4: -T).(eq T t4 (THead k u4 t2))) (\lambda (u4: T).(subst0 i2 u3 u0 u4))) (ex2 T -(\lambda (t5: T).(eq T t4 (THead k u0 t5))) (\lambda (t5: T).(subst0 (s k i2) -u3 t2 t5))) (ex3_2 T T (\lambda (u4: T).(\lambda (t5: T).(eq T t4 (THead k u4 -t5)))) (\lambda (u4: T).(\lambda (_: T).(subst0 i2 u3 u0 u4))) (\lambda (_: -T).(\lambda (t5: T).(subst0 (s k i2) u3 t2 t5)))) (ex2 T (\lambda (t: -T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v t4 t))) -(\lambda (H6: (ex2 T (\lambda (u4: T).(eq T t4 (THead k u4 t2))) (\lambda -(u4: T).(subst0 i2 u3 u0 u4)))).(ex2_ind T (\lambda (u4: T).(eq T t4 (THead k -u4 t2))) (\lambda (u4: T).(subst0 i2 u3 u0 u4)) (ex2 T (\lambda (t: -T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v t4 t))) -(\lambda (x: T).(\lambda (H7: (eq T t4 (THead k x t2))).(\lambda (H8: (subst0 -i2 u3 u0 x)).(eq_ind_r T (THead k x t2) (\lambda (t: T).(ex2 T (\lambda (t5: -T).(subst0 i2 u3 (THead k u2 t3) t5)) (\lambda (t5: T).(subst0 i v t t5)))) -(ex2_ind T (\lambda (t: T).(subst0 i2 u3 u2 t)) (\lambda (t: T).(subst0 i v x -t)) (ex2 T (\lambda (t: T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: -T).(subst0 i v (THead k x t2) t))) (\lambda (x0: T).(\lambda (H9: (subst0 i2 -u3 u2 x0)).(\lambda (H10: (subst0 i v x x0)).(ex_intro2 T (\lambda (t: -T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v (THead k x -t2) t)) (THead k x0 t3) (subst0_fst u3 x0 u2 i2 H9 t3 k) (subst0_both v x x0 -i H10 k t2 t3 H2))))) (H1 x u3 i2 H8 H5)) t4 H7)))) H6)) (\lambda (H6: (ex2 T -(\lambda (t5: T).(eq T t4 (THead k u0 t5))) (\lambda (t5: T).(subst0 (s k i2) -u3 t2 t5)))).(ex2_ind T (\lambda (t5: T).(eq T t4 (THead k u0 t5))) (\lambda -(t5: T).(subst0 (s k i2) u3 t2 t5)) (ex2 T (\lambda (t: T).(subst0 i2 u3 -(THead k u2 t3) t)) (\lambda (t: T).(subst0 i v t4 t))) (\lambda (x: -T).(\lambda (H7: (eq T t4 (THead k u0 x))).(\lambda (H8: (subst0 (s k i2) u3 -t2 x)).(eq_ind_r T (THead k u0 x) (\lambda (t: T).(ex2 T (\lambda (t5: -T).(subst0 i2 u3 (THead k u2 t3) t5)) (\lambda (t5: T).(subst0 i v t t5)))) -(ex2_ind T (\lambda (t: T).(subst0 (s k i2) u3 t3 t)) (\lambda (t: T).(subst0 -(s k i) v x t)) (ex2 T (\lambda (t: T).(subst0 i2 u3 (THead k u2 t3) t)) -(\lambda (t: T).(subst0 i v (THead k u0 x) t))) (\lambda (x0: T).(\lambda -(H9: (subst0 (s k i2) u3 t3 x0)).(\lambda (H10: (subst0 (s k i) v x -x0)).(ex_intro2 T (\lambda (t: T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda -(t: T).(subst0 i v (THead k u0 x) t)) (THead k u2 x0) (subst0_snd k u3 x0 t3 -i2 H9 u2) (subst0_both v u0 u2 i H0 k x x0 H10))))) (H3 x u3 (s k i2) H8 -(ex2_ind T (\lambda (t: T).(subst0 (s k i2) u3 t3 t)) (\lambda (t: T).(subst0 -(s k i) v x t)) ((eq nat (s k i) (s k i2)) \to False) (\lambda (x0: -T).(\lambda (_: (subst0 (s k i2) u3 t3 x0)).(\lambda (_: (subst0 (s k i) v x -x0)).(\lambda (H11: (eq nat (s k i) (s k i2))).(H5 (s_inj k i i2 H11)))))) -(H3 x u3 (s k i2) H8 (\lambda (H9: (eq nat (s k i) (s k i2))).(H5 (s_inj k i -i2 H9))))))) t4 H7)))) H6)) (\lambda (H6: (ex3_2 T T (\lambda (u4: -T).(\lambda (t5: T).(eq T t4 (THead k u4 t5)))) (\lambda (u4: T).(\lambda (_: -T).(subst0 i2 u3 u0 u4))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s k i2) -u3 t2 t5))))).(ex3_2_ind T T (\lambda (u4: T).(\lambda (t5: T).(eq T t4 -(THead k u4 t5)))) (\lambda (u4: T).(\lambda (_: T).(subst0 i2 u3 u0 u4))) -(\lambda (_: T).(\lambda (t5: T).(subst0 (s k i2) u3 t2 t5))) (ex2 T (\lambda -(t: T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v t4 t))) -(\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: (eq T t4 (THead k x0 -x1))).(\lambda (H8: (subst0 i2 u3 u0 x0)).(\lambda (H9: (subst0 (s k i2) u3 -t2 x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t: T).(ex2 T (\lambda (t5: -T).(subst0 i2 u3 (THead k u2 t3) t5)) (\lambda (t5: T).(subst0 i v t t5)))) -(ex2_ind T (\lambda (t: T).(subst0 i2 u3 u2 t)) (\lambda (t: T).(subst0 i v -x0 t)) (ex2 T (\lambda (t: T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: -T).(subst0 i v (THead k x0 x1) t))) (\lambda (x: T).(\lambda (H10: (subst0 i2 -u3 u2 x)).(\lambda (H11: (subst0 i v x0 x)).(ex2_ind T (\lambda (t: -T).(subst0 (s k i2) u3 t3 t)) (\lambda (t: T).(subst0 (s k i) v x1 t)) (ex2 T -(\lambda (t: T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v -(THead k x0 x1) t))) (\lambda (x2: T).(\lambda (H12: (subst0 (s k i2) u3 t3 -x2)).(\lambda (H13: (subst0 (s k i) v x1 x2)).(ex_intro2 T (\lambda (t: -T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v (THead k x0 -x1) t)) (THead k x x2) (subst0_both u3 u2 x i2 H10 k t3 x2 H12) (subst0_both -v x0 x i H11 k x1 x2 H13))))) (H3 x1 u3 (s k i2) H9 (ex2_ind T (\lambda (t: -T).(subst0 (s k i2) u3 t3 t)) (\lambda (t: T).(subst0 (s k i) v x1 t)) ((eq -nat (s k i) (s k i2)) \to False) (\lambda (x2: T).(\lambda (_: (subst0 (s k -i2) u3 t3 x2)).(\lambda (_: (subst0 (s k i) v x1 x2)).(\lambda (H14: (eq nat -(s k i) (s k i2))).(H5 (s_inj k i i2 H14)))))) (H3 x1 u3 (s k i2) H9 (\lambda -(H12: (eq nat (s k i) (s k i2))).(H5 (s_inj k i i2 H12)))))))))) (H1 x0 u3 i2 -H8 H5)) t4 H7)))))) H6)) (subst0_gen_head k u3 u0 t2 t4 i2 -H4)))))))))))))))))) i1 u1 t0 t1 H))))). -(* COMMENTS -Initial nodes: 5375 -END *) - -theorem subst0_confluence_eq: - \forall (t0: T).(\forall (t1: T).(\forall (u: T).(\forall (i: nat).((subst0 -i u t0 t1) \to (\forall (t2: T).((subst0 i u t0 t2) \to (or4 (eq T t1 t2) -(ex2 T (\lambda (t: T).(subst0 i u t1 t)) (\lambda (t: T).(subst0 i u t2 t))) -(subst0 i u t1 t2) (subst0 i u t2 t1)))))))) -\def - \lambda (t0: T).(\lambda (t1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H: (subst0 i u t0 t1)).(subst0_ind (\lambda (n: nat).(\lambda (t: -T).(\lambda (t2: T).(\lambda (t3: T).(\forall (t4: T).((subst0 n t t2 t4) \to -(or4 (eq T t3 t4) (ex2 T (\lambda (t5: T).(subst0 n t t3 t5)) (\lambda (t5: -T).(subst0 n t t4 t5))) (subst0 n t t3 t4) (subst0 n t t4 t3)))))))) (\lambda -(v: T).(\lambda (i0: nat).(\lambda (t2: T).(\lambda (H0: (subst0 i0 v (TLRef -i0) t2)).(land_ind (eq nat i0 i0) (eq T t2 (lift (S i0) O v)) (or4 (eq T -(lift (S i0) O v) t2) (ex2 T (\lambda (t: T).(subst0 i0 v (lift (S i0) O v) -t)) (\lambda (t: T).(subst0 i0 v t2 t))) (subst0 i0 v (lift (S i0) O v) t2) -(subst0 i0 v t2 (lift (S i0) O v))) (\lambda (_: (eq nat i0 i0)).(\lambda -(H2: (eq T t2 (lift (S i0) O v))).(or4_intro0 (eq T (lift (S i0) O v) t2) -(ex2 T (\lambda (t: T).(subst0 i0 v (lift (S i0) O v) t)) (\lambda (t: -T).(subst0 i0 v t2 t))) (subst0 i0 v (lift (S i0) O v) t2) (subst0 i0 v t2 -(lift (S i0) O v)) (sym_eq T t2 (lift (S i0) O v) H2)))) (subst0_gen_lref v -t2 i0 i0 H0)))))) (\lambda (v: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda -(i0: nat).(\lambda (H0: (subst0 i0 v u1 u2)).(\lambda (H1: ((\forall (t2: -T).((subst0 i0 v u1 t2) \to (or4 (eq T u2 t2) (ex2 T (\lambda (t: T).(subst0 -i0 v u2 t)) (\lambda (t: T).(subst0 i0 v t2 t))) (subst0 i0 v u2 t2) (subst0 -i0 v t2 u2)))))).(\lambda (t: T).(\lambda (k: K).(\lambda (t2: T).(\lambda -(H2: (subst0 i0 v (THead k u1 t) t2)).(or3_ind (ex2 T (\lambda (u3: T).(eq T -t2 (THead k u3 t))) (\lambda (u3: T).(subst0 i0 v u1 u3))) (ex2 T (\lambda -(t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v t -t3))) (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead k u3 -t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v u1 u3))) (\lambda (_: -T).(\lambda (t3: T).(subst0 (s k i0) v t t3)))) (or4 (eq T (THead k u2 t) t2) -(ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: -T).(subst0 i0 v t2 t3))) (subst0 i0 v (THead k u2 t) t2) (subst0 i0 v t2 -(THead k u2 t))) (\lambda (H3: (ex2 T (\lambda (u3: T).(eq T t2 (THead k u3 -t))) (\lambda (u3: T).(subst0 i0 v u1 u3)))).(ex2_ind T (\lambda (u3: T).(eq -T t2 (THead k u3 t))) (\lambda (u3: T).(subst0 i0 v u1 u3)) (or4 (eq T (THead -k u2 t) t2) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda -(t3: T).(subst0 i0 v t2 t3))) (subst0 i0 v (THead k u2 t) t2) (subst0 i0 v t2 -(THead k u2 t))) (\lambda (x: T).(\lambda (H4: (eq T t2 (THead k x -t))).(\lambda (H5: (subst0 i0 v u1 x)).(eq_ind_r T (THead k x t) (\lambda -(t3: T).(or4 (eq T (THead k u2 t) t3) (ex2 T (\lambda (t4: T).(subst0 i0 v -(THead k u2 t) t4)) (\lambda (t4: T).(subst0 i0 v t3 t4))) (subst0 i0 v -(THead k u2 t) t3) (subst0 i0 v t3 (THead k u2 t)))) (or4_ind (eq T u2 x) -(ex2 T (\lambda (t3: T).(subst0 i0 v u2 t3)) (\lambda (t3: T).(subst0 i0 v x -t3))) (subst0 i0 v u2 x) (subst0 i0 v x u2) (or4 (eq T (THead k u2 t) (THead -k x t)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda -(t3: T).(subst0 i0 v (THead k x t) t3))) (subst0 i0 v (THead k u2 t) (THead k -x t)) (subst0 i0 v (THead k x t) (THead k u2 t))) (\lambda (H6: (eq T u2 -x)).(eq_ind_r T x (\lambda (t3: T).(or4 (eq T (THead k t3 t) (THead k x t)) -(ex2 T (\lambda (t4: T).(subst0 i0 v (THead k t3 t) t4)) (\lambda (t4: -T).(subst0 i0 v (THead k x t) t4))) (subst0 i0 v (THead k t3 t) (THead k x -t)) (subst0 i0 v (THead k x t) (THead k t3 t)))) (or4_intro0 (eq T (THead k x -t) (THead k x t)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k x t) t3)) -(\lambda (t3: T).(subst0 i0 v (THead k x t) t3))) (subst0 i0 v (THead k x t) -(THead k x t)) (subst0 i0 v (THead k x t) (THead k x t)) (refl_equal T (THead -k x t))) u2 H6)) (\lambda (H6: (ex2 T (\lambda (t3: T).(subst0 i0 v u2 t3)) -(\lambda (t3: T).(subst0 i0 v x t3)))).(ex2_ind T (\lambda (t3: T).(subst0 i0 -v u2 t3)) (\lambda (t3: T).(subst0 i0 v x t3)) (or4 (eq T (THead k u2 t) -(THead k x t)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) -(\lambda (t3: T).(subst0 i0 v (THead k x t) t3))) (subst0 i0 v (THead k u2 t) -(THead k x t)) (subst0 i0 v (THead k x t) (THead k u2 t))) (\lambda (x0: -T).(\lambda (H7: (subst0 i0 v u2 x0)).(\lambda (H8: (subst0 i0 v x -x0)).(or4_intro1 (eq T (THead k u2 t) (THead k x t)) (ex2 T (\lambda (t3: -T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k x -t) t3))) (subst0 i0 v (THead k u2 t) (THead k x t)) (subst0 i0 v (THead k x -t) (THead k u2 t)) (ex_intro2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) -t3)) (\lambda (t3: T).(subst0 i0 v (THead k x t) t3)) (THead k x0 t) -(subst0_fst v x0 u2 i0 H7 t k) (subst0_fst v x0 x i0 H8 t k)))))) H6)) -(\lambda (H6: (subst0 i0 v u2 x)).(or4_intro2 (eq T (THead k u2 t) (THead k x -t)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: -T).(subst0 i0 v (THead k x t) t3))) (subst0 i0 v (THead k u2 t) (THead k x -t)) (subst0 i0 v (THead k x t) (THead k u2 t)) (subst0_fst v x u2 i0 H6 t -k))) (\lambda (H6: (subst0 i0 v x u2)).(or4_intro3 (eq T (THead k u2 t) -(THead k x t)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) -(\lambda (t3: T).(subst0 i0 v (THead k x t) t3))) (subst0 i0 v (THead k u2 t) -(THead k x t)) (subst0 i0 v (THead k x t) (THead k u2 t)) (subst0_fst v u2 x -i0 H6 t k))) (H1 x H5)) t2 H4)))) H3)) (\lambda (H3: (ex2 T (\lambda (t3: -T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v t -t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: -T).(subst0 (s k i0) v t t3)) (or4 (eq T (THead k u2 t) t2) (ex2 T (\lambda -(t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0 v t2 -t3))) (subst0 i0 v (THead k u2 t) t2) (subst0 i0 v t2 (THead k u2 t))) -(\lambda (x: T).(\lambda (H4: (eq T t2 (THead k u1 x))).(\lambda (H5: (subst0 -(s k i0) v t x)).(eq_ind_r T (THead k u1 x) (\lambda (t3: T).(or4 (eq T -(THead k u2 t) t3) (ex2 T (\lambda (t4: T).(subst0 i0 v (THead k u2 t) t4)) -(\lambda (t4: T).(subst0 i0 v t3 t4))) (subst0 i0 v (THead k u2 t) t3) -(subst0 i0 v t3 (THead k u2 t)))) (or4_ind (eq T u2 u2) (ex2 T (\lambda (t3: -T).(subst0 i0 v u2 t3)) (\lambda (t3: T).(subst0 i0 v u2 t3))) (subst0 i0 v -u2 u2) (subst0 i0 v u2 u2) (or4 (eq T (THead k u2 t) (THead k u1 x)) (ex2 T -(\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0 -v (THead k u1 x) t3))) (subst0 i0 v (THead k u2 t) (THead k u1 x)) (subst0 i0 -v (THead k u1 x) (THead k u2 t))) (\lambda (_: (eq T u2 u2)).(or4_intro1 (eq -T (THead k u2 t) (THead k u1 x)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead -k u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k u1 x) t3))) (subst0 i0 v -(THead k u2 t) (THead k u1 x)) (subst0 i0 v (THead k u1 x) (THead k u2 t)) -(ex_intro2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: -T).(subst0 i0 v (THead k u1 x) t3)) (THead k u2 x) (subst0_snd k v x t i0 H5 -u2) (subst0_fst v u2 u1 i0 H0 x k)))) (\lambda (H6: (ex2 T (\lambda (t3: -T).(subst0 i0 v u2 t3)) (\lambda (t3: T).(subst0 i0 v u2 t3)))).(ex2_ind T -(\lambda (t3: T).(subst0 i0 v u2 t3)) (\lambda (t3: T).(subst0 i0 v u2 t3)) -(or4 (eq T (THead k u2 t) (THead k u1 x)) (ex2 T (\lambda (t3: T).(subst0 i0 -v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k u1 x) t3))) -(subst0 i0 v (THead k u2 t) (THead k u1 x)) (subst0 i0 v (THead k u1 x) -(THead k u2 t))) (\lambda (x0: T).(\lambda (_: (subst0 i0 v u2 x0)).(\lambda -(_: (subst0 i0 v u2 x0)).(or4_intro1 (eq T (THead k u2 t) (THead k u1 x)) -(ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: -T).(subst0 i0 v (THead k u1 x) t3))) (subst0 i0 v (THead k u2 t) (THead k u1 -x)) (subst0 i0 v (THead k u1 x) (THead k u2 t)) (ex_intro2 T (\lambda (t3: -T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k u1 -x) t3)) (THead k u2 x) (subst0_snd k v x t i0 H5 u2) (subst0_fst v u2 u1 i0 -H0 x k)))))) H6)) (\lambda (_: (subst0 i0 v u2 u2)).(or4_intro1 (eq T (THead -k u2 t) (THead k u1 x)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) -t3)) (\lambda (t3: T).(subst0 i0 v (THead k u1 x) t3))) (subst0 i0 v (THead k -u2 t) (THead k u1 x)) (subst0 i0 v (THead k u1 x) (THead k u2 t)) (ex_intro2 -T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 -i0 v (THead k u1 x) t3)) (THead k u2 x) (subst0_snd k v x t i0 H5 u2) -(subst0_fst v u2 u1 i0 H0 x k)))) (\lambda (_: (subst0 i0 v u2 -u2)).(or4_intro1 (eq T (THead k u2 t) (THead k u1 x)) (ex2 T (\lambda (t3: -T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k u1 -x) t3))) (subst0 i0 v (THead k u2 t) (THead k u1 x)) (subst0 i0 v (THead k u1 -x) (THead k u2 t)) (ex_intro2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) -t3)) (\lambda (t3: T).(subst0 i0 v (THead k u1 x) t3)) (THead k u2 x) -(subst0_snd k v x t i0 H5 u2) (subst0_fst v u2 u1 i0 H0 x k)))) (H1 u2 H0)) -t2 H4)))) H3)) (\lambda (H3: (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq -T t2 (THead k u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v u1 -u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i0) v t -t3))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead k u3 -t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v u1 u3))) (\lambda (_: -T).(\lambda (t3: T).(subst0 (s k i0) v t t3))) (or4 (eq T (THead k u2 t) t2) -(ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: -T).(subst0 i0 v t2 t3))) (subst0 i0 v (THead k u2 t) t2) (subst0 i0 v t2 -(THead k u2 t))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t2 -(THead k x0 x1))).(\lambda (H5: (subst0 i0 v u1 x0)).(\lambda (H6: (subst0 (s -k i0) v t x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t3: T).(or4 (eq T (THead -k u2 t) t3) (ex2 T (\lambda (t4: T).(subst0 i0 v (THead k u2 t) t4)) (\lambda -(t4: T).(subst0 i0 v t3 t4))) (subst0 i0 v (THead k u2 t) t3) (subst0 i0 v t3 -(THead k u2 t)))) (or4_ind (eq T u2 x0) (ex2 T (\lambda (t3: T).(subst0 i0 v -u2 t3)) (\lambda (t3: T).(subst0 i0 v x0 t3))) (subst0 i0 v u2 x0) (subst0 i0 -v x0 u2) (or4 (eq T (THead k u2 t) (THead k x0 x1)) (ex2 T (\lambda (t3: -T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k x0 -x1) t3))) (subst0 i0 v (THead k u2 t) (THead k x0 x1)) (subst0 i0 v (THead k -x0 x1) (THead k u2 t))) (\lambda (H7: (eq T u2 x0)).(eq_ind_r T x0 (\lambda -(t3: T).(or4 (eq T (THead k t3 t) (THead k x0 x1)) (ex2 T (\lambda (t4: -T).(subst0 i0 v (THead k t3 t) t4)) (\lambda (t4: T).(subst0 i0 v (THead k x0 -x1) t4))) (subst0 i0 v (THead k t3 t) (THead k x0 x1)) (subst0 i0 v (THead k -x0 x1) (THead k t3 t)))) (or4_intro2 (eq T (THead k x0 t) (THead k x0 x1)) -(ex2 T (\lambda (t3: T).(subst0 i0 v (THead k x0 t) t3)) (\lambda (t3: -T).(subst0 i0 v (THead k x0 x1) t3))) (subst0 i0 v (THead k x0 t) (THead k x0 -x1)) (subst0 i0 v (THead k x0 x1) (THead k x0 t)) (subst0_snd k v x1 t i0 H6 -x0)) u2 H7)) (\lambda (H7: (ex2 T (\lambda (t3: T).(subst0 i0 v u2 t3)) -(\lambda (t3: T).(subst0 i0 v x0 t3)))).(ex2_ind T (\lambda (t3: T).(subst0 -i0 v u2 t3)) (\lambda (t3: T).(subst0 i0 v x0 t3)) (or4 (eq T (THead k u2 t) -(THead k x0 x1)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) -(\lambda (t3: T).(subst0 i0 v (THead k x0 x1) t3))) (subst0 i0 v (THead k u2 -t) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t))) (\lambda -(x: T).(\lambda (H8: (subst0 i0 v u2 x)).(\lambda (H9: (subst0 i0 v x0 -x)).(or4_intro1 (eq T (THead k u2 t) (THead k x0 x1)) (ex2 T (\lambda (t3: -T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k x0 -x1) t3))) (subst0 i0 v (THead k u2 t) (THead k x0 x1)) (subst0 i0 v (THead k -x0 x1) (THead k u2 t)) (ex_intro2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 -t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k x0 x1) t3)) (THead k x x1) -(subst0_both v u2 x i0 H8 k t x1 H6) (subst0_fst v x x0 i0 H9 x1 k)))))) H7)) -(\lambda (H7: (subst0 i0 v u2 x0)).(or4_intro2 (eq T (THead k u2 t) (THead k -x0 x1)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda -(t3: T).(subst0 i0 v (THead k x0 x1) t3))) (subst0 i0 v (THead k u2 t) (THead -k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t)) (subst0_both v u2 x0 -i0 H7 k t x1 H6))) (\lambda (H7: (subst0 i0 v x0 u2)).(or4_intro1 (eq T -(THead k u2 t) (THead k x0 x1)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k -u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k x0 x1) t3))) (subst0 i0 v -(THead k u2 t) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t)) -(ex_intro2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: -T).(subst0 i0 v (THead k x0 x1) t3)) (THead k u2 x1) (subst0_snd k v x1 t i0 -H6 u2) (subst0_fst v u2 x0 i0 H7 x1 k)))) (H1 x0 H5)) t2 H4)))))) H3)) -(subst0_gen_head k v u1 t t2 i0 H2)))))))))))) (\lambda (k: K).(\lambda (v: -T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (i0: nat).(\lambda (H0: (subst0 -(s k i0) v t3 t2)).(\lambda (H1: ((\forall (t4: T).((subst0 (s k i0) v t3 t4) -\to (or4 (eq T t2 t4) (ex2 T (\lambda (t: T).(subst0 (s k i0) v t2 t)) -(\lambda (t: T).(subst0 (s k i0) v t4 t))) (subst0 (s k i0) v t2 t4) (subst0 -(s k i0) v t4 t2)))))).(\lambda (u0: T).(\lambda (t4: T).(\lambda (H2: -(subst0 i0 v (THead k u0 t3) t4)).(or3_ind (ex2 T (\lambda (u2: T).(eq T t4 -(THead k u2 t3))) (\lambda (u2: T).(subst0 i0 v u0 u2))) (ex2 T (\lambda (t5: -T).(eq T t4 (THead k u0 t5))) (\lambda (t5: T).(subst0 (s k i0) v t3 t5))) -(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead k u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 i0 v u0 u2))) (\lambda (_: -T).(\lambda (t5: T).(subst0 (s k i0) v t3 t5)))) (or4 (eq T (THead k u0 t2) -t4) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u0 t2) t)) (\lambda (t: -T).(subst0 i0 v t4 t))) (subst0 i0 v (THead k u0 t2) t4) (subst0 i0 v t4 -(THead k u0 t2))) (\lambda (H3: (ex2 T (\lambda (u2: T).(eq T t4 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(subst0_snd k v t2 t3 i0 H0 x)))) (\lambda (_: (subst0 (s k i0) v t2 -t2)).(or4_intro1 (eq T (THead k u0 t2) (THead k x t3)) (ex2 T (\lambda (t: -T).(subst0 i0 v (THead k u0 t2) t)) (\lambda (t: T).(subst0 i0 v (THead k x -t3) t))) (subst0 i0 v (THead k u0 t2) (THead k x t3)) (subst0 i0 v (THead k x -t3) (THead k u0 t2)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k u0 -t2) t)) (\lambda (t: T).(subst0 i0 v (THead k x t3) t)) (THead k x t2) -(subst0_fst v x u0 i0 H5 t2 k) (subst0_snd k v t2 t3 i0 H0 x)))) (H1 t2 H0)) -t4 H4)))) H3)) (\lambda (H3: (ex2 T (\lambda (t5: T).(eq T t4 (THead k u0 -t5))) (\lambda (t5: T).(subst0 (s k i0) v t3 t5)))).(ex2_ind T (\lambda (t5: -T).(eq T t4 (THead k u0 t5))) (\lambda (t5: T).(subst0 (s k i0) v t3 t5)) -(or4 (eq T (THead k u0 t2) t4) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k -u0 t2) t)) (\lambda (t: T).(subst0 i0 v t4 t))) (subst0 i0 v (THead k u0 t2) -t4) (subst0 i0 v t4 (THead k u0 t2))) (\lambda (x: T).(\lambda (H4: (eq T t4 -(THead k u0 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(or4_intro0 (eq T (THead k u0 x) (THead k u0 x)) (ex2 T -(\lambda (t: T).(subst0 i0 v (THead k u0 x) t)) (\lambda (t: T).(subst0 i0 v -(THead k u0 x) t))) (subst0 i0 v (THead k u0 x) (THead k u0 x)) (subst0 i0 v -(THead k u0 x) (THead k u0 x)) (refl_equal T (THead k u0 x))) t2 H6)) -(\lambda (H6: (ex2 T (\lambda (t: T).(subst0 (s k i0) v t2 t)) (\lambda (t: -T).(subst0 (s k i0) v x t)))).(ex2_ind T (\lambda (t: T).(subst0 (s k i0) v -t2 t)) (\lambda (t: T).(subst0 (s k i0) v x t)) (or4 (eq T (THead k u0 t2) -(THead k u0 x)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u0 t2) t)) -(\lambda (t: T).(subst0 i0 v (THead k u0 x) t))) (subst0 i0 v (THead k u0 t2) -(THead k u0 x)) (subst0 i0 v (THead k u0 x) (THead k u0 t2))) (\lambda (x0: -T).(\lambda (H7: (subst0 (s k i0) v t2 x0)).(\lambda (H8: (subst0 (s k i0) v -x x0)).(or4_intro1 (eq T (THead k u0 t2) (THead k u0 x)) (ex2 T (\lambda (t: -T).(subst0 i0 v (THead k u0 t2) t)) (\lambda (t: T).(subst0 i0 v (THead k u0 -x) t))) (subst0 i0 v (THead k u0 t2) (THead k u0 x)) (subst0 i0 v (THead k u0 -x) (THead k u0 t2)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k u0 t2) -t)) (\lambda (t: T).(subst0 i0 v (THead k u0 x) t)) (THead k u0 x0) -(subst0_snd k v x0 t2 i0 H7 u0) (subst0_snd k v x0 x i0 H8 u0)))))) H6)) -(\lambda (H6: (subst0 (s k i0) v t2 x)).(or4_intro2 (eq T (THead k u0 t2) -(THead k u0 x)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u0 t2) t)) -(\lambda (t: T).(subst0 i0 v (THead k u0 x) t))) (subst0 i0 v (THead k u0 t2) -(THead k u0 x)) (subst0 i0 v (THead k u0 x) (THead k u0 t2)) (subst0_snd k v -x t2 i0 H6 u0))) (\lambda (H6: (subst0 (s k i0) v x t2)).(or4_intro3 (eq T -(THead k u0 t2) (THead k u0 x)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k -u0 t2) t)) (\lambda (t: T).(subst0 i0 v (THead k u0 x) t))) (subst0 i0 v -(THead k u0 t2) (THead k u0 x)) (subst0 i0 v (THead k u0 x) (THead k u0 t2)) -(subst0_snd k v t2 x i0 H6 u0))) (H1 x H5)) t4 H4)))) H3)) (\lambda (H3: -(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead k u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 i0 v u0 u2))) (\lambda (_: -T).(\lambda (t5: T).(subst0 (s k i0) v t3 t5))))).(ex3_2_ind T T (\lambda -(u2: T).(\lambda (t5: T).(eq T t4 (THead k u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i0 v u0 u2))) (\lambda (_: T).(\lambda (t5: -T).(subst0 (s k i0) v t3 t5))) (or4 (eq T (THead k u0 t2) t4) (ex2 T (\lambda -(t: T).(subst0 i0 v (THead k u0 t2) t)) (\lambda (t: T).(subst0 i0 v t4 t))) -(subst0 i0 v (THead k u0 t2) t4) (subst0 i0 v t4 (THead k u0 t2))) (\lambda -(x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t4 (THead k x0 x1))).(\lambda -(H5: (subst0 i0 v u0 x0)).(\lambda (H6: (subst0 (s k i0) v t3 x1)).(eq_ind_r -T (THead k x0 x1) (\lambda (t: T).(or4 (eq T (THead k u0 t2) t) (ex2 T -(\lambda (t5: T).(subst0 i0 v (THead k u0 t2) t5)) (\lambda (t5: T).(subst0 -i0 v t t5))) (subst0 i0 v (THead k u0 t2) t) (subst0 i0 v t (THead k u0 -t2)))) (or4_ind (eq T t2 x1) (ex2 T (\lambda (t: 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(\lambda (t: T).(subst0 (s k i0) v t2 t)) -(\lambda (t: T).(subst0 (s k i0) v x1 t)))).(ex2_ind T (\lambda (t: -T).(subst0 (s k i0) v t2 t)) (\lambda (t: T).(subst0 (s k i0) v x1 t)) (or4 -(eq T (THead k u0 t2) (THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v -(THead k u0 t2) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 -i0 v (THead k u0 t2) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k -u0 t2))) (\lambda (x: T).(\lambda (H8: (subst0 (s k i0) v t2 x)).(\lambda -(H9: (subst0 (s k i0) v x1 x)).(or4_intro1 (eq T (THead k u0 t2) (THead k x0 -x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u0 t2) t)) (\lambda (t: -T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u0 t2) (THead k x0 -x1)) (subst0 i0 v (THead k x0 x1) (THead k u0 t2)) (ex_intro2 T (\lambda (t: -T).(subst0 i0 v (THead k u0 t2) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 -x1) t)) (THead k x0 x) (subst0_both v u0 x0 i0 H5 k t2 x H8) (subst0_snd k v -x x1 i0 H9 x0)))))) H7)) (\lambda (H7: 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u2)).(\lambda (H1: ((\forall (t2: T).((subst0 i0 v u1 -t2) \to (or4 (eq T u2 t2) (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda -(t: T).(subst0 i0 v t2 t))) (subst0 i0 v u2 t2) (subst0 i0 v t2 -u2)))))).(\lambda (k: K).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H2: -(subst0 (s k i0) v t2 t3)).(\lambda (H3: ((\forall (t4: T).((subst0 (s k i0) -v t2 t4) \to (or4 (eq T t3 t4) (ex2 T (\lambda (t: T).(subst0 (s k i0) v t3 -t)) (\lambda (t: T).(subst0 (s k i0) v t4 t))) (subst0 (s k i0) v t3 t4) -(subst0 (s k i0) v t4 t3)))))).(\lambda (t4: T).(\lambda (H4: (subst0 i0 v -(THead k u1 t2) t4)).(or3_ind (ex2 T (\lambda (u3: T).(eq T t4 (THead k u3 -t2))) (\lambda (u3: T).(subst0 i0 v u1 u3))) (ex2 T (\lambda (t5: T).(eq T t4 -(THead k u1 t5))) (\lambda (t5: T).(subst0 (s k i0) v t2 t5))) (ex3_2 T T -(\lambda (u3: T).(\lambda (t5: T).(eq T t4 (THead k u3 t5)))) (\lambda (u3: -T).(\lambda (_: T).(subst0 i0 v u1 u3))) (\lambda (_: T).(\lambda (t5: -T).(subst0 (s k i0) v t2 t5)))) (or4 (eq T 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-t2)) (subst0 i0 v (THead k x t2) (THead k u2 t3)) (subst0_both v x u2 i0 H9 k -t2 t3 H2))) (H1 x H7))) (\lambda (H8: (ex2 T (\lambda (t: T).(subst0 (s k i0) -v t3 t)) (\lambda (t: T).(subst0 (s k i0) v t3 t)))).(ex2_ind T (\lambda (t: -T).(subst0 (s k i0) v t3 t)) (\lambda (t: T).(subst0 (s k i0) v t3 t)) (or4 -(eq T (THead k u2 t3) (THead k x t2)) (ex2 T (\lambda (t: T).(subst0 i0 v -(THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x t2) t))) (subst0 -i0 v (THead k u2 t3) (THead k x t2)) (subst0 i0 v (THead k x t2) (THead k u2 -t3))) (\lambda (x0: T).(\lambda (_: (subst0 (s k i0) v t3 x0)).(\lambda (_: -(subst0 (s k i0) v t3 x0)).(or4_ind (eq T u2 x) (ex2 T (\lambda (t: -T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0 i0 v x t))) (subst0 i0 v u2 x) -(subst0 i0 v x u2) (or4 (eq T (THead k u2 t3) (THead k x t2)) (ex2 T (\lambda -(t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k -x t2) t))) (subst0 i0 v (THead k u2 t3) (THead k x t2)) (subst0 i0 v (THead k -x t2) (THead k u2 t3))) (\lambda (H11: (eq T u2 x)).(eq_ind_r T x (\lambda -(t: T).(or4 (eq T (THead k t t3) (THead k x t2)) (ex2 T (\lambda (t5: -T).(subst0 i0 v (THead k t t3) t5)) (\lambda (t5: T).(subst0 i0 v (THead k x -t2) t5))) (subst0 i0 v (THead k t t3) (THead k x t2)) (subst0 i0 v (THead k x -t2) (THead k t t3)))) (or4_intro3 (eq T (THead k x t3) (THead k x t2)) (ex2 T -(\lambda (t: T).(subst0 i0 v (THead k x t3) t)) (\lambda (t: T).(subst0 i0 v -(THead k x t2) t))) (subst0 i0 v (THead k x t3) (THead k x t2)) (subst0 i0 v -(THead k x t2) (THead k x t3)) (subst0_snd k v t3 t2 i0 H2 x)) u2 H11)) -(\lambda (H11: (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: -T).(subst0 i0 v x t)))).(ex2_ind T (\lambda (t: T).(subst0 i0 v u2 t)) -(\lambda (t: T).(subst0 i0 v x t)) (or4 (eq T (THead k u2 t3) (THead k x t2)) -(ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: -T).(subst0 i0 v (THead k x t2) t))) (subst0 i0 v (THead k u2 t3) (THead k x -t2)) (subst0 i0 v (THead k x t2) (THead k u2 t3))) (\lambda (x1: T).(\lambda -(H12: (subst0 i0 v u2 x1)).(\lambda (H13: (subst0 i0 v x x1)).(or4_intro1 (eq -T (THead k u2 t3) (THead k x t2)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead -k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x t2) t))) (subst0 i0 v -(THead k u2 t3) (THead k x t2)) (subst0 i0 v (THead k x t2) (THead k u2 t3)) -(ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: -T).(subst0 i0 v (THead k x t2) t)) (THead k x1 t3) (subst0_fst v x1 u2 i0 H12 -t3 k) (subst0_both v x x1 i0 H13 k t2 t3 H2)))))) H11)) (\lambda (H11: -(subst0 i0 v u2 x)).(or4_intro1 (eq T (THead k u2 t3) (THead k x t2)) (ex2 T -(\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v -(THead k x t2) t))) (subst0 i0 v (THead k u2 t3) (THead k x t2)) (subst0 i0 v -(THead k x t2) (THead k u2 t3)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v -(THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x t2) t)) (THead k -x t3) (subst0_fst v x u2 i0 H11 t3 k) (subst0_snd k v t3 t2 i0 H2 x)))) -(\lambda (H11: (subst0 i0 v x u2)).(or4_intro3 (eq T (THead k u2 t3) (THead k -x t2)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: -T).(subst0 i0 v (THead k x t2) t))) (subst0 i0 v (THead k u2 t3) (THead k x -t2)) (subst0 i0 v (THead k x t2) (THead k u2 t3)) (subst0_both v x u2 i0 H11 -k t2 t3 H2))) (H1 x H7))))) H8)) (\lambda (_: (subst0 (s k i0) v t3 -t3)).(or4_ind (eq T u2 x) (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda -(t: T).(subst0 i0 v x t))) (subst0 i0 v u2 x) (subst0 i0 v x u2) (or4 (eq T -(THead k u2 t3) (THead k x t2)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k -u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x t2) t))) (subst0 i0 v -(THead k u2 t3) (THead k x t2)) (subst0 i0 v (THead k x t2) (THead k u2 t3))) -(\lambda (H9: (eq T u2 x)).(eq_ind_r T x (\lambda (t: T).(or4 (eq T (THead k -t t3) (THead k x t2)) (ex2 T (\lambda (t5: T).(subst0 i0 v (THead k t t3) -t5)) (\lambda (t5: T).(subst0 i0 v (THead k x t2) t5))) (subst0 i0 v (THead k -t t3) (THead k x t2)) (subst0 i0 v (THead k x t2) (THead k t t3)))) -(or4_intro3 (eq T (THead k x t3) (THead k x t2)) (ex2 T (\lambda (t: -T).(subst0 i0 v (THead k x t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x -t2) t))) (subst0 i0 v (THead k x t3) (THead k x t2)) (subst0 i0 v (THead k x -t2) (THead k x t3)) (subst0_snd k v t3 t2 i0 H2 x)) u2 H9)) (\lambda (H9: -(ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0 i0 v x -t)))).(ex2_ind T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0 -i0 v x t)) (or4 (eq T (THead k u2 t3) (THead k x t2)) (ex2 T (\lambda (t: -T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x -t2) t))) (subst0 i0 v (THead k u2 t3) (THead k x t2)) (subst0 i0 v (THead k x -t2) (THead k u2 t3))) (\lambda (x0: T).(\lambda (H10: (subst0 i0 v u2 -x0)).(\lambda (H11: (subst0 i0 v x x0)).(or4_intro1 (eq T (THead k u2 t3) -(THead k x t2)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) -(\lambda (t: T).(subst0 i0 v (THead k x t2) t))) (subst0 i0 v (THead k u2 t3) -(THead k x t2)) (subst0 i0 v (THead k x t2) (THead k u2 t3)) (ex_intro2 T -(\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v -(THead k x t2) t)) (THead k x0 t3) (subst0_fst v x0 u2 i0 H10 t3 k) -(subst0_both v x x0 i0 H11 k t2 t3 H2)))))) H9)) (\lambda (H9: (subst0 i0 v -u2 x)).(or4_intro1 (eq T (THead k u2 t3) (THead k x t2)) (ex2 T (\lambda (t: -T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x -t2) t))) (subst0 i0 v (THead k u2 t3) (THead k x t2)) (subst0 i0 v (THead k x -t2) (THead k u2 t3)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k u2 -t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x t2) t)) (THead k x t3) -(subst0_fst v x u2 i0 H9 t3 k) (subst0_snd k v t3 t2 i0 H2 x)))) (\lambda -(H9: (subst0 i0 v x u2)).(or4_intro3 (eq T (THead k u2 t3) (THead k x t2)) -(ex2 T (\lambda (t: T).(subst0 i0 v 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(\lambda (t: T).(subst0 i0 v (THead k u2 -t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x t2) t)) (THead k x0 t3) -(subst0_fst v x0 u2 i0 H10 t3 k) (subst0_both v x x0 i0 H11 k t2 t3 H2)))))) -H9)) (\lambda (H9: (subst0 i0 v u2 x)).(or4_intro1 (eq T (THead k u2 t3) -(THead k x t2)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) -(\lambda (t: T).(subst0 i0 v (THead k x t2) t))) (subst0 i0 v (THead k u2 t3) -(THead k x t2)) (subst0 i0 v (THead k x t2) (THead k u2 t3)) (ex_intro2 T -(\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v -(THead k x t2) t)) (THead k x t3) (subst0_fst v x u2 i0 H9 t3 k) (subst0_snd -k v t3 t2 i0 H2 x)))) (\lambda (H9: (subst0 i0 v x u2)).(or4_intro3 (eq T -(THead k u2 t3) (THead k x t2)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k -u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x t2) t))) (subst0 i0 v -(THead k u2 t3) (THead k x t2)) (subst0 i0 v (THead k x t2) (THead k u2 t3)) -(subst0_both v x u2 i0 H9 k t2 t3 H2))) (H1 x H7))) (H3 t3 H2)) t4 H6)))) -H5)) (\lambda (H5: (ex2 T (\lambda (t5: T).(eq T t4 (THead k u1 t5))) -(\lambda (t5: T).(subst0 (s k i0) v t2 t5)))).(ex2_ind T (\lambda (t5: T).(eq -T t4 (THead k u1 t5))) (\lambda (t5: T).(subst0 (s k i0) v t2 t5)) (or4 (eq T -(THead k u2 t3) t4) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) -(\lambda (t: T).(subst0 i0 v t4 t))) (subst0 i0 v (THead k u2 t3) t4) (subst0 -i0 v t4 (THead k u2 t3))) (\lambda (x: T).(\lambda (H6: (eq T t4 (THead k u1 -x))).(\lambda (H7: (subst0 (s k i0) v t2 x)).(eq_ind_r T (THead k u1 x) -(\lambda (t: T).(or4 (eq T (THead k u2 t3) t) (ex2 T (\lambda (t5: T).(subst0 -i0 v (THead k u2 t3) t5)) (\lambda (t5: T).(subst0 i0 v t t5))) (subst0 i0 v -(THead k u2 t3) t) (subst0 i0 v t (THead k u2 t3)))) (or4_ind (eq T t3 x) -(ex2 T (\lambda (t: T).(subst0 (s k i0) v t3 t)) (\lambda (t: T).(subst0 (s k -i0) v x t))) (subst0 (s k i0) v t3 x) (subst0 (s k i0) v x t3) (or4 (eq T -(THead k u2 t3) (THead k u1 x)) (ex2 T (\lambda 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(\lambda (t: -T).(subst0 i0 v (THead k u1 x) t))) (subst0 i0 v (THead k u2 x) (THead k u1 -x)) (subst0 i0 v (THead k u1 x) (THead k u2 x)) (subst0_fst v u2 u1 i0 H0 x -k))) (\lambda (H9: (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: -T).(subst0 i0 v u2 t)))).(ex2_ind T (\lambda (t: T).(subst0 i0 v u2 t)) -(\lambda (t: T).(subst0 i0 v u2 t)) (or4 (eq T (THead k u2 x) (THead k u1 x)) -(ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 x) t)) (\lambda (t: -T).(subst0 i0 v (THead k u1 x) t))) (subst0 i0 v (THead k u2 x) (THead k u1 -x)) (subst0 i0 v (THead k u1 x) (THead k u2 x))) (\lambda (x0: T).(\lambda -(_: (subst0 i0 v u2 x0)).(\lambda (_: (subst0 i0 v u2 x0)).(or4_intro3 (eq T -(THead k u2 x) (THead k u1 x)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k -u2 x) t)) (\lambda (t: T).(subst0 i0 v (THead k u1 x) t))) (subst0 i0 v -(THead k u2 x) (THead k u1 x)) (subst0 i0 v (THead k u1 x) (THead k u2 x)) -(subst0_fst v u2 u1 i0 H0 x k))))) H9)) (\lambda (_: (subst0 i0 v u2 -u2)).(or4_intro3 (eq T (THead k u2 x) (THead k u1 x)) (ex2 T (\lambda (t: -T).(subst0 i0 v (THead k u2 x) t)) (\lambda (t: T).(subst0 i0 v (THead k u1 -x) t))) (subst0 i0 v (THead k u2 x) (THead k u1 x)) (subst0 i0 v (THead k u1 -x) (THead k u2 x)) (subst0_fst v u2 u1 i0 H0 x k))) (\lambda (_: (subst0 i0 v -u2 u2)).(or4_intro3 (eq T (THead k u2 x) (THead k u1 x)) (ex2 T (\lambda (t: -T).(subst0 i0 v (THead k u2 x) t)) (\lambda (t: T).(subst0 i0 v (THead k u1 -x) t))) (subst0 i0 v (THead k u2 x) (THead k u1 x)) (subst0 i0 v (THead k u1 -x) (THead k u2 x)) (subst0_fst v u2 u1 i0 H0 x k))) (H1 u2 H0)) t3 H8)) -(\lambda (H8: (ex2 T (\lambda (t: T).(subst0 (s k i0) v t3 t)) (\lambda (t: -T).(subst0 (s k i0) v x t)))).(ex2_ind T (\lambda (t: T).(subst0 (s k i0) v -t3 t)) (\lambda (t: T).(subst0 (s k i0) v x t)) (or4 (eq T (THead k u2 t3) -(THead k u1 x)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) -(\lambda (t: T).(subst0 i0 v (THead k u1 x) t))) (subst0 i0 v (THead k u2 t3) -(THead k u1 x)) (subst0 i0 v (THead k u1 x) (THead k u2 t3))) (\lambda (x0: -T).(\lambda (H9: (subst0 (s k i0) v t3 x0)).(\lambda (H10: (subst0 (s k i0) v -x x0)).(or4_ind (eq T u2 u2) (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) -(\lambda (t: T).(subst0 i0 v u2 t))) (subst0 i0 v u2 u2) (subst0 i0 v u2 u2) -(or4 (eq T (THead k u2 t3) (THead k u1 x)) (ex2 T (\lambda (t: T).(subst0 i0 -v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k u1 x) t))) -(subst0 i0 v (THead k u2 t3) (THead k u1 x)) (subst0 i0 v (THead k u1 x) -(THead k u2 t3))) (\lambda (_: (eq T u2 u2)).(or4_intro1 (eq T (THead k u2 -t3) (THead k u1 x)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) -(\lambda (t: T).(subst0 i0 v (THead k u1 x) t))) (subst0 i0 v (THead k u2 t3) -(THead k u1 x)) (subst0 i0 v (THead k u1 x) (THead k u2 t3)) (ex_intro2 T -(\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v -(THead k u1 x) t)) (THead k u2 x0) (subst0_snd k v x0 t3 i0 H9 u2) -(subst0_both v u1 u2 i0 H0 k x x0 H10)))) (\lambda (H11: (ex2 T (\lambda (t: -T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0 i0 v u2 t)))).(ex2_ind T -(\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0 i0 v u2 t)) (or4 -(eq T (THead k u2 t3) (THead k u1 x)) (ex2 T (\lambda (t: T).(subst0 i0 v -(THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k u1 x) t))) (subst0 -i0 v (THead k u2 t3) (THead k u1 x)) (subst0 i0 v (THead k u1 x) (THead k u2 -t3))) (\lambda (x1: T).(\lambda (_: (subst0 i0 v u2 x1)).(\lambda (_: (subst0 -i0 v u2 x1)).(or4_intro1 (eq T (THead k u2 t3) (THead k u1 x)) (ex2 T -(\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v -(THead k u1 x) t))) (subst0 i0 v (THead k u2 t3) (THead k u1 x)) (subst0 i0 v -(THead k u1 x) (THead k u2 t3)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v -(THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k u1 x) t)) (THead k -u2 x0) (subst0_snd k v x0 t3 i0 H9 u2) (subst0_both v u1 u2 i0 H0 k x x0 -H10)))))) H11)) (\lambda (_: (subst0 i0 v u2 u2)).(or4_intro1 (eq T (THead k -u2 t3) (THead k u1 x)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) -t)) (\lambda (t: T).(subst0 i0 v (THead k u1 x) t))) (subst0 i0 v (THead k u2 -t3) (THead k u1 x)) (subst0 i0 v (THead k u1 x) (THead k u2 t3)) (ex_intro2 T -(\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v -(THead k u1 x) t)) (THead k u2 x0) (subst0_snd k v x0 t3 i0 H9 u2) -(subst0_both v u1 u2 i0 H0 k x x0 H10)))) (\lambda (_: (subst0 i0 v u2 -u2)).(or4_intro1 (eq T (THead k u2 t3) (THead k u1 x)) (ex2 T (\lambda (t: -T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k u1 -x) t))) (subst0 i0 v (THead k u2 t3) (THead k u1 x)) (subst0 i0 v (THead k u1 -x) (THead k u2 t3)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) -t)) (\lambda (t: T).(subst0 i0 v (THead k u1 x) t)) (THead k u2 x0) -(subst0_snd k v x0 t3 i0 H9 u2) (subst0_both v u1 u2 i0 H0 k x x0 H10)))) (H1 -u2 H0))))) H8)) (\lambda 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(\lambda (t: T).(subst0 i0 v u2 t)) -(\lambda (t: T).(subst0 i0 v u2 t)) (or4 (eq T (THead k u2 t3) (THead k u1 -x)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: -T).(subst0 i0 v (THead k u1 x) t))) (subst0 i0 v (THead k u2 t3) (THead k u1 -x)) (subst0 i0 v (THead k u1 x) (THead k u2 t3))) (\lambda (x0: T).(\lambda -(_: (subst0 i0 v u2 x0)).(\lambda (_: (subst0 i0 v u2 x0)).(or4_intro1 (eq T -(THead k u2 t3) (THead k u1 x)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k -u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k u1 x) t))) (subst0 i0 v -(THead k u2 t3) (THead k u1 x)) (subst0 i0 v (THead k u1 x) (THead k u2 t3)) -(ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: -T).(subst0 i0 v (THead k u1 x) t)) (THead k u2 x) (subst0_snd k v x t3 i0 H8 -u2) (subst0_fst v u2 u1 i0 H0 x k)))))) H9)) (\lambda (_: (subst0 i0 v u2 -u2)).(or4_intro1 (eq T (THead k u2 t3) (THead k u1 x)) (ex2 T (\lambda (t: -T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k u1 -x) t))) (subst0 i0 v (THead k u2 t3) (THead k u1 x)) (subst0 i0 v (THead k u1 -x) (THead k u2 t3)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) -t)) (\lambda (t: T).(subst0 i0 v (THead k u1 x) t)) (THead k u2 x) -(subst0_snd k v x t3 i0 H8 u2) (subst0_fst v u2 u1 i0 H0 x k)))) (\lambda (_: -(subst0 i0 v u2 u2)).(or4_intro1 (eq T (THead k u2 t3) (THead k u1 x)) (ex2 T -(\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v -(THead k u1 x) t))) (subst0 i0 v (THead k u2 t3) (THead k u1 x)) (subst0 i0 v -(THead k u1 x) (THead k u2 t3)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v -(THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k u1 x) t)) (THead k -u2 x) (subst0_snd k v x t3 i0 H8 u2) (subst0_fst v u2 u1 i0 H0 x k)))) (H1 u2 -H0))) (\lambda (H8: (subst0 (s k i0) v x t3)).(or4_ind (eq T u2 u2) (ex2 T -(\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0 i0 v u2 t))) -(subst0 i0 v u2 u2) (subst0 i0 v u2 u2) (or4 (eq T (THead k u2 t3) (THead k -u1 x)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: -T).(subst0 i0 v (THead k u1 x) t))) (subst0 i0 v (THead k u2 t3) (THead k u1 -x)) (subst0 i0 v (THead k u1 x) (THead k u2 t3))) (\lambda (_: (eq T u2 -u2)).(or4_intro3 (eq T (THead k u2 t3) (THead k u1 x)) (ex2 T (\lambda (t: -T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k u1 -x) t))) (subst0 i0 v (THead k u2 t3) (THead k u1 x)) (subst0 i0 v (THead k u1 -x) (THead k u2 t3)) (subst0_both v u1 u2 i0 H0 k x t3 H8))) (\lambda (H9: -(ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0 i0 v u2 -t)))).(ex2_ind T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0 -i0 v u2 t)) (or4 (eq T (THead k u2 t3) (THead k u1 x)) (ex2 T (\lambda (t: -T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k u1 -x) t))) (subst0 i0 v (THead k u2 t3) (THead k u1 x)) (subst0 i0 v (THead k u1 -x) (THead k u2 t3))) (\lambda (x0: T).(\lambda (_: (subst0 i0 v u2 -x0)).(\lambda (_: (subst0 i0 v u2 x0)).(or4_intro3 (eq T (THead k u2 t3) -(THead k u1 x)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) -(\lambda (t: T).(subst0 i0 v (THead k u1 x) t))) (subst0 i0 v (THead k u2 t3) -(THead k u1 x)) (subst0 i0 v (THead k u1 x) (THead k u2 t3)) (subst0_both v -u1 u2 i0 H0 k x t3 H8))))) H9)) (\lambda (_: (subst0 i0 v u2 u2)).(or4_intro3 -(eq T (THead k u2 t3) (THead k u1 x)) (ex2 T (\lambda (t: T).(subst0 i0 v -(THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k u1 x) t))) (subst0 -i0 v (THead k u2 t3) (THead k u1 x)) (subst0 i0 v (THead k u1 x) (THead k u2 -t3)) (subst0_both v u1 u2 i0 H0 k x t3 H8))) (\lambda (_: (subst0 i0 v u2 -u2)).(or4_intro3 (eq T (THead k u2 t3) (THead k u1 x)) (ex2 T (\lambda (t: -T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k u1 -x) t))) (subst0 i0 v (THead k u2 t3) (THead k u1 x)) (subst0 i0 v (THead k u1 -x) (THead k u2 t3)) (subst0_both v u1 u2 i0 H0 k x t3 H8))) (H1 u2 H0))) (H3 -x H7)) t4 H6)))) H5)) (\lambda (H5: (ex3_2 T T (\lambda (u3: T).(\lambda (t5: -T).(eq T t4 (THead k u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v -u1 u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s k i0) v t2 -t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t5: T).(eq T t4 (THead k u3 -t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v u1 u3))) (\lambda (_: -T).(\lambda (t5: T).(subst0 (s k i0) v t2 t5))) (or4 (eq T (THead k u2 t3) -t4) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: -T).(subst0 i0 v t4 t))) (subst0 i0 v (THead k u2 t3) t4) (subst0 i0 v t4 -(THead k u2 t3))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H6: (eq T t4 -(THead k x0 x1))).(\lambda (H7: (subst0 i0 v u1 x0)).(\lambda (H8: (subst0 (s -k i0) v t2 x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t: T).(or4 (eq T (THead -k u2 t3) t) (ex2 T (\lambda (t5: T).(subst0 i0 v (THead k u2 t3) t5)) -(\lambda (t5: T).(subst0 i0 v t t5))) (subst0 i0 v (THead k u2 t3) t) (subst0 -i0 v t (THead k u2 t3)))) (or4_ind (eq T t3 x1) (ex2 T (\lambda (t: -T).(subst0 (s k i0) v t3 t)) (\lambda (t: T).(subst0 (s k i0) v x1 t))) -(subst0 (s k i0) v t3 x1) (subst0 (s k i0) v x1 t3) (or4 (eq T (THead k u2 -t3) (THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) -(\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 -t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t3))) (\lambda -(H9: (eq T t3 x1)).(eq_ind_r T x1 (\lambda (t: T).(or4 (eq T (THead k u2 t) -(THead k x0 x1)) (ex2 T (\lambda (t5: T).(subst0 i0 v (THead k u2 t) t5)) -(\lambda (t5: T).(subst0 i0 v (THead k x0 x1) t5))) (subst0 i0 v (THead k u2 -t) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t)))) (or4_ind -(eq T u2 x0) (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: -T).(subst0 i0 v x0 t))) (subst0 i0 v u2 x0) (subst0 i0 v x0 u2) (or4 (eq T -(THead k u2 x1) (THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k -u2 x1) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v -(THead k u2 x1) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 -x1))) (\lambda (H10: (eq T u2 x0)).(eq_ind_r T x0 (\lambda (t: T).(or4 (eq T -(THead k t x1) (THead k x0 x1)) (ex2 T (\lambda (t5: T).(subst0 i0 v (THead k -t x1) t5)) (\lambda (t5: T).(subst0 i0 v (THead k x0 x1) t5))) (subst0 i0 v -(THead k t x1) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k t -x1)))) (or4_intro0 (eq T (THead k x0 x1) (THead k x0 x1)) (ex2 T (\lambda (t: -T).(subst0 i0 v (THead k x0 x1) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 -x1) t))) (subst0 i0 v (THead k x0 x1) (THead k x0 x1)) (subst0 i0 v (THead k -x0 x1) (THead k x0 x1)) (refl_equal T (THead k x0 x1))) u2 H10)) (\lambda -(H10: (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0 i0 v -x0 t)))).(ex2_ind T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: -T).(subst0 i0 v x0 t)) (or4 (eq T (THead k u2 x1) (THead k x0 x1)) (ex2 T -(\lambda (t: T).(subst0 i0 v (THead k u2 x1) t)) (\lambda (t: T).(subst0 i0 v -(THead k x0 x1) t))) (subst0 i0 v (THead k u2 x1) (THead k x0 x1)) (subst0 i0 -v (THead k x0 x1) (THead k u2 x1))) (\lambda (x: T).(\lambda (H11: (subst0 i0 -v u2 x)).(\lambda (H12: (subst0 i0 v x0 x)).(or4_intro1 (eq T (THead k u2 x1) -(THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 x1) t)) -(\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 -x1) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 x1)) (ex_intro2 -T (\lambda (t: T).(subst0 i0 v (THead k u2 x1) t)) (\lambda (t: T).(subst0 i0 -v (THead k x0 x1) t)) (THead k x x1) (subst0_fst v x u2 i0 H11 x1 k) -(subst0_fst v x x0 i0 H12 x1 k)))))) H10)) (\lambda (H10: (subst0 i0 v u2 -x0)).(or4_intro2 (eq T (THead k u2 x1) (THead k x0 x1)) (ex2 T (\lambda (t: -T).(subst0 i0 v (THead k u2 x1) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 -x1) t))) (subst0 i0 v (THead k u2 x1) (THead k x0 x1)) (subst0 i0 v (THead k -x0 x1) (THead k u2 x1)) (subst0_fst v x0 u2 i0 H10 x1 k))) (\lambda (H10: -(subst0 i0 v x0 u2)).(or4_intro3 (eq T (THead k u2 x1) (THead k x0 x1)) (ex2 -T (\lambda (t: T).(subst0 i0 v (THead k u2 x1) t)) (\lambda (t: T).(subst0 i0 -v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 x1) (THead k x0 x1)) (subst0 -i0 v (THead k x0 x1) (THead k u2 x1)) (subst0_fst v u2 x0 i0 H10 x1 k))) (H1 -x0 H7)) t3 H9)) (\lambda (H9: (ex2 T (\lambda (t: T).(subst0 (s k i0) v t3 -t)) (\lambda (t: T).(subst0 (s k i0) v x1 t)))).(ex2_ind T (\lambda (t: -T).(subst0 (s k i0) v t3 t)) (\lambda (t: T).(subst0 (s k i0) v x1 t)) (or4 -(eq T (THead k u2 t3) (THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v -(THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 -i0 v (THead k u2 t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k -u2 t3))) (\lambda (x: T).(\lambda (H10: (subst0 (s k i0) v t3 x)).(\lambda -(H11: (subst0 (s k i0) v x1 x)).(or4_ind (eq T u2 x0) (ex2 T (\lambda (t: -T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0 i0 v x0 t))) (subst0 i0 v u2 -x0) (subst0 i0 v x0 u2) (or4 (eq T (THead k u2 t3) (THead k x0 x1)) (ex2 T -(\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v -(THead k x0 x1) t))) (subst0 i0 v (THead k u2 t3) (THead k x0 x1)) (subst0 i0 -v (THead k x0 x1) (THead k u2 t3))) (\lambda (H12: (eq T u2 x0)).(eq_ind_r T -x0 (\lambda (t: T).(or4 (eq T (THead k t t3) (THead k x0 x1)) (ex2 T (\lambda -(t5: T).(subst0 i0 v (THead k t t3) t5)) (\lambda (t5: T).(subst0 i0 v (THead -k x0 x1) t5))) (subst0 i0 v (THead k t t3) (THead k x0 x1)) (subst0 i0 v -(THead k x0 x1) (THead k t t3)))) (or4_intro1 (eq T (THead k x0 t3) (THead k -x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k x0 t3) t)) (\lambda (t: -T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k x0 t3) (THead k x0 -x1)) (subst0 i0 v (THead k x0 x1) (THead k x0 t3)) (ex_intro2 T (\lambda (t: -T).(subst0 i0 v (THead k x0 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 -x1) t)) (THead k x0 x) (subst0_snd k v x t3 i0 H10 x0) (subst0_snd k v x x1 -i0 H11 x0))) u2 H12)) (\lambda (H12: (ex2 T (\lambda (t: T).(subst0 i0 v u2 -t)) (\lambda (t: T).(subst0 i0 v x0 t)))).(ex2_ind T (\lambda (t: T).(subst0 -i0 v u2 t)) (\lambda (t: T).(subst0 i0 v x0 t)) (or4 (eq T (THead k u2 t3) -(THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) -(\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 -t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t3))) (\lambda -(x2: T).(\lambda (H13: (subst0 i0 v u2 x2)).(\lambda (H14: (subst0 i0 v x0 -x2)).(or4_intro1 (eq T (THead k u2 t3) (THead k x0 x1)) (ex2 T (\lambda (t: -T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 -x1) t))) (subst0 i0 v (THead k u2 t3) (THead k x0 x1)) (subst0 i0 v (THead k -x0 x1) (THead k u2 t3)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k u2 -t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t)) (THead k x2 x) -(subst0_both v u2 x2 i0 H13 k t3 x H10) (subst0_both v x0 x2 i0 H14 k x1 x -H11)))))) H12)) (\lambda (H12: (subst0 i0 v u2 x0)).(or4_intro1 (eq T (THead -k u2 t3) (THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) -t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k -u2 t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t3)) -(ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: -T).(subst0 i0 v (THead k x0 x1) t)) (THead k x0 x) (subst0_both v u2 x0 i0 -H12 k t3 x H10) (subst0_snd k v x x1 i0 H11 x0)))) (\lambda (H12: (subst0 i0 -v x0 u2)).(or4_intro1 (eq T (THead k u2 t3) (THead k x0 x1)) (ex2 T (\lambda -(t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k -x0 x1) t))) (subst0 i0 v (THead k u2 t3) (THead k x0 x1)) (subst0 i0 v (THead -k x0 x1) (THead k u2 t3)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k -u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t)) (THead k u2 x) -(subst0_snd k v x t3 i0 H10 u2) (subst0_both v x0 u2 i0 H12 k x1 x H11)))) -(H1 x0 H7))))) H9)) (\lambda (H9: (subst0 (s k i0) v t3 x1)).(or4_ind (eq T -u2 x0) (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0 i0 -v x0 t))) (subst0 i0 v u2 x0) (subst0 i0 v x0 u2) (or4 (eq T (THead k u2 t3) -(THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) -(\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 -t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t3))) (\lambda -(H10: (eq T u2 x0)).(eq_ind_r T x0 (\lambda (t: T).(or4 (eq T (THead k t t3) -(THead k x0 x1)) (ex2 T (\lambda (t5: T).(subst0 i0 v (THead k t t3) t5)) -(\lambda (t5: T).(subst0 i0 v (THead k x0 x1) t5))) (subst0 i0 v (THead k t -t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k t t3)))) -(or4_intro2 (eq T (THead k x0 t3) (THead k x0 x1)) (ex2 T (\lambda (t: -T).(subst0 i0 v (THead k x0 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 -x1) t))) (subst0 i0 v (THead k x0 t3) (THead k x0 x1)) (subst0 i0 v (THead k -x0 x1) (THead k x0 t3)) (subst0_snd k v x1 t3 i0 H9 x0)) u2 H10)) (\lambda -(H10: (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0 i0 v -x0 t)))).(ex2_ind T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: -T).(subst0 i0 v x0 t)) (or4 (eq T (THead k u2 t3) (THead k x0 x1)) (ex2 T -(\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v -(THead k x0 x1) t))) (subst0 i0 v (THead k u2 t3) (THead k x0 x1)) (subst0 i0 -v (THead k x0 x1) (THead k u2 t3))) (\lambda (x: T).(\lambda (H11: (subst0 i0 -v u2 x)).(\lambda (H12: (subst0 i0 v x0 x)).(or4_intro1 (eq T (THead k u2 t3) -(THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) -(\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 -t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t3)) (ex_intro2 -T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 -v (THead k x0 x1) t)) (THead k x x1) (subst0_both v u2 x i0 H11 k t3 x1 H9) -(subst0_fst v x x0 i0 H12 x1 k)))))) H10)) (\lambda (H10: (subst0 i0 v u2 -x0)).(or4_intro2 (eq T (THead k u2 t3) (THead k x0 x1)) (ex2 T (\lambda (t: -T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 -x1) t))) (subst0 i0 v (THead k u2 t3) (THead k x0 x1)) (subst0 i0 v (THead k -x0 x1) (THead k u2 t3)) (subst0_both v u2 x0 i0 H10 k t3 x1 H9))) (\lambda -(H10: (subst0 i0 v x0 u2)).(or4_intro1 (eq T (THead k u2 t3) (THead k x0 x1)) -(ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: -T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 t3) (THead k x0 -x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t3)) (ex_intro2 T (\lambda (t: -T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 -x1) t)) (THead k u2 x1) (subst0_snd k v x1 t3 i0 H9 u2) (subst0_fst v u2 x0 -i0 H10 x1 k)))) (H1 x0 H7))) (\lambda (H9: (subst0 (s k i0) v x1 -t3)).(or4_ind (eq T u2 x0) (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) -(\lambda (t: T).(subst0 i0 v x0 t))) (subst0 i0 v u2 x0) (subst0 i0 v x0 u2) -(or4 (eq T (THead k u2 t3) (THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 -v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) -(subst0 i0 v (THead k u2 t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) -(THead k u2 t3))) (\lambda (H10: (eq T u2 x0)).(eq_ind_r T x0 (\lambda (t: -T).(or4 (eq T (THead k t t3) (THead k x0 x1)) (ex2 T (\lambda (t5: T).(subst0 -i0 v (THead k t t3) t5)) (\lambda (t5: T).(subst0 i0 v (THead k x0 x1) t5))) -(subst0 i0 v (THead k t t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) -(THead k t t3)))) (or4_intro3 (eq T (THead k x0 t3) (THead k x0 x1)) (ex2 T -(\lambda (t: T).(subst0 i0 v (THead k x0 t3) t)) (\lambda (t: T).(subst0 i0 v -(THead k x0 x1) t))) (subst0 i0 v (THead k x0 t3) (THead k x0 x1)) (subst0 i0 -v (THead k x0 x1) (THead k x0 t3)) (subst0_snd k v t3 x1 i0 H9 x0)) u2 H10)) -(\lambda (H10: (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: -T).(subst0 i0 v x0 t)))).(ex2_ind T (\lambda (t: T).(subst0 i0 v u2 t)) -(\lambda (t: T).(subst0 i0 v x0 t)) (or4 (eq T (THead k u2 t3) (THead k x0 -x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: -T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 t3) (THead k x0 -x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t3))) (\lambda (x: T).(\lambda -(H11: (subst0 i0 v u2 x)).(\lambda (H12: (subst0 i0 v x0 x)).(or4_intro1 (eq -T (THead k u2 t3) (THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead -k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v -(THead k u2 t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 -t3)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda -(t: T).(subst0 i0 v (THead k x0 x1) t)) (THead k x t3) (subst0_fst v x u2 i0 -H11 t3 k) (subst0_both v x0 x i0 H12 k x1 t3 H9)))))) H10)) (\lambda (H10: -(subst0 i0 v u2 x0)).(or4_intro1 (eq T (THead k u2 t3) (THead k x0 x1)) (ex2 -T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 -v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 t3) (THead k x0 x1)) (subst0 -i0 v (THead k x0 x1) (THead k u2 t3)) (ex_intro2 T (\lambda (t: T).(subst0 i0 -v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t)) (THead -k x0 t3) (subst0_fst v x0 u2 i0 H10 t3 k) (subst0_snd k v t3 x1 i0 H9 x0)))) -(\lambda (H10: (subst0 i0 v x0 u2)).(or4_intro3 (eq T (THead k u2 t3) (THead -k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda -(t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 t3) (THead -k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t3)) (subst0_both v x0 u2 -i0 H10 k x1 t3 H9))) (H1 x0 H7))) (H3 x1 H8)) t4 H6)))))) H5)) -(subst0_gen_head k v u1 t2 t4 i0 H4))))))))))))))) i u t0 t1 H))))). -(* COMMENTS -Initial nodes: 25595 -END *) - -theorem subst0_confluence_lift: - \forall (t0: T).(\forall (t1: T).(\forall (u: T).(\forall (i: nat).((subst0 -i u t0 (lift (S O) i t1)) \to (\forall (t2: T).((subst0 i u t0 (lift (S O) i -t2)) \to (eq T t1 t2))))))) -\def - \lambda (t0: T).(\lambda (t1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H: (subst0 i u t0 (lift (S O) i t1))).(\lambda (t2: T).(\lambda (H0: (subst0 -i u t0 (lift (S O) i t2))).(or4_ind (eq T (lift (S O) i t2) (lift (S O) i -t1)) (ex2 T (\lambda (t: T).(subst0 i u (lift (S O) i t2) t)) (\lambda (t: -T).(subst0 i u (lift (S O) i t1) t))) (subst0 i u (lift (S O) i t2) (lift (S -O) i t1)) (subst0 i u (lift (S O) i t1) (lift (S O) i t2)) (eq T t1 t2) -(\lambda (H1: (eq T (lift (S O) i t2) (lift (S O) i t1))).(let H2 \def -(sym_eq T (lift (S O) i t2) (lift (S O) i t1) H1) in (lift_inj t1 t2 (S O) i -H2))) (\lambda (H1: (ex2 T (\lambda (t: T).(subst0 i u (lift (S O) i t2) t)) -(\lambda (t: T).(subst0 i u (lift (S O) i t1) t)))).(ex2_ind T (\lambda (t: -T).(subst0 i u (lift (S O) i t2) t)) (\lambda (t: T).(subst0 i u (lift (S O) -i t1) t)) (eq T t1 t2) (\lambda (x: T).(\lambda (_: (subst0 i u (lift (S O) i -t2) x)).(\lambda (H3: (subst0 i u (lift (S O) i t1) -x)).(subst0_gen_lift_false t1 u x (S O) i i (le_n i) (eq_ind_r nat (plus (S -O) i) (\lambda (n: nat).(lt i n)) (le_n (plus (S O) i)) (plus i (S O)) -(plus_sym i (S O))) H3 (eq T t1 t2))))) H1)) (\lambda (H1: (subst0 i u (lift -(S O) i t2) (lift (S O) i t1))).(subst0_gen_lift_false t2 u (lift (S O) i t1) -(S O) i i (le_n i) (eq_ind_r nat (plus (S O) i) (\lambda (n: nat).(lt i n)) -(le_n (plus (S O) i)) (plus i (S O)) (plus_sym i (S O))) H1 (eq T t1 t2))) -(\lambda (H1: (subst0 i u (lift (S O) i t1) (lift (S O) i -t2))).(subst0_gen_lift_false t1 u (lift (S O) i t2) (S O) i i (le_n i) -(eq_ind_r nat (plus (S O) i) (\lambda (n: nat).(lt i n)) (le_n (plus (S O) -i)) (plus i (S O)) (plus_sym i (S O))) H1 (eq T t1 t2))) -(subst0_confluence_eq t0 (lift (S O) i t2) u i H0 (lift (S O) i t1) H)))))))). -(* COMMENTS -Initial nodes: 703 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/subst0/tlt.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/subst0/tlt.ma deleted file mode 100644 index c8e8420bb..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/subst0/tlt.ma +++ /dev/null @@ -1,468 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/subst0/defs.ma". - -include "Basic-1/lift/props.ma". - -include "Basic-1/lift/tlt.ma". - -theorem subst0_weight_le: - \forall (u: T).(\forall (t: T).(\forall (z: T).(\forall (d: nat).((subst0 d -u t z) \to (\forall (f: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S -d) O u)) (g d)) \to (le (weight_map f z) (weight_map g t)))))))))) -\def - \lambda (u: T).(\lambda (t: T).(\lambda (z: T).(\lambda (d: nat).(\lambda -(H: (subst0 d u t z)).(subst0_ind (\lambda (n: nat).(\lambda (t0: T).(\lambda -(t1: T).(\lambda (t2: T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S -n) O t0)) (g n)) \to (le (weight_map f t2) (weight_map g t1)))))))))) -(\lambda (v: T).(\lambda (i: nat).(\lambda (f: ((nat \to nat))).(\lambda (g: -((nat \to nat))).(\lambda (_: ((\forall (m: nat).(le (f m) (g m))))).(\lambda -(H1: (lt (weight_map f (lift (S i) O v)) (g i))).(le_S_n (weight_map f (lift -(S i) O v)) (weight_map g (TLRef i)) (le_S (S (weight_map f (lift (S i) O -v))) (weight_map g (TLRef i)) H1)))))))) (\lambda (v: T).(\lambda (u2: -T).(\lambda (u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1 -u2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S -i) O v)) (g i)) \to (le (weight_map f u2) (weight_map g u1)))))))).(\lambda -(t0: T).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (f: ((nat \to -nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) -\to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (le (weight_map f (THead -k0 u2 t0)) (weight_map g (THead k0 u1 t0)))))))) (\lambda (b: B).(B_ind -(\lambda (b0: B).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S -i) O v)) (g i)) \to (le (weight_map f (THead (Bind b0) u2 t0)) (weight_map g -(THead (Bind b0) u1 t0)))))))) (\lambda (f: ((nat \to nat))).(\lambda (g: -((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g -m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g i))).(le_n_S -(plus (weight_map f u2) (weight_map (wadd f (S (weight_map f u2))) t0)) (plus -(weight_map g u1) (weight_map (wadd g (S (weight_map g u1))) t0)) -(le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f (S -(weight_map f u2))) t0) (weight_map (wadd g (S (weight_map g u1))) t0) (H1 f -g H2 H3) (weight_le t0 (wadd f (S (weight_map f u2))) (wadd g (S (weight_map -g u1))) (\lambda (n: nat).(wadd_le f g H2 (S (weight_map f u2)) (S -(weight_map g u1)) (le_n_S (weight_map f u2) (weight_map g u1) (H1 f g H2 -H3)) n))))))))) (\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to -nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3: (lt -(weight_map f (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map f u2) -(weight_map (wadd f O) t0)) (plus (weight_map g u1) (weight_map (wadd g O) -t0)) (le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f O) -t0) (weight_map (wadd g O) t0) (H1 f g H2 H3) (weight_le t0 (wadd f O) (wadd -g O) (\lambda (n: nat).(wadd_le f g H2 O O (le_n O) n))))))))) (\lambda (f: -((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: -nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g -i))).(le_n_S (plus (weight_map f u2) (weight_map (wadd f O) t0)) (plus -(weight_map g u1) (weight_map (wadd g O) t0)) (le_plus_plus (weight_map f u2) -(weight_map g u1) (weight_map (wadd f O) t0) (weight_map (wadd g O) t0) (H1 f -g H2 H3) (weight_le t0 (wadd f O) (wadd g O) (\lambda (n: nat).(wadd_le f g -H2 O O (le_n O) n))))))))) b)) (\lambda (_: F).(\lambda (f0: ((nat \to -nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f0 -m) (g m))))).(\lambda (H3: (lt (weight_map f0 (lift (S i) O v)) (g -i))).(le_n_S (plus (weight_map f0 u2) (weight_map f0 t0)) (plus (weight_map g -u1) (weight_map g t0)) (le_plus_plus (weight_map f0 u2) (weight_map g u1) -(weight_map f0 t0) (weight_map g t0) (H1 f0 g H2 H3) (weight_le t0 f0 g -H2)))))))) k))))))))) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (v: -T).(\forall (t2: T).(\forall (t1: T).(\forall (i: nat).((subst0 (s k0 i) v t1 -t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S -(s k0 i)) O v)) (g (s k0 i))) \to (le (weight_map f t2) (weight_map g -t1))))))) \to (\forall (u0: T).(\forall (f: ((nat \to nat))).(\forall (g: -((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map -f (lift (S i) O v)) (g i)) \to (le (weight_map f (THead k0 u0 t2)) -(weight_map g (THead k0 u0 t1))))))))))))))) (\lambda (b: B).(B_ind (\lambda -(b0: B).(\forall (v: T).(\forall (t2: T).(\forall (t1: T).(\forall (i: -nat).((subst0 (s (Bind b0) i) v t1 t2) \to (((\forall (f: ((nat \to -nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) -\to ((lt (weight_map f (lift (S (s (Bind b0) i)) O v)) (g (s (Bind b0) i))) -\to (le (weight_map f t2) (weight_map g t1))))))) \to (\forall (u0: -T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: -nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to -(le (weight_map f (THead (Bind b0) u0 t2)) (weight_map g (THead (Bind b0) u0 -t1))))))))))))))) (\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda -(i: nat).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H1: ((\forall (f: -((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) -(g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (le -(weight_map f t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f: -((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: -nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g -i))).(le_n_S (plus (weight_map f u0) (weight_map (wadd f (S (weight_map f -u0))) t2)) (plus (weight_map g u0) (weight_map (wadd g (S (weight_map g u0))) -t1)) (le_plus_plus (weight_map f u0) (weight_map g u0) (weight_map (wadd f (S -(weight_map f u0))) t2) (weight_map (wadd g (S (weight_map g u0))) t1) -(weight_le u0 f g H2) (H1 (wadd f (S (weight_map f u0))) (wadd g (S -(weight_map g u0))) (\lambda (m: nat).(wadd_le f g H2 (S (weight_map f u0)) -(S (weight_map g u0)) (le_n_S (weight_map f u0) (weight_map g u0) (weight_le -u0 f g H2)) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda (n: -nat).(lt n (g i))) H3 (weight_map (wadd f (S (weight_map f u0))) (lift (S (S -i)) O v)) (lift_weight_add_O (S (weight_map f u0)) v (S i) f)))))))))))))))) -(\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda (i: nat).(\lambda -(_: (subst0 (S i) v t1 t2)).(\lambda (H1: ((\forall (f: ((nat \to -nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) -\to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (le (weight_map f -t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f: ((nat \to -nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f -m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g -i))).(le_n_S (plus (weight_map f u0) (weight_map (wadd f O) t2)) (plus -(weight_map g u0) (weight_map (wadd g O) t1)) (le_plus_plus (weight_map f u0) -(weight_map g u0) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1) -(weight_le u0 f g H2) (H1 (wadd f O) (wadd g O) (\lambda (m: nat).(wadd_le f -g H2 O O (le_n O) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda -(n: nat).(lt n (g i))) H3 (weight_map (wadd f O) (lift (S (S i)) O v)) -(lift_weight_add_O O v (S i) f)))))))))))))))) (\lambda (v: T).(\lambda (t2: -T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (_: (subst0 (S i) v t1 -t2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S -(S i)) O v)) (g (S i))) \to (le (weight_map f t2) (weight_map g -t1)))))))).(\lambda (u0: T).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat -\to nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3: -(lt (weight_map f (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map f u0) -(weight_map (wadd f O) t2)) (plus (weight_map g u0) (weight_map (wadd g O) -t1)) (le_plus_plus (weight_map f u0) (weight_map g u0) (weight_map (wadd f O) -t2) (weight_map (wadd g O) t1) (weight_le u0 f g H2) (H1 (wadd f O) (wadd g -O) (\lambda (m: nat).(wadd_le f g H2 O O (le_n O) m)) (eq_ind nat (weight_map -f (lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H3 (weight_map (wadd f O) -(lift (S (S i)) O v)) (lift_weight_add_O O v (S i) f)))))))))))))))) b)) -(\lambda (_: F).(\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda -(i: nat).(\lambda (_: (subst0 i v t1 t2)).(\lambda (H1: ((\forall (f0: ((nat -\to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f0 m) (g -m)))) \to ((lt (weight_map f0 (lift (S i) O v)) (g i)) \to (le (weight_map f0 -t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f0: ((nat \to -nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f0 -m) (g m))))).(\lambda (H3: (lt (weight_map f0 (lift (S i) O v)) (g -i))).(le_n_S (plus (weight_map f0 u0) (weight_map f0 t2)) (plus (weight_map g -u0) (weight_map g t1)) (le_plus_plus (weight_map f0 u0) (weight_map g u0) -(weight_map f0 t2) (weight_map g t1) (weight_le u0 f0 g H2) (H1 f0 g H2 -H3))))))))))))))) k)) (\lambda (v: T).(\lambda (u1: T).(\lambda (u2: -T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1 u2)).(\lambda (H1: ((\forall -(f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f -m) (g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (le -(weight_map f u2) (weight_map g u1)))))))).(\lambda (k: K).(K_ind (\lambda -(k0: K).(\forall (t1: T).(\forall (t2: T).((subst0 (s k0 i) v t1 t2) \to -(((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: -nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S (s k0 i)) O v)) (g (s -k0 i))) \to (le (weight_map f t2) (weight_map g t1))))))) \to (\forall (f: -((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) -(g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (le (weight_map -f (THead k0 u2 t2)) (weight_map g (THead k0 u1 t1)))))))))))) (\lambda (b: -B).(B_ind (\lambda (b0: B).(\forall (t1: T).(\forall (t2: T).((subst0 (s -(Bind b0) i) v t1 t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat -\to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f -(lift (S (s (Bind b0) i)) O v)) (g (s (Bind b0) i))) \to (le (weight_map f -t2) (weight_map g t1))))))) \to (\forall (f: ((nat \to nat))).(\forall (g: -((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map -f (lift (S i) O v)) (g i)) \to (le (weight_map f (THead (Bind b0) u2 t2)) -(weight_map g (THead (Bind b0) u1 t1)))))))))))) (\lambda (t1: T).(\lambda -(t2: T).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H3: ((\forall (f: -((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) -(g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (le -(weight_map f t2) (weight_map g t1)))))))).(\lambda (f: ((nat \to -nat))).(\lambda (g: ((nat \to nat))).(\lambda (H4: ((\forall (m: nat).(le (f -m) (g m))))).(\lambda (H5: (lt (weight_map f (lift (S i) O v)) (g -i))).(le_n_S (plus (weight_map f u2) (weight_map (wadd f (S (weight_map f -u2))) t2)) (plus (weight_map g u1) (weight_map (wadd g (S (weight_map g u1))) -t1)) (le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f (S -(weight_map f u2))) t2) (weight_map (wadd g (S (weight_map g u1))) t1) (H1 f -g H4 H5) (H3 (wadd f (S (weight_map f u2))) (wadd g (S (weight_map g u1))) -(\lambda (m: nat).(wadd_le f g H4 (S (weight_map f u2)) (S (weight_map g u1)) -(le_n_S (weight_map f u2) (weight_map g u1) (H1 f g H4 H5)) m)) (eq_ind nat -(weight_map f (lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H5 -(weight_map (wadd f (S (weight_map f u2))) (lift (S (S i)) O v)) -(lift_weight_add_O (S (weight_map f u2)) v (S i) f))))))))))))) (\lambda (t1: -T).(\lambda (t2: T).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H3: -((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: -nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S -i))) \to (le (weight_map f t2) (weight_map g t1)))))))).(\lambda (f: ((nat -\to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H4: ((\forall (m: nat).(le -(f m) (g m))))).(\lambda (H5: (lt (weight_map f (lift (S i) O v)) (g -i))).(le_n_S (plus (weight_map f u2) (weight_map (wadd f O) t2)) (plus -(weight_map g u1) (weight_map (wadd g O) t1)) (le_plus_plus (weight_map f u2) -(weight_map g u1) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1) (H1 f -g H4 H5) (H3 (wadd f O) (wadd g O) (\lambda (m: nat).(wadd_le f g H4 O O -(le_n O) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda (n: -nat).(lt n (g i))) H5 (weight_map (wadd f O) (lift (S (S i)) O v)) -(lift_weight_add_O O v (S i) f))))))))))))) (\lambda (t1: T).(\lambda (t2: -T).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H3: ((\forall (f: ((nat \to -nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) -\to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (le (weight_map f -t2) (weight_map g t1)))))))).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat -\to nat))).(\lambda (H4: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H5: -(lt (weight_map f (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map f u2) -(weight_map (wadd f O) t2)) (plus (weight_map g u1) (weight_map (wadd g O) -t1)) (le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f O) -t2) (weight_map (wadd g O) t1) (H1 f g H4 H5) (H3 (wadd f O) (wadd g O) -(\lambda (m: nat).(wadd_le f g H4 O O (le_n O) m)) (eq_ind nat (weight_map f -(lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H5 (weight_map (wadd f O) -(lift (S (S i)) O v)) (lift_weight_add_O O v (S i) f))))))))))))) b)) -(\lambda (_: F).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (subst0 i v t1 -t2)).(\lambda (H3: ((\forall (f0: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (m: nat).(le (f0 m) (g m)))) \to ((lt (weight_map f0 (lift -(S i) O v)) (g i)) \to (le (weight_map f0 t2) (weight_map g -t1)))))))).(\lambda (f0: ((nat \to nat))).(\lambda (g: ((nat \to -nat))).(\lambda (H4: ((\forall (m: nat).(le (f0 m) (g m))))).(\lambda (H5: -(lt (weight_map f0 (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map f0 u2) -(weight_map f0 t2)) (plus (weight_map g u1) (weight_map g t1)) (le_plus_plus -(weight_map f0 u2) (weight_map g u1) (weight_map f0 t2) (weight_map g t1) (H1 -f0 g H4 H5) (H3 f0 g H4 H5)))))))))))) k)))))))) d u t z H))))). -(* COMMENTS -Initial nodes: 4101 -END *) - -theorem subst0_weight_lt: - \forall (u: T).(\forall (t: T).(\forall (z: T).(\forall (d: nat).((subst0 d -u t z) \to (\forall (f: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S -d) O u)) (g d)) \to (lt (weight_map f z) (weight_map g t)))))))))) -\def - \lambda (u: T).(\lambda (t: T).(\lambda (z: T).(\lambda (d: nat).(\lambda -(H: (subst0 d u t z)).(subst0_ind (\lambda (n: nat).(\lambda (t0: T).(\lambda -(t1: T).(\lambda (t2: T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S -n) O t0)) (g n)) \to (lt (weight_map f t2) (weight_map g t1)))))))))) -(\lambda (v: T).(\lambda (i: nat).(\lambda (f: ((nat \to nat))).(\lambda (g: -((nat \to nat))).(\lambda (_: ((\forall (m: nat).(le (f m) (g m))))).(\lambda -(H1: (lt (weight_map f (lift (S i) O v)) (g i))).H1)))))) (\lambda (v: -T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i -v u1 u2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S -i) O v)) (g i)) \to (lt (weight_map f u2) (weight_map g u1)))))))).(\lambda -(t0: T).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (f: ((nat \to -nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) -\to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (lt (weight_map f (THead -k0 u2 t0)) (weight_map g (THead k0 u1 t0)))))))) (\lambda (b: B).(B_ind -(\lambda (b0: B).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S -i) O v)) (g i)) \to (lt (weight_map f (THead (Bind b0) u2 t0)) (weight_map g -(THead (Bind b0) u1 t0)))))))) (\lambda (f: ((nat \to nat))).(\lambda (g: -((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g -m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g i))).(lt_n_S -(plus (weight_map f u2) (weight_map (wadd f (S (weight_map f u2))) t0)) (plus -(weight_map g u1) (weight_map (wadd g (S (weight_map g u1))) t0)) -(lt_le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f (S -(weight_map f u2))) t0) (weight_map (wadd g (S (weight_map g u1))) t0) (H1 f -g H2 H3) (weight_le t0 (wadd f (S (weight_map f u2))) (wadd g (S (weight_map -g u1))) (\lambda (n: nat).(wadd_lt f g H2 (S (weight_map f u2)) (S -(weight_map g u1)) (lt_n_S (weight_map f u2) (weight_map g u1) (H1 f g H2 -H3)) n))))))))) (\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to -nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3: (lt -(weight_map f (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map f u2) -(weight_map (wadd f O) t0)) (plus (weight_map g u1) (weight_map (wadd g O) -t0)) (lt_le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f -O) t0) (weight_map (wadd g O) t0) (H1 f g H2 H3) (weight_le t0 (wadd f O) -(wadd g O) (\lambda (n: nat).(le_S_n (wadd f O n) (wadd g O n) (le_n_S (wadd -f O n) (wadd g O n) (wadd_le f g H2 O O (le_n O) n))))))))))) (\lambda (f: -((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: -nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g -i))).(lt_n_S (plus (weight_map f u2) (weight_map (wadd f O) t0)) (plus -(weight_map g u1) (weight_map (wadd g O) t0)) (lt_le_plus_plus (weight_map f -u2) (weight_map g u1) (weight_map (wadd f O) t0) (weight_map (wadd g O) t0) -(H1 f g H2 H3) (weight_le t0 (wadd f O) (wadd g O) (\lambda (n: nat).(le_S_n -(wadd f O n) (wadd g O n) (le_n_S (wadd f O n) (wadd g O n) (wadd_le f g H2 O -O (le_n O) n))))))))))) b)) (\lambda (_: F).(\lambda (f0: ((nat \to -nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f0 -m) (g m))))).(\lambda (H3: (lt (weight_map f0 (lift (S i) O v)) (g -i))).(lt_n_S (plus (weight_map f0 u2) (weight_map f0 t0)) (plus (weight_map g -u1) (weight_map g t0)) (lt_le_plus_plus (weight_map f0 u2) (weight_map g u1) -(weight_map f0 t0) (weight_map g t0) (H1 f0 g H2 H3) (weight_le t0 f0 g -H2)))))))) k))))))))) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (v: -T).(\forall (t2: T).(\forall (t1: T).(\forall (i: nat).((subst0 (s k0 i) v t1 -t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S -(s k0 i)) O v)) (g (s k0 i))) \to (lt (weight_map f t2) (weight_map g -t1))))))) \to (\forall (u0: T).(\forall (f: ((nat \to nat))).(\forall (g: -((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map -f (lift (S i) O v)) (g i)) \to (lt (weight_map f (THead k0 u0 t2)) -(weight_map g (THead k0 u0 t1))))))))))))))) (\lambda (b: B).(B_ind (\lambda -(b0: B).(\forall (v: T).(\forall (t2: T).(\forall (t1: T).(\forall (i: -nat).((subst0 (s (Bind b0) i) v t1 t2) \to (((\forall (f: ((nat \to -nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) -\to ((lt (weight_map f (lift (S (s (Bind b0) i)) O v)) (g (s (Bind b0) i))) -\to (lt (weight_map f t2) (weight_map g t1))))))) \to (\forall (u0: -T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: -nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to -(lt (weight_map f (THead (Bind b0) u0 t2)) (weight_map g (THead (Bind b0) u0 -t1))))))))))))))) (\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda -(i: nat).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H1: ((\forall (f: -((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) -(g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (lt -(weight_map f t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f: -((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: -nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g -i))).(lt_n_S (plus (weight_map f u0) (weight_map (wadd f (S (weight_map f -u0))) t2)) (plus (weight_map g u0) (weight_map (wadd g (S (weight_map g u0))) -t1)) (le_lt_plus_plus (weight_map f u0) (weight_map g u0) (weight_map (wadd f -(S (weight_map f u0))) t2) (weight_map (wadd g (S (weight_map g u0))) t1) -(weight_le u0 f g H2) (H1 (wadd f (S (weight_map f u0))) (wadd g (S -(weight_map g u0))) (\lambda (m: nat).(wadd_le f g H2 (S (weight_map f u0)) -(S (weight_map g u0)) (le_n_S (weight_map f u0) (weight_map g u0) (weight_le -u0 f g H2)) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda (n: -nat).(lt n (g i))) H3 (weight_map (wadd f (S (weight_map f u0))) (lift (S (S -i)) O v)) (lift_weight_add_O (S (weight_map f u0)) v (S i) f)))))))))))))))) -(\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda (i: nat).(\lambda -(_: (subst0 (S i) v t1 t2)).(\lambda (H1: ((\forall (f: ((nat \to -nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) -\to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (lt (weight_map f -t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f: ((nat \to -nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f -m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g -i))).(lt_n_S (plus (weight_map f u0) (weight_map (wadd f O) t2)) (plus -(weight_map g u0) (weight_map (wadd g O) t1)) (le_lt_plus_plus (weight_map f -u0) (weight_map g u0) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1) -(weight_le u0 f g H2) (H1 (wadd f O) (wadd g O) (\lambda (m: nat).(wadd_le f -g H2 O O (le_n O) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda -(n: nat).(lt n (g i))) H3 (weight_map (wadd f O) (lift (S (S i)) O v)) -(lift_weight_add_O O v (S i) f)))))))))))))))) (\lambda (v: T).(\lambda (t2: -T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (_: (subst0 (S i) v t1 -t2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S -(S i)) O v)) (g (S i))) \to (lt (weight_map f t2) (weight_map g -t1)))))))).(\lambda (u0: T).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat -\to nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3: -(lt (weight_map f (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map f u0) -(weight_map (wadd f O) t2)) (plus (weight_map g u0) (weight_map (wadd g O) -t1)) (le_lt_plus_plus (weight_map f u0) (weight_map g u0) (weight_map (wadd f -O) t2) (weight_map (wadd g O) t1) (weight_le u0 f g H2) (H1 (wadd f O) (wadd -g O) (\lambda (m: nat).(wadd_le f g H2 O O (le_n O) m)) (eq_ind nat -(weight_map f (lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H3 -(weight_map (wadd f O) (lift (S (S i)) O v)) (lift_weight_add_O O v (S i) -f)))))))))))))))) b)) (\lambda (_: F).(\lambda (v: T).(\lambda (t2: -T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v t1 -t2)).(\lambda (H1: ((\forall (f0: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (m: nat).(le (f0 m) (g m)))) \to ((lt (weight_map f0 (lift -(S i) O v)) (g i)) \to (lt (weight_map f0 t2) (weight_map g -t1)))))))).(\lambda (u0: T).(\lambda (f0: ((nat \to nat))).(\lambda (g: ((nat -\to nat))).(\lambda (H2: ((\forall (m: nat).(le (f0 m) (g m))))).(\lambda -(H3: (lt (weight_map f0 (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map -f0 u0) (weight_map f0 t2)) (plus (weight_map g u0) (weight_map g t1)) -(le_lt_plus_plus (weight_map f0 u0) (weight_map g u0) (weight_map f0 t2) -(weight_map g t1) (weight_le u0 f0 g H2) (H1 f0 g H2 H3))))))))))))))) k)) -(\lambda (v: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i: nat).(\lambda -(_: (subst0 i v u1 u2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall -(g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt -(weight_map f (lift (S i) O v)) (g i)) \to (lt (weight_map f u2) (weight_map -g u1)))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t1: -T).(\forall (t2: T).((subst0 (s k0 i) v t1 t2) \to (((\forall (f: ((nat \to -nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) -\to ((lt (weight_map f (lift (S (s k0 i)) O v)) (g (s k0 i))) \to (lt -(weight_map f t2) (weight_map g t1))))))) \to (\forall (f: ((nat \to -nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) -\to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (lt (weight_map f (THead -k0 u2 t2)) (weight_map g (THead k0 u1 t1)))))))))))) (\lambda (b: B).(B_ind -(\lambda (b0: B).(\forall (t1: T).(\forall (t2: T).((subst0 (s (Bind b0) i) v -t1 t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S -(s (Bind b0) i)) O v)) (g (s (Bind b0) i))) \to (lt (weight_map f t2) -(weight_map g t1))))))) \to (\forall (f: ((nat \to nat))).(\forall (g: ((nat -\to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f -(lift (S i) O v)) (g i)) \to (lt (weight_map f (THead (Bind b0) u2 t2)) -(weight_map g (THead (Bind b0) u1 t1)))))))))))) (\lambda (t1: T).(\lambda -(t2: T).(\lambda (H2: (subst0 (S i) v t1 t2)).(\lambda (_: ((\forall (f: -((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) -(g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (lt -(weight_map f t2) (weight_map g t1)))))))).(\lambda (f: ((nat \to -nat))).(\lambda (g: ((nat \to nat))).(\lambda (H4: ((\forall (m: nat).(le (f -m) (g m))))).(\lambda (H5: (lt (weight_map f (lift (S i) O v)) (g -i))).(lt_n_S (plus (weight_map f u2) (weight_map (wadd f (S (weight_map f -u2))) t2)) (plus (weight_map g u1) (weight_map (wadd g (S (weight_map g u1))) -t1)) (lt_le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f -(S (weight_map f u2))) t2) (weight_map (wadd g (S (weight_map g u1))) t1) (H1 -f g H4 H5) (subst0_weight_le v t1 t2 (S i) H2 (wadd f (S (weight_map f u2))) -(wadd g (S (weight_map g u1))) (\lambda (m: nat).(wadd_lt f g H4 (S -(weight_map f u2)) (S (weight_map g u1)) (lt_n_S (weight_map f u2) -(weight_map g u1) (H1 f g H4 H5)) m)) (eq_ind nat (weight_map f (lift (S i) O -v)) (\lambda (n: nat).(lt n (g i))) H5 (weight_map (wadd f (S (weight_map f -u2))) (lift (S (S i)) O v)) (lift_weight_add_O (S (weight_map f u2)) v (S i) -f))))))))))))) (\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (subst0 (S i) v -t1 t2)).(\lambda (H3: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S -(S i)) O v)) (g (S i))) \to (lt (weight_map f t2) (weight_map g -t1)))))))).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to -nat))).(\lambda (H4: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H5: (lt -(weight_map f (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map f u2) -(weight_map (wadd f O) t2)) (plus (weight_map g u1) (weight_map (wadd g O) -t1)) (lt_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f O) -t2) (weight_map (wadd g O) t1) (H1 f g H4 H5) (H3 (wadd f O) (wadd g O) -(\lambda (m: nat).(le_S_n (wadd f O m) (wadd g O m) (le_n_S (wadd f O m) -(wadd g O m) (wadd_le f g H4 O O (le_n O) m)))) (eq_ind nat (weight_map f -(lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H5 (weight_map (wadd f O) -(lift (S (S i)) O v)) (lift_weight_add_O O v (S i) f))))))))))))) (\lambda -(t1: T).(\lambda (t2: T).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H3: -((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: -nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S -i))) \to (lt (weight_map f t2) (weight_map g t1)))))))).(\lambda (f: ((nat -\to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H4: ((\forall (m: nat).(le -(f m) (g m))))).(\lambda (H5: (lt (weight_map f (lift (S i) O v)) (g -i))).(lt_n_S (plus (weight_map f u2) (weight_map (wadd f O) t2)) (plus -(weight_map g u1) (weight_map (wadd g O) t1)) (lt_plus_plus (weight_map f u2) -(weight_map g u1) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1) (H1 f -g H4 H5) (H3 (wadd f O) (wadd g O) (\lambda (m: nat).(le_S_n (wadd f O m) -(wadd g O m) (le_n_S (wadd f O m) (wadd g O m) (wadd_le f g H4 O O (le_n O) -m)))) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda (n: nat).(lt n (g -i))) H5 (weight_map (wadd f O) (lift (S (S i)) O v)) (lift_weight_add_O O v -(S i) f))))))))))))) b)) (\lambda (_: F).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (_: (subst0 i v t1 t2)).(\lambda (H3: ((\forall (f0: ((nat \to -nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f0 m) (g m)))) -\to ((lt (weight_map f0 (lift (S i) O v)) (g i)) \to (lt (weight_map f0 t2) -(weight_map g t1)))))))).(\lambda (f0: ((nat \to nat))).(\lambda (g: ((nat -\to nat))).(\lambda (H4: ((\forall (m: nat).(le (f0 m) (g m))))).(\lambda -(H5: (lt (weight_map f0 (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map -f0 u2) (weight_map f0 t2)) (plus (weight_map g u1) (weight_map g t1)) -(lt_plus_plus (weight_map f0 u2) (weight_map g u1) (weight_map f0 t2) -(weight_map g t1) (H1 f0 g H4 H5) (H3 f0 g H4 H5)))))))))))) k)))))))) d u t -z H))))). -(* COMMENTS -Initial nodes: 4207 -END *) - -theorem subst0_tlt_head: - \forall (u: T).(\forall (t: T).(\forall (z: T).((subst0 O u t z) \to (tlt -(THead (Bind Abbr) u z) (THead (Bind Abbr) u t))))) -\def - \lambda (u: T).(\lambda (t: T).(\lambda (z: T).(\lambda (H: (subst0 O u t -z)).(lt_n_S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd -(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) z)) (plus -(weight_map (\lambda (_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) (S -(weight_map (\lambda (_: nat).O) u))) t)) (le_lt_plus_plus (weight_map -(\lambda (_: nat).O) u) (weight_map (\lambda (_: nat).O) u) (weight_map (wadd -(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) z) (weight_map -(wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t) (le_n -(weight_map (\lambda (_: nat).O) u)) (subst0_weight_lt u t z O H (wadd -(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) (wadd (\lambda -(_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) (\lambda (m: nat).(le_n -(wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u)) m))) -(eq_ind nat (weight_map (\lambda (_: nat).O) (lift O O u)) (\lambda (n: -nat).(lt n (S (weight_map (\lambda (_: nat).O) u)))) (eq_ind_r T u (\lambda -(t0: T).(lt (weight_map (\lambda (_: nat).O) t0) (S (weight_map (\lambda (_: -nat).O) u)))) (le_n (S (weight_map (\lambda (_: nat).O) u))) (lift O O u) -(lift_r u O)) (weight_map (wadd (\lambda (_: nat).O) (S (weight_map (\lambda -(_: nat).O) u))) (lift (S O) O u)) (lift_weight_add_O (S (weight_map (\lambda -(_: nat).O) u)) u O (\lambda (_: nat).O))))))))). -(* COMMENTS -Initial nodes: 347 -END *) - -theorem subst0_tlt: - \forall (u: T).(\forall (t: T).(\forall (z: T).((subst0 O u t z) \to (tlt z -(THead (Bind Abbr) u t))))) -\def - \lambda (u: T).(\lambda (t: T).(\lambda (z: T).(\lambda (H: (subst0 O u t -z)).(tlt_trans (THead (Bind Abbr) u z) z (THead (Bind Abbr) u t) (tlt_head_dx -(Bind Abbr) u z) (subst0_tlt_head u t z H))))). -(* COMMENTS -Initial nodes: 59 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/subst1/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/subst1/defs.ma deleted file mode 100644 index 6a51bcfb5..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/subst1/defs.ma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/subst0/defs.ma". - -inductive subst1 (i: nat) (v: T) (t1: T): T \to Prop \def -| subst1_refl: subst1 i v t1 t1 -| subst1_single: \forall (t2: T).((subst0 i v t1 t2) \to (subst1 i v t1 t2)). - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/subst1/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/subst1/fwd.ma deleted file mode 100644 index a2bc1edd6..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/subst1/fwd.ma +++ /dev/null @@ -1,182 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/subst1/defs.ma". - -include "Basic-1/subst0/props.ma". - -theorem subst1_gen_sort: - \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst1 -i v (TSort n) x) \to (eq T x (TSort n)))))) -\def - \lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda -(H: (subst1 i v (TSort n) x)).(subst1_ind i v (TSort n) (\lambda (t: T).(eq T -t (TSort n))) (refl_equal T (TSort n)) (\lambda (t2: T).(\lambda (H0: (subst0 -i v (TSort n) t2)).(subst0_gen_sort v t2 i n H0 (eq T t2 (TSort n))))) x -H))))). -(* COMMENTS -Initial nodes: 89 -END *) - -theorem subst1_gen_lref: - \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst1 -i v (TLRef n) x) \to (or (eq T x (TLRef n)) (land (eq nat n i) (eq T x (lift -(S n) O v)))))))) -\def - \lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda -(H: (subst1 i v (TLRef n) x)).(subst1_ind i v (TLRef n) (\lambda (t: T).(or -(eq T t (TLRef n)) (land (eq nat n i) (eq T t (lift (S n) O v))))) (or_introl -(eq T (TLRef n) (TLRef n)) (land (eq nat n i) (eq T (TLRef n) (lift (S n) O -v))) (refl_equal T (TLRef n))) (\lambda (t2: T).(\lambda (H0: (subst0 i v -(TLRef n) t2)).(land_ind (eq nat n i) (eq T t2 (lift (S n) O v)) (or (eq T t2 -(TLRef n)) (land (eq nat n i) (eq T t2 (lift (S n) O v)))) (\lambda (H1: (eq -nat n i)).(\lambda (H2: (eq T t2 (lift (S n) O v))).(or_intror (eq T t2 -(TLRef n)) (land (eq nat n i) (eq T t2 (lift (S n) O v))) (conj (eq nat n i) -(eq T t2 (lift (S n) O v)) H1 H2)))) (subst0_gen_lref v t2 i n H0)))) x -H))))). -(* COMMENTS -Initial nodes: 305 -END *) - -theorem subst1_gen_head: - \forall (k: K).(\forall (v: T).(\forall (u1: T).(\forall (t1: T).(\forall -(x: T).(\forall (i: nat).((subst1 i v (THead k u1 t1) x) \to (ex3_2 T T -(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (t2: -T).(subst1 (s k i) v t1 t2)))))))))) -\def - \lambda (k: K).(\lambda (v: T).(\lambda (u1: T).(\lambda (t1: T).(\lambda -(x: T).(\lambda (i: nat).(\lambda (H: (subst1 i v (THead k u1 t1) -x)).(subst1_ind i v (THead k u1 t1) (\lambda (t: T).(ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T t (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (t2: T).(subst1 (s k i) v t1 -t2))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead k u1 -t1) (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2))) -(\lambda (_: T).(\lambda (t2: T).(subst1 (s k i) v t1 t2))) u1 t1 (refl_equal -T (THead k u1 t1)) (subst1_refl i v u1) (subst1_refl (s k i) v t1)) (\lambda -(t2: T).(\lambda (H0: (subst0 i v (THead k u1 t1) t2)).(or3_ind (ex2 T -(\lambda (u2: T).(eq T t2 (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 -u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: -T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: -T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v -u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3)))) (ex3_2 -T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) (\lambda -(u2: T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (t3: -T).(subst1 (s k i) v t1 t3)))) (\lambda (H1: (ex2 T (\lambda (u2: T).(eq T t2 -(THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2)))).(ex2_ind T (\lambda -(u2: T).(eq T t2 (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2)) -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(subst1 (s k i) v t1 t3)))) (\lambda (x0: T).(\lambda -(H2: (eq T t2 (THead k x0 t1))).(\lambda (H3: (subst0 i v u1 -x0)).(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(subst1 (s k i) v t1 t3))) x0 t1 H2 (subst1_single i v u1 -x0 H3) (subst1_refl (s k i) v t1))))) H1)) (\lambda (H1: (ex2 T (\lambda (t3: -T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 -t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: -T).(subst0 (s k i) v t1 t3)) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq -T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2))) -(\lambda (_: T).(\lambda (t3: T).(subst1 (s k i) v t1 t3)))) (\lambda (x0: -T).(\lambda (H2: (eq T t2 (THead k u1 x0))).(\lambda (H3: (subst0 (s k i) v -t1 x0)).(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(subst1 (s k i) v t1 t3))) u1 x0 H2 (subst1_refl i v u1) -(subst1_single (s k i) v t1 x0 H3))))) H1)) (\lambda (H1: (ex3_2 T T (\lambda -(u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t3: -T).(subst0 (s k i) v t1 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: -T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v -u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T -T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (t3: -T).(subst1 (s k i) v t1 t3)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda -(H2: (eq T t2 (THead k x0 x1))).(\lambda (H3: (subst0 i v u1 x0)).(\lambda -(H4: (subst0 (s k i) v t1 x1)).(ex3_2_intro T T (\lambda (u2: T).(\lambda -(t3: T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1 -i v u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst1 (s k i) v t1 t3))) x0 -x1 H2 (subst1_single i v u1 x0 H3) (subst1_single (s k i) v t1 x1 H4))))))) -H1)) (subst0_gen_head k v u1 t1 t2 i H0)))) x H))))))). -(* COMMENTS -Initial nodes: 1199 -END *) - -theorem subst1_gen_lift_lt: - \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall -(h: nat).(\forall (d: nat).((subst1 i (lift h d u) (lift h (S (plus i d)) t1) -x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda -(t2: T).(subst1 i u t1 t2))))))))) -\def - \lambda (u: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (i: nat).(\lambda -(h: nat).(\lambda (d: nat).(\lambda (H: (subst1 i (lift h d u) (lift h (S -(plus i d)) t1) x)).(subst1_ind i (lift h d u) (lift h (S (plus i d)) t1) -(\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h (S (plus i d)) t2))) -(\lambda (t2: T).(subst1 i u t1 t2)))) (ex_intro2 T (\lambda (t2: T).(eq T -(lift h (S (plus i d)) t1) (lift h (S (plus i d)) t2))) (\lambda (t2: -T).(subst1 i u t1 t2)) t1 (refl_equal T (lift h (S (plus i d)) t1)) -(subst1_refl i u t1)) (\lambda (t2: T).(\lambda (H0: (subst0 i (lift h d u) -(lift h (S (plus i d)) t1) t2)).(ex2_ind T (\lambda (t3: T).(eq T t2 (lift h -(S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u t1 t3)) (ex2 T (\lambda -(t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst1 i u t1 -t3))) (\lambda (x0: T).(\lambda (H1: (eq T t2 (lift h (S (plus i d)) -x0))).(\lambda (H2: (subst0 i u t1 x0)).(ex_intro2 T (\lambda (t3: T).(eq T -t2 (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst1 i u t1 t3)) x0 H1 -(subst1_single i u t1 x0 H2))))) (subst0_gen_lift_lt u t1 t2 i h d H0)))) x -H))))))). -(* COMMENTS -Initial nodes: 395 -END *) - -theorem subst1_gen_lift_eq: - \forall (t: T).(\forall (u: T).(\forall (x: T).(\forall (h: nat).(\forall -(d: nat).(\forall (i: nat).((le d i) \to ((lt i (plus d h)) \to ((subst1 i u -(lift h d t) x) \to (eq T x (lift h d t)))))))))) -\def - \lambda (t: T).(\lambda (u: T).(\lambda (x: T).(\lambda (h: nat).(\lambda -(d: nat).(\lambda (i: nat).(\lambda (H: (le d i)).(\lambda (H0: (lt i (plus d -h))).(\lambda (H1: (subst1 i u (lift h d t) x)).(subst1_ind i u (lift h d t) -(\lambda (t0: T).(eq T t0 (lift h d t))) (refl_equal T (lift h d t)) (\lambda -(t2: T).(\lambda (H2: (subst0 i u (lift h d t) t2)).(subst0_gen_lift_false t -u t2 h d i H H0 H2 (eq T t2 (lift h d t))))) x H1))))))))). -(* COMMENTS -Initial nodes: 141 -END *) - -theorem subst1_gen_lift_ge: - \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall -(h: nat).(\forall (d: nat).((subst1 i u (lift h d t1) x) \to ((le (plus d h) -i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: -T).(subst1 (minus i h) u t1 t2)))))))))) -\def - \lambda (u: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (i: nat).(\lambda -(h: nat).(\lambda (d: nat).(\lambda (H: (subst1 i u (lift h d t1) -x)).(\lambda (H0: (le (plus d h) i)).(subst1_ind i u (lift h d t1) (\lambda -(t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h d t2))) (\lambda (t2: -T).(subst1 (minus i h) u t1 t2)))) (ex_intro2 T (\lambda (t2: T).(eq T (lift -h d t1) (lift h d t2))) (\lambda (t2: T).(subst1 (minus i h) u t1 t2)) t1 -(refl_equal T (lift h d t1)) (subst1_refl (minus i h) u t1)) (\lambda (t2: -T).(\lambda (H1: (subst0 i u (lift h d t1) t2)).(ex2_ind T (\lambda (t3: -T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u t1 t3)) -(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst1 -(minus i h) u t1 t3))) (\lambda (x0: T).(\lambda (H2: (eq T t2 (lift h d -x0))).(\lambda (H3: (subst0 (minus i h) u t1 x0)).(ex_intro2 T (\lambda (t3: -T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst1 (minus i h) u t1 t3)) x0 -H2 (subst1_single (minus i h) u t1 x0 H3))))) (subst0_gen_lift_ge u t1 t2 i h -d H1 H0)))) x H)))))))). -(* COMMENTS -Initial nodes: 355 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/subst1/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/subst1/props.ma deleted file mode 100644 index ac8dea954..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/subst1/props.ma +++ /dev/null @@ -1,179 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/subst1/defs.ma". - -include "Basic-1/subst0/props.ma". - -theorem subst1_head: - \forall (v: T).(\forall (u1: T).(\forall (u2: T).(\forall (i: nat).((subst1 -i v u1 u2) \to (\forall (k: K).(\forall (t1: T).(\forall (t2: T).((subst1 (s -k i) v t1 t2) \to (subst1 i v (THead k u1 t1) (THead k u2 t2)))))))))) -\def - \lambda (v: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i: nat).(\lambda -(H: (subst1 i v u1 u2)).(subst1_ind i v u1 (\lambda (t: T).(\forall (k: -K).(\forall (t1: T).(\forall (t2: T).((subst1 (s k i) v t1 t2) \to (subst1 i -v (THead k u1 t1) (THead k t t2))))))) (\lambda (k: K).(\lambda (t1: -T).(\lambda (t2: T).(\lambda (H0: (subst1 (s k i) v t1 t2)).(subst1_ind (s k -i) v t1 (\lambda (t: T).(subst1 i v (THead k u1 t1) (THead k u1 t))) -(subst1_refl i v (THead k u1 t1)) (\lambda (t3: T).(\lambda (H1: (subst0 (s k -i) v t1 t3)).(subst1_single i v (THead k u1 t1) (THead k u1 t3) (subst0_snd k -v t3 t1 i H1 u1)))) t2 H0))))) (\lambda (t2: T).(\lambda (H0: (subst0 i v u1 -t2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t0: T).(\lambda (H1: (subst1 -(s k i) v t1 t0)).(subst1_ind (s k i) v t1 (\lambda (t: T).(subst1 i v (THead -k u1 t1) (THead k t2 t))) (subst1_single i v (THead k u1 t1) (THead k t2 t1) -(subst0_fst v t2 u1 i H0 t1 k)) (\lambda (t3: T).(\lambda (H2: (subst0 (s k -i) v t1 t3)).(subst1_single i v (THead k u1 t1) (THead k t2 t3) (subst0_both -v u1 t2 i H0 k t1 t3 H2)))) t0 H1))))))) u2 H))))). -(* COMMENTS -Initial nodes: 369 -END *) - -theorem subst1_lift_lt: - \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).((subst1 -i u t1 t2) \to (\forall (d: nat).((lt i d) \to (\forall (h: nat).(subst1 i -(lift h (minus d (S i)) u) (lift h d t1) (lift h d t2))))))))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H: (subst1 i u t1 t2)).(subst1_ind i u t1 (\lambda (t: T).(\forall (d: -nat).((lt i d) \to (\forall (h: nat).(subst1 i (lift h (minus d (S i)) u) -(lift h d t1) (lift h d t)))))) (\lambda (d: nat).(\lambda (_: (lt i -d)).(\lambda (h: nat).(subst1_refl i (lift h (minus d (S i)) u) (lift h d -t1))))) (\lambda (t3: T).(\lambda (H0: (subst0 i u t1 t3)).(\lambda (d: -nat).(\lambda (H1: (lt i d)).(\lambda (h: nat).(subst1_single i (lift h -(minus d (S i)) u) (lift h d t1) (lift h d t3) (subst0_lift_lt t1 t3 u i H0 d -H1 h))))))) t2 H))))). -(* COMMENTS -Initial nodes: 185 -END *) - -theorem subst1_lift_ge: - \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).(\forall -(h: nat).((subst1 i u t1 t2) \to (\forall (d: nat).((le d i) \to (subst1 -(plus i h) u (lift h d t1) (lift h d t2))))))))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda -(h: nat).(\lambda (H: (subst1 i u t1 t2)).(subst1_ind i u t1 (\lambda (t: -T).(\forall (d: nat).((le d i) \to (subst1 (plus i h) u (lift h d t1) (lift h -d t))))) (\lambda (d: nat).(\lambda (_: (le d i)).(subst1_refl (plus i h) u -(lift h d t1)))) (\lambda (t3: T).(\lambda (H0: (subst0 i u t1 t3)).(\lambda -(d: nat).(\lambda (H1: (le d i)).(subst1_single (plus i h) u (lift h d t1) -(lift h d t3) (subst0_lift_ge t1 t3 u i h H0 d H1)))))) t2 H)))))). -(* COMMENTS -Initial nodes: 157 -END *) - -theorem subst1_ex: - \forall (u: T).(\forall (t1: T).(\forall (d: nat).(ex T (\lambda (t2: -T).(subst1 d u t1 (lift (S O) d t2)))))) -\def - \lambda (u: T).(\lambda (t1: T).(T_ind (\lambda (t: T).(\forall (d: nat).(ex -T (\lambda (t2: T).(subst1 d u t (lift (S O) d t2)))))) (\lambda (n: -nat).(\lambda (d: nat).(ex_intro T (\lambda (t2: T).(subst1 d u (TSort n) -(lift (S O) d t2))) (TSort n) (eq_ind_r T (TSort n) (\lambda (t: T).(subst1 d -u (TSort n) t)) (subst1_refl d u (TSort n)) (lift (S O) d (TSort n)) -(lift_sort n (S O) d))))) (\lambda (n: nat).(\lambda (d: nat).(lt_eq_gt_e n d -(ex T (\lambda (t2: T).(subst1 d u (TLRef n) (lift (S O) d t2)))) (\lambda -(H: (lt n d)).(ex_intro T (\lambda (t2: T).(subst1 d u (TLRef n) (lift (S O) -d t2))) (TLRef n) (eq_ind_r T (TLRef n) (\lambda (t: T).(subst1 d u (TLRef n) -t)) (subst1_refl d u (TLRef n)) (lift (S O) d (TLRef n)) (lift_lref_lt n (S -O) d H)))) (\lambda (H: (eq nat n d)).(eq_ind nat n (\lambda (n0: nat).(ex T -(\lambda (t2: T).(subst1 n0 u (TLRef n) (lift (S O) n0 t2))))) (ex_intro T -(\lambda (t2: T).(subst1 n u (TLRef n) (lift (S O) n t2))) (lift n O u) -(eq_ind_r T (lift (plus (S O) n) O u) (\lambda (t: T).(subst1 n u (TLRef n) -t)) (subst1_single n u (TLRef n) (lift (S n) O u) (subst0_lref u n)) (lift (S -O) n (lift n O u)) (lift_free u n (S O) O n (le_n (plus O n)) (le_O_n n)))) d -H)) (\lambda (H: (lt d n)).(ex_intro T (\lambda (t2: T).(subst1 d u (TLRef n) -(lift (S O) d t2))) (TLRef (pred n)) (eq_ind_r T (TLRef n) (\lambda (t: -T).(subst1 d u (TLRef n) t)) (subst1_refl d u (TLRef n)) (lift (S O) d (TLRef -(pred n))) (lift_lref_gt d n H))))))) (\lambda (k: K).(\lambda (t: -T).(\lambda (H: ((\forall (d: nat).(ex T (\lambda (t2: T).(subst1 d u t (lift -(S O) d t2))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (d: nat).(ex T -(\lambda (t2: T).(subst1 d u t0 (lift (S O) d t2))))))).(\lambda (d: -nat).(let H_x \def (H d) in (let H1 \def H_x in (ex_ind T (\lambda (t2: -T).(subst1 d u t (lift (S O) d t2))) (ex T (\lambda (t2: T).(subst1 d u -(THead k t t0) (lift (S O) d t2)))) (\lambda (x: T).(\lambda (H2: (subst1 d u -t (lift (S O) d x))).(let H_x0 \def (H0 (s k d)) in (let H3 \def H_x0 in -(ex_ind T (\lambda (t2: T).(subst1 (s k d) u t0 (lift (S O) (s k d) t2))) (ex -T (\lambda (t2: T).(subst1 d u (THead k t t0) (lift (S O) d t2)))) (\lambda -(x0: T).(\lambda (H4: (subst1 (s k d) u t0 (lift (S O) (s k d) -x0))).(ex_intro T (\lambda (t2: T).(subst1 d u (THead k t t0) (lift (S O) d -t2))) (THead k x x0) (eq_ind_r T (THead k (lift (S O) d x) (lift (S O) (s k -d) x0)) (\lambda (t2: T).(subst1 d u (THead k t t0) t2)) (subst1_head u t -(lift (S O) d x) d H2 k t0 (lift (S O) (s k d) x0) H4) (lift (S O) d (THead k -x x0)) (lift_head k x x0 (S O) d))))) H3))))) H1))))))))) t1)). -(* COMMENTS -Initial nodes: 925 -END *) - -theorem subst1_lift_S: - \forall (u: T).(\forall (i: nat).(\forall (h: nat).((le h i) \to (subst1 i -(TLRef h) (lift (S h) (S i) u) (lift (S h) i u))))) -\def - \lambda (u: T).(T_ind (\lambda (t: T).(\forall (i: nat).(\forall (h: -nat).((le h i) \to (subst1 i (TLRef h) (lift (S h) (S i) t) (lift (S h) i -t)))))) (\lambda (n: nat).(\lambda (i: nat).(\lambda (h: nat).(\lambda (_: -(le h i)).(eq_ind_r T (TSort n) (\lambda (t: T).(subst1 i (TLRef h) t (lift -(S h) i (TSort n)))) (eq_ind_r T (TSort n) (\lambda (t: T).(subst1 i (TLRef -h) (TSort n) t)) (subst1_refl i (TLRef h) (TSort n)) (lift (S h) i (TSort n)) -(lift_sort n (S h) i)) (lift (S h) (S i) (TSort n)) (lift_sort n (S h) (S -i))))))) (\lambda (n: nat).(\lambda (i: nat).(\lambda (h: nat).(\lambda (H: -(le h i)).(lt_eq_gt_e n i (subst1 i (TLRef h) (lift (S h) (S i) (TLRef n)) -(lift (S h) i (TLRef n))) (\lambda (H0: (lt n i)).(eq_ind_r T (TLRef n) -(\lambda (t: T).(subst1 i (TLRef h) t (lift (S h) i (TLRef n)))) (eq_ind_r T -(TLRef n) (\lambda (t: T).(subst1 i (TLRef h) (TLRef n) t)) (subst1_refl i -(TLRef h) (TLRef n)) (lift (S h) i (TLRef n)) (lift_lref_lt n (S h) i H0)) -(lift (S h) (S i) (TLRef n)) (lift_lref_lt n (S h) (S i) (le_S (S n) i H0)))) -(\lambda (H0: (eq nat n i)).(let H1 \def (eq_ind_r nat i (\lambda (n0: -nat).(le h n0)) H n H0) in (eq_ind nat n (\lambda (n0: nat).(subst1 n0 (TLRef -h) (lift (S h) (S n0) (TLRef n)) (lift (S h) n0 (TLRef n)))) (eq_ind_r T -(TLRef n) (\lambda (t: T).(subst1 n (TLRef h) t (lift (S h) n (TLRef n)))) -(eq_ind_r T (TLRef (plus n (S h))) (\lambda (t: T).(subst1 n (TLRef h) (TLRef -n) t)) (eq_ind nat (S (plus n h)) (\lambda (n0: nat).(subst1 n (TLRef h) -(TLRef n) (TLRef n0))) (eq_ind_r nat (plus h n) (\lambda (n0: nat).(subst1 n -(TLRef h) (TLRef n) (TLRef (S n0)))) (eq_ind nat (plus h (S n)) (\lambda (n0: -nat).(subst1 n (TLRef h) (TLRef n) (TLRef n0))) (eq_ind T (lift (S n) O -(TLRef h)) (\lambda (t: T).(subst1 n (TLRef h) (TLRef n) t)) (subst1_single n -(TLRef h) (TLRef n) (lift (S n) O (TLRef h)) (subst0_lref (TLRef h) n)) -(TLRef (plus h (S n))) (lift_lref_ge h (S n) O (le_O_n h))) (S (plus h n)) -(sym_eq nat (S (plus h n)) (plus h (S n)) (plus_n_Sm h n))) (plus n h) -(plus_sym n h)) (plus n (S h)) (plus_n_Sm n h)) (lift (S h) n (TLRef n)) -(lift_lref_ge n (S h) n (le_n n))) (lift (S h) (S n) (TLRef n)) (lift_lref_lt -n (S h) (S n) (le_n (S n)))) i H0))) (\lambda (H0: (lt i n)).(eq_ind_r T -(TLRef (plus n (S h))) (\lambda (t: T).(subst1 i (TLRef h) t (lift (S h) i -(TLRef n)))) (eq_ind_r T (TLRef (plus n (S h))) (\lambda (t: T).(subst1 i -(TLRef h) (TLRef (plus n (S h))) t)) (subst1_refl i (TLRef h) (TLRef (plus n -(S h)))) (lift (S h) i (TLRef n)) (lift_lref_ge n (S h) i (le_S_n i n (le_S -(S i) n H0)))) (lift (S h) (S i) (TLRef n)) (lift_lref_ge n (S h) (S i) -H0)))))))) (\lambda (k: K).(\lambda (t: T).(\lambda (H: ((\forall (i: -nat).(\forall (h: nat).((le h i) \to (subst1 i (TLRef h) (lift (S h) (S i) t) -(lift (S h) i t))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (i: -nat).(\forall (h: nat).((le h i) \to (subst1 i (TLRef h) (lift (S h) (S i) -t0) (lift (S h) i t0))))))).(\lambda (i: nat).(\lambda (h: nat).(\lambda (H1: -(le h i)).(eq_ind_r T (THead k (lift (S h) (S i) t) (lift (S h) (s k (S i)) -t0)) (\lambda (t1: T).(subst1 i (TLRef h) t1 (lift (S h) i (THead k t t0)))) -(eq_ind_r T (THead k (lift (S h) i t) (lift (S h) (s k i) t0)) (\lambda (t1: -T).(subst1 i (TLRef h) (THead k (lift (S h) (S i) t) (lift (S h) (s k (S i)) -t0)) t1)) (subst1_head (TLRef h) (lift (S h) (S i) t) (lift (S h) i t) i (H i -h H1) k (lift (S h) (s k (S i)) t0) (lift (S h) (s k i) t0) (eq_ind_r nat (S -(s k i)) (\lambda (n: nat).(subst1 (s k i) (TLRef h) (lift (S h) n t0) (lift -(S h) (s k i) t0))) (H0 (s k i) h (le_trans h i (s k i) H1 (s_inc k i))) (s k -(S i)) (s_S k i))) (lift (S h) i (THead k t t0)) (lift_head k t t0 (S h) i)) -(lift (S h) (S i) (THead k t t0)) (lift_head k t t0 (S h) (S i))))))))))) u). -(* COMMENTS -Initial nodes: 1421 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/subst1/subst1.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/subst1/subst1.ma deleted file mode 100644 index e6dae0dc9..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/subst1/subst1.ma +++ /dev/null @@ -1,214 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/subst1/fwd.ma". - -include "Basic-1/subst0/subst0.ma". - -theorem subst1_subst1: - \forall (t1: T).(\forall (t2: T).(\forall (u2: T).(\forall (j: nat).((subst1 -j u2 t1 t2) \to (\forall (u1: T).(\forall (u: T).(\forall (i: nat).((subst1 i -u u1 u2) \to (ex2 T (\lambda (t: T).(subst1 j u1 t1 t)) (\lambda (t: -T).(subst1 (S (plus i j)) u t t2))))))))))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (u2: T).(\lambda (j: nat).(\lambda -(H: (subst1 j u2 t1 t2)).(subst1_ind j u2 t1 (\lambda (t: T).(\forall (u1: -T).(\forall (u: T).(\forall (i: nat).((subst1 i u u1 u2) \to (ex2 T (\lambda -(t0: T).(subst1 j u1 t1 t0)) (\lambda (t0: T).(subst1 (S (plus i j)) u t0 -t)))))))) (\lambda (u1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda (_: -(subst1 i u u1 u2)).(ex_intro2 T (\lambda (t: T).(subst1 j u1 t1 t)) (\lambda -(t: T).(subst1 (S (plus i j)) u t t1)) t1 (subst1_refl j u1 t1) (subst1_refl -(S (plus i j)) u t1)))))) (\lambda (t3: T).(\lambda (H0: (subst0 j u2 t1 -t3)).(\lambda (u1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda (H1: (subst1 -i u u1 u2)).(insert_eq T u2 (\lambda (t: T).(subst1 i u u1 t)) (\lambda (_: -T).(ex2 T (\lambda (t0: T).(subst1 j u1 t1 t0)) (\lambda (t0: T).(subst1 (S -(plus i j)) u t0 t3)))) (\lambda (y: T).(\lambda (H2: (subst1 i u u1 -y)).(subst1_ind i u u1 (\lambda (t: T).((eq T t u2) \to (ex2 T (\lambda (t0: -T).(subst1 j u1 t1 t0)) (\lambda (t0: T).(subst1 (S (plus i j)) u t0 t3))))) -(\lambda (H3: (eq T u1 u2)).(eq_ind_r T u2 (\lambda (t: T).(ex2 T (\lambda -(t0: T).(subst1 j t t1 t0)) (\lambda (t0: T).(subst1 (S (plus i j)) u t0 -t3)))) (ex_intro2 T (\lambda (t: T).(subst1 j u2 t1 t)) (\lambda (t: -T).(subst1 (S (plus i j)) u t t3)) t3 (subst1_single j u2 t1 t3 H0) -(subst1_refl (S (plus i j)) u t3)) u1 H3)) (\lambda (t0: T).(\lambda (H3: -(subst0 i u u1 t0)).(\lambda (H4: (eq T t0 u2)).(let H5 \def (eq_ind T t0 -(\lambda (t: T).(subst0 i u u1 t)) H3 u2 H4) in (ex2_ind T (\lambda (t: -T).(subst0 j u1 t1 t)) (\lambda (t: T).(subst0 (S (plus i j)) u t t3)) (ex2 T -(\lambda (t: T).(subst1 j u1 t1 t)) (\lambda (t: T).(subst1 (S (plus i j)) u -t t3))) (\lambda (x: T).(\lambda (H6: (subst0 j u1 t1 x)).(\lambda (H7: -(subst0 (S (plus i j)) u x t3)).(ex_intro2 T (\lambda (t: T).(subst1 j u1 t1 -t)) (\lambda (t: T).(subst1 (S (plus i j)) u t t3)) x (subst1_single j u1 t1 -x H6) (subst1_single (S (plus i j)) u x t3 H7))))) (subst0_subst0 t1 t3 u2 j -H0 u1 u i H5)))))) y H2))) H1))))))) t2 H))))). -(* COMMENTS -Initial nodes: 649 -END *) - -theorem subst1_subst1_back: - \forall (t1: T).(\forall (t2: T).(\forall (u2: T).(\forall (j: nat).((subst1 -j u2 t1 t2) \to (\forall (u1: T).(\forall (u: T).(\forall (i: nat).((subst1 i -u u2 u1) \to (ex2 T (\lambda (t: T).(subst1 j u1 t1 t)) (\lambda (t: -T).(subst1 (S (plus i j)) u t2 t))))))))))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (u2: T).(\lambda (j: nat).(\lambda -(H: (subst1 j u2 t1 t2)).(subst1_ind j u2 t1 (\lambda (t: T).(\forall (u1: -T).(\forall (u: T).(\forall (i: nat).((subst1 i u u2 u1) \to (ex2 T (\lambda -(t0: T).(subst1 j u1 t1 t0)) (\lambda (t0: T).(subst1 (S (plus i j)) u t -t0)))))))) (\lambda (u1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda (_: -(subst1 i u u2 u1)).(ex_intro2 T (\lambda (t: T).(subst1 j u1 t1 t)) (\lambda -(t: T).(subst1 (S (plus i j)) u t1 t)) t1 (subst1_refl j u1 t1) (subst1_refl -(S (plus i j)) u t1)))))) (\lambda (t3: T).(\lambda (H0: (subst0 j u2 t1 -t3)).(\lambda (u1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda (H1: (subst1 -i u u2 u1)).(subst1_ind i u u2 (\lambda (t: T).(ex2 T (\lambda (t0: -T).(subst1 j t t1 t0)) (\lambda (t0: T).(subst1 (S (plus i j)) u t3 t0)))) -(ex_intro2 T (\lambda (t: T).(subst1 j u2 t1 t)) (\lambda (t: T).(subst1 (S -(plus i j)) u t3 t)) t3 (subst1_single j u2 t1 t3 H0) (subst1_refl (S (plus i -j)) u t3)) (\lambda (t0: T).(\lambda (H2: (subst0 i u u2 t0)).(ex2_ind T -(\lambda (t: T).(subst0 j t0 t1 t)) (\lambda (t: T).(subst0 (S (plus i j)) u -t3 t)) (ex2 T (\lambda (t: T).(subst1 j t0 t1 t)) (\lambda (t: T).(subst1 (S -(plus i j)) u t3 t))) (\lambda (x: T).(\lambda (H3: (subst0 j t0 t1 -x)).(\lambda (H4: (subst0 (S (plus i j)) u t3 x)).(ex_intro2 T (\lambda (t: -T).(subst1 j t0 t1 t)) (\lambda (t: T).(subst1 (S (plus i j)) u t3 t)) x -(subst1_single j t0 t1 x H3) (subst1_single (S (plus i j)) u t3 x H4))))) -(subst0_subst0_back t1 t3 u2 j H0 t0 u i H2)))) u1 H1))))))) t2 H))))). -(* COMMENTS -Initial nodes: 487 -END *) - -theorem subst1_trans: - \forall (t2: T).(\forall (t1: T).(\forall (v: T).(\forall (i: nat).((subst1 -i v t1 t2) \to (\forall (t3: T).((subst1 i v t2 t3) \to (subst1 i v t1 -t3))))))) -\def - \lambda (t2: T).(\lambda (t1: T).(\lambda (v: T).(\lambda (i: nat).(\lambda -(H: (subst1 i v t1 t2)).(subst1_ind i v t1 (\lambda (t: T).(\forall (t3: -T).((subst1 i v t t3) \to (subst1 i v t1 t3)))) (\lambda (t3: T).(\lambda -(H0: (subst1 i v t1 t3)).H0)) (\lambda (t3: T).(\lambda (H0: (subst0 i v t1 -t3)).(\lambda (t4: T).(\lambda (H1: (subst1 i v t3 t4)).(subst1_ind i v t3 -(\lambda (t: T).(subst1 i v t1 t)) (subst1_single i v t1 t3 H0) (\lambda (t0: -T).(\lambda (H2: (subst0 i v t3 t0)).(subst1_single i v t1 t0 (subst0_trans -t3 t1 v i H0 t0 H2)))) t4 H1))))) t2 H))))). -(* COMMENTS -Initial nodes: 165 -END *) - -theorem subst1_confluence_neq: - \forall (t0: T).(\forall (t1: T).(\forall (u1: T).(\forall (i1: -nat).((subst1 i1 u1 t0 t1) \to (\forall (t2: T).(\forall (u2: T).(\forall -(i2: nat).((subst1 i2 u2 t0 t2) \to ((not (eq nat i1 i2)) \to (ex2 T (\lambda -(t: T).(subst1 i2 u2 t1 t)) (\lambda (t: T).(subst1 i1 u1 t2 t)))))))))))) -\def - \lambda (t0: T).(\lambda (t1: T).(\lambda (u1: T).(\lambda (i1: -nat).(\lambda (H: (subst1 i1 u1 t0 t1)).(subst1_ind i1 u1 t0 (\lambda (t: -T).(\forall (t2: T).(\forall (u2: T).(\forall (i2: nat).((subst1 i2 u2 t0 t2) -\to ((not (eq nat i1 i2)) \to (ex2 T (\lambda (t3: T).(subst1 i2 u2 t t3)) -(\lambda (t3: T).(subst1 i1 u1 t2 t3))))))))) (\lambda (t2: T).(\lambda (u2: -T).(\lambda (i2: nat).(\lambda (H0: (subst1 i2 u2 t0 t2)).(\lambda (_: (not -(eq nat i1 i2))).(ex_intro2 T (\lambda (t: T).(subst1 i2 u2 t0 t)) (\lambda -(t: T).(subst1 i1 u1 t2 t)) t2 H0 (subst1_refl i1 u1 t2))))))) (\lambda (t2: -T).(\lambda (H0: (subst0 i1 u1 t0 t2)).(\lambda (t3: T).(\lambda (u2: -T).(\lambda (i2: nat).(\lambda (H1: (subst1 i2 u2 t0 t3)).(\lambda (H2: (not -(eq nat i1 i2))).(subst1_ind i2 u2 t0 (\lambda (t: T).(ex2 T (\lambda (t4: -T).(subst1 i2 u2 t2 t4)) (\lambda (t4: T).(subst1 i1 u1 t t4)))) (ex_intro2 T -(\lambda (t: T).(subst1 i2 u2 t2 t)) (\lambda (t: T).(subst1 i1 u1 t0 t)) t2 -(subst1_refl i2 u2 t2) (subst1_single i1 u1 t0 t2 H0)) (\lambda (t4: -T).(\lambda (H3: (subst0 i2 u2 t0 t4)).(ex2_ind T (\lambda (t: T).(subst0 i1 -u1 t4 t)) (\lambda (t: T).(subst0 i2 u2 t2 t)) (ex2 T (\lambda (t: T).(subst1 -i2 u2 t2 t)) (\lambda (t: T).(subst1 i1 u1 t4 t))) (\lambda (x: T).(\lambda -(H4: (subst0 i1 u1 t4 x)).(\lambda (H5: (subst0 i2 u2 t2 x)).(ex_intro2 T -(\lambda (t: T).(subst1 i2 u2 t2 t)) (\lambda (t: T).(subst1 i1 u1 t4 t)) x -(subst1_single i2 u2 t2 x H5) (subst1_single i1 u1 t4 x H4))))) -(subst0_confluence_neq t0 t4 u2 i2 H3 t2 u1 i1 H0 (sym_not_eq nat i1 i2 -H2))))) t3 H1)))))))) t1 H))))). -(* COMMENTS -Initial nodes: 455 -END *) - -theorem subst1_confluence_eq: - \forall (t0: T).(\forall (t1: T).(\forall (u: T).(\forall (i: nat).((subst1 -i u t0 t1) \to (\forall (t2: T).((subst1 i u t0 t2) \to (ex2 T (\lambda (t: -T).(subst1 i u t1 t)) (\lambda (t: T).(subst1 i u t2 t))))))))) -\def - \lambda (t0: T).(\lambda (t1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H: (subst1 i u t0 t1)).(subst1_ind i u t0 (\lambda (t: T).(\forall (t2: -T).((subst1 i u t0 t2) \to (ex2 T (\lambda (t3: T).(subst1 i u t t3)) -(\lambda (t3: T).(subst1 i u t2 t3)))))) (\lambda (t2: T).(\lambda (H0: -(subst1 i u t0 t2)).(ex_intro2 T (\lambda (t: T).(subst1 i u t0 t)) (\lambda -(t: T).(subst1 i u t2 t)) t2 H0 (subst1_refl i u t2)))) (\lambda (t2: -T).(\lambda (H0: (subst0 i u t0 t2)).(\lambda (t3: T).(\lambda (H1: (subst1 i -u t0 t3)).(subst1_ind i u t0 (\lambda (t: T).(ex2 T (\lambda (t4: T).(subst1 -i u t2 t4)) (\lambda (t4: T).(subst1 i u t t4)))) (ex_intro2 T (\lambda (t: -T).(subst1 i u t2 t)) (\lambda (t: T).(subst1 i u t0 t)) t2 (subst1_refl i u -t2) (subst1_single i u t0 t2 H0)) (\lambda (t4: T).(\lambda (H2: (subst0 i u -t0 t4)).(or4_ind (eq T t4 t2) (ex2 T (\lambda (t: T).(subst0 i u t4 t)) -(\lambda (t: T).(subst0 i u t2 t))) (subst0 i u t4 t2) (subst0 i u t2 t4) -(ex2 T (\lambda (t: T).(subst1 i u t2 t)) (\lambda (t: T).(subst1 i u t4 t))) -(\lambda (H3: (eq T t4 t2)).(eq_ind_r T t2 (\lambda (t: T).(ex2 T (\lambda -(t5: T).(subst1 i u t2 t5)) (\lambda (t5: T).(subst1 i u t t5)))) (ex_intro2 -T (\lambda (t: T).(subst1 i u t2 t)) (\lambda (t: T).(subst1 i u t2 t)) t2 -(subst1_refl i u t2) (subst1_refl i u t2)) t4 H3)) (\lambda (H3: (ex2 T -(\lambda (t: T).(subst0 i u t4 t)) (\lambda (t: T).(subst0 i u t2 -t)))).(ex2_ind T (\lambda (t: T).(subst0 i u t4 t)) (\lambda (t: T).(subst0 i -u t2 t)) (ex2 T (\lambda (t: T).(subst1 i u t2 t)) (\lambda (t: T).(subst1 i -u t4 t))) (\lambda (x: T).(\lambda (H4: (subst0 i u t4 x)).(\lambda (H5: -(subst0 i u t2 x)).(ex_intro2 T (\lambda (t: T).(subst1 i u t2 t)) (\lambda -(t: T).(subst1 i u t4 t)) x (subst1_single i u t2 x H5) (subst1_single i u t4 -x H4))))) H3)) (\lambda (H3: (subst0 i u t4 t2)).(ex_intro2 T (\lambda (t: -T).(subst1 i u t2 t)) (\lambda (t: T).(subst1 i u t4 t)) t2 (subst1_refl i u -t2) (subst1_single i u t4 t2 H3))) (\lambda (H3: (subst0 i u t2 -t4)).(ex_intro2 T (\lambda (t: T).(subst1 i u t2 t)) (\lambda (t: T).(subst1 -i u t4 t)) t4 (subst1_single i u t2 t4 H3) (subst1_refl i u t4))) -(subst0_confluence_eq t0 t4 u i H2 t2 H0)))) t3 H1))))) t1 H))))). -(* COMMENTS -Initial nodes: 729 -END *) - -theorem subst1_confluence_lift: - \forall (t0: T).(\forall (t1: T).(\forall (u: T).(\forall (i: nat).((subst1 -i u t0 (lift (S O) i t1)) \to (\forall (t2: T).((subst1 i u t0 (lift (S O) i -t2)) \to (eq T t1 t2))))))) -\def - \lambda (t0: T).(\lambda (t1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H: (subst1 i u t0 (lift (S O) i t1))).(insert_eq T (lift (S O) i t1) -(\lambda (t: T).(subst1 i u t0 t)) (\lambda (_: T).(\forall (t2: T).((subst1 -i u t0 (lift (S O) i t2)) \to (eq T t1 t2)))) (\lambda (y: T).(\lambda (H0: -(subst1 i u t0 y)).(subst1_ind i u t0 (\lambda (t: T).((eq T t (lift (S O) i -t1)) \to (\forall (t2: T).((subst1 i u t0 (lift (S O) i t2)) \to (eq T t1 -t2))))) (\lambda (H1: (eq T t0 (lift (S O) i t1))).(\lambda (t2: T).(\lambda -(H2: (subst1 i u t0 (lift (S O) i t2))).(let H3 \def (eq_ind T t0 (\lambda -(t: T).(subst1 i u t (lift (S O) i t2))) H2 (lift (S O) i t1) H1) in (let H4 -\def (sym_eq T (lift (S O) i t2) (lift (S O) i t1) (subst1_gen_lift_eq t1 u -(lift (S O) i t2) (S O) i i (le_n i) (eq_ind_r nat (plus (S O) i) (\lambda -(n: nat).(lt i n)) (le_n (plus (S O) i)) (plus i (S O)) (plus_sym i (S O))) -H3)) in (lift_inj t1 t2 (S O) i H4)))))) (\lambda (t2: T).(\lambda (H1: -(subst0 i u t0 t2)).(\lambda (H2: (eq T t2 (lift (S O) i t1))).(\lambda (t3: -T).(\lambda (H3: (subst1 i u t0 (lift (S O) i t3))).(let H4 \def (eq_ind T t2 -(\lambda (t: T).(subst0 i u t0 t)) H1 (lift (S O) i t1) H2) in (insert_eq T -(lift (S O) i t3) (\lambda (t: T).(subst1 i u t0 t)) (\lambda (_: T).(eq T t1 -t3)) (\lambda (y0: T).(\lambda (H5: (subst1 i u t0 y0)).(subst1_ind i u t0 -(\lambda (t: T).((eq T t (lift (S O) i t3)) \to (eq T t1 t3))) (\lambda (H6: -(eq T t0 (lift (S O) i t3))).(let H7 \def (eq_ind T t0 (\lambda (t: -T).(subst0 i u t (lift (S O) i t1))) H4 (lift (S O) i t3) H6) in -(subst0_gen_lift_false t3 u (lift (S O) i t1) (S O) i i (le_n i) (eq_ind_r -nat (plus (S O) i) (\lambda (n: nat).(lt i n)) (le_n (plus (S O) i)) (plus i -(S O)) (plus_sym i (S O))) H7 (eq T t1 t3)))) (\lambda (t4: T).(\lambda (H6: -(subst0 i u t0 t4)).(\lambda (H7: (eq T t4 (lift (S O) i t3))).(let H8 \def -(eq_ind T t4 (\lambda (t: T).(subst0 i u t0 t)) H6 (lift (S O) i t3) H7) in -(sym_eq T t3 t1 (subst0_confluence_lift t0 t3 u i H8 t1 H4)))))) y0 H5))) -H3))))))) y H0))) H))))). -(* COMMENTS -Initial nodes: 735 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/theory.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/theory.ma deleted file mode 100644 index f405be588..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/theory.ma +++ /dev/null @@ -1,42 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/subst0/tlt.ma". - -include "Basic-1/subst/props.ma". - -include "Basic-1/sty1/cnt.ma". - -include "Basic-1/ex0/props.ma". - -include "Basic-1/wcpr0/fwd.ma". - -include "Basic-1/pr3/wcpr0.ma". - -include "Basic-1/ex2/props.ma". - -include "Basic-1/ex1/props.ma". - -include "Basic-1/ty3/sty0.ma". - -include "Basic-1/csubt/csuba.ma". - -include "Basic-1/ty3/fwd_nf2.ma". - -include "Basic-1/ty3/nf2.ma". - -include "Basic-1/wf3/props.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/tlist/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/tlist/defs.ma deleted file mode 100644 index 9445013d4..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/tlist/defs.ma +++ /dev/null @@ -1,47 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/T/defs.ma". - -inductive TList: Set \def -| TNil: TList -| TCons: T \to (TList \to TList). - -definition THeads: - K \to (TList \to (T \to T)) -\def - let rec THeads (k: K) (us: TList) on us: (T \to T) \def (\lambda (t: -T).(match us with [TNil \Rightarrow t | (TCons u ul) \Rightarrow (THead k u -(THeads k ul t))])) in THeads. - -definition TApp: - TList \to (T \to TList) -\def - let rec TApp (ts: TList) on ts: (T \to TList) \def (\lambda (v: T).(match ts -with [TNil \Rightarrow (TCons v TNil) | (TCons t ts0) \Rightarrow (TCons t -(TApp ts0 v))])) in TApp. - -definition tslen: - TList \to nat -\def - let rec tslen (ts: TList) on ts: nat \def (match ts with [TNil \Rightarrow O -| (TCons _ ts0) \Rightarrow (S (tslen ts0))]) in tslen. - -definition tslt: - TList \to (TList \to Prop) -\def - \lambda (ts1: TList).(\lambda (ts2: TList).(lt (tslen ts1) (tslen ts2))). - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/tlist/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/tlist/props.ma deleted file mode 100644 index 92b0d05ff..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/tlist/props.ma +++ /dev/null @@ -1,131 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/tlist/defs.ma". - -theorem tslt_wf__q_ind: - \forall (P: ((TList \to Prop))).(((\forall (n: nat).((\lambda (P0: ((TList -\to Prop))).(\lambda (n0: nat).(\forall (ts: TList).((eq nat (tslen ts) n0) -\to (P0 ts))))) P n))) \to (\forall (ts: TList).(P ts))) -\def - let Q \def (\lambda (P: ((TList \to Prop))).(\lambda (n: nat).(\forall (ts: -TList).((eq nat (tslen ts) n) \to (P ts))))) in (\lambda (P: ((TList \to -Prop))).(\lambda (H: ((\forall (n: nat).(\forall (ts: TList).((eq nat (tslen -ts) n) \to (P ts)))))).(\lambda (ts: TList).(H (tslen ts) ts (refl_equal nat -(tslen ts)))))). -(* COMMENTS -Initial nodes: 61 -END *) - -theorem tslt_wf_ind: - \forall (P: ((TList \to Prop))).(((\forall (ts2: TList).(((\forall (ts1: -TList).((tslt ts1 ts2) \to (P ts1)))) \to (P ts2)))) \to (\forall (ts: -TList).(P ts))) -\def - let Q \def (\lambda (P: ((TList \to Prop))).(\lambda (n: nat).(\forall (ts: -TList).((eq nat (tslen ts) n) \to (P ts))))) in (\lambda (P: ((TList \to -Prop))).(\lambda (H: ((\forall (ts2: TList).(((\forall (ts1: TList).((lt -(tslen ts1) (tslen ts2)) \to (P ts1)))) \to (P ts2))))).(\lambda (ts: -TList).(tslt_wf__q_ind (\lambda (t: TList).(P t)) (\lambda (n: -nat).(lt_wf_ind n (Q (\lambda (t: TList).(P t))) (\lambda (n0: nat).(\lambda -(H0: ((\forall (m: nat).((lt m n0) \to (Q (\lambda (t: TList).(P t)) -m))))).(\lambda (ts0: TList).(\lambda (H1: (eq nat (tslen ts0) n0)).(let H2 -\def (eq_ind_r nat n0 (\lambda (n1: nat).(\forall (m: nat).((lt m n1) \to -(\forall (ts1: TList).((eq nat (tslen ts1) m) \to (P ts1)))))) H0 (tslen ts0) -H1) in (H ts0 (\lambda (ts1: TList).(\lambda (H3: (lt (tslen ts1) (tslen -ts0))).(H2 (tslen ts1) H3 ts1 (refl_equal nat (tslen ts1))))))))))))) ts)))). -(* COMMENTS -Initial nodes: 179 -END *) - -theorem theads_tapp: - \forall (k: K).(\forall (v: T).(\forall (t: T).(\forall (vs: TList).(eq T -(THeads k (TApp vs v) t) (THeads k vs (THead k v t)))))) -\def - \lambda (k: K).(\lambda (v: T).(\lambda (t: T).(\lambda (vs: -TList).(TList_ind (\lambda (t0: TList).(eq T (THeads k (TApp t0 v) t) (THeads -k t0 (THead k v t)))) (refl_equal T (THead k v t)) (\lambda (t0: T).(\lambda -(t1: TList).(\lambda (H: (eq T (THeads k (TApp t1 v) t) (THeads k t1 (THead k -v t)))).(eq_ind T (THeads k (TApp t1 v) t) (\lambda (t2: T).(eq T (THead k t0 -(THeads k (TApp t1 v) t)) (THead k t0 t2))) (refl_equal T (THead k t0 (THeads -k (TApp t1 v) t))) (THeads k t1 (THead k v t)) H)))) vs)))). -(* COMMENTS -Initial nodes: 175 -END *) - -theorem tcons_tapp_ex: - \forall (ts1: TList).(\forall (t1: T).(ex2_2 TList T (\lambda (ts2: -TList).(\lambda (t2: T).(eq TList (TCons t1 ts1) (TApp ts2 t2)))) (\lambda -(ts2: TList).(\lambda (_: T).(eq nat (tslen ts1) (tslen ts2)))))) -\def - \lambda (ts1: TList).(TList_ind (\lambda (t: TList).(\forall (t1: T).(ex2_2 -TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons t1 t) (TApp -ts2 t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (tslen t) (tslen -ts2))))))) (\lambda (t1: T).(ex2_2_intro TList T (\lambda (ts2: -TList).(\lambda (t2: T).(eq TList (TCons t1 TNil) (TApp ts2 t2)))) (\lambda -(ts2: TList).(\lambda (_: T).(eq nat O (tslen ts2)))) TNil t1 (refl_equal -TList (TApp TNil t1)) (refl_equal nat (tslen TNil)))) (\lambda (t: -T).(\lambda (t0: TList).(\lambda (H: ((\forall (t1: T).(ex2_2 TList T -(\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons t1 t0) (TApp ts2 -t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (tslen t0) (tslen -ts2)))))))).(\lambda (t1: T).(let H_x \def (H t) in (let H0 \def H_x in -(ex2_2_ind TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons t -t0) (TApp ts2 t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (tslen t0) -(tslen ts2)))) (ex2_2 TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq -TList (TCons t1 (TCons t t0)) (TApp ts2 t2)))) (\lambda (ts2: TList).(\lambda -(_: T).(eq nat (S (tslen t0)) (tslen ts2))))) (\lambda (x0: TList).(\lambda -(x1: T).(\lambda (H1: (eq TList (TCons t t0) (TApp x0 x1))).(\lambda (H2: (eq -nat (tslen t0) (tslen x0))).(eq_ind_r TList (TApp x0 x1) (\lambda (t2: -TList).(ex2_2 TList T (\lambda (ts2: TList).(\lambda (t3: T).(eq TList (TCons -t1 t2) (TApp ts2 t3)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (S -(tslen t0)) (tslen ts2)))))) (eq_ind_r nat (tslen x0) (\lambda (n: -nat).(ex2_2 TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons -t1 (TApp x0 x1)) (TApp ts2 t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq -nat (S n) (tslen ts2)))))) (ex2_2_intro TList T (\lambda (ts2: -TList).(\lambda (t2: T).(eq TList (TCons t1 (TApp x0 x1)) (TApp ts2 t2)))) -(\lambda (ts2: TList).(\lambda (_: T).(eq nat (S (tslen x0)) (tslen ts2)))) -(TCons t1 x0) x1 (refl_equal TList (TApp (TCons t1 x0) x1)) (refl_equal nat -(tslen (TCons t1 x0)))) (tslen t0) H2) (TCons t t0) H1))))) H0))))))) ts1). -(* COMMENTS -Initial nodes: 503 -END *) - -theorem tlist_ind_rev: - \forall (P: ((TList \to Prop))).((P TNil) \to (((\forall (ts: -TList).(\forall (t: T).((P ts) \to (P (TApp ts t)))))) \to (\forall (ts: -TList).(P ts)))) -\def - \lambda (P: ((TList \to Prop))).(\lambda (H: (P TNil)).(\lambda (H0: -((\forall (ts: TList).(\forall (t: T).((P ts) \to (P (TApp ts -t))))))).(\lambda (ts: TList).(tslt_wf_ind (\lambda (t: TList).(P t)) -(\lambda (ts2: TList).(TList_ind (\lambda (t: TList).(((\forall (ts1: -TList).((tslt ts1 t) \to (P ts1)))) \to (P t))) (\lambda (_: ((\forall (ts1: -TList).((tslt ts1 TNil) \to (P ts1))))).H) (\lambda (t: T).(\lambda (t0: -TList).(\lambda (_: ((((\forall (ts1: TList).((tslt ts1 t0) \to (P ts1)))) -\to (P t0)))).(\lambda (H2: ((\forall (ts1: TList).((tslt ts1 (TCons t t0)) -\to (P ts1))))).(let H_x \def (tcons_tapp_ex t0 t) in (let H3 \def H_x in -(ex2_2_ind TList T (\lambda (ts3: TList).(\lambda (t2: T).(eq TList (TCons t -t0) (TApp ts3 t2)))) (\lambda (ts3: TList).(\lambda (_: T).(eq nat (tslen t0) -(tslen ts3)))) (P (TCons t t0)) (\lambda (x0: TList).(\lambda (x1: -T).(\lambda (H4: (eq TList (TCons t t0) (TApp x0 x1))).(\lambda (H5: (eq nat -(tslen t0) (tslen x0))).(eq_ind_r TList (TApp x0 x1) (\lambda (t1: TList).(P -t1)) (H0 x0 x1 (H2 x0 (eq_ind nat (tslen t0) (\lambda (n: nat).(lt n (tslen -(TCons t t0)))) (le_n (tslen (TCons t t0))) (tslen x0) H5))) (TCons t t0) -H4))))) H3))))))) ts2)) ts)))). -(* COMMENTS -Initial nodes: 273 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/tlt/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/tlt/defs.ma deleted file mode 100644 index 546bdb78d..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/tlt/defs.ma +++ /dev/null @@ -1,46 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/T/defs.ma". - -definition wadd: - ((nat \to nat)) \to (nat \to (nat \to nat)) -\def - \lambda (f: ((nat \to nat))).(\lambda (w: nat).(\lambda (n: nat).(match n -with [O \Rightarrow w | (S m) \Rightarrow (f m)]))). - -definition weight_map: - ((nat \to nat)) \to (T \to nat) -\def - let rec weight_map (f: ((nat \to nat))) (t: T) on t: nat \def (match t with -[(TSort _) \Rightarrow O | (TLRef n) \Rightarrow (f n) | (THead k u t0) -\Rightarrow (match k with [(Bind b) \Rightarrow (match b with [Abbr -\Rightarrow (S (plus (weight_map f u) (weight_map (wadd f (S (weight_map f -u))) t0))) | Abst \Rightarrow (S (plus (weight_map f u) (weight_map (wadd f -O) t0))) | Void \Rightarrow (S (plus (weight_map f u) (weight_map (wadd f O) -t0)))]) | (Flat _) \Rightarrow (S (plus (weight_map f u) (weight_map f -t0)))])]) in weight_map. - -definition weight: - T \to nat -\def - weight_map (\lambda (_: nat).O). - -definition tlt: - T \to (T \to Prop) -\def - \lambda (t1: T).(\lambda (t2: T).(lt (weight t1) (weight t2))). - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/tlt/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/tlt/props.ma deleted file mode 100644 index 0b4f16d41..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/tlt/props.ma +++ /dev/null @@ -1,300 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/tlt/defs.ma". - -theorem wadd_le: - \forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (n: -nat).(le (f n) (g n)))) \to (\forall (v: nat).(\forall (w: nat).((le v w) \to -(\forall (n: nat).(le (wadd f v n) (wadd g w n)))))))) -\def - \lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H: -((\forall (n: nat).(le (f n) (g n))))).(\lambda (v: nat).(\lambda (w: -nat).(\lambda (H0: (le v w)).(\lambda (n: nat).(nat_ind (\lambda (n0: -nat).(le (wadd f v n0) (wadd g w n0))) H0 (\lambda (n0: nat).(\lambda (_: (le -(wadd f v n0) (wadd g w n0))).(H n0))) n))))))). -(* COMMENTS -Initial nodes: 81 -END *) - -theorem wadd_lt: - \forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (n: -nat).(le (f n) (g n)))) \to (\forall (v: nat).(\forall (w: nat).((lt v w) \to -(\forall (n: nat).(le (wadd f v n) (wadd g w n)))))))) -\def - \lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H: -((\forall (n: nat).(le (f n) (g n))))).(\lambda (v: nat).(\lambda (w: -nat).(\lambda (H0: (lt v w)).(\lambda (n: nat).(nat_ind (\lambda (n0: -nat).(le (wadd f v n0) (wadd g w n0))) (le_S_n v w (le_S (S v) w H0)) -(\lambda (n0: nat).(\lambda (_: (le (wadd f v n0) (wadd g w n0))).(H n0))) -n))))))). -(* COMMENTS -Initial nodes: 95 -END *) - -theorem wadd_O: - \forall (n: nat).(eq nat (wadd (\lambda (_: nat).O) O n) O) -\def - \lambda (n: nat).(nat_ind (\lambda (n0: nat).(eq nat (wadd (\lambda (_: -nat).O) O n0) O)) (refl_equal nat O) (\lambda (n0: nat).(\lambda (_: (eq nat -(wadd (\lambda (_: nat).O) O n0) O)).(refl_equal nat O))) n). -(* COMMENTS -Initial nodes: 53 -END *) - -theorem weight_le: - \forall (t: T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (n: nat).(le (f n) (g n)))) \to (le (weight_map f t) -(weight_map g t))))) -\def - \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (f: ((nat \to -nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g n)))) -\to (le (weight_map f t0) (weight_map g t0)))))) (\lambda (n: nat).(\lambda -(f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (_: ((\forall -(n0: nat).(le (f n0) (g n0))))).(le_n (weight_map g (TSort n))))))) (\lambda -(n: nat).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda -(H: ((\forall (n0: nat).(le (f n0) (g n0))))).(H n))))) (\lambda (k: -K).(K_ind (\lambda (k0: K).(\forall (t0: T).(((\forall (f: ((nat \to -nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g n)))) -\to (le (weight_map f t0) (weight_map g t0)))))) \to (\forall (t1: -T).(((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall -(n: nat).(le (f n) (g n)))) \to (le (weight_map f t1) (weight_map g t1)))))) -\to (\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall -(n: nat).(le (f n) (g n)))) \to (le (weight_map f (THead k0 t0 t1)) -(weight_map g (THead k0 t0 t1))))))))))) (\lambda (b: B).(B_ind (\lambda (b0: -B).(\forall (t0: T).(((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (n: nat).(le (f n) (g n)))) \to (le (weight_map f t0) -(weight_map g t0)))))) \to (\forall (t1: T).(((\forall (f: ((nat \to -nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g n)))) -\to (le (weight_map f t1) (weight_map g t1)))))) \to (\forall (f: ((nat \to -nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g n)))) -\to (le (match b0 with [Abbr \Rightarrow (S (plus (weight_map f t0) -(weight_map (wadd f (S (weight_map f t0))) t1))) | Abst \Rightarrow (S (plus -(weight_map f t0) (weight_map (wadd f O) t1))) | Void \Rightarrow (S (plus -(weight_map f t0) (weight_map (wadd f O) t1)))]) (match b0 with [Abbr -\Rightarrow (S (plus (weight_map g t0) (weight_map (wadd g (S (weight_map g -t0))) t1))) | Abst \Rightarrow (S (plus (weight_map g t0) (weight_map (wadd g -O) t1))) | Void \Rightarrow (S (plus (weight_map g t0) (weight_map (wadd g O) -t1)))])))))))))) (\lambda (t0: T).(\lambda (H: ((\forall (f: ((nat \to -nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g n)))) -\to (le (weight_map f t0) (weight_map g t0))))))).(\lambda (t1: T).(\lambda -(H0: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall -(n: nat).(le (f n) (g n)))) \to (le (weight_map f t1) (weight_map g -t1))))))).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to -nat))).(\lambda (H1: ((\forall (n: nat).(le (f n) (g n))))).(le_n_S (plus -(weight_map f t0) (weight_map (wadd f (S (weight_map f t0))) t1)) (plus -(weight_map g t0) (weight_map (wadd g (S (weight_map g t0))) t1)) -(le_plus_plus (weight_map f t0) (weight_map g t0) (weight_map (wadd f (S -(weight_map f t0))) t1) (weight_map (wadd g (S (weight_map g t0))) t1) (H f g -H1) (H0 (wadd f (S (weight_map f t0))) (wadd g (S (weight_map g t0))) -(\lambda (n: nat).(wadd_le f g H1 (S (weight_map f t0)) (S (weight_map g t0)) -(le_n_S (weight_map f t0) (weight_map g t0) (H f g H1)) n)))))))))))) -(\lambda (t0: T).(\lambda (H: ((\forall (f: ((nat \to nat))).(\forall (g: -((nat \to nat))).(((\forall (n: nat).(le (f n) (g n)))) \to (le (weight_map f -t0) (weight_map g t0))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (f: -((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) -(g n)))) \to (le (weight_map f t1) (weight_map g t1))))))).(\lambda (f: ((nat -\to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H1: ((\forall (n: nat).(le -(f n) (g n))))).(le_n_S (plus (weight_map f t0) (weight_map (wadd f O) t1)) -(plus (weight_map g t0) (weight_map (wadd g O) t1)) (le_plus_plus (weight_map -f t0) (weight_map g t0) (weight_map (wadd f O) t1) (weight_map (wadd g O) t1) -(H f g H1) (H0 (wadd f O) (wadd g O) (\lambda (n: nat).(wadd_le f g H1 O O -(le_n O) n)))))))))))) (\lambda (t0: T).(\lambda (H: ((\forall (f: ((nat \to -nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g n)))) -\to (le (weight_map f t0) (weight_map g t0))))))).(\lambda (t1: T).(\lambda -(H0: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall -(n: nat).(le (f n) (g n)))) \to (le (weight_map f t1) (weight_map g -t1))))))).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to -nat))).(\lambda (H1: ((\forall (n: nat).(le (f n) (g n))))).(le_n_S (plus -(weight_map f t0) (weight_map (wadd f O) t1)) (plus (weight_map g t0) -(weight_map (wadd g O) t1)) (le_plus_plus (weight_map f t0) (weight_map g t0) -(weight_map (wadd f O) t1) (weight_map (wadd g O) t1) (H f g H1) (H0 (wadd f -O) (wadd g O) (\lambda (n: nat).(wadd_le f g H1 O O (le_n O) n)))))))))))) -b)) (\lambda (_: F).(\lambda (t0: T).(\lambda (H: ((\forall (f0: ((nat \to -nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f0 n) (g n)))) -\to (le (weight_map f0 t0) (weight_map g t0))))))).(\lambda (t1: T).(\lambda -(H0: ((\forall (f0: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall -(n: nat).(le (f0 n) (g n)))) \to (le (weight_map f0 t1) (weight_map g -t1))))))).(\lambda (f0: ((nat \to nat))).(\lambda (g: ((nat \to -nat))).(\lambda (H1: ((\forall (n: nat).(le (f0 n) (g n))))).(le_n_S (plus -(weight_map f0 t0) (weight_map f0 t1)) (plus (weight_map g t0) (weight_map g -t1)) (le_plus_plus (weight_map f0 t0) (weight_map g t0) (weight_map f0 t1) -(weight_map g t1) (H f0 g H1) (H0 f0 g H1))))))))))) k)) t). -(* COMMENTS -Initial nodes: 1309 -END *) - -theorem weight_eq: - \forall (t: T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (n: nat).(eq nat (f n) (g n)))) \to (eq nat (weight_map f -t) (weight_map g t))))) -\def - \lambda (t: T).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to -nat))).(\lambda (H: ((\forall (n: nat).(eq nat (f n) (g n))))).(le_antisym -(weight_map f t) (weight_map g t) (weight_le t f g (\lambda (n: -nat).(eq_ind_r nat (g n) (\lambda (n0: nat).(le n0 (g n))) (le_n (g n)) (f n) -(H n)))) (weight_le t g f (\lambda (n: nat).(eq_ind_r nat (g n) (\lambda (n0: -nat).(le (g n) n0)) (le_n (g n)) (f n) (H n)))))))). -(* COMMENTS -Initial nodes: 121 -END *) - -theorem weight_add_O: - \forall (t: T).(eq nat (weight_map (wadd (\lambda (_: nat).O) O) t) -(weight_map (\lambda (_: nat).O) t)) -\def - \lambda (t: T).(weight_eq t (wadd (\lambda (_: nat).O) O) (\lambda (_: -nat).O) (\lambda (n: nat).(wadd_O n))). -(* COMMENTS -Initial nodes: 23 -END *) - -theorem weight_add_S: - \forall (t: T).(\forall (m: nat).(le (weight_map (wadd (\lambda (_: nat).O) -O) t) (weight_map (wadd (\lambda (_: nat).O) (S m)) t))) -\def - \lambda (t: T).(\lambda (m: nat).(weight_le t (wadd (\lambda (_: nat).O) O) -(wadd (\lambda (_: nat).O) (S m)) (\lambda (n: nat).(wadd_le (\lambda (_: -nat).O) (\lambda (_: nat).O) (\lambda (_: nat).(le_n O)) O (S m) (le_S O m -(le_O_n m)) n)))). -(* COMMENTS -Initial nodes: 61 -END *) - -theorem tlt_trans: - \forall (v: T).(\forall (u: T).(\forall (t: T).((tlt u v) \to ((tlt v t) \to -(tlt u t))))) -\def - \lambda (v: T).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (lt (weight u) -(weight v))).(\lambda (H0: (lt (weight v) (weight t))).(lt_trans (weight u) -(weight v) (weight t) H H0))))). -(* COMMENTS -Initial nodes: 43 -END *) - -theorem tlt_head_sx: - \forall (k: K).(\forall (u: T).(\forall (t: T).(tlt u (THead k u t)))) -\def - \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (u: T).(\forall (t: T).(lt -(weight_map (\lambda (_: nat).O) u) (weight_map (\lambda (_: nat).O) (THead -k0 u t)))))) (\lambda (b: B).(B_ind (\lambda (b0: B).(\forall (u: T).(\forall -(t: T).(lt (weight_map (\lambda (_: nat).O) u) (match b0 with [Abbr -\Rightarrow (S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd -(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t))) | Abst -\Rightarrow (S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd -(\lambda (_: nat).O) O) t))) | Void \Rightarrow (S (plus (weight_map (\lambda -(_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) O) t)))]))))) (\lambda -(u: T).(\lambda (t: T).(le_n_S (weight_map (\lambda (_: nat).O) u) (plus -(weight_map (\lambda (_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) (S -(weight_map (\lambda (_: nat).O) u))) t)) (le_plus_l (weight_map (\lambda (_: -nat).O) u) (weight_map (wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: -nat).O) u))) t))))) (\lambda (u: T).(\lambda (t: T).(le_n_S (weight_map -(\lambda (_: nat).O) u) (plus (weight_map (\lambda (_: nat).O) u) (weight_map -(wadd (\lambda (_: nat).O) O) t)) (le_plus_l (weight_map (\lambda (_: nat).O) -u) (weight_map (wadd (\lambda (_: nat).O) O) t))))) (\lambda (u: T).(\lambda -(t: T).(le_n_S (weight_map (\lambda (_: nat).O) u) (plus (weight_map (\lambda -(_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) O) t)) (le_plus_l -(weight_map (\lambda (_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) O) -t))))) b)) (\lambda (_: F).(\lambda (u: T).(\lambda (t: T).(le_n_S -(weight_map (\lambda (_: nat).O) u) (plus (weight_map (\lambda (_: nat).O) u) -(weight_map (\lambda (_: nat).O) t)) (le_plus_l (weight_map (\lambda (_: -nat).O) u) (weight_map (\lambda (_: nat).O) t)))))) k). -(* COMMENTS -Initial nodes: 379 -END *) - -theorem tlt_head_dx: - \forall (k: K).(\forall (u: T).(\forall (t: T).(tlt t (THead k u t)))) -\def - \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (u: T).(\forall (t: T).(lt -(weight_map (\lambda (_: nat).O) t) (weight_map (\lambda (_: nat).O) (THead -k0 u t)))))) (\lambda (b: B).(B_ind (\lambda (b0: B).(\forall (u: T).(\forall -(t: T).(lt (weight_map (\lambda (_: nat).O) t) (match b0 with [Abbr -\Rightarrow (S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd -(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t))) | Abst -\Rightarrow (S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd -(\lambda (_: nat).O) O) t))) | Void \Rightarrow (S (plus (weight_map (\lambda -(_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) O) t)))]))))) (\lambda -(u: T).(\lambda (t: T).(lt_le_trans (weight_map (\lambda (_: nat).O) t) (S -(weight_map (\lambda (_: nat).O) t)) (S (plus (weight_map (\lambda (_: -nat).O) u) (weight_map (wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: -nat).O) u))) t))) (lt_n_Sn (weight_map (\lambda (_: nat).O) t)) (le_n_S -(weight_map (\lambda (_: nat).O) t) (plus (weight_map (\lambda (_: nat).O) u) -(weight_map (wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) -u))) t)) (le_trans (weight_map (\lambda (_: nat).O) t) (weight_map (wadd -(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t) (plus -(weight_map (\lambda (_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) (S -(weight_map (\lambda (_: nat).O) u))) t)) (eq_ind nat (weight_map (wadd -(\lambda (_: nat).O) O) t) (\lambda (n: nat).(le n (weight_map (wadd (\lambda -(_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t))) (weight_add_S t -(weight_map (\lambda (_: nat).O) u)) (weight_map (\lambda (_: nat).O) t) -(weight_add_O t)) (le_plus_r (weight_map (\lambda (_: nat).O) u) (weight_map -(wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t))))))) -(\lambda (u: T).(\lambda (t: T).(eq_ind_r nat (weight_map (\lambda (_: -nat).O) t) (\lambda (n: nat).(lt (weight_map (\lambda (_: nat).O) t) (S (plus -(weight_map (\lambda (_: nat).O) u) n)))) (le_n_S (weight_map (\lambda (_: -nat).O) t) (plus (weight_map (\lambda (_: nat).O) u) (weight_map (\lambda (_: -nat).O) t)) (le_plus_r (weight_map (\lambda (_: nat).O) u) (weight_map -(\lambda (_: nat).O) t))) (weight_map (wadd (\lambda (_: nat).O) O) t) -(weight_add_O t)))) (\lambda (u: T).(\lambda (t: T).(eq_ind_r nat (weight_map -(\lambda (_: nat).O) t) (\lambda (n: nat).(lt (weight_map (\lambda (_: -nat).O) t) (S (plus (weight_map (\lambda (_: nat).O) u) n)))) (le_n_S -(weight_map (\lambda (_: nat).O) t) (plus (weight_map (\lambda (_: nat).O) u) -(weight_map (\lambda (_: nat).O) t)) (le_plus_r (weight_map (\lambda (_: -nat).O) u) (weight_map (\lambda (_: nat).O) t))) (weight_map (wadd (\lambda -(_: nat).O) O) t) (weight_add_O t)))) b)) (\lambda (_: F).(\lambda (u: -T).(\lambda (t: T).(le_n_S (weight_map (\lambda (_: nat).O) t) (plus -(weight_map (\lambda (_: nat).O) u) (weight_map (\lambda (_: nat).O) t)) -(le_plus_r (weight_map (\lambda (_: nat).O) u) (weight_map (\lambda (_: -nat).O) t)))))) k). -(* COMMENTS -Initial nodes: 659 -END *) - -theorem tlt_wf__q_ind: - \forall (P: ((T \to Prop))).(((\forall (n: nat).((\lambda (P0: ((T \to -Prop))).(\lambda (n0: nat).(\forall (t: T).((eq nat (weight t) n0) \to (P0 -t))))) P n))) \to (\forall (t: T).(P t))) -\def - let Q \def (\lambda (P: ((T \to Prop))).(\lambda (n: nat).(\forall (t: -T).((eq nat (weight t) n) \to (P t))))) in (\lambda (P: ((T \to -Prop))).(\lambda (H: ((\forall (n: nat).(\forall (t: T).((eq nat (weight t) -n) \to (P t)))))).(\lambda (t: T).(H (weight t) t (refl_equal nat (weight -t)))))). -(* COMMENTS -Initial nodes: 61 -END *) - -theorem tlt_wf_ind: - \forall (P: ((T \to Prop))).(((\forall (t: T).(((\forall (v: T).((tlt v t) -\to (P v)))) \to (P t)))) \to (\forall (t: T).(P t))) -\def - let Q \def (\lambda (P: ((T \to Prop))).(\lambda (n: nat).(\forall (t: -T).((eq nat (weight t) n) \to (P t))))) in (\lambda (P: ((T \to -Prop))).(\lambda (H: ((\forall (t: T).(((\forall (v: T).((lt (weight v) -(weight t)) \to (P v)))) \to (P t))))).(\lambda (t: T).(tlt_wf__q_ind -(\lambda (t0: T).(P t0)) (\lambda (n: nat).(lt_wf_ind n (Q (\lambda (t0: -T).(P t0))) (\lambda (n0: nat).(\lambda (H0: ((\forall (m: nat).((lt m n0) -\to (Q (\lambda (t0: T).(P t0)) m))))).(\lambda (t0: T).(\lambda (H1: (eq nat -(weight t0) n0)).(let H2 \def (eq_ind_r nat n0 (\lambda (n1: nat).(\forall -(m: nat).((lt m n1) \to (\forall (t1: T).((eq nat (weight t1) m) \to (P -t1)))))) H0 (weight t0) H1) in (H t0 (\lambda (v: T).(\lambda (H3: (lt -(weight v) (weight t0))).(H2 (weight v) H3 v (refl_equal nat (weight -v))))))))))))) t)))). -(* COMMENTS -Initial nodes: 179 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/arity.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/arity.ma deleted file mode 100644 index 4400e1540..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/arity.ma +++ /dev/null @@ -1,186 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/ty3/pr3_props.ma". - -include "Basic-1/arity/pr3.ma". - -include "Basic-1/asucc/fwd.ma". - -theorem ty3_arity: - \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c -t1 t2) \to (ex2 A (\lambda (a1: A).(arity g c t1 a1)) (\lambda (a1: A).(arity -g c t2 (asucc g a1)))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H: (ty3 g c t1 t2)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda -(t0: T).(ex2 A (\lambda (a1: A).(arity g c0 t a1)) (\lambda (a1: A).(arity g -c0 t0 (asucc g a1))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t: -T).(\lambda (_: (ty3 g c0 t3 t)).(\lambda (H1: (ex2 A (\lambda (a1: A).(arity -g c0 t3 a1)) (\lambda (a1: A).(arity g c0 t (asucc g a1))))).(\lambda (u: -T).(\lambda (t4: T).(\lambda (_: (ty3 g c0 u t4)).(\lambda (H3: (ex2 A -(\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t4 (asucc g -a1))))).(\lambda (H4: (pc3 c0 t4 t3)).(let H5 \def H1 in (ex2_ind A (\lambda -(a1: A).(arity g c0 t3 a1)) (\lambda (a1: A).(arity g c0 t (asucc g a1))) -(ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t3 -(asucc g a1)))) (\lambda (x: A).(\lambda (H6: (arity g c0 t3 x)).(\lambda (_: -(arity g c0 t (asucc g x))).(let H8 \def H3 in (ex2_ind A (\lambda (a1: -A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t4 (asucc g a1))) (ex2 A -(\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t3 (asucc g -a1)))) (\lambda (x0: A).(\lambda (H9: (arity g c0 u x0)).(\lambda (H10: -(arity g c0 t4 (asucc g x0))).(let H11 \def H4 in (ex2_ind T (\lambda (t0: -T).(pr3 c0 t4 t0)) (\lambda (t0: T).(pr3 c0 t3 t0)) (ex2 A (\lambda (a1: -A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t3 (asucc g a1)))) -(\lambda (x1: T).(\lambda (H12: (pr3 c0 t4 x1)).(\lambda (H13: (pr3 c0 t3 -x1)).(ex_intro2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity -g c0 t3 (asucc g a1))) x0 H9 (arity_repl g c0 t3 x H6 (asucc g x0) (leq_sym g -(asucc g x0) x (arity_mono g c0 x1 (asucc g x0) (arity_sred_pr3 c0 t4 x1 H12 -g (asucc g x0) H10) x (arity_sred_pr3 c0 t3 x1 H13 g x H6)))))))) H11))))) -H8))))) H5)))))))))))) (\lambda (c0: C).(\lambda (m: nat).(ex_intro2 A -(\lambda (a1: A).(arity g c0 (TSort m) a1)) (\lambda (a1: A).(arity g c0 -(TSort (next g m)) (asucc g a1))) (ASort O m) (arity_sort g c0 m) (arity_sort -g c0 (next g m))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: -C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abbr) -u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2: (ex2 A -(\lambda (a1: A).(arity g d u a1)) (\lambda (a1: A).(arity g d t (asucc g -a1))))).(let H3 \def H2 in (ex2_ind A (\lambda (a1: A).(arity g d u a1)) -(\lambda (a1: A).(arity g d t (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g -c0 (TLRef n) a1)) (\lambda (a1: A).(arity g c0 (lift (S n) O t) (asucc g -a1)))) (\lambda (x: A).(\lambda (H4: (arity g d u x)).(\lambda (H5: (arity g -d t (asucc g x))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (TLRef n) a1)) -(\lambda (a1: A).(arity g c0 (lift (S n) O t) (asucc g a1))) x (arity_abbr g -c0 d u n H0 x H4) (arity_lift g d t (asucc g x) H5 c0 (S n) O (getl_drop Abbr -c0 d u n H0)))))) H3)))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda -(d: C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abst) -u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2: (ex2 A -(\lambda (a1: A).(arity g d u a1)) (\lambda (a1: A).(arity g d t (asucc g -a1))))).(let H3 \def H2 in (ex2_ind A (\lambda (a1: A).(arity g d u a1)) -(\lambda (a1: A).(arity g d t (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g -c0 (TLRef n) a1)) (\lambda (a1: A).(arity g c0 (lift (S n) O u) (asucc g -a1)))) (\lambda (x: A).(\lambda (H4: (arity g d u x)).(\lambda (_: (arity g d -t (asucc g x))).(let H_x \def (leq_asucc g x) in (let H6 \def H_x in (ex_ind -A (\lambda (a0: A).(leq g x (asucc g a0))) (ex2 A (\lambda (a1: A).(arity g -c0 (TLRef n) a1)) (\lambda (a1: A).(arity g c0 (lift (S n) O u) (asucc g -a1)))) (\lambda (x0: A).(\lambda (H7: (leq g x (asucc g x0))).(ex_intro2 A -(\lambda (a1: A).(arity g c0 (TLRef n) a1)) (\lambda (a1: A).(arity g c0 -(lift (S n) O u) (asucc g a1))) x0 (arity_abst g c0 d u n H0 x0 (arity_repl g -d u x H4 (asucc g x0) H7)) (arity_repl g c0 (lift (S n) O u) x (arity_lift g -d u x H4 c0 (S n) O (getl_drop Abst c0 d u n H0)) (asucc g x0) H7)))) -H6)))))) H3)))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (t: -T).(\lambda (_: (ty3 g c0 u t)).(\lambda (H1: (ex2 A (\lambda (a1: A).(arity -g c0 u a1)) (\lambda (a1: A).(arity g c0 t (asucc g a1))))).(\lambda (b: -B).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) -u) t3 t4)).(\lambda (H3: (ex2 A (\lambda (a1: A).(arity g (CHead c0 (Bind b) -u) t3 a1)) (\lambda (a1: A).(arity g (CHead c0 (Bind b) u) t4 (asucc g -a1))))).(let H4 \def H1 in (ex2_ind A (\lambda (a1: A).(arity g c0 u a1)) -(\lambda (a1: A).(arity g c0 t (asucc g a1))) (ex2 A (\lambda (a1: A).(arity -g c0 (THead (Bind b) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind b) -u t4) (asucc g a1)))) (\lambda (x: A).(\lambda (H5: (arity g c0 u -x)).(\lambda (_: (arity g c0 t (asucc g x))).(let H7 \def H3 in (ex2_ind A -(\lambda (a1: A).(arity g (CHead c0 (Bind b) u) t3 a1)) (\lambda (a1: -A).(arity g (CHead c0 (Bind b) u) t4 (asucc g a1))) (ex2 A (\lambda (a1: -A).(arity g c0 (THead (Bind b) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead -(Bind b) u t4) (asucc g a1)))) (\lambda (x0: A).(\lambda (H8: (arity g (CHead -c0 (Bind b) u) t3 x0)).(\lambda (H9: (arity g (CHead c0 (Bind b) u) t4 (asucc -g x0))).(let H_x \def (leq_asucc g x) in (let H10 \def H_x in (ex_ind A -(\lambda (a0: A).(leq g x (asucc g a0))) (ex2 A (\lambda (a1: A).(arity g c0 -(THead (Bind b) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind b) u t4) -(asucc g a1)))) (\lambda (x1: A).(\lambda (H11: (leq g x (asucc g -x1))).(B_ind (\lambda (b0: B).((arity g (CHead c0 (Bind b0) u) t3 x0) \to -((arity g (CHead c0 (Bind b0) u) t4 (asucc g x0)) \to (ex2 A (\lambda (a1: -A).(arity g c0 (THead (Bind b0) u t3) a1)) (\lambda (a1: A).(arity g c0 -(THead (Bind b0) u t4) (asucc g a1))))))) (\lambda (H12: (arity g (CHead c0 -(Bind Abbr) u) t3 x0)).(\lambda (H13: (arity g (CHead c0 (Bind Abbr) u) t4 -(asucc g x0))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Bind Abbr) u -t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind Abbr) u t4) (asucc g a1))) -x0 (arity_bind g Abbr not_abbr_abst c0 u x H5 t3 x0 H12) (arity_bind g Abbr -not_abbr_abst c0 u x H5 t4 (asucc g x0) H13)))) (\lambda (H12: (arity g -(CHead c0 (Bind Abst) u) t3 x0)).(\lambda (H13: (arity g (CHead c0 (Bind -Abst) u) t4 (asucc g x0))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (THead -(Bind Abst) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind Abst) u t4) -(asucc g a1))) (AHead x1 x0) (arity_head g c0 u x1 (arity_repl g c0 u x H5 -(asucc g x1) H11) t3 x0 H12) (arity_repl g c0 (THead (Bind Abst) u t4) (AHead -x1 (asucc g x0)) (arity_head g c0 u x1 (arity_repl g c0 u x H5 (asucc g x1) -H11) t4 (asucc g x0) H13) (asucc g (AHead x1 x0)) (leq_refl g (asucc g (AHead -x1 x0))))))) (\lambda (H12: (arity g (CHead c0 (Bind Void) u) t3 -x0)).(\lambda (H13: (arity g (CHead c0 (Bind Void) u) t4 (asucc g -x0))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Bind Void) u t3) a1)) -(\lambda (a1: A).(arity g c0 (THead (Bind Void) u t4) (asucc g a1))) x0 -(arity_bind g Void (sym_not_eq B Abst Void not_abst_void) c0 u x H5 t3 x0 -H12) (arity_bind g Void (sym_not_eq B Abst Void not_abst_void) c0 u x H5 t4 -(asucc g x0) H13)))) b H8 H9))) H10)))))) H7))))) H4)))))))))))) (\lambda -(c0: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda -(H1: (ex2 A (\lambda (a1: A).(arity g c0 w a1)) (\lambda (a1: A).(arity g c0 -u (asucc g a1))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v -(THead (Bind Abst) u t))).(\lambda (H3: (ex2 A (\lambda (a1: A).(arity g c0 v -a1)) (\lambda (a1: A).(arity g c0 (THead (Bind Abst) u t) (asucc g -a1))))).(let H4 \def H1 in (ex2_ind A (\lambda (a1: A).(arity g c0 w a1)) -(\lambda (a1: A).(arity g c0 u (asucc g a1))) (ex2 A (\lambda (a1: A).(arity -g c0 (THead (Flat Appl) w v) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat -Appl) w (THead (Bind Abst) u t)) (asucc g a1)))) (\lambda (x: A).(\lambda -(H5: (arity g c0 w x)).(\lambda (H6: (arity g c0 u (asucc g x))).(let H7 \def -H3 in (ex2_ind A (\lambda (a1: A).(arity g c0 v a1)) (\lambda (a1: A).(arity -g c0 (THead (Bind Abst) u t) (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g -c0 (THead (Flat Appl) w v) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat -Appl) w (THead (Bind Abst) u t)) (asucc g a1)))) (\lambda (x0: A).(\lambda -(H8: (arity g c0 v x0)).(\lambda (H9: (arity g c0 (THead (Bind Abst) u t) -(asucc g x0))).(let H10 \def (arity_gen_abst g c0 u t (asucc g x0) H9) in -(ex3_2_ind A A (\lambda (a1: A).(\lambda (a2: A).(eq A (asucc g x0) (AHead a1 -a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g c0 u (asucc g a1)))) -(\lambda (_: A).(\lambda (a2: A).(arity g (CHead c0 (Bind Abst) u) t a2))) -(ex2 A (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w v) a1)) (\lambda -(a1: A).(arity g c0 (THead (Flat Appl) w (THead (Bind Abst) u t)) (asucc g -a1)))) (\lambda (x1: A).(\lambda (x2: A).(\lambda (H11: (eq A (asucc g x0) -(AHead x1 x2))).(\lambda (H12: (arity g c0 u (asucc g x1))).(\lambda (H13: -(arity g (CHead c0 (Bind Abst) u) t x2)).(let H14 \def (sym_eq A (asucc g x0) -(AHead x1 x2) H11) in (let H15 \def (asucc_gen_head g x1 x2 x0 H14) in -(ex2_ind A (\lambda (a0: A).(eq A x0 (AHead x1 a0))) (\lambda (a0: A).(eq A -x2 (asucc g a0))) (ex2 A (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w v) -a1)) (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w (THead (Bind Abst) u -t)) (asucc g a1)))) (\lambda (x3: A).(\lambda (H16: (eq A x0 (AHead x1 -x3))).(\lambda (H17: (eq A x2 (asucc g x3))).(let H18 \def (eq_ind A x2 -(\lambda (a: A).(arity g (CHead c0 (Bind Abst) u) t a)) H13 (asucc g x3) H17) -in (let H19 \def (eq_ind A x0 (\lambda (a: A).(arity g c0 v a)) H8 (AHead x1 -x3) H16) in (ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w v) -a1)) (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w (THead (Bind Abst) u -t)) (asucc g a1))) x3 (arity_appl g c0 w x1 (arity_repl g c0 w x H5 x1 -(leq_sym g x1 x (asucc_inj g x1 x (arity_mono g c0 u (asucc g x1) H12 (asucc -g x) H6)))) v x3 H19) (arity_appl g c0 w x H5 (THead (Bind Abst) u t) (asucc -g x3) (arity_head g c0 u x H6 t (asucc g x3) H18)))))))) H15)))))))) H10))))) -H7))))) H4))))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: -T).(\lambda (_: (ty3 g c0 t3 t4)).(\lambda (H1: (ex2 A (\lambda (a1: -A).(arity g c0 t3 a1)) (\lambda (a1: A).(arity g c0 t4 (asucc g -a1))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t4 t0)).(\lambda (H3: (ex2 A -(\lambda (a1: A).(arity g c0 t4 a1)) (\lambda (a1: A).(arity g c0 t0 (asucc g -a1))))).(let H4 \def H1 in (ex2_ind A (\lambda (a1: A).(arity g c0 t3 a1)) -(\lambda (a1: A).(arity g c0 t4 (asucc g a1))) (ex2 A (\lambda (a1: A).(arity -g c0 (THead (Flat Cast) t4 t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat -Cast) t0 t4) (asucc g a1)))) (\lambda (x: A).(\lambda (H5: (arity g c0 t3 -x)).(\lambda (H6: (arity g c0 t4 (asucc g x))).(let H7 \def H3 in (ex2_ind A -(\lambda (a1: A).(arity g c0 t4 a1)) (\lambda (a1: A).(arity g c0 t0 (asucc g -a1))) (ex2 A (\lambda (a1: A).(arity g c0 (THead (Flat Cast) t4 t3) a1)) -(\lambda (a1: A).(arity g c0 (THead (Flat Cast) t0 t4) (asucc g a1)))) -(\lambda (x0: A).(\lambda (H8: (arity g c0 t4 x0)).(\lambda (H9: (arity g c0 -t0 (asucc g x0))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Flat -Cast) t4 t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat Cast) t0 t4) -(asucc g a1))) x (arity_cast g c0 t4 x H6 t3 H5) (arity_cast g c0 t0 (asucc g -x) (arity_repl g c0 t0 (asucc g x0) H9 (asucc g (asucc g x)) (asucc_repl g x0 -(asucc g x) (arity_mono g c0 t4 x0 H8 (asucc g x) H6))) t4 H6))))) H7))))) -H4)))))))))) c t1 t2 H))))). -(* COMMENTS -Initial nodes: 3761 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/arity_props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/arity_props.ma deleted file mode 100644 index 49d6c0572..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/arity_props.ma +++ /dev/null @@ -1,117 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/ty3/arity.ma". - -include "Basic-1/sc3/arity.ma". - -theorem ty3_predicative: - \forall (g: G).(\forall (c: C).(\forall (v: T).(\forall (t: T).(\forall (u: -T).((ty3 g c (THead (Bind Abst) v t) u) \to ((pc3 c u v) \to (\forall (P: -Prop).P))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (u: -T).(\lambda (H: (ty3 g c (THead (Bind Abst) v t) u)).(\lambda (H0: (pc3 c u -v)).(\lambda (P: Prop).(let H1 \def H in (ex3_2_ind T T (\lambda (t2: -T).(\lambda (_: T).(pc3 c (THead (Bind Abst) v t2) u))) (\lambda (_: -T).(\lambda (t0: T).(ty3 g c v t0))) (\lambda (t2: T).(\lambda (_: T).(ty3 g -(CHead c (Bind Abst) v) t t2))) P (\lambda (x0: T).(\lambda (x1: T).(\lambda -(_: (pc3 c (THead (Bind Abst) v x0) u)).(\lambda (H3: (ty3 g c v -x1)).(\lambda (_: (ty3 g (CHead c (Bind Abst) v) t x0)).(let H_y \def -(ty3_conv g c v x1 H3 (THead (Bind Abst) v t) u H H0) in (let H_x \def -(ty3_arity g c (THead (Bind Abst) v t) v H_y) in (let H5 \def H_x in (ex2_ind -A (\lambda (a1: A).(arity g c (THead (Bind Abst) v t) a1)) (\lambda (a1: -A).(arity g c v (asucc g a1))) P (\lambda (x: A).(\lambda (H6: (arity g c -(THead (Bind Abst) v t) x)).(\lambda (H7: (arity g c v (asucc g x))).(let H8 -\def (arity_gen_abst g c v t x H6) in (ex3_2_ind A A (\lambda (a1: -A).(\lambda (a2: A).(eq A x (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: -A).(arity g c v (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g -(CHead c (Bind Abst) v) t a2))) P (\lambda (x2: A).(\lambda (x3: A).(\lambda -(H9: (eq A x (AHead x2 x3))).(\lambda (H10: (arity g c v (asucc g -x2))).(\lambda (_: (arity g (CHead c (Bind Abst) v) t x3)).(let H12 \def -(eq_ind A x (\lambda (a: A).(arity g c v (asucc g a))) H7 (AHead x2 x3) H9) -in (leq_ahead_asucc_false g x2 (asucc g x3) (arity_mono g c v (asucc g (AHead -x2 x3)) H12 (asucc g x2) H10) P))))))) H8))))) H5))))))))) (ty3_gen_bind g -Abst c v t u H1)))))))))). -(* COMMENTS -Initial nodes: 497 -END *) - -theorem ty3_repellent: - \forall (g: G).(\forall (c: C).(\forall (w: T).(\forall (t: T).(\forall (u1: -T).((ty3 g c (THead (Bind Abst) w t) u1) \to (\forall (u2: T).((ty3 g (CHead -c (Bind Abst) w) t (lift (S O) O u2)) \to ((pc3 c u1 u2) \to (\forall (P: -Prop).P))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (w: T).(\lambda (t: T).(\lambda (u1: -T).(\lambda (H: (ty3 g c (THead (Bind Abst) w t) u1)).(\lambda (u2: -T).(\lambda (H0: (ty3 g (CHead c (Bind Abst) w) t (lift (S O) O -u2))).(\lambda (H1: (pc3 c u1 u2)).(\lambda (P: Prop).(ex_ind T (\lambda (t0: -T).(ty3 g (CHead c (Bind Abst) w) (lift (S O) O u2) t0)) P (\lambda (x: -T).(\lambda (H2: (ty3 g (CHead c (Bind Abst) w) (lift (S O) O u2) x)).(let H3 -\def (ty3_gen_lift g (CHead c (Bind Abst) w) u2 x (S O) O H2 c (drop_drop -(Bind Abst) O c c (drop_refl c) w)) in (ex2_ind T (\lambda (t2: T).(pc3 -(CHead c (Bind Abst) w) (lift (S O) O t2) x)) (\lambda (t2: T).(ty3 g c u2 -t2)) P (\lambda (x0: T).(\lambda (_: (pc3 (CHead c (Bind Abst) w) (lift (S O) -O x0) x)).(\lambda (H5: (ty3 g c u2 x0)).(let H_y \def (ty3_conv g c u2 x0 H5 -(THead (Bind Abst) w t) u1 H H1) in (let H_x \def (ty3_arity g (CHead c (Bind -Abst) w) t (lift (S O) O u2) H0) in (let H6 \def H_x in (ex2_ind A (\lambda -(a1: A).(arity g (CHead c (Bind Abst) w) t a1)) (\lambda (a1: A).(arity g -(CHead c (Bind Abst) w) (lift (S O) O u2) (asucc g a1))) P (\lambda (x1: -A).(\lambda (H7: (arity g (CHead c (Bind Abst) w) t x1)).(\lambda (H8: (arity -g (CHead c (Bind Abst) w) (lift (S O) O u2) (asucc g x1))).(let H_x0 \def -(ty3_arity g c (THead (Bind Abst) w t) u2 H_y) in (let H9 \def H_x0 in -(ex2_ind A (\lambda (a1: A).(arity g c (THead (Bind Abst) w t) a1)) (\lambda -(a1: A).(arity g c u2 (asucc g a1))) P (\lambda (x2: A).(\lambda (H10: (arity -g c (THead (Bind Abst) w t) x2)).(\lambda (H11: (arity g c u2 (asucc g -x2))).(arity_repellent g c w t x1 H7 x2 H10 (asucc_inj g x1 x2 (arity_mono g -c u2 (asucc g x1) (arity_gen_lift g (CHead c (Bind Abst) w) u2 (asucc g x1) -(S O) O H8 c (drop_drop (Bind Abst) O c c (drop_refl c) w)) (asucc g x2) -H11)) P)))) H9)))))) H6))))))) H3)))) (ty3_correct g (CHead c (Bind Abst) w) -t (lift (S O) O u2) H0))))))))))). -(* COMMENTS -Initial nodes: 651 -END *) - -theorem ty3_acyclic: - \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t -u) \to ((pc3 c u t) \to (\forall (P: Prop).P)))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H: -(ty3 g c t u)).(\lambda (H0: (pc3 c u t)).(\lambda (P: Prop).(let H_y \def -(ty3_conv g c t u H t u H H0) in (let H_x \def (ty3_arity g c t t H_y) in -(let H1 \def H_x in (ex2_ind A (\lambda (a1: A).(arity g c t a1)) (\lambda -(a1: A).(arity g c t (asucc g a1))) P (\lambda (x: A).(\lambda (H2: (arity g -c t x)).(\lambda (H3: (arity g c t (asucc g x))).(leq_asucc_false g x -(arity_mono g c t (asucc g x) H3 x H2) P)))) H1)))))))))). -(* COMMENTS -Initial nodes: 151 -END *) - -theorem ty3_sn3: - \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t -u) \to (sn3 c t))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H: -(ty3 g c t u)).(let H_x \def (ty3_arity g c t u H) in (let H0 \def H_x in -(ex2_ind A (\lambda (a1: A).(arity g c t a1)) (\lambda (a1: A).(arity g c u -(asucc g a1))) (sn3 c t) (\lambda (x: A).(\lambda (H1: (arity g c t -x)).(\lambda (_: (arity g c u (asucc g x))).(sc3_sn3 g x c t (sc3_arity g c t -x H1))))) H0))))))). -(* COMMENTS -Initial nodes: 119 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/dec.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/dec.ma deleted file mode 100644 index c176d635e..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/dec.ma +++ /dev/null @@ -1,438 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pc3/dec.ma". - -include "Basic-1/getl/flt.ma". - -include "Basic-1/getl/dec.ma". - -theorem ty3_inference: - \forall (g: G).(\forall (c: C).(\forall (t1: T).(or (ex T (\lambda (t2: -T).(ty3 g c t1 t2))) (\forall (t2: T).((ty3 g c t1 t2) \to False))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(flt_wf_ind (\lambda (c0: -C).(\lambda (t: T).(or (ex T (\lambda (t2: T).(ty3 g c0 t t2))) (\forall (t2: -T).((ty3 g c0 t t2) \to False))))) (\lambda (c2: C).(\lambda (t2: T).(T_ind -(\lambda (t: T).(((\forall (c1: C).(\forall (t3: T).((flt c1 t3 c2 t) \to (or -(ex T (\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4) -\to False))))))) \to (or (ex T (\lambda (t3: T).(ty3 g c2 t t3))) (\forall -(t3: T).((ty3 g c2 t t3) \to False))))) (\lambda (n: nat).(\lambda (_: -((\forall (c1: C).(\forall (t3: T).((flt c1 t3 c2 (TSort n)) \to (or (ex T -(\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to -False)))))))).(or_introl (ex T (\lambda (t3: T).(ty3 g c2 (TSort n) t3))) -(\forall (t3: T).((ty3 g c2 (TSort n) t3) \to False)) (ex_intro T (\lambda -(t3: T).(ty3 g c2 (TSort n) t3)) (TSort (next g n)) (ty3_sort g c2 n))))) -(\lambda (n: nat).(\lambda (H: ((\forall (c1: C).(\forall (t3: T).((flt c1 t3 -c2 (TLRef n)) \to (or (ex T (\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: -T).((ty3 g c1 t3 t4) \to False)))))))).(let H_x \def (getl_dec c2 n) in (let -H0 \def H_x in (or_ind (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda -(v: T).(getl n c2 (CHead e (Bind b) v)))))) (\forall (d: C).((getl n c2 d) -\to (\forall (P: Prop).P))) (or (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) -t3))) (\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to False))) (\lambda (H1: -(ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl n c2 (CHead -e (Bind b) v))))))).(ex_3_ind C B T (\lambda (e: C).(\lambda (b: B).(\lambda -(v: T).(getl n c2 (CHead e (Bind b) v))))) (or (ex T (\lambda (t3: T).(ty3 g -c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to False))) -(\lambda (x0: C).(\lambda (x1: B).(\lambda (x2: T).(\lambda (H2: (getl n c2 -(CHead x0 (Bind x1) x2))).(let H3 \def (H x0 x2 (getl_flt x1 c2 x0 x2 n H2)) -in (or_ind (ex T (\lambda (t3: T).(ty3 g x0 x2 t3))) (\forall (t3: T).((ty3 g -x0 x2 t3) \to False)) (or (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) -(\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to False))) (\lambda (H4: (ex T -(\lambda (t3: T).(ty3 g x0 x2 t3)))).(ex_ind T (\lambda (t3: T).(ty3 g x0 x2 -t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: -T).((ty3 g c2 (TLRef n) t3) \to False))) (\lambda (x: T).(\lambda (H5: (ty3 g -x0 x2 x)).(B_ind (\lambda (b: B).((getl n c2 (CHead x0 (Bind b) x2)) \to (or -(ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 -(TLRef n) t3) \to False))))) (\lambda (H6: (getl n c2 (CHead x0 (Bind Abbr) -x2))).(or_introl (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall -(t3: T).((ty3 g c2 (TLRef n) t3) \to False)) (ex_intro T (\lambda (t3: -T).(ty3 g c2 (TLRef n) t3)) (lift (S n) O x) (ty3_abbr g n c2 x0 x2 H6 x -H5)))) (\lambda (H6: (getl n c2 (CHead x0 (Bind Abst) x2))).(or_introl (ex T -(\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef -n) t3) \to False)) (ex_intro T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3)) -(lift (S n) O x2) (ty3_abst g n c2 x0 x2 H6 x H5)))) (\lambda (H6: (getl n c2 -(CHead x0 (Bind Void) x2))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 -(TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to False)) -(\lambda (t3: T).(\lambda (H7: (ty3 g c2 (TLRef n) t3)).(or_ind (ex3_3 C T T -(\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) -t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e -(Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u -t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 -(lift (S n) O u) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl -n c2 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: -T).(ty3 g e u t))))) False (\lambda (H8: (ex3_3 C T T (\lambda (_: -C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda -(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u))))) -(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind -C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O -t) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e -(Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u -t)))) False (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: -(pc3 c2 (lift (S n) O x5) t3)).(\lambda (H10: (getl n c2 (CHead x3 (Bind -Abbr) x4))).(\lambda (_: (ty3 g x3 x4 x5)).(let H12 \def (eq_ind C (CHead x0 -(Bind Void) x2) (\lambda (c0: C).(getl n c2 c0)) H6 (CHead x3 (Bind Abbr) x4) -(getl_mono c2 (CHead x0 (Bind Void) x2) n H6 (CHead x3 (Bind Abbr) x4) H10)) -in (let H13 \def (eq_ind C (CHead x0 (Bind Void) x2) (\lambda (ee: C).(match -ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | -(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr -\Rightarrow False | Abst \Rightarrow False | Void \Rightarrow True]) | (Flat -_) \Rightarrow False])])) I (CHead x3 (Bind Abbr) x4) (getl_mono c2 (CHead x0 -(Bind Void) x2) n H6 (CHead x3 (Bind Abbr) x4) H10)) in (False_ind False -H13))))))))) H8)) (\lambda (H8: (ex3_3 C T T (\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e: C).(\lambda -(u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u))))) (\lambda (e: -C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T -(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) -t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e -(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u -t)))) False (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: -(pc3 c2 (lift (S n) O x4) t3)).(\lambda (H10: (getl n c2 (CHead x3 (Bind -Abst) x4))).(\lambda (_: (ty3 g x3 x4 x5)).(let H12 \def (eq_ind C (CHead x0 -(Bind Void) x2) (\lambda (c0: C).(getl n c2 c0)) H6 (CHead x3 (Bind Abst) x4) -(getl_mono c2 (CHead x0 (Bind Void) x2) n H6 (CHead x3 (Bind Abst) x4) H10)) -in (let H13 \def (eq_ind C (CHead x0 (Bind Void) x2) (\lambda (ee: C).(match -ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | -(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr -\Rightarrow False | Abst \Rightarrow False | Void \Rightarrow True]) | (Flat -_) \Rightarrow False])])) I (CHead x3 (Bind Abst) x4) (getl_mono c2 (CHead x0 -(Bind Void) x2) n H6 (CHead x3 (Bind Abst) x4) H10)) in (False_ind False -H13))))))))) H8)) (ty3_gen_lref g c2 t3 n H7)))))) x1 H2))) H4)) (\lambda -(H4: ((\forall (t3: T).((ty3 g x0 x2 t3) \to False)))).(or_intror (ex T -(\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef -n) t3) \to False)) (\lambda (t3: T).(\lambda (H5: (ty3 g c2 (TLRef n) -t3)).(or_ind (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: -T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda -(_: T).(getl n c2 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: -T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e: -C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u))))) -(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))) False -(\lambda (H6: (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: -T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda -(_: T).(getl n c2 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: -T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T (\lambda (_: -C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda -(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u))))) -(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))) False -(\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (pc3 c2 (lift -(S n) O x5) t3)).(\lambda (H8: (getl n c2 (CHead x3 (Bind Abbr) -x4))).(\lambda (H9: (ty3 g x3 x4 x5)).(let H10 \def (eq_ind C (CHead x0 (Bind -x1) x2) (\lambda (c0: C).(getl n c2 c0)) H2 (CHead x3 (Bind Abbr) x4) -(getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind Abbr) x4) H8)) in -(let H11 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: -C).C) with [(CSort _) \Rightarrow x0 | (CHead c0 _ _) \Rightarrow c0])) -(CHead x0 (Bind x1) x2) (CHead x3 (Bind Abbr) x4) (getl_mono c2 (CHead x0 -(Bind x1) x2) n H2 (CHead x3 (Bind Abbr) x4) H8)) in ((let H12 \def (f_equal -C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with [(CSort _) -\Rightarrow x1 | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: -K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow x1])])) (CHead x0 -(Bind x1) x2) (CHead x3 (Bind Abbr) x4) (getl_mono c2 (CHead x0 (Bind x1) x2) -n H2 (CHead x3 (Bind Abbr) x4) H8)) in ((let H13 \def (f_equal C T (\lambda -(e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow x2 -| (CHead _ _ t) \Rightarrow t])) (CHead x0 (Bind x1) x2) (CHead x3 (Bind -Abbr) x4) (getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind Abbr) -x4) H8)) in (\lambda (_: (eq B x1 Abbr)).(\lambda (H15: (eq C x0 x3)).(let -H16 \def (eq_ind_r T x4 (\lambda (t: T).(getl n c2 (CHead x3 (Bind Abbr) t))) -H10 x2 H13) in (let H17 \def (eq_ind_r T x4 (\lambda (t: T).(ty3 g x3 t x5)) -H9 x2 H13) in (let H18 \def (eq_ind_r C x3 (\lambda (c0: C).(getl n c2 (CHead -c0 (Bind Abbr) x2))) H16 x0 H15) in (let H19 \def (eq_ind_r C x3 (\lambda -(c0: C).(ty3 g c0 x2 x5)) H17 x0 H15) in (H4 x5 H19)))))))) H12)) -H11))))))))) H6)) (\lambda (H6: (ex3_3 C T T (\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e: C).(\lambda -(u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u))))) (\lambda (e: -C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T -(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) -t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e -(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u -t)))) False (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H7: -(pc3 c2 (lift (S n) O x4) t3)).(\lambda (H8: (getl n c2 (CHead x3 (Bind Abst) -x4))).(\lambda (H9: (ty3 g x3 x4 x5)).(let H10 \def (eq_ind C (CHead x0 (Bind -x1) x2) (\lambda (c0: C).(getl n c2 c0)) H2 (CHead x3 (Bind Abst) x4) -(getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind Abst) x4) H8)) in -(let H11 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: -C).C) with [(CSort _) \Rightarrow x0 | (CHead c0 _ _) \Rightarrow c0])) -(CHead x0 (Bind x1) x2) (CHead x3 (Bind Abst) x4) (getl_mono c2 (CHead x0 -(Bind x1) x2) n H2 (CHead x3 (Bind Abst) x4) H8)) in ((let H12 \def (f_equal -C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with [(CSort _) -\Rightarrow x1 | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: -K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow x1])])) (CHead x0 -(Bind x1) x2) (CHead x3 (Bind Abst) x4) (getl_mono c2 (CHead x0 (Bind x1) x2) -n H2 (CHead x3 (Bind Abst) x4) H8)) in ((let H13 \def (f_equal C T (\lambda -(e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow x2 -| (CHead _ _ t) \Rightarrow t])) (CHead x0 (Bind x1) x2) (CHead x3 (Bind -Abst) x4) (getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind Abst) -x4) H8)) in (\lambda (_: (eq B x1 Abst)).(\lambda (H15: (eq C x0 x3)).(let -H16 \def (eq_ind_r T x4 (\lambda (t: T).(getl n c2 (CHead x3 (Bind Abst) t))) -H10 x2 H13) in (let H17 \def (eq_ind_r T x4 (\lambda (t: T).(ty3 g x3 t x5)) -H9 x2 H13) in (let H18 \def (eq_ind_r T x4 (\lambda (t: T).(pc3 c2 (lift (S -n) O t) t3)) H7 x2 H13) in (let H19 \def (eq_ind_r C x3 (\lambda (c0: -C).(getl n c2 (CHead c0 (Bind Abst) x2))) H16 x0 H15) in (let H20 \def -(eq_ind_r C x3 (\lambda (c0: C).(ty3 g c0 x2 x5)) H17 x0 H15) in (H4 x5 -H20))))))))) H12)) H11))))))))) H6)) (ty3_gen_lref g c2 t3 n H5)))))) -H3)))))) H1)) (\lambda (H1: ((\forall (d: C).((getl n c2 d) \to (\forall (P: -Prop).P))))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) -(\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to False)) (\lambda (t3: -T).(\lambda (H2: (ty3 g c2 (TLRef n) t3)).(or_ind (ex3_3 C T T (\lambda (_: -C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda -(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u))))) -(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T -T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) -t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e -(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u -t))))) False (\lambda (H3: (ex3_3 C T T (\lambda (_: C).(\lambda (_: -T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda (e: C).(\lambda -(u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u))))) (\lambda (e: -C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T -(\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) -t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e -(Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u -t)))) False (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: -(pc3 c2 (lift (S n) O x2) t3)).(\lambda (H5: (getl n c2 (CHead x0 (Bind Abbr) -x1))).(\lambda (_: (ty3 g x0 x1 x2)).(H1 (CHead x0 (Bind Abbr) x1) H5 -False))))))) H3)) (\lambda (H3: (ex3_3 C T T (\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e: C).(\lambda -(u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u))))) (\lambda (e: -C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T -(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) -t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e -(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u -t)))) False (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: -(pc3 c2 (lift (S n) O x1) t3)).(\lambda (H5: (getl n c2 (CHead x0 (Bind Abst) -x1))).(\lambda (_: (ty3 g x0 x1 x2)).(H1 (CHead x0 (Bind Abst) x1) H5 -False))))))) H3)) (ty3_gen_lref g c2 t3 n H2)))))) H0))))) (\lambda (k: -K).(\lambda (t: T).(\lambda (_: ((((\forall (c1: C).(\forall (t3: T).((flt c1 -t3 c2 t) \to (or (ex T (\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: -T).((ty3 g c1 t3 t4) \to False))))))) \to (or (ex T (\lambda (t3: T).(ty3 g -c2 t t3))) (\forall (t3: T).((ty3 g c2 t t3) \to False)))))).(\lambda (t0: -T).(\lambda (_: ((((\forall (c1: C).(\forall (t3: T).((flt c1 t3 c2 t0) \to -(or (ex T (\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3 g c1 t3 -t4) \to False))))))) \to (or (ex T (\lambda (t3: T).(ty3 g c2 t0 t3))) -(\forall (t3: T).((ty3 g c2 t0 t3) \to False)))))).(\lambda (H1: ((\forall -(c1: C).(\forall (t3: T).((flt c1 t3 c2 (THead k t t0)) \to (or (ex T -(\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to -False)))))))).(K_ind (\lambda (k0: K).(((\forall (c1: C).(\forall (t3: -T).((flt c1 t3 c2 (THead k0 t t0)) \to (or (ex T (\lambda (t4: T).(ty3 g c1 -t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to False))))))) \to (or (ex T -(\lambda (t3: T).(ty3 g c2 (THead k0 t t0) t3))) (\forall (t3: T).((ty3 g c2 -(THead k0 t t0) t3) \to False))))) (\lambda (b: B).(\lambda (H2: ((\forall -(c1: C).(\forall (t3: T).((flt c1 t3 c2 (THead (Bind b) t t0)) \to (or (ex T -(\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to -False)))))))).(let H3 \def (H2 c2 t (flt_thead_sx (Bind b) c2 t t0)) in -(or_ind (ex T (\lambda (t3: T).(ty3 g c2 t t3))) (\forall (t3: T).((ty3 g c2 -t t3) \to False)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Bind b) t t0) -t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0) t3) \to False))) -(\lambda (H4: (ex T (\lambda (t3: T).(ty3 g c2 t t3)))).(ex_ind T (\lambda -(t3: T).(ty3 g c2 t t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Bind b) -t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0) t3) \to -False))) (\lambda (x: T).(\lambda (H5: (ty3 g c2 t x)).(let H6 \def (H2 -(CHead c2 (Bind b) t) t0 (flt_shift (Bind b) c2 t t0)) in (or_ind (ex T -(\lambda (t3: T).(ty3 g (CHead c2 (Bind b) t) t0 t3))) (\forall (t3: T).((ty3 -g (CHead c2 (Bind b) t) t0 t3) \to False)) (or (ex T (\lambda (t3: T).(ty3 g -c2 (THead (Bind b) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t -t0) t3) \to False))) (\lambda (H7: (ex T (\lambda (t3: T).(ty3 g (CHead c2 -(Bind b) t) t0 t3)))).(ex_ind T (\lambda (t3: T).(ty3 g (CHead c2 (Bind b) t) -t0 t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Bind b) t t0) t3))) -(\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0) t3) \to False))) (\lambda -(x0: T).(\lambda (H8: (ty3 g (CHead c2 (Bind b) t) t0 x0)).(or_introl (ex T -(\lambda (t3: T).(ty3 g c2 (THead (Bind b) t t0) t3))) (\forall (t3: T).((ty3 -g c2 (THead (Bind b) t t0) t3) \to False)) (ex_intro T (\lambda (t3: T).(ty3 -g c2 (THead (Bind b) t t0) t3)) (THead (Bind b) t x0) (ty3_bind g c2 t x H5 b -t0 x0 H8))))) H7)) (\lambda (H7: ((\forall (t3: T).((ty3 g (CHead c2 (Bind b) -t) t0 t3) \to False)))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead -(Bind b) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0) t3) -\to False)) (\lambda (t3: T).(\lambda (H8: (ty3 g c2 (THead (Bind b) t t0) -t3)).(ex3_2_ind T T (\lambda (t4: T).(\lambda (_: T).(pc3 c2 (THead (Bind b) -t t4) t3))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c2 t t5))) (\lambda (t4: -T).(\lambda (_: T).(ty3 g (CHead c2 (Bind b) t) t0 t4))) False (\lambda (x0: -T).(\lambda (x1: T).(\lambda (_: (pc3 c2 (THead (Bind b) t x0) t3)).(\lambda -(_: (ty3 g c2 t x1)).(\lambda (H11: (ty3 g (CHead c2 (Bind b) t) t0 x0)).(H7 -x0 H11)))))) (ty3_gen_bind g b c2 t t0 t3 H8)))))) H6)))) H4)) (\lambda (H4: -((\forall (t3: T).((ty3 g c2 t t3) \to False)))).(or_intror (ex T (\lambda -(t3: T).(ty3 g c2 (THead (Bind b) t t0) t3))) (\forall (t3: T).((ty3 g c2 -(THead (Bind b) t t0) t3) \to False)) (\lambda (t3: T).(\lambda (H5: (ty3 g -c2 (THead (Bind b) t t0) t3)).(ex3_2_ind T T (\lambda (t4: T).(\lambda (_: -T).(pc3 c2 (THead (Bind b) t t4) t3))) (\lambda (_: T).(\lambda (t5: T).(ty3 -g c2 t t5))) (\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead c2 (Bind b) t) t0 -t4))) False (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 c2 (THead -(Bind b) t x0) t3)).(\lambda (H7: (ty3 g c2 t x1)).(\lambda (_: (ty3 g (CHead -c2 (Bind b) t) t0 x0)).(H4 x1 H7)))))) (ty3_gen_bind g b c2 t t0 t3 H5)))))) -H3)))) (\lambda (f: F).(\lambda (H2: ((\forall (c1: C).(\forall (t3: T).((flt -c1 t3 c2 (THead (Flat f) t t0)) \to (or (ex T (\lambda (t4: T).(ty3 g c1 t3 -t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to False)))))))).(F_ind (\lambda -(f0: F).(((\forall (c1: C).(\forall (t3: T).((flt c1 t3 c2 (THead (Flat f0) t -t0)) \to (or (ex T (\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3 -g c1 t3 t4) \to False))))))) \to (or (ex T (\lambda (t3: T).(ty3 g c2 (THead -(Flat f0) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat f0) t t0) t3) -\to False))))) (\lambda (H3: ((\forall (c1: C).(\forall (t3: T).((flt c1 t3 -c2 (THead (Flat Appl) t t0)) \to (or (ex T (\lambda (t4: T).(ty3 g c1 t3 -t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to False)))))))).(let H4 \def (H3 -c2 t (flt_thead_sx (Flat Appl) c2 t t0)) in (or_ind (ex T (\lambda (t3: -T).(ty3 g c2 t t3))) (\forall (t3: T).((ty3 g c2 t t3) \to False)) (or (ex T -(\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: -T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False))) (\lambda (H5: (ex T -(\lambda (t3: T).(ty3 g c2 t t3)))).(ex_ind T (\lambda (t3: T).(ty3 g c2 t -t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) -(\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False))) -(\lambda (x: T).(\lambda (H6: (ty3 g c2 t x)).(let H7 \def (H3 c2 t0 -(flt_thead_dx (Flat Appl) c2 t t0)) in (or_ind (ex T (\lambda (t3: T).(ty3 g -c2 t0 t3))) (\forall (t3: T).((ty3 g c2 t0 t3) \to False)) (or (ex T (\lambda -(t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 -(THead (Flat Appl) t t0) t3) \to False))) (\lambda (H8: (ex T (\lambda (t3: -T).(ty3 g c2 t0 t3)))).(ex_ind T (\lambda (t3: T).(ty3 g c2 t0 t3)) (or (ex T -(\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: -T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False))) (\lambda (x0: -T).(\lambda (H9: (ty3 g c2 t0 x0)).(ex_ind T (\lambda (t3: T).(ty3 g c2 x0 -t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) -(\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False))) -(\lambda (x1: T).(\lambda (H10: (ty3 g c2 x0 x1)).(ex_ind T (\lambda (t3: -T).(ty3 g c2 x t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t -t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to -False))) (\lambda (x2: T).(\lambda (H11: (ty3 g c2 x x2)).(let H12 \def -(ty3_sn3 g c2 x x2 H11) in (let H_x \def (nf2_sn3 c2 x H12) in (let H13 \def -H_x in (ex2_ind T (\lambda (u: T).(pr3 c2 x u)) (\lambda (u: T).(nf2 c2 u)) -(or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall -(t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False))) (\lambda (x3: -T).(\lambda (H14: (pr3 c2 x x3)).(\lambda (H15: (nf2 c2 x3)).(let H16 \def -(ty3_sred_pr3 c2 x x3 H14 g x2 H11) in (let H_x0 \def (pc3_abst_dec g c2 x0 -x1 H10 x3 x2 H16) in (let H17 \def H_x0 in (or_ind (ex4_2 T T (\lambda (u: -T).(\lambda (_: T).(pc3 c2 x0 (THead (Bind Abst) x3 u)))) (\lambda (u: -T).(\lambda (v2: T).(ty3 g c2 (THead (Bind Abst) v2 u) x1))) (\lambda (_: -T).(\lambda (v2: T).(pr3 c2 x3 v2))) (\lambda (_: T).(\lambda (v2: T).(nf2 c2 -v2)))) (\forall (u: T).((pc3 c2 x0 (THead (Bind Abst) x3 u)) \to False)) (or -(ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: -T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False))) (\lambda (H18: (ex4_2 -T T (\lambda (u: T).(\lambda (_: T).(pc3 c2 x0 (THead (Bind Abst) x3 u)))) -(\lambda (u: T).(\lambda (v2: T).(ty3 g c2 (THead (Bind Abst) v2 u) x1))) -(\lambda (_: T).(\lambda (v2: T).(pr3 c2 x3 v2))) (\lambda (_: T).(\lambda -(v2: T).(nf2 c2 v2))))).(ex4_2_ind T T (\lambda (u: T).(\lambda (_: T).(pc3 -c2 x0 (THead (Bind Abst) x3 u)))) (\lambda (u: T).(\lambda (v2: T).(ty3 g c2 -(THead (Bind Abst) v2 u) x1))) (\lambda (_: T).(\lambda (v2: T).(pr3 c2 x3 -v2))) (\lambda (_: T).(\lambda (v2: T).(nf2 c2 v2))) (or (ex T (\lambda (t3: -T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 -(THead (Flat Appl) t t0) t3) \to False))) (\lambda (x4: T).(\lambda (x5: -T).(\lambda (H19: (pc3 c2 x0 (THead (Bind Abst) x3 x4))).(\lambda (H20: (ty3 -g c2 (THead (Bind Abst) x5 x4) x1)).(\lambda (H21: (pr3 c2 x3 x5)).(\lambda -(_: (nf2 c2 x5)).(let H_y \def (nf2_pr3_unfold c2 x3 x5 H21 H15) in (let H23 -\def (eq_ind_r T x5 (\lambda (t3: T).(pr3 c2 x3 t3)) H21 x3 H_y) in (let H24 -\def (eq_ind_r T x5 (\lambda (t3: T).(ty3 g c2 (THead (Bind Abst) t3 x4) x1)) -H20 x3 H_y) in (or_introl (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) -t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to -False)) (ex_intro T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3)) -(THead (Flat Appl) t (THead (Bind Abst) x3 x4)) (ty3_appl g c2 t x3 (ty3_tred -g c2 t x H6 x3 H14) t0 x4 (ty3_conv g c2 (THead (Bind Abst) x3 x4) x1 H24 t0 -x0 H9 H19))))))))))))) H18)) (\lambda (H18: ((\forall (u: T).((pc3 c2 x0 -(THead (Bind Abst) x3 u)) \to False)))).(or_intror (ex T (\lambda (t3: -T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 -(THead (Flat Appl) t t0) t3) \to False)) (\lambda (t3: T).(\lambda (H19: (ty3 -g c2 (THead (Flat Appl) t t0) t3)).(ex3_2_ind T T (\lambda (u: T).(\lambda -(t4: T).(pc3 c2 (THead (Flat Appl) t (THead (Bind Abst) u t4)) t3))) (\lambda -(u: T).(\lambda (t4: T).(ty3 g c2 t0 (THead (Bind Abst) u t4)))) (\lambda (u: -T).(\lambda (_: T).(ty3 g c2 t u))) False (\lambda (x4: T).(\lambda (x5: -T).(\lambda (_: (pc3 c2 (THead (Flat Appl) t (THead (Bind Abst) x4 x5)) -t3)).(\lambda (H21: (ty3 g c2 t0 (THead (Bind Abst) x4 x5))).(\lambda (H22: -(ty3 g c2 t x4)).(let H_y \def (ty3_unique g c2 t x4 H22 x H6) in (let H_y0 -\def (ty3_unique g c2 t0 (THead (Bind Abst) x4 x5) H21 x0 H9) in (H18 x5 -(pc3_t (THead (Bind Abst) x4 x5) c2 x0 (pc3_s c2 x0 (THead (Bind Abst) x4 x5) -H_y0) (THead (Bind Abst) x3 x5) (pc3_head_1 c2 x4 x3 (pc3_t x c2 x4 H_y x3 -(pc3_pr3_r c2 x x3 H14)) (Bind Abst) x5)))))))))) (ty3_gen_appl g c2 t t0 t3 -H19)))))) H17))))))) H13)))))) (ty3_correct g c2 t x H6)))) (ty3_correct g c2 -t0 x0 H9)))) H8)) (\lambda (H8: ((\forall (t3: T).((ty3 g c2 t0 t3) \to -False)))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t -t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to -False)) (\lambda (t3: T).(\lambda (H9: (ty3 g c2 (THead (Flat Appl) t t0) -t3)).(ex3_2_ind T T (\lambda (u: T).(\lambda (t4: T).(pc3 c2 (THead (Flat -Appl) t (THead (Bind Abst) u t4)) t3))) (\lambda (u: T).(\lambda (t4: T).(ty3 -g c2 t0 (THead (Bind Abst) u t4)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c2 -t u))) False (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 c2 (THead -(Flat Appl) t (THead (Bind Abst) x0 x1)) t3)).(\lambda (H11: (ty3 g c2 t0 -(THead (Bind Abst) x0 x1))).(\lambda (_: (ty3 g c2 t x0)).(H8 (THead (Bind -Abst) x0 x1) H11)))))) (ty3_gen_appl g c2 t t0 t3 H9)))))) H7)))) H5)) -(\lambda (H5: ((\forall (t3: T).((ty3 g c2 t t3) \to False)))).(or_intror (ex -T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: -T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False)) (\lambda (t3: -T).(\lambda (H6: (ty3 g c2 (THead (Flat Appl) t t0) t3)).(ex3_2_ind T T -(\lambda (u: T).(\lambda (t4: T).(pc3 c2 (THead (Flat Appl) t (THead (Bind -Abst) u t4)) t3))) (\lambda (u: T).(\lambda (t4: T).(ty3 g c2 t0 (THead (Bind -Abst) u t4)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c2 t u))) False -(\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 c2 (THead (Flat Appl) t -(THead (Bind Abst) x0 x1)) t3)).(\lambda (_: (ty3 g c2 t0 (THead (Bind Abst) -x0 x1))).(\lambda (H9: (ty3 g c2 t x0)).(H5 x0 H9)))))) (ty3_gen_appl g c2 t -t0 t3 H6)))))) H4))) (\lambda (H3: ((\forall (c1: C).(\forall (t3: T).((flt -c1 t3 c2 (THead (Flat Cast) t t0)) \to (or (ex T (\lambda (t4: T).(ty3 g c1 -t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to False)))))))).(let H4 \def -(H3 c2 t (flt_thead_sx (Flat Cast) c2 t t0)) in (or_ind (ex T (\lambda (t3: -T).(ty3 g c2 t t3))) (\forall (t3: T).((ty3 g c2 t t3) \to False)) (or (ex T -(\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3: -T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False))) (\lambda (H5: (ex T -(\lambda (t3: T).(ty3 g c2 t t3)))).(ex_ind T (\lambda (t3: T).(ty3 g c2 t -t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) -(\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False))) -(\lambda (x: T).(\lambda (H6: (ty3 g c2 t x)).(let H7 \def (H3 c2 t0 -(flt_thead_dx (Flat Cast) c2 t t0)) in (or_ind (ex T (\lambda (t3: T).(ty3 g -c2 t0 t3))) (\forall (t3: T).((ty3 g c2 t0 t3) \to False)) (or (ex T (\lambda -(t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3: T).((ty3 g c2 -(THead (Flat Cast) t t0) t3) \to False))) (\lambda (H8: (ex T (\lambda (t3: -T).(ty3 g c2 t0 t3)))).(ex_ind T (\lambda (t3: T).(ty3 g c2 t0 t3)) (or (ex T -(\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3: -T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False))) (\lambda (x0: -T).(\lambda (H9: (ty3 g c2 t0 x0)).(ex_ind T (\lambda (t3: T).(ty3 g c2 x0 -t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) -(\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False))) -(\lambda (x1: T).(\lambda (H10: (ty3 g c2 x0 x1)).(let H_x \def (pc3_dec g c2 -x0 x1 H10 t x H6) in (let H11 \def H_x in (or_ind (pc3 c2 x0 t) ((pc3 c2 x0 -t) \to False) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) -t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False))) -(\lambda (H12: (pc3 c2 x0 t)).(or_introl (ex T (\lambda (t3: T).(ty3 g c2 -(THead (Flat Cast) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) -t t0) t3) \to False)) (ex_intro T (\lambda (t3: T).(ty3 g c2 (THead (Flat -Cast) t t0) t3)) (THead (Flat Cast) x t) (ty3_cast g c2 t0 t (ty3_conv g c2 t -x H6 t0 x0 H9 H12) x H6)))) (\lambda (H12: (((pc3 c2 x0 t) \to -False))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) -t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False)) -(\lambda (t3: T).(\lambda (H13: (ty3 g c2 (THead (Flat Cast) t t0) -t3)).(ex3_ind T (\lambda (t4: T).(pc3 c2 (THead (Flat Cast) t4 t) t3)) -(\lambda (_: T).(ty3 g c2 t0 t)) (\lambda (t4: T).(ty3 g c2 t t4)) False -(\lambda (x2: T).(\lambda (_: (pc3 c2 (THead (Flat Cast) x2 t) t3)).(\lambda -(H15: (ty3 g c2 t0 t)).(\lambda (H16: (ty3 g c2 t x2)).(let H_y \def -(ty3_unique g c2 t x2 H16 x H6) in (let H_y0 \def (ty3_unique g c2 t0 t H15 -x0 H9) in (H12 (ex2_sym T (pr3 c2 t) (pr3 c2 x0) H_y0)))))))) (ty3_gen_cast g -c2 t0 t t3 H13)))))) H11))))) (ty3_correct g c2 t0 x0 H9)))) H8)) (\lambda -(H8: ((\forall (t3: T).((ty3 g c2 t0 t3) \to False)))).(or_intror (ex T -(\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3: -T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False)) (\lambda (t3: -T).(\lambda (H9: (ty3 g c2 (THead (Flat Cast) t t0) t3)).(ex3_ind T (\lambda -(t4: T).(pc3 c2 (THead (Flat Cast) t4 t) t3)) (\lambda (_: T).(ty3 g c2 t0 -t)) (\lambda (t4: T).(ty3 g c2 t t4)) False (\lambda (x0: T).(\lambda (_: -(pc3 c2 (THead (Flat Cast) x0 t) t3)).(\lambda (H11: (ty3 g c2 t0 -t)).(\lambda (_: (ty3 g c2 t x0)).(H8 t H11))))) (ty3_gen_cast g c2 t0 t t3 -H9)))))) H7)))) H5)) (\lambda (H5: ((\forall (t3: T).((ty3 g c2 t t3) \to -False)))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t -t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to -False)) (\lambda (t3: T).(\lambda (H6: (ty3 g c2 (THead (Flat Cast) t t0) -t3)).(ex3_ind T (\lambda (t4: T).(pc3 c2 (THead (Flat Cast) t4 t) t3)) -(\lambda (_: T).(ty3 g c2 t0 t)) (\lambda (t4: T).(ty3 g c2 t t4)) False -(\lambda (x0: T).(\lambda (_: (pc3 c2 (THead (Flat Cast) x0 t) t3)).(\lambda -(_: (ty3 g c2 t0 t)).(\lambda (H9: (ty3 g c2 t x0)).(ex_ind T (\lambda (t4: -T).(ty3 g c2 x0 t4)) False (\lambda (x: T).(\lambda (_: (ty3 g c2 x0 x)).(H5 -x0 H9))) (ty3_correct g c2 t x0 H9)))))) (ty3_gen_cast g c2 t0 t t3 H6)))))) -H4))) f H2))) k H1))))))) t2))) c t1))). -(* COMMENTS -Initial nodes: 9001 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/defs.ma deleted file mode 100644 index d1f500cdf..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/defs.ma +++ /dev/null @@ -1,49 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/G/defs.ma". - -include "Basic-1/pc3/defs.ma". - -inductive ty3 (g: G): C \to (T \to (T \to Prop)) \def -| ty3_conv: \forall (c: C).(\forall (t2: T).(\forall (t: T).((ty3 g c t2 t) -\to (\forall (u: T).(\forall (t1: T).((ty3 g c u t1) \to ((pc3 c t1 t2) \to -(ty3 g c u t2)))))))) -| ty3_sort: \forall (c: C).(\forall (m: nat).(ty3 g c (TSort m) (TSort (next -g m)))) -| ty3_abbr: \forall (n: nat).(\forall (c: C).(\forall (d: C).(\forall (u: -T).((getl n c (CHead d (Bind Abbr) u)) \to (\forall (t: T).((ty3 g d u t) \to -(ty3 g c (TLRef n) (lift (S n) O t)))))))) -| ty3_abst: \forall (n: nat).(\forall (c: C).(\forall (d: C).(\forall (u: -T).((getl n c (CHead d (Bind Abst) u)) \to (\forall (t: T).((ty3 g d u t) \to -(ty3 g c (TLRef n) (lift (S n) O u)))))))) -| ty3_bind: \forall (c: C).(\forall (u: T).(\forall (t: T).((ty3 g c u t) \to -(\forall (b: B).(\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c (Bind b) -u) t1 t2) \to (ty3 g c (THead (Bind b) u t1) (THead (Bind b) u t2))))))))) -| ty3_appl: \forall (c: C).(\forall (w: T).(\forall (u: T).((ty3 g c w u) \to -(\forall (v: T).(\forall (t: T).((ty3 g c v (THead (Bind Abst) u t)) \to (ty3 -g c (THead (Flat Appl) w v) (THead (Flat Appl) w (THead (Bind Abst) u -t))))))))) -| ty3_cast: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c t1 t2) -\to (\forall (t0: T).((ty3 g c t2 t0) \to (ty3 g c (THead (Flat Cast) t2 t1) -(THead (Flat Cast) t0 t2))))))). - -inductive tys3 (g: G) (c: C): TList \to (T \to Prop) \def -| tys3_nil: \forall (u: T).(\forall (u0: T).((ty3 g c u u0) \to (tys3 g c -TNil u))) -| tys3_cons: \forall (t: T).(\forall (u: T).((ty3 g c t u) \to (\forall (ts: -TList).((tys3 g c ts u) \to (tys3 g c (TCons t ts) u))))). - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/fsubst0.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/fsubst0.ma deleted file mode 100644 index f92366a26..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/fsubst0.ma +++ /dev/null @@ -1,995 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/ty3/props.ma". - -include "Basic-1/pc3/fsubst0.ma". - -include "Basic-1/getl/getl.ma". - -theorem ty3_fsubst0: - \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t: T).((ty3 g c1 -t1 t) \to (\forall (i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t2: -T).((fsubst0 i u c1 t1 c2 t2) \to (\forall (e: C).((getl i c1 (CHead e (Bind -Abbr) u)) \to (ty3 g c2 t2 t)))))))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t: T).(\lambda -(H: (ty3 g c1 t1 t)).(ty3_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda -(t2: T).(\forall (i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t3: -T).((fsubst0 i u c t0 c2 t3) \to (\forall (e: C).((getl i c (CHead e (Bind -Abbr) u)) \to (ty3 g c2 t3 t2))))))))))) (\lambda (c: C).(\lambda (t2: -T).(\lambda (t0: T).(\lambda (H0: (ty3 g c t2 t0)).(\lambda (H1: ((\forall -(i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t3: T).((fsubst0 i u c t2 -c2 t3) \to (\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) \to (ty3 g c2 -t3 t0)))))))))).(\lambda (u: T).(\lambda (t3: T).(\lambda (_: (ty3 g c u -t3)).(\lambda (H3: ((\forall (i: nat).(\forall (u0: T).(\forall (c2: -C).(\forall (t4: T).((fsubst0 i u0 c u c2 t4) \to (\forall (e: C).((getl i c -(CHead e (Bind Abbr) u0)) \to (ty3 g c2 t4 t3)))))))))).(\lambda (H4: (pc3 c -t3 t2)).(\lambda (i: nat).(\lambda (u0: T).(\lambda (c2: C).(\lambda (t4: -T).(\lambda (H5: (fsubst0 i u0 c u c2 t4)).(fsubst0_ind i u0 c u (\lambda -(c0: C).(\lambda (t5: T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u0)) -\to (ty3 g c0 t5 t2))))) (\lambda (t5: T).(\lambda (H6: (subst0 i u0 u -t5)).(\lambda (e: C).(\lambda (H7: (getl i c (CHead e (Bind Abbr) -u0))).(ty3_conv g c t2 t0 H0 t5 t3 (H3 i u0 c t5 (fsubst0_snd i u0 c u t5 H6) -e H7) H4))))) (\lambda (c3: C).(\lambda (H6: (csubst0 i u0 c c3)).(\lambda -(e: C).(\lambda (H7: (getl i c (CHead e (Bind Abbr) u0))).(ty3_conv g c3 t2 -t0 (H1 i u0 c3 t2 (fsubst0_fst i u0 c t2 c3 H6) e H7) u t3 (H3 i u0 c3 u -(fsubst0_fst i u0 c u c3 H6) e H7) (pc3_fsubst0 c t3 t2 H4 i u0 c3 t3 -(fsubst0_fst i u0 c t3 c3 H6) e H7)))))) (\lambda (t5: T).(\lambda (H6: -(subst0 i u0 u t5)).(\lambda (c3: C).(\lambda (H7: (csubst0 i u0 c -c3)).(\lambda (e: C).(\lambda (H8: (getl i c (CHead e (Bind Abbr) -u0))).(ty3_conv g c3 t2 t0 (H1 i u0 c3 t2 (fsubst0_fst i u0 c t2 c3 H7) e H8) -t5 t3 (H3 i u0 c3 t5 (fsubst0_both i u0 c u t5 H6 c3 H7) e H8) (pc3_fsubst0 c -t3 t2 H4 i u0 c3 t3 (fsubst0_fst i u0 c t3 c3 H7) e H8)))))))) c2 t4 -H5)))))))))))))))) (\lambda (c: C).(\lambda (m: nat).(\lambda (i: -nat).(\lambda (u: T).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H0: (fsubst0 -i u c (TSort m) c2 t2)).(fsubst0_ind i u c (TSort m) (\lambda (c0: -C).(\lambda (t0: T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) \to -(ty3 g c0 t0 (TSort (next g m))))))) (\lambda (t3: T).(\lambda (H1: (subst0 i -u (TSort m) t3)).(\lambda (e: C).(\lambda (_: (getl i c (CHead e (Bind Abbr) -u))).(subst0_gen_sort u t3 i m H1 (ty3 g c t3 (TSort (next g m)))))))) -(\lambda (c3: C).(\lambda (_: (csubst0 i u c c3)).(\lambda (e: C).(\lambda -(_: (getl i c (CHead e (Bind Abbr) u))).(ty3_sort g c3 m))))) (\lambda (t3: -T).(\lambda (H1: (subst0 i u (TSort m) t3)).(\lambda (c3: C).(\lambda (_: -(csubst0 i u c c3)).(\lambda (e: C).(\lambda (_: (getl i c (CHead e (Bind -Abbr) u))).(subst0_gen_sort u t3 i m H1 (ty3 g c3 t3 (TSort (next g -m)))))))))) c2 t2 H0)))))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d: -C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind Abbr) u))).(\lambda -(t0: T).(\lambda (H1: (ty3 g d u t0)).(\lambda (H2: ((\forall (i: -nat).(\forall (u0: T).(\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 d u c2 -t2) \to (\forall (e: C).((getl i d (CHead e (Bind Abbr) u0)) \to (ty3 g c2 t2 -t0)))))))))).(\lambda (i: nat).(\lambda (u0: T).(\lambda (c2: C).(\lambda -(t2: T).(\lambda (H3: (fsubst0 i u0 c (TLRef n) c2 t2)).(fsubst0_ind i u0 c -(TLRef n) (\lambda (c0: C).(\lambda (t3: T).(\forall (e: C).((getl i c (CHead -e (Bind Abbr) u0)) \to (ty3 g c0 t3 (lift (S n) O t0)))))) (\lambda (t3: -T).(\lambda (H4: (subst0 i u0 (TLRef n) t3)).(\lambda (e: C).(\lambda (H5: -(getl i c (CHead e (Bind Abbr) u0))).(land_ind (eq nat n i) (eq T t3 (lift (S -n) O u0)) (ty3 g c t3 (lift (S n) O t0)) (\lambda (H6: (eq nat n i)).(\lambda -(H7: (eq T t3 (lift (S n) O u0))).(eq_ind_r T (lift (S n) O u0) (\lambda (t4: -T).(ty3 g c t4 (lift (S n) O t0))) (let H8 \def (eq_ind_r nat i (\lambda (n0: -nat).(getl n0 c (CHead e (Bind Abbr) u0))) H5 n H6) in (let H9 \def (eq_ind C -(CHead d (Bind Abbr) u) (\lambda (c0: C).(getl n c c0)) H0 (CHead e (Bind -Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) n H0 (CHead e (Bind Abbr) u0) -H8)) in (let H10 \def (f_equal C C (\lambda (e0: C).(match e0 in C return -(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow -c0])) (CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono c (CHead d -(Bind Abbr) u) n H0 (CHead e (Bind Abbr) u0) H8)) in ((let H11 \def (f_equal -C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) -\Rightarrow u | (CHead _ _ t4) \Rightarrow t4])) (CHead d (Bind Abbr) u) -(CHead e (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) n H0 (CHead e -(Bind Abbr) u0) H8)) in (\lambda (H12: (eq C d e)).(let H13 \def (eq_ind_r C -e (\lambda (c0: C).(getl n c (CHead c0 (Bind Abbr) u0))) H9 d H12) in (let -H14 \def (eq_ind_r T u0 (\lambda (t4: T).(getl n c (CHead d (Bind Abbr) t4))) -H13 u H11) in (eq_ind T u (\lambda (t4: T).(ty3 g c (lift (S n) O t4) (lift -(S n) O t0))) (ty3_lift g d u t0 H1 c O (S n) (getl_drop Abbr c d u n H14)) -u0 H11))))) H10)))) t3 H7))) (subst0_gen_lref u0 t3 i n H4)))))) (\lambda -(c3: C).(\lambda (H4: (csubst0 i u0 c c3)).(\lambda (e: C).(\lambda (H5: -(getl i c (CHead e (Bind Abbr) u0))).(lt_le_e n i (ty3 g c3 (TLRef n) (lift -(S n) O t0)) (\lambda (H6: (lt n i)).(let H7 \def (csubst0_getl_lt i n H6 c -c3 u0 H4 (CHead d (Bind Abbr) u) H0) in (or4_ind (getl n c3 (CHead d (Bind -Abbr) u)) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b) u1)))))) -(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 -(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: -T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w)))))) (ex3_4 B C C T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C -(CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda -(_: C).(\lambda (e2: C).(\lambda (u1: T).(getl n c3 (CHead e2 (Bind b) -u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus i (S n)) u0 e1 e2)))))) (ex4_5 B C C T T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C -(CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e2 -(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -(minus i (S n)) u0 e1 e2))))))) (ty3 g c3 (TLRef n) (lift (S n) O t0)) -(\lambda (H8: (getl n c3 (CHead d (Bind Abbr) u))).(ty3_abbr g n c3 d u H8 t0 -H1)) (\lambda (H8: (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda -(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b) -u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(getl n c3 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 -w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b) u1)))))) -(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 -(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: -T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w))))) (ty3 g c3 (TLRef n) -(lift (S n) O t0)) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: -T).(\lambda (x3: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead x1 -(Bind x0) x2))).(\lambda (H10: (getl n c3 (CHead x1 (Bind x0) x3))).(\lambda -(H11: (subst0 (minus i (S n)) u0 x2 x3)).(let H12 \def (f_equal C C (\lambda -(e0: C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow -d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abbr) u) (CHead x1 (Bind -x0) x2) H9) in ((let H13 \def (f_equal C B (\lambda (e0: C).(match e0 in C -return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) -\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead -x1 (Bind x0) x2) H9) in ((let H14 \def (f_equal C T (\lambda (e0: C).(match -e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ -t3) \Rightarrow t3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H9) in -(\lambda (H15: (eq B Abbr x0)).(\lambda (H16: (eq C d x1)).(let H17 \def -(eq_ind_r T x2 (\lambda (t3: T).(subst0 (minus i (S n)) u0 t3 x3)) H11 u H14) -in (let H18 \def (eq_ind_r C x1 (\lambda (c0: C).(getl n c3 (CHead c0 (Bind -x0) x3))) H10 d H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b: B).(getl n -c3 (CHead d (Bind b) x3))) H18 Abbr H15) in (let H20 \def (eq_ind nat (minus -i n) (\lambda (n0: nat).(getl n0 (CHead d (Bind Abbr) x3) (CHead e (Bind -Abbr) u0))) (getl_conf_le i (CHead e (Bind Abbr) u0) c3 (csubst0_getl_ge i i -(le_n i) c c3 u0 H4 (CHead e (Bind Abbr) u0) H5) (CHead d (Bind Abbr) x3) n -H19 (le_S_n n i (le_S (S n) i H6))) (S (minus i (S n))) (minus_x_Sy i n H6)) -in (ty3_abbr g n c3 d x3 H19 t0 (H2 (minus i (S n)) u0 d x3 (fsubst0_snd -(minus i (S n)) u0 d u x3 H17) e (getl_gen_S (Bind Abbr) d (CHead e (Bind -Abbr) u0) x3 (minus i (S n)) H20)))))))))) H13)) H12))))))))) H8)) (\lambda -(H8: (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda -(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl n c3 (CHead e2 -(Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 (minus i (S n)) u0 e1 e2))))))).(ex3_4_ind B C C T (\lambda -(b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind -Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u1: T).(getl n c3 (CHead e2 (Bind b) u1)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) -u0 e1 e2))))) (ty3 g c3 (TLRef n) (lift (S n) O t0)) (\lambda (x0: -B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H9: (eq C -(CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3))).(\lambda (H10: (getl n c3 -(CHead x2 (Bind x0) x3))).(\lambda (H11: (csubst0 (minus i (S n)) u0 x1 -x2)).(let H12 \def (f_equal C C (\lambda (e0: C).(match e0 in C return -(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow -c0])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H13 \def -(f_equal C B (\lambda (e0: C).(match e0 in C return (\lambda (_: C).B) with -[(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k in K return -(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow -Abbr])])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H14 -\def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow u | (CHead _ _ t3) \Rightarrow t3])) (CHead d -(Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in (\lambda (H15: (eq B Abbr -x0)).(\lambda (H16: (eq C d x1)).(let H17 \def (eq_ind_r T x3 (\lambda (t3: -T).(getl n c3 (CHead x2 (Bind x0) t3))) H10 u H14) in (let H18 \def (eq_ind_r -C x1 (\lambda (c0: C).(csubst0 (minus i (S n)) u0 c0 x2)) H11 d H16) in (let -H19 \def (eq_ind_r B x0 (\lambda (b: B).(getl n c3 (CHead x2 (Bind b) u))) -H17 Abbr H15) in (let H20 \def (eq_ind nat (minus i n) (\lambda (n0: -nat).(getl n0 (CHead x2 (Bind Abbr) u) (CHead e (Bind Abbr) u0))) -(getl_conf_le i (CHead e (Bind Abbr) u0) c3 (csubst0_getl_ge i i (le_n i) c -c3 u0 H4 (CHead e (Bind Abbr) u0) H5) (CHead x2 (Bind Abbr) u) n H19 (le_S_n -n i (le_S (S n) i H6))) (S (minus i (S n))) (minus_x_Sy i n H6)) in (ty3_abbr -g n c3 x2 u H19 t0 (H2 (minus i (S n)) u0 x2 u (fsubst0_fst (minus i (S n)) -u0 d u x2 H18) e (csubst0_getl_ge_back (minus i (S n)) (minus i (S n)) (le_n -(minus i (S n))) d x2 u0 H18 (CHead e (Bind Abbr) u0) (getl_gen_S (Bind Abbr) -x2 (CHead e (Bind Abbr) u0) u (minus i (S n)) H20))))))))))) H13)) -H12))))))))) H8)) (\lambda (H8: (ex4_5 B C C T T (\lambda (b: B).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind -Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e2 -(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -(minus i (S n)) u0 e1 e2)))))))).(ex4_5_ind B C C T T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C -(CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e2 -(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -(minus i (S n)) u0 e1 e2)))))) (ty3 g c3 (TLRef n) (lift (S n) O t0)) -(\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda -(x4: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) -x3))).(\lambda (H10: (getl n c3 (CHead x2 (Bind x0) x4))).(\lambda (H11: -(subst0 (minus i (S n)) u0 x3 x4)).(\lambda (H12: (csubst0 (minus i (S n)) u0 -x1 x2)).(let H13 \def (f_equal C C (\lambda (e0: C).(match e0 in C return -(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow -c0])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H14 \def -(f_equal C B (\lambda (e0: C).(match e0 in C return (\lambda (_: C).B) with -[(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k in K return -(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow -Abbr])])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H15 -\def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow u | (CHead _ _ t3) \Rightarrow t3])) (CHead d -(Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in (\lambda (H16: (eq B Abbr -x0)).(\lambda (H17: (eq C d x1)).(let H18 \def (eq_ind_r T x3 (\lambda (t3: -T).(subst0 (minus i (S n)) u0 t3 x4)) H11 u H15) in (let H19 \def (eq_ind_r C -x1 (\lambda (c0: C).(csubst0 (minus i (S n)) u0 c0 x2)) H12 d H17) in (let -H20 \def (eq_ind_r B x0 (\lambda (b: B).(getl n c3 (CHead x2 (Bind b) x4))) -H10 Abbr H16) in (let H21 \def (eq_ind nat (minus i n) (\lambda (n0: -nat).(getl n0 (CHead x2 (Bind Abbr) x4) (CHead e (Bind Abbr) u0))) -(getl_conf_le i (CHead e (Bind Abbr) u0) c3 (csubst0_getl_ge i i (le_n i) c -c3 u0 H4 (CHead e (Bind Abbr) u0) H5) (CHead x2 (Bind Abbr) x4) n H20 (le_S_n -n i (le_S (S n) i H6))) (S (minus i (S n))) (minus_x_Sy i n H6)) in (ty3_abbr -g n c3 x2 x4 H20 t0 (H2 (minus i (S n)) u0 x2 x4 (fsubst0_both (minus i (S -n)) u0 d u x4 H18 x2 H19) e (csubst0_getl_ge_back (minus i (S n)) (minus i (S -n)) (le_n (minus i (S n))) d x2 u0 H19 (CHead e (Bind Abbr) u0) (getl_gen_S -(Bind Abbr) x2 (CHead e (Bind Abbr) u0) x4 (minus i (S n)) H21))))))))))) -H14)) H13))))))))))) H8)) H7))) (\lambda (H6: (le i n)).(ty3_abbr g n c3 d u -(csubst0_getl_ge i n H6 c c3 u0 H4 (CHead d (Bind Abbr) u) H0) t0 H1))))))) -(\lambda (t3: T).(\lambda (H4: (subst0 i u0 (TLRef n) t3)).(\lambda (c3: -C).(\lambda (H5: (csubst0 i u0 c c3)).(\lambda (e: C).(\lambda (H6: (getl i c -(CHead e (Bind Abbr) u0))).(land_ind (eq nat n i) (eq T t3 (lift (S n) O u0)) -(ty3 g c3 t3 (lift (S n) O t0)) (\lambda (H7: (eq nat n i)).(\lambda (H8: (eq -T t3 (lift (S n) O u0))).(eq_ind_r T (lift (S n) O u0) (\lambda (t4: T).(ty3 -g c3 t4 (lift (S n) O t0))) (let H9 \def (eq_ind_r nat i (\lambda (n0: -nat).(getl n0 c (CHead e (Bind Abbr) u0))) H6 n H7) in (let H10 \def -(eq_ind_r nat i (\lambda (n0: nat).(csubst0 n0 u0 c c3)) H5 n H7) in (let H11 -\def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c0: C).(getl n c c0)) H0 -(CHead e (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) n H0 (CHead e -(Bind Abbr) u0) H9)) in (let H12 \def (f_equal C C (\lambda (e0: C).(match e0 -in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) -\Rightarrow c0])) (CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono -c (CHead d (Bind Abbr) u) n H0 (CHead e (Bind Abbr) u0) H9)) in ((let H13 -\def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow u | (CHead _ _ t4) \Rightarrow t4])) (CHead d -(Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) -n H0 (CHead e (Bind Abbr) u0) H9)) in (\lambda (H14: (eq C d e)).(let H15 -\def (eq_ind_r C e (\lambda (c0: C).(getl n c (CHead c0 (Bind Abbr) u0))) H11 -d H14) in (let H16 \def (eq_ind_r T u0 (\lambda (t4: T).(getl n c (CHead d -(Bind Abbr) t4))) H15 u H13) in (let H17 \def (eq_ind_r T u0 (\lambda (t4: -T).(csubst0 n t4 c c3)) H10 u H13) in (eq_ind T u (\lambda (t4: T).(ty3 g c3 -(lift (S n) O t4) (lift (S n) O t0))) (ty3_lift g d u t0 H1 c3 O (S n) -(getl_drop Abbr c3 d u n (csubst0_getl_ge n n (le_n n) c c3 u H17 (CHead d -(Bind Abbr) u) H16))) u0 H13)))))) H12))))) t3 H8))) (subst0_gen_lref u0 t3 i -n H4)))))))) c2 t2 H3)))))))))))))) (\lambda (n: nat).(\lambda (c: -C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind -Abst) u))).(\lambda (t0: T).(\lambda (H1: (ty3 g d u t0)).(\lambda (H2: -((\forall (i: nat).(\forall (u0: T).(\forall (c2: C).(\forall (t2: -T).((fsubst0 i u0 d u c2 t2) \to (\forall (e: C).((getl i d (CHead e (Bind -Abbr) u0)) \to (ty3 g c2 t2 t0)))))))))).(\lambda (i: nat).(\lambda (u0: -T).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H3: (fsubst0 i u0 c (TLRef n) -c2 t2)).(fsubst0_ind i u0 c (TLRef n) (\lambda (c0: C).(\lambda (t3: -T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u0)) \to (ty3 g c0 t3 -(lift (S n) O u)))))) (\lambda (t3: T).(\lambda (H4: (subst0 i u0 (TLRef n) -t3)).(\lambda (e: C).(\lambda (H5: (getl i c (CHead e (Bind Abbr) -u0))).(land_ind (eq nat n i) (eq T t3 (lift (S n) O u0)) (ty3 g c t3 (lift (S -n) O u)) (\lambda (H6: (eq nat n i)).(\lambda (H7: (eq T t3 (lift (S n) O -u0))).(eq_ind_r T (lift (S n) O u0) (\lambda (t4: T).(ty3 g c t4 (lift (S n) -O u))) (let H8 \def (eq_ind_r nat i (\lambda (n0: nat).(getl n0 c (CHead e -(Bind Abbr) u0))) H5 n H6) in (let H9 \def (eq_ind C (CHead d (Bind Abst) u) -(\lambda (c0: C).(getl n c c0)) H0 (CHead e (Bind Abbr) u0) (getl_mono c -(CHead d (Bind Abst) u) n H0 (CHead e (Bind Abbr) u0) H8)) in (let H10 \def -(eq_ind C (CHead d (Bind Abst) u) (\lambda (ee: C).(match ee in C return -(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b) -\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow -False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) -\Rightarrow False])])) I (CHead e (Bind Abbr) u0) (getl_mono c (CHead d (Bind -Abst) u) n H0 (CHead e (Bind Abbr) u0) H8)) in (False_ind (ty3 g c (lift (S -n) O u0) (lift (S n) O u)) H10)))) t3 H7))) (subst0_gen_lref u0 t3 i n -H4)))))) (\lambda (c3: C).(\lambda (H4: (csubst0 i u0 c c3)).(\lambda (e: -C).(\lambda (H5: (getl i c (CHead e (Bind Abbr) u0))).(lt_le_e n i (ty3 g c3 -(TLRef n) (lift (S n) O u)) (\lambda (H6: (lt n i)).(let H7 \def -(csubst0_getl_lt i n H6 c c3 u0 H4 (CHead d (Bind Abst) u) H0) in (or4_ind -(getl n c3 (CHead d (Bind Abst) u)) (ex3_4 B C T T (\lambda (b: B).(\lambda -(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead -e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(getl n c3 (CHead e0 (Bind b) w)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i (S n)) -u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1)))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl n c3 -(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus i (S n)) u0 e1 e2)))))) (ex4_5 B C C T T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1))))))) (\lambda -(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl -n c3 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus i (S n)) u0 e1 e2))))))) (ty3 g c3 (TLRef n) (lift (S -n) O u)) (\lambda (H8: (getl n c3 (CHead d (Bind Abst) u))).(ty3_abst g n c3 -d u H8 t0 H1)) (\lambda (H8: (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead e0 -(Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda -(w: T).(getl n c3 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 -w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead e0 (Bind b) u1)))))) -(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 -(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: -T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w))))) (ty3 g c3 (TLRef n) -(lift (S n) O u)) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda -(x3: T).(\lambda (H9: (eq C (CHead d (Bind Abst) u) (CHead x1 (Bind x0) -x2))).(\lambda (H10: (getl n c3 (CHead x1 (Bind x0) x3))).(\lambda (H11: -(subst0 (minus i (S n)) u0 x2 x3)).(let H12 \def (f_equal C C (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | -(CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) -x2) H9) in ((let H13 \def (f_equal C B (\lambda (e0: C).(match e0 in C return -(\lambda (_: C).B) with [(CSort _) \Rightarrow Abst | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) -\Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead d (Bind Abst) u) (CHead -x1 (Bind x0) x2) H9) in ((let H14 \def (f_equal C T (\lambda (e0: C).(match -e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ -t3) \Rightarrow t3])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x2) H9) in -(\lambda (H15: (eq B Abst x0)).(\lambda (H16: (eq C d x1)).(let H17 \def -(eq_ind_r T x2 (\lambda (t3: T).(subst0 (minus i (S n)) u0 t3 x3)) H11 u H14) -in (let H18 \def (eq_ind_r C x1 (\lambda (c0: C).(getl n c3 (CHead c0 (Bind -x0) x3))) H10 d H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b: B).(getl n -c3 (CHead d (Bind b) x3))) H18 Abst H15) in (let H20 \def (eq_ind nat (minus -i n) (\lambda (n0: nat).(getl n0 (CHead d (Bind Abst) x3) (CHead e (Bind -Abbr) u0))) (getl_conf_le i (CHead e (Bind Abbr) u0) c3 (csubst0_getl_ge i i -(le_n i) c c3 u0 H4 (CHead e (Bind Abbr) u0) H5) (CHead d (Bind Abst) x3) n -H19 (le_S_n n i (le_S (S n) i H6))) (S (minus i (S n))) (minus_x_Sy i n H6)) -in (ty3_conv g c3 (lift (S n) O u) (lift (S n) O t0) (ty3_lift g d u t0 H1 c3 -O (S n) (getl_drop Abst c3 d x3 n H19)) (TLRef n) (lift (S n) O x3) (ty3_abst -g n c3 d x3 H19 t0 (H2 (minus i (S n)) u0 d x3 (fsubst0_snd (minus i (S n)) -u0 d u x3 H17) e (getl_gen_S (Bind Abst) d (CHead e (Bind Abbr) u0) x3 (minus -i (S n)) H20))) (pc3_lift c3 d (S n) O (getl_drop Abst c3 d x3 n H19) x3 u -(pc3_pr2_x d x3 u (pr2_delta d e u0 (r (Bind Abst) (minus i (S n))) -(getl_gen_S (Bind Abst) d (CHead e (Bind Abbr) u0) x3 (minus i (S n)) H20) u -u (pr0_refl u) x3 H17))))))))))) H13)) H12))))))))) H8)) (\lambda (H8: (ex3_4 -B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq -C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl n c3 (CHead e2 -(Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 (minus i (S n)) u0 e1 e2))))))).(ex3_4_ind B C C T (\lambda -(b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind -Abst) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u1: T).(getl n c3 (CHead e2 (Bind b) u1)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) -u0 e1 e2))))) (ty3 g c3 (TLRef n) (lift (S n) O u)) (\lambda (x0: B).(\lambda -(x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H9: (eq C (CHead d (Bind -Abst) u) (CHead x1 (Bind x0) x3))).(\lambda (H10: (getl n c3 (CHead x2 (Bind -x0) x3))).(\lambda (H11: (csubst0 (minus i (S n)) u0 x1 x2)).(let H12 \def -(f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with -[(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind -Abst) u) (CHead x1 (Bind x0) x3) H9) in ((let H13 \def (f_equal C B (\lambda -(e0: C).(match e0 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow -Abst | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with -[(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead d (Bind Abst) -u) (CHead x1 (Bind x0) x3) H9) in ((let H14 \def (f_equal C T (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | -(CHead _ _ t3) \Rightarrow t3])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) -x3) H9) in (\lambda (H15: (eq B Abst x0)).(\lambda (H16: (eq C d x1)).(let -H17 \def (eq_ind_r T x3 (\lambda (t3: T).(getl n c3 (CHead x2 (Bind x0) t3))) -H10 u H14) in (let H18 \def (eq_ind_r C x1 (\lambda (c0: C).(csubst0 (minus i -(S n)) u0 c0 x2)) H11 d H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b: -B).(getl n c3 (CHead x2 (Bind b) u))) H17 Abst H15) in (let H20 \def (eq_ind -nat (minus i n) (\lambda (n0: nat).(getl n0 (CHead x2 (Bind Abst) u) (CHead e -(Bind Abbr) u0))) (getl_conf_le i (CHead e (Bind Abbr) u0) c3 -(csubst0_getl_ge i i (le_n i) c c3 u0 H4 (CHead e (Bind Abbr) u0) H5) (CHead -x2 (Bind Abst) u) n H19 (le_S_n n i (le_S (S n) i H6))) (S (minus i (S n))) -(minus_x_Sy i n H6)) in (ty3_abst g n c3 x2 u H19 t0 (H2 (minus i (S n)) u0 -x2 u (fsubst0_fst (minus i (S n)) u0 d u x2 H18) e (csubst0_getl_ge_back -(minus i (S n)) (minus i (S n)) (le_n (minus i (S n))) d x2 u0 H18 (CHead e -(Bind Abbr) u0) (getl_gen_S (Bind Abst) x2 (CHead e (Bind Abbr) u0) u (minus -i (S n)) H20))))))))))) H13)) H12))))))))) H8)) (\lambda (H8: (ex4_5 B C C T -T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1))))))) (\lambda -(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl -n c3 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus i (S n)) u0 e1 e2)))))))).(ex4_5_ind B C C T T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1))))))) (\lambda -(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl -n c3 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus i (S n)) u0 e1 e2)))))) (ty3 g c3 (TLRef n) (lift (S -n) O u)) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: -T).(\lambda (x4: T).(\lambda (H9: (eq C (CHead d (Bind Abst) u) (CHead x1 -(Bind x0) x3))).(\lambda (H10: (getl n c3 (CHead x2 (Bind x0) x4))).(\lambda -(H11: (subst0 (minus i (S n)) u0 x3 x4)).(\lambda (H12: (csubst0 (minus i (S -n)) u0 x1 x2)).(let H13 \def (f_equal C C (\lambda (e0: C).(match e0 in C -return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) -\Rightarrow c0])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3) H9) in -((let H14 \def (f_equal C B (\lambda (e0: C).(match e0 in C return (\lambda -(_: C).B) with [(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match -k in K return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) -\Rightarrow Abst])])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3) H9) in -((let H15 \def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda -(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t3) \Rightarrow t3])) -(CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3) H9) in (\lambda (H16: (eq B -Abst x0)).(\lambda (H17: (eq C d x1)).(let H18 \def (eq_ind_r T x3 (\lambda -(t3: T).(subst0 (minus i (S n)) u0 t3 x4)) H11 u H15) in (let H19 \def -(eq_ind_r C x1 (\lambda (c0: C).(csubst0 (minus i (S n)) u0 c0 x2)) H12 d -H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b: B).(getl n c3 (CHead x2 -(Bind b) x4))) H10 Abst H16) in (let H21 \def (eq_ind nat (minus i n) -(\lambda (n0: nat).(getl n0 (CHead x2 (Bind Abst) x4) (CHead e (Bind Abbr) -u0))) (getl_conf_le i (CHead e (Bind Abbr) u0) c3 (csubst0_getl_ge i i (le_n -i) c c3 u0 H4 (CHead e (Bind Abbr) u0) H5) (CHead x2 (Bind Abst) x4) n H20 -(le_S_n n i (le_S (S n) i H6))) (S (minus i (S n))) (minus_x_Sy i n H6)) in -(ty3_conv g c3 (lift (S n) O u) (lift (S n) O t0) (ty3_lift g x2 u t0 (H2 -(minus i (S n)) u0 x2 u (fsubst0_fst (minus i (S n)) u0 d u x2 H19) e -(csubst0_getl_ge_back (minus i (S n)) (minus i (S n)) (le_n (minus i (S n))) -d x2 u0 H19 (CHead e (Bind Abbr) u0) (getl_gen_S (Bind Abst) x2 (CHead e -(Bind Abbr) u0) x4 (minus i (S n)) H21))) c3 O (S n) (getl_drop Abst c3 x2 x4 -n H20)) (TLRef n) (lift (S n) O x4) (ty3_abst g n c3 x2 x4 H20 t0 (H2 (minus -i (S n)) u0 x2 x4 (fsubst0_both (minus i (S n)) u0 d u x4 H18 x2 H19) e -(csubst0_getl_ge_back (minus i (S n)) (minus i (S n)) (le_n (minus i (S n))) -d x2 u0 H19 (CHead e (Bind Abbr) u0) (getl_gen_S (Bind Abst) x2 (CHead e -(Bind Abbr) u0) x4 (minus i (S n)) H21)))) (pc3_lift c3 x2 (S n) O (getl_drop -Abst c3 x2 x4 n H20) x4 u (pc3_fsubst0 d u u (pc3_refl d u) (minus i (S n)) -u0 x2 x4 (fsubst0_both (minus i (S n)) u0 d u x4 H18 x2 H19) e -(csubst0_getl_ge_back (minus i (S n)) (minus i (S n)) (le_n (minus i (S n))) -d x2 u0 H19 (CHead e (Bind Abbr) u0) (getl_gen_S (Bind Abst) x2 (CHead e -(Bind Abbr) u0) x4 (minus i (S n)) H21)))))))))))) H14)) H13))))))))))) H8)) -H7))) (\lambda (H6: (le i n)).(ty3_abst g n c3 d u (csubst0_getl_ge i n H6 c -c3 u0 H4 (CHead d (Bind Abst) u) H0) t0 H1))))))) (\lambda (t3: T).(\lambda -(H4: (subst0 i u0 (TLRef n) t3)).(\lambda (c3: C).(\lambda (H5: (csubst0 i u0 -c c3)).(\lambda (e: C).(\lambda (H6: (getl i c (CHead e (Bind Abbr) -u0))).(land_ind (eq nat n i) (eq T t3 (lift (S n) O u0)) (ty3 g c3 t3 (lift -(S n) O u)) (\lambda (H7: (eq nat n i)).(\lambda (H8: (eq T t3 (lift (S n) O -u0))).(eq_ind_r T (lift (S n) O u0) (\lambda (t4: T).(ty3 g c3 t4 (lift (S n) -O u))) (let H9 \def (eq_ind_r nat i (\lambda (n0: nat).(getl n0 c (CHead e -(Bind Abbr) u0))) H6 n H7) in (let H10 \def (eq_ind_r nat i (\lambda (n0: -nat).(csubst0 n0 u0 c c3)) H5 n H7) in (let H11 \def (eq_ind C (CHead d (Bind -Abst) u) (\lambda (c0: C).(getl n c c0)) H0 (CHead e (Bind Abbr) u0) -(getl_mono c (CHead d (Bind Abst) u) n H0 (CHead e (Bind Abbr) u0) H9)) in -(let H12 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (ee: C).(match ee in -C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k -_) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b) -\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow -False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) -\Rightarrow False])])) I (CHead e (Bind Abbr) u0) (getl_mono c (CHead d (Bind -Abst) u) n H0 (CHead e (Bind Abbr) u0) H9)) in (False_ind (ty3 g c3 (lift (S -n) O u0) (lift (S n) O u)) H12))))) t3 H8))) (subst0_gen_lref u0 t3 i n -H4)))))))) c2 t2 H3)))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda -(t0: T).(\lambda (H0: (ty3 g c u t0)).(\lambda (H1: ((\forall (i: -nat).(\forall (u0: T).(\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 c u c2 -t2) \to (\forall (e: C).((getl i c (CHead e (Bind Abbr) u0)) \to (ty3 g c2 t2 -t0)))))))))).(\lambda (b: B).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H2: -(ty3 g (CHead c (Bind b) u) t2 t3)).(\lambda (H3: ((\forall (i: nat).(\forall -(u0: T).(\forall (c2: C).(\forall (t4: T).((fsubst0 i u0 (CHead c (Bind b) u) -t2 c2 t4) \to (\forall (e: C).((getl i (CHead c (Bind b) u) (CHead e (Bind -Abbr) u0)) \to (ty3 g c2 t4 t3)))))))))).(\lambda (i: nat).(\lambda (u0: -T).(\lambda (c2: C).(\lambda (t4: T).(\lambda (H4: (fsubst0 i u0 c (THead -(Bind b) u t2) c2 t4)).(fsubst0_ind i u0 c (THead (Bind b) u t2) (\lambda -(c0: C).(\lambda (t5: T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u0)) -\to (ty3 g c0 t5 (THead (Bind b) u t3)))))) (\lambda (t5: T).(\lambda (H5: -(subst0 i u0 (THead (Bind b) u t2) t5)).(\lambda (e: C).(\lambda (H6: (getl i -c (CHead e (Bind Abbr) u0))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t5 (THead -(Bind b) u2 t2))) (\lambda (u2: T).(subst0 i u0 u u2))) (ex2 T (\lambda (t6: -T).(eq T t5 (THead (Bind b) u t6))) (\lambda (t6: T).(subst0 (s (Bind b) i) -u0 t2 t6))) (ex3_2 T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead -(Bind b) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) -(\lambda (_: T).(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 t6)))) (ty3 g c -t5 (THead (Bind b) u t3)) (\lambda (H7: (ex2 T (\lambda (u2: T).(eq T t5 -(THead (Bind b) u2 t2))) (\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T -(\lambda (u2: T).(eq T t5 (THead (Bind b) u2 t2))) (\lambda (u2: T).(subst0 i -u0 u u2)) (ty3 g c t5 (THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H8: -(eq T t5 (THead (Bind b) x t2))).(\lambda (H9: (subst0 i u0 u x)).(eq_ind_r T -(THead (Bind b) x t2) (\lambda (t6: T).(ty3 g c t6 (THead (Bind b) u t3))) -(ex_ind T (\lambda (t6: T).(ty3 g (CHead c (Bind b) u) t3 t6)) (ty3 g c -(THead (Bind b) x t2) (THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (H10: -(ty3 g (CHead c (Bind b) u) t3 x0)).(ex_ind T (\lambda (t6: T).(ty3 g (CHead -c (Bind b) x) t3 t6)) (ty3 g c (THead (Bind b) x t2) (THead (Bind b) u t3)) -(\lambda (x1: T).(\lambda (_: (ty3 g (CHead c (Bind b) x) t3 x1)).(ty3_conv g -c (THead (Bind b) u t3) (THead (Bind b) u x0) (ty3_bind g c u t0 H0 b t3 x0 -H10) (THead (Bind b) x t2) (THead (Bind b) x t3) (ty3_bind g c x t0 (H1 i u0 -c x (fsubst0_snd i u0 c u x H9) e H6) b t2 t3 (H3 (S i) u0 (CHead c (Bind b) -x) t2 (fsubst0_fst (S i) u0 (CHead c (Bind b) u) t2 (CHead c (Bind b) x) -(csubst0_snd_bind b i u0 u x H9 c)) e (getl_head (Bind b) i c (CHead e (Bind -Abbr) u0) H6 u))) (pc3_fsubst0 c (THead (Bind b) u t3) (THead (Bind b) u t3) -(pc3_refl c (THead (Bind b) u t3)) i u0 c (THead (Bind b) x t3) (fsubst0_snd -i u0 c (THead (Bind b) u t3) (THead (Bind b) x t3) (subst0_fst u0 x u i H9 t3 -(Bind b))) e H6)))) (ty3_correct g (CHead c (Bind b) x) t2 t3 (H3 (S i) u0 -(CHead c (Bind b) x) t2 (fsubst0_fst (S i) u0 (CHead c (Bind b) u) t2 (CHead -c (Bind b) x) (csubst0_snd_bind b i u0 u x H9 c)) e (getl_head (Bind b) i c -(CHead e (Bind Abbr) u0) H6 u)))))) (ty3_correct g (CHead c (Bind b) u) t2 t3 -H2)) t5 H8)))) H7)) (\lambda (H7: (ex2 T (\lambda (t6: T).(eq T t5 (THead -(Bind b) u t6))) (\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 -t6)))).(ex2_ind T (\lambda (t6: T).(eq T t5 (THead (Bind b) u t6))) (\lambda -(t6: T).(subst0 (s (Bind b) i) u0 t2 t6)) (ty3 g c t5 (THead (Bind b) u t3)) -(\lambda (x: T).(\lambda (H8: (eq T t5 (THead (Bind b) u x))).(\lambda (H9: -(subst0 (s (Bind b) i) u0 t2 x)).(eq_ind_r T (THead (Bind b) u x) (\lambda -(t6: T).(ty3 g c t6 (THead (Bind b) u t3))) (ex_ind T (\lambda (t6: T).(ty3 g -(CHead c (Bind b) u) t3 t6)) (ty3 g c (THead (Bind b) u x) (THead (Bind b) u -t3)) (\lambda (x0: T).(\lambda (_: (ty3 g (CHead c (Bind b) u) t3 -x0)).(ty3_bind g c u t0 H0 b x t3 (H3 (S i) u0 (CHead c (Bind b) u) x -(fsubst0_snd (S i) u0 (CHead c (Bind b) u) t2 x H9) e (getl_head (Bind b) i c -(CHead e (Bind Abbr) u0) H6 u))))) (ty3_correct g (CHead c (Bind b) u) x t3 -(H3 (S i) u0 (CHead c (Bind b) u) x (fsubst0_snd (S i) u0 (CHead c (Bind b) -u) t2 x H9) e (getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H6 u)))) t5 -H8)))) H7)) (\lambda (H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t6: T).(eq T -t5 (THead (Bind b) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u -u2))) (\lambda (_: T).(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 -t6))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead -(Bind b) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) -(\lambda (_: T).(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 t6))) (ty3 g c -t5 (THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H8: (eq -T t5 (THead (Bind b) x0 x1))).(\lambda (H9: (subst0 i u0 u x0)).(\lambda -(H10: (subst0 (s (Bind b) i) u0 t2 x1)).(eq_ind_r T (THead (Bind b) x0 x1) -(\lambda (t6: T).(ty3 g c t6 (THead (Bind b) u t3))) (ex_ind T (\lambda (t6: -T).(ty3 g (CHead c (Bind b) u) t3 t6)) (ty3 g c (THead (Bind b) x0 x1) (THead -(Bind b) u t3)) (\lambda (x: T).(\lambda (H11: (ty3 g (CHead c (Bind b) u) t3 -x)).(ex_ind T (\lambda (t6: T).(ty3 g (CHead c (Bind b) x0) t3 t6)) (ty3 g c -(THead (Bind b) x0 x1) (THead (Bind b) u t3)) (\lambda (x2: T).(\lambda (_: -(ty3 g (CHead c (Bind b) x0) t3 x2)).(ty3_conv g c (THead (Bind b) u t3) -(THead (Bind b) u x) (ty3_bind g c u t0 H0 b t3 x H11) (THead (Bind b) x0 x1) -(THead (Bind b) x0 t3) (ty3_bind g c x0 t0 (H1 i u0 c x0 (fsubst0_snd i u0 c -u x0 H9) e H6) b x1 t3 (H3 (S i) u0 (CHead c (Bind b) x0) x1 (fsubst0_both (S -i) u0 (CHead c (Bind b) u) t2 x1 H10 (CHead c (Bind b) x0) (csubst0_snd_bind -b i u0 u x0 H9 c)) e (getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H6 u))) -(pc3_fsubst0 c (THead (Bind b) u t3) (THead (Bind b) u t3) (pc3_refl c (THead -(Bind b) u t3)) i u0 c (THead (Bind b) x0 t3) (fsubst0_snd i u0 c (THead -(Bind b) u t3) (THead (Bind b) x0 t3) (subst0_fst u0 x0 u i H9 t3 (Bind b))) -e H6)))) (ty3_correct g (CHead c (Bind b) x0) x1 t3 (H3 (S i) u0 (CHead c -(Bind b) x0) x1 (fsubst0_both (S i) u0 (CHead c (Bind b) u) t2 x1 H10 (CHead -c (Bind b) x0) (csubst0_snd_bind b i u0 u x0 H9 c)) e (getl_head (Bind b) i c -(CHead e (Bind Abbr) u0) H6 u)))))) (ty3_correct g (CHead c (Bind b) u) t2 t3 -H2)) t5 H8)))))) H7)) (subst0_gen_head (Bind b) u0 u t2 t5 i H5)))))) -(\lambda (c3: C).(\lambda (H5: (csubst0 i u0 c c3)).(\lambda (e: C).(\lambda -(H6: (getl i c (CHead e (Bind Abbr) u0))).(ex_ind T (\lambda (t5: T).(ty3 g -(CHead c3 (Bind b) u) t3 t5)) (ty3 g c3 (THead (Bind b) u t2) (THead (Bind b) -u t3)) (\lambda (x: T).(\lambda (_: (ty3 g (CHead c3 (Bind b) u) t3 -x)).(ty3_bind g c3 u t0 (H1 i u0 c3 u (fsubst0_fst i u0 c u c3 H5) e H6) b t2 -t3 (H3 (S i) u0 (CHead c3 (Bind b) u) t2 (fsubst0_fst (S i) u0 (CHead c (Bind -b) u) t2 (CHead c3 (Bind b) u) (csubst0_fst_bind b i c c3 u0 H5 u)) e -(getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H6 u))))) (ty3_correct g -(CHead c3 (Bind b) u) t2 t3 (H3 (S i) u0 (CHead c3 (Bind b) u) t2 -(fsubst0_fst (S i) u0 (CHead c (Bind b) u) t2 (CHead c3 (Bind b) u) -(csubst0_fst_bind b i c c3 u0 H5 u)) e (getl_head (Bind b) i c (CHead e (Bind -Abbr) u0) H6 u)))))))) (\lambda (t5: T).(\lambda (H5: (subst0 i u0 (THead -(Bind b) u t2) t5)).(\lambda (c3: C).(\lambda (H6: (csubst0 i u0 c -c3)).(\lambda (e: C).(\lambda (H7: (getl i c (CHead e (Bind Abbr) -u0))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t5 (THead (Bind b) u2 t2))) -(\lambda (u2: T).(subst0 i u0 u u2))) (ex2 T (\lambda (t6: T).(eq T t5 (THead -(Bind b) u t6))) (\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 t6))) (ex3_2 T -T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead (Bind b) u2 t6)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: -T).(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 t6)))) (ty3 g c3 t5 (THead -(Bind b) u t3)) (\lambda (H8: (ex2 T (\lambda (u2: T).(eq T t5 (THead (Bind -b) u2 t2))) (\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T (\lambda (u2: -T).(eq T t5 (THead (Bind b) u2 t2))) (\lambda (u2: T).(subst0 i u0 u u2)) -(ty3 g c3 t5 (THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H9: (eq T t5 -(THead (Bind b) x t2))).(\lambda (H10: (subst0 i u0 u x)).(eq_ind_r T (THead -(Bind b) x t2) (\lambda (t6: T).(ty3 g c3 t6 (THead (Bind b) u t3))) (ex_ind -T (\lambda (t6: T).(ty3 g (CHead c3 (Bind b) u) t3 t6)) (ty3 g c3 (THead -(Bind b) x t2) (THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (H11: (ty3 g -(CHead c3 (Bind b) u) t3 x0)).(ex_ind T (\lambda (t6: T).(ty3 g (CHead c3 -(Bind b) u) x0 t6)) (ty3 g c3 (THead (Bind b) x t2) (THead (Bind b) u t3)) -(\lambda (x1: T).(\lambda (_: (ty3 g (CHead c3 (Bind b) u) x0 x1)).(ex_ind T -(\lambda (t6: T).(ty3 g (CHead c3 (Bind b) x) t3 t6)) (ty3 g c3 (THead (Bind -b) x t2) (THead (Bind b) u t3)) (\lambda (x2: T).(\lambda (_: (ty3 g (CHead -c3 (Bind b) x) t3 x2)).(ty3_conv g c3 (THead (Bind b) u t3) (THead (Bind b) u -x0) (ty3_bind g c3 u t0 (H1 i u0 c3 u (fsubst0_fst i u0 c u c3 H6) e H7) b t3 -x0 H11) (THead (Bind b) x t2) (THead (Bind b) x t3) (ty3_bind g c3 x t0 (H1 i -u0 c3 x (fsubst0_both i u0 c u x H10 c3 H6) e H7) b t2 t3 (H3 (S i) u0 (CHead -c3 (Bind b) x) t2 (fsubst0_fst (S i) u0 (CHead c (Bind b) u) t2 (CHead c3 -(Bind b) x) (csubst0_both_bind b i u0 u x H10 c c3 H6)) e (getl_head (Bind b) -i c (CHead e (Bind Abbr) u0) H7 u))) (pc3_fsubst0 c (THead (Bind b) u t3) -(THead (Bind b) u t3) (pc3_refl c (THead (Bind b) u t3)) i u0 c3 (THead (Bind -b) x t3) (fsubst0_both i u0 c (THead (Bind b) u t3) (THead (Bind b) x t3) -(subst0_fst u0 x u i H10 t3 (Bind b)) c3 H6) e H7)))) (ty3_correct g (CHead -c3 (Bind b) x) t2 t3 (H3 (S i) u0 (CHead c3 (Bind b) x) t2 (fsubst0_fst (S i) -u0 (CHead c (Bind b) u) t2 (CHead c3 (Bind b) x) (csubst0_both_bind b i u0 u -x H10 c c3 H6)) e (getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H7 u)))))) -(ty3_correct g (CHead c3 (Bind b) u) t3 x0 H11)))) (ty3_correct g (CHead c3 -(Bind b) u) t2 t3 (H3 (S i) u0 (CHead c3 (Bind b) u) t2 (fsubst0_fst (S i) u0 -(CHead c (Bind b) u) t2 (CHead c3 (Bind b) u) (csubst0_fst_bind b i c c3 u0 -H6 u)) e (getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H7 u)))) t5 H9)))) -H8)) (\lambda (H8: (ex2 T (\lambda (t6: T).(eq T t5 (THead (Bind b) u t6))) -(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 t6)))).(ex2_ind T (\lambda (t6: -T).(eq T t5 (THead (Bind b) u t6))) (\lambda (t6: T).(subst0 (s (Bind b) i) -u0 t2 t6)) (ty3 g c3 t5 (THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H9: -(eq T t5 (THead (Bind b) u x))).(\lambda (H10: (subst0 (s (Bind b) i) u0 t2 -x)).(eq_ind_r T (THead (Bind b) u x) (\lambda (t6: T).(ty3 g c3 t6 (THead -(Bind b) u t3))) (ex_ind T (\lambda (t6: T).(ty3 g (CHead c3 (Bind b) u) t3 -t6)) (ty3 g c3 (THead (Bind b) u x) (THead (Bind b) u t3)) (\lambda (x0: -T).(\lambda (_: (ty3 g (CHead c3 (Bind b) u) t3 x0)).(ty3_bind g c3 u t0 (H1 -i u0 c3 u (fsubst0_fst i u0 c u c3 H6) e H7) b x t3 (H3 (S i) u0 (CHead c3 -(Bind b) u) x (fsubst0_both (S i) u0 (CHead c (Bind b) u) t2 x H10 (CHead c3 -(Bind b) u) (csubst0_fst_bind b i c c3 u0 H6 u)) e (getl_head (Bind b) i c -(CHead e (Bind Abbr) u0) H7 u))))) (ty3_correct g (CHead c3 (Bind b) u) x t3 -(H3 (S i) u0 (CHead c3 (Bind b) u) x (fsubst0_both (S i) u0 (CHead c (Bind b) -u) t2 x H10 (CHead c3 (Bind b) u) (csubst0_fst_bind b i c c3 u0 H6 u)) e -(getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H7 u)))) t5 H9)))) H8)) -(\lambda (H8: (ex3_2 T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead -(Bind b) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) -(\lambda (_: T).(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 -t6))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead -(Bind b) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) -(\lambda (_: T).(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 t6))) (ty3 g c3 -t5 (THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H9: (eq -T t5 (THead (Bind b) x0 x1))).(\lambda (H10: (subst0 i u0 u x0)).(\lambda -(H11: (subst0 (s (Bind b) i) u0 t2 x1)).(eq_ind_r T (THead (Bind b) x0 x1) -(\lambda (t6: T).(ty3 g c3 t6 (THead (Bind b) u t3))) (ex_ind T (\lambda (t6: -T).(ty3 g (CHead c3 (Bind b) u) t3 t6)) (ty3 g c3 (THead (Bind b) x0 x1) -(THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H12: (ty3 g (CHead c3 (Bind -b) u) t3 x)).(ex_ind T (\lambda (t6: T).(ty3 g (CHead c3 (Bind b) u) x t6)) -(ty3 g c3 (THead (Bind b) x0 x1) (THead (Bind b) u t3)) (\lambda (x2: -T).(\lambda (_: (ty3 g (CHead c3 (Bind b) u) x x2)).(ex_ind T (\lambda (t6: -T).(ty3 g (CHead c3 (Bind b) x0) t3 t6)) (ty3 g c3 (THead (Bind b) x0 x1) -(THead (Bind b) u t3)) (\lambda (x3: T).(\lambda (_: (ty3 g (CHead c3 (Bind -b) x0) t3 x3)).(ty3_conv g c3 (THead (Bind b) u t3) (THead (Bind b) u x) -(ty3_bind g c3 u t0 (H1 i u0 c3 u (fsubst0_fst i u0 c u c3 H6) e H7) b t3 x -H12) (THead (Bind b) x0 x1) (THead (Bind b) x0 t3) (ty3_bind g c3 x0 t0 (H1 i -u0 c3 x0 (fsubst0_both i u0 c u x0 H10 c3 H6) e H7) b x1 t3 (H3 (S i) u0 -(CHead c3 (Bind b) x0) x1 (fsubst0_both (S i) u0 (CHead c (Bind b) u) t2 x1 -H11 (CHead c3 (Bind b) x0) (csubst0_both_bind b i u0 u x0 H10 c c3 H6)) e -(getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H7 u))) (pc3_fsubst0 c -(THead (Bind b) u t3) (THead (Bind b) u t3) (pc3_refl c (THead (Bind b) u -t3)) i u0 c3 (THead (Bind b) x0 t3) (fsubst0_both i u0 c (THead (Bind b) u -t3) (THead (Bind b) x0 t3) (subst0_fst u0 x0 u i H10 t3 (Bind b)) c3 H6) e -H7)))) (ty3_correct g (CHead c3 (Bind b) x0) x1 t3 (H3 (S i) u0 (CHead c3 -(Bind b) x0) x1 (fsubst0_both (S i) u0 (CHead c (Bind b) u) t2 x1 H11 (CHead -c3 (Bind b) x0) (csubst0_both_bind b i u0 u x0 H10 c c3 H6)) e (getl_head -(Bind b) i c (CHead e (Bind Abbr) u0) H7 u)))))) (ty3_correct g (CHead c3 -(Bind b) u) t3 x H12)))) (ty3_correct g (CHead c3 (Bind b) u) t2 t3 (H3 (S i) -u0 (CHead c3 (Bind b) u) t2 (fsubst0_fst (S i) u0 (CHead c (Bind b) u) t2 -(CHead c3 (Bind b) u) (csubst0_fst_bind b i c c3 u0 H6 u)) e (getl_head (Bind -b) i c (CHead e (Bind Abbr) u0) H7 u)))) t5 H9)))))) H8)) (subst0_gen_head -(Bind b) u0 u t2 t5 i H5)))))))) c2 t4 H4)))))))))))))))) (\lambda (c: -C).(\lambda (w: T).(\lambda (u: T).(\lambda (H0: (ty3 g c w u)).(\lambda (H1: -((\forall (i: nat).(\forall (u0: T).(\forall (c2: C).(\forall (t2: -T).((fsubst0 i u0 c w c2 t2) \to (\forall (e: C).((getl i c (CHead e (Bind -Abbr) u0)) \to (ty3 g c2 t2 u)))))))))).(\lambda (v: T).(\lambda (t0: -T).(\lambda (H2: (ty3 g c v (THead (Bind Abst) u t0))).(\lambda (H3: -((\forall (i: nat).(\forall (u0: T).(\forall (c2: C).(\forall (t2: -T).((fsubst0 i u0 c v c2 t2) \to (\forall (e: C).((getl i c (CHead e (Bind -Abbr) u0)) \to (ty3 g c2 t2 (THead (Bind Abst) u t0))))))))))).(\lambda (i: -nat).(\lambda (u0: T).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H4: -(fsubst0 i u0 c (THead (Flat Appl) w v) c2 t2)).(fsubst0_ind i u0 c (THead -(Flat Appl) w v) (\lambda (c0: C).(\lambda (t3: T).(\forall (e: C).((getl i c -(CHead e (Bind Abbr) u0)) \to (ty3 g c0 t3 (THead (Flat Appl) w (THead (Bind -Abst) u t0))))))) (\lambda (t3: T).(\lambda (H5: (subst0 i u0 (THead (Flat -Appl) w v) t3)).(\lambda (e: C).(\lambda (H6: (getl i c (CHead e (Bind Abbr) -u0))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t3 (THead (Flat Appl) u2 v))) -(\lambda (u2: T).(subst0 i u0 w u2))) (ex2 T (\lambda (t4: T).(eq T t3 (THead -(Flat Appl) w t4))) (\lambda (t4: T).(subst0 (s (Flat Appl) i) u0 v t4))) -(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 -t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 w u2))) (\lambda (_: -T).(\lambda (t4: T).(subst0 (s (Flat Appl) i) u0 v t4)))) (ty3 g c t3 (THead -(Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (H7: (ex2 T (\lambda (u2: -T).(eq T t3 (THead (Flat Appl) u2 v))) (\lambda (u2: T).(subst0 i u0 w -u2)))).(ex2_ind T (\lambda (u2: T).(eq T t3 (THead (Flat Appl) u2 v))) -(\lambda (u2: T).(subst0 i u0 w u2)) (ty3 g c t3 (THead (Flat Appl) w (THead -(Bind Abst) u t0))) (\lambda (x: T).(\lambda (H8: (eq T t3 (THead (Flat Appl) -x v))).(\lambda (H9: (subst0 i u0 w x)).(eq_ind_r T (THead (Flat Appl) x v) -(\lambda (t4: T).(ty3 g c t4 (THead (Flat Appl) w (THead (Bind Abst) u t0)))) -(ex_ind T (\lambda (t4: T).(ty3 g c (THead (Bind Abst) u t0) t4)) (ty3 g c -(THead (Flat Appl) x v) (THead (Flat Appl) w (THead (Bind Abst) u t0))) -(\lambda (x0: T).(\lambda (H10: (ty3 g c (THead (Bind Abst) u t0) -x0)).(ex3_2_ind T T (\lambda (t4: T).(\lambda (_: T).(pc3 c (THead (Bind -Abst) u t4) x0))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c u t5))) (\lambda -(t4: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t0 t4))) (ty3 g c -(THead (Flat Appl) x v) (THead (Flat Appl) w (THead (Bind Abst) u t0))) -(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 c (THead (Bind Abst) u -x1) x0)).(\lambda (_: (ty3 g c u x2)).(\lambda (H13: (ty3 g (CHead c (Bind -Abst) u) t0 x1)).(ex_ind T (\lambda (t4: T).(ty3 g c u t4)) (ty3 g c (THead -(Flat Appl) x v) (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda -(x3: T).(\lambda (H14: (ty3 g c u x3)).(ty3_conv g c (THead (Flat Appl) w -(THead (Bind Abst) u t0)) (THead (Flat Appl) w (THead (Bind Abst) u x1)) -(ty3_appl g c w u H0 (THead (Bind Abst) u t0) x1 (ty3_bind g c u x3 H14 Abst -t0 x1 H13)) (THead (Flat Appl) x v) (THead (Flat Appl) x (THead (Bind Abst) u -t0)) (ty3_appl g c x u (H1 i u0 c x (fsubst0_snd i u0 c w x H9) e H6) v t0 -H2) (pc3_fsubst0 c (THead (Flat Appl) w (THead (Bind Abst) u t0)) (THead -(Flat Appl) w (THead (Bind Abst) u t0)) (pc3_refl c (THead (Flat Appl) w -(THead (Bind Abst) u t0))) i u0 c (THead (Flat Appl) x (THead (Bind Abst) u -t0)) (fsubst0_snd i u0 c (THead (Flat Appl) w (THead (Bind Abst) u t0)) -(THead (Flat Appl) x (THead (Bind Abst) u t0)) (subst0_fst u0 x w i H9 (THead -(Bind Abst) u t0) (Flat Appl))) e H6)))) (ty3_correct g c x u (H1 i u0 c x -(fsubst0_snd i u0 c w x H9) e H6)))))))) (ty3_gen_bind g Abst c u t0 x0 -H10)))) (ty3_correct g c v (THead (Bind Abst) u t0) H2)) t3 H8)))) H7)) -(\lambda (H7: (ex2 T (\lambda (t4: T).(eq T t3 (THead (Flat Appl) w t4))) -(\lambda (t4: T).(subst0 (s (Flat Appl) i) u0 v t4)))).(ex2_ind T (\lambda -(t4: T).(eq T t3 (THead (Flat Appl) w t4))) (\lambda (t4: T).(subst0 (s (Flat -Appl) i) u0 v t4)) (ty3 g c t3 (THead (Flat Appl) w (THead (Bind Abst) u -t0))) (\lambda (x: T).(\lambda (H8: (eq T t3 (THead (Flat Appl) w -x))).(\lambda (H9: (subst0 (s (Flat Appl) i) u0 v x)).(eq_ind_r T (THead -(Flat Appl) w x) (\lambda (t4: T).(ty3 g c t4 (THead (Flat Appl) w (THead -(Bind Abst) u t0)))) (ty3_appl g c w u H0 x t0 (H3 (s (Flat Appl) i) u0 c x -(fsubst0_snd (s (Flat Appl) i) u0 c v x H9) e H6)) t3 H8)))) H7)) (\lambda -(H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) -u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 w u2))) (\lambda (_: -T).(\lambda (t4: T).(subst0 (s (Flat Appl) i) u0 v t4))))).(ex3_2_ind T T -(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 i u0 w u2))) (\lambda (_: -T).(\lambda (t4: T).(subst0 (s (Flat Appl) i) u0 v t4))) (ty3 g c t3 (THead -(Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H8: (eq T t3 (THead (Flat Appl) x0 x1))).(\lambda (H9: (subst0 i -u0 w x0)).(\lambda (H10: (subst0 (s (Flat Appl) i) u0 v x1)).(eq_ind_r T -(THead (Flat Appl) x0 x1) (\lambda (t4: T).(ty3 g c t4 (THead (Flat Appl) w -(THead (Bind Abst) u t0)))) (ex_ind T (\lambda (t4: T).(ty3 g c (THead (Bind -Abst) u t0) t4)) (ty3 g c (THead (Flat Appl) x0 x1) (THead (Flat Appl) w -(THead (Bind Abst) u t0))) (\lambda (x: T).(\lambda (H11: (ty3 g c (THead -(Bind Abst) u t0) x)).(ex3_2_ind T T (\lambda (t4: T).(\lambda (_: T).(pc3 c -(THead (Bind Abst) u t4) x))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c u -t5))) (\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t0 -t4))) (ty3 g c (THead (Flat Appl) x0 x1) (THead (Flat Appl) w (THead (Bind -Abst) u t0))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (_: (pc3 c (THead -(Bind Abst) u x2) x)).(\lambda (_: (ty3 g c u x3)).(\lambda (H14: (ty3 g -(CHead c (Bind Abst) u) t0 x2)).(ex_ind T (\lambda (t4: T).(ty3 g c u t4)) -(ty3 g c (THead (Flat Appl) x0 x1) (THead (Flat Appl) w (THead (Bind Abst) u -t0))) (\lambda (x4: T).(\lambda (H15: (ty3 g c u x4)).(ty3_conv g c (THead -(Flat Appl) w (THead (Bind Abst) u t0)) (THead (Flat Appl) w (THead (Bind -Abst) u x2)) (ty3_appl g c w u H0 (THead (Bind Abst) u t0) x2 (ty3_bind g c u -x4 H15 Abst t0 x2 H14)) (THead (Flat Appl) x0 x1) (THead (Flat Appl) x0 -(THead (Bind Abst) u t0)) (ty3_appl g c x0 u (H1 i u0 c x0 (fsubst0_snd i u0 -c w x0 H9) e H6) x1 t0 (H3 (s (Flat Appl) i) u0 c x1 (fsubst0_snd (s (Flat -Appl) i) u0 c v x1 H10) e H6)) (pc3_fsubst0 c (THead (Flat Appl) w (THead -(Bind Abst) u t0)) (THead (Flat Appl) w (THead (Bind Abst) u t0)) (pc3_refl c -(THead 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(u2: T).(subst0 i u0 w u2))) (ex2 T (\lambda -(t4: T).(eq T t3 (THead (Flat Appl) w t4))) (\lambda (t4: T).(subst0 (s (Flat -Appl) i) u0 v t4))) (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 -(THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 w -u2))) (\lambda (_: T).(\lambda (t4: T).(subst0 (s (Flat Appl) i) u0 v t4)))) -(ty3 g c3 t3 (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (H8: -(ex2 T (\lambda (u2: T).(eq T t3 (THead (Flat Appl) u2 v))) (\lambda (u2: -T).(subst0 i u0 w u2)))).(ex2_ind T (\lambda (u2: T).(eq T t3 (THead (Flat -Appl) u2 v))) (\lambda (u2: T).(subst0 i u0 w u2)) (ty3 g c3 t3 (THead (Flat -Appl) w (THead (Bind Abst) u t0))) (\lambda (x: T).(\lambda (H9: (eq T t3 -(THead (Flat Appl) x v))).(\lambda (H10: (subst0 i u0 w x)).(eq_ind_r T -(THead (Flat Appl) x v) (\lambda (t4: T).(ty3 g c3 t4 (THead (Flat Appl) w -(THead (Bind Abst) u t0)))) (ex_ind T (\lambda (t4: T).(ty3 g c3 (THead (Bind -Abst) u t0) t4)) (ty3 g c3 (THead (Flat Appl) x v) (THead (Flat Appl) w -(THead (Bind Abst) u t0))) (\lambda (x0: T).(\lambda (H11: (ty3 g c3 (THead -(Bind Abst) u t0) x0)).(ex3_2_ind T T (\lambda (t4: T).(\lambda (_: T).(pc3 -c3 (THead (Bind Abst) u t4) x0))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c3 -u t5))) (\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead c3 (Bind Abst) u) t0 -t4))) (ty3 g c3 (THead (Flat Appl) x v) (THead (Flat Appl) w (THead (Bind -Abst) u t0))) (\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 c3 (THead -(Bind Abst) u x1) x0)).(\lambda (H13: (ty3 g c3 u x2)).(\lambda (H14: (ty3 g -(CHead c3 (Bind Abst) u) t0 x1)).(ty3_conv g c3 (THead (Flat Appl) w (THead -(Bind Abst) u t0)) (THead (Flat Appl) w (THead (Bind Abst) u x1)) (ty3_appl g -c3 w u (H1 i u0 c3 w (fsubst0_fst i u0 c w c3 H6) e H7) (THead (Bind Abst) u -t0) x1 (ty3_bind g c3 u x2 H13 Abst t0 x1 H14)) (THead (Flat Appl) x v) -(THead (Flat Appl) x (THead (Bind Abst) u t0)) (ty3_appl g c3 x u (H1 i u0 c3 -x (fsubst0_both i u0 c w x H10 c3 H6) e H7) v t0 (H3 i u0 c3 v (fsubst0_fst i -u0 c v c3 H6) e H7)) (pc3_fsubst0 c (THead (Flat Appl) w (THead (Bind Abst) u -t0)) (THead (Flat Appl) w (THead (Bind Abst) u t0)) (pc3_refl c (THead (Flat -Appl) w (THead (Bind Abst) u t0))) i u0 c3 (THead (Flat Appl) x (THead (Bind -Abst) u t0)) (fsubst0_both i u0 c (THead (Flat Appl) w (THead (Bind Abst) u -t0)) (THead (Flat Appl) x (THead (Bind Abst) u t0)) (subst0_fst u0 x w i H10 -(THead (Bind Abst) u t0) (Flat Appl)) c3 H6) e H7))))))) (ty3_gen_bind g Abst -c3 u t0 x0 H11)))) (ty3_correct g c3 v (THead (Bind Abst) u t0) (H3 i u0 c3 v -(fsubst0_fst i u0 c v c3 H6) e H7))) t3 H9)))) H8)) (\lambda (H8: (ex2 T -(\lambda (t4: T).(eq T t3 (THead (Flat Appl) w t4))) (\lambda (t4: T).(subst0 -(s (Flat Appl) i) u0 v t4)))).(ex2_ind T (\lambda (t4: T).(eq T t3 (THead -(Flat Appl) w t4))) (\lambda (t4: T).(subst0 (s (Flat Appl) i) u0 v t4)) (ty3 -g c3 t3 (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x: -T).(\lambda (H9: (eq T t3 (THead (Flat Appl) w x))).(\lambda (H10: (subst0 (s -(Flat Appl) i) u0 v x)).(eq_ind_r T (THead (Flat Appl) w x) (\lambda (t4: -T).(ty3 g c3 t4 (THead (Flat Appl) w (THead (Bind Abst) u t0)))) (ty3_appl g -c3 w u (H1 i u0 c3 w (fsubst0_fst i u0 c w c3 H6) e H7) x t0 (H3 i u0 c3 x -(fsubst0_both i u0 c v x H10 c3 H6) e H7)) t3 H9)))) H8)) (\lambda (H8: -(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 -t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 w u2))) (\lambda (_: -T).(\lambda (t4: T).(subst0 (s (Flat Appl) i) u0 v t4))))).(ex3_2_ind T T -(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 i u0 w u2))) (\lambda (_: -T).(\lambda (t4: T).(subst0 (s (Flat Appl) i) u0 v t4))) (ty3 g c3 t3 (THead -(Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H9: (eq T t3 (THead (Flat Appl) x0 x1))).(\lambda (H10: (subst0 -i u0 w x0)).(\lambda (H11: (subst0 (s (Flat 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c3 u x4)).(ty3_conv g c3 (THead -(Flat Appl) w (THead (Bind Abst) u t0)) (THead (Flat Appl) w (THead (Bind -Abst) u x2)) (ty3_appl g c3 w u (H1 i u0 c3 w (fsubst0_fst i u0 c w c3 H6) e -H7) (THead (Bind Abst) u t0) x2 (ty3_bind g c3 u x4 H16 Abst t0 x2 H15)) -(THead (Flat Appl) x0 x1) (THead (Flat Appl) x0 (THead (Bind Abst) u t0)) -(ty3_appl g c3 x0 u (H1 i u0 c3 x0 (fsubst0_both i u0 c w x0 H10 c3 H6) e H7) -x1 t0 (H3 i u0 c3 x1 (fsubst0_both i u0 c v x1 H11 c3 H6) e H7)) (pc3_fsubst0 -c (THead (Flat Appl) w (THead (Bind Abst) u t0)) (THead (Flat Appl) w (THead -(Bind Abst) u t0)) (pc3_refl c (THead (Flat Appl) w (THead (Bind Abst) u -t0))) i u0 c3 (THead (Flat Appl) x0 (THead (Bind Abst) u t0)) (fsubst0_both i -u0 c (THead (Flat Appl) w (THead (Bind Abst) u t0)) (THead (Flat Appl) x0 -(THead (Bind Abst) u t0)) (subst0_fst u0 x0 w i H10 (THead (Bind Abst) u t0) -(Flat Appl)) c3 H6) e H7)))) (ty3_correct g c3 w u (H1 i u0 c3 w (fsubst0_fst -i u0 c w c3 H6) e H7)))))))) (ty3_gen_bind g Abst c3 u t0 x H12)))) -(ty3_correct g c3 v (THead (Bind Abst) u t0) (H3 i u0 c3 v (fsubst0_fst i u0 -c v c3 H6) e H7))) t3 H9)))))) H8)) (subst0_gen_head (Flat Appl) u0 w v t3 i -H5)))))))) c2 t2 H4))))))))))))))) (\lambda (c: C).(\lambda (t2: T).(\lambda -(t3: T).(\lambda (H0: (ty3 g c t2 t3)).(\lambda (H1: ((\forall (i: -nat).(\forall (u: T).(\forall (c2: C).(\forall (t4: T).((fsubst0 i u c t2 c2 -t4) \to (\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) \to (ty3 g c2 t4 -t3)))))))))).(\lambda (t0: T).(\lambda (H2: (ty3 g c t3 t0)).(\lambda (H3: -((\forall (i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t4: -T).((fsubst0 i u c t3 c2 t4) \to (\forall (e: C).((getl i c (CHead e (Bind -Abbr) u)) \to (ty3 g c2 t4 t0)))))))))).(\lambda (i: nat).(\lambda (u: -T).(\lambda (c2: C).(\lambda (t4: T).(\lambda (H4: (fsubst0 i u c (THead -(Flat Cast) t3 t2) c2 t4)).(fsubst0_ind i u c (THead (Flat Cast) t3 t2) -(\lambda (c0: C).(\lambda (t5: T).(\forall (e: C).((getl i c (CHead e (Bind -Abbr) u)) \to (ty3 g c0 t5 (THead (Flat Cast) t0 t3)))))) (\lambda (t5: -T).(\lambda (H5: (subst0 i u (THead (Flat Cast) t3 t2) t5)).(\lambda (e: -C).(\lambda (H6: (getl i c (CHead e (Bind Abbr) u))).(or3_ind (ex2 T (\lambda -(u2: T).(eq T t5 (THead (Flat Cast) u2 t2))) (\lambda (u2: T).(subst0 i u t3 -u2))) (ex2 T (\lambda (t6: T).(eq T t5 (THead (Flat Cast) t3 t6))) (\lambda -(t6: T).(subst0 (s (Flat Cast) i) u t2 t6))) (ex3_2 T T (\lambda (u2: -T).(\lambda (t6: T).(eq T t5 (THead (Flat Cast) u2 t6)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i u t3 u2))) (\lambda (_: T).(\lambda (t6: -T).(subst0 (s (Flat Cast) i) u t2 t6)))) (ty3 g c t5 (THead (Flat Cast) t0 -t3)) (\lambda (H7: (ex2 T (\lambda (u2: T).(eq T t5 (THead (Flat Cast) u2 -t2))) (\lambda (u2: T).(subst0 i u t3 u2)))).(ex2_ind T (\lambda (u2: T).(eq -T t5 (THead (Flat Cast) u2 t2))) (\lambda (u2: T).(subst0 i u t3 u2)) (ty3 g -c t5 (THead (Flat Cast) t0 t3)) (\lambda (x: T).(\lambda (H8: (eq T t5 (THead -(Flat Cast) x t2))).(\lambda (H9: (subst0 i u t3 x)).(eq_ind_r T (THead (Flat -Cast) x t2) (\lambda (t6: T).(ty3 g c t6 (THead (Flat Cast) t0 t3))) (ex_ind -T (\lambda (t6: T).(ty3 g c t0 t6)) (ty3 g c (THead (Flat Cast) x t2) (THead -(Flat Cast) t0 t3)) (\lambda (x0: T).(\lambda (H10: (ty3 g c t0 -x0)).(ty3_conv g c (THead (Flat Cast) t0 t3) (THead (Flat Cast) x0 t0) -(ty3_cast g c t3 t0 H2 x0 H10) (THead (Flat Cast) x t2) (THead (Flat Cast) t0 -x) (ty3_cast g c t2 x (ty3_conv g c x t0 (H3 i u c x (fsubst0_snd i u c t3 x -H9) e H6) t2 t3 H0 (pc3_s c t3 x (pc3_fsubst0 c t3 t3 (pc3_refl c t3) i u c x -(fsubst0_snd i u c t3 x H9) e H6))) t0 (H3 i u c x (fsubst0_snd i u c t3 x -H9) e H6)) (pc3_fsubst0 c (THead (Flat Cast) t0 t3) (THead (Flat Cast) t0 t3) -(pc3_refl c (THead (Flat Cast) t0 t3)) i u c (THead (Flat Cast) t0 x) -(fsubst0_snd i u c (THead (Flat Cast) t0 t3) (THead (Flat Cast) t0 x) -(subst0_snd (Flat Cast) u x t3 i H9 t0)) e H6)))) (ty3_correct g c x t0 (H3 i -u c x (fsubst0_snd i u c t3 x H9) e H6))) t5 H8)))) H7)) (\lambda (H7: (ex2 T -(\lambda (t6: T).(eq T t5 (THead (Flat Cast) t3 t6))) (\lambda (t6: -T).(subst0 (s (Flat Cast) i) u t2 t6)))).(ex2_ind T (\lambda (t6: T).(eq T t5 -(THead (Flat Cast) t3 t6))) (\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 -t6)) (ty3 g c t5 (THead (Flat Cast) t0 t3)) (\lambda (x: T).(\lambda (H8: (eq -T t5 (THead (Flat Cast) t3 x))).(\lambda (H9: (subst0 (s (Flat Cast) i) u t2 -x)).(eq_ind_r T (THead (Flat Cast) t3 x) (\lambda (t6: T).(ty3 g c t6 (THead -(Flat Cast) t0 t3))) (ty3_cast g c x t3 (H1 (s (Flat Cast) i) u c x -(fsubst0_snd (s (Flat Cast) i) u c t2 x H9) e H6) t0 H2) t5 H8)))) H7)) -(\lambda (H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead -(Flat Cast) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u t3 u2))) -(\lambda (_: T).(\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 -t6))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead -(Flat Cast) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u t3 u2))) -(\lambda (_: T).(\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 t6))) (ty3 g -c t5 (THead (Flat Cast) t0 t3)) (\lambda (x0: T).(\lambda (x1: T).(\lambda -(H8: (eq T t5 (THead (Flat Cast) x0 x1))).(\lambda (H9: (subst0 i u t3 -x0)).(\lambda (H10: (subst0 (s (Flat Cast) i) u t2 x1)).(eq_ind_r T (THead -(Flat Cast) x0 x1) (\lambda (t6: T).(ty3 g c t6 (THead (Flat Cast) t0 t3))) -(ex_ind T (\lambda (t6: T).(ty3 g c t0 t6)) (ty3 g c (THead (Flat Cast) x0 -x1) (THead (Flat Cast) t0 t3)) (\lambda (x: T).(\lambda (H11: (ty3 g c t0 -x)).(ty3_conv g c (THead (Flat Cast) t0 t3) (THead (Flat Cast) x t0) -(ty3_cast g c t3 t0 H2 x H11) (THead (Flat Cast) x0 x1) (THead (Flat Cast) t0 -x0) (ty3_cast g c x1 x0 (ty3_conv g c x0 t0 (H3 i u c x0 (fsubst0_snd i u c -t3 x0 H9) e H6) x1 t3 (H1 (s (Flat Cast) i) u c x1 (fsubst0_snd (s (Flat -Cast) i) u c t2 x1 H10) e H6) (pc3_s c t3 x0 (pc3_fsubst0 c t3 t3 (pc3_refl c -t3) i u c x0 (fsubst0_snd i u c t3 x0 H9) e H6))) t0 (H3 i u c x0 -(fsubst0_snd i u c t3 x0 H9) e H6)) (pc3_fsubst0 c (THead (Flat Cast) t0 t3) -(THead (Flat Cast) t0 t3) (pc3_refl c (THead (Flat Cast) t0 t3)) i u c (THead -(Flat Cast) t0 x0) (fsubst0_snd i u c (THead (Flat Cast) t0 t3) (THead (Flat -Cast) t0 x0) (subst0_snd (Flat Cast) u x0 t3 i H9 t0)) e H6)))) (ty3_correct -g c x0 t0 (H3 i u c x0 (fsubst0_snd i u c t3 x0 H9) e H6))) t5 H8)))))) H7)) -(subst0_gen_head (Flat Cast) u t3 t2 t5 i H5)))))) (\lambda (c3: C).(\lambda -(H5: (csubst0 i u c c3)).(\lambda (e: C).(\lambda (H6: (getl i c (CHead e -(Bind Abbr) u))).(ty3_cast g c3 t2 t3 (H1 i u c3 t2 (fsubst0_fst i u c t2 c3 -H5) e H6) t0 (H3 i u c3 t3 (fsubst0_fst i u c t3 c3 H5) e H6)))))) (\lambda -(t5: T).(\lambda (H5: (subst0 i u (THead (Flat Cast) t3 t2) t5)).(\lambda -(c3: C).(\lambda (H6: (csubst0 i u c c3)).(\lambda (e: C).(\lambda (H7: (getl -i c (CHead e (Bind Abbr) u))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t5 -(THead (Flat Cast) u2 t2))) (\lambda (u2: T).(subst0 i u t3 u2))) (ex2 T -(\lambda (t6: T).(eq T t5 (THead (Flat Cast) t3 t6))) (\lambda (t6: -T).(subst0 (s (Flat Cast) i) u t2 t6))) (ex3_2 T T (\lambda (u2: T).(\lambda -(t6: T).(eq T t5 (THead (Flat Cast) u2 t6)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 i u t3 u2))) (\lambda (_: T).(\lambda (t6: T).(subst0 (s (Flat -Cast) i) u t2 t6)))) (ty3 g c3 t5 (THead (Flat Cast) t0 t3)) (\lambda (H8: -(ex2 T (\lambda (u2: T).(eq T t5 (THead (Flat Cast) u2 t2))) (\lambda (u2: -T).(subst0 i u t3 u2)))).(ex2_ind T (\lambda (u2: T).(eq T t5 (THead (Flat -Cast) u2 t2))) (\lambda (u2: T).(subst0 i u t3 u2)) (ty3 g c3 t5 (THead (Flat -Cast) t0 t3)) (\lambda (x: T).(\lambda (H9: (eq T t5 (THead (Flat Cast) x -t2))).(\lambda (H10: (subst0 i u t3 x)).(eq_ind_r T (THead (Flat Cast) x t2) -(\lambda (t6: T).(ty3 g c3 t6 (THead (Flat Cast) t0 t3))) (ex_ind T (\lambda -(t6: T).(ty3 g c3 t0 t6)) (ty3 g c3 (THead (Flat Cast) x t2) (THead (Flat -Cast) t0 t3)) (\lambda (x0: T).(\lambda (H11: (ty3 g c3 t0 x0)).(ty3_conv g -c3 (THead (Flat Cast) t0 t3) (THead (Flat Cast) x0 t0) (ty3_cast g c3 t3 t0 -(H3 i u c3 t3 (fsubst0_fst i u c t3 c3 H6) e H7) x0 H11) (THead (Flat Cast) x -t2) (THead (Flat Cast) t0 x) (ty3_cast g c3 t2 x (ty3_conv g c3 x t0 (H3 i u -c3 x (fsubst0_both i u c t3 x H10 c3 H6) e H7) t2 t3 (H1 i u c3 t2 -(fsubst0_fst i u c t2 c3 H6) e H7) (pc3_s c3 t3 x (pc3_fsubst0 c t3 t3 -(pc3_refl c t3) i u c3 x (fsubst0_both i u c t3 x H10 c3 H6) e H7))) t0 (H3 i -u c3 x (fsubst0_both i u c t3 x H10 c3 H6) e H7)) (pc3_fsubst0 c (THead (Flat -Cast) t0 t3) (THead (Flat Cast) t0 t3) (pc3_refl c (THead (Flat Cast) t0 t3)) -i u c3 (THead (Flat Cast) t0 x) (fsubst0_both i u c (THead (Flat Cast) t0 t3) -(THead (Flat Cast) t0 x) (subst0_snd (Flat Cast) u x t3 i H10 t0) c3 H6) e -H7)))) (ty3_correct g c3 t3 t0 (H3 i u c3 t3 (fsubst0_fst i u c t3 c3 H6) e -H7))) t5 H9)))) H8)) (\lambda (H8: (ex2 T (\lambda (t6: T).(eq T t5 (THead -(Flat Cast) t3 t6))) (\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 -t6)))).(ex2_ind T (\lambda (t6: T).(eq T t5 (THead (Flat Cast) t3 t6))) -(\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 t6)) (ty3 g c3 t5 (THead -(Flat Cast) t0 t3)) (\lambda (x: T).(\lambda (H9: (eq T t5 (THead (Flat Cast) -t3 x))).(\lambda (H10: (subst0 (s (Flat Cast) i) u t2 x)).(eq_ind_r T (THead -(Flat Cast) t3 x) (\lambda (t6: T).(ty3 g c3 t6 (THead (Flat Cast) t0 t3))) -(ty3_cast g c3 x t3 (H1 i u c3 x (fsubst0_both i u c t2 x H10 c3 H6) e H7) t0 -(H3 i u c3 t3 (fsubst0_fst i u c t3 c3 H6) e H7)) t5 H9)))) H8)) (\lambda -(H8: (ex3_2 T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead (Flat Cast) -u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u t3 u2))) (\lambda (_: -T).(\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 t6))))).(ex3_2_ind T T -(\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead (Flat Cast) u2 t6)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 i u t3 u2))) (\lambda (_: -T).(\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 t6))) (ty3 g c3 t5 (THead -(Flat Cast) t0 t3)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H9: (eq T t5 -(THead (Flat Cast) x0 x1))).(\lambda (H10: (subst0 i u t3 x0)).(\lambda (H11: -(subst0 (s (Flat Cast) i) u t2 x1)).(eq_ind_r T (THead (Flat Cast) x0 x1) -(\lambda (t6: T).(ty3 g c3 t6 (THead (Flat Cast) t0 t3))) (ex_ind T (\lambda -(t6: T).(ty3 g c3 t0 t6)) (ty3 g c3 (THead (Flat Cast) x0 x1) (THead (Flat -Cast) t0 t3)) (\lambda (x: T).(\lambda (H12: (ty3 g c3 t0 x)).(ty3_conv g c3 -(THead (Flat Cast) t0 t3) (THead (Flat Cast) x t0) (ty3_cast g c3 t3 t0 (H3 i -u c3 t3 (fsubst0_fst i u c t3 c3 H6) e H7) x H12) (THead (Flat Cast) x0 x1) -(THead (Flat Cast) t0 x0) (ty3_cast g c3 x1 x0 (ty3_conv g c3 x0 t0 (H3 i u -c3 x0 (fsubst0_both i u c t3 x0 H10 c3 H6) e H7) x1 t3 (H1 i u c3 x1 -(fsubst0_both i u c t2 x1 H11 c3 H6) e H7) (pc3_s c3 t3 x0 (pc3_fsubst0 c t3 -t3 (pc3_refl c t3) i u c3 x0 (fsubst0_both i u c t3 x0 H10 c3 H6) e H7))) t0 -(H3 i u c3 x0 (fsubst0_both i u c t3 x0 H10 c3 H6) e H7)) (pc3_fsubst0 c -(THead (Flat Cast) t0 t3) (THead (Flat Cast) t0 t3) (pc3_refl c (THead (Flat -Cast) t0 t3)) i u c3 (THead (Flat Cast) t0 x0) (fsubst0_both i u c (THead -(Flat Cast) t0 t3) (THead (Flat Cast) t0 x0) (subst0_snd (Flat Cast) u x0 t3 -i H10 t0) c3 H6) e H7)))) (ty3_correct g c3 t3 t0 (H3 i u c3 t3 (fsubst0_fst -i u c t3 c3 H6) e H7))) t5 H9)))))) H8)) (subst0_gen_head (Flat Cast) u t3 t2 -t5 i H5)))))))) c2 t4 H4)))))))))))))) c1 t1 t H))))). -(* COMMENTS -Initial nodes: 23439 -END *) - -theorem ty3_csubst0: - \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c1 -t1 t2) \to (\forall (e: C).(\forall (u: T).(\forall (i: nat).((getl i c1 -(CHead e (Bind Abbr) u)) \to (\forall (c2: C).((csubst0 i u c1 c2) \to (ty3 g -c2 t1 t2))))))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H: (ty3 g c1 t1 t2)).(\lambda (e: C).(\lambda (u: T).(\lambda (i: -nat).(\lambda (H0: (getl i c1 (CHead e (Bind Abbr) u))).(\lambda (c2: -C).(\lambda (H1: (csubst0 i u c1 c2)).(ty3_fsubst0 g c1 t1 t2 H i u c2 t1 -(fsubst0_fst i u c1 t1 c2 H1) e H0))))))))))). -(* COMMENTS -Initial nodes: 89 -END *) - -theorem ty3_subst0: - \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((ty3 g c t1 -t) \to (\forall (e: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead e -(Bind Abbr) u)) \to (\forall (t2: T).((subst0 i u t1 t2) \to (ty3 g c t2 -t))))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H: -(ty3 g c t1 t)).(\lambda (e: C).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H0: (getl i c (CHead e (Bind Abbr) u))).(\lambda (t2: T).(\lambda (H1: -(subst0 i u t1 t2)).(ty3_fsubst0 g c t1 t H i u c t2 (fsubst0_snd i u c t1 t2 -H1) e H0))))))))))). -(* COMMENTS -Initial nodes: 89 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/fwd.ma deleted file mode 100644 index bf6634e45..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/fwd.ma +++ /dev/null @@ -1,922 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/ty3/defs.ma". - -include "Basic-1/pc3/props.ma". - -theorem ty3_gen_sort: - \forall (g: G).(\forall (c: C).(\forall (x: T).(\forall (n: nat).((ty3 g c -(TSort n) x) \to (pc3 c (TSort (next g n)) x))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda -(H: (ty3 g c (TSort n) x)).(insert_eq T (TSort n) (\lambda (t: T).(ty3 g c t -x)) (\lambda (_: T).(pc3 c (TSort (next g n)) x)) (\lambda (y: T).(\lambda -(H0: (ty3 g c y x)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: -T).((eq T t (TSort n)) \to (pc3 c0 (TSort (next g n)) t0))))) (\lambda (c0: -C).(\lambda (t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda -(_: (((eq T t2 (TSort n)) \to (pc3 c0 (TSort (next g n)) t)))).(\lambda (u: -T).(\lambda (t1: T).(\lambda (H3: (ty3 g c0 u t1)).(\lambda (H4: (((eq T u -(TSort n)) \to (pc3 c0 (TSort (next g n)) t1)))).(\lambda (H5: (pc3 c0 t1 -t2)).(\lambda (H6: (eq T u (TSort n))).(let H7 \def (f_equal T T (\lambda (e: -T).e) u (TSort n) H6) in (let H8 \def (eq_ind T u (\lambda (t0: T).((eq T t0 -(TSort n)) \to (pc3 c0 (TSort (next g n)) t1))) H4 (TSort n) H7) in (let H9 -\def (eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 t1)) H3 (TSort n) H7) in -(pc3_t t1 c0 (TSort (next g n)) (H8 (refl_equal T (TSort n))) t2 -H5))))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq T -(TSort m) (TSort n))).(let H2 \def (f_equal T nat (\lambda (e: T).(match e in -T return (\lambda (_: T).nat) with [(TSort n0) \Rightarrow n0 | (TLRef _) -\Rightarrow m | (THead _ _ _) \Rightarrow m])) (TSort m) (TSort n) H1) in -(eq_ind_r nat n (\lambda (n0: nat).(pc3 c0 (TSort (next g n)) (TSort (next g -n0)))) (pc3_refl c0 (TSort (next g n))) m H2))))) (\lambda (n0: nat).(\lambda -(c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (_: (getl n0 c0 (CHead d -(Bind Abbr) u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (_: -(((eq T u (TSort n)) \to (pc3 d (TSort (next g n)) t)))).(\lambda (H4: (eq T -(TLRef n0) (TSort n))).(let H5 \def (eq_ind T (TLRef n0) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I -(TSort n) H4) in (False_ind (pc3 c0 (TSort (next g n)) (lift (S n0) O t)) -H5))))))))))) (\lambda (n0: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda -(u: T).(\lambda (_: (getl n0 c0 (CHead d (Bind Abst) u))).(\lambda (t: -T).(\lambda (_: (ty3 g d u t)).(\lambda (_: (((eq T u (TSort n)) \to (pc3 d -(TSort (next g n)) t)))).(\lambda (H4: (eq T (TLRef n0) (TSort n))).(let H5 -\def (eq_ind T (TLRef n0) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | -(THead _ _ _) \Rightarrow False])) I (TSort n) H4) in (False_ind (pc3 c0 -(TSort (next g n)) (lift (S n0) O u)) H5))))))))))) (\lambda (c0: C).(\lambda -(u: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((eq T u -(TSort n)) \to (pc3 c0 (TSort (next g n)) t)))).(\lambda (b: B).(\lambda (t1: -T).(\lambda (t2: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t1 -t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (pc3 (CHead c0 (Bind b) u) (TSort -(next g n)) t2)))).(\lambda (H5: (eq T (THead (Bind b) u t1) (TSort n))).(let -H6 \def (eq_ind T (THead (Bind b) u t1) (\lambda (ee: T).(match ee in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H5) in -(False_ind (pc3 c0 (TSort (next g n)) (THead (Bind b) u t2)) H6))))))))))))) -(\lambda (c0: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w -u)).(\lambda (_: (((eq T w (TSort n)) \to (pc3 c0 (TSort (next g n)) -u)))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead (Bind -Abst) u t))).(\lambda (_: (((eq T v (TSort n)) \to (pc3 c0 (TSort (next g n)) -(THead (Bind Abst) u t))))).(\lambda (H5: (eq T (THead (Flat Appl) w v) -(TSort n))).(let H6 \def (eq_ind T (THead (Flat Appl) w v) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I -(TSort n) H5) in (False_ind (pc3 c0 (TSort (next g n)) (THead (Flat Appl) w -(THead (Bind Abst) u t))) H6)))))))))))) (\lambda (c0: C).(\lambda (t1: -T).(\lambda (t2: T).(\lambda (_: (ty3 g c0 t1 t2)).(\lambda (_: (((eq T t1 -(TSort n)) \to (pc3 c0 (TSort (next g n)) t2)))).(\lambda (t0: T).(\lambda -(_: (ty3 g c0 t2 t0)).(\lambda (_: (((eq T t2 (TSort n)) \to (pc3 c0 (TSort -(next g n)) t0)))).(\lambda (H5: (eq T (THead (Flat Cast) t2 t1) (TSort -n))).(let H6 \def (eq_ind T (THead (Flat Cast) t2 t1) (\lambda (ee: T).(match -ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | -(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) -H5) in (False_ind (pc3 c0 (TSort (next g n)) (THead (Flat Cast) t0 t2)) -H6))))))))))) c y x H0))) H))))). -(* COMMENTS -Initial nodes: 1179 -END *) - -theorem ty3_gen_lref: - \forall (g: G).(\forall (c: C).(\forall (x: T).(\forall (n: nat).((ty3 g c -(TLRef n) x) \to (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda -(t: T).(pc3 c (lift (S n) O t) x)))) (\lambda (e: C).(\lambda (u: T).(\lambda -(_: T).(getl n c (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: -T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(pc3 c (lift (S n) O u) x)))) (\lambda (e: C).(\lambda -(u: T).(\lambda (_: T).(getl n c (CHead e (Bind Abst) u))))) (\lambda (e: -C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda -(H: (ty3 g c (TLRef n) x)).(insert_eq T (TLRef n) (\lambda (t: T).(ty3 g c t -x)) (\lambda (_: T).(or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda -(t0: T).(pc3 c (lift (S n) O t0) x)))) (\lambda (e: C).(\lambda (u: -T).(\lambda (_: T).(getl n c (CHead e (Bind Abbr) u))))) (\lambda (e: -C).(\lambda (u: T).(\lambda (t0: T).(ty3 g e u t0))))) (ex3_3 C T T (\lambda -(_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c (lift (S n) O u) x)))) (\lambda -(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead e (Bind Abst) u))))) -(\lambda (e: C).(\lambda (u: T).(\lambda (t0: T).(ty3 g e u t0))))))) -(\lambda (y: T).(\lambda (H0: (ty3 g c y x)).(ty3_ind g (\lambda (c0: -C).(\lambda (t: T).(\lambda (t0: T).((eq T t (TLRef n)) \to (or (ex3_3 C T T -(\lambda (_: C).(\lambda (_: T).(\lambda (t1: T).(pc3 c0 (lift (S n) O t1) -t0)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e -(Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t1: T).(ty3 g e -u t1))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 -c0 (lift (S n) O u) t0)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: -T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: -T).(\lambda (t1: T).(ty3 g e u t1)))))))))) (\lambda (c0: C).(\lambda (t2: -T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda (_: (((eq T t2 -(TLRef n)) \to (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: -T).(pc3 c0 (lift (S n) O t0) t)))) (\lambda (e: C).(\lambda (u: T).(\lambda -(_: T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: -T).(\lambda (t0: T).(ty3 g e u t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u) t)))) (\lambda (e: -C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) -(\lambda (e: C).(\lambda (u: T).(\lambda (t0: T).(ty3 g e u -t0))))))))).(\lambda (u: T).(\lambda (t1: T).(\lambda (H3: (ty3 g c0 u -t1)).(\lambda (H4: (((eq T u (TLRef n)) \to (or (ex3_3 C T T (\lambda (_: -C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t1)))) (\lambda -(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) -u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) -(ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift -(S n) O u0) t1)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n -c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(t0: T).(ty3 g e u0 t0))))))))).(\lambda (H5: (pc3 c0 t1 t2)).(\lambda (H6: -(eq T u (TLRef n))).(let H7 \def (f_equal T T (\lambda (e: T).e) u (TLRef n) -H6) in (let H8 \def (eq_ind T u (\lambda (t0: T).((eq T t0 (TLRef n)) \to (or -(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t3: T).(pc3 c0 (lift -(S n) O t3) t1)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n -c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(t3: T).(ty3 g e u0 t3))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: -T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) t1)))) (\lambda (e: C).(\lambda -(u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: -C).(\lambda (u0: T).(\lambda (t3: T).(ty3 g e u0 t3)))))))) H4 (TLRef n) H7) -in (let H9 \def (eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 t1)) H3 (TLRef n) -H7) in (let H10 \def (H8 (refl_equal T (TLRef n))) in (or_ind (ex3_3 C T T -(\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) -t1)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e -(Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g -e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: -T).(pc3 c0 (lift (S n) O u0) t1)))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (t0: T).(ty3 g e u0 t0))))) (or (ex3_3 C T T (\lambda (_: -C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda -(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) -u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) -(ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift -(S n) O u0) t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n -c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(t0: T).(ty3 g e u0 t0)))))) (\lambda (H11: (ex3_3 C T T (\lambda (_: -C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t1)))) (\lambda -(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) -u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 -t0)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: -T).(pc3 c0 (lift (S n) O t0) t1)))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(_: T).(getl n c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (t0: T).(ty3 g e u0 t0)))) (or (ex3_3 C T T (\lambda (_: -C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda -(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) -u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) -(ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift -(S n) O u0) t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n -c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(t0: T).(ty3 g e u0 t0)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: -T).(\lambda (H12: (pc3 c0 (lift (S n) O x2) t1)).(\lambda (H13: (getl n c0 -(CHead x0 (Bind Abbr) x1))).(\lambda (H14: (ty3 g x0 x1 x2)).(or_introl -(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift -(S n) O t0) t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n -c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: -T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) t2)))) (\lambda (e: C).(\lambda -(u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: -C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3_intro C T T -(\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) -t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e -(Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g -e u0 t0)))) x0 x1 x2 (pc3_t t1 c0 (lift (S n) O x2) H12 t2 H5) H13 -H14)))))))) H11)) (\lambda (H11: (ex3_3 C T T (\lambda (_: C).(\lambda (u0: -T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) t1)))) (\lambda (e: C).(\lambda -(u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: -C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))))).(ex3_3_ind C T T -(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) -t1)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e -(Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g -e u0 t0)))) (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: -T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(_: T).(getl n c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda -(u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) t2)))) (\lambda (e: -C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) -(\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))))) -(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H12: (pc3 c0 -(lift (S n) O x1) t1)).(\lambda (H13: (getl n c0 (CHead x0 (Bind Abst) -x1))).(\lambda (H14: (ty3 g x0 x1 x2)).(or_intror (ex3_3 C T T (\lambda (_: -C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda -(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) -u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) -(ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift -(S n) O u0) t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n -c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(t0: T).(ty3 g e u0 t0))))) (ex3_3_intro C T T (\lambda (_: C).(\lambda (u0: -T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) t2)))) (\lambda (e: C).(\lambda -(u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: -C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))) x0 x1 x2 (pc3_t t1 c0 -(lift (S n) O x1) H12 t2 H5) H13 H14)))))))) H11)) H10)))))))))))))))) -(\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq T (TSort m) (TLRef -n))).(let H2 \def (eq_ind T (TSort m) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef n) H1) in -(False_ind (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: -T).(pc3 c0 (lift (S n) O t) (TSort (next g m)))))) (\lambda (e: C).(\lambda -(u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: -C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T T (\lambda -(_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u) (TSort (next -g m)))))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e -(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u -t)))))) H2))))) (\lambda (n0: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda -(u: T).(\lambda (H1: (getl n0 c0 (CHead d (Bind Abbr) u))).(\lambda (t: -T).(\lambda (H2: (ty3 g d u t)).(\lambda (_: (((eq T u (TLRef n)) \to (or -(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 d (lift (S -n) O t0) t)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n d -(CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: -T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda -(_: T).(pc3 d (lift (S n) O u0) t)))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (_: T).(getl n d (CHead e (Bind Abst) u0))))) (\lambda (e: -C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))))))).(\lambda (H4: -(eq T (TLRef n0) (TLRef n))).(let H5 \def (f_equal T nat (\lambda (e: -T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow n0 | -(TLRef n1) \Rightarrow n1 | (THead _ _ _) \Rightarrow n0])) (TLRef n0) (TLRef -n) H4) in (let H6 \def (eq_ind nat n0 (\lambda (n1: nat).(getl n1 c0 (CHead d -(Bind Abbr) u))) H1 n H5) in (eq_ind_r nat n (\lambda (n1: nat).(or (ex3_3 C -T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O -t0) (lift (S n1) O t))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: -T).(getl n c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda -(u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) (lift (S n1) O t))))) -(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind -Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 -t0))))))) (or_introl (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda -(t0: T).(pc3 c0 (lift (S n) O t0) (lift (S n) O t))))) (\lambda (e: -C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u0))))) -(\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3 -C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O -u0) (lift (S n) O t))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: -T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3_intro C T T (\lambda (_: -C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) (lift (S n) O -t))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e -(Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g -e u0 t0)))) d u t (pc3_refl c0 (lift (S n) O t)) H6 H2)) n0 H5)))))))))))) -(\lambda (n0: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda -(H1: (getl n0 c0 (CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (H2: (ty3 -g d u t)).(\lambda (_: (((eq T u (TLRef n)) \to (or (ex3_3 C T T (\lambda (_: -C).(\lambda (_: T).(\lambda (t0: T).(pc3 d (lift (S n) O t0) t)))) (\lambda -(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n d (CHead e (Bind Abbr) -u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) -(ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 d (lift (S -n) O u0) t)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n d -(CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: -T).(ty3 g e u0 t0))))))))).(\lambda (H4: (eq T (TLRef n0) (TLRef n))).(let H5 -\def (f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) -with [(TSort _) \Rightarrow n0 | (TLRef n1) \Rightarrow n1 | (THead _ _ _) -\Rightarrow n0])) (TLRef n0) (TLRef n) H4) in (let H6 \def (eq_ind nat n0 -(\lambda (n1: nat).(getl n1 c0 (CHead d (Bind Abst) u))) H1 n H5) in -(eq_ind_r nat n (\lambda (n1: nat).(or (ex3_3 C T T (\lambda (_: C).(\lambda -(_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) (lift (S n1) O u))))) -(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind -Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 -t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 -(lift (S n) O u0) (lift (S n1) O u))))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: -C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))))) (or_intror (ex3_3 -C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O -t0) (lift (S n) O u))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: -T).(getl n c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda -(u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) (lift (S n) O u))))) -(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind -Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 -t0))))) (ex3_3_intro C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: -T).(pc3 c0 (lift (S n) O u0) (lift (S n) O u))))) (\lambda (e: C).(\lambda -(u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: -C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))) d u t (pc3_refl c0 -(lift (S n) O u)) H6 H2)) n0 H5)))))))))))) (\lambda (c0: C).(\lambda (u: -T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((eq T u (TLRef -n)) \to (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: -T).(pc3 c0 (lift (S n) O t0) t)))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(_: T).(getl n c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda -(u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) t)))) (\lambda (e: -C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) -(\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 -t0))))))))).(\lambda (b: B).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: -(ty3 g (CHead c0 (Bind b) u) t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to -(or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 (CHead -c0 (Bind b) u) (lift (S n) O t0) t2)))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (_: T).(getl n (CHead c0 (Bind b) u) (CHead e (Bind Abbr) u0))))) -(\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3 -C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 (CHead c0 (Bind -b) u) (lift (S n) O u0) t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: -T).(getl n (CHead c0 (Bind b) u) (CHead e (Bind Abst) u0))))) (\lambda (e: -C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))))))).(\lambda (H5: -(eq T (THead (Bind b) u t1) (TLRef n))).(let H6 \def (eq_ind T (THead (Bind -b) u t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) -\Rightarrow True])) I (TLRef n) H5) in (False_ind (or (ex3_3 C T T (\lambda -(_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) (THead -(Bind b) u t2))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n -c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: -T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) (THead (Bind b) u t2))))) -(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind -Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 -t0)))))) H6))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda (u: -T).(\lambda (_: (ty3 g c0 w u)).(\lambda (_: (((eq T w (TLRef n)) \to (or -(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c0 (lift (S -n) O t) u)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 -(CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t: -T).(ty3 g e u0 t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda -(_: T).(pc3 c0 (lift (S n) O u0) u)))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: -C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t))))))))).(\lambda (v: -T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead (Bind Abst) u -t))).(\lambda (_: (((eq T v (TLRef n)) \to (or (ex3_3 C T T (\lambda (_: -C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) (THead (Bind -Abst) u t))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 -(CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: -T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda -(_: T).(pc3 c0 (lift (S n) O u0) (THead (Bind Abst) u t))))) (\lambda (e: -C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) -(\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 -t0))))))))).(\lambda (H5: (eq T (THead (Flat Appl) w v) (TLRef n))).(let H6 -\def (eq_ind T (THead (Flat Appl) w v) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H5) in -(False_ind (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: -T).(pc3 c0 (lift (S n) O t0) (THead (Flat Appl) w (THead (Bind Abst) u -t)))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e -(Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g -e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: -T).(pc3 c0 (lift (S n) O u0) (THead (Flat Appl) w (THead (Bind Abst) u -t)))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e -(Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g -e u0 t0)))))) H6)))))))))))) (\lambda (c0: C).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (_: (ty3 g c0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (or -(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c0 (lift (S -n) O t) t2)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 -(CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: -T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda -(_: T).(pc3 c0 (lift (S n) O u) t2)))) (\lambda (e: C).(\lambda (u: -T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: -C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))))))).(\lambda (t0: -T).(\lambda (_: (ty3 g c0 t2 t0)).(\lambda (_: (((eq T t2 (TLRef n)) \to (or -(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c0 (lift (S -n) O t) t0)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 -(CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: -T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda -(_: T).(pc3 c0 (lift (S n) O u) t0)))) (\lambda (e: C).(\lambda (u: -T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: -C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))))))).(\lambda (H5: (eq T -(THead (Flat Cast) t2 t1) (TLRef n))).(let H6 \def (eq_ind T (THead (Flat -Cast) t2 t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) -with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ -_) \Rightarrow True])) I (TLRef n) H5) in (False_ind (or (ex3_3 C T T -(\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c0 (lift (S n) O t) -(THead (Flat Cast) t0 t2))))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: -T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: -T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u) (THead (Flat Cast) t0 t2))))) -(\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind -Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u -t)))))) H6))))))))))) c y x H0))) H))))). -(* COMMENTS -Initial nodes: 5569 -END *) - -theorem ty3_gen_bind: - \forall (g: G).(\forall (b: B).(\forall (c: C).(\forall (u: T).(\forall (t1: -T).(\forall (x: T).((ty3 g c (THead (Bind b) u t1) x) \to (ex3_2 T T (\lambda -(t2: T).(\lambda (_: T).(pc3 c (THead (Bind b) u t2) x))) (\lambda (_: -T).(\lambda (t: T).(ty3 g c u t))) (\lambda (t2: T).(\lambda (_: T).(ty3 g -(CHead c (Bind b) u) t1 t2)))))))))) -\def - \lambda (g: G).(\lambda (b: B).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: -T).(\lambda (x: T).(\lambda (H: (ty3 g c (THead (Bind b) u t1) x)).(insert_eq -T (THead (Bind b) u t1) (\lambda (t: T).(ty3 g c t x)) (\lambda (_: T).(ex3_2 -T T (\lambda (t2: T).(\lambda (_: T).(pc3 c (THead (Bind b) u t2) x))) -(\lambda (_: T).(\lambda (t0: T).(ty3 g c u t0))) (\lambda (t2: T).(\lambda -(_: T).(ty3 g (CHead c (Bind b) u) t1 t2))))) (\lambda (y: T).(\lambda (H0: -(ty3 g c y x)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: -T).((eq T t (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda (t2: T).(\lambda -(_: T).(pc3 c0 (THead (Bind b) u t2) t0))) (\lambda (_: T).(\lambda (t3: -T).(ty3 g c0 u t3))) (\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind -b) u) t1 t2)))))))) (\lambda (c0: C).(\lambda (t2: T).(\lambda (t: -T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda (_: (((eq T t2 (THead (Bind b) u -t1)) \to (ex3_2 T T (\lambda (t3: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) -u t3) t))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u t0))) (\lambda (t3: -T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3))))))).(\lambda (u0: -T).(\lambda (t0: T).(\lambda (H3: (ty3 g c0 u0 t0)).(\lambda (H4: (((eq T u0 -(THead (Bind b) u t1)) \to (ex3_2 T T (\lambda (t3: T).(\lambda (_: T).(pc3 -c0 (THead (Bind b) u t3) t0))) (\lambda (_: T).(\lambda (t4: T).(ty3 g c0 u -t4))) (\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 -t3))))))).(\lambda (H5: (pc3 c0 t0 t2)).(\lambda (H6: (eq T u0 (THead (Bind -b) u t1))).(let H7 \def (f_equal T T (\lambda (e: T).e) u0 (THead (Bind b) u -t1) H6) in (let H8 \def (eq_ind T u0 (\lambda (t3: T).((eq T t3 (THead (Bind -b) u t1)) \to (ex3_2 T T (\lambda (t4: T).(\lambda (_: T).(pc3 c0 (THead -(Bind b) u t4) t0))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c0 u t5))) -(\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t4)))))) H4 -(THead (Bind b) u t1) H7) in (let H9 \def (eq_ind T u0 (\lambda (t3: T).(ty3 -g c0 t3 t0)) H3 (THead (Bind b) u t1) H7) in (let H10 \def (H8 (refl_equal T -(THead (Bind b) u t1))) in (ex3_2_ind T T (\lambda (t3: T).(\lambda (_: -T).(pc3 c0 (THead (Bind b) u t3) t0))) (\lambda (_: T).(\lambda (t4: T).(ty3 -g c0 u t4))) (\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 -t3))) (ex3_2 T T (\lambda (t3: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u -t3) t2))) (\lambda (_: T).(\lambda (t4: T).(ty3 g c0 u t4))) (\lambda (t3: -T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3)))) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H11: (pc3 c0 (THead (Bind b) u x0) -t0)).(\lambda (H12: (ty3 g c0 u x1)).(\lambda (H13: (ty3 g (CHead c0 (Bind b) -u) t1 x0)).(ex3_2_intro T T (\lambda (t3: T).(\lambda (_: T).(pc3 c0 (THead -(Bind b) u t3) t2))) (\lambda (_: T).(\lambda (t4: T).(ty3 g c0 u t4))) -(\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3))) x0 x1 -(pc3_t t0 c0 (THead (Bind b) u x0) H11 t2 H5) H12 H13)))))) -H10)))))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq T -(TSort m) (THead (Bind b) u t1))).(let H2 \def (eq_ind T (TSort m) (\lambda -(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow -False])) I (THead (Bind b) u t1) H1) in (False_ind (ex3_2 T T (\lambda (t2: -T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t2) (TSort (next g m))))) -(\lambda (_: T).(\lambda (t: T).(ty3 g c0 u t))) (\lambda (t2: T).(\lambda -(_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2)))) H2))))) (\lambda (n: -nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (_: (getl n -c0 (CHead d (Bind Abbr) u0))).(\lambda (t: T).(\lambda (_: (ty3 g d u0 -t)).(\lambda (_: (((eq T u0 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda -(t2: T).(\lambda (_: T).(pc3 d (THead (Bind b) u t2) t))) (\lambda (_: -T).(\lambda (t0: T).(ty3 g d u t0))) (\lambda (t2: T).(\lambda (_: T).(ty3 g -(CHead d (Bind b) u) t1 t2))))))).(\lambda (H4: (eq T (TLRef n) (THead (Bind -b) u t1))).(let H5 \def (eq_ind T (TLRef n) (\lambda (ee: T).(match ee in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t1) -H4) in (False_ind (ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3 c0 (THead -(Bind b) u t2) (lift (S n) O t)))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c0 -u t0))) (\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 -t2)))) H5))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: -C).(\lambda (u0: T).(\lambda (_: (getl n c0 (CHead d (Bind Abst) -u0))).(\lambda (t: T).(\lambda (_: (ty3 g d u0 t)).(\lambda (_: (((eq T u0 -(THead (Bind b) u t1)) \to (ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3 d -(THead (Bind b) u t2) t))) (\lambda (_: T).(\lambda (t0: T).(ty3 g d u t0))) -(\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead d (Bind b) u) t1 -t2))))))).(\lambda (H4: (eq T (TLRef n) (THead (Bind b) u t1))).(let H5 \def -(eq_ind T (TLRef n) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | -(THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t1) H4) in (False_ind -(ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t2) -(lift (S n) O u0)))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u t0))) -(\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2)))) -H5))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (t: T).(\lambda (H1: -(ty3 g c0 u0 t)).(\lambda (H2: (((eq T u0 (THead (Bind b) u t1)) \to (ex3_2 T -T (\lambda (t2: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t2) t))) -(\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u t0))) (\lambda (t2: T).(\lambda -(_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2))))))).(\lambda (b0: B).(\lambda -(t0: T).(\lambda (t2: T).(\lambda (H3: (ty3 g (CHead c0 (Bind b0) u0) t0 -t2)).(\lambda (H4: (((eq T t0 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda -(t3: T).(\lambda (_: T).(pc3 (CHead c0 (Bind b0) u0) (THead (Bind b) u t3) -t2))) (\lambda (_: T).(\lambda (t4: T).(ty3 g (CHead c0 (Bind b0) u0) u t4))) -(\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead (CHead c0 (Bind b0) u0) (Bind -b) u) t1 t3))))))).(\lambda (H5: (eq T (THead (Bind b0) u0 t0) (THead (Bind -b) u t1))).(let H6 \def (f_equal T B (\lambda (e: T).(match e in T return -(\lambda (_: T).B) with [(TSort _) \Rightarrow b0 | (TLRef _) \Rightarrow b0 -| (THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).B) with -[(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow b0])])) (THead (Bind b0) u0 -t0) (THead (Bind b) u t1) H5) in ((let H7 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | -(TLRef _) \Rightarrow u0 | (THead _ t3 _) \Rightarrow t3])) (THead (Bind b0) -u0 t0) (THead (Bind b) u t1) H5) in ((let H8 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | -(TLRef _) \Rightarrow t0 | (THead _ _ t3) \Rightarrow t3])) (THead (Bind b0) -u0 t0) (THead (Bind b) u t1) H5) in (\lambda (H9: (eq T u0 u)).(\lambda (H10: -(eq B b0 b)).(let H11 \def (eq_ind T t0 (\lambda (t3: T).((eq T t3 (THead -(Bind b) u t1)) \to (ex3_2 T T (\lambda (t4: T).(\lambda (_: T).(pc3 (CHead -c0 (Bind b0) u0) (THead (Bind b) u t4) t2))) (\lambda (_: T).(\lambda (t5: -T).(ty3 g (CHead c0 (Bind b0) u0) u t5))) (\lambda (t4: T).(\lambda (_: -T).(ty3 g (CHead (CHead c0 (Bind b0) u0) (Bind b) u) t1 t4)))))) H4 t1 H8) in -(let H12 \def (eq_ind T t0 (\lambda (t3: T).(ty3 g (CHead c0 (Bind b0) u0) t3 -t2)) H3 t1 H8) in (let H13 \def (eq_ind B b0 (\lambda (b1: B).((eq T t1 -(THead (Bind b) u t1)) \to (ex3_2 T T (\lambda (t3: T).(\lambda (_: T).(pc3 -(CHead c0 (Bind b1) u0) (THead (Bind b) u t3) t2))) (\lambda (_: T).(\lambda -(t4: T).(ty3 g (CHead c0 (Bind b1) u0) u t4))) (\lambda (t3: T).(\lambda (_: -T).(ty3 g (CHead (CHead c0 (Bind b1) u0) (Bind b) u) t1 t3)))))) H11 b H10) -in (let H14 \def (eq_ind B b0 (\lambda (b1: B).(ty3 g (CHead c0 (Bind b1) u0) -t1 t2)) H12 b H10) in (eq_ind_r B b (\lambda (b1: B).(ex3_2 T T (\lambda (t3: -T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t3) (THead (Bind b1) u0 t2)))) -(\lambda (_: T).(\lambda (t4: T).(ty3 g c0 u t4))) (\lambda (t3: T).(\lambda -(_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3))))) (let H15 \def (eq_ind T u0 -(\lambda (t3: T).((eq T t1 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda -(t4: T).(\lambda (_: T).(pc3 (CHead c0 (Bind b) t3) (THead (Bind b) u t4) -t2))) (\lambda (_: T).(\lambda (t5: T).(ty3 g (CHead c0 (Bind b) t3) u t5))) -(\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead (CHead c0 (Bind b) t3) (Bind -b) u) t1 t4)))))) H13 u H9) in (let H16 \def (eq_ind T u0 (\lambda (t3: -T).(ty3 g (CHead c0 (Bind b) t3) t1 t2)) H14 u H9) in (let H17 \def (eq_ind T -u0 (\lambda (t3: T).((eq T t3 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda -(t4: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t4) t))) (\lambda (_: -T).(\lambda (t5: T).(ty3 g c0 u t5))) (\lambda (t4: T).(\lambda (_: T).(ty3 g -(CHead c0 (Bind b) u) t1 t4)))))) H2 u H9) in (let H18 \def (eq_ind T u0 -(\lambda (t3: T).(ty3 g c0 t3 t)) H1 u H9) in (eq_ind_r T u (\lambda (t3: -T).(ex3_2 T T (\lambda (t4: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t4) -(THead (Bind b) t3 t2)))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c0 u t5))) -(\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t4))))) -(ex3_2_intro T T (\lambda (t3: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u -t3) (THead (Bind b) u t2)))) (\lambda (_: T).(\lambda (t4: T).(ty3 g c0 u -t4))) (\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3))) -t2 t (pc3_refl c0 (THead (Bind b) u t2)) H18 H16) u0 H9))))) b0 H10)))))))) -H7)) H6))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda (u0: -T).(\lambda (_: (ty3 g c0 w u0)).(\lambda (_: (((eq T w (THead (Bind b) u -t1)) \to (ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) -u t2) u0))) (\lambda (_: T).(\lambda (t: T).(ty3 g c0 u t))) (\lambda (t2: -T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2))))))).(\lambda (v: -T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead (Bind Abst) u0 -t))).(\lambda (_: (((eq T v (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda -(t2: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t2) (THead (Bind Abst) u0 -t)))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u t0))) (\lambda (t2: -T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2))))))).(\lambda (H5: -(eq T (THead (Flat Appl) w v) (THead (Bind b) u t1))).(let H6 \def (eq_ind T -(THead (Flat Appl) w v) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind -b) u t1) H5) in (False_ind (ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3 -c0 (THead (Bind b) u t2) (THead (Flat Appl) w (THead (Bind Abst) u0 t))))) -(\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u t0))) (\lambda (t2: T).(\lambda -(_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2)))) H6)))))))))))) (\lambda (c0: -C).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (ty3 g c0 t0 t2)).(\lambda -(_: (((eq T t0 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda (t3: -T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t3) t2))) (\lambda (_: -T).(\lambda (t: T).(ty3 g c0 u t))) (\lambda (t3: T).(\lambda (_: T).(ty3 g -(CHead c0 (Bind b) u) t1 t3))))))).(\lambda (t3: T).(\lambda (_: (ty3 g c0 t2 -t3)).(\lambda (_: (((eq T t2 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda -(t4: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t4) t3))) (\lambda (_: -T).(\lambda (t: T).(ty3 g c0 u t))) (\lambda (t4: T).(\lambda (_: T).(ty3 g -(CHead c0 (Bind b) u) t1 t4))))))).(\lambda (H5: (eq T (THead (Flat Cast) t2 -t0) (THead (Bind b) u t1))).(let H6 \def (eq_ind T (THead (Flat Cast) t2 t0) -(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow -(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | -(Flat _) \Rightarrow True])])) I (THead (Bind b) u t1) H5) in (False_ind -(ex3_2 T T (\lambda (t4: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t4) -(THead (Flat Cast) t3 t2)))) (\lambda (_: T).(\lambda (t: T).(ty3 g c0 u t))) -(\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t4)))) -H6))))))))))) c y x H0))) H))))))). -(* COMMENTS -Initial nodes: 3389 -END *) - -theorem ty3_gen_appl: - \forall (g: G).(\forall (c: C).(\forall (w: T).(\forall (v: T).(\forall (x: -T).((ty3 g c (THead (Flat Appl) w v) x) \to (ex3_2 T T (\lambda (u: -T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x))) -(\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) -(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (w: T).(\lambda (v: T).(\lambda (x: -T).(\lambda (H: (ty3 g c (THead (Flat Appl) w v) x)).(insert_eq T (THead -(Flat Appl) w v) (\lambda (t: T).(ty3 g c t x)) (\lambda (_: T).(ex3_2 T T -(\lambda (u: T).(\lambda (t0: T).(pc3 c (THead (Flat Appl) w (THead (Bind -Abst) u t0)) x))) (\lambda (u: T).(\lambda (t0: T).(ty3 g c v (THead (Bind -Abst) u t0)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c w u))))) (\lambda (y: -T).(\lambda (H0: (ty3 g c y x)).(ty3_ind g (\lambda (c0: C).(\lambda (t: -T).(\lambda (t0: T).((eq T t (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda -(u: T).(\lambda (t1: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u -t1)) t0))) (\lambda (u: T).(\lambda (t1: T).(ty3 g c0 v (THead (Bind Abst) u -t1)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c0 w u)))))))) (\lambda (c0: -C).(\lambda (t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda -(_: (((eq T t2 (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda (u: -T).(\lambda (t0: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u t0)) -t))) (\lambda (u: T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u t0)))) -(\lambda (u: T).(\lambda (_: T).(ty3 g c0 w u))))))).(\lambda (u: T).(\lambda -(t1: T).(\lambda (H3: (ty3 g c0 u t1)).(\lambda (H4: (((eq T u (THead (Flat -Appl) w v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0 (THead -(Flat Appl) w (THead (Bind Abst) u0 t0)) t1))) (\lambda (u0: T).(\lambda (t0: -T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_: -T).(ty3 g c0 w u0))))))).(\lambda (H5: (pc3 c0 t1 t2)).(\lambda (H6: (eq T u -(THead (Flat Appl) w v))).(let H7 \def (f_equal T T (\lambda (e: T).e) u -(THead (Flat Appl) w v) H6) in (let H8 \def (eq_ind T u (\lambda (t0: T).((eq -T t0 (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t3: -T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t3)) t1))) (\lambda -(u0: T).(\lambda (t3: T).(ty3 g c0 v (THead (Bind Abst) u0 t3)))) (\lambda -(u0: T).(\lambda (_: T).(ty3 g c0 w u0)))))) H4 (THead (Flat Appl) w v) H7) -in (let H9 \def (eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 t1)) H3 (THead -(Flat Appl) w v) H7) in (let H10 \def (H8 (refl_equal T (THead (Flat Appl) w -v))) in (ex3_2_ind T T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0 (THead (Flat -Appl) w (THead (Bind Abst) u0 t0)) t1))) (\lambda (u0: T).(\lambda (t0: -T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_: -T).(ty3 g c0 w u0))) (ex3_2 T T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0 -(THead (Flat Appl) w (THead (Bind Abst) u0 t0)) t2))) (\lambda (u0: -T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: -T).(\lambda (_: T).(ty3 g c0 w u0)))) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H11: (pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) x0 x1)) -t1)).(\lambda (H12: (ty3 g c0 v (THead (Bind Abst) x0 x1))).(\lambda (H13: -(ty3 g c0 w x0)).(ex3_2_intro T T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0 -(THead (Flat Appl) w (THead (Bind Abst) u0 t0)) t2))) (\lambda (u0: -T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: -T).(\lambda (_: T).(ty3 g c0 w u0))) x0 x1 (pc3_t t1 c0 (THead (Flat Appl) w -(THead (Bind Abst) x0 x1)) H11 t2 H5) H12 H13)))))) H10)))))))))))))))) -(\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq T (TSort m) (THead (Flat -Appl) w v))).(let H2 \def (eq_ind T (TSort m) (\lambda (ee: T).(match ee in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) w -v) H1) in (False_ind (ex3_2 T T (\lambda (u: T).(\lambda (t: T).(pc3 c0 -(THead (Flat Appl) w (THead (Bind Abst) u t)) (TSort (next g m))))) (\lambda -(u: T).(\lambda (t: T).(ty3 g c0 v (THead (Bind Abst) u t)))) (\lambda (u: -T).(\lambda (_: T).(ty3 g c0 w u)))) H2))))) (\lambda (n: nat).(\lambda (c0: -C).(\lambda (d: C).(\lambda (u: T).(\lambda (_: (getl n c0 (CHead d (Bind -Abbr) u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (_: (((eq T u -(THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t0: -T).(pc3 d (THead (Flat Appl) w (THead (Bind Abst) u0 t0)) t))) (\lambda (u0: -T).(\lambda (t0: T).(ty3 g d v (THead (Bind Abst) u0 t0)))) (\lambda (u0: -T).(\lambda (_: T).(ty3 g d w u0))))))).(\lambda (H4: (eq T (TLRef n) (THead -(Flat Appl) w v))).(let H5 \def (eq_ind T (TLRef n) (\lambda (ee: T).(match -ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | -(TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead -(Flat Appl) w v) H4) in (False_ind (ex3_2 T T (\lambda (u0: T).(\lambda (t0: -T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t0)) (lift (S n) O -t)))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u0 -t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w u0)))) H5))))))))))) -(\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda -(_: (getl n c0 (CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (_: (ty3 g -d u t)).(\lambda (_: (((eq T u (THead (Flat Appl) w v)) \to (ex3_2 T T -(\lambda (u0: T).(\lambda (t0: T).(pc3 d (THead (Flat Appl) w (THead (Bind -Abst) u0 t0)) t))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g d v (THead (Bind -Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g d w -u0))))))).(\lambda (H4: (eq T (TLRef n) (THead (Flat Appl) w v))).(let H5 -\def (eq_ind T (TLRef n) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | -(THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) w v) H4) in -(False_ind (ex3_2 T T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0 (THead (Flat -Appl) w (THead (Bind Abst) u0 t0)) (lift (S n) O u)))) (\lambda (u0: -T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: -T).(\lambda (_: T).(ty3 g c0 w u0)))) H5))))))))))) (\lambda (c0: C).(\lambda -(u: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((eq T u -(THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t0: -T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t0)) t))) (\lambda (u0: -T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: -T).(\lambda (_: T).(ty3 g c0 w u0))))))).(\lambda (b: B).(\lambda (t1: -T).(\lambda (t2: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t1 -t2)).(\lambda (_: (((eq T t1 (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda -(u0: T).(\lambda (t0: T).(pc3 (CHead c0 (Bind b) u) (THead (Flat Appl) w -(THead (Bind Abst) u0 t0)) t2))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g -(CHead c0 (Bind b) u) v (THead (Bind Abst) u0 t0)))) (\lambda (u0: -T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) w u0))))))).(\lambda (H5: (eq -T (THead (Bind b) u t1) (THead (Flat Appl) w v))).(let H6 \def (eq_ind T -(THead (Bind b) u t1) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat -Appl) w v) H5) in (False_ind (ex3_2 T T (\lambda (u0: T).(\lambda (t0: -T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t0)) (THead (Bind b) u -t2)))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u0 -t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w u0)))) H6))))))))))))) -(\lambda (c0: C).(\lambda (w0: T).(\lambda (u: T).(\lambda (H1: (ty3 g c0 w0 -u)).(\lambda (H2: (((eq T w0 (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda -(u0: T).(\lambda (t: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 -t)) u))) (\lambda (u0: T).(\lambda (t: T).(ty3 g c0 v (THead (Bind Abst) u0 -t)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w u0))))))).(\lambda (v0: -T).(\lambda (t: T).(\lambda (H3: (ty3 g c0 v0 (THead (Bind Abst) u -t))).(\lambda (H4: (((eq T v0 (THead (Flat Appl) w v)) \to (ex3_2 T T -(\lambda (u0: T).(\lambda (t0: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind -Abst) u0 t0)) (THead (Bind Abst) u t)))) (\lambda (u0: T).(\lambda (t0: -T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_: -T).(ty3 g c0 w u0))))))).(\lambda (H5: (eq T (THead (Flat Appl) w0 v0) (THead -(Flat Appl) w v))).(let H6 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow w0 | (TLRef _) -\Rightarrow w0 | (THead _ t0 _) \Rightarrow t0])) (THead (Flat Appl) w0 v0) -(THead (Flat Appl) w v) H5) in ((let H7 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v0 | -(TLRef _) \Rightarrow v0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat -Appl) w0 v0) (THead (Flat Appl) w v) H5) in (\lambda (H8: (eq T w0 w)).(let -H9 \def (eq_ind T v0 (\lambda (t0: T).((eq T t0 (THead (Flat Appl) w v)) \to -(ex3_2 T T (\lambda (u0: T).(\lambda (t1: T).(pc3 c0 (THead (Flat Appl) w -(THead (Bind Abst) u0 t1)) (THead (Bind Abst) u t)))) (\lambda (u0: -T).(\lambda (t1: T).(ty3 g c0 v (THead (Bind Abst) u0 t1)))) (\lambda (u0: -T).(\lambda (_: T).(ty3 g c0 w u0)))))) H4 v H7) in (let H10 \def (eq_ind T -v0 (\lambda (t0: T).(ty3 g c0 t0 (THead (Bind Abst) u t))) H3 v H7) in (let -H11 \def (eq_ind T w0 (\lambda (t0: T).((eq T t0 (THead (Flat Appl) w v)) \to -(ex3_2 T T (\lambda (u0: T).(\lambda (t1: T).(pc3 c0 (THead (Flat Appl) w -(THead (Bind Abst) u0 t1)) u))) (\lambda (u0: T).(\lambda (t1: T).(ty3 g c0 v -(THead (Bind Abst) u0 t1)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w -u0)))))) H2 w H8) in (let H12 \def (eq_ind T w0 (\lambda (t0: T).(ty3 g c0 t0 -u)) H1 w H8) in (eq_ind_r T w (\lambda (t0: T).(ex3_2 T T (\lambda (u0: -T).(\lambda (t1: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t1)) -(THead (Flat Appl) t0 (THead (Bind Abst) u t))))) (\lambda (u0: T).(\lambda -(t1: T).(ty3 g c0 v (THead (Bind Abst) u0 t1)))) (\lambda (u0: T).(\lambda -(_: T).(ty3 g c0 w u0))))) (ex3_2_intro T T (\lambda (u0: T).(\lambda (t0: -T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t0)) (THead (Flat Appl) -w (THead (Bind Abst) u t))))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g c0 v -(THead (Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w -u0))) u t (pc3_refl c0 (THead (Flat Appl) w (THead (Bind Abst) u t))) H10 -H12) w0 H8))))))) H6)))))))))))) (\lambda (c0: C).(\lambda (t1: T).(\lambda -(t2: T).(\lambda (_: (ty3 g c0 t1 t2)).(\lambda (_: (((eq T t1 (THead (Flat -Appl) w v)) \to (ex3_2 T T (\lambda (u: T).(\lambda (t: T).(pc3 c0 (THead -(Flat Appl) w (THead (Bind Abst) u t)) t2))) (\lambda (u: T).(\lambda (t: -T).(ty3 g c0 v (THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (_: -T).(ty3 g c0 w u))))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t2 -t0)).(\lambda (_: (((eq T t2 (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda -(u: T).(\lambda (t: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u t)) -t0))) (\lambda (u: T).(\lambda (t: T).(ty3 g c0 v (THead (Bind Abst) u t)))) -(\lambda (u: T).(\lambda (_: T).(ty3 g c0 w u))))))).(\lambda (H5: (eq T -(THead (Flat Cast) t2 t1) (THead (Flat Appl) w v))).(let H6 \def (eq_ind T -(THead (Flat Cast) t2 t1) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f in F return -(\lambda (_: F).Prop) with [Appl \Rightarrow False | Cast \Rightarrow -True])])])) I (THead (Flat Appl) w v) H5) in (False_ind (ex3_2 T T (\lambda -(u: T).(\lambda (t: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u t)) -(THead (Flat Cast) t0 t2)))) (\lambda (u: T).(\lambda (t: T).(ty3 g c0 v -(THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c0 w u)))) -H6))))))))))) c y x H0))) H)))))). -(* COMMENTS -Initial nodes: 3171 -END *) - -theorem ty3_gen_cast: - \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).(\forall -(x: T).((ty3 g c (THead (Flat Cast) t2 t1) x) \to (ex3 T (\lambda (t0: -T).(pc3 c (THead (Flat Cast) t0 t2) x)) (\lambda (_: T).(ty3 g c t1 t2)) -(\lambda (t0: T).(ty3 g c t2 t0)))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(x: T).(\lambda (H: (ty3 g c (THead (Flat Cast) t2 t1) x)).(insert_eq T -(THead (Flat Cast) t2 t1) (\lambda (t: T).(ty3 g c t x)) (\lambda (_: T).(ex3 -T (\lambda (t0: T).(pc3 c (THead (Flat Cast) t0 t2) x)) (\lambda (_: T).(ty3 -g c t1 t2)) (\lambda (t0: T).(ty3 g c t2 t0)))) (\lambda (y: T).(\lambda (H0: -(ty3 g c y x)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: -T).((eq T t (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t3: T).(pc3 c0 -(THead (Flat Cast) t3 t2) t0)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda -(t3: T).(ty3 g c0 t2 t3))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t: -T).(\lambda (_: (ty3 g c0 t0 t)).(\lambda (_: (((eq T t0 (THead (Flat Cast) -t2 t1)) \to (ex3 T (\lambda (t3: T).(pc3 c0 (THead (Flat Cast) t3 t2) t)) -(\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t3: T).(ty3 g c0 t2 -t3)))))).(\lambda (u: T).(\lambda (t3: T).(\lambda (H3: (ty3 g c0 u -t3)).(\lambda (H4: (((eq T u (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda -(t4: T).(pc3 c0 (THead (Flat Cast) t4 t2) t3)) (\lambda (_: T).(ty3 g c0 t1 -t2)) (\lambda (t4: T).(ty3 g c0 t2 t4)))))).(\lambda (H5: (pc3 c0 t3 -t0)).(\lambda (H6: (eq T u (THead (Flat Cast) t2 t1))).(let H7 \def (f_equal -T T (\lambda (e: T).e) u (THead (Flat Cast) t2 t1) H6) in (let H8 \def -(eq_ind T u (\lambda (t4: T).((eq T t4 (THead (Flat Cast) t2 t1)) \to (ex3 T -(\lambda (t5: T).(pc3 c0 (THead (Flat Cast) t5 t2) t3)) (\lambda (_: T).(ty3 -g c0 t1 t2)) (\lambda (t5: T).(ty3 g c0 t2 t5))))) H4 (THead (Flat Cast) t2 -t1) H7) in (let H9 \def (eq_ind T u (\lambda (t4: T).(ty3 g c0 t4 t3)) H3 -(THead (Flat Cast) t2 t1) H7) in (let H10 \def (H8 (refl_equal T (THead (Flat -Cast) t2 t1))) in (ex3_ind T (\lambda (t4: T).(pc3 c0 (THead (Flat Cast) t4 -t2) t3)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t4: T).(ty3 g c0 t2 t4)) -(ex3 T (\lambda (t4: T).(pc3 c0 (THead (Flat Cast) t4 t2) t0)) (\lambda (_: -T).(ty3 g c0 t1 t2)) (\lambda (t4: T).(ty3 g c0 t2 t4))) (\lambda (x0: -T).(\lambda (H11: (pc3 c0 (THead (Flat Cast) x0 t2) t3)).(\lambda (H12: (ty3 -g c0 t1 t2)).(\lambda (H13: (ty3 g c0 t2 x0)).(ex3_intro T (\lambda (t4: -T).(pc3 c0 (THead (Flat Cast) t4 t2) t0)) (\lambda (_: T).(ty3 g c0 t1 t2)) -(\lambda (t4: T).(ty3 g c0 t2 t4)) x0 (pc3_t t3 c0 (THead (Flat Cast) x0 t2) -H11 t0 H5) H12 H13))))) H10)))))))))))))))) (\lambda (c0: C).(\lambda (m: -nat).(\lambda (H1: (eq T (TSort m) (THead (Flat Cast) t2 t1))).(let H2 \def -(eq_ind T (TSort m) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | -(THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) t2 t1) H1) in -(False_ind (ex3 T (\lambda (t0: T).(pc3 c0 (THead (Flat Cast) t0 t2) (TSort -(next g m)))) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0 t2 -t0))) H2))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: -T).(\lambda (_: (getl n c0 (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda -(_: (ty3 g d u t)).(\lambda (_: (((eq T u (THead (Flat Cast) t2 t1)) \to (ex3 -T (\lambda (t0: T).(pc3 d (THead (Flat Cast) t0 t2) t)) (\lambda (_: T).(ty3 -g d t1 t2)) (\lambda (t0: T).(ty3 g d t2 t0)))))).(\lambda (H4: (eq T (TLRef -n) (THead (Flat Cast) t2 t1))).(let H5 \def (eq_ind T (TLRef n) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I -(THead (Flat Cast) t2 t1) H4) in (False_ind (ex3 T (\lambda (t0: T).(pc3 c0 -(THead (Flat Cast) t0 t2) (lift (S n) O t))) (\lambda (_: T).(ty3 g c0 t1 -t2)) (\lambda (t0: T).(ty3 g c0 t2 t0))) H5))))))))))) (\lambda (n: -nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (_: (getl n c0 -(CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (_: (ty3 g d u -t)).(\lambda (_: (((eq T u (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda -(t0: T).(pc3 d (THead (Flat Cast) t0 t2) t)) (\lambda (_: T).(ty3 g d t1 t2)) -(\lambda (t0: T).(ty3 g d t2 t0)))))).(\lambda (H4: (eq T (TLRef n) (THead -(Flat Cast) t2 t1))).(let H5 \def (eq_ind T (TLRef n) (\lambda (ee: T).(match -ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | -(TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead -(Flat Cast) t2 t1) H4) in (False_ind (ex3 T (\lambda (t0: T).(pc3 c0 (THead -(Flat Cast) t0 t2) (lift (S n) O u))) (\lambda (_: T).(ty3 g c0 t1 t2)) -(\lambda (t0: T).(ty3 g c0 t2 t0))) H5))))))))))) (\lambda (c0: C).(\lambda -(u: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((eq T u -(THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t0: T).(pc3 c0 (THead (Flat -Cast) t0 t2) t)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0 -t2 t0)))))).(\lambda (b: B).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: -(ty3 g (CHead c0 (Bind b) u) t0 t3)).(\lambda (_: (((eq T t0 (THead (Flat -Cast) t2 t1)) \to (ex3 T (\lambda (t4: T).(pc3 (CHead c0 (Bind b) u) (THead -(Flat Cast) t4 t2) t3)) (\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2)) -(\lambda (t4: T).(ty3 g (CHead c0 (Bind b) u) t2 t4)))))).(\lambda (H5: (eq T -(THead (Bind b) u t0) (THead (Flat Cast) t2 t1))).(let H6 \def (eq_ind T -(THead (Bind b) u t0) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat -Cast) t2 t1) H5) in (False_ind (ex3 T (\lambda (t4: T).(pc3 c0 (THead (Flat -Cast) t4 t2) (THead (Bind b) u t3))) (\lambda (_: T).(ty3 g c0 t1 t2)) -(\lambda (t4: T).(ty3 g c0 t2 t4))) H6))))))))))))) (\lambda (c0: C).(\lambda -(w: T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda (_: (((eq T w -(THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t0: T).(pc3 c0 (THead (Flat -Cast) t0 t2) u)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0 -t2 t0)))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead -(Bind Abst) u t))).(\lambda (_: (((eq T v (THead (Flat Cast) t2 t1)) \to (ex3 -T (\lambda (t0: T).(pc3 c0 (THead (Flat Cast) t0 t2) (THead (Bind Abst) u -t))) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0 t2 -t0)))))).(\lambda (H5: (eq T (THead (Flat Appl) w v) (THead (Flat Cast) t2 -t1))).(let H6 \def (eq_ind T (THead (Flat Appl) w v) (\lambda (ee: T).(match -ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | -(TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return -(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow -(match f in F return (\lambda (_: F).Prop) with [Appl \Rightarrow True | Cast -\Rightarrow False])])])) I (THead (Flat Cast) t2 t1) H5) in (False_ind (ex3 T -(\lambda (t0: T).(pc3 c0 (THead (Flat Cast) t0 t2) (THead (Flat Appl) w -(THead (Bind Abst) u t)))) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0: -T).(ty3 g c0 t2 t0))) H6)))))))))))) (\lambda (c0: C).(\lambda (t0: -T).(\lambda (t3: T).(\lambda (H1: (ty3 g c0 t0 t3)).(\lambda (H2: (((eq T t0 -(THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t4: T).(pc3 c0 (THead (Flat -Cast) t4 t2) t3)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t4: T).(ty3 g -c0 t2 t4)))))).(\lambda (t4: T).(\lambda (H3: (ty3 g c0 t3 t4)).(\lambda (H4: -(((eq T t3 (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t5: T).(pc3 c0 -(THead (Flat Cast) t5 t2) t4)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda -(t5: T).(ty3 g c0 t2 t5)))))).(\lambda (H5: (eq T (THead (Flat Cast) t3 t0) -(THead (Flat Cast) t2 t1))).(let H6 \def (f_equal T T (\lambda (e: T).(match -e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) -\Rightarrow t3 | (THead _ t _) \Rightarrow t])) (THead (Flat Cast) t3 t0) -(THead (Flat Cast) t2 t1) H5) in ((let H7 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | -(TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Flat Cast) -t3 t0) (THead (Flat Cast) t2 t1) H5) in (\lambda (H8: (eq T t3 t2)).(let H9 -\def (eq_ind T t3 (\lambda (t: T).((eq T t (THead (Flat Cast) t2 t1)) \to -(ex3 T (\lambda (t5: T).(pc3 c0 (THead (Flat Cast) t5 t2) t4)) (\lambda (_: -T).(ty3 g c0 t1 t2)) (\lambda (t5: T).(ty3 g c0 t2 t5))))) H4 t2 H8) in (let -H10 \def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 t t4)) H3 t2 H8) in (let H11 -\def (eq_ind T t3 (\lambda (t: T).((eq T t0 (THead (Flat Cast) t2 t1)) \to -(ex3 T (\lambda (t5: T).(pc3 c0 (THead (Flat Cast) t5 t2) t)) (\lambda (_: -T).(ty3 g c0 t1 t2)) (\lambda (t5: T).(ty3 g c0 t2 t5))))) H2 t2 H8) in (let -H12 \def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 t0 t)) H1 t2 H8) in (eq_ind_r -T t2 (\lambda (t: T).(ex3 T (\lambda (t5: T).(pc3 c0 (THead (Flat Cast) t5 -t2) (THead (Flat Cast) t4 t))) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda -(t5: T).(ty3 g c0 t2 t5)))) (let H13 \def (eq_ind T t0 (\lambda (t: T).((eq T -t (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t5: T).(pc3 c0 (THead (Flat -Cast) t5 t2) t2)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t5: T).(ty3 g -c0 t2 t5))))) H11 t1 H7) in (let H14 \def (eq_ind T t0 (\lambda (t: T).(ty3 g -c0 t t2)) H12 t1 H7) in (ex3_intro T (\lambda (t5: T).(pc3 c0 (THead (Flat -Cast) t5 t2) (THead (Flat Cast) t4 t2))) (\lambda (_: T).(ty3 g c0 t1 t2)) -(\lambda (t5: T).(ty3 g c0 t2 t5)) t4 (pc3_refl c0 (THead (Flat Cast) t4 t2)) -H14 H10))) t3 H8))))))) H6))))))))))) c y x H0))) H)))))). -(* COMMENTS -Initial nodes: 2609 -END *) - -theorem tys3_gen_nil: - \forall (g: G).(\forall (c: C).(\forall (u: T).((tys3 g c TNil u) \to (ex T -(\lambda (u0: T).(ty3 g c u u0)))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (H: (tys3 g c TNil -u)).(insert_eq TList TNil (\lambda (t: TList).(tys3 g c t u)) (\lambda (_: -TList).(ex T (\lambda (u0: T).(ty3 g c u u0)))) (\lambda (y: TList).(\lambda -(H0: (tys3 g c y u)).(tys3_ind g c (\lambda (t: TList).(\lambda (t0: T).((eq -TList t TNil) \to (ex T (\lambda (u0: T).(ty3 g c t0 u0)))))) (\lambda (u0: -T).(\lambda (u1: T).(\lambda (H1: (ty3 g c u0 u1)).(\lambda (_: (eq TList -TNil TNil)).(ex_intro T (\lambda (u2: T).(ty3 g c u0 u2)) u1 H1))))) (\lambda -(t: T).(\lambda (u0: T).(\lambda (_: (ty3 g c t u0)).(\lambda (ts: -TList).(\lambda (_: (tys3 g c ts u0)).(\lambda (_: (((eq TList ts TNil) \to -(ex T (\lambda (u1: T).(ty3 g c u0 u1)))))).(\lambda (H4: (eq TList (TCons t -ts) TNil)).(let H5 \def (eq_ind TList (TCons t ts) (\lambda (ee: -TList).(match ee in TList return (\lambda (_: TList).Prop) with [TNil -\Rightarrow False | (TCons _ _) \Rightarrow True])) I TNil H4) in (False_ind -(ex T (\lambda (u1: T).(ty3 g c u0 u1))) H5))))))))) y u H0))) H)))). -(* COMMENTS -Initial nodes: 255 -END *) - -theorem tys3_gen_cons: - \forall (g: G).(\forall (c: C).(\forall (ts: TList).(\forall (t: T).(\forall -(u: T).((tys3 g c (TCons t ts) u) \to (land (ty3 g c t u) (tys3 g c ts -u))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (ts: TList).(\lambda (t: T).(\lambda -(u: T).(\lambda (H: (tys3 g c (TCons t ts) u)).(insert_eq TList (TCons t ts) -(\lambda (t0: TList).(tys3 g c t0 u)) (\lambda (_: TList).(land (ty3 g c t u) -(tys3 g c ts u))) (\lambda (y: TList).(\lambda (H0: (tys3 g c y u)).(tys3_ind -g c (\lambda (t0: TList).(\lambda (t1: T).((eq TList t0 (TCons t ts)) \to -(land (ty3 g c t t1) (tys3 g c ts t1))))) (\lambda (u0: T).(\lambda (u1: -T).(\lambda (_: (ty3 g c u0 u1)).(\lambda (H2: (eq TList TNil (TCons t -ts))).(let H3 \def (eq_ind TList TNil (\lambda (ee: TList).(match ee in TList -return (\lambda (_: TList).Prop) with [TNil \Rightarrow True | (TCons _ _) -\Rightarrow False])) I (TCons t ts) H2) in (False_ind (land (ty3 g c t u0) -(tys3 g c ts u0)) H3)))))) (\lambda (t0: T).(\lambda (u0: T).(\lambda (H1: -(ty3 g c t0 u0)).(\lambda (ts0: TList).(\lambda (H2: (tys3 g c ts0 -u0)).(\lambda (H3: (((eq TList ts0 (TCons t ts)) \to (land (ty3 g c t u0) -(tys3 g c ts u0))))).(\lambda (H4: (eq TList (TCons t0 ts0) (TCons t -ts))).(let H5 \def (f_equal TList T (\lambda (e: TList).(match e in TList -return (\lambda (_: TList).T) with [TNil \Rightarrow t0 | (TCons t1 _) -\Rightarrow t1])) (TCons t0 ts0) (TCons t ts) H4) in ((let H6 \def (f_equal -TList TList (\lambda (e: TList).(match e in TList return (\lambda (_: -TList).TList) with [TNil \Rightarrow ts0 | (TCons _ t1) \Rightarrow t1])) -(TCons t0 ts0) (TCons t ts) H4) in (\lambda (H7: (eq T t0 t)).(let H8 \def -(eq_ind TList ts0 (\lambda (t1: TList).((eq TList t1 (TCons t ts)) \to (land -(ty3 g c t u0) (tys3 g c ts u0)))) H3 ts H6) in (let H9 \def (eq_ind TList -ts0 (\lambda (t1: TList).(tys3 g c t1 u0)) H2 ts H6) in (let H10 \def (eq_ind -T t0 (\lambda (t1: T).(ty3 g c t1 u0)) H1 t H7) in (conj (ty3 g c t u0) (tys3 -g c ts u0) H10 H9)))))) H5))))))))) y u H0))) H)))))). -(* COMMENTS -Initial nodes: 479 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/fwd_nf2.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/fwd_nf2.ma deleted file mode 100644 index ca4e40c7b..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/fwd_nf2.ma +++ /dev/null @@ -1,301 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/ty3/arity_props.ma". - -include "Basic-1/pc3/nf2.ma". - -include "Basic-1/nf2/fwd.ma". - -theorem ty3_gen_appl_nf2: - \forall (g: G).(\forall (c: C).(\forall (w: T).(\forall (v: T).(\forall (x: -T).((ty3 g c (THead (Flat Appl) w v) x) \to (ex4_2 T T (\lambda (u: -T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x))) -(\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) -(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t: -T).(nf2 c (THead (Bind Abst) u t)))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (w: T).(\lambda (v: T).(\lambda (x: -T).(\lambda (H: (ty3 g c (THead (Flat Appl) w v) x)).(ex3_2_ind T T (\lambda -(u: T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) -x))) (\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) -(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (ex4_2 T T (\lambda (u: -T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x))) -(\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) -(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t: -T).(nf2 c (THead (Bind Abst) u t))))) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H0: (pc3 c (THead (Flat Appl) w (THead (Bind Abst) x0 x1)) -x)).(\lambda (H1: (ty3 g c v (THead (Bind Abst) x0 x1))).(\lambda (H2: (ty3 g -c w x0)).(let H_x \def (ty3_correct g c v (THead (Bind Abst) x0 x1) H1) in -(let H3 \def H_x in (ex_ind T (\lambda (t: T).(ty3 g c (THead (Bind Abst) x0 -x1) t)) (ex4_2 T T (\lambda (u: T).(\lambda (t: T).(pc3 c (THead (Flat Appl) -w (THead (Bind Abst) u t)) x))) (\lambda (u: T).(\lambda (t: T).(ty3 g c v -(THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) -(\lambda (u: T).(\lambda (t: T).(nf2 c (THead (Bind Abst) u t))))) (\lambda -(x2: T).(\lambda (H4: (ty3 g c (THead (Bind Abst) x0 x1) x2)).(let H_x0 \def -(ty3_correct g c w x0 H2) in (let H5 \def H_x0 in (ex_ind T (\lambda (t: -T).(ty3 g c x0 t)) (ex4_2 T T (\lambda (u: T).(\lambda (t: T).(pc3 c (THead -(Flat Appl) w (THead (Bind Abst) u t)) x))) (\lambda (u: T).(\lambda (t: -T).(ty3 g c v (THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3 -g c w u))) (\lambda (u: T).(\lambda (t: T).(nf2 c (THead (Bind Abst) u t))))) -(\lambda (x3: T).(\lambda (H6: (ty3 g c x0 x3)).(let H7 \def (ty3_sn3 g c -(THead (Bind Abst) x0 x1) x2 H4) in (let H_x1 \def (nf2_sn3 c (THead (Bind -Abst) x0 x1) H7) in (let H8 \def H_x1 in (ex2_ind T (\lambda (u: T).(pr3 c -(THead (Bind Abst) x0 x1) u)) (\lambda (u: T).(nf2 c u)) (ex4_2 T T (\lambda -(u: T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) -x))) (\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) -(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t: -T).(nf2 c (THead (Bind Abst) u t))))) (\lambda (x4: T).(\lambda (H9: (pr3 c -(THead (Bind Abst) x0 x1) x4)).(\lambda (H10: (nf2 c x4)).(let H11 \def -(pr3_gen_abst c x0 x1 x4 H9) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: -T).(eq T x4 (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall -(u: T).(pr3 (CHead c (Bind b) u) x1 t2))))) (ex4_2 T T (\lambda (u: -T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x))) -(\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) -(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t: -T).(nf2 c (THead (Bind Abst) u t))))) (\lambda (x5: T).(\lambda (x6: -T).(\lambda (H12: (eq T x4 (THead (Bind Abst) x5 x6))).(\lambda (H13: (pr3 c -x0 x5)).(\lambda (H14: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind -b) u) x1 x6))))).(let H15 \def (eq_ind T x4 (\lambda (t: T).(nf2 c t)) H10 -(THead (Bind Abst) x5 x6) H12) in (let H16 \def (pr3_head_12 c x0 x5 H13 -(Bind Abst) x1 x6 (H14 Abst x5)) in (ex4_2_intro T T (\lambda (u: T).(\lambda -(t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x))) (\lambda (u: -T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) (\lambda (u: -T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t: T).(nf2 c -(THead (Bind Abst) u t)))) x5 x6 (pc3_pr3_conf c (THead (Flat Appl) w (THead -(Bind Abst) x0 x1)) x H0 (THead (Flat Appl) w (THead (Bind Abst) x5 x6)) -(pr3_thin_dx c (THead (Bind Abst) x0 x1) (THead (Bind Abst) x5 x6) H16 w -Appl)) (ty3_conv g c (THead (Bind Abst) x5 x6) x2 (ty3_sred_pr3 c (THead -(Bind Abst) x0 x1) (THead (Bind Abst) x5 x6) H16 g x2 H4) v (THead (Bind -Abst) x0 x1) H1 (pc3_pr3_r c (THead (Bind Abst) x0 x1) (THead (Bind Abst) x5 -x6) H16)) (ty3_conv g c x5 x3 (ty3_sred_pr3 c x0 x5 H13 g x3 H6) w x0 H2 -(pc3_pr3_r c x0 x5 H13)) H15)))))))) H11))))) H8)))))) H5))))) H3)))))))) -(ty3_gen_appl g c w v x H))))))). -(* COMMENTS -Initial nodes: 1289 -END *) - -theorem ty3_inv_lref_nf2_pc3: - \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (i: nat).((ty3 g c -(TLRef i) u1) \to ((nf2 c (TLRef i)) \to (\forall (u2: T).((nf2 c u2) \to -((pc3 c u1 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift (S i) O u)))))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (u1: T).(\lambda (i: nat).(\lambda -(H: (ty3 g c (TLRef i) u1)).(insert_eq T (TLRef i) (\lambda (t: T).(ty3 g c t -u1)) (\lambda (t: T).((nf2 c t) \to (\forall (u2: T).((nf2 c u2) \to ((pc3 c -u1 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift (S i) O u))))))))) (\lambda -(y: T).(\lambda (H0: (ty3 g c y u1)).(ty3_ind g (\lambda (c0: C).(\lambda (t: -T).(\lambda (t0: T).((eq T t (TLRef i)) \to ((nf2 c0 t) \to (\forall (u2: -T).((nf2 c0 u2) \to ((pc3 c0 t0 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift -(S i) O u)))))))))))) (\lambda (c0: C).(\lambda (t2: T).(\lambda (t: -T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda (_: (((eq T t2 (TLRef i)) \to ((nf2 -c0 t2) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 t u2) \to (ex T -(\lambda (u: T).(eq T u2 (lift (S i) O u))))))))))).(\lambda (u: T).(\lambda -(t1: T).(\lambda (H3: (ty3 g c0 u t1)).(\lambda (H4: (((eq T u (TLRef i)) \to -((nf2 c0 u) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 t1 u2) \to (ex T -(\lambda (u0: T).(eq T u2 (lift (S i) O u0))))))))))).(\lambda (H5: (pc3 c0 -t1 t2)).(\lambda (H6: (eq T u (TLRef i))).(\lambda (H7: (nf2 c0 u)).(\lambda -(u2: T).(\lambda (H8: (nf2 c0 u2)).(\lambda (H9: (pc3 c0 t2 u2)).(let H10 -\def (eq_ind T u (\lambda (t0: T).(nf2 c0 t0)) H7 (TLRef i) H6) in (let H11 -\def (eq_ind T u (\lambda (t0: T).((eq T t0 (TLRef i)) \to ((nf2 c0 t0) \to -(\forall (u3: T).((nf2 c0 u3) \to ((pc3 c0 t1 u3) \to (ex T (\lambda (u0: -T).(eq T u3 (lift (S i) O u0)))))))))) H4 (TLRef i) H6) in (let H12 \def -(eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 t1)) H3 (TLRef i) H6) in (let H_y -\def (H11 (refl_equal T (TLRef i)) H10 u2 H8) in (H_y (pc3_t t2 c0 t1 H5 u2 -H9))))))))))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq -T (TSort m) (TLRef i))).(\lambda (_: (nf2 c0 (TSort m))).(\lambda (u2: -T).(\lambda (_: (nf2 c0 u2)).(\lambda (_: (pc3 c0 (TSort (next g m)) -u2)).(let H5 \def (eq_ind T (TSort m) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef i) H1) in -(False_ind (ex T (\lambda (u: T).(eq T u2 (lift (S i) O u)))) H5))))))))) -(\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda -(H1: (getl n c0 (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda (_: (ty3 g -d u t)).(\lambda (_: (((eq T u (TLRef i)) \to ((nf2 d u) \to (\forall (u2: -T).((nf2 d u2) \to ((pc3 d t u2) \to (ex T (\lambda (u0: T).(eq T u2 (lift (S -i) O u0))))))))))).(\lambda (H4: (eq T (TLRef n) (TLRef i))).(\lambda (H5: -(nf2 c0 (TLRef n))).(\lambda (u2: T).(\lambda (_: (nf2 c0 u2)).(\lambda (H7: -(pc3 c0 (lift (S n) O t) u2)).(let H8 \def (f_equal T nat (\lambda (e: -T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow n | -(TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef -i) H4) in (let H9 \def (eq_ind nat n (\lambda (n0: nat).(pc3 c0 (lift (S n0) -O t) u2)) H7 i H8) in (let H10 \def (eq_ind nat n (\lambda (n0: nat).(nf2 c0 -(TLRef n0))) H5 i H8) in (let H11 \def (eq_ind nat n (\lambda (n0: nat).(getl -n0 c0 (CHead d (Bind Abbr) u))) H1 i H8) in (nf2_gen_lref c0 d u i H11 H10 -(ex T (\lambda (u0: T).(eq T u2 (lift (S i) O u0)))))))))))))))))))))) -(\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda -(H1: (getl n c0 (CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (_: (ty3 g -d u t)).(\lambda (_: (((eq T u (TLRef i)) \to ((nf2 d u) \to (\forall (u2: -T).((nf2 d u2) \to ((pc3 d t u2) \to (ex T (\lambda (u0: T).(eq T u2 (lift (S -i) O u0))))))))))).(\lambda (H4: (eq T (TLRef n) (TLRef i))).(\lambda (H5: -(nf2 c0 (TLRef n))).(\lambda (u2: T).(\lambda (H6: (nf2 c0 u2)).(\lambda (H7: -(pc3 c0 (lift (S n) O u) u2)).(let H8 \def (f_equal T nat (\lambda (e: -T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow n | -(TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef -i) H4) in (let H9 \def (eq_ind nat n (\lambda (n0: nat).(pc3 c0 (lift (S n0) -O u) u2)) H7 i H8) in (let H10 \def (eq_ind nat n (\lambda (n0: nat).(nf2 c0 -(TLRef n0))) H5 i H8) in (let H11 \def (eq_ind nat n (\lambda (n0: nat).(getl -n0 c0 (CHead d (Bind Abst) u))) H1 i H8) in (let H_y \def (pc3_nf2_unfold c0 -(lift (S i) O u) u2 H9 H6) in (let H12 \def (pr3_gen_lift c0 u u2 (S i) O H_y -d (getl_drop Abst c0 d u i H11)) in (ex2_ind T (\lambda (t2: T).(eq T u2 -(lift (S i) O t2))) (\lambda (t2: T).(pr3 d u t2)) (ex T (\lambda (u0: T).(eq -T u2 (lift (S i) O u0)))) (\lambda (x: T).(\lambda (H13: (eq T u2 (lift (S i) -O x))).(\lambda (_: (pr3 d u x)).(eq_ind_r T (lift (S i) O x) (\lambda (t0: -T).(ex T (\lambda (u0: T).(eq T t0 (lift (S i) O u0))))) (ex_intro T (\lambda -(u0: T).(eq T (lift (S i) O x) (lift (S i) O u0))) x (refl_equal T (lift (S -i) O x))) u2 H13)))) H12)))))))))))))))))))) (\lambda (c0: C).(\lambda (u: -T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((eq T u (TLRef -i)) \to ((nf2 c0 u) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 t u2) \to -(ex T (\lambda (u0: T).(eq T u2 (lift (S i) O u0))))))))))).(\lambda (b: -B).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) -u) t1 t2)).(\lambda (_: (((eq T t1 (TLRef i)) \to ((nf2 (CHead c0 (Bind b) u) -t1) \to (\forall (u2: T).((nf2 (CHead c0 (Bind b) u) u2) \to ((pc3 (CHead c0 -(Bind b) u) t2 u2) \to (ex T (\lambda (u0: T).(eq T u2 (lift (S i) O -u0))))))))))).(\lambda (H5: (eq T (THead (Bind b) u t1) (TLRef i))).(\lambda -(_: (nf2 c0 (THead (Bind b) u t1))).(\lambda (u2: T).(\lambda (_: (nf2 c0 -u2)).(\lambda (_: (pc3 c0 (THead (Bind b) u t2) u2)).(let H9 \def (eq_ind T -(THead (Bind b) u t1) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead _ _ _) \Rightarrow True])) I (TLRef i) H5) in (False_ind (ex T -(\lambda (u0: T).(eq T u2 (lift (S i) O u0)))) H9))))))))))))))))) (\lambda -(c0: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda -(_: (((eq T w (TLRef i)) \to ((nf2 c0 w) \to (\forall (u2: T).((nf2 c0 u2) -\to ((pc3 c0 u u2) \to (ex T (\lambda (u0: T).(eq T u2 (lift (S i) O -u0))))))))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead -(Bind Abst) u t))).(\lambda (_: (((eq T v (TLRef i)) \to ((nf2 c0 v) \to -(\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 (THead (Bind Abst) u t) u2) \to -(ex T (\lambda (u0: T).(eq T u2 (lift (S i) O u0))))))))))).(\lambda (H5: (eq -T (THead (Flat Appl) w v) (TLRef i))).(\lambda (_: (nf2 c0 (THead (Flat Appl) -w v))).(\lambda (u2: T).(\lambda (_: (nf2 c0 u2)).(\lambda (_: (pc3 c0 (THead -(Flat Appl) w (THead (Bind Abst) u t)) u2)).(let H9 \def (eq_ind T (THead -(Flat Appl) w v) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) -with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ -_) \Rightarrow True])) I (TLRef i) H5) in (False_ind (ex T (\lambda (u0: -T).(eq T u2 (lift (S i) O u0)))) H9)))))))))))))))) (\lambda (c0: C).(\lambda -(t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g c0 t1 t2)).(\lambda (_: (((eq T -t1 (TLRef i)) \to ((nf2 c0 t1) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 -t2 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift (S i) O u))))))))))).(\lambda -(t0: T).(\lambda (_: (ty3 g c0 t2 t0)).(\lambda (_: (((eq T t2 (TLRef i)) \to -((nf2 c0 t2) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 t0 u2) \to (ex T -(\lambda (u: T).(eq T u2 (lift (S i) O u))))))))))).(\lambda (H5: (eq T -(THead (Flat Cast) t2 t1) (TLRef i))).(\lambda (_: (nf2 c0 (THead (Flat Cast) -t2 t1))).(\lambda (u2: T).(\lambda (_: (nf2 c0 u2)).(\lambda (_: (pc3 c0 -(THead (Flat Cast) t0 t2) u2)).(let H9 \def (eq_ind T (THead (Flat Cast) t2 -t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort -_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) -\Rightarrow True])) I (TLRef i) H5) in (False_ind (ex T (\lambda (u: T).(eq T -u2 (lift (S i) O u)))) H9))))))))))))))) c y u1 H0))) H))))). -(* COMMENTS -Initial nodes: 2175 -END *) - -theorem ty3_inv_lref_nf2: - \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (i: nat).((ty3 g c -(TLRef i) u) \to ((nf2 c (TLRef i)) \to ((nf2 c u) \to (ex T (\lambda (u0: -T).(eq T u (lift (S i) O u0)))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H: (ty3 g c (TLRef i) u)).(\lambda (H0: (nf2 c (TLRef i))).(\lambda (H1: -(nf2 c u)).(ty3_inv_lref_nf2_pc3 g c u i H H0 u H1 (pc3_refl c u)))))))). -(* COMMENTS -Initial nodes: 57 -END *) - -theorem ty3_inv_appls_lref_nf2: - \forall (g: G).(\forall (c: C).(\forall (vs: TList).(\forall (u1: -T).(\forall (i: nat).((ty3 g c (THeads (Flat Appl) vs (TLRef i)) u1) \to -((nf2 c (TLRef i)) \to ((nf2 c u1) \to (ex2 T (\lambda (u: T).(nf2 c (lift (S -i) O u))) (\lambda (u: T).(pc3 c (THeads (Flat Appl) vs (lift (S i) O u)) -u1)))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (vs: TList).(TList_ind (\lambda (t: -TList).(\forall (u1: T).(\forall (i: nat).((ty3 g c (THeads (Flat Appl) t -(TLRef i)) u1) \to ((nf2 c (TLRef i)) \to ((nf2 c u1) \to (ex2 T (\lambda (u: -T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3 c (THeads (Flat Appl) t -(lift (S i) O u)) u1))))))))) (\lambda (u1: T).(\lambda (i: nat).(\lambda (H: -(ty3 g c (TLRef i) u1)).(\lambda (H0: (nf2 c (TLRef i))).(\lambda (H1: (nf2 c -u1)).(let H_x \def (ty3_inv_lref_nf2 g c u1 i H H0 H1) in (let H2 \def H_x in -(ex_ind T (\lambda (u0: T).(eq T u1 (lift (S i) O u0))) (ex2 T (\lambda (u: -T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3 c (lift (S i) O u) u1))) -(\lambda (x: T).(\lambda (H3: (eq T u1 (lift (S i) O x))).(let H4 \def -(eq_ind T u1 (\lambda (t: T).(nf2 c t)) H1 (lift (S i) O x) H3) in (eq_ind_r -T (lift (S i) O x) (\lambda (t: T).(ex2 T (\lambda (u: T).(nf2 c (lift (S i) -O u))) (\lambda (u: T).(pc3 c (lift (S i) O u) t)))) (ex_intro2 T (\lambda -(u: T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3 c (lift (S i) O u) -(lift (S i) O x))) x H4 (pc3_refl c (lift (S i) O x))) u1 H3)))) H2)))))))) -(\lambda (t: T).(\lambda (t0: TList).(\lambda (H: ((\forall (u1: T).(\forall -(i: nat).((ty3 g c (THeads (Flat Appl) t0 (TLRef i)) u1) \to ((nf2 c (TLRef -i)) \to ((nf2 c u1) \to (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O u))) -(\lambda (u: T).(pc3 c (THeads (Flat Appl) t0 (lift (S i) O u)) -u1)))))))))).(\lambda (u1: T).(\lambda (i: nat).(\lambda (H0: (ty3 g c (THead -(Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) u1)).(\lambda (H1: (nf2 c -(TLRef i))).(\lambda (_: (nf2 c u1)).(let H_x \def (ty3_gen_appl_nf2 g c t -(THeads (Flat Appl) t0 (TLRef i)) u1 H0) in (let H3 \def H_x in (ex4_2_ind T -T (\lambda (u: T).(\lambda (t1: T).(pc3 c (THead (Flat Appl) t (THead (Bind -Abst) u t1)) u1))) (\lambda (u: T).(\lambda (t1: T).(ty3 g c (THeads (Flat -Appl) t0 (TLRef i)) (THead (Bind Abst) u t1)))) (\lambda (u: T).(\lambda (_: -T).(ty3 g c t u))) (\lambda (u: T).(\lambda (t1: T).(nf2 c (THead (Bind Abst) -u t1)))) (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O u))) (\lambda (u: -T).(pc3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O u))) -u1))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (pc3 c (THead (Flat -Appl) t (THead (Bind Abst) x0 x1)) u1)).(\lambda (H5: (ty3 g c (THeads (Flat -Appl) t0 (TLRef i)) (THead (Bind Abst) x0 x1))).(\lambda (_: (ty3 g c t -x0)).(\lambda (H7: (nf2 c (THead (Bind Abst) x0 x1))).(let H8 \def -(nf2_gen_abst c x0 x1 H7) in (land_ind (nf2 c x0) (nf2 (CHead c (Bind Abst) -x0) x1) (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3 -c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O u))) u1))) -(\lambda (H9: (nf2 c x0)).(\lambda (H10: (nf2 (CHead c (Bind Abst) x0) -x1)).(let H_y \def (H (THead (Bind Abst) x0 x1) i H5 H1) in (let H11 \def -(H_y (nf2_abst_shift c x0 H9 x1 H10)) in (ex2_ind T (\lambda (u: T).(nf2 c -(lift (S i) O u))) (\lambda (u: T).(pc3 c (THeads (Flat Appl) t0 (lift (S i) -O u)) (THead (Bind Abst) x0 x1))) (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O -u))) (\lambda (u: T).(pc3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift -(S i) O u))) u1))) (\lambda (x: T).(\lambda (H12: (nf2 c (lift (S i) O -x))).(\lambda (H13: (pc3 c (THeads (Flat Appl) t0 (lift (S i) O x)) (THead -(Bind Abst) x0 x1))).(ex_intro2 T (\lambda (u: T).(nf2 c (lift (S i) O u))) -(\lambda (u: T).(pc3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S -i) O u))) u1)) x H12 (pc3_t (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) c -(THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O x))) (pc3_thin_dx c -(THeads (Flat Appl) t0 (lift (S i) O x)) (THead (Bind Abst) x0 x1) H13 t -Appl) u1 H4))))) H11))))) H8)))))))) H3))))))))))) vs))). -(* COMMENTS -Initial nodes: 1213 -END *) - -theorem ty3_inv_lref_lref_nf2: - \forall (g: G).(\forall (c: C).(\forall (i: nat).(\forall (j: nat).((ty3 g c -(TLRef i) (TLRef j)) \to ((nf2 c (TLRef i)) \to ((nf2 c (TLRef j)) \to (lt i -j))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (i: nat).(\lambda (j: nat).(\lambda -(H: (ty3 g c (TLRef i) (TLRef j))).(\lambda (H0: (nf2 c (TLRef i))).(\lambda -(H1: (nf2 c (TLRef j))).(let H_x \def (ty3_inv_lref_nf2 g c (TLRef j) i H H0 -H1) in (let H2 \def H_x in (ex_ind T (\lambda (u0: T).(eq T (TLRef j) (lift -(S i) O u0))) (lt i j) (\lambda (x: T).(\lambda (H3: (eq T (TLRef j) (lift (S -i) O x))).(let H_x0 \def (lift_gen_lref x O (S i) j H3) in (let H4 \def H_x0 -in (or_ind (land (lt j O) (eq T x (TLRef j))) (land (le (S i) j) (eq T x -(TLRef (minus j (S i))))) (lt i j) (\lambda (H5: (land (lt j O) (eq T x -(TLRef j)))).(land_ind (lt j O) (eq T x (TLRef j)) (lt i j) (\lambda (H6: (lt -j O)).(\lambda (_: (eq T x (TLRef j))).(lt_x_O j H6 (lt i j)))) H5)) (\lambda -(H5: (land (le (S i) j) (eq T x (TLRef (minus j (S i)))))).(land_ind (le (S -i) j) (eq T x (TLRef (minus j (S i)))) (lt i j) (\lambda (H6: (le (S i) -j)).(\lambda (_: (eq T x (TLRef (minus j (S i))))).H6)) H5)) H4))))) -H2))))))))). -(* COMMENTS -Initial nodes: 337 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/nf2.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/nf2.ma deleted file mode 100644 index 47b675663..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/nf2.ma +++ /dev/null @@ -1,472 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/ty3/arity.ma". - -include "Basic-1/pc3/nf2.ma". - -include "Basic-1/nf2/arity.ma". - -definition ty3_nf2_inv_abst_premise: - C \to (T \to (T \to Prop)) -\def - \lambda (c: C).(\lambda (w: T).(\lambda (u: T).(\forall (d: C).(\forall (wi: -T).(\forall (i: nat).((getl i c (CHead d (Bind Abst) wi)) \to (\forall (vs: -TList).((pc3 c (THeads (Flat Appl) vs (lift (S i) O wi)) (THead (Bind Abst) w -u)) \to False)))))))). - -theorem ty3_nf2_inv_abst_premise_csort: - \forall (w: T).(\forall (u: T).(\forall (m: nat).(ty3_nf2_inv_abst_premise -(CSort m) w u))) -\def - \lambda (w: T).(\lambda (u: T).(\lambda (m: nat).(\lambda (d: C).(\lambda -(wi: T).(\lambda (i: nat).(\lambda (H: (getl i (CSort m) (CHead d (Bind Abst) -wi))).(\lambda (vs: TList).(\lambda (_: (pc3 (CSort m) (THeads (Flat Appl) vs -(lift (S i) O wi)) (THead (Bind Abst) w u))).(getl_gen_sort m i (CHead d -(Bind Abst) wi) H False))))))))). -(* COMMENTS -Initial nodes: 85 -END *) - -theorem ty3_nf2_inv_all: - \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t -u) \to ((nf2 c t) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T -t (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c w))) -(\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c (Bind Abst) w) u0)))) (ex nat -(\lambda (n: nat).(eq T t (TSort n)))) (ex3_2 TList nat (\lambda (ws: -TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i))))) -(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c (TLRef i))))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H: -(ty3 g c t u)).(\lambda (H0: (nf2 c t)).(let H_x \def (ty3_arity g c t u H) -in (let H1 \def H_x in (ex2_ind A (\lambda (a1: A).(arity g c t a1)) (\lambda -(a1: A).(arity g c u (asucc g a1))) (or3 (ex3_2 T T (\lambda (w: T).(\lambda -(u0: T).(eq T t (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: -T).(nf2 c w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c (Bind Abst) w) -u0)))) (ex nat (\lambda (n: nat).(eq T t (TSort n)))) (ex3_2 TList nat -(\lambda (ws: TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef -i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c (TLRef i)))))) (\lambda (x: A).(\lambda (H2: -(arity g c t x)).(\lambda (_: (arity g c u (asucc g x))).(arity_nf2_inv_all g -c t x H2 H0)))) H1)))))))). -(* COMMENTS -Initial nodes: 233 -END *) - -theorem ty3_nf2_inv_sort: - \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (m: nat).((ty3 g c t -(TSort m)) \to ((nf2 c t) \to (or (ex2 nat (\lambda (n: nat).(eq T t (TSort -n))) (\lambda (n: nat).(eq nat m (next g n)))) (ex3_2 TList nat (\lambda (ws: -TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i))))) -(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c (TLRef i))))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (m: nat).(\lambda -(H: (ty3 g c t (TSort m))).(\lambda (H0: (nf2 c t)).(let H_x \def -(ty3_nf2_inv_all g c t (TSort m) H H0) in (let H1 \def H_x in (or3_ind (ex3_2 -T T (\lambda (w: T).(\lambda (u: T).(eq T t (THead (Bind Abst) w u)))) -(\lambda (w: T).(\lambda (_: T).(nf2 c w))) (\lambda (w: T).(\lambda (u: -T).(nf2 (CHead c (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t -(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t -(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef i))))) -(or (ex2 nat (\lambda (n: nat).(eq T t (TSort n))) (\lambda (n: nat).(eq nat -m (next g n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T -t (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef -i)))))) (\lambda (H2: (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t -(THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c w))) -(\lambda (w: T).(\lambda (u: T).(nf2 (CHead c (Bind Abst) w) -u))))).(ex3_2_ind T T (\lambda (w: T).(\lambda (u: T).(eq T t (THead (Bind -Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c w))) (\lambda (w: -T).(\lambda (u: T).(nf2 (CHead c (Bind Abst) w) u))) (or (ex2 nat (\lambda -(n: nat).(eq T t (TSort n))) (\lambda (n: nat).(eq nat m (next g n)))) (ex3_2 -TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) -ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) -(\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef i)))))) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H3: (eq T t (THead (Bind Abst) x0 -x1))).(\lambda (_: (nf2 c x0)).(\lambda (_: (nf2 (CHead c (Bind Abst) x0) -x1)).(let H6 \def (eq_ind T t (\lambda (t0: T).(ty3 g c t0 (TSort m))) H -(THead (Bind Abst) x0 x1) H3) in (eq_ind_r T (THead (Bind Abst) x0 x1) -(\lambda (t0: T).(or (ex2 nat (\lambda (n: nat).(eq T t0 (TSort n))) (\lambda -(n: nat).(eq nat m (next g n)))) (ex3_2 TList nat (\lambda (ws: -TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i))))) -(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c (TLRef i))))))) (ex3_2_ind T T (\lambda (t2: -T).(\lambda (_: T).(pc3 c (THead (Bind Abst) x0 t2) (TSort m)))) (\lambda (_: -T).(\lambda (t0: T).(ty3 g c x0 t0))) (\lambda (t2: T).(\lambda (_: T).(ty3 g -(CHead c (Bind Abst) x0) x1 t2))) (or (ex2 nat (\lambda (n: nat).(eq T (THead -(Bind Abst) x0 x1) (TSort n))) (\lambda (n: nat).(eq nat m (next g n)))) -(ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Bind -Abst) x0 x1) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: -TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: -nat).(nf2 c (TLRef i)))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: -(pc3 c (THead (Bind Abst) x0 x2) (TSort m))).(\lambda (_: (ty3 g c x0 -x3)).(\lambda (_: (ty3 g (CHead c (Bind Abst) x0) x1 x2)).(pc3_gen_sort_abst -c x0 x2 m (pc3_s c (TSort m) (THead (Bind Abst) x0 x2) H7) (or (ex2 nat -(\lambda (n: nat).(eq T (THead (Bind Abst) x0 x1) (TSort n))) (\lambda (n: -nat).(eq nat m (next g n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda -(i: nat).(eq T (THead (Bind Abst) x0 x1) (THeads (Flat Appl) ws (TLRef i))))) -(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c (TLRef i)))))))))))) (ty3_gen_bind g Abst c -x0 x1 (TSort m) H6)) t H3))))))) H2)) (\lambda (H2: (ex nat (\lambda (n: -nat).(eq T t (TSort n))))).(ex_ind nat (\lambda (n: nat).(eq T t (TSort n))) -(or (ex2 nat (\lambda (n: nat).(eq T t (TSort n))) (\lambda (n: nat).(eq nat -m (next g n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T -t (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef -i)))))) (\lambda (x: nat).(\lambda (H3: (eq T t (TSort x))).(let H4 \def -(eq_ind T t (\lambda (t0: T).(ty3 g c t0 (TSort m))) H (TSort x) H3) in -(eq_ind_r T (TSort x) (\lambda (t0: T).(or (ex2 nat (\lambda (n: nat).(eq T -t0 (TSort n))) (\lambda (n: nat).(eq nat m (next g n)))) (ex3_2 TList nat -(\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef -i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c (TLRef i))))))) (eq_ind nat (next g x) -(\lambda (n: nat).(or (ex2 nat (\lambda (n0: nat).(eq T (TSort x) (TSort -n0))) (\lambda (n0: nat).(eq nat n (next g n0)))) (ex3_2 TList nat (\lambda -(ws: TList).(\lambda (i: nat).(eq T (TSort x) (THeads (Flat Appl) ws (TLRef -i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c (TLRef i))))))) (or_introl (ex2 nat (\lambda -(n: nat).(eq T (TSort x) (TSort n))) (\lambda (n: nat).(eq nat (next g x) -(next g n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T -(TSort x) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda -(_: nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef -i))))) (ex_intro2 nat (\lambda (n: nat).(eq T (TSort x) (TSort n))) (\lambda -(n: nat).(eq nat (next g x) (next g n))) x (refl_equal T (TSort x)) -(refl_equal nat (next g x)))) m (pc3_gen_sort c (next g x) m (ty3_gen_sort g -c (TSort m) x H4))) t H3)))) H2)) (\lambda (H2: (ex3_2 TList nat (\lambda -(ws: TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i))))) -(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c (TLRef i)))))).(ex3_2_ind TList nat (\lambda -(ws: TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i))))) -(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c (TLRef i)))) (or (ex2 nat (\lambda (n: -nat).(eq T t (TSort n))) (\lambda (n: nat).(eq nat m (next g n)))) (ex3_2 -TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) -ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) -(\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef i)))))) (\lambda (x0: -TList).(\lambda (x1: nat).(\lambda (H3: (eq T t (THeads (Flat Appl) x0 (TLRef -x1)))).(\lambda (H4: (nfs2 c x0)).(\lambda (H5: (nf2 c (TLRef x1))).(let H6 -\def (eq_ind T t (\lambda (t0: T).(ty3 g c t0 (TSort m))) H (THeads (Flat -Appl) x0 (TLRef x1)) H3) in (eq_ind_r T (THeads (Flat Appl) x0 (TLRef x1)) -(\lambda (t0: T).(or (ex2 nat (\lambda (n: nat).(eq T t0 (TSort n))) (\lambda -(n: nat).(eq nat m (next g n)))) (ex3_2 TList nat (\lambda (ws: -TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i))))) -(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c (TLRef i))))))) (or_intror (ex2 nat (\lambda -(n: nat).(eq T (THeads (Flat Appl) x0 (TLRef x1)) (TSort n))) (\lambda (n: -nat).(eq nat m (next g n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda -(i: nat).(eq T (THeads (Flat Appl) x0 (TLRef x1)) (THeads (Flat Appl) ws -(TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda -(_: TList).(\lambda (i: nat).(nf2 c (TLRef i))))) (ex3_2_intro TList nat -(\lambda (ws: TList).(\lambda (i: nat).(eq T (THeads (Flat Appl) x0 (TLRef -x1)) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef i)))) -x0 x1 (refl_equal T (THeads (Flat Appl) x0 (TLRef x1))) H4 H5)) t H3))))))) -H2)) H1)))))))). -(* COMMENTS -Initial nodes: 2045 -END *) - -theorem ty3_nf2_gen__ty3_nf2_inv_abst_aux: - \forall (c: C).(\forall (w1: T).(\forall (u1: T).((ty3_nf2_inv_abst_premise -c w1 u1) \to (\forall (t: T).(\forall (w2: T).(\forall (u2: T).((pc3 c (THead -(Flat Appl) t (THead (Bind Abst) w2 u2)) (THead (Bind Abst) w1 u1)) \to -(ty3_nf2_inv_abst_premise c w2 u2)))))))) -\def - \lambda (c: C).(\lambda (w1: T).(\lambda (u1: T).(\lambda (H: ((\forall (d: -C).(\forall (wi: T).(\forall (i: nat).((getl i c (CHead d (Bind Abst) wi)) -\to (\forall (vs: TList).((pc3 c (THeads (Flat Appl) vs (lift (S i) O wi)) -(THead (Bind Abst) w1 u1)) \to False)))))))).(\lambda (t: T).(\lambda (w2: -T).(\lambda (u2: T).(\lambda (H0: (pc3 c (THead (Flat Appl) t (THead (Bind -Abst) w2 u2)) (THead (Bind Abst) w1 u1))).(\lambda (d: C).(\lambda (wi: -T).(\lambda (i: nat).(\lambda (H1: (getl i c (CHead d (Bind Abst) -wi))).(\lambda (vs: TList).(\lambda (H2: (pc3 c (THeads (Flat Appl) vs (lift -(S i) O wi)) (THead (Bind Abst) w2 u2))).(H d wi i H1 (TCons t vs) (pc3_t -(THead (Flat Appl) t (THead (Bind Abst) w2 u2)) c (THead (Flat Appl) t -(THeads (Flat Appl) vs (lift (S i) O wi))) (pc3_thin_dx c (THeads (Flat Appl) -vs (lift (S i) O wi)) (THead (Bind Abst) w2 u2) H2 t Appl) (THead (Bind Abst) -w1 u1) H0))))))))))))))). -(* COMMENTS -Initial nodes: 271 -END *) - -theorem ty3_nf2_inv_abst: - \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (w: T).(\forall (u: -T).((ty3 g c t (THead (Bind Abst) w u)) \to ((nf2 c t) \to ((nf2 c w) \to -((ty3_nf2_inv_abst_premise c w u) \to (ex4_2 T T (\lambda (v: T).(\lambda (_: -T).(eq T t (THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g -c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v -u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) -v)))))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (w: T).(\lambda (u: -T).(\lambda (H: (ty3 g c t (THead (Bind Abst) w u))).(\lambda (H0: (nf2 c -t)).(\lambda (H1: (nf2 c w)).(\lambda (H2: (ty3_nf2_inv_abst_premise c w -u)).(let H_x \def (ty3_nf2_inv_all g c t (THead (Bind Abst) w u) H H0) in -(let H3 \def H_x in (or3_ind (ex3_2 T T (\lambda (w0: T).(\lambda (u0: T).(eq -T t (THead (Bind Abst) w0 u0)))) (\lambda (w0: T).(\lambda (_: T).(nf2 c -w0))) (\lambda (w0: T).(\lambda (u0: T).(nf2 (CHead c (Bind Abst) w0) u0)))) -(ex nat (\lambda (n: nat).(eq T t (TSort n)))) (ex3_2 TList nat (\lambda (ws: -TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i))))) -(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c (TLRef i))))) (ex4_2 T T (\lambda (v: -T).(\lambda (_: T).(eq T t (THead (Bind Abst) w v)))) (\lambda (_: -T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g -(CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c -(Bind Abst) w) v)))) (\lambda (H4: (ex3_2 T T (\lambda (w0: T).(\lambda (u0: -T).(eq T t (THead (Bind Abst) w0 u0)))) (\lambda (w0: T).(\lambda (_: T).(nf2 -c w0))) (\lambda (w0: T).(\lambda (u0: T).(nf2 (CHead c (Bind Abst) w0) -u0))))).(ex3_2_ind T T (\lambda (w0: T).(\lambda (u0: T).(eq T t (THead (Bind -Abst) w0 u0)))) (\lambda (w0: T).(\lambda (_: T).(nf2 c w0))) (\lambda (w0: -T).(\lambda (u0: T).(nf2 (CHead c (Bind Abst) w0) u0))) (ex4_2 T T (\lambda -(v: T).(\lambda (_: T).(eq T t (THead (Bind Abst) w v)))) (\lambda (_: -T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g -(CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c -(Bind Abst) w) v)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T t -(THead (Bind Abst) x0 x1))).(\lambda (H6: (nf2 c x0)).(\lambda (H7: (nf2 -(CHead c (Bind Abst) x0) x1)).(let H8 \def (eq_ind T t (\lambda (t0: T).(ty3 -g c t0 (THead (Bind Abst) w u))) H (THead (Bind Abst) x0 x1) H5) in (eq_ind_r -T (THead (Bind Abst) x0 x1) (\lambda (t0: T).(ex4_2 T T (\lambda (v: -T).(\lambda (_: T).(eq T t0 (THead (Bind Abst) w v)))) (\lambda (_: -T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g -(CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c -(Bind Abst) w) v))))) (ex_ind T (\lambda (t0: T).(ty3 g c (THead (Bind Abst) -w u) t0)) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (THead (Bind Abst) -x0 x1) (THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w -w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u))) -(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v)))) (\lambda -(x: T).(\lambda (H9: (ty3 g c (THead (Bind Abst) w u) x)).(ex3_2_ind T T -(\lambda (t2: T).(\lambda (_: T).(pc3 c (THead (Bind Abst) w t2) x))) -(\lambda (_: T).(\lambda (t0: T).(ty3 g c w t0))) (\lambda (t2: T).(\lambda -(_: T).(ty3 g (CHead c (Bind Abst) w) u t2))) (ex4_2 T T (\lambda (v: -T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) w v)))) -(\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda -(_: T).(ty3 g (CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: -T).(nf2 (CHead c (Bind Abst) w) v)))) (\lambda (x2: T).(\lambda (x3: -T).(\lambda (_: (pc3 c (THead (Bind Abst) w x2) x)).(\lambda (H11: (ty3 g c w -x3)).(\lambda (H12: (ty3 g (CHead c (Bind Abst) w) u x2)).(ex3_2_ind T T -(\lambda (t2: T).(\lambda (_: T).(pc3 c (THead (Bind Abst) x0 t2) (THead -(Bind Abst) w u)))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c x0 t0))) -(\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x0) x1 t2))) -(ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) -(THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) -(\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u))) -(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v)))) (\lambda -(x4: T).(\lambda (x5: T).(\lambda (H13: (pc3 c (THead (Bind Abst) x0 x4) -(THead (Bind Abst) w u))).(\lambda (_: (ty3 g c x0 x5)).(\lambda (H15: (ty3 g -(CHead c (Bind Abst) x0) x1 x4)).(land_ind (pc3 c x0 w) (\forall (b: -B).(\forall (u0: T).(pc3 (CHead c (Bind b) u0) x4 u))) (ex4_2 T T (\lambda -(v: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) w -v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: -T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u))) (\lambda (v: -T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v)))) (\lambda (H16: (pc3 c -x0 w)).(\lambda (H17: ((\forall (b: B).(\forall (u0: T).(pc3 (CHead c (Bind -b) u0) x4 u))))).(let H_y \def (pc3_nf2 c x0 w H16 H6 H1) in (let H18 \def -(eq_ind T x0 (\lambda (t0: T).(ty3 g (CHead c (Bind Abst) t0) x1 x4)) H15 w -H_y) in (let H19 \def (eq_ind T x0 (\lambda (t0: T).(nf2 (CHead c (Bind Abst) -t0) x1)) H7 w H_y) in (eq_ind_r T w (\lambda (t0: T).(ex4_2 T T (\lambda (v: -T).(\lambda (_: T).(eq T (THead (Bind Abst) t0 x1) (THead (Bind Abst) w v)))) -(\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda -(_: T).(ty3 g (CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: -T).(nf2 (CHead c (Bind Abst) w) v))))) (ex4_2_intro T T (\lambda (v: -T).(\lambda (_: T).(eq T (THead (Bind Abst) w x1) (THead (Bind Abst) w v)))) -(\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda -(_: T).(ty3 g (CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: -T).(nf2 (CHead c (Bind Abst) w) v))) x1 x3 (refl_equal T (THead (Bind Abst) w -x1)) H11 (ty3_conv g (CHead c (Bind Abst) w) u x2 H12 x1 x4 H18 (H17 Abst w)) -H19) x0 H_y)))))) (pc3_gen_abst c x0 w x4 u H13))))))) (ty3_gen_bind g Abst c -x0 x1 (THead (Bind Abst) w u) H8))))))) (ty3_gen_bind g Abst c w u x H9)))) -(ty3_correct g c (THead (Bind Abst) x0 x1) (THead (Bind Abst) w u) H8)) t -H5))))))) H4)) (\lambda (H4: (ex nat (\lambda (n: nat).(eq T t (TSort -n))))).(ex_ind nat (\lambda (n: nat).(eq T t (TSort n))) (ex4_2 T T (\lambda -(v: T).(\lambda (_: T).(eq T t (THead (Bind Abst) w v)))) (\lambda (_: -T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g -(CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c -(Bind Abst) w) v)))) (\lambda (x: nat).(\lambda (H5: (eq T t (TSort x))).(let -H6 \def (eq_ind T t (\lambda (t0: T).(ty3 g c t0 (THead (Bind Abst) w u))) H -(TSort x) H5) in (eq_ind_r T (TSort x) (\lambda (t0: T).(ex4_2 T T (\lambda -(v: T).(\lambda (_: T).(eq T t0 (THead (Bind Abst) w v)))) (\lambda (_: -T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g -(CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c -(Bind Abst) w) v))))) (pc3_gen_sort_abst c w u (next g x) (ty3_gen_sort g c -(THead (Bind Abst) w u) x H6) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq -T (TSort x) (THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 -g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v -u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v))))) t -H5)))) H4)) (\lambda (H4: (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: -nat).(eq T t (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: -TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: -nat).(nf2 c (TLRef i)))))).(ex3_2_ind TList nat (\lambda (ws: TList).(\lambda -(i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: -TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: -nat).(nf2 c (TLRef i)))) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T t -(THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) -(\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u))) -(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v)))) (\lambda -(x0: TList).(\lambda (x1: nat).(\lambda (H5: (eq T t (THeads (Flat Appl) x0 -(TLRef x1)))).(\lambda (_: (nfs2 c x0)).(\lambda (H7: (nf2 c (TLRef -x1))).(let H8 \def (eq_ind T t (\lambda (t0: T).(ty3 g c t0 (THead (Bind -Abst) w u))) H (THeads (Flat Appl) x0 (TLRef x1)) H5) in (eq_ind_r T (THeads -(Flat Appl) x0 (TLRef x1)) (\lambda (t0: T).(ex4_2 T T (\lambda (v: -T).(\lambda (_: T).(eq T t0 (THead (Bind Abst) w v)))) (\lambda (_: -T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g -(CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c -(Bind Abst) w) v))))) (let H9 \def H2 in ((let H10 \def H8 in (unintro T u -(\lambda (t0: T).((ty3 g c (THeads (Flat Appl) x0 (TLRef x1)) (THead (Bind -Abst) w t0)) \to ((ty3_nf2_inv_abst_premise c w t0) \to (ex4_2 T T (\lambda -(v: T).(\lambda (_: T).(eq T (THeads (Flat Appl) x0 (TLRef x1)) (THead (Bind -Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: -T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v t0))) (\lambda (v: -T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v))))))) (unintro T w -(\lambda (t0: T).(\forall (x: T).((ty3 g c (THeads (Flat Appl) x0 (TLRef x1)) -(THead (Bind Abst) t0 x)) \to ((ty3_nf2_inv_abst_premise c t0 x) \to (ex4_2 T -T (\lambda (v: T).(\lambda (_: T).(eq T (THeads (Flat Appl) x0 (TLRef x1)) -(THead (Bind Abst) t0 v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c t0 -w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) t0) v x))) -(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) t0) v)))))))) -(TList_ind (\lambda (t0: TList).(\forall (x: T).(\forall (x2: T).((ty3 g c -(THeads (Flat Appl) t0 (TLRef x1)) (THead (Bind Abst) x x2)) \to -((ty3_nf2_inv_abst_premise c x x2) \to (ex4_2 T T (\lambda (v: T).(\lambda -(_: T).(eq T (THeads (Flat Appl) t0 (TLRef x1)) (THead (Bind Abst) x v)))) -(\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda -(_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_: -T).(nf2 (CHead c (Bind Abst) x) v))))))))) (\lambda (x: T).(\lambda (x2: -T).(\lambda (H11: (ty3 g c (TLRef x1) (THead (Bind Abst) x x2))).(\lambda -(H12: (ty3_nf2_inv_abst_premise c x x2)).(or_ind (ex3_3 C T T (\lambda (_: -C).(\lambda (_: T).(\lambda (t0: T).(pc3 c (lift (S x1) O t0) (THead (Bind -Abst) x x2))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl x1 c -(CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: -T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda -(_: T).(pc3 c (lift (S x1) O u0) (THead (Bind Abst) x x2))))) (\lambda (e: -C).(\lambda (u0: T).(\lambda (_: T).(getl x1 c (CHead e (Bind Abst) u0))))) -(\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex4_2 -T T (\lambda (v: T).(\lambda (_: T).(eq T (TLRef x1) (THead (Bind Abst) x -v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: -T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v: -T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) v)))) (\lambda (H13: (ex3_3 C -T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c (lift (S x1) O -t0) (THead (Bind Abst) x x2))))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(_: T).(getl x1 c (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (t0: T).(ty3 g e u0 t0)))))).(ex3_3_ind C T T (\lambda (_: -C).(\lambda (_: T).(\lambda (t0: T).(pc3 c (lift (S x1) O t0) (THead (Bind -Abst) x x2))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl x1 c -(CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: -T).(ty3 g e u0 t0)))) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (TLRef -x1) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c x -w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) -(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) v)))) (\lambda -(x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (pc3 c (lift (S x1) O -x5) (THead (Bind Abst) x x2))).(\lambda (H15: (getl x1 c (CHead x3 (Bind -Abbr) x4))).(\lambda (_: (ty3 g x3 x4 x5)).(nf2_gen_lref c x3 x4 x1 H15 H7 -(ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (TLRef x1) (THead (Bind -Abst) x v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: -T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v: -T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) v))))))))))) H13)) (\lambda -(H13: (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c -(lift (S x1) O u0) (THead (Bind Abst) x x2))))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (_: T).(getl x1 c (CHead e (Bind Abst) u0))))) (\lambda (e: -C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))))).(ex3_3_ind C T T -(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c (lift (S x1) O u0) -(THead (Bind Abst) x x2))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: -T).(getl x1 c (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (t0: T).(ty3 g e u0 t0)))) (ex4_2 T T (\lambda (v: T).(\lambda -(_: T).(eq T (TLRef x1) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda -(w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c -(Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind -Abst) x) v)))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda -(H14: (pc3 c (lift (S x1) O x4) (THead (Bind Abst) x x2))).(\lambda (H15: -(getl x1 c (CHead x3 (Bind Abst) x4))).(\lambda (_: (ty3 g x3 x4 x5)).(let -H_x0 \def (H12 x3 x4 x1 H15 TNil H14) in (let H17 \def H_x0 in (False_ind -(ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (TLRef x1) (THead (Bind -Abst) x v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: -T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v: -T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) v)))) H17))))))))) H13)) -(ty3_gen_lref g c (THead (Bind Abst) x x2) x1 H11)))))) (\lambda (t0: -T).(\lambda (t1: TList).(\lambda (H11: ((\forall (x: T).(\forall (x2: -T).((ty3 g c (THeads (Flat Appl) t1 (TLRef x1)) (THead (Bind Abst) x x2)) \to -((ty3_nf2_inv_abst_premise c x x2) \to (ex4_2 T T (\lambda (v: T).(\lambda -(_: T).(eq T (THeads (Flat Appl) t1 (TLRef x1)) (THead (Bind Abst) x v)))) -(\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda -(_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_: -T).(nf2 (CHead c (Bind Abst) x) v)))))))))).(\lambda (x: T).(\lambda (x2: -T).(\lambda (H12: (ty3 g c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 -(TLRef x1))) (THead (Bind Abst) x x2))).(\lambda (H13: -(ty3_nf2_inv_abst_premise c x x2)).(ex3_2_ind T T (\lambda (u0: T).(\lambda -(t2: T).(pc3 c (THead (Flat Appl) t0 (THead (Bind Abst) u0 t2)) (THead (Bind -Abst) x x2)))) (\lambda (u0: T).(\lambda (t2: T).(ty3 g c (THeads (Flat Appl) -t1 (TLRef x1)) (THead (Bind Abst) u0 t2)))) (\lambda (u0: T).(\lambda (_: -T).(ty3 g c t0 u0))) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (THead -(Flat Appl) t0 (THeads (Flat Appl) t1 (TLRef x1))) (THead (Bind Abst) x v)))) -(\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda -(_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_: -T).(nf2 (CHead c (Bind Abst) x) v)))) (\lambda (x3: T).(\lambda (x4: -T).(\lambda (H14: (pc3 c (THead (Flat Appl) t0 (THead (Bind Abst) x3 x4)) -(THead (Bind Abst) x x2))).(\lambda (H15: (ty3 g c (THeads (Flat Appl) t1 -(TLRef x1)) (THead (Bind Abst) x3 x4))).(\lambda (_: (ty3 g c t0 x3)).(let -H_y \def (ty3_nf2_gen__ty3_nf2_inv_abst_aux c x x2 H13 t0 x3 x4 H14) in (let -H_x0 \def (H11 x3 x4 H15 H_y) in (let H17 \def H_x0 in (ex4_2_ind T T -(\lambda (v: T).(\lambda (_: T).(eq T (THeads (Flat Appl) t1 (TLRef x1)) -(THead (Bind Abst) x3 v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c x3 -w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x3) v x4))) -(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x3) v))) (ex4_2 T T -(\lambda (v: T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (THeads (Flat -Appl) t1 (TLRef x1))) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda -(w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c -(Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind -Abst) x) v)))) (\lambda (x5: T).(\lambda (x6: T).(\lambda (H18: (eq T (THeads -(Flat Appl) t1 (TLRef x1)) (THead (Bind Abst) x3 x5))).(\lambda (_: (ty3 g c -x3 x6)).(\lambda (_: (ty3 g (CHead c (Bind Abst) x3) x5 x4)).(\lambda (_: -(nf2 (CHead c (Bind Abst) x3) x5)).(TList_ind (\lambda (t2: TList).((eq T -(THeads (Flat Appl) t2 (TLRef x1)) (THead (Bind Abst) x3 x5)) \to (ex4_2 T T -(\lambda (v: T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (THeads (Flat -Appl) t2 (TLRef x1))) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda -(w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c -(Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind -Abst) x) v)))))) (\lambda (H22: (eq T (THeads (Flat Appl) TNil (TLRef x1)) -(THead (Bind Abst) x3 x5))).(let H23 \def (eq_ind T (TLRef x1) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I -(THead (Bind Abst) x3 x5) H22) in (False_ind (ex4_2 T T (\lambda (v: -T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (THeads (Flat Appl) TNil -(TLRef x1))) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 -g c x w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v -x2))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) v)))) -H23))) (\lambda (t2: T).(\lambda (t3: TList).(\lambda (_: (((eq T (THeads -(Flat Appl) t3 (TLRef x1)) (THead (Bind Abst) x3 x5)) \to (ex4_2 T T (\lambda -(v: T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (THeads (Flat Appl) t3 -(TLRef x1))) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 -g c x w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v -x2))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) -v))))))).(\lambda (H22: (eq T (THeads (Flat Appl) (TCons t2 t3) (TLRef x1)) -(THead (Bind Abst) x3 x5))).(let H23 \def (eq_ind T (THead (Flat Appl) t2 -(THeads (Flat Appl) t3 (TLRef x1))) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind Abst) x3 x5) H22) in (False_ind (ex4_2 T T (\lambda -(v: T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (THeads (Flat Appl) (TCons -t2 t3) (TLRef x1))) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda (w0: -T).(ty3 g c x w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind -Abst) x) v x2))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) -v)))) H23)))))) t1 H18))))))) H17))))))))) (ty3_gen_appl g c t0 (THeads (Flat -Appl) t1 (TLRef x1)) (THead (Bind Abst) x x2) H12))))))))) x0)) H10)) H9)) t -H5))))))) H4)) H3))))))))))). -(* COMMENTS -Initial nodes: 5333 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/pr3.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/pr3.ma deleted file mode 100644 index 8d184aa58..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/pr3.ma +++ /dev/null @@ -1,728 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubt/ty3.ma". - -include "Basic-1/ty3/subst1.ma". - -include "Basic-1/ty3/fsubst0.ma". - -include "Basic-1/pc3/pc1.ma". - -include "Basic-1/pc3/wcpr0.ma". - -include "Basic-1/pc1/props.ma". - -theorem ty3_sred_wcpr0_pr0: - \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t: T).((ty3 g c1 -t1 t) \to (\forall (c2: C).((wcpr0 c1 c2) \to (\forall (t2: T).((pr0 t1 t2) -\to (ty3 g c2 t2 t))))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t: T).(\lambda -(H: (ty3 g c1 t1 t)).(ty3_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda -(t2: T).(\forall (c2: C).((wcpr0 c c2) \to (\forall (t3: T).((pr0 t0 t3) \to -(ty3 g c2 t3 t2)))))))) (\lambda (c: C).(\lambda (t2: T).(\lambda (t0: -T).(\lambda (_: (ty3 g c t2 t0)).(\lambda (H1: ((\forall (c2: C).((wcpr0 c -c2) \to (\forall (t3: T).((pr0 t2 t3) \to (ty3 g c2 t3 t0))))))).(\lambda (u: -T).(\lambda (t3: T).(\lambda (_: (ty3 g c u t3)).(\lambda (H3: ((\forall (c2: -C).((wcpr0 c c2) \to (\forall (t4: T).((pr0 u t4) \to (ty3 g c2 t4 -t3))))))).(\lambda (H4: (pc3 c t3 t2)).(\lambda (c2: C).(\lambda (H5: (wcpr0 -c c2)).(\lambda (t4: T).(\lambda (H6: (pr0 u t4)).(ty3_conv g c2 t2 t0 (H1 c2 -H5 t2 (pr0_refl t2)) t4 t3 (H3 c2 H5 t4 H6) (pc3_wcpr0 c c2 H5 t3 t2 -H4)))))))))))))))) (\lambda (c: C).(\lambda (m: nat).(\lambda (c2: -C).(\lambda (_: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H1: (pr0 (TSort m) -t2)).(eq_ind_r T (TSort m) (\lambda (t0: T).(ty3 g c2 t0 (TSort (next g m)))) -(ty3_sort g c2 m) t2 (pr0_gen_sort t2 m H1)))))))) (\lambda (n: nat).(\lambda -(c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind -Abbr) u))).(\lambda (t0: T).(\lambda (_: (ty3 g d u t0)).(\lambda (H2: -((\forall (c2: C).((wcpr0 d c2) \to (\forall (t2: T).((pr0 u t2) \to (ty3 g -c2 t2 t0))))))).(\lambda (c2: C).(\lambda (H3: (wcpr0 c c2)).(\lambda (t2: -T).(\lambda (H4: (pr0 (TLRef n) t2)).(eq_ind_r T (TLRef n) (\lambda (t3: -T).(ty3 g c2 t3 (lift (S n) O t0))) (ex3_2_ind C T (\lambda (e2: C).(\lambda -(u2: T).(getl n c2 (CHead e2 (Bind Abbr) u2)))) (\lambda (e2: C).(\lambda (_: -T).(wcpr0 d e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u u2))) (ty3 g c2 -(TLRef n) (lift (S n) O t0)) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: -(getl n c2 (CHead x0 (Bind Abbr) x1))).(\lambda (H6: (wcpr0 d x0)).(\lambda -(H7: (pr0 u x1)).(ty3_abbr g n c2 x0 x1 H5 t0 (H2 x0 H6 x1 H7))))))) -(wcpr0_getl c c2 H3 n d u (Bind Abbr) H0)) t2 (pr0_gen_lref t2 n -H4)))))))))))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d: C).(\lambda -(u: T).(\lambda (H0: (getl n c (CHead d (Bind Abst) u))).(\lambda (t0: -T).(\lambda (_: (ty3 g d u t0)).(\lambda (H2: ((\forall (c2: C).((wcpr0 d c2) -\to (\forall (t2: T).((pr0 u t2) \to (ty3 g c2 t2 t0))))))).(\lambda (c2: -C).(\lambda (H3: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H4: (pr0 (TLRef n) -t2)).(eq_ind_r T (TLRef n) (\lambda (t3: T).(ty3 g c2 t3 (lift (S n) O u))) -(ex3_2_ind C T (\lambda (e2: C).(\lambda (u2: T).(getl n c2 (CHead e2 (Bind -Abst) u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 d e2))) (\lambda (_: -C).(\lambda (u2: T).(pr0 u u2))) (ty3 g c2 (TLRef n) (lift (S n) O u)) -(\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: (getl n c2 (CHead x0 (Bind -Abst) x1))).(\lambda (H6: (wcpr0 d x0)).(\lambda (H7: (pr0 u x1)).(ty3_conv g -c2 (lift (S n) O u) (lift (S n) O t0) (ty3_lift g x0 u t0 (H2 x0 H6 u -(pr0_refl u)) c2 O (S n) (getl_drop Abst c2 x0 x1 n H5)) (TLRef n) (lift (S -n) O x1) (ty3_abst g n c2 x0 x1 H5 t0 (H2 x0 H6 x1 H7)) (pc3_lift c2 x0 (S n) -O (getl_drop Abst c2 x0 x1 n H5) x1 u (pc3_pr2_x x0 x1 u (pr2_free x0 u x1 -H7))))))))) (wcpr0_getl c c2 H3 n d u (Bind Abst) H0)) t2 (pr0_gen_lref t2 n -H4)))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (t0: T).(\lambda -(_: (ty3 g c u t0)).(\lambda (H1: ((\forall (c2: C).((wcpr0 c c2) \to -(\forall (t2: T).((pr0 u t2) \to (ty3 g c2 t2 t0))))))).(\lambda (b: -B).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H2: (ty3 g (CHead c (Bind b) -u) t2 t3)).(\lambda (H3: ((\forall (c2: C).((wcpr0 (CHead c (Bind b) u) c2) -\to (\forall (t4: T).((pr0 t2 t4) \to (ty3 g c2 t4 t3))))))).(\lambda (c2: -C).(\lambda (H4: (wcpr0 c c2)).(\lambda (t4: T).(\lambda (H5: (pr0 (THead -(Bind b) u t2) t4)).(let H6 \def (match H5 in pr0 return (\lambda (t5: -T).(\lambda (t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 (THead (Bind b) u -t2)) \to ((eq T t6 t4) \to (ty3 g c2 t4 (THead (Bind b) u t3))))))) with -[(pr0_refl t5) \Rightarrow (\lambda (H6: (eq T t5 (THead (Bind b) u -t2))).(\lambda (H7: (eq T t5 t4)).(eq_ind T (THead (Bind b) u t2) (\lambda -(t6: T).((eq T t6 t4) \to (ty3 g c2 t4 (THead (Bind b) u t3)))) (\lambda (H8: -(eq T (THead (Bind b) u t2) t4)).(eq_ind T (THead (Bind b) u t2) (\lambda -(t6: T).(ty3 g c2 t6 (THead (Bind b) u t3))) (ty3_bind g c2 u t0 (H1 c2 H4 u -(pr0_refl u)) b t2 t3 (H3 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H4 u u -(pr0_refl u) (Bind b)) t2 (pr0_refl t2))) t4 H8)) t5 (sym_eq T t5 (THead -(Bind b) u t2) H6) H7))) | (pr0_comp u1 u2 H6 t5 t6 H7 k) \Rightarrow -(\lambda (H8: (eq T (THead k u1 t5) (THead (Bind b) u t2))).(\lambda (H9: (eq -T (THead k u2 t6) t4)).((let H10 \def (f_equal T T (\lambda (e: T).(match e -in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t5 | (TLRef _) -\Rightarrow t5 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t5) (THead -(Bind b) u t2) H8) in ((let H11 \def (f_equal T T (\lambda (e: T).(match e in -T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) -\Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t5) (THead -(Bind b) u t2) H8) in ((let H12 \def (f_equal T K (\lambda (e: T).(match e in -T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) -\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t5) (THead (Bind -b) u t2) H8) in (eq_ind K (Bind b) (\lambda (k0: K).((eq T u1 u) \to ((eq T -t5 t2) \to ((eq T (THead k0 u2 t6) t4) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to -(ty3 g c2 t4 (THead (Bind b) u t3)))))))) (\lambda (H13: (eq T u1 u)).(eq_ind -T u (\lambda (t7: T).((eq T t5 t2) \to ((eq T (THead (Bind b) u2 t6) t4) \to -((pr0 t7 u2) \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3))))))) -(\lambda (H14: (eq T t5 t2)).(eq_ind T t2 (\lambda (t7: T).((eq T (THead -(Bind b) u2 t6) t4) \to ((pr0 u u2) \to ((pr0 t7 t6) \to (ty3 g c2 t4 (THead -(Bind b) u t3)))))) (\lambda (H15: (eq T (THead (Bind b) u2 t6) t4)).(eq_ind -T (THead (Bind b) u2 t6) (\lambda (t7: T).((pr0 u u2) \to ((pr0 t2 t6) \to -(ty3 g c2 t7 (THead (Bind b) u t3))))) (\lambda (H16: (pr0 u u2)).(\lambda -(H17: (pr0 t2 t6)).(ex_ind T (\lambda (t7: T).(ty3 g (CHead c2 (Bind b) u) t3 -t7)) (ty3 g c2 (THead (Bind b) u2 t6) (THead (Bind b) u t3)) (\lambda (x: -T).(\lambda (H18: (ty3 g (CHead c2 (Bind b) u) t3 x)).(ex_ind T (\lambda (t7: -T).(ty3 g (CHead c2 (Bind b) u2) t3 t7)) (ty3 g c2 (THead (Bind b) u2 t6) -(THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (_: (ty3 g (CHead c2 (Bind -b) u2) t3 x0)).(ty3_conv g c2 (THead (Bind b) u t3) (THead (Bind b) u x) -(ty3_bind g c2 u t0 (H1 c2 H4 u (pr0_refl u)) b t3 x H18) (THead (Bind b) u2 -t6) (THead (Bind b) u2 t3) (ty3_bind g c2 u2 t0 (H1 c2 H4 u2 H16) b t6 t3 (H3 -(CHead c2 (Bind b) u2) (wcpr0_comp c c2 H4 u u2 H16 (Bind b)) t6 H17)) -(pc3_pr2_x c2 (THead (Bind b) u2 t3) (THead (Bind b) u t3) (pr2_head_1 c2 u -u2 (pr2_free c2 u u2 H16) (Bind b) t3))))) (ty3_correct g (CHead c2 (Bind b) -u2) t6 t3 (H3 (CHead c2 (Bind b) u2) (wcpr0_comp c c2 H4 u u2 H16 (Bind b)) -t6 H17))))) (ty3_correct g (CHead c2 (Bind b) u) t2 t3 (H3 (CHead c2 (Bind b) -u) (wcpr0_comp c c2 H4 u u (pr0_refl u) (Bind b)) t2 (pr0_refl t2)))))) t4 -H15)) t5 (sym_eq T t5 t2 H14))) u1 (sym_eq T u1 u H13))) k (sym_eq K k (Bind -b) H12))) H11)) H10)) H9 H6 H7))) | (pr0_beta u0 v1 v2 H6 t5 t6 H7) -\Rightarrow (\lambda (H8: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u0 -t5)) (THead (Bind b) u t2))).(\lambda (H9: (eq T (THead (Bind Abbr) v2 t6) -t4)).((let H10 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u0 -t5)) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort -_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) -\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u t2) -H8) in (False_ind ((eq T (THead (Bind Abbr) v2 t6) t4) \to ((pr0 v1 v2) \to -((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3))))) H10)) H9 H6 H7))) | -(pr0_upsilon b0 H6 v1 v2 H7 u1 u2 H8 t5 t6 H9) \Rightarrow (\lambda (H10: (eq -T (THead (Flat Appl) v1 (THead (Bind b0) u1 t5)) (THead (Bind b) u -t2))).(\lambda (H11: (eq T (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) -O v2) t6)) t4)).((let H12 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind -b0) u1 t5)) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) -\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u t2) -H10) in (False_ind ((eq T (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) -O v2) t6)) t4) \to ((not (eq B b0 Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) -\to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3))))))) H12)) H11 H6 H7 -H8 H9))) | (pr0_delta u1 u2 H6 t5 t6 H7 w H8) \Rightarrow (\lambda (H9: (eq T -(THead (Bind Abbr) u1 t5) (THead (Bind b) u t2))).(\lambda (H10: (eq T (THead -(Bind Abbr) u2 w) t4)).((let H11 \def (f_equal T T (\lambda (e: T).(match e -in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t5 | (TLRef _) -\Rightarrow t5 | (THead _ _ t7) \Rightarrow t7])) (THead (Bind Abbr) u1 t5) -(THead (Bind b) u t2) H9) in ((let H12 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | -(TLRef _) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead (Bind -Abbr) u1 t5) (THead (Bind b) u t2) H9) in ((let H13 \def (f_equal T B -(\lambda (e: T).(match e in T return (\lambda (_: T).B) with [(TSort _) -\Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | (THead k _ _) \Rightarrow -(match k in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | -(Flat _) \Rightarrow Abbr])])) (THead (Bind Abbr) u1 t5) (THead (Bind b) u -t2) H9) in (eq_ind B Abbr (\lambda (b0: B).((eq T u1 u) \to ((eq T t5 t2) \to -((eq T (THead (Bind Abbr) u2 w) t4) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to -((subst0 O u2 t6 w) \to (ty3 g c2 t4 (THead (Bind b0) u t3))))))))) (\lambda -(H14: (eq T u1 u)).(eq_ind T u (\lambda (t7: T).((eq T t5 t2) \to ((eq T -(THead (Bind Abbr) u2 w) t4) \to ((pr0 t7 u2) \to ((pr0 t5 t6) \to ((subst0 O -u2 t6 w) \to (ty3 g c2 t4 (THead (Bind Abbr) u t3)))))))) (\lambda (H15: (eq -T t5 t2)).(eq_ind T t2 (\lambda (t7: T).((eq T (THead (Bind Abbr) u2 w) t4) -\to ((pr0 u u2) \to ((pr0 t7 t6) \to ((subst0 O u2 t6 w) \to (ty3 g c2 t4 -(THead (Bind Abbr) u t3))))))) (\lambda (H16: (eq T (THead (Bind Abbr) u2 w) -t4)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t7: T).((pr0 u u2) \to -((pr0 t2 t6) \to ((subst0 O u2 t6 w) \to (ty3 g c2 t7 (THead (Bind Abbr) u -t3)))))) (\lambda (H17: (pr0 u u2)).(\lambda (H18: (pr0 t2 t6)).(\lambda -(H19: (subst0 O u2 t6 w)).(let H20 \def (eq_ind_r B b (\lambda (b0: -B).(\forall (c3: C).((wcpr0 (CHead c (Bind b0) u) c3) \to (\forall (t7: -T).((pr0 t2 t7) \to (ty3 g c3 t7 t3)))))) H3 Abbr H13) in (let H21 \def -(eq_ind_r B b (\lambda (b0: B).(ty3 g (CHead c (Bind b0) u) t2 t3)) H2 Abbr -H13) in (ex_ind T (\lambda (t7: T).(ty3 g (CHead c2 (Bind Abbr) u) t3 t7)) -(ty3 g c2 (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u t3)) (\lambda (x: -T).(\lambda (H22: (ty3 g (CHead c2 (Bind Abbr) u) t3 x)).(ex_ind T (\lambda -(t7: T).(ty3 g (CHead c2 (Bind Abbr) u2) t3 t7)) (ty3 g c2 (THead (Bind Abbr) -u2 w) (THead (Bind Abbr) u t3)) (\lambda (x0: T).(\lambda (_: (ty3 g (CHead -c2 (Bind Abbr) u2) t3 x0)).(ty3_conv g c2 (THead (Bind Abbr) u t3) (THead -(Bind Abbr) u x) (ty3_bind g c2 u t0 (H1 c2 H4 u (pr0_refl u)) Abbr t3 x H22) -(THead (Bind Abbr) u2 w) (THead (Bind Abbr) u2 t3) (ty3_bind g c2 u2 t0 (H1 -c2 H4 u2 H17) Abbr w t3 (ty3_subst0 g (CHead c2 (Bind Abbr) u2) t6 t3 (H20 -(CHead c2 (Bind Abbr) u2) (wcpr0_comp c c2 H4 u u2 H17 (Bind Abbr)) t6 H18) -c2 u2 O (getl_refl Abbr c2 u2) w H19)) (pc3_pr2_x c2 (THead (Bind Abbr) u2 -t3) (THead (Bind Abbr) u t3) (pr2_head_1 c2 u u2 (pr2_free c2 u u2 H17) (Bind -Abbr) t3))))) (ty3_correct g (CHead c2 (Bind Abbr) u2) t6 t3 (H20 (CHead c2 -(Bind Abbr) u2) (wcpr0_comp c c2 H4 u u2 H17 (Bind Abbr)) t6 H18))))) -(ty3_correct g (CHead c2 (Bind Abbr) u) t2 t3 (H20 (CHead c2 (Bind Abbr) u) -(wcpr0_comp c c2 H4 u u (pr0_refl u) (Bind Abbr)) t2 (pr0_refl t2))))))))) t4 -H16)) t5 (sym_eq T t5 t2 H15))) u1 (sym_eq T u1 u H14))) b H13)) H12)) H11)) -H10 H6 H7 H8))) | (pr0_zeta b0 H6 t5 t6 H7 u0) \Rightarrow (\lambda (H8: (eq -T (THead (Bind b0) u0 (lift (S O) O t5)) (THead (Bind b) u t2))).(\lambda -(H9: (eq T t6 t4)).((let H10 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: -((nat \to nat))) (d: nat) (t7: T) on t7: T \def (match t7 with [(TSort n) -\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with -[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u1 t8) -\Rightarrow (THead k (lref_map f d u1) (lref_map f (s k d) t8))]) in -lref_map) (\lambda (x: nat).(plus x (S O))) O t5) | (TLRef _) \Rightarrow -((let rec lref_map (f: ((nat \to nat))) (d: nat) (t7: T) on t7: T \def (match -t7 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef -(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | -(THead k u1 t8) \Rightarrow (THead k (lref_map f d u1) (lref_map f (s k d) -t8))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t5) | (THead _ _ t7) -\Rightarrow t7])) (THead (Bind b0) u0 (lift (S O) O t5)) (THead (Bind b) u -t2) H8) in ((let H11 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 -| (THead _ t7 _) \Rightarrow t7])) (THead (Bind b0) u0 (lift (S O) O t5)) -(THead (Bind b) u t2) H8) in ((let H12 \def (f_equal T B (\lambda (e: -T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b0 | -(TLRef _) \Rightarrow b0 | (THead k _ _) \Rightarrow (match k in K return -(\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow -b0])])) (THead (Bind b0) u0 (lift (S O) O t5)) (THead (Bind b) u t2) H8) in -(eq_ind B b (\lambda (b1: B).((eq T u0 u) \to ((eq T (lift (S O) O t5) t2) -\to ((eq T t6 t4) \to ((not (eq B b1 Abst)) \to ((pr0 t5 t6) \to (ty3 g c2 t4 -(THead (Bind b) u t3)))))))) (\lambda (H13: (eq T u0 u)).(eq_ind T u (\lambda -(_: T).((eq T (lift (S O) O t5) t2) \to ((eq T t6 t4) \to ((not (eq B b -Abst)) \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3))))))) (\lambda -(H14: (eq T (lift (S O) O t5) t2)).(eq_ind T (lift (S O) O t5) (\lambda (_: -T).((eq T t6 t4) \to ((not (eq B b Abst)) \to ((pr0 t5 t6) \to (ty3 g c2 t4 -(THead (Bind b) u t3)))))) (\lambda (H15: (eq T t6 t4)).(eq_ind T t4 (\lambda -(t7: T).((not (eq B b Abst)) \to ((pr0 t5 t7) \to (ty3 g c2 t4 (THead (Bind -b) u t3))))) (\lambda (H16: (not (eq B b Abst))).(\lambda (H17: (pr0 t5 -t4)).(let H18 \def (eq_ind_r T t2 (\lambda (t7: T).(\forall (c3: C).((wcpr0 -(CHead c (Bind b) u) c3) \to (\forall (t8: T).((pr0 t7 t8) \to (ty3 g c3 t8 -t3)))))) H3 (lift (S O) O t5) H14) in (let H19 \def (eq_ind_r T t2 (\lambda -(t7: T).(ty3 g (CHead c (Bind b) u) t7 t3)) H2 (lift (S O) O t5) H14) in -(ex_ind T (\lambda (t7: T).(ty3 g (CHead c2 (Bind b) u) t3 t7)) (ty3 g c2 t4 -(THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H20: (ty3 g (CHead c2 (Bind -b) u) t3 x)).(B_ind (\lambda (b1: B).((not (eq B b1 Abst)) \to ((ty3 g (CHead -c2 (Bind b1) u) t3 x) \to ((ty3 g (CHead c2 (Bind b1) u) (lift (S O) O t4) -t3) \to (ty3 g c2 t4 (THead (Bind b1) u t3)))))) (\lambda (H21: (not (eq B -Abbr Abst))).(\lambda (H22: (ty3 g (CHead c2 (Bind Abbr) u) t3 x)).(\lambda -(H23: (ty3 g (CHead c2 (Bind Abbr) u) (lift (S O) O t4) t3)).(let H24 \def -(ty3_gen_cabbr g (CHead c2 (Bind Abbr) u) (lift (S O) O t4) t3 H23 c2 u O -(getl_refl Abbr c2 u) (CHead c2 (Bind Abbr) u) (csubst1_refl O u (CHead c2 -(Bind Abbr) u)) c2 (drop_drop (Bind Abbr) O c2 c2 (drop_refl c2) u)) in -(ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(subst1 O u (lift (S O) O t4) -(lift (S O) O y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 O u t3 (lift (S -O) O y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g c2 y1 y2))) (ty3 g c2 t4 -(THead (Bind Abbr) u t3)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H25: -(subst1 O u (lift (S O) O t4) (lift (S O) O x0))).(\lambda (H26: (subst1 O u -t3 (lift (S O) O x1))).(\lambda (H27: (ty3 g c2 x0 x1)).(let H28 \def (eq_ind -T x0 (\lambda (t7: T).(ty3 g c2 t7 x1)) H27 t4 (lift_inj x0 t4 (S O) O -(subst1_gen_lift_eq t4 u (lift (S O) O x0) (S O) O O (le_n O) (eq_ind_r nat -(plus (S O) O) (\lambda (n: nat).(lt O n)) (le_n (plus (S O) O)) (plus O (S -O)) (plus_sym O (S O))) H25))) in (ty3_conv g c2 (THead (Bind Abbr) u t3) -(THead (Bind Abbr) u x) (ty3_bind g c2 u t0 (H1 c2 H4 u (pr0_refl u)) Abbr t3 -x H22) t4 x1 H28 (pc3_pr3_x c2 x1 (THead (Bind Abbr) u t3) (pr3_t (THead -(Bind Abbr) u (lift (S O) O x1)) (THead (Bind Abbr) u t3) c2 (pr3_pr2 c2 -(THead (Bind Abbr) u t3) (THead (Bind Abbr) u (lift (S O) O x1)) (pr2_free c2 -(THead (Bind Abbr) u t3) (THead (Bind Abbr) u (lift (S O) O x1)) (pr0_delta1 -u u (pr0_refl u) t3 t3 (pr0_refl t3) (lift (S O) O x1) H26))) x1 (pr3_pr2 c2 -(THead (Bind Abbr) u (lift (S O) O x1)) x1 (pr2_free c2 (THead (Bind Abbr) u -(lift (S O) O x1)) x1 (pr0_zeta Abbr H21 x1 x1 (pr0_refl x1) u)))))))))))) -H24))))) (\lambda (H21: (not (eq B Abst Abst))).(\lambda (_: (ty3 g (CHead c2 -(Bind Abst) u) t3 x)).(\lambda (_: (ty3 g (CHead c2 (Bind Abst) u) (lift (S -O) O t4) t3)).(let H24 \def (match (H21 (refl_equal B Abst)) in False return -(\lambda (_: False).(ty3 g c2 t4 (THead (Bind Abst) u t3))) with []) in -H24)))) (\lambda (H21: (not (eq B Void Abst))).(\lambda (H22: (ty3 g (CHead -c2 (Bind Void) u) t3 x)).(\lambda (H23: (ty3 g (CHead c2 (Bind Void) u) (lift -(S O) O t4) t3)).(let H24 \def (ty3_gen_cvoid g (CHead c2 (Bind Void) u) -(lift (S O) O t4) t3 H23 c2 u O (getl_refl Void c2 u) c2 (drop_drop (Bind -Void) O c2 c2 (drop_refl c2) u)) in (ex3_2_ind T T (\lambda (y1: T).(\lambda -(_: T).(eq T (lift (S O) O t4) (lift (S O) O y1)))) (\lambda (_: T).(\lambda -(y2: T).(eq T t3 (lift (S O) O y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 -g c2 y1 y2))) (ty3 g c2 t4 (THead (Bind Void) u t3)) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H25: (eq T (lift (S O) O t4) (lift (S O) O -x0))).(\lambda (H26: (eq T t3 (lift (S O) O x1))).(\lambda (H27: (ty3 g c2 x0 -x1)).(let H28 \def (eq_ind T t3 (\lambda (t7: T).(ty3 g (CHead c2 (Bind Void) -u) t7 x)) H22 (lift (S O) O x1) H26) in (eq_ind_r T (lift (S O) O x1) -(\lambda (t7: T).(ty3 g c2 t4 (THead (Bind Void) u t7))) (let H29 \def -(eq_ind_r T x0 (\lambda (t7: T).(ty3 g c2 t7 x1)) H27 t4 (lift_inj t4 x0 (S -O) O H25)) in (ty3_conv g c2 (THead (Bind Void) u (lift (S O) O x1)) (THead -(Bind Void) u x) (ty3_bind g c2 u t0 (H1 c2 H4 u (pr0_refl u)) Void (lift (S -O) O x1) x H28) t4 x1 H29 (pc3_s c2 x1 (THead (Bind Void) u (lift (S O) O -x1)) (pc3_pr2_r c2 (THead (Bind Void) u (lift (S O) O x1)) x1 (pr2_free c2 -(THead (Bind Void) u (lift (S O) O x1)) x1 (pr0_zeta Void H21 x1 x1 (pr0_refl -x1) u)))))) t3 H26))))))) H24))))) b H16 H20 (H18 (CHead c2 (Bind b) u) -(wcpr0_comp c c2 H4 u u (pr0_refl u) (Bind b)) (lift (S O) O t4) (pr0_lift t5 -t4 H17 (S O) O))))) (ty3_correct g (CHead c2 (Bind b) u) (lift (S O) O t4) t3 -(H18 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H4 u u (pr0_refl u) (Bind b)) -(lift (S O) O t4) (pr0_lift t5 t4 H17 (S O) O)))))))) t6 (sym_eq T t6 t4 -H15))) t2 H14)) u0 (sym_eq T u0 u H13))) b0 (sym_eq B b0 b H12))) H11)) H10)) -H9 H6 H7))) | (pr0_tau t5 t6 H6 u0) \Rightarrow (\lambda (H7: (eq T (THead -(Flat Cast) u0 t5) (THead (Bind b) u t2))).(\lambda (H8: (eq T t6 t4)).((let -H9 \def (eq_ind T (THead (Flat Cast) u0 t5) (\lambda (e: T).(match e in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind b) u t2) H7) in (False_ind ((eq T t6 t4) \to ((pr0 -t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3)))) H9)) H8 H6)))]) in (H6 -(refl_equal T (THead (Bind b) u t2)) (refl_equal T t4))))))))))))))))) -(\lambda (c: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c w -u)).(\lambda (H1: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 -w t2) \to (ty3 g c2 t2 u))))))).(\lambda (v: T).(\lambda (t0: T).(\lambda -(H2: (ty3 g c v (THead (Bind Abst) u t0))).(\lambda (H3: ((\forall (c2: -C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 v t2) \to (ty3 g c2 t2 (THead -(Bind Abst) u t0)))))))).(\lambda (c2: C).(\lambda (H4: (wcpr0 c -c2)).(\lambda (t2: T).(\lambda (H5: (pr0 (THead (Flat Appl) w v) t2)).(let H6 -\def (match H5 in pr0 return (\lambda (t3: T).(\lambda (t4: T).(\lambda (_: -(pr0 t3 t4)).((eq T t3 (THead (Flat Appl) w v)) \to ((eq T t4 t2) \to (ty3 g -c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u t0)))))))) with [(pr0_refl -t3) \Rightarrow (\lambda (H6: (eq T t3 (THead (Flat Appl) w v))).(\lambda -(H7: (eq T t3 t2)).(eq_ind T (THead (Flat Appl) w v) (\lambda (t4: T).((eq T -t4 t2) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u t0))))) -(\lambda (H8: (eq T (THead (Flat Appl) w v) t2)).(eq_ind T (THead (Flat Appl) -w v) (\lambda (t4: T).(ty3 g c2 t4 (THead (Flat Appl) w (THead (Bind Abst) u -t0)))) (ty3_appl g c2 w u (H1 c2 H4 w (pr0_refl w)) v t0 (H3 c2 H4 v -(pr0_refl v))) t2 H8)) t3 (sym_eq T t3 (THead (Flat Appl) w v) H6) H7))) | -(pr0_comp u1 u2 H6 t3 t4 H7 k) \Rightarrow (\lambda (H8: (eq T (THead k u1 -t3) (THead (Flat Appl) w v))).(\lambda (H9: (eq T (THead k u2 t4) t2)).((let -H10 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t5) -\Rightarrow t5])) (THead k u1 t3) (THead (Flat Appl) w v) H8) in ((let H11 -\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t5 _) -\Rightarrow t5])) (THead k u1 t3) (THead (Flat Appl) w v) H8) in ((let H12 -\def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) -with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) -\Rightarrow k0])) (THead k u1 t3) (THead (Flat Appl) w v) H8) in (eq_ind K -(Flat Appl) (\lambda (k0: K).((eq T u1 w) \to ((eq T t3 v) \to ((eq T (THead -k0 u2 t4) t2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat -Appl) w (THead (Bind Abst) u t0))))))))) (\lambda (H13: (eq T u1 w)).(eq_ind -T w (\lambda (t5: T).((eq T t3 v) \to ((eq T (THead (Flat Appl) u2 t4) t2) -\to ((pr0 t5 u2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w -(THead (Bind Abst) u t0)))))))) (\lambda (H14: (eq T t3 v)).(eq_ind T v -(\lambda (t5: T).((eq T (THead (Flat Appl) u2 t4) t2) \to ((pr0 w u2) \to -((pr0 t5 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u -t0))))))) (\lambda (H15: (eq T (THead (Flat Appl) u2 t4) t2)).(eq_ind T -(THead (Flat Appl) u2 t4) (\lambda (t5: T).((pr0 w u2) \to ((pr0 v t4) \to -(ty3 g c2 t5 (THead (Flat Appl) w (THead (Bind Abst) u t0)))))) (\lambda -(H16: (pr0 w u2)).(\lambda (H17: (pr0 v t4)).(ex_ind T (\lambda (t5: T).(ty3 -g c2 (THead (Bind Abst) u t0) t5)) (ty3 g c2 (THead (Flat Appl) u2 t4) (THead -(Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x: T).(\lambda (H18: (ty3 -g c2 (THead (Bind Abst) u t0) x)).(ex3_2_ind T T (\lambda (t5: T).(\lambda -(_: T).(pc3 c2 (THead (Bind Abst) u t5) x))) (\lambda (_: T).(\lambda (t6: -T).(ty3 g c2 u t6))) (\lambda (t5: T).(\lambda (_: T).(ty3 g (CHead c2 (Bind -Abst) u) t0 t5))) (ty3 g c2 (THead (Flat Appl) u2 t4) (THead (Flat Appl) w -(THead (Bind Abst) u t0))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: -(pc3 c2 (THead (Bind Abst) u x0) x)).(\lambda (H20: (ty3 g c2 u x1)).(\lambda -(H21: (ty3 g (CHead c2 (Bind Abst) u) t0 x0)).(ty3_conv g c2 (THead (Flat -Appl) w (THead (Bind Abst) u t0)) (THead (Flat Appl) w (THead (Bind Abst) u -x0)) (ty3_appl g c2 w u (H1 c2 H4 w (pr0_refl w)) (THead (Bind Abst) u t0) x0 -(ty3_bind g c2 u x1 H20 Abst t0 x0 H21)) (THead (Flat Appl) u2 t4) (THead -(Flat Appl) u2 (THead (Bind Abst) u t0)) (ty3_appl g c2 u2 u (H1 c2 H4 u2 -H16) t4 t0 (H3 c2 H4 t4 H17)) (pc3_pr2_x c2 (THead (Flat Appl) u2 (THead -(Bind Abst) u t0)) (THead (Flat Appl) w (THead (Bind Abst) u t0)) (pr2_head_1 -c2 w u2 (pr2_free c2 w u2 H16) (Flat Appl) (THead (Bind Abst) u t0))))))))) -(ty3_gen_bind g Abst c2 u t0 x H18)))) (ty3_correct g c2 v (THead (Bind Abst) -u t0) (H3 c2 H4 v (pr0_refl v)))))) t2 H15)) t3 (sym_eq T t3 v H14))) u1 -(sym_eq T u1 w H13))) k (sym_eq K k (Flat Appl) H12))) H11)) H10)) H9 H6 -H7))) | (pr0_beta u0 v1 v2 H6 t3 t4 H7) \Rightarrow (\lambda (H8: (eq T -(THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) (THead (Flat Appl) w -v))).(\lambda (H9: (eq T (THead (Bind Abbr) v2 t4) t2)).((let H10 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow (THead (Bind Abst) u0 t3) | (TLRef _) \Rightarrow -(THead (Bind Abst) u0 t3) | (THead _ _ t5) \Rightarrow t5])) (THead (Flat -Appl) v1 (THead (Bind Abst) u0 t3)) (THead (Flat Appl) w v) H8) in ((let H11 -\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t5 _) -\Rightarrow t5])) (THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) (THead -(Flat Appl) w v) H8) in (eq_ind T w (\lambda (t5: T).((eq T (THead (Bind -Abst) u0 t3) v) \to ((eq T (THead (Bind Abbr) v2 t4) t2) \to ((pr0 t5 v2) \to -((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u -t0)))))))) (\lambda (H12: (eq T (THead (Bind Abst) u0 t3) v)).(eq_ind T -(THead (Bind Abst) u0 t3) (\lambda (_: T).((eq T (THead (Bind Abbr) v2 t4) -t2) \to ((pr0 w v2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w -(THead (Bind Abst) u t0))))))) (\lambda (H13: (eq T (THead (Bind Abbr) v2 t4) -t2)).(eq_ind T (THead (Bind Abbr) v2 t4) (\lambda (t5: T).((pr0 w v2) \to -((pr0 t3 t4) \to (ty3 g c2 t5 (THead (Flat Appl) w (THead (Bind Abst) u -t0)))))) (\lambda (H14: (pr0 w v2)).(\lambda (H15: (pr0 t3 t4)).(let H16 \def -(eq_ind_r T v (\lambda (t5: T).(\forall (c3: C).((wcpr0 c c3) \to (\forall -(t6: T).((pr0 t5 t6) \to (ty3 g c3 t6 (THead (Bind Abst) u t0))))))) H3 -(THead (Bind Abst) u0 t3) H12) in (let H17 \def (eq_ind_r T v (\lambda (t5: -T).(ty3 g c t5 (THead (Bind Abst) u t0))) H2 (THead (Bind Abst) u0 t3) H12) -in (ex_ind T (\lambda (t5: T).(ty3 g c2 (THead (Bind Abst) u t0) t5)) (ty3 g -c2 (THead (Bind Abbr) v2 t4) (THead (Flat Appl) w (THead (Bind Abst) u t0))) -(\lambda (x: T).(\lambda (H18: (ty3 g c2 (THead (Bind Abst) u t0) -x)).(ex3_2_ind T T (\lambda (t5: T).(\lambda (_: T).(pc3 c2 (THead (Bind -Abst) u t5) x))) (\lambda (_: T).(\lambda (t6: T).(ty3 g c2 u t6))) (\lambda -(t5: T).(\lambda (_: T).(ty3 g (CHead c2 (Bind Abst) u) t0 t5))) (ty3 g c2 -(THead (Bind Abbr) v2 t4) (THead (Flat Appl) w (THead (Bind Abst) u t0))) -(\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 c2 (THead (Bind Abst) u -x0) x)).(\lambda (H20: (ty3 g c2 u x1)).(\lambda (H21: (ty3 g (CHead c2 (Bind -Abst) u) t0 x0)).(ex3_2_ind T T (\lambda (t5: T).(\lambda (_: T).(pc3 c2 -(THead (Bind Abst) u0 t5) (THead (Bind Abst) u t0)))) (\lambda (_: -T).(\lambda (t6: T).(ty3 g c2 u0 t6))) (\lambda (t5: T).(\lambda (_: T).(ty3 -g (CHead c2 (Bind Abst) u0) t4 t5))) (ty3 g c2 (THead (Bind Abbr) v2 t4) -(THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x2: T).(\lambda -(x3: T).(\lambda (H22: (pc3 c2 (THead (Bind Abst) u0 x2) (THead (Bind Abst) u -t0))).(\lambda (H23: (ty3 g c2 u0 x3)).(\lambda (H24: (ty3 g (CHead c2 (Bind -Abst) u0) t4 x2)).(land_ind (pc3 c2 u0 u) (\forall (b: B).(\forall (u1: -T).(pc3 (CHead c2 (Bind b) u1) x2 t0))) (ty3 g c2 (THead (Bind Abbr) v2 t4) -(THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (H25: (pc3 c2 u0 -u)).(\lambda (H26: ((\forall (b: B).(\forall (u1: T).(pc3 (CHead c2 (Bind b) -u1) x2 t0))))).(ty3_conv g c2 (THead (Flat Appl) w (THead (Bind Abst) u t0)) -(THead (Flat Appl) w (THead (Bind Abst) u x0)) (ty3_appl g c2 w u (H1 c2 H4 w -(pr0_refl w)) (THead (Bind Abst) u t0) x0 (ty3_bind g c2 u x1 H20 Abst t0 x0 -H21)) (THead (Bind Abbr) v2 t4) (THead (Bind Abbr) v2 x2) (ty3_bind g c2 v2 u -(H1 c2 H4 v2 H14) Abbr t4 x2 (csubt_ty3_ld g c2 v2 u0 (ty3_conv g c2 u0 x3 -H23 v2 u (H1 c2 H4 v2 H14) (pc3_s c2 u u0 H25)) t4 x2 H24)) (pc3_t (THead -(Bind Abbr) v2 t0) c2 (THead (Bind Abbr) v2 x2) (pc3_head_2 c2 v2 x2 t0 (Bind -Abbr) (H26 Abbr v2)) (THead (Flat Appl) w (THead (Bind Abst) u t0)) -(pc3_pr2_x c2 (THead (Bind Abbr) v2 t0) (THead (Flat Appl) w (THead (Bind -Abst) u t0)) (pr2_free c2 (THead (Flat Appl) w (THead (Bind Abst) u t0)) -(THead (Bind Abbr) v2 t0) (pr0_beta u w v2 H14 t0 t0 (pr0_refl t0)))))))) -(pc3_gen_abst c2 u0 u x2 t0 H22))))))) (ty3_gen_bind g Abst c2 u0 t4 (THead -(Bind Abst) u t0) (H16 c2 H4 (THead (Bind Abst) u0 t4) (pr0_comp u0 u0 -(pr0_refl u0) t3 t4 H15 (Bind Abst)))))))))) (ty3_gen_bind g Abst c2 u t0 x -H18)))) (ty3_correct g c2 (THead (Bind Abst) u0 t3) (THead (Bind Abst) u t0) -(H16 c2 H4 (THead (Bind Abst) u0 t3) (pr0_refl (THead (Bind Abst) u0 -t3))))))))) t2 H13)) v H12)) v1 (sym_eq T v1 w H11))) H10)) H9 H6 H7))) | -(pr0_upsilon b H6 v1 v2 H7 u1 u2 H8 t3 t4 H9) \Rightarrow (\lambda (H10: (eq -T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) w -v))).(\lambda (H11: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O -v2) t4)) t2)).((let H12 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow (THead (Bind b) u1 t3) -| (TLRef _) \Rightarrow (THead (Bind b) u1 t3) | (THead _ _ t5) \Rightarrow -t5])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) w v) -H10) in ((let H13 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 -| (THead _ t5 _) \Rightarrow t5])) (THead (Flat Appl) v1 (THead (Bind b) u1 -t3)) (THead (Flat Appl) w v) H10) in (eq_ind T w (\lambda (t5: T).((eq T -(THead (Bind b) u1 t3) v) \to ((eq T (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4)) t2) \to ((not (eq B b Abst)) \to ((pr0 t5 v2) \to -((pr0 u1 u2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead -(Bind Abst) u t0)))))))))) (\lambda (H14: (eq T (THead (Bind b) u1 t3) -v)).(eq_ind T (THead (Bind b) u1 t3) (\lambda (_: T).((eq T (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t2) \to ((not (eq B b Abst)) \to -((pr0 w v2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat -Appl) w (THead (Bind Abst) u t0))))))))) (\lambda (H15: (eq T (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t2)).(eq_ind T (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t4)) (\lambda (t5: T).((not (eq B b -Abst)) \to ((pr0 w v2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ty3 g c2 t5 -(THead (Flat Appl) w (THead (Bind Abst) u t0)))))))) (\lambda (H16: (not (eq -B b Abst))).(\lambda (H17: (pr0 w v2)).(\lambda (H18: (pr0 u1 u2)).(\lambda -(H19: (pr0 t3 t4)).(let H20 \def (eq_ind_r T v (\lambda (t5: T).(\forall (c3: -C).((wcpr0 c c3) \to (\forall (t6: T).((pr0 t5 t6) \to (ty3 g c3 t6 (THead -(Bind Abst) u t0))))))) H3 (THead (Bind b) u1 t3) H14) in (let H21 \def -(eq_ind_r T v (\lambda (t5: T).(ty3 g c t5 (THead (Bind Abst) u t0))) H2 -(THead (Bind b) u1 t3) H14) in (ex_ind T (\lambda (t5: T).(ty3 g c2 (THead -(Bind Abst) u t0) t5)) (ty3 g c2 (THead (Bind b) u2 (THead (Flat Appl) (lift -(S O) O v2) t4)) (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x: -T).(\lambda (H22: (ty3 g c2 (THead (Bind Abst) u t0) x)).(let H23 \def H22 in -(ex3_2_ind T T (\lambda (t5: T).(\lambda (_: T).(pc3 c2 (THead (Bind Abst) u -t5) x))) (\lambda (_: T).(\lambda (t6: T).(ty3 g c2 u t6))) (\lambda (t5: -T).(\lambda (_: T).(ty3 g (CHead c2 (Bind Abst) u) t0 t5))) (ty3 g c2 (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) (THead (Flat Appl) w -(THead (Bind Abst) u t0))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: -(pc3 c2 (THead (Bind Abst) u x0) x)).(\lambda (H25: (ty3 g c2 u x1)).(\lambda -(H26: (ty3 g (CHead c2 (Bind Abst) u) t0 x0)).(ex3_2_ind T T (\lambda (t5: -T).(\lambda (_: T).(pc3 c2 (THead (Bind b) u2 t5) (THead (Bind Abst) u t0)))) -(\lambda (_: T).(\lambda (t6: T).(ty3 g c2 u2 t6))) (\lambda (t5: T).(\lambda -(_: T).(ty3 g (CHead c2 (Bind b) u2) t4 t5))) (ty3 g c2 (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t4)) (THead (Flat Appl) w (THead (Bind -Abst) u t0))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H27: (pc3 c2 (THead -(Bind b) u2 x2) (THead (Bind Abst) u t0))).(\lambda (H28: (ty3 g c2 u2 -x3)).(\lambda (H29: (ty3 g (CHead c2 (Bind b) u2) t4 x2)).(let H30 \def -(eq_ind T (lift (S O) O (THead (Bind Abst) u t0)) (\lambda (t5: T).(pc3 -(CHead c2 (Bind b) u2) x2 t5)) (pc3_gen_not_abst b H16 c2 x2 t0 u2 u H27) -(THead (Bind Abst) (lift (S O) O u) (lift (S O) (S O) t0)) (lift_bind Abst u -t0 (S O) O)) in (let H31 \def (eq_ind T (lift (S O) O (THead (Bind Abst) u -t0)) (\lambda (t5: T).(ty3 g (CHead c2 (Bind b) u2) t5 (lift (S O) O x))) -(ty3_lift g c2 (THead (Bind Abst) u t0) x H22 (CHead c2 (Bind b) u2) O (S O) -(drop_drop (Bind b) O c2 c2 (drop_refl c2) u2)) (THead (Bind Abst) (lift (S -O) O u) (lift (S O) (S O) t0)) (lift_bind Abst u t0 (S O) O)) in (ex3_2_ind T -T (\lambda (t5: T).(\lambda (_: T).(pc3 (CHead c2 (Bind b) u2) (THead (Bind -Abst) (lift (S O) O u) t5) (lift (S O) O x)))) (\lambda (_: T).(\lambda (t6: -T).(ty3 g (CHead c2 (Bind b) u2) (lift (S O) O u) t6))) (\lambda (t5: -T).(\lambda (_: T).(ty3 g (CHead (CHead c2 (Bind b) u2) (Bind Abst) (lift (S -O) O u)) (lift (S O) (S O) t0) t5))) (ty3 g c2 (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t4)) (THead (Flat Appl) w (THead (Bind Abst) u -t0))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (pc3 (CHead c2 (Bind b) -u2) (THead (Bind Abst) (lift (S O) O u) x4) (lift (S O) O x))).(\lambda (H33: -(ty3 g (CHead c2 (Bind b) u2) (lift (S O) O u) x5)).(\lambda (H34: (ty3 g -(CHead (CHead c2 (Bind b) u2) (Bind Abst) (lift (S O) O u)) (lift (S O) (S O) -t0) x4)).(ty3_conv g c2 (THead (Flat Appl) w (THead (Bind Abst) u t0)) (THead -(Flat Appl) w (THead (Bind Abst) u x0)) (ty3_appl g c2 w u (H1 c2 H4 w -(pr0_refl w)) (THead (Bind Abst) u t0) x0 (ty3_bind g c2 u x1 H25 Abst t0 x0 -H26)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (THead (Bind Abst) (lift (S -O) O u) (lift (S O) (S O) t0)))) (ty3_bind g c2 u2 x3 H28 b (THead (Flat -Appl) (lift (S O) O v2) t4) (THead (Flat Appl) (lift (S O) O v2) (THead (Bind -Abst) (lift (S O) O u) (lift (S O) (S O) t0))) (ty3_appl g (CHead c2 (Bind b) -u2) (lift (S O) O v2) (lift (S O) O u) (ty3_lift g c2 v2 u (H1 c2 H4 v2 H17) -(CHead c2 (Bind b) u2) O (S O) (drop_drop (Bind b) O c2 c2 (drop_refl c2) -u2)) t4 (lift (S O) (S O) t0) (ty3_conv g (CHead c2 (Bind b) u2) (THead (Bind -Abst) (lift (S O) O u) (lift (S O) (S O) t0)) (THead (Bind Abst) (lift (S O) -O u) x4) (ty3_bind g (CHead c2 (Bind b) u2) (lift (S O) O u) x5 H33 Abst -(lift (S O) (S O) t0) x4 H34) t4 x2 H29 H30))) (eq_ind T (lift (S O) O (THead -(Bind Abst) u t0)) (\lambda (t5: T).(pc3 c2 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t5)) (THead (Flat Appl) w (THead (Bind Abst) u t0)))) -(pc3_pc1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (lift (S O) -O (THead (Bind Abst) u t0)))) (THead (Flat Appl) w (THead (Bind Abst) u t0)) -(pc1_pr0_u2 (THead (Flat Appl) v2 (THead (Bind b) u2 (lift (S O) O (THead -(Bind Abst) u t0)))) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) -(lift (S O) O (THead (Bind Abst) u t0)))) (pr0_upsilon b H16 v2 v2 (pr0_refl -v2) u2 u2 (pr0_refl u2) (lift (S O) O (THead (Bind Abst) u t0)) (lift (S O) O -(THead (Bind Abst) u t0)) (pr0_refl (lift (S O) O (THead (Bind Abst) u t0)))) -(THead (Flat Appl) w (THead (Bind Abst) u t0)) (pc1_head v2 w (pc1_pr0_x v2 w -H17) (THead (Bind b) u2 (lift (S O) O (THead (Bind Abst) u t0))) (THead (Bind -Abst) u t0) (pc1_pr0_r (THead (Bind b) u2 (lift (S O) O (THead (Bind Abst) u -t0))) (THead (Bind Abst) u t0) (pr0_zeta b H16 (THead (Bind Abst) u t0) -(THead (Bind Abst) u t0) (pr0_refl (THead (Bind Abst) u t0)) u2)) (Flat -Appl))) c2) (THead (Bind Abst) (lift (S O) O u) (lift (S O) (S O) t0)) -(lift_bind Abst u t0 (S O) O)))))))) (ty3_gen_bind g Abst (CHead c2 (Bind b) -u2) (lift (S O) O u) (lift (S O) (S O) t0) (lift (S O) O x) H31))))))))) -(ty3_gen_bind g b c2 u2 t4 (THead (Bind Abst) u t0) (H20 c2 H4 (THead (Bind -b) u2 t4) (pr0_comp u1 u2 H18 t3 t4 H19 (Bind b)))))))))) (ty3_gen_bind g -Abst c2 u t0 x H23))))) (ty3_correct g c2 (THead (Bind b) u2 t4) (THead (Bind -Abst) u t0) (H20 c2 H4 (THead (Bind b) u2 t4) (pr0_comp u1 u2 H18 t3 t4 H19 -(Bind b))))))))))) t2 H15)) v H14)) v1 (sym_eq T v1 w H13))) H12)) H11 H6 H7 -H8 H9))) | (pr0_delta u1 u2 H6 t3 t4 H7 w0 H8) \Rightarrow (\lambda (H9: (eq -T (THead (Bind Abbr) u1 t3) (THead (Flat Appl) w v))).(\lambda (H10: (eq T -(THead (Bind Abbr) u2 w0) t2)).((let H11 \def (eq_ind T (THead (Bind Abbr) u1 -t3) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort -_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) -\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) w v) -H9) in (False_ind ((eq T (THead (Bind Abbr) u2 w0) t2) \to ((pr0 u1 u2) \to -((pr0 t3 t4) \to ((subst0 O u2 t4 w0) \to (ty3 g c2 t2 (THead (Flat Appl) w -(THead (Bind Abst) u t0))))))) H11)) H10 H6 H7 H8))) | (pr0_zeta b H6 t3 t4 -H7 u0) \Rightarrow (\lambda (H8: (eq T (THead (Bind b) u0 (lift (S O) O t3)) -(THead (Flat Appl) w v))).(\lambda (H9: (eq T t4 t2)).((let H10 \def (eq_ind -T (THead (Bind b) u0 (lift (S O) O t3)) (\lambda (e: T).(match e in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow -False])])) I (THead (Flat Appl) w v) H8) in (False_ind ((eq T t4 t2) \to -((not (eq B b Abst)) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w -(THead (Bind Abst) u t0)))))) H10)) H9 H6 H7))) | (pr0_tau t3 t4 H6 u0) -\Rightarrow (\lambda (H7: (eq T (THead (Flat Cast) u0 t3) (THead (Flat Appl) -w v))).(\lambda (H8: (eq T t4 t2)).((let H9 \def (eq_ind T (THead (Flat Cast) -u0 t3) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) -\Rightarrow False | (Flat f) \Rightarrow (match f in F return (\lambda (_: -F).Prop) with [Appl \Rightarrow False | Cast \Rightarrow True])])])) I (THead -(Flat Appl) w v) H7) in (False_ind ((eq T t4 t2) \to ((pr0 t3 t4) \to (ty3 g -c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u t0))))) H9)) H8 H6)))]) in -(H6 (refl_equal T (THead (Flat Appl) w v)) (refl_equal T t2)))))))))))))))) -(\lambda (c: C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (ty3 g c t2 -t3)).(\lambda (H1: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t4: T).((pr0 -t2 t4) \to (ty3 g c2 t4 t3))))))).(\lambda (t0: T).(\lambda (_: (ty3 g c t3 -t0)).(\lambda (H3: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t4: T).((pr0 -t3 t4) \to (ty3 g c2 t4 t0))))))).(\lambda (c2: C).(\lambda (H4: (wcpr0 c -c2)).(\lambda (t4: T).(\lambda (H5: (pr0 (THead (Flat Cast) t3 t2) t4)).(let -H6 \def (match H5 in pr0 return (\lambda (t5: T).(\lambda (t6: T).(\lambda -(_: (pr0 t5 t6)).((eq T t5 (THead (Flat Cast) t3 t2)) \to ((eq T t6 t4) \to -(ty3 g c2 t4 (THead (Flat Cast) t0 t3))))))) with [(pr0_refl t5) \Rightarrow -(\lambda (H6: (eq T t5 (THead (Flat Cast) t3 t2))).(\lambda (H7: (eq T t5 -t4)).(eq_ind T (THead (Flat Cast) t3 t2) (\lambda (t6: T).((eq T t6 t4) \to -(ty3 g c2 t4 (THead (Flat Cast) t0 t3)))) (\lambda (H8: (eq T (THead (Flat -Cast) t3 t2) t4)).(eq_ind T (THead (Flat Cast) t3 t2) (\lambda (t6: T).(ty3 g -c2 t6 (THead (Flat Cast) t0 t3))) (ty3_cast g c2 t2 t3 (H1 c2 H4 t2 (pr0_refl -t2)) t0 (H3 c2 H4 t3 (pr0_refl t3))) t4 H8)) t5 (sym_eq T t5 (THead (Flat -Cast) t3 t2) H6) H7))) | (pr0_comp u1 u2 H6 t5 t6 H7 k) \Rightarrow (\lambda -(H8: (eq T (THead k u1 t5) (THead (Flat Cast) t3 t2))).(\lambda (H9: (eq T -(THead k u2 t6) t4)).((let H10 \def (f_equal T T (\lambda (e: T).(match e in -T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t5 | (TLRef _) -\Rightarrow t5 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t5) (THead -(Flat Cast) t3 t2) H8) in ((let H11 \def (f_equal T T (\lambda (e: T).(match -e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) -\Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t5) (THead -(Flat Cast) t3 t2) H8) in ((let H12 \def (f_equal T K (\lambda (e: T).(match -e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) -\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t5) (THead (Flat -Cast) t3 t2) H8) in (eq_ind K (Flat Cast) (\lambda (k0: K).((eq T u1 t3) \to -((eq T t5 t2) \to ((eq T (THead k0 u2 t6) t4) \to ((pr0 u1 u2) \to ((pr0 t5 -t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3)))))))) (\lambda (H13: (eq T u1 -t3)).(eq_ind T t3 (\lambda (t7: T).((eq T t5 t2) \to ((eq T (THead (Flat -Cast) u2 t6) t4) \to ((pr0 t7 u2) \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead -(Flat Cast) t0 t3))))))) (\lambda (H14: (eq T t5 t2)).(eq_ind T t2 (\lambda -(t7: T).((eq T (THead (Flat Cast) u2 t6) t4) \to ((pr0 t3 u2) \to ((pr0 t7 -t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3)))))) (\lambda (H15: (eq T -(THead (Flat Cast) u2 t6) t4)).(eq_ind T (THead (Flat Cast) u2 t6) (\lambda -(t7: T).((pr0 t3 u2) \to ((pr0 t2 t6) \to (ty3 g c2 t7 (THead (Flat Cast) t0 -t3))))) (\lambda (H16: (pr0 t3 u2)).(\lambda (H17: (pr0 t2 t6)).(ex_ind T -(\lambda (t7: T).(ty3 g c2 t0 t7)) (ty3 g c2 (THead (Flat Cast) u2 t6) (THead -(Flat Cast) t0 t3)) (\lambda (x: T).(\lambda (H18: (ty3 g c2 t0 x)).(ty3_conv -g c2 (THead (Flat Cast) t0 t3) (THead (Flat Cast) x t0) (ty3_cast g c2 t3 t0 -(H3 c2 H4 t3 (pr0_refl t3)) x H18) (THead (Flat Cast) u2 t6) (THead (Flat -Cast) t0 u2) (ty3_cast g c2 t6 u2 (ty3_conv g c2 u2 t0 (H3 c2 H4 u2 H16) t6 -t3 (H1 c2 H4 t6 H17) (pc3_pr2_r c2 t3 u2 (pr2_free c2 t3 u2 H16))) t0 (H3 c2 -H4 u2 H16)) (pc3_s c2 (THead (Flat Cast) t0 u2) (THead (Flat Cast) t0 t3) -(pc3_pr2_r c2 (THead (Flat Cast) t0 t3) (THead (Flat Cast) t0 u2) -(pr2_thin_dx c2 t3 u2 (pr2_free c2 t3 u2 H16) t0 Cast)))))) (ty3_correct g c2 -t3 t0 (H3 c2 H4 t3 (pr0_refl t3)))))) t4 H15)) t5 (sym_eq T t5 t2 H14))) u1 -(sym_eq T u1 t3 H13))) k (sym_eq K k (Flat Cast) H12))) H11)) H10)) H9 H6 -H7))) | (pr0_beta u v1 v2 H6 t5 t6 H7) \Rightarrow (\lambda (H8: (eq T (THead -(Flat Appl) v1 (THead (Bind Abst) u t5)) (THead (Flat Cast) t3 t2))).(\lambda -(H9: (eq T (THead (Bind Abbr) v2 t6) t4)).((let H10 \def (eq_ind T (THead -(Flat Appl) v1 (THead (Bind Abst) u t5)) (\lambda (e: T).(match e in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f -in F return (\lambda (_: F).Prop) with [Appl \Rightarrow True | Cast -\Rightarrow False])])])) I (THead (Flat Cast) t3 t2) H8) in (False_ind ((eq T -(THead (Bind Abbr) v2 t6) t4) \to ((pr0 v1 v2) \to ((pr0 t5 t6) \to (ty3 g c2 -t4 (THead (Flat Cast) t0 t3))))) H10)) H9 H6 H7))) | (pr0_upsilon b H6 v1 v2 -H7 u1 u2 H8 t5 t6 H9) \Rightarrow (\lambda (H10: (eq T (THead (Flat Appl) v1 -(THead (Bind b) u1 t5)) (THead (Flat Cast) t3 t2))).(\lambda (H11: (eq T -(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t6)) t4)).((let H12 -\def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t5)) (\lambda (e: -T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K -return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) -\Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl \Rightarrow -True | Cast \Rightarrow False])])])) I (THead (Flat Cast) t3 t2) H10) in -(False_ind ((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) -t6)) t4) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 -t5 t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3))))))) H12)) H11 H6 H7 H8 -H9))) | (pr0_delta u1 u2 H6 t5 t6 H7 w H8) \Rightarrow (\lambda (H9: (eq T -(THead (Bind Abbr) u1 t5) (THead (Flat Cast) t3 t2))).(\lambda (H10: (eq T -(THead (Bind Abbr) u2 w) t4)).((let H11 \def (eq_ind T (THead (Bind Abbr) u1 -t5) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort -_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) -\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Cast) t3 -t2) H9) in (False_ind ((eq T (THead (Bind Abbr) u2 w) t4) \to ((pr0 u1 u2) -\to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ty3 g c2 t4 (THead (Flat Cast) -t0 t3)))))) H11)) H10 H6 H7 H8))) | (pr0_zeta b H6 t5 t6 H7 u) \Rightarrow -(\lambda (H8: (eq T (THead (Bind b) u (lift (S O) O t5)) (THead (Flat Cast) -t3 t2))).(\lambda (H9: (eq T t6 t4)).((let H10 \def (eq_ind T (THead (Bind b) -u (lift (S O) O t5)) (\lambda (e: T).(match e in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat -Cast) t3 t2) H8) in (False_ind ((eq T t6 t4) \to ((not (eq B b Abst)) \to -((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3))))) H10)) H9 H6 H7))) -| (pr0_tau t5 t6 H6 u) \Rightarrow (\lambda (H7: (eq T (THead (Flat Cast) u -t5) (THead (Flat Cast) t3 t2))).(\lambda (H8: (eq T t6 t4)).((let H9 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow t5 | (TLRef _) \Rightarrow t5 | (THead _ _ t7) -\Rightarrow t7])) (THead (Flat Cast) u t5) (THead (Flat Cast) t3 t2) H7) in -((let H10 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: -T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t7 -_) \Rightarrow t7])) (THead (Flat Cast) u t5) (THead (Flat Cast) t3 t2) H7) -in (eq_ind T t3 (\lambda (_: T).((eq T t5 t2) \to ((eq T t6 t4) \to ((pr0 t5 -t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3)))))) (\lambda (H11: (eq T t5 -t2)).(eq_ind T t2 (\lambda (t7: T).((eq T t6 t4) \to ((pr0 t7 t6) \to (ty3 g -c2 t4 (THead (Flat Cast) t0 t3))))) (\lambda (H12: (eq T t6 t4)).(eq_ind T t4 -(\lambda (t7: T).((pr0 t2 t7) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3)))) -(\lambda (H13: (pr0 t2 t4)).(ex_ind T (\lambda (t7: T).(ty3 g c2 t0 t7)) (ty3 -g c2 t4 (THead (Flat Cast) t0 t3)) (\lambda (x: T).(\lambda (H14: (ty3 g c2 -t0 x)).(ty3_conv g c2 (THead (Flat Cast) t0 t3) (THead (Flat Cast) x t0) -(ty3_cast g c2 t3 t0 (H3 c2 H4 t3 (pr0_refl t3)) x H14) t4 t3 (H1 c2 H4 t4 -H13) (pc3_pr2_x c2 t3 (THead (Flat Cast) t0 t3) (pr2_free c2 (THead (Flat -Cast) t0 t3) t3 (pr0_tau t3 t3 (pr0_refl t3) t0)))))) (ty3_correct g c2 t3 t0 -(H3 c2 H4 t3 (pr0_refl t3))))) t6 (sym_eq T t6 t4 H12))) t5 (sym_eq T t5 t2 -H11))) u (sym_eq T u t3 H10))) H9)) H8 H6)))]) in (H6 (refl_equal T (THead -(Flat Cast) t3 t2)) (refl_equal T t4))))))))))))))) c1 t1 t H))))). -(* COMMENTS -Initial nodes: 14710 -END *) - -theorem ty3_sred_pr0: - \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (g: G).(\forall -(c: C).(\forall (t: T).((ty3 g c t1 t) \to (ty3 g c t2 t))))))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(\lambda (g: -G).(\lambda (c: C).(\lambda (t: T).(\lambda (H0: (ty3 g c t1 -t)).(ty3_sred_wcpr0_pr0 g c t1 t H0 c (wcpr0_refl c) t2 H))))))). -(* COMMENTS -Initial nodes: 47 -END *) - -theorem ty3_sred_pr1: - \forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall (g: G).(\forall -(c: C).(\forall (t: T).((ty3 g c t1 t) \to (ty3 g c t2 t))))))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr1 t1 t2)).(pr1_ind (\lambda -(t: T).(\lambda (t0: T).(\forall (g: G).(\forall (c: C).(\forall (t3: -T).((ty3 g c t t3) \to (ty3 g c t0 t3))))))) (\lambda (t: T).(\lambda (g: -G).(\lambda (c: C).(\lambda (t0: T).(\lambda (H0: (ty3 g c t t0)).H0))))) -(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t4 t3)).(\lambda (t5: -T).(\lambda (_: (pr1 t3 t5)).(\lambda (H2: ((\forall (g: G).(\forall (c: -C).(\forall (t: T).((ty3 g c t3 t) \to (ty3 g c t5 t))))))).(\lambda (g: -G).(\lambda (c: C).(\lambda (t: T).(\lambda (H3: (ty3 g c t4 t)).(H2 g c t -(ty3_sred_pr0 t4 t3 H0 g c t H3)))))))))))) t1 t2 H))). -(* COMMENTS -Initial nodes: 151 -END *) - -theorem ty3_sred_pr2: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall -(g: G).(\forall (t: T).((ty3 g c t1 t) \to (ty3 g c t2 t))))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 -t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (g: -G).(\forall (t3: T).((ty3 g c0 t t3) \to (ty3 g c0 t0 t3))))))) (\lambda (c0: -C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(\lambda (g: -G).(\lambda (t: T).(\lambda (H1: (ty3 g c0 t3 t)).(ty3_sred_wcpr0_pr0 g c0 t3 -t H1 c0 (wcpr0_refl c0) t4 H0)))))))) (\lambda (c0: C).(\lambda (d: -C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind -Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 -t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (g: -G).(\lambda (t0: T).(\lambda (H3: (ty3 g c0 t3 t0)).(ty3_subst0 g c0 t4 t0 -(ty3_sred_wcpr0_pr0 g c0 t3 t0 H3 c0 (wcpr0_refl c0) t4 H1) d u i H0 t -H2)))))))))))))) c t1 t2 H)))). -(* COMMENTS -Initial nodes: 205 -END *) - -theorem ty3_sred_pr3: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall -(g: G).(\forall (t: T).((ty3 g c t1 t) \to (ty3 g c t2 t))))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 -t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (g: G).(\forall -(t3: T).((ty3 g c t t3) \to (ty3 g c t0 t3)))))) (\lambda (t: T).(\lambda (g: -G).(\lambda (t0: T).(\lambda (H0: (ty3 g c t t0)).H0)))) (\lambda (t3: -T).(\lambda (t4: T).(\lambda (H0: (pr2 c t4 t3)).(\lambda (t5: T).(\lambda -(_: (pr3 c t3 t5)).(\lambda (H2: ((\forall (g: G).(\forall (t: T).((ty3 g c -t3 t) \to (ty3 g c t5 t)))))).(\lambda (g: G).(\lambda (t: T).(\lambda (H3: -(ty3 g c t4 t)).(H2 g t (ty3_sred_pr2 c t4 t3 H0 g t H3))))))))))) t1 t2 -H)))). -(* COMMENTS -Initial nodes: 151 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/pr3_props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/pr3_props.ma deleted file mode 100644 index b5bb1fb69..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/pr3_props.ma +++ /dev/null @@ -1,513 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/ty3/pr3.ma". - -theorem ty3_cred_pr2: - \forall (g: G).(\forall (c: C).(\forall (v1: T).(\forall (v2: T).((pr2 c v1 -v2) \to (\forall (b: B).(\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c -(Bind b) v1) t1 t2) \to (ty3 g (CHead c (Bind b) v2) t1 t2))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda -(H: (pr2 c v1 v2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: -T).(\forall (b: B).(\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c0 (Bind -b) t) t1 t2) \to (ty3 g (CHead c0 (Bind b) t0) t1 t2)))))))) (\lambda (c0: -C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr0 t1 t2)).(\lambda (b: -B).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H1: (ty3 g (CHead c0 (Bind b) -t1) t0 t3)).(ty3_sred_wcpr0_pr0 g (CHead c0 (Bind b) t1) t0 t3 H1 (CHead c0 -(Bind b) t2) (wcpr0_comp c0 c0 (wcpr0_refl c0) t1 t2 H0 (Bind b)) t0 -(pr0_refl t0)))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: -T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr) -u))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H1: (pr0 t1 t2)).(\lambda -(t: T).(\lambda (H2: (subst0 i u t2 t)).(\lambda (b: B).(\lambda (t0: -T).(\lambda (t3: T).(\lambda (H3: (ty3 g (CHead c0 (Bind b) t1) t0 -t3)).(ty3_csubst0 g (CHead c0 (Bind b) t2) t0 t3 (ty3_sred_wcpr0_pr0 g (CHead -c0 (Bind b) t1) t0 t3 H3 (CHead c0 (Bind b) t2) (wcpr0_comp c0 c0 (wcpr0_refl -c0) t1 t2 H1 (Bind b)) t0 (pr0_refl t0)) d u (S i) (getl_clear_bind b (CHead -c0 (Bind b) t2) c0 t2 (clear_bind b c0 t2) (CHead d (Bind Abbr) u) i H0) -(CHead c0 (Bind b) t) (csubst0_snd_bind b i u t2 t H2 c0)))))))))))))))) c v1 -v2 H))))). -(* COMMENTS -Initial nodes: 383 -END *) - -theorem ty3_cred_pr3: - \forall (g: G).(\forall (c: C).(\forall (v1: T).(\forall (v2: T).((pr3 c v1 -v2) \to (\forall (b: B).(\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c -(Bind b) v1) t1 t2) \to (ty3 g (CHead c (Bind b) v2) t1 t2))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda -(H: (pr3 c v1 v2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (b: -B).(\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c (Bind b) t) t1 t2) \to -(ty3 g (CHead c (Bind b) t0) t1 t2))))))) (\lambda (t: T).(\lambda (b: -B).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (ty3 g (CHead c (Bind b) -t) t1 t2)).H0))))) (\lambda (t2: T).(\lambda (t1: T).(\lambda (H0: (pr2 c t1 -t2)).(\lambda (t3: T).(\lambda (_: (pr3 c t2 t3)).(\lambda (H2: ((\forall (b: -B).(\forall (t4: T).(\forall (t5: T).((ty3 g (CHead c (Bind b) t2) t4 t5) \to -(ty3 g (CHead c (Bind b) t3) t4 t5))))))).(\lambda (b: B).(\lambda (t0: -T).(\lambda (t4: T).(\lambda (H3: (ty3 g (CHead c (Bind b) t1) t0 t4)).(H2 b -t0 t4 (ty3_cred_pr2 g c t1 t2 H0 b t0 t4 H3)))))))))))) v1 v2 H))))). -(* COMMENTS -Initial nodes: 215 -END *) - -theorem ty3_gen_lift: - \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (x: T).(\forall (h: -nat).(\forall (d: nat).((ty3 g c (lift h d t1) x) \to (\forall (e: C).((drop -h d c e) \to (ex2 T (\lambda (t2: T).(pc3 c (lift h d t2) x)) (\lambda (t2: -T).(ty3 g e t1 t2))))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (x: T).(\lambda (h: -nat).(\lambda (d: nat).(\lambda (H: (ty3 g c (lift h d t1) x)).(insert_eq T -(lift h d t1) (\lambda (t: T).(ty3 g c t x)) (\lambda (_: T).(\forall (e: -C).((drop h d c e) \to (ex2 T (\lambda (t2: T).(pc3 c (lift h d t2) x)) -(\lambda (t2: T).(ty3 g e t1 t2)))))) (\lambda (y: T).(\lambda (H0: (ty3 g c -y x)).(unintro nat d (\lambda (n: nat).((eq T y (lift h n t1)) \to (\forall -(e: C).((drop h n c e) \to (ex2 T (\lambda (t2: T).(pc3 c (lift h n t2) x)) -(\lambda (t2: T).(ty3 g e t1 t2))))))) (unintro T t1 (\lambda (t: T).(\forall -(x0: nat).((eq T y (lift h x0 t)) \to (\forall (e: C).((drop h x0 c e) \to -(ex2 T (\lambda (t2: T).(pc3 c (lift h x0 t2) x)) (\lambda (t2: T).(ty3 g e t -t2)))))))) (ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: -T).(\forall (x0: T).(\forall (x1: nat).((eq T t (lift h x1 x0)) \to (\forall -(e: C).((drop h x1 c0 e) \to (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) -t0)) (\lambda (t2: T).(ty3 g e x0 t2))))))))))) (\lambda (c0: C).(\lambda -(t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda (_: ((\forall -(x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to (\forall (e: -C).((drop h x1 c0 e) \to (ex2 T (\lambda (t3: T).(pc3 c0 (lift h x1 t3) t)) -(\lambda (t3: T).(ty3 g e x0 t3)))))))))).(\lambda (u: T).(\lambda (t3: -T).(\lambda (H3: (ty3 g c0 u t3)).(\lambda (H4: ((\forall (x0: T).(\forall -(x1: nat).((eq T u (lift h x1 x0)) \to (\forall (e: C).((drop h x1 c0 e) \to -(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) t3)) (\lambda (t4: T).(ty3 g e -x0 t4)))))))))).(\lambda (H5: (pc3 c0 t3 t2)).(\lambda (x0: T).(\lambda (x1: -nat).(\lambda (H6: (eq T u (lift h x1 x0))).(\lambda (e: C).(\lambda (H7: -(drop h x1 c0 e)).(let H8 \def (eq_ind T u (\lambda (t0: T).(\forall (x2: -T).(\forall (x3: nat).((eq T t0 (lift h x3 x2)) \to (\forall (e0: C).((drop h -x3 c0 e0) \to (ex2 T (\lambda (t4: T).(pc3 c0 (lift h x3 t4) t3)) (\lambda -(t4: T).(ty3 g e0 x2 t4))))))))) H4 (lift h x1 x0) H6) in (let H9 \def -(eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 t3)) H3 (lift h x1 x0) H6) in (let -H10 \def (H8 x0 x1 (refl_equal T (lift h x1 x0)) e H7) in (ex2_ind T (\lambda -(t4: T).(pc3 c0 (lift h x1 t4) t3)) (\lambda (t4: T).(ty3 g e x0 t4)) (ex2 T -(\lambda (t4: T).(pc3 c0 (lift h x1 t4) t2)) (\lambda (t4: T).(ty3 g e x0 -t4))) (\lambda (x2: T).(\lambda (H11: (pc3 c0 (lift h x1 x2) t3)).(\lambda -(H12: (ty3 g e x0 x2)).(ex_intro2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) -t2)) (\lambda (t4: T).(ty3 g e x0 t4)) x2 (pc3_t t3 c0 (lift h x1 x2) H11 t2 -H5) H12)))) H10))))))))))))))))))) (\lambda (c0: C).(\lambda (m: -nat).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H1: (eq T (TSort m) (lift -h x1 x0))).(\lambda (e: C).(\lambda (_: (drop h x1 c0 e)).(eq_ind_r T (TSort -m) (\lambda (t: T).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (TSort -(next g m)))) (\lambda (t2: T).(ty3 g e t t2)))) (ex_intro2 T (\lambda (t2: -T).(pc3 c0 (lift h x1 t2) (TSort (next g m)))) (\lambda (t2: T).(ty3 g e -(TSort m) t2)) (TSort (next g m)) (eq_ind_r T (TSort (next g m)) (\lambda (t: -T).(pc3 c0 t (TSort (next g m)))) (pc3_refl c0 (TSort (next g m))) (lift h x1 -(TSort (next g m))) (lift_sort (next g m) h x1)) (ty3_sort g e m)) x0 -(lift_gen_sort h x1 m x0 H1))))))))) (\lambda (n: nat).(\lambda (c0: -C).(\lambda (d0: C).(\lambda (u: T).(\lambda (H1: (getl n c0 (CHead d0 (Bind -Abbr) u))).(\lambda (t: T).(\lambda (H2: (ty3 g d0 u t)).(\lambda (H3: -((\forall (x0: T).(\forall (x1: nat).((eq T u (lift h x1 x0)) \to (\forall -(e: C).((drop h x1 d0 e) \to (ex2 T (\lambda (t2: T).(pc3 d0 (lift h x1 t2) -t)) (\lambda (t2: T).(ty3 g e x0 t2)))))))))).(\lambda (x0: T).(\lambda (x1: -nat).(\lambda (H4: (eq T (TLRef n) (lift h x1 x0))).(\lambda (e: C).(\lambda -(H5: (drop h x1 c0 e)).(let H_x \def (lift_gen_lref x0 x1 h n H4) in (let H6 -\def H_x in (or_ind (land (lt n x1) (eq T x0 (TLRef n))) (land (le (plus x1 -h) n) (eq T x0 (TLRef (minus n h)))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h -x1 t2) (lift (S n) O t))) (\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (H7: -(land (lt n x1) (eq T x0 (TLRef n)))).(land_ind (lt n x1) (eq T x0 (TLRef n)) -(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t))) (\lambda -(t2: T).(ty3 g e x0 t2))) (\lambda (H8: (lt n x1)).(\lambda (H9: (eq T x0 -(TLRef n))).(eq_ind_r T (TLRef n) (\lambda (t0: T).(ex2 T (\lambda (t2: -T).(pc3 c0 (lift h x1 t2) (lift (S n) O t))) (\lambda (t2: T).(ty3 g e t0 -t2)))) (let H10 \def (eq_ind nat x1 (\lambda (n0: nat).(drop h n0 c0 e)) H5 -(S (plus n (minus x1 (S n)))) (lt_plus_minus n x1 H8)) in (ex3_2_ind T C -(\lambda (v: T).(\lambda (_: C).(eq T u (lift h (minus x1 (S n)) v)))) -(\lambda (v: T).(\lambda (e0: C).(getl n e (CHead e0 (Bind Abbr) v)))) -(\lambda (_: T).(\lambda (e0: C).(drop h (minus x1 (S n)) d0 e0))) (ex2 T -(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t))) (\lambda (t2: -T).(ty3 g e (TLRef n) t2))) (\lambda (x2: T).(\lambda (x3: C).(\lambda (H11: -(eq T u (lift h (minus x1 (S n)) x2))).(\lambda (H12: (getl n e (CHead x3 -(Bind Abbr) x2))).(\lambda (H13: (drop h (minus x1 (S n)) d0 x3)).(let H14 -\def (eq_ind T u (\lambda (t0: T).(\forall (x4: T).(\forall (x5: nat).((eq T -t0 (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 d0 e0) \to (ex2 T -(\lambda (t2: T).(pc3 d0 (lift h x5 t2) t)) (\lambda (t2: T).(ty3 g e0 x4 -t2))))))))) H3 (lift h (minus x1 (S n)) x2) H11) in (let H15 \def (eq_ind T u -(\lambda (t0: T).(ty3 g d0 t0 t)) H2 (lift h (minus x1 (S n)) x2) H11) in -(let H16 \def (H14 x2 (minus x1 (S n)) (refl_equal T (lift h (minus x1 (S n)) -x2)) x3 H13) in (ex2_ind T (\lambda (t2: T).(pc3 d0 (lift h (minus x1 (S n)) -t2) t)) (\lambda (t2: T).(ty3 g x3 x2 t2)) (ex2 T (\lambda (t2: T).(pc3 c0 -(lift h x1 t2) (lift (S n) O t))) (\lambda (t2: T).(ty3 g e (TLRef n) t2))) -(\lambda (x4: T).(\lambda (H17: (pc3 d0 (lift h (minus x1 (S n)) x4) -t)).(\lambda (H18: (ty3 g x3 x2 x4)).(eq_ind_r nat (plus (S n) (minus x1 (S -n))) (\lambda (n0: nat).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h n0 t2) (lift -(S n) O t))) (\lambda (t2: T).(ty3 g e (TLRef n) t2)))) (ex_intro2 T (\lambda -(t2: T).(pc3 c0 (lift h (plus (S n) (minus x1 (S n))) t2) (lift (S n) O t))) -(\lambda (t2: T).(ty3 g e (TLRef n) t2)) (lift (S n) O x4) (eq_ind_r T (lift -(S n) O (lift h (minus x1 (S n)) x4)) (\lambda (t0: T).(pc3 c0 t0 (lift (S n) -O t))) (pc3_lift c0 d0 (S n) O (getl_drop Abbr c0 d0 u n H1) (lift h (minus -x1 (S n)) x4) t H17) (lift h (plus (S n) (minus x1 (S n))) (lift (S n) O x4)) -(lift_d x4 h (S n) (minus x1 (S n)) O (le_O_n (minus x1 (S n))))) (ty3_abbr g -n e x3 x2 H12 x4 H18)) x1 (le_plus_minus (S n) x1 H8))))) H16))))))))) -(getl_drop_conf_lt Abbr c0 d0 u n H1 e h (minus x1 (S n)) H10))) x0 H9))) -H7)) (\lambda (H7: (land (le (plus x1 h) n) (eq T x0 (TLRef (minus n -h))))).(land_ind (le (plus x1 h) n) (eq T x0 (TLRef (minus n h))) (ex2 T -(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t))) (\lambda (t2: -T).(ty3 g e x0 t2))) (\lambda (H8: (le (plus x1 h) n)).(\lambda (H9: (eq T x0 -(TLRef (minus n h)))).(eq_ind_r T (TLRef (minus n h)) (\lambda (t0: T).(ex2 T -(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t))) (\lambda (t2: -T).(ty3 g e t0 t2)))) (ex_intro2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) -(lift (S n) O t))) (\lambda (t2: T).(ty3 g e (TLRef (minus n h)) t2)) (lift -(S (minus n h)) O t) (eq_ind_r T (lift (plus h (S (minus n h))) O t) (\lambda -(t0: T).(pc3 c0 t0 (lift (S n) O t))) (eq_ind nat (S (plus h (minus n h))) -(\lambda (n0: nat).(pc3 c0 (lift n0 O t) (lift (S n) O t))) (eq_ind nat n -(\lambda (n0: nat).(pc3 c0 (lift (S n0) O t) (lift (S n) O t))) (pc3_refl c0 -(lift (S n) O t)) (plus h (minus n h)) (le_plus_minus h n (le_trans h (plus -x1 h) n (le_plus_r x1 h) H8))) (plus h (S (minus n h))) (plus_n_Sm h (minus n -h))) (lift h x1 (lift (S (minus n h)) O t)) (lift_free t (S (minus n h)) h O -x1 (le_trans x1 (S (minus n h)) (plus O (S (minus n h))) (le_S_minus x1 h n -H8) (le_n (plus O (S (minus n h))))) (le_O_n x1))) (ty3_abbr g (minus n h) e -d0 u (getl_drop_conf_ge n (CHead d0 (Bind Abbr) u) c0 H1 e h x1 H5 H8) t H2)) -x0 H9))) H7)) H6)))))))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda -(d0: C).(\lambda (u: T).(\lambda (H1: (getl n c0 (CHead d0 (Bind Abst) -u))).(\lambda (t: T).(\lambda (H2: (ty3 g d0 u t)).(\lambda (H3: ((\forall -(x0: T).(\forall (x1: nat).((eq T u (lift h x1 x0)) \to (\forall (e: -C).((drop h x1 d0 e) \to (ex2 T (\lambda (t2: T).(pc3 d0 (lift h x1 t2) t)) -(\lambda (t2: T).(ty3 g e x0 t2)))))))))).(\lambda (x0: T).(\lambda (x1: -nat).(\lambda (H4: (eq T (TLRef n) (lift h x1 x0))).(\lambda (e: C).(\lambda -(H5: (drop h x1 c0 e)).(let H_x \def (lift_gen_lref x0 x1 h n H4) in (let H6 -\def H_x in (or_ind (land (lt n x1) (eq T x0 (TLRef n))) (land (le (plus x1 -h) n) (eq T x0 (TLRef (minus n h)))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h -x1 t2) (lift (S n) O u))) (\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (H7: -(land (lt n x1) (eq T x0 (TLRef n)))).(land_ind (lt n x1) (eq T x0 (TLRef n)) -(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O u))) (\lambda -(t2: T).(ty3 g e x0 t2))) (\lambda (H8: (lt n x1)).(\lambda (H9: (eq T x0 -(TLRef n))).(eq_ind_r T (TLRef n) (\lambda (t0: T).(ex2 T (\lambda (t2: -T).(pc3 c0 (lift h x1 t2) (lift (S n) O u))) (\lambda (t2: T).(ty3 g e t0 -t2)))) (let H10 \def (eq_ind nat x1 (\lambda (n0: nat).(drop h n0 c0 e)) H5 -(S (plus n (minus x1 (S n)))) (lt_plus_minus n x1 H8)) in (ex3_2_ind T C -(\lambda (v: T).(\lambda (_: C).(eq T u (lift h (minus x1 (S n)) v)))) -(\lambda (v: T).(\lambda (e0: C).(getl n e (CHead e0 (Bind Abst) v)))) -(\lambda (_: T).(\lambda (e0: C).(drop h (minus x1 (S n)) d0 e0))) (ex2 T -(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O u))) (\lambda (t2: -T).(ty3 g e (TLRef n) t2))) (\lambda (x2: T).(\lambda (x3: C).(\lambda (H11: -(eq T u (lift h (minus x1 (S n)) x2))).(\lambda (H12: (getl n e (CHead x3 -(Bind Abst) x2))).(\lambda (H13: (drop h (minus x1 (S n)) d0 x3)).(let H14 -\def (eq_ind T u (\lambda (t0: T).(\forall (x4: T).(\forall (x5: nat).((eq T -t0 (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 d0 e0) \to (ex2 T -(\lambda (t2: T).(pc3 d0 (lift h x5 t2) t)) (\lambda (t2: T).(ty3 g e0 x4 -t2))))))))) H3 (lift h (minus x1 (S n)) x2) H11) in (let H15 \def (eq_ind T u -(\lambda (t0: T).(ty3 g d0 t0 t)) H2 (lift h (minus x1 (S n)) x2) H11) in -(eq_ind_r T (lift h (minus x1 (S n)) x2) (\lambda (t0: T).(ex2 T (\lambda -(t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t0))) (\lambda (t2: T).(ty3 g e -(TLRef n) t2)))) (let H16 \def (H14 x2 (minus x1 (S n)) (refl_equal T (lift h -(minus x1 (S n)) x2)) x3 H13) in (ex2_ind T (\lambda (t2: T).(pc3 d0 (lift h -(minus x1 (S n)) t2) t)) (\lambda (t2: T).(ty3 g x3 x2 t2)) (ex2 T (\lambda -(t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O (lift h (minus x1 (S n)) x2)))) -(\lambda (t2: T).(ty3 g e (TLRef n) t2))) (\lambda (x4: T).(\lambda (_: (pc3 -d0 (lift h (minus x1 (S n)) x4) t)).(\lambda (H18: (ty3 g x3 x2 -x4)).(eq_ind_r nat (plus (S n) (minus x1 (S n))) (\lambda (n0: nat).(ex2 T -(\lambda (t2: T).(pc3 c0 (lift h n0 t2) (lift (S n) O (lift h (minus n0 (S -n)) x2)))) (\lambda (t2: T).(ty3 g e (TLRef n) t2)))) (ex_intro2 T (\lambda -(t2: T).(pc3 c0 (lift h (plus (S n) (minus x1 (S n))) t2) (lift (S n) O (lift -h (minus (plus (S n) (minus x1 (S n))) (S n)) x2)))) (\lambda (t2: T).(ty3 g -e (TLRef n) t2)) (lift (S n) O x2) (eq_ind_r T (lift (S n) O (lift h (minus -x1 (S n)) x2)) (\lambda (t0: T).(pc3 c0 t0 (lift (S n) O (lift h (minus (plus -(S n) (minus x1 (S n))) (S n)) x2)))) (eq_ind nat x1 (\lambda (n0: nat).(pc3 -c0 (lift (S n) O (lift h (minus x1 (S n)) x2)) (lift (S n) O (lift h (minus -n0 (S n)) x2)))) (pc3_refl c0 (lift (S n) O (lift h (minus x1 (S n)) x2))) -(plus (S n) (minus x1 (S n))) (le_plus_minus (S n) x1 H8)) (lift h (plus (S -n) (minus x1 (S n))) (lift (S n) O x2)) (lift_d x2 h (S n) (minus x1 (S n)) O -(le_O_n (minus x1 (S n))))) (ty3_abst g n e x3 x2 H12 x4 H18)) x1 -(le_plus_minus (S n) x1 H8))))) H16)) u H11)))))))) (getl_drop_conf_lt Abst -c0 d0 u n H1 e h (minus x1 (S n)) H10))) x0 H9))) H7)) (\lambda (H7: (land -(le (plus x1 h) n) (eq T x0 (TLRef (minus n h))))).(land_ind (le (plus x1 h) -n) (eq T x0 (TLRef (minus n h))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 -t2) (lift (S n) O u))) (\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (H8: (le -(plus x1 h) n)).(\lambda (H9: (eq T x0 (TLRef (minus n h)))).(eq_ind_r T -(TLRef (minus n h)) (\lambda (t0: T).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h -x1 t2) (lift (S n) O u))) (\lambda (t2: T).(ty3 g e t0 t2)))) (ex_intro2 T -(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O u))) (\lambda (t2: -T).(ty3 g e (TLRef (minus n h)) t2)) (lift (S (minus n h)) O u) (eq_ind_r T -(lift (plus h (S (minus n h))) O u) (\lambda (t0: T).(pc3 c0 t0 (lift (S n) O -u))) (eq_ind nat (S (plus h (minus n h))) (\lambda (n0: nat).(pc3 c0 (lift n0 -O u) (lift (S n) O u))) (eq_ind nat n (\lambda (n0: nat).(pc3 c0 (lift (S n0) -O u) (lift (S n) O u))) (pc3_refl c0 (lift (S n) O u)) (plus h (minus n h)) -(le_plus_minus h n (le_trans h (plus x1 h) n (le_plus_r x1 h) H8))) (plus h -(S (minus n h))) (plus_n_Sm h (minus n h))) (lift h x1 (lift (S (minus n h)) -O u)) (lift_free u (S (minus n h)) h O x1 (le_trans x1 (S (minus n h)) (plus -O (S (minus n h))) (le_S_minus x1 h n H8) (le_n (plus O (S (minus n h))))) -(le_O_n x1))) (ty3_abst g (minus n h) e d0 u (getl_drop_conf_ge n (CHead d0 -(Bind Abst) u) c0 H1 e h x1 H5 H8) t H2)) x0 H9))) H7)) H6)))))))))))))))) -(\lambda (c0: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H1: (ty3 g c0 u -t)).(\lambda (H2: ((\forall (x0: T).(\forall (x1: nat).((eq T u (lift h x1 -x0)) \to (\forall (e: C).((drop h x1 c0 e) \to (ex2 T (\lambda (t2: T).(pc3 -c0 (lift h x1 t2) t)) (\lambda (t2: T).(ty3 g e x0 t2)))))))))).(\lambda (b: -B).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H3: (ty3 g (CHead c0 (Bind b) -u) t2 t3)).(\lambda (H4: ((\forall (x0: T).(\forall (x1: nat).((eq T t2 (lift -h x1 x0)) \to (\forall (e: C).((drop h x1 (CHead c0 (Bind b) u) e) \to (ex2 T -(\lambda (t4: T).(pc3 (CHead c0 (Bind b) u) (lift h x1 t4) t3)) (\lambda (t4: -T).(ty3 g e x0 t4)))))))))).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H5: -(eq T (THead (Bind b) u t2) (lift h x1 x0))).(\lambda (e: C).(\lambda (H6: -(drop h x1 c0 e)).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 -(THead (Bind b) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u (lift h x1 -y0)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h (S x1) z)))) (ex2 T -(\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b) u t3))) (\lambda (t4: -T).(ty3 g e x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq T x0 -(THead (Bind b) x2 x3))).(\lambda (H8: (eq T u (lift h x1 x2))).(\lambda (H9: -(eq T t2 (lift h (S x1) x3))).(eq_ind_r T (THead (Bind b) x2 x3) (\lambda -(t0: T).(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b) u -t3))) (\lambda (t4: T).(ty3 g e t0 t4)))) (let H10 \def (eq_ind T t2 (\lambda -(t0: T).(\forall (x4: T).(\forall (x5: nat).((eq T t0 (lift h x5 x4)) \to -(\forall (e0: C).((drop h x5 (CHead c0 (Bind b) u) e0) \to (ex2 T (\lambda -(t4: T).(pc3 (CHead c0 (Bind b) u) (lift h x5 t4) t3)) (\lambda (t4: T).(ty3 -g e0 x4 t4))))))))) H4 (lift h (S x1) x3) H9) in (let H11 \def (eq_ind T t2 -(\lambda (t0: T).(ty3 g (CHead c0 (Bind b) u) t0 t3)) H3 (lift h (S x1) x3) -H9) in (let H12 \def (eq_ind T u (\lambda (t0: T).(ty3 g (CHead c0 (Bind b) -t0) (lift h (S x1) x3) t3)) H11 (lift h x1 x2) H8) in (let H13 \def (eq_ind T -u (\lambda (t0: T).(\forall (x4: T).(\forall (x5: nat).((eq T (lift h (S x1) -x3) (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 (CHead c0 (Bind b) t0) -e0) \to (ex2 T (\lambda (t4: T).(pc3 (CHead c0 (Bind b) t0) (lift h x5 t4) -t3)) (\lambda (t4: T).(ty3 g e0 x4 t4))))))))) H10 (lift h x1 x2) H8) in (let -H14 \def (eq_ind T u (\lambda (t0: T).(\forall (x4: T).(\forall (x5: -nat).((eq T t0 (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 c0 e0) \to -(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x5 t4) t)) (\lambda (t4: T).(ty3 g e0 -x4 t4))))))))) H2 (lift h x1 x2) H8) in (let H15 \def (eq_ind T u (\lambda -(t0: T).(ty3 g c0 t0 t)) H1 (lift h x1 x2) H8) in (eq_ind_r T (lift h x1 x2) -(\lambda (t0: T).(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind -b) t0 t3))) (\lambda (t4: T).(ty3 g e (THead (Bind b) x2 x3) t4)))) (let H16 -\def (H14 x2 x1 (refl_equal T (lift h x1 x2)) e H6) in (ex2_ind T (\lambda -(t4: T).(pc3 c0 (lift h x1 t4) t)) (\lambda (t4: T).(ty3 g e x2 t4)) (ex2 T -(\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b) (lift h x1 x2) t3))) -(\lambda (t4: T).(ty3 g e (THead (Bind b) x2 x3) t4))) (\lambda (x4: -T).(\lambda (_: (pc3 c0 (lift h x1 x4) t)).(\lambda (H18: (ty3 g e x2 -x4)).(let H19 \def (H13 x3 (S x1) (refl_equal T (lift h (S x1) x3)) (CHead e -(Bind b) x2) (drop_skip_bind h x1 c0 e H6 b x2)) in (ex2_ind T (\lambda (t4: -T).(pc3 (CHead c0 (Bind b) (lift h x1 x2)) (lift h (S x1) t4) t3)) (\lambda -(t4: T).(ty3 g (CHead e (Bind b) x2) x3 t4)) (ex2 T (\lambda (t4: T).(pc3 c0 -(lift h x1 t4) (THead (Bind b) (lift h x1 x2) t3))) (\lambda (t4: T).(ty3 g e -(THead (Bind b) x2 x3) t4))) (\lambda (x5: T).(\lambda (H20: (pc3 (CHead c0 -(Bind b) (lift h x1 x2)) (lift h (S x1) x5) t3)).(\lambda (H21: (ty3 g (CHead -e (Bind b) x2) x3 x5)).(ex_ind T (\lambda (t0: T).(ty3 g (CHead e (Bind b) -x2) x5 t0)) (ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b) -(lift h x1 x2) t3))) (\lambda (t4: T).(ty3 g e (THead (Bind b) x2 x3) t4))) -(\lambda (x6: T).(\lambda (_: (ty3 g (CHead e (Bind b) x2) x5 x6)).(ex_intro2 -T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b) (lift h x1 x2) -t3))) (\lambda (t4: T).(ty3 g e (THead (Bind b) x2 x3) t4)) (THead (Bind b) -x2 x5) (eq_ind_r T (THead (Bind b) (lift h x1 x2) (lift h (S x1) x5)) -(\lambda (t0: T).(pc3 c0 t0 (THead (Bind b) (lift h x1 x2) t3))) (pc3_head_2 -c0 (lift h x1 x2) (lift h (S x1) x5) t3 (Bind b) H20) (lift h x1 (THead (Bind -b) x2 x5)) (lift_bind b x2 x5 h x1)) (ty3_bind g e x2 x4 H18 b x3 x5 H21)))) -(ty3_correct g (CHead e (Bind b) x2) x3 x5 H21))))) H19))))) H16)) u -H8))))))) x0 H7)))))) (lift_gen_bind b u t2 x0 h x1 H5))))))))))))))))) -(\lambda (c0: C).(\lambda (w: T).(\lambda (u: T).(\lambda (H1: (ty3 g c0 w -u)).(\lambda (H2: ((\forall (x0: T).(\forall (x1: nat).((eq T w (lift h x1 -x0)) \to (\forall (e: C).((drop h x1 c0 e) \to (ex2 T (\lambda (t2: T).(pc3 -c0 (lift h x1 t2) u)) (\lambda (t2: T).(ty3 g e x0 t2)))))))))).(\lambda (v: -T).(\lambda (t: T).(\lambda (H3: (ty3 g c0 v (THead (Bind Abst) u -t))).(\lambda (H4: ((\forall (x0: T).(\forall (x1: nat).((eq T v (lift h x1 -x0)) \to (\forall (e: C).((drop h x1 c0 e) \to (ex2 T (\lambda (t2: T).(pc3 -c0 (lift h x1 t2) (THead (Bind Abst) u t))) (\lambda (t2: T).(ty3 g e x0 -t2)))))))))).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H5: (eq T (THead -(Flat Appl) w v) (lift h x1 x0))).(\lambda (e: C).(\lambda (H6: (drop h x1 c0 -e)).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Flat -Appl) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T w (lift h x1 y0)))) -(\lambda (_: T).(\lambda (z: T).(eq T v (lift h x1 z)))) (ex2 T (\lambda (t2: -T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) w (THead (Bind Abst) u t)))) -(\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (x2: T).(\lambda (x3: T).(\lambda -(H7: (eq T x0 (THead (Flat Appl) x2 x3))).(\lambda (H8: (eq T w (lift h x1 -x2))).(\lambda (H9: (eq T v (lift h x1 x3))).(eq_ind_r T (THead (Flat Appl) -x2 x3) (\lambda (t0: T).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead -(Flat Appl) w (THead (Bind Abst) u t)))) (\lambda (t2: T).(ty3 g e t0 t2)))) -(let H10 \def (eq_ind T v (\lambda (t0: T).(\forall (x4: T).(\forall (x5: -nat).((eq T t0 (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 c0 e0) \to -(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x5 t2) (THead (Bind Abst) u t))) -(\lambda (t2: T).(ty3 g e0 x4 t2))))))))) H4 (lift h x1 x3) H9) in (let H11 -\def (eq_ind T v (\lambda (t0: T).(ty3 g c0 t0 (THead (Bind Abst) u t))) H3 -(lift h x1 x3) H9) in (let H12 \def (eq_ind T w (\lambda (t0: T).(\forall -(x4: T).(\forall (x5: nat).((eq T t0 (lift h x5 x4)) \to (\forall (e0: -C).((drop h x5 c0 e0) \to (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x5 t2) u)) -(\lambda (t2: T).(ty3 g e0 x4 t2))))))))) H2 (lift h x1 x2) H8) in (let H13 -\def (eq_ind T w (\lambda (t0: T).(ty3 g c0 t0 u)) H1 (lift h x1 x2) H8) in -(eq_ind_r T (lift h x1 x2) (\lambda (t0: T).(ex2 T (\lambda (t2: T).(pc3 c0 -(lift h x1 t2) (THead (Flat Appl) t0 (THead (Bind Abst) u t)))) (\lambda (t2: -T).(ty3 g e (THead (Flat Appl) x2 x3) t2)))) (let H14 \def (H12 x2 x1 -(refl_equal T (lift h x1 x2)) e H6) in (ex2_ind T (\lambda (t2: T).(pc3 c0 -(lift h x1 t2) u)) (\lambda (t2: T).(ty3 g e x2 t2)) (ex2 T (\lambda (t2: -T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) (lift h x1 x2) (THead (Bind -Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead (Flat Appl) x2 x3) t2))) -(\lambda (x4: T).(\lambda (H15: (pc3 c0 (lift h x1 x4) u)).(\lambda (H16: -(ty3 g e x2 x4)).(let H17 \def (H10 x3 x1 (refl_equal T (lift h x1 x3)) e H6) -in (ex2_ind T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead (Bind Abst) u -t))) (\lambda (t2: T).(ty3 g e x3 t2)) (ex2 T (\lambda (t2: T).(pc3 c0 (lift -h x1 t2) (THead (Flat Appl) (lift h x1 x2) (THead (Bind Abst) u t)))) -(\lambda (t2: T).(ty3 g e (THead (Flat Appl) x2 x3) t2))) (\lambda (x5: -T).(\lambda (H18: (pc3 c0 (lift h x1 x5) (THead (Bind Abst) u t))).(\lambda -(H19: (ty3 g e x3 x5)).(ex3_2_ind T T (\lambda (u1: T).(\lambda (t2: T).(pr3 -e x5 (THead (Bind Abst) u1 t2)))) (\lambda (u1: T).(\lambda (_: T).(pr3 c0 u -(lift h x1 u1)))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall -(u0: T).(pr3 (CHead c0 (Bind b) u0) t (lift h (S x1) t2)))))) (ex2 T (\lambda -(t2: T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) (lift h x1 x2) (THead (Bind -Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead (Flat Appl) x2 x3) t2))) -(\lambda (x6: T).(\lambda (x7: T).(\lambda (H20: (pr3 e x5 (THead (Bind Abst) -x6 x7))).(\lambda (H21: (pr3 c0 u (lift h x1 x6))).(\lambda (H22: ((\forall -(b: B).(\forall (u0: T).(pr3 (CHead c0 (Bind b) u0) t (lift h (S x1) -x7)))))).(ex_ind T (\lambda (t0: T).(ty3 g e x5 t0)) (ex2 T (\lambda (t2: -T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) (lift h x1 x2) (THead (Bind -Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead (Flat Appl) x2 x3) t2))) -(\lambda (x8: T).(\lambda (H23: (ty3 g e x5 x8)).(let H_y \def (ty3_sred_pr3 -e x5 (THead (Bind Abst) x6 x7) H20 g x8 H23) in (ex3_2_ind T T (\lambda (t2: -T).(\lambda (_: T).(pc3 e (THead (Bind Abst) x6 t2) x8))) (\lambda (_: -T).(\lambda (t0: T).(ty3 g e x6 t0))) (\lambda (t2: T).(\lambda (_: T).(ty3 g -(CHead e (Bind Abst) x6) x7 t2))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 -t2) (THead (Flat Appl) (lift h x1 x2) (THead (Bind Abst) u t)))) (\lambda -(t2: T).(ty3 g e (THead (Flat Appl) x2 x3) t2))) (\lambda (x9: T).(\lambda -(x10: T).(\lambda (_: (pc3 e (THead (Bind Abst) x6 x9) x8)).(\lambda (H25: -(ty3 g e x6 x10)).(\lambda (H26: (ty3 g (CHead e (Bind Abst) x6) x7 -x9)).(ex_intro2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) -(lift h x1 x2) (THead (Bind Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead -(Flat Appl) x2 x3) t2)) (THead (Flat Appl) x2 (THead (Bind Abst) x6 x7)) -(eq_ind_r T (THead (Flat Appl) (lift h x1 x2) (lift h x1 (THead (Bind Abst) -x6 x7))) (\lambda (t0: T).(pc3 c0 t0 (THead (Flat Appl) (lift h x1 x2) (THead -(Bind Abst) u t)))) (pc3_thin_dx c0 (lift h x1 (THead (Bind Abst) x6 x7)) -(THead (Bind Abst) u t) (eq_ind_r T (THead (Bind Abst) (lift h x1 x6) (lift h -(S x1) x7)) (\lambda (t0: T).(pc3 c0 t0 (THead (Bind Abst) u t))) -(pc3_head_21 c0 (lift h x1 x6) u (pc3_pr3_x c0 (lift h x1 x6) u H21) (Bind -Abst) (lift h (S x1) x7) t (pc3_pr3_x (CHead c0 (Bind Abst) (lift h x1 x6)) -(lift h (S x1) x7) t (H22 Abst (lift h x1 x6)))) (lift h x1 (THead (Bind -Abst) x6 x7)) (lift_bind Abst x6 x7 h x1)) (lift h x1 x2) Appl) (lift h x1 -(THead (Flat Appl) x2 (THead (Bind Abst) x6 x7))) (lift_flat Appl x2 (THead -(Bind Abst) x6 x7) h x1)) (ty3_appl g e x2 x6 (ty3_conv g e x6 x10 H25 x2 x4 -H16 (pc3_gen_lift c0 x4 x6 h x1 (pc3_t u c0 (lift h x1 x4) H15 (lift h x1 x6) -(pc3_pr3_r c0 u (lift h x1 x6) H21)) e H6)) x3 x7 (ty3_conv g e (THead (Bind -Abst) x6 x7) (THead (Bind Abst) x6 x9) (ty3_bind g e x6 x10 H25 Abst x7 x9 -H26) x3 x5 H19 (pc3_pr3_r e x5 (THead (Bind Abst) x6 x7) H20))))))))) -(ty3_gen_bind g Abst e x6 x7 x8 H_y))))) (ty3_correct g e x3 x5 H19))))))) -(pc3_gen_lift_abst c0 x5 t u h x1 H18 e H6))))) H17))))) H14)) w H8))))) x0 -H7)))))) (lift_gen_flat Appl w v x0 h x1 H5)))))))))))))))) (\lambda (c0: -C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (ty3 g c0 t2 t3)).(\lambda -(H2: ((\forall (x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to -(\forall (e: C).((drop h x1 c0 e) \to (ex2 T (\lambda (t4: T).(pc3 c0 (lift h -x1 t4) t3)) (\lambda (t4: T).(ty3 g e x0 t4)))))))))).(\lambda (t0: -T).(\lambda (H3: (ty3 g c0 t3 t0)).(\lambda (H4: ((\forall (x0: T).(\forall -(x1: nat).((eq T t3 (lift h x1 x0)) \to (\forall (e: C).((drop h x1 c0 e) \to -(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) t0)) (\lambda (t4: T).(ty3 g e -x0 t4)))))))))).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H5: (eq T -(THead (Flat Cast) t3 t2) (lift h x1 x0))).(\lambda (e: C).(\lambda (H6: -(drop h x1 c0 e)).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 -(THead (Flat Cast) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T t3 (lift h -x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h x1 z)))) (ex2 T -(\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Flat Cast) t0 t3))) (\lambda -(t4: T).(ty3 g e x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq -T x0 (THead (Flat Cast) x2 x3))).(\lambda (H8: (eq T t3 (lift h x1 -x2))).(\lambda (H9: (eq T t2 (lift h x1 x3))).(eq_ind_r T (THead (Flat Cast) -x2 x3) (\lambda (t: T).(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead -(Flat Cast) t0 t3))) (\lambda (t4: T).(ty3 g e t t4)))) (let H10 \def (eq_ind -T t3 (\lambda (t: T).(\forall (x4: T).(\forall (x5: nat).((eq T t (lift h x5 -x4)) \to (\forall (e0: C).((drop h x5 c0 e0) \to (ex2 T (\lambda (t4: T).(pc3 -c0 (lift h x5 t4) t0)) (\lambda (t4: T).(ty3 g e0 x4 t4))))))))) H4 (lift h -x1 x2) H8) in (let H11 \def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 t t0)) H3 -(lift h x1 x2) H8) in (let H12 \def (eq_ind T t3 (\lambda (t: T).(\forall -(x4: T).(\forall (x5: nat).((eq T t2 (lift h x5 x4)) \to (\forall (e0: -C).((drop h x5 c0 e0) \to (ex2 T (\lambda (t4: T).(pc3 c0 (lift h x5 t4) t)) -(\lambda (t4: T).(ty3 g e0 x4 t4))))))))) H2 (lift h x1 x2) H8) in (let H13 -\def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 t2 t)) H1 (lift h x1 x2) H8) in -(eq_ind_r T (lift h x1 x2) (\lambda (t: T).(ex2 T (\lambda (t4: T).(pc3 c0 -(lift h x1 t4) (THead (Flat Cast) t0 t))) (\lambda (t4: T).(ty3 g e (THead -(Flat Cast) x2 x3) t4)))) (let H14 \def (eq_ind T t2 (\lambda (t: T).(ty3 g -c0 t (lift h x1 x2))) H13 (lift h x1 x3) H9) in (let H15 \def (eq_ind T t2 -(\lambda (t: T).(\forall (x4: T).(\forall (x5: nat).((eq T t (lift h x5 x4)) -\to (\forall (e0: C).((drop h x5 c0 e0) \to (ex2 T (\lambda (t4: T).(pc3 c0 -(lift h x5 t4) (lift h x1 x2))) (\lambda (t4: T).(ty3 g e0 x4 t4))))))))) H12 -(lift h x1 x3) H9) in (let H16 \def (H15 x3 x1 (refl_equal T (lift h x1 x3)) -e H6) in (ex2_ind T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (lift h x1 x2))) -(\lambda (t4: T).(ty3 g e x3 t4)) (ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 -t4) (THead (Flat Cast) t0 (lift h x1 x2)))) (\lambda (t4: T).(ty3 g e (THead -(Flat Cast) x2 x3) t4))) (\lambda (x4: T).(\lambda (H17: (pc3 c0 (lift h x1 -x4) (lift h x1 x2))).(\lambda (H18: (ty3 g e x3 x4)).(let H19 \def (H10 x2 x1 -(refl_equal T (lift h x1 x2)) e H6) in (ex2_ind T (\lambda (t4: T).(pc3 c0 -(lift h x1 t4) t0)) (\lambda (t4: T).(ty3 g e x2 t4)) (ex2 T (\lambda (t4: -T).(pc3 c0 (lift h x1 t4) (THead (Flat Cast) t0 (lift h x1 x2)))) (\lambda -(t4: T).(ty3 g e (THead (Flat Cast) x2 x3) t4))) (\lambda (x5: T).(\lambda -(H20: (pc3 c0 (lift h x1 x5) t0)).(\lambda (H21: (ty3 g e x2 x5)).(ex_intro2 -T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Flat Cast) t0 (lift h x1 -x2)))) (\lambda (t4: T).(ty3 g e (THead (Flat Cast) x2 x3) t4)) (THead (Flat -Cast) x5 x2) (eq_ind_r T (THead (Flat Cast) (lift h x1 x5) (lift h x1 x2)) -(\lambda (t: T).(pc3 c0 t (THead (Flat Cast) t0 (lift h x1 x2)))) (pc3_head_1 -c0 (lift h x1 x5) t0 H20 (Flat Cast) (lift h x1 x2)) (lift h x1 (THead (Flat -Cast) x5 x2)) (lift_flat Cast x5 x2 h x1)) (ty3_cast g e x3 x2 (ty3_conv g e -x2 x5 H21 x3 x4 H18 (pc3_gen_lift c0 x4 x2 h x1 H17 e H6)) x5 H21))))) -H19))))) H16)))) t3 H8))))) x0 H7)))))) (lift_gen_flat Cast t3 t2 x0 h x1 -H5))))))))))))))) c y x H0))))) H))))))). -(* COMMENTS -Initial nodes: 9781 -END *) - -theorem ty3_tred: - \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).((ty3 g c u -t1) \to (\forall (t2: T).((pr3 c t1 t2) \to (ty3 g c u t2))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (H: -(ty3 g c u t1)).(\lambda (t2: T).(\lambda (H0: (pr3 c t1 t2)).(ex_ind T -(\lambda (t: T).(ty3 g c t1 t)) (ty3 g c u t2) (\lambda (x: T).(\lambda (H1: -(ty3 g c t1 x)).(let H_y \def (ty3_sred_pr3 c t1 t2 H0 g x H1) in (ty3_conv g -c t2 x H_y u t1 H (pc3_pr3_r c t1 t2 H0))))) (ty3_correct g c u t1 H)))))))). -(* COMMENTS -Initial nodes: 121 -END *) - -theorem ty3_sconv_pc3: - \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (t1: T).((ty3 g c -u1 t1) \to (\forall (u2: T).(\forall (t2: T).((ty3 g c u2 t2) \to ((pc3 c u1 -u2) \to (pc3 c t1 t2))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda -(H: (ty3 g c u1 t1)).(\lambda (u2: T).(\lambda (t2: T).(\lambda (H0: (ty3 g c -u2 t2)).(\lambda (H1: (pc3 c u1 u2)).(let H2 \def H1 in (ex2_ind T (\lambda -(t: T).(pr3 c u1 t)) (\lambda (t: T).(pr3 c u2 t)) (pc3 c t1 t2) (\lambda (x: -T).(\lambda (H3: (pr3 c u1 x)).(\lambda (H4: (pr3 c u2 x)).(let H_y \def -(ty3_sred_pr3 c u2 x H4 g t2 H0) in (let H_y0 \def (ty3_sred_pr3 c u1 x H3 g -t1 H) in (ty3_unique g c x t1 H_y0 t2 H_y)))))) H2)))))))))). -(* COMMENTS -Initial nodes: 141 -END *) - -theorem ty3_sred_back: - \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t0: T).((ty3 g c -t1 t0) \to (\forall (t2: T).((pr3 c t1 t2) \to (\forall (t: T).((ty3 g c t2 -t) \to (ty3 g c t1 t))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t0: T).(\lambda -(H: (ty3 g c t1 t0)).(\lambda (t2: T).(\lambda (H0: (pr3 c t1 t2)).(\lambda -(t: T).(\lambda (H1: (ty3 g c t2 t)).(ex_ind T (\lambda (t3: T).(ty3 g c t -t3)) (ty3 g c t1 t) (\lambda (x: T).(\lambda (H2: (ty3 g c t x)).(ty3_conv g -c t x H2 t1 t0 H (ty3_unique g c t2 t0 (ty3_sred_pr3 c t1 t2 H0 g t0 H) t -H1)))) (ty3_correct g c t2 t H1)))))))))). -(* COMMENTS -Initial nodes: 137 -END *) - -theorem ty3_sconv: - \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (t1: T).((ty3 g c -u1 t1) \to (\forall (u2: T).(\forall (t2: T).((ty3 g c u2 t2) \to ((pc3 c u1 -u2) \to (ty3 g c u1 t2))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda -(H: (ty3 g c u1 t1)).(\lambda (u2: T).(\lambda (t2: T).(\lambda (H0: (ty3 g c -u2 t2)).(\lambda (H1: (pc3 c u1 u2)).(let H2 \def H1 in (ex2_ind T (\lambda -(t: T).(pr3 c u1 t)) (\lambda (t: T).(pr3 c u2 t)) (ty3 g c u1 t2) (\lambda -(x: T).(\lambda (H3: (pr3 c u1 x)).(\lambda (H4: (pr3 c u2 x)).(ty3_sred_back -g c u1 t1 H x H3 t2 (ty3_sred_pr3 c u2 x H4 g t2 H0))))) H2)))))))))). -(* COMMENTS -Initial nodes: 129 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/props.ma deleted file mode 100644 index 579a8dd69..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/props.ma +++ /dev/null @@ -1,691 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/ty3/fwd.ma". - -include "Basic-1/pc3/fwd.ma". - -theorem ty3_lift: - \forall (g: G).(\forall (e: C).(\forall (t1: T).(\forall (t2: T).((ty3 g e -t1 t2) \to (\forall (c: C).(\forall (d: nat).(\forall (h: nat).((drop h d c -e) \to (ty3 g c (lift h d t1) (lift h d t2)))))))))) -\def - \lambda (g: G).(\lambda (e: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H: (ty3 g e t1 t2)).(ty3_ind g (\lambda (c: C).(\lambda (t: T).(\lambda (t0: -T).(\forall (c0: C).(\forall (d: nat).(\forall (h: nat).((drop h d c0 c) \to -(ty3 g c0 (lift h d t) (lift h d t0))))))))) (\lambda (c: C).(\lambda (t0: -T).(\lambda (t: T).(\lambda (_: (ty3 g c t0 t)).(\lambda (H1: ((\forall (c0: -C).(\forall (d: nat).(\forall (h: nat).((drop h d c0 c) \to (ty3 g c0 (lift h -d t0) (lift h d t)))))))).(\lambda (u: T).(\lambda (t3: T).(\lambda (_: (ty3 -g c u t3)).(\lambda (H3: ((\forall (c0: C).(\forall (d: nat).(\forall (h: -nat).((drop h d c0 c) \to (ty3 g c0 (lift h d u) (lift h d -t3)))))))).(\lambda (H4: (pc3 c t3 t0)).(\lambda (c0: C).(\lambda (d: -nat).(\lambda (h: nat).(\lambda (H5: (drop h d c0 c)).(ty3_conv g c0 (lift h -d t0) (lift h d t) (H1 c0 d h H5) (lift h d u) (lift h d t3) (H3 c0 d h H5) -(pc3_lift c0 c h d H5 t3 t0 H4)))))))))))))))) (\lambda (c: C).(\lambda (m: -nat).(\lambda (c0: C).(\lambda (d: nat).(\lambda (h: nat).(\lambda (_: (drop -h d c0 c)).(eq_ind_r T (TSort m) (\lambda (t: T).(ty3 g c0 t (lift h d (TSort -(next g m))))) (eq_ind_r T (TSort (next g m)) (\lambda (t: T).(ty3 g c0 -(TSort m) t)) (ty3_sort g c0 m) (lift h d (TSort (next g m))) (lift_sort -(next g m) h d)) (lift h d (TSort m)) (lift_sort m h d)))))))) (\lambda (n: -nat).(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c -(CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u -t)).(\lambda (H2: ((\forall (c0: C).(\forall (d0: nat).(\forall (h: -nat).((drop h d0 c0 d) \to (ty3 g c0 (lift h d0 u) (lift h d0 -t)))))))).(\lambda (c0: C).(\lambda (d0: nat).(\lambda (h: nat).(\lambda (H3: -(drop h d0 c0 c)).(lt_le_e n d0 (ty3 g c0 (lift h d0 (TLRef n)) (lift h d0 -(lift (S n) O t))) (\lambda (H4: (lt n d0)).(let H5 \def (drop_getl_trans_le -n d0 (le_S_n n d0 (le_S (S n) d0 H4)) c0 c h H3 (CHead d (Bind Abbr) u) H0) -in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop n O c0 e0))) -(\lambda (e0: C).(\lambda (e1: C).(drop h (minus d0 n) e0 e1))) (\lambda (_: -C).(\lambda (e1: C).(clear e1 (CHead d (Bind Abbr) u)))) (ty3 g c0 (lift h d0 -(TLRef n)) (lift h d0 (lift (S n) O t))) (\lambda (x0: C).(\lambda (x1: -C).(\lambda (H6: (drop n O c0 x0)).(\lambda (H7: (drop h (minus d0 n) x0 -x1)).(\lambda (H8: (clear x1 (CHead d (Bind Abbr) u))).(let H9 \def (eq_ind -nat (minus d0 n) (\lambda (n0: nat).(drop h n0 x0 x1)) H7 (S (minus d0 (S -n))) (minus_x_Sy d0 n H4)) in (let H10 \def (drop_clear_S x1 x0 h (minus d0 -(S n)) H9 Abbr d u H8) in (ex2_ind C (\lambda (c1: C).(clear x0 (CHead c1 -(Bind Abbr) (lift h (minus d0 (S n)) u)))) (\lambda (c1: C).(drop h (minus d0 -(S n)) c1 d)) (ty3 g c0 (lift h d0 (TLRef n)) (lift h d0 (lift (S n) O t))) -(\lambda (x: C).(\lambda (H11: (clear x0 (CHead x (Bind Abbr) (lift h (minus -d0 (S n)) u)))).(\lambda (H12: (drop h (minus d0 (S n)) x d)).(eq_ind_r T -(TLRef n) (\lambda (t0: T).(ty3 g c0 t0 (lift h d0 (lift (S n) O t)))) -(eq_ind nat (plus (S n) (minus d0 (S n))) (\lambda (n0: nat).(ty3 g c0 (TLRef -n) (lift h n0 (lift (S n) O t)))) (eq_ind_r T (lift (S n) O (lift h (minus d0 -(S n)) t)) (\lambda (t0: T).(ty3 g c0 (TLRef n) t0)) (eq_ind nat d0 (\lambda -(_: nat).(ty3 g c0 (TLRef n) (lift (S n) O (lift h (minus d0 (S n)) t)))) -(ty3_abbr g n c0 x (lift h (minus d0 (S n)) u) (getl_intro n c0 (CHead x -(Bind Abbr) (lift h (minus d0 (S n)) u)) x0 H6 H11) (lift h (minus d0 (S n)) -t) (H2 x (minus d0 (S n)) h H12)) (plus (S n) (minus d0 (S n))) -(le_plus_minus (S n) d0 H4)) (lift h (plus (S n) (minus d0 (S n))) (lift (S -n) O t)) (lift_d t h (S n) (minus d0 (S n)) O (le_O_n (minus d0 (S n))))) d0 -(le_plus_minus_r (S n) d0 H4)) (lift h d0 (TLRef n)) (lift_lref_lt n h d0 -H4))))) H10)))))))) H5))) (\lambda (H4: (le d0 n)).(eq_ind_r T (TLRef (plus n -h)) (\lambda (t0: T).(ty3 g c0 t0 (lift h d0 (lift (S n) O t)))) (eq_ind nat -(S n) (\lambda (_: nat).(ty3 g c0 (TLRef (plus n h)) (lift h d0 (lift (S n) O -t)))) (eq_ind_r T (lift (plus h (S n)) O t) (\lambda (t0: T).(ty3 g c0 (TLRef -(plus n h)) t0)) (eq_ind_r nat (plus (S n) h) (\lambda (n0: nat).(ty3 g c0 -(TLRef (plus n h)) (lift n0 O t))) (ty3_abbr g (plus n h) c0 d u -(drop_getl_trans_ge n c0 c d0 h H3 (CHead d (Bind Abbr) u) H0 H4) t H1) (plus -h (S n)) (plus_sym h (S n))) (lift h d0 (lift (S n) O t)) (lift_free t (S n) -h O d0 (le_S d0 n H4) (le_O_n d0))) (plus n (S O)) (eq_ind_r nat (plus (S O) -n) (\lambda (n0: nat).(eq nat (S n) n0)) (refl_equal nat (plus (S O) n)) -(plus n (S O)) (plus_sym n (S O)))) (lift h d0 (TLRef n)) (lift_lref_ge n h -d0 H4)))))))))))))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d: -C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind Abst) u))).(\lambda -(t: T).(\lambda (H1: (ty3 g d u t)).(\lambda (H2: ((\forall (c0: C).(\forall -(d0: nat).(\forall (h: nat).((drop h d0 c0 d) \to (ty3 g c0 (lift h d0 u) -(lift h d0 t)))))))).(\lambda (c0: C).(\lambda (d0: nat).(\lambda (h: -nat).(\lambda (H3: (drop h d0 c0 c)).(lt_le_e n d0 (ty3 g c0 (lift h d0 -(TLRef n)) (lift h d0 (lift (S n) O u))) (\lambda (H4: (lt n d0)).(let H5 -\def (drop_getl_trans_le n d0 (le_S_n n d0 (le_S (S n) d0 H4)) c0 c h H3 -(CHead d (Bind Abst) u) H0) in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: -C).(drop n O c0 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d0 n) -e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear e1 (CHead d (Bind Abst) -u)))) (ty3 g c0 (lift h d0 (TLRef n)) (lift h d0 (lift (S n) O u))) (\lambda -(x0: C).(\lambda (x1: C).(\lambda (H6: (drop n O c0 x0)).(\lambda (H7: (drop -h (minus d0 n) x0 x1)).(\lambda (H8: (clear x1 (CHead d (Bind Abst) u))).(let -H9 \def (eq_ind nat (minus d0 n) (\lambda (n0: nat).(drop h n0 x0 x1)) H7 (S -(minus d0 (S n))) (minus_x_Sy d0 n H4)) in (let H10 \def (drop_clear_S x1 x0 -h (minus d0 (S n)) H9 Abst d u H8) in (ex2_ind C (\lambda (c1: C).(clear x0 -(CHead c1 (Bind Abst) (lift h (minus d0 (S n)) u)))) (\lambda (c1: C).(drop h -(minus d0 (S n)) c1 d)) (ty3 g c0 (lift h d0 (TLRef n)) (lift h d0 (lift (S -n) O u))) (\lambda (x: C).(\lambda (H11: (clear x0 (CHead x (Bind Abst) (lift -h (minus d0 (S n)) u)))).(\lambda (H12: (drop h (minus d0 (S n)) x -d)).(eq_ind_r T (TLRef n) (\lambda (t0: T).(ty3 g c0 t0 (lift h d0 (lift (S -n) O u)))) (eq_ind nat (plus (S n) (minus d0 (S n))) (\lambda (n0: nat).(ty3 -g c0 (TLRef n) (lift h n0 (lift (S n) O u)))) (eq_ind_r T (lift (S n) O (lift -h (minus d0 (S n)) u)) (\lambda (t0: T).(ty3 g c0 (TLRef n) t0)) (eq_ind nat -d0 (\lambda (_: nat).(ty3 g c0 (TLRef n) (lift (S n) O (lift h (minus d0 (S -n)) u)))) (ty3_abst g n c0 x (lift h (minus d0 (S n)) u) (getl_intro n c0 -(CHead x (Bind Abst) (lift h (minus d0 (S n)) u)) x0 H6 H11) (lift h (minus -d0 (S n)) t) (H2 x (minus d0 (S n)) h H12)) (plus (S n) (minus d0 (S n))) -(le_plus_minus (S n) d0 H4)) (lift h (plus (S n) (minus d0 (S n))) (lift (S -n) O u)) (lift_d u h (S n) (minus d0 (S n)) O (le_O_n (minus d0 (S n))))) d0 -(le_plus_minus_r (S n) d0 H4)) (lift h d0 (TLRef n)) (lift_lref_lt n h d0 -H4))))) H10)))))))) H5))) (\lambda (H4: (le d0 n)).(eq_ind_r T (TLRef (plus n -h)) (\lambda (t0: T).(ty3 g c0 t0 (lift h d0 (lift (S n) O u)))) (eq_ind nat -(S n) (\lambda (_: nat).(ty3 g c0 (TLRef (plus n h)) (lift h d0 (lift (S n) O -u)))) (eq_ind_r T (lift (plus h (S n)) O u) (\lambda (t0: T).(ty3 g c0 (TLRef -(plus n h)) t0)) (eq_ind_r nat (plus (S n) h) (\lambda (n0: nat).(ty3 g c0 -(TLRef (plus n h)) (lift n0 O u))) (ty3_abst g (plus n h) c0 d u -(drop_getl_trans_ge n c0 c d0 h H3 (CHead d (Bind Abst) u) H0 H4) t H1) (plus -h (S n)) (plus_sym h (S n))) (lift h d0 (lift (S n) O u)) (lift_free u (S n) -h O d0 (le_S d0 n H4) (le_O_n d0))) (plus n (S O)) (eq_ind_r nat (plus (S O) -n) (\lambda (n0: nat).(eq nat (S n) n0)) (refl_equal nat (plus (S O) n)) -(plus n (S O)) (plus_sym n (S O)))) (lift h d0 (TLRef n)) (lift_lref_ge n h -d0 H4)))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (t: -T).(\lambda (_: (ty3 g c u t)).(\lambda (H1: ((\forall (c0: C).(\forall (d: -nat).(\forall (h: nat).((drop h d c0 c) \to (ty3 g c0 (lift h d u) (lift h d -t)))))))).(\lambda (b: B).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (ty3 -g (CHead c (Bind b) u) t0 t3)).(\lambda (H3: ((\forall (c0: C).(\forall (d: -nat).(\forall (h: nat).((drop h d c0 (CHead c (Bind b) u)) \to (ty3 g c0 -(lift h d t0) (lift h d t3)))))))).(\lambda (c0: C).(\lambda (d: -nat).(\lambda (h: nat).(\lambda (H4: (drop h d c0 c)).(eq_ind_r T (THead -(Bind b) (lift h d u) (lift h (s (Bind b) d) t0)) (\lambda (t4: T).(ty3 g c0 -t4 (lift h d (THead (Bind b) u t3)))) (eq_ind_r T (THead (Bind b) (lift h d -u) (lift h (s (Bind b) d) t3)) (\lambda (t4: T).(ty3 g c0 (THead (Bind b) -(lift h d u) (lift h (s (Bind b) d) t0)) t4)) (ty3_bind g c0 (lift h d u) -(lift h d t) (H1 c0 d h H4) b (lift h (S d) t0) (lift h (S d) t3) (H3 (CHead -c0 (Bind b) (lift h d u)) (S d) h (drop_skip_bind h d c0 c H4 b u))) (lift h -d (THead (Bind b) u t3)) (lift_head (Bind b) u t3 h d)) (lift h d (THead -(Bind b) u t0)) (lift_head (Bind b) u t0 h d)))))))))))))))) (\lambda (c: -C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c w u)).(\lambda (H1: -((\forall (c0: C).(\forall (d: nat).(\forall (h: nat).((drop h d c0 c) \to -(ty3 g c0 (lift h d w) (lift h d u)))))))).(\lambda (v: T).(\lambda (t: -T).(\lambda (_: (ty3 g c v (THead (Bind Abst) u t))).(\lambda (H3: ((\forall -(c0: C).(\forall (d: nat).(\forall (h: nat).((drop h d c0 c) \to (ty3 g c0 -(lift h d v) (lift h d (THead (Bind Abst) u t))))))))).(\lambda (c0: -C).(\lambda (d: nat).(\lambda (h: nat).(\lambda (H4: (drop h d c0 -c)).(eq_ind_r T (THead (Flat Appl) (lift h d w) (lift h (s (Flat Appl) d) v)) -(\lambda (t0: T).(ty3 g c0 t0 (lift h d (THead (Flat Appl) w (THead (Bind -Abst) u t))))) (eq_ind_r T (THead (Flat Appl) (lift h d w) (lift h (s (Flat -Appl) d) (THead (Bind Abst) u t))) (\lambda (t0: T).(ty3 g c0 (THead (Flat -Appl) (lift h d w) (lift h (s (Flat Appl) d) v)) t0)) (eq_ind_r T (THead -(Bind Abst) (lift h (s (Flat Appl) d) u) (lift h (s (Bind Abst) (s (Flat -Appl) d)) t)) (\lambda (t0: T).(ty3 g c0 (THead (Flat Appl) (lift h d w) -(lift h (s (Flat Appl) d) v)) (THead (Flat Appl) (lift h d w) t0))) (ty3_appl -g c0 (lift h d w) (lift h d u) (H1 c0 d h H4) (lift h d v) (lift h (S d) t) -(eq_ind T (lift h d (THead (Bind Abst) u t)) (\lambda (t0: T).(ty3 g c0 (lift -h d v) t0)) (H3 c0 d h H4) (THead (Bind Abst) (lift h d u) (lift h (S d) t)) -(lift_bind Abst u t h d))) (lift h (s (Flat Appl) d) (THead (Bind Abst) u t)) -(lift_head (Bind Abst) u t h (s (Flat Appl) d))) (lift h d (THead (Flat Appl) -w (THead (Bind Abst) u t))) (lift_head (Flat Appl) w (THead (Bind Abst) u t) -h d)) (lift h d (THead (Flat Appl) w v)) (lift_head (Flat Appl) w v h -d))))))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda (t3: T).(\lambda -(_: (ty3 g c t0 t3)).(\lambda (H1: ((\forall (c0: C).(\forall (d: -nat).(\forall (h: nat).((drop h d c0 c) \to (ty3 g c0 (lift h d t0) (lift h d -t3)))))))).(\lambda (t4: T).(\lambda (_: (ty3 g c t3 t4)).(\lambda (H3: -((\forall (c0: C).(\forall (d: nat).(\forall (h: nat).((drop h d c0 c) \to -(ty3 g c0 (lift h d t3) (lift h d t4)))))))).(\lambda (c0: C).(\lambda (d: -nat).(\lambda (h: nat).(\lambda (H4: (drop h d c0 c)).(eq_ind_r T (THead -(Flat Cast) (lift h d t3) (lift h (s (Flat Cast) d) t0)) (\lambda (t: T).(ty3 -g c0 t (lift h d (THead (Flat Cast) t4 t3)))) (eq_ind_r T (THead (Flat Cast) -(lift h d t4) (lift h (s (Flat Cast) d) t3)) (\lambda (t: T).(ty3 g c0 (THead -(Flat Cast) (lift h d t3) (lift h (s (Flat Cast) d) t0)) t)) (ty3_cast g c0 -(lift h (s (Flat Cast) d) t0) (lift h (s (Flat Cast) d) t3) (H1 c0 (s (Flat -Cast) d) h H4) (lift h d t4) (H3 c0 d h H4)) (lift h d (THead (Flat Cast) t4 -t3)) (lift_head (Flat Cast) t4 t3 h d)) (lift h d (THead (Flat Cast) t3 t0)) -(lift_head (Flat Cast) t3 t0 h d)))))))))))))) e t1 t2 H))))). -(* COMMENTS -Initial nodes: 4253 -END *) - -theorem ty3_correct: - \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c -t1 t2) \to (ex T (\lambda (t: T).(ty3 g c t2 t))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H: (ty3 g c t1 t2)).(ty3_ind g (\lambda (c0: C).(\lambda (_: T).(\lambda -(t0: T).(ex T (\lambda (t3: T).(ty3 g c0 t0 t3)))))) (\lambda (c0: -C).(\lambda (t0: T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 t0 t)).(\lambda -(_: (ex T (\lambda (t3: T).(ty3 g c0 t t3)))).(\lambda (u: T).(\lambda (t3: -T).(\lambda (_: (ty3 g c0 u t3)).(\lambda (_: (ex T (\lambda (t4: T).(ty3 g -c0 t3 t4)))).(\lambda (_: (pc3 c0 t3 t0)).(ex_intro T (\lambda (t4: T).(ty3 g -c0 t0 t4)) t H0))))))))))) (\lambda (c0: C).(\lambda (m: nat).(ex_intro T -(\lambda (t: T).(ty3 g c0 (TSort (next g m)) t)) (TSort (next g (next g m))) -(ty3_sort g c0 (next g m))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: -C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abbr) -u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2: (ex T (\lambda -(t0: T).(ty3 g d t t0)))).(let H3 \def H2 in (ex_ind T (\lambda (t0: T).(ty3 -g d t t0)) (ex T (\lambda (t0: T).(ty3 g c0 (lift (S n) O t) t0))) (\lambda -(x: T).(\lambda (H4: (ty3 g d t x)).(ex_intro T (\lambda (t0: T).(ty3 g c0 -(lift (S n) O t) t0)) (lift (S n) O x) (ty3_lift g d t x H4 c0 O (S n) -(getl_drop Abbr c0 d u n H0))))) H3)))))))))) (\lambda (n: nat).(\lambda (c0: -C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind -Abst) u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u t)).(\lambda (_: (ex T -(\lambda (t0: T).(ty3 g d t t0)))).(ex_intro T (\lambda (t0: T).(ty3 g c0 -(lift (S n) O u) t0)) (lift (S n) O t) (ty3_lift g d u t H1 c0 O (S n) -(getl_drop Abst c0 d u n H0))))))))))) (\lambda (c0: C).(\lambda (u: -T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 u t)).(\lambda (_: (ex T (\lambda -(t0: T).(ty3 g c0 t t0)))).(\lambda (b: B).(\lambda (t0: T).(\lambda (t3: -T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t0 t3)).(\lambda (H3: (ex T -(\lambda (t4: T).(ty3 g (CHead c0 (Bind b) u) t3 t4)))).(let H4 \def H3 in -(ex_ind T (\lambda (t4: T).(ty3 g (CHead c0 (Bind b) u) t3 t4)) (ex T -(\lambda (t4: T).(ty3 g c0 (THead (Bind b) u t3) t4))) (\lambda (x: -T).(\lambda (H5: (ty3 g (CHead c0 (Bind b) u) t3 x)).(ex_intro T (\lambda -(t4: T).(ty3 g c0 (THead (Bind b) u t3) t4)) (THead (Bind b) u x) (ty3_bind g -c0 u t H0 b t3 x H5)))) H4)))))))))))) (\lambda (c0: C).(\lambda (w: -T).(\lambda (u: T).(\lambda (H0: (ty3 g c0 w u)).(\lambda (H1: (ex T (\lambda -(t: T).(ty3 g c0 u t)))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g -c0 v (THead (Bind Abst) u t))).(\lambda (H3: (ex T (\lambda (t0: T).(ty3 g c0 -(THead (Bind Abst) u t) t0)))).(let H4 \def H1 in (ex_ind T (\lambda (t0: -T).(ty3 g c0 u t0)) (ex T (\lambda (t0: T).(ty3 g c0 (THead (Flat Appl) w -(THead (Bind Abst) u t)) t0))) (\lambda (x: T).(\lambda (_: (ty3 g c0 u -x)).(let H6 \def H3 in (ex_ind T (\lambda (t0: T).(ty3 g c0 (THead (Bind -Abst) u t) t0)) (ex T (\lambda (t0: T).(ty3 g c0 (THead (Flat Appl) w (THead -(Bind Abst) u t)) t0))) (\lambda (x0: T).(\lambda (H7: (ty3 g c0 (THead (Bind -Abst) u t) x0)).(ex3_2_ind T T (\lambda (t3: T).(\lambda (_: T).(pc3 c0 -(THead (Bind Abst) u t3) x0))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u -t0))) (\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind Abst) u) t -t3))) (ex T (\lambda (t0: T).(ty3 g c0 (THead (Flat Appl) w (THead (Bind -Abst) u t)) t0))) (\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 c0 -(THead (Bind Abst) u x1) x0)).(\lambda (H9: (ty3 g c0 u x2)).(\lambda (H10: -(ty3 g (CHead c0 (Bind Abst) u) t x1)).(ex_intro T (\lambda (t0: T).(ty3 g c0 -(THead (Flat Appl) w (THead (Bind Abst) u t)) t0)) (THead (Flat Appl) w -(THead (Bind Abst) u x1)) (ty3_appl g c0 w u H0 (THead (Bind Abst) u t) x1 -(ty3_bind g c0 u x2 H9 Abst t x1 H10)))))))) (ty3_gen_bind g Abst c0 u t x0 -H7)))) H6)))) H4))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t3: -T).(\lambda (_: (ty3 g c0 t0 t3)).(\lambda (_: (ex T (\lambda (t: T).(ty3 g -c0 t3 t)))).(\lambda (t4: T).(\lambda (H2: (ty3 g c0 t3 t4)).(\lambda (H3: -(ex T (\lambda (t: T).(ty3 g c0 t4 t)))).(let H4 \def H3 in (ex_ind T -(\lambda (t: T).(ty3 g c0 t4 t)) (ex T (\lambda (t: T).(ty3 g c0 (THead (Flat -Cast) t4 t3) t))) (\lambda (x: T).(\lambda (H5: (ty3 g c0 t4 x)).(ex_intro T -(\lambda (t: T).(ty3 g c0 (THead (Flat Cast) t4 t3) t)) (THead (Flat Cast) x -t4) (ty3_cast g c0 t3 t4 H2 x H5)))) H4)))))))))) c t1 t2 H))))). -(* COMMENTS -Initial nodes: 1333 -END *) - -theorem ty3_unique: - \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).((ty3 g c u -t1) \to (\forall (t2: T).((ty3 g c u t2) \to (pc3 c t1 t2))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (H: -(ty3 g c u t1)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: -T).(\forall (t2: T).((ty3 g c0 t t2) \to (pc3 c0 t0 t2)))))) (\lambda (c0: -C).(\lambda (t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda -(_: ((\forall (t3: T).((ty3 g c0 t2 t3) \to (pc3 c0 t t3))))).(\lambda (u0: -T).(\lambda (t0: T).(\lambda (_: (ty3 g c0 u0 t0)).(\lambda (H3: ((\forall -(t3: T).((ty3 g c0 u0 t3) \to (pc3 c0 t0 t3))))).(\lambda (H4: (pc3 c0 t0 -t2)).(\lambda (t3: T).(\lambda (H5: (ty3 g c0 u0 t3)).(pc3_t t0 c0 t2 (pc3_s -c0 t2 t0 H4) t3 (H3 t3 H5)))))))))))))) (\lambda (c0: C).(\lambda (m: -nat).(\lambda (t2: T).(\lambda (H0: (ty3 g c0 (TSort m) t2)).(ty3_gen_sort g -c0 t2 m H0))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda -(u0: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abbr) u0))).(\lambda (t: -T).(\lambda (_: (ty3 g d u0 t)).(\lambda (H2: ((\forall (t2: T).((ty3 g d u0 -t2) \to (pc3 d t t2))))).(\lambda (t2: T).(\lambda (H3: (ty3 g c0 (TLRef n) -t2)).(or_ind (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: -T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda -(_: T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: C).(\lambda (u1: -T).(\lambda (t0: T).(ty3 g e u1 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda -(u1: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u1) t2)))) (\lambda (e: -C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u1))))) -(\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 t0))))) (pc3 c0 -(lift (S n) O t) t2) (\lambda (H4: (ex3_3 C T T (\lambda (_: C).(\lambda (_: -T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda (e: C).(\lambda -(u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: -C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 t0)))))).(ex3_3_ind C T T -(\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) -t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e -(Bind Abbr) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g -e u1 t0)))) (pc3 c0 (lift (S n) O t) t2) (\lambda (x0: C).(\lambda (x1: -T).(\lambda (x2: T).(\lambda (H5: (pc3 c0 (lift (S n) O x2) t2)).(\lambda -(H6: (getl n c0 (CHead x0 (Bind Abbr) x1))).(\lambda (H7: (ty3 g x0 x1 -x2)).(let H8 \def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda (c1: C).(getl n -c0 c1)) H0 (CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind Abbr) u0) n -H0 (CHead x0 (Bind Abbr) x1) H6)) in (let H9 \def (f_equal C C (\lambda (e: -C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | -(CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u0) (CHead x0 (Bind -Abbr) x1) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead x0 (Bind Abbr) -x1) H6)) in ((let H10 \def (f_equal C T (\lambda (e: C).(match e in C return -(\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t0) -\Rightarrow t0])) (CHead d (Bind Abbr) u0) (CHead x0 (Bind Abbr) x1) -(getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in -(\lambda (H11: (eq C d x0)).(let H12 \def (eq_ind_r T x1 (\lambda (t0: -T).(getl n c0 (CHead x0 (Bind Abbr) t0))) H8 u0 H10) in (let H13 \def -(eq_ind_r T x1 (\lambda (t0: T).(ty3 g x0 t0 x2)) H7 u0 H10) in (let H14 \def -(eq_ind_r C x0 (\lambda (c1: C).(getl n c0 (CHead c1 (Bind Abbr) u0))) H12 d -H11) in (let H15 \def (eq_ind_r C x0 (\lambda (c1: C).(ty3 g c1 u0 x2)) H13 d -H11) in (pc3_t (lift (S n) O x2) c0 (lift (S n) O t) (pc3_lift c0 d (S n) O -(getl_drop Abbr c0 d u0 n H14) t x2 (H2 x2 H15)) t2 H5))))))) H9))))))))) -H4)) (\lambda (H4: (ex3_3 C T T (\lambda (_: C).(\lambda (u1: T).(\lambda (_: -T).(pc3 c0 (lift (S n) O u1) t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda -(_: T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1: -T).(\lambda (t0: T).(ty3 g e u1 t0)))))).(ex3_3_ind C T T (\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u1) t2)))) (\lambda -(e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) -u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 t0)))) -(pc3 c0 (lift (S n) O t) t2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: -T).(\lambda (_: (pc3 c0 (lift (S n) O x1) t2)).(\lambda (H6: (getl n c0 -(CHead x0 (Bind Abst) x1))).(\lambda (_: (ty3 g x0 x1 x2)).(let H8 \def -(eq_ind C (CHead d (Bind Abbr) u0) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead -x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead x0 -(Bind Abst) x1) H6)) in (let H9 \def (eq_ind C (CHead d (Bind Abbr) u0) -(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind b) \Rightarrow (match b in B return (\lambda (_: -B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | Void -\Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead x0 (Bind Abst) -x1) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead x0 (Bind Abst) x1) -H6)) in (False_ind (pc3 c0 (lift (S n) O t) t2) H9))))))))) H4)) -(ty3_gen_lref g c0 t2 n H3)))))))))))) (\lambda (n: nat).(\lambda (c0: -C).(\lambda (d: C).(\lambda (u0: T).(\lambda (H0: (getl n c0 (CHead d (Bind -Abst) u0))).(\lambda (t: T).(\lambda (_: (ty3 g d u0 t)).(\lambda (_: -((\forall (t2: T).((ty3 g d u0 t2) \to (pc3 d t t2))))).(\lambda (t2: -T).(\lambda (H3: (ty3 g c0 (TLRef n) t2)).(or_ind (ex3_3 C T T (\lambda (_: -C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda -(e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) -u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 t0))))) -(ex3_3 C T T (\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(pc3 c0 (lift -(S n) O u1) t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n -c0 (CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda -(t0: T).(ty3 g e u1 t0))))) (pc3 c0 (lift (S n) O u0) t2) (\lambda (H4: -(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift -(S n) O t0) t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n -c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda -(t0: T).(ty3 g e u1 t0)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (_: -T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda (e: C).(\lambda -(u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: -C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 t0)))) (pc3 c0 (lift (S n) O -u0) t2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 -c0 (lift (S n) O x2) t2)).(\lambda (H6: (getl n c0 (CHead x0 (Bind Abbr) -x1))).(\lambda (_: (ty3 g x0 x1 x2)).(let H8 \def (eq_ind C (CHead d (Bind -Abst) u0) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead x0 (Bind Abbr) x1) -(getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in -(let H9 \def (eq_ind C (CHead d (Bind Abst) u0) (\lambda (ee: C).(match ee in -C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k -_) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b) -\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow -False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) -\Rightarrow False])])) I (CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d -(Bind Abst) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in (False_ind (pc3 c0 -(lift (S n) O u0) t2) H9))))))))) H4)) (\lambda (H4: (ex3_3 C T T (\lambda -(_: C).(\lambda (u1: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u1) t2)))) -(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind -Abst) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 -t0)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (u1: T).(\lambda (_: -T).(pc3 c0 (lift (S n) O u1) t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda -(_: T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1: -T).(\lambda (t0: T).(ty3 g e u1 t0)))) (pc3 c0 (lift (S n) O u0) t2) (\lambda -(x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H5: (pc3 c0 (lift (S n) O -x1) t2)).(\lambda (H6: (getl n c0 (CHead x0 (Bind Abst) x1))).(\lambda (H7: -(ty3 g x0 x1 x2)).(let H8 \def (eq_ind C (CHead d (Bind Abst) u0) (\lambda -(c1: C).(getl n c0 c1)) H0 (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d -(Bind Abst) u0) n H0 (CHead x0 (Bind Abst) x1) H6)) in (let H9 \def (f_equal -C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abst) u0) -(CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead -x0 (Bind Abst) x1) H6)) in ((let H10 \def (f_equal C T (\lambda (e: C).(match -e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ -t0) \Rightarrow t0])) (CHead d (Bind Abst) u0) (CHead x0 (Bind Abst) x1) -(getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead x0 (Bind Abst) x1) H6)) in -(\lambda (H11: (eq C d x0)).(let H12 \def (eq_ind_r T x1 (\lambda (t0: -T).(getl n c0 (CHead x0 (Bind Abst) t0))) H8 u0 H10) in (let H13 \def -(eq_ind_r T x1 (\lambda (t0: T).(ty3 g x0 t0 x2)) H7 u0 H10) in (let H14 \def -(eq_ind_r T x1 (\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)) H5 u0 H10) in -(let H15 \def (eq_ind_r C x0 (\lambda (c1: C).(getl n c0 (CHead c1 (Bind -Abst) u0))) H12 d H11) in (let H16 \def (eq_ind_r C x0 (\lambda (c1: C).(ty3 -g c1 u0 x2)) H13 d H11) in H14))))))) H9))))))))) H4)) (ty3_gen_lref g c0 t2 -n H3)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (t: T).(\lambda -(_: (ty3 g c0 u0 t)).(\lambda (_: ((\forall (t2: T).((ty3 g c0 u0 t2) \to -(pc3 c0 t t2))))).(\lambda (b: B).(\lambda (t0: T).(\lambda (t2: T).(\lambda -(_: (ty3 g (CHead c0 (Bind b) u0) t0 t2)).(\lambda (H3: ((\forall (t3: -T).((ty3 g (CHead c0 (Bind b) u0) t0 t3) \to (pc3 (CHead c0 (Bind b) u0) t2 -t3))))).(\lambda (t3: T).(\lambda (H4: (ty3 g c0 (THead (Bind b) u0 t0) -t3)).(ex3_2_ind T T (\lambda (t4: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) -u0 t4) t3))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c0 u0 t5))) (\lambda -(t4: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u0) t0 t4))) (pc3 c0 (THead -(Bind b) u0 t2) t3) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (pc3 c0 -(THead (Bind b) u0 x0) t3)).(\lambda (_: (ty3 g c0 u0 x1)).(\lambda (H7: (ty3 -g (CHead c0 (Bind b) u0) t0 x0)).(pc3_t (THead (Bind b) u0 x0) c0 (THead -(Bind b) u0 t2) (pc3_head_2 c0 u0 t2 x0 (Bind b) (H3 x0 H7)) t3 H5)))))) -(ty3_gen_bind g b c0 u0 t0 t3 H4)))))))))))))) (\lambda (c0: C).(\lambda (w: -T).(\lambda (u0: T).(\lambda (_: (ty3 g c0 w u0)).(\lambda (_: ((\forall (t2: -T).((ty3 g c0 w t2) \to (pc3 c0 u0 t2))))).(\lambda (v: T).(\lambda (t: -T).(\lambda (_: (ty3 g c0 v (THead (Bind Abst) u0 t))).(\lambda (H3: -((\forall (t2: T).((ty3 g c0 v t2) \to (pc3 c0 (THead (Bind Abst) u0 t) -t2))))).(\lambda (t2: T).(\lambda (H4: (ty3 g c0 (THead (Flat Appl) w v) -t2)).(ex3_2_ind T T (\lambda (u1: T).(\lambda (t0: T).(pc3 c0 (THead (Flat -Appl) w (THead (Bind Abst) u1 t0)) t2))) (\lambda (u1: T).(\lambda (t0: -T).(ty3 g c0 v (THead (Bind Abst) u1 t0)))) (\lambda (u1: T).(\lambda (_: -T).(ty3 g c0 w u1))) (pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t)) -t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (pc3 c0 (THead (Flat -Appl) w (THead (Bind Abst) x0 x1)) t2)).(\lambda (H6: (ty3 g c0 v (THead -(Bind Abst) x0 x1))).(\lambda (_: (ty3 g c0 w x0)).(pc3_t (THead (Flat Appl) -w (THead (Bind Abst) x0 x1)) c0 (THead (Flat Appl) w (THead (Bind Abst) u0 -t)) (pc3_thin_dx c0 (THead (Bind Abst) u0 t) (THead (Bind Abst) x0 x1) (H3 -(THead (Bind Abst) x0 x1) H6) w Appl) t2 H5)))))) (ty3_gen_appl g c0 w v t2 -H4))))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: T).(\lambda -(_: (ty3 g c0 t0 t2)).(\lambda (_: ((\forall (t3: T).((ty3 g c0 t0 t3) \to -(pc3 c0 t2 t3))))).(\lambda (t3: T).(\lambda (_: (ty3 g c0 t2 t3)).(\lambda -(H3: ((\forall (t4: T).((ty3 g c0 t2 t4) \to (pc3 c0 t3 t4))))).(\lambda (t4: -T).(\lambda (H4: (ty3 g c0 (THead (Flat Cast) t2 t0) t4)).(ex3_ind T (\lambda -(t5: T).(pc3 c0 (THead (Flat Cast) t5 t2) t4)) (\lambda (_: T).(ty3 g c0 t0 -t2)) (\lambda (t5: T).(ty3 g c0 t2 t5)) (pc3 c0 (THead (Flat Cast) t3 t2) t4) -(\lambda (x0: T).(\lambda (H5: (pc3 c0 (THead (Flat Cast) x0 t2) -t4)).(\lambda (_: (ty3 g c0 t0 t2)).(\lambda (H7: (ty3 g c0 t2 x0)).(pc3_t -(THead (Flat Cast) x0 t2) c0 (THead (Flat Cast) t3 t2) (pc3_head_1 c0 t3 x0 -(H3 x0 H7) (Flat Cast) t2) t4 H5))))) (ty3_gen_cast g c0 t0 t2 t4 -H4)))))))))))) c u t1 H))))). -(* COMMENTS -Initial nodes: 3459 -END *) - -theorem ty3_gen_abst_abst: - \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall -(t2: T).((ty3 g c (THead (Bind Abst) u t1) (THead (Bind Abst) u t2)) \to (ex2 -T (\lambda (w: T).(ty3 g c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst) -u) t1 t2)))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda -(t2: T).(\lambda (H: (ty3 g c (THead (Bind Abst) u t1) (THead (Bind Abst) u -t2))).(ex_ind T (\lambda (t: T).(ty3 g c (THead (Bind Abst) u t2) t)) (ex2 T -(\lambda (w: T).(ty3 g c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) -t1 t2))) (\lambda (x: T).(\lambda (H0: (ty3 g c (THead (Bind Abst) u t2) -x)).(ex3_2_ind T T (\lambda (t3: T).(\lambda (_: T).(pc3 c (THead (Bind Abst) -u t3) x))) (\lambda (_: T).(\lambda (t: T).(ty3 g c u t))) (\lambda (t3: -T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t2 t3))) (ex2 T (\lambda -(w: T).(ty3 g c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t1 t2))) -(\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 c (THead (Bind Abst) u -x0) x)).(\lambda (_: (ty3 g c u x1)).(\lambda (H3: (ty3 g (CHead c (Bind -Abst) u) t2 x0)).(ex3_2_ind T T (\lambda (t3: T).(\lambda (_: T).(pc3 c -(THead (Bind Abst) u t3) (THead (Bind Abst) u t2)))) (\lambda (_: T).(\lambda -(t: T).(ty3 g c u t))) (\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c (Bind -Abst) u) t1 t3))) (ex2 T (\lambda (w: T).(ty3 g c u w)) (\lambda (_: T).(ty3 -g (CHead c (Bind Abst) u) t1 t2))) (\lambda (x2: T).(\lambda (x3: T).(\lambda -(H4: (pc3 c (THead (Bind Abst) u x2) (THead (Bind Abst) u t2))).(\lambda (H5: -(ty3 g c u x3)).(\lambda (H6: (ty3 g (CHead c (Bind Abst) u) t1 x2)).(let H_y -\def (pc3_gen_abst_shift c u x2 t2 H4) in (ex_intro2 T (\lambda (w: T).(ty3 g -c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t1 t2)) x3 H5 -(ty3_conv g (CHead c (Bind Abst) u) t2 x0 H3 t1 x2 H6 H_y)))))))) -(ty3_gen_bind g Abst c u t1 (THead (Bind Abst) u t2) H))))))) (ty3_gen_bind g -Abst c u t2 x H0)))) (ty3_correct g c (THead (Bind Abst) u t1) (THead (Bind -Abst) u t2) H))))))). -(* COMMENTS -Initial nodes: 571 -END *) - -theorem ty3_typecheck: - \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (v: T).((ty3 g c t -v) \to (ex T (\lambda (u: T).(ty3 g c (THead (Flat Cast) v t) u))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (v: T).(\lambda (H: -(ty3 g c t v)).(ex_ind T (\lambda (t0: T).(ty3 g c v t0)) (ex T (\lambda (u: -T).(ty3 g c (THead (Flat Cast) v t) u))) (\lambda (x: T).(\lambda (H0: (ty3 g -c v x)).(ex_intro T (\lambda (u: T).(ty3 g c (THead (Flat Cast) v t) u)) -(THead (Flat Cast) x v) (ty3_cast g c t v H x H0)))) (ty3_correct g c t v -H)))))). -(* COMMENTS -Initial nodes: 131 -END *) - -theorem ty3_getl_subst0: - \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t -u) \to (\forall (v0: T).(\forall (t0: T).(\forall (i: nat).((subst0 i v0 t -t0) \to (\forall (b: B).(\forall (d: C).(\forall (v: T).((getl i c (CHead d -(Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v w))))))))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H: -(ty3 g c t u)).(ty3_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (_: -T).(\forall (v0: T).(\forall (t2: T).(\forall (i: nat).((subst0 i v0 t0 t2) -\to (\forall (b: B).(\forall (d: C).(\forall (v: T).((getl i c0 (CHead d -(Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v w)))))))))))))) (\lambda -(c0: C).(\lambda (t2: T).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t2 -t0)).(\lambda (_: ((\forall (v0: T).(\forall (t1: T).(\forall (i: -nat).((subst0 i v0 t2 t1) \to (\forall (b: B).(\forall (d: C).(\forall (v: -T).((getl i c0 (CHead d (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v -w))))))))))))).(\lambda (u0: T).(\lambda (t1: T).(\lambda (_: (ty3 g c0 u0 -t1)).(\lambda (H3: ((\forall (v0: T).(\forall (t3: T).(\forall (i: -nat).((subst0 i v0 u0 t3) \to (\forall (b: B).(\forall (d: C).(\forall (v: -T).((getl i c0 (CHead d (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v -w))))))))))))).(\lambda (_: (pc3 c0 t1 t2)).(\lambda (v0: T).(\lambda (t3: -T).(\lambda (i: nat).(\lambda (H5: (subst0 i v0 u0 t3)).(\lambda (b: -B).(\lambda (d: C).(\lambda (v: T).(\lambda (H6: (getl i c0 (CHead d (Bind b) -v))).(H3 v0 t3 i H5 b d v H6))))))))))))))))))) (\lambda (c0: C).(\lambda (m: -nat).(\lambda (v0: T).(\lambda (t0: T).(\lambda (i: nat).(\lambda (H0: -(subst0 i v0 (TSort m) t0)).(\lambda (b: B).(\lambda (d: C).(\lambda (v: -T).(\lambda (_: (getl i c0 (CHead d (Bind b) v))).(subst0_gen_sort v0 t0 i m -H0 (ex T (\lambda (w: T).(ty3 g d v w)))))))))))))) (\lambda (n: -nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (H0: (getl n -c0 (CHead d (Bind Abbr) u0))).(\lambda (t0: T).(\lambda (H1: (ty3 g d u0 -t0)).(\lambda (_: ((\forall (v0: T).(\forall (t1: T).(\forall (i: -nat).((subst0 i v0 u0 t1) \to (\forall (b: B).(\forall (d0: C).(\forall (v: -T).((getl i d (CHead d0 (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d0 v -w))))))))))))).(\lambda (v0: T).(\lambda (t1: T).(\lambda (i: nat).(\lambda -(H3: (subst0 i v0 (TLRef n) t1)).(\lambda (b: B).(\lambda (d0: C).(\lambda -(v: T).(\lambda (H4: (getl i c0 (CHead d0 (Bind b) v))).(land_ind (eq nat n -i) (eq T t1 (lift (S n) O v0)) (ex T (\lambda (w: T).(ty3 g d0 v w))) -(\lambda (H5: (eq nat n i)).(\lambda (_: (eq T t1 (lift (S n) O v0))).(let H7 -\def (eq_ind_r nat i (\lambda (n0: nat).(getl n0 c0 (CHead d0 (Bind b) v))) -H4 n H5) in (let H8 \def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda (c1: -C).(getl n c0 c1)) H0 (CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind -Abbr) u0) n H0 (CHead d0 (Bind b) v) H7)) in (let H9 \def (f_equal C C -(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u0) -(CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead d0 -(Bind b) v) H7)) in ((let H10 \def (f_equal C B (\lambda (e: C).(match e in C -return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b0) -\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u0) -(CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead d0 -(Bind b) v) H7)) in ((let H11 \def (f_equal C T (\lambda (e: C).(match e in C -return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t2) -\Rightarrow t2])) (CHead d (Bind Abbr) u0) (CHead d0 (Bind b) v) (getl_mono -c0 (CHead d (Bind Abbr) u0) n H0 (CHead d0 (Bind b) v) H7)) in (\lambda (H12: -(eq B Abbr b)).(\lambda (H13: (eq C d d0)).(let H14 \def (eq_ind_r T v -(\lambda (t2: T).(getl n c0 (CHead d0 (Bind b) t2))) H8 u0 H11) in (eq_ind T -u0 (\lambda (t2: T).(ex T (\lambda (w: T).(ty3 g d0 t2 w)))) (let H15 \def -(eq_ind_r C d0 (\lambda (c1: C).(getl n c0 (CHead c1 (Bind b) u0))) H14 d -H13) in (eq_ind C d (\lambda (c1: C).(ex T (\lambda (w: T).(ty3 g c1 u0 w)))) -(let H16 \def (eq_ind_r B b (\lambda (b0: B).(getl n c0 (CHead d (Bind b0) -u0))) H15 Abbr H12) in (ex_intro T (\lambda (w: T).(ty3 g d u0 w)) t0 H1)) d0 -H13)) v H11))))) H10)) H9)))))) (subst0_gen_lref v0 t1 i n -H3)))))))))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: -C).(\lambda (u0: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abst) -u0))).(\lambda (t0: T).(\lambda (H1: (ty3 g d u0 t0)).(\lambda (_: ((\forall -(v0: T).(\forall (t1: T).(\forall (i: nat).((subst0 i v0 u0 t1) \to (\forall -(b: B).(\forall (d0: C).(\forall (v: T).((getl i d (CHead d0 (Bind b) v)) \to -(ex T (\lambda (w: T).(ty3 g d0 v w))))))))))))).(\lambda (v0: T).(\lambda -(t1: T).(\lambda (i: nat).(\lambda (H3: (subst0 i v0 (TLRef n) t1)).(\lambda -(b: B).(\lambda (d0: C).(\lambda (v: T).(\lambda (H4: (getl i c0 (CHead d0 -(Bind b) v))).(land_ind (eq nat n i) (eq T t1 (lift (S n) O v0)) (ex T -(\lambda (w: T).(ty3 g d0 v w))) (\lambda (H5: (eq nat n i)).(\lambda (_: (eq -T t1 (lift (S n) O v0))).(let H7 \def (eq_ind_r nat i (\lambda (n0: -nat).(getl n0 c0 (CHead d0 (Bind b) v))) H4 n H5) in (let H8 \def (eq_ind C -(CHead d (Bind Abst) u0) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead d0 (Bind -b) v) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead d0 (Bind b) v) H7)) -in (let H9 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda -(_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) -(CHead d (Bind Abst) u0) (CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind -Abst) u0) n H0 (CHead d0 (Bind b) v) H7)) in ((let H10 \def (f_equal C B -(\lambda (e: C).(match e in C return (\lambda (_: C).B) with [(CSort _) -\Rightarrow Abst | (CHead _ k _) \Rightarrow (match k in K return (\lambda -(_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abst])])) -(CHead d (Bind Abst) u0) (CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind -Abst) u0) n H0 (CHead d0 (Bind b) v) H7)) in ((let H11 \def (f_equal C T -(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) -\Rightarrow u0 | (CHead _ _ t2) \Rightarrow t2])) (CHead d (Bind Abst) u0) -(CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead d0 -(Bind b) v) H7)) in (\lambda (H12: (eq B Abst b)).(\lambda (H13: (eq C d -d0)).(let H14 \def (eq_ind_r T v (\lambda (t2: T).(getl n c0 (CHead d0 (Bind -b) t2))) H8 u0 H11) in (eq_ind T u0 (\lambda (t2: T).(ex T (\lambda (w: -T).(ty3 g d0 t2 w)))) (let H15 \def (eq_ind_r C d0 (\lambda (c1: C).(getl n -c0 (CHead c1 (Bind b) u0))) H14 d H13) in (eq_ind C d (\lambda (c1: C).(ex T -(\lambda (w: T).(ty3 g c1 u0 w)))) (let H16 \def (eq_ind_r B b (\lambda (b0: -B).(getl n c0 (CHead d (Bind b0) u0))) H15 Abst H12) in (ex_intro T (\lambda -(w: T).(ty3 g d u0 w)) t0 H1)) d0 H13)) v H11))))) H10)) H9)))))) -(subst0_gen_lref v0 t1 i n H3)))))))))))))))))) (\lambda (c0: C).(\lambda -(u0: T).(\lambda (t0: T).(\lambda (_: (ty3 g c0 u0 t0)).(\lambda (H1: -((\forall (v0: T).(\forall (t1: T).(\forall (i: nat).((subst0 i v0 u0 t1) \to -(\forall (b: B).(\forall (d: C).(\forall (v: T).((getl i c0 (CHead d (Bind b) -v)) \to (ex T (\lambda (w: T).(ty3 g d v w))))))))))))).(\lambda (b: -B).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) -u0) t1 t2)).(\lambda (H3: ((\forall (v0: T).(\forall (t3: T).(\forall (i: -nat).((subst0 i v0 t1 t3) \to (\forall (b0: B).(\forall (d: C).(\forall (v: -T).((getl i (CHead c0 (Bind b) u0) (CHead d (Bind b0) v)) \to (ex T (\lambda -(w: T).(ty3 g d v w))))))))))))).(\lambda (v0: T).(\lambda (t3: T).(\lambda -(i: nat).(\lambda (H4: (subst0 i v0 (THead (Bind b) u0 t1) t3)).(\lambda (b0: -B).(\lambda (d: C).(\lambda (v: T).(\lambda (H5: (getl i c0 (CHead d (Bind -b0) v))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t3 (THead (Bind b) u2 t1))) -(\lambda (u2: T).(subst0 i v0 u0 u2))) (ex2 T (\lambda (t4: T).(eq T t3 -(THead (Bind b) u0 t4))) (\lambda (t4: T).(subst0 (s (Bind b) i) v0 t1 t4))) -(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind b) u2 -t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 u0 u2))) (\lambda (_: -T).(\lambda (t4: T).(subst0 (s (Bind b) i) v0 t1 t4)))) (ex T (\lambda (w: -T).(ty3 g d v w))) (\lambda (H6: (ex2 T (\lambda (u2: T).(eq T t3 (THead -(Bind b) u2 t1))) (\lambda (u2: T).(subst0 i v0 u0 u2)))).(ex2_ind T (\lambda -(u2: T).(eq T t3 (THead (Bind b) u2 t1))) (\lambda (u2: T).(subst0 i v0 u0 -u2)) (ex T (\lambda (w: T).(ty3 g d v w))) (\lambda (x: T).(\lambda (_: (eq T -t3 (THead (Bind b) x t1))).(\lambda (H8: (subst0 i v0 u0 x)).(H1 v0 x i H8 b0 -d v H5)))) H6)) (\lambda (H6: (ex2 T (\lambda (t4: T).(eq T t3 (THead (Bind -b) u0 t4))) (\lambda (t4: T).(subst0 (s (Bind b) i) v0 t1 t4)))).(ex2_ind T -(\lambda (t4: T).(eq T t3 (THead (Bind b) u0 t4))) (\lambda (t4: T).(subst0 -(s (Bind b) i) v0 t1 t4)) (ex T (\lambda (w: T).(ty3 g d v w))) (\lambda (x: -T).(\lambda (_: (eq T t3 (THead (Bind b) u0 x))).(\lambda (H8: (subst0 (s -(Bind b) i) v0 t1 x)).(H3 v0 x (S i) H8 b0 d v (getl_head (Bind b) i c0 -(CHead d (Bind b0) v) H5 u0))))) H6)) (\lambda (H6: (ex3_2 T T (\lambda (u2: -T).(\lambda (t4: T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i v0 u0 u2))) (\lambda (_: T).(\lambda (t4: -T).(subst0 (s (Bind b) i) v0 t1 t4))))).(ex3_2_ind T T (\lambda (u2: -T).(\lambda (t4: T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i v0 u0 u2))) (\lambda (_: T).(\lambda (t4: -T).(subst0 (s (Bind b) i) v0 t1 t4))) (ex T (\lambda (w: T).(ty3 g d v w))) -(\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (eq T t3 (THead (Bind b) x0 -x1))).(\lambda (H8: (subst0 i v0 u0 x0)).(\lambda (_: (subst0 (s (Bind b) i) -v0 t1 x1)).(H1 v0 x0 i H8 b0 d v H5)))))) H6)) (subst0_gen_head (Bind b) v0 -u0 t1 t3 i H4)))))))))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda -(u0: T).(\lambda (_: (ty3 g c0 w u0)).(\lambda (H1: ((\forall (v0: -T).(\forall (t0: T).(\forall (i: nat).((subst0 i v0 w t0) \to (\forall (b: -B).(\forall (d: C).(\forall (v: T).((getl i c0 (CHead d (Bind b) v)) \to (ex -T (\lambda (w0: T).(ty3 g d v w0))))))))))))).(\lambda (v: T).(\lambda (t0: -T).(\lambda (_: (ty3 g c0 v (THead (Bind Abst) u0 t0))).(\lambda (H3: -((\forall (v0: T).(\forall (t1: T).(\forall (i: nat).((subst0 i v0 v t1) \to -(\forall (b: B).(\forall (d: C).(\forall (v1: T).((getl i c0 (CHead d (Bind -b) v1)) \to (ex T (\lambda (w0: T).(ty3 g d v1 w0))))))))))))).(\lambda (v0: -T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (H4: (subst0 i v0 (THead (Flat -Appl) w v) t1)).(\lambda (b: B).(\lambda (d: C).(\lambda (v1: T).(\lambda -(H5: (getl i c0 (CHead d (Bind b) v1))).(or3_ind (ex2 T (\lambda (u2: T).(eq -T t1 (THead (Flat Appl) u2 v))) (\lambda (u2: T).(subst0 i v0 w u2))) (ex2 T -(\lambda (t2: T).(eq T t1 (THead (Flat Appl) w t2))) (\lambda (t2: T).(subst0 -(s (Flat Appl) i) v0 v t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq -T t1 (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i -v0 w u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s (Flat Appl) i) v0 v -t2)))) (ex T (\lambda (w0: T).(ty3 g d v1 w0))) (\lambda (H6: (ex2 T (\lambda -(u2: T).(eq T t1 (THead (Flat Appl) u2 v))) (\lambda (u2: T).(subst0 i v0 w -u2)))).(ex2_ind T (\lambda (u2: T).(eq T t1 (THead (Flat Appl) u2 v))) -(\lambda (u2: T).(subst0 i v0 w u2)) (ex T (\lambda (w0: T).(ty3 g d v1 w0))) -(\lambda (x: T).(\lambda (_: (eq T t1 (THead (Flat Appl) x v))).(\lambda (H8: -(subst0 i v0 w x)).(H1 v0 x i H8 b d v1 H5)))) H6)) (\lambda (H6: (ex2 T -(\lambda (t2: T).(eq T t1 (THead (Flat Appl) w t2))) (\lambda (t2: T).(subst0 -(s (Flat Appl) i) v0 v t2)))).(ex2_ind T (\lambda (t2: T).(eq T t1 (THead -(Flat Appl) w t2))) (\lambda (t2: T).(subst0 (s (Flat Appl) i) v0 v t2)) (ex -T (\lambda (w0: T).(ty3 g d v1 w0))) (\lambda (x: T).(\lambda (_: (eq T t1 -(THead (Flat Appl) w x))).(\lambda (H8: (subst0 (s (Flat Appl) i) v0 v -x)).(H3 v0 x (s (Flat Appl) i) H8 b d v1 H5)))) H6)) (\lambda (H6: (ex3_2 T T -(\lambda (u2: T).(\lambda (t2: T).(eq T t1 (THead (Flat Appl) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 i v0 w u2))) (\lambda (_: -T).(\lambda (t2: T).(subst0 (s (Flat Appl) i) v0 v t2))))).(ex3_2_ind T T -(\lambda (u2: T).(\lambda (t2: T).(eq T t1 (THead (Flat Appl) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 i v0 w u2))) (\lambda (_: -T).(\lambda (t2: T).(subst0 (s (Flat Appl) i) v0 v t2))) (ex T (\lambda (w0: -T).(ty3 g d v1 w0))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (eq T t1 -(THead (Flat Appl) x0 x1))).(\lambda (_: (subst0 i v0 w x0)).(\lambda (H9: -(subst0 (s (Flat Appl) i) v0 v x1)).(H3 v0 x1 (s (Flat Appl) i) H9 b d v1 -H5)))))) H6)) (subst0_gen_head (Flat Appl) v0 w v t1 i H4))))))))))))))))))) -(\lambda (c0: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g c0 t1 -t2)).(\lambda (H1: ((\forall (v0: T).(\forall (t0: T).(\forall (i: -nat).((subst0 i v0 t1 t0) \to (\forall (b: B).(\forall (d: C).(\forall (v: -T).((getl i c0 (CHead d (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v -w))))))))))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t2 t0)).(\lambda (H3: -((\forall (v0: T).(\forall (t3: T).(\forall (i: nat).((subst0 i v0 t2 t3) \to -(\forall (b: B).(\forall (d: C).(\forall (v: T).((getl i c0 (CHead d (Bind b) -v)) \to (ex T (\lambda (w: T).(ty3 g d v w))))))))))))).(\lambda (v0: -T).(\lambda (t3: T).(\lambda (i: nat).(\lambda (H4: (subst0 i v0 (THead (Flat -Cast) t2 t1) t3)).(\lambda (b: B).(\lambda (d: C).(\lambda (v: T).(\lambda -(H5: (getl i c0 (CHead d (Bind b) v))).(or3_ind (ex2 T (\lambda (u2: T).(eq T -t3 (THead (Flat Cast) u2 t1))) (\lambda (u2: T).(subst0 i v0 t2 u2))) (ex2 T -(\lambda (t4: T).(eq T t3 (THead (Flat Cast) t2 t4))) (\lambda (t4: -T).(subst0 (s (Flat Cast) i) v0 t1 t4))) (ex3_2 T T (\lambda (u2: T).(\lambda -(t4: T).(eq T t3 (THead (Flat Cast) u2 t4)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 i v0 t2 u2))) (\lambda (_: T).(\lambda (t4: T).(subst0 (s (Flat -Cast) i) v0 t1 t4)))) (ex T (\lambda (w: T).(ty3 g d v w))) (\lambda (H6: -(ex2 T (\lambda (u2: T).(eq T t3 (THead (Flat Cast) u2 t1))) (\lambda (u2: -T).(subst0 i v0 t2 u2)))).(ex2_ind T (\lambda (u2: T).(eq T t3 (THead (Flat -Cast) u2 t1))) (\lambda (u2: T).(subst0 i v0 t2 u2)) (ex T (\lambda (w: -T).(ty3 g d v w))) (\lambda (x: T).(\lambda (_: (eq T t3 (THead (Flat Cast) x -t1))).(\lambda (H8: (subst0 i v0 t2 x)).(H3 v0 x i H8 b d v H5)))) H6)) -(\lambda (H6: (ex2 T (\lambda (t4: T).(eq T t3 (THead (Flat Cast) t2 t4))) -(\lambda (t4: T).(subst0 (s (Flat Cast) i) v0 t1 t4)))).(ex2_ind T (\lambda -(t4: T).(eq T t3 (THead (Flat Cast) t2 t4))) (\lambda (t4: T).(subst0 (s -(Flat Cast) i) v0 t1 t4)) (ex T (\lambda (w: T).(ty3 g d v w))) (\lambda (x: -T).(\lambda (_: (eq T t3 (THead (Flat Cast) t2 x))).(\lambda (H8: (subst0 (s -(Flat Cast) i) v0 t1 x)).(H1 v0 x (s (Flat Cast) i) H8 b d v H5)))) H6)) -(\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead -(Flat Cast) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 t2 u2))) -(\lambda (_: T).(\lambda (t4: T).(subst0 (s (Flat Cast) i) v0 t1 -t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead -(Flat Cast) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 t2 u2))) -(\lambda (_: T).(\lambda (t4: T).(subst0 (s (Flat Cast) i) v0 t1 t4))) (ex T -(\lambda (w: T).(ty3 g d v w))) (\lambda (x0: T).(\lambda (x1: T).(\lambda -(_: (eq T t3 (THead (Flat Cast) x0 x1))).(\lambda (H8: (subst0 i v0 t2 -x0)).(\lambda (_: (subst0 (s (Flat Cast) i) v0 t1 x1)).(H3 v0 x0 i H8 b d v -H5)))))) H6)) (subst0_gen_head (Flat Cast) v0 t2 t1 t3 i H4)))))))))))))))))) -c t u H))))). -(* COMMENTS -Initial nodes: 4343 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/sty0.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/sty0.ma deleted file mode 100644 index 3c630662c..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/sty0.ma +++ /dev/null @@ -1,237 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/ty3/pr3_props.ma". - -include "Basic-1/sty0/fwd.ma". - -theorem ty3_sty0: - \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).((ty3 g c u -t1) \to (\forall (t2: T).((sty0 g c u t2) \to (ty3 g c u t2))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (H: -(ty3 g c u t1)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (_: -T).(\forall (t2: T).((sty0 g c0 t t2) \to (ty3 g c0 t t2)))))) (\lambda (c0: -C).(\lambda (t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda -(_: ((\forall (t3: T).((sty0 g c0 t2 t3) \to (ty3 g c0 t2 t3))))).(\lambda -(u0: T).(\lambda (t3: T).(\lambda (_: (ty3 g c0 u0 t3)).(\lambda (H3: -((\forall (t4: T).((sty0 g c0 u0 t4) \to (ty3 g c0 u0 t4))))).(\lambda (_: -(pc3 c0 t3 t2)).(\lambda (t0: T).(\lambda (H5: (sty0 g c0 u0 t0)).(H3 t0 -H5))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (t2: T).(\lambda -(H0: (sty0 g c0 (TSort m) t2)).(let H_y \def (sty0_gen_sort g c0 t2 m H0) in -(let H1 \def (f_equal T T (\lambda (e: T).e) t2 (TSort (next g m)) H_y) in -(eq_ind_r T (TSort (next g m)) (\lambda (t: T).(ty3 g c0 (TSort m) t)) -(ty3_sort g c0 m) t2 H1))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda -(d: C).(\lambda (u0: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abbr) -u0))).(\lambda (t: T).(\lambda (_: (ty3 g d u0 t)).(\lambda (H2: ((\forall -(t2: T).((sty0 g d u0 t2) \to (ty3 g d u0 t2))))).(\lambda (t2: T).(\lambda -(H3: (sty0 g c0 (TLRef n) t2)).(let H_x \def (sty0_gen_lref g c0 t2 n H3) in -(let H4 \def H_x in (or_ind (ex3_3 C T T (\lambda (e: C).(\lambda (u1: -T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: -C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (t0: T).(eq T t2 (lift (S n) O t0)))))) (ex3_3 C -T T (\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e -(Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g -e u1 t0)))) (\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq T t2 (lift -(S n) O u1)))))) (ty3 g c0 (TLRef n) t2) (\lambda (H5: (ex3_3 C T T (\lambda -(e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) -u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g e u1 t0)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(eq T t2 (lift (S n) O -t0))))))).(ex3_3_ind C T T (\lambda (e: C).(\lambda (u1: T).(\lambda (_: -T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: C).(\lambda (u1: -T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (t0: T).(eq T t2 (lift (S n) O t0))))) (ty3 g c0 (TLRef n) t2) -(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H6: (getl n c0 -(CHead x0 (Bind Abbr) x1))).(\lambda (H7: (sty0 g x0 x1 x2)).(\lambda (H8: -(eq T t2 (lift (S n) O x2))).(let H9 \def (f_equal T T (\lambda (e: T).e) t2 -(lift (S n) O x2) H8) in (eq_ind_r T (lift (S n) O x2) (\lambda (t0: T).(ty3 -g c0 (TLRef n) t0)) (let H10 \def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda -(c1: C).(getl n c0 c1)) H0 (CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d -(Bind Abbr) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in (let H11 \def (f_equal -C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u0) -(CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead -x0 (Bind Abbr) x1) H6)) in ((let H12 \def (f_equal C T (\lambda (e: C).(match -e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ -t0) \Rightarrow t0])) (CHead d (Bind Abbr) u0) (CHead x0 (Bind Abbr) x1) -(getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in -(\lambda (H13: (eq C d x0)).(let H14 \def (eq_ind_r T x1 (\lambda (t0: -T).(getl n c0 (CHead x0 (Bind Abbr) t0))) H10 u0 H12) in (let H15 \def -(eq_ind_r T x1 (\lambda (t0: T).(sty0 g x0 t0 x2)) H7 u0 H12) in (let H16 -\def (eq_ind_r C x0 (\lambda (c1: C).(getl n c0 (CHead c1 (Bind Abbr) u0))) -H14 d H13) in (let H17 \def (eq_ind_r C x0 (\lambda (c1: C).(sty0 g c1 u0 -x2)) H15 d H13) in (ty3_abbr g n c0 d u0 H16 x2 (H2 x2 H17)))))))) H11))) t2 -H9)))))))) H5)) (\lambda (H5: (ex3_3 C T T (\lambda (e: C).(\lambda (u1: -T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e: -C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq T t2 (lift (S n) O -u1))))))).(ex3_3_ind C T T (\lambda (e: C).(\lambda (u1: T).(\lambda (_: -T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1: -T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_: C).(\lambda (u1: -T).(\lambda (_: T).(eq T t2 (lift (S n) O u1))))) (ty3 g c0 (TLRef n) t2) -(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H6: (getl n c0 -(CHead x0 (Bind Abst) x1))).(\lambda (_: (sty0 g x0 x1 x2)).(\lambda (H8: (eq -T t2 (lift (S n) O x1))).(let H9 \def (f_equal T T (\lambda (e: T).e) t2 -(lift (S n) O x1) H8) in (eq_ind_r T (lift (S n) O x1) (\lambda (t0: T).(ty3 -g c0 (TLRef n) t0)) (let H10 \def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda -(c1: C).(getl n c0 c1)) H0 (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d -(Bind Abbr) u0) n H0 (CHead x0 (Bind Abst) x1) H6)) in (let H11 \def (eq_ind -C (CHead d (Bind Abbr) u0) (\lambda (ee: C).(match ee in C return (\lambda -(_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow -(match k in K return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match -b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst -\Rightarrow False | Void \Rightarrow False]) | (Flat _) \Rightarrow -False])])) I (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abbr) u0) -n H0 (CHead x0 (Bind Abst) x1) H6)) in (False_ind (ty3 g c0 (TLRef n) (lift -(S n) O x1)) H11))) t2 H9)))))))) H5)) H4))))))))))))) (\lambda (n: -nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (H0: (getl n -c0 (CHead d (Bind Abst) u0))).(\lambda (t: T).(\lambda (H1: (ty3 g d u0 -t)).(\lambda (_: ((\forall (t2: T).((sty0 g d u0 t2) \to (ty3 g d u0 -t2))))).(\lambda (t2: T).(\lambda (H3: (sty0 g c0 (TLRef n) t2)).(let H_x -\def (sty0_gen_lref g c0 t2 n H3) in (let H4 \def H_x in (or_ind (ex3_3 C T T -(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind -Abbr) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g e u1 -t0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(eq T t2 (lift (S n) -O t0)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u1: T).(\lambda (_: -T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1: -T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_: C).(\lambda (u1: -T).(\lambda (_: T).(eq T t2 (lift (S n) O u1)))))) (ty3 g c0 (TLRef n) t2) -(\lambda (H5: (ex3_3 C T T (\lambda (e: C).(\lambda (u1: T).(\lambda (_: -T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: C).(\lambda (u1: -T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (t0: T).(eq T t2 (lift (S n) O t0))))))).(ex3_3_ind C T T -(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind -Abbr) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g e u1 -t0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(eq T t2 (lift (S n) -O t0))))) (ty3 g c0 (TLRef n) t2) (\lambda (x0: C).(\lambda (x1: T).(\lambda -(x2: T).(\lambda (H6: (getl n c0 (CHead x0 (Bind Abbr) x1))).(\lambda (_: -(sty0 g x0 x1 x2)).(\lambda (H8: (eq T t2 (lift (S n) O x2))).(let H9 \def -(f_equal T T (\lambda (e: T).e) t2 (lift (S n) O x2) H8) in (eq_ind_r T (lift -(S n) O x2) (\lambda (t0: T).(ty3 g c0 (TLRef n) t0)) (let H10 \def (eq_ind C -(CHead d (Bind Abst) u0) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead x0 (Bind -Abbr) x1) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead x0 (Bind Abbr) -x1) H6)) in (let H11 \def (eq_ind C (CHead d (Bind Abst) u0) (\lambda (ee: -C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow -False | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) -with [(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with -[Abbr \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | -(Flat _) \Rightarrow False])])) I (CHead x0 (Bind Abbr) x1) (getl_mono c0 -(CHead d (Bind Abst) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in (False_ind -(ty3 g c0 (TLRef n) (lift (S n) O x2)) H11))) t2 H9)))))))) H5)) (\lambda -(H5: (ex3_3 C T T (\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 -(CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: -T).(sty0 g e u1 t0)))) (\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq T -t2 (lift (S n) O u1))))))).(ex3_3_ind C T T (\lambda (e: C).(\lambda (u1: -T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e: -C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq T t2 (lift (S n) O u1))))) (ty3 g c0 -(TLRef n) t2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda -(H6: (getl n c0 (CHead x0 (Bind Abst) x1))).(\lambda (H7: (sty0 g x0 x1 -x2)).(\lambda (H8: (eq T t2 (lift (S n) O x1))).(let H9 \def (f_equal T T -(\lambda (e: T).e) t2 (lift (S n) O x1) H8) in (eq_ind_r T (lift (S n) O x1) -(\lambda (t0: T).(ty3 g c0 (TLRef n) t0)) (let H10 \def (eq_ind C (CHead d -(Bind Abst) u0) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead x0 (Bind Abst) x1) -(getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead x0 (Bind Abst) x1) H6)) in -(let H11 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: -C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead -d (Bind Abst) u0) (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind -Abst) u0) n H0 (CHead x0 (Bind Abst) x1) H6)) in ((let H12 \def (f_equal C T -(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) -\Rightarrow u0 | (CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind Abst) u0) -(CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead -x0 (Bind Abst) x1) H6)) in (\lambda (H13: (eq C d x0)).(let H14 \def -(eq_ind_r T x1 (\lambda (t0: T).(getl n c0 (CHead x0 (Bind Abst) t0))) H10 u0 -H12) in (let H15 \def (eq_ind_r T x1 (\lambda (t0: T).(sty0 g x0 t0 x2)) H7 -u0 H12) in (eq_ind T u0 (\lambda (t0: T).(ty3 g c0 (TLRef n) (lift (S n) O -t0))) (let H16 \def (eq_ind_r C x0 (\lambda (c1: C).(getl n c0 (CHead c1 -(Bind Abst) u0))) H14 d H13) in (let H17 \def (eq_ind_r C x0 (\lambda (c1: -C).(sty0 g c1 u0 x2)) H15 d H13) in (ty3_abst g n c0 d u0 H16 t H1))) x1 -H12))))) H11))) t2 H9)))))))) H5)) H4))))))))))))) (\lambda (c0: C).(\lambda -(u0: T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 u0 t)).(\lambda (_: ((\forall -(t2: T).((sty0 g c0 u0 t2) \to (ty3 g c0 u0 t2))))).(\lambda (b: B).(\lambda -(t2: T).(\lambda (t3: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u0) t2 -t3)).(\lambda (H3: ((\forall (t4: T).((sty0 g (CHead c0 (Bind b) u0) t2 t4) -\to (ty3 g (CHead c0 (Bind b) u0) t2 t4))))).(\lambda (t0: T).(\lambda (H4: -(sty0 g c0 (THead (Bind b) u0 t2) t0)).(let H_x \def (sty0_gen_bind g b c0 u0 -t2 t0 H4) in (let H5 \def H_x in (ex2_ind T (\lambda (t4: T).(sty0 g (CHead -c0 (Bind b) u0) t2 t4)) (\lambda (t4: T).(eq T t0 (THead (Bind b) u0 t4))) -(ty3 g c0 (THead (Bind b) u0 t2) t0) (\lambda (x: T).(\lambda (H6: (sty0 g -(CHead c0 (Bind b) u0) t2 x)).(\lambda (H7: (eq T t0 (THead (Bind b) u0 -x))).(let H8 \def (f_equal T T (\lambda (e: T).e) t0 (THead (Bind b) u0 x) -H7) in (eq_ind_r T (THead (Bind b) u0 x) (\lambda (t4: T).(ty3 g c0 (THead -(Bind b) u0 t2) t4)) (ty3_bind g c0 u0 t H0 b t2 x (H3 x H6)) t0 H8))))) -H5))))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda (u0: T).(\lambda -(H0: (ty3 g c0 w u0)).(\lambda (_: ((\forall (t2: T).((sty0 g c0 w t2) \to -(ty3 g c0 w t2))))).(\lambda (v: T).(\lambda (t: T).(\lambda (H2: (ty3 g c0 v -(THead (Bind Abst) u0 t))).(\lambda (H3: ((\forall (t2: T).((sty0 g c0 v t2) -\to (ty3 g c0 v t2))))).(\lambda (t2: T).(\lambda (H4: (sty0 g c0 (THead -(Flat Appl) w v) t2)).(let H_x \def (sty0_gen_appl g c0 w v t2 H4) in (let H5 -\def H_x in (ex2_ind T (\lambda (t3: T).(sty0 g c0 v t3)) (\lambda (t3: -T).(eq T t2 (THead (Flat Appl) w t3))) (ty3 g c0 (THead (Flat Appl) w v) t2) -(\lambda (x: T).(\lambda (H6: (sty0 g c0 v x)).(\lambda (H7: (eq T t2 (THead -(Flat Appl) w x))).(let H8 \def (f_equal T T (\lambda (e: T).e) t2 (THead -(Flat Appl) w x) H7) in (eq_ind_r T (THead (Flat Appl) w x) (\lambda (t0: -T).(ty3 g c0 (THead (Flat Appl) w v) t0)) (let H_y \def (H3 x H6) in (let H9 -\def (ty3_unique g c0 v x H_y (THead (Bind Abst) u0 t) H2) in (ex_ind T -(\lambda (t0: T).(ty3 g c0 x t0)) (ty3 g c0 (THead (Flat Appl) w v) (THead -(Flat Appl) w x)) (\lambda (x0: T).(\lambda (H10: (ty3 g c0 x x0)).(ex_ind T -(\lambda (t0: T).(ty3 g c0 u0 t0)) (ty3 g c0 (THead (Flat Appl) w v) (THead -(Flat Appl) w x)) (\lambda (x1: T).(\lambda (_: (ty3 g c0 u0 x1)).(ex_ind T -(\lambda (t0: T).(ty3 g c0 (THead (Bind Abst) u0 t) t0)) (ty3 g c0 (THead -(Flat Appl) w v) (THead (Flat Appl) w x)) (\lambda (x2: T).(\lambda (H12: -(ty3 g c0 (THead (Bind Abst) u0 t) x2)).(ex3_2_ind T T (\lambda (t3: -T).(\lambda (_: T).(pc3 c0 (THead (Bind Abst) u0 t3) x2))) (\lambda (_: -T).(\lambda (t0: T).(ty3 g c0 u0 t0))) (\lambda (t3: T).(\lambda (_: T).(ty3 -g (CHead c0 (Bind Abst) u0) t t3))) (ty3 g c0 (THead (Flat Appl) w v) (THead -(Flat Appl) w x)) (\lambda (x3: T).(\lambda (x4: T).(\lambda (_: (pc3 c0 -(THead (Bind Abst) u0 x3) x2)).(\lambda (H14: (ty3 g c0 u0 x4)).(\lambda -(H15: (ty3 g (CHead c0 (Bind Abst) u0) t x3)).(ty3_conv g c0 (THead (Flat -Appl) w x) (THead (Flat Appl) w (THead (Bind Abst) u0 x3)) (ty3_appl g c0 w -u0 H0 x x3 (ty3_sconv g c0 x x0 H10 (THead (Bind Abst) u0 t) (THead (Bind -Abst) u0 x3) (ty3_bind g c0 u0 x4 H14 Abst t x3 H15) H9)) (THead (Flat Appl) -w v) (THead (Flat Appl) w (THead (Bind Abst) u0 t)) (ty3_appl g c0 w u0 H0 v -t H2) (pc3_thin_dx c0 (THead (Bind Abst) u0 t) x (ty3_unique g c0 v (THead -(Bind Abst) u0 t) H2 x H_y) w Appl))))))) (ty3_gen_bind g Abst c0 u0 t x2 -H12)))) (ty3_correct g c0 v (THead (Bind Abst) u0 t) H2)))) (ty3_correct g c0 -w u0 H0)))) (ty3_correct g c0 v x H_y)))) t2 H8))))) H5)))))))))))))) -(\lambda (c0: C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H0: (ty3 g c0 t2 -t3)).(\lambda (H1: ((\forall (t4: T).((sty0 g c0 t2 t4) \to (ty3 g c0 t2 -t4))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t3 t0)).(\lambda (H3: -((\forall (t4: T).((sty0 g c0 t3 t4) \to (ty3 g c0 t3 t4))))).(\lambda (t4: -T).(\lambda (H4: (sty0 g c0 (THead (Flat Cast) t3 t2) t4)).(let H_x \def -(sty0_gen_cast g c0 t3 t2 t4 H4) in (let H5 \def H_x in (ex3_2_ind T T -(\lambda (v2: T).(\lambda (_: T).(sty0 g c0 t3 v2))) (\lambda (_: T).(\lambda -(t5: T).(sty0 g c0 t2 t5))) (\lambda (v2: T).(\lambda (t5: T).(eq T t4 (THead -(Flat Cast) v2 t5)))) (ty3 g c0 (THead (Flat Cast) t3 t2) t4) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H6: (sty0 g c0 t3 x0)).(\lambda (H7: (sty0 g c0 -t2 x1)).(\lambda (H8: (eq T t4 (THead (Flat Cast) x0 x1))).(let H9 \def -(f_equal T T (\lambda (e: T).e) t4 (THead (Flat Cast) x0 x1) H8) in (eq_ind_r -T (THead (Flat Cast) x0 x1) (\lambda (t: T).(ty3 g c0 (THead (Flat Cast) t3 -t2) t)) (let H_y \def (H1 x1 H7) in (let H_y0 \def (H3 x0 H6) in (let H10 -\def (ty3_unique g c0 t2 x1 H_y t3 H0) in (ex_ind T (\lambda (t: T).(ty3 g c0 -x0 t)) (ty3 g c0 (THead (Flat Cast) t3 t2) (THead (Flat Cast) x0 x1)) -(\lambda (x: T).(\lambda (H11: (ty3 g c0 x0 x)).(ex_ind T (\lambda (t: -T).(ty3 g c0 x1 t)) (ty3 g c0 (THead (Flat Cast) t3 t2) (THead (Flat Cast) x0 -x1)) (\lambda (x2: T).(\lambda (H12: (ty3 g c0 x1 x2)).(ty3_conv g c0 (THead -(Flat Cast) x0 x1) (THead (Flat Cast) x x0) (ty3_cast g c0 x1 x0 (ty3_sconv g -c0 x1 x2 H12 t3 x0 H_y0 H10) x H11) (THead (Flat Cast) t3 t2) (THead (Flat -Cast) x0 t3) (ty3_cast g c0 t2 t3 H0 x0 H_y0) (pc3_thin_dx c0 t3 x1 -(ty3_unique g c0 t2 t3 H0 x1 H_y) x0 Cast)))) (ty3_correct g c0 t2 x1 H_y)))) -(ty3_correct g c0 t3 x0 H_y0))))) t4 H9))))))) H5))))))))))))) c u t1 H))))). -(* COMMENTS -Initial nodes: 4539 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/subst1.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/subst1.ma deleted file mode 100644 index ca9516e58..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/subst1.ma +++ /dev/null @@ -1,1102 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/ty3/props.ma". - -include "Basic-1/pc3/subst1.ma". - -include "Basic-1/getl/getl.ma". - -theorem ty3_gen_cabbr: - \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c -t1 t2) \to (\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c -(CHead e (Bind Abbr) u)) \to (\forall (a0: C).((csubst1 d u c a0) \to -(\forall (a: C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda -(_: T).(subst1 d u t1 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: -T).(subst1 d u t2 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 -g a y1 y2)))))))))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H: (ty3 g c t1 t2)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda -(t0: T).(\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead -e (Bind Abbr) u)) \to (\forall (a0: C).((csubst1 d u c0 a0) \to (\forall (a: -C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: -T).(subst1 d u t (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: -T).(subst1 d u t0 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 -g a y1 y2))))))))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t: -T).(\lambda (_: (ty3 g c0 t3 t)).(\lambda (H1: ((\forall (e: C).(\forall (u: -T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u)) \to (\forall (a0: -C).((csubst1 d u c0 a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (ex3_2 T -T (\lambda (y1: T).(\lambda (_: T).(subst1 d u t3 (lift (S O) d y1)))) -(\lambda (_: T).(\lambda (y2: T).(subst1 d u t (lift (S O) d y2)))) (\lambda -(y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))))).(\lambda (u: -T).(\lambda (t4: T).(\lambda (_: (ty3 g c0 u t4)).(\lambda (H3: ((\forall (e: -C).(\forall (u0: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u0)) -\to (\forall (a0: C).((csubst1 d u0 c0 a0) \to (\forall (a: C).((drop (S O) d -a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u (lift (S -O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t4 (lift (S O) d -y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 -y2)))))))))))))).(\lambda (H4: (pc3 c0 t4 t3)).(\lambda (e: C).(\lambda (u0: -T).(\lambda (d: nat).(\lambda (H5: (getl d c0 (CHead e (Bind Abbr) -u0))).(\lambda (a0: C).(\lambda (H6: (csubst1 d u0 c0 a0)).(\lambda (a: -C).(\lambda (H7: (drop (S O) d a0 a)).(let H8 \def (H3 e u0 d H5 a0 H6 a H7) -in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u (lift (S O) -d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t4 (lift (S O) d -y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T -(\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u (lift (S O) d y1)))) (\lambda -(_: T).(\lambda (y2: T).(subst1 d u0 t3 (lift (S O) d y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H9: (subst1 d u0 u (lift (S O) d x0))).(\lambda (H10: (subst1 d -u0 t4 (lift (S O) d x1))).(\lambda (H11: (ty3 g a x0 x1)).(let H12 \def (H1 e -u0 d H5 a0 H6 a H7) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: -T).(subst1 d u0 t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: -T).(subst1 d u0 t (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 -g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u (lift -(S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t3 (lift (S O) d -y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: -T).(\lambda (x3: T).(\lambda (H13: (subst1 d u0 t3 (lift (S O) d -x2))).(\lambda (_: (subst1 d u0 t (lift (S O) d x3))).(\lambda (H15: (ty3 g a -x2 x3)).(ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u -(lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t3 (lift -(S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) x0 x2 H9 -H13 (ty3_conv g a x2 x3 H15 x0 x1 H11 (pc3_gen_cabbr c0 t4 t3 H4 e u0 d H5 a0 -H6 a H7 x1 H10 x2 H13)))))))) H12))))))) H8)))))))))))))))))))) (\lambda (c0: -C).(\lambda (m: nat).(\lambda (e: C).(\lambda (u: T).(\lambda (d: -nat).(\lambda (_: (getl d c0 (CHead e (Bind Abbr) u))).(\lambda (a0: -C).(\lambda (_: (csubst1 d u c0 a0)).(\lambda (a: C).(\lambda (_: (drop (S O) -d a0 a)).(ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u (TSort -m) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u (TSort -(next g m)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a -y1 y2))) (TSort m) (TSort (next g m)) (eq_ind_r T (TSort m) (\lambda (t: -T).(subst1 d u (TSort m) t)) (subst1_refl d u (TSort m)) (lift (S O) d (TSort -m)) (lift_sort m (S O) d)) (eq_ind_r T (TSort (next g m)) (\lambda (t: -T).(subst1 d u (TSort (next g m)) t)) (subst1_refl d u (TSort (next g m))) -(lift (S O) d (TSort (next g m))) (lift_sort (next g m) (S O) d)) (ty3_sort g -a m)))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda -(u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abbr) u))).(\lambda (t: -T).(\lambda (H1: (ty3 g d u t)).(\lambda (H2: ((\forall (e: C).(\forall (u0: -T).(\forall (d0: nat).((getl d0 d (CHead e (Bind Abbr) u0)) \to (\forall (a0: -C).((csubst1 d0 u0 d a0) \to (\forall (a: C).((drop (S O) d0 a0 a) \to (ex3_2 -T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 u (lift (S O) d0 y1)))) -(\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 t (lift (S O) d0 y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))))).(\lambda (e: -C).(\lambda (u0: T).(\lambda (d0: nat).(\lambda (H3: (getl d0 c0 (CHead e -(Bind Abbr) u0))).(\lambda (a0: C).(\lambda (H4: (csubst1 d0 u0 c0 -a0)).(\lambda (a: C).(\lambda (H5: (drop (S O) d0 a0 a)).(lt_eq_gt_e n d0 -(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S -O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O t) -(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) -(\lambda (H6: (lt n d0)).(let H7 \def (eq_ind nat (minus d0 n) (\lambda (n0: -nat).(getl n0 (CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0))) -(getl_conf_le d0 (CHead e (Bind Abbr) u0) c0 H3 (CHead d (Bind Abbr) u) n H0 -(le_S_n n d0 (le_S (S n) d0 H6))) (S (minus d0 (S n))) (minus_x_Sy d0 n H6)) -in (ex2_ind C (\lambda (e2: C).(csubst1 (minus d0 n) u0 (CHead d (Bind Abbr) -u) e2)) (\lambda (e2: C).(getl n a0 e2)) (ex3_2 T T (\lambda (y1: T).(\lambda -(_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda -(y2: T).(subst1 d0 u0 (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x: C).(\lambda (H8: (csubst1 -(minus d0 n) u0 (CHead d (Bind Abbr) u) x)).(\lambda (H9: (getl n a0 x)).(let -H10 \def (eq_ind nat (minus d0 n) (\lambda (n0: nat).(csubst1 n0 u0 (CHead d -(Bind Abbr) u) x)) H8 (S (minus d0 (S n))) (minus_x_Sy d0 n H6)) in (let H11 -\def (csubst1_gen_head (Bind Abbr) d x u u0 (minus d0 (S n)) H10) in -(ex3_2_ind T C (\lambda (u2: T).(\lambda (c2: C).(eq C x (CHead c2 (Bind -Abbr) u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 (minus d0 (S n)) u0 u -u2))) (\lambda (_: T).(\lambda (c2: C).(csubst1 (minus d0 (S n)) u0 d c2))) -(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S -O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O t) -(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) -(\lambda (x0: T).(\lambda (x1: C).(\lambda (H12: (eq C x (CHead x1 (Bind -Abbr) x0))).(\lambda (H13: (subst1 (minus d0 (S n)) u0 u x0)).(\lambda (H14: -(csubst1 (minus d0 (S n)) u0 d x1)).(let H15 \def (eq_ind C x (\lambda (c1: -C).(getl n a0 c1)) H9 (CHead x1 (Bind Abbr) x0) H12) in (let H16 \def (eq_ind -nat d0 (\lambda (n0: nat).(drop (S O) n0 a0 a)) H5 (S (plus n (minus d0 (S -n)))) (lt_plus_minus n d0 H6)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: -C).(eq T x0 (lift (S O) (minus d0 (S n)) v)))) (\lambda (v: T).(\lambda (e0: -C).(getl n a (CHead e0 (Bind Abbr) v)))) (\lambda (_: T).(\lambda (e0: -C).(drop (S O) (minus d0 (S n)) x1 e0))) (ex3_2 T T (\lambda (y1: T).(\lambda -(_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda -(y2: T).(subst1 d0 u0 (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: T).(\lambda (x3: -C).(\lambda (H17: (eq T x0 (lift (S O) (minus d0 (S n)) x2))).(\lambda (H18: -(getl n a (CHead x3 (Bind Abbr) x2))).(\lambda (H19: (drop (S O) (minus d0 (S -n)) x1 x3)).(let H20 \def (eq_ind T x0 (\lambda (t0: T).(subst1 (minus d0 (S -n)) u0 u t0)) H13 (lift (S O) (minus d0 (S n)) x2) H17) in (let H21 \def (H2 -e u0 (minus d0 (S n)) (getl_gen_S (Bind Abbr) d (CHead e (Bind Abbr) u0) u -(minus d0 (S n)) H7) x1 H14 x3 H19) in (ex3_2_ind T T (\lambda (y1: -T).(\lambda (_: T).(subst1 (minus d0 (S n)) u0 u (lift (S O) (minus d0 (S n)) -y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 (minus d0 (S n)) u0 t (lift -(S O) (minus d0 (S n)) y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g x3 y1 -y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) -(lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S -n) O t) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 -y2)))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H22: (subst1 (minus d0 (S -n)) u0 u (lift (S O) (minus d0 (S n)) x4))).(\lambda (H23: (subst1 (minus d0 -(S n)) u0 t (lift (S O) (minus d0 (S n)) x5))).(\lambda (H24: (ty3 g x3 x4 -x5)).(let H25 \def (eq_ind T x4 (\lambda (t0: T).(ty3 g x3 t0 x5)) H24 x2 -(subst1_confluence_lift u x4 u0 (minus d0 (S n)) H22 x2 H20)) in (eq_ind_r -nat (plus (minus d0 (S n)) (S n)) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: -T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: -T).(\lambda (y2: T).(subst1 n0 u0 (lift (S n) O t) (lift (S O) d0 y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r nat (plus (S -n) (minus d0 (S n))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda -(_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda -(y2: T).(subst1 (plus (minus d0 (S n)) (S n)) u0 (lift (S n) O t) (lift (S O) -n0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro -T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 -y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 (plus (minus d0 (S n)) (S n)) -u0 (lift (S n) O t) (lift (S O) (plus (S n) (minus d0 (S n))) y2)))) (\lambda -(y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (TLRef n) (lift (S n) O x5) -(eq_ind_r T (TLRef n) (\lambda (t0: T).(subst1 d0 u0 (TLRef n) t0)) -(subst1_refl d0 u0 (TLRef n)) (lift (S O) d0 (TLRef n)) (lift_lref_lt n (S O) -d0 H6)) (eq_ind_r T (lift (S n) O (lift (S O) (minus d0 (S n)) x5)) (\lambda -(t0: T).(subst1 (plus (minus d0 (S n)) (S n)) u0 (lift (S n) O t) t0)) -(subst1_lift_ge t (lift (S O) (minus d0 (S n)) x5) u0 (minus d0 (S n)) (S n) -H23 O (le_O_n (minus d0 (S n)))) (lift (S O) (plus (S n) (minus d0 (S n))) -(lift (S n) O x5)) (lift_d x5 (S O) (S n) (minus d0 (S n)) O (le_O_n (minus -d0 (S n))))) (ty3_abbr g n a x3 x2 H18 x5 H25)) d0 (le_plus_minus (S n) d0 -H6)) d0 (le_plus_minus_sym (S n) d0 H6)))))))) H21)))))))) (getl_drop_conf_lt -Abbr a0 x1 x0 n H15 a (S O) (minus d0 (S n)) H16))))))))) H11)))))) -(csubst1_getl_lt d0 n H6 c0 a0 u0 H4 (CHead d (Bind Abbr) u) H0)))) (\lambda -(H6: (eq nat n d0)).(let H7 \def (eq_ind_r nat d0 (\lambda (n0: nat).(drop (S -O) n0 a0 a)) H5 n H6) in (let H8 \def (eq_ind_r nat d0 (\lambda (n0: -nat).(csubst1 n0 u0 c0 a0)) H4 n H6) in (let H9 \def (eq_ind_r nat d0 -(\lambda (n0: nat).(getl n0 c0 (CHead e (Bind Abbr) u0))) H3 n H6) in (eq_ind -nat n (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 -n0 u0 (TLRef n) (lift (S O) n0 y1)))) (\lambda (_: T).(\lambda (y2: -T).(subst1 n0 u0 (lift (S n) O t) (lift (S O) n0 y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H10 \def (eq_ind C (CHead d -(Bind Abbr) u) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead e (Bind Abbr) u0) -(getl_mono c0 (CHead d (Bind Abbr) u) n H0 (CHead e (Bind Abbr) u0) H9)) in -(let H11 \def (f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda -(_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) -(CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d (Bind -Abbr) u) n H0 (CHead e (Bind Abbr) u0) H9)) in ((let H12 \def (f_equal C T -(\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) -\Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind Abbr) u) -(CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d (Bind Abbr) u) n H0 (CHead e -(Bind Abbr) u0) H9)) in (\lambda (H13: (eq C d e)).(let H14 \def (eq_ind_r T -u0 (\lambda (t0: T).(getl n c0 (CHead e (Bind Abbr) t0))) H10 u H12) in (let -H15 \def (eq_ind_r T u0 (\lambda (t0: T).(csubst1 n t0 c0 a0)) H8 u H12) in -(eq_ind T u (\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: -T).(subst1 n t0 (TLRef n) (lift (S O) n y1)))) (\lambda (_: T).(\lambda (y2: -T).(subst1 n t0 (lift (S n) O t) (lift (S O) n y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H16 \def (eq_ind_r C e (\lambda -(c1: C).(getl n c0 (CHead c1 (Bind Abbr) u))) H14 d H13) in (ex3_2_intro T T -(\lambda (y1: T).(\lambda (_: T).(subst1 n u (TLRef n) (lift (S O) n y1)))) -(\lambda (_: T).(\lambda (y2: T).(subst1 n u (lift (S n) O t) (lift (S O) n -y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (lift n O u) (lift -n O t) (subst1_single n u (TLRef n) (lift (S O) n (lift n O u)) (eq_ind_r T -(lift (plus (S O) n) O u) (\lambda (t0: T).(subst0 n u (TLRef n) t0)) -(subst0_lref u n) (lift (S O) n (lift n O u)) (lift_free u n (S O) O n (le_n -(plus O n)) (le_O_n n)))) (eq_ind_r T (lift (plus (S O) n) O t) (\lambda (t0: -T).(subst1 n u (lift (S n) O t) t0)) (subst1_refl n u (lift (S n) O t)) (lift -(S O) n (lift n O t)) (lift_free t n (S O) O n (le_n (plus O n)) (le_O_n n))) -(ty3_lift g d u t H1 a O n (getl_conf_ge_drop Abbr a0 d u n (csubst1_getl_ge -n n (le_n n) c0 a0 u H15 (CHead d (Bind Abbr) u) H16) a H7)))) u0 H12))))) -H11))) d0 H6))))) (\lambda (H6: (lt d0 n)).(eq_ind_r nat (S (plus O (minus n -(S O)))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: -T).(subst1 d0 u0 (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda -(y2: T).(subst1 d0 u0 (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind nat (plus (S O) (minus n (S -O))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 -d0 u0 (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: -T).(subst1 d0 u0 (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r nat (plus (minus n (S O)) -(S O)) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 -d0 u0 (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: -T).(subst1 d0 u0 (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1: -T).(\lambda (_: T).(subst1 d0 u0 (TLRef (plus (minus n (S O)) (S O))) (lift -(S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O -t) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) -(TLRef (minus n (S O))) (lift n O t) (eq_ind_r T (TLRef (plus (minus n (S O)) -(S O))) (\lambda (t0: T).(subst1 d0 u0 (TLRef (plus (minus n (S O)) (S O))) -t0)) (subst1_refl d0 u0 (TLRef (plus (minus n (S O)) (S O)))) (lift (S O) d0 -(TLRef (minus n (S O)))) (lift_lref_ge (minus n (S O)) (S O) d0 (lt_le_minus -d0 n H6))) (eq_ind_r T (lift (plus (S O) n) O t) (\lambda (t0: T).(subst1 d0 -u0 (lift (S n) O t) t0)) (subst1_refl d0 u0 (lift (S n) O t)) (lift (S O) d0 -(lift n O t)) (lift_free t n (S O) O d0 (le_S_n d0 (plus O n) (le_S (S d0) -(plus O n) H6)) (le_O_n d0))) (eq_ind_r nat (S (minus n (S O))) (\lambda (n0: -nat).(ty3 g a (TLRef (minus n (S O))) (lift n0 O t))) (ty3_abbr g (minus n (S -O)) a d u (getl_drop_conf_ge n (CHead d (Bind Abbr) u) a0 (csubst1_getl_ge d0 -n (le_S_n d0 n (le_S (S d0) n H6)) c0 a0 u0 H4 (CHead d (Bind Abbr) u) H0) a -(S O) d0 H5 (eq_ind_r nat (plus (S O) d0) (\lambda (n0: nat).(le n0 n)) H6 -(plus d0 (S O)) (plus_sym d0 (S O)))) t H1) n (minus_x_SO n (le_lt_trans O d0 -n (le_O_n d0) H6)))) (plus (S O) (minus n (S O))) (plus_sym (S O) (minus n (S -O)))) (S (plus O (minus n (S O)))) (refl_equal nat (S (plus O (minus n (S -O)))))) n (lt_plus_minus O n (le_lt_trans O d0 n (le_O_n d0) -H6))))))))))))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: -C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abst) -u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u t)).(\lambda (H2: ((\forall (e: -C).(\forall (u0: T).(\forall (d0: nat).((getl d0 d (CHead e (Bind Abbr) u0)) -\to (\forall (a0: C).((csubst1 d0 u0 d a0) \to (\forall (a: C).((drop (S O) -d0 a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 u -(lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 t (lift -(S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 -y2)))))))))))))).(\lambda (e: C).(\lambda (u0: T).(\lambda (d0: nat).(\lambda -(H3: (getl d0 c0 (CHead e (Bind Abbr) u0))).(\lambda (a0: C).(\lambda (H4: -(csubst1 d0 u0 c0 a0)).(\lambda (a: C).(\lambda (H5: (drop (S O) d0 a0 -a)).(lt_eq_gt_e n d0 (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 -u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 -d0 u0 (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: -T).(ty3 g a y1 y2)))) (\lambda (H6: (lt n d0)).(let H7 \def (eq_ind nat -(minus d0 n) (\lambda (n0: nat).(getl n0 (CHead d (Bind Abst) u) (CHead e -(Bind Abbr) u0))) (getl_conf_le d0 (CHead e (Bind Abbr) u0) c0 H3 (CHead d -(Bind Abst) u) n H0 (le_S_n n d0 (le_S (S n) d0 H6))) (S (minus d0 (S n))) -(minus_x_Sy d0 n H6)) in (ex2_ind C (\lambda (e2: C).(csubst1 (minus d0 n) u0 -(CHead d (Bind Abst) u) e2)) (\lambda (e2: C).(getl n a0 e2)) (ex3_2 T T -(\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 -y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O u) (lift -(S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda -(x: C).(\lambda (H8: (csubst1 (minus d0 n) u0 (CHead d (Bind Abst) u) -x)).(\lambda (H9: (getl n a0 x)).(let H10 \def (eq_ind nat (minus d0 n) -(\lambda (n0: nat).(csubst1 n0 u0 (CHead d (Bind Abst) u) x)) H8 (S (minus d0 -(S n))) (minus_x_Sy d0 n H6)) in (let H11 \def (csubst1_gen_head (Bind Abst) -d x u u0 (minus d0 (S n)) H10) in (ex3_2_ind T C (\lambda (u2: T).(\lambda -(c2: C).(eq C x (CHead c2 (Bind Abst) u2)))) (\lambda (u2: T).(\lambda (_: -C).(subst1 (minus d0 (S n)) u0 u u2))) (\lambda (_: T).(\lambda (c2: -C).(csubst1 (minus d0 (S n)) u0 d c2))) (ex3_2 T T (\lambda (y1: T).(\lambda -(_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda -(y2: T).(subst1 d0 u0 (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1: -C).(\lambda (H12: (eq C x (CHead x1 (Bind Abst) x0))).(\lambda (H13: (subst1 -(minus d0 (S n)) u0 u x0)).(\lambda (H14: (csubst1 (minus d0 (S n)) u0 d -x1)).(let H15 \def (eq_ind C x (\lambda (c1: C).(getl n a0 c1)) H9 (CHead x1 -(Bind Abst) x0) H12) in (let H16 \def (eq_ind nat d0 (\lambda (n0: nat).(drop -(S O) n0 a0 a)) H5 (S (plus n (minus d0 (S n)))) (lt_plus_minus n d0 H6)) in -(ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T x0 (lift (S O) (minus d0 -(S n)) v)))) (\lambda (v: T).(\lambda (e0: C).(getl n a (CHead e0 (Bind Abst) -v)))) (\lambda (_: T).(\lambda (e0: C).(drop (S O) (minus d0 (S n)) x1 e0))) -(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S -O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O u) -(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) -(\lambda (x2: T).(\lambda (x3: C).(\lambda (H17: (eq T x0 (lift (S O) (minus -d0 (S n)) x2))).(\lambda (H18: (getl n a (CHead x3 (Bind Abst) x2))).(\lambda -(H19: (drop (S O) (minus d0 (S n)) x1 x3)).(let H20 \def (eq_ind T x0 -(\lambda (t0: T).(subst1 (minus d0 (S n)) u0 u t0)) H13 (lift (S O) (minus d0 -(S n)) x2) H17) in (let H21 \def (H2 e u0 (minus d0 (S n)) (getl_gen_S (Bind -Abst) d (CHead e (Bind Abbr) u0) u (minus d0 (S n)) H7) x1 H14 x3 H19) in -(ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(subst1 (minus d0 (S n)) u0 u -(lift (S O) (minus d0 (S n)) y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 -(minus d0 (S n)) u0 t (lift (S O) (minus d0 (S n)) y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g x3 y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda -(_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda -(y2: T).(subst1 d0 u0 (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x4: T).(\lambda (x5: -T).(\lambda (H22: (subst1 (minus d0 (S n)) u0 u (lift (S O) (minus d0 (S n)) -x4))).(\lambda (_: (subst1 (minus d0 (S n)) u0 t (lift (S O) (minus d0 (S n)) -x5))).(\lambda (H24: (ty3 g x3 x4 x5)).(let H25 \def (eq_ind T x4 (\lambda -(t0: T).(ty3 g x3 t0 x5)) H24 x2 (subst1_confluence_lift u x4 u0 (minus d0 (S -n)) H22 x2 H20)) in (eq_ind_r nat (plus (minus d0 (S n)) (S n)) (\lambda (n0: -nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) -(lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 n0 u0 (lift (S -n) O u) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 -y2))))) (eq_ind_r nat (plus (S n) (minus d0 (S n))) (\lambda (n0: nat).(ex3_2 -T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 -y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 (plus (minus d0 (S n)) (S n)) -u0 (lift (S n) O u) (lift (S O) n0 y2)))) (\lambda (y1: T).(\lambda (y2: -T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: -T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda -(y2: T).(subst1 (plus (minus d0 (S n)) (S n)) u0 (lift (S n) O u) (lift (S O) -(plus (S n) (minus d0 (S n))) y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g -a y1 y2))) (TLRef n) (lift (S n) O x2) (eq_ind_r T (TLRef n) (\lambda (t0: -T).(subst1 d0 u0 (TLRef n) t0)) (subst1_refl d0 u0 (TLRef n)) (lift (S O) d0 -(TLRef n)) (lift_lref_lt n (S O) d0 H6)) (eq_ind_r T (lift (S n) O (lift (S -O) (minus d0 (S n)) x2)) (\lambda (t0: T).(subst1 (plus (minus d0 (S n)) (S -n)) u0 (lift (S n) O u) t0)) (subst1_lift_ge u (lift (S O) (minus d0 (S n)) -x2) u0 (minus d0 (S n)) (S n) H20 O (le_O_n (minus d0 (S n)))) (lift (S O) -(plus (S n) (minus d0 (S n))) (lift (S n) O x2)) (lift_d x2 (S O) (S n) -(minus d0 (S n)) O (le_O_n (minus d0 (S n))))) (ty3_abst g n a x3 x2 H18 x5 -H25)) d0 (le_plus_minus (S n) d0 H6)) d0 (le_plus_minus_sym (S n) d0 -H6)))))))) H21)))))))) (getl_drop_conf_lt Abst a0 x1 x0 n H15 a (S O) (minus -d0 (S n)) H16))))))))) H11)))))) (csubst1_getl_lt d0 n H6 c0 a0 u0 H4 (CHead -d (Bind Abst) u) H0)))) (\lambda (H6: (eq nat n d0)).(let H7 \def (eq_ind_r -nat d0 (\lambda (n0: nat).(drop (S O) n0 a0 a)) H5 n H6) in (let H8 \def -(eq_ind_r nat d0 (\lambda (n0: nat).(csubst1 n0 u0 c0 a0)) H4 n H6) in (let -H9 \def (eq_ind_r nat d0 (\lambda (n0: nat).(getl n0 c0 (CHead e (Bind Abbr) -u0))) H3 n H6) in (eq_ind nat n (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: -T).(\lambda (_: T).(subst1 n0 u0 (TLRef n) (lift (S O) n0 y1)))) (\lambda (_: -T).(\lambda (y2: T).(subst1 n0 u0 (lift (S n) O u) (lift (S O) n0 y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H10 \def (eq_ind C -(CHead d (Bind Abst) u) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead e (Bind -Abbr) u0) (getl_mono c0 (CHead d (Bind Abst) u) n H0 (CHead e (Bind Abbr) u0) -H9)) in (let H11 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (ee: -C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow -False | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) -with [(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with -[Abbr \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | -(Flat _) \Rightarrow False])])) I (CHead e (Bind Abbr) u0) (getl_mono c0 -(CHead d (Bind Abst) u) n H0 (CHead e (Bind Abbr) u0) H9)) in (False_ind -(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 n u0 (TLRef n) (lift (S -O) n y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 n u0 (lift (S n) O u) -(lift (S O) n y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) -H11))) d0 H6))))) (\lambda (H6: (lt d0 n)).(eq_ind_r nat (S (plus O (minus n -(S O)))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: -T).(subst1 d0 u0 (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda -(y2: T).(subst1 d0 u0 (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind nat (plus (S O) (minus n (S -O))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 -d0 u0 (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: -T).(subst1 d0 u0 (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r nat (plus (minus n (S O)) -(S O)) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 -d0 u0 (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: -T).(subst1 d0 u0 (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1: -T).(\lambda (_: T).(subst1 d0 u0 (TLRef (plus (minus n (S O)) (S O))) (lift -(S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O -u) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) -(TLRef (minus n (S O))) (lift n O u) (eq_ind_r T (TLRef (plus (minus n (S O)) -(S O))) (\lambda (t0: T).(subst1 d0 u0 (TLRef (plus (minus n (S O)) (S O))) -t0)) (subst1_refl d0 u0 (TLRef (plus (minus n (S O)) (S O)))) (lift (S O) d0 -(TLRef (minus n (S O)))) (lift_lref_ge (minus n (S O)) (S O) d0 (lt_le_minus -d0 n H6))) (eq_ind_r T (lift (plus (S O) n) O u) (\lambda (t0: T).(subst1 d0 -u0 (lift (S n) O u) t0)) (subst1_refl d0 u0 (lift (S n) O u)) (lift (S O) d0 -(lift n O u)) (lift_free u n (S O) O d0 (le_S_n d0 (plus O n) (le_S (S d0) -(plus O n) H6)) (le_O_n d0))) (eq_ind_r nat (S (minus n (S O))) (\lambda (n0: -nat).(ty3 g a (TLRef (minus n (S O))) (lift n0 O u))) (ty3_abst g (minus n (S -O)) a d u (getl_drop_conf_ge n (CHead d (Bind Abst) u) a0 (csubst1_getl_ge d0 -n (le_S_n d0 n (le_S (S d0) n H6)) c0 a0 u0 H4 (CHead d (Bind Abst) u) H0) a -(S O) d0 H5 (eq_ind_r nat (plus (S O) d0) (\lambda (n0: nat).(le n0 n)) H6 -(plus d0 (S O)) (plus_sym d0 (S O)))) t H1) n (minus_x_SO n (le_lt_trans O d0 -n (le_O_n d0) H6)))) (plus (S O) (minus n (S O))) (plus_sym (S O) (minus n (S -O)))) (S (plus O (minus n (S O)))) (refl_equal nat (S (plus O (minus n (S -O)))))) n (lt_plus_minus O n (le_lt_trans O d0 n (le_O_n d0) -H6))))))))))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (t: -T).(\lambda (_: (ty3 g c0 u t)).(\lambda (H1: ((\forall (e: C).(\forall (u0: -T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u0)) \to (\forall (a0: -C).((csubst1 d u0 c0 a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (ex3_2 -T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u (lift (S O) d y1)))) -(\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t (lift (S O) d y2)))) (\lambda -(y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))))).(\lambda (b: -B).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) -u) t3 t4)).(\lambda (H3: ((\forall (e: C).(\forall (u0: T).(\forall (d: -nat).((getl d (CHead c0 (Bind b) u) (CHead e (Bind Abbr) u0)) \to (\forall -(a0: C).((csubst1 d u0 (CHead c0 (Bind b) u) a0) \to (\forall (a: C).((drop -(S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 t3 -(lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t4 (lift -(S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 -y2)))))))))))))).(\lambda (e: C).(\lambda (u0: T).(\lambda (d: nat).(\lambda -(H4: (getl d c0 (CHead e (Bind Abbr) u0))).(\lambda (a0: C).(\lambda (H5: -(csubst1 d u0 c0 a0)).(\lambda (a: C).(\lambda (H6: (drop (S O) d a0 a)).(let -H7 \def (H1 e u0 d H4 a0 H5 a H6) in (ex3_2_ind T T (\lambda (y1: T).(\lambda -(_: T).(subst1 d u0 u (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: -T).(subst1 d u0 t (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 -g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 (THead -(Bind b) u t3) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 -d u0 (THead (Bind b) u t4) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda -(y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H8: -(subst1 d u0 u (lift (S O) d x0))).(\lambda (_: (subst1 d u0 t (lift (S O) d -x1))).(\lambda (H10: (ty3 g a x0 x1)).(let H11 \def (H3 e u0 (S d) (getl_head -(Bind b) d c0 (CHead e (Bind Abbr) u0) H4 u) (CHead a0 (Bind b) (lift (S O) d -x0)) (csubst1_bind b d u0 u (lift (S O) d x0) H8 c0 a0 H5) (CHead a (Bind b) -x0) (drop_skip_bind (S O) d a0 a H6 b x0)) in (ex3_2_ind T T (\lambda (y1: -T).(\lambda (_: T).(subst1 (S d) u0 t3 (lift (S O) (S d) y1)))) (\lambda (_: -T).(\lambda (y2: T).(subst1 (S d) u0 t4 (lift (S O) (S d) y2)))) (\lambda -(y1: T).(\lambda (y2: T).(ty3 g (CHead a (Bind b) x0) y1 y2))) (ex3_2 T T -(\lambda (y1: T).(\lambda (_: T).(subst1 d u0 (THead (Bind b) u t3) (lift (S -O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Bind b) u -t4) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) -(\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (subst1 (S d) u0 t3 (lift (S -O) (S d) x2))).(\lambda (H13: (subst1 (S d) u0 t4 (lift (S O) (S d) -x3))).(\lambda (H14: (ty3 g (CHead a (Bind b) x0) x2 x3)).(ex3_2_intro T T -(\lambda (y1: T).(\lambda (_: T).(subst1 d u0 (THead (Bind b) u t3) (lift (S -O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Bind b) u -t4) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) -(THead (Bind b) x0 x2) (THead (Bind b) x0 x3) (eq_ind_r T (THead (Bind b) -(lift (S O) d x0) (lift (S O) (S d) x2)) (\lambda (t0: T).(subst1 d u0 (THead -(Bind b) u t3) t0)) (subst1_head u0 u (lift (S O) d x0) d H8 (Bind b) t3 -(lift (S O) (S d) x2) H12) (lift (S O) d (THead (Bind b) x0 x2)) (lift_bind b -x0 x2 (S O) d)) (eq_ind_r T (THead (Bind b) (lift (S O) d x0) (lift (S O) (S -d) x3)) (\lambda (t0: T).(subst1 d u0 (THead (Bind b) u t4) t0)) (subst1_head -u0 u (lift (S O) d x0) d H8 (Bind b) t4 (lift (S O) (S d) x3) H13) (lift (S -O) d (THead (Bind b) x0 x3)) (lift_bind b x0 x3 (S O) d)) (ty3_bind g a x0 x1 -H10 b x2 x3 H14))))))) H11))))))) H7)))))))))))))))))))) (\lambda (c0: -C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda (H1: -((\forall (e: C).(\forall (u0: T).(\forall (d: nat).((getl d c0 (CHead e -(Bind Abbr) u0)) \to (\forall (a0: C).((csubst1 d u0 c0 a0) \to (\forall (a: -C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: -T).(subst1 d u0 w (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: -T).(subst1 d u0 u (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 -g a y1 y2)))))))))))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g -c0 v (THead (Bind Abst) u t))).(\lambda (H3: ((\forall (e: C).(\forall (u0: -T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u0)) \to (\forall (a0: -C).((csubst1 d u0 c0 a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (ex3_2 -T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 v (lift (S O) d y1)))) -(\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Bind Abst) u t) (lift -(S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 -y2)))))))))))))).(\lambda (e: C).(\lambda (u0: T).(\lambda (d: nat).(\lambda -(H4: (getl d c0 (CHead e (Bind Abbr) u0))).(\lambda (a0: C).(\lambda (H5: -(csubst1 d u0 c0 a0)).(\lambda (a: C).(\lambda (H6: (drop (S O) d a0 a)).(let -H7 \def (H3 e u0 d H4 a0 H5 a H6) in (ex3_2_ind T T (\lambda (y1: T).(\lambda -(_: T).(subst1 d u0 v (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: -T).(subst1 d u0 (THead (Bind Abst) u t) (lift (S O) d y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda -(_: T).(subst1 d u0 (THead (Flat Appl) w v) (lift (S O) d y1)))) (\lambda (_: -T).(\lambda (y2: T).(subst1 d u0 (THead (Flat Appl) w (THead (Bind Abst) u -t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) -(\lambda (x0: T).(\lambda (x1: T).(\lambda (H8: (subst1 d u0 v (lift (S O) d -x0))).(\lambda (H9: (subst1 d u0 (THead (Bind Abst) u t) (lift (S O) d -x1))).(\lambda (H10: (ty3 g a x0 x1)).(let H11 \def (H1 e u0 d H4 a0 H5 a H6) -in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 w (lift (S O) -d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 u (lift (S O) d y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: -T).(\lambda (_: T).(subst1 d u0 (THead (Flat Appl) w v) (lift (S O) d y1)))) -(\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Flat Appl) w (THead -(Bind Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 -g a y1 y2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (subst1 d u0 w -(lift (S O) d x2))).(\lambda (H13: (subst1 d u0 u (lift (S O) d -x3))).(\lambda (H14: (ty3 g a x2 x3)).(let H_x \def (subst1_gen_head (Bind -Abst) u0 u t (lift (S O) d x1) d H9) in (let H15 \def H_x in (ex3_2_ind T T -(\lambda (u2: T).(\lambda (t3: T).(eq T (lift (S O) d x1) (THead (Bind Abst) -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1 d u0 u u2))) (\lambda (_: -T).(\lambda (t3: T).(subst1 (S d) u0 t t3))) (ex3_2 T T (\lambda (y1: -T).(\lambda (_: T).(subst1 d u0 (THead (Flat Appl) w v) (lift (S O) d y1)))) -(\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Flat Appl) w (THead -(Bind Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 -g a y1 y2)))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H16: (eq T (lift (S -O) d x1) (THead (Bind Abst) x4 x5))).(\lambda (H17: (subst1 d u0 u -x4)).(\lambda (H18: (subst1 (S d) u0 t x5)).(let H19 \def (sym_eq T (lift (S -O) d x1) (THead (Bind Abst) x4 x5) H16) in (ex3_2_ind T T (\lambda (y: -T).(\lambda (z: T).(eq T x1 (THead (Bind Abst) y z)))) (\lambda (y: -T).(\lambda (_: T).(eq T x4 (lift (S O) d y)))) (\lambda (_: T).(\lambda (z: -T).(eq T x5 (lift (S O) (S d) z)))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: -T).(subst1 d u0 (THead (Flat Appl) w v) (lift (S O) d y1)))) (\lambda (_: -T).(\lambda (y2: T).(subst1 d u0 (THead (Flat Appl) w (THead (Bind Abst) u -t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) -(\lambda (x6: T).(\lambda (x7: T).(\lambda (H20: (eq T x1 (THead (Bind Abst) -x6 x7))).(\lambda (H21: (eq T x4 (lift (S O) d x6))).(\lambda (H22: (eq T x5 -(lift (S O) (S d) x7))).(let H23 \def (eq_ind T x5 (\lambda (t0: T).(subst1 -(S d) u0 t t0)) H18 (lift (S O) (S d) x7) H22) in (let H24 \def (eq_ind T x4 -(\lambda (t0: T).(subst1 d u0 u t0)) H17 (lift (S O) d x6) H21) in (let H25 -\def (eq_ind T x1 (\lambda (t0: T).(ty3 g a x0 t0)) H10 (THead (Bind Abst) x6 -x7) H20) in (let H26 \def (eq_ind T x6 (\lambda (t0: T).(ty3 g a x0 (THead -(Bind Abst) t0 x7))) H25 x3 (subst1_confluence_lift u x6 u0 d H24 x3 H13)) in -(ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 (THead (Flat -Appl) w v) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 -(THead (Flat Appl) w (THead (Bind Abst) u t)) (lift (S O) d y2)))) (\lambda -(y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (THead (Flat Appl) x2 x0) (THead -(Flat Appl) x2 (THead (Bind Abst) x3 x7)) (eq_ind_r T (THead (Flat Appl) -(lift (S O) d x2) (lift (S O) d x0)) (\lambda (t0: T).(subst1 d u0 (THead -(Flat Appl) w v) t0)) (subst1_head u0 w (lift (S O) d x2) d H12 (Flat Appl) v -(lift (S O) d x0) H8) (lift (S O) d (THead (Flat Appl) x2 x0)) (lift_flat -Appl x2 x0 (S O) d)) (eq_ind_r T (THead (Flat Appl) (lift (S O) d x2) (lift -(S O) d (THead (Bind Abst) x3 x7))) (\lambda (t0: T).(subst1 d u0 (THead -(Flat Appl) w (THead (Bind Abst) u t)) t0)) (subst1_head u0 w (lift (S O) d -x2) d H12 (Flat Appl) (THead (Bind Abst) u t) (lift (S O) d (THead (Bind -Abst) x3 x7)) (eq_ind_r T (THead (Bind Abst) (lift (S O) d x3) (lift (S O) (S -d) x7)) (\lambda (t0: T).(subst1 (s (Flat Appl) d) u0 (THead (Bind Abst) u t) -t0)) (subst1_head u0 u (lift (S O) d x3) (s (Flat Appl) d) H13 (Bind Abst) t -(lift (S O) (S d) x7) H23) (lift (S O) d (THead (Bind Abst) x3 x7)) -(lift_bind Abst x3 x7 (S O) d))) (lift (S O) d (THead (Flat Appl) x2 (THead -(Bind Abst) x3 x7))) (lift_flat Appl x2 (THead (Bind Abst) x3 x7) (S O) d)) -(ty3_appl g a x2 x3 H14 x0 x7 H26))))))))))) (lift_gen_bind Abst x4 x5 x1 (S -O) d H19)))))))) H15)))))))) H11))))))) H7))))))))))))))))))) (\lambda (c0: -C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (ty3 g c0 t3 t4)).(\lambda -(H1: ((\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead e -(Bind Abbr) u)) \to (\forall (a0: C).((csubst1 d u c0 a0) \to (\forall (a: -C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: -T).(subst1 d u t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: -T).(subst1 d u t4 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 -g a y1 y2)))))))))))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t4 -t0)).(\lambda (H3: ((\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl -d c0 (CHead e (Bind Abbr) u)) \to (\forall (a0: C).((csubst1 d u c0 a0) \to -(\forall (a: C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda -(_: T).(subst1 d u t4 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: -T).(subst1 d u t0 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 -g a y1 y2)))))))))))))).(\lambda (e: C).(\lambda (u: T).(\lambda (d: -nat).(\lambda (H4: (getl d c0 (CHead e (Bind Abbr) u))).(\lambda (a0: -C).(\lambda (H5: (csubst1 d u c0 a0)).(\lambda (a: C).(\lambda (H6: (drop (S -O) d a0 a)).(let H7 \def (H3 e u d H4 a0 H5 a H6) in (ex3_2_ind T T (\lambda -(y1: T).(\lambda (_: T).(subst1 d u t4 (lift (S O) d y1)))) (\lambda (_: -T).(\lambda (y2: T).(subst1 d u t0 (lift (S O) d y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda -(_: T).(subst1 d u (THead (Flat Cast) t4 t3) (lift (S O) d y1)))) (\lambda -(_: T).(\lambda (y2: T).(subst1 d u (THead (Flat Cast) t0 t4) (lift (S O) d -y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H8: (subst1 d u t4 (lift (S O) d x0))).(\lambda -(H9: (subst1 d u t0 (lift (S O) d x1))).(\lambda (H10: (ty3 g a x0 x1)).(let -H11 \def (H1 e u d H4 a0 H5 a H6) in (ex3_2_ind T T (\lambda (y1: T).(\lambda -(_: T).(subst1 d u t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: -T).(subst1 d u t4 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 -g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u (THead -(Flat Cast) t4 t3) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: -T).(subst1 d u (THead (Flat Cast) t0 t4) (lift (S O) d y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: T).(\lambda (x3: -T).(\lambda (H12: (subst1 d u t3 (lift (S O) d x2))).(\lambda (H13: (subst1 d -u t4 (lift (S O) d x3))).(\lambda (H14: (ty3 g a x2 x3)).(let H15 \def -(eq_ind T x3 (\lambda (t: T).(ty3 g a x2 t)) H14 x0 (subst1_confluence_lift -t4 x3 u d H13 x0 H8)) in (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: -T).(subst1 d u (THead (Flat Cast) t4 t3) (lift (S O) d y1)))) (\lambda (_: -T).(\lambda (y2: T).(subst1 d u (THead (Flat Cast) t0 t4) (lift (S O) d -y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (THead (Flat Cast) -x0 x2) (THead (Flat Cast) x1 x0) (eq_ind_r T (THead (Flat Cast) (lift (S O) d -x0) (lift (S O) d x2)) (\lambda (t: T).(subst1 d u (THead (Flat Cast) t4 t3) -t)) (subst1_head u t4 (lift (S O) d x0) d H8 (Flat Cast) t3 (lift (S O) d x2) -H12) (lift (S O) d (THead (Flat Cast) x0 x2)) (lift_flat Cast x0 x2 (S O) d)) -(eq_ind_r T (THead (Flat Cast) (lift (S O) d x1) (lift (S O) d x0)) (\lambda -(t: T).(subst1 d u (THead (Flat Cast) t0 t4) t)) (subst1_head u t0 (lift (S -O) d x1) d H9 (Flat Cast) t4 (lift (S O) d x0) H8) (lift (S O) d (THead (Flat -Cast) x1 x0)) (lift_flat Cast x1 x0 (S O) d)) (ty3_cast g a x2 x0 H15 x1 -H10)))))))) H11))))))) H7)))))))))))))))))) c t1 t2 H))))). -(* COMMENTS -Initial nodes: 12848 -END *) - -theorem ty3_gen_cvoid: - \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c -t1 t2) \to (\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c -(CHead e (Bind Void) u)) \to (\forall (a: C).((drop (S O) d c a) \to (ex3_2 T -T (\lambda (y1: T).(\lambda (_: T).(eq T t1 (lift (S O) d y1)))) (\lambda (_: -T).(\lambda (y2: T).(eq T t2 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda -(y2: T).(ty3 g a y1 y2)))))))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H: (ty3 g c t1 t2)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda -(t0: T).(\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead -e (Bind Void) u)) \to (\forall (a: C).((drop (S O) d c0 a) \to (ex3_2 T T -(\lambda (y1: T).(\lambda (_: T).(eq T t (lift (S O) d y1)))) (\lambda (_: -T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda -(y2: T).(ty3 g a y1 y2))))))))))))) (\lambda (c0: C).(\lambda (t3: -T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 t3 t)).(\lambda (H1: ((\forall (e: -C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Void) u)) \to -(\forall (a: C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda -(_: T).(eq T t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t -(lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 -y2)))))))))))).(\lambda (u: T).(\lambda (t4: T).(\lambda (H2: (ty3 g c0 u -t4)).(\lambda (H3: ((\forall (e: C).(\forall (u0: T).(\forall (d: nat).((getl -d c0 (CHead e (Bind Void) u0)) \to (\forall (a: C).((drop (S O) d c0 a) \to -(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T u (lift (S O) d y1)))) -(\lambda (_: T).(\lambda (y2: T).(eq T t4 (lift (S O) d y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (H4: (pc3 c0 t4 -t3)).(\lambda (e: C).(\lambda (u0: T).(\lambda (d: nat).(\lambda (H5: (getl d -c0 (CHead e (Bind Void) u0))).(\lambda (a: C).(\lambda (H6: (drop (S O) d c0 -a)).(let H7 \def (H3 e u0 d H5 a H6) in (ex3_2_ind T T (\lambda (y1: -T).(\lambda (_: T).(eq T u (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: -T).(eq T t4 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a -y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T u (lift (S O) d -y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t3 (lift (S O) d y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H8: (eq T u (lift (S O) d x0))).(\lambda (H9: -(eq T t4 (lift (S O) d x1))).(\lambda (H10: (ty3 g a x0 x1)).(let H11 \def -(eq_ind T t4 (\lambda (t0: T).(pc3 c0 t0 t3)) H4 (lift (S O) d x1) H9) in -(let H12 \def (eq_ind T t4 (\lambda (t0: T).(ty3 g c0 u t0)) H2 (lift (S O) d -x1) H9) in (let H13 \def (eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 (lift (S -O) d x1))) H12 (lift (S O) d x0) H8) in (eq_ind_r T (lift (S O) d x0) -(\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t0 (lift -(S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t3 (lift (S O) d y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H14 \def (H1 e u0 -d H5 a H6) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T t3 (lift -(S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: -T).(\lambda (_: T).(eq T (lift (S O) d x0) (lift (S O) d y1)))) (\lambda (_: -T).(\lambda (y2: T).(eq T t3 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda -(y2: T).(ty3 g a y1 y2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H15: -(eq T t3 (lift (S O) d x2))).(\lambda (H16: (eq T t (lift (S O) d -x3))).(\lambda (H17: (ty3 g a x2 x3)).(let H18 \def (eq_ind T t (\lambda (t0: -T).(ty3 g c0 t3 t0)) H0 (lift (S O) d x3) H16) in (let H19 \def (eq_ind T t3 -(\lambda (t0: T).(ty3 g c0 t0 (lift (S O) d x3))) H18 (lift (S O) d x2) H15) -in (let H20 \def (eq_ind T t3 (\lambda (t0: T).(pc3 c0 (lift (S O) d x1) t0)) -H11 (lift (S O) d x2) H15) in (eq_ind_r T (lift (S O) d x2) (\lambda (t0: -T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (lift (S O) d x0) (lift -(S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T -(\lambda (y1: T).(\lambda (_: T).(eq T (lift (S O) d x0) (lift (S O) d y1)))) -(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d x2) (lift (S O) d y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) x0 x2 (refl_equal T (lift -(S O) d x0)) (refl_equal T (lift (S O) d x2)) (ty3_conv g a x2 x3 H17 x0 x1 -H10 (pc3_gen_lift c0 x1 x2 (S O) d H20 a H6))) t3 H15))))))))) H14)) u -H8))))))))) H7)))))))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda -(e: C).(\lambda (u: T).(\lambda (d: nat).(\lambda (_: (getl d c0 (CHead e -(Bind Void) u))).(\lambda (a: C).(\lambda (_: (drop (S O) d c0 -a)).(ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(eq T (TSort m) (lift -(S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (TSort (next g m)) -(lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) -(TSort m) (TSort (next g m)) (eq_ind_r T (TSort m) (\lambda (t: T).(eq T -(TSort m) t)) (refl_equal T (TSort m)) (lift (S O) d (TSort m)) (lift_sort m -(S O) d)) (eq_ind_r T (TSort (next g m)) (\lambda (t: T).(eq T (TSort (next g -m)) t)) (refl_equal T (TSort (next g m))) (lift (S O) d (TSort (next g m))) -(lift_sort (next g m) (S O) d)) (ty3_sort g a m)))))))))) (\lambda (n: -nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n -c0 (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u -t)).(\lambda (H2: ((\forall (e: C).(\forall (u0: T).(\forall (d0: nat).((getl -d0 d (CHead e (Bind Void) u0)) \to (\forall (a: C).((drop (S O) d0 d a) \to -(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T u (lift (S O) d0 y1)))) -(\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d0 y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (e: C).(\lambda (u0: -T).(\lambda (d0: nat).(\lambda (H3: (getl d0 c0 (CHead e (Bind Void) -u0))).(\lambda (a: C).(\lambda (H4: (drop (S O) d0 c0 a)).(lt_eq_gt_e n d0 -(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 -y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O t) (lift (S O) d0 -y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (H5: (lt -n d0)).(let H6 \def (eq_ind nat (minus d0 n) (\lambda (n0: nat).(getl n0 -(CHead d (Bind Abbr) u) (CHead e (Bind Void) u0))) (getl_conf_le d0 (CHead e -(Bind Void) u0) c0 H3 (CHead d (Bind Abbr) u) n H0 (le_S_n n d0 (le_S (S n) -d0 H5))) (S (minus d0 (S n))) (minus_x_Sy d0 n H5)) in (let H7 \def (eq_ind -nat d0 (\lambda (n0: nat).(drop (S O) n0 c0 a)) H4 (S (plus n (minus d0 (S -n)))) (lt_plus_minus n d0 H5)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: -C).(eq T u (lift (S O) (minus d0 (S n)) v)))) (\lambda (v: T).(\lambda (e0: -C).(getl n a (CHead e0 (Bind Abbr) v)))) (\lambda (_: T).(\lambda (e0: -C).(drop (S O) (minus d0 (S n)) d e0))) (ex3_2 T T (\lambda (y1: T).(\lambda -(_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: -T).(eq T (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda -(y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1: C).(\lambda (H8: -(eq T u (lift (S O) (minus d0 (S n)) x0))).(\lambda (H9: (getl n a (CHead x1 -(Bind Abbr) x0))).(\lambda (H10: (drop (S O) (minus d0 (S n)) d x1)).(let H11 -\def (eq_ind T u (\lambda (t0: T).(\forall (e0: C).(\forall (u1: T).(\forall -(d1: nat).((getl d1 d (CHead e0 (Bind Void) u1)) \to (\forall (a0: C).((drop -(S O) d1 d a0) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t0 (lift -(S O) d1 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d1 -y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a0 y1 y2))))))))))) H2 (lift -(S O) (minus d0 (S n)) x0) H8) in (let H12 \def (eq_ind T u (\lambda (t0: -T).(ty3 g d t0 t)) H1 (lift (S O) (minus d0 (S n)) x0) H8) in (let H13 \def -(H11 e u0 (minus d0 (S n)) (getl_gen_S (Bind Abbr) d (CHead e (Bind Void) u0) -u (minus d0 (S n)) H6) x1 H10) in (ex3_2_ind T T (\lambda (y1: T).(\lambda -(_: T).(eq T (lift (S O) (minus d0 (S n)) x0) (lift (S O) (minus d0 (S n)) -y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) (minus d0 (S n)) -y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g x1 y1 y2))) (ex3_2 T T -(\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) -(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O t) (lift (S O) d0 y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: -T).(\lambda (x3: T).(\lambda (H14: (eq T (lift (S O) (minus d0 (S n)) x0) -(lift (S O) (minus d0 (S n)) x2))).(\lambda (H15: (eq T t (lift (S O) (minus -d0 (S n)) x3))).(\lambda (H16: (ty3 g x1 x2 x3)).(let H17 \def (eq_ind T t -(\lambda (t0: T).(ty3 g d (lift (S O) (minus d0 (S n)) x0) t0)) H12 (lift (S -O) (minus d0 (S n)) x3) H15) in (eq_ind_r T (lift (S O) (minus d0 (S n)) x3) -(\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) -(lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O -t0) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 -y2))))) (let H18 \def (eq_ind_r T x2 (\lambda (t0: T).(ty3 g x1 t0 x3)) H16 -x0 (lift_inj x0 x2 (S O) (minus d0 (S n)) H14)) in (eq_ind T (lift (S O) -(plus (S n) (minus d0 (S n))) (lift (S n) O x3)) (\lambda (t0: T).(ex3_2 T T -(\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) -(\lambda (_: T).(\lambda (y2: T).(eq T t0 (lift (S O) d0 y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind nat d0 (\lambda (n0: -nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) -d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) n0 (lift (S n) O -x3)) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 -y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) -(lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d0 -(lift (S n) O x3)) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: -T).(ty3 g a y1 y2))) (TLRef n) (lift (S n) O x3) (eq_ind_r T (TLRef n) -(\lambda (t0: T).(eq T (TLRef n) t0)) (refl_equal T (TLRef n)) (lift (S O) d0 -(TLRef n)) (lift_lref_lt n (S O) d0 H5)) (refl_equal T (lift (S O) d0 (lift -(S n) O x3))) (ty3_abbr g n a x1 x0 H9 x3 H18)) (plus (S n) (minus d0 (S n))) -(le_plus_minus (S n) d0 H5)) (lift (S n) O (lift (S O) (minus d0 (S n)) x3)) -(lift_d x3 (S O) (S n) (minus d0 (S n)) O (le_O_n (minus d0 (S n)))))) t -H15))))))) H13))))))))) (getl_drop_conf_lt Abbr c0 d u n H0 a (S O) (minus d0 -(S n)) H7))))) (\lambda (H5: (eq nat n d0)).(let H6 \def (eq_ind_r nat d0 -(\lambda (n0: nat).(drop (S O) n0 c0 a)) H4 n H5) in (let H7 \def (eq_ind_r -nat d0 (\lambda (n0: nat).(getl n0 c0 (CHead e (Bind Void) u0))) H3 n H5) in -(eq_ind nat n (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: -T).(eq T (TLRef n) (lift (S O) n0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq -T (lift (S n) O t) (lift (S O) n0 y2)))) (\lambda (y1: T).(\lambda (y2: -T).(ty3 g a y1 y2))))) (let H8 \def (eq_ind C (CHead d (Bind Abbr) u) -(\lambda (c1: C).(getl n c0 c1)) H0 (CHead e (Bind Void) u0) (getl_mono c0 -(CHead d (Bind Abbr) u) n H0 (CHead e (Bind Void) u0) H7)) in (let H9 \def -(eq_ind C (CHead d (Bind Abbr) u) (\lambda (ee: C).(match ee in C return -(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b) -\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow -True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _) -\Rightarrow False])])) I (CHead e (Bind Void) u0) (getl_mono c0 (CHead d -(Bind Abbr) u) n H0 (CHead e (Bind Void) u0) H7)) in (False_ind (ex3_2 T T -(\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) n y1)))) -(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O t) (lift (S O) n y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) H9))) d0 H5)))) (\lambda -(H5: (lt d0 n)).(eq_ind_r nat (S (plus O (minus n (S O)))) (\lambda (n0: -nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n0) (lift (S O) -d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O t) (lift (S O) -d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind nat -(plus (S O) (minus n (S O))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: -T).(\lambda (_: T).(eq T (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: -T).(\lambda (y2: T).(eq T (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda -(y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r nat (plus (minus n (S -O)) (S O)) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq -T (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T -(lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 -g a y1 y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef -(plus (minus n (S O)) (S O))) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda -(y2: T).(eq T (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2))) (TLRef (minus n (S O))) (lift n O t) -(eq_ind_r T (TLRef (plus (minus n (S O)) (S O))) (\lambda (t0: T).(eq T -(TLRef (plus (minus n (S O)) (S O))) t0)) (refl_equal T (TLRef (plus (minus n -(S O)) (S O)))) (lift (S O) d0 (TLRef (minus n (S O)))) (lift_lref_ge (minus -n (S O)) (S O) d0 (lt_le_minus d0 n H5))) (eq_ind_r T (lift (plus (S O) n) O -t) (\lambda (t0: T).(eq T (lift (S n) O t) t0)) (refl_equal T (lift (S n) O -t)) (lift (S O) d0 (lift n O t)) (lift_free t n (S O) O d0 (le_S_n d0 (plus O -n) (le_S (S d0) (plus O n) H5)) (le_O_n d0))) (eq_ind_r nat (S (minus n (S -O))) (\lambda (n0: nat).(ty3 g a (TLRef (minus n (S O))) (lift n0 O t))) -(ty3_abbr g (minus n (S O)) a d u (getl_drop_conf_ge n (CHead d (Bind Abbr) -u) c0 H0 a (S O) d0 H4 (eq_ind_r nat (plus (S O) d0) (\lambda (n0: nat).(le -n0 n)) H5 (plus d0 (S O)) (plus_sym d0 (S O)))) t H1) n (minus_x_SO n -(le_lt_trans O d0 n (le_O_n d0) H5)))) (plus (S O) (minus n (S O))) (plus_sym -(S O) (minus n (S O)))) (S (plus O (minus n (S O)))) (refl_equal nat (S (plus -O (minus n (S O)))))) n (lt_plus_minus O n (le_lt_trans O d0 n (le_O_n d0) -H5))))))))))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: -C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abst) -u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u t)).(\lambda (H2: ((\forall (e: -C).(\forall (u0: T).(\forall (d0: nat).((getl d0 d (CHead e (Bind Void) u0)) -\to (\forall (a: C).((drop (S O) d0 d a) \to (ex3_2 T T (\lambda (y1: -T).(\lambda (_: T).(eq T u (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda -(y2: T).(eq T t (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 -g a y1 y2)))))))))))).(\lambda (e: C).(\lambda (u0: T).(\lambda (d0: -nat).(\lambda (H3: (getl d0 c0 (CHead e (Bind Void) u0))).(\lambda (a: -C).(\lambda (H4: (drop (S O) d0 c0 a)).(lt_eq_gt_e n d0 (ex3_2 T T (\lambda -(y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: -T).(\lambda (y2: T).(eq T (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda -(y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (H5: (lt n d0)).(let H6 -\def (eq_ind nat (minus d0 n) (\lambda (n0: nat).(getl n0 (CHead d (Bind -Abst) u) (CHead e (Bind Void) u0))) (getl_conf_le d0 (CHead e (Bind Void) u0) -c0 H3 (CHead d (Bind Abst) u) n H0 (le_S_n n d0 (le_S (S n) d0 H5))) (S -(minus d0 (S n))) (minus_x_Sy d0 n H5)) in (let H7 \def (eq_ind nat d0 -(\lambda (n0: nat).(drop (S O) n0 c0 a)) H4 (S (plus n (minus d0 (S n)))) -(lt_plus_minus n d0 H5)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: -C).(eq T u (lift (S O) (minus d0 (S n)) v)))) (\lambda (v: T).(\lambda (e0: -C).(getl n a (CHead e0 (Bind Abst) v)))) (\lambda (_: T).(\lambda (e0: -C).(drop (S O) (minus d0 (S n)) d e0))) (ex3_2 T T (\lambda (y1: T).(\lambda -(_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: -T).(eq T (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda -(y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1: C).(\lambda (H8: -(eq T u (lift (S O) (minus d0 (S n)) x0))).(\lambda (H9: (getl n a (CHead x1 -(Bind Abst) x0))).(\lambda (H10: (drop (S O) (minus d0 (S n)) d x1)).(let H11 -\def (eq_ind T u (\lambda (t0: T).(\forall (e0: C).(\forall (u1: T).(\forall -(d1: nat).((getl d1 d (CHead e0 (Bind Void) u1)) \to (\forall (a0: C).((drop -(S O) d1 d a0) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t0 (lift -(S O) d1 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d1 -y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a0 y1 y2))))))))))) H2 (lift -(S O) (minus d0 (S n)) x0) H8) in (let H12 \def (eq_ind T u (\lambda (t0: -T).(ty3 g d t0 t)) H1 (lift (S O) (minus d0 (S n)) x0) H8) in (eq_ind_r T -(lift (S O) (minus d0 (S n)) x0) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: -T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: -T).(\lambda (y2: T).(eq T (lift (S n) O t0) (lift (S O) d0 y2)))) (\lambda -(y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H13 \def (H11 e u0 (minus -d0 (S n)) (getl_gen_S (Bind Abst) d (CHead e (Bind Void) u0) u (minus d0 (S -n)) H6) x1 H10) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T -(lift (S O) (minus d0 (S n)) x0) (lift (S O) (minus d0 (S n)) y1)))) (\lambda -(_: T).(\lambda (y2: T).(eq T t (lift (S O) (minus d0 (S n)) y2)))) (\lambda -(y1: T).(\lambda (y2: T).(ty3 g x1 y1 y2))) (ex3_2 T T (\lambda (y1: -T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: -T).(\lambda (y2: T).(eq T (lift (S n) O (lift (S O) (minus d0 (S n)) x0)) -(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) -(\lambda (x2: T).(\lambda (x3: T).(\lambda (H14: (eq T (lift (S O) (minus d0 -(S n)) x0) (lift (S O) (minus d0 (S n)) x2))).(\lambda (H15: (eq T t (lift (S -O) (minus d0 (S n)) x3))).(\lambda (H16: (ty3 g x1 x2 x3)).(let H17 \def -(eq_ind T t (\lambda (t0: T).(ty3 g d (lift (S O) (minus d0 (S n)) x0) t0)) -H12 (lift (S O) (minus d0 (S n)) x3) H15) in (let H18 \def (eq_ind_r T x2 -(\lambda (t0: T).(ty3 g x1 t0 x3)) H16 x0 (lift_inj x0 x2 (S O) (minus d0 (S -n)) H14)) in (eq_ind T (lift (S O) (plus (S n) (minus d0 (S n))) (lift (S n) -O x0)) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T -(TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t0 -(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) -(eq_ind nat d0 (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: -T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq -T (lift (S O) n0 (lift (S n) O x0)) (lift (S O) d0 y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1: -T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: -T).(\lambda (y2: T).(eq T (lift (S O) d0 (lift (S n) O x0)) (lift (S O) d0 -y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (TLRef n) (lift (S -n) O x0) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (TLRef n) t0)) -(refl_equal T (TLRef n)) (lift (S O) d0 (TLRef n)) (lift_lref_lt n (S O) d0 -H5)) (refl_equal T (lift (S O) d0 (lift (S n) O x0))) (ty3_abst g n a x1 x0 -H9 x3 H18)) (plus (S n) (minus d0 (S n))) (le_plus_minus (S n) d0 H5)) (lift -(S n) O (lift (S O) (minus d0 (S n)) x0)) (lift_d x0 (S O) (S n) (minus d0 (S -n)) O (le_O_n (minus d0 (S n)))))))))))) H13)) u H8)))))))) -(getl_drop_conf_lt Abst c0 d u n H0 a (S O) (minus d0 (S n)) H7))))) (\lambda -(H5: (eq nat n d0)).(let H6 \def (eq_ind_r nat d0 (\lambda (n0: nat).(drop (S -O) n0 c0 a)) H4 n H5) in (let H7 \def (eq_ind_r nat d0 (\lambda (n0: -nat).(getl n0 c0 (CHead e (Bind Void) u0))) H3 n H5) in (eq_ind nat n -(\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef -n) (lift (S O) n0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O -u) (lift (S O) n0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 -y2))))) (let H8 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (c1: C).(getl -n c0 c1)) H0 (CHead e (Bind Void) u0) (getl_mono c0 (CHead d (Bind Abst) u) n -H0 (CHead e (Bind Void) u0) H7)) in (let H9 \def (eq_ind C (CHead d (Bind -Abst) u) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with -[(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return -(\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B return -(\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow True | -Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead e (Bind -Void) u0) (getl_mono c0 (CHead d (Bind Abst) u) n H0 (CHead e (Bind Void) u0) -H7)) in (False_ind (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef -n) (lift (S O) n y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O -u) (lift (S O) n y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) -H9))) d0 H5)))) (\lambda (H5: (lt d0 n)).(eq_ind_r nat (S (plus O (minus n (S -O)))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T -(TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift -(S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a -y1 y2))))) (eq_ind nat (plus (S O) (minus n (S O))) (\lambda (n0: nat).(ex3_2 -T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n0) (lift (S O) d0 y1)))) -(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O u) (lift (S O) d0 y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r nat (plus -(minus n (S O)) (S O)) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: -T).(\lambda (_: T).(eq T (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: -T).(\lambda (y2: T).(eq T (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda -(y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1: -T).(\lambda (_: T).(eq T (TLRef (plus (minus n (S O)) (S O))) (lift (S O) d0 -y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O u) (lift (S O) d0 -y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (TLRef (minus n (S -O))) (lift n O u) (eq_ind_r T (TLRef (plus (minus n (S O)) (S O))) (\lambda -(t0: T).(eq T (TLRef (plus (minus n (S O)) (S O))) t0)) (refl_equal T (TLRef -(plus (minus n (S O)) (S O)))) (lift (S O) d0 (TLRef (minus n (S O)))) -(lift_lref_ge (minus n (S O)) (S O) d0 (lt_le_minus d0 n H5))) (eq_ind_r T -(lift (plus (S O) n) O u) (\lambda (t0: T).(eq T (lift (S n) O u) t0)) -(refl_equal T (lift (S n) O u)) (lift (S O) d0 (lift n O u)) (lift_free u n -(S O) O d0 (le_S_n d0 (plus O n) (le_S (S d0) (plus O n) H5)) (le_O_n d0))) -(eq_ind_r nat (S (minus n (S O))) (\lambda (n0: nat).(ty3 g a (TLRef (minus n -(S O))) (lift n0 O u))) (ty3_abst g (minus n (S O)) a d u (getl_drop_conf_ge -n (CHead d (Bind Abst) u) c0 H0 a (S O) d0 H4 (eq_ind_r nat (plus (S O) d0) -(\lambda (n0: nat).(le n0 n)) H5 (plus d0 (S O)) (plus_sym d0 (S O)))) t H1) -n (minus_x_SO n (le_lt_trans O d0 n (le_O_n d0) H5)))) (plus (S O) (minus n -(S O))) (plus_sym (S O) (minus n (S O)))) (S (plus O (minus n (S O)))) -(refl_equal nat (S (plus O (minus n (S O)))))) n (lt_plus_minus O n -(le_lt_trans O d0 n (le_O_n d0) H5))))))))))))))))))) (\lambda (c0: -C).(\lambda (u: T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 u t)).(\lambda -(H1: ((\forall (e: C).(\forall (u0: T).(\forall (d: nat).((getl d c0 (CHead e -(Bind Void) u0)) \to (\forall (a: C).((drop (S O) d c0 a) \to (ex3_2 T T -(\lambda (y1: T).(\lambda (_: T).(eq T u (lift (S O) d y1)))) (\lambda (_: -T).(\lambda (y2: T).(eq T t (lift (S O) d y2)))) (\lambda (y1: T).(\lambda -(y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (b: B).(\lambda (t3: T).(\lambda -(t4: T).(\lambda (H2: (ty3 g (CHead c0 (Bind b) u) t3 t4)).(\lambda (H3: -((\forall (e: C).(\forall (u0: T).(\forall (d: nat).((getl d (CHead c0 (Bind -b) u) (CHead e (Bind Void) u0)) \to (\forall (a: C).((drop (S O) d (CHead c0 -(Bind b) u) a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t3 (lift -(S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t4 (lift (S O) d y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (e: -C).(\lambda (u0: T).(\lambda (d: nat).(\lambda (H4: (getl d c0 (CHead e (Bind -Void) u0))).(\lambda (a: C).(\lambda (H5: (drop (S O) d c0 a)).(let H6 \def -(H1 e u0 d H4 a H5) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T -u (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d -y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T -(\lambda (y1: T).(\lambda (_: T).(eq T (THead (Bind b) u t3) (lift (S O) d -y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Bind b) u t4) (lift (S -O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda -(x0: T).(\lambda (x1: T).(\lambda (H7: (eq T u (lift (S O) d x0))).(\lambda -(H8: (eq T t (lift (S O) d x1))).(\lambda (H9: (ty3 g a x0 x1)).(let H10 \def -(eq_ind T t (\lambda (t0: T).(ty3 g c0 u t0)) H0 (lift (S O) d x1) H8) in -(let H11 \def (eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 (lift (S O) d x1))) -H10 (lift (S O) d x0) H7) in (let H12 \def (eq_ind T u (\lambda (t0: -T).(\forall (e0: C).(\forall (u1: T).(\forall (d0: nat).((getl d0 (CHead c0 -(Bind b) t0) (CHead e0 (Bind Void) u1)) \to (\forall (a0: C).((drop (S O) d0 -(CHead c0 (Bind b) t0) a0) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: -T).(eq T t3 (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t4 -(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a0 y1 -y2))))))))))) H3 (lift (S O) d x0) H7) in (let H13 \def (eq_ind T u (\lambda -(t0: T).(ty3 g (CHead c0 (Bind b) t0) t3 t4)) H2 (lift (S O) d x0) H7) in -(eq_ind_r T (lift (S O) d x0) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: -T).(\lambda (_: T).(eq T (THead (Bind b) t0 t3) (lift (S O) d y1)))) (\lambda -(_: T).(\lambda (y2: T).(eq T (THead (Bind b) t0 t4) (lift (S O) d y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H14 \def (H12 e u0 -(S d) (getl_head (Bind b) d c0 (CHead e (Bind Void) u0) H4 (lift (S O) d x0)) -(CHead a (Bind b) x0) (drop_skip_bind (S O) d c0 a H5 b x0)) in (ex3_2_ind T -T (\lambda (y1: T).(\lambda (_: T).(eq T t3 (lift (S O) (S d) y1)))) (\lambda -(_: T).(\lambda (y2: T).(eq T t4 (lift (S O) (S d) y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g (CHead a (Bind b) x0) y1 y2))) (ex3_2 T T (\lambda -(y1: T).(\lambda (_: T).(eq T (THead (Bind b) (lift (S O) d x0) t3) (lift (S -O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Bind b) (lift (S -O) d x0) t4) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a -y1 y2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H15: (eq T t3 (lift (S -O) (S d) x2))).(\lambda (H16: (eq T t4 (lift (S O) (S d) x3))).(\lambda (H17: -(ty3 g (CHead a (Bind b) x0) x2 x3)).(eq_ind_r T (lift (S O) (S d) x3) -(\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead -(Bind b) (lift (S O) d x0) t3) (lift (S O) d y1)))) (\lambda (_: T).(\lambda -(y2: T).(eq T (THead (Bind b) (lift (S O) d x0) t0) (lift (S O) d y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r T (lift (S O) -(S d) x2) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T -(THead (Bind b) (lift (S O) d x0) t0) (lift (S O) d y1)))) (\lambda (_: -T).(\lambda (y2: T).(eq T (THead (Bind b) (lift (S O) d x0) (lift (S O) (S d) -x3)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 -y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Bind -b) (lift (S O) d x0) (lift (S O) (S d) x2)) (lift (S O) d y1)))) (\lambda (_: -T).(\lambda (y2: T).(eq T (THead (Bind b) (lift (S O) d x0) (lift (S O) (S d) -x3)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) -(THead (Bind b) x0 x2) (THead (Bind b) x0 x3) (sym_eq T (lift (S O) d (THead -(Bind b) x0 x2)) (THead (Bind b) (lift (S O) d x0) (lift (S O) (S d) x2)) -(lift_bind b x0 x2 (S O) d)) (sym_eq T (lift (S O) d (THead (Bind b) x0 x3)) -(THead (Bind b) (lift (S O) d x0) (lift (S O) (S d) x3)) (lift_bind b x0 x3 -(S O) d)) (ty3_bind g a x0 x1 H9 b x2 x3 H17)) t3 H15) t4 H16)))))) H14)) u -H7)))))))))) H6)))))))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda -(u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda (H1: ((\forall (e: C).(\forall -(u0: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Void) u0)) \to (\forall -(a: C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: -T).(eq T w (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T u -(lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 -y2)))))))))))).(\lambda (v: T).(\lambda (t: T).(\lambda (H2: (ty3 g c0 v -(THead (Bind Abst) u t))).(\lambda (H3: ((\forall (e: C).(\forall (u0: -T).(\forall (d: nat).((getl d c0 (CHead e (Bind Void) u0)) \to (\forall (a: -C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T -v (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Bind -Abst) u t) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 -y2)))))))))))).(\lambda (e: C).(\lambda (u0: T).(\lambda (d: nat).(\lambda -(H4: (getl d c0 (CHead e (Bind Void) u0))).(\lambda (a: C).(\lambda (H5: -(drop (S O) d c0 a)).(let H6 \def (H3 e u0 d H4 a H5) in (ex3_2_ind T T -(\lambda (y1: T).(\lambda (_: T).(eq T v (lift (S O) d y1)))) (\lambda (_: -T).(\lambda (y2: T).(eq T (THead (Bind Abst) u t) (lift (S O) d y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: -T).(\lambda (_: T).(eq T (THead (Flat Appl) w v) (lift (S O) d y1)))) -(\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) w (THead (Bind -Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a -y1 y2)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: (eq T v (lift (S O) -d x0))).(\lambda (H8: (eq T (THead (Bind Abst) u t) (lift (S O) d -x1))).(\lambda (H9: (ty3 g a x0 x1)).(let H10 \def (eq_ind T v (\lambda (t0: -T).(ty3 g c0 t0 (THead (Bind Abst) u t))) H2 (lift (S O) d x0) H7) in -(eq_ind_r T (lift (S O) d x0) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: -T).(\lambda (_: T).(eq T (THead (Flat Appl) w t0) (lift (S O) d y1)))) -(\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) w (THead (Bind -Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a -y1 y2))))) (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x1 (THead -(Bind Abst) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift (S O) d -y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift (S O) (S d) z)))) (ex3_2 -T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Appl) w (lift (S O) d -x0)) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat -Appl) w (THead (Bind Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: T).(\lambda (x3: -T).(\lambda (H11: (eq T x1 (THead (Bind Abst) x2 x3))).(\lambda (H12: (eq T u -(lift (S O) d x2))).(\lambda (H13: (eq T t (lift (S O) (S d) x3))).(let H14 -\def (eq_ind T x1 (\lambda (t0: T).(ty3 g a x0 t0)) H9 (THead (Bind Abst) x2 -x3) H11) in (eq_ind_r T (lift (S O) (S d) x3) (\lambda (t0: T).(ex3_2 T T -(\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Appl) w (lift (S O) d -x0)) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat -Appl) w (THead (Bind Abst) u t0)) (lift (S O) d y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H15 \def (eq_ind T u (\lambda -(t0: T).(\forall (e0: C).(\forall (u1: T).(\forall (d0: nat).((getl d0 c0 -(CHead e0 (Bind Void) u1)) \to (\forall (a0: C).((drop (S O) d0 c0 a0) \to -(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T w (lift (S O) d0 y1)))) -(\lambda (_: T).(\lambda (y2: T).(eq T t0 (lift (S O) d0 y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a0 y1 y2))))))))))) H1 (lift (S O) d x2) H12) in -(eq_ind_r T (lift (S O) d x2) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: -T).(\lambda (_: T).(eq T (THead (Flat Appl) w (lift (S O) d x0)) (lift (S O) -d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) w (THead -(Bind Abst) t0 (lift (S O) (S d) x3))) (lift (S O) d y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H16 \def (H15 e u0 d H4 a H5) in -(ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T w (lift (S O) d y1)))) -(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d x2) (lift (S O) d y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: -T).(\lambda (_: T).(eq T (THead (Flat Appl) w (lift (S O) d x0)) (lift (S O) -d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) w (THead -(Bind Abst) (lift (S O) d x2) (lift (S O) (S d) x3))) (lift (S O) d y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x4: -T).(\lambda (x5: T).(\lambda (H17: (eq T w (lift (S O) d x4))).(\lambda (H18: -(eq T (lift (S O) d x2) (lift (S O) d x5))).(\lambda (H19: (ty3 g a x4 -x5)).(eq_ind_r T (lift (S O) d x4) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: -T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (lift (S O) d x0)) (lift (S O) -d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) t0 (THead -(Bind Abst) (lift (S O) d x2) (lift (S O) (S d) x3))) (lift (S O) d y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H20 \def (eq_ind_r -T x5 (\lambda (t0: T).(ty3 g a x4 t0)) H19 x2 (lift_inj x2 x5 (S O) d H18)) -in (eq_ind T (lift (S O) d (THead (Bind Abst) x2 x3)) (\lambda (t0: T).(ex3_2 -T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Appl) (lift (S O) d -x4) (lift (S O) d x0)) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: -T).(eq T (THead (Flat Appl) (lift (S O) d x4) t0) (lift (S O) d y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind T (lift (S O) d -(THead (Flat Appl) x4 x0)) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: -T).(\lambda (_: T).(eq T t0 (lift (S O) d y1)))) (\lambda (_: T).(\lambda -(y2: T).(eq T (THead (Flat Appl) (lift (S O) d x4) (lift (S O) d (THead (Bind -Abst) x2 x3))) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g -a y1 y2))))) (eq_ind T (lift (S O) d (THead (Flat Appl) x4 (THead (Bind Abst) -x2 x3))) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T -(lift (S O) d (THead (Flat Appl) x4 x0)) (lift (S O) d y1)))) (\lambda (_: -T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda -(y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: -T).(eq T (lift (S O) d (THead (Flat Appl) x4 x0)) (lift (S O) d y1)))) -(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d (THead (Flat Appl) x4 -(THead (Bind Abst) x2 x3))) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda -(y2: T).(ty3 g a y1 y2))) (THead (Flat Appl) x4 x0) (THead (Flat Appl) x4 -(THead (Bind Abst) x2 x3)) (refl_equal T (lift (S O) d (THead (Flat Appl) x4 -x0))) (refl_equal T (lift (S O) d (THead (Flat Appl) x4 (THead (Bind Abst) x2 -x3)))) (ty3_appl g a x4 x2 H20 x0 x3 H14)) (THead (Flat Appl) (lift (S O) d -x4) (lift (S O) d (THead (Bind Abst) x2 x3))) (lift_flat Appl x4 (THead (Bind -Abst) x2 x3) (S O) d)) (THead (Flat Appl) (lift (S O) d x4) (lift (S O) d -x0)) (lift_flat Appl x4 x0 (S O) d)) (THead (Bind Abst) (lift (S O) d x2) -(lift (S O) (S d) x3)) (lift_bind Abst x2 x3 (S O) d))) w H17)))))) H16)) u -H12)) t H13))))))) (lift_gen_bind Abst u t x1 (S O) d H8)) v H7))))))) -H6))))))))))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: -T).(\lambda (H0: (ty3 g c0 t3 t4)).(\lambda (H1: ((\forall (e: C).(\forall -(u: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Void) u)) \to (\forall -(a: C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: -T).(eq T t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t4 -(lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 -y2)))))))))))).(\lambda (t0: T).(\lambda (H2: (ty3 g c0 t4 t0)).(\lambda (H3: -((\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead e (Bind -Void) u)) \to (\forall (a: C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda -(y1: T).(\lambda (_: T).(eq T t4 (lift (S O) d y1)))) (\lambda (_: -T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda -(y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (e: C).(\lambda (u: T).(\lambda -(d: nat).(\lambda (H4: (getl d c0 (CHead e (Bind Void) u))).(\lambda (a: -C).(\lambda (H5: (drop (S O) d c0 a)).(let H6 \def (H3 e u d H4 a H5) in -(ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T t4 (lift (S O) d y1)))) -(\lambda (_: T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda -(_: T).(eq T (THead (Flat Cast) t4 t3) (lift (S O) d y1)))) (\lambda (_: -T).(\lambda (y2: T).(eq T (THead (Flat Cast) t0 t4) (lift (S O) d y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H7: (eq T t4 (lift (S O) d x0))).(\lambda (H8: -(eq T t0 (lift (S O) d x1))).(\lambda (H9: (ty3 g a x0 x1)).(let H10 \def -(eq_ind T t0 (\lambda (t: T).(ty3 g c0 t4 t)) H2 (lift (S O) d x1) H8) in -(eq_ind_r T (lift (S O) d x1) (\lambda (t: T).(ex3_2 T T (\lambda (y1: -T).(\lambda (_: T).(eq T (THead (Flat Cast) t4 t3) (lift (S O) d y1)))) -(\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Cast) t t4) (lift (S O) d -y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H11 \def -(eq_ind T t4 (\lambda (t: T).(ty3 g c0 t (lift (S O) d x1))) H10 (lift (S O) -d x0) H7) in (let H12 \def (eq_ind T t4 (\lambda (t: T).(\forall (e0: -C).(\forall (u0: T).(\forall (d0: nat).((getl d0 c0 (CHead e0 (Bind Void) -u0)) \to (\forall (a0: C).((drop (S O) d0 c0 a0) \to (ex3_2 T T (\lambda (y1: -T).(\lambda (_: T).(eq T t3 (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda -(y2: T).(eq T t (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 -g a0 y1 y2))))))))))) H1 (lift (S O) d x0) H7) in (let H13 \def (eq_ind T t4 -(\lambda (t: T).(ty3 g c0 t3 t)) H0 (lift (S O) d x0) H7) in (eq_ind_r T -(lift (S O) d x0) (\lambda (t: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: -T).(eq T (THead (Flat Cast) t t3) (lift (S O) d y1)))) (\lambda (_: -T).(\lambda (y2: T).(eq T (THead (Flat Cast) (lift (S O) d x1) t) (lift (S O) -d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H14 \def -(H12 e u d H4 a H5) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T -t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d -x0) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) -(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Cast) (lift (S -O) d x0) t3) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T -(THead (Flat Cast) (lift (S O) d x1) (lift (S O) d x0)) (lift (S O) d y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: -T).(\lambda (x3: T).(\lambda (H15: (eq T t3 (lift (S O) d x2))).(\lambda -(H16: (eq T (lift (S O) d x0) (lift (S O) d x3))).(\lambda (H17: (ty3 g a x2 -x3)).(let H18 \def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 t (lift (S O) d -x0))) H13 (lift (S O) d x2) H15) in (eq_ind_r T (lift (S O) d x2) (\lambda -(t: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Cast) -(lift (S O) d x0) t) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: -T).(eq T (THead (Flat Cast) (lift (S O) d x1) (lift (S O) d x0)) (lift (S O) -d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H19 \def -(eq_ind_r T x3 (\lambda (t: T).(ty3 g a x2 t)) H17 x0 (lift_inj x0 x3 (S O) d -H16)) in (eq_ind T (lift (S O) d (THead (Flat Cast) x0 x2)) (\lambda (t: -T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t (lift (S O) d y1)))) -(\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Cast) (lift (S O) d x1) -(lift (S O) d x0)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: -T).(ty3 g a y1 y2))))) (eq_ind T (lift (S O) d (THead (Flat Cast) x1 x0)) -(\lambda (t: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (lift (S O) -d (THead (Flat Cast) x0 x2)) (lift (S O) d y1)))) (\lambda (_: T).(\lambda -(y2: T).(eq T t (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g -a y1 y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(eq T (lift (S -O) d (THead (Flat Cast) x0 x2)) (lift (S O) d y1)))) (\lambda (_: T).(\lambda -(y2: T).(eq T (lift (S O) d (THead (Flat Cast) x1 x0)) (lift (S O) d y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (THead (Flat Cast) x0 x2) -(THead (Flat Cast) x1 x0) (refl_equal T (lift (S O) d (THead (Flat Cast) x0 -x2))) (refl_equal T (lift (S O) d (THead (Flat Cast) x1 x0))) (ty3_cast g a -x2 x0 H19 x1 H9)) (THead (Flat Cast) (lift (S O) d x1) (lift (S O) d x0)) -(lift_flat Cast x1 x0 (S O) d)) (THead (Flat Cast) (lift (S O) d x0) (lift (S -O) d x2)) (lift_flat Cast x0 x2 (S O) d))) t3 H15))))))) H14)) t4 H7)))) t0 -H8))))))) H6)))))))))))))))) c t1 t2 H))))). -(* COMMENTS -Initial nodes: 13105 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/wcpr0/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/wcpr0/defs.ma deleted file mode 100644 index cdc9bf3a4..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/wcpr0/defs.ma +++ /dev/null @@ -1,26 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr0/defs.ma". - -include "Basic-1/C/defs.ma". - -inductive wcpr0: C \to (C \to Prop) \def -| wcpr0_refl: \forall (c: C).(wcpr0 c c) -| wcpr0_comp: \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall -(u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (k: K).(wcpr0 (CHead c1 k -u1) (CHead c2 k u2)))))))). - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/wcpr0/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/wcpr0/fwd.ma deleted file mode 100644 index 2b0531a8a..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/wcpr0/fwd.ma +++ /dev/null @@ -1,105 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/wcpr0/defs.ma". - -theorem wcpr0_gen_sort: - \forall (x: C).(\forall (n: nat).((wcpr0 (CSort n) x) \to (eq C x (CSort -n)))) -\def - \lambda (x: C).(\lambda (n: nat).(\lambda (H: (wcpr0 (CSort n) -x)).(insert_eq C (CSort n) (\lambda (c: C).(wcpr0 c x)) (\lambda (c: C).(eq C -x c)) (\lambda (y: C).(\lambda (H0: (wcpr0 y x)).(wcpr0_ind (\lambda (c: -C).(\lambda (c0: C).((eq C c (CSort n)) \to (eq C c0 c)))) (\lambda (c: -C).(\lambda (H1: (eq C c (CSort n))).(let H2 \def (f_equal C C (\lambda (e: -C).e) c (CSort n) H1) in (eq_ind_r C (CSort n) (\lambda (c0: C).(eq C c0 c0)) -(refl_equal C (CSort n)) c H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda -(_: (wcpr0 c1 c2)).(\lambda (_: (((eq C c1 (CSort n)) \to (eq C c2 -c1)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda -(k: K).(\lambda (H4: (eq C (CHead c1 k u1) (CSort n))).(let H5 \def (eq_ind C -(CHead c1 k u1) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) -with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I -(CSort n) H4) in (False_ind (eq C (CHead c2 k u2) (CHead c1 k u1)) -H5))))))))))) y x H0))) H))). -(* COMMENTS -Initial nodes: 249 -END *) - -theorem wcpr0_gen_head: - \forall (k: K).(\forall (c1: C).(\forall (x: C).(\forall (u1: T).((wcpr0 -(CHead c1 k u1) x) \to (or (eq C x (CHead c1 k u1)) (ex3_2 C T (\lambda (c2: -C).(\lambda (u2: T).(eq C x (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: -T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2))))))))) -\def - \lambda (k: K).(\lambda (c1: C).(\lambda (x: C).(\lambda (u1: T).(\lambda -(H: (wcpr0 (CHead c1 k u1) x)).(insert_eq C (CHead c1 k u1) (\lambda (c: -C).(wcpr0 c x)) (\lambda (c: C).(or (eq C x c) (ex3_2 C T (\lambda (c2: -C).(\lambda (u2: T).(eq C x (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: -T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))) (\lambda -(y: C).(\lambda (H0: (wcpr0 y x)).(wcpr0_ind (\lambda (c: C).(\lambda (c0: -C).((eq C c (CHead c1 k u1)) \to (or (eq C c0 c) (ex3_2 C T (\lambda (c2: -C).(\lambda (u2: T).(eq C c0 (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: -T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))))) -(\lambda (c: C).(\lambda (H1: (eq C c (CHead c1 k u1))).(let H2 \def (f_equal -C C (\lambda (e: C).e) c (CHead c1 k u1) H1) in (eq_ind_r C (CHead c1 k u1) -(\lambda (c0: C).(or (eq C c0 c0) (ex3_2 C T (\lambda (c2: C).(\lambda (u2: -T).(eq C c0 (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: T).(wcpr0 c1 -c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))) (or_introl (eq C -(CHead c1 k u1) (CHead c1 k u1)) (ex3_2 C T (\lambda (c2: C).(\lambda (u2: -T).(eq C (CHead c1 k u1) (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: -T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))) -(refl_equal C (CHead c1 k u1))) c H2)))) (\lambda (c0: C).(\lambda (c2: -C).(\lambda (H1: (wcpr0 c0 c2)).(\lambda (H2: (((eq C c0 (CHead c1 k u1)) \to -(or (eq C c2 c0) (ex3_2 C T (\lambda (c3: C).(\lambda (u2: T).(eq C c2 (CHead -c3 k u2)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: -C).(\lambda (u2: T).(pr0 u1 u2)))))))).(\lambda (u0: T).(\lambda (u2: -T).(\lambda (H3: (pr0 u0 u2)).(\lambda (k0: K).(\lambda (H4: (eq C (CHead c0 -k0 u0) (CHead c1 k u1))).(let H5 \def (f_equal C C (\lambda (e: C).(match e -in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) -\Rightarrow c])) (CHead c0 k0 u0) (CHead c1 k u1) H4) in ((let H6 \def -(f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with -[(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead c0 k0 u0) -(CHead c1 k u1) H4) in ((let H7 \def (f_equal C T (\lambda (e: C).(match e in -C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) -\Rightarrow t])) (CHead c0 k0 u0) (CHead c1 k u1) H4) in (\lambda (H8: (eq K -k0 k)).(\lambda (H9: (eq C c0 c1)).(eq_ind_r K k (\lambda (k1: K).(or (eq C -(CHead c2 k1 u2) (CHead c0 k1 u0)) (ex3_2 C T (\lambda (c3: C).(\lambda (u3: -T).(eq C (CHead c2 k1 u2) (CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_: -T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3)))))) (let H10 -\def (eq_ind T u0 (\lambda (t: T).(pr0 t u2)) H3 u1 H7) in (eq_ind_r T u1 -(\lambda (t: T).(or (eq C (CHead c2 k u2) (CHead c0 k t)) (ex3_2 C T (\lambda -(c3: C).(\lambda (u3: T).(eq C (CHead c2 k u2) (CHead c3 k u3)))) (\lambda -(c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 -u1 u3)))))) (let H11 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 k -u1)) \to (or (eq C c2 c) (ex3_2 C T (\lambda (c3: C).(\lambda (u3: T).(eq C -c2 (CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) -(\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))))))) H2 c1 H9) in (let H12 \def -(eq_ind C c0 (\lambda (c: C).(wcpr0 c c2)) H1 c1 H9) in (eq_ind_r C c1 -(\lambda (c: C).(or (eq C (CHead c2 k u2) (CHead c k u1)) (ex3_2 C T (\lambda -(c3: C).(\lambda (u3: T).(eq C (CHead c2 k u2) (CHead c3 k u3)))) (\lambda -(c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 -u1 u3)))))) (or_intror (eq C (CHead c2 k u2) (CHead c1 k u1)) (ex3_2 C T -(\lambda (c3: C).(\lambda (u3: T).(eq C (CHead c2 k u2) (CHead c3 k u3)))) -(\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda -(u3: T).(pr0 u1 u3)))) (ex3_2_intro C T (\lambda (c3: C).(\lambda (u3: T).(eq -C (CHead c2 k u2) (CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 -c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))) c2 u2 (refl_equal C -(CHead c2 k u2)) H12 H10)) c0 H9))) u0 H7)) k0 H8)))) H6)) H5))))))))))) y x -H0))) H))))). -(* COMMENTS -Initial nodes: 1133 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/wcpr0/getl.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/wcpr0/getl.ma deleted file mode 100644 index d3a109e0d..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/wcpr0/getl.ma +++ /dev/null @@ -1,464 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/wcpr0/defs.ma". - -include "Basic-1/getl/props.ma". - -theorem wcpr0_drop: - \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (h: -nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((drop h O c1 (CHead -e1 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c2 -(CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda -(_: C).(\lambda (u2: T).(pr0 u1 u2))))))))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(wcpr0_ind -(\lambda (c: C).(\lambda (c0: C).(\forall (h: nat).(\forall (e1: C).(\forall -(u1: T).(\forall (k: K).((drop h O c (CHead e1 k u1)) \to (ex3_2 C T (\lambda -(e2: C).(\lambda (u2: T).(drop h O c0 (CHead e2 k u2)))) (\lambda (e2: -C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 -u2))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda -(u1: T).(\lambda (k: K).(\lambda (H0: (drop h O c (CHead e1 k -u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c (CHead -e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: -C).(\lambda (u2: T).(pr0 u1 u2))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl -u1)))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wcpr0 c3 -c4)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1: -T).(\forall (k: K).((drop h O c3 (CHead e1 k u1)) \to (ex3_2 C T (\lambda -(e2: C).(\lambda (u2: T).(drop h O c4 (CHead e2 k u2)))) (\lambda (e2: -C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 -u2))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 -u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall -(e1: C).(\forall (u3: T).(\forall (k0: K).((drop n O (CHead c3 k u1) (CHead -e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead -c4 k u2) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) -(\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))) (\lambda (e1: -C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (drop O O (CHead c3 k u1) -(CHead e1 k0 u0))).(let H4 \def (f_equal C C (\lambda (e: C).(match e in C -return (\lambda (_: C).C) with [(CSort _) \Rightarrow c3 | (CHead c _ _) -\Rightarrow c])) (CHead c3 k u1) (CHead e1 k0 u0) (drop_gen_refl (CHead c3 k -u1) (CHead e1 k0 u0) H3)) in ((let H5 \def (f_equal C K (\lambda (e: -C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | -(CHead _ k1 _) \Rightarrow k1])) (CHead c3 k u1) (CHead e1 k0 u0) -(drop_gen_refl (CHead c3 k u1) (CHead e1 k0 u0) H3)) in ((let H6 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead c3 k u1) -(CHead e1 k0 u0) (drop_gen_refl (CHead c3 k u1) (CHead e1 k0 u0) H3)) in -(\lambda (H7: (eq K k k0)).(\lambda (H8: (eq C c3 e1)).(eq_ind K k (\lambda -(k1: K).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c4 k -u2) (CHead e2 k1 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) -(\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))))) (eq_ind T u1 (\lambda (t: -T).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c4 k u2) -(CHead e2 k u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda -(_: C).(\lambda (u3: T).(pr0 t u3))))) (eq_ind C c3 (\lambda (c: C).(ex3_2 C -T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c4 k u2) (CHead e2 k -u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 c e2))) (\lambda (_: -C).(\lambda (u3: T).(pr0 u1 u3))))) (ex3_2_intro C T (\lambda (e2: -C).(\lambda (u3: T).(drop O O (CHead c4 k u2) (CHead e2 k u3)))) (\lambda -(e2: C).(\lambda (_: T).(wcpr0 c3 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 -u1 u3))) c4 u2 (drop_refl (CHead c4 k u2)) H0 H2) e1 H8) u0 H6) k0 H7)))) -H5)) H4)))))) (K_ind (\lambda (k0: K).(\forall (n: nat).(((\forall (e1: -C).(\forall (u3: T).(\forall (k1: K).((drop n O (CHead c3 k0 u1) (CHead e1 k1 -u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c4 k0 -u2) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) -(\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))) \to (\forall (e1: -C).(\forall (u3: T).(\forall (k1: K).((drop (S n) O (CHead c3 k0 u1) (CHead -e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop (S n) O -(CHead c4 k0 u2) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 -e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))))) (\lambda (b: -B).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall -(k0: K).((drop n O (CHead c3 (Bind b) u1) (CHead e1 k0 u3)) \to (ex3_2 C T -(\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c4 (Bind b) u2) (CHead e2 -k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: -C).(\lambda (u4: T).(pr0 u3 u4)))))))))).(\lambda (e1: C).(\lambda (u0: -T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c3 (Bind b) u1) (CHead -e1 k0 u0))).(let H5 \def (H1 n e1 u0 k0 (drop_gen_drop (Bind b) c3 (CHead e1 -k0 u0) u1 n H4)) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop n -O c4 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) -(\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) (ex3_2 C T (\lambda (e2: -C).(\lambda (u3: T).(drop (S n) O (CHead c4 (Bind b) u2) (CHead e2 k0 u3)))) -(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda -(u3: T).(pr0 u0 u3)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (drop -n O c4 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 e1 x0)).(\lambda (H8: (pr0 u0 -x1)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead -c4 (Bind b) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 -e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) x0 x1 (drop_drop -(Bind b) n c4 (CHead x0 k0 x1) H6 u2) H7 H8)))))) H5))))))))) (\lambda (f: -F).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall -(k0: K).((drop n O (CHead c3 (Flat f) u1) (CHead e1 k0 u3)) \to (ex3_2 C T -(\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c4 (Flat f) u2) (CHead e2 -k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: -C).(\lambda (u4: T).(pr0 u3 u4)))))))))).(\lambda (e1: C).(\lambda (u0: -T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c3 (Flat f) u1) (CHead -e1 k0 u0))).(let H5 \def (H1 (S n) e1 u0 k0 (drop_gen_drop (Flat f) c3 (CHead -e1 k0 u0) u1 n H4)) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop -(S n) O c4 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 -e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) (ex3_2 C T (\lambda (e2: -C).(\lambda (u3: T).(drop (S n) O (CHead c4 (Flat f) u2) (CHead e2 k0 u3)))) -(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda -(u3: T).(pr0 u0 u3)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (drop -(S n) O c4 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 e1 x0)).(\lambda (H8: (pr0 -u0 x1)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O -(CHead c4 (Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: -T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) x0 x1 -(drop_drop (Flat f) n c4 (CHead x0 k0 x1) H6 u2) H7 H8)))))) H5))))))))) k) -h)))))))))) c1 c2 H))). -(* COMMENTS -Initial nodes: 1755 -END *) - -theorem wcpr0_drop_back: - \forall (c1: C).(\forall (c2: C).((wcpr0 c2 c1) \to (\forall (h: -nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((drop h O c1 (CHead -e1 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c2 -(CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda -(_: C).(\lambda (u2: T).(pr0 u2 u1))))))))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c2 c1)).(wcpr0_ind -(\lambda (c: C).(\lambda (c0: C).(\forall (h: nat).(\forall (e1: C).(\forall -(u1: T).(\forall (k: K).((drop h O c0 (CHead e1 k u1)) \to (ex3_2 C T -(\lambda (e2: C).(\lambda (u2: T).(drop h O c (CHead e2 k u2)))) (\lambda -(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 -u2 u1))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda -(u1: T).(\lambda (k: K).(\lambda (H0: (drop h O c (CHead e1 k -u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c (CHead -e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: -C).(\lambda (u2: T).(pr0 u2 u1))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl -u1)))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wcpr0 c3 -c4)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1: -T).(\forall (k: K).((drop h O c4 (CHead e1 k u1)) \to (ex3_2 C T (\lambda -(e2: C).(\lambda (u2: T).(drop h O c3 (CHead e2 k u2)))) (\lambda (e2: -C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 -u1))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 -u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall -(e1: C).(\forall (u3: T).(\forall (k0: K).((drop n O (CHead c4 k u2) (CHead -e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead -c3 k u1) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) -(\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))) (\lambda (e1: -C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (drop O O (CHead c4 k u2) -(CHead e1 k0 u0))).(let H4 \def (f_equal C C (\lambda (e: C).(match e in C -return (\lambda (_: C).C) with [(CSort _) \Rightarrow c4 | (CHead c _ _) -\Rightarrow c])) (CHead c4 k u2) (CHead e1 k0 u0) (drop_gen_refl (CHead c4 k -u2) (CHead e1 k0 u0) H3)) in ((let H5 \def (f_equal C K (\lambda (e: -C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | -(CHead _ k1 _) \Rightarrow k1])) (CHead c4 k u2) (CHead e1 k0 u0) -(drop_gen_refl (CHead c4 k u2) (CHead e1 k0 u0) H3)) in ((let H6 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c4 k u2) -(CHead e1 k0 u0) (drop_gen_refl (CHead c4 k u2) (CHead e1 k0 u0) H3)) in -(\lambda (H7: (eq K k k0)).(\lambda (H8: (eq C c4 e1)).(eq_ind K k (\lambda -(k1: K).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c3 k -u1) (CHead e2 k1 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) -(\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))))) (eq_ind T u2 (\lambda (t: -T).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c3 k u1) -(CHead e2 k u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda -(_: C).(\lambda (u3: T).(pr0 u3 t))))) (eq_ind C c4 (\lambda (c: C).(ex3_2 C -T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c3 k u1) (CHead e2 k -u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 c))) (\lambda (_: -C).(\lambda (u3: T).(pr0 u3 u2))))) (ex3_2_intro C T (\lambda (e2: -C).(\lambda (u3: T).(drop O O (CHead c3 k u1) (CHead e2 k u3)))) (\lambda -(e2: C).(\lambda (_: T).(wcpr0 e2 c4))) (\lambda (_: C).(\lambda (u3: T).(pr0 -u3 u2))) c3 u1 (drop_refl (CHead c3 k u1)) H0 H2) e1 H8) u0 H6) k0 H7)))) -H5)) H4)))))) (K_ind (\lambda (k0: K).(\forall (n: nat).(((\forall (e1: -C).(\forall (u3: T).(\forall (k1: K).((drop n O (CHead c4 k0 u2) (CHead e1 k1 -u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c3 k0 -u1) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) -(\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))) \to (\forall (e1: -C).(\forall (u3: T).(\forall (k1: K).((drop (S n) O (CHead c4 k0 u2) (CHead -e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop (S n) O -(CHead c3 k0 u1) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 -e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))))) (\lambda (b: -B).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall -(k0: K).((drop n O (CHead c4 (Bind b) u2) (CHead e1 k0 u3)) \to (ex3_2 C T -(\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c3 (Bind b) u1) (CHead e2 -k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: -C).(\lambda (u4: T).(pr0 u4 u3)))))))))).(\lambda (e1: C).(\lambda (u0: -T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c4 (Bind b) u2) (CHead -e1 k0 u0))).(let H5 \def (H1 n e1 u0 k0 (drop_gen_drop (Bind b) c4 (CHead e1 -k0 u0) u2 n H4)) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop n -O c3 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) -(\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) (ex3_2 C T (\lambda (e2: -C).(\lambda (u3: T).(drop (S n) O (CHead c3 (Bind b) u1) (CHead e2 k0 u3)))) -(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda -(u3: T).(pr0 u3 u0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (drop -n O c3 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 x0 e1)).(\lambda (H8: (pr0 x1 -u0)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead -c3 (Bind b) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 -e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) x0 x1 (drop_drop -(Bind b) n c3 (CHead x0 k0 x1) H6 u1) H7 H8)))))) H5))))))))) (\lambda (f: -F).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall -(k0: K).((drop n O (CHead c4 (Flat f) u2) (CHead e1 k0 u3)) \to (ex3_2 C T -(\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c3 (Flat f) u1) (CHead e2 -k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: -C).(\lambda (u4: T).(pr0 u4 u3)))))))))).(\lambda (e1: C).(\lambda (u0: -T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c4 (Flat f) u2) (CHead -e1 k0 u0))).(let H5 \def (H1 (S n) e1 u0 k0 (drop_gen_drop (Flat f) c4 (CHead -e1 k0 u0) u2 n H4)) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop -(S n) O c3 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 -e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) (ex3_2 C T (\lambda (e2: -C).(\lambda (u3: T).(drop (S n) O (CHead c3 (Flat f) u1) (CHead e2 k0 u3)))) -(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda -(u3: T).(pr0 u3 u0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (drop -(S n) O c3 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 x0 e1)).(\lambda (H8: (pr0 -x1 u0)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O -(CHead c3 (Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: -T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) x0 x1 -(drop_drop (Flat f) n c3 (CHead x0 k0 x1) H6 u1) H7 H8)))))) H5))))))))) k) -h)))))))))) c2 c1 H))). -(* COMMENTS -Initial nodes: 1755 -END *) - -theorem wcpr0_getl: - \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (h: -nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((getl h c1 (CHead e1 -k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(getl h c2 (CHead e2 -k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: -C).(\lambda (u2: T).(pr0 u1 u2))))))))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(wcpr0_ind -(\lambda (c: C).(\lambda (c0: C).(\forall (h: nat).(\forall (e1: C).(\forall -(u1: T).(\forall (k: K).((getl h c (CHead e1 k u1)) \to (ex3_2 C T (\lambda -(e2: C).(\lambda (u2: T).(getl h c0 (CHead e2 k u2)))) (\lambda (e2: -C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 -u2))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda -(u1: T).(\lambda (k: K).(\lambda (H0: (getl h c (CHead e1 k -u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(getl h c (CHead e2 -k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: -C).(\lambda (u2: T).(pr0 u1 u2))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl -u1)))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wcpr0 c3 -c4)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1: -T).(\forall (k: K).((getl h c3 (CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2: -C).(\lambda (u2: T).(getl h c4 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda -(_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 -u2))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 -u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall -(e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c3 k u1) (CHead e1 -k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c4 k -u2) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) -(\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))) (\lambda (e1: -C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (getl O (CHead c3 k u1) -(CHead e1 k0 u0))).(K_ind (\lambda (k1: K).((clear (CHead c3 k1 u1) (CHead e1 -k0 u0)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c4 k1 -u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) -(\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))))) (\lambda (b: B).(\lambda -(H4: (clear (CHead c3 (Bind b) u1) (CHead e1 k0 u0))).(let H5 \def (f_equal C -C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow e1 | (CHead c _ _) \Rightarrow c])) (CHead e1 k0 u0) (CHead c3 -(Bind b) u1) (clear_gen_bind b c3 (CHead e1 k0 u0) u1 H4)) in ((let H6 \def -(f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with -[(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead e1 k0 u0) -(CHead c3 (Bind b) u1) (clear_gen_bind b c3 (CHead e1 k0 u0) u1 H4)) in ((let -H7 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead e1 k0 -u0) (CHead c3 (Bind b) u1) (clear_gen_bind b c3 (CHead e1 k0 u0) u1 H4)) in -(\lambda (H8: (eq K k0 (Bind b))).(\lambda (H9: (eq C e1 c3)).(eq_ind_r K -(Bind b) (\lambda (k1: K).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl -O (CHead c4 (Bind b) u2) (CHead e2 k1 u3)))) (\lambda (e2: C).(\lambda (_: -T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))))) (eq_ind_r -T u1 (\lambda (t: T).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O -(CHead c4 (Bind b) u2) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda -(_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 t u3))))) -(eq_ind_r C c3 (\lambda (c: C).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: -T).(getl O (CHead c4 (Bind b) u2) (CHead e2 (Bind b) u3)))) (\lambda (e2: -C).(\lambda (_: T).(wcpr0 c e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 -u3))))) (ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c4 -(Bind b) u2) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda (_: -T).(wcpr0 c3 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))) c4 u2 -(getl_refl b c4 u2) H0 H2) e1 H9) u0 H7) k0 H8)))) H6)) H5)))) (\lambda (f: -F).(\lambda (H4: (clear (CHead c3 (Flat f) u1) (CHead e1 k0 u0))).(let H5 -\def (H1 O e1 u0 k0 (getl_intro O c3 (CHead e1 k0 u0) c3 (drop_refl c3) -(clear_gen_flat f c3 (CHead e1 k0 u0) u1 H4))) in (ex3_2_ind C T (\lambda -(e2: C).(\lambda (u3: T).(getl O c4 (CHead e2 k0 u3)))) (\lambda (e2: -C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 -u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c4 (Flat f) -u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) -(\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))) (\lambda (x0: C).(\lambda -(x1: T).(\lambda (H6: (getl O c4 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 e1 -x0)).(\lambda (H8: (pr0 u0 x1)).(ex3_2_intro C T (\lambda (e2: C).(\lambda -(u3: T).(getl O (CHead c4 (Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2: -C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 -u3))) x0 x1 (getl_flat c4 (CHead x0 k0 x1) O H6 f u2) H7 H8)))))) H5)))) k -(getl_gen_O (CHead c3 k u1) (CHead e1 k0 u0) H3)))))) (K_ind (\lambda (k0: -K).(\forall (n: nat).(((\forall (e1: C).(\forall (u3: T).(\forall (k1: -K).((getl n (CHead c3 k0 u1) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: -C).(\lambda (u4: T).(getl n (CHead c4 k0 u2) (CHead e2 k1 u4)))) (\lambda -(e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 -u3 u4))))))))) \to (\forall (e1: C).(\forall (u3: T).(\forall (k1: K).((getl -(S n) (CHead c3 k0 u1) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: -C).(\lambda (u4: T).(getl (S n) (CHead c4 k0 u2) (CHead e2 k1 u4)))) (\lambda -(e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 -u3 u4))))))))))) (\lambda (b: B).(\lambda (n: nat).(\lambda (_: ((\forall -(e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c3 (Bind b) u1) -(CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n -(CHead c4 (Bind b) u2) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: -T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 u3 -u4)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4: -(getl (S n) (CHead c3 (Bind b) u1) (CHead e1 k0 u0))).(let H5 \def (H1 n e1 -u0 k0 (getl_gen_S (Bind b) c3 (CHead e1 k0 u0) u1 n H4)) in (ex3_2_ind C T -(\lambda (e2: C).(\lambda (u3: T).(getl n c4 (CHead e2 k0 u3)))) (\lambda -(e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 -u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c4 -(Bind b) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 -e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (H6: (getl n c4 (CHead x0 k0 x1))).(\lambda (H7: -(wcpr0 e1 x0)).(\lambda (H8: (pr0 u0 x1)).(ex3_2_intro C T (\lambda (e2: -C).(\lambda (u3: T).(getl (S n) (CHead c4 (Bind b) u2) (CHead e2 k0 u3)))) -(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda -(u3: T).(pr0 u0 u3))) x0 x1 (getl_head (Bind b) n c4 (CHead x0 k0 x1) H6 u2) -H7 H8)))))) H5))))))))) (\lambda (f: F).(\lambda (n: nat).(\lambda (_: -((\forall (e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c3 (Flat -f) u1) (CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: -T).(getl n (CHead c4 (Flat f) u2) (CHead e2 k0 u4)))) (\lambda (e2: -C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 u3 -u4)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4: -(getl (S n) (CHead c3 (Flat f) u1) (CHead e1 k0 u0))).(let H5 \def (H1 (S n) -e1 u0 k0 (getl_gen_S (Flat f) c3 (CHead e1 k0 u0) u1 n H4)) in (ex3_2_ind C T -(\lambda (e2: C).(\lambda (u3: T).(getl (S n) c4 (CHead e2 k0 u3)))) (\lambda -(e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 -u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c4 -(Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 -e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (H6: (getl (S n) c4 (CHead x0 k0 x1))).(\lambda -(H7: (wcpr0 e1 x0)).(\lambda (H8: (pr0 u0 x1)).(ex3_2_intro C T (\lambda (e2: -C).(\lambda (u3: T).(getl (S n) (CHead c4 (Flat f) u2) (CHead e2 k0 u3)))) -(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda -(u3: T).(pr0 u0 u3))) x0 x1 (getl_head (Flat f) n c4 (CHead x0 k0 x1) H6 u2) -H7 H8)))))) H5))))))))) k) h)))))))))) c1 c2 H))). -(* COMMENTS -Initial nodes: 2103 -END *) - -theorem wcpr0_getl_back: - \forall (c1: C).(\forall (c2: C).((wcpr0 c2 c1) \to (\forall (h: -nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((getl h c1 (CHead e1 -k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(getl h c2 (CHead e2 -k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: -C).(\lambda (u2: T).(pr0 u2 u1))))))))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c2 c1)).(wcpr0_ind -(\lambda (c: C).(\lambda (c0: C).(\forall (h: nat).(\forall (e1: C).(\forall -(u1: T).(\forall (k: K).((getl h c0 (CHead e1 k u1)) \to (ex3_2 C T (\lambda -(e2: C).(\lambda (u2: T).(getl h c (CHead e2 k u2)))) (\lambda (e2: -C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 -u1))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda -(u1: T).(\lambda (k: K).(\lambda (H0: (getl h c (CHead e1 k -u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(getl h c (CHead e2 -k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: -C).(\lambda (u2: T).(pr0 u2 u1))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl -u1)))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wcpr0 c3 -c4)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1: -T).(\forall (k: K).((getl h c4 (CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2: -C).(\lambda (u2: T).(getl h c3 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda -(_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 -u1))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 -u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall -(e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c4 k u2) (CHead e1 -k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c3 k -u1) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) -(\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))) (\lambda (e1: -C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (getl O (CHead c4 k u2) -(CHead e1 k0 u0))).(K_ind (\lambda (k1: K).((clear (CHead c4 k1 u2) (CHead e1 -k0 u0)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 k1 -u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) -(\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))))) (\lambda (b: B).(\lambda -(H4: (clear (CHead c4 (Bind b) u2) (CHead e1 k0 u0))).(let H5 \def (f_equal C -C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow e1 | (CHead c _ _) \Rightarrow c])) (CHead e1 k0 u0) (CHead c4 -(Bind b) u2) (clear_gen_bind b c4 (CHead e1 k0 u0) u2 H4)) in ((let H6 \def -(f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with -[(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead e1 k0 u0) -(CHead c4 (Bind b) u2) (clear_gen_bind b c4 (CHead e1 k0 u0) u2 H4)) in ((let -H7 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead e1 k0 -u0) (CHead c4 (Bind b) u2) (clear_gen_bind b c4 (CHead e1 k0 u0) u2 H4)) in -(\lambda (H8: (eq K k0 (Bind b))).(\lambda (H9: (eq C e1 c4)).(eq_ind_r K -(Bind b) (\lambda (k1: K).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl -O (CHead c3 (Bind b) u1) (CHead e2 k1 u3)))) (\lambda (e2: C).(\lambda (_: -T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))))) (eq_ind_r -T u2 (\lambda (t: T).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O -(CHead c3 (Bind b) u1) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda -(_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 t))))) -(eq_ind_r C c4 (\lambda (c: C).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: -T).(getl O (CHead c3 (Bind b) u1) (CHead e2 (Bind b) u3)))) (\lambda (e2: -C).(\lambda (_: T).(wcpr0 e2 c))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 -u2))))) (ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 -(Bind b) u1) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda (_: -T).(wcpr0 e2 c4))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u2))) c3 u1 -(getl_refl b c3 u1) H0 H2) e1 H9) u0 H7) k0 H8)))) H6)) H5)))) (\lambda (f: -F).(\lambda (H4: (clear (CHead c4 (Flat f) u2) (CHead e1 k0 u0))).(let H5 -\def (H1 O e1 u0 k0 (getl_intro O c4 (CHead e1 k0 u0) c4 (drop_refl c4) -(clear_gen_flat f c4 (CHead e1 k0 u0) u2 H4))) in (ex3_2_ind C T (\lambda -(e2: C).(\lambda (u3: T).(getl O c3 (CHead e2 k0 u3)))) (\lambda (e2: -C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 -u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 (Flat f) -u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) -(\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))) (\lambda (x0: C).(\lambda -(x1: T).(\lambda (H6: (getl O c3 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 x0 -e1)).(\lambda (H8: (pr0 x1 u0)).(ex3_2_intro C T (\lambda (e2: C).(\lambda -(u3: T).(getl O (CHead c3 (Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2: -C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 -u0))) x0 x1 (getl_flat c3 (CHead x0 k0 x1) O H6 f u1) H7 H8)))))) H5)))) k -(getl_gen_O (CHead c4 k u2) (CHead e1 k0 u0) H3)))))) (K_ind (\lambda (k0: -K).(\forall (n: nat).(((\forall (e1: C).(\forall (u3: T).(\forall (k1: -K).((getl n (CHead c4 k0 u2) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: -C).(\lambda (u4: T).(getl n (CHead c3 k0 u1) (CHead e2 k1 u4)))) (\lambda -(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 -u4 u3))))))))) \to (\forall (e1: C).(\forall (u3: T).(\forall (k1: K).((getl -(S n) (CHead c4 k0 u2) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: -C).(\lambda (u4: T).(getl (S n) (CHead c3 k0 u1) (CHead e2 k1 u4)))) (\lambda -(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 -u4 u3))))))))))) (\lambda (b: B).(\lambda (n: nat).(\lambda (_: ((\forall -(e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c4 (Bind b) u2) -(CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n -(CHead c3 (Bind b) u1) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: -T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 u4 -u3)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4: -(getl (S n) (CHead c4 (Bind b) u2) (CHead e1 k0 u0))).(let H5 \def (H1 n e1 -u0 k0 (getl_gen_S (Bind b) c4 (CHead e1 k0 u0) u2 n H4)) in (ex3_2_ind C T -(\lambda (e2: C).(\lambda (u3: T).(getl n c3 (CHead e2 k0 u3)))) (\lambda -(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 -u3 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c3 -(Bind b) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 -e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (H6: (getl n c3 (CHead x0 k0 x1))).(\lambda (H7: -(wcpr0 x0 e1)).(\lambda (H8: (pr0 x1 u0)).(ex3_2_intro C T (\lambda (e2: -C).(\lambda (u3: T).(getl (S n) (CHead c3 (Bind b) u1) (CHead e2 k0 u3)))) -(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda -(u3: T).(pr0 u3 u0))) x0 x1 (getl_head (Bind b) n c3 (CHead x0 k0 x1) H6 u1) -H7 H8)))))) H5))))))))) (\lambda (f: F).(\lambda (n: nat).(\lambda (_: -((\forall (e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c4 (Flat -f) u2) (CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: -T).(getl n (CHead c3 (Flat f) u1) (CHead e2 k0 u4)))) (\lambda (e2: -C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 u4 -u3)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4: -(getl (S n) (CHead c4 (Flat f) u2) (CHead e1 k0 u0))).(let H5 \def (H1 (S n) -e1 u0 k0 (getl_gen_S (Flat f) c4 (CHead e1 k0 u0) u2 n H4)) in (ex3_2_ind C T -(\lambda (e2: C).(\lambda (u3: T).(getl (S n) c3 (CHead e2 k0 u3)))) (\lambda -(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 -u3 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c3 -(Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 -e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (H6: (getl (S n) c3 (CHead x0 k0 x1))).(\lambda -(H7: (wcpr0 x0 e1)).(\lambda (H8: (pr0 x1 u0)).(ex3_2_intro C T (\lambda (e2: -C).(\lambda (u3: T).(getl (S n) (CHead c3 (Flat f) u1) (CHead e2 k0 u3)))) -(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda -(u3: T).(pr0 u3 u0))) x0 x1 (getl_head (Flat f) n c3 (CHead x0 k0 x1) H6 u1) -H7 H8)))))) H5))))))))) k) h)))))))))) c2 c1 H))). -(* COMMENTS -Initial nodes: 2103 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/wf3/clear.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/wf3/clear.ma deleted file mode 100644 index b2ddd47af..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/wf3/clear.ma +++ /dev/null @@ -1,91 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/wf3/fwd.ma". - -theorem wf3_clear_conf: - \forall (c1: C).(\forall (c: C).((clear c1 c) \to (\forall (g: G).(\forall -(c2: C).((wf3 g c1 c2) \to (wf3 g c c2)))))) -\def - \lambda (c1: C).(\lambda (c: C).(\lambda (H: (clear c1 c)).(clear_ind -(\lambda (c0: C).(\lambda (c2: C).(\forall (g: G).(\forall (c3: C).((wf3 g c0 -c3) \to (wf3 g c2 c3)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (u: -T).(\lambda (g: G).(\lambda (c2: C).(\lambda (H0: (wf3 g (CHead e (Bind b) u) -c2)).H0)))))) (\lambda (e: C).(\lambda (c0: C).(\lambda (_: (clear e -c0)).(\lambda (H1: ((\forall (g: G).(\forall (c2: C).((wf3 g e c2) \to (wf3 g -c0 c2)))))).(\lambda (f: F).(\lambda (u: T).(\lambda (g: G).(\lambda (c2: -C).(\lambda (H2: (wf3 g (CHead e (Flat f) u) c2)).(let H_y \def -(wf3_gen_flat1 g e c2 u f H2) in (H1 g c2 H_y))))))))))) c1 c H))). -(* COMMENTS -Initial nodes: 145 -END *) - -theorem clear_wf3_trans: - \forall (c1: C).(\forall (d1: C).((clear c1 d1) \to (\forall (g: G).(\forall -(d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda -(c2: C).(clear c2 d2)))))))) -\def - \lambda (c1: C).(\lambda (d1: C).(\lambda (H: (clear c1 d1)).(clear_ind -(\lambda (c: C).(\lambda (c0: C).(\forall (g: G).(\forall (d2: C).((wf3 g c0 -d2) \to (ex2 C (\lambda (c2: C).(wf3 g c c2)) (\lambda (c2: C).(clear c2 -d2)))))))) (\lambda (b: B).(\lambda (e: C).(\lambda (u: T).(\lambda (g: -G).(\lambda (d2: C).(\lambda (H0: (wf3 g (CHead e (Bind b) u) d2)).(let H_x -\def (wf3_gen_bind1 g e d2 u b H0) in (let H1 \def H_x in (or_ind (ex3_2 C T -(\lambda (c2: C).(\lambda (_: T).(eq C d2 (CHead c2 (Bind b) u)))) (\lambda -(c2: C).(\lambda (_: T).(wf3 g e c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g -e u w)))) (ex3 C (\lambda (c2: C).(eq C d2 (CHead c2 (Bind Void) (TSort O)))) -(\lambda (c2: C).(wf3 g e c2)) (\lambda (_: C).(\forall (w: T).((ty3 g e u w) -\to False)))) (ex2 C (\lambda (c2: C).(wf3 g (CHead e (Bind b) u) c2)) -(\lambda (c2: C).(clear c2 d2))) (\lambda (H2: (ex3_2 C T (\lambda (c2: -C).(\lambda (_: T).(eq C d2 (CHead c2 (Bind b) u)))) (\lambda (c2: -C).(\lambda (_: T).(wf3 g e c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g e u -w))))).(ex3_2_ind C T (\lambda (c2: C).(\lambda (_: T).(eq C d2 (CHead c2 -(Bind b) u)))) (\lambda (c2: C).(\lambda (_: T).(wf3 g e c2))) (\lambda (_: -C).(\lambda (w: T).(ty3 g e u w))) (ex2 C (\lambda (c2: C).(wf3 g (CHead e -(Bind b) u) c2)) (\lambda (c2: C).(clear c2 d2))) (\lambda (x0: C).(\lambda -(x1: T).(\lambda (H3: (eq C d2 (CHead x0 (Bind b) u))).(\lambda (H4: (wf3 g e -x0)).(\lambda (H5: (ty3 g e u x1)).(eq_ind_r C (CHead x0 (Bind b) u) (\lambda -(c: C).(ex2 C (\lambda (c2: C).(wf3 g (CHead e (Bind b) u) c2)) (\lambda (c2: -C).(clear c2 c)))) (ex_intro2 C (\lambda (c2: C).(wf3 g (CHead e (Bind b) u) -c2)) (\lambda (c2: C).(clear c2 (CHead x0 (Bind b) u))) (CHead x0 (Bind b) u) -(wf3_bind g e x0 H4 u x1 H5 b) (clear_bind b x0 u)) d2 H3)))))) H2)) (\lambda -(H2: (ex3 C (\lambda (c2: C).(eq C d2 (CHead c2 (Bind Void) (TSort O)))) -(\lambda (c2: C).(wf3 g e c2)) (\lambda (_: C).(\forall (w: T).((ty3 g e u w) -\to False))))).(ex3_ind C (\lambda (c2: C).(eq C d2 (CHead c2 (Bind Void) -(TSort O)))) (\lambda (c2: C).(wf3 g e c2)) (\lambda (_: C).(\forall (w: -T).((ty3 g e u w) \to False))) (ex2 C (\lambda (c2: C).(wf3 g (CHead e (Bind -b) u) c2)) (\lambda (c2: C).(clear c2 d2))) (\lambda (x0: C).(\lambda (H3: -(eq C d2 (CHead x0 (Bind Void) (TSort O)))).(\lambda (H4: (wf3 g e -x0)).(\lambda (H5: ((\forall (w: T).((ty3 g e u w) \to False)))).(eq_ind_r C -(CHead x0 (Bind Void) (TSort O)) (\lambda (c: C).(ex2 C (\lambda (c2: C).(wf3 -g (CHead e (Bind b) u) c2)) (\lambda (c2: C).(clear c2 c)))) (ex_intro2 C -(\lambda (c2: C).(wf3 g (CHead e (Bind b) u) c2)) (\lambda (c2: C).(clear c2 -(CHead x0 (Bind Void) (TSort O)))) (CHead x0 (Bind Void) (TSort O)) (wf3_void -g e x0 H4 u H5 b) (clear_bind Void x0 (TSort O))) d2 H3))))) H2)) H1))))))))) -(\lambda (e: C).(\lambda (c: C).(\lambda (_: (clear e c)).(\lambda (H1: -((\forall (g: G).(\forall (d2: C).((wf3 g c d2) \to (ex2 C (\lambda (c2: -C).(wf3 g e c2)) (\lambda (c2: C).(clear c2 d2)))))))).(\lambda (f: -F).(\lambda (u: T).(\lambda (g: G).(\lambda (d2: C).(\lambda (H2: (wf3 g c -d2)).(let H_x \def (H1 g d2 H2) in (let H3 \def H_x in (ex2_ind C (\lambda -(c2: C).(wf3 g e c2)) (\lambda (c2: C).(clear c2 d2)) (ex2 C (\lambda (c2: -C).(wf3 g (CHead e (Flat f) u) c2)) (\lambda (c2: C).(clear c2 d2))) (\lambda -(x: C).(\lambda (H4: (wf3 g e x)).(\lambda (H5: (clear x d2)).(ex_intro2 C -(\lambda (c2: C).(wf3 g (CHead e (Flat f) u) c2)) (\lambda (c2: C).(clear c2 -d2)) x (wf3_flat g e x H4 u f) H5)))) H3)))))))))))) c1 d1 H))). -(* COMMENTS -Initial nodes: 1023 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/wf3/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/wf3/defs.ma deleted file mode 100644 index a99661941..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/wf3/defs.ma +++ /dev/null @@ -1,29 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/ty3/defs.ma". - -inductive wf3 (g: G): C \to (C \to Prop) \def -| wf3_sort: \forall (m: nat).(wf3 g (CSort m) (CSort m)) -| wf3_bind: \forall (c1: C).(\forall (c2: C).((wf3 g c1 c2) \to (\forall (u: -T).(\forall (t: T).((ty3 g c1 u t) \to (\forall (b: B).(wf3 g (CHead c1 (Bind -b) u) (CHead c2 (Bind b) u)))))))) -| wf3_void: \forall (c1: C).(\forall (c2: C).((wf3 g c1 c2) \to (\forall (u: -T).(((\forall (t: T).((ty3 g c1 u t) \to False))) \to (\forall (b: B).(wf3 g -(CHead c1 (Bind b) u) (CHead c2 (Bind Void) (TSort O)))))))) -| wf3_flat: \forall (c1: C).(\forall (c2: C).((wf3 g c1 c2) \to (\forall (u: -T).(\forall (f: F).(wf3 g (CHead c1 (Flat f) u) c2))))). - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/wf3/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/wf3/fwd.ma deleted file mode 100644 index 71c903f06..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/wf3/fwd.ma +++ /dev/null @@ -1,311 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/wf3/defs.ma". - -theorem wf3_gen_sort1: - \forall (g: G).(\forall (x: C).(\forall (m: nat).((wf3 g (CSort m) x) \to -(eq C x (CSort m))))) -\def - \lambda (g: G).(\lambda (x: C).(\lambda (m: nat).(\lambda (H: (wf3 g (CSort -m) x)).(insert_eq C (CSort m) (\lambda (c: C).(wf3 g c x)) (\lambda (c: -C).(eq C x c)) (\lambda (y: C).(\lambda (H0: (wf3 g y x)).(wf3_ind g (\lambda -(c: C).(\lambda (c0: C).((eq C c (CSort m)) \to (eq C c0 c)))) (\lambda (m0: -nat).(\lambda (H1: (eq C (CSort m0) (CSort m))).(let H2 \def (f_equal C nat -(\lambda (e: C).(match e in C return (\lambda (_: C).nat) with [(CSort n) -\Rightarrow n | (CHead _ _ _) \Rightarrow m0])) (CSort m0) (CSort m) H1) in -(eq_ind_r nat m (\lambda (n: nat).(eq C (CSort n) (CSort n))) (refl_equal C -(CSort m)) m0 H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (wf3 g c1 -c2)).(\lambda (_: (((eq C c1 (CSort m)) \to (eq C c2 c1)))).(\lambda (u: -T).(\lambda (t: T).(\lambda (_: (ty3 g c1 u t)).(\lambda (b: B).(\lambda (H4: -(eq C (CHead c1 (Bind b) u) (CSort m))).(let H5 \def (eq_ind C (CHead c1 -(Bind b) u) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with -[(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort m) -H4) in (False_ind (eq C (CHead c2 (Bind b) u) (CHead c1 (Bind b) u)) -H5))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (wf3 g c1 -c2)).(\lambda (_: (((eq C c1 (CSort m)) \to (eq C c2 c1)))).(\lambda (u: -T).(\lambda (_: ((\forall (t: T).((ty3 g c1 u t) \to False)))).(\lambda (b: -B).(\lambda (H4: (eq C (CHead c1 (Bind b) u) (CSort m))).(let H5 \def (eq_ind -C (CHead c1 (Bind b) u) (\lambda (ee: C).(match ee in C return (\lambda (_: -C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow -True])) I (CSort m) H4) in (False_ind (eq C (CHead c2 (Bind Void) (TSort O)) -(CHead c1 (Bind b) u)) H5)))))))))) (\lambda (c1: C).(\lambda (c2: -C).(\lambda (_: (wf3 g c1 c2)).(\lambda (_: (((eq C c1 (CSort m)) \to (eq C -c2 c1)))).(\lambda (u: T).(\lambda (f: F).(\lambda (H3: (eq C (CHead c1 (Flat -f) u) (CSort m))).(let H4 \def (eq_ind C (CHead c1 (Flat f) u) (\lambda (ee: -C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow -False | (CHead _ _ _) \Rightarrow True])) I (CSort m) H3) in (False_ind (eq C -c2 (CHead c1 (Flat f) u)) H4))))))))) y x H0))) H)))). -(* COMMENTS -Initial nodes: 523 -END *) - -theorem wf3_gen_bind1: - \forall (g: G).(\forall (c1: C).(\forall (x: C).(\forall (v: T).(\forall (b: -B).((wf3 g (CHead c1 (Bind b) v) x) \to (or (ex3_2 C T (\lambda (c2: -C).(\lambda (_: T).(eq C x (CHead c2 (Bind b) v)))) (\lambda (c2: C).(\lambda -(_: T).(wf3 g c1 c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 -C (\lambda (c2: C).(eq C x (CHead c2 (Bind Void) (TSort O)))) (\lambda (c2: -C).(wf3 g c1 c2)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to -False)))))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (x: C).(\lambda (v: T).(\lambda (b: -B).(\lambda (H: (wf3 g (CHead c1 (Bind b) v) x)).(insert_eq C (CHead c1 (Bind -b) v) (\lambda (c: C).(wf3 g c x)) (\lambda (_: C).(or (ex3_2 C T (\lambda -(c2: C).(\lambda (_: T).(eq C x (CHead c2 (Bind b) v)))) (\lambda (c2: -C).(\lambda (_: T).(wf3 g c1 c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 -v w)))) (ex3 C (\lambda (c2: C).(eq C x (CHead c2 (Bind Void) (TSort O)))) -(\lambda (c2: C).(wf3 g c1 c2)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v -w) \to False)))))) (\lambda (y: C).(\lambda (H0: (wf3 g y x)).(wf3_ind g -(\lambda (c: C).(\lambda (c0: C).((eq C c (CHead c1 (Bind b) v)) \to (or -(ex3_2 C T (\lambda (c2: C).(\lambda (_: T).(eq C c0 (CHead c2 (Bind b) v)))) -(\lambda (c2: C).(\lambda (_: T).(wf3 g c1 c2))) (\lambda (_: C).(\lambda (w: -T).(ty3 g c1 v w)))) (ex3 C (\lambda (c2: C).(eq C c0 (CHead c2 (Bind Void) -(TSort O)))) (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (_: C).(\forall (w: -T).((ty3 g c1 v w) \to False)))))))) (\lambda (m: nat).(\lambda (H1: (eq C -(CSort m) (CHead c1 (Bind b) v))).(let H2 \def (eq_ind C (CSort m) (\lambda -(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead c1 (Bind b) v) -H1) in (False_ind (or (ex3_2 C T (\lambda (c2: C).(\lambda (_: T).(eq C -(CSort m) (CHead c2 (Bind b) v)))) (\lambda (c2: C).(\lambda (_: T).(wf3 g c1 -c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c2: -C).(eq C (CSort m) (CHead c2 (Bind Void) (TSort O)))) (\lambda (c2: C).(wf3 g -c1 c2)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to False))))) H2)))) -(\lambda (c0: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c0 c2)).(\lambda (H2: -(((eq C c0 (CHead c1 (Bind b) v)) \to (or (ex3_2 C T (\lambda (c3: -C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda (c3: -C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 -v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort O)))) -(\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v -w) \to False)))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H3: (ty3 g c0 -u t)).(\lambda (b0: B).(\lambda (H4: (eq C (CHead c0 (Bind b0) u) (CHead c1 -(Bind b) v))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return -(\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow -c])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) H4) in ((let H6 \def -(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with -[(CSort _) \Rightarrow b0 | (CHead _ k _) \Rightarrow (match k in K return -(\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow -b0])])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) H4) in ((let H7 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead c0 (Bind -b0) u) (CHead c1 (Bind b) v) H4) in (\lambda (H8: (eq B b0 b)).(\lambda (H9: -(eq C c0 c1)).(eq_ind_r B b (\lambda (b1: B).(or (ex3_2 C T (\lambda (c3: -C).(\lambda (_: T).(eq C (CHead c2 (Bind b1) u) (CHead c3 (Bind b) v)))) -(\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: -T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C (CHead c2 (Bind b1) u) -(CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda -(_: C).(\forall (w: T).((ty3 g c1 v w) \to False)))))) (let H10 \def (eq_ind -T u (\lambda (t0: T).(ty3 g c0 t0 t)) H3 v H7) in (eq_ind_r T v (\lambda (t0: -T).(or (ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C (CHead c2 (Bind b) -t0) (CHead c3 (Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) -(\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq -C (CHead c2 (Bind b) t0) (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: -C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to -False)))))) (let H11 \def (eq_ind C c0 (\lambda (c: C).(ty3 g c v t)) H10 c1 -H9) in (let H12 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 (Bind b) -v)) \to (or (ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 -(Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: -C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead -c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: -C).(\forall (w: T).((ty3 g c1 v w) \to False))))))) H2 c1 H9) in (let H13 -\def (eq_ind C c0 (\lambda (c: C).(wf3 g c c2)) H1 c1 H9) in (or_introl -(ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C (CHead c2 (Bind b) v) -(CHead c3 (Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) -(\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq -C (CHead c2 (Bind b) v) (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: -C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to -False)))) (ex3_2_intro C T (\lambda (c3: C).(\lambda (_: T).(eq C (CHead c2 -(Bind b) v) (CHead c3 (Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g -c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w))) c2 t (refl_equal C -(CHead c2 (Bind b) v)) H13 H11))))) u H7)) b0 H8)))) H6)) H5))))))))))) -(\lambda (c0: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c0 c2)).(\lambda (H2: -(((eq C c0 (CHead c1 (Bind b) v)) \to (or (ex3_2 C T (\lambda (c3: -C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda (c3: -C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 -v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort O)))) -(\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v -w) \to False)))))))).(\lambda (u: T).(\lambda (H3: ((\forall (t: T).((ty3 g -c0 u t) \to False)))).(\lambda (b0: B).(\lambda (H4: (eq C (CHead c0 (Bind -b0) u) (CHead c1 (Bind b) v))).(let H5 \def (f_equal C C (\lambda (e: -C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | -(CHead c _ _) \Rightarrow c])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) -H4) in ((let H6 \def (f_equal C B (\lambda (e: C).(match e in C return -(\lambda (_: C).B) with [(CSort _) \Rightarrow b0 | (CHead _ k _) \Rightarrow -(match k in K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | -(Flat _) \Rightarrow b0])])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) H4) -in ((let H7 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda -(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) -(CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) H4) in (\lambda (_: (eq B b0 -b)).(\lambda (H9: (eq C c0 c1)).(let H10 \def (eq_ind T u (\lambda (t: -T).(\forall (t0: T).((ty3 g c0 t t0) \to False))) H3 v H7) in (let H11 \def -(eq_ind C c0 (\lambda (c: C).(\forall (t: T).((ty3 g c v t) \to False))) H10 -c1 H9) in (let H12 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 (Bind -b) v)) \to (or (ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 -(Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: -C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead -c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: -C).(\forall (w: T).((ty3 g c1 v w) \to False))))))) H2 c1 H9) in (let H13 -\def (eq_ind C c0 (\lambda (c: C).(wf3 g c c2)) H1 c1 H9) in (or_intror -(ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C (CHead c2 (Bind Void) -(TSort O)) (CHead c3 (Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g -c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda -(c3: C).(eq C (CHead c2 (Bind Void) (TSort O)) (CHead c3 (Bind Void) (TSort -O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g -c1 v w) \to False)))) (ex3_intro C (\lambda (c3: C).(eq C (CHead c2 (Bind -Void) (TSort O)) (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g -c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to False))) c2 -(refl_equal C (CHead c2 (Bind Void) (TSort O))) H13 H11))))))))) H6)) -H5)))))))))) (\lambda (c0: C).(\lambda (c2: C).(\lambda (_: (wf3 g c0 -c2)).(\lambda (_: (((eq C c0 (CHead c1 (Bind b) v)) \to (or (ex3_2 C T -(\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda -(c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 -g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort -O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g -c1 v w) \to False)))))))).(\lambda (u: T).(\lambda (f: F).(\lambda (H3: (eq C -(CHead c0 (Flat f) u) (CHead c1 (Bind b) v))).(let H4 \def (eq_ind C (CHead -c0 (Flat f) u) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) -with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K -return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) -\Rightarrow True])])) I (CHead c1 (Bind b) v) H3) in (False_ind (or (ex3_2 C -T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda -(c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 -g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort -O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g -c1 v w) \to False))))) H4))))))))) y x H0))) H)))))). -(* COMMENTS -Initial nodes: 2507 -END *) - -theorem wf3_gen_flat1: - \forall (g: G).(\forall (c1: C).(\forall (x: C).(\forall (v: T).(\forall (f: -F).((wf3 g (CHead c1 (Flat f) v) x) \to (wf3 g c1 x)))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (x: C).(\lambda (v: T).(\lambda (f: -F).(\lambda (H: (wf3 g (CHead c1 (Flat f) v) x)).(insert_eq C (CHead c1 (Flat -f) v) (\lambda (c: C).(wf3 g c x)) (\lambda (_: C).(wf3 g c1 x)) (\lambda (y: -C).(\lambda (H0: (wf3 g y x)).(wf3_ind g (\lambda (c: C).(\lambda (c0: -C).((eq C c (CHead c1 (Flat f) v)) \to (wf3 g c1 c0)))) (\lambda (m: -nat).(\lambda (H1: (eq C (CSort m) (CHead c1 (Flat f) v))).(let H2 \def -(eq_ind C (CSort m) (\lambda (ee: C).(match ee in C return (\lambda (_: -C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow -False])) I (CHead c1 (Flat f) v) H1) in (False_ind (wf3 g c1 (CSort m)) -H2)))) (\lambda (c0: C).(\lambda (c2: C).(\lambda (_: (wf3 g c0 c2)).(\lambda -(_: (((eq C c0 (CHead c1 (Flat f) v)) \to (wf3 g c1 c2)))).(\lambda (u: -T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (b: B).(\lambda (H4: -(eq C (CHead c0 (Bind b) u) (CHead c1 (Flat f) v))).(let H5 \def (eq_ind C -(CHead c0 (Bind b) u) (\lambda (ee: C).(match ee in C return (\lambda (_: -C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match -k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat -_) \Rightarrow False])])) I (CHead c1 (Flat f) v) H4) in (False_ind (wf3 g c1 -(CHead c2 (Bind b) u)) H5))))))))))) (\lambda (c0: C).(\lambda (c2: -C).(\lambda (_: (wf3 g c0 c2)).(\lambda (_: (((eq C c0 (CHead c1 (Flat f) v)) -\to (wf3 g c1 c2)))).(\lambda (u: T).(\lambda (_: ((\forall (t: T).((ty3 g c0 -u t) \to False)))).(\lambda (b: B).(\lambda (H4: (eq C (CHead c0 (Bind b) u) -(CHead c1 (Flat f) v))).(let H5 \def (eq_ind C (CHead c0 (Bind b) u) (\lambda -(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow -False])])) I (CHead c1 (Flat f) v) H4) in (False_ind (wf3 g c1 (CHead c2 -(Bind Void) (TSort O))) H5)))))))))) (\lambda (c0: C).(\lambda (c2: -C).(\lambda (H1: (wf3 g c0 c2)).(\lambda (H2: (((eq C c0 (CHead c1 (Flat f) -v)) \to (wf3 g c1 c2)))).(\lambda (u: T).(\lambda (f0: F).(\lambda (H3: (eq C -(CHead c0 (Flat f0) u) (CHead c1 (Flat f) v))).(let H4 \def (f_equal C C -(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 (Flat f0) u) (CHead -c1 (Flat f) v) H3) in ((let H5 \def (f_equal C F (\lambda (e: C).(match e in -C return (\lambda (_: C).F) with [(CSort _) \Rightarrow f0 | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).F) with [(Bind _) -\Rightarrow f0 | (Flat f1) \Rightarrow f1])])) (CHead c0 (Flat f0) u) (CHead -c1 (Flat f) v) H3) in ((let H6 \def (f_equal C T (\lambda (e: C).(match e in -C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) -\Rightarrow t])) (CHead c0 (Flat f0) u) (CHead c1 (Flat f) v) H3) in (\lambda -(_: (eq F f0 f)).(\lambda (H8: (eq C c0 c1)).(let H9 \def (eq_ind C c0 -(\lambda (c: C).((eq C c (CHead c1 (Flat f) v)) \to (wf3 g c1 c2))) H2 c1 H8) -in (let H10 \def (eq_ind C c0 (\lambda (c: C).(wf3 g c c2)) H1 c1 H8) in -H10))))) H5)) H4))))))))) y x H0))) H)))))). -(* COMMENTS -Initial nodes: 737 -END *) - -theorem wf3_gen_head2: - \forall (g: G).(\forall (x: C).(\forall (c: C).(\forall (v: T).(\forall (k: -K).((wf3 g x (CHead c k v)) \to (ex B (\lambda (b: B).(eq K k (Bind b))))))))) -\def - \lambda (g: G).(\lambda (x: C).(\lambda (c: C).(\lambda (v: T).(\lambda (k: -K).(\lambda (H: (wf3 g x (CHead c k v))).(insert_eq C (CHead c k v) (\lambda -(c0: C).(wf3 g x c0)) (\lambda (_: C).(ex B (\lambda (b: B).(eq K k (Bind -b))))) (\lambda (y: C).(\lambda (H0: (wf3 g x y)).(wf3_ind g (\lambda (_: -C).(\lambda (c1: C).((eq C c1 (CHead c k v)) \to (ex B (\lambda (b: B).(eq K -k (Bind b))))))) (\lambda (m: nat).(\lambda (H1: (eq C (CSort m) (CHead c k -v))).(let H2 \def (eq_ind C (CSort m) (\lambda (ee: C).(match ee in C return -(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) -\Rightarrow False])) I (CHead c k v) H1) in (False_ind (ex B (\lambda (b: -B).(eq K k (Bind b)))) H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: -(wf3 g c1 c2)).(\lambda (H2: (((eq C c2 (CHead c k v)) \to (ex B (\lambda (b: -B).(eq K k (Bind b))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H3: (ty3 -g c1 u t)).(\lambda (b: B).(\lambda (H4: (eq C (CHead c2 (Bind b) u) (CHead c -k v))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return -(\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ _) -\Rightarrow c0])) (CHead c2 (Bind b) u) (CHead c k v) H4) in ((let H6 \def -(f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with -[(CSort _) \Rightarrow (Bind b) | (CHead _ k0 _) \Rightarrow k0])) (CHead c2 -(Bind b) u) (CHead c k v) H4) in ((let H7 \def (f_equal C T (\lambda (e: -C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | -(CHead _ _ t0) \Rightarrow t0])) (CHead c2 (Bind b) u) (CHead c k v) H4) in -(\lambda (H8: (eq K (Bind b) k)).(\lambda (H9: (eq C c2 c)).(let H10 \def -(eq_ind T u (\lambda (t0: T).(ty3 g c1 t0 t)) H3 v H7) in (let H11 \def -(eq_ind C c2 (\lambda (c0: C).((eq C c0 (CHead c k v)) \to (ex B (\lambda -(b0: B).(eq K k (Bind b0)))))) H2 c H9) in (let H12 \def (eq_ind C c2 -(\lambda (c0: C).(wf3 g c1 c0)) H1 c H9) in (let H13 \def (eq_ind_r K k -(\lambda (k0: K).((eq C c (CHead c k0 v)) \to (ex B (\lambda (b0: B).(eq K k0 -(Bind b0)))))) H11 (Bind b) H8) in (eq_ind K (Bind b) (\lambda (k0: K).(ex B -(\lambda (b0: B).(eq K k0 (Bind b0))))) (ex_intro B (\lambda (b0: B).(eq K -(Bind b) (Bind b0))) b (refl_equal K (Bind b))) k H8)))))))) H6)) -H5))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c1 -c2)).(\lambda (H2: (((eq C c2 (CHead c k v)) \to (ex B (\lambda (b: B).(eq K -k (Bind b))))))).(\lambda (u: T).(\lambda (_: ((\forall (t: T).((ty3 g c1 u -t) \to False)))).(\lambda (_: B).(\lambda (H4: (eq C (CHead c2 (Bind Void) -(TSort O)) (CHead c k v))).(let H5 \def (f_equal C C (\lambda (e: C).(match e -in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ -_) \Rightarrow c0])) (CHead c2 (Bind Void) (TSort O)) (CHead c k v) H4) in -((let H6 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: -C).K) with [(CSort _) \Rightarrow (Bind Void) | (CHead _ k0 _) \Rightarrow -k0])) (CHead c2 (Bind Void) (TSort O)) (CHead c k v) H4) in ((let H7 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow (TSort O) | (CHead _ _ t) \Rightarrow t])) (CHead c2 -(Bind Void) (TSort O)) (CHead c k v) H4) in (\lambda (H8: (eq K (Bind Void) -k)).(\lambda (H9: (eq C c2 c)).(let H10 \def (eq_ind C c2 (\lambda (c0: -C).((eq C c0 (CHead c k v)) \to (ex B (\lambda (b0: B).(eq K k (Bind b0)))))) -H2 c H9) in (let H11 \def (eq_ind C c2 (\lambda (c0: C).(wf3 g c1 c0)) H1 c -H9) in (let H12 \def (eq_ind_r K k (\lambda (k0: K).((eq C c (CHead c k0 v)) -\to (ex B (\lambda (b0: B).(eq K k0 (Bind b0)))))) H10 (Bind Void) H8) in -(eq_ind K (Bind Void) (\lambda (k0: K).(ex B (\lambda (b0: B).(eq K k0 (Bind -b0))))) (let H13 \def (eq_ind_r T v (\lambda (t: T).((eq C c (CHead c (Bind -Void) t)) \to (ex B (\lambda (b0: B).(eq K (Bind Void) (Bind b0)))))) H12 -(TSort O) H7) in (ex_intro B (\lambda (b0: B).(eq K (Bind Void) (Bind b0))) -Void (refl_equal K (Bind Void)))) k H8))))))) H6)) H5)))))))))) (\lambda (c1: -C).(\lambda (c2: C).(\lambda (H1: (wf3 g c1 c2)).(\lambda (H2: (((eq C c2 -(CHead c k v)) \to (ex B (\lambda (b: B).(eq K k (Bind b))))))).(\lambda (_: -T).(\lambda (_: F).(\lambda (H3: (eq C c2 (CHead c k v))).(let H4 \def -(f_equal C C (\lambda (e: C).e) c2 (CHead c k v) H3) in (let H5 \def (eq_ind -C c2 (\lambda (c0: C).((eq C c0 (CHead c k v)) \to (ex B (\lambda (b: B).(eq -K k (Bind b)))))) H2 (CHead c k v) H4) in (let H6 \def (eq_ind C c2 (\lambda -(c0: C).(wf3 g c1 c0)) H1 (CHead c k v) H4) in (H5 (refl_equal C (CHead c k -v))))))))))))) x y H0))) H)))))). -(* COMMENTS -Initial nodes: 1225 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/wf3/getl.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/wf3/getl.ma deleted file mode 100644 index 8a8f13780..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/wf3/getl.ma +++ /dev/null @@ -1,205 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/wf3/clear.ma". - -include "Basic-1/ty3/dec.ma". - -theorem wf3_getl_conf: - \forall (b: B).(\forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall -(v: T).((getl i c1 (CHead d1 (Bind b) v)) \to (\forall (g: G).(\forall (c2: -C).((wf3 g c1 c2) \to (\forall (w: T).((ty3 g d1 v w) \to (ex2 C (\lambda -(d2: C).(getl i c2 (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 -d2))))))))))))) -\def - \lambda (b: B).(\lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: -C).(\forall (d1: C).(\forall (v: T).((getl n c1 (CHead d1 (Bind b) v)) \to -(\forall (g: G).(\forall (c2: C).((wf3 g c1 c2) \to (\forall (w: T).((ty3 g -d1 v w) \to (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind b) v))) -(\lambda (d2: C).(wf3 g d1 d2))))))))))))) (\lambda (c1: C).(\lambda (d1: -C).(\lambda (v: T).(\lambda (H: (getl O c1 (CHead d1 (Bind b) v))).(\lambda -(g: G).(\lambda (c2: C).(\lambda (H0: (wf3 g c1 c2)).(\lambda (w: T).(\lambda -(H1: (ty3 g d1 v w)).(let H_y \def (wf3_clear_conf c1 (CHead d1 (Bind b) v) -(getl_gen_O c1 (CHead d1 (Bind b) v) H) g c2 H0) in (let H_x \def -(wf3_gen_bind1 g d1 c2 v b H_y) in (let H2 \def H_x in (or_ind (ex3_2 C T -(\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda -(c3: C).(\lambda (_: T).(wf3 g d1 c3))) (\lambda (_: C).(\lambda (w0: T).(ty3 -g d1 v w0)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort -O)))) (\lambda (c3: C).(wf3 g d1 c3)) (\lambda (_: C).(\forall (w0: T).((ty3 -g d1 v w0) \to False)))) (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind -b) v))) (\lambda (d2: C).(wf3 g d1 d2))) (\lambda (H3: (ex3_2 C T (\lambda -(c3: C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda (c3: -C).(\lambda (_: T).(wf3 g d1 c3))) (\lambda (_: C).(\lambda (w0: T).(ty3 g d1 -v w0))))).(ex3_2_ind C T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 -(Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g d1 c3))) (\lambda (_: -C).(\lambda (w0: T).(ty3 g d1 v w0))) (ex2 C (\lambda (d2: C).(getl O c2 -(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (H4: (eq C c2 (CHead x0 (Bind b) v))).(\lambda -(H5: (wf3 g d1 x0)).(\lambda (_: (ty3 g d1 v x1)).(eq_ind_r C (CHead x0 (Bind -b) v) (\lambda (c: C).(ex2 C (\lambda (d2: C).(getl O c (CHead d2 (Bind b) -v))) (\lambda (d2: C).(wf3 g d1 d2)))) (ex_intro2 C (\lambda (d2: C).(getl O -(CHead x0 (Bind b) v) (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)) -x0 (getl_refl b x0 v) H5) c2 H4)))))) H3)) (\lambda (H3: (ex3 C (\lambda (c3: -C).(eq C c2 (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g d1 -c3)) (\lambda (_: C).(\forall (w0: T).((ty3 g d1 v w0) \to -False))))).(ex3_ind C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort -O)))) (\lambda (c3: C).(wf3 g d1 c3)) (\lambda (_: C).(\forall (w0: T).((ty3 -g d1 v w0) \to False))) (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind b) -v))) (\lambda (d2: C).(wf3 g d1 d2))) (\lambda (x0: C).(\lambda (H4: (eq C c2 -(CHead x0 (Bind Void) (TSort O)))).(\lambda (_: (wf3 g d1 x0)).(\lambda (H6: -((\forall (w0: T).((ty3 g d1 v w0) \to False)))).(eq_ind_r C (CHead x0 (Bind -Void) (TSort O)) (\lambda (c: C).(ex2 C (\lambda (d2: C).(getl O c (CHead d2 -(Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)))) (let H_x0 \def (H6 w H1) in -(let H7 \def H_x0 in (False_ind (ex2 C (\lambda (d2: C).(getl O (CHead x0 -(Bind Void) (TSort O)) (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 -d2))) H7))) c2 H4))))) H3)) H2))))))))))))) (\lambda (n: nat).(\lambda (H: -((\forall (c1: C).(\forall (d1: C).(\forall (v: T).((getl n c1 (CHead d1 -(Bind b) v)) \to (\forall (g: G).(\forall (c2: C).((wf3 g c1 c2) \to (\forall -(w: T).((ty3 g d1 v w) \to (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind -b) v))) (\lambda (d2: C).(wf3 g d1 d2)))))))))))))).(\lambda (c1: C).(C_ind -(\lambda (c: C).(\forall (d1: C).(\forall (v: T).((getl (S n) c (CHead d1 -(Bind b) v)) \to (\forall (g: G).(\forall (c2: C).((wf3 g c c2) \to (\forall -(w: T).((ty3 g d1 v w) \to (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 -(Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)))))))))))) (\lambda (n0: -nat).(\lambda (d1: C).(\lambda (v: T).(\lambda (H0: (getl (S n) (CSort n0) -(CHead d1 (Bind b) v))).(\lambda (g: G).(\lambda (c2: C).(\lambda (_: (wf3 g -(CSort n0) c2)).(\lambda (w: T).(\lambda (_: (ty3 g d1 v w)).(getl_gen_sort -n0 (S n) (CHead d1 (Bind b) v) H0 (ex2 C (\lambda (d2: C).(getl (S n) c2 -(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2))))))))))))) (\lambda -(c: C).(\lambda (H0: ((\forall (d1: C).(\forall (v: T).((getl (S n) c (CHead -d1 (Bind b) v)) \to (\forall (g: G).(\forall (c2: C).((wf3 g c c2) \to -(\forall (w: T).((ty3 g d1 v w) \to (ex2 C (\lambda (d2: C).(getl (S n) c2 -(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2))))))))))))).(\lambda -(k: K).(\lambda (t: T).(\lambda (d1: C).(\lambda (v: T).(\lambda (H1: (getl -(S n) (CHead c k t) (CHead d1 (Bind b) v))).(\lambda (g: G).(\lambda (c2: -C).(\lambda (H2: (wf3 g (CHead c k t) c2)).(\lambda (w: T).(\lambda (H3: (ty3 -g d1 v w)).(K_ind (\lambda (k0: K).((wf3 g (CHead c k0 t) c2) \to ((getl (r -k0 n) c (CHead d1 (Bind b) v)) \to (ex2 C (\lambda (d2: C).(getl (S n) c2 -(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)))))) (\lambda (b0: -B).(\lambda (H4: (wf3 g (CHead c (Bind b0) t) c2)).(\lambda (H5: (getl (r -(Bind b0) n) c (CHead d1 (Bind b) v))).(let H_x \def (wf3_gen_bind1 g c c2 t -b0 H4) in (let H6 \def H_x in (or_ind (ex3_2 C T (\lambda (c3: C).(\lambda -(_: T).(eq C c2 (CHead c3 (Bind b0) t)))) (\lambda (c3: C).(\lambda (_: -T).(wf3 g c c3))) (\lambda (_: C).(\lambda (w0: T).(ty3 g c t w0)))) (ex3 C -(\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: -C).(wf3 g c c3)) (\lambda (_: C).(\forall (w0: T).((ty3 g c t w0) \to -False)))) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind b) v))) -(\lambda (d2: C).(wf3 g d1 d2))) (\lambda (H7: (ex3_2 C T (\lambda (c3: -C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b0) t)))) (\lambda (c3: -C).(\lambda (_: T).(wf3 g c c3))) (\lambda (_: C).(\lambda (w0: T).(ty3 g c t -w0))))).(ex3_2_ind C T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 -(Bind b0) t)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c c3))) (\lambda (_: -C).(\lambda (w0: T).(ty3 g c t w0))) (ex2 C (\lambda (d2: C).(getl (S n) c2 -(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (H8: (eq C c2 (CHead x0 (Bind b0) t))).(\lambda -(H9: (wf3 g c x0)).(\lambda (_: (ty3 g c t x1)).(eq_ind_r C (CHead x0 (Bind -b0) t) (\lambda (c0: C).(ex2 C (\lambda (d2: C).(getl (S n) c0 (CHead d2 -(Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)))) (let H_x0 \def (H c d1 v H5 g -x0 H9 w H3) in (let H11 \def H_x0 in (ex2_ind C (\lambda (d2: C).(getl n x0 -(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)) (ex2 C (\lambda (d2: -C).(getl (S n) (CHead x0 (Bind b0) t) (CHead d2 (Bind b) v))) (\lambda (d2: -C).(wf3 g d1 d2))) (\lambda (x: C).(\lambda (H12: (getl n x0 (CHead x (Bind -b) v))).(\lambda (H13: (wf3 g d1 x)).(ex_intro2 C (\lambda (d2: C).(getl (S -n) (CHead x0 (Bind b0) t) (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 -d2)) x (getl_head (Bind b0) n x0 (CHead x (Bind b) v) H12 t) H13)))) H11))) -c2 H8)))))) H7)) (\lambda (H7: (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 -(Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c c3)) (\lambda (_: -C).(\forall (w0: T).((ty3 g c t w0) \to False))))).(ex3_ind C (\lambda (c3: -C).(eq C c2 (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c c3)) -(\lambda (_: C).(\forall (w0: T).((ty3 g c t w0) \to False))) (ex2 C (\lambda -(d2: C).(getl (S n) c2 (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 -d2))) (\lambda (x0: C).(\lambda (H8: (eq C c2 (CHead x0 (Bind Void) (TSort -O)))).(\lambda (H9: (wf3 g c x0)).(\lambda (_: ((\forall (w0: T).((ty3 g c t -w0) \to False)))).(eq_ind_r C (CHead x0 (Bind Void) (TSort O)) (\lambda (c0: -C).(ex2 C (\lambda (d2: C).(getl (S n) c0 (CHead d2 (Bind b) v))) (\lambda -(d2: C).(wf3 g d1 d2)))) (let H_x0 \def (H c d1 v H5 g x0 H9 w H3) in (let -H11 \def H_x0 in (ex2_ind C (\lambda (d2: C).(getl n x0 (CHead d2 (Bind b) -v))) (\lambda (d2: C).(wf3 g d1 d2)) (ex2 C (\lambda (d2: C).(getl (S n) -(CHead x0 (Bind Void) (TSort O)) (CHead d2 (Bind b) v))) (\lambda (d2: -C).(wf3 g d1 d2))) (\lambda (x: C).(\lambda (H12: (getl n x0 (CHead x (Bind -b) v))).(\lambda (H13: (wf3 g d1 x)).(ex_intro2 C (\lambda (d2: C).(getl (S -n) (CHead x0 (Bind Void) (TSort O)) (CHead d2 (Bind b) v))) (\lambda (d2: -C).(wf3 g d1 d2)) x (getl_head (Bind Void) n x0 (CHead x (Bind b) v) H12 -(TSort O)) H13)))) H11))) c2 H8))))) H7)) H6)))))) (\lambda (f: F).(\lambda -(H4: (wf3 g (CHead c (Flat f) t) c2)).(\lambda (H5: (getl (r (Flat f) n) c -(CHead d1 (Bind b) v))).(let H_y \def (wf3_gen_flat1 g c c2 t f H4) in (H0 d1 -v H5 g c2 H_y w H3))))) k H2 (getl_gen_S k c (CHead d1 (Bind b) v) t n -H1)))))))))))))) c1)))) i)). -(* COMMENTS -Initial nodes: 2531 -END *) - -theorem getl_wf3_trans: - \forall (i: nat).(\forall (c1: C).(\forall (d1: C).((getl i c1 d1) \to -(\forall (g: G).(\forall (d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda (c2: -C).(wf3 g c1 c2)) (\lambda (c2: C).(getl i c2 d2))))))))) -\def - \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (d1: -C).((getl n c1 d1) \to (\forall (g: G).(\forall (d2: C).((wf3 g d1 d2) \to -(ex2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(getl n c2 -d2)))))))))) (\lambda (c1: C).(\lambda (d1: C).(\lambda (H: (getl O c1 -d1)).(\lambda (g: G).(\lambda (d2: C).(\lambda (H0: (wf3 g d1 d2)).(let H_x -\def (clear_wf3_trans c1 d1 (getl_gen_O c1 d1 H) g d2 H0) in (let H1 \def H_x -in (ex2_ind C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(clear c2 d2)) -(ex2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(getl O c2 d2))) -(\lambda (x: C).(\lambda (H2: (wf3 g c1 x)).(\lambda (H3: (clear x -d2)).(ex_intro2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(getl O c2 -d2)) x H2 (getl_intro O x d2 x (drop_refl x) H3))))) H1))))))))) (\lambda (n: -nat).(\lambda (H: ((\forall (c1: C).(\forall (d1: C).((getl n c1 d1) \to -(\forall (g: G).(\forall (d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda (c2: -C).(wf3 g c1 c2)) (\lambda (c2: C).(getl n c2 d2))))))))))).(\lambda (c1: -C).(C_ind (\lambda (c: C).(\forall (d1: C).((getl (S n) c d1) \to (\forall -(g: G).(\forall (d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda (c2: C).(wf3 g c -c2)) (\lambda (c2: C).(getl (S n) c2 d2))))))))) (\lambda (n0: nat).(\lambda -(d1: C).(\lambda (H0: (getl (S n) (CSort n0) d1)).(\lambda (g: G).(\lambda -(d2: C).(\lambda (_: (wf3 g d1 d2)).(getl_gen_sort n0 (S n) d1 H0 (ex2 C -(\lambda (c2: C).(wf3 g (CSort n0) c2)) (\lambda (c2: C).(getl (S n) c2 -d2)))))))))) (\lambda (c: C).(\lambda (H0: ((\forall (d1: C).((getl (S n) c -d1) \to (\forall (g: G).(\forall (d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda -(c2: C).(wf3 g c c2)) (\lambda (c2: C).(getl (S n) c2 d2)))))))))).(\lambda -(k: K).(\lambda (t: T).(\lambda (d1: C).(\lambda (H1: (getl (S n) (CHead c k -t) d1)).(\lambda (g: G).(\lambda (d2: C).(\lambda (H2: (wf3 g d1 d2)).(K_ind -(\lambda (k0: K).((getl (r k0 n) c d1) \to (ex2 C (\lambda (c2: C).(wf3 g -(CHead c k0 t) c2)) (\lambda (c2: C).(getl (S n) c2 d2))))) (\lambda (b: -B).(\lambda (H3: (getl (r (Bind b) n) c d1)).(let H_x \def (H c d1 H3 g d2 -H2) in (let H4 \def H_x in (ex2_ind C (\lambda (c2: C).(wf3 g c c2)) (\lambda -(c2: C).(getl n c2 d2)) (ex2 C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) -c2)) (\lambda (c2: C).(getl (S n) c2 d2))) (\lambda (x: C).(\lambda (H5: (wf3 -g c x)).(\lambda (H6: (getl n x d2)).(let H_x0 \def (ty3_inference g c t) in -(let H7 \def H_x0 in (or_ind (ex T (\lambda (t2: T).(ty3 g c t t2))) (\forall -(t2: T).((ty3 g c t t2) \to False)) (ex2 C (\lambda (c2: C).(wf3 g (CHead c -(Bind b) t) c2)) (\lambda (c2: C).(getl (S n) c2 d2))) (\lambda (H8: (ex T -(\lambda (t2: T).(ty3 g c t t2)))).(ex_ind T (\lambda (t2: T).(ty3 g c t t2)) -(ex2 C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2)) (\lambda (c2: -C).(getl (S n) c2 d2))) (\lambda (x0: T).(\lambda (H9: (ty3 g c t -x0)).(ex_intro2 C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2)) (\lambda -(c2: C).(getl (S n) c2 d2)) (CHead x (Bind b) t) (wf3_bind g c x H5 t x0 H9 -b) (getl_head (Bind b) n x d2 H6 t)))) H8)) (\lambda (H8: ((\forall (t2: -T).((ty3 g c t t2) \to False)))).(ex_intro2 C (\lambda (c2: C).(wf3 g (CHead -c (Bind b) t) c2)) (\lambda (c2: C).(getl (S n) c2 d2)) (CHead x (Bind Void) -(TSort O)) (wf3_void g c x H5 t H8 b) (getl_head (Bind Void) n x d2 H6 (TSort -O)))) H7)))))) H4))))) (\lambda (f: F).(\lambda (H3: (getl (r (Flat f) n) c -d1)).(let H_x \def (H0 d1 H3 g d2 H2) in (let H4 \def H_x in (ex2_ind C -(\lambda (c2: C).(wf3 g c c2)) (\lambda (c2: C).(getl (S n) c2 d2)) (ex2 C -(\lambda (c2: C).(wf3 g (CHead c (Flat f) t) c2)) (\lambda (c2: C).(getl (S -n) c2 d2))) (\lambda (x: C).(\lambda (H5: (wf3 g c x)).(\lambda (H6: (getl (S -n) x d2)).(ex_intro2 C (\lambda (c2: C).(wf3 g (CHead c (Flat f) t) c2)) -(\lambda (c2: C).(getl (S n) c2 d2)) x (wf3_flat g c x H5 t f) H6)))) H4))))) -k (getl_gen_S k c d1 t n H1))))))))))) c1)))) i). -(* COMMENTS -Initial nodes: 1139 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/wf3/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/wf3/props.ma deleted file mode 100644 index 98a05c637..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/wf3/props.ma +++ /dev/null @@ -1,248 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/wf3/ty3.ma". - -include "Basic-1/app/defs.ma". - -theorem wf3_mono: - \forall (g: G).(\forall (c: C).(\forall (c1: C).((wf3 g c c1) \to (\forall -(c2: C).((wf3 g c c2) \to (eq C c1 c2)))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (c1: C).(\lambda (H: (wf3 g c -c1)).(wf3_ind g (\lambda (c0: C).(\lambda (c2: C).(\forall (c3: C).((wf3 g c0 -c3) \to (eq C c2 c3))))) (\lambda (m: nat).(\lambda (c2: C).(\lambda (H0: -(wf3 g (CSort m) c2)).(let H_y \def (wf3_gen_sort1 g c2 m H0) in (eq_ind_r C -(CSort m) (\lambda (c0: C).(eq C (CSort m) c0)) (refl_equal C (CSort m)) c2 -H_y))))) (\lambda (c2: C).(\lambda (c3: C).(\lambda (_: (wf3 g c2 -c3)).(\lambda (H1: ((\forall (c4: C).((wf3 g c2 c4) \to (eq C c3 -c4))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H2: (ty3 g c2 u -t)).(\lambda (b: B).(\lambda (c0: C).(\lambda (H3: (wf3 g (CHead c2 (Bind b) -u) c0)).(let H_x \def (wf3_gen_bind1 g c2 c0 u b H3) in (let H4 \def H_x in -(or_ind (ex3_2 C T (\lambda (c4: C).(\lambda (_: T).(eq C c0 (CHead c4 (Bind -b) u)))) (\lambda (c4: C).(\lambda (_: T).(wf3 g c2 c4))) (\lambda (_: -C).(\lambda (w: T).(ty3 g c2 u w)))) (ex3 C (\lambda (c4: C).(eq C c0 (CHead -c4 (Bind Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda (_: -C).(\forall (w: T).((ty3 g c2 u w) \to False)))) (eq C (CHead c3 (Bind b) u) -c0) (\lambda (H5: (ex3_2 C T (\lambda (c4: C).(\lambda (_: T).(eq C c0 (CHead -c4 (Bind b) u)))) (\lambda (c4: C).(\lambda (_: T).(wf3 g c2 c4))) (\lambda -(_: C).(\lambda (w: T).(ty3 g c2 u w))))).(ex3_2_ind C T (\lambda (c4: -C).(\lambda (_: T).(eq C c0 (CHead c4 (Bind b) u)))) (\lambda (c4: -C).(\lambda (_: T).(wf3 g c2 c4))) (\lambda (_: C).(\lambda (w: T).(ty3 g c2 -u w))) (eq C (CHead c3 (Bind b) u) c0) (\lambda (x0: C).(\lambda (x1: -T).(\lambda (H6: (eq C c0 (CHead x0 (Bind b) u))).(\lambda (H7: (wf3 g c2 -x0)).(\lambda (_: (ty3 g c2 u x1)).(eq_ind_r C (CHead x0 (Bind b) u) (\lambda -(c4: C).(eq C (CHead c3 (Bind b) u) c4)) (f_equal3 C K T C CHead c3 x0 (Bind -b) (Bind b) u u (H1 x0 H7) (refl_equal K (Bind b)) (refl_equal T u)) c0 -H6)))))) H5)) (\lambda (H5: (ex3 C (\lambda (c4: C).(eq C c0 (CHead c4 (Bind -Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda (_: C).(\forall -(w: T).((ty3 g c2 u w) \to False))))).(ex3_ind C (\lambda (c4: C).(eq C c0 -(CHead c4 (Bind Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda -(_: C).(\forall (w: T).((ty3 g c2 u w) \to False))) (eq C (CHead c3 (Bind b) -u) c0) (\lambda (x0: C).(\lambda (H6: (eq C c0 (CHead x0 (Bind Void) (TSort -O)))).(\lambda (_: (wf3 g c2 x0)).(\lambda (H8: ((\forall (w: T).((ty3 g c2 u -w) \to False)))).(eq_ind_r C (CHead x0 (Bind Void) (TSort O)) (\lambda (c4: -C).(eq C (CHead c3 (Bind b) u) c4)) (let H_x0 \def (H8 t H2) in (let H9 \def -H_x0 in (False_ind (eq C (CHead c3 (Bind b) u) (CHead x0 (Bind Void) (TSort -O))) H9))) c0 H6))))) H5)) H4))))))))))))) (\lambda (c2: C).(\lambda (c3: -C).(\lambda (_: (wf3 g c2 c3)).(\lambda (H1: ((\forall (c4: C).((wf3 g c2 c4) -\to (eq C c3 c4))))).(\lambda (u: T).(\lambda (H2: ((\forall (t: T).((ty3 g -c2 u t) \to False)))).(\lambda (b: B).(\lambda (c0: C).(\lambda (H3: (wf3 g -(CHead c2 (Bind b) u) c0)).(let H_x \def (wf3_gen_bind1 g c2 c0 u b H3) in -(let H4 \def H_x in (or_ind (ex3_2 C T (\lambda (c4: C).(\lambda (_: T).(eq C -c0 (CHead c4 (Bind b) u)))) (\lambda (c4: C).(\lambda (_: T).(wf3 g c2 c4))) -(\lambda (_: C).(\lambda (w: T).(ty3 g c2 u w)))) (ex3 C (\lambda (c4: C).(eq -C c0 (CHead c4 (Bind Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) -(\lambda (_: C).(\forall (w: T).((ty3 g c2 u w) \to False)))) (eq C (CHead c3 -(Bind Void) (TSort O)) c0) (\lambda (H5: (ex3_2 C T (\lambda (c4: C).(\lambda -(_: T).(eq C c0 (CHead c4 (Bind b) u)))) (\lambda (c4: C).(\lambda (_: -T).(wf3 g c2 c4))) (\lambda (_: C).(\lambda (w: T).(ty3 g c2 u -w))))).(ex3_2_ind C T (\lambda (c4: C).(\lambda (_: T).(eq C c0 (CHead c4 -(Bind b) u)))) (\lambda (c4: C).(\lambda (_: T).(wf3 g c2 c4))) (\lambda (_: -C).(\lambda (w: T).(ty3 g c2 u w))) (eq C (CHead c3 (Bind Void) (TSort O)) -c0) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C c0 (CHead x0 (Bind -b) u))).(\lambda (_: (wf3 g c2 x0)).(\lambda (H8: (ty3 g c2 u x1)).(eq_ind_r -C (CHead x0 (Bind b) u) (\lambda (c4: C).(eq C (CHead c3 (Bind Void) (TSort -O)) c4)) (let H_x0 \def (H2 x1 H8) in (let H9 \def H_x0 in (False_ind (eq C -(CHead c3 (Bind Void) (TSort O)) (CHead x0 (Bind b) u)) H9))) c0 H6)))))) -H5)) (\lambda (H5: (ex3 C (\lambda (c4: C).(eq C c0 (CHead c4 (Bind Void) -(TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda (_: C).(\forall (w: -T).((ty3 g c2 u w) \to False))))).(ex3_ind C (\lambda (c4: C).(eq C c0 (CHead -c4 (Bind Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda (_: -C).(\forall (w: T).((ty3 g c2 u w) \to False))) (eq C (CHead c3 (Bind Void) -(TSort O)) c0) (\lambda (x0: C).(\lambda (H6: (eq C c0 (CHead x0 (Bind Void) -(TSort O)))).(\lambda (H7: (wf3 g c2 x0)).(\lambda (_: ((\forall (w: T).((ty3 -g c2 u w) \to False)))).(eq_ind_r C (CHead x0 (Bind Void) (TSort O)) (\lambda -(c4: C).(eq C (CHead c3 (Bind Void) (TSort O)) c4)) (f_equal3 C K T C CHead -c3 x0 (Bind Void) (Bind Void) (TSort O) (TSort O) (H1 x0 H7) (refl_equal K -(Bind Void)) (refl_equal T (TSort O))) c0 H6))))) H5)) H4)))))))))))) -(\lambda (c2: C).(\lambda (c3: C).(\lambda (_: (wf3 g c2 c3)).(\lambda (H1: -((\forall (c4: C).((wf3 g c2 c4) \to (eq C c3 c4))))).(\lambda (u: -T).(\lambda (f: F).(\lambda (c0: C).(\lambda (H2: (wf3 g (CHead c2 (Flat f) -u) c0)).(let H_y \def (wf3_gen_flat1 g c2 c0 u f H2) in (H1 c0 H_y)))))))))) -c c1 H)))). -(* COMMENTS -Initial nodes: 1555 -END *) - -theorem wf3_total: - \forall (g: G).(\forall (c1: C).(ex C (\lambda (c2: C).(wf3 g c1 c2)))) -\def - \lambda (g: G).(\lambda (c1: C).(C_ind (\lambda (c: C).(ex C (\lambda (c2: -C).(wf3 g c c2)))) (\lambda (n: nat).(ex_intro C (\lambda (c2: C).(wf3 g -(CSort n) c2)) (CSort n) (wf3_sort g n))) (\lambda (c: C).(\lambda (H: (ex C -(\lambda (c2: C).(wf3 g c c2)))).(\lambda (k: K).(\lambda (t: T).(let H0 \def -H in (ex_ind C (\lambda (c2: C).(wf3 g c c2)) (ex C (\lambda (c2: C).(wf3 g -(CHead c k t) c2))) (\lambda (x: C).(\lambda (H1: (wf3 g c x)).(K_ind -(\lambda (k0: K).(ex C (\lambda (c2: C).(wf3 g (CHead c k0 t) c2)))) (\lambda -(b: B).(let H_x \def (ty3_inference g c t) in (let H2 \def H_x in (or_ind (ex -T (\lambda (t2: T).(ty3 g c t t2))) (\forall (t2: T).((ty3 g c t t2) \to -False)) (ex C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2))) (\lambda -(H3: (ex T (\lambda (t2: T).(ty3 g c t t2)))).(ex_ind T (\lambda (t2: T).(ty3 -g c t t2)) (ex C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2))) (\lambda -(x0: T).(\lambda (H4: (ty3 g c t x0)).(ex_intro C (\lambda (c2: C).(wf3 g -(CHead c (Bind b) t) c2)) (CHead x (Bind b) t) (wf3_bind g c x H1 t x0 H4 -b)))) H3)) (\lambda (H3: ((\forall (t2: T).((ty3 g c t t2) \to -False)))).(ex_intro C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2)) -(CHead x (Bind Void) (TSort O)) (wf3_void g c x H1 t H3 b))) H2)))) (\lambda -(f: F).(ex_intro C (\lambda (c2: C).(wf3 g (CHead c (Flat f) t) c2)) x -(wf3_flat g c x H1 t f))) k))) H0)))))) c1)). -(* COMMENTS -Initial nodes: 435 -END *) - -theorem ty3_shift1: - \forall (g: G).(\forall (c: C).((wf3 g c c) \to (\forall (t1: T).(\forall -(t2: T).((ty3 g c t1 t2) \to (ty3 g (CSort (cbk c)) (app1 c t1) (app1 c -t2))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (H: (wf3 g c c)).(insert_eq C c -(\lambda (c0: C).(wf3 g c0 c)) (\lambda (c0: C).(\forall (t1: T).(\forall -(t2: T).((ty3 g c0 t1 t2) \to (ty3 g (CSort (cbk c0)) (app1 c0 t1) (app1 c0 -t2)))))) (\lambda (y: C).(\lambda (H0: (wf3 g y c)).(wf3_ind g (\lambda (c0: -C).(\lambda (c1: C).((eq C c0 c1) \to (\forall (t1: T).(\forall (t2: T).((ty3 -g c0 t1 t2) \to (ty3 g (CSort (cbk c0)) (app1 c0 t1) (app1 c0 t2)))))))) -(\lambda (m: nat).(\lambda (_: (eq C (CSort m) (CSort m))).(\lambda (t1: -T).(\lambda (t2: T).(\lambda (H2: (ty3 g (CSort m) t1 t2)).H2))))) (\lambda -(c1: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c1 c2)).(\lambda (H2: (((eq C -c1 c2) \to (\forall (t1: T).(\forall (t2: T).((ty3 g c1 t1 t2) \to (ty3 g -(CSort (cbk c1)) (app1 c1 t1) (app1 c1 t2)))))))).(\lambda (u: T).(\lambda -(t: T).(\lambda (H3: (ty3 g c1 u t)).(\lambda (b: B).(\lambda (H4: (eq C -(CHead c1 (Bind b) u) (CHead c2 (Bind b) u))).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (H5: (ty3 g (CHead c1 (Bind b) u) t1 t2)).(let H6 \def (f_equal C -C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow c1 | (CHead c0 _ _) \Rightarrow c0])) (CHead c1 (Bind b) u) -(CHead c2 (Bind b) u) H4) in (let H7 \def (eq_ind_r C c2 (\lambda (c0: -C).((eq C c1 c0) \to (\forall (t3: T).(\forall (t4: T).((ty3 g c1 t3 t4) \to -(ty3 g (CSort (cbk c1)) (app1 c1 t3) (app1 c1 t4))))))) H2 c1 H6) in (let H8 -\def (eq_ind_r C c2 (\lambda (c0: C).(wf3 g c1 c0)) H1 c1 H6) in (ex_ind T -(\lambda (t0: T).(ty3 g (CHead c1 (Bind b) u) t2 t0)) (ty3 g (CSort (cbk c1)) -(app1 c1 (THead (Bind b) u t1)) (app1 c1 (THead (Bind b) u t2))) (\lambda (x: -T).(\lambda (_: (ty3 g (CHead c1 (Bind b) u) t2 x)).(H7 (refl_equal C c1) -(THead (Bind b) u t1) (THead (Bind b) u t2) (ty3_bind g c1 u t H3 b t1 t2 -H5)))) (ty3_correct g (CHead c1 (Bind b) u) t1 t2 H5))))))))))))))))) -(\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c1 c2)).(\lambda (H2: -(((eq C c1 c2) \to (\forall (t1: T).(\forall (t2: T).((ty3 g c1 t1 t2) \to -(ty3 g (CSort (cbk c1)) (app1 c1 t1) (app1 c1 t2)))))))).(\lambda (u: -T).(\lambda (H3: ((\forall (t: T).((ty3 g c1 u t) \to False)))).(\lambda (b: -B).(\lambda (H4: (eq C (CHead c1 (Bind b) u) (CHead c2 (Bind Void) (TSort -O)))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H5: (ty3 g (CHead c1 (Bind -b) u) t1 t2)).(let H6 \def (f_equal C C (\lambda (e: C).(match e in C return -(\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _ _) -\Rightarrow c0])) (CHead c1 (Bind b) u) (CHead c2 (Bind Void) (TSort O)) H4) -in ((let H7 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda -(_: C).B) with [(CSort _) \Rightarrow b | (CHead _ k _) \Rightarrow (match k -in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) -\Rightarrow b])])) (CHead c1 (Bind b) u) (CHead c2 (Bind Void) (TSort O)) H4) -in ((let H8 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda -(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) -(CHead c1 (Bind b) u) (CHead c2 (Bind Void) (TSort O)) H4) in (\lambda (H9: -(eq B b Void)).(\lambda (H10: (eq C c1 c2)).(let H11 \def (eq_ind B b -(\lambda (b0: B).(ty3 g (CHead c1 (Bind b0) u) t1 t2)) H5 Void H9) in -(eq_ind_r B Void (\lambda (b0: B).(ty3 g (CSort (cbk (CHead c1 (Bind b0) u))) -(app1 (CHead c1 (Bind b0) u) t1) (app1 (CHead c1 (Bind b0) u) t2))) (let H12 -\def (eq_ind T u (\lambda (t: T).(ty3 g (CHead c1 (Bind Void) t) t1 t2)) H11 -(TSort O) H8) in (let H13 \def (eq_ind T u (\lambda (t: T).(\forall (t0: -T).((ty3 g c1 t t0) \to False))) H3 (TSort O) H8) in (eq_ind_r T (TSort O) -(\lambda (t: T).(ty3 g (CSort (cbk (CHead c1 (Bind Void) t))) (app1 (CHead c1 -(Bind Void) t) t1) (app1 (CHead c1 (Bind Void) t) t2))) (let H14 \def -(eq_ind_r C c2 (\lambda (c0: C).((eq C c1 c0) \to (\forall (t3: T).(\forall -(t4: T).((ty3 g c1 t3 t4) \to (ty3 g (CSort (cbk c1)) (app1 c1 t3) (app1 c1 -t4))))))) H2 c1 H10) in (let H15 \def (eq_ind_r C c2 (\lambda (c0: C).(wf3 g -c1 c0)) H1 c1 H10) in (ex_ind T (\lambda (t: T).(ty3 g (CHead c1 (Bind Void) -(TSort O)) t2 t)) (ty3 g (CSort (cbk c1)) (app1 c1 (THead (Bind Void) (TSort -O) t1)) (app1 c1 (THead (Bind Void) (TSort O) t2))) (\lambda (x: T).(\lambda -(_: (ty3 g (CHead c1 (Bind Void) (TSort O)) t2 x)).(H14 (refl_equal C c1) -(THead (Bind Void) (TSort O) t1) (THead (Bind Void) (TSort O) t2) (ty3_bind g -c1 (TSort O) (TSort (next g O)) (ty3_sort g c1 O) Void t1 t2 H12)))) -(ty3_correct g (CHead c1 (Bind Void) (TSort O)) t1 t2 H12)))) u H8))) b -H9))))) H7)) H6))))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: -(wf3 g c1 c2)).(\lambda (H2: (((eq C c1 c2) \to (\forall (t1: T).(\forall -(t2: T).((ty3 g c1 t1 t2) \to (ty3 g (CSort (cbk c1)) (app1 c1 t1) (app1 c1 -t2)))))))).(\lambda (u: T).(\lambda (f: F).(\lambda (H3: (eq C (CHead c1 -(Flat f) u) c2)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g (CHead -c1 (Flat f) u) t1 t2)).(let H5 \def (f_equal C C (\lambda (e: C).e) (CHead c1 -(Flat f) u) c2 H3) in (let H6 \def (eq_ind_r C c2 (\lambda (c0: C).((eq C c1 -c0) \to (\forall (t3: T).(\forall (t4: T).((ty3 g c1 t3 t4) \to (ty3 g (CSort -(cbk c1)) (app1 c1 t3) (app1 c1 t4))))))) H2 (CHead c1 (Flat f) u) H5) in -(let H7 \def (eq_ind_r C c2 (\lambda (c0: C).(wf3 g c1 c0)) H1 (CHead c1 -(Flat f) u) H5) in (let H_x \def (wf3_gen_head2 g c1 c1 u (Flat f) H7) in -(let H8 \def H_x in (ex_ind B (\lambda (b: B).(eq K (Flat f) (Bind b))) (ty3 -g (CSort (cbk c1)) (app1 c1 (THead (Flat f) u t1)) (app1 c1 (THead (Flat f) u -t2))) (\lambda (x: B).(\lambda (H9: (eq K (Flat f) (Bind x))).(let H10 \def -(eq_ind K (Flat f) (\lambda (ee: K).(match ee in K return (\lambda (_: -K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])) I -(Bind x) H9) in (False_ind (ty3 g (CSort (cbk c1)) (app1 c1 (THead (Flat f) u -t1)) (app1 c1 (THead (Flat f) u t2))) H10)))) H8)))))))))))))))) y c H0))) -H))). -(* COMMENTS -Initial nodes: 1677 -END *) - -theorem wf3_idem: - \forall (g: G).(\forall (c1: C).(\forall (c2: C).((wf3 g c1 c2) \to (wf3 g -c2 c2)))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wf3 g c1 -c2)).(wf3_ind g (\lambda (_: C).(\lambda (c0: C).(wf3 g c0 c0))) (\lambda (m: -nat).(wf3_sort g m)) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wf3 g -c3 c4)).(\lambda (H1: (wf3 g c4 c4)).(\lambda (u: T).(\lambda (t: T).(\lambda -(H2: (ty3 g c3 u t)).(\lambda (b: B).(wf3_bind g c4 c4 H1 u t (wf3_ty3_conf g -c3 u t H2 c4 H0) b))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (_: -(wf3 g c3 c4)).(\lambda (H1: (wf3 g c4 c4)).(\lambda (u: T).(\lambda (_: -((\forall (t: T).((ty3 g c3 u t) \to False)))).(\lambda (_: B).(wf3_bind g c4 -c4 H1 (TSort O) (TSort (next g O)) (ty3_sort g c4 O) Void)))))))) (\lambda -(c3: C).(\lambda (c4: C).(\lambda (_: (wf3 g c3 c4)).(\lambda (H1: (wf3 g c4 -c4)).(\lambda (_: T).(\lambda (_: F).H1)))))) c1 c2 H)))). -(* COMMENTS -Initial nodes: 207 -END *) - -theorem wf3_ty3: - \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (u: T).((ty3 g c1 t -u) \to (ex2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(ty3 g c2 t -u))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H: -(ty3 g c1 t u)).(let H_x \def (wf3_total g c1) in (let H0 \def H_x in (ex_ind -C (\lambda (c2: C).(wf3 g c1 c2)) (ex2 C (\lambda (c2: C).(wf3 g c1 c2)) -(\lambda (c2: C).(ty3 g c2 t u))) (\lambda (x: C).(\lambda (H1: (wf3 g c1 -x)).(ex_intro2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(ty3 g c2 t -u)) x H1 (wf3_ty3_conf g c1 t u H x H1)))) H0))))))). -(* COMMENTS -Initial nodes: 123 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/wf3/ty3.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/wf3/ty3.ma deleted file mode 100644 index e47d4a2e0..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-1/wf3/ty3.ma +++ /dev/null @@ -1,138 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/wf3/getl.ma". - -theorem wf3_pr2_conf: - \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pr2 c1 -t1 t2) \to (\forall (c2: C).((wf3 g c1 c2) \to (\forall (u: T).((ty3 g c1 t1 -u) \to (pr2 c2 t1 t2))))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H: (pr2 c1 t1 t2)).(pr2_ind (\lambda (c: C).(\lambda (t: T).(\lambda (t0: -T).(\forall (c2: C).((wf3 g c c2) \to (\forall (u: T).((ty3 g c t u) \to (pr2 -c2 t t0)))))))) (\lambda (c: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda -(H0: (pr0 t3 t4)).(\lambda (c2: C).(\lambda (_: (wf3 g c c2)).(\lambda (u: -T).(\lambda (_: (ty3 g c t3 u)).(pr2_free c2 t3 t4 H0))))))))) (\lambda (c: -C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c -(CHead d (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: -(pr0 t3 t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (c2: -C).(\lambda (H3: (wf3 g c c2)).(\lambda (u0: T).(\lambda (H4: (ty3 g c t3 -u0)).(let H_y \def (ty3_sred_pr0 t3 t4 H1 g c u0 H4) in (let H_x \def -(ty3_getl_subst0 g c t4 u0 H_y u t i H2 Abbr d u H0) in (let H5 \def H_x in -(ex_ind T (\lambda (w: T).(ty3 g d u w)) (pr2 c2 t3 t) (\lambda (x: -T).(\lambda (H6: (ty3 g d u x)).(let H_x0 \def (wf3_getl_conf Abbr i c d u H0 -g c2 H3 x H6) in (let H7 \def H_x0 in (ex2_ind C (\lambda (d2: C).(getl i c2 -(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(wf3 g d d2)) (pr2 c2 t3 t) -(\lambda (x0: C).(\lambda (H8: (getl i c2 (CHead x0 (Bind Abbr) u))).(\lambda -(_: (wf3 g d x0)).(pr2_delta c2 x0 u i H8 t3 t4 H1 t H2)))) H7))))) -H5)))))))))))))))))) c1 t1 t2 H))))). -(* COMMENTS -Initial nodes: 373 -END *) - -theorem wf3_pr3_conf: - \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pr3 c1 -t1 t2) \to (\forall (c2: C).((wf3 g c1 c2) \to (\forall (u: T).((ty3 g c1 t1 -u) \to (pr3 c2 t1 t2))))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H: (pr3 c1 t1 t2)).(pr3_ind c1 (\lambda (t: T).(\lambda (t0: T).(\forall -(c2: C).((wf3 g c1 c2) \to (\forall (u: T).((ty3 g c1 t u) \to (pr3 c2 t -t0))))))) (\lambda (t: T).(\lambda (c2: C).(\lambda (_: (wf3 g c1 -c2)).(\lambda (u: T).(\lambda (_: (ty3 g c1 t u)).(pr3_refl c2 t)))))) -(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr2 c1 t4 t3)).(\lambda (t5: -T).(\lambda (_: (pr3 c1 t3 t5)).(\lambda (H2: ((\forall (c2: C).((wf3 g c1 -c2) \to (\forall (u: T).((ty3 g c1 t3 u) \to (pr3 c2 t3 t5))))))).(\lambda -(c2: C).(\lambda (H3: (wf3 g c1 c2)).(\lambda (u: T).(\lambda (H4: (ty3 g c1 -t4 u)).(pr3_sing c2 t3 t4 (wf3_pr2_conf g c1 t4 t3 H0 c2 H3 u H4) t5 (H2 c2 -H3 u (ty3_sred_pr2 c1 t4 t3 H0 g u H4))))))))))))) t1 t2 H))))). -(* COMMENTS -Initial nodes: 217 -END *) - -theorem wf3_pc3_conf: - \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pc3 c1 -t1 t2) \to (\forall (c2: C).((wf3 g c1 c2) \to (\forall (u1: T).((ty3 g c1 t1 -u1) \to (\forall (u2: T).((ty3 g c1 t2 u2) \to (pc3 c2 t1 t2))))))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H: (pc3 c1 t1 t2)).(\lambda (c2: C).(\lambda (H0: (wf3 g c1 c2)).(\lambda -(u1: T).(\lambda (H1: (ty3 g c1 t1 u1)).(\lambda (u2: T).(\lambda (H2: (ty3 g -c1 t2 u2)).(let H3 \def H in (ex2_ind T (\lambda (t: T).(pr3 c1 t1 t)) -(\lambda (t: T).(pr3 c1 t2 t)) (pc3 c2 t1 t2) (\lambda (x: T).(\lambda (H4: -(pr3 c1 t1 x)).(\lambda (H5: (pr3 c1 t2 x)).(pc3_pr3_t c2 t1 x (wf3_pr3_conf -g c1 t1 x H4 c2 H0 u1 H1) t2 (wf3_pr3_conf g c1 t2 x H5 c2 H0 u2 H2))))) -H3)))))))))))). -(* COMMENTS -Initial nodes: 153 -END *) - -theorem wf3_ty3_conf: - \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c1 -t1 t2) \to (\forall (c2: C).((wf3 g c1 c2) \to (ty3 g c2 t1 t2))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H: (ty3 g c1 t1 t2)).(ty3_ind g (\lambda (c: C).(\lambda (t: T).(\lambda -(t0: T).(\forall (c2: C).((wf3 g c c2) \to (ty3 g c2 t t0)))))) (\lambda (c: -C).(\lambda (t3: T).(\lambda (t: T).(\lambda (H0: (ty3 g c t3 t)).(\lambda -(H1: ((\forall (c2: C).((wf3 g c c2) \to (ty3 g c2 t3 t))))).(\lambda (u: -T).(\lambda (t4: T).(\lambda (H2: (ty3 g c u t4)).(\lambda (H3: ((\forall -(c2: C).((wf3 g c c2) \to (ty3 g c2 u t4))))).(\lambda (H4: (pc3 c t4 -t3)).(\lambda (c2: C).(\lambda (H5: (wf3 g c c2)).(ex_ind T (\lambda (t0: -T).(ty3 g c t4 t0)) (ty3 g c2 u t3) (\lambda (x: T).(\lambda (H6: (ty3 g c t4 -x)).(ty3_conv g c2 t3 t (H1 c2 H5) u t4 (H3 c2 H5) (wf3_pc3_conf g c t4 t3 H4 -c2 H5 x H6 t H0)))) (ty3_correct g c u t4 H2)))))))))))))) (\lambda (c: -C).(\lambda (m: nat).(\lambda (c2: C).(\lambda (_: (wf3 g c c2)).(ty3_sort g -c2 m))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d: C).(\lambda (u: -T).(\lambda (H0: (getl n c (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda -(H1: (ty3 g d u t)).(\lambda (H2: ((\forall (c2: C).((wf3 g d c2) \to (ty3 g -c2 u t))))).(\lambda (c2: C).(\lambda (H3: (wf3 g c c2)).(let H_x \def -(wf3_getl_conf Abbr n c d u H0 g c2 H3 t H1) in (let H4 \def H_x in (ex2_ind -C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: -C).(wf3 g d d2)) (ty3 g c2 (TLRef n) (lift (S n) O t)) (\lambda (x: -C).(\lambda (H5: (getl n c2 (CHead x (Bind Abbr) u))).(\lambda (H6: (wf3 g d -x)).(ty3_abbr g n c2 x u H5 t (H2 x H6))))) H4))))))))))))) (\lambda (n: -nat).(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c -(CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u -t)).(\lambda (H2: ((\forall (c2: C).((wf3 g d c2) \to (ty3 g c2 u -t))))).(\lambda (c2: C).(\lambda (H3: (wf3 g c c2)).(let H_x \def -(wf3_getl_conf Abst n c d u H0 g c2 H3 t H1) in (let H4 \def H_x in (ex2_ind -C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(wf3 g d d2)) (ty3 g c2 (TLRef n) (lift (S n) O u)) (\lambda (x: -C).(\lambda (H5: (getl n c2 (CHead x (Bind Abst) u))).(\lambda (H6: (wf3 g d -x)).(ty3_abst g n c2 x u H5 t (H2 x H6))))) H4))))))))))))) (\lambda (c: -C).(\lambda (u: T).(\lambda (t: T).(\lambda (H0: (ty3 g c u t)).(\lambda (H1: -((\forall (c2: C).((wf3 g c c2) \to (ty3 g c2 u t))))).(\lambda (b: -B).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (ty3 g (CHead c (Bind b) u) -t3 t4)).(\lambda (H3: ((\forall (c2: C).((wf3 g (CHead c (Bind b) u) c2) \to -(ty3 g c2 t3 t4))))).(\lambda (c2: C).(\lambda (H4: (wf3 g c c2)).(ty3_bind g -c2 u t (H1 c2 H4) b t3 t4 (H3 (CHead c2 (Bind b) u) (wf3_bind g c c2 H4 u t -H0 b))))))))))))))) (\lambda (c: C).(\lambda (w: T).(\lambda (u: T).(\lambda -(_: (ty3 g c w u)).(\lambda (H1: ((\forall (c2: C).((wf3 g c c2) \to (ty3 g -c2 w u))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c v (THead -(Bind Abst) u t))).(\lambda (H3: ((\forall (c2: C).((wf3 g c c2) \to (ty3 g -c2 v (THead (Bind Abst) u t)))))).(\lambda (c2: C).(\lambda (H4: (wf3 g c -c2)).(ty3_appl g c2 w u (H1 c2 H4) v t (H3 c2 H4))))))))))))) (\lambda (c: -C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (ty3 g c t3 t4)).(\lambda -(H1: ((\forall (c2: C).((wf3 g c c2) \to (ty3 g c2 t3 t4))))).(\lambda (t0: -T).(\lambda (_: (ty3 g c t4 t0)).(\lambda (H3: ((\forall (c2: C).((wf3 g c -c2) \to (ty3 g c2 t4 t0))))).(\lambda (c2: C).(\lambda (H4: (wf3 g c -c2)).(ty3_cast g c2 t3 t4 (H1 c2 H4) t0 (H3 c2 H4)))))))))))) c1 t1 t2 H))))). -(* COMMENTS -Initial nodes: 1027 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/A/defs.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/A/defs.mma deleted file mode 100644 index 626dbb793..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/A/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/preamble.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/C/defs.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/C/defs.mma deleted file mode 100644 index e86d73b04..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/C/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/T/defs.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/C/props.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/C/props.mma deleted file mode 100644 index 7cbed8e15..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/C/props.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/C/defs.ma". - -include "Basic-2/T/props.ma". - -inline "Basic-1/C/props.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/G/defs.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/G/defs.mma deleted file mode 100644 index 626dbb793..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/G/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/preamble.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/T/dec.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/T/dec.mma deleted file mode 100644 index 85ff33c1b..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/T/dec.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/T/defs.ma". - -inline "Basic-1/T/dec.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/T/defs.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/T/defs.mma deleted file mode 100644 index 626dbb793..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/T/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/preamble.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/T/props.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/T/props.mma deleted file mode 100644 index 58ffc668a..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/T/props.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/T/defs.ma". - -inline "Basic-1/T/props.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/aplus/defs.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/aplus/defs.mma deleted file mode 100644 index c0cf937b3..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/aplus/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/asucc/defs.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/aplus/props.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/aplus/props.mma deleted file mode 100644 index 9e2234b3b..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/aplus/props.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/aplus/defs.ma". - -include "Basic-2/next_plus/props.ma". - -inline "Basic-1/aplus/props.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/app/defs.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/app/defs.mma deleted file mode 100644 index 8132fde52..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/app/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/C/defs.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/aprem/defs.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/aprem/defs.mma deleted file mode 100644 index 8e2b3a9f6..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/aprem/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/A/defs.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/aprem/fwd.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/aprem/fwd.mma deleted file mode 100644 index 757c62bde..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/aprem/fwd.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/aprem/defs.ma". - -inline "Basic-1/aprem/fwd.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/aprem/props.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/aprem/props.mma deleted file mode 100644 index 11509ac4b..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/aprem/props.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/aprem/fwd.ma". - -include "Basic-2/leq/defs.ma". - -inline "Basic-1/aprem/props.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/arity/aprem.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/arity/aprem.mma deleted file mode 100644 index 7a368efa9..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/arity/aprem.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/arity/props.ma". - -include "Basic-2/arity/cimp.ma". - -include "Basic-2/aprem/props.ma". - -inline "Basic-1/arity/aprem.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/arity/cimp.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/arity/cimp.mma deleted file mode 100644 index ca1bf3cad..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/arity/cimp.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/arity/defs.ma". - -include "Basic-2/cimp/props.ma". - -inline "Basic-1/arity/cimp.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/arity/defs.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/arity/defs.mma deleted file mode 100644 index 4619a73fb..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/arity/defs.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/leq/defs.ma". - -include "Basic-2/getl/defs.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/arity/fwd.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/arity/fwd.mma deleted file mode 100644 index 7ef60bef2..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/arity/fwd.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/arity/defs.ma". - -include "Basic-2/leq/asucc.ma". - -include "Basic-2/getl/drop.ma". - -inline "Basic-1/arity/fwd.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/arity/lift1.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/arity/lift1.mma deleted file mode 100644 index adde30893..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/arity/lift1.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/arity/props.ma". - -include "Basic-2/drop1/fwd.ma". - -inline "Basic-1/arity/lift1.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/arity/pr3.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/arity/pr3.mma deleted file mode 100644 index 8459c2a35..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/arity/pr3.mma +++ /dev/null @@ -1,30 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csuba/arity.ma". - -include "Basic-2/pr3/defs.ma". - -include "Basic-2/pr1/defs.ma". - -include "Basic-2/wcpr0/getl.ma". - -include "Basic-2/pr0/fwd.ma". - -include "Basic-2/arity/subst0.ma". - -inline "Basic-1/arity/pr3.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/arity/props.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/arity/props.mma deleted file mode 100644 index 2d5c30d0d..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/arity/props.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/arity/fwd.ma". - -inline "Basic-1/arity/props.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/arity/subst0.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/arity/subst0.mma deleted file mode 100644 index ff090fc82..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/arity/subst0.mma +++ /dev/null @@ -1,30 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/arity/props.ma". - -include "Basic-2/fsubst0/fwd.ma". - -include "Basic-2/csubst0/getl.ma". - -include "Basic-2/subst0/dec.ma". - -include "Basic-2/subst0/fwd.ma". - -include "Basic-2/getl/getl.ma". - -inline "Basic-1/arity/subst0.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/asucc/defs.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/asucc/defs.mma deleted file mode 100644 index 067d1947d..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/asucc/defs.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/A/defs.ma". - -include "Basic-2/G/defs.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/asucc/fwd.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/asucc/fwd.mma deleted file mode 100644 index d0e22947a..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/asucc/fwd.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/asucc/defs.ma". - -inline "Basic-1/asucc/fwd.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/cimp/defs.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/cimp/defs.mma deleted file mode 100644 index 0d4047fa3..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/cimp/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/getl/defs.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/cimp/props.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/cimp/props.mma deleted file mode 100644 index 288bce2f2..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/cimp/props.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/cimp/defs.ma". - -include "Basic-2/getl/getl.ma". - -inline "Basic-1/cimp/props.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/clear/defs.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/clear/defs.mma deleted file mode 100644 index 8132fde52..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/clear/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/C/defs.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/clear/drop.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/clear/drop.mma deleted file mode 100644 index d78b3941a..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/clear/drop.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/clear/fwd.ma". - -include "Basic-2/drop/fwd.ma". - -inline "Basic-1/clear/drop.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/clear/fwd.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/clear/fwd.mma deleted file mode 100644 index e56fb51b1..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/clear/fwd.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/clear/defs.ma". - -inline "Basic-1/clear/fwd.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/clear/props.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/clear/props.mma deleted file mode 100644 index 0bd37aee0..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/clear/props.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/clear/fwd.ma". - -inline "Basic-1/clear/props.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/clen/defs.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/clen/defs.mma deleted file mode 100644 index d5e42a147..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/clen/defs.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/C/defs.ma". - -include "Basic-2/s/defs.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/clen/getl.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/clen/getl.mma deleted file mode 100644 index 3d2d8f86c..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/clen/getl.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/clen/defs.ma". - -include "Basic-2/getl/props.ma". - -inline "Basic-1/clen/getl.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/cnt/defs.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/cnt/defs.mma deleted file mode 100644 index e86d73b04..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/cnt/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/T/defs.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/cnt/props.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/cnt/props.mma deleted file mode 100644 index 8b288f51f..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/cnt/props.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/cnt/defs.ma". - -include "Basic-2/lift/fwd.ma". - -inline "Basic-1/cnt/props.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csuba/arity.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csuba/arity.mma deleted file mode 100644 index 2d7cdb86d..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csuba/arity.mma +++ /dev/null @@ -1,26 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csuba/getl.ma". - -include "Basic-2/csuba/props.ma". - -include "Basic-2/arity/props.ma". - -include "Basic-2/csubv/getl.ma". - -inline "Basic-1/csuba/arity.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csuba/clear.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csuba/clear.mma deleted file mode 100644 index 15098af70..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csuba/clear.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csuba/defs.ma". - -include "Basic-2/clear/fwd.ma". - -inline "Basic-1/csuba/clear.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csuba/defs.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csuba/defs.mma deleted file mode 100644 index b9af9219e..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csuba/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/arity/defs.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csuba/drop.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csuba/drop.mma deleted file mode 100644 index 4aec6957d..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csuba/drop.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csuba/fwd.ma". - -include "Basic-2/drop/fwd.ma". - -inline "Basic-1/csuba/drop.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csuba/fwd.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csuba/fwd.mma deleted file mode 100644 index 2c5a16f23..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csuba/fwd.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csuba/defs.ma". - -inline "Basic-1/csuba/fwd.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csuba/getl.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csuba/getl.mma deleted file mode 100644 index e52f58df9..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csuba/getl.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csuba/drop.ma". - -include "Basic-2/csuba/clear.ma". - -include "Basic-2/getl/clear.ma". - -inline "Basic-1/csuba/getl.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csuba/props.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csuba/props.mma deleted file mode 100644 index d1ab31945..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csuba/props.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csuba/defs.ma". - -inline "Basic-1/csuba/props.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubc/arity.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubc/arity.mma deleted file mode 100644 index 63fff2532..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubc/arity.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubc/csuba.ma". - -inline "Basic-1/csubc/arity.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubc/clear.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubc/clear.mma deleted file mode 100644 index 1a0cbcf35..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubc/clear.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubc/fwd.ma". - -inline "Basic-1/csubc/clear.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubc/csuba.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubc/csuba.mma deleted file mode 100644 index f3465bf35..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubc/csuba.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubc/defs.ma". - -include "Basic-2/sc3/props.ma". - -inline "Basic-1/csubc/csuba.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubc/defs.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubc/defs.mma deleted file mode 100644 index 537c3eb52..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubc/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/sc3/defs.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubc/drop.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubc/drop.mma deleted file mode 100644 index 2a5d715f5..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubc/drop.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubc/fwd.ma". - -include "Basic-2/sc3/props.ma". - -inline "Basic-1/csubc/drop.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubc/drop1.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubc/drop1.mma deleted file mode 100644 index 9863063cf..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubc/drop1.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubc/drop.ma". - -inline "Basic-1/csubc/drop1.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubc/fwd.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubc/fwd.mma deleted file mode 100644 index 1d0dd32ab..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubc/fwd.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubc/defs.ma". - -inline "Basic-1/csubc/fwd.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubc/getl.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubc/getl.mma deleted file mode 100644 index 60462e153..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubc/getl.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubc/drop.ma". - -include "Basic-2/csubc/clear.ma". - -inline "Basic-1/csubc/getl.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubc/props.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubc/props.mma deleted file mode 100644 index ff7a8f4e5..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubc/props.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubc/defs.ma". - -include "Basic-2/sc3/props.ma". - -inline "Basic-1/csubc/props.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubst0/clear.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubst0/clear.mma deleted file mode 100644 index 0b3c214d7..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubst0/clear.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubst0/props.ma". - -include "Basic-2/csubst0/fwd.ma". - -include "Basic-2/clear/fwd.ma". - -inline "Basic-1/csubst0/clear.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubst0/defs.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubst0/defs.mma deleted file mode 100644 index 912ad35a1..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubst0/defs.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/subst0/defs.ma". - -include "Basic-2/C/defs.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubst0/drop.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubst0/drop.mma deleted file mode 100644 index 8ec4de71c..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubst0/drop.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubst0/fwd.ma". - -include "Basic-2/drop/fwd.ma". - -include "Basic-2/s/props.ma". - -inline "Basic-1/csubst0/drop.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubst0/fwd.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubst0/fwd.mma deleted file mode 100644 index 85ddb329e..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubst0/fwd.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubst0/defs.ma". - -inline "Basic-1/csubst0/fwd.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubst0/getl.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubst0/getl.mma deleted file mode 100644 index 679c49e9e..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubst0/getl.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubst0/clear.ma". - -include "Basic-2/csubst0/drop.ma". - -include "Basic-2/getl/fwd.ma". - -inline "Basic-1/csubst0/getl.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubst0/props.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubst0/props.mma deleted file mode 100644 index fee62c651..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubst0/props.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubst0/defs.ma". - -inline "Basic-1/csubst0/props.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubst1/defs.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubst1/defs.mma deleted file mode 100644 index 16b76c403..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubst1/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubst0/defs.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubst1/fwd.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubst1/fwd.mma deleted file mode 100644 index 2c9d5bedb..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubst1/fwd.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubst1/defs.ma". - -include "Basic-2/csubst0/fwd.ma". - -include "Basic-2/subst1/props.ma". - -inline "Basic-1/csubst1/fwd.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubst1/getl.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubst1/getl.mma deleted file mode 100644 index 05b80d444..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubst1/getl.mma +++ /dev/null @@ -1,26 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubst1/props.ma". - -include "Basic-2/csubst0/getl.ma". - -include "Basic-2/subst1/props.ma". - -include "Basic-2/drop/props.ma". - -inline "Basic-1/csubst1/getl.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubst1/props.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubst1/props.mma deleted file mode 100644 index 7480c122d..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubst1/props.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubst1/defs.ma". - -include "Basic-2/subst1/defs.ma". - -inline "Basic-1/csubst1/props.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubt/clear.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubt/clear.mma deleted file mode 100644 index c35e253b4..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubt/clear.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubt/defs.ma". - -include "Basic-2/clear/fwd.ma". - -inline "Basic-1/csubt/clear.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubt/csuba.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubt/csuba.mma deleted file mode 100644 index 3145d010e..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubt/csuba.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/ty3/arity.ma". - -inline "Basic-1/csubt/csuba.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubt/defs.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubt/defs.mma deleted file mode 100644 index 2dde8c752..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubt/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/ty3/defs.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubt/drop.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubt/drop.mma deleted file mode 100644 index 7d8d90b33..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubt/drop.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubt/fwd.ma". - -include "Basic-2/drop/fwd.ma". - -inline "Basic-1/csubt/drop.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubt/fwd.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubt/fwd.mma deleted file mode 100644 index 195de62cb..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubt/fwd.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubt/defs.ma". - -inline "Basic-1/csubt/fwd.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubt/getl.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubt/getl.mma deleted file mode 100644 index 14299efa2..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubt/getl.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubt/clear.ma". - -include "Basic-2/csubt/drop.ma". - -include "Basic-2/getl/clear.ma". - -inline "Basic-1/csubt/getl.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubt/pc3.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubt/pc3.mma deleted file mode 100644 index 48152a367..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubt/pc3.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubt/getl.ma". - -include "Basic-2/pc3/left.ma". - -inline "Basic-1/csubt/pc3.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubt/props.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubt/props.mma deleted file mode 100644 index 59d3c5188..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubt/props.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubt/defs.ma". - -inline "Basic-1/csubt/props.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubt/ty3.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubt/ty3.mma deleted file mode 100644 index 8bb74616a..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubt/ty3.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubt/pc3.ma". - -include "Basic-2/csubt/props.ma". - -inline "Basic-1/csubt/ty3.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubv/clear.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubv/clear.mma deleted file mode 100644 index 77768e993..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubv/clear.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubv/defs.ma". - -include "Basic-2/clear/fwd.ma". - -inline "Basic-1/csubv/clear.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubv/defs.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubv/defs.mma deleted file mode 100644 index 8132fde52..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubv/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/C/defs.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubv/drop.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubv/drop.mma deleted file mode 100644 index d0d2b3f76..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubv/drop.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubv/props.ma". - -include "Basic-2/drop/fwd.ma". - -inline "Basic-1/csubv/drop.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubv/getl.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubv/getl.mma deleted file mode 100644 index 66c63a639..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubv/getl.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubv/clear.ma". - -include "Basic-2/csubv/drop.ma". - -include "Basic-2/getl/fwd.ma". - -inline "Basic-1/csubv/getl.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubv/props.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubv/props.mma deleted file mode 100644 index 44d2edf27..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/csubv/props.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubv/defs.ma". - -include "Basic-2/T/props.ma". - -inline "Basic-1/csubv/props.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/drop/defs.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/drop/defs.mma deleted file mode 100644 index 8837d010f..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/drop/defs.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/C/defs.ma". - -include "Basic-2/lift/defs.ma". - -include "Basic-2/r/defs.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/drop/fwd.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/drop/fwd.mma deleted file mode 100644 index d88936b9c..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/drop/fwd.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/drop/defs.ma". - -inline "Basic-1/drop/fwd.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/drop/props.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/drop/props.mma deleted file mode 100644 index 9af06df85..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/drop/props.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/drop/fwd.ma". - -include "Basic-2/lift/props.ma". - -include "Basic-2/r/props.ma". - -inline "Basic-1/drop/props.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/drop1/defs.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/drop1/defs.mma deleted file mode 100644 index 6d92e123e..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/drop1/defs.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/drop/defs.ma". - -include "Basic-2/lift1/defs.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/drop1/fwd.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/drop1/fwd.mma deleted file mode 100644 index 78b91b349..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/drop1/fwd.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/drop1/defs.ma". - -inline "Basic-1/drop1/fwd.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/drop1/getl.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/drop1/getl.mma deleted file mode 100644 index 0b9e358d2..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/drop1/getl.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/drop1/fwd.ma". - -include "Basic-2/getl/drop.ma". - -inline "Basic-1/drop1/getl.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/drop1/props.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/drop1/props.mma deleted file mode 100644 index f4f2204ac..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/drop1/props.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/drop1/fwd.ma". - -include "Basic-2/drop/props.ma". - -include "Basic-2/getl/defs.ma". - -inline "Basic-1/drop1/props.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/ex0/defs.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/ex0/defs.mma deleted file mode 100644 index 067d1947d..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/ex0/defs.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/A/defs.ma". - -include "Basic-2/G/defs.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/ex0/props.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/ex0/props.mma deleted file mode 100644 index 80204275f..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/ex0/props.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/ex0/defs.ma". - -include "Basic-2/leq/defs.ma". - -include "Basic-2/aplus/props.ma". - -inline "Basic-1/ex0/props.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/ex1/defs.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/ex1/defs.mma deleted file mode 100644 index 8132fde52..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/ex1/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/C/defs.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/ex1/props.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/ex1/props.mma deleted file mode 100644 index 8ec7ebe3e..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/ex1/props.mma +++ /dev/null @@ -1,32 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/ex1/defs.ma". - -include "Basic-2/ty3/fwd.ma". - -include "Basic-2/pc3/fwd.ma". - -include "Basic-2/nf2/pr3.ma". - -include "Basic-2/nf2/props.ma". - -include "Basic-2/arity/defs.ma". - -include "Basic-2/leq/props.ma". - -inline "Basic-1/ex1/props.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/ex2/defs.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/ex2/defs.mma deleted file mode 100644 index 8132fde52..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/ex2/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/C/defs.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/ex2/props.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/ex2/props.mma deleted file mode 100644 index 56d0d8557..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/ex2/props.mma +++ /dev/null @@ -1,26 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/ex2/defs.ma". - -include "Basic-2/nf2/defs.ma". - -include "Basic-2/pr2/fwd.ma". - -include "Basic-2/arity/fwd.ma". - -inline "Basic-1/ex2/props.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/flt/defs.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/flt/defs.mma deleted file mode 100644 index 8132fde52..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/flt/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/C/defs.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/flt/props.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/flt/props.mma deleted file mode 100644 index e5b76348f..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/flt/props.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/flt/defs.ma". - -include "Basic-2/C/props.ma". - -inline "Basic-1/flt/props.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/fsubst0/defs.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/fsubst0/defs.mma deleted file mode 100644 index 16b76c403..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/fsubst0/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubst0/defs.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/fsubst0/fwd.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/fsubst0/fwd.mma deleted file mode 100644 index 4a60d7191..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/fsubst0/fwd.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/fsubst0/defs.ma". - -inline "Basic-1/fsubst0/fwd.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/getl/clear.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/getl/clear.mma deleted file mode 100644 index 3f69bde5d..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/getl/clear.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/getl/props.ma". - -include "Basic-2/clear/drop.ma". - -inline "Basic-1/getl/clear.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/getl/dec.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/getl/dec.mma deleted file mode 100644 index 4085b140e..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/getl/dec.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/getl/props.ma". - -inline "Basic-1/getl/dec.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/getl/defs.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/getl/defs.mma deleted file mode 100644 index 818f4f3ae..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/getl/defs.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/drop/defs.ma". - -include "Basic-2/clear/defs.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/getl/drop.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/getl/drop.mma deleted file mode 100644 index ff4bb646f..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/getl/drop.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/getl/props.ma". - -include "Basic-2/clear/drop.ma". - -inline "Basic-1/getl/drop.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/getl/flt.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/getl/flt.mma deleted file mode 100644 index c030693f5..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/getl/flt.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/getl/fwd.ma". - -include "Basic-2/clear/props.ma". - -include "Basic-2/flt/props.ma". - -inline "Basic-1/getl/flt.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/getl/fwd.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/getl/fwd.mma deleted file mode 100644 index dfcee9576..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/getl/fwd.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/getl/defs.ma". - -include "Basic-2/drop/fwd.ma". - -include "Basic-2/clear/fwd.ma". - -inline "Basic-1/getl/fwd.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/getl/getl.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/getl/getl.mma deleted file mode 100644 index b5e5163d7..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/getl/getl.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/getl/drop.ma". - -include "Basic-2/getl/clear.ma". - -inline "Basic-1/getl/getl.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/getl/props.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/getl/props.mma deleted file mode 100644 index 00ff97539..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/getl/props.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/getl/fwd.ma". - -include "Basic-2/drop/props.ma". - -include "Basic-2/clear/props.ma". - -inline "Basic-1/getl/props.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/iso/defs.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/iso/defs.mma deleted file mode 100644 index e86d73b04..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/iso/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/T/defs.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/iso/fwd.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/iso/fwd.mma deleted file mode 100644 index 8ae8aaad1..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/iso/fwd.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/iso/defs.ma". - -include "Basic-2/tlist/defs.ma". - -inline "Basic-1/iso/fwd.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/iso/props.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/iso/props.mma deleted file mode 100644 index c689ea1fc..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/iso/props.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/iso/fwd.ma". - -inline "Basic-1/iso/props.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/leq/asucc.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/leq/asucc.mma deleted file mode 100644 index a89191b08..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/leq/asucc.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/leq/props.ma". - -inline "Basic-1/leq/asucc.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/leq/defs.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/leq/defs.mma deleted file mode 100644 index dadf5769f..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/leq/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/aplus/defs.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/leq/fwd.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/leq/fwd.mma deleted file mode 100644 index 750437135..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/leq/fwd.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/leq/defs.ma". - -inline "Basic-1/leq/fwd.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/leq/props.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/leq/props.mma deleted file mode 100644 index b2509dd31..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/leq/props.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/leq/fwd.ma". - -include "Basic-2/aplus/props.ma". - -inline "Basic-1/leq/props.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/lift/defs.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/lift/defs.mma deleted file mode 100644 index 85138ec00..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/lift/defs.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/tlist/defs.ma". - -include "Basic-2/s/defs.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/lift/fwd.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/lift/fwd.mma deleted file mode 100644 index 95a454dbe..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/lift/fwd.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/lift/defs.ma". - -inline "Basic-1/lift/fwd.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/lift/props.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/lift/props.mma deleted file mode 100644 index 88f85f5cb..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/lift/props.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/lift/fwd.ma". - -include "Basic-2/s/props.ma". - -inline "Basic-1/lift/props.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/lift/tlt.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/lift/tlt.mma deleted file mode 100644 index 51c972987..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/lift/tlt.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/lift/fwd.ma". - -include "Basic-2/tlt/props.ma". - -inline "Basic-1/lift/tlt.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/lift1/defs.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/lift1/defs.mma deleted file mode 100644 index 2948c111d..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/lift1/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/lift/defs.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/lift1/fwd.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/lift1/fwd.mma deleted file mode 100644 index 3aabe676f..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/lift1/fwd.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/lift1/defs.ma". - -include "Basic-2/lift/fwd.ma". - -inline "Basic-1/lift1/fwd.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/lift1/props.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/lift1/props.mma deleted file mode 100644 index 89bed2166..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/lift1/props.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/lift/props.ma". - -include "Basic-2/drop1/defs.ma". - -inline "Basic-1/lift1/props.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/llt/defs.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/llt/defs.mma deleted file mode 100644 index 8e2b3a9f6..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/llt/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/A/defs.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/llt/props.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/llt/props.mma deleted file mode 100644 index 462e5c575..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/llt/props.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/llt/defs.ma". - -include "Basic-2/leq/defs.ma". - -inline "Basic-1/llt/props.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/next_plus/defs.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/next_plus/defs.mma deleted file mode 100644 index 33d7fdddc..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/next_plus/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/G/defs.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/next_plus/props.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/next_plus/props.mma deleted file mode 100644 index 8e6ff559c..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/next_plus/props.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/next_plus/defs.ma". - -inline "Basic-1/next_plus/props.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/nf2/arity.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/nf2/arity.mma deleted file mode 100644 index 356dfa591..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/nf2/arity.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/nf2/fwd.ma". - -include "Basic-2/arity/subst0.ma". - -inline "Basic-1/nf2/arity.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/nf2/dec.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/nf2/dec.mma deleted file mode 100644 index 179da42ff..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/nf2/dec.mma +++ /dev/null @@ -1,28 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/nf2/defs.ma". - -include "Basic-2/pr2/clen.ma". - -include "Basic-2/pr2/fwd.ma". - -include "Basic-2/pr0/dec.ma". - -include "Basic-2/C/props.ma". - -inline "Basic-1/nf2/dec.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/nf2/defs.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/nf2/defs.mma deleted file mode 100644 index 4f97474c7..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/nf2/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pr2/defs.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/nf2/fwd.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/nf2/fwd.mma deleted file mode 100644 index 0f941386e..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/nf2/fwd.mma +++ /dev/null @@ -1,26 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/nf2/defs.ma". - -include "Basic-2/pr2/clen.ma". - -include "Basic-2/subst0/dec.ma". - -include "Basic-2/T/props.ma". - -inline "Basic-1/nf2/fwd.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/nf2/iso.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/nf2/iso.mma deleted file mode 100644 index 0c6309e1c..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/nf2/iso.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/nf2/pr3.ma". - -include "Basic-2/pr3/fwd.ma". - -include "Basic-2/iso/props.ma". - -inline "Basic-1/nf2/iso.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/nf2/lift1.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/nf2/lift1.mma deleted file mode 100644 index 250c8dd78..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/nf2/lift1.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/nf2/props.ma". - -include "Basic-2/drop1/fwd.ma". - -inline "Basic-1/nf2/lift1.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/nf2/pr3.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/nf2/pr3.mma deleted file mode 100644 index 00ba3ffab..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/nf2/pr3.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/nf2/defs.ma". - -include "Basic-2/pr3/pr3.ma". - -inline "Basic-1/nf2/pr3.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/nf2/props.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/nf2/props.mma deleted file mode 100644 index 6977bd069..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/nf2/props.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/nf2/defs.ma". - -include "Basic-2/pr2/fwd.ma". - -inline "Basic-1/nf2/props.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pc1/defs.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pc1/defs.mma deleted file mode 100644 index 9818026a3..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pc1/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pr1/defs.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pc1/props.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pc1/props.mma deleted file mode 100644 index f031a2362..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pc1/props.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pc1/defs.ma". - -include "Basic-2/pr1/pr1.ma". - -inline "Basic-1/pc1/props.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pc3/dec.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pc3/dec.mma deleted file mode 100644 index d5ed4097b..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pc3/dec.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/ty3/arity_props.ma". - -include "Basic-2/nf2/fwd.ma". - -inline "Basic-1/pc3/dec.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pc3/defs.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pc3/defs.mma deleted file mode 100644 index ac791ab6d..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pc3/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pr3/defs.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pc3/fsubst0.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pc3/fsubst0.mma deleted file mode 100644 index 478c285af..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pc3/fsubst0.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pc3/left.ma". - -include "Basic-2/fsubst0/defs.ma". - -include "Basic-2/csubst0/getl.ma". - -inline "Basic-1/pc3/fsubst0.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pc3/fwd.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pc3/fwd.mma deleted file mode 100644 index 83816ceac..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pc3/fwd.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pc3/props.ma". - -include "Basic-2/pr3/fwd.ma". - -inline "Basic-1/pc3/fwd.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pc3/left.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pc3/left.mma deleted file mode 100644 index a0102b201..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pc3/left.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pc3/props.ma". - -inline "Basic-1/pc3/left.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pc3/nf2.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pc3/nf2.mma deleted file mode 100644 index 38a00ff31..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pc3/nf2.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pc3/defs.ma". - -include "Basic-2/nf2/pr3.ma". - -inline "Basic-1/pc3/nf2.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pc3/pc1.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pc3/pc1.mma deleted file mode 100644 index 0a82a8adc..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pc3/pc1.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pc3/defs.ma". - -include "Basic-2/pc1/defs.ma". - -include "Basic-2/pr3/pr1.ma". - -inline "Basic-1/pc3/pc1.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pc3/props.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pc3/props.mma deleted file mode 100644 index 01c5ee4e8..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pc3/props.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pc3/defs.ma". - -include "Basic-2/pr3/pr3.ma". - -inline "Basic-1/pc3/props.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pc3/subst1.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pc3/subst1.mma deleted file mode 100644 index 1ec4a10d5..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pc3/subst1.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pc3/props.ma". - -include "Basic-2/pr3/subst1.ma". - -inline "Basic-1/pc3/subst1.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pc3/wcpr0.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pc3/wcpr0.mma deleted file mode 100644 index 98ae77c9b..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pc3/wcpr0.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pc3/props.ma". - -include "Basic-2/wcpr0/getl.ma". - -inline "Basic-1/pc3/wcpr0.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr0/dec.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr0/dec.mma deleted file mode 100644 index c88b2b725..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr0/dec.mma +++ /dev/null @@ -1,26 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pr0/fwd.ma". - -include "Basic-2/subst0/dec.ma". - -include "Basic-2/T/dec.ma". - -include "Basic-2/T/props.ma". - -inline "Basic-1/pr0/dec.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr0/defs.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr0/defs.mma deleted file mode 100644 index e94eab620..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr0/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/subst0/defs.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr0/fwd.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr0/fwd.mma deleted file mode 100644 index 2349a5127..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr0/fwd.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pr0/props.ma". - -inline "Basic-1/pr0/fwd.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr0/pr0.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr0/pr0.mma deleted file mode 100644 index cc88c47e0..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr0/pr0.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pr0/fwd.ma". - -include "Basic-2/lift/tlt.ma". - -inline "Basic-1/pr0/pr0.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr0/props.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr0/props.mma deleted file mode 100644 index 9b8e83a73..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr0/props.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pr0/defs.ma". - -include "Basic-2/subst0/subst0.ma". - -inline "Basic-1/pr0/props.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr0/subst1.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr0/subst1.mma deleted file mode 100644 index a4af1a695..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr0/subst1.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pr0/props.ma". - -include "Basic-2/subst1/defs.ma". - -inline "Basic-1/pr0/subst1.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr1/defs.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr1/defs.mma deleted file mode 100644 index 9fb12774d..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr1/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pr0/defs.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr1/pr1.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr1/pr1.mma deleted file mode 100644 index f05ca1cd3..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr1/pr1.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pr1/props.ma". - -include "Basic-2/pr0/pr0.ma". - -inline "Basic-1/pr1/pr1.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr1/props.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr1/props.mma deleted file mode 100644 index 5ec9c7f3e..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr1/props.mma +++ /dev/null @@ -1,26 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pr1/defs.ma". - -include "Basic-2/pr0/subst1.ma". - -include "Basic-2/subst1/props.ma". - -include "Basic-2/T/props.ma". - -inline "Basic-1/pr1/props.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr2/clen.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr2/clen.mma deleted file mode 100644 index e3ea2f078..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr2/clen.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pr2/props.ma". - -include "Basic-2/clen/getl.ma". - -inline "Basic-1/pr2/clen.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr2/defs.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr2/defs.mma deleted file mode 100644 index b43baa983..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr2/defs.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pr0/defs.ma". - -include "Basic-2/getl/defs.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr2/fwd.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr2/fwd.mma deleted file mode 100644 index 9ec452e99..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr2/fwd.mma +++ /dev/null @@ -1,26 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pr2/defs.ma". - -include "Basic-2/pr0/fwd.ma". - -include "Basic-2/getl/drop.ma". - -include "Basic-2/getl/clear.ma". - -inline "Basic-1/pr2/fwd.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr2/pr2.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr2/pr2.mma deleted file mode 100644 index 8879c71e8..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr2/pr2.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pr2/defs.ma". - -include "Basic-2/pr0/pr0.ma". - -include "Basic-2/getl/props.ma". - -inline "Basic-1/pr2/pr2.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr2/props.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr2/props.mma deleted file mode 100644 index b0e27c8fc..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr2/props.mma +++ /dev/null @@ -1,26 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pr2/defs.ma". - -include "Basic-2/pr0/props.ma". - -include "Basic-2/getl/drop.ma". - -include "Basic-2/getl/clear.ma". - -inline "Basic-1/pr2/props.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr2/subst1.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr2/subst1.mma deleted file mode 100644 index 61b161222..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr2/subst1.mma +++ /dev/null @@ -1,32 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pr2/defs.ma". - -include "Basic-2/pr0/subst1.ma". - -include "Basic-2/pr0/fwd.ma". - -include "Basic-2/csubst1/getl.ma". - -include "Basic-2/csubst1/fwd.ma". - -include "Basic-2/subst1/subst1.ma". - -include "Basic-2/getl/drop.ma". - -inline "Basic-1/pr2/subst1.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr3/defs.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr3/defs.mma deleted file mode 100644 index 4f97474c7..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr3/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pr2/defs.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr3/fwd.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr3/fwd.mma deleted file mode 100644 index a29ddcee0..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr3/fwd.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pr3/props.ma". - -include "Basic-2/pr2/fwd.ma". - -inline "Basic-1/pr3/fwd.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr3/iso.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr3/iso.mma deleted file mode 100644 index a401d94dc..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr3/iso.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pr3/fwd.ma". - -include "Basic-2/iso/props.ma". - -include "Basic-2/tlist/props.ma". - -inline "Basic-1/pr3/iso.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr3/pr1.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr3/pr1.mma deleted file mode 100644 index 6827dfddc..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr3/pr1.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pr3/defs.ma". - -include "Basic-2/pr1/defs.ma". - -inline "Basic-1/pr3/pr1.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr3/pr3.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr3/pr3.mma deleted file mode 100644 index a393080b6..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr3/pr3.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pr3/props.ma". - -include "Basic-2/pr2/pr2.ma". - -inline "Basic-1/pr3/pr3.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr3/props.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr3/props.mma deleted file mode 100644 index c87af259a..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr3/props.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pr3/pr1.ma". - -include "Basic-2/pr2/props.ma". - -include "Basic-2/pr1/props.ma". - -inline "Basic-1/pr3/props.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr3/subst1.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr3/subst1.mma deleted file mode 100644 index 73288e883..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr3/subst1.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pr3/defs.ma". - -include "Basic-2/pr2/subst1.ma". - -inline "Basic-1/pr3/subst1.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr3/wcpr0.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr3/wcpr0.mma deleted file mode 100644 index 6c30039f1..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/pr3/wcpr0.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pr3/props.ma". - -include "Basic-2/wcpr0/getl.ma". - -inline "Basic-1/pr3/wcpr0.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/preamble.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/preamble.ma deleted file mode 100644 index d0ddca34a..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/preamble.ma +++ /dev/null @@ -1,16 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "Ground-2/theory.ma". -include "Basic-1/definitions.ma". diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/r/defs.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/r/defs.mma deleted file mode 100644 index e86d73b04..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/r/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/T/defs.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/r/props.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/r/props.mma deleted file mode 100644 index 79edc95f4..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/r/props.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/r/defs.ma". - -include "Basic-2/s/defs.ma". - -inline "Basic-1/r/props.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/s/defs.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/s/defs.mma deleted file mode 100644 index e86d73b04..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/s/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/T/defs.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/s/props.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/s/props.mma deleted file mode 100644 index 35f1ad31b..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/s/props.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/s/defs.ma". - -inline "Basic-1/s/props.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/sc3/arity.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/sc3/arity.mma deleted file mode 100644 index 9fa3eaf93..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/sc3/arity.mma +++ /dev/null @@ -1,26 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubc/arity.ma". - -include "Basic-2/csubc/getl.ma". - -include "Basic-2/csubc/drop1.ma". - -include "Basic-2/csubc/props.ma". - -inline "Basic-1/sc3/arity.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/sc3/defs.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/sc3/defs.mma deleted file mode 100644 index d1dde1884..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/sc3/defs.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/sn3/defs.ma". - -include "Basic-2/arity/defs.ma". - -include "Basic-2/drop1/defs.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/sc3/props.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/sc3/props.mma deleted file mode 100644 index 859458266..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/sc3/props.mma +++ /dev/null @@ -1,38 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/sc3/defs.ma". - -include "Basic-2/sn3/lift1.ma". - -include "Basic-2/nf2/lift1.ma". - -include "Basic-2/csuba/arity.ma". - -include "Basic-2/arity/lift1.ma". - -include "Basic-2/arity/aprem.ma". - -include "Basic-2/llt/props.ma". - -include "Basic-2/drop1/getl.ma". - -include "Basic-2/drop1/props.ma". - -include "Basic-2/lift1/props.ma". - -inline "Basic-1/sc3/props.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/sn3/defs.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/sn3/defs.mma deleted file mode 100644 index ac791ab6d..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/sn3/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pr3/defs.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/sn3/fwd.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/sn3/fwd.mma deleted file mode 100644 index a4662eefa..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/sn3/fwd.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/sn3/defs.ma". - -include "Basic-2/pr3/props.ma". - -inline "Basic-1/sn3/fwd.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/sn3/lift1.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/sn3/lift1.mma deleted file mode 100644 index e8c0c4de2..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/sn3/lift1.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/sn3/props.ma". - -include "Basic-2/drop1/fwd.ma". - -include "Basic-2/lift1/fwd.ma". - -inline "Basic-1/sn3/lift1.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/sn3/nf2.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/sn3/nf2.mma deleted file mode 100644 index 81de1d480..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/sn3/nf2.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/sn3/defs.ma". - -include "Basic-2/nf2/dec.ma". - -include "Basic-2/nf2/pr3.ma". - -inline "Basic-1/sn3/nf2.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/sn3/props.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/sn3/props.mma deleted file mode 100644 index 3244a5237..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/sn3/props.mma +++ /dev/null @@ -1,26 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/sn3/nf2.ma". - -include "Basic-2/sn3/fwd.ma". - -include "Basic-2/nf2/iso.ma". - -include "Basic-2/pr3/iso.ma". - -inline "Basic-1/sn3/props.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/sty0/defs.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/sty0/defs.mma deleted file mode 100644 index 24df71ad7..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/sty0/defs.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/G/defs.ma". - -include "Basic-2/getl/defs.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/sty0/fwd.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/sty0/fwd.mma deleted file mode 100644 index a2600b078..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/sty0/fwd.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/sty0/defs.ma". - -inline "Basic-1/sty0/fwd.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/sty0/props.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/sty0/props.mma deleted file mode 100644 index 403332ea0..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/sty0/props.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/sty0/defs.ma". - -include "Basic-2/getl/drop.ma". - -inline "Basic-1/sty0/props.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/sty1/cnt.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/sty1/cnt.mma deleted file mode 100644 index 062ba4446..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/sty1/cnt.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/sty1/props.ma". - -include "Basic-2/cnt/props.ma". - -inline "Basic-1/sty1/cnt.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/sty1/defs.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/sty1/defs.mma deleted file mode 100644 index d2245411f..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/sty1/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/sty0/defs.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/sty1/props.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/sty1/props.mma deleted file mode 100644 index f764f229a..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/sty1/props.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/sty1/defs.ma". - -include "Basic-2/sty0/props.ma". - -inline "Basic-1/sty1/props.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/subst/defs.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/subst/defs.mma deleted file mode 100644 index 2948c111d..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/subst/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/lift/defs.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/subst/fwd.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/subst/fwd.mma deleted file mode 100644 index 27b37067f..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/subst/fwd.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/subst/defs.ma". - -inline "Basic-1/subst/fwd.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/subst/props.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/subst/props.mma deleted file mode 100644 index 1a45f22b5..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/subst/props.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/subst/fwd.ma". - -include "Basic-2/subst0/defs.ma". - -include "Basic-2/lift/props.ma". - -inline "Basic-1/subst/props.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/subst0/dec.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/subst0/dec.mma deleted file mode 100644 index b6595349c..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/subst0/dec.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/subst0/defs.ma". - -include "Basic-2/lift/props.ma". - -inline "Basic-1/subst0/dec.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/subst0/defs.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/subst0/defs.mma deleted file mode 100644 index 2948c111d..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/subst0/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/lift/defs.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/subst0/fwd.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/subst0/fwd.mma deleted file mode 100644 index 3da14bb5e..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/subst0/fwd.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/subst0/defs.ma". - -include "Basic-2/lift/props.ma". - -inline "Basic-1/subst0/fwd.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/subst0/props.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/subst0/props.mma deleted file mode 100644 index 155ba9e4f..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/subst0/props.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/subst0/fwd.ma". - -inline "Basic-1/subst0/props.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/subst0/subst0.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/subst0/subst0.mma deleted file mode 100644 index 3e87d4560..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/subst0/subst0.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/subst0/props.ma". - -inline "Basic-1/subst0/subst0.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/subst0/tlt.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/subst0/tlt.mma deleted file mode 100644 index 6583c3c92..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/subst0/tlt.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/subst0/defs.ma". - -include "Basic-2/lift/props.ma". - -include "Basic-2/lift/tlt.ma". - -inline "Basic-1/subst0/tlt.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/subst1/defs.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/subst1/defs.mma deleted file mode 100644 index e94eab620..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/subst1/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/subst0/defs.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/subst1/fwd.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/subst1/fwd.mma deleted file mode 100644 index 78669f892..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/subst1/fwd.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/subst1/defs.ma". - -include "Basic-2/subst0/props.ma". - -inline "Basic-1/subst1/fwd.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/subst1/props.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/subst1/props.mma deleted file mode 100644 index 35ae5d43d..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/subst1/props.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/subst1/defs.ma". - -include "Basic-2/subst0/props.ma". - -inline "Basic-1/subst1/props.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/subst1/subst1.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/subst1/subst1.mma deleted file mode 100644 index af3f35761..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/subst1/subst1.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/subst1/fwd.ma". - -include "Basic-2/subst0/subst0.ma". - -inline "Basic-1/subst1/subst1.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/theory.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/theory.ma deleted file mode 100644 index 9ab93dc9b..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/theory.ma +++ /dev/null @@ -1,42 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/subst0/tlt.ma". - -include "Basic-2/subst/props.ma". - -include "Basic-2/sty1/cnt.ma". - -include "Basic-2/ex0/props.ma". - -include "Basic-2/wcpr0/fwd.ma". - -include "Basic-2/pr3/wcpr0.ma". - -include "Basic-2/ex2/props.ma". - -include "Basic-2/ex1/props.ma". - -include "Basic-2/ty3/sty0.ma". - -include "Basic-2/csubt/csuba.ma". - -include "Basic-2/ty3/fwd_nf2.ma". - -include "Basic-2/ty3/nf2.ma". - -include "Basic-2/wf3/props.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/tlist/defs.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/tlist/defs.mma deleted file mode 100644 index e86d73b04..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/tlist/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/T/defs.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/tlist/props.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/tlist/props.mma deleted file mode 100644 index 4fcfcb1fb..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/tlist/props.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/tlist/defs.ma". - -inline "Basic-1/tlist/props.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/tlt/defs.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/tlt/defs.mma deleted file mode 100644 index e86d73b04..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/tlt/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/T/defs.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/tlt/props.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/tlt/props.mma deleted file mode 100644 index 75aff34af..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/tlt/props.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/tlt/defs.ma". - -inline "Basic-1/tlt/props.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/arity.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/arity.mma deleted file mode 100644 index a50d25340..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/arity.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/ty3/pr3_props.ma". - -include "Basic-2/arity/pr3.ma". - -include "Basic-2/asucc/fwd.ma". - -inline "Basic-1/ty3/arity.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/arity_props.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/arity_props.mma deleted file mode 100644 index 65475e81c..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/arity_props.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/ty3/arity.ma". - -include "Basic-2/sc3/arity.ma". - -inline "Basic-1/ty3/arity_props.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/dec.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/dec.mma deleted file mode 100644 index da5291990..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/dec.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pc3/dec.ma". - -include "Basic-2/getl/flt.ma". - -include "Basic-2/getl/dec.ma". - -inline "Basic-1/ty3/dec.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/defs.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/defs.mma deleted file mode 100644 index a1a47d554..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/defs.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/G/defs.ma". - -include "Basic-2/pc3/defs.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/fsubst0.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/fsubst0.mma deleted file mode 100644 index 11b8873da..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/fsubst0.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/ty3/props.ma". - -include "Basic-2/pc3/fsubst0.ma". - -include "Basic-2/getl/getl.ma". - -inline "Basic-1/ty3/fsubst0.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/fwd.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/fwd.mma deleted file mode 100644 index c019d001a..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/fwd.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/ty3/defs.ma". - -include "Basic-2/pc3/props.ma". - -inline "Basic-1/ty3/fwd.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/fwd_nf2.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/fwd_nf2.mma deleted file mode 100644 index 6894ae466..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/fwd_nf2.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/ty3/arity_props.ma". - -include "Basic-2/pc3/nf2.ma". - -include "Basic-2/nf2/fwd.ma". - -inline "Basic-1/ty3/fwd_nf2.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/nf2.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/nf2.mma deleted file mode 100644 index af7960855..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/nf2.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/ty3/arity.ma". - -include "Basic-2/pc3/nf2.ma". - -include "Basic-2/nf2/arity.ma". - -inline "Basic-1/ty3/nf2.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/pr3.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/pr3.mma deleted file mode 100644 index dd43847bd..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/pr3.mma +++ /dev/null @@ -1,30 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubt/ty3.ma". - -include "Basic-2/ty3/subst1.ma". - -include "Basic-2/ty3/fsubst0.ma". - -include "Basic-2/pc3/pc1.ma". - -include "Basic-2/pc3/wcpr0.ma". - -include "Basic-2/pc1/props.ma". - -inline "Basic-1/ty3/pr3.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/pr3_props.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/pr3_props.mma deleted file mode 100644 index a352ace49..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/pr3_props.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/ty3/pr3.ma". - -inline "Basic-1/ty3/pr3_props.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/props.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/props.mma deleted file mode 100644 index c5e6cb33e..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/props.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/ty3/fwd.ma". - -include "Basic-2/pc3/fwd.ma". - -inline "Basic-1/ty3/props.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/sty0.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/sty0.mma deleted file mode 100644 index 0c5ebc151..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/sty0.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/ty3/pr3_props.ma". - -include "Basic-2/sty0/fwd.ma". - -inline "Basic-1/ty3/sty0.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/subst1.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/subst1.mma deleted file mode 100644 index 97f45a1ec..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/subst1.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/ty3/props.ma". - -include "Basic-2/pc3/subst1.ma". - -include "Basic-2/getl/getl.ma". - -inline "Basic-1/ty3/subst1.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/wcpr0/defs.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/wcpr0/defs.mma deleted file mode 100644 index 65ae9a1d5..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/wcpr0/defs.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pr0/defs.ma". - -include "Basic-2/C/defs.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/wcpr0/fwd.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/wcpr0/fwd.mma deleted file mode 100644 index 0a9a87638..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/wcpr0/fwd.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/wcpr0/defs.ma". - -inline "Basic-1/wcpr0/fwd.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/wcpr0/getl.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/wcpr0/getl.mma deleted file mode 100644 index e412692e5..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/wcpr0/getl.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/wcpr0/defs.ma". - -include "Basic-2/getl/props.ma". - -inline "Basic-1/wcpr0/getl.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/wf3/clear.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/wf3/clear.mma deleted file mode 100644 index e63bb3e82..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/wf3/clear.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/wf3/fwd.ma". - -inline "Basic-1/wf3/clear.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/wf3/defs.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/wf3/defs.mma deleted file mode 100644 index 2dde8c752..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/wf3/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/ty3/defs.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/wf3/fwd.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/wf3/fwd.mma deleted file mode 100644 index 73bf245a7..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/wf3/fwd.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/wf3/defs.ma". - -inline "Basic-1/wf3/fwd.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/wf3/getl.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/wf3/getl.mma deleted file mode 100644 index fe6e74042..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/wf3/getl.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/wf3/clear.ma". - -include "Basic-2/ty3/dec.ma". - -inline "Basic-1/wf3/getl.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/wf3/props.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/wf3/props.mma deleted file mode 100644 index 35a6a6474..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/wf3/props.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/wf3/ty3.ma". - -include "Basic-2/app/defs.ma". - -inline "Basic-1/wf3/props.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/wf3/ty3.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/wf3/ty3.mma deleted file mode 100644 index 8af324db5..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Basic-2/wf3/ty3.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/wf3/getl.ma". - -inline "Basic-1/wf3/ty3.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Ground-1/blt/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Ground-1/blt/defs.ma deleted file mode 100644 index 009627a3b..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Ground-1/blt/defs.ma +++ /dev/null @@ -1,25 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Ground-1/preamble.ma". - -definition blt: - nat \to (nat \to bool) -\def - let rec blt (m: nat) (n: nat) on n: bool \def (match n with [O \Rightarrow -false | (S n0) \Rightarrow (match m with [O \Rightarrow true | (S m0) -\Rightarrow (blt m0 n0)])]) in blt. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Ground-1/blt/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Ground-1/blt/props.ma deleted file mode 100644 index e5b569925..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Ground-1/blt/props.ma +++ /dev/null @@ -1,112 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Ground-1/blt/defs.ma". - -theorem lt_blt: - \forall (x: nat).(\forall (y: nat).((lt y x) \to (eq bool (blt y x) true))) -\def - \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((lt y n) \to -(eq bool (blt y n) true)))) (\lambda (y: nat).(\lambda (H: (lt y O)).(let H0 -\def (match H in le return (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat -n O) \to (eq bool (blt y O) true)))) with [le_n \Rightarrow (\lambda (H0: (eq -nat (S y) O)).(let H1 \def (eq_ind nat (S y) (\lambda (e: nat).(match e in -nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) -\Rightarrow True])) I O H0) in (False_ind (eq bool (blt y O) true) H1))) | -(le_S m H0) \Rightarrow (\lambda (H1: (eq nat (S m) O)).((let H2 \def (eq_ind -nat (S m) (\lambda (e: nat).(match e in nat return (\lambda (_: nat).Prop) -with [O \Rightarrow False | (S _) \Rightarrow True])) I O H1) in (False_ind -((le (S y) m) \to (eq bool (blt y O) true)) H2)) H0))]) in (H0 (refl_equal -nat O))))) (\lambda (n: nat).(\lambda (H: ((\forall (y: nat).((lt y n) \to -(eq bool (blt y n) true))))).(\lambda (y: nat).(nat_ind (\lambda (n0: -nat).((lt n0 (S n)) \to (eq bool (blt n0 (S n)) true))) (\lambda (_: (lt O (S -n))).(refl_equal bool true)) (\lambda (n0: nat).(\lambda (_: (((lt n0 (S n)) -\to (eq bool (match n0 with [O \Rightarrow true | (S m) \Rightarrow (blt m -n)]) true)))).(\lambda (H1: (lt (S n0) (S n))).(H n0 (le_S_n (S n0) n H1))))) -y)))) x). -(* COMMENTS -Initial nodes: 291 -END *) - -theorem le_bge: - \forall (x: nat).(\forall (y: nat).((le x y) \to (eq bool (blt y x) false))) -\def - \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((le n y) \to -(eq bool (blt y n) false)))) (\lambda (y: nat).(\lambda (_: (le O -y)).(refl_equal bool false))) (\lambda (n: nat).(\lambda (H: ((\forall (y: -nat).((le n y) \to (eq bool (blt y n) false))))).(\lambda (y: nat).(nat_ind -(\lambda (n0: nat).((le (S n) n0) \to (eq bool (blt n0 (S n)) false))) -(\lambda (H0: (le (S n) O)).(let H1 \def (match H0 in le return (\lambda (n0: -nat).(\lambda (_: (le ? n0)).((eq nat n0 O) \to (eq bool (blt O (S n)) -false)))) with [le_n \Rightarrow (\lambda (H1: (eq nat (S n) O)).(let H2 \def -(eq_ind nat (S n) (\lambda (e: nat).(match e in nat return (\lambda (_: -nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H1) in -(False_ind (eq bool (blt O (S n)) false) H2))) | (le_S m H1) \Rightarrow -(\lambda (H2: (eq nat (S m) O)).((let H3 \def (eq_ind nat (S m) (\lambda (e: -nat).(match e in nat return (\lambda (_: nat).Prop) with [O \Rightarrow False -| (S _) \Rightarrow True])) I O H2) in (False_ind ((le (S n) m) \to (eq bool -(blt O (S n)) false)) H3)) H1))]) in (H1 (refl_equal nat O)))) (\lambda (n0: -nat).(\lambda (_: (((le (S n) n0) \to (eq bool (blt n0 (S n)) -false)))).(\lambda (H1: (le (S n) (S n0))).(H n0 (le_S_n n n0 H1))))) y)))) -x). -(* COMMENTS -Initial nodes: 293 -END *) - -theorem blt_lt: - \forall (x: nat).(\forall (y: nat).((eq bool (blt y x) true) \to (lt y x))) -\def - \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((eq bool (blt -y n) true) \to (lt y n)))) (\lambda (y: nat).(\lambda (H: (eq bool (blt y O) -true)).(let H0 \def (match H in eq return (\lambda (b: bool).(\lambda (_: (eq -? ? b)).((eq bool b true) \to (lt y O)))) with [refl_equal \Rightarrow -(\lambda (H0: (eq bool (blt y O) true)).(let H1 \def (eq_ind bool (blt y O) -(\lambda (e: bool).(match e in bool return (\lambda (_: bool).Prop) with -[true \Rightarrow False | false \Rightarrow True])) I true H0) in (False_ind -(lt y O) H1)))]) in (H0 (refl_equal bool true))))) (\lambda (n: nat).(\lambda -(H: ((\forall (y: nat).((eq bool (blt y n) true) \to (lt y n))))).(\lambda -(y: nat).(nat_ind (\lambda (n0: nat).((eq bool (blt n0 (S n)) true) \to (lt -n0 (S n)))) (\lambda (_: (eq bool true true)).(le_S_n (S O) (S n) (le_n_S (S -O) (S n) (le_n_S O n (le_O_n n))))) (\lambda (n0: nat).(\lambda (_: (((eq -bool (match n0 with [O \Rightarrow true | (S m) \Rightarrow (blt m n)]) true) -\to (lt n0 (S n))))).(\lambda (H1: (eq bool (blt n0 n) true)).(lt_n_S n0 n (H -n0 H1))))) y)))) x). -(* COMMENTS -Initial nodes: 252 -END *) - -theorem bge_le: - \forall (x: nat).(\forall (y: nat).((eq bool (blt y x) false) \to (le x y))) -\def - \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((eq bool (blt -y n) false) \to (le n y)))) (\lambda (y: nat).(\lambda (_: (eq bool (blt y O) -false)).(le_O_n y))) (\lambda (n: nat).(\lambda (H: ((\forall (y: nat).((eq -bool (blt y n) false) \to (le n y))))).(\lambda (y: nat).(nat_ind (\lambda -(n0: nat).((eq bool (blt n0 (S n)) false) \to (le (S n) n0))) (\lambda (H0: -(eq bool (blt O (S n)) false)).(let H1 \def (match H0 in eq return (\lambda -(b: bool).(\lambda (_: (eq ? ? b)).((eq bool b false) \to (le (S n) O)))) -with [refl_equal \Rightarrow (\lambda (H1: (eq bool (blt O (S n)) -false)).(let H2 \def (eq_ind bool (blt O (S n)) (\lambda (e: bool).(match e -in bool return (\lambda (_: bool).Prop) with [true \Rightarrow True | false -\Rightarrow False])) I false H1) in (False_ind (le (S n) O) H2)))]) in (H1 -(refl_equal bool false)))) (\lambda (n0: nat).(\lambda (_: (((eq bool (blt n0 -(S n)) false) \to (le (S n) n0)))).(\lambda (H1: (eq bool (blt (S n0) (S n)) -false)).(le_S_n (S n) (S n0) (le_n_S (S n) (S n0) (le_n_S n n0 (H n0 -H1))))))) y)))) x). -(* COMMENTS -Initial nodes: 262 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Ground-1/definitions.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Ground-1/definitions.ma deleted file mode 100644 index df31468aa..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Ground-1/definitions.ma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Ground-1/types/defs.ma". - -include "Ground-1/blt/defs.ma". - -include "Ground-1/plist/defs.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Ground-1/ext/arith.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Ground-1/ext/arith.ma deleted file mode 100644 index f9796e7fd..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Ground-1/ext/arith.ma +++ /dev/null @@ -1,737 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Ground-1/preamble.ma". - -theorem nat_dec: - \forall (n1: nat).(\forall (n2: nat).(or (eq nat n1 n2) ((eq nat n1 n2) \to -(\forall (P: Prop).P)))) -\def - \lambda (n1: nat).(nat_ind (\lambda (n: nat).(\forall (n2: nat).(or (eq nat -n n2) ((eq nat n n2) \to (\forall (P: Prop).P))))) (\lambda (n2: -nat).(nat_ind (\lambda (n: nat).(or (eq nat O n) ((eq nat O n) \to (\forall -(P: Prop).P)))) (or_introl (eq nat O O) ((eq nat O O) \to (\forall (P: -Prop).P)) (refl_equal nat O)) (\lambda (n: nat).(\lambda (_: (or (eq nat O n) -((eq nat O n) \to (\forall (P: Prop).P)))).(or_intror (eq nat O (S n)) ((eq -nat O (S n)) \to (\forall (P: Prop).P)) (\lambda (H0: (eq nat O (S -n))).(\lambda (P: Prop).(let H1 \def (eq_ind nat O (\lambda (ee: nat).(match -ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) -\Rightarrow False])) I (S n) H0) in (False_ind P H1))))))) n2)) (\lambda (n: -nat).(\lambda (H: ((\forall (n2: nat).(or (eq nat n n2) ((eq nat n n2) \to -(\forall (P: Prop).P)))))).(\lambda (n2: nat).(nat_ind (\lambda (n0: nat).(or -(eq nat (S n) n0) ((eq nat (S n) n0) \to (\forall (P: Prop).P)))) (or_intror -(eq nat (S n) O) ((eq nat (S n) O) \to (\forall (P: Prop).P)) (\lambda (H0: -(eq nat (S n) O)).(\lambda (P: Prop).(let H1 \def (eq_ind nat (S n) (\lambda -(ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow -False | (S _) \Rightarrow True])) I O H0) in (False_ind P H1))))) (\lambda -(n0: nat).(\lambda (H0: (or (eq nat (S n) n0) ((eq nat (S n) n0) \to (\forall -(P: Prop).P)))).(or_ind (eq nat n n0) ((eq nat n n0) \to (\forall (P: -Prop).P)) (or (eq nat (S n) (S n0)) ((eq nat (S n) (S n0)) \to (\forall (P: -Prop).P))) (\lambda (H1: (eq nat n n0)).(let H2 \def (eq_ind_r nat n0 -(\lambda (n3: nat).(or (eq nat (S n) n3) ((eq nat (S n) n3) \to (\forall (P: -Prop).P)))) H0 n H1) in (eq_ind nat n (\lambda (n3: nat).(or (eq nat (S n) (S -n3)) ((eq nat (S n) (S n3)) \to (\forall (P: Prop).P)))) (or_introl (eq nat -(S n) (S n)) ((eq nat (S n) (S n)) \to (\forall (P: Prop).P)) (refl_equal nat -(S n))) n0 H1))) (\lambda (H1: (((eq nat n n0) \to (\forall (P: -Prop).P)))).(or_intror (eq nat (S n) (S n0)) ((eq nat (S n) (S n0)) \to -(\forall (P: Prop).P)) (\lambda (H2: (eq nat (S n) (S n0))).(\lambda (P: -Prop).(let H3 \def (f_equal nat nat (\lambda (e: nat).(match e in nat return -(\lambda (_: nat).nat) with [O \Rightarrow n | (S n3) \Rightarrow n3])) (S n) -(S n0) H2) in (let H4 \def (eq_ind_r nat n0 (\lambda (n3: nat).((eq nat n n3) -\to (\forall (P0: Prop).P0))) H1 n H3) in (let H5 \def (eq_ind_r nat n0 -(\lambda (n3: nat).(or (eq nat (S n) n3) ((eq nat (S n) n3) \to (\forall (P0: -Prop).P0)))) H0 n H3) in (H4 (refl_equal nat n) P)))))))) (H n0)))) n2)))) -n1). -(* COMMENTS -Initial nodes: 676 -END *) - -theorem simpl_plus_r: - \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((eq nat (plus m n) -(plus p n)) \to (eq nat m p)))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (eq nat -(plus m n) (plus p n))).(simpl_plus_l n m p (eq_ind_r nat (plus m n) (\lambda -(n0: nat).(eq nat n0 (plus n p))) (eq_ind_r nat (plus p n) (\lambda (n0: -nat).(eq nat n0 (plus n p))) (sym_eq nat (plus n p) (plus p n) (plus_sym n -p)) (plus m n) H) (plus n m) (plus_sym n m)))))). -(* COMMENTS -Initial nodes: 119 -END *) - -theorem minus_Sx_Sy: - \forall (x: nat).(\forall (y: nat).(eq nat (minus (S x) (S y)) (minus x y))) -\def - \lambda (x: nat).(\lambda (y: nat).(refl_equal nat (minus x y))). -(* COMMENTS -Initial nodes: 13 -END *) - -theorem minus_plus_r: - \forall (m: nat).(\forall (n: nat).(eq nat (minus (plus m n) n) m)) -\def - \lambda (m: nat).(\lambda (n: nat).(eq_ind_r nat (plus n m) (\lambda (n0: -nat).(eq nat (minus n0 n) m)) (minus_plus n m) (plus m n) (plus_sym m n))). -(* COMMENTS -Initial nodes: 45 -END *) - -theorem plus_permute_2_in_3: - \forall (x: nat).(\forall (y: nat).(\forall (z: nat).(eq nat (plus (plus x -y) z) (plus (plus x z) y)))) -\def - \lambda (x: nat).(\lambda (y: nat).(\lambda (z: nat).(eq_ind_r nat (plus x -(plus y z)) (\lambda (n: nat).(eq nat n (plus (plus x z) y))) (eq_ind_r nat -(plus z y) (\lambda (n: nat).(eq nat (plus x n) (plus (plus x z) y))) (eq_ind -nat (plus (plus x z) y) (\lambda (n: nat).(eq nat n (plus (plus x z) y))) -(refl_equal nat (plus (plus x z) y)) (plus x (plus z y)) (plus_assoc_r x z -y)) (plus y z) (plus_sym y z)) (plus (plus x y) z) (plus_assoc_r x y z)))). -(* COMMENTS -Initial nodes: 163 -END *) - -theorem plus_permute_2_in_3_assoc: - \forall (n: nat).(\forall (h: nat).(\forall (k: nat).(eq nat (plus (plus n -h) k) (plus n (plus k h))))) -\def - \lambda (n: nat).(\lambda (h: nat).(\lambda (k: nat).(eq_ind_r nat (plus -(plus n k) h) (\lambda (n0: nat).(eq nat n0 (plus n (plus k h)))) (eq_ind_r -nat (plus (plus n k) h) (\lambda (n0: nat).(eq nat (plus (plus n k) h) n0)) -(refl_equal nat (plus (plus n k) h)) (plus n (plus k h)) (plus_assoc_l n k -h)) (plus (plus n h) k) (plus_permute_2_in_3 n h k)))). -(* COMMENTS -Initial nodes: 119 -END *) - -theorem plus_O: - \forall (x: nat).(\forall (y: nat).((eq nat (plus x y) O) \to (land (eq nat -x O) (eq nat y O)))) -\def - \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((eq nat (plus -n y) O) \to (land (eq nat n O) (eq nat y O))))) (\lambda (y: nat).(\lambda -(H: (eq nat (plus O y) O)).(conj (eq nat O O) (eq nat y O) (refl_equal nat O) -H))) (\lambda (n: nat).(\lambda (_: ((\forall (y: nat).((eq nat (plus n y) O) -\to (land (eq nat n O) (eq nat y O)))))).(\lambda (y: nat).(\lambda (H0: (eq -nat (plus (S n) y) O)).(let H1 \def (match H0 in eq return (\lambda (n0: -nat).(\lambda (_: (eq ? ? n0)).((eq nat n0 O) \to (land (eq nat (S n) O) (eq -nat y O))))) with [refl_equal \Rightarrow (\lambda (H1: (eq nat (plus (S n) -y) O)).(let H2 \def (eq_ind nat (plus (S n) y) (\lambda (e: nat).(match e in -nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) -\Rightarrow True])) I O H1) in (False_ind (land (eq nat (S n) O) (eq nat y -O)) H2)))]) in (H1 (refl_equal nat O))))))) x). -(* COMMENTS -Initial nodes: 233 -END *) - -theorem minus_Sx_SO: - \forall (x: nat).(eq nat (minus (S x) (S O)) x) -\def - \lambda (x: nat).(eq_ind nat x (\lambda (n: nat).(eq nat n x)) (refl_equal -nat x) (minus x O) (minus_n_O x)). -(* COMMENTS -Initial nodes: 33 -END *) - -theorem eq_nat_dec: - \forall (i: nat).(\forall (j: nat).(or (not (eq nat i j)) (eq nat i j))) -\def - \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (j: nat).(or (not (eq -nat n j)) (eq nat n j)))) (\lambda (j: nat).(nat_ind (\lambda (n: nat).(or -(not (eq nat O n)) (eq nat O n))) (or_intror (not (eq nat O O)) (eq nat O O) -(refl_equal nat O)) (\lambda (n: nat).(\lambda (_: (or (not (eq nat O n)) (eq -nat O n))).(or_introl (not (eq nat O (S n))) (eq nat O (S n)) (O_S n)))) j)) -(\lambda (n: nat).(\lambda (H: ((\forall (j: nat).(or (not (eq nat n j)) (eq -nat n j))))).(\lambda (j: nat).(nat_ind (\lambda (n0: nat).(or (not (eq nat -(S n) n0)) (eq nat (S n) n0))) (or_introl (not (eq nat (S n) O)) (eq nat (S -n) O) (sym_not_eq nat O (S n) (O_S n))) (\lambda (n0: nat).(\lambda (_: (or -(not (eq nat (S n) n0)) (eq nat (S n) n0))).(or_ind (not (eq nat n n0)) (eq -nat n n0) (or (not (eq nat (S n) (S n0))) (eq nat (S n) (S n0))) (\lambda -(H1: (not (eq nat n n0))).(or_introl (not (eq nat (S n) (S n0))) (eq nat (S -n) (S n0)) (not_eq_S n n0 H1))) (\lambda (H1: (eq nat n n0)).(or_intror (not -(eq nat (S n) (S n0))) (eq nat (S n) (S n0)) (f_equal nat nat S n n0 H1))) (H -n0)))) j)))) i). -(* COMMENTS -Initial nodes: 401 -END *) - -theorem neq_eq_e: - \forall (i: nat).(\forall (j: nat).(\forall (P: Prop).((((not (eq nat i j)) -\to P)) \to ((((eq nat i j) \to P)) \to P)))) -\def - \lambda (i: nat).(\lambda (j: nat).(\lambda (P: Prop).(\lambda (H: (((not -(eq nat i j)) \to P))).(\lambda (H0: (((eq nat i j) \to P))).(let o \def -(eq_nat_dec i j) in (or_ind (not (eq nat i j)) (eq nat i j) P H H0 o)))))). -(* COMMENTS -Initial nodes: 61 -END *) - -theorem le_false: - \forall (m: nat).(\forall (n: nat).(\forall (P: Prop).((le m n) \to ((le (S -n) m) \to P)))) -\def - \lambda (m: nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).(\forall (P: -Prop).((le n n0) \to ((le (S n0) n) \to P))))) (\lambda (n: nat).(\lambda (P: -Prop).(\lambda (_: (le O n)).(\lambda (H0: (le (S n) O)).(let H1 \def (match -H0 in le return (\lambda (n0: nat).(\lambda (_: (le ? n0)).((eq nat n0 O) \to -P))) with [le_n \Rightarrow (\lambda (H1: (eq nat (S n) O)).(let H2 \def -(eq_ind nat (S n) (\lambda (e: nat).(match e in nat return (\lambda (_: -nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H1) in -(False_ind P H2))) | (le_S m0 H1) \Rightarrow (\lambda (H2: (eq nat (S m0) -O)).((let H3 \def (eq_ind nat (S m0) (\lambda (e: nat).(match e in nat return -(\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) -I O H2) in (False_ind ((le (S n) m0) \to P) H3)) H1))]) in (H1 (refl_equal -nat O))))))) (\lambda (n: nat).(\lambda (H: ((\forall (n0: nat).(\forall (P: -Prop).((le n n0) \to ((le (S n0) n) \to P)))))).(\lambda (n0: nat).(nat_ind -(\lambda (n1: nat).(\forall (P: Prop).((le (S n) n1) \to ((le (S n1) (S n)) -\to P)))) (\lambda (P: Prop).(\lambda (H0: (le (S n) O)).(\lambda (_: (le (S -O) (S n))).(let H2 \def (match H0 in le return (\lambda (n1: nat).(\lambda -(_: (le ? n1)).((eq nat n1 O) \to P))) with [le_n \Rightarrow (\lambda (H2: -(eq nat (S n) O)).(let H3 \def (eq_ind nat (S n) (\lambda (e: nat).(match e -in nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) -\Rightarrow True])) I O H2) in (False_ind P H3))) | (le_S m0 H2) \Rightarrow -(\lambda (H3: (eq nat (S m0) O)).((let H4 \def (eq_ind nat (S m0) (\lambda -(e: nat).(match e in nat return (\lambda (_: nat).Prop) with [O \Rightarrow -False | (S _) \Rightarrow True])) I O H3) in (False_ind ((le (S n) m0) \to P) -H4)) H2))]) in (H2 (refl_equal nat O)))))) (\lambda (n1: nat).(\lambda (_: -((\forall (P: Prop).((le (S n) n1) \to ((le (S n1) (S n)) \to P))))).(\lambda -(P: Prop).(\lambda (H1: (le (S n) (S n1))).(\lambda (H2: (le (S (S n1)) (S -n))).(H n1 P (le_S_n n n1 H1) (le_S_n (S n1) n H2))))))) n0)))) m). -(* COMMENTS -Initial nodes: 409 -END *) - -theorem le_Sx_x: - \forall (x: nat).((le (S x) x) \to (\forall (P: Prop).P)) -\def - \lambda (x: nat).(\lambda (H: (le (S x) x)).(\lambda (P: Prop).(let H0 \def -le_Sn_n in (False_ind P (H0 x H))))). -(* COMMENTS -Initial nodes: 23 -END *) - -theorem le_n_pred: - \forall (n: nat).(\forall (m: nat).((le n m) \to (le (pred n) (pred m)))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(le_ind n (\lambda -(n0: nat).(le (pred n) (pred n0))) (le_n (pred n)) (\lambda (m0: -nat).(\lambda (_: (le n m0)).(\lambda (H1: (le (pred n) (pred m0))).(le_trans -(pred n) (pred m0) m0 H1 (le_pred_n m0))))) m H))). -(* COMMENTS -Initial nodes: 71 -END *) - -theorem minus_le: - \forall (x: nat).(\forall (y: nat).(le (minus x y) x)) -\def - \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).(le (minus n -y) n))) (\lambda (_: nat).(le_n O)) (\lambda (n: nat).(\lambda (H: ((\forall -(y: nat).(le (minus n y) n)))).(\lambda (y: nat).(nat_ind (\lambda (n0: -nat).(le (minus (S n) n0) (S n))) (le_n (S n)) (\lambda (n0: nat).(\lambda -(_: (le (match n0 with [O \Rightarrow (S n) | (S l) \Rightarrow (minus n l)]) -(S n))).(le_S (minus n n0) n (H n0)))) y)))) x). -(* COMMENTS -Initial nodes: 101 -END *) - -theorem le_plus_minus_sym: - \forall (n: nat).(\forall (m: nat).((le n m) \to (eq nat m (plus (minus m n) -n)))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(eq_ind_r nat -(plus n (minus m n)) (\lambda (n0: nat).(eq nat m n0)) (le_plus_minus n m H) -(plus (minus m n) n) (plus_sym (minus m n) n)))). -(* COMMENTS -Initial nodes: 61 -END *) - -theorem le_minus_minus: - \forall (x: nat).(\forall (y: nat).((le x y) \to (\forall (z: nat).((le y z) -\to (le (minus y x) (minus z x)))))) -\def - \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (le x y)).(\lambda (z: -nat).(\lambda (H0: (le y z)).(simpl_le_plus_l x (minus y x) (minus z x) -(eq_ind_r nat y (\lambda (n: nat).(le n (plus x (minus z x)))) (eq_ind_r nat -z (\lambda (n: nat).(le y n)) H0 (plus x (minus z x)) (le_plus_minus_r x z -(le_trans x y z H H0))) (plus x (minus y x)) (le_plus_minus_r x y H))))))). -(* COMMENTS -Initial nodes: 117 -END *) - -theorem le_minus_plus: - \forall (z: nat).(\forall (x: nat).((le z x) \to (\forall (y: nat).(eq nat -(minus (plus x y) z) (plus (minus x z) y))))) -\def - \lambda (z: nat).(nat_ind (\lambda (n: nat).(\forall (x: nat).((le n x) \to -(\forall (y: nat).(eq nat (minus (plus x y) n) (plus (minus x n) y)))))) -(\lambda (x: nat).(\lambda (H: (le O x)).(let H0 \def (match H in le return -(\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat n x) \to (\forall (y: -nat).(eq nat (minus (plus x y) O) (plus (minus x O) y)))))) with [le_n -\Rightarrow (\lambda (H0: (eq nat O x)).(eq_ind nat O (\lambda (n: -nat).(\forall (y: nat).(eq nat (minus (plus n y) O) (plus (minus n O) y)))) -(\lambda (y: nat).(sym_eq nat (plus (minus O O) y) (minus (plus O y) O) -(minus_n_O (plus O y)))) x H0)) | (le_S m H0) \Rightarrow (\lambda (H1: (eq -nat (S m) x)).(eq_ind nat (S m) (\lambda (n: nat).((le O m) \to (\forall (y: -nat).(eq nat (minus (plus n y) O) (plus (minus n O) y))))) (\lambda (_: (le O -m)).(\lambda (y: nat).(refl_equal nat (plus (minus (S m) O) y)))) x H1 H0))]) -in (H0 (refl_equal nat x))))) (\lambda (z0: nat).(\lambda (H: ((\forall (x: -nat).((le z0 x) \to (\forall (y: nat).(eq nat (minus (plus x y) z0) (plus -(minus x z0) y))))))).(\lambda (x: nat).(nat_ind (\lambda (n: nat).((le (S -z0) n) \to (\forall (y: nat).(eq nat (minus (plus n y) (S z0)) (plus (minus n -(S z0)) y))))) (\lambda (H0: (le (S z0) O)).(\lambda (y: nat).(let H1 \def -(match H0 in le return (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat n O) -\to (eq nat (minus (plus O y) (S z0)) (plus (minus O (S z0)) y))))) with -[le_n \Rightarrow (\lambda (H1: (eq nat (S z0) O)).(let H2 \def (eq_ind nat -(S z0) (\lambda (e: nat).(match e in nat return (\lambda (_: nat).Prop) with -[O \Rightarrow False | (S _) \Rightarrow True])) I O H1) in (False_ind (eq -nat (minus (plus O y) (S z0)) (plus (minus O (S z0)) y)) H2))) | (le_S m H1) -\Rightarrow (\lambda (H2: (eq nat (S m) O)).((let H3 \def (eq_ind nat (S m) -(\lambda (e: nat).(match e in nat return (\lambda (_: nat).Prop) with [O -\Rightarrow False | (S _) \Rightarrow True])) I O H2) in (False_ind ((le (S -z0) m) \to (eq nat (minus (plus O y) (S z0)) (plus (minus O (S z0)) y))) H3)) -H1))]) in (H1 (refl_equal nat O))))) (\lambda (n: nat).(\lambda (_: (((le (S -z0) n) \to (\forall (y: nat).(eq nat (minus (plus n y) (S z0)) (plus (minus n -(S z0)) y)))))).(\lambda (H1: (le (S z0) (S n))).(\lambda (y: nat).(H n -(le_S_n z0 n H1) y))))) x)))) z). -(* COMMENTS -Initial nodes: 603 -END *) - -theorem le_minus: - \forall (x: nat).(\forall (z: nat).(\forall (y: nat).((le (plus x y) z) \to -(le x (minus z y))))) -\def - \lambda (x: nat).(\lambda (z: nat).(\lambda (y: nat).(\lambda (H: (le (plus -x y) z)).(eq_ind nat (minus (plus x y) y) (\lambda (n: nat).(le n (minus z -y))) (le_minus_minus y (plus x y) (le_plus_r x y) z H) x (minus_plus_r x -y))))). -(* COMMENTS -Initial nodes: 69 -END *) - -theorem le_trans_plus_r: - \forall (x: nat).(\forall (y: nat).(\forall (z: nat).((le (plus x y) z) \to -(le y z)))) -\def - \lambda (x: nat).(\lambda (y: nat).(\lambda (z: nat).(\lambda (H: (le (plus -x y) z)).(le_trans y (plus x y) z (le_plus_r x y) H)))). -(* COMMENTS -Initial nodes: 35 -END *) - -theorem lt_x_O: - \forall (x: nat).((lt x O) \to (\forall (P: Prop).P)) -\def - \lambda (x: nat).(\lambda (H: (le (S x) O)).(\lambda (P: Prop).(let H_y \def -(le_n_O_eq (S x) H) in (let H0 \def (eq_ind nat O (\lambda (ee: nat).(match -ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) -\Rightarrow False])) I (S x) H_y) in (False_ind P H0))))). -(* COMMENTS -Initial nodes: 48 -END *) - -theorem le_gen_S: - \forall (m: nat).(\forall (x: nat).((le (S m) x) \to (ex2 nat (\lambda (n: -nat).(eq nat x (S n))) (\lambda (n: nat).(le m n))))) -\def - \lambda (m: nat).(\lambda (x: nat).(\lambda (H: (le (S m) x)).(let H0 \def -(match H in le return (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat n x) -\to (ex2 nat (\lambda (n0: nat).(eq nat x (S n0))) (\lambda (n0: nat).(le m -n0)))))) with [le_n \Rightarrow (\lambda (H0: (eq nat (S m) x)).(eq_ind nat -(S m) (\lambda (n: nat).(ex2 nat (\lambda (n0: nat).(eq nat n (S n0))) -(\lambda (n0: nat).(le m n0)))) (ex_intro2 nat (\lambda (n: nat).(eq nat (S -m) (S n))) (\lambda (n: nat).(le m n)) m (refl_equal nat (S m)) (le_n m)) x -H0)) | (le_S m0 H0) \Rightarrow (\lambda (H1: (eq nat (S m0) x)).(eq_ind nat -(S m0) (\lambda (n: nat).((le (S m) m0) \to (ex2 nat (\lambda (n0: nat).(eq -nat n (S n0))) (\lambda (n0: nat).(le m n0))))) (\lambda (H2: (le (S m) -m0)).(ex_intro2 nat (\lambda (n: nat).(eq nat (S m0) (S n))) (\lambda (n: -nat).(le m n)) m0 (refl_equal nat (S m0)) (le_S_n m m0 (le_S (S m) m0 H2)))) -x H1 H0))]) in (H0 (refl_equal nat x))))). -(* COMMENTS -Initial nodes: 261 -END *) - -theorem lt_x_plus_x_Sy: - \forall (x: nat).(\forall (y: nat).(lt x (plus x (S y)))) -\def - \lambda (x: nat).(\lambda (y: nat).(eq_ind_r nat (plus (S y) x) (\lambda (n: -nat).(lt x n)) (le_S_n (S x) (S (plus y x)) (le_n_S (S x) (S (plus y x)) -(le_n_S x (plus y x) (le_plus_r y x)))) (plus x (S y)) (plus_sym x (S y)))). -(* COMMENTS -Initial nodes: 83 -END *) - -theorem simpl_lt_plus_r: - \forall (p: nat).(\forall (n: nat).(\forall (m: nat).((lt (plus n p) (plus m -p)) \to (lt n m)))) -\def - \lambda (p: nat).(\lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt (plus -n p) (plus m p))).(simpl_lt_plus_l n m p (let H0 \def (eq_ind nat (plus n p) -(\lambda (n0: nat).(lt n0 (plus m p))) H (plus p n) (plus_sym n p)) in (let -H1 \def (eq_ind nat (plus m p) (\lambda (n0: nat).(lt (plus p n) n0)) H0 -(plus p m) (plus_sym m p)) in H1)))))). -(* COMMENTS -Initial nodes: 101 -END *) - -theorem minus_x_Sy: - \forall (x: nat).(\forall (y: nat).((lt y x) \to (eq nat (minus x y) (S -(minus x (S y)))))) -\def - \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((lt y n) \to -(eq nat (minus n y) (S (minus n (S y))))))) (\lambda (y: nat).(\lambda (H: -(lt y O)).(let H0 \def (match H in le return (\lambda (n: nat).(\lambda (_: -(le ? n)).((eq nat n O) \to (eq nat (minus O y) (S (minus O (S y))))))) with -[le_n \Rightarrow (\lambda (H0: (eq nat (S y) O)).(let H1 \def (eq_ind nat (S -y) (\lambda (e: nat).(match e in nat return (\lambda (_: nat).Prop) with [O -\Rightarrow False | (S _) \Rightarrow True])) I O H0) in (False_ind (eq nat -(minus O y) (S (minus O (S y)))) H1))) | (le_S m H0) \Rightarrow (\lambda -(H1: (eq nat (S m) O)).((let H2 \def (eq_ind nat (S m) (\lambda (e: -nat).(match e in nat return (\lambda (_: nat).Prop) with [O \Rightarrow False -| (S _) \Rightarrow True])) I O H1) in (False_ind ((le (S y) m) \to (eq nat -(minus O y) (S (minus O (S y))))) H2)) H0))]) in (H0 (refl_equal nat O))))) -(\lambda (n: nat).(\lambda (H: ((\forall (y: nat).((lt y n) \to (eq nat -(minus n y) (S (minus n (S y)))))))).(\lambda (y: nat).(nat_ind (\lambda (n0: -nat).((lt n0 (S n)) \to (eq nat (minus (S n) n0) (S (minus (S n) (S n0)))))) -(\lambda (_: (lt O (S n))).(eq_ind nat n (\lambda (n0: nat).(eq nat (S n) (S -n0))) (refl_equal nat (S n)) (minus n O) (minus_n_O n))) (\lambda (n0: -nat).(\lambda (_: (((lt n0 (S n)) \to (eq nat (minus (S n) n0) (S (minus (S -n) (S n0))))))).(\lambda (H1: (lt (S n0) (S n))).(let H2 \def (le_S_n (S n0) -n H1) in (H n0 H2))))) y)))) x). -(* COMMENTS -Initial nodes: 383 -END *) - -theorem lt_plus_minus: - \forall (x: nat).(\forall (y: nat).((lt x y) \to (eq nat y (S (plus x (minus -y (S x))))))) -\def - \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(le_plus_minus (S -x) y H))). -(* COMMENTS -Initial nodes: 19 -END *) - -theorem lt_plus_minus_r: - \forall (x: nat).(\forall (y: nat).((lt x y) \to (eq nat y (S (plus (minus y -(S x)) x))))) -\def - \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(eq_ind_r nat -(plus x (minus y (S x))) (\lambda (n: nat).(eq nat y (S n))) (lt_plus_minus x -y H) (plus (minus y (S x)) x) (plus_sym (minus y (S x)) x)))). -(* COMMENTS -Initial nodes: 69 -END *) - -theorem minus_x_SO: - \forall (x: nat).((lt O x) \to (eq nat x (S (minus x (S O))))) -\def - \lambda (x: nat).(\lambda (H: (lt O x)).(eq_ind nat (minus x O) (\lambda (n: -nat).(eq nat x n)) (eq_ind nat x (\lambda (n: nat).(eq nat x n)) (refl_equal -nat x) (minus x O) (minus_n_O x)) (S (minus x (S O))) (minus_x_Sy x O H))). -(* COMMENTS -Initial nodes: 77 -END *) - -theorem le_x_pred_y: - \forall (y: nat).(\forall (x: nat).((lt x y) \to (le x (pred y)))) -\def - \lambda (y: nat).(nat_ind (\lambda (n: nat).(\forall (x: nat).((lt x n) \to -(le x (pred n))))) (\lambda (x: nat).(\lambda (H: (lt x O)).(let H0 \def -(match H in le return (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat n O) -\to (le x O)))) with [le_n \Rightarrow (\lambda (H0: (eq nat (S x) O)).(let -H1 \def (eq_ind nat (S x) (\lambda (e: nat).(match e in nat return (\lambda -(_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H0) -in (False_ind (le x O) H1))) | (le_S m H0) \Rightarrow (\lambda (H1: (eq nat -(S m) O)).((let H2 \def (eq_ind nat (S m) (\lambda (e: nat).(match e in nat -return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow -True])) I O H1) in (False_ind ((le (S x) m) \to (le x O)) H2)) H0))]) in (H0 -(refl_equal nat O))))) (\lambda (n: nat).(\lambda (_: ((\forall (x: nat).((lt -x n) \to (le x (pred n)))))).(\lambda (x: nat).(\lambda (H0: (lt x (S -n))).(le_S_n x n H0))))) y). -(* COMMENTS -Initial nodes: 189 -END *) - -theorem lt_le_minus: - \forall (x: nat).(\forall (y: nat).((lt x y) \to (le x (minus y (S O))))) -\def - \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(le_minus x y (S -O) (eq_ind_r nat (plus (S O) x) (\lambda (n: nat).(le n y)) H (plus x (S O)) -(plus_sym x (S O)))))). -(* COMMENTS -Initial nodes: 57 -END *) - -theorem lt_le_e: - \forall (n: nat).(\forall (d: nat).(\forall (P: Prop).((((lt n d) \to P)) -\to ((((le d n) \to P)) \to P)))) -\def - \lambda (n: nat).(\lambda (d: nat).(\lambda (P: Prop).(\lambda (H: (((lt n -d) \to P))).(\lambda (H0: (((le d n) \to P))).(let H1 \def (le_or_lt d n) in -(or_ind (le d n) (lt n d) P H0 H H1)))))). -(* COMMENTS -Initial nodes: 49 -END *) - -theorem lt_eq_e: - \forall (x: nat).(\forall (y: nat).(\forall (P: Prop).((((lt x y) \to P)) -\to ((((eq nat x y) \to P)) \to ((le x y) \to P))))) -\def - \lambda (x: nat).(\lambda (y: nat).(\lambda (P: Prop).(\lambda (H: (((lt x -y) \to P))).(\lambda (H0: (((eq nat x y) \to P))).(\lambda (H1: (le x -y)).(or_ind (lt x y) (eq nat x y) P H H0 (le_lt_or_eq x y H1))))))). -(* COMMENTS -Initial nodes: 59 -END *) - -theorem lt_eq_gt_e: - \forall (x: nat).(\forall (y: nat).(\forall (P: Prop).((((lt x y) \to P)) -\to ((((eq nat x y) \to P)) \to ((((lt y x) \to P)) \to P))))) -\def - \lambda (x: nat).(\lambda (y: nat).(\lambda (P: Prop).(\lambda (H: (((lt x -y) \to P))).(\lambda (H0: (((eq nat x y) \to P))).(\lambda (H1: (((lt y x) -\to P))).(lt_le_e x y P H (\lambda (H2: (le y x)).(lt_eq_e y x P H1 (\lambda -(H3: (eq nat y x)).(H0 (sym_eq nat y x H3))) H2)))))))). -(* COMMENTS -Initial nodes: 79 -END *) - -theorem lt_gen_xS: - \forall (x: nat).(\forall (n: nat).((lt x (S n)) \to (or (eq nat x O) (ex2 -nat (\lambda (m: nat).(eq nat x (S m))) (\lambda (m: nat).(lt m n)))))) -\def - \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).((lt n (S -n0)) \to (or (eq nat n O) (ex2 nat (\lambda (m: nat).(eq nat n (S m))) -(\lambda (m: nat).(lt m n0))))))) (\lambda (n: nat).(\lambda (_: (lt O (S -n))).(or_introl (eq nat O O) (ex2 nat (\lambda (m: nat).(eq nat O (S m))) -(\lambda (m: nat).(lt m n))) (refl_equal nat O)))) (\lambda (n: nat).(\lambda -(_: ((\forall (n0: nat).((lt n (S n0)) \to (or (eq nat n O) (ex2 nat (\lambda -(m: nat).(eq nat n (S m))) (\lambda (m: nat).(lt m n0)))))))).(\lambda (n0: -nat).(\lambda (H0: (lt (S n) (S n0))).(or_intror (eq nat (S n) O) (ex2 nat -(\lambda (m: nat).(eq nat (S n) (S m))) (\lambda (m: nat).(lt m n0))) -(ex_intro2 nat (\lambda (m: nat).(eq nat (S n) (S m))) (\lambda (m: nat).(lt -m n0)) n (refl_equal nat (S n)) (le_S_n (S n) n0 H0))))))) x). -(* COMMENTS -Initial nodes: 243 -END *) - -theorem le_lt_false: - \forall (x: nat).(\forall (y: nat).((le x y) \to ((lt y x) \to (\forall (P: -Prop).P)))) -\def - \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (le x y)).(\lambda (H0: (lt -y x)).(\lambda (P: Prop).(False_ind P (le_not_lt x y H H0)))))). -(* COMMENTS -Initial nodes: 31 -END *) - -theorem lt_neq: - \forall (x: nat).(\forall (y: nat).((lt x y) \to (not (eq nat x y)))) -\def - \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(\lambda (H0: (eq -nat x y)).(let H1 \def (eq_ind nat x (\lambda (n: nat).(lt n y)) H y H0) in -(lt_n_n y H1))))). -(* COMMENTS -Initial nodes: 43 -END *) - -theorem arith0: - \forall (h2: nat).(\forall (d2: nat).(\forall (n: nat).((le (plus d2 h2) n) -\to (\forall (h1: nat).(le (plus d2 h1) (minus (plus n h1) h2)))))) -\def - \lambda (h2: nat).(\lambda (d2: nat).(\lambda (n: nat).(\lambda (H: (le -(plus d2 h2) n)).(\lambda (h1: nat).(eq_ind nat (minus (plus h2 (plus d2 h1)) -h2) (\lambda (n0: nat).(le n0 (minus (plus n h1) h2))) (le_minus_minus h2 -(plus h2 (plus d2 h1)) (le_plus_l h2 (plus d2 h1)) (plus n h1) (eq_ind_r nat -(plus (plus h2 d2) h1) (\lambda (n0: nat).(le n0 (plus n h1))) (eq_ind_r nat -(plus d2 h2) (\lambda (n0: nat).(le (plus n0 h1) (plus n h1))) (le_S_n (plus -(plus d2 h2) h1) (plus n h1) (le_n_S (plus (plus d2 h2) h1) (plus n h1) -(le_plus_plus (plus d2 h2) n h1 h1 H (le_n h1)))) (plus h2 d2) (plus_sym h2 -d2)) (plus h2 (plus d2 h1)) (plus_assoc_l h2 d2 h1))) (plus d2 h1) -(minus_plus h2 (plus d2 h1))))))). -(* COMMENTS -Initial nodes: 235 -END *) - -theorem O_minus: - \forall (x: nat).(\forall (y: nat).((le x y) \to (eq nat (minus x y) O))) -\def - \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((le n y) \to -(eq nat (minus n y) O)))) (\lambda (y: nat).(\lambda (_: (le O -y)).(refl_equal nat O))) (\lambda (x0: nat).(\lambda (H: ((\forall (y: -nat).((le x0 y) \to (eq nat (minus x0 y) O))))).(\lambda (y: nat).(nat_ind -(\lambda (n: nat).((le (S x0) n) \to (eq nat (match n with [O \Rightarrow (S -x0) | (S l) \Rightarrow (minus x0 l)]) O))) (\lambda (H0: (le (S x0) -O)).(ex2_ind nat (\lambda (n: nat).(eq nat O (S n))) (\lambda (n: nat).(le x0 -n)) (eq nat (S x0) O) (\lambda (x1: nat).(\lambda (H1: (eq nat O (S -x1))).(\lambda (_: (le x0 x1)).(let H3 \def (eq_ind nat O (\lambda (ee: -nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True -| (S _) \Rightarrow False])) I (S x1) H1) in (False_ind (eq nat (S x0) O) -H3))))) (le_gen_S x0 O H0))) (\lambda (n: nat).(\lambda (_: (((le (S x0) n) -\to (eq nat (match n with [O \Rightarrow (S x0) | (S l) \Rightarrow (minus x0 -l)]) O)))).(\lambda (H1: (le (S x0) (S n))).(H n (le_S_n x0 n H1))))) y)))) -x). -(* COMMENTS -Initial nodes: 252 -END *) - -theorem minus_minus: - \forall (z: nat).(\forall (x: nat).(\forall (y: nat).((le z x) \to ((le z y) -\to ((eq nat (minus x z) (minus y z)) \to (eq nat x y)))))) -\def - \lambda (z: nat).(nat_ind (\lambda (n: nat).(\forall (x: nat).(\forall (y: -nat).((le n x) \to ((le n y) \to ((eq nat (minus x n) (minus y n)) \to (eq -nat x y))))))) (\lambda (x: nat).(\lambda (y: nat).(\lambda (_: (le O -x)).(\lambda (_: (le O y)).(\lambda (H1: (eq nat (minus x O) (minus y -O))).(let H2 \def (eq_ind_r nat (minus x O) (\lambda (n: nat).(eq nat n -(minus y O))) H1 x (minus_n_O x)) in (let H3 \def (eq_ind_r nat (minus y O) -(\lambda (n: nat).(eq nat x n)) H2 y (minus_n_O y)) in H3))))))) (\lambda -(z0: nat).(\lambda (IH: ((\forall (x: nat).(\forall (y: nat).((le z0 x) \to -((le z0 y) \to ((eq nat (minus x z0) (minus y z0)) \to (eq nat x -y)))))))).(\lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((le -(S z0) n) \to ((le (S z0) y) \to ((eq nat (minus n (S z0)) (minus y (S z0))) -\to (eq nat n y)))))) (\lambda (y: nat).(\lambda (H: (le (S z0) O)).(\lambda -(_: (le (S z0) y)).(\lambda (_: (eq nat (minus O (S z0)) (minus y (S -z0)))).(ex2_ind nat (\lambda (n: nat).(eq nat O (S n))) (\lambda (n: nat).(le -z0 n)) (eq nat O y) (\lambda (x0: nat).(\lambda (H2: (eq nat O (S -x0))).(\lambda (_: (le z0 x0)).(let H4 \def (eq_ind nat O (\lambda (ee: -nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True -| (S _) \Rightarrow False])) I (S x0) H2) in (False_ind (eq nat O y) H4))))) -(le_gen_S z0 O H)))))) (\lambda (x0: nat).(\lambda (_: ((\forall (y: -nat).((le (S z0) x0) \to ((le (S z0) y) \to ((eq nat (minus x0 (S z0)) (minus -y (S z0))) \to (eq nat x0 y))))))).(\lambda (y: nat).(nat_ind (\lambda (n: -nat).((le (S z0) (S x0)) \to ((le (S z0) n) \to ((eq nat (minus (S x0) (S -z0)) (minus n (S z0))) \to (eq nat (S x0) n))))) (\lambda (H: (le (S z0) (S -x0))).(\lambda (H0: (le (S z0) O)).(\lambda (_: (eq nat (minus (S x0) (S z0)) -(minus O (S z0)))).(let H_y \def (le_S_n z0 x0 H) in (ex2_ind nat (\lambda -(n: nat).(eq nat O (S n))) (\lambda (n: nat).(le z0 n)) (eq nat (S x0) O) -(\lambda (x1: nat).(\lambda (H2: (eq nat O (S x1))).(\lambda (_: (le z0 -x1)).(let H4 \def (eq_ind nat O (\lambda (ee: nat).(match ee in nat return -(\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) -I (S x1) H2) in (False_ind (eq nat (S x0) O) H4))))) (le_gen_S z0 O H0)))))) -(\lambda (y0: nat).(\lambda (_: (((le (S z0) (S x0)) \to ((le (S z0) y0) \to -((eq nat (minus (S x0) (S z0)) (minus y0 (S z0))) \to (eq nat (S x0) -y0)))))).(\lambda (H: (le (S z0) (S x0))).(\lambda (H0: (le (S z0) (S -y0))).(\lambda (H1: (eq nat (minus (S x0) (S z0)) (minus (S y0) (S -z0)))).(f_equal nat nat S x0 y0 (IH x0 y0 (le_S_n z0 x0 H) (le_S_n z0 y0 H0) -H1))))))) y)))) x)))) z). -(* COMMENTS -Initial nodes: 751 -END *) - -theorem plus_plus: - \forall (z: nat).(\forall (x1: nat).(\forall (x2: nat).(\forall (y1: -nat).(\forall (y2: nat).((le x1 z) \to ((le x2 z) \to ((eq nat (plus (minus z -x1) y1) (plus (minus z x2) y2)) \to (eq nat (plus x1 y2) (plus x2 y1))))))))) -\def - \lambda (z: nat).(nat_ind (\lambda (n: nat).(\forall (x1: nat).(\forall (x2: -nat).(\forall (y1: nat).(\forall (y2: nat).((le x1 n) \to ((le x2 n) \to ((eq -nat (plus (minus n x1) y1) (plus (minus n x2) y2)) \to (eq nat (plus x1 y2) -(plus x2 y1)))))))))) (\lambda (x1: nat).(\lambda (x2: nat).(\lambda (y1: -nat).(\lambda (y2: nat).(\lambda (H: (le x1 O)).(\lambda (H0: (le x2 -O)).(\lambda (H1: (eq nat y1 y2)).(eq_ind nat y1 (\lambda (n: nat).(eq nat -(plus x1 n) (plus x2 y1))) (let H_y \def (le_n_O_eq x2 H0) in (eq_ind nat O -(\lambda (n: nat).(eq nat (plus x1 y1) (plus n y1))) (let H_y0 \def -(le_n_O_eq x1 H) in (eq_ind nat O (\lambda (n: nat).(eq nat (plus n y1) (plus -O y1))) (refl_equal nat (plus O y1)) x1 H_y0)) x2 H_y)) y2 H1)))))))) -(\lambda (z0: nat).(\lambda (IH: ((\forall (x1: nat).(\forall (x2: -nat).(\forall (y1: nat).(\forall (y2: nat).((le x1 z0) \to ((le x2 z0) \to -((eq nat (plus (minus z0 x1) y1) (plus (minus z0 x2) y2)) \to (eq nat (plus -x1 y2) (plus x2 y1))))))))))).(\lambda (x1: nat).(nat_ind (\lambda (n: -nat).(\forall (x2: nat).(\forall (y1: nat).(\forall (y2: nat).((le n (S z0)) -\to ((le x2 (S z0)) \to ((eq nat (plus (minus (S z0) n) y1) (plus (minus (S -z0) x2) y2)) \to (eq nat (plus n y2) (plus x2 y1))))))))) (\lambda (x2: -nat).(nat_ind (\lambda (n: nat).(\forall (y1: nat).(\forall (y2: nat).((le O -(S z0)) \to ((le n (S z0)) \to ((eq nat (plus (minus (S z0) O) y1) (plus -(minus (S z0) n) y2)) \to (eq nat (plus O y2) (plus n y1)))))))) (\lambda -(y1: nat).(\lambda (y2: nat).(\lambda (_: (le O (S z0))).(\lambda (_: (le O -(S z0))).(\lambda (H1: (eq nat (S (plus z0 y1)) (S (plus z0 y2)))).(let H_y -\def (IH O O) in (let H2 \def (eq_ind_r nat (minus z0 O) (\lambda (n: -nat).(\forall (y3: nat).(\forall (y4: nat).((le O z0) \to ((le O z0) \to ((eq -nat (plus n y3) (plus n y4)) \to (eq nat y4 y3))))))) H_y z0 (minus_n_O z0)) -in (H2 y1 y2 (le_O_n z0) (le_O_n z0) (eq_add_S (plus z0 y1) (plus z0 y2) -H1))))))))) (\lambda (x3: nat).(\lambda (_: ((\forall (y1: nat).(\forall (y2: -nat).((le O (S z0)) \to ((le x3 (S z0)) \to ((eq nat (S (plus z0 y1)) (plus -(match x3 with [O \Rightarrow (S z0) | (S l) \Rightarrow (minus z0 l)]) y2)) -\to (eq nat y2 (plus x3 y1))))))))).(\lambda (y1: nat).(\lambda (y2: -nat).(\lambda (_: (le O (S z0))).(\lambda (H0: (le (S x3) (S z0))).(\lambda -(H1: (eq nat (S (plus z0 y1)) (plus (minus z0 x3) y2))).(let H_y \def (IH O -x3 (S y1)) in (let H2 \def (eq_ind_r nat (minus z0 O) (\lambda (n: -nat).(\forall (y3: nat).((le O z0) \to ((le x3 z0) \to ((eq nat (plus n (S -y1)) (plus (minus z0 x3) y3)) \to (eq nat y3 (plus x3 (S y1)))))))) H_y z0 -(minus_n_O z0)) in (let H3 \def (eq_ind_r nat (plus z0 (S y1)) (\lambda (n: -nat).(\forall (y3: nat).((le O z0) \to ((le x3 z0) \to ((eq nat n (plus -(minus z0 x3) y3)) \to (eq nat y3 (plus x3 (S y1)))))))) H2 (S (plus z0 y1)) -(plus_n_Sm z0 y1)) in (let H4 \def (eq_ind_r nat (plus x3 (S y1)) (\lambda -(n: nat).(\forall (y3: nat).((le O z0) \to ((le x3 z0) \to ((eq nat (S (plus -z0 y1)) (plus (minus z0 x3) y3)) \to (eq nat y3 n)))))) H3 (S (plus x3 y1)) -(plus_n_Sm x3 y1)) in (H4 y2 (le_O_n z0) (le_S_n x3 z0 H0) H1)))))))))))) -x2)) (\lambda (x2: nat).(\lambda (_: ((\forall (x3: nat).(\forall (y1: -nat).(\forall (y2: nat).((le x2 (S z0)) \to ((le x3 (S z0)) \to ((eq nat -(plus (minus (S z0) x2) y1) (plus (minus (S z0) x3) y2)) \to (eq nat (plus x2 -y2) (plus x3 y1)))))))))).(\lambda (x3: nat).(nat_ind (\lambda (n: -nat).(\forall (y1: nat).(\forall (y2: nat).((le (S x2) (S z0)) \to ((le n (S -z0)) \to ((eq nat (plus (minus (S z0) (S x2)) y1) (plus (minus (S z0) n) y2)) -\to (eq nat (plus (S x2) y2) (plus n y1)))))))) (\lambda (y1: nat).(\lambda -(y2: nat).(\lambda (H: (le (S x2) (S z0))).(\lambda (_: (le O (S -z0))).(\lambda (H1: (eq nat (plus (minus z0 x2) y1) (S (plus z0 y2)))).(let -H_y \def (IH x2 O y1 (S y2)) in (let H2 \def (eq_ind_r nat (minus z0 O) -(\lambda (n: nat).((le x2 z0) \to ((le O z0) \to ((eq nat (plus (minus z0 x2) -y1) (plus n (S y2))) \to (eq nat (plus x2 (S y2)) y1))))) H_y z0 (minus_n_O -z0)) in (let H3 \def (eq_ind_r nat (plus z0 (S y2)) (\lambda (n: nat).((le x2 -z0) \to ((le O z0) \to ((eq nat (plus (minus z0 x2) y1) n) \to (eq nat (plus -x2 (S y2)) y1))))) H2 (S (plus z0 y2)) (plus_n_Sm z0 y2)) in (let H4 \def -(eq_ind_r nat (plus x2 (S y2)) (\lambda (n: nat).((le x2 z0) \to ((le O z0) -\to ((eq nat (plus (minus z0 x2) y1) (S (plus z0 y2))) \to (eq nat n y1))))) -H3 (S (plus x2 y2)) (plus_n_Sm x2 y2)) in (H4 (le_S_n x2 z0 H) (le_O_n z0) -H1)))))))))) (\lambda (x4: nat).(\lambda (_: ((\forall (y1: nat).(\forall -(y2: nat).((le (S x2) (S z0)) \to ((le x4 (S z0)) \to ((eq nat (plus (minus -z0 x2) y1) (plus (match x4 with [O \Rightarrow (S z0) | (S l) \Rightarrow -(minus z0 l)]) y2)) \to (eq nat (S (plus x2 y2)) (plus x4 -y1))))))))).(\lambda (y1: nat).(\lambda (y2: nat).(\lambda (H: (le (S x2) (S -z0))).(\lambda (H0: (le (S x4) (S z0))).(\lambda (H1: (eq nat (plus (minus z0 -x2) y1) (plus (minus z0 x4) y2))).(f_equal nat nat S (plus x2 y2) (plus x4 -y1) (IH x2 x4 y1 y2 (le_S_n x2 z0 H) (le_S_n x4 z0 H0) H1))))))))) x3)))) -x1)))) z). -(* COMMENTS -Initial nodes: 1495 -END *) - -theorem le_S_minus: - \forall (d: nat).(\forall (h: nat).(\forall (n: nat).((le (plus d h) n) \to -(le d (S (minus n h)))))) -\def - \lambda (d: nat).(\lambda (h: nat).(\lambda (n: nat).(\lambda (H: (le (plus -d h) n)).(let H0 \def (le_trans d (plus d h) n (le_plus_l d h) H) in (let H1 -\def (eq_ind nat n (\lambda (n0: nat).(le d n0)) H0 (plus (minus n h) h) -(le_plus_minus_sym h n (le_trans h (plus d h) n (le_plus_r d h) H))) in (le_S -d (minus n h) (le_minus d n h H))))))). -(* COMMENTS -Initial nodes: 107 -END *) - -theorem lt_x_pred_y: - \forall (x: nat).(\forall (y: nat).((lt x (pred y)) \to (lt (S x) y))) -\def - \lambda (x: nat).(\lambda (y: nat).(nat_ind (\lambda (n: nat).((lt x (pred -n)) \to (lt (S x) n))) (\lambda (H: (lt x O)).(lt_x_O x H (lt (S x) O))) -(\lambda (n: nat).(\lambda (_: (((lt x (pred n)) \to (lt (S x) n)))).(\lambda -(H0: (lt x n)).(le_S_n (S (S x)) (S n) (le_n_S (S (S x)) (S n) (le_n_S (S x) -n H0)))))) y)). -(* COMMENTS -Initial nodes: 103 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Ground-1/ext/tactics.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Ground-1/ext/tactics.ma deleted file mode 100644 index 766db9f1f..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Ground-1/ext/tactics.ma +++ /dev/null @@ -1,50 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Ground-1/preamble.ma". - -theorem insert_eq: - \forall (S: Set).(\forall (x: S).(\forall (P: ((S \to Prop))).(\forall (G: -((S \to Prop))).(((\forall (y: S).((P y) \to ((eq S y x) \to (G y))))) \to -((P x) \to (G x)))))) -\def - \lambda (S: Set).(\lambda (x: S).(\lambda (P: ((S \to Prop))).(\lambda (G: -((S \to Prop))).(\lambda (H: ((\forall (y: S).((P y) \to ((eq S y x) \to (G -y)))))).(\lambda (H0: (P x)).(H x H0 (refl_equal S x))))))). -(* COMMENTS -Initial nodes: 45 -END *) - -theorem unintro: - \forall (A: Set).(\forall (a: A).(\forall (P: ((A \to Prop))).(((\forall (x: -A).(P x))) \to (P a)))) -\def - \lambda (A: Set).(\lambda (a: A).(\lambda (P: ((A \to Prop))).(\lambda (H: -((\forall (x: A).(P x)))).(H a)))). -(* COMMENTS -Initial nodes: 17 -END *) - -theorem xinduction: - \forall (A: Set).(\forall (t: A).(\forall (P: ((A \to Prop))).(((\forall (x: -A).((eq A t x) \to (P x)))) \to (P t)))) -\def - \lambda (A: Set).(\lambda (t: A).(\lambda (P: ((A \to Prop))).(\lambda (H: -((\forall (x: A).((eq A t x) \to (P x))))).(H t (refl_equal A t))))). -(* COMMENTS -Initial nodes: 31 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Ground-1/plist/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Ground-1/plist/defs.ma deleted file mode 100644 index 8f9c1d3cc..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Ground-1/plist/defs.ma +++ /dev/null @@ -1,43 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Ground-1/preamble.ma". - -inductive PList: Set \def -| PNil: PList -| PCons: nat \to (nat \to (PList \to PList)). - -definition PConsTail: - PList \to (nat \to (nat \to PList)) -\def - let rec PConsTail (hds: PList) on hds: (nat \to (nat \to PList)) \def -(\lambda (h0: nat).(\lambda (d0: nat).(match hds with [PNil \Rightarrow -(PCons h0 d0 PNil) | (PCons h d hds0) \Rightarrow (PCons h d (PConsTail hds0 -h0 d0))]))) in PConsTail. - -definition Ss: - PList \to PList -\def - let rec Ss (hds: PList) on hds: PList \def (match hds with [PNil \Rightarrow -PNil | (PCons h d hds0) \Rightarrow (PCons h (S d) (Ss hds0))]) in Ss. - -definition papp: - PList \to (PList \to PList) -\def - let rec papp (a: PList) on a: (PList \to PList) \def (\lambda (b: -PList).(match a with [PNil \Rightarrow b | (PCons h d a0) \Rightarrow (PCons -h d (papp a0 b))])) in papp. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Ground-1/plist/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Ground-1/plist/props.ma deleted file mode 100644 index 990397229..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Ground-1/plist/props.ma +++ /dev/null @@ -1,34 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Ground-1/plist/defs.ma". - -theorem papp_ss: - \forall (is1: PList).(\forall (is2: PList).(eq PList (papp (Ss is1) (Ss -is2)) (Ss (papp is1 is2)))) -\def - \lambda (is1: PList).(PList_ind (\lambda (p: PList).(\forall (is2: -PList).(eq PList (papp (Ss p) (Ss is2)) (Ss (papp p is2))))) (\lambda (is2: -PList).(refl_equal PList (Ss is2))) (\lambda (n: nat).(\lambda (n0: -nat).(\lambda (p: PList).(\lambda (H: ((\forall (is2: PList).(eq PList (papp -(Ss p) (Ss is2)) (Ss (papp p is2)))))).(\lambda (is2: PList).(eq_ind_r PList -(Ss (papp p is2)) (\lambda (p0: PList).(eq PList (PCons n (S n0) p0) (PCons n -(S n0) (Ss (papp p is2))))) (refl_equal PList (PCons n (S n0) (Ss (papp p -is2)))) (papp (Ss p) (Ss is2)) (H is2))))))) is1). -(* COMMENTS -Initial nodes: 151 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Ground-1/preamble.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Ground-1/preamble.ma deleted file mode 100644 index 16ff2dc44..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Ground-1/preamble.ma +++ /dev/null @@ -1,15 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "Legacy-1/theory.ma". diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Ground-1/spare.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Ground-1/spare.ma deleted file mode 100644 index a966c2ae2..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Ground-1/spare.ma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Ground-1/theory.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Ground-1/theory.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Ground-1/theory.ma deleted file mode 100644 index 3b59dc6e8..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Ground-1/theory.ma +++ /dev/null @@ -1,26 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Ground-1/ext/tactics.ma". - -include "Ground-1/ext/arith.ma". - -include "Ground-1/types/props.ma". - -include "Ground-1/blt/props.ma". - -include "Ground-1/plist/props.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Ground-1/types/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Ground-1/types/defs.ma deleted file mode 100644 index f94969d88..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Ground-1/types/defs.ma +++ /dev/null @@ -1,172 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Ground-1/preamble.ma". - -inductive and3 (P0: Prop) (P1: Prop) (P2: Prop): Prop \def -| and3_intro: P0 \to (P1 \to (P2 \to (and3 P0 P1 P2))). - -inductive and4 (P0: Prop) (P1: Prop) (P2: Prop) (P3: Prop): Prop \def -| and4_intro: P0 \to (P1 \to (P2 \to (P3 \to (and4 P0 P1 P2 P3)))). - -inductive and5 (P0: Prop) (P1: Prop) (P2: Prop) (P3: Prop) (P4: Prop): Prop -\def -| and5_intro: P0 \to (P1 \to (P2 \to (P3 \to (P4 \to (and5 P0 P1 P2 P3 -P4))))). - -inductive or3 (P0: Prop) (P1: Prop) (P2: Prop): Prop \def -| or3_intro0: P0 \to (or3 P0 P1 P2) -| or3_intro1: P1 \to (or3 P0 P1 P2) -| or3_intro2: P2 \to (or3 P0 P1 P2). - -inductive or4 (P0: Prop) (P1: Prop) (P2: Prop) (P3: Prop): Prop \def -| or4_intro0: P0 \to (or4 P0 P1 P2 P3) -| or4_intro1: P1 \to (or4 P0 P1 P2 P3) -| or4_intro2: P2 \to (or4 P0 P1 P2 P3) -| or4_intro3: P3 \to (or4 P0 P1 P2 P3). - -inductive or5 (P0: Prop) (P1: Prop) (P2: Prop) (P3: Prop) (P4: Prop): Prop -\def -| or5_intro0: P0 \to (or5 P0 P1 P2 P3 P4) -| or5_intro1: P1 \to (or5 P0 P1 P2 P3 P4) -| or5_intro2: P2 \to (or5 P0 P1 P2 P3 P4) -| or5_intro3: P3 \to (or5 P0 P1 P2 P3 P4) -| or5_intro4: P4 \to (or5 P0 P1 P2 P3 P4). - -inductive ex3 (A0: Set) (P0: A0 \to Prop) (P1: A0 \to Prop) (P2: A0 \to -Prop): Prop \def -| ex3_intro: \forall (x0: A0).((P0 x0) \to ((P1 x0) \to ((P2 x0) \to (ex3 A0 -P0 P1 P2)))). - -inductive ex4 (A0: Set) (P0: A0 \to Prop) (P1: A0 \to Prop) (P2: A0 \to Prop) -(P3: A0 \to Prop): Prop \def -| ex4_intro: \forall (x0: A0).((P0 x0) \to ((P1 x0) \to ((P2 x0) \to ((P3 x0) -\to (ex4 A0 P0 P1 P2 P3))))). - -inductive ex_2 (A0: Set) (A1: Set) (P0: A0 \to (A1 \to Prop)): Prop \def -| ex_2_intro: \forall (x0: A0).(\forall (x1: A1).((P0 x0 x1) \to (ex_2 A0 A1 -P0))). - -inductive ex2_2 (A0: Set) (A1: Set) (P0: A0 \to (A1 \to Prop)) (P1: A0 \to -(A1 \to Prop)): Prop \def -| ex2_2_intro: \forall (x0: A0).(\forall (x1: A1).((P0 x0 x1) \to ((P1 x0 x1) -\to (ex2_2 A0 A1 P0 P1)))). - -inductive ex3_2 (A0: Set) (A1: Set) (P0: A0 \to (A1 \to Prop)) (P1: A0 \to -(A1 \to Prop)) (P2: A0 \to (A1 \to Prop)): Prop \def -| ex3_2_intro: \forall (x0: A0).(\forall (x1: A1).((P0 x0 x1) \to ((P1 x0 x1) -\to ((P2 x0 x1) \to (ex3_2 A0 A1 P0 P1 P2))))). - -inductive ex4_2 (A0: Set) (A1: Set) (P0: A0 \to (A1 \to Prop)) (P1: A0 \to -(A1 \to Prop)) (P2: A0 \to (A1 \to Prop)) (P3: A0 \to (A1 \to Prop)): Prop -\def -| ex4_2_intro: \forall (x0: A0).(\forall (x1: A1).((P0 x0 x1) \to ((P1 x0 x1) -\to ((P2 x0 x1) \to ((P3 x0 x1) \to (ex4_2 A0 A1 P0 P1 P2 P3)))))). - -inductive ex_3 (A0: Set) (A1: Set) (A2: Set) (P0: A0 \to (A1 \to (A2 \to -Prop))): Prop \def -| ex_3_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).((P0 x0 x1 -x2) \to (ex_3 A0 A1 A2 P0)))). - -inductive ex2_3 (A0: Set) (A1: Set) (A2: Set) (P0: A0 \to (A1 \to (A2 \to -Prop))) (P1: A0 \to (A1 \to (A2 \to Prop))): Prop \def -| ex2_3_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).((P0 x0 -x1 x2) \to ((P1 x0 x1 x2) \to (ex2_3 A0 A1 A2 P0 P1))))). - -inductive ex3_3 (A0: Set) (A1: Set) (A2: Set) (P0: A0 \to (A1 \to (A2 \to -Prop))) (P1: A0 \to (A1 \to (A2 \to Prop))) (P2: A0 \to (A1 \to (A2 \to -Prop))): Prop \def -| ex3_3_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).((P0 x0 -x1 x2) \to ((P1 x0 x1 x2) \to ((P2 x0 x1 x2) \to (ex3_3 A0 A1 A2 P0 P1 -P2)))))). - -inductive ex4_3 (A0: Set) (A1: Set) (A2: Set) (P0: A0 \to (A1 \to (A2 \to -Prop))) (P1: A0 \to (A1 \to (A2 \to Prop))) (P2: A0 \to (A1 \to (A2 \to -Prop))) (P3: A0 \to (A1 \to (A2 \to Prop))): Prop \def -| ex4_3_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).((P0 x0 -x1 x2) \to ((P1 x0 x1 x2) \to ((P2 x0 x1 x2) \to ((P3 x0 x1 x2) \to (ex4_3 A0 -A1 A2 P0 P1 P2 P3))))))). - -inductive ex5_3 (A0: Set) (A1: Set) (A2: Set) (P0: A0 \to (A1 \to (A2 \to -Prop))) (P1: A0 \to (A1 \to (A2 \to Prop))) (P2: A0 \to (A1 \to (A2 \to -Prop))) (P3: A0 \to (A1 \to (A2 \to Prop))) (P4: A0 \to (A1 \to (A2 \to -Prop))): Prop \def -| ex5_3_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).((P0 x0 -x1 x2) \to ((P1 x0 x1 x2) \to ((P2 x0 x1 x2) \to ((P3 x0 x1 x2) \to ((P4 x0 -x1 x2) \to (ex5_3 A0 A1 A2 P0 P1 P2 P3 P4)))))))). - -inductive ex3_4 (A0: Set) (A1: Set) (A2: Set) (A3: Set) (P0: A0 \to (A1 \to -(A2 \to (A3 \to Prop)))) (P1: A0 \to (A1 \to (A2 \to (A3 \to Prop)))) (P2: A0 -\to (A1 \to (A2 \to (A3 \to Prop)))): Prop \def -| ex3_4_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall -(x3: A3).((P0 x0 x1 x2 x3) \to ((P1 x0 x1 x2 x3) \to ((P2 x0 x1 x2 x3) \to -(ex3_4 A0 A1 A2 A3 P0 P1 P2))))))). - -inductive ex4_4 (A0: Set) (A1: Set) (A2: Set) (A3: Set) (P0: A0 \to (A1 \to -(A2 \to (A3 \to Prop)))) (P1: A0 \to (A1 \to (A2 \to (A3 \to Prop)))) (P2: A0 -\to (A1 \to (A2 \to (A3 \to Prop)))) (P3: A0 \to (A1 \to (A2 \to (A3 \to -Prop)))): Prop \def -| ex4_4_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall -(x3: A3).((P0 x0 x1 x2 x3) \to ((P1 x0 x1 x2 x3) \to ((P2 x0 x1 x2 x3) \to -((P3 x0 x1 x2 x3) \to (ex4_4 A0 A1 A2 A3 P0 P1 P2 P3)))))))). - -inductive ex4_5 (A0: Set) (A1: Set) (A2: Set) (A3: Set) (A4: Set) (P0: A0 \to -(A1 \to (A2 \to (A3 \to (A4 \to Prop))))) (P1: A0 \to (A1 \to (A2 \to (A3 \to -(A4 \to Prop))))) (P2: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to Prop))))) (P3: -A0 \to (A1 \to (A2 \to (A3 \to (A4 \to Prop))))): Prop \def -| ex4_5_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall -(x3: A3).(\forall (x4: A4).((P0 x0 x1 x2 x3 x4) \to ((P1 x0 x1 x2 x3 x4) \to -((P2 x0 x1 x2 x3 x4) \to ((P3 x0 x1 x2 x3 x4) \to (ex4_5 A0 A1 A2 A3 A4 P0 P1 -P2 P3))))))))). - -inductive ex5_5 (A0: Set) (A1: Set) (A2: Set) (A3: Set) (A4: Set) (P0: A0 \to -(A1 \to (A2 \to (A3 \to (A4 \to Prop))))) (P1: A0 \to (A1 \to (A2 \to (A3 \to -(A4 \to Prop))))) (P2: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to Prop))))) (P3: -A0 \to (A1 \to (A2 \to (A3 \to (A4 \to Prop))))) (P4: A0 \to (A1 \to (A2 \to -(A3 \to (A4 \to Prop))))): Prop \def -| ex5_5_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall -(x3: A3).(\forall (x4: A4).((P0 x0 x1 x2 x3 x4) \to ((P1 x0 x1 x2 x3 x4) \to -((P2 x0 x1 x2 x3 x4) \to ((P3 x0 x1 x2 x3 x4) \to ((P4 x0 x1 x2 x3 x4) \to -(ex5_5 A0 A1 A2 A3 A4 P0 P1 P2 P3 P4)))))))))). - -inductive ex6_6 (A0: Set) (A1: Set) (A2: Set) (A3: Set) (A4: Set) (A5: Set) -(P0: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to Prop)))))) (P1: A0 \to -(A1 \to (A2 \to (A3 \to (A4 \to (A5 \to Prop)))))) (P2: A0 \to (A1 \to (A2 -\to (A3 \to (A4 \to (A5 \to Prop)))))) (P3: A0 \to (A1 \to (A2 \to (A3 \to -(A4 \to (A5 \to Prop)))))) (P4: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 -\to Prop)))))) (P5: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to -Prop)))))): Prop \def -| ex6_6_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall -(x3: A3).(\forall (x4: A4).(\forall (x5: A5).((P0 x0 x1 x2 x3 x4 x5) \to ((P1 -x0 x1 x2 x3 x4 x5) \to ((P2 x0 x1 x2 x3 x4 x5) \to ((P3 x0 x1 x2 x3 x4 x5) -\to ((P4 x0 x1 x2 x3 x4 x5) \to ((P5 x0 x1 x2 x3 x4 x5) \to (ex6_6 A0 A1 A2 -A3 A4 A5 P0 P1 P2 P3 P4 P5)))))))))))). - -inductive ex6_7 (A0: Set) (A1: Set) (A2: Set) (A3: Set) (A4: Set) (A5: Set) -(A6: Set) (P0: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 \to -Prop))))))) (P1: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 \to -Prop))))))) (P2: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 \to -Prop))))))) (P3: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 \to -Prop))))))) (P4: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 \to -Prop))))))) (P5: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 \to -Prop))))))): Prop \def -| ex6_7_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall -(x3: A3).(\forall (x4: A4).(\forall (x5: A5).(\forall (x6: A6).((P0 x0 x1 x2 -x3 x4 x5 x6) \to ((P1 x0 x1 x2 x3 x4 x5 x6) \to ((P2 x0 x1 x2 x3 x4 x5 x6) -\to ((P3 x0 x1 x2 x3 x4 x5 x6) \to ((P4 x0 x1 x2 x3 x4 x5 x6) \to ((P5 x0 x1 -x2 x3 x4 x5 x6) \to (ex6_7 A0 A1 A2 A3 A4 A5 A6 P0 P1 P2 P3 P4 -P5))))))))))))). - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Ground-1/types/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Ground-1/types/props.ma deleted file mode 100644 index 9c326f44c..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Ground-1/types/props.ma +++ /dev/null @@ -1,33 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Ground-1/types/defs.ma". - -theorem ex2_sym: - \forall (A: Set).(\forall (P: ((A \to Prop))).(\forall (Q: ((A \to -Prop))).((ex2 A (\lambda (x: A).(P x)) (\lambda (x: A).(Q x))) \to (ex2 A -(\lambda (x: A).(Q x)) (\lambda (x: A).(P x)))))) -\def - \lambda (A: Set).(\lambda (P: ((A \to Prop))).(\lambda (Q: ((A \to -Prop))).(\lambda (H: (ex2 A (\lambda (x: A).(P x)) (\lambda (x: A).(Q -x)))).(ex2_ind A (\lambda (x: A).(P x)) (\lambda (x: A).(Q x)) (ex2 A -(\lambda (x: A).(Q x)) (\lambda (x: A).(P x))) (\lambda (x: A).(\lambda (H0: -(P x)).(\lambda (H1: (Q x)).(ex_intro2 A (\lambda (x0: A).(Q x0)) (\lambda -(x0: A).(P x0)) x H1 H0)))) H)))). -(* COMMENTS -Initial nodes: 91 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Ground-2/blt/defs.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Ground-2/blt/defs.mma deleted file mode 100644 index 028f98a42..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Ground-2/blt/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Ground-2/preamble.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Ground-2/blt/props.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Ground-2/blt/props.mma deleted file mode 100644 index 24fd85f96..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Ground-2/blt/props.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Ground-2/blt/defs.ma". - -inline "Ground-1/blt/props.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Ground-2/ext/arith.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Ground-2/ext/arith.mma deleted file mode 100644 index ab8a4c76e..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Ground-2/ext/arith.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Ground-2/preamble.ma". - -inline "Ground-1/ext/arith.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Ground-2/ext/tactics.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Ground-2/ext/tactics.mma deleted file mode 100644 index 72cacebfc..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Ground-2/ext/tactics.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Ground-2/preamble.ma". - -inline "Ground-1/ext/tactics.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Ground-2/plist/defs.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Ground-2/plist/defs.mma deleted file mode 100644 index 028f98a42..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Ground-2/plist/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Ground-2/preamble.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Ground-2/plist/props.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Ground-2/plist/props.mma deleted file mode 100644 index 196cf33a7..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Ground-2/plist/props.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Ground-2/plist/defs.ma". - -inline "Ground-1/plist/props.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Ground-2/preamble.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Ground-2/preamble.ma deleted file mode 100644 index ffe3dce3a..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Ground-2/preamble.ma +++ /dev/null @@ -1,16 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "Ground-1/definitions.ma". -include "Legacy-2/theory.ma". diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Ground-2/theory.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Ground-2/theory.ma deleted file mode 100644 index 25de9e4c3..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Ground-2/theory.ma +++ /dev/null @@ -1,26 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Ground-2/ext/tactics.ma". - -include "Ground-2/ext/arith.ma". - -include "Ground-2/types/props.ma". - -include "Ground-2/blt/props.ma". - -include "Ground-2/plist/props.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Ground-2/types/defs.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Ground-2/types/defs.mma deleted file mode 100644 index 028f98a42..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Ground-2/types/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Ground-2/preamble.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Ground-2/types/props.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Ground-2/types/props.mma deleted file mode 100644 index 49444e73b..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Ground-2/types/props.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Ground-2/types/defs.ma". - -inline "Ground-1/types/props.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Legacy-1/coq/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Legacy-1/coq/defs.ma deleted file mode 100644 index 7d4696229..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Legacy-1/coq/defs.ma +++ /dev/null @@ -1,99 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Legacy-1/preamble.ma". - -inductive eq (A: Set) (x: A): A \to Prop \def -| refl_equal: eq A x x. - -inductive True: Prop \def -| I: True. - -inductive land (A: Prop) (B: Prop): Prop \def -| conj: A \to (B \to (land A B)). - -inductive or (A: Prop) (B: Prop): Prop \def -| or_introl: A \to (or A B) -| or_intror: B \to (or A B). - -inductive ex (A: Set) (P: A \to Prop): Prop \def -| ex_intro: \forall (x: A).((P x) \to (ex A P)). - -inductive ex2 (A: Set) (P: A \to Prop) (Q: A \to Prop): Prop \def -| ex_intro2: \forall (x: A).((P x) \to ((Q x) \to (ex2 A P Q))). - -definition not: - Prop \to Prop -\def - \lambda (A: Prop).(A \to False). - -inductive bool: Set \def -| true: bool -| false: bool. - -inductive nat: Set \def -| O: nat -| S: nat \to nat. - -inductive le (n: nat): nat \to Prop \def -| le_n: le n n -| le_S: \forall (m: nat).((le n m) \to (le n (S m))). - -definition lt: - nat \to (nat \to Prop) -\def - \lambda (n: nat).(\lambda (m: nat).(le (S n) m)). - -definition IsSucc: - nat \to Prop -\def - \lambda (n: nat).(match n with [O \Rightarrow False | (S _) \Rightarrow -True]). - -definition pred: - nat \to nat -\def - \lambda (n: nat).(match n with [O \Rightarrow O | (S u) \Rightarrow u]). - -definition plus: - nat \to (nat \to nat) -\def - let rec plus (n: nat) on n: (nat \to nat) \def (\lambda (m: nat).(match n -with [O \Rightarrow m | (S p) \Rightarrow (S (plus p m))])) in plus. - -definition minus: - nat \to (nat \to nat) -\def - let rec minus (n: nat) on n: (nat \to nat) \def (\lambda (m: nat).(match n -with [O \Rightarrow O | (S k) \Rightarrow (match m with [O \Rightarrow (S k) -| (S l) \Rightarrow (minus k l)])])) in minus. - -inductive Acc (A: Set) (R: A \to (A \to Prop)): A \to Prop \def -| Acc_intro: \forall (x: A).(((\forall (y: A).((R y x) \to (Acc A R y)))) \to -(Acc A R x)). - -definition well_founded: - \forall (A: Set).(((A \to (A \to Prop))) \to Prop) -\def - \lambda (A: Set).(\lambda (R: ((A \to (A \to Prop)))).(\forall (a: A).(Acc A -R a))). - -definition ltof: - \forall (A: Set).(((A \to nat)) \to (A \to (A \to Prop))) -\def - \lambda (A: Set).(\lambda (f: ((A \to nat))).(\lambda (a: A).(\lambda (b: -A).(lt (f a) (f b))))). - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Legacy-1/coq/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Legacy-1/coq/props.ma deleted file mode 100644 index 0b9d97b42..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Legacy-1/coq/props.ma +++ /dev/null @@ -1,805 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Legacy-1/coq/defs.ma". - -theorem f_equal: - \forall (A: Set).(\forall (B: Set).(\forall (f: ((A \to B))).(\forall (x: -A).(\forall (y: A).((eq A x y) \to (eq B (f x) (f y))))))) -\def - \lambda (A: Set).(\lambda (B: Set).(\lambda (f: ((A \to B))).(\lambda (x: -A).(\lambda (y: A).(\lambda (H: (eq A x y)).(eq_ind A x (\lambda (a: A).(eq B -(f x) (f a))) (refl_equal B (f x)) y H)))))). -(* COMMENTS -Initial nodes: 51 -END *) - -theorem f_equal2: - \forall (A1: Set).(\forall (A2: Set).(\forall (B: Set).(\forall (f: ((A1 \to -(A2 \to B)))).(\forall (x1: A1).(\forall (y1: A1).(\forall (x2: A2).(\forall -(y2: A2).((eq A1 x1 y1) \to ((eq A2 x2 y2) \to (eq B (f x1 x2) (f y1 -y2))))))))))) -\def - \lambda (A1: Set).(\lambda (A2: Set).(\lambda (B: Set).(\lambda (f: ((A1 \to -(A2 \to B)))).(\lambda (x1: A1).(\lambda (y1: A1).(\lambda (x2: A2).(\lambda -(y2: A2).(\lambda (H: (eq A1 x1 y1)).(eq_ind A1 x1 (\lambda (a: A1).((eq A2 -x2 y2) \to (eq B (f x1 x2) (f a y2)))) (\lambda (H0: (eq A2 x2 y2)).(eq_ind -A2 x2 (\lambda (a: A2).(eq B (f x1 x2) (f x1 a))) (refl_equal B (f x1 x2)) y2 -H0)) y1 H))))))))). -(* COMMENTS -Initial nodes: 109 -END *) - -theorem f_equal3: - \forall (A1: Set).(\forall (A2: Set).(\forall (A3: Set).(\forall (B: -Set).(\forall (f: ((A1 \to (A2 \to (A3 \to B))))).(\forall (x1: A1).(\forall -(y1: A1).(\forall (x2: A2).(\forall (y2: A2).(\forall (x3: A3).(\forall (y3: -A3).((eq A1 x1 y1) \to ((eq A2 x2 y2) \to ((eq A3 x3 y3) \to (eq B (f x1 x2 -x3) (f y1 y2 y3))))))))))))))) -\def - \lambda (A1: Set).(\lambda (A2: Set).(\lambda (A3: Set).(\lambda (B: -Set).(\lambda (f: ((A1 \to (A2 \to (A3 \to B))))).(\lambda (x1: A1).(\lambda -(y1: A1).(\lambda (x2: A2).(\lambda (y2: A2).(\lambda (x3: A3).(\lambda (y3: -A3).(\lambda (H: (eq A1 x1 y1)).(eq_ind A1 x1 (\lambda (a: A1).((eq A2 x2 y2) -\to ((eq A3 x3 y3) \to (eq B (f x1 x2 x3) (f a y2 y3))))) (\lambda (H0: (eq -A2 x2 y2)).(eq_ind A2 x2 (\lambda (a: A2).((eq A3 x3 y3) \to (eq B (f x1 x2 -x3) (f x1 a y3)))) (\lambda (H1: (eq A3 x3 y3)).(eq_ind A3 x3 (\lambda (a: -A3).(eq B (f x1 x2 x3) (f x1 x2 a))) (refl_equal B (f x1 x2 x3)) y3 H1)) y2 -H0)) y1 H)))))))))))). -(* COMMENTS -Initial nodes: 183 -END *) - -theorem sym_eq: - \forall (A: Set).(\forall (x: A).(\forall (y: A).((eq A x y) \to (eq A y -x)))) -\def - \lambda (A: Set).(\lambda (x: A).(\lambda (y: A).(\lambda (H: (eq A x -y)).(eq_ind A x (\lambda (a: A).(eq A a x)) (refl_equal A x) y H)))). -(* COMMENTS -Initial nodes: 39 -END *) - -theorem eq_ind_r: - \forall (A: Set).(\forall (x: A).(\forall (P: ((A \to Prop))).((P x) \to -(\forall (y: A).((eq A y x) \to (P y)))))) -\def - \lambda (A: Set).(\lambda (x: A).(\lambda (P: ((A \to Prop))).(\lambda (H: -(P x)).(\lambda (y: A).(\lambda (H0: (eq A y x)).(match (sym_eq A y x H0) in -eq return (\lambda (a: A).(\lambda (_: (eq ? ? a)).(P a))) with [refl_equal -\Rightarrow H])))))). -(* COMMENTS -Initial nodes: 38 -END *) - -theorem trans_eq: - \forall (A: Set).(\forall (x: A).(\forall (y: A).(\forall (z: A).((eq A x y) -\to ((eq A y z) \to (eq A x z)))))) -\def - \lambda (A: Set).(\lambda (x: A).(\lambda (y: A).(\lambda (z: A).(\lambda -(H: (eq A x y)).(\lambda (H0: (eq A y z)).(eq_ind A y (\lambda (a: A).(eq A x -a)) H z H0)))))). -(* COMMENTS -Initial nodes: 45 -END *) - -theorem sym_not_eq: - \forall (A: Set).(\forall (x: A).(\forall (y: A).((not (eq A x y)) \to (not -(eq A y x))))) -\def - \lambda (A: Set).(\lambda (x: A).(\lambda (y: A).(\lambda (h1: (not (eq A x -y))).(\lambda (h2: (eq A y x)).(h1 (eq_ind A y (\lambda (a: A).(eq A a y)) -(refl_equal A y) x h2)))))). -(* COMMENTS -Initial nodes: 51 -END *) - -theorem nat_double_ind: - \forall (R: ((nat \to (nat \to Prop)))).(((\forall (n: nat).(R O n))) \to -(((\forall (n: nat).(R (S n) O))) \to (((\forall (n: nat).(\forall (m: -nat).((R n m) \to (R (S n) (S m)))))) \to (\forall (n: nat).(\forall (m: -nat).(R n m)))))) -\def - \lambda (R: ((nat \to (nat \to Prop)))).(\lambda (H: ((\forall (n: nat).(R O -n)))).(\lambda (H0: ((\forall (n: nat).(R (S n) O)))).(\lambda (H1: ((\forall -(n: nat).(\forall (m: nat).((R n m) \to (R (S n) (S m))))))).(\lambda (n: -nat).(nat_ind (\lambda (n0: nat).(\forall (m: nat).(R n0 m))) H (\lambda (n0: -nat).(\lambda (H2: ((\forall (m: nat).(R n0 m)))).(\lambda (m: nat).(nat_ind -(\lambda (n1: nat).(R (S n0) n1)) (H0 n0) (\lambda (n1: nat).(\lambda (_: (R -(S n0) n1)).(H1 n0 n1 (H2 n1)))) m)))) n))))). -(* COMMENTS -Initial nodes: 111 -END *) - -theorem eq_add_S: - \forall (n: nat).(\forall (m: nat).((eq nat (S n) (S m)) \to (eq nat n m))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (eq nat (S n) (S -m))).(f_equal nat nat pred (S n) (S m) H))). -(* COMMENTS -Initial nodes: 33 -END *) - -theorem O_S: - \forall (n: nat).(not (eq nat O (S n))) -\def - \lambda (n: nat).(\lambda (H: (eq nat O (S n))).(eq_ind nat (S n) (\lambda -(n0: nat).(IsSucc n0)) I O (sym_eq nat O (S n) H))). -(* COMMENTS -Initial nodes: 41 -END *) - -theorem not_eq_S: - \forall (n: nat).(\forall (m: nat).((not (eq nat n m)) \to (not (eq nat (S -n) (S m))))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (not (eq nat n m))).(\lambda -(H0: (eq nat (S n) (S m))).(H (eq_add_S n m H0))))). -(* COMMENTS -Initial nodes: 35 -END *) - -theorem pred_Sn: - \forall (m: nat).(eq nat m (pred (S m))) -\def - \lambda (m: nat).(refl_equal nat (pred (S m))). -(* COMMENTS -Initial nodes: 11 -END *) - -theorem S_pred: - \forall (n: nat).(\forall (m: nat).((lt m n) \to (eq nat n (S (pred n))))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt m n)).(le_ind (S m) -(\lambda (n0: nat).(eq nat n0 (S (pred n0)))) (refl_equal nat (S (pred (S -m)))) (\lambda (m0: nat).(\lambda (_: (le (S m) m0)).(\lambda (_: (eq nat m0 -(S (pred m0)))).(refl_equal nat (S (pred (S m0))))))) n H))). -(* COMMENTS -Initial nodes: 79 -END *) - -theorem le_trans: - \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((le n m) \to ((le m p) -\to (le n p))))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (le n -m)).(\lambda (H0: (le m p)).(le_ind m (\lambda (n0: nat).(le n n0)) H -(\lambda (m0: nat).(\lambda (_: (le m m0)).(\lambda (IHle: (le n m0)).(le_S n -m0 IHle)))) p H0))))). -(* COMMENTS -Initial nodes: 57 -END *) - -theorem le_trans_S: - \forall (n: nat).(\forall (m: nat).((le (S n) m) \to (le n m))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le (S n) m)).(le_trans n (S -n) m (le_S n n (le_n n)) H))). -(* COMMENTS -Initial nodes: 33 -END *) - -theorem le_n_S: - \forall (n: nat).(\forall (m: nat).((le n m) \to (le (S n) (S m)))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(le_ind n (\lambda -(n0: nat).(le (S n) (S n0))) (le_n (S n)) (\lambda (m0: nat).(\lambda (_: (le -n m0)).(\lambda (IHle: (le (S n) (S m0))).(le_S (S n) (S m0) IHle)))) m H))). -(* COMMENTS -Initial nodes: 65 -END *) - -theorem le_O_n: - \forall (n: nat).(le O n) -\def - \lambda (n: nat).(nat_ind (\lambda (n0: nat).(le O n0)) (le_n O) (\lambda -(n0: nat).(\lambda (IHn: (le O n0)).(le_S O n0 IHn))) n). -(* COMMENTS -Initial nodes: 33 -END *) - -theorem le_S_n: - \forall (n: nat).(\forall (m: nat).((le (S n) (S m)) \to (le n m))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le (S n) (S m))).(le_ind (S -n) (\lambda (n0: nat).(le (pred (S n)) (pred n0))) (le_n n) (\lambda (m0: -nat).(\lambda (H0: (le (S n) m0)).(\lambda (_: (le n (pred m0))).(le_trans_S -n m0 H0)))) (S m) H))). -(* COMMENTS -Initial nodes: 69 -END *) - -theorem le_Sn_O: - \forall (n: nat).(not (le (S n) O)) -\def - \lambda (n: nat).(\lambda (H: (le (S n) O)).(le_ind (S n) (\lambda (n0: -nat).(IsSucc n0)) I (\lambda (m: nat).(\lambda (_: (le (S n) m)).(\lambda (_: -(IsSucc m)).I))) O H)). -(* COMMENTS -Initial nodes: 43 -END *) - -theorem le_Sn_n: - \forall (n: nat).(not (le (S n) n)) -\def - \lambda (n: nat).(nat_ind (\lambda (n0: nat).(not (le (S n0) n0))) (le_Sn_O -O) (\lambda (n0: nat).(\lambda (IHn: (not (le (S n0) n0))).(\lambda (H: (le -(S (S n0)) (S n0))).(IHn (le_S_n (S n0) n0 H))))) n). -(* COMMENTS -Initial nodes: 57 -END *) - -theorem le_antisym: - \forall (n: nat).(\forall (m: nat).((le n m) \to ((le m n) \to (eq nat n -m)))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (h: (le n m)).(le_ind n (\lambda -(n0: nat).((le n0 n) \to (eq nat n n0))) (\lambda (_: (le n n)).(refl_equal -nat n)) (\lambda (m0: nat).(\lambda (H: (le n m0)).(\lambda (_: (((le m0 n) -\to (eq nat n m0)))).(\lambda (H1: (le (S m0) n)).(False_ind (eq nat n (S -m0)) (let H2 \def (le_trans (S m0) n m0 H1 H) in ((let H3 \def (le_Sn_n m0) -in (\lambda (H4: (le (S m0) m0)).(H3 H4))) H2))))))) m h))). -(* COMMENTS -Initial nodes: 119 -END *) - -theorem le_n_O_eq: - \forall (n: nat).((le n O) \to (eq nat O n)) -\def - \lambda (n: nat).(\lambda (H: (le n O)).(le_antisym O n (le_O_n n) H)). -(* COMMENTS -Initial nodes: 19 -END *) - -theorem le_elim_rel: - \forall (P: ((nat \to (nat \to Prop)))).(((\forall (p: nat).(P O p))) \to -(((\forall (p: nat).(\forall (q: nat).((le p q) \to ((P p q) \to (P (S p) (S -q))))))) \to (\forall (n: nat).(\forall (m: nat).((le n m) \to (P n m)))))) -\def - \lambda (P: ((nat \to (nat \to Prop)))).(\lambda (H: ((\forall (p: nat).(P O -p)))).(\lambda (H0: ((\forall (p: nat).(\forall (q: nat).((le p q) \to ((P p -q) \to (P (S p) (S q)))))))).(\lambda (n: nat).(nat_ind (\lambda (n0: -nat).(\forall (m: nat).((le n0 m) \to (P n0 m)))) (\lambda (m: nat).(\lambda -(_: (le O m)).(H m))) (\lambda (n0: nat).(\lambda (IHn: ((\forall (m: -nat).((le n0 m) \to (P n0 m))))).(\lambda (m: nat).(\lambda (Le: (le (S n0) -m)).(le_ind (S n0) (\lambda (n1: nat).(P (S n0) n1)) (H0 n0 n0 (le_n n0) (IHn -n0 (le_n n0))) (\lambda (m0: nat).(\lambda (H1: (le (S n0) m0)).(\lambda (_: -(P (S n0) m0)).(H0 n0 m0 (le_trans_S n0 m0 H1) (IHn m0 (le_trans_S n0 m0 -H1)))))) m Le))))) n)))). -(* COMMENTS -Initial nodes: 181 -END *) - -theorem lt_n_n: - \forall (n: nat).(not (lt n n)) -\def - le_Sn_n. -(* COMMENTS -Initial nodes: 1 -END *) - -theorem lt_n_S: - \forall (n: nat).(\forall (m: nat).((lt n m) \to (lt (S n) (S m)))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt n m)).(le_n_S (S n) m -H))). -(* COMMENTS -Initial nodes: 19 -END *) - -theorem lt_n_Sn: - \forall (n: nat).(lt n (S n)) -\def - \lambda (n: nat).(le_n (S n)). -(* COMMENTS -Initial nodes: 7 -END *) - -theorem lt_S_n: - \forall (n: nat).(\forall (m: nat).((lt (S n) (S m)) \to (lt n m))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt (S n) (S m))).(le_S_n (S -n) m H))). -(* COMMENTS -Initial nodes: 23 -END *) - -theorem lt_n_O: - \forall (n: nat).(not (lt n O)) -\def - le_Sn_O. -(* COMMENTS -Initial nodes: 1 -END *) - -theorem lt_trans: - \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((lt n m) \to ((lt m p) -\to (lt n p))))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (lt n -m)).(\lambda (H0: (lt m p)).(le_ind (S m) (\lambda (n0: nat).(lt n n0)) (le_S -(S n) m H) (\lambda (m0: nat).(\lambda (_: (le (S m) m0)).(\lambda (IHle: (lt -n m0)).(le_S (S n) m0 IHle)))) p H0))))). -(* COMMENTS -Initial nodes: 71 -END *) - -theorem lt_O_Sn: - \forall (n: nat).(lt O (S n)) -\def - \lambda (n: nat).(le_n_S O n (le_O_n n)). -(* COMMENTS -Initial nodes: 11 -END *) - -theorem lt_le_S: - \forall (n: nat).(\forall (p: nat).((lt n p) \to (le (S n) p))) -\def - \lambda (n: nat).(\lambda (p: nat).(\lambda (H: (lt n p)).H)). -(* COMMENTS -Initial nodes: 11 -END *) - -theorem le_not_lt: - \forall (n: nat).(\forall (m: nat).((le n m) \to (not (lt m n)))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(le_ind n (\lambda -(n0: nat).(not (lt n0 n))) (lt_n_n n) (\lambda (m0: nat).(\lambda (_: (le n -m0)).(\lambda (IHle: (not (lt m0 n))).(\lambda (H1: (lt (S m0) n)).(IHle -(le_trans_S (S m0) n H1)))))) m H))). -(* COMMENTS -Initial nodes: 67 -END *) - -theorem le_lt_n_Sm: - \forall (n: nat).(\forall (m: nat).((le n m) \to (lt n (S m)))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(le_n_S n m H))). -(* COMMENTS -Initial nodes: 17 -END *) - -theorem le_lt_trans: - \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((le n m) \to ((lt m p) -\to (lt n p))))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (le n -m)).(\lambda (H0: (lt m p)).(le_ind (S m) (\lambda (n0: nat).(lt n n0)) -(le_n_S n m H) (\lambda (m0: nat).(\lambda (_: (le (S m) m0)).(\lambda (IHle: -(lt n m0)).(le_S (S n) m0 IHle)))) p H0))))). -(* COMMENTS -Initial nodes: 69 -END *) - -theorem lt_le_trans: - \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((lt n m) \to ((le m p) -\to (lt n p))))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (lt n -m)).(\lambda (H0: (le m p)).(le_ind m (\lambda (n0: nat).(lt n n0)) H -(\lambda (m0: nat).(\lambda (_: (le m m0)).(\lambda (IHle: (lt n m0)).(le_S -(S n) m0 IHle)))) p H0))))). -(* COMMENTS -Initial nodes: 59 -END *) - -theorem lt_le_weak: - \forall (n: nat).(\forall (m: nat).((lt n m) \to (le n m))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt n m)).(le_trans_S n m -H))). -(* COMMENTS -Initial nodes: 17 -END *) - -theorem lt_n_Sm_le: - \forall (n: nat).(\forall (m: nat).((lt n (S m)) \to (le n m))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt n (S m))).(le_S_n n m -H))). -(* COMMENTS -Initial nodes: 19 -END *) - -theorem le_lt_or_eq: - \forall (n: nat).(\forall (m: nat).((le n m) \to (or (lt n m) (eq nat n m)))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(le_ind n (\lambda -(n0: nat).(or (lt n n0) (eq nat n n0))) (or_intror (lt n n) (eq nat n n) -(refl_equal nat n)) (\lambda (m0: nat).(\lambda (H0: (le n m0)).(\lambda (_: -(or (lt n m0) (eq nat n m0))).(or_introl (lt n (S m0)) (eq nat n (S m0)) -(le_n_S n m0 H0))))) m H))). -(* COMMENTS -Initial nodes: 109 -END *) - -theorem le_or_lt: - \forall (n: nat).(\forall (m: nat).(or (le n m) (lt m n))) -\def - \lambda (n: nat).(\lambda (m: nat).(nat_double_ind (\lambda (n0: -nat).(\lambda (n1: nat).(or (le n0 n1) (lt n1 n0)))) (\lambda (n0: -nat).(or_introl (le O n0) (lt n0 O) (le_O_n n0))) (\lambda (n0: -nat).(or_intror (le (S n0) O) (lt O (S n0)) (lt_le_S O (S n0) (lt_O_Sn n0)))) -(\lambda (n0: nat).(\lambda (m0: nat).(\lambda (H: (or (le n0 m0) (lt m0 -n0))).(or_ind (le n0 m0) (lt m0 n0) (or (le (S n0) (S m0)) (lt (S m0) (S -n0))) (\lambda (H0: (le n0 m0)).(or_introl (le (S n0) (S m0)) (lt (S m0) (S -n0)) (le_n_S n0 m0 H0))) (\lambda (H0: (lt m0 n0)).(or_intror (le (S n0) (S -m0)) (lt (S m0) (S n0)) (le_n_S (S m0) n0 H0))) H)))) n m)). -(* COMMENTS -Initial nodes: 209 -END *) - -theorem plus_n_O: - \forall (n: nat).(eq nat n (plus n O)) -\def - \lambda (n: nat).(nat_ind (\lambda (n0: nat).(eq nat n0 (plus n0 O))) -(refl_equal nat O) (\lambda (n0: nat).(\lambda (H: (eq nat n0 (plus n0 -O))).(f_equal nat nat S n0 (plus n0 O) H))) n). -(* COMMENTS -Initial nodes: 57 -END *) - -theorem plus_n_Sm: - \forall (n: nat).(\forall (m: nat).(eq nat (S (plus n m)) (plus n (S m)))) -\def - \lambda (m: nat).(\lambda (n: nat).(nat_ind (\lambda (n0: nat).(eq nat (S -(plus n0 n)) (plus n0 (S n)))) (refl_equal nat (S n)) (\lambda (n0: -nat).(\lambda (H: (eq nat (S (plus n0 n)) (plus n0 (S n)))).(f_equal nat nat -S (S (plus n0 n)) (plus n0 (S n)) H))) m)). -(* COMMENTS -Initial nodes: 85 -END *) - -theorem plus_sym: - \forall (n: nat).(\forall (m: nat).(eq nat (plus n m) (plus m n))) -\def - \lambda (n: nat).(\lambda (m: nat).(nat_ind (\lambda (n0: nat).(eq nat (plus -n0 m) (plus m n0))) (plus_n_O m) (\lambda (y: nat).(\lambda (H: (eq nat (plus -y m) (plus m y))).(eq_ind nat (S (plus m y)) (\lambda (n0: nat).(eq nat (S -(plus y m)) n0)) (f_equal nat nat S (plus y m) (plus m y) H) (plus m (S y)) -(plus_n_Sm m y)))) n)). -(* COMMENTS -Initial nodes: 111 -END *) - -theorem plus_Snm_nSm: - \forall (n: nat).(\forall (m: nat).(eq nat (plus (S n) m) (plus n (S m)))) -\def - \lambda (n: nat).(\lambda (m: nat).(eq_ind_r nat (plus m n) (\lambda (n0: -nat).(eq nat (S n0) (plus n (S m)))) (eq_ind_r nat (plus (S m) n) (\lambda -(n0: nat).(eq nat (S (plus m n)) n0)) (refl_equal nat (plus (S m) n)) (plus n -(S m)) (plus_sym n (S m))) (plus n m) (plus_sym n m))). -(* COMMENTS -Initial nodes: 99 -END *) - -theorem plus_assoc_l: - \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(eq nat (plus n (plus m -p)) (plus (plus n m) p)))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(nat_ind (\lambda (n0: -nat).(eq nat (plus n0 (plus m p)) (plus (plus n0 m) p))) (refl_equal nat -(plus m p)) (\lambda (n0: nat).(\lambda (H: (eq nat (plus n0 (plus m p)) -(plus (plus n0 m) p))).(f_equal nat nat S (plus n0 (plus m p)) (plus (plus n0 -m) p) H))) n))). -(* COMMENTS -Initial nodes: 101 -END *) - -theorem plus_assoc_r: - \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(eq nat (plus (plus n -m) p) (plus n (plus m p))))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(sym_eq nat (plus n -(plus m p)) (plus (plus n m) p) (plus_assoc_l n m p)))). -(* COMMENTS -Initial nodes: 37 -END *) - -theorem simpl_plus_l: - \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((eq nat (plus n m) -(plus n p)) \to (eq nat m p)))) -\def - \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (m: nat).(\forall (p: -nat).((eq nat (plus n0 m) (plus n0 p)) \to (eq nat m p))))) (\lambda (m: -nat).(\lambda (p: nat).(\lambda (H: (eq nat m p)).H))) (\lambda (n0: -nat).(\lambda (IHn: ((\forall (m: nat).(\forall (p: nat).((eq nat (plus n0 m) -(plus n0 p)) \to (eq nat m p)))))).(\lambda (m: nat).(\lambda (p: -nat).(\lambda (H: (eq nat (S (plus n0 m)) (S (plus n0 p)))).(IHn m p (IHn -(plus n0 m) (plus n0 p) (f_equal nat nat (plus n0) (plus n0 m) (plus n0 p) -(eq_add_S (plus n0 m) (plus n0 p) H))))))))) n). -(* COMMENTS -Initial nodes: 161 -END *) - -theorem minus_n_O: - \forall (n: nat).(eq nat n (minus n O)) -\def - \lambda (n: nat).(nat_ind (\lambda (n0: nat).(eq nat n0 (minus n0 O))) -(refl_equal nat O) (\lambda (n0: nat).(\lambda (_: (eq nat n0 (minus n0 -O))).(refl_equal nat (S n0)))) n). -(* COMMENTS -Initial nodes: 47 -END *) - -theorem minus_n_n: - \forall (n: nat).(eq nat O (minus n n)) -\def - \lambda (n: nat).(nat_ind (\lambda (n0: nat).(eq nat O (minus n0 n0))) -(refl_equal nat O) (\lambda (n0: nat).(\lambda (IHn: (eq nat O (minus n0 -n0))).IHn)) n). -(* COMMENTS -Initial nodes: 41 -END *) - -theorem minus_Sn_m: - \forall (n: nat).(\forall (m: nat).((le m n) \to (eq nat (S (minus n m)) -(minus (S n) m)))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (Le: (le m n)).(le_elim_rel -(\lambda (n0: nat).(\lambda (n1: nat).(eq nat (S (minus n1 n0)) (minus (S n1) -n0)))) (\lambda (p: nat).(f_equal nat nat S (minus p O) p (sym_eq nat p -(minus p O) (minus_n_O p)))) (\lambda (p: nat).(\lambda (q: nat).(\lambda (_: -(le p q)).(\lambda (H0: (eq nat (S (minus q p)) (match p with [O \Rightarrow -(S q) | (S l) \Rightarrow (minus q l)]))).H0)))) m n Le))). -(* COMMENTS -Initial nodes: 111 -END *) - -theorem plus_minus: - \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((eq nat n (plus m p)) -\to (eq nat p (minus n m))))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(nat_double_ind -(\lambda (n0: nat).(\lambda (n1: nat).((eq nat n1 (plus n0 p)) \to (eq nat p -(minus n1 n0))))) (\lambda (n0: nat).(\lambda (H: (eq nat n0 p)).(eq_ind nat -n0 (\lambda (n1: nat).(eq nat p n1)) (sym_eq nat n0 p H) (minus n0 O) -(minus_n_O n0)))) (\lambda (n0: nat).(\lambda (H: (eq nat O (S (plus n0 -p)))).(False_ind (eq nat p O) (let H0 \def H in ((let H1 \def (O_S (plus n0 -p)) in (\lambda (H2: (eq nat O (S (plus n0 p)))).(H1 H2))) H0))))) (\lambda -(n0: nat).(\lambda (m0: nat).(\lambda (H: (((eq nat m0 (plus n0 p)) \to (eq -nat p (minus m0 n0))))).(\lambda (H0: (eq nat (S m0) (S (plus n0 p)))).(H -(eq_add_S m0 (plus n0 p) H0)))))) m n))). -(* COMMENTS -Initial nodes: 199 -END *) - -theorem minus_plus: - \forall (n: nat).(\forall (m: nat).(eq nat (minus (plus n m) n) m)) -\def - \lambda (n: nat).(\lambda (m: nat).(sym_eq nat m (minus (plus n m) n) -(plus_minus (plus n m) n m (refl_equal nat (plus n m))))). -(* COMMENTS -Initial nodes: 41 -END *) - -theorem le_pred_n: - \forall (n: nat).(le (pred n) n) -\def - \lambda (n: nat).(nat_ind (\lambda (n0: nat).(le (pred n0) n0)) (le_n O) -(\lambda (n0: nat).(\lambda (_: (le (pred n0) n0)).(le_S (pred (S n0)) n0 -(le_n n0)))) n). -(* COMMENTS -Initial nodes: 43 -END *) - -theorem le_plus_l: - \forall (n: nat).(\forall (m: nat).(le n (plus n m))) -\def - \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (m: nat).(le n0 (plus -n0 m)))) (\lambda (m: nat).(le_O_n m)) (\lambda (n0: nat).(\lambda (IHn: -((\forall (m: nat).(le n0 (plus n0 m))))).(\lambda (m: nat).(le_n_S n0 (plus -n0 m) (IHn m))))) n). -(* COMMENTS -Initial nodes: 55 -END *) - -theorem le_plus_r: - \forall (n: nat).(\forall (m: nat).(le m (plus n m))) -\def - \lambda (n: nat).(\lambda (m: nat).(nat_ind (\lambda (n0: nat).(le m (plus -n0 m))) (le_n m) (\lambda (n0: nat).(\lambda (H: (le m (plus n0 m))).(le_S m -(plus n0 m) H))) n)). -(* COMMENTS -Initial nodes: 47 -END *) - -theorem simpl_le_plus_l: - \forall (p: nat).(\forall (n: nat).(\forall (m: nat).((le (plus p n) (plus p -m)) \to (le n m)))) -\def - \lambda (p: nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).(\forall (m: -nat).((le (plus n n0) (plus n m)) \to (le n0 m))))) (\lambda (n: -nat).(\lambda (m: nat).(\lambda (H: (le n m)).H))) (\lambda (p0: -nat).(\lambda (IHp: ((\forall (n: nat).(\forall (m: nat).((le (plus p0 n) -(plus p0 m)) \to (le n m)))))).(\lambda (n: nat).(\lambda (m: nat).(\lambda -(H: (le (S (plus p0 n)) (S (plus p0 m)))).(IHp n m (le_S_n (plus p0 n) (plus -p0 m) H))))))) p). -(* COMMENTS -Initial nodes: 113 -END *) - -theorem le_plus_trans: - \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((le n m) \to (le n -(plus m p))))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (le n -m)).(le_trans n m (plus m p) H (le_plus_l m p))))). -(* COMMENTS -Initial nodes: 31 -END *) - -theorem le_reg_l: - \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((le n m) \to (le (plus -p n) (plus p m))))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(nat_ind (\lambda (n0: -nat).((le n m) \to (le (plus n0 n) (plus n0 m)))) (\lambda (H: (le n m)).H) -(\lambda (p0: nat).(\lambda (IHp: (((le n m) \to (le (plus p0 n) (plus p0 -m))))).(\lambda (H: (le n m)).(le_n_S (plus p0 n) (plus p0 m) (IHp H))))) -p))). -(* COMMENTS -Initial nodes: 85 -END *) - -theorem le_plus_plus: - \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(\forall (q: nat).((le -n m) \to ((le p q) \to (le (plus n p) (plus m q))))))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (q: -nat).(\lambda (H: (le n m)).(\lambda (H0: (le p q)).(le_ind n (\lambda (n0: -nat).(le (plus n p) (plus n0 q))) (le_reg_l p q n H0) (\lambda (m0: -nat).(\lambda (_: (le n m0)).(\lambda (H2: (le (plus n p) (plus m0 q))).(le_S -(plus n p) (plus m0 q) H2)))) m H)))))). -(* COMMENTS -Initial nodes: 91 -END *) - -theorem le_plus_minus: - \forall (n: nat).(\forall (m: nat).((le n m) \to (eq nat m (plus n (minus m -n))))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (Le: (le n m)).(le_elim_rel -(\lambda (n0: nat).(\lambda (n1: nat).(eq nat n1 (plus n0 (minus n1 n0))))) -(\lambda (p: nat).(minus_n_O p)) (\lambda (p: nat).(\lambda (q: nat).(\lambda -(_: (le p q)).(\lambda (H0: (eq nat q (plus p (minus q p)))).(f_equal nat nat -S q (plus p (minus q p)) H0))))) n m Le))). -(* COMMENTS -Initial nodes: 91 -END *) - -theorem le_plus_minus_r: - \forall (n: nat).(\forall (m: nat).((le n m) \to (eq nat (plus n (minus m -n)) m))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(sym_eq nat m -(plus n (minus m n)) (le_plus_minus n m H)))). -(* COMMENTS -Initial nodes: 33 -END *) - -theorem simpl_lt_plus_l: - \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((lt (plus p n) (plus p -m)) \to (lt n m)))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(nat_ind (\lambda (n0: -nat).((lt (plus n0 n) (plus n0 m)) \to (lt n m))) (\lambda (H: (lt n m)).H) -(\lambda (p0: nat).(\lambda (IHp: (((lt (plus p0 n) (plus p0 m)) \to (lt n -m)))).(\lambda (H: (lt (S (plus p0 n)) (S (plus p0 m)))).(IHp (le_S_n (S -(plus p0 n)) (plus p0 m) H))))) p))). -(* COMMENTS -Initial nodes: 99 -END *) - -theorem lt_reg_l: - \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((lt n m) \to (lt (plus -p n) (plus p m))))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(nat_ind (\lambda (n0: -nat).((lt n m) \to (lt (plus n0 n) (plus n0 m)))) (\lambda (H: (lt n m)).H) -(\lambda (p0: nat).(\lambda (IHp: (((lt n m) \to (lt (plus p0 n) (plus p0 -m))))).(\lambda (H: (lt n m)).(lt_n_S (plus p0 n) (plus p0 m) (IHp H))))) -p))). -(* COMMENTS -Initial nodes: 85 -END *) - -theorem lt_reg_r: - \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((lt n m) \to (lt (plus -n p) (plus m p))))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (lt n -m)).(eq_ind_r nat (plus p n) (\lambda (n0: nat).(lt n0 (plus m p))) (eq_ind_r -nat (plus p m) (\lambda (n0: nat).(lt (plus p n) n0)) (nat_ind (\lambda (n0: -nat).(lt (plus n0 n) (plus n0 m))) H (\lambda (n0: nat).(\lambda (_: (lt -(plus n0 n) (plus n0 m))).(lt_reg_l n m (S n0) H))) p) (plus m p) (plus_sym m -p)) (plus n p) (plus_sym n p))))). -(* COMMENTS -Initial nodes: 129 -END *) - -theorem le_lt_plus_plus: - \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(\forall (q: nat).((le -n m) \to ((lt p q) \to (lt (plus n p) (plus m q))))))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (q: -nat).(\lambda (H: (le n m)).(\lambda (H0: (le (S p) q)).(eq_ind_r nat (plus n -(S p)) (\lambda (n0: nat).(le n0 (plus m q))) (le_plus_plus n m (S p) q H H0) -(plus (S n) p) (plus_Snm_nSm n p))))))). -(* COMMENTS -Initial nodes: 75 -END *) - -theorem lt_le_plus_plus: - \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(\forall (q: nat).((lt -n m) \to ((le p q) \to (lt (plus n p) (plus m q))))))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (q: -nat).(\lambda (H: (le (S n) m)).(\lambda (H0: (le p q)).(le_plus_plus (S n) m -p q H H0)))))). -(* COMMENTS -Initial nodes: 37 -END *) - -theorem lt_plus_plus: - \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(\forall (q: nat).((lt -n m) \to ((lt p q) \to (lt (plus n p) (plus m q))))))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (q: -nat).(\lambda (H: (lt n m)).(\lambda (H0: (lt p q)).(lt_le_plus_plus n m p q -H (lt_le_weak p q H0))))))). -(* COMMENTS -Initial nodes: 39 -END *) - -theorem well_founded_ltof: - \forall (A: Set).(\forall (f: ((A \to nat))).(well_founded A (ltof A f))) -\def - \lambda (A: Set).(\lambda (f: ((A \to nat))).(let H \def (\lambda (n: -nat).(nat_ind (\lambda (n0: nat).(\forall (a: A).((lt (f a) n0) \to (Acc A -(ltof A f) a)))) (\lambda (a: A).(\lambda (H: (lt (f a) O)).(False_ind (Acc A -(ltof A f) a) (let H0 \def H in ((let H1 \def (lt_n_O (f a)) in (\lambda (H2: -(lt (f a) O)).(H1 H2))) H0))))) (\lambda (n0: nat).(\lambda (IHn: ((\forall -(a: A).((lt (f a) n0) \to (Acc A (ltof A f) a))))).(\lambda (a: A).(\lambda -(ltSma: (lt (f a) (S n0))).(Acc_intro A (ltof A f) a (\lambda (b: A).(\lambda -(ltfafb: (lt (f b) (f a))).(IHn b (lt_le_trans (f b) (f a) n0 ltfafb -(lt_n_Sm_le (f a) n0 ltSma)))))))))) n)) in (\lambda (a: A).(H (S (f a)) a -(le_n (S (f a))))))). -(* COMMENTS -Initial nodes: 189 -END *) - -theorem lt_wf: - well_founded nat lt -\def - well_founded_ltof nat (\lambda (m: nat).m). -(* COMMENTS -Initial nodes: 7 -END *) - -theorem lt_wf_ind: - \forall (p: nat).(\forall (P: ((nat \to Prop))).(((\forall (n: -nat).(((\forall (m: nat).((lt m n) \to (P m)))) \to (P n)))) \to (P p))) -\def - \lambda (p: nat).(\lambda (P: ((nat \to Prop))).(\lambda (H: ((\forall (n: -nat).(((\forall (m: nat).((lt m n) \to (P m)))) \to (P n))))).(Acc_ind nat lt -(\lambda (n: nat).(P n)) (\lambda (x: nat).(\lambda (_: ((\forall (y: -nat).((lt y x) \to (Acc nat lt y))))).(\lambda (H1: ((\forall (y: nat).((lt y -x) \to (P y))))).(H x H1)))) p (lt_wf p)))). -(* COMMENTS -Initial nodes: 77 -END *) - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Legacy-1/definitions.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Legacy-1/definitions.ma deleted file mode 100644 index 63fc85890..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Legacy-1/definitions.ma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Legacy-1/coq/defs.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Legacy-1/preamble.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Legacy-1/preamble.ma deleted file mode 100644 index 96c1bc1fa..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Legacy-1/preamble.ma +++ /dev/null @@ -1,15 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -inductive False: Prop \def . diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Legacy-1/spare.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Legacy-1/spare.ma deleted file mode 100644 index 77939a1b6..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Legacy-1/spare.ma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Legacy-1/theory.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Legacy-1/theory.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Legacy-1/theory.ma deleted file mode 100644 index 4ee597e09..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Legacy-1/theory.ma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Legacy-1/coq/props.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Legacy-2/coq/defs.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Legacy-2/coq/defs.mma deleted file mode 100644 index 175a39cdd..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Legacy-2/coq/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Legacy-2/preamble.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Legacy-2/coq/props.mma b/helm/software/matita/contribs/LAMBDA-TYPES/Legacy-2/coq/props.mma deleted file mode 100644 index e214da177..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Legacy-2/coq/props.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Legacy-2/coq/defs.ma". - -inline "Legacy-1/coq/props.ma" procedural. - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Legacy-2/preamble.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Legacy-2/preamble.ma deleted file mode 100644 index 00c38151d..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Legacy-2/preamble.ma +++ /dev/null @@ -1,62 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "Legacy-1/theory.ma". - -default "equality" - cic:/matita/LAMBDA-TYPES/Legacy-1/coq/defs/eq.ind - cic:/matita/LAMBDA-TYPES/Legacy-1/coq/props/sym_eq.con - cic:/matita/LAMBDA-TYPES/Legacy-1/coq/props/trans_eq.con - cic:/matita/LAMBDA-TYPES/Legacy-1/coq/defs/eq_ind.con - cic:/matita/LAMBDA-TYPES/Legacy-1/coq/props/eq_ind_r.con - cic:/matita/LAMBDA-TYPES/Legacy-1/coq/defs/eq_rec.con - cic:/matita/LAMBDA-TYPES/Legacy-1/coq/props/eq_rec_r.con - cic:/matita/LAMBDA-TYPES/Legacy-1/coq/defs/eq_rect.con - cic:/matita/LAMBDA-TYPES/Legacy-1/coq/props/eq_rect_r.con - cic:/matita/LAMBDA-TYPES/Legacy-1/coq/props/f_equal.con - cic:/matita/LAMBDA-TYPES/Legacy-2/preamble/f_equal_sym.con. - -default "true" - cic:/matita/LAMBDA-TYPES/Legacy-1/coq/defs/True.ind. -default "false" - cic:/matita/LAMBDA-TYPES/Legacy-1/preamble/False.ind. -default "absurd" - cic:/matita/LAMBDA-TYPES/Legacy-1/coq/props/absurd.con. - -interpretation "Coq 7.3.1 natural plus" 'plus x y = (cic:/matita/LAMBDA-TYPES/Legacy-1/coq/defs/plus.con x y). -interpretation "Coq 7.3.1 natural minus" 'minus x y = (cic:/matita/LAMBDA-TYPES/Legacy-1/coq/defs/minus.con x y). -interpretation "Coq 7.3.1 logical and" 'and x y = (cic:/matita/LAMBDA-TYPES/Legacy-1/coq/defs/land.ind#xpointer(1/1) x y). -interpretation "Coq 7.3.1 logical or" 'or x y = (cic:/matita/LAMBDA-TYPES/Legacy-1/coq/defs/or.ind#xpointer(1/1) x y). -interpretation "Coq 7.3.1 logical not" 'not x = (cic:/matita/LAMBDA-TYPES/Legacy-1/coq/defs/not.con x). -interpretation "Coq 7.3.1 exists" 'exists \eta.x = (cic:/matita/LAMBDA-TYPES/Legacy-1/coq/defs/ex.ind#xpointer(1/1) ? x). -interpretation "Coq 7.3.1 natural 'less or equal to'" 'leq x y = (cic:/matita/LAMBDA-TYPES/Legacy-1/coq/defs/le.ind#xpointer(1/1) x y). -interpretation "Coq 7.3.1 natural 'less than'" 'lt x y = (cic:/matita/LAMBDA-TYPES/Legacy-1/coq/defs/lt.con x y). -interpretation "Coq 7.3.1 leibnitz's equality" 'eq t x y = (cic:/matita/LAMBDA-TYPES/Legacy-1/coq/defs/eq.ind#xpointer(1/1) t x y). - -alias symbol "plus" = "Coq 7.3.1 natural plus". -alias symbol "minus" = "Coq 7.3.1 natural minus". -alias symbol "and" = "Coq 7.3.1 logical and". -alias symbol "or" = "Coq 7.3.1 logical or". -alias symbol "not" = "Coq 7.3.1 logical not". -alias symbol "exists" = "Coq 7.3.1 exists". -alias symbol "leq" = "Coq 7.3.1 natural 'less or equal to'". -alias symbol "lt" = "Coq 7.3.1 natural 'less than'". -alias symbol "eq" = "Coq 7.3.1 leibnitz's equality". - -theorem f_equal_sym: \forall A,B:Set.\forall f:A\to B.\forall x,y. - x = y \to (f y) = (f x). - intros; symmetry. - apply cic:/matita/LAMBDA-TYPES/Legacy-1/coq/props/f_equal.con. - assumption. -qed. diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Legacy-2/theory.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Legacy-2/theory.ma deleted file mode 100644 index 5cd562260..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Legacy-2/theory.ma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Legacy-2/coq/props.ma". - diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Makefile b/helm/software/matita/contribs/LAMBDA-TYPES/Makefile deleted file mode 100644 index 3e42a4ea1..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Makefile +++ /dev/null @@ -1,50 +0,0 @@ -include ../Makefile.defs - -DIR=$(shell basename $$PWD) - -H=@ - -MATITAOPTIONS=$(MATITAUSEROPTIONS) -onepass - -LOG = log.txt - -MMAS = $(shell find -name "*.mma") -MAS = $(MMAS:%.mma=%.ma) - -$(DIR) all: - $(H)$(RM) $(LOG) - $(H)$(BIN)matitac $(MATITAOPTIONS) 2>> $(LOG) -$(DIR).opt opt all.opt: - $(H)$(RM) $(LOG) - $(H)$(BIN)matitac.opt $(MATITAOPTIONS) 2>> $(LOG) - -%.ma %.mma: - $(H)$(RM) $(LOG) - $(H)$(BIN)matitac $(MATITAOPTIONS) $@ 2>> $(LOG) -%.ma.opt: - $(H)$(RM) $(LOG) - $(H)$(BIN)matitac.opt $(MATITAOPTIONS) $*.ma 2>> $(LOG) -%.mma.opt: - $(H)$(RM) $(LOG) - $(H)$(BIN)matitac.opt $(MATITAOPTIONS) $*.mma 2>> $(LOG) - -clean: - $(H)$(BIN)matitaclean $(MATITAOPTIONS) - $(H)$(RM) $(MAS) -clean.opt: - $(H)$(BIN)matitaclean.opt $(MATITAOPTIONS) - $(H)$(RM) $(MAS) - -depend: - $$(H)(BIN)matitadep $(MATITAOPTIONS) -depend.opt: - $(H)$(BIN)matitadep.opt $(MATITAOPTIONS) - -ifneq ($(strip $(MAS)),) -clean.ma: - $(H)$(BIN)matitaclean.opt $(MATITAOPTIONS) $(MAS) - $(H)$(RM) $(MAS) -else -clean.ma: - $(H)echo no files to clean -endif diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/depends b/helm/software/matita/contribs/LAMBDA-TYPES/depends deleted file mode 100644 index d205172eb..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/depends +++ /dev/null @@ -1,696 +0,0 @@ -Basic-2/clear/defs.ma Basic-2/clear/defs.mma -Basic-1/lift/defs.ma Basic-1/s/defs.ma Basic-1/tlist/defs.ma -Basic-1/subst0/dec.ma Basic-1/lift/props.ma Basic-1/subst0/defs.ma -Basic-2/sc3/props.mma Basic-1/sc3/props.ma Basic-2/arity/aprem.ma Basic-2/arity/lift1.ma Basic-2/csuba/arity.ma Basic-2/drop1/getl.ma Basic-2/drop1/props.ma Basic-2/lift1/props.ma Basic-2/llt/props.ma Basic-2/nf2/lift1.ma Basic-2/sc3/defs.ma Basic-2/sn3/lift1.ma -Basic-2/pr3/subst1.ma Basic-2/pr3/subst1.mma -Basic-2/csubt/csuba.mma Basic-1/csubt/csuba.ma Basic-2/ty3/arity.ma -Basic-2/sn3/props.mma Basic-1/sn3/props.ma Basic-2/nf2/iso.ma Basic-2/pr3/iso.ma Basic-2/sn3/fwd.ma Basic-2/sn3/nf2.ma -Basic-2/pr1/defs.ma Basic-2/pr1/defs.mma -Basic-2/arity/props.mma Basic-1/arity/props.ma Basic-2/arity/fwd.ma -Basic-2/flt/defs.mma Basic-2/C/defs.ma -Basic-2/clen/getl.ma Basic-2/clen/getl.mma -Basic-2/drop/defs.ma Basic-2/drop/defs.mma -Legacy-1/theory.ma Legacy-1/coq/props.ma -Basic-1/csubt/defs.ma Basic-1/ty3/defs.ma -Basic-1/clear/drop.ma Basic-1/clear/fwd.ma Basic-1/drop/fwd.ma -Basic-1/lift1/props.ma Basic-1/drop1/defs.ma Basic-1/lift/props.ma -Basic-1/nf2/lift1.ma Basic-1/drop1/fwd.ma Basic-1/nf2/props.ma -Basic-1/flt/defs.ma Basic-1/C/defs.ma -Basic-2/getl/props.ma Basic-2/getl/props.mma -Basic-1/pc3/subst1.ma Basic-1/pc3/props.ma Basic-1/pr3/subst1.ma -Basic-2/pr0/defs.mma Basic-2/subst0/defs.ma -Basic-1/drop1/fwd.ma Basic-1/drop1/defs.ma -Basic-2/arity/cimp.mma Basic-1/arity/cimp.ma Basic-2/arity/defs.ma Basic-2/cimp/props.ma -Basic-2/ty3/props.ma Basic-2/ty3/props.mma -Basic-2/drop1/getl.mma Basic-1/drop1/getl.ma Basic-2/drop1/fwd.ma Basic-2/getl/drop.ma -Basic-2/subst/defs.ma Basic-2/subst/defs.mma -Basic-2/lift1/defs.mma Basic-2/lift/defs.ma -Basic-2/A/defs.mma Basic-2/preamble.ma -Basic-2/csubt/clear.ma Basic-2/csubt/clear.mma -Basic-1/csuba/defs.ma Basic-1/arity/defs.ma -Basic-2/csuba/props.ma Basic-2/csuba/props.mma -Basic-1/csubc/fwd.ma Basic-1/csubc/defs.ma -Basic-2/pc3/wcpr0.mma Basic-1/pc3/wcpr0.ma Basic-2/pc3/props.ma Basic-2/wcpr0/getl.ma -Basic-2/subst/props.mma Basic-1/subst/props.ma Basic-2/lift/props.ma Basic-2/subst/fwd.ma Basic-2/subst0/defs.ma -Legacy-1/coq/defs.ma Legacy-1/preamble.ma -Basic-1/tlt/props.ma Basic-1/tlt/defs.ma -Basic-2/pr1/pr1.ma Basic-2/pr1/pr1.mma -Ground-1/ext/arith.ma Ground-1/preamble.ma -Basic-2/subst1/props.ma Basic-2/subst1/props.mma -Basic-2/llt/props.mma Basic-1/llt/props.ma Basic-2/leq/defs.ma Basic-2/llt/defs.ma -Basic-2/sn3/nf2.mma Basic-1/sn3/nf2.ma Basic-2/nf2/dec.ma Basic-2/nf2/pr3.ma Basic-2/sn3/defs.ma -Basic-1/pr3/pr3.ma Basic-1/pr2/pr2.ma Basic-1/pr3/props.ma -Basic-1/lift1/defs.ma Basic-1/lift/defs.ma -Basic-2/sty1/cnt.ma Basic-2/sty1/cnt.mma -Basic-1/wf3/props.ma Basic-1/app/defs.ma Basic-1/wf3/ty3.ma -Basic-2/wf3/getl.ma Basic-2/wf3/getl.mma -Basic-2/arity/subst0.ma Basic-2/arity/subst0.mma -Basic-2/pr0/props.mma Basic-1/pr0/props.ma Basic-2/pr0/defs.ma Basic-2/subst0/subst0.ma -Basic-2/csubt/csuba.ma Basic-2/csubt/csuba.mma -Basic-2/csubst0/clear.mma Basic-1/csubst0/clear.ma Basic-2/clear/fwd.ma Basic-2/csubst0/fwd.ma Basic-2/csubst0/props.ma -Basic-2/nf2/arity.mma Basic-1/nf2/arity.ma Basic-2/arity/subst0.ma Basic-2/nf2/fwd.ma -Basic-1/fsubst0/defs.ma Basic-1/csubst0/defs.ma -Basic-1/G/defs.ma Basic-1/preamble.ma -Basic-2/tlist/props.ma Basic-2/tlist/props.mma -Basic-2/subst0/props.ma Basic-2/subst0/props.mma -Basic-2/ex2/defs.mma Basic-2/C/defs.ma -Basic-1/ty3/defs.ma Basic-1/G/defs.ma Basic-1/pc3/defs.ma -Basic-2/clear/props.ma Basic-2/clear/props.mma -Basic-2/nf2/iso.ma Basic-2/nf2/iso.mma -Basic-1/theory.ma Basic-1/csubt/csuba.ma Basic-1/ex0/props.ma Basic-1/ex1/props.ma Basic-1/ex2/props.ma Basic-1/pr3/wcpr0.ma Basic-1/sty1/cnt.ma Basic-1/subst/props.ma Basic-1/subst0/tlt.ma Basic-1/ty3/fwd_nf2.ma Basic-1/ty3/nf2.ma Basic-1/ty3/sty0.ma Basic-1/wcpr0/fwd.ma Basic-1/wf3/props.ma -Basic-2/wf3/getl.mma Basic-1/wf3/getl.ma Basic-2/ty3/dec.ma Basic-2/wf3/clear.ma -Basic-2/csubst1/fwd.mma Basic-1/csubst1/fwd.ma Basic-2/csubst0/fwd.ma Basic-2/csubst1/defs.ma Basic-2/subst1/props.ma -Basic-2/app/defs.mma Basic-2/C/defs.ma -Basic-1/pr0/defs.ma Basic-1/subst0/defs.ma -Basic-1/nf2/defs.ma Basic-1/pr2/defs.ma -Basic-2/asucc/defs.mma Basic-2/A/defs.ma Basic-2/G/defs.ma -Basic-2/lift/fwd.ma Basic-2/lift/fwd.mma -Basic-2/csubst1/props.ma Basic-2/csubst1/props.mma -Basic-2/sty1/props.ma Basic-2/sty1/props.mma -Basic-2/ex1/defs.mma Basic-2/C/defs.ma -Basic-2/subst1/subst1.mma Basic-1/subst1/subst1.ma Basic-2/subst0/subst0.ma Basic-2/subst1/fwd.ma -Basic-2/pr2/props.mma Basic-1/pr2/props.ma Basic-2/getl/clear.ma Basic-2/getl/drop.ma Basic-2/pr0/props.ma Basic-2/pr2/defs.ma -Basic-2/arity/pr3.ma Basic-2/arity/pr3.mma -Basic-1/pr2/clen.ma Basic-1/clen/getl.ma Basic-1/pr2/props.ma -Basic-2/wcpr0/fwd.ma Basic-2/wcpr0/fwd.mma -Basic-2/ty3/dec.mma Basic-1/ty3/dec.ma Basic-2/getl/dec.ma Basic-2/getl/flt.ma Basic-2/pc3/dec.ma -Basic-2/sty0/props.mma Basic-1/sty0/props.ma Basic-2/getl/drop.ma Basic-2/sty0/defs.ma -Basic-2/T/props.mma Basic-1/T/props.ma Basic-2/T/defs.ma -Basic-1/wf3/defs.ma Basic-1/ty3/defs.ma -Basic-2/csubst0/props.ma Basic-2/csubst0/props.mma -Basic-2/pr0/dec.mma Basic-1/pr0/dec.ma Basic-2/T/dec.ma Basic-2/T/props.ma Basic-2/pr0/fwd.ma Basic-2/subst0/dec.ma -Basic-1/drop1/getl.ma Basic-1/drop1/fwd.ma Basic-1/getl/drop.ma -Basic-2/arity/props.ma Basic-2/arity/props.mma -Basic-2/C/defs.ma Basic-2/C/defs.mma -Basic-1/getl/flt.ma Basic-1/clear/props.ma Basic-1/flt/props.ma Basic-1/getl/fwd.ma -Basic-2/llt/props.ma Basic-2/llt/props.mma -Basic-2/drop/defs.mma Basic-2/C/defs.ma Basic-2/lift/defs.ma Basic-2/r/defs.ma -Basic-2/leq/asucc.mma Basic-1/leq/asucc.ma Basic-2/leq/props.ma -Basic-2/csubt/getl.ma Basic-2/csubt/getl.mma -Basic-2/clen/getl.mma Basic-1/clen/getl.ma Basic-2/clen/defs.ma Basic-2/getl/props.ma -Basic-2/sty0/props.ma Basic-2/sty0/props.mma -Basic-2/tlist/props.mma Basic-1/tlist/props.ma Basic-2/tlist/defs.ma -Ground-1/blt/defs.ma Ground-1/preamble.ma -Basic-1/sc3/defs.ma Basic-1/arity/defs.ma Basic-1/drop1/defs.ma Basic-1/sn3/defs.ma -Basic-2/ex0/defs.mma Basic-2/A/defs.ma Basic-2/G/defs.ma -Basic-2/nf2/dec.mma Basic-1/nf2/dec.ma Basic-2/C/props.ma Basic-2/nf2/defs.ma Basic-2/pr0/dec.ma Basic-2/pr2/clen.ma Basic-2/pr2/fwd.ma -Basic-1/subst/fwd.ma Basic-1/subst/defs.ma -Basic-2/iso/defs.ma Basic-2/iso/defs.mma -Basic-2/aplus/props.ma Basic-2/aplus/props.mma -Basic-2/s/props.mma Basic-1/s/props.ma Basic-2/s/defs.ma -Basic-2/pr2/subst1.mma Basic-1/pr2/subst1.ma Basic-2/csubst1/fwd.ma Basic-2/csubst1/getl.ma Basic-2/getl/drop.ma Basic-2/pr0/fwd.ma Basic-2/pr0/subst1.ma Basic-2/pr2/defs.ma Basic-2/subst1/subst1.ma -Basic-1/llt/defs.ma Basic-1/A/defs.ma -Basic-2/ty3/fsubst0.mma Basic-1/ty3/fsubst0.ma Basic-2/getl/getl.ma Basic-2/pc3/fsubst0.ma Basic-2/ty3/props.ma -Basic-2/pc3/defs.ma Basic-2/pc3/defs.mma -Basic-1/ty3/fsubst0.ma Basic-1/getl/getl.ma Basic-1/pc3/fsubst0.ma Basic-1/ty3/props.ma -Basic-1/subst1/props.ma Basic-1/subst0/props.ma Basic-1/subst1/defs.ma -Basic-1/wcpr0/defs.ma Basic-1/C/defs.ma Basic-1/pr0/defs.ma -Basic-2/csuba/getl.ma Basic-2/csuba/getl.mma -Basic-2/nf2/defs.mma Basic-2/pr2/defs.ma -Ground-1/preamble.ma Legacy-1/theory.ma -Basic-2/asucc/fwd.mma Basic-1/asucc/fwd.ma Basic-2/asucc/defs.ma -Basic-2/pr3/props.ma Basic-2/pr3/props.mma -Basic-2/nf2/props.mma Basic-1/nf2/props.ma Basic-2/nf2/defs.ma Basic-2/pr2/fwd.ma -Basic-1/s/props.ma Basic-1/s/defs.ma -Basic-1/tlist/defs.ma Basic-1/T/defs.ma -Basic-1/arity/subst0.ma Basic-1/arity/props.ma Basic-1/csubst0/getl.ma Basic-1/fsubst0/fwd.ma Basic-1/getl/getl.ma Basic-1/subst0/dec.ma Basic-1/subst0/fwd.ma -Basic-2/csubv/drop.ma Basic-2/csubv/drop.mma -Basic-2/nf2/iso.mma Basic-1/nf2/iso.ma Basic-2/iso/props.ma Basic-2/nf2/pr3.ma Basic-2/pr3/fwd.ma -Legacy-1/definitions.ma Legacy-1/coq/defs.ma -Basic-2/getl/defs.ma Basic-2/getl/defs.mma -Basic-2/cimp/defs.mma Basic-2/getl/defs.ma -Basic-2/wf3/clear.ma Basic-2/wf3/clear.mma -Basic-1/csubv/getl.ma Basic-1/csubv/clear.ma Basic-1/csubv/drop.ma Basic-1/getl/fwd.ma -Basic-2/clear/fwd.ma Basic-2/clear/fwd.mma -Basic-1/subst0/props.ma Basic-1/subst0/fwd.ma -Basic-1/cnt/props.ma Basic-1/cnt/defs.ma Basic-1/lift/fwd.ma -Basic-2/getl/fwd.ma Basic-2/getl/fwd.mma -Basic-1/r/props.ma Basic-1/r/defs.ma 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Basic-1/pr3/pr3.ma -Basic-1/csubc/defs.ma Basic-1/sc3/defs.ma -Basic-2/cnt/props.ma Basic-2/cnt/props.mma -Basic-2/pr0/pr0.mma Basic-1/pr0/pr0.ma Basic-2/lift/tlt.ma Basic-2/pr0/fwd.ma -Basic-2/pc3/nf2.mma Basic-1/pc3/nf2.ma Basic-2/nf2/pr3.ma Basic-2/pc3/defs.ma -Basic-1/ty3/sty0.ma Basic-1/sty0/fwd.ma Basic-1/ty3/pr3_props.ma -Basic-1/wf3/ty3.ma Basic-1/wf3/getl.ma -Basic-1/csubv/clear.ma Basic-1/clear/fwd.ma Basic-1/csubv/defs.ma -Basic-1/csubc/props.ma Basic-1/csubc/defs.ma Basic-1/sc3/props.ma -Basic-1/iso/props.ma Basic-1/iso/fwd.ma -Basic-2/csubc/defs.mma Basic-2/sc3/defs.ma -Basic-1/pr2/subst1.ma Basic-1/csubst1/fwd.ma Basic-1/csubst1/getl.ma Basic-1/getl/drop.ma Basic-1/pr0/fwd.ma Basic-1/pr0/subst1.ma Basic-1/pr2/defs.ma Basic-1/subst1/subst1.ma -Basic-2/lift/tlt.ma Basic-2/lift/tlt.mma -Basic-2/r/defs.mma Basic-2/T/defs.ma -Basic-2/drop1/getl.ma Basic-2/drop1/getl.mma -Basic-2/csubt/pc3.ma Basic-2/csubt/pc3.mma -Basic-2/sc3/arity.ma Basic-2/sc3/arity.mma -Basic-2/ex0/defs.ma 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-Basic-1/pc1/props.ma Basic-1/pc1/defs.ma Basic-1/pr1/pr1.ma -Basic-2/drop/fwd.mma Basic-1/drop/fwd.ma Basic-2/drop/defs.ma -Basic-2/getl/defs.mma Basic-2/clear/defs.ma Basic-2/drop/defs.ma -Basic-2/wcpr0/defs.ma Basic-2/wcpr0/defs.mma -Basic-1/pr0/dec.ma Basic-1/T/dec.ma Basic-1/T/props.ma Basic-1/pr0/fwd.ma Basic-1/subst0/dec.ma -Basic-1/pc1/defs.ma Basic-1/pr1/defs.ma -Basic-1/sn3/defs.ma Basic-1/pr3/defs.ma -Basic-2/csubst0/fwd.ma Basic-2/csubst0/fwd.mma -Basic-1/T/defs.ma Basic-1/preamble.ma -Basic-1/csubt/clear.ma Basic-1/clear/fwd.ma Basic-1/csubt/defs.ma -Basic-1/csuba/props.ma Basic-1/csuba/defs.ma -Basic-2/sn3/lift1.ma Basic-2/sn3/lift1.mma -Basic-2/T/dec.ma Basic-2/T/dec.mma -Basic-2/next_plus/props.mma Basic-1/next_plus/props.ma Basic-2/next_plus/defs.ma -Basic-2/csubv/getl.ma Basic-2/csubv/getl.mma -Basic-2/subst1/defs.ma Basic-2/subst1/defs.mma -Basic-2/csuba/defs.mma Basic-2/arity/defs.ma -Basic-2/tlist/defs.ma Basic-2/tlist/defs.mma -Basic-2/sn3/lift1.mma Basic-1/sn3/lift1.ma Basic-2/drop1/fwd.ma Basic-2/lift1/fwd.ma Basic-2/sn3/props.ma -Basic-2/arity/lift1.mma Basic-1/arity/lift1.ma Basic-2/arity/props.ma Basic-2/drop1/fwd.ma -Basic-2/ty3/defs.mma Basic-2/G/defs.ma Basic-2/pc3/defs.ma -Basic-2/sc3/props.ma Basic-2/sc3/props.mma -Basic-2/subst/fwd.mma Basic-1/subst/fwd.ma Basic-2/subst/defs.ma -Basic-2/subst1/defs.mma Basic-2/subst0/defs.ma -Basic-1/r/defs.ma Basic-1/T/defs.ma -Basic-1/csubt/csuba.ma Basic-1/ty3/arity.ma -Basic-2/ex2/props.mma Basic-1/ex2/props.ma Basic-2/arity/fwd.ma Basic-2/ex2/defs.ma Basic-2/nf2/defs.ma Basic-2/pr2/fwd.ma -Basic-2/lift/tlt.mma Basic-1/lift/tlt.ma Basic-2/lift/fwd.ma Basic-2/tlt/props.ma -Basic-2/getl/drop.ma Basic-2/getl/drop.mma -Basic-1/tlist/props.ma Basic-1/tlist/defs.ma -Basic-1/A/defs.ma Basic-1/preamble.ma -Basic-1/clear/props.ma Basic-1/clear/fwd.ma -Basic-2/tlist/defs.mma Basic-2/T/defs.ma -Basic-1/T/dec.ma Basic-1/T/defs.ma -Basic-1/app/defs.ma Basic-1/C/defs.ma -Basic-2/pr3/pr3.ma Basic-2/pr3/pr3.mma -Basic-2/csubc/getl.ma Basic-2/csubc/getl.mma -Basic-1/pr2/fwd.ma Basic-1/getl/clear.ma Basic-1/getl/drop.ma Basic-1/pr0/fwd.ma Basic-1/pr2/defs.ma -Basic-1/sn3/nf2.ma Basic-1/nf2/dec.ma Basic-1/nf2/pr3.ma Basic-1/sn3/defs.ma -Basic-2/csubt/fwd.ma Basic-2/csubt/fwd.mma -Basic-2/subst0/defs.mma Basic-2/lift/defs.ma -Basic-2/clear/defs.mma Basic-2/C/defs.ma -Basic-2/G/defs.mma Basic-2/preamble.ma -Basic-2/csubst1/defs.mma Basic-2/csubst0/defs.ma -Basic-1/ty3/nf2.ma Basic-1/nf2/arity.ma Basic-1/pc3/nf2.ma Basic-1/ty3/arity.ma -Basic-2/csubst0/defs.ma Basic-2/csubst0/defs.mma -Basic-1/csuba/fwd.ma Basic-1/csuba/defs.ma -Legacy-1/coq/props.ma Legacy-1/coq/defs.ma -Basic-2/sty1/defs.mma Basic-2/sty0/defs.ma -Basic-1/subst0/subst0.ma Basic-1/subst0/props.ma -Basic-1/arity/props.ma Basic-1/arity/fwd.ma -Basic-2/csubst1/fwd.ma Basic-2/csubst1/fwd.mma -Basic-2/csubv/getl.mma Basic-1/csubv/getl.ma Basic-2/csubv/clear.ma Basic-2/csubv/drop.ma Basic-2/getl/fwd.ma -Legacy-1/spare.ma Legacy-1/theory.ma -Basic-2/sc3/arity.mma Basic-1/sc3/arity.ma Basic-2/csubc/arity.ma Basic-2/csubc/drop1.ma Basic-2/csubc/getl.ma Basic-2/csubc/props.ma -Basic-1/subst0/fwd.ma Basic-1/lift/props.ma Basic-1/subst0/defs.ma -Basic-1/pc3/nf2.ma Basic-1/nf2/pr3.ma Basic-1/pc3/defs.ma -Basic-2/csubt/pc3.mma Basic-1/csubt/pc3.ma Basic-2/csubt/getl.ma Basic-2/pc3/left.ma -Basic-1/sn3/props.ma Basic-1/nf2/iso.ma Basic-1/pr3/iso.ma Basic-1/sn3/fwd.ma Basic-1/sn3/nf2.ma -Ground-2/blt/props.ma Ground-2/blt/props.mma -Basic-1/aplus/props.ma Basic-1/aplus/defs.ma Basic-1/next_plus/props.ma -Basic-2/pc3/left.ma Basic-2/pc3/left.mma -Basic-2/csubst0/defs.mma Basic-2/C/defs.ma Basic-2/subst0/defs.ma -Basic-2/sty0/defs.mma Basic-2/G/defs.ma Basic-2/getl/defs.ma -Basic-2/aprem/props.ma Basic-2/aprem/props.mma -Basic-2/arity/defs.mma Basic-2/getl/defs.ma Basic-2/leq/defs.ma -Basic-2/aplus/defs.mma Basic-2/asucc/defs.ma -Basic-2/llt/defs.mma Basic-2/A/defs.ma -Basic-2/csubc/arity.ma Basic-2/csubc/arity.mma -Ground-2/types/defs.ma Ground-2/types/defs.mma -Basic-2/iso/fwd.mma Basic-1/iso/fwd.ma Basic-2/iso/defs.ma Basic-2/tlist/defs.ma -Basic-1/T/props.ma Basic-1/T/defs.ma -Basic-2/pc3/fwd.mma Basic-1/pc3/fwd.ma Basic-2/pc3/props.ma Basic-2/pr3/fwd.ma -Basic-2/aprem/defs.ma Basic-2/aprem/defs.mma -Basic-1/sty0/fwd.ma Basic-1/sty0/defs.ma -Basic-2/subst0/dec.mma Basic-1/subst0/dec.ma Basic-2/lift/props.ma Basic-2/subst0/defs.ma -Basic-2/pr3/subst1.mma Basic-1/pr3/subst1.ma Basic-2/pr2/subst1.ma Basic-2/pr3/defs.ma -Basic-1/aplus/defs.ma Basic-1/asucc/defs.ma -Basic-2/csubst0/props.mma Basic-1/csubst0/props.ma Basic-2/csubst0/defs.ma -Basic-1/getl/getl.ma Basic-1/getl/clear.ma Basic-1/getl/drop.ma -Basic-2/sty1/defs.ma Basic-2/sty1/defs.mma -Basic-1/pr3/fwd.ma Basic-1/pr2/fwd.ma Basic-1/pr3/props.ma -Basic-2/arity/defs.ma Basic-2/arity/defs.mma -Basic-2/pr3/defs.mma Basic-2/pr2/defs.ma -Basic-1/lift/props.ma Basic-1/lift/fwd.ma Basic-1/s/props.ma -Basic-2/csubt/getl.mma Basic-1/csubt/getl.ma Basic-2/csubt/clear.ma Basic-2/csubt/drop.ma Basic-2/getl/clear.ma -Basic-1/leq/props.ma Basic-1/aplus/props.ma Basic-1/leq/fwd.ma -Basic-2/s/props.ma Basic-2/s/props.mma -Basic-2/drop1/props.mma Basic-1/drop1/props.ma Basic-2/drop/props.ma Basic-2/drop1/fwd.ma Basic-2/getl/defs.ma -Basic-2/arity/aprem.ma Basic-2/arity/aprem.mma -Basic-1/sty0/defs.ma Basic-1/G/defs.ma Basic-1/getl/defs.ma -Basic-2/getl/fwd.mma Basic-1/getl/fwd.ma Basic-2/clear/fwd.ma Basic-2/drop/fwd.ma Basic-2/getl/defs.ma -Basic-1/subst1/fwd.ma Basic-1/subst0/props.ma Basic-1/subst1/defs.ma -Basic-2/ty3/subst1.ma Basic-2/ty3/subst1.mma -Basic-2/pr2/defs.mma Basic-2/getl/defs.ma Basic-2/pr0/defs.ma -Ground-2/theory.ma Ground-2/blt/props.ma Ground-2/ext/arith.ma Ground-2/ext/tactics.ma Ground-2/plist/props.ma Ground-2/types/props.ma -Basic-1/preamble.ma Ground-1/theory.ma -Basic-2/csuba/arity.ma Basic-2/csuba/arity.mma -Basic-1/cimp/defs.ma Basic-1/getl/defs.ma -Basic-2/nf2/lift1.mma Basic-1/nf2/lift1.ma Basic-2/drop1/fwd.ma Basic-2/nf2/props.ma -Basic-1/ex2/defs.ma Basic-1/C/defs.ma -Basic-2/r/props.ma Basic-2/r/props.mma -Basic-1/csubst0/defs.ma Basic-1/C/defs.ma Basic-1/subst0/defs.ma -Basic-2/ty3/arity.ma Basic-2/ty3/arity.mma -Basic-2/sty1/cnt.mma Basic-1/sty1/cnt.ma Basic-2/cnt/props.ma Basic-2/sty1/props.ma -Basic-2/csubv/clear.ma Basic-2/csubv/clear.mma -Basic-2/fsubst0/defs.ma Basic-2/fsubst0/defs.mma -Basic-2/C/defs.mma Basic-2/T/defs.ma -Basic-2/csubc/props.ma Basic-2/csubc/props.mma -Basic-2/cnt/defs.ma Basic-2/cnt/defs.mma -Basic-2/arity/fwd.ma Basic-2/arity/fwd.mma -Basic-2/pr1/defs.mma Basic-2/pr0/defs.ma -Basic-1/clen/defs.ma Basic-1/C/defs.ma Basic-1/s/defs.ma -Basic-2/arity/subst0.mma Basic-1/arity/subst0.ma Basic-2/arity/props.ma Basic-2/csubst0/getl.ma Basic-2/fsubst0/fwd.ma Basic-2/getl/getl.ma Basic-2/subst0/dec.ma Basic-2/subst0/fwd.ma diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/root b/helm/software/matita/contribs/LAMBDA-TYPES/root deleted file mode 100644 index ca1729d65..000000000 --- a/helm/software/matita/contribs/LAMBDA-TYPES/root +++ /dev/null @@ -1 +0,0 @@ -baseuri=cic:/matita/LAMBDA-TYPES diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/A/defs.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/A/defs.ma deleted file mode 100644 index 2290c3de4..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/A/defs.ma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/preamble.ma". - -inductive A: Set \def -| ASort: nat \to (nat \to A) -| AHead: A \to (A \to A). - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/C/defs.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/C/defs.ma deleted file mode 100644 index cecc08267..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/C/defs.ma +++ /dev/null @@ -1,45 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/T/defs.ma". - -inductive C: Set \def -| CSort: nat \to C -| CHead: C \to (K \to (T \to C)). - -definition cweight: - C \to nat -\def - let rec cweight (c: C) on c: nat \def (match c with [(CSort _) \Rightarrow O -| (CHead c0 _ t) \Rightarrow (plus (cweight c0) (tweight t))]) in cweight. - -definition clt: - C \to (C \to Prop) -\def - \lambda (c1: C).(\lambda (c2: C).(lt (cweight c1) (cweight c2))). - -definition cle: - C \to (C \to Prop) -\def - \lambda (c1: C).(\lambda (c2: C).(le (cweight c1) (cweight c2))). - -definition CTail: - K \to (T \to (C \to C)) -\def - let rec CTail (k: K) (t: T) (c: C) on c: C \def (match c with [(CSort n) -\Rightarrow (CHead (CSort n) k t) | (CHead d h u) \Rightarrow (CHead (CTail k -t d) h u)]) in CTail. - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/C/props.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/C/props.ma deleted file mode 100644 index 878abb3bc..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/C/props.ma +++ /dev/null @@ -1,139 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/C/defs.ma". - -include "Basic-1/T/props.ma". - -theorem clt_cong: - \forall (c: C).(\forall (d: C).((clt c d) \to (\forall (k: K).(\forall (t: -T).(clt (CHead c k t) (CHead d k t)))))) -\def - \lambda (c: C).(\lambda (d: C).(\lambda (H: (lt (cweight c) (cweight -d))).(\lambda (_: K).(\lambda (t: T).(lt_reg_r (cweight c) (cweight d) -(tweight t) H))))). -(* COMMENTS -Initial nodes: 33 -END *) - -theorem clt_head: - \forall (k: K).(\forall (c: C).(\forall (u: T).(clt c (CHead c k u)))) -\def - \lambda (_: K).(\lambda (c: C).(\lambda (u: T).(eq_ind_r nat (plus (cweight -c) O) (\lambda (n: nat).(lt n (plus (cweight c) (tweight u)))) -(le_lt_plus_plus (cweight c) (cweight c) O (tweight u) (le_n (cweight c)) -(tweight_lt u)) (cweight c) (plus_n_O (cweight c))))). -(* COMMENTS -Initial nodes: 69 -END *) - -theorem clt_wf__q_ind: - \forall (P: ((C \to Prop))).(((\forall (n: nat).((\lambda (P0: ((C \to -Prop))).(\lambda (n0: nat).(\forall (c: C).((eq nat (cweight c) n0) \to (P0 -c))))) P n))) \to (\forall (c: C).(P c))) -\def - let Q \def (\lambda (P: ((C \to Prop))).(\lambda (n: nat).(\forall (c: -C).((eq nat (cweight c) n) \to (P c))))) in (\lambda (P: ((C \to -Prop))).(\lambda (H: ((\forall (n: nat).(\forall (c: C).((eq nat (cweight c) -n) \to (P c)))))).(\lambda (c: C).(H (cweight c) c (refl_equal nat (cweight -c)))))). -(* COMMENTS -Initial nodes: 61 -END *) - -theorem clt_wf_ind: - \forall (P: ((C \to Prop))).(((\forall (c: C).(((\forall (d: C).((clt d c) -\to (P d)))) \to (P c)))) \to (\forall (c: C).(P c))) -\def - let Q \def (\lambda (P: ((C \to Prop))).(\lambda (n: nat).(\forall (c: -C).((eq nat (cweight c) n) \to (P c))))) in (\lambda (P: ((C \to -Prop))).(\lambda (H: ((\forall (c: C).(((\forall (d: C).((lt (cweight d) -(cweight c)) \to (P d)))) \to (P c))))).(\lambda (c: C).(clt_wf__q_ind -(\lambda (c0: C).(P c0)) (\lambda (n: nat).(lt_wf_ind n (Q (\lambda (c0: -C).(P c0))) (\lambda (n0: nat).(\lambda (H0: ((\forall (m: nat).((lt m n0) -\to (Q (\lambda (c0: C).(P c0)) m))))).(\lambda (c0: C).(\lambda (H1: (eq nat -(cweight c0) n0)).(let H2 \def (eq_ind_r nat n0 (\lambda (n1: nat).(\forall -(m: nat).((lt m n1) \to (\forall (c1: C).((eq nat (cweight c1) m) \to (P -c1)))))) H0 (cweight c0) H1) in (H c0 (\lambda (d: C).(\lambda (H3: (lt -(cweight d) (cweight c0))).(H2 (cweight d) H3 d (refl_equal nat (cweight -d))))))))))))) c)))). -(* COMMENTS -Initial nodes: 179 -END *) - -theorem chead_ctail: - \forall (c: C).(\forall (t: T).(\forall (k: K).(ex_3 K C T (\lambda (h: -K).(\lambda (d: C).(\lambda (u: T).(eq C (CHead c k t) (CTail h u d)))))))) -\def - \lambda (c: C).(C_ind (\lambda (c0: C).(\forall (t: T).(\forall (k: K).(ex_3 -K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C (CHead c0 k t) -(CTail h u d))))))))) (\lambda (n: nat).(\lambda (t: T).(\lambda (k: -K).(ex_3_intro K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C -(CHead (CSort n) k t) (CTail h u d))))) k (CSort n) t (refl_equal C (CHead -(CSort n) k t)))))) (\lambda (c0: C).(\lambda (H: ((\forall (t: T).(\forall -(k: K).(ex_3 K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C -(CHead c0 k t) (CTail h u d)))))))))).(\lambda (k: K).(\lambda (t: -T).(\lambda (t0: T).(\lambda (k0: K).(let H_x \def (H t k) in (let H0 \def -H_x in (ex_3_ind K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C -(CHead c0 k t) (CTail h u d))))) (ex_3 K C T (\lambda (h: K).(\lambda (d: -C).(\lambda (u: T).(eq C (CHead (CHead c0 k t) k0 t0) (CTail h u d)))))) -(\lambda (x0: K).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H1: (eq C (CHead -c0 k t) (CTail x0 x2 x1))).(eq_ind_r C (CTail x0 x2 x1) (\lambda (c1: -C).(ex_3 K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C (CHead -c1 k0 t0) (CTail h u d))))))) (ex_3_intro K C T (\lambda (h: K).(\lambda (d: -C).(\lambda (u: T).(eq C (CHead (CTail x0 x2 x1) k0 t0) (CTail h u d))))) x0 -(CHead x1 k0 t0) x2 (refl_equal C (CHead (CTail x0 x2 x1) k0 t0))) (CHead c0 -k t) H1))))) H0))))))))) c). -(* COMMENTS -Initial nodes: 395 -END *) - -theorem clt_thead: - \forall (k: K).(\forall (u: T).(\forall (c: C).(clt c (CTail k u c)))) -\def - \lambda (k: K).(\lambda (u: T).(\lambda (c: C).(C_ind (\lambda (c0: C).(clt -c0 (CTail k u c0))) (\lambda (n: nat).(clt_head k (CSort n) u)) (\lambda (c0: -C).(\lambda (H: (clt c0 (CTail k u c0))).(\lambda (k0: K).(\lambda (t: -T).(clt_cong c0 (CTail k u c0) H k0 t))))) c))). -(* COMMENTS -Initial nodes: 71 -END *) - -theorem c_tail_ind: - \forall (P: ((C \to Prop))).(((\forall (n: nat).(P (CSort n)))) \to -(((\forall (c: C).((P c) \to (\forall (k: K).(\forall (t: T).(P (CTail k t -c))))))) \to (\forall (c: C).(P c)))) -\def - \lambda (P: ((C \to Prop))).(\lambda (H: ((\forall (n: nat).(P (CSort -n))))).(\lambda (H0: ((\forall (c: C).((P c) \to (\forall (k: K).(\forall (t: -T).(P (CTail k t c)))))))).(\lambda (c: C).(clt_wf_ind (\lambda (c0: C).(P -c0)) (\lambda (c0: C).(C_ind (\lambda (c1: C).(((\forall (d: C).((clt d c1) -\to (P d)))) \to (P c1))) (\lambda (n: nat).(\lambda (_: ((\forall (d: -C).((clt d (CSort n)) \to (P d))))).(H n))) (\lambda (c1: C).(\lambda (_: -((((\forall (d: C).((clt d c1) \to (P d)))) \to (P c1)))).(\lambda (k: -K).(\lambda (t: T).(\lambda (H2: ((\forall (d: C).((clt d (CHead c1 k t)) \to -(P d))))).(let H_x \def (chead_ctail c1 t k) in (let H3 \def H_x in (ex_3_ind -K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C (CHead c1 k t) -(CTail h u d))))) (P (CHead c1 k t)) (\lambda (x0: K).(\lambda (x1: -C).(\lambda (x2: T).(\lambda (H4: (eq C (CHead c1 k t) (CTail x0 x2 -x1))).(eq_ind_r C (CTail x0 x2 x1) (\lambda (c2: C).(P c2)) (let H5 \def -(eq_ind C (CHead c1 k t) (\lambda (c2: C).(\forall (d: C).((clt d c2) \to (P -d)))) H2 (CTail x0 x2 x1) H4) in (H0 x1 (H5 x1 (clt_thead x0 x2 x1)) x0 x2)) -(CHead c1 k t) H4))))) H3)))))))) c0)) c)))). -(* COMMENTS -Initial nodes: 295 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/G/defs.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/G/defs.ma deleted file mode 100644 index 23edf5789..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/G/defs.ma +++ /dev/null @@ -1,23 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/preamble.ma". - -record G : Set \def { - next: (nat \to nat); - next_lt: (\forall (n: nat).(lt n (next n))) -}. - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/T/dec.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/T/dec.ma deleted file mode 100644 index a088c40e3..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/T/dec.ma +++ /dev/null @@ -1,446 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/T/defs.ma". - -theorem terms_props__bind_dec: - \forall (b1: B).(\forall (b2: B).(or (eq B b1 b2) ((eq B b1 b2) \to (\forall -(P: Prop).P)))) -\def - \lambda (b1: B).(B_ind (\lambda (b: B).(\forall (b2: B).(or (eq B b b2) ((eq -B b b2) \to (\forall (P: Prop).P))))) (\lambda (b2: B).(B_ind (\lambda (b: -B).(or (eq B Abbr b) ((eq B Abbr b) \to (\forall (P: Prop).P)))) (or_introl -(eq B Abbr Abbr) ((eq B Abbr Abbr) \to (\forall (P: Prop).P)) (refl_equal B -Abbr)) (or_intror (eq B Abbr Abst) ((eq B Abbr Abst) \to (\forall (P: -Prop).P)) (\lambda (H: (eq B Abbr Abst)).(\lambda (P: Prop).(let H0 \def -(eq_ind B Abbr (\lambda (ee: B).(match ee in B return (\lambda (_: B).Prop) -with [Abbr \Rightarrow True | Abst \Rightarrow False | Void \Rightarrow -False])) I Abst H) in (False_ind P H0))))) (or_intror (eq B Abbr Void) ((eq B -Abbr Void) \to (\forall (P: Prop).P)) (\lambda (H: (eq B Abbr Void)).(\lambda -(P: Prop).(let H0 \def (eq_ind B Abbr (\lambda (ee: B).(match ee in B return -(\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | -Void \Rightarrow False])) I Void H) in (False_ind P H0))))) b2)) (\lambda -(b2: B).(B_ind (\lambda (b: B).(or (eq B Abst b) ((eq B Abst b) \to (\forall -(P: Prop).P)))) (or_intror (eq B Abst Abbr) ((eq B Abst Abbr) \to (\forall -(P: Prop).P)) (\lambda (H: (eq B Abst Abbr)).(\lambda (P: Prop).(let H0 \def -(eq_ind B Abst (\lambda (ee: B).(match ee in B return (\lambda (_: B).Prop) -with [Abbr \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow -False])) I Abbr H) in (False_ind P H0))))) (or_introl (eq B Abst Abst) ((eq B -Abst Abst) \to (\forall (P: Prop).P)) (refl_equal B Abst)) (or_intror (eq B -Abst Void) ((eq B Abst Void) \to (\forall (P: Prop).P)) (\lambda (H: (eq B -Abst Void)).(\lambda (P: Prop).(let H0 \def (eq_ind B Abst (\lambda (ee: -B).(match ee in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | -Abst \Rightarrow True | Void \Rightarrow False])) I Void H) in (False_ind P -H0))))) b2)) (\lambda (b2: B).(B_ind (\lambda (b: B).(or (eq B Void b) ((eq B -Void b) \to (\forall (P: Prop).P)))) (or_intror (eq B Void Abbr) ((eq B Void -Abbr) \to (\forall (P: Prop).P)) (\lambda (H: (eq B Void Abbr)).(\lambda (P: -Prop).(let H0 \def (eq_ind B Void (\lambda (ee: B).(match ee in B return -(\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False | -Void \Rightarrow True])) I Abbr H) in (False_ind P H0))))) (or_intror (eq B -Void Abst) ((eq B Void Abst) \to (\forall (P: Prop).P)) (\lambda (H: (eq B -Void Abst)).(\lambda (P: Prop).(let H0 \def (eq_ind B Void (\lambda (ee: -B).(match ee in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | -Abst \Rightarrow False | Void \Rightarrow True])) I Abst H) in (False_ind P -H0))))) (or_introl (eq B Void Void) ((eq B Void Void) \to (\forall (P: -Prop).P)) (refl_equal B Void)) b2)) b1). -(* COMMENTS -Initial nodes: 559 -END *) - -theorem bind_dec_not: - \forall (b1: B).(\forall (b2: B).(or (eq B b1 b2) (not (eq B b1 b2)))) -\def - \lambda (b1: B).(\lambda (b2: B).(let H_x \def (terms_props__bind_dec b1 b2) -in (let H \def H_x in (or_ind (eq B b1 b2) ((eq B b1 b2) \to (\forall (P: -Prop).P)) (or (eq B b1 b2) ((eq B b1 b2) \to False)) (\lambda (H0: (eq B b1 -b2)).(or_introl (eq B b1 b2) ((eq B b1 b2) \to False) H0)) (\lambda (H0: -(((eq B b1 b2) \to (\forall (P: Prop).P)))).(or_intror (eq B b1 b2) ((eq B b1 -b2) \to False) (\lambda (H1: (eq B b1 b2)).(H0 H1 False)))) H)))). -(* COMMENTS -Initial nodes: 131 -END *) - -theorem terms_props__flat_dec: - \forall (f1: F).(\forall (f2: F).(or (eq F f1 f2) ((eq F f1 f2) \to (\forall -(P: Prop).P)))) -\def - \lambda (f1: F).(F_ind (\lambda (f: F).(\forall (f2: F).(or (eq F f f2) ((eq -F f f2) \to (\forall (P: Prop).P))))) (\lambda (f2: F).(F_ind (\lambda (f: -F).(or (eq F Appl f) ((eq F Appl f) \to (\forall (P: Prop).P)))) (or_introl -(eq F Appl Appl) ((eq F Appl Appl) \to (\forall (P: Prop).P)) (refl_equal F -Appl)) (or_intror (eq F Appl Cast) ((eq F Appl Cast) \to (\forall (P: -Prop).P)) (\lambda (H: (eq F Appl Cast)).(\lambda (P: Prop).(let H0 \def -(eq_ind F Appl (\lambda (ee: F).(match ee in F return (\lambda (_: F).Prop) -with [Appl \Rightarrow True | Cast \Rightarrow False])) I Cast H) in -(False_ind P H0))))) f2)) (\lambda (f2: F).(F_ind (\lambda (f: F).(or (eq F -Cast f) ((eq F Cast f) \to (\forall (P: Prop).P)))) (or_intror (eq F Cast -Appl) ((eq F Cast Appl) \to (\forall (P: Prop).P)) (\lambda (H: (eq F Cast -Appl)).(\lambda (P: Prop).(let H0 \def (eq_ind F Cast (\lambda (ee: F).(match -ee in F return (\lambda (_: F).Prop) with [Appl \Rightarrow False | Cast -\Rightarrow True])) I Appl H) in (False_ind P H0))))) (or_introl (eq F Cast -Cast) ((eq F Cast Cast) \to (\forall (P: Prop).P)) (refl_equal F Cast)) f2)) -f1). -(* COMMENTS -Initial nodes: 263 -END *) - -theorem terms_props__kind_dec: - \forall (k1: K).(\forall (k2: K).(or (eq K k1 k2) ((eq K k1 k2) \to (\forall -(P: Prop).P)))) -\def - \lambda (k1: K).(K_ind (\lambda (k: K).(\forall (k2: K).(or (eq K k k2) ((eq -K k k2) \to (\forall (P: Prop).P))))) (\lambda (b: B).(\lambda (k2: K).(K_ind -(\lambda (k: K).(or (eq K (Bind b) k) ((eq K (Bind b) k) \to (\forall (P: -Prop).P)))) (\lambda (b0: B).(let H_x \def (terms_props__bind_dec b b0) in -(let H \def H_x in (or_ind (eq B b b0) ((eq B b b0) \to (\forall (P: -Prop).P)) (or (eq K (Bind b) (Bind b0)) ((eq K (Bind b) (Bind b0)) \to -(\forall (P: Prop).P))) (\lambda (H0: (eq B b b0)).(eq_ind B b (\lambda (b1: -B).(or (eq K (Bind b) (Bind b1)) ((eq K (Bind b) (Bind b1)) \to (\forall (P: -Prop).P)))) (or_introl (eq K (Bind b) (Bind b)) ((eq K (Bind b) (Bind b)) \to -(\forall (P: Prop).P)) (refl_equal K (Bind b))) b0 H0)) (\lambda (H0: (((eq B -b b0) \to (\forall (P: Prop).P)))).(or_intror (eq K (Bind b) (Bind b0)) ((eq -K (Bind b) (Bind b0)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq K (Bind b) -(Bind b0))).(\lambda (P: Prop).(let H2 \def (f_equal K B (\lambda (e: -K).(match e in K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | -(Flat _) \Rightarrow b])) (Bind b) (Bind b0) H1) in (let H3 \def (eq_ind_r B -b0 (\lambda (b1: B).((eq B b b1) \to (\forall (P0: Prop).P0))) H0 b H2) in -(H3 (refl_equal B b) P))))))) H)))) (\lambda (f: F).(or_intror (eq K (Bind b) -(Flat f)) ((eq K (Bind b) (Flat f)) \to (\forall (P: Prop).P)) (\lambda (H: -(eq K (Bind b) (Flat f))).(\lambda (P: Prop).(let H0 \def (eq_ind K (Bind b) -(\lambda (ee: K).(match ee in K return (\lambda (_: K).Prop) with [(Bind _) -\Rightarrow True | (Flat _) \Rightarrow False])) I (Flat f) H) in (False_ind -P H0)))))) k2))) (\lambda (f: F).(\lambda (k2: K).(K_ind (\lambda (k: K).(or -(eq K (Flat f) k) ((eq K (Flat f) k) \to (\forall (P: Prop).P)))) (\lambda -(b: B).(or_intror (eq K (Flat f) (Bind b)) ((eq K (Flat f) (Bind b)) \to -(\forall (P: Prop).P)) (\lambda (H: (eq K (Flat f) (Bind b))).(\lambda (P: -Prop).(let H0 \def (eq_ind K (Flat f) (\lambda (ee: K).(match ee in K return -(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])) I (Bind b) H) in (False_ind P H0)))))) (\lambda (f0: F).(let H_x \def -(terms_props__flat_dec f f0) in (let H \def H_x in (or_ind (eq F f f0) ((eq F -f f0) \to (\forall (P: Prop).P)) (or (eq K (Flat f) (Flat f0)) ((eq K (Flat -f) (Flat f0)) \to (\forall (P: Prop).P))) (\lambda (H0: (eq F f f0)).(eq_ind -F f (\lambda (f1: F).(or (eq K (Flat f) (Flat f1)) ((eq K (Flat f) (Flat f1)) -\to (\forall (P: Prop).P)))) (or_introl (eq K (Flat f) (Flat f)) ((eq K (Flat -f) (Flat f)) \to (\forall (P: Prop).P)) (refl_equal K (Flat f))) f0 H0)) -(\lambda (H0: (((eq F f f0) \to (\forall (P: Prop).P)))).(or_intror (eq K -(Flat f) (Flat f0)) ((eq K (Flat f) (Flat f0)) \to (\forall (P: Prop).P)) -(\lambda (H1: (eq K (Flat f) (Flat f0))).(\lambda (P: Prop).(let H2 \def -(f_equal K F (\lambda (e: K).(match e in K return (\lambda (_: K).F) with -[(Bind _) \Rightarrow f | (Flat f1) \Rightarrow f1])) (Flat f) (Flat f0) H1) -in (let H3 \def (eq_ind_r F f0 (\lambda (f1: F).((eq F f f1) \to (\forall -(P0: Prop).P0))) H0 f H2) in (H3 (refl_equal F f) P))))))) H)))) k2))) k1). -(* COMMENTS -Initial nodes: 799 -END *) - -theorem term_dec: - \forall (t1: T).(\forall (t2: T).(or (eq T t1 t2) ((eq T t1 t2) \to (\forall -(P: Prop).P)))) -\def - \lambda (t1: T).(T_ind (\lambda (t: T).(\forall (t2: T).(or (eq T t t2) ((eq -T t t2) \to (\forall (P: Prop).P))))) (\lambda (n: nat).(\lambda (t2: -T).(T_ind (\lambda (t: T).(or (eq T (TSort n) t) ((eq T (TSort n) t) \to -(\forall (P: Prop).P)))) (\lambda (n0: nat).(let H_x \def (nat_dec n n0) in -(let H \def H_x in (or_ind (eq nat n n0) ((eq nat n n0) \to (\forall (P: -Prop).P)) (or (eq T (TSort n) (TSort n0)) ((eq T (TSort n) (TSort n0)) \to -(\forall (P: Prop).P))) (\lambda (H0: (eq nat n n0)).(eq_ind nat n (\lambda -(n1: nat).(or (eq T (TSort n) (TSort n1)) ((eq T (TSort n) (TSort n1)) \to -(\forall (P: Prop).P)))) (or_introl (eq T (TSort n) (TSort n)) ((eq T (TSort -n) (TSort n)) \to (\forall (P: Prop).P)) (refl_equal T (TSort n))) n0 H0)) -(\lambda (H0: (((eq nat n n0) \to (\forall (P: Prop).P)))).(or_intror (eq T -(TSort n) (TSort n0)) ((eq T (TSort n) (TSort n0)) \to (\forall (P: Prop).P)) -(\lambda (H1: (eq T (TSort n) (TSort n0))).(\lambda (P: Prop).(let H2 \def -(f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with -[(TSort n1) \Rightarrow n1 | (TLRef _) \Rightarrow n | (THead _ _ _) -\Rightarrow n])) (TSort n) (TSort n0) H1) in (let H3 \def (eq_ind_r nat n0 -(\lambda (n1: nat).((eq nat n n1) \to (\forall (P0: Prop).P0))) H0 n H2) in -(H3 (refl_equal nat n) P))))))) H)))) (\lambda (n0: nat).(or_intror (eq T -(TSort n) (TLRef n0)) ((eq T (TSort n) (TLRef n0)) \to (\forall (P: Prop).P)) -(\lambda (H: (eq T (TSort n) (TLRef n0))).(\lambda (P: Prop).(let H0 \def -(eq_ind T (TSort n) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | -(THead _ _ _) \Rightarrow False])) I (TLRef n0) H) in (False_ind P H0)))))) -(\lambda (k: K).(\lambda (t: T).(\lambda (_: (or (eq T (TSort n) t) ((eq T -(TSort n) t) \to (\forall (P: Prop).P)))).(\lambda (t0: T).(\lambda (_: (or -(eq T (TSort n) t0) ((eq T (TSort n) t0) \to (\forall (P: -Prop).P)))).(or_intror (eq T (TSort n) (THead k t t0)) ((eq T (TSort n) -(THead k t t0)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq T (TSort n) -(THead k t t0))).(\lambda (P: Prop).(let H2 \def (eq_ind T (TSort n) (\lambda -(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow -False])) I (THead k t t0) H1) in (False_ind P H2)))))))))) t2))) (\lambda (n: -nat).(\lambda (t2: T).(T_ind (\lambda (t: T).(or (eq T (TLRef n) t) ((eq T -(TLRef n) t) \to (\forall (P: Prop).P)))) (\lambda (n0: nat).(or_intror (eq T -(TLRef n) (TSort n0)) ((eq T (TLRef n) (TSort n0)) \to (\forall (P: Prop).P)) -(\lambda (H: (eq T (TLRef n) (TSort n0))).(\lambda (P: Prop).(let H0 \def -(eq_ind T (TLRef n) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | -(THead _ _ _) \Rightarrow False])) I (TSort n0) H) in (False_ind P H0)))))) -(\lambda (n0: nat).(let H_x \def (nat_dec n n0) in (let H \def H_x in (or_ind -(eq nat n n0) ((eq nat n n0) \to (\forall (P: Prop).P)) (or (eq T (TLRef n) -(TLRef n0)) ((eq T (TLRef n) (TLRef n0)) \to (\forall (P: Prop).P))) (\lambda -(H0: (eq nat n n0)).(eq_ind nat n (\lambda (n1: nat).(or (eq T (TLRef n) -(TLRef n1)) ((eq T (TLRef n) (TLRef n1)) \to (\forall (P: Prop).P)))) -(or_introl (eq T (TLRef n) (TLRef n)) ((eq T (TLRef n) (TLRef n)) \to -(\forall (P: Prop).P)) (refl_equal T (TLRef n))) n0 H0)) (\lambda (H0: (((eq -nat n n0) \to (\forall (P: Prop).P)))).(or_intror (eq T (TLRef n) (TLRef n0)) -((eq T (TLRef n) (TLRef n0)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq T -(TLRef n) (TLRef n0))).(\lambda (P: Prop).(let H2 \def (f_equal T nat -(\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort _) -\Rightarrow n | (TLRef n1) \Rightarrow n1 | (THead _ _ _) \Rightarrow n])) -(TLRef n) (TLRef n0) H1) in (let H3 \def (eq_ind_r nat n0 (\lambda (n1: -nat).((eq nat n n1) \to (\forall (P0: Prop).P0))) H0 n H2) in (H3 (refl_equal -nat n) P))))))) H)))) (\lambda (k: K).(\lambda (t: T).(\lambda (_: (or (eq T -(TLRef n) t) ((eq T (TLRef n) t) \to (\forall (P: Prop).P)))).(\lambda (t0: -T).(\lambda (_: (or (eq T (TLRef n) t0) ((eq T (TLRef n) t0) \to (\forall (P: -Prop).P)))).(or_intror (eq T (TLRef n) (THead k t t0)) ((eq T (TLRef n) -(THead k t t0)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq T (TLRef n) -(THead k t t0))).(\lambda (P: Prop).(let H2 \def (eq_ind T (TLRef n) (\lambda -(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow -False])) I (THead k t t0) H1) in (False_ind P H2)))))))))) t2))) (\lambda (k: -K).(\lambda (t: T).(\lambda (H: ((\forall (t2: T).(or (eq T t t2) ((eq T t -t2) \to (\forall (P: Prop).P)))))).(\lambda (t0: T).(\lambda (H0: ((\forall -(t2: T).(or (eq T t0 t2) ((eq T t0 t2) \to (\forall (P: -Prop).P)))))).(\lambda (t2: T).(T_ind (\lambda (t3: T).(or (eq T (THead k t -t0) t3) ((eq T (THead k t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (n: -nat).(or_intror (eq T (THead k t t0) (TSort n)) ((eq T (THead k t t0) (TSort -n)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq T (THead k t t0) (TSort -n))).(\lambda (P: Prop).(let H2 \def (eq_ind T (THead k t t0) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I -(TSort n) H1) in (False_ind P H2)))))) (\lambda (n: nat).(or_intror (eq T -(THead k t t0) (TLRef n)) ((eq T (THead k t t0) (TLRef n)) \to (\forall (P: -Prop).P)) (\lambda (H1: (eq T (THead k t t0) (TLRef n))).(\lambda (P: -Prop).(let H2 \def (eq_ind T (THead k t t0) (\lambda (ee: T).(match ee in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H1) in -(False_ind P H2)))))) (\lambda (k0: K).(\lambda (t3: T).(\lambda (H1: (or (eq -T (THead k t t0) t3) ((eq T (THead k t t0) t3) \to (\forall (P: -Prop).P)))).(\lambda (t4: T).(\lambda (H2: (or (eq T (THead k t t0) t4) ((eq -T (THead k t t0) t4) \to (\forall (P: Prop).P)))).(let H_x \def (H t3) in -(let H3 \def H_x in (or_ind (eq T t t3) ((eq T t t3) \to (\forall (P: -Prop).P)) (or (eq T (THead k t t0) (THead k0 t3 t4)) ((eq T (THead k t t0) -(THead k0 t3 t4)) \to (\forall (P: Prop).P))) (\lambda (H4: (eq T t t3)).(let -H5 \def (eq_ind_r T t3 (\lambda (t5: T).(or (eq T (THead k t t0) t5) ((eq T -(THead k t t0) t5) \to (\forall (P: Prop).P)))) H1 t H4) in (eq_ind T t -(\lambda (t5: T).(or (eq T (THead k t t0) (THead k0 t5 t4)) ((eq T (THead k t -t0) (THead k0 t5 t4)) \to (\forall (P: Prop).P)))) (let H_x0 \def (H0 t4) in -(let H6 \def H_x0 in (or_ind (eq T t0 t4) ((eq T t0 t4) \to (\forall (P: -Prop).P)) (or (eq T (THead k t t0) (THead k0 t t4)) ((eq T (THead k t t0) -(THead k0 t t4)) \to (\forall (P: Prop).P))) (\lambda (H7: (eq T t0 t4)).(let -H8 \def (eq_ind_r T t4 (\lambda (t5: T).(or (eq T (THead k t t0) t5) ((eq T -(THead k t t0) t5) \to (\forall (P: Prop).P)))) H2 t0 H7) in (eq_ind T t0 -(\lambda (t5: T).(or (eq T (THead k t t0) (THead k0 t t5)) ((eq T (THead k t -t0) (THead k0 t t5)) \to (\forall (P: Prop).P)))) (let H_x1 \def -(terms_props__kind_dec k k0) in (let H9 \def H_x1 in (or_ind (eq K k k0) ((eq -K k k0) \to (\forall (P: Prop).P)) (or (eq T (THead k t t0) (THead k0 t t0)) -((eq T (THead k t t0) (THead k0 t t0)) \to (\forall (P: Prop).P))) (\lambda -(H10: (eq K k k0)).(eq_ind K k (\lambda (k1: K).(or (eq T (THead k t t0) -(THead k1 t t0)) ((eq T (THead k t t0) (THead k1 t t0)) \to (\forall (P: -Prop).P)))) (or_introl (eq T (THead k t t0) (THead k t t0)) ((eq T (THead k t -t0) (THead k t t0)) \to (\forall (P: Prop).P)) (refl_equal T (THead k t t0))) -k0 H10)) (\lambda (H10: (((eq K k k0) \to (\forall (P: Prop).P)))).(or_intror -(eq T (THead k t t0) (THead k0 t t0)) ((eq T (THead k t t0) (THead k0 t t0)) -\to (\forall (P: Prop).P)) (\lambda (H11: (eq T (THead k t t0) (THead k0 t -t0))).(\lambda (P: Prop).(let H12 \def (f_equal T K (\lambda (e: T).(match e -in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) -\Rightarrow k | (THead k1 _ _) \Rightarrow k1])) (THead k t t0) (THead k0 t -t0) H11) in (let H13 \def (eq_ind_r K k0 (\lambda (k1: K).((eq K k k1) \to -(\forall (P0: Prop).P0))) H10 k H12) in (H13 (refl_equal K k) P))))))) H9))) -t4 H7))) (\lambda (H7: (((eq T t0 t4) \to (\forall (P: Prop).P)))).(or_intror -(eq T (THead k t t0) (THead k0 t t4)) ((eq T (THead k t t0) (THead k0 t t4)) -\to (\forall (P: Prop).P)) (\lambda (H8: (eq T (THead k t t0) (THead k0 t -t4))).(\lambda (P: Prop).(let H9 \def (f_equal T K (\lambda (e: T).(match e -in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) -\Rightarrow k | (THead k1 _ _) \Rightarrow k1])) (THead k t t0) (THead k0 t -t4) H8) in ((let H10 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 -| (THead _ _ t5) \Rightarrow t5])) (THead k t t0) (THead k0 t t4) H8) in -(\lambda (_: (eq K k k0)).(let H12 \def (eq_ind_r T t4 (\lambda (t5: T).((eq -T t0 t5) \to (\forall (P0: Prop).P0))) H7 t0 H10) in (let H13 \def (eq_ind_r -T t4 (\lambda (t5: T).(or (eq T (THead k t t0) t5) ((eq T (THead k t t0) t5) -\to (\forall (P0: Prop).P0)))) H2 t0 H10) in (H12 (refl_equal T t0) P))))) -H9)))))) H6))) t3 H4))) (\lambda (H4: (((eq T t t3) \to (\forall (P: -Prop).P)))).(or_intror (eq T (THead k t t0) (THead k0 t3 t4)) ((eq T (THead k -t t0) (THead k0 t3 t4)) \to (\forall (P: Prop).P)) (\lambda (H5: (eq T (THead -k t t0) (THead k0 t3 t4))).(\lambda (P: Prop).(let H6 \def (f_equal T K -(\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) -\Rightarrow k | (TLRef _) \Rightarrow k | (THead k1 _ _) \Rightarrow k1])) -(THead k t t0) (THead k0 t3 t4) H5) in ((let H7 \def (f_equal T T (\lambda -(e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t -| (TLRef _) \Rightarrow t | (THead _ t5 _) \Rightarrow t5])) (THead k t t0) -(THead k0 t3 t4) H5) in ((let H8 \def (f_equal T T (\lambda (e: T).(match e -in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) -\Rightarrow t0 | (THead _ _ t5) \Rightarrow t5])) (THead k t t0) (THead k0 t3 -t4) H5) in (\lambda (H9: (eq T t t3)).(\lambda (_: (eq K k k0)).(let H11 \def -(eq_ind_r T t4 (\lambda (t5: T).(or (eq T (THead k t t0) t5) ((eq T (THead k -t t0) t5) \to (\forall (P0: Prop).P0)))) H2 t0 H8) in (let H12 \def (eq_ind_r -T t3 (\lambda (t5: T).((eq T t t5) \to (\forall (P0: Prop).P0))) H4 t H9) in -(let H13 \def (eq_ind_r T t3 (\lambda (t5: T).(or (eq T (THead k t t0) t5) -((eq T (THead k t t0) t5) \to (\forall (P0: Prop).P0)))) H1 t H9) in (H12 -(refl_equal T t) P))))))) H7)) H6)))))) H3)))))))) t2))))))) t1). -(* COMMENTS -Initial nodes: 2821 -END *) - -theorem binder_dec: - \forall (t: T).(or (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: -T).(eq T t (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall -(u: T).((eq T t (THead (Bind b) w u)) \to (\forall (P: Prop).P)))))) -\def - \lambda (t: T).(T_ind (\lambda (t0: T).(or (ex_3 B T T (\lambda (b: -B).(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w u)))))) -(\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t0 (THead (Bind b) w -u)) \to (\forall (P: Prop).P))))))) (\lambda (n: nat).(or_intror (ex_3 B T T -(\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T (TSort n) (THead (Bind -b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T (TSort n) -(THead (Bind b) w u)) \to (\forall (P: Prop).P))))) (\lambda (b: B).(\lambda -(w: T).(\lambda (u: T).(\lambda (H: (eq T (TSort n) (THead (Bind b) w -u))).(\lambda (P: Prop).(let H0 \def (eq_ind T (TSort n) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I -(THead (Bind b) w u) H) in (False_ind P H0))))))))) (\lambda (n: -nat).(or_intror (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: -T).(eq T (TLRef n) (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: -T).(\forall (u: T).((eq T (TLRef n) (THead (Bind b) w u)) \to (\forall (P: -Prop).P))))) (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(\lambda (H: (eq -T (TLRef n) (THead (Bind b) w u))).(\lambda (P: Prop).(let H0 \def (eq_ind T -(TLRef n) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) -\Rightarrow False])) I (THead (Bind b) w u) H) in (False_ind P H0))))))))) -(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t0: T).((or (ex_3 B T T -(\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w -u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t0 (THead -(Bind b) w u)) \to (\forall (P: Prop).P)))))) \to (\forall (t1: T).((or (ex_3 -B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t1 (THead (Bind -b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t1 (THead -(Bind b) w u)) \to (\forall (P: Prop).P)))))) \to (or (ex_3 B T T (\lambda -(b: B).(\lambda (w: T).(\lambda (u: T).(eq T (THead k0 t0 t1) (THead (Bind b) -w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T (THead k0 t0 -t1) (THead (Bind b) w u)) \to (\forall (P: Prop).P))))))))))) (\lambda (b: -B).(\lambda (t0: T).(\lambda (_: (or (ex_3 B T T (\lambda (b0: B).(\lambda -(w: T).(\lambda (u: T).(eq T t0 (THead (Bind b0) w u)))))) (\forall (b0: -B).(\forall (w: T).(\forall (u: T).((eq T t0 (THead (Bind b0) w u)) \to -(\forall (P: Prop).P))))))).(\lambda (t1: T).(\lambda (_: (or (ex_3 B T T -(\lambda (b0: B).(\lambda (w: T).(\lambda (u: T).(eq T t1 (THead (Bind b0) w -u)))))) (\forall (b0: B).(\forall (w: T).(\forall (u: T).((eq T t1 (THead -(Bind b0) w u)) \to (\forall (P: Prop).P))))))).(or_introl (ex_3 B T T -(\lambda (b0: B).(\lambda (w: T).(\lambda (u: T).(eq T (THead (Bind b) t0 t1) -(THead (Bind b0) w u)))))) (\forall (b0: B).(\forall (w: T).(\forall (u: -T).((eq T (THead (Bind b) t0 t1) (THead (Bind b0) w u)) \to (\forall (P: -Prop).P))))) (ex_3_intro B T T (\lambda (b0: B).(\lambda (w: T).(\lambda (u: -T).(eq T (THead (Bind b) t0 t1) (THead (Bind b0) w u))))) b t0 t1 (refl_equal -T (THead (Bind b) t0 t1))))))))) (\lambda (f: F).(\lambda (t0: T).(\lambda -(_: (or (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t0 -(THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: -T).((eq T t0 (THead (Bind b) w u)) \to (\forall (P: Prop).P))))))).(\lambda -(t1: T).(\lambda (_: (or (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda -(u: T).(eq T t1 (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: -T).(\forall (u: T).((eq T t1 (THead (Bind b) w u)) \to (\forall (P: -Prop).P))))))).(or_intror (ex_3 B T T (\lambda (b: B).(\lambda (w: -T).(\lambda (u: T).(eq T (THead (Flat f) t0 t1) (THead (Bind b) w u)))))) -(\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T (THead (Flat f) t0 t1) -(THead (Bind b) w u)) \to (\forall (P: Prop).P))))) (\lambda (b: B).(\lambda -(w: T).(\lambda (u: T).(\lambda (H1: (eq T (THead (Flat f) t0 t1) (THead -(Bind b) w u))).(\lambda (P: Prop).(let H2 \def (eq_ind T (THead (Flat f) t0 -t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort -_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _) -\Rightarrow (match k0 in K return (\lambda (_: K).Prop) with [(Bind _) -\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) w u) H1) -in (False_ind P H2))))))))))))) k)) t). -(* COMMENTS -Initial nodes: 1063 -END *) - -theorem abst_dec: - \forall (u: T).(\forall (v: T).(or (ex T (\lambda (t: T).(eq T u (THead -(Bind Abst) v t)))) (\forall (t: T).((eq T u (THead (Bind Abst) v t)) \to -(\forall (P: Prop).P))))) -\def - \lambda (u: T).(T_ind (\lambda (t: T).(\forall (v: T).(or (ex T (\lambda -(t0: T).(eq T t (THead (Bind Abst) v t0)))) (\forall (t0: T).((eq T t (THead -(Bind Abst) v t0)) \to (\forall (P: Prop).P)))))) (\lambda (n: nat).(\lambda -(v: T).(or_intror (ex T (\lambda (t: T).(eq T (TSort n) (THead (Bind Abst) v -t)))) (\forall (t: T).((eq T (TSort n) (THead (Bind Abst) v t)) \to (\forall -(P: Prop).P))) (\lambda (t: T).(\lambda (H: (eq T (TSort n) (THead (Bind -Abst) v t))).(\lambda (P: Prop).(let H0 \def (eq_ind T (TSort n) (\lambda -(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow -False])) I (THead (Bind Abst) v t) H) in (False_ind P H0)))))))) (\lambda (n: -nat).(\lambda (v: T).(or_intror (ex T (\lambda (t: T).(eq T (TLRef n) (THead -(Bind Abst) v t)))) (\forall (t: T).((eq T (TLRef n) (THead (Bind Abst) v t)) -\to (\forall (P: Prop).P))) (\lambda (t: T).(\lambda (H: (eq T (TLRef n) -(THead (Bind Abst) v t))).(\lambda (P: Prop).(let H0 \def (eq_ind T (TLRef n) -(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow -False])) I (THead (Bind Abst) v t) H) in (False_ind P H0)))))))) (\lambda (k: -K).(\lambda (t: T).(\lambda (_: ((\forall (v: T).(or (ex T (\lambda (t0: -T).(eq T t (THead (Bind Abst) v t0)))) (\forall (t0: T).((eq T t (THead (Bind -Abst) v t0)) \to (\forall (P: Prop).P))))))).(\lambda (t0: T).(\lambda (_: -((\forall (v: T).(or (ex T (\lambda (t1: T).(eq T t0 (THead (Bind Abst) v -t1)))) (\forall (t1: T).((eq T t0 (THead (Bind Abst) v t1)) \to (\forall (P: -Prop).P))))))).(\lambda (v: T).(let H_x \def (terms_props__kind_dec k (Bind -Abst)) in (let H1 \def H_x in (or_ind (eq K k (Bind Abst)) ((eq K k (Bind -Abst)) \to (\forall (P: Prop).P)) (or (ex T (\lambda (t1: T).(eq T (THead k t -t0) (THead (Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead k t t0) (THead -(Bind Abst) v t1)) \to (\forall (P: Prop).P)))) (\lambda (H2: (eq K k (Bind -Abst))).(eq_ind_r K (Bind Abst) (\lambda (k0: K).(or (ex T (\lambda (t1: -T).(eq T (THead k0 t t0) (THead (Bind Abst) v t1)))) (\forall (t1: T).((eq T -(THead k0 t t0) (THead (Bind Abst) v t1)) \to (\forall (P: Prop).P))))) (let -H_x0 \def (term_dec t v) in (let H3 \def H_x0 in (or_ind (eq T t v) ((eq T t -v) \to (\forall (P: Prop).P)) (or (ex T (\lambda (t1: T).(eq T (THead (Bind -Abst) t t0) (THead (Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead (Bind -Abst) t t0) (THead (Bind Abst) v t1)) \to (\forall (P: Prop).P)))) (\lambda -(H4: (eq T t v)).(eq_ind T t (\lambda (t1: T).(or (ex T (\lambda (t2: T).(eq -T (THead (Bind Abst) t t0) (THead (Bind Abst) t1 t2)))) (\forall (t2: T).((eq -T (THead (Bind Abst) t t0) (THead (Bind Abst) t1 t2)) \to (\forall (P: -Prop).P))))) (or_introl (ex T (\lambda (t1: T).(eq T (THead (Bind Abst) t t0) -(THead (Bind Abst) t t1)))) (\forall (t1: T).((eq T (THead (Bind Abst) t t0) -(THead (Bind Abst) t t1)) \to (\forall (P: Prop).P))) (ex_intro T (\lambda -(t1: T).(eq T (THead (Bind Abst) t t0) (THead (Bind Abst) t t1))) t0 -(refl_equal T (THead (Bind Abst) t t0)))) v H4)) (\lambda (H4: (((eq T t v) -\to (\forall (P: Prop).P)))).(or_intror (ex T (\lambda (t1: T).(eq T (THead -(Bind Abst) t t0) (THead (Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead -(Bind Abst) t t0) (THead (Bind Abst) v t1)) \to (\forall (P: Prop).P))) -(\lambda (t1: T).(\lambda (H5: (eq T (THead (Bind Abst) t t0) (THead (Bind -Abst) v t1))).(\lambda (P: Prop).(let H6 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t | -(TLRef _) \Rightarrow t | (THead _ t2 _) \Rightarrow t2])) (THead (Bind Abst) -t t0) (THead (Bind Abst) v t1) H5) in ((let H7 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | -(TLRef _) \Rightarrow t0 | (THead _ _ t2) \Rightarrow t2])) (THead (Bind -Abst) t t0) (THead (Bind Abst) v t1) H5) in (\lambda (H8: (eq T t v)).(H4 H8 -P))) H6))))))) H3))) k H2)) (\lambda (H2: (((eq K k (Bind Abst)) \to (\forall -(P: Prop).P)))).(or_intror (ex T (\lambda (t1: T).(eq T (THead k t t0) (THead -(Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead k t t0) (THead (Bind -Abst) v t1)) \to (\forall (P: Prop).P))) (\lambda (t1: T).(\lambda (H3: (eq T -(THead k t t0) (THead (Bind Abst) v t1))).(\lambda (P: Prop).(let H4 \def -(f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with -[(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) -\Rightarrow k0])) (THead k t t0) (THead (Bind Abst) v t1) H3) in ((let H5 -\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ t2 _) -\Rightarrow t2])) (THead k t t0) (THead (Bind Abst) v t1) H3) in ((let H6 -\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t2) -\Rightarrow t2])) (THead k t t0) (THead (Bind Abst) v t1) H3) in (\lambda (_: -(eq T t v)).(\lambda (H8: (eq K k (Bind Abst))).(H2 H8 P)))) H5)) H4))))))) -H1))))))))) u). -(* COMMENTS -Initial nodes: 1305 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/T/defs.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/T/defs.ma deleted file mode 100644 index 6ddbe6d0d..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/T/defs.ma +++ /dev/null @@ -1,43 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/preamble.ma". - -inductive B: Set \def -| Abbr: B -| Abst: B -| Void: B. - -inductive F: Set \def -| Appl: F -| Cast: F. - -inductive K: Set \def -| Bind: B \to K -| Flat: F \to K. - -inductive T: Set \def -| TSort: nat \to T -| TLRef: nat \to T -| THead: K \to (T \to (T \to T)). - -definition tweight: - T \to nat -\def - let rec tweight (t: T) on t: nat \def (match t with [(TSort _) \Rightarrow -(S O) | (TLRef _) \Rightarrow (S O) | (THead _ u t0) \Rightarrow (S (plus -(tweight u) (tweight t0)))]) in tweight. - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/T/props.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/T/props.ma deleted file mode 100644 index faa9ed95d..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/T/props.ma +++ /dev/null @@ -1,111 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/T/defs.ma". - -theorem not_abbr_abst: - not (eq B Abbr Abst) -\def - \lambda (H: (eq B Abbr Abst)).(let H0 \def (eq_ind B Abbr (\lambda (ee: -B).(match ee in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow True | -Abst \Rightarrow False | Void \Rightarrow False])) I Abst H) in (False_ind -False H0)). -(* COMMENTS -Initial nodes: 34 -END *) - -theorem not_void_abst: - not (eq B Void Abst) -\def - \lambda (H: (eq B Void Abst)).(let H0 \def (eq_ind B Void (\lambda (ee: -B).(match ee in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | -Abst \Rightarrow False | Void \Rightarrow True])) I Abst H) in (False_ind -False H0)). -(* COMMENTS -Initial nodes: 34 -END *) - -theorem not_abbr_void: - not (eq B Abbr Void) -\def - \lambda (H: (eq B Abbr Void)).(let H0 \def (eq_ind B Abbr (\lambda (ee: -B).(match ee in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow True | -Abst \Rightarrow False | Void \Rightarrow False])) I Void H) in (False_ind -False H0)). -(* COMMENTS -Initial nodes: 34 -END *) - -theorem not_abst_void: - not (eq B Abst Void) -\def - \lambda (H: (eq B Abst Void)).(let H0 \def (eq_ind B Abst (\lambda (ee: -B).(match ee in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | -Abst \Rightarrow True | Void \Rightarrow False])) I Void H) in (False_ind -False H0)). -(* COMMENTS -Initial nodes: 34 -END *) - -theorem thead_x_y_y: - \forall (k: K).(\forall (v: T).(\forall (t: T).((eq T (THead k v t) t) \to -(\forall (P: Prop).P)))) -\def - \lambda (k: K).(\lambda (v: T).(\lambda (t: T).(T_ind (\lambda (t0: T).((eq -T (THead k v t0) t0) \to (\forall (P: Prop).P))) (\lambda (n: nat).(\lambda -(H: (eq T (THead k v (TSort n)) (TSort n))).(\lambda (P: Prop).(let H0 \def -(eq_ind T (THead k v (TSort n)) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H) in -(False_ind P H0))))) (\lambda (n: nat).(\lambda (H: (eq T (THead k v (TLRef -n)) (TLRef n))).(\lambda (P: Prop).(let H0 \def (eq_ind T (THead k v (TLRef -n)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort -_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) -\Rightarrow True])) I (TLRef n) H) in (False_ind P H0))))) (\lambda (k0: -K).(\lambda (t0: T).(\lambda (_: (((eq T (THead k v t0) t0) \to (\forall (P: -Prop).P)))).(\lambda (t1: T).(\lambda (H0: (((eq T (THead k v t1) t1) \to -(\forall (P: Prop).P)))).(\lambda (H1: (eq T (THead k v (THead k0 t0 t1)) -(THead k0 t0 t1))).(\lambda (P: Prop).(let H2 \def (f_equal T K (\lambda (e: -T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | -(TLRef _) \Rightarrow k | (THead k1 _ _) \Rightarrow k1])) (THead k v (THead -k0 t0 t1)) (THead k0 t0 t1) H1) in ((let H3 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v | -(TLRef _) \Rightarrow v | (THead _ t2 _) \Rightarrow t2])) (THead k v (THead -k0 t0 t1)) (THead k0 t0 t1) H1) in ((let H4 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow (THead -k0 t0 t1) | (TLRef _) \Rightarrow (THead k0 t0 t1) | (THead _ _ t2) -\Rightarrow t2])) (THead k v (THead k0 t0 t1)) (THead k0 t0 t1) H1) in -(\lambda (H5: (eq T v t0)).(\lambda (H6: (eq K k k0)).(let H7 \def (eq_ind T -v (\lambda (t2: T).((eq T (THead k t2 t1) t1) \to (\forall (P0: Prop).P0))) -H0 t0 H5) in (let H8 \def (eq_ind K k (\lambda (k1: K).((eq T (THead k1 t0 -t1) t1) \to (\forall (P0: Prop).P0))) H7 k0 H6) in (H8 H4 P)))))) H3)) -H2))))))))) t))). -(* COMMENTS -Initial nodes: 461 -END *) - -theorem tweight_lt: - \forall (t: T).(lt O (tweight t)) -\def - \lambda (t: T).(T_ind (\lambda (t0: T).(lt O (tweight t0))) (\lambda (_: -nat).(le_n (S O))) (\lambda (_: nat).(le_n (S O))) (\lambda (_: K).(\lambda -(t0: T).(\lambda (H: (lt O (tweight t0))).(\lambda (t1: T).(\lambda (_: (lt O -(tweight t1))).(le_S (S O) (plus (tweight t0) (tweight t1)) (le_plus_trans (S -O) (tweight t0) (tweight t1) H))))))) t). -(* COMMENTS -Initial nodes: 85 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/aplus/defs.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/aplus/defs.ma deleted file mode 100644 index 4095b163b..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/aplus/defs.ma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/asucc/defs.ma". - -definition aplus: - G \to (A \to (nat \to A)) -\def - let rec aplus (g: G) (a: A) (n: nat) on n: A \def (match n with [O -\Rightarrow a | (S n0) \Rightarrow (asucc g (aplus g a n0))]) in aplus. - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/aplus/props.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/aplus/props.ma deleted file mode 100644 index 94bb9a069..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/aplus/props.ma +++ /dev/null @@ -1,282 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/aplus/defs.ma". - -include "Basic-1/next_plus/props.ma". - -theorem aplus_reg_r: - \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (h1: nat).(\forall -(h2: nat).((eq A (aplus g a1 h1) (aplus g a2 h2)) \to (\forall (h: nat).(eq A -(aplus g a1 (plus h h1)) (aplus g a2 (plus h h2))))))))) -\def - \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (h1: nat).(\lambda -(h2: nat).(\lambda (H: (eq A (aplus g a1 h1) (aplus g a2 h2))).(\lambda (h: -nat).(nat_ind (\lambda (n: nat).(eq A (aplus g a1 (plus n h1)) (aplus g a2 -(plus n h2)))) H (\lambda (n: nat).(\lambda (H0: (eq A (aplus g a1 (plus n -h1)) (aplus g a2 (plus n h2)))).(f_equal2 G A A asucc g g (aplus g a1 (plus n -h1)) (aplus g a2 (plus n h2)) (refl_equal G g) H0))) h))))))). -(* COMMENTS -Initial nodes: 143 -END *) - -theorem aplus_assoc: - \forall (g: G).(\forall (a: A).(\forall (h1: nat).(\forall (h2: nat).(eq A -(aplus g (aplus g a h1) h2) (aplus g a (plus h1 h2)))))) -\def - \lambda (g: G).(\lambda (a: A).(\lambda (h1: nat).(nat_ind (\lambda (n: -nat).(\forall (h2: nat).(eq A (aplus g (aplus g a n) h2) (aplus g a (plus n -h2))))) (\lambda (h2: nat).(refl_equal A (aplus g a h2))) (\lambda (n: -nat).(\lambda (_: ((\forall (h2: nat).(eq A (aplus g (aplus g a n) h2) (aplus -g a (plus n h2)))))).(\lambda (h2: nat).(nat_ind (\lambda (n0: nat).(eq A -(aplus g (asucc g (aplus g a n)) n0) (asucc g (aplus g a (plus n n0))))) -(eq_ind nat n (\lambda (n0: nat).(eq A (asucc g (aplus g a n)) (asucc g -(aplus g a n0)))) (refl_equal A (asucc g (aplus g a n))) (plus n O) (plus_n_O -n)) (\lambda (n0: nat).(\lambda (H0: (eq A (aplus g (asucc g (aplus g a n)) -n0) (asucc g (aplus g a (plus n n0))))).(eq_ind nat (S (plus n n0)) (\lambda -(n1: nat).(eq A (asucc g (aplus g (asucc g (aplus g a n)) n0)) (asucc g -(aplus g a n1)))) (f_equal2 G A A asucc g g (aplus g (asucc g (aplus g a n)) -n0) (asucc g (aplus g a (plus n n0))) (refl_equal G g) H0) (plus n (S n0)) -(plus_n_Sm n n0)))) h2)))) h1))). -(* COMMENTS -Initial nodes: 361 -END *) - -theorem aplus_asucc: - \forall (g: G).(\forall (h: nat).(\forall (a: A).(eq A (aplus g (asucc g a) -h) (asucc g (aplus g a h))))) -\def - \lambda (g: G).(\lambda (h: nat).(\lambda (a: A).(eq_ind_r A (aplus g a -(plus (S O) h)) (\lambda (a0: A).(eq A a0 (asucc g (aplus g a h)))) -(refl_equal A (asucc g (aplus g a h))) (aplus g (aplus g a (S O)) h) -(aplus_assoc g a (S O) h)))). -(* COMMENTS -Initial nodes: 87 -END *) - -theorem aplus_sort_O_S_simpl: - \forall (g: G).(\forall (n: nat).(\forall (k: nat).(eq A (aplus g (ASort O -n) (S k)) (aplus g (ASort O (next g n)) k)))) -\def - \lambda (g: G).(\lambda (n: nat).(\lambda (k: nat).(eq_ind A (aplus g (asucc -g (ASort O n)) k) (\lambda (a: A).(eq A a (aplus g (ASort O (next g n)) k))) -(refl_equal A (aplus g (ASort O (next g n)) k)) (asucc g (aplus g (ASort O n) -k)) (aplus_asucc g k (ASort O n))))). -(* COMMENTS -Initial nodes: 97 -END *) - -theorem aplus_sort_S_S_simpl: - \forall (g: G).(\forall (n: nat).(\forall (h: nat).(\forall (k: nat).(eq A -(aplus g (ASort (S h) n) (S k)) (aplus g (ASort h n) k))))) -\def - \lambda (g: G).(\lambda (n: nat).(\lambda (h: nat).(\lambda (k: nat).(eq_ind -A (aplus g (asucc g (ASort (S h) n)) k) (\lambda (a: A).(eq A a (aplus g -(ASort h n) k))) (refl_equal A (aplus g (ASort h n) k)) (asucc g (aplus g -(ASort (S h) n) k)) (aplus_asucc g k (ASort (S h) n)))))). -(* COMMENTS -Initial nodes: 97 -END *) - -theorem aplus_asort_O_simpl: - \forall (g: G).(\forall (h: nat).(\forall (n: nat).(eq A (aplus g (ASort O -n) h) (ASort O (next_plus g n h))))) -\def - \lambda (g: G).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall (n0: -nat).(eq A (aplus g (ASort O n0) n) (ASort O (next_plus g n0 n))))) (\lambda -(n: nat).(refl_equal A (ASort O n))) (\lambda (n: nat).(\lambda (H: ((\forall -(n0: nat).(eq A (aplus g (ASort O n0) n) (ASort O (next_plus g n0 -n)))))).(\lambda (n0: nat).(eq_ind A (aplus g (asucc g (ASort O n0)) n) -(\lambda (a: A).(eq A a (ASort O (next g (next_plus g n0 n))))) (eq_ind nat -(next_plus g (next g n0) n) (\lambda (n1: nat).(eq A (aplus g (ASort O (next -g n0)) n) (ASort O n1))) (H (next g n0)) (next g (next_plus g n0 n)) -(next_plus_next g n0 n)) (asucc g (aplus g (ASort O n0) n)) (aplus_asucc g n -(ASort O n0)))))) h)). -(* COMMENTS -Initial nodes: 229 -END *) - -theorem aplus_asort_le_simpl: - \forall (g: G).(\forall (h: nat).(\forall (k: nat).(\forall (n: nat).((le h -k) \to (eq A (aplus g (ASort k n) h) (ASort (minus k h) n)))))) -\def - \lambda (g: G).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall (k: -nat).(\forall (n0: nat).((le n k) \to (eq A (aplus g (ASort k n0) n) (ASort -(minus k n) n0)))))) (\lambda (k: nat).(\lambda (n: nat).(\lambda (_: (le O -k)).(eq_ind nat k (\lambda (n0: nat).(eq A (ASort k n) (ASort n0 n))) -(refl_equal A (ASort k n)) (minus k O) (minus_n_O k))))) (\lambda (h0: -nat).(\lambda (H: ((\forall (k: nat).(\forall (n: nat).((le h0 k) \to (eq A -(aplus g (ASort k n) h0) (ASort (minus k h0) n))))))).(\lambda (k: -nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).((le (S h0) n) \to (eq A -(asucc g (aplus g (ASort n n0) h0)) (ASort (minus n (S h0)) n0))))) (\lambda -(n: nat).(\lambda (H0: (le (S h0) O)).(ex2_ind nat (\lambda (n0: nat).(eq nat -O (S n0))) (\lambda (n0: nat).(le h0 n0)) (eq A (asucc g (aplus g (ASort O n) -h0)) (ASort (minus O (S h0)) n)) (\lambda (x: nat).(\lambda (H1: (eq nat O (S -x))).(\lambda (_: (le h0 x)).(let H3 \def (eq_ind nat O (\lambda (ee: -nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True -| (S _) \Rightarrow False])) I (S x) H1) in (False_ind (eq A (asucc g (aplus -g (ASort O n) h0)) (ASort (minus O (S h0)) n)) H3))))) (le_gen_S h0 O H0)))) -(\lambda (n: nat).(\lambda (_: ((\forall (n0: nat).((le (S h0) n) \to (eq A -(asucc g (aplus g (ASort n n0) h0)) (ASort (minus n (S h0)) n0)))))).(\lambda -(n0: nat).(\lambda (H1: (le (S h0) (S n))).(eq_ind A (aplus g (asucc g (ASort -(S n) n0)) h0) (\lambda (a: A).(eq A a (ASort (minus (S n) (S h0)) n0))) (H n -n0 (le_S_n h0 n H1)) (asucc g (aplus g (ASort (S n) n0) h0)) (aplus_asucc g -h0 (ASort (S n) n0))))))) k)))) h)). -(* COMMENTS -Initial nodes: 484 -END *) - -theorem aplus_asort_simpl: - \forall (g: G).(\forall (h: nat).(\forall (k: nat).(\forall (n: nat).(eq A -(aplus g (ASort k n) h) (ASort (minus k h) (next_plus g n (minus h k))))))) -\def - \lambda (g: G).(\lambda (h: nat).(\lambda (k: nat).(\lambda (n: -nat).(lt_le_e k h (eq A (aplus g (ASort k n) h) (ASort (minus k h) (next_plus -g n (minus h k)))) (\lambda (H: (lt k h)).(eq_ind_r nat (plus k (minus h k)) -(\lambda (n0: nat).(eq A (aplus g (ASort k n) n0) (ASort (minus k h) -(next_plus g n (minus h k))))) (eq_ind A (aplus g (aplus g (ASort k n) k) -(minus h k)) (\lambda (a: A).(eq A a (ASort (minus k h) (next_plus g n (minus -h k))))) (eq_ind_r A (ASort (minus k k) n) (\lambda (a: A).(eq A (aplus g a -(minus h k)) (ASort (minus k h) (next_plus g n (minus h k))))) (eq_ind nat O -(\lambda (n0: nat).(eq A (aplus g (ASort n0 n) (minus h k)) (ASort (minus k -h) (next_plus g n (minus h k))))) (eq_ind_r nat O (\lambda (n0: nat).(eq A -(aplus g (ASort O n) (minus h k)) (ASort n0 (next_plus g n (minus h k))))) -(aplus_asort_O_simpl g (minus h k) n) (minus k h) (O_minus k h (le_S_n k h -(le_S (S k) h H)))) (minus k k) (minus_n_n k)) (aplus g (ASort k n) k) -(aplus_asort_le_simpl g k k n (le_n k))) (aplus g (ASort k n) (plus k (minus -h k))) (aplus_assoc g (ASort k n) k (minus h k))) h (le_plus_minus k h -(le_S_n k h (le_S (S k) h H))))) (\lambda (H: (le h k)).(eq_ind_r A (ASort -(minus k h) n) (\lambda (a: A).(eq A a (ASort (minus k h) (next_plus g n -(minus h k))))) (eq_ind_r nat O (\lambda (n0: nat).(eq A (ASort (minus k h) -n) (ASort (minus k h) (next_plus g n n0)))) (refl_equal A (ASort (minus k h) -(next_plus g n O))) (minus h k) (O_minus h k H)) (aplus g (ASort k n) h) -(aplus_asort_le_simpl g h k n H))))))). -(* COMMENTS -Initial nodes: 587 -END *) - -theorem aplus_ahead_simpl: - \forall (g: G).(\forall (h: nat).(\forall (a1: A).(\forall (a2: A).(eq A -(aplus g (AHead a1 a2) h) (AHead a1 (aplus g a2 h)))))) -\def - \lambda (g: G).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall (a1: -A).(\forall (a2: A).(eq A (aplus g (AHead a1 a2) n) (AHead a1 (aplus g a2 -n)))))) (\lambda (a1: A).(\lambda (a2: A).(refl_equal A (AHead a1 a2)))) -(\lambda (n: nat).(\lambda (H: ((\forall (a1: A).(\forall (a2: A).(eq A -(aplus g (AHead a1 a2) n) (AHead a1 (aplus g a2 n))))))).(\lambda (a1: -A).(\lambda (a2: A).(eq_ind A (aplus g (asucc g (AHead a1 a2)) n) (\lambda -(a: A).(eq A a (AHead a1 (asucc g (aplus g a2 n))))) (eq_ind A (aplus g -(asucc g a2) n) (\lambda (a: A).(eq A (aplus g (asucc g (AHead a1 a2)) n) -(AHead a1 a))) (H a1 (asucc g a2)) (asucc g (aplus g a2 n)) (aplus_asucc g n -a2)) (asucc g (aplus g (AHead a1 a2) n)) (aplus_asucc g n (AHead a1 a2))))))) -h)). -(* COMMENTS -Initial nodes: 239 -END *) - -theorem aplus_asucc_false: - \forall (g: G).(\forall (a: A).(\forall (h: nat).((eq A (aplus g (asucc g a) -h) a) \to (\forall (P: Prop).P)))) -\def - \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(\forall (h: -nat).((eq A (aplus g (asucc g a0) h) a0) \to (\forall (P: Prop).P)))) -(\lambda (n: nat).(\lambda (n0: nat).(\lambda (h: nat).(\lambda (H: (eq A -(aplus g (match n with [O \Rightarrow (ASort O (next g n0)) | (S h0) -\Rightarrow (ASort h0 n0)]) h) (ASort n n0))).(\lambda (P: Prop).(nat_ind -(\lambda (n1: nat).((eq A (aplus g (match n1 with [O \Rightarrow (ASort O -(next g n0)) | (S h0) \Rightarrow (ASort h0 n0)]) h) (ASort n1 n0)) \to P)) -(\lambda (H0: (eq A (aplus g (ASort O (next g n0)) h) (ASort O n0))).(let H1 -\def (eq_ind A (aplus g (ASort O (next g n0)) h) (\lambda (a0: A).(eq A a0 -(ASort O n0))) H0 (ASort (minus O h) (next_plus g (next g n0) (minus h O))) -(aplus_asort_simpl g h O (next g n0))) in (let H2 \def (f_equal A nat -(\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort _ n1) -\Rightarrow n1 | (AHead _ _) \Rightarrow ((let rec next_plus (g0: G) (n1: -nat) (i: nat) on i: nat \def (match i with [O \Rightarrow n1 | (S i0) -\Rightarrow (next g0 (next_plus g0 n1 i0))]) in next_plus) g (next g n0) -(minus h O))])) (ASort (minus O h) (next_plus g (next g n0) (minus h O))) -(ASort O n0) H1) in (let H3 \def (eq_ind_r nat (minus h O) (\lambda (n1: -nat).(eq nat (next_plus g (next g n0) n1) n0)) H2 h (minus_n_O h)) in -(le_lt_false (next_plus g (next g n0) h) n0 (eq_ind nat (next_plus g (next g -n0) h) (\lambda (n1: nat).(le (next_plus g (next g n0) h) n1)) (le_n -(next_plus g (next g n0) h)) n0 H3) (next_plus_lt g h n0) P))))) (\lambda -(n1: nat).(\lambda (_: (((eq A (aplus g (match n1 with [O \Rightarrow (ASort -O (next g n0)) | (S h0) \Rightarrow (ASort h0 n0)]) h) (ASort n1 n0)) \to -P))).(\lambda (H0: (eq A (aplus g (ASort n1 n0) h) (ASort (S n1) n0))).(let -H1 \def (eq_ind A (aplus g (ASort n1 n0) h) (\lambda (a0: A).(eq A a0 (ASort -(S n1) n0))) H0 (ASort (minus n1 h) (next_plus g n0 (minus h n1))) -(aplus_asort_simpl g h n1 n0)) in (let H2 \def (f_equal A nat (\lambda (e: -A).(match e in A return (\lambda (_: A).nat) with [(ASort n2 _) \Rightarrow -n2 | (AHead _ _) \Rightarrow ((let rec minus (n2: nat) on n2: (nat \to nat) -\def (\lambda (m: nat).(match n2 with [O \Rightarrow O | (S k) \Rightarrow -(match m with [O \Rightarrow (S k) | (S l) \Rightarrow (minus k l)])])) in -minus) n1 h)])) (ASort (minus n1 h) (next_plus g n0 (minus h n1))) (ASort (S -n1) n0) H1) in ((let H3 \def (f_equal A nat (\lambda (e: A).(match e in A -return (\lambda (_: A).nat) with [(ASort _ n2) \Rightarrow n2 | (AHead _ _) -\Rightarrow ((let rec next_plus (g0: G) (n2: nat) (i: nat) on i: nat \def -(match i with [O \Rightarrow n2 | (S i0) \Rightarrow (next g0 (next_plus g0 -n2 i0))]) in next_plus) g n0 (minus h n1))])) (ASort (minus n1 h) (next_plus -g n0 (minus h n1))) (ASort (S n1) n0) H1) in (\lambda (H4: (eq nat (minus n1 -h) (S n1))).(le_Sx_x n1 (eq_ind nat (minus n1 h) (\lambda (n2: nat).(le n2 -n1)) (minus_le n1 h) (S n1) H4) P))) H2)))))) n H)))))) (\lambda (a0: -A).(\lambda (_: ((\forall (h: nat).((eq A (aplus g (asucc g a0) h) a0) \to -(\forall (P: Prop).P))))).(\lambda (a1: A).(\lambda (H0: ((\forall (h: -nat).((eq A (aplus g (asucc g a1) h) a1) \to (\forall (P: -Prop).P))))).(\lambda (h: nat).(\lambda (H1: (eq A (aplus g (AHead a0 (asucc -g a1)) h) (AHead a0 a1))).(\lambda (P: Prop).(let H2 \def (eq_ind A (aplus g -(AHead a0 (asucc g a1)) h) (\lambda (a2: A).(eq A a2 (AHead a0 a1))) H1 -(AHead a0 (aplus g (asucc g a1) h)) (aplus_ahead_simpl g h a0 (asucc g a1))) -in (let H3 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda -(_: A).A) with [(ASort _ _) \Rightarrow ((let rec aplus (g0: G) (a2: A) (n: -nat) on n: A \def (match n with [O \Rightarrow a2 | (S n0) \Rightarrow (asucc -g0 (aplus g0 a2 n0))]) in aplus) g (asucc g a1) h) | (AHead _ a2) \Rightarrow -a2])) (AHead a0 (aplus g (asucc g a1) h)) (AHead a0 a1) H2) in (H0 h H3 -P)))))))))) a)). -(* COMMENTS -Initial nodes: 977 -END *) - -theorem aplus_inj: - \forall (g: G).(\forall (h1: nat).(\forall (h2: nat).(\forall (a: A).((eq A -(aplus g a h1) (aplus g a h2)) \to (eq nat h1 h2))))) -\def - \lambda (g: G).(\lambda (h1: nat).(nat_ind (\lambda (n: nat).(\forall (h2: -nat).(\forall (a: A).((eq A (aplus g a n) (aplus g a h2)) \to (eq nat n -h2))))) (\lambda (h2: nat).(nat_ind (\lambda (n: nat).(\forall (a: A).((eq A -(aplus g a O) (aplus g a n)) \to (eq nat O n)))) (\lambda (a: A).(\lambda (_: -(eq A a a)).(refl_equal nat O))) (\lambda (n: nat).(\lambda (_: ((\forall (a: -A).((eq A a (aplus g a n)) \to (eq nat O n))))).(\lambda (a: A).(\lambda (H0: -(eq A a (asucc g (aplus g a n)))).(let H1 \def (eq_ind_r A (asucc g (aplus g -a n)) (\lambda (a0: A).(eq A a a0)) H0 (aplus g (asucc g a) n) (aplus_asucc g -n a)) in (aplus_asucc_false g a n (sym_eq A a (aplus g (asucc g a) n) H1) (eq -nat O (S n)))))))) h2)) (\lambda (n: nat).(\lambda (H: ((\forall (h2: -nat).(\forall (a: A).((eq A (aplus g a n) (aplus g a h2)) \to (eq nat n -h2)))))).(\lambda (h2: nat).(nat_ind (\lambda (n0: nat).(\forall (a: A).((eq -A (aplus g a (S n)) (aplus g a n0)) \to (eq nat (S n) n0)))) (\lambda (a: -A).(\lambda (H0: (eq A (asucc g (aplus g a n)) a)).(let H1 \def (eq_ind_r A -(asucc g (aplus g a n)) (\lambda (a0: A).(eq A a0 a)) H0 (aplus g (asucc g a) -n) (aplus_asucc g n a)) in (aplus_asucc_false g a n H1 (eq nat (S n) O))))) -(\lambda (n0: nat).(\lambda (_: ((\forall (a: A).((eq A (asucc g (aplus g a -n)) (aplus g a n0)) \to (eq nat (S n) n0))))).(\lambda (a: A).(\lambda (H1: -(eq A (asucc g (aplus g a n)) (asucc g (aplus g a n0)))).(let H2 \def -(eq_ind_r A (asucc g (aplus g a n)) (\lambda (a0: A).(eq A a0 (asucc g (aplus -g a n0)))) H1 (aplus g (asucc g a) n) (aplus_asucc g n a)) in (let H3 \def -(eq_ind_r A (asucc g (aplus g a n0)) (\lambda (a0: A).(eq A (aplus g (asucc g -a) n) a0)) H2 (aplus g (asucc g a) n0) (aplus_asucc g n0 a)) in (f_equal nat -nat S n n0 (H n0 (asucc g a) H3)))))))) h2)))) h1)). -(* COMMENTS -Initial nodes: 599 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/app/defs.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/app/defs.ma deleted file mode 100644 index 9cf1e37d1..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/app/defs.ma +++ /dev/null @@ -1,31 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/C/defs.ma". - -definition cbk: - C \to nat -\def - let rec cbk (c: C) on c: nat \def (match c with [(CSort m) \Rightarrow m | -(CHead c0 _ _) \Rightarrow (cbk c0)]) in cbk. - -definition app1: - C \to (T \to T) -\def - let rec app1 (c: C) on c: (T \to T) \def (\lambda (t: T).(match c with -[(CSort _) \Rightarrow t | (CHead c0 k u) \Rightarrow (app1 c0 (THead k u -t))])) in app1. - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/aprem/defs.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/aprem/defs.ma deleted file mode 100644 index 78b49e920..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/aprem/defs.ma +++ /dev/null @@ -1,23 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/A/defs.ma". - -inductive aprem: nat \to (A \to (A \to Prop)) \def -| aprem_zero: \forall (a1: A).(\forall (a2: A).(aprem O (AHead a1 a2) a1)) -| aprem_succ: \forall (a2: A).(\forall (a: A).(\forall (i: nat).((aprem i a2 -a) \to (\forall (a1: A).(aprem (S i) (AHead a1 a2) a))))). - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/aprem/fwd.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/aprem/fwd.ma deleted file mode 100644 index ed48846a8..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/aprem/fwd.ma +++ /dev/null @@ -1,120 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/aprem/defs.ma". - -theorem aprem_gen_sort: - \forall (x: A).(\forall (i: nat).(\forall (h: nat).(\forall (n: nat).((aprem -i (ASort h n) x) \to False)))) -\def - \lambda (x: A).(\lambda (i: nat).(\lambda (h: nat).(\lambda (n: -nat).(\lambda (H: (aprem i (ASort h n) x)).(insert_eq A (ASort h n) (\lambda -(a: A).(aprem i a x)) (\lambda (_: A).False) (\lambda (y: A).(\lambda (H0: -(aprem i y x)).(aprem_ind (\lambda (_: nat).(\lambda (a: A).(\lambda (_: -A).((eq A a (ASort h n)) \to False)))) (\lambda (a1: A).(\lambda (a2: -A).(\lambda (H1: (eq A (AHead a1 a2) (ASort h n))).(let H2 \def (eq_ind A -(AHead a1 a2) (\lambda (ee: A).(match ee in A return (\lambda (_: A).Prop) -with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I -(ASort h n) H1) in (False_ind False H2))))) (\lambda (a2: A).(\lambda (a: -A).(\lambda (i0: nat).(\lambda (_: (aprem i0 a2 a)).(\lambda (_: (((eq A a2 -(ASort h n)) \to False))).(\lambda (a1: A).(\lambda (H3: (eq A (AHead a1 a2) -(ASort h n))).(let H4 \def (eq_ind A (AHead a1 a2) (\lambda (ee: A).(match ee -in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | -(AHead _ _) \Rightarrow True])) I (ASort h n) H3) in (False_ind False -H4))))))))) i y x H0))) H))))). -(* COMMENTS -Initial nodes: 227 -END *) - -theorem aprem_gen_head_O: - \forall (a1: A).(\forall (a2: A).(\forall (x: A).((aprem O (AHead a1 a2) x) -\to (eq A x a1)))) -\def - \lambda (a1: A).(\lambda (a2: A).(\lambda (x: A).(\lambda (H: (aprem O -(AHead a1 a2) x)).(insert_eq A (AHead a1 a2) (\lambda (a: A).(aprem O a x)) -(\lambda (_: A).(eq A x a1)) (\lambda (y: A).(\lambda (H0: (aprem O y -x)).(insert_eq nat O (\lambda (n: nat).(aprem n y x)) (\lambda (_: nat).((eq -A y (AHead a1 a2)) \to (eq A x a1))) (\lambda (y0: nat).(\lambda (H1: (aprem -y0 y x)).(aprem_ind (\lambda (n: nat).(\lambda (a: A).(\lambda (a0: A).((eq -nat n O) \to ((eq A a (AHead a1 a2)) \to (eq A a0 a1)))))) (\lambda (a0: -A).(\lambda (a3: A).(\lambda (_: (eq nat O O)).(\lambda (H3: (eq A (AHead a0 -a3) (AHead a1 a2))).(let H4 \def (f_equal A A (\lambda (e: A).(match e in A -return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead a _) -\Rightarrow a])) (AHead a0 a3) (AHead a1 a2) H3) in ((let H5 \def (f_equal A -A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) -\Rightarrow a3 | (AHead _ a) \Rightarrow a])) (AHead a0 a3) (AHead a1 a2) H3) -in (\lambda (H6: (eq A a0 a1)).H6)) H4)))))) (\lambda (a0: A).(\lambda (a: -A).(\lambda (i: nat).(\lambda (H2: (aprem i a0 a)).(\lambda (H3: (((eq nat i -O) \to ((eq A a0 (AHead a1 a2)) \to (eq A a a1))))).(\lambda (a3: A).(\lambda -(H4: (eq nat (S i) O)).(\lambda (H5: (eq A (AHead a3 a0) (AHead a1 a2))).(let -H6 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) -with [(ASort _ _) \Rightarrow a3 | (AHead a4 _) \Rightarrow a4])) (AHead a3 -a0) (AHead a1 a2) H5) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e -in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead _ -a4) \Rightarrow a4])) (AHead a3 a0) (AHead a1 a2) H5) in (\lambda (_: (eq A -a3 a1)).(let H9 \def (eq_ind A a0 (\lambda (a4: A).((eq nat i O) \to ((eq A -a4 (AHead a1 a2)) \to (eq A a a1)))) H3 a2 H7) in (let H10 \def (eq_ind A a0 -(\lambda (a4: A).(aprem i a4 a)) H2 a2 H7) in (let H11 \def (eq_ind nat (S i) -(\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O -\Rightarrow False | (S _) \Rightarrow True])) I O H4) in (False_ind (eq A a -a1) H11)))))) H6)))))))))) y0 y x H1))) H0))) H)))). -(* COMMENTS -Initial nodes: 500 -END *) - -theorem aprem_gen_head_S: - \forall (a1: A).(\forall (a2: A).(\forall (x: A).(\forall (i: nat).((aprem -(S i) (AHead a1 a2) x) \to (aprem i a2 x))))) -\def - \lambda (a1: A).(\lambda (a2: A).(\lambda (x: A).(\lambda (i: nat).(\lambda -(H: (aprem (S i) (AHead a1 a2) x)).(insert_eq A (AHead a1 a2) (\lambda (a: -A).(aprem (S i) a x)) (\lambda (_: A).(aprem i a2 x)) (\lambda (y: -A).(\lambda (H0: (aprem (S i) y x)).(insert_eq nat (S i) (\lambda (n: -nat).(aprem n y x)) (\lambda (_: nat).((eq A y (AHead a1 a2)) \to (aprem i a2 -x))) (\lambda (y0: nat).(\lambda (H1: (aprem y0 y x)).(aprem_ind (\lambda (n: -nat).(\lambda (a: A).(\lambda (a0: A).((eq nat n (S i)) \to ((eq A a (AHead -a1 a2)) \to (aprem i a2 a0)))))) (\lambda (a0: A).(\lambda (a3: A).(\lambda -(H2: (eq nat O (S i))).(\lambda (H3: (eq A (AHead a0 a3) (AHead a1 a2))).(let -H4 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) -with [(ASort _ _) \Rightarrow a0 | (AHead a _) \Rightarrow a])) (AHead a0 a3) -(AHead a1 a2) H3) in ((let H5 \def (f_equal A A (\lambda (e: A).(match e in A -return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a3 | (AHead _ a) -\Rightarrow a])) (AHead a0 a3) (AHead a1 a2) H3) in (\lambda (H6: (eq A a0 -a1)).(eq_ind_r A a1 (\lambda (a: A).(aprem i a2 a)) (let H7 \def (eq_ind nat -O (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O -\Rightarrow True | (S _) \Rightarrow False])) I (S i) H2) in (False_ind -(aprem i a2 a1) H7)) a0 H6))) H4)))))) (\lambda (a0: A).(\lambda (a: -A).(\lambda (i0: nat).(\lambda (H2: (aprem i0 a0 a)).(\lambda (H3: (((eq nat -i0 (S i)) \to ((eq A a0 (AHead a1 a2)) \to (aprem i a2 a))))).(\lambda (a3: -A).(\lambda (H4: (eq nat (S i0) (S i))).(\lambda (H5: (eq A (AHead a3 a0) -(AHead a1 a2))).(let H6 \def (f_equal A A (\lambda (e: A).(match e in A -return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a3 | (AHead a4 _) -\Rightarrow a4])) (AHead a3 a0) (AHead a1 a2) H5) in ((let H7 \def (f_equal A -A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) -\Rightarrow a0 | (AHead _ a4) \Rightarrow a4])) (AHead a3 a0) (AHead a1 a2) -H5) in (\lambda (_: (eq A a3 a1)).(let H9 \def (eq_ind A a0 (\lambda (a4: -A).((eq nat i0 (S i)) \to ((eq A a4 (AHead a1 a2)) \to (aprem i a2 a)))) H3 -a2 H7) in (let H10 \def (eq_ind A a0 (\lambda (a4: A).(aprem i0 a4 a)) H2 a2 -H7) in (let H11 \def (f_equal nat nat (\lambda (e: nat).(match e in nat -return (\lambda (_: nat).nat) with [O \Rightarrow i0 | (S n) \Rightarrow n])) -(S i0) (S i) H4) in (let H12 \def (eq_ind nat i0 (\lambda (n: nat).((eq nat n -(S i)) \to ((eq A a2 (AHead a1 a2)) \to (aprem i a2 a)))) H9 i H11) in (let -H13 \def (eq_ind nat i0 (\lambda (n: nat).(aprem n a2 a)) H10 i H11) in -H13))))))) H6)))))))))) y0 y x H1))) H0))) H))))). -(* COMMENTS -Initial nodes: 631 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/aprem/props.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/aprem/props.ma deleted file mode 100644 index fb8321062..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/aprem/props.ma +++ /dev/null @@ -1,76 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/aprem/fwd.ma". - -include "Basic-1/leq/defs.ma". - -theorem aprem_repl: - \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall -(i: nat).(\forall (b2: A).((aprem i a2 b2) \to (ex2 A (\lambda (b1: A).(leq g -b1 b2)) (\lambda (b1: A).(aprem i a1 b1))))))))) -\def - \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1 -a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(\forall (i: nat).(\forall -(b2: A).((aprem i a0 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda -(b1: A).(aprem i a b1)))))))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda -(n1: nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (_: (eq A (aplus g -(ASort h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (i: nat).(\lambda (b2: -A).(\lambda (H1: (aprem i (ASort h2 n2) b2)).(let H_x \def (aprem_gen_sort b2 -i h2 n2 H1) in (let H2 \def H_x in (False_ind (ex2 A (\lambda (b1: A).(leq g -b1 b2)) (\lambda (b1: A).(aprem i (ASort h1 n1) b1))) H2)))))))))))) (\lambda -(a0: A).(\lambda (a3: A).(\lambda (H0: (leq g a0 a3)).(\lambda (_: ((\forall -(i: nat).(\forall (b2: A).((aprem i a3 b2) \to (ex2 A (\lambda (b1: A).(leq g -b1 b2)) (\lambda (b1: A).(aprem i a0 b1)))))))).(\lambda (a4: A).(\lambda -(a5: A).(\lambda (_: (leq g a4 a5)).(\lambda (H3: ((\forall (i: nat).(\forall -(b2: A).((aprem i a5 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda -(b1: A).(aprem i a4 b1)))))))).(\lambda (i: nat).(\lambda (b2: A).(\lambda -(H4: (aprem i (AHead a3 a5) b2)).(nat_ind (\lambda (n: nat).((aprem n (AHead -a3 a5) b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem -n (AHead a0 a4) b1))))) (\lambda (H5: (aprem O (AHead a3 a5) b2)).(let H_y -\def (aprem_gen_head_O a3 a5 b2 H5) in (eq_ind_r A a3 (\lambda (a: A).(ex2 A -(\lambda (b1: A).(leq g b1 a)) (\lambda (b1: A).(aprem O (AHead a0 a4) b1)))) -(ex_intro2 A (\lambda (b1: A).(leq g b1 a3)) (\lambda (b1: A).(aprem O (AHead -a0 a4) b1)) a0 H0 (aprem_zero a0 a4)) b2 H_y))) (\lambda (i0: nat).(\lambda -(_: (((aprem i0 (AHead a3 a5) b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) -(\lambda (b1: A).(aprem i0 (AHead a0 a4) b1)))))).(\lambda (H5: (aprem (S i0) -(AHead a3 a5) b2)).(let H_y \def (aprem_gen_head_S a3 a5 b2 i0 H5) in (let -H_x \def (H3 i0 b2 H_y) in (let H6 \def H_x in (ex2_ind A (\lambda (b1: -A).(leq g b1 b2)) (\lambda (b1: A).(aprem i0 a4 b1)) (ex2 A (\lambda (b1: -A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S i0) (AHead a0 a4) b1))) (\lambda -(x: A).(\lambda (H7: (leq g x b2)).(\lambda (H8: (aprem i0 a4 x)).(ex_intro2 -A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S i0) (AHead a0 -a4) b1)) x H7 (aprem_succ a4 x i0 H8 a0))))) H6))))))) i H4)))))))))))) a1 a2 -H)))). -(* COMMENTS -Initial nodes: 621 -END *) - -theorem aprem_asucc: - \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (i: nat).((aprem i -a1 a2) \to (aprem i (asucc g a1) a2))))) -\def - \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (i: nat).(\lambda -(H: (aprem i a1 a2)).(aprem_ind (\lambda (n: nat).(\lambda (a: A).(\lambda -(a0: A).(aprem n (asucc g a) a0)))) (\lambda (a0: A).(\lambda (a3: -A).(aprem_zero a0 (asucc g a3)))) (\lambda (a0: A).(\lambda (a: A).(\lambda -(i0: nat).(\lambda (_: (aprem i0 a0 a)).(\lambda (H1: (aprem i0 (asucc g a0) -a)).(\lambda (a3: A).(aprem_succ (asucc g a0) a i0 H1 a3))))))) i a1 a2 -H))))). -(* COMMENTS -Initial nodes: 101 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/arity/aprem.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/arity/aprem.ma deleted file mode 100644 index 35e8f58e0..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/arity/aprem.ma +++ /dev/null @@ -1,260 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/arity/props.ma". - -include "Basic-1/arity/cimp.ma". - -include "Basic-1/aprem/props.ma". - -theorem arity_aprem: - \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t -a) \to (\forall (i: nat).(\forall (b: A).((aprem i a b) \to (ex2_3 C T nat -(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c)))) -(\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g -b))))))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: -(arity g c t a)).(arity_ind g (\lambda (c0: C).(\lambda (_: T).(\lambda (a0: -A).(\forall (i: nat).(\forall (b: A).((aprem i a0 b) \to (ex2_3 C T nat -(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) -(\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g -b)))))))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (i: nat).(\lambda -(b: A).(\lambda (H0: (aprem i (ASort O n) b)).(let H_x \def (aprem_gen_sort b -i O n H0) in (let H1 \def H_x in (False_ind (ex2_3 C T nat (\lambda (d: -C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: -C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b)))))) H1)))))))) -(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H0: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (a0: A).(\lambda (_: -(arity g d u a0)).(\lambda (H2: ((\forall (i0: nat).(\forall (b: A).((aprem -i0 a0 b) \to (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: -nat).(drop (plus i0 j) O d0 d)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda -(_: nat).(arity g d0 u0 (asucc g b))))))))))).(\lambda (i0: nat).(\lambda (b: -A).(\lambda (H3: (aprem i0 a0 b)).(let H_x \def (H2 i0 b H3) in (let H4 \def -H_x in (ex2_3_ind C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: -nat).(drop (plus i0 j) O d0 d)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda -(_: nat).(arity g d0 u0 (asucc g b))))) (ex2_3 C T nat (\lambda (d0: -C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d0 c0)))) (\lambda -(d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b)))))) -(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H5: (drop -(plus i0 x2) O x0 d)).(\lambda (H6: (arity g x0 x1 (asucc g b))).(let H_x0 -\def (getl_drop_conf_rev (plus i0 x2) x0 d H5 Abbr c0 u i H0) in (let H7 \def -H_x0 in (ex2_ind C (\lambda (c1: C).(drop (plus i0 x2) O c1 c0)) (\lambda -(c1: C).(drop (S i) (plus i0 x2) c1 x0)) (ex2_3 C T nat (\lambda (d0: -C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d0 c0)))) (\lambda -(d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b)))))) -(\lambda (x: C).(\lambda (H8: (drop (plus i0 x2) O x c0)).(\lambda (H9: (drop -(S i) (plus i0 x2) x x0)).(ex2_3_intro C T nat (\lambda (d0: C).(\lambda (_: -T).(\lambda (j: nat).(drop (plus i0 j) O d0 c0)))) (\lambda (d0: C).(\lambda -(u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b))))) x (lift (S i) (plus -i0 x2) x1) x2 H8 (arity_lift g x0 x1 (asucc g b) H6 x (S i) (plus i0 x2) -H9))))) H7)))))))) H4)))))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda -(u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abst) -u))).(\lambda (a0: A).(\lambda (_: (arity g d u (asucc g a0))).(\lambda (H2: -((\forall (i0: nat).(\forall (b: A).((aprem i0 (asucc g a0) b) \to (ex2_3 C T -nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d0 -d)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 -(asucc g b))))))))))).(\lambda (i0: nat).(\lambda (b: A).(\lambda (H3: (aprem -i0 a0 b)).(let H4 \def (H2 i0 b (aprem_asucc g a0 b i0 H3)) in (ex2_3_ind C T -nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d0 -d)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 -(asucc g b))))) (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: -nat).(drop (plus i0 j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda -(_: nat).(arity g d0 u0 (asucc g b)))))) (\lambda (x0: C).(\lambda (x1: -T).(\lambda (x2: nat).(\lambda (H5: (drop (plus i0 x2) O x0 d)).(\lambda (H6: -(arity g x0 x1 (asucc g b))).(let H_x \def (getl_drop_conf_rev (plus i0 x2) -x0 d H5 Abst c0 u i H0) in (let H7 \def H_x in (ex2_ind C (\lambda (c1: -C).(drop (plus i0 x2) O c1 c0)) (\lambda (c1: C).(drop (S i) (plus i0 x2) c1 -x0)) (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop -(plus i0 j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: -nat).(arity g d0 u0 (asucc g b)))))) (\lambda (x: C).(\lambda (H8: (drop -(plus i0 x2) O x c0)).(\lambda (H9: (drop (S i) (plus i0 x2) x -x0)).(ex2_3_intro C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: -nat).(drop (plus i0 j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda -(_: nat).(arity g d0 u0 (asucc g b))))) x (lift (S i) (plus i0 x2) x1) x2 H8 -(arity_lift g x0 x1 (asucc g b) H6 x (S i) (plus i0 x2) H9))))) H7)))))))) -H4))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda -(c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u -a1)).(\lambda (_: ((\forall (i: nat).(\forall (b0: A).((aprem i a1 b0) \to -(ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus -i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d -u0 (asucc g b0))))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: -(arity g (CHead c0 (Bind b) u) t0 a2)).(\lambda (H4: ((\forall (i: -nat).(\forall (b0: A).((aprem i a2 b0) \to (ex2_3 C T nat (\lambda (d: -C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d (CHead c0 (Bind b) -u))))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 -(asucc g b0))))))))))).(\lambda (i: nat).(\lambda (b0: A).(\lambda (H5: -(aprem i a2 b0)).(let H_x \def (H4 i b0 H5) in (let H6 \def H_x in (ex2_3_ind -C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O -d (CHead c0 (Bind b) u))))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: -nat).(arity g d u0 (asucc g b0))))) (ex2_3 C T nat (\lambda (d: C).(\lambda -(_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda -(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b0)))))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H7: (drop (plus i x2) O x0 -(CHead c0 (Bind b) u))).(\lambda (H8: (arity g x0 x1 (asucc g b0))).(let H9 -\def (eq_ind nat (S (plus i x2)) (\lambda (n: nat).(drop n O x0 c0)) (drop_S -b x0 c0 u (plus i x2) H7) (plus i (S x2)) (plus_n_Sm i x2)) in (ex2_3_intro C -T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d -c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 -(asucc g b0))))) x0 x1 (S x2) H9 H8))))))) H6))))))))))))))))) (\lambda (c0: -C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H0: (arity g c0 u (asucc g -a1))).(\lambda (_: ((\forall (i: nat).(\forall (b: A).((aprem i (asucc g a1) -b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop -(plus i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: -nat).(arity g d u0 (asucc g b))))))))))).(\lambda (t0: T).(\lambda (a2: -A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u) t0 a2)).(\lambda (H3: -((\forall (i: nat).(\forall (b: A).((aprem i a2 b) \to (ex2_3 C T nat -(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d (CHead -c0 (Bind Abst) u))))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: -nat).(arity g d u0 (asucc g b))))))))))).(\lambda (i: nat).(\lambda (b: -A).(\lambda (H4: (aprem i (AHead a1 a2) b)).(nat_ind (\lambda (n: -nat).((aprem n (AHead a1 a2) b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda -(_: T).(\lambda (j: nat).(drop (plus n j) O d c0)))) (\lambda (d: C).(\lambda -(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b)))))))) (\lambda (H5: -(aprem O (AHead a1 a2) b)).(let H_y \def (aprem_gen_head_O a1 a2 b H5) in -(eq_ind_r A a1 (\lambda (a0: A).(ex2_3 C T nat (\lambda (d: C).(\lambda (_: -T).(\lambda (j: nat).(drop (plus O j) O d c0)))) (\lambda (d: C).(\lambda -(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g a0))))))) (ex2_3_intro C T -nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus O j) O d -c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 -(asucc g a1))))) c0 u O (drop_refl c0) H0) b H_y))) (\lambda (i0: -nat).(\lambda (_: (((aprem i0 (AHead a1 a2) b) \to (ex2_3 C T nat (\lambda -(d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d c0)))) -(\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g -b))))))))).(\lambda (H5: (aprem (S i0) (AHead a1 a2) b)).(let H_y \def -(aprem_gen_head_S a1 a2 b i0 H5) in (let H_x \def (H3 i0 b H_y) in (let H6 -\def H_x in (ex2_3_ind C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: -nat).(drop (plus i0 j) O d (CHead c0 (Bind Abst) u))))) (\lambda (d: -C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))) (ex2_3 C -T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus (S i0) j) -O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 -(asucc g b)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: -nat).(\lambda (H7: (drop (plus i0 x2) O x0 (CHead c0 (Bind Abst) -u))).(\lambda (H8: (arity g x0 x1 (asucc g b))).(ex2_3_intro C T nat (\lambda -(d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus (S i0) j) O d c0)))) -(\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g -b))))) x0 x1 x2 (drop_S Abst x0 c0 u (plus i0 x2) H7) H8)))))) H6))))))) i -H4))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda -(_: (arity g c0 u a1)).(\lambda (_: ((\forall (i: nat).(\forall (b: -A).((aprem i a1 b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: -T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda -(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))))))))).(\lambda (t0: -T).(\lambda (a2: A).(\lambda (_: (arity g c0 t0 (AHead a1 a2))).(\lambda (H3: -((\forall (i: nat).(\forall (b: A).((aprem i (AHead a1 a2) b) \to (ex2_3 C T -nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d -c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 -(asucc g b))))))))))).(\lambda (i: nat).(\lambda (b: A).(\lambda (H4: (aprem -i a2 b)).(let H5 \def (H3 (S i) b (aprem_succ a2 b i H4 a1)) in (ex2_3_ind C -T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (S (plus i j)) -O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 -(asucc g b))))) (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: -nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda -(_: nat).(arity g d u0 (asucc g b)))))) (\lambda (x0: C).(\lambda (x1: -T).(\lambda (x2: nat).(\lambda (H6: (drop (S (plus i x2)) O x0 c0)).(\lambda -(H7: (arity g x0 x1 (asucc g b))).(C_ind (\lambda (c1: C).((drop (S (plus i -x2)) O c1 c0) \to ((arity g c1 x1 (asucc g b)) \to (ex2_3 C T nat (\lambda -(d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda -(d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))))))) -(\lambda (n: nat).(\lambda (H8: (drop (S (plus i x2)) O (CSort n) -c0)).(\lambda (_: (arity g (CSort n) x1 (asucc g b))).(and3_ind (eq C c0 -(CSort n)) (eq nat (S (plus i x2)) O) (eq nat O O) (ex2_3 C T nat (\lambda -(d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda -(d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b)))))) -(\lambda (_: (eq C c0 (CSort n))).(\lambda (H11: (eq nat (S (plus i x2)) -O)).(\lambda (_: (eq nat O O)).(let H13 \def (eq_ind nat (S (plus i x2)) -(\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O -\Rightarrow False | (S _) \Rightarrow True])) I O H11) in (False_ind (ex2_3 C -T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d -c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 -(asucc g b)))))) H13))))) (drop_gen_sort n (S (plus i x2)) O c0 H8))))) -(\lambda (d: C).(\lambda (IHd: (((drop (S (plus i x2)) O d c0) \to ((arity g -d x1 (asucc g b)) \to (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: -T).(\lambda (j: nat).(drop (plus i j) O d0 c0)))) (\lambda (d0: C).(\lambda -(u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b)))))))))).(\lambda (k: -K).(\lambda (t1: T).(\lambda (H8: (drop (S (plus i x2)) O (CHead d k t1) -c0)).(\lambda (H9: (arity g (CHead d k t1) x1 (asucc g b))).(K_ind (\lambda -(k0: K).((arity g (CHead d k0 t1) x1 (asucc g b)) \to ((drop (r k0 (plus i -x2)) O d c0) \to (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: -nat).(drop (plus i j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda -(_: nat).(arity g d0 u0 (asucc g b))))))))) (\lambda (b0: B).(\lambda (H10: -(arity g (CHead d (Bind b0) t1) x1 (asucc g b))).(\lambda (H11: (drop (r -(Bind b0) (plus i x2)) O d c0)).(ex2_3_intro C T nat (\lambda (d0: -C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d0 c0)))) (\lambda -(d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b))))) -(CHead d (Bind b0) t1) x1 (S x2) (eq_ind nat (S (plus i x2)) (\lambda (n: -nat).(drop n O (CHead d (Bind b0) t1) c0)) (drop_drop (Bind b0) (plus i x2) d -c0 H11 t1) (plus i (S x2)) (plus_n_Sm i x2)) H10)))) (\lambda (f: F).(\lambda -(H10: (arity g (CHead d (Flat f) t1) x1 (asucc g b))).(\lambda (H11: (drop (r -(Flat f) (plus i x2)) O d c0)).(let H12 \def (IHd H11 (arity_cimp_conf g -(CHead d (Flat f) t1) x1 (asucc g b) H10 d (cimp_flat_sx f d t1))) in -(ex2_3_ind C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop -(plus i j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: -nat).(arity g d0 u0 (asucc g b))))) (ex2_3 C T nat (\lambda (d0: C).(\lambda -(_: T).(\lambda (j: nat).(drop (plus i j) O d0 c0)))) (\lambda (d0: -C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b)))))) -(\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: nat).(\lambda (H13: (drop -(plus i x5) O x3 c0)).(\lambda (H14: (arity g x3 x4 (asucc g -b))).(ex2_3_intro C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: -nat).(drop (plus i j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda -(_: nat).(arity g d0 u0 (asucc g b))))) x3 x4 x5 H13 H14)))))) H12))))) k H9 -(drop_gen_drop k d c0 t1 (plus i x2) H8)))))))) x0 H6 H7)))))) -H5)))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a0: A).(\lambda -(_: (arity g c0 u (asucc g a0))).(\lambda (_: ((\forall (i: nat).(\forall (b: -A).((aprem i (asucc g a0) b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: -T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda -(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))))))))).(\lambda (t0: -T).(\lambda (_: (arity g c0 t0 a0)).(\lambda (H3: ((\forall (i: nat).(\forall -(b: A).((aprem i a0 b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: -T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda -(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))))))))).(\lambda (i: -nat).(\lambda (b: A).(\lambda (H4: (aprem i a0 b)).(let H_x \def (H3 i b H4) -in (let H5 \def H_x in (ex2_3_ind C T nat (\lambda (d: C).(\lambda (_: -T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda -(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))) (ex2_3 C T nat -(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) -(\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g -b)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H6: -(drop (plus i x2) O x0 c0)).(\lambda (H7: (arity g x0 x1 (asucc g -b))).(ex2_3_intro C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: -nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda -(_: nat).(arity g d u0 (asucc g b))))) x0 x1 x2 H6 H7)))))) H5)))))))))))))) -(\lambda (c0: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda (_: (arity g c0 -t0 a1)).(\lambda (H1: ((\forall (i: nat).(\forall (b: A).((aprem i a1 b) \to -(ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus -i j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d -u (asucc g b))))))))))).(\lambda (a2: A).(\lambda (H2: (leq g a1 -a2)).(\lambda (i: nat).(\lambda (b: A).(\lambda (H3: (aprem i a2 b)).(let H_x -\def (aprem_repl g a1 a2 H2 i b H3) in (let H4 \def H_x in (ex2_ind A -(\lambda (b1: A).(leq g b1 b)) (\lambda (b1: A).(aprem i a1 b1)) (ex2_3 C T -nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d -c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc -g b)))))) (\lambda (x: A).(\lambda (H5: (leq g x b)).(\lambda (H6: (aprem i -a1 x)).(let H_x0 \def (H1 i x H6) in (let H7 \def H_x0 in (ex2_3_ind C T nat -(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) -(\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g -x))))) (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop -(plus i j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: -nat).(arity g d u (asucc g b)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda -(x2: nat).(\lambda (H8: (drop (plus i x2) O x0 c0)).(\lambda (H9: (arity g x0 -x1 (asucc g x))).(ex2_3_intro C T nat (\lambda (d: C).(\lambda (_: -T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u: -T).(\lambda (_: nat).(arity g d u (asucc g b))))) x0 x1 x2 H8 (arity_repl g -x0 x1 (asucc g x) H9 (asucc g b) (asucc_repl g x b H5)))))))) H7)))))) -H4))))))))))))) c t a H))))). -(* COMMENTS -Initial nodes: 4526 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/arity/cimp.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/arity/cimp.ma deleted file mode 100644 index c6212ac15..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/arity/cimp.ma +++ /dev/null @@ -1,102 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/arity/defs.ma". - -include "Basic-1/cimp/props.ma". - -theorem arity_cimp_conf: - \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1 -t a) \to (\forall (c2: C).((cimp c1 c2) \to (arity g c2 t a))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: -(arity g c1 t a)).(arity_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda (a0: -A).(\forall (c2: C).((cimp c c2) \to (arity g c2 t0 a0)))))) (\lambda (c: -C).(\lambda (n: nat).(\lambda (c2: C).(\lambda (_: (cimp c c2)).(arity_sort g -c2 n))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: -nat).(\lambda (H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (a0: -A).(\lambda (_: (arity g d u a0)).(\lambda (H2: ((\forall (c2: C).((cimp d -c2) \to (arity g c2 u a0))))).(\lambda (c2: C).(\lambda (H3: (cimp c -c2)).(let H_x \def (H3 Abbr d u i H0) in (let H4 \def H_x in (ex_ind C -(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (arity g c2 (TLRef i) -a0) (\lambda (x: C).(\lambda (H5: (getl i c2 (CHead x (Bind Abbr) u))).(let -H_x0 \def (cimp_getl_conf c c2 H3 Abbr d u i H0) in (let H6 \def H_x0 in -(ex2_ind C (\lambda (d2: C).(cimp d d2)) (\lambda (d2: C).(getl i c2 (CHead -d2 (Bind Abbr) u))) (arity g c2 (TLRef i) a0) (\lambda (x0: C).(\lambda (H7: -(cimp d x0)).(\lambda (H8: (getl i c2 (CHead x0 (Bind Abbr) u))).(let H9 \def -(eq_ind C (CHead x (Bind Abbr) u) (\lambda (c0: C).(getl i c2 c0)) H5 (CHead -x0 (Bind Abbr) u) (getl_mono c2 (CHead x (Bind Abbr) u) i H5 (CHead x0 (Bind -Abbr) u) H8)) in (let H10 \def (f_equal C C (\lambda (e: C).(match e in C -return (\lambda (_: C).C) with [(CSort _) \Rightarrow x | (CHead c0 _ _) -\Rightarrow c0])) (CHead x (Bind Abbr) u) (CHead x0 (Bind Abbr) u) (getl_mono -c2 (CHead x (Bind Abbr) u) i H5 (CHead x0 (Bind Abbr) u) H8)) in (let H11 -\def (eq_ind_r C x0 (\lambda (c0: C).(getl i c2 (CHead c0 (Bind Abbr) u))) H9 -x H10) in (let H12 \def (eq_ind_r C x0 (\lambda (c0: C).(cimp d c0)) H7 x -H10) in (arity_abbr g c2 x u i H11 a0 (H2 x H12))))))))) H6))))) -H4))))))))))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: -nat).(\lambda (H0: (getl i c (CHead d (Bind Abst) u))).(\lambda (a0: -A).(\lambda (_: (arity g d u (asucc g a0))).(\lambda (H2: ((\forall (c2: -C).((cimp d c2) \to (arity g c2 u (asucc g a0)))))).(\lambda (c2: C).(\lambda -(H3: (cimp c c2)).(let H_x \def (H3 Abst d u i H0) in (let H4 \def H_x in -(ex_ind C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) (arity g c2 -(TLRef i) a0) (\lambda (x: C).(\lambda (H5: (getl i c2 (CHead x (Bind Abst) -u))).(let H_x0 \def (cimp_getl_conf c c2 H3 Abst d u i H0) in (let H6 \def -H_x0 in (ex2_ind C (\lambda (d2: C).(cimp d d2)) (\lambda (d2: C).(getl i c2 -(CHead d2 (Bind Abst) u))) (arity g c2 (TLRef i) a0) (\lambda (x0: -C).(\lambda (H7: (cimp d x0)).(\lambda (H8: (getl i c2 (CHead x0 (Bind Abst) -u))).(let H9 \def (eq_ind C (CHead x (Bind Abst) u) (\lambda (c0: C).(getl i -c2 c0)) H5 (CHead x0 (Bind Abst) u) (getl_mono c2 (CHead x (Bind Abst) u) i -H5 (CHead x0 (Bind Abst) u) H8)) in (let H10 \def (f_equal C C (\lambda (e: -C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow x | -(CHead c0 _ _) \Rightarrow c0])) (CHead x (Bind Abst) u) (CHead x0 (Bind -Abst) u) (getl_mono c2 (CHead x (Bind Abst) u) i H5 (CHead x0 (Bind Abst) u) -H8)) in (let H11 \def (eq_ind_r C x0 (\lambda (c0: C).(getl i c2 (CHead c0 -(Bind Abst) u))) H9 x H10) in (let H12 \def (eq_ind_r C x0 (\lambda (c0: -C).(cimp d c0)) H7 x H10) in (arity_abst g c2 x u i H11 a0 (H2 x H12))))))))) -H6))))) H4))))))))))))) (\lambda (b: B).(\lambda (H0: (not (eq B b -Abst))).(\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity -g c u a1)).(\lambda (H2: ((\forall (c2: C).((cimp c c2) \to (arity g c2 u -a1))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c -(Bind b) u) t0 a2)).(\lambda (H4: ((\forall (c2: C).((cimp (CHead c (Bind b) -u) c2) \to (arity g c2 t0 a2))))).(\lambda (c2: C).(\lambda (H5: (cimp c -c2)).(arity_bind g b H0 c2 u a1 (H2 c2 H5) t0 a2 (H4 (CHead c2 (Bind b) u) -(cimp_bind c c2 H5 b u)))))))))))))))) (\lambda (c: C).(\lambda (u: -T).(\lambda (a1: A).(\lambda (_: (arity g c u (asucc g a1))).(\lambda (H1: -((\forall (c2: C).((cimp c c2) \to (arity g c2 u (asucc g a1)))))).(\lambda -(t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c (Bind Abst) u) t0 -a2)).(\lambda (H3: ((\forall (c2: C).((cimp (CHead c (Bind Abst) u) c2) \to -(arity g c2 t0 a2))))).(\lambda (c2: C).(\lambda (H4: (cimp c -c2)).(arity_head g c2 u a1 (H1 c2 H4) t0 a2 (H3 (CHead c2 (Bind Abst) u) -(cimp_bind c c2 H4 Abst u)))))))))))))) (\lambda (c: C).(\lambda (u: -T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H1: ((\forall -(c2: C).((cimp c c2) \to (arity g c2 u a1))))).(\lambda (t0: T).(\lambda (a2: -A).(\lambda (_: (arity g c t0 (AHead a1 a2))).(\lambda (H3: ((\forall (c2: -C).((cimp c c2) \to (arity g c2 t0 (AHead a1 a2)))))).(\lambda (c2: -C).(\lambda (H4: (cimp c c2)).(arity_appl g c2 u a1 (H1 c2 H4) t0 a2 (H3 c2 -H4))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a0: A).(\lambda (_: -(arity g c u (asucc g a0))).(\lambda (H1: ((\forall (c2: C).((cimp c c2) \to -(arity g c2 u (asucc g a0)))))).(\lambda (t0: T).(\lambda (_: (arity g c t0 -a0)).(\lambda (H3: ((\forall (c2: C).((cimp c c2) \to (arity g c2 t0 -a0))))).(\lambda (c2: C).(\lambda (H4: (cimp c c2)).(arity_cast g c2 u a0 (H1 -c2 H4) t0 (H3 c2 H4)))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda -(a1: A).(\lambda (_: (arity g c t0 a1)).(\lambda (H1: ((\forall (c2: -C).((cimp c c2) \to (arity g c2 t0 a1))))).(\lambda (a2: A).(\lambda (H2: -(leq g a1 a2)).(\lambda (c2: C).(\lambda (H3: (cimp c c2)).(arity_repl g c2 -t0 a1 (H1 c2 H3) a2 H2)))))))))) c1 t a H))))). -(* COMMENTS -Initial nodes: 1505 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/arity/defs.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/arity/defs.ma deleted file mode 100644 index 99324dcbc..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/arity/defs.ma +++ /dev/null @@ -1,45 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/leq/defs.ma". - -include "Basic-1/getl/defs.ma". - -inductive arity (g: G): C \to (T \to (A \to Prop)) \def -| arity_sort: \forall (c: C).(\forall (n: nat).(arity g c (TSort n) (ASort O -n))) -| arity_abbr: \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: -nat).((getl i c (CHead d (Bind Abbr) u)) \to (\forall (a: A).((arity g d u a) -\to (arity g c (TLRef i) a))))))) -| arity_abst: \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: -nat).((getl i c (CHead d (Bind Abst) u)) \to (\forall (a: A).((arity g d u -(asucc g a)) \to (arity g c (TLRef i) a))))))) -| arity_bind: \forall (b: B).((not (eq B b Abst)) \to (\forall (c: -C).(\forall (u: T).(\forall (a1: A).((arity g c u a1) \to (\forall (t: -T).(\forall (a2: A).((arity g (CHead c (Bind b) u) t a2) \to (arity g c -(THead (Bind b) u t) a2))))))))) -| arity_head: \forall (c: C).(\forall (u: T).(\forall (a1: A).((arity g c u -(asucc g a1)) \to (\forall (t: T).(\forall (a2: A).((arity g (CHead c (Bind -Abst) u) t a2) \to (arity g c (THead (Bind Abst) u t) (AHead a1 a2)))))))) -| arity_appl: \forall (c: C).(\forall (u: T).(\forall (a1: A).((arity g c u -a1) \to (\forall (t: T).(\forall (a2: A).((arity g c t (AHead a1 a2)) \to -(arity g c (THead (Flat Appl) u t) a2))))))) -| arity_cast: \forall (c: C).(\forall (u: T).(\forall (a: A).((arity g c u -(asucc g a)) \to (\forall (t: T).((arity g c t a) \to (arity g c (THead (Flat -Cast) u t) a)))))) -| arity_repl: \forall (c: C).(\forall (t: T).(\forall (a1: A).((arity g c t -a1) \to (\forall (a2: A).((leq g a1 a2) \to (arity g c t a2)))))). - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/arity/fwd.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/arity/fwd.ma deleted file mode 100644 index 31fa35f3c..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/arity/fwd.ma +++ /dev/null @@ -1,1163 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/arity/defs.ma". - -include "Basic-1/leq/asucc.ma". - -include "Basic-1/getl/drop.ma". - -theorem arity_gen_sort: - \forall (g: G).(\forall (c: C).(\forall (n: nat).(\forall (a: A).((arity g c -(TSort n) a) \to (leq g a (ASort O n)))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (n: nat).(\lambda (a: A).(\lambda -(H: (arity g c (TSort n) a)).(insert_eq T (TSort n) (\lambda (t: T).(arity g -c t a)) (\lambda (_: T).(leq g a (ASort O n))) (\lambda (y: T).(\lambda (H0: -(arity g c y a)).(arity_ind g (\lambda (_: C).(\lambda (t: T).(\lambda (a0: -A).((eq T t (TSort n)) \to (leq g a0 (ASort O n)))))) (\lambda (_: -C).(\lambda (n0: nat).(\lambda (H1: (eq T (TSort n0) (TSort n))).(let H2 \def -(f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with -[(TSort n1) \Rightarrow n1 | (TLRef _) \Rightarrow n0 | (THead _ _ _) -\Rightarrow n0])) (TSort n0) (TSort n) H1) in (eq_ind_r nat n (\lambda (n1: -nat).(leq g (ASort O n1) (ASort O n))) (leq_refl g (ASort O n)) n0 H2))))) -(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda -(_: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (a0: A).(\lambda (_: (arity -g d u a0)).(\lambda (_: (((eq T u (TSort n)) \to (leq g a0 (ASort O -n))))).(\lambda (H4: (eq T (TLRef i) (TSort n))).(let H5 \def (eq_ind T -(TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) -\Rightarrow False])) I (TSort n) H4) in (False_ind (leq g a0 (ASort O n)) -H5))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: -nat).(\lambda (_: (getl i c0 (CHead d (Bind Abst) u))).(\lambda (a0: -A).(\lambda (_: (arity g d u (asucc g a0))).(\lambda (_: (((eq T u (TSort n)) -\to (leq g (asucc g a0) (ASort O n))))).(\lambda (H4: (eq T (TLRef i) (TSort -n))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (TSort n) H4) in -(False_ind (leq g a0 (ASort O n)) H5))))))))))) (\lambda (b: B).(\lambda (_: -(not (eq B b Abst))).(\lambda (c0: C).(\lambda (u: T).(\lambda (a1: -A).(\lambda (_: (arity g c0 u a1)).(\lambda (_: (((eq T u (TSort n)) \to (leq -g a1 (ASort O n))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (_: (arity g -(CHead c0 (Bind b) u) t a2)).(\lambda (_: (((eq T t (TSort n)) \to (leq g a2 -(ASort O n))))).(\lambda (H6: (eq T (THead (Bind b) u t) (TSort n))).(let H7 -\def (eq_ind T (THead (Bind b) u t) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H6) in -(False_ind (leq g a2 (ASort O n)) H7)))))))))))))) (\lambda (c0: C).(\lambda -(u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u (asucc g a1))).(\lambda -(_: (((eq T u (TSort n)) \to (leq g (asucc g a1) (ASort O n))))).(\lambda (t: -T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u) t -a2)).(\lambda (_: (((eq T t (TSort n)) \to (leq g a2 (ASort O n))))).(\lambda -(H5: (eq T (THead (Bind Abst) u t) (TSort n))).(let H6 \def (eq_ind T (THead -(Bind Abst) u t) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) -with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ -_) \Rightarrow True])) I (TSort n) H5) in (False_ind (leq g (AHead a1 a2) -(ASort O n)) H6)))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a1: -A).(\lambda (_: (arity g c0 u a1)).(\lambda (_: (((eq T u (TSort n)) \to (leq -g a1 (ASort O n))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (_: (arity g -c0 t (AHead a1 a2))).(\lambda (_: (((eq T t (TSort n)) \to (leq g (AHead a1 -a2) (ASort O n))))).(\lambda (H5: (eq T (THead (Flat Appl) u t) (TSort -n))).(let H6 \def (eq_ind T (THead (Flat Appl) u t) (\lambda (ee: T).(match -ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | -(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) -H5) in (False_ind (leq g a2 (ASort O n)) H6)))))))))))) (\lambda (c0: -C).(\lambda (u: T).(\lambda (a0: A).(\lambda (_: (arity g c0 u (asucc g -a0))).(\lambda (_: (((eq T u (TSort n)) \to (leq g (asucc g a0) (ASort O -n))))).(\lambda (t: T).(\lambda (_: (arity g c0 t a0)).(\lambda (_: (((eq T t -(TSort n)) \to (leq g a0 (ASort O n))))).(\lambda (H5: (eq T (THead (Flat -Cast) u t) (TSort n))).(let H6 \def (eq_ind T (THead (Flat Cast) u t) -(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow -True])) I (TSort n) H5) in (False_ind (leq g a0 (ASort O n)) H6))))))))))) -(\lambda (c0: C).(\lambda (t: T).(\lambda (a1: A).(\lambda (H1: (arity g c0 t -a1)).(\lambda (H2: (((eq T t (TSort n)) \to (leq g a1 (ASort O -n))))).(\lambda (a2: A).(\lambda (H3: (leq g a1 a2)).(\lambda (H4: (eq T t -(TSort n))).(let H5 \def (f_equal T T (\lambda (e: T).e) t (TSort n) H4) in -(let H6 \def (eq_ind T t (\lambda (t0: T).((eq T t0 (TSort n)) \to (leq g a1 -(ASort O n)))) H2 (TSort n) H5) in (let H7 \def (eq_ind T t (\lambda (t0: -T).(arity g c0 t0 a1)) H1 (TSort n) H5) in (leq_trans g a2 a1 (leq_sym g a1 -a2 H3) (ASort O n) (H6 (refl_equal T (TSort n))))))))))))))) c y a H0))) -H))))). -(* COMMENTS -Initial nodes: 1235 -END *) - -theorem arity_gen_lref: - \forall (g: G).(\forall (c: C).(\forall (i: nat).(\forall (a: A).((arity g c -(TLRef i) a) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c -(CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u a)))) -(ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c (CHead d (Bind Abst) -u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a)))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (i: nat).(\lambda (a: A).(\lambda -(H: (arity g c (TLRef i) a)).(insert_eq T (TLRef i) (\lambda (t: T).(arity g -c t a)) (\lambda (_: T).(or (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl -i c (CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u -a)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c (CHead d (Bind -Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a))))))) -(\lambda (y: T).(\lambda (H0: (arity g c y a)).(arity_ind g (\lambda (c0: -C).(\lambda (t: T).(\lambda (a0: A).((eq T t (TLRef i)) \to (or (ex2_2 C T -(\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u)))) -(\lambda (d: C).(\lambda (u: T).(arity g d u a0)))) (ex2_2 C T (\lambda (d: -C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d: -C).(\lambda (u: T).(arity g d u (asucc g a0)))))))))) (\lambda (c0: -C).(\lambda (n: nat).(\lambda (H1: (eq T (TSort n) (TLRef i))).(let H2 \def -(eq_ind T (TSort n) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | -(THead _ _ _) \Rightarrow False])) I (TLRef i) H1) in (False_ind (or (ex2_2 C -T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u)))) -(\lambda (d: C).(\lambda (u: T).(arity g d u (ASort O n))))) (ex2_2 C T -(\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) -(\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g (ASort O n))))))) -H2))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i0: -nat).(\lambda (H1: (getl i0 c0 (CHead d (Bind Abbr) u))).(\lambda (a0: -A).(\lambda (H2: (arity g d u a0)).(\lambda (_: (((eq T u (TLRef i)) \to (or -(ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl i d (CHead d0 (Bind Abbr) -u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a0)))) (ex2_2 C T -(\lambda (d0: C).(\lambda (u0: T).(getl i d (CHead d0 (Bind Abst) u0)))) -(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g -a0))))))))).(\lambda (H4: (eq T (TLRef i0) (TLRef i))).(let H5 \def (f_equal -T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort -_) \Rightarrow i0 | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow i0])) -(TLRef i0) (TLRef i) H4) in (let H6 \def (eq_ind nat i0 (\lambda (n: -nat).(getl n c0 (CHead d (Bind Abbr) u))) H1 i H5) in (or_introl (ex2_2 C T -(\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abbr) u0)))) -(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a0)))) (ex2_2 C T (\lambda -(d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0)))) (\lambda -(d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a0))))) (ex2_2_intro C T -(\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abbr) u0)))) -(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a0))) d u H6 H2))))))))))))) -(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i0: nat).(\lambda -(H1: (getl i0 c0 (CHead d (Bind Abst) u))).(\lambda (a0: A).(\lambda (H2: -(arity g d u (asucc g a0))).(\lambda (_: (((eq T u (TLRef i)) \to (or (ex2_2 -C T (\lambda (d0: C).(\lambda (u0: T).(getl i d (CHead d0 (Bind Abbr) u0)))) -(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a0))))) (ex2_2 C T -(\lambda (d0: C).(\lambda (u0: T).(getl i d (CHead d0 (Bind Abst) u0)))) -(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g (asucc g -a0)))))))))).(\lambda (H4: (eq T (TLRef i0) (TLRef i))).(let H5 \def (f_equal -T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort -_) \Rightarrow i0 | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow i0])) -(TLRef i0) (TLRef i) H4) in (let H6 \def (eq_ind nat i0 (\lambda (n: -nat).(getl n c0 (CHead d (Bind Abst) u))) H1 i H5) in (or_intror (ex2_2 C T -(\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abbr) u0)))) -(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a0)))) (ex2_2 C T (\lambda -(d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0)))) (\lambda -(d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a0))))) (ex2_2_intro C T -(\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0)))) -(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a0)))) d u H6 -H2))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda -(c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u -a1)).(\lambda (_: (((eq T u (TLRef i)) \to (or (ex2_2 C T (\lambda (d: -C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d: -C).(\lambda (u0: T).(arity g d u0 a1)))) (ex2_2 C T (\lambda (d: C).(\lambda -(u0: T).(getl i c0 (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0: -T).(arity g d u0 (asucc g a1))))))))).(\lambda (t: T).(\lambda (a2: -A).(\lambda (_: (arity g (CHead c0 (Bind b) u) t a2)).(\lambda (_: (((eq T t -(TLRef i)) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i (CHead -c0 (Bind b) u) (CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: -T).(arity g d u0 a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i -(CHead c0 (Bind b) u) (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda -(u0: T).(arity g d u0 (asucc g a2))))))))).(\lambda (H6: (eq T (THead (Bind -b) u t) (TLRef i))).(let H7 \def (eq_ind T (THead (Bind b) u t) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I -(TLRef i) H6) in (False_ind (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: -T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: -T).(arity g d u0 a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i -c0 (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 -(asucc g a2)))))) H7)))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda -(a1: A).(\lambda (_: (arity g c0 u (asucc g a1))).(\lambda (_: (((eq T u -(TLRef i)) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 -(CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 -(asucc g a1))))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 -(CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 -(asucc g (asucc g a1)))))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (_: -(arity g (CHead c0 (Bind Abst) u) t a2)).(\lambda (_: (((eq T t (TLRef i)) -\to (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i (CHead c0 (Bind -Abst) u) (CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity -g d u0 a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i (CHead c0 -(Bind Abst) u) (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0: -T).(arity g d u0 (asucc g a2))))))))).(\lambda (H5: (eq T (THead (Bind Abst) -u t) (TLRef i))).(let H6 \def (eq_ind T (THead (Bind Abst) u t) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I -(TLRef i) H5) in (False_ind (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: -T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: -T).(arity g d u0 (AHead a1 a2))))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: -T).(getl i c0 (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0: -T).(arity g d u0 (asucc g (AHead a1 a2))))))) H6)))))))))))) (\lambda (c0: -C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u a1)).(\lambda -(_: (((eq T u (TLRef i)) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: -T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: -T).(arity g d u0 a1)))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i -c0 (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 -(asucc g a1))))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (_: (arity g -c0 t (AHead a1 a2))).(\lambda (_: (((eq T t (TLRef i)) \to (or (ex2_2 C T -(\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abbr) u0)))) -(\lambda (d: C).(\lambda (u0: T).(arity g d u0 (AHead a1 a2))))) (ex2_2 C T -(\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abst) u0)))) -(\lambda (d: C).(\lambda (u0: T).(arity g d u0 (asucc g (AHead a1 -a2)))))))))).(\lambda (H5: (eq T (THead (Flat Appl) u t) (TLRef i))).(let H6 -\def (eq_ind T (THead (Flat Appl) u t) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef i) H5) in -(False_ind (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead -d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 a2)))) -(ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abst) -u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 (asucc g a2)))))) -H6)))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a0: A).(\lambda (_: -(arity g c0 u (asucc g a0))).(\lambda (_: (((eq T u (TLRef i)) \to (or (ex2_2 -C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abbr) u0)))) -(\lambda (d: C).(\lambda (u0: T).(arity g d u0 (asucc g a0))))) (ex2_2 C T -(\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abst) u0)))) -(\lambda (d: C).(\lambda (u0: T).(arity g d u0 (asucc g (asucc g -a0)))))))))).(\lambda (t: T).(\lambda (_: (arity g c0 t a0)).(\lambda (_: -(((eq T t (TLRef i)) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: -T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: -T).(arity g d u0 a0)))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i -c0 (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 -(asucc g a0))))))))).(\lambda (H5: (eq T (THead (Flat Cast) u t) (TLRef -i))).(let H6 \def (eq_ind T (THead (Flat Cast) u t) (\lambda (ee: T).(match -ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | -(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef i) -H5) in (False_ind (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 -(CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 -a0)))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind -Abst) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 (asucc g a0)))))) -H6))))))))))) (\lambda (c0: C).(\lambda (t: T).(\lambda (a1: A).(\lambda (H1: -(arity g c0 t a1)).(\lambda (H2: (((eq T t (TLRef i)) \to (or (ex2_2 C T -(\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u)))) -(\lambda (d: C).(\lambda (u: T).(arity g d u a1)))) (ex2_2 C T (\lambda (d: -C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d: -C).(\lambda (u: T).(arity g d u (asucc g a1))))))))).(\lambda (a2: -A).(\lambda (H3: (leq g a1 a2)).(\lambda (H4: (eq T t (TLRef i))).(let H5 -\def (f_equal T T (\lambda (e: T).e) t (TLRef i) H4) in (let H6 \def (eq_ind -T t (\lambda (t0: T).((eq T t0 (TLRef i)) \to (or (ex2_2 C T (\lambda (d: -C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u)))) (\lambda (d: -C).(\lambda (u: T).(arity g d u a1)))) (ex2_2 C T (\lambda (d: C).(\lambda -(u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u: -T).(arity g d u (asucc g a1)))))))) H2 (TLRef i) H5) in (let H7 \def (eq_ind -T t (\lambda (t0: T).(arity g c0 t0 a1)) H1 (TLRef i) H5) in (let H8 \def (H6 -(refl_equal T (TLRef i))) in (or_ind (ex2_2 C T (\lambda (d: C).(\lambda (u: -T).(getl i c0 (CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: -T).(arity g d u a1)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 -(CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u -(asucc g a1))))) (or (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 -(CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u -a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind -Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a2)))))) -(\lambda (H9: (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d -(Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u -a1))))).(ex2_2_ind C T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d -(Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u a1))) (or -(ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) -u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u a2)))) (ex2_2 C T (\lambda -(d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d: -C).(\lambda (u: T).(arity g d u (asucc g a2)))))) (\lambda (x0: C).(\lambda -(x1: T).(\lambda (H10: (getl i c0 (CHead x0 (Bind Abbr) x1))).(\lambda (H11: -(arity g x0 x1 a1)).(or_introl (ex2_2 C T (\lambda (d: C).(\lambda (u: -T).(getl i c0 (CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: -T).(arity g d u a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 -(CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u -(asucc g a2))))) (ex2_2_intro C T (\lambda (d: C).(\lambda (u: T).(getl i c0 -(CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u a2))) -x0 x1 H10 (arity_repl g x0 x1 a1 H11 a2 H3))))))) H9)) (\lambda (H9: (ex2_2 C -T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) -(\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a1)))))).(ex2_2_ind C T -(\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) -(\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a1)))) (or (ex2_2 C T -(\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u)))) -(\lambda (d: C).(\lambda (u: T).(arity g d u a2)))) (ex2_2 C T (\lambda (d: -C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d: -C).(\lambda (u: T).(arity g d u (asucc g a2)))))) (\lambda (x0: C).(\lambda -(x1: T).(\lambda (H10: (getl i c0 (CHead x0 (Bind Abst) x1))).(\lambda (H11: -(arity g x0 x1 (asucc g a1))).(or_intror (ex2_2 C T (\lambda (d: C).(\lambda -(u: T).(getl i c0 (CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: -T).(arity g d u a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 -(CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u -(asucc g a2))))) (ex2_2_intro C T (\lambda (d: C).(\lambda (u: T).(getl i c0 -(CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u -(asucc g a2)))) x0 x1 H10 (arity_repl g x0 x1 (asucc g a1) H11 (asucc g a2) -(asucc_repl g a1 a2 H3)))))))) H9)) H8))))))))))))) c y a H0))) H))))). -(* COMMENTS -Initial nodes: 3853 -END *) - -theorem arity_gen_bind: - \forall (b: B).((not (eq B b Abst)) \to (\forall (g: G).(\forall (c: -C).(\forall (u: T).(\forall (t: T).(\forall (a2: A).((arity g c (THead (Bind -b) u t) a2) \to (ex2 A (\lambda (a1: A).(arity g c u a1)) (\lambda (_: -A).(arity g (CHead c (Bind b) u) t a2)))))))))) -\def - \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (g: G).(\lambda -(c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (a2: A).(\lambda (H0: (arity -g c (THead (Bind b) u t) a2)).(insert_eq T (THead (Bind b) u t) (\lambda (t0: -T).(arity g c t0 a2)) (\lambda (_: T).(ex2 A (\lambda (a1: A).(arity g c u -a1)) (\lambda (_: A).(arity g (CHead c (Bind b) u) t a2)))) (\lambda (y: -T).(\lambda (H1: (arity g c y a2)).(arity_ind g (\lambda (c0: C).(\lambda -(t0: T).(\lambda (a: A).((eq T t0 (THead (Bind b) u t)) \to (ex2 A (\lambda -(a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t -a))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (H2: (eq T (TSort n) -(THead (Bind b) u t))).(let H3 \def (eq_ind T (TSort n) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I -(THead (Bind b) u t) H2) in (False_ind (ex2 A (\lambda (a1: A).(arity g c0 u -a1)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t (ASort O n)))) H3))))) -(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda -(_: (getl i c0 (CHead d (Bind Abbr) u0))).(\lambda (a: A).(\lambda (_: (arity -g d u0 a)).(\lambda (_: (((eq T u0 (THead (Bind b) u t)) \to (ex2 A (\lambda -(a1: A).(arity g d u a1)) (\lambda (_: A).(arity g (CHead d (Bind b) u) t -a)))))).(\lambda (H5: (eq T (TLRef i) (THead (Bind b) u t))).(let H6 \def -(eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | -(THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t) H5) in (False_ind -(ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g (CHead c0 -(Bind b) u) t a))) H6))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda -(u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind Abst) -u0))).(\lambda (a: A).(\lambda (_: (arity g d u0 (asucc g a))).(\lambda (_: -(((eq T u0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a1: A).(arity g d u -a1)) (\lambda (_: A).(arity g (CHead d (Bind b) u) t (asucc g -a))))))).(\lambda (H5: (eq T (TLRef i) (THead (Bind b) u t))).(let H6 \def -(eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | -(THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t) H5) in (False_ind -(ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g (CHead c0 -(Bind b) u) t a))) H6))))))))))) (\lambda (b0: B).(\lambda (H2: (not (eq B b0 -Abst))).(\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (H3: -(arity g c0 u0 a1)).(\lambda (H4: (((eq T u0 (THead (Bind b) u t)) \to (ex2 A -(\lambda (a3: A).(arity g c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind -b) u) t a1)))))).(\lambda (t0: T).(\lambda (a0: A).(\lambda (H5: (arity g -(CHead c0 (Bind b0) u0) t0 a0)).(\lambda (H6: (((eq T t0 (THead (Bind b) u -t)) \to (ex2 A (\lambda (a3: A).(arity g (CHead c0 (Bind b0) u0) u a3)) -(\lambda (_: A).(arity g (CHead (CHead c0 (Bind b0) u0) (Bind b) u) t -a0)))))).(\lambda (H7: (eq T (THead (Bind b0) u0 t0) (THead (Bind b) u -t))).(let H8 \def (f_equal T B (\lambda (e: T).(match e in T return (\lambda -(_: T).B) with [(TSort _) \Rightarrow b0 | (TLRef _) \Rightarrow b0 | (THead -k _ _) \Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b1) -\Rightarrow b1 | (Flat _) \Rightarrow b0])])) (THead (Bind b0) u0 t0) (THead -(Bind b) u t) H7) in ((let H9 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) -\Rightarrow u0 | (THead _ t1 _) \Rightarrow t1])) (THead (Bind b0) u0 t0) -(THead (Bind b) u t) H7) in ((let H10 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | -(TLRef _) \Rightarrow t0 | (THead _ _ t1) \Rightarrow t1])) (THead (Bind b0) -u0 t0) (THead (Bind b) u t) H7) in (\lambda (H11: (eq T u0 u)).(\lambda (H12: -(eq B b0 b)).(let H13 \def (eq_ind T t0 (\lambda (t1: T).((eq T t1 (THead -(Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g (CHead c0 (Bind b0) u0) u -a3)) (\lambda (_: A).(arity g (CHead (CHead c0 (Bind b0) u0) (Bind b) u) t -a0))))) H6 t H10) in (let H14 \def (eq_ind T t0 (\lambda (t1: T).(arity g -(CHead c0 (Bind b0) u0) t1 a0)) H5 t H10) in (let H15 \def (eq_ind T u0 -(\lambda (t1: T).((eq T t (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: -A).(arity g (CHead c0 (Bind b0) t1) u a3)) (\lambda (_: A).(arity g (CHead -(CHead c0 (Bind b0) t1) (Bind b) u) t a0))))) H13 u H11) in (let H16 \def -(eq_ind T u0 (\lambda (t1: T).(arity g (CHead c0 (Bind b0) t1) t a0)) H14 u -H11) in (let H17 \def (eq_ind T u0 (\lambda (t1: T).((eq T t1 (THead (Bind b) -u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (_: A).(arity g -(CHead c0 (Bind b) u) t a1))))) H4 u H11) in (let H18 \def (eq_ind T u0 -(\lambda (t1: T).(arity g c0 t1 a1)) H3 u H11) in (let H19 \def (eq_ind B b0 -(\lambda (b1: B).((eq T t (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: -A).(arity g (CHead c0 (Bind b1) u) u a3)) (\lambda (_: A).(arity g (CHead -(CHead c0 (Bind b1) u) (Bind b) u) t a0))))) H15 b H12) in (let H20 \def -(eq_ind B b0 (\lambda (b1: B).(arity g (CHead c0 (Bind b1) u) t a0)) H16 b -H12) in (let H21 \def (eq_ind B b0 (\lambda (b1: B).(not (eq B b1 Abst))) H2 -b H12) in (ex_intro2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (_: -A).(arity g (CHead c0 (Bind b) u) t a0)) a1 H18 H20))))))))))))) H9)) -H8)))))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: A).(\lambda -(H2: (arity g c0 u0 (asucc g a1))).(\lambda (H3: (((eq T u0 (THead (Bind b) u -t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (_: A).(arity g -(CHead c0 (Bind b) u) t (asucc g a1))))))).(\lambda (t0: T).(\lambda (a0: -A).(\lambda (H4: (arity g (CHead c0 (Bind Abst) u0) t0 a0)).(\lambda (H5: -(((eq T t0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g (CHead -c0 (Bind Abst) u0) u a3)) (\lambda (_: A).(arity g (CHead (CHead c0 (Bind -Abst) u0) (Bind b) u) t a0)))))).(\lambda (H6: (eq T (THead (Bind Abst) u0 -t0) (THead (Bind b) u t))).(let H7 \def (f_equal T B (\lambda (e: T).(match e -in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow Abst | (TLRef _) -\Rightarrow Abst | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abst])])) -(THead (Bind Abst) u0 t0) (THead (Bind b) u t) H6) in ((let H8 \def (f_equal -T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t1 _) \Rightarrow t1])) -(THead (Bind Abst) u0 t0) (THead (Bind b) u t) H6) in ((let H9 \def (f_equal -T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t1) \Rightarrow t1])) -(THead (Bind Abst) u0 t0) (THead (Bind b) u t) H6) in (\lambda (H10: (eq T u0 -u)).(\lambda (H11: (eq B Abst b)).(let H12 \def (eq_ind T t0 (\lambda (t1: -T).((eq T t1 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g -(CHead c0 (Bind Abst) u0) u a3)) (\lambda (_: A).(arity g (CHead (CHead c0 -(Bind Abst) u0) (Bind b) u) t a0))))) H5 t H9) in (let H13 \def (eq_ind T t0 -(\lambda (t1: T).(arity g (CHead c0 (Bind Abst) u0) t1 a0)) H4 t H9) in (let -H14 \def (eq_ind T u0 (\lambda (t1: T).((eq T t (THead (Bind b) u t)) \to -(ex2 A (\lambda (a3: A).(arity g (CHead c0 (Bind Abst) t1) u a3)) (\lambda -(_: A).(arity g (CHead (CHead c0 (Bind Abst) t1) (Bind b) u) t a0))))) H12 u -H10) in (let H15 \def (eq_ind T u0 (\lambda (t1: T).(arity g (CHead c0 (Bind -Abst) t1) t a0)) H13 u H10) in (let H16 \def (eq_ind T u0 (\lambda (t1: -T).((eq T t1 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u -a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t (asucc g a1)))))) H3 u -H10) in (let H17 \def (eq_ind T u0 (\lambda (t1: T).(arity g c0 t1 (asucc g -a1))) H2 u H10) in (let H18 \def (eq_ind_r B b (\lambda (b0: B).((eq T t -(THead (Bind b0) u t)) \to (ex2 A (\lambda (a3: A).(arity g (CHead c0 (Bind -Abst) u) u a3)) (\lambda (_: A).(arity g (CHead (CHead c0 (Bind Abst) u) -(Bind b0) u) t a0))))) H14 Abst H11) in (let H19 \def (eq_ind_r B b (\lambda -(b0: B).((eq T u (THead (Bind b0) u t)) \to (ex2 A (\lambda (a3: A).(arity g -c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b0) u) t (asucc g a1)))))) -H16 Abst H11) in (let H20 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 -Abst))) H Abst H11) in (eq_ind B Abst (\lambda (b0: B).(ex2 A (\lambda (a3: -A).(arity g c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b0) u) t -(AHead a1 a0))))) (let H21 \def (match (H20 (refl_equal B Abst)) in False -return (\lambda (_: False).(ex2 A (\lambda (a3: A).(arity g c0 u a3)) -(\lambda (_: A).(arity g (CHead c0 (Bind Abst) u) t (AHead a1 a0))))) with -[]) in H21) b H11))))))))))))) H8)) H7)))))))))))) (\lambda (c0: C).(\lambda -(u0: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u0 a1)).(\lambda (_: (((eq -T u0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) -(\lambda (_: A).(arity g (CHead c0 (Bind b) u) t a1)))))).(\lambda (t0: -T).(\lambda (a0: A).(\lambda (_: (arity g c0 t0 (AHead a1 a0))).(\lambda (_: -(((eq T t0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u -a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t (AHead a1 -a0))))))).(\lambda (H6: (eq T (THead (Flat Appl) u0 t0) (THead (Bind b) u -t))).(let H7 \def (eq_ind T (THead (Flat Appl) u0 t0) (\lambda (ee: T).(match -ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | -(TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return -(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind b) u t) H6) in (False_ind (ex2 A (\lambda (a3: -A).(arity g c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t a0))) -H7)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a: A).(\lambda (_: -(arity g c0 u0 (asucc g a))).(\lambda (_: (((eq T u0 (THead (Bind b) u t)) -\to (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g -(CHead c0 (Bind b) u) t (asucc g a))))))).(\lambda (t0: T).(\lambda (_: -(arity g c0 t0 a)).(\lambda (_: (((eq T t0 (THead (Bind b) u t)) \to (ex2 A -(\lambda (a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g (CHead c0 (Bind -b) u) t a)))))).(\lambda (H6: (eq T (THead (Flat Cast) u0 t0) (THead (Bind b) -u t))).(let H7 \def (eq_ind T (THead (Flat Cast) u0 t0) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K -return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) -\Rightarrow True])])) I (THead (Bind b) u t) H6) in (False_ind (ex2 A -(\lambda (a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g (CHead c0 (Bind -b) u) t a))) H7))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (a1: -A).(\lambda (H2: (arity g c0 t0 a1)).(\lambda (H3: (((eq T t0 (THead (Bind b) -u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (_: A).(arity g -(CHead c0 (Bind b) u) t a1)))))).(\lambda (a0: A).(\lambda (H4: (leq g a1 -a0)).(\lambda (H5: (eq T t0 (THead (Bind b) u t))).(let H6 \def (f_equal T T -(\lambda (e: T).e) t0 (THead (Bind b) u t) H5) in (let H7 \def (eq_ind T t0 -(\lambda (t1: T).((eq T t1 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: -A).(arity g c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t -a1))))) H3 (THead (Bind b) u t) H6) in (let H8 \def (eq_ind T t0 (\lambda -(t1: T).(arity g c0 t1 a1)) H2 (THead (Bind b) u t) H6) in (let H9 \def (H7 -(refl_equal T (THead (Bind b) u t))) in (ex2_ind A (\lambda (a3: A).(arity g -c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t a1)) (ex2 A -(\lambda (a3: A).(arity g c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind -b) u) t a0))) (\lambda (x: A).(\lambda (H10: (arity g c0 u x)).(\lambda (H11: -(arity g (CHead c0 (Bind b) u) t a1)).(ex_intro2 A (\lambda (a3: A).(arity g -c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t a0)) x H10 -(arity_repl g (CHead c0 (Bind b) u) t a1 H11 a0 H4))))) H9))))))))))))) c y -a2 H1))) H0)))))))). -(* COMMENTS -Initial nodes: 3365 -END *) - -theorem arity_gen_abst: - \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (a: -A).((arity g c (THead (Bind Abst) u t) a) \to (ex3_2 A A (\lambda (a1: -A).(\lambda (a2: A).(eq A a (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: -A).(arity g c u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g -(CHead c (Bind Abst) u) t a2))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (a: -A).(\lambda (H: (arity g c (THead (Bind Abst) u t) a)).(insert_eq T (THead -(Bind Abst) u t) (\lambda (t0: T).(arity g c t0 a)) (\lambda (_: T).(ex3_2 A -A (\lambda (a1: A).(\lambda (a2: A).(eq A a (AHead a1 a2)))) (\lambda (a1: -A).(\lambda (_: A).(arity g c u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: -A).(arity g (CHead c (Bind Abst) u) t a2))))) (\lambda (y: T).(\lambda (H0: -(arity g c y a)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (a0: -A).((eq T t0 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a1: -A).(\lambda (a2: A).(eq A a0 (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: -A).(arity g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g -(CHead c0 (Bind Abst) u) t a2)))))))) (\lambda (c0: C).(\lambda (n: -nat).(\lambda (H1: (eq T (TSort n) (THead (Bind Abst) u t))).(let H2 \def -(eq_ind T (TSort n) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | -(THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) u t) H1) in -(False_ind (ex3_2 A A (\lambda (a1: A).(\lambda (a2: A).(eq A (ASort O n) -(AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g c0 u (asucc g -a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g (CHead c0 (Bind Abst) u) t -a2)))) H2))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: -nat).(\lambda (_: (getl i c0 (CHead d (Bind Abbr) u0))).(\lambda (a0: -A).(\lambda (_: (arity g d u0 a0)).(\lambda (_: (((eq T u0 (THead (Bind Abst) -u t)) \to (ex3_2 A A (\lambda (a1: A).(\lambda (a2: A).(eq A a0 (AHead a1 -a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g d u (asucc g a1)))) (\lambda -(_: A).(\lambda (a2: A).(arity g (CHead d (Bind Abst) u) t a2))))))).(\lambda -(H4: (eq T (TLRef i) (THead (Bind Abst) u t))).(let H5 \def (eq_ind T (TLRef -i) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort -_) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow -False])) I (THead (Bind Abst) u t) H4) in (False_ind (ex3_2 A A (\lambda (a1: -A).(\lambda (a2: A).(eq A a0 (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: -A).(arity g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g -(CHead c0 (Bind Abst) u) t a2)))) H5))))))))))) (\lambda (c0: C).(\lambda (d: -C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind -Abst) u0))).(\lambda (a0: A).(\lambda (_: (arity g d u0 (asucc g -a0))).(\lambda (_: (((eq T u0 (THead (Bind Abst) u t)) \to (ex3_2 A A -(\lambda (a1: A).(\lambda (a2: A).(eq A (asucc g a0) (AHead a1 a2)))) -(\lambda (a1: A).(\lambda (_: A).(arity g d u (asucc g a1)))) (\lambda (_: -A).(\lambda (a2: A).(arity g (CHead d (Bind Abst) u) t a2))))))).(\lambda -(H4: (eq T (TLRef i) (THead (Bind Abst) u t))).(let H5 \def (eq_ind T (TLRef -i) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort -_) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow -False])) I (THead (Bind Abst) u t) H4) in (False_ind (ex3_2 A A (\lambda (a1: -A).(\lambda (a2: A).(eq A a0 (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: -A).(arity g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g -(CHead c0 (Bind Abst) u) t a2)))) H5))))))))))) (\lambda (b: B).(\lambda (H1: -(not (eq B b Abst))).(\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: -A).(\lambda (H2: (arity g c0 u0 a1)).(\lambda (H3: (((eq T u0 (THead (Bind -Abst) u t)) \to (ex3_2 A A (\lambda (a2: A).(\lambda (a3: A).(eq A a1 (AHead -a2 a3)))) (\lambda (a2: A).(\lambda (_: A).(arity g c0 u (asucc g a2)))) -(\lambda (_: A).(\lambda (a3: A).(arity g (CHead c0 (Bind Abst) u) t -a3))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (H4: (arity g (CHead c0 -(Bind b) u0) t0 a2)).(\lambda (H5: (((eq T t0 (THead (Bind Abst) u t)) \to -(ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) -(\lambda (a3: A).(\lambda (_: A).(arity g (CHead c0 (Bind b) u0) u (asucc g -a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead (CHead c0 (Bind b) -u0) (Bind Abst) u) t a4))))))).(\lambda (H6: (eq T (THead (Bind b) u0 t0) -(THead (Bind Abst) u t))).(let H7 \def (f_equal T B (\lambda (e: T).(match e -in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | (TLRef _) -\Rightarrow b | (THead k _ _) \Rightarrow (match k in K return (\lambda (_: -K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow b])])) (THead -(Bind b) u0 t0) (THead (Bind Abst) u t) H6) in ((let H8 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t1 _) \Rightarrow t1])) -(THead (Bind b) u0 t0) (THead (Bind Abst) u t) H6) in ((let H9 \def (f_equal -T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t1) \Rightarrow t1])) -(THead (Bind b) u0 t0) (THead (Bind Abst) u t) H6) in (\lambda (H10: (eq T u0 -u)).(\lambda (H11: (eq B b Abst)).(let H12 \def (eq_ind T t0 (\lambda (t1: -T).((eq T t1 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3: -A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: -A).(arity g (CHead c0 (Bind b) u0) u (asucc g a3)))) (\lambda (_: A).(\lambda -(a4: A).(arity g (CHead (CHead c0 (Bind b) u0) (Bind Abst) u) t a4)))))) H5 t -H9) in (let H13 \def (eq_ind T t0 (\lambda (t1: T).(arity g (CHead c0 (Bind -b) u0) t1 a2)) H4 t H9) in (let H14 \def (eq_ind T u0 (\lambda (t1: T).((eq T -t (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: -A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g (CHead -c0 (Bind b) t1) u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g -(CHead (CHead c0 (Bind b) t1) (Bind Abst) u) t a4)))))) H12 u H10) in (let -H15 \def (eq_ind T u0 (\lambda (t1: T).(arity g (CHead c0 (Bind b) t1) t a2)) -H13 u H10) in (let H16 \def (eq_ind T u0 (\lambda (t1: T).((eq T t1 (THead -(Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a1 -(AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g -a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t -a4)))))) H3 u H10) in (let H17 \def (eq_ind T u0 (\lambda (t1: T).(arity g c0 -t1 a1)) H2 u H10) in (let H18 \def (eq_ind B b (\lambda (b0: B).((eq T t -(THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq -A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g (CHead c0 -(Bind b0) u) u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g -(CHead (CHead c0 (Bind b0) u) (Bind Abst) u) t a4)))))) H14 Abst H11) in (let -H19 \def (eq_ind B b (\lambda (b0: B).(arity g (CHead c0 (Bind b0) u) t a2)) -H15 Abst H11) in (let H20 \def (eq_ind B b (\lambda (b0: B).(not (eq B b0 -Abst))) H1 Abst H11) in (let H21 \def (match (H20 (refl_equal B Abst)) in -False return (\lambda (_: False).(ex3_2 A A (\lambda (a3: A).(\lambda (a4: -A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u -(asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind -Abst) u) t a4))))) with []) in H21))))))))))))) H8)) H7)))))))))))))) -(\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (H1: (arity g c0 -u0 (asucc g a1))).(\lambda (H2: (((eq T u0 (THead (Bind Abst) u t)) \to -(ex3_2 A A (\lambda (a2: A).(\lambda (a3: A).(eq A (asucc g a1) (AHead a2 -a3)))) (\lambda (a2: A).(\lambda (_: A).(arity g c0 u (asucc g a2)))) -(\lambda (_: A).(\lambda (a3: A).(arity g (CHead c0 (Bind Abst) u) t -a3))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (H3: (arity g (CHead c0 -(Bind Abst) u0) t0 a2)).(\lambda (H4: (((eq T t0 (THead (Bind Abst) u t)) \to -(ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) -(\lambda (a3: A).(\lambda (_: A).(arity g (CHead c0 (Bind Abst) u0) u (asucc -g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead (CHead c0 (Bind -Abst) u0) (Bind Abst) u) t a4))))))).(\lambda (H5: (eq T (THead (Bind Abst) -u0 t0) (THead (Bind Abst) u t))).(let H6 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | -(TLRef _) \Rightarrow u0 | (THead _ t1 _) \Rightarrow t1])) (THead (Bind -Abst) u0 t0) (THead (Bind Abst) u t) H5) in ((let H7 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t1) \Rightarrow t1])) -(THead (Bind Abst) u0 t0) (THead (Bind Abst) u t) H5) in (\lambda (H8: (eq T -u0 u)).(let H9 \def (eq_ind T t0 (\lambda (t1: T).((eq T t1 (THead (Bind -Abst) u t)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead -a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g (CHead c0 (Bind Abst) u0) -u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead (CHead c0 -(Bind Abst) u0) (Bind Abst) u) t a4)))))) H4 t H7) in (let H10 \def (eq_ind T -t0 (\lambda (t1: T).(arity g (CHead c0 (Bind Abst) u0) t1 a2)) H3 t H7) in -(let H11 \def (eq_ind T u0 (\lambda (t1: T).((eq T t (THead (Bind Abst) u t)) -\to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) -(\lambda (a3: A).(\lambda (_: A).(arity g (CHead c0 (Bind Abst) t1) u (asucc -g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead (CHead c0 (Bind -Abst) t1) (Bind Abst) u) t a4)))))) H9 u H8) in (let H12 \def (eq_ind T u0 -(\lambda (t1: T).(arity g (CHead c0 (Bind Abst) t1) t a2)) H10 u H8) in (let -H13 \def (eq_ind T u0 (\lambda (t1: T).((eq T t1 (THead (Bind Abst) u t)) \to -(ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A (asucc g a1) (AHead a3 -a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g a3)))) -(\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t a4)))))) -H2 u H8) in (let H14 \def (eq_ind T u0 (\lambda (t1: T).(arity g c0 t1 (asucc -g a1))) H1 u H8) in (ex3_2_intro A A (\lambda (a3: A).(\lambda (a4: A).(eq A -(AHead a1 a2) (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u -(asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind -Abst) u) t a4))) a1 a2 (refl_equal A (AHead a1 a2)) H14 H12))))))))) -H6)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: A).(\lambda -(_: (arity g c0 u0 a1)).(\lambda (_: (((eq T u0 (THead (Bind Abst) u t)) \to -(ex3_2 A A (\lambda (a2: A).(\lambda (a3: A).(eq A a1 (AHead a2 a3)))) -(\lambda (a2: A).(\lambda (_: A).(arity g c0 u (asucc g a2)))) (\lambda (_: -A).(\lambda (a3: A).(arity g (CHead c0 (Bind Abst) u) t a3))))))).(\lambda -(t0: T).(\lambda (a2: A).(\lambda (_: (arity g c0 t0 (AHead a1 a2))).(\lambda -(_: (((eq T t0 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3: -A).(\lambda (a4: A).(eq A (AHead a1 a2) (AHead a3 a4)))) (\lambda (a3: -A).(\lambda (_: A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda -(a4: A).(arity g (CHead c0 (Bind Abst) u) t a4))))))).(\lambda (H5: (eq T -(THead (Flat Appl) u0 t0) (THead (Bind Abst) u t))).(let H6 \def (eq_ind T -(THead (Flat Appl) u0 t0) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind -Abst) u t) H5) in (False_ind (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq -A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g -a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t -a4)))) H6)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a0: -A).(\lambda (_: (arity g c0 u0 (asucc g a0))).(\lambda (_: (((eq T u0 (THead -(Bind Abst) u t)) \to (ex3_2 A A (\lambda (a1: A).(\lambda (a2: A).(eq A -(asucc g a0) (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g c0 u -(asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g (CHead c0 (Bind -Abst) u) t a2))))))).(\lambda (t0: T).(\lambda (_: (arity g c0 t0 -a0)).(\lambda (_: (((eq T t0 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda -(a1: A).(\lambda (a2: A).(eq A a0 (AHead a1 a2)))) (\lambda (a1: A).(\lambda -(_: A).(arity g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity -g (CHead c0 (Bind Abst) u) t a2))))))).(\lambda (H5: (eq T (THead (Flat Cast) -u0 t0) (THead (Bind Abst) u t))).(let H6 \def (eq_ind T (THead (Flat Cast) u0 -t0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort -_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) -\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u t) -H5) in (False_ind (ex3_2 A A (\lambda (a1: A).(\lambda (a2: A).(eq A a0 -(AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g c0 u (asucc g -a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g (CHead c0 (Bind Abst) u) t -a2)))) H6))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (a1: -A).(\lambda (H1: (arity g c0 t0 a1)).(\lambda (H2: (((eq T t0 (THead (Bind -Abst) u t)) \to (ex3_2 A A (\lambda (a2: A).(\lambda (a3: A).(eq A a1 (AHead -a2 a3)))) (\lambda (a2: A).(\lambda (_: A).(arity g c0 u (asucc g a2)))) -(\lambda (_: A).(\lambda (a3: A).(arity g (CHead c0 (Bind Abst) u) t -a3))))))).(\lambda (a2: A).(\lambda (H3: (leq g a1 a2)).(\lambda (H4: (eq T -t0 (THead (Bind Abst) u t))).(let H5 \def (f_equal T T (\lambda (e: T).e) t0 -(THead (Bind Abst) u t) H4) in (let H6 \def (eq_ind T t0 (\lambda (t1: -T).((eq T t1 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3: -A).(\lambda (a4: A).(eq A a1 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: -A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g -(CHead c0 (Bind Abst) u) t a4)))))) H2 (THead (Bind Abst) u t) H5) in (let H7 -\def (eq_ind T t0 (\lambda (t1: T).(arity g c0 t1 a1)) H1 (THead (Bind Abst) -u t) H5) in (let H8 \def (H6 (refl_equal T (THead (Bind Abst) u t))) in -(ex3_2_ind A A (\lambda (a3: A).(\lambda (a4: A).(eq A a1 (AHead a3 a4)))) -(\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g a3)))) (\lambda (_: -A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t a4))) (ex3_2 A A -(\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: -A).(\lambda (_: A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda -(a4: A).(arity g (CHead c0 (Bind Abst) u) t a4)))) (\lambda (x0: A).(\lambda -(x1: A).(\lambda (H9: (eq A a1 (AHead x0 x1))).(\lambda (H10: (arity g c0 u -(asucc g x0))).(\lambda (H11: (arity g (CHead c0 (Bind Abst) u) t x1)).(let -H12 \def (eq_ind A a1 (\lambda (a0: A).(leq g a0 a2)) H3 (AHead x0 x1) H9) in -(let H13 \def (eq_ind A a1 (\lambda (a0: A).(arity g c0 (THead (Bind Abst) u -t) a0)) H7 (AHead x0 x1) H9) in (let H_x \def (leq_gen_head1 g x0 x1 a2 H12) -in (let H14 \def H_x in (ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq -g x0 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g x1 a4))) (\lambda (a3: -A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) (ex3_2 A A (\lambda (a3: -A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: -A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g -(CHead c0 (Bind Abst) u) t a4)))) (\lambda (x2: A).(\lambda (x3: A).(\lambda -(H15: (leq g x0 x2)).(\lambda (H16: (leq g x1 x3)).(\lambda (H17: (eq A a2 -(AHead x2 x3))).(let H18 \def (f_equal A A (\lambda (e: A).e) a2 (AHead x2 -x3) H17) in (eq_ind_r A (AHead x2 x3) (\lambda (a0: A).(ex3_2 A A (\lambda -(a3: A).(\lambda (a4: A).(eq A a0 (AHead a3 a4)))) (\lambda (a3: A).(\lambda -(_: A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity -g (CHead c0 (Bind Abst) u) t a4))))) (ex3_2_intro A A (\lambda (a3: -A).(\lambda (a4: A).(eq A (AHead x2 x3) (AHead a3 a4)))) (\lambda (a3: -A).(\lambda (_: A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda -(a4: A).(arity g (CHead c0 (Bind Abst) u) t a4))) x2 x3 (refl_equal A (AHead -x2 x3)) (arity_repl g c0 u (asucc g x0) H10 (asucc g x2) (asucc_repl g x0 x2 -H15)) (arity_repl g (CHead c0 (Bind Abst) u) t x1 H11 x3 H16)) a2 H18))))))) -H14)))))))))) H8))))))))))))) c y a H0))) H)))))). -(* COMMENTS -Initial nodes: 4265 -END *) - -theorem arity_gen_appl: - \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (a2: -A).((arity g c (THead (Flat Appl) u t) a2) \to (ex2 A (\lambda (a1: A).(arity -g c u a1)) (\lambda (a1: A).(arity g c t (AHead a1 a2))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (a2: -A).(\lambda (H: (arity g c (THead (Flat Appl) u t) a2)).(insert_eq T (THead -(Flat Appl) u t) (\lambda (t0: T).(arity g c t0 a2)) (\lambda (_: T).(ex2 A -(\lambda (a1: A).(arity g c u a1)) (\lambda (a1: A).(arity g c t (AHead a1 -a2))))) (\lambda (y: T).(\lambda (H0: (arity g c y a2)).(arity_ind g (\lambda -(c0: C).(\lambda (t0: T).(\lambda (a: A).((eq T t0 (THead (Flat Appl) u t)) -\to (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t -(AHead a1 a)))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (H1: (eq T -(TSort n) (THead (Flat Appl) u t))).(let H2 \def (eq_ind T (TSort n) (\lambda -(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow -False])) I (THead (Flat Appl) u t) H1) in (False_ind (ex2 A (\lambda (a1: -A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t (AHead a1 (ASort O -n))))) H2))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: -nat).(\lambda (_: (getl i c0 (CHead d (Bind Abbr) u0))).(\lambda (a: -A).(\lambda (_: (arity g d u0 a)).(\lambda (_: (((eq T u0 (THead (Flat Appl) -u t)) \to (ex2 A (\lambda (a1: A).(arity g d u a1)) (\lambda (a1: A).(arity g -d t (AHead a1 a))))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat Appl) u -t))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) u -t) H4) in (False_ind (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: -A).(arity g c0 t (AHead a1 a)))) H5))))))))))) (\lambda (c0: C).(\lambda (d: -C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind -Abst) u0))).(\lambda (a: A).(\lambda (_: (arity g d u0 (asucc g a))).(\lambda -(_: (((eq T u0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda (a1: A).(arity g -d u a1)) (\lambda (a1: A).(arity g d t (AHead a1 (asucc g a)))))))).(\lambda -(H4: (eq T (TLRef i) (THead (Flat Appl) u t))).(let H5 \def (eq_ind T (TLRef -i) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort -_) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow -False])) I (THead (Flat Appl) u t) H4) in (False_ind (ex2 A (\lambda (a1: -A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t (AHead a1 a)))) -H5))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (c0: -C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u0 -a1)).(\lambda (_: (((eq T u0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda -(a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3 -a1))))))).(\lambda (t0: T).(\lambda (a0: A).(\lambda (_: (arity g (CHead c0 -(Bind b) u0) t0 a0)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u t)) \to -(ex2 A (\lambda (a3: A).(arity g (CHead c0 (Bind b) u0) u a3)) (\lambda (a3: -A).(arity g (CHead c0 (Bind b) u0) t (AHead a3 a0))))))).(\lambda (H6: (eq T -(THead (Bind b) u0 t0) (THead (Flat Appl) u t))).(let H7 \def (eq_ind T -(THead (Bind b) u0 t0) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat -Appl) u t) H6) in (False_ind (ex2 A (\lambda (a3: A).(arity g c0 u a3)) -(\lambda (a3: A).(arity g c0 t (AHead a3 a0)))) H7)))))))))))))) (\lambda -(c0: C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u0 (asucc -g a1))).(\lambda (_: (((eq T u0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda -(a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3 (asucc g -a1)))))))).(\lambda (t0: T).(\lambda (a0: A).(\lambda (_: (arity g (CHead c0 -(Bind Abst) u0) t0 a0)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u t)) \to -(ex2 A (\lambda (a3: A).(arity g (CHead c0 (Bind Abst) u0) u a3)) (\lambda -(a3: A).(arity g (CHead c0 (Bind Abst) u0) t (AHead a3 a0))))))).(\lambda -(H5: (eq T (THead (Bind Abst) u0 t0) (THead (Flat Appl) u t))).(let H6 \def -(eq_ind T (THead (Bind Abst) u0 t0) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow -False])])) I (THead (Flat Appl) u t) H5) in (False_ind (ex2 A (\lambda (a3: -A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3 (AHead a1 -a0))))) H6)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: -A).(\lambda (H1: (arity g c0 u0 a1)).(\lambda (H2: (((eq T u0 (THead (Flat -Appl) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3: -A).(arity g c0 t (AHead a3 a1))))))).(\lambda (t0: T).(\lambda (a0: -A).(\lambda (H3: (arity g c0 t0 (AHead a1 a0))).(\lambda (H4: (((eq T t0 -(THead (Flat Appl) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) -(\lambda (a3: A).(arity g c0 t (AHead a3 (AHead a1 a0)))))))).(\lambda (H5: -(eq T (THead (Flat Appl) u0 t0) (THead (Flat Appl) u t))).(let H6 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t1 _) -\Rightarrow t1])) (THead (Flat Appl) u0 t0) (THead (Flat Appl) u t) H5) in -((let H7 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: -T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ -t1) \Rightarrow t1])) (THead (Flat Appl) u0 t0) (THead (Flat Appl) u t) H5) -in (\lambda (H8: (eq T u0 u)).(let H9 \def (eq_ind T t0 (\lambda (t1: T).((eq -T t1 (THead (Flat Appl) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) -(\lambda (a3: A).(arity g c0 t (AHead a3 (AHead a1 a0))))))) H4 t H7) in (let -H10 \def (eq_ind T t0 (\lambda (t1: T).(arity g c0 t1 (AHead a1 a0))) H3 t -H7) in (let H11 \def (eq_ind T u0 (\lambda (t1: T).((eq T t1 (THead (Flat -Appl) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3: -A).(arity g c0 t (AHead a3 a1)))))) H2 u H8) in (let H12 \def (eq_ind T u0 -(\lambda (t1: T).(arity g c0 t1 a1)) H1 u H8) in (ex_intro2 A (\lambda (a3: -A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3 a0))) a1 H12 -H10))))))) H6)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a: -A).(\lambda (_: (arity g c0 u0 (asucc g a))).(\lambda (_: (((eq T u0 (THead -(Flat Appl) u t)) \to (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda -(a1: A).(arity g c0 t (AHead a1 (asucc g a)))))))).(\lambda (t0: T).(\lambda -(_: (arity g c0 t0 a)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u t)) \to -(ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t -(AHead a1 a))))))).(\lambda (H5: (eq T (THead (Flat Cast) u0 t0) (THead (Flat -Appl) u t))).(let H6 \def (eq_ind T (THead (Flat Cast) u0 t0) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K -return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) -\Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl \Rightarrow -False | Cast \Rightarrow True])])])) I (THead (Flat Appl) u t) H5) in -(False_ind (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity -g c0 t (AHead a1 a)))) H6))))))))))) (\lambda (c0: C).(\lambda (t0: -T).(\lambda (a1: A).(\lambda (H1: (arity g c0 t0 a1)).(\lambda (H2: (((eq T -t0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) -(\lambda (a3: A).(arity g c0 t (AHead a3 a1))))))).(\lambda (a0: A).(\lambda -(H3: (leq g a1 a0)).(\lambda (H4: (eq T t0 (THead (Flat Appl) u t))).(let H5 -\def (f_equal T T (\lambda (e: T).e) t0 (THead (Flat Appl) u t) H4) in (let -H6 \def (eq_ind T t0 (\lambda (t1: T).((eq T t1 (THead (Flat Appl) u t)) \to -(ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t -(AHead a3 a1)))))) H2 (THead (Flat Appl) u t) H5) in (let H7 \def (eq_ind T -t0 (\lambda (t1: T).(arity g c0 t1 a1)) H1 (THead (Flat Appl) u t) H5) in -(let H8 \def (H6 (refl_equal T (THead (Flat Appl) u t))) in (ex2_ind A -(\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3 -a1))) (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 -t (AHead a3 a0)))) (\lambda (x: A).(\lambda (H9: (arity g c0 u x)).(\lambda -(H10: (arity g c0 t (AHead x a1))).(ex_intro2 A (\lambda (a3: A).(arity g c0 -u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3 a0))) x H9 (arity_repl g c0 t -(AHead x a1) H10 (AHead x a0) (leq_head g x x (leq_refl g x) a1 a0 H3)))))) -H8))))))))))))) c y a2 H0))) H)))))). -(* COMMENTS -Initial nodes: 2277 -END *) - -theorem arity_gen_cast: - \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (a: -A).((arity g c (THead (Flat Cast) u t) a) \to (land (arity g c u (asucc g a)) -(arity g c t a))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (a: -A).(\lambda (H: (arity g c (THead (Flat Cast) u t) a)).(insert_eq T (THead -(Flat Cast) u t) (\lambda (t0: T).(arity g c t0 a)) (\lambda (_: T).(land -(arity g c u (asucc g a)) (arity g c t a))) (\lambda (y: T).(\lambda (H0: -(arity g c y a)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (a0: -A).((eq T t0 (THead (Flat Cast) u t)) \to (land (arity g c0 u (asucc g a0)) -(arity g c0 t a0)))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (H1: (eq T -(TSort n) (THead (Flat Cast) u t))).(let H2 \def (eq_ind T (TSort n) (\lambda -(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow -False])) I (THead (Flat Cast) u t) H1) in (False_ind (land (arity g c0 u -(asucc g (ASort O n))) (arity g c0 t (ASort O n))) H2))))) (\lambda (c0: -C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0 -(CHead d (Bind Abbr) u0))).(\lambda (a0: A).(\lambda (_: (arity g d u0 -a0)).(\lambda (_: (((eq T u0 (THead (Flat Cast) u t)) \to (land (arity g d u -(asucc g a0)) (arity g d t a0))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat -Cast) u t))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) u -t) H4) in (False_ind (land (arity g c0 u (asucc g a0)) (arity g c0 t a0)) -H5))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: -nat).(\lambda (_: (getl i c0 (CHead d (Bind Abst) u0))).(\lambda (a0: -A).(\lambda (_: (arity g d u0 (asucc g a0))).(\lambda (_: (((eq T u0 (THead -(Flat Cast) u t)) \to (land (arity g d u (asucc g (asucc g a0))) (arity g d t -(asucc g a0)))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat Cast) u -t))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) u -t) H4) in (False_ind (land (arity g c0 u (asucc g a0)) (arity g c0 t a0)) -H5))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (c0: -C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u0 -a1)).(\lambda (_: (((eq T u0 (THead (Flat Cast) u t)) \to (land (arity g c0 u -(asucc g a1)) (arity g c0 t a1))))).(\lambda (t0: T).(\lambda (a2: -A).(\lambda (_: (arity g (CHead c0 (Bind b) u0) t0 a2)).(\lambda (_: (((eq T -t0 (THead (Flat Cast) u t)) \to (land (arity g (CHead c0 (Bind b) u0) u -(asucc g a2)) (arity g (CHead c0 (Bind b) u0) t a2))))).(\lambda (H6: (eq T -(THead (Bind b) u0 t0) (THead (Flat Cast) u t))).(let H7 \def (eq_ind T -(THead (Bind b) u0 t0) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat -Cast) u t) H6) in (False_ind (land (arity g c0 u (asucc g a2)) (arity g c0 t -a2)) H7)))))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: -A).(\lambda (_: (arity g c0 u0 (asucc g a1))).(\lambda (_: (((eq T u0 (THead -(Flat Cast) u t)) \to (land (arity g c0 u (asucc g (asucc g a1))) (arity g c0 -t (asucc g a1)))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g -(CHead c0 (Bind Abst) u0) t0 a2)).(\lambda (_: (((eq T t0 (THead (Flat Cast) -u t)) \to (land (arity g (CHead c0 (Bind Abst) u0) u (asucc g a2)) (arity g -(CHead c0 (Bind Abst) u0) t a2))))).(\lambda (H5: (eq T (THead (Bind Abst) u0 -t0) (THead (Flat Cast) u t))).(let H6 \def (eq_ind T (THead (Bind Abst) u0 -t0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort -_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) -\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Cast) u t) -H5) in (False_ind (land (arity g c0 u (asucc g (AHead a1 a2))) (arity g c0 t -(AHead a1 a2))) H6)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda -(a1: A).(\lambda (_: (arity g c0 u0 a1)).(\lambda (_: (((eq T u0 (THead (Flat -Cast) u t)) \to (land (arity g c0 u (asucc g a1)) (arity g c0 t -a1))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g c0 t0 (AHead -a1 a2))).(\lambda (_: (((eq T t0 (THead (Flat Cast) u t)) \to (land (arity g -c0 u (asucc g (AHead a1 a2))) (arity g c0 t (AHead a1 a2)))))).(\lambda (H5: -(eq T (THead (Flat Appl) u0 t0) (THead (Flat Cast) u t))).(let H6 \def -(eq_ind T (THead (Flat Appl) u0 t0) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f -in F return (\lambda (_: F).Prop) with [Appl \Rightarrow True | Cast -\Rightarrow False])])])) I (THead (Flat Cast) u t) H5) in (False_ind (land -(arity g c0 u (asucc g a2)) (arity g c0 t a2)) H6)))))))))))) (\lambda (c0: -C).(\lambda (u0: T).(\lambda (a0: A).(\lambda (H1: (arity g c0 u0 (asucc g -a0))).(\lambda (H2: (((eq T u0 (THead (Flat Cast) u t)) \to (land (arity g c0 -u (asucc g (asucc g a0))) (arity g c0 t (asucc g a0)))))).(\lambda (t0: -T).(\lambda (H3: (arity g c0 t0 a0)).(\lambda (H4: (((eq T t0 (THead (Flat -Cast) u t)) \to (land (arity g c0 u (asucc g a0)) (arity g c0 t -a0))))).(\lambda (H5: (eq T (THead (Flat Cast) u0 t0) (THead (Flat Cast) u -t))).(let H6 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda -(_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead -_ t1 _) \Rightarrow t1])) (THead (Flat Cast) u0 t0) (THead (Flat Cast) u t) -H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 -| (THead _ _ t1) \Rightarrow t1])) (THead (Flat Cast) u0 t0) (THead (Flat -Cast) u t) H5) in (\lambda (H8: (eq T u0 u)).(let H9 \def (eq_ind T t0 -(\lambda (t1: T).((eq T t1 (THead (Flat Cast) u t)) \to (land (arity g c0 u -(asucc g a0)) (arity g c0 t a0)))) H4 t H7) in (let H10 \def (eq_ind T t0 -(\lambda (t1: T).(arity g c0 t1 a0)) H3 t H7) in (let H11 \def (eq_ind T u0 -(\lambda (t1: T).((eq T t1 (THead (Flat Cast) u t)) \to (land (arity g c0 u -(asucc g (asucc g a0))) (arity g c0 t (asucc g a0))))) H2 u H8) in (let H12 -\def (eq_ind T u0 (\lambda (t1: T).(arity g c0 t1 (asucc g a0))) H1 u H8) in -(conj (arity g c0 u (asucc g a0)) (arity g c0 t a0) H12 H10))))))) -H6))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda -(H1: (arity g c0 t0 a1)).(\lambda (H2: (((eq T t0 (THead (Flat Cast) u t)) -\to (land (arity g c0 u (asucc g a1)) (arity g c0 t a1))))).(\lambda (a2: -A).(\lambda (H3: (leq g a1 a2)).(\lambda (H4: (eq T t0 (THead (Flat Cast) u -t))).(let H5 \def (f_equal T T (\lambda (e: T).e) t0 (THead (Flat Cast) u t) -H4) in (let H6 \def (eq_ind T t0 (\lambda (t1: T).((eq T t1 (THead (Flat -Cast) u t)) \to (land (arity g c0 u (asucc g a1)) (arity g c0 t a1)))) H2 -(THead (Flat Cast) u t) H5) in (let H7 \def (eq_ind T t0 (\lambda (t1: -T).(arity g c0 t1 a1)) H1 (THead (Flat Cast) u t) H5) in (let H8 \def (H6 -(refl_equal T (THead (Flat Cast) u t))) in (land_ind (arity g c0 u (asucc g -a1)) (arity g c0 t a1) (land (arity g c0 u (asucc g a2)) (arity g c0 t a2)) -(\lambda (H9: (arity g c0 u (asucc g a1))).(\lambda (H10: (arity g c0 t -a1)).(conj (arity g c0 u (asucc g a2)) (arity g c0 t a2) (arity_repl g c0 u -(asucc g a1) H9 (asucc g a2) (asucc_repl g a1 a2 H3)) (arity_repl g c0 t a1 -H10 a2 H3)))) H8))))))))))))) c y a H0))) H)))))). -(* COMMENTS -Initial nodes: 2147 -END *) - -theorem arity_gen_appls: - \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (vs: TList).(\forall -(a2: A).((arity g c (THeads (Flat Appl) vs t) a2) \to (ex A (\lambda (a: -A).(arity g c t a)))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (vs: -TList).(TList_ind (\lambda (t0: TList).(\forall (a2: A).((arity g c (THeads -(Flat Appl) t0 t) a2) \to (ex A (\lambda (a: A).(arity g c t a)))))) (\lambda -(a2: A).(\lambda (H: (arity g c t a2)).(ex_intro A (\lambda (a: A).(arity g c -t a)) a2 H))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H: ((\forall -(a2: A).((arity g c (THeads (Flat Appl) t1 t) a2) \to (ex A (\lambda (a: -A).(arity g c t a))))))).(\lambda (a2: A).(\lambda (H0: (arity g c (THead -(Flat Appl) t0 (THeads (Flat Appl) t1 t)) a2)).(let H1 \def (arity_gen_appl g -c t0 (THeads (Flat Appl) t1 t) a2 H0) in (ex2_ind A (\lambda (a1: A).(arity g -c t0 a1)) (\lambda (a1: A).(arity g c (THeads (Flat Appl) t1 t) (AHead a1 -a2))) (ex A (\lambda (a: A).(arity g c t a))) (\lambda (x: A).(\lambda (_: -(arity g c t0 x)).(\lambda (H3: (arity g c (THeads (Flat Appl) t1 t) (AHead x -a2))).(let H_x \def (H (AHead x a2) H3) in (let H4 \def H_x in (ex_ind A -(\lambda (a: A).(arity g c t a)) (ex A (\lambda (a: A).(arity g c t a))) -(\lambda (x0: A).(\lambda (H5: (arity g c t x0)).(ex_intro A (\lambda (a: -A).(arity g c t a)) x0 H5))) H4)))))) H1))))))) vs)))). -(* COMMENTS -Initial nodes: 341 -END *) - -theorem arity_gen_lift: - \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).(\forall (h: -nat).(\forall (d: nat).((arity g c1 (lift h d t) a) \to (\forall (c2: -C).((drop h d c1 c2) \to (arity g c2 t a))))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (a: A).(\lambda (h: -nat).(\lambda (d: nat).(\lambda (H: (arity g c1 (lift h d t) a)).(insert_eq T -(lift h d t) (\lambda (t0: T).(arity g c1 t0 a)) (\lambda (_: T).(\forall -(c2: C).((drop h d c1 c2) \to (arity g c2 t a)))) (\lambda (y: T).(\lambda -(H0: (arity g c1 y a)).(unintro T t (\lambda (t0: T).((eq T y (lift h d t0)) -\to (\forall (c2: C).((drop h d c1 c2) \to (arity g c2 t0 a))))) (unintro nat -d (\lambda (n: nat).(\forall (x: T).((eq T y (lift h n x)) \to (\forall (c2: -C).((drop h n c1 c2) \to (arity g c2 x a)))))) (arity_ind g (\lambda (c: -C).(\lambda (t0: T).(\lambda (a0: A).(\forall (x: nat).(\forall (x0: T).((eq -T t0 (lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to (arity g c2 x0 -a0))))))))) (\lambda (c: C).(\lambda (n: nat).(\lambda (x: nat).(\lambda (x0: -T).(\lambda (H1: (eq T (TSort n) (lift h x x0))).(\lambda (c2: C).(\lambda -(_: (drop h x c c2)).(eq_ind_r T (TSort n) (\lambda (t0: T).(arity g c2 t0 -(ASort O n))) (arity_sort g c2 n) x0 (lift_gen_sort h x n x0 H1))))))))) -(\lambda (c: C).(\lambda (d0: C).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H1: (getl i c (CHead d0 (Bind Abbr) u))).(\lambda (a0: A).(\lambda (H2: -(arity g d0 u a0)).(\lambda (H3: ((\forall (x: nat).(\forall (x0: T).((eq T u -(lift h x x0)) \to (\forall (c2: C).((drop h x d0 c2) \to (arity g c2 x0 -a0)))))))).(\lambda (x: nat).(\lambda (x0: T).(\lambda (H4: (eq T (TLRef i) -(lift h x x0))).(\lambda (c2: C).(\lambda (H5: (drop h x c c2)).(let H_x \def -(lift_gen_lref x0 x h i H4) in (let H6 \def H_x in (or_ind (land (lt i x) (eq -T x0 (TLRef i))) (land (le (plus x h) i) (eq T x0 (TLRef (minus i h)))) -(arity g c2 x0 a0) (\lambda (H7: (land (lt i x) (eq T x0 (TLRef -i)))).(land_ind (lt i x) (eq T x0 (TLRef i)) (arity g c2 x0 a0) (\lambda (H8: -(lt i x)).(\lambda (H9: (eq T x0 (TLRef i))).(eq_ind_r T (TLRef i) (\lambda -(t0: T).(arity g c2 t0 a0)) (let H10 \def (eq_ind nat x (\lambda (n: -nat).(drop h n c c2)) H5 (S (plus i (minus x (S i)))) (lt_plus_minus i x H8)) -in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (minus x (S -i)) v)))) (\lambda (v: T).(\lambda (e0: C).(getl i c2 (CHead e0 (Bind Abbr) -v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (minus x (S i)) d0 e0))) -(arity g c2 (TLRef i) a0) (\lambda (x1: T).(\lambda (x2: C).(\lambda (H11: -(eq T u (lift h (minus x (S i)) x1))).(\lambda (H12: (getl i c2 (CHead x2 -(Bind Abbr) x1))).(\lambda (H13: (drop h (minus x (S i)) d0 x2)).(let H14 -\def (eq_ind T u (\lambda (t0: T).(\forall (x3: nat).(\forall (x4: T).((eq T -t0 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 d0 c3) \to (arity g c3 x4 -a0))))))) H3 (lift h (minus x (S i)) x1) H11) in (let H15 \def (eq_ind T u -(\lambda (t0: T).(arity g d0 t0 a0)) H2 (lift h (minus x (S i)) x1) H11) in -(arity_abbr g c2 x2 x1 i H12 a0 (H14 (minus x (S i)) x1 (refl_equal T (lift h -(minus x (S i)) x1)) x2 H13))))))))) (getl_drop_conf_lt Abbr c d0 u i H1 c2 h -(minus x (S i)) H10))) x0 H9))) H7)) (\lambda (H7: (land (le (plus x h) i) -(eq T x0 (TLRef (minus i h))))).(land_ind (le (plus x h) i) (eq T x0 (TLRef -(minus i h))) (arity g c2 x0 a0) (\lambda (H8: (le (plus x h) i)).(\lambda -(H9: (eq T x0 (TLRef (minus i h)))).(eq_ind_r T (TLRef (minus i h)) (\lambda -(t0: T).(arity g c2 t0 a0)) (arity_abbr g c2 d0 u (minus i h) -(getl_drop_conf_ge i (CHead d0 (Bind Abbr) u) c H1 c2 h x H5 H8) a0 H2) x0 -H9))) H7)) H6)))))))))))))))) (\lambda (c: C).(\lambda (d0: C).(\lambda (u: -T).(\lambda (i: nat).(\lambda (H1: (getl i c (CHead d0 (Bind Abst) -u))).(\lambda (a0: A).(\lambda (H2: (arity g d0 u (asucc g a0))).(\lambda -(H3: ((\forall (x: nat).(\forall (x0: T).((eq T u (lift h x x0)) \to (\forall -(c2: C).((drop h x d0 c2) \to (arity g c2 x0 (asucc g a0))))))))).(\lambda -(x: nat).(\lambda (x0: T).(\lambda (H4: (eq T (TLRef i) (lift h x -x0))).(\lambda (c2: C).(\lambda (H5: (drop h x c c2)).(let H_x \def -(lift_gen_lref x0 x h i H4) in (let H6 \def H_x in (or_ind (land (lt i x) (eq -T x0 (TLRef i))) (land (le (plus x h) i) (eq T x0 (TLRef (minus i h)))) -(arity g c2 x0 a0) (\lambda (H7: (land (lt i x) (eq T x0 (TLRef -i)))).(land_ind (lt i x) (eq T x0 (TLRef i)) (arity g c2 x0 a0) (\lambda (H8: -(lt i x)).(\lambda (H9: (eq T x0 (TLRef i))).(eq_ind_r T (TLRef i) (\lambda -(t0: T).(arity g c2 t0 a0)) (let H10 \def (eq_ind nat x (\lambda (n: -nat).(drop h n c c2)) H5 (S (plus i (minus x (S i)))) (lt_plus_minus i x H8)) -in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (minus x (S -i)) v)))) (\lambda (v: T).(\lambda (e0: C).(getl i c2 (CHead e0 (Bind Abst) -v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (minus x (S i)) d0 e0))) -(arity g c2 (TLRef i) a0) (\lambda (x1: T).(\lambda (x2: C).(\lambda (H11: -(eq T u (lift h (minus x (S i)) x1))).(\lambda (H12: (getl i c2 (CHead x2 -(Bind Abst) x1))).(\lambda (H13: (drop h (minus x (S i)) d0 x2)).(let H14 -\def (eq_ind T u (\lambda (t0: T).(\forall (x3: nat).(\forall (x4: T).((eq T -t0 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 d0 c3) \to (arity g c3 x4 -(asucc g a0)))))))) H3 (lift h (minus x (S i)) x1) H11) in (let H15 \def -(eq_ind T u (\lambda (t0: T).(arity g d0 t0 (asucc g a0))) H2 (lift h (minus -x (S i)) x1) H11) in (arity_abst g c2 x2 x1 i H12 a0 (H14 (minus x (S i)) x1 -(refl_equal T (lift h (minus x (S i)) x1)) x2 H13))))))))) (getl_drop_conf_lt -Abst c d0 u i H1 c2 h (minus x (S i)) H10))) x0 H9))) H7)) (\lambda (H7: -(land (le (plus x h) i) (eq T x0 (TLRef (minus i h))))).(land_ind (le (plus x -h) i) (eq T x0 (TLRef (minus i h))) (arity g c2 x0 a0) (\lambda (H8: (le -(plus x h) i)).(\lambda (H9: (eq T x0 (TLRef (minus i h)))).(eq_ind_r T -(TLRef (minus i h)) (\lambda (t0: T).(arity g c2 t0 a0)) (arity_abst g c2 d0 -u (minus i h) (getl_drop_conf_ge i (CHead d0 (Bind Abst) u) c H1 c2 h x H5 -H8) a0 H2) x0 H9))) H7)) H6)))))))))))))))) (\lambda (b: B).(\lambda (H1: -(not (eq B b Abst))).(\lambda (c: C).(\lambda (u: T).(\lambda (a1: -A).(\lambda (H2: (arity g c u a1)).(\lambda (H3: ((\forall (x: nat).(\forall -(x0: T).((eq T u (lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to -(arity g c2 x0 a1)))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (H4: -(arity g (CHead c (Bind b) u) t0 a2)).(\lambda (H5: ((\forall (x: -nat).(\forall (x0: T).((eq T t0 (lift h x x0)) \to (\forall (c2: C).((drop h -x (CHead c (Bind b) u) c2) \to (arity g c2 x0 a2)))))))).(\lambda (x: -nat).(\lambda (x0: T).(\lambda (H6: (eq T (THead (Bind b) u t0) (lift h x -x0))).(\lambda (c2: C).(\lambda (H7: (drop h x c c2)).(ex3_2_ind T T (\lambda -(y0: T).(\lambda (z: T).(eq T x0 (THead (Bind b) y0 z)))) (\lambda (y0: -T).(\lambda (_: T).(eq T u (lift h x y0)))) (\lambda (_: T).(\lambda (z: -T).(eq T t0 (lift h (S x) z)))) (arity g c2 x0 a2) (\lambda (x1: T).(\lambda -(x2: T).(\lambda (H8: (eq T x0 (THead (Bind b) x1 x2))).(\lambda (H9: (eq T u -(lift h x x1))).(\lambda (H10: (eq T t0 (lift h (S x) x2))).(eq_ind_r T -(THead (Bind b) x1 x2) (\lambda (t1: T).(arity g c2 t1 a2)) (let H11 \def -(eq_ind T t0 (\lambda (t1: T).(\forall (x3: nat).(\forall (x4: T).((eq T t1 -(lift h x3 x4)) \to (\forall (c3: C).((drop h x3 (CHead c (Bind b) u) c3) \to -(arity g c3 x4 a2))))))) H5 (lift h (S x) x2) H10) in (let H12 \def (eq_ind T -t0 (\lambda (t1: T).(arity g (CHead c (Bind b) u) t1 a2)) H4 (lift h (S x) -x2) H10) in (let H13 \def (eq_ind T u (\lambda (t1: T).(arity g (CHead c -(Bind b) t1) (lift h (S x) x2) a2)) H12 (lift h x x1) H9) in (let H14 \def -(eq_ind T u (\lambda (t1: T).(\forall (x3: nat).(\forall (x4: T).((eq T (lift -h (S x) x2) (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 (CHead c (Bind -b) t1) c3) \to (arity g c3 x4 a2))))))) H11 (lift h x x1) H9) in (let H15 -\def (eq_ind T u (\lambda (t1: T).(\forall (x3: nat).(\forall (x4: T).((eq T -t1 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 c c3) \to (arity g c3 x4 -a1))))))) H3 (lift h x x1) H9) in (let H16 \def (eq_ind T u (\lambda (t1: -T).(arity g c t1 a1)) H2 (lift h x x1) H9) in (arity_bind g b H1 c2 x1 a1 -(H15 x x1 (refl_equal T (lift h x x1)) c2 H7) x2 a2 (H14 (S x) x2 (refl_equal -T (lift h (S x) x2)) (CHead c2 (Bind b) x1) (drop_skip_bind h x c c2 H7 b -x1))))))))) x0 H8)))))) (lift_gen_bind b u t0 x0 h x H6)))))))))))))))))) -(\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H1: (arity g c u -(asucc g a1))).(\lambda (H2: ((\forall (x: nat).(\forall (x0: T).((eq T u -(lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to (arity g c2 x0 -(asucc g a1))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (H3: (arity g -(CHead c (Bind Abst) u) t0 a2)).(\lambda (H4: ((\forall (x: nat).(\forall -(x0: T).((eq T t0 (lift h x x0)) \to (\forall (c2: C).((drop h x (CHead c -(Bind Abst) u) c2) \to (arity g c2 x0 a2)))))))).(\lambda (x: nat).(\lambda -(x0: T).(\lambda (H5: (eq T (THead (Bind Abst) u t0) (lift h x x0))).(\lambda -(c2: C).(\lambda (H6: (drop h x c c2)).(ex3_2_ind T T (\lambda (y0: -T).(\lambda (z: T).(eq T x0 (THead (Bind Abst) y0 z)))) (\lambda (y0: -T).(\lambda (_: T).(eq T u (lift h x y0)))) (\lambda (_: T).(\lambda (z: -T).(eq T t0 (lift h (S x) z)))) (arity g c2 x0 (AHead a1 a2)) (\lambda (x1: -T).(\lambda (x2: T).(\lambda (H7: (eq T x0 (THead (Bind Abst) x1 -x2))).(\lambda (H8: (eq T u (lift h x x1))).(\lambda (H9: (eq T t0 (lift h (S -x) x2))).(eq_ind_r T (THead (Bind Abst) x1 x2) (\lambda (t1: T).(arity g c2 -t1 (AHead a1 a2))) (let H10 \def (eq_ind T t0 (\lambda (t1: T).(\forall (x3: -nat).(\forall (x4: T).((eq T t1 (lift h x3 x4)) \to (\forall (c3: C).((drop h -x3 (CHead c (Bind Abst) u) c3) \to (arity g c3 x4 a2))))))) H4 (lift h (S x) -x2) H9) in (let H11 \def (eq_ind T t0 (\lambda (t1: T).(arity g (CHead c -(Bind Abst) u) t1 a2)) H3 (lift h (S x) x2) H9) in (let H12 \def (eq_ind T u -(\lambda (t1: T).(arity g (CHead c (Bind Abst) t1) (lift h (S x) x2) a2)) H11 -(lift h x x1) H8) in (let H13 \def (eq_ind T u (\lambda (t1: T).(\forall (x3: -nat).(\forall (x4: T).((eq T (lift h (S x) x2) (lift h x3 x4)) \to (\forall -(c3: C).((drop h x3 (CHead c (Bind Abst) t1) c3) \to (arity g c3 x4 a2))))))) -H10 (lift h x x1) H8) in (let H14 \def (eq_ind T u (\lambda (t1: T).(\forall -(x3: nat).(\forall (x4: T).((eq T t1 (lift h x3 x4)) \to (\forall (c3: -C).((drop h x3 c c3) \to (arity g c3 x4 (asucc g a1)))))))) H2 (lift h x x1) -H8) in (let H15 \def (eq_ind T u (\lambda (t1: T).(arity g c t1 (asucc g -a1))) H1 (lift h x x1) H8) in (arity_head g c2 x1 a1 (H14 x x1 (refl_equal T -(lift h x x1)) c2 H6) x2 a2 (H13 (S x) x2 (refl_equal T (lift h (S x) x2)) -(CHead c2 (Bind Abst) x1) (drop_skip_bind h x c c2 H6 Abst x1))))))))) x0 -H7)))))) (lift_gen_bind Abst u t0 x0 h x H5)))))))))))))))) (\lambda (c: -C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H1: (arity g c u a1)).(\lambda -(H2: ((\forall (x: nat).(\forall (x0: T).((eq T u (lift h x x0)) \to (\forall -(c2: C).((drop h x c c2) \to (arity g c2 x0 a1)))))))).(\lambda (t0: -T).(\lambda (a2: A).(\lambda (H3: (arity g c t0 (AHead a1 a2))).(\lambda (H4: -((\forall (x: nat).(\forall (x0: T).((eq T t0 (lift h x x0)) \to (\forall -(c2: C).((drop h x c c2) \to (arity g c2 x0 (AHead a1 a2))))))))).(\lambda -(x: nat).(\lambda (x0: T).(\lambda (H5: (eq T (THead (Flat Appl) u t0) (lift -h x x0))).(\lambda (c2: C).(\lambda (H6: (drop h x c c2)).(ex3_2_ind T T -(\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Flat Appl) y0 z)))) -(\lambda (y0: T).(\lambda (_: T).(eq T u (lift h x y0)))) (\lambda (_: -T).(\lambda (z: T).(eq T t0 (lift h x z)))) (arity g c2 x0 a2) (\lambda (x1: -T).(\lambda (x2: T).(\lambda (H7: (eq T x0 (THead (Flat Appl) x1 -x2))).(\lambda (H8: (eq T u (lift h x x1))).(\lambda (H9: (eq T t0 (lift h x -x2))).(eq_ind_r T (THead (Flat Appl) x1 x2) (\lambda (t1: T).(arity g c2 t1 -a2)) (let H10 \def (eq_ind T t0 (\lambda (t1: T).(\forall (x3: nat).(\forall -(x4: T).((eq T t1 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 c c3) \to -(arity g c3 x4 (AHead a1 a2)))))))) H4 (lift h x x2) H9) in (let H11 \def -(eq_ind T t0 (\lambda (t1: T).(arity g c t1 (AHead a1 a2))) H3 (lift h x x2) -H9) in (let H12 \def (eq_ind T u (\lambda (t1: T).(\forall (x3: nat).(\forall -(x4: T).((eq T t1 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 c c3) \to -(arity g c3 x4 a1))))))) H2 (lift h x x1) H8) in (let H13 \def (eq_ind T u -(\lambda (t1: T).(arity g c t1 a1)) H1 (lift h x x1) H8) in (arity_appl g c2 -x1 a1 (H12 x x1 (refl_equal T (lift h x x1)) c2 H6) x2 a2 (H10 x x2 -(refl_equal T (lift h x x2)) c2 H6)))))) x0 H7)))))) (lift_gen_flat Appl u t0 -x0 h x H5)))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a0: -A).(\lambda (H1: (arity g c u (asucc g a0))).(\lambda (H2: ((\forall (x: -nat).(\forall (x0: T).((eq T u (lift h x x0)) \to (\forall (c2: C).((drop h x -c c2) \to (arity g c2 x0 (asucc g a0))))))))).(\lambda (t0: T).(\lambda (H3: -(arity g c t0 a0)).(\lambda (H4: ((\forall (x: nat).(\forall (x0: T).((eq T -t0 (lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to (arity g c2 x0 -a0)))))))).(\lambda (x: nat).(\lambda (x0: T).(\lambda (H5: (eq T (THead -(Flat Cast) u t0) (lift h x x0))).(\lambda (c2: C).(\lambda (H6: (drop h x c -c2)).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Flat -Cast) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u (lift h x y0)))) -(\lambda (_: T).(\lambda (z: T).(eq T t0 (lift h x z)))) (arity g c2 x0 a0) -(\lambda (x1: T).(\lambda (x2: T).(\lambda (H7: (eq T x0 (THead (Flat Cast) -x1 x2))).(\lambda (H8: (eq T u (lift h x x1))).(\lambda (H9: (eq T t0 (lift h -x x2))).(eq_ind_r T (THead (Flat Cast) x1 x2) (\lambda (t1: T).(arity g c2 t1 -a0)) (let H10 \def (eq_ind T t0 (\lambda (t1: T).(\forall (x3: nat).(\forall -(x4: T).((eq T t1 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 c c3) \to -(arity g c3 x4 a0))))))) H4 (lift h x x2) H9) in (let H11 \def (eq_ind T t0 -(\lambda (t1: T).(arity g c t1 a0)) H3 (lift h x x2) H9) in (let H12 \def -(eq_ind T u (\lambda (t1: T).(\forall (x3: nat).(\forall (x4: T).((eq T t1 -(lift h x3 x4)) \to (\forall (c3: C).((drop h x3 c c3) \to (arity g c3 x4 -(asucc g a0)))))))) H2 (lift h x x1) H8) in (let H13 \def (eq_ind T u -(\lambda (t1: T).(arity g c t1 (asucc g a0))) H1 (lift h x x1) H8) in -(arity_cast g c2 x1 a0 (H12 x x1 (refl_equal T (lift h x x1)) c2 H6) x2 (H10 -x x2 (refl_equal T (lift h x x2)) c2 H6)))))) x0 H7)))))) (lift_gen_flat Cast -u t0 x0 h x H5))))))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda (a1: -A).(\lambda (_: (arity g c t0 a1)).(\lambda (H2: ((\forall (x: nat).(\forall -(x0: T).((eq T t0 (lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to -(arity g c2 x0 a1)))))))).(\lambda (a2: A).(\lambda (H3: (leq g a1 -a2)).(\lambda (x: nat).(\lambda (x0: T).(\lambda (H4: (eq T t0 (lift h x -x0))).(\lambda (c2: C).(\lambda (H5: (drop h x c c2)).(arity_repl g c2 x0 a1 -(H2 x x0 H4 c2 H5) a2 H3))))))))))))) c1 y a H0))))) H))))))). -(* COMMENTS -Initial nodes: 4693 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/arity/lift1.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/arity/lift1.ma deleted file mode 100644 index 4077a8a1c..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/arity/lift1.ma +++ /dev/null @@ -1,44 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/arity/props.ma". - -include "Basic-1/drop1/fwd.ma". - -theorem arity_lift1: - \forall (g: G).(\forall (a: A).(\forall (c2: C).(\forall (hds: -PList).(\forall (c1: C).(\forall (t: T).((drop1 hds c1 c2) \to ((arity g c2 t -a) \to (arity g c1 (lift1 hds t) a)))))))) -\def - \lambda (g: G).(\lambda (a: A).(\lambda (c2: C).(\lambda (hds: -PList).(PList_ind (\lambda (p: PList).(\forall (c1: C).(\forall (t: -T).((drop1 p c1 c2) \to ((arity g c2 t a) \to (arity g c1 (lift1 p t) a)))))) -(\lambda (c1: C).(\lambda (t: T).(\lambda (H: (drop1 PNil c1 c2)).(\lambda -(H0: (arity g c2 t a)).(let H_y \def (drop1_gen_pnil c1 c2 H) in (eq_ind_r C -c2 (\lambda (c: C).(arity g c t a)) H0 c1 H_y)))))) (\lambda (n: -nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c1: -C).(\forall (t: T).((drop1 p c1 c2) \to ((arity g c2 t a) \to (arity g c1 -(lift1 p t) a))))))).(\lambda (c1: C).(\lambda (t: T).(\lambda (H0: (drop1 -(PCons n n0 p) c1 c2)).(\lambda (H1: (arity g c2 t a)).(let H_x \def -(drop1_gen_pcons c1 c2 p n n0 H0) in (let H2 \def H_x in (ex2_ind C (\lambda -(c3: C).(drop n n0 c1 c3)) (\lambda (c3: C).(drop1 p c3 c2)) (arity g c1 -(lift n n0 (lift1 p t)) a) (\lambda (x: C).(\lambda (H3: (drop n n0 c1 -x)).(\lambda (H4: (drop1 p x c2)).(arity_lift g x (lift1 p t) a (H x t H4 H1) -c1 n n0 H3)))) H2))))))))))) hds)))). -(* COMMENTS -Initial nodes: 289 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/arity/pr3.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/arity/pr3.ma deleted file mode 100644 index f8952088e..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/arity/pr3.ma +++ /dev/null @@ -1,635 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csuba/arity.ma". - -include "Basic-1/pr3/defs.ma". - -include "Basic-1/pr1/defs.ma". - -include "Basic-1/wcpr0/getl.ma". - -include "Basic-1/pr0/fwd.ma". - -include "Basic-1/arity/subst0.ma". - -theorem arity_sred_wcpr0_pr0: - \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (a: A).((arity g -c1 t1 a) \to (\forall (c2: C).((wcpr0 c1 c2) \to (\forall (t2: T).((pr0 t1 -t2) \to (arity g c2 t2 a))))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (a: A).(\lambda -(H: (arity g c1 t1 a)).(arity_ind g (\lambda (c: C).(\lambda (t: T).(\lambda -(a0: A).(\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 t t2) \to -(arity g c2 t2 a0)))))))) (\lambda (c: C).(\lambda (n: nat).(\lambda (c2: -C).(\lambda (_: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H1: (pr0 (TSort n) -t2)).(eq_ind_r T (TSort n) (\lambda (t: T).(arity g c2 t (ASort O n))) -(arity_sort g c2 n) t2 (pr0_gen_sort t2 n H1)))))))) (\lambda (c: C).(\lambda -(d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d -(Bind Abbr) u))).(\lambda (a0: A).(\lambda (_: (arity g d u a0)).(\lambda -(H2: ((\forall (c2: C).((wcpr0 d c2) \to (\forall (t2: T).((pr0 u t2) \to -(arity g c2 t2 a0))))))).(\lambda (c2: C).(\lambda (H3: (wcpr0 c -c2)).(\lambda (t2: T).(\lambda (H4: (pr0 (TLRef i) t2)).(eq_ind_r T (TLRef i) -(\lambda (t: T).(arity g c2 t a0)) (ex3_2_ind C T (\lambda (e2: C).(\lambda -(u2: T).(getl i c2 (CHead e2 (Bind Abbr) u2)))) (\lambda (e2: C).(\lambda (_: -T).(wcpr0 d e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u u2))) (arity g c2 -(TLRef i) a0) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: (getl i c2 -(CHead x0 (Bind Abbr) x1))).(\lambda (H6: (wcpr0 d x0)).(\lambda (H7: (pr0 u -x1)).(arity_abbr g c2 x0 x1 i H5 a0 (H2 x0 H6 x1 H7))))))) (wcpr0_getl c c2 -H3 i d u (Bind Abbr) H0)) t2 (pr0_gen_lref t2 i H4)))))))))))))) (\lambda (c: -C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c -(CHead d (Bind Abst) u))).(\lambda (a0: A).(\lambda (_: (arity g d u (asucc g -a0))).(\lambda (H2: ((\forall (c2: C).((wcpr0 d c2) \to (\forall (t2: -T).((pr0 u t2) \to (arity g c2 t2 (asucc g a0)))))))).(\lambda (c2: -C).(\lambda (H3: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H4: (pr0 (TLRef i) -t2)).(eq_ind_r T (TLRef i) (\lambda (t: T).(arity g c2 t a0)) (ex3_2_ind C T -(\lambda (e2: C).(\lambda (u2: T).(getl i c2 (CHead e2 (Bind Abst) u2)))) -(\lambda (e2: C).(\lambda (_: T).(wcpr0 d e2))) (\lambda (_: C).(\lambda (u2: -T).(pr0 u u2))) (arity g c2 (TLRef i) a0) (\lambda (x0: C).(\lambda (x1: -T).(\lambda (H5: (getl i c2 (CHead x0 (Bind Abst) x1))).(\lambda (H6: (wcpr0 -d x0)).(\lambda (H7: (pr0 u x1)).(arity_abst g c2 x0 x1 i H5 a0 (H2 x0 H6 x1 -H7))))))) (wcpr0_getl c c2 H3 i d u (Bind Abst) H0)) t2 (pr0_gen_lref t2 i -H4)))))))))))))) (\lambda (b: B).(\lambda (H0: (not (eq B b Abst))).(\lambda -(c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c u -a1)).(\lambda (H2: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 -u t2) \to (arity g c2 t2 a1))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda -(H3: (arity g (CHead c (Bind b) u) t a2)).(\lambda (H4: ((\forall (c2: -C).((wcpr0 (CHead c (Bind b) u) c2) \to (\forall (t2: T).((pr0 t t2) \to -(arity g c2 t2 a2))))))).(\lambda (c2: C).(\lambda (H5: (wcpr0 c -c2)).(\lambda (t2: T).(\lambda (H6: (pr0 (THead (Bind b) u t) t2)).(insert_eq -T (THead (Bind b) u t) (\lambda (t0: T).(pr0 t0 t2)) (\lambda (_: T).(arity g -c2 t2 a2)) (\lambda (y: T).(\lambda (H7: (pr0 y t2)).(pr0_ind (\lambda (t0: -T).(\lambda (t3: T).((eq T t0 (THead (Bind b) u t)) \to (arity g c2 t3 a2)))) -(\lambda (t0: T).(\lambda (H8: (eq T t0 (THead (Bind b) u t))).(let H9 \def -(f_equal T T (\lambda (e: T).e) t0 (THead (Bind b) u t) H8) in (eq_ind_r T -(THead (Bind b) u t) (\lambda (t3: T).(arity g c2 t3 a2)) (arity_bind g b H0 -c2 u a1 (H2 c2 H5 u (pr0_refl u)) t a2 (H4 (CHead c2 (Bind b) u) (wcpr0_comp -c c2 H5 u u (pr0_refl u) (Bind b)) t (pr0_refl t))) t0 H9)))) (\lambda (u1: -T).(\lambda (u2: T).(\lambda (H8: (pr0 u1 u2)).(\lambda (H9: (((eq T u1 -(THead (Bind b) u t)) \to (arity g c2 u2 a2)))).(\lambda (t3: T).(\lambda -(t4: T).(\lambda (H10: (pr0 t3 t4)).(\lambda (H11: (((eq T t3 (THead (Bind b) -u t)) \to (arity g c2 t4 a2)))).(\lambda (k: K).(\lambda (H12: (eq T (THead k -u1 t3) (THead (Bind b) u t))).(let H13 \def (f_equal T K (\lambda (e: -T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | -(TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) -(THead (Bind b) u t) H12) in ((let H14 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | -(TLRef _) \Rightarrow u1 | (THead _ t0 _) \Rightarrow t0])) (THead k u1 t3) -(THead (Bind b) u t) H12) in ((let H15 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | -(TLRef _) \Rightarrow t3 | (THead _ _ t0) \Rightarrow t0])) (THead k u1 t3) -(THead (Bind b) u t) H12) in (\lambda (H16: (eq T u1 u)).(\lambda (H17: (eq K -k (Bind b))).(eq_ind_r K (Bind b) (\lambda (k0: K).(arity g c2 (THead k0 u2 -t4) a2)) (let H18 \def (eq_ind T t3 (\lambda (t0: T).((eq T t0 (THead (Bind -b) u t)) \to (arity g c2 t4 a2))) H11 t H15) in (let H19 \def (eq_ind T t3 -(\lambda (t0: T).(pr0 t0 t4)) H10 t H15) in (let H20 \def (eq_ind T u1 -(\lambda (t0: T).((eq T t0 (THead (Bind b) u t)) \to (arity g c2 u2 a2))) H9 -u H16) in (let H21 \def (eq_ind T u1 (\lambda (t0: T).(pr0 t0 u2)) H8 u H16) -in (arity_bind g b H0 c2 u2 a1 (H2 c2 H5 u2 H21) t4 a2 (H4 (CHead c2 (Bind b) -u2) (wcpr0_comp c c2 H5 u u2 H21 (Bind b)) t4 H19)))))) k H17)))) H14)) -H13)))))))))))) (\lambda (u0: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda -(_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind b) u t)) \to (arity g -c2 v2 a2)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 -t4)).(\lambda (_: (((eq T t3 (THead (Bind b) u t)) \to (arity g c2 t4 -a2)))).(\lambda (H12: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) -(THead (Bind b) u t))).(let H13 \def (eq_ind T (THead (Flat Appl) v1 (THead -(Bind Abst) u0 t3)) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind -b) u t) H12) in (False_ind (arity g c2 (THead (Bind Abbr) v2 t4) a2) -H13)))))))))))) (\lambda (b0: B).(\lambda (_: (not (eq B b0 Abst))).(\lambda -(v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 -(THead (Bind b) u t)) \to (arity g c2 v2 a2)))).(\lambda (u1: T).(\lambda -(u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 (THead (Bind b) u -t)) \to (arity g c2 u2 a2)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: -(pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead (Bind b) u t)) \to (arity g c2 t4 -a2)))).(\lambda (H15: (eq T (THead (Flat Appl) v1 (THead (Bind b0) u1 t3)) -(THead (Bind b) u t))).(let H16 \def (eq_ind T (THead (Flat Appl) v1 (THead -(Bind b0) u1 t3)) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind -b) u t) H15) in (False_ind (arity g c2 (THead (Bind b0) u2 (THead (Flat Appl) -(lift (S O) O v2) t4)) a2) H16))))))))))))))))) (\lambda (u1: T).(\lambda -(u2: T).(\lambda (H8: (pr0 u1 u2)).(\lambda (H9: (((eq T u1 (THead (Bind b) u -t)) \to (arity g c2 u2 a2)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda -(H10: (pr0 t3 t4)).(\lambda (H11: (((eq T t3 (THead (Bind b) u t)) \to (arity -g c2 t4 a2)))).(\lambda (w: T).(\lambda (H12: (subst0 O u2 t4 w)).(\lambda -(H13: (eq T (THead (Bind Abbr) u1 t3) (THead (Bind b) u t))).(let H14 \def -(f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with -[(TSort _) \Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | (THead k _ _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b0) -\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (THead (Bind Abbr) u1 t3) -(THead (Bind b) u t) H13) in ((let H15 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | -(TLRef _) \Rightarrow u1 | (THead _ t0 _) \Rightarrow t0])) (THead (Bind -Abbr) u1 t3) (THead (Bind b) u t) H13) in ((let H16 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t0) \Rightarrow t0])) -(THead (Bind Abbr) u1 t3) (THead (Bind b) u t) H13) in (\lambda (H17: (eq T -u1 u)).(\lambda (H18: (eq B Abbr b)).(let H19 \def (eq_ind T t3 (\lambda (t0: -T).((eq T t0 (THead (Bind b) u t)) \to (arity g c2 t4 a2))) H11 t H16) in -(let H20 \def (eq_ind T t3 (\lambda (t0: T).(pr0 t0 t4)) H10 t H16) in (let -H21 \def (eq_ind T u1 (\lambda (t0: T).((eq T t0 (THead (Bind b) u t)) \to -(arity g c2 u2 a2))) H9 u H17) in (let H22 \def (eq_ind T u1 (\lambda (t0: -T).(pr0 t0 u2)) H8 u H17) in (let H23 \def (eq_ind_r B b (\lambda (b0: -B).((eq T t (THead (Bind b0) u t)) \to (arity g c2 t4 a2))) H19 Abbr H18) in -(let H24 \def (eq_ind_r B b (\lambda (b0: B).((eq T u (THead (Bind b0) u t)) -\to (arity g c2 u2 a2))) H21 Abbr H18) in (let H25 \def (eq_ind_r B b -(\lambda (b0: B).(\forall (c3: C).((wcpr0 (CHead c (Bind b0) u) c3) \to -(\forall (t5: T).((pr0 t t5) \to (arity g c3 t5 a2)))))) H4 Abbr H18) in (let -H26 \def (eq_ind_r B b (\lambda (b0: B).(arity g (CHead c (Bind b0) u) t a2)) -H3 Abbr H18) in (let H27 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 -Abst))) H0 Abbr H18) in (arity_bind g Abbr H27 c2 u2 a1 (H2 c2 H5 u2 H22) w -a2 (arity_subst0 g (CHead c2 (Bind Abbr) u2) t4 a2 (H25 (CHead c2 (Bind Abbr) -u2) (wcpr0_comp c c2 H5 u u2 H22 (Bind Abbr)) t4 H20) c2 u2 O (getl_refl Abbr -c2 u2) w H12)))))))))))))) H15)) H14))))))))))))) (\lambda (b0: B).(\lambda -(H8: (not (eq B b0 Abst))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H9: -(pr0 t3 t4)).(\lambda (H10: (((eq T t3 (THead (Bind b) u t)) \to (arity g c2 -t4 a2)))).(\lambda (u0: T).(\lambda (H11: (eq T (THead (Bind b0) u0 (lift (S -O) O t3)) (THead (Bind b) u t))).(let H12 \def (f_equal T B (\lambda (e: -T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b0 | -(TLRef _) \Rightarrow b0 | (THead k _ _) \Rightarrow (match k in K return -(\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow -b0])])) (THead (Bind b0) u0 (lift (S O) O t3)) (THead (Bind b) u t) H11) in -((let H13 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: -T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t0 -_) \Rightarrow t0])) (THead (Bind b0) u0 (lift (S O) O t3)) (THead (Bind b) u -t) H11) in ((let H14 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat -\to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n) -\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with -[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u1 t5) -\Rightarrow (THead k (lref_map f d u1) (lref_map f (s k d) t5))]) in -lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (TLRef _) \Rightarrow -((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match -t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef -(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | -(THead k u1 t5) \Rightarrow (THead k (lref_map f d u1) (lref_map f (s k d) -t5))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (THead _ _ t0) -\Rightarrow t0])) (THead (Bind b0) u0 (lift (S O) O t3)) (THead (Bind b) u t) -H11) in (\lambda (_: (eq T u0 u)).(\lambda (H16: (eq B b0 b)).(let H17 \def -(eq_ind B b0 (\lambda (b1: B).(not (eq B b1 Abst))) H8 b H16) in (let H18 -\def (eq_ind_r T t (\lambda (t0: T).((eq T t3 (THead (Bind b) u t0)) \to -(arity g c2 t4 a2))) H10 (lift (S O) O t3) H14) in (let H19 \def (eq_ind_r T -t (\lambda (t0: T).(\forall (c3: C).((wcpr0 (CHead c (Bind b) u) c3) \to -(\forall (t5: T).((pr0 t0 t5) \to (arity g c3 t5 a2)))))) H4 (lift (S O) O -t3) H14) in (let H20 \def (eq_ind_r T t (\lambda (t0: T).(arity g (CHead c -(Bind b) u) t0 a2)) H3 (lift (S O) O t3) H14) in (arity_gen_lift g (CHead c2 -(Bind b) u) t4 a2 (S O) O (H19 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H5 u u -(pr0_refl u) (Bind b)) (lift (S O) O t4) (pr0_lift t3 t4 H9 (S O) O)) c2 -(drop_drop (Bind b) O c2 c2 (drop_refl c2) u))))))))) H13)) H12)))))))))) -(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: -(((eq T t3 (THead (Bind b) u t)) \to (arity g c2 t4 a2)))).(\lambda (u0: -T).(\lambda (H10: (eq T (THead (Flat Cast) u0 t3) (THead (Bind b) u t))).(let -H11 \def (eq_ind T (THead (Flat Cast) u0 t3) (\lambda (ee: T).(match ee in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind b) u t) H10) in (False_ind (arity g c2 t4 a2) -H11)))))))) y t2 H7))) H6)))))))))))))))) (\lambda (c: C).(\lambda (u: -T).(\lambda (a1: A).(\lambda (_: (arity g c u (asucc g a1))).(\lambda (H1: -((\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 u t2) \to (arity g -c2 t2 (asucc g a1)))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (H2: -(arity g (CHead c (Bind Abst) u) t a2)).(\lambda (H3: ((\forall (c2: -C).((wcpr0 (CHead c (Bind Abst) u) c2) \to (\forall (t2: T).((pr0 t t2) \to -(arity g c2 t2 a2))))))).(\lambda (c2: C).(\lambda (H4: (wcpr0 c -c2)).(\lambda (t2: T).(\lambda (H5: (pr0 (THead (Bind Abst) u t) -t2)).(insert_eq T (THead (Bind Abst) u t) (\lambda (t0: T).(pr0 t0 t2)) -(\lambda (_: T).(arity g c2 t2 (AHead a1 a2))) (\lambda (y: T).(\lambda (H6: -(pr0 y t2)).(pr0_ind (\lambda (t0: T).(\lambda (t3: T).((eq T t0 (THead (Bind -Abst) u t)) \to (arity g c2 t3 (AHead a1 a2))))) (\lambda (t0: T).(\lambda -(H7: (eq T t0 (THead (Bind Abst) u t))).(let H8 \def (f_equal T T (\lambda -(e: T).e) t0 (THead (Bind Abst) u t) H7) in (eq_ind_r T (THead (Bind Abst) u -t) (\lambda (t3: T).(arity g c2 t3 (AHead a1 a2))) (arity_head g c2 u a1 (H1 -c2 H4 u (pr0_refl u)) t a2 (H3 (CHead c2 (Bind Abst) u) (wcpr0_comp c c2 H4 u -u (pr0_refl u) (Bind Abst)) t (pr0_refl t))) t0 H8)))) (\lambda (u1: -T).(\lambda (u2: T).(\lambda (H7: (pr0 u1 u2)).(\lambda (H8: (((eq T u1 -(THead (Bind Abst) u t)) \to (arity g c2 u2 (AHead a1 a2))))).(\lambda (t3: -T).(\lambda (t4: T).(\lambda (H9: (pr0 t3 t4)).(\lambda (H10: (((eq T t3 -(THead (Bind Abst) u t)) \to (arity g c2 t4 (AHead a1 a2))))).(\lambda (k: -K).(\lambda (H11: (eq T (THead k u1 t3) (THead (Bind Abst) u t))).(let H12 -\def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) -with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) -\Rightarrow k0])) (THead k u1 t3) (THead (Bind Abst) u t) H11) in ((let H13 -\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t0 _) -\Rightarrow t0])) (THead k u1 t3) (THead (Bind Abst) u t) H11) in ((let H14 -\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t0) -\Rightarrow t0])) (THead k u1 t3) (THead (Bind Abst) u t) H11) in (\lambda -(H15: (eq T u1 u)).(\lambda (H16: (eq K k (Bind Abst))).(eq_ind_r K (Bind -Abst) (\lambda (k0: K).(arity g c2 (THead k0 u2 t4) (AHead a1 a2))) (let H17 -\def (eq_ind T t3 (\lambda (t0: T).((eq T t0 (THead (Bind Abst) u t)) \to -(arity g c2 t4 (AHead a1 a2)))) H10 t H14) in (let H18 \def (eq_ind T t3 -(\lambda (t0: T).(pr0 t0 t4)) H9 t H14) in (let H19 \def (eq_ind T u1 -(\lambda (t0: T).((eq T t0 (THead (Bind Abst) u t)) \to (arity g c2 u2 (AHead -a1 a2)))) H8 u H15) in (let H20 \def (eq_ind T u1 (\lambda (t0: T).(pr0 t0 -u2)) H7 u H15) in (arity_head g c2 u2 a1 (H1 c2 H4 u2 H20) t4 a2 (H3 (CHead -c2 (Bind Abst) u2) (wcpr0_comp c c2 H4 u u2 H20 (Bind Abst)) t4 H18)))))) k -H16)))) H13)) H12)))))))))))) (\lambda (u0: T).(\lambda (v1: T).(\lambda (v2: -T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind Abst) u t)) -\to (arity g c2 v2 (AHead a1 a2))))).(\lambda (t3: T).(\lambda (t4: -T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead (Bind Abst) u t)) -\to (arity g c2 t4 (AHead a1 a2))))).(\lambda (H11: (eq T (THead (Flat Appl) -v1 (THead (Bind Abst) u0 t3)) (THead (Bind Abst) u t))).(let H12 \def (eq_ind -T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) (\lambda (ee: T).(match ee -in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef -_) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return -(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind Abst) u t) H11) in (False_ind (arity g c2 (THead -(Bind Abbr) v2 t4) (AHead a1 a2)) H12)))))))))))) (\lambda (b: B).(\lambda -(_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 -v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind Abst) u t)) \to (arity g c2 v2 -(AHead a1 a2))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 -u2)).(\lambda (_: (((eq T u1 (THead (Bind Abst) u t)) \to (arity g c2 u2 -(AHead a1 a2))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 -t4)).(\lambda (_: (((eq T t3 (THead (Bind Abst) u t)) \to (arity g c2 t4 -(AHead a1 a2))))).(\lambda (H14: (eq T (THead (Flat Appl) v1 (THead (Bind b) -u1 t3)) (THead (Bind Abst) u t))).(let H15 \def (eq_ind T (THead (Flat Appl) -v1 (THead (Bind b) u1 t3)) (\lambda (ee: T).(match ee in T return (\lambda -(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False -| (THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind -Abst) u t) H14) in (False_ind (arity g c2 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) (AHead a1 a2)) H15))))))))))))))))) (\lambda -(u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 -(THead (Bind Abst) u t)) \to (arity g c2 u2 (AHead a1 a2))))).(\lambda (t3: -T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead -(Bind Abst) u t)) \to (arity g c2 t4 (AHead a1 a2))))).(\lambda (w: -T).(\lambda (_: (subst0 O u2 t4 w)).(\lambda (H12: (eq T (THead (Bind Abbr) -u1 t3) (THead (Bind Abst) u t))).(let H13 \def (eq_ind T (THead (Bind Abbr) -u1 t3) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b) -\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow -True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _) -\Rightarrow False])])) I (THead (Bind Abst) u t) H12) in (False_ind (arity g -c2 (THead (Bind Abbr) u2 w) (AHead a1 a2)) H13))))))))))))) (\lambda (b: -B).(\lambda (H7: (not (eq B b Abst))).(\lambda (t3: T).(\lambda (t4: -T).(\lambda (_: (pr0 t3 t4)).(\lambda (H9: (((eq T t3 (THead (Bind Abst) u -t)) \to (arity g c2 t4 (AHead a1 a2))))).(\lambda (u0: T).(\lambda (H10: (eq -T (THead (Bind b) u0 (lift (S O) O t3)) (THead (Bind Abst) u t))).(let H11 -\def (f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) -with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b0) -\Rightarrow b0 | (Flat _) \Rightarrow b])])) (THead (Bind b) u0 (lift (S O) O -t3)) (THead (Bind Abst) u t) H10) in ((let H12 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | -(TLRef _) \Rightarrow u0 | (THead _ t0 _) \Rightarrow t0])) (THead (Bind b) -u0 (lift (S O) O t3)) (THead (Bind Abst) u t) H10) in ((let H13 \def (f_equal -T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T -\def (match t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow -(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) -| (THead k u1 t5) \Rightarrow (THead k (lref_map f d u1) (lref_map f (s k d) -t5))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (TLRef _) -\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T -\def (match t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow -(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) -| (THead k u1 t5) \Rightarrow (THead k (lref_map f d u1) (lref_map f (s k d) -t5))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (THead _ _ t0) -\Rightarrow t0])) (THead (Bind b) u0 (lift (S O) O t3)) (THead (Bind Abst) u -t) H10) in (\lambda (_: (eq T u0 u)).(\lambda (H15: (eq B b Abst)).(let H16 -\def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H7 Abst H15) in (let -H17 \def (eq_ind_r T t (\lambda (t0: T).((eq T t3 (THead (Bind Abst) u t0)) -\to (arity g c2 t4 (AHead a1 a2)))) H9 (lift (S O) O t3) H13) in (let H18 -\def (eq_ind_r T t (\lambda (t0: T).(\forall (c3: C).((wcpr0 (CHead c (Bind -Abst) u) c3) \to (\forall (t5: T).((pr0 t0 t5) \to (arity g c3 t5 a2)))))) H3 -(lift (S O) O t3) H13) in (let H19 \def (eq_ind_r T t (\lambda (t0: T).(arity -g (CHead c (Bind Abst) u) t0 a2)) H2 (lift (S O) O t3) H13) in (let H20 \def -(match (H16 (refl_equal B Abst)) in False return (\lambda (_: False).(arity g -c2 t4 (AHead a1 a2))) with []) in H20)))))))) H12)) H11)))))))))) (\lambda -(t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 -(THead (Bind Abst) u t)) \to (arity g c2 t4 (AHead a1 a2))))).(\lambda (u0: -T).(\lambda (H9: (eq T (THead (Flat Cast) u0 t3) (THead (Bind Abst) u -t))).(let H10 \def (eq_ind T (THead (Flat Cast) u0 t3) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K -return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) -\Rightarrow True])])) I (THead (Bind Abst) u t) H9) in (False_ind (arity g c2 -t4 (AHead a1 a2)) H10)))))))) y t2 H6))) H5)))))))))))))) (\lambda (c: -C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda -(H1: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 u t2) \to -(arity g c2 t2 a1))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (H2: -(arity g c t (AHead a1 a2))).(\lambda (H3: ((\forall (c2: C).((wcpr0 c c2) -\to (\forall (t2: T).((pr0 t t2) \to (arity g c2 t2 (AHead a1 -a2)))))))).(\lambda (c2: C).(\lambda (H4: (wcpr0 c c2)).(\lambda (t2: -T).(\lambda (H5: (pr0 (THead (Flat Appl) u t) t2)).(insert_eq T (THead (Flat -Appl) u t) (\lambda (t0: T).(pr0 t0 t2)) (\lambda (_: T).(arity g c2 t2 a2)) -(\lambda (y: T).(\lambda (H6: (pr0 y t2)).(pr0_ind (\lambda (t0: T).(\lambda -(t3: T).((eq T t0 (THead (Flat Appl) u t)) \to (arity g c2 t3 a2)))) (\lambda -(t0: T).(\lambda (H7: (eq T t0 (THead (Flat Appl) u t))).(let H8 \def -(f_equal T T (\lambda (e: T).e) t0 (THead (Flat Appl) u t) H7) in (eq_ind_r T -(THead (Flat Appl) u t) (\lambda (t3: T).(arity g c2 t3 a2)) (arity_appl g c2 -u a1 (H1 c2 H4 u (pr0_refl u)) t a2 (H3 c2 H4 t (pr0_refl t))) t0 H8)))) -(\lambda (u1: T).(\lambda (u2: T).(\lambda (H7: (pr0 u1 u2)).(\lambda (H8: -(((eq T u1 (THead (Flat Appl) u t)) \to (arity g c2 u2 a2)))).(\lambda (t3: -T).(\lambda (t4: T).(\lambda (H9: (pr0 t3 t4)).(\lambda (H10: (((eq T t3 -(THead (Flat Appl) u t)) \to (arity g c2 t4 a2)))).(\lambda (k: K).(\lambda -(H11: (eq T (THead k u1 t3) (THead (Flat Appl) u t))).(let H12 \def (f_equal -T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) -\Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) -(THead k u1 t3) (THead (Flat Appl) u t) H11) in ((let H13 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t0 _) \Rightarrow t0])) -(THead k u1 t3) (THead (Flat Appl) u t) H11) in ((let H14 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t0) \Rightarrow t0])) -(THead k u1 t3) (THead (Flat Appl) u t) H11) in (\lambda (H15: (eq T u1 -u)).(\lambda (H16: (eq K k (Flat Appl))).(eq_ind_r K (Flat Appl) (\lambda -(k0: K).(arity g c2 (THead k0 u2 t4) a2)) (let H17 \def (eq_ind T t3 (\lambda -(t0: T).((eq T t0 (THead (Flat Appl) u t)) \to (arity g c2 t4 a2))) H10 t -H14) in (let H18 \def (eq_ind T t3 (\lambda (t0: T).(pr0 t0 t4)) H9 t H14) in -(let H19 \def (eq_ind T u1 (\lambda (t0: T).((eq T t0 (THead (Flat Appl) u -t)) \to (arity g c2 u2 a2))) H8 u H15) in (let H20 \def (eq_ind T u1 (\lambda -(t0: T).(pr0 t0 u2)) H7 u H15) in (arity_appl g c2 u2 a1 (H1 c2 H4 u2 H20) t4 -a2 (H3 c2 H4 t4 H18)))))) k H16)))) H13)) H12)))))))))))) (\lambda (u0: -T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H7: (pr0 v1 v2)).(\lambda (H8: -(((eq T v1 (THead (Flat Appl) u t)) \to (arity g c2 v2 a2)))).(\lambda (t3: -T).(\lambda (t4: T).(\lambda (H9: (pr0 t3 t4)).(\lambda (H10: (((eq T t3 -(THead (Flat Appl) u t)) \to (arity g c2 t4 a2)))).(\lambda (H11: (eq T -(THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) (THead (Flat Appl) u -t))).(let H12 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda -(_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead -_ t0 _) \Rightarrow t0])) (THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) -(THead (Flat Appl) u t) H11) in ((let H13 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow (THead -(Bind Abst) u0 t3) | (TLRef _) \Rightarrow (THead (Bind Abst) u0 t3) | (THead -_ _ t0) \Rightarrow t0])) (THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) -(THead (Flat Appl) u t) H11) in (\lambda (H14: (eq T v1 u)).(let H15 \def -(eq_ind T v1 (\lambda (t0: T).((eq T t0 (THead (Flat Appl) u t)) \to (arity g -c2 v2 a2))) H8 u H14) in (let H16 \def (eq_ind T v1 (\lambda (t0: T).(pr0 t0 -v2)) H7 u H14) in (let H17 \def (eq_ind_r T t (\lambda (t0: T).((eq T t3 -(THead (Flat Appl) u t0)) \to (arity g c2 t4 a2))) H10 (THead (Bind Abst) u0 -t3) H13) in (let H18 \def (eq_ind_r T t (\lambda (t0: T).((eq T u (THead -(Flat Appl) u t0)) \to (arity g c2 v2 a2))) H15 (THead (Bind Abst) u0 t3) -H13) in (let H19 \def (eq_ind_r T t (\lambda (t0: T).(\forall (c3: C).((wcpr0 -c c3) \to (\forall (t5: T).((pr0 t0 t5) \to (arity g c3 t5 (AHead a1 -a2))))))) H3 (THead (Bind Abst) u0 t3) H13) in (let H20 \def (eq_ind_r T t -(\lambda (t0: T).(arity g c t0 (AHead a1 a2))) H2 (THead (Bind Abst) u0 t3) -H13) in (let H21 \def (H1 c2 H4 v2 H16) in (let H22 \def (H19 c2 H4 (THead -(Bind Abst) u0 t4) (pr0_comp u0 u0 (pr0_refl u0) t3 t4 H9 (Bind Abst))) in -(let H23 \def (arity_gen_abst g c2 u0 t4 (AHead a1 a2) H22) in (ex3_2_ind A A -(\lambda (a3: A).(\lambda (a4: A).(eq A (AHead a1 a2) (AHead a3 a4)))) -(\lambda (a3: A).(\lambda (_: A).(arity g c2 u0 (asucc g a3)))) (\lambda (_: -A).(\lambda (a4: A).(arity g (CHead c2 (Bind Abst) u0) t4 a4))) (arity g c2 -(THead (Bind Abbr) v2 t4) a2) (\lambda (x0: A).(\lambda (x1: A).(\lambda -(H24: (eq A (AHead a1 a2) (AHead x0 x1))).(\lambda (H25: (arity g c2 u0 -(asucc g x0))).(\lambda (H26: (arity g (CHead c2 (Bind Abst) u0) t4 x1)).(let -H27 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) -with [(ASort _ _) \Rightarrow a1 | (AHead a0 _) \Rightarrow a0])) (AHead a1 -a2) (AHead x0 x1) H24) in ((let H28 \def (f_equal A A (\lambda (e: A).(match -e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a2 | (AHead _ -a0) \Rightarrow a0])) (AHead a1 a2) (AHead x0 x1) H24) in (\lambda (H29: (eq -A a1 x0)).(let H30 \def (eq_ind_r A x1 (\lambda (a0: A).(arity g (CHead c2 -(Bind Abst) u0) t4 a0)) H26 a2 H28) in (let H31 \def (eq_ind_r A x0 (\lambda -(a0: A).(arity g c2 u0 (asucc g a0))) H25 a1 H29) in (arity_bind g Abbr -not_abbr_abst c2 v2 a1 H21 t4 a2 (csuba_arity g (CHead c2 (Bind Abst) u0) t4 -a2 H30 (CHead c2 (Bind Abbr) v2) (csuba_abst g c2 c2 (csuba_refl g c2) u0 a1 -H31 v2 H21))))))) H27))))))) H23)))))))))))) H12)))))))))))) (\lambda (b: -B).(\lambda (H7: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: -T).(\lambda (H8: (pr0 v1 v2)).(\lambda (H9: (((eq T v1 (THead (Flat Appl) u -t)) \to (arity g c2 v2 a2)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda -(H10: (pr0 u1 u2)).(\lambda (H11: (((eq T u1 (THead (Flat Appl) u t)) \to -(arity g c2 u2 a2)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H12: (pr0 -t3 t4)).(\lambda (H13: (((eq T t3 (THead (Flat Appl) u t)) \to (arity g c2 t4 -a2)))).(\lambda (H14: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) -(THead (Flat Appl) u t))).(let H15 \def (f_equal T T (\lambda (e: T).(match e -in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) -\Rightarrow v1 | (THead _ t0 _) \Rightarrow t0])) (THead (Flat Appl) v1 -(THead (Bind b) u1 t3)) (THead (Flat Appl) u t) H14) in ((let H16 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow (THead (Bind b) u1 t3) | (TLRef _) \Rightarrow (THead -(Bind b) u1 t3) | (THead _ _ t0) \Rightarrow t0])) (THead (Flat Appl) v1 -(THead (Bind b) u1 t3)) (THead (Flat Appl) u t) H14) in (\lambda (H17: (eq T -v1 u)).(let H18 \def (eq_ind T v1 (\lambda (t0: T).((eq T t0 (THead (Flat -Appl) u t)) \to (arity g c2 v2 a2))) H9 u H17) in (let H19 \def (eq_ind T v1 -(\lambda (t0: T).(pr0 t0 v2)) H8 u H17) in (let H20 \def (eq_ind_r T t -(\lambda (t0: T).((eq T t3 (THead (Flat Appl) u t0)) \to (arity g c2 t4 a2))) -H13 (THead (Bind b) u1 t3) H16) in (let H21 \def (eq_ind_r T t (\lambda (t0: -T).((eq T u1 (THead (Flat Appl) u t0)) \to (arity g c2 u2 a2))) H11 (THead -(Bind b) u1 t3) H16) in (let H22 \def (eq_ind_r T t (\lambda (t0: T).((eq T u -(THead (Flat Appl) u t0)) \to (arity g c2 v2 a2))) H18 (THead (Bind b) u1 t3) -H16) in (let H23 \def (eq_ind_r T t (\lambda (t0: T).(\forall (c3: C).((wcpr0 -c c3) \to (\forall (t5: T).((pr0 t0 t5) \to (arity g c3 t5 (AHead a1 -a2))))))) H3 (THead (Bind b) u1 t3) H16) in (let H24 \def (eq_ind_r T t -(\lambda (t0: T).(arity g c t0 (AHead a1 a2))) H2 (THead (Bind b) u1 t3) H16) -in (let H25 \def (H1 c2 H4 v2 H19) in (let H26 \def (H23 c2 H4 (THead (Bind -b) u2 t4) (pr0_comp u1 u2 H10 t3 t4 H12 (Bind b))) in (let H27 \def -(arity_gen_bind b H7 g c2 u2 t4 (AHead a1 a2) H26) in (ex2_ind A (\lambda -(a3: A).(arity g c2 u2 a3)) (\lambda (_: A).(arity g (CHead c2 (Bind b) u2) -t4 (AHead a1 a2))) (arity g c2 (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t4)) a2) (\lambda (x: A).(\lambda (H28: (arity g c2 u2 x)).(\lambda -(H29: (arity g (CHead c2 (Bind b) u2) t4 (AHead a1 a2))).(arity_bind g b H7 -c2 u2 x H28 (THead (Flat Appl) (lift (S O) O v2) t4) a2 (arity_appl g (CHead -c2 (Bind b) u2) (lift (S O) O v2) a1 (arity_lift g c2 v2 a1 H25 (CHead c2 -(Bind b) u2) (S O) O (drop_drop (Bind b) O c2 c2 (drop_refl c2) u2)) t4 a2 -H29))))) H27))))))))))))) H15))))))))))))))))) (\lambda (u1: T).(\lambda (u2: -T).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 (THead (Flat Appl) u t)) -\to (arity g c2 u2 a2)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 -t3 t4)).(\lambda (_: (((eq T t3 (THead (Flat Appl) u t)) \to (arity g c2 t4 -a2)))).(\lambda (w: T).(\lambda (_: (subst0 O u2 t4 w)).(\lambda (H12: (eq T -(THead (Bind Abbr) u1 t3) (THead (Flat Appl) u t))).(let H13 \def (eq_ind T -(THead (Bind Abbr) u1 t3) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat -Appl) u t) H12) in (False_ind (arity g c2 (THead (Bind Abbr) u2 w) a2) -H13))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda -(t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 -(THead (Flat Appl) u t)) \to (arity g c2 t4 a2)))).(\lambda (u0: T).(\lambda -(H10: (eq T (THead (Bind b) u0 (lift (S O) O t3)) (THead (Flat Appl) u -t))).(let H11 \def (eq_ind T (THead (Bind b) u0 (lift (S O) O t3)) (\lambda -(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow -(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | -(Flat _) \Rightarrow False])])) I (THead (Flat Appl) u t) H10) in (False_ind -(arity g c2 t4 a2) H11)))))))))) (\lambda (t3: T).(\lambda (t4: T).(\lambda -(_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead (Flat Appl) u t)) \to (arity -g c2 t4 a2)))).(\lambda (u0: T).(\lambda (H9: (eq T (THead (Flat Cast) u0 t3) -(THead (Flat Appl) u t))).(let H10 \def (eq_ind T (THead (Flat Cast) u0 t3) -(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow -(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | -(Flat f) \Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl -\Rightarrow False | Cast \Rightarrow True])])])) I (THead (Flat Appl) u t) -H9) in (False_ind (arity g c2 t4 a2) H10)))))))) y t2 H6))) H5)))))))))))))) -(\lambda (c: C).(\lambda (u: T).(\lambda (a0: A).(\lambda (_: (arity g c u -(asucc g a0))).(\lambda (H1: ((\forall (c2: C).((wcpr0 c c2) \to (\forall -(t2: T).((pr0 u t2) \to (arity g c2 t2 (asucc g a0)))))))).(\lambda (t: -T).(\lambda (_: (arity g c t a0)).(\lambda (H3: ((\forall (c2: C).((wcpr0 c -c2) \to (\forall (t2: T).((pr0 t t2) \to (arity g c2 t2 a0))))))).(\lambda -(c2: C).(\lambda (H4: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H5: (pr0 -(THead (Flat Cast) u t) t2)).(insert_eq T (THead (Flat Cast) u t) (\lambda -(t0: T).(pr0 t0 t2)) (\lambda (_: T).(arity g c2 t2 a0)) (\lambda (y: -T).(\lambda (H6: (pr0 y t2)).(pr0_ind (\lambda (t0: T).(\lambda (t3: T).((eq -T t0 (THead (Flat Cast) u t)) \to (arity g c2 t3 a0)))) (\lambda (t0: -T).(\lambda (H7: (eq T t0 (THead (Flat Cast) u t))).(let H8 \def (f_equal T T -(\lambda (e: T).e) t0 (THead (Flat Cast) u t) H7) in (eq_ind_r T (THead (Flat -Cast) u t) (\lambda (t3: T).(arity g c2 t3 a0)) (arity_cast g c2 u a0 (H1 c2 -H4 u (pr0_refl u)) t (H3 c2 H4 t (pr0_refl t))) t0 H8)))) (\lambda (u1: -T).(\lambda (u2: T).(\lambda (H7: (pr0 u1 u2)).(\lambda (H8: (((eq T u1 -(THead (Flat Cast) u t)) \to (arity g c2 u2 a0)))).(\lambda (t3: T).(\lambda -(t4: T).(\lambda (H9: (pr0 t3 t4)).(\lambda (H10: (((eq T t3 (THead (Flat -Cast) u t)) \to (arity g c2 t4 a0)))).(\lambda (k: K).(\lambda (H11: (eq T -(THead k u1 t3) (THead (Flat Cast) u t))).(let H12 \def (f_equal T K (\lambda -(e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k -| (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) -(THead (Flat Cast) u t) H11) in ((let H13 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | -(TLRef _) \Rightarrow u1 | (THead _ t0 _) \Rightarrow t0])) (THead k u1 t3) -(THead (Flat Cast) u t) H11) in ((let H14 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | -(TLRef _) \Rightarrow t3 | (THead _ _ t0) \Rightarrow t0])) (THead k u1 t3) -(THead (Flat Cast) u t) H11) in (\lambda (H15: (eq T u1 u)).(\lambda (H16: -(eq K k (Flat Cast))).(eq_ind_r K (Flat Cast) (\lambda (k0: K).(arity g c2 -(THead k0 u2 t4) a0)) (let H17 \def (eq_ind T t3 (\lambda (t0: T).((eq T t0 -(THead (Flat Cast) u t)) \to (arity g c2 t4 a0))) H10 t H14) in (let H18 \def -(eq_ind T t3 (\lambda (t0: T).(pr0 t0 t4)) H9 t H14) in (let H19 \def (eq_ind -T u1 (\lambda (t0: T).((eq T t0 (THead (Flat Cast) u t)) \to (arity g c2 u2 -a0))) H8 u H15) in (let H20 \def (eq_ind T u1 (\lambda (t0: T).(pr0 t0 u2)) -H7 u H15) in (arity_cast g c2 u2 a0 (H1 c2 H4 u2 H20) t4 (H3 c2 H4 t4 -H18)))))) k H16)))) H13)) H12)))))))))))) (\lambda (u0: T).(\lambda (v1: -T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (THead -(Flat Cast) u t)) \to (arity g c2 v2 a0)))).(\lambda (t3: T).(\lambda (t4: -T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead (Flat Cast) u t)) -\to (arity g c2 t4 a0)))).(\lambda (H11: (eq T (THead (Flat Appl) v1 (THead -(Bind Abst) u0 t3)) (THead (Flat Cast) u t))).(let H12 \def (eq_ind T (THead -(Flat Appl) v1 (THead (Bind Abst) u0 t3)) (\lambda (ee: T).(match ee in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f -in F return (\lambda (_: F).Prop) with [Appl \Rightarrow True | Cast -\Rightarrow False])])])) I (THead (Flat Cast) u t) H11) in (False_ind (arity -g c2 (THead (Bind Abbr) v2 t4) a0) H12)))))))))))) (\lambda (b: B).(\lambda -(_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 -v1 v2)).(\lambda (_: (((eq T v1 (THead (Flat Cast) u t)) \to (arity g c2 v2 -a0)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda -(_: (((eq T u1 (THead (Flat Cast) u t)) \to (arity g c2 u2 a0)))).(\lambda -(t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 -(THead (Flat Cast) u t)) \to (arity g c2 t4 a0)))).(\lambda (H14: (eq T -(THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Cast) u t))).(let -H15 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (\lambda -(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow -(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | -(Flat f) \Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl -\Rightarrow True | Cast \Rightarrow False])])])) I (THead (Flat Cast) u t) -H14) in (False_ind (arity g c2 (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t4)) a0) H15))))))))))))))))) (\lambda (u1: T).(\lambda (u2: -T).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 (THead (Flat Cast) u t)) -\to (arity g c2 u2 a0)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 -t3 t4)).(\lambda (_: (((eq T t3 (THead (Flat Cast) u t)) \to (arity g c2 t4 -a0)))).(\lambda (w: T).(\lambda (_: (subst0 O u2 t4 w)).(\lambda (H12: (eq T -(THead (Bind Abbr) u1 t3) (THead (Flat Cast) u t))).(let H13 \def (eq_ind T -(THead (Bind Abbr) u1 t3) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat -Cast) u t) H12) in (False_ind (arity g c2 (THead (Bind Abbr) u2 w) a0) -H13))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda -(t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 -(THead (Flat Cast) u t)) \to (arity g c2 t4 a0)))).(\lambda (u0: T).(\lambda -(H10: (eq T (THead (Bind b) u0 (lift (S O) O t3)) (THead (Flat Cast) u -t))).(let H11 \def (eq_ind T (THead (Bind b) u0 (lift (S O) O t3)) (\lambda -(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow -(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | -(Flat _) \Rightarrow False])])) I (THead (Flat Cast) u t) H10) in (False_ind -(arity g c2 t4 a0) H11)))))))))) (\lambda (t3: T).(\lambda (t4: T).(\lambda -(H7: (pr0 t3 t4)).(\lambda (H8: (((eq T t3 (THead (Flat Cast) u t)) \to -(arity g c2 t4 a0)))).(\lambda (u0: T).(\lambda (H9: (eq T (THead (Flat Cast) -u0 t3) (THead (Flat Cast) u t))).(let H10 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | -(TLRef _) \Rightarrow u0 | (THead _ t0 _) \Rightarrow t0])) (THead (Flat -Cast) u0 t3) (THead (Flat Cast) u t) H9) in ((let H11 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t0) \Rightarrow t0])) -(THead (Flat Cast) u0 t3) (THead (Flat Cast) u t) H9) in (\lambda (_: (eq T -u0 u)).(let H13 \def (eq_ind T t3 (\lambda (t0: T).((eq T t0 (THead (Flat -Cast) u t)) \to (arity g c2 t4 a0))) H8 t H11) in (let H14 \def (eq_ind T t3 -(\lambda (t0: T).(pr0 t0 t4)) H7 t H11) in (H3 c2 H4 t4 H14))))) H10)))))))) -y t2 H6))) H5))))))))))))) (\lambda (c: C).(\lambda (t: T).(\lambda (a1: -A).(\lambda (_: (arity g c t a1)).(\lambda (H1: ((\forall (c2: C).((wcpr0 c -c2) \to (\forall (t2: T).((pr0 t t2) \to (arity g c2 t2 a1))))))).(\lambda -(a2: A).(\lambda (H2: (leq g a1 a2)).(\lambda (c2: C).(\lambda (H3: (wcpr0 c -c2)).(\lambda (t2: T).(\lambda (H4: (pr0 t t2)).(arity_repl g c2 t2 a1 (H1 c2 -H3 t2 H4) a2 H2)))))))))))) c1 t1 a H))))). -(* COMMENTS -Initial nodes: 10246 -END *) - -theorem arity_sred_wcpr0_pr1: - \forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall (g: G).(\forall -(c1: C).(\forall (a: A).((arity g c1 t1 a) \to (\forall (c2: C).((wcpr0 c1 -c2) \to (arity g c2 t2 a))))))))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr1 t1 t2)).(pr1_ind (\lambda -(t: T).(\lambda (t0: T).(\forall (g: G).(\forall (c1: C).(\forall (a: -A).((arity g c1 t a) \to (\forall (c2: C).((wcpr0 c1 c2) \to (arity g c2 t0 -a))))))))) (\lambda (t: T).(\lambda (g: G).(\lambda (c1: C).(\lambda (a: -A).(\lambda (H0: (arity g c1 t a)).(\lambda (c2: C).(\lambda (H1: (wcpr0 c1 -c2)).(arity_sred_wcpr0_pr0 g c1 t a H0 c2 H1 t (pr0_refl t))))))))) (\lambda -(t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t4 t3)).(\lambda (t5: T).(\lambda -(_: (pr1 t3 t5)).(\lambda (H2: ((\forall (g: G).(\forall (c1: C).(\forall (a: -A).((arity g c1 t3 a) \to (\forall (c2: C).((wcpr0 c1 c2) \to (arity g c2 t5 -a))))))))).(\lambda (g: G).(\lambda (c1: C).(\lambda (a: A).(\lambda (H3: -(arity g c1 t4 a)).(\lambda (c2: C).(\lambda (H4: (wcpr0 c1 c2)).(H2 g c2 a -(arity_sred_wcpr0_pr0 g c1 t4 a H3 c2 H4 t3 H0) c2 (wcpr0_refl -c2)))))))))))))) t1 t2 H))). -(* COMMENTS -Initial nodes: 213 -END *) - -theorem arity_sred_pr2: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall -(g: G).(\forall (a: A).((arity g c t1 a) \to (arity g c t2 a))))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 -t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (g: -G).(\forall (a: A).((arity g c0 t a) \to (arity g c0 t0 a))))))) (\lambda -(c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(\lambda -(g: G).(\lambda (a: A).(\lambda (H1: (arity g c0 t3 a)).(arity_sred_wcpr0_pr0 -g c0 t3 a H1 c0 (wcpr0_refl c0) t4 H0)))))))) (\lambda (c0: C).(\lambda (d: -C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind -Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 -t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (g: -G).(\lambda (a: A).(\lambda (H3: (arity g c0 t3 a)).(arity_subst0 g c0 t4 a -(arity_sred_wcpr0_pr0 g c0 t3 a H3 c0 (wcpr0_refl c0) t4 H1) d u i H0 t -H2)))))))))))))) c t1 t2 H)))). -(* COMMENTS -Initial nodes: 205 -END *) - -theorem arity_sred_pr3: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall -(g: G).(\forall (a: A).((arity g c t1 a) \to (arity g c t2 a))))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 -t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (g: G).(\forall (a: -A).((arity g c t a) \to (arity g c t0 a)))))) (\lambda (t: T).(\lambda (g: -G).(\lambda (a: A).(\lambda (H0: (arity g c t a)).H0)))) (\lambda (t3: -T).(\lambda (t4: T).(\lambda (H0: (pr2 c t4 t3)).(\lambda (t5: T).(\lambda -(_: (pr3 c t3 t5)).(\lambda (H2: ((\forall (g: G).(\forall (a: A).((arity g c -t3 a) \to (arity g c t5 a)))))).(\lambda (g: G).(\lambda (a: A).(\lambda (H3: -(arity g c t4 a)).(H2 g a (arity_sred_pr2 c t4 t3 H0 g a H3))))))))))) t1 t2 -H)))). -(* COMMENTS -Initial nodes: 151 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/arity/props.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/arity/props.ma deleted file mode 100644 index 6f131df09..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/arity/props.ma +++ /dev/null @@ -1,434 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/arity/fwd.ma". - -theorem node_inh: - \forall (g: G).(\forall (n: nat).(\forall (k: nat).(ex_2 C T (\lambda (c: -C).(\lambda (t: T).(arity g c t (ASort k n))))))) -\def - \lambda (g: G).(\lambda (n: nat).(\lambda (k: nat).(nat_ind (\lambda (n0: -nat).(ex_2 C T (\lambda (c: C).(\lambda (t: T).(arity g c t (ASort n0 n)))))) -(ex_2_intro C T (\lambda (c: C).(\lambda (t: T).(arity g c t (ASort O n)))) -(CSort O) (TSort n) (arity_sort g (CSort O) n)) (\lambda (n0: nat).(\lambda -(H: (ex_2 C T (\lambda (c: C).(\lambda (t: T).(arity g c t (ASort n0 -n)))))).(let H0 \def H in (ex_2_ind C T (\lambda (c: C).(\lambda (t: -T).(arity g c t (ASort n0 n)))) (ex_2 C T (\lambda (c: C).(\lambda (t: -T).(arity g c t (ASort (S n0) n))))) (\lambda (x0: C).(\lambda (x1: -T).(\lambda (H1: (arity g x0 x1 (ASort n0 n))).(ex_2_intro C T (\lambda (c: -C).(\lambda (t: T).(arity g c t (ASort (S n0) n)))) (CHead x0 (Bind Abst) x1) -(TLRef O) (arity_abst g (CHead x0 (Bind Abst) x1) x0 x1 O (getl_refl Abst x0 -x1) (ASort (S n0) n) H1))))) H0)))) k))). -(* COMMENTS -Initial nodes: 253 -END *) - -theorem arity_lift: - \forall (g: G).(\forall (c2: C).(\forall (t: T).(\forall (a: A).((arity g c2 -t a) \to (\forall (c1: C).(\forall (h: nat).(\forall (d: nat).((drop h d c1 -c2) \to (arity g c1 (lift h d t) a))))))))) -\def - \lambda (g: G).(\lambda (c2: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: -(arity g c2 t a)).(arity_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda (a0: -A).(\forall (c1: C).(\forall (h: nat).(\forall (d: nat).((drop h d c1 c) \to -(arity g c1 (lift h d t0) a0)))))))) (\lambda (c: C).(\lambda (n: -nat).(\lambda (c1: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (_: (drop -h d c1 c)).(eq_ind_r T (TSort n) (\lambda (t0: T).(arity g c1 t0 (ASort O -n))) (arity_sort g c1 n) (lift h d (TSort n)) (lift_sort n h d)))))))) -(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (a0: A).(\lambda (H1: -(arity g d u a0)).(\lambda (H2: ((\forall (c1: C).(\forall (h: nat).(\forall -(d0: nat).((drop h d0 c1 d) \to (arity g c1 (lift h d0 u) a0))))))).(\lambda -(c1: C).(\lambda (h: nat).(\lambda (d0: nat).(\lambda (H3: (drop h d0 c1 -c)).(lt_le_e i d0 (arity g c1 (lift h d0 (TLRef i)) a0) (\lambda (H4: (lt i -d0)).(eq_ind_r T (TLRef i) (\lambda (t0: T).(arity g c1 t0 a0)) (let H5 \def -(drop_getl_trans_le i d0 (le_S_n i d0 (le_S (S i) d0 H4)) c1 c h H3 (CHead d -(Bind Abbr) u) H0) in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop i -O c1 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d0 i) e0 e1))) -(\lambda (_: C).(\lambda (e1: C).(clear e1 (CHead d (Bind Abbr) u)))) (arity -g c1 (TLRef i) a0) (\lambda (x0: C).(\lambda (x1: C).(\lambda (H6: (drop i O -c1 x0)).(\lambda (H7: (drop h (minus d0 i) x0 x1)).(\lambda (H8: (clear x1 -(CHead d (Bind Abbr) u))).(let H9 \def (eq_ind nat (minus d0 i) (\lambda (n: -nat).(drop h n x0 x1)) H7 (S (minus d0 (S i))) (minus_x_Sy d0 i H4)) in (let -H10 \def (drop_clear_S x1 x0 h (minus d0 (S i)) H9 Abbr d u H8) in (ex2_ind C -(\lambda (c3: C).(clear x0 (CHead c3 (Bind Abbr) (lift h (minus d0 (S i)) -u)))) (\lambda (c3: C).(drop h (minus d0 (S i)) c3 d)) (arity g c1 (TLRef i) -a0) (\lambda (x: C).(\lambda (H11: (clear x0 (CHead x (Bind Abbr) (lift h -(minus d0 (S i)) u)))).(\lambda (H12: (drop h (minus d0 (S i)) x -d)).(arity_abbr g c1 x (lift h (minus d0 (S i)) u) i (getl_intro i c1 (CHead -x (Bind Abbr) (lift h (minus d0 (S i)) u)) x0 H6 H11) a0 (H2 x h (minus d0 (S -i)) H12))))) H10)))))))) H5)) (lift h d0 (TLRef i)) (lift_lref_lt i h d0 -H4))) (\lambda (H4: (le d0 i)).(eq_ind_r T (TLRef (plus i h)) (\lambda (t0: -T).(arity g c1 t0 a0)) (arity_abbr g c1 d u (plus i h) (drop_getl_trans_ge i -c1 c d0 h H3 (CHead d (Bind Abbr) u) H0 H4) a0 H1) (lift h d0 (TLRef i)) -(lift_lref_ge i h d0 H4)))))))))))))))) (\lambda (c: C).(\lambda (d: -C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d (Bind -Abst) u))).(\lambda (a0: A).(\lambda (H1: (arity g d u (asucc g -a0))).(\lambda (H2: ((\forall (c1: C).(\forall (h: nat).(\forall (d0: -nat).((drop h d0 c1 d) \to (arity g c1 (lift h d0 u) (asucc g -a0)))))))).(\lambda (c1: C).(\lambda (h: nat).(\lambda (d0: nat).(\lambda -(H3: (drop h d0 c1 c)).(lt_le_e i d0 (arity g c1 (lift h d0 (TLRef i)) a0) -(\lambda (H4: (lt i d0)).(eq_ind_r T (TLRef i) (\lambda (t0: T).(arity g c1 -t0 a0)) (let H5 \def (drop_getl_trans_le i d0 (le_S_n i d0 (le_S (S i) d0 -H4)) c1 c h H3 (CHead d (Bind Abst) u) H0) in (ex3_2_ind C C (\lambda (e0: -C).(\lambda (_: C).(drop i O c1 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop -h (minus d0 i) e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear e1 (CHead d -(Bind Abst) u)))) (arity g c1 (TLRef i) a0) (\lambda (x0: C).(\lambda (x1: -C).(\lambda (H6: (drop i O c1 x0)).(\lambda (H7: (drop h (minus d0 i) x0 -x1)).(\lambda (H8: (clear x1 (CHead d (Bind Abst) u))).(let H9 \def (eq_ind -nat (minus d0 i) (\lambda (n: nat).(drop h n x0 x1)) H7 (S (minus d0 (S i))) -(minus_x_Sy d0 i H4)) in (let H10 \def (drop_clear_S x1 x0 h (minus d0 (S i)) -H9 Abst d u H8) in (ex2_ind C (\lambda (c3: C).(clear x0 (CHead c3 (Bind -Abst) (lift h (minus d0 (S i)) u)))) (\lambda (c3: C).(drop h (minus d0 (S -i)) c3 d)) (arity g c1 (TLRef i) a0) (\lambda (x: C).(\lambda (H11: (clear x0 -(CHead x (Bind Abst) (lift h (minus d0 (S i)) u)))).(\lambda (H12: (drop h -(minus d0 (S i)) x d)).(arity_abst g c1 x (lift h (minus d0 (S i)) u) i -(getl_intro i c1 (CHead x (Bind Abst) (lift h (minus d0 (S i)) u)) x0 H6 H11) -a0 (H2 x h (minus d0 (S i)) H12))))) H10)))))))) H5)) (lift h d0 (TLRef i)) -(lift_lref_lt i h d0 H4))) (\lambda (H4: (le d0 i)).(eq_ind_r T (TLRef (plus -i h)) (\lambda (t0: T).(arity g c1 t0 a0)) (arity_abst g c1 d u (plus i h) -(drop_getl_trans_ge i c1 c d0 h H3 (CHead d (Bind Abst) u) H0 H4) a0 H1) -(lift h d0 (TLRef i)) (lift_lref_ge i h d0 H4)))))))))))))))) (\lambda (b: -B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c: C).(\lambda (u: -T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H2: ((\forall -(c1: C).(\forall (h: nat).(\forall (d: nat).((drop h d c1 c) \to (arity g c1 -(lift h d u) a1))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity -g (CHead c (Bind b) u) t0 a2)).(\lambda (H4: ((\forall (c1: C).(\forall (h: -nat).(\forall (d: nat).((drop h d c1 (CHead c (Bind b) u)) \to (arity g c1 -(lift h d t0) a2))))))).(\lambda (c1: C).(\lambda (h: nat).(\lambda (d: -nat).(\lambda (H5: (drop h d c1 c)).(eq_ind_r T (THead (Bind b) (lift h d u) -(lift h (s (Bind b) d) t0)) (\lambda (t1: T).(arity g c1 t1 a2)) (arity_bind -g b H0 c1 (lift h d u) a1 (H2 c1 h d H5) (lift h (s (Bind b) d) t0) a2 (H4 -(CHead c1 (Bind b) (lift h d u)) h (s (Bind b) d) (drop_skip_bind h d c1 c H5 -b u))) (lift h d (THead (Bind b) u t0)) (lift_head (Bind b) u t0 h -d))))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda -(_: (arity g c u (asucc g a1))).(\lambda (H1: ((\forall (c1: C).(\forall (h: -nat).(\forall (d: nat).((drop h d c1 c) \to (arity g c1 (lift h d u) (asucc g -a1)))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c -(Bind Abst) u) t0 a2)).(\lambda (H3: ((\forall (c1: C).(\forall (h: -nat).(\forall (d: nat).((drop h d c1 (CHead c (Bind Abst) u)) \to (arity g c1 -(lift h d t0) a2))))))).(\lambda (c1: C).(\lambda (h: nat).(\lambda (d: -nat).(\lambda (H4: (drop h d c1 c)).(eq_ind_r T (THead (Bind Abst) (lift h d -u) (lift h (s (Bind Abst) d) t0)) (\lambda (t1: T).(arity g c1 t1 (AHead a1 -a2))) (arity_head g c1 (lift h d u) a1 (H1 c1 h d H4) (lift h (s (Bind Abst) -d) t0) a2 (H3 (CHead c1 (Bind Abst) (lift h d u)) h (s (Bind Abst) d) -(drop_skip_bind h d c1 c H4 Abst u))) (lift h d (THead (Bind Abst) u t0)) -(lift_head (Bind Abst) u t0 h d))))))))))))))) (\lambda (c: C).(\lambda (u: -T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H1: ((\forall -(c1: C).(\forall (h: nat).(\forall (d: nat).((drop h d c1 c) \to (arity g c1 -(lift h d u) a1))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity -g c t0 (AHead a1 a2))).(\lambda (H3: ((\forall (c1: C).(\forall (h: -nat).(\forall (d: nat).((drop h d c1 c) \to (arity g c1 (lift h d t0) (AHead -a1 a2)))))))).(\lambda (c1: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda -(H4: (drop h d c1 c)).(eq_ind_r T (THead (Flat Appl) (lift h d u) (lift h (s -(Flat Appl) d) t0)) (\lambda (t1: T).(arity g c1 t1 a2)) (arity_appl g c1 -(lift h d u) a1 (H1 c1 h d H4) (lift h (s (Flat Appl) d) t0) a2 (H3 c1 h (s -(Flat Appl) d) H4)) (lift h d (THead (Flat Appl) u t0)) (lift_head (Flat -Appl) u t0 h d))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a0: -A).(\lambda (_: (arity g c u (asucc g a0))).(\lambda (H1: ((\forall (c1: -C).(\forall (h: nat).(\forall (d: nat).((drop h d c1 c) \to (arity g c1 (lift -h d u) (asucc g a0)))))))).(\lambda (t0: T).(\lambda (_: (arity g c t0 -a0)).(\lambda (H3: ((\forall (c1: C).(\forall (h: nat).(\forall (d: -nat).((drop h d c1 c) \to (arity g c1 (lift h d t0) a0))))))).(\lambda (c1: -C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H4: (drop h d c1 -c)).(eq_ind_r T (THead (Flat Cast) (lift h d u) (lift h (s (Flat Cast) d) -t0)) (\lambda (t1: T).(arity g c1 t1 a0)) (arity_cast g c1 (lift h d u) a0 -(H1 c1 h d H4) (lift h (s (Flat Cast) d) t0) (H3 c1 h (s (Flat Cast) d) H4)) -(lift h d (THead (Flat Cast) u t0)) (lift_head (Flat Cast) u t0 h -d)))))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda -(_: (arity g c t0 a1)).(\lambda (H1: ((\forall (c1: C).(\forall (h: -nat).(\forall (d: nat).((drop h d c1 c) \to (arity g c1 (lift h d t0) -a1))))))).(\lambda (a2: A).(\lambda (H2: (leq g a1 a2)).(\lambda (c1: -C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H3: (drop h d c1 -c)).(arity_repl g c1 (lift h d t0) a1 (H1 c1 h d H3) a2 H2)))))))))))) c2 t a -H))))). -(* COMMENTS -Initial nodes: 2661 -END *) - -theorem arity_mono: - \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a1: A).((arity g c -t a1) \to (\forall (a2: A).((arity g c t a2) \to (leq g a1 a2))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a1: A).(\lambda (H: -(arity g c t a1)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (a: -A).(\forall (a2: A).((arity g c0 t0 a2) \to (leq g a a2)))))) (\lambda (c0: -C).(\lambda (n: nat).(\lambda (a2: A).(\lambda (H0: (arity g c0 (TSort n) -a2)).(leq_sym g a2 (ASort O n) (arity_gen_sort g c0 n a2 H0)))))) (\lambda -(c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl -i c0 (CHead d (Bind Abbr) u))).(\lambda (a: A).(\lambda (_: (arity g d u -a)).(\lambda (H2: ((\forall (a2: A).((arity g d u a2) \to (leq g a -a2))))).(\lambda (a2: A).(\lambda (H3: (arity g c0 (TLRef i) a2)).(let H4 -\def (arity_gen_lref g c0 i a2 H3) in (or_ind (ex2_2 C T (\lambda (d0: -C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abbr) u0)))) (\lambda (d0: -C).(\lambda (u0: T).(arity g d0 u0 a2)))) (ex2_2 C T (\lambda (d0: -C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0)))) (\lambda (d0: -C).(\lambda (u0: T).(arity g d0 u0 (asucc g a2))))) (leq g a a2) (\lambda -(H5: (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind -Abbr) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 -a2))))).(ex2_2_ind C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 -(Bind Abbr) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a2))) -(leq g a a2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl i c0 -(CHead x0 (Bind Abbr) x1))).(\lambda (H7: (arity g x0 x1 a2)).(let H8 \def -(eq_ind C (CHead d (Bind Abbr) u) (\lambda (c1: C).(getl i c0 c1)) H0 (CHead -x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead x0 (Bind -Abbr) x1) H6)) in (let H9 \def (f_equal C C (\lambda (e: C).(match e in C -return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) -\Rightarrow c1])) (CHead d (Bind Abbr) u) (CHead x0 (Bind Abbr) x1) -(getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead x0 (Bind Abbr) x1) H6)) in -((let H10 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: -C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead -d (Bind Abbr) u) (CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind Abbr) -u) i H0 (CHead x0 (Bind Abbr) x1) H6)) in (\lambda (H11: (eq C d x0)).(let -H12 \def (eq_ind_r T x1 (\lambda (t0: T).(getl i c0 (CHead x0 (Bind Abbr) -t0))) H8 u H10) in (let H13 \def (eq_ind_r T x1 (\lambda (t0: T).(arity g x0 -t0 a2)) H7 u H10) in (let H14 \def (eq_ind_r C x0 (\lambda (c1: C).(getl i c0 -(CHead c1 (Bind Abbr) u))) H12 d H11) in (let H15 \def (eq_ind_r C x0 -(\lambda (c1: C).(arity g c1 u a2)) H13 d H11) in (H2 a2 H15))))))) H9))))))) -H5)) (\lambda (H5: (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 -(CHead d0 (Bind Abst) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 -(asucc g a2)))))).(ex2_2_ind C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 -(CHead d0 (Bind Abst) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 -(asucc g a2)))) (leq g a a2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: -(getl i c0 (CHead x0 (Bind Abst) x1))).(\lambda (_: (arity g x0 x1 (asucc g -a2))).(let H8 \def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c1: C).(getl i -c0 c1)) H0 (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abbr) u) i -H0 (CHead x0 (Bind Abst) x1) H6)) in (let H9 \def (eq_ind C (CHead d (Bind -Abbr) u) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with -[(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return -(\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B return -(\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | -Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead x0 (Bind -Abst) x1) (getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead x0 (Bind Abst) -x1) H6)) in (False_ind (leq g a a2) H9))))))) H5)) H4)))))))))))) (\lambda -(c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl -i c0 (CHead d (Bind Abst) u))).(\lambda (a: A).(\lambda (_: (arity g d u -(asucc g a))).(\lambda (H2: ((\forall (a2: A).((arity g d u a2) \to (leq g -(asucc g a) a2))))).(\lambda (a2: A).(\lambda (H3: (arity g c0 (TLRef i) -a2)).(let H4 \def (arity_gen_lref g c0 i a2 H3) in (or_ind (ex2_2 C T -(\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abbr) u0)))) -(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a2)))) (ex2_2 C T (\lambda -(d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0)))) (\lambda -(d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a2))))) (leq g a a2) -(\lambda (H5: (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead -d0 (Bind Abbr) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 -a2))))).(ex2_2_ind C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 -(Bind Abbr) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a2))) -(leq g a a2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl i c0 -(CHead x0 (Bind Abbr) x1))).(\lambda (_: (arity g x0 x1 a2)).(let H8 \def -(eq_ind C (CHead d (Bind Abst) u) (\lambda (c1: C).(getl i c0 c1)) H0 (CHead -x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind Abst) u) i H0 (CHead x0 (Bind -Abbr) x1) H6)) in (let H9 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda -(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind b) \Rightarrow (match b in B return (\lambda (_: -B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow True | Void -\Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead x0 (Bind Abbr) -x1) (getl_mono c0 (CHead d (Bind Abst) u) i H0 (CHead x0 (Bind Abbr) x1) H6)) -in (False_ind (leq g a a2) H9))))))) H5)) (\lambda (H5: (ex2_2 C T (\lambda -(d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0)))) (\lambda -(d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a2)))))).(ex2_2_ind C T -(\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0)))) -(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a2)))) (leq g a a2) -(\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl i c0 (CHead x0 (Bind -Abst) x1))).(\lambda (H7: (arity g x0 x1 (asucc g a2))).(let H8 \def (eq_ind -C (CHead d (Bind Abst) u) (\lambda (c1: C).(getl i c0 c1)) H0 (CHead x0 (Bind -Abst) x1) (getl_mono c0 (CHead d (Bind Abst) u) i H0 (CHead x0 (Bind Abst) -x1) H6)) in (let H9 \def (f_equal C C (\lambda (e: C).(match e in C return -(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow -c1])) (CHead d (Bind Abst) u) (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead -d (Bind Abst) u) i H0 (CHead x0 (Bind Abst) x1) H6)) in ((let H10 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind -Abst) u) (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abst) u) i H0 -(CHead x0 (Bind Abst) x1) H6)) in (\lambda (H11: (eq C d x0)).(let H12 \def -(eq_ind_r T x1 (\lambda (t0: T).(getl i c0 (CHead x0 (Bind Abst) t0))) H8 u -H10) in (let H13 \def (eq_ind_r T x1 (\lambda (t0: T).(arity g x0 t0 (asucc g -a2))) H7 u H10) in (let H14 \def (eq_ind_r C x0 (\lambda (c1: C).(getl i c0 -(CHead c1 (Bind Abst) u))) H12 d H11) in (let H15 \def (eq_ind_r C x0 -(\lambda (c1: C).(arity g c1 u (asucc g a2))) H13 d H11) in (asucc_inj g a a2 -(H2 (asucc g a2) H15)))))))) H9))))))) H5)) H4)))))))))))) (\lambda (b: -B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c0: C).(\lambda (u: -T).(\lambda (a2: A).(\lambda (_: (arity g c0 u a2)).(\lambda (_: ((\forall -(a3: A).((arity g c0 u a3) \to (leq g a2 a3))))).(\lambda (t0: T).(\lambda -(a3: A).(\lambda (_: (arity g (CHead c0 (Bind b) u) t0 a3)).(\lambda (H4: -((\forall (a4: A).((arity g (CHead c0 (Bind b) u) t0 a4) \to (leq g a3 -a4))))).(\lambda (a0: A).(\lambda (H5: (arity g c0 (THead (Bind b) u t0) -a0)).(let H6 \def (arity_gen_bind b H0 g c0 u t0 a0 H5) in (ex2_ind A -(\lambda (a4: A).(arity g c0 u a4)) (\lambda (_: A).(arity g (CHead c0 (Bind -b) u) t0 a0)) (leq g a3 a0) (\lambda (x: A).(\lambda (_: (arity g c0 u -x)).(\lambda (H8: (arity g (CHead c0 (Bind b) u) t0 a0)).(H4 a0 H8)))) -H6))))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a2: A).(\lambda -(_: (arity g c0 u (asucc g a2))).(\lambda (H1: ((\forall (a3: A).((arity g c0 -u a3) \to (leq g (asucc g a2) a3))))).(\lambda (t0: T).(\lambda (a3: -A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u) t0 a3)).(\lambda (H3: -((\forall (a4: A).((arity g (CHead c0 (Bind Abst) u) t0 a4) \to (leq g a3 -a4))))).(\lambda (a0: A).(\lambda (H4: (arity g c0 (THead (Bind Abst) u t0) -a0)).(let H5 \def (arity_gen_abst g c0 u t0 a0 H4) in (ex3_2_ind A A (\lambda -(a4: A).(\lambda (a5: A).(eq A a0 (AHead a4 a5)))) (\lambda (a4: A).(\lambda -(_: A).(arity g c0 u (asucc g a4)))) (\lambda (_: A).(\lambda (a5: A).(arity -g (CHead c0 (Bind Abst) u) t0 a5))) (leq g (AHead a2 a3) a0) (\lambda (x0: -A).(\lambda (x1: A).(\lambda (H6: (eq A a0 (AHead x0 x1))).(\lambda (H7: -(arity g c0 u (asucc g x0))).(\lambda (H8: (arity g (CHead c0 (Bind Abst) u) -t0 x1)).(eq_ind_r A (AHead x0 x1) (\lambda (a: A).(leq g (AHead a2 a3) a)) -(leq_head g a2 x0 (asucc_inj g a2 x0 (H1 (asucc g x0) H7)) a3 x1 (H3 x1 H8)) -a0 H6)))))) H5))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a2: -A).(\lambda (_: (arity g c0 u a2)).(\lambda (_: ((\forall (a3: A).((arity g -c0 u a3) \to (leq g a2 a3))))).(\lambda (t0: T).(\lambda (a3: A).(\lambda (_: -(arity g c0 t0 (AHead a2 a3))).(\lambda (H3: ((\forall (a4: A).((arity g c0 -t0 a4) \to (leq g (AHead a2 a3) a4))))).(\lambda (a0: A).(\lambda (H4: (arity -g c0 (THead (Flat Appl) u t0) a0)).(let H5 \def (arity_gen_appl g c0 u t0 a0 -H4) in (ex2_ind A (\lambda (a4: A).(arity g c0 u a4)) (\lambda (a4: A).(arity -g c0 t0 (AHead a4 a0))) (leq g a3 a0) (\lambda (x: A).(\lambda (_: (arity g -c0 u x)).(\lambda (H7: (arity g c0 t0 (AHead x a0))).(ahead_inj_snd g a2 a3 x -a0 (H3 (AHead x a0) H7))))) H5))))))))))))) (\lambda (c0: C).(\lambda (u: -T).(\lambda (a: A).(\lambda (_: (arity g c0 u (asucc g a))).(\lambda (_: -((\forall (a2: A).((arity g c0 u a2) \to (leq g (asucc g a) a2))))).(\lambda -(t0: T).(\lambda (_: (arity g c0 t0 a)).(\lambda (H3: ((\forall (a2: -A).((arity g c0 t0 a2) \to (leq g a a2))))).(\lambda (a2: A).(\lambda (H4: -(arity g c0 (THead (Flat Cast) u t0) a2)).(let H5 \def (arity_gen_cast g c0 u -t0 a2 H4) in (land_ind (arity g c0 u (asucc g a2)) (arity g c0 t0 a2) (leq g -a a2) (\lambda (_: (arity g c0 u (asucc g a2))).(\lambda (H7: (arity g c0 t0 -a2)).(H3 a2 H7))) H5)))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda -(a2: A).(\lambda (_: (arity g c0 t0 a2)).(\lambda (H1: ((\forall (a3: -A).((arity g c0 t0 a3) \to (leq g a2 a3))))).(\lambda (a3: A).(\lambda (H2: -(leq g a2 a3)).(\lambda (a0: A).(\lambda (H3: (arity g c0 t0 a0)).(leq_trans -g a3 a2 (leq_sym g a2 a3 H2) a0 (H1 a0 H3))))))))))) c t a1 H))))). -(* COMMENTS -Initial nodes: 2947 -END *) - -theorem arity_repellent: - \forall (g: G).(\forall (c: C).(\forall (w: T).(\forall (t: T).(\forall (a1: -A).((arity g (CHead c (Bind Abst) w) t a1) \to (\forall (a2: A).((arity g c -(THead (Bind Abst) w t) a2) \to ((leq g a1 a2) \to (\forall (P: -Prop).P))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (w: T).(\lambda (t: T).(\lambda (a1: -A).(\lambda (H: (arity g (CHead c (Bind Abst) w) t a1)).(\lambda (a2: -A).(\lambda (H0: (arity g c (THead (Bind Abst) w t) a2)).(\lambda (H1: (leq g -a1 a2)).(\lambda (P: Prop).(let H_y \def (arity_repl g (CHead c (Bind Abst) -w) t a1 H a2 H1) in (let H2 \def (arity_gen_abst g c w t a2 H0) in (ex3_2_ind -A A (\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: -A).(\lambda (_: A).(arity g c w (asucc g a3)))) (\lambda (_: A).(\lambda (a4: -A).(arity g (CHead c (Bind Abst) w) t a4))) P (\lambda (x0: A).(\lambda (x1: -A).(\lambda (H3: (eq A a2 (AHead x0 x1))).(\lambda (_: (arity g c w (asucc g -x0))).(\lambda (H5: (arity g (CHead c (Bind Abst) w) t x1)).(let H6 \def -(eq_ind A a2 (\lambda (a: A).(arity g (CHead c (Bind Abst) w) t a)) H_y -(AHead x0 x1) H3) in (leq_ahead_false_2 g x1 x0 (arity_mono g (CHead c (Bind -Abst) w) t (AHead x0 x1) H6 x1 H5) P))))))) H2)))))))))))). -(* COMMENTS -Initial nodes: 283 -END *) - -theorem arity_appls_cast: - \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (vs: -TList).(\forall (a: A).((arity g c (THeads (Flat Appl) vs u) (asucc g a)) \to -((arity g c (THeads (Flat Appl) vs t) a) \to (arity g c (THeads (Flat Appl) -vs (THead (Flat Cast) u t)) a)))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (vs: -TList).(TList_ind (\lambda (t0: TList).(\forall (a: A).((arity g c (THeads -(Flat Appl) t0 u) (asucc g a)) \to ((arity g c (THeads (Flat Appl) t0 t) a) -\to (arity g c (THeads (Flat Appl) t0 (THead (Flat Cast) u t)) a))))) -(\lambda (a: A).(\lambda (H: (arity g c u (asucc g a))).(\lambda (H0: (arity -g c t a)).(arity_cast g c u a H t H0)))) (\lambda (t0: T).(\lambda (t1: -TList).(\lambda (H: ((\forall (a: A).((arity g c (THeads (Flat Appl) t1 u) -(asucc g a)) \to ((arity g c (THeads (Flat Appl) t1 t) a) \to (arity g c -(THeads (Flat Appl) t1 (THead (Flat Cast) u t)) a)))))).(\lambda (a: -A).(\lambda (H0: (arity g c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 u)) -(asucc g a))).(\lambda (H1: (arity g c (THead (Flat Appl) t0 (THeads (Flat -Appl) t1 t)) a)).(let H2 \def (arity_gen_appl g c t0 (THeads (Flat Appl) t1 -t) a H1) in (ex2_ind A (\lambda (a1: A).(arity g c t0 a1)) (\lambda (a1: -A).(arity g c (THeads (Flat Appl) t1 t) (AHead a1 a))) (arity g c (THead -(Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Flat Cast) u t))) a) (\lambda -(x: A).(\lambda (H3: (arity g c t0 x)).(\lambda (H4: (arity g c (THeads (Flat -Appl) t1 t) (AHead x a))).(let H5 \def (arity_gen_appl g c t0 (THeads (Flat -Appl) t1 u) (asucc g a) H0) in (ex2_ind A (\lambda (a1: A).(arity g c t0 a1)) -(\lambda (a1: A).(arity g c (THeads (Flat Appl) t1 u) (AHead a1 (asucc g -a)))) (arity g c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Flat -Cast) u t))) a) (\lambda (x0: A).(\lambda (H6: (arity g c t0 x0)).(\lambda -(H7: (arity g c (THeads (Flat Appl) t1 u) (AHead x0 (asucc g -a)))).(arity_appl g c t0 x H3 (THeads (Flat Appl) t1 (THead (Flat Cast) u t)) -a (H (AHead x a) (arity_repl g c (THeads (Flat Appl) t1 u) (AHead x (asucc g -a)) (arity_repl g c (THeads (Flat Appl) t1 u) (AHead x0 (asucc g a)) H7 -(AHead x (asucc g a)) (leq_head g x0 x (arity_mono g c t0 x0 H6 x H3) (asucc -g a) (asucc g a) (leq_refl g (asucc g a)))) (asucc g (AHead x a)) (leq_refl g -(asucc g (AHead x a)))) H4))))) H5))))) H2)))))))) vs))))). -(* COMMENTS -Initial nodes: 707 -END *) - -theorem arity_appls_abbr: - \forall (g: G).(\forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: -nat).((getl i c (CHead d (Bind Abbr) v)) \to (\forall (vs: TList).(\forall -(a: A).((arity g c (THeads (Flat Appl) vs (lift (S i) O v)) a) \to (arity g c -(THeads (Flat Appl) vs (TLRef i)) a))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: -nat).(\lambda (H: (getl i c (CHead d (Bind Abbr) v))).(\lambda (vs: -TList).(TList_ind (\lambda (t: TList).(\forall (a: A).((arity g c (THeads -(Flat Appl) t (lift (S i) O v)) a) \to (arity g c (THeads (Flat Appl) t -(TLRef i)) a)))) (\lambda (a: A).(\lambda (H0: (arity g c (lift (S i) O v) -a)).(arity_abbr g c d v i H a (arity_gen_lift g c v a (S i) O H0 d (getl_drop -Abbr c d v i H))))) (\lambda (t: T).(\lambda (t0: TList).(\lambda (H0: -((\forall (a: A).((arity g c (THeads (Flat Appl) t0 (lift (S i) O v)) a) \to -(arity g c (THeads (Flat Appl) t0 (TLRef i)) a))))).(\lambda (a: A).(\lambda -(H1: (arity g c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O -v))) a)).(let H2 \def (arity_gen_appl g c t (THeads (Flat Appl) t0 (lift (S -i) O v)) a H1) in (ex2_ind A (\lambda (a1: A).(arity g c t a1)) (\lambda (a1: -A).(arity g c (THeads (Flat Appl) t0 (lift (S i) O v)) (AHead a1 a))) (arity -g c (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) a) (\lambda (x: -A).(\lambda (H3: (arity g c t x)).(\lambda (H4: (arity g c (THeads (Flat -Appl) t0 (lift (S i) O v)) (AHead x a))).(arity_appl g c t x H3 (THeads (Flat -Appl) t0 (TLRef i)) a (H0 (AHead x a) H4))))) H2))))))) vs))))))). -(* COMMENTS -Initial nodes: 425 -END *) - -theorem arity_appls_bind: - \forall (g: G).(\forall (b: B).((not (eq B b Abst)) \to (\forall (c: -C).(\forall (v: T).(\forall (a1: A).((arity g c v a1) \to (\forall (t: -T).(\forall (vs: TList).(\forall (a2: A).((arity g (CHead c (Bind b) v) -(THeads (Flat Appl) (lifts (S O) O vs) t) a2) \to (arity g c (THeads (Flat -Appl) vs (THead (Bind b) v t)) a2))))))))))) -\def - \lambda (g: G).(\lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda -(c: C).(\lambda (v: T).(\lambda (a1: A).(\lambda (H0: (arity g c v -a1)).(\lambda (t: T).(\lambda (vs: TList).(TList_ind (\lambda (t0: -TList).(\forall (a2: A).((arity g (CHead c (Bind b) v) (THeads (Flat Appl) -(lifts (S O) O t0) t) a2) \to (arity g c (THeads (Flat Appl) t0 (THead (Bind -b) v t)) a2)))) (\lambda (a2: A).(\lambda (H1: (arity g (CHead c (Bind b) v) -t a2)).(arity_bind g b H c v a1 H0 t a2 H1))) (\lambda (t0: T).(\lambda (t1: -TList).(\lambda (H1: ((\forall (a2: A).((arity g (CHead c (Bind b) v) (THeads -(Flat Appl) (lifts (S O) O t1) t) a2) \to (arity g c (THeads (Flat Appl) t1 -(THead (Bind b) v t)) a2))))).(\lambda (a2: A).(\lambda (H2: (arity g (CHead -c (Bind b) v) (THead (Flat Appl) (lift (S O) O t0) (THeads (Flat Appl) (lifts -(S O) O t1) t)) a2)).(let H3 \def (arity_gen_appl g (CHead c (Bind b) v) -(lift (S O) O t0) (THeads (Flat Appl) (lifts (S O) O t1) t) a2 H2) in -(ex2_ind A (\lambda (a3: A).(arity g (CHead c (Bind b) v) (lift (S O) O t0) -a3)) (\lambda (a3: A).(arity g (CHead c (Bind b) v) (THeads (Flat Appl) -(lifts (S O) O t1) t) (AHead a3 a2))) (arity g c (THead (Flat Appl) t0 -(THeads (Flat Appl) t1 (THead (Bind b) v t))) a2) (\lambda (x: A).(\lambda -(H4: (arity g (CHead c (Bind b) v) (lift (S O) O t0) x)).(\lambda (H5: (arity -g (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O t1) t) (AHead x -a2))).(arity_appl g c t0 x (arity_gen_lift g (CHead c (Bind b) v) t0 x (S O) -O H4 c (drop_drop (Bind b) O c c (drop_refl c) v)) (THeads (Flat Appl) t1 -(THead (Bind b) v t)) a2 (H1 (AHead x a2) H5))))) H3))))))) vs))))))))). -(* COMMENTS -Initial nodes: 567 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/arity/subst0.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/arity/subst0.ma deleted file mode 100644 index 16046993b..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/arity/subst0.ma +++ /dev/null @@ -1,1137 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/arity/props.ma". - -include "Basic-1/fsubst0/fwd.ma". - -include "Basic-1/csubst0/getl.ma". - -include "Basic-1/subst0/dec.ma". - -include "Basic-1/subst0/fwd.ma". - -include "Basic-1/getl/getl.ma". - -theorem arity_gen_cvoid_subst0: - \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t -a) \to (\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d -(Bind Void) u)) \to (\forall (w: T).(\forall (v: T).((subst0 i w t v) \to -(\forall (P: Prop).P)))))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: -(arity g c t a)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (_: -A).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c0 (CHead d -(Bind Void) u)) \to (\forall (w: T).(\forall (v: T).((subst0 i w t0 v) \to -(\forall (P: Prop).P))))))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda -(d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d -(Bind Void) u))).(\lambda (w: T).(\lambda (v: T).(\lambda (H1: (subst0 i w -(TSort n) v)).(\lambda (P: Prop).(subst0_gen_sort w v i n H1 P))))))))))) -(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H0: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (a0: A).(\lambda (_: -(arity g d u a0)).(\lambda (_: ((\forall (d0: C).(\forall (u0: T).(\forall -(i0: nat).((getl i0 d (CHead d0 (Bind Void) u0)) \to (\forall (w: T).(\forall -(v: T).((subst0 i0 w u v) \to (\forall (P: Prop).P)))))))))).(\lambda (d0: -C).(\lambda (u0: T).(\lambda (i0: nat).(\lambda (H3: (getl i0 c0 (CHead d0 -(Bind Void) u0))).(\lambda (w: T).(\lambda (v: T).(\lambda (H4: (subst0 i0 w -(TLRef i) v)).(\lambda (P: Prop).(land_ind (eq nat i i0) (eq T v (lift (S i) -O w)) P (\lambda (H5: (eq nat i i0)).(\lambda (_: (eq T v (lift (S i) O -w))).(let H7 \def (eq_ind_r nat i0 (\lambda (n: nat).(getl n c0 (CHead d0 -(Bind Void) u0))) H3 i H5) in (let H8 \def (eq_ind C (CHead d (Bind Abbr) u) -(\lambda (c1: C).(getl i c0 c1)) H0 (CHead d0 (Bind Void) u0) (getl_mono c0 -(CHead d (Bind Abbr) u) i H0 (CHead d0 (Bind Void) u0) H7)) in (let H9 \def -(eq_ind C (CHead d (Bind Abbr) u) (\lambda (ee: C).(match ee in C return -(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b) -\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow -True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _) -\Rightarrow False])])) I (CHead d0 (Bind Void) u0) (getl_mono c0 (CHead d -(Bind Abbr) u) i H0 (CHead d0 (Bind Void) u0) H7)) in (False_ind P H9)))))) -(subst0_gen_lref w v i0 i H4)))))))))))))))))) (\lambda (c0: C).(\lambda (d: -C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind -Abst) u))).(\lambda (a0: A).(\lambda (_: (arity g d u (asucc g a0))).(\lambda -(_: ((\forall (d0: C).(\forall (u0: T).(\forall (i0: nat).((getl i0 d (CHead -d0 (Bind Void) u0)) \to (\forall (w: T).(\forall (v: T).((subst0 i0 w u v) -\to (\forall (P: Prop).P)))))))))).(\lambda (d0: C).(\lambda (u0: T).(\lambda -(i0: nat).(\lambda (H3: (getl i0 c0 (CHead d0 (Bind Void) u0))).(\lambda (w: -T).(\lambda (v: T).(\lambda (H4: (subst0 i0 w (TLRef i) v)).(\lambda (P: -Prop).(land_ind (eq nat i i0) (eq T v (lift (S i) O w)) P (\lambda (H5: (eq -nat i i0)).(\lambda (_: (eq T v (lift (S i) O w))).(let H7 \def (eq_ind_r nat -i0 (\lambda (n: nat).(getl n c0 (CHead d0 (Bind Void) u0))) H3 i H5) in (let -H8 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (c1: C).(getl i c0 c1)) H0 -(CHead d0 (Bind Void) u0) (getl_mono c0 (CHead d (Bind Abst) u) i H0 (CHead -d0 (Bind Void) u0) H7)) in (let H9 \def (eq_ind C (CHead d (Bind Abst) u) -(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind b) \Rightarrow (match b in B return (\lambda (_: -B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow True | Void -\Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead d0 (Bind Void) -u0) (getl_mono c0 (CHead d (Bind Abst) u) i H0 (CHead d0 (Bind Void) u0) H7)) -in (False_ind P H9)))))) (subst0_gen_lref w v i0 i H4)))))))))))))))))) -(\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (c0: C).(\lambda -(u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u a1)).(\lambda (H2: -((\forall (d: C).(\forall (u0: T).(\forall (i: nat).((getl i c0 (CHead d -(Bind Void) u0)) \to (\forall (w: T).(\forall (v: T).((subst0 i w u v) \to -(\forall (P: Prop).P)))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: -(arity g (CHead c0 (Bind b) u) t0 a2)).(\lambda (H4: ((\forall (d: -C).(\forall (u0: T).(\forall (i: nat).((getl i (CHead c0 (Bind b) u) (CHead d -(Bind Void) u0)) \to (\forall (w: T).(\forall (v: T).((subst0 i w t0 v) \to -(\forall (P: Prop).P)))))))))).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: -nat).(\lambda (H5: (getl i c0 (CHead d (Bind Void) u0))).(\lambda (w: -T).(\lambda (v: T).(\lambda (H6: (subst0 i w (THead (Bind b) u t0) -v)).(\lambda (P: Prop).(or3_ind (ex2 T (\lambda (u2: T).(eq T v (THead (Bind -b) u2 t0))) (\lambda (u2: T).(subst0 i w u u2))) (ex2 T (\lambda (t2: T).(eq -T v (THead (Bind b) u t2))) (\lambda (t2: T).(subst0 (s (Bind b) i) w t0 -t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Bind b) u2 -t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: -T).(\lambda (t2: T).(subst0 (s (Bind b) i) w t0 t2)))) P (\lambda (H7: (ex2 T -(\lambda (u2: T).(eq T v (THead (Bind b) u2 t0))) (\lambda (u2: T).(subst0 i -w u u2)))).(ex2_ind T (\lambda (u2: T).(eq T v (THead (Bind b) u2 t0))) -(\lambda (u2: T).(subst0 i w u u2)) P (\lambda (x: T).(\lambda (_: (eq T v -(THead (Bind b) x t0))).(\lambda (H9: (subst0 i w u x)).(H2 d u0 i H5 w x H9 -P)))) H7)) (\lambda (H7: (ex2 T (\lambda (t2: T).(eq T v (THead (Bind b) u -t2))) (\lambda (t2: T).(subst0 (s (Bind b) i) w t0 t2)))).(ex2_ind T (\lambda -(t2: T).(eq T v (THead (Bind b) u t2))) (\lambda (t2: T).(subst0 (s (Bind b) -i) w t0 t2)) P (\lambda (x: T).(\lambda (_: (eq T v (THead (Bind b) u -x))).(\lambda (H9: (subst0 (s (Bind b) i) w t0 x)).(H4 d u0 (S i) -(getl_clear_bind b (CHead c0 (Bind b) u) c0 u (clear_bind b c0 u) (CHead d -(Bind Void) u0) i H5) w x H9 P)))) H7)) (\lambda (H7: (ex3_2 T T (\lambda -(u2: T).(\lambda (t2: T).(eq T v (THead (Bind b) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda (t2: -T).(subst0 (s (Bind b) i) w t0 t2))))).(ex3_2_ind T T (\lambda (u2: -T).(\lambda (t2: T).(eq T v (THead (Bind b) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda (t2: -T).(subst0 (s (Bind b) i) w t0 t2))) P (\lambda (x0: T).(\lambda (x1: -T).(\lambda (_: (eq T v (THead (Bind b) x0 x1))).(\lambda (H9: (subst0 i w u -x0)).(\lambda (_: (subst0 (s (Bind b) i) w t0 x1)).(H2 d u0 i H5 w x0 H9 -P)))))) H7)) (subst0_gen_head (Bind b) w u t0 v i H6))))))))))))))))))))) -(\lambda (c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u -(asucc g a1))).(\lambda (H1: ((\forall (d: C).(\forall (u0: T).(\forall (i: -nat).((getl i c0 (CHead d (Bind Void) u0)) \to (\forall (w: T).(\forall (v: -T).((subst0 i w u v) \to (\forall (P: Prop).P)))))))))).(\lambda (t0: -T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u) t0 -a2)).(\lambda (H3: ((\forall (d: C).(\forall (u0: T).(\forall (i: nat).((getl -i (CHead c0 (Bind Abst) u) (CHead d (Bind Void) u0)) \to (\forall (w: -T).(\forall (v: T).((subst0 i w t0 v) \to (\forall (P: -Prop).P)))))))))).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda -(H4: (getl i c0 (CHead d (Bind Void) u0))).(\lambda (w: T).(\lambda (v: -T).(\lambda (H5: (subst0 i w (THead (Bind Abst) u t0) v)).(\lambda (P: -Prop).(or3_ind (ex2 T (\lambda (u2: T).(eq T v (THead (Bind Abst) u2 t0))) -(\lambda (u2: T).(subst0 i w u u2))) (ex2 T (\lambda (t2: T).(eq T v (THead -(Bind Abst) u t2))) (\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0 t2))) -(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Bind Abst) u2 -t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: -T).(\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0 t2)))) P (\lambda (H6: -(ex2 T (\lambda (u2: T).(eq T v (THead (Bind Abst) u2 t0))) (\lambda (u2: -T).(subst0 i w u u2)))).(ex2_ind T (\lambda (u2: T).(eq T v (THead (Bind -Abst) u2 t0))) (\lambda (u2: T).(subst0 i w u u2)) P (\lambda (x: T).(\lambda -(_: (eq T v (THead (Bind Abst) x t0))).(\lambda (H8: (subst0 i w u x)).(H1 d -u0 i H4 w x H8 P)))) H6)) (\lambda (H6: (ex2 T (\lambda (t2: T).(eq T v -(THead (Bind Abst) u t2))) (\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0 -t2)))).(ex2_ind T (\lambda (t2: T).(eq T v (THead (Bind Abst) u t2))) -(\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0 t2)) P (\lambda (x: -T).(\lambda (_: (eq T v (THead (Bind Abst) u x))).(\lambda (H8: (subst0 (s -(Bind Abst) i) w t0 x)).(H3 d u0 (S i) (getl_clear_bind Abst (CHead c0 (Bind -Abst) u) c0 u (clear_bind Abst c0 u) (CHead d (Bind Void) u0) i H4) w x H8 -P)))) H6)) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v -(THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w u -u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0 -t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Bind -Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda -(_: T).(\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0 t2))) P (\lambda (x0: -T).(\lambda (x1: T).(\lambda (_: (eq T v (THead (Bind Abst) x0 x1))).(\lambda -(H8: (subst0 i w u x0)).(\lambda (_: (subst0 (s (Bind Abst) i) w t0 x1)).(H1 -d u0 i H4 w x0 H8 P)))))) H6)) (subst0_gen_head (Bind Abst) w u t0 v i -H5))))))))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a1: -A).(\lambda (_: (arity g c0 u a1)).(\lambda (H1: ((\forall (d: C).(\forall -(u0: T).(\forall (i: nat).((getl i c0 (CHead d (Bind Void) u0)) \to (\forall -(w: T).(\forall (v: T).((subst0 i w u v) \to (\forall (P: -Prop).P)))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g c0 -t0 (AHead a1 a2))).(\lambda (H3: ((\forall (d: C).(\forall (u0: T).(\forall -(i: nat).((getl i c0 (CHead d (Bind Void) u0)) \to (\forall (w: T).(\forall -(v: T).((subst0 i w t0 v) \to (\forall (P: Prop).P)))))))))).(\lambda (d: -C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H4: (getl i c0 (CHead d (Bind -Void) u0))).(\lambda (w: T).(\lambda (v: T).(\lambda (H5: (subst0 i w (THead -(Flat Appl) u t0) v)).(\lambda (P: Prop).(or3_ind (ex2 T (\lambda (u2: T).(eq -T v (THead (Flat Appl) u2 t0))) (\lambda (u2: T).(subst0 i w u u2))) (ex2 T -(\lambda (t2: T).(eq T v (THead (Flat Appl) u t2))) (\lambda (t2: T).(subst0 -(s (Flat Appl) i) w t0 t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq -T v (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w -u u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s (Flat Appl) i) w t0 -t2)))) P (\lambda (H6: (ex2 T (\lambda (u2: T).(eq T v (THead (Flat Appl) u2 -t0))) (\lambda (u2: T).(subst0 i w u u2)))).(ex2_ind T (\lambda (u2: T).(eq T -v (THead (Flat Appl) u2 t0))) (\lambda (u2: T).(subst0 i w u u2)) P (\lambda -(x: T).(\lambda (_: (eq T v (THead (Flat Appl) x t0))).(\lambda (H8: (subst0 -i w u x)).(H1 d u0 i H4 w x H8 P)))) H6)) (\lambda (H6: (ex2 T (\lambda (t2: -T).(eq T v (THead (Flat Appl) u t2))) (\lambda (t2: T).(subst0 (s (Flat Appl) -i) w t0 t2)))).(ex2_ind T (\lambda (t2: T).(eq T v (THead (Flat Appl) u t2))) -(\lambda (t2: T).(subst0 (s (Flat Appl) i) w t0 t2)) P (\lambda (x: -T).(\lambda (_: (eq T v (THead (Flat Appl) u x))).(\lambda (H8: (subst0 (s -(Flat Appl) i) w t0 x)).(H3 d u0 i H4 w x H8 P)))) H6)) (\lambda (H6: (ex3_2 -T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Flat Appl) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda -(t2: T).(subst0 (s (Flat Appl) i) w t0 t2))))).(ex3_2_ind T T (\lambda (u2: -T).(\lambda (t2: T).(eq T v (THead (Flat Appl) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda (t2: -T).(subst0 (s (Flat Appl) i) w t0 t2))) P (\lambda (x0: T).(\lambda (x1: -T).(\lambda (_: (eq T v (THead (Flat Appl) x0 x1))).(\lambda (H8: (subst0 i w -u x0)).(\lambda (_: (subst0 (s (Flat Appl) i) w t0 x1)).(H1 d u0 i H4 w x0 H8 -P)))))) H6)) (subst0_gen_head (Flat Appl) w u t0 v i H5))))))))))))))))))) -(\lambda (c0: C).(\lambda (u: T).(\lambda (a0: A).(\lambda (_: (arity g c0 u -(asucc g a0))).(\lambda (H1: ((\forall (d: C).(\forall (u0: T).(\forall (i: -nat).((getl i c0 (CHead d (Bind Void) u0)) \to (\forall (w: T).(\forall (v: -T).((subst0 i w u v) \to (\forall (P: Prop).P)))))))))).(\lambda (t0: -T).(\lambda (_: (arity g c0 t0 a0)).(\lambda (H3: ((\forall (d: C).(\forall -(u0: T).(\forall (i: nat).((getl i c0 (CHead d (Bind Void) u0)) \to (\forall -(w: T).(\forall (v: T).((subst0 i w t0 v) \to (\forall (P: -Prop).P)))))))))).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda -(H4: (getl i c0 (CHead d (Bind Void) u0))).(\lambda (w: T).(\lambda (v: -T).(\lambda (H5: (subst0 i w (THead (Flat Cast) u t0) v)).(\lambda (P: -Prop).(or3_ind (ex2 T (\lambda (u2: T).(eq T v (THead (Flat Cast) u2 t0))) -(\lambda (u2: T).(subst0 i w u u2))) (ex2 T (\lambda (t2: T).(eq T v (THead -(Flat Cast) u t2))) (\lambda (t2: T).(subst0 (s (Flat Cast) i) w t0 t2))) -(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Flat Cast) u2 -t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: -T).(\lambda (t2: T).(subst0 (s (Flat Cast) i) w t0 t2)))) P (\lambda (H6: -(ex2 T (\lambda (u2: T).(eq T v (THead (Flat Cast) u2 t0))) (\lambda (u2: -T).(subst0 i w u u2)))).(ex2_ind T (\lambda (u2: T).(eq T v (THead (Flat -Cast) u2 t0))) (\lambda (u2: T).(subst0 i w u u2)) P (\lambda (x: T).(\lambda -(_: (eq T v (THead (Flat Cast) x t0))).(\lambda (H8: (subst0 i w u x)).(H1 d -u0 i H4 w x H8 P)))) H6)) (\lambda (H6: (ex2 T (\lambda (t2: T).(eq T v -(THead (Flat Cast) u t2))) (\lambda (t2: T).(subst0 (s (Flat Cast) i) w t0 -t2)))).(ex2_ind T (\lambda (t2: T).(eq T v (THead (Flat Cast) u t2))) -(\lambda (t2: T).(subst0 (s (Flat Cast) i) w t0 t2)) P (\lambda (x: -T).(\lambda (_: (eq T v (THead (Flat Cast) u x))).(\lambda (H8: (subst0 (s -(Flat Cast) i) w t0 x)).(H3 d u0 i H4 w x H8 P)))) H6)) (\lambda (H6: (ex3_2 -T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Flat Cast) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda -(t2: T).(subst0 (s (Flat Cast) i) w t0 t2))))).(ex3_2_ind T T (\lambda (u2: -T).(\lambda (t2: T).(eq T v (THead (Flat Cast) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda (t2: -T).(subst0 (s (Flat Cast) i) w t0 t2))) P (\lambda (x0: T).(\lambda (x1: -T).(\lambda (_: (eq T v (THead (Flat Cast) x0 x1))).(\lambda (H8: (subst0 i w -u x0)).(\lambda (_: (subst0 (s (Flat Cast) i) w t0 x1)).(H1 d u0 i H4 w x0 H8 -P)))))) H6)) (subst0_gen_head (Flat Cast) w u t0 v i H5)))))))))))))))))) -(\lambda (c0: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda (_: (arity g c0 -t0 a1)).(\lambda (H1: ((\forall (d: C).(\forall (u: T).(\forall (i: -nat).((getl i c0 (CHead d (Bind Void) u)) \to (\forall (w: T).(\forall (v: -T).((subst0 i w t0 v) \to (\forall (P: Prop).P)))))))))).(\lambda (a2: -A).(\lambda (_: (leq g a1 a2)).(\lambda (d: C).(\lambda (u: T).(\lambda (i: -nat).(\lambda (H3: (getl i c0 (CHead d (Bind Void) u))).(\lambda (w: -T).(\lambda (v: T).(\lambda (H4: (subst0 i w t0 v)).(\lambda (P: Prop).(H1 d -u i H3 w v H4 P)))))))))))))))) c t a H))))). -(* COMMENTS -Initial nodes: 4131 -END *) - -theorem arity_gen_cvoid: - \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t -a) \to (\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d -(Bind Void) u)) \to (ex T (\lambda (v: T).(eq T t (lift (S O) i v)))))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: -(arity g c t a)).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H0: (getl i c (CHead d (Bind Void) u))).(let H_x \def (dnf_dec u t i) in -(let H1 \def H_x in (ex_ind T (\lambda (v: T).(or (subst0 i u t (lift (S O) i -v)) (eq T t (lift (S O) i v)))) (ex T (\lambda (v: T).(eq T t (lift (S O) i -v)))) (\lambda (x: T).(\lambda (H2: (or (subst0 i u t (lift (S O) i x)) (eq T -t (lift (S O) i x)))).(or_ind (subst0 i u t (lift (S O) i x)) (eq T t (lift -(S O) i x)) (ex T (\lambda (v: T).(eq T t (lift (S O) i v)))) (\lambda (H3: -(subst0 i u t (lift (S O) i x))).(arity_gen_cvoid_subst0 g c t a H d u i H0 u -(lift (S O) i x) H3 (ex T (\lambda (v: T).(eq T t (lift (S O) i v)))))) -(\lambda (H3: (eq T t (lift (S O) i x))).(let H4 \def (eq_ind T t (\lambda -(t0: T).(arity g c t0 a)) H (lift (S O) i x) H3) in (eq_ind_r T (lift (S O) i -x) (\lambda (t0: T).(ex T (\lambda (v: T).(eq T t0 (lift (S O) i v))))) -(ex_intro T (\lambda (v: T).(eq T (lift (S O) i x) (lift (S O) i v))) x -(refl_equal T (lift (S O) i x))) t H3))) H2))) H1))))))))))). -(* COMMENTS -Initial nodes: 423 -END *) - -theorem arity_fsubst0: - \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (a: A).((arity g -c1 t1 a) \to (\forall (d1: C).(\forall (u: T).(\forall (i: nat).((getl i c1 -(CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u -c1 t1 c2 t2) \to (arity g c2 t2 a)))))))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (a: A).(\lambda -(H: (arity g c1 t1 a)).(arity_ind g (\lambda (c: C).(\lambda (t: T).(\lambda -(a0: A).(\forall (d1: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead -d1 (Bind Abbr) u)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u c t c2 -t2) \to (arity g c2 t2 a0))))))))))) (\lambda (c: C).(\lambda (n: -nat).(\lambda (d1: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (_: (getl i -c (CHead d1 (Bind Abbr) u))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H1: -(fsubst0 i u c (TSort n) c2 t2)).(let H_x \def (fsubst0_gen_base c c2 (TSort -n) t2 u i H1) in (let H2 \def H_x in (or3_ind (land (eq C c c2) (subst0 i u -(TSort n) t2)) (land (eq T (TSort n) t2) (csubst0 i u c c2)) (land (subst0 i -u (TSort n) t2) (csubst0 i u c c2)) (arity g c2 t2 (ASort O n)) (\lambda (H3: -(land (eq C c c2) (subst0 i u (TSort n) t2))).(land_ind (eq C c c2) (subst0 i -u (TSort n) t2) (arity g c2 t2 (ASort O n)) (\lambda (H4: (eq C c -c2)).(\lambda (H5: (subst0 i u (TSort n) t2)).(eq_ind C c (\lambda (c0: -C).(arity g c0 t2 (ASort O n))) (subst0_gen_sort u t2 i n H5 (arity g c t2 -(ASort O n))) c2 H4))) H3)) (\lambda (H3: (land (eq T (TSort n) t2) (csubst0 -i u c c2))).(land_ind (eq T (TSort n) t2) (csubst0 i u c c2) (arity g c2 t2 -(ASort O n)) (\lambda (H4: (eq T (TSort n) t2)).(\lambda (_: (csubst0 i u c -c2)).(eq_ind T (TSort n) (\lambda (t: T).(arity g c2 t (ASort O n))) -(arity_sort g c2 n) t2 H4))) H3)) (\lambda (H3: (land (subst0 i u (TSort n) -t2) (csubst0 i u c c2))).(land_ind (subst0 i u (TSort n) t2) (csubst0 i u c -c2) (arity g c2 t2 (ASort O n)) (\lambda (H4: (subst0 i u (TSort n) -t2)).(\lambda (_: (csubst0 i u c c2)).(subst0_gen_sort u t2 i n H4 (arity g -c2 t2 (ASort O n))))) H3)) H2)))))))))))) (\lambda (c: C).(\lambda (d: -C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d (Bind -Abbr) u))).(\lambda (a0: A).(\lambda (H1: (arity g d u a0)).(\lambda (H2: -((\forall (d1: C).(\forall (u0: T).(\forall (i0: nat).((getl i0 d (CHead d1 -(Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i0 u0 d u c2 -t2) \to (arity g c2 t2 a0)))))))))).(\lambda (d1: C).(\lambda (u0: -T).(\lambda (i0: nat).(\lambda (H3: (getl i0 c (CHead d1 (Bind Abbr) -u0))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H4: (fsubst0 i0 u0 c (TLRef -i) c2 t2)).(let H_x \def (fsubst0_gen_base c c2 (TLRef i) t2 u0 i0 H4) in -(let H5 \def H_x in (or3_ind (land (eq C c c2) (subst0 i0 u0 (TLRef i) t2)) -(land (eq T (TLRef i) t2) (csubst0 i0 u0 c c2)) (land (subst0 i0 u0 (TLRef i) -t2) (csubst0 i0 u0 c c2)) (arity g c2 t2 a0) (\lambda (H6: (land (eq C c c2) -(subst0 i0 u0 (TLRef i) t2))).(land_ind (eq C c c2) (subst0 i0 u0 (TLRef i) -t2) (arity g c2 t2 a0) (\lambda (H7: (eq C c c2)).(\lambda (H8: (subst0 i0 u0 -(TLRef i) t2)).(eq_ind C c (\lambda (c0: C).(arity g c0 t2 a0)) (land_ind (eq -nat i i0) (eq T t2 (lift (S i) O u0)) (arity g c t2 a0) (\lambda (H9: (eq nat -i i0)).(\lambda (H10: (eq T t2 (lift (S i) O u0))).(eq_ind_r T (lift (S i) O -u0) (\lambda (t: T).(arity g c t a0)) (let H11 \def (eq_ind_r nat i0 (\lambda -(n: nat).(getl n c (CHead d1 (Bind Abbr) u0))) H3 i H9) in (let H12 \def -(eq_ind C (CHead d (Bind Abbr) u) (\lambda (c0: C).(getl i c c0)) H0 (CHead -d1 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) i H0 (CHead d1 (Bind -Abbr) u0) H11)) in (let H13 \def (f_equal C C (\lambda (e: C).(match e in C -return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) -\Rightarrow c0])) (CHead d (Bind Abbr) u) (CHead d1 (Bind Abbr) u0) -(getl_mono c (CHead d (Bind Abbr) u) i H0 (CHead d1 (Bind Abbr) u0) H11)) in -((let H14 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: -C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d -(Bind Abbr) u) (CHead d1 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) -i H0 (CHead d1 (Bind Abbr) u0) H11)) in (\lambda (H15: (eq C d d1)).(let H16 -\def (eq_ind_r T u0 (\lambda (t: T).(getl i c (CHead d1 (Bind Abbr) t))) H12 -u H14) in (eq_ind T u (\lambda (t: T).(arity g c (lift (S i) O t) a0)) (let -H17 \def (eq_ind_r C d1 (\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr) u))) -H16 d H15) in (arity_lift g d u a0 H1 c (S i) O (getl_drop Abbr c d u i -H17))) u0 H14)))) H13)))) t2 H10))) (subst0_gen_lref u0 t2 i0 i H8)) c2 H7))) -H6)) (\lambda (H6: (land (eq T (TLRef i) t2) (csubst0 i0 u0 c c2))).(land_ind -(eq T (TLRef i) t2) (csubst0 i0 u0 c c2) (arity g c2 t2 a0) (\lambda (H7: (eq -T (TLRef i) t2)).(\lambda (H8: (csubst0 i0 u0 c c2)).(eq_ind T (TLRef i) -(\lambda (t: T).(arity g c2 t a0)) (lt_le_e i i0 (arity g c2 (TLRef i) a0) -(\lambda (H9: (lt i i0)).(let H10 \def (csubst0_getl_lt i0 i H9 c c2 u0 H8 -(CHead d (Bind Abbr) u) H0) in (or4_ind (getl i c2 (CHead d (Bind Abbr) u)) -(ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: -B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl i c2 (CHead e0 -(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda -(w: T).(subst0 (minus i0 (S i)) u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind -Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u1: T).(getl i c2 (CHead e2 (Bind b) u1)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S -i)) u0 e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead -e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(getl i c2 (CHead e2 (Bind b) w))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: -T).(subst0 (minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) -u0 e1 e2))))))) (arity g c2 (TLRef i) a0) (\lambda (H11: (getl i c2 (CHead d -(Bind Abbr) u))).(let H12 \def (eq_ind nat (minus i0 i) (\lambda (n: -nat).(getl n (CHead d (Bind Abbr) u) (CHead d1 (Bind Abbr) u0))) -(getl_conf_le i0 (CHead d1 (Bind Abbr) u0) c H3 (CHead d (Bind Abbr) u) i H0 -(le_S_n i i0 (le_S (S i) i0 H9))) (S (minus i0 (S i))) (minus_x_Sy i0 i H9)) -in (arity_abbr g c2 d u i H11 a0 H1))) (\lambda (H11: (ex3_4 B C T T (\lambda -(b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind -Abbr) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(getl i c2 (CHead e0 (Bind b) w)))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 -(minus i0 (S i)) u0 u1 w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda -(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead -e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(getl i c2 (CHead e0 (Bind b) w)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) -u0 u1 w))))) (arity g c2 (TLRef i) a0) (\lambda (x0: B).(\lambda (x1: -C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (eq C (CHead d (Bind -Abbr) u) (CHead x1 (Bind x0) x2))).(\lambda (H13: (getl i c2 (CHead x1 (Bind -x0) x3))).(\lambda (H14: (subst0 (minus i0 (S i)) u0 x2 x3)).(let H15 \def -(eq_ind nat (minus i0 i) (\lambda (n: nat).(getl n (CHead d (Bind Abbr) u) -(CHead d1 (Bind Abbr) u0))) (getl_conf_le i0 (CHead d1 (Bind Abbr) u0) c H3 -(CHead d (Bind Abbr) u) i H0 (le_S_n i i0 (le_S (S i) i0 H9))) (S (minus i0 -(S i))) (minus_x_Sy i0 i H9)) in (let H16 \def (f_equal C C (\lambda (e: -C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | -(CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) -x2) H12) in ((let H17 \def (f_equal C B (\lambda (e: C).(match e in C return -(\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) -\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead -x1 (Bind x0) x2) H12) in ((let H18 \def (f_equal C T (\lambda (e: C).(match e -in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) -\Rightarrow t])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H12) in -(\lambda (H19: (eq B Abbr x0)).(\lambda (H20: (eq C d x1)).(let H21 \def -(eq_ind_r T x2 (\lambda (t: T).(subst0 (minus i0 (S i)) u0 t x3)) H14 u H18) -in (let H22 \def (eq_ind_r C x1 (\lambda (c0: C).(getl i c2 (CHead c0 (Bind -x0) x3))) H13 d H20) in (let H23 \def (eq_ind_r B x0 (\lambda (b: B).(getl i -c2 (CHead d (Bind b) x3))) H22 Abbr H19) in (arity_abbr g c2 d x3 i H23 a0 -(H2 d1 u0 (r (Bind Abbr) (minus i0 (S i))) (getl_gen_S (Bind Abbr) d (CHead -d1 (Bind Abbr) u0) u (minus i0 (S i)) H15) d x3 (fsubst0_snd (r (Bind Abbr) -(minus i0 (S i))) u0 d u x3 H21))))))))) H17)) H16)))))))))) H11)) (\lambda -(H11: (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c2 -(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))).(ex3_4_ind B C C -T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C -(CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda -(_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c2 (CHead e2 (Bind b) -u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus i0 (S i)) u0 e1 e2))))) (arity g c2 (TLRef i) a0) (\lambda -(x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H12: (eq -C (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3))).(\lambda (H13: (getl i c2 -(CHead x2 (Bind x0) x3))).(\lambda (H14: (csubst0 (minus i0 (S i)) u0 x1 -x2)).(let H15 \def (eq_ind nat (minus i0 i) (\lambda (n: nat).(getl n (CHead -d (Bind Abbr) u) (CHead d1 (Bind Abbr) u0))) (getl_conf_le i0 (CHead d1 (Bind -Abbr) u0) c H3 (CHead d (Bind Abbr) u) i H0 (le_S_n i i0 (le_S (S i) i0 H9))) -(S (minus i0 (S i))) (minus_x_Sy i0 i H9)) in (let H16 \def (f_equal C C -(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abbr) u) -(CHead x1 (Bind x0) x3) H12) in ((let H17 \def (f_equal C B (\lambda (e: -C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | -(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind -b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) -(CHead x1 (Bind x0) x3) H12) in ((let H18 \def (f_equal C T (\lambda (e: -C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | -(CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) -x3) H12) in (\lambda (H19: (eq B Abbr x0)).(\lambda (H20: (eq C d x1)).(let -H21 \def (eq_ind_r T x3 (\lambda (t: T).(getl i c2 (CHead x2 (Bind x0) t))) -H13 u H18) in (let H22 \def (eq_ind_r C x1 (\lambda (c0: C).(csubst0 (minus -i0 (S i)) u0 c0 x2)) H14 d H20) in (let H23 \def (eq_ind_r B x0 (\lambda (b: -B).(getl i c2 (CHead x2 (Bind b) u))) H21 Abbr H19) in (arity_abbr g c2 x2 u -i H23 a0 (H2 d1 u0 (r (Bind Abbr) (minus i0 (S i))) (getl_gen_S (Bind Abbr) d -(CHead d1 (Bind Abbr) u0) u (minus i0 (S i)) H15) x2 u (fsubst0_fst (r (Bind -Abbr) (minus i0 (S i))) u0 d u x2 H22))))))))) H17)) H16)))))))))) H11)) -(\lambda (H11: (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 -(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(getl i c2 (CHead e2 (Bind b) w))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 -(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 -e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 -(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(getl i c2 (CHead e2 (Bind b) w))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 -(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) -(arity g c2 (TLRef i) a0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: -C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H12: (eq C (CHead d (Bind -Abbr) u) (CHead x1 (Bind x0) x3))).(\lambda (H13: (getl i c2 (CHead x2 (Bind -x0) x4))).(\lambda (H14: (subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H15: -(csubst0 (minus i0 (S i)) u0 x1 x2)).(let H16 \def (eq_ind nat (minus i0 i) -(\lambda (n: nat).(getl n (CHead d (Bind Abbr) u) (CHead d1 (Bind Abbr) u0))) -(getl_conf_le i0 (CHead d1 (Bind Abbr) u0) c H3 (CHead d (Bind Abbr) u) i H0 -(le_S_n i i0 (le_S (S i) i0 H9))) (S (minus i0 (S i))) (minus_x_Sy i0 i H9)) -in (let H17 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda -(_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) -(CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H12) in ((let H18 \def -(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with -[(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k in K return -(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow -Abbr])])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H12) in ((let H19 -\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind -Abbr) u) (CHead x1 (Bind x0) x3) H12) in (\lambda (H20: (eq B Abbr -x0)).(\lambda (H21: (eq C d x1)).(let H22 \def (eq_ind_r T x3 (\lambda (t: -T).(subst0 (minus i0 (S i)) u0 t x4)) H14 u H19) in (let H23 \def (eq_ind_r C -x1 (\lambda (c0: C).(csubst0 (minus i0 (S i)) u0 c0 x2)) H15 d H21) in (let -H24 \def (eq_ind_r B x0 (\lambda (b: B).(getl i c2 (CHead x2 (Bind b) x4))) -H13 Abbr H20) in (arity_abbr g c2 x2 x4 i H24 a0 (H2 d1 u0 (r (Bind Abbr) -(minus i0 (S i))) (getl_gen_S (Bind Abbr) d (CHead d1 (Bind Abbr) u0) u -(minus i0 (S i)) H16) x2 x4 (fsubst0_both (r (Bind Abbr) (minus i0 (S i))) u0 -d u x4 H22 x2 H23))))))))) H18)) H17)))))))))))) H11)) H10))) (\lambda (H9: -(le i0 i)).(arity_abbr g c2 d u i (csubst0_getl_ge i0 i H9 c c2 u0 H8 (CHead -d (Bind Abbr) u) H0) a0 H1))) t2 H7))) H6)) (\lambda (H6: (land (subst0 i0 u0 -(TLRef i) t2) (csubst0 i0 u0 c c2))).(land_ind (subst0 i0 u0 (TLRef i) t2) -(csubst0 i0 u0 c c2) (arity g c2 t2 a0) (\lambda (H7: (subst0 i0 u0 (TLRef i) -t2)).(\lambda (H8: (csubst0 i0 u0 c c2)).(land_ind (eq nat i i0) (eq T t2 -(lift (S i) O u0)) (arity g c2 t2 a0) (\lambda (H9: (eq nat i i0)).(\lambda -(H10: (eq T t2 (lift (S i) O u0))).(eq_ind_r T (lift (S i) O u0) (\lambda (t: -T).(arity g c2 t a0)) (let H11 \def (eq_ind_r nat i0 (\lambda (n: -nat).(csubst0 n u0 c c2)) H8 i H9) in (let H12 \def (eq_ind_r nat i0 (\lambda -(n: nat).(getl n c (CHead d1 (Bind Abbr) u0))) H3 i H9) in (let H13 \def -(eq_ind C (CHead d (Bind Abbr) u) (\lambda (c0: C).(getl i c c0)) H0 (CHead -d1 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) i H0 (CHead d1 (Bind -Abbr) u0) H12)) in (let H14 \def (f_equal C C (\lambda (e: C).(match e in C -return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) -\Rightarrow c0])) (CHead d (Bind Abbr) u) (CHead d1 (Bind Abbr) u0) -(getl_mono c (CHead d (Bind Abbr) u) i H0 (CHead d1 (Bind Abbr) u0) H12)) in -((let H15 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: -C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d -(Bind Abbr) u) (CHead d1 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) -i H0 (CHead d1 (Bind Abbr) u0) H12)) in (\lambda (H16: (eq C d d1)).(let H17 -\def (eq_ind_r T u0 (\lambda (t: T).(getl i c (CHead d1 (Bind Abbr) t))) H13 -u H15) in (let H18 \def (eq_ind_r T u0 (\lambda (t: T).(csubst0 i t c c2)) -H11 u H15) in (eq_ind T u (\lambda (t: T).(arity g c2 (lift (S i) O t) a0)) -(let H19 \def (eq_ind_r C d1 (\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr) -u))) H17 d H16) in (arity_lift g d u a0 H1 c2 (S i) O (getl_drop Abbr c2 d u -i (csubst0_getl_ge i i (le_n i) c c2 u H18 (CHead d (Bind Abbr) u) H19)))) u0 -H15))))) H14))))) t2 H10))) (subst0_gen_lref u0 t2 i0 i H7)))) H6)) -H5)))))))))))))))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda -(i: nat).(\lambda (H0: (getl i c (CHead d (Bind Abst) u))).(\lambda (a0: -A).(\lambda (H1: (arity g d u (asucc g a0))).(\lambda (H2: ((\forall (d1: -C).(\forall (u0: T).(\forall (i0: nat).((getl i0 d (CHead d1 (Bind Abbr) u0)) -\to (\forall (c2: C).(\forall (t2: T).((fsubst0 i0 u0 d u c2 t2) \to (arity g -c2 t2 (asucc g a0))))))))))).(\lambda (d1: C).(\lambda (u0: T).(\lambda (i0: -nat).(\lambda (H3: (getl i0 c (CHead d1 (Bind Abbr) u0))).(\lambda (c2: -C).(\lambda (t2: T).(\lambda (H4: (fsubst0 i0 u0 c (TLRef i) c2 t2)).(let H_x -\def (fsubst0_gen_base c c2 (TLRef i) t2 u0 i0 H4) in (let H5 \def H_x in -(or3_ind (land (eq C c c2) (subst0 i0 u0 (TLRef i) t2)) (land (eq T (TLRef i) -t2) (csubst0 i0 u0 c c2)) (land (subst0 i0 u0 (TLRef i) t2) (csubst0 i0 u0 c -c2)) (arity g c2 t2 a0) (\lambda (H6: (land (eq C c c2) (subst0 i0 u0 (TLRef -i) t2))).(land_ind (eq C c c2) (subst0 i0 u0 (TLRef i) t2) (arity g c2 t2 a0) -(\lambda (H7: (eq C c c2)).(\lambda (H8: (subst0 i0 u0 (TLRef i) t2)).(eq_ind -C c (\lambda (c0: C).(arity g c0 t2 a0)) (land_ind (eq nat i i0) (eq T t2 -(lift (S i) O u0)) (arity g c t2 a0) (\lambda (H9: (eq nat i i0)).(\lambda -(H10: (eq T t2 (lift (S i) O u0))).(eq_ind_r T (lift (S i) O u0) (\lambda (t: -T).(arity g c t a0)) (let H11 \def (eq_ind_r nat i0 (\lambda (n: nat).(getl n -c (CHead d1 (Bind Abbr) u0))) H3 i H9) in (let H12 \def (eq_ind C (CHead d -(Bind Abst) u) (\lambda (c0: C).(getl i c c0)) H0 (CHead d1 (Bind Abbr) u0) -(getl_mono c (CHead d (Bind Abst) u) i H0 (CHead d1 (Bind Abbr) u0) H11)) in -(let H13 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (ee: C).(match ee in -C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k -_) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b) -\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow -False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) -\Rightarrow False])])) I (CHead d1 (Bind Abbr) u0) (getl_mono c (CHead d -(Bind Abst) u) i H0 (CHead d1 (Bind Abbr) u0) H11)) in (False_ind (arity g c -(lift (S i) O u0) a0) H13)))) t2 H10))) (subst0_gen_lref u0 t2 i0 i H8)) c2 -H7))) H6)) (\lambda (H6: (land (eq T (TLRef i) t2) (csubst0 i0 u0 c -c2))).(land_ind (eq T (TLRef i) t2) (csubst0 i0 u0 c c2) (arity g c2 t2 a0) -(\lambda (H7: (eq T (TLRef i) t2)).(\lambda (H8: (csubst0 i0 u0 c -c2)).(eq_ind T (TLRef i) (\lambda (t: T).(arity g c2 t a0)) (lt_le_e i i0 -(arity g c2 (TLRef i) a0) (\lambda (H9: (lt i i0)).(let H10 \def -(csubst0_getl_lt i0 i H9 c c2 u0 H8 (CHead d (Bind Abst) u) H0) in (or4_ind -(getl i c2 (CHead d (Bind Abst) u)) (ex3_4 B C T T (\lambda (b: B).(\lambda -(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead -e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(getl i c2 (CHead e0 (Bind b) w)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) -u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1)))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c2 -(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) (ex4_5 B C C T T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1))))))) (\lambda -(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl -i c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))) (arity g c2 (TLRef i) a0) -(\lambda (H11: (getl i c2 (CHead d (Bind Abst) u))).(let H12 \def (eq_ind nat -(minus i0 i) (\lambda (n: nat).(getl n (CHead d (Bind Abst) u) (CHead d1 -(Bind Abbr) u0))) (getl_conf_le i0 (CHead d1 (Bind Abbr) u0) c H3 (CHead d -(Bind Abst) u) i H0 (le_S_n i i0 (le_S (S i) i0 H9))) (S (minus i0 (S i))) -(minus_x_Sy i0 i H9)) in (arity_abst g c2 d u i H11 a0 H1))) (\lambda (H11: -(ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(eq C (CHead d (Bind Abst) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: -B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl i c2 (CHead e0 -(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda -(w: T).(subst0 (minus i0 (S i)) u0 u1 w))))))).(ex3_4_ind B C T T (\lambda -(b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind -Abst) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(getl i c2 (CHead e0 (Bind b) w)))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 -(minus i0 (S i)) u0 u1 w))))) (arity g c2 (TLRef i) a0) (\lambda (x0: -B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (eq C -(CHead d (Bind Abst) u) (CHead x1 (Bind x0) x2))).(\lambda (H13: (getl i c2 -(CHead x1 (Bind x0) x3))).(\lambda (H14: (subst0 (minus i0 (S i)) u0 x2 -x3)).(let H15 \def (eq_ind nat (minus i0 i) (\lambda (n: nat).(getl n (CHead -d (Bind Abst) u) (CHead d1 (Bind Abbr) u0))) (getl_conf_le i0 (CHead d1 (Bind -Abbr) u0) c H3 (CHead d (Bind Abst) u) i H0 (le_S_n i i0 (le_S (S i) i0 H9))) -(S (minus i0 (S i))) (minus_x_Sy i0 i H9)) in (let H16 \def (f_equal C C -(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abst) u) -(CHead x1 (Bind x0) x2) H12) in ((let H17 \def (f_equal C B (\lambda (e: -C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abst | -(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind -b) \Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead d (Bind Abst) u) -(CHead x1 (Bind x0) x2) H12) in ((let H18 \def (f_equal C T (\lambda (e: -C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | -(CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) -x2) H12) in (\lambda (H19: (eq B Abst x0)).(\lambda (H20: (eq C d x1)).(let -H21 \def (eq_ind_r T x2 (\lambda (t: T).(subst0 (minus i0 (S i)) u0 t x3)) -H14 u H18) in (let H22 \def (eq_ind_r C x1 (\lambda (c0: C).(getl i c2 (CHead -c0 (Bind x0) x3))) H13 d H20) in (let H23 \def (eq_ind_r B x0 (\lambda (b: -B).(getl i c2 (CHead d (Bind b) x3))) H22 Abst H19) in (arity_abst g c2 d x3 -i H23 a0 (H2 d1 u0 (r (Bind Abst) (minus i0 (S i))) (getl_gen_S (Bind Abst) d -(CHead d1 (Bind Abbr) u0) u (minus i0 (S i)) H15) d x3 (fsubst0_snd (r (Bind -Abst) (minus i0 (S i))) u0 d u x3 H21))))))))) H17)) H16)))))))))) H11)) -(\lambda (H11: (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1)))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c2 -(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))).(ex3_4_ind B C C -T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C -(CHead d (Bind Abst) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda -(_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c2 (CHead e2 (Bind b) -u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus i0 (S i)) u0 e1 e2))))) (arity g c2 (TLRef i) a0) (\lambda -(x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H12: (eq -C (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3))).(\lambda (H13: (getl i c2 -(CHead x2 (Bind x0) x3))).(\lambda (H14: (csubst0 (minus i0 (S i)) u0 x1 -x2)).(let H15 \def (eq_ind nat (minus i0 i) (\lambda (n: nat).(getl n (CHead -d (Bind Abst) u) (CHead d1 (Bind Abbr) u0))) (getl_conf_le i0 (CHead d1 (Bind -Abbr) u0) c H3 (CHead d (Bind Abst) u) i H0 (le_S_n i i0 (le_S (S i) i0 H9))) -(S (minus i0 (S i))) (minus_x_Sy i0 i H9)) in (let H16 \def (f_equal C C -(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abst) u) -(CHead x1 (Bind x0) x3) H12) in ((let H17 \def (f_equal C B (\lambda (e: -C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abst | -(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind -b) \Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead d (Bind Abst) u) -(CHead x1 (Bind x0) x3) H12) in ((let H18 \def (f_equal C T (\lambda (e: -C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | -(CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) -x3) H12) in (\lambda (H19: (eq B Abst x0)).(\lambda (H20: (eq C d x1)).(let -H21 \def (eq_ind_r T x3 (\lambda (t: T).(getl i c2 (CHead x2 (Bind x0) t))) -H13 u H18) in (let H22 \def (eq_ind_r C x1 (\lambda (c0: C).(csubst0 (minus -i0 (S i)) u0 c0 x2)) H14 d H20) in (let H23 \def (eq_ind_r B x0 (\lambda (b: -B).(getl i c2 (CHead x2 (Bind b) u))) H21 Abst H19) in (arity_abst g c2 x2 u -i H23 a0 (H2 d1 u0 (r (Bind Abst) (minus i0 (S i))) (getl_gen_S (Bind Abst) d -(CHead d1 (Bind Abbr) u0) u (minus i0 (S i)) H15) x2 u (fsubst0_fst (r (Bind -Abst) (minus i0 (S i))) u0 d u x2 H22))))))))) H17)) H16)))))))))) H11)) -(\lambda (H11: (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead e1 -(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(getl i c2 (CHead e2 (Bind b) w))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 -(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 -e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead e1 -(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(getl i c2 (CHead e2 (Bind b) w))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 -(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) -(arity g c2 (TLRef i) a0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: -C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H12: (eq C (CHead d (Bind -Abst) u) (CHead x1 (Bind x0) x3))).(\lambda (H13: (getl i c2 (CHead x2 (Bind -x0) x4))).(\lambda (H14: (subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H15: -(csubst0 (minus i0 (S i)) u0 x1 x2)).(let H16 \def (eq_ind nat (minus i0 i) -(\lambda (n: nat).(getl n (CHead d (Bind Abst) u) (CHead d1 (Bind Abbr) u0))) -(getl_conf_le i0 (CHead d1 (Bind Abbr) u0) c H3 (CHead d (Bind Abst) u) i H0 -(le_S_n i i0 (le_S (S i) i0 H9))) (S (minus i0 (S i))) (minus_x_Sy i0 i H9)) -in (let H17 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda -(_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) -(CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3) H12) in ((let H18 \def -(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with -[(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match k in K return -(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow -Abst])])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3) H12) in ((let H19 -\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind -Abst) u) (CHead x1 (Bind x0) x3) H12) in (\lambda (H20: (eq B Abst -x0)).(\lambda (H21: (eq C d x1)).(let H22 \def (eq_ind_r T x3 (\lambda (t: -T).(subst0 (minus i0 (S i)) u0 t x4)) H14 u H19) in (let H23 \def (eq_ind_r C -x1 (\lambda (c0: C).(csubst0 (minus i0 (S i)) u0 c0 x2)) H15 d H21) in (let -H24 \def (eq_ind_r B x0 (\lambda (b: B).(getl i c2 (CHead x2 (Bind b) x4))) -H13 Abst H20) in (arity_abst g c2 x2 x4 i H24 a0 (H2 d1 u0 (r (Bind Abst) -(minus i0 (S i))) (getl_gen_S (Bind Abst) d (CHead d1 (Bind Abbr) u0) u -(minus i0 (S i)) H16) x2 x4 (fsubst0_both (r (Bind Abst) (minus i0 (S i))) u0 -d u x4 H22 x2 H23))))))))) H18)) H17)))))))))))) H11)) H10))) (\lambda (H9: -(le i0 i)).(arity_abst g c2 d u i (csubst0_getl_ge i0 i H9 c c2 u0 H8 (CHead -d (Bind Abst) u) H0) a0 H1))) t2 H7))) H6)) (\lambda (H6: (land (subst0 i0 u0 -(TLRef i) t2) (csubst0 i0 u0 c c2))).(land_ind (subst0 i0 u0 (TLRef i) t2) -(csubst0 i0 u0 c c2) (arity g c2 t2 a0) (\lambda (H7: (subst0 i0 u0 (TLRef i) -t2)).(\lambda (H8: (csubst0 i0 u0 c c2)).(land_ind (eq nat i i0) (eq T t2 -(lift (S i) O u0)) (arity g c2 t2 a0) (\lambda (H9: (eq nat i i0)).(\lambda -(H10: (eq T t2 (lift (S i) O u0))).(eq_ind_r T (lift (S i) O u0) (\lambda (t: -T).(arity g c2 t a0)) (let H11 \def (eq_ind_r nat i0 (\lambda (n: -nat).(csubst0 n u0 c c2)) H8 i H9) in (let H12 \def (eq_ind_r nat i0 (\lambda -(n: nat).(getl n c (CHead d1 (Bind Abbr) u0))) H3 i H9) in (let H13 \def -(eq_ind C (CHead d (Bind Abst) u) (\lambda (c0: C).(getl i c c0)) H0 (CHead -d1 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abst) u) i H0 (CHead d1 (Bind -Abbr) u0) H12)) in (let H14 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda -(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind b) \Rightarrow (match b in B return (\lambda (_: -B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow True | Void -\Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead d1 (Bind Abbr) -u0) (getl_mono c (CHead d (Bind Abst) u) i H0 (CHead d1 (Bind Abbr) u0) H12)) -in (False_ind (arity g c2 (lift (S i) O u0) a0) H14))))) t2 H10))) -(subst0_gen_lref u0 t2 i0 i H7)))) H6)) H5)))))))))))))))))) (\lambda (b: -B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c: C).(\lambda (u: -T).(\lambda (a1: A).(\lambda (H1: (arity g c u a1)).(\lambda (H2: ((\forall -(d1: C).(\forall (u0: T).(\forall (i: nat).((getl i c (CHead d1 (Bind Abbr) -u0)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 c u c2 t2) \to -(arity g c2 t2 a1)))))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (_: -(arity g (CHead c (Bind b) u) t a2)).(\lambda (H4: ((\forall (d1: C).(\forall -(u0: T).(\forall (i: nat).((getl i (CHead c (Bind b) u) (CHead d1 (Bind Abbr) -u0)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 (CHead c (Bind b) -u) t c2 t2) \to (arity g c2 t2 a2)))))))))).(\lambda (d1: C).(\lambda (u0: -T).(\lambda (i: nat).(\lambda (H5: (getl i c (CHead d1 (Bind Abbr) -u0))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H6: (fsubst0 i u0 c (THead -(Bind b) u t) c2 t2)).(let H_x \def (fsubst0_gen_base c c2 (THead (Bind b) u -t) t2 u0 i H6) in (let H7 \def H_x in (or3_ind (land (eq C c c2) (subst0 i u0 -(THead (Bind b) u t) t2)) (land (eq T (THead (Bind b) u t) t2) (csubst0 i u0 -c c2)) (land (subst0 i u0 (THead (Bind b) u t) t2) (csubst0 i u0 c c2)) -(arity g c2 t2 a2) (\lambda (H8: (land (eq C c c2) (subst0 i u0 (THead (Bind -b) u t) t2))).(land_ind (eq C c c2) (subst0 i u0 (THead (Bind b) u t) t2) -(arity g c2 t2 a2) (\lambda (H9: (eq C c c2)).(\lambda (H10: (subst0 i u0 -(THead (Bind b) u t) t2)).(eq_ind C c (\lambda (c0: C).(arity g c0 t2 a2)) -(or3_ind (ex2 T (\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t))) (\lambda -(u2: T).(subst0 i u0 u u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead (Bind b) -u t3))) (\lambda (t3: T).(subst0 (s (Bind b) i) u0 t t3))) (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) u2 t3)))) (\lambda -(u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: -T).(subst0 (s (Bind b) i) u0 t t3)))) (arity g c t2 a2) (\lambda (H11: (ex2 T -(\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t))) (\lambda (u2: T).(subst0 i -u0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t))) -(\lambda (u2: T).(subst0 i u0 u u2)) (arity g c t2 a2) (\lambda (x: -T).(\lambda (H12: (eq T t2 (THead (Bind b) x t))).(\lambda (H13: (subst0 i u0 -u x)).(eq_ind_r T (THead (Bind b) x t) (\lambda (t0: T).(arity g c t0 a2)) -(arity_bind g b H0 c x a1 (H2 d1 u0 i H5 c x (fsubst0_snd i u0 c u x H13)) t -a2 (H4 d1 u0 (S i) (getl_clear_bind b (CHead c (Bind b) u) c u (clear_bind b -c u) (CHead d1 (Bind Abbr) u0) i H5) (CHead c (Bind b) x) t (fsubst0_fst (S -i) u0 (CHead c (Bind b) u) t (CHead c (Bind b) x) (csubst0_snd_bind b i u0 u -x H13 c)))) t2 H12)))) H11)) (\lambda (H11: (ex2 T (\lambda (t3: T).(eq T t2 -(THead (Bind b) u t3))) (\lambda (t3: T).(subst0 (s (Bind b) i) u0 t -t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead (Bind b) u t3))) (\lambda -(t3: T).(subst0 (s (Bind b) i) u0 t t3)) (arity g c t2 a2) (\lambda (x: -T).(\lambda (H12: (eq T t2 (THead (Bind b) u x))).(\lambda (H13: (subst0 (s -(Bind b) i) u0 t x)).(eq_ind_r T (THead (Bind b) u x) (\lambda (t0: T).(arity -g c t0 a2)) (arity_bind g b H0 c u a1 H1 x a2 (H4 d1 u0 (S i) -(getl_clear_bind b (CHead c (Bind b) u) c u (clear_bind b c u) (CHead d1 -(Bind Abbr) u0) i H5) (CHead c (Bind b) u) x (fsubst0_snd (S i) u0 (CHead c -(Bind b) u) t x H13))) t2 H12)))) H11)) (\lambda (H11: (ex3_2 T T (\lambda -(u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: -T).(subst0 (s (Bind b) i) u0 t t3))))).(ex3_2_ind T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Bind b) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: -T).(subst0 (s (Bind b) i) u0 t t3))) (arity g c t2 a2) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H12: (eq T t2 (THead (Bind b) x0 x1))).(\lambda -(H13: (subst0 i u0 u x0)).(\lambda (H14: (subst0 (s (Bind b) i) u0 t -x1)).(eq_ind_r T (THead (Bind b) x0 x1) (\lambda (t0: T).(arity g c t0 a2)) -(arity_bind g b H0 c x0 a1 (H2 d1 u0 i H5 c x0 (fsubst0_snd i u0 c u x0 H13)) -x1 a2 (H4 d1 u0 (S i) (getl_clear_bind b (CHead c (Bind b) u) c u (clear_bind -b c u) (CHead d1 (Bind Abbr) u0) i H5) (CHead c (Bind b) x0) x1 (fsubst0_both -(S i) u0 (CHead c (Bind b) u) t x1 H14 (CHead c (Bind b) x0) -(csubst0_snd_bind b i u0 u x0 H13 c)))) t2 H12)))))) H11)) (subst0_gen_head -(Bind b) u0 u t t2 i H10)) c2 H9))) H8)) (\lambda (H8: (land (eq T (THead -(Bind b) u t) t2) (csubst0 i u0 c c2))).(land_ind (eq T (THead (Bind b) u t) -t2) (csubst0 i u0 c c2) (arity g c2 t2 a2) (\lambda (H9: (eq T (THead (Bind -b) u t) t2)).(\lambda (H10: (csubst0 i u0 c c2)).(eq_ind T (THead (Bind b) u -t) (\lambda (t0: T).(arity g c2 t0 a2)) (arity_bind g b H0 c2 u a1 (H2 d1 u0 -i H5 c2 u (fsubst0_fst i u0 c u c2 H10)) t a2 (H4 d1 u0 (S i) -(getl_clear_bind b (CHead c (Bind b) u) c u (clear_bind b c u) (CHead d1 -(Bind Abbr) u0) i H5) (CHead c2 (Bind b) u) t (fsubst0_fst (S i) u0 (CHead c -(Bind b) u) t (CHead c2 (Bind b) u) (csubst0_fst_bind b i c c2 u0 H10 u)))) -t2 H9))) H8)) (\lambda (H8: (land (subst0 i u0 (THead (Bind b) u t) t2) -(csubst0 i u0 c c2))).(land_ind (subst0 i u0 (THead (Bind b) u t) t2) -(csubst0 i u0 c c2) (arity g c2 t2 a2) (\lambda (H9: (subst0 i u0 (THead -(Bind b) u t) t2)).(\lambda (H10: (csubst0 i u0 c c2)).(or3_ind (ex2 T -(\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t))) (\lambda (u2: T).(subst0 i -u0 u u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead (Bind b) u t3))) (\lambda -(t3: T).(subst0 (s (Bind b) i) u0 t t3))) (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Bind b) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: -T).(subst0 (s (Bind b) i) u0 t t3)))) (arity g c2 t2 a2) (\lambda (H11: (ex2 -T (\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t))) (\lambda (u2: T).(subst0 -i u0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t))) -(\lambda (u2: T).(subst0 i u0 u u2)) (arity g c2 t2 a2) (\lambda (x: -T).(\lambda (H12: (eq T t2 (THead (Bind b) x t))).(\lambda (H13: (subst0 i u0 -u x)).(eq_ind_r T (THead (Bind b) x t) (\lambda (t0: T).(arity g c2 t0 a2)) -(arity_bind g b H0 c2 x a1 (H2 d1 u0 i H5 c2 x (fsubst0_both i u0 c u x H13 -c2 H10)) t a2 (H4 d1 u0 (S i) (getl_clear_bind b (CHead c (Bind b) u) c u -(clear_bind b c u) (CHead d1 (Bind Abbr) u0) i H5) (CHead c2 (Bind b) x) t -(fsubst0_fst (S i) u0 (CHead c (Bind b) u) t (CHead c2 (Bind b) x) -(csubst0_both_bind b i u0 u x H13 c c2 H10)))) t2 H12)))) H11)) (\lambda -(H11: (ex2 T (\lambda (t3: T).(eq T t2 (THead (Bind b) u t3))) (\lambda (t3: -T).(subst0 (s (Bind b) i) u0 t t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 -(THead (Bind b) u t3))) (\lambda (t3: T).(subst0 (s (Bind b) i) u0 t t3)) -(arity g c2 t2 a2) (\lambda (x: T).(\lambda (H12: (eq T t2 (THead (Bind b) u -x))).(\lambda (H13: (subst0 (s (Bind b) i) u0 t x)).(eq_ind_r T (THead (Bind -b) u x) (\lambda (t0: T).(arity g c2 t0 a2)) (arity_bind g b H0 c2 u a1 (H2 -d1 u0 i H5 c2 u (fsubst0_fst i u0 c u c2 H10)) x a2 (H4 d1 u0 (S i) -(getl_clear_bind b (CHead c (Bind b) u) c u (clear_bind b c u) (CHead d1 -(Bind Abbr) u0) i H5) (CHead c2 (Bind b) u) x (fsubst0_both (S i) u0 (CHead c -(Bind b) u) t x H13 (CHead c2 (Bind b) u) (csubst0_fst_bind b i c c2 u0 H10 -u)))) t2 H12)))) H11)) (\lambda (H11: (ex3_2 T T (\lambda (u2: T).(\lambda -(t3: T).(eq T t2 (THead (Bind b) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind b) -i) u0 t t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 -(THead (Bind b) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u -u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind b) i) u0 t t3))) -(arity g c2 t2 a2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H12: (eq T t2 -(THead (Bind b) x0 x1))).(\lambda (H13: (subst0 i u0 u x0)).(\lambda (H14: -(subst0 (s (Bind b) i) u0 t x1)).(eq_ind_r T (THead (Bind b) x0 x1) (\lambda -(t0: T).(arity g c2 t0 a2)) (arity_bind g b H0 c2 x0 a1 (H2 d1 u0 i H5 c2 x0 -(fsubst0_both i u0 c u x0 H13 c2 H10)) x1 a2 (H4 d1 u0 (S i) (getl_clear_bind -b (CHead c (Bind b) u) c u (clear_bind b c u) (CHead d1 (Bind Abbr) u0) i H5) -(CHead c2 (Bind b) x0) x1 (fsubst0_both (S i) u0 (CHead c (Bind b) u) t x1 -H14 (CHead c2 (Bind b) x0) (csubst0_both_bind b i u0 u x0 H13 c c2 H10)))) t2 -H12)))))) H11)) (subst0_gen_head (Bind b) u0 u t t2 i H9)))) H8)) -H7))))))))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a1: -A).(\lambda (H0: (arity g c u (asucc g a1))).(\lambda (H1: ((\forall (d1: -C).(\forall (u0: T).(\forall (i: nat).((getl i c (CHead d1 (Bind Abbr) u0)) -\to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 c u c2 t2) \to (arity g -c2 t2 (asucc g a1))))))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (_: -(arity g (CHead c (Bind Abst) u) t a2)).(\lambda (H3: ((\forall (d1: -C).(\forall (u0: T).(\forall (i: nat).((getl i (CHead c (Bind Abst) u) (CHead -d1 (Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 -(CHead c (Bind Abst) u) t c2 t2) \to (arity g c2 t2 a2)))))))))).(\lambda -(d1: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H4: (getl i c (CHead d1 -(Bind Abbr) u0))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H5: (fsubst0 i -u0 c (THead (Bind Abst) u t) c2 t2)).(let H_x \def (fsubst0_gen_base c c2 -(THead (Bind Abst) u t) t2 u0 i H5) in (let H6 \def H_x in (or3_ind (land (eq -C c c2) (subst0 i u0 (THead (Bind Abst) u t) t2)) (land (eq T (THead (Bind -Abst) u t) t2) (csubst0 i u0 c c2)) (land (subst0 i u0 (THead (Bind Abst) u -t) t2) (csubst0 i u0 c c2)) (arity g c2 t2 (AHead a1 a2)) (\lambda (H7: (land -(eq C c c2) (subst0 i u0 (THead (Bind Abst) u t) t2))).(land_ind (eq C c c2) -(subst0 i u0 (THead (Bind Abst) u t) t2) (arity g c2 t2 (AHead a1 a2)) -(\lambda (H8: (eq C c c2)).(\lambda (H9: (subst0 i u0 (THead (Bind Abst) u t) -t2)).(eq_ind C c (\lambda (c0: C).(arity g c0 t2 (AHead a1 a2))) (or3_ind -(ex2 T (\lambda (u2: T).(eq T t2 (THead (Bind Abst) u2 t))) (\lambda (u2: -T).(subst0 i u0 u u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead (Bind Abst) u -t3))) (\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3))) (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: -T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3)))) (arity g c t2 -(AHead a1 a2)) (\lambda (H10: (ex2 T (\lambda (u2: T).(eq T t2 (THead (Bind -Abst) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T (\lambda (u2: -T).(eq T t2 (THead (Bind Abst) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)) -(arity g c t2 (AHead a1 a2)) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead -(Bind Abst) x t))).(\lambda (H12: (subst0 i u0 u x)).(eq_ind_r T (THead (Bind -Abst) x t) (\lambda (t0: T).(arity g c t0 (AHead a1 a2))) (arity_head g c x -a1 (H1 d1 u0 i H4 c x (fsubst0_snd i u0 c u x H12)) t a2 (H3 d1 u0 (S i) -(getl_clear_bind Abst (CHead c (Bind Abst) u) c u (clear_bind Abst c u) -(CHead d1 (Bind Abbr) u0) i H4) (CHead c (Bind Abst) x) t (fsubst0_fst (S i) -u0 (CHead c (Bind Abst) u) t (CHead c (Bind Abst) x) (csubst0_snd_bind Abst i -u0 u x H12 c)))) t2 H11)))) H10)) (\lambda (H10: (ex2 T (\lambda (t3: T).(eq -T t2 (THead (Bind Abst) u t3))) (\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 -t t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead (Bind Abst) u t3))) -(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3)) (arity g c t2 (AHead a1 -a2)) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Bind Abst) u -x))).(\lambda (H12: (subst0 (s (Bind Abst) i) u0 t x)).(eq_ind_r T (THead -(Bind Abst) u x) (\lambda (t0: T).(arity g c t0 (AHead a1 a2))) (arity_head g -c u a1 H0 x a2 (H3 d1 u0 (S i) (getl_clear_bind Abst (CHead c (Bind Abst) u) -c u (clear_bind Abst c u) (CHead d1 (Bind Abbr) u0) i H4) (CHead c (Bind -Abst) u) x (fsubst0_snd (S i) u0 (CHead c (Bind Abst) u) t x H12))) t2 -H11)))) H10)) (\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq -T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i -u0 u u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t -t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead -(Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) -(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3))) (arity -g c t2 (AHead a1 a2)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq T -t2 (THead (Bind Abst) x0 x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda -(H13: (subst0 (s (Bind Abst) i) u0 t x1)).(eq_ind_r T (THead (Bind Abst) x0 -x1) (\lambda (t0: T).(arity g c t0 (AHead a1 a2))) (arity_head g c x0 a1 (H1 -d1 u0 i H4 c x0 (fsubst0_snd i u0 c u x0 H12)) x1 a2 (H3 d1 u0 (S i) -(getl_clear_bind Abst (CHead c (Bind Abst) u) c u (clear_bind Abst c u) -(CHead d1 (Bind Abbr) u0) i H4) (CHead c (Bind Abst) x0) x1 (fsubst0_both (S -i) u0 (CHead c (Bind Abst) u) t x1 H13 (CHead c (Bind Abst) x0) -(csubst0_snd_bind Abst i u0 u x0 H12 c)))) t2 H11)))))) H10)) -(subst0_gen_head (Bind Abst) u0 u t t2 i H9)) c2 H8))) H7)) (\lambda (H7: -(land (eq T (THead (Bind Abst) u t) t2) (csubst0 i u0 c c2))).(land_ind (eq T -(THead (Bind Abst) u t) t2) (csubst0 i u0 c c2) (arity g c2 t2 (AHead a1 a2)) -(\lambda (H8: (eq T (THead (Bind Abst) u t) t2)).(\lambda 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T).(\lambda -(t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind -Abst) i) u0 t t3)))) (arity g c2 t2 (AHead a1 a2)) (\lambda (H10: (ex2 T -(\lambda (u2: T).(eq T t2 (THead (Bind Abst) u2 t))) (\lambda (u2: T).(subst0 -i u0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T t2 (THead (Bind Abst) u2 t))) -(\lambda (u2: T).(subst0 i u0 u u2)) (arity g c2 t2 (AHead a1 a2)) (\lambda -(x: T).(\lambda (H11: (eq T t2 (THead (Bind Abst) x t))).(\lambda (H12: -(subst0 i u0 u x)).(eq_ind_r T (THead (Bind Abst) x t) (\lambda (t0: -T).(arity g c2 t0 (AHead a1 a2))) (arity_head g c2 x a1 (H1 d1 u0 i H4 c2 x -(fsubst0_both i u0 c u x H12 c2 H9)) t a2 (H3 d1 u0 (S i) (getl_clear_bind -Abst (CHead c (Bind Abst) u) c u (clear_bind Abst c u) (CHead d1 (Bind Abbr) -u0) i H4) (CHead c2 (Bind Abst) x) t (fsubst0_fst (S i) u0 (CHead c (Bind -Abst) u) t (CHead c2 (Bind Abst) x) (csubst0_both_bind Abst i u0 u x H12 c c2 -H9)))) t2 H11)))) H10)) (\lambda (H10: (ex2 T (\lambda (t3: T).(eq T t2 -(THead (Bind Abst) u t3))) (\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t -t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead (Bind Abst) u t3))) -(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3)) (arity g c2 t2 (AHead a1 -a2)) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Bind Abst) u -x))).(\lambda (H12: (subst0 (s (Bind Abst) i) u0 t x)).(eq_ind_r T (THead -(Bind Abst) u x) (\lambda (t0: T).(arity g c2 t0 (AHead a1 a2))) (arity_head -g c2 u a1 (H1 d1 u0 i H4 c2 u (fsubst0_fst i u0 c u c2 H9)) x a2 (H3 d1 u0 (S -i) (getl_clear_bind Abst (CHead c (Bind Abst) u) c u (clear_bind Abst c u) -(CHead d1 (Bind Abbr) u0) i H4) (CHead c2 (Bind Abst) u) x (fsubst0_both (S -i) u0 (CHead c (Bind Abst) u) t x H12 (CHead c2 (Bind Abst) u) -(csubst0_fst_bind Abst i c c2 u0 H9 u)))) t2 H11)))) H10)) (\lambda (H10: -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: -T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3))))).(ex3_2_ind T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: -T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3))) (arity g c2 t2 -(AHead a1 a2)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq T t2 -(THead (Bind Abst) x0 x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda (H13: -(subst0 (s (Bind Abst) i) u0 t x1)).(eq_ind_r T (THead (Bind Abst) x0 x1) -(\lambda (t0: T).(arity g c2 t0 (AHead a1 a2))) (arity_head g c2 x0 a1 (H1 d1 -u0 i H4 c2 x0 (fsubst0_both i u0 c u x0 H12 c2 H9)) x1 a2 (H3 d1 u0 (S i) -(getl_clear_bind Abst (CHead c (Bind Abst) u) c u (clear_bind Abst c u) -(CHead d1 (Bind Abbr) u0) i H4) (CHead c2 (Bind Abst) x0) x1 (fsubst0_both (S -i) u0 (CHead c (Bind Abst) u) t x1 H13 (CHead c2 (Bind Abst) x0) -(csubst0_both_bind Abst i u0 u x0 H12 c c2 H9)))) t2 H11)))))) H10)) -(subst0_gen_head (Bind Abst) u0 u t t2 i H8)))) H7)) H6))))))))))))))))))) -(\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H0: (arity g c u -a1)).(\lambda (H1: ((\forall (d1: C).(\forall (u0: T).(\forall (i: -nat).((getl i c (CHead d1 (Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2: -T).((fsubst0 i u0 c u c2 t2) \to (arity g c2 t2 a1)))))))))).(\lambda (t: -T).(\lambda (a2: A).(\lambda (H2: (arity g c t (AHead a1 a2))).(\lambda (H3: -((\forall (d1: C).(\forall (u0: T).(\forall (i: nat).((getl i c (CHead d1 -(Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 c t c2 -t2) \to (arity g c2 t2 (AHead a1 a2))))))))))).(\lambda (d1: C).(\lambda (u0: -T).(\lambda (i: nat).(\lambda (H4: (getl i c (CHead d1 (Bind Abbr) -u0))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H5: (fsubst0 i u0 c (THead -(Flat Appl) u t) c2 t2)).(let H_x \def (fsubst0_gen_base c c2 (THead (Flat -Appl) u t) t2 u0 i H5) in (let H6 \def H_x in (or3_ind (land (eq C c c2) -(subst0 i u0 (THead (Flat Appl) u t) t2)) (land (eq T (THead (Flat Appl) u t) -t2) (csubst0 i u0 c c2)) (land (subst0 i u0 (THead (Flat Appl) u t) t2) -(csubst0 i u0 c c2)) (arity g c2 t2 a2) (\lambda (H7: (land (eq C c c2) -(subst0 i u0 (THead (Flat Appl) u t) t2))).(land_ind (eq C c c2) (subst0 i u0 -(THead (Flat Appl) u t) t2) (arity g c2 t2 a2) (\lambda (H8: (eq C c -c2)).(\lambda (H9: (subst0 i u0 (THead (Flat Appl) u t) t2)).(eq_ind C c -(\lambda (c0: C).(arity g c0 t2 a2)) (or3_ind (ex2 T (\lambda (u2: T).(eq T -t2 (THead (Flat Appl) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2))) (ex2 T -(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u t3))) (\lambda (t3: T).(subst0 -(s (Flat Appl) i) u0 t t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq -T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i -u0 u u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 t -t3)))) (arity g c t2 a2) (\lambda (H10: (ex2 T (\lambda (u2: T).(eq T t2 -(THead (Flat 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-(\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead -(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) -(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 t -t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead -(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) -(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 t t3))) (arity -g c t2 a2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq T t2 (THead -(Flat Appl) x0 x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda (H13: -(subst0 (s (Flat Appl) i) u0 t x1)).(eq_ind_r T (THead (Flat Appl) x0 x1) -(\lambda (t0: T).(arity g c t0 a2)) (arity_appl g c x0 a1 (H1 d1 u0 i H4 c x0 -(fsubst0_snd i u0 c u x0 H12)) x1 a2 (H3 d1 u0 i H4 c x1 (fsubst0_snd i u0 c -t x1 H13))) t2 H11)))))) H10)) (subst0_gen_head (Flat Appl) u0 u t t2 i H9)) -c2 H8))) H7)) (\lambda (H7: (land (eq T (THead (Flat Appl) u t) t2) (csubst0 -i u0 c 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T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: -T).(subst0 (s (Flat Appl) i) u0 t t3)))) (arity g c2 t2 a2) (\lambda (H10: -(ex2 T (\lambda (u2: T).(eq T t2 (THead (Flat Appl) u2 t))) (\lambda (u2: -T).(subst0 i u0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T t2 (THead (Flat -Appl) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)) (arity g c2 t2 a2) -(\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat Appl) x t))).(\lambda -(H12: (subst0 i u0 u x)).(eq_ind_r T (THead (Flat Appl) x t) (\lambda (t0: -T).(arity g c2 t0 a2)) (arity_appl g c2 x a1 (H1 d1 u0 i H4 c2 x -(fsubst0_both i u0 c u x H12 c2 H9)) t a2 (H3 d1 u0 i H4 c2 t (fsubst0_fst i -u0 c t c2 H9))) t2 H11)))) H10)) (\lambda (H10: (ex2 T (\lambda (t3: T).(eq T -t2 (THead (Flat Appl) u t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 t -t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead (Flat Appl) u t3))) -(\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 t t3)) (arity g c2 t2 a2) -(\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat Appl) u x))).(\lambda -(H12: (subst0 (s (Flat Appl) i) u0 t x)).(eq_ind_r T (THead (Flat Appl) u x) -(\lambda (t0: T).(arity g c2 t0 a2)) (arity_appl g c2 u a1 (H1 d1 u0 i H4 c2 -u (fsubst0_fst i u0 c u c2 H9)) x a2 (H3 d1 u0 i H4 c2 x (fsubst0_both i u0 c -t x H12 c2 H9))) t2 H11)))) H10)) (\lambda (H10: (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: -T).(subst0 (s (Flat Appl) i) u0 t t3))))).(ex3_2_ind T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: -T).(subst0 (s (Flat Appl) i) u0 t t3))) (arity g c2 t2 a2) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H11: (eq T t2 (THead (Flat Appl) x0 -x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda (H13: (subst0 (s (Flat -Appl) i) u0 t x1)).(eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda (t0: -T).(arity g c2 t0 a2)) (arity_appl g c2 x0 a1 (H1 d1 u0 i H4 c2 x0 -(fsubst0_both i u0 c u x0 H12 c2 H9)) x1 a2 (H3 d1 u0 i H4 c2 x1 -(fsubst0_both i u0 c t x1 H13 c2 H9))) t2 H11)))))) H10)) (subst0_gen_head -(Flat Appl) u0 u t t2 i H8)))) H7)) H6))))))))))))))))))) (\lambda (c: -C).(\lambda (u: T).(\lambda (a0: A).(\lambda (H0: (arity g c u (asucc g -a0))).(\lambda (H1: ((\forall (d1: C).(\forall (u0: T).(\forall (i: -nat).((getl i c (CHead d1 (Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2: -T).((fsubst0 i u0 c u c2 t2) \to (arity g c2 t2 (asucc g -a0))))))))))).(\lambda (t: T).(\lambda (H2: (arity g c t a0)).(\lambda (H3: -((\forall (d1: C).(\forall (u0: T).(\forall (i: nat).((getl i c (CHead d1 -(Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 c t c2 -t2) \to (arity g c2 t2 a0)))))))))).(\lambda (d1: C).(\lambda (u0: -T).(\lambda (i: nat).(\lambda (H4: (getl i c (CHead d1 (Bind Abbr) -u0))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H5: (fsubst0 i u0 c (THead -(Flat Cast) u t) c2 t2)).(let H_x \def (fsubst0_gen_base c c2 (THead (Flat -Cast) u t) t2 u0 i H5) in (let H6 \def H_x in (or3_ind (land (eq C c c2) -(subst0 i u0 (THead (Flat Cast) u t) t2)) (land (eq T (THead (Flat Cast) u t) -t2) (csubst0 i u0 c c2)) (land (subst0 i u0 (THead (Flat Cast) u t) t2) -(csubst0 i u0 c c2)) (arity g c2 t2 a0) (\lambda (H7: (land (eq C c c2) -(subst0 i u0 (THead (Flat Cast) u t) t2))).(land_ind (eq C c c2) (subst0 i u0 -(THead (Flat Cast) u t) t2) (arity g c2 t2 a0) (\lambda (H8: (eq C c -c2)).(\lambda (H9: (subst0 i u0 (THead (Flat Cast) u t) t2)).(eq_ind C c -(\lambda (c0: C).(arity g c0 t2 a0)) (or3_ind (ex2 T (\lambda (u2: T).(eq T -t2 (THead (Flat Cast) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2))) (ex2 T -(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u t3))) (\lambda (t3: T).(subst0 -(s (Flat Cast) i) u0 t t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq -T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i -u0 u u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t -t3)))) (arity g c t2 a0) (\lambda (H10: (ex2 T (\lambda (u2: T).(eq T t2 -(THead (Flat Cast) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T -(\lambda (u2: T).(eq T t2 (THead (Flat Cast) u2 t))) (\lambda (u2: T).(subst0 -i u0 u u2)) (arity g c t2 a0) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead -(Flat Cast) x t))).(\lambda (H12: (subst0 i u0 u x)).(eq_ind_r T (THead (Flat -Cast) x t) (\lambda (t0: T).(arity g c t0 a0)) (arity_cast g c x a0 (H1 d1 u0 -i H4 c x (fsubst0_snd i u0 c u x H12)) t H2) t2 H11)))) H10)) (\lambda (H10: -(ex2 T (\lambda (t3: T).(eq T t2 (THead (Flat Cast) u t3))) (\lambda (t3: -T).(subst0 (s (Flat Cast) i) u0 t t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 -(THead (Flat Cast) u t3))) (\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t -t3)) (arity g c t2 a0) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat -Cast) u x))).(\lambda (H12: (subst0 (s (Flat Cast) i) u0 t x)).(eq_ind_r T -(THead (Flat Cast) u x) (\lambda (t0: T).(arity g c t0 a0)) (arity_cast g c u -a0 H0 x (H3 d1 u0 i H4 c x (fsubst0_snd i u0 c t x H12))) t2 H11)))) H10)) -(\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead -(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) -(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t -t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead -(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) -(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t t3))) (arity -g c t2 a0) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq T t2 (THead -(Flat Cast) x0 x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda (H13: -(subst0 (s (Flat Cast) i) u0 t x1)).(eq_ind_r T (THead (Flat Cast) x0 x1) -(\lambda (t0: T).(arity g c t0 a0)) (arity_cast g c x0 a0 (H1 d1 u0 i H4 c x0 -(fsubst0_snd i u0 c u x0 H12)) x1 (H3 d1 u0 i H4 c x1 (fsubst0_snd i u0 c t -x1 H13))) t2 H11)))))) H10)) (subst0_gen_head (Flat Cast) u0 u t t2 i H9)) c2 -H8))) H7)) (\lambda (H7: (land (eq T (THead (Flat Cast) u t) t2) (csubst0 i -u0 c c2))).(land_ind (eq T (THead (Flat Cast) u t) t2) (csubst0 i u0 c c2) -(arity g c2 t2 a0) (\lambda (H8: (eq T (THead (Flat Cast) u t) t2)).(\lambda -(H9: (csubst0 i u0 c c2)).(eq_ind T (THead (Flat Cast) u t) (\lambda (t0: -T).(arity g c2 t0 a0)) (arity_cast g c2 u a0 (H1 d1 u0 i H4 c2 u (fsubst0_fst -i u0 c u c2 H9)) t (H3 d1 u0 i H4 c2 t (fsubst0_fst i u0 c t c2 H9))) t2 -H8))) H7)) (\lambda (H7: (land (subst0 i u0 (THead (Flat Cast) u t) t2) -(csubst0 i u0 c c2))).(land_ind (subst0 i u0 (THead (Flat Cast) u t) t2) -(csubst0 i u0 c c2) (arity g c2 t2 a0) (\lambda (H8: (subst0 i u0 (THead -(Flat Cast) u t) t2)).(\lambda (H9: (csubst0 i u0 c c2)).(or3_ind (ex2 T -(\lambda (u2: T).(eq T t2 (THead (Flat Cast) u2 t))) (\lambda (u2: T).(subst0 -i u0 u u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead (Flat Cast) u t3))) -(\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t t3))) (ex3_2 T T (\lambda -(u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: -T).(subst0 (s (Flat Cast) i) u0 t t3)))) (arity g c2 t2 a0) (\lambda (H10: -(ex2 T (\lambda (u2: T).(eq T t2 (THead (Flat Cast) u2 t))) (\lambda (u2: -T).(subst0 i u0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T t2 (THead (Flat -Cast) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)) (arity g c2 t2 a0) -(\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat Cast) x t))).(\lambda -(H12: (subst0 i u0 u x)).(eq_ind_r T (THead (Flat Cast) x t) (\lambda (t0: -T).(arity g c2 t0 a0)) (arity_cast g c2 x a0 (H1 d1 u0 i H4 c2 x -(fsubst0_both i u0 c u x H12 c2 H9)) t (H3 d1 u0 i H4 c2 t (fsubst0_fst i u0 -c t c2 H9))) t2 H11)))) H10)) (\lambda (H10: (ex2 T (\lambda (t3: T).(eq T t2 -(THead (Flat Cast) u t3))) (\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t -t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead (Flat Cast) u t3))) -(\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t t3)) (arity g c2 t2 a0) -(\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat Cast) u x))).(\lambda -(H12: (subst0 (s (Flat Cast) i) u0 t x)).(eq_ind_r T (THead (Flat Cast) u x) -(\lambda (t0: T).(arity g c2 t0 a0)) (arity_cast g c2 u a0 (H1 d1 u0 i H4 c2 -u (fsubst0_fst i u0 c u c2 H9)) x (H3 d1 u0 i H4 c2 x (fsubst0_both i u0 c t -x H12 c2 H9))) t2 H11)))) H10)) (\lambda (H10: (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: -T).(subst0 (s (Flat Cast) i) u0 t t3))))).(ex3_2_ind T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: -T).(subst0 (s (Flat Cast) i) u0 t t3))) (arity g c2 t2 a0) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H11: (eq T t2 (THead (Flat Cast) x0 -x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda (H13: (subst0 (s (Flat -Cast) i) u0 t x1)).(eq_ind_r T (THead (Flat Cast) x0 x1) (\lambda (t0: -T).(arity g c2 t0 a0)) (arity_cast g c2 x0 a0 (H1 d1 u0 i H4 c2 x0 -(fsubst0_both i u0 c u x0 H12 c2 H9)) x1 (H3 d1 u0 i H4 c2 x1 (fsubst0_both i -u0 c t x1 H13 c2 H9))) t2 H11)))))) H10)) (subst0_gen_head (Flat Cast) u0 u t -t2 i H8)))) H7)) H6)))))))))))))))))) (\lambda (c: C).(\lambda (t: -T).(\lambda (a1: A).(\lambda (_: (arity g c t a1)).(\lambda (H1: ((\forall -(d1: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d1 (Bind Abbr) -u)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u c t c2 t2) \to (arity -g c2 t2 a1)))))))))).(\lambda (a2: A).(\lambda (H2: (leq g a1 a2)).(\lambda -(d1: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H3: (getl i c (CHead d1 -(Bind Abbr) u))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H4: (fsubst0 i u -c t c2 t2)).(let H_x \def (fsubst0_gen_base c c2 t t2 u i H4) in (let H5 \def -H_x in (or3_ind (land (eq C c c2) (subst0 i u t t2)) (land (eq T t t2) -(csubst0 i u c c2)) (land (subst0 i u t t2) (csubst0 i u c c2)) (arity g c2 -t2 a2) (\lambda (H6: (land (eq C c c2) (subst0 i u t t2))).(land_ind (eq C c -c2) (subst0 i u t t2) (arity g c2 t2 a2) (\lambda (H7: (eq C c c2)).(\lambda -(H8: (subst0 i u t t2)).(eq_ind C c (\lambda (c0: C).(arity g c0 t2 a2)) -(arity_repl g c t2 a1 (H1 d1 u i H3 c t2 (fsubst0_snd i u c t t2 H8)) a2 H2) -c2 H7))) H6)) (\lambda (H6: (land (eq T t t2) (csubst0 i u c c2))).(land_ind -(eq T t t2) (csubst0 i u c c2) (arity g c2 t2 a2) (\lambda (H7: (eq T t -t2)).(\lambda (H8: (csubst0 i u c c2)).(eq_ind T t (\lambda (t0: T).(arity g -c2 t0 a2)) (arity_repl g c2 t a1 (H1 d1 u i H3 c2 t (fsubst0_fst i u c t c2 -H8)) a2 H2) t2 H7))) H6)) (\lambda (H6: (land (subst0 i u t t2) (csubst0 i u -c c2))).(land_ind (subst0 i u t t2) (csubst0 i u c c2) (arity g c2 t2 a2) -(\lambda (H7: (subst0 i u t t2)).(\lambda (H8: (csubst0 i u c -c2)).(arity_repl g c2 t2 a1 (H1 d1 u i H3 c2 t2 (fsubst0_both i u c t t2 H7 -c2 H8)) a2 H2))) H6)) H5))))))))))))))))) c1 t1 a H))))). -(* COMMENTS -Initial nodes: 20387 -END *) - -theorem arity_subst0: - \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (a: A).((arity g c -t1 a) \to (\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead -d (Bind Abbr) u)) \to (\forall (t2: T).((subst0 i u t1 t2) \to (arity g c t2 -a))))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (a: A).(\lambda (H: -(arity g c t1 a)).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (t2: T).(\lambda (H1: -(subst0 i u t1 t2)).(arity_fsubst0 g c t1 a H d u i H0 c t2 (fsubst0_snd i u -c t1 t2 H1)))))))))))). -(* COMMENTS -Initial nodes: 89 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/asucc/defs.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/asucc/defs.ma deleted file mode 100644 index 2f1af0279..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/asucc/defs.ma +++ /dev/null @@ -1,28 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/A/defs.ma". - -include "Basic-1/G/defs.ma". - -definition asucc: - G \to (A \to A) -\def - let rec asucc (g: G) (l: A) on l: A \def (match l with [(ASort n0 n) -\Rightarrow (match n0 with [O \Rightarrow (ASort O (next g n)) | (S h) -\Rightarrow (ASort h n)]) | (AHead a1 a2) \Rightarrow (AHead a1 (asucc g -a2))]) in asucc. - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/asucc/fwd.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/asucc/fwd.ma deleted file mode 100644 index 61fcb5799..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/asucc/fwd.ma +++ /dev/null @@ -1,99 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/asucc/defs.ma". - -theorem asucc_gen_sort: - \forall (g: G).(\forall (h: nat).(\forall (n: nat).(\forall (a: A).((eq A -(ASort h n) (asucc g a)) \to (ex_2 nat nat (\lambda (h0: nat).(\lambda (n0: -nat).(eq A a (ASort h0 n0))))))))) -\def - \lambda (g: G).(\lambda (h: nat).(\lambda (n: nat).(\lambda (a: A).(A_ind -(\lambda (a0: A).((eq A (ASort h n) (asucc g a0)) \to (ex_2 nat nat (\lambda -(h0: nat).(\lambda (n0: nat).(eq A a0 (ASort h0 n0))))))) (\lambda (n0: -nat).(\lambda (n1: nat).(\lambda (H: (eq A (ASort h n) (asucc g (ASort n0 -n1)))).(let H0 \def (f_equal A A (\lambda (e: A).e) (ASort h n) (match n0 -with [O \Rightarrow (ASort O (next g n1)) | (S h0) \Rightarrow (ASort h0 -n1)]) H) in (ex_2_intro nat nat (\lambda (h0: nat).(\lambda (n2: nat).(eq A -(ASort n0 n1) (ASort h0 n2)))) n0 n1 (refl_equal A (ASort n0 n1))))))) -(\lambda (a0: A).(\lambda (_: (((eq A (ASort h n) (asucc g a0)) \to (ex_2 nat -nat (\lambda (h0: nat).(\lambda (n0: nat).(eq A a0 (ASort h0 -n0)))))))).(\lambda (a1: A).(\lambda (_: (((eq A (ASort h n) (asucc g a1)) -\to (ex_2 nat nat (\lambda (h0: nat).(\lambda (n0: nat).(eq A a1 (ASort h0 -n0)))))))).(\lambda (H1: (eq A (ASort h n) (asucc g (AHead a0 a1)))).(let H2 -\def (eq_ind A (ASort h n) (\lambda (ee: A).(match ee in A return (\lambda -(_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow -False])) I (asucc g (AHead a0 a1)) H1) in (False_ind (ex_2 nat nat (\lambda -(h0: nat).(\lambda (n0: nat).(eq A (AHead a0 a1) (ASort h0 n0))))) H2))))))) -a)))). -(* COMMENTS -Initial nodes: 317 -END *) - -theorem asucc_gen_head: - \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a: A).((eq A -(AHead a1 a2) (asucc g a)) \to (ex2 A (\lambda (a0: A).(eq A a (AHead a1 -a0))) (\lambda (a0: A).(eq A a2 (asucc g a0)))))))) -\def - \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a: A).(A_ind -(\lambda (a0: A).((eq A (AHead a1 a2) (asucc g a0)) \to (ex2 A (\lambda (a3: -A).(eq A a0 (AHead a1 a3))) (\lambda (a3: A).(eq A a2 (asucc g a3)))))) -(\lambda (n: nat).(\lambda (n0: nat).(\lambda (H: (eq A (AHead a1 a2) (asucc -g (ASort n n0)))).(nat_ind (\lambda (n1: nat).((eq A (AHead a1 a2) (asucc g -(ASort n1 n0))) \to (ex2 A (\lambda (a0: A).(eq A (ASort n1 n0) (AHead a1 -a0))) (\lambda (a0: A).(eq A a2 (asucc g a0)))))) (\lambda (H0: (eq A (AHead -a1 a2) (asucc g (ASort O n0)))).(let H1 \def (eq_ind A (AHead a1 a2) (\lambda -(ee: A).(match ee in A return (\lambda (_: A).Prop) with [(ASort _ _) -\Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort O (next g n0)) -H0) in (False_ind (ex2 A (\lambda (a0: A).(eq A (ASort O n0) (AHead a1 a0))) -(\lambda (a0: A).(eq A a2 (asucc g a0)))) H1))) (\lambda (n1: nat).(\lambda -(_: (((eq A (AHead a1 a2) (asucc g (ASort n1 n0))) \to (ex2 A (\lambda (a0: -A).(eq A (ASort n1 n0) (AHead a1 a0))) (\lambda (a0: A).(eq A a2 (asucc g -a0))))))).(\lambda (H0: (eq A (AHead a1 a2) (asucc g (ASort (S n1) -n0)))).(let H1 \def (eq_ind A (AHead a1 a2) (\lambda (ee: A).(match ee in A -return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ -_) \Rightarrow True])) I (ASort n1 n0) H0) in (False_ind (ex2 A (\lambda (a0: -A).(eq A (ASort (S n1) n0) (AHead a1 a0))) (\lambda (a0: A).(eq A a2 (asucc g -a0)))) H1))))) n H)))) (\lambda (a0: A).(\lambda (H: (((eq A (AHead a1 a2) -(asucc g a0)) \to (ex2 A (\lambda (a3: A).(eq A a0 (AHead a1 a3))) (\lambda -(a3: A).(eq A a2 (asucc g a3))))))).(\lambda (a3: A).(\lambda (H0: (((eq A -(AHead a1 a2) (asucc g a3)) \to (ex2 A (\lambda (a4: A).(eq A a3 (AHead a1 -a4))) (\lambda (a4: A).(eq A a2 (asucc g a4))))))).(\lambda (H1: (eq A (AHead -a1 a2) (asucc g (AHead a0 a3)))).(let H2 \def (f_equal A A (\lambda (e: -A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a1 | -(AHead a4 _) \Rightarrow a4])) (AHead a1 a2) (AHead a0 (asucc g a3)) H1) in -((let H3 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: -A).A) with [(ASort _ _) \Rightarrow a2 | (AHead _ a4) \Rightarrow a4])) -(AHead a1 a2) (AHead a0 (asucc g a3)) H1) in (\lambda (H4: (eq A a1 a0)).(let -H5 \def (eq_ind_r A a0 (\lambda (a4: A).((eq A (AHead a1 a2) (asucc g a4)) -\to (ex2 A (\lambda (a5: A).(eq A a4 (AHead a1 a5))) (\lambda (a5: A).(eq A -a2 (asucc g a5)))))) H a1 H4) in (eq_ind A a1 (\lambda (a4: A).(ex2 A -(\lambda (a5: A).(eq A (AHead a4 a3) (AHead a1 a5))) (\lambda (a5: A).(eq A -a2 (asucc g a5))))) (let H6 \def (eq_ind A a2 (\lambda (a4: A).((eq A (AHead -a1 a4) (asucc g a3)) \to (ex2 A (\lambda (a5: A).(eq A a3 (AHead a1 a5))) -(\lambda (a5: A).(eq A a4 (asucc g a5)))))) H0 (asucc g a3) H3) in (let H7 -\def (eq_ind A a2 (\lambda (a4: A).((eq A (AHead a1 a4) (asucc g a1)) \to -(ex2 A (\lambda (a5: A).(eq A a1 (AHead a1 a5))) (\lambda (a5: A).(eq A a4 -(asucc g a5)))))) H5 (asucc g a3) H3) in (eq_ind_r A (asucc g a3) (\lambda -(a4: A).(ex2 A (\lambda (a5: A).(eq A (AHead a1 a3) (AHead a1 a5))) (\lambda -(a5: A).(eq A a4 (asucc g a5))))) (ex_intro2 A (\lambda (a4: A).(eq A (AHead -a1 a3) (AHead a1 a4))) (\lambda (a4: A).(eq A (asucc g a3) (asucc g a4))) a3 -(refl_equal A (AHead a1 a3)) (refl_equal A (asucc g a3))) a2 H3))) a0 H4)))) -H2))))))) a)))). -(* COMMENTS -Initial nodes: 957 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/cimp/defs.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/cimp/defs.ma deleted file mode 100644 index b45b64b59..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/cimp/defs.ma +++ /dev/null @@ -1,25 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/getl/defs.ma". - -definition cimp: - C \to (C \to Prop) -\def - \lambda (c1: C).(\lambda (c2: C).(\forall (b: B).(\forall (d1: C).(\forall -(w: T).(\forall (h: nat).((getl h c1 (CHead d1 (Bind b) w)) \to (ex C -(\lambda (d2: C).(getl h c2 (CHead d2 (Bind b) w)))))))))). - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/cimp/props.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/cimp/props.ma deleted file mode 100644 index 0834a7afe..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/cimp/props.ma +++ /dev/null @@ -1,139 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/cimp/defs.ma". - -include "Basic-1/getl/getl.ma". - -theorem cimp_flat_sx: - \forall (f: F).(\forall (c: C).(\forall (v: T).(cimp (CHead c (Flat f) v) -c))) -\def - \lambda (f: F).(\lambda (c: C).(\lambda (v: T).(\lambda (b: B).(\lambda (d1: -C).(\lambda (w: T).(\lambda (h: nat).(\lambda (H: (getl h (CHead c (Flat f) -v) (CHead d1 (Bind b) w))).(nat_ind (\lambda (n: nat).((getl n (CHead c (Flat -f) v) (CHead d1 (Bind b) w)) \to (ex C (\lambda (d2: C).(getl n c (CHead d2 -(Bind b) w)))))) (\lambda (H0: (getl O (CHead c (Flat f) v) (CHead d1 (Bind -b) w))).(ex_intro C (\lambda (d2: C).(getl O c (CHead d2 (Bind b) w))) d1 -(getl_intro O c (CHead d1 (Bind b) w) c (drop_refl c) (clear_gen_flat f c -(CHead d1 (Bind b) w) v (getl_gen_O (CHead c (Flat f) v) (CHead d1 (Bind b) -w) H0))))) (\lambda (h0: nat).(\lambda (_: (((getl h0 (CHead c (Flat f) v) -(CHead d1 (Bind b) w)) \to (ex C (\lambda (d2: C).(getl h0 c (CHead d2 (Bind -b) w))))))).(\lambda (H0: (getl (S h0) (CHead c (Flat f) v) (CHead d1 (Bind -b) w))).(ex_intro C (\lambda (d2: C).(getl (S h0) c (CHead d2 (Bind b) w))) -d1 (getl_gen_S (Flat f) c (CHead d1 (Bind b) w) v h0 H0))))) h H)))))))). -(* COMMENTS -Initial nodes: 327 -END *) - -theorem cimp_flat_dx: - \forall (f: F).(\forall (c: C).(\forall (v: T).(cimp c (CHead c (Flat f) -v)))) -\def - \lambda (f: F).(\lambda (c: C).(\lambda (v: T).(\lambda (b: B).(\lambda (d1: -C).(\lambda (w: T).(\lambda (h: nat).(\lambda (H: (getl h c (CHead d1 (Bind -b) w))).(ex_intro C (\lambda (d2: C).(getl h (CHead c (Flat f) v) (CHead d2 -(Bind b) w))) d1 (getl_flat c (CHead d1 (Bind b) w) h H f v))))))))). -(* COMMENTS -Initial nodes: 83 -END *) - -theorem cimp_bind: - \forall (c1: C).(\forall (c2: C).((cimp c1 c2) \to (\forall (b: B).(\forall -(v: T).(cimp (CHead c1 (Bind b) v) (CHead c2 (Bind b) v)))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (H: ((\forall (b: B).(\forall (d1: -C).(\forall (w: T).(\forall (h: nat).((getl h c1 (CHead d1 (Bind b) w)) \to -(ex C (\lambda (d2: C).(getl h c2 (CHead d2 (Bind b) w))))))))))).(\lambda -(b: B).(\lambda (v: T).(\lambda (b0: B).(\lambda (d1: C).(\lambda (w: -T).(\lambda (h: nat).(\lambda (H0: (getl h (CHead c1 (Bind b) v) (CHead d1 -(Bind b0) w))).(nat_ind (\lambda (n: nat).((getl n (CHead c1 (Bind b) v) -(CHead d1 (Bind b0) w)) \to (ex C (\lambda (d2: C).(getl n (CHead c2 (Bind b) -v) (CHead d2 (Bind b0) w)))))) (\lambda (H1: (getl O (CHead c1 (Bind b) v) -(CHead d1 (Bind b0) w))).(let H2 \def (f_equal C C (\lambda (e: C).(match e -in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d1 | (CHead c _ _) -\Rightarrow c])) (CHead d1 (Bind b0) w) (CHead c1 (Bind b) v) (clear_gen_bind -b c1 (CHead d1 (Bind b0) w) v (getl_gen_O (CHead c1 (Bind b) v) (CHead d1 -(Bind b0) w) H1))) in ((let H3 \def (f_equal C B (\lambda (e: C).(match e in -C return (\lambda (_: C).B) with [(CSort _) \Rightarrow b0 | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b1) -\Rightarrow b1 | (Flat _) \Rightarrow b0])])) (CHead d1 (Bind b0) w) (CHead -c1 (Bind b) v) (clear_gen_bind b c1 (CHead d1 (Bind b0) w) v (getl_gen_O -(CHead c1 (Bind b) v) (CHead d1 (Bind b0) w) H1))) in ((let H4 \def (f_equal -C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) -\Rightarrow w | (CHead _ _ t) \Rightarrow t])) (CHead d1 (Bind b0) w) (CHead -c1 (Bind b) v) (clear_gen_bind b c1 (CHead d1 (Bind b0) w) v (getl_gen_O -(CHead c1 (Bind b) v) (CHead d1 (Bind b0) w) H1))) in (\lambda (H5: (eq B b0 -b)).(\lambda (_: (eq C d1 c1)).(eq_ind_r T v (\lambda (t: T).(ex C (\lambda -(d2: C).(getl O (CHead c2 (Bind b) v) (CHead d2 (Bind b0) t))))) (eq_ind_r B -b (\lambda (b1: B).(ex C (\lambda (d2: C).(getl O (CHead c2 (Bind b) v) -(CHead d2 (Bind b1) v))))) (ex_intro C (\lambda (d2: C).(getl O (CHead c2 -(Bind b) v) (CHead d2 (Bind b) v))) c2 (getl_refl b c2 v)) b0 H5) w H4)))) -H3)) H2))) (\lambda (h0: nat).(\lambda (_: (((getl h0 (CHead c1 (Bind b) v) -(CHead d1 (Bind b0) w)) \to (ex C (\lambda (d2: C).(getl h0 (CHead c2 (Bind -b) v) (CHead d2 (Bind b0) w))))))).(\lambda (H1: (getl (S h0) (CHead c1 (Bind -b) v) (CHead d1 (Bind b0) w))).(let H_x \def (H b0 d1 w (r (Bind b) h0) -(getl_gen_S (Bind b) c1 (CHead d1 (Bind b0) w) v h0 H1)) in (let H2 \def H_x -in (ex_ind C (\lambda (d2: C).(getl h0 c2 (CHead d2 (Bind b0) w))) (ex C -(\lambda (d2: C).(getl (S h0) (CHead c2 (Bind b) v) (CHead d2 (Bind b0) w)))) -(\lambda (x: C).(\lambda (H3: (getl h0 c2 (CHead x (Bind b0) w))).(ex_intro C -(\lambda (d2: C).(getl (S h0) (CHead c2 (Bind b) v) (CHead d2 (Bind b0) w))) -x (getl_head (Bind b) h0 c2 (CHead x (Bind b0) w) H3 v)))) H2)))))) h -H0)))))))))). -(* COMMENTS -Initial nodes: 817 -END *) - -theorem cimp_getl_conf: - \forall (c1: C).(\forall (c2: C).((cimp c1 c2) \to (\forall (b: B).(\forall -(d1: C).(\forall (w: T).(\forall (i: nat).((getl i c1 (CHead d1 (Bind b) w)) -\to (ex2 C (\lambda (d2: C).(cimp d1 d2)) (\lambda (d2: C).(getl i c2 (CHead -d2 (Bind b) w))))))))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (H: ((\forall (b: B).(\forall (d1: -C).(\forall (w: T).(\forall (h: nat).((getl h c1 (CHead d1 (Bind b) w)) \to -(ex C (\lambda (d2: C).(getl h c2 (CHead d2 (Bind b) w))))))))))).(\lambda -(b: B).(\lambda (d1: C).(\lambda (w: T).(\lambda (i: nat).(\lambda (H0: (getl -i c1 (CHead d1 (Bind b) w))).(let H_x \def (H b d1 w i H0) in (let H1 \def -H_x in (ex_ind C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind b) w))) (ex2 C -(\lambda (d2: C).(\forall (b0: B).(\forall (d3: C).(\forall (w0: T).(\forall -(h: nat).((getl h d1 (CHead d3 (Bind b0) w0)) \to (ex C (\lambda (d4: -C).(getl h d2 (CHead d4 (Bind b0) w0)))))))))) (\lambda (d2: C).(getl i c2 -(CHead d2 (Bind b) w)))) (\lambda (x: C).(\lambda (H2: (getl i c2 (CHead x -(Bind b) w))).(ex_intro2 C (\lambda (d2: C).(\forall (b0: B).(\forall (d3: -C).(\forall (w0: T).(\forall (h: nat).((getl h d1 (CHead d3 (Bind b0) w0)) -\to (ex C (\lambda (d4: C).(getl h d2 (CHead d4 (Bind b0) w0)))))))))) -(\lambda (d2: C).(getl i c2 (CHead d2 (Bind b) w))) x (\lambda (b0: -B).(\lambda (d0: C).(\lambda (w0: T).(\lambda (h: nat).(\lambda (H3: (getl h -d1 (CHead d0 (Bind b0) w0))).(let H_y \def (getl_trans (S h) c1 (CHead d1 -(Bind b) w) i H0) in (let H_x0 \def (H b0 d0 w0 (plus (S h) i) (H_y (CHead d0 -(Bind b0) w0) (getl_head (Bind b) h d1 (CHead d0 (Bind b0) w0) H3 w))) in -(let H4 \def H_x0 in (ex_ind C (\lambda (d2: C).(getl (S (plus h i)) c2 -(CHead d2 (Bind b0) w0))) (ex C (\lambda (d2: C).(getl h x (CHead d2 (Bind -b0) w0)))) (\lambda (x0: C).(\lambda (H5: (getl (S (plus h i)) c2 (CHead x0 -(Bind b0) w0))).(let H_y0 \def (getl_conf_le (S (plus h i)) (CHead x0 (Bind -b0) w0) c2 H5 (CHead x (Bind b) w) i H2) in (let H6 \def (refl_equal nat -(plus (S h) i)) in (let H7 \def (eq_ind nat (S (plus h i)) (\lambda (n: -nat).(getl (minus n i) (CHead x (Bind b) w) (CHead x0 (Bind b0) w0))) (H_y0 -(le_S i (plus h i) (le_plus_r h i))) (plus (S h) i) H6) in (let H8 \def -(eq_ind nat (minus (plus (S h) i) i) (\lambda (n: nat).(getl n (CHead x (Bind -b) w) (CHead x0 (Bind b0) w0))) H7 (S h) (minus_plus_r (S h) i)) in (ex_intro -C (\lambda (d2: C).(getl h x (CHead d2 (Bind b0) w0))) x0 (getl_gen_S (Bind -b) x (CHead x0 (Bind b0) w0) w h H8)))))))) H4))))))))) H2))) H1)))))))))). -(* COMMENTS -Initial nodes: 673 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/clear/defs.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/clear/defs.ma deleted file mode 100644 index 30607a4fc..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/clear/defs.ma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/C/defs.ma". - -inductive clear: C \to (C \to Prop) \def -| clear_bind: \forall (b: B).(\forall (e: C).(\forall (u: T).(clear (CHead e -(Bind b) u) (CHead e (Bind b) u)))) -| clear_flat: \forall (e: C).(\forall (c: C).((clear e c) \to (\forall (f: -F).(\forall (u: T).(clear (CHead e (Flat f) u) c))))). - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/clear/drop.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/clear/drop.ma deleted file mode 100644 index aae3fedc0..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/clear/drop.ma +++ /dev/null @@ -1,181 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/clear/fwd.ma". - -include "Basic-1/drop/fwd.ma". - -theorem drop_clear: - \forall (c1: C).(\forall (c2: C).(\forall (i: nat).((drop (S i) O c1 c2) \to -(ex2_3 B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear c1 (CHead -e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e -c2)))))))) -\def - \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (i: -nat).((drop (S i) O c c2) \to (ex2_3 B C T (\lambda (b: B).(\lambda (e: -C).(\lambda (v: T).(clear c (CHead e (Bind b) v))))) (\lambda (_: B).(\lambda -(e: C).(\lambda (_: T).(drop i O e c2))))))))) (\lambda (n: nat).(\lambda -(c2: C).(\lambda (i: nat).(\lambda (H: (drop (S i) O (CSort n) c2)).(and3_ind -(eq C c2 (CSort n)) (eq nat (S i) O) (eq nat O O) (ex2_3 B C T (\lambda (b: -B).(\lambda (e: C).(\lambda (v: T).(clear (CSort n) (CHead e (Bind b) v))))) -(\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2))))) (\lambda -(_: (eq C c2 (CSort n))).(\lambda (H1: (eq nat (S i) O)).(\lambda (_: (eq nat -O O)).(let H3 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee in nat -return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow -True])) I O H1) in (False_ind (ex2_3 B C T (\lambda (b: B).(\lambda (e: -C).(\lambda (v: T).(clear (CSort n) (CHead e (Bind b) v))))) (\lambda (_: -B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2))))) H3))))) (drop_gen_sort -n (S i) O c2 H)))))) (\lambda (c: C).(\lambda (H: ((\forall (c2: C).(\forall -(i: nat).((drop (S i) O c c2) \to (ex2_3 B C T (\lambda (b: B).(\lambda (e: -C).(\lambda (v: T).(clear c (CHead e (Bind b) v))))) (\lambda (_: B).(\lambda -(e: C).(\lambda (_: T).(drop i O e c2)))))))))).(\lambda (k: K).(\lambda (t: -T).(\lambda (c2: C).(\lambda (i: nat).(\lambda (H0: (drop (S i) O (CHead c k -t) c2)).(K_ind (\lambda (k0: K).((drop (r k0 i) O c c2) \to (ex2_3 B C T -(\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear (CHead c k0 t) (CHead -e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e -c2))))))) (\lambda (b: B).(\lambda (H1: (drop (r (Bind b) i) O c -c2)).(ex2_3_intro B C T (\lambda (b0: B).(\lambda (e: C).(\lambda (v: -T).(clear (CHead c (Bind b) t) (CHead e (Bind b0) v))))) (\lambda (_: -B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2)))) b c t (clear_bind b c -t) H1))) (\lambda (f: F).(\lambda (H1: (drop (r (Flat f) i) O c c2)).(let H2 -\def (H c2 i H1) in (ex2_3_ind B C T (\lambda (b: B).(\lambda (e: C).(\lambda -(v: T).(clear c (CHead e (Bind b) v))))) (\lambda (_: B).(\lambda (e: -C).(\lambda (_: T).(drop i O e c2)))) (ex2_3 B C T (\lambda (b: B).(\lambda -(e: C).(\lambda (v: T).(clear (CHead c (Flat f) t) (CHead e (Bind b) v))))) -(\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2))))) (\lambda -(x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H3: (clear c (CHead x1 -(Bind x0) x2))).(\lambda (H4: (drop i O x1 c2)).(ex2_3_intro B C T (\lambda -(b: B).(\lambda (e: C).(\lambda (v: T).(clear (CHead c (Flat f) t) (CHead e -(Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e -c2)))) x0 x1 x2 (clear_flat c (CHead x1 (Bind x0) x2) H3 f t) H4)))))) H2)))) -k (drop_gen_drop k c c2 t i H0))))))))) c1). -(* COMMENTS -Initial nodes: 770 -END *) - -theorem drop_clear_O: - \forall (b: B).(\forall (c: C).(\forall (e1: C).(\forall (u: T).((clear c -(CHead e1 (Bind b) u)) \to (\forall (e2: C).(\forall (i: nat).((drop i O e1 -e2) \to (drop (S i) O c e2)))))))) -\def - \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e1: -C).(\forall (u: T).((clear c0 (CHead e1 (Bind b) u)) \to (\forall (e2: -C).(\forall (i: nat).((drop i O e1 e2) \to (drop (S i) O c0 e2)))))))) -(\lambda (n: nat).(\lambda (e1: C).(\lambda (u: T).(\lambda (H: (clear (CSort -n) (CHead e1 (Bind b) u))).(\lambda (e2: C).(\lambda (i: nat).(\lambda (_: -(drop i O e1 e2)).(clear_gen_sort (CHead e1 (Bind b) u) n H (drop (S i) O -(CSort n) e2))))))))) (\lambda (c0: C).(\lambda (H: ((\forall (e1: -C).(\forall (u: T).((clear c0 (CHead e1 (Bind b) u)) \to (\forall (e2: -C).(\forall (i: nat).((drop i O e1 e2) \to (drop (S i) O c0 -e2))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e1: C).(\lambda (u: -T).(\lambda (H0: (clear (CHead c0 k t) (CHead e1 (Bind b) u))).(\lambda (e2: -C).(\lambda (i: nat).(\lambda (H1: (drop i O e1 e2)).(K_ind (\lambda (k0: -K).((clear (CHead c0 k0 t) (CHead e1 (Bind b) u)) \to (drop (S i) O (CHead c0 -k0 t) e2))) (\lambda (b0: B).(\lambda (H2: (clear (CHead c0 (Bind b0) t) -(CHead e1 (Bind b) u))).(let H3 \def (f_equal C C (\lambda (e: C).(match e in -C return (\lambda (_: C).C) with [(CSort _) \Rightarrow e1 | (CHead c1 _ _) -\Rightarrow c1])) (CHead e1 (Bind b) u) (CHead c0 (Bind b0) t) -(clear_gen_bind b0 c0 (CHead e1 (Bind b) u) t H2)) in ((let H4 \def (f_equal -C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with [(CSort _) -\Rightarrow b | (CHead _ k0 _) \Rightarrow (match k0 in K return (\lambda (_: -K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow b])])) (CHead e1 -(Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e1 (Bind b) -u) t H2)) in ((let H5 \def (f_equal C T (\lambda (e: C).(match e in C return -(\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow -t0])) (CHead e1 (Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 -(CHead e1 (Bind b) u) t H2)) in (\lambda (H6: (eq B b b0)).(\lambda (H7: (eq -C e1 c0)).(let H8 \def (eq_ind C e1 (\lambda (c1: C).(drop i O c1 e2)) H1 c0 -H7) in (eq_ind B b (\lambda (b1: B).(drop (S i) O (CHead c0 (Bind b1) t) e2)) -(drop_drop (Bind b) i c0 e2 H8 t) b0 H6))))) H4)) H3)))) (\lambda (f: -F).(\lambda (H2: (clear (CHead c0 (Flat f) t) (CHead e1 (Bind b) -u))).(drop_drop (Flat f) i c0 e2 (H e1 u (clear_gen_flat f c0 (CHead e1 (Bind -b) u) t H2) e2 i H1) t))) k H0))))))))))) c)). -(* COMMENTS -Initial nodes: 619 -END *) - -theorem drop_clear_S: - \forall (x2: C).(\forall (x1: C).(\forall (h: nat).(\forall (d: nat).((drop -h (S d) x1 x2) \to (\forall (b: B).(\forall (c2: C).(\forall (u: T).((clear -x2 (CHead c2 (Bind b) u)) \to (ex2 C (\lambda (c1: C).(clear x1 (CHead c1 -(Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2))))))))))) -\def - \lambda (x2: C).(C_ind (\lambda (c: C).(\forall (x1: C).(\forall (h: -nat).(\forall (d: nat).((drop h (S d) x1 c) \to (\forall (b: B).(\forall (c2: -C).(\forall (u: T).((clear c (CHead c2 (Bind b) u)) \to (ex2 C (\lambda (c1: -C).(clear x1 (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 -c2)))))))))))) (\lambda (n: nat).(\lambda (x1: C).(\lambda (h: nat).(\lambda -(d: nat).(\lambda (_: (drop h (S d) x1 (CSort n))).(\lambda (b: B).(\lambda -(c2: C).(\lambda (u: T).(\lambda (H0: (clear (CSort n) (CHead c2 (Bind b) -u))).(clear_gen_sort (CHead c2 (Bind b) u) n H0 (ex2 C (\lambda (c1: -C).(clear x1 (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 -c2))))))))))))) (\lambda (c: C).(\lambda (H: ((\forall (x1: C).(\forall (h: -nat).(\forall (d: nat).((drop h (S d) x1 c) \to (\forall (b: B).(\forall (c2: -C).(\forall (u: T).((clear c (CHead c2 (Bind b) u)) \to (ex2 C (\lambda (c1: -C).(clear x1 (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 -c2))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (x1: C).(\lambda (h: -nat).(\lambda (d: nat).(\lambda (H0: (drop h (S d) x1 (CHead c k -t))).(\lambda (b: B).(\lambda (c2: C).(\lambda (u: T).(\lambda (H1: (clear -(CHead c k t) (CHead c2 (Bind b) u))).(ex2_ind C (\lambda (e: C).(eq C x1 -(CHead e k (lift h (r k d) t)))) (\lambda (e: C).(drop h (r k d) e c)) (ex2 C -(\lambda (c1: C).(clear x1 (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: -C).(drop h d c1 c2))) (\lambda (x: C).(\lambda (H2: (eq C x1 (CHead x k (lift -h (r k d) t)))).(\lambda (H3: (drop h (r k d) x c)).(eq_ind_r C (CHead x k -(lift h (r k d) t)) (\lambda (c0: C).(ex2 C (\lambda (c1: C).(clear c0 (CHead -c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2)))) (K_ind -(\lambda (k0: K).((clear (CHead c k0 t) (CHead c2 (Bind b) u)) \to ((drop h -(r k0 d) x c) \to (ex2 C (\lambda (c1: C).(clear (CHead x k0 (lift h (r k0 d) -t)) (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2)))))) -(\lambda (b0: B).(\lambda (H4: (clear (CHead c (Bind b0) t) (CHead c2 (Bind -b) u))).(\lambda (H5: (drop h (r (Bind b0) d) x c)).(let H6 \def (f_equal C C -(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) (CHead c2 (Bind b) u) -(CHead c (Bind b0) t) (clear_gen_bind b0 c (CHead c2 (Bind b) u) t H4)) in -((let H7 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: -C).B) with [(CSort _) \Rightarrow b | (CHead _ k0 _) \Rightarrow (match k0 in -K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) -\Rightarrow b])])) (CHead c2 (Bind b) u) (CHead c (Bind b0) t) -(clear_gen_bind b0 c (CHead c2 (Bind b) u) t H4)) in ((let H8 \def (f_equal C -T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) -\Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead c2 (Bind b) u) (CHead -c (Bind b0) t) (clear_gen_bind b0 c (CHead c2 (Bind b) u) t H4)) in (\lambda -(H9: (eq B b b0)).(\lambda (H10: (eq C c2 c)).(eq_ind_r T t (\lambda (t0: -T).(ex2 C (\lambda (c1: C).(clear (CHead x (Bind b0) (lift h (r (Bind b0) d) -t)) (CHead c1 (Bind b) (lift h d t0)))) (\lambda (c1: C).(drop h d c1 c2)))) -(eq_ind_r C c (\lambda (c0: C).(ex2 C (\lambda (c1: C).(clear (CHead x (Bind -b0) (lift h (r (Bind b0) d) t)) (CHead c1 (Bind b) (lift h d t)))) (\lambda -(c1: C).(drop h d c1 c0)))) (eq_ind_r B b0 (\lambda (b1: B).(ex2 C (\lambda -(c1: C).(clear (CHead x (Bind b0) (lift h (r (Bind b0) d) t)) (CHead c1 (Bind -b1) (lift h d t)))) (\lambda (c1: C).(drop h d c1 c)))) (ex_intro2 C (\lambda -(c1: C).(clear (CHead x (Bind b0) (lift h (r (Bind b0) d) t)) (CHead c1 (Bind -b0) (lift h d t)))) (\lambda (c1: C).(drop h d c1 c)) x (clear_bind b0 x -(lift h d t)) H5) b H9) c2 H10) u H8)))) H7)) H6))))) (\lambda (f: -F).(\lambda (H4: (clear (CHead c (Flat f) t) (CHead c2 (Bind b) u))).(\lambda -(H5: (drop h (r (Flat f) d) x c)).(let H6 \def (H x h d H5 b c2 u -(clear_gen_flat f c (CHead c2 (Bind b) u) t H4)) in (ex2_ind C (\lambda (c1: -C).(clear x (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 -c2)) (ex2 C (\lambda (c1: C).(clear (CHead x (Flat f) (lift h (r (Flat f) d) -t)) (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2))) -(\lambda (x0: C).(\lambda (H7: (clear x (CHead x0 (Bind b) (lift h d -u)))).(\lambda (H8: (drop h d x0 c2)).(ex_intro2 C (\lambda (c1: C).(clear -(CHead x (Flat f) (lift h (r (Flat f) d) t)) (CHead c1 (Bind b) (lift h d -u)))) (\lambda (c1: C).(drop h d c1 c2)) x0 (clear_flat x (CHead x0 (Bind b) -(lift h d u)) H7 f (lift h (r (Flat f) d) t)) H8)))) H6))))) k H1 H3) x1 -H2)))) (drop_gen_skip_r c x1 t h d k H0)))))))))))))) x2). -(* COMMENTS -Initial nodes: 1449 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/clear/fwd.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/clear/fwd.ma deleted file mode 100644 index d64ff77be..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/clear/fwd.ma +++ /dev/null @@ -1,164 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/clear/defs.ma". - -theorem clear_gen_sort: - \forall (x: C).(\forall (n: nat).((clear (CSort n) x) \to (\forall (P: -Prop).P))) -\def - \lambda (x: C).(\lambda (n: nat).(\lambda (H: (clear (CSort n) x)).(\lambda -(P: Prop).(insert_eq C (CSort n) (\lambda (c: C).(clear c x)) (\lambda (_: -C).P) (\lambda (y: C).(\lambda (H0: (clear y x)).(clear_ind (\lambda (c: -C).(\lambda (_: C).((eq C c (CSort n)) \to P))) (\lambda (b: B).(\lambda (e: -C).(\lambda (u: T).(\lambda (H1: (eq C (CHead e (Bind b) u) (CSort n))).(let -H2 \def (eq_ind C (CHead e (Bind b) u) (\lambda (ee: C).(match ee in C return -(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) -\Rightarrow True])) I (CSort n) H1) in (False_ind P H2)))))) (\lambda (e: -C).(\lambda (c: C).(\lambda (_: (clear e c)).(\lambda (_: (((eq C e (CSort -n)) \to P))).(\lambda (f: F).(\lambda (u: T).(\lambda (H3: (eq C (CHead e -(Flat f) u) (CSort n))).(let H4 \def (eq_ind C (CHead e (Flat f) u) (\lambda -(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H3) in -(False_ind P H4))))))))) y x H0))) H)))). -(* COMMENTS -Initial nodes: 215 -END *) - -theorem clear_gen_bind: - \forall (b: B).(\forall (e: C).(\forall (x: C).(\forall (u: T).((clear -(CHead e (Bind b) u) x) \to (eq C x (CHead e (Bind b) u)))))) -\def - \lambda (b: B).(\lambda (e: C).(\lambda (x: C).(\lambda (u: T).(\lambda (H: -(clear (CHead e (Bind b) u) x)).(insert_eq C (CHead e (Bind b) u) (\lambda -(c: C).(clear c x)) (\lambda (c: C).(eq C x c)) (\lambda (y: C).(\lambda (H0: -(clear y x)).(clear_ind (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead e -(Bind b) u)) \to (eq C c0 c)))) (\lambda (b0: B).(\lambda (e0: C).(\lambda -(u0: T).(\lambda (H1: (eq C (CHead e0 (Bind b0) u0) (CHead e (Bind b) -u))).(let H2 \def (f_equal C C (\lambda (e1: C).(match e1 in C return -(\lambda (_: C).C) with [(CSort _) \Rightarrow e0 | (CHead c _ _) \Rightarrow -c])) (CHead e0 (Bind b0) u0) (CHead e (Bind b) u) H1) in ((let H3 \def -(f_equal C B (\lambda (e1: C).(match e1 in C return (\lambda (_: C).B) with -[(CSort _) \Rightarrow b0 | (CHead _ k _) \Rightarrow (match k in K return -(\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow -b0])])) (CHead e0 (Bind b0) u0) (CHead e (Bind b) u) H1) in ((let H4 \def -(f_equal C T (\lambda (e1: C).(match e1 in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead e0 (Bind -b0) u0) (CHead e (Bind b) u) H1) in (\lambda (H5: (eq B b0 b)).(\lambda (H6: -(eq C e0 e)).(eq_ind_r T u (\lambda (t: T).(eq C (CHead e0 (Bind b0) t) -(CHead e0 (Bind b0) t))) (eq_ind_r C e (\lambda (c: C).(eq C (CHead c (Bind -b0) u) (CHead c (Bind b0) u))) (eq_ind_r B b (\lambda (b1: B).(eq C (CHead e -(Bind b1) u) (CHead e (Bind b1) u))) (refl_equal C (CHead e (Bind b) u)) b0 -H5) e0 H6) u0 H4)))) H3)) H2)))))) (\lambda (e0: C).(\lambda (c: C).(\lambda -(_: (clear e0 c)).(\lambda (_: (((eq C e0 (CHead e (Bind b) u)) \to (eq C c -e0)))).(\lambda (f: F).(\lambda (u0: T).(\lambda (H3: (eq C (CHead e0 (Flat -f) u0) (CHead e (Bind b) u))).(let H4 \def (eq_ind C (CHead e0 (Flat f) u0) -(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (CHead e (Bind b) u) H3) in (False_ind (eq C c (CHead e0 (Flat f) -u0)) H4))))))))) y x H0))) H))))). -(* COMMENTS -Initial nodes: 525 -END *) - -theorem clear_gen_flat: - \forall (f: F).(\forall (e: C).(\forall (x: C).(\forall (u: T).((clear -(CHead e (Flat f) u) x) \to (clear e x))))) -\def - \lambda (f: F).(\lambda (e: C).(\lambda (x: C).(\lambda (u: T).(\lambda (H: -(clear (CHead e (Flat f) u) x)).(insert_eq C (CHead e (Flat f) u) (\lambda -(c: C).(clear c x)) (\lambda (_: C).(clear e x)) (\lambda (y: C).(\lambda -(H0: (clear y x)).(clear_ind (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead -e (Flat f) u)) \to (clear e c0)))) (\lambda (b: B).(\lambda (e0: C).(\lambda -(u0: T).(\lambda (H1: (eq C (CHead e0 (Bind b) u0) (CHead e (Flat f) -u))).(let H2 \def (eq_ind C (CHead e0 (Bind b) u0) (\lambda (ee: C).(match ee -in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead -_ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) -\Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead e (Flat f) u) H1) -in (False_ind (clear e (CHead e0 (Bind b) u0)) H2)))))) (\lambda (e0: -C).(\lambda (c: C).(\lambda (H1: (clear e0 c)).(\lambda (H2: (((eq C e0 -(CHead e (Flat f) u)) \to (clear e c)))).(\lambda (f0: F).(\lambda (u0: -T).(\lambda (H3: (eq C (CHead e0 (Flat f0) u0) (CHead e (Flat f) u))).(let H4 -\def (f_equal C C (\lambda (e1: C).(match e1 in C return (\lambda (_: C).C) -with [(CSort _) \Rightarrow e0 | (CHead c0 _ _) \Rightarrow c0])) (CHead e0 -(Flat f0) u0) (CHead e (Flat f) u) H3) in ((let H5 \def (f_equal C F (\lambda -(e1: C).(match e1 in C return (\lambda (_: C).F) with [(CSort _) \Rightarrow -f0 | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).F) with -[(Bind _) \Rightarrow f0 | (Flat f1) \Rightarrow f1])])) (CHead e0 (Flat f0) -u0) (CHead e (Flat f) u) H3) in ((let H6 \def (f_equal C T (\lambda (e1: -C).(match e1 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | -(CHead _ _ t) \Rightarrow t])) (CHead e0 (Flat f0) u0) (CHead e (Flat f) u) -H3) in (\lambda (_: (eq F f0 f)).(\lambda (H8: (eq C e0 e)).(let H9 \def -(eq_ind C e0 (\lambda (c0: C).((eq C c0 (CHead e (Flat f) u)) \to (clear e -c))) H2 e H8) in (let H10 \def (eq_ind C e0 (\lambda (c0: C).(clear c0 c)) H1 -e H8) in H10))))) H5)) H4))))))))) y x H0))) H))))). -(* COMMENTS -Initial nodes: 453 -END *) - -theorem clear_gen_flat_r: - \forall (f: F).(\forall (x: C).(\forall (e: C).(\forall (u: T).((clear x -(CHead e (Flat f) u)) \to (\forall (P: Prop).P))))) -\def - \lambda (f: F).(\lambda (x: C).(\lambda (e: C).(\lambda (u: T).(\lambda (H: -(clear x (CHead e (Flat f) u))).(\lambda (P: Prop).(insert_eq C (CHead e -(Flat f) u) (\lambda (c: C).(clear x c)) (\lambda (_: C).P) (\lambda (y: -C).(\lambda (H0: (clear x y)).(clear_ind (\lambda (_: C).(\lambda (c0: -C).((eq C c0 (CHead e (Flat f) u)) \to P))) (\lambda (b: B).(\lambda (e0: -C).(\lambda (u0: T).(\lambda (H1: (eq C (CHead e0 (Bind b) u0) (CHead e (Flat -f) u))).(let H2 \def (eq_ind C (CHead e0 (Bind b) u0) (\lambda (ee: C).(match -ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | -(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead e (Flat -f) u) H1) in (False_ind P H2)))))) (\lambda (e0: C).(\lambda (c: C).(\lambda -(H1: (clear e0 c)).(\lambda (H2: (((eq C c (CHead e (Flat f) u)) \to -P))).(\lambda (_: F).(\lambda (_: T).(\lambda (H3: (eq C c (CHead e (Flat f) -u))).(let H4 \def (eq_ind C c (\lambda (c0: C).((eq C c0 (CHead e (Flat f) -u)) \to P)) H2 (CHead e (Flat f) u) H3) in (let H5 \def (eq_ind C c (\lambda -(c0: C).(clear e0 c0)) H1 (CHead e (Flat f) u) H3) in (H4 (refl_equal C -(CHead e (Flat f) u)))))))))))) x y H0))) H)))))). -(* COMMENTS -Initial nodes: 303 -END *) - -theorem clear_gen_all: - \forall (c1: C).(\forall (c2: C).((clear c1 c2) \to (ex_3 B C T (\lambda (b: -B).(\lambda (e: C).(\lambda (u: T).(eq C c2 (CHead e (Bind b) u)))))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (clear c1 c2)).(clear_ind -(\lambda (_: C).(\lambda (c0: C).(ex_3 B C T (\lambda (b: B).(\lambda (e: -C).(\lambda (u: T).(eq C c0 (CHead e (Bind b) u)))))))) (\lambda (b: -B).(\lambda (e: C).(\lambda (u: T).(ex_3_intro B C T (\lambda (b0: -B).(\lambda (e0: C).(\lambda (u0: T).(eq C (CHead e (Bind b) u) (CHead e0 -(Bind b0) u0))))) b e u (refl_equal C (CHead e (Bind b) u)))))) (\lambda (e: -C).(\lambda (c: C).(\lambda (H0: (clear e c)).(\lambda (H1: (ex_3 B C T -(\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(eq C c (CHead e0 (Bind b) -u))))))).(\lambda (_: F).(\lambda (_: T).(let H2 \def H1 in (ex_3_ind B C T -(\lambda (b: B).(\lambda (e0: C).(\lambda (u0: T).(eq C c (CHead e0 (Bind b) -u0))))) (ex_3 B C T (\lambda (b: B).(\lambda (e0: C).(\lambda (u0: T).(eq C c -(CHead e0 (Bind b) u0)))))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: -T).(\lambda (H3: (eq C c (CHead x1 (Bind x0) x2))).(let H4 \def (eq_ind C c -(\lambda (c0: C).(clear e c0)) H0 (CHead x1 (Bind x0) x2) H3) in (eq_ind_r C -(CHead x1 (Bind x0) x2) (\lambda (c0: C).(ex_3 B C T (\lambda (b: B).(\lambda -(e0: C).(\lambda (u0: T).(eq C c0 (CHead e0 (Bind b) u0))))))) (ex_3_intro B -C T (\lambda (b: B).(\lambda (e0: C).(\lambda (u0: T).(eq C (CHead x1 (Bind -x0) x2) (CHead e0 (Bind b) u0))))) x0 x1 x2 (refl_equal C (CHead x1 (Bind x0) -x2))) c H3)))))) H2)))))))) c1 c2 H))). -(* COMMENTS -Initial nodes: 381 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/clear/props.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/clear/props.ma deleted file mode 100644 index 68e250d76..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/clear/props.ma +++ /dev/null @@ -1,152 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/clear/fwd.ma". - -theorem clear_clear: - \forall (c1: C).(\forall (c2: C).((clear c1 c2) \to (clear c2 c2))) -\def - \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).((clear c c2) \to -(clear c2 c2)))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (H: (clear -(CSort n) c2)).(clear_gen_sort c2 n H (clear c2 c2))))) (\lambda (c: -C).(\lambda (H: ((\forall (c2: C).((clear c c2) \to (clear c2 -c2))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (H0: (clear -(CHead c k t) c2)).(K_ind (\lambda (k0: K).((clear (CHead c k0 t) c2) \to -(clear c2 c2))) (\lambda (b: B).(\lambda (H1: (clear (CHead c (Bind b) t) -c2)).(eq_ind_r C (CHead c (Bind b) t) (\lambda (c0: C).(clear c0 c0)) -(clear_bind b c t) c2 (clear_gen_bind b c c2 t H1)))) (\lambda (f: -F).(\lambda (H1: (clear (CHead c (Flat f) t) c2)).(H c2 (clear_gen_flat f c -c2 t H1)))) k H0))))))) c1). -(* COMMENTS -Initial nodes: 199 -END *) - -theorem clear_mono: - \forall (c: C).(\forall (c1: C).((clear c c1) \to (\forall (c2: C).((clear c -c2) \to (eq C c1 c2))))) -\def - \lambda (c: C).(C_ind (\lambda (c0: C).(\forall (c1: C).((clear c0 c1) \to -(\forall (c2: C).((clear c0 c2) \to (eq C c1 c2)))))) (\lambda (n: -nat).(\lambda (c1: C).(\lambda (_: (clear (CSort n) c1)).(\lambda (c2: -C).(\lambda (H0: (clear (CSort n) c2)).(clear_gen_sort c2 n H0 (eq C c1 -c2))))))) (\lambda (c0: C).(\lambda (H: ((\forall (c1: C).((clear c0 c1) \to -(\forall (c2: C).((clear c0 c2) \to (eq C c1 c2))))))).(\lambda (k: -K).(\lambda (t: T).(\lambda (c1: C).(\lambda (H0: (clear (CHead c0 k t) -c1)).(\lambda (c2: C).(\lambda (H1: (clear (CHead c0 k t) c2)).(K_ind -(\lambda (k0: K).((clear (CHead c0 k0 t) c1) \to ((clear (CHead c0 k0 t) c2) -\to (eq C c1 c2)))) (\lambda (b: B).(\lambda (H2: (clear (CHead c0 (Bind b) -t) c1)).(\lambda (H3: (clear (CHead c0 (Bind b) t) c2)).(eq_ind_r C (CHead c0 -(Bind b) t) (\lambda (c3: C).(eq C c1 c3)) (eq_ind_r C (CHead c0 (Bind b) t) -(\lambda (c3: C).(eq C c3 (CHead c0 (Bind b) t))) (refl_equal C (CHead c0 -(Bind b) t)) c1 (clear_gen_bind b c0 c1 t H2)) c2 (clear_gen_bind b c0 c2 t -H3))))) (\lambda (f: F).(\lambda (H2: (clear (CHead c0 (Flat f) t) -c1)).(\lambda (H3: (clear (CHead c0 (Flat f) t) c2)).(H c1 (clear_gen_flat f -c0 c1 t H2) c2 (clear_gen_flat f c0 c2 t H3))))) k H0 H1))))))))) c). -(* COMMENTS -Initial nodes: 357 -END *) - -theorem clear_trans: - \forall (c1: C).(\forall (c: C).((clear c1 c) \to (\forall (c2: C).((clear c -c2) \to (clear c1 c2))))) -\def - \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c0: C).((clear c c0) \to -(\forall (c2: C).((clear c0 c2) \to (clear c c2)))))) (\lambda (n: -nat).(\lambda (c: C).(\lambda (H: (clear (CSort n) c)).(\lambda (c2: -C).(\lambda (_: (clear c c2)).(clear_gen_sort c n H (clear (CSort n) -c2))))))) (\lambda (c: C).(\lambda (H: ((\forall (c0: C).((clear c c0) \to -(\forall (c2: C).((clear c0 c2) \to (clear c c2))))))).(\lambda (k: -K).(\lambda (t: T).(\lambda (c0: C).(\lambda (H0: (clear (CHead c k t) -c0)).(\lambda (c2: C).(\lambda (H1: (clear c0 c2)).(K_ind (\lambda (k0: -K).((clear (CHead c k0 t) c0) \to (clear (CHead c k0 t) c2))) (\lambda (b: -B).(\lambda (H2: (clear (CHead c (Bind b) t) c0)).(let H3 \def (eq_ind C c0 -(\lambda (c3: C).(clear c3 c2)) H1 (CHead c (Bind b) t) (clear_gen_bind b c -c0 t H2)) in (eq_ind_r C (CHead c (Bind b) t) (\lambda (c3: C).(clear (CHead -c (Bind b) t) c3)) (clear_bind b c t) c2 (clear_gen_bind b c c2 t H3))))) -(\lambda (f: F).(\lambda (H2: (clear (CHead c (Flat f) t) c0)).(clear_flat c -c2 (H c0 (clear_gen_flat f c c0 t H2) c2 H1) f t))) k H0))))))))) c1). -(* COMMENTS -Initial nodes: 299 -END *) - -theorem clear_ctail: - \forall (b: B).(\forall (c1: C).(\forall (c2: C).(\forall (u2: T).((clear c1 -(CHead c2 (Bind b) u2)) \to (\forall (k: K).(\forall (u1: T).(clear (CTail k -u1 c1) (CHead (CTail k u1 c2) (Bind b) u2)))))))) -\def - \lambda (b: B).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: -C).(\forall (u2: T).((clear c (CHead c2 (Bind b) u2)) \to (\forall (k: -K).(\forall (u1: T).(clear (CTail k u1 c) (CHead (CTail k u1 c2) (Bind b) -u2)))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (u2: T).(\lambda (H: -(clear (CSort n) (CHead c2 (Bind b) u2))).(\lambda (k: K).(\lambda (u1: -T).(K_ind (\lambda (k0: K).(clear (CHead (CSort n) k0 u1) (CHead (CTail k0 u1 -c2) (Bind b) u2))) (\lambda (b0: B).(clear_gen_sort (CHead c2 (Bind b) u2) n -H (clear (CHead (CSort n) (Bind b0) u1) (CHead (CTail (Bind b0) u1 c2) (Bind -b) u2)))) (\lambda (f: F).(clear_gen_sort (CHead c2 (Bind b) u2) n H (clear -(CHead (CSort n) (Flat f) u1) (CHead (CTail (Flat f) u1 c2) (Bind b) u2)))) -k))))))) (\lambda (c: C).(\lambda (H: ((\forall (c2: C).(\forall (u2: -T).((clear c (CHead c2 (Bind b) u2)) \to (\forall (k: K).(\forall (u1: -T).(clear (CTail k u1 c) (CHead (CTail k u1 c2) (Bind b) u2))))))))).(\lambda -(k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (u2: T).(\lambda (H0: (clear -(CHead c k t) (CHead c2 (Bind b) u2))).(\lambda (k0: K).(\lambda (u1: -T).(K_ind (\lambda (k1: K).((clear (CHead c k1 t) (CHead c2 (Bind b) u2)) \to -(clear (CHead (CTail k0 u1 c) k1 t) (CHead (CTail k0 u1 c2) (Bind b) u2)))) -(\lambda (b0: B).(\lambda (H1: (clear (CHead c (Bind b0) t) (CHead c2 (Bind -b) u2))).(let H2 \def (f_equal C C (\lambda (e: C).(match e in C return -(\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ _) -\Rightarrow c0])) (CHead c2 (Bind b) u2) (CHead c (Bind b0) t) -(clear_gen_bind b0 c (CHead c2 (Bind b) u2) t H1)) in ((let H3 \def (f_equal -C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with [(CSort _) -\Rightarrow b | (CHead _ k1 _) \Rightarrow (match k1 in K return (\lambda (_: -K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow b])])) (CHead c2 -(Bind b) u2) (CHead c (Bind b0) t) (clear_gen_bind b0 c (CHead c2 (Bind b) -u2) t H1)) in ((let H4 \def (f_equal C T (\lambda (e: C).(match e in C return -(\lambda (_: C).T) with [(CSort _) \Rightarrow u2 | (CHead _ _ t0) -\Rightarrow t0])) (CHead c2 (Bind b) u2) (CHead c (Bind b0) t) -(clear_gen_bind b0 c (CHead c2 (Bind b) u2) t H1)) in (\lambda (H5: (eq B b -b0)).(\lambda (H6: (eq C c2 c)).(eq_ind_r T t (\lambda (t0: T).(clear (CHead -(CTail k0 u1 c) (Bind b0) t) (CHead (CTail k0 u1 c2) (Bind b) t0))) (eq_ind_r -C c (\lambda (c0: C).(clear (CHead (CTail k0 u1 c) (Bind b0) t) (CHead (CTail -k0 u1 c0) (Bind b) t))) (eq_ind B b (\lambda (b1: B).(clear (CHead (CTail k0 -u1 c) (Bind b1) t) (CHead (CTail k0 u1 c) (Bind b) t))) (clear_bind b (CTail -k0 u1 c) t) b0 H5) c2 H6) u2 H4)))) H3)) H2)))) (\lambda (f: F).(\lambda (H1: -(clear (CHead c (Flat f) t) (CHead c2 (Bind b) u2))).(clear_flat (CTail k0 u1 -c) (CHead (CTail k0 u1 c2) (Bind b) u2) (H c2 u2 (clear_gen_flat f c (CHead -c2 (Bind b) u2) t H1) k0 u1) f t))) k H0)))))))))) c1)). -(* COMMENTS -Initial nodes: 819 -END *) - -theorem clear_cle: - \forall (c1: C).(\forall (c2: C).((clear c1 c2) \to (cle c2 c1))) -\def - \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).((clear c c2) \to -(le (cweight c2) (cweight c))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda -(H: (clear (CSort n) c2)).(clear_gen_sort c2 n H (le (cweight c2) O))))) -(\lambda (c: C).(\lambda (H: ((\forall (c2: C).((clear c c2) \to (le (cweight -c2) (cweight c)))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: -C).(\lambda (H0: (clear (CHead c k t) c2)).(K_ind (\lambda (k0: K).((clear -(CHead c k0 t) c2) \to (le (cweight c2) (plus (cweight c) (tweight t))))) -(\lambda (b: B).(\lambda (H1: (clear (CHead c (Bind b) t) c2)).(eq_ind_r C -(CHead c (Bind b) t) (\lambda (c0: C).(le (cweight c0) (plus (cweight c) -(tweight t)))) (le_n (plus (cweight c) (tweight t))) c2 (clear_gen_bind b c -c2 t H1)))) (\lambda (f: F).(\lambda (H1: (clear (CHead c (Flat f) t) -c2)).(le_plus_trans (cweight c2) (cweight c) (tweight t) (H c2 -(clear_gen_flat f c c2 t H1))))) k H0))))))) c1). -(* COMMENTS -Initial nodes: 247 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/clen/defs.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/clen/defs.ma deleted file mode 100644 index 07539ffad..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/clen/defs.ma +++ /dev/null @@ -1,26 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/C/defs.ma". - -include "Basic-1/s/defs.ma". - -definition clen: - C \to nat -\def - let rec clen (c: C) on c: nat \def (match c with [(CSort _) \Rightarrow O | -(CHead c0 k _) \Rightarrow (s k (clen c0))]) in clen. - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/clen/getl.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/clen/getl.ma deleted file mode 100644 index af8a96cf5..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/clen/getl.ma +++ /dev/null @@ -1,361 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/clen/defs.ma". - -include "Basic-1/getl/props.ma". - -theorem getl_ctail_clen: - \forall (b: B).(\forall (t: T).(\forall (c: C).(ex nat (\lambda (n: -nat).(getl (clen c) (CTail (Bind b) t c) (CHead (CSort n) (Bind b) t)))))) -\def - \lambda (b: B).(\lambda (t: T).(\lambda (c: C).(C_ind (\lambda (c0: C).(ex -nat (\lambda (n: nat).(getl (clen c0) (CTail (Bind b) t c0) (CHead (CSort n) -(Bind b) t))))) (\lambda (n: nat).(ex_intro nat (\lambda (n0: nat).(getl O -(CHead (CSort n) (Bind b) t) (CHead (CSort n0) (Bind b) t))) n (getl_refl b -(CSort n) t))) (\lambda (c0: C).(\lambda (H: (ex nat (\lambda (n: nat).(getl -(clen c0) (CTail (Bind b) t c0) (CHead (CSort n) (Bind b) t))))).(\lambda (k: -K).(\lambda (t0: T).(let H0 \def H in (ex_ind nat (\lambda (n: nat).(getl -(clen c0) (CTail (Bind b) t c0) (CHead (CSort n) (Bind b) t))) (ex nat -(\lambda (n: nat).(getl (s k (clen c0)) (CHead (CTail (Bind b) t c0) k t0) -(CHead (CSort n) (Bind b) t)))) (\lambda (x: nat).(\lambda (H1: (getl (clen -c0) (CTail (Bind b) t c0) (CHead (CSort x) (Bind b) t))).(K_ind (\lambda (k0: -K).(ex nat (\lambda (n: nat).(getl (s k0 (clen c0)) (CHead (CTail (Bind b) t -c0) k0 t0) (CHead (CSort n) (Bind b) t))))) (\lambda (b0: B).(ex_intro nat -(\lambda (n: nat).(getl (S (clen c0)) (CHead (CTail (Bind b) t c0) (Bind b0) -t0) (CHead (CSort n) (Bind b) t))) x (getl_head (Bind b0) (clen c0) (CTail -(Bind b) t c0) (CHead (CSort x) (Bind b) t) H1 t0))) (\lambda (f: -F).(ex_intro nat (\lambda (n: nat).(getl (clen c0) (CHead (CTail (Bind b) t -c0) (Flat f) t0) (CHead (CSort n) (Bind b) t))) x (getl_flat (CTail (Bind b) -t c0) (CHead (CSort x) (Bind b) t) (clen c0) H1 f t0))) k))) H0)))))) c))). -(* COMMENTS -Initial nodes: 459 -END *) - -theorem getl_gen_tail: - \forall (k: K).(\forall (b: B).(\forall (u1: T).(\forall (u2: T).(\forall -(c2: C).(\forall (c1: C).(\forall (i: nat).((getl i (CTail k u1 c1) (CHead c2 -(Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) -(\lambda (e: C).(getl i c1 (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: -nat).(eq nat i (clen c1))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: -nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n)))))))))))) -\def - \lambda (k: K).(\lambda (b: B).(\lambda (u1: T).(\lambda (u2: T).(\lambda -(c2: C).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (i: nat).((getl i -(CTail k u1 c) (CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C -c2 (CTail k u1 e))) (\lambda (e: C).(getl i c (CHead e (Bind b) u2)))) (ex4 -nat (\lambda (_: nat).(eq nat i (clen c))) (\lambda (_: nat).(eq K k (Bind -b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort -n)))))))) (\lambda (n: nat).(\lambda (i: nat).(nat_ind (\lambda (n0: -nat).((getl n0 (CTail k u1 (CSort n)) (CHead c2 (Bind b) u2)) \to (or (ex2 C -(\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl n0 (CSort n) -(CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat n0 (clen (CSort -n)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) -(\lambda (n1: nat).(eq C c2 (CSort n1))))))) (\lambda (H: (getl O (CHead -(CSort n) k u1) (CHead c2 (Bind b) u2))).(K_ind (\lambda (k0: K).((clear -(CHead (CSort n) k0 u1) (CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda (e: -C).(eq C c2 (CTail k0 u1 e))) (\lambda (e: C).(getl O (CSort n) (CHead e -(Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O O)) (\lambda (_: -nat).(eq K k0 (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: -nat).(eq C c2 (CSort n0))))))) (\lambda (b0: B).(\lambda (H0: (clear (CHead -(CSort n) (Bind b0) u1) (CHead c2 (Bind b) u2))).(let H1 \def (f_equal C C -(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow c2 | (CHead c _ _) \Rightarrow c])) (CHead c2 (Bind b) u2) (CHead -(CSort n) (Bind b0) u1) (clear_gen_bind b0 (CSort n) (CHead c2 (Bind b) u2) -u1 H0)) in ((let H2 \def (f_equal C B (\lambda (e: C).(match e in C return -(\lambda (_: C).B) with [(CSort _) \Rightarrow b | (CHead _ k0 _) \Rightarrow -(match k0 in K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | -(Flat _) \Rightarrow b])])) (CHead c2 (Bind b) u2) (CHead (CSort n) (Bind b0) -u1) (clear_gen_bind b0 (CSort n) (CHead c2 (Bind b) u2) u1 H0)) in ((let H3 -\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c2 -(Bind b) u2) (CHead (CSort n) (Bind b0) u1) (clear_gen_bind b0 (CSort n) -(CHead c2 (Bind b) u2) u1 H0)) in (\lambda (H4: (eq B b b0)).(\lambda (H5: -(eq C c2 (CSort n))).(eq_ind_r C (CSort n) (\lambda (c: C).(or (ex2 C -(\lambda (e: C).(eq C c (CTail (Bind b0) u1 e))) (\lambda (e: C).(getl O -(CSort n) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O O)) -(\lambda (_: nat).(eq K (Bind b0) (Bind b))) (\lambda (_: nat).(eq T u1 u2)) -(\lambda (n0: nat).(eq C c (CSort n0)))))) (eq_ind_r T u1 (\lambda (t: T).(or -(ex2 C (\lambda (e: C).(eq C (CSort n) (CTail (Bind b0) u1 e))) (\lambda (e: -C).(getl O (CSort n) (CHead e (Bind b) t)))) (ex4 nat (\lambda (_: nat).(eq -nat O O)) (\lambda (_: nat).(eq K (Bind b0) (Bind b))) (\lambda (_: nat).(eq -T u1 t)) (\lambda (n0: nat).(eq C (CSort n) (CSort n0)))))) (eq_ind_r B b0 -(\lambda (b1: B).(or (ex2 C (\lambda (e: C).(eq C (CSort n) (CTail (Bind b0) -u1 e))) (\lambda (e: C).(getl O (CSort n) (CHead e (Bind b1) u1)))) (ex4 nat -(\lambda (_: nat).(eq nat O O)) (\lambda (_: nat).(eq K (Bind b0) (Bind b1))) -(\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: nat).(eq C (CSort n) (CSort -n0)))))) (or_intror (ex2 C (\lambda (e: C).(eq C (CSort n) (CTail (Bind b0) -u1 e))) (\lambda (e: C).(getl O (CSort n) (CHead e (Bind b0) u1)))) (ex4 nat -(\lambda (_: nat).(eq nat O O)) (\lambda (_: nat).(eq K (Bind b0) (Bind b0))) -(\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: nat).(eq C (CSort n) (CSort -n0)))) (ex4_intro nat (\lambda (_: nat).(eq nat O O)) (\lambda (_: nat).(eq K -(Bind b0) (Bind b0))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: nat).(eq -C (CSort n) (CSort n0))) n (refl_equal nat O) (refl_equal K (Bind b0)) -(refl_equal T u1) (refl_equal C (CSort n)))) b H4) u2 H3) c2 H5)))) H2)) -H1)))) (\lambda (f: F).(\lambda (H0: (clear (CHead (CSort n) (Flat f) u1) -(CHead c2 (Bind b) u2))).(clear_gen_sort (CHead c2 (Bind b) u2) n -(clear_gen_flat f (CSort n) (CHead c2 (Bind b) u2) u1 H0) (or (ex2 C (\lambda -(e: C).(eq C c2 (CTail (Flat f) u1 e))) (\lambda (e: C).(getl O (CSort n) -(CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O O)) (\lambda -(_: nat).(eq K (Flat f) (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda -(n0: nat).(eq C c2 (CSort n0)))))))) k (getl_gen_O (CHead (CSort n) k u1) -(CHead c2 (Bind b) u2) H))) (\lambda (n0: nat).(\lambda (_: (((getl n0 (CHead -(CSort n) k u1) (CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C -c2 (CTail k u1 e))) (\lambda (e: C).(getl n0 (CSort n) (CHead e (Bind b) -u2)))) (ex4 nat (\lambda (_: nat).(eq nat n0 O)) (\lambda (_: nat).(eq K k -(Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n1: nat).(eq C c2 (CSort -n1)))))))).(\lambda (H0: (getl (S n0) (CHead (CSort n) k u1) (CHead c2 (Bind -b) u2))).(getl_gen_sort n (r k n0) (CHead c2 (Bind b) u2) (getl_gen_S k -(CSort n) (CHead c2 (Bind b) u2) u1 n0 H0) (or (ex2 C (\lambda (e: C).(eq C -c2 (CTail k u1 e))) (\lambda (e: C).(getl (S n0) (CSort n) (CHead e (Bind b) -u2)))) (ex4 nat (\lambda (_: nat).(eq nat (S n0) O)) (\lambda (_: nat).(eq K -k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n1: nat).(eq C c2 -(CSort n1))))))))) i))) (\lambda (c: C).(\lambda (H: ((\forall (i: -nat).((getl i (CTail k u1 c) (CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda -(e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl i c (CHead e (Bind b) -u2)))) (ex4 nat (\lambda (_: nat).(eq nat i (clen c))) (\lambda (_: nat).(eq -K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 -(CSort n))))))))).(\lambda (k0: K).(\lambda (t: T).(\lambda (i: nat).(nat_ind -(\lambda (n: nat).((getl n (CTail k u1 (CHead c k0 t)) (CHead c2 (Bind b) -u2)) \to (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: -C).(getl n (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: -nat).(eq nat n (clen (CHead c k0 t)))) (\lambda (_: nat).(eq K k (Bind b))) -(\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0))))))) -(\lambda (H0: (getl O (CHead (CTail k u1 c) k0 t) (CHead c2 (Bind b) -u2))).(K_ind (\lambda (k1: K).((clear (CHead (CTail k u1 c) k1 t) (CHead c2 -(Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) -(\lambda (e: C).(getl O (CHead c k1 t) (CHead e (Bind b) u2)))) (ex4 nat -(\lambda (_: nat).(eq nat O (s k1 (clen c)))) (\lambda (_: nat).(eq K k (Bind -b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort -n))))))) (\lambda (b0: B).(\lambda (H1: (clear (CHead (CTail k u1 c) (Bind -b0) t) (CHead c2 (Bind b) u2))).(let H2 \def (f_equal C C (\lambda (e: -C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | -(CHead c0 _ _) \Rightarrow c0])) (CHead c2 (Bind b) u2) (CHead (CTail k u1 c) -(Bind b0) t) (clear_gen_bind b0 (CTail k u1 c) (CHead c2 (Bind b) u2) t H1)) -in ((let H3 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda -(_: C).B) with [(CSort _) \Rightarrow b | (CHead _ k1 _) \Rightarrow (match -k1 in K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) -\Rightarrow b])])) (CHead c2 (Bind b) u2) (CHead (CTail k u1 c) (Bind b0) t) -(clear_gen_bind b0 (CTail k u1 c) (CHead c2 (Bind b) u2) t H1)) in ((let H4 -\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow u2 | (CHead _ _ t0) \Rightarrow t0])) (CHead c2 -(Bind b) u2) (CHead (CTail k u1 c) (Bind b0) t) (clear_gen_bind b0 (CTail k -u1 c) (CHead c2 (Bind b) u2) t H1)) in (\lambda (H5: (eq B b b0)).(\lambda -(H6: (eq C c2 (CTail k u1 c))).(eq_ind T u2 (\lambda (t0: T).(or (ex2 C -(\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c -(Bind b0) t0) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O -(s (Bind b0) (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: -nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n)))))) (eq_ind B b -(\lambda (b1: B).(or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) -(\lambda (e: C).(getl O (CHead c (Bind b1) u2) (CHead e (Bind b) u2)))) (ex4 -nat (\lambda (_: nat).(eq nat O (s (Bind b1) (clen c)))) (\lambda (_: -nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq -C c2 (CSort n)))))) (let H7 \def (eq_ind C c2 (\lambda (c0: C).(\forall (i0: -nat).((getl i0 (CTail k u1 c) (CHead c0 (Bind b) u2)) \to (or (ex2 C (\lambda -(e: C).(eq C c0 (CTail k u1 e))) (\lambda (e: C).(getl i0 c (CHead e (Bind b) -u2)))) (ex4 nat (\lambda (_: nat).(eq nat i0 (clen c))) (\lambda (_: nat).(eq -K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c0 -(CSort n)))))))) H (CTail k u1 c) H6) in (eq_ind_r C (CTail k u1 c) (\lambda -(c0: C).(or (ex2 C (\lambda (e: C).(eq C c0 (CTail k u1 e))) (\lambda (e: -C).(getl O (CHead c (Bind b) u2) (CHead e (Bind b) u2)))) (ex4 nat (\lambda -(_: nat).(eq nat O (s (Bind b) (clen c)))) (\lambda (_: nat).(eq K k (Bind -b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c0 (CSort -n)))))) (or_introl (ex2 C (\lambda (e: C).(eq C (CTail k u1 c) (CTail k u1 -e))) (\lambda (e: C).(getl O (CHead c (Bind b) u2) (CHead e (Bind b) u2)))) -(ex4 nat (\lambda (_: nat).(eq nat O (s (Bind b) (clen c)))) (\lambda (_: -nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq -C (CTail k u1 c) (CSort n)))) (ex_intro2 C (\lambda (e: C).(eq C (CTail k u1 -c) (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Bind b) u2) (CHead e -(Bind b) u2))) c (refl_equal C (CTail k u1 c)) (getl_refl b c u2))) c2 H6)) -b0 H5) t H4)))) H3)) H2)))) (\lambda (f: F).(\lambda (H1: (clear (CHead -(CTail k u1 c) (Flat f) t) (CHead c2 (Bind b) u2))).(let H2 \def (H O -(getl_intro O (CTail k u1 c) (CHead c2 (Bind b) u2) (CTail k u1 c) (drop_refl -(CTail k u1 c)) (clear_gen_flat f (CTail k u1 c) (CHead c2 (Bind b) u2) t -H1))) in (or_ind (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda -(e: C).(getl O c (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat -O (clen c))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 -u2)) (\lambda (n: nat).(eq C c2 (CSort n)))) (or (ex2 C (\lambda (e: C).(eq C -c2 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e -(Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) -(\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda -(n: nat).(eq C c2 (CSort n))))) (\lambda (H3: (ex2 C (\lambda (e: C).(eq C c2 -(CTail k u1 e))) (\lambda (e: C).(getl O c (CHead e (Bind b) u2))))).(ex2_ind -C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl O c (CHead -e (Bind b) u2))) (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) -(\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind b) u2)))) (ex4 -nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq -K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 -(CSort n))))) (\lambda (x: C).(\lambda (H4: (eq C c2 (CTail k u1 -x))).(\lambda (H5: (getl O c (CHead x (Bind b) u2))).(eq_ind_r C (CTail k u1 -x) (\lambda (c0: C).(or (ex2 C (\lambda (e: C).(eq C c0 (CTail k u1 e))) -(\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind b) u2)))) (ex4 -nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq -K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c0 -(CSort n)))))) (or_introl (ex2 C (\lambda (e: C).(eq C (CTail k u1 x) (CTail -k u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind b) -u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) (\lambda -(_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: -nat).(eq C (CTail k u1 x) (CSort n)))) (ex_intro2 C (\lambda (e: C).(eq C -(CTail k u1 x) (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) -(CHead e (Bind b) u2))) x (refl_equal C (CTail k u1 x)) (getl_flat c (CHead x -(Bind b) u2) O H5 f t))) c2 H4)))) H3)) (\lambda (H3: (ex4 nat (\lambda (_: -nat).(eq nat O (clen c))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: -nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n))))).(ex4_ind nat -(\lambda (_: nat).(eq nat O (clen c))) (\lambda (_: nat).(eq K k (Bind b))) -(\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n))) (or -(ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl O -(CHead c (Flat f) t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq -nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda -(_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n))))) (\lambda (x0: -nat).(\lambda (H4: (eq nat O (clen c))).(\lambda (H5: (eq K k (Bind -b))).(\lambda (H6: (eq T u1 u2)).(\lambda (H7: (eq C c2 (CSort -x0))).(eq_ind_r C (CSort x0) (\lambda (c0: C).(or (ex2 C (\lambda (e: C).(eq -C c0 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e -(Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) -(\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda -(n: nat).(eq C c0 (CSort n)))))) (eq_ind T u1 (\lambda (t0: T).(or (ex2 C -(\lambda (e: C).(eq C (CSort x0) (CTail k u1 e))) (\lambda (e: C).(getl O -(CHead c (Flat f) t) (CHead e (Bind b) t0)))) (ex4 nat (\lambda (_: nat).(eq -nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda -(_: nat).(eq T u1 t0)) (\lambda (n: nat).(eq C (CSort x0) (CSort n)))))) -(eq_ind_r K (Bind b) (\lambda (k1: K).(or (ex2 C (\lambda (e: C).(eq C (CSort -x0) (CTail k1 u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e -(Bind b) u1)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) -(\lambda (_: nat).(eq K k1 (Bind b))) (\lambda (_: nat).(eq T u1 u1)) -(\lambda (n: nat).(eq C (CSort x0) (CSort n)))))) (or_intror (ex2 C (\lambda -(e: C).(eq C (CSort x0) (CTail (Bind b) u1 e))) (\lambda (e: C).(getl O -(CHead c (Flat f) t) (CHead e (Bind b) u1)))) (ex4 nat (\lambda (_: nat).(eq -nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq K (Bind b) (Bind b))) -(\lambda (_: nat).(eq T u1 u1)) (\lambda (n: nat).(eq C (CSort x0) (CSort -n)))) (ex4_intro nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) -(\lambda (_: nat).(eq K (Bind b) (Bind b))) (\lambda (_: nat).(eq T u1 u1)) -(\lambda (n: nat).(eq C (CSort x0) (CSort n))) x0 H4 (refl_equal K (Bind b)) -(refl_equal T u1) (refl_equal C (CSort x0)))) k H5) u2 H6) c2 H7)))))) H3)) -H2)))) k0 (getl_gen_O (CHead (CTail k u1 c) k0 t) (CHead c2 (Bind b) u2) -H0))) (\lambda (n: nat).(\lambda (H0: (((getl n (CHead (CTail k u1 c) k0 t) -(CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 -e))) (\lambda (e: C).(getl n (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat -(\lambda (_: nat).(eq nat n (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind -b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort -n0)))))))).(\lambda (H1: (getl (S n) (CHead (CTail k u1 c) k0 t) (CHead c2 -(Bind b) u2))).(let H_x \def (H (r k0 n) (getl_gen_S k0 (CTail k u1 c) (CHead -c2 (Bind b) u2) t n H1)) in (let H2 \def H_x in (or_ind (ex2 C (\lambda (e: -C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl (r k0 n) c (CHead e (Bind -b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat (r k0 n) (clen c))) (\lambda (_: -nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: -nat).(eq C c2 (CSort n0)))) (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 -e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 -nat (\lambda (_: nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K -k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 -(CSort n0))))) (\lambda (H3: (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) -(\lambda (e: C).(getl (r k0 n) c (CHead e (Bind b) u2))))).(ex2_ind C -(\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl (r k0 n) c -(CHead e (Bind b) u2))) (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) -(\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat -(\lambda (_: nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k -(Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort -n0))))) (\lambda (x: C).(\lambda (H4: (eq C c2 (CTail k u1 x))).(\lambda (H5: -(getl (r k0 n) c (CHead x (Bind b) u2))).(let H6 \def (eq_ind C c2 (\lambda -(c0: C).(getl (r k0 n) (CTail k u1 c) (CHead c0 (Bind b) u2))) (getl_gen_S k0 -(CTail k u1 c) (CHead c2 (Bind b) u2) t n H1) (CTail k u1 x) H4) in (let H7 -\def (eq_ind C c2 (\lambda (c0: C).((getl n (CHead (CTail k u1 c) k0 t) -(CHead c0 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c0 (CTail k u1 -e))) (\lambda (e: C).(getl n (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat -(\lambda (_: nat).(eq nat n (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind -b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c0 (CSort -n0))))))) H0 (CTail k u1 x) H4) in (eq_ind_r C (CTail k u1 x) (\lambda (c0: -C).(or (ex2 C (\lambda (e: C).(eq C c0 (CTail k u1 e))) (\lambda (e: C).(getl -(S n) (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq -nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: -nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c0 (CSort n0)))))) (or_introl -(ex2 C (\lambda (e: C).(eq C (CTail k u1 x) (CTail k u1 e))) (\lambda (e: -C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: -nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) -(\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C (CTail k u1 x) -(CSort n0)))) (ex_intro2 C (\lambda (e: C).(eq C (CTail k u1 x) (CTail k u1 -e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u2))) x -(refl_equal C (CTail k u1 x)) (getl_head k0 n c (CHead x (Bind b) u2) H5 t))) -c2 H4)))))) H3)) (\lambda (H3: (ex4 nat (\lambda (_: nat).(eq nat (r k0 n) -(clen c))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 -u2)) (\lambda (n0: nat).(eq C c2 (CSort n0))))).(ex4_ind nat (\lambda (_: -nat).(eq nat (r k0 n) (clen c))) (\lambda (_: nat).(eq K k (Bind b))) -(\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0))) (or -(ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl (S n) -(CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat (S -n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: -nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0))))) (\lambda (x0: -nat).(\lambda (H4: (eq nat (r k0 n) (clen c))).(\lambda (H5: (eq K k (Bind -b))).(\lambda (H6: (eq T u1 u2)).(\lambda (H7: (eq C c2 (CSort x0))).(let H8 -\def (eq_ind C c2 (\lambda (c0: C).(getl (r k0 n) (CTail k u1 c) (CHead c0 -(Bind b) u2))) (getl_gen_S k0 (CTail k u1 c) (CHead c2 (Bind b) u2) t n H1) -(CSort x0) H7) in (let H9 \def (eq_ind C c2 (\lambda (c0: C).((getl n (CHead -(CTail k u1 c) k0 t) (CHead c0 (Bind b) u2)) \to (or (ex2 C (\lambda (e: -C).(eq C c0 (CTail k u1 e))) (\lambda (e: C).(getl n (CHead c k0 t) (CHead e -(Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat n (s k0 (clen c)))) -(\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda -(n0: nat).(eq C c0 (CSort n0))))))) H0 (CSort x0) H7) in (eq_ind_r C (CSort -x0) (\lambda (c0: C).(or (ex2 C (\lambda (e: C).(eq C c0 (CTail k u1 e))) -(\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat -(\lambda (_: nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k -(Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c0 (CSort -n0)))))) (let H10 \def (eq_ind_r T u2 (\lambda (t0: T).((getl n (CHead (CTail -k u1 c) k0 t) (CHead (CSort x0) (Bind b) t0)) \to (or (ex2 C (\lambda (e: -C).(eq C (CSort x0) (CTail k u1 e))) (\lambda (e: C).(getl n (CHead c k0 t) -(CHead e (Bind b) t0)))) (ex4 nat (\lambda (_: nat).(eq nat n (s k0 (clen -c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 t0)) -(\lambda (n0: nat).(eq C (CSort x0) (CSort n0))))))) H9 u1 H6) in (let H11 -\def (eq_ind_r T u2 (\lambda (t0: T).(getl (r k0 n) (CTail k u1 c) (CHead -(CSort x0) (Bind b) t0))) H8 u1 H6) in (eq_ind T u1 (\lambda (t0: T).(or (ex2 -C (\lambda (e: C).(eq C (CSort x0) (CTail k u1 e))) (\lambda (e: C).(getl (S -n) (CHead c k0 t) (CHead e (Bind b) t0)))) (ex4 nat (\lambda (_: nat).(eq nat -(S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: -nat).(eq T u1 t0)) (\lambda (n0: nat).(eq C (CSort x0) (CSort n0)))))) (let -H12 \def (eq_ind K k (\lambda (k1: K).((getl n (CHead (CTail k1 u1 c) k0 t) -(CHead (CSort x0) (Bind b) u1)) \to (or (ex2 C (\lambda (e: C).(eq C (CSort -x0) (CTail k1 u1 e))) (\lambda (e: C).(getl n (CHead c k0 t) (CHead e (Bind -b) u1)))) (ex4 nat (\lambda (_: nat).(eq nat n (s k0 (clen c)))) (\lambda (_: -nat).(eq K k1 (Bind b))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: -nat).(eq C (CSort x0) (CSort n0))))))) H10 (Bind b) H5) in (let H13 \def -(eq_ind K k (\lambda (k1: K).(getl (r k0 n) (CTail k1 u1 c) (CHead (CSort x0) -(Bind b) u1))) H11 (Bind b) H5) in (eq_ind_r K (Bind b) (\lambda (k1: K).(or -(ex2 C (\lambda (e: C).(eq C (CSort x0) (CTail k1 u1 e))) (\lambda (e: -C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u1)))) (ex4 nat (\lambda (_: -nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k1 (Bind b))) -(\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: nat).(eq C (CSort x0) (CSort -n0)))))) (eq_ind nat (r k0 n) (\lambda (n0: nat).(or (ex2 C (\lambda (e: -C).(eq C (CSort x0) (CTail (Bind b) u1 e))) (\lambda (e: C).(getl (S n) -(CHead c k0 t) (CHead e (Bind b) u1)))) (ex4 nat (\lambda (_: nat).(eq nat (S -n) (s k0 n0))) (\lambda (_: nat).(eq K (Bind b) (Bind b))) (\lambda (_: -nat).(eq T u1 u1)) (\lambda (n1: nat).(eq C (CSort x0) (CSort n1)))))) -(eq_ind_r nat (S n) (\lambda (n0: nat).(or (ex2 C (\lambda (e: C).(eq C -(CSort x0) (CTail (Bind b) u1 e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) -(CHead e (Bind b) u1)))) (ex4 nat (\lambda (_: nat).(eq nat (S n) n0)) -(\lambda (_: nat).(eq K (Bind b) (Bind b))) (\lambda (_: nat).(eq T u1 u1)) -(\lambda (n1: nat).(eq C (CSort x0) (CSort n1)))))) (or_intror (ex2 C -(\lambda (e: C).(eq C (CSort x0) (CTail (Bind b) u1 e))) (\lambda (e: -C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u1)))) (ex4 nat (\lambda (_: -nat).(eq nat (S n) (S n))) (\lambda (_: nat).(eq K (Bind b) (Bind b))) -(\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: nat).(eq C (CSort x0) (CSort -n0)))) (ex4_intro nat (\lambda (_: nat).(eq nat (S n) (S n))) (\lambda (_: -nat).(eq K (Bind b) (Bind b))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: -nat).(eq C (CSort x0) (CSort n0))) x0 (refl_equal nat (S n)) (refl_equal K -(Bind b)) (refl_equal T u1) (refl_equal C (CSort x0)))) (s k0 (r k0 n)) (s_r -k0 n)) (clen c) H4) k H5))) u2 H6))) c2 H7)))))))) H3)) H2)))))) i)))))) -c1)))))). -(* COMMENTS -Initial nodes: 7489 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/cnt/defs.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/cnt/defs.ma deleted file mode 100644 index d6ab61d78..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/cnt/defs.ma +++ /dev/null @@ -1,23 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/T/defs.ma". - -inductive cnt: T \to Prop \def -| cnt_sort: \forall (n: nat).(cnt (TSort n)) -| cnt_head: \forall (t: T).((cnt t) \to (\forall (k: K).(\forall (v: T).(cnt -(THead k v t))))). - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/cnt/props.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/cnt/props.ma deleted file mode 100644 index 7fcd315eb..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/cnt/props.ma +++ /dev/null @@ -1,37 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/cnt/defs.ma". - -include "Basic-1/lift/fwd.ma". - -theorem cnt_lift: - \forall (t: T).((cnt t) \to (\forall (i: nat).(\forall (d: nat).(cnt (lift i -d t))))) -\def - \lambda (t: T).(\lambda (H: (cnt t)).(cnt_ind (\lambda (t0: T).(\forall (i: -nat).(\forall (d: nat).(cnt (lift i d t0))))) (\lambda (n: nat).(\lambda (i: -nat).(\lambda (d: nat).(eq_ind_r T (TSort n) (\lambda (t0: T).(cnt t0)) -(cnt_sort n) (lift i d (TSort n)) (lift_sort n i d))))) (\lambda (t0: -T).(\lambda (_: (cnt t0)).(\lambda (H1: ((\forall (i: nat).(\forall (d: -nat).(cnt (lift i d t0)))))).(\lambda (k: K).(\lambda (v: T).(\lambda (i: -nat).(\lambda (d: nat).(eq_ind_r T (THead k (lift i d v) (lift i (s k d) t0)) -(\lambda (t1: T).(cnt t1)) (cnt_head (lift i (s k d) t0) (H1 i (s k d)) k -(lift i d v)) (lift i d (THead k v t0)) (lift_head k v t0 i d))))))))) t H)). -(* COMMENTS -Initial nodes: 191 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/arity.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/arity.ma deleted file mode 100644 index fe021cd0f..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/arity.ma +++ /dev/null @@ -1,336 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csuba/getl.ma". - -include "Basic-1/csuba/props.ma". - -include "Basic-1/arity/props.ma". - -include "Basic-1/csubv/getl.ma". - -theorem csuba_arity: - \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1 -t a) \to (\forall (c2: C).((csuba g c1 c2) \to (arity g c2 t a))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: -(arity g c1 t a)).(arity_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda (a0: -A).(\forall (c2: C).((csuba g c c2) \to (arity g c2 t0 a0)))))) (\lambda (c: -C).(\lambda (n: nat).(\lambda (c2: C).(\lambda (_: (csuba g c -c2)).(arity_sort g c2 n))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: -T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d (Bind Abbr) -u))).(\lambda (a0: A).(\lambda (_: (arity g d u a0)).(\lambda (H2: ((\forall -(c2: C).((csuba g d c2) \to (arity g c2 u a0))))).(\lambda (c2: C).(\lambda -(H3: (csuba g c c2)).(let H4 \def (csuba_getl_abbr g c d u i H0 c2 H3) in -(ex2_ind C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda -(d2: C).(csuba g d d2)) (arity g c2 (TLRef i) a0) (\lambda (x: C).(\lambda -(H5: (getl i c2 (CHead x (Bind Abbr) u))).(\lambda (H6: (csuba g d -x)).(arity_abbr g c2 x u i H5 a0 (H2 x H6))))) H4)))))))))))) (\lambda (c: -C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c -(CHead d (Bind Abst) u))).(\lambda (a0: A).(\lambda (H1: (arity g d u (asucc -g a0))).(\lambda (H2: ((\forall (c2: C).((csuba g d c2) \to (arity g c2 u -(asucc g a0)))))).(\lambda (c2: C).(\lambda (H3: (csuba g c c2)).(let H4 \def -(csuba_getl_abst g c d u i H0 c2 H3) in (or_ind (ex2 C (\lambda (d2: C).(getl -i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d d2))) (ex4_3 C T -A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 -(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d u (asucc -g a1))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2 -a1))))) (arity g c2 (TLRef i) a0) (\lambda (H5: (ex2 C (\lambda (d2: C).(getl -i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d d2)))).(ex2_ind C -(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d d2)) (arity g c2 (TLRef i) a0) (\lambda (x: C).(\lambda (H6: -(getl i c2 (CHead x (Bind Abst) u))).(\lambda (H7: (csuba g d x)).(arity_abst -g c2 x u i H6 a0 (H2 x H7))))) H5)) (\lambda (H5: (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d d2)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d u (asucc g a1))))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2 -a1)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a1: A).(arity g d u (asucc g a1))))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a1: A).(arity g d2 u2 a1)))) (arity g c2 (TLRef i) a0) (\lambda -(x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H6: (getl i c2 (CHead x0 -(Bind Abbr) x1))).(\lambda (_: (csuba g d x0)).(\lambda (H8: (arity g d u -(asucc g x2))).(\lambda (H9: (arity g x0 x1 x2)).(arity_repl g c2 (TLRef i) -x2 (arity_abbr g c2 x0 x1 i H6 x2 H9) a0 (asucc_inj g x2 a0 (arity_mono g d u -(asucc g x2) H8 (asucc g a0) H1)))))))))) H5)) H4)))))))))))) (\lambda (b: -B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c: C).(\lambda (u: -T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H2: ((\forall -(c2: C).((csuba g c c2) \to (arity g c2 u a1))))).(\lambda (t0: T).(\lambda -(a2: A).(\lambda (_: (arity g (CHead c (Bind b) u) t0 a2)).(\lambda (H4: -((\forall (c2: C).((csuba g (CHead c (Bind b) u) c2) \to (arity g c2 t0 -a2))))).(\lambda (c2: C).(\lambda (H5: (csuba g c c2)).(arity_bind g b H0 c2 -u a1 (H2 c2 H5) t0 a2 (H4 (CHead c2 (Bind b) u) (csuba_head g c c2 H5 (Bind -b) u)))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a1: -A).(\lambda (_: (arity g c u (asucc g a1))).(\lambda (H1: ((\forall (c2: -C).((csuba g c c2) \to (arity g c2 u (asucc g a1)))))).(\lambda (t0: -T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c (Bind Abst) u) t0 -a2)).(\lambda (H3: ((\forall (c2: C).((csuba g (CHead c (Bind Abst) u) c2) -\to (arity g c2 t0 a2))))).(\lambda (c2: C).(\lambda (H4: (csuba g c -c2)).(arity_head g c2 u a1 (H1 c2 H4) t0 a2 (H3 (CHead c2 (Bind Abst) u) -(csuba_head g c c2 H4 (Bind Abst) u)))))))))))))) (\lambda (c: C).(\lambda -(u: T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H1: -((\forall (c2: C).((csuba g c c2) \to (arity g c2 u a1))))).(\lambda (t0: -T).(\lambda (a2: A).(\lambda (_: (arity g c t0 (AHead a1 a2))).(\lambda (H3: -((\forall (c2: C).((csuba g c c2) \to (arity g c2 t0 (AHead a1 -a2)))))).(\lambda (c2: C).(\lambda (H4: (csuba g c c2)).(arity_appl g c2 u a1 -(H1 c2 H4) t0 a2 (H3 c2 H4))))))))))))) (\lambda (c: C).(\lambda (u: -T).(\lambda (a0: A).(\lambda (_: (arity g c u (asucc g a0))).(\lambda (H1: -((\forall (c2: C).((csuba g c c2) \to (arity g c2 u (asucc g -a0)))))).(\lambda (t0: T).(\lambda (_: (arity g c t0 a0)).(\lambda (H3: -((\forall (c2: C).((csuba g c c2) \to (arity g c2 t0 a0))))).(\lambda (c2: -C).(\lambda (H4: (csuba g c c2)).(arity_cast g c2 u a0 (H1 c2 H4) t0 (H3 c2 -H4)))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda (_: -(arity g c t0 a1)).(\lambda (H1: ((\forall (c2: C).((csuba g c c2) \to (arity -g c2 t0 a1))))).(\lambda (a2: A).(\lambda (H2: (leq g a1 a2)).(\lambda (c2: -C).(\lambda (H3: (csuba g c c2)).(arity_repl g c2 t0 a1 (H1 c2 H3) a2 -H2)))))))))) c1 t a H))))). -(* COMMENTS -Initial nodes: 1505 -END *) - -theorem csuba_arity_rev: - \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1 -t a) \to (\forall (c2: C).((csuba g c2 c1) \to ((csubv c2 c1) \to (arity g c2 -t a)))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: -(arity g c1 t a)).(arity_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda (a0: -A).(\forall (c2: C).((csuba g c2 c) \to ((csubv c2 c) \to (arity g c2 t0 -a0))))))) (\lambda (c: C).(\lambda (n: nat).(\lambda (c2: C).(\lambda (_: -(csuba g c2 c)).(\lambda (_: (csubv c2 c)).(arity_sort g c2 n)))))) (\lambda -(c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl -i c (CHead d (Bind Abbr) u))).(\lambda (a0: A).(\lambda (H1: (arity g d u -a0)).(\lambda (H2: ((\forall (c2: C).((csuba g c2 d) \to ((csubv c2 d) \to -(arity g c2 u a0)))))).(\lambda (c2: C).(\lambda (H3: (csuba g c2 -c)).(\lambda (H4: (csubv c2 c)).(let H_x \def (csuba_getl_abbr_rev g c d u i -H0 c2 H3) in (let H5 \def H_x in (or3_ind (ex2 C (\lambda (d2: C).(getl i c2 -(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d2 d))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -d)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2 -(asucc g a1))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d -u a1))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d)))) (arity -g c2 (TLRef i) a0) (\lambda (H6: (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 -(Bind Abbr) u))) (\lambda (d2: C).(csuba g d2 d)))).(ex2_ind C (\lambda (d2: -C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d2 d)) -(arity g c2 (TLRef i) a0) (\lambda (x: C).(\lambda (H7: (getl i c2 (CHead x -(Bind Abbr) u))).(\lambda (H8: (csuba g x d)).(let H_x0 \def (csubv_getl_conf -c2 c H4 Abbr x u i H7) in (let H9 \def H_x0 in (ex2_3_ind B C T (\lambda (_: -B).(\lambda (d2: C).(\lambda (_: T).(csubv x d2)))) (\lambda (b2: B).(\lambda -(d2: C).(\lambda (v2: T).(getl i c (CHead d2 (Bind b2) v2))))) (arity g c2 -(TLRef i) a0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda -(H10: (csubv x x1)).(\lambda (H11: (getl i c (CHead x1 (Bind x0) x2))).(let -H12 \def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c0: C).(getl i c c0)) H0 -(CHead x1 (Bind x0) x2) (getl_mono c (CHead d (Bind Abbr) u) i H0 (CHead x1 -(Bind x0) x2) H11)) in (let H13 \def (f_equal C C (\lambda (e: C).(match e in -C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) -\Rightarrow c0])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) (getl_mono -c (CHead d (Bind Abbr) u) i H0 (CHead x1 (Bind x0) x2) H11)) in ((let H14 -\def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) -with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k in K -return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow -Abbr])])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) (getl_mono c (CHead -d (Bind Abbr) u) i H0 (CHead x1 (Bind x0) x2) H11)) in ((let H15 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind -Abbr) u) (CHead x1 (Bind x0) x2) (getl_mono c (CHead d (Bind Abbr) u) i H0 -(CHead x1 (Bind x0) x2) H11)) in (\lambda (H16: (eq B Abbr x0)).(\lambda -(H17: (eq C d x1)).(let H18 \def (eq_ind_r T x2 (\lambda (t0: T).(getl i c -(CHead x1 (Bind x0) t0))) H12 u H15) in (let H19 \def (eq_ind_r C x1 (\lambda -(c0: C).(getl i c (CHead c0 (Bind x0) u))) H18 d H17) in (let H20 \def -(eq_ind_r C x1 (\lambda (c0: C).(csubv x c0)) H10 d H17) in (let H21 \def -(eq_ind_r B x0 (\lambda (b: B).(getl i c (CHead d (Bind b) u))) H19 Abbr H16) -in (arity_abbr g c2 x u i H7 a0 (H2 x H8 H20))))))))) H14)) H13)))))))) -H9)))))) H6)) (\lambda (H6: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2 (asucc g a1))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d u a1)))))).(ex4_3_ind C T -A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 -(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d2 d)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2 -(asucc g a1))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d -u a1)))) (arity g c2 (TLRef i) a0) (\lambda (x0: C).(\lambda (x1: T).(\lambda -(x2: A).(\lambda (H7: (getl i c2 (CHead x0 (Bind Abst) x1))).(\lambda (_: -(csuba g x0 d)).(\lambda (H9: (arity g x0 x1 (asucc g x2))).(\lambda (H10: -(arity g d u x2)).(arity_repl g c2 (TLRef i) x2 (arity_abst g c2 x0 x1 i H7 -x2 H9) a0 (arity_mono g d u x2 H10 a0 H1))))))))) H6)) (\lambda (H6: (ex2_2 C -T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d))))).(ex2_2_ind C T (\lambda -(d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d))) (arity g c2 (TLRef i) a0) (\lambda -(x0: C).(\lambda (x1: T).(\lambda (H7: (getl i c2 (CHead x0 (Bind Void) -x1))).(\lambda (_: (csuba g x0 d)).(let H_x0 \def (csubv_getl_conf_void c2 c -H4 x0 x1 i H7) in (let H9 \def H_x0 in (ex2_2_ind C T (\lambda (d2: -C).(\lambda (_: T).(csubv x0 d2))) (\lambda (d2: C).(\lambda (v2: T).(getl i -c (CHead d2 (Bind Void) v2)))) (arity g c2 (TLRef i) a0) (\lambda (x2: -C).(\lambda (x3: T).(\lambda (_: (csubv x0 x2)).(\lambda (H11: (getl i c -(CHead x2 (Bind Void) x3))).(let H12 \def (eq_ind C (CHead d (Bind Abbr) u) -(\lambda (c0: C).(getl i c c0)) H0 (CHead x2 (Bind Void) x3) (getl_mono c -(CHead d (Bind Abbr) u) i H0 (CHead x2 (Bind Void) x3) H11)) in (let H13 \def -(eq_ind C (CHead d (Bind Abbr) u) (\lambda (ee: C).(match ee in C return -(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b) -\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow -True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _) -\Rightarrow False])])) I (CHead x2 (Bind Void) x3) (getl_mono c (CHead d -(Bind Abbr) u) i H0 (CHead x2 (Bind Void) x3) H11)) in (False_ind (arity g c2 -(TLRef i) a0) H13))))))) H9))))))) H6)) H5)))))))))))))) (\lambda (c: -C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c -(CHead d (Bind Abst) u))).(\lambda (a0: A).(\lambda (_: (arity g d u (asucc g -a0))).(\lambda (H2: ((\forall (c2: C).((csuba g c2 d) \to ((csubv c2 d) \to -(arity g c2 u (asucc g a0))))))).(\lambda (c2: C).(\lambda (H3: (csuba g c2 -c)).(\lambda (H4: (csubv c2 c)).(let H_x \def (csuba_getl_abst_rev g c d u i -H0 c2 H3) in (let H5 \def H_x in (or_ind (ex2 C (\lambda (d2: C).(getl i c2 -(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d)))) (arity g c2 (TLRef i) a0) -(\lambda (H6: (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d)))).(ex2_ind C (\lambda (d2: C).(getl i c2 -(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d)) (arity g c2 -(TLRef i) a0) (\lambda (x: C).(\lambda (H7: (getl i c2 (CHead x (Bind Abst) -u))).(\lambda (H8: (csuba g x d)).(let H_x0 \def (csubv_getl_conf c2 c H4 -Abst x u i H7) in (let H9 \def H_x0 in (ex2_3_ind B C T (\lambda (_: -B).(\lambda (d2: C).(\lambda (_: T).(csubv x d2)))) (\lambda (b2: B).(\lambda -(d2: C).(\lambda (v2: T).(getl i c (CHead d2 (Bind b2) v2))))) (arity g c2 -(TLRef i) a0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda -(H10: (csubv x x1)).(\lambda (H11: (getl i c (CHead x1 (Bind x0) x2))).(let -H12 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (c0: C).(getl i c c0)) H0 -(CHead x1 (Bind x0) x2) (getl_mono c (CHead d (Bind Abst) u) i H0 (CHead x1 -(Bind x0) x2) H11)) in (let H13 \def (f_equal C C (\lambda (e: C).(match e in -C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) -\Rightarrow c0])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x2) (getl_mono -c (CHead d (Bind Abst) u) i H0 (CHead x1 (Bind x0) x2) H11)) in ((let H14 -\def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) -with [(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match k in K -return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow -Abst])])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x2) (getl_mono c (CHead -d (Bind Abst) u) i H0 (CHead x1 (Bind x0) x2) H11)) in ((let H15 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind -Abst) u) (CHead x1 (Bind x0) x2) (getl_mono c (CHead d (Bind Abst) u) i H0 -(CHead x1 (Bind x0) x2) H11)) in (\lambda (H16: (eq B Abst x0)).(\lambda -(H17: (eq C d x1)).(let H18 \def (eq_ind_r T x2 (\lambda (t0: T).(getl i c -(CHead x1 (Bind x0) t0))) H12 u H15) in (let H19 \def (eq_ind_r C x1 (\lambda -(c0: C).(getl i c (CHead c0 (Bind x0) u))) H18 d H17) in (let H20 \def -(eq_ind_r C x1 (\lambda (c0: C).(csubv x c0)) H10 d H17) in (let H21 \def -(eq_ind_r B x0 (\lambda (b: B).(getl i c (CHead d (Bind b) u))) H19 Abst H16) -in (arity_abst g c2 x u i H7 a0 (H2 x H8 H20))))))))) H14)) H13)))))))) -H9)))))) H6)) (\lambda (H6: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(getl -i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g -d2 d))) (arity g c2 (TLRef i) a0) (\lambda (x0: C).(\lambda (x1: T).(\lambda -(H7: (getl i c2 (CHead x0 (Bind Void) x1))).(\lambda (_: (csuba g x0 d)).(let -H_x0 \def (csubv_getl_conf_void c2 c H4 x0 x1 i H7) in (let H9 \def H_x0 in -(ex2_2_ind C T (\lambda (d2: C).(\lambda (_: T).(csubv x0 d2))) (\lambda (d2: -C).(\lambda (v2: T).(getl i c (CHead d2 (Bind Void) v2)))) (arity g c2 (TLRef -i) a0) (\lambda (x2: C).(\lambda (x3: T).(\lambda (_: (csubv x0 x2)).(\lambda -(H11: (getl i c (CHead x2 (Bind Void) x3))).(let H12 \def (eq_ind C (CHead d -(Bind Abst) u) (\lambda (c0: C).(getl i c c0)) H0 (CHead x2 (Bind Void) x3) -(getl_mono c (CHead d (Bind Abst) u) i H0 (CHead x2 (Bind Void) x3) H11)) in -(let H13 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (ee: C).(match ee in -C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k -_) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b) -\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow -False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) -\Rightarrow False])])) I (CHead x2 (Bind Void) x3) (getl_mono c (CHead d -(Bind Abst) u) i H0 (CHead x2 (Bind Void) x3) H11)) in (False_ind (arity g c2 -(TLRef i) a0) H13))))))) H9))))))) H6)) H5)))))))))))))) (\lambda (b: -B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c: C).(\lambda (u: -T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H2: ((\forall -(c2: C).((csuba g c2 c) \to ((csubv c2 c) \to (arity g c2 u a1)))))).(\lambda -(t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c (Bind b) u) t0 -a2)).(\lambda (H4: ((\forall (c2: C).((csuba g c2 (CHead c (Bind b) u)) \to -((csubv c2 (CHead c (Bind b) u)) \to (arity g c2 t0 a2)))))).(\lambda (c2: -C).(\lambda (H5: (csuba g c2 c)).(\lambda (H6: (csubv c2 c)).(arity_bind g b -H0 c2 u a1 (H2 c2 H5 H6) t0 a2 (H4 (CHead c2 (Bind b) u) (csuba_head g c2 c -H5 (Bind b) u) (csubv_bind_same c2 c H6 b u u))))))))))))))))) (\lambda (c: -C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c u (asucc g -a1))).(\lambda (H1: ((\forall (c2: C).((csuba g c2 c) \to ((csubv c2 c) \to -(arity g c2 u (asucc g a1))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda -(_: (arity g (CHead c (Bind Abst) u) t0 a2)).(\lambda (H3: ((\forall (c2: -C).((csuba g c2 (CHead c (Bind Abst) u)) \to ((csubv c2 (CHead c (Bind Abst) -u)) \to (arity g c2 t0 a2)))))).(\lambda (c2: C).(\lambda (H4: (csuba g c2 -c)).(\lambda (H5: (csubv c2 c)).(arity_head g c2 u a1 (H1 c2 H4 H5) t0 a2 (H3 -(CHead c2 (Bind Abst) u) (csuba_head g c2 c H4 (Bind Abst) u) -(csubv_bind_same c2 c H5 Abst u u))))))))))))))) (\lambda (c: C).(\lambda (u: -T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H1: ((\forall -(c2: C).((csuba g c2 c) \to ((csubv c2 c) \to (arity g c2 u a1)))))).(\lambda -(t0: T).(\lambda (a2: A).(\lambda (_: (arity g c t0 (AHead a1 a2))).(\lambda -(H3: ((\forall (c2: C).((csuba g c2 c) \to ((csubv c2 c) \to (arity g c2 t0 -(AHead a1 a2))))))).(\lambda (c2: C).(\lambda (H4: (csuba g c2 c)).(\lambda -(H5: (csubv c2 c)).(arity_appl g c2 u a1 (H1 c2 H4 H5) t0 a2 (H3 c2 H4 -H5)))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a0: A).(\lambda -(_: (arity g c u (asucc g a0))).(\lambda (H1: ((\forall (c2: C).((csuba g c2 -c) \to ((csubv c2 c) \to (arity g c2 u (asucc g a0))))))).(\lambda (t0: -T).(\lambda (_: (arity g c t0 a0)).(\lambda (H3: ((\forall (c2: C).((csuba g -c2 c) \to ((csubv c2 c) \to (arity g c2 t0 a0)))))).(\lambda (c2: C).(\lambda -(H4: (csuba g c2 c)).(\lambda (H5: (csubv c2 c)).(arity_cast g c2 u a0 (H1 c2 -H4 H5) t0 (H3 c2 H4 H5))))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda -(a1: A).(\lambda (_: (arity g c t0 a1)).(\lambda (H1: ((\forall (c2: -C).((csuba g c2 c) \to ((csubv c2 c) \to (arity g c2 t0 a1)))))).(\lambda -(a2: A).(\lambda (H2: (leq g a1 a2)).(\lambda (c2: C).(\lambda (H3: (csuba g -c2 c)).(\lambda (H4: (csubv c2 c)).(arity_repl g c2 t0 a1 (H1 c2 H3 H4) a2 -H2))))))))))) c1 t a H))))). -(* COMMENTS -Initial nodes: 3597 -END *) - -theorem arity_appls_appl: - \forall (g: G).(\forall (c: C).(\forall (v: T).(\forall (a1: A).((arity g c -v a1) \to (\forall (u: T).((arity g c u (asucc g a1)) \to (\forall (t: -T).(\forall (vs: TList).(\forall (a2: A).((arity g c (THeads (Flat Appl) vs -(THead (Bind Abbr) v t)) a2) \to (arity g c (THeads (Flat Appl) vs (THead -(Flat Appl) v (THead (Bind Abst) u t))) a2))))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (v: T).(\lambda (a1: A).(\lambda (H: -(arity g c v a1)).(\lambda (u: T).(\lambda (H0: (arity g c u (asucc g -a1))).(\lambda (t: T).(\lambda (vs: TList).(TList_ind (\lambda (t0: -TList).(\forall (a2: A).((arity g c (THeads (Flat Appl) t0 (THead (Bind Abbr) -v t)) a2) \to (arity g c (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead -(Bind Abst) u t))) a2)))) (\lambda (a2: A).(\lambda (H1: (arity g c (THead -(Bind Abbr) v t) a2)).(let H_x \def (arity_gen_bind Abbr (\lambda (H2: (eq B -Abbr Abst)).(not_abbr_abst H2)) g c v t a2 H1) in (let H2 \def H_x in -(ex2_ind A (\lambda (a3: A).(arity g c v a3)) (\lambda (_: A).(arity g (CHead -c (Bind Abbr) v) t a2)) (arity g c (THead (Flat Appl) v (THead (Bind Abst) u -t)) a2) (\lambda (x: A).(\lambda (_: (arity g c v x)).(\lambda (H4: (arity g -(CHead c (Bind Abbr) v) t a2)).(arity_appl g c v a1 H (THead (Bind Abst) u t) -a2 (arity_head g c u a1 H0 t a2 (csuba_arity_rev g (CHead c (Bind Abbr) v) t -a2 H4 (CHead c (Bind Abst) u) (csuba_abst g c c (csuba_refl g c) u a1 H0 v H) -(csubv_bind c c (csubv_refl c) Abst (sym_not_eq B Void Abst not_void_abst) -Abbr u v))))))) H2))))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H1: -((\forall (a2: A).((arity g c (THeads (Flat Appl) t1 (THead (Bind Abbr) v t)) -a2) \to (arity g c (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind -Abst) u t))) a2))))).(\lambda (a2: A).(\lambda (H2: (arity g c (THead (Flat -Appl) t0 (THeads (Flat Appl) t1 (THead (Bind Abbr) v t))) a2)).(let H3 \def -(arity_gen_appl g c t0 (THeads (Flat Appl) t1 (THead (Bind Abbr) v t)) a2 H2) -in (ex2_ind A (\lambda (a3: A).(arity g c t0 a3)) (\lambda (a3: A).(arity g c -(THeads (Flat Appl) t1 (THead (Bind Abbr) v t)) (AHead a3 a2))) (arity g c -(THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead -(Bind Abst) u t)))) a2) (\lambda (x: A).(\lambda (H4: (arity g c t0 -x)).(\lambda (H5: (arity g c (THeads (Flat Appl) t1 (THead (Bind Abbr) v t)) -(AHead x a2))).(arity_appl g c t0 x H4 (THeads (Flat Appl) t1 (THead (Flat -Appl) v (THead (Bind Abst) u t))) a2 (H1 (AHead x a2) H5))))) H3))))))) -vs))))))))). -(* COMMENTS -Initial nodes: 687 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/clear.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/clear.ma deleted file mode 100644 index 2350fd1e8..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/clear.ma +++ /dev/null @@ -1,128 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csuba/defs.ma". - -include "Basic-1/clear/fwd.ma". - -theorem csuba_clear_conf: - \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csuba g c1 c2) \to -(\forall (e1: C).((clear c1 e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) -(\lambda (e2: C).(clear c2 e2)))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csuba g c1 -c2)).(csuba_ind g (\lambda (c: C).(\lambda (c0: C).(\forall (e1: C).((clear c -e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2: C).(clear c0 -e2))))))) (\lambda (n: nat).(\lambda (e1: C).(\lambda (H0: (clear (CSort n) -e1)).(clear_gen_sort e1 n H0 (ex2 C (\lambda (e2: C).(csuba g e1 e2)) -(\lambda (e2: C).(clear (CSort n) e2))))))) (\lambda (c3: C).(\lambda (c4: -C).(\lambda (H0: (csuba g c3 c4)).(\lambda (H1: ((\forall (e1: C).((clear c3 -e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2: C).(clear c4 -e2))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (e1: C).(\lambda (H2: -(clear (CHead c3 k u) e1)).(K_ind (\lambda (k0: K).((clear (CHead c3 k0 u) -e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2: C).(clear -(CHead c4 k0 u) e2))))) (\lambda (b: B).(\lambda (H3: (clear (CHead c3 (Bind -b) u) e1)).(eq_ind_r C (CHead c3 (Bind b) u) (\lambda (c: C).(ex2 C (\lambda -(e2: C).(csuba g c e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b) u) e2)))) -(ex_intro2 C (\lambda (e2: C).(csuba g (CHead c3 (Bind b) u) e2)) (\lambda -(e2: C).(clear (CHead c4 (Bind b) u) e2)) (CHead c4 (Bind b) u) (csuba_head g -c3 c4 H0 (Bind b) u) (clear_bind b c4 u)) e1 (clear_gen_bind b c3 e1 u H3)))) -(\lambda (f: F).(\lambda (H3: (clear (CHead c3 (Flat f) u) e1)).(let H4 \def -(H1 e1 (clear_gen_flat f c3 e1 u H3)) in (ex2_ind C (\lambda (e2: C).(csuba g -e1 e2)) (\lambda (e2: C).(clear c4 e2)) (ex2 C (\lambda (e2: C).(csuba g e1 -e2)) (\lambda (e2: C).(clear (CHead c4 (Flat f) u) e2))) (\lambda (x: -C).(\lambda (H5: (csuba g e1 x)).(\lambda (H6: (clear c4 x)).(ex_intro2 C -(\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2: C).(clear (CHead c4 (Flat f) -u) e2)) x H5 (clear_flat c4 x H6 f u))))) H4)))) k H2))))))))) (\lambda (c3: -C).(\lambda (c4: C).(\lambda (H0: (csuba g c3 c4)).(\lambda (_: ((\forall -(e1: C).((clear c3 e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) (\lambda -(e2: C).(clear c4 e2))))))).(\lambda (b: B).(\lambda (H2: (not (eq B b -Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (e1: C).(\lambda (H3: -(clear (CHead c3 (Bind Void) u1) e1)).(eq_ind_r C (CHead c3 (Bind Void) u1) -(\lambda (c: C).(ex2 C (\lambda (e2: C).(csuba g c e2)) (\lambda (e2: -C).(clear (CHead c4 (Bind b) u2) e2)))) (ex_intro2 C (\lambda (e2: C).(csuba -g (CHead c3 (Bind Void) u1) e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b) -u2) e2)) (CHead c4 (Bind b) u2) (csuba_void g c3 c4 H0 b H2 u1 u2) -(clear_bind b c4 u2)) e1 (clear_gen_bind Void c3 e1 u1 H3)))))))))))) -(\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csuba g c3 c4)).(\lambda (_: -((\forall (e1: C).((clear c3 e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) -(\lambda (e2: C).(clear c4 e2))))))).(\lambda (t: T).(\lambda (a: A).(\lambda -(H2: (arity g c3 t (asucc g a))).(\lambda (u: T).(\lambda (H3: (arity g c4 u -a)).(\lambda (e1: C).(\lambda (H4: (clear (CHead c3 (Bind Abst) t) -e1)).(eq_ind_r C (CHead c3 (Bind Abst) t) (\lambda (c: C).(ex2 C (\lambda -(e2: C).(csuba g c e2)) (\lambda (e2: C).(clear (CHead c4 (Bind Abbr) u) -e2)))) (ex_intro2 C (\lambda (e2: C).(csuba g (CHead c3 (Bind Abst) t) e2)) -(\lambda (e2: C).(clear (CHead c4 (Bind Abbr) u) e2)) (CHead c4 (Bind Abbr) -u) (csuba_abst g c3 c4 H0 t a H2 u H3) (clear_bind Abbr c4 u)) e1 -(clear_gen_bind Abst c3 e1 t H4))))))))))))) c1 c2 H)))). -(* COMMENTS -Initial nodes: 937 -END *) - -theorem csuba_clear_trans: - \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csuba g c2 c1) \to -(\forall (e1: C).((clear c1 e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) -(\lambda (e2: C).(clear c2 e2)))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csuba g c2 -c1)).(csuba_ind g (\lambda (c: C).(\lambda (c0: C).(\forall (e1: C).((clear -c0 e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) (\lambda (e2: C).(clear c -e2))))))) (\lambda (n: nat).(\lambda (e1: C).(\lambda (H0: (clear (CSort n) -e1)).(clear_gen_sort e1 n H0 (ex2 C (\lambda (e2: C).(csuba g e2 e1)) -(\lambda (e2: C).(clear (CSort n) e2))))))) (\lambda (c3: C).(\lambda (c4: -C).(\lambda (H0: (csuba g c3 c4)).(\lambda (H1: ((\forall (e1: C).((clear c4 -e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) (\lambda (e2: C).(clear c3 -e2))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (e1: C).(\lambda (H2: -(clear (CHead c4 k u) e1)).(K_ind (\lambda (k0: K).((clear (CHead c4 k0 u) -e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) (\lambda (e2: C).(clear -(CHead c3 k0 u) e2))))) (\lambda (b: B).(\lambda (H3: (clear (CHead c4 (Bind -b) u) e1)).(eq_ind_r C (CHead c4 (Bind b) u) (\lambda (c: C).(ex2 C (\lambda -(e2: C).(csuba g e2 c)) (\lambda (e2: C).(clear (CHead c3 (Bind b) u) e2)))) -(ex_intro2 C (\lambda (e2: C).(csuba g e2 (CHead c4 (Bind b) u))) (\lambda -(e2: C).(clear (CHead c3 (Bind b) u) e2)) (CHead c3 (Bind b) u) (csuba_head g -c3 c4 H0 (Bind b) u) (clear_bind b c3 u)) e1 (clear_gen_bind b c4 e1 u H3)))) -(\lambda (f: F).(\lambda (H3: (clear (CHead c4 (Flat f) u) e1)).(let H4 \def -(H1 e1 (clear_gen_flat f c4 e1 u H3)) in (ex2_ind C (\lambda (e2: C).(csuba g -e2 e1)) (\lambda (e2: C).(clear c3 e2)) (ex2 C (\lambda (e2: C).(csuba g e2 -e1)) (\lambda (e2: C).(clear (CHead c3 (Flat f) u) e2))) (\lambda (x: -C).(\lambda (H5: (csuba g x e1)).(\lambda (H6: (clear c3 x)).(ex_intro2 C -(\lambda (e2: C).(csuba g e2 e1)) (\lambda (e2: C).(clear (CHead c3 (Flat f) -u) e2)) x H5 (clear_flat c3 x H6 f u))))) H4)))) k H2))))))))) (\lambda (c3: -C).(\lambda (c4: C).(\lambda (H0: (csuba g c3 c4)).(\lambda (_: ((\forall -(e1: C).((clear c4 e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) (\lambda -(e2: C).(clear c3 e2))))))).(\lambda (b: B).(\lambda (H2: (not (eq B b -Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (e1: C).(\lambda (H3: -(clear (CHead c4 (Bind b) u2) e1)).(eq_ind_r C (CHead c4 (Bind b) u2) -(\lambda (c: C).(ex2 C (\lambda (e2: C).(csuba g e2 c)) (\lambda (e2: -C).(clear (CHead c3 (Bind Void) u1) e2)))) (ex_intro2 C (\lambda (e2: -C).(csuba g e2 (CHead c4 (Bind b) u2))) (\lambda (e2: C).(clear (CHead c3 -(Bind Void) u1) e2)) (CHead c3 (Bind Void) u1) (csuba_void g c3 c4 H0 b H2 u1 -u2) (clear_bind Void c3 u1)) e1 (clear_gen_bind b c4 e1 u2 H3)))))))))))) -(\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csuba g c3 c4)).(\lambda (_: -((\forall (e1: C).((clear c4 e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) -(\lambda (e2: C).(clear c3 e2))))))).(\lambda (t: T).(\lambda (a: A).(\lambda -(H2: (arity g c3 t (asucc g a))).(\lambda (u: T).(\lambda (H3: (arity g c4 u -a)).(\lambda (e1: C).(\lambda (H4: (clear (CHead c4 (Bind Abbr) u) -e1)).(eq_ind_r C (CHead c4 (Bind Abbr) u) (\lambda (c: C).(ex2 C (\lambda -(e2: C).(csuba g e2 c)) (\lambda (e2: C).(clear (CHead c3 (Bind Abst) t) -e2)))) (ex_intro2 C (\lambda (e2: C).(csuba g e2 (CHead c4 (Bind Abbr) u))) -(\lambda (e2: C).(clear (CHead c3 (Bind Abst) t) e2)) (CHead c3 (Bind Abst) -t) (csuba_abst g c3 c4 H0 t a H2 u H3) (clear_bind Abst c3 t)) e1 -(clear_gen_bind Abbr c4 e1 u H4))))))))))))) c2 c1 H)))). -(* COMMENTS -Initial nodes: 937 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/defs.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/defs.ma deleted file mode 100644 index cc6a46cd8..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/defs.ma +++ /dev/null @@ -1,30 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/arity/defs.ma". - -inductive csuba (g: G): C \to (C \to Prop) \def -| csuba_sort: \forall (n: nat).(csuba g (CSort n) (CSort n)) -| csuba_head: \forall (c1: C).(\forall (c2: C).((csuba g c1 c2) \to (\forall -(k: K).(\forall (u: T).(csuba g (CHead c1 k u) (CHead c2 k u)))))) -| csuba_void: \forall (c1: C).(\forall (c2: C).((csuba g c1 c2) \to (\forall -(b: B).((not (eq B b Void)) \to (\forall (u1: T).(\forall (u2: T).(csuba g -(CHead c1 (Bind Void) u1) (CHead c2 (Bind b) u2)))))))) -| csuba_abst: \forall (c1: C).(\forall (c2: C).((csuba g c1 c2) \to (\forall -(t: T).(\forall (a: A).((arity g c1 t (asucc g a)) \to (\forall (u: -T).((arity g c2 u a) \to (csuba g (CHead c1 (Bind Abst) t) (CHead c2 (Bind -Abbr) u))))))))). - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/drop.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/drop.ma deleted file mode 100644 index 1047ac13d..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/drop.ma +++ /dev/null @@ -1,2468 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csuba/fwd.ma". - -include "Basic-1/drop/fwd.ma". - -theorem csuba_drop_abbr: - \forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).((drop i -O c1 (CHead d1 (Bind Abbr) u)) \to (\forall (g: G).(\forall (c2: C).((csuba g -c1 c2) \to (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr) u))) -(\lambda (d2: C).(csuba g d1 d2)))))))))) -\def - \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (d1: -C).(\forall (u: T).((drop n O c1 (CHead d1 (Bind Abbr) u)) \to (\forall (g: -G).(\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(drop n O c2 -(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))))))))) -(\lambda (c1: C).(\lambda (d1: C).(\lambda (u: T).(\lambda (H: (drop O O c1 -(CHead d1 (Bind Abbr) u))).(\lambda (g: G).(\lambda (c2: C).(\lambda (H0: -(csuba g c1 c2)).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csuba g c c2)) H0 -(CHead d1 (Bind Abbr) u) (drop_gen_refl c1 (CHead d1 (Bind Abbr) u) H)) in -(let H_x \def (csuba_gen_abbr g d1 c2 u H1) in (let H2 \def H_x in (ex2_ind C -(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba -g d1 d2)) (ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abbr) u))) -(\lambda (d2: C).(csuba g d1 d2))) (\lambda (x: C).(\lambda (H3: (eq C c2 -(CHead x (Bind Abbr) u))).(\lambda (H4: (csuba g d1 x)).(eq_ind_r C (CHead x -(Bind Abbr) u) (\lambda (c: C).(ex2 C (\lambda (d2: C).(drop O O c (CHead d2 -(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (ex_intro2 C (\lambda -(d2: C).(drop O O (CHead x (Bind Abbr) u) (CHead d2 (Bind Abbr) u))) (\lambda -(d2: C).(csuba g d1 d2)) x (drop_refl (CHead x (Bind Abbr) u)) H4) c2 H3)))) -H2))))))))))) (\lambda (n: nat).(\lambda (H: ((\forall (c1: C).(\forall (d1: -C).(\forall (u: T).((drop n O c1 (CHead d1 (Bind Abbr) u)) \to (\forall (g: -G).(\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(drop n O c2 -(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 -d2)))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (d1: -C).(\forall (u: T).((drop (S n) O c (CHead d1 (Bind Abbr) u)) \to (\forall -(g: G).(\forall (c2: C).((csuba g c c2) \to (ex2 C (\lambda (d2: C).(drop (S -n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))))))) -(\lambda (n0: nat).(\lambda (d1: C).(\lambda (u: T).(\lambda (H0: (drop (S n) -O (CSort n0) (CHead d1 (Bind Abbr) u))).(\lambda (g: G).(\lambda (c2: -C).(\lambda (_: (csuba g (CSort n0) c2)).(and3_ind (eq C (CHead d1 (Bind -Abbr) u) (CSort n0)) (eq nat (S n) O) (eq nat O O) (ex2 C (\lambda (d2: -C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 -d2))) (\lambda (_: (eq C (CHead d1 (Bind Abbr) u) (CSort n0))).(\lambda (H3: -(eq nat (S n) O)).(\lambda (_: (eq nat O O)).(let H5 \def (eq_ind nat (S n) -(\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O -\Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind (ex2 C -(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: -C).(csuba g d1 d2))) H5))))) (drop_gen_sort n0 (S n) O (CHead d1 (Bind Abbr) -u) H0))))))))) (\lambda (c: C).(\lambda (H0: ((\forall (d1: C).(\forall (u: -T).((drop (S n) O c (CHead d1 (Bind Abbr) u)) \to (\forall (g: G).(\forall -(c2: C).((csuba g c c2) \to (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead -d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))))))))).(\lambda (k: -K).(\lambda (t: T).(\lambda (d1: C).(\lambda (u: T).(\lambda (H1: (drop (S n) -O (CHead c k t) (CHead d1 (Bind Abbr) u))).(\lambda (g: G).(\lambda (c2: -C).(\lambda (H2: (csuba g (CHead c k t) c2)).(K_ind (\lambda (k0: K).((csuba -g (CHead c k0 t) c2) \to ((drop (r k0 n) O c (CHead d1 (Bind Abbr) u)) \to -(ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda -(d2: C).(csuba g d1 d2)))))) (\lambda (b: B).(\lambda (H3: (csuba g (CHead c -(Bind b) t) c2)).(\lambda (H4: (drop (r (Bind b) n) O c (CHead d1 (Bind Abbr) -u))).(B_ind (\lambda (b0: B).((csuba g (CHead c (Bind b0) t) c2) \to ((drop -(r (Bind b0) n) O c (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: -C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 -d2)))))) (\lambda (H5: (csuba g (CHead c (Bind Abbr) t) c2)).(\lambda (H6: -(drop (r (Bind Abbr) n) O c (CHead d1 (Bind Abbr) u))).(let H_x \def -(csuba_gen_abbr g c c2 t H5) in (let H7 \def H_x in (ex2_ind C (\lambda (d2: -C).(eq C c2 (CHead d2 (Bind Abbr) t))) (\lambda (d2: C).(csuba g c d2)) (ex2 -C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: -C).(csuba g d1 d2))) (\lambda (x: C).(\lambda (H8: (eq C c2 (CHead x (Bind -Abbr) t))).(\lambda (H9: (csuba g c x)).(eq_ind_r C (CHead x (Bind Abbr) t) -(\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind -Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H10 \def (H c d1 u H6 g x -H9) in (ex2_ind C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u))) -(\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(drop (S n) O -(CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g -d1 d2))) (\lambda (x0: C).(\lambda (H11: (drop n O x (CHead x0 (Bind Abbr) -u))).(\lambda (H12: (csuba g d1 x0)).(let H13 \def (refl_equal nat (r (Bind -Abbr) n)) in (let H14 \def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x -(CHead x0 (Bind Abbr) u))) H11 (r (Bind Abbr) n) H13) in (ex_intro2 C -(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) -u))) (\lambda (d2: C).(csuba g d1 d2)) x0 (drop_drop (Bind Abbr) n x (CHead -x0 (Bind Abbr) u) H14 t) H12)))))) H10)) c2 H8)))) H7))))) (\lambda (H5: -(csuba g (CHead c (Bind Abst) t) c2)).(\lambda (H6: (drop (r (Bind Abst) n) O -c (CHead d1 (Bind Abbr) u))).(let H_x \def (csuba_gen_abst g c c2 t H5) in -(let H7 \def H_x in (or_ind (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind -Abst) t))) (\lambda (d2: C).(csuba g c d2))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g c d2)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex2 C -(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: -C).(csuba g d1 d2))) (\lambda (H8: (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 -(Bind Abst) t))) (\lambda (d2: C).(csuba g c d2)))).(ex2_ind C (\lambda (d2: -C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba g c d2)) (ex2 -C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: -C).(csuba g d1 d2))) (\lambda (x: C).(\lambda (H9: (eq C c2 (CHead x (Bind -Abst) t))).(\lambda (H10: (csuba g c x)).(eq_ind_r C (CHead x (Bind Abst) t) -(\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind -Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H11 \def (H c d1 u H6 g x -H10) in (ex2_ind C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u))) -(\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(drop (S n) O -(CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g -d1 d2))) (\lambda (x0: C).(\lambda (H12: (drop n O x (CHead x0 (Bind Abbr) -u))).(\lambda (H13: (csuba g d1 x0)).(let H14 \def (refl_equal nat (r (Bind -Abbr) n)) in (let H15 \def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x -(CHead x0 (Bind Abbr) u))) H12 (r (Bind Abbr) n) H14) in (ex_intro2 C -(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) -u))) (\lambda (d2: C).(csuba g d1 d2)) x0 (drop_drop (Bind Abst) n x (CHead -x0 (Bind Abbr) u) H15 t) H13)))))) H11)) c2 H9)))) H8)) (\lambda (H8: (ex4_3 -C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 -(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -c d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t (asucc -g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g c d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g c t (asucc g a))))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 a)))) (ex2 C (\lambda (d2: C).(drop (S n) O -c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda -(x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H9: (eq C c2 (CHead x0 -(Bind Abbr) x1))).(\lambda (H10: (csuba g c x0)).(\lambda (_: (arity g c t -(asucc g x2))).(\lambda (_: (arity g x0 x1 x2)).(eq_ind_r C (CHead x0 (Bind -Abbr) x1) (\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 -(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H13 \def (H c d1 u -H6 g x0 H10) in (ex2_ind C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind -Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(drop (S -n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abbr) u))) (\lambda (d2: -C).(csuba g d1 d2))) (\lambda (x: C).(\lambda (H14: (drop n O x0 (CHead x -(Bind Abbr) u))).(\lambda (H15: (csuba g d1 x)).(let H16 \def (refl_equal nat -(r (Bind Abbr) n)) in (let H17 \def (eq_ind nat n (\lambda (n0: nat).(drop n0 -O x0 (CHead x (Bind Abbr) u))) H14 (r (Bind Abbr) n) H16) in (ex_intro2 C -(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind -Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x (drop_drop (Bind Abbr) n x0 -(CHead x (Bind Abbr) u) H17 x1) H15)))))) H13)) c2 H9)))))))) H8)) H7))))) -(\lambda (H5: (csuba g (CHead c (Bind Void) t) c2)).(\lambda (H6: (drop (r -(Bind Void) n) O c (CHead d1 (Bind Abbr) u))).(let H_x \def (csuba_gen_void g -c c2 t H5) in (let H7 \def H_x in (ex2_3_ind B C T (\lambda (b0: B).(\lambda -(d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind b0) u2))))) (\lambda (_: -B).(\lambda (d2: C).(\lambda (_: T).(csuba g c d2)))) (ex2 C (\lambda (d2: -C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 -d2))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H8: (eq C -c2 (CHead x1 (Bind x0) x2))).(\lambda (H9: (csuba g c x1)).(eq_ind_r C (CHead -x1 (Bind x0) x2) (\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0 -(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H10 \def -(H c d1 u H6 g x1 H9) in (ex2_ind C (\lambda (d2: C).(drop n O x1 (CHead d2 -(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abbr) u))) (\lambda -(d2: C).(csuba g d1 d2))) (\lambda (x: C).(\lambda (H11: (drop n O x1 (CHead -x (Bind Abbr) u))).(\lambda (H12: (csuba g d1 x)).(let H13 \def (refl_equal -nat (r (Bind Abbr) n)) in (let H14 \def (eq_ind nat n (\lambda (n0: -nat).(drop n0 O x1 (CHead x (Bind Abbr) u))) H11 (r (Bind Abbr) n) H13) in -(ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2 -(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x (drop_drop (Bind x0) n -x1 (CHead x (Bind Abbr) u) H14 x2) H12)))))) H10)) c2 H8)))))) H7))))) b H3 -H4)))) (\lambda (f: F).(\lambda (H3: (csuba g (CHead c (Flat f) t) -c2)).(\lambda (H4: (drop (r (Flat f) n) O c (CHead d1 (Bind Abbr) u))).(let -H_x \def (csuba_gen_flat g c c2 t f H3) in (let H5 \def H_x in (ex2_2_ind C T -(\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Flat f) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g c d2))) (ex2 C (\lambda (d2: C).(drop (S n) -O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda -(x0: C).(\lambda (x1: T).(\lambda (H6: (eq C c2 (CHead x0 (Flat f) -x1))).(\lambda (H7: (csuba g c x0)).(eq_ind_r C (CHead x0 (Flat f) x1) -(\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind -Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H8 \def (H0 d1 u H4 g x0 -H7) in (ex2_ind C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind Abbr) -u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(drop (S n) O -(CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g -d1 d2))) (\lambda (x: C).(\lambda (H9: (drop (S n) O x0 (CHead x (Bind Abbr) -u))).(\lambda (H10: (csuba g d1 x)).(ex_intro2 C (\lambda (d2: C).(drop (S n) -O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g -d1 d2)) x (drop_drop (Flat f) n x0 (CHead x (Bind Abbr) u) H9 x1) H10)))) -H8)) c2 H6))))) H5)))))) k H2 (drop_gen_drop k c (CHead d1 (Bind Abbr) u) t n -H1)))))))))))) c1)))) i). -(* COMMENTS -Initial nodes: 3648 -END *) - -theorem csuba_drop_abst: - \forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u1: T).((drop i -O c1 (CHead d1 (Bind Abst) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba -g c1 c2) \to (or (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst) -u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a))))))))))))) -\def - \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (d1: -C).(\forall (u1: T).((drop n O c1 (CHead d1 (Bind Abst) u1)) \to (\forall (g: -G).(\forall (c2: C).((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(drop n -O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C -T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O c2 (CHead d2 -(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 -(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 -u2 a)))))))))))))) (\lambda (c1: C).(\lambda (d1: C).(\lambda (u1: -T).(\lambda (H: (drop O O c1 (CHead d1 (Bind Abst) u1))).(\lambda (g: -G).(\lambda (c2: C).(\lambda (H0: (csuba g c1 c2)).(let H1 \def (eq_ind C c1 -(\lambda (c: C).(csuba g c c2)) H0 (CHead d1 (Bind Abst) u1) (drop_gen_refl -c1 (CHead d1 (Bind Abst) u1) H)) in (let H_x \def (csuba_gen_abst g d1 c2 u1 -H1) in (let H2 \def H_x in (or_ind (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 -(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a))))) (or (ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) u1))) -(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(drop O O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (H3: -(ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: -C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind -Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: -C).(drop O O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 -d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O -O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: -A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda -(a: A).(arity g d2 u2 a)))))) (\lambda (x: C).(\lambda (H4: (eq C c2 (CHead x -(Bind Abst) u1))).(\lambda (H5: (csuba g d1 x)).(eq_ind_r C (CHead x (Bind -Abst) u1) (\lambda (c: C).(or (ex2 C (\lambda (d2: C).(drop O O c (CHead d2 -(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O c (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a))))))) (or_introl (ex2 C (\lambda (d2: C).(drop O O (CHead x (Bind Abst) -u1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T -A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O (CHead x (Bind -Abst) u1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: -C).(drop O O (CHead x (Bind Abst) u1) (CHead d2 (Bind Abst) u1))) (\lambda -(d2: C).(csuba g d1 d2)) x (drop_refl (CHead x (Bind Abst) u1)) H5)) c2 -H4)))) H3)) (\lambda (H3: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))).(ex4_3_ind C -T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 -(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 -(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 -u2 a)))) (or (ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) u1))) -(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(drop O O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H4: (eq C c2 (CHead x0 (Bind -Abbr) x1))).(\lambda (H5: (csuba g d1 x0)).(\lambda (H6: (arity g d1 u1 -(asucc g x2))).(\lambda (H7: (arity g x0 x1 x2)).(eq_ind_r C (CHead x0 (Bind -Abbr) x1) (\lambda (c: C).(or (ex2 C (\lambda (d2: C).(drop O O c (CHead d2 -(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O c (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a))))))) (or_intror (ex2 C (\lambda (d2: C).(drop O O (CHead x0 (Bind Abbr) -x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T -A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O (CHead x0 (Bind -Abbr) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O (CHead x0 (Bind Abbr) x1) -(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity -g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 a)))) x0 x1 x2 (drop_refl (CHead x0 (Bind Abbr) x1)) H5 H6 -H7)) c2 H4)))))))) H3)) H2))))))))))) (\lambda (n: nat).(\lambda (H: -((\forall (c1: C).(\forall (d1: C).(\forall (u1: T).((drop n O c1 (CHead d1 -(Bind Abst) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba g c1 c2) \to -(or (ex2 C (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abst) u1))) (\lambda -(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop n O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a))))))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (d1: -C).(\forall (u1: T).((drop (S n) O c (CHead d1 (Bind Abst) u1)) \to (\forall -(g: G).(\forall (c2: C).((csuba g c c2) \to (or (ex2 C (\lambda (d2: C).(drop -(S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O -c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: -A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda -(a: A).(arity g d2 u2 a))))))))))))) (\lambda (n0: nat).(\lambda (d1: -C).(\lambda (u1: T).(\lambda (H0: (drop (S n) O (CSort n0) (CHead d1 (Bind -Abst) u1))).(\lambda (g: G).(\lambda (c2: C).(\lambda (_: (csuba g (CSort n0) -c2)).(and3_ind (eq C (CHead d1 (Bind Abst) u1) (CSort n0)) (eq nat (S n) O) -(eq nat O O) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind -Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))) (\lambda (_: (eq C (CHead d1 (Bind Abst) u1) (CSort n0))).(\lambda -(H3: (eq nat (S n) O)).(\lambda (_: (eq nat O O)).(let H5 \def (eq_ind nat (S -n) (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O -\Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind (or (ex2 -C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: -C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) H5))))) -(drop_gen_sort n0 (S n) O (CHead d1 (Bind Abst) u1) H0))))))))) (\lambda (c: -C).(\lambda (H0: ((\forall (d1: C).(\forall (u1: T).((drop (S n) O c (CHead -d1 (Bind Abst) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba g c c2) \to -(or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) -(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (d1: C).(\lambda -(u1: T).(\lambda (H1: (drop (S n) O (CHead c k t) (CHead d1 (Bind Abst) -u1))).(\lambda (g: G).(\lambda (c2: C).(\lambda (H2: (csuba g (CHead c k t) -c2)).(K_ind (\lambda (k0: K).((csuba g (CHead c k0 t) c2) \to ((drop (r k0 n) -O c (CHead d1 (Bind Abst) u1)) \to (or (ex2 C (\lambda (d2: C).(drop (S n) O -c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T -A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead -d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity -g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 a))))))))) (\lambda (b: B).(\lambda (H3: (csuba g (CHead c -(Bind b) t) c2)).(\lambda (H4: (drop (r (Bind b) n) O c (CHead d1 (Bind Abst) -u1))).(B_ind (\lambda (b0: B).((csuba g (CHead c (Bind b0) t) c2) \to ((drop -(r (Bind b0) n) O c (CHead d1 (Bind Abst) u1)) \to (or (ex2 C (\lambda (d2: -C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 -d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S -n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))) (\lambda (H5: (csuba g -(CHead c (Bind Abbr) t) c2)).(\lambda (H6: (drop (r (Bind Abbr) n) O c (CHead -d1 (Bind Abst) u1))).(let H_x \def (csuba_gen_abbr g c c2 t H5) in (let H7 -\def H_x in (ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) -(\lambda (d2: C).(csuba g c d2)) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 -(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 -(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 -(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 -u2 a)))))) (\lambda (x: C).(\lambda (H8: (eq C c2 (CHead x (Bind Abbr) -t))).(\lambda (H9: (csuba g c x)).(eq_ind_r C (CHead x (Bind Abbr) t) -(\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind -Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a))))))) (let H10 \def (H c d1 u1 H6 g x H9) in (or_ind (ex2 C (\lambda (d2: -C).(drop n O x (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x -(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity -g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x -(Bind Abbr) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O -(CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda -(H11: (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) u1))) -(\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(drop n O x -(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C -(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) -u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t) -(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity -g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 a)))))) (\lambda (x0: C).(\lambda (H12: (drop n O x (CHead -x0 (Bind Abst) u1))).(\lambda (H13: (csuba g d1 x0)).(let H14 \def -(refl_equal nat (r (Bind Abbr) n)) in (let H15 \def (eq_ind nat n (\lambda -(n0: nat).(drop n0 O x (CHead x0 (Bind Abst) u1))) H12 (r (Bind Abbr) n) H14) -in (or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) -(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x -(Bind Abbr) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: -C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u1))) (\lambda -(d2: C).(csuba g d1 d2)) x0 (drop_drop (Bind Abbr) n x (CHead x0 (Bind Abst) -u1) H15 t) H13))))))) H11)) (\lambda (H11: (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abbr) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop n O x (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u1))) (\lambda -(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind -Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 -d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc -g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H12: -(drop n O x (CHead x0 (Bind Abbr) x1))).(\lambda (H13: (csuba g d1 -x0)).(\lambda (H14: (arity g d1 u1 (asucc g x2))).(\lambda (H15: (arity g x0 -x1 x2)).(let H16 \def (refl_equal nat (r (Bind Abbr) n)) in (let H17 \def -(eq_ind nat n (\lambda (n0: nat).(drop n0 O x (CHead x0 (Bind Abbr) x1))) H12 -(r (Bind Abbr) n) H16) in (or_intror (ex2 C (\lambda (d2: C).(drop (S n) O -(CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g -d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop -(S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C -T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x -(Bind Abbr) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x0 x1 x2 (drop_drop (Bind Abbr) -n x (CHead x0 (Bind Abbr) x1) H17 t) H13 H14 H15))))))))))) H11)) H10)) c2 -H8)))) H7))))) (\lambda (H5: (csuba g (CHead c (Bind Abst) t) c2)).(\lambda -(H6: (drop (r (Bind Abst) n) O c (CHead d1 (Bind Abst) u1))).(let H_x \def -(csuba_gen_abst g c c2 t H5) in (let H7 \def H_x in (or_ind (ex2 C (\lambda -(d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba g c d2))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 -(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g c d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g -c t (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity -g d2 u2 a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind -Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))) (\lambda (H8: (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) -t))) (\lambda (d2: C).(csuba g c d2)))).(ex2_ind C (\lambda (d2: C).(eq C c2 -(CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba g c d2)) (or (ex2 C -(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: -C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x: -C).(\lambda (H9: (eq C c2 (CHead x (Bind Abst) t))).(\lambda (H10: (csuba g c -x)).(eq_ind_r C (CHead x (Bind Abst) t) (\lambda (c0: C).(or (ex2 C (\lambda -(d2: C).(drop (S n) O c0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba -g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop (S n) O c0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda -(_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a))))))) (let H11 \def (H c d1 u1 H6 g -x H10) in (or_ind (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) -u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abbr) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (or -(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind -Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) -(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity -g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 a)))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x -(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C -(\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) u1))) (\lambda (d2: -C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind -Abst) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O -(CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x0: -C).(\lambda (H13: (drop n O x (CHead x0 (Bind Abst) u1))).(\lambda (H14: -(csuba g d1 x0)).(let H15 \def (refl_equal nat (r (Bind Abbr) n)) in (let H16 -\def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x (CHead x0 (Bind Abst) -u1))) H13 (r (Bind Abbr) n) H15) in (or_introl (ex2 C (\lambda (d2: C).(drop -(S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: -C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) -(ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 -(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) x0 (drop_drop (Bind Abst) -n x (CHead x0 (Bind Abst) u1) H16 t) H14))))))) H12)) (\lambda (H12: (ex4_3 C -T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 -(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 -(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 -u2 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop n O x (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u1))) (\lambda -(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind -Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 -d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc -g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H13: -(drop n O x (CHead x0 (Bind Abbr) x1))).(\lambda (H14: (csuba g d1 -x0)).(\lambda (H15: (arity g d1 u1 (asucc g x2))).(\lambda (H16: (arity g x0 -x1 x2)).(let H17 \def (refl_equal nat (r (Bind Abbr) n)) in (let H18 \def -(eq_ind nat n (\lambda (n0: nat).(drop n0 O x (CHead x0 (Bind Abbr) x1))) H13 -(r (Bind Abbr) n) H17) in (or_intror (ex2 C (\lambda (d2: C).(drop (S n) O -(CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g -d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop -(S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C -T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x -(Bind Abst) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x0 x1 x2 (drop_drop (Bind Abst) -n x (CHead x0 (Bind Abbr) x1) H18 t) H14 H15 H16))))))))))) H12)) H11)) c2 -H9)))) H8)) (\lambda (H8: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g c d2)))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g c t (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))).(ex4_3_ind C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g c d2)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C -(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: -C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H9: (eq C c2 (CHead x0 (Bind -Abbr) x1))).(\lambda (H10: (csuba g c x0)).(\lambda (_: (arity g c t (asucc g -x2))).(\lambda (_: (arity g x0 x1 x2)).(eq_ind_r C (CHead x0 (Bind Abbr) x1) -(\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind -Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a))))))) (let H13 \def (H c d1 u1 H6 g x0 H10) in (or_ind (ex2 C (\lambda -(d2: C).(drop n O x0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 -d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n -O x0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: -A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda -(a: A).(arity g d2 u2 a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead -x0 (Bind Abbr) x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 -d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S -n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda -(H14: (ex2 C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind Abst) u1))) -(\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(drop n O x0 -(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C -(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind -Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abbr) x1) -(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity -g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 a)))))) (\lambda (x: C).(\lambda (H15: (drop n O x0 (CHead -x (Bind Abst) u1))).(\lambda (H16: (csuba g d1 x)).(let H17 \def (refl_equal -nat (r (Bind Abbr) n)) in (let H18 \def (eq_ind nat n (\lambda (n0: -nat).(drop n0 O x0 (CHead x (Bind Abst) u1))) H15 (r (Bind Abbr) n) H17) in -(or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abbr) x1) -(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 -(Bind Abbr) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abst) u1))) -(\lambda (d2: C).(csuba g d1 d2)) x (drop_drop (Bind Abbr) n x0 (CHead x -(Bind Abst) u1) H18 x1) H16))))))) H14)) (\lambda (H14: (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x0 (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop n O x0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abst) u1))) -(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 -(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 -(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 -u2 a)))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: A).(\lambda (H15: -(drop n O x0 (CHead x3 (Bind Abbr) x4))).(\lambda (H16: (csuba g d1 -x3)).(\lambda (H17: (arity g d1 u1 (asucc g x5))).(\lambda (H18: (arity g x3 -x4 x5)).(let H19 \def (refl_equal nat (r (Bind Abbr) n)) in (let H20 \def -(eq_ind nat n (\lambda (n0: nat).(drop n0 O x0 (CHead x3 (Bind Abbr) x4))) -H15 (r (Bind Abbr) n) H19) in (or_intror (ex2 C (\lambda (d2: C).(drop (S n) -O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: -C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abbr) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) -(ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S -n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x3 x4 x5 -(drop_drop (Bind Abbr) n x0 (CHead x3 (Bind Abbr) x4) H20 x1) H16 H17 -H18))))))))))) H14)) H13)) c2 H9)))))))) H8)) H7))))) (\lambda (H5: (csuba g -(CHead c (Bind Void) t) c2)).(\lambda (H6: (drop (r (Bind Void) n) O c (CHead -d1 (Bind Abst) u1))).(let H_x \def (csuba_gen_void g c c2 t H5) in (let H7 -\def H_x in (ex2_3_ind B C T (\lambda (b0: B).(\lambda (d2: C).(\lambda (u2: -T).(eq C c2 (CHead d2 (Bind b0) u2))))) (\lambda (_: B).(\lambda (d2: -C).(\lambda (_: T).(csuba g c d2)))) (or (ex2 C (\lambda (d2: C).(drop (S n) -O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C -T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead -d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity -g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 a)))))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: -T).(\lambda (H8: (eq C c2 (CHead x1 (Bind x0) x2))).(\lambda (H9: (csuba g c -x1)).(eq_ind_r C (CHead x1 (Bind x0) x2) (\lambda (c0: C).(or (ex2 C (\lambda -(d2: C).(drop (S n) O c0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba -g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop (S n) O c0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda -(_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a))))))) (let H10 \def (H c d1 u1 H6 g -x1 H9) in (or_ind (ex2 C (\lambda (d2: C).(drop n O x1 (CHead d2 (Bind Abst) -u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop n O x1 (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x1 (Bind x0) x2) -(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x1 -(Bind x0) x2) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (H11: (ex2 C (\lambda -(d2: C).(drop n O x1 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 -d2)))).(ex2_ind C (\lambda (d2: C).(drop n O x1 (CHead d2 (Bind Abst) u1))) -(\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: C).(drop (S n) O -(CHead x1 (Bind x0) x2) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g -d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop -(S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x: -C).(\lambda (H12: (drop n O x1 (CHead x (Bind Abst) u1))).(\lambda (H13: -(csuba g d1 x)).(let H14 \def (refl_equal nat (r (Bind Abbr) n)) in (let H15 -\def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x1 (CHead x (Bind Abst) -u1))) H12 (r (Bind Abbr) n) H14) in (or_introl (ex2 C (\lambda (d2: C).(drop -(S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abst) u1))) (\lambda (d2: -C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abbr) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) -(ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2 -(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) x (drop_drop (Bind x0) n -x1 (CHead x (Bind Abst) u1) H15 x2) H13))))))) H11)) (\lambda (H11: (ex4_3 C -T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x1 (CHead d2 -(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 -(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 -u2 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop n O x1 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abst) u1))) (\lambda -(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind -Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 -d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc -g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: A).(\lambda (H12: -(drop n O x1 (CHead x3 (Bind Abbr) x4))).(\lambda (H13: (csuba g d1 -x3)).(\lambda (H14: (arity g d1 u1 (asucc g x5))).(\lambda (H15: (arity g x3 -x4 x5)).(let H16 \def (refl_equal nat (r (Bind Abbr) n)) in (let H17 \def -(eq_ind nat n (\lambda (n0: nat).(drop n0 O x1 (CHead x3 (Bind Abbr) x4))) -H12 (r (Bind Abbr) n) H16) in (or_intror (ex2 C (\lambda (d2: C).(drop (S n) -O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba -g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abbr) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) -(ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S -n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x3 x4 x5 -(drop_drop (Bind x0) n x1 (CHead x3 (Bind Abbr) x4) H17 x2) H13 H14 -H15))))))))))) H11)) H10)) c2 H8)))))) H7))))) b H3 H4)))) (\lambda (f: -F).(\lambda (H3: (csuba g (CHead c (Flat f) t) c2)).(\lambda (H4: (drop (r -(Flat f) n) O c (CHead d1 (Bind Abst) u1))).(let H_x \def (csuba_gen_flat g c -c2 t f H3) in (let H5 \def H_x in (ex2_2_ind C T (\lambda (d2: C).(\lambda -(u2: T).(eq C c2 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g c d2))) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 -(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind -Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 -d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc -g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C c2 (CHead x0 -(Flat f) x1))).(\lambda (H7: (csuba g c x0)).(eq_ind_r C (CHead x0 (Flat f) -x1) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 -(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind -Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 -d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc -g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a))))))) (let H8 \def (H0 d1 u1 H4 g x0 H7) in (or_ind (ex2 C (\lambda (d2: -C).(drop (S n) O x0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 -d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S -n) O x0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (or (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u1))) (\lambda -(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))) (\lambda (H9: (ex2 C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 -(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda -(d2: C).(drop (S n) O x0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba -g d1 d2)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) -(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 -(Flat f) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x: C).(\lambda (H10: -(drop (S n) O x0 (CHead x (Bind Abst) u1))).(\lambda (H11: (csuba g d1 -x)).(or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) -(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 -(Flat f) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u1))) (\lambda -(d2: C).(csuba g d1 d2)) x (drop_drop (Flat f) n x0 (CHead x (Bind Abst) u1) -H10 x1) H11))))) H9)) (\lambda (H9: (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(drop (S n) O x0 (CHead d2 (Bind Abbr) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop (S n) O x0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda -(_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u1))) (\lambda -(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))) (\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: A).(\lambda (H10: -(drop (S n) O x0 (CHead x2 (Bind Abbr) x3))).(\lambda (H11: (csuba g d1 -x2)).(\lambda (H12: (arity g d1 u1 (asucc g x4))).(\lambda (H13: (arity g x2 -x3 x4)).(or_intror (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) -x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T -A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 -(Flat f) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) -(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity -g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 a)))) x2 x3 x4 (drop_drop (Flat f) n x0 (CHead x2 (Bind -Abbr) x3) H10 x1) H11 H12 H13))))))))) H9)) H8)) c2 H6))))) H5)))))) k H2 -(drop_gen_drop k c (CHead d1 (Bind Abst) u1) t n H1)))))))))))) c1)))) i). -(* COMMENTS -Initial nodes: 12528 -END *) - -theorem csuba_drop_abst_rev: - \forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).((drop i -O c1 (CHead d1 (Bind Abst) u)) \to (\forall (g: G).(\forall (c2: C).((csuba g -c2 c1) \to (or (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst) -u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda -(u2: T).(drop i O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda -(_: T).(csuba g d2 d1)))))))))))) -\def - \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (d1: -C).(\forall (u: T).((drop n O c1 (CHead d1 (Bind Abst) u)) \to (\forall (g: -G).(\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: C).(drop n -O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop n O c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))))) (\lambda (c1: -C).(\lambda (d1: C).(\lambda (u: T).(\lambda (H: (drop O O c1 (CHead d1 (Bind -Abst) u))).(\lambda (g: G).(\lambda (c2: C).(\lambda (H0: (csuba g c2 -c1)).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csuba g c2 c)) H0 (CHead d1 -(Bind Abst) u) (drop_gen_refl c1 (CHead d1 (Bind Abst) u) H)) in (let H_x -\def (csuba_gen_abst_rev g d1 c2 u H1) in (let H2 \def H_x in (or_ind (ex2 C -(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba -g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or -(ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O -c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))) (\lambda (H3: (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) -u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(eq C c2 -(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C -(\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O -c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))) (\lambda (x: C).(\lambda (H4: (eq C c2 (CHead x (Bind Abst) -u))).(\lambda (H5: (csuba g x d1)).(eq_ind_r C (CHead x (Bind Abst) u) -(\lambda (c: C).(or (ex2 C (\lambda (d2: C).(drop O O c (CHead d2 (Bind Abst) -u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda -(u2: T).(drop O O c (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda -(_: T).(csuba g d2 d1)))))) (or_introl (ex2 C (\lambda (d2: C).(drop O O -(CHead x (Bind Abst) u) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g -d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O (CHead x -(Bind Abst) u) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop O O (CHead x (Bind -Abst) u) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x -(drop_refl (CHead x (Bind Abst) u)) H5)) c2 H4)))) H3)) (\lambda (H3: (ex2_2 -C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda -(d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1))) (or (ex2 C (\lambda (d2: C).(drop O O c2 -(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop O O c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (H4: (eq C c2 (CHead x0 (Bind Void) -x1))).(\lambda (H5: (csuba g x0 d1)).(eq_ind_r C (CHead x0 (Bind Void) x1) -(\lambda (c: C).(or (ex2 C (\lambda (d2: C).(drop O O c (CHead d2 (Bind Abst) -u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda -(u2: T).(drop O O c (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda -(_: T).(csuba g d2 d1)))))) (or_intror (ex2 C (\lambda (d2: C).(drop O O -(CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba -g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O (CHead x0 -(Bind Void) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: -T).(drop O O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1 (drop_refl (CHead x0 (Bind -Void) x1)) H5)) c2 H4))))) H3)) H2))))))))))) (\lambda (n: nat).(\lambda (H: -((\forall (c1: C).(\forall (d1: C).(\forall (u: T).((drop n O c1 (CHead d1 -(Bind Abst) u)) \to (\forall (g: G).(\forall (c2: C).((csuba g c2 c1) \to (or -(ex2 C (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O -c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (d1: -C).(\forall (u: T).((drop (S n) O c (CHead d1 (Bind Abst) u)) \to (\forall -(g: G).(\forall (c2: C).((csuba g c2 c) \to (or (ex2 C (\lambda (d2: C).(drop -(S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))))))) -(\lambda (n0: nat).(\lambda (d1: C).(\lambda (u: T).(\lambda (H0: (drop (S n) -O (CSort n0) (CHead d1 (Bind Abst) u))).(\lambda (g: G).(\lambda (c2: -C).(\lambda (_: (csuba g c2 (CSort n0))).(and3_ind (eq C (CHead d1 (Bind -Abst) u) (CSort n0)) (eq nat (S n) O) (eq nat O O) (or (ex2 C (\lambda (d2: -C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 -d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) -(\lambda (_: (eq C (CHead d1 (Bind Abst) u) (CSort n0))).(\lambda (H3: (eq -nat (S n) O)).(\lambda (_: (eq nat O O)).(let H5 \def (eq_ind nat (S n) -(\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O -\Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind (or (ex2 -C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) -O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g -d2 d1))))) H5))))) (drop_gen_sort n0 (S n) O (CHead d1 (Bind Abst) u) -H0))))))))) (\lambda (c: C).(\lambda (H0: ((\forall (d1: C).(\forall (u: -T).((drop (S n) O c (CHead d1 (Bind Abst) u)) \to (\forall (g: G).(\forall -(c2: C).((csuba g c2 c) \to (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 -(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))))).(\lambda -(k: K).(\lambda (t: T).(\lambda (d1: C).(\lambda (u: T).(\lambda (H1: (drop -(S n) O (CHead c k t) (CHead d1 (Bind Abst) u))).(\lambda (g: G).(\lambda -(c2: C).(\lambda (H2: (csuba g c2 (CHead c k t))).(K_ind (\lambda (k0: -K).((csuba g c2 (CHead c k0 t)) \to ((drop (r k0 n) O c (CHead d1 (Bind Abst) -u)) \to (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) -u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda -(u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))))))) (\lambda (b: B).(\lambda (H3: -(csuba g c2 (CHead c (Bind b) t))).(\lambda (H4: (drop (r (Bind b) n) O c -(CHead d1 (Bind Abst) u))).(B_ind (\lambda (b0: B).((csuba g c2 (CHead c -(Bind b0) t)) \to ((drop (r (Bind b0) n) O c (CHead d1 (Bind Abst) u)) \to -(or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda -(_: T).(csuba g d2 d1)))))))) (\lambda (H5: (csuba g c2 (CHead c (Bind Abbr) -t))).(\lambda (H6: (drop (r (Bind Abbr) n) O c (CHead d1 (Bind Abst) -u))).(let H_x \def (csuba_gen_abbr_rev g c c2 t H5) in (let H7 \def H_x in -(or3_ind (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) (\lambda -(d2: C).(csuba g d2 c))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 c)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 c)))) (or (ex2 C (\lambda (d2: -C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 -d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) -(\lambda (H8: (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) -(\lambda (d2: C).(csuba g d2 c)))).(ex2_ind C (\lambda (d2: C).(eq C c2 -(CHead d2 (Bind Abbr) t))) (\lambda (d2: C).(csuba g d2 c)) (or (ex2 C -(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) -O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g -d2 d1))))) (\lambda (x: C).(\lambda (H9: (eq C c2 (CHead x (Bind Abbr) -t))).(\lambda (H10: (csuba g x c)).(eq_ind_r C (CHead x (Bind Abbr) t) -(\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind -Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 (Bind Void) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H11 \def (H c d1 u H6 g x -H10) in (or_ind (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) -u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda -(u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda -(_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x -(Bind Abbr) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind -Abbr) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba -g d2 d1))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 -(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: -C).(drop n O x (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) -(or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 -(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda -(x0: C).(\lambda (H13: (drop n O x (CHead x0 (Bind Abst) u))).(\lambda (H14: -(csuba g x0 d1)).(or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x -(Bind Abbr) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind -Abbr) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba -g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) -t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x0 (drop_drop -(Bind Abbr) n x (CHead x0 (Bind Abst) u) H13 t) H14))))) H12)) (\lambda (H12: -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind -C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or (ex2 C (\lambda -(d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (H13: (drop n O x (CHead x0 (Bind Void) -x1))).(\lambda (H14: (csuba g x0 d1)).(or_intror (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u))) (\lambda -(d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop -(S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1 -(drop_drop (Bind Abbr) n x (CHead x0 (Bind Void) x1) H13 t) H14)))))) H12)) -H11)) c2 H9)))) H8)) (\lambda (H8: (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 c)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t a)))))).(ex4_3_ind C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -c)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc -g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t a)))) -(or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda -(_: T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: -A).(\lambda (H9: (eq C c2 (CHead x0 (Bind Abst) x1))).(\lambda (H10: (csuba g -x0 c)).(\lambda (_: (arity g x0 x1 (asucc g x2))).(\lambda (_: (arity g c t -x2)).(eq_ind_r C (CHead x0 (Bind Abst) x1) (\lambda (c0: C).(or (ex2 C -(\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) -O c0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g -d2 d1)))))) (let H13 \def (H c d1 u H6 g x0 H10) in (or_ind (ex2 C (\lambda -(d2: C).(drop n O x0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 -d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or -(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 -(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda -(H14: (ex2 C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop n O x0 -(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C -(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind -Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda -(x: C).(\lambda (H15: (drop n O x0 (CHead x (Bind Abst) u))).(\lambda (H16: -(csuba g x d1)).(or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 -(Bind Abst) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind -Abst) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x0 -(Bind Abst) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) -x (drop_drop (Bind Abst) n x0 (CHead x (Bind Abst) u) H15 x1) H16))))) H14)) -(\lambda (H14: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead -d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or -(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 -(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda -(x3: C).(\lambda (x4: T).(\lambda (H15: (drop n O x0 (CHead x3 (Bind Void) -x4))).(\lambda (H16: (csuba g x3 d1)).(or_intror (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda -(d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x3 x4 -(drop_drop (Bind Abst) n x0 (CHead x3 (Bind Void) x4) H15 x1) H16)))))) H14)) -H13)) c2 H9)))))))) H8)) (\lambda (H8: (ex2_2 C T (\lambda (d2: C).(\lambda -(u2: T).(eq C c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 c))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(eq C -c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -c))) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda -(_: T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H9: -(eq C c2 (CHead x0 (Bind Void) x1))).(\lambda (H10: (csuba g x0 c)).(eq_ind_r -C (CHead x0 (Bind Void) x1) (\lambda (c0: C).(or (ex2 C (\lambda (d2: -C).(drop (S n) O c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 -d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let -H11 \def (H c d1 u H6 g x0 H10) in (or_ind (ex2 C (\lambda (d2: C).(drop n O -x0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H12: (ex2 C -(\lambda (d2: C).(drop n O x0 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop n O x0 (CHead d2 -(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda -(H13: (drop n O x0 (CHead x (Bind Abst) u))).(\lambda (H14: (csuba g x -d1)).(or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) -x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) -(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop -(Bind Void) n x0 (CHead x (Bind Abst) u) H13 x1) H14))))) H12)) (\lambda -(H12: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead -d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or -(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 -(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda -(x2: C).(\lambda (x3: T).(\lambda (H13: (drop n O x0 (CHead x2 (Bind Void) -x3))).(\lambda (H14: (csuba g x2 d1)).(or_intror (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda -(d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x2 x3 -(drop_drop (Bind Void) n x0 (CHead x2 (Bind Void) x3) H13 x1) H14)))))) H12)) -H11)) c2 H9))))) H8)) H7))))) (\lambda (H5: (csuba g c2 (CHead c (Bind Abst) -t))).(\lambda (H6: (drop (r (Bind Abst) n) O c (CHead d1 (Bind Abst) -u))).(let H_x \def (csuba_gen_abst_rev g c c2 t H5) in (let H7 \def H_x in -(or_ind (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda -(d2: C).(csuba g d2 c))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C -c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -c)))) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda -(_: T).(csuba g d2 d1))))) (\lambda (H8: (ex2 C (\lambda (d2: C).(eq C c2 -(CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba g d2 c)))).(ex2_ind C -(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba -g d2 c)) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) -u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda -(u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda (H9: (eq C c2 -(CHead x (Bind Abst) t))).(\lambda (H10: (csuba g x c)).(eq_ind_r C (CHead x -(Bind Abst) t) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 -(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H11 \def (H -c d1 u H6 g x H10) in (or_ind (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 -(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: C).(drop (S n) -O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba -g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead -x (Bind Abst) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x -(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C -(\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind -Abst) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 -C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abst) t) -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))) (\lambda (x0: C).(\lambda (H13: (drop n O x (CHead x0 (Bind Abst) -u))).(\lambda (H14: (csuba g x0 d1)).(or_introl (ex2 C (\lambda (d2: C).(drop -(S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) -O (CHead x (Bind Abst) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S -n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1)) x0 (drop_drop (Bind Abst) n x (CHead x0 (Bind Abst) u) -H13 t) H14))))) H12)) (\lambda (H12: (ex2_2 C T (\lambda (d2: C).(\lambda -(u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda -(_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: -T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind -Abst) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 -C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abst) t) -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H13: (drop n O x (CHead -x0 (Bind Void) x1))).(\lambda (H14: (csuba g x0 d1)).(or_intror (ex2 C -(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) -u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda -(u2: T).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda -(d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1 -(drop_drop (Bind Abst) n x (CHead x0 (Bind Void) x1) H13 t) H14)))))) H12)) -H11)) c2 H9)))) H8)) (\lambda (H8: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(eq C c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 c))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(eq C -c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -c))) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda -(_: T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H9: -(eq C c2 (CHead x0 (Bind Void) x1))).(\lambda (H10: (csuba g x0 c)).(eq_ind_r -C (CHead x0 (Bind Void) x1) (\lambda (c0: C).(or (ex2 C (\lambda (d2: -C).(drop (S n) O c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 -d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let -H11 \def (H c d1 u H6 g x0 H10) in (or_ind (ex2 C (\lambda (d2: C).(drop n O -x0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H12: (ex2 C -(\lambda (d2: C).(drop n O x0 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop n O x0 (CHead d2 -(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda -(H13: (drop n O x0 (CHead x (Bind Abst) u))).(\lambda (H14: (csuba g x -d1)).(or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) -x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) -(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop -(Bind Void) n x0 (CHead x (Bind Abst) u) H13 x1) H14))))) H12)) (\lambda -(H12: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead -d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or -(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 -(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda -(x2: C).(\lambda (x3: T).(\lambda (H13: (drop n O x0 (CHead x2 (Bind Void) -x3))).(\lambda (H14: (csuba g x2 d1)).(or_intror (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda -(d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x2 x3 -(drop_drop (Bind Void) n x0 (CHead x2 (Bind Void) x3) H13 x1) H14)))))) H12)) -H11)) c2 H9))))) H8)) H7))))) (\lambda (H5: (csuba g c2 (CHead c (Bind Void) -t))).(\lambda (H6: (drop (r (Bind Void) n) O c (CHead d1 (Bind Abst) -u))).(let H_x \def (csuba_gen_void_rev g c c2 t H5) in (let H7 \def H_x in -(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Void) t))) (\lambda (d2: -C).(csuba g d2 c)) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 -(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda (H8: (eq -C c2 (CHead x (Bind Void) t))).(\lambda (H9: (csuba g x c)).(eq_ind_r C -(CHead x (Bind Void) t) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S -n) O c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 -C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H10 \def (H -c d1 u H6 g x H9) in (or_ind (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 -(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: C).(drop (S n) -O (CHead x (Bind Void) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba -g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead -x (Bind Void) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))) (\lambda (H11: (ex2 C (\lambda (d2: C).(drop n O x -(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C -(\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind -Void) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 -C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Void) t) -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))) (\lambda (x0: C).(\lambda (H12: (drop n O x (CHead x0 (Bind Abst) -u))).(\lambda (H13: (csuba g x0 d1)).(or_introl (ex2 C (\lambda (d2: C).(drop -(S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) -O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S -n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1)) x0 (drop_drop (Bind Void) n x (CHead x0 (Bind Abst) u) -H12 t) H13))))) H11)) (\lambda (H11: (ex2_2 C T (\lambda (d2: C).(\lambda -(u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda -(_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: -T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind -Void) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 -C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Void) t) -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H12: (drop n O x (CHead -x0 (Bind Void) x1))).(\lambda (H13: (csuba g x0 d1)).(or_intror (ex2 C -(\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abst) -u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda -(u2: T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda -(d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1 -(drop_drop (Bind Void) n x (CHead x0 (Bind Void) x1) H12 t) H13)))))) H11)) -H10)) c2 H8)))) H7))))) b H3 H4)))) (\lambda (f: F).(\lambda (H3: (csuba g c2 -(CHead c (Flat f) t))).(\lambda (H4: (drop (r (Flat f) n) O c (CHead d1 (Bind -Abst) u))).(let H_x \def (csuba_gen_flat_rev g c c2 t f H3) in (let H5 \def -H_x in (ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 -(Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 c))) (or (ex2 C -(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) -O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g -d2 d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C c2 (CHead x0 -(Flat f) x1))).(\lambda (H7: (csuba g x0 c)).(eq_ind_r C (CHead x0 (Flat f) -x1) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 -(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 (Bind Void) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H8 \def (H0 d1 u H4 g x0 -H7) in (or_ind (ex2 C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind Abst) -u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda -(u2: T).(drop (S n) O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: C).(drop (S n) -O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g -d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 -(Flat f) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))) (\lambda (H9: (ex2 C (\lambda (d2: C).(drop (S n) O x0 -(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C -(\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat -f) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C -T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f) x1) -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))) (\lambda (x: C).(\lambda (H10: (drop (S n) O x0 (CHead x (Bind Abst) -u))).(\lambda (H11: (csuba g x d1)).(or_introl (ex2 C (\lambda (d2: C).(drop -(S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) -O (CHead x0 (Flat f) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S -n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1)) x (drop_drop (Flat f) n x0 (CHead x (Bind Abst) u) H10 -x1) H11))))) H9)) (\lambda (H9: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda -(_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda -(_: T).(csuba g d2 d1))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 -(Flat f) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f) -x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))) (\lambda (x2: C).(\lambda (x3: T).(\lambda (H10: (drop (S n) O x0 -(CHead x2 (Bind Void) x3))).(\lambda (H11: (csuba g x2 d1)).(or_intror (ex2 C -(\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) -u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda -(u2: T).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda -(d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x2 x3 -(drop_drop (Flat f) n x0 (CHead x2 (Bind Void) x3) H10 x1) H11)))))) H9)) -H8)) c2 H6))))) H5)))))) k H2 (drop_gen_drop k c (CHead d1 (Bind Abst) u) t n -H1)))))))))))) c1)))) i). -(* COMMENTS -Initial nodes: 11438 -END *) - -theorem csuba_drop_abbr_rev: - \forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u1: T).((drop i -O c1 (CHead d1 (Bind Abbr) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba -g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop i O c2 (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))))))) -\def - \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (d1: -C).(\forall (u1: T).((drop n O c1 (CHead d1 (Bind Abbr) u1)) \to (\forall (g: -G).(\forall (c2: C).((csuba g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(drop n -O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C -T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O c2 (CHead d2 -(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O c2 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))))))))))) (\lambda (c1: C).(\lambda (d1: C).(\lambda (u1: T).(\lambda -(H: (drop O O c1 (CHead d1 (Bind Abbr) u1))).(\lambda (g: G).(\lambda (c2: -C).(\lambda (H0: (csuba g c2 c1)).(let H1 \def (eq_ind C c1 (\lambda (c: -C).(csuba g c2 c)) H0 (CHead d1 (Bind Abbr) u1) (drop_gen_refl c1 (CHead d1 -(Bind Abbr) u1) H)) in (let H_x \def (csuba_gen_abbr_rev g d1 c2 u1 H1) in -(let H2 \def H_x in (or3_ind (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 -C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: -C).(drop O O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O -O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop O O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))) (\lambda (H3: (ex2 C (\lambda (d2: C).(eq C c2 (CHead -d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda -(d2: C).(eq C c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1)) (or3 (ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abbr) u1))) -(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(drop O O c2 (CHead d2 (Bind Abst) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop O O c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda -(H4: (eq C c2 (CHead x (Bind Abbr) u1))).(\lambda (H5: (csuba g x -d1)).(eq_ind_r C (CHead x (Bind Abbr) u1) (\lambda (c: C).(or3 (ex2 C -(\lambda (d2: C).(drop O O c (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(drop O O c (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda -(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop O O c (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))))) (or3_intro0 (ex2 C (\lambda (d2: -C).(drop O O (CHead x (Bind Abbr) u1) (CHead d2 (Bind Abbr) u1))) (\lambda -(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop O O (CHead x (Bind Abbr) u1) (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O (CHead x (Bind Abbr) -u1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))) (ex_intro2 C (\lambda (d2: C).(drop O O (CHead x (Bind Abbr) u1) -(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_refl -(CHead x (Bind Abbr) u1)) H5)) c2 H4)))) H3)) (\lambda (H3: (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(eq C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C (\lambda (d2: -C).(drop O O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O -O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop O O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: -A).(\lambda (H4: (eq C c2 (CHead x0 (Bind Abst) x1))).(\lambda (H5: (csuba g -x0 d1)).(\lambda (H6: (arity g x0 x1 (asucc g x2))).(\lambda (H7: (arity g d1 -u1 x2)).(eq_ind_r C (CHead x0 (Bind Abst) x1) (\lambda (c: C).(or3 (ex2 C -(\lambda (d2: C).(drop O O c (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(drop O O c (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda -(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop O O c (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))))) (or3_intro1 (ex2 C (\lambda (d2: -C).(drop O O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abbr) u1))) (\lambda -(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop O O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O (CHead x0 (Bind Abst) -x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop O O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x0 x1 x2 -(drop_refl (CHead x0 (Bind Abst) x1)) H5 H6 H7)) c2 H4)))))))) H3)) (\lambda -(H3: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind -C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C (\lambda (d2: -C).(drop O O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O -O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop O O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H4: (eq C -c2 (CHead x0 (Bind Void) x1))).(\lambda (H5: (csuba g x0 d1)).(eq_ind_r C -(CHead x0 (Bind Void) x1) (\lambda (c: C).(or3 (ex2 C (\lambda (d2: C).(drop -O O c (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C -T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O c (CHead d2 -(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O c (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) -(or3_intro2 (ex2 C (\lambda (d2: C).(drop O O (CHead x0 (Bind Void) x1) -(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O (CHead x0 (Bind -Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop O O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T -(\lambda (d2: C).(\lambda (u2: T).(drop O O (CHead x0 (Bind Void) x1) (CHead -d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 -x1 (drop_refl (CHead x0 (Bind Void) x1)) H5)) c2 H4))))) H3)) H2))))))))))) -(\lambda (n: nat).(\lambda (H: ((\forall (c1: C).(\forall (d1: C).(\forall -(u1: T).((drop n O c1 (CHead d1 (Bind Abbr) u1)) \to (\forall (g: G).(\forall -(c2: C).((csuba g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(drop n O c2 (CHead -d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O c2 (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O c2 (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (d1: -C).(\forall (u1: T).((drop (S n) O c (CHead d1 (Bind Abbr) u1)) \to (\forall -(g: G).(\forall (c2: C).((csuba g c2 c) \to (or3 (ex2 C (\lambda (d2: -C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S -n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1)))))))))))) (\lambda (n0: -nat).(\lambda (d1: C).(\lambda (u1: T).(\lambda (H0: (drop (S n) O (CSort n0) -(CHead d1 (Bind Abbr) u1))).(\lambda (g: G).(\lambda (c2: C).(\lambda (_: -(csuba g c2 (CSort n0))).(and3_ind (eq C (CHead d1 (Bind Abbr) u1) (CSort -n0)) (eq nat (S n) O) (eq nat O O) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O -c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T -A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead -d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda -(_: T).(csuba g d2 d1))))) (\lambda (_: (eq C (CHead d1 (Bind Abbr) u1) -(CSort n0))).(\lambda (H3: (eq nat (S n) O)).(\lambda (_: (eq nat O O)).(let -H5 \def (eq_ind nat (S n) (\lambda (ee: nat).(match ee in nat return (\lambda -(_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H3) -in (False_ind (or3 (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) H5))))) -(drop_gen_sort n0 (S n) O (CHead d1 (Bind Abbr) u1) H0))))))))) (\lambda (c: -C).(\lambda (H0: ((\forall (d1: C).(\forall (u1: T).((drop (S n) O c (CHead -d1 (Bind Abbr) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba g c2 c) \to -(or3 (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) -(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 -C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))))).(\lambda -(k: K).(\lambda (t: T).(\lambda (d1: C).(\lambda (u1: T).(\lambda (H1: (drop -(S n) O (CHead c k t) (CHead d1 (Bind Abbr) u1))).(\lambda (g: G).(\lambda -(c2: C).(\lambda (H2: (csuba g c2 (CHead c k t))).(K_ind (\lambda (k0: -K).((csuba g c2 (CHead c k0 t)) \to ((drop (r k0 n) O c (CHead d1 (Bind Abbr) -u1)) \to (or3 (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))) (\lambda -(b: B).(\lambda (H3: (csuba g c2 (CHead c (Bind b) t))).(\lambda (H4: (drop -(r (Bind b) n) O c (CHead d1 (Bind Abbr) u1))).(B_ind (\lambda (b0: -B).((csuba g c2 (CHead c (Bind b0) t)) \to ((drop (r (Bind b0) n) O c (CHead -d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead -d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))))))) (\lambda (H5: (csuba g c2 (CHead c (Bind Abbr) t))).(\lambda (H6: -(drop (r (Bind Abbr) n) O c (CHead d1 (Bind Abbr) u1))).(let H_x \def -(csuba_gen_abbr_rev g c c2 t H5) in (let H7 \def H_x in (or3_ind (ex2 C -(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) (\lambda (d2: C).(csuba -g d2 c))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq -C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 c)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g c t a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C -c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -c)))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda -(H8: (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) (\lambda -(d2: C).(csuba g d2 c)))).(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 -(Bind Abbr) t))) (\lambda (d2: C).(csuba g d2 c)) (or3 (ex2 C (\lambda (d2: -C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S -n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda (H9: (eq -C c2 (CHead x (Bind Abbr) t))).(\lambda (H10: (csuba g x c)).(eq_ind_r C -(CHead x (Bind Abbr) t) (\lambda (c0: C).(or3 (ex2 C (\lambda (d2: C).(drop -(S n) O c0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O -c0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O c0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda -(_: T).(csuba g d2 d1)))))) (let H11 \def (H c d1 u1 H6 g x H10) in (or3_ind -(ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(drop n O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda -(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(drop (S -n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 -C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abbr) t) -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: -C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) -(or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 -(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) -t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: -C).(\lambda (H13: (drop n O x (CHead x0 (Bind Abbr) u1))).(\lambda (H14: -(csuba g x0 d1)).(or3_intro0 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x -(Bind Abbr) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O -(CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abbr) t) -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) -(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x0 (drop_drop -(Bind Abbr) n x (CHead x0 (Bind Abbr) u1) H13 t) H14))))) H12)) (\lambda -(H12: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n -O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C -(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t) -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H13: (drop n O x (CHead x0 -(Bind Abst) x1))).(\lambda (H14: (csuba g x0 d1)).(\lambda (H15: (arity g x0 -x1 (asucc g x2))).(\lambda (H16: (arity g d1 u1 x2)).(or3_intro1 (ex2 C -(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t) -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex4_3_intro C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x -(Bind Abbr) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a)))) x0 x1 x2 (drop_drop (Bind Abbr) n -x (CHead x0 (Bind Abst) x1) H13 t) H14 H15 H16))))))))) H12)) (\lambda (H12: -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind -C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C -(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t) -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (H13: (drop n O x (CHead x0 (Bind Void) -x1))).(\lambda (H14: (csuba g x0 d1)).(or3_intro2 (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u1))) (\lambda -(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x -(Bind Abbr) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1 (drop_drop (Bind -Abbr) n x (CHead x0 (Bind Void) x1) H13 t) H14)))))) H12)) H11)) c2 H9)))) -H8)) (\lambda (H8: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(eq C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 c)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g c t a)))))).(ex4_3_ind C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 c)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t a)))) (or3 (ex2 -C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H9: (eq C c2 (CHead x0 (Bind -Abst) x1))).(\lambda (H10: (csuba g x0 c)).(\lambda (_: (arity g x0 x1 (asucc -g x2))).(\lambda (_: (arity g c t x2)).(eq_ind_r C (CHead x0 (Bind Abst) x1) -(\lambda (c0: C).(or3 (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 -(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0 -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))))) (let H13 \def (H c d1 u1 H6 g x0 H10) in (or3_ind (ex2 C (\lambda -(d2: C).(drop n O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n -O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 -(Bind Abst) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S -n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))) (\lambda (H14: (ex2 C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: -C).(drop n O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) -(or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead -d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abst) -x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda -(H15: (drop n O x0 (CHead x (Bind Abbr) u1))).(\lambda (H16: (csuba g x -d1)).(or3_intro0 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) -x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T -A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 -(Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C -(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop (Bind Abst) n x0 -(CHead x (Bind Abbr) u1) H15 x1) H16))))) H14)) (\lambda (H14: (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x0 (CHead d2 -(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop n O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abbr) u1))) -(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 -(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead -x0 (Bind Abst) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: -A).(\lambda (H15: (drop n O x0 (CHead x3 (Bind Abst) x4))).(\lambda (H16: -(csuba g x3 d1)).(\lambda (H17: (arity g x3 x4 (asucc g x5))).(\lambda (H18: -(arity g d1 u1 x5)).(or3_intro1 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead -x0 (Bind Abst) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S -n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x3 x4 x5 -(drop_drop (Bind Abst) n x0 (CHead x3 (Bind Abst) x4) H15 x1) H16 H17 -H18))))))))) H14)) (\lambda (H14: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(drop -n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g -d2 d1))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) -(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 -(Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda -(x3: C).(\lambda (x4: T).(\lambda (H15: (drop n O x0 (CHead x3 (Bind Void) -x4))).(\lambda (H16: (csuba g x3 d1)).(or3_intro2 (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abbr) u1))) -(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 -(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead -x0 (Bind Abst) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x3 x4 (drop_drop (Bind -Abst) n x0 (CHead x3 (Bind Void) x4) H15 x1) H16)))))) H14)) H13)) c2 -H9)))))))) H8)) (\lambda (H8: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(eq C c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 c))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(eq C -c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -c))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda -(x0: C).(\lambda (x1: T).(\lambda (H9: (eq C c2 (CHead x0 (Bind Void) -x1))).(\lambda (H10: (csuba g x0 c)).(eq_ind_r C (CHead x0 (Bind Void) x1) -(\lambda (c0: C).(or3 (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 -(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0 -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))))) (let H11 \def (H c d1 u1 H6 g x0 H10) in (or3_ind (ex2 C (\lambda -(d2: C).(drop n O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n -O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 -(Bind Void) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S -n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: -C).(drop n O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) -(or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead -d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Void) -x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda -(H13: (drop n O x0 (CHead x (Bind Abbr) u1))).(\lambda (H14: (csuba g x -d1)).(or3_intro0 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) -x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T -A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 -(Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C -(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop (Bind Void) n x0 -(CHead x (Bind Abbr) u1) H13 x1) H14))))) H12)) (\lambda (H12: (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x0 (CHead d2 -(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop n O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abbr) u1))) -(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 -(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead -x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))) (\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: -A).(\lambda (H13: (drop n O x0 (CHead x2 (Bind Abst) x3))).(\lambda (H14: -(csuba g x2 d1)).(\lambda (H15: (arity g x2 x3 (asucc g x4))).(\lambda (H16: -(arity g d1 u1 x4)).(or3_intro1 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead -x0 (Bind Void) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S -n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x2 x3 x4 -(drop_drop (Bind Void) n x0 (CHead x2 (Bind Abst) x3) H13 x1) H14 H15 -H16))))))))) H12)) (\lambda (H12: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(drop -n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g -d2 d1))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) -(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 -(Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda -(x2: C).(\lambda (x3: T).(\lambda (H13: (drop n O x0 (CHead x2 (Bind Void) -x3))).(\lambda (H14: (csuba g x2 d1)).(or3_intro2 (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abbr) u1))) -(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 -(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead -x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x2 x3 (drop_drop (Bind -Void) n x0 (CHead x2 (Bind Void) x3) H13 x1) H14)))))) H12)) H11)) c2 H9))))) -H8)) H7))))) (\lambda (H5: (csuba g c2 (CHead c (Bind Abst) t))).(\lambda -(H6: (drop (r (Bind Abst) n) O c (CHead d1 (Bind Abbr) u1))).(let H_x \def -(csuba_gen_abst_rev g c c2 t H5) in (let H7 \def H_x in (or_ind (ex2 C -(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba -g d2 c))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 c)))) (or3 -(ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda -(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H8: -(ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2: -C).(csuba g d2 c)))).(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind -Abst) t))) (\lambda (d2: C).(csuba g d2 c)) (or3 (ex2 C (\lambda (d2: -C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S -n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda (H9: (eq -C c2 (CHead x (Bind Abst) t))).(\lambda (H10: (csuba g x c)).(eq_ind_r C -(CHead x (Bind Abst) t) (\lambda (c0: C).(or3 (ex2 C (\lambda (d2: C).(drop -(S n) O c0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O -c0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O c0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda -(_: T).(csuba g d2 d1)))))) (let H11 \def (H c d1 u1 H6 g x H10) in (or3_ind -(ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(drop n O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda -(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(drop (S -n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 -C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abst) t) -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: -C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) -(or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 -(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) -t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: -C).(\lambda (H13: (drop n O x (CHead x0 (Bind Abbr) u1))).(\lambda (H14: -(csuba g x0 d1)).(or3_intro0 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x -(Bind Abst) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O -(CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abst) t) -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) -(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x0 (drop_drop -(Bind Abst) n x (CHead x0 (Bind Abbr) u1) H13 t) H14))))) H12)) (\lambda -(H12: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n -O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C -(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H13: (drop n O x (CHead x0 -(Bind Abst) x1))).(\lambda (H14: (csuba g x0 d1)).(\lambda (H15: (arity g x0 -x1 (asucc g x2))).(\lambda (H16: (arity g d1 u1 x2)).(or3_intro1 (ex2 C -(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex4_3_intro C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x -(Bind Abst) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a)))) x0 x1 x2 (drop_drop (Bind Abst) n -x (CHead x0 (Bind Abst) x1) H13 t) H14 H15 H16))))))))) H12)) (\lambda (H12: -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind -C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C -(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (H13: (drop n O x (CHead x0 (Bind Void) -x1))).(\lambda (H14: (csuba g x0 d1)).(or3_intro2 (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u1))) (\lambda -(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x -(Bind Abst) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1 (drop_drop (Bind -Abst) n x (CHead x0 (Bind Void) x1) H13 t) H14)))))) H12)) H11)) c2 H9)))) -H8)) (\lambda (H8: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -c))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 c))) (or3 -(ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda -(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (H9: (eq C c2 (CHead x0 (Bind Void) -x1))).(\lambda (H10: (csuba g x0 c)).(eq_ind_r C (CHead x0 (Bind Void) x1) -(\lambda (c0: C).(or3 (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 -(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0 -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))))) (let H11 \def (H c d1 u1 H6 g x0 H10) in (or3_ind (ex2 C (\lambda -(d2: C).(drop n O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n -O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 -(Bind Void) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S -n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: -C).(drop n O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) -(or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead -d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Void) -x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda -(H13: (drop n O x0 (CHead x (Bind Abbr) u1))).(\lambda (H14: (csuba g x -d1)).(or3_intro0 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) -x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T -A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 -(Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C -(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop (Bind Void) n x0 -(CHead x (Bind Abbr) u1) H13 x1) H14))))) H12)) (\lambda (H12: (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x0 (CHead d2 -(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop n O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abbr) u1))) -(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 -(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead -x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))) (\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: -A).(\lambda (H13: (drop n O x0 (CHead x2 (Bind Abst) x3))).(\lambda (H14: -(csuba g x2 d1)).(\lambda (H15: (arity g x2 x3 (asucc g x4))).(\lambda (H16: -(arity g d1 u1 x4)).(or3_intro1 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead -x0 (Bind Void) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S -n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x2 x3 x4 -(drop_drop (Bind Void) n x0 (CHead x2 (Bind Abst) x3) H13 x1) H14 H15 -H16))))))))) H12)) (\lambda (H12: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(drop -n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g -d2 d1))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) -(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 -(Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda -(x2: C).(\lambda (x3: T).(\lambda (H13: (drop n O x0 (CHead x2 (Bind Void) -x3))).(\lambda (H14: (csuba g x2 d1)).(or3_intro2 (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abbr) u1))) -(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 -(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead -x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x2 x3 (drop_drop (Bind -Void) n x0 (CHead x2 (Bind Void) x3) H13 x1) H14)))))) H12)) H11)) c2 H9))))) -H8)) H7))))) (\lambda (H5: (csuba g c2 (CHead c (Bind Void) t))).(\lambda -(H6: (drop (r (Bind Void) n) O c (CHead d1 (Bind Abbr) u1))).(let H_x \def -(csuba_gen_void_rev g c c2 t H5) in (let H7 \def H_x in (ex2_ind C (\lambda -(d2: C).(eq C c2 (CHead d2 (Bind Void) t))) (\lambda (d2: C).(csuba g d2 c)) -(or3 (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) -(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 -C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: -C).(\lambda (H8: (eq C c2 (CHead x (Bind Void) t))).(\lambda (H9: (csuba g x -c)).(eq_ind_r C (CHead x (Bind Void) t) (\lambda (c0: C).(or3 (ex2 C (\lambda -(d2: C).(drop (S n) O c0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba -g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop (S n) O c0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda -(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 (Bind Void) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H10 \def (H c d1 u1 H6 g x -H9) in (or3_ind (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 -C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C -(\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H11: (ex2 C -(\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop n O x (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) (or3 (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) u1))) (\lambda -(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x -(Bind Void) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (H12: (drop n O x (CHead x0 -(Bind Abbr) u1))).(\lambda (H13: (csuba g x0 d1)).(or3_intro0 (ex2 C (\lambda -(d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) u1))) -(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x -(Bind Void) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x -(Bind Void) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) -x0 (drop_drop (Bind Void) n x (CHead x0 (Bind Abbr) u1) H12 t) H13))))) H11)) -(\lambda (H11: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop n O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 -(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H12: (drop n O x (CHead x0 -(Bind Abst) x1))).(\lambda (H13: (csuba g x0 d1)).(\lambda (H14: (arity g x0 -x1 (asucc g x2))).(\lambda (H15: (arity g d1 u1 x2)).(or3_intro1 (ex2 C -(\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex4_3_intro C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x -(Bind Void) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a)))) x0 x1 x2 (drop_drop (Bind Void) n -x (CHead x0 (Bind Abst) x1) H12 t) H13 H14 H15))))))))) H11)) (\lambda (H11: -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind -C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C -(\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (H12: (drop n O x (CHead x0 (Bind Void) -x1))).(\lambda (H13: (csuba g x0 d1)).(or3_intro2 (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) u1))) (\lambda -(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x -(Bind Void) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1 (drop_drop (Bind -Void) n x (CHead x0 (Bind Void) x1) H12 t) H13)))))) H11)) H10)) c2 H8)))) -H7))))) b H3 H4)))) (\lambda (f: F).(\lambda (H3: (csuba g c2 (CHead c (Flat -f) t))).(\lambda (H4: (drop (r (Flat f) n) O c (CHead d1 (Bind Abbr) -u1))).(let H_x \def (csuba_gen_flat_rev g c c2 t f H3) in (let H5 \def H_x in -(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Flat f) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 c))) (or3 (ex2 C (\lambda -(d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba -g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda -(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (x1: -T).(\lambda (H6: (eq C c2 (CHead x0 (Flat f) x1))).(\lambda (H7: (csuba g x0 -c)).(eq_ind_r C (CHead x0 (Flat f) x1) (\lambda (c0: C).(or3 (ex2 C (\lambda -(d2: C).(drop (S n) O c0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba -g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop (S n) O c0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda -(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 (Bind Void) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H8 \def (H0 d1 u1 H4 g x0 -H7) in (or3_ind (ex2 C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O x0 (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O x0 (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C -(\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Void) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H9: (ex2 C (\lambda -(d2: C).(drop (S n) O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba -g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) (or3 (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u1))) (\lambda -(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f) -x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))) (\lambda (x: C).(\lambda (H10: (drop (S n) O x0 (CHead x (Bind Abbr) -u1))).(\lambda (H11: (csuba g x d1)).(or3_intro0 (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u1))) (\lambda -(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f) -x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) -(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop -(Flat f) n x0 (CHead x (Bind Abbr) u1) H10 x1) H11))))) H9)) (\lambda (H9: -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O -x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop (S n) O x0 (CHead d2 (Bind Abst) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C -(\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Void) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x2: C).(\lambda (x3: -T).(\lambda (x4: A).(\lambda (H10: (drop (S n) O x0 (CHead x2 (Bind Abst) -x3))).(\lambda (H11: (csuba g x2 d1)).(\lambda (H12: (arity g x2 x3 (asucc g -x4))).(\lambda (H13: (arity g d1 u1 x4)).(or3_intro1 (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u1))) (\lambda -(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f) -x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x2 x3 x4 -(drop_drop (Flat f) n x0 (CHead x2 (Bind Abst) x3) H10 x1) H11 H12 -H13))))))))) H9)) (\lambda (H9: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda -(_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda -(_: T).(csuba g d2 d1))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 -(Flat f) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O -(CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f) x1) (CHead -d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) -(\lambda (x2: C).(\lambda (x3: T).(\lambda (H10: (drop (S n) O x0 (CHead x2 -(Bind Void) x3))).(\lambda (H11: (csuba g x2 d1)).(or3_intro2 (ex2 C (\lambda -(d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u1))) -(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead -x0 (Flat f) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Void) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1))) x2 x3 (drop_drop (Flat f) n x0 -(CHead x2 (Bind Void) x3) H10 x1) H11)))))) H9)) H8)) c2 H6))))) H5)))))) k -H2 (drop_gen_drop k c (CHead d1 (Bind Abbr) u1) t n H1)))))))))))) c1)))) i). -(* COMMENTS -Initial nodes: 23852 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/fwd.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/fwd.ma deleted file mode 100644 index a618761fe..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/fwd.ma +++ /dev/null @@ -1,1083 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csuba/defs.ma". - -theorem csuba_gen_abbr: - \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g -(CHead d1 (Bind Abbr) u) c) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 -(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))))) -\def - \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u: T).(\lambda (H: -(csuba g (CHead d1 (Bind Abbr) u) c)).(insert_eq C (CHead d1 (Bind Abbr) u) -(\lambda (c0: C).(csuba g c0 c)) (\lambda (_: C).(ex2 C (\lambda (d2: C).(eq -C c (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (\lambda -(y: C).(\lambda (H0: (csuba g y c)).(csuba_ind g (\lambda (c0: C).(\lambda -(c1: C).((eq C c0 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(eq C -c1 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))) (\lambda -(n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Abbr) u))).(let H2 -\def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return (\lambda (_: -C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow -False])) I (CHead d1 (Bind Abbr) u) H1) in (False_ind (ex2 C (\lambda (d2: -C).(eq C (CSort n) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 -d2))) H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 -c2)).(\lambda (H2: (((eq C c1 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda -(d2: C).(eq C c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 -d2)))))).(\lambda (k: K).(\lambda (u0: T).(\lambda (H3: (eq C (CHead c1 k u0) -(CHead d1 (Bind Abbr) u))).(let H4 \def (f_equal C C (\lambda (e: C).(match e -in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _ -_) \Rightarrow c0])) (CHead c1 k u0) (CHead d1 (Bind Abbr) u) H3) in ((let H5 -\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) -with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c1 k -u0) (CHead d1 (Bind Abbr) u) H3) in ((let H6 \def (f_equal C T (\lambda (e: -C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | -(CHead _ _ t) \Rightarrow t])) (CHead c1 k u0) (CHead d1 (Bind Abbr) u) H3) -in (\lambda (H7: (eq K k (Bind Abbr))).(\lambda (H8: (eq C c1 d1)).(eq_ind_r -T u (\lambda (t: T).(ex2 C (\lambda (d2: C).(eq C (CHead c2 k t) (CHead d2 -(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (eq_ind_r K (Bind Abbr) -(\lambda (k0: K).(ex2 C (\lambda (d2: C).(eq C (CHead c2 k0 u) (CHead d2 -(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H9 \def (eq_ind C -c1 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda -(d2: C).(eq C c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 -d2))))) H2 d1 H8) in (let H10 \def (eq_ind C c1 (\lambda (c0: C).(csuba g c0 -c2)) H1 d1 H8) in (ex_intro2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Abbr) -u) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) c2 -(refl_equal C (CHead c2 (Bind Abbr) u)) H10))) k H7) u0 H6)))) H5)) -H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1 -c2)).(\lambda (_: (((eq C c1 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda -(d2: C).(eq C c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 -d2)))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: -T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u1) (CHead d1 -(Bind Abbr) u))).(let H5 \def (eq_ind C (CHead c1 (Bind Void) u1) (\lambda -(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind b0) \Rightarrow (match b0 in B return (\lambda (_: -B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False | Void -\Rightarrow True]) | (Flat _) \Rightarrow False])])) I (CHead d1 (Bind Abbr) -u) H4) in (False_ind (ex2 C (\lambda (d2: C).(eq C (CHead c2 (Bind b) u2) -(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) H5))))))))))) -(\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1 c2)).(\lambda (_: -(((eq C c1 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(eq C c2 -(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))).(\lambda (t: -T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g a))).(\lambda (u0: -T).(\lambda (_: (arity g c2 u0 a)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) -t) (CHead d1 (Bind Abbr) u))).(let H6 \def (eq_ind C (CHead c1 (Bind Abst) t) -(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind b) \Rightarrow (match b in B return (\lambda (_: -B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow True | Void -\Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead d1 (Bind Abbr) -u) H5) in (False_ind (ex2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Abbr) u0) -(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) H6)))))))))))) -y c H0))) H))))). -(* COMMENTS -Initial nodes: 1117 -END *) - -theorem csuba_gen_void: - \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).((csuba g -(CHead d1 (Bind Void) u1) c) \to (ex2_3 B C T (\lambda (b: B).(\lambda (d2: -C).(\lambda (u2: T).(eq C c (CHead d2 (Bind b) u2))))) (\lambda (_: -B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))))))))) -\def - \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda -(H: (csuba g (CHead d1 (Bind Void) u1) c)).(insert_eq C (CHead d1 (Bind Void) -u1) (\lambda (c0: C).(csuba g c0 c)) (\lambda (_: C).(ex2_3 B C T (\lambda -(b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Bind b) u2))))) -(\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))))) -(\lambda (y: C).(\lambda (H0: (csuba g y c)).(csuba_ind g (\lambda (c0: -C).(\lambda (c1: C).((eq C c0 (CHead d1 (Bind Void) u1)) \to (ex2_3 B C T -(\lambda (b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 (Bind b) -u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 -d2)))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind -Void) u1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C -return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) -\Rightarrow False])) I (CHead d1 (Bind Void) u1) H1) in (False_ind (ex2_3 B C -T (\lambda (b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C (CSort n) (CHead d2 -(Bind b) u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 -d2))))) H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 -c2)).(\lambda (H2: (((eq C c1 (CHead d1 (Bind Void) u1)) \to (ex2_3 B C T -(\lambda (b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind b) -u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 -d2)))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c1 k u) -(CHead d1 (Bind Void) u1))).(let H4 \def (f_equal C C (\lambda (e: C).(match -e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _ -_) \Rightarrow c0])) (CHead c1 k u) (CHead d1 (Bind Void) u1) H3) in ((let H5 -\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) -with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c1 k -u) (CHead d1 (Bind Void) u1) H3) in ((let H6 \def (f_equal C T (\lambda (e: -C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | -(CHead _ _ t) \Rightarrow t])) (CHead c1 k u) (CHead d1 (Bind Void) u1) H3) -in (\lambda (H7: (eq K k (Bind Void))).(\lambda (H8: (eq C c1 d1)).(eq_ind_r -T u1 (\lambda (t: T).(ex2_3 B C T (\lambda (b: B).(\lambda (d2: C).(\lambda -(u2: T).(eq C (CHead c2 k t) (CHead d2 (Bind b) u2))))) (\lambda (_: -B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))))) (eq_ind_r K (Bind -Void) (\lambda (k0: K).(ex2_3 B C T (\lambda (b: B).(\lambda (d2: C).(\lambda -(u2: T).(eq C (CHead c2 k0 u1) (CHead d2 (Bind b) u2))))) (\lambda (_: -B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))))) (let H9 \def (eq_ind -C c1 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Void) u1)) \to (ex2_3 B C T -(\lambda (b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind b) -u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 -d2))))))) H2 d1 H8) in (let H10 \def (eq_ind C c1 (\lambda (c0: C).(csuba g -c0 c2)) H1 d1 H8) in (ex2_3_intro B C T (\lambda (b: B).(\lambda (d2: -C).(\lambda (u2: T).(eq C (CHead c2 (Bind Void) u1) (CHead d2 (Bind b) -u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))) -Void c2 u1 (refl_equal C (CHead c2 (Bind Void) u1)) H10))) k H7) u H6)))) -H5)) H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 -c2)).(\lambda (H2: (((eq C c1 (CHead d1 (Bind Void) u1)) \to (ex2_3 B C T -(\lambda (b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind b) -u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 -d2)))))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u0: -T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u0) (CHead d1 -(Bind Void) u1))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C -return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _ _) -\Rightarrow c0])) (CHead c1 (Bind Void) u0) (CHead d1 (Bind Void) u1) H4) in -((let H6 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: -C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead -c1 (Bind Void) u0) (CHead d1 (Bind Void) u1) H4) in (\lambda (H7: (eq C c1 -d1)).(let H8 \def (eq_ind C c1 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind -Void) u1)) \to (ex2_3 B C T (\lambda (b0: B).(\lambda (d2: C).(\lambda (u3: -T).(eq C c2 (CHead d2 (Bind b0) u3))))) (\lambda (_: B).(\lambda (d2: -C).(\lambda (_: T).(csuba g d1 d2))))))) H2 d1 H7) in (let H9 \def (eq_ind C -c1 (\lambda (c0: C).(csuba g c0 c2)) H1 d1 H7) in (ex2_3_intro B C T (\lambda -(b0: B).(\lambda (d2: C).(\lambda (u3: T).(eq C (CHead c2 (Bind b) u2) (CHead -d2 (Bind b0) u3))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba -g d1 d2)))) b c2 u2 (refl_equal C (CHead c2 (Bind b) u2)) H9))))) -H5))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1 -c2)).(\lambda (_: (((eq C c1 (CHead d1 (Bind Void) u1)) \to (ex2_3 B C T -(\lambda (b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind b) -u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 -d2)))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc -g a))).(\lambda (u: T).(\lambda (_: (arity g c2 u a)).(\lambda (H5: (eq C -(CHead c1 (Bind Abst) t) (CHead d1 (Bind Void) u1))).(let H6 \def (eq_ind C -(CHead c1 (Bind Abst) t) (\lambda (ee: C).(match ee in C return (\lambda (_: -C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match -k in K return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B -return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow -True | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead d1 -(Bind Void) u1) H5) in (False_ind (ex2_3 B C T (\lambda (b: B).(\lambda (d2: -C).(\lambda (u2: T).(eq C (CHead c2 (Bind Abbr) u) (CHead d2 (Bind b) u2))))) -(\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))))) -H6)))))))))))) y c H0))) H))))). -(* COMMENTS -Initial nodes: 1418 -END *) - -theorem csuba_gen_abst: - \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).((csuba g -(CHead d1 (Bind Abst) u1) c) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead -d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))))))) -\def - \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda -(H: (csuba g (CHead d1 (Bind Abst) u1) c)).(insert_eq C (CHead d1 (Bind Abst) -u1) (\lambda (c0: C).(csuba g c0 c)) (\lambda (_: C).(or (ex2 C (\lambda (d2: -C).(eq C c (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead -d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity -g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 a))))))) (\lambda (y: C).(\lambda (H0: (csuba g y -c)).(csuba_ind g (\lambda (c0: C).(\lambda (c1: C).((eq C c0 (CHead d1 (Bind -Abst) u1)) \to (or (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abst) -u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind Abbr) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))) -(\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Abst) -u1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return -(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) -\Rightarrow False])) I (CHead d1 (Bind Abst) u1) H1) in (False_ind (or (ex2 C -(\lambda (d2: C).(eq C (CSort n) (CHead d2 (Bind Abst) u1))) (\lambda (d2: -C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(eq C (CSort n) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) H2)))) -(\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda -(H2: (((eq C c1 (CHead d1 (Bind Abst) u1)) \to (or (ex2 C (\lambda (d2: -C).(eq C c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 -(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity -g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 a))))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: -(eq C (CHead c1 k u) (CHead d1 (Bind Abst) u1))).(let H4 \def (f_equal C C -(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow c1 | (CHead c0 _ _) \Rightarrow c0])) (CHead c1 k u) (CHead d1 -(Bind Abst) u1) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e in -C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) -\Rightarrow k0])) (CHead c1 k u) (CHead d1 (Bind Abst) u1) H3) in ((let H6 -\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u) -(CHead d1 (Bind Abst) u1) H3) in (\lambda (H7: (eq K k (Bind Abst))).(\lambda -(H8: (eq C c1 d1)).(eq_ind_r T u1 (\lambda (t: T).(or (ex2 C (\lambda (d2: -C).(eq C (CHead c2 k t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g -d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C -(CHead c2 k t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a))))))) (eq_ind_r K (Bind Abst) -(\lambda (k0: K).(or (ex2 C (\lambda (d2: C).(eq C (CHead c2 k0 u1) (CHead d2 -(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c2 k0 u1) (CHead d2 -(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 -(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 -u2 a))))))) (let H9 \def (eq_ind C c1 (\lambda (c0: C).((eq C c0 (CHead d1 -(Bind Abst) u1)) \to (or (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind -Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))))) H2 -d1 H8) in (let H10 \def (eq_ind C c1 (\lambda (c0: C).(csuba g c0 c2)) H1 d1 -H8) in (or_introl (ex2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Abst) u1) -(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abst) -u1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: -A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda -(a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: C).(eq C (CHead c2 -(Bind Abst) u1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) -c2 (refl_equal C (CHead c2 (Bind Abst) u1)) H10)))) k H7) u H6)))) H5)) -H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1 -c2)).(\lambda (_: (((eq C c1 (CHead d1 (Bind Abst) u1)) \to (or (ex2 C -(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba -g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq -C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: -A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda -(a: A).(arity g d2 u2 a))))))))).(\lambda (b: B).(\lambda (_: (not (eq B b -Void))).(\lambda (u0: T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind -Void) u0) (CHead d1 (Bind Abst) u1))).(let H5 \def (eq_ind C (CHead c1 (Bind -Void) u0) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with -[(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return -(\lambda (_: K).Prop) with [(Bind b0) \Rightarrow (match b0 in B return -(\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False | -Void \Rightarrow True]) | (Flat _) \Rightarrow False])])) I (CHead d1 (Bind -Abst) u1) H4) in (False_ind (or (ex2 C (\lambda (d2: C).(eq C (CHead c2 (Bind -b) u2) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 -C T A (\lambda (d2: C).(\lambda (u3: T).(\lambda (_: A).(eq C (CHead c2 (Bind -b) u2) (CHead d2 (Bind Abbr) u3))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u3: T).(\lambda (a: A).(arity g d2 u3 a)))))) H5))))))))))) (\lambda (c1: -C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c1 -(CHead d1 (Bind Abst) u1)) \to (or (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 -(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a))))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (H3: (arity g c1 t (asucc -g a))).(\lambda (u: T).(\lambda (H4: (arity g c2 u a)).(\lambda (H5: (eq C -(CHead c1 (Bind Abst) t) (CHead d1 (Bind Abst) u1))).(let H6 \def (f_equal C -C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow c1 | (CHead c0 _ _) \Rightarrow c0])) (CHead c1 (Bind Abst) t) -(CHead d1 (Bind Abst) u1) H5) in ((let H7 \def (f_equal C T (\lambda (e: -C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow t | -(CHead _ _ t0) \Rightarrow t0])) (CHead c1 (Bind Abst) t) (CHead d1 (Bind -Abst) u1) H5) in (\lambda (H8: (eq C c1 d1)).(let H9 \def (eq_ind T t -(\lambda (t0: T).(arity g c1 t0 (asucc g a))) H3 u1 H7) in (let H10 \def -(eq_ind C c1 (\lambda (c0: C).(arity g c0 u1 (asucc g a))) H9 d1 H8) in (let -H11 \def (eq_ind C c1 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abst) u1)) -\to (or (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) u1))) (\lambda -(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 (asucc g a0))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 a0)))))))) H2 d1 H8) -in (let H12 \def (eq_ind C c1 (\lambda (c0: C).(csuba g c0 c2)) H1 d1 H8) in -(or_intror (ex2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Abbr) u) (CHead d2 -(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) u) -(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity -g d1 u1 (asucc g a0))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: -A).(arity g d2 u2 a0))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) u) (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 (asucc g -a0))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 -a0)))) c2 u a (refl_equal C (CHead c2 (Bind Abbr) u)) H12 H10 H4)))))))) -H6)))))))))))) y c H0))) H))))). -(* COMMENTS -Initial nodes: 2550 -END *) - -theorem csuba_gen_flat: - \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).(\forall -(f: F).((csuba g (CHead d1 (Flat f) u1) c) \to (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d1 d2))))))))) -\def - \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda -(f: F).(\lambda (H: (csuba g (CHead d1 (Flat f) u1) c)).(insert_eq C (CHead -d1 (Flat f) u1) (\lambda (c0: C).(csuba g c0 c)) (\lambda (_: C).(ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d1 d2))))) (\lambda (y: C).(\lambda (H0: -(csuba g y c)).(csuba_ind g (\lambda (c0: C).(\lambda (c1: C).((eq C c0 -(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq -C c1 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 -d2))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Flat f) -u1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return -(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) -\Rightarrow False])) I (CHead d1 (Flat f) u1) H1) in (False_ind (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(eq C (CSort n) (CHead d2 (Flat f) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))) H2)))) (\lambda (c1: -C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c1 -(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq -C c2 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 -d2))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c1 k u) -(CHead d1 (Flat f) u1))).(let H4 \def (f_equal C C (\lambda (e: C).(match e -in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _ -_) \Rightarrow c0])) (CHead c1 k u) (CHead d1 (Flat f) u1) H3) in ((let H5 -\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) -with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c1 k -u) (CHead d1 (Flat f) u1) H3) in ((let H6 \def (f_equal C T (\lambda (e: -C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | -(CHead _ _ t) \Rightarrow t])) (CHead c1 k u) (CHead d1 (Flat f) u1) H3) in -(\lambda (H7: (eq K k (Flat f))).(\lambda (H8: (eq C c1 d1)).(eq_ind_r T u1 -(\lambda (t: T).(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c2 -k t) (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 -d2))))) (eq_ind_r K (Flat f) (\lambda (k0: K).(ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(eq C (CHead c2 k0 u1) (CHead d2 (Flat f) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d1 d2))))) (let H9 \def (eq_ind C c1 -(\lambda (c0: C).((eq C c0 (CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda -(d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Flat f) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d1 d2)))))) H2 d1 H8) in (let H10 \def (eq_ind C -c1 (\lambda (c0: C).(csuba g c0 c2)) H1 d1 H8) in (ex2_2_intro C T (\lambda -(d2: C).(\lambda (u2: T).(eq C (CHead c2 (Flat f) u1) (CHead d2 (Flat f) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))) c2 u1 (refl_equal C -(CHead c2 (Flat f) u1)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: -C).(\lambda (c2: C).(\lambda (_: (csuba g c1 c2)).(\lambda (_: (((eq C c1 -(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq -C c2 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 -d2))))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u0: -T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u0) (CHead d1 -(Flat f) u1))).(let H5 \def (eq_ind C (CHead c1 (Bind Void) u0) (\lambda (ee: -C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow -False | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) -with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead d1 -(Flat f) u1) H4) in (False_ind (ex2_2 C T (\lambda (d2: C).(\lambda (u3: -T).(eq C (CHead c2 (Bind b) u2) (CHead d2 (Flat f) u3)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d1 d2)))) H5))))))))))) (\lambda (c1: C).(\lambda -(c2: C).(\lambda (_: (csuba g c1 c2)).(\lambda (_: (((eq C c1 (CHead d1 (Flat -f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 -(Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 -d2))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g -a))).(\lambda (u: T).(\lambda (_: (arity g c2 u a)).(\lambda (H5: (eq C -(CHead c1 (Bind Abst) t) (CHead d1 (Flat f) u1))).(let H6 \def (eq_ind C -(CHead c1 (Bind Abst) t) (\lambda (ee: C).(match ee in C return (\lambda (_: -C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match -k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat -_) \Rightarrow False])])) I (CHead d1 (Flat f) u1) H5) in (False_ind (ex2_2 C -T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c2 (Bind Abbr) u) (CHead d2 -(Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))) -H6)))))))))))) y c H0))) H)))))). -(* COMMENTS -Initial nodes: 1183 -END *) - -theorem csuba_gen_bind: - \forall (g: G).(\forall (b1: B).(\forall (e1: C).(\forall (c2: C).(\forall -(v1: T).((csuba g (CHead e1 (Bind b1) v1) c2) \to (ex2_3 B C T (\lambda (b2: -B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) -(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2)))))))))) -\def - \lambda (g: G).(\lambda (b1: B).(\lambda (e1: C).(\lambda (c2: C).(\lambda -(v1: T).(\lambda (H: (csuba g (CHead e1 (Bind b1) v1) c2)).(insert_eq C -(CHead e1 (Bind b1) v1) (\lambda (c: C).(csuba g c c2)) (\lambda (_: -C).(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 -(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: -T).(csuba g e1 e2)))))) (\lambda (y: C).(\lambda (H0: (csuba g y -c2)).(csuba_ind g (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead e1 (Bind -b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: -T).(eq C c0 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: -C).(\lambda (_: T).(csuba g e1 e2)))))))) (\lambda (n: nat).(\lambda (H1: (eq -C (CSort n) (CHead e1 (Bind b1) v1))).(let H2 \def (eq_ind C (CSort n) -(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 (Bind b1) -v1) H1) in (False_ind (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda -(v2: T).(eq C (CSort n) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda -(e2: C).(\lambda (_: T).(csuba g e1 e2))))) H2)))) (\lambda (c1: C).(\lambda -(c3: C).(\lambda (H1: (csuba g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 -(Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda -(v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: -C).(\lambda (_: T).(csuba g e1 e2)))))))).(\lambda (k: K).(\lambda (u: -T).(\lambda (H3: (eq C (CHead c1 k u) (CHead e1 (Bind b1) v1))).(let H4 \def -(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with -[(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u) -(CHead e1 (Bind b1) v1) H3) in ((let H5 \def (f_equal C K (\lambda (e: -C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | -(CHead _ k0 _) \Rightarrow k0])) (CHead c1 k u) (CHead e1 (Bind b1) v1) H3) -in ((let H6 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda -(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) -(CHead c1 k u) (CHead e1 (Bind b1) v1) H3) in (\lambda (H7: (eq K k (Bind -b1))).(\lambda (H8: (eq C c1 e1)).(eq_ind_r T v1 (\lambda (t: T).(ex2_3 B C T -(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 k t) -(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: -T).(csuba g e1 e2)))))) (eq_ind_r K (Bind b1) (\lambda (k0: K).(ex2_3 B C T -(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 k0 v1) -(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: -T).(csuba g e1 e2)))))) (let H9 \def (eq_ind C c1 (\lambda (c: C).((eq C c -(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: -C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: -B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2))))))) H2 e1 H8) in (let -H10 \def (eq_ind C c1 (\lambda (c: C).(csuba g c c3)) H1 e1 H8) in -(ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C -(CHead c3 (Bind b1) v1) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda -(e2: C).(\lambda (_: T).(csuba g e1 e2)))) b1 c3 v1 (refl_equal C (CHead c3 -(Bind b1) v1)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: -C).(\lambda (c3: C).(\lambda (H1: (csuba g c1 c3)).(\lambda (H2: (((eq C c1 -(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: -C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: -B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2)))))))).(\lambda (b: -B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: -T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) -v1))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda -(_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) -(CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) v1) H4) in ((let H6 \def -(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with -[(CSort _) \Rightarrow Void | (CHead _ k _) \Rightarrow (match k in K return -(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow -Void])])) (CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) v1) H4) in ((let H7 -\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead c1 -(Bind Void) u1) (CHead e1 (Bind b1) v1) H4) in (\lambda (H8: (eq B Void -b1)).(\lambda (H9: (eq C c1 e1)).(let H10 \def (eq_ind C c1 (\lambda (c: -C).((eq C c (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: -B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) -(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2))))))) H2 e1 -H9) in (let H11 \def (eq_ind C c1 (\lambda (c: C).(csuba g c c3)) H1 e1 H9) -in (let H12 \def (eq_ind_r B b1 (\lambda (b0: B).((eq C e1 (CHead e1 (Bind -b0) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: -T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: -C).(\lambda (_: T).(csuba g e1 e2))))))) H10 Void H8) in (ex2_3_intro B C T -(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind b) -u2) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: -T).(csuba g e1 e2)))) b c3 u2 (refl_equal C (CHead c3 (Bind b) u2)) -H11))))))) H6)) H5))))))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1: -(csuba g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind b1) v1)) \to (ex2_3 -B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 -(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g -e1 e2)))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (H3: (arity g c1 t -(asucc g a))).(\lambda (u: T).(\lambda (_: (arity g c3 u a)).(\lambda (H5: -(eq C (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1))).(let H6 \def -(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with -[(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 (Bind -Abst) t) (CHead e1 (Bind b1) v1) H5) in ((let H7 \def (f_equal C B (\lambda -(e: C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow -Abst | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with -[(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead c1 (Bind -Abst) t) (CHead e1 (Bind b1) v1) H5) in ((let H8 \def (f_equal C T (\lambda -(e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow t -| (CHead _ _ t0) \Rightarrow t0])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind -b1) v1) H5) in (\lambda (H9: (eq B Abst b1)).(\lambda (H10: (eq C c1 -e1)).(let H11 \def (eq_ind T t (\lambda (t0: T).(arity g c1 t0 (asucc g a))) -H3 v1 H8) in (let H12 \def (eq_ind C c1 (\lambda (c: C).(arity g c v1 (asucc -g a))) H11 e1 H10) in (let H13 \def (eq_ind C c1 (\lambda (c: C).((eq C c -(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: -C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: -B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2))))))) H2 e1 H10) in (let -H14 \def (eq_ind C c1 (\lambda (c: C).(csuba g c c3)) H1 e1 H10) in (let H15 -\def (eq_ind_r B b1 (\lambda (b: B).((eq C e1 (CHead e1 (Bind b) v1)) \to -(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 -(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: -T).(csuba g e1 e2))))))) H13 Abst H9) in (ex2_3_intro B C T (\lambda (b2: -B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 -(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g -e1 e2)))) Abbr c3 u (refl_equal C (CHead c3 (Bind Abbr) u)) H14))))))))) H7)) -H6)))))))))))) y c2 H0))) H)))))). -(* COMMENTS -Initial nodes: 1889 -END *) - -theorem csuba_gen_abst_rev: - \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g c -(CHead d1 (Bind Abst) u)) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead d2 -(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(eq C c (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1))))))))) -\def - \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u: T).(\lambda (H: -(csuba g c (CHead d1 (Bind Abst) u))).(insert_eq C (CHead d1 (Bind Abst) u) -(\lambda (c0: C).(csuba g c c0)) (\lambda (_: C).(or (ex2 C (\lambda (d2: -C).(eq C c (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (\lambda (y: -C).(\lambda (H0: (csuba g c y)).(csuba_ind g (\lambda (c0: C).(\lambda (c1: -C).((eq C c1 (CHead d1 (Bind Abst) u)) \to (or (ex2 C (\lambda (d2: C).(eq C -c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(eq C c0 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))) (\lambda (n: -nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Abst) u))).(let H2 \def -(eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return (\lambda (_: -C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow -False])) I (CHead d1 (Bind Abst) u) H1) in (False_ind (or (ex2 C (\lambda -(d2: C).(eq C (CSort n) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g -d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CSort n) (CHead -d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) -H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 -c2)).(\lambda (H2: (((eq C c2 (CHead d1 (Bind Abst) u)) \to (or (ex2 C -(\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba -g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))))))).(\lambda (k: K).(\lambda (u0: T).(\lambda (H3: (eq C (CHead c2 k -u0) (CHead d1 (Bind Abst) u))).(let H4 \def (f_equal C C (\lambda (e: -C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | -(CHead c0 _ _) \Rightarrow c0])) (CHead c2 k u0) (CHead d1 (Bind Abst) u) H3) -in ((let H5 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda -(_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) -(CHead c2 k u0) (CHead d1 (Bind Abst) u) H3) in ((let H6 \def (f_equal C T -(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) -\Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead c2 k u0) (CHead d1 -(Bind Abst) u) H3) in (\lambda (H7: (eq K k (Bind Abst))).(\lambda (H8: (eq C -c2 d1)).(eq_ind_r T u (\lambda (t: T).(or (ex2 C (\lambda (d2: C).(eq C -(CHead c1 k t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 k t) (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) -(eq_ind_r K (Bind Abst) (\lambda (k0: K).(or (ex2 C (\lambda (d2: C).(eq C -(CHead c1 k0 u) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 k0 u) (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let -H9 \def (eq_ind C c2 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abst) u)) \to -(or (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))))) H2 d1 H8) in (let H10 \def (eq_ind C c2 (\lambda (c0: C).(csuba g -c1 c0)) H1 d1 H8) in (or_introl (ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind -Abst) u) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 -C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 (Bind Abst) u) (CHead -d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) -(ex_intro2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abst) u) (CHead d2 (Bind -Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) c1 (refl_equal C (CHead c1 (Bind -Abst) u)) H10)))) k H7) u0 H6)))) H5)) H4))))))))) (\lambda (c1: C).(\lambda -(c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c2 (CHead d1 -(Bind Abst) u)) \to (or (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind -Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(eq C c1 (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))))))).(\lambda (b: B).(\lambda (H3: (not -(eq B b Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead -c2 (Bind b) u2) (CHead d1 (Bind Abst) u))).(let H5 \def (f_equal C C (\lambda -(e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 -| (CHead c0 _ _) \Rightarrow c0])) (CHead c2 (Bind b) u2) (CHead d1 (Bind -Abst) u) H4) in ((let H6 \def (f_equal C B (\lambda (e: C).(match e in C -return (\lambda (_: C).B) with [(CSort _) \Rightarrow b | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b0) -\Rightarrow b0 | (Flat _) \Rightarrow b])])) (CHead c2 (Bind b) u2) (CHead d1 -(Bind Abst) u) H4) in ((let H7 \def (f_equal C T (\lambda (e: C).(match e in -C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u2 | (CHead _ _ t) -\Rightarrow t])) (CHead c2 (Bind b) u2) (CHead d1 (Bind Abst) u) H4) in -(\lambda (H8: (eq B b Abst)).(\lambda (H9: (eq C c2 d1)).(let H10 \def -(eq_ind B b (\lambda (b0: B).(not (eq B b0 Void))) H3 Abst H8) in (let H11 -\def (eq_ind C c2 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abst) u)) \to -(or (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u3: T).(eq C c1 -(CHead d2 (Bind Void) u3)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))))) H2 d1 H9) in (let H12 \def (eq_ind C c2 (\lambda (c0: C).(csuba g -c1 c0)) H1 d1 H9) in (or_intror (ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind -Void) u1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u3: T).(eq C (CHead c1 (Bind Void) u1) -(CHead d2 (Bind Void) u3)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u3: T).(eq C (CHead c1 -(Bind Void) u1) (CHead d2 (Bind Void) u3)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))) c1 u1 (refl_equal C (CHead c1 (Bind Void) u1)) -H12)))))))) H6)) H5))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: -(csuba g c1 c2)).(\lambda (_: (((eq C c2 (CHead d1 (Bind Abst) u)) \to (or -(ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc -g a))).(\lambda (u0: T).(\lambda (_: (arity g c2 u0 a)).(\lambda (H5: (eq C -(CHead c2 (Bind Abbr) u0) (CHead d1 (Bind Abst) u))).(let H6 \def (eq_ind C -(CHead c2 (Bind Abbr) u0) (\lambda (ee: C).(match ee in C return (\lambda (_: -C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match -k in K return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B -return (\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow -False | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead -d1 (Bind Abst) u) H5) in (False_ind (or (ex2 C (\lambda (d2: C).(eq C (CHead -c1 (Bind Abst) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 -d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 (Bind -Abst) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba -g d2 d1))))) H6)))))))))))) c y H0))) H))))). -(* COMMENTS -Initial nodes: 1980 -END *) - -theorem csuba_gen_void_rev: - \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g c -(CHead d1 (Bind Void) u)) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind -Void) u))) (\lambda (d2: C).(csuba g d2 d1))))))) -\def - \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u: T).(\lambda (H: -(csuba g c (CHead d1 (Bind Void) u))).(insert_eq C (CHead d1 (Bind Void) u) -(\lambda (c0: C).(csuba g c c0)) (\lambda (_: C).(ex2 C (\lambda (d2: C).(eq -C c (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))) (\lambda -(y: C).(\lambda (H0: (csuba g c y)).(csuba_ind g (\lambda (c0: C).(\lambda -(c1: C).((eq C c1 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda (d2: C).(eq C -c0 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))))) (\lambda -(n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Void) u))).(let H2 -\def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return (\lambda (_: -C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow -False])) I (CHead d1 (Bind Void) u) H1) in (False_ind (ex2 C (\lambda (d2: -C).(eq C (CSort n) (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 -d1))) H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 -c2)).(\lambda (H2: (((eq C c2 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda -(d2: C).(eq C c1 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 -d1)))))).(\lambda (k: K).(\lambda (u0: T).(\lambda (H3: (eq C (CHead c2 k u0) -(CHead d1 (Bind Void) u))).(let H4 \def (f_equal C C (\lambda (e: C).(match e -in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ -_) \Rightarrow c0])) (CHead c2 k u0) (CHead d1 (Bind Void) u) H3) in ((let H5 -\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) -with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c2 k -u0) (CHead d1 (Bind Void) u) H3) in ((let H6 \def (f_equal C T (\lambda (e: -C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | -(CHead _ _ t) \Rightarrow t])) (CHead c2 k u0) (CHead d1 (Bind Void) u) H3) -in (\lambda (H7: (eq K k (Bind Void))).(\lambda (H8: (eq C c2 d1)).(eq_ind_r -T u (\lambda (t: T).(ex2 C (\lambda (d2: C).(eq C (CHead c1 k t) (CHead d2 -(Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))) (eq_ind_r K (Bind Void) -(\lambda (k0: K).(ex2 C (\lambda (d2: C).(eq C (CHead c1 k0 u) (CHead d2 -(Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))) (let H9 \def (eq_ind C -c2 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda -(d2: C).(eq C c1 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 -d1))))) H2 d1 H8) in (let H10 \def (eq_ind C c2 (\lambda (c0: C).(csuba g c1 -c0)) H1 d1 H8) in (ex_intro2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Void) -u) (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)) c1 -(refl_equal C (CHead c1 (Bind Void) u)) H10))) k H7) u0 H6)))) H5)) -H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 -c2)).(\lambda (H2: (((eq C c2 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda -(d2: C).(eq C c1 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 -d1)))))).(\lambda (b: B).(\lambda (H3: (not (eq B b Void))).(\lambda (u1: -T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c2 (Bind b) u2) (CHead d1 -(Bind Void) u))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C -return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ _) -\Rightarrow c0])) (CHead c2 (Bind b) u2) (CHead d1 (Bind Void) u) H4) in -((let H6 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: -C).B) with [(CSort _) \Rightarrow b | (CHead _ k _) \Rightarrow (match k in K -return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) -\Rightarrow b])])) (CHead c2 (Bind b) u2) (CHead d1 (Bind Void) u) H4) in -((let H7 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: -C).T) with [(CSort _) \Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead -c2 (Bind b) u2) (CHead d1 (Bind Void) u) H4) in (\lambda (H8: (eq B b -Void)).(\lambda (H9: (eq C c2 d1)).(let H10 \def (eq_ind B b (\lambda (b0: -B).(not (eq B b0 Void))) H3 Void H8) in (let H11 \def (eq_ind C c2 (\lambda -(c0: C).((eq C c0 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda (d2: C).(eq C -c1 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1))))) H2 d1 H9) -in (let H12 \def (eq_ind C c2 (\lambda (c0: C).(csuba g c1 c0)) H1 d1 H9) in -(let H13 \def (match (H10 (refl_equal B Void)) in False return (\lambda (_: -False).(ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Void) u1) (CHead d2 -(Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))) with []) in H13))))))) -H6)) H5))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1 -c2)).(\lambda (_: (((eq C c2 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda -(d2: C).(eq C c1 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 -d1)))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g -a))).(\lambda (u0: T).(\lambda (_: (arity g c2 u0 a)).(\lambda (H5: (eq C -(CHead c2 (Bind Abbr) u0) (CHead d1 (Bind Void) u))).(let H6 \def (eq_ind C -(CHead c2 (Bind Abbr) u0) (\lambda (ee: C).(match ee in C return (\lambda (_: -C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match -k in K return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B -return (\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow -False | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead -d1 (Bind Void) u) H5) in (False_ind (ex2 C (\lambda (d2: C).(eq C (CHead c1 -(Bind Abst) t) (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1))) -H6)))))))))))) c y H0))) H))))). -(* COMMENTS -Initial nodes: 1326 -END *) - -theorem csuba_gen_abbr_rev: - \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).((csuba g c -(CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(eq C c (CHead d2 -(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))) -\def - \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda -(H: (csuba g c (CHead d1 (Bind Abbr) u1))).(insert_eq C (CHead d1 (Bind Abbr) -u1) (\lambda (c0: C).(csuba g c c0)) (\lambda (_: C).(or3 (ex2 C (\lambda -(d2: C).(eq C c (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq -C c (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))))) (\lambda (y: C).(\lambda (H0: (csuba g c y)).(csuba_ind g (\lambda -(c0: C).(\lambda (c1: C).((eq C c1 (CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C -(\lambda (d2: C).(eq C c0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba -g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq -C c0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq -C c0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g -d2 d1)))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind -Abbr) u1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C -return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) -\Rightarrow False])) I (CHead d1 (Bind Abbr) u1) H1) in (False_ind (or3 (ex2 -C (\lambda (d2: C).(eq C (CSort n) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(eq C (CSort n) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(eq C (CSort n) (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) H2)))) (\lambda -(c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C -c2 (CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(eq C c1 -(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c2 k u) -(CHead d1 (Bind Abbr) u1))).(let H4 \def (f_equal C C (\lambda (e: C).(match -e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ -_) \Rightarrow c0])) (CHead c2 k u) (CHead d1 (Bind Abbr) u1) H3) in ((let H5 -\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) -with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c2 k -u) (CHead d1 (Bind Abbr) u1) H3) in ((let H6 \def (f_equal C T (\lambda (e: -C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | -(CHead _ _ t) \Rightarrow t])) (CHead c2 k u) (CHead d1 (Bind Abbr) u1) H3) -in (\lambda (H7: (eq K k (Bind Abbr))).(\lambda (H8: (eq C c2 d1)).(eq_ind_r -T u1 (\lambda (t: T).(or3 (ex2 C (\lambda (d2: C).(eq C (CHead c1 k t) (CHead -d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c1 k t) (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 k t) -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))))) (eq_ind_r K (Bind Abbr) (\lambda (k0: K).(or3 (ex2 C (\lambda (d2: -C).(eq C (CHead c1 k0 u1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba -g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq -C (CHead c1 k0 u1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda -(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(eq C (CHead c1 k0 u1) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H9 \def (eq_ind C -c2 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C -(\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba -g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq -C c1 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq -C c1 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g -d2 d1))))))) H2 d1 H8) in (let H10 \def (eq_ind C c2 (\lambda (c0: C).(csuba -g c1 c0)) H1 d1 H8) in (or3_intro0 (ex2 C (\lambda (d2: C).(eq C (CHead c1 -(Bind Abbr) u1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C -(CHead c1 (Bind Abbr) u1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 (Bind Abbr) u1) (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) -(ex_intro2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abbr) u1) (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) c1 (refl_equal C (CHead c1 -(Bind Abbr) u1)) H10)))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: -C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c2 -(CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(eq C c1 (CHead -d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))).(\lambda (b: -B).(\lambda (H3: (not (eq B b Void))).(\lambda (u0: T).(\lambda (u2: -T).(\lambda (H4: (eq C (CHead c2 (Bind b) u2) (CHead d1 (Bind Abbr) -u1))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda -(_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) -(CHead c2 (Bind b) u2) (CHead d1 (Bind Abbr) u1) H4) in ((let H6 \def -(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with -[(CSort _) \Rightarrow b | (CHead _ k _) \Rightarrow (match k in K return -(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow -b])])) (CHead c2 (Bind b) u2) (CHead d1 (Bind Abbr) u1) H4) in ((let H7 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c2 (Bind b) -u2) (CHead d1 (Bind Abbr) u1) H4) in (\lambda (H8: (eq B b Abbr)).(\lambda -(H9: (eq C c2 d1)).(let H10 \def (eq_ind B b (\lambda (b0: B).(not (eq B b0 -Void))) H3 Abbr H8) in (let H11 \def (eq_ind C c2 (\lambda (c0: C).((eq C c0 -(CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(eq C c1 (CHead -d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u3: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind Abst) -u3))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u3: T).(\lambda (a: A).(arity g d2 u3 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u3: T).(eq C c1 (CHead d2 (Bind Void) -u3)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))) H2 d1 H9) in -(let H12 \def (eq_ind C c2 (\lambda (c0: C).(csuba g c1 c0)) H1 d1 H9) in -(or3_intro2 (ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Void) u0) (CHead d2 -(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u3: T).(\lambda (_: A).(eq C (CHead c1 (Bind Void) u0) -(CHead d2 (Bind Abst) u3))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u3: T).(\lambda (a: -A).(arity g d2 u3 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u3: T).(eq -C (CHead c1 (Bind Void) u0) (CHead d2 (Bind Void) u3)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: -C).(\lambda (u3: T).(eq C (CHead c1 (Bind Void) u0) (CHead d2 (Bind Void) -u3)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) c1 u0 (refl_equal C -(CHead c1 (Bind Void) u0)) H12)))))))) H6)) H5))))))))))) (\lambda (c1: -C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c2 -(CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(eq C c1 (CHead -d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))).(\lambda (t: -T).(\lambda (a: A).(\lambda (H3: (arity g c1 t (asucc g a))).(\lambda (u: -T).(\lambda (H4: (arity g c2 u a)).(\lambda (H5: (eq C (CHead c2 (Bind Abbr) -u) (CHead d1 (Bind Abbr) u1))).(let H6 \def (f_equal C C (\lambda (e: -C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | -(CHead c0 _ _) \Rightarrow c0])) (CHead c2 (Bind Abbr) u) (CHead d1 (Bind -Abbr) u1) H5) in ((let H7 \def (f_equal C T (\lambda (e: C).(match e in C -return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) -\Rightarrow t0])) (CHead c2 (Bind Abbr) u) (CHead d1 (Bind Abbr) u1) H5) in -(\lambda (H8: (eq C c2 d1)).(let H9 \def (eq_ind T u (\lambda (t0: T).(arity -g c2 t0 a)) H4 u1 H7) in (let H10 \def (eq_ind C c2 (\lambda (c0: C).(arity g -c0 u1 a)) H9 d1 H8) in (let H11 \def (eq_ind C c2 (\lambda (c0: C).((eq C c0 -(CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(eq C c1 (CHead -d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 (asucc g -a0))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 -a0))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))) H2 d1 H8) -in (let H12 \def (eq_ind C c2 (\lambda (c0: C).(csuba g c1 c0)) H1 d1 H8) in -(or3_intro1 (ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abst) t) (CHead d2 -(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) t) -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: -A).(arity g d2 u2 (asucc g a0))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a0: A).(arity g d1 u1 a0))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(eq C (CHead c1 (Bind Abst) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))) (ex4_3_intro C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) t) (CHead d2 -(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 -(asucc g a0))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 -u1 a0)))) c1 t a (refl_equal C (CHead c1 (Bind Abst) t)) H12 H3 H10)))))))) -H6)))))))))))) c y H0))) H))))). -(* COMMENTS -Initial nodes: 3459 -END *) - -theorem csuba_gen_flat_rev: - \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).(\forall -(f: F).((csuba g c (CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1))))))))) -\def - \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda -(f: F).(\lambda (H: (csuba g c (CHead d1 (Flat f) u1))).(insert_eq C (CHead -d1 (Flat f) u1) (\lambda (c0: C).(csuba g c c0)) (\lambda (_: C).(ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (y: C).(\lambda (H0: -(csuba g c y)).(csuba_ind g (\lambda (c0: C).(\lambda (c1: C).((eq C c1 -(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq -C c0 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Flat f) -u1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return -(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) -\Rightarrow False])) I (CHead d1 (Flat f) u1) H1) in (False_ind (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(eq C (CSort n) (CHead d2 (Flat f) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) H2)))) (\lambda (c1: -C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c2 -(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq -C c1 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c2 k u) -(CHead d1 (Flat f) u1))).(let H4 \def (f_equal C C (\lambda (e: C).(match e -in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ -_) \Rightarrow c0])) (CHead c2 k u) (CHead d1 (Flat f) u1) H3) in ((let H5 -\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) -with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c2 k -u) (CHead d1 (Flat f) u1) H3) in ((let H6 \def (f_equal C T (\lambda (e: -C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | -(CHead _ _ t) \Rightarrow t])) (CHead c2 k u) (CHead d1 (Flat f) u1) H3) in -(\lambda (H7: (eq K k (Flat f))).(\lambda (H8: (eq C c2 d1)).(eq_ind_r T u1 -(\lambda (t: T).(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 -k t) (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))) (eq_ind_r K (Flat f) (\lambda (k0: K).(ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(eq C (CHead c1 k0 u1) (CHead d2 (Flat f) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (let H9 \def (eq_ind C c2 -(\lambda (c0: C).((eq C c0 (CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda -(d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 (Flat f) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))))) H2 d1 H8) in (let H10 \def (eq_ind C -c2 (\lambda (c0: C).(csuba g c1 c0)) H1 d1 H8) in (ex2_2_intro C T (\lambda -(d2: C).(\lambda (u2: T).(eq C (CHead c1 (Flat f) u1) (CHead d2 (Flat f) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) c1 u1 (refl_equal C -(CHead c1 (Flat f) u1)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: -C).(\lambda (c2: C).(\lambda (_: (csuba g c1 c2)).(\lambda (_: (((eq C c2 -(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq -C c1 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u0: -T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c2 (Bind b) u2) (CHead d1 -(Flat f) u1))).(let H5 \def (eq_ind C (CHead c2 (Bind b) u2) (\lambda (ee: -C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow -False | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) -with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead d1 -(Flat f) u1) H4) in (False_ind (ex2_2 C T (\lambda (d2: C).(\lambda (u3: -T).(eq C (CHead c1 (Bind Void) u0) (CHead d2 (Flat f) u3)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))) H5))))))))))) (\lambda (c1: C).(\lambda -(c2: C).(\lambda (_: (csuba g c1 c2)).(\lambda (_: (((eq C c2 (CHead d1 (Flat -f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 -(Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g -a))).(\lambda (u: T).(\lambda (_: (arity g c2 u a)).(\lambda (H5: (eq C -(CHead c2 (Bind Abbr) u) (CHead d1 (Flat f) u1))).(let H6 \def (eq_ind C -(CHead c2 (Bind Abbr) u) (\lambda (ee: C).(match ee in C return (\lambda (_: -C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match -k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat -_) \Rightarrow False])])) I (CHead d1 (Flat f) u1) H5) in (False_ind (ex2_2 C -T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 (Bind Abst) t) (CHead d2 -(Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) -H6)))))))))))) c y H0))) H)))))). -(* COMMENTS -Initial nodes: 1183 -END *) - -theorem csuba_gen_bind_rev: - \forall (g: G).(\forall (b1: B).(\forall (e1: C).(\forall (c2: C).(\forall -(v1: T).((csuba g c2 (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: -B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) -(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1)))))))))) -\def - \lambda (g: G).(\lambda (b1: B).(\lambda (e1: C).(\lambda (c2: C).(\lambda -(v1: T).(\lambda (H: (csuba g c2 (CHead e1 (Bind b1) v1))).(insert_eq C -(CHead e1 (Bind b1) v1) (\lambda (c: C).(csuba g c2 c)) (\lambda (_: -C).(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 -(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: -T).(csuba g e2 e1)))))) (\lambda (y: C).(\lambda (H0: (csuba g c2 -y)).(csuba_ind g (\lambda (c: C).(\lambda (c0: C).((eq C c0 (CHead e1 (Bind -b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: -T).(eq C c (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: -C).(\lambda (_: T).(csuba g e2 e1)))))))) (\lambda (n: nat).(\lambda (H1: (eq -C (CSort n) (CHead e1 (Bind b1) v1))).(let H2 \def (eq_ind C (CSort n) -(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 (Bind b1) -v1) H1) in (False_ind (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda -(v2: T).(eq C (CSort n) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda -(e2: C).(\lambda (_: T).(csuba g e2 e1))))) H2)))) (\lambda (c1: C).(\lambda -(c3: C).(\lambda (H1: (csuba g c1 c3)).(\lambda (H2: (((eq C c3 (CHead e1 -(Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda -(v2: T).(eq C c1 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: -C).(\lambda (_: T).(csuba g e2 e1)))))))).(\lambda (k: K).(\lambda (u: -T).(\lambda (H3: (eq C (CHead c3 k u) (CHead e1 (Bind b1) v1))).(let H4 \def -(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with -[(CSort _) \Rightarrow c3 | (CHead c _ _) \Rightarrow c])) (CHead c3 k u) -(CHead e1 (Bind b1) v1) H3) in ((let H5 \def (f_equal C K (\lambda (e: -C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | -(CHead _ k0 _) \Rightarrow k0])) (CHead c3 k u) (CHead e1 (Bind b1) v1) H3) -in ((let H6 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda -(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) -(CHead c3 k u) (CHead e1 (Bind b1) v1) H3) in (\lambda (H7: (eq K k (Bind -b1))).(\lambda (H8: (eq C c3 e1)).(eq_ind_r T v1 (\lambda (t: T).(ex2_3 B C T -(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c1 k t) -(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: -T).(csuba g e2 e1)))))) (eq_ind_r K (Bind b1) (\lambda (k0: K).(ex2_3 B C T -(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c1 k0 v1) -(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: -T).(csuba g e2 e1)))))) (let H9 \def (eq_ind C c3 (\lambda (c: C).((eq C c -(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: -C).(\lambda (v2: T).(eq C c1 (CHead e2 (Bind b2) v2))))) (\lambda (_: -B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1))))))) H2 e1 H8) in (let -H10 \def (eq_ind C c3 (\lambda (c: C).(csuba g c1 c)) H1 e1 H8) in -(ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C -(CHead c1 (Bind b1) v1) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda -(e2: C).(\lambda (_: T).(csuba g e2 e1)))) b1 c1 v1 (refl_equal C (CHead c1 -(Bind b1) v1)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: -C).(\lambda (c3: C).(\lambda (H1: (csuba g c1 c3)).(\lambda (H2: (((eq C c3 -(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: -C).(\lambda (v2: T).(eq C c1 (CHead e2 (Bind b2) v2))))) (\lambda (_: -B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1)))))))).(\lambda (b: -B).(\lambda (H3: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: -T).(\lambda (H4: (eq C (CHead c3 (Bind b) u2) (CHead e1 (Bind b1) v1))).(let -H5 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) -with [(CSort _) \Rightarrow c3 | (CHead c _ _) \Rightarrow c])) (CHead c3 -(Bind b) u2) (CHead e1 (Bind b1) v1) H4) in ((let H6 \def (f_equal C B -(\lambda (e: C).(match e in C return (\lambda (_: C).B) with [(CSort _) -\Rightarrow b | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: -K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow b])])) (CHead c3 -(Bind b) u2) (CHead e1 (Bind b1) v1) H4) in ((let H7 \def (f_equal C T -(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) -\Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c3 (Bind b) u2) (CHead -e1 (Bind b1) v1) H4) in (\lambda (H8: (eq B b b1)).(\lambda (H9: (eq C c3 -e1)).(let H10 \def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Void))) H3 b1 -H8) in (let H11 \def (eq_ind C c3 (\lambda (c: C).((eq C c (CHead e1 (Bind -b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: -T).(eq C c1 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: -C).(\lambda (_: T).(csuba g e2 e1))))))) H2 e1 H9) in (let H12 \def (eq_ind C -c3 (\lambda (c: C).(csuba g c1 c)) H1 e1 H9) in (ex2_3_intro B C T (\lambda -(b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c1 (Bind Void) u1) -(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: -T).(csuba g e2 e1)))) Void c1 u1 (refl_equal C (CHead c1 (Bind Void) u1)) -H12))))))) H6)) H5))))))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1: -(csuba g c1 c3)).(\lambda (H2: (((eq C c3 (CHead e1 (Bind b1) v1)) \to (ex2_3 -B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c1 (CHead e2 -(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g -e2 e1)))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t -(asucc g a))).(\lambda (u: T).(\lambda (H4: (arity g c3 u a)).(\lambda (H5: -(eq C (CHead c3 (Bind Abbr) u) (CHead e1 (Bind b1) v1))).(let H6 \def -(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with -[(CSort _) \Rightarrow c3 | (CHead c _ _) \Rightarrow c])) (CHead c3 (Bind -Abbr) u) (CHead e1 (Bind b1) v1) H5) in ((let H7 \def (f_equal C B (\lambda -(e: C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow -Abbr | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with -[(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead c3 (Bind -Abbr) u) (CHead e1 (Bind b1) v1) H5) in ((let H8 \def (f_equal C T (\lambda -(e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u -| (CHead _ _ t0) \Rightarrow t0])) (CHead c3 (Bind Abbr) u) (CHead e1 (Bind -b1) v1) H5) in (\lambda (H9: (eq B Abbr b1)).(\lambda (H10: (eq C c3 -e1)).(let H11 \def (eq_ind T u (\lambda (t0: T).(arity g c3 t0 a)) H4 v1 H8) -in (let H12 \def (eq_ind C c3 (\lambda (c: C).(arity g c v1 a)) H11 e1 H10) -in (let H13 \def (eq_ind C c3 (\lambda (c: C).((eq C c (CHead e1 (Bind b1) -v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq -C c1 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda -(_: T).(csuba g e2 e1))))))) H2 e1 H10) in (let H14 \def (eq_ind C c3 -(\lambda (c: C).(csuba g c1 c)) H1 e1 H10) in (let H15 \def (eq_ind_r B b1 -(\lambda (b: B).((eq C e1 (CHead e1 (Bind b) v1)) \to (ex2_3 B C T (\lambda -(b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c1 (CHead e2 (Bind b2) -v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 -e1))))))) H13 Abbr H9) in (ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: -C).(\lambda (v2: T).(eq C (CHead c1 (Bind Abst) t) (CHead e2 (Bind b2) -v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1)))) -Abst c1 t (refl_equal C (CHead c1 (Bind Abst) t)) H14))))))))) H7)) -H6)))))))))))) c2 y H0))) H)))))). -(* COMMENTS -Initial nodes: 1831 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/getl.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/getl.ma deleted file mode 100644 index b49d9b7c2..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/getl.ma +++ /dev/null @@ -1,1178 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csuba/drop.ma". - -include "Basic-1/csuba/clear.ma". - -include "Basic-1/getl/clear.ma". - -theorem csuba_getl_abbr: - \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).(\forall -(i: nat).((getl i c1 (CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csuba g -c1 c2) \to (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) -(\lambda (d2: C).(csuba g d1 d2)))))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u: T).(\lambda -(i: nat).(\lambda (H: (getl i c1 (CHead d1 (Bind Abbr) u))).(let H0 \def -(getl_gen_all c1 (CHead d1 (Bind Abbr) u) i H) in (ex2_ind C (\lambda (e: -C).(drop i O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abbr) u))) -(\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(getl i c2 -(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))) (\lambda (x: -C).(\lambda (H1: (drop i O c1 x)).(\lambda (H2: (clear x (CHead d1 (Bind -Abbr) u))).(C_ind (\lambda (c: C).((drop i O c1 c) \to ((clear c (CHead d1 -(Bind Abbr) u)) \to (\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda -(d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 -d2)))))))) (\lambda (n: nat).(\lambda (_: (drop i O c1 (CSort n))).(\lambda -(H4: (clear (CSort n) (CHead d1 (Bind Abbr) u))).(clear_gen_sort (CHead d1 -(Bind Abbr) u) n H4 (\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda -(d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 -d2))))))))) (\lambda (x0: C).(\lambda (_: (((drop i O c1 x0) \to ((clear x0 -(CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csuba g c1 c2) \to (ex2 C -(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: -C).(csuba g d1 d2))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (H3: -(drop i O c1 (CHead x0 k t))).(\lambda (H4: (clear (CHead x0 k t) (CHead d1 -(Bind Abbr) u))).(K_ind (\lambda (k0: K).((drop i O c1 (CHead x0 k0 t)) \to -((clear (CHead x0 k0 t) (CHead d1 (Bind Abbr) u)) \to (\forall (c2: -C).((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind -Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))))) (\lambda (b: B).(\lambda -(H5: (drop i O c1 (CHead x0 (Bind b) t))).(\lambda (H6: (clear (CHead x0 -(Bind b) t) (CHead d1 (Bind Abbr) u))).(let H7 \def (f_equal C C (\lambda (e: -C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d1 | -(CHead c _ _) \Rightarrow c])) (CHead d1 (Bind Abbr) u) (CHead x0 (Bind b) t) -(clear_gen_bind b x0 (CHead d1 (Bind Abbr) u) t H6)) in ((let H8 \def -(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with -[(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow (match k0 in K -return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) -\Rightarrow Abbr])])) (CHead d1 (Bind Abbr) u) (CHead x0 (Bind b) t) -(clear_gen_bind b x0 (CHead d1 (Bind Abbr) u) t H6)) in ((let H9 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d1 (Bind -Abbr) u) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abbr) u) -t H6)) in (\lambda (H10: (eq B Abbr b)).(\lambda (H11: (eq C d1 x0)).(\lambda -(c2: C).(\lambda (H12: (csuba g c1 c2)).(let H13 \def (eq_ind_r T t (\lambda -(t0: T).(drop i O c1 (CHead x0 (Bind b) t0))) H5 u H9) in (let H14 \def -(eq_ind_r B b (\lambda (b0: B).(drop i O c1 (CHead x0 (Bind b0) u))) H13 Abbr -H10) in (let H15 \def (eq_ind_r C x0 (\lambda (c: C).(drop i O c1 (CHead c -(Bind Abbr) u))) H14 d1 H11) in (let H16 \def (csuba_drop_abbr i c1 d1 u H15 -g c2 H12) in (ex2_ind C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr) -u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(getl i c2 -(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x1: -C).(\lambda (H17: (drop i O c2 (CHead x1 (Bind Abbr) u))).(\lambda (H18: -(csuba g d1 x1)).(ex_intro2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind -Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x1 (getl_intro i c2 (CHead x1 -(Bind Abbr) u) (CHead x1 (Bind Abbr) u) H17 (clear_bind Abbr x1 u)) H18)))) -H16)))))))))) H8)) H7))))) (\lambda (f: F).(\lambda (H5: (drop i O c1 (CHead -x0 (Flat f) t))).(\lambda (H6: (clear (CHead x0 (Flat f) t) (CHead d1 (Bind -Abbr) u))).(let H7 \def H5 in (unintro C c1 (\lambda (c: C).((drop i O c -(CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g c c2) \to (ex2 C -(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: -C).(csuba g d1 d2))))))) (nat_ind (\lambda (n: nat).(\forall (x1: C).((drop n -O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g x1 c2) \to (ex2 C -(\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: -C).(csuba g d1 d2)))))))) (\lambda (x1: C).(\lambda (H8: (drop O O x1 (CHead -x0 (Flat f) t))).(\lambda (c2: C).(\lambda (H9: (csuba g x1 c2)).(let H10 -\def (eq_ind C x1 (\lambda (c: C).(csuba g c c2)) H9 (CHead x0 (Flat f) t) -(drop_gen_refl x1 (CHead x0 (Flat f) t) H8)) in (let H_y \def (clear_flat x0 -(CHead d1 (Bind Abbr) u) (clear_gen_flat f x0 (CHead d1 (Bind Abbr) u) t H6) -f t) in (let H11 \def (csuba_clear_conf g (CHead x0 (Flat f) t) c2 H10 (CHead -d1 (Bind Abbr) u) H_y) in (ex2_ind C (\lambda (e2: C).(csuba g (CHead d1 -(Bind Abbr) u) e2)) (\lambda (e2: C).(clear c2 e2)) (ex2 C (\lambda (d2: -C).(getl O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) -(\lambda (x2: C).(\lambda (H12: (csuba g (CHead d1 (Bind Abbr) u) -x2)).(\lambda (H13: (clear c2 x2)).(let H_x \def (csuba_gen_abbr g d1 x2 u -H12) in (let H14 \def H_x in (ex2_ind C (\lambda (d2: C).(eq C x2 (CHead d2 -(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: -C).(getl O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) -(\lambda (x3: C).(\lambda (H15: (eq C x2 (CHead x3 (Bind Abbr) u))).(\lambda -(H16: (csuba g d1 x3)).(let H17 \def (eq_ind C x2 (\lambda (c: C).(clear c2 -c)) H13 (CHead x3 (Bind Abbr) u) H15) in (ex_intro2 C (\lambda (d2: C).(getl -O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x3 -(getl_intro O c2 (CHead x3 (Bind Abbr) u) c2 (drop_refl c2) H17) H16))))) -H14)))))) H11)))))))) (\lambda (n: nat).(\lambda (H8: ((\forall (x1: -C).((drop n O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g x1 c2) -\to (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))) (\lambda -(d2: C).(csuba g d1 d2))))))))).(\lambda (x1: C).(\lambda (H9: (drop (S n) O -x1 (CHead x0 (Flat f) t))).(\lambda (c2: C).(\lambda (H10: (csuba g x1 -c2)).(let H11 \def (drop_clear x1 (CHead x0 (Flat f) t) n H9) in (ex2_3_ind B -C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear x1 (CHead e (Bind -b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop n O e (CHead -x0 (Flat f) t))))) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind -Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x2: B).(\lambda (x3: -C).(\lambda (x4: T).(\lambda (H12: (clear x1 (CHead x3 (Bind x2) -x4))).(\lambda (H13: (drop n O x3 (CHead x0 (Flat f) t))).(let H14 \def -(csuba_clear_conf g x1 c2 H10 (CHead x3 (Bind x2) x4) H12) in (ex2_ind C -(\lambda (e2: C).(csuba g (CHead x3 (Bind x2) x4) e2)) (\lambda (e2: -C).(clear c2 e2)) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind -Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x5: C).(\lambda (H15: -(csuba g (CHead x3 (Bind x2) x4) x5)).(\lambda (H16: (clear c2 x5)).(let H_x -\def (csuba_gen_bind g x2 x3 x5 x4 H15) in (let H17 \def H_x in (ex2_3_ind B -C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C x5 (CHead e2 -(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g -x3 e2)))) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) u))) -(\lambda (d2: C).(csuba g d1 d2))) (\lambda (x6: B).(\lambda (x7: C).(\lambda -(x8: T).(\lambda (H18: (eq C x5 (CHead x7 (Bind x6) x8))).(\lambda (H19: -(csuba g x3 x7)).(let H20 \def (eq_ind C x5 (\lambda (c: C).(clear c2 c)) H16 -(CHead x7 (Bind x6) x8) H18) in (let H21 \def (H8 x3 H13 x7 H19) in (ex2_ind -C (\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abbr) u))) (\lambda (d2: -C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind -Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x9: C).(\lambda (H22: -(getl n x7 (CHead x9 (Bind Abbr) u))).(\lambda (H23: (csuba g d1 -x9)).(ex_intro2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) u))) -(\lambda (d2: C).(csuba g d1 d2)) x9 (getl_clear_bind x6 c2 x7 x8 H20 (CHead -x9 (Bind Abbr) u) n H22) H23)))) H21)))))))) H17)))))) H14))))))) H11)))))))) -i) H7))))) k H3 H4))))))) x H1 H2)))) H0))))))). -(* COMMENTS -Initial nodes: 2319 -END *) - -theorem csuba_getl_abst: - \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u1: T).(\forall -(i: nat).((getl i c1 (CHead d1 (Bind Abst) u1)) \to (\forall (c2: C).((csuba -g c1 c2) \to (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) -u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a))))))))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u1: T).(\lambda -(i: nat).(\lambda (H: (getl i c1 (CHead d1 (Bind Abst) u1))).(let H0 \def -(getl_gen_all c1 (CHead d1 (Bind Abst) u1) i H) in (ex2_ind C (\lambda (e: -C).(drop i O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abst) u1))) -(\forall (c2: C).((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(getl i c2 -(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind -Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 -d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc -g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))))) (\lambda (x: C).(\lambda (H1: (drop i O c1 x)).(\lambda (H2: (clear -x (CHead d1 (Bind Abst) u1))).(C_ind (\lambda (c: C).((drop i O c1 c) \to -((clear c (CHead d1 (Bind Abst) u1)) \to (\forall (c2: C).((csuba g c1 c2) -\to (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) -(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))))) (\lambda -(n: nat).(\lambda (_: (drop i O c1 (CSort n))).(\lambda (H4: (clear (CSort n) -(CHead d1 (Bind Abst) u1))).(clear_gen_sort (CHead d1 (Bind Abst) u1) n H4 -(\forall (c2: C).((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(getl i c2 -(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind -Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 -d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc -g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))))))))) (\lambda (x0: C).(\lambda (_: (((drop i O c1 x0) \to ((clear x0 -(CHead d1 (Bind Abst) u1)) \to (\forall (c2: C).((csuba g c1 c2) \to (or (ex2 -C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: -C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))))))))).(\lambda (k: K).(\lambda -(t: T).(\lambda (H3: (drop i O c1 (CHead x0 k t))).(\lambda (H4: (clear -(CHead x0 k t) (CHead d1 (Bind Abst) u1))).(K_ind (\lambda (k0: K).((drop i O -c1 (CHead x0 k0 t)) \to ((clear (CHead x0 k0 t) (CHead d1 (Bind Abst) u1)) -\to (\forall (c2: C).((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(getl i -c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T -A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 -(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 -(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 -u2 a))))))))))) (\lambda (b: B).(\lambda (H5: (drop i O c1 (CHead x0 (Bind b) -t))).(\lambda (H6: (clear (CHead x0 (Bind b) t) (CHead d1 (Bind Abst) -u1))).(let H7 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda -(_: C).C) with [(CSort _) \Rightarrow d1 | (CHead c _ _) \Rightarrow c])) -(CHead d1 (Bind Abst) u1) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead -d1 (Bind Abst) u1) t H6)) in ((let H8 \def (f_equal C B (\lambda (e: -C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abst | -(CHead _ k0 _) \Rightarrow (match k0 in K return (\lambda (_: K).B) with -[(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abst])])) (CHead d1 (Bind -Abst) u1) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abst) -u1) t H6)) in ((let H9 \def (f_equal C T (\lambda (e: C).(match e in C return -(\lambda (_: C).T) with [(CSort _) \Rightarrow u1 | (CHead _ _ t0) -\Rightarrow t0])) (CHead d1 (Bind Abst) u1) (CHead x0 (Bind b) t) -(clear_gen_bind b x0 (CHead d1 (Bind Abst) u1) t H6)) in (\lambda (H10: (eq B -Abst b)).(\lambda (H11: (eq C d1 x0)).(\lambda (c2: C).(\lambda (H12: (csuba -g c1 c2)).(let H13 \def (eq_ind_r T t (\lambda (t0: T).(drop i O c1 (CHead x0 -(Bind b) t0))) H5 u1 H9) in (let H14 \def (eq_ind_r B b (\lambda (b0: -B).(drop i O c1 (CHead x0 (Bind b0) u1))) H13 Abst H10) in (let H15 \def -(eq_ind_r C x0 (\lambda (c: C).(drop i O c1 (CHead c (Bind Abst) u1))) H14 d1 -H11) in (let H16 \def (csuba_drop_abst i c1 d1 u1 H15 g c2 H12) in (or_ind -(ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst) u1))) (\lambda -(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (or (ex2 C -(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: -C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (H17: (ex2 C (\lambda -(d2: C).(drop i O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 -d2)))).(ex2_ind C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst) u1))) -(\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: C).(getl i c2 -(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind -Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 -d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc -g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))) (\lambda (x1: C).(\lambda (H18: (drop i O c2 (CHead x1 (Bind Abst) -u1))).(\lambda (H19: (csuba g d1 x1)).(or_introl (ex2 C (\lambda (d2: -C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 -(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity -g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: C).(getl i c2 (CHead d2 -(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) x1 (getl_intro i c2 -(CHead x1 (Bind Abst) u1) (CHead x1 (Bind Abst) u1) H18 (clear_bind Abst x1 -u1)) H19))))) H17)) (\lambda (H17: (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))).(ex4_3_ind C -T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 -(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 -(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 -u2 a)))) (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) -(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x1: -C).(\lambda (x2: T).(\lambda (x3: A).(\lambda (H18: (drop i O c2 (CHead x1 -(Bind Abbr) x2))).(\lambda (H19: (csuba g d1 x1)).(\lambda (H20: (arity g d1 -u1 (asucc g x3))).(\lambda (H21: (arity g x1 x2 x3)).(or_intror (ex2 C -(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: -C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) -x1 x2 x3 (getl_intro i c2 (CHead x1 (Bind Abbr) x2) (CHead x1 (Bind Abbr) x2) -H18 (clear_bind Abbr x1 x2)) H19 H20 H21))))))))) H17)) H16)))))))))) H8)) -H7))))) (\lambda (f: F).(\lambda (H5: (drop i O c1 (CHead x0 (Flat f) -t))).(\lambda (H6: (clear (CHead x0 (Flat f) t) (CHead d1 (Bind Abst) -u1))).(let H7 \def H5 in (unintro C c1 (\lambda (c: C).((drop i O c (CHead x0 -(Flat f) t)) \to (\forall (c2: C).((csuba g c c2) \to (or (ex2 C (\lambda -(d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 -d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i -c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: -A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda -(a: A).(arity g d2 u2 a)))))))))) (nat_ind (\lambda (n: nat).(\forall (x1: -C).((drop n O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g x1 c2) -\to (or (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) u1))) -(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(getl n c2 (CHead d2 (Bind Abbr) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))))) (\lambda -(x1: C).(\lambda (H8: (drop O O x1 (CHead x0 (Flat f) t))).(\lambda (c2: -C).(\lambda (H9: (csuba g x1 c2)).(let H10 \def (eq_ind C x1 (\lambda (c: -C).(csuba g c c2)) H9 (CHead x0 (Flat f) t) (drop_gen_refl x1 (CHead x0 (Flat -f) t) H8)) in (let H_y \def (clear_flat x0 (CHead d1 (Bind Abst) u1) -(clear_gen_flat f x0 (CHead d1 (Bind Abst) u1) t H6) f t) in (let H11 \def -(csuba_clear_conf g (CHead x0 (Flat f) t) c2 H10 (CHead d1 (Bind Abst) u1) -H_y) in (ex2_ind C (\lambda (e2: C).(csuba g (CHead d1 (Bind Abst) u1) e2)) -(\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(getl O c2 (CHead -d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))) (\lambda (x2: C).(\lambda (H12: (csuba g (CHead d1 (Bind Abst) u1) -x2)).(\lambda (H13: (clear c2 x2)).(let H_x \def (csuba_gen_abst g d1 x2 u1 -H12) in (let H14 \def H_x in (or_ind (ex2 C (\lambda (d2: C).(eq C x2 (CHead -d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C x2 (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a))))) (or (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u1))) -(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda -(H15: (ex2 C (\lambda (d2: C).(eq C x2 (CHead d2 (Bind Abst) u1))) (\lambda -(d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(eq C x2 (CHead d2 -(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: -C).(getl O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 -(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity -g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 a)))))) (\lambda (x3: C).(\lambda (H16: (eq C x2 (CHead x3 -(Bind Abst) u1))).(\lambda (H17: (csuba g d1 x3)).(let H18 \def (eq_ind C x2 -(\lambda (c: C).(clear c2 c)) H13 (CHead x3 (Bind Abst) u1) H16) in -(or_introl (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u1))) -(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C -(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: -C).(csuba g d1 d2)) x3 (getl_intro O c2 (CHead x3 (Bind Abst) u1) c2 -(drop_refl c2) H18) H17)))))) H15)) (\lambda (H15: (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(eq C x2 (CHead d2 (Bind Abbr) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(eq C x2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2: -C).(getl O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 -(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity -g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 a)))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: -A).(\lambda (H16: (eq C x2 (CHead x3 (Bind Abbr) x4))).(\lambda (H17: (csuba -g d1 x3)).(\lambda (H18: (arity g d1 u1 (asucc g x5))).(\lambda (H19: (arity -g x3 x4 x5)).(let H20 \def (eq_ind C x2 (\lambda (c: C).(clear c2 c)) H13 -(CHead x3 (Bind Abbr) x4) H16) in (or_intror (ex2 C (\lambda (d2: C).(getl O -c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T -A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 -(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 -(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 -u2 a))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(getl O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x3 x4 x5 (getl_intro O c2 (CHead -x3 (Bind Abbr) x4) c2 (drop_refl c2) H20) H17 H18 H19)))))))))) H15)) -H14)))))) H11)))))))) (\lambda (n: nat).(\lambda (H8: ((\forall (x1: -C).((drop n O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g x1 c2) -\to (or (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) u1))) -(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(getl n c2 (CHead d2 (Bind Abbr) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))))))))).(\lambda (x1: C).(\lambda (H9: (drop (S n) O x1 (CHead x0 (Flat -f) t))).(\lambda (c2: C).(\lambda (H10: (csuba g x1 c2)).(let H11 \def -(drop_clear x1 (CHead x0 (Flat f) t) n H9) in (ex2_3_ind B C T (\lambda (b: -B).(\lambda (e: C).(\lambda (v: T).(clear x1 (CHead e (Bind b) v))))) -(\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop n O e (CHead x0 (Flat -f) t))))) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) -u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: T).(\lambda (H12: -(clear x1 (CHead x3 (Bind x2) x4))).(\lambda (H13: (drop n O x3 (CHead x0 -(Flat f) t))).(let H14 \def (csuba_clear_conf g x1 c2 H10 (CHead x3 (Bind x2) -x4) H12) in (ex2_ind C (\lambda (e2: C).(csuba g (CHead x3 (Bind x2) x4) e2)) -(\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 -(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 -(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 -(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 -u2 a)))))) (\lambda (x5: C).(\lambda (H15: (csuba g (CHead x3 (Bind x2) x4) -x5)).(\lambda (H16: (clear c2 x5)).(let H_x \def (csuba_gen_bind g x2 x3 x5 -x4 H15) in (let H17 \def H_x in (ex2_3_ind B C T (\lambda (b2: B).(\lambda -(e2: C).(\lambda (v2: T).(eq C x5 (CHead e2 (Bind b2) v2))))) (\lambda (_: -B).(\lambda (e2: C).(\lambda (_: T).(csuba g x3 e2)))) (or (ex2 C (\lambda -(d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g -d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl -(S n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x6: B).(\lambda (x7: -C).(\lambda (x8: T).(\lambda (H18: (eq C x5 (CHead x7 (Bind x6) -x8))).(\lambda (H19: (csuba g x3 x7)).(let H20 \def (eq_ind C x5 (\lambda (c: -C).(clear c2 c)) H16 (CHead x7 (Bind x6) x8) H18) in (let H21 \def (H8 x3 H13 -x7 H19) in (or_ind (ex2 C (\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abst) -u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(getl n x7 (CHead d2 (Bind Abbr) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (or -(ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u1))) (\lambda -(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda -(H22: (ex2 C (\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abst) u1))) (\lambda -(d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(getl n x7 (CHead d2 -(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: -C).(getl (S n) c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 -d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S -n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: -A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda -(a: A).(arity g d2 u2 a)))))) (\lambda (x9: C).(\lambda (H23: (getl n x7 -(CHead x9 (Bind Abst) u1))).(\lambda (H24: (csuba g d1 x9)).(or_introl (ex2 C -(\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: -C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(getl (S n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda -(_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: -C).(getl (S n) c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 -d2)) x9 (getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Abst) u1) n H23) -H24))))) H22)) (\lambda (H22: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(getl n x7 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))).(ex4_3_ind C -T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl n x7 (CHead d2 -(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 -(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 -u2 a)))) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) -u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))) (\lambda (x9: C).(\lambda (x10: T).(\lambda (x11: A).(\lambda (H23: -(getl n x7 (CHead x9 (Bind Abbr) x10))).(\lambda (H24: (csuba g d1 -x9)).(\lambda (H25: (arity g d1 u1 (asucc g x11))).(\lambda (H26: (arity g x9 -x10 x11)).(or_intror (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind -Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(getl (S n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x9 x10 x11 (getl_clear_bind x6 -c2 x7 x8 H20 (CHead x9 (Bind Abbr) x10) n H23) H24 H25 H26))))))))) H22)) -H21)))))))) H17)))))) H14))))))) H11)))))))) i) H7))))) k H3 H4))))))) x H1 -H2)))) H0))))))). -(* COMMENTS -Initial nodes: 6437 -END *) - -theorem csuba_getl_abst_rev: - \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).(\forall -(i: nat).((getl i c1 (CHead d1 (Bind Abst) u)) \to (\forall (c2: C).((csuba g -c2 c1) \to (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1)))))))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u: T).(\lambda -(i: nat).(\lambda (H: (getl i c1 (CHead d1 (Bind Abst) u))).(let H0 \def -(getl_gen_all c1 (CHead d1 (Bind Abst) u) i H) in (ex2_ind C (\lambda (e: -C).(drop i O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abst) u))) -(\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: C).(getl i c2 -(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))) (\lambda (x: -C).(\lambda (H1: (drop i O c1 x)).(\lambda (H2: (clear x (CHead d1 (Bind -Abst) u))).(C_ind (\lambda (c: C).((drop i O c1 c) \to ((clear c (CHead d1 -(Bind Abst) u)) \to (\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda -(d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 -d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))))) -(\lambda (n: nat).(\lambda (_: (drop i O c1 (CSort n))).(\lambda (H4: (clear -(CSort n) (CHead d1 (Bind Abst) u))).(clear_gen_sort (CHead d1 (Bind Abst) u) -n H4 (\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: C).(getl -i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))) (\lambda (x0: -C).(\lambda (_: (((drop i O c1 x0) \to ((clear x0 (CHead d1 (Bind Abst) u)) -\to (\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: C).(getl i -c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))).(\lambda (k: -K).(\lambda (t: T).(\lambda (H3: (drop i O c1 (CHead x0 k t))).(\lambda (H4: -(clear (CHead x0 k t) (CHead d1 (Bind Abst) u))).(K_ind (\lambda (k0: -K).((drop i O c1 (CHead x0 k0 t)) \to ((clear (CHead x0 k0 t) (CHead d1 (Bind -Abst) u)) \to (\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: -C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))))) (\lambda (b: -B).(\lambda (H5: (drop i O c1 (CHead x0 (Bind b) t))).(\lambda (H6: (clear -(CHead x0 (Bind b) t) (CHead d1 (Bind Abst) u))).(let H7 \def (f_equal C C -(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow d1 | (CHead c _ _) \Rightarrow c])) (CHead d1 (Bind Abst) u) -(CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abst) u) t H6)) in -((let H8 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: -C).B) with [(CSort _) \Rightarrow Abst | (CHead _ k0 _) \Rightarrow (match k0 -in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) -\Rightarrow Abst])])) (CHead d1 (Bind Abst) u) (CHead x0 (Bind b) t) -(clear_gen_bind b x0 (CHead d1 (Bind Abst) u) t H6)) in ((let H9 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d1 (Bind -Abst) u) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abst) u) -t H6)) in (\lambda (H10: (eq B Abst b)).(\lambda (H11: (eq C d1 x0)).(\lambda -(c2: C).(\lambda (H12: (csuba g c2 c1)).(let H13 \def (eq_ind_r T t (\lambda -(t0: T).(drop i O c1 (CHead x0 (Bind b) t0))) H5 u H9) in (let H14 \def -(eq_ind_r B b (\lambda (b0: B).(drop i O c1 (CHead x0 (Bind b0) u))) H13 Abst -H10) in (let H15 \def (eq_ind_r C x0 (\lambda (c: C).(drop i O c1 (CHead c -(Bind Abst) u))) H14 d1 H11) in (let H16 \def (csuba_drop_abst_rev i c1 d1 u -H15 g c2 H12) in (or_ind (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind -Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop i O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: C).(getl i c2 -(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H17: (ex2 C -(\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop i O c2 (CHead d2 -(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: -C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x1: -C).(\lambda (H18: (drop i O c2 (CHead x1 (Bind Abst) u))).(\lambda (H19: -(csuba g x1 d1)).(or_introl (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 -(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(getl i -c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x1 -(getl_intro i c2 (CHead x1 (Bind Abst) u) (CHead x1 (Bind Abst) u) H18 -(clear_bind Abst x1 u)) H19))))) H17)) (\lambda (H17: (ex2_2 C T (\lambda -(d2: C).(\lambda (u2: T).(drop i O c2 (CHead d2 (Bind Void) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: -C).(\lambda (u2: T).(drop i O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1))) (or (ex2 C (\lambda (d2: C).(getl i c2 -(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x1: -C).(\lambda (x2: T).(\lambda (H18: (drop i O c2 (CHead x1 (Bind Void) -x2))).(\lambda (H19: (csuba g x1 d1)).(or_intror (ex2 C (\lambda (d2: -C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T -(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x1 x2 (getl_intro i c2 -(CHead x1 (Bind Void) x2) (CHead x1 (Bind Void) x2) H18 (clear_bind Void x1 -x2)) H19)))))) H17)) H16)))))))))) H8)) H7))))) (\lambda (f: F).(\lambda (H5: -(drop i O c1 (CHead x0 (Flat f) t))).(\lambda (H6: (clear (CHead x0 (Flat f) -t) (CHead d1 (Bind Abst) u))).(let H7 \def H5 in (unintro C c1 (\lambda (c: -C).((drop i O c (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g c2 c) -\to (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))))))) (nat_ind (\lambda (n: nat).(\forall (x1: C).((drop -n O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g c2 x1) \to (or -(ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl n c2 -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))))))))) (\lambda (x1: C).(\lambda (H8: (drop O O x1 (CHead x0 (Flat f) -t))).(\lambda (c2: C).(\lambda (H9: (csuba g c2 x1)).(let H10 \def (eq_ind C -x1 (\lambda (c: C).(csuba g c2 c)) H9 (CHead x0 (Flat f) t) (drop_gen_refl x1 -(CHead x0 (Flat f) t) H8)) in (let H_y \def (clear_flat x0 (CHead d1 (Bind -Abst) u) (clear_gen_flat f x0 (CHead d1 (Bind Abst) u) t H6) f t) in (let H11 -\def (csuba_clear_trans g (CHead x0 (Flat f) t) c2 H10 (CHead d1 (Bind Abst) -u) H_y) in (ex2_ind C (\lambda (e2: C).(csuba g e2 (CHead d1 (Bind Abst) u))) -(\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(getl O c2 (CHead -d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda -(d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 (Bind Void) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x2: C).(\lambda (H12: -(csuba g x2 (CHead d1 (Bind Abst) u))).(\lambda (H13: (clear c2 x2)).(let H_x -\def (csuba_gen_abst_rev g d1 x2 u H12) in (let H14 \def H_x in (or_ind (ex2 -C (\lambda (d2: C).(eq C x2 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C x2 -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))) (or (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(getl O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))) (\lambda (H15: (ex2 C (\lambda (d2: C).(eq C x2 (CHead -d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda -(d2: C).(eq C x2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) -(or (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u))) (\lambda -(d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl -O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g -d2 d1))))) (\lambda (x3: C).(\lambda (H16: (eq C x2 (CHead x3 (Bind Abst) -u))).(\lambda (H17: (csuba g x3 d1)).(let H18 \def (eq_ind C x2 (\lambda (c: -C).(clear c2 c)) H13 (CHead x3 (Bind Abst) u) H16) in (or_introl (ex2 C -(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl O c2 -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))) (ex_intro2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1)) x3 (getl_intro O c2 (CHead x3 (Bind Abst) -u) c2 (drop_refl c2) H18) H17)))))) H15)) (\lambda (H15: (ex2_2 C T (\lambda -(d2: C).(\lambda (u2: T).(eq C x2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: -C).(\lambda (u2: T).(eq C x2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1))) (or (ex2 C (\lambda (d2: C).(getl O c2 -(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x3: -C).(\lambda (x4: T).(\lambda (H16: (eq C x2 (CHead x3 (Bind Void) -x4))).(\lambda (H17: (csuba g x3 d1)).(let H18 \def (eq_ind C x2 (\lambda (c: -C).(clear c2 c)) H13 (CHead x3 (Bind Void) x4) H16) in (or_intror (ex2 C -(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl O c2 -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead -d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x3 -x4 (getl_intro O c2 (CHead x3 (Bind Void) x4) c2 (drop_refl c2) H18) -H17))))))) H15)) H14)))))) H11)))))))) (\lambda (n: nat).(\lambda (H8: -((\forall (x1: C).((drop n O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: -C).((csuba g c2 x1) \to (or (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 -(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(getl n c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1))))))))))).(\lambda (x1: C).(\lambda (H9: -(drop (S n) O x1 (CHead x0 (Flat f) t))).(\lambda (c2: C).(\lambda (H10: -(csuba g c2 x1)).(let H11 \def (drop_clear x1 (CHead x0 (Flat f) t) n H9) in -(ex2_3_ind B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear x1 -(CHead e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: -T).(drop n O e (CHead x0 (Flat f) t))))) (or (ex2 C (\lambda (d2: C).(getl (S -n) c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C -T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x2: -B).(\lambda (x3: C).(\lambda (x4: T).(\lambda (H12: (clear x1 (CHead x3 (Bind -x2) x4))).(\lambda (H13: (drop n O x3 (CHead x0 (Flat f) t))).(let H14 \def -(csuba_clear_trans g x1 c2 H10 (CHead x3 (Bind x2) x4) H12) in (ex2_ind C -(\lambda (e2: C).(csuba g e2 (CHead x3 (Bind x2) x4))) (\lambda (e2: -C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind -Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x5: C).(\lambda (H15: (csuba -g x5 (CHead x3 (Bind x2) x4))).(\lambda (H16: (clear c2 x5)).(let H_x \def -(csuba_gen_bind_rev g x2 x3 x5 x4 H15) in (let H17 \def H_x in (ex2_3_ind B C -T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C x5 (CHead e2 (Bind -b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 -x3)))) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))) (\lambda (x6: B).(\lambda (x7: C).(\lambda (x8: -T).(\lambda (H18: (eq C x5 (CHead x7 (Bind x6) x8))).(\lambda (H19: (csuba g -x7 x3)).(let H20 \def (eq_ind C x5 (\lambda (c: C).(clear c2 c)) H16 (CHead -x7 (Bind x6) x8) H18) in (let H21 \def (H8 x3 H13 x7 H19) in (or_ind (ex2 C -(\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl n x7 -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))) (\lambda (H22: (ex2 C (\lambda (d2: C).(getl n x7 -(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C -(\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 -(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x9: C).(\lambda (H23: (getl -n x7 (CHead x9 (Bind Abst) u))).(\lambda (H24: (csuba g x9 d1)).(or_introl -(ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u))) (\lambda -(d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl -(S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba -g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind -Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x9 (getl_clear_bind x6 c2 x7 x8 -H20 (CHead x9 (Bind Abst) u) n H23) H24))))) H22)) (\lambda (H22: (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(getl n x7 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda -(d2: C).(\lambda (u2: T).(getl n x7 (CHead d2 (Bind Void) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1))) (or (ex2 C (\lambda (d2: C).(getl -(S n) c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 -C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x9: -C).(\lambda (x10: T).(\lambda (H23: (getl n x7 (CHead x9 (Bind Void) -x10))).(\lambda (H24: (csuba g x9 d1)).(or_intror (ex2 C (\lambda (d2: -C).(getl (S n) c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 -d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) -(ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x9 x10 -(getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Void) x10) n H23) H24)))))) -H22)) H21)))))))) H17)))))) H14))))))) H11)))))))) i) H7))))) k H3 H4))))))) -x H1 H2)))) H0))))))). -(* COMMENTS -Initial nodes: 4703 -END *) - -theorem csuba_getl_abbr_rev: - \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u1: T).(\forall -(i: nat).((getl i c1 (CHead d1 (Bind Abbr) u1)) \to (\forall (c2: C).((csuba -g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 -C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u1: T).(\lambda -(i: nat).(\lambda (H: (getl i c1 (CHead d1 (Bind Abbr) u1))).(let H0 \def -(getl_gen_all c1 (CHead d1 (Bind Abbr) u1) i H) in (ex2_ind C (\lambda (e: -C).(drop i O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abbr) u1))) -(\forall (c2: C).((csuba g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(getl i c2 -(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))) -(\lambda (x: C).(\lambda (H1: (drop i O c1 x)).(\lambda (H2: (clear x (CHead -d1 (Bind Abbr) u1))).(C_ind (\lambda (c: C).((drop i O c1 c) \to ((clear c -(CHead d1 (Bind Abbr) u1)) \to (\forall (c2: C).((csuba g c2 c1) \to (or3 -(ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))))))))) (\lambda (n: nat).(\lambda (_: -(drop i O c1 (CSort n))).(\lambda (H4: (clear (CSort n) (CHead d1 (Bind Abbr) -u1))).(clear_gen_sort (CHead d1 (Bind Abbr) u1) n H4 (\forall (c2: C).((csuba -g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 -C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))) (\lambda (x0: -C).(\lambda (_: (((drop i O c1 x0) \to ((clear x0 (CHead d1 (Bind Abbr) u1)) -\to (\forall (c2: C).((csuba g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(getl -i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C -T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 -(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (H3: (drop i O c1 -(CHead x0 k t))).(\lambda (H4: (clear (CHead x0 k t) (CHead d1 (Bind Abbr) -u1))).(K_ind (\lambda (k0: K).((drop i O c1 (CHead x0 k0 t)) \to ((clear -(CHead x0 k0 t) (CHead d1 (Bind Abbr) u1)) \to (\forall (c2: C).((csuba g c2 -c1) \to (or3 (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) -(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))))) (\lambda (b: -B).(\lambda (H5: (drop i O c1 (CHead x0 (Bind b) t))).(\lambda (H6: (clear -(CHead x0 (Bind b) t) (CHead d1 (Bind Abbr) u1))).(let H7 \def (f_equal C C -(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow d1 | (CHead c _ _) \Rightarrow c])) (CHead d1 (Bind Abbr) u1) -(CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abbr) u1) t H6)) -in ((let H8 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda -(_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow -(match k0 in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | -(Flat _) \Rightarrow Abbr])])) (CHead d1 (Bind Abbr) u1) (CHead x0 (Bind b) -t) (clear_gen_bind b x0 (CHead d1 (Bind Abbr) u1) t H6)) in ((let H9 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow u1 | (CHead _ _ t0) \Rightarrow t0])) (CHead d1 (Bind -Abbr) u1) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abbr) -u1) t H6)) in (\lambda (H10: (eq B Abbr b)).(\lambda (H11: (eq C d1 -x0)).(\lambda (c2: C).(\lambda (H12: (csuba g c2 c1)).(let H13 \def (eq_ind_r -T t (\lambda (t0: T).(drop i O c1 (CHead x0 (Bind b) t0))) H5 u1 H9) in (let -H14 \def (eq_ind_r B b (\lambda (b0: B).(drop i O c1 (CHead x0 (Bind b0) -u1))) H13 Abbr H10) in (let H15 \def (eq_ind_r C x0 (\lambda (c: C).(drop i O -c1 (CHead c (Bind Abbr) u1))) H14 d1 H11) in (let H16 \def -(csuba_drop_abbr_rev i c1 d1 u1 H15 g c2 H12) in (or3_ind (ex2 C (\lambda -(d2: C).(drop i O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop i -O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop i O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 -C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H17: (ex2 C -(\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop i O c2 (CHead d2 -(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) (or3 (ex2 C (\lambda (d2: -C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))) (\lambda (x1: C).(\lambda (H18: (drop i O c2 (CHead x1 -(Bind Abbr) u1))).(\lambda (H19: (csuba g x1 d1)).(or3_intro0 (ex2 C (\lambda -(d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i -c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(getl i c2 (CHead d2 -(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x1 (getl_intro i c2 -(CHead x1 (Bind Abbr) u1) (CHead x1 (Bind Abbr) u1) H18 (clear_bind Abbr x1 -u1)) H19))))) H17)) (\lambda (H17: (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abst) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T -A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 -(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a)))) (or3 (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) -(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x1: -C).(\lambda (x2: T).(\lambda (x3: A).(\lambda (H18: (drop i O c2 (CHead x1 -(Bind Abst) x2))).(\lambda (H19: (csuba g x1 d1)).(\lambda (H20: (arity g x1 -x2 (asucc g x3))).(\lambda (H21: (arity g d1 u1 x3)).(or3_intro1 (ex2 C -(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))) (ex4_3_intro C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x1 x2 x3 -(getl_intro i c2 (CHead x1 (Bind Abst) x2) (CHead x1 (Bind Abst) x2) H18 -(clear_bind Abst x1 x2)) H19 H20 H21))))))))) H17)) (\lambda (H17: (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop i O c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda -(d2: C).(\lambda (u2: T).(drop i O c2 (CHead d2 (Bind Void) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C (\lambda (d2: C).(getl -i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C -T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 -(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) -(\lambda (x1: C).(\lambda (x2: T).(\lambda (H18: (drop i O c2 (CHead x1 (Bind -Void) x2))).(\lambda (H19: (csuba g x1 d1)).(or3_intro2 (ex2 C (\lambda (d2: -C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: -T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))) x1 x2 (getl_intro i c2 (CHead x1 (Bind Void) x2) (CHead -x1 (Bind Void) x2) H18 (clear_bind Void x1 x2)) H19)))))) H17)) H16)))))))))) -H8)) H7))))) (\lambda (f: F).(\lambda (H5: (drop i O c1 (CHead x0 (Flat f) -t))).(\lambda (H6: (clear (CHead x0 (Flat f) t) (CHead d1 (Bind Abbr) -u1))).(let H7 \def H5 in (unintro C c1 (\lambda (c: C).((drop i O c (CHead x0 -(Flat f) t)) \to (\forall (c2: C).((csuba g c2 c) \to (or3 (ex2 C (\lambda -(d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i -c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))))))) (nat_ind (\lambda (n: nat).(\forall (x1: C).((drop -n O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g c2 x1) \to (or3 -(ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(getl n c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(getl n c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))))))))) (\lambda (x1: C).(\lambda (H8: -(drop O O x1 (CHead x0 (Flat f) t))).(\lambda (c2: C).(\lambda (H9: (csuba g -c2 x1)).(let H10 \def (eq_ind C x1 (\lambda (c: C).(csuba g c2 c)) H9 (CHead -x0 (Flat f) t) (drop_gen_refl x1 (CHead x0 (Flat f) t) H8)) in (let H_y \def -(clear_flat x0 (CHead d1 (Bind Abbr) u1) (clear_gen_flat f x0 (CHead d1 (Bind -Abbr) u1) t H6) f t) in (let H11 \def (csuba_clear_trans g (CHead x0 (Flat f) -t) c2 H10 (CHead d1 (Bind Abbr) u1) H_y) in (ex2_ind C (\lambda (e2: -C).(csuba g e2 (CHead d1 (Bind Abbr) u1))) (\lambda (e2: C).(clear c2 e2)) -(or3 (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u1))) (\lambda -(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x2: -C).(\lambda (H12: (csuba g x2 (CHead d1 (Bind Abbr) u1))).(\lambda (H13: -(clear c2 x2)).(let H_x \def (csuba_gen_abbr_rev g d1 x2 u1 H12) in (let H14 -\def H_x in (or3_ind (ex2 C (\lambda (d2: C).(eq C x2 (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(eq C x2 (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 -C T (\lambda (d2: C).(\lambda (u2: T).(eq C x2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: -C).(getl O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(getl O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))) (\lambda (H15: (ex2 C (\lambda (d2: C).(eq C x2 (CHead -d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda -(d2: C).(eq C x2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1)) (or3 (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u1))) -(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abst) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x3: -C).(\lambda (H16: (eq C x2 (CHead x3 (Bind Abbr) u1))).(\lambda (H17: (csuba -g x3 d1)).(let H18 \def (eq_ind C x2 (\lambda (c: C).(clear c2 c)) H13 (CHead -x3 (Bind Abbr) u1) H16) in (or3_intro0 (ex2 C (\lambda (d2: C).(getl O c2 -(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) -(ex_intro2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u1))) (\lambda -(d2: C).(csuba g d2 d1)) x3 (getl_intro O c2 (CHead x3 (Bind Abbr) u1) c2 -(drop_refl c2) H18) H17)))))) H15)) (\lambda (H15: (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(eq C x2 (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 -a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(eq C x2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C (\lambda (d2: -C).(getl O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(getl O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: -A).(\lambda (H16: (eq C x2 (CHead x3 (Bind Abst) x4))).(\lambda (H17: (csuba -g x3 d1)).(\lambda (H18: (arity g x3 x4 (asucc g x5))).(\lambda (H19: (arity -g d1 u1 x5)).(let H20 \def (eq_ind C x2 (\lambda (c: C).(clear c2 c)) H13 -(CHead x3 (Bind Abst) x4) H16) in (or3_intro1 (ex2 C (\lambda (d2: C).(getl O -c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T -A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 -(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) -(ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O -c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a)))) x3 x4 x5 (getl_intro O c2 (CHead x3 (Bind Abst) -x4) c2 (drop_refl c2) H20) H17 H18 H19)))))))))) H15)) (\lambda (H15: (ex2_2 -C T (\lambda (d2: C).(\lambda (u2: T).(eq C x2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda -(d2: C).(\lambda (u2: T).(eq C x2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C (\lambda (d2: C).(getl O c2 -(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) -(\lambda (x3: C).(\lambda (x4: T).(\lambda (H16: (eq C x2 (CHead x3 (Bind -Void) x4))).(\lambda (H17: (csuba g x3 d1)).(let H18 \def (eq_ind C x2 -(\lambda (c: C).(clear c2 c)) H13 (CHead x3 (Bind Void) x4) H16) in -(or3_intro2 (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u1))) -(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abst) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda -(d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 (Bind Void) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1))) x3 x4 (getl_intro O c2 (CHead x3 -(Bind Void) x4) c2 (drop_refl c2) H18) H17))))))) H15)) H14)))))) H11)))))))) -(\lambda (n: nat).(\lambda (H8: ((\forall (x1: C).((drop n O x1 (CHead x0 -(Flat f) t)) \to (\forall (c2: C).((csuba g c2 x1) \to (or3 (ex2 C (\lambda -(d2: C).(getl n c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl n -c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(getl n c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))))))))).(\lambda (x1: C).(\lambda (H9: (drop (S n) O x1 -(CHead x0 (Flat f) t))).(\lambda (c2: C).(\lambda (H10: (csuba g c2 x1)).(let -H11 \def (drop_clear x1 (CHead x0 (Flat f) t) n H9) in (ex2_3_ind B C T -(\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear x1 (CHead e (Bind b) -v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop n O e (CHead x0 -(Flat f) t))))) (or3 (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda -(x2: B).(\lambda (x3: C).(\lambda (x4: T).(\lambda (H12: (clear x1 (CHead x3 -(Bind x2) x4))).(\lambda (H13: (drop n O x3 (CHead x0 (Flat f) t))).(let H14 -\def (csuba_clear_trans g x1 c2 H10 (CHead x3 (Bind x2) x4) H12) in (ex2_ind -C (\lambda (e2: C).(csuba g e2 (CHead x3 (Bind x2) x4))) (\lambda (e2: -C).(clear c2 e2)) (or3 (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda -(x5: C).(\lambda (H15: (csuba g x5 (CHead x3 (Bind x2) x4))).(\lambda (H16: -(clear c2 x5)).(let H_x \def (csuba_gen_bind_rev g x2 x3 x5 x4 H15) in (let -H17 \def H_x in (ex2_3_ind B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda -(v2: T).(eq C x5 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: -C).(\lambda (_: T).(csuba g e2 x3)))) (or3 (ex2 C (\lambda (d2: C).(getl (S -n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C -T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead -d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))) (\lambda (x6: B).(\lambda (x7: C).(\lambda (x8: -T).(\lambda (H18: (eq C x5 (CHead x7 (Bind x6) x8))).(\lambda (H19: (csuba g -x7 x3)).(let H20 \def (eq_ind C x5 (\lambda (c: C).(clear c2 c)) H16 (CHead -x7 (Bind x6) x8) H18) in (let H21 \def (H8 x3 H13 x7 H19) in (or3_ind (ex2 C -(\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(getl n x7 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(getl n x7 (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(getl (S -n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C -T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead -d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))) (\lambda (H22: (ex2 C (\lambda (d2: C).(getl n x7 -(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C -(\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1)) (or3 (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 -(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda -(x9: C).(\lambda (H23: (getl n x7 (CHead x9 (Bind Abbr) u1))).(\lambda (H24: -(csuba g x9 d1)).(or3_intro0 (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 -(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C -(\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1)) x9 (getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Abbr) -u1) n H23) H24))))) H22)) (\lambda (H22: (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(getl n x7 (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 -a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(getl n x7 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C (\lambda (d2: -C).(getl (S n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S -n) c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))) (\lambda (x9: C).(\lambda (x10: T).(\lambda (x11: -A).(\lambda (H23: (getl n x7 (CHead x9 (Bind Abst) x10))).(\lambda (H24: -(csuba g x9 d1)).(\lambda (H25: (arity g x9 x10 (asucc g x11))).(\lambda -(H26: (arity g d1 u1 x11)).(or3_intro1 (ex2 C (\lambda (d2: C).(getl (S n) c2 -(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 -(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead -d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) -(ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S -n) c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a)))) x9 x10 x11 (getl_clear_bind x6 c2 x7 x8 H20 -(CHead x9 (Bind Abst) x10) n H23) H24 H25 H26))))))))) H22)) (\lambda (H22: -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl n x7 (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind C T -(\lambda (d2: C).(\lambda (u2: T).(getl n x7 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C (\lambda (d2: -C).(getl (S n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S -n) c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))) (\lambda (x9: C).(\lambda (x10: T).(\lambda (H23: -(getl n x7 (CHead x9 (Bind Void) x10))).(\lambda (H24: (csuba g x9 -d1)).(or3_intro2 (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro -C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x9 x10 -(getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Void) x10) n H23) H24)))))) -H22)) H21)))))))) H17)))))) H14))))))) H11)))))))) i) H7))))) k H3 H4))))))) -x H1 H2)))) H0))))))). -(* COMMENTS -Initial nodes: 9091 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/props.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/props.ma deleted file mode 100644 index 823e88825..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/props.ma +++ /dev/null @@ -1,28 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csuba/defs.ma". - -theorem csuba_refl: - \forall (g: G).(\forall (c: C).(csuba g c c)) -\def - \lambda (g: G).(\lambda (c: C).(C_ind (\lambda (c0: C).(csuba g c0 c0)) -(\lambda (n: nat).(csuba_sort g n)) (\lambda (c0: C).(\lambda (H: (csuba g c0 -c0)).(\lambda (k: K).(\lambda (t: T).(csuba_head g c0 c0 H k t))))) c)). -(* COMMENTS -Initial nodes: 53 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/arity.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/arity.ma deleted file mode 100644 index 62922d311..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/arity.ma +++ /dev/null @@ -1,42 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubc/csuba.ma". - -theorem csubc_arity_conf: - \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to -(\forall (t: T).(\forall (a: A).((arity g c1 t a) \to (arity g c2 t a))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubc g c1 -c2)).(\lambda (t: T).(\lambda (a: A).(\lambda (H0: (arity g c1 t -a)).(csuba_arity g c1 t a H0 c2 (csubc_csuba g c1 c2 H)))))))). -(* COMMENTS -Initial nodes: 51 -END *) - -theorem csubc_arity_trans: - \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to -((csubv c1 c2) \to (\forall (t: T).(\forall (a: A).((arity g c2 t a) \to -(arity g c1 t a)))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubc g c1 -c2)).(\lambda (H0: (csubv c1 c2)).(\lambda (t: T).(\lambda (a: A).(\lambda -(H1: (arity g c2 t a)).(csuba_arity_rev g c2 t a H1 c1 (csubc_csuba g c1 c2 -H) H0)))))))). -(* COMMENTS -Initial nodes: 59 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/clear.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/clear.ma deleted file mode 100644 index 185bdff75..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/clear.ma +++ /dev/null @@ -1,170 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubc/fwd.ma". - -theorem csubc_clear_conf: - \forall (g: G).(\forall (c1: C).(\forall (e1: C).((clear c1 e1) \to (\forall -(c2: C).((csubc g c1 c2) \to (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda -(e2: C).(csubc g e1 e2)))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (e1: C).(\lambda (H: (clear c1 -e1)).(clear_ind (\lambda (c: C).(\lambda (c0: C).(\forall (c2: C).((csubc g c -c2) \to (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2: C).(csubc g c0 -e2))))))) (\lambda (b: B).(\lambda (e: C).(\lambda (u: T).(\lambda (c2: -C).(\lambda (H0: (csubc g (CHead e (Bind b) u) c2)).(let H_x \def -(csubc_gen_head_l g e c2 u (Bind b) H0) in (let H1 \def H_x in (or3_ind (ex2 -C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind b) u))) (\lambda (c3: C).(csubc g -e c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K -(Bind b) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq -C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda -(_: A).(csubc g e c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 -g (asucc g a) e u)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g -a c3 w))))) (ex4_3 B C T (\lambda (b0: B).(\lambda (c3: C).(\lambda (v2: -T).(eq C c2 (CHead c3 (Bind b0) v2))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: T).(eq K (Bind b) (Bind Void))))) (\lambda (b0: B).(\lambda -(_: C).(\lambda (_: T).(not (eq B b0 Void))))) (\lambda (_: B).(\lambda (c3: -C).(\lambda (_: T).(csubc g e c3))))) (ex2 C (\lambda (e2: C).(clear c2 e2)) -(\lambda (e2: C).(csubc g (CHead e (Bind b) u) e2))) (\lambda (H2: (ex2 C -(\lambda (c3: C).(eq C c2 (CHead c3 (Bind b) u))) (\lambda (c3: C).(csubc g e -c3)))).(ex2_ind C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind b) u))) (\lambda -(c3: C).(csubc g e c3)) (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2: -C).(csubc g (CHead e (Bind b) u) e2))) (\lambda (x: C).(\lambda (H3: (eq C c2 -(CHead x (Bind b) u))).(\lambda (H4: (csubc g e x)).(eq_ind_r C (CHead x -(Bind b) u) (\lambda (c: C).(ex2 C (\lambda (e2: C).(clear c e2)) (\lambda -(e2: C).(csubc g (CHead e (Bind b) u) e2)))) (ex_intro2 C (\lambda (e2: -C).(clear (CHead x (Bind b) u) e2)) (\lambda (e2: C).(csubc g (CHead e (Bind -b) u) e2)) (CHead x (Bind b) u) (clear_bind b x u) (csubc_head g e x H4 (Bind -b) u)) c2 H3)))) H2)) (\lambda (H2: (ex5_3 C T A (\lambda (_: C).(\lambda (_: -T).(\lambda (_: A).(eq K (Bind b) (Bind Abst))))) (\lambda (c3: C).(\lambda -(w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: -C).(\lambda (_: T).(\lambda (_: A).(csubc g e c3)))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(sc3 g (asucc g a) e u)))) (\lambda (c3: C).(\lambda -(w: T).(\lambda (a: A).(sc3 g a c3 w)))))).(ex5_3_ind C T A (\lambda (_: -C).(\lambda (_: T).(\lambda (_: A).(eq K (Bind b) (Bind Abst))))) (\lambda -(c3: C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) -(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g e c3)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) e u)))) (\lambda -(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w)))) (ex2 C (\lambda -(e2: C).(clear c2 e2)) (\lambda (e2: C).(csubc g (CHead e (Bind b) u) e2))) -(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H3: (eq K (Bind -b) (Bind Abst))).(\lambda (H4: (eq C c2 (CHead x0 (Bind Abbr) x1))).(\lambda -(H5: (csubc g e x0)).(\lambda (H6: (sc3 g (asucc g x2) e u)).(\lambda (H7: -(sc3 g x2 x0 x1)).(eq_ind_r C (CHead x0 (Bind Abbr) x1) (\lambda (c: C).(ex2 -C (\lambda (e2: C).(clear c e2)) (\lambda (e2: C).(csubc g (CHead e (Bind b) -u) e2)))) (let H8 \def (f_equal K B (\lambda (e0: K).(match e0 in K return -(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow b])) -(Bind b) (Bind Abst) H3) in (eq_ind_r B Abst (\lambda (b0: B).(ex2 C (\lambda -(e2: C).(clear (CHead x0 (Bind Abbr) x1) e2)) (\lambda (e2: C).(csubc g -(CHead e (Bind b0) u) e2)))) (ex_intro2 C (\lambda (e2: C).(clear (CHead x0 -(Bind Abbr) x1) e2)) (\lambda (e2: C).(csubc g (CHead e (Bind Abst) u) e2)) -(CHead x0 (Bind Abbr) x1) (clear_bind Abbr x0 x1) (csubc_abst g e x0 H5 u x2 -H6 x1 H7)) b H8)) c2 H4))))))))) H2)) (\lambda (H2: (ex4_3 B C T (\lambda -(b0: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b0) -v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K (Bind b) (Bind -Void))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b0 -Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g e -c3)))))).(ex4_3_ind B C T (\lambda (b0: B).(\lambda (c3: C).(\lambda (v2: -T).(eq C c2 (CHead c3 (Bind b0) v2))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: T).(eq K (Bind b) (Bind Void))))) (\lambda (b0: B).(\lambda -(_: C).(\lambda (_: T).(not (eq B b0 Void))))) (\lambda (_: B).(\lambda (c3: -C).(\lambda (_: T).(csubc g e c3)))) (ex2 C (\lambda (e2: C).(clear c2 e2)) -(\lambda (e2: C).(csubc g (CHead e (Bind b) u) e2))) (\lambda (x0: -B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H3: (eq C c2 (CHead x1 (Bind -x0) x2))).(\lambda (H4: (eq K (Bind b) (Bind Void))).(\lambda (H5: (not (eq B -x0 Void))).(\lambda (H6: (csubc g e x1)).(eq_ind_r C (CHead x1 (Bind x0) x2) -(\lambda (c: C).(ex2 C (\lambda (e2: C).(clear c e2)) (\lambda (e2: C).(csubc -g (CHead e (Bind b) u) e2)))) (let H7 \def (f_equal K B (\lambda (e0: -K).(match e0 in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | -(Flat _) \Rightarrow b])) (Bind b) (Bind Void) H4) in (eq_ind_r B Void -(\lambda (b0: B).(ex2 C (\lambda (e2: C).(clear (CHead x1 (Bind x0) x2) e2)) -(\lambda (e2: C).(csubc g (CHead e (Bind b0) u) e2)))) (ex_intro2 C (\lambda -(e2: C).(clear (CHead x1 (Bind x0) x2) e2)) (\lambda (e2: C).(csubc g (CHead -e (Bind Void) u) e2)) (CHead x1 (Bind x0) x2) (clear_bind x0 x1 x2) -(csubc_void g e x1 H6 x0 H5 u x2)) b H7)) c2 H3)))))))) H2)) H1)))))))) -(\lambda (e: C).(\lambda (c: C).(\lambda (_: (clear e c)).(\lambda (H1: -((\forall (c2: C).((csubc g e c2) \to (ex2 C (\lambda (e2: C).(clear c2 e2)) -(\lambda (e2: C).(csubc g c e2))))))).(\lambda (f: F).(\lambda (u: -T).(\lambda (c2: C).(\lambda (H2: (csubc g (CHead e (Flat f) u) c2)).(let H_x -\def (csubc_gen_head_l g e c2 u (Flat f) H2) in (let H3 \def H_x in (or3_ind -(ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 (Flat f) u))) (\lambda (c3: -C).(csubc g e c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: -A).(eq K (Flat f) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda -(_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: -T).(\lambda (_: A).(csubc g e c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(sc3 g (asucc g a) e u)))) (\lambda (c3: C).(\lambda (w: T).(\lambda -(a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: -C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: T).(eq K (Flat f) (Bind Void))))) (\lambda -(b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: -B).(\lambda (c3: C).(\lambda (_: T).(csubc g e c3))))) (ex2 C (\lambda (e2: -C).(clear c2 e2)) (\lambda (e2: C).(csubc g c e2))) (\lambda (H4: (ex2 C -(\lambda (c3: C).(eq C c2 (CHead c3 (Flat f) u))) (\lambda (c3: C).(csubc g e -c3)))).(ex2_ind C (\lambda (c3: C).(eq C c2 (CHead c3 (Flat f) u))) (\lambda -(c3: C).(csubc g e c3)) (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2: -C).(csubc g c e2))) (\lambda (x: C).(\lambda (H5: (eq C c2 (CHead x (Flat f) -u))).(\lambda (H6: (csubc g e x)).(eq_ind_r C (CHead x (Flat f) u) (\lambda -(c0: C).(ex2 C (\lambda (e2: C).(clear c0 e2)) (\lambda (e2: C).(csubc g c -e2)))) (let H_x0 \def (H1 x H6) in (let H7 \def H_x0 in (ex2_ind C (\lambda -(e2: C).(clear x e2)) (\lambda (e2: C).(csubc g c e2)) (ex2 C (\lambda (e2: -C).(clear (CHead x (Flat f) u) e2)) (\lambda (e2: C).(csubc g c e2))) -(\lambda (x0: C).(\lambda (H8: (clear x x0)).(\lambda (H9: (csubc g c -x0)).(ex_intro2 C (\lambda (e2: C).(clear (CHead x (Flat f) u) e2)) (\lambda -(e2: C).(csubc g c e2)) x0 (clear_flat x x0 H8 f u) H9)))) H7))) c2 H5)))) -H4)) (\lambda (H4: (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: -A).(eq K (Flat f) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda -(_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: -T).(\lambda (_: A).(csubc g e c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(sc3 g (asucc g a) e u)))) (\lambda (c3: C).(\lambda (w: T).(\lambda -(a: A).(sc3 g a c3 w)))))).(ex5_3_ind C T A (\lambda (_: C).(\lambda (_: -T).(\lambda (_: A).(eq K (Flat f) (Bind Abst))))) (\lambda (c3: C).(\lambda -(w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: -C).(\lambda (_: T).(\lambda (_: A).(csubc g e c3)))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(sc3 g (asucc g a) e u)))) (\lambda (c3: C).(\lambda -(w: T).(\lambda (a: A).(sc3 g a c3 w)))) (ex2 C (\lambda (e2: C).(clear c2 -e2)) (\lambda (e2: C).(csubc g c e2))) (\lambda (x0: C).(\lambda (x1: -T).(\lambda (x2: A).(\lambda (H5: (eq K (Flat f) (Bind Abst))).(\lambda (H6: -(eq C c2 (CHead x0 (Bind Abbr) x1))).(\lambda (_: (csubc g e x0)).(\lambda -(_: (sc3 g (asucc g x2) e u)).(\lambda (_: (sc3 g x2 x0 x1)).(eq_ind_r C -(CHead x0 (Bind Abbr) x1) (\lambda (c0: C).(ex2 C (\lambda (e2: C).(clear c0 -e2)) (\lambda (e2: C).(csubc g c e2)))) (let H10 \def (eq_ind K (Flat f) -(\lambda (ee: K).(match ee in K return (\lambda (_: K).Prop) with [(Bind _) -\Rightarrow False | (Flat _) \Rightarrow True])) I (Bind Abst) H5) in -(False_ind (ex2 C (\lambda (e2: C).(clear (CHead x0 (Bind Abbr) x1) e2)) -(\lambda (e2: C).(csubc g c e2))) H10)) c2 H6))))))))) H4)) (\lambda (H4: -(ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 -(CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -T).(eq K (Flat f) (Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda -(_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: -T).(csubc g e c3)))))).(ex4_3_ind B C T (\lambda (b: B).(\lambda (c3: -C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: T).(eq K (Flat f) (Bind Void))))) (\lambda -(b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: -B).(\lambda (c3: C).(\lambda (_: T).(csubc g e c3)))) (ex2 C (\lambda (e2: -C).(clear c2 e2)) (\lambda (e2: C).(csubc g c e2))) (\lambda (x0: B).(\lambda -(x1: C).(\lambda (x2: T).(\lambda (H5: (eq C c2 (CHead x1 (Bind x0) -x2))).(\lambda (H6: (eq K (Flat f) (Bind Void))).(\lambda (_: (not (eq B x0 -Void))).(\lambda (_: (csubc g e x1)).(eq_ind_r C (CHead x1 (Bind x0) x2) -(\lambda (c0: C).(ex2 C (\lambda (e2: C).(clear c0 e2)) (\lambda (e2: -C).(csubc g c e2)))) (let H9 \def (eq_ind K (Flat f) (\lambda (ee: K).(match -ee in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat -_) \Rightarrow True])) I (Bind Void) H6) in (False_ind (ex2 C (\lambda (e2: -C).(clear (CHead x1 (Bind x0) x2) e2)) (\lambda (e2: C).(csubc g c e2))) H9)) -c2 H5)))))))) H4)) H3))))))))))) c1 e1 H)))). -(* COMMENTS -Initial nodes: 2837 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/csuba.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/csuba.ma deleted file mode 100644 index 2abfe2a1f..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/csuba.ma +++ /dev/null @@ -1,40 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubc/defs.ma". - -include "Basic-1/sc3/props.ma". - -theorem csubc_csuba: - \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to (csuba -g c1 c2)))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubc g c1 -c2)).(csubc_ind g (\lambda (c: C).(\lambda (c0: C).(csuba g c c0))) (\lambda -(n: nat).(csuba_refl g (CSort n))) (\lambda (c3: C).(\lambda (c4: C).(\lambda -(_: (csubc g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (k: K).(\lambda -(v: T).(csuba_head g c3 c4 H1 k v))))))) (\lambda (c3: C).(\lambda (c4: -C).(\lambda (_: (csubc g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (b: -B).(\lambda (H2: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: -T).(csuba_void g c3 c4 H1 b H2 u1 u2))))))))) (\lambda (c3: C).(\lambda (c4: -C).(\lambda (_: (csubc g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (v: -T).(\lambda (a: A).(\lambda (H2: (sc3 g (asucc g a) c3 v)).(\lambda (w: -T).(\lambda (H3: (sc3 g a c4 w)).(csuba_abst g c3 c4 H1 v a (sc3_arity_gen g -c3 v (asucc g a) H2) w (sc3_arity_gen g c4 w a H3))))))))))) c1 c2 H)))). -(* COMMENTS -Initial nodes: 231 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/defs.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/defs.ma deleted file mode 100644 index 73a4c5625..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/defs.ma +++ /dev/null @@ -1,30 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/sc3/defs.ma". - -inductive csubc (g: G): C \to (C \to Prop) \def -| csubc_sort: \forall (n: nat).(csubc g (CSort n) (CSort n)) -| csubc_head: \forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to (\forall -(k: K).(\forall (v: T).(csubc g (CHead c1 k v) (CHead c2 k v)))))) -| csubc_void: \forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to (\forall -(b: B).((not (eq B b Void)) \to (\forall (u1: T).(\forall (u2: T).(csubc g -(CHead c1 (Bind Void) u1) (CHead c2 (Bind b) u2)))))))) -| csubc_abst: \forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to (\forall -(v: T).(\forall (a: A).((sc3 g (asucc g a) c1 v) \to (\forall (w: T).((sc3 g -a c2 w) \to (csubc g (CHead c1 (Bind Abst) v) (CHead c2 (Bind Abbr) -w))))))))). - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/drop.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/drop.ma deleted file mode 100644 index 195d36491..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/drop.ma +++ /dev/null @@ -1,475 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubc/fwd.ma". - -include "Basic-1/sc3/props.ma". - -theorem csubc_drop_conf_O: - \forall (g: G).(\forall (c1: C).(\forall (e1: C).(\forall (h: nat).((drop h -O c1 e1) \to (\forall (c2: C).((csubc g c1 c2) \to (ex2 C (\lambda (e2: -C).(drop h O c2 e2)) (\lambda (e2: C).(csubc g e1 e2))))))))) -\def - \lambda (g: G).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (e1: -C).(\forall (h: nat).((drop h O c e1) \to (\forall (c2: C).((csubc g c c2) -\to (ex2 C (\lambda (e2: C).(drop h O c2 e2)) (\lambda (e2: C).(csubc g e1 -e2))))))))) (\lambda (n: nat).(\lambda (e1: C).(\lambda (h: nat).(\lambda (H: -(drop h O (CSort n) e1)).(\lambda (c2: C).(\lambda (H0: (csubc g (CSort n) -c2)).(and3_ind (eq C e1 (CSort n)) (eq nat h O) (eq nat O O) (ex2 C (\lambda -(e2: C).(drop h O c2 e2)) (\lambda (e2: C).(csubc g e1 e2))) (\lambda (H1: -(eq C e1 (CSort n))).(\lambda (H2: (eq nat h O)).(\lambda (_: (eq nat O -O)).(eq_ind_r nat O (\lambda (n0: nat).(ex2 C (\lambda (e2: C).(drop n0 O c2 -e2)) (\lambda (e2: C).(csubc g e1 e2)))) (eq_ind_r C (CSort n) (\lambda (c: -C).(ex2 C (\lambda (e2: C).(drop O O c2 e2)) (\lambda (e2: C).(csubc g c -e2)))) (ex_intro2 C (\lambda (e2: C).(drop O O c2 e2)) (\lambda (e2: -C).(csubc g (CSort n) e2)) c2 (drop_refl c2) H0) e1 H1) h H2)))) -(drop_gen_sort n h O e1 H)))))))) (\lambda (c: C).(\lambda (H: ((\forall (e1: -C).(\forall (h: nat).((drop h O c e1) \to (\forall (c2: C).((csubc g c c2) -\to (ex2 C (\lambda (e2: C).(drop h O c2 e2)) (\lambda (e2: C).(csubc g e1 -e2)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e1: C).(\lambda (h: -nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c k t) e1) \to (\forall -(c2: C).((csubc g (CHead c k t) c2) \to (ex2 C (\lambda (e2: C).(drop n O c2 -e2)) (\lambda (e2: C).(csubc g e1 e2))))))) (\lambda (H0: (drop O O (CHead c -k t) e1)).(\lambda (c2: C).(\lambda (H1: (csubc g (CHead c k t) c2)).(eq_ind -C (CHead c k t) (\lambda (c0: C).(ex2 C (\lambda (e2: C).(drop O O c2 e2)) -(\lambda (e2: C).(csubc g c0 e2)))) (ex_intro2 C (\lambda (e2: C).(drop O O -c2 e2)) (\lambda (e2: C).(csubc g (CHead c k t) e2)) c2 (drop_refl c2) H1) e1 -(drop_gen_refl (CHead c k t) e1 H0))))) (\lambda (n: nat).(\lambda (H0: -(((drop n O (CHead c k t) e1) \to (\forall (c2: C).((csubc g (CHead c k t) -c2) \to (ex2 C (\lambda (e2: C).(drop n O c2 e2)) (\lambda (e2: C).(csubc g -e1 e2)))))))).(\lambda (H1: (drop (S n) O (CHead c k t) e1)).(\lambda (c2: -C).(\lambda (H2: (csubc g (CHead c k t) c2)).(let H_x \def (csubc_gen_head_l -g c c2 t k H2) in (let H3 \def H_x in (or3_ind (ex2 C (\lambda (c3: C).(eq C -c2 (CHead c3 k t))) (\lambda (c3: C).(csubc g c c3))) (ex5_3 C T A (\lambda -(_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3: -C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) -(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c c3)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c t)))) (\lambda -(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T -(\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b) -v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind -Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b -Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c c3))))) -(ex2 C (\lambda (e2: C).(drop (S n) O c2 e2)) (\lambda (e2: C).(csubc g e1 -e2))) (\lambda (H4: (ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 k t))) -(\lambda (c3: C).(csubc g c c3)))).(ex2_ind C (\lambda (c3: C).(eq C c2 -(CHead c3 k t))) (\lambda (c3: C).(csubc g c c3)) (ex2 C (\lambda (e2: -C).(drop (S n) O c2 e2)) (\lambda (e2: C).(csubc g e1 e2))) (\lambda (x: -C).(\lambda (H5: (eq C c2 (CHead x k t))).(\lambda (H6: (csubc g c -x)).(eq_ind_r C (CHead x k t) (\lambda (c0: C).(ex2 C (\lambda (e2: C).(drop -(S n) O c0 e2)) (\lambda (e2: C).(csubc g e1 e2)))) (let H_x0 \def (H e1 (r k -n) (drop_gen_drop k c e1 t n H1) x H6) in (let H7 \def H_x0 in (ex2_ind C -(\lambda (e2: C).(drop (r k n) O x e2)) (\lambda (e2: C).(csubc g e1 e2)) -(ex2 C (\lambda (e2: C).(drop (S n) O (CHead x k t) e2)) (\lambda (e2: -C).(csubc g e1 e2))) (\lambda (x0: C).(\lambda (H8: (drop (r k n) O x -x0)).(\lambda (H9: (csubc g e1 x0)).(ex_intro2 C (\lambda (e2: C).(drop (S n) -O (CHead x k t) e2)) (\lambda (e2: C).(csubc g e1 e2)) x0 (drop_drop k n x x0 -H8 t) H9)))) H7))) c2 H5)))) H4)) (\lambda (H4: (ex5_3 C T A (\lambda (_: -C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3: -C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) -(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c c3)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c t)))) (\lambda -(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w)))))).(ex5_3_ind C T A -(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) -(\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind -Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c -c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c -t)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w)))) (ex2 -C (\lambda (e2: C).(drop (S n) O c2 e2)) (\lambda (e2: C).(csubc g e1 e2))) -(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H5: (eq K k -(Bind Abst))).(\lambda (H6: (eq C c2 (CHead x0 (Bind Abbr) x1))).(\lambda -(H7: (csubc g c x0)).(\lambda (_: (sc3 g (asucc g x2) c t)).(\lambda (_: (sc3 -g x2 x0 x1)).(eq_ind_r C (CHead x0 (Bind Abbr) x1) (\lambda (c0: C).(ex2 C -(\lambda (e2: C).(drop (S n) O c0 e2)) (\lambda (e2: C).(csubc g e1 e2)))) -(let H10 \def (eq_ind K k (\lambda (k0: K).(drop (r k0 n) O c e1)) -(drop_gen_drop k c e1 t n H1) (Bind Abst) H5) in (let H11 \def (eq_ind K k -(\lambda (k0: K).((drop n O (CHead c k0 t) e1) \to (\forall (c3: C).((csubc g -(CHead c k0 t) c3) \to (ex2 C (\lambda (e2: C).(drop n O c3 e2)) (\lambda -(e2: C).(csubc g e1 e2))))))) H0 (Bind Abst) H5) in (let H_x0 \def (H e1 (r -(Bind Abst) n) H10 x0 H7) in (let H12 \def H_x0 in (ex2_ind C (\lambda (e2: -C).(drop n O x0 e2)) (\lambda (e2: C).(csubc g e1 e2)) (ex2 C (\lambda (e2: -C).(drop (S n) O (CHead x0 (Bind Abbr) x1) e2)) (\lambda (e2: C).(csubc g e1 -e2))) (\lambda (x: C).(\lambda (H13: (drop n O x0 x)).(\lambda (H14: (csubc g -e1 x)).(ex_intro2 C (\lambda (e2: C).(drop (S n) O (CHead x0 (Bind Abbr) x1) -e2)) (\lambda (e2: C).(csubc g e1 e2)) x (drop_drop (Bind Abbr) n x0 x H13 -x1) H14)))) H12))))) c2 H6))))))))) H4)) (\lambda (H4: (ex4_3 B C T (\lambda -(b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b) v2))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind Void))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) -(\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c -c3)))))).(ex4_3_ind B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: -T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: -C).(\lambda (_: T).(csubc g c c3)))) (ex2 C (\lambda (e2: C).(drop (S n) O c2 -e2)) (\lambda (e2: C).(csubc g e1 e2))) (\lambda (x0: B).(\lambda (x1: -C).(\lambda (x2: T).(\lambda (H5: (eq C c2 (CHead x1 (Bind x0) x2))).(\lambda -(H6: (eq K k (Bind Void))).(\lambda (_: (not (eq B x0 Void))).(\lambda (H8: -(csubc g c x1)).(eq_ind_r C (CHead x1 (Bind x0) x2) (\lambda (c0: C).(ex2 C -(\lambda (e2: C).(drop (S n) O c0 e2)) (\lambda (e2: C).(csubc g e1 e2)))) -(let H9 \def (eq_ind K k (\lambda (k0: K).(drop (r k0 n) O c e1)) -(drop_gen_drop k c e1 t n H1) (Bind Void) H6) in (let H10 \def (eq_ind K k -(\lambda (k0: K).((drop n O (CHead c k0 t) e1) \to (\forall (c3: C).((csubc g -(CHead c k0 t) c3) \to (ex2 C (\lambda (e2: C).(drop n O c3 e2)) (\lambda -(e2: C).(csubc g e1 e2))))))) H0 (Bind Void) H6) in (let H_x0 \def (H e1 (r -(Bind Void) n) H9 x1 H8) in (let H11 \def H_x0 in (ex2_ind C (\lambda (e2: -C).(drop n O x1 e2)) (\lambda (e2: C).(csubc g e1 e2)) (ex2 C (\lambda (e2: -C).(drop (S n) O (CHead x1 (Bind x0) x2) e2)) (\lambda (e2: C).(csubc g e1 -e2))) (\lambda (x: C).(\lambda (H12: (drop n O x1 x)).(\lambda (H13: (csubc g -e1 x)).(ex_intro2 C (\lambda (e2: C).(drop (S n) O (CHead x1 (Bind x0) x2) -e2)) (\lambda (e2: C).(csubc g e1 e2)) x (drop_drop (Bind x0) n x1 x H12 x2) -H13)))) H11))))) c2 H5)))))))) H4)) H3)))))))) h))))))) c1)). -(* COMMENTS -Initial nodes: 2389 -END *) - -theorem drop_csubc_trans: - \forall (g: G).(\forall (c2: C).(\forall (e2: C).(\forall (d: nat).(\forall -(h: nat).((drop h d c2 e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C -(\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c2 c1)))))))))) -\def - \lambda (g: G).(\lambda (c2: C).(C_ind (\lambda (c: C).(\forall (e2: -C).(\forall (d: nat).(\forall (h: nat).((drop h d c e2) \to (\forall (e1: -C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda -(c1: C).(csubc g c c1)))))))))) (\lambda (n: nat).(\lambda (e2: C).(\lambda -(d: nat).(\lambda (h: nat).(\lambda (H: (drop h d (CSort n) e2)).(\lambda -(e1: C).(\lambda (H0: (csubc g e2 e1)).(and3_ind (eq C e2 (CSort n)) (eq nat -h O) (eq nat d O) (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: -C).(csubc g (CSort n) c1))) (\lambda (H1: (eq C e2 (CSort n))).(\lambda (H2: -(eq nat h O)).(\lambda (H3: (eq nat d O)).(eq_ind_r nat O (\lambda (n0: -nat).(ex2 C (\lambda (c1: C).(drop n0 d c1 e1)) (\lambda (c1: C).(csubc g -(CSort n) c1)))) (eq_ind_r nat O (\lambda (n0: nat).(ex2 C (\lambda (c1: -C).(drop O n0 c1 e1)) (\lambda (c1: C).(csubc g (CSort n) c1)))) (let H4 \def -(eq_ind C e2 (\lambda (c: C).(csubc g c e1)) H0 (CSort n) H1) in (ex_intro2 C -(\lambda (c1: C).(drop O O c1 e1)) (\lambda (c1: C).(csubc g (CSort n) c1)) -e1 (drop_refl e1) H4)) d H3) h H2)))) (drop_gen_sort n h d e2 H))))))))) -(\lambda (c: C).(\lambda (H: ((\forall (e2: C).(\forall (d: nat).(\forall (h: -nat).((drop h d c e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C -(\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c -c1))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e2: C).(\lambda (d: -nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n (CHead c k t) -e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop h -n c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1)))))))) (\lambda (h: -nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c k t) e2) \to (\forall -(e1: C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop n O c1 e1)) -(\lambda (c1: C).(csubc g (CHead c k t) c1))))))) (\lambda (H0: (drop O O -(CHead c k t) e2)).(\lambda (e1: C).(\lambda (H1: (csubc g e2 e1)).(let H2 -\def (eq_ind_r C e2 (\lambda (c0: C).(csubc g c0 e1)) H1 (CHead c k t) -(drop_gen_refl (CHead c k t) e2 H0)) in (ex_intro2 C (\lambda (c1: C).(drop O -O c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1)) e1 (drop_refl e1) -H2))))) (\lambda (n: nat).(\lambda (_: (((drop n O (CHead c k t) e2) \to -(\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop n O c1 -e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1)))))))).(\lambda (H1: (drop -(S n) O (CHead c k t) e2)).(\lambda (e1: C).(\lambda (H2: (csubc g e2 -e1)).(let H_x \def (H e2 O (r k n) (drop_gen_drop k c e2 t n H1) e1 H2) in -(let H3 \def H_x in (ex2_ind C (\lambda (c1: C).(drop (r k n) O c1 e1)) -(\lambda (c1: C).(csubc g c c1)) (ex2 C (\lambda (c1: C).(drop (S n) O c1 -e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1))) (\lambda (x: C).(\lambda -(H4: (drop (r k n) O x e1)).(\lambda (H5: (csubc g c x)).(ex_intro2 C -(\lambda (c1: C).(drop (S n) O c1 e1)) (\lambda (c1: C).(csubc g (CHead c k -t) c1)) (CHead x k t) (drop_drop k n x e1 H4 t) (csubc_head g c x H5 k t))))) -H3)))))))) h)) (\lambda (n: nat).(\lambda (H0: ((\forall (h: nat).((drop h n -(CHead c k t) e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda -(c1: C).(drop h n c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t) -c1))))))))).(\lambda (h: nat).(\lambda (H1: (drop h (S n) (CHead c k t) -e2)).(\lambda (e1: C).(\lambda (H2: (csubc g e2 e1)).(ex3_2_ind C T (\lambda -(e: C).(\lambda (v: T).(eq C e2 (CHead e k v)))) (\lambda (_: C).(\lambda (v: -T).(eq T t (lift h (r k n) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r k -n) c e))) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1: -C).(csubc g (CHead c k t) c1))) (\lambda (x0: C).(\lambda (x1: T).(\lambda -(H3: (eq C e2 (CHead x0 k x1))).(\lambda (H4: (eq T t (lift h (r k n) -x1))).(\lambda (H5: (drop h (r k n) c x0)).(let H6 \def (eq_ind C e2 (\lambda -(c0: C).(csubc g c0 e1)) H2 (CHead x0 k x1) H3) in (let H7 \def (eq_ind C e2 -(\lambda (c0: C).(\forall (h0: nat).((drop h0 n (CHead c k t) c0) \to -(\forall (e3: C).((csubc g c0 e3) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 -e3)) (\lambda (c1: C).(csubc g (CHead c k t) c1)))))))) H0 (CHead x0 k x1) -H3) in (let H8 \def (eq_ind T t (\lambda (t0: T).(\forall (h0: nat).((drop h0 -n (CHead c k t0) (CHead x0 k x1)) \to (\forall (e3: C).((csubc g (CHead x0 k -x1) e3) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 e3)) (\lambda (c1: -C).(csubc g (CHead c k t0) c1)))))))) H7 (lift h (r k n) x1) H4) in (eq_ind_r -T (lift h (r k n) x1) (\lambda (t0: T).(ex2 C (\lambda (c1: C).(drop h (S n) -c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t0) c1)))) (let H_x \def -(csubc_gen_head_l g x0 e1 x1 k H6) in (let H9 \def H_x in (or3_ind (ex2 C -(\lambda (c3: C).(eq C e1 (CHead c3 k x1))) (\lambda (c3: C).(csubc g x0 -c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k -(Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C e1 -(CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: -A).(csubc g x0 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g -(asucc g a) x0 x1)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g -a c3 w))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: -T).(eq C e1 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: -C).(\lambda (_: T).(csubc g x0 c3))))) (ex2 C (\lambda (c1: C).(drop h (S n) -c1 e1)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1))) -(\lambda (H10: (ex2 C (\lambda (c3: C).(eq C e1 (CHead c3 k x1))) (\lambda -(c3: C).(csubc g x0 c3)))).(ex2_ind C (\lambda (c3: C).(eq C e1 (CHead c3 k -x1))) (\lambda (c3: C).(csubc g x0 c3)) (ex2 C (\lambda (c1: C).(drop h (S n) -c1 e1)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1))) -(\lambda (x: C).(\lambda (H11: (eq C e1 (CHead x k x1))).(\lambda (H12: -(csubc g x0 x)).(eq_ind_r C (CHead x k x1) (\lambda (c0: C).(ex2 C (\lambda -(c1: C).(drop h (S n) c1 c0)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r -k n) x1)) c1)))) (let H_x0 \def (H x0 (r k n) h H5 x H12) in (let H13 \def -H_x0 in (ex2_ind C (\lambda (c1: C).(drop h (r k n) c1 x)) (\lambda (c1: -C).(csubc g c c1)) (ex2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x k x1))) -(\lambda (c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1))) (\lambda (x2: -C).(\lambda (H14: (drop h (r k n) x2 x)).(\lambda (H15: (csubc g c -x2)).(ex_intro2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x k x1))) (\lambda -(c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1)) (CHead x2 k (lift h (r -k n) x1)) (drop_skip k h n x2 x H14 x1) (csubc_head g c x2 H15 k (lift h (r k -n) x1)))))) H13))) e1 H11)))) H10)) (\lambda (H10: (ex5_3 C T A (\lambda (_: -C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3: -C).(\lambda (w: T).(\lambda (_: A).(eq C e1 (CHead c3 (Bind Abbr) w))))) -(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g x0 c3)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) x0 x1)))) (\lambda -(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w)))))).(ex5_3_ind C T A -(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) -(\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C e1 (CHead c3 (Bind -Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g x0 -c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) x0 -x1)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w)))) -(ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1: C).(csubc g -(CHead c k (lift h (r k n) x1)) c1))) (\lambda (x2: C).(\lambda (x3: -T).(\lambda (x4: A).(\lambda (H11: (eq K k (Bind Abst))).(\lambda (H12: (eq C -e1 (CHead x2 (Bind Abbr) x3))).(\lambda (H13: (csubc g x0 x2)).(\lambda (H14: -(sc3 g (asucc g x4) x0 x1)).(\lambda (H15: (sc3 g x4 x2 x3)).(eq_ind_r C -(CHead x2 (Bind Abbr) x3) (\lambda (c0: C).(ex2 C (\lambda (c1: C).(drop h (S -n) c1 c0)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1)))) -(let H16 \def (eq_ind K k (\lambda (k0: K).(\forall (h0: nat).((drop h0 n -(CHead c k0 (lift h (r k0 n) x1)) (CHead x0 k0 x1)) \to (\forall (e3: -C).((csubc g (CHead x0 k0 x1) e3) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 -e3)) (\lambda (c1: C).(csubc g (CHead c k0 (lift h (r k0 n) x1)) c1)))))))) -H8 (Bind Abst) H11) in (let H17 \def (eq_ind K k (\lambda (k0: K).(drop h (r -k0 n) c x0)) H5 (Bind Abst) H11) in (eq_ind_r K (Bind Abst) (\lambda (k0: -K).(ex2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x2 (Bind Abbr) x3))) -(\lambda (c1: C).(csubc g (CHead c k0 (lift h (r k0 n) x1)) c1)))) (let H_x0 -\def (H x0 (r (Bind Abst) n) h H17 x2 H13) in (let H18 \def H_x0 in (ex2_ind -C (\lambda (c1: C).(drop h n c1 x2)) (\lambda (c1: C).(csubc g c c1)) (ex2 C -(\lambda (c1: C).(drop h (S n) c1 (CHead x2 (Bind Abbr) x3))) (\lambda (c1: -C).(csubc g (CHead c (Bind Abst) (lift h (r (Bind Abst) n) x1)) c1))) -(\lambda (x: C).(\lambda (H19: (drop h n x x2)).(\lambda (H20: (csubc g c -x)).(ex_intro2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x2 (Bind Abbr) -x3))) (\lambda (c1: C).(csubc g (CHead c (Bind Abst) (lift h (r (Bind Abst) -n) x1)) c1)) (CHead x (Bind Abbr) (lift h n x3)) (drop_skip_bind h n x x2 H19 -Abbr x3) (csubc_abst g c x H20 (lift h (r (Bind Abst) n) x1) x4 (sc3_lift g -(asucc g x4) x0 x1 H14 c h (r (Bind Abst) n) H17) (lift h n x3) (sc3_lift g -x4 x2 x3 H15 x h n H19)))))) H18))) k H11))) e1 H12))))))))) H10)) (\lambda -(H10: (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C e1 -(CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: -T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: -T).(csubc g x0 c3)))))).(ex4_3_ind B C T (\lambda (b: B).(\lambda (c3: -C).(\lambda (v2: T).(eq C e1 (CHead c3 (Bind b) v2))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: -B).(\lambda (c3: C).(\lambda (_: T).(csubc g x0 c3)))) (ex2 C (\lambda (c1: -C).(drop h (S n) c1 e1)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r k n) -x1)) c1))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: T).(\lambda (H11: -(eq C e1 (CHead x3 (Bind x2) x4))).(\lambda (H12: (eq K k (Bind -Void))).(\lambda (H13: (not (eq B x2 Void))).(\lambda (H14: (csubc g x0 -x3)).(eq_ind_r C (CHead x3 (Bind x2) x4) (\lambda (c0: C).(ex2 C (\lambda -(c1: C).(drop h (S n) c1 c0)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r -k n) x1)) c1)))) (let H15 \def (eq_ind K k (\lambda (k0: K).(\forall (h0: -nat).((drop h0 n (CHead c k0 (lift h (r k0 n) x1)) (CHead x0 k0 x1)) \to -(\forall (e3: C).((csubc g (CHead x0 k0 x1) e3) \to (ex2 C (\lambda (c1: -C).(drop h0 n c1 e3)) (\lambda (c1: C).(csubc g (CHead c k0 (lift h (r k0 n) -x1)) c1)))))))) H8 (Bind Void) H12) in (let H16 \def (eq_ind K k (\lambda -(k0: K).(drop h (r k0 n) c x0)) H5 (Bind Void) H12) in (eq_ind_r K (Bind -Void) (\lambda (k0: K).(ex2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x3 -(Bind x2) x4))) (\lambda (c1: C).(csubc g (CHead c k0 (lift h (r k0 n) x1)) -c1)))) (let H_x0 \def (H x0 (r (Bind Void) n) h H16 x3 H14) in (let H17 \def -H_x0 in (ex2_ind C (\lambda (c1: C).(drop h n c1 x3)) (\lambda (c1: C).(csubc -g c c1)) (ex2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x3 (Bind x2) x4))) -(\lambda (c1: C).(csubc g (CHead c (Bind Void) (lift h (r (Bind Void) n) x1)) -c1))) (\lambda (x: C).(\lambda (H18: (drop h n x x3)).(\lambda (H19: (csubc g -c x)).(ex_intro2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x3 (Bind x2) -x4))) (\lambda (c1: C).(csubc g (CHead c (Bind Void) (lift h (r (Bind Void) -n) x1)) c1)) (CHead x (Bind x2) (lift h n x4)) (drop_skip_bind h n x x3 H18 -x2 x4) (csubc_void g c x H19 x2 H13 (lift h (r (Bind Void) n) x1) (lift h n -x4)))))) H17))) k H12))) e1 H11)))))))) H10)) H9))) t H4))))))))) -(drop_gen_skip_l c e2 t h n k H1)))))))) d))))))) c2)). -(* COMMENTS -Initial nodes: 3747 -END *) - -theorem csubc_drop_conf_rev: - \forall (g: G).(\forall (c2: C).(\forall (e2: C).(\forall (d: nat).(\forall -(h: nat).((drop h d c2 e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C -(\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c1 c2)))))))))) -\def - \lambda (g: G).(\lambda (c2: C).(C_ind (\lambda (c: C).(\forall (e2: -C).(\forall (d: nat).(\forall (h: nat).((drop h d c e2) \to (\forall (e1: -C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda -(c1: C).(csubc g c1 c)))))))))) (\lambda (n: nat).(\lambda (e2: C).(\lambda -(d: nat).(\lambda (h: nat).(\lambda (H: (drop h d (CSort n) e2)).(\lambda -(e1: C).(\lambda (H0: (csubc g e1 e2)).(and3_ind (eq C e2 (CSort n)) (eq nat -h O) (eq nat d O) (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: -C).(csubc g c1 (CSort n)))) (\lambda (H1: (eq C e2 (CSort n))).(\lambda (H2: -(eq nat h O)).(\lambda (H3: (eq nat d O)).(eq_ind_r nat O (\lambda (n0: -nat).(ex2 C (\lambda (c1: C).(drop n0 d c1 e1)) (\lambda (c1: C).(csubc g c1 -(CSort n))))) (eq_ind_r nat O (\lambda (n0: nat).(ex2 C (\lambda (c1: -C).(drop O n0 c1 e1)) (\lambda (c1: C).(csubc g c1 (CSort n))))) (let H4 \def -(eq_ind C e2 (\lambda (c: C).(csubc g e1 c)) H0 (CSort n) H1) in (ex_intro2 C -(\lambda (c1: C).(drop O O c1 e1)) (\lambda (c1: C).(csubc g c1 (CSort n))) -e1 (drop_refl e1) H4)) d H3) h H2)))) (drop_gen_sort n h d e2 H))))))))) -(\lambda (c: C).(\lambda (H: ((\forall (e2: C).(\forall (d: nat).(\forall (h: -nat).((drop h d c e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C -(\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c1 -c))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e2: C).(\lambda (d: -nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n (CHead c k t) -e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop h -n c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t))))))))) (\lambda (h: -nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c k t) e2) \to (\forall -(e1: C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop n O c1 e1)) -(\lambda (c1: C).(csubc g c1 (CHead c k t)))))))) (\lambda (H0: (drop O O -(CHead c k t) e2)).(\lambda (e1: C).(\lambda (H1: (csubc g e1 e2)).(let H2 -\def (eq_ind_r C e2 (\lambda (c0: C).(csubc g e1 c0)) H1 (CHead c k t) -(drop_gen_refl (CHead c k t) e2 H0)) in (ex_intro2 C (\lambda (c1: C).(drop O -O c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t))) e1 (drop_refl e1) -H2))))) (\lambda (n: nat).(\lambda (_: (((drop n O (CHead c k t) e2) \to -(\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop n O c1 -e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t))))))))).(\lambda (H1: (drop -(S n) O (CHead c k t) e2)).(\lambda (e1: C).(\lambda (H2: (csubc g e1 -e2)).(let H_x \def (H e2 O (r k n) (drop_gen_drop k c e2 t n H1) e1 H2) in -(let H3 \def H_x in (ex2_ind C (\lambda (c1: C).(drop (r k n) O c1 e1)) -(\lambda (c1: C).(csubc g c1 c)) (ex2 C (\lambda (c1: C).(drop (S n) O c1 -e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t)))) (\lambda (x: C).(\lambda -(H4: (drop (r k n) O x e1)).(\lambda (H5: (csubc g x c)).(ex_intro2 C -(\lambda (c1: C).(drop (S n) O c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c -k t))) (CHead x k t) (drop_drop k n x e1 H4 t) (csubc_head g x c H5 k t))))) -H3)))))))) h)) (\lambda (n: nat).(\lambda (H0: ((\forall (h: nat).((drop h n -(CHead c k t) e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda -(c1: C).(drop h n c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k -t)))))))))).(\lambda (h: nat).(\lambda (H1: (drop h (S n) (CHead c k t) -e2)).(\lambda (e1: C).(\lambda (H2: (csubc g e1 e2)).(ex3_2_ind C T (\lambda -(e: C).(\lambda (v: T).(eq C e2 (CHead e k v)))) (\lambda (_: C).(\lambda (v: -T).(eq T t (lift h (r k n) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r k -n) c e))) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1: -C).(csubc g c1 (CHead c k t)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda -(H3: (eq C e2 (CHead x0 k x1))).(\lambda (H4: (eq T t (lift h (r k n) -x1))).(\lambda (H5: (drop h (r k n) c x0)).(let H6 \def (eq_ind C e2 (\lambda -(c0: C).(csubc g e1 c0)) H2 (CHead x0 k x1) H3) in (let H7 \def (eq_ind C e2 -(\lambda (c0: C).(\forall (h0: nat).((drop h0 n (CHead c k t) c0) \to -(\forall (e3: C).((csubc g e3 c0) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 -e3)) (\lambda (c1: C).(csubc g c1 (CHead c k t))))))))) H0 (CHead x0 k x1) -H3) in (let H8 \def (eq_ind T t (\lambda (t0: T).(\forall (h0: nat).((drop h0 -n (CHead c k t0) (CHead x0 k x1)) \to (\forall (e3: C).((csubc g e3 (CHead x0 -k x1)) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 e3)) (\lambda (c1: C).(csubc -g c1 (CHead c k t0))))))))) H7 (lift h (r k n) x1) H4) in (eq_ind_r T (lift h -(r k n) x1) (\lambda (t0: T).(ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) -(\lambda (c1: C).(csubc g c1 (CHead c k t0))))) (let H_x \def -(csubc_gen_head_r g x0 e1 x1 k H6) in (let H9 \def H_x in (or3_ind (ex2 C -(\lambda (c1: C).(eq C e1 (CHead c1 k x1))) (\lambda (c1: C).(csubc g c1 -x0))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k -(Bind Abbr))))) (\lambda (c1: C).(\lambda (v: T).(\lambda (_: A).(eq C e1 -(CHead c1 (Bind Abst) v))))) (\lambda (c1: C).(\lambda (_: T).(\lambda (_: -A).(csubc g c1 x0)))) (\lambda (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g -(asucc g a) c1 v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a -x0 x1))))) (ex4_3 B C T (\lambda (_: B).(\lambda (c1: C).(\lambda (v1: T).(eq -C e1 (CHead c1 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda -(_: T).(eq K k (Bind b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: -T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c1: C).(\lambda (_: -T).(csubc g c1 x0))))) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda -(c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1))))) (\lambda (H10: (ex2 C -(\lambda (c1: C).(eq C e1 (CHead c1 k x1))) (\lambda (c1: C).(csubc g c1 -x0)))).(ex2_ind C (\lambda (c1: C).(eq C e1 (CHead c1 k x1))) (\lambda (c1: -C).(csubc g c1 x0)) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda -(c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1))))) (\lambda (x: -C).(\lambda (H11: (eq C e1 (CHead x k x1))).(\lambda (H12: (csubc g x -x0)).(eq_ind_r C (CHead x k x1) (\lambda (c0: C).(ex2 C (\lambda (c1: -C).(drop h (S n) c1 c0)) (\lambda (c1: C).(csubc g c1 (CHead c k (lift h (r k -n) x1)))))) (let H_x0 \def (H x0 (r k n) h H5 x H12) in (let H13 \def H_x0 in -(ex2_ind C (\lambda (c1: C).(drop h (r k n) c1 x)) (\lambda (c1: C).(csubc g -c1 c)) (ex2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x k x1))) (\lambda -(c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1))))) (\lambda (x2: -C).(\lambda (H14: (drop h (r k n) x2 x)).(\lambda (H15: (csubc g x2 -c)).(ex_intro2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x k x1))) (\lambda -(c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1)))) (CHead x2 k (lift h (r -k n) x1)) (drop_skip k h n x2 x H14 x1) (csubc_head g x2 c H15 k (lift h (r k -n) x1)))))) H13))) e1 H11)))) H10)) (\lambda (H10: (ex5_3 C T A (\lambda (_: -C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c1: -C).(\lambda (v: T).(\lambda (_: A).(eq C e1 (CHead c1 (Bind Abst) v))))) -(\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 x0)))) (\lambda -(c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a x0 x1)))))).(ex5_3_ind C T A -(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) -(\lambda (c1: C).(\lambda (v: T).(\lambda (_: A).(eq C e1 (CHead c1 (Bind -Abst) v))))) (\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 -x0)))) (\lambda (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1 -v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a x0 x1)))) (ex2 -C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead -c k (lift h (r k n) x1))))) (\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: -A).(\lambda (H11: (eq K k (Bind Abbr))).(\lambda (H12: (eq C e1 (CHead x2 -(Bind Abst) x3))).(\lambda (H13: (csubc g x2 x0)).(\lambda (H14: (sc3 g -(asucc g x4) x2 x3)).(\lambda (H15: (sc3 g x4 x0 x1)).(eq_ind_r C (CHead x2 -(Bind Abst) x3) (\lambda (c0: C).(ex2 C (\lambda (c1: C).(drop h (S n) c1 -c0)) (\lambda (c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1)))))) (let -H16 \def (eq_ind K k (\lambda (k0: K).(\forall (h0: nat).((drop h0 n (CHead c -k0 (lift h (r k0 n) x1)) (CHead x0 k0 x1)) \to (\forall (e3: C).((csubc g e3 -(CHead x0 k0 x1)) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 e3)) (\lambda -(c1: C).(csubc g c1 (CHead c k0 (lift h (r k0 n) x1)))))))))) H8 (Bind Abbr) -H11) in (let H17 \def (eq_ind K k (\lambda (k0: K).(drop h (r k0 n) c x0)) H5 -(Bind Abbr) H11) in (eq_ind_r K (Bind Abbr) (\lambda (k0: K).(ex2 C (\lambda -(c1: C).(drop h (S n) c1 (CHead x2 (Bind Abst) x3))) (\lambda (c1: C).(csubc -g c1 (CHead c k0 (lift h (r k0 n) x1)))))) (let H_x0 \def (H x0 (r (Bind -Abbr) n) h H17 x2 H13) in (let H18 \def H_x0 in (ex2_ind C (\lambda (c1: -C).(drop h n c1 x2)) (\lambda (c1: C).(csubc g c1 c)) (ex2 C (\lambda (c1: -C).(drop h (S n) c1 (CHead x2 (Bind Abst) x3))) (\lambda (c1: C).(csubc g c1 -(CHead c (Bind Abbr) (lift h (r (Bind Abbr) n) x1))))) (\lambda (x: -C).(\lambda (H19: (drop h n x x2)).(\lambda (H20: (csubc g x c)).(ex_intro2 C -(\lambda (c1: C).(drop h (S n) c1 (CHead x2 (Bind Abst) x3))) (\lambda (c1: -C).(csubc g c1 (CHead c (Bind Abbr) (lift h (r (Bind Abbr) n) x1)))) (CHead x -(Bind Abst) (lift h n x3)) (drop_skip_bind h n x x2 H19 Abst x3) (csubc_abst -g x c H20 (lift h n x3) x4 (sc3_lift g (asucc g x4) x2 x3 H14 x h n H19) -(lift h (r (Bind Abbr) n) x1) (sc3_lift g x4 x0 x1 H15 c h (r (Bind Abbr) n) -H17)))))) H18))) k H11))) e1 H12))))))))) H10)) (\lambda (H10: (ex4_3 B C T -(\lambda (_: B).(\lambda (c1: C).(\lambda (v1: T).(eq C e1 (CHead c1 (Bind -Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind -b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) -(\lambda (_: B).(\lambda (c1: C).(\lambda (_: T).(csubc g c1 -x0)))))).(ex4_3_ind B C T (\lambda (_: B).(\lambda (c1: C).(\lambda (v1: -T).(eq C e1 (CHead c1 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c1: -C).(\lambda (_: T).(csubc g c1 x0)))) (ex2 C (\lambda (c1: C).(drop h (S n) -c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1))))) -(\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: T).(\lambda (H11: (eq C e1 -(CHead x3 (Bind Void) x4))).(\lambda (H12: (eq K k (Bind x2))).(\lambda (H13: -(not (eq B x2 Void))).(\lambda (H14: (csubc g x3 x0)).(eq_ind_r C (CHead x3 -(Bind Void) x4) (\lambda (c0: C).(ex2 C (\lambda (c1: C).(drop h (S n) c1 -c0)) (\lambda (c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1)))))) (let -H15 \def (eq_ind K k (\lambda (k0: K).(\forall (h0: nat).((drop h0 n (CHead c -k0 (lift h (r k0 n) x1)) (CHead x0 k0 x1)) \to (\forall (e3: C).((csubc g e3 -(CHead x0 k0 x1)) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 e3)) (\lambda -(c1: C).(csubc g c1 (CHead c k0 (lift h (r k0 n) x1)))))))))) H8 (Bind x2) -H12) in (let H16 \def (eq_ind K k (\lambda (k0: K).(drop h (r k0 n) c x0)) H5 -(Bind x2) H12) in (eq_ind_r K (Bind x2) (\lambda (k0: K).(ex2 C (\lambda (c1: -C).(drop h (S n) c1 (CHead x3 (Bind Void) x4))) (\lambda (c1: C).(csubc g c1 -(CHead c k0 (lift h (r k0 n) x1)))))) (let H_x0 \def (H x0 (r (Bind x2) n) h -H16 x3 H14) in (let H17 \def H_x0 in (ex2_ind C (\lambda (c1: C).(drop h n c1 -x3)) (\lambda (c1: C).(csubc g c1 c)) (ex2 C (\lambda (c1: C).(drop h (S n) -c1 (CHead x3 (Bind Void) x4))) (\lambda (c1: C).(csubc g c1 (CHead c (Bind -x2) (lift h (r (Bind x2) n) x1))))) (\lambda (x: C).(\lambda (H18: (drop h n -x x3)).(\lambda (H19: (csubc g x c)).(ex_intro2 C (\lambda (c1: C).(drop h (S -n) c1 (CHead x3 (Bind Void) x4))) (\lambda (c1: C).(csubc g c1 (CHead c (Bind -x2) (lift h (r (Bind x2) n) x1)))) (CHead x (Bind Void) (lift h n x4)) -(drop_skip_bind h n x x3 H18 Void x4) (csubc_void g x c H19 x2 H13 (lift h n -x4) (lift h (r (Bind x2) n) x1)))))) H17))) k H12))) e1 H11)))))))) H10)) -H9))) t H4))))))))) (drop_gen_skip_l c e2 t h n k H1)))))))) d))))))) c2)). -(* COMMENTS -Initial nodes: 3747 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/drop1.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/drop1.ma deleted file mode 100644 index 7c539cb4a..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/drop1.ma +++ /dev/null @@ -1,92 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubc/drop.ma". - -theorem drop1_csubc_trans: - \forall (g: G).(\forall (hds: PList).(\forall (c2: C).(\forall (e2: -C).((drop1 hds c2 e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C -(\lambda (c1: C).(drop1 hds c1 e1)) (\lambda (c1: C).(csubc g c2 c1))))))))) -\def - \lambda (g: G).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall -(c2: C).(\forall (e2: C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e2 -e1) \to (ex2 C (\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c2 -c1))))))))) (\lambda (c2: C).(\lambda (e2: C).(\lambda (H: (drop1 PNil c2 -e2)).(\lambda (e1: C).(\lambda (H0: (csubc g e2 e1)).(let H_y \def -(drop1_gen_pnil c2 e2 H) in (let H1 \def (eq_ind_r C e2 (\lambda (c: -C).(csubc g c e1)) H0 c2 H_y) in (ex_intro2 C (\lambda (c1: C).(drop1 PNil c1 -e1)) (\lambda (c1: C).(csubc g c2 c1)) e1 (drop1_nil e1) H1)))))))) (\lambda -(n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c2: -C).(\forall (e2: C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e2 e1) -\to (ex2 C (\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c2 -c1)))))))))).(\lambda (c2: C).(\lambda (e2: C).(\lambda (H0: (drop1 (PCons n -n0 p) c2 e2)).(\lambda (e1: C).(\lambda (H1: (csubc g e2 e1)).(let H_x \def -(drop1_gen_pcons c2 e2 p n n0 H0) in (let H2 \def H_x in (ex2_ind C (\lambda -(c3: C).(drop n n0 c2 c3)) (\lambda (c3: C).(drop1 p c3 e2)) (ex2 C (\lambda -(c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c2 c1))) -(\lambda (x: C).(\lambda (H3: (drop n n0 c2 x)).(\lambda (H4: (drop1 p x -e2)).(let H_x0 \def (H x e2 H4 e1 H1) in (let H5 \def H_x0 in (ex2_ind C -(\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g x c1)) (ex2 C -(\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c2 -c1))) (\lambda (x0: C).(\lambda (H6: (drop1 p x0 e1)).(\lambda (H7: (csubc g -x x0)).(let H_x1 \def (drop_csubc_trans g c2 x n0 n H3 x0 H7) in (let H8 \def -H_x1 in (ex2_ind C (\lambda (c1: C).(drop n n0 c1 x0)) (\lambda (c1: -C).(csubc g c2 c1)) (ex2 C (\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) -(\lambda (c1: C).(csubc g c2 c1))) (\lambda (x1: C).(\lambda (H9: (drop n n0 -x1 x0)).(\lambda (H10: (csubc g c2 x1)).(ex_intro2 C (\lambda (c1: C).(drop1 -(PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c2 c1)) x1 (drop1_cons x1 x0 -n n0 H9 e1 p H6) H10)))) H8)))))) H5)))))) H2)))))))))))) hds)). -(* COMMENTS -Initial nodes: 551 -END *) - -theorem csubc_drop1_conf_rev: - \forall (g: G).(\forall (hds: PList).(\forall (c2: C).(\forall (e2: -C).((drop1 hds c2 e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C -(\lambda (c1: C).(drop1 hds c1 e1)) (\lambda (c1: C).(csubc g c1 c2))))))))) -\def - \lambda (g: G).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall -(c2: C).(\forall (e2: C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e1 -e2) \to (ex2 C (\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c1 -c2))))))))) (\lambda (c2: C).(\lambda (e2: C).(\lambda (H: (drop1 PNil c2 -e2)).(\lambda (e1: C).(\lambda (H0: (csubc g e1 e2)).(let H_y \def -(drop1_gen_pnil c2 e2 H) in (let H1 \def (eq_ind_r C e2 (\lambda (c: -C).(csubc g e1 c)) H0 c2 H_y) in (ex_intro2 C (\lambda (c1: C).(drop1 PNil c1 -e1)) (\lambda (c1: C).(csubc g c1 c2)) e1 (drop1_nil e1) H1)))))))) (\lambda -(n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c2: -C).(\forall (e2: C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e1 e2) -\to (ex2 C (\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c1 -c2)))))))))).(\lambda (c2: C).(\lambda (e2: C).(\lambda (H0: (drop1 (PCons n -n0 p) c2 e2)).(\lambda (e1: C).(\lambda (H1: (csubc g e1 e2)).(let H_x \def -(drop1_gen_pcons c2 e2 p n n0 H0) in (let H2 \def H_x in (ex2_ind C (\lambda -(c3: C).(drop n n0 c2 c3)) (\lambda (c3: C).(drop1 p c3 e2)) (ex2 C (\lambda -(c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c1 c2))) -(\lambda (x: C).(\lambda (H3: (drop n n0 c2 x)).(\lambda (H4: (drop1 p x -e2)).(let H_x0 \def (H x e2 H4 e1 H1) in (let H5 \def H_x0 in (ex2_ind C -(\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c1 x)) (ex2 C -(\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c1 -c2))) (\lambda (x0: C).(\lambda (H6: (drop1 p x0 e1)).(\lambda (H7: (csubc g -x0 x)).(let H_x1 \def (csubc_drop_conf_rev g c2 x n0 n H3 x0 H7) in (let H8 -\def H_x1 in (ex2_ind C (\lambda (c1: C).(drop n n0 c1 x0)) (\lambda (c1: -C).(csubc g c1 c2)) (ex2 C (\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) -(\lambda (c1: C).(csubc g c1 c2))) (\lambda (x1: C).(\lambda (H9: (drop n n0 -x1 x0)).(\lambda (H10: (csubc g x1 c2)).(ex_intro2 C (\lambda (c1: C).(drop1 -(PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c1 c2)) x1 (drop1_cons x1 x0 -n n0 H9 e1 p H6) H10)))) H8)))))) H5)))))) H2)))))))))))) hds)). -(* COMMENTS -Initial nodes: 551 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/fwd.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/fwd.ma deleted file mode 100644 index fe04ddd45..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/fwd.ma +++ /dev/null @@ -1,673 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubc/defs.ma". - -theorem csubc_gen_sort_l: - \forall (g: G).(\forall (x: C).(\forall (n: nat).((csubc g (CSort n) x) \to -(eq C x (CSort n))))) -\def - \lambda (g: G).(\lambda (x: C).(\lambda (n: nat).(\lambda (H: (csubc g -(CSort n) x)).(insert_eq C (CSort n) (\lambda (c: C).(csubc g c x)) (\lambda -(c: C).(eq C x c)) (\lambda (y: C).(\lambda (H0: (csubc g y x)).(csubc_ind g -(\lambda (c: C).(\lambda (c0: C).((eq C c (CSort n)) \to (eq C c0 c)))) -(\lambda (n0: nat).(\lambda (H1: (eq C (CSort n0) (CSort n))).(let H2 \def -(f_equal C nat (\lambda (e: C).(match e in C return (\lambda (_: C).nat) with -[(CSort n1) \Rightarrow n1 | (CHead _ _ _) \Rightarrow n0])) (CSort n0) -(CSort n) H1) in (eq_ind_r nat n (\lambda (n1: nat).(eq C (CSort n1) (CSort -n1))) (refl_equal C (CSort n)) n0 H2)))) (\lambda (c1: C).(\lambda (c2: -C).(\lambda (_: (csubc g c1 c2)).(\lambda (_: (((eq C c1 (CSort n)) \to (eq C -c2 c1)))).(\lambda (k: K).(\lambda (v: T).(\lambda (H3: (eq C (CHead c1 k v) -(CSort n))).(let H4 \def (eq_ind C (CHead c1 k v) (\lambda (ee: C).(match ee -in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead -_ _ _) \Rightarrow True])) I (CSort n) H3) in (False_ind (eq C (CHead c2 k v) -(CHead c1 k v)) H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: -(csubc g c1 c2)).(\lambda (_: (((eq C c1 (CSort n)) \to (eq C c2 -c1)))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: -T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u1) (CSort -n))).(let H5 \def (eq_ind C (CHead c1 (Bind Void) u1) (\lambda (ee: C).(match -ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | -(CHead _ _ _) \Rightarrow True])) I (CSort n) H4) in (False_ind (eq C (CHead -c2 (Bind b) u2) (CHead c1 (Bind Void) u1)) H5))))))))))) (\lambda (c1: -C).(\lambda (c2: C).(\lambda (_: (csubc g c1 c2)).(\lambda (_: (((eq C c1 -(CSort n)) \to (eq C c2 c1)))).(\lambda (v: T).(\lambda (a: A).(\lambda (_: -(sc3 g (asucc g a) c1 v)).(\lambda (w: T).(\lambda (_: (sc3 g a c2 -w)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) v) (CSort n))).(let H6 \def -(eq_ind C (CHead c1 (Bind Abst) v) (\lambda (ee: C).(match ee in C return -(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) -\Rightarrow True])) I (CSort n) H5) in (False_ind (eq C (CHead c2 (Bind Abbr) -w) (CHead c1 (Bind Abst) v)) H6)))))))))))) y x H0))) H)))). -(* COMMENTS -Initial nodes: 533 -END *) - -theorem csubc_gen_head_l: - \forall (g: G).(\forall (c1: C).(\forall (x: C).(\forall (v: T).(\forall (k: -K).((csubc g (CHead c1 k v) x) \to (or3 (ex2 C (\lambda (c2: C).(eq C x -(CHead c2 k v))) (\lambda (c2: C).(csubc g c1 c2))) (ex5_3 C T A (\lambda (_: -C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c2: -C).(\lambda (w: T).(\lambda (_: A).(eq C x (CHead c2 (Bind Abbr) w))))) -(\lambda (c2: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c2)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda -(c2: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T -(\lambda (b: B).(\lambda (c2: C).(\lambda (v2: T).(eq C x (CHead c2 (Bind b) -v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind -Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b -Void))))) (\lambda (_: B).(\lambda (c2: C).(\lambda (_: T).(csubc g c1 -c2))))))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (x: C).(\lambda (v: T).(\lambda (k: -K).(\lambda (H: (csubc g (CHead c1 k v) x)).(insert_eq C (CHead c1 k v) -(\lambda (c: C).(csubc g c x)) (\lambda (_: C).(or3 (ex2 C (\lambda (c2: -C).(eq C x (CHead c2 k v))) (\lambda (c2: C).(csubc g c1 c2))) (ex5_3 C T A -(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) -(\lambda (c2: C).(\lambda (w: T).(\lambda (_: A).(eq C x (CHead c2 (Bind -Abbr) w))))) (\lambda (c2: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 -c2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 -v)))) (\lambda (c2: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c2 w))))) -(ex4_3 B C T (\lambda (b: B).(\lambda (c2: C).(\lambda (v2: T).(eq C x (CHead -c2 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k -(Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b -Void))))) (\lambda (_: B).(\lambda (c2: C).(\lambda (_: T).(csubc g c1 -c2))))))) (\lambda (y: C).(\lambda (H0: (csubc g y x)).(csubc_ind g (\lambda -(c: C).(\lambda (c0: C).((eq C c (CHead c1 k v)) \to (or3 (ex2 C (\lambda -(c2: C).(eq C c0 (CHead c2 k v))) (\lambda (c2: C).(csubc g c1 c2))) (ex5_3 C -T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) -(\lambda (c2: C).(\lambda (w: T).(\lambda (_: A).(eq C c0 (CHead c2 (Bind -Abbr) w))))) (\lambda (c2: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 -c2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 -v)))) (\lambda (c2: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c2 w))))) -(ex4_3 B C T (\lambda (b: B).(\lambda (c2: C).(\lambda (v2: T).(eq C c0 -(CHead c2 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: -T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c2: C).(\lambda (_: -T).(csubc g c1 c2))))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) -(CHead c1 k v))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee -in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ -_ _) \Rightarrow False])) I (CHead c1 k v) H1) in (False_ind (or3 (ex2 C -(\lambda (c2: C).(eq C (CSort n) (CHead c2 k v))) (\lambda (c2: C).(csubc g -c1 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k -(Bind Abst))))) (\lambda (c2: C).(\lambda (w: T).(\lambda (_: A).(eq C (CSort -n) (CHead c2 (Bind Abbr) w))))) (\lambda (c2: C).(\lambda (_: T).(\lambda (_: -A).(csubc g c1 c2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g -(asucc g a) c1 v)))) (\lambda (c2: C).(\lambda (w: T).(\lambda (a: A).(sc3 g -a c2 w))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c2: C).(\lambda (v2: -T).(eq C (CSort n) (CHead c2 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c2: -C).(\lambda (_: T).(csubc g c1 c2)))))) H2)))) (\lambda (c0: C).(\lambda (c2: -C).(\lambda (H1: (csubc g c0 c2)).(\lambda (H2: (((eq C c0 (CHead c1 k v)) -\to (or3 (ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 k v))) (\lambda (c3: -C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda -(_: A).(eq K k (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: -A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: -T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (c3: C).(\lambda (w: -T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda (b: B).(\lambda -(c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: -B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 c3))))))))).(\lambda (k0: -K).(\lambda (v0: T).(\lambda (H3: (eq C (CHead c0 k0 v0) (CHead c1 k -v))).(let H4 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda -(_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) -(CHead c0 k0 v0) (CHead c1 k v) H3) in ((let H5 \def (f_equal C K (\lambda -(e: C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k0 -| (CHead _ k1 _) \Rightarrow k1])) (CHead c0 k0 v0) (CHead c1 k v) H3) in -((let H6 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: -C).T) with [(CSort _) \Rightarrow v0 | (CHead _ _ t) \Rightarrow t])) (CHead -c0 k0 v0) (CHead c1 k v) H3) in (\lambda (H7: (eq K k0 k)).(\lambda (H8: (eq -C c0 c1)).(eq_ind_r T v (\lambda (t: T).(or3 (ex2 C (\lambda (c3: C).(eq C -(CHead c2 k0 t) (CHead c3 k v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C -T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) -(\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C (CHead c2 k0 t) (CHead -c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc -g c1 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g -a) c1 v)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 -w))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C -(CHead c2 k0 t) (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: -C).(\lambda (_: T).(csubc g c1 c3))))))) (eq_ind_r K k (\lambda (k1: K).(or3 -(ex2 C (\lambda (c3: C).(eq C (CHead c2 k1 v) (CHead c3 k v))) (\lambda (c3: -C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda -(_: A).(eq K k (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: -A).(eq C (CHead c2 k1 v) (CHead c3 (Bind Abbr) w))))) (\lambda (c3: -C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (c3: -C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda -(b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C (CHead c2 k1 v) (CHead c3 -(Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k -(Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b -Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 -c3))))))) (let H9 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 k v)) -\to (or3 (ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 k v))) (\lambda (c3: -C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda -(_: A).(eq K k (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: -A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: -T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (c3: C).(\lambda (w: -T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda (b: B).(\lambda -(c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: -B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 c3)))))))) H2 c1 H8) in (let -H10 \def (eq_ind C c0 (\lambda (c: C).(csubc g c c2)) H1 c1 H8) in -(or3_intro0 (ex2 C (\lambda (c3: C).(eq C (CHead c2 k v) (CHead c3 k v))) -(\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: -T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3: C).(\lambda (w: -T).(\lambda (_: A).(eq C (CHead c2 k v) (CHead c3 (Bind Abbr) w))))) (\lambda -(c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (c3: -C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda -(b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C (CHead c2 k v) (CHead c3 (Bind -b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind -Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b -Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 -c3))))) (ex_intro2 C (\lambda (c3: C).(eq C (CHead c2 k v) (CHead c3 k v))) -(\lambda (c3: C).(csubc g c1 c3)) c2 (refl_equal C (CHead c2 k v)) H10)))) k0 -H7) v0 H6)))) H5)) H4))))))))) (\lambda (c0: C).(\lambda (c2: C).(\lambda -(H1: (csubc g c0 c2)).(\lambda (H2: (((eq C c0 (CHead c1 k v)) \to (or3 (ex2 -C (\lambda (c3: C).(eq C c2 (CHead c3 k v))) (\lambda (c3: C).(csubc g c1 -c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k -(Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C c2 -(CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: -A).(csubc g c1 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g -(asucc g a) c1 v)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g -a c3 w))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: -T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: -C).(\lambda (_: T).(csubc g c1 c3))))))))).(\lambda (b: B).(\lambda (H3: (not -(eq B b Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead -c0 (Bind Void) u1) (CHead c1 k v))).(let H5 \def (f_equal C C (\lambda (e: -C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | -(CHead c _ _) \Rightarrow c])) (CHead c0 (Bind Void) u1) (CHead c1 k v) H4) -in ((let H6 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda -(_: C).K) with [(CSort _) \Rightarrow (Bind Void) | (CHead _ k0 _) -\Rightarrow k0])) (CHead c0 (Bind Void) u1) (CHead c1 k v) H4) in ((let H7 -\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead c0 -(Bind Void) u1) (CHead c1 k v) H4) in (\lambda (H8: (eq K (Bind Void) -k)).(\lambda (H9: (eq C c0 c1)).(let H10 \def (eq_ind C c0 (\lambda (c: -C).((eq C c (CHead c1 k v)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c2 (CHead -c3 k v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: -C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3: -C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) -(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda -(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T -(\lambda (b0: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind -b0) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind -Void))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b0 -Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 -c3)))))))) H2 c1 H9) in (let H11 \def (eq_ind C c0 (\lambda (c: C).(csubc g c -c2)) H1 c1 H9) in (let H12 \def (eq_ind_r K k (\lambda (k0: K).((eq C c1 -(CHead c1 k0 v)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 k0 v))) -(\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: -T).(\lambda (_: A).(eq K k0 (Bind Abst))))) (\lambda (c3: C).(\lambda (w: -T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: -C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (c3: -C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda -(b0: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b0) -v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k0 (Bind -Void))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b0 -Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 -c3)))))))) H10 (Bind Void) H8) in (eq_ind K (Bind Void) (\lambda (k0: K).(or3 -(ex2 C (\lambda (c3: C).(eq C (CHead c2 (Bind b) u2) (CHead c3 k0 v))) -(\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: -T).(\lambda (_: A).(eq K k0 (Bind Abst))))) (\lambda (c3: C).(\lambda (w: -T).(\lambda (_: A).(eq C (CHead c2 (Bind b) u2) (CHead c3 (Bind Abbr) w))))) -(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda -(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T -(\lambda (b0: B).(\lambda (c3: C).(\lambda (v2: T).(eq C (CHead c2 (Bind b) -u2) (CHead c3 (Bind b0) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -T).(eq K k0 (Bind Void))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: -T).(not (eq B b0 Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: -T).(csubc g c1 c3))))))) (or3_intro2 (ex2 C (\lambda (c3: C).(eq C (CHead c2 -(Bind b) u2) (CHead c3 (Bind Void) v))) (\lambda (c3: C).(csubc g c1 c3))) -(ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K (Bind -Void) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C -(CHead c2 (Bind b) u2) (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda -(_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (c3: C).(\lambda (w: -T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda (b0: B).(\lambda -(c3: C).(\lambda (v2: T).(eq C (CHead c2 (Bind b) u2) (CHead c3 (Bind b0) -v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K (Bind Void) -(Bind Void))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B -b0 Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 -c3))))) (ex4_3_intro B C T (\lambda (b0: B).(\lambda (c3: C).(\lambda (v2: -T).(eq C (CHead c2 (Bind b) u2) (CHead c3 (Bind b0) v2))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: T).(eq K (Bind Void) (Bind Void))))) (\lambda -(b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b0 Void))))) (\lambda (_: -B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 c3)))) b c2 u2 (refl_equal C -(CHead c2 (Bind b) u2)) (refl_equal K (Bind Void)) H3 H11)) k H8))))))) H6)) -H5))))))))))) (\lambda (c0: C).(\lambda (c2: C).(\lambda (H1: (csubc g c0 -c2)).(\lambda (H2: (((eq C c0 (CHead c1 k v)) \to (or3 (ex2 C (\lambda (c3: -C).(eq C c2 (CHead c3 k v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A -(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) -(\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind -Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 -c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 -v)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) -(ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 -(CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: -T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: -T).(csubc g c1 c3))))))))).(\lambda (v0: T).(\lambda (a: A).(\lambda (H3: -(sc3 g (asucc g a) c0 v0)).(\lambda (w: T).(\lambda (H4: (sc3 g a c2 -w)).(\lambda (H5: (eq C (CHead c0 (Bind Abst) v0) (CHead c1 k v))).(let H6 -\def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) -with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 -(Bind Abst) v0) (CHead c1 k v) H5) in ((let H7 \def (f_equal C K (\lambda (e: -C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow (Bind -Abst) | (CHead _ k0 _) \Rightarrow k0])) (CHead c0 (Bind Abst) v0) (CHead c1 -k v) H5) in ((let H8 \def (f_equal C T (\lambda (e: C).(match e in C return -(\lambda (_: C).T) with [(CSort _) \Rightarrow v0 | (CHead _ _ t) \Rightarrow -t])) (CHead c0 (Bind Abst) v0) (CHead c1 k v) H5) in (\lambda (H9: (eq K -(Bind Abst) k)).(\lambda (H10: (eq C c0 c1)).(let H11 \def (eq_ind T v0 -(\lambda (t: T).(sc3 g (asucc g a) c0 t)) H3 v H8) in (let H12 \def (eq_ind C -c0 (\lambda (c: C).(sc3 g (asucc g a) c v)) H11 c1 H10) in (let H13 \def -(eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 k v)) \to (or3 (ex2 C -(\lambda (c3: C).(eq C c2 (CHead c3 k v))) (\lambda (c3: C).(csubc g c1 c3))) -(ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind -Abst))))) (\lambda (c3: C).(\lambda (w0: T).(\lambda (_: A).(eq C c2 (CHead -c3 (Bind Abbr) w0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: -A).(csubc g c1 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g -(asucc g a0) c1 v)))) (\lambda (c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 -g a0 c3 w0))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: -T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: -C).(\lambda (_: T).(csubc g c1 c3)))))))) H2 c1 H10) in (let H14 \def (eq_ind -C c0 (\lambda (c: C).(csubc g c c2)) H1 c1 H10) in (let H15 \def (eq_ind_r K -k (\lambda (k0: K).((eq C c1 (CHead c1 k0 v)) \to (or3 (ex2 C (\lambda (c3: -C).(eq C c2 (CHead c3 k0 v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A -(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k0 (Bind Abst))))) -(\lambda (c3: C).(\lambda (w0: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind -Abbr) w0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 -c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g (asucc g a0) -c1 v)))) (\lambda (c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 g a0 c3 -w0))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C -c2 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -T).(eq K k0 (Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: -T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: -T).(csubc g c1 c3)))))))) H13 (Bind Abst) H9) in (eq_ind K (Bind Abst) -(\lambda (k0: K).(or3 (ex2 C (\lambda (c3: C).(eq C (CHead c2 (Bind Abbr) w) -(CHead c3 k0 v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda -(_: C).(\lambda (_: T).(\lambda (_: A).(eq K k0 (Bind Abst))))) (\lambda (c3: -C).(\lambda (w0: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) w) (CHead c3 -(Bind Abbr) w0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g -c1 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g (asucc g -a0) c1 v)))) (\lambda (c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 g a0 c3 -w0))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C -(CHead c2 (Bind Abbr) w) (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda -(_: C).(\lambda (_: T).(eq K k0 (Bind Void))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: -C).(\lambda (_: T).(csubc g c1 c3))))))) (or3_intro1 (ex2 C (\lambda (c3: -C).(eq C (CHead c2 (Bind Abbr) w) (CHead c3 (Bind Abst) v))) (\lambda (c3: -C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda -(_: A).(eq K (Bind Abst) (Bind Abst))))) (\lambda (c3: C).(\lambda (w0: -T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) w) (CHead c3 (Bind Abbr) -w0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g (asucc g a0) c1 v)))) -(\lambda (c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 g a0 c3 w0))))) -(ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C (CHead -c2 (Bind Abbr) w) (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: T).(eq K (Bind Abst) (Bind Void))))) (\lambda (b: B).(\lambda -(_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: -C).(\lambda (_: T).(csubc g c1 c3))))) (ex5_3_intro C T A (\lambda (_: -C).(\lambda (_: T).(\lambda (_: A).(eq K (Bind Abst) (Bind Abst))))) (\lambda -(c3: C).(\lambda (w0: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) w) -(CHead c3 (Bind Abbr) w0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: -A).(csubc g c1 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g -(asucc g a0) c1 v)))) (\lambda (c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 -g a0 c3 w0)))) c2 w a (refl_equal K (Bind Abst)) (refl_equal C (CHead c2 -(Bind Abbr) w)) H14 H12 H4)) k H9))))))))) H7)) H6)))))))))))) y x H0))) -H)))))). -(* COMMENTS -Initial nodes: 5205 -END *) - -theorem csubc_gen_sort_r: - \forall (g: G).(\forall (x: C).(\forall (n: nat).((csubc g x (CSort n)) \to -(eq C x (CSort n))))) -\def - \lambda (g: G).(\lambda (x: C).(\lambda (n: nat).(\lambda (H: (csubc g x -(CSort n))).(insert_eq C (CSort n) (\lambda (c: C).(csubc g x c)) (\lambda -(c: C).(eq C x c)) (\lambda (y: C).(\lambda (H0: (csubc g x y)).(csubc_ind g -(\lambda (c: C).(\lambda (c0: C).((eq C c0 (CSort n)) \to (eq C c c0)))) -(\lambda (n0: nat).(\lambda (H1: (eq C (CSort n0) (CSort n))).(let H2 \def -(f_equal C nat (\lambda (e: C).(match e in C return (\lambda (_: C).nat) with -[(CSort n1) \Rightarrow n1 | (CHead _ _ _) \Rightarrow n0])) (CSort n0) -(CSort n) H1) in (eq_ind_r nat n (\lambda (n1: nat).(eq C (CSort n1) (CSort -n1))) (refl_equal C (CSort n)) n0 H2)))) (\lambda (c1: C).(\lambda (c2: -C).(\lambda (_: (csubc g c1 c2)).(\lambda (_: (((eq C c2 (CSort n)) \to (eq C -c1 c2)))).(\lambda (k: K).(\lambda (v: T).(\lambda (H3: (eq C (CHead c2 k v) -(CSort n))).(let H4 \def (eq_ind C (CHead c2 k v) (\lambda (ee: C).(match ee -in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead -_ _ _) \Rightarrow True])) I (CSort n) H3) in (False_ind (eq C (CHead c1 k v) -(CHead c2 k v)) H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: -(csubc g c1 c2)).(\lambda (_: (((eq C c2 (CSort n)) \to (eq C c1 -c2)))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: -T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c2 (Bind b) u2) (CSort -n))).(let H5 \def (eq_ind C (CHead c2 (Bind b) u2) (\lambda (ee: C).(match ee -in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead -_ _ _) \Rightarrow True])) I (CSort n) H4) in (False_ind (eq C (CHead c1 -(Bind Void) u1) (CHead c2 (Bind b) u2)) H5))))))))))) (\lambda (c1: -C).(\lambda (c2: C).(\lambda (_: (csubc g c1 c2)).(\lambda (_: (((eq C c2 -(CSort n)) \to (eq C c1 c2)))).(\lambda (v: T).(\lambda (a: A).(\lambda (_: -(sc3 g (asucc g a) c1 v)).(\lambda (w: T).(\lambda (_: (sc3 g a c2 -w)).(\lambda (H5: (eq C (CHead c2 (Bind Abbr) w) (CSort n))).(let H6 \def -(eq_ind C (CHead c2 (Bind Abbr) w) (\lambda (ee: C).(match ee in C return -(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) -\Rightarrow True])) I (CSort n) H5) in (False_ind (eq C (CHead c1 (Bind Abst) -v) (CHead c2 (Bind Abbr) w)) H6)))))))))))) x y H0))) H)))). -(* COMMENTS -Initial nodes: 533 -END *) - -theorem csubc_gen_head_r: - \forall (g: G).(\forall (c2: C).(\forall (x: C).(\forall (w: T).(\forall (k: -K).((csubc g x (CHead c2 k w)) \to (or3 (ex2 C (\lambda (c1: C).(eq C x -(CHead c1 k w))) (\lambda (c1: C).(csubc g c1 c2))) (ex5_3 C T A (\lambda (_: -C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c1: -C).(\lambda (v: T).(\lambda (_: A).(eq C x (CHead c1 (Bind Abst) v))))) -(\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c2)))) (\lambda -(c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T -(\lambda (_: B).(\lambda (c1: C).(\lambda (v1: T).(eq C x (CHead c1 (Bind -Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind -b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) -(\lambda (_: B).(\lambda (c1: C).(\lambda (_: T).(csubc g c1 c2))))))))))) -\def - \lambda (g: G).(\lambda (c2: C).(\lambda (x: C).(\lambda (w: T).(\lambda (k: -K).(\lambda (H: (csubc g x (CHead c2 k w))).(insert_eq C (CHead c2 k w) -(\lambda (c: C).(csubc g x c)) (\lambda (_: C).(or3 (ex2 C (\lambda (c1: -C).(eq C x (CHead c1 k w))) (\lambda (c1: C).(csubc g c1 c2))) (ex5_3 C T A -(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) -(\lambda (c1: C).(\lambda (v: T).(\lambda (_: A).(eq C x (CHead c1 (Bind -Abst) v))))) (\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 -c2)))) (\lambda (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1 -v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) -(ex4_3 B C T (\lambda (_: B).(\lambda (c1: C).(\lambda (v1: T).(eq C x (CHead -c1 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(eq K -k (Bind b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b -Void))))) (\lambda (_: B).(\lambda (c1: C).(\lambda (_: T).(csubc g c1 -c2))))))) (\lambda (y: C).(\lambda (H0: (csubc g x y)).(csubc_ind g (\lambda -(c: C).(\lambda (c0: C).((eq C c0 (CHead c2 k w)) \to (or3 (ex2 C (\lambda -(c1: C).(eq C c (CHead c1 k w))) (\lambda (c1: C).(csubc g c1 c2))) (ex5_3 C -T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) -(\lambda (c1: C).(\lambda (v: T).(\lambda (_: A).(eq C c (CHead c1 (Bind -Abst) v))))) (\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 -c2)))) (\lambda (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1 -v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) -(ex4_3 B C T (\lambda (_: B).(\lambda (c1: C).(\lambda (v1: T).(eq C c (CHead -c1 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(eq K -k (Bind b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b -Void))))) (\lambda (_: B).(\lambda (c1: C).(\lambda (_: T).(csubc g c1 -c2))))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead c2 k -w))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return -(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) -\Rightarrow False])) I (CHead c2 k w) H1) in (False_ind (or3 (ex2 C (\lambda -(c1: C).(eq C (CSort n) (CHead c1 k w))) (\lambda (c1: C).(csubc g c1 c2))) -(ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind -Abbr))))) (\lambda (c1: C).(\lambda (v: T).(\lambda (_: A).(eq C (CSort n) -(CHead c1 (Bind Abst) v))))) (\lambda (c1: C).(\lambda (_: T).(\lambda (_: -A).(csubc g c1 c2)))) (\lambda (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g -(asucc g a) c1 v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a -c2 w))))) (ex4_3 B C T (\lambda (_: B).(\lambda (c1: C).(\lambda (v1: T).(eq -C (CSort n) (CHead c1 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c1: -C).(\lambda (_: T).(csubc g c1 c2)))))) H2)))) (\lambda (c1: C).(\lambda (c0: -C).(\lambda (H1: (csubc g c1 c0)).(\lambda (H2: (((eq C c0 (CHead c2 k w)) -\to (or3 (ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k w))) (\lambda (c3: -C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda -(_: A).(eq K k (Bind Abbr))))) (\lambda (c3: C).(\lambda (v: T).(\lambda (_: -A).(eq C c1 (CHead c3 (Bind Abst) v))))) (\lambda (c3: C).(\lambda (_: -T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v: -T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T (\lambda (_: B).(\lambda -(c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind Void) v1))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: -B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 c2))))))))).(\lambda (k0: -K).(\lambda (v: T).(\lambda (H3: (eq C (CHead c0 k0 v) (CHead c2 k w))).(let -H4 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) -with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k0 -v) (CHead c2 k w) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e -in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k1 -_) \Rightarrow k1])) (CHead c0 k0 v) (CHead c2 k w) H3) in ((let H6 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow v | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 v) -(CHead c2 k w) H3) in (\lambda (H7: (eq K k0 k)).(\lambda (H8: (eq C c0 -c2)).(eq_ind_r T w (\lambda (t: T).(or3 (ex2 C (\lambda (c3: C).(eq C (CHead -c1 k0 t) (CHead c3 k w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A -(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) -(\lambda (c3: C).(\lambda (v0: T).(\lambda (_: A).(eq C (CHead c1 k0 t) -(CHead c3 (Bind Abst) v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: -A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (a: A).(sc3 -g (asucc g a) c3 v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 -g a c2 w))))) (ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: -T).(eq C (CHead c1 k0 t) (CHead c3 (Bind Void) v1))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: -B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 c2))))))) (eq_ind_r K k -(\lambda (k1: K).(or3 (ex2 C (\lambda (c3: C).(eq C (CHead c1 k1 w) (CHead c3 -k w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: -C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c3: -C).(\lambda (v0: T).(\lambda (_: A).(eq C (CHead c1 k1 w) (CHead c3 (Bind -Abst) v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 -c2)))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (a: A).(sc3 g (asucc g a) -c3 v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) -(ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead -c1 k1 w) (CHead c3 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: -C).(\lambda (_: T).(csubc g c3 c2))))))) (let H9 \def (eq_ind C c0 (\lambda -(c: C).((eq C c (CHead c2 k w)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c1 -(CHead c3 k w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: -C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c3: -C).(\lambda (v0: T).(\lambda (_: A).(eq C c1 (CHead c3 (Bind Abst) v0))))) -(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda -(c3: C).(\lambda (v0: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v0)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C -T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind -Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind -b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) -(\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 c2)))))))) H2 c2 -H8) in (let H10 \def (eq_ind C c0 (\lambda (c: C).(csubc g c1 c)) H1 c2 H8) -in (or3_intro0 (ex2 C (\lambda (c3: C).(eq C (CHead c1 k w) (CHead c3 k w))) -(\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: -T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c3: C).(\lambda (v0: -T).(\lambda (_: A).(eq C (CHead c1 k w) (CHead c3 (Bind Abst) v0))))) -(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda -(c3: C).(\lambda (v0: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v0)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C -T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead c1 k w) -(CHead c3 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: -T).(eq K k (Bind b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not -(eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g -c3 c2))))) (ex_intro2 C (\lambda (c3: C).(eq C (CHead c1 k w) (CHead c3 k -w))) (\lambda (c3: C).(csubc g c3 c2)) c1 (refl_equal C (CHead c1 k w)) -H10)))) k0 H7) v H6)))) H5)) H4))))))))) (\lambda (c1: C).(\lambda (c0: -C).(\lambda (H1: (csubc g c1 c0)).(\lambda (H2: (((eq C c0 (CHead c2 k w)) -\to (or3 (ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k w))) (\lambda (c3: -C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda -(_: A).(eq K k (Bind Abbr))))) (\lambda (c3: C).(\lambda (v: T).(\lambda (_: -A).(eq C c1 (CHead c3 (Bind Abst) v))))) (\lambda (c3: C).(\lambda (_: -T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v: -T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T (\lambda (_: B).(\lambda -(c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind Void) v1))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: -B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 c2))))))))).(\lambda (b: -B).(\lambda (H3: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: -T).(\lambda (H4: (eq C (CHead c0 (Bind b) u2) (CHead c2 k w))).(let H5 \def -(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with -[(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 (Bind b) -u2) (CHead c2 k w) H4) in ((let H6 \def (f_equal C K (\lambda (e: C).(match e -in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow (Bind b) | (CHead -_ k0 _) \Rightarrow k0])) (CHead c0 (Bind b) u2) (CHead c2 k w) H4) in ((let -H7 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c0 -(Bind b) u2) (CHead c2 k w) H4) in (\lambda (H8: (eq K (Bind b) k)).(\lambda -(H9: (eq C c0 c2)).(let H10 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead -c2 k w)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k w))) (\lambda -(c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: -T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c3: C).(\lambda (v: -T).(\lambda (_: A).(eq C c1 (CHead c3 (Bind Abst) v))))) (\lambda (c3: -C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: -C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T (\lambda -(_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind Void) -v1))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind -b0))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b0 -Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 -c2)))))))) H2 c2 H9) in (let H11 \def (eq_ind C c0 (\lambda (c: C).(csubc g -c1 c)) H1 c2 H9) in (let H12 \def (eq_ind_r K k (\lambda (k0: K).((eq C c2 -(CHead c2 k0 w)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k0 w))) -(\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: -T).(\lambda (_: A).(eq K k0 (Bind Abbr))))) (\lambda (c3: C).(\lambda (v: -T).(\lambda (_: A).(eq C c1 (CHead c3 (Bind Abst) v))))) (\lambda (c3: -C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: -C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T (\lambda -(_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind Void) -v1))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(eq K k0 (Bind -b0))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b0 -Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 -c2)))))))) H10 (Bind b) H8) in (eq_ind K (Bind b) (\lambda (k0: K).(or3 (ex2 -C (\lambda (c3: C).(eq C (CHead c1 (Bind Void) u1) (CHead c3 k0 w))) (\lambda -(c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: -T).(\lambda (_: A).(eq K k0 (Bind Abbr))))) (\lambda (c3: C).(\lambda (v: -T).(\lambda (_: A).(eq C (CHead c1 (Bind Void) u1) (CHead c3 (Bind Abst) -v))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) -(\lambda (c3: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C -T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead c1 (Bind -Void) u1) (CHead c3 (Bind Void) v1))))) (\lambda (b0: B).(\lambda (_: -C).(\lambda (_: T).(eq K k0 (Bind b0))))) (\lambda (b0: B).(\lambda (_: -C).(\lambda (_: T).(not (eq B b0 Void))))) (\lambda (_: B).(\lambda (c3: -C).(\lambda (_: T).(csubc g c3 c2))))))) (or3_intro2 (ex2 C (\lambda (c3: -C).(eq C (CHead c1 (Bind Void) u1) (CHead c3 (Bind b) w))) (\lambda (c3: -C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda -(_: A).(eq K (Bind b) (Bind Abbr))))) (\lambda (c3: C).(\lambda (v: -T).(\lambda (_: A).(eq C (CHead c1 (Bind Void) u1) (CHead c3 (Bind Abst) -v))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) -(\lambda (c3: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C -T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead c1 (Bind -Void) u1) (CHead c3 (Bind Void) v1))))) (\lambda (b0: B).(\lambda (_: -C).(\lambda (_: T).(eq K (Bind b) (Bind b0))))) (\lambda (b0: B).(\lambda (_: -C).(\lambda (_: T).(not (eq B b0 Void))))) (\lambda (_: B).(\lambda (c3: -C).(\lambda (_: T).(csubc g c3 c2))))) (ex4_3_intro B C T (\lambda (_: -B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead c1 (Bind Void) u1) (CHead -c3 (Bind Void) v1))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(eq K -(Bind b) (Bind b0))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not -(eq B b0 Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g -c3 c2)))) b c1 u1 (refl_equal C (CHead c1 (Bind Void) u1)) (refl_equal K -(Bind b)) H3 H11)) k H8))))))) H6)) H5))))))))))) (\lambda (c1: C).(\lambda -(c0: C).(\lambda (H1: (csubc g c1 c0)).(\lambda (H2: (((eq C c0 (CHead c2 k -w)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k w))) (\lambda (c3: -C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda -(_: A).(eq K k (Bind Abbr))))) (\lambda (c3: C).(\lambda (v: T).(\lambda (_: -A).(eq C c1 (CHead c3 (Bind Abst) v))))) (\lambda (c3: C).(\lambda (_: -T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v: -T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T (\lambda (_: B).(\lambda -(c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind Void) v1))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: -B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 c2))))))))).(\lambda (v: -T).(\lambda (a: A).(\lambda (H3: (sc3 g (asucc g a) c1 v)).(\lambda (w0: -T).(\lambda (H4: (sc3 g a c0 w0)).(\lambda (H5: (eq C (CHead c0 (Bind Abbr) -w0) (CHead c2 k w))).(let H6 \def (f_equal C C (\lambda (e: C).(match e in C -return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) -\Rightarrow c])) (CHead c0 (Bind Abbr) w0) (CHead c2 k w) H5) in ((let H7 -\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) -with [(CSort _) \Rightarrow (Bind Abbr) | (CHead _ k0 _) \Rightarrow k0])) -(CHead c0 (Bind Abbr) w0) (CHead c2 k w) H5) in ((let H8 \def (f_equal C T -(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) -\Rightarrow w0 | (CHead _ _ t) \Rightarrow t])) (CHead c0 (Bind Abbr) w0) -(CHead c2 k w) H5) in (\lambda (H9: (eq K (Bind Abbr) k)).(\lambda (H10: (eq -C c0 c2)).(let H11 \def (eq_ind T w0 (\lambda (t: T).(sc3 g a c0 t)) H4 w H8) -in (let H12 \def (eq_ind C c0 (\lambda (c: C).(sc3 g a c w)) H11 c2 H10) in -(let H13 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c2 k w)) \to (or3 -(ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k w))) (\lambda (c3: C).(csubc g -c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k -(Bind Abbr))))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (_: A).(eq C c1 -(CHead c3 (Bind Abst) v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: -A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (a0: A).(sc3 -g (asucc g a0) c3 v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: -A).(sc3 g a0 c2 w))))) (ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda -(v1: T).(eq C c1 (CHead c3 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: -C).(\lambda (_: T).(csubc g c3 c2)))))))) H2 c2 H10) in (let H14 \def (eq_ind -C c0 (\lambda (c: C).(csubc g c1 c)) H1 c2 H10) in (let H15 \def (eq_ind_r K -k (\lambda (k0: K).((eq C c2 (CHead c2 k0 w)) \to (or3 (ex2 C (\lambda (c3: -C).(eq C c1 (CHead c3 k0 w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A -(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k0 (Bind Abbr))))) -(\lambda (c3: C).(\lambda (v0: T).(\lambda (_: A).(eq C c1 (CHead c3 (Bind -Abst) v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 -c2)))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (a0: A).(sc3 g (asucc g a0) -c3 v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g a0 c2 -w))))) (ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C -c1 (CHead c3 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: -T).(eq K k0 (Bind b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not -(eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g -c3 c2)))))))) H13 (Bind Abbr) H9) in (eq_ind K (Bind Abbr) (\lambda (k0: -K).(or3 (ex2 C (\lambda (c3: C).(eq C (CHead c1 (Bind Abst) v) (CHead c3 k0 -w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda -(_: T).(\lambda (_: A).(eq K k0 (Bind Abbr))))) (\lambda (c3: C).(\lambda -(v0: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) v) (CHead c3 (Bind Abst) -v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) -(\lambda (c3: C).(\lambda (v0: T).(\lambda (a0: A).(sc3 g (asucc g a0) c3 -v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g a0 c2 w))))) -(ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead -c1 (Bind Abst) v) (CHead c3 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (_: T).(eq K k0 (Bind b))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: -C).(\lambda (_: T).(csubc g c3 c2))))))) (or3_intro1 (ex2 C (\lambda (c3: -C).(eq C (CHead c1 (Bind Abst) v) (CHead c3 (Bind Abbr) w))) (\lambda (c3: -C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda -(_: A).(eq K (Bind Abbr) (Bind Abbr))))) (\lambda (c3: C).(\lambda (v0: -T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) v) (CHead c3 (Bind Abst) -v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) -(\lambda (c3: C).(\lambda (v0: T).(\lambda (a0: A).(sc3 g (asucc g a0) c3 -v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g a0 c2 w))))) -(ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead -c1 (Bind Abst) v) (CHead c3 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (_: T).(eq K (Bind Abbr) (Bind b))))) (\lambda (b: B).(\lambda -(_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: -C).(\lambda (_: T).(csubc g c3 c2))))) (ex5_3_intro C T A (\lambda (_: -C).(\lambda (_: T).(\lambda (_: A).(eq K (Bind Abbr) (Bind Abbr))))) (\lambda -(c3: C).(\lambda (v0: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) v) -(CHead c3 (Bind Abst) v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: -A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (a0: A).(sc3 -g (asucc g a0) c3 v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: -A).(sc3 g a0 c2 w)))) c1 v a (refl_equal K (Bind Abbr)) (refl_equal C (CHead -c1 (Bind Abst) v)) H14 H3 H12)) k H9))))))))) H7)) H6)))))))))))) x y H0))) -H)))))). -(* COMMENTS -Initial nodes: 5197 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/getl.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/getl.ma deleted file mode 100644 index 244c84927..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/getl.ma +++ /dev/null @@ -1,45 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubc/drop.ma". - -include "Basic-1/csubc/clear.ma". - -theorem csubc_getl_conf: - \forall (g: G).(\forall (c1: C).(\forall (e1: C).(\forall (i: nat).((getl i -c1 e1) \to (\forall (c2: C).((csubc g c1 c2) \to (ex2 C (\lambda (e2: -C).(getl i c2 e2)) (\lambda (e2: C).(csubc g e1 e2))))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (e1: C).(\lambda (i: nat).(\lambda -(H: (getl i c1 e1)).(\lambda (c2: C).(\lambda (H0: (csubc g c1 c2)).(let H1 -\def (getl_gen_all c1 e1 i H) in (ex2_ind C (\lambda (e: C).(drop i O c1 e)) -(\lambda (e: C).(clear e e1)) (ex2 C (\lambda (e2: C).(getl i c2 e2)) -(\lambda (e2: C).(csubc g e1 e2))) (\lambda (x: C).(\lambda (H2: (drop i O c1 -x)).(\lambda (H3: (clear x e1)).(let H_x \def (csubc_drop_conf_O g c1 x i H2 -c2 H0) in (let H4 \def H_x in (ex2_ind C (\lambda (e2: C).(drop i O c2 e2)) -(\lambda (e2: C).(csubc g x e2)) (ex2 C (\lambda (e2: C).(getl i c2 e2)) -(\lambda (e2: C).(csubc g e1 e2))) (\lambda (x0: C).(\lambda (H5: (drop i O -c2 x0)).(\lambda (H6: (csubc g x x0)).(let H_x0 \def (csubc_clear_conf g x e1 -H3 x0 H6) in (let H7 \def H_x0 in (ex2_ind C (\lambda (e2: C).(clear x0 e2)) -(\lambda (e2: C).(csubc g e1 e2)) (ex2 C (\lambda (e2: C).(getl i c2 e2)) -(\lambda (e2: C).(csubc g e1 e2))) (\lambda (x1: C).(\lambda (H8: (clear x0 -x1)).(\lambda (H9: (csubc g e1 x1)).(ex_intro2 C (\lambda (e2: C).(getl i c2 -e2)) (\lambda (e2: C).(csubc g e1 e2)) x1 (getl_intro i c2 x1 x0 H5 H8) -H9)))) H7)))))) H4)))))) H1)))))))). -(* COMMENTS -Initial nodes: 315 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/props.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/props.ma deleted file mode 100644 index d6399181c..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/props.ma +++ /dev/null @@ -1,30 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubc/defs.ma". - -include "Basic-1/sc3/props.ma". - -theorem csubc_refl: - \forall (g: G).(\forall (c: C).(csubc g c c)) -\def - \lambda (g: G).(\lambda (c: C).(C_ind (\lambda (c0: C).(csubc g c0 c0)) -(\lambda (n: nat).(csubc_sort g n)) (\lambda (c0: C).(\lambda (H: (csubc g c0 -c0)).(\lambda (k: K).(\lambda (t: T).(csubc_head g c0 c0 H k t))))) c)). -(* COMMENTS -Initial nodes: 53 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubst0/clear.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubst0/clear.ma deleted file mode 100644 index 0700d7d48..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubst0/clear.ma +++ /dev/null @@ -1,1139 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubst0/props.ma". - -include "Basic-1/csubst0/fwd.ma". - -include "Basic-1/clear/fwd.ma". - -theorem csubst0_clear_O: - \forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 O v c1 c2) \to -(\forall (c: C).((clear c1 c) \to (clear c2 c)))))) -\def - \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: -T).((csubst0 O v c c2) \to (\forall (c0: C).((clear c c0) \to (clear c2 -c0))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (H: -(csubst0 O v (CSort n) c2)).(\lambda (c: C).(\lambda (_: (clear (CSort n) -c)).(csubst0_gen_sort c2 v O n H (clear c2 c)))))))) (\lambda (c: C).(\lambda -(H: ((\forall (c2: C).(\forall (v: T).((csubst0 O v c c2) \to (\forall (c0: -C).((clear c c0) \to (clear c2 c0)))))))).(\lambda (k: K).(\lambda (t: -T).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 O v (CHead c k t) -c2)).(\lambda (c0: C).(\lambda (H1: (clear (CHead c k t) c0)).(or3_ind (ex3_2 -T nat (\lambda (_: T).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (u2: -T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: -nat).(subst0 j v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq -nat O (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k -t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (ex4_3 T C nat -(\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j))))) -(\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k -u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t -u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c -c3))))) (clear c2 c0) (\lambda (H2: (ex3_2 T nat (\lambda (_: T).(\lambda (j: -nat).(eq nat O (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead -c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t -u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: nat).(eq nat O (s k -j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda -(u2: T).(\lambda (j: nat).(subst0 j v t u2))) (clear c2 c0) (\lambda (x0: -T).(\lambda (x1: nat).(\lambda (H3: (eq nat O (s k x1))).(\lambda (H4: (eq C -c2 (CHead c k x0))).(\lambda (H5: (subst0 x1 v t x0)).(eq_ind_r C (CHead c k -x0) (\lambda (c3: C).(clear c3 c0)) (K_ind (\lambda (k0: K).((clear (CHead c -k0 t) c0) \to ((eq nat O (s k0 x1)) \to (clear (CHead c k0 x0) c0)))) -(\lambda (b: B).(\lambda (_: (clear (CHead c (Bind b) t) c0)).(\lambda (H7: -(eq nat O (s (Bind b) x1))).(let H8 \def (eq_ind nat O (\lambda (ee: -nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True -| (S _) \Rightarrow False])) I (S x1) H7) in (False_ind (clear (CHead c (Bind -b) x0) c0) H8))))) (\lambda (f: F).(\lambda (H6: (clear (CHead c (Flat f) t) -c0)).(\lambda (H7: (eq nat O (s (Flat f) x1))).(let H8 \def (eq_ind_r nat x1 -(\lambda (n: nat).(subst0 n v t x0)) H5 O H7) in (clear_flat c c0 -(clear_gen_flat f c c0 t H6) f x0))))) k H1 H3) c2 H4)))))) H2)) (\lambda -(H2: (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j)))) -(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: -C).(\lambda (j: nat).(csubst0 j v c c3))))).(ex3_2_ind C nat (\lambda (_: -C).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (c3: C).(\lambda (_: -nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j -v c c3))) (clear c2 c0) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H3: (eq -nat O (s k x1))).(\lambda (H4: (eq C c2 (CHead x0 k t))).(\lambda (H5: -(csubst0 x1 v c x0)).(eq_ind_r C (CHead x0 k t) (\lambda (c3: C).(clear c3 -c0)) (K_ind (\lambda (k0: K).((clear (CHead c k0 t) c0) \to ((eq nat O (s k0 -x1)) \to (clear (CHead x0 k0 t) c0)))) (\lambda (b: B).(\lambda (_: (clear -(CHead c (Bind b) t) c0)).(\lambda (H7: (eq nat O (s (Bind b) x1))).(let H8 -\def (eq_ind nat O (\lambda (ee: nat).(match ee in nat return (\lambda (_: -nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x1) H7) -in (False_ind (clear (CHead x0 (Bind b) t) c0) H8))))) (\lambda (f: -F).(\lambda (H6: (clear (CHead c (Flat f) t) c0)).(\lambda (H7: (eq nat O (s -(Flat f) x1))).(let H8 \def (eq_ind_r nat x1 (\lambda (n: nat).(csubst0 n v c -x0)) H5 O H7) in (clear_flat x0 c0 (H x0 v H8 c0 (clear_gen_flat f c c0 t -H6)) f t))))) k H1 H3) c2 H4)))))) H2)) (\lambda (H2: (ex4_3 T C nat (\lambda -(_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j))))) (\lambda (u2: -T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda -(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: -T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))))).(ex4_3_ind T C -nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j))))) -(\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k -u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t -u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c -c3)))) (clear c2 c0) (\lambda (x0: T).(\lambda (x1: C).(\lambda (x2: -nat).(\lambda (H3: (eq nat O (s k x2))).(\lambda (H4: (eq C c2 (CHead x1 k -x0))).(\lambda (H5: (subst0 x2 v t x0)).(\lambda (H6: (csubst0 x2 v c -x1)).(eq_ind_r C (CHead x1 k x0) (\lambda (c3: C).(clear c3 c0)) (K_ind -(\lambda (k0: K).((clear (CHead c k0 t) c0) \to ((eq nat O (s k0 x2)) \to -(clear (CHead x1 k0 x0) c0)))) (\lambda (b: B).(\lambda (_: (clear (CHead c -(Bind b) t) c0)).(\lambda (H8: (eq nat O (s (Bind b) x2))).(let H9 \def -(eq_ind nat O (\lambda (ee: nat).(match ee in nat return (\lambda (_: -nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x2) H8) -in (False_ind (clear (CHead x1 (Bind b) x0) c0) H9))))) (\lambda (f: -F).(\lambda (H7: (clear (CHead c (Flat f) t) c0)).(\lambda (H8: (eq nat O (s -(Flat f) x2))).(let H9 \def (eq_ind_r nat x2 (\lambda (n: nat).(csubst0 n v c -x1)) H6 O H8) in (let H10 \def (eq_ind_r nat x2 (\lambda (n: nat).(subst0 n v -t x0)) H5 O H8) in (clear_flat x1 c0 (H x1 v H9 c0 (clear_gen_flat f c c0 t -H7)) f x0)))))) k H1 H3) c2 H4)))))))) H2)) (csubst0_gen_head k c c2 t v O -H0))))))))))) c1). -(* COMMENTS -Initial nodes: 1582 -END *) - -theorem csubst0_clear_O_back: - \forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 O v c1 c2) \to -(\forall (c: C).((clear c2 c) \to (clear c1 c)))))) -\def - \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: -T).((csubst0 O v c c2) \to (\forall (c0: C).((clear c2 c0) \to (clear c -c0))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (H: -(csubst0 O v (CSort n) c2)).(\lambda (c: C).(\lambda (_: (clear c2 -c)).(csubst0_gen_sort c2 v O n H (clear (CSort n) c)))))))) (\lambda (c: -C).(\lambda (H: ((\forall (c2: C).(\forall (v: T).((csubst0 O v c c2) \to -(\forall (c0: C).((clear c2 c0) \to (clear c c0)))))))).(\lambda (k: -K).(\lambda (t: T).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 O -v (CHead c k t) c2)).(\lambda (c0: C).(\lambda (H1: (clear c2 c0)).(or3_ind -(ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda -(u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: -T).(\lambda (j: nat).(subst0 j v t u2)))) (ex3_2 C nat (\lambda (_: -C).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (c3: C).(\lambda (_: -nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j -v c c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: -nat).(eq nat O (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: -nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda -(j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: -nat).(csubst0 j v c c3))))) (clear (CHead c k t) c0) (\lambda (H2: (ex3_2 T -nat (\lambda (_: T).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (u2: -T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: -nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: -nat).(eq nat O (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead -c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))) (clear -(CHead c k t) c0) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H3: (eq nat O -(s k x1))).(\lambda (H4: (eq C c2 (CHead c k x0))).(\lambda (H5: (subst0 x1 v -t x0)).(let H6 \def (eq_ind C c2 (\lambda (c3: C).(clear c3 c0)) H1 (CHead c -k x0) H4) in (K_ind (\lambda (k0: K).((eq nat O (s k0 x1)) \to ((clear (CHead -c k0 x0) c0) \to (clear (CHead c k0 t) c0)))) (\lambda (b: B).(\lambda (H7: -(eq nat O (s (Bind b) x1))).(\lambda (_: (clear (CHead c (Bind b) x0) -c0)).(let H9 \def (eq_ind nat O (\lambda (ee: nat).(match ee in nat return -(\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) -I (S x1) H7) in (False_ind (clear (CHead c (Bind b) t) c0) H9))))) (\lambda -(f: F).(\lambda (H7: (eq nat O (s (Flat f) x1))).(\lambda (H8: (clear (CHead -c (Flat f) x0) c0)).(let H9 \def (eq_ind_r nat x1 (\lambda (n: nat).(subst0 n -v t x0)) H5 O H7) in (clear_flat c c0 (clear_gen_flat f c c0 x0 H8) f t))))) -k H3 H6))))))) H2)) (\lambda (H2: (ex3_2 C nat (\lambda (_: C).(\lambda (j: -nat).(eq nat O (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead -c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c -c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat O (s k -j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda -(c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (clear (CHead c k t) c0) -(\lambda (x0: C).(\lambda (x1: nat).(\lambda (H3: (eq nat O (s k -x1))).(\lambda (H4: (eq C c2 (CHead x0 k t))).(\lambda (H5: (csubst0 x1 v c -x0)).(let H6 \def (eq_ind C c2 (\lambda (c3: C).(clear c3 c0)) H1 (CHead x0 k -t) H4) in (K_ind (\lambda (k0: K).((eq nat O (s k0 x1)) \to ((clear (CHead x0 -k0 t) c0) \to (clear (CHead c k0 t) c0)))) (\lambda (b: B).(\lambda (H7: (eq -nat O (s (Bind b) x1))).(\lambda (_: (clear (CHead x0 (Bind b) t) c0)).(let -H9 \def (eq_ind nat O (\lambda (ee: nat).(match ee in nat return (\lambda (_: -nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x1) H7) -in (False_ind (clear (CHead c (Bind b) t) c0) H9))))) (\lambda (f: -F).(\lambda (H7: (eq nat O (s (Flat f) x1))).(\lambda (H8: (clear (CHead x0 -(Flat f) t) c0)).(let H9 \def (eq_ind_r nat x1 (\lambda (n: nat).(csubst0 n v -c x0)) H5 O H7) in (clear_flat c c0 (H x0 v H9 c0 (clear_gen_flat f x0 c0 t -H8)) f t))))) k H3 H6))))))) H2)) (\lambda (H2: (ex4_3 T C nat (\lambda (_: -T).(\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j))))) (\lambda (u2: -T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda -(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: -T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))))).(ex4_3_ind T C -nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j))))) -(\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k -u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t -u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c -c3)))) (clear (CHead c k t) c0) (\lambda (x0: T).(\lambda (x1: C).(\lambda -(x2: nat).(\lambda (H3: (eq nat O (s k x2))).(\lambda (H4: (eq C c2 (CHead x1 -k x0))).(\lambda (H5: (subst0 x2 v t x0)).(\lambda (H6: (csubst0 x2 v c -x1)).(let H7 \def (eq_ind C c2 (\lambda (c3: C).(clear c3 c0)) H1 (CHead x1 k -x0) H4) in (K_ind (\lambda (k0: K).((eq nat O (s k0 x2)) \to ((clear (CHead -x1 k0 x0) c0) \to (clear (CHead c k0 t) c0)))) (\lambda (b: B).(\lambda (H8: -(eq nat O (s (Bind b) x2))).(\lambda (_: (clear (CHead x1 (Bind b) x0) -c0)).(let H10 \def (eq_ind nat O (\lambda (ee: nat).(match ee in nat return -(\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) -I (S x2) H8) in (False_ind (clear (CHead c (Bind b) t) c0) H10))))) (\lambda -(f: F).(\lambda (H8: (eq nat O (s (Flat f) x2))).(\lambda (H9: (clear (CHead -x1 (Flat f) x0) c0)).(let H10 \def (eq_ind_r nat x2 (\lambda (n: -nat).(csubst0 n v c x1)) H6 O H8) in (let H11 \def (eq_ind_r nat x2 (\lambda -(n: nat).(subst0 n v t x0)) H5 O H8) in (clear_flat c c0 (H x1 v H10 c0 -(clear_gen_flat f x1 c0 x0 H9)) f t)))))) k H3 H7))))))))) H2)) -(csubst0_gen_head k c c2 t v O H0))))))))))) c1). -(* COMMENTS -Initial nodes: 1606 -END *) - -theorem csubst0_clear_S: - \forall (c1: C).(\forall (c2: C).(\forall (v: T).(\forall (i: nat).((csubst0 -(S i) v c1 c2) \to (\forall (c: C).((clear c1 c) \to (or4 (clear c2 c) (ex3_4 -B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq -C c (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: -T).(\lambda (u2: T).(clear c2 (CHead e (Bind b) u2)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) -(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(clear c2 (CHead e2 (Bind b) u)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b) u1))))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 i v e1 e2)))))))))))))) -\def - \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: -T).(\forall (i: nat).((csubst0 (S i) v c c2) \to (\forall (c0: C).((clear c -c0) \to (or4 (clear c2 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: -C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) -(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 -(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind -b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(clear c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e2 -(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 -e2))))))))))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda -(i: nat).(\lambda (H: (csubst0 (S i) v (CSort n) c2)).(\lambda (c: -C).(\lambda (_: (clear (CSort n) c)).(csubst0_gen_sort c2 v (S i) n H (or4 -(clear c2 c) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: -T).(\lambda (_: T).(eq C c (CHead e (Bind b) u1)))))) (\lambda (b: -B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e (Bind -b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind b) u)))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c2 -(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c -(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))))))))))) -(\lambda (c: C).(\lambda (H: ((\forall (c2: C).(\forall (v: T).(\forall (i: -nat).((csubst0 (S i) v c c2) \to (\forall (c0: C).((clear c c0) \to (or4 -(clear c2 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: -T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: -B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e (Bind -b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c2 -(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 -(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 -e2)))))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda -(v: T).(\lambda (i: nat).(\lambda (H0: (csubst0 (S i) v (CHead c k t) -c2)).(\lambda (c0: C).(\lambda (H1: (clear (CHead c k t) c0)).(or3_ind (ex3_2 -T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (S i) (s k j)))) (\lambda -(u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: -T).(\lambda (j: nat).(subst0 j v t u2)))) (ex3_2 C nat (\lambda (_: -C).(\lambda (j: nat).(eq nat (S i) (s k j)))) (\lambda (c3: C).(\lambda (_: -nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j -v c c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: -nat).(eq nat (S i) (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: -nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda -(j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: -nat).(csubst0 j v c c3))))) (or4 (clear c2 c0) (ex3_4 B C T T (\lambda (b: -B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind -b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: -T).(clear c2 (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 -(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(clear c2 (CHead e2 (Bind b) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) -(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear -c2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -i v e1 e2)))))))) (\lambda (H2: (ex3_2 T nat (\lambda (_: T).(\lambda (j: -nat).(eq nat (S i) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 -(CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t -u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (S i) (s k -j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda -(u2: T).(\lambda (j: nat).(subst0 j v t u2))) (or4 (clear c2 c0) (ex3_4 B C T -T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 -(CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: -T).(\lambda (u2: T).(clear c2 (CHead e (Bind b) u2)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) -(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(clear c2 (CHead e2 (Bind b) u)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 i v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: -nat).(\lambda (H3: (eq nat (S i) (s k x1))).(\lambda (H4: (eq C c2 (CHead c k -x0))).(\lambda (H5: (subst0 x1 v t x0)).(eq_ind_r C (CHead c k x0) (\lambda -(c3: C).(or4 (clear c3 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: -C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) -(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c3 -(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind -b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(clear c3 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear c3 (CHead e2 -(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 -e2))))))))) (K_ind (\lambda (k0: K).((clear (CHead c k0 t) c0) \to ((eq nat -(S i) (s k0 x1)) \to (or4 (clear (CHead c k0 x0) c0) (ex3_4 B C T T (\lambda -(b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e -(Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda -(u2: T).(clear (CHead c k0 x0) (CHead e (Bind b) u2)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) -(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead c k0 x0) (CHead e2 (Bind b) -u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind -b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (u2: T).(clear (CHead c k0 x0) (CHead e2 (Bind b) u2))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))))))) -(\lambda (b: B).(\lambda (H6: (clear (CHead c (Bind b) t) c0)).(\lambda (H7: -(eq nat (S i) (s (Bind b) x1))).(let H8 \def (f_equal nat nat (\lambda (e: -nat).(match e in nat return (\lambda (_: nat).nat) with [O \Rightarrow i | (S -n) \Rightarrow n])) (S i) (S x1) H7) in (let H9 \def (eq_ind_r nat x1 -(\lambda (n: nat).(subst0 n v t x0)) H5 i H8) in (eq_ind_r C (CHead c (Bind -b) t) (\lambda (c3: C).(or4 (clear (CHead c (Bind b) x0) c3) (ex3_4 B C T T -(\lambda (b0: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c3 -(CHead e (Bind b0) u1)))))) (\lambda (b0: B).(\lambda (e: C).(\lambda (_: -T).(\lambda (u2: T).(clear (CHead c (Bind b) x0) (CHead e (Bind b0) u2)))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v -u1 u2)))))) (ex3_4 B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C c3 (CHead e1 (Bind b0) u)))))) (\lambda (b0: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead c (Bind b) -x0) (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda -(b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq -C c3 (CHead e1 (Bind b0) u1))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead c (Bind b) x0) (CHead -e2 (Bind b0) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -i v e1 e2))))))))) (or4_intro1 (clear (CHead c (Bind b) x0) (CHead c (Bind b) -t)) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e: C).(\lambda (u1: T).(\lambda -(_: T).(eq C (CHead c (Bind b) t) (CHead e (Bind b0) u1)))))) (\lambda (b0: -B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead c (Bind b) -x0) (CHead e (Bind b0) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b0: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c (Bind b) -t) (CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(clear (CHead c (Bind b) x0) (CHead e2 (Bind b0) u)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e1 -(Bind b0) u1))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead c (Bind b) x0) (CHead e2 -(Bind b0) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 -e2))))))) (ex3_4_intro B C T T (\lambda (b0: B).(\lambda (e: C).(\lambda (u1: -T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e (Bind b0) u1)))))) -(\lambda (b0: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear -(CHead c (Bind b) x0) (CHead e (Bind b0) u2)))))) (\lambda (_: B).(\lambda -(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))) b c t x0 -(refl_equal C (CHead c (Bind b) t)) (clear_bind b c x0) H9)) c0 -(clear_gen_bind b c c0 t H6))))))) (\lambda (f: F).(\lambda (H6: (clear -(CHead c (Flat f) t) c0)).(\lambda (H7: (eq nat (S i) (s (Flat f) x1))).(let -H8 \def (f_equal nat nat (\lambda (e: nat).e) (S i) (s (Flat f) x1) H7) in -(let H9 \def (eq_ind_r nat x1 (\lambda (n: nat).(subst0 n v t x0)) H5 (S i) -H8) in (or4_intro0 (clear (CHead c (Flat f) x0) c0) (ex3_4 B C T T (\lambda -(b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e -(Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda -(u2: T).(clear (CHead c (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) -(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead c (Flat f) x0) (CHead e2 -(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 -(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (u2: T).(clear (CHead c (Flat f) x0) (CHead e2 (Bind b) -u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) -(clear_flat c c0 (clear_gen_flat f c c0 t H6) f x0))))))) k H1 H3) c2 -H4)))))) H2)) (\lambda (H2: (ex3_2 C nat (\lambda (_: C).(\lambda (j: -nat).(eq nat (S i) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 -(CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c -c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S i) (s k -j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda -(c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (or4 (clear c2 c0) (ex3_4 B C -T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 -(CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: -T).(\lambda (u2: T).(clear c2 (CHead e (Bind b) u2)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) -(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(clear c2 (CHead e2 (Bind b) u)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 i v e1 e2)))))))) (\lambda (x0: C).(\lambda (x1: -nat).(\lambda (H3: (eq nat (S i) (s k x1))).(\lambda (H4: (eq C c2 (CHead x0 -k t))).(\lambda (H5: (csubst0 x1 v c x0)).(eq_ind_r C (CHead x0 k t) (\lambda -(c3: C).(or4 (clear c3 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: -C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) -(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c3 -(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind -b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(clear c3 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear c3 (CHead e2 -(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 -e2))))))))) (K_ind (\lambda (k0: K).((clear (CHead c k0 t) c0) \to ((eq nat -(S i) (s k0 x1)) \to (or4 (clear (CHead x0 k0 t) c0) (ex3_4 B C T T (\lambda -(b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e -(Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda -(u2: T).(clear (CHead x0 k0 t) (CHead e (Bind b) u2)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) -(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 k0 t) (CHead e2 (Bind b) -u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind -b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (u2: T).(clear (CHead x0 k0 t) (CHead e2 (Bind b) u2))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))))))) -(\lambda (b: B).(\lambda (H6: (clear (CHead c (Bind b) t) c0)).(\lambda (H7: -(eq nat (S i) (s (Bind b) x1))).(let H8 \def (f_equal nat nat (\lambda (e: -nat).(match e in nat return (\lambda (_: nat).nat) with [O \Rightarrow i | (S -n) \Rightarrow n])) (S i) (S x1) H7) in (let H9 \def (eq_ind_r nat x1 -(\lambda (n: nat).(csubst0 n v c x0)) H5 i H8) in (eq_ind_r C (CHead c (Bind -b) t) (\lambda (c3: C).(or4 (clear (CHead x0 (Bind b) t) c3) (ex3_4 B C T T -(\lambda (b0: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c3 -(CHead e (Bind b0) u1)))))) (\lambda (b0: B).(\lambda (e: C).(\lambda (_: -T).(\lambda (u2: T).(clear (CHead x0 (Bind b) t) (CHead e (Bind b0) u2)))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v -u1 u2)))))) (ex3_4 B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C c3 (CHead e1 (Bind b0) u)))))) (\lambda (b0: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Bind b) -t) (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c3 -(CHead e1 (Bind b0) u1))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Bind b) t) (CHead e2 -(Bind b0) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 -e2))))))))) (or4_intro2 (clear (CHead x0 (Bind b) t) (CHead c (Bind b) t)) -(ex3_4 B C T T (\lambda (b0: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: -T).(eq C (CHead c (Bind b) t) (CHead e (Bind b0) u1)))))) (\lambda (b0: -B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Bind b) -t) (CHead e (Bind b0) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b0: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c (Bind b) -t) (CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(clear (CHead x0 (Bind b) t) (CHead e2 (Bind b0) u)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e1 -(Bind b0) u1))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Bind b) t) (CHead e2 -(Bind b0) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 -e2))))))) (ex3_4_intro B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C (CHead c (Bind b) t) (CHead e1 (Bind b0) u)))))) -(\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear -(CHead x0 (Bind b) t) (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2))))) b c x0 t -(refl_equal C (CHead c (Bind b) t)) (clear_bind b x0 t) H9)) c0 -(clear_gen_bind b c c0 t H6))))))) (\lambda (f: F).(\lambda (H6: (clear -(CHead c (Flat f) t) c0)).(\lambda (H7: (eq nat (S i) (s (Flat f) x1))).(let -H8 \def (f_equal nat nat (\lambda (e: nat).e) (S i) (s (Flat f) x1) H7) in -(let H9 \def (eq_ind_r nat x1 (\lambda (n: nat).(csubst0 n v c x0)) H5 (S i) -H8) in (let H10 \def (H x0 v i H9 c0 (clear_gen_flat f c c0 t H6)) in -(or4_ind (clear x0 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: -C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) -(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x0 -(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind -b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(clear x0 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear x0 (CHead e2 -(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 -e2))))))) (or4 (clear (CHead x0 (Flat f) t) c0) (ex3_4 B C T T (\lambda (b: -B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind -b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: -T).(clear (CHead x0 (Flat f) t) (CHead e (Bind b) u2)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) -(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Flat f) t) (CHead e2 -(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 -(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) -u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 -e2)))))))) (\lambda (H11: (clear x0 c0)).(or4_intro0 (clear (CHead x0 (Flat -f) t) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: -T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: -B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) -t) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind -b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) -(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear -(CHead x0 (Flat f) t) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda -(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 -u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (clear_flat x0 c0 H11 f t))) -(\lambda (H11: (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: -T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: -B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x0 (CHead e (Bind -b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 i v u1 u2))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e: -C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) -(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x0 -(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 i v u1 u2))))) (or4 (clear (CHead x0 (Flat f) t) -c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda -(_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: -C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e -(Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear -(CHead x0 (Flat f) t) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C -C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear -(CHead x0 (Flat f) t) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda -(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 -u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 i v e1 e2)))))))) (\lambda (x2: B).(\lambda (x3: -C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H12: (eq C c0 (CHead x3 (Bind -x2) x4))).(\lambda (H13: (clear x0 (CHead x3 (Bind x2) x5))).(\lambda (H14: -(subst0 i v x4 x5)).(or4_intro1 (clear (CHead x0 (Flat f) t) c0) (ex3_4 B C T -T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 -(CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: -T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e (Bind b) u2)))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v -u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Flat f) -t) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 -(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 -(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 -e2))))))) (ex3_4_intro B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: -T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: -B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) -t) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 i v u1 u2))))) x2 x3 x4 x5 H12 (clear_flat x0 -(CHead x3 (Bind x2) x5) H13 f t) H14))))))))) H11)) (\lambda (H11: (ex3_4 B C -C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 -(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(clear x0 (CHead e2 (Bind b) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 -e2))))))).(ex3_4_ind B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x0 (CHead e2 (Bind -b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 i v e1 e2))))) (or4 (clear (CHead x0 (Flat f) t) c0) (ex3_4 B C T -T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 -(CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: -T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e (Bind b) u2)))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v -u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Flat f) -t) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 -(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 -(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 -e2)))))))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: -T).(\lambda (H12: (eq C c0 (CHead x3 (Bind x2) x5))).(\lambda (H13: (clear x0 -(CHead x4 (Bind x2) x5))).(\lambda (H14: (csubst0 i v x3 x4)).(or4_intro2 -(clear (CHead x0 (Flat f) t) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: -C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) -(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear -(CHead x0 (Flat f) t) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 -(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u2))))))) (\lambda -(_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (ex3_4_intro B C -C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 -(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v e1 e2))))) x2 x3 x4 x5 H12 (clear_flat x0 (CHead x4 (Bind x2) x5) H13 f t) -H14))))))))) H11)) (\lambda (H11: (ex4_5 B C C T T (\lambda (b: B).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 -(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (u2: T).(clear x0 (CHead e2 (Bind b) u2))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 -i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))).(ex4_5_ind B C C T T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear x0 (CHead e2 -(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 -e2)))))) (or4 (clear (CHead x0 (Flat f) t) c0) (ex3_4 B C T T (\lambda (b: -B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind -b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: -T).(clear (CHead x0 (Flat f) t) (CHead e (Bind b) u2)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) -(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Flat f) t) (CHead e2 -(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 -(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) -u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 -e2)))))))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: -T).(\lambda (x6: T).(\lambda (H12: (eq C c0 (CHead x3 (Bind x2) -x5))).(\lambda (H13: (clear x0 (CHead x4 (Bind x2) x6))).(\lambda (H14: -(subst0 i v x5 x6)).(\lambda (H15: (csubst0 i v x3 x4)).(or4_intro3 (clear -(CHead x0 (Flat f) t) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: -C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) -(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear -(CHead x0 (Flat f) t) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 -(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u2))))))) (\lambda -(_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (ex4_5_intro B C -C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear -(CHead x0 (Flat f) t) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda -(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 -u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 i v e1 e2)))))) x2 x3 x4 x5 x6 H12 (clear_flat x0 -(CHead x4 (Bind x2) x6) H13 f t) H14 H15))))))))))) H11)) H10))))))) k H1 H3) -c2 H4)))))) H2)) (\lambda (H2: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: -C).(\lambda (j: nat).(eq nat (S i) (s k j))))) (\lambda (u2: T).(\lambda (c3: -C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda -(_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: -C).(\lambda (j: nat).(csubst0 j v c c3)))))).(ex4_3_ind T C nat (\lambda (_: -T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S i) (s k j))))) (\lambda (u2: -T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda -(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: -T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (or4 (clear c2 -c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda -(_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: -C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e (Bind b) u2)))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v -u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c2 (CHead e2 (Bind -b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind -b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 -i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))) (\lambda (x0: T).(\lambda -(x1: C).(\lambda (x2: nat).(\lambda (H3: (eq nat (S i) (s k x2))).(\lambda -(H4: (eq C c2 (CHead x1 k x0))).(\lambda (H5: (subst0 x2 v t x0)).(\lambda -(H6: (csubst0 x2 v c x1)).(eq_ind_r C (CHead x1 k x0) (\lambda (c3: C).(or4 -(clear c3 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: -T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: -B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c3 (CHead e (Bind -b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c3 -(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 -(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(clear c3 (CHead e2 (Bind b) u2))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))))) (K_ind -(\lambda (k0: K).((clear (CHead c k0 t) c0) \to ((eq nat (S i) (s k0 x2)) \to -(or4 (clear (CHead x1 k0 x0) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: -C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) -(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear -(CHead x1 k0 x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 -(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(clear (CHead x1 k0 x0) (CHead e2 (Bind b) u)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(clear (CHead x1 k0 x0) (CHead e2 (Bind b) u2))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 -i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))))))) (\lambda (b: B).(\lambda -(H7: (clear (CHead c (Bind b) t) c0)).(\lambda (H8: (eq nat (S i) (s (Bind b) -x2))).(let H9 \def (f_equal nat nat (\lambda (e: nat).(match e in nat return -(\lambda (_: nat).nat) with [O \Rightarrow i | (S n) \Rightarrow n])) (S i) -(S x2) H8) in (let H10 \def (eq_ind_r nat x2 (\lambda (n: nat).(csubst0 n v c -x1)) H6 i H9) in (let H11 \def (eq_ind_r nat x2 (\lambda (n: nat).(subst0 n v -t x0)) H5 i H9) in (eq_ind_r C (CHead c (Bind b) t) (\lambda (c3: C).(or4 -(clear (CHead x1 (Bind b) x0) c3) (ex3_4 B C T T (\lambda (b0: B).(\lambda -(e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c3 (CHead e (Bind b0) u1)))))) -(\lambda (b0: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear -(CHead x1 (Bind b) x0) (CHead e (Bind b0) u2)))))) (\lambda (_: B).(\lambda -(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C -T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3 -(CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(clear (CHead x1 (Bind b) x0) (CHead e2 (Bind b0) u)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C c3 (CHead e1 (Bind b0) u1))))))) -(\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(clear (CHead x1 (Bind b) x0) (CHead e2 (Bind b0) u2))))))) (\lambda -(_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))))) (or4_intro3 -(clear (CHead x1 (Bind b) x0) (CHead c (Bind b) t)) (ex3_4 B C T T (\lambda -(b0: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead c (Bind -b) t) (CHead e (Bind b0) u1)))))) (\lambda (b0: B).(\lambda (e: C).(\lambda -(_: T).(\lambda (u2: T).(clear (CHead x1 (Bind b) x0) (CHead e (Bind b0) -u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b0: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c (Bind b) t) (CHead e1 (Bind -b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(clear (CHead x1 (Bind b) x0) (CHead e2 (Bind b0) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) -(ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e1 (Bind b0) -u1))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (u2: T).(clear (CHead x1 (Bind b) x0) (CHead e2 (Bind b0) -u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) -(ex4_5_intro B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e1 -(Bind b0) u1))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Bind b) x0) (CHead e2 -(Bind b0) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 -e2)))))) b c x1 t x0 (refl_equal C (CHead c (Bind b) t)) (clear_bind b x1 x0) -H11 H10)) c0 (clear_gen_bind b c c0 t H7)))))))) (\lambda (f: F).(\lambda -(H7: (clear (CHead c (Flat f) t) c0)).(\lambda (H8: (eq nat (S i) (s (Flat f) -x2))).(let H9 \def (f_equal nat nat (\lambda (e: nat).e) (S i) (s (Flat f) -x2) H8) in (let H10 \def (eq_ind_r nat x2 (\lambda (n: nat).(csubst0 n v c -x1)) H6 (S i) H9) in (let H11 \def (eq_ind_r nat x2 (\lambda (n: nat).(subst0 -n v t x0)) H5 (S i) H9) in (let H12 \def (H x1 v i H10 c0 (clear_gen_flat f c -c0 t H7)) in (or4_ind (clear x1 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda -(e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) -(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x1 -(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind -b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(clear x1 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear x1 (CHead e2 -(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 -e2))))))) (or4 (clear (CHead x1 (Flat f) x0) c0) (ex3_4 B C T T (\lambda (b: -B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind -b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: -T).(clear (CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) -(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2 -(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 -(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) -u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 -e2)))))))) (\lambda (H13: (clear x1 c0)).(or4_intro0 (clear (CHead x1 (Flat -f) x0) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: -T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: -B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) -x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind -b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) -(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear -(CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda -(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 -u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (clear_flat x1 c0 H13 f x0))) -(\lambda (H13: (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: -T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: -B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x1 (CHead e (Bind -b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 i v u1 u2))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e: -C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) -(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x1 -(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 i v u1 u2))))) (or4 (clear (CHead x1 (Flat f) x0) -c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda -(_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: -C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e -(Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear -(CHead x1 (Flat f) x0) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C -C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear -(CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda -(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 -u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 i v e1 e2)))))))) (\lambda (x3: B).(\lambda (x4: -C).(\lambda (x5: T).(\lambda (x6: T).(\lambda (H14: (eq C c0 (CHead x4 (Bind -x3) x5))).(\lambda (H15: (clear x1 (CHead x4 (Bind x3) x6))).(\lambda (H16: -(subst0 i v x5 x6)).(or4_intro1 (clear (CHead x1 (Flat f) x0) c0) (ex3_4 B C -T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 -(CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: -T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v -u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x1 (Flat f) -x0) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 -(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 -(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 -e2))))))) (ex3_4_intro B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: -T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: -B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) -x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 i v u1 u2))))) x3 x4 x5 x6 H14 (clear_flat x1 -(CHead x4 (Bind x3) x6) H15 f x0) H16))))))))) H13)) (\lambda (H13: (ex3_4 B -C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C -c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(clear x1 (CHead e2 (Bind b) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 -e2))))))).(ex3_4_ind B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x1 (CHead e2 (Bind -b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 i v e1 e2))))) (or4 (clear (CHead x1 (Flat f) x0) c0) (ex3_4 B C -T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 -(CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: -T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v -u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x1 (Flat f) -x0) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 -(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 -(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 -e2)))))))) (\lambda (x3: B).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: -T).(\lambda (H14: (eq C c0 (CHead x4 (Bind x3) x6))).(\lambda (H15: (clear x1 -(CHead x5 (Bind x3) x6))).(\lambda (H16: (csubst0 i v x4 x5)).(or4_intro2 -(clear (CHead x1 (Flat f) x0) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: -C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) -(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear -(CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda -(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C -T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 -(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda -(_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (ex3_4_intro B C -C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 -(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v e1 e2))))) x3 x4 x5 x6 H14 (clear_flat x1 (CHead x5 (Bind x3) x6) H15 f x0) -H16))))))))) H13)) (\lambda (H13: (ex4_5 B C C T T (\lambda (b: B).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 -(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (u2: T).(clear x1 (CHead e2 (Bind b) u2))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 -i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))).(ex4_5_ind B C C T T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear x1 (CHead e2 -(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 -e2)))))) (or4 (clear (CHead x1 (Flat f) x0) c0) (ex3_4 B C T T (\lambda (b: -B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind -b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: -T).(clear (CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) -(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2 -(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 -(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) -u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 -e2)))))))) (\lambda (x3: B).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: -T).(\lambda (x7: T).(\lambda (H14: (eq C c0 (CHead x4 (Bind x3) -x6))).(\lambda (H15: (clear x1 (CHead x5 (Bind x3) x7))).(\lambda (H16: -(subst0 i v x6 x7)).(\lambda (H17: (csubst0 i v x4 x5)).(or4_intro3 (clear -(CHead x1 (Flat f) x0) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: -C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) -(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear -(CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda -(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C -T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 -(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda -(_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (ex4_5_intro B C -C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear -(CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda -(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 -u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 i v e1 e2)))))) x3 x4 x5 x6 x7 H14 (clear_flat x1 -(CHead x5 (Bind x3) x7) H15 f x0) H16 H17))))))))))) H13)) H12)))))))) k H1 -H3) c2 H4)))))))) H2)) (csubst0_gen_head k c c2 t v (S i) H0)))))))))))) c1). -(* COMMENTS -Initial nodes: 14968 -END *) - -theorem csubst0_clear_trans: - \forall (c1: C).(\forall (c2: C).(\forall (v: T).(\forall (i: nat).((csubst0 -i v c1 c2) \to (\forall (e2: C).((clear c2 e2) \to (or (clear c1 e2) (ex2 C -(\lambda (e1: C).(csubst0 i v e1 e2)) (\lambda (e1: C).(clear c1 e1)))))))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda (i: nat).(\lambda -(H: (csubst0 i v c1 c2)).(csubst0_ind (\lambda (n: nat).(\lambda (t: -T).(\lambda (c: C).(\lambda (c0: C).(\forall (e2: C).((clear c0 e2) \to (or -(clear c e2) (ex2 C (\lambda (e1: C).(csubst0 n t e1 e2)) (\lambda (e1: -C).(clear c e1)))))))))) (\lambda (k: K).(\lambda (i0: nat).(\lambda (v0: -T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H0: (subst0 i0 v0 u1 -u2)).(\lambda (c: C).(\lambda (e2: C).(\lambda (H1: (clear (CHead c k u2) -e2)).(K_ind (\lambda (k0: K).((clear (CHead c k0 u2) e2) \to (or (clear -(CHead c k0 u1) e2) (ex2 C (\lambda (e1: C).(csubst0 (s k0 i0) v0 e1 e2)) -(\lambda (e1: C).(clear (CHead c k0 u1) e1)))))) (\lambda (b: B).(\lambda -(H2: (clear (CHead c (Bind b) u2) e2)).(eq_ind_r C (CHead c (Bind b) u2) -(\lambda (c0: C).(or (clear (CHead c (Bind b) u1) c0) (ex2 C (\lambda (e1: -C).(csubst0 (s (Bind b) i0) v0 e1 c0)) (\lambda (e1: C).(clear (CHead c (Bind -b) u1) e1))))) (or_intror (clear (CHead c (Bind b) u1) (CHead c (Bind b) u2)) -(ex2 C (\lambda (e1: C).(csubst0 (S i0) v0 e1 (CHead c (Bind b) u2))) -(\lambda (e1: C).(clear (CHead c (Bind b) u1) e1))) (ex_intro2 C (\lambda -(e1: C).(csubst0 (S i0) v0 e1 (CHead c (Bind b) u2))) (\lambda (e1: C).(clear -(CHead c (Bind b) u1) e1)) (CHead c (Bind b) u1) (csubst0_snd_bind b i0 v0 u1 -u2 H0 c) (clear_bind b c u1))) e2 (clear_gen_bind b c e2 u2 H2)))) (\lambda -(f: F).(\lambda (H2: (clear (CHead c (Flat f) u2) e2)).(or_introl (clear -(CHead c (Flat f) u1) e2) (ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2)) -(\lambda (e1: C).(clear (CHead c (Flat f) u1) e1))) (clear_flat c e2 -(clear_gen_flat f c e2 u2 H2) f u1)))) k H1)))))))))) (\lambda (k: -K).(\lambda (i0: nat).(\lambda (c3: C).(\lambda (c4: C).(\lambda (v0: -T).(\lambda (H0: (csubst0 i0 v0 c3 c4)).(\lambda (H1: ((\forall (e2: -C).((clear c4 e2) \to (or (clear c3 e2) (ex2 C (\lambda (e1: C).(csubst0 i0 -v0 e1 e2)) (\lambda (e1: C).(clear c3 e1)))))))).(\lambda (u: T).(\lambda -(e2: C).(\lambda (H2: (clear (CHead c4 k u) e2)).(K_ind (\lambda (k0: -K).((clear (CHead c4 k0 u) e2) \to (or (clear (CHead c3 k0 u) e2) (ex2 C -(\lambda (e1: C).(csubst0 (s k0 i0) v0 e1 e2)) (\lambda (e1: C).(clear (CHead -c3 k0 u) e1)))))) (\lambda (b: B).(\lambda (H3: (clear (CHead c4 (Bind b) u) -e2)).(eq_ind_r C (CHead c4 (Bind b) u) (\lambda (c: C).(or (clear (CHead c3 -(Bind b) u) c) (ex2 C (\lambda (e1: C).(csubst0 (s (Bind b) i0) v0 e1 c)) -(\lambda (e1: C).(clear (CHead c3 (Bind b) u) e1))))) (or_intror (clear -(CHead c3 (Bind b) u) (CHead c4 (Bind b) u)) (ex2 C (\lambda (e1: C).(csubst0 -(S i0) v0 e1 (CHead c4 (Bind b) u))) (\lambda (e1: C).(clear (CHead c3 (Bind -b) u) e1))) (ex_intro2 C (\lambda (e1: C).(csubst0 (S i0) v0 e1 (CHead c4 -(Bind b) u))) (\lambda (e1: C).(clear (CHead c3 (Bind b) u) e1)) (CHead c3 -(Bind b) u) (csubst0_fst_bind b i0 c3 c4 v0 H0 u) (clear_bind b c3 u))) e2 -(clear_gen_bind b c4 e2 u H3)))) (\lambda (f: F).(\lambda (H3: (clear (CHead -c4 (Flat f) u) e2)).(let H_x \def (H1 e2 (clear_gen_flat f c4 e2 u H3)) in -(let H4 \def H_x in (or_ind (clear c3 e2) (ex2 C (\lambda (e1: C).(csubst0 i0 -v0 e1 e2)) (\lambda (e1: C).(clear c3 e1))) (or (clear (CHead c3 (Flat f) u) -e2) (ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear -(CHead c3 (Flat f) u) e1)))) (\lambda (H5: (clear c3 e2)).(or_introl (clear -(CHead c3 (Flat f) u) e2) (ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2)) -(\lambda (e1: C).(clear (CHead c3 (Flat f) u) e1))) (clear_flat c3 e2 H5 f -u))) (\lambda (H5: (ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda -(e1: C).(clear c3 e1)))).(ex2_ind C (\lambda (e1: C).(csubst0 i0 v0 e1 e2)) -(\lambda (e1: C).(clear c3 e1)) (or (clear (CHead c3 (Flat f) u) e2) (ex2 C -(\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear (CHead c3 -(Flat f) u) e1)))) (\lambda (x: C).(\lambda (H6: (csubst0 i0 v0 x -e2)).(\lambda (H7: (clear c3 x)).(or_intror (clear (CHead c3 (Flat f) u) e2) -(ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear (CHead -c3 (Flat f) u) e1))) (ex_intro2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2)) -(\lambda (e1: C).(clear (CHead c3 (Flat f) u) e1)) x H6 (clear_flat c3 x H7 f -u)))))) H5)) H4))))) k H2))))))))))) (\lambda (k: K).(\lambda (i0: -nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H0: (subst0 -i0 v0 u1 u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (H1: (csubst0 i0 v0 -c3 c4)).(\lambda (H2: ((\forall (e2: C).((clear c4 e2) \to (or (clear c3 e2) -(ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear c3 -e1)))))))).(\lambda (e2: C).(\lambda (H3: (clear (CHead c4 k u2) e2)).(K_ind -(\lambda (k0: K).((clear (CHead c4 k0 u2) e2) \to (or (clear (CHead c3 k0 u1) -e2) (ex2 C (\lambda (e1: C).(csubst0 (s k0 i0) v0 e1 e2)) (\lambda (e1: -C).(clear (CHead c3 k0 u1) e1)))))) (\lambda (b: B).(\lambda (H4: (clear -(CHead c4 (Bind b) u2) e2)).(eq_ind_r C (CHead c4 (Bind b) u2) (\lambda (c: -C).(or (clear (CHead c3 (Bind b) u1) c) (ex2 C (\lambda (e1: C).(csubst0 (s -(Bind b) i0) v0 e1 c)) (\lambda (e1: C).(clear (CHead c3 (Bind b) u1) e1))))) -(or_intror (clear (CHead c3 (Bind b) u1) (CHead c4 (Bind b) u2)) (ex2 C -(\lambda (e1: C).(csubst0 (S i0) v0 e1 (CHead c4 (Bind b) u2))) (\lambda (e1: -C).(clear (CHead c3 (Bind b) u1) e1))) (ex_intro2 C (\lambda (e1: C).(csubst0 -(S i0) v0 e1 (CHead c4 (Bind b) u2))) (\lambda (e1: C).(clear (CHead c3 (Bind -b) u1) e1)) (CHead c3 (Bind b) u1) (csubst0_both_bind b i0 v0 u1 u2 H0 c3 c4 -H1) (clear_bind b c3 u1))) e2 (clear_gen_bind b c4 e2 u2 H4)))) (\lambda (f: -F).(\lambda (H4: (clear (CHead c4 (Flat f) u2) e2)).(let H_x \def (H2 e2 -(clear_gen_flat f c4 e2 u2 H4)) in (let H5 \def H_x in (or_ind (clear c3 e2) -(ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear c3 -e1))) (or (clear (CHead c3 (Flat f) u1) e2) (ex2 C (\lambda (e1: C).(csubst0 -i0 v0 e1 e2)) (\lambda (e1: C).(clear (CHead c3 (Flat f) u1) e1)))) (\lambda -(H6: (clear c3 e2)).(or_introl (clear (CHead c3 (Flat f) u1) e2) (ex2 C -(\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear (CHead c3 -(Flat f) u1) e1))) (clear_flat c3 e2 H6 f u1))) (\lambda (H6: (ex2 C (\lambda -(e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear c3 e1)))).(ex2_ind C -(\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear c3 e1)) (or -(clear (CHead c3 (Flat f) u1) e2) (ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 -e2)) (\lambda (e1: C).(clear (CHead c3 (Flat f) u1) e1)))) (\lambda (x: -C).(\lambda (H7: (csubst0 i0 v0 x e2)).(\lambda (H8: (clear c3 x)).(or_intror -(clear (CHead c3 (Flat f) u1) e2) (ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 -e2)) (\lambda (e1: C).(clear (CHead c3 (Flat f) u1) e1))) (ex_intro2 C -(\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear (CHead c3 -(Flat f) u1) e1)) x H7 (clear_flat c3 x H8 f u1)))))) H6)) H5))))) k -H3))))))))))))) i v c1 c2 H))))). -(* COMMENTS -Initial nodes: 2085 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubst0/defs.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubst0/defs.ma deleted file mode 100644 index 0068e19f6..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubst0/defs.ma +++ /dev/null @@ -1,32 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/subst0/defs.ma". - -include "Basic-1/C/defs.ma". - -inductive csubst0: nat \to (T \to (C \to (C \to Prop))) \def -| csubst0_snd: \forall (k: K).(\forall (i: nat).(\forall (v: T).(\forall (u1: -T).(\forall (u2: T).((subst0 i v u1 u2) \to (\forall (c: C).(csubst0 (s k i) -v (CHead c k u1) (CHead c k u2)))))))) -| csubst0_fst: \forall (k: K).(\forall (i: nat).(\forall (c1: C).(\forall -(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (u: T).(csubst0 (s -k i) v (CHead c1 k u) (CHead c2 k u)))))))) -| csubst0_both: \forall (k: K).(\forall (i: nat).(\forall (v: T).(\forall -(u1: T).(\forall (u2: T).((subst0 i v u1 u2) \to (\forall (c1: C).(\forall -(c2: C).((csubst0 i v c1 c2) \to (csubst0 (s k i) v (CHead c1 k u1) (CHead c2 -k u2)))))))))). - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubst0/drop.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubst0/drop.ma deleted file mode 100644 index b6bc6cae7..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubst0/drop.ma +++ /dev/null @@ -1,6294 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubst0/fwd.ma". - -include "Basic-1/drop/fwd.ma". - -include "Basic-1/s/props.ma". - -theorem csubst0_drop_gt: - \forall (n: nat).(\forall (i: nat).((lt i n) \to (\forall (c1: C).(\forall -(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n O -c1 e) \to (drop n O c2 e))))))))) -\def - \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (i: nat).((lt i n0) -\to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) -\to (\forall (e: C).((drop n0 O c1 e) \to (drop n0 O c2 e)))))))))) (\lambda -(i: nat).(\lambda (H: (lt i O)).(\lambda (c1: C).(\lambda (c2: C).(\lambda -(v: T).(\lambda (_: (csubst0 i v c1 c2)).(\lambda (e: C).(\lambda (_: (drop O -O c1 e)).(lt_x_O i H (drop O O c2 e)))))))))) (\lambda (n0: nat).(\lambda (H: -((\forall (i: nat).((lt i n0) \to (\forall (c1: C).(\forall (c2: C).(\forall -(v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n0 O c1 e) \to (drop -n0 O c2 e))))))))))).(\lambda (i: nat).(\lambda (H0: (lt i (S n0))).(\lambda -(c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v -c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (drop (S n0) O c2 e))))))) -(\lambda (n1: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (_: (csubst0 i v -(CSort n1) c2)).(\lambda (e: C).(\lambda (H2: (drop (S n0) O (CSort n1) -e)).(and3_ind (eq C e (CSort n1)) (eq nat (S n0) O) (eq nat O O) (drop (S n0) -O c2 e) (\lambda (H3: (eq C e (CSort n1))).(\lambda (H4: (eq nat (S n0) -O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n1) (\lambda (c: C).(drop -(S n0) O c2 c)) (let H6 \def (eq_ind nat (S n0) (\lambda (ee: nat).(match ee -in nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) -\Rightarrow True])) I O H4) in (False_ind (drop (S n0) O c2 (CSort n1)) H6)) -e H3)))) (drop_gen_sort n1 (S n0) O e H2)))))))) (\lambda (c: C).(\lambda -(H1: ((\forall (c2: C).(\forall (v: T).((csubst0 i v c c2) \to (\forall (e: -C).((drop (S n0) O c e) \to (drop (S n0) O c2 e)))))))).(\lambda (k: -K).(\lambda (t: T).(\lambda (c2: C).(\lambda (v: T).(\lambda (H2: (csubst0 i -v (CHead c k t) c2)).(\lambda (e: C).(\lambda (H3: (drop (S n0) O (CHead c k -t) e)).(or3_ind (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k -j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda -(u2: T).(\lambda (j: nat).(subst0 j v t u2)))) (ex3_2 C nat (\lambda (_: -C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_: -nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j -v c c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: -nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: -nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda -(j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: -nat).(csubst0 j v c c3))))) (drop (S n0) O c2 e) (\lambda (H4: (ex3_2 T nat -(\lambda (_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: -T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: -nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: -nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead -c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))) (drop (S -n0) O c2 e) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H5: (eq nat i (s k -x1))).(\lambda (H6: (eq C c2 (CHead c k x0))).(\lambda (_: (subst0 x1 v t -x0)).(eq_ind_r C (CHead c k x0) (\lambda (c0: C).(drop (S n0) O c0 e)) (let -H8 \def (eq_ind nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0: -T).((csubst0 n1 v0 c c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (drop -(S n0) O c3 e0))))))) H1 (s k x1) H5) in (let H9 \def (eq_ind nat i (\lambda -(n1: nat).(lt n1 (S n0))) H0 (s k x1) H5) in (K_ind (\lambda (k0: K).((drop -(r k0 n0) O c e) \to (((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1) -v0 c c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3 -e0))))))) \to ((lt (s k0 x1) (S n0)) \to (drop (S n0) O (CHead c k0 x0) -e))))) (\lambda (b: B).(\lambda (H10: (drop (r (Bind b) n0) O c e)).(\lambda -(_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to -(\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3 -e0)))))))).(\lambda (_: (lt (s (Bind b) x1) (S n0))).(drop_drop (Bind b) n0 c -e H10 x0))))) (\lambda (f: F).(\lambda (H10: (drop (r (Flat f) n0) O c -e)).(\lambda (_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Flat f) x1) -v0 c c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3 -e0)))))))).(\lambda (H12: (lt (s (Flat f) x1) (S n0))).(or_ind (eq nat x1 O) -(ex2 nat (\lambda (m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0))) -(drop (S n0) O (CHead c (Flat f) x0) e) (\lambda (_: (eq nat x1 -O)).(drop_drop (Flat f) n0 c e H10 x0)) (\lambda (H13: (ex2 nat (\lambda (m: -nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda -(m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O -(CHead c (Flat f) x0) e) (\lambda (x: nat).(\lambda (_: (eq nat x1 (S -x))).(\lambda (_: (lt x n0)).(drop_drop (Flat f) n0 c e H10 x0)))) H13)) -(lt_gen_xS x1 n0 H12)))))) k (drop_gen_drop k c e t n0 H3) H8 H9))) c2 -H6)))))) H4)) (\lambda (H4: (ex3_2 C nat (\lambda (_: C).(\lambda (j: -nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead -c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c -c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k -j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda -(c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (drop (S n0) O c2 e) (\lambda -(x0: C).(\lambda (x1: nat).(\lambda (H5: (eq nat i (s k x1))).(\lambda (H6: -(eq C c2 (CHead x0 k t))).(\lambda (H7: (csubst0 x1 v c x0)).(eq_ind_r C -(CHead x0 k t) (\lambda (c0: C).(drop (S n0) O c0 e)) (let H8 \def (eq_ind -nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c -c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3 -e0))))))) H1 (s k x1) H5) in (let H9 \def (eq_ind nat i (\lambda (n1: -nat).(lt n1 (S n0))) H0 (s k x1) H5) in (K_ind (\lambda (k0: K).((drop (r k0 -n0) O c e) \to (((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c -c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3 -e0))))))) \to ((lt (s k0 x1) (S n0)) \to (drop (S n0) O (CHead x0 k0 t) -e))))) (\lambda (b: B).(\lambda (H10: (drop (r (Bind b) n0) O c e)).(\lambda -(_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to -(\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3 -e0)))))))).(\lambda (H12: (lt (s (Bind b) x1) (S n0))).(drop_drop (Bind b) n0 -x0 e (H x1 (lt_S_n x1 n0 H12) c x0 v H7 e H10) t))))) (\lambda (f: -F).(\lambda (H10: (drop (r (Flat f) n0) O c e)).(\lambda (H11: ((\forall (c3: -C).(\forall (v0: T).((csubst0 (s (Flat f) x1) v0 c c3) \to (\forall (e0: -C).((drop (S n0) O c e0) \to (drop (S n0) O c3 e0)))))))).(\lambda (H12: (lt -(s (Flat f) x1) (S n0))).(or_ind (eq nat x1 O) (ex2 nat (\lambda (m: nat).(eq -nat x1 (S m))) (\lambda (m: nat).(lt m n0))) (drop (S n0) O (CHead x0 (Flat -f) t) e) (\lambda (_: (eq nat x1 O)).(drop_drop (Flat f) n0 x0 e (H11 x0 v H7 -e H10) t)) (\lambda (H13: (ex2 nat (\lambda (m: nat).(eq nat x1 (S m))) -(\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda (m: nat).(eq nat x1 (S -m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O (CHead x0 (Flat f) t) e) -(\lambda (x: nat).(\lambda (_: (eq nat x1 (S x))).(\lambda (_: (lt x -n0)).(drop_drop (Flat f) n0 x0 e (H11 x0 v H7 e H10) t)))) H13)) (lt_gen_xS -x1 n0 H12)))))) k (drop_gen_drop k c e t n0 H3) H8 H9))) c2 H6)))))) H4)) -(\lambda (H4: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: -nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: -nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda -(j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: -nat).(csubst0 j v c c3)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_: -C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: -C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda -(_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: -C).(\lambda (j: nat).(csubst0 j v c c3)))) (drop (S n0) O c2 e) (\lambda (x0: -T).(\lambda (x1: C).(\lambda (x2: nat).(\lambda (H5: (eq nat i (s k -x2))).(\lambda (H6: (eq C c2 (CHead x1 k x0))).(\lambda (_: (subst0 x2 v t -x0)).(\lambda (H8: (csubst0 x2 v c x1)).(eq_ind_r C (CHead x1 k x0) (\lambda -(c0: C).(drop (S n0) O c0 e)) (let H9 \def (eq_ind nat i (\lambda (n1: -nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall -(e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3 e0))))))) H1 (s k x2) H5) -in (let H10 \def (eq_ind nat i (\lambda (n1: nat).(lt n1 (S n0))) H0 (s k x2) -H5) in (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to (((\forall (c3: -C).(\forall (v0: T).((csubst0 (s k0 x2) v0 c c3) \to (\forall (e0: C).((drop -(S n0) O c e0) \to (drop (S n0) O c3 e0))))))) \to ((lt (s k0 x2) (S n0)) \to -(drop (S n0) O (CHead x1 k0 x0) e))))) (\lambda (b: B).(\lambda (H11: (drop -(r (Bind b) n0) O c e)).(\lambda (_: ((\forall (c3: C).(\forall (v0: -T).((csubst0 (s (Bind b) x2) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c -e0) \to (drop (S n0) O c3 e0)))))))).(\lambda (H13: (lt (s (Bind b) x2) (S -n0))).(drop_drop (Bind b) n0 x1 e (H x2 (lt_S_n x2 n0 H13) c x1 v H8 e H11) -x0))))) (\lambda (f: F).(\lambda (H11: (drop (r (Flat f) n0) O c e)).(\lambda -(H12: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Flat f) x2) v0 c c3) -\to (\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3 -e0)))))))).(\lambda (H13: (lt (s (Flat f) x2) (S n0))).(or_ind (eq nat x2 O) -(ex2 nat (\lambda (m: nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m n0))) -(drop (S n0) O (CHead x1 (Flat f) x0) e) (\lambda (_: (eq nat x2 -O)).(drop_drop (Flat f) n0 x1 e (H12 x1 v H8 e H11) x0)) (\lambda (H14: (ex2 -nat (\lambda (m: nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m -n0)))).(ex2_ind nat (\lambda (m: nat).(eq nat x2 (S m))) (\lambda (m: -nat).(lt m n0)) (drop (S n0) O (CHead x1 (Flat f) x0) e) (\lambda (x: -nat).(\lambda (_: (eq nat x2 (S x))).(\lambda (_: (lt x n0)).(drop_drop (Flat -f) n0 x1 e (H12 x1 v H8 e H11) x0)))) H14)) (lt_gen_xS x2 n0 H13)))))) k -(drop_gen_drop k c e t n0 H3) H9 H10))) c2 H6)))))))) H4)) (csubst0_gen_head -k c c2 t v i H2))))))))))) c1)))))) n). -(* COMMENTS -Initial nodes: 3092 -END *) - -theorem csubst0_drop_gt_back: - \forall (n: nat).(\forall (i: nat).((lt i n) \to (\forall (c1: C).(\forall -(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n O -c2 e) \to (drop n O c1 e))))))))) -\def - \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (i: nat).((lt i n0) -\to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) -\to (\forall (e: C).((drop n0 O c2 e) \to (drop n0 O c1 e)))))))))) (\lambda -(i: nat).(\lambda (H: (lt i O)).(\lambda (c1: C).(\lambda (c2: C).(\lambda -(v: T).(\lambda (_: (csubst0 i v c1 c2)).(\lambda (e: C).(\lambda (_: (drop O -O c2 e)).(lt_x_O i H (drop O O c1 e)))))))))) (\lambda (n0: nat).(\lambda (H: -((\forall (i: nat).((lt i n0) \to (\forall (c1: C).(\forall (c2: C).(\forall -(v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n0 O c2 e) \to (drop -n0 O c1 e))))))))))).(\lambda (i: nat).(\lambda (H0: (lt i (S n0))).(\lambda -(c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v -c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c e))))))) -(\lambda (n1: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (H1: (csubst0 i -v (CSort n1) c2)).(\lambda (e: C).(\lambda (_: (drop (S n0) O c2 -e)).(csubst0_gen_sort c2 v i n1 H1 (drop (S n0) O (CSort n1) e)))))))) -(\lambda (c: C).(\lambda (H1: ((\forall (c2: C).(\forall (v: T).((csubst0 i v -c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c -e)))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (v: -T).(\lambda (H2: (csubst0 i v (CHead c k t) c2)).(\lambda (e: C).(\lambda -(H3: (drop (S n0) O c2 e)).(or3_ind (ex3_2 T nat (\lambda (_: T).(\lambda (j: -nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead -c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2)))) (ex3_2 C -nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: -C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: -nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: -C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: -C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda -(_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: -C).(\lambda (j: nat).(csubst0 j v c c3))))) (drop (S n0) O (CHead c k t) e) -(\lambda (H4: (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k -j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda -(u2: T).(\lambda (j: nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: -T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: -nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j -v t u2))) (drop (S n0) O (CHead c k t) e) (\lambda (x0: T).(\lambda (x1: -nat).(\lambda (H5: (eq nat i (s k x1))).(\lambda (H6: (eq C c2 (CHead c k -x0))).(\lambda (_: (subst0 x1 v t x0)).(let H8 \def (eq_ind C c2 (\lambda -(c0: C).(drop (S n0) O c0 e)) H3 (CHead c k x0) H6) in (let H9 \def (eq_ind -nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c -c3) \to (\forall (e0: C).((drop (S n0) O c3 e0) \to (drop (S n0) O c -e0))))))) H1 (s k x1) H5) in (let H10 \def (eq_ind nat i (\lambda (n1: -nat).(lt n1 (S n0))) H0 (s k x1) H5) in (K_ind (\lambda (k0: K).(((\forall -(c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c c3) \to (\forall (e0: -C).((drop (S n0) O c3 e0) \to (drop (S n0) O c e0))))))) \to ((lt (s k0 x1) -(S n0)) \to ((drop (r k0 n0) O c e) \to (drop (S n0) O (CHead c k0 t) e))))) -(\lambda (b: B).(\lambda (_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s -(Bind b) x1) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3 e0) \to (drop -(S n0) O c e0)))))))).(\lambda (_: (lt (s (Bind b) x1) (S n0))).(\lambda -(H13: (drop (r (Bind b) n0) O c e)).(drop_drop (Bind b) n0 c e H13 t))))) -(\lambda (f: F).(\lambda (_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s -(Flat f) x1) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3 e0) \to (drop -(S n0) O c e0)))))))).(\lambda (H12: (lt (s (Flat f) x1) (S n0))).(\lambda -(H13: (drop (r (Flat f) n0) O c e)).(or_ind (eq nat x1 O) (ex2 nat (\lambda -(m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0))) (drop (S n0) O -(CHead c (Flat f) t) e) (\lambda (_: (eq nat x1 O)).(drop_drop (Flat f) n0 c -e H13 t)) (\lambda (H14: (ex2 nat (\lambda (m: nat).(eq nat x1 (S m))) -(\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda (m: nat).(eq nat x1 (S -m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O (CHead c (Flat f) t) e) -(\lambda (x: nat).(\lambda (_: (eq nat x1 (S x))).(\lambda (_: (lt x -n0)).(drop_drop (Flat f) n0 c e H13 t)))) H14)) (lt_gen_xS x1 n0 H12)))))) k -H9 H10 (drop_gen_drop k c e x0 n0 H8)))))))))) H4)) (\lambda (H4: (ex3_2 C -nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: -C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: -nat).(csubst0 j v c c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: -nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead -c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (drop (S -n0) O (CHead c k t) e) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H5: (eq -nat i (s k x1))).(\lambda (H6: (eq C c2 (CHead x0 k t))).(\lambda (H7: -(csubst0 x1 v c x0)).(let H8 \def (eq_ind C c2 (\lambda (c0: C).(drop (S n0) -O c0 e)) H3 (CHead x0 k t) H6) in (let H9 \def (eq_ind nat i (\lambda (n1: -nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall -(e0: C).((drop (S n0) O c3 e0) \to (drop (S n0) O c e0))))))) H1 (s k x1) H5) -in (let H10 \def (eq_ind nat i (\lambda (n1: nat).(lt n1 (S n0))) H0 (s k x1) -H5) in (K_ind (\lambda (k0: K).(((\forall (c3: C).(\forall (v0: T).((csubst0 -(s k0 x1) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3 e0) \to (drop (S -n0) O c e0))))))) \to ((lt (s k0 x1) (S n0)) \to ((drop (r k0 n0) O x0 e) \to -(drop (S n0) O (CHead c k0 t) e))))) (\lambda (b: B).(\lambda (_: ((\forall -(c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to (\forall (e0: -C).((drop (S n0) O c3 e0) \to (drop (S n0) O c e0)))))))).(\lambda (H12: (lt -(s (Bind b) x1) (S n0))).(\lambda (H13: (drop (r (Bind b) n0) O x0 -e)).(drop_drop (Bind b) n0 c e (H x1 (lt_S_n x1 n0 H12) c x0 v H7 e H13) -t))))) (\lambda (f: F).(\lambda (H11: ((\forall (c3: C).(\forall (v0: -T).((csubst0 (s (Flat f) x1) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3 -e0) \to (drop (S n0) O c e0)))))))).(\lambda (H12: (lt (s (Flat f) x1) (S -n0))).(\lambda (H13: (drop (r (Flat f) n0) O x0 e)).(or_ind (eq nat x1 O) -(ex2 nat (\lambda (m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0))) -(drop (S n0) O (CHead c (Flat f) t) e) (\lambda (_: (eq nat x1 O)).(drop_drop -(Flat f) n0 c e (H11 x0 v H7 e H13) t)) (\lambda (H14: (ex2 nat (\lambda (m: -nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda -(m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O -(CHead c (Flat f) t) e) (\lambda (x: nat).(\lambda (_: (eq nat x1 (S -x))).(\lambda (_: (lt x n0)).(drop_drop (Flat f) n0 c e (H11 x0 v H7 e H13) -t)))) H14)) (lt_gen_xS x1 n0 H12)))))) k H9 H10 (drop_gen_drop k x0 e t n0 -H8)))))))))) H4)) (\lambda (H4: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: -C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: -C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda -(_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: -C).(\lambda (j: nat).(csubst0 j v c c3)))))).(ex4_3_ind T C nat (\lambda (_: -T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: -T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda -(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: -T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (drop (S n0) O -(CHead c k t) e) (\lambda (x0: T).(\lambda (x1: C).(\lambda (x2: -nat).(\lambda (H5: (eq nat i (s k x2))).(\lambda (H6: (eq C c2 (CHead x1 k -x0))).(\lambda (_: (subst0 x2 v t x0)).(\lambda (H8: (csubst0 x2 v c -x1)).(let H9 \def (eq_ind C c2 (\lambda (c0: C).(drop (S n0) O c0 e)) H3 -(CHead x1 k x0) H6) in (let H10 \def (eq_ind nat i (\lambda (n1: -nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall -(e0: C).((drop (S n0) O c3 e0) \to (drop (S n0) O c e0))))))) H1 (s k x2) H5) -in (let H11 \def (eq_ind nat i (\lambda (n1: nat).(lt n1 (S n0))) H0 (s k x2) -H5) in (K_ind (\lambda (k0: K).(((\forall (c3: C).(\forall (v0: T).((csubst0 -(s k0 x2) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3 e0) \to (drop (S -n0) O c e0))))))) \to ((lt (s k0 x2) (S n0)) \to ((drop (r k0 n0) O x1 e) \to -(drop (S n0) O (CHead c k0 t) e))))) (\lambda (b: B).(\lambda (_: ((\forall -(c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x2) v0 c c3) \to (\forall (e0: -C).((drop (S n0) O c3 e0) \to (drop (S n0) O c e0)))))))).(\lambda (H13: (lt -(s (Bind b) x2) (S n0))).(\lambda (H14: (drop (r (Bind b) n0) O x1 -e)).(drop_drop (Bind b) n0 c e (H x2 (lt_S_n x2 n0 H13) c x1 v H8 e H14) -t))))) (\lambda (f: F).(\lambda (H12: ((\forall (c3: C).(\forall (v0: -T).((csubst0 (s (Flat f) x2) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3 -e0) \to (drop (S n0) O c e0)))))))).(\lambda (H13: (lt (s (Flat f) x2) (S -n0))).(\lambda (H14: (drop (r (Flat f) n0) O x1 e)).(or_ind (eq nat x2 O) -(ex2 nat (\lambda (m: nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m n0))) -(drop (S n0) O (CHead c (Flat f) t) e) (\lambda (_: (eq nat x2 O)).(drop_drop -(Flat f) n0 c e (H12 x1 v H8 e H14) t)) (\lambda (H15: (ex2 nat (\lambda (m: -nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda -(m: nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O -(CHead c (Flat f) t) e) (\lambda (x: nat).(\lambda (_: (eq nat x2 (S -x))).(\lambda (_: (lt x n0)).(drop_drop (Flat f) n0 c e (H12 x1 v H8 e H14) -t)))) H15)) (lt_gen_xS x2 n0 H13)))))) k H10 H11 (drop_gen_drop k x1 e x0 n0 -H9)))))))))))) H4)) (csubst0_gen_head k c c2 t v i H2))))))))))) c1)))))) n). -(* COMMENTS -Initial nodes: 2989 -END *) - -theorem csubst0_drop_lt: - \forall (n: nat).(\forall (i: nat).((lt n i) \to (\forall (c1: C).(\forall -(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n O -c1 e) \to (or4 (drop n O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: -K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e0 k -w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (s k n)) v u w)))))) (ex3_4 K C C T (\lambda (k: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k -u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop n O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1 e2)))))) (ex4_5 K C C -T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda -(_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e2 k w))))))) -(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (s k n)) v u w)))))) (\lambda (k: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k -n)) v e1 e2)))))))))))))))) -\def - \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (i: nat).((lt n0 i) -\to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) -\to (\forall (e: C).((drop n0 O c1 e) \to (or4 (drop n0 O c2 e) (ex3_4 K C T -T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda -(w: T).(drop n0 O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n0)) v u w)))))) -(ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop n0 O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k n0)) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 -O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n0)) v u w)))))) -(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus i (s k n0)) v e1 e2))))))))))))))))) (\lambda (i: -nat).(\lambda (_: (lt O i)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (v: -T).(\lambda (H0: (csubst0 i v c1 c2)).(\lambda (e: C).(\lambda (H1: (drop O O -c1 e)).(eq_ind C c1 (\lambda (c: C).(or4 (drop O O c2 c) (ex3_4 K C T T -(\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c -(CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda -(w: T).(drop O O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k O)) v u w)))))) -(ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop O O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k O)) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 k u))))))) (\lambda (k: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O -c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k O)) v u w)))))) -(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus i (s k O)) v e1 e2))))))))) (csubst0_ind (\lambda (n0: -nat).(\lambda (t: T).(\lambda (c: C).(\lambda (c0: C).(or4 (drop O O c0 c) -(ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: -T).(eq C c (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop O O c0 (CHead e0 k w)))))) (\lambda (k: K).(\lambda -(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n0 (s k O)) t u w)))))) -(ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop O O c0 (CHead e2 k u)))))) (\lambda (k: K).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n0 (s k O)) t e1 -e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 k u))))))) (\lambda (k: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O -c0 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n0 (s k O)) t u w)))))) -(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus n0 (s k O)) t e1 e2)))))))))))) (\lambda (k: -K).(\lambda (i0: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: -T).(\lambda (H2: (subst0 i0 v0 u1 u2)).(\lambda (c: C).(let H3 \def (eq_ind_r -nat i0 (\lambda (n0: nat).(subst0 n0 v0 u1 u2)) H2 (minus (s k i0) (s k O)) -(s_arith0 k i0)) in (or4_intro1 (drop O O (CHead c k u2) (CHead c k u1)) -(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: -T).(eq C (CHead c k u1) (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c k u2) (CHead e0 k0 -w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus (s k i0) (s k0 O)) v0 u w)))))) (ex3_4 K C C T (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c k u1) -(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop O O (CHead c k u2) (CHead e2 k0 u)))))) (\lambda -(k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s -k i0) (s k0 O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c k u1) -(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c k u2) (CHead e2 k0 -w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 O)) v0 u w)))))) (\lambda -(k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: -T).(csubst0 (minus (s k i0) (s k0 O)) v0 e1 e2))))))) (ex3_4_intro K C T T -(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C -(CHead c k u1) (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c k u2) (CHead e0 k0 -w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus (s k i0) (s k0 O)) v0 u w))))) k c u1 u2 (refl_equal C -(CHead c k u1)) (drop_refl (CHead c k u2)) H3)))))))))) (\lambda (k: -K).(\lambda (i0: nat).(\lambda (c3: C).(\lambda (c4: C).(\lambda (v0: -T).(\lambda (H2: (csubst0 i0 v0 c3 c4)).(\lambda (H3: (or4 (drop O O c4 c3) -(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: -T).(eq C c3 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda -(_: T).(\lambda (w: T).(drop O O c4 (CHead e0 k0 w)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k0 -O)) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u: T).(eq C c3 (CHead e1 k0 u)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O c4 (CHead e2 k0 -u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus i0 (s k0 O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 -(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e2 k0 w))))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i0 (s k0 O)) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k0 -O)) v0 e1 e2))))))))).(\lambda (u: T).(let H4 \def (eq_ind_r nat i0 (\lambda -(n0: nat).(csubst0 n0 v0 c3 c4)) H2 (minus (s k i0) (s k O)) (s_arith0 k i0)) -in (let H5 \def (eq_ind_r nat i0 (\lambda (n0: nat).(or4 (drop O O c4 c3) -(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda -(_: T).(eq C c3 (CHead e0 k0 u0)))))) (\lambda (k0: K).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e0 k0 w)))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus n0 (s -k0 O)) v0 u0 w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u0: T).(eq C c3 (CHead e1 k0 u0)))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop O O c4 (CHead -e2 k0 u0)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 (minus n0 (s k0 O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq -C c3 (CHead e1 k0 u0))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e2 k0 w))))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda -(w: T).(subst0 (minus n0 (s k0 O)) v0 u0 w)))))) (\lambda (k0: K).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n0 -(s k0 O)) v0 e1 e2))))))))) H3 (minus (s k i0) (s k O)) (s_arith0 k i0)) in -(or4_intro2 (drop O O (CHead c4 k u) (CHead c3 k u)) (ex3_4 K C T T (\lambda -(k0: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 k -u) (CHead e0 k0 u0)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop O O (CHead c4 k u) (CHead e0 k0 w)))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k -i0) (s k0 O)) v0 u0 w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead c3 k u) (CHead e1 k0 -u0)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: -T).(drop O O (CHead c4 k u) (CHead e2 k0 u0)))))) (\lambda (k0: K).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 O)) -v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 k u) (CHead e1 k0 -u0))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop O O (CHead c4 k u) (CHead e2 k0 w))))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: -T).(subst0 (minus (s k i0) (s k0 O)) v0 u0 w)))))) (\lambda (k0: K).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k -i0) (s k0 O)) v0 e1 e2))))))) (ex3_4_intro K C C T (\lambda (k0: K).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead c3 k u) (CHead e1 k0 -u0)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: -T).(drop O O (CHead c4 k u) (CHead e2 k0 u0)))))) (\lambda (k0: K).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 O)) -v0 e1 e2))))) k c3 c4 u (refl_equal C (CHead c3 k u)) (drop_refl (CHead c4 k -u)) H4)))))))))))) (\lambda (k: K).(\lambda (i0: nat).(\lambda (v0: -T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 i0 v0 u1 -u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (H3: (csubst0 i0 v0 c3 -c4)).(\lambda (_: (or4 (drop O O c4 c3) (ex3_4 K C T T (\lambda (k0: -K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e0 k0 -u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(drop O O c4 (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k0 O)) v0 u w)))))) -(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c3 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u: T).(drop O O c4 (CHead e2 k0 u)))))) (\lambda (k0: -K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (s k0 -O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 k0 -u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop O O c4 (CHead e2 k0 w))))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus i0 (s k0 O)) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k0 O)) v0 e1 -e2))))))))).(let H5 \def (eq_ind_r nat i0 (\lambda (n0: nat).(subst0 n0 v0 u1 -u2)) H2 (minus (s k i0) (s k O)) (s_arith0 k i0)) in (let H6 \def (eq_ind_r -nat i0 (\lambda (n0: nat).(csubst0 n0 v0 c3 c4)) H3 (minus (s k i0) (s k O)) -(s_arith0 k i0)) in (or4_intro3 (drop O O (CHead c4 k u2) (CHead c3 k u1)) -(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: -T).(eq C (CHead c3 k u1) (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 k u2) (CHead e0 k0 -w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus (s k i0) (s k0 O)) v0 u w)))))) (ex3_4 K C C T (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c3 k -u1) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop O O (CHead c4 k u2) (CHead e2 k0 u)))))) (\lambda -(k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s -k i0) (s k0 O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 k u1) -(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 k u2) (CHead e2 k0 -w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 O)) v0 u w)))))) (\lambda -(k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: -T).(csubst0 (minus (s k i0) (s k0 O)) v0 e1 e2))))))) (ex4_5_intro K C C T T -(\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda -(_: T).(eq C (CHead c3 k u1) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 k -u2) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 O)) v0 u -w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 O)) v0 e1 e2)))))) k c3 c4 -u1 u2 (refl_equal C (CHead c3 k u1)) (drop_refl (CHead c4 k u2)) H5 -H6)))))))))))))) i v c1 c2 H0) e (drop_gen_refl c1 e H1)))))))))) (\lambda -(n0: nat).(\lambda (IHn: ((\forall (i: nat).((lt n0 i) \to (\forall (c1: -C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: -C).((drop n0 O c1 e) \to (or4 (drop n0 O c2 e) (ex3_4 K C T T (\lambda (k: -K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k -u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(drop n0 O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda -(u: T).(\lambda (w: T).(subst0 (minus i (s k n0)) v u w)))))) (ex3_4 K C C T -(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda -(u: T).(drop n0 O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k n0)) v e1 e2)))))) -(ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c2 (CHead -e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus i (s k n0)) v u w)))))) (\lambda (k: -K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -(minus i (s k n0)) v e1 e2)))))))))))))))))).(\lambda (i: nat).(\lambda (H: -(lt (S n0) i)).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: -C).(\forall (v: T).((csubst0 i v c c2) \to (\forall (e: C).((drop (S n0) O c -e) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: -K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead -e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k -u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w)))))) -(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus i (s k (S n0))) v e1 e2)))))))))))))) (\lambda (n1: -nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (_: (csubst0 i v (CSort n1) -c2)).(\lambda (e: C).(\lambda (H1: (drop (S n0) O (CSort n1) e)).(and3_ind -(eq C e (CSort n1)) (eq nat (S n0) O) (eq nat O O) (or4 (drop (S n0) O c2 e) -(ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: -T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: -K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S -n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda -(_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) -(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 -(minus i (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) -(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k (S -n0))) v e1 e2)))))))) (\lambda (H2: (eq C e (CSort n1))).(\lambda (H3: (eq -nat (S n0) O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n1) (\lambda (c: -C).(or4 (drop (S n0) O c2 c) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 k u)))))) (\lambda (k: -K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead -e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 k -u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 k u))))))) (\lambda (k: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w)))))) -(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus i (s k (S n0))) v e1 e2))))))))) (let H5 \def (eq_ind -nat (S n0) (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) -with [O \Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind -(or4 (drop (S n0) O c2 (CSort n1)) (ex3_4 K C T T (\lambda (k: K).(\lambda -(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CSort n1) (CHead e0 k u)))))) -(\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) -O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w)))))) (ex3_4 K C C T -(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CSort -n1) (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: -K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k -(S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CSort n1) (CHead e1 -k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k: -K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus i (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1 -e2)))))))) H5)) e H2)))) (drop_gen_sort n1 (S n0) O e H1)))))))) (\lambda (c: -C).(\lambda (H0: ((\forall (c2: C).(\forall (v: T).((csubst0 i v c c2) \to -(\forall (e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C -T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda -(w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w)))))) -(ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: -K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k -(S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k -u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k: -K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus i (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1 -e2))))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda -(v: T).(\lambda (H1: (csubst0 i v (CHead c k t) c2)).(\lambda (e: C).(\lambda -(H2: (drop (S n0) O (CHead c k t) e)).(or3_ind (ex3_2 T nat (\lambda (_: -T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: -nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j -v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k -j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda -(c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_: -T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: -T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda -(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: -T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))) (or4 (drop (S -n0) O c2 e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda -(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k0 -w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 -u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O c2 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k0 (S n0))) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O c2 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S -n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 (minus i (s k0 (S n0))) v e1 e2)))))))) -(\lambda (H3: (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k -j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda -(u2: T).(\lambda (j: nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: -T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: -nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j -v t u2))) (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k0: K).(\lambda -(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda -(k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 -(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (ex3_4 K C C T -(\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k0 u)))))) (\lambda (k0: -K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k0 -(S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 -u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k0 w))))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus i (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k0 -(S n0))) v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H4: -(eq nat i (s k x1))).(\lambda (H5: (eq C c2 (CHead c k x0))).(\lambda (_: -(subst0 x1 v t x0)).(eq_ind_r C (CHead c k x0) (\lambda (c0: C).(or4 (drop (S -n0) O c0 e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda -(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e0 k0 -w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 -u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O c0 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k0 (S n0))) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O c0 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S -n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 (minus i (s k0 (S n0))) v e1 e2))))))))) (let -H7 \def (eq_ind nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0: -T).((csubst0 n1 v0 c c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (or4 -(drop (S n0) O c3 e0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c3 -(CHead e1 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus n1 (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C -T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e0 -(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O c3 (CHead e2 k0 u)))))) (\lambda (k0: -K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n1 (s k0 -(S n0))) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 -u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus n1 (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 (s k0 -(S n0))) v0 e1 e2)))))))))))))) H0 (s k x1) H4) in (let H8 \def (eq_ind nat i -(\lambda (n1: nat).(lt (S n0) n1)) H (s k x1) H4) in (eq_ind_r nat (s k x1) -(\lambda (n1: nat).(or4 (drop (S n0) O (CHead c k x0) e) (ex3_4 K C T T -(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead c k x0) (CHead e0 k0 w)))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus n1 (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O -(CHead c k x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n1 (s k0 (S n0))) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead c k x0) (CHead e2 k0 w))))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus n1 (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 (s k0 -(S n0))) v e1 e2))))))))) (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to -(((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c c3) \to (\forall -(e0: C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T -(\lambda (k1: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 -(CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k1 w)))))) (\lambda (k1: -K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 x1) -(s k1 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k1 u)))))) (\lambda -(k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 -(CHead e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus (s k0 x1) (s k1 (S n0))) v0 e1 e2)))))) -(ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq C e0 (CHead e1 k1 u))))))) (\lambda (k1: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O c3 (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 x1) (s k1 (S n0))) v0 -u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (minus (s k0 x1) (s k1 (S n0))) v0 e1 -e2)))))))))))))) \to ((lt (S n0) (s k0 x1)) \to (or4 (drop (S n0) O (CHead c -k0 x0) e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda -(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c k0 x0) (CHead -e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus (s k0 x1) (s k1 (S n0))) v u w)))))) (ex3_4 K C C T -(\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O (CHead c k0 x0) (CHead e2 k1 u)))))) -(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 -(minus (s k0 x1) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c k0 x0) (CHead e2 -k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus (s k0 x1) (s k1 (S n0))) v u w)))))) -(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus (s k0 x1) (s k1 (S n0))) v e1 e2)))))))))))) (\lambda -(b: B).(\lambda (H9: (drop (r (Bind b) n0) O c e)).(\lambda (_: ((\forall -(c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to (\forall (e0: -C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T -(\lambda (k0: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 -(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) -x1) (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 -(CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v0 e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u))))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) -x1) (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 -(S n0))) v0 e1 e2))))))))))))))).(\lambda (_: (lt (S n0) (s (Bind b) -x1))).(or4_intro0 (drop (S n0) O (CHead c (Bind b) x0) e) (ex3_4 K C T T -(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead c (Bind b) x0) (CHead e0 k0 w)))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 -u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O (CHead c (Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0: -K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind -b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c (Bind b) x0) -(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S -n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1 -e2))))))) (drop_drop (Bind b) n0 c e H9 x0)))))) (\lambda (f: F).(\lambda -(H9: (drop (r (Flat f) n0) O c e)).(\lambda (_: ((\forall (c3: C).(\forall -(v0: T).((csubst0 (s (Flat f) x1) v0 c c3) \to (\forall (e0: C).((drop (S n0) -O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T (\lambda (k0: -K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 -u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S -n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 (CHead -e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v0 e1 e2)))))) (ex4_5 K C C -T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c3 -(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S -n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S -n0))) v0 e1 e2))))))))))))))).(\lambda (_: (lt (S n0) (s (Flat f) -x1))).(or4_intro0 (drop (S n0) O (CHead c (Flat f) x0) e) (ex3_4 K C T T -(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead c (Flat f) x0) (CHead e0 k0 w)))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 -u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O (CHead c (Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: -K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat -f) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c (Flat f) x0) -(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S -n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1 -e2))))))) (drop_drop (Flat f) n0 c e H9 x0)))))) k (drop_gen_drop k c e t n0 -H2) H7 H8) i H4))) c2 H5)))))) H3)) (\lambda (H3: (ex3_2 C nat (\lambda (_: -C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_: -nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j -v c c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k -j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda -(c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (or4 (drop (S n0) O c2 e) -(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: -T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda -(_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k0 w)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S -n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda -(_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k0 -u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus i (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq -C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k0 w))))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k0 -(S n0))) v e1 e2)))))))) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H4: -(eq nat i (s k x1))).(\lambda (H5: (eq C c2 (CHead x0 k t))).(\lambda (H6: -(csubst0 x1 v c x0)).(eq_ind_r C (CHead x0 k t) (\lambda (c0: C).(or4 (drop -(S n0) O c0 e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda -(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e0 k0 -w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 -u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O c0 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k0 (S n0))) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O c0 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S -n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 (minus i (s k0 (S n0))) v e1 e2))))))))) (let -H7 \def (eq_ind nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0: -T).((csubst0 n1 v0 c c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (or4 -(drop (S n0) O c3 e0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c3 -(CHead e1 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus n1 (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C -T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e0 -(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O c3 (CHead e2 k0 u)))))) (\lambda (k0: -K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n1 (s k0 -(S n0))) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 -u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus n1 (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 (s k0 -(S n0))) v0 e1 e2)))))))))))))) H0 (s k x1) H4) in (let H8 \def (eq_ind nat i -(\lambda (n1: nat).(lt (S n0) n1)) H (s k x1) H4) in (eq_ind_r nat (s k x1) -(\lambda (n1: nat).(or4 (drop (S n0) O (CHead x0 k t) e) (ex3_4 K C T T -(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x0 k t) (CHead e0 k0 w)))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus n1 (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O -(CHead x0 k t) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n1 (s k0 (S n0))) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x0 k t) (CHead e2 k0 w))))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus n1 (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 (s k0 -(S n0))) v e1 e2))))))))) (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to -(((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c c3) \to (\forall -(e0: C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T -(\lambda (k1: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 -(CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k1 w)))))) (\lambda (k1: -K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 x1) -(s k1 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k1 u)))))) (\lambda -(k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 -(CHead e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus (s k0 x1) (s k1 (S n0))) v0 e1 e2)))))) -(ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq C e0 (CHead e1 k1 u))))))) (\lambda (k1: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O c3 (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 x1) (s k1 (S n0))) v0 -u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (minus (s k0 x1) (s k1 (S n0))) v0 e1 -e2)))))))))))))) \to ((lt (S n0) (s k0 x1)) \to (or4 (drop (S n0) O (CHead x0 -k0 t) e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda -(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 k0 t) (CHead -e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus (s k0 x1) (s k1 (S n0))) v u w)))))) (ex3_4 K C C T -(\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O (CHead x0 k0 t) (CHead e2 k1 u)))))) -(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 -(minus (s k0 x1) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 k0 t) (CHead e2 -k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus (s k0 x1) (s k1 (S n0))) v u w)))))) -(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus (s k0 x1) (s k1 (S n0))) v e1 e2)))))))))))) (\lambda -(b: B).(\lambda (H9: (drop (r (Bind b) n0) O c e)).(\lambda (_: ((\forall -(c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to (\forall (e0: -C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T -(\lambda (k0: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 -(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) -x1) (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 -(CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v0 e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u))))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) -x1) (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 -(S n0))) v0 e1 e2))))))))))))))).(\lambda (H11: (lt (S n0) (s (Bind b) -x1))).(let H12 \def (IHn x1 (le_S_n (S n0) x1 H11) c x0 v H6 e H9) in -(or4_ind (drop n0 O x0 e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: -K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x0 (CHead e0 -k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus x1 (s k0 n0)) v u w)))))) (ex3_4 K C C T (\lambda (k0: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 -u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop n0 O x0 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus x1 (s k0 n0)) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop n0 O x0 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x1 (s k0 -n0)) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 (minus x1 (s k0 n0)) v e1 e2))))))) (or4 -(drop (S n0) O (CHead x0 (Bind b) t) e) (ex3_4 K C T T (\lambda (k0: -K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 -u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 k0 w)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) -x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 -(Bind b) t) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s -(Bind b) x1) (s k0 (S n0))) v e1 e2)))))))) (\lambda (H13: (drop n0 O x0 -e)).(or4_intro0 (drop (S n0) O (CHead x0 (Bind b) t) e) (ex3_4 K C T T -(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 k0 w)))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 -u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 u)))))) (\lambda (k0: -K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind -b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) -(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S -n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1 -e2))))))) (drop_drop (Bind b) n0 x0 e H13 t))) (\lambda (H13: (ex3_4 K C T T -(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop n0 O x0 (CHead e0 k0 w)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x1 (s k0 -n0)) v u w))))))).(ex3_4_ind K C T T (\lambda (k0: K).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: -K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x0 (CHead e0 -k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus x1 (s k0 n0)) v u w))))) (or4 (drop (S n0) O (CHead x0 -(Bind b) t) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda -(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) -(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) -(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 -u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C -T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead -x0 (Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) -x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 -(S n0))) v e1 e2)))))))) (\lambda (x2: K).(\lambda (x3: C).(\lambda (x4: -T).(\lambda (x5: T).(\lambda (H14: (eq C e (CHead x3 x2 x4))).(\lambda (H15: -(drop n0 O x0 (CHead x3 x2 x5))).(\lambda (H16: (subst0 (minus x1 (s x2 n0)) -v x4 x5)).(eq_ind_r C (CHead x3 x2 x4) (\lambda (c0: C).(or4 (drop (S n0) O -(CHead x0 (Bind b) t) c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 k0 u)))))) (\lambda -(k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O -(CHead x0 (Bind b) t) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S -n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u: T).(eq C c0 (CHead e1 k0 u)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 -(Bind b) t) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 k0 u))))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s -(Bind b) x1) (s k0 (S n0))) v e1 e2))))))))) (or4_intro1 (drop (S n0) O -(CHead x0 (Bind b) t) (CHead x3 x2 x4)) (ex3_4 K C T T (\lambda (k0: -K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x4) -(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 k0 w)))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 x2 -x4) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 u)))))) -(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 -(minus (s (Bind b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq -C (CHead x3 x2 x4) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 -(Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda -(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 -(S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S -n0))) v e1 e2))))))) (ex3_4_intro K C T T (\lambda (k0: K).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x4) (CHead e0 k0 -u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 k0 w)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) -x1) (s k0 (S n0))) v u w))))) x2 x3 x4 x5 (refl_equal C (CHead x3 x2 x4)) -(drop_drop (Bind b) n0 x0 (CHead x3 x2 x5) H15 t) (eq_ind_r nat (S (s x2 n0)) -(\lambda (n1: nat).(subst0 (minus (s (Bind b) x1) n1) v x4 x5)) H16 (s x2 (S -n0)) (s_S x2 n0)))) e H14)))))))) H13)) (\lambda (H13: (ex3_4 K C C T -(\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 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(\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 x2 -x5) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 u)))))) -(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 -(minus (s (Bind b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq -C (CHead x3 x2 x5) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 -(Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda -(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 -(S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S -n0))) v e1 e2))))))) (ex3_4_intro K C C T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 x2 x5) (CHead e1 k0 -u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 u)))))) (\lambda (k0: -K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind -b) x1) (s k0 (S n0))) v e1 e2))))) x2 x3 x4 x5 (refl_equal C (CHead x3 x2 -x5)) (drop_drop (Bind b) n0 x0 (CHead x4 x2 x5) H15 t) (eq_ind_r nat (S (s x2 -n0)) (\lambda (n1: nat).(csubst0 (minus (s (Bind b) x1) n1) v x3 x4)) H16 (s -x2 (S n0)) (s_S x2 n0)))) e H14)))))))) H13)) (\lambda (H13: (ex4_5 K C C T T -(\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda -(_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x0 (CHead e2 -k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus x1 (s k0 n0)) v u w)))))) (\lambda (k0: -K).(\lambda (e1: 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v u w)))))) (ex3_4 K C C T (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 -u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 u)))))) (\lambda (k0: -K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind -b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) -(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S -n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1 -e2)))))))) (\lambda (x2: K).(\lambda (x3: C).(\lambda (x4: 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b) x1) (s k0 (S n0))) v e1 -e2))))))) (ex4_5_intro K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x5) (CHead e1 k0 -u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 w))))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: -K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -(minus (s (Bind b) x1) (s k0 (S n0))) v e1 e2)))))) x2 x3 x4 x5 x6 -(refl_equal C (CHead x3 x2 x5)) (drop_drop (Bind b) n0 x0 (CHead x4 x2 x6) -H15 t) (eq_ind_r nat (S (s x2 n0)) (\lambda (n1: nat).(subst0 (minus (s (Bind -b) x1) n1) v x5 x6)) H16 (s x2 (S n0)) (s_S x2 n0)) (eq_ind_r nat (S (s x2 -n0)) (\lambda (n1: nat).(csubst0 (minus (s (Bind b) x1) n1) v x3 x4)) H17 (s -x2 (S n0)) (s_S x2 n0)))) e H14)))))))))) H13)) H12)))))) (\lambda (f: -F).(\lambda (H9: (drop (r (Flat f) n0) O c e)).(\lambda (H10: ((\forall (c3: -C).(\forall (v0: T).((csubst0 (s (Flat f) x1) v0 c c3) \to (\forall (e0: -C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T -(\lambda (k0: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 -(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) -x1) (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 -(CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v0 e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u))))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) -x1) (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 -(S n0))) v0 e1 e2))))))))))))))).(\lambda (_: (lt (S n0) (s (Flat f) -x1))).(let H12 \def (H10 x0 v H6 e H9) in (or4_ind (drop (S n0) O x0 e) -(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: -T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda -(_: T).(\lambda (w: T).(drop (S n0) O x0 (CHead e0 k0 w)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x1 (s k0 (S -n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda -(_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O x0 (CHead e2 k0 -u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus x1 (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq -C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x0 (CHead e2 k0 w))))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus x1 (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus x1 -(s k0 (S n0))) v e1 e2))))))) (or4 (drop (S n0) O (CHead x0 (Flat f) t) e) -(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: -T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda -(_: 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C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 w))))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s -(Flat f) x1) (s k0 (S n0))) v e1 e2))))))) (drop_drop (Flat f) n0 x0 e H13 -t))) (\lambda (H13: (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda -(u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda -(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x0 (CHead e0 k0 -w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus x1 (s k0 (S n0))) v u w))))))).(ex3_4_ind K C T T (\lambda -(k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 -u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O x0 (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x1 (s k0 (S n0))) v u -w))))) (or4 (drop (S n0) O (CHead x0 (Flat f) t) e) (ex3_4 K C T T (\lambda -(k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 -u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 k0 w)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) -x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 -(Flat f) t) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1 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(CHead x3 x2 x4)) (drop_drop (Flat f) n0 x0 (CHead -x3 x2 x5) H15 t) H16)) e H14)))))))) H13)) (\lambda (H13: (ex3_4 K C C T -(\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O x0 (CHead e2 k0 u)))))) (\lambda (k0: -K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus x1 (s k0 -(S n0))) v e1 e2))))))).(ex3_4_ind K C C T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O x0 (CHead -e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus x1 (s k0 (S n0))) v e1 e2))))) (or4 (drop (S n0) O (CHead -x0 (Flat f) t) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda -(u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda -(e0: C).(\lambda (_: T).(\lambda 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(w: T).(subst0 -(minus x1 (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus x1 (s k0 -(S n0))) v e1 e2)))))))).(ex4_5_ind K C C T T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 -u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O x0 (CHead e2 k0 w))))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus x1 (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus x1 (s k0 -(S n0))) v e1 e2)))))) (or4 (drop (S n0) O (CHead x0 (Flat f) t) e) (ex3_4 K -C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C -e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 k0 w)))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 -u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 u)))))) (\lambda (k0: -K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat -f) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) -(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S -n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1 -e2)))))))) (\lambda (x2: K).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: -T).(\lambda (x6: T).(\lambda (H14: (eq C e (CHead x3 x2 x5))).(\lambda (H15: -(drop (S n0) O x0 (CHead x4 x2 x6))).(\lambda (H16: (subst0 (minus x1 (s x2 -(S n0))) v x5 x6)).(\lambda (H17: (csubst0 (minus x1 (s x2 (S n0))) v x3 -x4)).(eq_ind_r C (CHead x3 x2 x5) (\lambda (c0: C).(or4 (drop (S n0) O (CHead -x0 (Flat f) t) c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda -(u: T).(\lambda (_: T).(eq C c0 (CHead e0 k0 u)))))) (\lambda (k0: -K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 -(Flat f) t) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda -(u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S n0))) v u -w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C c0 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda -(_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) -(CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 k0 u))))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 w))))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s -(Flat f) x1) (s k0 (S n0))) v e1 e2))))))))) (or4_intro3 (drop (S n0) O -(CHead x0 (Flat f) t) (CHead x3 x2 x5)) (ex3_4 K C T T (\lambda (k0: -K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x5) -(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 k0 w)))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 x2 -x5) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 u)))))) -(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 -(minus (s (Flat f) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq -C (CHead x3 x2 x5) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 -(Flat f) t) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda -(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 -(S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S -n0))) v e1 e2))))))) (ex4_5_intro K C C T T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x5) -(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) -(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S -n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1 -e2)))))) x2 x3 x4 x5 x6 (refl_equal C (CHead x3 x2 x5)) (drop_drop (Flat f) -n0 x0 (CHead x4 x2 x6) H15 t) H16 H17)) e H14)))))))))) H13)) H12)))))) k -(drop_gen_drop k c e t n0 H2) H7 H8) i H4))) c2 H5)))))) H3)) (\lambda (H3: -(ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s -k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead -c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t -u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c -c3)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: -nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: -nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda -(j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: -nat).(csubst0 j v c c3)))) (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda -(k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 -u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O c2 (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S n0))) v u -w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k0 u)))))) -(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 -(minus i (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k0 w))))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k0 -(S n0))) v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: C).(\lambda (x2: -nat).(\lambda (H4: (eq nat i (s k x2))).(\lambda (H5: (eq C c2 (CHead x1 k -x0))).(\lambda (_: (subst0 x2 v t x0)).(\lambda (H7: (csubst0 x2 v c -x1)).(eq_ind_r C (CHead x1 k x0) (\lambda (c0: C).(or4 (drop (S n0) O c0 e) -(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: -T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda -(_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e0 k0 w)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S -n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda -(_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c0 (CHead e2 k0 -u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus i (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq -C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 k0 w))))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k0 -(S n0))) v e1 e2))))))))) (let H8 \def (eq_ind nat i (\lambda (n1: -nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall -(e0: C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T -(\lambda (k0: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 -(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n1 (s k0 (S -n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 (CHead -e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus n1 (s k0 (S n0))) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq -C e0 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c3 (CHead e2 k0 w))))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus n1 (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 -(s k0 (S n0))) v0 e1 e2)))))))))))))) H0 (s k x2) H4) in (let H9 \def (eq_ind -nat i (\lambda (n1: nat).(lt (S n0) n1)) H (s k x2) H4) in (eq_ind_r nat (s k -x2) (\lambda (n1: nat).(or4 (drop (S n0) O (CHead x1 k x0) e) (ex3_4 K C T T -(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 k x0) (CHead e0 k0 w)))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus n1 (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O -(CHead x1 k x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n1 (s k0 (S n0))) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x1 k x0) (CHead e2 k0 w))))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus n1 (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 (s k0 -(S n0))) v e1 e2))))))))) (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to -(((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x2) v0 c c3) \to (\forall -(e0: C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T -(\lambda (k1: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 -(CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k1 w)))))) (\lambda (k1: -K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 x2) -(s k1 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k1 u)))))) (\lambda -(k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 -(CHead e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus (s k0 x2) (s k1 (S n0))) v0 e1 e2)))))) -(ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq C e0 (CHead e1 k1 u))))))) (\lambda (k1: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O c3 (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 x2) (s k1 (S n0))) v0 -u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (minus (s k0 x2) (s k1 (S n0))) v0 e1 -e2)))))))))))))) \to ((lt (S n0) (s k0 x2)) \to (or4 (drop (S n0) O (CHead x1 -k0 x0) e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda -(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 k0 x0) -(CHead e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus (s k0 x2) (s k1 (S n0))) v u w)))))) (ex3_4 -K C C T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq -C e (CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O (CHead x1 k0 x0) (CHead e2 k1 u)))))) -(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 -(minus (s k0 x2) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 k0 x0) (CHead e2 -k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus (s k0 x2) (s k1 (S n0))) v u w)))))) -(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus (s k0 x2) (s k1 (S n0))) v e1 e2)))))))))))) (\lambda -(b: B).(\lambda (H10: (drop (r (Bind b) n0) O c e)).(\lambda (_: ((\forall -(c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x2) v0 c c3) \to (\forall (e0: -C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T -(\lambda (k0: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 -(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) -x2) (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 -(CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v0 e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u))))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) -x2) (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 -(S n0))) v0 e1 e2))))))))))))))).(\lambda (H12: (lt (S n0) (s (Bind b) -x2))).(let H13 \def (IHn x2 (le_S_n (S n0) x2 H12) c x1 v H7 e H10) in -(or4_ind (drop n0 O x1 e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: -K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x1 (CHead e0 -k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus x2 (s k0 n0)) v u w)))))) (ex3_4 K C C T (\lambda (k0: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 -u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop n0 O x1 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda 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K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 -(Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s -(Bind b) x2) (s k0 (S n0))) v e1 e2)))))))) (\lambda (H14: (drop n0 O x1 -e)).(or4_intro0 (drop (S n0) O (CHead x1 (Bind b) x0) e) (ex3_4 K C T T -(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 -u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0: -K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind -b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) -(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S -n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 -e2))))))) (drop_drop (Bind b) n0 x1 e H14 x0))) (\lambda (H14: (ex3_4 K C T T -(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop n0 O x1 (CHead e0 k0 w)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x2 (s k0 -n0)) v u w))))))).(ex3_4_ind K C T T (\lambda (k0: K).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: -K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x1 (CHead e0 -k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus x2 (s k0 n0)) v u w))))) (or4 (drop (S n0) O (CHead x1 -(Bind b) x0) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda -(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) -(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) -(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 -u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C -T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead -x1 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) -x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 -(S n0))) v e1 e2)))))))) (\lambda (x3: K).(\lambda (x4: C).(\lambda (x5: -T).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x4 x3 x5))).(\lambda (H16: -(drop n0 O x1 (CHead x4 x3 x6))).(\lambda (H17: (subst0 (minus x2 (s x3 n0)) -v x5 x6)).(eq_ind_r C (CHead x4 x3 x5) (\lambda (c0: C).(or4 (drop (S n0) O -(CHead x1 (Bind b) x0) c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 k0 u)))))) (\lambda -(k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O -(CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S -n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u: T).(eq C c0 (CHead e1 k0 u)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 -(Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 k0 u))))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s -(Bind b) x2) (s k0 (S n0))) v e1 e2))))))))) (or4_intro1 (drop (S n0) O -(CHead x1 (Bind b) x0) (CHead 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K C C T -(\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop n0 O x1 (CHead e2 k0 u)))))) (\lambda (k0: -K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus x2 (s k0 -n0)) v e1 e2))))))).(ex3_4_ind K C C T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O x1 (CHead e2 -k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus x2 (s k0 n0)) v e1 e2))))) (or4 (drop (S n0) O (CHead x1 -(Bind b) x0) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda -(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) -(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) -(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 -u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C -T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead -x1 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) -x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 -(S n0))) v e1 e2)))))))) (\lambda (x3: K).(\lambda (x4: C).(\lambda (x5: -C).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x4 x3 x6))).(\lambda (H16: -(drop n0 O x1 (CHead x5 x3 x6))).(\lambda (H17: (csubst0 (minus x2 (s x3 n0)) -v x4 x5)).(eq_ind_r C (CHead x4 x3 x6) (\lambda (c0: C).(or4 (drop (S n0) O -(CHead x1 (Bind b) x0) c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 k0 u)))))) (\lambda -(k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O -(CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S -n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u: T).(eq C c0 (CHead e1 k0 u)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 -(Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 (minus (s 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T).(subst0 -(minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x4 x3 -x6) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 u)))))) -(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 -(minus (s (Bind b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq -C (CHead x4 x3 x6) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 -(Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda -(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 -(S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S -n0))) v e1 e2))))))) (ex3_4_intro K C C T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x4 x3 x6) (CHead e1 k0 -u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0: -K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind -b) x2) (s k0 (S n0))) v e1 e2))))) x3 x4 x5 x6 (refl_equal C (CHead x4 x3 -x6)) (drop_drop (Bind b) n0 x1 (CHead x5 x3 x6) H16 x0) (eq_ind_r nat (S (s -x3 n0)) (\lambda (n1: nat).(csubst0 (minus (s (Bind b) x2) n1) v x4 x5)) H17 -(s x3 (S n0)) (s_S x3 n0)))) e H15)))))))) H14)) (\lambda (H14: (ex4_5 K C C -T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x1 (CHead -e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: 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(S n0))) v e1 -e2)))))))) (\lambda (x3: K).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: -T).(\lambda (x7: T).(\lambda (H15: (eq C e (CHead x4 x3 x6))).(\lambda (H16: -(drop n0 O x1 (CHead x5 x3 x7))).(\lambda (H17: (subst0 (minus x2 (s x3 n0)) -v x6 x7)).(\lambda (H18: (csubst0 (minus x2 (s x3 n0)) v x4 x5)).(eq_ind_r C -(CHead x4 x3 x6) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Bind b) x0) -c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda -(_: T).(eq C c0 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) -(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) -(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c0 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 -u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C -T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(eq C c0 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead -x1 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) -x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 -(S n0))) v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead x1 (Bind b) x0) -(CHead x4 x3 x6)) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda -(u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6) (CHead e0 k0 u)))))) (\lambda -(k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O -(CHead x1 (Bind 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C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 -e2))))))) (ex4_5_intro K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6) (CHead e1 k0 -u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 -w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) -(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 e2)))))) x3 x4 x5 -x6 x7 (refl_equal C (CHead x4 x3 x6)) (drop_drop (Bind b) n0 x1 (CHead x5 x3 -x7) H16 x0) (eq_ind_r nat (S (s x3 n0)) (\lambda (n1: nat).(subst0 (minus (s -(Bind b) x2) n1) v x6 x7)) H17 (s x3 (S n0)) (s_S x3 n0)) (eq_ind_r nat (S (s -x3 n0)) (\lambda (n1: nat).(csubst0 (minus (s (Bind b) x2) n1) v x4 x5)) H18 -(s x3 (S n0)) (s_S x3 n0)))) e H15)))))))))) H14)) H13)))))) (\lambda (f: -F).(\lambda (H10: (drop (r (Flat f) n0) O c e)).(\lambda (H11: ((\forall (c3: -C).(\forall (v0: T).((csubst0 (s (Flat f) x2) v0 c c3) \to (\forall (e0: -C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T -(\lambda (k0: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 -(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) -x2) (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 -(CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v0 e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u))))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) -x2) (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 -(S n0))) v0 e1 e2))))))))))))))).(\lambda (_: (lt (S n0) (s (Flat f) -x2))).(let H13 \def (H11 x1 v H7 e H10) in (or4_ind (drop (S n0) O x1 e) -(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: -T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda -(_: T).(\lambda (w: T).(drop (S n0) O x1 (CHead e0 k0 w)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x2 (s k0 (S -n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda -(_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O x1 (CHead e2 k0 -u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus x2 (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq -C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x1 (CHead e2 k0 w))))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus x2 (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus x2 -(s k0 (S n0))) v e1 e2))))))) (or4 (drop (S n0) O (CHead x1 (Flat f) x0) e) -(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: -T).(eq C e 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T).(subst0 (minus (s (Flat f) x2) (s k0 (S -n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 -e2)))))))) (\lambda (H14: (drop (S n0) O x1 e)).(or4_intro0 (drop (S n0) O -(CHead x1 (Flat f) x0) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: -K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 -(Flat f) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda -(u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u -w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0) -(CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus (s (Flat f) 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(\lambda -(k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 -u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O x1 (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x2 (s k0 (S n0))) v u -w))))) (or4 (drop (S n0) O (CHead x1 (Flat f) x0) e) (ex3_4 K C T T (\lambda -(k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 -u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 k0 w)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) -x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 -(Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 w))))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s -(Flat f) x2) (s k0 (S n0))) v e1 e2)))))))) (\lambda (x3: K).(\lambda (x4: -C).(\lambda (x5: T).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x4 x3 -x5))).(\lambda (H16: (drop (S n0) O x1 (CHead x4 x3 x6))).(\lambda (H17: -(subst0 (minus x2 (s x3 (S n0))) v x5 x6)).(eq_ind_r C (CHead x4 x3 x5) -(\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Flat f) x0) c0) (ex3_4 K C T -T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 -(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 k0 w)))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 -k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: -K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat -f) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 -(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) -(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S -n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 -e2))))))))) (or4_intro1 (drop (S n0) O (CHead x1 (Flat f) x0) (CHead x4 x3 -x5)) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C (CHead x4 x3 x5) (CHead e0 k0 u)))))) (\lambda (k0: -K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 -(Flat f) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda -(u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u -w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C (CHead x4 x3 x5) (CHead e1 k0 u)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 -(Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x5) (CHead e1 k0 -u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 -w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) -(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2))))))) -(ex3_4_intro K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C (CHead x4 x3 x5) (CHead e0 k0 u)))))) (\lambda (k0: -K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 -(Flat f) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda -(u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u -w))))) x3 x4 x5 x6 (refl_equal C (CHead x4 x3 x5)) (drop_drop (Flat f) n0 x1 -(CHead x4 x3 x6) H16 x0) H17)) e H15)))))))) H14)) (\lambda (H14: (ex3_4 K C -C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O x1 (CHead e2 k0 u)))))) (\lambda (k0: -K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus x2 (s k0 -(S n0))) v e1 e2))))))).(ex3_4_ind K C C T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O x1 (CHead -e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus x2 (s k0 (S n0))) v e1 e2))))) (or4 (drop (S n0) O (CHead -x1 (Flat f) x0) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda -(u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda -(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) -(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) -(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 -u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C -T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead -x1 (Flat f) x0) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) -x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 -(S n0))) v e1 e2)))))))) (\lambda (x3: K).(\lambda (x4: C).(\lambda (x5: -C).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x4 x3 x6))).(\lambda (H16: -(drop (S n0) O x1 (CHead x5 x3 x6))).(\lambda (H17: (csubst0 (minus x2 (s x3 -(S n0))) v x4 x5)).(eq_ind_r C (CHead x4 x3 x6) (\lambda (c0: C).(or4 (drop -(S n0) O (CHead x1 (Flat f) x0) c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda -(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 k0 u)))))) -(\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O (CHead x1 (Flat f) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda -(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 -(S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 k0 u)))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O -(CHead x1 (Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S -n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 k0 -u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 -w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) -(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2))))))))) -(or4_intro2 (drop (S n0) O (CHead x1 (Flat f) x0) (CHead x4 x3 x6)) (ex3_4 K -C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C -(CHead x4 x3 x6) (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) -(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) -(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C (CHead x4 x3 x6) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0) -(CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6) (CHead e1 k0 -u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 -w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) -(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2))))))) -(ex3_4_intro K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C (CHead x4 x3 x6) (CHead e1 k0 u)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 -(Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 -e2))))) x3 x4 x5 x6 (refl_equal C (CHead x4 x3 x6)) (drop_drop (Flat f) n0 x1 -(CHead x5 x3 x6) H16 x0) H17)) e H15)))))))) H14)) (\lambda (H14: (ex4_5 K C -C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x1 -(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x2 (s k0 (S n0))) v u -w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (minus x2 (s k0 (S n0))) v e1 -e2)))))))).(ex4_5_ind K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O x1 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x2 (s k0 (S -n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 (minus x2 (s k0 (S n0))) v e1 e2)))))) (or4 -(drop (S n0) O (CHead x1 (Flat f) x0) e) (ex3_4 K C T T (\lambda (k0: -K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 -u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 k0 w)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) -x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 -(Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 w))))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s -(Flat f) x2) (s k0 (S n0))) v e1 e2)))))))) (\lambda (x3: K).(\lambda (x4: -C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (x7: T).(\lambda (H15: (eq C e -(CHead x4 x3 x6))).(\lambda (H16: (drop (S n0) O x1 (CHead x5 x3 -x7))).(\lambda (H17: (subst0 (minus x2 (s x3 (S n0))) v x6 x7)).(\lambda -(H18: (csubst0 (minus x2 (s x3 (S n0))) v x4 x5)).(eq_ind_r C (CHead x4 x3 -x6) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Flat f) x0) c0) (ex3_4 K -C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C -c0 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 k0 w)))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 -k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: -K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat -f) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 -(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) -(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S -n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 -e2))))))))) (or4_intro3 (drop (S n0) O (CHead x1 (Flat f) x0) (CHead x4 x3 -x6)) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C (CHead x4 x3 x6) (CHead e0 k0 u)))))) (\lambda (k0: -K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 -(Flat f) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda -(u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u -w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C (CHead x4 x3 x6) (CHead e1 k0 u)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 -(Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6) (CHead e1 k0 -u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 -w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) -(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2))))))) -(ex4_5_intro K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6) (CHead e1 k0 -u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 -w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) -(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2)))))) x3 x4 x5 -x6 x7 (refl_equal C (CHead x4 x3 x6)) (drop_drop (Flat f) n0 x1 (CHead x5 x3 -x7) H16 x0) H17 H18)) e H15)))))))))) H14)) H13)))))) k (drop_gen_drop k c e -t n0 H2) H8 H9) i H4))) c2 H5)))))))) H3)) (csubst0_gen_head k c c2 t v i -H1))))))))))) c1)))))) n). -(* COMMENTS -Initial nodes: 39886 -END *) - -theorem csubst0_drop_eq: - \forall (n: nat).(\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 -n v c1 c2) \to (\forall (e: C).((drop n O c1 e) \to (or4 (drop n O c2 e) -(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: -T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e0 (Flat f) w)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u -w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n O c2 (CHead e2 -(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 -(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(drop n O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O -v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))) -\def - \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (c1: C).(\forall (c2: -C).(\forall (v: T).((csubst0 n0 v c1 c2) \to (\forall (e: C).((drop n0 O c1 -e) \to (or4 (drop n0 O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O -c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat -f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop n0 O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: -T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c2 (CHead e2 -(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2))))))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda -(H: (csubst0 O v c1 c2)).(\lambda (e: C).(\lambda (H0: (drop O O c1 -e)).(eq_ind C c1 (\lambda (c: C).(or4 (drop O O c2 c) (ex3_4 F C T T (\lambda -(f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 -(Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda -(w: T).(drop O O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c -(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop O O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) -(ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq C c (CHead e1 (Flat f) u))))))) (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O -c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2))))))))) (insert_eq nat O (\lambda (n0: nat).(csubst0 n0 v c1 c2)) -(\lambda (n0: nat).(or4 (drop n0 n0 c2 c1) (ex3_4 F C T T (\lambda (f: -F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c1 (CHead e0 (Flat -f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(drop n0 n0 c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v u w)))))) (ex3_4 F C C T -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c1 -(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop n0 n0 c2 (CHead e2 (Flat f) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 v e1 -e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C c1 (CHead e1 (Flat f) u))))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop n0 n0 c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v u w)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 n0 v e1 e2))))))))) (\lambda (y: nat).(\lambda (H1: (csubst0 -y v c1 c2)).(csubst0_ind (\lambda (n0: nat).(\lambda (t: T).(\lambda (c: -C).(\lambda (c0: C).((eq nat n0 O) \to (or4 (drop n0 n0 c0 c) (ex3_4 F C T T -(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c -(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop n0 n0 c0 (CHead e0 (Flat f) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 t u w)))))) -(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop n0 n0 c0 (CHead e2 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 -t e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Flat f) u))))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop n0 n0 c0 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 t u w)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 n0 t e1 e2))))))))))))) (\lambda (k: K).(K_ind (\lambda (k0: -K).(\forall (i: nat).(\forall (v0: T).(\forall (u1: T).(\forall (u2: -T).((subst0 i v0 u1 u2) \to (\forall (c: C).((eq nat (s k0 i) O) \to (or4 -(drop (s k0 i) (s k0 i) (CHead c k0 u2) (CHead c k0 u1)) (ex3_4 F C T T -(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead -c k0 u1) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda -(_: T).(\lambda (w: T).(drop (s k0 i) (s k0 i) (CHead c k0 u2) (CHead e0 -(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (s k0 i) v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c k0 u1) (CHead e1 (Flat f) -u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (s k0 i) (s k0 i) (CHead c k0 u2) (CHead e2 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (s -k0 i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c k0 u1) -(CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) (s k0 i) (CHead c k0 u2) -(CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (s k0 i) v0 u w)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -(s k0 i) v0 e1 e2)))))))))))))))) (\lambda (b: B).(\lambda (i: nat).(\lambda -(v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0 i v0 u1 -u2)).(\lambda (c: C).(\lambda (H3: (eq nat (S i) O)).(let H4 \def (eq_ind nat -(S i) (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with -[O \Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind (or4 -(drop (S i) (S i) (CHead c (Bind b) u2) (CHead c (Bind b) u1)) (ex3_4 F C T T -(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead -c (Bind b) u1) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(drop (S i) (S i) (CHead c (Bind b) u2) -(CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (S i) v0 u w)))))) (ex3_4 F C C T (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c (Bind b) -u1) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S i) (S i) (CHead c (Bind b) u2) (CHead e2 (Flat f) -u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (S i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c (Bind -b) u1) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S i) (S i) (CHead c (Bind b) -u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (S i) v0 u w)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -(S i) v0 e1 e2)))))))) H4)))))))))) (\lambda (f: F).(\lambda (i: -nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 -i v0 u1 u2)).(\lambda (c: C).(\lambda (H3: (eq nat i O)).(let H4 \def (eq_ind -nat i (\lambda (n0: nat).(subst0 n0 v0 u1 u2)) H2 O H3) in (eq_ind_r nat O -(\lambda (n0: nat).(or4 (drop n0 n0 (CHead c (Flat f) u2) (CHead c (Flat f) -u1)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C (CHead c (Flat f) u1) (CHead e0 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0 -(CHead c (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda -(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v0 u w)))))) (ex3_4 F C C T -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C -(CHead c (Flat f) u1) (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda -(_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 n0 (CHead c (Flat f) u2) -(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C -(CHead c (Flat f) u1) (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0 (CHead c -(Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v0 u w)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 n0 v0 e1 e2))))))))) (or4_intro1 (drop O O (CHead c (Flat f) -u2) (CHead c (Flat f) u1)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c (Flat f) u1) (CHead e0 -(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda -(w: T).(drop O O (CHead c (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) -(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C (CHead c (Flat f) u1) (CHead e1 (Flat f0) u)))))) (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O (CHead c (Flat -f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda -(f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq -C (CHead c (Flat f) u1) (CHead e1 (Flat f0) u))))))) (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O -(CHead c (Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda -(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v0 e1 e2))))))) (ex3_4_intro F C T T (\lambda (f0: -F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c (Flat f) -u1) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda -(_: T).(\lambda (w: T).(drop O O (CHead c (Flat f) u2) (CHead e0 (Flat f0) -w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v0 u w))))) f c u1 u2 (refl_equal C (CHead c (Flat f) u1)) -(drop_refl (CHead c (Flat f) u2)) H4)) i H3)))))))))) k)) (\lambda (k: -K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (c3: C).(\forall (c4: -C).(\forall (v0: T).((csubst0 i v0 c3 c4) \to ((((eq nat i O) \to (or4 (drop -i i c4 c3) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C c3 (CHead e0 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop i i c4 (CHead e0 -(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 i v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 (Flat f) u)))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop i i c4 -(CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 i v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 -(CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop i i c4 (CHead e2 (Flat f) w))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 i v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v0 e1 e2)))))))))) \to (\forall -(u: T).((eq nat (s k0 i) O) \to (or4 (drop (s k0 i) (s k0 i) (CHead c4 k0 u) -(CHead c3 k0 u)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda -(u0: T).(\lambda (_: T).(eq C (CHead c3 k0 u) (CHead e0 (Flat f) u0)))))) -(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 -i) (s k0 i) (CHead c4 k0 u) (CHead e0 (Flat f) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (s k0 i) v0 u0 -w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u0: T).(eq C (CHead c3 k0 u) (CHead e1 (Flat f) u0)))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s k0 -i) (s k0 i) (CHead c4 k0 u) (CHead e2 (Flat f) u0)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (s k0 i) v0 e1 -e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 k0 u) (CHead e1 (Flat f) -u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (s k0 i) (s k0 i) (CHead c4 k0 u) (CHead e2 (Flat f) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: -T).(\lambda (w: T).(subst0 (s k0 i) v0 u0 w)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (s k0 i) v0 -e1 e2))))))))))))))))) (\lambda (b: B).(\lambda (i: nat).(\lambda (c3: -C).(\lambda (c4: C).(\lambda (v0: T).(\lambda (_: (csubst0 i v0 c3 -c4)).(\lambda (_: (((eq nat i O) \to (or4 (drop i i c4 c3) (ex3_4 F C T T -(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 -(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop i i c4 (CHead e0 (Flat f) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w)))))) -(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c3 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop i i c4 (CHead e2 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f) u))))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop i i c4 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (u: T).(\lambda (H4: (eq nat -(S i) O)).(let H5 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee in nat -return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow -True])) I O H4) in (False_ind (or4 (drop (S i) (S i) (CHead c4 (Bind b) u) -(CHead c3 (Bind b) u)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: -C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 (Bind b) u) (CHead e0 -(Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda -(w: T).(drop (S i) (S i) (CHead c4 (Bind b) u) (CHead e0 (Flat f) w)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (S -i) v0 u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u0: T).(eq C (CHead c3 (Bind b) u) (CHead e1 (Flat f) u0)))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S i) -(S i) (CHead c4 (Bind b) u) (CHead e2 (Flat f) u0)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (S i) v0 e1 -e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 (Bind b) u) (CHead e1 -(Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(drop (S i) (S i) (CHead c4 (Bind b) u) (CHead e2 -(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda -(u0: T).(\lambda (w: T).(subst0 (S i) v0 u0 w)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (S i) v0 e1 -e2)))))))) H5))))))))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (c3: -C).(\lambda (c4: C).(\lambda (v0: T).(\lambda (H2: (csubst0 i v0 c3 -c4)).(\lambda (H3: (((eq nat i O) \to (or4 (drop i i c4 c3) (ex3_4 F C T T -(\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 -(CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop i i c4 (CHead e0 (Flat f0) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w)))))) -(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c3 (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop i i c4 (CHead e2 (Flat f0) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f0) u))))))) -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(w: T).(drop i i c4 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (u: T).(\lambda (H4: (eq nat i -O)).(let H5 \def (eq_ind nat i (\lambda (n0: nat).((eq nat n0 O) \to (or4 -(drop n0 n0 c4 c3) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda -(u0: T).(\lambda (_: T).(eq C c3 (CHead e0 (Flat f0) u0)))))) (\lambda (f0: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0 c4 (CHead e0 -(Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda -(w: T).(subst0 n0 v0 u0 w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u0: T).(eq C c3 (CHead e1 (Flat f0) u0)))))) -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop n0 -n0 c4 (CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda -(f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq -C c3 (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0 c4 (CHead e2 (Flat f0) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: -T).(\lambda (w: T).(subst0 n0 v0 u0 w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 n0 v0 e1 -e2)))))))))) H3 O H4) in (let H6 \def (eq_ind nat i (\lambda (n0: -nat).(csubst0 n0 v0 c3 c4)) H2 O H4) in (eq_ind_r nat O (\lambda (n0: -nat).(or4 (drop n0 n0 (CHead c4 (Flat f) u) (CHead c3 (Flat f) u)) (ex3_4 F C -T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C -(CHead c3 (Flat f) u) (CHead e0 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda -(e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0 (CHead c4 (Flat f) u) -(CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: -T).(\lambda (w: T).(subst0 n0 v0 u0 w)))))) (ex3_4 F C C T (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead c3 (Flat f) -u) (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u0: T).(drop n0 n0 (CHead c4 (Flat f) u) (CHead e2 (Flat -f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 (Flat f) -u) (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0 (CHead c4 (Flat f) u) -(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u0: T).(\lambda (w: T).(subst0 n0 v0 u0 w)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -n0 v0 e1 e2))))))))) (or4_intro2 (drop O O (CHead c4 (Flat f) u) (CHead c3 -(Flat f) u)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: -T).(\lambda (_: T).(eq C (CHead c3 (Flat f) u) (CHead e0 (Flat f0) u0)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O -(CHead c4 (Flat f) u) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda -(_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v0 u0 w)))))) (ex3_4 F C C -T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C -(CHead c3 (Flat f) u) (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda -(_: C).(\lambda (e2: C).(\lambda (u0: T).(drop O O (CHead c4 (Flat f) u) -(CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C -(CHead c3 (Flat f) u) (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 -(Flat f) u) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v0 u0 w)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v0 e1 e2))))))) (ex3_4_intro F C C T (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead c3 (Flat f) -u) (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u0: T).(drop O O (CHead c4 (Flat f) u) (CHead e2 (Flat f0) -u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v0 e1 e2))))) f c3 c4 u (refl_equal C (CHead c3 (Flat f) u)) -(drop_refl (CHead c4 (Flat f) u)) H6)) i H4)))))))))))) k)) (\lambda (k: -K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (v0: T).(\forall (u1: -T).(\forall (u2: T).((subst0 i v0 u1 u2) \to (\forall (c3: C).(\forall (c4: -C).((csubst0 i v0 c3 c4) \to ((((eq nat i O) \to (or4 (drop i i c4 c3) (ex3_4 -F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq -C c3 (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop i i c4 (CHead e0 (Flat f) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w)))))) -(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c3 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop i i c4 (CHead e2 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f) u))))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop i i c4 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 i v0 e1 e2)))))))))) \to ((eq nat (s k0 i) O) \to (or4 (drop -(s k0 i) (s k0 i) (CHead c4 k0 u2) (CHead c3 k0 u1)) (ex3_4 F C T T (\lambda -(f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 k0 -u1) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (s k0 i) (s k0 i) (CHead c4 k0 u2) (CHead e0 (Flat -f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (s k0 i) v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c3 k0 u1) (CHead e1 (Flat f) -u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (s k0 i) (s k0 i) (CHead c4 k0 u2) (CHead e2 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (s -k0 i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 k0 u1) -(CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) (s k0 i) (CHead c4 k0 u2) -(CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (s k0 i) v0 u w)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -(s k0 i) v0 e1 e2))))))))))))))))))) (\lambda (b: B).(\lambda (i: -nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0 -i v0 u1 u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (_: (csubst0 i v0 c3 -c4)).(\lambda (_: (((eq nat i O) \to (or4 (drop i i c4 c3) (ex3_4 F C T T -(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 -(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop i i c4 (CHead e0 (Flat f) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w)))))) -(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c3 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop i i c4 (CHead e2 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f) u))))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop i i c4 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (H5: (eq nat (S i) O)).(let H6 -\def (eq_ind nat (S i) (\lambda (ee: nat).(match ee in nat return (\lambda -(_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H5) -in (False_ind (or4 (drop (S i) (S i) (CHead c4 (Bind b) u2) (CHead c3 (Bind -b) u1)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C (CHead c3 (Bind b) u1) (CHead e0 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S i) -(S i) (CHead c4 (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (S i) v0 u w)))))) -(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C (CHead c3 (Bind b) u1) (CHead e1 (Flat f) u)))))) (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S i) (S i) (CHead -c4 (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (S i) v0 e1 e2)))))) (ex4_5 F C -C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(eq C (CHead c3 (Bind b) u1) (CHead e1 (Flat f) u))))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S i) (S i) (CHead c4 (Bind b) u2) (CHead e2 (Flat f) w))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (S i) v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (S i) v0 e1 e2)))))))) -H6))))))))))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (v0: T).(\lambda -(u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 i v0 u1 u2)).(\lambda (c3: -C).(\lambda (c4: C).(\lambda (H3: (csubst0 i v0 c3 c4)).(\lambda (H4: (((eq -nat i O) \to (or4 (drop i i c4 c3) (ex3_4 F C T T (\lambda (f0: F).(\lambda -(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e0 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop i i -c4 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 i v0 u w)))))) (ex3_4 F C C T (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 (Flat -f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop i i c4 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v0 e1 e2)))))) (ex4_5 F C C T -T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda -(_: T).(eq C c3 (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop i i c4 (CHead e2 -(Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (w: T).(subst0 i v0 u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v0 e1 -e2))))))))))).(\lambda (H5: (eq nat i O)).(let H6 \def (eq_ind nat i (\lambda -(n0: nat).((eq nat n0 O) \to (or4 (drop n0 n0 c4 c3) (ex3_4 F C T T (\lambda -(f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e0 -(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda -(w: T).(drop n0 n0 c4 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda -(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v0 u w)))))) (ex3_4 F C C T -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3 -(CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop n0 n0 c4 (CHead e2 (Flat f0) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 v0 e1 -e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f0) u))))))) -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(w: T).(drop n0 n0 c4 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda -(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v0 u -w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))))))) H4 O H5) in (let H7 \def -(eq_ind nat i (\lambda (n0: nat).(csubst0 n0 v0 c3 c4)) H3 O H5) in (let H8 -\def (eq_ind nat i (\lambda (n0: nat).(subst0 n0 v0 u1 u2)) H2 O H5) in -(eq_ind_r nat O (\lambda (n0: nat).(or4 (drop n0 n0 (CHead c4 (Flat f) u2) -(CHead c3 (Flat f) u1)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 (Flat f) u1) (CHead e0 -(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda -(w: T).(drop n0 n0 (CHead c4 (Flat f) u2) (CHead e0 (Flat f0) w)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v0 -u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 n0 -(CHead c4 (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F -C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(eq C (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u))))))) -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(w: T).(drop n0 n0 (CHead c4 (Flat f) u2) (CHead e2 (Flat f0) w))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 n0 v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 n0 v0 e1 e2))))))))) (or4_intro3 -(drop O O (CHead c4 (Flat f) u2) (CHead c3 (Flat f) u1)) (ex3_4 F C T T -(\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C -(CHead c3 (Flat f) u1) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda -(e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 (Flat f) u2) -(CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v0 u w)))))) (ex3_4 F C C T (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c3 (Flat f) -u1) (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u: T).(drop O O (CHead c4 (Flat f) u2) (CHead e2 (Flat f0) -u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 (Flat f) -u1) (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 (Flat f) u2) -(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v0 e1 e2))))))) (ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 (Flat f) -u1) (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 (Flat f) u2) -(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v0 e1 e2)))))) f c3 c4 u1 u2 (refl_equal C (CHead c3 (Flat f) u1)) -(drop_refl (CHead c4 (Flat f) u2)) H8 H7)) i H5))))))))))))))) k)) y v c1 c2 -H1))) H) e (drop_gen_refl c1 e H0)))))))) (\lambda (n0: nat).(\lambda (IHn: -((\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 n0 v c1 c2) \to -(\forall (e: C).((drop n0 O c1 e) \to (or4 (drop n0 O c2 e) (ex3_4 F C T T -(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop n0 O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O c2 (CHead e2 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O -v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop n0 O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2)))))))))))))))).(\lambda (c1: C).(C_ind (\lambda -(c: C).(\forall (c2: C).(\forall (v: T).((csubst0 (S n0) v c c2) \to (\forall -(e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 F C T T -(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 (Flat f) -u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) -u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O -v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))) (\lambda (n1: -nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (_: (csubst0 (S n0) v (CSort -n1) c2)).(\lambda (e: C).(\lambda (H0: (drop (S n0) O (CSort n1) -e)).(and3_ind (eq C e (CSort n1)) (eq nat (S n0) O) (eq nat O O) (or4 (drop -(S n0) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead -e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda -(w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) -O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda -(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C -e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2)))))))) (\lambda (H1: (eq C e (CSort n1))).(\lambda (H2: (eq nat (S n0) -O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n1) (\lambda (c: C).(or4 -(drop (S n0) O c2 c) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda -(u: T).(\lambda (_: T).(eq C c (CHead e0 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead -e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda -(w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Flat f) u)))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) -O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda -(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C -c (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2))))))))) (let H4 \def (eq_ind nat (S n0) (\lambda (ee: nat).(match ee in -nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) -\Rightarrow True])) I O H2) in (False_ind (or4 (drop (S n0) O c2 (CSort n1)) -(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: -T).(eq C (CSort n1) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 (Flat f) -w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C (CSort n1) (CHead e1 (Flat f) -u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: -T).(eq C (CSort n1) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead -e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2)))))))) H4)) e H1)))) (drop_gen_sort n1 (S n0) O e H0)))))))) (\lambda (c: -C).(\lambda (H: ((\forall (c2: C).(\forall (v: T).((csubst0 (S n0) v c c2) -\to (\forall (e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 -F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq -C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 (Flat f) -u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) -u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O -v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))))))).(\lambda (k: K).(\lambda -(t: T).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 (S n0) v -(CHead c k t) c2)).(\lambda (e: C).(\lambda (H1: (drop (S n0) O (CHead c k t) -e)).(or3_ind (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (S n0) (s -k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) -(\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2)))) (ex3_2 C nat (\lambda -(_: C).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (c3: C).(\lambda -(_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: -nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: -C).(\lambda (j: nat).(eq nat (S n0) (s k j))))) (\lambda (u2: T).(\lambda -(c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: -T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: -T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))) (or4 (drop (S -n0) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead -e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda -(w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) -O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda -(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C -e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2)))))))) (\lambda (H2: (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq -nat (S n0) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k -u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))))).(ex3_2_ind T -nat (\lambda (_: T).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (u2: -T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: -nat).(subst0 j v t u2))) (or4 (drop (S n0) O c2 e) (ex3_4 F C T T (\lambda -(f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 -(Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda -(w: T).(drop (S n0) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda -(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) -(ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda -(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H3: (eq nat -(S n0) (s k x1))).(\lambda (H4: (eq C c2 (CHead c k x0))).(\lambda (H5: -(subst0 x1 v t x0)).(eq_ind_r C (CHead c k x0) (\lambda (c0: C).(or4 (drop (S -n0) O c0 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead -e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda -(w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) -O c0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda -(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C -e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 (Flat f) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2))))))))) (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to ((eq nat (S -n0) (s k0 x1)) \to (or4 (drop (S n0) O (CHead c k0 x0) e) (ex3_4 F C T T -(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead c k0 x0) (CHead e0 (Flat f) w)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u -w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c -k0 x0) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda -(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C -e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c k0 x0) (CHead e2 -(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2))))))))))) (\lambda (b: B).(\lambda (H6: (drop (r (Bind b) n0) O c -e)).(\lambda (H7: (eq nat (S n0) (s (Bind b) x1))).(let H8 \def (f_equal nat -nat (\lambda (e0: nat).(match e0 in nat return (\lambda (_: nat).nat) with [O -\Rightarrow n0 | (S n1) \Rightarrow n1])) (S n0) (S x1) H7) in (let H9 \def -(eq_ind_r nat x1 (\lambda (n1: nat).(subst0 n1 v t x0)) H5 n0 H8) in -(or4_intro0 (drop (S n0) O (CHead c (Bind b) x0) e) (ex3_4 F C T T (\lambda -(f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 -(Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda -(w: T).(drop (S n0) O (CHead c (Bind b) x0) (CHead e0 (Flat f) w)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u -w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c -(Bind b) x0) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: -T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c -(Bind b) x0) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2))))))) (drop_drop (Bind b) n0 c e H6 x0))))))) -(\lambda (f: F).(\lambda (H6: (drop (r (Flat f) n0) O c e)).(\lambda (H7: (eq -nat (S n0) (s (Flat f) x1))).(let H8 \def (f_equal nat nat (\lambda (e0: -nat).e0) (S n0) (s (Flat f) x1) H7) in (let H9 \def (eq_ind_r nat x1 (\lambda -(n1: nat).(subst0 n1 v t x0)) H5 (S n0) H8) in (or4_intro0 (drop (S n0) O -(CHead c (Flat f) x0) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O (CHead c (Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) x0) -(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c (Flat f) x0) -(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2))))))) (drop_drop (Flat f) n0 c e H6 x0))))))) k (drop_gen_drop k c -e t n0 H1) H3) c2 H4)))))) H2)) (\lambda (H2: (ex3_2 C nat (\lambda (_: -C).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (c3: C).(\lambda (_: -nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j -v c c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) -(s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) -(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (or4 (drop (S n0) O -c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead -e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda -(w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) -O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda -(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C -e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2)))))))) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H3: (eq nat (S n0) -(s k x1))).(\lambda (H4: (eq C c2 (CHead x0 k t))).(\lambda (H5: (csubst0 x1 -v c x0)).(eq_ind_r C (CHead x0 k t) (\lambda (c0: C).(or4 (drop (S n0) O c0 -e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda -(_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e0 (Flat f) -w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) -O c0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda -(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C -e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 (Flat f) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2))))))))) (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to ((eq nat (S -n0) (s k0 x1)) \to (or4 (drop (S n0) O (CHead x0 k0 t) e) (ex3_4 F C T T -(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x0 k0 t) (CHead e0 (Flat f) w)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u -w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 -k0 t) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda -(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C -e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 k0 t) (CHead e2 -(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2))))))))))) (\lambda (b: B).(\lambda (H6: (drop (r (Bind b) n0) O c -e)).(\lambda (H7: (eq nat (S n0) (s (Bind b) x1))).(let H8 \def (f_equal nat -nat (\lambda (e0: nat).(match e0 in nat return (\lambda (_: nat).nat) with [O -\Rightarrow n0 | (S n1) \Rightarrow n1])) (S n0) (S x1) H7) in (let H9 \def -(eq_ind_r nat x1 (\lambda (n1: nat).(csubst0 n1 v c x0)) H5 n0 H8) in (let -H10 \def (IHn c x0 v H9 e H6) in (or4_ind (drop n0 O x0 e) (ex3_4 F C T T -(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop n0 O x0 (CHead e0 (Flat f) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O x0 (CHead e2 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O -v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop n0 O x0 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead x0 (Bind b) t) e) -(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: -T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) -(CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat -f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) u)))))) (\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 -e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H11: -(drop n0 O x0 e)).(or4_intro0 (drop (S n0) O (CHead x0 (Bind b) t) e) (ex3_4 -F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq -C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f) -w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) -O (CHead x0 (Bind b) t) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C -C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O -v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Bind b) n0 x0 e H11 -t))) (\lambda (H11: (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda -(u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x0 (CHead e0 -(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w))))))).(ex3_4_ind F C T T (\lambda (f: F).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O -x0 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w))))) (or4 (drop (S n0) O (CHead x0 (Bind -b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 -(Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) -u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) -u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: -T).(\lambda (H12: (eq C e (CHead x3 (Flat x2) x4))).(\lambda (H13: (drop n0 O -x0 (CHead x3 (Flat x2) x5))).(\lambda (H14: (subst0 O v x4 x5)).(eq_ind_r C -(CHead x3 (Flat x2) x4) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x0 (Bind -b) t) c0) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 -(Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 -(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) -u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat -f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 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(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C -(CHead x3 (Flat x2) x4) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead -x0 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C T T (\lambda (f: -F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) -x4) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f) -w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w))))) x2 x3 x4 x5 (refl_equal C (CHead x3 (Flat x2) x4)) -(drop_drop (Bind b) n0 x0 (CHead x3 (Flat x2) x5) H13 t) H14)) e H12)))))))) -H11)) (\lambda (H11: (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O x0 (CHead e2 -(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O -x0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2))))) (or4 (drop (S n0) O (CHead x0 -(Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 -(Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) -u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) -u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: -T).(\lambda (H12: (eq C e (CHead x3 (Flat x2) x5))).(\lambda (H13: (drop n0 O -x0 (CHead x4 (Flat x2) x5))).(\lambda (H14: (csubst0 O v x3 x4)).(eq_ind_r C -(CHead x3 (Flat x2) x5) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x0 (Bind -b) t) c0) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 -(Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 -(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) -u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat -f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2))))))))) (or4_intro2 (drop (S n0) O (CHead x0 (Bind b) t) (CHead x3 (Flat -x2) x5)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) -O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda -(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead -x3 (Flat x2) x5) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) -(CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C -(CHead x3 (Flat x2) x5) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead -x0 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C C T (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 (Flat x2) -x5) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) -u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v e1 e2))))) x2 x3 x4 x5 (refl_equal C (CHead x3 (Flat x2) x5)) -(drop_drop (Bind b) n0 x0 (CHead x4 (Flat x2) x5) H13 t) H14)) e H12)))))))) -H11)) (\lambda (H11: (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) -u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop n0 O x0 (CHead e2 (Flat f) w))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O -v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda -(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C -e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x0 (CHead e2 (Flat f) w))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) -O (CHead x0 (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) -O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda -(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) -u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) -u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: -T).(\lambda (x6: T).(\lambda (H12: (eq C e (CHead x3 (Flat x2) x5))).(\lambda -(H13: (drop n0 O x0 (CHead x4 (Flat x2) x6))).(\lambda (H14: (subst0 O v x5 -x6)).(\lambda (H15: (csubst0 O v x3 x4)).(eq_ind_r C (CHead x3 (Flat x2) x5) -(\lambda (c0: C).(or4 (drop (S n0) O (CHead x0 (Bind b) t) c0) (ex3_4 F C T T -(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 -(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f) -w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Flat f) u)))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) -O (CHead x0 (Bind b) t) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C -C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat f) u))))))) (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O -v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O -(CHead x0 (Bind b) t) (CHead x3 (Flat x2) x5)) (ex3_4 F C T T (\lambda (f: -F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) -x5) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f) -w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1 -(Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O -v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1 -(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat -f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) -(ex4_5_intro F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1 -(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat -f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) -x2 x3 x4 x5 x6 (refl_equal C (CHead x3 (Flat x2) x5)) (drop_drop (Bind b) n0 -x0 (CHead x4 (Flat x2) x6) H13 t) H14 H15)) e H12)))))))))) H11)) H10))))))) -(\lambda (f: F).(\lambda (H6: (drop (r (Flat f) n0) O c e)).(\lambda (H7: (eq -nat (S n0) (s (Flat f) x1))).(let H8 \def (f_equal nat nat (\lambda (e0: -nat).e0) (S n0) (s (Flat f) x1) H7) in (let H9 \def (eq_ind_r nat x1 (\lambda -(n1: nat).(csubst0 n1 v c x0)) H5 (S n0) H8) in (let H10 \def (H x0 v H9 e -H6) in (or4_ind (drop (S n0) O x0 e) (ex3_4 F C T T (\lambda (f0: F).(\lambda -(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O x0 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda -(u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat -f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O x0 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C -C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O x0 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead x0 (Flat f) t) e) -(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: -T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) -(CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat -f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u)))))) (\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 -e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H11: -(drop (S n0) O x0 e)).(or4_intro0 (drop (S n0) O (CHead x0 (Flat f) t) e) -(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: -T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) -(CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat -f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u)))))) (\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 -e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Flat -f) n0 x0 e H11 t))) (\lambda (H11: (ex3_4 F C T T (\lambda (f0: F).(\lambda -(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O x0 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda -(u: T).(\lambda (w: T).(subst0 O v u w))))))).(ex3_4_ind F C T T (\lambda -(f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 -(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda -(w: T).(drop (S n0) O x0 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda -(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))) (or4 (drop (S n0) -O (CHead x0 (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) -(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) -(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: T).(\lambda -(x5: T).(\lambda (H12: (eq C e (CHead x3 (Flat x2) x4))).(\lambda (H13: (drop -(S n0) O x0 (CHead x3 (Flat x2) x5))).(\lambda (H14: (subst0 O v x4 -x5)).(eq_ind_r C (CHead x3 (Flat x2) x4) (\lambda (c0: C).(or4 (drop (S n0) O -(CHead x0 (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c0 (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) -(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 -(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) -(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead x0 (Flat f) t) (CHead x3 -(Flat x2) x4)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x4) (CHead e0 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C (CHead x3 (Flat x2) x4) (CHead e1 (Flat f0) u)))))) (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 -(Flat f) t) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda -(_: T).(eq C (CHead x3 (Flat x2) x4) (CHead e1 (Flat f0) u))))))) (\lambda -(f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C -T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C -(CHead x3 (Flat x2) x4) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda -(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) -(CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w))))) x2 x3 x4 x5 (refl_equal C (CHead x3 -(Flat x2) x4)) (drop_drop (Flat f) n0 x0 (CHead x3 (Flat x2) x5) H13 t) H14)) -e H12)))))))) H11)) (\lambda (H11: (ex3_4 F C C T (\lambda (f0: F).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S -n0) O x0 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C -C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O x0 (CHead e2 (Flat f0) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) -(or4 (drop (S n0) O (CHead x0 (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: -F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat -f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) -(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) -(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda -(x5: T).(\lambda (H12: (eq C e (CHead x3 (Flat x2) x5))).(\lambda (H13: (drop -(S n0) O x0 (CHead x4 (Flat x2) x5))).(\lambda (H14: (csubst0 O v x3 -x4)).(eq_ind_r C (CHead x3 (Flat x2) x5) (\lambda (c0: C).(or4 (drop (S n0) O -(CHead x0 (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c0 (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) -(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 -(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) -(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead x0 (Flat f) t) (CHead x3 -(Flat x2) x5)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C (CHead x3 (Flat x2) x5) (CHead e1 (Flat f0) u)))))) (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 -(Flat f) t) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda -(_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1 (Flat f0) u))))))) (\lambda -(f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C -C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C -(CHead x3 (Flat x2) x5) (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda -(_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) -(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2))))) x2 x3 x4 x5 (refl_equal C (CHead -x3 (Flat x2) x5)) (drop_drop (Flat f) n0 x0 (CHead x4 (Flat x2) x5) H13 t) -H14)) e H12)))))))) H11)) (\lambda (H11: (ex4_5 F C C T T (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x0 (CHead e2 (Flat f0) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2)))))))).(ex4_5_ind F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(w: T).(drop (S n0) O x0 (CHead e2 (Flat f0) w))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O -v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) O (CHead x0 -(Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 -(Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) -u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat -f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: -T).(\lambda (x6: T).(\lambda (H12: (eq C e (CHead x3 (Flat x2) x5))).(\lambda -(H13: (drop (S n0) O x0 (CHead x4 (Flat x2) x6))).(\lambda (H14: (subst0 O v -x5 x6)).(\lambda (H15: (csubst0 O v x3 x4)).(eq_ind_r C (CHead x3 (Flat x2) -x5) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x0 (Flat f) t) c0) (ex3_4 F C -T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C -c0 (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) -w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S -n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) -(ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat f0) u))))))) (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O -v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O -(CHead x0 (Flat f) t) (CHead x3 (Flat x2) x5)) (ex3_4 F C T T (\lambda (f0: -F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) -x5) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda -(_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat -f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1 -(Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(u: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O -v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1 -(Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat -f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) -(ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1 -(Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat -f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) -x2 x3 x4 x5 x6 (refl_equal C (CHead x3 (Flat x2) x5)) (drop_drop (Flat f) n0 -x0 (CHead x4 (Flat x2) x6) H13 t) H14 H15)) e H12)))))))))) H11)) H10))))))) -k (drop_gen_drop k c e t n0 H1) H3) c2 H4)))))) H2)) (\lambda (H2: (ex4_3 T C -nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k -j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 -k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t -u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c -c3)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: -nat).(eq nat (S n0) (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda -(_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: -C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: -C).(\lambda (j: nat).(csubst0 j v c c3)))) (or4 (drop (S n0) O c2 e) (ex3_4 F -C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C -e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 (Flat f) -u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) -u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O -v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: -C).(\lambda (x2: nat).(\lambda (H3: (eq nat (S n0) (s k x2))).(\lambda (H4: -(eq C c2 (CHead x1 k x0))).(\lambda (H5: (subst0 x2 v t x0)).(\lambda (H6: -(csubst0 x2 v c x1)).(eq_ind_r C (CHead x1 k x0) (\lambda (c0: C).(or4 (drop -(S n0) O c0 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead -e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda -(w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) -O c0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda -(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C -e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 (Flat f) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2))))))))) (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to ((eq nat (S -n0) (s k0 x2)) \to (or4 (drop (S n0) O (CHead x1 k0 x0) e) (ex3_4 F C T T -(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 k0 x0) (CHead e0 (Flat f) w)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u -w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 -k0 x0) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda -(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C -e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 k0 x0) (CHead e2 -(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2))))))))))) (\lambda (b: B).(\lambda (H7: (drop (r (Bind b) n0) O c -e)).(\lambda (H8: (eq nat (S n0) (s (Bind b) x2))).(let H9 \def (f_equal nat -nat (\lambda (e0: nat).(match e0 in nat return (\lambda (_: nat).nat) with [O -\Rightarrow n0 | (S n1) \Rightarrow n1])) (S n0) (S x2) H8) in (let H10 \def -(eq_ind_r nat x2 (\lambda (n1: nat).(csubst0 n1 v c x1)) H6 n0 H9) in (let -H11 \def (eq_ind_r nat x2 (\lambda (n1: nat).(subst0 n1 v t x0)) H5 n0 H9) in -(let H12 \def (IHn c x1 v H10 e H7) in (or4_ind (drop n0 O x1 e) (ex3_4 F C T -T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop n0 O x1 (CHead e0 (Flat f) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O x1 (CHead e2 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O -v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop n0 O x1 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead x1 (Bind b) x0) -e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda -(_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) -(CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat -f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u)))))) (\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 -e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) w))))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H13: -(drop n0 O x1 e)).(or4_intro0 (drop (S n0) O (CHead x1 (Bind b) x0) e) (ex3_4 -F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq -C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) -w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) -O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C -C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) w))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O -v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Bind b) n0 x1 e H13 -x0))) (\lambda (H13: (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda -(u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x1 (CHead e0 -(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w))))))).(ex3_4_ind F C T T (\lambda (f: F).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O -x1 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w))))) (or4 (drop (S n0) O (CHead x1 (Bind -b) x0) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 -(Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) -u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) -u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: T).(\lambda (x6: -T).(\lambda (H14: (eq C e (CHead x4 (Flat x3) x5))).(\lambda (H15: (drop n0 O -x1 (CHead x4 (Flat x3) x6))).(\lambda (H16: (subst0 O v x5 x6)).(eq_ind_r C -(CHead x4 (Flat x3) x5) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Bind -b) x0) c0) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 -(Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 -(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) -u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat -f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2))))))))) (or4_intro1 (drop (S n0) O (CHead x1 (Bind b) x0) (CHead x4 (Flat -x3) x5)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x5) (CHead e0 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) -O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda -(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead -x4 (Flat x3) x5) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) -(CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C -(CHead x4 (Flat x3) x5) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead -x1 (Bind b) x0) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C T T (\lambda (f: -F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) -x5) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) -w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w))))) x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x3) x5)) -(drop_drop (Bind b) n0 x1 (CHead x4 (Flat x3) x6) H15 x0) H16)) e H14)))))))) -H13)) (\lambda (H13: (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O x1 (CHead e2 -(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O -x1 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2))))) (or4 (drop (S n0) O (CHead x1 -(Bind b) x0) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 -(Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) -u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) -u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: -T).(\lambda (H14: (eq C e (CHead x4 (Flat x3) x6))).(\lambda (H15: (drop n0 O -x1 (CHead x5 (Flat x3) x6))).(\lambda (H16: (csubst0 O v x4 x5)).(eq_ind_r C -(CHead x4 (Flat x3) x6) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Bind -b) x0) c0) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 -(Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 -(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) -u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat -f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2))))))))) (or4_intro2 (drop (S n0) O (CHead x1 (Bind b) x0) (CHead x4 (Flat -x3) x6)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) -O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda -(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead -x4 (Flat x3) x6) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) -(CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C -(CHead x4 (Flat x3) x6) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead -x1 (Bind b) x0) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C C T (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x4 (Flat x3) -x6) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) -u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v e1 e2))))) x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x3) x6)) -(drop_drop (Bind b) n0 x1 (CHead x5 (Flat x3) x6) H15 x0) H16)) e H14)))))))) -H13)) (\lambda (H13: (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) -u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop n0 O x1 (CHead e2 (Flat f) w))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O -v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda -(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C -e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x1 (CHead e2 (Flat f) w))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) -O (CHead x1 (Bind b) x0) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) -O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda -(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) -u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) -u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: -T).(\lambda (x7: T).(\lambda (H14: (eq C e (CHead x4 (Flat x3) x6))).(\lambda -(H15: (drop n0 O x1 (CHead x5 (Flat x3) x7))).(\lambda (H16: (subst0 O v x6 -x7)).(\lambda (H17: (csubst0 O v x4 x5)).(eq_ind_r C (CHead x4 (Flat x3) x6) -(\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Bind b) x0) c0) (ex3_4 F C T -T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 -(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) -w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Flat f) u)))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) -O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C -C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat f) u))))))) (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) w))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O -v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O -(CHead x1 (Bind b) x0) (CHead x4 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f: -F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) -x6) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) -w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 -(Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O -v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 -(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat -f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) -(ex4_5_intro F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 -(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat -f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) -x3 x4 x5 x6 x7 (refl_equal C (CHead x4 (Flat x3) x6)) (drop_drop (Bind b) n0 -x1 (CHead x5 (Flat x3) x7) H15 x0) H16 H17)) e H14)))))))))) H13)) -H12)))))))) (\lambda (f: F).(\lambda (H7: (drop (r (Flat f) n0) O c -e)).(\lambda (H8: (eq nat (S n0) (s (Flat f) x2))).(let H9 \def (f_equal nat -nat (\lambda (e0: nat).e0) (S n0) (s (Flat f) x2) H8) in (let H10 \def -(eq_ind_r nat x2 (\lambda (n1: nat).(csubst0 n1 v c x1)) H6 (S n0) H9) in -(let H11 \def (eq_ind_r nat x2 (\lambda (n1: nat).(subst0 n1 v t x0)) H5 (S -n0) H9) in (let H12 \def (H x1 v H10 e H7) in (or4_ind (drop (S n0) O x1 e) -(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: -T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x1 (CHead e0 (Flat f0) -w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S -n0) O x1 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda -(_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x1 (CHead -e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2))))))) (or4 (drop (S n0) O (CHead x1 (Flat f) x0) e) (ex3_4 F C T T -(\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) -w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S -n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) -(ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O -v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H13: (drop (S n0) O -x1 e)).(or4_intro0 (drop (S n0) O (CHead x1 (Flat f) x0) e) (ex3_4 F C T T -(\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) -w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S -n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) -(ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O -v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Flat f) n0 x1 e H13 -x0))) (\lambda (H13: (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O x1 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda -(u: T).(\lambda (w: T).(subst0 O v u w))))))).(ex3_4_ind F C T T (\lambda -(f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 -(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda -(w: T).(drop (S n0) O x1 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda -(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))) (or4 (drop (S n0) -O (CHead x1 (Flat f) x0) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0) -(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) -(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: T).(\lambda -(x6: T).(\lambda (H14: (eq C e (CHead x4 (Flat x3) x5))).(\lambda (H15: (drop -(S n0) O x1 (CHead x4 (Flat x3) x6))).(\lambda (H16: (subst0 O v x5 -x6)).(eq_ind_r C (CHead x4 (Flat x3) x5) (\lambda (c0: C).(or4 (drop (S n0) O -(CHead x1 (Flat f) x0) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c0 (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0) -(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 -(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) -(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead x1 (Flat f) x0) (CHead -x4 (Flat x3) x5)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda -(u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x5) (CHead e0 (Flat f0) -u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C (CHead x4 (Flat x3) x5) (CHead e1 (Flat f0) u)))))) (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 -(Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda -(_: T).(eq C (CHead x4 (Flat x3) x5) (CHead e1 (Flat f0) u))))))) (\lambda -(f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C -T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C -(CHead x4 (Flat x3) x5) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda -(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) -(CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w))))) x3 x4 x5 x6 (refl_equal C (CHead x4 -(Flat x3) x5)) (drop_drop (Flat f) n0 x1 (CHead x4 (Flat x3) x6) H15 x0) -H16)) e H14)))))))) H13)) (\lambda (H13: (ex3_4 F C C T (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat -f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O x1 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind -F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq -C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u: T).(drop (S n0) O x1 (CHead e2 (Flat f0) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O -v e1 e2))))) (or4 (drop (S n0) O (CHead x1 (Flat f) x0) e) (ex3_4 F C T T -(\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) -w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S -n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) -(ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O -v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: -C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (H14: (eq C e (CHead x4 (Flat -x3) x6))).(\lambda (H15: (drop (S n0) O x1 (CHead x5 (Flat x3) x6))).(\lambda -(H16: (csubst0 O v x4 x5)).(eq_ind_r C (CHead x4 (Flat x3) x6) (\lambda (c0: -C).(or4 (drop (S n0) O (CHead x1 (Flat f) x0) c0) (ex3_4 F C T T (\lambda -(f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 -(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda -(w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u -w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C c0 (CHead e1 (Flat f0) u)))))) (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 -(Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda -(_: T).(eq C c0 (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 -(Flat f) x0) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead x1 (Flat -f) x0) (CHead x4 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0 -(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda -(w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u -w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S -n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) -(ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) -u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) -(ex3_4_intro F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S -n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) -x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x3) x6)) (drop_drop (Flat f) n0 x1 -(CHead x5 (Flat x3) x6) H15 x0) H16)) e H14)))))))) H13)) (\lambda (H13: -(ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O x1 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x1 (CHead e2 (Flat f0) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) -(or4 (drop (S n0) O (CHead x1 (Flat f) x0) e) (ex3_4 F C T T (\lambda (f0: -F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat -f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0) -(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) -(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda -(x6: T).(\lambda (x7: T).(\lambda (H14: (eq C e (CHead x4 (Flat x3) -x6))).(\lambda (H15: (drop (S n0) O x1 (CHead x5 (Flat x3) x7))).(\lambda -(H16: (subst0 O v x6 x7)).(\lambda (H17: (csubst0 O v x4 x5)).(eq_ind_r C -(CHead x4 (Flat x3) x6) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Flat -f) x0) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f0) u)))))) (\lambda (f0: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 -(Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 -(CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) -u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat -f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2))))))))) (or4_intro3 (drop (S n0) O (CHead x1 (Flat f) x0) (CHead x4 (Flat -x3) x6)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) u)))))) (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 -(Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda -(_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) u))))))) (\lambda -(f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex4_5_intro F C -C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) u))))))) -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) x3 x4 x5 x6 x7 -(refl_equal C (CHead x4 (Flat x3) x6)) (drop_drop (Flat f) n0 x1 (CHead x5 -(Flat x3) x7) H15 x0) H16 H17)) e H14)))))))))) H13)) H12)))))))) k -(drop_gen_drop k c e t n0 H1) H3) c2 H4)))))))) H2)) (csubst0_gen_head k c c2 -t v (S n0) H0))))))))))) c1)))) n). -(* COMMENTS -Initial nodes: 36162 -END *) - -theorem csubst0_drop_eq_back: - \forall (n: nat).(\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 -n v c1 c2) \to (\forall (e: C).((drop n O c2 e) \to (or4 (drop n O c1 e) -(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: -T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(drop n O c1 (CHead e0 (Flat f) u1)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v -u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n O c1 (CHead e1 -(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat -f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (_: T).(drop n O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 -O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))) -\def - \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (c1: C).(\forall (c2: -C).(\forall (v: T).((csubst0 n0 v c1 c2) \to (\forall (e: C).((drop n0 O c2 -e) \to (or4 (drop n0 O c1 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) -(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O -c1 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat -f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(drop n0 O c1 (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c1 (CHead e1 -(Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2))))))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda -(H: (csubst0 O v c1 c2)).(\lambda (e: C).(\lambda (H0: (drop O O c2 -e)).(eq_ind C c2 (\lambda (c: C).(or4 (drop O O c1 c) (ex3_4 F C T T (\lambda -(f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c (CHead e0 -(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda -(_: T).(drop O O c1 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c -(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(drop O O c1 (CHead e1 (Flat f) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) -(ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (u2: T).(eq C c (CHead e2 (Flat f) u2))))))) (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop O -O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda -(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2))))))))) (insert_eq nat O (\lambda (n0: nat).(csubst0 n0 v c1 c2)) -(\lambda (n0: nat).(or4 (drop n0 n0 c1 c2) (ex3_4 F C T T (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c2 (CHead e0 (Flat -f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(drop n0 n0 c1 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 n0 v u1 u2)))))) (ex3_4 F C C T -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c2 -(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(drop n0 n0 c1 (CHead e1 (Flat f) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 v e1 -e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C c2 (CHead e2 (Flat f) u2))))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop n0 n0 c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda -(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 n0 v u1 -u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 n0 v e1 e2))))))))) (\lambda (y: nat).(\lambda -(H1: (csubst0 y v c1 c2)).(csubst0_ind (\lambda (n0: nat).(\lambda (t: -T).(\lambda (c: C).(\lambda (c0: C).((eq nat n0 O) \to (or4 (drop n0 n0 c c0) -(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: -T).(eq C c0 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(drop n0 n0 c (CHead e0 (Flat f) u1)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 n0 -t u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 n0 c (CHead e1 -(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 n0 t e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e2 -(Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (_: T).(drop n0 n0 c (CHead e1 (Flat f) u1))))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 n0 t u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 n0 t e1 e2))))))))))))) (\lambda -(k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (v0: T).(\forall -(u1: T).(\forall (u2: T).((subst0 i v0 u1 u2) \to (\forall (c: C).((eq nat (s -k0 i) O) \to (or4 (drop (s k0 i) (s k0 i) (CHead c k0 u1) (CHead c k0 u2)) -(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: -T).(eq C (CHead c k0 u2) (CHead e0 (Flat f) u4)))))) (\lambda (f: F).(\lambda -(e0: C).(\lambda (u3: T).(\lambda (_: T).(drop (s k0 i) (s k0 i) (CHead c k0 -u1) (CHead e0 (Flat f) u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: -T).(\lambda (u4: T).(subst0 (s k0 i) v0 u3 u4)))))) (ex3_4 F C C T (\lambda -(f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c k0 u2) -(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(drop (s k0 i) (s k0 i) (CHead c k0 u1) (CHead e1 (Flat f) -u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (s k0 i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c k0 -u2) (CHead e2 (Flat f) u4))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u3: T).(\lambda (_: T).(drop (s k0 i) (s k0 i) (CHead c k0 -u1) (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u3: T).(\lambda (u4: T).(subst0 (s k0 i) v0 u3 u4)))))) (\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: -T).(csubst0 (s k0 i) v0 e1 e2)))))))))))))))) (\lambda (b: B).(\lambda (i: -nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0 -i v0 u1 u2)).(\lambda (c: C).(\lambda (H3: (eq nat (S i) O)).(let H4 \def -(eq_ind nat (S i) (\lambda (ee: nat).(match ee in nat return (\lambda (_: -nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H3) in -(False_ind (or4 (drop (S i) (S i) (CHead c (Bind b) u1) (CHead c (Bind b) -u2)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda -(u4: T).(eq C (CHead c (Bind b) u2) (CHead e0 (Flat f) u4)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop (S i) (S i) (CHead -c (Bind b) u1) (CHead e0 (Flat f) u3)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u3: T).(\lambda (u4: T).(subst0 (S i) v0 u3 u4)))))) (ex3_4 F C -C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C -(CHead c (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S i) (S i) (CHead c (Bind b) -u1) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (S i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda -(f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq -C (CHead c (Bind b) u2) (CHead e2 (Flat f) u4))))))) (\lambda (f: F).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (_: T).(drop (S i) (S i) -(CHead c (Bind b) u1) (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda -(_: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 (S i) v0 u3 -u4)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (S i) v0 e1 e2)))))))) H4)))))))))) (\lambda (f: -F).(\lambda (i: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: -T).(\lambda (H2: (subst0 i v0 u1 u2)).(\lambda (c: C).(\lambda (H3: (eq nat i -O)).(let H4 \def (eq_ind nat i (\lambda (n0: nat).(subst0 n0 v0 u1 u2)) H2 O -H3) in (eq_ind_r nat O (\lambda (n0: nat).(or4 (drop n0 n0 (CHead c (Flat f) -u1) (CHead c (Flat f) u2)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c (Flat f) u2) (CHead e0 -(Flat f0) u4)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u3: -T).(\lambda (_: T).(drop n0 n0 (CHead c (Flat f) u1) (CHead e0 (Flat f0) -u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: -T).(subst0 n0 v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c (Flat f) u2) (CHead e2 -(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(drop n0 n0 (CHead c (Flat f) u1) (CHead e1 (Flat f0) u)))))) (\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 v0 e1 -e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c (Flat f) u2) (CHead e2 -(Flat f0) u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u3: T).(\lambda (_: T).(drop n0 n0 (CHead c (Flat f) u1) (CHead -e1 (Flat f0) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u3: T).(\lambda (u4: T).(subst0 n0 v0 u3 u4)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -n0 v0 e1 e2))))))))) (or4_intro1 (drop O O (CHead c (Flat f) u1) (CHead c -(Flat f) u2)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (u4: T).(eq C (CHead c (Flat f) u2) (CHead e0 (Flat f0) u4)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop O O -(CHead c (Flat f) u1) (CHead e0 (Flat f0) u3)))))) (\lambda (_: F).(\lambda -(_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (ex3_4 F C -C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C -(CHead c (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u: T).(drop O O (CHead c (Flat f) u1) -(CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C -(CHead c (Flat f) u2) (CHead e2 (Flat f0) u4))))))) (\lambda (f0: F).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c -(Flat f) u1) (CHead e1 (Flat f0) u3))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v0 e1 e2))))))) (ex3_4_intro F C T T (\lambda (f0: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c (Flat f) -u2) (CHead e0 (Flat f0) u4)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda -(u3: T).(\lambda (_: T).(drop O O (CHead c (Flat f) u1) (CHead e0 (Flat f0) -u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: -T).(subst0 O v0 u3 u4))))) f c u1 u2 (refl_equal C (CHead c (Flat f) u2)) -(drop_refl (CHead c (Flat f) u1)) H4)) i H3)))))))))) k)) (\lambda (k: -K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (c3: C).(\forall (c4: -C).(\forall (v0: T).((csubst0 i v0 c3 c4) \to ((((eq nat i O) \to (or4 (drop -i i c3 c4) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (u2: T).(eq C c4 (CHead e0 (Flat f) u2)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop i i c3 (CHead e0 -(Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 i v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(eq C c4 (CHead e2 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop i i c3 -(CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 i v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 -(CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(drop i i c3 (CHead e1 (Flat f) u1))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 i v0 u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v0 e1 e2)))))))))) \to -(\forall (u: T).((eq nat (s k0 i) O) \to (or4 (drop (s k0 i) (s k0 i) (CHead -c3 k0 u) (CHead c4 k0 u)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 k0 u) (CHead e0 (Flat f) -u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(drop (s k0 i) (s k0 i) (CHead c3 k0 u) (CHead e0 (Flat f) u1)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (s -k0 i) v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u0: T).(eq C (CHead c4 k0 u) (CHead e2 (Flat f) u0)))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(drop (s k0 -i) (s k0 i) (CHead c3 k0 u) (CHead e1 (Flat f) u0)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (s k0 i) v0 e1 -e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 k0 u) (CHead e2 (Flat f) -u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (_: T).(drop (s k0 i) (s k0 i) (CHead c3 k0 u) (CHead e1 (Flat f) -u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 (s k0 i) v0 u1 u2)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (s k0 i) v0 -e1 e2))))))))))))))))) (\lambda (b: B).(\lambda (i: nat).(\lambda (c3: -C).(\lambda (c4: C).(\lambda (v0: T).(\lambda (_: (csubst0 i v0 c3 -c4)).(\lambda (_: (((eq nat i O) \to (or4 (drop i i c3 c4) (ex3_4 F C T T -(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 -(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(drop i i c3 (CHead e0 (Flat f) u1)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v0 u1 u2)))))) -(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(eq C c4 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(drop i i c3 (CHead e1 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e2 (Flat f) u2))))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop i i c3 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v0 u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (u: T).(\lambda (H4: (eq nat -(S i) O)).(let H5 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee in nat -return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow -True])) I O H4) in (False_ind (or4 (drop (S i) (S i) (CHead c3 (Bind b) u) -(CHead c4 (Bind b) u)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 (Bind b) u) (CHead e0 -(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S i) (S i) (CHead c3 (Bind b) u) (CHead e0 (Flat f) u1)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (S -i) v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u0: T).(eq C (CHead c4 (Bind b) u) (CHead e2 (Flat f) -u0)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: -T).(drop (S i) (S i) (CHead c3 (Bind b) u) (CHead e1 (Flat f) u0)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (S -i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 (Bind b) u) (CHead -e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S i) (S i) (CHead c3 (Bind b) u) -(CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (S i) v0 u1 u2)))))) (\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: -T).(csubst0 (S i) v0 e1 e2)))))))) H5))))))))))) (\lambda (f: F).(\lambda (i: -nat).(\lambda (c3: C).(\lambda (c4: C).(\lambda (v0: T).(\lambda (H2: -(csubst0 i v0 c3 c4)).(\lambda (H3: (((eq nat i O) \to (or4 (drop i i c3 c4) -(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda -(u2: T).(eq C c4 (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(drop i i c3 (CHead e0 (Flat f0) u1)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i -v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C c4 (CHead e2 (Flat f0) u)))))) (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop i i c3 (CHead e1 -(Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 i v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e2 -(Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(drop i i c3 (CHead e1 (Flat f0) -u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 i v0 u1 u2)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v0 e1 -e2))))))))))).(\lambda (u: T).(\lambda (H4: (eq nat i O)).(let H5 \def -(eq_ind nat i (\lambda (n0: nat).((eq nat n0 O) \to (or4 (drop n0 n0 c3 c4) -(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda -(u2: T).(eq C c4 (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(drop n0 n0 c3 (CHead e0 (Flat f0) -u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 n0 v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u0: T).(eq C c4 (CHead e2 (Flat f0) u0)))))) -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(drop n0 -n0 c3 (CHead e1 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda -(f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq -C c4 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 n0 c3 (CHead e1 (Flat f0) -u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 n0 v0 u1 u2)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 n0 v0 e1 -e2)))))))))) H3 O H4) in (let H6 \def (eq_ind nat i (\lambda (n0: -nat).(csubst0 n0 v0 c3 c4)) H2 O H4) in (eq_ind_r nat O (\lambda (n0: -nat).(or4 (drop n0 n0 (CHead c3 (Flat f) u) (CHead c4 (Flat f) u)) (ex3_4 F C -T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C -(CHead c4 (Flat f) u) (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda -(e0: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 n0 (CHead c3 (Flat f) u) -(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 n0 v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(eq C (CHead c4 (Flat f) -u) (CHead e2 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u0: T).(drop n0 n0 (CHead c3 (Flat f) u) (CHead e1 (Flat f0) -u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 (Flat f) -u) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 n0 (CHead c3 (Flat f) u) -(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 n0 v0 u1 u2)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -n0 v0 e1 e2))))))))) (or4_intro2 (drop O O (CHead c3 (Flat f) u) (CHead c4 -(Flat f) u)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (u2: T).(eq C (CHead c4 (Flat f) u) (CHead e0 (Flat f0) u2)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop O O -(CHead c3 (Flat f) u) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda -(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (ex3_4 F C -C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(eq C -(CHead c4 (Flat f) u) (CHead e2 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u0: T).(drop O O (CHead c3 (Flat f) u) -(CHead e1 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C -(CHead c4 (Flat f) u) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop O O (CHead c3 -(Flat f) u) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v0 e1 e2))))))) (ex3_4_intro F C C T (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(eq C (CHead c4 (Flat f) -u) (CHead e2 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u0: T).(drop O O (CHead c3 (Flat f) u) (CHead e1 (Flat f0) -u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v0 e1 e2))))) f c3 c4 u (refl_equal C (CHead c4 (Flat f) u)) -(drop_refl (CHead c3 (Flat f) u)) H6)) i H4)))))))))))) k)) (\lambda (k: -K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (v0: T).(\forall (u1: -T).(\forall (u2: T).((subst0 i v0 u1 u2) \to (\forall (c3: C).(\forall (c4: -C).((csubst0 i v0 c3 c4) \to ((((eq nat i O) \to (or4 (drop i i c3 c4) (ex3_4 -F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq -C c4 (CHead e0 (Flat f) u4)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda -(u3: T).(\lambda (_: T).(drop i i c3 (CHead e0 (Flat f) u3)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 i v0 u3 u4)))))) -(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(eq C c4 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(drop i i c3 (CHead e1 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u4: T).(eq C c4 (CHead e2 (Flat f) u4))))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda -(_: T).(drop i i c3 (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 i v0 u3 u4)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 i v0 e1 e2)))))))))) \to ((eq nat (s k0 i) O) \to (or4 (drop -(s k0 i) (s k0 i) (CHead c3 k0 u1) (CHead c4 k0 u2)) (ex3_4 F C T T (\lambda -(f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 k0 -u2) (CHead e0 (Flat f) u4)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda -(u3: T).(\lambda (_: T).(drop (s k0 i) (s k0 i) (CHead c3 k0 u1) (CHead e0 -(Flat f) u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda -(u4: T).(subst0 (s k0 i) v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c4 k0 u2) -(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(drop (s k0 i) (s k0 i) (CHead c3 k0 u1) (CHead e1 (Flat -f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (s k0 i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 k0 -u2) (CHead e2 (Flat f) u4))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u3: T).(\lambda (_: T).(drop (s k0 i) (s k0 i) (CHead c3 k0 -u1) (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u3: T).(\lambda (u4: T).(subst0 (s k0 i) v0 u3 u4)))))) (\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: -T).(csubst0 (s k0 i) v0 e1 e2))))))))))))))))))) (\lambda (b: B).(\lambda (i: -nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0 -i v0 u1 u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (_: (csubst0 i v0 c3 -c4)).(\lambda (_: (((eq nat i O) \to (or4 (drop i i c3 c4) (ex3_4 F C T T -(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C c4 -(CHead e0 (Flat f) u4)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u3: -T).(\lambda (_: T).(drop i i c3 (CHead e0 (Flat f) u3)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 i v0 u3 u4)))))) -(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(eq C c4 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(drop i i c3 (CHead e1 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u4: T).(eq C c4 (CHead e2 (Flat f) u4))))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda -(_: T).(drop i i c3 (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 i v0 u3 u4)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (H5: (eq nat (S i) O)).(let H6 -\def (eq_ind nat (S i) (\lambda (ee: nat).(match ee in nat return (\lambda -(_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H5) -in (False_ind (or4 (drop (S i) (S i) (CHead c3 (Bind b) u1) (CHead c4 (Bind -b) u2)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (u4: T).(eq C (CHead c4 (Bind b) u2) (CHead e0 (Flat f) u4)))))) -(\lambda (f: F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop (S i) -(S i) (CHead c3 (Bind b) u1) (CHead e0 (Flat f) u3)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 (S i) v0 u3 -u4)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C (CHead c4 (Bind b) u2) (CHead e2 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S i) -(S i) (CHead c3 (Bind b) u1) (CHead e1 (Flat f) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (S i) v0 e1 -e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 (Bind b) u2) (CHead e2 -(Flat f) u4))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u3: T).(\lambda (_: T).(drop (S i) (S i) (CHead c3 (Bind b) u1) (CHead e1 -(Flat f) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda -(u3: T).(\lambda (u4: T).(subst0 (S i) v0 u3 u4)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -(S i) v0 e1 e2)))))))) H6))))))))))))) (\lambda (f: F).(\lambda (i: -nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 -i v0 u1 u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (H3: (csubst0 i v0 c3 -c4)).(\lambda (H4: (((eq nat i O) \to (or4 (drop i i c3 c4) (ex3_4 F C T T -(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C c4 -(CHead e0 (Flat f0) u4)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u3: -T).(\lambda (_: T).(drop i i c3 (CHead e0 (Flat f0) u3)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 i v0 u3 u4)))))) -(ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(eq C c4 (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(drop i i c3 (CHead e1 (Flat f0) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C c4 (CHead e2 (Flat f0) -u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: -T).(\lambda (_: T).(drop i i c3 (CHead e1 (Flat f0) u3))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 -i v0 u3 u4)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (H5: (eq nat i -O)).(let H6 \def (eq_ind nat i (\lambda (n0: nat).((eq nat n0 O) \to (or4 -(drop n0 n0 c3 c4) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda -(_: T).(\lambda (u4: T).(eq C c4 (CHead e0 (Flat f0) u4)))))) (\lambda (f0: -F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop n0 n0 c3 (CHead e0 -(Flat f0) u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda -(u4: T).(subst0 n0 v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda -(_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c4 (CHead e2 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 n0 -c3 (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C c4 -(CHead e2 (Flat f0) u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u3: T).(\lambda (_: T).(drop n0 n0 c3 (CHead e1 (Flat f0) -u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u3: -T).(\lambda (u4: T).(subst0 n0 v0 u3 u4)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 n0 v0 e1 -e2)))))))))) H4 O H5) in (let H7 \def (eq_ind nat i (\lambda (n0: -nat).(csubst0 n0 v0 c3 c4)) H3 O H5) in (let H8 \def (eq_ind nat i (\lambda -(n0: nat).(subst0 n0 v0 u1 u2)) H2 O H5) in (eq_ind_r nat O (\lambda (n0: -nat).(or4 (drop n0 n0 (CHead c3 (Flat f) u1) (CHead c4 (Flat f) u2)) (ex3_4 F -C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq -C (CHead c4 (Flat f) u2) (CHead e0 (Flat f0) u4)))))) (\lambda (f0: -F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop n0 n0 (CHead c3 -(Flat f) u1) (CHead e0 (Flat f0) u3)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u3: T).(\lambda (u4: T).(subst0 n0 v0 u3 u4)))))) (ex3_4 F C C T -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C -(CHead c4 (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 n0 (CHead c3 (Flat f) u1) -(CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C -(CHead c4 (Flat f) u2) (CHead e2 (Flat f0) u4))))))) (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (_: T).(drop n0 -n0 (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u3))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 -n0 v0 u3 u4)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 n0 v0 e1 e2))))))))) (or4_intro3 (drop O O -(CHead c3 (Flat f) u1) (CHead c4 (Flat f) u2)) (ex3_4 F C T T (\lambda (f0: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 (Flat f) -u2) (CHead e0 (Flat f0) u4)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda -(u3: T).(\lambda (_: T).(drop O O (CHead c3 (Flat f) u1) (CHead e0 (Flat f0) -u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: -T).(subst0 O v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c4 (Flat f) u2) (CHead e2 -(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(drop O O (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u)))))) (\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 -e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 (Flat f) u2) (CHead e2 -(Flat f0) u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c3 (Flat f) u1) (CHead -e1 (Flat f0) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v0 e1 e2))))))) (ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 (Flat f) -u2) (CHead e2 (Flat f0) u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c3 (Flat f) u1) -(CHead e1 (Flat f0) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v0 e1 e2)))))) f c3 c4 u1 u2 (refl_equal C (CHead c4 (Flat f) u2)) -(drop_refl (CHead c3 (Flat f) u1)) H8 H7)) i H5))))))))))))))) k)) y v c1 c2 -H1))) H) e (drop_gen_refl c2 e H0)))))))) (\lambda (n0: nat).(\lambda (IHn: -((\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 n0 v c1 c2) \to -(\forall (e: C).((drop n0 O c2 e) \to (or4 (drop n0 O c1 e) (ex3_4 F C T T -(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e -(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(drop n0 O c1 (CHead e0 (Flat f) u1)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c1 (CHead e1 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O -v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop n0 O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda -(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 -u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))))).(\lambda (c1: C).(C_ind -(\lambda (c: C).(\forall (c2: C).(\forall (v: T).((csubst0 (S n0) v c c2) \to -(\forall (e: C).((drop (S n0) O c2 e) \to (or4 (drop (S n0) O c e) (ex3_4 F C -T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e -(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f) u1)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O c (CHead e1 (Flat f) -u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat -f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f) u1))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 -O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))) (\lambda (n1: -nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (H: (csubst0 (S n0) v (CSort -n1) c2)).(\lambda (e: C).(\lambda (_: (drop (S n0) O c2 e)).(csubst0_gen_sort -c2 v (S n0) n1 H (or4 (drop (S n0) O (CSort n1) e) (ex3_4 F C T T (\lambda -(f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 -(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S n0) O (CSort n1) (CHead e0 (Flat f) u1)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CSort n1) (CHead e1 (Flat -f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat -f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O (CSort n1) (CHead e1 (Flat f) u1))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))))))) -(\lambda (c: C).(\lambda (H: ((\forall (c2: C).(\forall (v: T).((csubst0 (S -n0) v c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (or4 (drop (S n0) O -c e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda -(u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f) -u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) -O c (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e -(CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f) -u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2))))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda -(v: T).(\lambda (H0: (csubst0 (S n0) v (CHead c k t) c2)).(\lambda (e: -C).(\lambda (H1: (drop (S n0) O c2 e)).(or3_ind (ex3_2 T nat (\lambda (_: -T).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (u2: T).(\lambda (_: -nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j -v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k -j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda -(c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_: -T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k j))))) (\lambda (u2: -T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda -(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: -T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))) (or4 (drop (S -n0) O (CHead c k t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) -(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O (CHead c k t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e -(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O -v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u1))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 -O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H2: (ex3_2 T nat -(\lambda (_: T).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (u2: -T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: -nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: -nat).(eq nat (S n0) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 -(CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))) (or4 -(drop (S n0) O (CHead c k t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) -(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O (CHead c k t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e -(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O -v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u1))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 -O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: T).(\lambda -(x1: nat).(\lambda (H3: (eq nat (S n0) (s k x1))).(\lambda (H4: (eq C c2 -(CHead c k x0))).(\lambda (H5: (subst0 x1 v t x0)).(let H6 \def (eq_ind C c2 -(\lambda (c0: C).(drop (S n0) O c0 e)) H1 (CHead c k x0) H4) in (K_ind -(\lambda (k0: K).((eq nat (S n0) (s k0 x1)) \to ((drop (r k0 n0) O c e) \to -(or4 (drop (S n0) O (CHead c k0 t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda -(e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) -(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O (CHead c k0 t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e -(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(drop (S n0) O (CHead c k0 t) (CHead e1 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O -v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S n0) O (CHead c k0 t) (CHead e1 (Flat f) u1))))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))) (\lambda (b: -B).(\lambda (H7: (eq nat (S n0) (s (Bind b) x1))).(\lambda (H8: (drop (r -(Bind b) n0) O c e)).(let H9 \def (f_equal nat nat (\lambda (e0: nat).(match -e0 in nat return (\lambda (_: nat).nat) with [O \Rightarrow n0 | (S n1) -\Rightarrow n1])) (S n0) (S x1) H7) in (let H10 \def (eq_ind_r nat x1 -(\lambda (n1: nat).(subst0 n1 v t x0)) H5 n0 H9) in (or4_intro0 (drop (S n0) -O (CHead c (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) -(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead -e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e -(CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) -(CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2))))))) (drop_drop (Bind b) n0 c e H8 t))))))) (\lambda (f: -F).(\lambda (H7: (eq nat (S n0) (s (Flat f) x1))).(\lambda (H8: (drop (r -(Flat f) n0) O c e)).(let H9 \def (f_equal nat nat (\lambda (e0: nat).e0) (S -n0) x1 H7) in (let H10 \def (eq_ind_r nat x1 (\lambda (n1: nat).(subst0 n1 v -t x0)) H5 (S n0) H9) in (or4_intro0 (drop (S n0) O (CHead c (Flat f) t) e) -(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda -(u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) -(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat -f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 -e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop -(Flat f) n0 c e H8 t))))))) k H3 (drop_gen_drop k c e x0 n0 H6)))))))) H2)) -(\lambda (H2: (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) -(s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) -(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))).(ex3_2_ind C nat -(\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (c3: -C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: -nat).(csubst0 j v c c3))) (or4 (drop (S n0) O (CHead c k t) e) (ex3_4 F C T T -(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e -(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O (CHead c k t) (CHead e0 (Flat f) u1)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v -u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c k -t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e -(CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k t) (CHead e1 -(Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2)))))))) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H3: (eq nat (S n0) -(s k x1))).(\lambda (H4: (eq C c2 (CHead x0 k t))).(\lambda (H5: (csubst0 x1 -v c x0)).(let H6 \def (eq_ind C c2 (\lambda (c0: C).(drop (S n0) O c0 e)) H1 -(CHead x0 k t) H4) in (K_ind (\lambda (k0: K).((eq nat (S n0) (s k0 x1)) \to -((drop (r k0 n0) O x0 e) \to (or4 (drop (S n0) O (CHead c k0 t) e) (ex3_4 F C -T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e -(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O (CHead c k0 t) (CHead e0 (Flat f) u1)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v -u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c -k0 t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda -(f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq -C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k0 t) (CHead -e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2))))))))))) (\lambda (b: B).(\lambda (H7: (eq nat (S n0) (s (Bind b) -x1))).(\lambda (H8: (drop (r (Bind b) n0) O x0 e)).(let H9 \def (f_equal nat -nat (\lambda (e0: nat).(match e0 in nat return (\lambda (_: nat).nat) with [O -\Rightarrow n0 | (S n1) \Rightarrow n1])) (S n0) (S x1) H7) in (let H10 \def -(eq_ind_r nat x1 (\lambda (n1: nat).(csubst0 n1 v c x0)) H5 n0 H9) in (let -H11 \def (IHn c x0 v H10 e H8) in (or4_ind (drop n0 O c e) (ex3_4 F C T T -(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e -(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 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f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat -f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 -e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda -(H12: (drop n0 O c e)).(or4_intro0 (drop (S n0) O (CHead c (Bind b) t) e) -(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: -T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) -(CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat -f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 -e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop -(Bind b) n0 c e H12 t))) (\lambda (H12: (ex3_4 F C T T (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat -f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))).(ex3_4_ind F C -T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e -(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))) -(or4 (drop (S n0) O (CHead c (Bind b) t) e) (ex3_4 F C T T (\lambda (f: -F).(\lambda (e0: 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F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v -u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C (CHead x3 (Flat x2) x5) (CHead e2 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) -O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C -C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e2 (Flat f) -u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) -u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) -(ex3_4_intro F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0 (Flat f) u2)))))) -(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))) -x2 x3 x4 x5 (refl_equal C (CHead x3 (Flat x2) x5)) (drop_drop (Bind b) n0 c -(CHead x3 (Flat x2) x4) H14 t) H15)) e H13)))))))) H12)) (\lambda (H12: -(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O -v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1 -(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 O v e1 e2))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e) -(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: -T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) -(CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat -f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 -e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda -(x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: T).(\lambda (H13: (eq -C e (CHead x4 (Flat x2) x5))).(\lambda (H14: (drop n0 O c (CHead x3 (Flat x2) -x5))).(\lambda (H15: (csubst0 O v x3 x4)).(eq_ind_r C (CHead x4 (Flat x2) x5) -(\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T -(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 -(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) -u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) -O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C -C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 -O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro2 (drop (S n0) O -(CHead c (Bind b) t) (CHead x4 (Flat x2) x5)) (ex3_4 F C T T (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat -x2) x5) (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda -(u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat -f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 -(Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O -v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 -(Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat -f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) -(ex3_4_intro F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) -O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) x2 x3 x4 x5 -(refl_equal C (CHead x4 (Flat x2) x5)) (drop_drop (Bind b) n0 c (CHead x3 -(Flat x2) x5) H14 t) H15)) e H13)))))))) H12)) (\lambda (H12: (ex4_5 F C C T -T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c (CHead e1 -(Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2)))))))).(ex4_5_ind F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop n0 O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e) -(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: -T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) -(CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat -f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 -e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda -(x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: T).(\lambda (x6: -T).(\lambda (H13: (eq C e (CHead x4 (Flat x2) x6))).(\lambda (H14: (drop n0 O -c (CHead x3 (Flat x2) x5))).(\lambda (H15: (subst0 O v x5 x6)).(\lambda (H16: -(csubst0 O v x3 x4)).(eq_ind_r C (CHead x4 (Flat x2) x6) (\lambda (c0: -C).(or4 (drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat -f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 -u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c -(Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c -(Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead c (Bind -b) t) (CHead x4 (Flat x2) x6)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e0 -(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v -u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) -O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C -C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 (Flat f) -u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) -u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) -(ex4_5_intro F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 -(Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat -f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) -x2 x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x2) x6)) (drop_drop (Bind b) n0 -c (CHead x3 (Flat x2) x5) H14 t) H15 H16)) e H13)))))))))) H12)) H11))))))) -(\lambda (f: F).(\lambda (H7: (eq nat (S n0) (s (Flat f) x1))).(\lambda (H8: -(drop (r (Flat f) n0) O x0 e)).(let H9 \def (f_equal nat nat (\lambda (e0: -nat).e0) (S n0) x1 H7) in (let H10 \def (eq_ind_r nat x1 (\lambda (n1: -nat).(csubst0 n1 v c x0)) H5 (S n0) H9) in (let H11 \def (H x0 v H10 e H8) in -(or4_ind (drop (S n0) O c e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O c (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat -f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(drop (S n0) O c (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead -e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2))))))) (or4 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T -(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e -(CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) -u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S -n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) -(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 -O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H12: (drop (S n0) -O c e)).(or4_intro0 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T -(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e -(CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) -u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S -n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) -(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 -O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Flat f) n0 c e -H12 t))) (\lambda (H12: (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O c (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))).(ex3_4_ind F C T T (\lambda -(f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 -(Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f0) u1)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))) -(or4 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat -f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 -u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c -(Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c -(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda -(x4: T).(\lambda (x5: T).(\lambda (H13: (eq C e (CHead x3 (Flat x2) -x5))).(\lambda (H14: (drop (S n0) O c (CHead x3 (Flat x2) x4))).(\lambda -(H15: (subst0 O v x4 x5)).(eq_ind_r C (CHead x3 (Flat x2) x5) (\lambda (c0: -C).(or4 (drop (S n0) O (CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat -f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 -u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c -(Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(eq C c0 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c -(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead c (Flat -f) t) (CHead x3 (Flat x2) x5)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0 -(Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) -u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x3 (Flat x2) x5) (CHead e2 -(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O -v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e2 -(Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) -(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2))))))) (ex3_4_intro F C T T (\lambda (f0: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0 -(Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) -u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 O v u1 u2))))) x2 x3 x4 x5 (refl_equal C (CHead x3 (Flat x2) x5)) -(drop_drop (Flat f) n0 c (CHead x3 (Flat x2) x4) H14 t) H15)) e H13)))))))) -H12)) (\lambda (H12: (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O c (CHead -e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat -f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(drop (S n0) O c (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) (or4 (drop (S n0) -O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead -e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e -(CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) -(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda -(x5: T).(\lambda (H13: (eq C e (CHead x4 (Flat x2) x5))).(\lambda (H14: (drop -(S n0) O c (CHead x3 (Flat x2) x5))).(\lambda (H15: (csubst0 O v x3 -x4)).(eq_ind_r C (CHead x4 (Flat x2) x5) (\lambda (c0: C).(or4 (drop (S n0) O -(CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat f0) u2)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead -e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 -(CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) -(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead c (Flat f) t) (CHead x4 -(Flat x2) x5)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x5) (CHead e0 (Flat f0) -u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 -u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S -n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) -(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 (Flat f0) -u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) -u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) -(ex3_4_intro F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S -n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) -x2 x3 x4 x5 (refl_equal C (CHead x4 (Flat x2) x5)) (drop_drop (Flat f) n0 c -(CHead x3 (Flat x2) x5) H14 t) H15)) e H13)))))))) H12)) (\lambda (H12: -(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O c (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e -(CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f0) -u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) -(or4 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat -f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 -u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c -(Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c -(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda -(x4: C).(\lambda (x5: T).(\lambda (x6: T).(\lambda (H13: (eq C e (CHead x4 -(Flat x2) x6))).(\lambda (H14: (drop (S n0) O c (CHead x3 (Flat x2) -x5))).(\lambda (H15: (subst0 O v x5 x6)).(\lambda (H16: (csubst0 O v x3 -x4)).(eq_ind_r C (CHead x4 (Flat x2) x6) (\lambda (c0: C).(or4 (drop (S n0) O -(CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat f0) u2)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead -e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 -(CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) -(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead c (Flat f) t) (CHead x4 -(Flat x2) x6)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e0 (Flat f0) -u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 -u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S -n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) -(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 (Flat f0) -u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) -u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) -(ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 -(Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) -(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2)))))) x2 x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x2) x6)) -(drop_drop (Flat f) n0 c (CHead x3 (Flat x2) x5) H14 t) H15 H16)) e -H13)))))))))) H12)) H11))))))) k H3 (drop_gen_drop k x0 e t n0 H6)))))))) -H2)) (\lambda (H2: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda -(j: nat).(eq nat (S n0) (s k j))))) (\lambda (u2: T).(\lambda (c3: -C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda -(_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: -C).(\lambda (j: nat).(csubst0 j v c c3)))))).(ex4_3_ind T C nat (\lambda (_: -T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k j))))) (\lambda (u2: -T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda -(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: -T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (or4 (drop (S n0) -O (CHead c k t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda -(_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c -k t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat -f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) -(ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O (CHead c k t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: C).(\lambda -(x2: nat).(\lambda (H3: (eq nat (S n0) (s k x2))).(\lambda (H4: (eq C c2 -(CHead x1 k x0))).(\lambda (H5: (subst0 x2 v t x0)).(\lambda (H6: (csubst0 x2 -v c x1)).(let H7 \def (eq_ind C c2 (\lambda (c0: C).(drop (S n0) O c0 e)) H1 -(CHead x1 k x0) H4) in (K_ind (\lambda (k0: K).((eq nat (S n0) (s k0 x2)) \to -((drop (r k0 n0) O x1 e) \to (or4 (drop (S n0) O (CHead c k0 t) e) (ex3_4 F C -T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e -(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O (CHead c k0 t) (CHead e0 (Flat f) u1)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v -u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c -k0 t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda -(f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq -C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k0 t) (CHead -e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2))))))))))) (\lambda (b: B).(\lambda (H8: (eq nat (S n0) (s (Bind b) -x2))).(\lambda (H9: (drop (r (Bind b) n0) O x1 e)).(let H10 \def (f_equal nat -nat (\lambda (e0: nat).(match e0 in nat return (\lambda (_: nat).nat) with [O -\Rightarrow n0 | (S n1) \Rightarrow n1])) (S n0) (S x2) H8) in (let H11 \def -(eq_ind_r nat x2 (\lambda (n1: nat).(csubst0 n1 v c x1)) H6 n0 H10) in (let -H12 \def (eq_ind_r nat x2 (\lambda (n1: nat).(subst0 n1 v t x0)) H5 n0 H10) -in (let H13 \def (IHn c x1 v H11 e H9) in (or4_ind (drop n0 O c e) (ex3_4 F C -T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e -(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O -v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop n0 O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e) -(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: -T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) -(CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat -f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 -e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda -(H14: (drop n0 O c e)).(or4_intro0 (drop (S n0) O (CHead c (Bind b) t) e) -(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: -T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) -(CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat -f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 -e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop -(Bind b) n0 c e H14 t))) (\lambda (H14: (ex3_4 F C T T (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat -f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))).(ex3_4_ind F C -T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e -(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))) -(or4 (drop (S n0) O (CHead c (Bind b) t) e) (ex3_4 F C T T (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat -f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 -u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c -(Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c -(Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda -(x5: T).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x4 (Flat x3) -x6))).(\lambda (H16: (drop n0 O c (CHead x4 (Flat x3) x5))).(\lambda (H17: -(subst0 O v x5 x6)).(eq_ind_r C (CHead x4 (Flat x3) x6) (\lambda (c0: C).(or4 -(drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat -f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 -u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c -(Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c -(Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead c (Bind -b) t) (CHead x4 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0 -(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v -u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C (CHead x4 (Flat x3) x6) (CHead e2 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) -O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C -C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (u2: T).(eq C (CHead x4 (Flat x3) x6) (CHead e2 (Flat f) -u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) -u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) -(ex3_4_intro F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (u2: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0 (Flat f) u2)))))) -(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))) -x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x3) x6)) (drop_drop (Bind b) n0 c -(CHead x4 (Flat x3) x5) H16 t) H17)) e H15)))))))) H14)) (\lambda (H14: -(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O -v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1 -(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 O v e1 e2))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e) -(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: -T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) -(CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat -f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 -e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda -(x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (H15: (eq -C e (CHead x5 (Flat x3) x6))).(\lambda (H16: (drop n0 O c (CHead x4 (Flat x3) -x6))).(\lambda (H17: (csubst0 O v x4 x5)).(eq_ind_r C (CHead x5 (Flat x3) x6) -(\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T -(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 -(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) -u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) -O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C -C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 -O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro2 (drop (S n0) O -(CHead c (Bind b) t) (CHead x5 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat -x3) x6) (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda -(u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat -f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2 -(Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O -v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2 -(Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat -f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) -(ex3_4_intro F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) -O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) x3 x4 x5 x6 -(refl_equal C (CHead x5 (Flat x3) x6)) (drop_drop (Bind b) n0 c (CHead x4 -(Flat x3) x6) H16 t) H17)) e H15)))))))) H14)) (\lambda (H14: (ex4_5 F C C T -T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c (CHead e1 -(Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2)))))))).(ex4_5_ind F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop n0 O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e) -(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: -T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) -(CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat -f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 -e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda -(x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (x7: -T).(\lambda (H15: (eq C e (CHead x5 (Flat x3) x7))).(\lambda (H16: (drop n0 O -c (CHead x4 (Flat x3) x6))).(\lambda (H17: (subst0 O v x6 x7)).(\lambda (H18: -(csubst0 O v x4 x5)).(eq_ind_r C (CHead x5 (Flat x3) x7) (\lambda (c0: -C).(or4 (drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat -f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 -u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c -(Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c -(Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead c (Bind -b) t) (CHead x5 (Flat x3) x7)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e0 -(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v -u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) -O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C -C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2 (Flat f) -u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) -u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) -(ex4_5_intro F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2 -(Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat -f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) -x3 x4 x5 x6 x7 (refl_equal C (CHead x5 (Flat x3) x7)) (drop_drop (Bind b) n0 -c (CHead x4 (Flat x3) x6) H16 t) H17 H18)) e H15)))))))))) H14)) H13)))))))) -(\lambda (f: F).(\lambda (H8: (eq nat (S n0) (s (Flat f) x2))).(\lambda (H9: -(drop (r (Flat f) n0) O x1 e)).(let H10 \def (f_equal nat nat (\lambda (e0: -nat).e0) (S n0) x2 H8) in (let H11 \def (eq_ind_r nat x2 (\lambda (n1: -nat).(csubst0 n1 v c x1)) H6 (S n0) H10) in (let H12 \def (eq_ind_r nat x2 -(\lambda (n1: nat).(subst0 n1 v t x0)) H5 (S n0) H10) in (let H13 \def (H x1 -v H11 e H9) in (or4_ind (drop (S n0) O c e) (ex3_4 F C T T (\lambda (f0: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat -f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(drop (S n0) O c (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e -(CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(drop (S n0) O c (CHead e1 (Flat f0) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) -(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O c (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead c (Flat f) t) e) -(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda -(u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) -(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat -f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 -e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda -(H14: (drop (S n0) O c e)).(or4_intro0 (drop (S n0) O (CHead c (Flat f) t) e) -(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda -(u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) -(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat -f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 -e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop -(Flat f) n0 c e H14 t))) (\lambda (H14: (ex3_4 F C T T (\lambda (f0: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat -f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(drop (S n0) O c (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))).(ex3_4_ind F C -T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C -e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda -(u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f0) u1)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v -u1 u2))))) (or4 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T -(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e -(CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) -u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S -n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) -(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 -O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x3: F).(\lambda -(x4: C).(\lambda (x5: T).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x4 -(Flat x3) x6))).(\lambda (H16: (drop (S n0) O c (CHead x4 (Flat x3) -x5))).(\lambda (H17: (subst0 O v x5 x6)).(eq_ind_r C (CHead x4 (Flat x3) x6) -(\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Flat f) t) c0) (ex3_4 F C T T -(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 -(CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) -u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S -n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) -(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 -O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro1 (drop (S n0) O -(CHead c (Flat f) t) (CHead x4 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f0: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat -x3) x6) (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) -(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x4 (Flat x3) -x6) (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat -f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat -x3) x6) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c -(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C T T (\lambda (f0: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat -x3) x6) (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) -(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2))))) x3 x4 x5 x6 (refl_equal C (CHead -x4 (Flat x3) x6)) (drop_drop (Flat f) n0 c (CHead x4 (Flat x3) x5) H16 t) -H17)) e H15)))))))) H14)) (\lambda (H14: (ex3_4 F C C T (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat -f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(drop (S n0) O c (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C -C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e -(CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(drop (S n0) O c (CHead e1 (Flat f0) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) -(or4 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat -f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 -u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c -(Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c -(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda -(x5: C).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x5 (Flat x3) -x6))).(\lambda (H16: (drop (S n0) O c (CHead x4 (Flat x3) x6))).(\lambda -(H17: (csubst0 O v x4 x5)).(eq_ind_r C (CHead x5 (Flat x3) x6) (\lambda (c0: -C).(or4 (drop (S n0) O (CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat -f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 -u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c -(Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(eq C c0 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c -(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead c (Flat -f) t) (CHead x5 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x6) (CHead e0 -(Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) -u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2 -(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O -v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2 -(Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) -(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2))))))) (ex3_4_intro F C C T (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2 -(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O -v e1 e2))))) x3 x4 x5 x6 (refl_equal C (CHead x5 (Flat x3) x6)) (drop_drop -(Flat f) n0 c (CHead x4 (Flat x3) x6) H16 t) H17)) e H15)))))))) H14)) -(\lambda (H14: (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) -u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f0) u1))))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C -C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O c (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) O (CHead c (Flat f) t) e) -(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda -(u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) -(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat -f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 -e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda -(x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (x7: -T).(\lambda (H15: (eq C e (CHead x5 (Flat x3) x7))).(\lambda (H16: (drop (S -n0) O c (CHead x4 (Flat x3) x6))).(\lambda (H17: (subst0 O v x6 x7)).(\lambda -(H18: (csubst0 O v x4 x5)).(eq_ind_r C (CHead x5 (Flat x3) x7) (\lambda (c0: -C).(or4 (drop (S n0) O (CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat -f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 -u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c -(Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(eq C c0 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c -(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead c (Flat -f) t) (CHead x5 (Flat x3) x7)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e0 -(Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) -u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2 -(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O -v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2 -(Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) -(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2))))))) (ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat -x3) x7) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c -(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2)))))) x3 x4 x5 x6 x7 (refl_equal C (CHead x5 (Flat -x3) x7)) (drop_drop (Flat f) n0 c (CHead x4 (Flat x3) x6) H16 t) H17 H18)) e -H15)))))))))) H14)) H13)))))))) k H3 (drop_gen_drop k x1 e x0 n0 H7)))))))))) -H2)) (csubst0_gen_head k c c2 t v (S n0) H0))))))))))) c1)))) n). -(* COMMENTS -Initial nodes: 34765 -END *) - -theorem csubst0_drop_lt_back: - \forall (n: nat).(\forall (i: nat).((lt n i) \to (\forall (c1: C).(\forall -(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e2: C).((drop n O -c2 e2) \to (or (drop n O c1 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) -v e1 e2)) (\lambda (e1: C).(drop n O c1 e1)))))))))))) -\def - \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (i: nat).((lt n0 i) -\to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) -\to (\forall (e2: C).((drop n0 O c2 e2) \to (or (drop n0 O c1 e2) (ex2 C -(\lambda (e1: C).(csubst0 (minus i n0) v e1 e2)) (\lambda (e1: C).(drop n0 O -c1 e1))))))))))))) (\lambda (i: nat).(\lambda (_: (lt O i)).(\lambda (c1: -C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 i v c1 -c2)).(\lambda (e2: C).(\lambda (H1: (drop O O c2 e2)).(eq_ind C c2 (\lambda -(c: C).(or (drop O O c1 c) (ex2 C (\lambda (e1: C).(csubst0 (minus i O) v e1 -c)) (\lambda (e1: C).(drop O O c1 e1))))) (eq_ind nat i (\lambda (n0: -nat).(or (drop O O c1 c2) (ex2 C (\lambda (e1: C).(csubst0 n0 v e1 c2)) -(\lambda (e1: C).(drop O O c1 e1))))) (or_intror (drop O O c1 c2) (ex2 C -(\lambda (e1: C).(csubst0 i v e1 c2)) (\lambda (e1: C).(drop O O c1 e1))) -(ex_intro2 C (\lambda (e1: C).(csubst0 i v e1 c2)) (\lambda (e1: C).(drop O O -c1 e1)) c1 H0 (drop_refl c1))) (minus i O) (minus_n_O i)) e2 (drop_gen_refl -c2 e2 H1)))))))))) (\lambda (n0: nat).(\lambda (IHn: ((\forall (i: nat).((lt -n0 i) \to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 -c2) \to (\forall (e2: C).((drop n0 O c2 e2) \to (or (drop n0 O c1 e2) (ex2 C -(\lambda (e1: C).(csubst0 (minus i n0) v e1 e2)) (\lambda (e1: C).(drop n0 O -c1 e1)))))))))))))).(\lambda (i: nat).(\lambda (H: (lt (S n0) i)).(\lambda -(c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v -c c2) \to (\forall (e2: C).((drop (S n0) O c2 e2) \to (or (drop (S n0) O c -e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i (S n0)) v e1 e2)) (\lambda (e1: -C).(drop (S n0) O c e1)))))))))) (\lambda (n1: nat).(\lambda (c2: C).(\lambda -(v: T).(\lambda (H0: (csubst0 i v (CSort n1) c2)).(\lambda (e2: C).(\lambda -(_: (drop (S n0) O c2 e2)).(csubst0_gen_sort c2 v i n1 H0 (or (drop (S n0) O -(CSort n1) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i (S n0)) v e1 e2)) -(\lambda (e1: C).(drop (S n0) O (CSort n1) e1))))))))))) (\lambda (c: -C).(\lambda (H0: ((\forall (c2: C).(\forall (v: T).((csubst0 i v c c2) \to -(\forall (e2: C).((drop (S n0) O c2 e2) \to (or (drop (S n0) O c e2) (ex2 C -(\lambda (e1: C).(csubst0 (minus i (S n0)) v e1 e2)) (\lambda (e1: C).(drop -(S n0) O c e1))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: -C).(\lambda (v: T).(\lambda (H1: (csubst0 i v (CHead c k t) c2)).(\lambda -(e2: C).(\lambda (H2: (drop (S n0) O c2 e2)).(or3_ind (ex3_2 T nat (\lambda -(_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: -nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j -v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k -j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda -(c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_: -T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: -T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda -(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: -T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))) (or (drop (S n0) -O (CHead c k t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i (S n0)) v e1 -e2)) (\lambda (e1: C).(drop (S n0) O (CHead c k t) e1)))) (\lambda (H3: -(ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda -(u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: -T).(\lambda (j: nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: -T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: -nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j -v t u2))) (or (drop (S n0) O (CHead c k t) e2) (ex2 C (\lambda (e1: -C).(csubst0 (minus i (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead -c k t) e1)))) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H4: (eq nat i (s -k x1))).(\lambda (H5: (eq C c2 (CHead c k x0))).(\lambda (_: (subst0 x1 v t -x0)).(let H7 \def (eq_ind C c2 (\lambda (c0: C).(drop (S n0) O c0 e2)) H2 -(CHead c k x0) H5) in (let H8 \def (eq_ind nat i (\lambda (n1: nat).(\forall -(c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall (e3: C).((drop (S -n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: C).(csubst0 -(minus n1 (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop (S n0) O c e1)))))))))) -H0 (s k x1) H4) in (let H9 \def (eq_ind nat i (\lambda (n1: nat).(lt (S n0) -n1)) H (s k x1) H4) in (eq_ind_r nat (s k x1) (\lambda (n1: nat).(or (drop (S -n0) O (CHead c k t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus n1 (S n0)) v -e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c k t) e1))))) (K_ind (\lambda -(k0: K).(((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c c3) \to -(\forall (e3: C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C -(\lambda (e1: C).(csubst0 (minus (s k0 x1) (S n0)) v0 e1 e3)) (\lambda (e1: -C).(drop (S n0) O c e1)))))))))) \to ((lt (S n0) (s k0 x1)) \to ((drop (r k0 -n0) O c e2) \to (or (drop (S n0) O (CHead c k0 t) e2) (ex2 C (\lambda (e1: -C).(csubst0 (minus (s k0 x1) (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) -O (CHead c k0 t) e1)))))))) (\lambda (b: B).(\lambda (_: ((\forall (c3: -C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to (\forall (e3: -C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: -C).(csubst0 (minus (s (Bind b) x1) (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop -(S n0) O c e1))))))))))).(\lambda (_: (lt (S n0) (s (Bind b) x1))).(\lambda -(H12: (drop (r (Bind b) n0) O c e2)).(or_introl (drop (S n0) O (CHead c (Bind -b) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x1 n0) v e1 e2)) (\lambda -(e1: C).(drop (S n0) O (CHead c (Bind b) t) e1))) (drop_drop (Bind b) n0 c e2 -H12 t)))))) (\lambda (f: F).(\lambda (_: ((\forall (c3: C).(\forall (v0: -T).((csubst0 (s (Flat f) x1) v0 c c3) \to (\forall (e3: C).((drop (S n0) O c3 -e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: C).(csubst0 (minus (s -(Flat f) x1) (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop (S n0) O c -e1))))))))))).(\lambda (_: (lt (S n0) (s (Flat f) x1))).(\lambda (H12: (drop -(r (Flat f) n0) O c e2)).(or_introl (drop (S n0) O (CHead c (Flat f) t) e2) -(ex2 C (\lambda (e1: C).(csubst0 (minus x1 (S n0)) v e1 e2)) (\lambda (e1: -C).(drop (S n0) O (CHead c (Flat f) t) e1))) (drop_drop (Flat f) n0 c e2 H12 -t)))))) k H8 H9 (drop_gen_drop k c e2 x0 n0 H7)) i H4))))))))) H3)) (\lambda -(H3: (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j)))) -(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: -C).(\lambda (j: nat).(csubst0 j v c c3))))).(ex3_2_ind C nat (\lambda (_: -C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_: -nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j -v c c3))) (or (drop (S n0) O (CHead c k t) e2) (ex2 C (\lambda (e1: -C).(csubst0 (minus i (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead -c k t) e1)))) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H4: (eq nat i (s -k x1))).(\lambda (H5: (eq C c2 (CHead x0 k t))).(\lambda (H6: (csubst0 x1 v c -x0)).(let H7 \def (eq_ind C c2 (\lambda (c0: C).(drop (S n0) O c0 e2)) H2 -(CHead x0 k t) H5) in (let H8 \def (eq_ind nat i (\lambda (n1: nat).(\forall -(c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall (e3: C).((drop (S -n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: C).(csubst0 -(minus n1 (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop (S n0) O c e1)))))))))) -H0 (s k x1) H4) in (let H9 \def (eq_ind nat i (\lambda (n1: nat).(lt (S n0) -n1)) H (s k x1) H4) in (eq_ind_r nat (s k x1) (\lambda (n1: nat).(or (drop (S -n0) O (CHead c k t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus n1 (S n0)) v -e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c k t) e1))))) (K_ind (\lambda -(k0: K).(((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c c3) \to -(\forall (e3: C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C -(\lambda (e1: C).(csubst0 (minus (s k0 x1) (S n0)) v0 e1 e3)) (\lambda (e1: -C).(drop (S n0) O c e1)))))))))) \to ((lt (S n0) (s k0 x1)) \to ((drop (r k0 -n0) O x0 e2) \to (or (drop (S n0) O (CHead c k0 t) e2) (ex2 C (\lambda (e1: -C).(csubst0 (minus (s k0 x1) (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) -O (CHead c k0 t) e1)))))))) (\lambda (b: B).(\lambda (_: ((\forall (c3: -C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to (\forall (e3: -C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: -C).(csubst0 (minus (s (Bind b) x1) (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop -(S n0) O c e1))))))))))).(\lambda (H11: (lt (S n0) (s (Bind b) x1))).(\lambda -(H12: (drop (r (Bind b) n0) O x0 e2)).(let H_x \def (IHn x1 (lt_S_n n0 x1 -H11) c x0 v H6 e2 H12) in (let H13 \def H_x in (or_ind (drop n0 O c e2) (ex2 -C (\lambda (e1: C).(csubst0 (minus x1 n0) v e1 e2)) (\lambda (e1: C).(drop n0 -O c e1))) (or (drop (S n0) O (CHead c (Bind b) t) e2) (ex2 C (\lambda (e1: -C).(csubst0 (minus x1 n0) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c -(Bind b) t) e1)))) (\lambda (H14: (drop n0 O c e2)).(or_introl (drop (S n0) O -(CHead c (Bind b) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x1 n0) v e1 -e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Bind b) t) e1))) (drop_drop -(Bind b) n0 c e2 H14 t))) (\lambda (H14: (ex2 C (\lambda (e1: C).(csubst0 -(minus x1 n0) v e1 e2)) (\lambda (e1: C).(drop n0 O c e1)))).(ex2_ind C -(\lambda (e1: C).(csubst0 (minus x1 n0) v e1 e2)) (\lambda (e1: C).(drop n0 O -c e1)) (or (drop (S n0) O (CHead c (Bind b) t) e2) (ex2 C (\lambda (e1: -C).(csubst0 (minus x1 n0) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c -(Bind b) t) e1)))) (\lambda (x: C).(\lambda (H15: (csubst0 (minus x1 n0) v x -e2)).(\lambda (H16: (drop n0 O c x)).(or_intror (drop (S n0) O (CHead c (Bind -b) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x1 n0) v e1 e2)) (\lambda -(e1: C).(drop (S n0) O (CHead c (Bind b) t) e1))) (ex_intro2 C (\lambda (e1: -C).(csubst0 (minus x1 n0) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c -(Bind b) t) e1)) x H15 (drop_drop (Bind b) n0 c x H16 t)))))) H14)) -H13))))))) (\lambda (f: F).(\lambda (H10: ((\forall (c3: C).(\forall (v0: -T).((csubst0 (s (Flat f) x1) v0 c c3) \to (\forall (e3: C).((drop (S n0) O c3 -e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: C).(csubst0 (minus (s -(Flat f) x1) (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop (S n0) O c -e1))))))))))).(\lambda (_: (lt (S n0) (s (Flat f) x1))).(\lambda (H12: (drop -(r (Flat f) n0) O x0 e2)).(let H_x \def (H10 x0 v H6 e2 H12) in (let H13 \def -H_x in (or_ind (drop (S n0) O c e2) (ex2 C (\lambda (e1: C).(csubst0 (minus -x1 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O c e1))) (or (drop (S n0) -O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x1 (S n0)) -v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Flat f) t) e1)))) -(\lambda (H14: (drop (S n0) O c e2)).(or_introl (drop (S n0) O (CHead c (Flat -f) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x1 (S n0)) v e1 e2)) -(\lambda (e1: C).(drop (S n0) O (CHead c (Flat f) t) e1))) (drop_drop (Flat -f) n0 c e2 H14 t))) (\lambda (H14: (ex2 C (\lambda (e1: C).(csubst0 (minus x1 -(S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O c e1)))).(ex2_ind C -(\lambda (e1: C).(csubst0 (minus x1 (S n0)) v e1 e2)) (\lambda (e1: C).(drop -(S n0) O c e1)) (or (drop (S n0) O (CHead c (Flat f) t) e2) (ex2 C (\lambda -(e1: C).(csubst0 (minus x1 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O -(CHead c (Flat f) t) e1)))) (\lambda (x: C).(\lambda (H15: (csubst0 (minus x1 -(S n0)) v x e2)).(\lambda (H16: (drop (S n0) O c x)).(or_intror (drop (S n0) -O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x1 (S n0)) -v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Flat f) t) e1))) -(ex_intro2 C (\lambda (e1: C).(csubst0 (minus x1 (S n0)) v e1 e2)) (\lambda -(e1: C).(drop (S n0) O (CHead c (Flat f) t) e1)) x H15 (drop_drop (Flat f) n0 -c x H16 t)))))) H14)) H13))))))) k H8 H9 (drop_gen_drop k x0 e2 t n0 H7)) i -H4))))))))) H3)) (\lambda (H3: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: -C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: -C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda -(_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: -C).(\lambda (j: nat).(csubst0 j v c c3)))))).(ex4_3_ind T C nat (\lambda (_: -T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: -T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda -(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: -T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (or (drop (S n0) -O (CHead c k t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i (S n0)) v e1 -e2)) (\lambda (e1: C).(drop (S n0) O (CHead c k t) e1)))) (\lambda (x0: -T).(\lambda (x1: C).(\lambda (x2: nat).(\lambda (H4: (eq nat i (s k -x2))).(\lambda (H5: (eq C c2 (CHead x1 k x0))).(\lambda (_: (subst0 x2 v t -x0)).(\lambda (H7: (csubst0 x2 v c x1)).(let H8 \def (eq_ind C c2 (\lambda -(c0: C).(drop (S n0) O c0 e2)) H2 (CHead x1 k x0) H5) in (let H9 \def (eq_ind -nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c -c3) \to (\forall (e3: C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) -(ex2 C (\lambda (e1: C).(csubst0 (minus n1 (S n0)) v0 e1 e3)) (\lambda (e1: -C).(drop (S n0) O c e1)))))))))) H0 (s k x2) H4) in (let H10 \def (eq_ind nat -i (\lambda (n1: nat).(lt (S n0) n1)) H (s k x2) H4) in (eq_ind_r nat (s k x2) -(\lambda (n1: nat).(or (drop (S n0) O (CHead c k t) e2) (ex2 C (\lambda (e1: -C).(csubst0 (minus n1 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O -(CHead c k t) e1))))) (K_ind (\lambda (k0: K).(((\forall (c3: C).(\forall -(v0: T).((csubst0 (s k0 x2) v0 c c3) \to (\forall (e3: C).((drop (S n0) O c3 -e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: C).(csubst0 (minus (s -k0 x2) (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop (S n0) O c e1)))))))))) \to -((lt (S n0) (s k0 x2)) \to ((drop (r k0 n0) O x1 e2) \to (or (drop (S n0) O -(CHead c k0 t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus (s k0 x2) (S n0)) -v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c k0 t) e1)))))))) (\lambda -(b: B).(\lambda (_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Bind b) -x2) v0 c c3) \to (\forall (e3: C).((drop (S n0) O c3 e3) \to (or (drop (S n0) -O c e3) (ex2 C (\lambda (e1: C).(csubst0 (minus (s (Bind b) x2) (S n0)) v0 e1 -e3)) (\lambda (e1: C).(drop (S n0) O c e1))))))))))).(\lambda (H12: (lt (S -n0) (s (Bind b) x2))).(\lambda (H13: (drop (r (Bind b) n0) O x1 e2)).(let H_x -\def (IHn x2 (lt_S_n n0 x2 H12) c x1 v H7 e2 H13) in (let H14 \def H_x in -(or_ind (drop n0 O c e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x2 n0) v e1 -e2)) (\lambda (e1: C).(drop n0 O c e1))) (or (drop (S n0) O (CHead c (Bind b) -t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x2 n0) v e1 e2)) (\lambda (e1: -C).(drop (S n0) O (CHead c (Bind b) t) e1)))) (\lambda (H15: (drop n0 O c -e2)).(or_introl (drop (S n0) O (CHead c (Bind b) t) e2) (ex2 C (\lambda (e1: -C).(csubst0 (minus x2 n0) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c -(Bind b) t) e1))) (drop_drop (Bind b) n0 c e2 H15 t))) (\lambda (H15: (ex2 C -(\lambda (e1: C).(csubst0 (minus x2 n0) v e1 e2)) (\lambda (e1: C).(drop n0 O -c e1)))).(ex2_ind C (\lambda (e1: C).(csubst0 (minus x2 n0) v e1 e2)) -(\lambda (e1: C).(drop n0 O c e1)) (or (drop (S n0) O (CHead c (Bind b) t) -e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x2 n0) v e1 e2)) (\lambda (e1: -C).(drop (S n0) O (CHead c (Bind b) t) e1)))) (\lambda (x: C).(\lambda (H16: -(csubst0 (minus x2 n0) v x e2)).(\lambda (H17: (drop n0 O c x)).(or_intror -(drop (S n0) O (CHead c (Bind b) t) e2) (ex2 C (\lambda (e1: C).(csubst0 -(minus x2 n0) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Bind b) t) -e1))) (ex_intro2 C (\lambda (e1: C).(csubst0 (minus x2 n0) v e1 e2)) (\lambda -(e1: C).(drop (S n0) O (CHead c (Bind b) t) e1)) x H16 (drop_drop (Bind b) n0 -c x H17 t)))))) H15)) H14))))))) (\lambda (f: F).(\lambda (H11: ((\forall -(c3: C).(\forall (v0: T).((csubst0 (s (Flat f) x2) v0 c c3) \to (\forall (e3: -C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: -C).(csubst0 (minus (s (Flat f) x2) (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop -(S n0) O c e1))))))))))).(\lambda (_: (lt (S n0) (s (Flat f) x2))).(\lambda -(H13: (drop (r (Flat f) n0) O x1 e2)).(let H_x \def (H11 x1 v H7 e2 H13) in -(let H14 \def H_x in (or_ind (drop (S n0) O c e2) (ex2 C (\lambda (e1: -C).(csubst0 (minus x2 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O c -e1))) (or (drop (S n0) O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1: -C).(csubst0 (minus x2 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O -(CHead c (Flat f) t) e1)))) (\lambda (H15: (drop (S n0) O c e2)).(or_introl -(drop (S n0) O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1: C).(csubst0 -(minus x2 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Flat f) -t) e1))) (drop_drop (Flat f) n0 c e2 H15 t))) (\lambda (H15: (ex2 C (\lambda -(e1: C).(csubst0 (minus x2 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O -c e1)))).(ex2_ind C (\lambda (e1: C).(csubst0 (minus x2 (S n0)) v e1 e2)) -(\lambda (e1: C).(drop (S n0) O c e1)) (or (drop (S n0) O (CHead c (Flat f) -t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x2 (S n0)) v e1 e2)) (\lambda -(e1: C).(drop (S n0) O (CHead c (Flat f) t) e1)))) (\lambda (x: C).(\lambda -(H16: (csubst0 (minus x2 (S n0)) v x e2)).(\lambda (H17: (drop (S n0) O c -x)).(or_intror (drop (S n0) O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1: -C).(csubst0 (minus x2 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O -(CHead c (Flat f) t) e1))) (ex_intro2 C (\lambda (e1: C).(csubst0 (minus x2 -(S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Flat f) t) e1)) x -H16 (drop_drop (Flat f) n0 c x H17 t)))))) H15)) H14))))))) k H9 H10 -(drop_gen_drop k x1 e2 x0 n0 H8)) i H4))))))))))) H3)) (csubst0_gen_head k c -c2 t v i H1))))))))))) c1)))))) n). -(* COMMENTS -Initial nodes: 5939 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubst0/fwd.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubst0/fwd.ma deleted file mode 100644 index 9b3de2983..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubst0/fwd.ma +++ /dev/null @@ -1,462 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubst0/defs.ma". - -theorem csubst0_gen_sort: - \forall (x: C).(\forall (v: T).(\forall (i: nat).(\forall (n: nat).((csubst0 -i v (CSort n) x) \to (\forall (P: Prop).P))))) -\def - \lambda (x: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda -(H: (csubst0 i v (CSort n) x)).(\lambda (P: Prop).(insert_eq C (CSort n) -(\lambda (c: C).(csubst0 i v c x)) (\lambda (_: C).P) (\lambda (y: -C).(\lambda (H0: (csubst0 i v y x)).(csubst0_ind (\lambda (_: nat).(\lambda -(_: T).(\lambda (c: C).(\lambda (_: C).((eq C c (CSort n)) \to P))))) -(\lambda (k: K).(\lambda (i0: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda -(u2: T).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (c: C).(\lambda (H2: (eq -C (CHead c k u1) (CSort n))).(let H3 \def (eq_ind C (CHead c k u1) (\lambda -(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H2) in -(False_ind P H3)))))))))) (\lambda (k: K).(\lambda (i0: nat).(\lambda (c1: -C).(\lambda (c2: C).(\lambda (v0: T).(\lambda (_: (csubst0 i0 v0 c1 -c2)).(\lambda (_: (((eq C c1 (CSort n)) \to P))).(\lambda (u: T).(\lambda -(H3: (eq C (CHead c1 k u) (CSort n))).(let H4 \def (eq_ind C (CHead c1 k u) -(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H3) in -(False_ind P H4))))))))))) (\lambda (k: K).(\lambda (i0: nat).(\lambda (v0: -T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0 i0 v0 u1 -u2)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csubst0 i0 v0 c1 -c2)).(\lambda (_: (((eq C c1 (CSort n)) \to P))).(\lambda (H4: (eq C (CHead -c1 k u1) (CSort n))).(let H5 \def (eq_ind C (CHead c1 k u1) (\lambda (ee: -C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow -False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H4) in (False_ind P -H5))))))))))))) i v y x H0))) H)))))). -(* COMMENTS -Initial nodes: 355 -END *) - -theorem csubst0_gen_head: - \forall (k: K).(\forall (c1: C).(\forall (x: C).(\forall (u1: T).(\forall -(v: T).(\forall (i: nat).((csubst0 i v (CHead c1 k u1) x) \to (or3 (ex3_2 T -nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: -T).(\lambda (_: nat).(eq C x (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: -nat).(subst0 j v u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq -nat i (s k j)))) (\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2 k -u1)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 c2)))) (ex4_3 T C -nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) -(\lambda (u2: T).(\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2 k -u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 -u2)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 -c2)))))))))))) -\def - \lambda (k: K).(\lambda (c1: C).(\lambda (x: C).(\lambda (u1: T).(\lambda -(v: T).(\lambda (i: nat).(\lambda (H: (csubst0 i v (CHead c1 k u1) -x)).(insert_eq C (CHead c1 k u1) (\lambda (c: C).(csubst0 i v c x)) (\lambda -(_: C).(or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k -j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C x (CHead c1 k u2)))) (\lambda -(u2: T).(\lambda (j: nat).(subst0 j v u1 u2)))) (ex3_2 C nat (\lambda (_: -C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c2: C).(\lambda (_: -nat).(eq C x (CHead c2 k u1)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j -v c1 c2)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: -nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c2: C).(\lambda (_: -nat).(eq C x (CHead c2 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda -(j: nat).(subst0 j v u1 u2)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: -nat).(csubst0 j v c1 c2))))))) (\lambda (y: C).(\lambda (H0: (csubst0 i v y -x)).(csubst0_ind (\lambda (n: nat).(\lambda (t: T).(\lambda (c: C).(\lambda -(c0: C).((eq C c (CHead c1 k u1)) \to (or3 (ex3_2 T nat (\lambda (_: -T).(\lambda (j: nat).(eq nat n (s k j)))) (\lambda (u2: T).(\lambda (_: -nat).(eq C c0 (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j -t u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat n (s k -j)))) (\lambda (c2: C).(\lambda (_: nat).(eq C c0 (CHead c2 k u1)))) (\lambda -(c2: C).(\lambda (j: nat).(csubst0 j t c1 c2)))) (ex4_3 T C nat (\lambda (_: -T).(\lambda (_: C).(\lambda (j: nat).(eq nat n (s k j))))) (\lambda (u2: -T).(\lambda (c2: C).(\lambda (_: nat).(eq C c0 (CHead c2 k u2))))) (\lambda -(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j t u1 u2)))) (\lambda (_: -T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j t c1 c2))))))))))) (\lambda -(k0: K).(\lambda (i0: nat).(\lambda (v0: T).(\lambda (u0: T).(\lambda (u2: -T).(\lambda (H1: (subst0 i0 v0 u0 u2)).(\lambda (c: C).(\lambda (H2: (eq C -(CHead c k0 u0) (CHead c1 k u1))).(let H3 \def (f_equal C C (\lambda (e: -C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c | -(CHead c0 _ _) \Rightarrow c0])) (CHead c k0 u0) (CHead c1 k u1) H2) in ((let -H4 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) -with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead c k0 -u0) (CHead c1 k u1) H2) in ((let H5 \def (f_equal C T (\lambda (e: C).(match -e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ -t) \Rightarrow t])) (CHead c k0 u0) (CHead c1 k u1) H2) in (\lambda (H6: (eq -K k0 k)).(\lambda (H7: (eq C c c1)).(eq_ind_r C c1 (\lambda (c0: C).(or3 -(ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) -(\lambda (u3: T).(\lambda (_: nat).(eq C (CHead c0 k0 u2) (CHead c1 k u3)))) -(\lambda (u3: T).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (ex3_2 C nat -(\lambda (_: C).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) (\lambda (c2: -C).(\lambda (_: nat).(eq C (CHead c0 k0 u2) (CHead c2 k u1)))) (\lambda (c2: -C).(\lambda (j: nat).(csubst0 j v0 c1 c2)))) (ex4_3 T C nat (\lambda (_: -T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k0 i0) (s k j))))) (\lambda -(u3: T).(\lambda (c2: C).(\lambda (_: nat).(eq C (CHead c0 k0 u2) (CHead c2 k -u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 -u3)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v0 c1 -c2))))))) (let H8 \def (eq_ind T u0 (\lambda (t: T).(subst0 i0 v0 t u2)) H1 -u1 H5) in (eq_ind_r K k (\lambda (k1: K).(or3 (ex3_2 T nat (\lambda (_: -T).(\lambda (j: nat).(eq nat (s k1 i0) (s k j)))) (\lambda (u3: T).(\lambda -(_: nat).(eq C (CHead c1 k1 u2) (CHead c1 k u3)))) (\lambda (u3: T).(\lambda -(j: nat).(subst0 j v0 u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: -nat).(eq nat (s k1 i0) (s k j)))) (\lambda (c2: C).(\lambda (_: nat).(eq C -(CHead c1 k1 u2) (CHead c2 k u1)))) (\lambda (c2: C).(\lambda (j: -nat).(csubst0 j v0 c1 c2)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: -C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j))))) (\lambda (u3: T).(\lambda -(c2: C).(\lambda (_: nat).(eq C (CHead c1 k1 u2) (CHead c2 k u3))))) (\lambda -(u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (\lambda (_: -T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v0 c1 c2))))))) (or3_intro0 -(ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) -(\lambda (u3: T).(\lambda (_: nat).(eq C (CHead c1 k u2) (CHead c1 k u3)))) -(\lambda (u3: T).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (ex3_2 C nat -(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (c2: -C).(\lambda (_: nat).(eq C (CHead c1 k u2) (CHead c2 k u1)))) (\lambda (c2: -C).(\lambda (j: nat).(csubst0 j v0 c1 c2)))) (ex4_3 T C nat (\lambda (_: -T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))) (\lambda -(u3: T).(\lambda (c2: C).(\lambda (_: nat).(eq C (CHead c1 k u2) (CHead c2 k -u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 -u3)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v0 c1 -c2))))) (ex3_2_intro T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i0) -(s k j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C (CHead c1 k u2) (CHead c1 -k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j v0 u1 u3))) u2 i0 -(refl_equal nat (s k i0)) (refl_equal C (CHead c1 k u2)) H8)) k0 H6)) c -H7)))) H4)) H3)))))))))) (\lambda (k0: K).(\lambda (i0: nat).(\lambda (c0: -C).(\lambda (c2: C).(\lambda (v0: T).(\lambda (H1: (csubst0 i0 v0 c0 -c2)).(\lambda (H2: (((eq C c0 (CHead c1 k u1)) \to (or3 (ex3_2 T nat (\lambda -(_: T).(\lambda (j: nat).(eq nat i0 (s k j)))) (\lambda (u2: T).(\lambda (_: -nat).(eq C c2 (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j -v0 u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i0 (s k -j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u1)))) (\lambda -(c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: -T).(\lambda (_: C).(\lambda (j: nat).(eq nat i0 (s k j))))) (\lambda (u2: -T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda -(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 u2)))) (\lambda (_: -T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3))))))))).(\lambda -(u: T).(\lambda (H3: (eq C (CHead c0 k0 u) (CHead c1 k u1))).(let H4 \def -(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with -[(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k0 u) -(CHead c1 k u1) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e in -C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) -\Rightarrow k1])) (CHead c0 k0 u) (CHead c1 k u1) H3) in ((let H6 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 u) -(CHead c1 k u1) H3) in (\lambda (H7: (eq K k0 k)).(\lambda (H8: (eq C c0 -c1)).(eq_ind_r T u1 (\lambda (t: T).(or3 (ex3_2 T nat (\lambda (_: -T).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) (\lambda (u2: T).(\lambda -(_: nat).(eq C (CHead c2 k0 t) (CHead c1 k u2)))) (\lambda (u2: T).(\lambda -(j: nat).(subst0 j v0 u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: -nat).(eq nat (s k0 i0) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C -(CHead c2 k0 t) (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: -nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: -C).(\lambda (j: nat).(eq nat (s k0 i0) (s k j))))) (\lambda (u2: T).(\lambda -(c3: C).(\lambda (_: nat).(eq C (CHead c2 k0 t) (CHead c3 k u2))))) (\lambda -(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 u2)))) (\lambda (_: -T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3))))))) (let H9 \def -(eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 k u1)) \to (or3 (ex3_2 T nat -(\lambda (_: T).(\lambda (j: nat).(eq nat i0 (s k j)))) (\lambda (u2: -T).(\lambda (_: nat).(eq C c2 (CHead c1 k u2)))) (\lambda (u2: T).(\lambda -(j: nat).(subst0 j v0 u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: -nat).(eq nat i0 (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead -c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 -T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat i0 (s k -j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 -k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 -u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 -c3)))))))) H2 c1 H8) in (let H10 \def (eq_ind C c0 (\lambda (c: C).(csubst0 -i0 v0 c c2)) H1 c1 H8) in (eq_ind_r K k (\lambda (k1: K).(or3 (ex3_2 T nat -(\lambda (_: T).(\lambda (j: nat).(eq nat (s k1 i0) (s k j)))) (\lambda (u2: -T).(\lambda (_: nat).(eq C (CHead c2 k1 u1) (CHead c1 k u2)))) (\lambda (u2: -T).(\lambda (j: nat).(subst0 j v0 u1 u2)))) (ex3_2 C nat (\lambda (_: -C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j)))) (\lambda (c3: C).(\lambda -(_: nat).(eq C (CHead c2 k1 u1) (CHead c3 k u1)))) (\lambda (c3: C).(\lambda -(j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: -C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j))))) (\lambda (u2: T).(\lambda -(c3: C).(\lambda (_: nat).(eq C (CHead c2 k1 u1) (CHead c3 k u2))))) (\lambda -(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 u2)))) (\lambda (_: -T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3))))))) (or3_intro1 -(ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) -(\lambda (u2: T).(\lambda (_: nat).(eq C (CHead c2 k u1) (CHead c1 k u2)))) -(\lambda (u2: T).(\lambda (j: nat).(subst0 j v0 u1 u2)))) (ex3_2 C nat -(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (c3: -C).(\lambda (_: nat).(eq C (CHead c2 k u1) (CHead c3 k u1)))) (\lambda (c3: -C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: -T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))) (\lambda -(u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k u1) (CHead c3 k -u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 -u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 -c3))))) (ex3_2_intro C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) -(s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k u1) (CHead c3 -k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3))) c2 i0 -(refl_equal nat (s k i0)) (refl_equal C (CHead c2 k u1)) H10)) k0 H7))) u -H6)))) H5)) H4))))))))))) (\lambda (k0: K).(\lambda (i0: nat).(\lambda (v0: -T).(\lambda (u0: T).(\lambda (u2: T).(\lambda (H1: (subst0 i0 v0 u0 -u2)).(\lambda (c0: C).(\lambda (c2: C).(\lambda (H2: (csubst0 i0 v0 c0 -c2)).(\lambda (H3: (((eq C c0 (CHead c1 k u1)) \to (or3 (ex3_2 T nat (\lambda -(_: T).(\lambda (j: nat).(eq nat i0 (s k j)))) (\lambda (u3: T).(\lambda (_: -nat).(eq C c2 (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j -v0 u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i0 (s k -j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u1)))) (\lambda -(c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: -T).(\lambda (_: C).(\lambda (j: nat).(eq nat i0 (s k j))))) (\lambda (u3: -T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u3))))) (\lambda -(u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (\lambda (_: -T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3))))))))).(\lambda -(H4: (eq C (CHead c0 k0 u0) (CHead c1 k u1))).(let H5 \def (f_equal C C -(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k0 u0) (CHead c1 k -u1) H4) in ((let H6 \def (f_equal C K (\lambda (e: C).(match e in C return -(\lambda (_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) -\Rightarrow k1])) (CHead c0 k0 u0) (CHead c1 k u1) H4) in ((let H7 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 u0) -(CHead c1 k u1) H4) in (\lambda (H8: (eq K k0 k)).(\lambda (H9: (eq C c0 -c1)).(let H10 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 k u1)) \to -(or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i0 (s k j)))) -(\lambda (u3: T).(\lambda (_: nat).(eq C c2 (CHead c1 k u3)))) (\lambda (u3: -T).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (ex3_2 C nat (\lambda (_: -C).(\lambda (j: nat).(eq nat i0 (s k j)))) (\lambda (c3: C).(\lambda (_: -nat).(eq C c2 (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 -j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: -nat).(eq nat i0 (s k j))))) (\lambda (u3: T).(\lambda (c3: C).(\lambda (_: -nat).(eq C c2 (CHead c3 k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda -(j: nat).(subst0 j v0 u1 u3)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: -nat).(csubst0 j v0 c1 c3)))))))) H3 c1 H9) in (let H11 \def (eq_ind C c0 -(\lambda (c: C).(csubst0 i0 v0 c c2)) H2 c1 H9) in (let H12 \def (eq_ind T u0 -(\lambda (t: T).(subst0 i0 v0 t u2)) H1 u1 H7) in (eq_ind_r K k (\lambda (k1: -K).(or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k1 i0) (s k -j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C (CHead c2 k1 u2) (CHead c1 k -u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (ex3_2 C nat -(\lambda (_: C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j)))) (\lambda (c3: -C).(\lambda (_: nat).(eq C (CHead c2 k1 u2) (CHead c3 k u1)))) (\lambda (c3: -C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: -T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j))))) (\lambda -(u3: T).(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k1 u2) (CHead c3 k -u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 -u3)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 -c3))))))) (or3_intro2 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat -(s k i0) (s k j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C (CHead c2 k u2) -(CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j v0 u1 u3)))) -(ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) -(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k u2) (CHead c3 k u1)))) -(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat -(\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k -j))))) (\lambda (u3: T).(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k -u2) (CHead c3 k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: -nat).(subst0 j v0 u1 u3)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: -nat).(csubst0 j v0 c1 c3))))) (ex4_3_intro T C nat (\lambda (_: T).(\lambda -(_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))) (\lambda (u3: -T).(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k u2) (CHead c3 k -u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 -u3)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 -c3)))) u2 c2 i0 (refl_equal nat (s k i0)) (refl_equal C (CHead c2 k u2)) H12 -H11)) k0 H8))))))) H6)) H5))))))))))))) i v y x H0))) H))))))). -(* COMMENTS -Initial nodes: 4039 -END *) - -theorem csubst0_gen_S_bind_2: - \forall (b: B).(\forall (x: C).(\forall (c2: C).(\forall (v: T).(\forall -(v2: T).(\forall (i: nat).((csubst0 (S i) v x (CHead c2 (Bind b) v2)) \to -(or3 (ex2 T (\lambda (v1: T).(subst0 i v v1 v2)) (\lambda (v1: T).(eq C x -(CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i v c1 c2)) -(\lambda (c1: C).(eq C x (CHead c1 (Bind b) v2)))) (ex3_2 C T (\lambda (_: -C).(\lambda (v1: T).(subst0 i v v1 v2))) (\lambda (c1: C).(\lambda (_: -T).(csubst0 i v c1 c2))) (\lambda (c1: C).(\lambda (v1: T).(eq C x (CHead c1 -(Bind b) v1)))))))))))) -\def - \lambda (b: B).(\lambda (x: C).(\lambda (c2: C).(\lambda (v: T).(\lambda -(v2: T).(\lambda (i: nat).(\lambda (H: (csubst0 (S i) v x (CHead c2 (Bind b) -v2))).(insert_eq C (CHead c2 (Bind b) v2) (\lambda (c: C).(csubst0 (S i) v x -c)) (\lambda (_: C).(or3 (ex2 T (\lambda (v1: T).(subst0 i v v1 v2)) (\lambda -(v1: T).(eq C x (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i -v c1 c2)) (\lambda (c1: C).(eq C x (CHead c1 (Bind b) v2)))) (ex3_2 C T -(\lambda (_: C).(\lambda (v1: T).(subst0 i v v1 v2))) (\lambda (c1: -C).(\lambda (_: T).(csubst0 i v c1 c2))) (\lambda (c1: C).(\lambda (v1: -T).(eq C x (CHead c1 (Bind b) v1))))))) (\lambda (y: C).(\lambda (H0: -(csubst0 (S i) v x y)).(insert_eq nat (S i) (\lambda (n: nat).(csubst0 n v x -y)) (\lambda (_: nat).((eq C y (CHead c2 (Bind b) v2)) \to (or3 (ex2 T -(\lambda (v1: T).(subst0 i v v1 v2)) (\lambda (v1: T).(eq C x (CHead c2 (Bind -b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i v c1 c2)) (\lambda (c1: C).(eq C -x (CHead c1 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: -T).(subst0 i v v1 v2))) (\lambda (c1: C).(\lambda (_: T).(csubst0 i v c1 -c2))) (\lambda (c1: C).(\lambda (v1: T).(eq C x (CHead c1 (Bind b) v1)))))))) -(\lambda (y0: nat).(\lambda (H1: (csubst0 y0 v x y)).(csubst0_ind (\lambda -(n: nat).(\lambda (t: T).(\lambda (c: C).(\lambda (c0: C).((eq nat n (S i)) -\to ((eq C c0 (CHead c2 (Bind b) v2)) \to (or3 (ex2 T (\lambda (v1: -T).(subst0 i t v1 v2)) (\lambda (v1: T).(eq C c (CHead c2 (Bind b) v1)))) -(ex2 C (\lambda (c1: C).(csubst0 i t c1 c2)) (\lambda (c1: C).(eq C c (CHead -c1 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i t v1 -v2))) (\lambda (c1: C).(\lambda (_: T).(csubst0 i t c1 c2))) (\lambda (c1: -C).(\lambda (v1: T).(eq C c (CHead c1 (Bind b) v1)))))))))))) (\lambda (k: -K).(\lambda (i0: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: -T).(\lambda (H2: (subst0 i0 v0 u1 u2)).(\lambda (c: C).(\lambda (H3: (eq nat -(s k i0) (S i))).(\lambda (H4: (eq C (CHead c k u2) (CHead c2 (Bind b) -v2))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda -(_: C).C) with [(CSort _) \Rightarrow c | (CHead c0 _ _) \Rightarrow c0])) -(CHead c k u2) (CHead c2 (Bind b) v2) H4) in ((let H6 \def (f_equal C K -(\lambda (e: C).(match e in C return (\lambda (_: C).K) with [(CSort _) -\Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c k u2) (CHead c2 -(Bind b) v2) H4) in ((let H7 \def (f_equal C T (\lambda (e: C).(match e in C -return (\lambda (_: C).T) with [(CSort _) \Rightarrow u2 | (CHead _ _ t) -\Rightarrow t])) (CHead c k u2) (CHead c2 (Bind b) v2) H4) in (\lambda (H8: -(eq K k (Bind b))).(\lambda (H9: (eq C c c2)).(eq_ind_r C c2 (\lambda (c0: -C).(or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C -(CHead c0 k u1) (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i -v0 c1 c2)) (\lambda (c1: C).(eq C (CHead c0 k u1) (CHead c1 (Bind b) v2)))) -(ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda -(c1: C).(\lambda (_: T).(csubst0 i v0 c1 c2))) (\lambda (c1: C).(\lambda (v1: -T).(eq C (CHead c0 k u1) (CHead c1 (Bind b) v1))))))) (let H10 \def (eq_ind T -u2 (\lambda (t: T).(subst0 i0 v0 u1 t)) H2 v2 H7) in (let H11 \def (eq_ind K -k (\lambda (k0: K).(eq nat (s k0 i0) (S i))) H3 (Bind b) H8) in (eq_ind_r K -(Bind b) (\lambda (k0: K).(or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) -(\lambda (v1: T).(eq C (CHead c2 k0 u1) (CHead c2 (Bind b) v1)))) (ex2 C -(\lambda (c1: C).(csubst0 i v0 c1 c2)) (\lambda (c1: C).(eq C (CHead c2 k0 -u1) (CHead c1 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: -T).(subst0 i v0 v1 v2))) (\lambda (c1: C).(\lambda (_: T).(csubst0 i v0 c1 -c2))) (\lambda (c1: C).(\lambda (v1: T).(eq C (CHead c2 k0 u1) (CHead c1 -(Bind b) v1))))))) (let H12 \def (f_equal nat nat (\lambda (e: nat).(match e -in nat return (\lambda (_: nat).nat) with [O \Rightarrow i0 | (S n) -\Rightarrow n])) (S i0) (S i) H11) in (let H13 \def (eq_ind nat i0 (\lambda -(n: nat).(subst0 n v0 u1 v2)) H10 i H12) in (or3_intro0 (ex2 T (\lambda (v1: -T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C (CHead c2 (Bind b) u1) (CHead -c2 (Bind b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i v0 c1 c2)) (\lambda -(c1: C).(eq C (CHead c2 (Bind b) u1) (CHead c1 (Bind b) v2)))) (ex3_2 C T -(\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda (c1: -C).(\lambda (_: T).(csubst0 i v0 c1 c2))) (\lambda (c1: C).(\lambda (v1: -T).(eq C (CHead c2 (Bind b) u1) (CHead c1 (Bind b) v1))))) (ex_intro2 T -(\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C (CHead c2 (Bind -b) u1) (CHead c2 (Bind b) v1))) u1 H13 (refl_equal C (CHead c2 (Bind b) -u1)))))) k H8))) c H9)))) H6)) H5))))))))))) (\lambda (k: K).(\lambda (i0: -nat).(\lambda (c1: C).(\lambda (c0: C).(\lambda (v0: T).(\lambda (H2: -(csubst0 i0 v0 c1 c0)).(\lambda (H3: (((eq nat i0 (S i)) \to ((eq C c0 (CHead -c2 (Bind b) v2)) \to (or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) -(\lambda (v1: T).(eq C c1 (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3: -C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C c1 (CHead c3 (Bind b) v2)))) -(ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda -(c3: C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1: -T).(eq C c1 (CHead c3 (Bind b) v1)))))))))).(\lambda (u: T).(\lambda (H4: (eq -nat (s k i0) (S i))).(\lambda (H5: (eq C (CHead c0 k u) (CHead c2 (Bind b) -v2))).(let H6 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda -(_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) -(CHead c0 k u) (CHead c2 (Bind b) v2) H5) in ((let H7 \def (f_equal C K -(\lambda (e: C).(match e in C return (\lambda (_: C).K) with [(CSort _) -\Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c0 k u) (CHead c2 -(Bind b) v2) H5) in ((let H8 \def (f_equal C T (\lambda (e: C).(match e in C -return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) -\Rightarrow t])) (CHead c0 k u) (CHead c2 (Bind b) v2) H5) in (\lambda (H9: -(eq K k (Bind b))).(\lambda (H10: (eq C c0 c2)).(eq_ind_r T v2 (\lambda (t: -T).(or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C -(CHead c1 k t) (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3: C).(csubst0 i -v0 c3 c2)) (\lambda (c3: C).(eq C (CHead c1 k t) (CHead c3 (Bind b) v2)))) -(ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda -(c3: C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1: -T).(eq C (CHead c1 k t) (CHead c3 (Bind b) v1))))))) (let H11 \def (eq_ind C -c0 (\lambda (c: C).((eq nat i0 (S i)) \to ((eq C c (CHead c2 (Bind b) v2)) -\to (or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C -c1 (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3: C).(csubst0 i v0 c3 c2)) -(\lambda (c3: C).(eq C c1 (CHead c3 (Bind b) v2)))) (ex3_2 C T (\lambda (_: -C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda (c3: C).(\lambda (_: -T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1: T).(eq C c1 (CHead -c3 (Bind b) v1))))))))) H3 c2 H10) in (let H12 \def (eq_ind C c0 (\lambda (c: -C).(csubst0 i0 v0 c1 c)) H2 c2 H10) in (let H13 \def (eq_ind K k (\lambda -(k0: K).(eq nat (s k0 i0) (S i))) H4 (Bind b) H9) in (eq_ind_r K (Bind b) -(\lambda (k0: K).(or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda -(v1: T).(eq C (CHead c1 k0 v2) (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3: -C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C (CHead c1 k0 v2) (CHead c3 -(Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 -v2))) (\lambda (c3: C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3: -C).(\lambda (v1: T).(eq C (CHead c1 k0 v2) (CHead c3 (Bind b) v1))))))) (let -H14 \def (f_equal nat nat (\lambda (e: nat).(match e in nat return (\lambda -(_: nat).nat) with [O \Rightarrow i0 | (S n) \Rightarrow n])) (S i0) (S i) -H13) in (let H15 \def (eq_ind nat i0 (\lambda (n: nat).((eq nat n (S i)) \to -((eq C c2 (CHead c2 (Bind b) v2)) \to (or3 (ex2 T (\lambda (v1: T).(subst0 i -v0 v1 v2)) (\lambda (v1: T).(eq C c1 (CHead c2 (Bind b) v1)))) (ex2 C -(\lambda (c3: C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C c1 (CHead c3 -(Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 -v2))) (\lambda (c3: C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3: -C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind b) v1))))))))) H11 i H14) in -(let H16 \def (eq_ind nat i0 (\lambda (n: nat).(csubst0 n v0 c1 c2)) H12 i -H14) in (or3_intro1 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda -(v1: T).(eq C (CHead c1 (Bind b) v2) (CHead c2 (Bind b) v1)))) (ex2 C -(\lambda (c3: C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C (CHead c1 (Bind -b) v2) (CHead c3 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: -T).(subst0 i v0 v1 v2))) (\lambda (c3: C).(\lambda (_: T).(csubst0 i v0 c3 -c2))) (\lambda (c3: C).(\lambda (v1: T).(eq C (CHead c1 (Bind b) v2) (CHead -c3 (Bind b) v1))))) (ex_intro2 C (\lambda (c3: C).(csubst0 i v0 c3 c2)) -(\lambda (c3: C).(eq C (CHead c1 (Bind b) v2) (CHead c3 (Bind b) v2))) c1 H16 -(refl_equal C (CHead c1 (Bind b) v2))))))) k H9)))) u H8)))) H7)) -H6)))))))))))) (\lambda (k: K).(\lambda (i0: nat).(\lambda (v0: T).(\lambda -(u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 i0 v0 u1 u2)).(\lambda (c1: -C).(\lambda (c0: C).(\lambda (H3: (csubst0 i0 v0 c1 c0)).(\lambda (H4: (((eq -nat i0 (S i)) \to ((eq C c0 (CHead c2 (Bind b) v2)) \to (or3 (ex2 T (\lambda -(v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C c1 (CHead c2 (Bind b) -v1)))) (ex2 C (\lambda (c3: C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C -c1 (CHead c3 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: -T).(subst0 i v0 v1 v2))) (\lambda (c3: C).(\lambda (_: T).(csubst0 i v0 c3 -c2))) (\lambda (c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind b) -v1)))))))))).(\lambda (H5: (eq nat (s k i0) (S i))).(\lambda (H6: (eq C -(CHead c0 k u2) (CHead c2 (Bind b) v2))).(let H7 \def (f_equal C C (\lambda -(e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 -| (CHead c _ _) \Rightarrow c])) (CHead c0 k u2) (CHead c2 (Bind b) v2) H6) -in ((let H8 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda -(_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) -(CHead c0 k u2) (CHead c2 (Bind b) v2) H6) in ((let H9 \def (f_equal C T -(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) -\Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c0 k u2) (CHead c2 -(Bind b) v2) H6) in (\lambda (H10: (eq K k (Bind b))).(\lambda (H11: (eq C c0 -c2)).(let H12 \def (eq_ind C c0 (\lambda (c: C).((eq nat i0 (S i)) \to ((eq C -c (CHead c2 (Bind b) v2)) \to (or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 -v2)) (\lambda (v1: T).(eq C c1 (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3: -C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C c1 (CHead c3 (Bind b) v2)))) -(ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda -(c3: C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1: -T).(eq C c1 (CHead c3 (Bind b) v1))))))))) H4 c2 H11) in (let H13 \def -(eq_ind C c0 (\lambda (c: C).(csubst0 i0 v0 c1 c)) H3 c2 H11) in (let H14 -\def (eq_ind T u2 (\lambda (t: T).(subst0 i0 v0 u1 t)) H2 v2 H9) in (let H15 -\def (eq_ind K k (\lambda (k0: K).(eq nat (s k0 i0) (S i))) H5 (Bind b) H10) -in (eq_ind_r K (Bind b) (\lambda (k0: K).(or3 (ex2 T (\lambda (v1: T).(subst0 -i v0 v1 v2)) (\lambda (v1: T).(eq C (CHead c1 k0 u1) (CHead c2 (Bind b) -v1)))) (ex2 C (\lambda (c3: C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C -(CHead c1 k0 u1) (CHead c3 (Bind b) v2)))) (ex3_2 C T (\lambda (_: -C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda (c3: C).(\lambda (_: -T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1: T).(eq C (CHead c1 -k0 u1) (CHead c3 (Bind b) v1))))))) (let H16 \def (f_equal nat nat (\lambda -(e: nat).(match e in nat return (\lambda (_: nat).nat) with [O \Rightarrow i0 -| (S n) \Rightarrow n])) (S i0) (S i) H15) in (let H17 \def (eq_ind nat i0 -(\lambda (n: nat).((eq nat n (S i)) \to ((eq C c2 (CHead c2 (Bind b) v2)) \to -(or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C c1 -(CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3: C).(csubst0 i v0 c3 c2)) -(\lambda (c3: C).(eq C c1 (CHead c3 (Bind b) v2)))) (ex3_2 C T (\lambda (_: -C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda (c3: C).(\lambda (_: -T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1: T).(eq C c1 (CHead -c3 (Bind b) v1))))))))) H12 i H16) in (let H18 \def (eq_ind nat i0 (\lambda -(n: nat).(csubst0 n v0 c1 c2)) H13 i H16) in (let H19 \def (eq_ind nat i0 -(\lambda (n: nat).(subst0 n v0 u1 v2)) H14 i H16) in (or3_intro2 (ex2 T -(\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C (CHead c1 (Bind -b) u1) (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3: C).(csubst0 i v0 c3 -c2)) (\lambda (c3: C).(eq C (CHead c1 (Bind b) u1) (CHead c3 (Bind b) v2)))) -(ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda -(c3: C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1: -T).(eq C (CHead c1 (Bind b) u1) (CHead c3 (Bind b) v1))))) (ex3_2_intro C T -(\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda (c3: -C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1: -T).(eq C (CHead c1 (Bind b) u1) (CHead c3 (Bind b) v1)))) c1 u1 H19 H18 -(refl_equal C (CHead c1 (Bind b) u1)))))))) k H10)))))))) H8)) -H7)))))))))))))) y0 v x y H1))) H0))) H))))))). -(* COMMENTS -Initial nodes: 3878 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubst0/getl.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubst0/getl.ma deleted file mode 100644 index 2701af000..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubst0/getl.ma +++ /dev/null @@ -1,1157 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubst0/clear.ma". - -include "Basic-1/csubst0/drop.ma". - -include "Basic-1/getl/fwd.ma". - -theorem csubst0_getl_ge: - \forall (i: nat).(\forall (n: nat).((le i n) \to (\forall (c1: C).(\forall -(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((getl n c1 -e) \to (getl n c2 e))))))))) -\def - \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (le i n)).(\lambda (c1: -C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 i v c1 -c2)).(\lambda (e: C).(\lambda (H1: (getl n c1 e)).(let H2 \def (getl_gen_all -c1 e n H1) in (ex2_ind C (\lambda (e0: C).(drop n O c1 e0)) (\lambda (e0: -C).(clear e0 e)) (getl n c2 e) (\lambda (x: C).(\lambda (H3: (drop n O c1 -x)).(\lambda (H4: (clear x e)).(lt_eq_gt_e i n (getl n c2 e) (\lambda (H5: -(lt i n)).(getl_intro n c2 e x (csubst0_drop_gt n i H5 c1 c2 v H0 x H3) H4)) -(\lambda (H5: (eq nat i n)).(let H6 \def (eq_ind_r nat n (\lambda (n0: -nat).(drop n0 O c1 x)) H3 i H5) in (let H7 \def (eq_ind_r nat n (\lambda (n0: -nat).(le i n0)) H i H5) in (eq_ind nat i (\lambda (n0: nat).(getl n0 c2 e)) -(let H8 \def (csubst0_drop_eq i c1 c2 v H0 x H6) in (or4_ind (drop i O c2 x) -(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: -T).(eq C x (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(drop i O c2 (CHead e0 (Flat f) w)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u -w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C x (CHead e1 (Flat f) u)))))) (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop i O c2 (CHead e2 -(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C x (CHead e1 -(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(drop i O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O -v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (getl i c2 e) (\lambda (H9: -(drop i O c2 x)).(getl_intro i c2 e x H9 H4)) (\lambda (H9: (ex3_4 F C T T -(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C x -(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop i O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u -w))))))).(ex3_4_ind F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C x (CHead e0 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop i O c2 (CHead e0 -(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w))))) (getl i c2 e) (\lambda (x0: F).(\lambda (x1: -C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H10: (eq C x (CHead x1 (Flat -x0) x2))).(\lambda (H11: (drop i O c2 (CHead x1 (Flat x0) x3))).(\lambda (_: -(subst0 O v x2 x3)).(let H13 \def (eq_ind C x (\lambda (c: C).(clear c e)) H4 -(CHead x1 (Flat x0) x2) H10) in (getl_intro i c2 e (CHead x1 (Flat x0) x3) -H11 (clear_flat x1 e (clear_gen_flat x0 x1 e x2 H13) x0 x3)))))))))) H9)) -(\lambda (H9: (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C x (CHead e1 (Flat f) u)))))) (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop i O c2 (CHead e2 -(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C x (CHead e1 (Flat f) u)))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop i O c2 -(CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2))))) (getl i c2 e) (\lambda (x0: -F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H10: (eq C x -(CHead x1 (Flat x0) x3))).(\lambda (H11: (drop i O c2 (CHead x2 (Flat x0) -x3))).(\lambda (H12: (csubst0 O v x1 x2)).(let H13 \def (eq_ind C x (\lambda -(c: C).(clear c e)) H4 (CHead x1 (Flat x0) x3) H10) in (getl_intro i c2 e -(CHead x2 (Flat x0) x3) H11 (clear_flat x2 e (csubst0_clear_O x1 x2 v H12 e -(clear_gen_flat x0 x1 e x3 H13)) x0 x3)))))))))) H9)) (\lambda (H9: (ex4_5 F -C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(eq C x (CHead e1 (Flat f) u))))))) (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop i O -c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C x (CHead e1 (Flat f) -u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop i O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O -v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (getl i c2 e) (\lambda (x0: -F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: -T).(\lambda (H10: (eq C x (CHead x1 (Flat x0) x3))).(\lambda (H11: (drop i O -c2 (CHead x2 (Flat x0) x4))).(\lambda (_: (subst0 O v x3 x4)).(\lambda (H13: -(csubst0 O v x1 x2)).(let H14 \def (eq_ind C x (\lambda (c: C).(clear c e)) -H4 (CHead x1 (Flat x0) x3) H10) in (getl_intro i c2 e (CHead x2 (Flat x0) x4) -H11 (clear_flat x2 e (csubst0_clear_O x1 x2 v H13 e (clear_gen_flat x0 x1 e -x3 H14)) x0 x4)))))))))))) H9)) H8)) n H5)))) (\lambda (H5: (lt n -i)).(le_lt_false i n H H5 (getl n c2 e))))))) H2)))))))))). -(* COMMENTS -Initial nodes: 1525 -END *) - -theorem csubst0_getl_lt: - \forall (i: nat).(\forall (n: nat).((lt n i) \to (\forall (c1: C).(\forall -(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((getl n c1 -e) \to (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) -(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 -(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))))))))))) -\def - \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (lt n i)).(\lambda (c1: -C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 i v c1 -c2)).(\lambda (e: C).(\lambda (H1: (getl n c1 e)).(let H2 \def (getl_gen_all -c1 e n H1) in (ex2_ind C (\lambda (e0: C).(drop n O c1 e0)) (\lambda (e0: -C).(clear e0 e)) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda -(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) -(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 -(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x: -C).(\lambda (H3: (drop n O c1 x)).(\lambda (H4: (clear x e)).(let H5 \def -(csubst0_drop_lt n i H c1 c2 v H0 x H3) in (or4_ind (drop n O c2 x) (ex3_4 K -C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C -x (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop n O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda -(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n)) v u w)))))) -(ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C x (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop n O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C x (CHead e1 k u))))))) (\lambda (k: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n O -c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n)) v u w)))))) -(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus i (s k n)) v e1 e2))))))) (or4 (getl n c2 e) (ex3_4 B -C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C -e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 -(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))))) (\lambda (H6: (drop n O c2 x)).(or4_intro0 (getl n c2 e) -(ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: -T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 -(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: -T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 -(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -(minus i (S n)) v e1 e2))))))) (getl_intro n c2 e x H6 H4))) (\lambda (H6: -(ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: -T).(eq C x (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop n O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda -(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n)) v u -w))))))).(ex3_4_ind K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C x (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e0 k w)))))) (\lambda -(k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k -n)) v u w))))) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda -(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) -(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 -(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x0: -K).(\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq C x -(CHead x1 x0 x2))).(\lambda (H8: (drop n O c2 (CHead x1 x0 x3))).(\lambda -(H9: (subst0 (minus i (s x0 n)) v x2 x3)).(let H10 \def (eq_ind C x (\lambda -(c: C).(clear c e)) H4 (CHead x1 x0 x2) H7) in (K_ind (\lambda (k: K).((drop -n O c2 (CHead x1 k x3)) \to ((subst0 (minus i (s k n)) v x2 x3) \to ((clear -(CHead x1 k x2) e) \to (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: -B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind -b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 -B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq -C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))))))) (\lambda (b: -B).(\lambda (H11: (drop n O c2 (CHead x1 (Bind b) x3))).(\lambda (H12: -(subst0 (minus i (s (Bind b) n)) v x2 x3)).(\lambda (H13: (clear (CHead x1 -(Bind b) x2) e)).(eq_ind_r C (CHead x1 (Bind b) x2) (\lambda (c: C).(or4 -(getl n c2 c) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C c (CHead e0 (Bind b0) u)))))) (\lambda (b0: -B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 -(Bind b0) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda -(w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b0: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind -b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(getl n c2 (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) -(ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b0) u))))))) (\lambda (b0: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n -c2 (CHead e2 (Bind b0) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro1 (getl n c2 -(CHead x1 (Bind b) x2)) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Bind b) x2) (CHead e0 -(Bind b0) u)))))) (\lambda (b0: B).(\lambda (e0: C).(\lambda (_: T).(\lambda -(w: T).(getl n c2 (CHead e0 (Bind b0) w)))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 -B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq -C (CHead x1 (Bind b) x2) (CHead e1 (Bind b0) u)))))) (\lambda (b0: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 -(Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C -(CHead x1 (Bind b) x2) (CHead e1 (Bind b0) u))))))) (\lambda (b0: B).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 -(Bind b0) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -(minus i (S n)) v e1 e2))))))) (ex3_4_intro B C T T (\lambda (b0: B).(\lambda -(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Bind b) x2) (CHead -e0 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b0) w)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w))))) b x1 x2 x3 (refl_equal C (CHead x1 (Bind b) x2)) (getl_intro n c2 -(CHead x1 (Bind b) x3) (CHead x1 (Bind b) x3) H11 (clear_bind b x1 x3)) H12)) -e (clear_gen_bind b x1 e x2 H13)))))) (\lambda (f: F).(\lambda (H11: (drop n -O c2 (CHead x1 (Flat f) x3))).(\lambda (_: (subst0 (minus i (s (Flat f) n)) v -x2 x3)).(\lambda (H13: (clear (CHead x1 (Flat f) x2) e)).(or4_intro0 (getl n -c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: -B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 -(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind -b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) -(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n -c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (getl_intro n c2 e (CHead x1 -(Flat f) x3) H11 (clear_flat x1 e (clear_gen_flat f x1 e x2 H13) f x3))))))) -x0 H8 H9 H10))))))))) H6)) (\lambda (H6: (ex3_4 K C C T (\lambda (k: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C x (CHead e1 k -u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop n O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1 e2))))))).(ex3_4_ind -K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq -C x (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop n O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1 -e2))))) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) -(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 -(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x0: -K).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H7: (eq C x -(CHead x1 x0 x3))).(\lambda (H8: (drop n O c2 (CHead x2 x0 x3))).(\lambda -(H9: (csubst0 (minus i (s x0 n)) v x1 x2)).(let H10 \def (eq_ind C x (\lambda -(c: C).(clear c e)) H4 (CHead x1 x0 x3) H7) in (K_ind (\lambda (k: K).((drop -n O c2 (CHead x2 k x3)) \to ((csubst0 (minus i (s k n)) v x1 x2) \to ((clear -(CHead x1 k x3) e) \to (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: -B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind -b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 -B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq -C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))))))) (\lambda (b: -B).(\lambda (H11: (drop n O c2 (CHead x2 (Bind b) x3))).(\lambda (H12: -(csubst0 (minus i (s (Bind b) n)) v x1 x2)).(\lambda (H13: (clear (CHead x1 -(Bind b) x3) e)).(eq_ind_r C (CHead x1 (Bind b) x3) (\lambda (c: C).(or4 -(getl n c2 c) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C c (CHead e0 (Bind b0) u)))))) (\lambda (b0: -B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 -(Bind b0) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda -(w: 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(_: T).(eq C (CHead x1 (Bind b) x3) (CHead e0 -(Bind b0) u)))))) (\lambda (b0: B).(\lambda (e0: C).(\lambda (_: T).(\lambda -(w: T).(getl n c2 (CHead e0 (Bind b0) w)))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 -B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq -C (CHead x1 (Bind b) x3) (CHead e1 (Bind b0) u)))))) (\lambda (b0: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 -(Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C -(CHead x1 (Bind b) x3) (CHead e1 (Bind b0) u))))))) (\lambda (b0: B).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 -(Bind b0) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -(minus i (S n)) v e1 e2))))))) (ex3_4_intro B C C T (\lambda (b0: B).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x1 (Bind b) x3) (CHead -e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b0) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2))))) b x1 x2 x3 (refl_equal C (CHead x1 (Bind b) x3)) (getl_intro n -c2 (CHead x2 (Bind b) x3) (CHead x2 (Bind b) x3) H11 (clear_bind b x2 x3)) -H12)) e (clear_gen_bind b x1 e x3 H13)))))) (\lambda (f: F).(\lambda (H11: -(drop n O c2 (CHead x2 (Flat f) x3))).(\lambda (H12: (csubst0 (minus i (s -(Flat f) n)) v x1 x2)).(\lambda (H13: (clear (CHead x1 (Flat f) x3) e)).(let -H14 \def (eq_ind nat (minus i n) (\lambda (n0: nat).(csubst0 n0 v x1 x2)) H12 -(S (minus i (S n))) (minus_x_Sy i n H)) in (let H15 \def (csubst0_clear_S x1 -x2 v (minus i (S n)) H14 e (clear_gen_flat f x1 e x3 H13)) in (or4_ind (clear -x2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u1)))))) (\lambda (b: -B).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e0 -(Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 (minus i (S n)) v u1 u2)))))) (ex3_4 B C C T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind -b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(clear x2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) -(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u1))))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: 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(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))))) (\lambda (H16: (clear x2 e)).(or4_intro0 (getl n c2 e) (ex3_4 -B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq -C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: 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u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 -B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq -C c (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b) u))))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: 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(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(clear x2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 -e2))))))).(ex3_4_ind B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x2 (CHead e2 (Bind -b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus i (S n)) v e1 e2))))) (or4 (getl n c2 e) (ex3_4 B C T T -(\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 -(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))))) (\lambda (x4: B).(\lambda (x5: C).(\lambda (x6: C).(\lambda -(x7: T).(\lambda (H17: (eq C e (CHead x5 (Bind x4) x7))).(\lambda (H18: -(clear x2 (CHead x6 (Bind x4) x7))).(\lambda (H19: (csubst0 (minus i (S n)) v -x5 x6)).(eq_ind_r C (CHead x5 (Bind x4) x7) (\lambda (c: C).(or4 (getl n c2 -c) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda -(_: T).(eq C c (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind b) u)))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 -(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: -T).(eq C c (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 -(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -(minus i (S n)) v e1 e2))))))))) (or4_intro2 (getl n c2 (CHead x5 (Bind x4) -x7)) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda -(_: T).(eq C (CHead x5 (Bind x4) x7) (CHead e0 (Bind b) u)))))) (\lambda (b: -B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 -(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x5 (Bind x4) -x7) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) x7) (CHead e1 -(Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))) -(ex3_4_intro B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C (CHead x5 (Bind x4) x7) (CHead e1 (Bind b) u)))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 -(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))) x4 x5 x6 x7 -(refl_equal C (CHead x5 (Bind x4) x7)) (getl_intro n c2 (CHead x6 (Bind x4) -x7) (CHead x2 (Flat f) x3) H11 (clear_flat x2 (CHead x6 (Bind x4) x7) H18 f -x3)) H19)) e H17)))))))) H16)) (\lambda (H16: (ex4_5 B C C T T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C e -(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e2 (Bind b) u2))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 (minus i (S n)) v u1 u2)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C e (CHead e1 (Bind -b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (u2: T).(clear x2 (CHead e2 (Bind b) u2))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 -(minus i (S n)) v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) -(or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda -(u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: -B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 -(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind -b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) -(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n -c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x4: B).(\lambda -(x5: C).(\lambda (x6: C).(\lambda (x7: T).(\lambda (x8: T).(\lambda (H17: (eq -C e (CHead x5 (Bind x4) x7))).(\lambda (H18: (clear x2 (CHead x6 (Bind x4) -x8))).(\lambda (H19: (subst0 (minus i (S n)) v x7 x8)).(\lambda (H20: -(csubst0 (minus i (S n)) v x5 x6)).(eq_ind_r C (CHead x5 (Bind x4) x7) -(\lambda (c: C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda (b: B).(\lambda -(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Bind b) u)))))) -(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 -(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c -(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b) u))))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro3 -(getl n c2 (CHead x5 (Bind x4) x7)) (ex3_4 B C T T (\lambda (b: B).(\lambda -(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) x7) (CHead -e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda -(w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 -B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq -C (CHead x5 (Bind x4) x7) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda -(_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 -(minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) -x7) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2))))))) (ex4_5_intro B C C T T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) -x7) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))) x4 x5 x6 x7 x8 (refl_equal C (CHead x5 (Bind x4) x7)) -(getl_intro n c2 (CHead x6 (Bind x4) x8) (CHead x2 (Flat f) x3) H11 -(clear_flat x2 (CHead x6 (Bind x4) x8) H18 f x3)) H19 H20)) e H17)))))))))) -H16)) H15))))))) x0 H8 H9 H10))))))))) H6)) (\lambda (H6: (ex4_5 K C C T T -(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: -T).(eq C x (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e2 k w))))))) (\lambda -(k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (s k n)) v u w)))))) (\lambda (k: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k -n)) v e1 e2)))))))).(ex4_5_ind K C C T T (\lambda (k: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C x (CHead e1 k -u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop n O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda -(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k -n)) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1 e2)))))) (or4 (getl n -c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: -B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 -(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind -b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) -(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n -c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x0: K).(\lambda -(x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H7: (eq -C x (CHead x1 x0 x3))).(\lambda (H8: (drop n O c2 (CHead x2 x0 x4))).(\lambda -(H9: (subst0 (minus i (s x0 n)) v x3 x4)).(\lambda (H10: (csubst0 (minus i (s -x0 n)) v x1 x2)).(let H11 \def (eq_ind C x (\lambda (c: C).(clear c e)) H4 -(CHead x1 x0 x3) H7) in (K_ind (\lambda (k: K).((drop n O c2 (CHead x2 k x4)) -\to ((subst0 (minus i (s k n)) v x3 x4) \to ((csubst0 (minus i (s k n)) v x1 -x2) \to ((clear (CHead x1 k x3) e) \to (or4 (getl n c2 e) (ex3_4 B C T T -(\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 -(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2))))))))))))) (\lambda (b: B).(\lambda (H12: (drop n O c2 (CHead x2 -(Bind b) x4))).(\lambda (H13: (subst0 (minus i (s (Bind b) n)) v x3 -x4)).(\lambda (H14: (csubst0 (minus i (s (Bind b) n)) v x1 x2)).(\lambda -(H15: (clear (CHead x1 (Bind b) x3) e)).(eq_ind_r C (CHead x1 (Bind b) x3) -(\lambda (c: C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda (b0: B).(\lambda -(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Bind b0) u)))))) -(\lambda (b0: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 -(CHead e0 (Bind b0) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T -(\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c -(CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b0) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b0) u))))))) -(\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(w: T).(getl n c2 (CHead e2 (Bind b0) w))))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro3 -(getl n c2 (CHead x1 (Bind b) x3)) (ex3_4 B C T T (\lambda (b0: B).(\lambda -(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Bind b) x3) (CHead -e0 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b0) w)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (ex3_4 B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C (CHead x1 (Bind b) x3) (CHead e1 (Bind b0) u)))))) -(\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 -(CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T -(\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda -(_: T).(eq C (CHead x1 (Bind b) x3) (CHead e1 (Bind b0) u))))))) (\lambda -(b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(getl n c2 (CHead e2 (Bind b0) w))))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (ex4_5_intro B C C -T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(eq C (CHead x1 (Bind b) x3) (CHead e1 (Bind b0) u))))))) -(\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(w: T).(getl n c2 (CHead e2 (Bind b0) w))))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) b x1 x2 x3 x4 -(refl_equal C (CHead x1 (Bind b) x3)) (getl_intro n c2 (CHead x2 (Bind b) x4) -(CHead x2 (Bind b) x4) H12 (clear_bind b x2 x4)) H13 H14)) e (clear_gen_bind -b x1 e x3 H15))))))) (\lambda (f: F).(\lambda (H12: (drop n O c2 (CHead x2 -(Flat f) x4))).(\lambda (_: (subst0 (minus i (s (Flat f) n)) v x3 -x4)).(\lambda (H14: (csubst0 (minus i (s (Flat f) n)) v x1 x2)).(\lambda -(H15: (clear (CHead x1 (Flat f) x3) e)).(let H16 \def (eq_ind nat (minus i n) -(\lambda (n0: nat).(csubst0 n0 v x1 x2)) H14 (S (minus i (S n))) (minus_x_Sy -i n H)) in (let H17 \def (csubst0_clear_S x1 x2 v (minus i (S n)) H16 e -(clear_gen_flat f x1 e x3 H15)) in (or4_ind (clear x2 e) (ex3_4 B C T T -(\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C e -(CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: -T).(\lambda (u2: T).(clear x2 (CHead e0 (Bind b) u2)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (minus i (S n)) -v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x2 (CHead e2 (Bind -b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C e -(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e2 (Bind b) u2))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 (minus i (S n)) v u1 u2)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2))))))) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda -(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) -(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 -(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (H18: -(clear x2 e)).(or4_intro0 (getl n c2 e) (ex3_4 B C T T (\lambda (b: -B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind -b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 -B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq -C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (getl_intro n c2 e -(CHead x2 (Flat f) x4) H12 (clear_flat x2 e H18 f x4)))) (\lambda (H18: -(ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(eq C e (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e0 (Bind b) u2)))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 -(minus i (S n)) v u1 u2))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda -(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u1)))))) -(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(clear x2 -(CHead e0 (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 (minus i (S n)) v u1 u2))))) (or4 (getl n c2 e) -(ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: -T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 -(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: -T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 -(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -(minus i (S n)) v e1 e2)))))))) (\lambda (x5: B).(\lambda (x6: C).(\lambda -(x7: T).(\lambda (x8: T).(\lambda (H19: (eq C e (CHead x6 (Bind x5) -x7))).(\lambda (H20: (clear x2 (CHead x6 (Bind x5) x8))).(\lambda (H21: -(subst0 (minus i (S n)) v x7 x8)).(eq_ind_r C (CHead x6 (Bind x5) x7) -(\lambda (c: C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda (b: B).(\lambda -(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Bind b) u)))))) -(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 -(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c -(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b) u))))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro1 -(getl n c2 (CHead x6 (Bind x5) x7)) (ex3_4 B C T T (\lambda (b: B).(\lambda -(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) x7) (CHead -e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda -(w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 -B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq -C (CHead x6 (Bind x5) x7) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda -(_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 -(minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) -x7) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2))))))) (ex3_4_intro B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda -(u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) x7) (CHead e0 (Bind b) -u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w))))) x5 x6 -x7 x8 (refl_equal C (CHead x6 (Bind x5) x7)) (getl_intro n c2 (CHead x6 (Bind -x5) x8) (CHead x2 (Flat f) x4) H12 (clear_flat x2 (CHead x6 (Bind x5) x8) H20 -f x4)) H21)) e H19)))))))) H18)) (\lambda (H18: (ex3_4 B C C T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind -b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(clear x2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 -e2))))))).(ex3_4_ind B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x2 (CHead e2 (Bind -b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus i (S n)) v e1 e2))))) (or4 (getl n c2 e) (ex3_4 B C T T -(\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 -(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))))) (\lambda (x5: B).(\lambda (x6: C).(\lambda (x7: C).(\lambda -(x8: T).(\lambda (H19: (eq C e (CHead x6 (Bind x5) x8))).(\lambda (H20: -(clear x2 (CHead x7 (Bind x5) x8))).(\lambda (H21: (csubst0 (minus i (S n)) v -x6 x7)).(eq_ind_r C (CHead x6 (Bind x5) x8) (\lambda (c: C).(or4 (getl n c2 -c) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda -(_: T).(eq C c (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind b) u)))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 -(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: 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-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x6 (Bind x5) -x8) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) x8) (CHead e1 -(Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))) -(ex3_4_intro B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C (CHead x6 (Bind x5) x8) (CHead e1 (Bind b) u)))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 -(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))) x5 x6 x7 x8 -(refl_equal C (CHead x6 (Bind x5) x8)) (getl_intro n c2 (CHead x7 (Bind x5) -x8) (CHead x2 (Flat f) x4) H12 (clear_flat x2 (CHead x7 (Bind x5) x8) H20 f -x4)) H21)) e H19)))))))) H18)) (\lambda (H18: (ex4_5 B C C T T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C e -(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e2 (Bind b) u2))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 (minus i (S n)) v u1 u2)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C e (CHead e1 (Bind -b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (u2: T).(clear x2 (CHead e2 (Bind b) u2))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 -(minus i (S n)) v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) -(or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda -(u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: -B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 -(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind -b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) -(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n -c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x5: B).(\lambda -(x6: C).(\lambda (x7: C).(\lambda (x8: T).(\lambda (x9: T).(\lambda (H19: (eq -C e (CHead x6 (Bind x5) x8))).(\lambda (H20: (clear x2 (CHead x7 (Bind x5) -x9))).(\lambda (H21: (subst0 (minus i (S n)) v x8 x9)).(\lambda (H22: -(csubst0 (minus i (S n)) v x6 x7)).(eq_ind_r C (CHead x6 (Bind x5) x8) -(\lambda (c: C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda (b: B).(\lambda -(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Bind b) u)))))) -(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 -(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c -(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b) u))))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro3 -(getl n c2 (CHead x6 (Bind x5) x8)) (ex3_4 B C T T (\lambda (b: B).(\lambda -(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) x8) (CHead -e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda -(w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 -B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq -C (CHead x6 (Bind x5) x8) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda -(_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 -(minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) -x8) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2))))))) (ex4_5_intro B C C T T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) -x8) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))) x5 x6 x7 x8 x9 (refl_equal C (CHead x6 (Bind x5) x8)) -(getl_intro n c2 (CHead x7 (Bind x5) x9) (CHead x2 (Flat f) x4) H12 -(clear_flat x2 (CHead x7 (Bind x5) x9) H20 f x4)) H21 H22)) e H19)))))))))) -H18)) H17)))))))) x0 H8 H9 H10 H11))))))))))) H6)) H5))))) H2)))))))))). -(* COMMENTS -Initial nodes: 17179 -END *) - -theorem csubst0_getl_ge_back: - \forall (i: nat).(\forall (n: nat).((le i n) \to (\forall (c1: C).(\forall -(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((getl n c2 -e) \to (getl n c1 e))))))))) -\def - \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (le i n)).(\lambda (c1: -C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 i v c1 -c2)).(\lambda (e: C).(\lambda (H1: (getl n c2 e)).(let H2 \def (getl_gen_all -c2 e n H1) in (ex2_ind C (\lambda (e0: C).(drop n O c2 e0)) (\lambda (e0: -C).(clear e0 e)) (getl n c1 e) (\lambda (x: C).(\lambda (H3: (drop n O c2 -x)).(\lambda (H4: (clear x e)).(lt_eq_gt_e i n (getl n c1 e) (\lambda (H5: -(lt i n)).(getl_intro n c1 e x (csubst0_drop_gt_back n i H5 c1 c2 v H0 x H3) -H4)) (\lambda (H5: (eq nat i n)).(let H6 \def (eq_ind_r nat n (\lambda (n0: -nat).(drop n0 O c2 x)) H3 i H5) in (let H7 \def (eq_ind_r nat n (\lambda (n0: -nat).(le i n0)) H i H5) in (eq_ind nat i (\lambda (n0: nat).(getl n0 c1 e)) -(let H8 \def (csubst0_drop_eq_back i c1 c2 v H0 x H6) in (or4_ind (drop i O -c1 x) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (u2: T).(eq C x (CHead e0 (Flat f) u2)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop i O c1 (CHead e0 -(Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(eq C x (CHead e2 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop i O c1 -(CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C x -(CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(drop i O c1 (CHead e1 (Flat f) u1))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (getl i c1 -e) (\lambda (H9: (drop i O c1 x)).(getl_intro i c1 e x H9 H4)) (\lambda (H9: -(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: -T).(eq C x (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(drop i O c1 (CHead e0 (Flat f) u1)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v -u1 u2))))))).(ex3_4_ind F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (u2: T).(eq C x (CHead e0 (Flat f) u2)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop i O c1 (CHead e0 -(Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 O v u1 u2))))) (getl i c1 e) (\lambda (x0: F).(\lambda (x1: -C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H10: (eq C x (CHead x1 (Flat -x0) x3))).(\lambda (H11: (drop i O c1 (CHead x1 (Flat x0) x2))).(\lambda (_: -(subst0 O v x2 x3)).(let H13 \def (eq_ind C x (\lambda (c: C).(clear c e)) H4 -(CHead x1 (Flat x0) x3) H10) in (getl_intro i c1 e (CHead x1 (Flat x0) x2) -H11 (clear_flat x1 e (clear_gen_flat x0 x1 e x3 H13) x0 x2)))))))))) H9)) -(\lambda (H9: (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C x (CHead e2 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop i O c1 (CHead e1 -(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda -(_: C).(\lambda (e2: C).(\lambda (u: T).(eq C x (CHead e2 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop i O c1 -(CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2))))) (getl i c1 e) (\lambda (x0: -F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H10: (eq C x -(CHead x2 (Flat x0) x3))).(\lambda (H11: (drop i O c1 (CHead x1 (Flat x0) -x3))).(\lambda (H12: (csubst0 O v x1 x2)).(let H13 \def (eq_ind C x (\lambda -(c: C).(clear c e)) H4 (CHead x2 (Flat x0) x3) H10) in (getl_intro i c1 e -(CHead x1 (Flat x0) x3) H11 (clear_flat x1 e (csubst0_clear_O_back x1 x2 v -H12 e (clear_gen_flat x0 x2 e x3 H13)) x0 x3)))))))))) H9)) (\lambda (H9: -(ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (u2: T).(eq C x (CHead e2 (Flat f) u2))))))) (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop i -O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda -(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C x (CHead e2 (Flat -f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (_: T).(drop i O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 -O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (getl i c1 e) (\lambda (x0: -F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: -T).(\lambda (H10: (eq C x (CHead x2 (Flat x0) x4))).(\lambda (H11: (drop i O -c1 (CHead x1 (Flat x0) x3))).(\lambda (_: (subst0 O v x3 x4)).(\lambda (H13: -(csubst0 O v x1 x2)).(let H14 \def (eq_ind C x (\lambda (c: C).(clear c e)) -H4 (CHead x2 (Flat x0) x4) H10) in (getl_intro i c1 e (CHead x1 (Flat x0) x3) -H11 (clear_flat x1 e (csubst0_clear_O_back x1 x2 v H13 e (clear_gen_flat x0 -x2 e x4 H14)) x0 x3)))))))))))) H9)) H8)) n H5)))) (\lambda (H5: (lt n -i)).(le_lt_false i n H H5 (getl n c1 e))))))) H2)))))))))). -(* COMMENTS -Initial nodes: 1525 -END *) - -theorem csubst0_getl_lt_back: - \forall (n: nat).(\forall (i: nat).((lt n i) \to (\forall (c1: C).(\forall -(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e2: C).((getl n c2 -e2) \to (or (getl n c1 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) v e1 -e2)) (\lambda (e1: C).(getl n c1 e1)))))))))))) -\def - \lambda (n: nat).(\lambda (i: nat).(\lambda (H: (lt n i)).(\lambda (c1: -C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 i v c1 -c2)).(\lambda (e2: C).(\lambda (H1: (getl n c2 e2)).(let H2 \def -(getl_gen_all c2 e2 n H1) in (ex2_ind C (\lambda (e: C).(drop n O c2 e)) -(\lambda (e: C).(clear e e2)) (or (getl n c1 e2) (ex2 C (\lambda (e1: -C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(getl n c1 e1)))) (\lambda -(x: C).(\lambda (H3: (drop n O c2 x)).(\lambda (H4: (clear x e2)).(let H_x -\def (csubst0_drop_lt_back n i H c1 c2 v H0 x H3) in (let H5 \def H_x in -(or_ind (drop n O c1 x) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) v e1 x)) -(\lambda (e1: C).(drop n O c1 e1))) (or (getl n c1 e2) (ex2 C (\lambda (e1: -C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(getl n c1 e1)))) (\lambda -(H6: (drop n O c1 x)).(or_introl (getl n c1 e2) (ex2 C (\lambda (e1: -C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(getl n c1 e1))) -(getl_intro n c1 e2 x H6 H4))) (\lambda (H6: (ex2 C (\lambda (e1: C).(csubst0 -(minus i n) v e1 x)) (\lambda (e1: C).(drop n O c1 e1)))).(ex2_ind C (\lambda -(e1: C).(csubst0 (minus i n) v e1 x)) (\lambda (e1: C).(drop n O c1 e1)) (or -(getl n c1 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) v e1 e2)) -(\lambda (e1: C).(getl n c1 e1)))) (\lambda (x0: C).(\lambda (H7: (csubst0 -(minus i n) v x0 x)).(\lambda (H8: (drop n O c1 x0)).(let H_x0 \def -(csubst0_clear_trans x0 x v (minus i n) H7 e2 H4) in (let H9 \def H_x0 in -(or_ind (clear x0 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) v e1 e2)) -(\lambda (e1: C).(clear x0 e1))) (or (getl n c1 e2) (ex2 C (\lambda (e1: -C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(getl n c1 e1)))) (\lambda -(H10: (clear x0 e2)).(or_introl (getl n c1 e2) (ex2 C (\lambda (e1: -C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(getl n c1 e1))) -(getl_intro n c1 e2 x0 H8 H10))) (\lambda (H10: (ex2 C (\lambda (e1: -C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(clear x0 e1)))).(ex2_ind -C (\lambda (e1: C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(clear x0 -e1)) (or (getl n c1 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) v e1 -e2)) (\lambda (e1: C).(getl n c1 e1)))) (\lambda (x1: C).(\lambda (H11: -(csubst0 (minus i n) v x1 e2)).(\lambda (H12: (clear x0 x1)).(or_intror (getl -n c1 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: -C).(getl n c1 e1))) (ex_intro2 C (\lambda (e1: C).(csubst0 (minus i n) v e1 -e2)) (\lambda (e1: C).(getl n c1 e1)) x1 H11 (getl_intro n c1 x1 x0 H8 -H12)))))) H10)) H9)))))) H6)) H5)))))) H2)))))))))). -(* COMMENTS -Initial nodes: 801 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubst0/props.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubst0/props.ma deleted file mode 100644 index 28b75ec37..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubst0/props.ma +++ /dev/null @@ -1,61 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubst0/defs.ma". - -theorem csubst0_snd_bind: - \forall (b: B).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall -(u2: T).((subst0 i v u1 u2) \to (\forall (c: C).(csubst0 (S i) v (CHead c -(Bind b) u1) (CHead c (Bind b) u2)))))))) -\def - \lambda (b: B).(\lambda (i: nat).(\lambda (v: T).(\lambda (u1: T).(\lambda -(u2: T).(\lambda (H: (subst0 i v u1 u2)).(\lambda (c: C).(eq_ind nat (s (Bind -b) i) (\lambda (n: nat).(csubst0 n v (CHead c (Bind b) u1) (CHead c (Bind b) -u2))) (csubst0_snd (Bind b) i v u1 u2 H c) (S i) (refl_equal nat (S -i))))))))). -(* COMMENTS -Initial nodes: 91 -END *) - -theorem csubst0_fst_bind: - \forall (b: B).(\forall (i: nat).(\forall (c1: C).(\forall (c2: C).(\forall -(v: T).((csubst0 i v c1 c2) \to (\forall (u: T).(csubst0 (S i) v (CHead c1 -(Bind b) u) (CHead c2 (Bind b) u)))))))) -\def - \lambda (b: B).(\lambda (i: nat).(\lambda (c1: C).(\lambda (c2: C).(\lambda -(v: T).(\lambda (H: (csubst0 i v c1 c2)).(\lambda (u: T).(eq_ind nat (s (Bind -b) i) (\lambda (n: nat).(csubst0 n v (CHead c1 (Bind b) u) (CHead c2 (Bind b) -u))) (csubst0_fst (Bind b) i c1 c2 v H u) (S i) (refl_equal nat (S i))))))))). -(* COMMENTS -Initial nodes: 91 -END *) - -theorem csubst0_both_bind: - \forall (b: B).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall -(u2: T).((subst0 i v u1 u2) \to (\forall (c1: C).(\forall (c2: C).((csubst0 i -v c1 c2) \to (csubst0 (S i) v (CHead c1 (Bind b) u1) (CHead c2 (Bind b) -u2)))))))))) -\def - \lambda (b: B).(\lambda (i: nat).(\lambda (v: T).(\lambda (u1: T).(\lambda -(u2: T).(\lambda (H: (subst0 i v u1 u2)).(\lambda (c1: C).(\lambda (c2: -C).(\lambda (H0: (csubst0 i v c1 c2)).(eq_ind nat (s (Bind b) i) (\lambda (n: -nat).(csubst0 n v (CHead c1 (Bind b) u1) (CHead c2 (Bind b) u2))) -(csubst0_both (Bind b) i v u1 u2 H c1 c2 H0) (S i) (refl_equal nat (S -i))))))))))). -(* COMMENTS -Initial nodes: 107 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubst1/defs.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubst1/defs.ma deleted file mode 100644 index d6ba0a942..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubst1/defs.ma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubst0/defs.ma". - -inductive csubst1 (i: nat) (v: T) (c1: C): C \to Prop \def -| csubst1_refl: csubst1 i v c1 c1 -| csubst1_sing: \forall (c2: C).((csubst0 i v c1 c2) \to (csubst1 i v c1 c2)). - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubst1/fwd.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubst1/fwd.ma deleted file mode 100644 index fe71a56fd..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubst1/fwd.ma +++ /dev/null @@ -1,117 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubst1/defs.ma". - -include "Basic-1/csubst0/fwd.ma". - -include "Basic-1/subst1/props.ma". - -theorem csubst1_gen_head: - \forall (k: K).(\forall (c1: C).(\forall (x: C).(\forall (u1: T).(\forall -(v: T).(\forall (i: nat).((csubst1 (s k i) v (CHead c1 k u1) x) \to (ex3_2 T -C (\lambda (u2: T).(\lambda (c2: C).(eq C x (CHead c2 k u2)))) (\lambda (u2: -T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c2: -C).(csubst1 i v c1 c2)))))))))) -\def - \lambda (k: K).(\lambda (c1: C).(\lambda (x: C).(\lambda (u1: T).(\lambda -(v: T).(\lambda (i: nat).(\lambda (H: (csubst1 (s k i) v (CHead c1 k u1) -x)).(csubst1_ind (s k i) v (CHead c1 k u1) (\lambda (c: C).(ex3_2 T C -(\lambda (u2: T).(\lambda (c2: C).(eq C c (CHead c2 k u2)))) (\lambda (u2: -T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c2: -C).(csubst1 i v c1 c2))))) (ex3_2_intro T C (\lambda (u2: T).(\lambda (c2: -C).(eq C (CHead c1 k u1) (CHead c2 k u2)))) (\lambda (u2: T).(\lambda (_: -C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c2: C).(csubst1 i v c1 -c2))) u1 c1 (refl_equal C (CHead c1 k u1)) (subst1_refl i v u1) (csubst1_refl -i v c1)) (\lambda (c2: C).(\lambda (H0: (csubst0 (s k i) v (CHead c1 k u1) -c2)).(or3_ind (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i) -(s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c1 k u2)))) -(\lambda (u2: T).(\lambda (j: nat).(subst0 j v u1 u2)))) (ex3_2 C nat -(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i) (s k j)))) (\lambda (c3: -C).(\lambda (_: nat).(eq C c2 (CHead c3 k u1)))) (\lambda (c3: C).(\lambda -(j: nat).(csubst0 j v c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: -C).(\lambda (j: nat).(eq nat (s k i) (s k j))))) (\lambda (u2: T).(\lambda -(c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: -T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u2)))) (\lambda (_: -T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3))))) (ex3_2 T C -(\lambda (u2: T).(\lambda (c3: C).(eq C c2 (CHead c3 k u2)))) (\lambda (u2: -T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3: -C).(csubst1 i v c1 c3)))) (\lambda (H1: (ex3_2 T nat (\lambda (_: T).(\lambda -(j: nat).(eq nat (s k i) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C -c2 (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v u1 -u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i) (s -k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c1 k u2)))) -(\lambda (u2: T).(\lambda (j: nat).(subst0 j v u1 u2))) (ex3_2 T C (\lambda -(u2: T).(\lambda (c3: C).(eq C c2 (CHead c3 k u2)))) (\lambda (u2: -T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3: -C).(csubst1 i v c1 c3)))) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H2: -(eq nat (s k i) (s k x1))).(\lambda (H3: (eq C c2 (CHead c1 k x0))).(\lambda -(H4: (subst0 x1 v u1 x0)).(eq_ind_r C (CHead c1 k x0) (\lambda (c: C).(ex3_2 -T C (\lambda (u2: T).(\lambda (c3: C).(eq C c (CHead c3 k u2)))) (\lambda -(u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3: -C).(csubst1 i v c1 c3))))) (let H5 \def (eq_ind_r nat x1 (\lambda (n: -nat).(subst0 n v u1 x0)) H4 i (s_inj k i x1 H2)) in (ex3_2_intro T C (\lambda -(u2: T).(\lambda (c3: C).(eq C (CHead c1 k x0) (CHead c3 k u2)))) (\lambda -(u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3: -C).(csubst1 i v c1 c3))) x0 c1 (refl_equal C (CHead c1 k x0)) (subst1_single -i v u1 x0 H5) (csubst1_refl i v c1))) c2 H3)))))) H1)) (\lambda (H1: (ex3_2 C -nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i) (s k j)))) (\lambda -(c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u1)))) (\lambda (c3: -C).(\lambda (j: nat).(csubst0 j v c1 c3))))).(ex3_2_ind C nat (\lambda (_: -C).(\lambda (j: nat).(eq nat (s k i) (s k j)))) (\lambda (c3: C).(\lambda (_: -nat).(eq C c2 (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 -j v c1 c3))) (ex3_2 T C (\lambda (u2: T).(\lambda (c3: C).(eq C c2 (CHead c3 -k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: -T).(\lambda (c3: C).(csubst1 i v c1 c3)))) (\lambda (x0: C).(\lambda (x1: -nat).(\lambda (H2: (eq nat (s k i) (s k x1))).(\lambda (H3: (eq C c2 (CHead -x0 k u1))).(\lambda (H4: (csubst0 x1 v c1 x0)).(eq_ind_r C (CHead x0 k u1) -(\lambda (c: C).(ex3_2 T C (\lambda (u2: T).(\lambda (c3: C).(eq C c (CHead -c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda -(_: T).(\lambda (c3: C).(csubst1 i v c1 c3))))) (let H5 \def (eq_ind_r nat x1 -(\lambda (n: nat).(csubst0 n v c1 x0)) H4 i (s_inj k i x1 H2)) in -(ex3_2_intro T C (\lambda (u2: T).(\lambda (c3: C).(eq C (CHead x0 k u1) -(CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) -(\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1 c3))) u1 x0 (refl_equal C -(CHead x0 k u1)) (subst1_refl i v u1) (csubst1_sing i v c1 x0 H5))) c2 -H3)))))) H1)) (\lambda (H1: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: -C).(\lambda (j: nat).(eq nat (s k i) (s k j))))) (\lambda (u2: T).(\lambda -(c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: -T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u2)))) (\lambda (_: -T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))))).(ex4_3_ind T C -nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i) (s k -j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 -k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 -u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 -c3)))) (ex3_2 T C (\lambda (u2: T).(\lambda (c3: C).(eq C c2 (CHead c3 k -u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: -T).(\lambda (c3: C).(csubst1 i v c1 c3)))) (\lambda (x0: T).(\lambda (x1: -C).(\lambda (x2: nat).(\lambda (H2: (eq nat (s k i) (s k x2))).(\lambda (H3: -(eq C c2 (CHead x1 k x0))).(\lambda (H4: (subst0 x2 v u1 x0)).(\lambda (H5: -(csubst0 x2 v c1 x1)).(eq_ind_r C (CHead x1 k x0) (\lambda (c: C).(ex3_2 T C -(\lambda (u2: T).(\lambda (c3: C).(eq C c (CHead c3 k u2)))) (\lambda (u2: -T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3: -C).(csubst1 i v c1 c3))))) (let H6 \def (eq_ind_r nat x2 (\lambda (n: -nat).(csubst0 n v c1 x1)) H5 i (s_inj k i x2 H2)) in (let H7 \def (eq_ind_r -nat x2 (\lambda (n: nat).(subst0 n v u1 x0)) H4 i (s_inj k i x2 H2)) in -(ex3_2_intro T C (\lambda (u2: T).(\lambda (c3: C).(eq C (CHead x1 k x0) -(CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) -(\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1 c3))) x0 x1 (refl_equal C -(CHead x1 k x0)) (subst1_single i v u1 x0 H7) (csubst1_sing i v c1 x1 H6)))) -c2 H3)))))))) H1)) (csubst0_gen_head k c1 c2 u1 v (s k i) H0)))) x H))))))). -(* COMMENTS -Initial nodes: 1817 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubst1/getl.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubst1/getl.ma deleted file mode 100644 index e24ba9533..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubst1/getl.ma +++ /dev/null @@ -1,283 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubst1/props.ma". - -include "Basic-1/csubst0/getl.ma". - -include "Basic-1/subst1/props.ma". - -include "Basic-1/drop/props.ma". - -theorem csubst1_getl_ge: - \forall (i: nat).(\forall (n: nat).((le i n) \to (\forall (c1: C).(\forall -(c2: C).(\forall (v: T).((csubst1 i v c1 c2) \to (\forall (e: C).((getl n c1 -e) \to (getl n c2 e))))))))) -\def - \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (le i n)).(\lambda (c1: -C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst1 i v c1 -c2)).(csubst1_ind i v c1 (\lambda (c: C).(\forall (e: C).((getl n c1 e) \to -(getl n c e)))) (\lambda (e: C).(\lambda (H1: (getl n c1 e)).H1)) (\lambda -(c3: C).(\lambda (H1: (csubst0 i v c1 c3)).(\lambda (e: C).(\lambda (H2: -(getl n c1 e)).(csubst0_getl_ge i n H c1 c3 v H1 e H2))))) c2 H0))))))). -(* COMMENTS -Initial nodes: 111 -END *) - -theorem csubst1_getl_lt: - \forall (i: nat).(\forall (n: nat).((lt n i) \to (\forall (c1: C).(\forall -(c2: C).(\forall (v: T).((csubst1 i v c1 c2) \to (\forall (e1: C).((getl n c1 -e1) \to (ex2 C (\lambda (e2: C).(csubst1 (minus i n) v e1 e2)) (\lambda (e2: -C).(getl n c2 e2))))))))))) -\def - \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (lt n i)).(\lambda (c1: -C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst1 i v c1 -c2)).(csubst1_ind i v c1 (\lambda (c: C).(\forall (e1: C).((getl n c1 e1) \to -(ex2 C (\lambda (e2: C).(csubst1 (minus i n) v e1 e2)) (\lambda (e2: C).(getl -n c e2)))))) (\lambda (e1: C).(\lambda (H1: (getl n c1 e1)).(eq_ind_r nat (S -(minus i (S n))) (\lambda (n0: nat).(ex2 C (\lambda (e2: C).(csubst1 n0 v e1 -e2)) (\lambda (e2: C).(getl n c1 e2)))) (ex_intro2 C (\lambda (e2: -C).(csubst1 (S (minus i (S n))) v e1 e2)) (\lambda (e2: C).(getl n c1 e2)) e1 -(csubst1_refl (S (minus i (S n))) v e1) H1) (minus i n) (minus_x_Sy i n H)))) -(\lambda (c3: C).(\lambda (H1: (csubst0 i v c1 c3)).(\lambda (e1: C).(\lambda -(H2: (getl n c1 e1)).(eq_ind_r nat (S (minus i (S n))) (\lambda (n0: -nat).(ex2 C (\lambda (e2: C).(csubst1 n0 v e1 e2)) (\lambda (e2: C).(getl n -c3 e2)))) (let H3 \def (csubst0_getl_lt i n H c1 c3 v H1 e1 H2) in (or4_ind -(getl n c3 e1) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e1 (CHead e0 (Bind b) u)))))) (\lambda (b: -B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e0 -(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: -B).(\lambda (e2: C).(\lambda (_: C).(\lambda (u: T).(eq C e1 (CHead e2 (Bind -b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e3: C).(\lambda (u: -T).(getl n c3 (CHead e3 (Bind b) u)))))) (\lambda (_: B).(\lambda (e2: -C).(\lambda (e3: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e2 e3)))))) -(ex4_5 B C C T T (\lambda (b: B).(\lambda (e2: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq C e1 (CHead e2 (Bind b) u))))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e3: C).(\lambda (_: T).(\lambda (w: T).(getl n -c3 (CHead e3 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) -(\lambda (_: B).(\lambda (e2: C).(\lambda (e3: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus i (S n)) v e2 e3))))))) (ex2 C (\lambda (e2: -C).(csubst1 (S (minus i (S n))) v e1 e2)) (\lambda (e2: C).(getl n c3 e2))) -(\lambda (H4: (getl n c3 e1)).(ex_intro2 C (\lambda (e2: C).(csubst1 (S -(minus i (S n))) v e1 e2)) (\lambda (e2: C).(getl n c3 e2)) e1 (csubst1_refl -(S (minus i (S n))) v e1) H4)) (\lambda (H4: (ex3_4 B C T T (\lambda (b: -B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e1 (CHead e0 (Bind -b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(getl n c3 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u -w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e1 (CHead e0 (Bind b) u)))))) (\lambda (b: -B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e0 -(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (S n)) v u w))))) (ex2 C (\lambda (e2: C).(csubst1 (S -(minus i (S n))) v e1 e2)) (\lambda (e2: C).(getl n c3 e2))) (\lambda (x0: -B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H5: (eq C e1 -(CHead x1 (Bind x0) x2))).(\lambda (H6: (getl n c3 (CHead x1 (Bind x0) -x3))).(\lambda (H7: (subst0 (minus i (S n)) v x2 x3)).(eq_ind_r C (CHead x1 -(Bind x0) x2) (\lambda (c: C).(ex2 C (\lambda (e2: C).(csubst1 (S (minus i (S -n))) v c e2)) (\lambda (e2: C).(getl n c3 e2)))) (ex_intro2 C (\lambda (e2: -C).(csubst1 (S (minus i (S n))) v (CHead x1 (Bind x0) x2) e2)) (\lambda (e2: -C).(getl n c3 e2)) (CHead x1 (Bind x0) x3) (csubst1_sing (S (minus i (S n))) -v (CHead x1 (Bind x0) x2) (CHead x1 (Bind x0) x3) (csubst0_snd_bind x0 (minus -i (S n)) v x2 x3 H7 x1)) H6) e1 H5)))))))) H4)) (\lambda (H4: (ex3_4 B C C T -(\lambda (b: B).(\lambda (e2: C).(\lambda (_: C).(\lambda (u: T).(eq C e1 -(CHead e2 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e3: -C).(\lambda (u: T).(getl n c3 (CHead e3 (Bind b) u)))))) (\lambda (_: -B).(\lambda (e2: C).(\lambda (e3: C).(\lambda (_: T).(csubst0 (minus i (S n)) -v e2 e3))))))).(ex3_4_ind B C C T (\lambda (b: B).(\lambda (e2: C).(\lambda -(_: C).(\lambda (u: T).(eq C e1 (CHead e2 (Bind b) u)))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e3: C).(\lambda (u: T).(getl n c3 (CHead e3 -(Bind b) u)))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (e3: C).(\lambda -(_: T).(csubst0 (minus i (S n)) v e2 e3))))) (ex2 C (\lambda (e2: C).(csubst1 -(S (minus i (S n))) v e1 e2)) (\lambda (e2: C).(getl n c3 e2))) (\lambda (x0: -B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H5: (eq C e1 -(CHead x1 (Bind x0) x3))).(\lambda (H6: (getl n c3 (CHead x2 (Bind x0) -x3))).(\lambda (H7: (csubst0 (minus i (S n)) v x1 x2)).(eq_ind_r C (CHead x1 -(Bind x0) x3) (\lambda (c: C).(ex2 C (\lambda (e2: C).(csubst1 (S (minus i (S -n))) v c e2)) (\lambda (e2: C).(getl n c3 e2)))) (ex_intro2 C (\lambda (e2: -C).(csubst1 (S (minus i (S n))) v (CHead x1 (Bind x0) x3) e2)) (\lambda (e2: -C).(getl n c3 e2)) (CHead x2 (Bind x0) x3) (csubst1_sing (S (minus i (S n))) -v (CHead x1 (Bind x0) x3) (CHead x2 (Bind x0) x3) (csubst0_fst_bind x0 (minus -i (S n)) x1 x2 v H7 x3)) H6) e1 H5)))))))) H4)) (\lambda (H4: (ex4_5 B C C T -T (\lambda (b: B).(\lambda (e2: C).(\lambda (_: C).(\lambda (u: T).(\lambda -(_: T).(eq C e1 (CHead e2 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e3: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e3 -(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: -B).(\lambda (e2: C).(\lambda (e3: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -(minus i (S n)) v e2 e3)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda -(e2: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e1 (CHead e2 -(Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e3: C).(\lambda -(_: T).(\lambda (w: T).(getl n c3 (CHead e3 (Bind b) w))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (e3: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e2 e3)))))) -(ex2 C (\lambda (e2: C).(csubst1 (S (minus i (S n))) v e1 e2)) (\lambda (e2: -C).(getl n c3 e2))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: -C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H5: (eq C e1 (CHead x1 (Bind -x0) x3))).(\lambda (H6: (getl n c3 (CHead x2 (Bind x0) x4))).(\lambda (H7: -(subst0 (minus i (S n)) v x3 x4)).(\lambda (H8: (csubst0 (minus i (S n)) v x1 -x2)).(eq_ind_r C (CHead x1 (Bind x0) x3) (\lambda (c: C).(ex2 C (\lambda (e2: -C).(csubst1 (S (minus i (S n))) v c e2)) (\lambda (e2: C).(getl n c3 e2)))) -(ex_intro2 C (\lambda (e2: C).(csubst1 (S (minus i (S n))) v (CHead x1 (Bind -x0) x3) e2)) (\lambda (e2: C).(getl n c3 e2)) (CHead x2 (Bind x0) x4) -(csubst1_sing (S (minus i (S n))) v (CHead x1 (Bind x0) x3) (CHead x2 (Bind -x0) x4) (csubst0_both_bind x0 (minus i (S n)) v x3 x4 H7 x1 x2 H8)) H6) e1 -H5)))))))))) H4)) H3)) (minus i n) (minus_x_Sy i n H)))))) c2 H0))))))). -(* COMMENTS -Initial nodes: 2035 -END *) - -theorem csubst1_getl_ge_back: - \forall (i: nat).(\forall (n: nat).((le i n) \to (\forall (c1: C).(\forall -(c2: C).(\forall (v: T).((csubst1 i v c1 c2) \to (\forall (e: C).((getl n c2 -e) \to (getl n c1 e))))))))) -\def - \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (le i n)).(\lambda (c1: -C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst1 i v c1 -c2)).(csubst1_ind i v c1 (\lambda (c: C).(\forall (e: C).((getl n c e) \to -(getl n c1 e)))) (\lambda (e: C).(\lambda (H1: (getl n c1 e)).H1)) (\lambda -(c3: C).(\lambda (H1: (csubst0 i v c1 c3)).(\lambda (e: C).(\lambda (H2: -(getl n c3 e)).(csubst0_getl_ge_back i n H c1 c3 v H1 e H2))))) c2 H0))))))). -(* COMMENTS -Initial nodes: 111 -END *) - -theorem getl_csubst1: - \forall (d: nat).(\forall (c: C).(\forall (e: C).(\forall (u: T).((getl d c -(CHead e (Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0: C).(\lambda (_: -C).(csubst1 d u c a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) d a0 -a)))))))) -\def - \lambda (d: nat).(nat_ind (\lambda (n: nat).(\forall (c: C).(\forall (e: -C).(\forall (u: T).((getl n c (CHead e (Bind Abbr) u)) \to (ex2_2 C C -(\lambda (a0: C).(\lambda (_: C).(csubst1 n u c a0))) (\lambda (a0: -C).(\lambda (a: C).(drop (S O) n a0 a))))))))) (\lambda (c: C).(C_ind -(\lambda (c0: C).(\forall (e: C).(\forall (u: T).((getl O c0 (CHead e (Bind -Abbr) u)) \to (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O u c0 -a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0 a)))))))) (\lambda -(n: nat).(\lambda (e: C).(\lambda (u: T).(\lambda (H: (getl O (CSort n) -(CHead e (Bind Abbr) u))).(getl_gen_sort n O (CHead e (Bind Abbr) u) H (ex2_2 -C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O u (CSort n) a0))) (\lambda -(a0: C).(\lambda (a: C).(drop (S O) O a0 a))))))))) (\lambda (c0: C).(\lambda -(H: ((\forall (e: C).(\forall (u: T).((getl O c0 (CHead e (Bind Abbr) u)) \to -(ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O u c0 a0))) (\lambda -(a0: C).(\lambda (a: C).(drop (S O) O a0 a))))))))).(\lambda (k: K).(K_ind -(\lambda (k0: K).(\forall (t: T).(\forall (e: C).(\forall (u: T).((getl O -(CHead c0 k0 t) (CHead e (Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0: -C).(\lambda (_: C).(csubst1 O u (CHead c0 k0 t) a0))) (\lambda (a0: -C).(\lambda (a: C).(drop (S O) O a0 a))))))))) (\lambda (b: B).(\lambda (t: -T).(\lambda (e: C).(\lambda (u: T).(\lambda (H0: (getl O (CHead c0 (Bind b) -t) (CHead e (Bind Abbr) u))).(let H1 \def (f_equal C C (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow e | -(CHead c1 _ _) \Rightarrow c1])) (CHead e (Bind Abbr) u) (CHead c0 (Bind b) -t) (clear_gen_bind b c0 (CHead e (Bind Abbr) u) t (getl_gen_O (CHead c0 (Bind -b) t) (CHead e (Bind Abbr) u) H0))) in ((let H2 \def (f_equal C B (\lambda -(e0: C).(match e0 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow -Abbr | (CHead _ k0 _) \Rightarrow (match k0 in K return (\lambda (_: K).B) -with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead e -(Bind Abbr) u) (CHead c0 (Bind b) t) (clear_gen_bind b c0 (CHead e (Bind -Abbr) u) t (getl_gen_O (CHead c0 (Bind b) t) (CHead e (Bind Abbr) u) H0))) in -((let H3 \def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda -(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) -(CHead e (Bind Abbr) u) (CHead c0 (Bind b) t) (clear_gen_bind b c0 (CHead e -(Bind Abbr) u) t (getl_gen_O (CHead c0 (Bind b) t) (CHead e (Bind Abbr) u) -H0))) in (\lambda (H4: (eq B Abbr b)).(\lambda (_: (eq C e c0)).(eq_ind_r T t -(\lambda (t0: T).(ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O t0 -(CHead c0 (Bind b) t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0 -a))))) (eq_ind B Abbr (\lambda (b0: B).(ex2_2 C C (\lambda (a0: C).(\lambda -(_: C).(csubst1 O t (CHead c0 (Bind b0) t) a0))) (\lambda (a0: C).(\lambda -(a: C).(drop (S O) O a0 a))))) (ex2_2_intro C C (\lambda (a0: C).(\lambda (_: -C).(csubst1 O t (CHead c0 (Bind Abbr) t) a0))) (\lambda (a0: C).(\lambda (a: -C).(drop (S O) O a0 a))) (CHead c0 (Bind Abbr) t) c0 (csubst1_refl O t (CHead -c0 (Bind Abbr) t)) (drop_drop (Bind Abbr) O c0 c0 (drop_refl c0) t)) b H4) u -H3)))) H2)) H1))))))) (\lambda (f: F).(\lambda (t: T).(\lambda (e: -C).(\lambda (u: T).(\lambda (H0: (getl O (CHead c0 (Flat f) t) (CHead e (Bind -Abbr) u))).(let H_x \def (subst1_ex u t O) in (let H1 \def H_x in (ex_ind T -(\lambda (t2: T).(subst1 O u t (lift (S O) O t2))) (ex2_2 C C (\lambda (a0: -C).(\lambda (_: C).(csubst1 O u (CHead c0 (Flat f) t) a0))) (\lambda (a0: -C).(\lambda (a: C).(drop (S O) O a0 a)))) (\lambda (x: T).(\lambda (H2: -(subst1 O u t (lift (S O) O x))).(let H3 \def (H e u (getl_intro O c0 (CHead -e (Bind Abbr) u) c0 (drop_refl c0) (clear_gen_flat f c0 (CHead e (Bind Abbr) -u) t (getl_gen_O (CHead c0 (Flat f) t) (CHead e (Bind Abbr) u) H0)))) in -(ex2_2_ind C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O u c0 a0))) -(\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0 a))) (ex2_2 C C (\lambda -(a0: C).(\lambda (_: C).(csubst1 O u (CHead c0 (Flat f) t) a0))) (\lambda -(a0: C).(\lambda (a: C).(drop (S O) O a0 a)))) (\lambda (x0: C).(\lambda (x1: -C).(\lambda (H4: (csubst1 O u c0 x0)).(\lambda (H5: (drop (S O) O x0 -x1)).(ex2_2_intro C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O u (CHead c0 -(Flat f) t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0 a))) -(CHead x0 (Flat f) (lift (S O) O x)) x1 (csubst1_flat f O u t (lift (S O) O -x) H2 c0 x0 H4) (drop_drop (Flat f) O x0 x1 H5 (lift (S O) O x))))))) H3)))) -H1)))))))) k)))) c)) (\lambda (n: nat).(\lambda (H: ((\forall (c: C).(\forall -(e: C).(\forall (u: T).((getl n c (CHead e (Bind Abbr) u)) \to (ex2_2 C C -(\lambda (a0: C).(\lambda (_: C).(csubst1 n u c a0))) (\lambda (a0: -C).(\lambda (a: C).(drop (S O) n a0 a)))))))))).(\lambda (c: C).(C_ind -(\lambda (c0: C).(\forall (e: C).(\forall (u: T).((getl (S n) c0 (CHead e -(Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S -n) u c0 a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a)))))))) -(\lambda (n0: nat).(\lambda (e: C).(\lambda (u: T).(\lambda (H0: (getl (S n) -(CSort n0) (CHead e (Bind Abbr) u))).(getl_gen_sort n0 (S n) (CHead e (Bind -Abbr) u) H0 (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u -(CSort n0) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 -a))))))))) (\lambda (c0: C).(\lambda (H0: ((\forall (e: C).(\forall (u: -T).((getl (S n) c0 (CHead e (Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0: -C).(\lambda (_: C).(csubst1 (S n) u c0 a0))) (\lambda (a0: C).(\lambda (a: -C).(drop (S O) (S n) a0 a))))))))).(\lambda (k: K).(K_ind (\lambda (k0: -K).(\forall (t: T).(\forall (e: C).(\forall (u: T).((getl (S n) (CHead c0 k0 -t) (CHead e (Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0: C).(\lambda (_: -C).(csubst1 (S n) u (CHead c0 k0 t) a0))) (\lambda (a0: C).(\lambda (a: -C).(drop (S O) (S n) a0 a))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda -(e: C).(\lambda (u: T).(\lambda (H1: (getl (S n) (CHead c0 (Bind b) t) (CHead -e (Bind Abbr) u))).(let H_x \def (subst1_ex u t n) in (let H2 \def H_x in -(ex_ind T (\lambda (t2: T).(subst1 n u t (lift (S O) n t2))) (ex2_2 C C -(\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u (CHead c0 (Bind b) t) a0))) -(\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a)))) (\lambda (x: -T).(\lambda (H3: (subst1 n u t (lift (S O) n x))).(let H4 \def (H c0 e u -(getl_gen_S (Bind b) c0 (CHead e (Bind Abbr) u) t n H1)) in (ex2_2_ind C C -(\lambda (a0: C).(\lambda (_: C).(csubst1 n u c0 a0))) (\lambda (a0: -C).(\lambda (a: C).(drop (S O) n a0 a))) (ex2_2 C C (\lambda (a0: C).(\lambda -(_: C).(csubst1 (S n) u (CHead c0 (Bind b) t) a0))) (\lambda (a0: C).(\lambda -(a: C).(drop (S O) (S n) a0 a)))) (\lambda (x0: C).(\lambda (x1: C).(\lambda -(H5: (csubst1 n u c0 x0)).(\lambda (H6: (drop (S O) n x0 x1)).(ex2_2_intro C -C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u (CHead c0 (Bind b) t) -a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a))) (CHead x0 -(Bind b) (lift (S O) n x)) (CHead x1 (Bind b) x) (csubst1_bind b n u t (lift -(S O) n x) H3 c0 x0 H5) (drop_skip_bind (S O) n x0 x1 H6 b x)))))) H4)))) -H2)))))))) (\lambda (f: F).(\lambda (t: T).(\lambda (e: C).(\lambda (u: -T).(\lambda (H1: (getl (S n) (CHead c0 (Flat f) t) (CHead e (Bind Abbr) -u))).(let H_x \def (subst1_ex u t (S n)) in (let H2 \def H_x in (ex_ind T -(\lambda (t2: T).(subst1 (S n) u t (lift (S O) (S n) t2))) (ex2_2 C C -(\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u (CHead c0 (Flat f) t) a0))) -(\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a)))) (\lambda (x: -T).(\lambda (H3: (subst1 (S n) u t (lift (S O) (S n) x))).(let H4 \def (H0 e -u (getl_gen_S (Flat f) c0 (CHead e (Bind Abbr) u) t n H1)) in (ex2_2_ind C C -(\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u c0 a0))) (\lambda (a0: -C).(\lambda (a: C).(drop (S O) (S n) a0 a))) (ex2_2 C C (\lambda (a0: -C).(\lambda (_: C).(csubst1 (S n) u (CHead c0 (Flat f) t) a0))) (\lambda (a0: -C).(\lambda (a: C).(drop (S O) (S n) a0 a)))) (\lambda (x0: C).(\lambda (x1: -C).(\lambda (H5: (csubst1 (S n) u c0 x0)).(\lambda (H6: (drop (S O) (S n) x0 -x1)).(ex2_2_intro C C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u -(CHead c0 (Flat f) t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S -n) a0 a))) (CHead x0 (Flat f) (lift (S O) (S n) x)) (CHead x1 (Flat f) x) -(csubst1_flat f (S n) u t (lift (S O) (S n) x) H3 c0 x0 H5) (drop_skip_flat -(S O) n x0 x1 H6 f x)))))) H4)))) H2)))))))) k)))) c)))) d). -(* COMMENTS -Initial nodes: 2467 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubst1/props.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubst1/props.ma deleted file mode 100644 index 518a86564..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubst1/props.ma +++ /dev/null @@ -1,75 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubst1/defs.ma". - -include "Basic-1/subst1/defs.ma". - -theorem csubst1_head: - \forall (k: K).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall -(u2: T).((subst1 i v u1 u2) \to (\forall (c1: C).(\forall (c2: C).((csubst1 i -v c1 c2) \to (csubst1 (s k i) v (CHead c1 k u1) (CHead c2 k u2)))))))))) -\def - \lambda (k: K).(\lambda (i: nat).(\lambda (v: T).(\lambda (u1: T).(\lambda -(u2: T).(\lambda (H: (subst1 i v u1 u2)).(subst1_ind i v u1 (\lambda (t: -T).(\forall (c1: C).(\forall (c2: C).((csubst1 i v c1 c2) \to (csubst1 (s k -i) v (CHead c1 k u1) (CHead c2 k t)))))) (\lambda (c1: C).(\lambda (c2: -C).(\lambda (H0: (csubst1 i v c1 c2)).(csubst1_ind i v c1 (\lambda (c: -C).(csubst1 (s k i) v (CHead c1 k u1) (CHead c k u1))) (csubst1_refl (s k i) -v (CHead c1 k u1)) (\lambda (c3: C).(\lambda (H1: (csubst0 i v c1 -c3)).(csubst1_sing (s k i) v (CHead c1 k u1) (CHead c3 k u1) (csubst0_fst k i -c1 c3 v H1 u1)))) c2 H0)))) (\lambda (t2: T).(\lambda (H0: (subst0 i v u1 -t2)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csubst1 i v c1 -c2)).(csubst1_ind i v c1 (\lambda (c: C).(csubst1 (s k i) v (CHead c1 k u1) -(CHead c k t2))) (csubst1_sing (s k i) v (CHead c1 k u1) (CHead c1 k t2) -(csubst0_snd k i v u1 t2 H0 c1)) (\lambda (c3: C).(\lambda (H2: (csubst0 i v -c1 c3)).(csubst1_sing (s k i) v (CHead c1 k u1) (CHead c3 k t2) (csubst0_both -k i v u1 t2 H0 c1 c3 H2)))) c2 H1)))))) u2 H)))))). -(* COMMENTS -Initial nodes: 365 -END *) - -theorem csubst1_bind: - \forall (b: B).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall -(u2: T).((subst1 i v u1 u2) \to (\forall (c1: C).(\forall (c2: C).((csubst1 i -v c1 c2) \to (csubst1 (S i) v (CHead c1 (Bind b) u1) (CHead c2 (Bind b) -u2)))))))))) -\def - \lambda (b: B).(\lambda (i: nat).(\lambda (v: T).(\lambda (u1: T).(\lambda -(u2: T).(\lambda (H: (subst1 i v u1 u2)).(\lambda (c1: C).(\lambda (c2: -C).(\lambda (H0: (csubst1 i v c1 c2)).(eq_ind nat (s (Bind b) i) (\lambda (n: -nat).(csubst1 n v (CHead c1 (Bind b) u1) (CHead c2 (Bind b) u2))) -(csubst1_head (Bind b) i v u1 u2 H c1 c2 H0) (S i) (refl_equal nat (S -i))))))))))). -(* COMMENTS -Initial nodes: 107 -END *) - -theorem csubst1_flat: - \forall (f: F).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall -(u2: T).((subst1 i v u1 u2) \to (\forall (c1: C).(\forall (c2: C).((csubst1 i -v c1 c2) \to (csubst1 i v (CHead c1 (Flat f) u1) (CHead c2 (Flat f) -u2)))))))))) -\def - \lambda (f: F).(\lambda (i: nat).(\lambda (v: T).(\lambda (u1: T).(\lambda -(u2: T).(\lambda (H: (subst1 i v u1 u2)).(\lambda (c1: C).(\lambda (c2: -C).(\lambda (H0: (csubst1 i v c1 c2)).(eq_ind nat (s (Flat f) i) (\lambda (n: -nat).(csubst1 n v (CHead c1 (Flat f) u1) (CHead c2 (Flat f) u2))) -(csubst1_head (Flat f) i v u1 u2 H c1 c2 H0) i (refl_equal nat i)))))))))). -(* COMMENTS -Initial nodes: 103 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/clear.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/clear.ma deleted file mode 100644 index c60700147..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/clear.ma +++ /dev/null @@ -1,74 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubt/defs.ma". - -include "Basic-1/clear/fwd.ma". - -theorem csubt_clear_conf: - \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to -(\forall (e1: C).((clear c1 e1) \to (ex2 C (\lambda (e2: C).(csubt g e1 e2)) -(\lambda (e2: C).(clear c2 e2)))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubt g c1 -c2)).(csubt_ind g (\lambda (c: C).(\lambda (c0: C).(\forall (e1: C).((clear c -e1) \to (ex2 C (\lambda (e2: C).(csubt g e1 e2)) (\lambda (e2: C).(clear c0 -e2))))))) (\lambda (n: nat).(\lambda (e1: C).(\lambda (H0: (clear (CSort n) -e1)).(clear_gen_sort e1 n H0 (ex2 C (\lambda (e2: C).(csubt g e1 e2)) -(\lambda (e2: C).(clear (CSort n) e2))))))) (\lambda (c3: C).(\lambda (c4: -C).(\lambda (H0: (csubt g c3 c4)).(\lambda (H1: ((\forall (e1: C).((clear c3 -e1) \to (ex2 C (\lambda (e2: C).(csubt g e1 e2)) (\lambda (e2: C).(clear c4 -e2))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (e1: C).(\lambda (H2: -(clear (CHead c3 k u) e1)).(K_ind (\lambda (k0: K).((clear (CHead c3 k0 u) -e1) \to (ex2 C (\lambda (e2: C).(csubt g e1 e2)) (\lambda (e2: C).(clear -(CHead c4 k0 u) e2))))) (\lambda (b: B).(\lambda (H3: (clear (CHead c3 (Bind -b) u) e1)).(eq_ind_r C (CHead c3 (Bind b) u) (\lambda (c: C).(ex2 C (\lambda -(e2: C).(csubt g c e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b) u) e2)))) -(ex_intro2 C (\lambda (e2: C).(csubt g (CHead c3 (Bind b) u) e2)) (\lambda -(e2: C).(clear (CHead c4 (Bind b) u) e2)) (CHead c4 (Bind b) u) (csubt_head g -c3 c4 H0 (Bind b) u) (clear_bind b c4 u)) e1 (clear_gen_bind b c3 e1 u H3)))) -(\lambda (f: F).(\lambda (H3: (clear (CHead c3 (Flat f) u) e1)).(let H4 \def -(H1 e1 (clear_gen_flat f c3 e1 u H3)) in (ex2_ind C (\lambda (e2: C).(csubt g -e1 e2)) (\lambda (e2: C).(clear c4 e2)) (ex2 C (\lambda (e2: C).(csubt g e1 -e2)) (\lambda (e2: C).(clear (CHead c4 (Flat f) u) e2))) (\lambda (x: -C).(\lambda (H5: (csubt g e1 x)).(\lambda (H6: (clear c4 x)).(ex_intro2 C -(\lambda (e2: C).(csubt g e1 e2)) (\lambda (e2: C).(clear (CHead c4 (Flat f) -u) e2)) x H5 (clear_flat c4 x H6 f u))))) H4)))) k H2))))))))) (\lambda (c3: -C).(\lambda (c4: C).(\lambda (H0: (csubt g c3 c4)).(\lambda (_: ((\forall -(e1: C).((clear c3 e1) \to (ex2 C (\lambda (e2: C).(csubt g e1 e2)) (\lambda -(e2: C).(clear c4 e2))))))).(\lambda (b: B).(\lambda (H2: (not (eq B b -Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (e1: C).(\lambda (H3: -(clear (CHead c3 (Bind Void) u1) e1)).(eq_ind_r C (CHead c3 (Bind Void) u1) -(\lambda (c: C).(ex2 C (\lambda (e2: C).(csubt g c e2)) (\lambda (e2: -C).(clear (CHead c4 (Bind b) u2) e2)))) (ex_intro2 C (\lambda (e2: C).(csubt -g (CHead c3 (Bind Void) u1) e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b) -u2) e2)) (CHead c4 (Bind b) u2) (csubt_void g c3 c4 H0 b H2 u1 u2) -(clear_bind b c4 u2)) e1 (clear_gen_bind Void c3 e1 u1 H3)))))))))))) -(\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csubt g c3 c4)).(\lambda (_: -((\forall (e1: C).((clear c3 e1) \to (ex2 C (\lambda (e2: C).(csubt g e1 e2)) -(\lambda (e2: C).(clear c4 e2))))))).(\lambda (u: T).(\lambda (t: T).(\lambda -(H2: (ty3 g c3 u t)).(\lambda (H3: (ty3 g c4 u t)).(\lambda (e1: C).(\lambda -(H4: (clear (CHead c3 (Bind Abst) t) e1)).(eq_ind_r C (CHead c3 (Bind Abst) -t) (\lambda (c: C).(ex2 C (\lambda (e2: C).(csubt g c e2)) (\lambda (e2: -C).(clear (CHead c4 (Bind Abbr) u) e2)))) (ex_intro2 C (\lambda (e2: -C).(csubt g (CHead c3 (Bind Abst) t) e2)) (\lambda (e2: C).(clear (CHead c4 -(Bind Abbr) u) e2)) (CHead c4 (Bind Abbr) u) (csubt_abst g c3 c4 H0 u t H2 -H3) (clear_bind Abbr c4 u)) e1 (clear_gen_bind Abst c3 e1 t H4)))))))))))) c1 -c2 H)))). -(* COMMENTS -Initial nodes: 929 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/csuba.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/csuba.ma deleted file mode 100644 index b2cf1183f..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/csuba.ma +++ /dev/null @@ -1,42 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/ty3/arity.ma". - -theorem csubt_csuba: - \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to (csuba -g c1 c2)))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubt g c1 -c2)).(csubt_ind g (\lambda (c: C).(\lambda (c0: C).(csuba g c c0))) (\lambda -(n: nat).(csuba_refl g (CSort n))) (\lambda (c3: C).(\lambda (c4: C).(\lambda -(_: (csubt g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (k: K).(\lambda -(u: T).(csuba_head g c3 c4 H1 k u))))))) (\lambda (c3: C).(\lambda (c4: -C).(\lambda (_: (csubt g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (b: -B).(\lambda (H2: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: -T).(csuba_void g c3 c4 H1 b H2 u1 u2))))))))) (\lambda (c3: C).(\lambda (c4: -C).(\lambda (_: (csubt g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (u: -T).(\lambda (t: T).(\lambda (H2: (ty3 g c3 u t)).(\lambda (_: (ty3 g c4 u -t)).(let H_x \def (ty3_arity g c3 u t H2) in (let H4 \def H_x in (ex2_ind A -(\lambda (a1: A).(arity g c3 u a1)) (\lambda (a1: A).(arity g c3 t (asucc g -a1))) (csuba g (CHead c3 (Bind Abst) t) (CHead c4 (Bind Abbr) u)) (\lambda -(x: A).(\lambda (H5: (arity g c3 u x)).(\lambda (H6: (arity g c3 t (asucc g -x))).(csuba_abst g c3 c4 H1 t x H6 u (csuba_arity g c3 u x H5 c4 H1))))) -H4))))))))))) c1 c2 H)))). -(* COMMENTS -Initial nodes: 313 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/defs.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/defs.ma deleted file mode 100644 index a12c9f829..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/defs.ma +++ /dev/null @@ -1,29 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/ty3/defs.ma". - -inductive csubt (g: G): C \to (C \to Prop) \def -| csubt_sort: \forall (n: nat).(csubt g (CSort n) (CSort n)) -| csubt_head: \forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to (\forall -(k: K).(\forall (u: T).(csubt g (CHead c1 k u) (CHead c2 k u)))))) -| csubt_void: \forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to (\forall -(b: B).((not (eq B b Void)) \to (\forall (u1: T).(\forall (u2: T).(csubt g -(CHead c1 (Bind Void) u1) (CHead c2 (Bind b) u2)))))))) -| csubt_abst: \forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to (\forall -(u: T).(\forall (t: T).((ty3 g c1 u t) \to ((ty3 g c2 u t) \to (csubt g -(CHead c1 (Bind Abst) t) (CHead c2 (Bind Abbr) u)))))))). - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/drop.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/drop.ma deleted file mode 100644 index adaedcc9a..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/drop.ma +++ /dev/null @@ -1,590 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubt/fwd.ma". - -include "Basic-1/drop/fwd.ma". - -theorem csubt_drop_flat: - \forall (g: G).(\forall (f: F).(\forall (n: nat).(\forall (c1: C).(\forall -(c2: C).((csubt g c1 c2) \to (\forall (d1: C).(\forall (u: T).((drop n O c1 -(CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda -(d2: C).(drop n O c2 (CHead d2 (Flat f) u)))))))))))) -\def - \lambda (g: G).(\lambda (f: F).(\lambda (n: nat).(nat_ind (\lambda (n0: -nat).(\forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to (\forall (d1: -C).(\forall (u: T).((drop n0 O c1 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda -(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 (Flat f) -u))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubt g c1 -c2)).(\lambda (d1: C).(\lambda (u: T).(\lambda (H0: (drop O O c1 (CHead d1 -(Flat f) u))).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c2)) H -(CHead d1 (Flat f) u) (drop_gen_refl c1 (CHead d1 (Flat f) u) H0)) in (let -H_x \def (csubt_gen_flat g d1 c2 u f H1) in (let H2 \def H_x in (ex2_ind C -(\lambda (e2: C).(eq C c2 (CHead e2 (Flat f) u))) (\lambda (e2: C).(csubt g -d1 e2)) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O -c2 (CHead d2 (Flat f) u)))) (\lambda (x: C).(\lambda (H3: (eq C c2 (CHead x -(Flat f) u))).(\lambda (H4: (csubt g d1 x)).(eq_ind_r C (CHead x (Flat f) u) -(\lambda (c: C).(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(drop O O c (CHead d2 (Flat f) u))))) (ex_intro2 C (\lambda (d2: C).(csubt -g d1 d2)) (\lambda (d2: C).(drop O O (CHead x (Flat f) u) (CHead d2 (Flat f) -u))) x H4 (drop_refl (CHead x (Flat f) u))) c2 H3)))) H2)))))))))) (\lambda -(n0: nat).(\lambda (H: ((\forall (c1: C).(\forall (c2: C).((csubt g c1 c2) -\to (\forall (d1: C).(\forall (u: T).((drop n0 O c1 (CHead d1 (Flat f) u)) -\to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c2 -(CHead d2 (Flat f) u)))))))))))).(\lambda (c1: C).(\lambda (c2: C).(\lambda -(H0: (csubt g c1 c2)).(csubt_ind g (\lambda (c: C).(\lambda (c0: C).(\forall -(d1: C).(\forall (u: T).((drop (S n0) O c (CHead d1 (Flat f) u)) \to (ex2 C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O c0 (CHead -d2 (Flat f) u))))))))) (\lambda (n1: nat).(\lambda (d1: C).(\lambda (u: -T).(\lambda (H1: (drop (S n0) O (CSort n1) (CHead d1 (Flat f) u))).(and3_ind -(eq C (CHead d1 (Flat f) u) (CSort n1)) (eq nat (S n0) O) (eq nat O O) (ex2 C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) -(CHead d2 (Flat f) u)))) (\lambda (_: (eq C (CHead d1 (Flat f) u) (CSort -n1))).(\lambda (H3: (eq nat (S n0) O)).(\lambda (_: (eq nat O O)).(let H5 -\def (eq_ind nat (S n0) (\lambda (ee: nat).(match ee in nat return (\lambda -(_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H3) -in (False_ind (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop -(S n0) O (CSort n1) (CHead d2 (Flat f) u)))) H5))))) (drop_gen_sort n1 (S n0) -O (CHead d1 (Flat f) u) H1)))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda -(H1: (csubt g c0 c3)).(\lambda (H2: ((\forall (d1: C).(\forall (u: T).((drop -(S n0) O c0 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 -d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Flat f) -u))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (u: T).(\forall -(d1: C).(\forall (u0: T).((drop (S n0) O (CHead c0 k0 u) (CHead d1 (Flat f) -u0)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S -n0) O (CHead c3 k0 u) (CHead d2 (Flat f) u0))))))))) (\lambda (b: B).(\lambda -(u: T).(\lambda (d1: C).(\lambda (u0: T).(\lambda (H3: (drop (S n0) O (CHead -c0 (Bind b) u) (CHead d1 (Flat f) u0))).(ex2_ind C (\lambda (d2: C).(csubt g -d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Flat f) u0))) (ex2 C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 -(Bind b) u) (CHead d2 (Flat f) u0)))) (\lambda (x: C).(\lambda (H4: (csubt g -d1 x)).(\lambda (H5: (drop n0 O c3 (CHead x (Flat f) u0))).(ex_intro2 C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 -(Bind b) u) (CHead d2 (Flat f) u0))) x H4 (drop_drop (Bind b) n0 c3 (CHead x -(Flat f) u0) H5 u))))) (H c0 c3 H1 d1 u0 (drop_gen_drop (Bind b) c0 (CHead d1 -(Flat f) u0) u n0 H3)))))))) (\lambda (f0: F).(\lambda (u: T).(\lambda (d1: -C).(\lambda (u0: T).(\lambda (H3: (drop (S n0) O (CHead c0 (Flat f0) u) -(CHead d1 (Flat f) u0))).(ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Flat f) u0))) (ex2 C (\lambda -(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f0) -u) (CHead d2 (Flat f) u0)))) (\lambda (x: C).(\lambda (H4: (csubt g d1 -x)).(\lambda (H5: (drop (S n0) O c3 (CHead x (Flat f) u0))).(ex_intro2 C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 -(Flat f0) u) (CHead d2 (Flat f) u0))) x H4 (drop_drop (Flat f0) n0 c3 (CHead -x (Flat f) u0) H5 u))))) (H2 d1 u0 (drop_gen_drop (Flat f0) c0 (CHead d1 -(Flat f) u0) u n0 H3)))))))) k)))))) (\lambda (c0: C).(\lambda (c3: -C).(\lambda (H1: (csubt g c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (u: -T).((drop (S n0) O c0 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: -C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Flat f) -u))))))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: -T).(\lambda (u2: T).(\lambda (d1: C).(\lambda (u: T).(\lambda (H4: (drop (S -n0) O (CHead c0 (Bind Void) u1) (CHead d1 (Flat f) u))).(ex2_ind C (\lambda -(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Flat f) -u))) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O -(CHead c3 (Bind b) u2) (CHead d2 (Flat f) u)))) (\lambda (x: C).(\lambda (H5: -(csubt g d1 x)).(\lambda (H6: (drop n0 O c3 (CHead x (Flat f) u))).(ex_intro2 -C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 -(Bind b) u2) (CHead d2 (Flat f) u))) x H5 (drop_drop (Bind b) n0 c3 (CHead x -(Flat f) u) H6 u2))))) (H c0 c3 H1 d1 u (drop_gen_drop (Bind Void) c0 (CHead -d1 (Flat f) u) u1 n0 H4)))))))))))))) (\lambda (c0: C).(\lambda (c3: -C).(\lambda (H1: (csubt g c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (u: -T).((drop (S n0) O c0 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: -C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Flat f) -u))))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u -t)).(\lambda (_: (ty3 g c3 u t)).(\lambda (d1: C).(\lambda (u0: T).(\lambda -(H5: (drop (S n0) O (CHead c0 (Bind Abst) t) (CHead d1 (Flat f) -u0))).(ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 -O c3 (CHead d2 (Flat f) u0))) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Flat f) -u0)))) (\lambda (x: C).(\lambda (H6: (csubt g d1 x)).(\lambda (H7: (drop n0 O -c3 (CHead x (Flat f) u0))).(ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Flat f) -u0))) x H6 (drop_drop (Bind Abbr) n0 c3 (CHead x (Flat f) u0) H7 u))))) (H c0 -c3 H1 d1 u0 (drop_gen_drop (Bind Abst) c0 (CHead d1 (Flat f) u0) t n0 -H5)))))))))))))) c1 c2 H0)))))) n))). -(* COMMENTS -Initial nodes: 2090 -END *) - -theorem csubt_drop_abbr: - \forall (g: G).(\forall (n: nat).(\forall (c1: C).(\forall (c2: C).((csubt g -c1 c2) \to (\forall (d1: C).(\forall (u: T).((drop n O c1 (CHead d1 (Bind -Abbr) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop -n O c2 (CHead d2 (Bind Abbr) u))))))))))) -\def - \lambda (g: G).(\lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (c1: -C).(\forall (c2: C).((csubt g c1 c2) \to (\forall (d1: C).(\forall (u: -T).((drop n0 O c1 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: -C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 (Bind Abbr) -u))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubt g c1 -c2)).(\lambda (d1: C).(\lambda (u: T).(\lambda (H0: (drop O O c1 (CHead d1 -(Bind Abbr) u))).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c2)) H -(CHead d1 (Bind Abbr) u) (drop_gen_refl c1 (CHead d1 (Bind Abbr) u) H0)) in -(let H2 \def (csubt_gen_abbr g d1 c2 u H1) in (ex2_ind C (\lambda (e2: C).(eq -C c2 (CHead e2 (Bind Abbr) u))) (\lambda (e2: C).(csubt g d1 e2)) (ex2 C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 -(Bind Abbr) u)))) (\lambda (x: C).(\lambda (H3: (eq C c2 (CHead x (Bind Abbr) -u))).(\lambda (H4: (csubt g d1 x)).(eq_ind_r C (CHead x (Bind Abbr) u) -(\lambda (c: C).(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(drop O O c (CHead d2 (Bind Abbr) u))))) (ex_intro2 C (\lambda (d2: -C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O (CHead x (Bind Abbr) u) (CHead -d2 (Bind Abbr) u))) x H4 (drop_refl (CHead x (Bind Abbr) u))) c2 H3)))) -H2))))))))) (\lambda (n0: nat).(\lambda (H: ((\forall (c1: C).(\forall (c2: -C).((csubt g c1 c2) \to (\forall (d1: C).(\forall (u: T).((drop n0 O c1 -(CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(drop n0 O c2 (CHead d2 (Bind Abbr) u)))))))))))).(\lambda -(c1: C).(\lambda (c2: C).(\lambda (H0: (csubt g c1 c2)).(csubt_ind g (\lambda -(c: C).(\lambda (c0: C).(\forall (d1: C).(\forall (u: T).((drop (S n0) O c -(CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(drop (S n0) O c0 (CHead d2 (Bind Abbr) u))))))))) (\lambda -(n1: nat).(\lambda (d1: C).(\lambda (u: T).(\lambda (H1: (drop (S n0) O -(CSort n1) (CHead d1 (Bind Abbr) u))).(and3_ind (eq C (CHead d1 (Bind Abbr) -u) (CSort n1)) (eq nat (S n0) O) (eq nat O O) (ex2 C (\lambda (d2: C).(csubt -g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) -u)))) (\lambda (_: (eq C (CHead d1 (Bind Abbr) u) (CSort n1))).(\lambda (H3: -(eq nat (S n0) O)).(\lambda (_: (eq nat O O)).(let H5 \def (eq_ind nat (S n0) -(\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O -\Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind (ex2 C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) -(CHead d2 (Bind Abbr) u)))) H5))))) (drop_gen_sort n1 (S n0) O (CHead d1 -(Bind Abbr) u) H1)))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H1: -(csubt g c0 c3)).(\lambda (H2: ((\forall (d1: C).(\forall (u: T).((drop (S -n0) O c0 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 -d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abbr) -u))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (u: T).(\forall -(d1: C).(\forall (u0: T).((drop (S n0) O (CHead c0 k0 u) (CHead d1 (Bind -Abbr) u0)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(drop (S n0) O (CHead c3 k0 u) (CHead d2 (Bind Abbr) u0))))))))) (\lambda -(b: B).(\lambda (u: T).(\lambda (d1: C).(\lambda (u0: T).(\lambda (H3: (drop -(S n0) O (CHead c0 (Bind b) u) (CHead d1 (Bind Abbr) u0))).(ex2_ind C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 -(Bind Abbr) u0))) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abbr) u0)))) (\lambda -(x: C).(\lambda (H4: (csubt g d1 x)).(\lambda (H5: (drop n0 O c3 (CHead x -(Bind Abbr) u0))).(ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda -(d2: C).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abbr) u0))) x H4 -(drop_drop (Bind b) n0 c3 (CHead x (Bind Abbr) u0) H5 u))))) (H c0 c3 H1 d1 -u0 (drop_gen_drop (Bind b) c0 (CHead d1 (Bind Abbr) u0) u n0 H3)))))))) -(\lambda (f: F).(\lambda (u: T).(\lambda (d1: C).(\lambda (u0: T).(\lambda -(H3: (drop (S n0) O (CHead c0 (Flat f) u) (CHead d1 (Bind Abbr) -u0))).(ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S -n0) O c3 (CHead d2 (Bind Abbr) u0))) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind Abbr) -u0)))) (\lambda (x: C).(\lambda (H4: (csubt g d1 x)).(\lambda (H5: (drop (S -n0) O c3 (CHead x (Bind Abbr) u0))).(ex_intro2 C (\lambda (d2: C).(csubt g d1 -d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind -Abbr) u0))) x H4 (drop_drop (Flat f) n0 c3 (CHead x (Bind Abbr) u0) H5 u))))) -(H2 d1 u0 (drop_gen_drop (Flat f) c0 (CHead d1 (Bind Abbr) u0) u n0 -H3)))))))) k)))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H1: (csubt g -c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (u: T).((drop (S n0) O c0 -(CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u))))))))).(\lambda -(b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: -T).(\lambda (d1: C).(\lambda (u: T).(\lambda (H4: (drop (S n0) O (CHead c0 -(Bind Void) u1) (CHead d1 (Bind Abbr) u))).(ex2_ind C (\lambda (d2: C).(csubt -g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abbr) u))) (ex2 C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 -(Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda (x: C).(\lambda (H5: (csubt -g d1 x)).(\lambda (H6: (drop n0 O c3 (CHead x (Bind Abbr) u))).(ex_intro2 C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 -(Bind b) u2) (CHead d2 (Bind Abbr) u))) x H5 (drop_drop (Bind b) n0 c3 (CHead -x (Bind Abbr) u) H6 u2))))) (H c0 c3 H1 d1 u (drop_gen_drop (Bind Void) c0 -(CHead d1 (Bind Abbr) u) u1 n0 H4)))))))))))))) (\lambda (c0: C).(\lambda -(c3: C).(\lambda (H1: (csubt g c0 c3)).(\lambda (_: ((\forall (d1: -C).(\forall (u: T).((drop (S n0) O c0 (CHead d1 (Bind Abbr) u)) \to (ex2 C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead -d2 (Bind Abbr) u))))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (_: (ty3 g -c0 u t)).(\lambda (_: (ty3 g c3 u t)).(\lambda (d1: C).(\lambda (u0: -T).(\lambda (H5: (drop (S n0) O (CHead c0 (Bind Abst) t) (CHead d1 (Bind -Abbr) u0))).(ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(drop n0 O c3 (CHead d2 (Bind Abbr) u0))) (ex2 C (\lambda (d2: C).(csubt g -d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 -(Bind Abbr) u0)))) (\lambda (x: C).(\lambda (H6: (csubt g d1 x)).(\lambda -(H7: (drop n0 O c3 (CHead x (Bind Abbr) u0))).(ex_intro2 C (\lambda (d2: -C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) -(CHead d2 (Bind Abbr) u0))) x H6 (drop_drop (Bind Abbr) n0 c3 (CHead x (Bind -Abbr) u0) H7 u))))) (H c0 c3 H1 d1 u0 (drop_gen_drop (Bind Abst) c0 (CHead d1 -(Bind Abbr) u0) t n0 H5)))))))))))))) c1 c2 H0)))))) n)). -(* COMMENTS -Initial nodes: 2084 -END *) - -theorem csubt_drop_abst: - \forall (g: G).(\forall (n: nat).(\forall (c1: C).(\forall (c2: C).((csubt g -c1 c2) \to (\forall (d1: C).(\forall (t: T).((drop n O c1 (CHead d1 (Bind -Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(drop n O c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: -C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop n -O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u -t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))))) -\def - \lambda (g: G).(\lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (c1: -C).(\forall (c2: C).((csubt g c1 c2) \to (\forall (d1: C).(\forall (t: -T).((drop n0 O c1 (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: -C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 (Bind Abst) -t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda -(d2: C).(\lambda (u: T).(drop n0 O c2 (CHead d2 (Bind Abbr) u)))) (\lambda -(_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 -g d2 u t)))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubt g -c1 c2)).(\lambda (d1: C).(\lambda (t: T).(\lambda (H0: (drop O O c1 (CHead d1 -(Bind Abst) t))).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c2)) H -(CHead d1 (Bind Abst) t) (drop_gen_refl c1 (CHead d1 (Bind Abst) t) H0)) in -(let H2 \def (csubt_gen_abst g d1 c2 t H1) in (or_ind (ex2 C (\lambda (e2: -C).(eq C c2 (CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csubt g d1 e2))) -(ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) -v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g d1 e2))) (\lambda (_: -C).(\lambda (v2: T).(ty3 g d1 v2 t))) (\lambda (e2: C).(\lambda (v2: T).(ty3 -g e2 v2 t)))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(drop O O c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: -C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop O -O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u -t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (H3: (ex2 C -(\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csubt -g d1 e2)))).(ex2_ind C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abst) t))) -(\lambda (e2: C).(csubt g d1 e2)) (or (ex2 C (\lambda (d2: C).(csubt g d1 -d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T -(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda -(u: T).(drop O O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: -T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) -(\lambda (x: C).(\lambda (H4: (eq C c2 (CHead x (Bind Abst) t))).(\lambda -(H5: (csubt g d1 x)).(eq_ind_r C (CHead x (Bind Abst) t) (\lambda (c: C).(or -(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O c (CHead -d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 -d2))) (\lambda (d2: C).(\lambda (u: T).(drop O O c (CHead d2 (Bind Abbr) -u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: -C).(\lambda (u: T).(ty3 g d2 u t)))))) (or_introl (ex2 C (\lambda (d2: -C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O (CHead x (Bind Abst) t) (CHead -d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 -d2))) (\lambda (d2: C).(\lambda (u: T).(drop O O (CHead x (Bind Abst) t) -(CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) -(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: -C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O (CHead x (Bind Abst) t) (CHead -d2 (Bind Abst) t))) x H5 (drop_refl (CHead x (Bind Abst) t)))) c2 H4)))) H3)) -(\lambda (H3: (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 -(Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g d1 e2))) -(\lambda (_: C).(\lambda (v2: T).(ty3 g d1 v2 t))) (\lambda (e2: C).(\lambda -(v2: T).(ty3 g e2 v2 t))))).(ex4_2_ind C T (\lambda (e2: C).(\lambda (v2: -T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: -T).(csubt g d1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g d1 v2 t))) -(\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 t))) (or (ex2 C (\lambda (d2: -C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) t)))) -(ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: -C).(\lambda (u: T).(drop O O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: -C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g -d2 u t))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H4: (eq C c2 (CHead -x0 (Bind Abbr) x1))).(\lambda (H5: (csubt g d1 x0)).(\lambda (H6: (ty3 g d1 -x1 t)).(\lambda (H7: (ty3 g x0 x1 t)).(eq_ind_r C (CHead x0 (Bind Abbr) x1) -(\lambda (c: C).(or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(drop O O c (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: -C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop O -O c (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u -t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))) (or_intror (ex2 C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O (CHead x0 (Bind -Abbr) x1) (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda -(_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop O O (CHead x0 -(Bind Abbr) x1) (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: -T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) -(ex4_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda -(d2: C).(\lambda (u: T).(drop O O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind -Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: -C).(\lambda (u: T).(ty3 g d2 u t))) x0 x1 H5 (drop_refl (CHead x0 (Bind Abbr) -x1)) H6 H7)) c2 H4))))))) H3)) H2))))))))) (\lambda (n0: nat).(\lambda (H: -((\forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to (\forall (d1: -C).(\forall (t: T).((drop n0 O c1 (CHead d1 (Bind Abst) t)) \to (or (ex2 C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 -(Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 -d2))) (\lambda (d2: C).(\lambda (u: T).(drop n0 O c2 (CHead d2 (Bind Abbr) -u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: -C).(\lambda (u: T).(ty3 g d2 u t))))))))))))).(\lambda (c1: C).(\lambda (c2: -C).(\lambda (H0: (csubt g c1 c2)).(csubt_ind g (\lambda (c: C).(\lambda (c0: -C).(\forall (d1: C).(\forall (t: T).((drop (S n0) O c (CHead d1 (Bind Abst) -t)) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop -(S n0) O c0 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda -(_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O c0 -(CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) -(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))) (\lambda (n1: -nat).(\lambda (d1: C).(\lambda (t: T).(\lambda (H1: (drop (S n0) O (CSort n1) -(CHead d1 (Bind Abst) t))).(and3_ind (eq C (CHead d1 (Bind Abst) t) (CSort -n1)) (eq nat (S n0) O) (eq nat O O) (or (ex2 C (\lambda (d2: C).(csubt g d1 -d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abst) t)))) -(ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: -C).(\lambda (u: T).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) u)))) -(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda -(u: T).(ty3 g d2 u t))))) (\lambda (_: (eq C (CHead d1 (Bind Abst) t) (CSort -n1))).(\lambda (H3: (eq nat (S n0) O)).(\lambda (_: (eq nat O O)).(let H5 -\def (eq_ind nat (S n0) (\lambda (ee: nat).(match ee in nat return (\lambda -(_: nat).Prop) with [O \Rightarrow False | (S _) 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(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) -(\lambda (x: C).(\lambda (H6: (csubt g d1 x)).(\lambda (H7: (drop n0 O c3 -(CHead x (Bind Abst) t))).(or_introl (ex2 C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abst) -t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda -(d2: C).(\lambda (u: T).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind -Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: -C).(\lambda (u: T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csubt g -d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 -(Bind Abst) t))) x H6 (drop_drop (Bind b) n0 c3 (CHead x (Bind Abst) t) H7 -u2)))))) H5)) (\lambda (H5: (ex4_2 C T (\lambda (d2: C).(\lambda (_: -T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop n0 O c3 (CHead d2 -(Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda -(d2: C).(\lambda (u: T).(ty3 g d2 u t))))).(ex4_2_ind C T (\lambda (d2: -C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop -n0 O c3 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 -u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))) (or (ex2 C (\lambda -(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) -u2) (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: -T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O (CHead -c3 (Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: -T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) -(\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (csubt g d1 x0)).(\lambda -(H7: (drop n0 O c3 (CHead x0 (Bind Abbr) x1))).(\lambda (H8: (ty3 g d1 x1 -t)).(\lambda (H9: (ty3 g x0 x1 t)).(or_intror (ex2 C (\lambda (d2: C).(csubt -g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 -(Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 -d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O (CHead c3 (Bind b) u2) -(CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) -(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (ex4_2_intro C T (\lambda -(d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: -T).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda -(_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 -g d2 u t))) x0 x1 H6 (drop_drop (Bind b) n0 c3 (CHead x0 (Bind Abbr) x1) H7 -u2) H8 H9)))))))) H5)) (H c0 c3 H1 d1 t (drop_gen_drop (Bind Void) c0 (CHead -d1 (Bind Abst) t) u1 n0 H4)))))))))))))) (\lambda (c0: C).(\lambda (c3: -C).(\lambda (H1: (csubt g c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (t: -T).((drop (S n0) O c0 (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: -C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abst) -t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda -(d2: C).(\lambda (u: T).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u)))) -(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda -(u: T).(ty3 g d2 u t)))))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (_: -(ty3 g c0 u t)).(\lambda (_: (ty3 g c3 u t)).(\lambda (d1: C).(\lambda (t0: -T).(\lambda (H5: (drop (S n0) O (CHead c0 (Bind Abst) t) (CHead d1 (Bind -Abst) t0))).(or_ind (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(drop n0 O c3 (CHead d2 (Bind Abst) t0)))) (ex4_2 C T (\lambda (d2: -C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop -n0 O c3 (CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g -d1 u0 t0))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0)))) (or (ex2 C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 -(Bind Abbr) u) (CHead d2 (Bind Abst) t0)))) (ex4_2 C T (\lambda (d2: -C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop -(S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (_: -C).(\lambda (u0: T).(ty3 g d1 u0 t0))) (\lambda (d2: C).(\lambda (u0: T).(ty3 -g d2 u0 t0))))) (\lambda (H6: (ex2 C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abst) t0))))).(ex2_ind C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 -(Bind Abst) t0))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abst) t0)))) -(ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: -C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind -Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g d1 u0 t0))) (\lambda -(d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0))))) (\lambda (x: C).(\lambda (H7: -(csubt g d1 x)).(\lambda (H8: (drop n0 O c3 (CHead x (Bind Abst) -t0))).(or_introl (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abst) t0)))) -(ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: -C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind -Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g d1 u0 t0))) (\lambda -(d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0)))) (ex_intro2 C (\lambda (d2: -C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) -(CHead d2 (Bind Abst) t0))) x H7 (drop_drop (Bind Abbr) n0 c3 (CHead x (Bind -Abst) t0) H8 u)))))) H6)) (\lambda (H6: (ex4_2 C T (\lambda (d2: C).(\lambda -(_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop n0 O c3 -(CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g d1 u0 -t0))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0))))).(ex4_2_ind C T -(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda -(u0: T).(drop n0 O c3 (CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda -(u0: T).(ty3 g d1 u0 t0))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 -t0))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S -n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abst) t0)))) (ex4_2 C T -(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda -(u0: T).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) -(\lambda (_: C).(\lambda (u0: T).(ty3 g d1 u0 t0))) (\lambda (d2: C).(\lambda -(u0: T).(ty3 g d2 u0 t0))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H7: -(csubt g d1 x0)).(\lambda (H8: (drop n0 O c3 (CHead x0 (Bind Abbr) -x1))).(\lambda (H9: (ty3 g d1 x1 t0)).(\lambda (H10: (ty3 g x0 x1 -t0)).(or_intror (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abst) t0)))) -(ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: -C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind -Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g d1 u0 t0))) (\lambda -(d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0)))) (ex4_2_intro C T (\lambda (d2: -C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop -(S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (_: -C).(\lambda (u0: T).(ty3 g d1 u0 t0))) (\lambda (d2: C).(\lambda (u0: T).(ty3 -g d2 u0 t0))) x0 x1 H7 (drop_drop (Bind Abbr) n0 c3 (CHead x0 (Bind Abbr) x1) -H8 u) H9 H10)))))))) H6)) (H c0 c3 H1 d1 t0 (drop_gen_drop (Bind Abst) c0 -(CHead d1 (Bind Abst) t0) t n0 H5)))))))))))))) c1 c2 H0)))))) n)). -(* COMMENTS -Initial nodes: 7940 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/fwd.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/fwd.ma deleted file mode 100644 index 63a3eca47..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/fwd.ma +++ /dev/null @@ -1,398 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubt/defs.ma". - -theorem csubt_gen_abbr: - \forall (g: G).(\forall (e1: C).(\forall (c2: C).(\forall (v: T).((csubt g -(CHead e1 (Bind Abbr) v) c2) \to (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 -(Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2))))))) -\def - \lambda (g: G).(\lambda (e1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda -(H: (csubt g (CHead e1 (Bind Abbr) v) c2)).(insert_eq C (CHead e1 (Bind Abbr) -v) (\lambda (c: C).(csubt g c c2)) (\lambda (_: C).(ex2 C (\lambda (e2: -C).(eq C c2 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))) -(\lambda (y: C).(\lambda (H0: (csubt g y c2)).(csubt_ind g (\lambda (c: -C).(\lambda (c0: C).((eq C c (CHead e1 (Bind Abbr) v)) \to (ex2 C (\lambda -(e2: C).(eq C c0 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 -e2)))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead e1 (Bind -Abbr) v))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C -return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) -\Rightarrow False])) I (CHead e1 (Bind Abbr) v) H1) in (False_ind (ex2 C -(\lambda (e2: C).(eq C (CSort n) (CHead e2 (Bind Abbr) v))) (\lambda (e2: -C).(csubt g e1 e2))) H2)))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1: -(csubt g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind Abbr) v)) \to (ex2 -C (\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abbr) v))) (\lambda (e2: -C).(csubt g e1 e2)))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C -(CHead c1 k u) (CHead e1 (Bind Abbr) v))).(let H4 \def (f_equal C C (\lambda -(e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 -| (CHead c _ _) \Rightarrow c])) (CHead c1 k u) (CHead e1 (Bind Abbr) v) H3) -in ((let H5 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda -(_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) -(CHead c1 k u) (CHead e1 (Bind Abbr) v) H3) in ((let H6 \def (f_equal C T -(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) -\Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u) (CHead e1 (Bind -Abbr) v) H3) in (\lambda (H7: (eq K k (Bind Abbr))).(\lambda (H8: (eq C c1 -e1)).(eq_ind_r T v (\lambda (t: T).(ex2 C (\lambda (e2: C).(eq C (CHead c3 k -t) (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))) (eq_ind_r K -(Bind Abbr) (\lambda (k0: K).(ex2 C (\lambda (e2: C).(eq C (CHead c3 k0 v) -(CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))) (let H9 \def -(eq_ind C c1 (\lambda (c: C).((eq C c (CHead e1 (Bind Abbr) v)) \to (ex2 C -(\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt -g e1 e2))))) H2 e1 H8) in (let H10 \def (eq_ind C c1 (\lambda (c: C).(csubt g -c c3)) H1 e1 H8) in (ex_intro2 C (\lambda (e2: C).(eq C (CHead c3 (Bind Abbr) -v) (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)) c3 -(refl_equal C (CHead c3 (Bind Abbr) v)) H10))) k H7) u H6)))) H5)) -H4))))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (_: (csubt g c1 -c3)).(\lambda (_: (((eq C c1 (CHead e1 (Bind Abbr) v)) \to (ex2 C (\lambda -(e2: C).(eq C c3 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 -e2)))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: -T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u1) (CHead e1 -(Bind Abbr) v))).(let H5 \def (eq_ind C (CHead c1 (Bind Void) u1) (\lambda -(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind b0) \Rightarrow (match b0 in B return (\lambda (_: -B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False | Void -\Rightarrow True]) | (Flat _) \Rightarrow False])])) I (CHead e1 (Bind Abbr) -v) H4) in (False_ind (ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind b) u2) -(CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2))) H5))))))))))) -(\lambda (c1: C).(\lambda (c3: C).(\lambda (_: (csubt g c1 c3)).(\lambda (_: -(((eq C c1 (CHead e1 (Bind Abbr) v)) \to (ex2 C (\lambda (e2: C).(eq C c3 -(CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))))).(\lambda (u: -T).(\lambda (t: T).(\lambda (_: (ty3 g c1 u t)).(\lambda (_: (ty3 g c3 u -t)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) t) (CHead e1 (Bind Abbr) -v))).(let H6 \def (eq_ind C (CHead c1 (Bind Abst) t) (\lambda (ee: C).(match -ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | -(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr -\Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat -_) \Rightarrow False])])) I (CHead e1 (Bind Abbr) v) H5) in (False_ind (ex2 C -(\lambda (e2: C).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 (Bind Abbr) v))) -(\lambda (e2: C).(csubt g e1 e2))) H6))))))))))) y c2 H0))) H))))). -(* COMMENTS -Initial nodes: 1111 -END *) - -theorem csubt_gen_abst: - \forall (g: G).(\forall (e1: C).(\forall (c2: C).(\forall (v1: T).((csubt g -(CHead e1 (Bind Abst) v1) c2) \to (or (ex2 C (\lambda (e2: C).(eq C c2 (CHead -e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda -(e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: -C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g -e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))))))))) -\def - \lambda (g: G).(\lambda (e1: C).(\lambda (c2: C).(\lambda (v1: T).(\lambda -(H: (csubt g (CHead e1 (Bind Abst) v1) c2)).(insert_eq C (CHead e1 (Bind -Abst) v1) (\lambda (c: C).(csubt g c c2)) (\lambda (_: C).(or (ex2 C (\lambda -(e2: C).(eq C c2 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 -e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind -Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: -C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 -g e2 v2 v1)))))) (\lambda (y: C).(\lambda (H0: (csubt g y c2)).(csubt_ind g -(\lambda (c: C).(\lambda (c0: C).((eq C c (CHead e1 (Bind Abst) v1)) \to (or -(ex2 C (\lambda (e2: C).(eq C c0 (CHead e2 (Bind Abst) v1))) (\lambda (e2: -C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c0 -(CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 -e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: -C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))))) (\lambda (n: nat).(\lambda (H1: -(eq C (CSort n) (CHead e1 (Bind Abst) v1))).(let H2 \def (eq_ind C (CSort n) -(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 (Bind Abst) -v1) H1) in (False_ind (or (ex2 C (\lambda (e2: C).(eq C (CSort n) (CHead e2 -(Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: -C).(\lambda (v2: T).(eq C (CSort n) (CHead e2 (Bind Abbr) v2)))) (\lambda -(e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: -T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))))) -H2)))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1: (csubt g c1 -c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind Abst) v1)) \to (or (ex2 C -(\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt -g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 -(Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) -(\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda -(v2: T).(ty3 g e2 v2 v1)))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: -(eq C (CHead c1 k u) (CHead e1 (Bind Abst) v1))).(let H4 \def (f_equal C C -(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u) (CHead e1 -(Bind Abst) v1) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e in -C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) -\Rightarrow k0])) (CHead c1 k u) (CHead e1 (Bind Abst) v1) H3) in ((let H6 -\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u) -(CHead e1 (Bind Abst) v1) H3) in (\lambda (H7: (eq K k (Bind Abst))).(\lambda -(H8: (eq C c1 e1)).(eq_ind_r T v1 (\lambda (t: T).(or (ex2 C (\lambda (e2: -C).(eq C (CHead c3 k t) (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g -e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 k t) -(CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 -e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: -C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))) (eq_ind_r K (Bind Abst) (\lambda -(k0: K).(or (ex2 C (\lambda (e2: C).(eq C (CHead c3 k0 v1) (CHead e2 (Bind -Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: -C).(\lambda (v2: T).(eq C (CHead c3 k0 v1) (CHead e2 (Bind Abbr) v2)))) -(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda -(v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 -v1)))))) (let H9 \def (eq_ind C c1 (\lambda (c: C).((eq C c (CHead e1 (Bind -Abst) v1)) \to (or (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abst) -v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda -(v2: T).(eq C c3 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: -T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) -(\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))))))) H2 e1 H8) in (let -H10 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H8) in (or_introl -(ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind Abst) v1) (CHead e2 (Bind Abst) -v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda -(v2: T).(eq C (CHead c3 (Bind Abst) v1) (CHead e2 (Bind Abbr) v2)))) (\lambda -(e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: -T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))) -(ex_intro2 C (\lambda (e2: C).(eq C (CHead c3 (Bind Abst) v1) (CHead e2 (Bind -Abst) v1))) (\lambda (e2: C).(csubt g e1 e2)) c3 (refl_equal C (CHead c3 -(Bind Abst) v1)) H10)))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: -C).(\lambda (c3: C).(\lambda (_: (csubt g c1 c3)).(\lambda (_: (((eq C c1 -(CHead e1 (Bind Abst) v1)) \to (or (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 -(Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: -C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: -C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g -e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 -v1)))))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: -T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u1) (CHead e1 -(Bind Abst) v1))).(let H5 \def (eq_ind C (CHead c1 (Bind Void) u1) (\lambda -(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind b0) \Rightarrow (match b0 in B return (\lambda (_: -B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False | Void -\Rightarrow True]) | (Flat _) \Rightarrow False])])) I (CHead e1 (Bind Abst) -v1) H4) in (False_ind (or (ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind b) -u2) (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T -(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind b) u2) (CHead e2 -(Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) -(\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda -(v2: T).(ty3 g e2 v2 v1))))) H5))))))))))) (\lambda (c1: C).(\lambda (c3: -C).(\lambda (H1: (csubt g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind -Abst) v1)) \to (or (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abst) -v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda -(v2: T).(eq C c3 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: -T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) -(\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))))).(\lambda (u: -T).(\lambda (t: T).(\lambda (H3: (ty3 g c1 u t)).(\lambda (H4: (ty3 g c3 u -t)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) t) (CHead e1 (Bind Abst) -v1))).(let H6 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda -(_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) -(CHead c1 (Bind Abst) t) (CHead e1 (Bind Abst) v1) H5) in ((let H7 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow t | (CHead _ _ t0) \Rightarrow t0])) (CHead c1 (Bind -Abst) t) (CHead e1 (Bind Abst) v1) H5) in (\lambda (H8: (eq C c1 e1)).(let H9 -\def (eq_ind T t (\lambda (t0: T).(ty3 g c3 u t0)) H4 v1 H7) in (let H10 \def -(eq_ind T t (\lambda (t0: T).(ty3 g c1 u t0)) H3 v1 H7) in (let H11 \def -(eq_ind C c1 (\lambda (c: C).(ty3 g c u v1)) H10 e1 H8) in (let H12 \def -(eq_ind C c1 (\lambda (c: C).((eq C c (CHead e1 (Bind Abst) v1)) \to (or (ex2 -C (\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abst) v1))) (\lambda (e2: -C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c3 -(CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 -e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: -C).(\lambda (v2: T).(ty3 g e2 v2 v1))))))) H2 e1 H8) in (let H13 \def (eq_ind -C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H8) in (or_intror (ex2 C (\lambda -(e2: C).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 (Bind Abst) v1))) (\lambda -(e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C -(CHead c3 (Bind Abbr) u) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: -C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g -e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))) -(ex4_2_intro C T (\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind -Abbr) u) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt -g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: -C).(\lambda (v2: T).(ty3 g e2 v2 v1))) c3 u (refl_equal C (CHead c3 (Bind -Abbr) u)) H13 H11 H9))))))))) H6))))))))))) y c2 H0))) H))))). -(* COMMENTS -Initial nodes: 2362 -END *) - -theorem csubt_gen_flat: - \forall (g: G).(\forall (e1: C).(\forall (c2: C).(\forall (v: T).(\forall -(f: F).((csubt g (CHead e1 (Flat f) v) c2) \to (ex2 C (\lambda (e2: C).(eq C -c2 (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g e1 e2)))))))) -\def - \lambda (g: G).(\lambda (e1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda -(f: F).(\lambda (H: (csubt g (CHead e1 (Flat f) v) c2)).(insert_eq C (CHead -e1 (Flat f) v) (\lambda (c: C).(csubt g c c2)) (\lambda (_: C).(ex2 C -(\lambda (e2: C).(eq C c2 (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g -e1 e2)))) (\lambda (y: C).(\lambda (H0: (csubt g y c2)).(csubt_ind g (\lambda -(c: C).(\lambda (c0: C).((eq C c (CHead e1 (Flat f) v)) \to (ex2 C (\lambda -(e2: C).(eq C c0 (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g e1 -e2)))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead e1 (Flat f) -v))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return -(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) -\Rightarrow False])) I (CHead e1 (Flat f) v) H1) in (False_ind (ex2 C -(\lambda (e2: C).(eq C (CSort n) (CHead e2 (Flat f) v))) (\lambda (e2: -C).(csubt g e1 e2))) H2)))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1: -(csubt g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Flat f) v)) \to (ex2 C -(\lambda (e2: C).(eq C c3 (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g -e1 e2)))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c1 k -u) (CHead e1 (Flat f) v))).(let H4 \def (f_equal C C (\lambda (e: C).(match e -in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) -\Rightarrow c])) (CHead c1 k u) (CHead e1 (Flat f) v) H3) in ((let H5 \def -(f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with -[(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c1 k u) -(CHead e1 (Flat f) v) H3) in ((let H6 \def (f_equal C T (\lambda (e: -C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | -(CHead _ _ t) \Rightarrow t])) (CHead c1 k u) (CHead e1 (Flat f) v) H3) in -(\lambda (H7: (eq K k (Flat f))).(\lambda (H8: (eq C c1 e1)).(eq_ind_r T v -(\lambda (t: T).(ex2 C (\lambda (e2: C).(eq C (CHead c3 k t) (CHead e2 (Flat -f) v))) (\lambda (e2: C).(csubt g e1 e2)))) (eq_ind_r K (Flat f) (\lambda -(k0: K).(ex2 C (\lambda (e2: C).(eq C (CHead c3 k0 v) (CHead e2 (Flat f) v))) -(\lambda (e2: C).(csubt g e1 e2)))) (let H9 \def (eq_ind C c1 (\lambda (c: -C).((eq C c (CHead e1 (Flat f) v)) \to (ex2 C (\lambda (e2: C).(eq C c3 -(CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g e1 e2))))) H2 e1 H8) in -(let H10 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H8) in -(ex_intro2 C (\lambda (e2: C).(eq C (CHead c3 (Flat f) v) (CHead e2 (Flat f) -v))) (\lambda (e2: C).(csubt g e1 e2)) c3 (refl_equal C (CHead c3 (Flat f) -v)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: C).(\lambda (c3: -C).(\lambda (_: (csubt g c1 c3)).(\lambda (_: (((eq C c1 (CHead e1 (Flat f) -v)) \to (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 (Flat f) v))) (\lambda -(e2: C).(csubt g e1 e2)))))).(\lambda (b: B).(\lambda (_: (not (eq B b -Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind -Void) u1) (CHead e1 (Flat f) v))).(let H5 \def (eq_ind C (CHead c1 (Bind -Void) u1) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with -[(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return -(\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow -False])])) I (CHead e1 (Flat f) v) H4) in (False_ind (ex2 C (\lambda (e2: -C).(eq C (CHead c3 (Bind b) u2) (CHead e2 (Flat f) v))) (\lambda (e2: -C).(csubt g e1 e2))) H5))))))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda -(_: (csubt g c1 c3)).(\lambda (_: (((eq C c1 (CHead e1 (Flat f) v)) \to (ex2 -C (\lambda (e2: C).(eq C c3 (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g -e1 e2)))))).(\lambda (u: T).(\lambda (t: T).(\lambda (_: (ty3 g c1 u -t)).(\lambda (_: (ty3 g c3 u t)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) t) -(CHead e1 (Flat f) v))).(let H6 \def (eq_ind C (CHead c1 (Bind Abst) t) -(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow -False])])) I (CHead e1 (Flat f) v) H5) in (False_ind (ex2 C (\lambda (e2: -C).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 (Flat f) v))) (\lambda (e2: -C).(csubt g e1 e2))) H6))))))))))) y c2 H0))) H)))))). -(* COMMENTS -Initial nodes: 1103 -END *) - -theorem csubt_gen_bind: - \forall (g: G).(\forall (b1: B).(\forall (e1: C).(\forall (c2: C).(\forall -(v1: T).((csubt g (CHead e1 (Bind b1) v1) c2) \to (ex2_3 B C T (\lambda (b2: -B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) -(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2)))))))))) -\def - \lambda (g: G).(\lambda (b1: B).(\lambda (e1: C).(\lambda (c2: C).(\lambda -(v1: T).(\lambda (H: (csubt g (CHead e1 (Bind b1) v1) c2)).(insert_eq C -(CHead e1 (Bind b1) v1) (\lambda (c: C).(csubt g c c2)) (\lambda (_: -C).(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 -(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: -T).(csubt g e1 e2)))))) (\lambda (y: C).(\lambda (H0: (csubt g y -c2)).(csubt_ind g (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead e1 (Bind -b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: -T).(eq C c0 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: -C).(\lambda (_: T).(csubt g e1 e2)))))))) (\lambda (n: nat).(\lambda (H1: (eq -C (CSort n) (CHead e1 (Bind b1) v1))).(let H2 \def (eq_ind C (CSort n) -(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 (Bind b1) -v1) H1) in (False_ind (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda -(v2: T).(eq C (CSort n) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda -(e2: C).(\lambda (_: T).(csubt g e1 e2))))) H2)))) (\lambda (c1: C).(\lambda -(c3: C).(\lambda (H1: (csubt g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 -(Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda -(v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: -C).(\lambda (_: T).(csubt g e1 e2)))))))).(\lambda (k: K).(\lambda (u: -T).(\lambda (H3: (eq C (CHead c1 k u) (CHead e1 (Bind b1) v1))).(let H4 \def -(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with -[(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u) -(CHead e1 (Bind b1) v1) H3) in ((let H5 \def (f_equal C K (\lambda (e: -C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | -(CHead _ k0 _) \Rightarrow k0])) (CHead c1 k u) (CHead e1 (Bind b1) v1) H3) -in ((let H6 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda -(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) -(CHead c1 k u) (CHead e1 (Bind b1) v1) H3) in (\lambda (H7: (eq K k (Bind -b1))).(\lambda (H8: (eq C c1 e1)).(eq_ind_r T v1 (\lambda (t: T).(ex2_3 B C T -(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 k t) -(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: -T).(csubt g e1 e2)))))) (eq_ind_r K (Bind b1) (\lambda (k0: K).(ex2_3 B C T -(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 k0 v1) -(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: -T).(csubt g e1 e2)))))) (let H9 \def (eq_ind C c1 (\lambda (c: C).((eq C c -(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: -C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: -B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))))))) H2 e1 H8) in (let -H10 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H8) in -(ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C -(CHead c3 (Bind b1) v1) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda -(e2: C).(\lambda (_: T).(csubt g e1 e2)))) b1 c3 v1 (refl_equal C (CHead c3 -(Bind b1) v1)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: -C).(\lambda (c3: C).(\lambda (H1: (csubt g c1 c3)).(\lambda (H2: (((eq C c1 -(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: -C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: -B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2)))))))).(\lambda (b: -B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: -T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) -v1))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda -(_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) -(CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) v1) H4) in ((let H6 \def -(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with -[(CSort _) \Rightarrow Void | (CHead _ k _) \Rightarrow (match k in K return -(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow -Void])])) (CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) v1) H4) in ((let H7 -\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead c1 -(Bind Void) u1) (CHead e1 (Bind b1) v1) H4) in (\lambda (H8: (eq B Void -b1)).(\lambda (H9: (eq C c1 e1)).(let H10 \def (eq_ind C c1 (\lambda (c: -C).((eq C c (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: -B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) -(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))))))) H2 e1 -H9) in (let H11 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H9) -in (let H12 \def (eq_ind_r B b1 (\lambda (b0: B).((eq C e1 (CHead e1 (Bind -b0) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: -T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: -C).(\lambda (_: T).(csubt g e1 e2))))))) H10 Void H8) in (ex2_3_intro B C T -(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind b) -u2) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: -T).(csubt g e1 e2)))) b c3 u2 (refl_equal C (CHead c3 (Bind b) u2)) -H11))))))) H6)) H5))))))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1: -(csubt g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind b1) v1)) \to (ex2_3 -B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 -(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g -e1 e2)))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H3: (ty3 g c1 u -t)).(\lambda (H4: (ty3 g c3 u t)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) -t) (CHead e1 (Bind b1) v1))).(let H6 \def (f_equal C C (\lambda (e: C).(match -e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ -_) \Rightarrow c])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H5) in -((let H7 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: -C).B) with [(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match k -in K return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) -\Rightarrow Abst])])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H5) in -((let H8 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: -C).T) with [(CSort _) \Rightarrow t | (CHead _ _ t0) \Rightarrow t0])) (CHead -c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H5) in (\lambda (H9: (eq B Abst -b1)).(\lambda (H10: (eq C c1 e1)).(let H11 \def (eq_ind T t (\lambda (t0: -T).(ty3 g c3 u t0)) H4 v1 H8) in (let H12 \def (eq_ind T t (\lambda (t0: -T).(ty3 g c1 u t0)) H3 v1 H8) in (let H13 \def (eq_ind C c1 (\lambda (c: -C).(ty3 g c u v1)) H12 e1 H10) in (let H14 \def (eq_ind C c1 (\lambda (c: -C).((eq C c (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: -B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) -(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))))))) H2 e1 -H10) in (let H15 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H10) -in (let H16 \def (eq_ind_r B b1 (\lambda (b: B).((eq C e1 (CHead e1 (Bind b) -v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq -C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda -(_: T).(csubt g e1 e2))))))) H14 Abst H9) in (ex2_3_intro B C T (\lambda (b2: -B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 -(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g -e1 e2)))) Abbr c3 u (refl_equal C (CHead c3 (Bind Abbr) u)) H15)))))))))) -H7)) H6))))))))))) y c2 H0))) H)))))). -(* COMMENTS -Initial nodes: 1899 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/getl.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/getl.ma deleted file mode 100644 index df14528ed..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/getl.ma +++ /dev/null @@ -1,426 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubt/clear.ma". - -include "Basic-1/csubt/drop.ma". - -include "Basic-1/getl/clear.ma". - -theorem csubt_getl_abbr: - \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).(\forall -(n: nat).((getl n c1 (CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csubt g -c1 c2) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n -c2 (CHead d2 (Bind Abbr) u))))))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u: T).(\lambda -(n: nat).(\lambda (H: (getl n c1 (CHead d1 (Bind Abbr) u))).(let H0 \def -(getl_gen_all c1 (CHead d1 (Bind Abbr) u) n H) in (ex2_ind C (\lambda (e: -C).(drop n O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abbr) u))) -(\forall (c2: C).((csubt g c1 c2) \to (ex2 C (\lambda (d2: C).(csubt g d1 -d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u)))))) (\lambda (x: -C).(\lambda (H1: (drop n O c1 x)).(\lambda (H2: (clear x (CHead d1 (Bind -Abbr) u))).(C_ind (\lambda (c: C).((drop n O c1 c) \to ((clear c (CHead d1 -(Bind Abbr) u)) \to (\forall (c2: C).((csubt g c1 c2) \to (ex2 C (\lambda -(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) -u))))))))) (\lambda (n0: nat).(\lambda (_: (drop n O c1 (CSort n0))).(\lambda -(H4: (clear (CSort n0) (CHead d1 (Bind Abbr) u))).(clear_gen_sort (CHead d1 -(Bind Abbr) u) n0 H4 (\forall (c2: C).((csubt g c1 c2) \to (ex2 C (\lambda -(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) -u)))))))))) (\lambda (x0: C).(\lambda (_: (((drop n O c1 x0) \to ((clear x0 -(CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csubt g c1 c2) \to (ex2 C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind -Abbr) u)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (H3: (drop n O c1 -(CHead x0 k t))).(\lambda (H4: (clear (CHead x0 k t) (CHead d1 (Bind Abbr) -u))).(K_ind (\lambda (k0: K).((drop n O c1 (CHead x0 k0 t)) \to ((clear -(CHead x0 k0 t) (CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csubt g c1 -c2) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 -(CHead d2 (Bind Abbr) u))))))))) (\lambda (b: B).(\lambda (H5: (drop n O c1 -(CHead x0 (Bind b) t))).(\lambda (H6: (clear (CHead x0 (Bind b) t) (CHead d1 -(Bind Abbr) u))).(let H7 \def (f_equal C C (\lambda (e: C).(match e in C -return (\lambda (_: C).C) with [(CSort _) \Rightarrow d1 | (CHead c _ _) -\Rightarrow c])) (CHead d1 (Bind Abbr) u) (CHead x0 (Bind b) t) -(clear_gen_bind b x0 (CHead d1 (Bind Abbr) u) t H6)) in ((let H8 \def -(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with -[(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow (match k0 in K -return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) -\Rightarrow Abbr])])) (CHead d1 (Bind Abbr) u) (CHead x0 (Bind b) t) -(clear_gen_bind b x0 (CHead d1 (Bind Abbr) u) t H6)) in ((let H9 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d1 (Bind -Abbr) u) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abbr) u) -t H6)) in (\lambda (H10: (eq B Abbr b)).(\lambda (H11: (eq C d1 x0)).(\lambda -(c2: C).(\lambda (H12: (csubt g c1 c2)).(let H13 \def (eq_ind_r T t (\lambda -(t0: T).(drop n O c1 (CHead x0 (Bind b) t0))) H5 u H9) in (let H14 \def -(eq_ind_r B b (\lambda (b0: B).(drop n O c1 (CHead x0 (Bind b0) u))) H13 Abbr -H10) in (let H15 \def (eq_ind_r C x0 (\lambda (c: C).(drop n O c1 (CHead c -(Bind Abbr) u))) H14 d1 H11) in (ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abbr) u))) (ex2 C (\lambda (d2: -C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u)))) -(\lambda (x1: C).(\lambda (H16: (csubt g d1 x1)).(\lambda (H17: (drop n O c2 -(CHead x1 (Bind Abbr) u))).(ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))) x1 H16 (getl_intro n -c2 (CHead x1 (Bind Abbr) u) (CHead x1 (Bind Abbr) u) H17 (clear_bind Abbr x1 -u)))))) (csubt_drop_abbr g n c1 c2 H12 d1 u H15)))))))))) H8)) H7))))) -(\lambda (f: F).(\lambda (H5: (drop n O c1 (CHead x0 (Flat f) t))).(\lambda -(H6: (clear (CHead x0 (Flat f) t) (CHead d1 (Bind Abbr) u))).(let H7 \def H5 -in (unintro C c1 (\lambda (c: C).((drop n O c (CHead x0 (Flat f) t)) \to -(\forall (c2: C).((csubt g c c2) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u)))))))) (nat_ind (\lambda -(n0: nat).(\forall (x1: C).((drop n0 O x1 (CHead x0 (Flat f) t)) \to (\forall -(c2: C).((csubt g x1 c2) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(getl n0 c2 (CHead d2 (Bind Abbr) u))))))))) (\lambda (x1: -C).(\lambda (H8: (drop O O x1 (CHead x0 (Flat f) t))).(\lambda (c2: -C).(\lambda (H9: (csubt g x1 c2)).(let H10 \def (eq_ind C x1 (\lambda (c: -C).(csubt g c c2)) H9 (CHead x0 (Flat f) t) (drop_gen_refl x1 (CHead x0 (Flat -f) t) H8)) in (let H_y \def (clear_flat x0 (CHead d1 (Bind Abbr) u) -(clear_gen_flat f x0 (CHead d1 (Bind Abbr) u) t H6) f t) in (let H11 \def -(csubt_clear_conf g (CHead x0 (Flat f) t) c2 H10 (CHead d1 (Bind Abbr) u) -H_y) in (ex2_ind C (\lambda (e2: C).(csubt g (CHead d1 (Bind Abbr) u) e2)) -(\lambda (e2: C).(clear c2 e2)) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (x2: -C).(\lambda (H12: (csubt g (CHead d1 (Bind Abbr) u) x2)).(\lambda (H13: -(clear c2 x2)).(let H14 \def (csubt_gen_abbr g d1 x2 u H12) in (ex2_ind C -(\lambda (e2: C).(eq C x2 (CHead e2 (Bind Abbr) u))) (\lambda (e2: C).(csubt -g d1 e2)) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl O -c2 (CHead d2 (Bind Abbr) u)))) (\lambda (x3: C).(\lambda (H15: (eq C x2 -(CHead x3 (Bind Abbr) u))).(\lambda (H16: (csubt g d1 x3)).(let H17 \def -(eq_ind C x2 (\lambda (c: C).(clear c2 c)) H13 (CHead x3 (Bind Abbr) u) H15) -in (ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl O c2 -(CHead d2 (Bind Abbr) u))) x3 H16 (getl_intro O c2 (CHead x3 (Bind Abbr) u) -c2 (drop_refl c2) H17)))))) H14))))) H11)))))))) (\lambda (n0: nat).(\lambda -(H8: ((\forall (x1: C).((drop n0 O x1 (CHead x0 (Flat f) t)) \to (\forall -(c2: C).((csubt g x1 c2) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(getl n0 c2 (CHead d2 (Bind Abbr) u)))))))))).(\lambda (x1: -C).(\lambda (H9: (drop (S n0) O x1 (CHead x0 (Flat f) t))).(\lambda (c2: -C).(\lambda (H10: (csubt g x1 c2)).(let H11 \def (drop_clear x1 (CHead x0 -(Flat f) t) n0 H9) in (ex2_3_ind B C T (\lambda (b: B).(\lambda (e: -C).(\lambda (v: T).(clear x1 (CHead e (Bind b) v))))) (\lambda (_: -B).(\lambda (e: C).(\lambda (_: T).(drop n0 O e (CHead x0 (Flat f) t))))) -(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 -(CHead d2 (Bind Abbr) u)))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: -T).(\lambda (H12: (clear x1 (CHead x3 (Bind x2) x4))).(\lambda (H13: (drop n0 -O x3 (CHead x0 (Flat f) t))).(let H14 \def (csubt_clear_conf g x1 c2 H10 -(CHead x3 (Bind x2) x4) H12) in (ex2_ind C (\lambda (e2: C).(csubt g (CHead -x3 (Bind x2) x4) e2)) (\lambda (e2: C).(clear c2 e2)) (ex2 C (\lambda (d2: -C).(csubt g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr) -u)))) (\lambda (x5: C).(\lambda (H15: (csubt g (CHead x3 (Bind x2) x4) -x5)).(\lambda (H16: (clear c2 x5)).(let H17 \def (csubt_gen_bind g x2 x3 x5 -x4 H15) in (ex2_3_ind B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: -T).(eq C x5 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: -C).(\lambda (_: T).(csubt g x3 e2)))) (ex2 C (\lambda (d2: C).(csubt g d1 -d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda -(x6: B).(\lambda (x7: C).(\lambda (x8: T).(\lambda (H18: (eq C x5 (CHead x7 -(Bind x6) x8))).(\lambda (H19: (csubt g x3 x7)).(let H20 \def (eq_ind C x5 -(\lambda (c: C).(clear c2 c)) H16 (CHead x7 (Bind x6) x8) H18) in (let H21 -\def (H8 x3 H13 x7 H19) in (ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(getl n0 x7 (CHead d2 (Bind Abbr) u))) (ex2 C (\lambda (d2: -C).(csubt g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr) -u)))) (\lambda (x9: C).(\lambda (H22: (csubt g d1 x9)).(\lambda (H23: (getl -n0 x7 (CHead x9 (Bind Abbr) u))).(ex_intro2 C (\lambda (d2: C).(csubt g d1 -d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr) u))) x9 H22 -(getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Abbr) u) n0 H23))))) -H21)))))))) H17))))) H14))))))) H11)))))))) n) H7))))) k H3 H4))))))) x H1 -H2)))) H0))))))). -(* COMMENTS -Initial nodes: 2313 -END *) - -theorem csubt_getl_abst: - \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (t: T).(\forall -(n: nat).((getl n c1 (CHead d1 (Bind Abst) t)) \to (\forall (c2: C).((csubt g -c1 c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: -C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n -c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u -t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (t: T).(\lambda -(n: nat).(\lambda (H: (getl n c1 (CHead d1 (Bind Abst) t))).(let H0 \def -(getl_gen_all c1 (CHead d1 (Bind Abst) t) n H) in (ex2_ind C (\lambda (e: -C).(drop n O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abst) t))) -(\forall (c2: C).((csubt g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 -d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T -(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda -(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: -T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))))) -(\lambda (x: C).(\lambda (H1: (drop n O c1 x)).(\lambda (H2: (clear x (CHead -d1 (Bind Abst) t))).(C_ind (\lambda (c: C).((drop n O c1 c) \to ((clear c -(CHead d1 (Bind Abst) t)) \to (\forall (c2: C).((csubt g c1 c2) \to (or (ex2 -C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 -(Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 -d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) -(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda -(u: T).(ty3 g d2 u t)))))))))) (\lambda (n0: nat).(\lambda (_: (drop n O c1 -(CSort n0))).(\lambda (H4: (clear (CSort n0) (CHead d1 (Bind Abst) -t))).(clear_gen_sort (CHead d1 (Bind Abst) t) n0 H4 (\forall (c2: C).((csubt -g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: -C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n -c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u -t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))))))))) (\lambda (x0: -C).(\lambda (_: (((drop n O c1 x0) \to ((clear x0 (CHead d1 (Bind Abst) t)) -\to (\forall (c2: C).((csubt g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csubt -g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T -(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda -(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: -T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u -t))))))))))).(\lambda (k: K).(\lambda (t0: T).(\lambda (H3: (drop n O c1 -(CHead x0 k t0))).(\lambda (H4: (clear (CHead x0 k t0) (CHead d1 (Bind Abst) -t))).(K_ind (\lambda (k0: K).((drop n O c1 (CHead x0 k0 t0)) \to ((clear -(CHead x0 k0 t0) (CHead d1 (Bind Abst) t)) \to (\forall (c2: C).((csubt g c1 -c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n -c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: -T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 -(Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda -(d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))) (\lambda (b: B).(\lambda (H5: -(drop n O c1 (CHead x0 (Bind b) t0))).(\lambda (H6: (clear (CHead x0 (Bind b) -t0) (CHead d1 (Bind Abst) t))).(let H7 \def (f_equal C C (\lambda (e: -C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d1 | -(CHead c _ _) \Rightarrow c])) (CHead d1 (Bind Abst) t) (CHead x0 (Bind b) -t0) (clear_gen_bind b x0 (CHead d1 (Bind Abst) t) t0 H6)) in ((let H8 \def -(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with -[(CSort _) \Rightarrow Abst | (CHead _ k0 _) \Rightarrow (match k0 in K -return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) -\Rightarrow Abst])])) (CHead d1 (Bind Abst) t) (CHead x0 (Bind b) t0) -(clear_gen_bind b x0 (CHead d1 (Bind Abst) t) t0 H6)) in ((let H9 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow t | (CHead _ _ t1) \Rightarrow t1])) (CHead d1 (Bind -Abst) t) (CHead x0 (Bind b) t0) (clear_gen_bind b x0 (CHead d1 (Bind Abst) t) -t0 H6)) in (\lambda (H10: (eq B Abst b)).(\lambda (H11: (eq C d1 -x0)).(\lambda (c2: C).(\lambda (H12: (csubt g c1 c2)).(let H13 \def (eq_ind_r -T t0 (\lambda (t1: T).(drop n O c1 (CHead x0 (Bind b) t1))) H5 t H9) in (let -H14 \def (eq_ind_r B b (\lambda (b0: B).(drop n O c1 (CHead x0 (Bind b0) t))) -H13 Abst H10) in (let H15 \def (eq_ind_r C x0 (\lambda (c: C).(drop n O c1 -(CHead c (Bind Abst) t))) H14 d1 H11) in (or_ind (ex2 C (\lambda (d2: -C).(csubt g d1 d2)) (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abst) t)))) -(ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: -C).(\lambda (u: T).(drop n O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: -C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g -d2 u t)))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: -C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n -c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u -t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (H16: (ex2 -C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n O c2 (CHead d2 -(Bind Abst) t))))).(ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(drop n O c2 (CHead d2 (Bind Abst) t))) (or (ex2 C (\lambda (d2: C).(csubt -g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T -(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda -(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: -T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) -(\lambda (x1: C).(\lambda (H17: (csubt g d1 x1)).(\lambda (H18: (drop n O c2 -(CHead x1 (Bind Abst) t))).(or_introl (ex2 C (\lambda (d2: C).(csubt g d1 -d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T -(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda -(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: -T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) -(ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 -(CHead d2 (Bind Abst) t))) x1 H17 (getl_intro n c2 (CHead x1 (Bind Abst) t) -(CHead x1 (Bind Abst) t) H18 (clear_bind Abst x1 t))))))) H16)) (\lambda -(H16: (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda -(d2: C).(\lambda (u: T).(drop n O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: -C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g -d2 u t))))).(ex4_2_ind C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) -(\lambda (d2: C).(\lambda (u: T).(drop n O c2 (CHead d2 (Bind Abbr) u)))) -(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda -(u: T).(ty3 g d2 u t))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda -(d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: -C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n -c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u -t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x1: -C).(\lambda (x2: T).(\lambda (H17: (csubt g d1 x1)).(\lambda (H18: (drop n O -c2 (CHead x1 (Bind Abbr) x2))).(\lambda (H19: (ty3 g d1 x2 t)).(\lambda (H20: -(ty3 g x1 x2 t)).(or_intror (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda -(d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: -C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n -c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u -t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (ex4_2_intro C T -(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda -(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: -T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))) x1 x2 -H17 (getl_intro n c2 (CHead x1 (Bind Abbr) x2) (CHead x1 (Bind Abbr) x2) H18 -(clear_bind Abbr x1 x2)) H19 H20)))))))) H16)) (csubt_drop_abst g n c1 c2 H12 -d1 t H15)))))))))) H8)) H7))))) (\lambda (f: F).(\lambda (H5: (drop n O c1 -(CHead x0 (Flat f) t0))).(\lambda (H6: (clear (CHead x0 (Flat f) t0) (CHead -d1 (Bind Abst) t))).(let H7 \def H5 in (unintro C c1 (\lambda (c: C).((drop n -O c (CHead x0 (Flat f) t0)) \to (\forall (c2: C).((csubt g c c2) \to (or (ex2 -C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 -(Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 -d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) -(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda -(u: T).(ty3 g d2 u t))))))))) (nat_ind (\lambda (n0: nat).(\forall (x1: -C).((drop n0 O x1 (CHead x0 (Flat f) t0)) \to (\forall (c2: C).((csubt g x1 -c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl -n0 c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: -T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n0 c2 (CHead d2 -(Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda -(d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))) (\lambda (x1: C).(\lambda -(H8: (drop O O x1 (CHead x0 (Flat f) t0))).(\lambda (c2: C).(\lambda (H9: -(csubt g x1 c2)).(let H10 \def (eq_ind C x1 (\lambda (c: C).(csubt g c c2)) -H9 (CHead x0 (Flat f) t0) (drop_gen_refl x1 (CHead x0 (Flat f) t0) H8)) in -(let H_y \def (clear_flat x0 (CHead d1 (Bind Abst) t) (clear_gen_flat f x0 -(CHead d1 (Bind Abst) t) t0 H6) f t0) in (let H11 \def (csubt_clear_conf g -(CHead x0 (Flat f) t0) c2 H10 (CHead d1 (Bind Abst) t) H_y) in (ex2_ind C -(\lambda (e2: C).(csubt g (CHead d1 (Bind Abst) t) e2)) (\lambda (e2: -C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: -C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl O -c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u -t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x2: -C).(\lambda (H12: (csubt g (CHead d1 (Bind Abst) t) x2)).(\lambda (H13: -(clear c2 x2)).(let H14 \def (csubt_gen_abst g d1 x2 t H12) in (or_ind (ex2 C -(\lambda (e2: C).(eq C x2 (CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csubt -g d1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C x2 (CHead e2 -(Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g d1 e2))) -(\lambda (_: C).(\lambda (v2: T).(ty3 g d1 v2 t))) (\lambda (e2: C).(\lambda -(v2: T).(ty3 g e2 v2 t)))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda -(d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: -T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: -T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) -(\lambda (H15: (ex2 C (\lambda (e2: C).(eq C x2 (CHead e2 (Bind Abst) t))) -(\lambda (e2: C).(csubt g d1 e2)))).(ex2_ind C (\lambda (e2: C).(eq C x2 -(CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csubt g d1 e2)) (or (ex2 C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind -Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) -(\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr) u)))) -(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda -(u: T).(ty3 g d2 u t))))) (\lambda (x3: C).(\lambda (H16: (eq C x2 (CHead x3 -(Bind Abst) t))).(\lambda (H17: (csubt g d1 x3)).(let H18 \def (eq_ind C x2 -(\lambda (c: C).(clear c2 c)) H13 (CHead x3 (Bind Abst) t) H16) in (or_introl -(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead -d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 -d2))) (\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr) u)))) -(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda -(u: T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t))) x3 H17 (getl_intro O -c2 (CHead x3 (Bind Abst) t) c2 (drop_refl c2) H18))))))) H15)) (\lambda (H15: -(ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C x2 (CHead e2 (Bind Abbr) -v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g d1 e2))) (\lambda (_: -C).(\lambda (v2: T).(ty3 g d1 v2 t))) (\lambda (e2: C).(\lambda (v2: T).(ty3 -g e2 v2 t))))).(ex4_2_ind C T (\lambda (e2: C).(\lambda (v2: T).(eq C x2 -(CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g d1 -e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g d1 v2 t))) (\lambda (e2: -C).(\lambda (v2: T).(ty3 g e2 v2 t))) (or (ex2 C (\lambda (d2: C).(csubt g d1 -d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T -(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda -(u: T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: -T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) -(\lambda (x3: C).(\lambda (x4: T).(\lambda (H16: (eq C x2 (CHead x3 (Bind -Abbr) x4))).(\lambda (H17: (csubt g d1 x3)).(\lambda (H18: (ty3 g d1 x4 -t)).(\lambda (H19: (ty3 g x3 x4 t)).(let H20 \def (eq_ind C x2 (\lambda (c: -C).(clear c2 c)) H13 (CHead x3 (Bind Abbr) x4) H16) in (or_intror (ex2 C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind -Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) -(\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr) u)))) -(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda -(u: T).(ty3 g d2 u t)))) (ex4_2_intro C T (\lambda (d2: C).(\lambda (_: -T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 -(Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda -(d2: C).(\lambda (u: T).(ty3 g d2 u t))) x3 x4 H17 (getl_intro O c2 (CHead x3 -(Bind Abbr) x4) c2 (drop_refl c2) H20) H18 H19))))))))) H15)) H14))))) -H11)))))))) (\lambda (n0: nat).(\lambda (H8: ((\forall (x1: C).((drop n0 O x1 -(CHead x0 (Flat f) t0)) \to (\forall (c2: C).((csubt g x1 c2) \to (or (ex2 C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n0 c2 (CHead d2 -(Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 -d2))) (\lambda (d2: C).(\lambda (u: T).(getl n0 c2 (CHead d2 (Bind Abbr) -u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: -C).(\lambda (u: T).(ty3 g d2 u t))))))))))).(\lambda (x1: C).(\lambda (H9: -(drop (S n0) O x1 (CHead x0 (Flat f) t0))).(\lambda (c2: C).(\lambda (H10: -(csubt g x1 c2)).(let H11 \def (drop_clear x1 (CHead x0 (Flat f) t0) n0 H9) -in (ex2_3_ind B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear x1 -(CHead e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: -T).(drop n0 O e (CHead x0 (Flat f) t0))))) (or (ex2 C (\lambda (d2: C).(csubt -g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex4_2 -C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: -C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: -C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g -d2 u t))))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: T).(\lambda (H12: -(clear x1 (CHead x3 (Bind x2) x4))).(\lambda (H13: (drop n0 O x3 (CHead x0 -(Flat f) t0))).(let H14 \def (csubt_clear_conf g x1 c2 H10 (CHead x3 (Bind -x2) x4) H12) in (ex2_ind C (\lambda (e2: C).(csubt g (CHead x3 (Bind x2) x4) -e2)) (\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(csubt g d1 -d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T -(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda -(u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda -(u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) -(\lambda (x5: C).(\lambda (H15: (csubt g (CHead x3 (Bind x2) x4) -x5)).(\lambda (H16: (clear c2 x5)).(let H17 \def (csubt_gen_bind g x2 x3 x5 -x4 H15) in (ex2_3_ind B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: -T).(eq C x5 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: -C).(\lambda (_: T).(csubt g x3 e2)))) (or (ex2 C (\lambda (d2: C).(csubt g d1 -d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T -(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda -(u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda -(u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) -(\lambda (x6: B).(\lambda (x7: C).(\lambda (x8: T).(\lambda (H18: (eq C x5 -(CHead x7 (Bind x6) x8))).(\lambda (H19: (csubt g x3 x7)).(let H20 \def -(eq_ind C x5 (\lambda (c: C).(clear c2 c)) H16 (CHead x7 (Bind x6) x8) H18) -in (let H21 \def (H8 x3 H13 x7 H19) in (or_ind (ex2 C (\lambda (d2: C).(csubt -g d1 d2)) (\lambda (d2: C).(getl n0 x7 (CHead d2 (Bind Abst) t)))) (ex4_2 C T -(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda -(u: T).(getl n0 x7 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: -T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (or -(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 -(CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: -T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead -d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) -(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (H22: (ex2 C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n0 x7 (CHead d2 -(Bind Abst) t))))).(ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(getl n0 x7 (CHead d2 (Bind Abst) t))) (or (ex2 C (\lambda (d2: C).(csubt -g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex4_2 -C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: -C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: -C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g -d2 u t))))) (\lambda (x9: C).(\lambda (H23: (csubt g d1 x9)).(\lambda (H24: -(getl n0 x7 (CHead x9 (Bind Abst) t))).(or_introl (ex2 C (\lambda (d2: -C).(csubt g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) -t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda -(d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda -(_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 -g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(getl (S n0) c2 (CHead d2 (Bind Abst) t))) x9 H23 (getl_clear_bind x6 c2 -x7 x8 H20 (CHead x9 (Bind Abst) t) n0 H24)))))) H22)) (\lambda (H22: (ex4_2 C -T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: -C).(\lambda (u: T).(getl n0 x7 (CHead d2 (Bind Abbr) u)))) (\lambda (_: -C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g -d2 u t))))).(ex4_2_ind C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) -(\lambda (d2: C).(\lambda (u: T).(getl n0 x7 (CHead d2 (Bind Abbr) u)))) -(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda -(u: T).(ty3 g d2 u t))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda -(d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: -C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl -(S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g -d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x9: -C).(\lambda (x10: T).(\lambda (H23: (csubt g d1 x9)).(\lambda (H24: (getl n0 -x7 (CHead x9 (Bind Abbr) x10))).(\lambda (H25: (ty3 g d1 x10 t)).(\lambda -(H26: (ty3 g x9 x10 t)).(or_intror (ex2 C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T -(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda -(u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda -(u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) -(ex4_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda -(d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda -(_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 -g d2 u t))) x9 x10 H23 (getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Abbr) -x10) n0 H24) H25 H26)))))))) H22)) H21)))))))) H17))))) H14))))))) -H11)))))))) n) H7))))) k H3 H4))))))) x H1 H2)))) H0))))))). -(* COMMENTS -Initial nodes: 5861 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/pc3.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/pc3.ma deleted file mode 100644 index 900ba49e9..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/pc3.ma +++ /dev/null @@ -1,62 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubt/getl.ma". - -include "Basic-1/pc3/left.ma". - -theorem csubt_pr2: - \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pr2 c1 -t1 t2) \to (\forall (c2: C).((csubt g c1 c2) \to (pr2 c2 t1 t2))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H: (pr2 c1 t1 t2)).(pr2_ind (\lambda (c: C).(\lambda (t: T).(\lambda (t0: -T).(\forall (c2: C).((csubt g c c2) \to (pr2 c2 t t0)))))) (\lambda (c: -C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(\lambda (c2: -C).(\lambda (_: (csubt g c c2)).(pr2_free c2 t3 t4 H0))))))) (\lambda (c: -C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c -(CHead d (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: -(pr0 t3 t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (c2: -C).(\lambda (H3: (csubt g c c2)).(let H4 \def (csubt_getl_abbr g c d u i H0 -c2 H3) in (ex2_ind C (\lambda (d2: C).(csubt g d d2)) (\lambda (d2: C).(getl -i c2 (CHead d2 (Bind Abbr) u))) (pr2 c2 t3 t) (\lambda (x: C).(\lambda (_: -(csubt g d x)).(\lambda (H6: (getl i c2 (CHead x (Bind Abbr) u))).(pr2_delta -c2 x u i H6 t3 t4 H1 t H2)))) H4)))))))))))))) c1 t1 t2 H))))). -(* COMMENTS -Initial nodes: 245 -END *) - -theorem csubt_pc3: - \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pc3 c1 -t1 t2) \to (\forall (c2: C).((csubt g c1 c2) \to (pc3 c2 t1 t2))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H: (pc3 c1 t1 t2)).(pc3_ind_left c1 (\lambda (t: T).(\lambda (t0: -T).(\forall (c2: C).((csubt g c1 c2) \to (pc3 c2 t t0))))) (\lambda (t: -T).(\lambda (c2: C).(\lambda (_: (csubt g c1 c2)).(pc3_refl c2 t)))) (\lambda -(t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c1 t0 t3)).(\lambda (t4: -T).(\lambda (_: (pc3 c1 t3 t4)).(\lambda (H2: ((\forall (c2: C).((csubt g c1 -c2) \to (pc3 c2 t3 t4))))).(\lambda (c2: C).(\lambda (H3: (csubt g c1 -c2)).(pc3_t t3 c2 t0 (pc3_pr2_r c2 t0 t3 (csubt_pr2 g c1 t0 t3 H0 c2 H3)) t4 -(H2 c2 H3)))))))))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c1 -t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3 c1 t0 t4)).(\lambda (H2: ((\forall -(c2: C).((csubt g c1 c2) \to (pc3 c2 t0 t4))))).(\lambda (c2: C).(\lambda -(H3: (csubt g c1 c2)).(pc3_t t0 c2 t3 (pc3_pr2_x c2 t3 t0 (csubt_pr2 g c1 t0 -t3 H0 c2 H3)) t4 (H2 c2 H3)))))))))) t1 t2 H))))). -(* COMMENTS -Initial nodes: 245 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/props.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/props.ma deleted file mode 100644 index 2efc87ccb..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/props.ma +++ /dev/null @@ -1,28 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubt/defs.ma". - -theorem csubt_refl: - \forall (g: G).(\forall (c: C).(csubt g c c)) -\def - \lambda (g: G).(\lambda (c: C).(C_ind (\lambda (c0: C).(csubt g c0 c0)) -(\lambda (n: nat).(csubt_sort g n)) (\lambda (c0: C).(\lambda (H: (csubt g c0 -c0)).(\lambda (k: K).(\lambda (t: T).(csubt_head g c0 c0 H k t))))) c)). -(* COMMENTS -Initial nodes: 53 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/ty3.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/ty3.ma deleted file mode 100644 index e6199ac15..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/ty3.ma +++ /dev/null @@ -1,102 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubt/pc3.ma". - -include "Basic-1/csubt/props.ma". - -theorem csubt_ty3: - \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c1 -t1 t2) \to (\forall (c2: C).((csubt g c1 c2) \to (ty3 g c2 t1 t2))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H: (ty3 g c1 t1 t2)).(ty3_ind g (\lambda (c: C).(\lambda (t: T).(\lambda -(t0: T).(\forall (c2: C).((csubt g c c2) \to (ty3 g c2 t t0)))))) (\lambda -(c: C).(\lambda (t0: T).(\lambda (t: T).(\lambda (_: (ty3 g c t0 t)).(\lambda -(H1: ((\forall (c2: C).((csubt g c c2) \to (ty3 g c2 t0 t))))).(\lambda (u: -T).(\lambda (t3: T).(\lambda (_: (ty3 g c u t3)).(\lambda (H3: ((\forall (c2: -C).((csubt g c c2) \to (ty3 g c2 u t3))))).(\lambda (H4: (pc3 c t3 -t0)).(\lambda (c2: C).(\lambda (H5: (csubt g c c2)).(ty3_conv g c2 t0 t (H1 -c2 H5) u t3 (H3 c2 H5) (csubt_pc3 g c t3 t0 H4 c2 H5)))))))))))))) (\lambda -(c: C).(\lambda (m: nat).(\lambda (c2: C).(\lambda (_: (csubt g c -c2)).(ty3_sort g c2 m))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d: -C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind Abbr) u))).(\lambda -(t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2: ((\forall (c2: C).((csubt g -d c2) \to (ty3 g c2 u t))))).(\lambda (c2: C).(\lambda (H3: (csubt g c -c2)).(let H4 \def (csubt_getl_abbr g c d u n H0 c2 H3) in (ex2_ind C (\lambda -(d2: C).(csubt g d d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) -u))) (ty3 g c2 (TLRef n) (lift (S n) O t)) (\lambda (x: C).(\lambda (H5: -(csubt g d x)).(\lambda (H6: (getl n c2 (CHead x (Bind Abbr) u))).(ty3_abbr g -n c2 x u H6 t (H2 x H5))))) H4)))))))))))) (\lambda (n: nat).(\lambda (c: -C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind -Abst) u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2: -((\forall (c2: C).((csubt g d c2) \to (ty3 g c2 u t))))).(\lambda (c2: -C).(\lambda (H3: (csubt g c c2)).(let H4 \def (csubt_getl_abst g c d u n H0 -c2 H3) in (or_ind (ex2 C (\lambda (d2: C).(csubt g d d2)) (\lambda (d2: -C).(getl n c2 (CHead d2 (Bind Abst) u)))) (ex4_2 C T (\lambda (d2: -C).(\lambda (_: T).(csubt g d d2))) (\lambda (d2: C).(\lambda (u0: T).(getl n -c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g d u0 -u))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 u)))) (ty3 g c2 (TLRef n) -(lift (S n) O u)) (\lambda (H5: (ex2 C (\lambda (d2: C).(csubt g d d2)) -(\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) u))))).(ex2_ind C (\lambda -(d2: C).(csubt g d d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) -u))) (ty3 g c2 (TLRef n) (lift (S n) O u)) (\lambda (x: C).(\lambda (H6: -(csubt g d x)).(\lambda (H7: (getl n c2 (CHead x (Bind Abst) u))).(ty3_abst g -n c2 x u H7 t (H2 x H6))))) H5)) (\lambda (H5: (ex4_2 C T (\lambda (d2: -C).(\lambda (_: T).(csubt g d d2))) (\lambda (d2: C).(\lambda (u0: T).(getl n -c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g d u0 -u))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 u))))).(ex4_2_ind C T -(\lambda (d2: C).(\lambda (_: T).(csubt g d d2))) (\lambda (d2: C).(\lambda -(u0: T).(getl n c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: -T).(ty3 g d u0 u))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 u))) (ty3 -g c2 (TLRef n) (lift (S n) O u)) (\lambda (x0: C).(\lambda (x1: T).(\lambda -(_: (csubt g d x0)).(\lambda (H7: (getl n c2 (CHead x0 (Bind Abbr) -x1))).(\lambda (_: (ty3 g d x1 u)).(\lambda (H9: (ty3 g x0 x1 u)).(ty3_abbr g -n c2 x0 x1 H7 u H9))))))) H5)) H4)))))))))))) (\lambda (c: C).(\lambda (u: -T).(\lambda (t: T).(\lambda (_: (ty3 g c u t)).(\lambda (H1: ((\forall (c2: -C).((csubt g c c2) \to (ty3 g c2 u t))))).(\lambda (b: B).(\lambda (t0: -T).(\lambda (t3: T).(\lambda (_: (ty3 g (CHead c (Bind b) u) t0 t3)).(\lambda -(H3: ((\forall (c2: C).((csubt g (CHead c (Bind b) u) c2) \to (ty3 g c2 t0 -t3))))).(\lambda (c2: C).(\lambda (H4: (csubt g c c2)).(ty3_bind g c2 u t (H1 -c2 H4) b t0 t3 (H3 (CHead c2 (Bind b) u) (csubt_head g c c2 H4 (Bind b) -u))))))))))))))) (\lambda (c: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: -(ty3 g c w u)).(\lambda (H1: ((\forall (c2: C).((csubt g c c2) \to (ty3 g c2 -w u))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c v (THead (Bind -Abst) u t))).(\lambda (H3: ((\forall (c2: C).((csubt g c c2) \to (ty3 g c2 v -(THead (Bind Abst) u t)))))).(\lambda (c2: C).(\lambda (H4: (csubt g c -c2)).(ty3_appl g c2 w u (H1 c2 H4) v t (H3 c2 H4))))))))))))) (\lambda (c: -C).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (ty3 g c t0 t3)).(\lambda -(H1: ((\forall (c2: C).((csubt g c c2) \to (ty3 g c2 t0 t3))))).(\lambda (t4: -T).(\lambda (_: (ty3 g c t3 t4)).(\lambda (H3: ((\forall (c2: C).((csubt g c -c2) \to (ty3 g c2 t3 t4))))).(\lambda (c2: C).(\lambda (H4: (csubt g c -c2)).(ty3_cast g c2 t0 t3 (H1 c2 H4) t4 (H3 c2 H4)))))))))))) c1 t1 t2 H))))). -(* COMMENTS -Initial nodes: 1325 -END *) - -theorem csubt_ty3_ld: - \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (v: T).((ty3 g c u -v) \to (\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c (Bind Abst) v) t1 -t2) \to (ty3 g (CHead c (Bind Abbr) u) t1 t2)))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (v: T).(\lambda (H: -(ty3 g c u v)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (ty3 g (CHead -c (Bind Abst) v) t1 t2)).(csubt_ty3 g (CHead c (Bind Abst) v) t1 t2 H0 (CHead -c (Bind Abbr) u) (csubt_abst g c c (csubt_refl g c) u v H H))))))))). -(* COMMENTS -Initial nodes: 91 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubv/clear.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubv/clear.ma deleted file mode 100644 index 5c54f5b9a..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubv/clear.ma +++ /dev/null @@ -1,194 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubv/defs.ma". - -include "Basic-1/clear/fwd.ma". - -theorem csubv_clear_conf: - \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (b1: -B).(\forall (d1: C).(\forall (v1: T).((clear c1 (CHead d1 (Bind b1) v1)) \to -(ex2_3 B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 -d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda (v2: T).(clear c2 (CHead d2 -(Bind b2) v2)))))))))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubv c1 c2)).(csubv_ind -(\lambda (c: C).(\lambda (c0: C).(\forall (b1: B).(\forall (d1: C).(\forall -(v1: T).((clear c (CHead d1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (_: -B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2: -B).(\lambda (d2: C).(\lambda (v2: T).(clear c0 (CHead d2 (Bind b2) -v2)))))))))))) (\lambda (n: nat).(\lambda (b1: B).(\lambda (d1: C).(\lambda -(v1: T).(\lambda (H0: (clear (CSort n) (CHead d1 (Bind b1) -v1))).(clear_gen_sort (CHead d1 (Bind b1) v1) n H0 (ex2_3 B C T (\lambda (_: -B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2: -B).(\lambda (d2: C).(\lambda (v2: T).(clear (CSort n) (CHead d2 (Bind b2) -v2)))))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csubv c3 -c4)).(\lambda (_: ((\forall (b1: B).(\forall (d1: C).(\forall (v1: T).((clear -c3 (CHead d1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (_: B).(\lambda (d2: -C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2: B).(\lambda (d2: -C).(\lambda (v2: T).(clear c4 (CHead d2 (Bind b2) v2)))))))))))).(\lambda -(v1: T).(\lambda (v2: T).(\lambda (b1: B).(\lambda (d1: C).(\lambda (v0: -T).(\lambda (H2: (clear (CHead c3 (Bind Void) v1) (CHead d1 (Bind b1) -v0))).(let H3 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda -(_: C).C) with [(CSort _) \Rightarrow d1 | (CHead c _ _) \Rightarrow c])) -(CHead d1 (Bind b1) v0) (CHead c3 (Bind Void) v1) (clear_gen_bind Void c3 -(CHead d1 (Bind b1) v0) v1 H2)) in ((let H4 \def (f_equal C B (\lambda (e: -C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow b1 | -(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind -b) \Rightarrow b | (Flat _) \Rightarrow b1])])) (CHead d1 (Bind b1) v0) -(CHead c3 (Bind Void) v1) (clear_gen_bind Void c3 (CHead d1 (Bind b1) v0) v1 -H2)) in ((let H5 \def (f_equal C T (\lambda (e: C).(match e in C return -(\lambda (_: C).T) with [(CSort _) \Rightarrow v0 | (CHead _ _ t) \Rightarrow -t])) (CHead d1 (Bind b1) v0) (CHead c3 (Bind Void) v1) (clear_gen_bind Void -c3 (CHead d1 (Bind b1) v0) v1 H2)) in (\lambda (_: (eq B b1 Void)).(\lambda -(H7: (eq C d1 c3)).(eq_ind_r C c3 (\lambda (c: C).(ex2_3 B C T (\lambda (_: -B).(\lambda (d2: C).(\lambda (_: T).(csubv c d2)))) (\lambda (b2: B).(\lambda -(d2: C).(\lambda (v3: T).(clear (CHead c4 (Bind Void) v2) (CHead d2 (Bind b2) -v3))))))) (ex2_3_intro B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: -T).(csubv c3 d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda (v3: T).(clear -(CHead c4 (Bind Void) v2) (CHead d2 (Bind b2) v3))))) Void c4 v2 H0 -(clear_bind Void c4 v2)) d1 H7)))) H4)) H3)))))))))))) (\lambda (c3: -C).(\lambda (c4: C).(\lambda (H0: (csubv c3 c4)).(\lambda (_: ((\forall (b1: -B).(\forall (d1: C).(\forall (v1: T).((clear c3 (CHead d1 (Bind b1) v1)) \to -(ex2_3 B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 -d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda (v2: T).(clear c4 (CHead d2 -(Bind b2) v2)))))))))))).(\lambda (b1: B).(\lambda (_: (not (eq B b1 -Void))).(\lambda (b2: B).(\lambda (v1: T).(\lambda (v2: T).(\lambda (b0: -B).(\lambda (d1: C).(\lambda (v0: T).(\lambda (H3: (clear (CHead c3 (Bind b1) -v1) (CHead d1 (Bind b0) v0))).(let H4 \def (f_equal C C (\lambda (e: -C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d1 | -(CHead c _ _) \Rightarrow c])) (CHead d1 (Bind b0) v0) (CHead c3 (Bind b1) -v1) (clear_gen_bind b1 c3 (CHead d1 (Bind b0) v0) v1 H3)) in ((let H5 \def -(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with -[(CSort _) \Rightarrow b0 | (CHead _ k _) \Rightarrow (match k in K return -(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow -b0])])) (CHead d1 (Bind b0) v0) (CHead c3 (Bind b1) v1) (clear_gen_bind b1 c3 -(CHead d1 (Bind b0) v0) v1 H3)) in ((let H6 \def (f_equal C T (\lambda (e: -C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow v0 | -(CHead _ _ t) \Rightarrow t])) (CHead d1 (Bind b0) v0) (CHead c3 (Bind b1) -v1) (clear_gen_bind b1 c3 (CHead d1 (Bind b0) v0) v1 H3)) in (\lambda (_: (eq -B b0 b1)).(\lambda (H8: (eq C d1 c3)).(eq_ind_r C c3 (\lambda (c: C).(ex2_3 B -C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv c d2)))) (\lambda -(b3: B).(\lambda (d2: C).(\lambda (v3: T).(clear (CHead c4 (Bind b2) v2) -(CHead d2 (Bind b3) v3))))))) (ex2_3_intro B C T (\lambda (_: B).(\lambda -(d2: C).(\lambda (_: T).(csubv c3 d2)))) (\lambda (b3: B).(\lambda (d2: -C).(\lambda (v3: T).(clear (CHead c4 (Bind b2) v2) (CHead d2 (Bind b3) -v3))))) b2 c4 v2 H0 (clear_bind b2 c4 v2)) d1 H8)))) H5)) H4))))))))))))))) -(\lambda (c3: C).(\lambda (c4: C).(\lambda (_: (csubv c3 c4)).(\lambda (H1: -((\forall (b1: B).(\forall (d1: C).(\forall (v1: T).((clear c3 (CHead d1 -(Bind b1) v1)) \to (ex2_3 B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: -T).(csubv d1 d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda (v2: T).(clear -c4 (CHead d2 (Bind b2) v2)))))))))))).(\lambda (f1: F).(\lambda (f2: -F).(\lambda (v1: T).(\lambda (v2: T).(\lambda (b1: B).(\lambda (d1: -C).(\lambda (v0: T).(\lambda (H2: (clear (CHead c3 (Flat f1) v1) (CHead d1 -(Bind b1) v0))).(let H_x \def (H1 b1 d1 v0 (clear_gen_flat f1 c3 (CHead d1 -(Bind b1) v0) v1 H2)) in (let H3 \def H_x in (ex2_3_ind B C T (\lambda (_: -B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2: -B).(\lambda (d2: C).(\lambda (v3: T).(clear c4 (CHead d2 (Bind b2) v3))))) -(ex2_3 B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 -d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda (v3: T).(clear (CHead c4 -(Flat f2) v2) (CHead d2 (Bind b2) v3)))))) (\lambda (x0: B).(\lambda (x1: -C).(\lambda (x2: T).(\lambda (H4: (csubv d1 x1)).(\lambda (H5: (clear c4 -(CHead x1 (Bind x0) x2))).(ex2_3_intro B C T (\lambda (_: B).(\lambda (d2: -C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2: B).(\lambda (d2: -C).(\lambda (v3: T).(clear (CHead c4 (Flat f2) v2) (CHead d2 (Bind b2) -v3))))) x0 x1 x2 H4 (clear_flat c4 (CHead x1 (Bind x0) x2) H5 f2 v2))))))) -H3))))))))))))))) c1 c2 H))). -(* COMMENTS -Initial nodes: 1357 -END *) - -theorem csubv_clear_conf_void: - \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (d1: -C).(\forall (v1: T).((clear c1 (CHead d1 (Bind Void) v1)) \to (ex2_2 C T -(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2))) (\lambda (d2: C).(\lambda -(v2: T).(clear c2 (CHead d2 (Bind Void) v2)))))))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubv c1 c2)).(csubv_ind -(\lambda (c: C).(\lambda (c0: C).(\forall (d1: C).(\forall (v1: T).((clear c -(CHead d1 (Bind Void) v1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (_: -T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(clear c0 (CHead d2 -(Bind Void) v2)))))))))) (\lambda (n: nat).(\lambda (d1: C).(\lambda (v1: -T).(\lambda (H0: (clear (CSort n) (CHead d1 (Bind Void) v1))).(clear_gen_sort -(CHead d1 (Bind Void) v1) n H0 (ex2_2 C T (\lambda (d2: C).(\lambda (_: -T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(clear (CSort n) (CHead -d2 (Bind Void) v2)))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: -(csubv c3 c4)).(\lambda (_: ((\forall (d1: C).(\forall (v1: T).((clear c3 -(CHead d1 (Bind Void) v1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (_: -T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(clear c4 (CHead d2 -(Bind Void) v2)))))))))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (d1: -C).(\lambda (v0: T).(\lambda (H2: (clear (CHead c3 (Bind Void) v1) (CHead d1 -(Bind Void) v0))).(let H3 \def (f_equal C C (\lambda (e: C).(match e in C -return (\lambda (_: C).C) with [(CSort _) \Rightarrow d1 | (CHead c _ _) -\Rightarrow c])) (CHead d1 (Bind Void) v0) (CHead c3 (Bind Void) v1) -(clear_gen_bind Void c3 (CHead d1 (Bind Void) v0) v1 H2)) in ((let H4 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow v0 | (CHead _ _ t) \Rightarrow t])) (CHead d1 (Bind -Void) v0) (CHead c3 (Bind Void) v1) (clear_gen_bind Void c3 (CHead d1 (Bind -Void) v0) v1 H2)) in (\lambda (H5: (eq C d1 c3)).(eq_ind_r C c3 (\lambda (c: -C).(ex2_2 C T (\lambda (d2: C).(\lambda (_: T).(csubv c d2))) (\lambda (d2: -C).(\lambda (v3: T).(clear (CHead c4 (Bind Void) v2) (CHead d2 (Bind Void) -v3)))))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csubv c3 d2))) -(\lambda (d2: C).(\lambda (v3: T).(clear (CHead c4 (Bind Void) v2) (CHead d2 -(Bind Void) v3)))) c4 v2 H0 (clear_bind Void c4 v2)) d1 H5))) H3))))))))))) -(\lambda (c3: C).(\lambda (c4: C).(\lambda (_: (csubv c3 c4)).(\lambda (_: -((\forall (d1: C).(\forall (v1: T).((clear c3 (CHead d1 (Bind Void) v1)) \to -(ex2_2 C T (\lambda (d2: C).(\lambda (_: T).(csubv d1 d2))) (\lambda (d2: -C).(\lambda (v2: T).(clear c4 (CHead d2 (Bind Void) v2)))))))))).(\lambda -(b1: B).(\lambda (H2: (not (eq B b1 Void))).(\lambda (b2: B).(\lambda (v1: -T).(\lambda (v2: T).(\lambda (d1: C).(\lambda (v0: T).(\lambda (H3: (clear -(CHead c3 (Bind b1) v1) (CHead d1 (Bind Void) v0))).(let H4 \def (f_equal C C -(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow d1 | (CHead c _ _) \Rightarrow c])) (CHead d1 (Bind Void) v0) -(CHead c3 (Bind b1) v1) (clear_gen_bind b1 c3 (CHead d1 (Bind Void) v0) v1 -H3)) in ((let H5 \def (f_equal C B (\lambda (e: C).(match e in C return -(\lambda (_: C).B) with [(CSort _) \Rightarrow Void | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) -\Rightarrow b | (Flat _) \Rightarrow Void])])) (CHead d1 (Bind Void) v0) -(CHead c3 (Bind b1) v1) (clear_gen_bind b1 c3 (CHead d1 (Bind Void) v0) v1 -H3)) in ((let H6 \def (f_equal C T (\lambda (e: C).(match e in C return -(\lambda (_: C).T) with [(CSort _) \Rightarrow v0 | (CHead _ _ t) \Rightarrow -t])) (CHead d1 (Bind Void) v0) (CHead c3 (Bind b1) v1) (clear_gen_bind b1 c3 -(CHead d1 (Bind Void) v0) v1 H3)) in (\lambda (H7: (eq B Void b1)).(\lambda -(H8: (eq C d1 c3)).(eq_ind_r C c3 (\lambda (c: C).(ex2_2 C T (\lambda (d2: -C).(\lambda (_: T).(csubv c d2))) (\lambda (d2: C).(\lambda (v3: T).(clear -(CHead c4 (Bind b2) v2) (CHead d2 (Bind Void) v3)))))) (let H9 \def (eq_ind_r -B b1 (\lambda (b: B).(not (eq B b Void))) H2 Void H7) in (let H10 \def (match -(H9 (refl_equal B Void)) in False return (\lambda (_: False).(ex2_2 C T -(\lambda (d2: C).(\lambda (_: T).(csubv c3 d2))) (\lambda (d2: C).(\lambda -(v3: T).(clear (CHead c4 (Bind b2) v2) (CHead d2 (Bind Void) v3)))))) with -[]) in H10)) d1 H8)))) H5)) H4)))))))))))))) (\lambda (c3: C).(\lambda (c4: -C).(\lambda (_: (csubv c3 c4)).(\lambda (H1: ((\forall (d1: C).(\forall (v1: -T).((clear c3 (CHead d1 (Bind Void) v1)) \to (ex2_2 C T (\lambda (d2: -C).(\lambda (_: T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(clear -c4 (CHead d2 (Bind Void) v2)))))))))).(\lambda (f1: F).(\lambda (f2: -F).(\lambda (v1: T).(\lambda (v2: T).(\lambda (d1: C).(\lambda (v0: -T).(\lambda (H2: (clear (CHead c3 (Flat f1) v1) (CHead d1 (Bind Void) -v0))).(let H_x \def (H1 d1 v0 (clear_gen_flat f1 c3 (CHead d1 (Bind Void) v0) -v1 H2)) in (let H3 \def H_x in (ex2_2_ind C T (\lambda (d2: C).(\lambda (_: -T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v3: T).(clear c4 (CHead d2 -(Bind Void) v3)))) (ex2_2 C T (\lambda (d2: C).(\lambda (_: T).(csubv d1 -d2))) (\lambda (d2: C).(\lambda (v3: T).(clear (CHead c4 (Flat f2) v2) (CHead -d2 (Bind Void) v3))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H4: (csubv -d1 x0)).(\lambda (H5: (clear c4 (CHead x0 (Bind Void) x1))).(ex2_2_intro C T -(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2))) (\lambda (d2: C).(\lambda -(v3: T).(clear (CHead c4 (Flat f2) v2) (CHead d2 (Bind Void) v3)))) x0 x1 H4 -(clear_flat c4 (CHead x0 (Bind Void) x1) H5 f2 v2)))))) H3)))))))))))))) c1 -c2 H))). -(* COMMENTS -Initial nodes: 1205 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubv/defs.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubv/defs.ma deleted file mode 100644 index 6ffb63608..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubv/defs.ma +++ /dev/null @@ -1,30 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/C/defs.ma". - -inductive csubv: C \to (C \to Prop) \def -| csubv_sort: \forall (n: nat).(csubv (CSort n) (CSort n)) -| csubv_void: \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall -(v1: T).(\forall (v2: T).(csubv (CHead c1 (Bind Void) v1) (CHead c2 (Bind -Void) v2)))))) -| csubv_bind: \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall -(b1: B).((not (eq B b1 Void)) \to (\forall (b2: B).(\forall (v1: T).(\forall -(v2: T).(csubv (CHead c1 (Bind b1) v1) (CHead c2 (Bind b2) v2))))))))) -| csubv_flat: \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall -(f1: F).(\forall (f2: F).(\forall (v1: T).(\forall (v2: T).(csubv (CHead c1 -(Flat f1) v1) (CHead c2 (Flat f2) v2)))))))). - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubv/drop.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubv/drop.ma deleted file mode 100644 index 007670dbb..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubv/drop.ma +++ /dev/null @@ -1,115 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubv/props.ma". - -include "Basic-1/drop/fwd.ma". - -theorem csubv_drop_conf: - \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (e1: -C).(\forall (h: nat).((drop h O c1 e1) \to (ex2 C (\lambda (e2: C).(csubv e1 -e2)) (\lambda (e2: C).(drop h O c2 e2)))))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubv c1 c2)).(csubv_ind -(\lambda (c: C).(\lambda (c0: C).(\forall (e1: C).(\forall (h: nat).((drop h -O c e1) \to (ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop h O -c0 e2)))))))) (\lambda (n: nat).(\lambda (e1: C).(\lambda (h: nat).(\lambda -(H0: (drop h O (CSort n) e1)).(and3_ind (eq C e1 (CSort n)) (eq nat h O) (eq -nat O O) (ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop h O -(CSort n) e2))) (\lambda (H1: (eq C e1 (CSort n))).(\lambda (H2: (eq nat h -O)).(\lambda (_: (eq nat O O)).(eq_ind_r nat O (\lambda (n0: nat).(ex2 C -(\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop n0 O (CSort n) e2)))) -(eq_ind_r C (CSort n) (\lambda (c: C).(ex2 C (\lambda (e2: C).(csubv c e2)) -(\lambda (e2: C).(drop O O (CSort n) e2)))) (ex_intro2 C (\lambda (e2: -C).(csubv (CSort n) e2)) (\lambda (e2: C).(drop O O (CSort n) e2)) (CSort n) -(csubv_refl (CSort n)) (drop_refl (CSort n))) e1 H1) h H2)))) (drop_gen_sort -n h O e1 H0)))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csubv c3 -c4)).(\lambda (H1: ((\forall (e1: C).(\forall (h: nat).((drop h O c3 e1) \to -(ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop h O c4 -e2)))))))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (e1: C).(\lambda (h: -nat).(\lambda (H2: (drop h O (CHead c3 (Bind Void) v1) e1)).(nat_ind (\lambda -(n: nat).((drop n O (CHead c3 (Bind Void) v1) e1) \to (ex2 C (\lambda (e2: -C).(csubv e1 e2)) (\lambda (e2: C).(drop n O (CHead c4 (Bind Void) v2) -e2))))) (\lambda (H3: (drop O O (CHead c3 (Bind Void) v1) e1)).(eq_ind C -(CHead c3 (Bind Void) v1) (\lambda (c: C).(ex2 C (\lambda (e2: C).(csubv c -e2)) (\lambda (e2: C).(drop O O (CHead c4 (Bind Void) v2) e2)))) (ex_intro2 C -(\lambda (e2: C).(csubv (CHead c3 (Bind Void) v1) e2)) (\lambda (e2: C).(drop -O O (CHead c4 (Bind Void) v2) e2)) (CHead c4 (Bind Void) v2) (csubv_bind_same -c3 c4 H0 Void v1 v2) (drop_refl (CHead c4 (Bind Void) v2))) e1 (drop_gen_refl -(CHead c3 (Bind Void) v1) e1 H3))) (\lambda (h0: nat).(\lambda (_: (((drop h0 -O (CHead c3 (Bind Void) v1) e1) \to (ex2 C (\lambda (e2: C).(csubv e1 e2)) -(\lambda (e2: C).(drop h0 O (CHead c4 (Bind Void) v2) e2)))))).(\lambda (H3: -(drop (S h0) O (CHead c3 (Bind Void) v1) e1)).(let H_x \def (H1 e1 (r (Bind -Void) h0) (drop_gen_drop (Bind Void) c3 e1 v1 h0 H3)) in (let H4 \def H_x in -(ex2_ind C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop h0 O c4 -e2)) (ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop (S h0) O -(CHead c4 (Bind Void) v2) e2))) (\lambda (x: C).(\lambda (H5: (csubv e1 -x)).(\lambda (H6: (drop h0 O c4 x)).(ex_intro2 C (\lambda (e2: C).(csubv e1 -e2)) (\lambda (e2: C).(drop (S h0) O (CHead c4 (Bind Void) v2) e2)) x H5 -(drop_drop (Bind Void) h0 c4 x H6 v2))))) H4)))))) h H2)))))))))) (\lambda -(c3: C).(\lambda (c4: C).(\lambda (H0: (csubv c3 c4)).(\lambda (H1: ((\forall -(e1: C).(\forall (h: nat).((drop h O c3 e1) \to (ex2 C (\lambda (e2: -C).(csubv e1 e2)) (\lambda (e2: C).(drop h O c4 e2)))))))).(\lambda (b1: -B).(\lambda (H2: (not (eq B b1 Void))).(\lambda (b2: B).(\lambda (v1: -T).(\lambda (v2: T).(\lambda (e1: C).(\lambda (h: nat).(\lambda (H3: (drop h -O (CHead c3 (Bind b1) v1) e1)).(nat_ind (\lambda (n: nat).((drop n O (CHead -c3 (Bind b1) v1) e1) \to (ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: -C).(drop n O (CHead c4 (Bind b2) v2) e2))))) (\lambda (H4: (drop O O (CHead -c3 (Bind b1) v1) e1)).(eq_ind C (CHead c3 (Bind b1) v1) (\lambda (c: C).(ex2 -C (\lambda (e2: C).(csubv c e2)) (\lambda (e2: C).(drop O O (CHead c4 (Bind -b2) v2) e2)))) (ex_intro2 C (\lambda (e2: C).(csubv (CHead c3 (Bind b1) v1) -e2)) (\lambda (e2: C).(drop O O (CHead c4 (Bind b2) v2) e2)) (CHead c4 (Bind -b2) v2) (csubv_bind c3 c4 H0 b1 H2 b2 v1 v2) (drop_refl (CHead c4 (Bind b2) -v2))) e1 (drop_gen_refl (CHead c3 (Bind b1) v1) e1 H4))) (\lambda (h0: -nat).(\lambda (_: (((drop h0 O (CHead c3 (Bind b1) v1) e1) \to (ex2 C -(\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop h0 O (CHead c4 (Bind -b2) v2) e2)))))).(\lambda (H4: (drop (S h0) O (CHead c3 (Bind b1) v1) -e1)).(let H_x \def (H1 e1 (r (Bind b1) h0) (drop_gen_drop (Bind b1) c3 e1 v1 -h0 H4)) in (let H5 \def H_x in (ex2_ind C (\lambda (e2: C).(csubv e1 e2)) -(\lambda (e2: C).(drop h0 O c4 e2)) (ex2 C (\lambda (e2: C).(csubv e1 e2)) -(\lambda (e2: C).(drop (S h0) O (CHead c4 (Bind b2) v2) e2))) (\lambda (x: -C).(\lambda (H6: (csubv e1 x)).(\lambda (H7: (drop h0 O c4 x)).(ex_intro2 C -(\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop (S h0) O (CHead c4 -(Bind b2) v2) e2)) x H6 (drop_drop (Bind b2) h0 c4 x H7 v2))))) H5)))))) h -H3))))))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csubv c3 -c4)).(\lambda (H1: ((\forall (e1: C).(\forall (h: nat).((drop h O c3 e1) \to -(ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop h O c4 -e2)))))))).(\lambda (f1: F).(\lambda (f2: F).(\lambda (v1: T).(\lambda (v2: -T).(\lambda (e1: C).(\lambda (h: nat).(\lambda (H2: (drop h O (CHead c3 (Flat -f1) v1) e1)).(nat_ind (\lambda (n: nat).((drop n O (CHead c3 (Flat f1) v1) -e1) \to (ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop n O -(CHead c4 (Flat f2) v2) e2))))) (\lambda (H3: (drop O O (CHead c3 (Flat f1) -v1) e1)).(eq_ind C (CHead c3 (Flat f1) v1) (\lambda (c: C).(ex2 C (\lambda -(e2: C).(csubv c e2)) (\lambda (e2: C).(drop O O (CHead c4 (Flat f2) v2) -e2)))) (ex_intro2 C (\lambda (e2: C).(csubv (CHead c3 (Flat f1) v1) e2)) -(\lambda (e2: C).(drop O O (CHead c4 (Flat f2) v2) e2)) (CHead c4 (Flat f2) -v2) (csubv_flat c3 c4 H0 f1 f2 v1 v2) (drop_refl (CHead c4 (Flat f2) v2))) e1 -(drop_gen_refl (CHead c3 (Flat f1) v1) e1 H3))) (\lambda (h0: nat).(\lambda -(_: (((drop h0 O (CHead c3 (Flat f1) v1) e1) \to (ex2 C (\lambda (e2: -C).(csubv e1 e2)) (\lambda (e2: C).(drop h0 O (CHead c4 (Flat f2) v2) -e2)))))).(\lambda (H3: (drop (S h0) O (CHead c3 (Flat f1) v1) e1)).(let H_x -\def (H1 e1 (r (Flat f1) h0) (drop_gen_drop (Flat f1) c3 e1 v1 h0 H3)) in -(let H4 \def H_x in (ex2_ind C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: -C).(drop (S h0) O c4 e2)) (ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda -(e2: C).(drop (S h0) O (CHead c4 (Flat f2) v2) e2))) (\lambda (x: C).(\lambda -(H5: (csubv e1 x)).(\lambda (H6: (drop (S h0) O c4 x)).(ex_intro2 C (\lambda -(e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop (S h0) O (CHead c4 (Flat f2) -v2) e2)) x H5 (drop_drop (Flat f2) h0 c4 x H6 v2))))) H4)))))) h -H2)))))))))))) c1 c2 H))). -(* COMMENTS -Initial nodes: 1897 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubv/getl.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubv/getl.ma deleted file mode 100644 index bbba95084..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubv/getl.ma +++ /dev/null @@ -1,90 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubv/clear.ma". - -include "Basic-1/csubv/drop.ma". - -include "Basic-1/getl/fwd.ma". - -theorem csubv_getl_conf: - \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (b1: -B).(\forall (d1: C).(\forall (v1: T).(\forall (i: nat).((getl i c1 (CHead d1 -(Bind b1) v1)) \to (ex2_3 B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: -T).(csubv d1 d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda (v2: T).(getl -i c2 (CHead d2 (Bind b2) v2))))))))))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubv c1 c2)).(\lambda (b1: -B).(\lambda (d1: C).(\lambda (v1: T).(\lambda (i: nat).(\lambda (H0: (getl i -c1 (CHead d1 (Bind b1) v1))).(let H1 \def (getl_gen_all c1 (CHead d1 (Bind -b1) v1) i H0) in (ex2_ind C (\lambda (e: C).(drop i O c1 e)) (\lambda (e: -C).(clear e (CHead d1 (Bind b1) v1))) (ex2_3 B C T (\lambda (_: B).(\lambda -(d2: C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2: B).(\lambda (d2: -C).(\lambda (v2: T).(getl i c2 (CHead d2 (Bind b2) v2)))))) (\lambda (x: -C).(\lambda (H2: (drop i O c1 x)).(\lambda (H3: (clear x (CHead d1 (Bind b1) -v1))).(let H_x \def (csubv_drop_conf c1 c2 H x i H2) in (let H4 \def H_x in -(ex2_ind C (\lambda (e2: C).(csubv x e2)) (\lambda (e2: C).(drop i O c2 e2)) -(ex2_3 B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 -d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda (v2: T).(getl i c2 (CHead -d2 (Bind b2) v2)))))) (\lambda (x0: C).(\lambda (H5: (csubv x x0)).(\lambda -(H6: (drop i O c2 x0)).(let H_x0 \def (csubv_clear_conf x x0 H5 b1 d1 v1 H3) -in (let H7 \def H_x0 in (ex2_3_ind B C T (\lambda (_: B).(\lambda (d2: -C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2: B).(\lambda (d2: -C).(\lambda (v2: T).(clear x0 (CHead d2 (Bind b2) v2))))) (ex2_3 B C T -(\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2)))) (\lambda -(b2: B).(\lambda (d2: C).(\lambda (v2: T).(getl i c2 (CHead d2 (Bind b2) -v2)))))) (\lambda (x1: B).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H8: -(csubv d1 x2)).(\lambda (H9: (clear x0 (CHead x2 (Bind x1) x3))).(ex2_3_intro -B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2)))) -(\lambda (b2: B).(\lambda (d2: C).(\lambda (v2: T).(getl i c2 (CHead d2 (Bind -b2) v2))))) x1 x2 x3 H8 (getl_intro i c2 (CHead x2 (Bind x1) x3) x0 H6 -H9))))))) H7)))))) H4)))))) H1))))))))). -(* COMMENTS -Initial nodes: 455 -END *) - -theorem csubv_getl_conf_void: - \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (d1: -C).(\forall (v1: T).(\forall (i: nat).((getl i c1 (CHead d1 (Bind Void) v1)) -\to (ex2_2 C T (\lambda (d2: C).(\lambda (_: T).(csubv d1 d2))) (\lambda (d2: -C).(\lambda (v2: T).(getl i c2 (CHead d2 (Bind Void) v2))))))))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubv c1 c2)).(\lambda (d1: -C).(\lambda (v1: T).(\lambda (i: nat).(\lambda (H0: (getl i c1 (CHead d1 -(Bind Void) v1))).(let H1 \def (getl_gen_all c1 (CHead d1 (Bind Void) v1) i -H0) in (ex2_ind C (\lambda (e: C).(drop i O c1 e)) (\lambda (e: C).(clear e -(CHead d1 (Bind Void) v1))) (ex2_2 C T (\lambda (d2: C).(\lambda (_: -T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(getl i c2 (CHead d2 -(Bind Void) v2))))) (\lambda (x: C).(\lambda (H2: (drop i O c1 x)).(\lambda -(H3: (clear x (CHead d1 (Bind Void) v1))).(let H_x \def (csubv_drop_conf c1 -c2 H x i H2) in (let H4 \def H_x in (ex2_ind C (\lambda (e2: C).(csubv x e2)) -(\lambda (e2: C).(drop i O c2 e2)) (ex2_2 C T (\lambda (d2: C).(\lambda (_: -T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(getl i c2 (CHead d2 -(Bind Void) v2))))) (\lambda (x0: C).(\lambda (H5: (csubv x x0)).(\lambda -(H6: (drop i O c2 x0)).(let H_x0 \def (csubv_clear_conf_void x x0 H5 d1 v1 -H3) in (let H7 \def H_x0 in (ex2_2_ind C T (\lambda (d2: C).(\lambda (_: -T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(clear x0 (CHead d2 -(Bind Void) v2)))) (ex2_2 C T (\lambda (d2: C).(\lambda (_: T).(csubv d1 -d2))) (\lambda (d2: C).(\lambda (v2: T).(getl i c2 (CHead d2 (Bind Void) -v2))))) (\lambda (x1: C).(\lambda (x2: T).(\lambda (H8: (csubv d1 -x1)).(\lambda (H9: (clear x0 (CHead x1 (Bind Void) x2))).(ex2_2_intro C T -(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2))) (\lambda (d2: C).(\lambda -(v2: T).(getl i c2 (CHead d2 (Bind Void) v2)))) x1 x2 H8 (getl_intro i c2 -(CHead x1 (Bind Void) x2) x0 H6 H9)))))) H7)))))) H4)))))) H1)))))))). -(* COMMENTS -Initial nodes: 417 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubv/props.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubv/props.ma deleted file mode 100644 index 716922824..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/csubv/props.ma +++ /dev/null @@ -1,48 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubv/defs.ma". - -include "Basic-1/T/props.ma". - -theorem csubv_bind_same: - \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (b: B).(\forall -(v1: T).(\forall (v2: T).(csubv (CHead c1 (Bind b) v1) (CHead c2 (Bind b) -v2))))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubv c1 c2)).(\lambda (b: -B).(B_ind (\lambda (b0: B).(\forall (v1: T).(\forall (v2: T).(csubv (CHead c1 -(Bind b0) v1) (CHead c2 (Bind b0) v2))))) (\lambda (v1: T).(\lambda (v2: -T).(csubv_bind c1 c2 H Abbr (\lambda (H0: (eq B Abbr Void)).(not_abbr_void -H0)) Abbr v1 v2))) (\lambda (v1: T).(\lambda (v2: T).(csubv_bind c1 c2 H Abst -(sym_not_eq B Void Abst not_void_abst) Abst v1 v2))) (\lambda (v1: -T).(\lambda (v2: T).(csubv_void c1 c2 H v1 v2))) b)))). -(* COMMENTS -Initial nodes: 121 -END *) - -theorem csubv_refl: - \forall (c: C).(csubv c c) -\def - \lambda (c: C).(C_ind (\lambda (c0: C).(csubv c0 c0)) (\lambda (n: -nat).(csubv_sort n)) (\lambda (c0: C).(\lambda (H: (csubv c0 c0)).(\lambda -(k: K).(K_ind (\lambda (k0: K).(\forall (t: T).(csubv (CHead c0 k0 t) (CHead -c0 k0 t)))) (\lambda (b: B).(\lambda (t: T).(csubv_bind_same c0 c0 H b t t))) -(\lambda (f: F).(\lambda (t: T).(csubv_flat c0 c0 H f f t t))) k)))) c). -(* COMMENTS -Initial nodes: 93 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/definitions.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/definitions.ma deleted file mode 100644 index 5863b4243..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/definitions.ma +++ /dev/null @@ -1,68 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/tlt/defs.ma". - -include "Basic-1/iso/defs.ma". - -include "Basic-1/clen/defs.ma". - -include "Basic-1/flt/defs.ma". - -include "Basic-1/app/defs.ma". - -include "Basic-1/cnt/defs.ma". - -include "Basic-1/cimp/defs.ma". - -include "Basic-1/csubv/defs.ma". - -include "Basic-1/subst/defs.ma". - -include "Basic-1/subst1/defs.ma". - -include "Basic-1/csubst1/defs.ma". - -include "Basic-1/fsubst0/defs.ma". - -include "Basic-1/next_plus/defs.ma". - -include "Basic-1/sty1/defs.ma". - -include "Basic-1/llt/defs.ma". - -include "Basic-1/aprem/defs.ma". - -include "Basic-1/ex0/defs.ma". - -include "Basic-1/wcpr0/defs.ma". - -include "Basic-1/csuba/defs.ma". - -include "Basic-1/nf2/defs.ma". - -include "Basic-1/ex2/defs.ma". - -include "Basic-1/csubc/defs.ma". - -include "Basic-1/pc1/defs.ma". - -include "Basic-1/ex1/defs.ma". - -include "Basic-1/csubt/defs.ma". - -include "Basic-1/wf3/defs.ma". - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/drop/defs.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/drop/defs.ma deleted file mode 100644 index 987ae0ceb..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/drop/defs.ma +++ /dev/null @@ -1,31 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/C/defs.ma". - -include "Basic-1/lift/defs.ma". - -include "Basic-1/r/defs.ma". - -inductive drop: nat \to (nat \to (C \to (C \to Prop))) \def -| drop_refl: \forall (c: C).(drop O O c c) -| drop_drop: \forall (k: K).(\forall (h: nat).(\forall (c: C).(\forall (e: -C).((drop (r k h) O c e) \to (\forall (u: T).(drop (S h) O (CHead c k u) -e)))))) -| drop_skip: \forall (k: K).(\forall (h: nat).(\forall (d: nat).(\forall (c: -C).(\forall (e: C).((drop h (r k d) c e) \to (\forall (u: T).(drop h (S d) -(CHead c k (lift h (r k d) u)) (CHead e k u)))))))). - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/drop/fwd.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/drop/fwd.ma deleted file mode 100644 index 48495c148..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/drop/fwd.ma +++ /dev/null @@ -1,384 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/drop/defs.ma". - -theorem drop_gen_sort: - \forall (n: nat).(\forall (h: nat).(\forall (d: nat).(\forall (x: C).((drop -h d (CSort n) x) \to (and3 (eq C x (CSort n)) (eq nat h O) (eq nat d O)))))) -\def - \lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (x: -C).(\lambda (H: (drop h d (CSort n) x)).(insert_eq C (CSort n) (\lambda (c: -C).(drop h d c x)) (\lambda (c: C).(and3 (eq C x c) (eq nat h O) (eq nat d -O))) (\lambda (y: C).(\lambda (H0: (drop h d y x)).(drop_ind (\lambda (n0: -nat).(\lambda (n1: nat).(\lambda (c: C).(\lambda (c0: C).((eq C c (CSort n)) -\to (and3 (eq C c0 c) (eq nat n0 O) (eq nat n1 O))))))) (\lambda (c: -C).(\lambda (H1: (eq C c (CSort n))).(let H2 \def (f_equal C C (\lambda (e: -C).e) c (CSort n) H1) in (eq_ind_r C (CSort n) (\lambda (c0: C).(and3 (eq C -c0 c0) (eq nat O O) (eq nat O O))) (and3_intro (eq C (CSort n) (CSort n)) (eq -nat O O) (eq nat O O) (refl_equal C (CSort n)) (refl_equal nat O) (refl_equal -nat O)) c H2)))) (\lambda (k: K).(\lambda (h0: nat).(\lambda (c: C).(\lambda -(e: C).(\lambda (_: (drop (r k h0) O c e)).(\lambda (_: (((eq C c (CSort n)) -\to (and3 (eq C e c) (eq nat (r k h0) O) (eq nat O O))))).(\lambda (u: -T).(\lambda (H3: (eq C (CHead c k u) (CSort n))).(let H4 \def (eq_ind C -(CHead c k u) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) -with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I -(CSort n) H3) in (False_ind (and3 (eq C e (CHead c k u)) (eq nat (S h0) O) -(eq nat O O)) H4)))))))))) (\lambda (k: K).(\lambda (h0: nat).(\lambda (d0: -nat).(\lambda (c: C).(\lambda (e: C).(\lambda (_: (drop h0 (r k d0) c -e)).(\lambda (_: (((eq C c (CSort n)) \to (and3 (eq C e c) (eq nat h0 O) (eq -nat (r k d0) O))))).(\lambda (u: T).(\lambda (H3: (eq C (CHead c k (lift h0 -(r k d0) u)) (CSort n))).(let H4 \def (eq_ind C (CHead c k (lift h0 (r k d0) -u)) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort -_) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H3) in -(False_ind (and3 (eq C (CHead e k u) (CHead c k (lift h0 (r k d0) u))) (eq -nat h0 O) (eq nat (S d0) O)) H4))))))))))) h d y x H0))) H))))). -(* COMMENTS -Initial nodes: 595 -END *) - -theorem drop_gen_refl: - \forall (x: C).(\forall (e: C).((drop O O x e) \to (eq C x e))) -\def - \lambda (x: C).(\lambda (e: C).(\lambda (H: (drop O O x e)).(insert_eq nat O -(\lambda (n: nat).(drop n O x e)) (\lambda (_: nat).(eq C x e)) (\lambda (y: -nat).(\lambda (H0: (drop y O x e)).(insert_eq nat O (\lambda (n: nat).(drop y -n x e)) (\lambda (n: nat).((eq nat y n) \to (eq C x e))) (\lambda (y0: -nat).(\lambda (H1: (drop y y0 x e)).(drop_ind (\lambda (n: nat).(\lambda (n0: -nat).(\lambda (c: C).(\lambda (c0: C).((eq nat n0 O) \to ((eq nat n n0) \to -(eq C c c0))))))) (\lambda (c: C).(\lambda (_: (eq nat O O)).(\lambda (_: (eq -nat O O)).(refl_equal C c)))) (\lambda (k: K).(\lambda (h: nat).(\lambda (c: -C).(\lambda (e0: C).(\lambda (_: (drop (r k h) O c e0)).(\lambda (_: (((eq -nat O O) \to ((eq nat (r k h) O) \to (eq C c e0))))).(\lambda (u: T).(\lambda -(_: (eq nat O O)).(\lambda (H5: (eq nat (S h) O)).(let H6 \def (eq_ind nat (S -h) (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O -\Rightarrow False | (S _) \Rightarrow True])) I O H5) in (False_ind (eq C -(CHead c k u) e0) H6))))))))))) (\lambda (k: K).(\lambda (h: nat).(\lambda -(d: nat).(\lambda (c: C).(\lambda (e0: C).(\lambda (H2: (drop h (r k d) c -e0)).(\lambda (H3: (((eq nat (r k d) O) \to ((eq nat h (r k d)) \to (eq C c -e0))))).(\lambda (u: T).(\lambda (H4: (eq nat (S d) O)).(\lambda (H5: (eq nat -h (S d))).(let H6 \def (f_equal nat nat (\lambda (e1: nat).e1) h (S d) H5) in -(let H7 \def (eq_ind nat h (\lambda (n: nat).((eq nat (r k d) O) \to ((eq nat -n (r k d)) \to (eq C c e0)))) H3 (S d) H6) in (let H8 \def (eq_ind nat h -(\lambda (n: nat).(drop n (r k d) c e0)) H2 (S d) H6) in (eq_ind_r nat (S d) -(\lambda (n: nat).(eq C (CHead c k (lift n (r k d) u)) (CHead e0 k u))) (let -H9 \def (eq_ind nat (S d) (\lambda (ee: nat).(match ee in nat return (\lambda -(_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H4) -in (False_ind (eq C (CHead c k (lift (S d) (r k d) u)) (CHead e0 k u)) H9)) h -H6)))))))))))))) y y0 x e H1))) H0))) H))). -(* COMMENTS -Initial nodes: 561 -END *) - -theorem drop_gen_drop: - \forall (k: K).(\forall (c: C).(\forall (x: C).(\forall (u: T).(\forall (h: -nat).((drop (S h) O (CHead c k u) x) \to (drop (r k h) O c x)))))) -\def - \lambda (k: K).(\lambda (c: C).(\lambda (x: C).(\lambda (u: T).(\lambda (h: -nat).(\lambda (H: (drop (S h) O (CHead c k u) x)).(insert_eq C (CHead c k u) -(\lambda (c0: C).(drop (S h) O c0 x)) (\lambda (_: C).(drop (r k h) O c x)) -(\lambda (y: C).(\lambda (H0: (drop (S h) O y x)).(insert_eq nat O (\lambda -(n: nat).(drop (S h) n y x)) (\lambda (n: nat).((eq C y (CHead c k u)) \to -(drop (r k h) n c x))) (\lambda (y0: nat).(\lambda (H1: (drop (S h) y0 y -x)).(insert_eq nat (S h) (\lambda (n: nat).(drop n y0 y x)) (\lambda (_: -nat).((eq nat y0 O) \to ((eq C y (CHead c k u)) \to (drop (r k h) y0 c x)))) -(\lambda (y1: nat).(\lambda (H2: (drop y1 y0 y x)).(drop_ind (\lambda (n: -nat).(\lambda (n0: nat).(\lambda (c0: C).(\lambda (c1: C).((eq nat n (S h)) -\to ((eq nat n0 O) \to ((eq C c0 (CHead c k u)) \to (drop (r k h) n0 c -c1)))))))) (\lambda (c0: C).(\lambda (H3: (eq nat O (S h))).(\lambda (_: (eq -nat O O)).(\lambda (H5: (eq C c0 (CHead c k u))).(eq_ind_r C (CHead c k u) -(\lambda (c1: C).(drop (r k h) O c c1)) (let H6 \def (eq_ind nat O (\lambda -(ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow -True | (S _) \Rightarrow False])) I (S h) H3) in (False_ind (drop (r k h) O c -(CHead c k u)) H6)) c0 H5))))) (\lambda (k0: K).(\lambda (h0: nat).(\lambda -(c0: C).(\lambda (e: C).(\lambda (H3: (drop (r k0 h0) O c0 e)).(\lambda (H4: -(((eq nat (r k0 h0) (S h)) \to ((eq nat O O) \to ((eq C c0 (CHead c k u)) \to -(drop (r k h) O c e)))))).(\lambda (u0: T).(\lambda (H5: (eq nat (S h0) (S -h))).(\lambda (_: (eq nat O O)).(\lambda (H7: (eq C (CHead c0 k0 u0) (CHead c -k u))).(let H8 \def (f_equal C C (\lambda (e0: C).(match e0 in C return -(\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c1 _ _) -\Rightarrow c1])) (CHead c0 k0 u0) (CHead c k u) H7) in ((let H9 \def -(f_equal C K (\lambda (e0: C).(match e0 in C return (\lambda (_: C).K) with -[(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead c0 k0 u0) -(CHead c k u) H7) in ((let H10 \def (f_equal C T (\lambda (e0: C).(match e0 -in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) -\Rightarrow t])) (CHead c0 k0 u0) (CHead c k u) H7) in (\lambda (H11: (eq K -k0 k)).(\lambda (H12: (eq C c0 c)).(let H13 \def (eq_ind C c0 (\lambda (c1: -C).((eq nat (r k0 h0) (S h)) \to ((eq nat O O) \to ((eq C c1 (CHead c k u)) -\to (drop (r k h) O c e))))) H4 c H12) in (let H14 \def (eq_ind C c0 (\lambda -(c1: C).(drop (r k0 h0) O c1 e)) H3 c H12) in (let H15 \def (eq_ind K k0 -(\lambda (k1: K).((eq nat (r k1 h0) (S h)) \to ((eq nat O O) \to ((eq C c -(CHead c k u)) \to (drop (r k h) O c e))))) H13 k H11) in (let H16 \def -(eq_ind K k0 (\lambda (k1: K).(drop (r k1 h0) O c e)) H14 k H11) in (let H17 -\def (f_equal nat nat (\lambda (e0: nat).(match e0 in nat return (\lambda (_: -nat).nat) with [O \Rightarrow h0 | (S n) \Rightarrow n])) (S h0) (S h) H5) in -(let H18 \def (eq_ind nat h0 (\lambda (n: nat).((eq nat (r k n) (S h)) \to -((eq nat O O) \to ((eq C c (CHead c k u)) \to (drop (r k h) O c e))))) H15 h -H17) in (let H19 \def (eq_ind nat h0 (\lambda (n: nat).(drop (r k n) O c e)) -H16 h H17) in H19)))))))))) H9)) H8)))))))))))) (\lambda (k0: K).(\lambda -(h0: nat).(\lambda (d: nat).(\lambda (c0: C).(\lambda (e: C).(\lambda (H3: -(drop h0 (r k0 d) c0 e)).(\lambda (H4: (((eq nat h0 (S h)) \to ((eq nat (r k0 -d) O) \to ((eq C c0 (CHead c k u)) \to (drop (r k h) (r k0 d) c -e)))))).(\lambda (u0: T).(\lambda (H5: (eq nat h0 (S h))).(\lambda (H6: (eq -nat (S d) O)).(\lambda (H7: (eq C (CHead c0 k0 (lift h0 (r k0 d) u0)) (CHead -c k u))).(let H8 \def (eq_ind nat h0 (\lambda (n: nat).(eq C (CHead c0 k0 -(lift n (r k0 d) u0)) (CHead c k u))) H7 (S h) H5) in (let H9 \def (eq_ind -nat h0 (\lambda (n: nat).((eq nat n (S h)) \to ((eq nat (r k0 d) O) \to ((eq -C c0 (CHead c k u)) \to (drop (r k h) (r k0 d) c e))))) H4 (S h) H5) in (let -H10 \def (eq_ind nat h0 (\lambda (n: nat).(drop n (r k0 d) c0 e)) H3 (S h) -H5) in (let H11 \def (f_equal C C (\lambda (e0: C).(match e0 in C return -(\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c1 _ _) -\Rightarrow c1])) (CHead c0 k0 (lift (S h) (r k0 d) u0)) (CHead c k u) H8) in -((let H12 \def (f_equal C K (\lambda (e0: C).(match e0 in C return (\lambda -(_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) -(CHead c0 k0 (lift (S h) (r k0 d) u0)) (CHead c k u) H8) in ((let H13 \def -(f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d0: nat) (t: -T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) -\Rightarrow (TLRef (match (blt i d0) with [true \Rightarrow i | false -\Rightarrow (f i)])) | (THead k1 u1 t0) \Rightarrow (THead k1 (lref_map f d0 -u1) (lref_map f (s k1 d0) t0))]) in lref_map) (\lambda (x0: nat).(plus x0 (S -h))) (r k0 d) u0) | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 (lift (S h) -(r k0 d) u0)) (CHead c k u) H8) in (\lambda (H14: (eq K k0 k)).(\lambda (H15: -(eq C c0 c)).(let H16 \def (eq_ind C c0 (\lambda (c1: C).((eq nat (S h) (S -h)) \to ((eq nat (r k0 d) O) \to ((eq C c1 (CHead c k u)) \to (drop (r k h) -(r k0 d) c e))))) H9 c H15) in (let H17 \def (eq_ind C c0 (\lambda (c1: -C).(drop (S h) (r k0 d) c1 e)) H10 c H15) in (let H18 \def (eq_ind K k0 -(\lambda (k1: K).(eq T (lift (S h) (r k1 d) u0) u)) H13 k H14) in (let H19 -\def (eq_ind K k0 (\lambda (k1: K).((eq nat (S h) (S h)) \to ((eq nat (r k1 -d) O) \to ((eq C c (CHead c k u)) \to (drop (r k h) (r k1 d) c e))))) H16 k -H14) in (let H20 \def (eq_ind K k0 (\lambda (k1: K).(drop (S h) (r k1 d) c -e)) H17 k H14) in (eq_ind_r K k (\lambda (k1: K).(drop (r k h) (S d) c (CHead -e k1 u0))) (let H21 \def (eq_ind_r T u (\lambda (t: T).((eq nat (S h) (S h)) -\to ((eq nat (r k d) O) \to ((eq C c (CHead c k t)) \to (drop (r k h) (r k d) -c e))))) H19 (lift (S h) (r k d) u0) H18) in (let H22 \def (eq_ind nat (S d) -(\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O -\Rightarrow False | (S _) \Rightarrow True])) I O H6) in (False_ind (drop (r -k h) (S d) c (CHead e k u0)) H22))) k0 H14))))))))) H12)) H11)))))))))))))))) -y1 y0 y x H2))) H1))) H0))) H)))))). -(* COMMENTS -Initial nodes: 1856 -END *) - -theorem drop_gen_skip_r: - \forall (c: C).(\forall (x: C).(\forall (u: T).(\forall (h: nat).(\forall -(d: nat).(\forall (k: K).((drop h (S d) x (CHead c k u)) \to (ex2 C (\lambda -(e: C).(eq C x (CHead e k (lift h (r k d) u)))) (\lambda (e: C).(drop h (r k -d) e c))))))))) -\def - \lambda (c: C).(\lambda (x: C).(\lambda (u: T).(\lambda (h: nat).(\lambda -(d: nat).(\lambda (k: K).(\lambda (H: (drop h (S d) x (CHead c k -u))).(insert_eq C (CHead c k u) (\lambda (c0: C).(drop h (S d) x c0)) -(\lambda (_: C).(ex2 C (\lambda (e: C).(eq C x (CHead e k (lift h (r k d) -u)))) (\lambda (e: C).(drop h (r k d) e c)))) (\lambda (y: C).(\lambda (H0: -(drop h (S d) x y)).(insert_eq nat (S d) (\lambda (n: nat).(drop h n x y)) -(\lambda (_: nat).((eq C y (CHead c k u)) \to (ex2 C (\lambda (e: C).(eq C x -(CHead e k (lift h (r k d) u)))) (\lambda (e: C).(drop h (r k d) e c))))) -(\lambda (y0: nat).(\lambda (H1: (drop h y0 x y)).(drop_ind (\lambda (n: -nat).(\lambda (n0: nat).(\lambda (c0: C).(\lambda (c1: C).((eq nat n0 (S d)) -\to ((eq C c1 (CHead c k u)) \to (ex2 C (\lambda (e: C).(eq C c0 (CHead e k -(lift n (r k d) u)))) (\lambda (e: C).(drop n (r k d) e c))))))))) (\lambda -(c0: C).(\lambda (H2: (eq nat O (S d))).(\lambda (H3: (eq C c0 (CHead c k -u))).(eq_ind_r C (CHead c k u) (\lambda (c1: C).(ex2 C (\lambda (e: C).(eq C -c1 (CHead e k (lift O (r k d) u)))) (\lambda (e: C).(drop O (r k d) e c)))) -(let H4 \def (eq_ind nat O (\lambda (ee: nat).(match ee in nat return -(\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) -I (S d) H2) in (False_ind (ex2 C (\lambda (e: C).(eq C (CHead c k u) (CHead e -k (lift O (r k d) u)))) (\lambda (e: C).(drop O (r k d) e c))) H4)) c0 H3)))) -(\lambda (k0: K).(\lambda (h0: nat).(\lambda (c0: C).(\lambda (e: C).(\lambda -(H2: (drop (r k0 h0) O c0 e)).(\lambda (H3: (((eq nat O (S d)) \to ((eq C e -(CHead c k u)) \to (ex2 C (\lambda (e0: C).(eq C c0 (CHead e0 k (lift (r k0 -h0) (r k d) u)))) (\lambda (e0: C).(drop (r k0 h0) (r k d) e0 -c))))))).(\lambda (u0: T).(\lambda (H4: (eq nat O (S d))).(\lambda (H5: (eq C -e (CHead c k u))).(let H6 \def (eq_ind C e (\lambda (c1: C).((eq nat O (S d)) -\to ((eq C c1 (CHead c k u)) \to (ex2 C (\lambda (e0: C).(eq C c0 (CHead e0 k -(lift (r k0 h0) (r k d) u)))) (\lambda (e0: C).(drop (r k0 h0) (r k d) e0 -c)))))) H3 (CHead c k u) H5) in (let H7 \def (eq_ind C e (\lambda (c1: -C).(drop (r k0 h0) O c0 c1)) H2 (CHead c k u) H5) in (let H8 \def (eq_ind nat -O (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O -\Rightarrow True | (S _) \Rightarrow False])) I (S d) H4) in (False_ind (ex2 -C (\lambda (e0: C).(eq C (CHead c0 k0 u0) (CHead e0 k (lift (S h0) (r k d) -u)))) (\lambda (e0: C).(drop (S h0) (r k d) e0 c))) H8))))))))))))) (\lambda -(k0: K).(\lambda (h0: nat).(\lambda (d0: nat).(\lambda (c0: C).(\lambda (e: -C).(\lambda (H2: (drop h0 (r k0 d0) c0 e)).(\lambda (H3: (((eq nat (r k0 d0) -(S d)) \to ((eq C e (CHead c k u)) \to (ex2 C (\lambda (e0: C).(eq C c0 -(CHead e0 k (lift h0 (r k d) u)))) (\lambda (e0: C).(drop h0 (r k d) e0 -c))))))).(\lambda (u0: T).(\lambda (H4: (eq nat (S d0) (S d))).(\lambda (H5: -(eq C (CHead e k0 u0) (CHead c k u))).(let H6 \def (f_equal C C (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow e | -(CHead c1 _ _) \Rightarrow c1])) (CHead e k0 u0) (CHead c k u) H5) in ((let -H7 \def (f_equal C K (\lambda (e0: C).(match e0 in C return (\lambda (_: -C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) -(CHead e k0 u0) (CHead c k u) H5) in ((let H8 \def (f_equal C T (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | -(CHead _ _ t) \Rightarrow t])) (CHead e k0 u0) (CHead c k u) H5) in (\lambda -(H9: (eq K k0 k)).(\lambda (H10: (eq C e c)).(eq_ind_r T u (\lambda (t: -T).(ex2 C (\lambda (e0: C).(eq C (CHead c0 k0 (lift h0 (r k0 d0) t)) (CHead -e0 k (lift h0 (r k d) u)))) (\lambda (e0: C).(drop h0 (r k d) e0 c)))) (let -H11 \def (eq_ind C e (\lambda (c1: C).((eq nat (r k0 d0) (S d)) \to ((eq C c1 -(CHead c k u)) \to (ex2 C (\lambda (e0: C).(eq C c0 (CHead e0 k (lift h0 (r k -d) u)))) (\lambda (e0: C).(drop h0 (r k d) e0 c)))))) H3 c H10) in (let H12 -\def (eq_ind C e (\lambda (c1: C).(drop h0 (r k0 d0) c0 c1)) H2 c H10) in -(let H13 \def (eq_ind K k0 (\lambda (k1: K).((eq nat (r k1 d0) (S d)) \to -((eq C c (CHead c k u)) \to (ex2 C (\lambda (e0: C).(eq C c0 (CHead e0 k -(lift h0 (r k d) u)))) (\lambda (e0: C).(drop h0 (r k d) e0 c)))))) H11 k H9) -in (let H14 \def (eq_ind K k0 (\lambda (k1: K).(drop h0 (r k1 d0) c0 c)) H12 -k H9) in (eq_ind_r K k (\lambda (k1: K).(ex2 C (\lambda (e0: C).(eq C (CHead -c0 k1 (lift h0 (r k1 d0) u)) (CHead e0 k (lift h0 (r k d) u)))) (\lambda (e0: -C).(drop h0 (r k d) e0 c)))) (let H15 \def (f_equal nat nat (\lambda (e0: -nat).(match e0 in nat return (\lambda (_: nat).nat) with [O \Rightarrow d0 | -(S n) \Rightarrow n])) (S d0) (S d) H4) in (let H16 \def (eq_ind nat d0 -(\lambda (n: nat).((eq nat (r k n) (S d)) \to ((eq C c (CHead c k u)) \to -(ex2 C (\lambda (e0: C).(eq C c0 (CHead e0 k (lift h0 (r k d) u)))) (\lambda -(e0: C).(drop h0 (r k d) e0 c)))))) H13 d H15) in (let H17 \def (eq_ind nat -d0 (\lambda (n: nat).(drop h0 (r k n) c0 c)) H14 d H15) in (eq_ind_r nat d -(\lambda (n: nat).(ex2 C (\lambda (e0: C).(eq C (CHead c0 k (lift h0 (r k n) -u)) (CHead e0 k (lift h0 (r k d) u)))) (\lambda (e0: C).(drop h0 (r k d) e0 -c)))) (ex_intro2 C (\lambda (e0: C).(eq C (CHead c0 k (lift h0 (r k d) u)) -(CHead e0 k (lift h0 (r k d) u)))) (\lambda (e0: C).(drop h0 (r k d) e0 c)) -c0 (refl_equal C (CHead c0 k (lift h0 (r k d) u))) H17) d0 H15)))) k0 H9))))) -u0 H8)))) H7)) H6)))))))))))) h y0 x y H1))) H0))) H))))))). -(* COMMENTS -Initial nodes: 1758 -END *) - -theorem drop_gen_skip_l: - \forall (c: C).(\forall (x: C).(\forall (u: T).(\forall (h: nat).(\forall -(d: nat).(\forall (k: K).((drop h (S d) (CHead c k u) x) \to (ex3_2 C T -(\lambda (e: C).(\lambda (v: T).(eq C x (CHead e k v)))) (\lambda (_: -C).(\lambda (v: T).(eq T u (lift h (r k d) v)))) (\lambda (e: C).(\lambda (_: -T).(drop h (r k d) c e)))))))))) -\def - \lambda (c: C).(\lambda (x: C).(\lambda (u: T).(\lambda (h: nat).(\lambda -(d: nat).(\lambda (k: K).(\lambda (H: (drop h (S d) (CHead c k u) -x)).(insert_eq C (CHead c k u) (\lambda (c0: C).(drop h (S d) c0 x)) (\lambda -(_: C).(ex3_2 C T (\lambda (e: C).(\lambda (v: T).(eq C x (CHead e k v)))) -(\lambda (_: C).(\lambda (v: T).(eq T u (lift h (r k d) v)))) (\lambda (e: -C).(\lambda (_: T).(drop h (r k d) c e))))) (\lambda (y: C).(\lambda (H0: -(drop h (S d) y x)).(insert_eq nat (S d) (\lambda (n: nat).(drop h n y x)) -(\lambda (_: nat).((eq C y (CHead c k u)) \to (ex3_2 C T (\lambda (e: -C).(\lambda (v: T).(eq C x (CHead e k v)))) (\lambda (_: C).(\lambda (v: -T).(eq T u (lift h (r k d) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r k -d) c e)))))) (\lambda (y0: nat).(\lambda (H1: (drop h y0 y x)).(drop_ind -(\lambda (n: nat).(\lambda (n0: nat).(\lambda (c0: C).(\lambda (c1: C).((eq -nat n0 (S d)) \to ((eq C c0 (CHead c k u)) \to (ex3_2 C T (\lambda (e: -C).(\lambda (v: T).(eq C c1 (CHead e k v)))) (\lambda (_: C).(\lambda (v: -T).(eq T u (lift n (r k d) v)))) (\lambda (e: C).(\lambda (_: T).(drop n (r k -d) c e)))))))))) (\lambda (c0: C).(\lambda (H2: (eq nat O (S d))).(\lambda -(H3: (eq C c0 (CHead c k u))).(eq_ind_r C (CHead c k u) (\lambda (c1: -C).(ex3_2 C T (\lambda (e: C).(\lambda (v: T).(eq C c1 (CHead e k v)))) -(\lambda (_: C).(\lambda (v: T).(eq T u (lift O (r k d) v)))) (\lambda (e: -C).(\lambda (_: T).(drop O (r k d) c e))))) (let H4 \def (eq_ind nat O -(\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O -\Rightarrow True | (S _) \Rightarrow False])) I (S d) H2) in (False_ind -(ex3_2 C T (\lambda (e: C).(\lambda (v: T).(eq C (CHead c k u) (CHead e k -v)))) (\lambda (_: C).(\lambda (v: T).(eq T u (lift O (r k d) v)))) (\lambda -(e: C).(\lambda (_: T).(drop O (r k d) c e)))) H4)) c0 H3)))) (\lambda (k0: -K).(\lambda (h0: nat).(\lambda (c0: C).(\lambda (e: C).(\lambda (H2: (drop (r -k0 h0) O c0 e)).(\lambda (H3: (((eq nat O (S d)) \to ((eq C c0 (CHead c k u)) -\to (ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 k v)))) -(\lambda (_: C).(\lambda (v: T).(eq T u (lift (r k0 h0) (r k d) v)))) -(\lambda (e0: C).(\lambda (_: T).(drop (r k0 h0) (r k d) c -e0)))))))).(\lambda (u0: T).(\lambda (H4: (eq nat O (S d))).(\lambda (H5: (eq -C (CHead c0 k0 u0) (CHead c k u))).(let H6 \def (f_equal C C (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | -(CHead c1 _ _) \Rightarrow c1])) (CHead c0 k0 u0) (CHead c k u) H5) in ((let -H7 \def (f_equal C K (\lambda (e0: C).(match e0 in C return (\lambda (_: -C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) -(CHead c0 k0 u0) (CHead c k u) H5) in ((let H8 \def (f_equal C T (\lambda -(e0: C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow -u0 | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 u0) (CHead c k u) H5) in -(\lambda (H9: (eq K k0 k)).(\lambda (H10: (eq C c0 c)).(let H11 \def (eq_ind -C c0 (\lambda (c1: C).((eq nat O (S d)) \to ((eq C c1 (CHead c k u)) \to -(ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 k v)))) -(\lambda (_: C).(\lambda (v: T).(eq T u (lift (r k0 h0) (r k d) v)))) -(\lambda (e0: C).(\lambda (_: T).(drop (r k0 h0) (r k d) c e0))))))) H3 c -H10) in (let H12 \def (eq_ind C c0 (\lambda (c1: C).(drop (r k0 h0) O c1 e)) -H2 c H10) in (let H13 \def (eq_ind K k0 (\lambda (k1: K).((eq nat O (S d)) -\to ((eq C c (CHead c k u)) \to (ex3_2 C T (\lambda (e0: C).(\lambda (v: -T).(eq C e (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T u (lift (r -k1 h0) (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop (r k1 h0) (r k d) -c e0))))))) H11 k H9) in (let H14 \def (eq_ind K k0 (\lambda (k1: K).(drop (r -k1 h0) O c e)) H12 k H9) in (let H15 \def (eq_ind nat O (\lambda (ee: -nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True -| (S _) \Rightarrow False])) I (S d) H4) in (False_ind (ex3_2 C T (\lambda -(e0: C).(\lambda (v: T).(eq C e (CHead e0 k v)))) (\lambda (_: C).(\lambda -(v: T).(eq T u (lift (S h0) (r k d) v)))) (\lambda (e0: C).(\lambda (_: -T).(drop (S h0) (r k d) c e0)))) H15))))))))) H7)) H6))))))))))) (\lambda -(k0: K).(\lambda (h0: nat).(\lambda (d0: nat).(\lambda (c0: C).(\lambda (e: -C).(\lambda (H2: (drop h0 (r k0 d0) c0 e)).(\lambda (H3: (((eq nat (r k0 d0) -(S d)) \to ((eq C c0 (CHead c k u)) \to (ex3_2 C T (\lambda (e0: C).(\lambda -(v: T).(eq C e (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T u -(lift h0 (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h0 (r k d) c -e0)))))))).(\lambda (u0: T).(\lambda (H4: (eq nat (S d0) (S d))).(\lambda -(H5: (eq C (CHead c0 k0 (lift h0 (r k0 d0) u0)) (CHead c k u))).(let H6 \def -(f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with -[(CSort _) \Rightarrow c0 | (CHead c1 _ _) \Rightarrow c1])) (CHead c0 k0 -(lift h0 (r k0 d0) u0)) (CHead c k u) H5) in ((let H7 \def (f_equal C K -(\lambda (e0: C).(match e0 in C return (\lambda (_: C).K) with [(CSort _) -\Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead c0 k0 (lift h0 (r k0 -d0) u0)) (CHead c k u) H5) in ((let H8 \def (f_equal C T (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow ((let -rec lref_map (f: ((nat \to nat))) (d1: nat) (t: T) on t: T \def (match t with -[(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i -d1) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k1 u1 t0) -\Rightarrow (THead k1 (lref_map f d1 u1) (lref_map f (s k1 d1) t0))]) in -lref_map) (\lambda (x0: nat).(plus x0 h0)) (r k0 d0) u0) | (CHead _ _ t) -\Rightarrow t])) (CHead c0 k0 (lift h0 (r k0 d0) u0)) (CHead c k u) H5) in -(\lambda (H9: (eq K k0 k)).(\lambda (H10: (eq C c0 c)).(let H11 \def (eq_ind -C c0 (\lambda (c1: C).((eq nat (r k0 d0) (S d)) \to ((eq C c1 (CHead c k u)) -\to (ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 k v)))) -(\lambda (_: C).(\lambda (v: T).(eq T u (lift h0 (r k d) v)))) (\lambda (e0: -C).(\lambda (_: T).(drop h0 (r k d) c e0))))))) H3 c H10) in (let H12 \def -(eq_ind C c0 (\lambda (c1: C).(drop h0 (r k0 d0) c1 e)) H2 c H10) in (let H13 -\def (eq_ind K k0 (\lambda (k1: K).(eq T (lift h0 (r k1 d0) u0) u)) H8 k H9) -in (let H14 \def (eq_ind K k0 (\lambda (k1: K).((eq nat (r k1 d0) (S d)) \to -((eq C c (CHead c k u)) \to (ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C -e (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T u (lift h0 (r k d) -v)))) (\lambda (e0: C).(\lambda (_: T).(drop h0 (r k d) c e0))))))) H11 k H9) -in (let H15 \def (eq_ind K k0 (\lambda (k1: K).(drop h0 (r k1 d0) c e)) H12 k -H9) in (eq_ind_r K k (\lambda (k1: K).(ex3_2 C T (\lambda (e0: C).(\lambda -(v: T).(eq C (CHead e k1 u0) (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: -T).(eq T u (lift h0 (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h0 -(r k d) c e0))))) (let H16 \def (eq_ind_r T u (\lambda (t: T).((eq nat (r k -d0) (S d)) \to ((eq C c (CHead c k t)) \to (ex3_2 C T (\lambda (e0: -C).(\lambda (v: T).(eq C e (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: -T).(eq T t (lift h0 (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h0 -(r k d) c e0))))))) H14 (lift h0 (r k d0) u0) H13) in (eq_ind T (lift h0 (r k -d0) u0) (\lambda (t: T).(ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C -(CHead e k u0) (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T t -(lift h0 (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h0 (r k d) c -e0))))) (let H17 \def (f_equal nat nat (\lambda (e0: nat).(match e0 in nat -return (\lambda (_: nat).nat) with [O \Rightarrow d0 | (S n) \Rightarrow n])) -(S d0) (S d) H4) in (let H18 \def (eq_ind nat d0 (\lambda (n: nat).((eq nat -(r k n) (S d)) \to ((eq C c (CHead c k (lift h0 (r k n) u0))) \to (ex3_2 C T -(\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 k v)))) (\lambda (_: -C).(\lambda (v: T).(eq T (lift h0 (r k n) u0) (lift h0 (r k d) v)))) (\lambda -(e0: C).(\lambda (_: T).(drop h0 (r k d) c e0))))))) H16 d H17) in (let H19 -\def (eq_ind nat d0 (\lambda (n: nat).(drop h0 (r k n) c e)) H15 d H17) in -(eq_ind_r nat d (\lambda (n: nat).(ex3_2 C T (\lambda (e0: C).(\lambda (v: -T).(eq C (CHead e k u0) (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq -T (lift h0 (r k n) u0) (lift h0 (r k d) v)))) (\lambda (e0: C).(\lambda (_: -T).(drop h0 (r k d) c e0))))) (ex3_2_intro C T (\lambda (e0: C).(\lambda (v: -T).(eq C (CHead e k u0) (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq -T (lift h0 (r k d) u0) (lift h0 (r k d) v)))) (\lambda (e0: C).(\lambda (_: -T).(drop h0 (r k d) c e0))) e u0 (refl_equal C (CHead e k u0)) (refl_equal T -(lift h0 (r k d) u0)) H19) d0 H17)))) u H13)) k0 H9))))))))) H7)) -H6)))))))))))) h y0 y x H1))) H0))) H))))))). -(* COMMENTS -Initial nodes: 2574 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/drop/props.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/drop/props.ma deleted file mode 100644 index ac802b105..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/drop/props.ma +++ /dev/null @@ -1,737 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/drop/fwd.ma". - -include "Basic-1/lift/props.ma". - -include "Basic-1/r/props.ma". - -theorem drop_skip_bind: - \forall (h: nat).(\forall (d: nat).(\forall (c: C).(\forall (e: C).((drop h -d c e) \to (\forall (b: B).(\forall (u: T).(drop h (S d) (CHead c (Bind b) -(lift h d u)) (CHead e (Bind b) u)))))))) -\def - \lambda (h: nat).(\lambda (d: nat).(\lambda (c: C).(\lambda (e: C).(\lambda -(H: (drop h d c e)).(\lambda (b: B).(\lambda (u: T).(eq_ind nat (r (Bind b) -d) (\lambda (n: nat).(drop h (S d) (CHead c (Bind b) (lift h n u)) (CHead e -(Bind b) u))) (drop_skip (Bind b) h d c e H u) d (refl_equal nat d)))))))). -(* COMMENTS -Initial nodes: 95 -END *) - -theorem drop_skip_flat: - \forall (h: nat).(\forall (d: nat).(\forall (c: C).(\forall (e: C).((drop h -(S d) c e) \to (\forall (f: F).(\forall (u: T).(drop h (S d) (CHead c (Flat -f) (lift h (S d) u)) (CHead e (Flat f) u)))))))) -\def - \lambda (h: nat).(\lambda (d: nat).(\lambda (c: C).(\lambda (e: C).(\lambda -(H: (drop h (S d) c e)).(\lambda (f: F).(\lambda (u: T).(eq_ind nat (r (Flat -f) d) (\lambda (n: nat).(drop h (S d) (CHead c (Flat f) (lift h n u)) (CHead -e (Flat f) u))) (drop_skip (Flat f) h d c e H u) (S d) (refl_equal nat (S -d))))))))). -(* COMMENTS -Initial nodes: 101 -END *) - -theorem drop_S: - \forall (b: B).(\forall (c: C).(\forall (e: C).(\forall (u: T).(\forall (h: -nat).((drop h O c (CHead e (Bind b) u)) \to (drop (S h) O c e)))))) -\def - \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e: -C).(\forall (u: T).(\forall (h: nat).((drop h O c0 (CHead e (Bind b) u)) \to -(drop (S h) O c0 e)))))) (\lambda (n: nat).(\lambda (e: C).(\lambda (u: -T).(\lambda (h: nat).(\lambda (H: (drop h O (CSort n) (CHead e (Bind b) -u))).(and3_ind (eq C (CHead e (Bind b) u) (CSort n)) (eq nat h O) (eq nat O -O) (drop (S h) O (CSort n) e) (\lambda (H0: (eq C (CHead e (Bind b) u) (CSort -n))).(\lambda (H1: (eq nat h O)).(\lambda (_: (eq nat O O)).(eq_ind_r nat O -(\lambda (n0: nat).(drop (S n0) O (CSort n) e)) (let H3 \def (eq_ind C (CHead -e (Bind b) u) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) -with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I -(CSort n) H0) in (False_ind (drop (S O) O (CSort n) e) H3)) h H1)))) -(drop_gen_sort n h O (CHead e (Bind b) u) H))))))) (\lambda (c0: C).(\lambda -(H: ((\forall (e: C).(\forall (u: T).(\forall (h: nat).((drop h O c0 (CHead e -(Bind b) u)) \to (drop (S h) O c0 e))))))).(\lambda (k: K).(\lambda (t: -T).(\lambda (e: C).(\lambda (u: T).(\lambda (h: nat).(nat_ind (\lambda (n: -nat).((drop n O (CHead c0 k t) (CHead e (Bind b) u)) \to (drop (S n) O (CHead -c0 k t) e))) (\lambda (H0: (drop O O (CHead c0 k t) (CHead e (Bind b) -u))).(let H1 \def (f_equal C C (\lambda (e0: C).(match e0 in C return -(\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c1 _ _) -\Rightarrow c1])) (CHead c0 k t) (CHead e (Bind b) u) (drop_gen_refl (CHead -c0 k t) (CHead e (Bind b) u) H0)) in ((let H2 \def (f_equal C K (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | -(CHead _ k0 _) \Rightarrow k0])) (CHead c0 k t) (CHead e (Bind b) u) -(drop_gen_refl (CHead c0 k t) (CHead e (Bind b) u) H0)) in ((let H3 \def -(f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow t | (CHead _ _ t0) \Rightarrow t0])) (CHead c0 k t) -(CHead e (Bind b) u) (drop_gen_refl (CHead c0 k t) (CHead e (Bind b) u) H0)) -in (\lambda (H4: (eq K k (Bind b))).(\lambda (H5: (eq C c0 e)).(eq_ind C c0 -(\lambda (c1: C).(drop (S O) O (CHead c0 k t) c1)) (eq_ind_r K (Bind b) -(\lambda (k0: K).(drop (S O) O (CHead c0 k0 t) c0)) (drop_drop (Bind b) O c0 -c0 (drop_refl c0) t) k H4) e H5)))) H2)) H1))) (\lambda (n: nat).(\lambda (_: -(((drop n O (CHead c0 k t) (CHead e (Bind b) u)) \to (drop (S n) O (CHead c0 -k t) e)))).(\lambda (H1: (drop (S n) O (CHead c0 k t) (CHead e (Bind b) -u))).(drop_drop k (S n) c0 e (eq_ind_r nat (S (r k n)) (\lambda (n0: -nat).(drop n0 O c0 e)) (H e u (r k n) (drop_gen_drop k c0 (CHead e (Bind b) -u) t n H1)) (r k (S n)) (r_S k n)) t)))) h)))))))) c)). -(* COMMENTS -Initial nodes: 807 -END *) - -theorem drop_ctail: - \forall (c1: C).(\forall (c2: C).(\forall (d: nat).(\forall (h: nat).((drop -h d c1 c2) \to (\forall (k: K).(\forall (u: T).(drop h d (CTail k u c1) -(CTail k u c2)))))))) -\def - \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (d: -nat).(\forall (h: nat).((drop h d c c2) \to (\forall (k: K).(\forall (u: -T).(drop h d (CTail k u c) (CTail k u c2))))))))) (\lambda (n: nat).(\lambda -(c2: C).(\lambda (d: nat).(\lambda (h: nat).(\lambda (H: (drop h d (CSort n) -c2)).(\lambda (k: K).(\lambda (u: T).(and3_ind (eq C c2 (CSort n)) (eq nat h -O) (eq nat d O) (drop h d (CTail k u (CSort n)) (CTail k u c2)) (\lambda (H0: -(eq C c2 (CSort n))).(\lambda (H1: (eq nat h O)).(\lambda (H2: (eq nat d -O)).(eq_ind_r nat O (\lambda (n0: nat).(drop n0 d (CTail k u (CSort n)) -(CTail k u c2))) (eq_ind_r nat O (\lambda (n0: nat).(drop O n0 (CTail k u -(CSort n)) (CTail k u c2))) (eq_ind_r C (CSort n) (\lambda (c: C).(drop O O -(CTail k u (CSort n)) (CTail k u c))) (drop_refl (CTail k u (CSort n))) c2 -H0) d H2) h H1)))) (drop_gen_sort n h d c2 H))))))))) (\lambda (c2: -C).(\lambda (IHc: ((\forall (c3: C).(\forall (d: nat).(\forall (h: -nat).((drop h d c2 c3) \to (\forall (k: K).(\forall (u: T).(drop h d (CTail k -u c2) (CTail k u c3)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c3: -C).(\lambda (d: nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n -(CHead c2 k t) c3) \to (\forall (k0: K).(\forall (u: T).(drop h n (CTail k0 u -(CHead c2 k t)) (CTail k0 u c3))))))) (\lambda (h: nat).(nat_ind (\lambda (n: -nat).((drop n O (CHead c2 k t) c3) \to (\forall (k0: K).(\forall (u: T).(drop -n O (CTail k0 u (CHead c2 k t)) (CTail k0 u c3)))))) (\lambda (H: (drop O O -(CHead c2 k t) c3)).(\lambda (k0: K).(\lambda (u: T).(eq_ind C (CHead c2 k t) -(\lambda (c: C).(drop O O (CTail k0 u (CHead c2 k t)) (CTail k0 u c))) -(drop_refl (CTail k0 u (CHead c2 k t))) c3 (drop_gen_refl (CHead c2 k t) c3 -H))))) (\lambda (n: nat).(\lambda (_: (((drop n O (CHead c2 k t) c3) \to -(\forall (k0: K).(\forall (u: T).(drop n O (CTail k0 u (CHead c2 k t)) (CTail -k0 u c3))))))).(\lambda (H0: (drop (S n) O (CHead c2 k t) c3)).(\lambda (k0: -K).(\lambda (u: T).(drop_drop k n (CTail k0 u c2) (CTail k0 u c3) (IHc c3 O -(r k n) (drop_gen_drop k c2 c3 t n H0) k0 u) t)))))) h)) (\lambda (n: -nat).(\lambda (H: ((\forall (h: nat).((drop h n (CHead c2 k t) c3) \to -(\forall (k0: K).(\forall (u: T).(drop h n (CTail k0 u (CHead c2 k t)) (CTail -k0 u c3)))))))).(\lambda (h: nat).(\lambda (H0: (drop h (S n) (CHead c2 k t) -c3)).(\lambda (k0: K).(\lambda (u: T).(ex3_2_ind C T (\lambda (e: C).(\lambda -(v: T).(eq C c3 (CHead e k v)))) (\lambda (_: C).(\lambda (v: T).(eq T t -(lift h (r k n) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r k n) c2 e))) -(drop h (S n) (CTail k0 u (CHead c2 k t)) (CTail k0 u c3)) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (H1: (eq C c3 (CHead x0 k x1))).(\lambda (H2: -(eq T t (lift h (r k n) x1))).(\lambda (H3: (drop h (r k n) c2 x0)).(let H4 -\def (eq_ind C c3 (\lambda (c: C).(\forall (h0: nat).((drop h0 n (CHead c2 k -t) c) \to (\forall (k1: K).(\forall (u0: T).(drop h0 n (CTail k1 u0 (CHead c2 -k t)) (CTail k1 u0 c))))))) H (CHead x0 k x1) H1) in (eq_ind_r C (CHead x0 k -x1) (\lambda (c: C).(drop h (S n) (CTail k0 u (CHead c2 k t)) (CTail k0 u -c))) (let H5 \def (eq_ind T t (\lambda (t0: T).(\forall (h0: nat).((drop h0 n -(CHead c2 k t0) (CHead x0 k x1)) \to (\forall (k1: K).(\forall (u0: T).(drop -h0 n (CTail k1 u0 (CHead c2 k t0)) (CTail k1 u0 (CHead x0 k x1)))))))) H4 -(lift h (r k n) x1) H2) in (eq_ind_r T (lift h (r k n) x1) (\lambda (t0: -T).(drop h (S n) (CTail k0 u (CHead c2 k t0)) (CTail k0 u (CHead x0 k x1)))) -(drop_skip k h n (CTail k0 u c2) (CTail k0 u x0) (IHc x0 (r k n) h H3 k0 u) -x1) t H2)) c3 H1))))))) (drop_gen_skip_l c2 c3 t h n k H0)))))))) d))))))) -c1). -(* COMMENTS -Initial nodes: 1211 -END *) - -theorem drop_mono: - \forall (c: C).(\forall (x1: C).(\forall (d: nat).(\forall (h: nat).((drop h -d c x1) \to (\forall (x2: C).((drop h d c x2) \to (eq C x1 x2))))))) -\def - \lambda (c: C).(C_ind (\lambda (c0: C).(\forall (x1: C).(\forall (d: -nat).(\forall (h: nat).((drop h d c0 x1) \to (\forall (x2: C).((drop h d c0 -x2) \to (eq C x1 x2)))))))) (\lambda (n: nat).(\lambda (x1: C).(\lambda (d: -nat).(\lambda (h: nat).(\lambda (H: (drop h d (CSort n) x1)).(\lambda (x2: -C).(\lambda (H0: (drop h d (CSort n) x2)).(and3_ind (eq C x2 (CSort n)) (eq -nat h O) (eq nat d O) (eq C x1 x2) (\lambda (H1: (eq C x2 (CSort -n))).(\lambda (H2: (eq nat h O)).(\lambda (H3: (eq nat d O)).(and3_ind (eq C -x1 (CSort n)) (eq nat h O) (eq nat d O) (eq C x1 x2) (\lambda (H4: (eq C x1 -(CSort n))).(\lambda (H5: (eq nat h O)).(\lambda (H6: (eq nat d O)).(eq_ind_r -C (CSort n) (\lambda (c0: C).(eq C x1 c0)) (let H7 \def (eq_ind nat h -(\lambda (n0: nat).(eq nat n0 O)) H2 O H5) in (let H8 \def (eq_ind nat d -(\lambda (n0: nat).(eq nat n0 O)) H3 O H6) in (eq_ind_r C (CSort n) (\lambda -(c0: C).(eq C c0 (CSort n))) (refl_equal C (CSort n)) x1 H4))) x2 H1)))) -(drop_gen_sort n h d x1 H))))) (drop_gen_sort n h d x2 H0))))))))) (\lambda -(c0: C).(\lambda (H: ((\forall (x1: C).(\forall (d: nat).(\forall (h: -nat).((drop h d c0 x1) \to (\forall (x2: C).((drop h d c0 x2) \to (eq C x1 -x2))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (x1: C).(\lambda (d: -nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n (CHead c0 k t) -x1) \to (\forall (x2: C).((drop h n (CHead c0 k t) x2) \to (eq C x1 x2)))))) -(\lambda (h: nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c0 k t) x1) -\to (\forall (x2: C).((drop n O (CHead c0 k t) x2) \to (eq C x1 x2))))) -(\lambda (H0: (drop O O (CHead c0 k t) x1)).(\lambda (x2: C).(\lambda (H1: -(drop O O (CHead c0 k t) x2)).(eq_ind C (CHead c0 k t) (\lambda (c1: C).(eq C -x1 c1)) (eq_ind C (CHead c0 k t) (\lambda (c1: C).(eq C c1 (CHead c0 k t))) -(refl_equal C (CHead c0 k t)) x1 (drop_gen_refl (CHead c0 k t) x1 H0)) x2 -(drop_gen_refl (CHead c0 k t) x2 H1))))) (\lambda (n: nat).(\lambda (_: -(((drop n O (CHead c0 k t) x1) \to (\forall (x2: C).((drop n O (CHead c0 k t) -x2) \to (eq C x1 x2)))))).(\lambda (H1: (drop (S n) O (CHead c0 k t) -x1)).(\lambda (x2: C).(\lambda (H2: (drop (S n) O (CHead c0 k t) x2)).(H x1 O -(r k n) (drop_gen_drop k c0 x1 t n H1) x2 (drop_gen_drop k c0 x2 t n -H2))))))) h)) (\lambda (n: nat).(\lambda (H0: ((\forall (h: nat).((drop h n -(CHead c0 k t) x1) \to (\forall (x2: C).((drop h n (CHead c0 k t) x2) \to (eq -C x1 x2))))))).(\lambda (h: nat).(\lambda (H1: (drop h (S n) (CHead c0 k t) -x1)).(\lambda (x2: C).(\lambda (H2: (drop h (S n) (CHead c0 k t) -x2)).(ex3_2_ind C T (\lambda (e: C).(\lambda (v: T).(eq C x2 (CHead e k v)))) -(\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r k n) v)))) (\lambda (e: -C).(\lambda (_: T).(drop h (r k n) c0 e))) (eq C x1 x2) (\lambda (x0: -C).(\lambda (x3: T).(\lambda (H3: (eq C x2 (CHead x0 k x3))).(\lambda (H4: -(eq T t (lift h (r k n) x3))).(\lambda (H5: (drop h (r k n) c0 -x0)).(ex3_2_ind C T (\lambda (e: C).(\lambda (v: T).(eq C x1 (CHead e k v)))) -(\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r k n) v)))) (\lambda (e: -C).(\lambda (_: T).(drop h (r k n) c0 e))) (eq C x1 x2) (\lambda (x4: -C).(\lambda (x5: T).(\lambda (H6: (eq C x1 (CHead x4 k x5))).(\lambda (H7: -(eq T t (lift h (r k n) x5))).(\lambda (H8: (drop h (r k n) c0 x4)).(eq_ind_r -C (CHead x0 k x3) (\lambda (c1: C).(eq C x1 c1)) (let H9 \def (eq_ind C x1 -(\lambda (c1: C).(\forall (h0: nat).((drop h0 n (CHead c0 k t) c1) \to -(\forall (x6: C).((drop h0 n (CHead c0 k t) x6) \to (eq C c1 x6)))))) H0 -(CHead x4 k x5) H6) in (eq_ind_r C (CHead x4 k x5) (\lambda (c1: C).(eq C c1 -(CHead x0 k x3))) (let H10 \def (eq_ind T t (\lambda (t0: T).(\forall (h0: -nat).((drop h0 n (CHead c0 k t0) (CHead x4 k x5)) \to (\forall (x6: C).((drop -h0 n (CHead c0 k t0) x6) \to (eq C (CHead x4 k x5) x6)))))) H9 (lift h (r k -n) x5) H7) in (let H11 \def (eq_ind T t (\lambda (t0: T).(eq T t0 (lift h (r -k n) x3))) H4 (lift h (r k n) x5) H7) in (let H12 \def (eq_ind T x5 (\lambda -(t0: T).(\forall (h0: nat).((drop h0 n (CHead c0 k (lift h (r k n) t0)) -(CHead x4 k t0)) \to (\forall (x6: C).((drop h0 n (CHead c0 k (lift h (r k n) -t0)) x6) \to (eq C (CHead x4 k t0) x6)))))) H10 x3 (lift_inj x5 x3 h (r k n) -H11)) in (eq_ind_r T x3 (\lambda (t0: T).(eq C (CHead x4 k t0) (CHead x0 k -x3))) (f_equal3 C K T C CHead x4 x0 k k x3 x3 (sym_eq C x0 x4 (H x0 (r k n) h -H5 x4 H8)) (refl_equal K k) (refl_equal T x3)) x5 (lift_inj x5 x3 h (r k n) -H11))))) x1 H6)) x2 H3)))))) (drop_gen_skip_l c0 x1 t h n k H1))))))) -(drop_gen_skip_l c0 x2 t h n k H2)))))))) d))))))) c). -(* COMMENTS -Initial nodes: 1539 -END *) - -theorem drop_conf_lt: - \forall (k: K).(\forall (i: nat).(\forall (u: T).(\forall (c0: C).(\forall -(c: C).((drop i O c (CHead c0 k u)) \to (\forall (e: C).(\forall (h: -nat).(\forall (d: nat).((drop h (S (plus i d)) c e) \to (ex3_2 T C (\lambda -(v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda -(e0: C).(drop i O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop -h (r k d) c0 e0))))))))))))) -\def - \lambda (k: K).(\lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (u: -T).(\forall (c0: C).(\forall (c: C).((drop n O c (CHead c0 k u)) \to (\forall -(e: C).(\forall (h: nat).(\forall (d: nat).((drop h (S (plus n d)) c e) \to -(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) -(\lambda (v: T).(\lambda (e0: C).(drop n O e (CHead e0 k v)))) (\lambda (_: -T).(\lambda (e0: C).(drop h (r k d) c0 e0))))))))))))) (\lambda (u: -T).(\lambda (c0: C).(\lambda (c: C).(\lambda (H: (drop O O c (CHead c0 k -u))).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H0: (drop -h (S (plus O d)) c e)).(let H1 \def (eq_ind C c (\lambda (c1: C).(drop h (S -(plus O d)) c1 e)) H0 (CHead c0 k u) (drop_gen_refl c (CHead c0 k u) H)) in -(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 k v)))) -(\lambda (_: C).(\lambda (v: T).(eq T u (lift h (r k (plus O d)) v)))) -(\lambda (e0: C).(\lambda (_: T).(drop h (r k (plus O d)) c0 e0))) (ex3_2 T C -(\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: -T).(\lambda (e0: C).(drop O O e (CHead e0 k v)))) (\lambda (_: T).(\lambda -(e0: C).(drop h (r k d) c0 e0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda -(H2: (eq C e (CHead x0 k x1))).(\lambda (H3: (eq T u (lift h (r k (plus O d)) -x1))).(\lambda (H4: (drop h (r k (plus O d)) c0 x0)).(eq_ind_r C (CHead x0 k -x1) (\lambda (c1: C).(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift -h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop O O c1 (CHead e0 k -v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0))))) (eq_ind_r T -(lift h (r k (plus O d)) x1) (\lambda (t: T).(ex3_2 T C (\lambda (v: -T).(\lambda (_: C).(eq T t (lift h (r k d) v)))) (\lambda (v: T).(\lambda -(e0: C).(drop O O (CHead x0 k x1) (CHead e0 k v)))) (\lambda (_: T).(\lambda -(e0: C).(drop h (r k d) c0 e0))))) (ex3_2_intro T C (\lambda (v: T).(\lambda -(_: C).(eq T (lift h (r k (plus O d)) x1) (lift h (r k d) v)))) (\lambda (v: -T).(\lambda (e0: C).(drop O O (CHead x0 k x1) (CHead e0 k v)))) (\lambda (_: -T).(\lambda (e0: C).(drop h (r k d) c0 e0))) x1 x0 (refl_equal T (lift h (r k -d) x1)) (drop_refl (CHead x0 k x1)) H4) u H3) e H2)))))) (drop_gen_skip_l c0 -e u h (plus O d) k H1))))))))))) (\lambda (i0: nat).(\lambda (H: ((\forall -(u: T).(\forall (c0: C).(\forall (c: C).((drop i0 O c (CHead c0 k u)) \to -(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h (S (plus i0 d)) -c e) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) -v)))) (\lambda (v: T).(\lambda (e0: C).(drop i0 O e (CHead e0 k v)))) -(\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0)))))))))))))).(\lambda -(u: T).(\lambda (c0: C).(\lambda (c: C).(C_ind (\lambda (c1: C).((drop (S i0) -O c1 (CHead c0 k u)) \to (\forall (e: C).(\forall (h: nat).(\forall (d: -nat).((drop h (S (plus (S i0) d)) c1 e) \to (ex3_2 T C (\lambda (v: -T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda -(e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: -C).(drop h (r k d) c0 e0)))))))))) (\lambda (n: nat).(\lambda (_: (drop (S -i0) O (CSort n) (CHead c0 k u))).(\lambda (e: C).(\lambda (h: nat).(\lambda -(d: nat).(\lambda (H1: (drop h (S (plus (S i0) d)) (CSort n) e)).(and3_ind -(eq C e (CSort n)) (eq nat h O) (eq nat (S (plus (S i0) d)) O) (ex3_2 T C -(\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: -T).(\lambda (e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda (_: -T).(\lambda (e0: C).(drop h (r k d) c0 e0)))) (\lambda (_: (eq C e (CSort -n))).(\lambda (_: (eq nat h O)).(\lambda (H4: (eq nat (S (plus (S i0) d)) -O)).(let H5 \def (eq_ind nat (S (plus (S i0) d)) (\lambda (ee: nat).(match ee -in nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) -\Rightarrow True])) I O H4) in (False_ind (ex3_2 T C (\lambda (v: T).(\lambda -(_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop -(S i0) O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) -c0 e0)))) H5))))) (drop_gen_sort n h (S (plus (S i0) d)) e H1)))))))) -(\lambda (c1: C).(\lambda (H0: (((drop (S i0) O c1 (CHead c0 k u)) \to -(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h (S (plus (S i0) -d)) c1 e) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k -d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S i0) O e (CHead e0 k v)))) -(\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0))))))))))).(\lambda -(k0: K).(K_ind (\lambda (k1: K).(\forall (t: T).((drop (S i0) O (CHead c1 k1 -t) (CHead c0 k u)) \to (\forall (e: C).(\forall (h: nat).(\forall (d: -nat).((drop h (S (plus (S i0) d)) (CHead c1 k1 t) e) \to (ex3_2 T C (\lambda -(v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda -(e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: -C).(drop h (r k d) c0 e0))))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda -(H1: (drop (S i0) O (CHead c1 (Bind b) t) (CHead c0 k u))).(\lambda (e: -C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2: (drop h (S (plus (S i0) -d)) (CHead c1 (Bind b) t) e)).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: -T).(eq C e (CHead e0 (Bind b) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t -(lift h (r (Bind b) (plus (S i0) d)) v)))) (\lambda (e0: C).(\lambda (_: -T).(drop h (r (Bind b) (plus (S i0) d)) c1 e0))) (ex3_2 T C (\lambda (v: -T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda -(e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: -C).(drop h (r k d) c0 e0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H3: -(eq C e (CHead x0 (Bind b) x1))).(\lambda (_: (eq T t (lift h (r (Bind b) -(plus (S i0) d)) x1))).(\lambda (H5: (drop h (r (Bind b) (plus (S i0) d)) c1 -x0)).(eq_ind_r C (CHead x0 (Bind b) x1) (\lambda (c2: C).(ex3_2 T C (\lambda -(v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda -(e0: C).(drop (S i0) O c2 (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: -C).(drop h (r k d) c0 e0))))) (let H6 \def (H u c0 c1 (drop_gen_drop (Bind b) -c1 (CHead c0 k u) t i0 H1) x0 h d H5) in (ex3_2_ind T C (\lambda (v: -T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda -(e0: C).(drop i0 O x0 (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: -C).(drop h (r k d) c0 e0))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T -u (lift h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S i0) O -(CHead x0 (Bind b) x1) (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: -C).(drop h (r k d) c0 e0)))) (\lambda (x2: T).(\lambda (x3: C).(\lambda (H7: -(eq T u (lift h (r k d) x2))).(\lambda (H8: (drop i0 O x0 (CHead x3 k -x2))).(\lambda (H9: (drop h (r k d) c0 x3)).(ex3_2_intro T C (\lambda (v: -T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda -(e0: C).(drop (S i0) O (CHead x0 (Bind b) x1) (CHead e0 k v)))) (\lambda (_: -T).(\lambda (e0: C).(drop h (r k d) c0 e0))) x2 x3 H7 (drop_drop (Bind b) i0 -x0 (CHead x3 k x2) H8 x1) H9)))))) H6)) e H3)))))) (drop_gen_skip_l c1 e t h -(plus (S i0) d) (Bind b) H2))))))))) (\lambda (f: F).(\lambda (t: T).(\lambda -(H1: (drop (S i0) O (CHead c1 (Flat f) t) (CHead c0 k u))).(\lambda (e: -C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2: (drop h (S (plus (S i0) -d)) (CHead c1 (Flat f) t) e)).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: -T).(eq C e (CHead e0 (Flat f) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t -(lift h (r (Flat f) (plus (S i0) d)) v)))) (\lambda (e0: C).(\lambda (_: -T).(drop h (r (Flat f) (plus (S i0) d)) c1 e0))) (ex3_2 T C (\lambda (v: -T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda -(e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: -C).(drop h (r k d) c0 e0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H3: -(eq C e (CHead x0 (Flat f) x1))).(\lambda (_: (eq T t (lift h (r (Flat f) -(plus (S i0) d)) x1))).(\lambda (H5: (drop h (r (Flat f) (plus (S i0) d)) c1 -x0)).(eq_ind_r C (CHead x0 (Flat f) x1) (\lambda (c2: C).(ex3_2 T C (\lambda -(v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda -(e0: C).(drop (S i0) O c2 (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: -C).(drop h (r k d) c0 e0))))) (ex3_2_ind T C (\lambda (v: T).(\lambda (_: -C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S -i0) O x0 (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) -c0 e0))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) -v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S i0) O (CHead x0 (Flat f) x1) -(CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0)))) -(\lambda (x2: T).(\lambda (x3: C).(\lambda (H6: (eq T u (lift h (r k d) -x2))).(\lambda (H7: (drop (S i0) O x0 (CHead x3 k x2))).(\lambda (H8: (drop h -(r k d) c0 x3)).(ex3_2_intro T C (\lambda (v: T).(\lambda (_: C).(eq T u -(lift h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S i0) O (CHead -x0 (Flat f) x1) (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r -k d) c0 e0))) x2 x3 H6 (drop_drop (Flat f) i0 x0 (CHead x3 k x2) H7 x1) -H8)))))) (H0 (drop_gen_drop (Flat f) c1 (CHead c0 k u) t i0 H1) x0 h d H5)) e -H3)))))) (drop_gen_skip_l c1 e t h (plus (S i0) d) (Flat f) H2))))))))) -k0)))) c)))))) i)). -(* COMMENTS -Initial nodes: 2972 -END *) - -theorem drop_conf_ge: - \forall (i: nat).(\forall (a: C).(\forall (c: C).((drop i O c a) \to -(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to ((le -(plus d h) i) \to (drop (minus i h) O e a))))))))) -\def - \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (a: C).(\forall (c: -C).((drop n O c a) \to (\forall (e: C).(\forall (h: nat).(\forall (d: -nat).((drop h d c e) \to ((le (plus d h) n) \to (drop (minus n h) O e -a)))))))))) (\lambda (a: C).(\lambda (c: C).(\lambda (H: (drop O O c -a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H0: (drop h -d c e)).(\lambda (H1: (le (plus d h) O)).(let H2 \def (eq_ind C c (\lambda -(c0: C).(drop h d c0 e)) H0 a (drop_gen_refl c a H)) in (let H_y \def -(le_n_O_eq (plus d h) H1) in (land_ind (eq nat d O) (eq nat h O) (drop (minus -O h) O e a) (\lambda (H3: (eq nat d O)).(\lambda (H4: (eq nat h O)).(let H5 -\def (eq_ind nat d (\lambda (n: nat).(drop h n a e)) H2 O H3) in (let H6 \def -(eq_ind nat h (\lambda (n: nat).(drop n O a e)) H5 O H4) in (eq_ind_r nat O -(\lambda (n: nat).(drop (minus O n) O e a)) (eq_ind C a (\lambda (c0: -C).(drop (minus O O) O c0 a)) (drop_refl a) e (drop_gen_refl a e H6)) h -H4))))) (plus_O d h (sym_eq nat O (plus d h) H_y))))))))))))) (\lambda (i0: -nat).(\lambda (H: ((\forall (a: C).(\forall (c: C).((drop i0 O c a) \to -(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to ((le -(plus d h) i0) \to (drop (minus i0 h) O e a))))))))))).(\lambda (a: -C).(\lambda (c: C).(C_ind (\lambda (c0: C).((drop (S i0) O c0 a) \to (\forall -(e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c0 e) \to ((le (plus d -h) (S i0)) \to (drop (minus (S i0) h) O e a)))))))) (\lambda (n: -nat).(\lambda (H0: (drop (S i0) O (CSort n) a)).(\lambda (e: C).(\lambda (h: -nat).(\lambda (d: nat).(\lambda (H1: (drop h d (CSort n) e)).(\lambda (H2: -(le (plus d h) (S i0))).(and3_ind (eq C e (CSort n)) (eq nat h O) (eq nat d -O) (drop (minus (S i0) h) O e a) (\lambda (H3: (eq C e (CSort n))).(\lambda -(H4: (eq nat h O)).(\lambda (H5: (eq nat d O)).(and3_ind (eq C a (CSort n)) -(eq nat (S i0) O) (eq nat O O) (drop (minus (S i0) h) O e a) (\lambda (H6: -(eq C a (CSort n))).(\lambda (H7: (eq nat (S i0) O)).(\lambda (_: (eq nat O -O)).(let H9 \def (eq_ind nat d (\lambda (n0: nat).(le (plus n0 h) (S i0))) H2 -O H5) in (let H10 \def (eq_ind nat h (\lambda (n0: nat).(le (plus O n0) (S -i0))) H9 O H4) in (eq_ind_r nat O (\lambda (n0: nat).(drop (minus (S i0) n0) -O e a)) (eq_ind_r C (CSort n) (\lambda (c0: C).(drop (minus (S i0) O) O c0 -a)) (eq_ind_r C (CSort n) (\lambda (c0: C).(drop (minus (S i0) O) O (CSort n) -c0)) (let H11 \def (eq_ind nat (S i0) (\lambda (ee: nat).(match ee in nat -return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow -True])) I O H7) in (False_ind (drop (minus (S i0) O) O (CSort n) (CSort n)) -H11)) a H6) e H3) h H4)))))) (drop_gen_sort n (S i0) O a H0))))) -(drop_gen_sort n h d e H1))))))))) (\lambda (c0: C).(\lambda (H0: (((drop (S -i0) O c0 a) \to (\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h -d c0 e) \to ((le (plus d h) (S i0)) \to (drop (minus (S i0) h) O e -a))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t: T).((drop (S -i0) O (CHead c0 k0 t) a) \to (\forall (e: C).(\forall (h: nat).(\forall (d: -nat).((drop h d (CHead c0 k0 t) e) \to ((le (plus d h) (S i0)) \to (drop -(minus (S i0) h) O e a))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (H1: -(drop (S i0) O (CHead c0 (Bind b) t) a)).(\lambda (e: C).(\lambda (h: -nat).(\lambda (d: nat).(\lambda (H2: (drop h d (CHead c0 (Bind b) t) -e)).(\lambda (H3: (le (plus d h) (S i0))).(nat_ind (\lambda (n: nat).((drop h -n (CHead c0 (Bind b) t) e) \to ((le (plus n h) (S i0)) \to (drop (minus (S -i0) h) O e a)))) (\lambda (H4: (drop h O (CHead c0 (Bind b) t) e)).(\lambda -(H5: (le (plus O h) (S i0))).(nat_ind (\lambda (n: nat).((drop n O (CHead c0 -(Bind b) t) e) \to ((le (plus O n) (S i0)) \to (drop (minus (S i0) n) O e -a)))) (\lambda (H6: (drop O O (CHead c0 (Bind b) t) e)).(\lambda (_: (le -(plus O O) (S i0))).(eq_ind C (CHead c0 (Bind b) t) (\lambda (c1: C).(drop -(minus (S i0) O) O c1 a)) (drop_drop (Bind b) i0 c0 a (drop_gen_drop (Bind b) -c0 a t i0 H1) t) e (drop_gen_refl (CHead c0 (Bind b) t) e H6)))) (\lambda -(h0: nat).(\lambda (_: (((drop h0 O (CHead c0 (Bind b) t) e) \to ((le (plus O -h0) (S i0)) \to (drop (minus (S i0) h0) O e a))))).(\lambda (H6: (drop (S h0) -O (CHead c0 (Bind b) t) e)).(\lambda (H7: (le (plus O (S h0)) (S i0))).(H a -c0 (drop_gen_drop (Bind b) c0 a t i0 H1) e h0 O (drop_gen_drop (Bind b) c0 e -t h0 H6) (le_S_n (plus O h0) i0 H7)))))) h H4 H5))) (\lambda (d0: -nat).(\lambda (_: (((drop h d0 (CHead c0 (Bind b) t) e) \to ((le (plus d0 h) -(S i0)) \to (drop (minus (S i0) h) O e a))))).(\lambda (H4: (drop h (S d0) -(CHead c0 (Bind b) t) e)).(\lambda (H5: (le (plus (S d0) h) (S -i0))).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 (Bind -b) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r (Bind b) d0) -v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r (Bind b) d0) c0 e0))) (drop -(minus (S i0) h) O e a) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C -e (CHead x0 (Bind b) x1))).(\lambda (_: (eq T t (lift h (r (Bind b) d0) -x1))).(\lambda (H8: (drop h (r (Bind b) d0) c0 x0)).(eq_ind_r C (CHead x0 -(Bind b) x1) (\lambda (c1: C).(drop (minus (S i0) h) O c1 a)) (eq_ind nat (S -(minus i0 h)) (\lambda (n: nat).(drop n O (CHead x0 (Bind b) x1) a)) -(drop_drop (Bind b) (minus i0 h) x0 a (H a c0 (drop_gen_drop (Bind b) c0 a t -i0 H1) x0 h d0 H8 (le_S_n (plus d0 h) i0 H5)) x1) (minus (S i0) h) -(minus_Sn_m i0 h (le_trans_plus_r d0 h i0 (le_S_n (plus d0 h) i0 H5)))) e -H6)))))) (drop_gen_skip_l c0 e t h d0 (Bind b) H4)))))) d H2 H3))))))))) -(\lambda (f: F).(\lambda (t: T).(\lambda (H1: (drop (S i0) O (CHead c0 (Flat -f) t) a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2: -(drop h d (CHead c0 (Flat f) t) e)).(\lambda (H3: (le (plus d h) (S -i0))).(nat_ind (\lambda (n: nat).((drop h n (CHead c0 (Flat f) t) e) \to ((le -(plus n h) (S i0)) \to (drop (minus (S i0) h) O e a)))) (\lambda (H4: (drop h -O (CHead c0 (Flat f) t) e)).(\lambda (H5: (le (plus O h) (S i0))).(nat_ind -(\lambda (n: nat).((drop n O (CHead c0 (Flat f) t) e) \to ((le (plus O n) (S -i0)) \to (drop (minus (S i0) n) O e a)))) (\lambda (H6: (drop O O (CHead c0 -(Flat f) t) e)).(\lambda (_: (le (plus O O) (S i0))).(eq_ind C (CHead c0 -(Flat f) t) (\lambda (c1: C).(drop (minus (S i0) O) O c1 a)) (drop_drop (Flat -f) i0 c0 a (drop_gen_drop (Flat f) c0 a t i0 H1) t) e (drop_gen_refl (CHead -c0 (Flat f) t) e H6)))) (\lambda (h0: nat).(\lambda (_: (((drop h0 O (CHead -c0 (Flat f) t) e) \to ((le (plus O h0) (S i0)) \to (drop (minus (S i0) h0) O -e a))))).(\lambda (H6: (drop (S h0) O (CHead c0 (Flat f) t) e)).(\lambda (H7: -(le (plus O (S h0)) (S i0))).(H0 (drop_gen_drop (Flat f) c0 a t i0 H1) e (S -h0) O (drop_gen_drop (Flat f) c0 e t h0 H6) H7))))) h H4 H5))) (\lambda (d0: -nat).(\lambda (_: (((drop h d0 (CHead c0 (Flat f) t) e) \to ((le (plus d0 h) -(S i0)) \to (drop (minus (S i0) h) O e a))))).(\lambda (H4: (drop h (S d0) -(CHead c0 (Flat f) t) e)).(\lambda (H5: (le (plus (S d0) h) (S -i0))).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 (Flat -f) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r (Flat f) d0) -v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r (Flat f) d0) c0 e0))) (drop -(minus (S i0) h) O e a) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C -e (CHead x0 (Flat f) x1))).(\lambda (_: (eq T t (lift h (r (Flat f) d0) -x1))).(\lambda (H8: (drop h (r (Flat f) d0) c0 x0)).(eq_ind_r C (CHead x0 -(Flat f) x1) (\lambda (c1: C).(drop (minus (S i0) h) O c1 a)) (let H9 \def -(eq_ind_r nat (minus (S i0) h) (\lambda (n: nat).(drop n O x0 a)) (H0 -(drop_gen_drop (Flat f) c0 a t i0 H1) x0 h (S d0) H8 H5) (S (minus i0 h)) -(minus_Sn_m i0 h (le_trans_plus_r d0 h i0 (le_S_n (plus d0 h) i0 H5)))) in -(eq_ind nat (S (minus i0 h)) (\lambda (n: nat).(drop n O (CHead x0 (Flat f) -x1) a)) (drop_drop (Flat f) (minus i0 h) x0 a H9 x1) (minus (S i0) h) -(minus_Sn_m i0 h (le_trans_plus_r d0 h i0 (le_S_n (plus d0 h) i0 H5))))) e -H6)))))) (drop_gen_skip_l c0 e t h d0 (Flat f) H4)))))) d H2 H3))))))))) -k)))) c))))) i). -(* COMMENTS -Initial nodes: 2726 -END *) - -theorem drop_conf_rev: - \forall (j: nat).(\forall (e1: C).(\forall (e2: C).((drop j O e1 e2) \to -(\forall (c2: C).(\forall (i: nat).((drop i O c2 e2) \to (ex2 C (\lambda (c1: -C).(drop j O c1 c2)) (\lambda (c1: C).(drop i j c1 e1))))))))) -\def - \lambda (j: nat).(nat_ind (\lambda (n: nat).(\forall (e1: C).(\forall (e2: -C).((drop n O e1 e2) \to (\forall (c2: C).(\forall (i: nat).((drop i O c2 e2) -\to (ex2 C (\lambda (c1: C).(drop n O c1 c2)) (\lambda (c1: C).(drop i n c1 -e1)))))))))) (\lambda (e1: C).(\lambda (e2: C).(\lambda (H: (drop O O e1 -e2)).(\lambda (c2: C).(\lambda (i: nat).(\lambda (H0: (drop i O c2 e2)).(let -H1 \def (eq_ind_r C e2 (\lambda (c: C).(drop i O c2 c)) H0 e1 (drop_gen_refl -e1 e2 H)) in (ex_intro2 C (\lambda (c1: C).(drop O O c1 c2)) (\lambda (c1: -C).(drop i O c1 e1)) c2 (drop_refl c2) H1)))))))) (\lambda (j0: nat).(\lambda -(IHj: ((\forall (e1: C).(\forall (e2: C).((drop j0 O e1 e2) \to (\forall (c2: -C).(\forall (i: nat).((drop i O c2 e2) \to (ex2 C (\lambda (c1: C).(drop j0 O -c1 c2)) (\lambda (c1: C).(drop i j0 c1 e1))))))))))).(\lambda (e1: C).(C_ind -(\lambda (c: C).(\forall (e2: C).((drop (S j0) O c e2) \to (\forall (c2: -C).(\forall (i: nat).((drop i O c2 e2) \to (ex2 C (\lambda (c1: C).(drop (S -j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 c))))))))) (\lambda (n: -nat).(\lambda (e2: C).(\lambda (H: (drop (S j0) O (CSort n) e2)).(\lambda -(c2: C).(\lambda (i: nat).(\lambda (H0: (drop i O c2 e2)).(and3_ind (eq C e2 -(CSort n)) (eq nat (S j0) O) (eq nat O O) (ex2 C (\lambda (c1: C).(drop (S -j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 (CSort n)))) (\lambda (H1: -(eq C e2 (CSort n))).(\lambda (H2: (eq nat (S j0) O)).(\lambda (_: (eq nat O -O)).(let H4 \def (eq_ind C e2 (\lambda (c: C).(drop i O c2 c)) H0 (CSort n) -H1) in (let H5 \def (eq_ind nat (S j0) (\lambda (ee: nat).(match ee in nat -return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow -True])) I O H2) in (False_ind (ex2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) -(\lambda (c1: C).(drop i (S j0) c1 (CSort n)))) H5)))))) (drop_gen_sort n (S -j0) O e2 H)))))))) (\lambda (e2: C).(\lambda (IHe1: ((\forall (e3: C).((drop -(S j0) O e2 e3) \to (\forall (c2: C).(\forall (i: nat).((drop i O c2 e3) \to -(ex2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S -j0) c1 e2)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e3: C).(\lambda -(H: (drop (S j0) O (CHead e2 k t) e3)).(\lambda (c2: C).(\lambda (i: -nat).(\lambda (H0: (drop i O c2 e3)).(K_ind (\lambda (k0: K).((drop (r k0 j0) -O e2 e3) \to (ex2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: -C).(drop i (S j0) c1 (CHead e2 k0 t)))))) (\lambda (b: B).(\lambda (H1: (drop -(r (Bind b) j0) O e2 e3)).(let H_x \def (IHj e2 e3 H1 c2 i H0) in (let H2 -\def H_x in (ex2_ind C (\lambda (c1: C).(drop j0 O c1 c2)) (\lambda (c1: -C).(drop i j0 c1 e2)) (ex2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda -(c1: C).(drop i (S j0) c1 (CHead e2 (Bind b) t)))) (\lambda (x: C).(\lambda -(H3: (drop j0 O x c2)).(\lambda (H4: (drop i j0 x e2)).(ex_intro2 C (\lambda -(c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 (CHead e2 -(Bind b) t))) (CHead x (Bind b) (lift i (r (Bind b) j0) t)) (drop_drop (Bind -b) j0 x c2 H3 (lift i (r (Bind b) j0) t)) (drop_skip (Bind b) i j0 x e2 H4 -t))))) H2))))) (\lambda (f: F).(\lambda (H1: (drop (r (Flat f) j0) O e2 -e3)).(let H_x \def (IHe1 e3 H1 c2 i H0) in (let H2 \def H_x in (ex2_ind C -(\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 -e2)) (ex2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i -(S j0) c1 (CHead e2 (Flat f) t)))) (\lambda (x: C).(\lambda (H3: (drop (S j0) -O x c2)).(\lambda (H4: (drop i (S j0) x e2)).(ex_intro2 C (\lambda (c1: -C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 (CHead e2 (Flat -f) t))) (CHead x (Flat f) (lift i (r (Flat f) j0) t)) (drop_drop (Flat f) j0 -x c2 H3 (lift i (r (Flat f) j0) t)) (drop_skip (Flat f) i j0 x e2 H4 t))))) -H2))))) k (drop_gen_drop k e2 e3 t j0 H))))))))))) e1)))) j). -(* COMMENTS -Initial nodes: 1154 -END *) - -theorem drop_trans_le: - \forall (i: nat).(\forall (d: nat).((le i d) \to (\forall (c1: C).(\forall -(c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((drop i O -c2 e2) \to (ex2 C (\lambda (e1: C).(drop i O c1 e1)) (\lambda (e1: C).(drop h -(minus d i) e1 e2))))))))))) -\def - \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (d: nat).((le n d) \to -(\forall (c1: C).(\forall (c2: C).(\forall (h: nat).((drop h d c1 c2) \to -(\forall (e2: C).((drop n O c2 e2) \to (ex2 C (\lambda (e1: C).(drop n O c1 -e1)) (\lambda (e1: C).(drop h (minus d n) e1 e2)))))))))))) (\lambda (d: -nat).(\lambda (_: (le O d)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (h: -nat).(\lambda (H0: (drop h d c1 c2)).(\lambda (e2: C).(\lambda (H1: (drop O O -c2 e2)).(let H2 \def (eq_ind C c2 (\lambda (c: C).(drop h d c1 c)) H0 e2 -(drop_gen_refl c2 e2 H1)) in (eq_ind nat d (\lambda (n: nat).(ex2 C (\lambda -(e1: C).(drop O O c1 e1)) (\lambda (e1: C).(drop h n e1 e2)))) (ex_intro2 C -(\lambda (e1: C).(drop O O c1 e1)) (\lambda (e1: C).(drop h d e1 e2)) c1 -(drop_refl c1) H2) (minus d O) (minus_n_O d))))))))))) (\lambda (i0: -nat).(\lambda (IHi: ((\forall (d: nat).((le i0 d) \to (\forall (c1: -C).(\forall (c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: -C).((drop i0 O c2 e2) \to (ex2 C (\lambda (e1: C).(drop i0 O c1 e1)) (\lambda -(e1: C).(drop h (minus d i0) e1 e2))))))))))))).(\lambda (d: nat).(nat_ind -(\lambda (n: nat).((le (S i0) n) \to (\forall (c1: C).(\forall (c2: -C).(\forall (h: nat).((drop h n c1 c2) \to (\forall (e2: C).((drop (S i0) O -c2 e2) \to (ex2 C (\lambda (e1: C).(drop (S i0) O c1 e1)) (\lambda (e1: -C).(drop h (minus n (S i0)) e1 e2))))))))))) (\lambda (H: (le (S i0) -O)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (h: nat).(\lambda (_: (drop h -O c1 c2)).(\lambda (e2: C).(\lambda (_: (drop (S i0) O c2 e2)).(ex2_ind nat -(\lambda (n: nat).(eq nat O (S n))) (\lambda (n: nat).(le i0 n)) (ex2 C -(\lambda (e1: C).(drop (S i0) O c1 e1)) (\lambda (e1: C).(drop h (minus O (S -i0)) e1 e2))) (\lambda (x: nat).(\lambda (H2: (eq nat O (S x))).(\lambda (_: -(le i0 x)).(let H4 \def (eq_ind nat O (\lambda (ee: nat).(match ee in nat -return (\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow -False])) I (S x) H2) in (False_ind (ex2 C (\lambda (e1: C).(drop (S i0) O c1 -e1)) (\lambda (e1: C).(drop h (minus O (S i0)) e1 e2))) H4))))) (le_gen_S i0 -O H))))))))) (\lambda (d0: nat).(\lambda (_: (((le (S i0) d0) \to (\forall -(c1: C).(\forall (c2: C).(\forall (h: nat).((drop h d0 c1 c2) \to (\forall -(e2: C).((drop (S i0) O c2 e2) \to (ex2 C (\lambda (e1: C).(drop (S i0) O c1 -e1)) (\lambda (e1: C).(drop h (minus d0 (S i0)) e1 e2)))))))))))).(\lambda -(H: (le (S i0) (S d0))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: -C).(\forall (h: nat).((drop h (S d0) c c2) \to (\forall (e2: C).((drop (S i0) -O c2 e2) \to (ex2 C (\lambda (e1: C).(drop (S i0) O c e1)) (\lambda (e1: -C).(drop h (minus (S d0) (S i0)) e1 e2))))))))) (\lambda (n: nat).(\lambda -(c2: C).(\lambda (h: nat).(\lambda (H0: (drop h (S d0) (CSort n) -c2)).(\lambda (e2: C).(\lambda (H1: (drop (S i0) O c2 e2)).(and3_ind (eq C c2 -(CSort n)) (eq nat h O) (eq nat (S d0) O) (ex2 C (\lambda (e1: C).(drop (S -i0) O (CSort n) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2))) -(\lambda (H2: (eq C c2 (CSort n))).(\lambda (_: (eq nat h O)).(\lambda (_: -(eq nat (S d0) O)).(let H5 \def (eq_ind C c2 (\lambda (c: C).(drop (S i0) O c -e2)) H1 (CSort n) H2) in (and3_ind (eq C e2 (CSort n)) (eq nat (S i0) O) (eq -nat O O) (ex2 C (\lambda (e1: C).(drop (S i0) O (CSort n) e1)) (\lambda (e1: -C).(drop h (minus (S d0) (S i0)) e1 e2))) (\lambda (H6: (eq C e2 (CSort -n))).(\lambda (H7: (eq nat (S i0) O)).(\lambda (_: (eq nat O O)).(eq_ind_r C -(CSort n) (\lambda (c: C).(ex2 C (\lambda (e1: C).(drop (S i0) O (CSort n) -e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 c)))) (let H9 \def -(eq_ind nat (S i0) (\lambda (ee: nat).(match ee in nat return (\lambda (_: -nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H7) in -(False_ind (ex2 C (\lambda (e1: C).(drop (S i0) O (CSort n) e1)) (\lambda -(e1: C).(drop h (minus (S d0) (S i0)) e1 (CSort n)))) H9)) e2 H6)))) -(drop_gen_sort n (S i0) O e2 H5)))))) (drop_gen_sort n h (S d0) c2 H0)))))))) -(\lambda (c2: C).(\lambda (IHc: ((\forall (c3: C).(\forall (h: nat).((drop h -(S d0) c2 c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to (ex2 C (\lambda -(e1: C).(drop (S i0) O c2 e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) -e1 e2)))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t: -T).(\forall (c3: C).(\forall (h: nat).((drop h (S d0) (CHead c2 k0 t) c3) \to -(\forall (e2: C).((drop (S i0) O c3 e2) \to (ex2 C (\lambda (e1: C).(drop (S -i0) O (CHead c2 k0 t) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 -e2)))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (c3: C).(\lambda (h: -nat).(\lambda (H0: (drop h (S d0) (CHead c2 (Bind b) t) c3)).(\lambda (e2: -C).(\lambda (H1: (drop (S i0) O c3 e2)).(ex3_2_ind C T (\lambda (e: -C).(\lambda (v: T).(eq C c3 (CHead e (Bind b) v)))) (\lambda (_: C).(\lambda -(v: T).(eq T t (lift h (r (Bind b) d0) v)))) (\lambda (e: C).(\lambda (_: -T).(drop h (r (Bind b) d0) c2 e))) (ex2 C (\lambda (e1: C).(drop (S i0) O -(CHead c2 (Bind b) t) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 -e2))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H2: (eq C c3 (CHead x0 -(Bind b) x1))).(\lambda (H3: (eq T t (lift h (r (Bind b) d0) x1))).(\lambda -(H4: (drop h (r (Bind b) d0) c2 x0)).(let H5 \def (eq_ind C c3 (\lambda (c: -C).(drop (S i0) O c e2)) H1 (CHead x0 (Bind b) x1) H2) in (eq_ind_r T (lift h -(r (Bind b) d0) x1) (\lambda (t0: T).(ex2 C (\lambda (e1: C).(drop (S i0) O -(CHead c2 (Bind b) t0) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 -e2)))) (ex2_ind C (\lambda (e1: C).(drop i0 O c2 e1)) (\lambda (e1: C).(drop -h (minus d0 i0) e1 e2)) (ex2 C (\lambda (e1: C).(drop (S i0) O (CHead c2 -(Bind b) (lift h (r (Bind b) d0) x1)) e1)) (\lambda (e1: C).(drop h (minus (S -d0) (S i0)) e1 e2))) (\lambda (x: C).(\lambda (H6: (drop i0 O c2 x)).(\lambda -(H7: (drop h (minus d0 i0) x e2)).(ex_intro2 C (\lambda (e1: C).(drop (S i0) -O (CHead c2 (Bind b) (lift h (r (Bind b) d0) x1)) e1)) (\lambda (e1: C).(drop -h (minus (S d0) (S i0)) e1 e2)) x (drop_drop (Bind b) i0 c2 x H6 (lift h (r -(Bind b) d0) x1)) H7)))) (IHi d0 (le_S_n i0 d0 H) c2 x0 h H4 e2 -(drop_gen_drop (Bind b) x0 e2 x1 i0 H5))) t H3))))))) (drop_gen_skip_l c2 c3 -t h d0 (Bind b) H0))))))))) (\lambda (f: F).(\lambda (t: T).(\lambda (c3: -C).(\lambda (h: nat).(\lambda (H0: (drop h (S d0) (CHead c2 (Flat f) t) -c3)).(\lambda (e2: C).(\lambda (H1: (drop (S i0) O c3 e2)).(ex3_2_ind C T -(\lambda (e: C).(\lambda (v: T).(eq C c3 (CHead e (Flat f) v)))) (\lambda (_: -C).(\lambda (v: T).(eq T t (lift h (r (Flat f) d0) v)))) (\lambda (e: -C).(\lambda (_: T).(drop h (r (Flat f) d0) c2 e))) (ex2 C (\lambda (e1: -C).(drop (S i0) O (CHead c2 (Flat f) t) e1)) (\lambda (e1: C).(drop h (minus -(S d0) (S i0)) e1 e2))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H2: (eq C -c3 (CHead x0 (Flat f) x1))).(\lambda (H3: (eq T t (lift h (r (Flat f) d0) -x1))).(\lambda (H4: (drop h (r (Flat f) d0) c2 x0)).(let H5 \def (eq_ind C c3 -(\lambda (c: C).(drop (S i0) O c e2)) H1 (CHead x0 (Flat f) x1) H2) in -(eq_ind_r T (lift h (r (Flat f) d0) x1) (\lambda (t0: T).(ex2 C (\lambda (e1: -C).(drop (S i0) O (CHead c2 (Flat f) t0) e1)) (\lambda (e1: C).(drop h (minus -(S d0) (S i0)) e1 e2)))) (ex2_ind C (\lambda (e1: C).(drop (S i0) O c2 e1)) -(\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2)) (ex2 C (\lambda (e1: -C).(drop (S i0) O (CHead c2 (Flat f) (lift h (r (Flat f) d0) x1)) e1)) -(\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2))) (\lambda (x: -C).(\lambda (H6: (drop (S i0) O c2 x)).(\lambda (H7: (drop h (minus (S d0) (S -i0)) x e2)).(ex_intro2 C (\lambda (e1: C).(drop (S i0) O (CHead c2 (Flat f) -(lift h (r (Flat f) d0) x1)) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S -i0)) e1 e2)) x (drop_drop (Flat f) i0 c2 x H6 (lift h (r (Flat f) d0) x1)) -H7)))) (IHc x0 h H4 e2 (drop_gen_drop (Flat f) x0 e2 x1 i0 H5))) t H3))))))) -(drop_gen_skip_l c2 c3 t h d0 (Flat f) H0))))))))) k)))) c1))))) d)))) i). -(* COMMENTS -Initial nodes: 2453 -END *) - -theorem drop_trans_ge: - \forall (i: nat).(\forall (c1: C).(\forall (c2: C).(\forall (d: -nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((drop i O c2 -e2) \to ((le d i) \to (drop (plus i h) O c1 e2))))))))) -\def - \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (c2: -C).(\forall (d: nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: -C).((drop n O c2 e2) \to ((le d n) \to (drop (plus n h) O c1 e2)))))))))) -(\lambda (c1: C).(\lambda (c2: C).(\lambda (d: nat).(\lambda (h: -nat).(\lambda (H: (drop h d c1 c2)).(\lambda (e2: C).(\lambda (H0: (drop O O -c2 e2)).(\lambda (H1: (le d O)).(eq_ind C c2 (\lambda (c: C).(drop (plus O h) -O c1 c)) (let H_y \def (le_n_O_eq d H1) in (let H2 \def (eq_ind_r nat d -(\lambda (n: nat).(drop h n c1 c2)) H O H_y) in H2)) e2 (drop_gen_refl c2 e2 -H0)))))))))) (\lambda (i0: nat).(\lambda (IHi: ((\forall (c1: C).(\forall -(c2: C).(\forall (d: nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall -(e2: C).((drop i0 O c2 e2) \to ((le d i0) \to (drop (plus i0 h) O c1 -e2))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: -C).(\forall (d: nat).(\forall (h: nat).((drop h d c c2) \to (\forall (e2: -C).((drop (S i0) O c2 e2) \to ((le d (S i0)) \to (drop (plus (S i0) h) O c -e2))))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (d: nat).(\lambda (h: -nat).(\lambda (H: (drop h d (CSort n) c2)).(\lambda (e2: C).(\lambda (H0: -(drop (S i0) O c2 e2)).(\lambda (H1: (le d (S i0))).(and3_ind (eq C c2 (CSort -n)) (eq nat h O) (eq nat d O) (drop (S (plus i0 h)) O (CSort n) e2) (\lambda -(H2: (eq C c2 (CSort n))).(\lambda (H3: (eq nat h O)).(\lambda (H4: (eq nat d -O)).(eq_ind_r nat O (\lambda (n0: nat).(drop (S (plus i0 n0)) O (CSort n) -e2)) (let H5 \def (eq_ind nat d (\lambda (n0: nat).(le n0 (S i0))) H1 O H4) -in (let H6 \def (eq_ind C c2 (\lambda (c: C).(drop (S i0) O c e2)) H0 (CSort -n) H2) in (and3_ind (eq C e2 (CSort n)) (eq nat (S i0) O) (eq nat O O) (drop -(S (plus i0 O)) O (CSort n) e2) (\lambda (H7: (eq C e2 (CSort n))).(\lambda -(H8: (eq nat (S i0) O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n) -(\lambda (c: C).(drop (S (plus i0 O)) O (CSort n) c)) (let H10 \def (eq_ind -nat (S i0) (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) -with [O \Rightarrow False | (S _) \Rightarrow True])) I O H8) in (False_ind -(drop (S (plus i0 O)) O (CSort n) (CSort n)) H10)) e2 H7)))) (drop_gen_sort n -(S i0) O e2 H6)))) h H3)))) (drop_gen_sort n h d c2 H)))))))))) (\lambda (c2: -C).(\lambda (IHc: ((\forall (c3: C).(\forall (d: nat).(\forall (h: -nat).((drop h d c2 c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to ((le d -(S i0)) \to (drop (S (plus i0 h)) O c2 e2)))))))))).(\lambda (k: K).(\lambda -(t: T).(\lambda (c3: C).(\lambda (d: nat).(nat_ind (\lambda (n: nat).(\forall -(h: nat).((drop h n (CHead c2 k t) c3) \to (\forall (e2: C).((drop (S i0) O -c3 e2) \to ((le n (S i0)) \to (drop (S (plus i0 h)) O (CHead c2 k t) -e2))))))) (\lambda (h: nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c2 k -t) c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to ((le O (S i0)) \to -(drop (S (plus i0 n)) O (CHead c2 k t) e2)))))) (\lambda (H: (drop O O (CHead -c2 k t) c3)).(\lambda (e2: C).(\lambda (H0: (drop (S i0) O c3 e2)).(\lambda -(_: (le O (S i0))).(let H2 \def (eq_ind_r C c3 (\lambda (c: C).(drop (S i0) O -c e2)) H0 (CHead c2 k t) (drop_gen_refl (CHead c2 k t) c3 H)) in (eq_ind nat -i0 (\lambda (n: nat).(drop (S n) O (CHead c2 k t) e2)) (drop_drop k i0 c2 e2 -(drop_gen_drop k c2 e2 t i0 H2) t) (plus i0 O) (plus_n_O i0))))))) (\lambda -(n: nat).(\lambda (_: (((drop n O (CHead c2 k t) c3) \to (\forall (e2: -C).((drop (S i0) O c3 e2) \to ((le O (S i0)) \to (drop (S (plus i0 n)) O -(CHead c2 k t) e2))))))).(\lambda (H0: (drop (S n) O (CHead c2 k t) -c3)).(\lambda (e2: C).(\lambda (H1: (drop (S i0) O c3 e2)).(\lambda (H2: (le -O (S i0))).(eq_ind nat (S (plus i0 n)) (\lambda (n0: nat).(drop (S n0) O -(CHead c2 k t) e2)) (drop_drop k (S (plus i0 n)) c2 e2 (eq_ind_r nat (S (r k -(plus i0 n))) (\lambda (n0: nat).(drop n0 O c2 e2)) (eq_ind_r nat (plus i0 (r -k n)) (\lambda (n0: nat).(drop (S n0) O c2 e2)) (IHc c3 O (r k n) -(drop_gen_drop k c2 c3 t n H0) e2 H1 H2) (r k (plus i0 n)) (r_plus_sym k i0 -n)) (r k (S (plus i0 n))) (r_S k (plus i0 n))) t) (plus i0 (S n)) (plus_n_Sm -i0 n)))))))) h)) (\lambda (d0: nat).(\lambda (IHd: ((\forall (h: nat).((drop -h d0 (CHead c2 k t) c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to ((le -d0 (S i0)) \to (drop (S (plus i0 h)) O (CHead c2 k t) e2)))))))).(\lambda (h: -nat).(\lambda (H: (drop h (S d0) (CHead c2 k t) c3)).(\lambda (e2: -C).(\lambda (H0: (drop (S i0) O c3 e2)).(\lambda (H1: (le (S d0) (S -i0))).(ex3_2_ind C T (\lambda (e: C).(\lambda (v: T).(eq C c3 (CHead e k -v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r k d0) v)))) (\lambda -(e: C).(\lambda (_: T).(drop h (r k d0) c2 e))) (drop (S (plus i0 h)) O -(CHead c2 k t) e2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H2: (eq C c3 -(CHead x0 k x1))).(\lambda (H3: (eq T t (lift h (r k d0) x1))).(\lambda (H4: -(drop h (r k d0) c2 x0)).(let H5 \def (eq_ind C c3 (\lambda (c: C).(\forall -(h0: nat).((drop h0 d0 (CHead c2 k t) c) \to (\forall (e3: C).((drop (S i0) O -c e3) \to ((le d0 (S i0)) \to (drop (S (plus i0 h0)) O (CHead c2 k t) -e3))))))) IHd (CHead x0 k x1) H2) in (let H6 \def (eq_ind C c3 (\lambda (c: -C).(drop (S i0) O c e2)) H0 (CHead x0 k x1) H2) in (let H7 \def (eq_ind T t -(\lambda (t0: T).(\forall (h0: nat).((drop h0 d0 (CHead c2 k t0) (CHead x0 k -x1)) \to (\forall (e3: C).((drop (S i0) O (CHead x0 k x1) e3) \to ((le d0 (S -i0)) \to (drop (S (plus i0 h0)) O (CHead c2 k t0) e3))))))) H5 (lift h (r k -d0) x1) H3) in (eq_ind_r T (lift h (r k d0) x1) (\lambda (t0: T).(drop (S -(plus i0 h)) O (CHead c2 k t0) e2)) (drop_drop k (plus i0 h) c2 e2 (K_ind -(\lambda (k0: K).((drop h (r k0 d0) c2 x0) \to ((drop (r k0 i0) O x0 e2) \to -(drop (r k0 (plus i0 h)) O c2 e2)))) (\lambda (b: B).(\lambda (H8: (drop h (r -(Bind b) d0) c2 x0)).(\lambda (H9: (drop (r (Bind b) i0) O x0 e2)).(IHi c2 x0 -(r (Bind b) d0) h H8 e2 H9 (le_S_n (r (Bind b) d0) i0 H1))))) (\lambda (f: -F).(\lambda (H8: (drop h (r (Flat f) d0) c2 x0)).(\lambda (H9: (drop (r (Flat -f) i0) O x0 e2)).(IHc x0 (r (Flat f) d0) h H8 e2 H9 H1)))) k H4 -(drop_gen_drop k x0 e2 x1 i0 H6)) (lift h (r k d0) x1)) t H3))))))))) -(drop_gen_skip_l c2 c3 t h d0 k H))))))))) d))))))) c1)))) i). -(* COMMENTS -Initial nodes: 2020 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/drop1/defs.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/drop1/defs.ma deleted file mode 100644 index c0e14f438..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/drop1/defs.ma +++ /dev/null @@ -1,35 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/drop/defs.ma". - -include "Basic-1/lift1/defs.ma". - -inductive drop1: PList \to (C \to (C \to Prop)) \def -| drop1_nil: \forall (c: C).(drop1 PNil c c) -| drop1_cons: \forall (c1: C).(\forall (c2: C).(\forall (h: nat).(\forall (d: -nat).((drop h d c1 c2) \to (\forall (c3: C).(\forall (hds: PList).((drop1 hds -c2 c3) \to (drop1 (PCons h d hds) c1 c3)))))))). - -definition ptrans: - PList \to (nat \to PList) -\def - let rec ptrans (hds: PList) on hds: (nat \to PList) \def (\lambda (i: -nat).(match hds with [PNil \Rightarrow PNil | (PCons h d hds0) \Rightarrow -(let j \def (trans hds0 i) in (let q \def (ptrans hds0 i) in (match (blt j d) -with [true \Rightarrow (PCons h (minus d (S j)) q) | false \Rightarrow -q])))])) in ptrans. - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/drop1/fwd.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/drop1/fwd.ma deleted file mode 100644 index 6e4d1789e..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/drop1/fwd.ma +++ /dev/null @@ -1,81 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/drop1/defs.ma". - -theorem drop1_gen_pnil: - \forall (c1: C).(\forall (c2: C).((drop1 PNil c1 c2) \to (eq C c1 c2))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (drop1 PNil c1 c2)).(insert_eq -PList PNil (\lambda (p: PList).(drop1 p c1 c2)) (\lambda (_: PList).(eq C c1 -c2)) (\lambda (y: PList).(\lambda (H0: (drop1 y c1 c2)).(drop1_ind (\lambda -(p: PList).(\lambda (c: C).(\lambda (c0: C).((eq PList p PNil) \to (eq C c -c0))))) (\lambda (c: C).(\lambda (_: (eq PList PNil PNil)).(refl_equal C c))) -(\lambda (c3: C).(\lambda (c4: C).(\lambda (h: nat).(\lambda (d: -nat).(\lambda (_: (drop h d c3 c4)).(\lambda (c5: C).(\lambda (hds: -PList).(\lambda (_: (drop1 hds c4 c5)).(\lambda (_: (((eq PList hds PNil) \to -(eq C c4 c5)))).(\lambda (H4: (eq PList (PCons h d hds) PNil)).(let H5 \def -(eq_ind PList (PCons h d hds) (\lambda (ee: PList).(match ee in PList return -(\lambda (_: PList).Prop) with [PNil \Rightarrow False | (PCons _ _ _) -\Rightarrow True])) I PNil H4) in (False_ind (eq C c3 c5) H5)))))))))))) y c1 -c2 H0))) H))). -(* COMMENTS -Initial nodes: 198 -END *) - -theorem drop1_gen_pcons: - \forall (c1: C).(\forall (c3: C).(\forall (hds: PList).(\forall (h: -nat).(\forall (d: nat).((drop1 (PCons h d hds) c1 c3) \to (ex2 C (\lambda -(c2: C).(drop h d c1 c2)) (\lambda (c2: C).(drop1 hds c2 c3)))))))) -\def - \lambda (c1: C).(\lambda (c3: C).(\lambda (hds: PList).(\lambda (h: -nat).(\lambda (d: nat).(\lambda (H: (drop1 (PCons h d hds) c1 c3)).(insert_eq -PList (PCons h d hds) (\lambda (p: PList).(drop1 p c1 c3)) (\lambda (_: -PList).(ex2 C (\lambda (c2: C).(drop h d c1 c2)) (\lambda (c2: C).(drop1 hds -c2 c3)))) (\lambda (y: PList).(\lambda (H0: (drop1 y c1 c3)).(drop1_ind -(\lambda (p: PList).(\lambda (c: C).(\lambda (c0: C).((eq PList p (PCons h d -hds)) \to (ex2 C (\lambda (c2: C).(drop h d c c2)) (\lambda (c2: C).(drop1 -hds c2 c0))))))) (\lambda (c: C).(\lambda (H1: (eq PList PNil (PCons h d -hds))).(let H2 \def (eq_ind PList PNil (\lambda (ee: PList).(match ee in -PList return (\lambda (_: PList).Prop) with [PNil \Rightarrow True | (PCons _ -_ _) \Rightarrow False])) I (PCons h d hds) H1) in (False_ind (ex2 C (\lambda -(c2: C).(drop h d c c2)) (\lambda (c2: C).(drop1 hds c2 c))) H2)))) (\lambda -(c2: C).(\lambda (c4: C).(\lambda (h0: nat).(\lambda (d0: nat).(\lambda (H1: -(drop h0 d0 c2 c4)).(\lambda (c5: C).(\lambda (hds0: PList).(\lambda (H2: -(drop1 hds0 c4 c5)).(\lambda (H3: (((eq PList hds0 (PCons h d hds)) \to (ex2 -C (\lambda (c6: C).(drop h d c4 c6)) (\lambda (c6: C).(drop1 hds c6 -c5)))))).(\lambda (H4: (eq PList (PCons h0 d0 hds0) (PCons h d hds))).(let H5 -\def (f_equal PList nat (\lambda (e: PList).(match e in PList return (\lambda -(_: PList).nat) with [PNil \Rightarrow h0 | (PCons n _ _) \Rightarrow n])) -(PCons h0 d0 hds0) (PCons h d hds) H4) in ((let H6 \def (f_equal PList nat -(\lambda (e: PList).(match e in PList return (\lambda (_: PList).nat) with -[PNil \Rightarrow d0 | (PCons _ n _) \Rightarrow n])) (PCons h0 d0 hds0) -(PCons h d hds) H4) in ((let H7 \def (f_equal PList PList (\lambda (e: -PList).(match e in PList return (\lambda (_: PList).PList) with [PNil -\Rightarrow hds0 | (PCons _ _ p) \Rightarrow p])) (PCons h0 d0 hds0) (PCons h -d hds) H4) in (\lambda (H8: (eq nat d0 d)).(\lambda (H9: (eq nat h0 h)).(let -H10 \def (eq_ind PList hds0 (\lambda (p: PList).((eq PList p (PCons h d hds)) -\to (ex2 C (\lambda (c6: C).(drop h d c4 c6)) (\lambda (c6: C).(drop1 hds c6 -c5))))) H3 hds H7) in (let H11 \def (eq_ind PList hds0 (\lambda (p: -PList).(drop1 p c4 c5)) H2 hds H7) in (let H12 \def (eq_ind nat d0 (\lambda -(n: nat).(drop h0 n c2 c4)) H1 d H8) in (let H13 \def (eq_ind nat h0 (\lambda -(n: nat).(drop n d c2 c4)) H12 h H9) in (ex_intro2 C (\lambda (c6: C).(drop h -d c2 c6)) (\lambda (c6: C).(drop1 hds c6 c5)) c4 H13 H11)))))))) H6)) -H5)))))))))))) y c1 c3 H0))) H)))))). -(* COMMENTS -Initial nodes: 587 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/drop1/getl.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/drop1/getl.ma deleted file mode 100644 index dbee79850..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/drop1/getl.ma +++ /dev/null @@ -1,110 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/drop1/fwd.ma". - -include "Basic-1/getl/drop.ma". - -theorem drop1_getl_trans: - \forall (hds: PList).(\forall (c1: C).(\forall (c2: C).((drop1 hds c2 c1) -\to (\forall (b: B).(\forall (e1: C).(\forall (v: T).(\forall (i: nat).((getl -i c1 (CHead e1 (Bind b) v)) \to (ex2 C (\lambda (e2: C).(drop1 (ptrans hds i) -e2 e1)) (\lambda (e2: C).(getl (trans hds i) c2 (CHead e2 (Bind b) (lift1 -(ptrans hds i) v))))))))))))) -\def - \lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (c1: -C).(\forall (c2: C).((drop1 p c2 c1) \to (\forall (b: B).(\forall (e1: -C).(\forall (v: T).(\forall (i: nat).((getl i c1 (CHead e1 (Bind b) v)) \to -(ex2 C (\lambda (e2: C).(drop1 (ptrans p i) e2 e1)) (\lambda (e2: C).(getl -(trans p i) c2 (CHead e2 (Bind b) (lift1 (ptrans p i) v)))))))))))))) -(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (drop1 PNil c2 c1)).(\lambda -(b: B).(\lambda (e1: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H0: (getl -i c1 (CHead e1 (Bind b) v))).(let H_y \def (drop1_gen_pnil c2 c1 H) in -(eq_ind_r C c1 (\lambda (c: C).(ex2 C (\lambda (e2: C).(drop1 PNil e2 e1)) -(\lambda (e2: C).(getl i c (CHead e2 (Bind b) v))))) (ex_intro2 C (\lambda -(e2: C).(drop1 PNil e2 e1)) (\lambda (e2: C).(getl i c1 (CHead e2 (Bind b) -v))) e1 (drop1_nil e1) H0) c2 H_y)))))))))) (\lambda (h: nat).(\lambda (d: -nat).(\lambda (hds0: PList).(\lambda (H: ((\forall (c1: C).(\forall (c2: -C).((drop1 hds0 c2 c1) \to (\forall (b: B).(\forall (e1: C).(\forall (v: -T).(\forall (i: nat).((getl i c1 (CHead e1 (Bind b) v)) \to (ex2 C (\lambda -(e2: C).(drop1 (ptrans hds0 i) e2 e1)) (\lambda (e2: C).(getl (trans hds0 i) -c2 (CHead e2 (Bind b) (lift1 (ptrans hds0 i) v))))))))))))))).(\lambda (c1: -C).(\lambda (c2: C).(\lambda (H0: (drop1 (PCons h d hds0) c2 c1)).(\lambda -(b: B).(\lambda (e1: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H1: (getl -i c1 (CHead e1 (Bind b) v))).(let H_x \def (drop1_gen_pcons c2 c1 hds0 h d -H0) in (let H2 \def H_x in (ex2_ind C (\lambda (c3: C).(drop h d c2 c3)) -(\lambda (c3: C).(drop1 hds0 c3 c1)) (ex2 C (\lambda (e2: C).(drop1 (match -(blt (trans hds0 i) d) with [true \Rightarrow (PCons h (minus d (S (trans -hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)]) e2 e1)) -(\lambda (e2: C).(getl (match (blt (trans hds0 i) d) with [true \Rightarrow -(trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 -(Bind b) (lift1 (match (blt (trans hds0 i) d) with [true \Rightarrow (PCons h -(minus d (S (trans hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans -hds0 i)]) v))))) (\lambda (x: C).(\lambda (H3: (drop h d c2 x)).(\lambda (H4: -(drop1 hds0 x c1)).(xinduction bool (blt (trans hds0 i) d) (\lambda (b0: -bool).(ex2 C (\lambda (e2: C).(drop1 (match b0 with [true \Rightarrow (PCons -h (minus d (S (trans hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans -hds0 i)]) e2 e1)) (\lambda (e2: C).(getl (match b0 with [true \Rightarrow -(trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 -(Bind b) (lift1 (match b0 with [true \Rightarrow (PCons h (minus d (S (trans -hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)]) v)))))) -(\lambda (x_x: bool).(bool_ind (\lambda (b0: bool).((eq bool (blt (trans hds0 -i) d) b0) \to (ex2 C (\lambda (e2: C).(drop1 (match b0 with [true \Rightarrow -(PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) | false \Rightarrow -(ptrans hds0 i)]) e2 e1)) (\lambda (e2: C).(getl (match b0 with [true -\Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2 -(CHead e2 (Bind b) (lift1 (match b0 with [true \Rightarrow (PCons h (minus d -(S (trans hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)]) -v))))))) (\lambda (H5: (eq bool (blt (trans hds0 i) d) true)).(let H_x0 \def -(H c1 x H4 b e1 v i H1) in (let H6 \def H_x0 in (ex2_ind C (\lambda (e2: -C).(drop1 (ptrans hds0 i) e2 e1)) (\lambda (e2: C).(getl (trans hds0 i) x -(CHead e2 (Bind b) (lift1 (ptrans hds0 i) v)))) (ex2 C (\lambda (e2: -C).(drop1 (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) e2 e1)) -(\lambda (e2: C).(getl (trans hds0 i) c2 (CHead e2 (Bind b) (lift1 (PCons h -(minus d (S (trans hds0 i))) (ptrans hds0 i)) v))))) (\lambda (x0: -C).(\lambda (H7: (drop1 (ptrans hds0 i) x0 e1)).(\lambda (H8: (getl (trans -hds0 i) x (CHead x0 (Bind b) (lift1 (ptrans hds0 i) v)))).(let H_x1 \def -(drop_getl_trans_lt (trans hds0 i) d (blt_lt d (trans hds0 i) H5) c2 x h H3 b -x0 (lift1 (ptrans hds0 i) v) H8) in (let H9 \def H_x1 in (ex2_ind C (\lambda -(e2: C).(getl (trans hds0 i) c2 (CHead e2 (Bind b) (lift h (minus d (S (trans -hds0 i))) (lift1 (ptrans hds0 i) v))))) (\lambda (e2: C).(drop h (minus d (S -(trans hds0 i))) e2 x0)) (ex2 C (\lambda (e2: C).(drop1 (PCons h (minus d (S -(trans hds0 i))) (ptrans hds0 i)) e2 e1)) (\lambda (e2: C).(getl (trans hds0 -i) c2 (CHead e2 (Bind b) (lift1 (PCons h (minus d (S (trans hds0 i))) (ptrans -hds0 i)) v))))) (\lambda (x1: C).(\lambda (H10: (getl (trans hds0 i) c2 -(CHead x1 (Bind b) (lift h (minus d (S (trans hds0 i))) (lift1 (ptrans hds0 -i) v))))).(\lambda (H11: (drop h (minus d (S (trans hds0 i))) x1 -x0)).(ex_intro2 C (\lambda (e2: C).(drop1 (PCons h (minus d (S (trans hds0 -i))) (ptrans hds0 i)) e2 e1)) (\lambda (e2: C).(getl (trans hds0 i) c2 (CHead -e2 (Bind b) (lift1 (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) -v)))) x1 (drop1_cons x1 x0 h (minus d (S (trans hds0 i))) H11 e1 (ptrans hds0 -i) H7) H10)))) H9)))))) H6)))) (\lambda (H5: (eq bool (blt (trans hds0 i) d) -false)).(let H_x0 \def (H c1 x H4 b e1 v i H1) in (let H6 \def H_x0 in -(ex2_ind C (\lambda (e2: C).(drop1 (ptrans hds0 i) e2 e1)) (\lambda (e2: -C).(getl (trans hds0 i) x (CHead e2 (Bind b) (lift1 (ptrans hds0 i) v)))) -(ex2 C (\lambda (e2: C).(drop1 (ptrans hds0 i) e2 e1)) (\lambda (e2: C).(getl -(plus (trans hds0 i) h) c2 (CHead e2 (Bind b) (lift1 (ptrans hds0 i) v))))) -(\lambda (x0: C).(\lambda (H7: (drop1 (ptrans hds0 i) x0 e1)).(\lambda (H8: -(getl (trans hds0 i) x (CHead x0 (Bind b) (lift1 (ptrans hds0 i) v)))).(let -H9 \def (drop_getl_trans_ge (trans hds0 i) c2 x d h H3 (CHead x0 (Bind b) -(lift1 (ptrans hds0 i) v)) H8) in (ex_intro2 C (\lambda (e2: C).(drop1 -(ptrans hds0 i) e2 e1)) (\lambda (e2: C).(getl (plus (trans hds0 i) h) c2 -(CHead e2 (Bind b) (lift1 (ptrans hds0 i) v)))) x0 H7 (H9 (bge_le d (trans -hds0 i) H5))))))) H6)))) x_x)))))) H2))))))))))))))) hds). -(* COMMENTS -Initial nodes: 1674 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/drop1/props.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/drop1/props.ma deleted file mode 100644 index 240219b95..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/drop1/props.ma +++ /dev/null @@ -1,97 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/drop1/fwd.ma". - -include "Basic-1/drop/props.ma". - -include "Basic-1/getl/defs.ma". - -theorem drop1_skip_bind: - \forall (b: B).(\forall (e: C).(\forall (hds: PList).(\forall (c: -C).(\forall (u: T).((drop1 hds c e) \to (drop1 (Ss hds) (CHead c (Bind b) -(lift1 hds u)) (CHead e (Bind b) u))))))) -\def - \lambda (b: B).(\lambda (e: C).(\lambda (hds: PList).(PList_ind (\lambda (p: -PList).(\forall (c: C).(\forall (u: T).((drop1 p c e) \to (drop1 (Ss p) -(CHead c (Bind b) (lift1 p u)) (CHead e (Bind b) u)))))) (\lambda (c: -C).(\lambda (u: T).(\lambda (H: (drop1 PNil c e)).(let H_y \def -(drop1_gen_pnil c e H) in (eq_ind_r C e (\lambda (c0: C).(drop1 PNil (CHead -c0 (Bind b) u) (CHead e (Bind b) u))) (drop1_nil (CHead e (Bind b) u)) c -H_y))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda -(H: ((\forall (c: C).(\forall (u: T).((drop1 p c e) \to (drop1 (Ss p) (CHead -c (Bind b) (lift1 p u)) (CHead e (Bind b) u))))))).(\lambda (c: C).(\lambda -(u: T).(\lambda (H0: (drop1 (PCons n n0 p) c e)).(let H_x \def -(drop1_gen_pcons c e p n n0 H0) in (let H1 \def H_x in (ex2_ind C (\lambda -(c2: C).(drop n n0 c c2)) (\lambda (c2: C).(drop1 p c2 e)) (drop1 (PCons n (S -n0) (Ss p)) (CHead c (Bind b) (lift n n0 (lift1 p u))) (CHead e (Bind b) u)) -(\lambda (x: C).(\lambda (H2: (drop n n0 c x)).(\lambda (H3: (drop1 p x -e)).(drop1_cons (CHead c (Bind b) (lift n n0 (lift1 p u))) (CHead x (Bind b) -(lift1 p u)) n (S n0) (drop_skip_bind n n0 c x H2 b (lift1 p u)) (CHead e -(Bind b) u) (Ss p) (H x u H3))))) H1)))))))))) hds))). -(* COMMENTS -Initial nodes: 379 -END *) - -theorem drop1_cons_tail: - \forall (c2: C).(\forall (c3: C).(\forall (h: nat).(\forall (d: nat).((drop -h d c2 c3) \to (\forall (hds: PList).(\forall (c1: C).((drop1 hds c1 c2) \to -(drop1 (PConsTail hds h d) c1 c3)))))))) -\def - \lambda (c2: C).(\lambda (c3: C).(\lambda (h: nat).(\lambda (d: -nat).(\lambda (H: (drop h d c2 c3)).(\lambda (hds: PList).(PList_ind (\lambda -(p: PList).(\forall (c1: C).((drop1 p c1 c2) \to (drop1 (PConsTail p h d) c1 -c3)))) (\lambda (c1: C).(\lambda (H0: (drop1 PNil c1 c2)).(let H_y \def -(drop1_gen_pnil c1 c2 H0) in (eq_ind_r C c2 (\lambda (c: C).(drop1 (PCons h d -PNil) c c3)) (drop1_cons c2 c3 h d H c3 PNil (drop1_nil c3)) c1 H_y)))) -(\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H0: -((\forall (c1: C).((drop1 p c1 c2) \to (drop1 (PConsTail p h d) c1 -c3))))).(\lambda (c1: C).(\lambda (H1: (drop1 (PCons n n0 p) c1 c2)).(let H_x -\def (drop1_gen_pcons c1 c2 p n n0 H1) in (let H2 \def H_x in (ex2_ind C -(\lambda (c4: C).(drop n n0 c1 c4)) (\lambda (c4: C).(drop1 p c4 c2)) (drop1 -(PCons n n0 (PConsTail p h d)) c1 c3) (\lambda (x: C).(\lambda (H3: (drop n -n0 c1 x)).(\lambda (H4: (drop1 p x c2)).(drop1_cons c1 x n n0 H3 c3 -(PConsTail p h d) (H0 x H4))))) H2))))))))) hds)))))). -(* COMMENTS -Initial nodes: 271 -END *) - -theorem drop1_trans: - \forall (is1: PList).(\forall (c1: C).(\forall (c0: C).((drop1 is1 c1 c0) -\to (\forall (is2: PList).(\forall (c2: C).((drop1 is2 c0 c2) \to (drop1 -(papp is1 is2) c1 c2))))))) -\def - \lambda (is1: PList).(PList_ind (\lambda (p: PList).(\forall (c1: -C).(\forall (c0: C).((drop1 p c1 c0) \to (\forall (is2: PList).(\forall (c2: -C).((drop1 is2 c0 c2) \to (drop1 (papp p is2) c1 c2)))))))) (\lambda (c1: -C).(\lambda (c0: C).(\lambda (H: (drop1 PNil c1 c0)).(\lambda (is2: -PList).(\lambda (c2: C).(\lambda (H0: (drop1 is2 c0 c2)).(let H_y \def -(drop1_gen_pnil c1 c0 H) in (let H1 \def (eq_ind_r C c0 (\lambda (c: -C).(drop1 is2 c c2)) H0 c1 H_y) in H1)))))))) (\lambda (n: nat).(\lambda (n0: -nat).(\lambda (p: PList).(\lambda (H: ((\forall (c1: C).(\forall (c0: -C).((drop1 p c1 c0) \to (\forall (is2: PList).(\forall (c2: C).((drop1 is2 c0 -c2) \to (drop1 (papp p is2) c1 c2))))))))).(\lambda (c1: C).(\lambda (c0: -C).(\lambda (H0: (drop1 (PCons n n0 p) c1 c0)).(\lambda (is2: PList).(\lambda -(c2: C).(\lambda (H1: (drop1 is2 c0 c2)).(let H_x \def (drop1_gen_pcons c1 c0 -p n n0 H0) in (let H2 \def H_x in (ex2_ind C (\lambda (c3: C).(drop n n0 c1 -c3)) (\lambda (c3: C).(drop1 p c3 c0)) (drop1 (PCons n n0 (papp p is2)) c1 -c2) (\lambda (x: C).(\lambda (H3: (drop n n0 c1 x)).(\lambda (H4: (drop1 p x -c0)).(drop1_cons c1 x n n0 H3 c2 (papp p is2) (H x c0 H4 is2 c2 H1))))) -H2))))))))))))) is1). -(* COMMENTS -Initial nodes: 287 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/ex0/defs.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/ex0/defs.ma deleted file mode 100644 index 6ab98b1af..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/ex0/defs.ma +++ /dev/null @@ -1,32 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/A/defs.ma". - -include "Basic-1/G/defs.ma". - -definition gz: - G -\def - mk_G S lt_n_Sn. - -inductive leqz: A \to (A \to Prop) \def -| leqz_sort: \forall (h1: nat).(\forall (h2: nat).(\forall (n1: nat).(\forall -(n2: nat).((eq nat (plus h1 n2) (plus h2 n1)) \to (leqz (ASort h1 n1) (ASort -h2 n2)))))) -| leqz_head: \forall (a1: A).(\forall (a2: A).((leqz a1 a2) \to (\forall (a3: -A).(\forall (a4: A).((leqz a3 a4) \to (leqz (AHead a1 a3) (AHead a2 a4))))))). - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/ex0/props.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/ex0/props.ma deleted file mode 100644 index 96dd77da3..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/ex0/props.ma +++ /dev/null @@ -1,207 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/ex0/defs.ma". - -include "Basic-1/leq/defs.ma". - -include "Basic-1/aplus/props.ma". - -theorem aplus_gz_le: - \forall (k: nat).(\forall (h: nat).(\forall (n: nat).((le h k) \to (eq A -(aplus gz (ASort h n) k) (ASort O (plus (minus k h) n)))))) -\def - \lambda (k: nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).(\forall (n0: -nat).((le h n) \to (eq A (aplus gz (ASort h n0) n) (ASort O (plus (minus n h) -n0))))))) (\lambda (h: nat).(\lambda (n: nat).(\lambda (H: (le h O)).(let H_y -\def (le_n_O_eq h H) in (eq_ind nat O (\lambda (n0: nat).(eq A (ASort n0 n) -(ASort O n))) (refl_equal A (ASort O n)) h H_y))))) (\lambda (k0: -nat).(\lambda (IH: ((\forall (h: nat).(\forall (n: nat).((le h k0) \to (eq A -(aplus gz (ASort h n) k0) (ASort O (plus (minus k0 h) n)))))))).(\lambda (h: -nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).((le n (S k0)) \to (eq A -(asucc gz (aplus gz (ASort n n0) k0)) (ASort O (plus (match n with [O -\Rightarrow (S k0) | (S l) \Rightarrow (minus k0 l)]) n0)))))) (\lambda (n: -nat).(\lambda (_: (le O (S k0))).(eq_ind A (aplus gz (asucc gz (ASort O n)) -k0) (\lambda (a: A).(eq A a (ASort O (S (plus k0 n))))) (eq_ind_r A (ASort O -(plus (minus k0 O) (S n))) (\lambda (a: A).(eq A a (ASort O (S (plus k0 -n))))) (eq_ind nat k0 (\lambda (n0: nat).(eq A (ASort O (plus n0 (S n))) -(ASort O (S (plus k0 n))))) (eq_ind nat (S (plus k0 n)) (\lambda (n0: -nat).(eq A (ASort O n0) (ASort O (S (plus k0 n))))) (refl_equal A (ASort O (S -(plus k0 n)))) (plus k0 (S n)) (plus_n_Sm k0 n)) (minus k0 O) (minus_n_O k0)) -(aplus gz (ASort O (S n)) k0) (IH O (S n) (le_O_n k0))) (asucc gz (aplus gz -(ASort O n) k0)) (aplus_asucc gz k0 (ASort O n))))) (\lambda (n: -nat).(\lambda (_: ((\forall (n0: nat).((le n (S k0)) \to (eq A (asucc gz -(aplus gz (ASort n n0) k0)) (ASort O (plus (match n with [O \Rightarrow (S -k0) | (S l) \Rightarrow (minus k0 l)]) n0))))))).(\lambda (n0: nat).(\lambda -(H0: (le (S n) (S k0))).(let H_y \def (le_S_n n k0 H0) in (eq_ind A (aplus gz -(ASort n n0) k0) (\lambda (a: A).(eq A (asucc gz (aplus gz (ASort (S n) n0) -k0)) a)) (eq_ind A (aplus gz (asucc gz (ASort (S n) n0)) k0) (\lambda (a: -A).(eq A a (aplus gz (ASort n n0) k0))) (refl_equal A (aplus gz (ASort n n0) -k0)) (asucc gz (aplus gz (ASort (S n) n0) k0)) (aplus_asucc gz k0 (ASort (S -n) n0))) (ASort O (plus (minus k0 n) n0)) (IH n n0 H_y))))))) h)))) k). -(* COMMENTS -Initial nodes: 683 -END *) - -theorem aplus_gz_ge: - \forall (n: nat).(\forall (k: nat).(\forall (h: nat).((le k h) \to (eq A -(aplus gz (ASort h n) k) (ASort (minus h k) n))))) -\def - \lambda (n: nat).(\lambda (k: nat).(nat_ind (\lambda (n0: nat).(\forall (h: -nat).((le n0 h) \to (eq A (aplus gz (ASort h n) n0) (ASort (minus h n0) -n))))) (\lambda (h: nat).(\lambda (_: (le O h)).(eq_ind nat h (\lambda (n0: -nat).(eq A (ASort h n) (ASort n0 n))) (refl_equal A (ASort h n)) (minus h O) -(minus_n_O h)))) (\lambda (k0: nat).(\lambda (IH: ((\forall (h: nat).((le k0 -h) \to (eq A (aplus gz (ASort h n) k0) (ASort (minus h k0) n)))))).(\lambda -(h: nat).(nat_ind (\lambda (n0: nat).((le (S k0) n0) \to (eq A (asucc gz -(aplus gz (ASort n0 n) k0)) (ASort (minus n0 (S k0)) n)))) (\lambda (H: (le -(S k0) O)).(ex2_ind nat (\lambda (n0: nat).(eq nat O (S n0))) (\lambda (n0: -nat).(le k0 n0)) (eq A (asucc gz (aplus gz (ASort O n) k0)) (ASort O n)) -(\lambda (x: nat).(\lambda (H0: (eq nat O (S x))).(\lambda (_: (le k0 -x)).(let H2 \def (eq_ind nat O (\lambda (ee: nat).(match ee in nat return -(\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) -I (S x) H0) in (False_ind (eq A (asucc gz (aplus gz (ASort O n) k0)) (ASort O -n)) H2))))) (le_gen_S k0 O H))) (\lambda (n0: nat).(\lambda (_: (((le (S k0) -n0) \to (eq A (asucc gz (aplus gz (ASort n0 n) k0)) (ASort (minus n0 (S k0)) -n))))).(\lambda (H0: (le (S k0) (S n0))).(let H_y \def (le_S_n k0 n0 H0) in -(eq_ind A (aplus gz (ASort n0 n) k0) (\lambda (a: A).(eq A (asucc gz (aplus -gz (ASort (S n0) n) k0)) a)) (eq_ind A (aplus gz (asucc gz (ASort (S n0) n)) -k0) (\lambda (a: A).(eq A a (aplus gz (ASort n0 n) k0))) (refl_equal A (aplus -gz (ASort n0 n) k0)) (asucc gz (aplus gz (ASort (S n0) n) k0)) (aplus_asucc -gz k0 (ASort (S n0) n))) (ASort (minus n0 k0) n) (IH n0 H_y)))))) h)))) k)). -(* COMMENTS -Initial nodes: 524 -END *) - -theorem next_plus_gz: - \forall (n: nat).(\forall (h: nat).(eq nat (next_plus gz n h) (plus h n))) -\def - \lambda (n: nat).(\lambda (h: nat).(nat_ind (\lambda (n0: nat).(eq nat -(next_plus gz n n0) (plus n0 n))) (refl_equal nat n) (\lambda (n0: -nat).(\lambda (H: (eq nat (next_plus gz n n0) (plus n0 n))).(f_equal nat nat -S (next_plus gz n n0) (plus n0 n) H))) h)). -(* COMMENTS -Initial nodes: 77 -END *) - -theorem leqz_leq: - \forall (a1: A).(\forall (a2: A).((leq gz a1 a2) \to (leqz a1 a2))) -\def - \lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq gz a1 a2)).(leq_ind gz -(\lambda (a: A).(\lambda (a0: A).(leqz a a0))) (\lambda (h1: nat).(\lambda -(h2: nat).(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda -(H0: (eq A (aplus gz (ASort h1 n1) k) (aplus gz (ASort h2 n2) k))).(lt_le_e k -h1 (leqz (ASort h1 n1) (ASort h2 n2)) (\lambda (H1: (lt k h1)).(lt_le_e k h2 -(leqz (ASort h1 n1) (ASort h2 n2)) (\lambda (H2: (lt k h2)).(let H3 \def -(eq_ind A (aplus gz (ASort h1 n1) k) (\lambda (a: A).(eq A a (aplus gz (ASort -h2 n2) k))) H0 (ASort (minus h1 k) n1) (aplus_gz_ge n1 k h1 (le_S_n k h1 -(le_S (S k) h1 H1)))) in (let H4 \def (eq_ind A (aplus gz (ASort h2 n2) k) -(\lambda (a: A).(eq A (ASort (minus h1 k) n1) a)) H3 (ASort (minus h2 k) n2) -(aplus_gz_ge n2 k h2 (le_S_n k h2 (le_S (S k) h2 H2)))) in (let H5 \def -(f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with -[(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow ((let rec minus (n: nat) -on n: (nat \to nat) \def (\lambda (m: nat).(match n with [O \Rightarrow O | -(S k0) \Rightarrow (match m with [O \Rightarrow (S k0) | (S l) \Rightarrow -(minus k0 l)])])) in minus) h1 k)])) (ASort (minus h1 k) n1) (ASort (minus h2 -k) n2) H4) in ((let H6 \def (f_equal A nat (\lambda (e: A).(match e in A -return (\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) -\Rightarrow n1])) (ASort (minus h1 k) n1) (ASort (minus h2 k) n2) H4) in -(\lambda (H7: (eq nat (minus h1 k) (minus h2 k))).(eq_ind nat n1 (\lambda (n: -nat).(leqz (ASort h1 n1) (ASort h2 n))) (eq_ind nat h1 (\lambda (n: -nat).(leqz (ASort h1 n1) (ASort n n1))) (leqz_sort h1 h1 n1 n1 (refl_equal -nat (plus h1 n1))) h2 (minus_minus k h1 h2 (le_S_n k h1 (le_S (S k) h1 H1)) -(le_S_n k h2 (le_S (S k) h2 H2)) H7)) n2 H6))) H5))))) (\lambda (H2: (le h2 -k)).(let H3 \def (eq_ind A (aplus gz (ASort h1 n1) k) (\lambda (a: A).(eq A a -(aplus gz (ASort h2 n2) k))) H0 (ASort (minus h1 k) n1) (aplus_gz_ge n1 k h1 -(le_S_n k h1 (le_S (S k) h1 H1)))) in (let H4 \def (eq_ind A (aplus gz (ASort -h2 n2) k) (\lambda (a: A).(eq A (ASort (minus h1 k) n1) a)) H3 (ASort O (plus -(minus k h2) n2)) (aplus_gz_le k h2 n2 H2)) in (let H5 \def (eq_ind nat -(minus h1 k) (\lambda (n: nat).(eq A (ASort n n1) (ASort O (plus (minus k h2) -n2)))) H4 (S (minus h1 (S k))) (minus_x_Sy h1 k H1)) in (let H6 \def (eq_ind -A (ASort (S (minus h1 (S k))) n1) (\lambda (ee: A).(match ee in A return -(\lambda (_: A).Prop) with [(ASort n _) \Rightarrow (match n in nat return -(\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True]) -| (AHead _ _) \Rightarrow False])) I (ASort O (plus (minus k h2) n2)) H5) in -(False_ind (leqz (ASort h1 n1) (ASort h2 n2)) H6)))))))) (\lambda (H1: (le h1 -k)).(lt_le_e k h2 (leqz (ASort h1 n1) (ASort h2 n2)) (\lambda (H2: (lt k -h2)).(let H3 \def (eq_ind A (aplus gz (ASort h1 n1) k) (\lambda (a: A).(eq A -a (aplus gz (ASort h2 n2) k))) H0 (ASort O (plus (minus k h1) n1)) -(aplus_gz_le k h1 n1 H1)) in (let H4 \def (eq_ind A (aplus gz (ASort h2 n2) -k) (\lambda (a: A).(eq A (ASort O (plus (minus k h1) n1)) a)) H3 (ASort -(minus h2 k) n2) (aplus_gz_ge n2 k h2 (le_S_n k h2 (le_S (S k) h2 H2)))) in -(let H5 \def (sym_eq A (ASort O (plus (minus k h1) n1)) (ASort (minus h2 k) -n2) H4) in (let H6 \def (eq_ind nat (minus h2 k) (\lambda (n: nat).(eq A -(ASort n n2) (ASort O (plus (minus k h1) n1)))) H5 (S (minus h2 (S k))) -(minus_x_Sy h2 k H2)) in (let H7 \def (eq_ind A (ASort (S (minus h2 (S k))) -n2) (\lambda (ee: A).(match ee in A return (\lambda (_: A).Prop) with [(ASort -n _) \Rightarrow (match n in nat return (\lambda (_: nat).Prop) with [O -\Rightarrow False | (S _) \Rightarrow True]) | (AHead _ _) \Rightarrow -False])) I (ASort O (plus (minus k h1) n1)) H6) in (False_ind (leqz (ASort h1 -n1) (ASort h2 n2)) H7))))))) (\lambda (H2: (le h2 k)).(let H3 \def (eq_ind A -(aplus gz (ASort h1 n1) k) (\lambda (a: A).(eq A a (aplus gz (ASort h2 n2) -k))) H0 (ASort O (plus (minus k h1) n1)) (aplus_gz_le k h1 n1 H1)) in (let H4 -\def (eq_ind A (aplus gz (ASort h2 n2) k) (\lambda (a: A).(eq A (ASort O -(plus (minus k h1) n1)) a)) H3 (ASort O (plus (minus k h2) n2)) (aplus_gz_le -k h2 n2 H2)) in (let H5 \def (f_equal A nat (\lambda (e: A).(match e in A -return (\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) -\Rightarrow ((let rec plus (n: nat) on n: (nat \to nat) \def (\lambda (m: -nat).(match n with [O \Rightarrow m | (S p) \Rightarrow (S (plus p m))])) in -plus) (minus k h1) n1)])) (ASort O (plus (minus k h1) n1)) (ASort O (plus -(minus k h2) n2)) H4) in (let H_y \def (plus_plus k h1 h2 n1 n2 H1 H2 H5) in -(leqz_sort h1 h2 n1 n2 H_y))))))))))))))) (\lambda (a0: A).(\lambda (a3: -A).(\lambda (_: (leq gz a0 a3)).(\lambda (H1: (leqz a0 a3)).(\lambda (a4: -A).(\lambda (a5: A).(\lambda (_: (leq gz a4 a5)).(\lambda (H3: (leqz a4 -a5)).(leqz_head a0 a3 H1 a4 a5 H3))))))))) a1 a2 H))). -(* COMMENTS -Initial nodes: 1375 -END *) - -theorem leq_leqz: - \forall (a1: A).(\forall (a2: A).((leqz a1 a2) \to (leq gz a1 a2))) -\def - \lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leqz a1 a2)).(leqz_ind -(\lambda (a: A).(\lambda (a0: A).(leq gz a a0))) (\lambda (h1: nat).(\lambda -(h2: nat).(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (H0: (eq nat (plus -h1 n2) (plus h2 n1))).(leq_sort gz h1 h2 n1 n2 (plus h1 h2) (eq_ind_r A -(ASort (minus h1 (plus h1 h2)) (next_plus gz n1 (minus (plus h1 h2) h1))) -(\lambda (a: A).(eq A a (aplus gz (ASort h2 n2) (plus h1 h2)))) (eq_ind_r A -(ASort (minus h2 (plus h1 h2)) (next_plus gz n2 (minus (plus h1 h2) h2))) -(\lambda (a: A).(eq A (ASort (minus h1 (plus h1 h2)) (next_plus gz n1 (minus -(plus h1 h2) h1))) a)) (eq_ind_r nat h2 (\lambda (n: nat).(eq A (ASort (minus -h1 (plus h1 h2)) (next_plus gz n1 n)) (ASort (minus h2 (plus h1 h2)) -(next_plus gz n2 (minus (plus h1 h2) h2))))) (eq_ind_r nat h1 (\lambda (n: -nat).(eq A (ASort (minus h1 (plus h1 h2)) (next_plus gz n1 h2)) (ASort (minus -h2 (plus h1 h2)) (next_plus gz n2 n)))) (eq_ind_r nat O (\lambda (n: nat).(eq -A (ASort n (next_plus gz n1 h2)) (ASort (minus h2 (plus h1 h2)) (next_plus gz -n2 h1)))) (eq_ind_r nat O (\lambda (n: nat).(eq A (ASort O (next_plus gz n1 -h2)) (ASort n (next_plus gz n2 h1)))) (eq_ind_r nat (plus h2 n1) (\lambda (n: -nat).(eq A (ASort O n) (ASort O (next_plus gz n2 h1)))) (eq_ind_r nat (plus -h1 n2) (\lambda (n: nat).(eq A (ASort O (plus h2 n1)) (ASort O n))) (f_equal -nat A (ASort O) (plus h2 n1) (plus h1 n2) (sym_eq nat (plus h1 n2) (plus h2 -n1) H0)) (next_plus gz n2 h1) (next_plus_gz n2 h1)) (next_plus gz n1 h2) -(next_plus_gz n1 h2)) (minus h2 (plus h1 h2)) (O_minus h2 (plus h1 h2) -(le_plus_r h1 h2))) (minus h1 (plus h1 h2)) (O_minus h1 (plus h1 h2) -(le_plus_l h1 h2))) (minus (plus h1 h2) h2) (minus_plus_r h1 h2)) (minus -(plus h1 h2) h1) (minus_plus h1 h2)) (aplus gz (ASort h2 n2) (plus h1 h2)) -(aplus_asort_simpl gz (plus h1 h2) h2 n2)) (aplus gz (ASort h1 n1) (plus h1 -h2)) (aplus_asort_simpl gz (plus h1 h2) h1 n1)))))))) (\lambda (a0: -A).(\lambda (a3: A).(\lambda (_: (leqz a0 a3)).(\lambda (H1: (leq gz a0 -a3)).(\lambda (a4: A).(\lambda (a5: A).(\lambda (_: (leqz a4 a5)).(\lambda -(H3: (leq gz a4 a5)).(leq_head gz a0 a3 H1 a4 a5 H3))))))))) a1 a2 H))). -(* COMMENTS -Initial nodes: 717 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/ex1/defs.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/ex1/defs.ma deleted file mode 100644 index dc18f6fc2..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/ex1/defs.ma +++ /dev/null @@ -1,29 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/C/defs.ma". - -definition ex1_c: - C -\def - CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) -(Bind Abst) (TLRef O). - -definition ex1_t: - T -\def - THead (Flat Appl) (TLRef O) (THead (Bind Abst) (TLRef (S (S O))) (TSort O)). - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/ex1/props.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/ex1/props.ma deleted file mode 100644 index 5c442f5bb..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/ex1/props.ma +++ /dev/null @@ -1,536 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/ex1/defs.ma". - -include "Basic-1/ty3/fwd.ma". - -include "Basic-1/pc3/fwd.ma". - -include "Basic-1/nf2/pr3.ma". - -include "Basic-1/nf2/props.ma". - -include "Basic-1/arity/defs.ma". - -include "Basic-1/leq/props.ma". - -theorem ex1__leq_sort_SS: - \forall (g: G).(\forall (k: nat).(\forall (n: nat).(leq g (ASort k n) (asucc -g (asucc g (ASort (S (S k)) n)))))) -\def - \lambda (g: G).(\lambda (k: nat).(\lambda (n: nat).(leq_refl g (asucc g -(asucc g (ASort (S (S k)) n)))))). -(* COMMENTS -Initial nodes: 27 -END *) - -theorem ex1_arity: - \forall (g: G).(arity g ex1_c ex1_t (ASort O O)) -\def - \lambda (g: G).(arity_appl g (CHead (CHead (CHead (CSort O) (Bind Abst) -(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef O) (ASort (S -(S O)) O) (arity_abst g (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) -(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) (TLRef O) O (getl_refl Abst (CHead -(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (TLRef O)) -(ASort (S (S O)) O) (arity_abst g (CHead (CHead (CSort O) (Bind Abst) (TSort -O)) (Bind Abst) (TSort O)) (CHead (CSort O) (Bind Abst) (TSort O)) (TSort O) -O (getl_refl Abst (CHead (CSort O) (Bind Abst) (TSort O)) (TSort O)) (asucc g -(ASort (S (S O)) O)) (arity_repl g (CHead (CSort O) (Bind Abst) (TSort O)) -(TSort O) (ASort O O) (arity_sort g (CHead (CSort O) (Bind Abst) (TSort O)) -O) (asucc g (asucc g (ASort (S (S O)) O))) (ex1__leq_sort_SS g O O)))) (THead -(Bind Abst) (TLRef (S (S O))) (TSort O)) (ASort O O) (arity_head g (CHead -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind -Abst) (TLRef O)) (TLRef (S (S O))) (ASort (S (S O)) O) (arity_abst g (CHead -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind -Abst) (TLRef O)) (CSort O) (TSort O) (S (S O)) (getl_clear_bind Abst (CHead -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind -Abst) (TLRef O)) (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) -(TSort O)) (TLRef O) (clear_bind Abst (CHead (CHead (CSort O) (Bind Abst) -(TSort O)) (Bind Abst) (TSort O)) (TLRef O)) (CHead (CSort O) (Bind Abst) -(TSort O)) (S O) (getl_head (Bind Abst) O (CHead (CSort O) (Bind Abst) (TSort -O)) (CHead (CSort O) (Bind Abst) (TSort O)) (getl_refl Abst (CSort O) (TSort -O)) (TSort O))) (asucc g (ASort (S (S O)) O)) (arity_repl g (CSort O) (TSort -O) (ASort O O) (arity_sort g (CSort O) O) (asucc g (asucc g (ASort (S (S O)) -O))) (ex1__leq_sort_SS g O O))) (TSort O) (ASort O O) (arity_sort g (CHead -(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) -(Bind Abst) (TLRef O)) (Bind Abst) (TLRef (S (S O)))) O))). -(* COMMENTS -Initial nodes: 753 -END *) - -theorem ex1_ty3: - \forall (g: G).(\forall (u: T).((ty3 g ex1_c ex1_t u) \to (\forall (P: -Prop).P))) -\def - \lambda (g: G).(\lambda (u: T).(\lambda (H: (ty3 g (CHead (CHead (CHead -(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef -O)) (THead (Flat Appl) (TLRef O) (THead (Bind Abst) (TLRef (S (S O))) (TSort -O))) u)).(\lambda (P: Prop).(ex3_2_ind T T (\lambda (u0: T).(\lambda (t: -T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) -(TSort O)) (Bind Abst) (TLRef O)) (THead (Flat Appl) (TLRef O) (THead (Bind -Abst) u0 t)) u))) (\lambda (u0: T).(\lambda (t: T).(ty3 g (CHead (CHead -(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) -(TLRef O)) (THead (Bind Abst) (TLRef (S (S O))) (TSort O)) (THead (Bind Abst) -u0 t)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g (CHead (CHead (CHead (CSort -O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) -(TLRef O) u0))) P (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 (CHead -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind -Abst) (TLRef O)) (THead (Flat Appl) (TLRef O) (THead (Bind Abst) x0 x1)) -u)).(\lambda (H1: (ty3 g (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort -O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (THead (Bind Abst) (TLRef -(S (S O))) (TSort O)) (THead (Bind Abst) x0 x1))).(\lambda (H2: (ty3 g (CHead -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind -Abst) (TLRef O)) (TLRef O) x0)).(or_ind (ex3_3 C T T (\lambda (_: C).(\lambda -(_: T).(\lambda (t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort -O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S O) O t) x0)))) -(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl O (CHead (CHead (CHead -(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef -O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(t: T).(ty3 g e u0 t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: -T).(\lambda (_: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) -(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S O) O u0) x0)))) -(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl O (CHead (CHead (CHead -(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef -O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(t: T).(ty3 g e u0 t))))) P (\lambda (H3: (ex3_3 C T T (\lambda (_: -C).(\lambda (_: T).(\lambda (t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S O) O -t) x0)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl O (CHead -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind -Abst) (TLRef O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (t: T).(ty3 g e u0 t)))))).(ex3_3_ind C T T (\lambda (_: -C).(\lambda (_: T).(\lambda (t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S O) O -t) x0)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl O (CHead -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind -Abst) (TLRef O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (t: T).(ty3 g e u0 t)))) P (\lambda (x2: C).(\lambda (x3: -T).(\lambda (x4: T).(\lambda (_: (pc3 (CHead (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S O) O -x4) x0)).(\lambda (H5: (getl O (CHead (CHead (CHead (CSort O) (Bind Abst) -(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead x2 (Bind -Abbr) x3))).(\lambda (_: (ty3 g x2 x3 x4)).(ex3_2_ind T T (\lambda (t2: -T).(\lambda (_: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) -(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (THead (Bind Abst) (TLRef (S (S -O))) t2) (THead (Bind Abst) x0 x1)))) (\lambda (_: T).(\lambda (t: T).(ty3 g -(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) -(Bind Abst) (TLRef O)) (TLRef (S (S O))) t))) (\lambda (t2: T).(\lambda (_: -T).(ty3 g (CHead (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind -Abst) (TSort O)) (Bind Abst) (TLRef O)) (Bind Abst) (TLRef (S (S O)))) (TSort -O) t2))) P (\lambda (x5: T).(\lambda (x6: T).(\lambda (_: (pc3 (CHead (CHead -(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) -(TLRef O)) (THead (Bind Abst) (TLRef (S (S O))) x5) (THead (Bind Abst) x0 -x1))).(\lambda (H8: (ty3 g (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort -O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) -x6)).(\lambda (_: (ty3 g (CHead (CHead (CHead (CHead (CSort O) (Bind Abst) -(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (Bind Abst) (TLRef -(S (S O)))) (TSort O) x5)).(or_ind (ex3_3 C T T (\lambda (_: C).(\lambda (_: -T).(\lambda (t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) -(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O t) x6)))) -(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) (CHead -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind -Abst) (TLRef O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (t: T).(ty3 g e u0 t))))) (ex3_3 C T T (\lambda (_: C).(\lambda -(u0: T).(\lambda (_: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) -(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) -O u0) x6)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) -(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) -(Bind Abst) (TLRef O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda -(u0: T).(\lambda (t: T).(ty3 g e u0 t))))) P (\lambda (H10: (ex3_3 C T T -(\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 (CHead (CHead (CHead -(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef -O)) (lift (S (S (S O))) O t) x6)))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(_: T).(getl (S (S O)) (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) -(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead e (Bind Abbr) u0))))) -(\lambda (e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 -t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 -(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) -(Bind Abst) (TLRef O)) (lift (S (S (S O))) O t) x6)))) (\lambda (e: -C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) (CHead (CHead (CHead -(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef -O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(t: T).(ty3 g e u0 t)))) P (\lambda (x7: C).(\lambda (x8: T).(\lambda (x9: -T).(\lambda (_: (pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) -(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O x9) -x6)).(\lambda (H12: (getl (S (S O)) (CHead (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead x7 -(Bind Abbr) x8))).(\lambda (_: (ty3 g x7 x8 x9)).(let H14 \def (getl_gen_all -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead -x7 (Bind Abbr) x8) (r (Bind Abst) (S O)) (getl_gen_S (Bind Abst) (CHead -(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead x7 -(Bind Abbr) x8) (TLRef O) (S O) H12)) in (ex2_ind C (\lambda (e: C).(drop (S -O) O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) -e)) (\lambda (e: C).(clear e (CHead x7 (Bind Abbr) x8))) P (\lambda (x: -C).(\lambda (_: (drop (S O) O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) -(Bind Abst) (TSort O)) x)).(\lambda (_: (clear x (CHead x7 (Bind Abbr) -x8))).(let H17 \def (eq_ind C (CHead x2 (Bind Abbr) x3) (\lambda (ee: -C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow -False | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) -with [(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with -[Abbr \Rightarrow True | Abst \Rightarrow False | Void \Rightarrow False]) | -(Flat _) \Rightarrow False])])) I (CHead (CHead (CHead (CSort O) (Bind Abst) -(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (clear_gen_bind Abst -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead -x2 (Bind Abbr) x3) (TLRef O) (getl_gen_O (CHead (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead x2 -(Bind Abbr) x3) H5))) in (False_ind P H17))))) H14)))))))) H10)) (\lambda -(H10: (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 -(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) -(Bind Abst) (TLRef O)) (lift (S (S (S O))) O u0) x6)))) (\lambda (e: -C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) (CHead (CHead (CHead -(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef -O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(t: T).(ty3 g e u0 t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (u0: -T).(\lambda (_: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) -(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O u0) -x6)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) -(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) -(Bind Abst) (TLRef O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda -(u0: T).(\lambda (t: T).(ty3 g e u0 t)))) P (\lambda (x7: C).(\lambda (x8: -T).(\lambda (x9: T).(\lambda (_: (pc3 (CHead (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S -O))) O x8) x6)).(\lambda (H12: (getl (S (S O)) (CHead (CHead (CHead (CSort O) -(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead -x7 (Bind Abst) x8))).(\lambda (_: (ty3 g x7 x8 x9)).(let H14 \def -(getl_gen_all (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) -(TSort O)) (CHead x7 (Bind Abst) x8) (r (Bind Abst) (S O)) (getl_gen_S (Bind -Abst) (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) -(CHead x7 (Bind Abst) x8) (TLRef O) (S O) H12)) in (ex2_ind C (\lambda (e: -C).(drop (S O) O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) -(TSort O)) e)) (\lambda (e: C).(clear e (CHead x7 (Bind Abst) x8))) P -(\lambda (x: C).(\lambda (_: (drop (S O) O (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) x)).(\lambda (_: (clear x (CHead x7 -(Bind Abst) x8))).(let H17 \def (eq_ind C (CHead x2 (Bind Abbr) x3) (\lambda -(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind b) \Rightarrow (match b in B return (\lambda (_: -B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | Void -\Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead (CHead (CHead -(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef -O)) (clear_gen_bind Abst (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind -Abst) (TSort O)) (CHead x2 (Bind Abbr) x3) (TLRef O) (getl_gen_O (CHead -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind -Abst) (TLRef O)) (CHead x2 (Bind Abbr) x3) H5))) in (False_ind P H17))))) -H14)))))))) H10)) (ty3_gen_lref g (CHead (CHead (CHead (CSort O) (Bind Abst) -(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) x6 (S (S O)) -H8))))))) (ty3_gen_bind g Abst (CHead (CHead (CHead (CSort O) (Bind Abst) -(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) -(TSort O) (THead (Bind Abst) x0 x1) H1)))))))) H3)) (\lambda (H3: (ex3_3 C T -T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 (CHead (CHead (CHead -(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef -O)) (lift (S O) O u0) x0)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: -T).(getl O (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) -(TSort O)) (Bind Abst) (TLRef O)) (CHead e (Bind Abst) u0))))) (\lambda (e: -C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t)))))).(ex3_3_ind C T T -(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 (CHead (CHead (CHead -(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef -O)) (lift (S O) O u0) x0)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: -T).(getl O (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) -(TSort O)) (Bind Abst) (TLRef O)) (CHead e (Bind Abst) u0))))) (\lambda (e: -C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t)))) P (\lambda (x2: -C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H4: (pc3 (CHead (CHead (CHead -(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef -O)) (lift (S O) O x3) x0)).(\lambda (H5: (getl O (CHead (CHead (CHead (CSort -O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) -(CHead x2 (Bind Abst) x3))).(\lambda (H6: (ty3 g x2 x3 x4)).(ex3_2_ind T T -(\lambda (t2: T).(\lambda (_: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (THead (Bind -Abst) (TLRef (S (S O))) t2) (THead (Bind Abst) x0 x1)))) (\lambda (_: -T).(\lambda (t: T).(ty3 g (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort -O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) t))) -(\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead (CHead (CHead (CHead (CSort O) -(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (Bind -Abst) (TLRef (S (S O)))) (TSort O) t2))) P (\lambda (x5: T).(\lambda (x6: -T).(\lambda (H7: (pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) -(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (THead (Bind Abst) (TLRef (S (S -O))) x5) (THead (Bind Abst) x0 x1))).(\lambda (H8: (ty3 g (CHead (CHead -(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) -(TLRef O)) (TLRef (S (S O))) x6)).(\lambda (_: (ty3 g (CHead (CHead (CHead -(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) -(TLRef O)) (Bind Abst) (TLRef (S (S O)))) (TSort O) x5)).(or_ind (ex3_3 C T T -(\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 (CHead (CHead (CHead -(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef -O)) (lift (S (S (S O))) O t) x6)))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(_: T).(getl (S (S O)) (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) -(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead e (Bind Abbr) u0))))) -(\lambda (e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t))))) (ex3_3 C -T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 (CHead (CHead -(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) -(TLRef O)) (lift (S (S (S O))) O u0) x6)))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (_: T).(getl (S (S O)) (CHead (CHead (CHead (CSort O) (Bind Abst) -(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead e (Bind Abst) -u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t))))) P -(\lambda (H10: (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: -T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) -(TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O t) x6)))) (\lambda -(e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) (CHead (CHead (CHead -(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef -O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(t: T).(ty3 g e u0 t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (_: -T).(\lambda (t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) -(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O t) x6)))) -(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) (CHead -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind -Abst) (TLRef O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (t: T).(ty3 g e u0 t)))) P (\lambda (x7: C).(\lambda (x8: -T).(\lambda (x9: T).(\lambda (_: (pc3 (CHead (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S -O))) O x9) x6)).(\lambda (H12: (getl (S (S O)) (CHead (CHead (CHead (CSort O) -(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead -x7 (Bind Abbr) x8))).(\lambda (_: (ty3 g x7 x8 x9)).(let H14 \def -(getl_gen_all (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) -(TSort O)) (CHead x7 (Bind Abbr) x8) (r (Bind Abst) (S O)) (getl_gen_S (Bind -Abst) (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) -(CHead x7 (Bind Abbr) x8) (TLRef O) (S O) H12)) in (ex2_ind C (\lambda (e: -C).(drop (S O) O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) -(TSort O)) e)) (\lambda (e: C).(clear e (CHead x7 (Bind Abbr) x8))) P -(\lambda (x: C).(\lambda (H15: (drop (S O) O (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) x)).(\lambda (H16: (clear x (CHead x7 -(Bind Abbr) x8))).(let H17 \def (f_equal C C (\lambda (e: C).(match e in C -return (\lambda (_: C).C) with [(CSort _) \Rightarrow x2 | (CHead c _ _) -\Rightarrow c])) (CHead x2 (Bind Abst) x3) (CHead (CHead (CHead (CSort O) -(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) -(clear_gen_bind Abst (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind -Abst) (TSort O)) (CHead x2 (Bind Abst) x3) (TLRef O) (getl_gen_O (CHead -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind -Abst) (TLRef O)) (CHead x2 (Bind Abst) x3) H5))) in ((let H18 \def (f_equal C -T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) -\Rightarrow x3 | (CHead _ _ t) \Rightarrow t])) (CHead x2 (Bind Abst) x3) -(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) -(Bind Abst) (TLRef O)) (clear_gen_bind Abst (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead x2 (Bind Abst) x3) (TLRef O) -(getl_gen_O (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) -(TSort O)) (Bind Abst) (TLRef O)) (CHead x2 (Bind Abst) x3) H5))) in (\lambda -(H19: (eq C x2 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) -(TSort O)))).(let H20 \def (eq_ind T x3 (\lambda (t: T).(ty3 g x2 t x4)) H6 -(TLRef O) H18) in (let H21 \def (eq_ind T x3 (\lambda (t: T).(pc3 (CHead -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind -Abst) (TLRef O)) (lift (S O) O t) x0)) H4 (TLRef O) H18) in (let H22 \def -(eq_ind C x2 (\lambda (c: C).(ty3 g c (TLRef O) x4)) H20 (CHead (CHead (CSort -O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) H19) in (let H23 \def -(eq_ind_r C x (\lambda (c: C).(clear c (CHead x7 (Bind Abbr) x8))) H16 (CHead -(CSort O) (Bind Abst) (TSort O)) (drop_gen_refl (CHead (CSort O) (Bind Abst) -(TSort O)) x (drop_gen_drop (Bind Abst) (CHead (CSort O) (Bind Abst) (TSort -O)) x (TSort O) O H15))) in (let H24 \def (eq_ind C (CHead x7 (Bind Abbr) x8) -(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind b) \Rightarrow (match b in B return (\lambda (_: -B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | Void -\Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead (CSort O) -(Bind Abst) (TSort O)) (clear_gen_bind Abst (CSort O) (CHead x7 (Bind Abbr) -x8) (TSort O) H23)) in (False_ind P H24)))))))) H17))))) H14)))))))) H10)) -(\lambda (H10: (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: -T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) -(TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O u0) x6)))) (\lambda -(e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) (CHead (CHead (CHead -(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef -O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(t: T).(ty3 g e u0 t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (u0: -T).(\lambda (_: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) -(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O u0) -x6)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) -(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) -(Bind Abst) (TLRef O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda -(u0: T).(\lambda (t: T).(ty3 g e u0 t)))) P (\lambda (x7: C).(\lambda (x8: -T).(\lambda (x9: T).(\lambda (H11: (pc3 (CHead (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S -O))) O x8) x6)).(\lambda (H12: (getl (S (S O)) (CHead (CHead (CHead (CSort O) -(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead -x7 (Bind Abst) x8))).(\lambda (H13: (ty3 g x7 x8 x9)).(let H14 \def -(getl_gen_all (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) -(TSort O)) (CHead x7 (Bind Abst) x8) (r (Bind Abst) (S O)) (getl_gen_S (Bind -Abst) (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) -(CHead x7 (Bind Abst) x8) (TLRef O) (S O) H12)) in (ex2_ind C (\lambda (e: -C).(drop (S O) O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) -(TSort O)) e)) (\lambda (e: C).(clear e (CHead x7 (Bind Abst) x8))) P -(\lambda (x: C).(\lambda (H15: (drop (S O) O (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) x)).(\lambda (H16: (clear x (CHead x7 -(Bind Abst) x8))).(let H17 \def (f_equal C C (\lambda (e: C).(match e in C -return (\lambda (_: C).C) with [(CSort _) \Rightarrow x2 | (CHead c _ _) -\Rightarrow c])) (CHead x2 (Bind Abst) x3) (CHead (CHead (CHead (CSort O) -(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) -(clear_gen_bind Abst (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind -Abst) (TSort O)) (CHead x2 (Bind Abst) x3) (TLRef O) (getl_gen_O (CHead -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind -Abst) (TLRef O)) (CHead x2 (Bind Abst) x3) H5))) in ((let H18 \def (f_equal C -T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) -\Rightarrow x3 | (CHead _ _ t) \Rightarrow t])) (CHead x2 (Bind Abst) x3) -(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) -(Bind Abst) (TLRef O)) (clear_gen_bind Abst (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead x2 (Bind Abst) x3) (TLRef O) -(getl_gen_O (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) -(TSort O)) (Bind Abst) (TLRef O)) (CHead x2 (Bind Abst) x3) H5))) in (\lambda -(H19: (eq C x2 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) -(TSort O)))).(let H20 \def (eq_ind T x3 (\lambda (t: T).(ty3 g x2 t x4)) H6 -(TLRef O) H18) in (let H21 \def (eq_ind T x3 (\lambda (t: T).(pc3 (CHead -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind -Abst) (TLRef O)) (lift (S O) O t) x0)) H4 (TLRef O) H18) in (let H22 \def -(eq_ind C x2 (\lambda (c: C).(ty3 g c (TLRef O) x4)) H20 (CHead (CHead (CSort -O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) H19) in (let H23 \def -(eq_ind_r C x (\lambda (c: C).(clear c (CHead x7 (Bind Abst) x8))) H16 (CHead -(CSort O) (Bind Abst) (TSort O)) (drop_gen_refl (CHead (CSort O) (Bind Abst) -(TSort O)) x (drop_gen_drop (Bind Abst) (CHead (CSort O) (Bind Abst) (TSort -O)) x (TSort O) O H15))) in (let H24 \def (f_equal C C (\lambda (e: C).(match -e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow x7 | (CHead c _ -_) \Rightarrow c])) (CHead x7 (Bind Abst) x8) (CHead (CSort O) (Bind Abst) -(TSort O)) (clear_gen_bind Abst (CSort O) (CHead x7 (Bind Abst) x8) (TSort O) -H23)) in ((let H25 \def (f_equal C T (\lambda (e: C).(match e in C return -(\lambda (_: C).T) with [(CSort _) \Rightarrow x8 | (CHead _ _ t) \Rightarrow -t])) (CHead x7 (Bind Abst) x8) (CHead (CSort O) (Bind Abst) (TSort O)) -(clear_gen_bind Abst (CSort O) (CHead x7 (Bind Abst) x8) (TSort O) H23)) in -(\lambda (H26: (eq C x7 (CSort O))).(let H27 \def (eq_ind T x8 (\lambda (t: -T).(ty3 g x7 t x9)) H13 (TSort O) H25) in (let H28 \def (eq_ind T x8 (\lambda -(t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) -(TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O t) x6)) H11 (TSort O) -H25) in (let H29 \def (eq_ind C x7 (\lambda (c: C).(ty3 g c (TSort O) x9)) -H27 (CSort O) H26) in (or_ind (ex3_3 C T T (\lambda (_: C).(\lambda (_: -T).(\lambda (t: T).(pc3 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind -Abst) (TSort O)) (lift (S O) O t) x4)))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (_: T).(getl O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) -(Bind Abst) (TSort O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda -(u0: T).(\lambda (t: T).(ty3 g e u0 t))))) (ex3_3 C T T (\lambda (_: -C).(\lambda (u0: T).(\lambda (_: T).(pc3 (CHead (CHead (CSort O) (Bind Abst) -(TSort O)) (Bind Abst) (TSort O)) (lift (S O) O u0) x4)))) (\lambda (e: -C).(\lambda (u0: T).(\lambda (_: T).(getl O (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead e (Bind Abst) u0))))) (\lambda -(e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t))))) P (\lambda (H30: -(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 (CHead -(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (lift (S O) O -t) x4)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl O (CHead -(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead e (Bind -Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 -t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (lift -(S O) O t) x4)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl O -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead -e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g -e u0 t)))) P (\lambda (x10: C).(\lambda (x11: T).(\lambda (x12: T).(\lambda -(_: (pc3 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort -O)) (lift (S O) O x12) x4)).(\lambda (H32: (getl O (CHead (CHead (CSort O) -(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead x10 (Bind Abbr) -x11))).(\lambda (_: (ty3 g x10 x11 x12)).(let H34 \def (eq_ind C (CHead x10 -(Bind Abbr) x11) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) -with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K -return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B return -(\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | -Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead (CHead -(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (clear_gen_bind Abst -(CHead (CSort O) (Bind Abst) (TSort O)) (CHead x10 (Bind Abbr) x11) (TSort O) -(getl_gen_O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort -O)) (CHead x10 (Bind Abbr) x11) H32))) in (False_ind P H34)))))))) H30)) -(\lambda (H30: (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: -T).(pc3 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) -(lift (S O) O u0) x4)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: -T).(getl O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort -O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(t: T).(ty3 g e u0 t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (u0: -T).(\lambda (_: T).(pc3 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind -Abst) (TSort O)) (lift (S O) O u0) x4)))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (_: T).(getl O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) -(Bind Abst) (TSort O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda -(u0: T).(\lambda (t: T).(ty3 g e u0 t)))) P (\lambda (x10: C).(\lambda (x11: -T).(\lambda (x12: T).(\lambda (H31: (pc3 (CHead (CHead (CSort O) (Bind Abst) -(TSort O)) (Bind Abst) (TSort O)) (lift (S O) O x11) x4)).(\lambda (H32: -(getl O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) -(CHead x10 (Bind Abst) x11))).(\lambda (H33: (ty3 g x10 x11 x12)).(let H34 -\def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) -with [(CSort _) \Rightarrow x10 | (CHead c _ _) \Rightarrow c])) (CHead x10 -(Bind Abst) x11) (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) -(TSort O)) (clear_gen_bind Abst (CHead (CSort O) (Bind Abst) (TSort O)) -(CHead x10 (Bind Abst) x11) (TSort O) (getl_gen_O (CHead (CHead (CSort O) -(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead x10 (Bind Abst) x11) -H32))) in ((let H35 \def (f_equal C T (\lambda (e: C).(match e in C return -(\lambda (_: C).T) with [(CSort _) \Rightarrow x11 | (CHead _ _ t) -\Rightarrow t])) (CHead x10 (Bind Abst) x11) (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) (clear_gen_bind Abst (CHead (CSort O) -(Bind Abst) (TSort O)) (CHead x10 (Bind Abst) x11) (TSort O) (getl_gen_O -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead -x10 (Bind Abst) x11) H32))) in (\lambda (H36: (eq C x10 (CHead (CSort O) -(Bind Abst) (TSort O)))).(let H37 \def (eq_ind T x11 (\lambda (t: T).(ty3 g -x10 t x12)) H33 (TSort O) H35) in (let H38 \def (eq_ind T x11 (\lambda (t: -T).(pc3 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) -(lift (S O) O t) x4)) H31 (TSort O) H35) in (let H39 \def (eq_ind C x10 -(\lambda (c: C).(ty3 g c (TSort O) x12)) H37 (CHead (CSort O) (Bind Abst) -(TSort O)) H36) in (land_ind (pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) -(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) -x0) (\forall (b: B).(\forall (u0: T).(pc3 (CHead (CHead (CHead (CHead (CSort -O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (Bind -b) u0) x5 x1))) P (\lambda (H40: (pc3 (CHead (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S -O))) x0)).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pc3 (CHead (CHead -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind -Abst) (TLRef O)) (Bind b) u0) x5 x1))))).(let H42 \def (eq_ind T (lift (S O) -O (TLRef O)) (\lambda (t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) -(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) -t)) (pc3_t x0 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind -Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) H40 (lift (S O) O -(TLRef O)) (ex2_sym T (pr3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort -O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S O) O (TLRef O))) -(pr3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort -O)) (Bind Abst) (TLRef O)) x0) H21)) (TLRef (plus O (S O))) (lift_lref_ge O -(S O) O (le_n O))) in (let H43 \def H42 in (ex2_ind T (\lambda (t: T).(pr3 -(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) -(Bind Abst) (TLRef O)) (TLRef (S (S O))) t)) (\lambda (t: T).(pr3 (CHead -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind -Abst) (TLRef O)) (TLRef (S O)) t)) P (\lambda (x13: T).(\lambda (H44: (pr3 -(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) -(Bind Abst) (TLRef O)) (TLRef (S (S O))) x13)).(\lambda (H45: (pr3 (CHead -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind -Abst) (TLRef O)) (TLRef (S O)) x13)).(let H46 \def (eq_ind_r T x13 (\lambda -(t: T).(eq T (TLRef (S (S O))) t)) (nf2_pr3_unfold (CHead (CHead (CHead -(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef -O)) (TLRef (S (S O))) x13 H44 (nf2_lref_abst (CHead (CHead (CHead (CSort O) -(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CSort -O) (TSort O) (S (S O)) (getl_clear_bind Abst (CHead (CHead (CHead (CSort O) -(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead -(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (TLRef O) -(clear_bind Abst (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) -(TSort O)) (TLRef O)) (CHead (CSort O) (Bind Abst) (TSort O)) (S O) -(getl_head (Bind Abst) O (CHead (CSort O) (Bind Abst) (TSort O)) (CHead -(CSort O) (Bind Abst) (TSort O)) (getl_refl Abst (CSort O) (TSort O)) (TSort -O))))) (TLRef (S O)) (nf2_pr3_unfold (CHead (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S O)) -x13 H45 (nf2_lref_abst (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) -(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead (CSort O) (Bind Abst) -(TSort O)) (TSort O) (S O) (getl_head (Bind Abst) O (CHead (CHead (CSort O) -(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) (getl_refl Abst (CHead (CSort O) -(Bind Abst) (TSort O)) (TSort O)) (TLRef O))))) in (let H47 \def (eq_ind T -(TLRef (S (S O))) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef n) \Rightarrow (match n -in nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S n0) -\Rightarrow (match n0 in nat return (\lambda (_: nat).Prop) with [O -\Rightarrow False | (S _) \Rightarrow True])]) | (THead _ _ _) \Rightarrow -False])) I (TLRef (S O)) H46) in (False_ind P H47)))))) H43))))) -(pc3_gen_abst (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind -Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) x0 x5 x1 H7))))))) -H34)))))))) H30)) (ty3_gen_lref g (CHead (CHead (CSort O) (Bind Abst) (TSort -O)) (Bind Abst) (TSort O)) x4 O H22))))))) H24)))))))) H17))))) H14)))))))) -H10)) (ty3_gen_lref g (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) -(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) x6 (S (S O)) H8))))))) -(ty3_gen_bind g Abst (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) -(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) (TSort O) -(THead (Bind Abst) x0 x1) H1)))))))) H3)) (ty3_gen_lref g (CHead (CHead -(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) -(TLRef O)) x0 O H2))))))) (ty3_gen_appl g (CHead (CHead (CHead (CSort O) -(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef -O) (THead (Bind Abst) (TLRef (S (S O))) (TSort O)) u H))))). -(* COMMENTS -Initial nodes: 9973 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/ex2/defs.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/ex2/defs.ma deleted file mode 100644 index 35b5df73e..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/ex2/defs.ma +++ /dev/null @@ -1,28 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/C/defs.ma". - -definition ex2_c: - C -\def - CSort O. - -definition ex2_t: - T -\def - THead (Flat Appl) (TSort O) (TSort O). - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/ex2/props.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/ex2/props.ma deleted file mode 100644 index 4dfa6a582..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/ex2/props.ma +++ /dev/null @@ -1,159 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/ex2/defs.ma". - -include "Basic-1/nf2/defs.ma". - -include "Basic-1/pr2/fwd.ma". - -include "Basic-1/arity/fwd.ma". - -theorem ex2_nf2: - nf2 ex2_c ex2_t -\def - \lambda (t2: T).(\lambda (H: (pr2 (CSort O) (THead (Flat Appl) (TSort O) -(TSort O)) t2)).(let H0 \def (pr2_gen_appl (CSort O) (TSort O) (TSort O) t2 -H) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead -(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 (CSort O) (TSort -O) u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 (CSort O) (TSort O) t3)))) -(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(eq T (TSort O) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda -(_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 -t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: -T).(pr2 (CSort O) (TSort O) u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead (CSort O) -(Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B -b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (TSort O) (THead (Bind b) y1 -z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: -T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat -Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 (CSort O) (TSort -O) u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CSort O) y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead (CSort O) (Bind b) y2) z1 z2)))))))) (eq T (THead (Flat -Appl) (TSort O) (TSort O)) t2) (\lambda (H1: (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 (CSort O) (TSort O) u2))) (\lambda (_: T).(\lambda -(t3: T).(pr2 (CSort O) (TSort O) t3))))).(ex3_2_ind T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 (CSort O) (TSort O) u2))) (\lambda (_: T).(\lambda -(t3: T).(pr2 (CSort O) (TSort O) t3))) (eq T (THead (Flat Appl) (TSort O) -(TSort O)) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H2: (eq T t2 -(THead (Flat Appl) x0 x1))).(\lambda (H3: (pr2 (CSort O) (TSort O) -x0)).(\lambda (H4: (pr2 (CSort O) (TSort O) x1)).(let H5 \def (eq_ind T x1 -(\lambda (t: T).(eq T t2 (THead (Flat Appl) x0 t))) H2 (TSort O) -(pr2_gen_sort (CSort O) x1 O H4)) in (let H6 \def (eq_ind T x0 (\lambda (t: -T).(eq T t2 (THead (Flat Appl) t (TSort O)))) H5 (TSort O) (pr2_gen_sort -(CSort O) x0 O H3)) in (eq_ind_r T (THead (Flat Appl) (TSort O) (TSort O)) -(\lambda (t: T).(eq T (THead (Flat Appl) (TSort O) (TSort O)) t)) (refl_equal -T (THead (Flat Appl) (TSort O) (TSort O))) t2 H6)))))))) H1)) (\lambda (H1: -(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(eq T (TSort O) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda -(_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 -t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: -T).(pr2 (CSort O) (TSort O) u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead (CSort O) -(Bind b) u) z1 t3))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T (TSort O) (THead (Bind Abst) y1 -z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: -T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 (CSort O) (TSort O) u2))))) (\lambda -(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: -B).(\forall (u: T).(pr2 (CHead (CSort O) (Bind b) u) z1 t3))))))) (eq T -(THead (Flat Appl) (TSort O) (TSort O)) t2) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H2: (eq T (TSort O) (THead -(Bind Abst) x0 x1))).(\lambda (H3: (eq T t2 (THead (Bind Abbr) x2 -x3))).(\lambda (H4: (pr2 (CSort O) (TSort O) x2)).(\lambda (_: ((\forall (b: -B).(\forall (u: T).(pr2 (CHead (CSort O) (Bind b) u) x1 x3))))).(let H6 \def -(eq_ind T x2 (\lambda (t: T).(eq T t2 (THead (Bind Abbr) t x3))) H3 (TSort O) -(pr2_gen_sort (CSort O) x2 O H4)) in (eq_ind_r T (THead (Bind Abbr) (TSort O) -x3) (\lambda (t: T).(eq T (THead (Flat Appl) (TSort O) (TSort O)) t)) (let H7 -\def (eq_ind T (TSort O) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | -(THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) x0 x1) H2) in -(False_ind (eq T (THead (Flat Appl) (TSort O) (TSort O)) (THead (Bind Abbr) -(TSort O) x3)) H7)) t2 H6)))))))))) H1)) (\lambda (H1: (ex6_6 B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T -(TSort O) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T -t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 (CSort O) (TSort O) u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CSort O) y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda -(z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead (CSort -O) (Bind b) y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (TSort O) -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 (CSort O) (TSort O) u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 -(CSort O) y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead (CSort O) -(Bind b) y2) z1 z2))))))) (eq T (THead (Flat Appl) (TSort O) (TSort O)) t2) -(\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda -(x4: T).(\lambda (x5: T).(\lambda (_: (not (eq B x0 Abst))).(\lambda (H3: (eq -T (TSort O) (THead (Bind x0) x1 x2))).(\lambda (H4: (eq T t2 (THead (Bind x0) -x5 (THead (Flat Appl) (lift (S O) O x4) x3)))).(\lambda (H5: (pr2 (CSort O) -(TSort O) x4)).(\lambda (H6: (pr2 (CSort O) x1 x5)).(\lambda (_: (pr2 (CHead -(CSort O) (Bind x0) x5) x2 x3)).(let H_y \def (pr2_gen_csort x1 x5 O H6) in -(let H8 \def (eq_ind T x4 (\lambda (t: T).(eq T t2 (THead (Bind x0) x5 (THead -(Flat Appl) (lift (S O) O t) x3)))) H4 (TSort O) (pr2_gen_sort (CSort O) x4 O -H5)) in (eq_ind_r T (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O -(TSort O)) x3)) (\lambda (t: T).(eq T (THead (Flat Appl) (TSort O) (TSort O)) -t)) (let H9 \def (eq_ind T (TSort O) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Bind x0) x1 -x2) H3) in (False_ind (eq T (THead (Flat Appl) (TSort O) (TSort O)) (THead -(Bind x0) x5 (THead (Flat Appl) (lift (S O) O (TSort O)) x3))) H9)) t2 -H8))))))))))))))) H1)) H0))). -(* COMMENTS -Initial nodes: 1939 -END *) - -theorem ex2_arity: - \forall (g: G).(\forall (a: A).((arity g ex2_c ex2_t a) \to (\forall (P: -Prop).P))) -\def - \lambda (g: G).(\lambda (a: A).(\lambda (H: (arity g (CSort O) (THead (Flat -Appl) (TSort O) (TSort O)) a)).(\lambda (P: Prop).(let H0 \def -(arity_gen_appl g (CSort O) (TSort O) (TSort O) a H) in (ex2_ind A (\lambda -(a1: A).(arity g (CSort O) (TSort O) a1)) (\lambda (a1: A).(arity g (CSort O) -(TSort O) (AHead a1 a))) P (\lambda (x: A).(\lambda (_: (arity g (CSort O) -(TSort O) x)).(\lambda (H2: (arity g (CSort O) (TSort O) (AHead x a))).(let -H_x \def (leq_gen_head1 g x a (ASort O O) (arity_gen_sort g (CSort O) O -(AHead x a) H2)) in (let H3 \def H_x in (ex3_2_ind A A (\lambda (a3: -A).(\lambda (_: A).(leq g x a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a -a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort O O) (AHead a3 a4)))) P -(\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g x x0)).(\lambda (_: -(leq g a x1)).(\lambda (H6: (eq A (ASort O O) (AHead x0 x1))).(let H7 \def -(eq_ind A (ASort O O) (\lambda (ee: A).(match ee in A return (\lambda (_: -A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow -False])) I (AHead x0 x1) H6) in (False_ind P H7))))))) H3)))))) H0))))). -(* COMMENTS -Initial nodes: 289 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/flt/defs.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/flt/defs.ma deleted file mode 100644 index 3191e1e38..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/flt/defs.ma +++ /dev/null @@ -1,29 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/C/defs.ma". - -definition fweight: - C \to (T \to nat) -\def - \lambda (c: C).(\lambda (t: T).(plus (cweight c) (tweight t))). - -definition flt: - C \to (T \to (C \to (T \to Prop))) -\def - \lambda (c1: C).(\lambda (t1: T).(\lambda (c2: C).(\lambda (t2: T).(lt -(fweight c1 t1) (fweight c2 t2))))). - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/flt/props.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/flt/props.ma deleted file mode 100644 index 57df41528..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/flt/props.ma +++ /dev/null @@ -1,154 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/flt/defs.ma". - -include "Basic-1/C/props.ma". - -theorem flt_thead_sx: - \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).(flt c u c -(THead k u t))))) -\def - \lambda (_: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t: -T).(le_lt_plus_plus (cweight c) (cweight c) (tweight u) (S (plus (tweight u) -(tweight t))) (le_n (cweight c)) (le_n_S (tweight u) (plus (tweight u) -(tweight t)) (le_plus_l (tweight u) (tweight t))))))). -(* COMMENTS -Initial nodes: 65 -END *) - -theorem flt_thead_dx: - \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).(flt c t c -(THead k u t))))) -\def - \lambda (_: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t: -T).(le_lt_plus_plus (cweight c) (cweight c) (tweight t) (S (plus (tweight u) -(tweight t))) (le_n (cweight c)) (le_n_S (tweight t) (plus (tweight u) -(tweight t)) (le_plus_r (tweight u) (tweight t))))))). -(* COMMENTS -Initial nodes: 65 -END *) - -theorem flt_shift: - \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).(flt (CHead c -k u) t c (THead k u t))))) -\def - \lambda (_: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(eq_ind nat -(S (plus (cweight c) (plus (tweight u) (tweight t)))) (\lambda (n: nat).(lt -(plus (plus (cweight c) (tweight u)) (tweight t)) n)) (eq_ind_r nat (plus -(plus (cweight c) (tweight u)) (tweight t)) (\lambda (n: nat).(lt (plus (plus -(cweight c) (tweight u)) (tweight t)) (S n))) (le_n (S (plus (plus (cweight -c) (tweight u)) (tweight t)))) (plus (cweight c) (plus (tweight u) (tweight -t))) (plus_assoc_l (cweight c) (tweight u) (tweight t))) (plus (cweight c) (S -(plus (tweight u) (tweight t)))) (plus_n_Sm (cweight c) (plus (tweight u) -(tweight t))))))). -(* COMMENTS -Initial nodes: 179 -END *) - -theorem flt_arith0: - \forall (k: K).(\forall (c: C).(\forall (t: T).(\forall (i: nat).(flt c t -(CHead c k t) (TLRef i))))) -\def - \lambda (_: K).(\lambda (c: C).(\lambda (t: T).(\lambda (_: -nat).(lt_x_plus_x_Sy (plus (cweight c) (tweight t)) O)))). -(* COMMENTS -Initial nodes: 21 -END *) - -theorem flt_arith1: - \forall (k1: K).(\forall (c1: C).(\forall (c2: C).(\forall (t1: T).((cle -(CHead c1 k1 t1) c2) \to (\forall (k2: K).(\forall (t2: T).(\forall (i: -nat).(flt c1 t1 (CHead c2 k2 t2) (TLRef i))))))))) -\def - \lambda (_: K).(\lambda (c1: C).(\lambda (c2: C).(\lambda (t1: T).(\lambda -(H: (le (plus (cweight c1) (tweight t1)) (cweight c2))).(\lambda (_: -K).(\lambda (t2: T).(\lambda (_: nat).(le_lt_trans (plus (cweight c1) -(tweight t1)) (cweight c2) (plus (plus (cweight c2) (tweight t2)) (S O)) H -(eq_ind_r nat (plus (S O) (plus (cweight c2) (tweight t2))) (\lambda (n: -nat).(lt (cweight c2) n)) (le_lt_n_Sm (cweight c2) (plus (cweight c2) -(tweight t2)) (le_plus_l (cweight c2) (tweight t2))) (plus (plus (cweight c2) -(tweight t2)) (S O)) (plus_sym (plus (cweight c2) (tweight t2)) (S -O))))))))))). -(* COMMENTS -Initial nodes: 151 -END *) - -theorem flt_arith2: - \forall (c1: C).(\forall (c2: C).(\forall (t1: T).(\forall (i: nat).((flt c1 -t1 c2 (TLRef i)) \to (\forall (k2: K).(\forall (t2: T).(\forall (j: nat).(flt -c1 t1 (CHead c2 k2 t2) (TLRef j))))))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (t1: T).(\lambda (_: nat).(\lambda -(H: (lt (plus (cweight c1) (tweight t1)) (plus (cweight c2) (S O)))).(\lambda -(_: K).(\lambda (t2: T).(\lambda (_: nat).(lt_le_trans (plus (cweight c1) -(tweight t1)) (plus (cweight c2) (S O)) (plus (plus (cweight c2) (tweight -t2)) (S O)) H (le_plus_plus (cweight c2) (plus (cweight c2) (tweight t2)) (S -O) (S O) (le_plus_l (cweight c2) (tweight t2)) (le_n (S O))))))))))). -(* COMMENTS -Initial nodes: 115 -END *) - -theorem flt_trans: - \forall (c1: C).(\forall (c2: C).(\forall (t1: T).(\forall (t2: T).((flt c1 -t1 c2 t2) \to (\forall (c3: C).(\forall (t3: T).((flt c2 t2 c3 t3) \to (flt -c1 t1 c3 t3)))))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H: (lt (fweight c1 t1) (fweight c2 t2))).(\lambda (c3: C).(\lambda (t3: -T).(\lambda (H0: (lt (fweight c2 t2) (fweight c3 t3))).(lt_trans (fweight c1 -t1) (fweight c2 t2) (fweight c3 t3) H H0)))))))). -(* COMMENTS -Initial nodes: 63 -END *) - -theorem flt_wf__q_ind: - \forall (P: ((C \to (T \to Prop)))).(((\forall (n: nat).((\lambda (P0: ((C -\to (T \to Prop)))).(\lambda (n0: nat).(\forall (c: C).(\forall (t: T).((eq -nat (fweight c t) n0) \to (P0 c t)))))) P n))) \to (\forall (c: C).(\forall -(t: T).(P c t)))) -\def - let Q \def (\lambda (P: ((C \to (T \to Prop)))).(\lambda (n: nat).(\forall -(c: C).(\forall (t: T).((eq nat (fweight c t) n) \to (P c t)))))) in (\lambda -(P: ((C \to (T \to Prop)))).(\lambda (H: ((\forall (n: nat).(\forall (c: -C).(\forall (t: T).((eq nat (fweight c t) n) \to (P c t))))))).(\lambda (c: -C).(\lambda (t: T).(H (fweight c t) c t (refl_equal nat (fweight c t))))))). -(* COMMENTS -Initial nodes: 85 -END *) - -theorem flt_wf_ind: - \forall (P: ((C \to (T \to Prop)))).(((\forall (c2: C).(\forall (t2: -T).(((\forall (c1: C).(\forall (t1: T).((flt c1 t1 c2 t2) \to (P c1 t1))))) -\to (P c2 t2))))) \to (\forall (c: C).(\forall (t: T).(P c t)))) -\def - let Q \def (\lambda (P: ((C \to (T \to Prop)))).(\lambda (n: nat).(\forall -(c: C).(\forall (t: T).((eq nat (fweight c t) n) \to (P c t)))))) in (\lambda -(P: ((C \to (T \to Prop)))).(\lambda (H: ((\forall (c2: C).(\forall (t2: -T).(((\forall (c1: C).(\forall (t1: T).((flt c1 t1 c2 t2) \to (P c1 t1))))) -\to (P c2 t2)))))).(\lambda (c: C).(\lambda (t: T).(flt_wf__q_ind P (\lambda -(n: nat).(lt_wf_ind n (Q P) (\lambda (n0: nat).(\lambda (H0: ((\forall (m: -nat).((lt m n0) \to (Q P m))))).(\lambda (c0: C).(\lambda (t0: T).(\lambda -(H1: (eq nat (fweight c0 t0) n0)).(let H2 \def (eq_ind_r nat n0 (\lambda (n1: -nat).(\forall (m: nat).((lt m n1) \to (\forall (c1: C).(\forall (t1: T).((eq -nat (fweight c1 t1) m) \to (P c1 t1))))))) H0 (fweight c0 t0) H1) in (H c0 t0 -(\lambda (c1: C).(\lambda (t1: T).(\lambda (H3: (flt c1 t1 c0 t0)).(H2 -(fweight c1 t1) H3 c1 t1 (refl_equal nat (fweight c1 t1))))))))))))))) c -t))))). -(* COMMENTS -Initial nodes: 211 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/fsubst0/defs.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/fsubst0/defs.ma deleted file mode 100644 index f39baebf7..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/fsubst0/defs.ma +++ /dev/null @@ -1,26 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubst0/defs.ma". - -inductive fsubst0 (i: nat) (v: T) (c1: C) (t1: T): C \to (T \to Prop) \def -| fsubst0_snd: \forall (t2: T).((subst0 i v t1 t2) \to (fsubst0 i v c1 t1 c1 -t2)) -| fsubst0_fst: \forall (c2: C).((csubst0 i v c1 c2) \to (fsubst0 i v c1 t1 c2 -t1)) -| fsubst0_both: \forall (t2: T).((subst0 i v t1 t2) \to (\forall (c2: -C).((csubst0 i v c1 c2) \to (fsubst0 i v c1 t1 c2 t2)))). - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/fsubst0/fwd.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/fsubst0/fwd.ma deleted file mode 100644 index 3c1212510..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/fsubst0/fwd.ma +++ /dev/null @@ -1,43 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/fsubst0/defs.ma". - -theorem fsubst0_gen_base: - \forall (c1: C).(\forall (c2: C).(\forall (t1: T).(\forall (t2: T).(\forall -(v: T).(\forall (i: nat).((fsubst0 i v c1 t1 c2 t2) \to (or3 (land (eq C c1 -c2) (subst0 i v t1 t2)) (land (eq T t1 t2) (csubst0 i v c1 c2)) (land (subst0 -i v t1 t2) (csubst0 i v c1 c2))))))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(v: T).(\lambda (i: nat).(\lambda (H: (fsubst0 i v c1 t1 c2 t2)).(fsubst0_ind -i v c1 t1 (\lambda (c: C).(\lambda (t: T).(or3 (land (eq C c1 c) (subst0 i v -t1 t)) (land (eq T t1 t) (csubst0 i v c1 c)) (land (subst0 i v t1 t) (csubst0 -i v c1 c))))) (\lambda (t0: T).(\lambda (H0: (subst0 i v t1 t0)).(or3_intro0 -(land (eq C c1 c1) (subst0 i v t1 t0)) (land (eq T t1 t0) (csubst0 i v c1 -c1)) (land (subst0 i v t1 t0) (csubst0 i v c1 c1)) (conj (eq C c1 c1) (subst0 -i v t1 t0) (refl_equal C c1) H0)))) (\lambda (c0: C).(\lambda (H0: (csubst0 i -v c1 c0)).(or3_intro1 (land (eq C c1 c0) (subst0 i v t1 t1)) (land (eq T t1 -t1) (csubst0 i v c1 c0)) (land (subst0 i v t1 t1) (csubst0 i v c1 c0)) (conj -(eq T t1 t1) (csubst0 i v c1 c0) (refl_equal T t1) H0)))) (\lambda (t0: -T).(\lambda (H0: (subst0 i v t1 t0)).(\lambda (c0: C).(\lambda (H1: (csubst0 -i v c1 c0)).(or3_intro2 (land (eq C c1 c0) (subst0 i v t1 t0)) (land (eq T t1 -t0) (csubst0 i v c1 c0)) (land (subst0 i v t1 t0) (csubst0 i v c1 c0)) (conj -(subst0 i v t1 t0) (csubst0 i v c1 c0) H0 H1)))))) c2 t2 H))))))). -(* COMMENTS -Initial nodes: 431 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/getl/clear.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/getl/clear.ma deleted file mode 100644 index de0a85bd3..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/getl/clear.ma +++ /dev/null @@ -1,153 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/getl/props.ma". - -include "Basic-1/clear/drop.ma". - -theorem clear_getl_trans: - \forall (i: nat).(\forall (c2: C).(\forall (c3: C).((getl i c2 c3) \to -(\forall (c1: C).((clear c1 c2) \to (getl i c1 c3)))))) -\def - \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c2: C).(\forall (c3: -C).((getl n c2 c3) \to (\forall (c1: C).((clear c1 c2) \to (getl n c1 -c3))))))) (\lambda (c2: C).(\lambda (c3: C).(\lambda (H: (getl O c2 -c3)).(\lambda (c1: C).(\lambda (H0: (clear c1 c2)).(getl_intro O c1 c3 c1 -(drop_refl c1) (clear_trans c1 c2 H0 c3 (getl_gen_O c2 c3 H)))))))) (\lambda -(n: nat).(\lambda (_: ((\forall (c2: C).(\forall (c3: C).((getl n c2 c3) \to -(\forall (c1: C).((clear c1 c2) \to (getl n c1 c3)))))))).(\lambda (c2: -C).(C_ind (\lambda (c: C).(\forall (c3: C).((getl (S n) c c3) \to (\forall -(c1: C).((clear c1 c) \to (getl (S n) c1 c3)))))) (\lambda (n0: nat).(\lambda -(c3: C).(\lambda (H0: (getl (S n) (CSort n0) c3)).(\lambda (c1: C).(\lambda -(_: (clear c1 (CSort n0))).(getl_gen_sort n0 (S n) c3 H0 (getl (S n) c1 -c3))))))) (\lambda (c: C).(\lambda (_: ((\forall (c3: C).((getl (S n) c c3) -\to (\forall (c1: C).((clear c1 c) \to (getl (S n) c1 c3))))))).(\lambda (k: -K).(\lambda (t: T).(\lambda (c3: C).(\lambda (H1: (getl (S n) (CHead c k t) -c3)).(\lambda (c1: C).(\lambda (H2: (clear c1 (CHead c k t))).(K_ind (\lambda -(k0: K).((getl (S n) (CHead c k0 t) c3) \to ((clear c1 (CHead c k0 t)) \to -(getl (S n) c1 c3)))) (\lambda (b: B).(\lambda (H3: (getl (S n) (CHead c -(Bind b) t) c3)).(\lambda (H4: (clear c1 (CHead c (Bind b) t))).(let H5 \def -(getl_gen_all c c3 (r (Bind b) n) (getl_gen_S (Bind b) c c3 t n H3)) in -(ex2_ind C (\lambda (e: C).(drop n O c e)) (\lambda (e: C).(clear e c3)) -(getl (S n) c1 c3) (\lambda (x: C).(\lambda (H6: (drop n O c x)).(\lambda -(H7: (clear x c3)).(getl_intro (S n) c1 c3 x (drop_clear_O b c1 c t H4 x n -H6) H7)))) H5))))) (\lambda (f: F).(\lambda (_: (getl (S n) (CHead c (Flat f) -t) c3)).(\lambda (H4: (clear c1 (CHead c (Flat f) t))).(clear_gen_flat_r f c1 -c t H4 (getl (S n) c1 c3))))) k H1 H2))))))))) c2)))) i). -(* COMMENTS -Initial nodes: 525 -END *) - -theorem getl_clear_trans: - \forall (i: nat).(\forall (c1: C).(\forall (c2: C).((getl i c1 c2) \to -(\forall (c3: C).((clear c2 c3) \to (getl i c1 c3)))))) -\def - \lambda (i: nat).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (getl i c1 -c2)).(\lambda (c3: C).(\lambda (H0: (clear c2 c3)).(let H1 \def (getl_gen_all -c1 c2 i H) in (ex2_ind C (\lambda (e: C).(drop i O c1 e)) (\lambda (e: -C).(clear e c2)) (getl i c1 c3) (\lambda (x: C).(\lambda (H2: (drop i O c1 -x)).(\lambda (H3: (clear x c2)).(let H4 \def (clear_gen_all x c2 H3) in -(ex_3_ind B C T (\lambda (b: B).(\lambda (e: C).(\lambda (u: T).(eq C c2 -(CHead e (Bind b) u))))) (getl i c1 c3) (\lambda (x0: B).(\lambda (x1: -C).(\lambda (x2: T).(\lambda (H5: (eq C c2 (CHead x1 (Bind x0) x2))).(let H6 -\def (eq_ind C c2 (\lambda (c: C).(clear x c)) H3 (CHead x1 (Bind x0) x2) H5) -in (let H7 \def (eq_ind C c2 (\lambda (c: C).(clear c c3)) H0 (CHead x1 (Bind -x0) x2) H5) in (eq_ind_r C (CHead x1 (Bind x0) x2) (\lambda (c: C).(getl i c1 -c)) (getl_intro i c1 (CHead x1 (Bind x0) x2) x H2 H6) c3 (clear_gen_bind x0 -x1 c3 x2 H7)))))))) H4))))) H1))))))). -(* COMMENTS -Initial nodes: 269 -END *) - -theorem getl_clear_bind: - \forall (b: B).(\forall (c: C).(\forall (e1: C).(\forall (v: T).((clear c -(CHead e1 (Bind b) v)) \to (\forall (e2: C).(\forall (n: nat).((getl n e1 e2) -\to (getl (S n) c e2)))))))) -\def - \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e1: -C).(\forall (v: T).((clear c0 (CHead e1 (Bind b) v)) \to (\forall (e2: -C).(\forall (n: nat).((getl n e1 e2) \to (getl (S n) c0 e2)))))))) (\lambda -(n: nat).(\lambda (e1: C).(\lambda (v: T).(\lambda (H: (clear (CSort n) -(CHead e1 (Bind b) v))).(\lambda (e2: C).(\lambda (n0: nat).(\lambda (_: -(getl n0 e1 e2)).(clear_gen_sort (CHead e1 (Bind b) v) n H (getl (S n0) -(CSort n) e2))))))))) (\lambda (c0: C).(\lambda (H: ((\forall (e1: -C).(\forall (v: T).((clear c0 (CHead e1 (Bind b) v)) \to (\forall (e2: -C).(\forall (n: nat).((getl n e1 e2) \to (getl (S n) c0 e2))))))))).(\lambda -(k: K).(\lambda (t: T).(\lambda (e1: C).(\lambda (v: T).(\lambda (H0: (clear -(CHead c0 k t) (CHead e1 (Bind b) v))).(\lambda (e2: C).(\lambda (n: -nat).(\lambda (H1: (getl n e1 e2)).(K_ind (\lambda (k0: K).((clear (CHead c0 -k0 t) (CHead e1 (Bind b) v)) \to (getl (S n) (CHead c0 k0 t) e2))) (\lambda -(b0: B).(\lambda (H2: (clear (CHead c0 (Bind b0) t) (CHead e1 (Bind b) -v))).(let H3 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda -(_: C).C) with [(CSort _) \Rightarrow e1 | (CHead c1 _ _) \Rightarrow c1])) -(CHead e1 (Bind b) v) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e1 -(Bind b) v) t H2)) in ((let H4 \def (f_equal C B (\lambda (e: C).(match e in -C return (\lambda (_: C).B) with [(CSort _) \Rightarrow b | (CHead _ k0 _) -\Rightarrow (match k0 in K return (\lambda (_: K).B) with [(Bind b1) -\Rightarrow b1 | (Flat _) \Rightarrow b])])) (CHead e1 (Bind b) v) (CHead c0 -(Bind b0) t) (clear_gen_bind b0 c0 (CHead e1 (Bind b) v) t H2)) in ((let H5 -\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow v | (CHead _ _ t0) \Rightarrow t0])) (CHead e1 -(Bind b) v) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e1 (Bind b) -v) t H2)) in (\lambda (H6: (eq B b b0)).(\lambda (H7: (eq C e1 c0)).(let H8 -\def (eq_ind C e1 (\lambda (c1: C).(getl n c1 e2)) H1 c0 H7) in (eq_ind B b -(\lambda (b1: B).(getl (S n) (CHead c0 (Bind b1) t) e2)) (getl_head (Bind b) -n c0 e2 H8 t) b0 H6))))) H4)) H3)))) (\lambda (f: F).(\lambda (H2: (clear -(CHead c0 (Flat f) t) (CHead e1 (Bind b) v))).(getl_flat c0 e2 (S n) (H e1 v -(clear_gen_flat f c0 (CHead e1 (Bind b) v) t H2) e2 n H1) f t))) k -H0))))))))))) c)). -(* COMMENTS -Initial nodes: 599 -END *) - -theorem getl_clear_conf: - \forall (i: nat).(\forall (c1: C).(\forall (c3: C).((getl i c1 c3) \to -(\forall (c2: C).((clear c1 c2) \to (getl i c2 c3)))))) -\def - \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (c3: -C).((getl n c1 c3) \to (\forall (c2: C).((clear c1 c2) \to (getl n c2 -c3))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H: (getl O c1 -c3)).(\lambda (c2: C).(\lambda (H0: (clear c1 c2)).(eq_ind C c3 (\lambda (c: -C).(getl O c c3)) (let H1 \def (clear_gen_all c1 c3 (getl_gen_O c1 c3 H)) in -(ex_3_ind B C T (\lambda (b: B).(\lambda (e: C).(\lambda (u: T).(eq C c3 -(CHead e (Bind b) u))))) (getl O c3 c3) (\lambda (x0: B).(\lambda (x1: -C).(\lambda (x2: T).(\lambda (H2: (eq C c3 (CHead x1 (Bind x0) x2))).(let H3 -\def (eq_ind C c3 (\lambda (c: C).(clear c1 c)) (getl_gen_O c1 c3 H) (CHead -x1 (Bind x0) x2) H2) in (eq_ind_r C (CHead x1 (Bind x0) x2) (\lambda (c: -C).(getl O c c)) (getl_refl x0 x1 x2) c3 H2)))))) H1)) c2 (clear_mono c1 c3 -(getl_gen_O c1 c3 H) c2 H0))))))) (\lambda (n: nat).(\lambda (_: ((\forall -(c1: C).(\forall (c3: C).((getl n c1 c3) \to (\forall (c2: C).((clear c1 c2) -\to (getl n c2 c3)))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall -(c3: C).((getl (S n) c c3) \to (\forall (c2: C).((clear c c2) \to (getl (S n) -c2 c3)))))) (\lambda (n0: nat).(\lambda (c3: C).(\lambda (H0: (getl (S n) -(CSort n0) c3)).(\lambda (c2: C).(\lambda (_: (clear (CSort n0) -c2)).(getl_gen_sort n0 (S n) c3 H0 (getl (S n) c2 c3))))))) (\lambda (c: -C).(\lambda (H0: ((\forall (c3: C).((getl (S n) c c3) \to (\forall (c2: -C).((clear c c2) \to (getl (S n) c2 c3))))))).(\lambda (k: K).(\lambda (t: -T).(\lambda (c3: C).(\lambda (H1: (getl (S n) (CHead c k t) c3)).(\lambda -(c2: C).(\lambda (H2: (clear (CHead c k t) c2)).(K_ind (\lambda (k0: -K).((getl (S n) (CHead c k0 t) c3) \to ((clear (CHead c k0 t) c2) \to (getl -(S n) c2 c3)))) (\lambda (b: B).(\lambda (H3: (getl (S n) (CHead c (Bind b) -t) c3)).(\lambda (H4: (clear (CHead c (Bind b) t) c2)).(eq_ind_r C (CHead c -(Bind b) t) (\lambda (c0: C).(getl (S n) c0 c3)) (getl_head (Bind b) n c c3 -(getl_gen_S (Bind b) c c3 t n H3) t) c2 (clear_gen_bind b c c2 t H4))))) -(\lambda (f: F).(\lambda (H3: (getl (S n) (CHead c (Flat f) t) c3)).(\lambda -(H4: (clear (CHead c (Flat f) t) c2)).(H0 c3 (getl_gen_S (Flat f) c c3 t n -H3) c2 (clear_gen_flat f c c2 t H4))))) k H1 H2))))))))) c1)))) i). -(* COMMENTS -Initial nodes: 641 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/getl/dec.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/getl/dec.ma deleted file mode 100644 index 278362590..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/getl/dec.ma +++ /dev/null @@ -1,100 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/getl/props.ma". - -theorem getl_dec: - \forall (c: C).(\forall (i: nat).(or (ex_3 C B T (\lambda (e: C).(\lambda -(b: B).(\lambda (v: T).(getl i c (CHead e (Bind b) v)))))) (\forall (d: -C).((getl i c d) \to (\forall (P: Prop).P))))) -\def - \lambda (c: C).(C_ind (\lambda (c0: C).(\forall (i: nat).(or (ex_3 C B T -(\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl i c0 (CHead e (Bind b) -v)))))) (\forall (d: C).((getl i c0 d) \to (\forall (P: Prop).P)))))) -(\lambda (n: nat).(\lambda (i: nat).(or_intror (ex_3 C B T (\lambda (e: -C).(\lambda (b: B).(\lambda (v: T).(getl i (CSort n) (CHead e (Bind b) -v)))))) (\forall (d: C).((getl i (CSort n) d) \to (\forall (P: Prop).P))) -(\lambda (d: C).(\lambda (H: (getl i (CSort n) d)).(\lambda (P: -Prop).(getl_gen_sort n i d H P))))))) (\lambda (c0: C).(\lambda (H: ((\forall -(i: nat).(or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: -T).(getl i c0 (CHead e (Bind b) v)))))) (\forall (d: C).((getl i c0 d) \to -(\forall (P: Prop).P))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (i: -nat).(nat_ind (\lambda (n: nat).(or (ex_3 C B T (\lambda (e: C).(\lambda (b: -B).(\lambda (v: T).(getl n (CHead c0 k t) (CHead e (Bind b) v)))))) (\forall -(d: C).((getl n (CHead c0 k t) d) \to (\forall (P: Prop).P))))) (K_ind -(\lambda (k0: K).(or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: -T).(getl O (CHead c0 k0 t) (CHead e (Bind b) v)))))) (\forall (d: C).((getl O -(CHead c0 k0 t) d) \to (\forall (P: Prop).P))))) (\lambda (b: B).(or_introl -(ex_3 C B T (\lambda (e: C).(\lambda (b0: B).(\lambda (v: T).(getl O (CHead -c0 (Bind b) t) (CHead e (Bind b0) v)))))) (\forall (d: C).((getl O (CHead c0 -(Bind b) t) d) \to (\forall (P: Prop).P))) (ex_3_intro C B T (\lambda (e: -C).(\lambda (b0: B).(\lambda (v: T).(getl O (CHead c0 (Bind b) t) (CHead e -(Bind b0) v))))) c0 b t (getl_refl b c0 t)))) (\lambda (f: F).(let H_x \def -(H O) in (let H0 \def H_x in (or_ind (ex_3 C B T (\lambda (e: C).(\lambda (b: -B).(\lambda (v: T).(getl O c0 (CHead e (Bind b) v)))))) (\forall (d: -C).((getl O c0 d) \to (\forall (P: Prop).P))) (or (ex_3 C B T (\lambda (e: -C).(\lambda (b: B).(\lambda (v: T).(getl O (CHead c0 (Flat f) t) (CHead e -(Bind b) v)))))) (\forall (d: C).((getl O (CHead c0 (Flat f) t) d) \to -(\forall (P: Prop).P)))) (\lambda (H1: (ex_3 C B T (\lambda (e: C).(\lambda -(b: B).(\lambda (v: T).(getl O c0 (CHead e (Bind b) v))))))).(ex_3_ind C B T -(\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl O c0 (CHead e (Bind b) -v))))) (or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl -O (CHead c0 (Flat f) t) (CHead e (Bind b) v)))))) (\forall (d: C).((getl O -(CHead c0 (Flat f) t) d) \to (\forall (P: Prop).P)))) (\lambda (x0: -C).(\lambda (x1: B).(\lambda (x2: T).(\lambda (H2: (getl O c0 (CHead x0 (Bind -x1) x2))).(or_introl (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: -T).(getl O (CHead c0 (Flat f) t) (CHead e (Bind b) v)))))) (\forall (d: -C).((getl O (CHead c0 (Flat f) t) d) \to (\forall (P: Prop).P))) (ex_3_intro -C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl O (CHead c0 (Flat -f) t) (CHead e (Bind b) v))))) x0 x1 x2 (getl_flat c0 (CHead x0 (Bind x1) x2) -O H2 f t))))))) H1)) (\lambda (H1: ((\forall (d: C).((getl O c0 d) \to -(\forall (P: Prop).P))))).(or_intror (ex_3 C B T (\lambda (e: C).(\lambda (b: -B).(\lambda (v: T).(getl O (CHead c0 (Flat f) t) (CHead e (Bind b) v)))))) -(\forall (d: C).((getl O (CHead c0 (Flat f) t) d) \to (\forall (P: Prop).P))) -(\lambda (d: C).(\lambda (H2: (getl O (CHead c0 (Flat f) t) d)).(\lambda (P: -Prop).(H1 d (getl_intro O c0 d c0 (drop_refl c0) (clear_gen_flat f c0 d t -(getl_gen_O (CHead c0 (Flat f) t) d H2))) P)))))) H0)))) k) (\lambda (n: -nat).(\lambda (_: (or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda -(v: T).(getl n (CHead c0 k t) (CHead e (Bind b) v)))))) (\forall (d: -C).((getl n (CHead c0 k t) d) \to (\forall (P: Prop).P))))).(let H_x \def (H -(r k n)) in (let H1 \def H_x in (or_ind (ex_3 C B T (\lambda (e: C).(\lambda -(b: B).(\lambda (v: T).(getl (r k n) c0 (CHead e (Bind b) v)))))) (\forall -(d: C).((getl (r k n) c0 d) \to (\forall (P: Prop).P))) (or (ex_3 C B T -(\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl (S n) (CHead c0 k t) -(CHead e (Bind b) v)))))) (\forall (d: C).((getl (S n) (CHead c0 k t) d) \to -(\forall (P: Prop).P)))) (\lambda (H2: (ex_3 C B T (\lambda (e: C).(\lambda -(b: B).(\lambda (v: T).(getl (r k n) c0 (CHead e (Bind b) v))))))).(ex_3_ind -C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl (r k n) c0 (CHead -e (Bind b) v))))) (or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda -(v: T).(getl (S n) (CHead c0 k t) (CHead e (Bind b) v)))))) (\forall (d: -C).((getl (S n) (CHead c0 k t) d) \to (\forall (P: Prop).P)))) (\lambda (x0: -C).(\lambda (x1: B).(\lambda (x2: T).(\lambda (H3: (getl (r k n) c0 (CHead x0 -(Bind x1) x2))).(or_introl (ex_3 C B T (\lambda (e: C).(\lambda (b: -B).(\lambda (v: T).(getl (S n) (CHead c0 k t) (CHead e (Bind b) v)))))) -(\forall (d: C).((getl (S n) (CHead c0 k t) d) \to (\forall (P: Prop).P))) -(ex_3_intro C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl (S n) -(CHead c0 k t) (CHead e (Bind b) v))))) x0 x1 x2 (getl_head k n c0 (CHead x0 -(Bind x1) x2) H3 t))))))) H2)) (\lambda (H2: ((\forall (d: C).((getl (r k n) -c0 d) \to (\forall (P: Prop).P))))).(or_intror (ex_3 C B T (\lambda (e: -C).(\lambda (b: B).(\lambda (v: T).(getl (S n) (CHead c0 k t) (CHead e (Bind -b) v)))))) (\forall (d: C).((getl (S n) (CHead c0 k t) d) \to (\forall (P: -Prop).P))) (\lambda (d: C).(\lambda (H3: (getl (S n) (CHead c0 k t) -d)).(\lambda (P: Prop).(H2 d (getl_gen_S k c0 d t n H3) P)))))) H1))))) -i)))))) c). -(* COMMENTS -Initial nodes: 1563 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/getl/defs.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/getl/defs.ma deleted file mode 100644 index a6efd2eee..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/getl/defs.ma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/drop/defs.ma". - -include "Basic-1/clear/defs.ma". - -inductive getl (h: nat) (c1: C) (c2: C): Prop \def -| getl_intro: \forall (e: C).((drop h O c1 e) \to ((clear e c2) \to (getl h -c1 c2))). - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/getl/drop.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/getl/drop.ma deleted file mode 100644 index e4404e137..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/getl/drop.ma +++ /dev/null @@ -1,514 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/getl/props.ma". - -include "Basic-1/clear/drop.ma". - -theorem getl_drop: - \forall (b: B).(\forall (c: C).(\forall (e: C).(\forall (u: T).(\forall (h: -nat).((getl h c (CHead e (Bind b) u)) \to (drop (S h) O c e)))))) -\def - \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e: -C).(\forall (u: T).(\forall (h: nat).((getl h c0 (CHead e (Bind b) u)) \to -(drop (S h) O c0 e)))))) (\lambda (n: nat).(\lambda (e: C).(\lambda (u: -T).(\lambda (h: nat).(\lambda (H: (getl h (CSort n) (CHead e (Bind b) -u))).(getl_gen_sort n h (CHead e (Bind b) u) H (drop (S h) O (CSort n) -e))))))) (\lambda (c0: C).(\lambda (H: ((\forall (e: C).(\forall (u: -T).(\forall (h: nat).((getl h c0 (CHead e (Bind b) u)) \to (drop (S h) O c0 -e))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e: C).(\lambda (u: -T).(\lambda (h: nat).(nat_ind (\lambda (n: nat).((getl n (CHead c0 k t) -(CHead e (Bind b) u)) \to (drop (S n) O (CHead c0 k t) e))) (\lambda (H0: -(getl O (CHead c0 k t) (CHead e (Bind b) u))).(K_ind (\lambda (k0: K).((clear -(CHead c0 k0 t) (CHead e (Bind b) u)) \to (drop (S O) O (CHead c0 k0 t) e))) -(\lambda (b0: B).(\lambda (H1: (clear (CHead c0 (Bind b0) t) (CHead e (Bind -b) u))).(let H2 \def (f_equal C C (\lambda (e0: C).(match e0 in C return -(\lambda (_: C).C) with [(CSort _) \Rightarrow e | (CHead c1 _ _) \Rightarrow -c1])) (CHead e (Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 -(CHead e (Bind b) u) t H1)) in ((let H3 \def (f_equal C B (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow b | -(CHead _ k0 _) \Rightarrow (match k0 in K return (\lambda (_: K).B) with -[(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow b])])) (CHead e (Bind b) u) -(CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e (Bind b) u) t H1)) in -((let H4 \def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda -(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) -(CHead e (Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e -(Bind b) u) t H1)) in (\lambda (H5: (eq B b b0)).(\lambda (H6: (eq C e -c0)).(eq_ind_r C c0 (\lambda (c1: C).(drop (S O) O (CHead c0 (Bind b0) t) -c1)) (eq_ind B b (\lambda (b1: B).(drop (S O) O (CHead c0 (Bind b1) t) c0)) -(drop_drop (Bind b) O c0 c0 (drop_refl c0) t) b0 H5) e H6)))) H3)) H2)))) -(\lambda (f: F).(\lambda (H1: (clear (CHead c0 (Flat f) t) (CHead e (Bind b) -u))).(drop_clear_O b (CHead c0 (Flat f) t) e u (clear_flat c0 (CHead e (Bind -b) u) (clear_gen_flat f c0 (CHead e (Bind b) u) t H1) f t) e O (drop_refl -e)))) k (getl_gen_O (CHead c0 k t) (CHead e (Bind b) u) H0))) (\lambda (n: -nat).(\lambda (_: (((getl n (CHead c0 k t) (CHead e (Bind b) u)) \to (drop (S -n) O (CHead c0 k t) e)))).(\lambda (H1: (getl (S n) (CHead c0 k t) (CHead e -(Bind b) u))).(drop_drop k (S n) c0 e (eq_ind_r nat (S (r k n)) (\lambda (n0: -nat).(drop n0 O c0 e)) (H e u (r k n) (getl_gen_S k c0 (CHead e (Bind b) u) t -n H1)) (r k (S n)) (r_S k n)) t)))) h)))))))) c)). -(* COMMENTS -Initial nodes: 827 -END *) - -theorem getl_drop_conf_lt: - \forall (b: B).(\forall (c: C).(\forall (c0: C).(\forall (u: T).(\forall (i: -nat).((getl i c (CHead c0 (Bind b) u)) \to (\forall (e: C).(\forall (h: -nat).(\forall (d: nat).((drop h (S (plus i d)) c e) \to (ex3_2 T C (\lambda -(v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: -C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop -h d c0 e0))))))))))))) -\def - \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (c1: -C).(\forall (u: T).(\forall (i: nat).((getl i c0 (CHead c1 (Bind b) u)) \to -(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h (S (plus i d)) -c0 e) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) -(\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda -(_: T).(\lambda (e0: C).(drop h d c1 e0))))))))))))) (\lambda (n: -nat).(\lambda (c0: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H: (getl i -(CSort n) (CHead c0 (Bind b) u))).(\lambda (e: C).(\lambda (h: nat).(\lambda -(d: nat).(\lambda (_: (drop h (S (plus i d)) (CSort n) e)).(getl_gen_sort n i -(CHead c0 (Bind b) u) H (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u -(lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) -v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c0 e0)))))))))))))) (\lambda -(c0: C).(\lambda (H: ((\forall (c1: C).(\forall (u: T).(\forall (i: -nat).((getl i c0 (CHead c1 (Bind b) u)) \to (\forall (e: C).(\forall (h: -nat).(\forall (d: nat).((drop h (S (plus i d)) c0 e) \to (ex3_2 T C (\lambda -(v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: -C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop -h d c1 e0)))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c1: -C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i (CHead c0 k t) -(CHead c1 (Bind b) u))).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: -nat).(\lambda (H1: (drop h (S (plus i d)) (CHead c0 k t) e)).(let H2 \def -(getl_gen_all (CHead c0 k t) (CHead c1 (Bind b) u) i H0) in (ex2_ind C -(\lambda (e0: C).(drop i O (CHead c0 k t) e0)) (\lambda (e0: C).(clear e0 -(CHead c1 (Bind b) u))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u -(lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) -v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x: -C).(\lambda (H3: (drop i O (CHead c0 k t) x)).(\lambda (H4: (clear x (CHead -c1 (Bind b) u))).(C_ind (\lambda (c2: C).((drop i O (CHead c0 k t) c2) \to -((clear c2 (CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: -C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead -e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) -(\lambda (n: nat).(\lambda (_: (drop i O (CHead c0 k t) (CSort n))).(\lambda -(H6: (clear (CSort n) (CHead c1 (Bind b) u))).(clear_gen_sort (CHead c1 (Bind -b) u) n H6 (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) -(\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda -(_: T).(\lambda (e0: C).(drop h d c1 e0)))))))) (\lambda (x0: C).(\lambda -(IHx: (((drop i O (CHead c0 k t) x0) \to ((clear x0 (CHead c1 (Bind b) u)) -\to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) -(\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda -(_: T).(\lambda (e0: C).(drop h d c1 e0)))))))).(\lambda (k0: K).(\lambda -(t0: T).(\lambda (H5: (drop i O (CHead c0 k t) (CHead x0 k0 t0))).(\lambda -(H6: (clear (CHead x0 k0 t0) (CHead c1 (Bind b) u))).(K_ind (\lambda (k1: -K).((drop i O (CHead c0 k t) (CHead x0 k1 t0)) \to ((clear (CHead x0 k1 t0) -(CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u -(lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) -v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) (\lambda (b0: -B).(\lambda (H7: (drop i O (CHead c0 k t) (CHead x0 (Bind b0) t0))).(\lambda -(H8: (clear (CHead x0 (Bind b0) t0) (CHead c1 (Bind b) u))).(let H9 \def -(f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with -[(CSort _) \Rightarrow c1 | (CHead c2 _ _) \Rightarrow c2])) (CHead c1 (Bind -b) u) (CHead x0 (Bind b0) t0) (clear_gen_bind b0 x0 (CHead c1 (Bind b) u) t0 -H8)) in ((let H10 \def (f_equal C B (\lambda (e0: C).(match e0 in C return -(\lambda (_: C).B) with [(CSort _) \Rightarrow b | (CHead _ k1 _) \Rightarrow -(match k1 in K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | -(Flat _) \Rightarrow b])])) (CHead c1 (Bind b) u) (CHead x0 (Bind b0) t0) -(clear_gen_bind b0 x0 (CHead c1 (Bind b) u) t0 H8)) in ((let H11 \def -(f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow u | (CHead _ _ t1) \Rightarrow t1])) (CHead c1 (Bind -b) u) (CHead x0 (Bind b0) t0) (clear_gen_bind b0 x0 (CHead c1 (Bind b) u) t0 -H8)) in (\lambda (H12: (eq B b b0)).(\lambda (H13: (eq C c1 x0)).(let H14 -\def (eq_ind_r T t0 (\lambda (t1: T).(drop i O (CHead c0 k t) (CHead x0 (Bind -b0) t1))) H7 u H11) in (let H15 \def (eq_ind_r B b0 (\lambda (b1: B).(drop i -O (CHead c0 k t) (CHead x0 (Bind b1) u))) H14 b H12) in (let H16 \def -(eq_ind_r C x0 (\lambda (c2: C).((drop i O (CHead c0 k t) c2) \to ((clear c2 -(CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u -(lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) -v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) IHx c1 H13) in -(let H17 \def (eq_ind_r C x0 (\lambda (c2: C).(drop i O (CHead c0 k t) (CHead -c2 (Bind b) u))) H15 c1 H13) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: -C).(eq T u (lift h (r (Bind b) d) v)))) (\lambda (v: T).(\lambda (e0: -C).(drop i O e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: -C).(drop h (r (Bind b) d) c1 e0))) (ex3_2 T C (\lambda (v: T).(\lambda (_: -C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead -e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))) -(\lambda (x1: T).(\lambda (x2: C).(\lambda (H18: (eq T u (lift h (r (Bind b) -d) x1))).(\lambda (H19: (drop i O e (CHead x2 (Bind b) x1))).(\lambda (H20: -(drop h (r (Bind b) d) c1 x2)).(let H21 \def (eq_ind T u (\lambda (t1: -T).((drop i O (CHead c0 k t) c1) \to ((clear c1 (CHead c1 (Bind b) t1)) \to -(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T t1 (lift h d v)))) (\lambda -(v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: -T).(\lambda (e0: C).(drop h d c1 e0))))))) H16 (lift h (r (Bind b) d) x1) -H18) in (eq_ind_r T (lift h (r (Bind b) d) x1) (\lambda (t1: T).(ex3_2 T C -(\lambda (v: T).(\lambda (_: C).(eq T t1 (lift h d v)))) (\lambda (v: -T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: -T).(\lambda (e0: C).(drop h d c1 e0))))) (ex3_2_intro T C (\lambda (v: -T).(\lambda (_: C).(eq T (lift h (r (Bind b) d) x1) (lift h d v)))) (\lambda -(v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: -T).(\lambda (e0: C).(drop h d c1 e0))) x1 x2 (refl_equal T (lift h d x1)) -(getl_intro i e (CHead x2 (Bind b) x1) (CHead x2 (Bind b) x1) H19 (clear_bind -b x2 x1)) H20) u H18))))))) (drop_conf_lt (Bind b) i u c1 (CHead c0 k t) H17 -e h d H1))))))))) H10)) H9))))) (\lambda (f: F).(\lambda (H7: (drop i O -(CHead c0 k t) (CHead x0 (Flat f) t0))).(\lambda (H8: (clear (CHead x0 (Flat -f) t0) (CHead c1 (Bind b) u))).(nat_ind (\lambda (n: nat).((drop h (S (plus n -d)) (CHead c0 k t) e) \to ((drop n O (CHead c0 k t) (CHead x0 (Flat f) t0)) -\to ((((drop n O (CHead c0 k t) x0) \to ((clear x0 (CHead c1 (Bind b) u)) \to -(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda -(v: T).(\lambda (e0: C).(getl n e (CHead e0 (Bind b) v)))) (\lambda (_: -T).(\lambda (e0: C).(drop h d c1 e0))))))) \to (ex3_2 T C (\lambda (v: -T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: -C).(getl n e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop -h d c1 e0)))))))) (\lambda (H9: (drop h (S (plus O d)) (CHead c0 k t) -e)).(\lambda (H10: (drop O O (CHead c0 k t) (CHead x0 (Flat f) t0))).(\lambda -(IHx0: (((drop O O (CHead c0 k t) x0) \to ((clear x0 (CHead c1 (Bind b) u)) -\to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) -(\lambda (v: T).(\lambda (e0: C).(getl O e (CHead e0 (Bind b) v)))) (\lambda -(_: T).(\lambda (e0: C).(drop h d c1 e0)))))))).(let H11 \def (f_equal C C -(\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow c0 | (CHead c2 _ _) \Rightarrow c2])) (CHead c0 k t) (CHead x0 -(Flat f) t0) (drop_gen_refl (CHead c0 k t) (CHead x0 (Flat f) t0) H10)) in -((let H12 \def (f_equal C K (\lambda (e0: C).(match e0 in C return (\lambda -(_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k1 _) \Rightarrow k1])) -(CHead c0 k t) (CHead x0 (Flat f) t0) (drop_gen_refl (CHead c0 k t) (CHead x0 -(Flat f) t0) H10)) in ((let H13 \def (f_equal C T (\lambda (e0: C).(match e0 -in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow t | (CHead _ _ t1) -\Rightarrow t1])) (CHead c0 k t) (CHead x0 (Flat f) t0) (drop_gen_refl (CHead -c0 k t) (CHead x0 (Flat f) t0) H10)) in (\lambda (H14: (eq K k (Flat -f))).(\lambda (H15: (eq C c0 x0)).(let H16 \def (eq_ind_r C x0 (\lambda (c2: -C).(clear c2 (CHead c1 (Bind b) u))) (clear_gen_flat f x0 (CHead c1 (Bind b) -u) t0 H8) c0 H15) in (let H17 \def (eq_ind_r C x0 (\lambda (c2: C).((drop O O -(CHead c0 k t) c2) \to ((clear c2 (CHead c1 (Bind b) u)) \to (ex3_2 T C -(\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: -T).(\lambda (e0: C).(getl O e (CHead e0 (Bind b) v)))) (\lambda (_: -T).(\lambda (e0: C).(drop h d c1 e0))))))) IHx0 c0 H15) in (let H18 \def -(eq_ind K k (\lambda (k1: K).((drop O O (CHead c0 k1 t) c0) \to ((clear c0 -(CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u -(lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl O e (CHead e0 (Bind b) -v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) H17 (Flat f) -H14) in (let H19 \def (eq_ind K k (\lambda (k1: K).(drop h (S (plus O d)) -(CHead c0 k1 t) e)) H9 (Flat f) H14) in (ex3_2_ind C T (\lambda (e0: -C).(\lambda (v: T).(eq C e (CHead e0 (Flat f) v)))) (\lambda (_: C).(\lambda -(v: T).(eq T t (lift h (r (Flat f) (plus O d)) v)))) (\lambda (e0: -C).(\lambda (_: T).(drop h (r (Flat f) (plus O d)) c0 e0))) (ex3_2 T C -(\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: -T).(\lambda (e0: C).(getl O e (CHead e0 (Bind b) v)))) (\lambda (_: -T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x1: C).(\lambda (x2: -T).(\lambda (H20: (eq C e (CHead x1 (Flat f) x2))).(\lambda (H21: (eq T t -(lift h (r (Flat f) (plus O d)) x2))).(\lambda (H22: (drop h (r (Flat f) -(plus O d)) c0 x1)).(let H23 \def (f_equal T T (\lambda (e0: T).e0) t (lift h -(r (Flat f) (plus O d)) x2) H21) in (let H24 \def (eq_ind C e (\lambda (c2: -C).((drop O O (CHead c0 (Flat f) t) c0) \to ((clear c0 (CHead c1 (Bind b) u)) -\to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) -(\lambda (v: T).(\lambda (e0: C).(getl O c2 (CHead e0 (Bind b) v)))) (\lambda -(_: T).(\lambda (e0: C).(drop h d c1 e0))))))) H18 (CHead x1 (Flat f) x2) -H20) in (eq_ind_r C (CHead x1 (Flat f) x2) (\lambda (c2: C).(ex3_2 T C -(\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: -T).(\lambda (e0: C).(getl O c2 (CHead e0 (Bind b) v)))) (\lambda (_: -T).(\lambda (e0: C).(drop h d c1 e0))))) (let H25 \def (eq_ind T t (\lambda -(t1: T).((drop O O (CHead c0 (Flat f) t1) c0) \to ((clear c0 (CHead c1 (Bind -b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) -(\lambda (v: T).(\lambda (e0: C).(getl O (CHead x1 (Flat f) x2) (CHead e0 -(Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) H24 -(lift h (S d) x2) H23) in (let H26 \def (H c1 u O (getl_intro O c0 (CHead c1 -(Bind b) u) c0 (drop_refl c0) H16) x1 h d H22) in (ex3_2_ind T C (\lambda (v: -T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: -C).(getl O x1 (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop -h d c1 e0))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d -v)))) (\lambda (v: T).(\lambda (e0: C).(getl O (CHead x1 (Flat f) x2) (CHead -e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))) -(\lambda (x3: T).(\lambda (x4: C).(\lambda (H27: (eq T u (lift h d -x3))).(\lambda (H28: (getl O x1 (CHead x4 (Bind b) x3))).(\lambda (H29: (drop -h d c1 x4)).(let H30 \def (eq_ind T u (\lambda (t1: T).((drop O O (CHead c0 -(Flat f) (lift h (S d) x2)) c0) \to ((clear c0 (CHead c1 (Bind b) t1)) \to -(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T t1 (lift h d v)))) (\lambda -(v: T).(\lambda (e0: C).(getl O (CHead x1 (Flat f) x2) (CHead e0 (Bind b) -v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) H25 (lift h d -x3) H27) in (let H31 \def (eq_ind T u (\lambda (t1: T).(clear c0 (CHead c1 -(Bind b) t1))) H16 (lift h d x3) H27) in (eq_ind_r T (lift h d x3) (\lambda -(t1: T).(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T t1 (lift h d v)))) -(\lambda (v: T).(\lambda (e0: C).(getl O (CHead x1 (Flat f) x2) (CHead e0 -(Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))) -(ex3_2_intro T C (\lambda (v: T).(\lambda (_: C).(eq T (lift h d x3) (lift h -d v)))) (\lambda (v: T).(\lambda (e0: C).(getl O (CHead x1 (Flat f) x2) -(CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))) -x3 x4 (refl_equal T (lift h d x3)) (getl_flat x1 (CHead x4 (Bind b) x3) O H28 -f x2) H29) u H27)))))))) H26))) e H20)))))))) (drop_gen_skip_l c0 e t h (plus -O d) (Flat f) H19))))))))) H12)) H11))))) (\lambda (i0: nat).(\lambda (IHi: -(((drop h (S (plus i0 d)) (CHead c0 k t) e) \to ((drop i0 O (CHead c0 k t) -(CHead x0 (Flat f) t0)) \to ((((drop i0 O (CHead c0 k t) x0) \to ((clear x0 -(CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u -(lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i0 e (CHead e0 (Bind -b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) \to (ex3_2 T -C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: -T).(\lambda (e0: C).(getl i0 e (CHead e0 (Bind b) v)))) (\lambda (_: -T).(\lambda (e0: C).(drop h d c1 e0))))))))).(\lambda (H9: (drop h (S (plus -(S i0) d)) (CHead c0 k t) e)).(\lambda (H10: (drop (S i0) O (CHead c0 k t) -(CHead x0 (Flat f) t0))).(\lambda (IHx0: (((drop (S i0) O (CHead c0 k t) x0) -\to ((clear x0 (CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda -(_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl (S i0) -e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 -e0)))))))).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 -k v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r k (plus (S i0) d)) -v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r k (plus (S i0) d)) c0 e0))) -(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda -(v: T).(\lambda (e0: C).(getl (S i0) e (CHead e0 (Bind b) v)))) (\lambda (_: -T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x1: C).(\lambda (x2: -T).(\lambda (H11: (eq C e (CHead x1 k x2))).(\lambda (H12: (eq T t (lift h (r -k (plus (S i0) d)) x2))).(\lambda (H13: (drop h (r k (plus (S i0) d)) c0 -x1)).(let H14 \def (f_equal T T (\lambda (e0: T).e0) t (lift h (r k (plus (S -i0) d)) x2) H12) in (let H15 \def (eq_ind C e (\lambda (c2: C).((drop h (S -(plus i0 d)) (CHead c0 k t) c2) \to ((drop i0 O (CHead c0 k t) (CHead x0 -(Flat f) t0)) \to ((((drop i0 O (CHead c0 k t) x0) \to ((clear x0 (CHead c1 -(Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d -v)))) (\lambda (v: T).(\lambda (e0: C).(getl i0 c2 (CHead e0 (Bind b) v)))) -(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) \to (ex3_2 T C -(\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: -T).(\lambda (e0: C).(getl i0 c2 (CHead e0 (Bind b) v)))) (\lambda (_: -T).(\lambda (e0: C).(drop h d c1 e0)))))))) IHi (CHead x1 k x2) H11) in (let -H16 \def (eq_ind C e (\lambda (c2: C).((drop (S i0) O (CHead c0 k t) x0) \to -((clear x0 (CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: -C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl (S i0) c2 -(CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 -e0))))))) IHx0 (CHead x1 k x2) H11) in (eq_ind_r C (CHead x1 k x2) (\lambda -(c2: C).(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) -(\lambda (v: T).(\lambda (e0: C).(getl (S i0) c2 (CHead e0 (Bind b) v)))) -(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))) (let H17 \def (eq_ind T -t (\lambda (t1: T).((drop (S i0) O (CHead c0 k t1) x0) \to ((clear x0 (CHead -c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift -h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl (S i0) (CHead x1 k x2) -(CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 -e0))))))) H16 (lift h (r k (S (plus i0 d))) x2) H14) in (let H18 \def (eq_ind -T t (\lambda (t1: T).((drop h (S (plus i0 d)) (CHead c0 k t1) (CHead x1 k -x2)) \to ((drop i0 O (CHead c0 k t1) (CHead x0 (Flat f) t0)) \to ((((drop i0 -O (CHead c0 k t1) x0) \to ((clear x0 (CHead c1 (Bind b) u)) \to (ex3_2 T C -(\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: -T).(\lambda (e0: C).(getl i0 (CHead x1 k x2) (CHead e0 (Bind b) v)))) -(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) \to (ex3_2 T C -(\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: -T).(\lambda (e0: C).(getl i0 (CHead x1 k x2) (CHead e0 (Bind b) v)))) -(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))))))) H15 (lift h (r k (S -(plus i0 d))) x2) H14) in (let H19 \def (eq_ind nat (r k (plus (S i0) d)) -(\lambda (n: nat).(drop h n c0 x1)) H13 (plus (r k (S i0)) d) (r_plus k (S -i0) d)) in (let H20 \def (eq_ind nat (r k (S i0)) (\lambda (n: nat).(drop h -(plus n d) c0 x1)) H19 (S (r k i0)) (r_S k i0)) in (let H21 \def (H c1 u (r k -i0) (getl_intro (r k i0) c0 (CHead c1 (Bind b) u) (CHead x0 (Flat f) t0) -(drop_gen_drop k c0 (CHead x0 (Flat f) t0) t i0 H10) (clear_flat x0 (CHead c1 -(Bind b) u) (clear_gen_flat f x0 (CHead c1 (Bind b) u) t0 H8) f t0)) x1 h d -H20) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d -v)))) (\lambda (v: T).(\lambda (e0: C).(getl (r k i0) x1 (CHead e0 (Bind b) -v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))) (ex3_2 T C (\lambda -(v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: -C).(getl (S i0) (CHead x1 k x2) (CHead e0 (Bind b) v)))) (\lambda (_: -T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x3: T).(\lambda (x4: -C).(\lambda (H22: (eq T u (lift h d x3))).(\lambda (H23: (getl (r k i0) x1 -(CHead x4 (Bind b) x3))).(\lambda (H24: (drop h d c1 x4)).(let H25 \def -(eq_ind T u (\lambda (t1: T).((drop (S i0) O (CHead c0 k (lift h (r k (S -(plus i0 d))) x2)) x0) \to ((clear x0 (CHead c1 (Bind b) t1)) \to (ex3_2 T C -(\lambda (v: T).(\lambda (_: C).(eq T t1 (lift h d v)))) (\lambda (v: -T).(\lambda (e0: C).(getl (S i0) (CHead x1 k x2) (CHead e0 (Bind b) v)))) -(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) H17 (lift h d x3) -H22) in (let H26 \def (eq_ind T u (\lambda (t1: T).(clear x0 (CHead c1 (Bind -b) t1))) (clear_gen_flat f x0 (CHead c1 (Bind b) u) t0 H8) (lift h d x3) H22) -in (eq_ind_r T (lift h d x3) (\lambda (t1: T).(ex3_2 T C (\lambda (v: -T).(\lambda (_: C).(eq T t1 (lift h d v)))) (\lambda (v: T).(\lambda (e0: -C).(getl (S i0) (CHead x1 k x2) (CHead e0 (Bind b) v)))) (\lambda (_: -T).(\lambda (e0: C).(drop h d c1 e0))))) (ex3_2_intro T C (\lambda (v: -T).(\lambda (_: C).(eq T (lift h d x3) (lift h d v)))) (\lambda (v: -T).(\lambda (e0: C).(getl (S i0) (CHead x1 k x2) (CHead e0 (Bind b) v)))) -(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))) x3 x4 (refl_equal T (lift -h d x3)) (getl_head k i0 x1 (CHead x4 (Bind b) x3) H23 x2) H24) u H22)))))))) -H21)))))) e H11))))))))) (drop_gen_skip_l c0 e t h (plus (S i0) d) k -H9))))))) i H1 H7 IHx)))) k0 H5 H6))))))) x H3 H4)))) H2)))))))))))))) c)). -(* COMMENTS -Initial nodes: 6137 -END *) - -theorem getl_drop_conf_ge: - \forall (i: nat).(\forall (a: C).(\forall (c: C).((getl i c a) \to (\forall -(e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to ((le (plus d -h) i) \to (getl (minus i h) e a))))))))) -\def - \lambda (i: nat).(\lambda (a: C).(\lambda (c: C).(\lambda (H: (getl i c -a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H0: (drop h -d c e)).(\lambda (H1: (le (plus d h) i)).(let H2 \def (getl_gen_all c a i H) -in (ex2_ind C (\lambda (e0: C).(drop i O c e0)) (\lambda (e0: C).(clear e0 -a)) (getl (minus i h) e a) (\lambda (x: C).(\lambda (H3: (drop i O c -x)).(\lambda (H4: (clear x a)).(getl_intro (minus i h) e a x (drop_conf_ge i -x c H3 e h d H0 H1) H4)))) H2)))))))))). -(* COMMENTS -Initial nodes: 141 -END *) - -theorem getl_conf_ge_drop: - \forall (b: B).(\forall (c1: C).(\forall (e: C).(\forall (u: T).(\forall (i: -nat).((getl i c1 (CHead e (Bind b) u)) \to (\forall (c2: C).((drop (S O) i c1 -c2) \to (drop i O c2 e)))))))) -\def - \lambda (b: B).(\lambda (c1: C).(\lambda (e: C).(\lambda (u: T).(\lambda (i: -nat).(\lambda (H: (getl i c1 (CHead e (Bind b) u))).(\lambda (c2: C).(\lambda -(H0: (drop (S O) i c1 c2)).(let H3 \def (eq_ind nat (minus (S i) (S O)) -(\lambda (n: nat).(drop n O c2 e)) (drop_conf_ge (S i) e c1 (getl_drop b c1 e -u i H) c2 (S O) i H0 (eq_ind_r nat (plus (S O) i) (\lambda (n: nat).(le n (S -i))) (le_n (S i)) (plus i (S O)) (plus_sym i (S O)))) i (minus_Sx_SO i)) in -H3)))))))). -(* COMMENTS -Initial nodes: 151 -END *) - -theorem getl_drop_conf_rev: - \forall (j: nat).(\forall (e1: C).(\forall (e2: C).((drop j O e1 e2) \to -(\forall (b: B).(\forall (c2: C).(\forall (v2: T).(\forall (i: nat).((getl i -c2 (CHead e2 (Bind b) v2)) \to (ex2 C (\lambda (c1: C).(drop j O c1 c2)) -(\lambda (c1: C).(drop (S i) j c1 e1))))))))))) -\def - \lambda (j: nat).(\lambda (e1: C).(\lambda (e2: C).(\lambda (H: (drop j O e1 -e2)).(\lambda (b: B).(\lambda (c2: C).(\lambda (v2: T).(\lambda (i: -nat).(\lambda (H0: (getl i c2 (CHead e2 (Bind b) v2))).(drop_conf_rev j e1 e2 -H c2 (S i) (getl_drop b c2 e2 v2 i H0)))))))))). -(* COMMENTS -Initial nodes: 69 -END *) - -theorem drop_getl_trans_lt: - \forall (i: nat).(\forall (d: nat).((lt i d) \to (\forall (c1: C).(\forall -(c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (b: B).(\forall (e2: -C).(\forall (v: T).((getl i c2 (CHead e2 (Bind b) v)) \to (ex2 C (\lambda -(e1: C).(getl i c1 (CHead e1 (Bind b) (lift h (minus d (S i)) v)))) (\lambda -(e1: C).(drop h (minus d (S i)) e1 e2))))))))))))) -\def - \lambda (i: nat).(\lambda (d: nat).(\lambda (H: (lt i d)).(\lambda (c1: -C).(\lambda (c2: C).(\lambda (h: nat).(\lambda (H0: (drop h d c1 -c2)).(\lambda (b: B).(\lambda (e2: C).(\lambda (v: T).(\lambda (H1: (getl i -c2 (CHead e2 (Bind b) v))).(let H2 \def (getl_gen_all c2 (CHead e2 (Bind b) -v) i H1) in (ex2_ind C (\lambda (e: C).(drop i O c2 e)) (\lambda (e: -C).(clear e (CHead e2 (Bind b) v))) (ex2 C (\lambda (e1: C).(getl i c1 (CHead -e1 (Bind b) (lift h (minus d (S i)) v)))) (\lambda (e1: C).(drop h (minus d -(S i)) e1 e2))) (\lambda (x: C).(\lambda (H3: (drop i O c2 x)).(\lambda (H4: -(clear x (CHead e2 (Bind b) v))).(ex2_ind C (\lambda (e1: C).(drop i O c1 -e1)) (\lambda (e1: C).(drop h (minus d i) e1 x)) (ex2 C (\lambda (e1: -C).(getl i c1 (CHead e1 (Bind b) (lift h (minus d (S i)) v)))) (\lambda (e1: -C).(drop h (minus d (S i)) e1 e2))) (\lambda (x0: C).(\lambda (H5: (drop i O -c1 x0)).(\lambda (H6: (drop h (minus d i) x0 x)).(let H7 \def (eq_ind nat -(minus d i) (\lambda (n: nat).(drop h n x0 x)) H6 (S (minus d (S i))) -(minus_x_Sy d i H)) in (let H8 \def (drop_clear_S x x0 h (minus d (S i)) H7 b -e2 v H4) in (ex2_ind C (\lambda (c3: C).(clear x0 (CHead c3 (Bind b) (lift h -(minus d (S i)) v)))) (\lambda (c3: C).(drop h (minus d (S i)) c3 e2)) (ex2 C -(\lambda (e1: C).(getl i c1 (CHead e1 (Bind b) (lift h (minus d (S i)) v)))) -(\lambda (e1: C).(drop h (minus d (S i)) e1 e2))) (\lambda (x1: C).(\lambda -(H9: (clear x0 (CHead x1 (Bind b) (lift h (minus d (S i)) v)))).(\lambda -(H10: (drop h (minus d (S i)) x1 e2)).(ex_intro2 C (\lambda (e1: C).(getl i -c1 (CHead e1 (Bind b) (lift h (minus d (S i)) v)))) (\lambda (e1: C).(drop h -(minus d (S i)) e1 e2)) x1 (getl_intro i c1 (CHead x1 (Bind b) (lift h (minus -d (S i)) v)) x0 H5 H9) H10)))) H8)))))) (drop_trans_le i d (le_S_n i d (le_S -(S i) d H)) c1 c2 h H0 x H3))))) H2)))))))))))). -(* COMMENTS -Initial nodes: 627 -END *) - -theorem drop_getl_trans_le: - \forall (i: nat).(\forall (d: nat).((le i d) \to (\forall (c1: C).(\forall -(c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((getl i c2 -e2) \to (ex3_2 C C (\lambda (e0: C).(\lambda (_: C).(drop i O c1 e0))) -(\lambda (e0: C).(\lambda (e1: C).(drop h (minus d i) e0 e1))) (\lambda (_: -C).(\lambda (e1: C).(clear e1 e2)))))))))))) -\def - \lambda (i: nat).(\lambda (d: nat).(\lambda (H: (le i d)).(\lambda (c1: -C).(\lambda (c2: C).(\lambda (h: nat).(\lambda (H0: (drop h d c1 -c2)).(\lambda (e2: C).(\lambda (H1: (getl i c2 e2)).(let H2 \def -(getl_gen_all c2 e2 i H1) in (ex2_ind C (\lambda (e: C).(drop i O c2 e)) -(\lambda (e: C).(clear e e2)) (ex3_2 C C (\lambda (e0: C).(\lambda (_: -C).(drop i O c1 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d i) -e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear e1 e2)))) (\lambda (x: -C).(\lambda (H3: (drop i O c2 x)).(\lambda (H4: (clear x e2)).(let H5 \def -(drop_trans_le i d H c1 c2 h H0 x H3) in (ex2_ind C (\lambda (e1: C).(drop i -O c1 e1)) (\lambda (e1: C).(drop h (minus d i) e1 x)) (ex3_2 C C (\lambda -(e0: C).(\lambda (_: C).(drop i O c1 e0))) (\lambda (e0: C).(\lambda (e1: -C).(drop h (minus d i) e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear e1 -e2)))) (\lambda (x0: C).(\lambda (H6: (drop i O c1 x0)).(\lambda (H7: (drop h -(minus d i) x0 x)).(ex3_2_intro C C (\lambda (e0: C).(\lambda (_: C).(drop i -O c1 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d i) e0 e1))) -(\lambda (_: C).(\lambda (e1: C).(clear e1 e2))) x0 x H6 H7 H4)))) H5))))) -H2)))))))))). -(* COMMENTS -Initial nodes: 323 -END *) - -theorem drop_getl_trans_ge: - \forall (i: nat).(\forall (c1: C).(\forall (c2: C).(\forall (d: -nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((getl i c2 e2) -\to ((le d i) \to (getl (plus i h) c1 e2))))))))) -\def - \lambda (i: nat).(\lambda (c1: C).(\lambda (c2: C).(\lambda (d: -nat).(\lambda (h: nat).(\lambda (H: (drop h d c1 c2)).(\lambda (e2: -C).(\lambda (H0: (getl i c2 e2)).(\lambda (H1: (le d i)).(let H2 \def -(getl_gen_all c2 e2 i H0) in (ex2_ind C (\lambda (e: C).(drop i O c2 e)) -(\lambda (e: C).(clear e e2)) (getl (plus i h) c1 e2) (\lambda (x: -C).(\lambda (H3: (drop i O c2 x)).(\lambda (H4: (clear x e2)).(getl_intro -(plus i h) c1 e2 x (drop_trans_ge i c1 c2 d h H x H3 H1) H4)))) H2)))))))))). -(* COMMENTS -Initial nodes: 137 -END *) - -theorem getl_drop_trans: - \forall (c1: C).(\forall (c2: C).(\forall (h: nat).((getl h c1 c2) \to -(\forall (e2: C).(\forall (i: nat).((drop (S i) O c2 e2) \to (drop (S (plus i -h)) O c1 e2))))))) -\def - \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (h: -nat).((getl h c c2) \to (\forall (e2: C).(\forall (i: nat).((drop (S i) O c2 -e2) \to (drop (S (plus i h)) O c e2)))))))) (\lambda (n: nat).(\lambda (c2: -C).(\lambda (h: nat).(\lambda (H: (getl h (CSort n) c2)).(\lambda (e2: -C).(\lambda (i: nat).(\lambda (_: (drop (S i) O c2 e2)).(getl_gen_sort n h c2 -H (drop (S (plus i h)) O (CSort n) e2))))))))) (\lambda (c2: C).(\lambda -(IHc: ((\forall (c3: C).(\forall (h: nat).((getl h c2 c3) \to (\forall (e2: -C).(\forall (i: nat).((drop (S i) O c3 e2) \to (drop (S (plus i h)) O c2 -e2))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t: T).(\forall -(c3: C).(\forall (h: nat).((getl h (CHead c2 k0 t) c3) \to (\forall (e2: -C).(\forall (i: nat).((drop (S i) O c3 e2) \to (drop (S (plus i h)) O (CHead -c2 k0 t) e2))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (c3: -C).(\lambda (h: nat).(nat_ind (\lambda (n: nat).((getl n (CHead c2 (Bind b) -t) c3) \to (\forall (e2: C).(\forall (i: nat).((drop (S i) O c3 e2) \to (drop -(S (plus i n)) O (CHead c2 (Bind b) t) e2)))))) (\lambda (H: (getl O (CHead -c2 (Bind b) t) c3)).(\lambda (e2: C).(\lambda (i: nat).(\lambda (H0: (drop (S -i) O c3 e2)).(let H1 \def (eq_ind C c3 (\lambda (c: C).(drop (S i) O c e2)) -H0 (CHead c2 (Bind b) t) (clear_gen_bind b c2 c3 t (getl_gen_O (CHead c2 -(Bind b) t) c3 H))) in (eq_ind nat i (\lambda (n: nat).(drop (S n) O (CHead -c2 (Bind b) t) e2)) (drop_drop (Bind b) i c2 e2 (drop_gen_drop (Bind b) c2 e2 -t i H1) t) (plus i O) (plus_n_O i))))))) (\lambda (n: nat).(\lambda (_: -(((getl n (CHead c2 (Bind b) t) c3) \to (\forall (e2: C).(\forall (i: -nat).((drop (S i) O c3 e2) \to (drop (S (plus i n)) O (CHead c2 (Bind b) t) -e2))))))).(\lambda (H0: (getl (S n) (CHead c2 (Bind b) t) c3)).(\lambda (e2: -C).(\lambda (i: nat).(\lambda (H1: (drop (S i) O c3 e2)).(eq_ind nat (plus (S -i) n) (\lambda (n0: nat).(drop (S n0) O (CHead c2 (Bind b) t) e2)) (drop_drop -(Bind b) (plus (S i) n) c2 e2 (IHc c3 n (getl_gen_S (Bind b) c2 c3 t n H0) e2 -i H1) t) (plus i (S n)) (plus_Snm_nSm i n)))))))) h))))) (\lambda (f: -F).(\lambda (t: T).(\lambda (c3: C).(\lambda (h: nat).(nat_ind (\lambda (n: -nat).((getl n (CHead c2 (Flat f) t) c3) \to (\forall (e2: C).(\forall (i: -nat).((drop (S i) O c3 e2) \to (drop (S (plus i n)) O (CHead c2 (Flat f) t) -e2)))))) (\lambda (H: (getl O (CHead c2 (Flat f) t) c3)).(\lambda (e2: -C).(\lambda (i: nat).(\lambda (H0: (drop (S i) O c3 e2)).(drop_drop (Flat f) -(plus i O) c2 e2 (IHc c3 O (getl_intro O c2 c3 c2 (drop_refl c2) -(clear_gen_flat f c2 c3 t (getl_gen_O (CHead c2 (Flat f) t) c3 H))) e2 i H0) -t))))) (\lambda (n: nat).(\lambda (_: (((getl n (CHead c2 (Flat f) t) c3) \to -(\forall (e2: C).(\forall (i: nat).((drop (S i) O c3 e2) \to (drop (S (plus i -n)) O (CHead c2 (Flat f) t) e2))))))).(\lambda (H0: (getl (S n) (CHead c2 -(Flat f) t) c3)).(\lambda (e2: C).(\lambda (i: nat).(\lambda (H1: (drop (S i) -O c3 e2)).(drop_drop (Flat f) (plus i (S n)) c2 e2 (IHc c3 (S n) (getl_gen_S -(Flat f) c2 c3 t n H0) e2 i H1) t))))))) h))))) k)))) c1). -(* COMMENTS -Initial nodes: 953 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/getl/flt.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/getl/flt.ma deleted file mode 100644 index 3d6c90f73..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/getl/flt.ma +++ /dev/null @@ -1,67 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/getl/fwd.ma". - -include "Basic-1/clear/props.ma". - -include "Basic-1/flt/props.ma". - -theorem getl_flt: - \forall (b: B).(\forall (c: C).(\forall (e: C).(\forall (u: T).(\forall (i: -nat).((getl i c (CHead e (Bind b) u)) \to (flt e u c (TLRef i))))))) -\def - \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e: -C).(\forall (u: T).(\forall (i: nat).((getl i c0 (CHead e (Bind b) u)) \to -(flt e u c0 (TLRef i))))))) (\lambda (n: nat).(\lambda (e: C).(\lambda (u: -T).(\lambda (i: nat).(\lambda (H: (getl i (CSort n) (CHead e (Bind b) -u))).(getl_gen_sort n i (CHead e (Bind b) u) H (flt e u (CSort n) (TLRef -i)))))))) (\lambda (c0: C).(\lambda (H: ((\forall (e: C).(\forall (u: -T).(\forall (i: nat).((getl i c0 (CHead e (Bind b) u)) \to (flt e u c0 (TLRef -i)))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e: C).(\lambda (u: -T).(\lambda (i: nat).(nat_ind (\lambda (n: nat).((getl n (CHead c0 k t) -(CHead e (Bind b) u)) \to (flt e u (CHead c0 k t) (TLRef n)))) (\lambda (H0: -(getl O (CHead c0 k t) (CHead e (Bind b) u))).(K_ind (\lambda (k0: K).((clear -(CHead c0 k0 t) (CHead e (Bind b) u)) \to (flt e u (CHead c0 k0 t) (TLRef -O)))) (\lambda (b0: B).(\lambda (H1: (clear (CHead c0 (Bind b0) t) (CHead e -(Bind b) u))).(let H2 \def (f_equal C C (\lambda (e0: C).(match e0 in C -return (\lambda (_: C).C) with [(CSort _) \Rightarrow e | (CHead c1 _ _) -\Rightarrow c1])) (CHead e (Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind -b0 c0 (CHead e (Bind b) u) t H1)) in ((let H3 \def (f_equal C B (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow b | -(CHead _ k0 _) \Rightarrow (match k0 in K return (\lambda (_: K).B) with -[(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow b])])) (CHead e (Bind b) u) -(CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e (Bind b) u) t H1)) in -((let H4 \def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda -(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) -(CHead e (Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e -(Bind b) u) t H1)) in (\lambda (H5: (eq B b b0)).(\lambda (H6: (eq C e -c0)).(eq_ind_r T t (\lambda (t0: T).(flt e t0 (CHead c0 (Bind b0) t) (TLRef -O))) (eq_ind_r C c0 (\lambda (c1: C).(flt c1 t (CHead c0 (Bind b0) t) (TLRef -O))) (eq_ind B b (\lambda (b1: B).(flt c0 t (CHead c0 (Bind b1) t) (TLRef -O))) (flt_arith0 (Bind b) c0 t O) b0 H5) e H6) u H4)))) H3)) H2)))) (\lambda -(f: F).(\lambda (H1: (clear (CHead c0 (Flat f) t) (CHead e (Bind b) -u))).(flt_arith1 (Bind b) e c0 u (clear_cle c0 (CHead e (Bind b) u) -(clear_gen_flat f c0 (CHead e (Bind b) u) t H1)) (Flat f) t O))) k -(getl_gen_O (CHead c0 k t) (CHead e (Bind b) u) H0))) (\lambda (n: -nat).(\lambda (_: (((getl n (CHead c0 k t) (CHead e (Bind b) u)) \to (flt e u -(CHead c0 k t) (TLRef n))))).(\lambda (H1: (getl (S n) (CHead c0 k t) (CHead -e (Bind b) u))).(let H_y \def (H e u (r k n) (getl_gen_S k c0 (CHead e (Bind -b) u) t n H1)) in (flt_arith2 e c0 u (r k n) H_y k t (S n)))))) i)))))))) c)). -(* COMMENTS -Initial nodes: 815 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/getl/fwd.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/getl/fwd.ma deleted file mode 100644 index b378f61fc..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/getl/fwd.ma +++ /dev/null @@ -1,148 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/getl/defs.ma". - -include "Basic-1/drop/fwd.ma". - -include "Basic-1/clear/fwd.ma". - -theorem getl_gen_all: - \forall (c1: C).(\forall (c2: C).(\forall (i: nat).((getl i c1 c2) \to (ex2 -C (\lambda (e: C).(drop i O c1 e)) (\lambda (e: C).(clear e c2)))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (i: nat).(\lambda (H: (getl i c1 -c2)).(getl_ind i c1 c2 (ex2 C (\lambda (e: C).(drop i O c1 e)) (\lambda (e: -C).(clear e c2))) (\lambda (e: C).(\lambda (H0: (drop i O c1 e)).(\lambda -(H1: (clear e c2)).(ex_intro2 C (\lambda (e0: C).(drop i O c1 e0)) (\lambda -(e0: C).(clear e0 c2)) e H0 H1)))) H)))). -(* COMMENTS -Initial nodes: 95 -END *) - -theorem getl_gen_sort: - \forall (n: nat).(\forall (h: nat).(\forall (x: C).((getl h (CSort n) x) \to -(\forall (P: Prop).P)))) -\def - \lambda (n: nat).(\lambda (h: nat).(\lambda (x: C).(\lambda (H: (getl h -(CSort n) x)).(\lambda (P: Prop).(let H0 \def (getl_gen_all (CSort n) x h H) -in (ex2_ind C (\lambda (e: C).(drop h O (CSort n) e)) (\lambda (e: C).(clear -e x)) P (\lambda (x0: C).(\lambda (H1: (drop h O (CSort n) x0)).(\lambda (H2: -(clear x0 x)).(and3_ind (eq C x0 (CSort n)) (eq nat h O) (eq nat O O) P -(\lambda (H3: (eq C x0 (CSort n))).(\lambda (_: (eq nat h O)).(\lambda (_: -(eq nat O O)).(let H6 \def (eq_ind C x0 (\lambda (c: C).(clear c x)) H2 -(CSort n) H3) in (clear_gen_sort x n H6 P))))) (drop_gen_sort n h O x0 -H1))))) H0)))))). -(* COMMENTS -Initial nodes: 179 -END *) - -theorem getl_gen_O: - \forall (e: C).(\forall (x: C).((getl O e x) \to (clear e x))) -\def - \lambda (e: C).(\lambda (x: C).(\lambda (H: (getl O e x)).(let H0 \def -(getl_gen_all e x O H) in (ex2_ind C (\lambda (e0: C).(drop O O e e0)) -(\lambda (e0: C).(clear e0 x)) (clear e x) (\lambda (x0: C).(\lambda (H1: -(drop O O e x0)).(\lambda (H2: (clear x0 x)).(let H3 \def (eq_ind_r C x0 -(\lambda (c: C).(clear c x)) H2 e (drop_gen_refl e x0 H1)) in H3)))) H0)))). -(* COMMENTS -Initial nodes: 99 -END *) - -theorem getl_gen_S: - \forall (k: K).(\forall (c: C).(\forall (x: C).(\forall (u: T).(\forall (h: -nat).((getl (S h) (CHead c k u) x) \to (getl (r k h) c x)))))) -\def - \lambda (k: K).(\lambda (c: C).(\lambda (x: C).(\lambda (u: T).(\lambda (h: -nat).(\lambda (H: (getl (S h) (CHead c k u) x)).(let H0 \def (getl_gen_all -(CHead c k u) x (S h) H) in (ex2_ind C (\lambda (e: C).(drop (S h) O (CHead c -k u) e)) (\lambda (e: C).(clear e x)) (getl (r k h) c x) (\lambda (x0: -C).(\lambda (H1: (drop (S h) O (CHead c k u) x0)).(\lambda (H2: (clear x0 -x)).(getl_intro (r k h) c x x0 (drop_gen_drop k c x0 u h H1) H2)))) H0))))))). -(* COMMENTS -Initial nodes: 145 -END *) - -theorem getl_gen_2: - \forall (c1: C).(\forall (c2: C).(\forall (i: nat).((getl i c1 c2) \to (ex_3 -B C T (\lambda (b: B).(\lambda (c: C).(\lambda (v: T).(eq C c2 (CHead c (Bind -b) v))))))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (i: nat).(\lambda (H: (getl i c1 -c2)).(let H0 \def (getl_gen_all c1 c2 i H) in (ex2_ind C (\lambda (e: -C).(drop i O c1 e)) (\lambda (e: C).(clear e c2)) (ex_3 B C T (\lambda (b: -B).(\lambda (c: C).(\lambda (v: T).(eq C c2 (CHead c (Bind b) v)))))) -(\lambda (x: C).(\lambda (_: (drop i O c1 x)).(\lambda (H2: (clear x -c2)).(let H3 \def (clear_gen_all x c2 H2) in (ex_3_ind B C T (\lambda (b: -B).(\lambda (e: C).(\lambda (u: T).(eq C c2 (CHead e (Bind b) u))))) (ex_3 B -C T (\lambda (b: B).(\lambda (c: C).(\lambda (v: T).(eq C c2 (CHead c (Bind -b) v)))))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H4: -(eq C c2 (CHead x1 (Bind x0) x2))).(let H5 \def (eq_ind C c2 (\lambda (c: -C).(clear x c)) H2 (CHead x1 (Bind x0) x2) H4) in (eq_ind_r C (CHead x1 (Bind -x0) x2) (\lambda (c: C).(ex_3 B C T (\lambda (b: B).(\lambda (c0: C).(\lambda -(v: T).(eq C c (CHead c0 (Bind b) v))))))) (ex_3_intro B C T (\lambda (b: -B).(\lambda (c: C).(\lambda (v: T).(eq C (CHead x1 (Bind x0) x2) (CHead c -(Bind b) v))))) x0 x1 x2 (refl_equal C (CHead x1 (Bind x0) x2))) c2 H4)))))) -H3))))) H0))))). -(* COMMENTS -Initial nodes: 325 -END *) - -theorem getl_gen_flat: - \forall (f: F).(\forall (e: C).(\forall (d: C).(\forall (v: T).(\forall (i: -nat).((getl i (CHead e (Flat f) v) d) \to (getl i e d)))))) -\def - \lambda (f: F).(\lambda (e: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: -nat).(nat_ind (\lambda (n: nat).((getl n (CHead e (Flat f) v) d) \to (getl n -e d))) (\lambda (H: (getl O (CHead e (Flat f) v) d)).(getl_intro O e d e -(drop_refl e) (clear_gen_flat f e d v (getl_gen_O (CHead e (Flat f) v) d -H)))) (\lambda (n: nat).(\lambda (_: (((getl n (CHead e (Flat f) v) d) \to -(getl n e d)))).(\lambda (H0: (getl (S n) (CHead e (Flat f) v) -d)).(getl_gen_S (Flat f) e d v n H0)))) i))))). -(* COMMENTS -Initial nodes: 155 -END *) - -theorem getl_gen_bind: - \forall (b: B).(\forall (e: C).(\forall (d: C).(\forall (v: T).(\forall (i: -nat).((getl i (CHead e (Bind b) v) d) \to (or (land (eq nat i O) (eq C d -(CHead e (Bind b) v))) (ex2 nat (\lambda (j: nat).(eq nat i (S j))) (\lambda -(j: nat).(getl j e d))))))))) -\def - \lambda (b: B).(\lambda (e: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: -nat).(nat_ind (\lambda (n: nat).((getl n (CHead e (Bind b) v) d) \to (or -(land (eq nat n O) (eq C d (CHead e (Bind b) v))) (ex2 nat (\lambda (j: -nat).(eq nat n (S j))) (\lambda (j: nat).(getl j e d)))))) (\lambda (H: (getl -O (CHead e (Bind b) v) d)).(eq_ind_r C (CHead e (Bind b) v) (\lambda (c: -C).(or (land (eq nat O O) (eq C c (CHead e (Bind b) v))) (ex2 nat (\lambda -(j: nat).(eq nat O (S j))) (\lambda (j: nat).(getl j e c))))) (or_introl -(land (eq nat O O) (eq C (CHead e (Bind b) v) (CHead e (Bind b) v))) (ex2 nat -(\lambda (j: nat).(eq nat O (S j))) (\lambda (j: nat).(getl j e (CHead e -(Bind b) v)))) (conj (eq nat O O) (eq C (CHead e (Bind b) v) (CHead e (Bind -b) v)) (refl_equal nat O) (refl_equal C (CHead e (Bind b) v)))) d -(clear_gen_bind b e d v (getl_gen_O (CHead e (Bind b) v) d H)))) (\lambda (n: -nat).(\lambda (_: (((getl n (CHead e (Bind b) v) d) \to (or (land (eq nat n -O) (eq C d (CHead e (Bind b) v))) (ex2 nat (\lambda (j: nat).(eq nat n (S -j))) (\lambda (j: nat).(getl j e d))))))).(\lambda (H0: (getl (S n) (CHead e -(Bind b) v) d)).(or_intror (land (eq nat (S n) O) (eq C d (CHead e (Bind b) -v))) (ex2 nat (\lambda (j: nat).(eq nat (S n) (S j))) (\lambda (j: nat).(getl -j e d))) (ex_intro2 nat (\lambda (j: nat).(eq nat (S n) (S j))) (\lambda (j: -nat).(getl j e d)) n (refl_equal nat (S n)) (getl_gen_S (Bind b) e d v n -H0)))))) i))))). -(* COMMENTS -Initial nodes: 525 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/getl/getl.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/getl/getl.ma deleted file mode 100644 index 71327becf..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/getl/getl.ma +++ /dev/null @@ -1,57 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/getl/drop.ma". - -include "Basic-1/getl/clear.ma". - -theorem getl_conf_le: - \forall (i: nat).(\forall (a: C).(\forall (c: C).((getl i c a) \to (\forall -(e: C).(\forall (h: nat).((getl h c e) \to ((le h i) \to (getl (minus i h) e -a)))))))) -\def - \lambda (i: nat).(\lambda (a: C).(\lambda (c: C).(\lambda (H: (getl i c -a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (H0: (getl h c e)).(\lambda -(H1: (le h i)).(let H2 \def (getl_gen_all c e h H0) in (ex2_ind C (\lambda -(e0: C).(drop h O c e0)) (\lambda (e0: C).(clear e0 e)) (getl (minus i h) e -a) (\lambda (x: C).(\lambda (H3: (drop h O c x)).(\lambda (H4: (clear x -e)).(getl_clear_conf (minus i h) x a (getl_drop_conf_ge i a c H x h O H3 H1) -e H4)))) H2))))))))). -(* COMMENTS -Initial nodes: 133 -END *) - -theorem getl_trans: - \forall (i: nat).(\forall (c1: C).(\forall (c2: C).(\forall (h: nat).((getl -h c1 c2) \to (\forall (e2: C).((getl i c2 e2) \to (getl (plus i h) c1 -e2))))))) -\def - \lambda (i: nat).(\lambda (c1: C).(\lambda (c2: C).(\lambda (h: -nat).(\lambda (H: (getl h c1 c2)).(\lambda (e2: C).(\lambda (H0: (getl i c2 -e2)).(let H1 \def (getl_gen_all c2 e2 i H0) in (ex2_ind C (\lambda (e: -C).(drop i O c2 e)) (\lambda (e: C).(clear e e2)) (getl (plus i h) c1 e2) -(\lambda (x: C).(\lambda (H2: (drop i O c2 x)).(\lambda (H3: (clear x -e2)).(nat_ind (\lambda (n: nat).((drop n O c2 x) \to (getl (plus n h) c1 -e2))) (\lambda (H4: (drop O O c2 x)).(let H5 \def (eq_ind_r C x (\lambda (c: -C).(clear c e2)) H3 c2 (drop_gen_refl c2 x H4)) in (getl_clear_trans (plus O -h) c1 c2 H e2 H5))) (\lambda (i0: nat).(\lambda (_: (((drop i0 O c2 x) \to -(getl (plus i0 h) c1 e2)))).(\lambda (H4: (drop (S i0) O c2 x)).(let H_y \def -(getl_drop_trans c1 c2 h H x i0 H4) in (getl_intro (plus (S i0) h) c1 e2 x -H_y H3))))) i H2)))) H1)))))))). -(* COMMENTS -Initial nodes: 247 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/getl/props.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/getl/props.ma deleted file mode 100644 index 9399ad4ab..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/getl/props.ma +++ /dev/null @@ -1,104 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/getl/fwd.ma". - -include "Basic-1/drop/props.ma". - -include "Basic-1/clear/props.ma". - -theorem getl_refl: - \forall (b: B).(\forall (c: C).(\forall (u: T).(getl O (CHead c (Bind b) u) -(CHead c (Bind b) u)))) -\def - \lambda (b: B).(\lambda (c: C).(\lambda (u: T).(getl_intro O (CHead c (Bind -b) u) (CHead c (Bind b) u) (CHead c (Bind b) u) (drop_refl (CHead c (Bind b) -u)) (clear_bind b c u)))). -(* COMMENTS -Initial nodes: 59 -END *) - -theorem getl_head: - \forall (k: K).(\forall (h: nat).(\forall (c: C).(\forall (e: C).((getl (r k -h) c e) \to (\forall (u: T).(getl (S h) (CHead c k u) e)))))) -\def - \lambda (k: K).(\lambda (h: nat).(\lambda (c: C).(\lambda (e: C).(\lambda -(H: (getl (r k h) c e)).(\lambda (u: T).(let H0 \def (getl_gen_all c e (r k -h) H) in (ex2_ind C (\lambda (e0: C).(drop (r k h) O c e0)) (\lambda (e0: -C).(clear e0 e)) (getl (S h) (CHead c k u) e) (\lambda (x: C).(\lambda (H1: -(drop (r k h) O c x)).(\lambda (H2: (clear x e)).(getl_intro (S h) (CHead c k -u) e x (drop_drop k h c x H1 u) H2)))) H0))))))). -(* COMMENTS -Initial nodes: 137 -END *) - -theorem getl_flat: - \forall (c: C).(\forall (e: C).(\forall (h: nat).((getl h c e) \to (\forall -(f: F).(\forall (u: T).(getl h (CHead c (Flat f) u) e)))))) -\def - \lambda (c: C).(\lambda (e: C).(\lambda (h: nat).(\lambda (H: (getl h c -e)).(\lambda (f: F).(\lambda (u: T).(let H0 \def (getl_gen_all c e h H) in -(ex2_ind C (\lambda (e0: C).(drop h O c e0)) (\lambda (e0: C).(clear e0 e)) -(getl h (CHead c (Flat f) u) e) (\lambda (x: C).(\lambda (H1: (drop h O c -x)).(\lambda (H2: (clear x e)).(nat_ind (\lambda (n: nat).((drop n O c x) \to -(getl n (CHead c (Flat f) u) e))) (\lambda (H3: (drop O O c x)).(let H4 \def -(eq_ind_r C x (\lambda (c0: C).(clear c0 e)) H2 c (drop_gen_refl c x H3)) in -(getl_intro O (CHead c (Flat f) u) e (CHead c (Flat f) u) (drop_refl (CHead c -(Flat f) u)) (clear_flat c e H4 f u)))) (\lambda (h0: nat).(\lambda (_: -(((drop h0 O c x) \to (getl h0 (CHead c (Flat f) u) e)))).(\lambda (H3: (drop -(S h0) O c x)).(getl_intro (S h0) (CHead c (Flat f) u) e x (drop_drop (Flat -f) h0 c x H3 u) H2)))) h H1)))) H0))))))). -(* COMMENTS -Initial nodes: 285 -END *) - -theorem getl_ctail: - \forall (b: B).(\forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: -nat).((getl i c (CHead d (Bind b) u)) \to (\forall (k: K).(\forall (v: -T).(getl i (CTail k v c) (CHead (CTail k v d) (Bind b) u))))))))) -\def - \lambda (b: B).(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: -nat).(\lambda (H: (getl i c (CHead d (Bind b) u))).(\lambda (k: K).(\lambda -(v: T).(let H0 \def (getl_gen_all c (CHead d (Bind b) u) i H) in (ex2_ind C -(\lambda (e: C).(drop i O c e)) (\lambda (e: C).(clear e (CHead d (Bind b) -u))) (getl i (CTail k v c) (CHead (CTail k v d) (Bind b) u)) (\lambda (x: -C).(\lambda (H1: (drop i O c x)).(\lambda (H2: (clear x (CHead d (Bind b) -u))).(getl_intro i (CTail k v c) (CHead (CTail k v d) (Bind b) u) (CTail k v -x) (drop_ctail c x O i H1 k v) (clear_ctail b x d u H2 k v))))) H0))))))))). -(* COMMENTS -Initial nodes: 203 -END *) - -theorem getl_mono: - \forall (c: C).(\forall (x1: C).(\forall (h: nat).((getl h c x1) \to -(\forall (x2: C).((getl h c x2) \to (eq C x1 x2)))))) -\def - \lambda (c: C).(\lambda (x1: C).(\lambda (h: nat).(\lambda (H: (getl h c -x1)).(\lambda (x2: C).(\lambda (H0: (getl h c x2)).(let H1 \def (getl_gen_all -c x2 h H0) in (ex2_ind C (\lambda (e: C).(drop h O c e)) (\lambda (e: -C).(clear e x2)) (eq C x1 x2) (\lambda (x: C).(\lambda (H2: (drop h O c -x)).(\lambda (H3: (clear x x2)).(let H4 \def (getl_gen_all c x1 h H) in -(ex2_ind C (\lambda (e: C).(drop h O c e)) (\lambda (e: C).(clear e x1)) (eq -C x1 x2) (\lambda (x0: C).(\lambda (H5: (drop h O c x0)).(\lambda (H6: (clear -x0 x1)).(let H7 \def (eq_ind C x (\lambda (c0: C).(drop h O c c0)) H2 x0 -(drop_mono c x O h H2 x0 H5)) in (let H8 \def (eq_ind_r C x0 (\lambda (c0: -C).(drop h O c c0)) H7 x (drop_mono c x O h H2 x0 H5)) in (let H9 \def -(eq_ind_r C x0 (\lambda (c0: C).(clear c0 x1)) H6 x (drop_mono c x O h H2 x0 -H5)) in (clear_mono x x1 H9 x2 H3))))))) H4))))) H1))))))). -(* COMMENTS -Initial nodes: 269 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/iso/defs.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/iso/defs.ma deleted file mode 100644 index 202b8a300..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/iso/defs.ma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/T/defs.ma". - -inductive iso: T \to (T \to Prop) \def -| iso_sort: \forall (n1: nat).(\forall (n2: nat).(iso (TSort n1) (TSort n2))) -| iso_lref: \forall (i1: nat).(\forall (i2: nat).(iso (TLRef i1) (TLRef i2))) -| iso_head: \forall (v1: T).(\forall (v2: T).(\forall (t1: T).(\forall (t2: -T).(\forall (k: K).(iso (THead k v1 t1) (THead k v2 t2)))))). - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/iso/fwd.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/iso/fwd.ma deleted file mode 100644 index 761e982f7..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/iso/fwd.ma +++ /dev/null @@ -1,191 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/iso/defs.ma". - -include "Basic-1/tlist/defs.ma". - -theorem iso_gen_sort: - \forall (u2: T).(\forall (n1: nat).((iso (TSort n1) u2) \to (ex nat (\lambda -(n2: nat).(eq T u2 (TSort n2)))))) -\def - \lambda (u2: T).(\lambda (n1: nat).(\lambda (H: (iso (TSort n1) -u2)).(insert_eq T (TSort n1) (\lambda (t: T).(iso t u2)) (\lambda (_: T).(ex -nat (\lambda (n2: nat).(eq T u2 (TSort n2))))) (\lambda (y: T).(\lambda (H0: -(iso y u2)).(iso_ind (\lambda (t: T).(\lambda (t0: T).((eq T t (TSort n1)) -\to (ex nat (\lambda (n2: nat).(eq T t0 (TSort n2))))))) (\lambda (n0: -nat).(\lambda (n2: nat).(\lambda (H1: (eq T (TSort n0) (TSort n1))).(let H2 -\def (f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) -with [(TSort n) \Rightarrow n | (TLRef _) \Rightarrow n0 | (THead _ _ _) -\Rightarrow n0])) (TSort n0) (TSort n1) H1) in (ex_intro nat (\lambda (n3: -nat).(eq T (TSort n2) (TSort n3))) n2 (refl_equal T (TSort n2))))))) (\lambda -(i1: nat).(\lambda (i2: nat).(\lambda (H1: (eq T (TLRef i1) (TSort n1))).(let -H2 \def (eq_ind T (TLRef i1) (\lambda (ee: T).(match ee in T return (\lambda -(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | -(THead _ _ _) \Rightarrow False])) I (TSort n1) H1) in (False_ind (ex nat -(\lambda (n2: nat).(eq T (TLRef i2) (TSort n2)))) H2))))) (\lambda (v1: -T).(\lambda (v2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k: -K).(\lambda (H1: (eq T (THead k v1 t1) (TSort n1))).(let H2 \def (eq_ind T -(THead k v1 t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) -with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ -_) \Rightarrow True])) I (TSort n1) H1) in (False_ind (ex nat (\lambda (n2: -nat).(eq T (THead k v2 t2) (TSort n2)))) H2)))))))) y u2 H0))) H))). -(* COMMENTS -Initial nodes: 321 -END *) - -theorem iso_gen_lref: - \forall (u2: T).(\forall (n1: nat).((iso (TLRef n1) u2) \to (ex nat (\lambda -(n2: nat).(eq T u2 (TLRef n2)))))) -\def - \lambda (u2: T).(\lambda (n1: nat).(\lambda (H: (iso (TLRef n1) -u2)).(insert_eq T (TLRef n1) (\lambda (t: T).(iso t u2)) (\lambda (_: T).(ex -nat (\lambda (n2: nat).(eq T u2 (TLRef n2))))) (\lambda (y: T).(\lambda (H0: -(iso y u2)).(iso_ind (\lambda (t: T).(\lambda (t0: T).((eq T t (TLRef n1)) -\to (ex nat (\lambda (n2: nat).(eq T t0 (TLRef n2))))))) (\lambda (n0: -nat).(\lambda (n2: nat).(\lambda (H1: (eq T (TSort n0) (TLRef n1))).(let H2 -\def (eq_ind T (TSort n0) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | -(THead _ _ _) \Rightarrow False])) I (TLRef n1) H1) in (False_ind (ex nat -(\lambda (n3: nat).(eq T (TSort n2) (TLRef n3)))) H2))))) (\lambda (i1: -nat).(\lambda (i2: nat).(\lambda (H1: (eq T (TLRef i1) (TLRef n1))).(let H2 -\def (f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) -with [(TSort _) \Rightarrow i1 | (TLRef n) \Rightarrow n | (THead _ _ _) -\Rightarrow i1])) (TLRef i1) (TLRef n1) H1) in (ex_intro nat (\lambda (n2: -nat).(eq T (TLRef i2) (TLRef n2))) i2 (refl_equal T (TLRef i2))))))) (\lambda -(v1: T).(\lambda (v2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k: -K).(\lambda (H1: (eq T (THead k v1 t1) (TLRef n1))).(let H2 \def (eq_ind T -(THead k v1 t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) -with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ -_) \Rightarrow True])) I (TLRef n1) H1) in (False_ind (ex nat (\lambda (n2: -nat).(eq T (THead k v2 t2) (TLRef n2)))) H2)))))))) y u2 H0))) H))). -(* COMMENTS -Initial nodes: 321 -END *) - -theorem iso_gen_head: - \forall (k: K).(\forall (v1: T).(\forall (t1: T).(\forall (u2: T).((iso -(THead k v1 t1) u2) \to (ex_2 T T (\lambda (v2: T).(\lambda (t2: T).(eq T u2 -(THead k v2 t2))))))))) -\def - \lambda (k: K).(\lambda (v1: T).(\lambda (t1: T).(\lambda (u2: T).(\lambda -(H: (iso (THead k v1 t1) u2)).(insert_eq T (THead k v1 t1) (\lambda (t: -T).(iso t u2)) (\lambda (_: T).(ex_2 T T (\lambda (v2: T).(\lambda (t2: -T).(eq T u2 (THead k v2 t2)))))) (\lambda (y: T).(\lambda (H0: (iso y -u2)).(iso_ind (\lambda (t: T).(\lambda (t0: T).((eq T t (THead k v1 t1)) \to -(ex_2 T T (\lambda (v2: T).(\lambda (t2: T).(eq T t0 (THead k v2 t2)))))))) -(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (H1: (eq T (TSort n1) (THead k -v1 t1))).(let H2 \def (eq_ind T (TSort n1) (\lambda (ee: T).(match ee in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead k v1 t1) H1) -in (False_ind (ex_2 T T (\lambda (v2: T).(\lambda (t2: T).(eq T (TSort n2) -(THead k v2 t2))))) H2))))) (\lambda (i1: nat).(\lambda (i2: nat).(\lambda -(H1: (eq T (TLRef i1) (THead k v1 t1))).(let H2 \def (eq_ind T (TLRef i1) -(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow -False])) I (THead k v1 t1) H1) in (False_ind (ex_2 T T (\lambda (v2: -T).(\lambda (t2: T).(eq T (TLRef i2) (THead k v2 t2))))) H2))))) (\lambda -(v0: T).(\lambda (v2: T).(\lambda (t0: T).(\lambda (t2: T).(\lambda (k0: -K).(\lambda (H1: (eq T (THead k0 v0 t0) (THead k v1 t1))).(let H2 \def -(f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with -[(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _ _) -\Rightarrow k1])) (THead k0 v0 t0) (THead k v1 t1) H1) in ((let H3 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow v0 | (TLRef _) \Rightarrow v0 | (THead _ t _) -\Rightarrow t])) (THead k0 v0 t0) (THead k v1 t1) H1) in ((let H4 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) -\Rightarrow t])) (THead k0 v0 t0) (THead k v1 t1) H1) in (\lambda (_: (eq T -v0 v1)).(\lambda (H6: (eq K k0 k)).(eq_ind_r K k (\lambda (k1: K).(ex_2 T T -(\lambda (v3: T).(\lambda (t3: T).(eq T (THead k1 v2 t2) (THead k v3 t3)))))) -(ex_2_intro T T (\lambda (v3: T).(\lambda (t3: T).(eq T (THead k v2 t2) -(THead k v3 t3)))) v2 t2 (refl_equal T (THead k v2 t2))) k0 H6)))) H3)) -H2)))))))) y u2 H0))) H))))). -(* COMMENTS -Initial nodes: 545 -END *) - -theorem iso_flats_lref_bind_false: - \forall (f: F).(\forall (b: B).(\forall (i: nat).(\forall (v: T).(\forall -(t: T).(\forall (vs: TList).((iso (THeads (Flat f) vs (TLRef i)) (THead (Bind -b) v t)) \to (\forall (P: Prop).P))))))) -\def - \lambda (f: F).(\lambda (b: B).(\lambda (i: nat).(\lambda (v: T).(\lambda -(t: T).(\lambda (vs: TList).(TList_ind (\lambda (t0: TList).((iso (THeads -(Flat f) t0 (TLRef i)) (THead (Bind b) v t)) \to (\forall (P: Prop).P))) -(\lambda (H: (iso (TLRef i) (THead (Bind b) v t))).(\lambda (P: Prop).(let -H_x \def (iso_gen_lref (THead (Bind b) v t) i H) in (let H0 \def H_x in -(ex_ind nat (\lambda (n2: nat).(eq T (THead (Bind b) v t) (TLRef n2))) P -(\lambda (x: nat).(\lambda (H1: (eq T (THead (Bind b) v t) (TLRef x))).(let -H2 \def (eq_ind T (THead (Bind b) v t) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef x) H1) in -(False_ind P H2)))) H0))))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda -(_: (((iso (THeads (Flat f) t1 (TLRef i)) (THead (Bind b) v t)) \to (\forall -(P: Prop).P)))).(\lambda (H0: (iso (THead (Flat f) t0 (THeads (Flat f) t1 -(TLRef i))) (THead (Bind b) v t))).(\lambda (P: Prop).(let H_x \def -(iso_gen_head (Flat f) t0 (THeads (Flat f) t1 (TLRef i)) (THead (Bind b) v t) -H0) in (let H1 \def H_x in (ex_2_ind T T (\lambda (v2: T).(\lambda (t2: -T).(eq T (THead (Bind b) v t) (THead (Flat f) v2 t2)))) P (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H2: (eq T (THead (Bind b) v t) (THead (Flat f) -x0 x1))).(let H3 \def (eq_ind T (THead (Bind b) v t) (\lambda (ee: T).(match -ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | -(TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return -(\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow -False])])) I (THead (Flat f) x0 x1) H2) in (False_ind P H3))))) H1)))))))) -vs)))))). -(* COMMENTS -Initial nodes: 391 -END *) - -theorem iso_flats_flat_bind_false: - \forall (f1: F).(\forall (f2: F).(\forall (b: B).(\forall (v: T).(\forall -(v2: T).(\forall (t: T).(\forall (t2: T).(\forall (vs: TList).((iso (THeads -(Flat f1) vs (THead (Flat f2) v2 t2)) (THead (Bind b) v t)) \to (\forall (P: -Prop).P))))))))) -\def - \lambda (f1: F).(\lambda (f2: F).(\lambda (b: B).(\lambda (v: T).(\lambda -(v2: T).(\lambda (t: T).(\lambda (t2: T).(\lambda (vs: TList).(TList_ind -(\lambda (t0: TList).((iso (THeads (Flat f1) t0 (THead (Flat f2) v2 t2)) -(THead (Bind b) v t)) \to (\forall (P: Prop).P))) (\lambda (H: (iso (THead -(Flat f2) v2 t2) (THead (Bind b) v t))).(\lambda (P: Prop).(let H_x \def -(iso_gen_head (Flat f2) v2 t2 (THead (Bind b) v t) H) in (let H0 \def H_x in -(ex_2_ind T T (\lambda (v3: T).(\lambda (t3: T).(eq T (THead (Bind b) v t) -(THead (Flat f2) v3 t3)))) P (\lambda (x0: T).(\lambda (x1: T).(\lambda (H1: -(eq T (THead (Bind b) v t) (THead (Flat f2) x0 x1))).(let H2 \def (eq_ind T -(THead (Bind b) v t) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat -f2) x0 x1) H1) in (False_ind P H2))))) H0))))) (\lambda (t0: T).(\lambda (t1: -TList).(\lambda (_: (((iso (THeads (Flat f1) t1 (THead (Flat f2) v2 t2)) -(THead (Bind b) v t)) \to (\forall (P: Prop).P)))).(\lambda (H0: (iso (THead -(Flat f1) t0 (THeads (Flat f1) t1 (THead (Flat f2) v2 t2))) (THead (Bind b) v -t))).(\lambda (P: Prop).(let H_x \def (iso_gen_head (Flat f1) t0 (THeads -(Flat f1) t1 (THead (Flat f2) v2 t2)) (THead (Bind b) v t) H0) in (let H1 -\def H_x in (ex_2_ind T T (\lambda (v3: T).(\lambda (t3: T).(eq T (THead -(Bind b) v t) (THead (Flat f1) v3 t3)))) P (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H2: (eq T (THead (Bind b) v t) (THead (Flat f1) x0 x1))).(let H3 -\def (eq_ind T (THead (Bind b) v t) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow -False])])) I (THead (Flat f1) x0 x1) H2) in (False_ind P H3))))) H1)))))))) -vs)))))))). -(* COMMENTS -Initial nodes: 461 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/iso/props.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/iso/props.ma deleted file mode 100644 index af521d073..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/iso/props.ma +++ /dev/null @@ -1,56 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/iso/fwd.ma". - -theorem iso_refl: - \forall (t: T).(iso t t) -\def - \lambda (t: T).(T_ind (\lambda (t0: T).(iso t0 t0)) (\lambda (n: -nat).(iso_sort n n)) (\lambda (n: nat).(iso_lref n n)) (\lambda (k: -K).(\lambda (t0: T).(\lambda (_: (iso t0 t0)).(\lambda (t1: T).(\lambda (_: -(iso t1 t1)).(iso_head t0 t0 t1 t1 k)))))) t). -(* COMMENTS -Initial nodes: 59 -END *) - -theorem iso_trans: - \forall (t1: T).(\forall (t2: T).((iso t1 t2) \to (\forall (t3: T).((iso t2 -t3) \to (iso t1 t3))))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (iso t1 t2)).(iso_ind (\lambda -(t: T).(\lambda (t0: T).(\forall (t3: T).((iso t0 t3) \to (iso t t3))))) -(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (t3: T).(\lambda (H0: (iso -(TSort n2) t3)).(let H_x \def (iso_gen_sort t3 n2 H0) in (let H1 \def H_x in -(ex_ind nat (\lambda (n3: nat).(eq T t3 (TSort n3))) (iso (TSort n1) t3) -(\lambda (x: nat).(\lambda (H2: (eq T t3 (TSort x))).(eq_ind_r T (TSort x) -(\lambda (t: T).(iso (TSort n1) t)) (iso_sort n1 x) t3 H2))) H1))))))) -(\lambda (i1: nat).(\lambda (i2: nat).(\lambda (t3: T).(\lambda (H0: (iso -(TLRef i2) t3)).(let H_x \def (iso_gen_lref t3 i2 H0) in (let H1 \def H_x in -(ex_ind nat (\lambda (n2: nat).(eq T t3 (TLRef n2))) (iso (TLRef i1) t3) -(\lambda (x: nat).(\lambda (H2: (eq T t3 (TLRef x))).(eq_ind_r T (TLRef x) -(\lambda (t: T).(iso (TLRef i1) t)) (iso_lref i1 x) t3 H2))) H1))))))) -(\lambda (v1: T).(\lambda (v2: T).(\lambda (t3: T).(\lambda (t4: T).(\lambda -(k: K).(\lambda (t5: T).(\lambda (H0: (iso (THead k v2 t4) t5)).(let H_x \def -(iso_gen_head k v2 t4 t5 H0) in (let H1 \def H_x in (ex_2_ind T T (\lambda -(v3: T).(\lambda (t6: T).(eq T t5 (THead k v3 t6)))) (iso (THead k v1 t3) t5) -(\lambda (x0: T).(\lambda (x1: T).(\lambda (H2: (eq T t5 (THead k x0 -x1))).(eq_ind_r T (THead k x0 x1) (\lambda (t: T).(iso (THead k v1 t3) t)) -(iso_head v1 x0 t3 x1 k) t5 H2)))) H1)))))))))) t1 t2 H))). -(* COMMENTS -Initial nodes: 351 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/leq/asucc.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/leq/asucc.ma deleted file mode 100644 index fd9e7c1d3..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/leq/asucc.ma +++ /dev/null @@ -1,479 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/leq/props.ma". - -theorem asucc_repl: - \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (leq g -(asucc g a1) (asucc g a2))))) -\def - \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1 -a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(leq g (asucc g a) (asucc g -a0)))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: nat).(\lambda (n2: -nat).(\lambda (k: nat).(\lambda (H0: (eq A (aplus g (ASort h1 n1) k) (aplus g -(ASort h2 n2) k))).(nat_ind (\lambda (n: nat).((eq A (aplus g (ASort n n1) k) -(aplus g (ASort h2 n2) k)) \to (leq g (match n with [O \Rightarrow (ASort O -(next g n1)) | (S h) \Rightarrow (ASort h n1)]) (match h2 with [O \Rightarrow -(ASort O (next g n2)) | (S h) \Rightarrow (ASort h n2)])))) (\lambda (H1: (eq -A (aplus g (ASort O n1) k) (aplus g (ASort h2 n2) k))).(nat_ind (\lambda (n: -nat).((eq A (aplus g (ASort O n1) k) (aplus g (ASort n n2) k)) \to (leq g -(ASort O (next g n1)) (match n with [O \Rightarrow (ASort O (next g n2)) | (S -h) \Rightarrow (ASort h n2)])))) (\lambda (H2: (eq A (aplus g (ASort O n1) k) -(aplus g (ASort O n2) k))).(leq_sort g O O (next g n1) (next g n2) k (eq_ind -A (aplus g (ASort O n1) (S k)) (\lambda (a: A).(eq A a (aplus g (ASort O -(next g n2)) k))) (eq_ind A (aplus g (ASort O n2) (S k)) (\lambda (a: A).(eq -A (aplus g (ASort O n1) (S k)) a)) (eq_ind_r A (aplus g (ASort O n2) k) -(\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g (ASort O n2) k)))) -(refl_equal A (asucc g (aplus g (ASort O n2) k))) (aplus g (ASort O n1) k) -H2) (aplus g (ASort O (next g n2)) k) (aplus_sort_O_S_simpl g n2 k)) (aplus g -(ASort O (next g n1)) k) (aplus_sort_O_S_simpl g n1 k)))) (\lambda (h3: -nat).(\lambda (_: (((eq A (aplus g (ASort O n1) k) (aplus g (ASort h3 n2) k)) -\to (leq g (ASort O (next g n1)) (match h3 with [O \Rightarrow (ASort O (next -g n2)) | (S h) \Rightarrow (ASort h n2)]))))).(\lambda (H2: (eq A (aplus g -(ASort O n1) k) (aplus g (ASort (S h3) n2) k))).(leq_sort g O h3 (next g n1) -n2 k (eq_ind A (aplus g (ASort O n1) (S k)) (\lambda (a: A).(eq A a (aplus g -(ASort h3 n2) k))) (eq_ind A (aplus g (ASort (S h3) n2) (S k)) (\lambda (a: -A).(eq A (aplus g (ASort O n1) (S k)) a)) (eq_ind_r A (aplus g (ASort (S h3) -n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g (ASort (S h3) n2) -k)))) (refl_equal A (asucc g (aplus g (ASort (S h3) n2) k))) (aplus g (ASort -O n1) k) H2) (aplus g (ASort h3 n2) k) (aplus_sort_S_S_simpl g n2 h3 k)) -(aplus g (ASort O (next g n1)) k) (aplus_sort_O_S_simpl g n1 k)))))) h2 H1)) -(\lambda (h3: nat).(\lambda (IHh1: (((eq A (aplus g (ASort h3 n1) k) (aplus g -(ASort h2 n2) k)) \to (leq g (match h3 with [O \Rightarrow (ASort O (next g -n1)) | (S h) \Rightarrow (ASort h n1)]) (match h2 with [O \Rightarrow (ASort -O (next g n2)) | (S h) \Rightarrow (ASort h n2)]))))).(\lambda (H1: (eq A -(aplus g (ASort (S h3) n1) k) (aplus g (ASort h2 n2) k))).(nat_ind (\lambda -(n: nat).((eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort n n2) k)) \to -((((eq A (aplus g (ASort h3 n1) k) (aplus g (ASort n n2) k)) \to (leq g -(match h3 with [O \Rightarrow (ASort O (next g n1)) | (S h) \Rightarrow -(ASort h n1)]) (match n with [O \Rightarrow (ASort O (next g n2)) | (S h) -\Rightarrow (ASort h n2)])))) \to (leq g (ASort h3 n1) (match n with [O -\Rightarrow (ASort O (next g n2)) | (S h) \Rightarrow (ASort h n2)]))))) -(\lambda (H2: (eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort O n2) -k))).(\lambda (_: (((eq A (aplus g (ASort h3 n1) k) (aplus g (ASort O n2) k)) -\to (leq g (match h3 with [O \Rightarrow (ASort O (next g n1)) | (S h) -\Rightarrow (ASort h n1)]) (ASort O (next g n2)))))).(leq_sort g h3 O n1 -(next g n2) k (eq_ind A (aplus g (ASort O n2) (S k)) (\lambda (a: A).(eq A -(aplus g (ASort h3 n1) k) a)) (eq_ind A (aplus g (ASort (S h3) n1) (S k)) -(\lambda (a: A).(eq A a (aplus g (ASort O n2) (S k)))) (eq_ind_r A (aplus g -(ASort O n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g (ASort O -n2) k)))) (refl_equal A (asucc g (aplus g (ASort O n2) k))) (aplus g (ASort -(S h3) n1) k) H2) (aplus g (ASort h3 n1) k) (aplus_sort_S_S_simpl g n1 h3 k)) -(aplus g (ASort O (next g n2)) k) (aplus_sort_O_S_simpl g n2 k))))) (\lambda -(h4: nat).(\lambda (_: (((eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort -h4 n2) k)) \to ((((eq A (aplus g (ASort h3 n1) k) (aplus g (ASort h4 n2) k)) -\to (leq g (match h3 with [O \Rightarrow (ASort O (next g n1)) | (S h) -\Rightarrow (ASort h n1)]) (match h4 with [O \Rightarrow (ASort O (next g -n2)) | (S h) \Rightarrow (ASort h n2)])))) \to (leq g (ASort h3 n1) (match h4 -with [O \Rightarrow (ASort O (next g n2)) | (S h) \Rightarrow (ASort h -n2)])))))).(\lambda (H2: (eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort -(S h4) n2) k))).(\lambda (_: (((eq A (aplus g (ASort h3 n1) k) (aplus g -(ASort (S h4) n2) k)) \to (leq g (match h3 with [O \Rightarrow (ASort O (next -g n1)) | (S h) \Rightarrow (ASort h n1)]) (ASort h4 n2))))).(leq_sort g h3 h4 -n1 n2 k (eq_ind A (aplus g (ASort (S h3) n1) (S k)) (\lambda (a: A).(eq A a -(aplus g (ASort h4 n2) k))) (eq_ind A (aplus g (ASort (S h4) n2) (S k)) -(\lambda (a: A).(eq A (aplus g (ASort (S h3) n1) (S k)) a)) (eq_ind_r A -(aplus g (ASort (S h4) n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g -(aplus g (ASort (S h4) n2) k)))) (refl_equal A (asucc g (aplus g (ASort (S -h4) n2) k))) (aplus g (ASort (S h3) n1) k) H2) (aplus g (ASort h4 n2) k) -(aplus_sort_S_S_simpl g n2 h4 k)) (aplus g (ASort h3 n1) k) -(aplus_sort_S_S_simpl g n1 h3 k))))))) h2 H1 IHh1)))) h1 H0))))))) (\lambda -(a3: A).(\lambda (a4: A).(\lambda (H0: (leq g a3 a4)).(\lambda (_: (leq g -(asucc g a3) (asucc g a4))).(\lambda (a5: A).(\lambda (a6: A).(\lambda (_: -(leq g a5 a6)).(\lambda (H3: (leq g (asucc g a5) (asucc g a6))).(leq_head g -a3 a4 H0 (asucc g a5) (asucc g a6) H3))))))))) a1 a2 H)))). -(* COMMENTS -Initial nodes: 1907 -END *) - -theorem asucc_inj: - \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (asucc g a1) (asucc -g a2)) \to (leq g a1 a2)))) -\def - \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2: -A).((leq g (asucc g a) (asucc g a2)) \to (leq g a a2)))) (\lambda (n: -nat).(\lambda (n0: nat).(\lambda (a2: A).(A_ind (\lambda (a: A).((leq g -(asucc g (ASort n n0)) (asucc g a)) \to (leq g (ASort n n0) a))) (\lambda -(n1: nat).(\lambda (n2: nat).(\lambda (H: (leq g (asucc g (ASort n n0)) -(asucc g (ASort n1 n2)))).(nat_ind (\lambda (n3: nat).((leq g (asucc g (ASort -n3 n0)) (asucc g (ASort n1 n2))) \to (leq g (ASort n3 n0) (ASort n1 n2)))) -(\lambda (H0: (leq g (asucc g (ASort O n0)) (asucc g (ASort n1 -n2)))).(nat_ind (\lambda (n3: nat).((leq g (asucc g (ASort O n0)) (asucc g -(ASort n3 n2))) \to (leq g (ASort O n0) (ASort n3 n2)))) (\lambda (H1: (leq g -(asucc g (ASort O n0)) (asucc g (ASort O n2)))).(let H_x \def (leq_gen_sort1 -g O (next g n0) (ASort O (next g n2)) H1) in (let H2 \def H_x in (ex2_3_ind -nat nat nat (\lambda (n3: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A -(aplus g (ASort O (next g n0)) k) (aplus g (ASort h2 n3) k))))) (\lambda (n3: -nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A (ASort O (next g n2)) (ASort -h2 n3))))) (leq g (ASort O n0) (ASort O n2)) (\lambda (x0: nat).(\lambda (x1: -nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort O (next g n0)) -x2) (aplus g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort O (next g n2)) -(ASort x1 x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e in A -return (\lambda (_: A).nat) with [(ASort n3 _) \Rightarrow n3 | (AHead _ _) -\Rightarrow O])) (ASort O (next g n2)) (ASort x1 x0) H4) in ((let H6 \def -(f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with -[(ASort _ n3) \Rightarrow n3 | (AHead _ _) \Rightarrow ((match g with [(mk_G -next _) \Rightarrow next]) n2)])) (ASort O (next g n2)) (ASort x1 x0) H4) in -(\lambda (H7: (eq nat O x1)).(let H8 \def (eq_ind_r nat x1 (\lambda (n3: -nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort n3 x0) x2))) H3 -O H7) in (let H9 \def (eq_ind_r nat x0 (\lambda (n3: nat).(eq A (aplus g -(ASort O (next g n0)) x2) (aplus g (ASort O n3) x2))) H8 (next g n2) H6) in -(let H10 \def (eq_ind_r A (aplus g (ASort O (next g n0)) x2) (\lambda (a: -A).(eq A a (aplus g (ASort O (next g n2)) x2))) H9 (aplus g (ASort O n0) (S -x2)) (aplus_sort_O_S_simpl g n0 x2)) in (let H11 \def (eq_ind_r A (aplus g -(ASort O (next g n2)) x2) (\lambda (a: A).(eq A (aplus g (ASort O n0) (S x2)) -a)) H10 (aplus g (ASort O n2) (S x2)) (aplus_sort_O_S_simpl g n2 x2)) in -(leq_sort g O O n0 n2 (S x2) H11))))))) H5))))))) H2)))) (\lambda (n3: -nat).(\lambda (_: (((leq g (asucc g (ASort O n0)) (asucc g (ASort n3 n2))) -\to (leq g (ASort O n0) (ASort n3 n2))))).(\lambda (H1: (leq g (asucc g -(ASort O n0)) (asucc g (ASort (S n3) n2)))).(let H_x \def (leq_gen_sort1 g O -(next g n0) (ASort n3 n2) H1) in (let H2 \def H_x in (ex2_3_ind nat nat nat -(\lambda (n4: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort -O (next g n0)) k) (aplus g (ASort h2 n4) k))))) (\lambda (n4: nat).(\lambda -(h2: nat).(\lambda (_: nat).(eq A (ASort n3 n2) (ASort h2 n4))))) (leq g -(ASort O n0) (ASort (S n3) n2)) (\lambda (x0: nat).(\lambda (x1: -nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort O (next g n0)) -x2) (aplus g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort n3 n2) (ASort x1 -x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e in A return -(\lambda (_: A).nat) with [(ASort n4 _) \Rightarrow n4 | (AHead _ _) -\Rightarrow n3])) (ASort n3 n2) (ASort x1 x0) H4) in ((let H6 \def (f_equal A -nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort _ -n4) \Rightarrow n4 | (AHead _ _) \Rightarrow n2])) (ASort n3 n2) (ASort x1 -x0) H4) in (\lambda (H7: (eq nat n3 x1)).(let H8 \def (eq_ind_r nat x1 -(\lambda (n4: nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort -n4 x0) x2))) H3 n3 H7) in (let H9 \def (eq_ind_r nat x0 (\lambda (n4: -nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort n3 n4) x2))) H8 -n2 H6) in (let H10 \def (eq_ind_r A (aplus g (ASort O (next g n0)) x2) -(\lambda (a: A).(eq A a (aplus g (ASort n3 n2) x2))) H9 (aplus g (ASort O n0) -(S x2)) (aplus_sort_O_S_simpl g n0 x2)) in (let H11 \def (eq_ind_r A (aplus g -(ASort n3 n2) x2) (\lambda (a: A).(eq A (aplus g (ASort O n0) (S x2)) a)) H10 -(aplus g (ASort (S n3) n2) (S x2)) (aplus_sort_S_S_simpl g n2 n3 x2)) in -(leq_sort g O (S n3) n0 n2 (S x2) H11))))))) H5))))))) H2)))))) n1 H0)) -(\lambda (n3: nat).(\lambda (IHn: (((leq g (asucc g (ASort n3 n0)) (asucc g -(ASort n1 n2))) \to (leq g (ASort n3 n0) (ASort n1 n2))))).(\lambda (H0: (leq -g (asucc g (ASort (S n3) n0)) (asucc g (ASort n1 n2)))).(nat_ind (\lambda -(n4: nat).((leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort n4 n2))) \to -((((leq g (asucc g (ASort n3 n0)) (asucc g (ASort n4 n2))) \to (leq g (ASort -n3 n0) (ASort n4 n2)))) \to (leq g (ASort (S n3) n0) (ASort n4 n2))))) -(\lambda (H1: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort O -n2)))).(\lambda (_: (((leq g (asucc g (ASort n3 n0)) (asucc g (ASort O n2))) -\to (leq g (ASort n3 n0) (ASort O n2))))).(let H_x \def (leq_gen_sort1 g n3 -n0 (ASort O (next g n2)) H1) in (let H2 \def H_x in (ex2_3_ind nat nat nat -(\lambda (n4: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort -n3 n0) k) (aplus g (ASort h2 n4) k))))) (\lambda (n4: nat).(\lambda (h2: -nat).(\lambda (_: nat).(eq A (ASort O (next g n2)) (ASort h2 n4))))) (leq g -(ASort (S n3) n0) (ASort O n2)) (\lambda (x0: nat).(\lambda (x1: -nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort n3 n0) x2) (aplus -g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort O (next g n2)) (ASort x1 -x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e in A return -(\lambda (_: A).nat) with [(ASort n4 _) \Rightarrow n4 | (AHead _ _) -\Rightarrow O])) (ASort O (next g n2)) (ASort x1 x0) H4) in ((let H6 \def -(f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with -[(ASort _ n4) \Rightarrow n4 | (AHead _ _) \Rightarrow ((match g with [(mk_G -next _) \Rightarrow next]) n2)])) (ASort O (next g n2)) (ASort x1 x0) H4) in -(\lambda (H7: (eq nat O x1)).(let H8 \def (eq_ind_r nat x1 (\lambda (n4: -nat).(eq A (aplus g (ASort n3 n0) x2) (aplus g (ASort n4 x0) x2))) H3 O H7) -in (let H9 \def (eq_ind_r nat x0 (\lambda (n4: nat).(eq A (aplus g (ASort n3 -n0) x2) (aplus g (ASort O n4) x2))) H8 (next g n2) H6) in (let H10 \def -(eq_ind_r A (aplus g (ASort n3 n0) x2) (\lambda (a: A).(eq A a (aplus g -(ASort O (next g n2)) x2))) H9 (aplus g (ASort (S n3) n0) (S x2)) -(aplus_sort_S_S_simpl g n0 n3 x2)) in (let H11 \def (eq_ind_r A (aplus g -(ASort O (next g n2)) x2) (\lambda (a: A).(eq A (aplus g (ASort (S n3) n0) (S -x2)) a)) H10 (aplus g (ASort O n2) (S x2)) (aplus_sort_O_S_simpl g n2 x2)) in -(leq_sort g (S n3) O n0 n2 (S x2) H11))))))) H5))))))) H2))))) (\lambda (n4: -nat).(\lambda (_: (((leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort n4 -n2))) \to ((((leq g (asucc g (ASort n3 n0)) (asucc g (ASort n4 n2))) \to (leq -g (ASort n3 n0) (ASort n4 n2)))) \to (leq g (ASort (S n3) n0) (ASort n4 -n2)))))).(\lambda (H1: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort (S -n4) n2)))).(\lambda (_: (((leq g (asucc g (ASort n3 n0)) (asucc g (ASort (S -n4) n2))) \to (leq g (ASort n3 n0) (ASort (S n4) n2))))).(let H_x \def -(leq_gen_sort1 g n3 n0 (ASort n4 n2) H1) in (let H2 \def H_x in (ex2_3_ind -nat nat nat (\lambda (n5: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A -(aplus g (ASort n3 n0) k) (aplus g (ASort h2 n5) k))))) (\lambda (n5: -nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A (ASort n4 n2) (ASort h2 -n5))))) (leq g (ASort (S n3) n0) (ASort (S n4) n2)) (\lambda (x0: -nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g -(ASort n3 n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort n4 -n2) (ASort x1 x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e in A -return (\lambda (_: A).nat) with [(ASort n5 _) \Rightarrow n5 | (AHead _ _) -\Rightarrow n4])) (ASort n4 n2) (ASort x1 x0) H4) in ((let H6 \def (f_equal A -nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort _ -n5) \Rightarrow n5 | (AHead _ _) \Rightarrow n2])) (ASort n4 n2) (ASort x1 -x0) H4) in (\lambda (H7: (eq nat n4 x1)).(let H8 \def (eq_ind_r nat x1 -(\lambda (n5: nat).(eq A (aplus g (ASort n3 n0) x2) (aplus g (ASort n5 x0) -x2))) H3 n4 H7) in (let H9 \def (eq_ind_r nat x0 (\lambda (n5: nat).(eq A -(aplus g (ASort n3 n0) x2) (aplus g (ASort n4 n5) x2))) H8 n2 H6) in (let H10 -\def (eq_ind_r A (aplus g (ASort n3 n0) x2) (\lambda (a: A).(eq A a (aplus g -(ASort n4 n2) x2))) H9 (aplus g (ASort (S n3) n0) (S x2)) -(aplus_sort_S_S_simpl g n0 n3 x2)) in (let H11 \def (eq_ind_r A (aplus g -(ASort n4 n2) x2) (\lambda (a: A).(eq A (aplus g (ASort (S n3) n0) (S x2)) -a)) H10 (aplus g (ASort (S n4) n2) (S x2)) (aplus_sort_S_S_simpl g n2 n4 x2)) -in (leq_sort g (S n3) (S n4) n0 n2 (S x2) H11))))))) H5))))))) H2))))))) n1 -H0 IHn)))) n H)))) (\lambda (a: A).(\lambda (H: (((leq g (asucc g (ASort n -n0)) (asucc g a)) \to (leq g (ASort n n0) a)))).(\lambda (a0: A).(\lambda -(H0: (((leq g (asucc g (ASort n n0)) (asucc g a0)) \to (leq g (ASort n n0) -a0)))).(\lambda (H1: (leq g (asucc g (ASort n n0)) (asucc g (AHead a -a0)))).(nat_ind (\lambda (n1: nat).((((leq g (asucc g (ASort n1 n0)) (asucc g -a)) \to (leq g (ASort n1 n0) a))) \to ((((leq g (asucc g (ASort n1 n0)) -(asucc g a0)) \to (leq g (ASort n1 n0) a0))) \to ((leq g (asucc g (ASort n1 -n0)) (asucc g (AHead a a0))) \to (leq g (ASort n1 n0) (AHead a a0)))))) -(\lambda (_: (((leq g (asucc g (ASort O n0)) (asucc g a)) \to (leq g (ASort O -n0) a)))).(\lambda (_: (((leq g (asucc g (ASort O n0)) (asucc g a0)) \to (leq -g (ASort O n0) a0)))).(\lambda (H4: (leq g (asucc g (ASort O n0)) (asucc g -(AHead a a0)))).(let H_x \def (leq_gen_sort1 g O (next g n0) (AHead a (asucc -g a0)) H4) in (let H5 \def H_x in (ex2_3_ind nat nat nat (\lambda (n2: -nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort O (next g -n0)) k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: -nat).(\lambda (_: nat).(eq A (AHead a (asucc g a0)) (ASort h2 n2))))) (leq g -(ASort O n0) (AHead a a0)) (\lambda (x0: nat).(\lambda (x1: nat).(\lambda -(x2: nat).(\lambda (_: (eq A (aplus g (ASort O (next g n0)) x2) (aplus g -(ASort x1 x0) x2))).(\lambda (H7: (eq A (AHead a (asucc g a0)) (ASort x1 -x0))).(let H8 \def (eq_ind A (AHead a (asucc g a0)) (\lambda (ee: A).(match -ee in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | -(AHead _ _) \Rightarrow True])) I (ASort x1 x0) H7) in (False_ind (leq g -(ASort O n0) (AHead a a0)) H8))))))) H5)))))) (\lambda (n1: nat).(\lambda (_: -(((((leq g (asucc g (ASort n1 n0)) (asucc g a)) \to (leq g (ASort n1 n0) a))) -\to ((((leq g (asucc g (ASort n1 n0)) (asucc g a0)) \to (leq g (ASort n1 n0) -a0))) \to ((leq g (asucc g (ASort n1 n0)) (asucc g (AHead a a0))) \to (leq g -(ASort n1 n0) (AHead a a0))))))).(\lambda (_: (((leq g (asucc g (ASort (S n1) -n0)) (asucc g a)) \to (leq g (ASort (S n1) n0) a)))).(\lambda (_: (((leq g -(asucc g (ASort (S n1) n0)) (asucc g a0)) \to (leq g (ASort (S n1) n0) -a0)))).(\lambda (H4: (leq g (asucc g (ASort (S n1) n0)) (asucc g (AHead a -a0)))).(let H_x \def (leq_gen_sort1 g n1 n0 (AHead a (asucc g a0)) H4) in -(let H5 \def H_x in (ex2_3_ind nat nat nat (\lambda (n2: nat).(\lambda (h2: -nat).(\lambda (k: nat).(eq A (aplus g (ASort n1 n0) k) (aplus g (ASort h2 n2) -k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A (AHead a -(asucc g a0)) (ASort h2 n2))))) (leq g (ASort (S n1) n0) (AHead a a0)) -(\lambda (x0: nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (_: (eq A -(aplus g (ASort n1 n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H7: (eq A -(AHead a (asucc g a0)) (ASort x1 x0))).(let H8 \def (eq_ind A (AHead a (asucc -g a0)) (\lambda (ee: A).(match ee in A return (\lambda (_: A).Prop) with -[(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort x1 -x0) H7) in (False_ind (leq g (ASort (S n1) n0) (AHead a a0)) H8))))))) -H5)))))))) n H H0 H1)))))) a2)))) (\lambda (a: A).(\lambda (_: ((\forall (a2: -A).((leq g (asucc g a) (asucc g a2)) \to (leq g a a2))))).(\lambda (a0: -A).(\lambda (H0: ((\forall (a2: A).((leq g (asucc g a0) (asucc g a2)) \to -(leq g a0 a2))))).(\lambda (a2: A).(A_ind (\lambda (a3: A).((leq g (asucc g -(AHead a a0)) (asucc g a3)) \to (leq g (AHead a a0) a3))) (\lambda (n: -nat).(\lambda (n0: nat).(\lambda (H1: (leq g (asucc g (AHead a a0)) (asucc g -(ASort n n0)))).(nat_ind (\lambda (n1: nat).((leq g (asucc g (AHead a a0)) -(asucc g (ASort n1 n0))) \to (leq g (AHead a a0) (ASort n1 n0)))) (\lambda -(H2: (leq g (asucc g (AHead a a0)) (asucc g (ASort O n0)))).(let H_x \def -(leq_gen_head1 g a (asucc g a0) (ASort O (next g n0)) H2) in (let H3 \def H_x -in (ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq g a a3))) (\lambda -(_: A).(\lambda (a4: A).(leq g (asucc g a0) a4))) (\lambda (a3: A).(\lambda -(a4: A).(eq A (ASort O (next g n0)) (AHead a3 a4)))) (leq g (AHead a a0) -(ASort O n0)) (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g a -x0)).(\lambda (_: (leq g (asucc g a0) x1)).(\lambda (H6: (eq A (ASort O (next -g n0)) (AHead x0 x1))).(let H7 \def (eq_ind A (ASort O (next g n0)) (\lambda -(ee: A).(match ee in A return (\lambda (_: A).Prop) with [(ASort _ _) -\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H6) in -(False_ind (leq g (AHead a a0) (ASort O n0)) H7))))))) H3)))) (\lambda (n1: -nat).(\lambda (_: (((leq g (asucc g (AHead a a0)) (asucc g (ASort n1 n0))) -\to (leq g (AHead a a0) (ASort n1 n0))))).(\lambda (H2: (leq g (asucc g -(AHead a a0)) (asucc g (ASort (S n1) n0)))).(let H_x \def (leq_gen_head1 g a -(asucc g a0) (ASort n1 n0) H2) in (let H3 \def H_x in (ex3_2_ind A A (\lambda -(a3: A).(\lambda (_: A).(leq g a a3))) (\lambda (_: A).(\lambda (a4: A).(leq -g (asucc g a0) a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort n1 n0) -(AHead a3 a4)))) (leq g (AHead a a0) (ASort (S n1) n0)) (\lambda (x0: -A).(\lambda (x1: A).(\lambda (_: (leq g a x0)).(\lambda (_: (leq g (asucc g -a0) x1)).(\lambda (H6: (eq A (ASort n1 n0) (AHead x0 x1))).(let H7 \def -(eq_ind A (ASort n1 n0) (\lambda (ee: A).(match ee in A return (\lambda (_: -A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow -False])) I (AHead x0 x1) H6) in (False_ind (leq g (AHead a a0) (ASort (S n1) -n0)) H7))))))) H3)))))) n H1)))) (\lambda (a3: A).(\lambda (_: (((leq g -(asucc g (AHead a a0)) (asucc g a3)) \to (leq g (AHead a a0) a3)))).(\lambda -(a4: A).(\lambda (_: (((leq g (asucc g (AHead a a0)) (asucc g a4)) \to (leq g -(AHead a a0) a4)))).(\lambda (H3: (leq g (asucc g (AHead a a0)) (asucc g -(AHead a3 a4)))).(let H_x \def (leq_gen_head1 g a (asucc g a0) (AHead a3 -(asucc g a4)) H3) in (let H4 \def H_x in (ex3_2_ind A A (\lambda (a5: -A).(\lambda (_: A).(leq g a a5))) (\lambda (_: A).(\lambda (a6: A).(leq g -(asucc g a0) a6))) (\lambda (a5: A).(\lambda (a6: A).(eq A (AHead a3 (asucc g -a4)) (AHead a5 a6)))) (leq g (AHead a a0) (AHead a3 a4)) (\lambda (x0: -A).(\lambda (x1: A).(\lambda (H5: (leq g a x0)).(\lambda (H6: (leq g (asucc g -a0) x1)).(\lambda (H7: (eq A (AHead a3 (asucc g a4)) (AHead x0 x1))).(let H8 -\def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) -with [(ASort _ _) \Rightarrow a3 | (AHead a5 _) \Rightarrow a5])) (AHead a3 -(asucc g a4)) (AHead x0 x1) H7) in ((let H9 \def (f_equal A A (\lambda (e: -A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow -((let rec asucc (g0: G) (l: A) on l: A \def (match l with [(ASort n0 n) -\Rightarrow (match n0 with [O \Rightarrow (ASort O (next g0 n)) | (S h) -\Rightarrow (ASort h n)]) | (AHead a5 a6) \Rightarrow (AHead a5 (asucc g0 -a6))]) in asucc) g a4) | (AHead _ a5) \Rightarrow a5])) (AHead a3 (asucc g -a4)) (AHead x0 x1) H7) in (\lambda (H10: (eq A a3 x0)).(let H11 \def -(eq_ind_r A x1 (\lambda (a5: A).(leq g (asucc g a0) a5)) H6 (asucc g a4) H9) -in (let H12 \def (eq_ind_r A x0 (\lambda (a5: A).(leq g a a5)) H5 a3 H10) in -(leq_head g a a3 H12 a0 a4 (H0 a4 H11)))))) H8))))))) H4)))))))) a2)))))) -a1)). -(* COMMENTS -Initial nodes: 4697 -END *) - -theorem leq_asucc: - \forall (g: G).(\forall (a: A).(ex A (\lambda (a0: A).(leq g a (asucc g -a0))))) -\def - \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(ex A (\lambda (a1: -A).(leq g a0 (asucc g a1))))) (\lambda (n: nat).(\lambda (n0: nat).(ex_intro -A (\lambda (a0: A).(leq g (ASort n n0) (asucc g a0))) (ASort (S n) n0) -(leq_refl g (ASort n n0))))) (\lambda (a0: A).(\lambda (_: (ex A (\lambda -(a1: A).(leq g a0 (asucc g a1))))).(\lambda (a1: A).(\lambda (H0: (ex A -(\lambda (a2: A).(leq g a1 (asucc g a2))))).(let H1 \def H0 in (ex_ind A -(\lambda (a2: A).(leq g a1 (asucc g a2))) (ex A (\lambda (a2: A).(leq g -(AHead a0 a1) (asucc g a2)))) (\lambda (x: A).(\lambda (H2: (leq g a1 (asucc -g x))).(ex_intro A (\lambda (a2: A).(leq g (AHead a0 a1) (asucc g a2))) -(AHead a0 x) (leq_head g a0 a0 (leq_refl g a0) a1 (asucc g x) H2)))) H1)))))) -a)). -(* COMMENTS -Initial nodes: 221 -END *) - -theorem leq_ahead_asucc_false: - \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (AHead a1 a2) -(asucc g a1)) \to (\forall (P: Prop).P)))) -\def - \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2: -A).((leq g (AHead a a2) (asucc g a)) \to (\forall (P: Prop).P)))) (\lambda -(n: nat).(\lambda (n0: nat).(\lambda (a2: A).(\lambda (H: (leq g (AHead -(ASort n n0) a2) (match n with [O \Rightarrow (ASort O (next g n0)) | (S h) -\Rightarrow (ASort h n0)]))).(\lambda (P: Prop).(nat_ind (\lambda (n1: -nat).((leq g (AHead (ASort n1 n0) a2) (match n1 with [O \Rightarrow (ASort O -(next g n0)) | (S h) \Rightarrow (ASort h n0)])) \to P)) (\lambda (H0: (leq g -(AHead (ASort O n0) a2) (ASort O (next g n0)))).(let H_x \def (leq_gen_head1 -g (ASort O n0) a2 (ASort O (next g n0)) H0) in (let H1 \def H_x in (ex3_2_ind -A A (\lambda (a3: A).(\lambda (_: A).(leq g (ASort O n0) a3))) (\lambda (_: -A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A -(ASort O (next g n0)) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: -A).(\lambda (_: (leq g (ASort O n0) x0)).(\lambda (_: (leq g a2 x1)).(\lambda -(H4: (eq A (ASort O (next g n0)) (AHead x0 x1))).(let H5 \def (eq_ind A -(ASort O (next g n0)) (\lambda (ee: A).(match ee in A return (\lambda (_: -A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow -False])) I (AHead x0 x1) H4) in (False_ind P H5))))))) H1)))) (\lambda (n1: -nat).(\lambda (_: (((leq g (AHead (ASort n1 n0) a2) (match n1 with [O -\Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)])) \to -P))).(\lambda (H0: (leq g (AHead (ASort (S n1) n0) a2) (ASort n1 n0))).(let -H_x \def (leq_gen_head1 g (ASort (S n1) n0) a2 (ASort n1 n0) H0) in (let H1 -\def H_x in (ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq g (ASort (S -n1) n0) a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: -A).(\lambda (a4: A).(eq A (ASort n1 n0) (AHead a3 a4)))) P (\lambda (x0: -A).(\lambda (x1: A).(\lambda (_: (leq g (ASort (S n1) n0) x0)).(\lambda (_: -(leq g a2 x1)).(\lambda (H4: (eq A (ASort n1 n0) (AHead x0 x1))).(let H5 \def -(eq_ind A (ASort n1 n0) (\lambda (ee: A).(match ee in A return (\lambda (_: -A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow -False])) I (AHead x0 x1) H4) in (False_ind P H5))))))) H1)))))) n H)))))) -(\lambda (a: A).(\lambda (_: ((\forall (a2: A).((leq g (AHead a a2) (asucc g -a)) \to (\forall (P: Prop).P))))).(\lambda (a0: A).(\lambda (_: ((\forall -(a2: A).((leq g (AHead a0 a2) (asucc g a0)) \to (\forall (P: -Prop).P))))).(\lambda (a2: A).(\lambda (H1: (leq g (AHead (AHead a a0) a2) -(AHead a (asucc g a0)))).(\lambda (P: Prop).(let H_x \def (leq_gen_head1 g -(AHead a a0) a2 (AHead a (asucc g a0)) H1) in (let H2 \def H_x in (ex3_2_ind -A A (\lambda (a3: A).(\lambda (_: A).(leq g (AHead a a0) a3))) (\lambda (_: -A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A -(AHead a (asucc g a0)) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: -A).(\lambda (H3: (leq g (AHead a a0) x0)).(\lambda (H4: (leq g a2 -x1)).(\lambda (H5: (eq A (AHead a (asucc g a0)) (AHead x0 x1))).(let H6 \def -(f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with -[(ASort _ _) \Rightarrow a | (AHead a3 _) \Rightarrow a3])) (AHead a (asucc g -a0)) (AHead x0 x1) H5) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e -in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow ((let rec asucc -(g0: G) (l: A) on l: A \def (match l with [(ASort n0 n) \Rightarrow (match n0 -with [O \Rightarrow (ASort O (next g0 n)) | (S h) \Rightarrow (ASort h n)]) | -(AHead a3 a4) \Rightarrow (AHead a3 (asucc g0 a4))]) in asucc) g a0) | (AHead -_ a3) \Rightarrow a3])) (AHead a (asucc g a0)) (AHead x0 x1) H5) in (\lambda -(H8: (eq A a x0)).(let H9 \def (eq_ind_r A x1 (\lambda (a3: A).(leq g a2 a3)) -H4 (asucc g a0) H7) in (let H10 \def (eq_ind_r A x0 (\lambda (a3: A).(leq g -(AHead a a0) a3)) H3 a H8) in (leq_ahead_false_1 g a a0 H10 P))))) H6))))))) -H2)))))))))) a1)). -(* COMMENTS -Initial nodes: 927 -END *) - -theorem leq_asucc_false: - \forall (g: G).(\forall (a: A).((leq g (asucc g a) a) \to (\forall (P: -Prop).P))) -\def - \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).((leq g (asucc g a0) -a0) \to (\forall (P: Prop).P))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda -(H: (leq g (match n with [O \Rightarrow (ASort O (next g n0)) | (S h) -\Rightarrow (ASort h n0)]) (ASort n n0))).(\lambda (P: Prop).(nat_ind -(\lambda (n1: nat).((leq g (match n1 with [O \Rightarrow (ASort O (next g -n0)) | (S h) \Rightarrow (ASort h n0)]) (ASort n1 n0)) \to P)) (\lambda (H0: -(leq g (ASort O (next g n0)) (ASort O n0))).(let H_x \def (leq_gen_sort1 g O -(next g n0) (ASort O n0) H0) in (let H1 \def H_x in (ex2_3_ind nat nat nat -(\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort -O (next g n0)) k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda -(h2: nat).(\lambda (_: nat).(eq A (ASort O n0) (ASort h2 n2))))) P (\lambda -(x0: nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (H2: (eq A (aplus g -(ASort O (next g n0)) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H3: (eq A -(ASort O n0) (ASort x1 x0))).(let H4 \def (f_equal A nat (\lambda (e: -A).(match e in A return (\lambda (_: A).nat) with [(ASort n1 _) \Rightarrow -n1 | (AHead _ _) \Rightarrow O])) (ASort O n0) (ASort x1 x0) H3) in ((let H5 -\def (f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) -with [(ASort _ n1) \Rightarrow n1 | (AHead _ _) \Rightarrow n0])) (ASort O -n0) (ASort x1 x0) H3) in (\lambda (H6: (eq nat O x1)).(let H7 \def (eq_ind_r -nat x1 (\lambda (n1: nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g -(ASort n1 x0) x2))) H2 O H6) in (let H8 \def (eq_ind_r nat x0 (\lambda (n1: -nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort O n1) x2))) H7 -n0 H5) in (let H9 \def (eq_ind_r A (aplus g (ASort O (next g n0)) x2) -(\lambda (a0: A).(eq A a0 (aplus g (ASort O n0) x2))) H8 (aplus g (ASort O -n0) (S x2)) (aplus_sort_O_S_simpl g n0 x2)) in (let H_y \def (aplus_inj g (S -x2) x2 (ASort O n0) H9) in (le_Sx_x x2 (eq_ind_r nat x2 (\lambda (n1: -nat).(le n1 x2)) (le_n x2) (S x2) H_y) P))))))) H4))))))) H1)))) (\lambda -(n1: nat).(\lambda (_: (((leq g (match n1 with [O \Rightarrow (ASort O (next -g n0)) | (S h) \Rightarrow (ASort h n0)]) (ASort n1 n0)) \to P))).(\lambda -(H0: (leq g (ASort n1 n0) (ASort (S n1) n0))).(let H_x \def (leq_gen_sort1 g -n1 n0 (ASort (S n1) n0) H0) in (let H1 \def H_x in (ex2_3_ind nat nat nat -(\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort -n1 n0) k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: -nat).(\lambda (_: nat).(eq A (ASort (S n1) n0) (ASort h2 n2))))) P (\lambda -(x0: nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (H2: (eq A (aplus g -(ASort n1 n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H3: (eq A (ASort (S -n1) n0) (ASort x1 x0))).(let H4 \def (f_equal A nat (\lambda (e: A).(match e -in A return (\lambda (_: A).nat) with [(ASort n2 _) \Rightarrow n2 | (AHead _ -_) \Rightarrow (S n1)])) (ASort (S n1) n0) (ASort x1 x0) H3) in ((let H5 \def -(f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with -[(ASort _ n2) \Rightarrow n2 | (AHead _ _) \Rightarrow n0])) (ASort (S n1) -n0) (ASort x1 x0) H3) in (\lambda (H6: (eq nat (S n1) x1)).(let H7 \def -(eq_ind_r nat x1 (\lambda (n2: nat).(eq A (aplus g (ASort n1 n0) x2) (aplus g -(ASort n2 x0) x2))) H2 (S n1) H6) in (let H8 \def (eq_ind_r nat x0 (\lambda -(n2: nat).(eq A (aplus g (ASort n1 n0) x2) (aplus g (ASort (S n1) n2) x2))) -H7 n0 H5) in (let H9 \def (eq_ind_r A (aplus g (ASort n1 n0) x2) (\lambda -(a0: A).(eq A a0 (aplus g (ASort (S n1) n0) x2))) H8 (aplus g (ASort (S n1) -n0) (S x2)) (aplus_sort_S_S_simpl g n0 n1 x2)) in (let H_y \def (aplus_inj g -(S x2) x2 (ASort (S n1) n0) H9) in (le_Sx_x x2 (eq_ind_r nat x2 (\lambda (n2: -nat).(le n2 x2)) (le_n x2) (S x2) H_y) P))))))) H4))))))) H1)))))) n H))))) -(\lambda (a0: A).(\lambda (_: (((leq g (asucc g a0) a0) \to (\forall (P: -Prop).P)))).(\lambda (a1: A).(\lambda (H0: (((leq g (asucc g a1) a1) \to -(\forall (P: Prop).P)))).(\lambda (H1: (leq g (AHead a0 (asucc g a1)) (AHead -a0 a1))).(\lambda (P: Prop).(let H_x \def (leq_gen_head1 g a0 (asucc g a1) -(AHead a0 a1) H1) in (let H2 \def H_x in (ex3_2_ind A A (\lambda (a3: -A).(\lambda (_: A).(leq g a0 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g -(asucc g a1) a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (AHead a0 a1) -(AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (H3: (leq g a0 -x0)).(\lambda (H4: (leq g (asucc g a1) x1)).(\lambda (H5: (eq A (AHead a0 a1) -(AHead x0 x1))).(let H6 \def (f_equal A A (\lambda (e: A).(match e in A -return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead a2 _) -\Rightarrow a2])) (AHead a0 a1) (AHead x0 x1) H5) in ((let H7 \def (f_equal A -A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) -\Rightarrow a1 | (AHead _ a2) \Rightarrow a2])) (AHead a0 a1) (AHead x0 x1) -H5) in (\lambda (H8: (eq A a0 x0)).(let H9 \def (eq_ind_r A x1 (\lambda (a2: -A).(leq g (asucc g a1) a2)) H4 a1 H7) in (let H10 \def (eq_ind_r A x0 -(\lambda (a2: A).(leq g a0 a2)) H3 a0 H8) in (H0 H9 P))))) H6))))))) -H2))))))))) a)). -(* COMMENTS -Initial nodes: 1327 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/leq/defs.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/leq/defs.ma deleted file mode 100644 index 5a5308280..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/leq/defs.ma +++ /dev/null @@ -1,26 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/aplus/defs.ma". - -inductive leq (g: G): A \to (A \to Prop) \def -| leq_sort: \forall (h1: nat).(\forall (h2: nat).(\forall (n1: nat).(\forall -(n2: nat).(\forall (k: nat).((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort -h2 n2) k)) \to (leq g (ASort h1 n1) (ASort h2 n2))))))) -| leq_head: \forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall (a3: -A).(\forall (a4: A).((leq g a3 a4) \to (leq g (AHead a1 a3) (AHead a2 -a4))))))). - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/leq/fwd.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/leq/fwd.ma deleted file mode 100644 index e259fa19e..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/leq/fwd.ma +++ /dev/null @@ -1,244 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/leq/defs.ma". - -theorem leq_gen_sort1: - \forall (g: G).(\forall (h1: nat).(\forall (n1: nat).(\forall (a2: A).((leq -g (ASort h1 n1) a2) \to (ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: -nat).(\lambda (k: nat).(eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) -k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a2 -(ASort h2 n2)))))))))) -\def - \lambda (g: G).(\lambda (h1: nat).(\lambda (n1: nat).(\lambda (a2: -A).(\lambda (H: (leq g (ASort h1 n1) a2)).(insert_eq A (ASort h1 n1) (\lambda -(a: A).(leq g a a2)) (\lambda (a: A).(ex2_3 nat nat nat (\lambda (n2: -nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g a k) (aplus g (ASort -h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A -a2 (ASort h2 n2))))))) (\lambda (y: A).(\lambda (H0: (leq g y a2)).(leq_ind g -(\lambda (a: A).(\lambda (a0: A).((eq A a (ASort h1 n1)) \to (ex2_3 nat nat -nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g a -k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: -nat).(\lambda (_: nat).(eq A a0 (ASort h2 n2))))))))) (\lambda (h0: -nat).(\lambda (h2: nat).(\lambda (n0: nat).(\lambda (n2: nat).(\lambda (k: -nat).(\lambda (H1: (eq A (aplus g (ASort h0 n0) k) (aplus g (ASort h2 n2) -k))).(\lambda (H2: (eq A (ASort h0 n0) (ASort h1 n1))).(let H3 \def (f_equal -A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort -n _) \Rightarrow n | (AHead _ _) \Rightarrow h0])) (ASort h0 n0) (ASort h1 -n1) H2) in ((let H4 \def (f_equal A nat (\lambda (e: A).(match e in A return -(\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) -\Rightarrow n0])) (ASort h0 n0) (ASort h1 n1) H2) in (\lambda (H5: (eq nat h0 -h1)).(let H6 \def (eq_ind nat n0 (\lambda (n: nat).(eq A (aplus g (ASort h0 -n) k) (aplus g (ASort h2 n2) k))) H1 n1 H4) in (eq_ind_r nat n1 (\lambda (n: -nat).(ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (k0: -nat).(eq A (aplus g (ASort h0 n) k0) (aplus g (ASort h3 n3) k0))))) (\lambda -(n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A (ASort h2 n2) (ASort h3 -n3))))))) (let H7 \def (eq_ind nat h0 (\lambda (n: nat).(eq A (aplus g (ASort -n n1) k) (aplus g (ASort h2 n2) k))) H6 h1 H5) in (eq_ind_r nat h1 (\lambda -(n: nat).(ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda -(k0: nat).(eq A (aplus g (ASort n n1) k0) (aplus g (ASort h3 n3) k0))))) -(\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A (ASort h2 n2) -(ASort h3 n3))))))) (ex2_3_intro nat nat nat (\lambda (n3: nat).(\lambda (h3: -nat).(\lambda (k0: nat).(eq A (aplus g (ASort h1 n1) k0) (aplus g (ASort h3 -n3) k0))))) (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A -(ASort h2 n2) (ASort h3 n3))))) n2 h2 k H7 (refl_equal A (ASort h2 n2))) h0 -H5)) n0 H4)))) H3))))))))) (\lambda (a1: A).(\lambda (a3: A).(\lambda (_: -(leq g a1 a3)).(\lambda (_: (((eq A a1 (ASort h1 n1)) \to (ex2_3 nat nat nat -(\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g a1 k) -(aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda -(_: nat).(eq A a3 (ASort h2 n2))))))))).(\lambda (a4: A).(\lambda (a5: -A).(\lambda (_: (leq g a4 a5)).(\lambda (_: (((eq A a4 (ASort h1 n1)) \to -(ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: -nat).(eq A (aplus g a4 k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: -nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a5 (ASort h2 -n2))))))))).(\lambda (H5: (eq A (AHead a1 a4) (ASort h1 n1))).(let H6 \def -(eq_ind A (AHead a1 a4) (\lambda (ee: A).(match ee in A return (\lambda (_: -A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow -True])) I (ASort h1 n1) H5) in (False_ind (ex2_3 nat nat nat (\lambda (n2: -nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (AHead a1 a4) k) -(aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda -(_: nat).(eq A (AHead a3 a5) (ASort h2 n2)))))) H6))))))))))) y a2 H0))) -H))))). -(* COMMENTS -Initial nodes: 913 -END *) - -theorem leq_gen_head1: - \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a: A).((leq g -(AHead a1 a2) a) \to (ex3_2 A A (\lambda (a3: A).(\lambda (_: A).(leq g a1 -a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: -A).(\lambda (a4: A).(eq A a (AHead a3 a4))))))))) -\def - \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a: A).(\lambda -(H: (leq g (AHead a1 a2) a)).(insert_eq A (AHead a1 a2) (\lambda (a0: A).(leq -g a0 a)) (\lambda (_: A).(ex3_2 A A (\lambda (a3: A).(\lambda (_: A).(leq g -a1 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: -A).(\lambda (a4: A).(eq A a (AHead a3 a4)))))) (\lambda (y: A).(\lambda (H0: -(leq g y a)).(leq_ind g (\lambda (a0: A).(\lambda (a3: A).((eq A a0 (AHead a1 -a2)) \to (ex3_2 A A (\lambda (a4: A).(\lambda (_: A).(leq g a1 a4))) (\lambda -(_: A).(\lambda (a5: A).(leq g a2 a5))) (\lambda (a4: A).(\lambda (a5: A).(eq -A a3 (AHead a4 a5)))))))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: -nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (_: (eq A (aplus g (ASort -h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (H2: (eq A (ASort h1 n1) -(AHead a1 a2))).(let H3 \def (eq_ind A (ASort h1 n1) (\lambda (ee: A).(match -ee in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | -(AHead _ _) \Rightarrow False])) I (AHead a1 a2) H2) in (False_ind (ex3_2 A A -(\lambda (a3: A).(\lambda (_: A).(leq g a1 a3))) (\lambda (_: A).(\lambda -(a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort h2 n2) -(AHead a3 a4))))) H3))))))))) (\lambda (a0: A).(\lambda (a3: A).(\lambda (H1: -(leq g a0 a3)).(\lambda (H2: (((eq A a0 (AHead a1 a2)) \to (ex3_2 A A -(\lambda (a4: A).(\lambda (_: A).(leq g a1 a4))) (\lambda (_: A).(\lambda -(a5: A).(leq g a2 a5))) (\lambda (a4: A).(\lambda (a5: A).(eq A a3 (AHead a4 -a5)))))))).(\lambda (a4: A).(\lambda (a5: A).(\lambda (H3: (leq g a4 -a5)).(\lambda (H4: (((eq A a4 (AHead a1 a2)) \to (ex3_2 A A (\lambda (a6: -A).(\lambda (_: A).(leq g a1 a6))) (\lambda (_: A).(\lambda (a7: A).(leq g a2 -a7))) (\lambda (a6: A).(\lambda (a7: A).(eq A a5 (AHead a6 -a7)))))))).(\lambda (H5: (eq A (AHead a0 a4) (AHead a1 a2))).(let H6 \def -(f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with -[(ASort _ _) \Rightarrow a0 | (AHead a6 _) \Rightarrow a6])) (AHead a0 a4) -(AHead a1 a2) H5) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e in A -return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a4 | (AHead _ a6) -\Rightarrow a6])) (AHead a0 a4) (AHead a1 a2) H5) in (\lambda (H8: (eq A a0 -a1)).(let H9 \def (eq_ind A a4 (\lambda (a6: A).((eq A a6 (AHead a1 a2)) \to -(ex3_2 A A (\lambda (a7: A).(\lambda (_: A).(leq g a1 a7))) (\lambda (_: -A).(\lambda (a8: A).(leq g a2 a8))) (\lambda (a7: A).(\lambda (a8: A).(eq A -a5 (AHead a7 a8))))))) H4 a2 H7) in (let H10 \def (eq_ind A a4 (\lambda (a6: -A).(leq g a6 a5)) H3 a2 H7) in (let H11 \def (eq_ind A a0 (\lambda (a6: -A).((eq A a6 (AHead a1 a2)) \to (ex3_2 A A (\lambda (a7: A).(\lambda (_: -A).(leq g a1 a7))) (\lambda (_: A).(\lambda (a8: A).(leq g a2 a8))) (\lambda -(a7: A).(\lambda (a8: A).(eq A a3 (AHead a7 a8))))))) H2 a1 H8) in (let H12 -\def (eq_ind A a0 (\lambda (a6: A).(leq g a6 a3)) H1 a1 H8) in (ex3_2_intro A -A (\lambda (a6: A).(\lambda (_: A).(leq g a1 a6))) (\lambda (_: A).(\lambda -(a7: A).(leq g a2 a7))) (\lambda (a6: A).(\lambda (a7: A).(eq A (AHead a3 a5) -(AHead a6 a7)))) a3 a5 H12 H10 (refl_equal A (AHead a3 a5))))))))) -H6))))))))))) y a H0))) H))))). -(* COMMENTS -Initial nodes: 797 -END *) - -theorem leq_gen_sort2: - \forall (g: G).(\forall (h1: nat).(\forall (n1: nat).(\forall (a2: A).((leq -g a2 (ASort h1 n1)) \to (ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: -nat).(\lambda (k: nat).(eq A (aplus g (ASort h2 n2) k) (aplus g (ASort h1 n1) -k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a2 -(ASort h2 n2)))))))))) -\def - \lambda (g: G).(\lambda (h1: nat).(\lambda (n1: nat).(\lambda (a2: -A).(\lambda (H: (leq g a2 (ASort h1 n1))).(insert_eq A (ASort h1 n1) (\lambda -(a: A).(leq g a2 a)) (\lambda (a: A).(ex2_3 nat nat nat (\lambda (n2: -nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort h2 n2) k) -(aplus g a k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq -A a2 (ASort h2 n2))))))) (\lambda (y: A).(\lambda (H0: (leq g a2 y)).(leq_ind -g (\lambda (a: A).(\lambda (a0: A).((eq A a0 (ASort h1 n1)) \to (ex2_3 nat -nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus -g (ASort h2 n2) k) (aplus g a0 k))))) (\lambda (n2: nat).(\lambda (h2: -nat).(\lambda (_: nat).(eq A a (ASort h2 n2))))))))) (\lambda (h0: -nat).(\lambda (h2: nat).(\lambda (n0: nat).(\lambda (n2: nat).(\lambda (k: -nat).(\lambda (H1: (eq A (aplus g (ASort h0 n0) k) (aplus g (ASort h2 n2) -k))).(\lambda (H2: (eq A (ASort h2 n2) (ASort h1 n1))).(let H3 \def (f_equal -A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort -n _) \Rightarrow n | (AHead _ _) \Rightarrow h2])) (ASort h2 n2) (ASort h1 -n1) H2) in ((let H4 \def (f_equal A nat (\lambda (e: A).(match e in A return -(\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) -\Rightarrow n2])) (ASort h2 n2) (ASort h1 n1) H2) in (\lambda (H5: (eq nat h2 -h1)).(let H6 \def (eq_ind nat n2 (\lambda (n: nat).(eq A (aplus g (ASort h0 -n0) k) (aplus g (ASort h2 n) k))) H1 n1 H4) in (eq_ind_r nat n1 (\lambda (n: -nat).(ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (k0: -nat).(eq A (aplus g (ASort h3 n3) k0) (aplus g (ASort h2 n) k0))))) (\lambda -(n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A (ASort h0 n0) (ASort h3 -n3))))))) (let H7 \def (eq_ind nat h2 (\lambda (n: nat).(eq A (aplus g (ASort -h0 n0) k) (aplus g (ASort n n1) k))) H6 h1 H5) in (eq_ind_r nat h1 (\lambda -(n: nat).(ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda -(k0: nat).(eq A (aplus g (ASort h3 n3) k0) (aplus g (ASort n n1) k0))))) -(\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A (ASort h0 n0) -(ASort h3 n3))))))) (ex2_3_intro nat nat nat (\lambda (n3: nat).(\lambda (h3: -nat).(\lambda (k0: nat).(eq A (aplus g (ASort h3 n3) k0) (aplus g (ASort h1 -n1) k0))))) (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A -(ASort h0 n0) (ASort h3 n3))))) n0 h0 k H7 (refl_equal A (ASort h0 n0))) h2 -H5)) n2 H4)))) H3))))))))) (\lambda (a1: A).(\lambda (a3: A).(\lambda (_: -(leq g a1 a3)).(\lambda (_: (((eq A a3 (ASort h1 n1)) \to (ex2_3 nat nat nat -(\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort -h2 n2) k) (aplus g a3 k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda -(_: nat).(eq A a1 (ASort h2 n2))))))))).(\lambda (a4: A).(\lambda (a5: -A).(\lambda (_: (leq g a4 a5)).(\lambda (_: (((eq A a5 (ASort h1 n1)) \to -(ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: -nat).(eq A (aplus g (ASort h2 n2) k) (aplus g a5 k))))) (\lambda (n2: -nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a4 (ASort h2 -n2))))))))).(\lambda (H5: (eq A (AHead a3 a5) (ASort h1 n1))).(let H6 \def -(eq_ind A (AHead a3 a5) (\lambda (ee: A).(match ee in A return (\lambda (_: -A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow -True])) I (ASort h1 n1) H5) in (False_ind (ex2_3 nat nat nat (\lambda (n2: -nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort h2 n2) k) -(aplus g (AHead a3 a5) k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda -(_: nat).(eq A (AHead a1 a4) (ASort h2 n2)))))) H6))))))))))) a2 y H0))) -H))))). -(* COMMENTS -Initial nodes: 913 -END *) - -theorem leq_gen_head2: - \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a: A).((leq g a -(AHead a1 a2)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (_: A).(leq g a3 -a1))) (\lambda (_: A).(\lambda (a4: A).(leq g a4 a2))) (\lambda (a3: -A).(\lambda (a4: A).(eq A a (AHead a3 a4))))))))) -\def - \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a: A).(\lambda -(H: (leq g a (AHead a1 a2))).(insert_eq A (AHead a1 a2) (\lambda (a0: A).(leq -g a a0)) (\lambda (_: A).(ex3_2 A A (\lambda (a3: A).(\lambda (_: A).(leq g -a3 a1))) (\lambda (_: A).(\lambda (a4: A).(leq g a4 a2))) (\lambda (a3: -A).(\lambda (a4: A).(eq A a (AHead a3 a4)))))) (\lambda (y: A).(\lambda (H0: -(leq g a y)).(leq_ind g (\lambda (a0: A).(\lambda (a3: A).((eq A a3 (AHead a1 -a2)) \to (ex3_2 A A (\lambda (a4: A).(\lambda (_: A).(leq g a4 a1))) (\lambda -(_: A).(\lambda (a5: A).(leq g a5 a2))) (\lambda (a4: A).(\lambda (a5: A).(eq -A a0 (AHead a4 a5)))))))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: -nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (_: (eq A (aplus g (ASort -h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (H2: (eq A (ASort h2 n2) -(AHead a1 a2))).(let H3 \def (eq_ind A (ASort h2 n2) (\lambda (ee: A).(match -ee in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | -(AHead _ _) \Rightarrow False])) I (AHead a1 a2) H2) in (False_ind (ex3_2 A A -(\lambda (a3: A).(\lambda (_: A).(leq g a3 a1))) (\lambda (_: A).(\lambda -(a4: A).(leq g a4 a2))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort h1 n1) -(AHead a3 a4))))) H3))))))))) (\lambda (a0: A).(\lambda (a3: A).(\lambda (H1: -(leq g a0 a3)).(\lambda (H2: (((eq A a3 (AHead a1 a2)) \to (ex3_2 A A -(\lambda (a4: A).(\lambda (_: A).(leq g a4 a1))) (\lambda (_: A).(\lambda -(a5: A).(leq g a5 a2))) (\lambda (a4: A).(\lambda (a5: A).(eq A a0 (AHead a4 -a5)))))))).(\lambda (a4: A).(\lambda (a5: A).(\lambda (H3: (leq g a4 -a5)).(\lambda (H4: (((eq A a5 (AHead a1 a2)) \to (ex3_2 A A (\lambda (a6: -A).(\lambda (_: A).(leq g a6 a1))) (\lambda (_: A).(\lambda (a7: A).(leq g a7 -a2))) (\lambda (a6: A).(\lambda (a7: A).(eq A a4 (AHead a6 -a7)))))))).(\lambda (H5: (eq A (AHead a3 a5) (AHead a1 a2))).(let H6 \def -(f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with -[(ASort _ _) \Rightarrow a3 | (AHead a6 _) \Rightarrow a6])) (AHead a3 a5) -(AHead a1 a2) H5) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e in A -return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a5 | (AHead _ a6) -\Rightarrow a6])) (AHead a3 a5) (AHead a1 a2) H5) in (\lambda (H8: (eq A a3 -a1)).(let H9 \def (eq_ind A a5 (\lambda (a6: A).((eq A a6 (AHead a1 a2)) \to -(ex3_2 A A (\lambda (a7: A).(\lambda (_: A).(leq g a7 a1))) (\lambda (_: -A).(\lambda (a8: A).(leq g a8 a2))) (\lambda (a7: A).(\lambda (a8: A).(eq A -a4 (AHead a7 a8))))))) H4 a2 H7) in (let H10 \def (eq_ind A a5 (\lambda (a6: -A).(leq g a4 a6)) H3 a2 H7) in (let H11 \def (eq_ind A a3 (\lambda (a6: -A).((eq A a6 (AHead a1 a2)) \to (ex3_2 A A (\lambda (a7: A).(\lambda (_: -A).(leq g a7 a1))) (\lambda (_: A).(\lambda (a8: A).(leq g a8 a2))) (\lambda -(a7: A).(\lambda (a8: A).(eq A a0 (AHead a7 a8))))))) H2 a1 H8) in (let H12 -\def (eq_ind A a3 (\lambda (a6: A).(leq g a0 a6)) H1 a1 H8) in (ex3_2_intro A -A (\lambda (a6: A).(\lambda (_: A).(leq g a6 a1))) (\lambda (_: A).(\lambda -(a7: A).(leq g a7 a2))) (\lambda (a6: A).(\lambda (a7: A).(eq A (AHead a0 a4) -(AHead a6 a7)))) a0 a4 H12 H10 (refl_equal A (AHead a0 a4))))))))) -H6))))))))))) a y H0))) H))))). -(* COMMENTS -Initial nodes: 797 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/leq/props.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/leq/props.ma deleted file mode 100644 index b83fc503e..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/leq/props.ma +++ /dev/null @@ -1,233 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/leq/fwd.ma". - -include "Basic-1/aplus/props.ma". - -theorem ahead_inj_snd: - \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a3: A).(\forall -(a4: A).((leq g (AHead a1 a2) (AHead a3 a4)) \to (leq g a2 a4)))))) -\def - \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a3: A).(\lambda -(a4: A).(\lambda (H: (leq g (AHead a1 a2) (AHead a3 a4))).(let H_x \def -(leq_gen_head1 g a1 a2 (AHead a3 a4) H) in (let H0 \def H_x in (ex3_2_ind A A -(\lambda (a5: A).(\lambda (_: A).(leq g a1 a5))) (\lambda (_: A).(\lambda -(a6: A).(leq g a2 a6))) (\lambda (a5: A).(\lambda (a6: A).(eq A (AHead a3 a4) -(AHead a5 a6)))) (leq g a2 a4) (\lambda (x0: A).(\lambda (x1: A).(\lambda -(H1: (leq g a1 x0)).(\lambda (H2: (leq g a2 x1)).(\lambda (H3: (eq A (AHead -a3 a4) (AHead x0 x1))).(let H4 \def (f_equal A A (\lambda (e: A).(match e in -A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a3 | (AHead a _) -\Rightarrow a])) (AHead a3 a4) (AHead x0 x1) H3) in ((let H5 \def (f_equal A -A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) -\Rightarrow a4 | (AHead _ a) \Rightarrow a])) (AHead a3 a4) (AHead x0 x1) H3) -in (\lambda (H6: (eq A a3 x0)).(let H7 \def (eq_ind_r A x1 (\lambda (a: -A).(leq g a2 a)) H2 a4 H5) in (let H8 \def (eq_ind_r A x0 (\lambda (a: -A).(leq g a1 a)) H1 a3 H6) in H7)))) H4))))))) H0)))))))). -(* COMMENTS -Initial nodes: 259 -END *) - -theorem leq_refl: - \forall (g: G).(\forall (a: A).(leq g a a)) -\def - \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(leq g a0 a0)) -(\lambda (n: nat).(\lambda (n0: nat).(leq_sort g n n n0 n0 O (refl_equal A -(aplus g (ASort n n0) O))))) (\lambda (a0: A).(\lambda (H: (leq g a0 -a0)).(\lambda (a1: A).(\lambda (H0: (leq g a1 a1)).(leq_head g a0 a0 H a1 a1 -H0))))) a)). -(* COMMENTS -Initial nodes: 87 -END *) - -theorem leq_eq: - \forall (g: G).(\forall (a1: A).(\forall (a2: A).((eq A a1 a2) \to (leq g a1 -a2)))) -\def - \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (eq A a1 -a2)).(eq_ind A a1 (\lambda (a: A).(leq g a1 a)) (leq_refl g a1) a2 H)))). -(* COMMENTS -Initial nodes: 39 -END *) - -theorem leq_sym: - \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (leq g -a2 a1)))) -\def - \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1 -a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(leq g a0 a))) (\lambda (h1: -nat).(\lambda (h2: nat).(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (k: -nat).(\lambda (H0: (eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) -k))).(leq_sort g h2 h1 n2 n1 k (sym_eq A (aplus g (ASort h1 n1) k) (aplus g -(ASort h2 n2) k) H0)))))))) (\lambda (a3: A).(\lambda (a4: A).(\lambda (_: -(leq g a3 a4)).(\lambda (H1: (leq g a4 a3)).(\lambda (a5: A).(\lambda (a6: -A).(\lambda (_: (leq g a5 a6)).(\lambda (H3: (leq g a6 a5)).(leq_head g a4 a3 -H1 a6 a5 H3))))))))) a1 a2 H)))). -(* COMMENTS -Initial nodes: 173 -END *) - -theorem leq_trans: - \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall -(a3: A).((leq g a2 a3) \to (leq g a1 a3)))))) -\def - \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1 -a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(\forall (a3: A).((leq g a0 -a3) \to (leq g a a3))))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: -nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (H0: (eq A (aplus g (ASort -h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (a3: A).(\lambda (H1: (leq g -(ASort h2 n2) a3)).(let H_x \def (leq_gen_sort1 g h2 n2 a3 H1) in (let H2 -\def H_x in (ex2_3_ind nat nat nat (\lambda (n3: nat).(\lambda (h3: -nat).(\lambda (k0: nat).(eq A (aplus g (ASort h2 n2) k0) (aplus g (ASort h3 -n3) k0))))) (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A a3 -(ASort h3 n3))))) (leq g (ASort h1 n1) a3) (\lambda (x0: nat).(\lambda (x1: -nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort h2 n2) x2) (aplus -g (ASort x1 x0) x2))).(\lambda (H4: (eq A a3 (ASort x1 x0))).(let H5 \def -(f_equal A A (\lambda (e: A).e) a3 (ASort x1 x0) H4) in (eq_ind_r A (ASort x1 -x0) (\lambda (a: A).(leq g (ASort h1 n1) a)) (lt_le_e k x2 (leq g (ASort h1 -n1) (ASort x1 x0)) (\lambda (H6: (lt k x2)).(let H_y \def (aplus_reg_r g -(ASort h1 n1) (ASort h2 n2) k k H0 (minus x2 k)) in (let H7 \def (eq_ind_r -nat (plus (minus x2 k) k) (\lambda (n: nat).(eq A (aplus g (ASort h1 n1) n) -(aplus g (ASort h2 n2) n))) H_y x2 (le_plus_minus_sym k x2 (le_trans k (S k) -x2 (le_S k k (le_n k)) H6))) in (leq_sort g h1 x1 n1 x0 x2 (trans_eq A (aplus -g (ASort h1 n1) x2) (aplus g (ASort h2 n2) x2) (aplus g (ASort x1 x0) x2) H7 -H3))))) (\lambda (H6: (le x2 k)).(let H_y \def (aplus_reg_r g (ASort h2 n2) -(ASort x1 x0) x2 x2 H3 (minus k x2)) in (let H7 \def (eq_ind_r nat (plus -(minus k x2) x2) (\lambda (n: nat).(eq A (aplus g (ASort h2 n2) n) (aplus g -(ASort x1 x0) n))) H_y k (le_plus_minus_sym x2 k H6)) in (leq_sort g h1 x1 n1 -x0 k (trans_eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k) (aplus g -(ASort x1 x0) k) H0 H7)))))) a3 H5))))))) H2))))))))))) (\lambda (a3: -A).(\lambda (a4: A).(\lambda (_: (leq g a3 a4)).(\lambda (H1: ((\forall (a5: -A).((leq g a4 a5) \to (leq g a3 a5))))).(\lambda (a5: A).(\lambda (a6: -A).(\lambda (_: (leq g a5 a6)).(\lambda (H3: ((\forall (a7: A).((leq g a6 a7) -\to (leq g a5 a7))))).(\lambda (a0: A).(\lambda (H4: (leq g (AHead a4 a6) -a0)).(let H_x \def (leq_gen_head1 g a4 a6 a0 H4) in (let H5 \def H_x in -(ex3_2_ind A A (\lambda (a7: A).(\lambda (_: A).(leq g a4 a7))) (\lambda (_: -A).(\lambda (a8: A).(leq g a6 a8))) (\lambda (a7: A).(\lambda (a8: A).(eq A -a0 (AHead a7 a8)))) (leq g (AHead a3 a5) a0) (\lambda (x0: A).(\lambda (x1: -A).(\lambda (H6: (leq g a4 x0)).(\lambda (H7: (leq g a6 x1)).(\lambda (H8: -(eq A a0 (AHead x0 x1))).(let H9 \def (f_equal A A (\lambda (e: A).e) a0 -(AHead x0 x1) H8) in (eq_ind_r A (AHead x0 x1) (\lambda (a: A).(leq g (AHead -a3 a5) a)) (leq_head g a3 x0 (H1 x0 H6) a5 x1 (H3 x1 H7)) a0 H9))))))) -H5))))))))))))) a1 a2 H)))). -(* COMMENTS -Initial nodes: 869 -END *) - -theorem leq_ahead_false_1: - \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (AHead a1 a2) a1) -\to (\forall (P: Prop).P)))) -\def - \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2: -A).((leq g (AHead a a2) a) \to (\forall (P: Prop).P)))) (\lambda (n: -nat).(\lambda (n0: nat).(\lambda (a2: A).(\lambda (H: (leq g (AHead (ASort n -n0) a2) (ASort n n0))).(\lambda (P: Prop).(nat_ind (\lambda (n1: nat).((leq g -(AHead (ASort n1 n0) a2) (ASort n1 n0)) \to P)) (\lambda (H0: (leq g (AHead -(ASort O n0) a2) (ASort O n0))).(let H_x \def (leq_gen_head1 g (ASort O n0) -a2 (ASort O n0) H0) in (let H1 \def H_x in (ex3_2_ind A A (\lambda (a3: -A).(\lambda (_: A).(leq g (ASort O n0) a3))) (\lambda (_: A).(\lambda (a4: -A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort O n0) -(AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g -(ASort O n0) x0)).(\lambda (_: (leq g a2 x1)).(\lambda (H4: (eq A (ASort O -n0) (AHead x0 x1))).(let H5 \def (eq_ind A (ASort O n0) (\lambda (ee: -A).(match ee in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow -True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H4) in (False_ind P -H5))))))) H1)))) (\lambda (n1: nat).(\lambda (_: (((leq g (AHead (ASort n1 -n0) a2) (ASort n1 n0)) \to P))).(\lambda (H0: (leq g (AHead (ASort (S n1) n0) -a2) (ASort (S n1) n0))).(let H_x \def (leq_gen_head1 g (ASort (S n1) n0) a2 -(ASort (S n1) n0) H0) in (let H1 \def H_x in (ex3_2_ind A A (\lambda (a3: -A).(\lambda (_: A).(leq g (ASort (S n1) n0) a3))) (\lambda (_: A).(\lambda -(a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort (S n1) -n0) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g -(ASort (S n1) n0) x0)).(\lambda (_: (leq g a2 x1)).(\lambda (H4: (eq A (ASort -(S n1) n0) (AHead x0 x1))).(let H5 \def (eq_ind A (ASort (S n1) n0) (\lambda -(ee: A).(match ee in A return (\lambda (_: A).Prop) with [(ASort _ _) -\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H4) in -(False_ind P H5))))))) H1)))))) n H)))))) (\lambda (a: A).(\lambda (H: -((\forall (a2: A).((leq g (AHead a a2) a) \to (\forall (P: -Prop).P))))).(\lambda (a0: A).(\lambda (_: ((\forall (a2: A).((leq g (AHead -a0 a2) a0) \to (\forall (P: Prop).P))))).(\lambda (a2: A).(\lambda (H1: (leq -g (AHead (AHead a a0) a2) (AHead a a0))).(\lambda (P: Prop).(let H_x \def -(leq_gen_head1 g (AHead a a0) a2 (AHead a a0) H1) in (let H2 \def H_x in -(ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq g (AHead a a0) a3))) -(\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda -(a4: A).(eq A (AHead a a0) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: -A).(\lambda (H3: (leq g (AHead a a0) x0)).(\lambda (H4: (leq g a2 -x1)).(\lambda (H5: (eq A (AHead a a0) (AHead x0 x1))).(let H6 \def (f_equal A -A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) -\Rightarrow a | (AHead a3 _) \Rightarrow a3])) (AHead a a0) (AHead x0 x1) H5) -in ((let H7 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda -(_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead _ a3) \Rightarrow a3])) -(AHead a a0) (AHead x0 x1) H5) in (\lambda (H8: (eq A a x0)).(let H9 \def -(eq_ind_r A x1 (\lambda (a3: A).(leq g a2 a3)) H4 a0 H7) in (let H10 \def -(eq_ind_r A x0 (\lambda (a3: A).(leq g (AHead a a0) a3)) H3 a H8) in (H a0 -H10 P))))) H6))))))) H2)))))))))) a1)). -(* COMMENTS -Initial nodes: 797 -END *) - -theorem leq_ahead_false_2: - \forall (g: G).(\forall (a2: A).(\forall (a1: A).((leq g (AHead a1 a2) a2) -\to (\forall (P: Prop).P)))) -\def - \lambda (g: G).(\lambda (a2: A).(A_ind (\lambda (a: A).(\forall (a1: -A).((leq g (AHead a1 a) a) \to (\forall (P: Prop).P)))) (\lambda (n: -nat).(\lambda (n0: nat).(\lambda (a1: A).(\lambda (H: (leq g (AHead a1 (ASort -n n0)) (ASort n n0))).(\lambda (P: Prop).(nat_ind (\lambda (n1: nat).((leq g -(AHead a1 (ASort n1 n0)) (ASort n1 n0)) \to P)) (\lambda (H0: (leq g (AHead -a1 (ASort O n0)) (ASort O n0))).(let H_x \def (leq_gen_head1 g a1 (ASort O -n0) (ASort O n0) H0) in (let H1 \def H_x in (ex3_2_ind A A (\lambda (a3: -A).(\lambda (_: A).(leq g a1 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g -(ASort O n0) a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort O n0) -(AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g a1 -x0)).(\lambda (_: (leq g (ASort O n0) x1)).(\lambda (H4: (eq A (ASort O n0) -(AHead x0 x1))).(let H5 \def (eq_ind A (ASort O n0) (\lambda (ee: A).(match -ee in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | -(AHead _ _) \Rightarrow False])) I (AHead x0 x1) H4) in (False_ind P -H5))))))) H1)))) (\lambda (n1: nat).(\lambda (_: (((leq g (AHead a1 (ASort n1 -n0)) (ASort n1 n0)) \to P))).(\lambda (H0: (leq g (AHead a1 (ASort (S n1) -n0)) (ASort (S n1) n0))).(let H_x \def (leq_gen_head1 g a1 (ASort (S n1) n0) -(ASort (S n1) n0) H0) in (let H1 \def H_x in (ex3_2_ind A A (\lambda (a3: -A).(\lambda (_: A).(leq g a1 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g -(ASort (S n1) n0) a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort (S n1) -n0) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g -a1 x0)).(\lambda (_: (leq g (ASort (S n1) n0) x1)).(\lambda (H4: (eq A (ASort -(S n1) n0) (AHead x0 x1))).(let H5 \def (eq_ind A (ASort (S n1) n0) (\lambda -(ee: A).(match ee in A return (\lambda (_: A).Prop) with [(ASort _ _) -\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H4) in -(False_ind P H5))))))) H1)))))) n H)))))) (\lambda (a: A).(\lambda (_: -((\forall (a1: A).((leq g (AHead a1 a) a) \to (\forall (P: -Prop).P))))).(\lambda (a0: A).(\lambda (H0: ((\forall (a1: A).((leq g (AHead -a1 a0) a0) \to (\forall (P: Prop).P))))).(\lambda (a1: A).(\lambda (H1: (leq -g (AHead a1 (AHead a a0)) (AHead a a0))).(\lambda (P: Prop).(let H_x \def -(leq_gen_head1 g a1 (AHead a a0) (AHead a a0) H1) in (let H2 \def H_x in -(ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq g a1 a3))) (\lambda (_: -A).(\lambda (a4: A).(leq g (AHead a a0) a4))) (\lambda (a3: A).(\lambda (a4: -A).(eq A (AHead a a0) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: -A).(\lambda (H3: (leq g a1 x0)).(\lambda (H4: (leq g (AHead a a0) -x1)).(\lambda (H5: (eq A (AHead a a0) (AHead x0 x1))).(let H6 \def (f_equal A -A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) -\Rightarrow a | (AHead a3 _) \Rightarrow a3])) (AHead a a0) (AHead x0 x1) H5) -in ((let H7 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda -(_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead _ a3) \Rightarrow a3])) -(AHead a a0) (AHead x0 x1) H5) in (\lambda (H8: (eq A a x0)).(let H9 \def -(eq_ind_r A x1 (\lambda (a3: A).(leq g (AHead a a0) a3)) H4 a0 H7) in (let -H10 \def (eq_ind_r A x0 (\lambda (a3: A).(leq g a1 a3)) H3 a H8) in (H0 a H9 -P))))) H6))))))) H2)))))))))) a2)). -(* COMMENTS -Initial nodes: 797 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/lift/defs.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/lift/defs.ma deleted file mode 100644 index 046506672..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/lift/defs.ma +++ /dev/null @@ -1,42 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/tlist/defs.ma". - -include "Basic-1/s/defs.ma". - -definition lref_map: - ((nat \to nat)) \to (nat \to (T \to T)) -\def - let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t -with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match -(blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u -t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in -lref_map. - -definition lift: - nat \to (nat \to (T \to T)) -\def - \lambda (h: nat).(\lambda (i: nat).(\lambda (t: T).(lref_map (\lambda (x: -nat).(plus x h)) i t))). - -definition lifts: - nat \to (nat \to (TList \to TList)) -\def - let rec lifts (h: nat) (d: nat) (ts: TList) on ts: TList \def (match ts with -[TNil \Rightarrow TNil | (TCons t ts0) \Rightarrow (TCons (lift h d t) (lifts -h d ts0))]) in lifts. - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/lift/fwd.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/lift/fwd.ma deleted file mode 100644 index 324fed2fb..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/lift/fwd.ma +++ /dev/null @@ -1,434 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/lift/defs.ma". - -theorem lift_sort: - \forall (n: nat).(\forall (h: nat).(\forall (d: nat).(eq T (lift h d (TSort -n)) (TSort n)))) -\def - \lambda (n: nat).(\lambda (_: nat).(\lambda (_: nat).(refl_equal T (TSort -n)))). -(* COMMENTS -Initial nodes: 13 -END *) - -theorem lift_lref_lt: - \forall (n: nat).(\forall (h: nat).(\forall (d: nat).((lt n d) \to (eq T -(lift h d (TLRef n)) (TLRef n))))) -\def - \lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (lt n -d)).(eq_ind bool true (\lambda (b: bool).(eq T (TLRef (match b with [true -\Rightarrow n | false \Rightarrow (plus n h)])) (TLRef n))) (refl_equal T -(TLRef n)) (blt n d) (sym_eq bool (blt n d) true (lt_blt d n H)))))). -(* COMMENTS -Initial nodes: 72 -END *) - -theorem lift_lref_ge: - \forall (n: nat).(\forall (h: nat).(\forall (d: nat).((le d n) \to (eq T -(lift h d (TLRef n)) (TLRef (plus n h)))))) -\def - \lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (le d -n)).(eq_ind bool false (\lambda (b: bool).(eq T (TLRef (match b with [true -\Rightarrow n | false \Rightarrow (plus n h)])) (TLRef (plus n h)))) -(refl_equal T (TLRef (plus n h))) (blt n d) (sym_eq bool (blt n d) false -(le_bge d n H)))))). -(* COMMENTS -Initial nodes: 80 -END *) - -theorem lift_head: - \forall (k: K).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall -(d: nat).(eq T (lift h d (THead k u t)) (THead k (lift h d u) (lift h (s k d) -t))))))) -\def - \lambda (k: K).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda -(d: nat).(refl_equal T (THead k (lift h d u) (lift h (s k d) t))))))). -(* COMMENTS -Initial nodes: 37 -END *) - -theorem lift_bind: - \forall (b: B).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall -(d: nat).(eq T (lift h d (THead (Bind b) u t)) (THead (Bind b) (lift h d u) -(lift h (S d) t))))))) -\def - \lambda (b: B).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda -(d: nat).(refl_equal T (THead (Bind b) (lift h d u) (lift h (S d) t))))))). -(* COMMENTS -Initial nodes: 37 -END *) - -theorem lift_flat: - \forall (f: F).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall -(d: nat).(eq T (lift h d (THead (Flat f) u t)) (THead (Flat f) (lift h d u) -(lift h d t))))))) -\def - \lambda (f: F).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda -(d: nat).(refl_equal T (THead (Flat f) (lift h d u) (lift h d t))))))). -(* COMMENTS -Initial nodes: 35 -END *) - -theorem lift_gen_sort: - \forall (h: nat).(\forall (d: nat).(\forall (n: nat).(\forall (t: T).((eq T -(TSort n) (lift h d t)) \to (eq T t (TSort n)))))) -\def - \lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (t: T).(T_ind -(\lambda (t0: T).((eq T (TSort n) (lift h d t0)) \to (eq T t0 (TSort n)))) -(\lambda (n0: nat).(\lambda (H: (eq T (TSort n) (lift h d (TSort -n0)))).(sym_eq T (TSort n) (TSort n0) H))) (\lambda (n0: nat).(\lambda (H: -(eq T (TSort n) (lift h d (TLRef n0)))).(lt_le_e n0 d (eq T (TLRef n0) (TSort -n)) (\lambda (_: (lt n0 d)).(let H1 \def (eq_ind T (lift h d (TLRef n0)) -(\lambda (t0: T).(eq T (TSort n) t0)) H (TLRef n0) (lift_lref_lt n0 h d (let -H1 \def (eq_ind T (TSort n) (\lambda (ee: T).(match ee in T return (\lambda -(_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | -(THead _ _ _) \Rightarrow False])) I (lift h d (TLRef n0)) H) in (False_ind -(lt n0 d) H1)))) in (let H2 \def (eq_ind T (TSort n) (\lambda (ee: T).(match -ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | -(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef n0) -H1) in (False_ind (eq T (TLRef n0) (TSort n)) H2)))) (\lambda (_: (le d -n0)).(let H1 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t0: T).(eq T -(TSort n) t0)) H (TLRef (plus n0 h)) (lift_lref_ge n0 h d (let H1 \def -(eq_ind T (TSort n) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | -(THead _ _ _) \Rightarrow False])) I (lift h d (TLRef n0)) H) in (False_ind -(le d n0) H1)))) in (let H2 \def (eq_ind T (TSort n) (\lambda (ee: T).(match -ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | -(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef -(plus n0 h)) H1) in (False_ind (eq T (TLRef n0) (TSort n)) H2))))))) (\lambda -(k: K).(\lambda (t0: T).(\lambda (_: (((eq T (TSort n) (lift h d t0)) \to (eq -T t0 (TSort n))))).(\lambda (t1: T).(\lambda (_: (((eq T (TSort n) (lift h d -t1)) \to (eq T t1 (TSort n))))).(\lambda (H1: (eq T (TSort n) (lift h d -(THead k t0 t1)))).(let H2 \def (eq_ind T (lift h d (THead k t0 t1)) (\lambda -(t2: T).(eq T (TSort n) t2)) H1 (THead k (lift h d t0) (lift h (s k d) t1)) -(lift_head k t0 t1 h d)) in (let H3 \def (eq_ind T (TSort n) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I -(THead k (lift h d t0) (lift h (s k d) t1)) H2) in (False_ind (eq T (THead k -t0 t1) (TSort n)) H3))))))))) t)))). -(* COMMENTS -Initial nodes: 613 -END *) - -theorem lift_gen_lref: - \forall (t: T).(\forall (d: nat).(\forall (h: nat).(\forall (i: nat).((eq T -(TLRef i) (lift h d t)) \to (or (land (lt i d) (eq T t (TLRef i))) (land (le -(plus d h) i) (eq T t (TLRef (minus i h))))))))) -\def - \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d: nat).(\forall (h: -nat).(\forall (i: nat).((eq T (TLRef i) (lift h d t0)) \to (or (land (lt i d) -(eq T t0 (TLRef i))) (land (le (plus d h) i) (eq T t0 (TLRef (minus i -h)))))))))) (\lambda (n: nat).(\lambda (d: nat).(\lambda (h: nat).(\lambda -(i: nat).(\lambda (H: (eq T (TLRef i) (lift h d (TSort n)))).(let H0 \def -(eq_ind T (lift h d (TSort n)) (\lambda (t0: T).(eq T (TLRef i) t0)) H (TSort -n) (lift_sort n h d)) in (let H1 \def (eq_ind T (TLRef i) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I -(TSort n) H0) in (False_ind (or (land (lt i d) (eq T (TSort n) (TLRef i))) -(land (le (plus d h) i) (eq T (TSort n) (TLRef (minus i h))))) H1)))))))) -(\lambda (n: nat).(\lambda (d: nat).(\lambda (h: nat).(\lambda (i: -nat).(\lambda (H: (eq T (TLRef i) (lift h d (TLRef n)))).(lt_le_e n d (or -(land (lt i d) (eq T (TLRef n) (TLRef i))) (land (le (plus d h) i) (eq T -(TLRef n) (TLRef (minus i h))))) (\lambda (H0: (lt n d)).(let H1 \def (eq_ind -T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (TLRef i) t0)) H (TLRef n) -(lift_lref_lt n h d H0)) in (let H2 \def (f_equal T nat (\lambda (e: -T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow i | -(TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow i])) (TLRef i) (TLRef -n) H1) in (eq_ind_r nat n (\lambda (n0: nat).(or (land (lt n0 d) (eq T (TLRef -n) (TLRef n0))) (land (le (plus d h) n0) (eq T (TLRef n) (TLRef (minus n0 -h)))))) (or_introl (land (lt n d) (eq T (TLRef n) (TLRef n))) (land (le (plus -d h) n) (eq T (TLRef n) (TLRef (minus n h)))) (conj (lt n d) (eq T (TLRef n) -(TLRef n)) H0 (refl_equal T (TLRef n)))) i H2)))) (\lambda (H0: (le d -n)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (TLRef -i) t0)) H (TLRef (plus n h)) (lift_lref_ge n h d H0)) in (let H2 \def -(f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with -[(TSort _) \Rightarrow i | (TLRef n0) \Rightarrow n0 | (THead _ _ _) -\Rightarrow i])) (TLRef i) (TLRef (plus n h)) H1) in (eq_ind_r nat (plus n h) -(\lambda (n0: nat).(or (land (lt n0 d) (eq T (TLRef n) (TLRef n0))) (land (le -(plus d h) n0) (eq T (TLRef n) (TLRef (minus n0 h)))))) (eq_ind_r nat n -(\lambda (n0: nat).(or (land (lt (plus n h) d) (eq T (TLRef n) (TLRef (plus n -h)))) (land (le (plus d h) (plus n h)) (eq T (TLRef n) (TLRef n0))))) -(or_intror (land (lt (plus n h) d) (eq T (TLRef n) (TLRef (plus n h)))) (land -(le (plus d h) (plus n h)) (eq T (TLRef n) (TLRef n))) (conj (le (plus d h) -(plus n h)) (eq T (TLRef n) (TLRef n)) (le_plus_plus d n h h H0 (le_n h)) -(refl_equal T (TLRef n)))) (minus (plus n h) h) (minus_plus_r n h)) i -H2)))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (_: ((\forall (d: -nat).(\forall (h: nat).(\forall (i: nat).((eq T (TLRef i) (lift h d t0)) \to -(or (land (lt i d) (eq T t0 (TLRef i))) (land (le (plus d h) i) (eq T t0 -(TLRef (minus i h))))))))))).(\lambda (t1: T).(\lambda (_: ((\forall (d: -nat).(\forall (h: nat).(\forall (i: nat).((eq T (TLRef i) (lift h d t1)) \to -(or (land (lt i d) (eq T t1 (TLRef i))) (land (le (plus d h) i) (eq T t1 -(TLRef (minus i h))))))))))).(\lambda (d: nat).(\lambda (h: nat).(\lambda (i: -nat).(\lambda (H1: (eq T (TLRef i) (lift h d (THead k t0 t1)))).(let H2 \def -(eq_ind T (lift h d (THead k t0 t1)) (\lambda (t2: T).(eq T (TLRef i) t2)) H1 -(THead k (lift h d t0) (lift h (s k d) t1)) (lift_head k t0 t1 h d)) in (let -H3 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda -(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | -(THead _ _ _) \Rightarrow False])) I (THead k (lift h d t0) (lift h (s k d) -t1)) H2) in (False_ind (or (land (lt i d) (eq T (THead k t0 t1) (TLRef i))) -(land (le (plus d h) i) (eq T (THead k t0 t1) (TLRef (minus i h))))) -H3)))))))))))) t). -(* COMMENTS -Initial nodes: 1221 -END *) - -theorem lift_gen_lref_lt: - \forall (h: nat).(\forall (d: nat).(\forall (n: nat).((lt n d) \to (\forall -(t: T).((eq T (TLRef n) (lift h d t)) \to (eq T t (TLRef n))))))) -\def - \lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (H: (lt n -d)).(\lambda (t: T).(\lambda (H0: (eq T (TLRef n) (lift h d t))).(let H_x -\def (lift_gen_lref t d h n H0) in (let H1 \def H_x in (or_ind (land (lt n d) -(eq T t (TLRef n))) (land (le (plus d h) n) (eq T t (TLRef (minus n h)))) (eq -T t (TLRef n)) (\lambda (H2: (land (lt n d) (eq T t (TLRef n)))).(land_ind -(lt n d) (eq T t (TLRef n)) (eq T t (TLRef n)) (\lambda (_: (lt n -d)).(\lambda (H4: (eq T t (TLRef n))).(eq_ind_r T (TLRef n) (\lambda (t0: -T).(eq T t0 (TLRef n))) (refl_equal T (TLRef n)) t H4))) H2)) (\lambda (H2: -(land (le (plus d h) n) (eq T t (TLRef (minus n h))))).(land_ind (le (plus d -h) n) (eq T t (TLRef (minus n h))) (eq T t (TLRef n)) (\lambda (H3: (le (plus -d h) n)).(\lambda (H4: (eq T t (TLRef (minus n h)))).(eq_ind_r T (TLRef -(minus n h)) (\lambda (t0: T).(eq T t0 (TLRef n))) (le_false (plus d h) n (eq -T (TLRef (minus n h)) (TLRef n)) H3 (lt_le_S n (plus d h) (le_plus_trans (S -n) d h H))) t H4))) H2)) H1)))))))). -(* COMMENTS -Initial nodes: 363 -END *) - -theorem lift_gen_lref_false: - \forall (h: nat).(\forall (d: nat).(\forall (n: nat).((le d n) \to ((lt n -(plus d h)) \to (\forall (t: T).((eq T (TLRef n) (lift h d t)) \to (\forall -(P: Prop).P))))))) -\def - \lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (H: (le d -n)).(\lambda (H0: (lt n (plus d h))).(\lambda (t: T).(\lambda (H1: (eq T -(TLRef n) (lift h d t))).(\lambda (P: Prop).(let H_x \def (lift_gen_lref t d -h n H1) in (let H2 \def H_x in (or_ind (land (lt n d) (eq T t (TLRef n))) -(land (le (plus d h) n) (eq T t (TLRef (minus n h)))) P (\lambda (H3: (land -(lt n d) (eq T t (TLRef n)))).(land_ind (lt n d) (eq T t (TLRef n)) P -(\lambda (H4: (lt n d)).(\lambda (_: (eq T t (TLRef n))).(le_false d n P H -H4))) H3)) (\lambda (H3: (land (le (plus d h) n) (eq T t (TLRef (minus n -h))))).(land_ind (le (plus d h) n) (eq T t (TLRef (minus n h))) P (\lambda -(H4: (le (plus d h) n)).(\lambda (_: (eq T t (TLRef (minus n h)))).(le_false -(plus d h) n P H4 H0))) H3)) H2)))))))))). -(* COMMENTS -Initial nodes: 269 -END *) - -theorem lift_gen_lref_ge: - \forall (h: nat).(\forall (d: nat).(\forall (n: nat).((le d n) \to (\forall -(t: T).((eq T (TLRef (plus n h)) (lift h d t)) \to (eq T t (TLRef n))))))) -\def - \lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (H: (le d -n)).(\lambda (t: T).(\lambda (H0: (eq T (TLRef (plus n h)) (lift h d -t))).(let H_x \def (lift_gen_lref t d h (plus n h) H0) in (let H1 \def H_x in -(or_ind (land (lt (plus n h) d) (eq T t (TLRef (plus n h)))) (land (le (plus -d h) (plus n h)) (eq T t (TLRef (minus (plus n h) h)))) (eq T t (TLRef n)) -(\lambda (H2: (land (lt (plus n h) d) (eq T t (TLRef (plus n h))))).(land_ind -(lt (plus n h) d) (eq T t (TLRef (plus n h))) (eq T t (TLRef n)) (\lambda -(H3: (lt (plus n h) d)).(\lambda (H4: (eq T t (TLRef (plus n h)))).(eq_ind_r -T (TLRef (plus n h)) (\lambda (t0: T).(eq T t0 (TLRef n))) (le_false d n (eq -T (TLRef (plus n h)) (TLRef n)) H (lt_le_S n d (simpl_lt_plus_r h n d -(lt_le_trans (plus n h) d (plus d h) H3 (le_plus_l d h))))) t H4))) H2)) -(\lambda (H2: (land (le (plus d h) (plus n h)) (eq T t (TLRef (minus (plus n -h) h))))).(land_ind (le (plus d h) (plus n h)) (eq T t (TLRef (minus (plus n -h) h))) (eq T t (TLRef n)) (\lambda (_: (le (plus d h) (plus n h))).(\lambda -(H4: (eq T t (TLRef (minus (plus n h) h)))).(eq_ind_r T (TLRef (minus (plus n -h) h)) (\lambda (t0: T).(eq T t0 (TLRef n))) (f_equal nat T TLRef (minus -(plus n h) h) n (minus_plus_r n h)) t H4))) H2)) H1)))))))). -(* COMMENTS -Initial nodes: 473 -END *) - -theorem lift_gen_head: - \forall (k: K).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h: -nat).(\forall (d: nat).((eq T (THead k u t) (lift h d x)) \to (ex3_2 T T -(\lambda (y: T).(\lambda (z: T).(eq T x (THead k y z)))) (\lambda (y: -T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: -T).(eq T t (lift h (s k d) z))))))))))) -\def - \lambda (k: K).(\lambda (u: T).(\lambda (t: T).(\lambda (x: T).(T_ind -(\lambda (t0: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k u t) -(lift h d t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead -k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda -(_: T).(\lambda (z: T).(eq T t (lift h (s k d) z))))))))) (\lambda (n: -nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (eq T (THead k u t) -(lift h d (TSort n)))).(let H0 \def (eq_ind T (lift h d (TSort n)) (\lambda -(t0: T).(eq T (THead k u t) t0)) H (TSort n) (lift_sort n h d)) in (let H1 -\def (eq_ind T (THead k u t) (\lambda (ee: T).(match ee in T return (\lambda -(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False -| (THead _ _ _) \Rightarrow True])) I (TSort n) H0) in (False_ind (ex3_2 T T -(\lambda (y: T).(\lambda (z: T).(eq T (TSort n) (THead k y z)))) (\lambda (y: -T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: -T).(eq T t (lift h (s k d) z))))) H1))))))) (\lambda (n: nat).(\lambda (h: -nat).(\lambda (d: nat).(\lambda (H: (eq T (THead k u t) (lift h d (TLRef -n)))).(lt_le_e n d (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) -(THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) -(\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k d) z))))) (\lambda (H0: -(lt n d)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T -(THead k u t) t0)) H (TLRef n) (lift_lref_lt n h d H0)) in (let H2 \def -(eq_ind T (THead k u t) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead _ _ _) \Rightarrow True])) I (TLRef n) H1) in (False_ind (ex3_2 T T -(\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead k y z)))) (\lambda (y: -T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: -T).(eq T t (lift h (s k d) z))))) H2)))) (\lambda (H0: (le d n)).(let H1 \def -(eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (THead k u t) t0)) H -(TLRef (plus n h)) (lift_lref_ge n h d H0)) in (let H2 \def (eq_ind T (THead -k u t) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) -\Rightarrow True])) I (TLRef (plus n h)) H1) in (False_ind (ex3_2 T T -(\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead k y z)))) (\lambda (y: -T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: -T).(eq T t (lift h (s k d) z))))) H2))))))))) (\lambda (k0: K).(\lambda (t0: -T).(\lambda (H: ((\forall (h: nat).(\forall (d: nat).((eq T (THead k u t) -(lift h d t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead -k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda -(_: T).(\lambda (z: T).(eq T t (lift h (s k d) z)))))))))).(\lambda (t1: -T).(\lambda (H0: ((\forall (h: nat).(\forall (d: nat).((eq T (THead k u t) -(lift h d t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t1 (THead -k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda -(_: T).(\lambda (z: T).(eq T t (lift h (s k d) z)))))))))).(\lambda (h: -nat).(\lambda (d: nat).(\lambda (H1: (eq T (THead k u t) (lift h d (THead k0 -t0 t1)))).(let H2 \def (eq_ind T (lift h d (THead k0 t0 t1)) (\lambda (t2: -T).(eq T (THead k u t) t2)) H1 (THead k0 (lift h d t0) (lift h (s k0 d) t1)) -(lift_head k0 t0 t1 h d)) in (let H3 \def (f_equal T K (\lambda (e: T).(match -e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) -\Rightarrow k | (THead k1 _ _) \Rightarrow k1])) (THead k u t) (THead k0 -(lift h d t0) (lift h (s k0 d) t1)) H2) in ((let H4 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t2 _) \Rightarrow t2])) -(THead k u t) (THead k0 (lift h d t0) (lift h (s k0 d) t1)) H2) in ((let H5 -\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t2) -\Rightarrow t2])) (THead k u t) (THead k0 (lift h d t0) (lift h (s k0 d) t1)) -H2) in (\lambda (H6: (eq T u (lift h d t0))).(\lambda (H7: (eq K k k0)).(let -H8 \def (eq_ind_r K k0 (\lambda (k1: K).(eq T t (lift h (s k1 d) t1))) H5 k -H7) in (eq_ind K k (\lambda (k1: K).(ex3_2 T T (\lambda (y: T).(\lambda (z: -T).(eq T (THead k1 t0 t1) (THead k y z)))) (\lambda (y: T).(\lambda (_: -T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s -k d) z)))))) (let H9 \def (eq_ind T t (\lambda (t2: T).(\forall (h0: -nat).(\forall (d0: nat).((eq T (THead k u t2) (lift h0 d0 t1)) \to (ex3_2 T T -(\lambda (y: T).(\lambda (z: T).(eq T t1 (THead k y z)))) (\lambda (y: -T).(\lambda (_: T).(eq T u (lift h0 d0 y)))) (\lambda (_: T).(\lambda (z: -T).(eq T t2 (lift h0 (s k d0) z))))))))) H0 (lift h (s k d) t1) H8) in (let -H10 \def (eq_ind T t (\lambda (t2: T).(\forall (h0: nat).(\forall (d0: -nat).((eq T (THead k u t2) (lift h0 d0 t0)) \to (ex3_2 T T (\lambda (y: -T).(\lambda (z: T).(eq T t0 (THead k y z)))) (\lambda (y: T).(\lambda (_: -T).(eq T u (lift h0 d0 y)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift -h0 (s k d0) z))))))))) H (lift h (s k d) t1) H8) in (eq_ind_r T (lift h (s k -d) t1) (\lambda (t2: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T -(THead k t0 t1) (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u -(lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h (s k d) -z)))))) (let H11 \def (eq_ind T u (\lambda (t2: T).(\forall (h0: -nat).(\forall (d0: nat).((eq T (THead k t2 (lift h (s k d) t1)) (lift h0 d0 -t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead k y z)))) -(\lambda (y: T).(\lambda (_: T).(eq T t2 (lift h0 d0 y)))) (\lambda (_: -T).(\lambda (z: T).(eq T (lift h (s k d) t1) (lift h0 (s k d0) z))))))))) H10 -(lift h d t0) H6) in (let H12 \def (eq_ind T u (\lambda (t2: T).(\forall (h0: -nat).(\forall (d0: nat).((eq T (THead k t2 (lift h (s k d) t1)) (lift h0 d0 -t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t1 (THead k y z)))) -(\lambda (y: T).(\lambda (_: T).(eq T t2 (lift h0 d0 y)))) (\lambda (_: -T).(\lambda (z: T).(eq T (lift h (s k d) t1) (lift h0 (s k d0) z))))))))) H9 -(lift h d t0) H6) in (eq_ind_r T (lift h d t0) (\lambda (t2: T).(ex3_2 T T -(\lambda (y: T).(\lambda (z: T).(eq T (THead k t0 t1) (THead k y z)))) -(\lambda (y: T).(\lambda (_: T).(eq T t2 (lift h d y)))) (\lambda (_: -T).(\lambda (z: T).(eq T (lift h (s k d) t1) (lift h (s k d) z)))))) -(ex3_2_intro T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k t0 t1) (THead -k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t0) (lift h d y)))) -(\lambda (_: T).(\lambda (z: T).(eq T (lift h (s k d) t1) (lift h (s k d) -z)))) t0 t1 (refl_equal T (THead k t0 t1)) (refl_equal T (lift h d t0)) -(refl_equal T (lift h (s k d) t1))) u H6))) t H8))) k0 H7))))) H4)) -H3))))))))))) x)))). -(* COMMENTS -Initial nodes: 2083 -END *) - -theorem lift_gen_bind: - \forall (b: B).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h: -nat).(\forall (d: nat).((eq T (THead (Bind b) u t) (lift h d x)) \to (ex3_2 T -T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Bind b) y z)))) (\lambda -(y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: -T).(eq T t (lift h (S d) z))))))))))) -\def - \lambda (b: B).(\lambda (u: T).(\lambda (t: T).(\lambda (x: T).(\lambda (h: -nat).(\lambda (d: nat).(\lambda (H: (eq T (THead (Bind b) u t) (lift h d -x))).(let H_x \def (lift_gen_head (Bind b) u t x h d H) in (let H0 \def H_x -in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Bind b) y -z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: -T).(\lambda (z: T).(eq T t (lift h (S d) z)))) (ex3_2 T T (\lambda (y: -T).(\lambda (z: T).(eq T x (THead (Bind b) y z)))) (\lambda (y: T).(\lambda -(_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift -h (S d) z))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H1: (eq T x (THead -(Bind b) x0 x1))).(\lambda (H2: (eq T u (lift h d x0))).(\lambda (H3: (eq T t -(lift h (S d) x1))).(eq_ind_r T (THead (Bind b) x0 x1) (\lambda (t0: -T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead (Bind b) y -z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: -T).(\lambda (z: T).(eq T t (lift h (S d) z)))))) (eq_ind_r T (lift h (S d) -x1) (\lambda (t0: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead -(Bind b) x0 x1) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T -u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t0 (lift h (S d) -z)))))) (eq_ind_r T (lift h d x0) (\lambda (t0: T).(ex3_2 T T (\lambda (y: -T).(\lambda (z: T).(eq T (THead (Bind b) x0 x1) (THead (Bind b) y z)))) -(\lambda (y: T).(\lambda (_: T).(eq T t0 (lift h d y)))) (\lambda (_: -T).(\lambda (z: T).(eq T (lift h (S d) x1) (lift h (S d) z)))))) (ex3_2_intro -T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Bind b) x0 x1) (THead (Bind -b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d x0) (lift h d -y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h (S d) x1) (lift h (S d) -z)))) x0 x1 (refl_equal T (THead (Bind b) x0 x1)) (refl_equal T (lift h d -x0)) (refl_equal T (lift h (S d) x1))) u H2) t H3) x H1)))))) H0))))))))). -(* COMMENTS -Initial nodes: 637 -END *) - -theorem lift_gen_flat: - \forall (f: F).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h: -nat).(\forall (d: nat).((eq T (THead (Flat f) u t) (lift h d x)) \to (ex3_2 T -T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Flat f) y z)))) (\lambda -(y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: -T).(eq T t (lift h d z))))))))))) -\def - \lambda (f: F).(\lambda (u: T).(\lambda (t: T).(\lambda (x: T).(\lambda (h: -nat).(\lambda (d: nat).(\lambda (H: (eq T (THead (Flat f) u t) (lift h d -x))).(let H_x \def (lift_gen_head (Flat f) u t x h d H) in (let H0 \def H_x -in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Flat f) y -z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: -T).(\lambda (z: T).(eq T t (lift h d z)))) (ex3_2 T T (\lambda (y: -T).(\lambda (z: T).(eq T x (THead (Flat f) y z)))) (\lambda (y: T).(\lambda -(_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift -h d z))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H1: (eq T x (THead -(Flat f) x0 x1))).(\lambda (H2: (eq T u (lift h d x0))).(\lambda (H3: (eq T t -(lift h d x1))).(eq_ind_r T (THead (Flat f) x0 x1) (\lambda (t0: T).(ex3_2 T -T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead (Flat f) y z)))) (\lambda -(y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: -T).(eq T t (lift h d z)))))) (eq_ind_r T (lift h d x1) (\lambda (t0: -T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Flat f) x0 x1) -(THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d -y)))) (\lambda (_: T).(\lambda (z: T).(eq T t0 (lift h d z)))))) (eq_ind_r T -(lift h d x0) (\lambda (t0: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq -T (THead (Flat f) x0 x1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: -T).(eq T t0 (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h d -x1) (lift h d z)))))) (ex3_2_intro T T (\lambda (y: T).(\lambda (z: T).(eq T -(THead (Flat f) x0 x1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: -T).(eq T (lift h d x0) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T -(lift h d x1) (lift h d z)))) x0 x1 (refl_equal T (THead (Flat f) x0 x1)) -(refl_equal T (lift h d x0)) (refl_equal T (lift h d x1))) u H2) t H3) x -H1)))))) H0))))))))). -(* COMMENTS -Initial nodes: 615 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/lift/props.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/lift/props.ma deleted file mode 100644 index f0ed22451..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/lift/props.ma +++ /dev/null @@ -1,592 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/lift/fwd.ma". - -include "Basic-1/s/props.ma". - -theorem thead_x_lift_y_y: - \forall (k: K).(\forall (t: T).(\forall (v: T).(\forall (h: nat).(\forall -(d: nat).((eq T (THead k v (lift h d t)) t) \to (\forall (P: Prop).P)))))) -\def - \lambda (k: K).(\lambda (t: T).(T_ind (\lambda (t0: T).(\forall (v: -T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k v (lift h d t0)) t0) -\to (\forall (P: Prop).P)))))) (\lambda (n: nat).(\lambda (v: T).(\lambda (h: -nat).(\lambda (d: nat).(\lambda (H: (eq T (THead k v (lift h d (TSort n))) -(TSort n))).(\lambda (P: Prop).(let H0 \def (eq_ind T (THead k v (lift h d -(TSort n))) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) -\Rightarrow True])) I (TSort n) H) in (False_ind P H0)))))))) (\lambda (n: -nat).(\lambda (v: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (eq T -(THead k v (lift h d (TLRef n))) (TLRef n))).(\lambda (P: Prop).(let H0 \def -(eq_ind T (THead k v (lift h d (TLRef n))) (\lambda (ee: T).(match ee in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H) in -(False_ind P H0)))))))) (\lambda (k0: K).(\lambda (t0: T).(\lambda (_: -((\forall (v: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k v (lift -h d t0)) t0) \to (\forall (P: Prop).P))))))).(\lambda (t1: T).(\lambda (H0: -((\forall (v: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k v (lift -h d t1)) t1) \to (\forall (P: Prop).P))))))).(\lambda (v: T).(\lambda (h: -nat).(\lambda (d: nat).(\lambda (H1: (eq T (THead k v (lift h d (THead k0 t0 -t1))) (THead k0 t0 t1))).(\lambda (P: Prop).(let H2 \def (f_equal T K -(\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) -\Rightarrow k | (TLRef _) \Rightarrow k | (THead k1 _ _) \Rightarrow k1])) -(THead k v (lift h d (THead k0 t0 t1))) (THead k0 t0 t1) H1) in ((let H3 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow v | (TLRef _) \Rightarrow v | (THead _ t2 _) -\Rightarrow t2])) (THead k v (lift h d (THead k0 t0 t1))) (THead k0 t0 t1) -H1) in ((let H4 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow (THead k0 ((let rec lref_map -(f: ((nat \to nat))) (d0: nat) (t2: T) on t2: T \def (match t2 with [(TSort -n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) -with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k1 u t3) -\Rightarrow (THead k1 (lref_map f d0 u) (lref_map f (s k1 d0) t3))]) in -lref_map) (\lambda (x: nat).(plus x h)) d t0) ((let rec lref_map (f: ((nat -\to nat))) (d0: nat) (t2: T) on t2: T \def (match t2 with [(TSort n) -\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) with -[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k1 u t3) -\Rightarrow (THead k1 (lref_map f d0 u) (lref_map f (s k1 d0) t3))]) in -lref_map) (\lambda (x: nat).(plus x h)) (s k0 d) t1)) | (TLRef _) \Rightarrow -(THead k0 ((let rec lref_map (f: ((nat \to nat))) (d0: nat) (t2: T) on t2: T -\def (match t2 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow -(TLRef (match (blt i d0) with [true \Rightarrow i | false \Rightarrow (f -i)])) | (THead k1 u t3) \Rightarrow (THead k1 (lref_map f d0 u) (lref_map f -(s k1 d0) t3))]) in lref_map) (\lambda (x: nat).(plus x h)) d t0) ((let rec -lref_map (f: ((nat \to nat))) (d0: nat) (t2: T) on t2: T \def (match t2 with -[(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i -d0) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k1 u t3) -\Rightarrow (THead k1 (lref_map f d0 u) (lref_map f (s k1 d0) t3))]) in -lref_map) (\lambda (x: nat).(plus x h)) (s k0 d) t1)) | (THead _ _ t2) -\Rightarrow t2])) (THead k v (lift h d (THead k0 t0 t1))) (THead k0 t0 t1) -H1) in (\lambda (_: (eq T v t0)).(\lambda (H6: (eq K k k0)).(let H7 \def -(eq_ind K k (\lambda (k1: K).(\forall (v0: T).(\forall (h0: nat).(\forall -(d0: nat).((eq T (THead k1 v0 (lift h0 d0 t1)) t1) \to (\forall (P0: -Prop).P0)))))) H0 k0 H6) in (let H8 \def (eq_ind T (lift h d (THead k0 t0 -t1)) (\lambda (t2: T).(eq T t2 t1)) H4 (THead k0 (lift h d t0) (lift h (s k0 -d) t1)) (lift_head k0 t0 t1 h d)) in (H7 (lift h d t0) h (s k0 d) H8 P)))))) -H3)) H2)))))))))))) t)). -(* COMMENTS -Initial nodes: 887 -END *) - -theorem lift_r: - \forall (t: T).(\forall (d: nat).(eq T (lift O d t) t)) -\def - \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d: nat).(eq T (lift O d t0) -t0))) (\lambda (n: nat).(\lambda (_: nat).(refl_equal T (TSort n)))) (\lambda -(n: nat).(\lambda (d: nat).(lt_le_e n d (eq T (lift O d (TLRef n)) (TLRef n)) -(\lambda (H: (lt n d)).(eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 (TLRef -n))) (refl_equal T (TLRef n)) (lift O d (TLRef n)) (lift_lref_lt n O d H))) -(\lambda (H: (le d n)).(eq_ind_r T (TLRef (plus n O)) (\lambda (t0: T).(eq T -t0 (TLRef n))) (f_equal nat T TLRef (plus n O) n (sym_eq nat n (plus n O) -(plus_n_O n))) (lift O d (TLRef n)) (lift_lref_ge n O d H)))))) (\lambda (k: -K).(\lambda (t0: T).(\lambda (H: ((\forall (d: nat).(eq T (lift O d t0) -t0)))).(\lambda (t1: T).(\lambda (H0: ((\forall (d: nat).(eq T (lift O d t1) -t1)))).(\lambda (d: nat).(eq_ind_r T (THead k (lift O d t0) (lift O (s k d) -t1)) (\lambda (t2: T).(eq T t2 (THead k t0 t1))) (f_equal3 K T T T THead k k -(lift O d t0) t0 (lift O (s k d) t1) t1 (refl_equal K k) (H d) (H0 (s k d))) -(lift O d (THead k t0 t1)) (lift_head k t0 t1 O d)))))))) t). -(* COMMENTS -Initial nodes: 367 -END *) - -theorem lift_lref_gt: - \forall (d: nat).(\forall (n: nat).((lt d n) \to (eq T (lift (S O) d (TLRef -(pred n))) (TLRef n)))) -\def - \lambda (d: nat).(\lambda (n: nat).(\lambda (H: (lt d n)).(eq_ind_r T (TLRef -(plus (pred n) (S O))) (\lambda (t: T).(eq T t (TLRef n))) (eq_ind nat (plus -(S O) (pred n)) (\lambda (n0: nat).(eq T (TLRef n0) (TLRef n))) (eq_ind nat n -(\lambda (n0: nat).(eq T (TLRef n0) (TLRef n))) (refl_equal T (TLRef n)) (S -(pred n)) (S_pred n d H)) (plus (pred n) (S O)) (plus_sym (S O) (pred n))) -(lift (S O) d (TLRef (pred n))) (lift_lref_ge (pred n) (S O) d (le_S_n d -(pred n) (eq_ind nat n (\lambda (n0: nat).(le (S d) n0)) H (S (pred n)) -(S_pred n d H))))))). -(* COMMENTS -Initial nodes: 193 -END *) - -theorem lifts_tapp: - \forall (h: nat).(\forall (d: nat).(\forall (v: T).(\forall (vs: TList).(eq -TList (lifts h d (TApp vs v)) (TApp (lifts h d vs) (lift h d v)))))) -\def - \lambda (h: nat).(\lambda (d: nat).(\lambda (v: T).(\lambda (vs: -TList).(TList_ind (\lambda (t: TList).(eq TList (lifts h d (TApp t v)) (TApp -(lifts h d t) (lift h d v)))) (refl_equal TList (TCons (lift h d v) TNil)) -(\lambda (t: T).(\lambda (t0: TList).(\lambda (H: (eq TList (lifts h d (TApp -t0 v)) (TApp (lifts h d t0) (lift h d v)))).(eq_ind_r TList (TApp (lifts h d -t0) (lift h d v)) (\lambda (t1: TList).(eq TList (TCons (lift h d t) t1) -(TCons (lift h d t) (TApp (lifts h d t0) (lift h d v))))) (refl_equal TList -(TCons (lift h d t) (TApp (lifts h d t0) (lift h d v)))) (lifts h d (TApp t0 -v)) H)))) vs)))). -(* COMMENTS -Initial nodes: 215 -END *) - -theorem lift_inj: - \forall (x: T).(\forall (t: T).(\forall (h: nat).(\forall (d: nat).((eq T -(lift h d x) (lift h d t)) \to (eq T x t))))) -\def - \lambda (x: T).(T_ind (\lambda (t: T).(\forall (t0: T).(\forall (h: -nat).(\forall (d: nat).((eq T (lift h d t) (lift h d t0)) \to (eq T t -t0)))))) (\lambda (n: nat).(\lambda (t: T).(\lambda (h: nat).(\lambda (d: -nat).(\lambda (H: (eq T (lift h d (TSort n)) (lift h d t))).(let H0 \def -(eq_ind T (lift h d (TSort n)) (\lambda (t0: T).(eq T t0 (lift h d t))) H -(TSort n) (lift_sort n h d)) in (sym_eq T t (TSort n) (lift_gen_sort h d n t -H0)))))))) (\lambda (n: nat).(\lambda (t: T).(\lambda (h: nat).(\lambda (d: -nat).(\lambda (H: (eq T (lift h d (TLRef n)) (lift h d t))).(lt_le_e n d (eq -T (TLRef n) t) (\lambda (H0: (lt n d)).(let H1 \def (eq_ind T (lift h d -(TLRef n)) (\lambda (t0: T).(eq T t0 (lift h d t))) H (TLRef n) (lift_lref_lt -n h d H0)) in (sym_eq T t (TLRef n) (lift_gen_lref_lt h d n (lt_le_trans n d -d H0 (le_n d)) t H1)))) (\lambda (H0: (le d n)).(let H1 \def (eq_ind T (lift -h d (TLRef n)) (\lambda (t0: T).(eq T t0 (lift h d t))) H (TLRef (plus n h)) -(lift_lref_ge n h d H0)) in (sym_eq T t (TLRef n) (lift_gen_lref_ge h d n H0 -t H1)))))))))) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t: -T).(((\forall (t0: T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t) -(lift h d t0)) \to (eq T t t0)))))) \to (\forall (t0: T).(((\forall (t1: -T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t0) (lift h d t1)) -\to (eq T t0 t1)))))) \to (\forall (t1: T).(\forall (h: nat).(\forall (d: -nat).((eq T (lift h d (THead k0 t t0)) (lift h d t1)) \to (eq T (THead k0 t -t0) t1)))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (H: ((\forall (t0: -T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t) (lift h d t0)) \to -(eq T t t0))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (t1: T).(\forall -(h: nat).(\forall (d: nat).((eq T (lift h d t0) (lift h d t1)) \to (eq T t0 -t1))))))).(\lambda (t1: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: -(eq T (lift h d (THead (Bind b) t t0)) (lift h d t1))).(let H2 \def (eq_ind T -(lift h d (THead (Bind b) t t0)) (\lambda (t2: T).(eq T t2 (lift h d t1))) H1 -(THead (Bind b) (lift h d t) (lift h (S d) t0)) (lift_bind b t t0 h d)) in -(ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T t1 (THead (Bind b) y -z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t) (lift h d y)))) -(\lambda (_: T).(\lambda (z: T).(eq T (lift h (S d) t0) (lift h (S d) z)))) -(eq T (THead (Bind b) t t0) t1) (\lambda (x0: T).(\lambda (x1: T).(\lambda -(H3: (eq T t1 (THead (Bind b) x0 x1))).(\lambda (H4: (eq T (lift h d t) (lift -h d x0))).(\lambda (H5: (eq T (lift h (S d) t0) (lift h (S d) x1))).(eq_ind_r -T (THead (Bind b) x0 x1) (\lambda (t2: T).(eq T (THead (Bind b) t t0) t2)) -(f_equal3 K T T T THead (Bind b) (Bind b) t x0 t0 x1 (refl_equal K (Bind b)) -(H x0 h d H4) (H0 x1 h (S d) H5)) t1 H3)))))) (lift_gen_bind b (lift h d t) -(lift h (S d) t0) t1 h d H2)))))))))))) (\lambda (f: F).(\lambda (t: -T).(\lambda (H: ((\forall (t0: T).(\forall (h: nat).(\forall (d: nat).((eq T -(lift h d t) (lift h d t0)) \to (eq T t t0))))))).(\lambda (t0: T).(\lambda -(H0: ((\forall (t1: T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d -t0) (lift h d t1)) \to (eq T t0 t1))))))).(\lambda (t1: T).(\lambda (h: -nat).(\lambda (d: nat).(\lambda (H1: (eq T (lift h d (THead (Flat f) t t0)) -(lift h d t1))).(let H2 \def (eq_ind T (lift h d (THead (Flat f) t t0)) -(\lambda (t2: T).(eq T t2 (lift h d t1))) H1 (THead (Flat f) (lift h d t) -(lift h d t0)) (lift_flat f t t0 h d)) in (ex3_2_ind T T (\lambda (y: -T).(\lambda (z: T).(eq T t1 (THead (Flat f) y z)))) (\lambda (y: T).(\lambda -(_: T).(eq T (lift h d t) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq -T (lift h d t0) (lift h d z)))) (eq T (THead (Flat f) t t0) t1) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H3: (eq T t1 (THead (Flat f) x0 x1))).(\lambda -(H4: (eq T (lift h d t) (lift h d x0))).(\lambda (H5: (eq T (lift h d t0) -(lift h d x1))).(eq_ind_r T (THead (Flat f) x0 x1) (\lambda (t2: T).(eq T -(THead (Flat f) t t0) t2)) (f_equal3 K T T T THead (Flat f) (Flat f) t x0 t0 -x1 (refl_equal K (Flat f)) (H x0 h d H4) (H0 x1 h d H5)) t1 H3)))))) -(lift_gen_flat f (lift h d t) (lift h d t0) t1 h d H2)))))))))))) k)) x). -(* COMMENTS -Initial nodes: 1391 -END *) - -theorem lift_gen_lift: - \forall (t1: T).(\forall (x: T).(\forall (h1: nat).(\forall (h2: -nat).(\forall (d1: nat).(\forall (d2: nat).((le d1 d2) \to ((eq T (lift h1 d1 -t1) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1 -t2))) (\lambda (t2: T).(eq T t1 (lift h2 d2 t2))))))))))) -\def - \lambda (t1: T).(T_ind (\lambda (t: T).(\forall (x: T).(\forall (h1: -nat).(\forall (h2: nat).(\forall (d1: nat).(\forall (d2: nat).((le d1 d2) \to -((eq T (lift h1 d1 t) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2: -T).(eq T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T t (lift h2 d2 -t2)))))))))))) (\lambda (n: nat).(\lambda (x: T).(\lambda (h1: nat).(\lambda -(h2: nat).(\lambda (d1: nat).(\lambda (d2: nat).(\lambda (_: (le d1 -d2)).(\lambda (H0: (eq T (lift h1 d1 (TSort n)) (lift h2 (plus d2 h1) -x))).(let H1 \def (eq_ind T (lift h1 d1 (TSort n)) (\lambda (t: T).(eq T t -(lift h2 (plus d2 h1) x))) H0 (TSort n) (lift_sort n h1 d1)) in (eq_ind_r T -(TSort n) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h1 d1 t2))) -(\lambda (t2: T).(eq T (TSort n) (lift h2 d2 t2))))) (ex_intro2 T (\lambda -(t2: T).(eq T (TSort n) (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TSort n) -(lift h2 d2 t2))) (TSort n) (eq_ind_r T (TSort n) (\lambda (t: T).(eq T -(TSort n) t)) (refl_equal T (TSort n)) (lift h1 d1 (TSort n)) (lift_sort n h1 -d1)) (eq_ind_r T (TSort n) (\lambda (t: T).(eq T (TSort n) t)) (refl_equal T -(TSort n)) (lift h2 d2 (TSort n)) (lift_sort n h2 d2))) x (lift_gen_sort h2 -(plus d2 h1) n x H1))))))))))) (\lambda (n: nat).(\lambda (x: T).(\lambda -(h1: nat).(\lambda (h2: nat).(\lambda (d1: nat).(\lambda (d2: nat).(\lambda -(H: (le d1 d2)).(\lambda (H0: (eq T (lift h1 d1 (TLRef n)) (lift h2 (plus d2 -h1) x))).(lt_le_e n d1 (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1 t2))) -(\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2)))) (\lambda (H1: (lt n -d1)).(let H2 \def (eq_ind T (lift h1 d1 (TLRef n)) (\lambda (t: T).(eq T t -(lift h2 (plus d2 h1) x))) H0 (TLRef n) (lift_lref_lt n h1 d1 H1)) in -(eq_ind_r T (TLRef n) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift -h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2))))) (ex_intro2 T -(\lambda (t2: T).(eq T (TLRef n) (lift h1 d1 t2))) (\lambda (t2: T).(eq T -(TLRef n) (lift h2 d2 t2))) (TLRef n) (eq_ind_r T (TLRef n) (\lambda (t: -T).(eq T (TLRef n) t)) (refl_equal T (TLRef n)) (lift h1 d1 (TLRef n)) -(lift_lref_lt n h1 d1 H1)) (eq_ind_r T (TLRef n) (\lambda (t: T).(eq T (TLRef -n) t)) (refl_equal T (TLRef n)) (lift h2 d2 (TLRef n)) (lift_lref_lt n h2 d2 -(lt_le_trans n d1 d2 H1 H)))) x (lift_gen_lref_lt h2 (plus d2 h1) n -(lt_le_trans n d1 (plus d2 h1) H1 (le_plus_trans d1 d2 h1 H)) x H2)))) -(\lambda (H1: (le d1 n)).(let H2 \def (eq_ind T (lift h1 d1 (TLRef n)) -(\lambda (t: T).(eq T t (lift h2 (plus d2 h1) x))) H0 (TLRef (plus n h1)) -(lift_lref_ge n h1 d1 H1)) in (lt_le_e n d2 (ex2 T (\lambda (t2: T).(eq T x -(lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2)))) -(\lambda (H3: (lt n d2)).(eq_ind_r T (TLRef (plus n h1)) (\lambda (t: T).(ex2 -T (\lambda (t2: T).(eq T t (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) -(lift h2 d2 t2))))) (ex_intro2 T (\lambda (t2: T).(eq T (TLRef (plus n h1)) -(lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2))) (TLRef -n) (eq_ind_r T (TLRef (plus n h1)) (\lambda (t: T).(eq T (TLRef (plus n h1)) -t)) (refl_equal T (TLRef (plus n h1))) (lift h1 d1 (TLRef n)) (lift_lref_ge n -h1 d1 H1)) (eq_ind_r T (TLRef n) (\lambda (t: T).(eq T (TLRef n) t)) -(refl_equal T (TLRef n)) (lift h2 d2 (TLRef n)) (lift_lref_lt n h2 d2 H3))) x -(lift_gen_lref_lt h2 (plus d2 h1) (plus n h1) (lt_reg_r n d2 h1 H3) x H2))) -(\lambda (H3: (le d2 n)).(lt_le_e n (plus d2 h2) (ex2 T (\lambda (t2: T).(eq -T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2)))) -(\lambda (H4: (lt n (plus d2 h2))).(lift_gen_lref_false h2 (plus d2 h1) (plus -n h1) (le_plus_plus d2 n h1 h1 H3 (le_n h1)) (eq_ind_r nat (plus (plus d2 h2) -h1) (\lambda (n0: nat).(lt (plus n h1) n0)) (lt_reg_r n (plus d2 h2) h1 H4) -(plus (plus d2 h1) h2) (plus_permute_2_in_3 d2 h1 h2)) x H2 (ex2 T (\lambda -(t2: T).(eq T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 -d2 t2)))))) (\lambda (H4: (le (plus d2 h2) n)).(let H5 \def (eq_ind nat (plus -n h1) (\lambda (n0: nat).(eq T (TLRef n0) (lift h2 (plus d2 h1) x))) H2 (plus -(minus (plus n h1) h2) h2) (le_plus_minus_sym h2 (plus n h1) (le_plus_trans -h2 n h1 (le_trans h2 (plus d2 h2) n (le_plus_r d2 h2) H4)))) in (eq_ind_r T -(TLRef (minus (plus n h1) h2)) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T -t (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2))))) -(ex_intro2 T (\lambda (t2: T).(eq T (TLRef (minus (plus n h1) h2)) (lift h1 -d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2))) (TLRef (minus n -h2)) (eq_ind_r nat (plus (minus n h2) h1) (\lambda (n0: nat).(eq T (TLRef n0) -(lift h1 d1 (TLRef (minus n h2))))) (eq_ind_r T (TLRef (plus (minus n h2) -h1)) (\lambda (t: T).(eq T (TLRef (plus (minus n h2) h1)) t)) (refl_equal T -(TLRef (plus (minus n h2) h1))) (lift h1 d1 (TLRef (minus n h2))) -(lift_lref_ge (minus n h2) h1 d1 (le_trans d1 d2 (minus n h2) H (le_minus d2 -n h2 H4)))) (minus (plus n h1) h2) (le_minus_plus h2 n (le_trans h2 (plus d2 -h2) n (le_plus_r d2 h2) H4) h1)) (eq_ind_r nat (plus (minus n h2) h2) -(\lambda (n0: nat).(eq T (TLRef n0) (lift h2 d2 (TLRef (minus n0 h2))))) -(eq_ind_r T (TLRef (plus (minus (plus (minus n h2) h2) h2) h2)) (\lambda (t: -T).(eq T (TLRef (plus (minus n h2) h2)) t)) (f_equal nat T TLRef (plus (minus -n h2) h2) (plus (minus (plus (minus n h2) h2) h2) h2) (f_equal2 nat nat nat -plus (minus n h2) (minus (plus (minus n h2) h2) h2) h2 h2 (sym_eq nat (minus -(plus (minus n h2) h2) h2) (minus n h2) (minus_plus_r (minus n h2) h2)) -(refl_equal nat h2))) (lift h2 d2 (TLRef (minus (plus (minus n h2) h2) h2))) -(lift_lref_ge (minus (plus (minus n h2) h2) h2) h2 d2 (le_minus d2 (plus -(minus n h2) h2) h2 (le_plus_plus d2 (minus n h2) h2 h2 (le_minus d2 n h2 H4) -(le_n h2))))) n (le_plus_minus_sym h2 n (le_trans h2 (plus d2 h2) n -(le_plus_r d2 h2) H4)))) x (lift_gen_lref_ge h2 (plus d2 h1) (minus (plus n -h1) h2) (arith0 h2 d2 n H4 h1) x H5)))))))))))))))))) (\lambda (k: -K).(\lambda (t: T).(\lambda (H: ((\forall (x: T).(\forall (h1: nat).(\forall -(h2: nat).(\forall (d1: nat).(\forall (d2: nat).((le d1 d2) \to ((eq T (lift -h1 d1 t) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2: T).(eq T x (lift -h1 d1 t2))) (\lambda (t2: T).(eq T t (lift h2 d2 t2))))))))))))).(\lambda -(t0: T).(\lambda (H0: ((\forall (x: T).(\forall (h1: nat).(\forall (h2: -nat).(\forall (d1: nat).(\forall (d2: nat).((le d1 d2) \to ((eq T (lift h1 d1 -t0) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1 -t2))) (\lambda (t2: T).(eq T t0 (lift h2 d2 t2))))))))))))).(\lambda (x: -T).(\lambda (h1: nat).(\lambda (h2: nat).(\lambda (d1: nat).(\lambda (d2: -nat).(\lambda (H1: (le d1 d2)).(\lambda (H2: (eq T (lift h1 d1 (THead k t -t0)) (lift h2 (plus d2 h1) x))).(K_ind (\lambda (k0: K).((eq T (lift h1 d1 -(THead k0 t t0)) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2: T).(eq T -x (lift h1 d1 t2))) (\lambda (t2: T).(eq T (THead k0 t t0) (lift h2 d2 -t2)))))) (\lambda (b: B).(\lambda (H3: (eq T (lift h1 d1 (THead (Bind b) t -t0)) (lift h2 (plus d2 h1) x))).(let H4 \def (eq_ind T (lift h1 d1 (THead -(Bind b) t t0)) (\lambda (t2: T).(eq T t2 (lift h2 (plus d2 h1) x))) H3 -(THead (Bind b) (lift h1 d1 t) (lift h1 (S d1) t0)) (lift_bind b t t0 h1 d1)) -in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Bind b) y -z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h1 d1 t) (lift h2 (plus d2 -h1) y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h1 (S d1) t0) (lift h2 -(S (plus d2 h1)) z)))) (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1 t2))) -(\lambda (t2: T).(eq T (THead (Bind b) t t0) (lift h2 d2 t2)))) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H5: (eq T x (THead (Bind b) x0 x1))).(\lambda -(H6: (eq T (lift h1 d1 t) (lift h2 (plus d2 h1) x0))).(\lambda (H7: (eq T -(lift h1 (S d1) t0) (lift h2 (S (plus d2 h1)) x1))).(eq_ind_r T (THead (Bind -b) x0 x1) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h1 d1 t3))) -(\lambda (t3: T).(eq T (THead (Bind b) t t0) (lift h2 d2 t3))))) (ex2_ind T -(\lambda (t2: T).(eq T x0 (lift h1 d1 t2))) (\lambda (t2: T).(eq T t (lift h2 -d2 t2))) (ex2 T (\lambda (t2: T).(eq T (THead (Bind b) x0 x1) (lift h1 d1 -t2))) (\lambda (t2: T).(eq T (THead (Bind b) t t0) (lift h2 d2 t2)))) -(\lambda (x2: T).(\lambda (H8: (eq T x0 (lift h1 d1 x2))).(\lambda (H9: (eq T -t (lift h2 d2 x2))).(eq_ind_r T (lift h1 d1 x2) (\lambda (t2: T).(ex2 T -(\lambda (t3: T).(eq T (THead (Bind b) t2 x1) (lift h1 d1 t3))) (\lambda (t3: -T).(eq T (THead (Bind b) t t0) (lift h2 d2 t3))))) (eq_ind_r T (lift h2 d2 -x2) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead (Bind b) (lift h1 -d1 x2) x1) (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead (Bind b) t2 t0) -(lift h2 d2 t3))))) (let H10 \def (refl_equal nat (plus (S d2) h1)) in (let -H11 \def (eq_ind nat (S (plus d2 h1)) (\lambda (n: nat).(eq T (lift h1 (S d1) -t0) (lift h2 n x1))) H7 (plus (S d2) h1) H10) in (ex2_ind T (\lambda (t2: -T).(eq T x1 (lift h1 (S d1) t2))) (\lambda (t2: T).(eq T t0 (lift h2 (S d2) -t2))) (ex2 T (\lambda (t2: T).(eq T (THead (Bind b) (lift h1 d1 x2) x1) (lift -h1 d1 t2))) (\lambda (t2: T).(eq T (THead (Bind b) (lift h2 d2 x2) t0) (lift -h2 d2 t2)))) (\lambda (x3: T).(\lambda (H12: (eq T x1 (lift h1 (S d1) -x3))).(\lambda (H13: (eq T t0 (lift h2 (S d2) x3))).(eq_ind_r T (lift h1 (S -d1) x3) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead (Bind b) (lift -h1 d1 x2) t2) (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead (Bind b) (lift -h2 d2 x2) t0) (lift h2 d2 t3))))) (eq_ind_r T (lift h2 (S d2) x3) (\lambda -(t2: T).(ex2 T (\lambda (t3: T).(eq T (THead (Bind b) (lift h1 d1 x2) (lift -h1 (S d1) x3)) (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead (Bind b) (lift -h2 d2 x2) t2) (lift h2 d2 t3))))) (ex_intro2 T (\lambda (t2: T).(eq T (THead -(Bind b) (lift h1 d1 x2) (lift h1 (S d1) x3)) (lift h1 d1 t2))) (\lambda (t2: -T).(eq T (THead (Bind b) (lift h2 d2 x2) (lift h2 (S d2) x3)) (lift h2 d2 -t2))) (THead (Bind b) x2 x3) (eq_ind_r T (THead (Bind b) (lift h1 d1 x2) -(lift h1 (S d1) x3)) (\lambda (t2: T).(eq T (THead (Bind b) (lift h1 d1 x2) -(lift h1 (S d1) x3)) t2)) (refl_equal T (THead (Bind b) (lift h1 d1 x2) (lift -h1 (S d1) x3))) (lift h1 d1 (THead (Bind b) x2 x3)) (lift_bind b x2 x3 h1 -d1)) (eq_ind_r T (THead (Bind b) (lift h2 d2 x2) (lift h2 (S d2) x3)) -(\lambda (t2: T).(eq T (THead (Bind b) (lift h2 d2 x2) (lift h2 (S d2) x3)) -t2)) (refl_equal T (THead (Bind b) (lift h2 d2 x2) (lift h2 (S d2) x3))) -(lift h2 d2 (THead (Bind b) x2 x3)) (lift_bind b x2 x3 h2 d2))) t0 H13) x1 -H12)))) (H0 x1 h1 h2 (S d1) (S d2) (le_n_S d1 d2 H1) H11)))) t H9) x0 H8)))) -(H x0 h1 h2 d1 d2 H1 H6)) x H5)))))) (lift_gen_bind b (lift h1 d1 t) (lift h1 -(S d1) t0) x h2 (plus d2 h1) H4))))) (\lambda (f: F).(\lambda (H3: (eq T -(lift h1 d1 (THead (Flat f) t t0)) (lift h2 (plus d2 h1) x))).(let H4 \def -(eq_ind T (lift h1 d1 (THead (Flat f) t t0)) (\lambda (t2: T).(eq T t2 (lift -h2 (plus d2 h1) x))) H3 (THead (Flat f) (lift h1 d1 t) (lift h1 d1 t0)) -(lift_flat f t t0 h1 d1)) in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: -T).(eq T x (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T -(lift h1 d1 t) (lift h2 (plus d2 h1) y)))) (\lambda (_: T).(\lambda (z: -T).(eq T (lift h1 d1 t0) (lift h2 (plus d2 h1) z)))) (ex2 T (\lambda (t2: -T).(eq T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T (THead (Flat f) t t0) -(lift h2 d2 t2)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T x -(THead (Flat f) x0 x1))).(\lambda (H6: (eq T (lift h1 d1 t) (lift h2 (plus d2 -h1) x0))).(\lambda (H7: (eq T (lift h1 d1 t0) (lift h2 (plus d2 h1) -x1))).(eq_ind_r T (THead (Flat f) x0 x1) (\lambda (t2: T).(ex2 T (\lambda -(t3: T).(eq T t2 (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead (Flat f) t -t0) (lift h2 d2 t3))))) (ex2_ind T (\lambda (t2: T).(eq T x0 (lift h1 d1 -t2))) (\lambda (t2: T).(eq T t (lift h2 d2 t2))) (ex2 T (\lambda (t2: T).(eq -T (THead (Flat f) x0 x1) (lift h1 d1 t2))) (\lambda (t2: T).(eq T (THead -(Flat f) t t0) (lift h2 d2 t2)))) (\lambda (x2: T).(\lambda (H8: (eq T x0 -(lift h1 d1 x2))).(\lambda (H9: (eq T t (lift h2 d2 x2))).(eq_ind_r T (lift -h1 d1 x2) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead (Flat f) t2 -x1) (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead (Flat f) t t0) (lift h2 -d2 t3))))) (eq_ind_r T (lift h2 d2 x2) (\lambda (t2: T).(ex2 T (\lambda (t3: -T).(eq T (THead (Flat f) (lift h1 d1 x2) x1) (lift h1 d1 t3))) (\lambda (t3: -T).(eq T (THead (Flat f) t2 t0) (lift h2 d2 t3))))) (ex2_ind T (\lambda (t2: -T).(eq T x1 (lift h1 d1 t2))) (\lambda (t2: T).(eq T t0 (lift h2 d2 t2))) -(ex2 T (\lambda (t2: T).(eq T (THead (Flat f) (lift h1 d1 x2) x1) (lift h1 d1 -t2))) (\lambda (t2: T).(eq T (THead (Flat f) (lift h2 d2 x2) t0) (lift h2 d2 -t2)))) (\lambda (x3: T).(\lambda (H10: (eq T x1 (lift h1 d1 x3))).(\lambda -(H11: (eq T t0 (lift h2 d2 x3))).(eq_ind_r T (lift h1 d1 x3) (\lambda (t2: -T).(ex2 T (\lambda (t3: T).(eq T (THead (Flat f) (lift h1 d1 x2) t2) (lift h1 -d1 t3))) (\lambda (t3: T).(eq T (THead (Flat f) (lift h2 d2 x2) t0) (lift h2 -d2 t3))))) (eq_ind_r T (lift h2 d2 x3) (\lambda (t2: T).(ex2 T (\lambda (t3: -T).(eq T (THead (Flat f) (lift h1 d1 x2) (lift h1 d1 x3)) (lift h1 d1 t3))) -(\lambda (t3: T).(eq T (THead (Flat f) (lift h2 d2 x2) t2) (lift h2 d2 -t3))))) (ex_intro2 T (\lambda (t2: T).(eq T (THead (Flat f) (lift h1 d1 x2) -(lift h1 d1 x3)) (lift h1 d1 t2))) (\lambda (t2: T).(eq T (THead (Flat f) -(lift h2 d2 x2) (lift h2 d2 x3)) (lift h2 d2 t2))) (THead (Flat f) x2 x3) -(eq_ind_r T (THead (Flat f) (lift h1 d1 x2) (lift h1 d1 x3)) (\lambda (t2: -T).(eq T (THead (Flat f) (lift h1 d1 x2) (lift h1 d1 x3)) t2)) (refl_equal T -(THead (Flat f) (lift h1 d1 x2) (lift h1 d1 x3))) (lift h1 d1 (THead (Flat f) -x2 x3)) (lift_flat f x2 x3 h1 d1)) (eq_ind_r T (THead (Flat f) (lift h2 d2 -x2) (lift h2 d2 x3)) (\lambda (t2: T).(eq T (THead (Flat f) (lift h2 d2 x2) -(lift h2 d2 x3)) t2)) (refl_equal T (THead (Flat f) (lift h2 d2 x2) (lift h2 -d2 x3))) (lift h2 d2 (THead (Flat f) x2 x3)) (lift_flat f x2 x3 h2 d2))) t0 -H11) x1 H10)))) (H0 x1 h1 h2 d1 d2 H1 H7)) t H9) x0 H8)))) (H x0 h1 h2 d1 d2 -H1 H6)) x H5)))))) (lift_gen_flat f (lift h1 d1 t) (lift h1 d1 t0) x h2 (plus -d2 h1) H4))))) k H2))))))))))))) t1). -(* COMMENTS -Initial nodes: 5037 -END *) - -theorem lifts_inj: - \forall (xs: TList).(\forall (ts: TList).(\forall (h: nat).(\forall (d: -nat).((eq TList (lifts h d xs) (lifts h d ts)) \to (eq TList xs ts))))) -\def - \lambda (xs: TList).(TList_ind (\lambda (t: TList).(\forall (ts: -TList).(\forall (h: nat).(\forall (d: nat).((eq TList (lifts h d t) (lifts h -d ts)) \to (eq TList t ts)))))) (\lambda (ts: TList).(TList_ind (\lambda (t: -TList).(\forall (h: nat).(\forall (d: nat).((eq TList (lifts h d TNil) (lifts -h d t)) \to (eq TList TNil t))))) (\lambda (_: nat).(\lambda (_: -nat).(\lambda (_: (eq TList TNil TNil)).(refl_equal TList TNil)))) (\lambda -(t: T).(\lambda (t0: TList).(\lambda (_: ((\forall (h: nat).(\forall (d: -nat).((eq TList TNil (lifts h d t0)) \to (eq TList TNil t0)))))).(\lambda (h: -nat).(\lambda (d: nat).(\lambda (H0: (eq TList TNil (TCons (lift h d t) -(lifts h d t0)))).(let H1 \def (eq_ind TList TNil (\lambda (ee: TList).(match -ee in TList return (\lambda (_: TList).Prop) with [TNil \Rightarrow True | -(TCons _ _) \Rightarrow False])) I (TCons (lift h d t) (lifts h d t0)) H0) in -(False_ind (eq TList TNil (TCons t t0)) H1)))))))) ts)) (\lambda (t: -T).(\lambda (t0: TList).(\lambda (H: ((\forall (ts: TList).(\forall (h: -nat).(\forall (d: nat).((eq TList (lifts h d t0) (lifts h d ts)) \to (eq -TList t0 ts))))))).(\lambda (ts: TList).(TList_ind (\lambda (t1: -TList).(\forall (h: nat).(\forall (d: nat).((eq TList (lifts h d (TCons t -t0)) (lifts h d t1)) \to (eq TList (TCons t t0) t1))))) (\lambda (h: -nat).(\lambda (d: nat).(\lambda (H0: (eq TList (TCons (lift h d t) (lifts h d -t0)) TNil)).(let H1 \def (eq_ind TList (TCons (lift h d t) (lifts h d t0)) -(\lambda (ee: TList).(match ee in TList return (\lambda (_: TList).Prop) with -[TNil \Rightarrow False | (TCons _ _) \Rightarrow True])) I TNil H0) in -(False_ind (eq TList (TCons t t0) TNil) H1))))) (\lambda (t1: T).(\lambda -(t2: TList).(\lambda (_: ((\forall (h: nat).(\forall (d: nat).((eq TList -(TCons (lift h d t) (lifts h d t0)) (lifts h d t2)) \to (eq TList (TCons t -t0) t2)))))).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: (eq TList -(TCons (lift h d t) (lifts h d t0)) (TCons (lift h d t1) (lifts h d -t2)))).(let H2 \def (f_equal TList T (\lambda (e: TList).(match e in TList -return (\lambda (_: TList).T) with [TNil \Rightarrow ((let rec lref_map (f: -((nat \to nat))) (d0: nat) (t3: T) on t3: T \def (match t3 with [(TSort n) -\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) with -[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t4) \Rightarrow -(THead k (lref_map f d0 u) (lref_map f (s k d0) t4))]) in lref_map) (\lambda -(x: nat).(plus x h)) d t) | (TCons t3 _) \Rightarrow t3])) (TCons (lift h d -t) (lifts h d t0)) (TCons (lift h d t1) (lifts h d t2)) H1) in ((let H3 \def -(f_equal TList TList (\lambda (e: TList).(match e in TList return (\lambda -(_: TList).TList) with [TNil \Rightarrow ((let rec lifts (h0: nat) (d0: nat) -(ts0: TList) on ts0: TList \def (match ts0 with [TNil \Rightarrow TNil | -(TCons t3 ts1) \Rightarrow (TCons (lift h0 d0 t3) (lifts h0 d0 ts1))]) in -lifts) h d t0) | (TCons _ t3) \Rightarrow t3])) (TCons (lift h d t) (lifts h -d t0)) (TCons (lift h d t1) (lifts h d t2)) H1) in (\lambda (H4: (eq T (lift -h d t) (lift h d t1))).(eq_ind T t (\lambda (t3: T).(eq TList (TCons t t0) -(TCons t3 t2))) (f_equal2 T TList TList TCons t t t0 t2 (refl_equal T t) (H -t2 h d H3)) t1 (lift_inj t t1 h d H4)))) H2)))))))) ts))))) xs). -(* COMMENTS -Initial nodes: 772 -END *) - -theorem lift_free: - \forall (t: T).(\forall (h: nat).(\forall (k: nat).(\forall (d: -nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to (eq T (lift k e -(lift h d t)) (lift (plus k h) d t)))))))) -\def - \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (h: nat).(\forall (k: -nat).(\forall (d: nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to -(eq T (lift k e (lift h d t0)) (lift (plus k h) d t0))))))))) (\lambda (n: -nat).(\lambda (h: nat).(\lambda (k: nat).(\lambda (d: nat).(\lambda (e: -nat).(\lambda (_: (le e (plus d h))).(\lambda (_: (le d e)).(eq_ind_r T -(TSort n) (\lambda (t0: T).(eq T (lift k e t0) (lift (plus k h) d (TSort -n)))) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq T t0 (lift (plus k h) d -(TSort n)))) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq T (TSort n) t0)) -(refl_equal T (TSort n)) (lift (plus k h) d (TSort n)) (lift_sort n (plus k -h) d)) (lift k e (TSort n)) (lift_sort n k e)) (lift h d (TSort n)) -(lift_sort n h d))))))))) (\lambda (n: nat).(\lambda (h: nat).(\lambda (k: -nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (H: (le e (plus d -h))).(\lambda (H0: (le d e)).(lt_le_e n d (eq T (lift k e (lift h d (TLRef -n))) (lift (plus k h) d (TLRef n))) (\lambda (H1: (lt n d)).(eq_ind_r T -(TLRef n) (\lambda (t0: T).(eq T (lift k e t0) (lift (plus k h) d (TLRef -n)))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 (lift (plus k h) d -(TLRef n)))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (TLRef n) t0)) -(refl_equal T (TLRef n)) (lift (plus k h) d (TLRef n)) (lift_lref_lt n (plus -k h) d H1)) (lift k e (TLRef n)) (lift_lref_lt n k e (lt_le_trans n d e H1 -H0))) (lift h d (TLRef n)) (lift_lref_lt n h d H1))) (\lambda (H1: (le d -n)).(eq_ind_r T (TLRef (plus n h)) (\lambda (t0: T).(eq T (lift k e t0) (lift -(plus k h) d (TLRef n)))) (eq_ind_r T (TLRef (plus (plus n h) k)) (\lambda -(t0: T).(eq T t0 (lift (plus k h) d (TLRef n)))) (eq_ind_r T (TLRef (plus n -(plus k h))) (\lambda (t0: T).(eq T (TLRef (plus (plus n h) k)) t0)) (f_equal -nat T TLRef (plus (plus n h) k) (plus n (plus k h)) -(plus_permute_2_in_3_assoc n h k)) (lift (plus k h) d (TLRef n)) -(lift_lref_ge n (plus k h) d H1)) (lift k e (TLRef (plus n h))) (lift_lref_ge -(plus n h) k e (le_trans e (plus d h) (plus n h) H (le_plus_plus d n h h H1 -(le_n h))))) (lift h d (TLRef n)) (lift_lref_ge n h d H1))))))))))) (\lambda -(k: K).(\lambda (t0: T).(\lambda (H: ((\forall (h: nat).(\forall (k0: -nat).(\forall (d: nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to -(eq T (lift k0 e (lift h d t0)) (lift (plus k0 h) d t0)))))))))).(\lambda -(t1: T).(\lambda (H0: ((\forall (h: nat).(\forall (k0: nat).(\forall (d: -nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to (eq T (lift k0 e -(lift h d t1)) (lift (plus k0 h) d t1)))))))))).(\lambda (h: nat).(\lambda -(k0: nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (H1: (le e (plus d -h))).(\lambda (H2: (le d e)).(eq_ind_r T (THead k (lift h d t0) (lift h (s k -d) t1)) (\lambda (t2: T).(eq T (lift k0 e t2) (lift (plus k0 h) d (THead k t0 -t1)))) (eq_ind_r T (THead k (lift k0 e (lift h d t0)) (lift k0 (s k e) (lift -h (s k d) t1))) (\lambda (t2: T).(eq T t2 (lift (plus k0 h) d (THead k t0 -t1)))) (eq_ind_r T (THead k (lift (plus k0 h) d t0) (lift (plus k0 h) (s k d) -t1)) (\lambda (t2: T).(eq T (THead k (lift k0 e (lift h d t0)) (lift k0 (s k -e) (lift h (s k d) t1))) t2)) (f_equal3 K T T T THead k k (lift k0 e (lift h -d t0)) (lift (plus k0 h) d t0) (lift k0 (s k e) (lift h (s k d) t1)) (lift -(plus k0 h) (s k d) t1) (refl_equal K k) (H h k0 d e H1 H2) (H0 h k0 (s k d) -(s k e) (eq_ind nat (s k (plus d h)) (\lambda (n: nat).(le (s k e) n)) (s_le -k e (plus d h) H1) (plus (s k d) h) (s_plus k d h)) (s_le k d e H2))) (lift -(plus k0 h) d (THead k t0 t1)) (lift_head k t0 t1 (plus k0 h) d)) (lift k0 e -(THead k (lift h d t0) (lift h (s k d) t1))) (lift_head k (lift h d t0) (lift -h (s k d) t1) k0 e)) (lift h d (THead k t0 t1)) (lift_head k t0 t1 h -d))))))))))))) t). -(* COMMENTS -Initial nodes: 1407 -END *) - -theorem lift_d: - \forall (t: T).(\forall (h: nat).(\forall (k: nat).(\forall (d: -nat).(\forall (e: nat).((le e d) \to (eq T (lift h (plus k d) (lift k e t)) -(lift k e (lift h d t)))))))) -\def - \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (h: nat).(\forall (k: -nat).(\forall (d: nat).(\forall (e: nat).((le e d) \to (eq T (lift h (plus k -d) (lift k e t0)) (lift k e (lift h d t0))))))))) (\lambda (n: nat).(\lambda -(h: nat).(\lambda (k: nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (_: -(le e d)).(eq_ind_r T (TSort n) (\lambda (t0: T).(eq T (lift h (plus k d) t0) -(lift k e (lift h d (TSort n))))) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq -T t0 (lift k e (lift h d (TSort n))))) (eq_ind_r T (TSort n) (\lambda (t0: -T).(eq T (TSort n) (lift k e t0))) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq -T (TSort n) t0)) (refl_equal T (TSort n)) (lift k e (TSort n)) (lift_sort n k -e)) (lift h d (TSort n)) (lift_sort n h d)) (lift h (plus k d) (TSort n)) -(lift_sort n h (plus k d))) (lift k e (TSort n)) (lift_sort n k e)))))))) -(\lambda (n: nat).(\lambda (h: nat).(\lambda (k: nat).(\lambda (d: -nat).(\lambda (e: nat).(\lambda (H: (le e d)).(lt_le_e n e (eq T (lift h -(plus k d) (lift k e (TLRef n))) (lift k e (lift h d (TLRef n)))) (\lambda -(H0: (lt n e)).(let H1 \def (lt_le_trans n e d H0 H) in (eq_ind_r T (TLRef n) -(\lambda (t0: T).(eq T (lift h (plus k d) t0) (lift k e (lift h d (TLRef -n))))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 (lift k e (lift h d -(TLRef n))))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (TLRef n) (lift k -e t0))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (TLRef n) t0)) -(refl_equal T (TLRef n)) (lift k e (TLRef n)) (lift_lref_lt n k e H0)) (lift -h d (TLRef n)) (lift_lref_lt n h d H1)) (lift h (plus k d) (TLRef n)) -(lift_lref_lt n h (plus k d) (lt_le_trans n d (plus k d) H1 (le_plus_r k -d)))) (lift k e (TLRef n)) (lift_lref_lt n k e H0)))) (\lambda (H0: (le e -n)).(eq_ind_r T (TLRef (plus n k)) (\lambda (t0: T).(eq T (lift h (plus k d) -t0) (lift k e (lift h d (TLRef n))))) (eq_ind_r nat (plus d k) (\lambda (n0: -nat).(eq T (lift h n0 (TLRef (plus n k))) (lift k e (lift h d (TLRef n))))) -(lt_le_e n d (eq T (lift h (plus d k) (TLRef (plus n k))) (lift k e (lift h d -(TLRef n)))) (\lambda (H1: (lt n d)).(eq_ind_r T (TLRef (plus n k)) (\lambda -(t0: T).(eq T t0 (lift k e (lift h d (TLRef n))))) (eq_ind_r T (TLRef n) -(\lambda (t0: T).(eq T (TLRef (plus n k)) (lift k e t0))) (eq_ind_r T (TLRef -(plus n k)) (\lambda (t0: T).(eq T (TLRef (plus n k)) t0)) (refl_equal T -(TLRef (plus n k))) (lift k e (TLRef n)) (lift_lref_ge n k e H0)) (lift h d -(TLRef n)) (lift_lref_lt n h d H1)) (lift h (plus d k) (TLRef (plus n k))) -(lift_lref_lt (plus n k) h (plus d k) (lt_reg_r n d k H1)))) (\lambda (H1: -(le d n)).(eq_ind_r T (TLRef (plus (plus n k) h)) (\lambda (t0: T).(eq T t0 -(lift k e (lift h d (TLRef n))))) (eq_ind_r T (TLRef (plus n h)) (\lambda -(t0: T).(eq T (TLRef (plus (plus n k) h)) (lift k e t0))) (eq_ind_r T (TLRef -(plus (plus n h) k)) (\lambda (t0: T).(eq T (TLRef (plus (plus n k) h)) t0)) -(f_equal nat T TLRef (plus (plus n k) h) (plus (plus n h) k) (sym_eq nat -(plus (plus n h) k) (plus (plus n k) h) (plus_permute_2_in_3 n h k))) (lift k -e (TLRef (plus n h))) (lift_lref_ge (plus n h) k e (le_plus_trans e n h H0))) -(lift h d (TLRef n)) (lift_lref_ge n h d H1)) (lift h (plus d k) (TLRef (plus -n k))) (lift_lref_ge (plus n k) h (plus d k) (le_plus_plus d n k k H1 (le_n -k)))))) (plus k d) (plus_sym k d)) (lift k e (TLRef n)) (lift_lref_ge n k e -H0)))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (H: ((\forall (h: -nat).(\forall (k0: nat).(\forall (d: nat).(\forall (e: nat).((le e d) \to (eq -T (lift h (plus k0 d) (lift k0 e t0)) (lift k0 e (lift h d -t0)))))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (h: nat).(\forall (k0: -nat).(\forall (d: nat).(\forall (e: nat).((le e d) \to (eq T (lift h (plus k0 -d) (lift k0 e t1)) (lift k0 e (lift h d t1)))))))))).(\lambda (h: -nat).(\lambda (k0: nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (H1: (le -e d)).(eq_ind_r T (THead k (lift k0 e t0) (lift k0 (s k e) t1)) (\lambda (t2: -T).(eq T (lift h (plus k0 d) t2) (lift k0 e (lift h d (THead k t0 t1))))) -(eq_ind_r T (THead k (lift h (plus k0 d) (lift k0 e t0)) (lift h (s k (plus -k0 d)) (lift k0 (s k e) t1))) (\lambda (t2: T).(eq T t2 (lift k0 e (lift h d -(THead k t0 t1))))) (eq_ind_r T (THead k (lift h d t0) (lift h (s k d) t1)) -(\lambda (t2: T).(eq T (THead k (lift h (plus k0 d) (lift k0 e t0)) (lift h -(s k (plus k0 d)) (lift k0 (s k e) t1))) (lift k0 e t2))) (eq_ind_r T (THead -k (lift k0 e (lift h d t0)) (lift k0 (s k e) (lift h (s k d) t1))) (\lambda -(t2: T).(eq T (THead k (lift h (plus k0 d) (lift k0 e t0)) (lift h (s k (plus -k0 d)) (lift k0 (s k e) t1))) t2)) (eq_ind_r nat (plus k0 (s k d)) (\lambda -(n: nat).(eq T (THead k (lift h (plus k0 d) (lift k0 e t0)) (lift h n (lift -k0 (s k e) t1))) (THead k (lift k0 e (lift h d t0)) (lift k0 (s k e) (lift h -(s k d) t1))))) (f_equal3 K T T T THead k k (lift h (plus k0 d) (lift k0 e -t0)) (lift k0 e (lift h d t0)) (lift h (plus k0 (s k d)) (lift k0 (s k e) -t1)) (lift k0 (s k e) (lift h (s k d) t1)) (refl_equal K k) (H h k0 d e H1) -(H0 h k0 (s k d) (s k e) (s_le k e d H1))) (s k (plus k0 d)) (s_plus_sym k k0 -d)) (lift k0 e (THead k (lift h d t0) (lift h (s k d) t1))) (lift_head k -(lift h d t0) (lift h (s k d) t1) k0 e)) (lift h d (THead k t0 t1)) -(lift_head k t0 t1 h d)) (lift h (plus k0 d) (THead k (lift k0 e t0) (lift k0 -(s k e) t1))) (lift_head k (lift k0 e t0) (lift k0 (s k e) t1) h (plus k0 -d))) (lift k0 e (THead k t0 t1)) (lift_head k t0 t1 k0 e)))))))))))) t). -(* COMMENTS -Initial nodes: 2143 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/lift/tlt.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/lift/tlt.ma deleted file mode 100644 index 1d8edc7df..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/lift/tlt.ma +++ /dev/null @@ -1,299 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/lift/fwd.ma". - -include "Basic-1/tlt/props.ma". - -theorem lift_weight_map: - \forall (t: T).(\forall (h: nat).(\forall (d: nat).(\forall (f: ((nat \to -nat))).(((\forall (m: nat).((le d m) \to (eq nat (f m) O)))) \to (eq nat -(weight_map f (lift h d t)) (weight_map f t)))))) -\def - \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (h: nat).(\forall (d: -nat).(\forall (f: ((nat \to nat))).(((\forall (m: nat).((le d m) \to (eq nat -(f m) O)))) \to (eq nat (weight_map f (lift h d t0)) (weight_map f t0))))))) -(\lambda (n: nat).(\lambda (_: nat).(\lambda (d: nat).(\lambda (f: ((nat \to -nat))).(\lambda (_: ((\forall (m: nat).((le d m) \to (eq nat (f m) -O))))).(refl_equal nat (weight_map f (TSort n)))))))) (\lambda (n: -nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (f: ((nat \to -nat))).(\lambda (H: ((\forall (m: nat).((le d m) \to (eq nat (f m) -O))))).(lt_le_e n d (eq nat (weight_map f (lift h d (TLRef n))) (weight_map f -(TLRef n))) (\lambda (H0: (lt n d)).(eq_ind_r T (TLRef n) (\lambda (t0: -T).(eq nat (weight_map f t0) (weight_map f (TLRef n)))) (refl_equal nat -(weight_map f (TLRef n))) (lift h d (TLRef n)) (lift_lref_lt n h d H0))) -(\lambda (H0: (le d n)).(eq_ind_r T (TLRef (plus n h)) (\lambda (t0: T).(eq -nat (weight_map f t0) (weight_map f (TLRef n)))) (eq_ind_r nat O (\lambda -(n0: nat).(eq nat (f (plus n h)) n0)) (H (plus n h) (le_plus_trans d n h H0)) -(f n) (H n H0)) (lift h d (TLRef n)) (lift_lref_ge n h d H0))))))))) (\lambda -(k: K).(\lambda (t0: T).(\lambda (H: ((\forall (h: nat).(\forall (d: -nat).(\forall (f: ((nat \to nat))).(((\forall (m: nat).((le d m) \to (eq nat -(f m) O)))) \to (eq nat (weight_map f (lift h d t0)) (weight_map f -t0)))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (h: nat).(\forall (d: -nat).(\forall (f: ((nat \to nat))).(((\forall (m: nat).((le d m) \to (eq nat -(f m) O)))) \to (eq nat (weight_map f (lift h d t1)) (weight_map f -t1)))))))).(\lambda (h: nat).(\lambda (d: nat).(\lambda (f: ((nat \to -nat))).(\lambda (H1: ((\forall (m: nat).((le d m) \to (eq nat (f m) -O))))).(K_ind (\lambda (k0: K).(eq nat (weight_map f (lift h d (THead k0 t0 -t1))) (weight_map f (THead k0 t0 t1)))) (\lambda (b: B).(eq_ind_r T (THead -(Bind b) (lift h d t0) (lift h (s (Bind b) d) t1)) (\lambda (t2: T).(eq nat -(weight_map f t2) (weight_map f (THead (Bind b) t0 t1)))) (B_ind (\lambda -(b0: B).(eq nat (match b0 with [Abbr \Rightarrow (S (plus (weight_map f (lift -h d t0)) (weight_map (wadd f (S (weight_map f (lift h d t0)))) (lift h (S d) -t1)))) | Abst \Rightarrow (S (plus (weight_map f (lift h d t0)) (weight_map -(wadd f O) (lift h (S d) t1)))) | Void \Rightarrow (S (plus (weight_map f -(lift h d t0)) (weight_map (wadd f O) (lift h (S d) t1))))]) (match b0 with -[Abbr \Rightarrow (S (plus (weight_map f t0) (weight_map (wadd f (S -(weight_map f t0))) t1))) | Abst \Rightarrow (S (plus (weight_map f t0) -(weight_map (wadd f O) t1))) | Void \Rightarrow (S (plus (weight_map f t0) -(weight_map (wadd f O) t1)))]))) (eq_ind_r nat (weight_map f t0) (\lambda (n: -nat).(eq nat (S (plus n (weight_map (wadd f (S n)) (lift h (S d) t1)))) (S -(plus (weight_map f t0) (weight_map (wadd f (S (weight_map f t0))) t1))))) -(eq_ind_r nat (weight_map (wadd f (S (weight_map f t0))) t1) (\lambda (n: -nat).(eq nat (S (plus (weight_map f t0) n)) (S (plus (weight_map f t0) -(weight_map (wadd f (S (weight_map f t0))) t1))))) (refl_equal nat (S (plus -(weight_map f t0) (weight_map (wadd f (S (weight_map f t0))) t1)))) -(weight_map (wadd f (S (weight_map f t0))) (lift h (S d) t1)) (H0 h (S d) -(wadd f (S (weight_map f t0))) (\lambda (m: nat).(\lambda (H2: (le (S d) -m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S n))) (\lambda (n: nat).(le d -n)) (eq nat (wadd f (S (weight_map f t0)) m) O) (\lambda (x: nat).(\lambda -(H3: (eq nat m (S x))).(\lambda (H4: (le d x)).(eq_ind_r nat (S x) (\lambda -(n: nat).(eq nat (wadd f (S (weight_map f t0)) n) O)) (H1 x H4) m H3)))) -(le_gen_S d m H2)))))) (weight_map f (lift h d t0)) (H h d f H1)) (eq_ind_r -nat (weight_map (wadd f O) t1) (\lambda (n: nat).(eq nat (S (plus (weight_map -f (lift h d t0)) n)) (S (plus (weight_map f t0) (weight_map (wadd f O) -t1))))) (f_equal nat nat S (plus (weight_map f (lift h d t0)) (weight_map -(wadd f O) t1)) (plus (weight_map f t0) (weight_map (wadd f O) t1)) (f_equal2 -nat nat nat plus (weight_map f (lift h d t0)) (weight_map f t0) (weight_map -(wadd f O) t1) (weight_map (wadd f O) t1) (H h d f H1) (refl_equal nat -(weight_map (wadd f O) t1)))) (weight_map (wadd f O) (lift h (S d) t1)) (H0 h -(S d) (wadd f O) (\lambda (m: nat).(\lambda (H2: (le (S d) m)).(ex2_ind nat -(\lambda (n: nat).(eq nat m (S n))) (\lambda (n: nat).(le d n)) (eq nat (wadd -f O m) O) (\lambda (x: nat).(\lambda (H3: (eq nat m (S x))).(\lambda (H4: (le -d x)).(eq_ind_r nat (S x) (\lambda (n: nat).(eq nat (wadd f O n) O)) (H1 x -H4) m H3)))) (le_gen_S d m H2)))))) (eq_ind_r nat (weight_map (wadd f O) t1) -(\lambda (n: nat).(eq nat (S (plus (weight_map f (lift h d t0)) n)) (S (plus -(weight_map f t0) (weight_map (wadd f O) t1))))) (f_equal nat nat S (plus -(weight_map f (lift h d t0)) (weight_map (wadd f O) t1)) (plus (weight_map f -t0) (weight_map (wadd f O) t1)) (f_equal2 nat nat nat plus (weight_map f -(lift h d t0)) (weight_map f t0) (weight_map (wadd f O) t1) (weight_map (wadd -f O) t1) (H h d f H1) (refl_equal nat (weight_map (wadd f O) t1)))) -(weight_map (wadd f O) (lift h (S d) t1)) (H0 h (S d) (wadd f O) (\lambda (m: -nat).(\lambda (H2: (le (S d) m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S -n))) (\lambda (n: nat).(le d n)) (eq nat (wadd f O m) O) (\lambda (x: -nat).(\lambda (H3: (eq nat m (S x))).(\lambda (H4: (le d x)).(eq_ind_r nat (S -x) (\lambda (n: nat).(eq nat (wadd f O n) O)) (H1 x H4) m H3)))) (le_gen_S d -m H2)))))) b) (lift h d (THead (Bind b) t0 t1)) (lift_head (Bind b) t0 t1 h -d))) (\lambda (f0: F).(eq_ind_r T (THead (Flat f0) (lift h d t0) (lift h (s -(Flat f0) d) t1)) (\lambda (t2: T).(eq nat (weight_map f t2) (weight_map f -(THead (Flat f0) t0 t1)))) (f_equal nat nat S (plus (weight_map f (lift h d -t0)) (weight_map f (lift h d t1))) (plus (weight_map f t0) (weight_map f t1)) -(f_equal2 nat nat nat plus (weight_map f (lift h d t0)) (weight_map f t0) -(weight_map f (lift h d t1)) (weight_map f t1) (H h d f H1) (H0 h d f H1))) -(lift h d (THead (Flat f0) t0 t1)) (lift_head (Flat f0) t0 t1 h d))) -k)))))))))) t). -(* COMMENTS -Initial nodes: 1969 -END *) - -theorem lift_weight: - \forall (t: T).(\forall (h: nat).(\forall (d: nat).(eq nat (weight (lift h d -t)) (weight t)))) -\def - \lambda (t: T).(\lambda (h: nat).(\lambda (d: nat).(lift_weight_map t h d -(\lambda (_: nat).O) (\lambda (m: nat).(\lambda (_: (le d m)).(refl_equal nat -O)))))). -(* COMMENTS -Initial nodes: 31 -END *) - -theorem lift_weight_add: - \forall (w: nat).(\forall (t: T).(\forall (h: nat).(\forall (d: -nat).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall -(m: nat).((lt m d) \to (eq nat (g m) (f m))))) \to ((eq nat (g d) w) \to -(((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f m))))) \to (eq nat -(weight_map f (lift h d t)) (weight_map g (lift (S h) d t))))))))))) -\def - \lambda (w: nat).(\lambda (t: T).(T_ind (\lambda (t0: T).(\forall (h: -nat).(\forall (d: nat).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (m: nat).((lt m d) \to (eq nat (g m) (f m))))) \to ((eq nat -(g d) w) \to (((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f m))))) -\to (eq nat (weight_map f (lift h d t0)) (weight_map g (lift (S h) d -t0))))))))))) (\lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda -(f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (_: ((\forall (m: -nat).((lt m d) \to (eq nat (g m) (f m)))))).(\lambda (_: (eq nat (g d) -w)).(\lambda (_: ((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f -m)))))).(refl_equal nat (weight_map g (lift (S h) d (TSort n)))))))))))) -(\lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (f: ((nat \to -nat))).(\lambda (g: ((nat \to nat))).(\lambda (H: ((\forall (m: nat).((lt m -d) \to (eq nat (g m) (f m)))))).(\lambda (_: (eq nat (g d) w)).(\lambda (H1: -((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f m)))))).(lt_le_e n d -(eq nat (weight_map f (lift h d (TLRef n))) (weight_map g (lift (S h) d -(TLRef n)))) (\lambda (H2: (lt n d)).(eq_ind_r T (TLRef n) (\lambda (t0: -T).(eq nat (weight_map f t0) (weight_map g (lift (S h) d (TLRef n))))) -(eq_ind_r T (TLRef n) (\lambda (t0: T).(eq nat (weight_map f (TLRef n)) -(weight_map g t0))) (sym_eq nat (g n) (f n) (H n H2)) (lift (S h) d (TLRef -n)) (lift_lref_lt n (S h) d H2)) (lift h d (TLRef n)) (lift_lref_lt n h d -H2))) (\lambda (H2: (le d n)).(eq_ind_r T (TLRef (plus n h)) (\lambda (t0: -T).(eq nat (weight_map f t0) (weight_map g (lift (S h) d (TLRef n))))) -(eq_ind_r T (TLRef (plus n (S h))) (\lambda (t0: T).(eq nat (weight_map f -(TLRef (plus n h))) (weight_map g t0))) (eq_ind nat (S (plus n h)) (\lambda -(n0: nat).(eq nat (f (plus n h)) (g n0))) (sym_eq nat (g (S (plus n h))) (f -(plus n h)) (H1 (plus n h) (le_plus_trans d n h H2))) (plus n (S h)) -(plus_n_Sm n h)) (lift (S h) d (TLRef n)) (lift_lref_ge n (S h) d H2)) (lift -h d (TLRef n)) (lift_lref_ge n h d H2)))))))))))) (\lambda (k: K).(\lambda -(t0: T).(\lambda (H: ((\forall (h: nat).(\forall (d: nat).(\forall (f: ((nat -\to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).((lt m d) \to -(eq nat (g m) (f m))))) \to ((eq nat (g d) w) \to (((\forall (m: nat).((le d -m) \to (eq nat (g (S m)) (f m))))) \to (eq nat (weight_map f (lift h d t0)) -(weight_map g (lift (S h) d t0)))))))))))).(\lambda (t1: T).(\lambda (H0: -((\forall (h: nat).(\forall (d: nat).(\forall (f: ((nat \to nat))).(\forall -(g: ((nat \to nat))).(((\forall (m: nat).((lt m d) \to (eq nat (g m) (f -m))))) \to ((eq nat (g d) w) \to (((\forall (m: nat).((le d m) \to (eq nat (g -(S m)) (f m))))) \to (eq nat (weight_map f (lift h d t1)) (weight_map g (lift -(S h) d t1)))))))))))).(\lambda (h: nat).(\lambda (d: nat).(\lambda (f: ((nat -\to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H1: ((\forall (m: -nat).((lt m d) \to (eq nat (g m) (f m)))))).(\lambda (H2: (eq nat (g d) -w)).(\lambda (H3: ((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f -m)))))).(K_ind (\lambda (k0: K).(eq nat (weight_map f (lift h d (THead k0 t0 -t1))) (weight_map g (lift (S h) d (THead k0 t0 t1))))) (\lambda (b: -B).(eq_ind_r T (THead (Bind b) (lift h d t0) (lift h (s (Bind b) d) t1)) -(\lambda (t2: T).(eq nat (weight_map f t2) (weight_map g (lift (S h) d (THead -(Bind b) t0 t1))))) (eq_ind_r T (THead (Bind b) (lift (S h) d t0) (lift (S h) -(s (Bind b) d) t1)) (\lambda (t2: T).(eq nat (weight_map f (THead (Bind b) -(lift h d t0) (lift h (s (Bind b) d) t1))) (weight_map g t2))) (B_ind -(\lambda (b0: B).(eq nat (match b0 with [Abbr \Rightarrow (S (plus -(weight_map f (lift h d t0)) (weight_map (wadd f (S (weight_map f (lift h d -t0)))) (lift h (S d) t1)))) | Abst \Rightarrow (S (plus (weight_map f (lift h -d t0)) (weight_map (wadd f O) (lift h (S d) t1)))) | Void \Rightarrow (S -(plus (weight_map f (lift h d t0)) (weight_map (wadd f O) (lift h (S d) -t1))))]) (match b0 with [Abbr \Rightarrow (S (plus (weight_map g (lift (S h) -d t0)) (weight_map (wadd g (S (weight_map g (lift (S h) d t0)))) (lift (S h) -(S d) t1)))) | Abst \Rightarrow (S (plus (weight_map g (lift (S h) d t0)) -(weight_map (wadd g O) (lift (S h) (S d) t1)))) | Void \Rightarrow (S (plus -(weight_map g (lift (S h) d t0)) (weight_map (wadd g O) (lift (S h) (S d) -t1))))]))) (f_equal nat nat S (plus (weight_map f (lift h d t0)) (weight_map -(wadd f (S (weight_map f (lift h d t0)))) (lift h (S d) t1))) (plus -(weight_map g (lift (S h) d t0)) (weight_map (wadd g (S (weight_map g (lift -(S h) d t0)))) (lift (S h) (S d) t1))) (f_equal2 nat nat nat plus (weight_map -f (lift h d t0)) (weight_map g (lift (S h) d t0)) (weight_map (wadd f (S -(weight_map f (lift h d t0)))) (lift h (S d) t1)) (weight_map (wadd g (S -(weight_map g (lift (S h) d t0)))) (lift (S h) (S d) t1)) (H h d f g H1 H2 -H3) (H0 h (S d) (wadd f (S (weight_map f (lift h d t0)))) (wadd g (S -(weight_map g (lift (S h) d t0)))) (\lambda (m: nat).(\lambda (H4: (lt m (S -d))).(or_ind (eq nat m O) (ex2 nat (\lambda (m0: nat).(eq nat m (S m0))) -(\lambda (m0: nat).(lt m0 d))) (eq nat (wadd g (S (weight_map g (lift (S h) d -t0))) m) (wadd f (S (weight_map f (lift h d t0))) m)) (\lambda (H5: (eq nat m -O)).(eq_ind_r nat O (\lambda (n: nat).(eq nat (wadd g (S (weight_map g (lift -(S h) d t0))) n) (wadd f (S (weight_map f (lift h d t0))) n))) (f_equal nat -nat S (weight_map g (lift (S h) d t0)) (weight_map f (lift h d t0)) (sym_eq -nat (weight_map f (lift h d t0)) (weight_map g (lift (S h) d t0)) (H h d f g -H1 H2 H3))) m H5)) (\lambda (H5: (ex2 nat (\lambda (m0: nat).(eq nat m (S -m0))) (\lambda (m0: nat).(lt m0 d)))).(ex2_ind nat (\lambda (m0: nat).(eq nat -m (S m0))) (\lambda (m0: nat).(lt m0 d)) (eq nat (wadd g (S (weight_map g -(lift (S h) d t0))) m) (wadd f (S (weight_map f (lift h d t0))) m)) (\lambda -(x: nat).(\lambda (H6: (eq nat m (S x))).(\lambda (H7: (lt x d)).(eq_ind_r -nat (S x) (\lambda (n: nat).(eq nat (wadd g (S (weight_map g (lift (S h) d -t0))) n) (wadd f (S (weight_map f (lift h d t0))) n))) (H1 x H7) m H6)))) -H5)) (lt_gen_xS m d H4)))) H2 (\lambda (m: nat).(\lambda (H4: (le (S d) -m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S n))) (\lambda (n: nat).(le d -n)) (eq nat (g m) (wadd f (S (weight_map f (lift h d t0))) m)) (\lambda (x: -nat).(\lambda (H5: (eq nat m (S x))).(\lambda (H6: (le d x)).(eq_ind_r nat (S -x) (\lambda (n: nat).(eq nat (g n) (wadd f (S (weight_map f (lift h d t0))) -n))) (H3 x H6) m H5)))) (le_gen_S d m H4))))))) (f_equal nat nat S (plus -(weight_map f (lift h d t0)) (weight_map (wadd f O) (lift h (S d) t1))) (plus -(weight_map g (lift (S h) d t0)) (weight_map (wadd g O) (lift (S h) (S d) -t1))) (f_equal2 nat nat nat plus (weight_map f (lift h d t0)) (weight_map g -(lift (S h) d t0)) (weight_map (wadd f O) (lift h (S d) t1)) (weight_map -(wadd g O) (lift (S h) (S d) t1)) (H h d f g H1 H2 H3) (H0 h (S d) (wadd f O) -(wadd g O) (\lambda (m: nat).(\lambda (H4: (lt m (S d))).(or_ind (eq nat m O) -(ex2 nat (\lambda (m0: nat).(eq nat m (S m0))) (\lambda (m0: nat).(lt m0 d))) -(eq nat (wadd g O m) (wadd f O m)) (\lambda (H5: (eq nat m O)).(eq_ind_r nat -O (\lambda (n: nat).(eq nat (wadd g O n) (wadd f O n))) (refl_equal nat O) m -H5)) (\lambda (H5: (ex2 nat (\lambda (m0: nat).(eq nat m (S m0))) (\lambda -(m0: nat).(lt m0 d)))).(ex2_ind nat (\lambda (m0: nat).(eq nat m (S m0))) -(\lambda (m0: nat).(lt m0 d)) (eq nat (wadd g O m) (wadd f O m)) (\lambda (x: -nat).(\lambda (H6: (eq nat m (S x))).(\lambda (H7: (lt x d)).(eq_ind_r nat (S -x) (\lambda (n: nat).(eq nat (wadd g O n) (wadd f O n))) (H1 x H7) m H6)))) -H5)) (lt_gen_xS m d H4)))) H2 (\lambda (m: nat).(\lambda (H4: (le (S d) -m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S n))) (\lambda (n: nat).(le d -n)) (eq nat (g m) (wadd f O m)) (\lambda (x: nat).(\lambda (H5: (eq nat m (S -x))).(\lambda (H6: (le d x)).(eq_ind_r nat (S x) (\lambda (n: nat).(eq nat (g -n) (wadd f O n))) (H3 x H6) m H5)))) (le_gen_S d m H4))))))) (f_equal nat nat -S (plus (weight_map f (lift h d t0)) (weight_map (wadd f O) (lift h (S d) -t1))) (plus (weight_map g (lift (S h) d t0)) (weight_map (wadd g O) (lift (S -h) (S d) t1))) (f_equal2 nat nat nat plus (weight_map f (lift h d t0)) -(weight_map g (lift (S h) d t0)) (weight_map (wadd f O) (lift h (S d) t1)) -(weight_map (wadd g O) (lift (S h) (S d) t1)) (H h d f g H1 H2 H3) (H0 h (S -d) (wadd f O) (wadd g O) (\lambda (m: nat).(\lambda (H4: (lt m (S -d))).(or_ind (eq nat m O) (ex2 nat (\lambda (m0: nat).(eq nat m (S m0))) -(\lambda (m0: nat).(lt m0 d))) (eq nat (wadd g O m) (wadd f O m)) (\lambda -(H5: (eq nat m O)).(eq_ind_r nat O (\lambda (n: nat).(eq nat (wadd g O n) -(wadd f O n))) (refl_equal nat O) m H5)) (\lambda (H5: (ex2 nat (\lambda (m0: -nat).(eq nat m (S m0))) (\lambda (m0: nat).(lt m0 d)))).(ex2_ind nat (\lambda -(m0: nat).(eq nat m (S m0))) (\lambda (m0: nat).(lt m0 d)) (eq nat (wadd g O -m) (wadd f O m)) (\lambda (x: nat).(\lambda (H6: (eq nat m (S x))).(\lambda -(H7: (lt x d)).(eq_ind_r nat (S x) (\lambda (n: nat).(eq nat (wadd g O n) -(wadd f O n))) (H1 x H7) m H6)))) H5)) (lt_gen_xS m d H4)))) H2 (\lambda (m: -nat).(\lambda (H4: (le (S d) m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S -n))) (\lambda (n: nat).(le d n)) (eq nat (g m) (wadd f O m)) (\lambda (x: -nat).(\lambda (H5: (eq nat m (S x))).(\lambda (H6: (le d x)).(eq_ind_r nat (S -x) (\lambda (n: nat).(eq nat (g n) (wadd f O n))) (H3 x H6) m H5)))) -(le_gen_S d m H4))))))) b) (lift (S h) d (THead (Bind b) t0 t1)) (lift_head -(Bind b) t0 t1 (S h) d)) (lift h d (THead (Bind b) t0 t1)) (lift_head (Bind -b) t0 t1 h d))) (\lambda (f0: F).(eq_ind_r T (THead (Flat f0) (lift h d t0) -(lift h (s (Flat f0) d) t1)) (\lambda (t2: T).(eq nat (weight_map f t2) -(weight_map g (lift (S h) d (THead (Flat f0) t0 t1))))) (eq_ind_r T (THead -(Flat f0) (lift (S h) d t0) (lift (S h) (s (Flat f0) d) t1)) (\lambda (t2: -T).(eq nat (weight_map f (THead (Flat f0) (lift h d t0) (lift h (s (Flat f0) -d) t1))) (weight_map g t2))) (f_equal nat nat S (plus (weight_map f (lift h d -t0)) (weight_map f (lift h d t1))) (plus (weight_map g (lift (S h) d t0)) -(weight_map g (lift (S h) d t1))) (f_equal2 nat nat nat plus (weight_map f -(lift h d t0)) (weight_map g (lift (S h) d t0)) (weight_map f (lift h d t1)) -(weight_map g (lift (S h) d t1)) (H h d f g H1 H2 H3) (H0 h d f g H1 H2 H3))) -(lift (S h) d (THead (Flat f0) t0 t1)) (lift_head (Flat f0) t0 t1 (S h) d)) -(lift h d (THead (Flat f0) t0 t1)) (lift_head (Flat f0) t0 t1 h d))) -k))))))))))))) t)). -(* COMMENTS -Initial nodes: 3697 -END *) - -theorem lift_weight_add_O: - \forall (w: nat).(\forall (t: T).(\forall (h: nat).(\forall (f: ((nat \to -nat))).(eq nat (weight_map f (lift h O t)) (weight_map (wadd f w) (lift (S h) -O t)))))) -\def - \lambda (w: nat).(\lambda (t: T).(\lambda (h: nat).(\lambda (f: ((nat \to -nat))).(lift_weight_add (plus (wadd f w O) O) t h O f (wadd f w) (\lambda (m: -nat).(\lambda (H: (lt m O)).(lt_x_O m H (eq nat (wadd f w m) (f m))))) -(plus_n_O (wadd f w O)) (\lambda (m: nat).(\lambda (_: (le O m)).(refl_equal -nat (f m)))))))). -(* COMMENTS -Initial nodes: 93 -END *) - -theorem lift_tlt_dx: - \forall (k: K).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall -(d: nat).(tlt t (THead k u (lift h d t))))))) -\def - \lambda (k: K).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda -(d: nat).(eq_ind nat (weight (lift h d t)) (\lambda (n: nat).(lt n (weight -(THead k u (lift h d t))))) (tlt_head_dx k u (lift h d t)) (weight t) -(lift_weight t h d)))))). -(* COMMENTS -Initial nodes: 71 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/lift1/defs.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/lift1/defs.ma deleted file mode 100644 index 1f473cbce..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/lift1/defs.ma +++ /dev/null @@ -1,40 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/lift/defs.ma". - -definition trans: - PList \to (nat \to nat) -\def - let rec trans (hds: PList) on hds: (nat \to nat) \def (\lambda (i: -nat).(match hds with [PNil \Rightarrow i | (PCons h d hds0) \Rightarrow (let -j \def (trans hds0 i) in (match (blt j d) with [true \Rightarrow j | false -\Rightarrow (plus j h)]))])) in trans. - -definition lift1: - PList \to (T \to T) -\def - let rec lift1 (hds: PList) on hds: (T \to T) \def (\lambda (t: T).(match hds -with [PNil \Rightarrow t | (PCons h d hds0) \Rightarrow (lift h d (lift1 hds0 -t))])) in lift1. - -definition lifts1: - PList \to (TList \to TList) -\def - let rec lifts1 (hds: PList) (ts: TList) on ts: TList \def (match ts with -[TNil \Rightarrow TNil | (TCons t ts0) \Rightarrow (TCons (lift1 hds t) -(lifts1 hds ts0))]) in lifts1. - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/lift1/fwd.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/lift1/fwd.ma deleted file mode 100644 index e9ae2d11c..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/lift1/fwd.ma +++ /dev/null @@ -1,164 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/lift1/defs.ma". - -include "Basic-1/lift/fwd.ma". - -theorem lift1_sort: - \forall (n: nat).(\forall (is: PList).(eq T (lift1 is (TSort n)) (TSort n))) -\def - \lambda (n: nat).(\lambda (is: PList).(PList_ind (\lambda (p: PList).(eq T -(lift1 p (TSort n)) (TSort n))) (refl_equal T (TSort n)) (\lambda (n0: -nat).(\lambda (n1: nat).(\lambda (p: PList).(\lambda (H: (eq T (lift1 p -(TSort n)) (TSort n))).(eq_ind_r T (TSort n) (\lambda (t: T).(eq T (lift n0 -n1 t) (TSort n))) (refl_equal T (TSort n)) (lift1 p (TSort n)) H))))) is)). -(* COMMENTS -Initial nodes: 99 -END *) - -theorem lift1_lref: - \forall (hds: PList).(\forall (i: nat).(eq T (lift1 hds (TLRef i)) (TLRef -(trans hds i)))) -\def - \lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (i: nat).(eq T -(lift1 p (TLRef i)) (TLRef (trans p i))))) (\lambda (i: nat).(refl_equal T -(TLRef i))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda -(H: ((\forall (i: nat).(eq T (lift1 p (TLRef i)) (TLRef (trans p -i)))))).(\lambda (i: nat).(eq_ind_r T (TLRef (trans p i)) (\lambda (t: T).(eq -T (lift n n0 t) (TLRef (match (blt (trans p i) n0) with [true \Rightarrow -(trans p i) | false \Rightarrow (plus (trans p i) n)])))) (refl_equal T -(TLRef (match (blt (trans p i) n0) with [true \Rightarrow (trans p i) | false -\Rightarrow (plus (trans p i) n)]))) (lift1 p (TLRef i)) (H i))))))) hds). -(* COMMENTS -Initial nodes: 165 -END *) - -theorem lift1_bind: - \forall (b: B).(\forall (hds: PList).(\forall (u: T).(\forall (t: T).(eq T -(lift1 hds (THead (Bind b) u t)) (THead (Bind b) (lift1 hds u) (lift1 (Ss -hds) t)))))) -\def - \lambda (b: B).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall -(u: T).(\forall (t: T).(eq T (lift1 p (THead (Bind b) u t)) (THead (Bind b) -(lift1 p u) (lift1 (Ss p) t)))))) (\lambda (u: T).(\lambda (t: T).(refl_equal -T (THead (Bind b) u t)))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: -PList).(\lambda (H: ((\forall (u: T).(\forall (t: T).(eq T (lift1 p (THead -(Bind b) u t)) (THead (Bind b) (lift1 p u) (lift1 (Ss p) t))))))).(\lambda -(u: T).(\lambda (t: T).(eq_ind_r T (THead (Bind b) (lift1 p u) (lift1 (Ss p) -t)) (\lambda (t0: T).(eq T (lift n n0 t0) (THead (Bind b) (lift n n0 (lift1 p -u)) (lift n (S n0) (lift1 (Ss p) t))))) (eq_ind_r T (THead (Bind b) (lift n -n0 (lift1 p u)) (lift n (S n0) (lift1 (Ss p) t))) (\lambda (t0: T).(eq T t0 -(THead (Bind b) (lift n n0 (lift1 p u)) (lift n (S n0) (lift1 (Ss p) t))))) -(refl_equal T (THead (Bind b) (lift n n0 (lift1 p u)) (lift n (S n0) (lift1 -(Ss p) t)))) (lift n n0 (THead (Bind b) (lift1 p u) (lift1 (Ss p) t))) -(lift_bind b (lift1 p u) (lift1 (Ss p) t) n n0)) (lift1 p (THead (Bind b) u -t)) (H u t)))))))) hds)). -(* COMMENTS -Initial nodes: 379 -END *) - -theorem lift1_flat: - \forall (f: F).(\forall (hds: PList).(\forall (u: T).(\forall (t: T).(eq T -(lift1 hds (THead (Flat f) u t)) (THead (Flat f) (lift1 hds u) (lift1 hds -t)))))) -\def - \lambda (f: F).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall -(u: T).(\forall (t: T).(eq T (lift1 p (THead (Flat f) u t)) (THead (Flat f) -(lift1 p u) (lift1 p t)))))) (\lambda (u: T).(\lambda (t: T).(refl_equal T -(THead (Flat f) u t)))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: -PList).(\lambda (H: ((\forall (u: T).(\forall (t: T).(eq T (lift1 p (THead -(Flat f) u t)) (THead (Flat f) (lift1 p u) (lift1 p t))))))).(\lambda (u: -T).(\lambda (t: T).(eq_ind_r T (THead (Flat f) (lift1 p u) (lift1 p t)) -(\lambda (t0: T).(eq T (lift n n0 t0) (THead (Flat f) (lift n n0 (lift1 p u)) -(lift n n0 (lift1 p t))))) (eq_ind_r T (THead (Flat f) (lift n n0 (lift1 p -u)) (lift n n0 (lift1 p t))) (\lambda (t0: T).(eq T t0 (THead (Flat f) (lift -n n0 (lift1 p u)) (lift n n0 (lift1 p t))))) (refl_equal T (THead (Flat f) -(lift n n0 (lift1 p u)) (lift n n0 (lift1 p t)))) (lift n n0 (THead (Flat f) -(lift1 p u) (lift1 p t))) (lift_flat f (lift1 p u) (lift1 p t) n n0)) (lift1 -p (THead (Flat f) u t)) (H u t)))))))) hds)). -(* COMMENTS -Initial nodes: 353 -END *) - -theorem lift1_cons_tail: - \forall (t: T).(\forall (h: nat).(\forall (d: nat).(\forall (hds: PList).(eq -T (lift1 (PConsTail hds h d) t) (lift1 hds (lift h d t)))))) -\def - \lambda (t: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (hds: -PList).(PList_ind (\lambda (p: PList).(eq T (lift1 (PConsTail p h d) t) -(lift1 p (lift h d t)))) (refl_equal T (lift h d t)) (\lambda (n: -nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: (eq T (lift1 -(PConsTail p h d) t) (lift1 p (lift h d t)))).(eq_ind_r T (lift1 p (lift h d -t)) (\lambda (t0: T).(eq T (lift n n0 t0) (lift n n0 (lift1 p (lift h d -t))))) (refl_equal T (lift n n0 (lift1 p (lift h d t)))) (lift1 (PConsTail p -h d) t) H))))) hds)))). -(* COMMENTS -Initial nodes: 171 -END *) - -theorem lifts1_flat: - \forall (f: F).(\forall (hds: PList).(\forall (t: T).(\forall (ts: -TList).(eq T (lift1 hds (THeads (Flat f) ts t)) (THeads (Flat f) (lifts1 hds -ts) (lift1 hds t)))))) -\def - \lambda (f: F).(\lambda (hds: PList).(\lambda (t: T).(\lambda (ts: -TList).(TList_ind (\lambda (t0: TList).(eq T (lift1 hds (THeads (Flat f) t0 -t)) (THeads (Flat f) (lifts1 hds t0) (lift1 hds t)))) (refl_equal T (lift1 -hds t)) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H: (eq T (lift1 hds -(THeads (Flat f) t1 t)) (THeads (Flat f) (lifts1 hds t1) (lift1 hds -t)))).(eq_ind_r T (THead (Flat f) (lift1 hds t0) (lift1 hds (THeads (Flat f) -t1 t))) (\lambda (t2: T).(eq T t2 (THead (Flat f) (lift1 hds t0) (THeads -(Flat f) (lifts1 hds t1) (lift1 hds t))))) (eq_ind_r T (THeads (Flat f) -(lifts1 hds t1) (lift1 hds t)) (\lambda (t2: T).(eq T (THead (Flat f) (lift1 -hds t0) t2) (THead (Flat f) (lift1 hds t0) (THeads (Flat f) (lifts1 hds t1) -(lift1 hds t))))) (refl_equal T (THead (Flat f) (lift1 hds t0) (THeads (Flat -f) (lifts1 hds t1) (lift1 hds t)))) (lift1 hds (THeads (Flat f) t1 t)) H) -(lift1 hds (THead (Flat f) t0 (THeads (Flat f) t1 t))) (lift1_flat f hds t0 -(THeads (Flat f) t1 t)))))) ts)))). -(* COMMENTS -Initial nodes: 329 -END *) - -theorem lifts1_nil: - \forall (ts: TList).(eq TList (lifts1 PNil ts) ts) -\def - \lambda (ts: TList).(TList_ind (\lambda (t: TList).(eq TList (lifts1 PNil t) -t)) (refl_equal TList TNil) (\lambda (t: T).(\lambda (t0: TList).(\lambda (H: -(eq TList (lifts1 PNil t0) t0)).(eq_ind_r TList t0 (\lambda (t1: TList).(eq -TList (TCons t t1) (TCons t t0))) (refl_equal TList (TCons t t0)) (lifts1 -PNil t0) H)))) ts). -(* COMMENTS -Initial nodes: 83 -END *) - -theorem lifts1_cons: - \forall (h: nat).(\forall (d: nat).(\forall (hds: PList).(\forall (ts: -TList).(eq TList (lifts1 (PCons h d hds) ts) (lifts h d (lifts1 hds ts)))))) -\def - \lambda (h: nat).(\lambda (d: nat).(\lambda (hds: PList).(\lambda (ts: -TList).(TList_ind (\lambda (t: TList).(eq TList (lifts1 (PCons h d hds) t) -(lifts h d (lifts1 hds t)))) (refl_equal TList TNil) (\lambda (t: T).(\lambda -(t0: TList).(\lambda (H: (eq TList (lifts1 (PCons h d hds) t0) (lifts h d -(lifts1 hds t0)))).(eq_ind_r TList (lifts h d (lifts1 hds t0)) (\lambda (t1: -TList).(eq TList (TCons (lift h d (lift1 hds t)) t1) (TCons (lift h d (lift1 -hds t)) (lifts h d (lifts1 hds t0))))) (refl_equal TList (TCons (lift h d -(lift1 hds t)) (lifts h d (lifts1 hds t0)))) (lifts1 (PCons h d hds) t0) -H)))) ts)))). -(* COMMENTS -Initial nodes: 187 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/lift1/props.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/lift1/props.ma deleted file mode 100644 index ebda0267b..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/lift1/props.ma +++ /dev/null @@ -1,139 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/lift/props.ma". - -include "Basic-1/drop1/defs.ma". - -theorem lift1_lift1: - \forall (is1: PList).(\forall (is2: PList).(\forall (t: T).(eq T (lift1 is1 -(lift1 is2 t)) (lift1 (papp is1 is2) t)))) -\def - \lambda (is1: PList).(PList_ind (\lambda (p: PList).(\forall (is2: -PList).(\forall (t: T).(eq T (lift1 p (lift1 is2 t)) (lift1 (papp p is2) -t))))) (\lambda (is2: PList).(\lambda (t: T).(refl_equal T (lift1 is2 t)))) -(\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: -((\forall (is2: PList).(\forall (t: T).(eq T (lift1 p (lift1 is2 t)) (lift1 -(papp p is2) t)))))).(\lambda (is2: PList).(\lambda (t: T).(f_equal3 nat nat -T T lift n n n0 n0 (lift1 p (lift1 is2 t)) (lift1 (papp p is2) t) (refl_equal -nat n) (refl_equal nat n0) (H is2 t)))))))) is1). -(* COMMENTS -Initial nodes: 145 -END *) - -theorem lift1_xhg: - \forall (hds: PList).(\forall (t: T).(eq T (lift1 (Ss hds) (lift (S O) O t)) -(lift (S O) O (lift1 hds t)))) -\def - \lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (t: T).(eq T -(lift1 (Ss p) (lift (S O) O t)) (lift (S O) O (lift1 p t))))) (\lambda (t: -T).(refl_equal T (lift (S O) O t))) (\lambda (h: nat).(\lambda (d: -nat).(\lambda (p: PList).(\lambda (H: ((\forall (t: T).(eq T (lift1 (Ss p) -(lift (S O) O t)) (lift (S O) O (lift1 p t)))))).(\lambda (t: T).(eq_ind_r T -(lift (S O) O (lift1 p t)) (\lambda (t0: T).(eq T (lift h (S d) t0) (lift (S -O) O (lift h d (lift1 p t))))) (eq_ind nat (plus (S O) d) (\lambda (n: -nat).(eq T (lift h n (lift (S O) O (lift1 p t))) (lift (S O) O (lift h d -(lift1 p t))))) (eq_ind_r T (lift (S O) O (lift h d (lift1 p t))) (\lambda -(t0: T).(eq T t0 (lift (S O) O (lift h d (lift1 p t))))) (refl_equal T (lift -(S O) O (lift h d (lift1 p t)))) (lift h (plus (S O) d) (lift (S O) O (lift1 -p t))) (lift_d (lift1 p t) h (S O) d O (le_O_n d))) (S d) (refl_equal nat (S -d))) (lift1 (Ss p) (lift (S O) O t)) (H t))))))) hds). -(* COMMENTS -Initial nodes: 371 -END *) - -theorem lifts1_xhg: - \forall (hds: PList).(\forall (ts: TList).(eq TList (lifts1 (Ss hds) (lifts -(S O) O ts)) (lifts (S O) O (lifts1 hds ts)))) -\def - \lambda (hds: PList).(\lambda (ts: TList).(TList_ind (\lambda (t: TList).(eq -TList (lifts1 (Ss hds) (lifts (S O) O t)) (lifts (S O) O (lifts1 hds t)))) -(refl_equal TList TNil) (\lambda (t: T).(\lambda (t0: TList).(\lambda (H: (eq -TList (lifts1 (Ss hds) (lifts (S O) O t0)) (lifts (S O) O (lifts1 hds -t0)))).(eq_ind_r T (lift (S O) O (lift1 hds t)) (\lambda (t1: T).(eq TList -(TCons t1 (lifts1 (Ss hds) (lifts (S O) O t0))) (TCons (lift (S O) O (lift1 -hds t)) (lifts (S O) O (lifts1 hds t0))))) (eq_ind_r TList (lifts (S O) O -(lifts1 hds t0)) (\lambda (t1: TList).(eq TList (TCons (lift (S O) O (lift1 -hds t)) t1) (TCons (lift (S O) O (lift1 hds t)) (lifts (S O) O (lifts1 hds -t0))))) (refl_equal TList (TCons (lift (S O) O (lift1 hds t)) (lifts (S O) O -(lifts1 hds t0)))) (lifts1 (Ss hds) (lifts (S O) O t0)) H) (lift1 (Ss hds) -(lift (S O) O t)) (lift1_xhg hds t))))) ts)). -(* COMMENTS -Initial nodes: 307 -END *) - -theorem lift1_free: - \forall (hds: PList).(\forall (i: nat).(\forall (t: T).(eq T (lift1 hds -(lift (S i) O t)) (lift (S (trans hds i)) O (lift1 (ptrans hds i) t))))) -\def - \lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (i: -nat).(\forall (t: T).(eq T (lift1 p (lift (S i) O t)) (lift (S (trans p i)) O -(lift1 (ptrans p i) t)))))) (\lambda (i: nat).(\lambda (t: T).(refl_equal T -(lift (S i) O t)))) (\lambda (h: nat).(\lambda (d: nat).(\lambda (hds0: -PList).(\lambda (H: ((\forall (i: nat).(\forall (t: T).(eq T (lift1 hds0 -(lift (S i) O t)) (lift (S (trans hds0 i)) O (lift1 (ptrans hds0 i) -t))))))).(\lambda (i: nat).(\lambda (t: T).(eq_ind_r T (lift (S (trans hds0 -i)) O (lift1 (ptrans hds0 i) t)) (\lambda (t0: T).(eq T (lift h d t0) (lift -(S (match (blt (trans hds0 i) d) with [true \Rightarrow (trans hds0 i) | -false \Rightarrow (plus (trans hds0 i) h)])) O (lift1 (match (blt (trans hds0 -i) d) with [true \Rightarrow (PCons h (minus d (S (trans hds0 i))) (ptrans -hds0 i)) | false \Rightarrow (ptrans hds0 i)]) t)))) (xinduction bool (blt -(trans hds0 i) d) (\lambda (b: bool).(eq T (lift h d (lift (S (trans hds0 i)) -O (lift1 (ptrans hds0 i) t))) (lift (S (match b with [true \Rightarrow (trans -hds0 i) | false \Rightarrow (plus (trans hds0 i) h)])) O (lift1 (match b with -[true \Rightarrow (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) | -false \Rightarrow (ptrans hds0 i)]) t)))) (\lambda (x_x: bool).(bool_ind -(\lambda (b: bool).((eq bool (blt (trans hds0 i) d) b) \to (eq T (lift h d -(lift (S (trans hds0 i)) O (lift1 (ptrans hds0 i) t))) (lift (S (match b with -[true \Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i) -h)])) O (lift1 (match b with [true \Rightarrow (PCons h (minus d (S (trans -hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)]) t))))) -(\lambda (H0: (eq bool (blt (trans hds0 i) d) true)).(eq_ind_r nat (plus (S -(trans hds0 i)) (minus d (S (trans hds0 i)))) (\lambda (n: nat).(eq T (lift h -n (lift (S (trans hds0 i)) O (lift1 (ptrans hds0 i) t))) (lift (S (trans hds0 -i)) O (lift1 (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) t)))) -(eq_ind_r T (lift (S (trans hds0 i)) O (lift h (minus d (S (trans hds0 i))) -(lift1 (ptrans hds0 i) t))) (\lambda (t0: T).(eq T t0 (lift (S (trans hds0 -i)) O (lift1 (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) t)))) -(refl_equal T (lift (S (trans hds0 i)) O (lift1 (PCons h (minus d (S (trans -hds0 i))) (ptrans hds0 i)) t))) (lift h (plus (S (trans hds0 i)) (minus d (S -(trans hds0 i)))) (lift (S (trans hds0 i)) O (lift1 (ptrans hds0 i) t))) -(lift_d (lift1 (ptrans hds0 i) t) h (S (trans hds0 i)) (minus d (S (trans -hds0 i))) O (le_O_n (minus d (S (trans hds0 i)))))) d (le_plus_minus (S -(trans hds0 i)) d (bge_le (S (trans hds0 i)) d (le_bge (S (trans hds0 i)) d -(lt_le_S (trans hds0 i) d (blt_lt d (trans hds0 i) H0))))))) (\lambda (H0: -(eq bool (blt (trans hds0 i) d) false)).(eq_ind_r T (lift (plus h (S (trans -hds0 i))) O (lift1 (ptrans hds0 i) t)) (\lambda (t0: T).(eq T t0 (lift (S -(plus (trans hds0 i) h)) O (lift1 (ptrans hds0 i) t)))) (eq_ind nat (S (plus -h (trans hds0 i))) (\lambda (n: nat).(eq T (lift n O (lift1 (ptrans hds0 i) -t)) (lift (S (plus (trans hds0 i) h)) O (lift1 (ptrans hds0 i) t)))) -(eq_ind_r nat (plus (trans hds0 i) h) (\lambda (n: nat).(eq T (lift (S n) O -(lift1 (ptrans hds0 i) t)) (lift (S (plus (trans hds0 i) h)) O (lift1 (ptrans -hds0 i) t)))) (refl_equal T (lift (S (plus (trans hds0 i) h)) O (lift1 -(ptrans hds0 i) t))) (plus h (trans hds0 i)) (plus_sym h (trans hds0 i))) -(plus h (S (trans hds0 i))) (plus_n_Sm h (trans hds0 i))) (lift h d (lift (S -(trans hds0 i)) O (lift1 (ptrans hds0 i) t))) (lift_free (lift1 (ptrans hds0 -i) t) (S (trans hds0 i)) h O d (eq_ind nat (S (plus O (trans hds0 i))) -(\lambda (n: nat).(le d n)) (eq_ind_r nat (plus (trans hds0 i) O) (\lambda -(n: nat).(le d (S n))) (le_S d (plus (trans hds0 i) O) (le_plus_trans d -(trans hds0 i) O (bge_le d (trans hds0 i) H0))) (plus O (trans hds0 i)) -(plus_sym O (trans hds0 i))) (plus O (S (trans hds0 i))) (plus_n_Sm O (trans -hds0 i))) (le_O_n d)))) x_x))) (lift1 hds0 (lift (S i) O t)) (H i t)))))))) -hds). -(* COMMENTS -Initial nodes: 1339 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/llt/defs.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/llt/defs.ma deleted file mode 100644 index 96d869935..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/llt/defs.ma +++ /dev/null @@ -1,30 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/A/defs.ma". - -definition lweight: - A \to nat -\def - let rec lweight (a: A) on a: nat \def (match a with [(ASort _ _) \Rightarrow -O | (AHead a1 a2) \Rightarrow (S (plus (lweight a1) (lweight a2)))]) in -lweight. - -definition llt: - A \to (A \to Prop) -\def - \lambda (a1: A).(\lambda (a2: A).(lt (lweight a1) (lweight a2))). - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/llt/props.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/llt/props.ma deleted file mode 100644 index ef1f15a96..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/llt/props.ma +++ /dev/null @@ -1,114 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/llt/defs.ma". - -include "Basic-1/leq/defs.ma". - -theorem lweight_repl: - \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (eq nat -(lweight a1) (lweight a2))))) -\def - \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1 -a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(eq nat (lweight a) (lweight -a0)))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: nat).(\lambda (n2: -nat).(\lambda (k: nat).(\lambda (_: (eq A (aplus g (ASort h1 n1) k) (aplus g -(ASort h2 n2) k))).(refl_equal nat O))))))) (\lambda (a0: A).(\lambda (a3: -A).(\lambda (_: (leq g a0 a3)).(\lambda (H1: (eq nat (lweight a0) (lweight -a3))).(\lambda (a4: A).(\lambda (a5: A).(\lambda (_: (leq g a4 a5)).(\lambda -(H3: (eq nat (lweight a4) (lweight a5))).(f_equal nat nat S (plus (lweight -a0) (lweight a4)) (plus (lweight a3) (lweight a5)) (f_equal2 nat nat nat plus -(lweight a0) (lweight a3) (lweight a4) (lweight a5) H1 H3)))))))))) a1 a2 -H)))). -(* COMMENTS -Initial nodes: 189 -END *) - -theorem llt_repl: - \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall -(a3: A).((llt a1 a3) \to (llt a2 a3)))))) -\def - \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1 -a2)).(\lambda (a3: A).(\lambda (H0: (lt (lweight a1) (lweight a3))).(let H1 -\def (eq_ind nat (lweight a1) (\lambda (n: nat).(lt n (lweight a3))) H0 -(lweight a2) (lweight_repl g a1 a2 H)) in H1)))))). -(* COMMENTS -Initial nodes: 61 -END *) - -theorem llt_trans: - \forall (a1: A).(\forall (a2: A).(\forall (a3: A).((llt a1 a2) \to ((llt a2 -a3) \to (llt a1 a3))))) -\def - \lambda (a1: A).(\lambda (a2: A).(\lambda (a3: A).(\lambda (H: (lt (lweight -a1) (lweight a2))).(\lambda (H0: (lt (lweight a2) (lweight a3))).(lt_trans -(lweight a1) (lweight a2) (lweight a3) H H0))))). -(* COMMENTS -Initial nodes: 43 -END *) - -theorem llt_head_sx: - \forall (a1: A).(\forall (a2: A).(llt a1 (AHead a1 a2))) -\def - \lambda (a1: A).(\lambda (a2: A).(le_n_S (lweight a1) (plus (lweight a1) -(lweight a2)) (le_plus_l (lweight a1) (lweight a2)))). -(* COMMENTS -Initial nodes: 29 -END *) - -theorem llt_head_dx: - \forall (a1: A).(\forall (a2: A).(llt a2 (AHead a1 a2))) -\def - \lambda (a1: A).(\lambda (a2: A).(le_n_S (lweight a2) (plus (lweight a1) -(lweight a2)) (le_plus_r (lweight a1) (lweight a2)))). -(* COMMENTS -Initial nodes: 29 -END *) - -theorem llt_wf__q_ind: - \forall (P: ((A \to Prop))).(((\forall (n: nat).((\lambda (P0: ((A \to -Prop))).(\lambda (n0: nat).(\forall (a: A).((eq nat (lweight a) n0) \to (P0 -a))))) P n))) \to (\forall (a: A).(P a))) -\def - let Q \def (\lambda (P: ((A \to Prop))).(\lambda (n: nat).(\forall (a: -A).((eq nat (lweight a) n) \to (P a))))) in (\lambda (P: ((A \to -Prop))).(\lambda (H: ((\forall (n: nat).(\forall (a: A).((eq nat (lweight a) -n) \to (P a)))))).(\lambda (a: A).(H (lweight a) a (refl_equal nat (lweight -a)))))). -(* COMMENTS -Initial nodes: 61 -END *) - -theorem llt_wf_ind: - \forall (P: ((A \to Prop))).(((\forall (a2: A).(((\forall (a1: A).((llt a1 -a2) \to (P a1)))) \to (P a2)))) \to (\forall (a: A).(P a))) -\def - let Q \def (\lambda (P: ((A \to Prop))).(\lambda (n: nat).(\forall (a: -A).((eq nat (lweight a) n) \to (P a))))) in (\lambda (P: ((A \to -Prop))).(\lambda (H: ((\forall (a2: A).(((\forall (a1: A).((lt (lweight a1) -(lweight a2)) \to (P a1)))) \to (P a2))))).(\lambda (a: A).(llt_wf__q_ind -(\lambda (a0: A).(P a0)) (\lambda (n: nat).(lt_wf_ind n (Q (\lambda (a0: -A).(P a0))) (\lambda (n0: nat).(\lambda (H0: ((\forall (m: nat).((lt m n0) -\to (Q (\lambda (a0: A).(P a0)) m))))).(\lambda (a0: A).(\lambda (H1: (eq nat -(lweight a0) n0)).(let H2 \def (eq_ind_r nat n0 (\lambda (n1: nat).(\forall -(m: nat).((lt m n1) \to (\forall (a1: A).((eq nat (lweight a1) m) \to (P -a1)))))) H0 (lweight a0) H1) in (H a0 (\lambda (a1: A).(\lambda (H3: (lt -(lweight a1) (lweight a0))).(H2 (lweight a1) H3 a1 (refl_equal nat (lweight -a1))))))))))))) a)))). -(* COMMENTS -Initial nodes: 179 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/next_plus/defs.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/next_plus/defs.ma deleted file mode 100644 index 1ff7ecdff..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/next_plus/defs.ma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/G/defs.ma". - -definition next_plus: - G \to (nat \to (nat \to nat)) -\def - let rec next_plus (g: G) (n: nat) (i: nat) on i: nat \def (match i with [O -\Rightarrow n | (S i0) \Rightarrow (next g (next_plus g n i0))]) in next_plus. - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/next_plus/props.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/next_plus/props.ma deleted file mode 100644 index 258c3a711..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/next_plus/props.ma +++ /dev/null @@ -1,68 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/next_plus/defs.ma". - -theorem next_plus_assoc: - \forall (g: G).(\forall (n: nat).(\forall (h1: nat).(\forall (h2: nat).(eq -nat (next_plus g (next_plus g n h1) h2) (next_plus g n (plus h1 h2)))))) -\def - \lambda (g: G).(\lambda (n: nat).(\lambda (h1: nat).(nat_ind (\lambda (n0: -nat).(\forall (h2: nat).(eq nat (next_plus g (next_plus g n n0) h2) -(next_plus g n (plus n0 h2))))) (\lambda (h2: nat).(refl_equal nat (next_plus -g n h2))) (\lambda (n0: nat).(\lambda (_: ((\forall (h2: nat).(eq nat -(next_plus g (next_plus g n n0) h2) (next_plus g n (plus n0 h2)))))).(\lambda -(h2: nat).(nat_ind (\lambda (n1: nat).(eq nat (next_plus g (next g (next_plus -g n n0)) n1) (next g (next_plus g n (plus n0 n1))))) (eq_ind nat n0 (\lambda -(n1: nat).(eq nat (next g (next_plus g n n0)) (next g (next_plus g n n1)))) -(refl_equal nat (next g (next_plus g n n0))) (plus n0 O) (plus_n_O n0)) -(\lambda (n1: nat).(\lambda (H0: (eq nat (next_plus g (next g (next_plus g n -n0)) n1) (next g (next_plus g n (plus n0 n1))))).(eq_ind nat (S (plus n0 n1)) -(\lambda (n2: nat).(eq nat (next g (next_plus g (next g (next_plus g n n0)) -n1)) (next g (next_plus g n n2)))) (f_equal nat nat (next g) (next_plus g -(next g (next_plus g n n0)) n1) (next g (next_plus g n (plus n0 n1))) H0) -(plus n0 (S n1)) (plus_n_Sm n0 n1)))) h2)))) h1))). -(* COMMENTS -Initial nodes: 351 -END *) - -theorem next_plus_next: - \forall (g: G).(\forall (n: nat).(\forall (h: nat).(eq nat (next_plus g -(next g n) h) (next g (next_plus g n h))))) -\def - \lambda (g: G).(\lambda (n: nat).(\lambda (h: nat).(eq_ind_r nat (next_plus -g n (plus (S O) h)) (\lambda (n0: nat).(eq nat n0 (next g (next_plus g n -h)))) (refl_equal nat (next g (next_plus g n h))) (next_plus g (next_plus g n -(S O)) h) (next_plus_assoc g n (S O) h)))). -(* COMMENTS -Initial nodes: 87 -END *) - -theorem next_plus_lt: - \forall (g: G).(\forall (h: nat).(\forall (n: nat).(lt n (next_plus g (next -g n) h)))) -\def - \lambda (g: G).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall (n0: -nat).(lt n0 (next_plus g (next g n0) n)))) (\lambda (n: nat).(next_lt g n)) -(\lambda (n: nat).(\lambda (H: ((\forall (n0: nat).(lt n0 (next_plus g (next -g n0) n))))).(\lambda (n0: nat).(eq_ind nat (next_plus g (next g (next g n0)) -n) (\lambda (n1: nat).(lt n0 n1)) (lt_trans n0 (next g n0) (next_plus g (next -g (next g n0)) n) (next_lt g n0) (H (next g n0))) (next g (next_plus g (next -g n0) n)) (next_plus_next g (next g n0) n))))) h)). -(* COMMENTS -Initial nodes: 153 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/arity.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/arity.ma deleted file mode 100644 index 98770d9e9..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/arity.ma +++ /dev/null @@ -1,496 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/nf2/fwd.ma". - -include "Basic-1/arity/subst0.ma". - -theorem arity_nf2_inv_all: - \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t -a) \to ((nf2 c t) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t -(THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c w))) -(\lambda (w: T).(\lambda (u: T).(nf2 (CHead c (Bind Abst) w) u)))) (ex nat -(\lambda (n: nat).(eq T t (TSort n)))) (ex3_2 TList nat (\lambda (ws: -TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i))))) -(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c (TLRef i))))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: -(arity g c t a)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (_: -A).((nf2 c0 t0) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t0 -(THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) -(\lambda (w: T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat -(\lambda (n: nat).(eq T t0 (TSort n)))) (ex3_2 TList nat (\lambda (ws: -TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i))))) -(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))))))) (\lambda (c0: C).(\lambda -(n: nat).(\lambda (_: (nf2 c0 (TSort n))).(or3_intro1 (ex3_2 T T (\lambda (w: -T).(\lambda (u: T).(eq T (TSort n) (THead (Bind Abst) w u)))) (\lambda (w: -T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead -c0 (Bind Abst) w) u)))) (ex nat (\lambda (n0: nat).(eq T (TSort n) (TSort -n0)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (TSort -n) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef -i))))) (ex_intro nat (\lambda (n0: nat).(eq T (TSort n) (TSort n0))) n -(refl_equal T (TSort n))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: -T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr) -u))).(\lambda (a0: A).(\lambda (_: (arity g d u a0)).(\lambda (_: (((nf2 d u) -\to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u (THead (Bind -Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 d w))) (\lambda (w: -T).(\lambda (u0: T).(nf2 (CHead d (Bind Abst) w) u0)))) (ex nat (\lambda (n: -nat).(eq T u (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i0: -nat).(eq T u (THeads (Flat Appl) ws (TLRef i0))))) (\lambda (ws: -TList).(\lambda (_: nat).(nfs2 d ws))) (\lambda (_: TList).(\lambda (i0: -nat).(nf2 d (TLRef i0))))))))).(\lambda (H3: (nf2 c0 (TLRef -i))).(nf2_gen_lref c0 d u i H0 H3 (or3 (ex3_2 T T (\lambda (w: T).(\lambda -(u0: T).(eq T (TLRef i) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda -(_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind -Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (TLRef i) (TSort n)))) (ex3_2 -TList nat (\lambda (ws: TList).(\lambda (i0: nat).(eq T (TLRef i) (THeads -(Flat Appl) ws (TLRef i0))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 -ws))) (\lambda (_: TList).(\lambda (i0: nat).(nf2 c0 (TLRef -i0)))))))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda -(i: nat).(\lambda (_: (getl i c0 (CHead d (Bind Abst) u))).(\lambda (a0: -A).(\lambda (_: (arity g d u (asucc g a0))).(\lambda (_: (((nf2 d u) \to (or3 -(ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u (THead (Bind Abst) w -u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 d w))) (\lambda (w: T).(\lambda -(u0: T).(nf2 (CHead d (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T u -(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i0: nat).(eq T u -(THeads (Flat Appl) ws (TLRef i0))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 d ws))) (\lambda (_: TList).(\lambda (i0: nat).(nf2 d (TLRef -i0))))))))).(\lambda (H3: (nf2 c0 (TLRef i))).(or3_intro2 (ex3_2 T T (\lambda -(w: T).(\lambda (u0: T).(eq T (TLRef i) (THead (Bind Abst) w u0)))) (\lambda -(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 -(CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (TLRef i) -(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i0: nat).(eq T -(TLRef i) (THeads (Flat Appl) ws (TLRef i0))))) (\lambda (ws: TList).(\lambda -(_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i0: nat).(nf2 c0 (TLRef -i0))))) (ex3_2_intro TList nat (\lambda (ws: TList).(\lambda (i0: nat).(eq T -(TLRef i) (THeads (Flat Appl) ws (TLRef i0))))) (\lambda (ws: TList).(\lambda -(_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i0: nat).(nf2 c0 (TLRef -i0)))) TNil i (refl_equal T (TLRef i)) I H3))))))))))) (\lambda (b: -B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c0: C).(\lambda (u: -T).(\lambda (a1: A).(\lambda (_: (arity g c0 u a1)).(\lambda (_: (((nf2 c0 u) -\to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u (THead (Bind -Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: -T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: -nat).(eq T u (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: -nat).(eq T u (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: -TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: -nat).(nf2 c0 (TLRef i))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda -(H3: (arity g (CHead c0 (Bind b) u) t0 a2)).(\lambda (_: (((nf2 (CHead c0 -(Bind b) u) t0) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T t0 -(THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 (CHead c0 -(Bind b) u) w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead (CHead c0 (Bind -b) u) (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort n)))) -(ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads -(Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 -(CHead c0 (Bind b) u) ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 (CHead -c0 (Bind b) u) (TLRef i))))))))).(\lambda (H5: (nf2 c0 (THead (Bind b) u -t0))).(B_ind (\lambda (b0: B).((not (eq B b0 Abst)) \to ((arity g (CHead c0 -(Bind b0) u) t0 a2) \to ((nf2 c0 (THead (Bind b0) u t0)) \to (or3 (ex3_2 T T -(\lambda (w: T).(\lambda (u0: T).(eq T (THead (Bind b0) u t0) (THead (Bind -Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: -T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: -nat).(eq T (THead (Bind b0) u t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws: -TList).(\lambda (i: nat).(eq T (THead (Bind b0) u t0) (THeads (Flat Appl) ws -(TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda -(_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))))))) (\lambda (_: (not (eq -B Abbr Abst))).(\lambda (_: (arity g (CHead c0 (Bind Abbr) u) t0 -a2)).(\lambda (H8: (nf2 c0 (THead (Bind Abbr) u t0))).(nf2_gen_abbr c0 u t0 -H8 (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Bind Abbr) -u t0) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 -w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) -(ex nat (\lambda (n: nat).(eq T (THead (Bind Abbr) u t0) (TSort n)))) (ex3_2 -TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Bind Abbr) u -t0) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef -i)))))))))) (\lambda (H6: (not (eq B Abst Abst))).(\lambda (_: (arity g -(CHead c0 (Bind Abst) u) t0 a2)).(\lambda (_: (nf2 c0 (THead (Bind Abst) u -t0))).(let H9 \def (match (H6 (refl_equal B Abst)) in False return (\lambda -(_: False).(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead -(Bind Abst) u t0) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: -T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) -w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind Abst) u t0) (TSort -n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead -(Bind Abst) u t0) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: -TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: -nat).(nf2 c0 (TLRef i))))))) with []) in H9)))) (\lambda (_: (not (eq B Void -Abst))).(\lambda (H7: (arity g (CHead c0 (Bind Void) u) t0 a2)).(\lambda (H8: -(nf2 c0 (THead (Bind Void) u t0))).(let H9 \def (arity_gen_cvoid g (CHead c0 -(Bind Void) u) t0 a2 H7 c0 u O (getl_refl Void c0 u)) in (ex_ind T (\lambda -(v: T).(eq T t0 (lift (S O) O v))) (or3 (ex3_2 T T (\lambda (w: T).(\lambda -(u0: T).(eq T (THead (Bind Void) u t0) (THead (Bind Abst) w u0)))) (\lambda -(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 -(CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind -Void) u t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: -nat).(eq T (THead (Bind Void) u t0) (THeads (Flat Appl) ws (TLRef i))))) -(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))) (\lambda (x: T).(\lambda -(H10: (eq T t0 (lift (S O) O x))).(let H11 \def (eq_ind T t0 (\lambda (t1: -T).(nf2 c0 (THead (Bind Void) u t1))) H8 (lift (S O) O x) H10) in (eq_ind_r T -(lift (S O) O x) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (w: T).(\lambda -(u0: T).(eq T (THead (Bind Void) u t1) (THead (Bind Abst) w u0)))) (\lambda -(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 -(CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind -Void) u t1) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: -nat).(eq T (THead (Bind Void) u t1) (THeads (Flat Appl) ws (TLRef i))))) -(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))) (nf2_gen_void c0 u x H11 -(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Bind Void) u -(lift (S O) O x)) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: -T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) -w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind Void) u (lift (S O) O -x)) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq -T (THead (Bind Void) u (lift (S O) O x)) (THeads (Flat Appl) ws (TLRef i))))) -(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))) t0 H10)))) H9))))) b H0 H3 -H5))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda -(_: (arity g c0 u (asucc g a1))).(\lambda (_: (((nf2 c0 u) \to (or3 (ex3_2 T -T (\lambda (w: T).(\lambda (u0: T).(eq T u (THead (Bind Abst) w u0)))) -(\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: -T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T u -(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T u -(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef -i))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c0 -(Bind Abst) u) t0 a2)).(\lambda (_: (((nf2 (CHead c0 (Bind Abst) u) t0) \to -(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) -w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 (CHead c0 (Bind Abst) u) w))) -(\lambda (w: T).(\lambda (u0: T).(nf2 (CHead (CHead c0 (Bind Abst) u) (Bind -Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort n)))) (ex3_2 TList -nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws -(TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 (CHead c0 (Bind -Abst) u) ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 (CHead c0 (Bind -Abst) u) (TLRef i))))))))).(\lambda (H4: (nf2 c0 (THead (Bind Abst) u -t0))).(let H5 \def (nf2_gen_abst c0 u t0 H4) in (land_ind (nf2 c0 u) (nf2 -(CHead c0 (Bind Abst) u) t0) (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: -T).(eq T (THead (Bind Abst) u t0) (THead (Bind Abst) w u0)))) (\lambda (w: -T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead -c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind Abst) u -t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq -T (THead (Bind Abst) u t0) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: -TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: -nat).(nf2 c0 (TLRef i)))))) (\lambda (H6: (nf2 c0 u)).(\lambda (H7: (nf2 -(CHead c0 (Bind Abst) u) t0)).(or3_intro0 (ex3_2 T T (\lambda (w: T).(\lambda -(u0: T).(eq T (THead (Bind Abst) u t0) (THead (Bind Abst) w u0)))) (\lambda -(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 -(CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind -Abst) u t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: -nat).(eq T (THead (Bind Abst) u t0) (THeads (Flat Appl) ws (TLRef i))))) -(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))) (ex3_2_intro T T (\lambda (w: -T).(\lambda (u0: T).(eq T (THead (Bind Abst) u t0) (THead (Bind Abst) w -u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda -(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0))) u t0 (refl_equal T (THead (Bind -Abst) u t0)) H6 H7)))) H5)))))))))))) (\lambda (c0: C).(\lambda (u: -T).(\lambda (a1: A).(\lambda (_: (arity g c0 u a1)).(\lambda (_: (((nf2 c0 u) -\to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u (THead (Bind -Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: -T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: -nat).(eq T u (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: -nat).(eq T u (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: -TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: -nat).(nf2 c0 (TLRef i))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda -(H2: (arity g c0 t0 (AHead a1 a2))).(\lambda (H3: (((nf2 c0 t0) \to (or3 -(ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w -u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda -(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T -t0 (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T -t0 (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef -i))))))))).(\lambda (H4: (nf2 c0 (THead (Flat Appl) u t0))).(let H5 \def -(nf2_gen_flat Appl c0 u t0 H4) in (land_ind (nf2 c0 u) (nf2 c0 t0) (or3 -(ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t0) -(THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) -(\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat -(\lambda (n: nat).(eq T (THead (Flat Appl) u t0) (TSort n)))) (ex3_2 TList -nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u t0) -(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef -i)))))) (\lambda (H6: (nf2 c0 u)).(\lambda (H7: (nf2 c0 t0)).(let H_x \def -(H3 H7) in (let H8 \def H_x in (or3_ind (ex3_2 T T (\lambda (w: T).(\lambda -(u0: T).(eq T t0 (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: -T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) -w) u0)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort n)))) (ex3_2 TList nat -(\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef -i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))) (or3 (ex3_2 T T (\lambda (w: -T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t0) (THead (Bind Abst) w -u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda -(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T -(THead (Flat Appl) u t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws: -TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u t0) (THeads (Flat Appl) -ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) -(\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))) (\lambda (H9: -(ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w -u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda -(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0))))).(ex3_2_ind T T (\lambda (w: -T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w u0)))) (\lambda (w: -T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead -c0 (Bind Abst) w) u0))) (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq -T (THead (Flat Appl) u t0) (THead (Bind Abst) w u0)))) (\lambda (w: -T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead -c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u -t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq -T (THead (Flat Appl) u t0) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: -TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: -nat).(nf2 c0 (TLRef i)))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H10: -(eq T t0 (THead (Bind Abst) x0 x1))).(\lambda (_: (nf2 c0 x0)).(\lambda (_: -(nf2 (CHead c0 (Bind Abst) x0) x1)).(let H13 \def (eq_ind T t0 (\lambda (t1: -T).(nf2 c0 (THead (Flat Appl) u t1))) H4 (THead (Bind Abst) x0 x1) H10) in -(let H14 \def (eq_ind T t0 (\lambda (t1: T).(arity g c0 t1 (AHead a1 a2))) H2 -(THead (Bind Abst) x0 x1) H10) in (eq_ind_r T (THead (Bind Abst) x0 x1) -(\lambda (t1: T).(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T -(THead (Flat Appl) u t1) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda -(_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind -Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u t1) -(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T -(THead (Flat Appl) u t1) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: -TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: -nat).(nf2 c0 (TLRef i))))))) (nf2_gen_beta c0 u x0 x1 H13 (or3 (ex3_2 T T -(\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl) u (THead (Bind -Abst) x0 x1)) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: -T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) -w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u (THead (Bind -Abst) x0 x1)) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: -nat).(eq T (THead (Flat Appl) u (THead (Bind Abst) x0 x1)) (THeads (Flat -Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) -(\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))) t0 H10)))))))) -H9)) (\lambda (H9: (ex nat (\lambda (n: nat).(eq T t0 (TSort n))))).(ex_ind -nat (\lambda (n: nat).(eq T t0 (TSort n))) (or3 (ex3_2 T T (\lambda (w: -T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t0) (THead (Bind Abst) w -u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda -(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T -(THead (Flat Appl) u t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws: -TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u t0) (THeads (Flat Appl) -ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) -(\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))) (\lambda (x: -nat).(\lambda (H10: (eq T t0 (TSort x))).(let H11 \def (eq_ind T t0 (\lambda -(t1: T).(nf2 c0 (THead (Flat Appl) u t1))) H4 (TSort x) H10) in (let H12 \def -(eq_ind T t0 (\lambda (t1: T).(arity g c0 t1 (AHead a1 a2))) H2 (TSort x) -H10) in (eq_ind_r T (TSort x) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (w: -T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t1) (THead (Bind Abst) w -u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda -(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T -(THead (Flat Appl) u t1) (TSort n)))) (ex3_2 TList nat (\lambda (ws: -TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u t1) (THeads (Flat Appl) -ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) -(\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))) (let H_x0 \def -(leq_gen_head1 g a1 a2 (ASort O x) (arity_gen_sort g c0 x (AHead a1 a2) H12)) -in (let H13 \def H_x0 in (ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq -g a1 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: -A).(\lambda (a4: A).(eq A (ASort O x) (AHead a3 a4)))) (or3 (ex3_2 T T -(\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl) u (TSort x)) (THead -(Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda -(w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda -(n: nat).(eq T (THead (Flat Appl) u (TSort x)) (TSort n)))) (ex3_2 TList nat -(\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u (TSort x)) -(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef -i)))))) (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g a1 -x0)).(\lambda (_: (leq g a2 x1)).(\lambda (H16: (eq A (ASort O x) (AHead x0 -x1))).(let H17 \def (eq_ind A (ASort O x) (\lambda (ee: A).(match ee in A -return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) -\Rightarrow False])) I (AHead x0 x1) H16) in (False_ind (or3 (ex3_2 T T -(\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl) u (TSort x)) (THead -(Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda -(w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda -(n: nat).(eq T (THead (Flat Appl) u (TSort x)) (TSort n)))) (ex3_2 TList nat -(\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u (TSort x)) -(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef -i)))))) H17))))))) H13))) t0 H10))))) H9)) (\lambda (H9: (ex3_2 TList nat -(\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef -i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))).(ex3_2_ind TList nat (\lambda -(ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i))))) -(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))) (or3 (ex3_2 T T (\lambda (w: -T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t0) (THead (Bind Abst) w -u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda -(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T -(THead (Flat Appl) u t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws: -TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u t0) (THeads (Flat Appl) -ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) -(\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))) (\lambda (x0: -TList).(\lambda (x1: nat).(\lambda (H10: (eq T t0 (THeads (Flat Appl) x0 -(TLRef x1)))).(\lambda (H11: (nfs2 c0 x0)).(\lambda (H12: (nf2 c0 (TLRef -x1))).(let H13 \def (eq_ind T t0 (\lambda (t1: T).(nf2 c0 (THead (Flat Appl) -u t1))) H4 (THeads (Flat Appl) x0 (TLRef x1)) H10) in (let H14 \def (eq_ind T -t0 (\lambda (t1: T).(arity g c0 t1 (AHead a1 a2))) H2 (THeads (Flat Appl) x0 -(TLRef x1)) H10) in (eq_ind_r T (THeads (Flat Appl) x0 (TLRef x1)) (\lambda -(t1: T).(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat -Appl) u t1) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 -c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) -(ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u t1) (TSort n)))) (ex3_2 -TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u -t1) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef -i))))))) (or3_intro2 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead -(Flat Appl) u (THeads (Flat Appl) x0 (TLRef x1))) (THead (Bind Abst) w u0)))) -(\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: -T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T -(THead (Flat Appl) u (THeads (Flat Appl) x0 (TLRef x1))) (TSort n)))) (ex3_2 -TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u -(THeads (Flat Appl) x0 (TLRef x1))) (THeads (Flat Appl) ws (TLRef i))))) -(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))) (ex3_2_intro TList nat -(\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u (THeads -(Flat Appl) x0 (TLRef x1))) (THeads (Flat Appl) ws (TLRef i))))) (\lambda -(ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: -nat).(nf2 c0 (TLRef i)))) (TCons u x0) x1 (refl_equal T (THead (Flat Appl) u -(THeads (Flat Appl) x0 (TLRef x1)))) (conj (nf2 c0 u) (nfs2 c0 x0) H6 H11) -H12)) t0 H10)))))))) H9)) H8))))) H5)))))))))))) (\lambda (c0: C).(\lambda -(u: T).(\lambda (a0: A).(\lambda (_: (arity g c0 u (asucc g a0))).(\lambda -(_: (((nf2 c0 u) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u -(THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) -(\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat -(\lambda (n: nat).(eq T u (TSort n)))) (ex3_2 TList nat (\lambda (ws: -TList).(\lambda (i: nat).(eq T u (THeads (Flat Appl) ws (TLRef i))))) -(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))))).(\lambda (t0: T).(\lambda -(_: (arity g c0 t0 a0)).(\lambda (_: (((nf2 c0 t0) \to (or3 (ex3_2 T T -(\lambda (w: T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w u0)))) -(\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: -T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T t0 -(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 -(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef -i))))))))).(\lambda (H4: (nf2 c0 (THead (Flat Cast) u t0))).(nf2_gen_cast c0 -u t0 H4 (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat -Cast) u t0) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 -c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) -(ex nat (\lambda (n: nat).(eq T (THead (Flat Cast) u t0) (TSort n)))) (ex3_2 -TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Cast) u -t0) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef -i)))))))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda -(_: (arity g c0 t0 a1)).(\lambda (H1: (((nf2 c0 t0) \to (or3 (ex3_2 T T -(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda -(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 -(CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort -n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 -(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef -i))))))))).(\lambda (a2: A).(\lambda (_: (leq g a1 a2)).(\lambda (H3: (nf2 c0 -t0)).(let H_x \def (H1 H3) in (let H4 \def H_x in (or3_ind (ex3_2 T T -(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda -(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 -(CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort -n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 -(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef -i))))) (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind -Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: -T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: -nat).(eq T t0 (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: -nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: -TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: -nat).(nf2 c0 (TLRef i)))))) (\lambda (H5: (ex3_2 T T (\lambda (w: T).(\lambda -(u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: -T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) -u))))).(ex3_2_ind T T (\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind -Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: -T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u))) (or3 (ex3_2 T T -(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda -(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 -(CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort -n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 -(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef -i)))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H6: (eq T t0 (THead (Bind -Abst) x0 x1))).(\lambda (H7: (nf2 c0 x0)).(\lambda (H8: (nf2 (CHead c0 (Bind -Abst) x0) x1)).(eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t1: T).(or3 -(ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t1 (THead (Bind Abst) w -u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda -(u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t1 -(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t1 -(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef -i))))))) (or3_intro0 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T (THead -(Bind Abst) x0 x1) (THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: -T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) -u)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind Abst) x0 x1) (TSort n)))) -(ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Bind -Abst) x0 x1) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: -TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: -nat).(nf2 c0 (TLRef i))))) (ex3_2_intro T T (\lambda (w: T).(\lambda (u: -T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) w u)))) (\lambda (w: -T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead -c0 (Bind Abst) w) u))) x0 x1 (refl_equal T (THead (Bind Abst) x0 x1)) H7 H8)) -t0 H6)))))) H5)) (\lambda (H5: (ex nat (\lambda (n: nat).(eq T t0 (TSort -n))))).(ex_ind nat (\lambda (n: nat).(eq T t0 (TSort n))) (or3 (ex3_2 T T -(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda -(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 -(CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort -n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 -(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef -i)))))) (\lambda (x: nat).(\lambda (H6: (eq T t0 (TSort x))).(eq_ind_r T -(TSort x) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u: -T).(eq T t1 (THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 -c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) -(ex nat (\lambda (n: nat).(eq T t1 (TSort n)))) (ex3_2 TList nat (\lambda -(ws: TList).(\lambda (i: nat).(eq T t1 (THeads (Flat Appl) ws (TLRef i))))) -(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))) (or3_intro1 (ex3_2 T T -(\lambda (w: T).(\lambda (u: T).(eq T (TSort x) (THead (Bind Abst) w u)))) -(\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: -T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T (TSort -x) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T -(TSort x) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda -(_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef -i))))) (ex_intro nat (\lambda (n: nat).(eq T (TSort x) (TSort n))) x -(refl_equal T (TSort x)))) t0 H6))) H5)) (\lambda (H5: (ex3_2 TList nat -(\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef -i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))).(ex3_2_ind TList nat (\lambda -(ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i))))) -(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))) (or3 (ex3_2 T T (\lambda (w: -T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda (w: -T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead -c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort n)))) -(ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads -(Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 -ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))) (\lambda -(x0: TList).(\lambda (x1: nat).(\lambda (H6: (eq T t0 (THeads (Flat Appl) x0 -(TLRef x1)))).(\lambda (H7: (nfs2 c0 x0)).(\lambda (H8: (nf2 c0 (TLRef -x1))).(eq_ind_r T (THeads (Flat Appl) x0 (TLRef x1)) (\lambda (t1: T).(or3 -(ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t1 (THead (Bind Abst) w -u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda -(u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t1 -(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t1 -(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef -i))))))) (or3_intro2 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T (THeads -(Flat Appl) x0 (TLRef x1)) (THead (Bind Abst) w u)))) (\lambda (w: -T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead -c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T (THeads (Flat Appl) -x0 (TLRef x1)) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda -(i: nat).(eq T (THeads (Flat Appl) x0 (TLRef x1)) (THeads (Flat Appl) ws -(TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda -(_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))) (ex3_2_intro TList nat -(\lambda (ws: TList).(\lambda (i: nat).(eq T (THeads (Flat Appl) x0 (TLRef -x1)) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef -i)))) x0 x1 (refl_equal T (THeads (Flat Appl) x0 (TLRef x1))) H7 H8)) t0 -H6)))))) H5)) H4))))))))))) c t a H))))). -(* COMMENTS -Initial nodes: 9193 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/dec.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/dec.ma deleted file mode 100644 index 33b652baf..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/dec.ma +++ /dev/null @@ -1,200 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/nf2/defs.ma". - -include "Basic-1/pr2/clen.ma". - -include "Basic-1/pr2/fwd.ma". - -include "Basic-1/pr0/dec.ma". - -include "Basic-1/C/props.ma". - -theorem nf2_dec: - \forall (c: C).(\forall (t1: T).(or (nf2 c t1) (ex2 T (\lambda (t2: T).((eq -T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c t1 t2))))) -\def - \lambda (c: C).(c_tail_ind (\lambda (c0: C).(\forall (t1: T).(or (\forall -(t2: T).((pr2 c0 t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 -t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c0 t1 t2)))))) (\lambda -(n: nat).(\lambda (t1: T).(let H_x \def (nf0_dec t1) in (let H \def H_x in -(or_ind (\forall (t2: T).((pr0 t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: -T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t1 t2))) -(or (\forall (t2: T).((pr2 (CSort n) t1 t2) \to (eq T t1 t2))) (ex2 T -(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: -T).(pr2 (CSort n) t1 t2)))) (\lambda (H0: ((\forall (t2: T).((pr0 t1 t2) \to -(eq T t1 t2))))).(or_introl (\forall (t2: T).((pr2 (CSort n) t1 t2) \to (eq T -t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) -(\lambda (t2: T).(pr2 (CSort n) t1 t2))) (\lambda (t2: T).(\lambda (H1: (pr2 -(CSort n) t1 t2)).(let H_y \def (pr2_gen_csort t1 t2 n H1) in (H0 t2 -H_y)))))) (\lambda (H0: (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall -(P: Prop).P))) (\lambda (t2: T).(pr0 t1 t2)))).(ex2_ind T (\lambda (t2: -T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t1 t2)) -(or (\forall (t2: T).((pr2 (CSort n) t1 t2) \to (eq T t1 t2))) (ex2 T -(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: -T).(pr2 (CSort n) t1 t2)))) (\lambda (x: T).(\lambda (H1: (((eq T t1 x) \to -(\forall (P: Prop).P)))).(\lambda (H2: (pr0 t1 x)).(or_intror (\forall (t2: -T).((pr2 (CSort n) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T -t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CSort n) t1 t2))) -(ex_intro2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) -(\lambda (t2: T).(pr2 (CSort n) t1 t2)) x H1 (pr2_free (CSort n) t1 x -H2)))))) H0)) H))))) (\lambda (c0: C).(\lambda (H: ((\forall (t1: T).(or -(\forall (t2: T).((pr2 c0 t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: -T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c0 t1 -t2))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (t1: T).(let H_x \def (H -t1) in (let H0 \def H_x in (or_ind (\forall (t2: T).((pr2 c0 t1 t2) \to (eq T -t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) -(\lambda (t2: T).(pr2 c0 t1 t2))) (or (\forall (t2: T).((pr2 (CTail k t c0) -t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall -(P: Prop).P))) (\lambda (t2: T).(pr2 (CTail k t c0) t1 t2)))) (\lambda (H1: -((\forall (t2: T).((pr2 c0 t1 t2) \to (eq T t1 t2))))).(K_ind (\lambda (k0: -K).(or (\forall (t2: T).((pr2 (CTail k0 t c0) t1 t2) \to (eq T t1 t2))) (ex2 -T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: -T).(pr2 (CTail k0 t c0) t1 t2))))) (\lambda (b: B).(B_ind (\lambda (b0: -B).(or (\forall (t2: T).((pr2 (CTail (Bind b0) t c0) t1 t2) \to (eq T t1 -t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) -(\lambda (t2: T).(pr2 (CTail (Bind b0) t c0) t1 t2))))) (let H_x0 \def -(dnf_dec t t1 (clen c0)) in (let H2 \def H_x0 in (ex_ind T (\lambda (v: -T).(or (subst0 (clen c0) t t1 (lift (S O) (clen c0) v)) (eq T t1 (lift (S O) -(clen c0) v)))) (or (\forall (t2: T).((pr2 (CTail (Bind Abbr) t c0) t1 t2) -\to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) t1 t2)))) (\lambda -(x: T).(\lambda (H3: (or (subst0 (clen c0) t t1 (lift (S O) (clen c0) x)) (eq -T t1 (lift (S O) (clen c0) x)))).(or_ind (subst0 (clen c0) t t1 (lift (S O) -(clen c0) x)) (eq T t1 (lift (S O) (clen c0) x)) (or (\forall (t2: T).((pr2 -(CTail (Bind Abbr) t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: -T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail -(Bind Abbr) t c0) t1 t2)))) (\lambda (H4: (subst0 (clen c0) t t1 (lift (S O) -(clen c0) x))).(let H_x1 \def (getl_ctail_clen Abbr t c0) in (let H5 \def -H_x1 in (ex_ind nat (\lambda (n: nat).(getl (clen c0) (CTail (Bind Abbr) t -c0) (CHead (CSort n) (Bind Abbr) t))) (or (\forall (t2: T).((pr2 (CTail (Bind -Abbr) t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) -\to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) t1 -t2)))) (\lambda (x0: nat).(\lambda (H6: (getl (clen c0) (CTail (Bind Abbr) t -c0) (CHead (CSort x0) (Bind Abbr) t))).(or_intror (\forall (t2: T).((pr2 -(CTail (Bind Abbr) t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: -T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail -(Bind Abbr) t c0) t1 t2))) (ex_intro2 T (\lambda (t2: T).((eq T t1 t2) \to -(\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) t1 -t2)) (lift (S O) (clen c0) x) (\lambda (H7: (eq T t1 (lift (S O) (clen c0) -x))).(\lambda (P: Prop).(let H8 \def (eq_ind T t1 (\lambda (t0: T).(subst0 -(clen c0) t t0 (lift (S O) (clen c0) x))) H4 (lift (S O) (clen c0) x) H7) in -(subst0_gen_lift_false x t (lift (S O) (clen c0) x) (S O) (clen c0) (clen c0) -(le_n (clen c0)) (eq_ind_r nat (plus (S O) (clen c0)) (\lambda (n: nat).(lt -(clen c0) n)) (le_n (plus (S O) (clen c0))) (plus (clen c0) (S O)) (plus_sym -(clen c0) (S O))) H8 P)))) (pr2_delta (CTail (Bind Abbr) t c0) (CSort x0) t -(clen c0) H6 t1 t1 (pr0_refl t1) (lift (S O) (clen c0) x) H4))))) H5)))) -(\lambda (H4: (eq T t1 (lift (S O) (clen c0) x))).(let H5 \def (eq_ind T t1 -(\lambda (t0: T).(\forall (t2: T).((pr2 c0 t0 t2) \to (eq T t0 t2)))) H1 -(lift (S O) (clen c0) x) H4) in (eq_ind_r T (lift (S O) (clen c0) x) (\lambda -(t0: T).(or (\forall (t2: T).((pr2 (CTail (Bind Abbr) t c0) t0 t2) \to (eq T -t0 t2))) (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: Prop).P))) -(\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) t0 t2))))) (or_introl (\forall -(t2: T).((pr2 (CTail (Bind Abbr) t c0) (lift (S O) (clen c0) x) t2) \to (eq T -(lift (S O) (clen c0) x) t2))) (ex2 T (\lambda (t2: T).((eq T (lift (S O) -(clen c0) x) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail -(Bind Abbr) t c0) (lift (S O) (clen c0) x) t2))) (\lambda (t2: T).(\lambda -(H6: (pr2 (CTail (Bind Abbr) t c0) (lift (S O) (clen c0) x) t2)).(let H_x1 -\def (pr2_gen_ctail (Bind Abbr) c0 t (lift (S O) (clen c0) x) t2 H6) in (let -H7 \def H_x1 in (or_ind (pr2 c0 (lift (S O) (clen c0) x) t2) (ex3 T (\lambda -(_: T).(eq K (Bind Abbr) (Bind Abbr))) (\lambda (t0: T).(pr0 (lift (S O) -(clen c0) x) t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2))) (eq T (lift -(S O) (clen c0) x) t2) (\lambda (H8: (pr2 c0 (lift (S O) (clen c0) x) -t2)).(H5 t2 H8)) (\lambda (H8: (ex3 T (\lambda (_: T).(eq K (Bind Abbr) (Bind -Abbr))) (\lambda (t0: T).(pr0 (lift (S O) (clen c0) x) t0)) (\lambda (t0: -T).(subst0 (clen c0) t t0 t2)))).(ex3_ind T (\lambda (_: T).(eq K (Bind Abbr) -(Bind Abbr))) (\lambda (t0: T).(pr0 (lift (S O) (clen c0) x) t0)) (\lambda -(t0: T).(subst0 (clen c0) t t0 t2)) (eq T (lift (S O) (clen c0) x) t2) -(\lambda (x0: T).(\lambda (_: (eq K (Bind Abbr) (Bind Abbr))).(\lambda (H10: -(pr0 (lift (S O) (clen c0) x) x0)).(\lambda (H11: (subst0 (clen c0) t x0 -t2)).(ex2_ind T (\lambda (t3: T).(eq T x0 (lift (S O) (clen c0) t3))) -(\lambda (t3: T).(pr0 x t3)) (eq T (lift (S O) (clen c0) x) t2) (\lambda (x1: -T).(\lambda (H12: (eq T x0 (lift (S O) (clen c0) x1))).(\lambda (_: (pr0 x -x1)).(let H14 \def (eq_ind T x0 (\lambda (t0: T).(subst0 (clen c0) t t0 t2)) -H11 (lift (S O) (clen c0) x1) H12) in (subst0_gen_lift_false x1 t t2 (S O) -(clen c0) (clen c0) (le_n (clen c0)) (eq_ind_r nat (plus (S O) (clen c0)) -(\lambda (n: nat).(lt (clen c0) n)) (le_n (plus (S O) (clen c0))) (plus (clen -c0) (S O)) (plus_sym (clen c0) (S O))) H14 (eq T (lift (S O) (clen c0) x) -t2)))))) (pr0_gen_lift x x0 (S O) (clen c0) H10)))))) H8)) H7)))))) t1 H4))) -H3))) H2))) (or_introl (\forall (t2: T).((pr2 (CTail (Bind Abst) t c0) t1 t2) -\to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abst) t c0) t1 t2))) (\lambda -(t2: T).(\lambda (H2: (pr2 (CTail (Bind Abst) t c0) t1 t2)).(let H_x0 \def -(pr2_gen_ctail (Bind Abst) c0 t t1 t2 H2) in (let H3 \def H_x0 in (or_ind -(pr2 c0 t1 t2) (ex3 T (\lambda (_: T).(eq K (Bind Abst) (Bind Abbr))) -(\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2))) -(eq T t1 t2) (\lambda (H4: (pr2 c0 t1 t2)).(H1 t2 H4)) (\lambda (H4: (ex3 T -(\lambda (_: T).(eq K (Bind Abst) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) -(\lambda (t0: T).(subst0 (clen c0) t t0 t2)))).(ex3_ind T (\lambda (_: T).(eq -K (Bind Abst) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: -T).(subst0 (clen c0) t t0 t2)) (eq T t1 t2) (\lambda (x0: T).(\lambda (H5: -(eq K (Bind Abst) (Bind Abbr))).(\lambda (_: (pr0 t1 x0)).(\lambda (_: -(subst0 (clen c0) t x0 t2)).(let H8 \def (eq_ind K (Bind Abst) (\lambda (ee: -K).(match ee in K return (\lambda (_: K).Prop) with [(Bind b0) \Rightarrow -(match b0 in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | -Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) \Rightarrow -False])) I (Bind Abbr) H5) in (False_ind (eq T t1 t2) H8)))))) H4)) H3)))))) -(or_introl (\forall (t2: T).((pr2 (CTail (Bind Void) t c0) t1 t2) \to (eq T -t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) -(\lambda (t2: T).(pr2 (CTail (Bind Void) t c0) t1 t2))) (\lambda (t2: -T).(\lambda (H2: (pr2 (CTail (Bind Void) t c0) t1 t2)).(let H_x0 \def -(pr2_gen_ctail (Bind Void) c0 t t1 t2 H2) in (let H3 \def H_x0 in (or_ind -(pr2 c0 t1 t2) (ex3 T (\lambda (_: T).(eq K (Bind Void) (Bind Abbr))) -(\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2))) -(eq T t1 t2) (\lambda (H4: (pr2 c0 t1 t2)).(H1 t2 H4)) (\lambda (H4: (ex3 T -(\lambda (_: T).(eq K (Bind Void) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) -(\lambda (t0: T).(subst0 (clen c0) t t0 t2)))).(ex3_ind T (\lambda (_: T).(eq -K (Bind Void) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: -T).(subst0 (clen c0) t t0 t2)) (eq T t1 t2) (\lambda (x0: T).(\lambda (H5: -(eq K (Bind Void) (Bind Abbr))).(\lambda (_: (pr0 t1 x0)).(\lambda (_: -(subst0 (clen c0) t x0 t2)).(let H8 \def (eq_ind K (Bind Void) (\lambda (ee: -K).(match ee in K return (\lambda (_: K).Prop) with [(Bind b0) \Rightarrow -(match b0 in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | -Abst \Rightarrow False | Void \Rightarrow True]) | (Flat _) \Rightarrow -False])) I (Bind Abbr) H5) in (False_ind (eq T t1 t2) H8)))))) H4)) H3)))))) -b)) (\lambda (f: F).(or_introl (\forall (t2: T).((pr2 (CTail (Flat f) t c0) -t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall -(P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Flat f) t c0) t1 t2))) (\lambda -(t2: T).(\lambda (H2: (pr2 (CTail (Flat f) t c0) t1 t2)).(let H_x0 \def -(pr2_gen_ctail (Flat f) c0 t t1 t2 H2) in (let H3 \def H_x0 in (or_ind (pr2 -c0 t1 t2) (ex3 T (\lambda (_: T).(eq K (Flat f) (Bind Abbr))) (\lambda (t0: -T).(pr0 t1 t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2))) (eq T t1 t2) -(\lambda (H4: (pr2 c0 t1 t2)).(H1 t2 H4)) (\lambda (H4: (ex3 T (\lambda (_: -T).(eq K (Flat f) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: -T).(subst0 (clen c0) t t0 t2)))).(ex3_ind T (\lambda (_: T).(eq K (Flat f) -(Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(subst0 (clen -c0) t t0 t2)) (eq T t1 t2) (\lambda (x0: T).(\lambda (H5: (eq K (Flat f) -(Bind Abbr))).(\lambda (_: (pr0 t1 x0)).(\lambda (_: (subst0 (clen c0) t x0 -t2)).(let H8 \def (eq_ind K (Flat f) (\lambda (ee: K).(match ee in K return -(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])) I (Bind Abbr) H5) in (False_ind (eq T t1 t2) H8)))))) H4)) H3))))))) -k)) (\lambda (H1: (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr2 c0 t1 t2)))).(ex2_ind T (\lambda (t2: -T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c0 t1 t2)) -(or (\forall (t2: T).((pr2 (CTail k t c0) t1 t2) \to (eq T t1 t2))) (ex2 T -(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: -T).(pr2 (CTail k t c0) t1 t2)))) (\lambda (x: T).(\lambda (H2: (((eq T t1 x) -\to (\forall (P: Prop).P)))).(\lambda (H3: (pr2 c0 t1 x)).(or_intror (\forall -(t2: T).((pr2 (CTail k t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: -T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail k t -c0) t1 t2))) (ex_intro2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr2 (CTail k t c0) t1 t2)) x H2 (pr2_ctail c0 t1 -x H3 k t)))))) H1)) H0)))))))) c). -(* COMMENTS -Initial nodes: 3653 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/defs.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/defs.ma deleted file mode 100644 index 98e931c0b..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/defs.ma +++ /dev/null @@ -1,31 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr2/defs.ma". - -definition nf2: - C \to (T \to Prop) -\def - \lambda (c: C).(\lambda (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (eq T t1 -t2)))). - -definition nfs2: - C \to (TList \to Prop) -\def - let rec nfs2 (c: C) (ts: TList) on ts: Prop \def (match ts with [TNil -\Rightarrow True | (TCons t ts0) \Rightarrow (land (nf2 c t) (nfs2 c ts0))]) -in nfs2. - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/fwd.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/fwd.ma deleted file mode 100644 index 9138ff2fa..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/fwd.ma +++ /dev/null @@ -1,220 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/nf2/defs.ma". - -include "Basic-1/pr2/clen.ma". - -include "Basic-1/subst0/dec.ma". - -include "Basic-1/T/props.ma". - -theorem nf2_gen_lref: - \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c -(CHead d (Bind Abbr) u)) \to ((nf2 c (TLRef i)) \to (\forall (P: Prop).P)))))) -\def - \lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H0: ((\forall (t2: T).((pr2 -c (TLRef i) t2) \to (eq T (TLRef i) t2))))).(\lambda (P: -Prop).(lift_gen_lref_false (S i) O i (le_O_n i) (le_n (plus O (S i))) u (H0 -(lift (S i) O u) (pr2_delta c d u i H (TLRef i) (TLRef i) (pr0_refl (TLRef -i)) (lift (S i) O u) (subst0_lref u i))) P))))))). -(* COMMENTS -Initial nodes: 129 -END *) - -theorem nf2_gen_abst: - \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Bind Abst) u -t)) \to (land (nf2 c u) (nf2 (CHead c (Bind Abst) u) t))))) -\def - \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: ((\forall (t2: -T).((pr2 c (THead (Bind Abst) u t) t2) \to (eq T (THead (Bind Abst) u t) -t2))))).(conj (\forall (t2: T).((pr2 c u t2) \to (eq T u t2))) (\forall (t2: -T).((pr2 (CHead c (Bind Abst) u) t t2) \to (eq T t t2))) (\lambda (t2: -T).(\lambda (H0: (pr2 c u t2)).(let H1 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | -(TLRef _) \Rightarrow u | (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abst) -u t) (THead (Bind Abst) t2 t) (H (THead (Bind Abst) t2 t) (pr2_head_1 c u t2 -H0 (Bind Abst) t))) in (let H2 \def (eq_ind_r T t2 (\lambda (t0: T).(pr2 c u -t0)) H0 u H1) in (eq_ind T u (\lambda (t0: T).(eq T u t0)) (refl_equal T u) -t2 H1))))) (\lambda (t2: T).(\lambda (H0: (pr2 (CHead c (Bind Abst) u) t -t2)).(let H1 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda -(_: T).T) with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ -_ t0) \Rightarrow t0])) (THead (Bind Abst) u t) (THead (Bind Abst) u t2) (H -(THead (Bind Abst) u t2) (let H_y \def (pr2_gen_cbind Abst c u t t2 H0) in -H_y))) in (let H2 \def (eq_ind_r T t2 (\lambda (t0: T).(pr2 (CHead c (Bind -Abst) u) t t0)) H0 t H1) in (eq_ind T t (\lambda (t0: T).(eq T t t0)) -(refl_equal T t) t2 H1))))))))). -(* COMMENTS -Initial nodes: 353 -END *) - -theorem nf2_gen_cast: - \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Flat Cast) u -t)) \to (\forall (P: Prop).P)))) -\def - \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (nf2 c (THead -(Flat Cast) u t))).(\lambda (P: Prop).(thead_x_y_y (Flat Cast) u t (H t -(pr2_free c (THead (Flat Cast) u t) t (pr0_tau t t (pr0_refl t) u))) P))))). -(* COMMENTS -Initial nodes: 65 -END *) - -theorem nf2_gen_beta: - \forall (c: C).(\forall (u: T).(\forall (v: T).(\forall (t: T).((nf2 c -(THead (Flat Appl) u (THead (Bind Abst) v t))) \to (\forall (P: Prop).P))))) -\def - \lambda (c: C).(\lambda (u: T).(\lambda (v: T).(\lambda (t: T).(\lambda (H: -((\forall (t2: T).((pr2 c (THead (Flat Appl) u (THead (Bind Abst) v t)) t2) -\to (eq T (THead (Flat Appl) u (THead (Bind Abst) v t)) t2))))).(\lambda (P: -Prop).(let H0 \def (eq_ind T (THead (Flat Appl) u (THead (Bind Abst) v t)) -(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow -(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | -(Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u t) (H (THead (Bind -Abbr) u t) (pr2_free c (THead (Flat Appl) u (THead (Bind Abst) v t)) (THead -(Bind Abbr) u t) (pr0_beta v u u (pr0_refl u) t t (pr0_refl t))))) in -(False_ind P H0))))))). -(* COMMENTS -Initial nodes: 183 -END *) - -theorem nf2_gen_flat: - \forall (f: F).(\forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c -(THead (Flat f) u t)) \to (land (nf2 c u) (nf2 c t)))))) -\def - \lambda (f: F).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: -((\forall (t2: T).((pr2 c (THead (Flat f) u t) t2) \to (eq T (THead (Flat f) -u t) t2))))).(conj (\forall (t2: T).((pr2 c u t2) \to (eq T u t2))) (\forall -(t2: T).((pr2 c t t2) \to (eq T t t2))) (\lambda (t2: T).(\lambda (H0: (pr2 c -u t2)).(let H1 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | -(THead _ t0 _) \Rightarrow t0])) (THead (Flat f) u t) (THead (Flat f) t2 t) -(H (THead (Flat f) t2 t) (pr2_head_1 c u t2 H0 (Flat f) t))) in H1))) -(\lambda (t2: T).(\lambda (H0: (pr2 c t t2)).(let H1 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t0) \Rightarrow t0])) -(THead (Flat f) u t) (THead (Flat f) u t2) (H (THead (Flat f) u t2) -(pr2_head_2 c u t t2 (Flat f) (pr2_cflat c t t2 H0 f u)))) in H1)))))))). -(* COMMENTS -Initial nodes: 251 -END *) - -theorem nf2_gen__nf2_gen_aux: - \forall (b: B).(\forall (x: T).(\forall (u: T).(\forall (d: nat).((eq T -(THead (Bind b) u (lift (S O) d x)) x) \to (\forall (P: Prop).P))))) -\def - \lambda (b: B).(\lambda (x: T).(T_ind (\lambda (t: T).(\forall (u: -T).(\forall (d: nat).((eq T (THead (Bind b) u (lift (S O) d t)) t) \to -(\forall (P: Prop).P))))) (\lambda (n: nat).(\lambda (u: T).(\lambda (d: -nat).(\lambda (H: (eq T (THead (Bind b) u (lift (S O) d (TSort n))) (TSort -n))).(\lambda (P: Prop).(let H0 \def (eq_ind T (THead (Bind b) u (lift (S O) -d (TSort n))) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) -with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ -_) \Rightarrow True])) I (TSort n) H) in (False_ind P H0))))))) (\lambda (n: -nat).(\lambda (u: T).(\lambda (d: nat).(\lambda (H: (eq T (THead (Bind b) u -(lift (S O) d (TLRef n))) (TLRef n))).(\lambda (P: Prop).(let H0 \def (eq_ind -T (THead (Bind b) u (lift (S O) d (TLRef n))) (\lambda (ee: T).(match ee in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H) in -(False_ind P H0))))))) (\lambda (k: K).(\lambda (t: T).(\lambda (_: ((\forall -(u: T).(\forall (d: nat).((eq T (THead (Bind b) u (lift (S O) d t)) t) \to -(\forall (P: Prop).P)))))).(\lambda (t0: T).(\lambda (H0: ((\forall (u: -T).(\forall (d: nat).((eq T (THead (Bind b) u (lift (S O) d t0)) t0) \to -(\forall (P: Prop).P)))))).(\lambda (u: T).(\lambda (d: nat).(\lambda (H1: -(eq T (THead (Bind b) u (lift (S O) d (THead k t t0))) (THead k t -t0))).(\lambda (P: Prop).(let H2 \def (f_equal T K (\lambda (e: T).(match e -in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow (Bind b) | (TLRef -_) \Rightarrow (Bind b) | (THead k0 _ _) \Rightarrow k0])) (THead (Bind b) u -(lift (S O) d (THead k t t0))) (THead k t t0) H1) in ((let H3 \def (f_equal T -T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t1 _) \Rightarrow t1])) -(THead (Bind b) u (lift (S O) d (THead k t t0))) (THead k t t0) H1) in ((let -H4 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow (THead k ((let rec lref_map (f: ((nat \to nat))) -(d0: nat) (t1: T) on t1: T \def (match t1 with [(TSort n) \Rightarrow (TSort -n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) with [true \Rightarrow i -| false \Rightarrow (f i)])) | (THead k0 u0 t2) \Rightarrow (THead k0 -(lref_map f d0 u0) (lref_map f (s k0 d0) t2))]) in lref_map) (\lambda (x0: -nat).(plus x0 (S O))) d t) ((let rec lref_map (f: ((nat \to nat))) (d0: nat) -(t1: T) on t1: T \def (match t1 with [(TSort n) \Rightarrow (TSort n) | -(TLRef i) \Rightarrow (TLRef (match (blt i d0) with [true \Rightarrow i | -false \Rightarrow (f i)])) | (THead k0 u0 t2) \Rightarrow (THead k0 (lref_map -f d0 u0) (lref_map f (s k0 d0) t2))]) in lref_map) (\lambda (x0: nat).(plus -x0 (S O))) (s k d) t0)) | (TLRef _) \Rightarrow (THead k ((let rec lref_map -(f: ((nat \to nat))) (d0: nat) (t1: T) on t1: T \def (match t1 with [(TSort -n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) -with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k0 u0 t2) -\Rightarrow (THead k0 (lref_map f d0 u0) (lref_map f (s k0 d0) t2))]) in -lref_map) (\lambda (x0: nat).(plus x0 (S O))) d t) ((let rec lref_map (f: -((nat \to nat))) (d0: nat) (t1: T) on t1: T \def (match t1 with [(TSort n) -\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) with -[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k0 u0 t2) -\Rightarrow (THead k0 (lref_map f d0 u0) (lref_map f (s k0 d0) t2))]) in -lref_map) (\lambda (x0: nat).(plus x0 (S O))) (s k d) t0)) | (THead _ _ t1) -\Rightarrow t1])) (THead (Bind b) u (lift (S O) d (THead k t t0))) (THead k t -t0) H1) in (\lambda (_: (eq T u t)).(\lambda (H6: (eq K (Bind b) k)).(let H7 -\def (eq_ind_r K k (\lambda (k0: K).(eq T (lift (S O) d (THead k0 t t0)) t0)) -H4 (Bind b) H6) in (let H8 \def (eq_ind T (lift (S O) d (THead (Bind b) t -t0)) (\lambda (t1: T).(eq T t1 t0)) H7 (THead (Bind b) (lift (S O) d t) (lift -(S O) (S d) t0)) (lift_bind b t t0 (S O) d)) in (H0 (lift (S O) d t) (S d) H8 -P)))))) H3)) H2))))))))))) x)). -(* COMMENTS -Initial nodes: 935 -END *) - -theorem nf2_gen_abbr: - \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Bind Abbr) u -t)) \to (\forall (P: Prop).P)))) -\def - \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: ((\forall (t2: -T).((pr2 c (THead (Bind Abbr) u t) t2) \to (eq T (THead (Bind Abbr) u t) -t2))))).(\lambda (P: Prop).(let H_x \def (dnf_dec u t O) in (let H0 \def H_x -in (ex_ind T (\lambda (v: T).(or (subst0 O u t (lift (S O) O v)) (eq T t -(lift (S O) O v)))) P (\lambda (x: T).(\lambda (H1: (or (subst0 O u t (lift -(S O) O x)) (eq T t (lift (S O) O x)))).(or_ind (subst0 O u t (lift (S O) O -x)) (eq T t (lift (S O) O x)) P (\lambda (H2: (subst0 O u t (lift (S O) O -x))).(let H3 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda -(_: T).T) with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ -_ t0) \Rightarrow t0])) (THead (Bind Abbr) u t) (THead (Bind Abbr) u (lift (S -O) O x)) (H (THead (Bind Abbr) u (lift (S O) O x)) (pr2_free c (THead (Bind -Abbr) u t) (THead (Bind Abbr) u (lift (S O) O x)) (pr0_delta u u (pr0_refl u) -t t (pr0_refl t) (lift (S O) O x) H2)))) in (let H4 \def (eq_ind T t (\lambda -(t0: T).(subst0 O u t0 (lift (S O) O x))) H2 (lift (S O) O x) H3) in -(subst0_refl u (lift (S O) O x) O H4 P)))) (\lambda (H2: (eq T t (lift (S O) -O x))).(let H3 \def (eq_ind T t (\lambda (t0: T).(\forall (t2: T).((pr2 c -(THead (Bind Abbr) u t0) t2) \to (eq T (THead (Bind Abbr) u t0) t2)))) H -(lift (S O) O x) H2) in (nf2_gen__nf2_gen_aux Abbr x u O (H3 x (pr2_free c -(THead (Bind Abbr) u (lift (S O) O x)) x (pr0_zeta Abbr not_abbr_abst x x -(pr0_refl x) u))) P))) H1))) H0))))))). -(* COMMENTS -Initial nodes: 511 -END *) - -theorem nf2_gen_void: - \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Bind Void) u -(lift (S O) O t))) \to (\forall (P: Prop).P)))) -\def - \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: ((\forall (t2: -T).((pr2 c (THead (Bind Void) u (lift (S O) O t)) t2) \to (eq T (THead (Bind -Void) u (lift (S O) O t)) t2))))).(\lambda (P: Prop).(nf2_gen__nf2_gen_aux -Void t u O (H t (pr2_free c (THead (Bind Void) u (lift (S O) O t)) t -(pr0_zeta Void (sym_not_eq B Abst Void not_abst_void) t t (pr0_refl t) u))) -P))))). -(* COMMENTS -Initial nodes: 121 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/iso.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/iso.ma deleted file mode 100644 index 6a2ce00f8..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/iso.ma +++ /dev/null @@ -1,130 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/nf2/pr3.ma". - -include "Basic-1/pr3/fwd.ma". - -include "Basic-1/iso/props.ma". - -theorem nf2_iso_appls_lref: - \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (vs: -TList).(\forall (u: T).((pr3 c (THeads (Flat Appl) vs (TLRef i)) u) \to (iso -(THeads (Flat Appl) vs (TLRef i)) u)))))) -\def - \lambda (c: C).(\lambda (i: nat).(\lambda (H: (nf2 c (TLRef i))).(\lambda -(vs: TList).(TList_ind (\lambda (t: TList).(\forall (u: T).((pr3 c (THeads -(Flat Appl) t (TLRef i)) u) \to (iso (THeads (Flat Appl) t (TLRef i)) u)))) -(\lambda (u: T).(\lambda (H0: (pr3 c (TLRef i) u)).(let H_y \def -(nf2_pr3_unfold c (TLRef i) u H0 H) in (let H1 \def (eq_ind_r T u (\lambda -(t: T).(pr3 c (TLRef i) t)) H0 (TLRef i) H_y) in (eq_ind T (TLRef i) (\lambda -(t: T).(iso (TLRef i) t)) (iso_refl (TLRef i)) u H_y))))) (\lambda (t: -T).(\lambda (t0: TList).(\lambda (H0: ((\forall (u: T).((pr3 c (THeads (Flat -Appl) t0 (TLRef i)) u) \to (iso (THeads (Flat Appl) t0 (TLRef i)) -u))))).(\lambda (u: T).(\lambda (H1: (pr3 c (THead (Flat Appl) t (THeads -(Flat Appl) t0 (TLRef i))) u)).(let H2 \def (pr3_gen_appl c t (THeads (Flat -Appl) t0 (TLRef i)) u H1) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda -(t2: T).(eq T u (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c t u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) -t0 (TLRef i)) t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) u))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))) (\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat -Appl) t0 (TLRef i)) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u0: -T).(pr3 (CHead c (Bind b) u0) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat -Appl) t0 (TLRef i)) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda -(_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: -T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) -u))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) -y2) z1 z2)))))))) (iso (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef -i))) u) (\lambda (H3: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T u -(THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) -(\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) -t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T u (THead (Flat -Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_: -T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) t2))) (iso -(THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) u) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H4: (eq T u (THead (Flat Appl) x0 -x1))).(\lambda (_: (pr3 c t x0)).(\lambda (_: (pr3 c (THeads (Flat Appl) t0 -(TLRef i)) x1)).(eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda (t1: T).(iso -(THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) t1)) (iso_head t x0 -(THeads (Flat Appl) t0 (TLRef i)) x1 (Flat Appl)) u H4)))))) H3)) (\lambda -(H3: (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t2: T).(pr3 c (THead (Bind Abbr) u2 t2) u))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))) (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat -Appl) t0 (TLRef i)) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u0: -T).(pr3 (CHead c (Bind b) u0) z1 t2))))))))).(ex4_4_ind T T T T (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind -Abbr) u2 t2) u))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c t u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(t2: T).(\forall (b: B).(\forall (u0: T).(pr3 (CHead c (Bind b) u0) z1 -t2))))))) (iso (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) u) -(\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda -(_: (pr3 c (THead (Bind Abbr) x2 x3) u)).(\lambda (_: (pr3 c t x2)).(\lambda -(H6: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind Abst) x0 -x1))).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pr3 (CHead c (Bind b) -u0) x1 x3))))).(let H_y \def (H0 (THead (Bind Abst) x0 x1) H6) in -(iso_flats_lref_bind_false Appl Abst i x0 x1 t0 H_y (iso (THead (Flat Appl) t -(THeads (Flat Appl) t0 (TLRef i))) u))))))))))) H3)) (\lambda (H3: (ex6_6 B T -T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind b) y1 z1)))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda -(u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift -(S O) O u2) z2)) u))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 -(CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat -Appl) t0 (TLRef i)) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda -(_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: -T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) -u))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) -y2) z1 z2))))))) (iso (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) -u) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: -T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (not (eq B x0 -Abst))).(\lambda (H5: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind -x0) x1 x2))).(\lambda (_: (pr3 c (THead (Bind x0) x5 (THead (Flat Appl) (lift -(S O) O x4) x3)) u)).(\lambda (_: (pr3 c t x4)).(\lambda (_: (pr3 c x1 -x5)).(\lambda (_: (pr3 (CHead c (Bind x0) x5) x2 x3)).(let H_y \def (H0 -(THead (Bind x0) x1 x2) H5) in (iso_flats_lref_bind_false Appl x0 i x1 x2 t0 -H_y (iso (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) -u))))))))))))))) H3)) H2))))))) vs)))). -(* COMMENTS -Initial nodes: 1817 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/lift1.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/lift1.ma deleted file mode 100644 index d50790336..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/lift1.ma +++ /dev/null @@ -1,41 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/nf2/props.ma". - -include "Basic-1/drop1/fwd.ma". - -theorem nf2_lift1: - \forall (e: C).(\forall (hds: PList).(\forall (c: C).(\forall (t: T).((drop1 -hds c e) \to ((nf2 e t) \to (nf2 c (lift1 hds t))))))) -\def - \lambda (e: C).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall -(c: C).(\forall (t: T).((drop1 p c e) \to ((nf2 e t) \to (nf2 c (lift1 p -t))))))) (\lambda (c: C).(\lambda (t: T).(\lambda (H: (drop1 PNil c -e)).(\lambda (H0: (nf2 e t)).(let H_y \def (drop1_gen_pnil c e H) in -(eq_ind_r C e (\lambda (c0: C).(nf2 c0 t)) H0 c H_y)))))) (\lambda (n: -nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c: -C).(\forall (t: T).((drop1 p c e) \to ((nf2 e t) \to (nf2 c (lift1 p -t)))))))).(\lambda (c: C).(\lambda (t: T).(\lambda (H0: (drop1 (PCons n n0 p) -c e)).(\lambda (H1: (nf2 e t)).(let H_x \def (drop1_gen_pcons c e p n n0 H0) -in (let H2 \def H_x in (ex2_ind C (\lambda (c2: C).(drop n n0 c c2)) (\lambda -(c2: C).(drop1 p c2 e)) (nf2 c (lift n n0 (lift1 p t))) (\lambda (x: -C).(\lambda (H3: (drop n n0 c x)).(\lambda (H4: (drop1 p x e)).(nf2_lift x -(lift1 p t) (H x t H4 H1) c n n0 H3)))) H2))))))))))) hds)). -(* COMMENTS -Initial nodes: 249 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/pr3.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/pr3.ma deleted file mode 100644 index 3db24223e..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/pr3.ma +++ /dev/null @@ -1,56 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/nf2/defs.ma". - -include "Basic-1/pr3/pr3.ma". - -theorem nf2_pr3_unfold: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to ((nf2 c -t1) \to (eq T t1 t2))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 -t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).((nf2 c t) \to (eq T t -t0)))) (\lambda (t: T).(\lambda (H0: (nf2 c t)).(H0 t (pr2_free c t t -(pr0_refl t))))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c t3 -t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2: (((nf2 c t0) -\to (eq T t0 t4)))).(\lambda (H3: (nf2 c t3)).(let H4 \def H3 in (let H5 \def -(eq_ind T t3 (\lambda (t: T).(nf2 c t)) H3 t0 (H4 t0 H0)) in (let H6 \def -(eq_ind T t3 (\lambda (t: T).(pr2 c t t0)) H0 t0 (H4 t0 H0)) in (eq_ind_r T -t0 (\lambda (t: T).(eq T t t4)) (H2 H5) t3 (H4 t0 H0)))))))))))) t1 t2 H)))). -(* COMMENTS -Initial nodes: 187 -END *) - -theorem nf2_pr3_confluence: - \forall (c: C).(\forall (t1: T).((nf2 c t1) \to (\forall (t2: T).((nf2 c t2) -\to (\forall (t: T).((pr3 c t t1) \to ((pr3 c t t2) \to (eq T t1 t2)))))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (H: (nf2 c t1)).(\lambda (t2: -T).(\lambda (H0: (nf2 c t2)).(\lambda (t: T).(\lambda (H1: (pr3 c t -t1)).(\lambda (H2: (pr3 c t t2)).(ex2_ind T (\lambda (t0: T).(pr3 c t2 t0)) -(\lambda (t0: T).(pr3 c t1 t0)) (eq T t1 t2) (\lambda (x: T).(\lambda (H3: -(pr3 c t2 x)).(\lambda (H4: (pr3 c t1 x)).(let H_y \def (nf2_pr3_unfold c t1 -x H4 H) in (let H5 \def (eq_ind_r T x (\lambda (t0: T).(pr3 c t1 t0)) H4 t1 -H_y) in (let H6 \def (eq_ind_r T x (\lambda (t0: T).(pr3 c t2 t0)) H3 t1 H_y) -in (let H_y0 \def (nf2_pr3_unfold c t2 t1 H6 H0) in (let H7 \def (eq_ind T t2 -(\lambda (t0: T).(pr3 c t0 t1)) H6 t1 H_y0) in (eq_ind_r T t1 (\lambda (t0: -T).(eq T t1 t0)) (refl_equal T t1) t2 H_y0))))))))) (pr3_confluence c t t2 H2 -t1 H1))))))))). -(* COMMENTS -Initial nodes: 215 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/props.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/props.ma deleted file mode 100644 index 2f0f092f2..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/props.ma +++ /dev/null @@ -1,341 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/nf2/defs.ma". - -include "Basic-1/pr2/fwd.ma". - -theorem nf2_sort: - \forall (c: C).(\forall (n: nat).(nf2 c (TSort n))) -\def - \lambda (c: C).(\lambda (n: nat).(\lambda (t2: T).(\lambda (H: (pr2 c (TSort -n) t2)).(eq_ind_r T (TSort n) (\lambda (t: T).(eq T (TSort n) t)) (refl_equal -T (TSort n)) t2 (pr2_gen_sort c t2 n H))))). -(* COMMENTS -Initial nodes: 55 -END *) - -theorem nf2_csort_lref: - \forall (n: nat).(\forall (i: nat).(nf2 (CSort n) (TLRef i))) -\def - \lambda (n: nat).(\lambda (i: nat).(\lambda (t2: T).(\lambda (H: (pr2 (CSort -n) (TLRef i) t2)).(let H0 \def (pr2_gen_lref (CSort n) t2 i H) in (or_ind (eq -T t2 (TLRef i)) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i (CSort n) -(CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T t2 (lift (S -i) O u))))) (eq T (TLRef i) t2) (\lambda (H1: (eq T t2 (TLRef i))).(eq_ind_r -T (TLRef i) (\lambda (t: T).(eq T (TLRef i) t)) (refl_equal T (TLRef i)) t2 -H1)) (\lambda (H1: (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i (CSort -n) (CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T t2 (lift -(S i) O u)))))).(ex2_2_ind C T (\lambda (d: C).(\lambda (u: T).(getl i (CSort -n) (CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T t2 (lift -(S i) O u)))) (eq T (TLRef i) t2) (\lambda (x0: C).(\lambda (x1: T).(\lambda -(H2: (getl i (CSort n) (CHead x0 (Bind Abbr) x1))).(\lambda (H3: (eq T t2 -(lift (S i) O x1))).(eq_ind_r T (lift (S i) O x1) (\lambda (t: T).(eq T -(TLRef i) t)) (getl_gen_sort n i (CHead x0 (Bind Abbr) x1) H2 (eq T (TLRef i) -(lift (S i) O x1))) t2 H3))))) H1)) H0))))). -(* COMMENTS -Initial nodes: 355 -END *) - -theorem nf2_abst: - \forall (c: C).(\forall (u: T).((nf2 c u) \to (\forall (b: B).(\forall (v: -T).(\forall (t: T).((nf2 (CHead c (Bind b) v) t) \to (nf2 c (THead (Bind -Abst) u t)))))))) -\def - \lambda (c: C).(\lambda (u: T).(\lambda (H: ((\forall (t2: T).((pr2 c u t2) -\to (eq T u t2))))).(\lambda (b: B).(\lambda (v: T).(\lambda (t: T).(\lambda -(H0: ((\forall (t2: T).((pr2 (CHead c (Bind b) v) t t2) \to (eq T t -t2))))).(\lambda (t2: T).(\lambda (H1: (pr2 c (THead (Bind Abst) u t) -t2)).(let H2 \def (pr2_gen_abst c u t t2 H1) in (ex3_2_ind T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c u u2))) (\lambda (_: T).(\lambda (t3: T).(\forall -(b0: B).(\forall (u0: T).(pr2 (CHead c (Bind b0) u0) t t3))))) (eq T (THead -(Bind Abst) u t) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H3: (eq T t2 -(THead (Bind Abst) x0 x1))).(\lambda (H4: (pr2 c u x0)).(\lambda (H5: -((\forall (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind b0) u0) t -x1))))).(eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t0: T).(eq T (THead -(Bind Abst) u t) t0)) (f_equal3 K T T T THead (Bind Abst) (Bind Abst) u x0 t -x1 (refl_equal K (Bind Abst)) (H x0 H4) (H0 x1 (H5 b v))) t2 H3)))))) -H2)))))))))). -(* COMMENTS -Initial nodes: 299 -END *) - -theorem nf2_abst_shift: - \forall (c: C).(\forall (u: T).((nf2 c u) \to (\forall (t: T).((nf2 (CHead c -(Bind Abst) u) t) \to (nf2 c (THead (Bind Abst) u t)))))) -\def - \lambda (c: C).(\lambda (u: T).(\lambda (H: ((\forall (t2: T).((pr2 c u t2) -\to (eq T u t2))))).(\lambda (t: T).(\lambda (H0: ((\forall (t2: T).((pr2 -(CHead c (Bind Abst) u) t t2) \to (eq T t t2))))).(\lambda (t2: T).(\lambda -(H1: (pr2 c (THead (Bind Abst) u t) t2)).(let H2 \def (pr2_gen_abst c u t t2 -H1) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u u2))) (\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) -u0) t t3))))) (eq T (THead (Bind Abst) u t) t2) (\lambda (x0: T).(\lambda -(x1: T).(\lambda (H3: (eq T t2 (THead (Bind Abst) x0 x1))).(\lambda (H4: (pr2 -c u x0)).(\lambda (H5: ((\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind -b) u0) t x1))))).(eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t0: T).(eq T -(THead (Bind Abst) u t) t0)) (f_equal3 K T T T THead (Bind Abst) (Bind Abst) -u x0 t x1 (refl_equal K (Bind Abst)) (H x0 H4) (H0 x1 (H5 Abst u))) t2 -H3)))))) H2)))))))). -(* COMMENTS -Initial nodes: 295 -END *) - -theorem nfs2_tapp: - \forall (c: C).(\forall (t: T).(\forall (ts: TList).((nfs2 c (TApp ts t)) -\to (land (nfs2 c ts) (nf2 c t))))) -\def - \lambda (c: C).(\lambda (t: T).(\lambda (ts: TList).(TList_ind (\lambda (t0: -TList).((nfs2 c (TApp t0 t)) \to (land (nfs2 c t0) (nf2 c t)))) (\lambda (H: -(land (nf2 c t) True)).(let H0 \def H in (land_ind (nf2 c t) True (land True -(nf2 c t)) (\lambda (H1: (nf2 c t)).(\lambda (_: True).(conj True (nf2 c t) I -H1))) H0))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H: (((nfs2 c -(TApp t1 t)) \to (land (nfs2 c t1) (nf2 c t))))).(\lambda (H0: (land (nf2 c -t0) (nfs2 c (TApp t1 t)))).(let H1 \def H0 in (land_ind (nf2 c t0) (nfs2 c -(TApp t1 t)) (land (land (nf2 c t0) (nfs2 c t1)) (nf2 c t)) (\lambda (H2: -(nf2 c t0)).(\lambda (H3: (nfs2 c (TApp t1 t))).(let H_x \def (H H3) in (let -H4 \def H_x in (land_ind (nfs2 c t1) (nf2 c t) (land (land (nf2 c t0) (nfs2 c -t1)) (nf2 c t)) (\lambda (H5: (nfs2 c t1)).(\lambda (H6: (nf2 c t)).(conj -(land (nf2 c t0) (nfs2 c t1)) (nf2 c t) (conj (nf2 c t0) (nfs2 c t1) H2 H5) -H6))) H4))))) H1)))))) ts))). -(* COMMENTS -Initial nodes: 295 -END *) - -theorem nf2_appls_lref: - \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (vs: -TList).((nfs2 c vs) \to (nf2 c (THeads (Flat Appl) vs (TLRef i))))))) -\def - \lambda (c: C).(\lambda (i: nat).(\lambda (H: (nf2 c (TLRef i))).(\lambda -(vs: TList).(TList_ind (\lambda (t: TList).((nfs2 c t) \to (nf2 c (THeads -(Flat Appl) t (TLRef i))))) (\lambda (_: True).H) (\lambda (t: T).(\lambda -(t0: TList).(\lambda (H0: (((nfs2 c t0) \to (nf2 c (THeads (Flat Appl) t0 -(TLRef i)))))).(\lambda (H1: (land (nf2 c t) (nfs2 c t0))).(let H2 \def H1 in -(land_ind (nf2 c t) (nfs2 c t0) (nf2 c (THead (Flat Appl) t (THeads (Flat -Appl) t0 (TLRef i)))) (\lambda (H3: (nf2 c t)).(\lambda (H4: (nfs2 c -t0)).(let H_y \def (H0 H4) in (\lambda (t2: T).(\lambda (H5: (pr2 c (THead -(Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) t2)).(let H6 \def -(pr2_gen_appl c t (THeads (Flat Appl) t0 (TLRef i)) t2 H5) in (or3_ind (ex3_2 -T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c t u2))) (\lambda (_: T).(\lambda (t3: -T).(pr2 c (THeads (Flat Appl) t0 (TLRef i)) t3)))) (ex4_4 T T T T (\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THeads (Flat -Appl) t0 (TLRef i)) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda -(_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 -t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: -T).(pr2 c t u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 -t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) -(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(eq T (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead -(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t -u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (eq T (THead (Flat Appl) t -(THeads (Flat Appl) t0 (TLRef i))) t2) (\lambda (H7: (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c t u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c -(THeads (Flat Appl) t0 (TLRef i)) t3))))).(ex3_2_ind T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c t u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c -(THeads (Flat Appl) t0 (TLRef i)) t3))) (eq T (THead (Flat Appl) t (THeads -(Flat Appl) t0 (TLRef i))) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda -(H8: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H9: (pr2 c t -x0)).(\lambda (H10: (pr2 c (THeads (Flat Appl) t0 (TLRef i)) x1)).(eq_ind_r T -(THead (Flat Appl) x0 x1) (\lambda (t1: T).(eq T (THead (Flat Appl) t (THeads -(Flat Appl) t0 (TLRef i))) t1)) (let H11 \def (eq_ind_r T x1 (\lambda (t1: -T).(pr2 c (THeads (Flat Appl) t0 (TLRef i)) t1)) H10 (THeads (Flat Appl) t0 -(TLRef i)) (H_y x1 H10)) in (eq_ind T (THeads (Flat Appl) t0 (TLRef i)) -(\lambda (t1: T).(eq T (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef -i))) (THead (Flat Appl) x0 t1))) (let H12 \def (eq_ind_r T x0 (\lambda (t1: -T).(pr2 c t t1)) H9 t (H3 x0 H9)) in (eq_ind T t (\lambda (t1: T).(eq T -(THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) (THead (Flat Appl) t1 -(THeads (Flat Appl) t0 (TLRef i))))) (refl_equal T (THead (Flat Appl) t -(THeads (Flat Appl) t0 (TLRef i)))) x0 (H3 x0 H9))) x1 (H_y x1 H10))) t2 -H8)))))) H7)) (\lambda (H7: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THeads (Flat Appl) t0 (TLRef i)) -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t u2))))) (\lambda -(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: -B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3))))))))).(ex4_4_ind T T T -T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THeads (Flat Appl) t0 (TLRef i)) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c t u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) -z1 t3))))))) (eq T (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) -t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: -T).(\lambda (H8: (eq T (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind Abst) -x0 x1))).(\lambda (H9: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (_: (pr2 -c t x2)).(\lambda (_: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) -u) x1 x3))))).(eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda (t1: T).(eq T -(THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) t1)) (TList_ind -(\lambda (t1: TList).((nf2 c (THeads (Flat Appl) t1 (TLRef i))) \to ((eq T -(THeads (Flat Appl) t1 (TLRef i)) (THead (Bind Abst) x0 x1)) \to (eq T (THead -(Flat Appl) t (THeads (Flat Appl) t1 (TLRef i))) (THead (Bind Abbr) x2 -x3))))) (\lambda (_: (nf2 c (THeads (Flat Appl) TNil (TLRef i)))).(\lambda -(H13: (eq T (THeads (Flat Appl) TNil (TLRef i)) (THead (Bind Abst) x0 -x1))).(let H14 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) x0 -x1) H13) in (False_ind (eq T (THead (Flat Appl) t (THeads (Flat Appl) TNil -(TLRef i))) (THead (Bind Abbr) x2 x3)) H14)))) (\lambda (t1: T).(\lambda (t3: -TList).(\lambda (_: (((nf2 c (THeads (Flat Appl) t3 (TLRef i))) \to ((eq T -(THeads (Flat Appl) t3 (TLRef i)) (THead (Bind Abst) x0 x1)) \to (eq T (THead -(Flat Appl) t (THeads (Flat Appl) t3 (TLRef i))) (THead (Bind Abbr) x2 -x3)))))).(\lambda (_: (nf2 c (THeads (Flat Appl) (TCons t1 t3) (TLRef -i)))).(\lambda (H13: (eq T (THeads (Flat Appl) (TCons t1 t3) (TLRef i)) -(THead (Bind Abst) x0 x1))).(let H14 \def (eq_ind T (THead (Flat Appl) t1 -(THeads (Flat Appl) t3 (TLRef i))) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind Abst) x0 x1) H13) in (False_ind (eq T (THead (Flat -Appl) t (THeads (Flat Appl) (TCons t1 t3) (TLRef i))) (THead (Bind Abbr) x2 -x3)) H14))))))) t0 H_y H8) t2 H9))))))))) H7)) (\lambda (H7: (ex6_6 B T T T T -T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq -T (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind b) y1 z1)))))))) (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: -T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S -O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) -y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B -b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THeads (Flat Appl) t0 (TLRef i)) -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c t u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) (eq T (THead -(Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) t2) (\lambda (x0: -B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: -T).(\lambda (x5: T).(\lambda (_: (not (eq B x0 Abst))).(\lambda (H9: (eq T -(THeads (Flat Appl) t0 (TLRef i)) (THead (Bind x0) x1 x2))).(\lambda (H10: -(eq T t2 (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) -x3)))).(\lambda (_: (pr2 c t x4)).(\lambda (_: (pr2 c x1 x5)).(\lambda (_: -(pr2 (CHead c (Bind x0) x5) x2 x3)).(eq_ind_r T (THead (Bind x0) x5 (THead -(Flat Appl) (lift (S O) O x4) x3)) (\lambda (t1: T).(eq T (THead (Flat Appl) -t (THeads (Flat Appl) t0 (TLRef i))) t1)) (TList_ind (\lambda (t1: -TList).((nf2 c (THeads (Flat Appl) t1 (TLRef i))) \to ((eq T (THeads (Flat -Appl) t1 (TLRef i)) (THead (Bind x0) x1 x2)) \to (eq T (THead (Flat Appl) t -(THeads (Flat Appl) t1 (TLRef i))) (THead (Bind x0) x5 (THead (Flat Appl) -(lift (S O) O x4) x3)))))) (\lambda (_: (nf2 c (THeads (Flat Appl) TNil -(TLRef i)))).(\lambda (H15: (eq T (THeads (Flat Appl) TNil (TLRef i)) (THead -(Bind x0) x1 x2))).(let H16 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match -ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | -(TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead -(Bind x0) x1 x2) H15) in (False_ind (eq T (THead (Flat Appl) t (THeads (Flat -Appl) TNil (TLRef i))) (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O -x4) x3))) H16)))) (\lambda (t1: T).(\lambda (t3: TList).(\lambda (_: (((nf2 c -(THeads (Flat Appl) t3 (TLRef i))) \to ((eq T (THeads (Flat Appl) t3 (TLRef -i)) (THead (Bind x0) x1 x2)) \to (eq T (THead (Flat Appl) t (THeads (Flat -Appl) t3 (TLRef i))) (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) -x3))))))).(\lambda (_: (nf2 c (THeads (Flat Appl) (TCons t1 t3) (TLRef -i)))).(\lambda (H15: (eq T (THeads (Flat Appl) (TCons t1 t3) (TLRef i)) -(THead (Bind x0) x1 x2))).(let H16 \def (eq_ind T (THead (Flat Appl) t1 -(THeads (Flat Appl) t3 (TLRef i))) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind x0) x1 x2) H15) in (False_ind (eq T (THead (Flat -Appl) t (THeads (Flat Appl) (TCons t1 t3) (TLRef i))) (THead (Bind x0) x5 -(THead (Flat Appl) (lift (S O) O x4) x3))) H16))))))) t0 H_y H9) t2 -H10))))))))))))) H7)) H6))))))) H2)))))) vs)))). -(* COMMENTS -Initial nodes: 2915 -END *) - -theorem nf2_appl_lref: - \forall (c: C).(\forall (u: T).((nf2 c u) \to (\forall (i: nat).((nf2 c -(TLRef i)) \to (nf2 c (THead (Flat Appl) u (TLRef i))))))) -\def - \lambda (c: C).(\lambda (u: T).(\lambda (H: (nf2 c u)).(\lambda (i: -nat).(\lambda (H0: (nf2 c (TLRef i))).(let H_y \def (nf2_appls_lref c i H0 -(TCons u TNil)) in (H_y (conj (nf2 c u) True H I))))))). -(* COMMENTS -Initial nodes: 49 -END *) - -theorem nf2_lref_abst: - \forall (c: C).(\forall (e: C).(\forall (u: T).(\forall (i: nat).((getl i c -(CHead e (Bind Abst) u)) \to (nf2 c (TLRef i)))))) -\def - \lambda (c: C).(\lambda (e: C).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H: (getl i c (CHead e (Bind Abst) u))).(\lambda (t2: T).(\lambda (H0: (pr2 c -(TLRef i) t2)).(let H1 \def (pr2_gen_lref c t2 i H0) in (or_ind (eq T t2 -(TLRef i)) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c (CHead d -(Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T t2 (lift (S i) O -u0))))) (eq T (TLRef i) t2) (\lambda (H2: (eq T t2 (TLRef i))).(eq_ind_r T -(TLRef i) (\lambda (t: T).(eq T (TLRef i) t)) (refl_equal T (TLRef i)) t2 -H2)) (\lambda (H2: (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c -(CHead d (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T t2 (lift -(S i) O u0)))))).(ex2_2_ind C T (\lambda (d: C).(\lambda (u0: T).(getl i c -(CHead d (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T t2 (lift -(S i) O u0)))) (eq T (TLRef i) t2) (\lambda (x0: C).(\lambda (x1: T).(\lambda -(H3: (getl i c (CHead x0 (Bind Abbr) x1))).(\lambda (H4: (eq T t2 (lift (S i) -O x1))).(eq_ind_r T (lift (S i) O x1) (\lambda (t: T).(eq T (TLRef i) t)) -(let H5 \def (eq_ind C (CHead e (Bind Abst) u) (\lambda (c0: C).(getl i c -c0)) H (CHead x0 (Bind Abbr) x1) (getl_mono c (CHead e (Bind Abst) u) i H -(CHead x0 (Bind Abbr) x1) H3)) in (let H6 \def (eq_ind C (CHead e (Bind Abst) -u) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort -_) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return -(\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B return -(\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow True | -Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead x0 (Bind -Abbr) x1) (getl_mono c (CHead e (Bind Abst) u) i H (CHead x0 (Bind Abbr) x1) -H3)) in (False_ind (eq T (TLRef i) (lift (S i) O x1)) H6))) t2 H4))))) H2)) -H1)))))))). -(* COMMENTS -Initial nodes: 494 -END *) - -theorem nf2_lift: - \forall (d: C).(\forall (t: T).((nf2 d t) \to (\forall (c: C).(\forall (h: -nat).(\forall (i: nat).((drop h i c d) \to (nf2 c (lift h i t)))))))) -\def - \lambda (d: C).(\lambda (t: T).(\lambda (H: ((\forall (t2: T).((pr2 d t t2) -\to (eq T t t2))))).(\lambda (c: C).(\lambda (h: nat).(\lambda (i: -nat).(\lambda (H0: (drop h i c d)).(\lambda (t2: T).(\lambda (H1: (pr2 c -(lift h i t) t2)).(let H2 \def (pr2_gen_lift c t t2 h i H1 d H0) in (ex2_ind -T (\lambda (t3: T).(eq T t2 (lift h i t3))) (\lambda (t3: T).(pr2 d t t3)) -(eq T (lift h i t) t2) (\lambda (x: T).(\lambda (H3: (eq T t2 (lift h i -x))).(\lambda (H4: (pr2 d t x)).(eq_ind_r T (lift h i x) (\lambda (t0: T).(eq -T (lift h i t) t0)) (let H_y \def (H x H4) in (let H5 \def (eq_ind_r T x -(\lambda (t0: T).(pr2 d t t0)) H4 t H_y) in (eq_ind T t (\lambda (t0: T).(eq -T (lift h i t) (lift h i t0))) (refl_equal T (lift h i t)) x H_y))) t2 H3)))) -H2)))))))))). -(* COMMENTS -Initial nodes: 245 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pc1/defs.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pc1/defs.ma deleted file mode 100644 index 80726a188..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pc1/defs.ma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr1/defs.ma". - -definition pc1: - T \to (T \to Prop) -\def - \lambda (t1: T).(\lambda (t2: T).(ex2 T (\lambda (t: T).(pr1 t1 t)) (\lambda -(t: T).(pr1 t2 t)))). - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pc1/props.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pc1/props.ma deleted file mode 100644 index 0da1a5aca..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pc1/props.ma +++ /dev/null @@ -1,146 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pc1/defs.ma". - -include "Basic-1/pr1/pr1.ma". - -theorem pc1_pr0_r: - \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (pc1 t1 t2))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(ex_intro2 T -(\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)) t2 (pr1_pr0 t1 t2 H) -(pr1_refl t2)))). -(* COMMENTS -Initial nodes: 43 -END *) - -theorem pc1_pr0_x: - \forall (t1: T).(\forall (t2: T).((pr0 t2 t1) \to (pc1 t1 t2))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t2 t1)).(ex_intro2 T -(\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)) t1 (pr1_refl t1) -(pr1_pr0 t2 t1 H)))). -(* COMMENTS -Initial nodes: 43 -END *) - -theorem pc1_refl: - \forall (t: T).(pc1 t t) -\def - \lambda (t: T).(ex_intro2 T (\lambda (t0: T).(pr1 t t0)) (\lambda (t0: -T).(pr1 t t0)) t (pr1_refl t) (pr1_refl t)). -(* COMMENTS -Initial nodes: 31 -END *) - -theorem pc1_pr0_u: - \forall (t2: T).(\forall (t1: T).((pr0 t1 t2) \to (\forall (t3: T).((pc1 t2 -t3) \to (pc1 t1 t3))))) -\def - \lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pr0 t1 t2)).(\lambda (t3: -T).(\lambda (H0: (pc1 t2 t3)).(let H1 \def H0 in (ex2_ind T (\lambda (t: -T).(pr1 t2 t)) (\lambda (t: T).(pr1 t3 t)) (pc1 t1 t3) (\lambda (x: -T).(\lambda (H2: (pr1 t2 x)).(\lambda (H3: (pr1 t3 x)).(ex_intro2 T (\lambda -(t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t3 t)) x (pr1_sing t2 t1 H x H2) -H3)))) H1)))))). -(* COMMENTS -Initial nodes: 97 -END *) - -theorem pc1_s: - \forall (t2: T).(\forall (t1: T).((pc1 t1 t2) \to (pc1 t2 t1))) -\def - \lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pc1 t1 t2)).(let H0 \def H in -(ex2_ind T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)) (pc1 t2 -t1) (\lambda (x: T).(\lambda (H1: (pr1 t1 x)).(\lambda (H2: (pr1 t2 -x)).(ex_intro2 T (\lambda (t: T).(pr1 t2 t)) (\lambda (t: T).(pr1 t1 t)) x H2 -H1)))) H0)))). -(* COMMENTS -Initial nodes: 79 -END *) - -theorem pc1_head_1: - \forall (u1: T).(\forall (u2: T).((pc1 u1 u2) \to (\forall (t: T).(\forall -(k: K).(pc1 (THead k u1 t) (THead k u2 t)))))) -\def - \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pc1 u1 u2)).(\lambda (t: -T).(\lambda (k: K).(let H0 \def H in (ex2_ind T (\lambda (t0: T).(pr1 u1 t0)) -(\lambda (t0: T).(pr1 u2 t0)) (pc1 (THead k u1 t) (THead k u2 t)) (\lambda -(x: T).(\lambda (H1: (pr1 u1 x)).(\lambda (H2: (pr1 u2 x)).(ex_intro2 T -(\lambda (t0: T).(pr1 (THead k u1 t) t0)) (\lambda (t0: T).(pr1 (THead k u2 -t) t0)) (THead k x t) (pr1_head_1 u1 x H1 t k) (pr1_head_1 u2 x H2 t k))))) -H0)))))). -(* COMMENTS -Initial nodes: 133 -END *) - -theorem pc1_head_2: - \forall (t1: T).(\forall (t2: T).((pc1 t1 t2) \to (\forall (u: T).(\forall -(k: K).(pc1 (THead k u t1) (THead k u t2)))))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc1 t1 t2)).(\lambda (u: -T).(\lambda (k: K).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr1 t1 t)) -(\lambda (t: T).(pr1 t2 t)) (pc1 (THead k u t1) (THead k u t2)) (\lambda (x: -T).(\lambda (H1: (pr1 t1 x)).(\lambda (H2: (pr1 t2 x)).(ex_intro2 T (\lambda -(t: T).(pr1 (THead k u t1) t)) (\lambda (t: T).(pr1 (THead k u t2) t)) (THead -k u x) (pr1_head_2 t1 x H1 u k) (pr1_head_2 t2 x H2 u k))))) H0)))))). -(* COMMENTS -Initial nodes: 133 -END *) - -theorem pc1_t: - \forall (t2: T).(\forall (t1: T).((pc1 t1 t2) \to (\forall (t3: T).((pc1 t2 -t3) \to (pc1 t1 t3))))) -\def - \lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pc1 t1 t2)).(\lambda (t3: -T).(\lambda (H0: (pc1 t2 t3)).(let H1 \def H0 in (ex2_ind T (\lambda (t: -T).(pr1 t2 t)) (\lambda (t: T).(pr1 t3 t)) (pc1 t1 t3) (\lambda (x: -T).(\lambda (H2: (pr1 t2 x)).(\lambda (H3: (pr1 t3 x)).(let H4 \def H in -(ex2_ind T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)) (pc1 t1 -t3) (\lambda (x0: T).(\lambda (H5: (pr1 t1 x0)).(\lambda (H6: (pr1 t2 -x0)).(ex2_ind T (\lambda (t: T).(pr1 x0 t)) (\lambda (t: T).(pr1 x t)) (pc1 -t1 t3) (\lambda (x1: T).(\lambda (H7: (pr1 x0 x1)).(\lambda (H8: (pr1 x -x1)).(ex_intro2 T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t3 t)) x1 -(pr1_t x0 t1 H5 x1 H7) (pr1_t x t3 H3 x1 H8))))) (pr1_confluence t2 x0 H6 x -H2))))) H4))))) H1)))))). -(* COMMENTS -Initial nodes: 203 -END *) - -theorem pc1_pr0_u2: - \forall (t0: T).(\forall (t1: T).((pr0 t0 t1) \to (\forall (t2: T).((pc1 t0 -t2) \to (pc1 t1 t2))))) -\def - \lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr0 t0 t1)).(\lambda (t2: -T).(\lambda (H0: (pc1 t0 t2)).(pc1_t t0 t1 (pc1_pr0_x t1 t0 H) t2 H0))))). -(* COMMENTS -Initial nodes: 35 -END *) - -theorem pc1_head: - \forall (u1: T).(\forall (u2: T).((pc1 u1 u2) \to (\forall (t1: T).(\forall -(t2: T).((pc1 t1 t2) \to (\forall (k: K).(pc1 (THead k u1 t1) (THead k u2 -t2)))))))) -\def - \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pc1 u1 u2)).(\lambda (t1: -T).(\lambda (t2: T).(\lambda (H0: (pc1 t1 t2)).(\lambda (k: K).(pc1_t (THead -k u2 t1) (THead k u1 t1) (pc1_head_1 u1 u2 H t1 k) (THead k u2 t2) -(pc1_head_2 t1 t2 H0 u2 k)))))))). -(* COMMENTS -Initial nodes: 71 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/dec.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/dec.ma deleted file mode 100644 index aa7a4d89e..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/dec.ma +++ /dev/null @@ -1,152 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/ty3/arity_props.ma". - -include "Basic-1/nf2/fwd.ma". - -theorem pc3_dec: - \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (t1: T).((ty3 g c -u1 t1) \to (\forall (u2: T).(\forall (t2: T).((ty3 g c u2 t2) \to (or (pc3 c -u1 u2) ((pc3 c u1 u2) \to False))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda -(H: (ty3 g c u1 t1)).(\lambda (u2: T).(\lambda (t2: T).(\lambda (H0: (ty3 g c -u2 t2)).(let H_y \def (ty3_sn3 g c u1 t1 H) in (let H_y0 \def (ty3_sn3 g c u2 -t2 H0) in (let H_x \def (nf2_sn3 c u1 H_y) in (let H1 \def H_x in (ex2_ind T -(\lambda (u: T).(pr3 c u1 u)) (\lambda (u: T).(nf2 c u)) (or (pc3 c u1 u2) -((pc3 c u1 u2) \to False)) (\lambda (x: T).(\lambda (H2: (pr3 c u1 -x)).(\lambda (H3: (nf2 c x)).(let H_x0 \def (nf2_sn3 c u2 H_y0) in (let H4 -\def H_x0 in (ex2_ind T (\lambda (u: T).(pr3 c u2 u)) (\lambda (u: T).(nf2 c -u)) (or (pc3 c u1 u2) ((pc3 c u1 u2) \to False)) (\lambda (x0: T).(\lambda -(H5: (pr3 c u2 x0)).(\lambda (H6: (nf2 c x0)).(let H_x1 \def (term_dec x x0) -in (let H7 \def H_x1 in (or_ind (eq T x x0) ((eq T x x0) \to (\forall (P: -Prop).P)) (or (pc3 c u1 u2) ((pc3 c u1 u2) \to False)) (\lambda (H8: (eq T x -x0)).(let H9 \def (eq_ind_r T x0 (\lambda (t: T).(nf2 c t)) H6 x H8) in (let -H10 \def (eq_ind_r T x0 (\lambda (t: T).(pr3 c u2 t)) H5 x H8) in (or_introl -(pc3 c u1 u2) ((pc3 c u1 u2) \to False) (pc3_pr3_t c u1 x H2 u2 H10))))) -(\lambda (H8: (((eq T x x0) \to (\forall (P: Prop).P)))).(or_intror (pc3 c u1 -u2) ((pc3 c u1 u2) \to False) (\lambda (H9: (pc3 c u1 u2)).(let H10 \def H9 -in (ex2_ind T (\lambda (t: T).(pr3 c u1 t)) (\lambda (t: T).(pr3 c u2 t)) -False (\lambda (x1: T).(\lambda (H11: (pr3 c u1 x1)).(\lambda (H12: (pr3 c u2 -x1)).(let H_x2 \def (pr3_confluence c u2 x0 H5 x1 H12) in (let H13 \def H_x2 -in (ex2_ind T (\lambda (t: T).(pr3 c x0 t)) (\lambda (t: T).(pr3 c x1 t)) -False (\lambda (x2: T).(\lambda (H14: (pr3 c x0 x2)).(\lambda (H15: (pr3 c x1 -x2)).(let H_y1 \def (nf2_pr3_unfold c x0 x2 H14 H6) in (let H16 \def -(eq_ind_r T x2 (\lambda (t: T).(pr3 c x1 t)) H15 x0 H_y1) in (let H17 \def -(nf2_pr3_confluence c x H3 x0 H6 u1 H2) in (H8 (H17 (pr3_t x1 u1 c H11 x0 -H16)) False))))))) H13)))))) H10))))) H7)))))) H4)))))) H1)))))))))))). -(* COMMENTS -Initial nodes: 551 -END *) - -theorem pc3_abst_dec: - \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (t1: T).((ty3 g c -u1 t1) \to (\forall (u2: T).(\forall (t2: T).((ty3 g c u2 t2) \to (or (ex4_2 -T T (\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst) u2 u)))) -(\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u) t1))) -(\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_: T).(\lambda -(v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind Abst) u2 u)) -\to False)))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda -(H: (ty3 g c u1 t1)).(\lambda (u2: T).(\lambda (t2: T).(\lambda (H0: (ty3 g c -u2 t2)).(let H1 \def (ty3_sn3 g c u1 t1 H) in (let H2 \def (ty3_sn3 g c u2 t2 -H0) in (let H_x \def (nf2_sn3 c u1 H1) in (let H3 \def H_x in (ex2_ind T -(\lambda (u: T).(pr3 c u1 u)) (\lambda (u: T).(nf2 c u)) (or (ex4_2 T T -(\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst) u2 u)))) -(\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u) t1))) -(\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_: T).(\lambda -(v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind Abst) u2 u)) -\to False))) (\lambda (x: T).(\lambda (H4: (pr3 c u1 x)).(\lambda (H5: (nf2 c -x)).(let H_x0 \def (nf2_sn3 c u2 H2) in (let H6 \def H_x0 in (ex2_ind T -(\lambda (u: T).(pr3 c u2 u)) (\lambda (u: T).(nf2 c u)) (or (ex4_2 T T -(\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst) u2 u)))) -(\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u) t1))) -(\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_: T).(\lambda -(v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind Abst) u2 u)) -\to False))) (\lambda (x0: T).(\lambda (H7: (pr3 c u2 x0)).(\lambda (H8: (nf2 -c x0)).(let H_x1 \def (abst_dec x x0) in (let H9 \def H_x1 in (or_ind (ex T -(\lambda (t: T).(eq T x (THead (Bind Abst) x0 t)))) (\forall (t: T).((eq T x -(THead (Bind Abst) x0 t)) \to (\forall (P: Prop).P))) (or (ex4_2 T T (\lambda -(u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst) u2 u)))) (\lambda (u: -T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u) t1))) (\lambda (_: -T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_: T).(\lambda (v2: T).(nf2 c -v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind Abst) u2 u)) \to False))) -(\lambda (H10: (ex T (\lambda (t: T).(eq T x (THead (Bind Abst) x0 -t))))).(ex_ind T (\lambda (t: T).(eq T x (THead (Bind Abst) x0 t))) (or -(ex4_2 T T (\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst) u2 -u)))) (\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u) -t1))) (\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_: -T).(\lambda (v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind -Abst) u2 u)) \to False))) (\lambda (x1: T).(\lambda (H11: (eq T x (THead -(Bind Abst) x0 x1))).(let H12 \def (eq_ind T x (\lambda (t: T).(nf2 c t)) H5 -(THead (Bind Abst) x0 x1) H11) in (let H13 \def (eq_ind T x (\lambda (t: -T).(pr3 c u1 t)) H4 (THead (Bind Abst) x0 x1) H11) in (let H_y \def -(ty3_sred_pr3 c u1 (THead (Bind Abst) x0 x1) H13 g t1 H) in (or_introl (ex4_2 -T T (\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst) u2 u)))) -(\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u) t1))) -(\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_: T).(\lambda -(v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind Abst) u2 u)) -\to False)) (ex4_2_intro T T (\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead -(Bind Abst) u2 u)))) (\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind -Abst) v2 u) t1))) (\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda -(_: T).(\lambda (v2: T).(nf2 c v2))) x1 x0 (pc3_pr3_t c u1 (THead (Bind Abst) -x0 x1) H13 (THead (Bind Abst) u2 x1) (pr3_head_12 c u2 x0 H7 (Bind Abst) x1 -x1 (pr3_refl (CHead c (Bind Abst) x0) x1))) H_y H7 H8))))))) H10)) (\lambda -(H10: ((\forall (t: T).((eq T x (THead (Bind Abst) x0 t)) \to (\forall (P: -Prop).P))))).(or_intror (ex4_2 T T (\lambda (u: T).(\lambda (_: T).(pc3 c u1 -(THead (Bind Abst) u2 u)))) (\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead -(Bind Abst) v2 u) t1))) (\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) -(\lambda (_: T).(\lambda (v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1 -(THead (Bind Abst) u2 u)) \to False)) (\lambda (u: T).(\lambda (H11: (pc3 c -u1 (THead (Bind Abst) u2 u))).(let H12 \def H11 in (ex2_ind T (\lambda (t: -T).(pr3 c u1 t)) (\lambda (t: T).(pr3 c (THead (Bind Abst) u2 u) t)) False -(\lambda (x1: T).(\lambda (H13: (pr3 c u1 x1)).(\lambda (H14: (pr3 c (THead -(Bind Abst) u2 u) x1)).(ex2_ind T (\lambda (t: T).(pr3 c x1 t)) (\lambda (t: -T).(pr3 c x t)) False (\lambda (x2: T).(\lambda (H15: (pr3 c x1 x2)).(\lambda -(H16: (pr3 c x x2)).(let H_y \def (nf2_pr3_unfold c x x2 H16 H5) in (let H17 -\def (eq_ind_r T x2 (\lambda (t: T).(pr3 c x1 t)) H15 x H_y) in (let H18 \def -(pr3_gen_abst c u2 u x1 H14) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: -T).(eq T x1 (THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: -T).(pr3 c u2 u3))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall -(u0: T).(pr3 (CHead c (Bind b) u0) u t3))))) False (\lambda (x3: T).(\lambda -(x4: T).(\lambda (H19: (eq T x1 (THead (Bind Abst) x3 x4))).(\lambda (H20: -(pr3 c u2 x3)).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pr3 (CHead c -(Bind b) u0) u x4))))).(let H22 \def (eq_ind T x1 (\lambda (t: T).(pr3 c t -x)) H17 (THead (Bind Abst) x3 x4) H19) in (let H23 \def (pr3_gen_abst c x3 x4 -x H22) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead -(Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c x3 u3))) -(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr3 (CHead -c (Bind b) u0) x4 t3))))) False (\lambda (x5: T).(\lambda (x6: T).(\lambda -(H24: (eq T x (THead (Bind Abst) x5 x6))).(\lambda (H25: (pr3 c x3 -x5)).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pr3 (CHead c (Bind b) -u0) x4 x6))))).(let H27 \def (eq_ind T x (\lambda (t: T).(\forall (t0: -T).((eq T t (THead (Bind Abst) x0 t0)) \to (\forall (P: Prop).P)))) H10 -(THead (Bind Abst) x5 x6) H24) in (let H28 \def (eq_ind T x (\lambda (t: -T).(nf2 c t)) H5 (THead (Bind Abst) x5 x6) H24) in (let H29 \def -(nf2_gen_abst c x5 x6 H28) in (land_ind (nf2 c x5) (nf2 (CHead c (Bind Abst) -x5) x6) False (\lambda (H30: (nf2 c x5)).(\lambda (_: (nf2 (CHead c (Bind -Abst) x5) x6)).(let H32 \def (nf2_pr3_confluence c x0 H8 x5 H30 u2 H7) in -(H27 x6 (sym_eq T (THead (Bind Abst) x0 x6) (THead (Bind Abst) x5 x6) -(f_equal3 K T T T THead (Bind Abst) (Bind Abst) x0 x5 x6 x6 (refl_equal K -(Bind Abst)) (H32 (pr3_t x3 u2 c H20 x5 H25)) (refl_equal T x6))) False)))) -H29))))))))) H23)))))))) H18))))))) (pr3_confluence c u1 x1 H13 x H4))))) -H12)))))) H9)))))) H6)))))) H3)))))))))))). -(* COMMENTS -Initial nodes: 1759 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/defs.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/defs.ma deleted file mode 100644 index e7ea2b24f..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/defs.ma +++ /dev/null @@ -1,31 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr3/defs.ma". - -definition pc3: - C \to (T \to (T \to Prop)) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(ex2 T (\lambda (t: T).(pr3 -c t1 t)) (\lambda (t: T).(pr3 c t2 t))))). - -inductive pc3_left (c: C): T \to (T \to Prop) \def -| pc3_left_r: \forall (t: T).(pc3_left c t t) -| pc3_left_ur: \forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall -(t3: T).((pc3_left c t2 t3) \to (pc3_left c t1 t3))))) -| pc3_left_ux: \forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall -(t3: T).((pc3_left c t1 t3) \to (pc3_left c t2 t3))))). - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/fsubst0.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/fsubst0.ma deleted file mode 100644 index c563ca397..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/fsubst0.ma +++ /dev/null @@ -1,726 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pc3/left.ma". - -include "Basic-1/fsubst0/defs.ma". - -include "Basic-1/csubst0/getl.ma". - -theorem pc3_pr2_fsubst0: - \forall (c1: C).(\forall (t1: T).(\forall (t: T).((pr2 c1 t1 t) \to (\forall -(i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t2: T).((fsubst0 i u c1 -t1 c2 t2) \to (\forall (e: C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3 -c2 t2 t))))))))))) -\def - \lambda (c1: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H: (pr2 c1 t1 -t)).(pr2_ind (\lambda (c: C).(\lambda (t0: T).(\lambda (t2: T).(\forall (i: -nat).(\forall (u: T).(\forall (c2: C).(\forall (t3: T).((fsubst0 i u c t0 c2 -t3) \to (\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) \to (pc3 c2 t3 -t2))))))))))) (\lambda (c: C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H0: -(pr0 t2 t3)).(\lambda (i: nat).(\lambda (u: T).(\lambda (c2: C).(\lambda (t0: -T).(\lambda (H1: (fsubst0 i u c t2 c2 t0)).(fsubst0_ind i u c t2 (\lambda -(c0: C).(\lambda (t4: T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) -\to (pc3 c0 t4 t3))))) (\lambda (t4: T).(\lambda (H2: (subst0 i u t2 -t4)).(\lambda (e: C).(\lambda (H3: (getl i c (CHead e (Bind Abbr) -u))).(or_ind (pr0 t4 t3) (ex2 T (\lambda (w2: T).(pr0 t4 w2)) (\lambda (w2: -T).(subst0 i u t3 w2))) (pc3 c t4 t3) (\lambda (H4: (pr0 t4 t3)).(pc3_pr2_r c -t4 t3 (pr2_free c t4 t3 H4))) (\lambda (H4: (ex2 T (\lambda (w2: T).(pr0 t4 -w2)) (\lambda (w2: T).(subst0 i u t3 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 -t4 w2)) (\lambda (w2: T).(subst0 i u t3 w2)) (pc3 c t4 t3) (\lambda (x: -T).(\lambda (H5: (pr0 t4 x)).(\lambda (H6: (subst0 i u t3 x)).(pc3_pr2_u c x -t4 (pr2_free c t4 x H5) t3 (pc3_pr2_x c x t3 (pr2_delta c e u i H3 t3 t3 -(pr0_refl t3) x H6)))))) H4)) (pr0_subst0 t2 t3 H0 u t4 i H2 u (pr0_refl -u))))))) (\lambda (c0: C).(\lambda (_: (csubst0 i u c c0)).(\lambda (e: -C).(\lambda (_: (getl i c (CHead e (Bind Abbr) u))).(pc3_pr2_r c0 t2 t3 -(pr2_free c0 t2 t3 H0)))))) (\lambda (t4: T).(\lambda (H2: (subst0 i u t2 -t4)).(\lambda (c0: C).(\lambda (H3: (csubst0 i u c c0)).(\lambda (e: -C).(\lambda (H4: (getl i c (CHead e (Bind Abbr) u))).(or_ind (pr0 t4 t3) (ex2 -T (\lambda (w2: T).(pr0 t4 w2)) (\lambda (w2: T).(subst0 i u t3 w2))) (pc3 c0 -t4 t3) (\lambda (H5: (pr0 t4 t3)).(pc3_pr2_r c0 t4 t3 (pr2_free c0 t4 t3 -H5))) (\lambda (H5: (ex2 T (\lambda (w2: T).(pr0 t4 w2)) (\lambda (w2: -T).(subst0 i u t3 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t4 w2)) (\lambda -(w2: T).(subst0 i u t3 w2)) (pc3 c0 t4 t3) (\lambda (x: T).(\lambda (H6: (pr0 -t4 x)).(\lambda (H7: (subst0 i u t3 x)).(pc3_pr2_u c0 x t4 (pr2_free c0 t4 x -H6) t3 (pc3_pr2_x c0 x t3 (pr2_delta c0 e u i (csubst0_getl_ge i i (le_n i) c -c0 u H3 (CHead e (Bind Abbr) u) H4) t3 t3 (pr0_refl t3) x H7)))))) H5)) -(pr0_subst0 t2 t3 H0 u t4 i H2 u (pr0_refl u))))))))) c2 t0 H1)))))))))) -(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (t2: T).(\lambda (t3: -T).(\lambda (H1: (pr0 t2 t3)).(\lambda (t0: T).(\lambda (H2: (subst0 i u t3 -t0)).(\lambda (i0: nat).(\lambda (u0: T).(\lambda (c2: C).(\lambda (t4: -T).(\lambda (H3: (fsubst0 i0 u0 c t2 c2 t4)).(fsubst0_ind i0 u0 c t2 (\lambda -(c0: C).(\lambda (t5: T).(\forall (e: C).((getl i0 c (CHead e (Bind Abbr) -u0)) \to (pc3 c0 t5 t0))))) (\lambda (t5: T).(\lambda (H4: (subst0 i0 u0 t2 -t5)).(\lambda (e: C).(\lambda (H5: (getl i0 c (CHead e (Bind Abbr) -u0))).(pc3_t t2 c t5 (pc3_s c t5 t2 (pc3_pr2_r c t2 t5 (pr2_delta c e u0 i0 -H5 t2 t2 (pr0_refl t2) t5 H4))) t0 (pc3_pr2_r c t2 t0 (pr2_delta c d u i H0 -t2 t3 H1 t0 H2))))))) (\lambda (c0: C).(\lambda (H4: (csubst0 i0 u0 c -c0)).(\lambda (e: C).(\lambda (H5: (getl i0 c (CHead e (Bind Abbr) -u0))).(lt_le_e i i0 (pc3 c0 t2 t0) (\lambda (H6: (lt i i0)).(let H7 \def -(csubst0_getl_lt i0 i H6 c c0 u0 H4 (CHead d (Bind Abbr) u) H0) in (or4_ind -(getl i c0 (CHead d (Bind Abbr) u)) (ex3_4 B C T T (\lambda (b: B).(\lambda -(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead -e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) -u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c0 -(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) (ex4_5 B C C T T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda -(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl -i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))) (pc3 c0 t2 t0) (\lambda (H8: -(getl i c0 (CHead d (Bind Abbr) u))).(pc3_pr2_r c0 t2 t0 (pr2_delta c0 d u i -H8 t2 t3 H1 t0 H2))) (\lambda (H8: (ex3_4 B C T T (\lambda (b: B).(\lambda -(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead -e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) -u0 u1 w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda -(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b) -u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w))))) -(pc3 c0 t2 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda -(x3: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) -x2))).(\lambda (H10: (getl i c0 (CHead x1 (Bind x0) x3))).(\lambda (H11: -(subst0 (minus i0 (S i)) u0 x2 x3)).(let H12 \def (f_equal C C (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | -(CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) -x2) H9) in ((let H13 \def (f_equal C B (\lambda (e0: C).(match e0 in C return -(\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) -\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead -x1 (Bind x0) x2) H9) in ((let H14 \def (f_equal C T (\lambda (e0: C).(match -e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ -t5) \Rightarrow t5])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H9) in -(\lambda (H15: (eq B Abbr x0)).(\lambda (H16: (eq C d x1)).(let H17 \def -(eq_ind_r T x2 (\lambda (t5: T).(subst0 (minus i0 (S i)) u0 t5 x3)) H11 u -H14) in (let H18 \def (eq_ind_r C x1 (\lambda (c3: C).(getl i c0 (CHead c3 -(Bind x0) x3))) H10 d H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b: -B).(getl i c0 (CHead d (Bind b) x3))) H18 Abbr H15) in (ex2_ind T (\lambda -(t5: T).(subst0 i x3 t3 t5)) (\lambda (t5: T).(subst0 (S (plus (minus i0 (S -i)) i)) u0 t0 t5)) (pc3 c0 t2 t0) (\lambda (x: T).(\lambda (H20: (subst0 i x3 -t3 x)).(\lambda (H21: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let -H22 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n: -nat).(subst0 n u0 t0 x)) H21 i0 (lt_plus_minus_r i i0 H6)) in (pc3_pr2_u c0 x -t2 (pr2_delta c0 d x3 i H19 t2 t3 H1 x H20) t0 (pc3_pr2_x c0 x t0 (pr2_delta -c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H4 (CHead e (Bind Abbr) -u0) H5) t0 t0 (pr0_refl t0) x H22))))))) (subst0_subst0_back t3 t0 u i H2 x3 -u0 (minus i0 (S i)) H17)))))))) H13)) H12))))))))) H8)) (\lambda (H8: (ex3_4 -B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq -C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 -(Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))).(ex3_4_ind B C C T (\lambda -(b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind -Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 (Bind b) u1)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S -i)) u0 e1 e2))))) (pc3 c0 t2 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda -(x2: C).(\lambda (x3: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead -x1 (Bind x0) x3))).(\lambda (H10: (getl i c0 (CHead x2 (Bind x0) -x3))).(\lambda (H11: (csubst0 (minus i0 (S i)) u0 x1 x2)).(let H12 \def -(f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with -[(CSort _) \Rightarrow d | (CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind -Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H13 \def (f_equal C B (\lambda -(e0: C).(match e0 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow -Abbr | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with -[(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) -u) (CHead x1 (Bind x0) x3) H9) in ((let H14 \def (f_equal C T (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | -(CHead _ _ t5) \Rightarrow t5])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) -x3) H9) in (\lambda (H15: (eq B Abbr x0)).(\lambda (H16: (eq C d x1)).(let -H17 \def (eq_ind_r T x3 (\lambda (t5: T).(getl i c0 (CHead x2 (Bind x0) t5))) -H10 u H14) in (let H18 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus -i0 (S i)) u0 c3 x2)) H11 d H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b: -B).(getl i c0 (CHead x2 (Bind b) u))) H17 Abbr H15) in (pc3_pr2_r c0 t2 t0 -(pr2_delta c0 x2 u i H19 t2 t3 H1 t0 H2)))))))) H13)) H12))))))))) H8)) -(\lambda (H8: (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 -(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 -(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 -e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 -(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 -(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) -(pc3 c0 t2 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda -(x3: T).(\lambda (x4: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead -x1 (Bind x0) x3))).(\lambda (H10: (getl i c0 (CHead x2 (Bind x0) -x4))).(\lambda (H11: (subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H12: -(csubst0 (minus i0 (S i)) u0 x1 x2)).(let H13 \def (f_equal C C (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | -(CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) -x3) H9) in ((let H14 \def (f_equal C B (\lambda (e0: C).(match e0 in C return -(\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) -\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead -x1 (Bind x0) x3) H9) in ((let H15 \def (f_equal C T (\lambda (e0: C).(match -e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ -t5) \Rightarrow t5])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in -(\lambda (H16: (eq B Abbr x0)).(\lambda (H17: (eq C d x1)).(let H18 \def -(eq_ind_r T x3 (\lambda (t5: T).(subst0 (minus i0 (S i)) u0 t5 x4)) H11 u -H15) in (let H19 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus i0 (S -i)) u0 c3 x2)) H12 d H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b: -B).(getl i c0 (CHead x2 (Bind b) x4))) H10 Abbr H16) in (ex2_ind T (\lambda -(t5: T).(subst0 i x4 t3 t5)) (\lambda (t5: T).(subst0 (S (plus (minus i0 (S -i)) i)) u0 t0 t5)) (pc3 c0 t2 t0) (\lambda (x: T).(\lambda (H21: (subst0 i x4 -t3 x)).(\lambda (H22: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let -H23 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n: -nat).(subst0 n u0 t0 x)) H22 i0 (lt_plus_minus_r i i0 H6)) in (pc3_pr2_u c0 x -t2 (pr2_delta c0 x2 x4 i H20 t2 t3 H1 x H21) t0 (pc3_pr2_x c0 x t0 (pr2_delta -c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H4 (CHead e (Bind Abbr) -u0) H5) t0 t0 (pr0_refl t0) x H23))))))) (subst0_subst0_back t3 t0 u i H2 x4 -u0 (minus i0 (S i)) H18)))))))) H14)) H13))))))))))) H8)) H7))) (\lambda (H6: -(le i0 i)).(pc3_pr2_r c0 t2 t0 (pr2_delta c0 d u i (csubst0_getl_ge i0 i H6 c -c0 u0 H4 (CHead d (Bind Abbr) u) H0) t2 t3 H1 t0 H2)))))))) (\lambda (t5: -T).(\lambda (H4: (subst0 i0 u0 t2 t5)).(\lambda (c0: C).(\lambda (H5: -(csubst0 i0 u0 c c0)).(\lambda (e: C).(\lambda (H6: (getl i0 c (CHead e (Bind -Abbr) u0))).(lt_le_e i i0 (pc3 c0 t5 t0) (\lambda (H7: (lt i i0)).(let H8 -\def (csubst0_getl_lt i0 i H7 c c0 u0 H5 (CHead d (Bind Abbr) u) H0) in -(or4_ind (getl i c0 (CHead d (Bind Abbr) u)) (ex3_4 B C T T (\lambda (b: -B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind -Abbr) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 -(minus i0 (S i)) u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 -(Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(u1: T).(getl i c0 (CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) -(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) -u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 -(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))) -(pc3 c0 t5 t0) (\lambda (H9: (getl i c0 (CHead d (Bind Abbr) u))).(pc3_pr2_u2 -c0 t2 t5 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 -(CHead e (Bind Abbr) u0) H6) t2 t2 (pr0_refl t2) t5 H4) t0 (pc3_pr2_r c0 t2 -t0 (pr2_delta c0 d u i H9 t2 t3 H1 t0 H2)))) (\lambda (H9: (ex3_4 B C T T -(\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C -(CHead d (Bind Abbr) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda -(e0: C).(\lambda (_: T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 -(minus i0 (S i)) u0 u1 w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda -(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead -e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) -u0 u1 w))))) (pc3 c0 t5 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: -T).(\lambda (x3: T).(\lambda (H10: (eq C (CHead d (Bind Abbr) u) (CHead x1 -(Bind x0) x2))).(\lambda (H11: (getl i c0 (CHead x1 (Bind x0) x3))).(\lambda -(H12: (subst0 (minus i0 (S i)) u0 x2 x3)).(let H13 \def (f_equal C C (\lambda -(e0: C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow -d | (CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind -x0) x2) H10) in ((let H14 \def (f_equal C B (\lambda (e0: C).(match e0 in C -return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) -\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead -x1 (Bind x0) x2) H10) in ((let H15 \def (f_equal C T (\lambda (e0: C).(match -e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ -t6) \Rightarrow t6])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H10) in -(\lambda (H16: (eq B Abbr x0)).(\lambda (H17: (eq C d x1)).(let H18 \def -(eq_ind_r T x2 (\lambda (t6: T).(subst0 (minus i0 (S i)) u0 t6 x3)) H12 u -H15) in (let H19 \def (eq_ind_r C x1 (\lambda (c3: C).(getl i c0 (CHead c3 -(Bind x0) x3))) H11 d H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b: -B).(getl i c0 (CHead d (Bind b) x3))) H19 Abbr H16) in (ex2_ind T (\lambda -(t6: T).(subst0 i x3 t3 t6)) (\lambda (t6: T).(subst0 (S (plus (minus i0 (S -i)) i)) u0 t0 t6)) (pc3 c0 t5 t0) (\lambda (x: T).(\lambda (H21: (subst0 i x3 -t3 x)).(\lambda (H22: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let -H23 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n: -nat).(subst0 n u0 t0 x)) H22 i0 (lt_plus_minus_r i i0 H7)) in (pc3_pr2_u2 c0 -t2 t5 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 -(CHead e (Bind Abbr) u0) H6) t2 t2 (pr0_refl t2) t5 H4) t0 (pc3_pr2_u c0 x t2 -(pr2_delta c0 d x3 i H20 t2 t3 H1 x H21) t0 (pc3_pr2_x c0 x t0 (pr2_delta c0 -e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e (Bind Abbr) u0) -H6) t0 t0 (pr0_refl t0) x H23)))))))) (subst0_subst0_back t3 t0 u i H2 x3 u0 -(minus i0 (S i)) H18)))))))) H14)) H13))))))))) H9)) (\lambda (H9: (ex3_4 B C -C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C -(CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda -(_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 (Bind b) -u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))).(ex3_4_ind B C C T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind -Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 (Bind b) u1)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S -i)) u0 e1 e2))))) (pc3 c0 t5 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda -(x2: C).(\lambda (x3: T).(\lambda (H10: (eq C (CHead d (Bind Abbr) u) (CHead -x1 (Bind x0) x3))).(\lambda (H11: (getl i c0 (CHead x2 (Bind x0) -x3))).(\lambda (H12: (csubst0 (minus i0 (S i)) u0 x1 x2)).(let H13 \def -(f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with -[(CSort _) \Rightarrow d | (CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind -Abbr) u) (CHead x1 (Bind x0) x3) H10) in ((let H14 \def (f_equal C B (\lambda -(e0: C).(match e0 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow -Abbr | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with -[(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) -u) (CHead x1 (Bind x0) x3) H10) in ((let H15 \def (f_equal C T (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | -(CHead _ _ t6) \Rightarrow t6])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) -x3) H10) in (\lambda (H16: (eq B Abbr x0)).(\lambda (H17: (eq C d x1)).(let -H18 \def (eq_ind_r T x3 (\lambda (t6: T).(getl i c0 (CHead x2 (Bind x0) t6))) -H11 u H15) in (let H19 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus -i0 (S i)) u0 c3 x2)) H12 d H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b: -B).(getl i c0 (CHead x2 (Bind b) u))) H18 Abbr H16) in (pc3_pr2_u2 c0 t2 t5 -(pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e -(Bind Abbr) u0) H6) t2 t2 (pr0_refl t2) t5 H4) t0 (pc3_pr2_r c0 t2 t0 -(pr2_delta c0 x2 u i H20 t2 t3 H1 t0 H2))))))))) H14)) H13))))))))) H9)) -(\lambda (H9: (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 -(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 -(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 -e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 -(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 -(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) -(pc3 c0 t5 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda -(x3: T).(\lambda (x4: T).(\lambda (H10: (eq C (CHead d (Bind Abbr) u) (CHead -x1 (Bind x0) x3))).(\lambda (H11: (getl i c0 (CHead x2 (Bind x0) -x4))).(\lambda (H12: (subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H13: -(csubst0 (minus i0 (S i)) u0 x1 x2)).(let H14 \def (f_equal C C (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | -(CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) -x3) H10) in ((let H15 \def (f_equal C B (\lambda (e0: C).(match e0 in C -return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) -\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead -x1 (Bind x0) x3) H10) in ((let H16 \def (f_equal C T (\lambda (e0: C).(match -e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ -t6) \Rightarrow t6])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H10) in -(\lambda (H17: (eq B Abbr x0)).(\lambda (H18: (eq C d x1)).(let H19 \def -(eq_ind_r T x3 (\lambda (t6: T).(subst0 (minus i0 (S i)) u0 t6 x4)) H12 u -H16) in (let H20 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus i0 (S -i)) u0 c3 x2)) H13 d H18) in (let H21 \def (eq_ind_r B x0 (\lambda (b: -B).(getl i c0 (CHead x2 (Bind b) x4))) H11 Abbr H17) in (ex2_ind T (\lambda -(t6: T).(subst0 i x4 t3 t6)) (\lambda (t6: T).(subst0 (S (plus (minus i0 (S -i)) i)) u0 t0 t6)) (pc3 c0 t5 t0) (\lambda (x: T).(\lambda (H22: (subst0 i x4 -t3 x)).(\lambda (H23: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let -H24 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n: -nat).(subst0 n u0 t0 x)) H23 i0 (lt_plus_minus_r i i0 H7)) in (pc3_pr2_u2 c0 -t2 t5 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 -(CHead e (Bind Abbr) u0) H6) t2 t2 (pr0_refl t2) t5 H4) t0 (pc3_pr2_u c0 x t2 -(pr2_delta c0 x2 x4 i H21 t2 t3 H1 x H22) t0 (pc3_pr2_x c0 x t0 (pr2_delta c0 -e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e (Bind Abbr) u0) -H6) t0 t0 (pr0_refl t0) x H24)))))))) (subst0_subst0_back t3 t0 u i H2 x4 u0 -(minus i0 (S i)) H19)))))))) H15)) H14))))))))))) H9)) H8))) (\lambda (H7: -(le i0 i)).(pc3_pr2_u2 c0 t2 t5 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 -(le_n i0) c c0 u0 H5 (CHead e (Bind Abbr) u0) H6) t2 t2 (pr0_refl t2) t5 H4) -t0 (pc3_pr2_r c0 t2 t0 (pr2_delta c0 d u i (csubst0_getl_ge i0 i H7 c c0 u0 -H5 (CHead d (Bind Abbr) u) H0) t2 t3 H1 t0 H2))))))))))) c2 t4 -H3)))))))))))))))) c1 t1 t H)))). -(* COMMENTS -Initial nodes: 6455 -END *) - -theorem pc3_pr2_fsubst0_back: - \forall (c1: C).(\forall (t: T).(\forall (t1: T).((pr2 c1 t t1) \to (\forall -(i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t2: T).((fsubst0 i u c1 -t1 c2 t2) \to (\forall (e: C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3 -c2 t t2))))))))))) -\def - \lambda (c1: C).(\lambda (t: T).(\lambda (t1: T).(\lambda (H: (pr2 c1 t -t1)).(pr2_ind (\lambda (c: C).(\lambda (t0: T).(\lambda (t2: T).(\forall (i: -nat).(\forall (u: T).(\forall (c2: C).(\forall (t3: T).((fsubst0 i u c t2 c2 -t3) \to (\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) \to (pc3 c2 t0 -t3))))))))))) (\lambda (c: C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H0: -(pr0 t2 t3)).(\lambda (i: nat).(\lambda (u: T).(\lambda (c2: C).(\lambda (t0: -T).(\lambda (H1: (fsubst0 i u c t3 c2 t0)).(fsubst0_ind i u c t3 (\lambda -(c0: C).(\lambda (t4: T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) -\to (pc3 c0 t2 t4))))) (\lambda (t4: T).(\lambda (H2: (subst0 i u t3 -t4)).(\lambda (e: C).(\lambda (H3: (getl i c (CHead e (Bind Abbr) -u))).(pc3_pr2_u c t3 t2 (pr2_free c t2 t3 H0) t4 (pc3_pr2_r c t3 t4 -(pr2_delta c e u i H3 t3 t3 (pr0_refl t3) t4 H2))))))) (\lambda (c0: -C).(\lambda (_: (csubst0 i u c c0)).(\lambda (e: C).(\lambda (_: (getl i c -(CHead e (Bind Abbr) u))).(pc3_pr2_r c0 t2 t3 (pr2_free c0 t2 t3 H0)))))) -(\lambda (t4: T).(\lambda (H2: (subst0 i u t3 t4)).(\lambda (c0: C).(\lambda -(H3: (csubst0 i u c c0)).(\lambda (e: C).(\lambda (H4: (getl i c (CHead e -(Bind Abbr) u))).(pc3_pr2_u c0 t3 t2 (pr2_free c0 t2 t3 H0) t4 (pc3_pr2_r c0 -t3 t4 (pr2_delta c0 e u i (csubst0_getl_ge i i (le_n i) c c0 u H3 (CHead e -(Bind Abbr) u) H4) t3 t3 (pr0_refl t3) t4 H2))))))))) c2 t0 H1)))))))))) -(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (t2: T).(\lambda (t3: -T).(\lambda (H1: (pr0 t2 t3)).(\lambda (t0: T).(\lambda (H2: (subst0 i u t3 -t0)).(\lambda (i0: nat).(\lambda (u0: T).(\lambda (c2: C).(\lambda (t4: -T).(\lambda (H3: (fsubst0 i0 u0 c t0 c2 t4)).(fsubst0_ind i0 u0 c t0 (\lambda -(c0: C).(\lambda (t5: T).(\forall (e: C).((getl i0 c (CHead e (Bind Abbr) -u0)) \to (pc3 c0 t2 t5))))) (\lambda (t5: T).(\lambda (H4: (subst0 i0 u0 t0 -t5)).(\lambda (e: C).(\lambda (H5: (getl i0 c (CHead e (Bind Abbr) -u0))).(pc3_t t3 c t2 (pc3_pr3_r c t2 t3 (pr3_pr2 c t2 t3 (pr2_free c t2 t3 -H1))) t5 (pc3_pr3_r c t3 t5 (pr3_sing c t0 t3 (pr2_delta c d u i H0 t3 t3 -(pr0_refl t3) t0 H2) t5 (pr3_pr2 c t0 t5 (pr2_delta c e u0 i0 H5 t0 t0 -(pr0_refl t0) t5 H4))))))))) (\lambda (c0: C).(\lambda (H4: (csubst0 i0 u0 c -c0)).(\lambda (e: C).(\lambda (H5: (getl i0 c (CHead e (Bind Abbr) -u0))).(lt_le_e i i0 (pc3 c0 t2 t0) (\lambda (H6: (lt i i0)).(let H7 \def -(csubst0_getl_lt i0 i H6 c c0 u0 H4 (CHead d (Bind Abbr) u) H0) in (or4_ind -(getl i c0 (CHead d (Bind Abbr) u)) (ex3_4 B C T T (\lambda (b: B).(\lambda -(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead -e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) -u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c0 -(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) (ex4_5 B C C T T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda -(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl -i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))) (pc3 c0 t2 t0) (\lambda (H8: -(getl i c0 (CHead d (Bind Abbr) u))).(pc3_pr2_r c0 t2 t0 (pr2_delta c0 d u i -H8 t2 t3 H1 t0 H2))) (\lambda (H8: (ex3_4 B C T T (\lambda (b: B).(\lambda -(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead -e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) -u0 u1 w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda -(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b) -u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w))))) -(pc3 c0 t2 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda -(x3: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) -x2))).(\lambda (H10: (getl i c0 (CHead x1 (Bind x0) x3))).(\lambda (H11: -(subst0 (minus i0 (S i)) u0 x2 x3)).(let H12 \def (f_equal C C (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | -(CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) -x2) H9) in ((let H13 \def (f_equal C B (\lambda (e0: C).(match e0 in C return -(\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) -\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead -x1 (Bind x0) x2) H9) in ((let H14 \def (f_equal C T (\lambda (e0: C).(match -e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ -t5) \Rightarrow t5])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H9) in -(\lambda (H15: (eq B Abbr x0)).(\lambda (H16: (eq C d x1)).(let H17 \def -(eq_ind_r T x2 (\lambda (t5: T).(subst0 (minus i0 (S i)) u0 t5 x3)) H11 u -H14) in (let H18 \def (eq_ind_r C x1 (\lambda (c3: C).(getl i c0 (CHead c3 -(Bind x0) x3))) H10 d H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b: -B).(getl i c0 (CHead d (Bind b) x3))) H18 Abbr H15) in (ex2_ind T (\lambda -(t5: T).(subst0 i x3 t3 t5)) (\lambda (t5: T).(subst0 (S (plus (minus i0 (S -i)) i)) u0 t0 t5)) (pc3 c0 t2 t0) (\lambda (x: T).(\lambda (H20: (subst0 i x3 -t3 x)).(\lambda (H21: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let -H22 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n: -nat).(subst0 n u0 t0 x)) H21 i0 (lt_plus_minus_r i i0 H6)) in (pc3_pr2_u c0 x -t2 (pr2_delta c0 d x3 i H19 t2 t3 H1 x H20) t0 (pc3_pr2_x c0 x t0 (pr2_delta -c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H4 (CHead e (Bind Abbr) -u0) H5) t0 t0 (pr0_refl t0) x H22))))))) (subst0_subst0_back t3 t0 u i H2 x3 -u0 (minus i0 (S i)) H17)))))))) H13)) H12))))))))) H8)) (\lambda (H8: (ex3_4 -B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq -C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 -(Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))).(ex3_4_ind B C C T (\lambda -(b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind -Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 (Bind b) u1)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S -i)) u0 e1 e2))))) (pc3 c0 t2 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda -(x2: C).(\lambda (x3: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead -x1 (Bind x0) x3))).(\lambda (H10: (getl i c0 (CHead x2 (Bind x0) -x3))).(\lambda (H11: (csubst0 (minus i0 (S i)) u0 x1 x2)).(let H12 \def -(f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with -[(CSort _) \Rightarrow d | (CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind -Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H13 \def (f_equal C B (\lambda -(e0: C).(match e0 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow -Abbr | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with -[(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) -u) (CHead x1 (Bind x0) x3) H9) in ((let H14 \def (f_equal C T (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | -(CHead _ _ t5) \Rightarrow t5])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) -x3) H9) in (\lambda (H15: (eq B Abbr x0)).(\lambda (H16: (eq C d x1)).(let -H17 \def (eq_ind_r T x3 (\lambda (t5: T).(getl i c0 (CHead x2 (Bind x0) t5))) -H10 u H14) in (let H18 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus -i0 (S i)) u0 c3 x2)) H11 d H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b: -B).(getl i c0 (CHead x2 (Bind b) u))) H17 Abbr H15) in (pc3_pr2_r c0 t2 t0 -(pr2_delta c0 x2 u i H19 t2 t3 H1 t0 H2)))))))) H13)) H12))))))))) H8)) -(\lambda (H8: (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 -(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 -(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 -e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 -(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 -(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) -(pc3 c0 t2 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda -(x3: T).(\lambda (x4: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead -x1 (Bind x0) x3))).(\lambda (H10: (getl i c0 (CHead x2 (Bind x0) -x4))).(\lambda (H11: (subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H12: -(csubst0 (minus i0 (S i)) u0 x1 x2)).(let H13 \def (f_equal C C (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | -(CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) -x3) H9) in ((let H14 \def (f_equal C B (\lambda (e0: C).(match e0 in C return -(\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) -\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead -x1 (Bind x0) x3) H9) in ((let H15 \def (f_equal C T (\lambda (e0: C).(match -e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ -t5) \Rightarrow t5])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in -(\lambda (H16: (eq B Abbr x0)).(\lambda (H17: (eq C d x1)).(let H18 \def -(eq_ind_r T x3 (\lambda (t5: T).(subst0 (minus i0 (S i)) u0 t5 x4)) H11 u -H15) in (let H19 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus i0 (S -i)) u0 c3 x2)) H12 d H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b: -B).(getl i c0 (CHead x2 (Bind b) x4))) H10 Abbr H16) in (ex2_ind T (\lambda -(t5: T).(subst0 i x4 t3 t5)) (\lambda (t5: T).(subst0 (S (plus (minus i0 (S -i)) i)) u0 t0 t5)) (pc3 c0 t2 t0) (\lambda (x: T).(\lambda (H21: (subst0 i x4 -t3 x)).(\lambda (H22: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let -H23 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n: -nat).(subst0 n u0 t0 x)) H22 i0 (lt_plus_minus_r i i0 H6)) in (pc3_pr2_u c0 x -t2 (pr2_delta c0 x2 x4 i H20 t2 t3 H1 x H21) t0 (pc3_pr2_x c0 x t0 (pr2_delta -c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H4 (CHead e (Bind Abbr) -u0) H5) t0 t0 (pr0_refl t0) x H23))))))) (subst0_subst0_back t3 t0 u i H2 x4 -u0 (minus i0 (S i)) H18)))))))) H14)) H13))))))))))) H8)) H7))) (\lambda (H6: -(le i0 i)).(pc3_pr2_r c0 t2 t0 (pr2_delta c0 d u i (csubst0_getl_ge i0 i H6 c -c0 u0 H4 (CHead d (Bind Abbr) u) H0) t2 t3 H1 t0 H2)))))))) (\lambda (t5: -T).(\lambda (H4: (subst0 i0 u0 t0 t5)).(\lambda (c0: C).(\lambda (H5: -(csubst0 i0 u0 c c0)).(\lambda (e: C).(\lambda (H6: (getl i0 c (CHead e (Bind -Abbr) u0))).(lt_le_e i i0 (pc3 c0 t2 t5) (\lambda (H7: (lt i i0)).(let H8 -\def (csubst0_getl_lt i0 i H7 c c0 u0 H5 (CHead d (Bind Abbr) u) H0) in -(or4_ind (getl i c0 (CHead d (Bind Abbr) u)) (ex3_4 B C T T (\lambda (b: -B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind -Abbr) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 -(minus i0 (S i)) u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 -(Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(u1: T).(getl i c0 (CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) -(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) -u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 -(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))) -(pc3 c0 t2 t5) (\lambda (H9: (getl i c0 (CHead d (Bind Abbr) u))).(pc3_pr2_u -c0 t3 t2 (pr2_free c0 t2 t3 H1) t5 (pc3_pr3_r c0 t3 t5 (pr3_sing c0 t0 t3 -(pr2_delta c0 d u i H9 t3 t3 (pr0_refl t3) t0 H2) t5 (pr3_pr2 c0 t0 t5 -(pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e -(Bind Abbr) u0) H6) t0 t0 (pr0_refl t0) t5 H4)))))) (\lambda (H9: (ex3_4 B C -T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C -(CHead d (Bind Abbr) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda -(e0: C).(\lambda (_: T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 -(minus i0 (S i)) u0 u1 w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda -(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead -e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) -u0 u1 w))))) (pc3 c0 t2 t5) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: -T).(\lambda (x3: T).(\lambda (H10: (eq C (CHead d (Bind Abbr) u) (CHead x1 -(Bind x0) x2))).(\lambda (H11: (getl i c0 (CHead x1 (Bind x0) x3))).(\lambda -(H12: (subst0 (minus i0 (S i)) u0 x2 x3)).(let H13 \def (f_equal C C (\lambda -(e0: C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow -d | (CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind -x0) x2) H10) in ((let H14 \def (f_equal C B (\lambda (e0: C).(match e0 in C -return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) -\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead -x1 (Bind x0) x2) H10) in ((let H15 \def (f_equal C T (\lambda (e0: C).(match -e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ -t6) \Rightarrow t6])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H10) in -(\lambda (H16: (eq B Abbr x0)).(\lambda (H17: (eq C d x1)).(let H18 \def -(eq_ind_r T x2 (\lambda (t6: T).(subst0 (minus i0 (S i)) u0 t6 x3)) H12 u -H15) in (let H19 \def (eq_ind_r C x1 (\lambda (c3: C).(getl i c0 (CHead c3 -(Bind x0) x3))) H11 d H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b: -B).(getl i c0 (CHead d (Bind b) x3))) H19 Abbr H16) in (ex2_ind T (\lambda -(t6: T).(subst0 i x3 t3 t6)) (\lambda (t6: T).(subst0 (S (plus (minus i0 (S -i)) i)) u0 t0 t6)) (pc3 c0 t2 t5) (\lambda (x: T).(\lambda (H21: (subst0 i x3 -t3 x)).(\lambda (H22: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let -H23 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n: -nat).(subst0 n u0 t0 x)) H22 i0 (lt_plus_minus_r i i0 H7)) in (pc3_pr2_u c0 x -t2 (pr2_delta c0 d x3 i H20 t2 t3 H1 x H21) t5 (pc3_pr2_u2 c0 t0 x (pr2_delta -c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e (Bind Abbr) -u0) H6) t0 t0 (pr0_refl t0) x H23) t5 (pc3_pr2_r c0 t0 t5 (pr2_delta c0 e u0 -i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e (Bind Abbr) u0) H6) -t0 t0 (pr0_refl t0) t5 H4)))))))) (subst0_subst0_back t3 t0 u i H2 x3 u0 -(minus i0 (S i)) H18)))))))) H14)) H13))))))))) H9)) (\lambda (H9: (ex3_4 B C -C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C -(CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda -(_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 (Bind b) -u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))).(ex3_4_ind B C C T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind -Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 (Bind b) u1)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S -i)) u0 e1 e2))))) (pc3 c0 t2 t5) (\lambda (x0: B).(\lambda (x1: C).(\lambda -(x2: C).(\lambda (x3: T).(\lambda (H10: (eq C (CHead d (Bind Abbr) u) (CHead -x1 (Bind x0) x3))).(\lambda (H11: (getl i c0 (CHead x2 (Bind x0) -x3))).(\lambda (H12: (csubst0 (minus i0 (S i)) u0 x1 x2)).(let H13 \def -(f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with -[(CSort _) \Rightarrow d | (CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind -Abbr) u) (CHead x1 (Bind x0) x3) H10) in ((let H14 \def (f_equal C B (\lambda -(e0: C).(match e0 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow -Abbr | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with -[(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) -u) (CHead x1 (Bind x0) x3) H10) in ((let H15 \def (f_equal C T (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | -(CHead _ _ t6) \Rightarrow t6])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) -x3) H10) in (\lambda (H16: (eq B Abbr x0)).(\lambda (H17: (eq C d x1)).(let -H18 \def (eq_ind_r T x3 (\lambda (t6: T).(getl i c0 (CHead x2 (Bind x0) t6))) -H11 u H15) in (let H19 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus -i0 (S i)) u0 c3 x2)) H12 d H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b: -B).(getl i c0 (CHead x2 (Bind b) u))) H18 Abbr H16) in (pc3_pr2_u c0 t0 t2 -(pr2_delta c0 x2 u i H20 t2 t3 H1 t0 H2) t5 (pc3_pr2_r c0 t0 t5 (pr2_delta c0 -e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e (Bind Abbr) u0) -H6) t0 t0 (pr0_refl t0) t5 H4))))))))) H14)) H13))))))))) H9)) (\lambda (H9: -(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) -u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 -(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 -e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 -(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 -(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) -(pc3 c0 t2 t5) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda -(x3: T).(\lambda (x4: T).(\lambda (H10: (eq C (CHead d (Bind Abbr) u) (CHead -x1 (Bind x0) x3))).(\lambda (H11: (getl i c0 (CHead x2 (Bind x0) -x4))).(\lambda (H12: (subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H13: -(csubst0 (minus i0 (S i)) u0 x1 x2)).(let H14 \def (f_equal C C (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | -(CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) -x3) H10) in ((let H15 \def (f_equal C B (\lambda (e0: C).(match e0 in C -return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) -\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead -x1 (Bind x0) x3) H10) in ((let H16 \def (f_equal C T (\lambda (e0: C).(match -e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ -t6) \Rightarrow t6])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H10) in -(\lambda (H17: (eq B Abbr x0)).(\lambda (H18: (eq C d x1)).(let H19 \def -(eq_ind_r T x3 (\lambda (t6: T).(subst0 (minus i0 (S i)) u0 t6 x4)) H12 u -H16) in (let H20 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus i0 (S -i)) u0 c3 x2)) H13 d H18) in (let H21 \def (eq_ind_r B x0 (\lambda (b: -B).(getl i c0 (CHead x2 (Bind b) x4))) H11 Abbr H17) in (ex2_ind T (\lambda -(t6: T).(subst0 i x4 t3 t6)) (\lambda (t6: T).(subst0 (S (plus (minus i0 (S -i)) i)) u0 t0 t6)) (pc3 c0 t2 t5) (\lambda (x: T).(\lambda (H22: (subst0 i x4 -t3 x)).(\lambda (H23: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let -H24 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n: -nat).(subst0 n u0 t0 x)) H23 i0 (lt_plus_minus_r i i0 H7)) in (pc3_pr2_u c0 x -t2 (pr2_delta c0 x2 x4 i H21 t2 t3 H1 x H22) t5 (pc3_pr2_u2 c0 t0 x -(pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e -(Bind Abbr) u0) H6) t0 t0 (pr0_refl t0) x H24) t5 (pc3_pr2_r c0 t0 t5 -(pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e -(Bind Abbr) u0) H6) t0 t0 (pr0_refl t0) t5 H4)))))))) (subst0_subst0_back t3 -t0 u i H2 x4 u0 (minus i0 (S i)) H19)))))))) H15)) H14))))))))))) H9)) H8))) -(\lambda (H7: (le i0 i)).(pc3_pr2_u c0 t0 t2 (pr2_delta c0 d u i -(csubst0_getl_ge i0 i H7 c c0 u0 H5 (CHead d (Bind Abbr) u) H0) t2 t3 H1 t0 -H2) t5 (pc3_pr2_r c0 t0 t5 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n -i0) c c0 u0 H5 (CHead e (Bind Abbr) u0) H6) t0 t0 (pr0_refl t0) t5 -H4))))))))))) c2 t4 H3)))))))))))))))) c1 t t1 H)))). -(* COMMENTS -Initial nodes: 6191 -END *) - -theorem pc3_fsubst0: - \forall (c1: C).(\forall (t1: T).(\forall (t: T).((pc3 c1 t1 t) \to (\forall -(i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t2: T).((fsubst0 i u c1 -t1 c2 t2) \to (\forall (e: C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3 -c2 t2 t))))))))))) -\def - \lambda (c1: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H: (pc3 c1 t1 -t)).(pc3_ind_left c1 (\lambda (t0: T).(\lambda (t2: T).(\forall (i: -nat).(\forall (u: T).(\forall (c2: C).(\forall (t3: T).((fsubst0 i u c1 t0 c2 -t3) \to (\forall (e: C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3 c2 t3 -t2)))))))))) (\lambda (t0: T).(\lambda (i: nat).(\lambda (u: T).(\lambda (c2: -C).(\lambda (t2: T).(\lambda (H0: (fsubst0 i u c1 t0 c2 t2)).(fsubst0_ind i u -c1 t0 (\lambda (c: C).(\lambda (t3: T).(\forall (e: C).((getl i c1 (CHead e -(Bind Abbr) u)) \to (pc3 c t3 t0))))) (\lambda (t3: T).(\lambda (H1: (subst0 -i u t0 t3)).(\lambda (e: C).(\lambda (H2: (getl i c1 (CHead e (Bind Abbr) -u))).(pc3_pr2_x c1 t3 t0 (pr2_delta c1 e u i H2 t0 t0 (pr0_refl t0) t3 -H1)))))) (\lambda (c0: C).(\lambda (_: (csubst0 i u c1 c0)).(\lambda (e: -C).(\lambda (_: (getl i c1 (CHead e (Bind Abbr) u))).(pc3_refl c0 t0))))) -(\lambda (t3: T).(\lambda (H1: (subst0 i u t0 t3)).(\lambda (c0: C).(\lambda -(H2: (csubst0 i u c1 c0)).(\lambda (e: C).(\lambda (H3: (getl i c1 (CHead e -(Bind Abbr) u))).(pc3_pr2_x c0 t3 t0 (pr2_delta c0 e u i (csubst0_getl_ge i i -(le_n i) c1 c0 u H2 (CHead e (Bind Abbr) u) H3) t0 t0 (pr0_refl t0) t3 -H1)))))))) c2 t2 H0))))))) (\lambda (t0: T).(\lambda (t2: T).(\lambda (H0: -(pr2 c1 t0 t2)).(\lambda (t3: T).(\lambda (H1: (pc3 c1 t2 t3)).(\lambda (H2: -((\forall (i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t4: -T).((fsubst0 i u c1 t2 c2 t4) \to (\forall (e: C).((getl i c1 (CHead e (Bind -Abbr) u)) \to (pc3 c2 t4 t3)))))))))).(\lambda (i: nat).(\lambda (u: -T).(\lambda (c2: C).(\lambda (t4: T).(\lambda (H3: (fsubst0 i u c1 t0 c2 -t4)).(fsubst0_ind i u c1 t0 (\lambda (c: C).(\lambda (t5: T).(\forall (e: -C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3 c t5 t3))))) (\lambda (t5: -T).(\lambda (H4: (subst0 i u t0 t5)).(\lambda (e: C).(\lambda (H5: (getl i c1 -(CHead e (Bind Abbr) u))).(pc3_t t2 c1 t5 (pc3_pr2_fsubst0 c1 t0 t2 H0 i u c1 -t5 (fsubst0_snd i u c1 t0 t5 H4) e H5) t3 H1))))) (\lambda (c0: C).(\lambda -(H4: (csubst0 i u c1 c0)).(\lambda (e: C).(\lambda (H5: (getl i c1 (CHead e -(Bind Abbr) u))).(pc3_t t2 c0 t0 (pc3_pr2_fsubst0 c1 t0 t2 H0 i u c0 t0 -(fsubst0_fst i u c1 t0 c0 H4) e H5) t3 (H2 i u c0 t2 (fsubst0_fst i u c1 t2 -c0 H4) e H5)))))) (\lambda (t5: T).(\lambda (H4: (subst0 i u t0 t5)).(\lambda -(c0: C).(\lambda (H5: (csubst0 i u c1 c0)).(\lambda (e: C).(\lambda (H6: -(getl i c1 (CHead e (Bind Abbr) u))).(pc3_t t2 c0 t5 (pc3_pr2_fsubst0 c1 t0 -t2 H0 i u c0 t5 (fsubst0_both i u c1 t0 t5 H4 c0 H5) e H6) t3 (H2 i u c0 t2 -(fsubst0_fst i u c1 t2 c0 H5) e H6)))))))) c2 t4 H3)))))))))))) (\lambda (t0: -T).(\lambda (t2: T).(\lambda (H0: (pr2 c1 t0 t2)).(\lambda (t3: T).(\lambda -(H1: (pc3 c1 t0 t3)).(\lambda (H2: ((\forall (i: nat).(\forall (u: -T).(\forall (c2: C).(\forall (t4: T).((fsubst0 i u c1 t0 c2 t4) \to (\forall -(e: C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3 c2 t4 -t3)))))))))).(\lambda (i: nat).(\lambda (u: T).(\lambda (c2: C).(\lambda (t4: -T).(\lambda (H3: (fsubst0 i u c1 t2 c2 t4)).(fsubst0_ind i u c1 t2 (\lambda -(c: C).(\lambda (t5: T).(\forall (e: C).((getl i c1 (CHead e (Bind Abbr) u)) -\to (pc3 c t5 t3))))) (\lambda (t5: T).(\lambda (H4: (subst0 i u t2 -t5)).(\lambda (e: C).(\lambda (H5: (getl i c1 (CHead e (Bind Abbr) -u))).(pc3_t t0 c1 t5 (pc3_s c1 t5 t0 (pc3_pr2_fsubst0_back c1 t0 t2 H0 i u c1 -t5 (fsubst0_snd i u c1 t2 t5 H4) e H5)) t3 H1))))) (\lambda (c0: C).(\lambda -(H4: (csubst0 i u c1 c0)).(\lambda (e: C).(\lambda (H5: (getl i c1 (CHead e -(Bind Abbr) u))).(pc3_t t0 c0 t2 (pc3_s c0 t2 t0 (pc3_pr2_fsubst0_back c1 t0 -t2 H0 i u c0 t2 (fsubst0_fst i u c1 t2 c0 H4) e H5)) t3 (H2 i u c0 t0 -(fsubst0_fst i u c1 t0 c0 H4) e H5)))))) (\lambda (t5: T).(\lambda (H4: -(subst0 i u t2 t5)).(\lambda (c0: C).(\lambda (H5: (csubst0 i u c1 -c0)).(\lambda (e: C).(\lambda (H6: (getl i c1 (CHead e (Bind Abbr) -u))).(pc3_t t0 c0 t5 (pc3_s c0 t5 t0 (pc3_pr2_fsubst0_back c1 t0 t2 H0 i u c0 -t5 (fsubst0_both i u c1 t2 t5 H4 c0 H5) e H6)) t3 (H2 i u c0 t0 (fsubst0_fst -i u c1 t0 c0 H5) e H6)))))))) c2 t4 H3)))))))))))) t1 t H)))). -(* COMMENTS -Initial nodes: 1249 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/fwd.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/fwd.ma deleted file mode 100644 index c89e4563c..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/fwd.ma +++ /dev/null @@ -1,333 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pc3/props.ma". - -include "Basic-1/pr3/fwd.ma". - -theorem pc3_gen_sort: - \forall (c: C).(\forall (m: nat).(\forall (n: nat).((pc3 c (TSort m) (TSort -n)) \to (eq nat m n)))) -\def - \lambda (c: C).(\lambda (m: nat).(\lambda (n: nat).(\lambda (H: (pc3 c -(TSort m) (TSort n))).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr3 c -(TSort m) t)) (\lambda (t: T).(pr3 c (TSort n) t)) (eq nat m n) (\lambda (x: -T).(\lambda (H1: (pr3 c (TSort m) x)).(\lambda (H2: (pr3 c (TSort n) x)).(let -H3 \def (eq_ind T x (\lambda (t: T).(eq T t (TSort n))) (pr3_gen_sort c x n -H2) (TSort m) (pr3_gen_sort c x m H1)) in (let H4 \def (f_equal T nat -(\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort n0) -\Rightarrow n0 | (TLRef _) \Rightarrow m | (THead _ _ _) \Rightarrow m])) -(TSort m) (TSort n) H3) in H4))))) H0))))). -(* COMMENTS -Initial nodes: 153 -END *) - -theorem pc3_gen_abst: - \forall (c: C).(\forall (u1: T).(\forall (u2: T).(\forall (t1: T).(\forall -(t2: T).((pc3 c (THead (Bind Abst) u1 t1) (THead (Bind Abst) u2 t2)) \to -(land (pc3 c u1 u2) (\forall (b: B).(\forall (u: T).(pc3 (CHead c (Bind b) u) -t1 t2))))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (t1: T).(\lambda -(t2: T).(\lambda (H: (pc3 c (THead (Bind Abst) u1 t1) (THead (Bind Abst) u2 -t2))).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr3 c (THead (Bind Abst) -u1 t1) t)) (\lambda (t: T).(pr3 c (THead (Bind Abst) u2 t2) t)) (land (pc3 c -u1 u2) (\forall (b: B).(\forall (u: T).(pc3 (CHead c (Bind b) u) t1 t2)))) -(\lambda (x: T).(\lambda (H1: (pr3 c (THead (Bind Abst) u1 t1) x)).(\lambda -(H2: (pr3 c (THead (Bind Abst) u2 t2) x)).(let H3 \def (pr3_gen_abst c u2 t2 -x H2) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead -(Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c u2 u3))) -(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead -c (Bind b) u) t2 t3))))) (land (pc3 c u1 u2) (\forall (b: B).(\forall (u: -T).(pc3 (CHead c (Bind b) u) t1 t2)))) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H4: (eq T x (THead (Bind Abst) x0 x1))).(\lambda (H5: (pr3 c u2 -x0)).(\lambda (H6: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -t2 x1))))).(let H7 \def (pr3_gen_abst c u1 t1 x H1) in (ex3_2_ind T T -(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u3 t3)))) -(\lambda (u3: T).(\lambda (_: T).(pr3 c u1 u3))) (\lambda (_: T).(\lambda -(t3: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t1 t3))))) -(land (pc3 c u1 u2) (\forall (b: B).(\forall (u: T).(pc3 (CHead c (Bind b) u) -t1 t2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H8: (eq T x (THead -(Bind Abst) x2 x3))).(\lambda (H9: (pr3 c u1 x2)).(\lambda (H10: ((\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t1 x3))))).(let H11 \def -(eq_ind T x (\lambda (t: T).(eq T t (THead (Bind Abst) x0 x1))) H4 (THead -(Bind Abst) x2 x3) H8) in (let H12 \def (f_equal T T (\lambda (e: T).(match e -in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x2 | (TLRef _) -\Rightarrow x2 | (THead _ t _) \Rightarrow t])) (THead (Bind Abst) x2 x3) -(THead (Bind Abst) x0 x1) H11) in ((let H13 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x3 | -(TLRef _) \Rightarrow x3 | (THead _ _ t) \Rightarrow t])) (THead (Bind Abst) -x2 x3) (THead (Bind Abst) x0 x1) H11) in (\lambda (H14: (eq T x2 x0)).(let -H15 \def (eq_ind T x3 (\lambda (t: T).(\forall (b: B).(\forall (u: T).(pr3 -(CHead c (Bind b) u) t1 t)))) H10 x1 H13) in (let H16 \def (eq_ind T x2 -(\lambda (t: T).(pr3 c u1 t)) H9 x0 H14) in (conj (pc3 c u1 u2) (\forall (b: -B).(\forall (u: T).(pc3 (CHead c (Bind b) u) t1 t2))) (pc3_pr3_t c u1 x0 H16 -u2 H5) (\lambda (b: B).(\lambda (u: T).(pc3_pr3_t (CHead c (Bind b) u) t1 x1 -(H15 b u) t2 (H6 b u))))))))) H12)))))))) H7))))))) H3))))) H0))))))). -(* COMMENTS -Initial nodes: 715 -END *) - -theorem pc3_gen_abst_shift: - \forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).((pc3 c -(THead (Bind Abst) u t1) (THead (Bind Abst) u t2)) \to (pc3 (CHead c (Bind -Abst) u) t1 t2))))) -\def - \lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H: (pc3 c (THead (Bind Abst) u t1) (THead (Bind Abst) u t2))).(let H_x \def -(pc3_gen_abst c u u t1 t2 H) in (let H0 \def H_x in (land_ind (pc3 c u u) -(\forall (b: B).(\forall (u0: T).(pc3 (CHead c (Bind b) u0) t1 t2))) (pc3 -(CHead c (Bind Abst) u) t1 t2) (\lambda (_: (pc3 c u u)).(\lambda (H2: -((\forall (b: B).(\forall (u0: T).(pc3 (CHead c (Bind b) u0) t1 t2))))).(H2 -Abst u))) H0))))))). -(* COMMENTS -Initial nodes: 129 -END *) - -theorem pc3_gen_lift: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).(\forall (h: nat).(\forall -(d: nat).((pc3 c (lift h d t1) (lift h d t2)) \to (\forall (e: C).((drop h d -c e) \to (pc3 e t1 t2)))))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (h: nat).(\lambda -(d: nat).(\lambda (H: (pc3 c (lift h d t1) (lift h d t2))).(\lambda (e: -C).(\lambda (H0: (drop h d c e)).(let H1 \def H in (ex2_ind T (\lambda (t: -T).(pr3 c (lift h d t1) t)) (\lambda (t: T).(pr3 c (lift h d t2) t)) (pc3 e -t1 t2) (\lambda (x: T).(\lambda (H2: (pr3 c (lift h d t1) x)).(\lambda (H3: -(pr3 c (lift h d t2) x)).(let H4 \def (pr3_gen_lift c t2 x h d H3 e H0) in -(ex2_ind T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr3 e -t2 t3)) (pc3 e t1 t2) (\lambda (x0: T).(\lambda (H5: (eq T x (lift h d -x0))).(\lambda (H6: (pr3 e t2 x0)).(let H7 \def (pr3_gen_lift c t1 x h d H2 e -H0) in (ex2_ind T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: -T).(pr3 e t1 t3)) (pc3 e t1 t2) (\lambda (x1: T).(\lambda (H8: (eq T x (lift -h d x1))).(\lambda (H9: (pr3 e t1 x1)).(let H10 \def (eq_ind T x (\lambda (t: -T).(eq T t (lift h d x0))) H5 (lift h d x1) H8) in (let H11 \def (eq_ind T x1 -(\lambda (t: T).(pr3 e t1 t)) H9 x0 (lift_inj x1 x0 h d H10)) in (pc3_pr3_t e -t1 x0 H11 t2 H6)))))) H7))))) H4))))) H1))))))))). -(* COMMENTS -Initial nodes: 363 -END *) - -theorem pc3_gen_not_abst: - \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (t1: -T).(\forall (t2: T).(\forall (u1: T).(\forall (u2: T).((pc3 c (THead (Bind b) -u1 t1) (THead (Bind Abst) u2 t2)) \to (pc3 (CHead c (Bind b) u1) t1 (lift (S -O) O (THead (Bind Abst) u2 t2)))))))))) -\def - \lambda (b: B).(B_ind (\lambda (b0: B).((not (eq B b0 Abst)) \to (\forall -(c: C).(\forall (t1: T).(\forall (t2: T).(\forall (u1: T).(\forall (u2: -T).((pc3 c (THead (Bind b0) u1 t1) (THead (Bind Abst) u2 t2)) \to (pc3 (CHead -c (Bind b0) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2))))))))))) (\lambda -(_: (not (eq B Abbr Abst))).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H0: (pc3 c (THead (Bind Abbr) -u1 t1) (THead (Bind Abst) u2 t2))).(let H1 \def H0 in (ex2_ind T (\lambda (t: -T).(pr3 c (THead (Bind Abbr) u1 t1) t)) (\lambda (t: T).(pr3 c (THead (Bind -Abst) u2 t2) t)) (pc3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O (THead (Bind -Abst) u2 t2))) (\lambda (x: T).(\lambda (H2: (pr3 c (THead (Bind Abbr) u1 t1) -x)).(\lambda (H3: (pr3 c (THead (Bind Abst) u2 t2) x)).(let H4 \def -(pr3_gen_abbr c u1 t1 x H2) in (or_ind (ex3_2 T T (\lambda (u3: T).(\lambda -(t3: T).(eq T x (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: -T).(pr3 c u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr3 (CHead c (Bind Abbr) -u1) t1 t3)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O x)) (pc3 (CHead -c (Bind Abbr) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2))) (\lambda (H5: -(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3 -t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c u1 u3))) (\lambda (_: -T).(\lambda (t3: T).(pr3 (CHead c (Bind Abbr) u1) t1 t3))))).(ex3_2_ind T T -(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3 t3)))) -(\lambda (u3: T).(\lambda (_: T).(pr3 c u1 u3))) (\lambda (_: T).(\lambda -(t3: T).(pr3 (CHead c (Bind Abbr) u1) t1 t3))) (pc3 (CHead c (Bind Abbr) u1) -t1 (lift (S O) O (THead (Bind Abst) u2 t2))) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H6: (eq T x (THead (Bind Abbr) x0 x1))).(\lambda (_: (pr3 c u1 -x0)).(\lambda (_: (pr3 (CHead c (Bind Abbr) u1) t1 x1)).(let H9 \def -(pr3_gen_abst c u2 t2 x H3) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: -T).(eq T x (THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 -c u2 u3))) (\lambda (_: T).(\lambda (t3: T).(\forall (b0: B).(\forall (u: -T).(pr3 (CHead c (Bind b0) u) t2 t3))))) (pc3 (CHead c (Bind Abbr) u1) t1 -(lift (S O) O (THead (Bind Abst) u2 t2))) (\lambda (x2: T).(\lambda (x3: -T).(\lambda (H10: (eq T x (THead (Bind Abst) x2 x3))).(\lambda (_: (pr3 c u2 -x2)).(\lambda (_: ((\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) -u) t2 x3))))).(let H13 \def (eq_ind T x (\lambda (t: T).(eq T t (THead (Bind -Abbr) x0 x1))) H6 (THead (Bind Abst) x2 x3) H10) in (let H14 \def (eq_ind T -(THead (Bind Abst) x2 x3) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind b0) \Rightarrow (match b0 in B return (\lambda (_: B).Prop) with [Abbr -\Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat -_) \Rightarrow False])])) I (THead (Bind Abbr) x0 x1) H13) in (False_ind (pc3 -(CHead c (Bind Abbr) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2))) -H14)))))))) H9))))))) H5)) (\lambda (H5: (pr3 (CHead c (Bind Abbr) u1) t1 -(lift (S O) O x))).(let H6 \def (pr3_gen_abst c u2 t2 x H3) in (ex3_2_ind T T -(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u3 t3)))) -(\lambda (u3: T).(\lambda (_: T).(pr3 c u2 u3))) (\lambda (_: T).(\lambda -(t3: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t2 -t3))))) (pc3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O (THead (Bind Abst) u2 -t2))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: (eq T x (THead (Bind -Abst) x0 x1))).(\lambda (H8: (pr3 c u2 x0)).(\lambda (H9: ((\forall (b0: -B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t2 x1))))).(let H10 \def -(eq_ind T x (\lambda (t: T).(pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O -t))) H5 (THead (Bind Abst) x0 x1) H7) in (pc3_pr3_t (CHead c (Bind Abbr) u1) -t1 (lift (S O) O (THead (Bind Abst) x0 x1)) H10 (lift (S O) O (THead (Bind -Abst) u2 t2)) (pr3_lift (CHead c (Bind Abbr) u1) c (S O) O (drop_drop (Bind -Abbr) O c c (drop_refl c) u1) (THead (Bind Abst) u2 t2) (THead (Bind Abst) x0 -x1) (pr3_head_12 c u2 x0 H8 (Bind Abst) t2 x1 (H9 Abst x0)))))))))) H6))) -H4))))) H1))))))))) (\lambda (H: (not (eq B Abst Abst))).(\lambda (c: -C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (u1: T).(\lambda (u2: -T).(\lambda (_: (pc3 c (THead (Bind Abst) u1 t1) (THead (Bind Abst) u2 -t2))).(let H1 \def (match (H (refl_equal B Abst)) in False return (\lambda -(_: False).(pc3 (CHead c (Bind Abst) u1) t1 (lift (S O) O (THead (Bind Abst) -u2 t2)))) with []) in H1)))))))) (\lambda (_: (not (eq B Void -Abst))).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (u1: -T).(\lambda (u2: T).(\lambda (H0: (pc3 c (THead (Bind Void) u1 t1) (THead -(Bind Abst) u2 t2))).(let H1 \def H0 in (ex2_ind T (\lambda (t: T).(pr3 c -(THead (Bind Void) u1 t1) t)) (\lambda (t: T).(pr3 c (THead (Bind Abst) u2 -t2) t)) (pc3 (CHead c (Bind Void) u1) t1 (lift (S O) O (THead (Bind Abst) u2 -t2))) (\lambda (x: T).(\lambda (H2: (pr3 c (THead (Bind Void) u1 t1) -x)).(\lambda (H3: (pr3 c (THead (Bind Abst) u2 t2) x)).(let H4 \def -(pr3_gen_void c u1 t1 x H2) in (or_ind (ex3_2 T T (\lambda (u3: T).(\lambda -(t3: T).(eq T x (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: -T).(pr3 c u1 u3))) (\lambda (_: T).(\lambda (t3: T).(\forall (b0: B).(\forall -(u: T).(pr3 (CHead c (Bind b0) u) t1 t3)))))) (pr3 (CHead c (Bind Void) u1) -t1 (lift (S O) O x)) (pc3 (CHead c (Bind Void) u1) t1 (lift (S O) O (THead -(Bind Abst) u2 t2))) (\lambda (H5: (ex3_2 T T (\lambda (u3: T).(\lambda (t3: -T).(eq T x (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 -c u1 u3))) (\lambda (_: T).(\lambda (t3: T).(\forall (b0: B).(\forall (u: -T).(pr3 (CHead c (Bind b0) u) t1 t3))))))).(ex3_2_ind T T (\lambda (u3: -T).(\lambda (t3: T).(eq T x (THead (Bind Void) u3 t3)))) (\lambda (u3: -T).(\lambda (_: T).(pr3 c u1 u3))) (\lambda (_: T).(\lambda (t3: T).(\forall -(b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t1 t3))))) (pc3 (CHead c -(Bind Void) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2))) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H6: (eq T x (THead (Bind Void) x0 -x1))).(\lambda (_: (pr3 c u1 x0)).(\lambda (_: ((\forall (b0: B).(\forall (u: -T).(pr3 (CHead c (Bind b0) u) t1 x1))))).(let H9 \def (pr3_gen_abst c u2 t2 x -H3) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind -Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c u2 u3))) (\lambda (_: -T).(\lambda (t3: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) -u) t2 t3))))) (pc3 (CHead c (Bind Void) u1) t1 (lift (S O) O (THead (Bind -Abst) u2 t2))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H10: (eq T x -(THead (Bind Abst) x2 x3))).(\lambda (_: (pr3 c u2 x2)).(\lambda (_: -((\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t2 x3))))).(let -H13 \def (eq_ind T x (\lambda (t: T).(eq T t (THead (Bind Void) x0 x1))) H6 -(THead (Bind Abst) x2 x3) H10) in (let H14 \def (eq_ind T (THead (Bind Abst) -x2 x3) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b0) -\Rightarrow (match b0 in B return (\lambda (_: B).Prop) with [Abbr -\Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat -_) \Rightarrow False])])) I (THead (Bind Void) x0 x1) H13) in (False_ind (pc3 -(CHead c (Bind Void) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2))) -H14)))))))) H9))))))) H5)) (\lambda (H5: (pr3 (CHead c (Bind Void) u1) t1 -(lift (S O) O x))).(let H6 \def (pr3_gen_abst c u2 t2 x H3) in (ex3_2_ind T T -(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u3 t3)))) -(\lambda (u3: T).(\lambda (_: T).(pr3 c u2 u3))) (\lambda (_: T).(\lambda -(t3: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t2 -t3))))) (pc3 (CHead c (Bind Void) u1) t1 (lift (S O) O (THead (Bind Abst) u2 -t2))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: (eq T x (THead (Bind -Abst) x0 x1))).(\lambda (H8: (pr3 c u2 x0)).(\lambda (H9: ((\forall (b0: -B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t2 x1))))).(let H10 \def -(eq_ind T x (\lambda (t: T).(pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O -t))) H5 (THead (Bind Abst) x0 x1) H7) in (pc3_pr3_t (CHead c (Bind Void) u1) -t1 (lift (S O) O (THead (Bind Abst) x0 x1)) H10 (lift (S O) O (THead (Bind -Abst) u2 t2)) (pr3_lift (CHead c (Bind Void) u1) c (S O) O (drop_drop (Bind -Void) O c c (drop_refl c) u1) (THead (Bind Abst) u2 t2) (THead (Bind Abst) x0 -x1) (pr3_head_12 c u2 x0 H8 (Bind Abst) t2 x1 (H9 Abst x0)))))))))) H6))) -H4))))) H1))))))))) b). -(* COMMENTS -Initial nodes: 2427 -END *) - -theorem pc3_gen_lift_abst: - \forall (c: C).(\forall (t: T).(\forall (t2: T).(\forall (u2: T).(\forall -(h: nat).(\forall (d: nat).((pc3 c (lift h d t) (THead (Bind Abst) u2 t2)) -\to (\forall (e: C).((drop h d c e) \to (ex3_2 T T (\lambda (u1: T).(\lambda -(t1: T).(pr3 e t (THead (Bind Abst) u1 t1)))) (\lambda (u1: T).(\lambda (_: -T).(pr3 c u2 (lift h d u1)))) (\lambda (_: T).(\lambda (t1: T).(\forall (b: -B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t2 (lift h (S d) -t1))))))))))))))) -\def - \lambda (c: C).(\lambda (t: T).(\lambda (t2: T).(\lambda (u2: T).(\lambda -(h: nat).(\lambda (d: nat).(\lambda (H: (pc3 c (lift h d t) (THead (Bind -Abst) u2 t2))).(\lambda (e: C).(\lambda (H0: (drop h d c e)).(let H1 \def H -in (ex2_ind T (\lambda (t0: T).(pr3 c (lift h d t) t0)) (\lambda (t0: T).(pr3 -c (THead (Bind Abst) u2 t2) t0)) (ex3_2 T T (\lambda (u1: T).(\lambda (t1: -T).(pr3 e t (THead (Bind Abst) u1 t1)))) (\lambda (u1: T).(\lambda (_: -T).(pr3 c u2 (lift h d u1)))) (\lambda (_: T).(\lambda (t1: T).(\forall (b: -B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t2 (lift h (S d) t1))))))) -(\lambda (x: T).(\lambda (H2: (pr3 c (lift h d t) x)).(\lambda (H3: (pr3 c -(THead (Bind Abst) u2 t2) x)).(let H4 \def (pr3_gen_lift c t x h d H2 e H0) -in (ex2_ind T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr3 -e t t3)) (ex3_2 T T (\lambda (u1: T).(\lambda (t1: T).(pr3 e t (THead (Bind -Abst) u1 t1)))) (\lambda (u1: T).(\lambda (_: T).(pr3 c u2 (lift h d u1)))) -(\lambda (_: T).(\lambda (t1: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead -c (Bind b) u) t2 (lift h (S d) t1))))))) (\lambda (x0: T).(\lambda (H5: (eq T -x (lift h d x0))).(\lambda (H6: (pr3 e t x0)).(let H7 \def (pr3_gen_abst c u2 -t2 x H3) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead -(Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c u2 u3))) -(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead -c (Bind b) u) t2 t3))))) (ex3_2 T T (\lambda (u1: T).(\lambda (t1: T).(pr3 e -t (THead (Bind Abst) u1 t1)))) (\lambda (u1: T).(\lambda (_: T).(pr3 c u2 -(lift h d u1)))) (\lambda (_: T).(\lambda (t1: T).(\forall (b: B).(\forall -(u: T).(pr3 (CHead c (Bind b) u) t2 (lift h (S d) t1))))))) (\lambda (x1: -T).(\lambda (x2: T).(\lambda (H8: (eq T x (THead (Bind Abst) x1 -x2))).(\lambda (H9: (pr3 c u2 x1)).(\lambda (H10: ((\forall (b: B).(\forall -(u: T).(pr3 (CHead c (Bind b) u) t2 x2))))).(let H11 \def (eq_ind T x -(\lambda (t0: T).(eq T t0 (lift h d x0))) H5 (THead (Bind Abst) x1 x2) H8) in -(ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x0 (THead (Bind Abst) y -z)))) (\lambda (y: T).(\lambda (_: T).(eq T x1 (lift h d y)))) (\lambda (_: -T).(\lambda (z: T).(eq T x2 (lift h (S d) z)))) (ex3_2 T T (\lambda (u1: -T).(\lambda (t1: T).(pr3 e t (THead (Bind Abst) u1 t1)))) (\lambda (u1: -T).(\lambda (_: T).(pr3 c u2 (lift h d u1)))) (\lambda (_: T).(\lambda (t1: -T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t2 (lift h (S d) -t1))))))) (\lambda (x3: T).(\lambda (x4: T).(\lambda (H12: (eq T x0 (THead -(Bind Abst) x3 x4))).(\lambda (H13: (eq T x1 (lift h d x3))).(\lambda (H14: -(eq T x2 (lift h (S d) x4))).(let H15 \def (eq_ind T x2 (\lambda (t0: -T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t2 t0)))) H10 -(lift h (S d) x4) H14) in (let H16 \def (eq_ind T x1 (\lambda (t0: T).(pr3 c -u2 t0)) H9 (lift h d x3) H13) in (let H17 \def (eq_ind T x0 (\lambda (t0: -T).(pr3 e t t0)) H6 (THead (Bind Abst) x3 x4) H12) in (ex3_2_intro T T -(\lambda (u1: T).(\lambda (t1: T).(pr3 e t (THead (Bind Abst) u1 t1)))) -(\lambda (u1: T).(\lambda (_: T).(pr3 c u2 (lift h d u1)))) (\lambda (_: -T).(\lambda (t1: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -t2 (lift h (S d) t1)))))) x3 x4 H17 H16 H15))))))))) (lift_gen_bind Abst x1 -x2 x0 h d H11)))))))) H7))))) H4))))) H1)))))))))). -(* COMMENTS -Initial nodes: 973 -END *) - -theorem pc3_gen_sort_abst: - \forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (n: nat).((pc3 c -(TSort n) (THead (Bind Abst) u t)) \to (\forall (P: Prop).P))))) -\def - \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (n: nat).(\lambda -(H: (pc3 c (TSort n) (THead (Bind Abst) u t))).(\lambda (P: Prop).(let H0 -\def H in (ex2_ind T (\lambda (t0: T).(pr3 c (TSort n) t0)) (\lambda (t0: -T).(pr3 c (THead (Bind Abst) u t) t0)) P (\lambda (x: T).(\lambda (H1: (pr3 c -(TSort n) x)).(\lambda (H2: (pr3 c (THead (Bind Abst) u t) x)).(let H3 \def -(pr3_gen_abst c u t x H2) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: -T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 -c u u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u0: -T).(pr3 (CHead c (Bind b) u0) t t2))))) P (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H4: (eq T x (THead (Bind Abst) x0 x1))).(\lambda (_: (pr3 c u -x0)).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pr3 (CHead c (Bind b) -u0) t x1))))).(let H7 \def (eq_ind T x (\lambda (t0: T).(eq T t0 (TSort n))) -(pr3_gen_sort c x n H1) (THead (Bind Abst) x0 x1) H4) in (let H8 \def (eq_ind -T (THead (Bind Abst) x0 x1) (\lambda (ee: T).(match ee in T return (\lambda -(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False -| (THead _ _ _) \Rightarrow True])) I (TSort n) H7) in (False_ind P -H8)))))))) H3))))) H0))))))). -(* COMMENTS -Initial nodes: 303 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/left.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/left.ma deleted file mode 100644 index 12d22d10c..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/left.ma +++ /dev/null @@ -1,125 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pc3/props.ma". - -theorem pc3_ind_left__pc3_left_pr3: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to -(pc3_left c t1 t2)))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 -t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(pc3_left c t t0))) (\lambda -(t: T).(pc3_left_r c t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 -c t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2: -(pc3_left c t0 t4)).(pc3_left_ur c t3 t0 H0 t4 H2))))))) t1 t2 H)))). -(* COMMENTS -Initial nodes: 87 -END *) - -theorem pc3_ind_left__pc3_left_trans: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3_left c t1 t2) \to -(\forall (t3: T).((pc3_left c t2 t3) \to (pc3_left c t1 t3)))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3_left c t1 -t2)).(pc3_left_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (t3: -T).((pc3_left c t0 t3) \to (pc3_left c t t3))))) (\lambda (t: T).(\lambda -(t3: T).(\lambda (H0: (pc3_left c t t3)).H0))) (\lambda (t0: T).(\lambda (t3: -T).(\lambda (H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t3 -t4)).(\lambda (H2: ((\forall (t5: T).((pc3_left c t4 t5) \to (pc3_left c t3 -t5))))).(\lambda (t5: T).(\lambda (H3: (pc3_left c t4 t5)).(pc3_left_ur c t0 -t3 H0 t5 (H2 t5 H3)))))))))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: -(pr2 c t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t0 t4)).(\lambda -(H2: ((\forall (t5: T).((pc3_left c t4 t5) \to (pc3_left c t0 -t5))))).(\lambda (t5: T).(\lambda (H3: (pc3_left c t4 t5)).(pc3_left_ux c t0 -t3 H0 t5 (H2 t5 H3)))))))))) t1 t2 H)))). -(* COMMENTS -Initial nodes: 195 -END *) - -theorem pc3_ind_left__pc3_left_sym: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3_left c t1 t2) \to -(pc3_left c t2 t1)))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3_left c t1 -t2)).(pc3_left_ind c (\lambda (t: T).(\lambda (t0: T).(pc3_left c t0 t))) -(\lambda (t: T).(pc3_left_r c t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda -(H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t3 -t4)).(\lambda (H2: (pc3_left c t4 t3)).(pc3_ind_left__pc3_left_trans c t4 t3 -H2 t0 (pc3_left_ux c t0 t3 H0 t0 (pc3_left_r c t0))))))))) (\lambda (t0: -T).(\lambda (t3: T).(\lambda (H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda -(_: (pc3_left c t0 t4)).(\lambda (H2: (pc3_left c t4 -t0)).(pc3_ind_left__pc3_left_trans c t4 t0 H2 t3 (pc3_left_ur c t0 t3 H0 t3 -(pc3_left_r c t3))))))))) t1 t2 H)))). -(* COMMENTS -Initial nodes: 163 -END *) - -theorem pc3_ind_left__pc3_left_pc3: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to -(pc3_left c t1 t2)))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c t1 -t2)).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: -T).(pr3 c t2 t)) (pc3_left c t1 t2) (\lambda (x: T).(\lambda (H1: (pr3 c t1 -x)).(\lambda (H2: (pr3 c t2 x)).(pc3_ind_left__pc3_left_trans c t1 x -(pc3_ind_left__pc3_left_pr3 c t1 x H1) t2 (pc3_ind_left__pc3_left_sym c t2 x -(pc3_ind_left__pc3_left_pr3 c t2 x H2)))))) H0))))). -(* COMMENTS -Initial nodes: 105 -END *) - -theorem pc3_ind_left__pc3_pc3_left: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3_left c t1 t2) \to -(pc3 c t1 t2)))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3_left c t1 -t2)).(pc3_left_ind c (\lambda (t: T).(\lambda (t0: T).(pc3 c t t0))) (\lambda -(t: T).(pc3_refl c t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c -t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t3 t4)).(\lambda (H2: (pc3 -c t3 t4)).(pc3_t t3 c t0 (pc3_pr2_r c t0 t3 H0) t4 H2))))))) (\lambda (t0: -T).(\lambda (t3: T).(\lambda (H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda -(_: (pc3_left c t0 t4)).(\lambda (H2: (pc3 c t0 t4)).(pc3_t t0 c t3 -(pc3_pr2_x c t3 t0 H0) t4 H2))))))) t1 t2 H)))). -(* COMMENTS -Initial nodes: 147 -END *) - -theorem pc3_ind_left: - \forall (c: C).(\forall (P: ((T \to (T \to Prop)))).(((\forall (t: T).(P t -t))) \to (((\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall (t3: -T).((pc3 c t2 t3) \to ((P t2 t3) \to (P t1 t3)))))))) \to (((\forall (t1: -T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall (t3: T).((pc3 c t1 t3) \to -((P t1 t3) \to (P t2 t3)))))))) \to (\forall (t: T).(\forall (t0: T).((pc3 c -t t0) \to (P t t0)))))))) -\def - \lambda (c: C).(\lambda (P: ((T \to (T \to Prop)))).(\lambda (H: ((\forall -(t: T).(P t t)))).(\lambda (H0: ((\forall (t1: T).(\forall (t2: T).((pr2 c t1 -t2) \to (\forall (t3: T).((pc3 c t2 t3) \to ((P t2 t3) \to (P t1 -t3))))))))).(\lambda (H1: ((\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) -\to (\forall (t3: T).((pc3 c t1 t3) \to ((P t1 t3) \to (P t2 -t3))))))))).(\lambda (t: T).(\lambda (t0: T).(\lambda (H2: (pc3 c t -t0)).(pc3_left_ind c (\lambda (t1: T).(\lambda (t2: T).(P t1 t2))) H (\lambda -(t1: T).(\lambda (t2: T).(\lambda (H3: (pr2 c t1 t2)).(\lambda (t3: -T).(\lambda (H4: (pc3_left c t2 t3)).(\lambda (H5: (P t2 t3)).(H0 t1 t2 H3 t3 -(pc3_ind_left__pc3_pc3_left c t2 t3 H4) H5))))))) (\lambda (t1: T).(\lambda -(t2: T).(\lambda (H3: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda (H4: (pc3_left -c t1 t3)).(\lambda (H5: (P t1 t3)).(H1 t1 t2 H3 t3 -(pc3_ind_left__pc3_pc3_left c t1 t3 H4) H5))))))) t t0 -(pc3_ind_left__pc3_left_pc3 c t t0 H2))))))))). -(* COMMENTS -Initial nodes: 225 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/nf2.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/nf2.ma deleted file mode 100644 index ed930b85e..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/nf2.ma +++ /dev/null @@ -1,52 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pc3/defs.ma". - -include "Basic-1/nf2/pr3.ma". - -theorem pc3_nf2: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to ((nf2 c -t1) \to ((nf2 c t2) \to (eq T t1 t2)))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c t1 -t2)).(\lambda (H0: (nf2 c t1)).(\lambda (H1: (nf2 c t2)).(let H2 \def H in -(ex2_ind T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) (eq T -t1 t2) (\lambda (x: T).(\lambda (H3: (pr3 c t1 x)).(\lambda (H4: (pr3 c t2 -x)).(let H_y \def (nf2_pr3_unfold c t1 x H3 H0) in (let H5 \def (eq_ind_r T x -(\lambda (t: T).(pr3 c t2 t)) H4 t1 H_y) in (let H6 \def (eq_ind_r T x -(\lambda (t: T).(pr3 c t1 t)) H3 t1 H_y) in (let H_y0 \def (nf2_pr3_unfold c -t2 t1 H5 H1) in (let H7 \def (eq_ind T t2 (\lambda (t: T).(pr3 c t t1)) H5 t1 -H_y0) in (eq_ind_r T t1 (\lambda (t: T).(eq T t1 t)) (refl_equal T t1) t2 -H_y0))))))))) H2))))))). -(* COMMENTS -Initial nodes: 195 -END *) - -theorem pc3_nf2_unfold: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to ((nf2 c -t2) \to (pr3 c t1 t2))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c t1 -t2)).(\lambda (H0: (nf2 c t2)).(let H1 \def H in (ex2_ind T (\lambda (t: -T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) (pr3 c t1 t2) (\lambda (x: -T).(\lambda (H2: (pr3 c t1 x)).(\lambda (H3: (pr3 c t2 x)).(let H_y \def -(nf2_pr3_unfold c t2 x H3 H0) in (let H4 \def (eq_ind_r T x (\lambda (t: -T).(pr3 c t1 t)) H2 t2 H_y) in H4))))) H1)))))). -(* COMMENTS -Initial nodes: 109 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/pc1.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/pc1.ma deleted file mode 100644 index dd89f951d..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/pc1.ma +++ /dev/null @@ -1,36 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pc3/defs.ma". - -include "Basic-1/pc1/defs.ma". - -include "Basic-1/pr3/pr1.ma". - -theorem pc3_pc1: - \forall (t1: T).(\forall (t2: T).((pc1 t1 t2) \to (\forall (c: C).(pc3 c t1 -t2)))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc1 t1 t2)).(\lambda (c: -C).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: -T).(pr1 t2 t)) (pc3 c t1 t2) (\lambda (x: T).(\lambda (H1: (pr1 t1 -x)).(\lambda (H2: (pr1 t2 x)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) -(\lambda (t: T).(pr3 c t2 t)) x (pr3_pr1 t1 x H1 c) (pr3_pr1 t2 x H2 c))))) -H0))))). -(* COMMENTS -Initial nodes: 103 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/props.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/props.ma deleted file mode 100644 index b0a9a2f50..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/props.ma +++ /dev/null @@ -1,483 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pc3/defs.ma". - -include "Basic-1/pr3/pr3.ma". - -theorem clear_pc3_trans: - \forall (c2: C).(\forall (t1: T).(\forall (t2: T).((pc3 c2 t1 t2) \to -(\forall (c1: C).((clear c1 c2) \to (pc3 c1 t1 t2)))))) -\def - \lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c2 t1 -t2)).(\lambda (c1: C).(\lambda (H0: (clear c1 c2)).(let H1 \def H in (ex2_ind -T (\lambda (t: T).(pr3 c2 t1 t)) (\lambda (t: T).(pr3 c2 t2 t)) (pc3 c1 t1 -t2) (\lambda (x: T).(\lambda (H2: (pr3 c2 t1 x)).(\lambda (H3: (pr3 c2 t2 -x)).(ex_intro2 T (\lambda (t: T).(pr3 c1 t1 t)) (\lambda (t: T).(pr3 c1 t2 -t)) x (clear_pr3_trans c2 t1 x H2 c1 H0) (clear_pr3_trans c2 t2 x H3 c1 -H0))))) H1))))))). -(* COMMENTS -Initial nodes: 129 -END *) - -theorem pc3_pr2_r: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (pc3 c -t1 t2)))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 -t2)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) -t2 (pr3_pr2 c t1 t2 H) (pr3_refl c t2))))). -(* COMMENTS -Initial nodes: 55 -END *) - -theorem pc3_pr2_x: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t2 t1) \to (pc3 c -t1 t2)))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t2 -t1)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) -t1 (pr3_refl c t1) (pr3_pr2 c t2 t1 H))))). -(* COMMENTS -Initial nodes: 55 -END *) - -theorem pc3_pr3_r: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (pc3 c -t1 t2)))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 -t2)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) -t2 H (pr3_refl c t2))))). -(* COMMENTS -Initial nodes: 47 -END *) - -theorem pc3_pr3_x: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t2 t1) \to (pc3 c -t1 t2)))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t2 -t1)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) -t1 (pr3_refl c t1) H)))). -(* COMMENTS -Initial nodes: 47 -END *) - -theorem pc3_pr3_t: - \forall (c: C).(\forall (t1: T).(\forall (t0: T).((pr3 c t1 t0) \to (\forall -(t2: T).((pr3 c t2 t0) \to (pc3 c t1 t2)))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t0: T).(\lambda (H: (pr3 c t1 -t0)).(\lambda (t2: T).(\lambda (H0: (pr3 c t2 t0)).(ex_intro2 T (\lambda (t: -T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) t0 H H0)))))). -(* COMMENTS -Initial nodes: 53 -END *) - -theorem pc3_refl: - \forall (c: C).(\forall (t: T).(pc3 c t t)) -\def - \lambda (c: C).(\lambda (t: T).(ex_intro2 T (\lambda (t0: T).(pr3 c t t0)) -(\lambda (t0: T).(pr3 c t t0)) t (pr3_refl c t) (pr3_refl c t))). -(* COMMENTS -Initial nodes: 41 -END *) - -theorem pc3_s: - \forall (c: C).(\forall (t2: T).(\forall (t1: T).((pc3 c t1 t2) \to (pc3 c -t2 t1)))) -\def - \lambda (c: C).(\lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pc3 c t1 -t2)).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: -T).(pr3 c t2 t)) (pc3 c t2 t1) (\lambda (x: T).(\lambda (H1: (pr3 c t1 -x)).(\lambda (H2: (pr3 c t2 x)).(ex_intro2 T (\lambda (t: T).(pr3 c t2 t)) -(\lambda (t: T).(pr3 c t1 t)) x H2 H1)))) H0))))). -(* COMMENTS -Initial nodes: 97 -END *) - -theorem pc3_thin_dx: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to (\forall -(u: T).(\forall (f: F).(pc3 c (THead (Flat f) u t1) (THead (Flat f) u -t2))))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c t1 -t2)).(\lambda (u: T).(\lambda (f: F).(let H0 \def H in (ex2_ind T (\lambda -(t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) (pc3 c (THead (Flat f) u -t1) (THead (Flat f) u t2)) (\lambda (x: T).(\lambda (H1: (pr3 c t1 -x)).(\lambda (H2: (pr3 c t2 x)).(ex_intro2 T (\lambda (t: T).(pr3 c (THead -(Flat f) u t1) t)) (\lambda (t: T).(pr3 c (THead (Flat f) u t2) t)) (THead -(Flat f) u x) (pr3_thin_dx c t1 x H1 u f) (pr3_thin_dx c t2 x H2 u f))))) -H0))))))). -(* COMMENTS -Initial nodes: 165 -END *) - -theorem pc3_head_1: - \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pc3 c u1 u2) \to (\forall -(k: K).(\forall (t: T).(pc3 c (THead k u1 t) (THead k u2 t))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pc3 c u1 -u2)).(\lambda (k: K).(\lambda (t: T).(let H0 \def H in (ex2_ind T (\lambda -(t0: T).(pr3 c u1 t0)) (\lambda (t0: T).(pr3 c u2 t0)) (pc3 c (THead k u1 t) -(THead k u2 t)) (\lambda (x: T).(\lambda (H1: (pr3 c u1 x)).(\lambda (H2: -(pr3 c u2 x)).(ex_intro2 T (\lambda (t0: T).(pr3 c (THead k u1 t) t0)) -(\lambda (t0: T).(pr3 c (THead k u2 t) t0)) (THead k x t) (pr3_head_12 c u1 x -H1 k t t (pr3_refl (CHead c k x) t)) (pr3_head_12 c u2 x H2 k t t (pr3_refl -(CHead c k x) t)))))) H0))))))). -(* COMMENTS -Initial nodes: 183 -END *) - -theorem pc3_head_2: - \forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).(\forall -(k: K).((pc3 (CHead c k u) t1 t2) \to (pc3 c (THead k u t1) (THead k u -t2))))))) -\def - \lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(k: K).(\lambda (H: (pc3 (CHead c k u) t1 t2)).(let H0 \def H in (ex2_ind T -(\lambda (t: T).(pr3 (CHead c k u) t1 t)) (\lambda (t: T).(pr3 (CHead c k u) -t2 t)) (pc3 c (THead k u t1) (THead k u t2)) (\lambda (x: T).(\lambda (H1: -(pr3 (CHead c k u) t1 x)).(\lambda (H2: (pr3 (CHead c k u) t2 x)).(ex_intro2 -T (\lambda (t: T).(pr3 c (THead k u t1) t)) (\lambda (t: T).(pr3 c (THead k u -t2) t)) (THead k u x) (pr3_head_12 c u u (pr3_refl c u) k t1 x H1) -(pr3_head_12 c u u (pr3_refl c u) k t2 x H2))))) H0))))))). -(* COMMENTS -Initial nodes: 201 -END *) - -theorem pc3_pr2_u: - \forall (c: C).(\forall (t2: T).(\forall (t1: T).((pr2 c t1 t2) \to (\forall -(t3: T).((pc3 c t2 t3) \to (pc3 c t1 t3)))))) -\def - \lambda (c: C).(\lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pr2 c t1 -t2)).(\lambda (t3: T).(\lambda (H0: (pc3 c t2 t3)).(let H1 \def H0 in -(ex2_ind T (\lambda (t: T).(pr3 c t2 t)) (\lambda (t: T).(pr3 c t3 t)) (pc3 c -t1 t3) (\lambda (x: T).(\lambda (H2: (pr3 c t2 x)).(\lambda (H3: (pr3 c t3 -x)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t3 t)) -x (pr3_sing c t2 t1 H x H2) H3)))) H1))))))). -(* COMMENTS -Initial nodes: 119 -END *) - -theorem pc3_t: - \forall (t2: T).(\forall (c: C).(\forall (t1: T).((pc3 c t1 t2) \to (\forall -(t3: T).((pc3 c t2 t3) \to (pc3 c t1 t3)))))) -\def - \lambda (t2: T).(\lambda (c: C).(\lambda (t1: T).(\lambda (H: (pc3 c t1 -t2)).(\lambda (t3: T).(\lambda (H0: (pc3 c t2 t3)).(let H1 \def H0 in -(ex2_ind T (\lambda (t: T).(pr3 c t2 t)) (\lambda (t: T).(pr3 c t3 t)) (pc3 c -t1 t3) (\lambda (x: T).(\lambda (H2: (pr3 c t2 x)).(\lambda (H3: (pr3 c t3 -x)).(let H4 \def H in (ex2_ind T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: -T).(pr3 c t2 t)) (pc3 c t1 t3) (\lambda (x0: T).(\lambda (H5: (pr3 c t1 -x0)).(\lambda (H6: (pr3 c t2 x0)).(ex2_ind T (\lambda (t: T).(pr3 c x0 t)) -(\lambda (t: T).(pr3 c x t)) (pc3 c t1 t3) (\lambda (x1: T).(\lambda (H7: -(pr3 c x0 x1)).(\lambda (H8: (pr3 c x x1)).(pc3_pr3_t c t1 x1 (pr3_t x0 t1 c -H5 x1 H7) t3 (pr3_t x t3 c H3 x1 H8))))) (pr3_confluence c t2 x0 H6 x H2))))) -H4))))) H1))))))). -(* COMMENTS -Initial nodes: 233 -END *) - -theorem pc3_pr2_u2: - \forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr2 c t0 t1) \to (\forall -(t2: T).((pc3 c t0 t2) \to (pc3 c t1 t2)))))) -\def - \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr2 c t0 -t1)).(\lambda (t2: T).(\lambda (H0: (pc3 c t0 t2)).(pc3_t t0 c t1 (pc3_pr2_x -c t1 t0 H) t2 H0)))))). -(* COMMENTS -Initial nodes: 45 -END *) - -theorem pc3_pr3_conf: - \forall (c: C).(\forall (t: T).(\forall (t1: T).((pc3 c t t1) \to (\forall -(t2: T).((pr3 c t t2) \to (pc3 c t2 t1)))))) -\def - \lambda (c: C).(\lambda (t: T).(\lambda (t1: T).(\lambda (H: (pc3 c t -t1)).(\lambda (t2: T).(\lambda (H0: (pr3 c t t2)).(pc3_t t c t2 (pc3_pr3_x c -t2 t H0) t1 H)))))). -(* COMMENTS -Initial nodes: 45 -END *) - -theorem pc3_head_12: - \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pc3 c u1 u2) \to (\forall -(k: K).(\forall (t1: T).(\forall (t2: T).((pc3 (CHead c k u2) t1 t2) \to (pc3 -c (THead k u1 t1) (THead k u2 t2))))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pc3 c u1 -u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pc3 -(CHead c k u2) t1 t2)).(pc3_t (THead k u2 t1) c (THead k u1 t1) (pc3_head_1 c -u1 u2 H k t1) (THead k u2 t2) (pc3_head_2 c u2 t1 t2 k H0))))))))). -(* COMMENTS -Initial nodes: 89 -END *) - -theorem pc3_head_21: - \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pc3 c u1 u2) \to (\forall -(k: K).(\forall (t1: T).(\forall (t2: T).((pc3 (CHead c k u1) t1 t2) \to (pc3 -c (THead k u1 t1) (THead k u2 t2))))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pc3 c u1 -u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pc3 -(CHead c k u1) t1 t2)).(pc3_t (THead k u1 t2) c (THead k u1 t1) (pc3_head_2 c -u1 t1 t2 k H0) (THead k u2 t2) (pc3_head_1 c u1 u2 H k t2))))))))). -(* COMMENTS -Initial nodes: 89 -END *) - -theorem pc3_pr0_pr2_t: - \forall (u1: T).(\forall (u2: T).((pr0 u2 u1) \to (\forall (c: C).(\forall -(t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pc3 -(CHead c k u1) t1 t2)))))))) -\def - \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr0 u2 u1)).(\lambda (c: -C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k: K).(\lambda (H0: (pr2 -(CHead c k u2) t1 t2)).(insert_eq C (CHead c k u2) (\lambda (c0: C).(pr2 c0 -t1 t2)) (\lambda (_: C).(pc3 (CHead c k u1) t1 t2)) (\lambda (y: C).(\lambda -(H1: (pr2 y t1 t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: -T).((eq C c0 (CHead c k u2)) \to (pc3 (CHead c k u1) t t0))))) (\lambda (c0: -C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (H3: -(eq C c0 (CHead c k u2))).(let H4 \def (f_equal C C (\lambda (e: C).e) c0 -(CHead c k u2) H3) in (pc3_pr2_r (CHead c k u1) t3 t4 (pr2_free (CHead c k -u1) t3 t4 H2)))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: -T).(\lambda (i: nat).(\lambda (H2: (getl i c0 (CHead d (Bind Abbr) -u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda -(t: T).(\lambda (H4: (subst0 i u t4 t)).(\lambda (H5: (eq C c0 (CHead c k -u2))).(let H6 \def (f_equal C C (\lambda (e: C).e) c0 (CHead c k u2) H5) in -(let H7 \def (eq_ind C c0 (\lambda (c1: C).(getl i c1 (CHead d (Bind Abbr) -u))) H2 (CHead c k u2) H6) in (nat_ind (\lambda (n: nat).((getl n (CHead c k -u2) (CHead d (Bind Abbr) u)) \to ((subst0 n u t4 t) \to (pc3 (CHead c k u1) -t3 t)))) (\lambda (H8: (getl O (CHead c k u2) (CHead d (Bind Abbr) -u))).(\lambda (H9: (subst0 O u t4 t)).(K_ind (\lambda (k0: K).((clear (CHead -c k0 u2) (CHead d (Bind Abbr) u)) \to (pc3 (CHead c k0 u1) t3 t))) (\lambda -(b: B).(\lambda (H10: (clear (CHead c (Bind b) u2) (CHead d (Bind Abbr) -u))).(let H11 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda -(_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) -(CHead d (Bind Abbr) u) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d -(Bind Abbr) u) u2 H10)) in ((let H12 \def (f_equal C B (\lambda (e: C).(match -e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ -k0 _) \Rightarrow (match k0 in K return (\lambda (_: K).B) with [(Bind b0) -\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) -(CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 H10)) in -((let H13 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: -C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead -d (Bind Abbr) u) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind -Abbr) u) u2 H10)) in (\lambda (H14: (eq B Abbr b)).(\lambda (_: (eq C d -c)).(let H16 \def (eq_ind T u (\lambda (t0: T).(subst0 O t0 t4 t)) H9 u2 H13) -in (eq_ind B Abbr (\lambda (b0: B).(pc3 (CHead c (Bind b0) u1) t3 t)) -(ex2_ind T (\lambda (t0: T).(subst0 O u1 t4 t0)) (\lambda (t0: T).(pr0 t t0)) -(pc3 (CHead c (Bind Abbr) u1) t3 t) (\lambda (x: T).(\lambda (H17: (subst0 O -u1 t4 x)).(\lambda (H18: (pr0 t x)).(pc3_pr3_t (CHead c (Bind Abbr) u1) t3 x -(pr3_pr2 (CHead c (Bind Abbr) u1) t3 x (pr2_delta (CHead c (Bind Abbr) u1) c -u1 O (getl_refl Abbr c u1) t3 t4 H3 x H17)) t (pr3_pr2 (CHead c (Bind Abbr) -u1) t x (pr2_free (CHead c (Bind Abbr) u1) t x H18)))))) (pr0_subst0_fwd u2 -t4 t O H16 u1 H)) b H14))))) H12)) H11)))) (\lambda (f: F).(\lambda (H10: -(clear (CHead c (Flat f) u2) (CHead d (Bind Abbr) u))).(clear_pc3_trans -(CHead d (Bind Abbr) u) t3 t (pc3_pr2_r (CHead d (Bind Abbr) u) t3 t -(pr2_delta (CHead d (Bind Abbr) u) d u O (getl_refl Abbr d u) t3 t4 H3 t H9)) -(CHead c (Flat f) u1) (clear_flat c (CHead d (Bind Abbr) u) (clear_gen_flat f -c (CHead d (Bind Abbr) u) u2 H10) f u1)))) k (getl_gen_O (CHead c k u2) -(CHead d (Bind Abbr) u) H8)))) (\lambda (i0: nat).(\lambda (IHi: (((getl i0 -(CHead c k u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t4 t) \to (pc3 -(CHead c k u1) t3 t))))).(\lambda (H8: (getl (S i0) (CHead c k u2) (CHead d -(Bind Abbr) u))).(\lambda (H9: (subst0 (S i0) u t4 t)).(K_ind (\lambda (k0: -K).((((getl i0 (CHead c k0 u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t4 -t) \to (pc3 (CHead c k0 u1) t3 t)))) \to ((getl (r k0 i0) c (CHead d (Bind -Abbr) u)) \to (pc3 (CHead c k0 u1) t3 t)))) (\lambda (b: B).(\lambda (_: -(((getl i0 (CHead c (Bind b) u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u -t4 t) \to (pc3 (CHead c (Bind b) u1) t3 t))))).(\lambda (H10: (getl (r (Bind -b) i0) c (CHead d (Bind Abbr) u))).(pc3_pr2_r (CHead c (Bind b) u1) t3 t -(pr2_delta (CHead c (Bind b) u1) d u (S i0) (getl_head (Bind b) i0 c (CHead d -(Bind Abbr) u) H10 u1) t3 t4 H3 t H9))))) (\lambda (f: F).(\lambda (_: -(((getl i0 (CHead c (Flat f) u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u -t4 t) \to (pc3 (CHead c (Flat f) u1) t3 t))))).(\lambda (H10: (getl (r (Flat -f) i0) c (CHead d (Bind Abbr) u))).(pc3_pr2_r (CHead c (Flat f) u1) t3 t -(pr2_cflat c t3 t (pr2_delta c d u (r (Flat f) i0) H10 t3 t4 H3 t H9) f -u1))))) k IHi (getl_gen_S k c (CHead d (Bind Abbr) u) u2 i0 H8)))))) i H7 -H4)))))))))))))) y t1 t2 H1))) H0)))))))). -(* COMMENTS -Initial nodes: 1533 -END *) - -theorem pc3_pr2_pr2_t: - \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr2 c u2 u1) \to (\forall -(t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pc3 -(CHead c k u1) t1 t2)))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr2 c u2 -u1)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (t1: -T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c0 k t) t1 t2) \to (pc3 -(CHead c0 k t0) t1 t2)))))))) (\lambda (c0: C).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (H0: (pr0 t1 t2)).(\lambda (t0: T).(\lambda (t3: T).(\lambda (k: -K).(\lambda (H1: (pr2 (CHead c0 k t1) t0 t3)).(pc3_pr0_pr2_t t2 t1 H0 c0 t0 -t3 k H1))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda -(i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t1: -T).(\lambda (t2: T).(\lambda (H1: (pr0 t1 t2)).(\lambda (t: T).(\lambda (H2: -(subst0 i u t2 t)).(\lambda (t0: T).(\lambda (t3: T).(\lambda (k: K).(\lambda -(H3: (pr2 (CHead c0 k t1) t0 t3)).(insert_eq C (CHead c0 k t1) (\lambda (c1: -C).(pr2 c1 t0 t3)) (\lambda (_: C).(pc3 (CHead c0 k t) t0 t3)) (\lambda (y: -C).(\lambda (H4: (pr2 y t0 t3)).(pr2_ind (\lambda (c1: C).(\lambda (t4: -T).(\lambda (t5: T).((eq C c1 (CHead c0 k t1)) \to (pc3 (CHead c0 k t) t4 -t5))))) (\lambda (c1: C).(\lambda (t4: T).(\lambda (t5: T).(\lambda (H5: (pr0 -t4 t5)).(\lambda (_: (eq C c1 (CHead c0 k t1))).(pc3_pr2_r (CHead c0 k t) t4 -t5 (pr2_free (CHead c0 k t) t4 t5 H5))))))) (\lambda (c1: C).(\lambda (d0: -C).(\lambda (u0: T).(\lambda (i0: nat).(\lambda (H5: (getl i0 c1 (CHead d0 -(Bind Abbr) u0))).(\lambda (t4: T).(\lambda (t5: T).(\lambda (H6: (pr0 t4 -t5)).(\lambda (t6: T).(\lambda (H7: (subst0 i0 u0 t5 t6)).(\lambda (H8: (eq C -c1 (CHead c0 k t1))).(let H9 \def (eq_ind C c1 (\lambda (c2: C).(getl i0 c2 -(CHead d0 (Bind Abbr) u0))) H5 (CHead c0 k t1) H8) in (nat_ind (\lambda (n: -nat).((getl n (CHead c0 k t1) (CHead d0 (Bind Abbr) u0)) \to ((subst0 n u0 t5 -t6) \to (pc3 (CHead c0 k t) t4 t6)))) (\lambda (H10: (getl O (CHead c0 k t1) -(CHead d0 (Bind Abbr) u0))).(\lambda (H11: (subst0 O u0 t5 t6)).(K_ind -(\lambda (k0: K).((clear (CHead c0 k0 t1) (CHead d0 (Bind Abbr) u0)) \to (pc3 -(CHead c0 k0 t) t4 t6))) (\lambda (b: B).(\lambda (H12: (clear (CHead c0 -(Bind b) t1) (CHead d0 (Bind Abbr) u0))).(let H13 \def (f_equal C C (\lambda -(e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d0 -| (CHead c2 _ _) \Rightarrow c2])) (CHead d0 (Bind Abbr) u0) (CHead c0 (Bind -b) t1) (clear_gen_bind b c0 (CHead d0 (Bind Abbr) u0) t1 H12)) in ((let H14 -\def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) -with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow (match k0 in K -return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) -\Rightarrow Abbr])])) (CHead d0 (Bind Abbr) u0) (CHead c0 (Bind b) t1) -(clear_gen_bind b c0 (CHead d0 (Bind Abbr) u0) t1 H12)) in ((let H15 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow u0 | (CHead _ _ t7) \Rightarrow t7])) (CHead d0 (Bind -Abbr) u0) (CHead c0 (Bind b) t1) (clear_gen_bind b c0 (CHead d0 (Bind Abbr) -u0) t1 H12)) in (\lambda (H16: (eq B Abbr b)).(\lambda (_: (eq C d0 c0)).(let -H18 \def (eq_ind T u0 (\lambda (t7: T).(subst0 O t7 t5 t6)) H11 t1 H15) in -(eq_ind B Abbr (\lambda (b0: B).(pc3 (CHead c0 (Bind b0) t) t4 t6)) (ex2_ind -T (\lambda (t7: T).(subst0 O t2 t5 t7)) (\lambda (t7: T).(pr0 t6 t7)) (pc3 -(CHead c0 (Bind Abbr) t) t4 t6) (\lambda (x: T).(\lambda (H19: (subst0 O t2 -t5 x)).(\lambda (H20: (pr0 t6 x)).(ex2_ind T (\lambda (t7: T).(subst0 O t t5 -t7)) (\lambda (t7: T).(subst0 (S (plus i O)) u x t7)) (pc3 (CHead c0 (Bind -Abbr) t) t4 t6) (\lambda (x0: T).(\lambda (H21: (subst0 O t t5 x0)).(\lambda -(H22: (subst0 (S (plus i O)) u x x0)).(let H23 \def (f_equal nat nat S (plus -i O) i (sym_eq nat i (plus i O) (plus_n_O i))) in (let H24 \def (eq_ind nat -(S (plus i O)) (\lambda (n: nat).(subst0 n u x x0)) H22 (S i) H23) in -(pc3_pr2_u (CHead c0 (Bind Abbr) t) x0 t4 (pr2_delta (CHead c0 (Bind Abbr) t) -c0 t O (getl_refl Abbr c0 t) t4 t5 H6 x0 H21) t6 (pc3_pr2_x (CHead c0 (Bind -Abbr) t) x0 t6 (pr2_delta (CHead c0 (Bind Abbr) t) d u (S i) (getl_head (Bind -Abbr) i c0 (CHead d (Bind Abbr) u) H0 t) t6 x H20 x0 H24)))))))) -(subst0_subst0_back t5 x t2 O H19 t u i H2))))) (pr0_subst0_fwd t1 t5 t6 O -H18 t2 H1)) b H16))))) H14)) H13)))) (\lambda (f: F).(\lambda (H12: (clear -(CHead c0 (Flat f) t1) (CHead d0 (Bind Abbr) u0))).(clear_pc3_trans (CHead d0 -(Bind Abbr) u0) t4 t6 (pc3_pr2_r (CHead d0 (Bind Abbr) u0) t4 t6 (pr2_delta -(CHead d0 (Bind Abbr) u0) d0 u0 O (getl_refl Abbr d0 u0) t4 t5 H6 t6 H11)) -(CHead c0 (Flat f) t) (clear_flat c0 (CHead d0 (Bind Abbr) u0) -(clear_gen_flat f c0 (CHead d0 (Bind Abbr) u0) t1 H12) f t)))) k (getl_gen_O -(CHead c0 k t1) (CHead d0 (Bind Abbr) u0) H10)))) (\lambda (i1: nat).(\lambda -(_: (((getl i1 (CHead c0 k t1) (CHead d0 (Bind Abbr) u0)) \to ((subst0 i1 u0 -t5 t6) \to (pc3 (CHead c0 k t) t4 t6))))).(\lambda (H10: (getl (S i1) (CHead -c0 k t1) (CHead d0 (Bind Abbr) u0))).(\lambda (H11: (subst0 (S i1) u0 t5 -t6)).(K_ind (\lambda (k0: K).((getl (r k0 i1) c0 (CHead d0 (Bind Abbr) u0)) -\to (pc3 (CHead c0 k0 t) t4 t6))) (\lambda (b: B).(\lambda (H12: (getl (r -(Bind b) i1) c0 (CHead d0 (Bind Abbr) u0))).(pc3_pr2_r (CHead c0 (Bind b) t) -t4 t6 (pr2_delta (CHead c0 (Bind b) t) d0 u0 (S i1) (getl_head (Bind b) i1 c0 -(CHead d0 (Bind Abbr) u0) H12 t) t4 t5 H6 t6 H11)))) (\lambda (f: F).(\lambda -(H12: (getl (r (Flat f) i1) c0 (CHead d0 (Bind Abbr) u0))).(pc3_pr2_r (CHead -c0 (Flat f) t) t4 t6 (pr2_cflat c0 t4 t6 (pr2_delta c0 d0 u0 (r (Flat f) i1) -H12 t4 t5 H6 t6 H11) f t)))) k (getl_gen_S k c0 (CHead d0 (Bind Abbr) u0) t1 -i1 H10)))))) i0 H9 H7))))))))))))) y t0 t3 H4))) H3))))))))))))))) c u2 u1 -H)))). -(* COMMENTS -Initial nodes: 1671 -END *) - -theorem pc3_pr2_pr3_t: - \forall (c: C).(\forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall -(k: K).((pr3 (CHead c k u2) t1 t2) \to (\forall (u1: T).((pr2 c u2 u1) \to -(pc3 (CHead c k u1) t1 t2)))))))) -\def - \lambda (c: C).(\lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(k: K).(\lambda (H: (pr3 (CHead c k u2) t1 t2)).(pr3_ind (CHead c k u2) -(\lambda (t: T).(\lambda (t0: T).(\forall (u1: T).((pr2 c u2 u1) \to (pc3 -(CHead c k u1) t t0))))) (\lambda (t: T).(\lambda (u1: T).(\lambda (_: (pr2 c -u2 u1)).(pc3_refl (CHead c k u1) t)))) (\lambda (t0: T).(\lambda (t3: -T).(\lambda (H0: (pr2 (CHead c k u2) t3 t0)).(\lambda (t4: T).(\lambda (_: -(pr3 (CHead c k u2) t0 t4)).(\lambda (H2: ((\forall (u1: T).((pr2 c u2 u1) -\to (pc3 (CHead c k u1) t0 t4))))).(\lambda (u1: T).(\lambda (H3: (pr2 c u2 -u1)).(pc3_t t0 (CHead c k u1) t3 (pc3_pr2_pr2_t c u1 u2 H3 t3 t0 k H0) t4 (H2 -u1 H3)))))))))) t1 t2 H)))))). -(* COMMENTS -Initial nodes: 199 -END *) - -theorem pc3_pr3_pc3_t: - \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u2 u1) \to (\forall -(t1: T).(\forall (t2: T).(\forall (k: K).((pc3 (CHead c k u2) t1 t2) \to (pc3 -(CHead c k u1) t1 t2)))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u2 -u1)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (t1: T).(\forall -(t2: T).(\forall (k: K).((pc3 (CHead c k t) t1 t2) \to (pc3 (CHead c k t0) t1 -t2))))))) (\lambda (t: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k: -K).(\lambda (H0: (pc3 (CHead c k t) t1 t2)).H0))))) (\lambda (t2: T).(\lambda -(t1: T).(\lambda (H0: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda (_: (pr3 c t2 -t3)).(\lambda (H2: ((\forall (t4: T).(\forall (t5: T).(\forall (k: K).((pc3 -(CHead c k t2) t4 t5) \to (pc3 (CHead c k t3) t4 t5))))))).(\lambda (t0: -T).(\lambda (t4: T).(\lambda (k: K).(\lambda (H3: (pc3 (CHead c k t1) t0 -t4)).(H2 t0 t4 k (let H4 \def H3 in (ex2_ind T (\lambda (t: T).(pr3 (CHead c -k t1) t0 t)) (\lambda (t: T).(pr3 (CHead c k t1) t4 t)) (pc3 (CHead c k t2) -t0 t4) (\lambda (x: T).(\lambda (H5: (pr3 (CHead c k t1) t0 x)).(\lambda (H6: -(pr3 (CHead c k t1) t4 x)).(pc3_t x (CHead c k t2) t0 (pc3_pr2_pr3_t c t1 t0 -x k H5 t2 H0) t4 (pc3_s (CHead c k t2) x t4 (pc3_pr2_pr3_t c t1 t4 x k H6 t2 -H0)))))) H4))))))))))))) u2 u1 H)))). -(* COMMENTS -Initial nodes: 319 -END *) - -theorem pc3_lift: - \forall (c: C).(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h -d c e) \to (\forall (t1: T).(\forall (t2: T).((pc3 e t1 t2) \to (pc3 c (lift -h d t1) (lift h d t2))))))))) -\def - \lambda (c: C).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda -(H: (drop h d c e)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pc3 e t1 -t2)).(let H1 \def H0 in (ex2_ind T (\lambda (t: T).(pr3 e t1 t)) (\lambda (t: -T).(pr3 e t2 t)) (pc3 c (lift h d t1) (lift h d t2)) (\lambda (x: T).(\lambda -(H2: (pr3 e t1 x)).(\lambda (H3: (pr3 e t2 x)).(pc3_pr3_t c (lift h d t1) -(lift h d x) (pr3_lift c e h d H t1 x H2) (lift h d t2) (pr3_lift c e h d H -t2 x H3))))) H1))))))))). -(* COMMENTS -Initial nodes: 159 -END *) - -theorem pc3_eta: - \forall (c: C).(\forall (t: T).(\forall (w: T).(\forall (u: T).((pc3 c t -(THead (Bind Abst) w u)) \to (\forall (v: T).((pc3 c v w) \to (pc3 c (THead -(Bind Abst) v (THead (Flat Appl) (TLRef O) (lift (S O) O t))) t))))))) -\def - \lambda (c: C).(\lambda (t: T).(\lambda (w: T).(\lambda (u: T).(\lambda (H: -(pc3 c t (THead (Bind Abst) w u))).(\lambda (v: T).(\lambda (H0: (pc3 c v -w)).(pc3_t (THead (Bind Abst) w (THead (Flat Appl) (TLRef O) (lift (S O) O -(THead (Bind Abst) w u)))) c (THead (Bind Abst) v (THead (Flat Appl) (TLRef -O) (lift (S O) O t))) (pc3_head_21 c v w H0 (Bind Abst) (THead (Flat Appl) -(TLRef O) (lift (S O) O t)) (THead (Flat Appl) (TLRef O) (lift (S O) O (THead -(Bind Abst) w u))) (pc3_thin_dx (CHead c (Bind Abst) v) (lift (S O) O t) -(lift (S O) O (THead (Bind Abst) w u)) (pc3_lift (CHead c (Bind Abst) v) c (S -O) O (drop_drop (Bind Abst) O c c (drop_refl c) v) t (THead (Bind Abst) w u) -H) (TLRef O) Appl)) t (pc3_t (THead (Bind Abst) w u) c (THead (Bind Abst) w -(THead (Flat Appl) (TLRef O) (lift (S O) O (THead (Bind Abst) w u)))) -(pc3_pr3_r c (THead (Bind Abst) w (THead (Flat Appl) (TLRef O) (lift (S O) O -(THead (Bind Abst) w u)))) (THead (Bind Abst) w u) (pr3_eta c w u w (pr3_refl -c w))) t (pc3_s c (THead (Bind Abst) w u) t H))))))))). -(* COMMENTS -Initial nodes: 399 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/subst1.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/subst1.ma deleted file mode 100644 index 3af245b3a..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/subst1.ma +++ /dev/null @@ -1,48 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pc3/props.ma". - -include "Basic-1/pr3/subst1.ma". - -theorem pc3_gen_cabbr: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to (\forall -(e: C).(\forall (u: T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) -\to (\forall (a0: C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d -a0 a) \to (\forall (x1: T).((subst1 d u t1 (lift (S O) d x1)) \to (\forall -(x2: T).((subst1 d u t2 (lift (S O) d x2)) \to (pc3 a x1 x2)))))))))))))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c t1 -t2)).(\lambda (e: C).(\lambda (u: T).(\lambda (d: nat).(\lambda (H0: (getl d -c (CHead e (Bind Abbr) u))).(\lambda (a0: C).(\lambda (H1: (csubst1 d u c -a0)).(\lambda (a: C).(\lambda (H2: (drop (S O) d a0 a)).(\lambda (x1: -T).(\lambda (H3: (subst1 d u t1 (lift (S O) d x1))).(\lambda (x2: T).(\lambda -(H4: (subst1 d u t2 (lift (S O) d x2))).(let H5 \def H in (ex2_ind T (\lambda -(t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) (pc3 a x1 x2) (\lambda (x: -T).(\lambda (H6: (pr3 c t1 x)).(\lambda (H7: (pr3 c t2 x)).(ex2_ind T -(\lambda (x3: T).(subst1 d u x (lift (S O) d x3))) (\lambda (x3: T).(pr3 a x2 -x3)) (pc3 a x1 x2) (\lambda (x0: T).(\lambda (H8: (subst1 d u x (lift (S O) d -x0))).(\lambda (H9: (pr3 a x2 x0)).(ex2_ind T (\lambda (x3: T).(subst1 d u x -(lift (S O) d x3))) (\lambda (x3: T).(pr3 a x1 x3)) (pc3 a x1 x2) (\lambda -(x3: T).(\lambda (H10: (subst1 d u x (lift (S O) d x3))).(\lambda (H11: (pr3 -a x1 x3)).(let H12 \def (eq_ind T x3 (\lambda (t: T).(pr3 a x1 t)) H11 x0 -(subst1_confluence_lift x x3 u d H10 x0 H8)) in (pc3_pr3_t a x1 x0 H12 x2 -H9))))) (pr3_gen_cabbr c t1 x H6 e u d H0 a0 H1 a H2 x1 H3))))) -(pr3_gen_cabbr c t2 x H7 e u d H0 a0 H1 a H2 x2 H4))))) H5))))))))))))))))). -(* COMMENTS -Initial nodes: 405 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/wcpr0.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/wcpr0.ma deleted file mode 100644 index 2b51edf62..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/wcpr0.ma +++ /dev/null @@ -1,96 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pc3/props.ma". - -include "Basic-1/wcpr0/getl.ma". - -theorem pc3_wcpr0__pc3_wcpr0_t_aux: - \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (k: K).(\forall -(u: T).(\forall (t1: T).(\forall (t2: T).((pr3 (CHead c1 k u) t1 t2) \to (pc3 -(CHead c2 k u) t1 t2)))))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(\lambda (k: -K).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 -(CHead c1 k u) t1 t2)).(pr3_ind (CHead c1 k u) (\lambda (t: T).(\lambda (t0: -T).(pc3 (CHead c2 k u) t t0))) (\lambda (t: T).(pc3_refl (CHead c2 k u) t)) -(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr2 (CHead c1 k u) t4 -t3)).(\lambda (t5: T).(\lambda (_: (pr3 (CHead c1 k u) t3 t5)).(\lambda (H3: -(pc3 (CHead c2 k u) t3 t5)).(pc3_t t3 (CHead c2 k u) t4 (insert_eq C (CHead -c1 k u) (\lambda (c: C).(pr2 c t4 t3)) (\lambda (_: C).(pc3 (CHead c2 k u) t4 -t3)) (\lambda (y: C).(\lambda (H4: (pr2 y t4 t3)).(pr2_ind (\lambda (c: -C).(\lambda (t: T).(\lambda (t0: T).((eq C c (CHead c1 k u)) \to (pc3 (CHead -c2 k u) t t0))))) (\lambda (c: C).(\lambda (t6: T).(\lambda (t0: T).(\lambda -(H5: (pr0 t6 t0)).(\lambda (_: (eq C c (CHead c1 k u))).(pc3_pr2_r (CHead c2 -k u) t6 t0 (pr2_free (CHead c2 k u) t6 t0 H5))))))) (\lambda (c: C).(\lambda -(d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H5: (getl i c (CHead d -(Bind Abbr) u0))).(\lambda (t6: T).(\lambda (t0: T).(\lambda (H6: (pr0 t6 -t0)).(\lambda (t: T).(\lambda (H7: (subst0 i u0 t0 t)).(\lambda (H8: (eq C c -(CHead c1 k u))).(let H9 \def (eq_ind C c (\lambda (c0: C).(getl i c0 (CHead -d (Bind Abbr) u0))) H5 (CHead c1 k u) H8) in (ex3_2_ind C T (\lambda (e2: -C).(\lambda (u2: T).(getl i (CHead c2 k u) (CHead e2 (Bind Abbr) u2)))) -(\lambda (e2: C).(\lambda (_: T).(wcpr0 d e2))) (\lambda (_: C).(\lambda (u2: -T).(pr0 u0 u2))) (pc3 (CHead c2 k u) t6 t) (\lambda (x0: C).(\lambda (x1: -T).(\lambda (H10: (getl i (CHead c2 k u) (CHead x0 (Bind Abbr) x1))).(\lambda -(_: (wcpr0 d x0)).(\lambda (H12: (pr0 u0 x1)).(ex2_ind T (\lambda (t7: -T).(subst0 i x1 t0 t7)) (\lambda (t7: T).(pr0 t t7)) (pc3 (CHead c2 k u) t6 -t) (\lambda (x: T).(\lambda (H13: (subst0 i x1 t0 x)).(\lambda (H14: (pr0 t -x)).(pc3_pr2_u (CHead c2 k u) x t6 (pr2_delta (CHead c2 k u) x0 x1 i H10 t6 -t0 H6 x H13) t (pc3_pr2_x (CHead c2 k u) x t (pr2_free (CHead c2 k u) t x -H14)))))) (pr0_subst0_fwd u0 t0 t i H7 x1 H12))))))) (wcpr0_getl (CHead c1 k -u) (CHead c2 k u) (wcpr0_comp c1 c2 H u u (pr0_refl u) k) i d u0 (Bind Abbr) -H9)))))))))))))) y t4 t3 H4))) H1) t5 H3))))))) t1 t2 H0)))))))). -(* COMMENTS -Initial nodes: 689 -END *) - -theorem pc3_wcpr0_t: - \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (t1: -T).(\forall (t2: T).((pr3 c1 t1 t2) \to (pc3 c2 t1 t2)))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(wcpr0_ind -(\lambda (c: C).(\lambda (c0: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 -t2) \to (pc3 c0 t1 t2)))))) (\lambda (c: C).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (H0: (pr3 c t1 t2)).(pc3_pr3_r c t1 t2 H0))))) (\lambda (c0: -C).(\lambda (c3: C).(\lambda (H0: (wcpr0 c0 c3)).(\lambda (_: ((\forall (t1: -T).(\forall (t2: T).((pr3 c0 t1 t2) \to (pc3 c3 t1 t2)))))).(\lambda (u1: -T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 u2)).(\lambda (k: K).(\lambda (t1: -T).(\lambda (t2: T).(\lambda (H3: (pr3 (CHead c0 k u1) t1 t2)).(let H4 \def -(pc3_pr2_pr3_t c0 u1 t1 t2 k H3 u2 (pr2_free c0 u1 u2 H2)) in (ex2_ind T -(\lambda (t: T).(pr3 (CHead c0 k u2) t1 t)) (\lambda (t: T).(pr3 (CHead c0 k -u2) t2 t)) (pc3 (CHead c3 k u2) t1 t2) (\lambda (x: T).(\lambda (H5: (pr3 -(CHead c0 k u2) t1 x)).(\lambda (H6: (pr3 (CHead c0 k u2) t2 x)).(pc3_t x -(CHead c3 k u2) t1 (pc3_wcpr0__pc3_wcpr0_t_aux c0 c3 H0 k u2 t1 x H5) t2 -(pc3_s (CHead c3 k u2) x t2 (pc3_wcpr0__pc3_wcpr0_t_aux c0 c3 H0 k u2 t2 x -H6)))))) H4))))))))))))) c1 c2 H))). -(* COMMENTS -Initial nodes: 299 -END *) - -theorem pc3_wcpr0: - \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (t1: -T).(\forall (t2: T).((pc3 c1 t1 t2) \to (pc3 c2 t1 t2)))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(\lambda (t1: -T).(\lambda (t2: T).(\lambda (H0: (pc3 c1 t1 t2)).(let H1 \def H0 in (ex2_ind -T (\lambda (t: T).(pr3 c1 t1 t)) (\lambda (t: T).(pr3 c1 t2 t)) (pc3 c2 t1 -t2) (\lambda (x: T).(\lambda (H2: (pr3 c1 t1 x)).(\lambda (H3: (pr3 c1 t2 -x)).(pc3_t x c2 t1 (pc3_wcpr0_t c1 c2 H t1 x H2) t2 (pc3_s c2 x t2 -(pc3_wcpr0_t c1 c2 H t2 x H3)))))) H1))))))). -(* COMMENTS -Initial nodes: 121 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr0/dec.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr0/dec.ma deleted file mode 100644 index c28504bee..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr0/dec.ma +++ /dev/null @@ -1,529 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr0/fwd.ma". - -include "Basic-1/subst0/dec.ma". - -include "Basic-1/T/dec.ma". - -include "Basic-1/T/props.ma". - -theorem nf0_dec: - \forall (t1: T).(or (\forall (t2: T).((pr0 t1 t2) \to (eq T t1 t2))) (ex2 T -(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: -T).(pr0 t1 t2)))) -\def - \lambda (t1: T).(T_ind (\lambda (t: T).(or (\forall (t2: T).((pr0 t t2) \to -(eq T t t2))) (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 t t2))))) (\lambda (n: nat).(or_introl -(\forall (t2: T).((pr0 (TSort n) t2) \to (eq T (TSort n) t2))) (ex2 T -(\lambda (t2: T).((eq T (TSort n) t2) \to (\forall (P: Prop).P))) (\lambda -(t2: T).(pr0 (TSort n) t2))) (\lambda (t2: T).(\lambda (H: (pr0 (TSort n) -t2)).(eq_ind_r T (TSort n) (\lambda (t: T).(eq T (TSort n) t)) (refl_equal T -(TSort n)) t2 (pr0_gen_sort t2 n H)))))) (\lambda (n: nat).(or_introl -(\forall (t2: T).((pr0 (TLRef n) t2) \to (eq T (TLRef n) t2))) (ex2 T -(\lambda (t2: T).((eq T (TLRef n) t2) \to (\forall (P: Prop).P))) (\lambda -(t2: T).(pr0 (TLRef n) t2))) (\lambda (t2: T).(\lambda (H: (pr0 (TLRef n) -t2)).(eq_ind_r T (TLRef n) (\lambda (t: T).(eq T (TLRef n) t)) (refl_equal T -(TLRef n)) t2 (pr0_gen_lref t2 n H)))))) (\lambda (k: K).(\lambda (t: -T).(\lambda (H: (or (\forall (t2: T).((pr0 t t2) \to (eq T t t2))) (ex2 T -(\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: -T).(pr0 t t2))))).(\lambda (t0: T).(\lambda (H0: (or (\forall (t2: T).((pr0 -t0 t2) \to (eq T t0 t2))) (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall -(P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2))))).(K_ind (\lambda (k0: K).(or -(\forall (t2: T).((pr0 (THead k0 t t0) t2) \to (eq T (THead k0 t t0) t2))) -(ex2 T (\lambda (t2: T).((eq T (THead k0 t t0) t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 (THead k0 t t0) t2))))) (\lambda (b: -B).(B_ind (\lambda (b0: B).(or (\forall (t2: T).((pr0 (THead (Bind b0) t t0) -t2) \to (eq T (THead (Bind b0) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T -(THead (Bind b0) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 -(THead (Bind b0) t t0) t2))))) (or_intror (\forall (t2: T).((pr0 (THead (Bind -Abbr) t t0) t2) \to (eq T (THead (Bind Abbr) t t0) t2))) (ex2 T (\lambda (t2: -T).((eq T (THead (Bind Abbr) t t0) t2) \to (\forall (P: Prop).P))) (\lambda -(t2: T).(pr0 (THead (Bind Abbr) t t0) t2))) (let H_x \def (dnf_dec t t0 O) in -(let H1 \def H_x in (ex_ind T (\lambda (v: T).(or (subst0 O t t0 (lift (S O) -O v)) (eq T t0 (lift (S O) O v)))) (ex2 T (\lambda (t2: T).((eq T (THead -(Bind Abbr) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 -(THead (Bind Abbr) t t0) t2))) (\lambda (x: T).(\lambda (H2: (or (subst0 O t -t0 (lift (S O) O x)) (eq T t0 (lift (S O) O x)))).(or_ind (subst0 O t t0 -(lift (S O) O x)) (eq T t0 (lift (S O) O x)) (ex2 T (\lambda (t2: T).((eq T -(THead (Bind Abbr) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: -T).(pr0 (THead (Bind Abbr) t t0) t2))) (\lambda (H3: (subst0 O t t0 (lift (S -O) O x))).(ex_intro2 T (\lambda (t2: T).((eq T (THead (Bind Abbr) t t0) t2) -\to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abbr) t t0) -t2)) (THead (Bind Abbr) t (lift (S O) O x)) (\lambda (H4: (eq T (THead (Bind -Abbr) t t0) (THead (Bind Abbr) t (lift (S O) O x)))).(\lambda (P: Prop).(let -H5 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t2) -\Rightarrow t2])) (THead (Bind Abbr) t t0) (THead (Bind Abbr) t (lift (S O) O -x)) H4) in (let H6 \def (eq_ind T t0 (\lambda (t2: T).(subst0 O t t2 (lift (S -O) O x))) H3 (lift (S O) O x) H5) in (subst0_refl t (lift (S O) O x) O H6 -P))))) (pr0_delta t t (pr0_refl t) t0 t0 (pr0_refl t0) (lift (S O) O x) H3))) -(\lambda (H3: (eq T t0 (lift (S O) O x))).(eq_ind_r T (lift (S O) O x) -(\lambda (t2: T).(ex2 T (\lambda (t3: T).((eq T (THead (Bind Abbr) t t2) t3) -\to (\forall (P: Prop).P))) (\lambda (t3: T).(pr0 (THead (Bind Abbr) t t2) -t3)))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Bind Abbr) t (lift (S O) -O x)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind -Abbr) t (lift (S O) O x)) t2)) x (\lambda (H4: (eq T (THead (Bind Abbr) t -(lift (S O) O x)) x)).(\lambda (P: Prop).(thead_x_lift_y_y (Bind Abbr) x t (S -O) O H4 P))) (pr0_zeta Abbr not_abbr_abst x x (pr0_refl x) t)) t0 H3)) H2))) -H1)))) (let H1 \def H in (or_ind (\forall (t2: T).((pr0 t t2) \to (eq T t -t2))) (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) -(\lambda (t2: T).(pr0 t t2))) (or (\forall (t2: T).((pr0 (THead (Bind Abst) t -t0) t2) \to (eq T (THead (Bind Abst) t t0) t2))) (ex2 T (\lambda (t2: T).((eq -T (THead (Bind Abst) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: -T).(pr0 (THead (Bind Abst) t t0) t2)))) (\lambda (H2: ((\forall (t2: T).((pr0 -t t2) \to (eq T t t2))))).(let H3 \def H0 in (or_ind (\forall (t2: T).((pr0 -t0 t2) \to (eq T t0 t2))) (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall -(P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2))) (or (\forall (t2: T).((pr0 -(THead (Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0) t2))) (ex2 T -(\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2)))) (\lambda -(H4: ((\forall (t2: T).((pr0 t0 t2) \to (eq T t0 t2))))).(or_introl (\forall -(t2: T).((pr0 (THead (Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0) -t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to -(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2))) -(\lambda (t2: T).(\lambda (H5: (pr0 (THead (Bind Abst) t t0) t2)).(ex3_2_ind -T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t0 t3))) (eq T (THead (Bind Abst) t t0) t2) (\lambda (x0: T).(\lambda -(x1: T).(\lambda (H6: (eq T t2 (THead (Bind Abst) x0 x1))).(\lambda (H7: (pr0 -t x0)).(\lambda (H8: (pr0 t0 x1)).(let H_y \def (H4 x1 H8) in (let H_y0 \def -(H2 x0 H7) in (let H9 \def (eq_ind_r T x1 (\lambda (t3: T).(pr0 t0 t3)) H8 t0 -H_y) in (let H10 \def (eq_ind_r T x1 (\lambda (t3: T).(eq T t2 (THead (Bind -Abst) x0 t3))) H6 t0 H_y) in (let H11 \def (eq_ind_r T x0 (\lambda (t3: -T).(pr0 t t3)) H7 t H_y0) in (let H12 \def (eq_ind_r T x0 (\lambda (t3: -T).(eq T t2 (THead (Bind Abst) t3 t0))) H10 t H_y0) in (eq_ind_r T (THead -(Bind Abst) t t0) (\lambda (t3: T).(eq T (THead (Bind Abst) t t0) t3)) -(refl_equal T (THead (Bind Abst) t t0)) t2 H12)))))))))))) (pr0_gen_abst t t0 -t2 H5)))))) (\lambda (H4: (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall -(P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2)))).(ex2_ind T (\lambda (t2: -T).((eq T t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2)) -(or (\forall (t2: T).((pr0 (THead (Bind Abst) t t0) t2) \to (eq T (THead -(Bind Abst) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abst) t -t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) -t t0) t2)))) (\lambda (x: T).(\lambda (H5: (((eq T t0 x) \to (\forall (P: -Prop).P)))).(\lambda (H6: (pr0 t0 x)).(or_intror (\forall (t2: T).((pr0 -(THead (Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0) t2))) (ex2 T -(\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2))) (ex_intro2 T -(\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2)) (THead (Bind -Abst) t x) (\lambda (H7: (eq T (THead (Bind Abst) t t0) (THead (Bind Abst) t -x))).(\lambda (P: Prop).(let H8 \def (f_equal T T (\lambda (e: T).(match e in -T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) -\Rightarrow t0 | (THead _ _ t2) \Rightarrow t2])) (THead (Bind Abst) t t0) -(THead (Bind Abst) t x) H7) in (let H9 \def (eq_ind_r T x (\lambda (t2: -T).(pr0 t0 t2)) H6 t0 H8) in (let H10 \def (eq_ind_r T x (\lambda (t2: -T).((eq T t0 t2) \to (\forall (P0: Prop).P0))) H5 t0 H8) in (H10 (refl_equal -T t0) P)))))) (pr0_comp t t (pr0_refl t) t0 x H6 (Bind Abst))))))) H4)) H3))) -(\lambda (H2: (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 t t2)))).(ex2_ind T (\lambda (t2: T).((eq T -t t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2)) (or (\forall -(t2: T).((pr0 (THead (Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0) -t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to -(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2)))) -(\lambda (x: T).(\lambda (H3: (((eq T t x) \to (\forall (P: -Prop).P)))).(\lambda (H4: (pr0 t x)).(or_intror (\forall (t2: T).((pr0 (THead -(Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0) t2))) (ex2 T -(\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2))) (ex_intro2 T -(\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2)) (THead (Bind -Abst) x t0) (\lambda (H5: (eq T (THead (Bind Abst) t t0) (THead (Bind Abst) x -t0))).(\lambda (P: Prop).(let H6 \def (f_equal T T (\lambda (e: T).(match e -in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t | (TLRef _) -\Rightarrow t | (THead _ t2 _) \Rightarrow t2])) (THead (Bind Abst) t t0) -(THead (Bind Abst) x t0) H5) in (let H7 \def (eq_ind_r T x (\lambda (t2: -T).(pr0 t t2)) H4 t H6) in (let H8 \def (eq_ind_r T x (\lambda (t2: T).((eq T -t t2) \to (\forall (P0: Prop).P0))) H3 t H6) in (H8 (refl_equal T t) P)))))) -(pr0_comp t x H4 t0 t0 (pr0_refl t0) (Bind Abst))))))) H2)) H1)) (let H_x -\def (dnf_dec t t0 O) in (let H1 \def H_x in (ex_ind T (\lambda (v: T).(or -(subst0 O t t0 (lift (S O) O v)) (eq T t0 (lift (S O) O v)))) (or (\forall -(t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) -t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to -(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)))) -(\lambda (x: T).(\lambda (H2: (or (subst0 O t t0 (lift (S O) O x)) (eq T t0 -(lift (S O) O x)))).(or_ind (subst0 O t t0 (lift (S O) O x)) (eq T t0 (lift -(S O) O x)) (or (\forall (t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T -(THead (Bind Void) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind -Void) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead -(Bind Void) t t0) t2)))) (\lambda (H3: (subst0 O t t0 (lift (S O) O x))).(let -H4 \def H in (or_ind (\forall (t2: T).((pr0 t t2) \to (eq T t t2))) (ex2 T -(\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: -T).(pr0 t t2))) (or (\forall (t2: T).((pr0 (THead (Bind Void) t t0) t2) \to -(eq T (THead (Bind Void) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead -(Bind Void) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 -(THead (Bind Void) t t0) t2)))) (\lambda (H5: ((\forall (t2: T).((pr0 t t2) -\to (eq T t t2))))).(let H6 \def H0 in (or_ind (\forall (t2: T).((pr0 t0 t2) -\to (eq T t0 t2))) (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 t0 t2))) (or (\forall (t2: T).((pr0 (THead -(Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) t2))) (ex2 T -(\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)))) (\lambda -(H7: ((\forall (t2: T).((pr0 t0 t2) \to (eq T t0 t2))))).(or_introl (\forall -(t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) -t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to -(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2))) -(\lambda (t2: T).(\lambda (H8: (pr0 (THead (Bind Void) t t0) t2)).(or_ind -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda -(t3: T).(pr0 t0 t3)))) (pr0 t0 (lift (S O) O t2)) (eq T (THead (Bind Void) t -t0) t2) (\lambda (H9: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 -(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) -(\lambda (_: T).(\lambda (t3: T).(pr0 t0 t3))))).(ex3_2_ind T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t0 -t3))) (eq T (THead (Bind Void) t t0) t2) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H10: (eq T t2 (THead (Bind Void) x0 x1))).(\lambda (H11: (pr0 t -x0)).(\lambda (H12: (pr0 t0 x1)).(let H_y \def (H7 x1 H12) in (let H_y0 \def -(H5 x0 H11) in (let H13 \def (eq_ind_r T x1 (\lambda (t3: T).(pr0 t0 t3)) H12 -t0 H_y) in (let H14 \def (eq_ind_r T x1 (\lambda (t3: T).(eq T t2 (THead -(Bind Void) x0 t3))) H10 t0 H_y) in (let H15 \def (eq_ind_r T x0 (\lambda -(t3: T).(pr0 t t3)) H11 t H_y0) in (let H16 \def (eq_ind_r T x0 (\lambda (t3: -T).(eq T t2 (THead (Bind Void) t3 t0))) H14 t H_y0) in (eq_ind_r T (THead -(Bind Void) t t0) (\lambda (t3: T).(eq T (THead (Bind Void) t t0) t3)) -(refl_equal T (THead (Bind Void) t t0)) t2 H16)))))))))))) H9)) (\lambda (H9: -(pr0 t0 (lift (S O) O t2))).(let H_y \def (H7 (lift (S O) O t2) H9) in (let -H10 \def (eq_ind T t0 (\lambda (t3: T).(subst0 O t t3 (lift (S O) O x))) H3 -(lift (S O) O t2) H_y) in (eq_ind_r T (lift (S O) O t2) (\lambda (t3: T).(eq -T (THead (Bind Void) t t3) t2)) (subst0_gen_lift_false t2 t (lift (S O) O x) -(S O) O O (le_n O) (eq_ind_r nat (plus (S O) O) (\lambda (n: nat).(lt O n)) -(le_n (plus (S O) O)) (plus O (S O)) (plus_sym O (S O))) H10 (eq T (THead -(Bind Void) t (lift (S O) O t2)) t2)) t0 H_y)))) (pr0_gen_void t t0 t2 -H8)))))) (\lambda (H7: (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 t0 t2)))).(ex2_ind T (\lambda (t2: T).((eq T -t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2)) (or (\forall -(t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) -t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to -(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)))) -(\lambda (x0: T).(\lambda (H8: (((eq T t0 x0) \to (\forall (P: -Prop).P)))).(\lambda (H9: (pr0 t0 x0)).(or_intror (\forall (t2: T).((pr0 -(THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) t2))) (ex2 T -(\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2))) (ex_intro2 T -(\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)) (THead (Bind -Void) t x0) (\lambda (H10: (eq T (THead (Bind Void) t t0) (THead (Bind Void) -t x0))).(\lambda (P: Prop).(let H11 \def (f_equal T T (\lambda (e: T).(match -e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) -\Rightarrow t0 | (THead _ _ t2) \Rightarrow t2])) (THead (Bind Void) t t0) -(THead (Bind Void) t x0) H10) in (let H12 \def (eq_ind_r T x0 (\lambda (t2: -T).(pr0 t0 t2)) H9 t0 H11) in (let H13 \def (eq_ind_r T x0 (\lambda (t2: -T).((eq T t0 t2) \to (\forall (P0: Prop).P0))) H8 t0 H11) in (H13 (refl_equal -T t0) P)))))) (pr0_comp t t (pr0_refl t) t0 x0 H9 (Bind Void))))))) H7)) -H6))) (\lambda (H5: (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 t t2)))).(ex2_ind T (\lambda (t2: T).((eq T -t t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2)) (or (\forall -(t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) -t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to -(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)))) -(\lambda (x0: T).(\lambda (H6: (((eq T t x0) \to (\forall (P: -Prop).P)))).(\lambda (H7: (pr0 t x0)).(or_intror (\forall (t2: T).((pr0 -(THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) t2))) (ex2 T -(\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2))) (ex_intro2 T -(\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)) (THead (Bind -Void) x0 t0) (\lambda (H8: (eq T (THead (Bind Void) t t0) (THead (Bind Void) -x0 t0))).(\lambda (P: Prop).(let H9 \def (f_equal T T (\lambda (e: T).(match -e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t | (TLRef _) -\Rightarrow t | (THead _ t2 _) \Rightarrow t2])) (THead (Bind Void) t t0) -(THead (Bind Void) x0 t0) H8) in (let H10 \def (eq_ind_r T x0 (\lambda (t2: -T).(pr0 t t2)) H7 t H9) in (let H11 \def (eq_ind_r T x0 (\lambda (t2: T).((eq -T t t2) \to (\forall (P0: Prop).P0))) H6 t H9) in (H11 (refl_equal T t) -P)))))) (pr0_comp t x0 H7 t0 t0 (pr0_refl t0) (Bind Void))))))) H5)) H4))) -(\lambda (H3: (eq T t0 (lift (S O) O x))).(let H4 \def (eq_ind T t0 (\lambda -(t2: T).(or (\forall (t3: T).((pr0 t2 t3) \to (eq T t2 t3))) (ex2 T (\lambda -(t3: T).((eq T t2 t3) \to (\forall (P: Prop).P))) (\lambda (t3: T).(pr0 t2 -t3))))) H0 (lift (S O) O x) H3) in (eq_ind_r T (lift (S O) O x) (\lambda (t2: -T).(or (\forall (t3: T).((pr0 (THead (Bind Void) t t2) t3) \to (eq T (THead -(Bind Void) t t2) t3))) (ex2 T (\lambda (t3: T).((eq T (THead (Bind Void) t -t2) t3) \to (\forall (P: Prop).P))) (\lambda (t3: T).(pr0 (THead (Bind Void) -t t2) t3))))) (or_intror (\forall (t2: T).((pr0 (THead (Bind Void) t (lift (S -O) O x)) t2) \to (eq T (THead (Bind Void) t (lift (S O) O x)) t2))) (ex2 T -(\lambda (t2: T).((eq T (THead (Bind Void) t (lift (S O) O x)) t2) \to -(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t (lift (S -O) O x)) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Bind Void) t -(lift (S O) O x)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 -(THead (Bind Void) t (lift (S O) O x)) t2)) x (\lambda (H5: (eq T (THead -(Bind Void) t (lift (S O) O x)) x)).(\lambda (P: Prop).(thead_x_lift_y_y -(Bind Void) x t (S O) O H5 P))) (pr0_zeta Void (sym_not_eq B Abst Void -not_abst_void) x x (pr0_refl x) t))) t0 H3))) H2))) H1))) b)) (\lambda (f: -F).(F_ind (\lambda (f0: F).(or (\forall (t2: T).((pr0 (THead (Flat f0) t t0) -t2) \to (eq T (THead (Flat f0) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T -(THead (Flat f0) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 -(THead (Flat f0) t t0) t2))))) (let H_x \def (binder_dec t0) in (let H1 \def -H_x in (or_ind (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: -T).(eq T t0 (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: -T).(\forall (u: T).((eq T t0 (THead (Bind b) w u)) \to (\forall (P: -Prop).P))))) (or (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq -T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat -Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead -(Flat Appl) t t0) t2)))) (\lambda (H2: (ex_3 B T T (\lambda (b: B).(\lambda -(w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w u))))))).(ex_3_ind B T T -(\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w -u))))) (or (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T -(THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat -Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead -(Flat Appl) t t0) t2)))) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: -T).(\lambda (H3: (eq T t0 (THead (Bind x0) x1 x2))).(let H4 \def (eq_ind T t0 -(\lambda (t2: T).(or (\forall (t3: T).((pr0 t2 t3) \to (eq T t2 t3))) (ex2 T -(\lambda (t3: T).((eq T t2 t3) \to (\forall (P: Prop).P))) (\lambda (t3: -T).(pr0 t2 t3))))) H0 (THead (Bind x0) x1 x2) H3) in (eq_ind_r T (THead (Bind -x0) x1 x2) (\lambda (t2: T).(or (\forall (t3: T).((pr0 (THead (Flat Appl) t -t2) t3) \to (eq T (THead (Flat Appl) t t2) t3))) (ex2 T (\lambda (t3: T).((eq -T (THead (Flat Appl) t t2) t3) \to (\forall (P: Prop).P))) (\lambda (t3: -T).(pr0 (THead (Flat Appl) t t2) t3))))) (B_ind (\lambda (b: B).((or (\forall -(t2: T).((pr0 (THead (Bind b) x1 x2) t2) \to (eq T (THead (Bind b) x1 x2) -t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind b) x1 x2) t2) \to (\forall -(P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind b) x1 x2) t2)))) \to (or -(\forall (t2: T).((pr0 (THead (Flat Appl) t (THead (Bind b) x1 x2)) t2) \to -(eq T (THead (Flat Appl) t (THead (Bind b) x1 x2)) t2))) (ex2 T (\lambda (t2: -T).((eq T (THead (Flat Appl) t (THead (Bind b) x1 x2)) t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead (Bind b) x1 x2)) -t2)))))) (\lambda (_: (or (\forall (t2: T).((pr0 (THead (Bind Abbr) x1 x2) -t2) \to (eq T (THead (Bind Abbr) x1 x2) t2))) (ex2 T (\lambda (t2: T).((eq T -(THead (Bind Abbr) x1 x2) t2) \to (\forall (P: Prop).P))) (\lambda (t2: -T).(pr0 (THead (Bind Abbr) x1 x2) t2))))).(or_intror (\forall (t2: T).((pr0 -(THead (Flat Appl) t (THead (Bind Abbr) x1 x2)) t2) \to (eq T (THead (Flat -Appl) t (THead (Bind Abbr) x1 x2)) t2))) (ex2 T (\lambda (t2: T).((eq T -(THead (Flat Appl) t (THead (Bind Abbr) x1 x2)) t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead (Bind Abbr) x1 -x2)) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t (THead -(Bind Abbr) x1 x2)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 -(THead (Flat Appl) t (THead (Bind Abbr) x1 x2)) t2)) (THead (Bind Abbr) x1 -(THead (Flat Appl) (lift (S O) O t) x2)) (\lambda (H6: (eq T (THead (Flat -Appl) t (THead (Bind Abbr) x1 x2)) (THead (Bind Abbr) x1 (THead (Flat Appl) -(lift (S O) O t) x2)))).(\lambda (P: Prop).(let H7 \def (eq_ind T (THead -(Flat Appl) t (THead (Bind Abbr) x1 x2)) (\lambda (ee: T).(match ee in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead _ _ t2) \Rightarrow (match t2 in T return (\lambda -(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False -| (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])])) I (THead (Bind -Abbr) x1 (THead (Flat Appl) (lift (S O) O t) x2)) H6) in (False_ind P H7)))) -(pr0_upsilon Abbr not_abbr_abst t t (pr0_refl t) x1 x1 (pr0_refl x1) x2 x2 -(pr0_refl x2))))) (\lambda (_: (or (\forall (t2: T).((pr0 (THead (Bind Abst) -x1 x2) t2) \to (eq T (THead (Bind Abst) x1 x2) t2))) (ex2 T (\lambda (t2: -T).((eq T (THead (Bind Abst) x1 x2) t2) \to (\forall (P: Prop).P))) (\lambda -(t2: T).(pr0 (THead (Bind Abst) x1 x2) t2))))).(or_intror (\forall (t2: -T).((pr0 (THead (Flat Appl) t (THead (Bind Abst) x1 x2)) t2) \to (eq T (THead -(Flat Appl) t (THead (Bind Abst) x1 x2)) t2))) (ex2 T (\lambda (t2: T).((eq T -(THead (Flat Appl) t (THead (Bind Abst) x1 x2)) t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead (Bind Abst) x1 -x2)) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t (THead -(Bind Abst) x1 x2)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 -(THead (Flat Appl) t (THead (Bind Abst) x1 x2)) t2)) (THead (Bind Abbr) t x2) -(\lambda (H6: (eq T (THead (Flat Appl) t (THead (Bind Abst) x1 x2)) (THead -(Bind Abbr) t x2))).(\lambda (P: Prop).(let H7 \def (eq_ind T (THead (Flat -Appl) t (THead (Bind Abst) x1 x2)) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind Abbr) t x2) H6) in (False_ind P H7)))) (pr0_beta x1 -t t (pr0_refl t) x2 x2 (pr0_refl x2))))) (\lambda (_: (or (\forall (t2: -T).((pr0 (THead (Bind Void) x1 x2) t2) \to (eq T (THead (Bind Void) x1 x2) -t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) x1 x2) t2) \to -(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) x1 x2) -t2))))).(or_intror (\forall (t2: T).((pr0 (THead (Flat Appl) t (THead (Bind -Void) x1 x2)) t2) \to (eq T (THead (Flat Appl) t (THead (Bind Void) x1 x2)) -t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t (THead (Bind Void) -x1 x2)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat -Appl) t (THead (Bind Void) x1 x2)) t2))) (ex_intro2 T (\lambda (t2: T).((eq T -(THead (Flat Appl) t (THead (Bind Void) x1 x2)) t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead (Bind Void) x1 -x2)) t2)) (THead (Bind Void) x1 (THead (Flat Appl) (lift (S O) O t) x2)) -(\lambda (H6: (eq T (THead (Flat Appl) t (THead (Bind Void) x1 x2)) (THead -(Bind Void) x1 (THead (Flat Appl) (lift (S O) O t) x2)))).(\lambda (P: -Prop).(let H7 \def (eq_ind T (THead (Flat Appl) t (THead (Bind Void) x1 x2)) -(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ t2) \Rightarrow -(match t2 in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False -| (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K -return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) -\Rightarrow False])])])) I (THead (Bind Void) x1 (THead (Flat Appl) (lift (S -O) O t) x2)) H6) in (False_ind P H7)))) (pr0_upsilon Void (sym_not_eq B Abst -Void not_abst_void) t t (pr0_refl t) x1 x1 (pr0_refl x1) x2 x2 (pr0_refl -x2))))) x0 H4) t0 H3)))))) H2)) (\lambda (H2: ((\forall (b: B).(\forall (w: -T).(\forall (u: T).((eq T t0 (THead (Bind b) w u)) \to (\forall (P: -Prop).P))))))).(let H3 \def H in (or_ind (\forall (t2: T).((pr0 t t2) \to (eq -T t t2))) (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) -(\lambda (t2: T).(pr0 t t2))) (or (\forall (t2: T).((pr0 (THead (Flat Appl) t -t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq -T (THead (Flat Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: -T).(pr0 (THead (Flat Appl) t t0) t2)))) (\lambda (H4: ((\forall (t2: T).((pr0 -t t2) \to (eq T t t2))))).(let H5 \def H0 in (or_ind (\forall (t2: T).((pr0 -t0 t2) \to (eq T t0 t2))) (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall -(P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2))) (or (\forall (t2: T).((pr0 -(THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T -(\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)))) (\lambda -(H6: ((\forall (t2: T).((pr0 t0 t2) \to (eq T t0 t2))))).(or_introl (\forall -(t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) -t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to -(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2))) -(\lambda (t2: T).(\lambda (H7: (pr0 (THead (Flat Appl) t t0) t2)).(or3_ind -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda -(t3: T).(pr0 t0 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T t0 (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 -(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr0 t u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t0 (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead -(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 t -u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(t3: T).(pr0 z1 t3)))))))) (eq T (THead (Flat Appl) t t0) t2) (\lambda (H8: -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda -(t3: T).(pr0 t0 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq -T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t -u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t0 t3))) (eq T (THead (Flat Appl) -t t0) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H9: (eq T t2 (THead -(Flat Appl) x0 x1))).(\lambda (H10: (pr0 t x0)).(\lambda (H11: (pr0 t0 -x1)).(let H_y \def (H6 x1 H11) in (let H_y0 \def (H4 x0 H10) in (let H12 \def -(eq_ind_r T x1 (\lambda (t3: T).(pr0 t0 t3)) H11 t0 H_y) in (let H13 \def -(eq_ind_r T x1 (\lambda (t3: T).(eq T t2 (THead (Flat Appl) x0 t3))) H9 t0 -H_y) in (let H14 \def (eq_ind_r T x0 (\lambda (t3: T).(pr0 t t3)) H10 t H_y0) -in (let H15 \def (eq_ind_r T x0 (\lambda (t3: T).(eq T t2 (THead (Flat Appl) -t3 t0))) H13 t H_y0) in (eq_ind_r T (THead (Flat Appl) t t0) (\lambda (t3: -T).(eq T (THead (Flat Appl) t t0) t3)) (refl_equal T (THead (Flat Appl) t -t0)) t2 H15)))))))))))) H8)) (\lambda (H8: (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t0 (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr0 t u2))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))).(ex4_4_ind T T T T -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t0 -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 t u2))))) (\lambda -(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))) (eq -T (THead (Flat Appl) t t0) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda -(x2: T).(\lambda (x3: T).(\lambda (H9: (eq T t0 (THead (Bind Abst) x0 -x1))).(\lambda (H10: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (_: (pr0 t -x2)).(\lambda (_: (pr0 x1 x3)).(eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda -(t3: T).(eq T (THead (Flat Appl) t t0) t3)) (let H13 \def (eq_ind T t0 -(\lambda (t3: T).(\forall (t4: T).((pr0 t3 t4) \to (eq T t3 t4)))) H6 (THead -(Bind Abst) x0 x1) H9) in (let H14 \def (eq_ind T t0 (\lambda (t3: -T).(\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t3 (THead (Bind b) -w u)) \to (\forall (P: Prop).P)))))) H2 (THead (Bind Abst) x0 x1) H9) in -(eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t3: T).(eq T (THead (Flat -Appl) t t3) (THead (Bind Abbr) x2 x3))) (H14 Abst x0 x1 (H13 (THead (Bind -Abst) x0 x1) (pr0_refl (THead (Bind Abst) x0 x1))) (eq T (THead (Flat Appl) t -(THead (Bind Abst) x0 x1)) (THead (Bind Abbr) x2 x3))) t0 H9))) t2 -H10))))))))) H8)) (\lambda (H8: (ex6_6 B T T T T T (\lambda (b: B).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not -(eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t0 (THead (Bind b) -y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead (Flat -Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 t u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 -t3))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T t0 (THead (Bind b) y1 z1)))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead (Flat Appl) (lift -(S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 t u2))))))) (\lambda -(_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: -T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))) -(eq T (THead (Flat Appl) t t0) t2) (\lambda (x0: B).(\lambda (x1: T).(\lambda -(x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (not -(eq B x0 Abst))).(\lambda (H10: (eq T t0 (THead (Bind x0) x1 x2))).(\lambda -(H11: (eq T t2 (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) -x5)))).(\lambda (_: (pr0 t x3)).(\lambda (_: (pr0 x1 x4)).(\lambda (_: (pr0 -x2 x5)).(eq_ind_r T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) -x5)) (\lambda (t3: T).(eq T (THead (Flat Appl) t t0) t3)) (let H15 \def -(eq_ind T t0 (\lambda (t3: T).(\forall (t4: T).((pr0 t3 t4) \to (eq T t3 -t4)))) H6 (THead (Bind x0) x1 x2) H10) in (let H16 \def (eq_ind T t0 (\lambda -(t3: T).(\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t3 (THead -(Bind b) w u)) \to (\forall (P: Prop).P)))))) H2 (THead (Bind x0) x1 x2) H10) -in (eq_ind_r T (THead (Bind x0) x1 x2) (\lambda (t3: T).(eq T (THead (Flat -Appl) t t3) (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)))) -(H16 x0 x1 x2 (H15 (THead (Bind x0) x1 x2) (pr0_refl (THead (Bind x0) x1 -x2))) (eq T (THead (Flat Appl) t (THead (Bind x0) x1 x2)) (THead (Bind x0) x4 -(THead (Flat Appl) (lift (S O) O x3) x5)))) t0 H10))) t2 H11))))))))))))) -H8)) (pr0_gen_appl t t0 t2 H7)))))) (\lambda (H6: (ex2 T (\lambda (t2: -T).((eq T t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t0 -t2)))).(ex2_ind T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: Prop).P))) -(\lambda (t2: T).(pr0 t0 t2)) (or (\forall (t2: T).((pr0 (THead (Flat Appl) t -t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq -T (THead (Flat Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: -T).(pr0 (THead (Flat Appl) t t0) t2)))) (\lambda (x: T).(\lambda (H7: (((eq T -t0 x) \to (\forall (P: Prop).P)))).(\lambda (H8: (pr0 t0 x)).(or_intror -(\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat -Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) -\to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) -t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to -(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)) -(THead (Flat Appl) t x) (\lambda (H9: (eq T (THead (Flat Appl) t t0) (THead -(Flat Appl) t x))).(\lambda (P: Prop).(let H10 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | -(TLRef _) \Rightarrow t0 | (THead _ _ t2) \Rightarrow t2])) (THead (Flat -Appl) t t0) (THead (Flat Appl) t x) H9) in (let H11 \def (eq_ind_r T x -(\lambda (t2: T).(pr0 t0 t2)) H8 t0 H10) in (let H12 \def (eq_ind_r T x -(\lambda (t2: T).((eq T t0 t2) \to (\forall (P0: Prop).P0))) H7 t0 H10) in -(H12 (refl_equal T t0) P)))))) (pr0_comp t t (pr0_refl t) t0 x H8 (Flat -Appl))))))) H6)) H5))) (\lambda (H4: (ex2 T (\lambda (t2: T).((eq T t t2) \to -(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2)))).(ex2_ind T (\lambda -(t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2)) -(or (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead -(Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t -t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) -t t0) t2)))) (\lambda (x: T).(\lambda (H5: (((eq T t x) \to (\forall (P: -Prop).P)))).(\lambda (H6: (pr0 t x)).(or_intror (\forall (t2: T).((pr0 (THead -(Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T -(\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2))) (ex_intro2 T -(\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)) (THead (Flat -Appl) x t0) (\lambda (H7: (eq T (THead (Flat Appl) t t0) (THead (Flat Appl) x -t0))).(\lambda (P: Prop).(let H8 \def (f_equal T T (\lambda (e: T).(match e -in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t | (TLRef _) -\Rightarrow t | (THead _ t2 _) \Rightarrow t2])) (THead (Flat Appl) t t0) -(THead (Flat Appl) x t0) H7) in (let H9 \def (eq_ind_r T x (\lambda (t2: -T).(pr0 t t2)) H6 t H8) in (let H10 \def (eq_ind_r T x (\lambda (t2: T).((eq -T t t2) \to (\forall (P0: Prop).P0))) H5 t H8) in (H10 (refl_equal T t) -P)))))) (pr0_comp t x H6 t0 t0 (pr0_refl t0) (Flat Appl))))))) H4)) H3))) -H1))) (or_intror (\forall (t2: T).((pr0 (THead (Flat Cast) t t0) t2) \to (eq -T (THead (Flat Cast) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat -Cast) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead -(Flat Cast) t t0) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Flat -Cast) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead -(Flat Cast) t t0) t2)) t0 (\lambda (H1: (eq T (THead (Flat Cast) t t0) -t0)).(\lambda (P: Prop).(thead_x_y_y (Flat Cast) t t0 H1 P))) (pr0_tau t0 t0 -(pr0_refl t0) t))) f)) k)))))) t1). -(* COMMENTS -Initial nodes: 10459 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr0/defs.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr0/defs.ma deleted file mode 100644 index 0568e070c..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr0/defs.ma +++ /dev/null @@ -1,40 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/subst0/defs.ma". - -inductive pr0: T \to (T \to Prop) \def -| pr0_refl: \forall (t: T).(pr0 t t) -| pr0_comp: \forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (t1: -T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (k: K).(pr0 (THead k u1 t1) -(THead k u2 t2)))))))) -| pr0_beta: \forall (u: T).(\forall (v1: T).(\forall (v2: T).((pr0 v1 v2) \to -(\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (pr0 (THead (Flat Appl) v1 -(THead (Bind Abst) u t1)) (THead (Bind Abbr) v2 t2)))))))) -| pr0_upsilon: \forall (b: B).((not (eq B b Abst)) \to (\forall (v1: -T).(\forall (v2: T).((pr0 v1 v2) \to (\forall (u1: T).(\forall (u2: T).((pr0 -u1 u2) \to (\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (pr0 (THead -(Flat Appl) v1 (THead (Bind b) u1 t1)) (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t2))))))))))))) -| pr0_delta: \forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (t1: -T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (w: T).((subst0 O u2 t2 w) \to -(pr0 (THead (Bind Abbr) u1 t1) (THead (Bind Abbr) u2 w))))))))) -| pr0_zeta: \forall (b: B).((not (eq B b Abst)) \to (\forall (t1: T).(\forall -(t2: T).((pr0 t1 t2) \to (\forall (u: T).(pr0 (THead (Bind b) u (lift (S O) O -t1)) t2)))))) -| pr0_tau: \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (u: -T).(pr0 (THead (Flat Cast) u t1) t2)))). - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr0/fwd.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr0/fwd.ma deleted file mode 100644 index 46caceab4..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr0/fwd.ma +++ /dev/null @@ -1,2018 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr0/props.ma". - -theorem pr0_gen_sort: - \forall (x: T).(\forall (n: nat).((pr0 (TSort n) x) \to (eq T x (TSort n)))) -\def - \lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr0 (TSort n) x)).(insert_eq -T (TSort n) (\lambda (t: T).(pr0 t x)) (\lambda (t: T).(eq T x t)) (\lambda -(y: T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: T).(\lambda (t0: -T).((eq T t (TSort n)) \to (eq T t0 t)))) (\lambda (t: T).(\lambda (H1: (eq T -t (TSort n))).(let H2 \def (f_equal T T (\lambda (e: T).e) t (TSort n) H1) in -(eq_ind_r T (TSort n) (\lambda (t0: T).(eq T t0 t0)) (refl_equal T (TSort n)) -t H2)))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda -(_: (((eq T u1 (TSort n)) \to (eq T u2 u1)))).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2 -t1)))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u1 t1) (TSort n))).(let -H6 \def (eq_ind T (THead k u1 t1) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H5) in -(False_ind (eq T (THead k u2 t2) (THead k u1 t1)) H6)))))))))))) (\lambda (u: -T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: -(((eq T v1 (TSort n)) \to (eq T v2 v1)))).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2 -t1)))).(\lambda (H5: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t1)) -(TSort n))).(let H6 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u -t1)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) -\Rightarrow True])) I (TSort n) H5) in (False_ind (eq T (THead (Bind Abbr) v2 -t2) (THead (Flat Appl) v1 (THead (Bind Abst) u t1))) H6)))))))))))) (\lambda -(b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: -T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (TSort n)) \to (eq T v2 -v1)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda -(_: (((eq T u1 (TSort n)) \to (eq T u2 u1)))).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2 -t1)))).(\lambda (H8: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t1)) -(TSort n))).(let H9 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 -t1)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) -\Rightarrow True])) I (TSort n) H8) in (False_ind (eq T (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Flat Appl) v1 (THead (Bind -b) u1 t1))) H9))))))))))))))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda -(_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 (TSort n)) \to (eq T u2 -u1)))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda -(_: (((eq T t1 (TSort n)) \to (eq T t2 t1)))).(\lambda (w: T).(\lambda (_: -(subst0 O u2 t2 w)).(\lambda (H6: (eq T (THead (Bind Abbr) u1 t1) (TSort -n))).(let H7 \def (eq_ind T (THead (Bind Abbr) u1 t1) (\lambda (ee: T).(match -ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | -(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) -H6) in (False_ind (eq T (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u1 t1)) -H7))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda -(t1: T).(\lambda (t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 -(TSort n)) \to (eq T t2 t1)))).(\lambda (u: T).(\lambda (H4: (eq T (THead -(Bind b) u (lift (S O) O t1)) (TSort n))).(let H5 \def (eq_ind T (THead (Bind -b) u (lift (S O) O t1)) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead _ _ _) \Rightarrow True])) I (TSort n) H4) in (False_ind (eq T t2 -(THead (Bind b) u (lift (S O) O t1))) H5)))))))))) (\lambda (t1: T).(\lambda -(t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq -T t2 t1)))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u t1) -(TSort n))).(let H4 \def (eq_ind T (THead (Flat Cast) u t1) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I -(TSort n) H3) in (False_ind (eq T t2 (THead (Flat Cast) u t1)) H4)))))))) y x -H0))) H))). -(* COMMENTS -Initial nodes: 1045 -END *) - -theorem pr0_gen_lref: - \forall (x: T).(\forall (n: nat).((pr0 (TLRef n) x) \to (eq T x (TLRef n)))) -\def - \lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr0 (TLRef n) x)).(insert_eq -T (TLRef n) (\lambda (t: T).(pr0 t x)) (\lambda (t: T).(eq T x t)) (\lambda -(y: T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: T).(\lambda (t0: -T).((eq T t (TLRef n)) \to (eq T t0 t)))) (\lambda (t: T).(\lambda (H1: (eq T -t (TLRef n))).(let H2 \def (f_equal T T (\lambda (e: T).e) t (TLRef n) H1) in -(eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 t0)) (refl_equal T (TLRef n)) -t H2)))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda -(_: (((eq T u1 (TLRef n)) \to (eq T u2 u1)))).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (eq T t2 -t1)))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u1 t1) (TLRef n))).(let -H6 \def (eq_ind T (THead k u1 t1) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H5) in -(False_ind (eq T (THead k u2 t2) (THead k u1 t1)) H6)))))))))))) (\lambda (u: -T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: -(((eq T v1 (TLRef n)) \to (eq T v2 v1)))).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (eq T t2 -t1)))).(\lambda (H5: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t1)) -(TLRef n))).(let H6 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u -t1)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) -\Rightarrow True])) I (TLRef n) H5) in (False_ind (eq T (THead (Bind Abbr) v2 -t2) (THead (Flat Appl) v1 (THead (Bind Abst) u t1))) H6)))))))))))) (\lambda -(b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: -T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (TLRef n)) \to (eq T v2 -v1)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda -(_: (((eq T u1 (TLRef n)) \to (eq T u2 u1)))).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (eq T t2 -t1)))).(\lambda (H8: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t1)) -(TLRef n))).(let H9 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 -t1)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) -\Rightarrow True])) I (TLRef n) H8) in (False_ind (eq T (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Flat Appl) v1 (THead (Bind -b) u1 t1))) H9))))))))))))))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda -(_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 (TLRef n)) \to (eq T u2 -u1)))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda -(_: (((eq T t1 (TLRef n)) \to (eq T t2 t1)))).(\lambda (w: T).(\lambda (_: -(subst0 O u2 t2 w)).(\lambda (H6: (eq T (THead (Bind Abbr) u1 t1) (TLRef -n))).(let H7 \def (eq_ind T (THead (Bind Abbr) u1 t1) (\lambda (ee: T).(match -ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | -(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) -H6) in (False_ind (eq T (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u1 t1)) -H7))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda -(t1: T).(\lambda (t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 -(TLRef n)) \to (eq T t2 t1)))).(\lambda (u: T).(\lambda (H4: (eq T (THead -(Bind b) u (lift (S O) O t1)) (TLRef n))).(let H5 \def (eq_ind T (THead (Bind -b) u (lift (S O) O t1)) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead _ _ _) \Rightarrow True])) I (TLRef n) H4) in (False_ind (eq T t2 -(THead (Bind b) u (lift (S O) O t1))) H5)))))))))) (\lambda (t1: T).(\lambda -(t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (eq -T t2 t1)))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u t1) -(TLRef n))).(let H4 \def (eq_ind T (THead (Flat Cast) u t1) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I -(TLRef n) H3) in (False_ind (eq T t2 (THead (Flat Cast) u t1)) H4)))))))) y x -H0))) H))). -(* COMMENTS -Initial nodes: 1045 -END *) - -theorem pr0_gen_abst: - \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Bind Abst) u1 -t1) x) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind -Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: -T).(\lambda (t2: T).(pr0 t1 t2))))))) -\def - \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead -(Bind Abst) u1 t1) x)).(insert_eq T (THead (Bind Abst) u1 t1) (\lambda (t: -T).(pr0 t x)) (\lambda (_: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 -u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))))) (\lambda (y: -T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: T).(\lambda (t0: T).((eq T -t (THead (Bind Abst) u1 t1)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T t0 (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))))))) (\lambda -(t: T).(\lambda (H1: (eq T t (THead (Bind Abst) u1 t1))).(let H2 \def -(f_equal T T (\lambda (e: T).e) t (THead (Bind Abst) u1 t1) H1) in (eq_ind_r -T (THead (Bind Abst) u1 t1) (\lambda (t0: T).(ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T t0 (THead (Bind Abst) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 -t2))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Bind -Abst) u1 t1) (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))) u1 t1 -(refl_equal T (THead (Bind Abst) u1 t1)) (pr0_refl u1) (pr0_refl t1)) t -H2)))) (\lambda (u0: T).(\lambda (u2: T).(\lambda (H1: (pr0 u0 u2)).(\lambda -(H2: (((eq T u0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T (\lambda (u3: -T).(\lambda (t2: T).(eq T u2 (THead (Bind Abst) u3 t2)))) (\lambda (u3: -T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 -t2))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H3: (pr0 t0 -t2)).(\lambda (H4: (((eq T t0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T -(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u3 t3)))) -(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3))))))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u0 t0) -(THead (Bind Abst) u1 t1))).(let H6 \def (f_equal T K (\lambda (e: T).(match -e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) -\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u0 t0) (THead (Bind -Abst) u1 t1) H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) -\Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead k u0 t0) (THead (Bind -Abst) u1 t1) H5) in ((let H8 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) -\Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k u0 t0) (THead (Bind -Abst) u1 t1) H5) in (\lambda (H9: (eq T u0 u1)).(\lambda (H10: (eq K k (Bind -Abst))).(eq_ind_r K (Bind Abst) (\lambda (k0: K).(ex3_2 T T (\lambda (u3: -T).(\lambda (t3: T).(eq T (THead k0 u2 t2) (THead (Bind Abst) u3 t3)))) -(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3))))) (let H11 \def (eq_ind T t0 (\lambda (t: T).((eq T t (THead -(Bind Abst) u1 t1)) \to (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 -(THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) -(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))))) H4 t1 H8) in (let H12 \def -(eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H3 t1 H8) in (let H13 \def (eq_ind T -u0 (\lambda (t: T).((eq T t (THead (Bind Abst) u1 t1)) \to (ex3_2 T T -(\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead (Bind Abst) u3 t3)))) -(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3)))))) H2 u1 H9) in (let H14 \def (eq_ind T u0 (\lambda (t: -T).(pr0 t u2)) H1 u1 H9) in (ex3_2_intro T T (\lambda (u3: T).(\lambda (t3: -T).(eq T (THead (Bind Abst) u2 t2) (THead (Bind Abst) u3 t3)))) (\lambda (u3: -T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 -t3))) u2 t2 (refl_equal T (THead (Bind Abst) u2 t2)) H14 H12))))) k H10)))) -H7)) H6)))))))))))) (\lambda (u: T).(\lambda (v1: T).(\lambda (v2: -T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind Abst) u1 -t1)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind -Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: -T).(\lambda (t2: T).(pr0 t1 t2))))))).(\lambda (t0: T).(\lambda (t2: -T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Abst) u1 -t1)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t1 t3))))))).(\lambda (H5: (eq T (THead (Flat Appl) -v1 (THead (Bind Abst) u t0)) (THead (Bind Abst) u1 t1))).(let H6 \def (eq_ind -T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (ee: T).(match ee -in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef -_) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return -(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind Abst) u1 t1) H5) in (False_ind (ex3_2 T T (\lambda -(u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) (THead (Bind Abst) -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t1 t3)))) H6)))))))))))) (\lambda (b: B).(\lambda -(_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 -v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T -(\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind Abst) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: -T).(pr0 t1 t2))))))).(\lambda (u0: T).(\lambda (u2: T).(\lambda (_: (pr0 u0 -u2)).(\lambda (_: (((eq T u0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T -(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Bind Abst) u3 t2)))) -(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: -T).(pr0 t1 t2))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 -t2)).(\lambda (_: (((eq T t0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T -(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u3 t3)))) -(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3))))))).(\lambda (H8: (eq T (THead (Flat Appl) v1 (THead (Bind -b) u0 t0)) (THead (Bind Abst) u1 t1))).(let H9 \def (eq_ind T (THead (Flat -Appl) v1 (THead (Bind b) u0 t0)) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind Abst) u1 t1) H8) in (False_ind (ex3_2 T T (\lambda -(u3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t2)) (THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: -T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) -H9))))))))))))))))) (\lambda (u0: T).(\lambda (u2: T).(\lambda (_: (pr0 u0 -u2)).(\lambda (_: (((eq T u0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T -(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Bind Abst) u3 t2)))) -(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: -T).(pr0 t1 t2))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 -t2)).(\lambda (_: (((eq T t0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T -(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u3 t3)))) -(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3))))))).(\lambda (w: T).(\lambda (_: (subst0 O u2 t2 -w)).(\lambda (H6: (eq T (THead (Bind Abbr) u0 t0) (THead (Bind Abst) u1 -t1))).(let H7 \def (eq_ind T (THead (Bind Abbr) u0 t0) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K -return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B return -(\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | -Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (THead (Bind -Abst) u1 t1) H6) in (False_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t3: -T).(eq T (THead (Bind Abbr) u2 w) (THead (Bind Abst) u3 t3)))) (\lambda (u3: -T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 -t3)))) H7))))))))))))) (\lambda (b: B).(\lambda (H1: (not (eq B b -Abst))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda -(H3: (((eq T t0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 -t3))))))).(\lambda (u: T).(\lambda (H4: (eq T (THead (Bind b) u (lift (S O) O -t0)) (THead (Bind Abst) u1 t1))).(let H5 \def (f_equal T B (\lambda (e: -T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | -(TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k in K return -(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow -b])])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abst) u1 t1) H4) in -((let H6 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: -T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) -\Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abst) u1 -t1) H4) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat -\to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow -(TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true -\Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t3) \Rightarrow -(THead k (lref_map f d u0) (lref_map f (s k d) t3))]) in lref_map) (\lambda -(x0: nat).(plus x0 (S O))) O t0) | (TLRef _) \Rightarrow ((let rec lref_map -(f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) -\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with -[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t3) -\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t3))]) in -lref_map) (\lambda (x0: nat).(plus x0 (S O))) O t0) | (THead _ _ t) -\Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abst) u1 -t1) H4) in (\lambda (_: (eq T u u1)).(\lambda (H9: (eq B b Abst)).(let H10 -\def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H1 Abst H9) in (let -H11 \def (eq_ind_r T t1 (\lambda (t: T).((eq T t0 (THead (Bind Abst) u1 t)) -\to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t t3)))))) H3 (lift (S O) O t0) H7) in (eq_ind T -(lift (S O) O t0) (\lambda (t: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t3: -T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t t3))))) (let H12 -\def (match (H10 (refl_equal B Abst)) in False return (\lambda (_: -False).(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr0 (lift (S O) O t0) t3))))) with []) in H12) t1 -H7)))))) H6)) H5)))))))))) (\lambda (t0: T).(\lambda (t2: T).(\lambda (_: -(pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T -T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3))))))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u -t0) (THead (Bind Abst) u1 t1))).(let H4 \def (eq_ind T (THead (Flat Cast) u -t0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort -_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) -\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u1 -t1) H3) in (False_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 -(THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) -(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) H4)))))))) y x H0))) H)))). -(* COMMENTS -Initial nodes: 2838 -END *) - -theorem pr0_gen_appl: - \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Flat Appl) u1 -t1) x) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead -(Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda -(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T x (THead (Bind b) -v2 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: B).(\lambda -(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 -u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(t2: T).(pr0 z1 t2)))))))))))) -\def - \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead -(Flat Appl) u1 t1) x)).(insert_eq T (THead (Flat Appl) u1 t1) (\lambda (t: -T).(pr0 t x)) (\lambda (_: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 -u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda -(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) -(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T x -(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda -(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: -T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 -v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))))))) (\lambda (y: -T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: T).(\lambda (t0: T).((eq T -t (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda -(t2: T).(eq T t0 (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T -T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda -(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) -(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T -t0 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: -T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 -v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))))))))) (\lambda (t: -T).(\lambda (H1: (eq T t (THead (Flat Appl) u1 t1))).(let H2 \def (f_equal T -T (\lambda (e: T).e) t (THead (Flat Appl) u1 t1) H1) in (eq_ind_r T (THead -(Flat Appl) u1 t1) (\lambda (t0: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda -(t2: T).(eq T t0 (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T -T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda -(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) -(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T -t0 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: -T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 -v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))))))) (or3_intro0 (ex3_2 T -T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Appl) u1 t1) (THead -(Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda -(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t2: T).(eq T (THead (Flat Appl) u1 t1) (THead (Bind Abbr) u2 t2)))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 -u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: -T).(pr0 z1 t2)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B -b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(v2: T).(\lambda (t2: T).(eq T (THead (Flat Appl) u1 t1) (THead (Bind b) v2 -(THead (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 -u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(t2: T).(pr0 z1 t2)))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: -T).(eq T (THead (Flat Appl) u1 t1) (THead (Flat Appl) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 -t2))) u1 t1 (refl_equal T (THead (Flat Appl) u1 t1)) (pr0_refl u1) (pr0_refl -t1))) t H2)))) (\lambda (u0: T).(\lambda (u2: T).(\lambda (H1: (pr0 u0 -u2)).(\lambda (H2: (((eq T u0 (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T -(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) -(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: -T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Bind -Abbr) u3 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda -(_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(u3: T).(\lambda (v2: T).(\lambda (t2: T).(eq T u2 (THead (Bind b) v2 (THead -(Flat Appl) (lift (S O) O u3) t2))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 -u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(t2: T).(pr0 z1 t2)))))))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda -(H3: (pr0 t0 t2)).(\lambda (H4: (((eq T t0 (THead (Flat Appl) u1 t1)) \to -(or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) -u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 -z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: -T).(eq T t2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -b) v2 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda -(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3)))))))))))).(\lambda (k: K).(\lambda (H5: (eq -T (THead k u0 t0) (THead (Flat Appl) u1 t1))).(let H6 \def (f_equal T K -(\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) -\Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) -(THead k u0 t0) (THead (Flat Appl) u1 t1) H5) in ((let H7 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) -(THead k u0 t0) (THead (Flat Appl) u1 t1) H5) in ((let H8 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) -(THead k u0 t0) (THead (Flat Appl) u1 t1) H5) in (\lambda (H9: (eq T u0 -u1)).(\lambda (H10: (eq K k (Flat Appl))).(eq_ind_r K (Flat Appl) (\lambda -(k0: K).(or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead k0 u2 -t2) (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 -u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead -(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: -T).(\lambda (t3: T).(eq T (THead k0 u2 t2) (THead (Bind Abbr) u3 t3)))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 -u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: -T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B -b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda -(v2: T).(\lambda (t3: T).(eq T (THead k0 u2 t2) (THead (Bind b) v2 (THead -(Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 -u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(t3: T).(pr0 z1 t3)))))))))) (let H11 \def (eq_ind T t0 (\lambda (t: T).((eq -T t (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda -(t3: T).(eq T t2 (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: -T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T -T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: -T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda -(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) -(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T -t2 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: -T).(\lambda (_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 -v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))))))) H4 t1 H8) in (let -H12 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H3 t1 H8) in (let H13 \def -(eq_ind T u0 (\lambda (t: T).((eq T t (THead (Flat Appl) u1 t1)) \to (or3 -(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead (Flat Appl) u3 -t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 -z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: -T).(eq T u2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T u2 (THead (Bind -b) v2 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda -(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3))))))))))) H2 u1 H9) in (let H14 \def (eq_ind -T u0 (\lambda (t: T).(pr0 t u2)) H1 u1 H9) in (or3_intro0 (ex3_2 T T (\lambda -(u3: T).(\lambda (t3: T).(eq T (THead (Flat Appl) u2 t2) (THead (Flat Appl) -u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 -z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: -T).(eq T (THead (Flat Appl) u2 t2) (THead (Bind Abbr) u3 t3)))))) (\lambda -(_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 -t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) -(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (v2: T).(\lambda -(t3: T).(eq T (THead (Flat Appl) u2 t2) (THead (Bind b) v2 (THead (Flat Appl) -(lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 -t3)))))))) (ex3_2_intro T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead -(Flat Appl) u2 t2) (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: -T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) u2 t2 -(refl_equal T (THead (Flat Appl) u2 t2)) H14 H12)))))) k H10)))) H7)) -H6)))))))))))) (\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda -(H1: (pr0 v1 v2)).(\lambda (H2: (((eq T v1 (THead (Flat Appl) u1 t1)) \to -(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Flat Appl) -u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: -T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 -z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: -T).(eq T v2 (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (v3: T).(\lambda (t2: T).(eq T v2 (THead (Bind -b) v3 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda -(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (t2: T).(pr0 z1 t2)))))))))))).(\lambda (t0: T).(\lambda (t2: -T).(\lambda (H3: (pr0 t0 t2)).(\lambda (H4: (((eq T t0 (THead (Flat Appl) u1 -t1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead -(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda -(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T t2 (THead (Bind -b) v3 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda -(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3)))))))))))).(\lambda (H5: (eq T (THead (Flat -Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat Appl) u1 t1))).(let H6 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t _) -\Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat -Appl) u1 t1) H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow (THead (Bind Abst) u -t0) | (TLRef _) \Rightarrow (THead (Bind Abst) u t0) | (THead _ _ t) -\Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat -Appl) u1 t1) H5) in (\lambda (H8: (eq T v1 u1)).(let H9 \def (eq_ind T v1 -(\lambda (t: T).((eq T t (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T v2 (THead (Flat Appl) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T v2 (THead (Bind -Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T v2 (THead (Bind b) v3 (THead -(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 -u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(t3: T).(pr0 z1 t3))))))))))) H2 u1 H8) in (let H10 \def (eq_ind T v1 -(\lambda (t: T).(pr0 t v2)) H1 u1 H8) in (let H11 \def (eq_ind_r T t1 -(\lambda (t: T).((eq T t0 (THead (Flat Appl) u1 t)) \to (or3 (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v3 (THead -(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 -u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(t3: T).(pr0 z1 t3))))))))))) H4 (THead (Bind Abst) u t0) H7) in (let H12 -\def (eq_ind_r T t1 (\lambda (t: T).((eq T u1 (THead (Flat Appl) u1 t)) \to -(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T v2 (THead (Flat Appl) -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 -z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: -T).(eq T v2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T v2 (THead (Bind -b) v3 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda -(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3))))))))))) H9 (THead (Bind Abst) u t0) H7) in -(eq_ind T (THead (Bind Abst) u t0) (\lambda (t: T).(or3 (ex3_2 T T (\lambda -(u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) (THead (Flat Appl) -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 -z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: -T).(eq T (THead (Bind Abbr) v2 t2) (THead (Bind Abbr) u2 t3)))))) (\lambda -(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 -t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) -(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(eq T t (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v3: T).(\lambda -(t3: T).(eq T (THead (Bind Abbr) v2 t2) (THead (Bind b) v3 (THead (Flat Appl) -(lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 -t3)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T -(THead (Bind Abbr) v2 t2) (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 (THead -(Bind Abst) u t0) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) u t0) (THead -(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) (THead (Bind Abbr) u2 -t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: -T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B -b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) u t0) (THead -(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T (THead (Bind -Abbr) v2 t2) (THead (Bind b) v3 (THead (Flat Appl) (lift (S O) O u2) -t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda -(y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 -y1 v3))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))))) (ex4_4_intro T T T T -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind Abst) u t0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) -v2 t2) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))) u t0 v2 t2 -(refl_equal T (THead (Bind Abst) u t0)) (refl_equal T (THead (Bind Abbr) v2 -t2)) H10 H3)) t1 H7))))))) H6)))))))))))) (\lambda (b: B).(\lambda (H1: (not -(eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H2: (pr0 v1 -v2)).(\lambda (H3: (((eq T v1 (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T -(\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Flat Appl) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: -T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind -Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T (\lambda (b0: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (v3: T).(\lambda (t2: T).(eq T v2 (THead (Bind b0) v3 (THead -(Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 -u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(t2: T).(pr0 z1 t2)))))))))))).(\lambda (u0: T).(\lambda (u2: T).(\lambda -(H4: (pr0 u0 u2)).(\lambda (H5: (((eq T u0 (THead (Flat Appl) u1 t1)) \to -(or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) -u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: -T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 -z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: -T).(eq T u2 (THead (Bind Abbr) u3 t2)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T -(\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t2: T).(eq T u2 (THead (Bind -b0) v3 (THead (Flat Appl) (lift (S O) O u3) t2))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda -(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (t2: T).(pr0 z1 t2)))))))))))).(\lambda (t0: T).(\lambda (t2: -T).(\lambda (H6: (pr0 t0 t2)).(\lambda (H7: (((eq T t0 (THead (Flat Appl) u1 -t1)) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead -(Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda -(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda -(t3: T).(eq T t2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T -(\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T t2 (THead (Bind -b0) v3 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda -(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3)))))))))))).(\lambda (H8: (eq T (THead (Flat -Appl) v1 (THead (Bind b) u0 t0)) (THead (Flat Appl) u1 t1))).(let H9 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t _) -\Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead (Flat -Appl) u1 t1) H8) in ((let H10 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow (THead (Bind b) u0 t0) -| (TLRef _) \Rightarrow (THead (Bind b) u0 t0) | (THead _ _ t) \Rightarrow -t])) (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead (Flat Appl) u1 t1) -H8) in (\lambda (H11: (eq T v1 u1)).(let H12 \def (eq_ind T v1 (\lambda (t: -T).((eq T t (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T (\lambda (u3: -T).(\lambda (t3: T).(eq T v2 (THead (Flat Appl) u3 t3)))) (\lambda (u3: -T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 -t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T v2 (THead (Bind -Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda -(_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b0: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T v2 (THead (Bind b0) v3 (THead -(Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 -u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(t3: T).(pr0 z1 t3))))))))))) H3 u1 H11) in (let H13 \def (eq_ind T v1 -(\lambda (t: T).(pr0 t v2)) H2 u1 H11) in (let H14 \def (eq_ind_r T t1 -(\lambda (t: T).((eq T t0 (THead (Flat Appl) u1 t)) \to (or3 (ex3_2 T T -(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u3 t3)))) -(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind -Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda -(_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b0: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind -b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T t2 (THead (Bind b0) v3 (THead -(Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 -u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(t3: T).(pr0 z1 t3))))))))))) H7 (THead (Bind b) u0 t0) H10) in (let H15 \def -(eq_ind_r T t1 (\lambda (t: T).((eq T u0 (THead (Flat Appl) u1 t)) \to (or3 -(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead (Flat Appl) u3 -t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 -z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: -T).(eq T u2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T -(\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t -(THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T u2 (THead (Bind -b0) v3 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda -(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3))))))))))) H5 (THead (Bind b) u0 t0) H10) in -(let H16 \def (eq_ind_r T t1 (\lambda (t: T).((eq T u1 (THead (Flat Appl) u1 -t)) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T v2 (THead -(Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda -(_: T).(\lambda (t3: T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda -(t3: T).(eq T v2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T -(\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t -(THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T v2 (THead (Bind -b0) v3 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda -(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3))))))))))) H12 (THead (Bind b) u0 t0) H10) in -(eq_ind T (THead (Bind b) u0 t0) (\lambda (t: T).(or3 (ex3_2 T T (\lambda -(u3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t2)) (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: -T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t t3)))) (ex4_4 T T T -T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: -T).(\lambda (t3: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O -v2) t2)) (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T -(\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t -(THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Bind b0) v3 (THead (Flat -Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 -t3)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T -(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Flat -Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: -T).(\lambda (t3: T).(pr0 (THead (Bind b) u0 t0) t3)))) (ex4_4 T T T T -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind b) u0 t0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Bind Abbr) u3 t3)))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 -u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: -T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B -b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) u0 t0) (THead -(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Bind b0) v3 (THead (Flat -Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 -t3)))))))) (ex6_6_intro B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 -Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) u0 t0) (THead (Bind -b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t2)) (THead (Bind b0) v3 (THead (Flat Appl) -(lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 -t3))))))) b u0 t0 v2 u2 t2 H1 (refl_equal T (THead (Bind b) u0 t0)) -(refl_equal T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2))) -H13 H4 H6)) t1 H10)))))))) H9))))))))))))))))) (\lambda (u0: T).(\lambda (u2: -T).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (((eq T u0 (THead (Flat Appl) u1 -t1)) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead -(Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda -(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda -(t2: T).(eq T u2 (THead (Bind Abbr) u3 t2)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t2: T).(eq T u2 (THead (Bind -b) v2 (THead (Flat Appl) (lift (S O) O u3) t2))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda -(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (t2: T).(pr0 z1 t2)))))))))))).(\lambda (t0: T).(\lambda (t2: -T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u1 -t1)) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead -(Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda -(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda -(t3: T).(eq T t2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -b) v2 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda -(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3)))))))))))).(\lambda (w: T).(\lambda (_: -(subst0 O u2 t2 w)).(\lambda (H6: (eq T (THead (Bind Abbr) u0 t0) (THead -(Flat Appl) u1 t1))).(let H7 \def (eq_ind T (THead (Bind Abbr) u0 t0) -(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow -(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | -(Flat _) \Rightarrow False])])) I (THead (Flat Appl) u1 t1) H6) in (False_ind -(or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 -w) (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 -u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead -(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: -T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u3 -t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: -T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B -b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda -(v2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 w) (THead (Bind b) v2 -(THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 -u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(t3: T).(pr0 z1 t3))))))))) H7))))))))))))) (\lambda (b: B).(\lambda (_: (not -(eq B b Abst))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 -t2)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b0: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b0) v2 (THead -(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 -u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(t3: T).(pr0 z1 t3)))))))))))).(\lambda (u: T).(\lambda (H4: (eq T (THead -(Bind b) u (lift (S O) O t0)) (THead (Flat Appl) u1 t1))).(let H5 \def -(eq_ind T (THead (Bind b) u (lift (S O) O t0)) (\lambda (ee: T).(match ee in -T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow -False])])) I (THead (Flat Appl) u1 t1) H4) in (False_ind (or3 (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b0: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b0) v2 (THead -(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 -u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(t3: T).(pr0 z1 t3))))))))) H5)))))))))) (\lambda (t0: T).(\lambda (t2: -T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u1 -t1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead -(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda -(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda -(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3)))))))))))).(\lambda (u: T).(\lambda (H3: (eq -T (THead (Flat Cast) u t0) (THead (Flat Appl) u1 t1))).(let H4 \def (eq_ind T -(THead (Flat Cast) u t0) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f in F return -(\lambda (_: F).Prop) with [Appl \Rightarrow False | Cast \Rightarrow -True])])])) I (THead (Flat Appl) u1 t1) H3) in (False_ind (or3 (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead -(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 -u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(t3: T).(pr0 z1 t3))))))))) H4)))))))) y x H0))) H)))). -(* COMMENTS -Initial nodes: 12299 -END *) - -theorem pr0_gen_cast: - \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Flat Cast) u1 -t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead -(Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda -(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 x))))) -\def - \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead -(Flat Cast) u1 t1) x)).(insert_eq T (THead (Flat Cast) u1 t1) (\lambda (t: -T).(pr0 t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 -u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 x))) -(\lambda (y: T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: T).(\lambda -(t0: T).((eq T t (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T t0 (THead (Flat Cast) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 -t2)))) (pr0 t1 t0))))) (\lambda (t: T).(\lambda (H1: (eq T t (THead (Flat -Cast) u1 t1))).(let H2 \def (f_equal T T (\lambda (e: T).e) t (THead (Flat -Cast) u1 t1) H1) in (eq_ind_r T (THead (Flat Cast) u1 t1) (\lambda (t0: -T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Flat -Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: -T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 t0))) (or_introl (ex3_2 T T -(\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Cast) u1 t1) (THead -(Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda -(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (THead (Flat Cast) u1 t1)) -(ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Cast) -u1 t1) (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 -u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))) u1 t1 (refl_equal T -(THead (Flat Cast) u1 t1)) (pr0_refl u1) (pr0_refl t1))) t H2)))) (\lambda -(u0: T).(\lambda (u2: T).(\lambda (H1: (pr0 u0 u2)).(\lambda (H2: (((eq T u0 -(THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: -T).(eq T u2 (THead (Flat Cast) u3 t2)))) (\lambda (u3: T).(\lambda (_: -T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 -u2))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H3: (pr0 t0 t2)).(\lambda -(H4: (((eq T t0 (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: -T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u3 t3)))) (\lambda (u3: -T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 -t3)))) (pr0 t1 t2))))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u0 t0) -(THead (Flat Cast) u1 t1))).(let H6 \def (f_equal T K (\lambda (e: T).(match -e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) -\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u0 t0) (THead (Flat -Cast) u1 t1) H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) -\Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead k u0 t0) (THead (Flat -Cast) u1 t1) H5) in ((let H8 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) -\Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k u0 t0) (THead (Flat -Cast) u1 t1) H5) in (\lambda (H9: (eq T u0 u1)).(\lambda (H10: (eq K k (Flat -Cast))).(eq_ind_r K (Flat Cast) (\lambda (k0: K).(or (ex3_2 T T (\lambda (u3: -T).(\lambda (t3: T).(eq T (THead k0 u2 t2) (THead (Flat Cast) u3 t3)))) -(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3)))) (pr0 t1 (THead k0 u2 t2)))) (let H11 \def (eq_ind T t0 -(\lambda (t: T).((eq T t (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T -(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u3 t3)))) -(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3)))) (pr0 t1 t2)))) H4 t1 H8) in (let H12 \def (eq_ind T t0 -(\lambda (t: T).(pr0 t t2)) H3 t1 H8) in (let H13 \def (eq_ind T u0 (\lambda -(t: T).((eq T t (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: -T).(\lambda (t3: T).(eq T u2 (THead (Flat Cast) u3 t3)))) (\lambda (u3: -T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 -t3)))) (pr0 t1 u2)))) H2 u1 H9) in (let H14 \def (eq_ind T u0 (\lambda (t: -T).(pr0 t u2)) H1 u1 H9) in (or_introl (ex3_2 T T (\lambda (u3: T).(\lambda -(t3: T).(eq T (THead (Flat Cast) u2 t2) (THead (Flat Cast) u3 t3)))) (\lambda -(u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 -t1 t3)))) (pr0 t1 (THead (Flat Cast) u2 t2)) (ex3_2_intro T T (\lambda (u3: -T).(\lambda (t3: T).(eq T (THead (Flat Cast) u2 t2) (THead (Flat Cast) u3 -t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t1 t3))) u2 t2 (refl_equal T (THead (Flat Cast) u2 -t2)) H14 H12)))))) k H10)))) H7)) H6)))))))))))) (\lambda (u: T).(\lambda -(v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 -(THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T v2 (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 -v2))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda -(_: (((eq T t0 (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 -t3)))) (pr0 t1 t2))))).(\lambda (H5: (eq T (THead (Flat Appl) v1 (THead (Bind -Abst) u t0)) (THead (Flat Cast) u1 t1))).(let H6 \def (eq_ind T (THead (Flat -Appl) v1 (THead (Bind Abst) u t0)) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f -in F return (\lambda (_: F).Prop) with [Appl \Rightarrow True | Cast -\Rightarrow False])])])) I (THead (Flat Cast) u1 t1) H5) in (False_ind (or -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) -(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) -(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (THead (Bind Abbr) v2 -t2))) H6)))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b -Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda -(_: (((eq T v1 (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T v2 (THead (Flat Cast) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 -t2)))) (pr0 t1 v2))))).(\lambda (u0: T).(\lambda (u2: T).(\lambda (_: (pr0 u0 -u2)).(\lambda (_: (((eq T u0 (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T -(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Cast) u3 t2)))) -(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: -T).(pr0 t1 t2)))) (pr0 t1 u2))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda -(_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Flat Cast) u1 t1)) \to (or -(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u3 -t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2))))).(\lambda (H8: (eq T (THead -(Flat Appl) v1 (THead (Bind b) u0 t0)) (THead (Flat Cast) u1 t1))).(let H9 -\def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K -return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) -\Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl \Rightarrow -True | Cast \Rightarrow False])])])) I (THead (Flat Cast) u1 t1) H8) in -(False_ind (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Flat Cast) u3 -t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t2)))) H9))))))))))))))))) (\lambda (u0: T).(\lambda -(u2: T).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (((eq T u0 (THead (Flat Cast) -u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead -(Flat Cast) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda -(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 u2))))).(\lambda (t0: -T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead -(Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq -T t2 (THead (Flat Cast) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 -u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 -t2))))).(\lambda (w: T).(\lambda (_: (subst0 O u2 t2 w)).(\lambda (H6: (eq T -(THead (Bind Abbr) u0 t0) (THead (Flat Cast) u1 t1))).(let H7 \def (eq_ind T -(THead (Bind Abbr) u0 t0) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat -Cast) u1 t1) H6) in (False_ind (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: -T).(eq T (THead (Bind Abbr) u2 w) (THead (Flat Cast) u3 t3)))) (\lambda (u3: -T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 -t3)))) (pr0 t1 (THead (Bind Abbr) u2 w))) H7))))))))))))) (\lambda (b: -B).(\lambda (_: (not (eq B b Abst))).(\lambda (t0: T).(\lambda (t2: -T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Flat Cast) u1 -t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead -(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda -(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2))))).(\lambda (u: -T).(\lambda (H4: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead (Flat -Cast) u1 t1))).(let H5 \def (eq_ind T (THead (Bind b) u (lift (S O) O t0)) -(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow -(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | -(Flat _) \Rightarrow False])])) I (THead (Flat Cast) u1 t1) H4) in (False_ind -(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2)) H5)))))))))) (\lambda (t0: -T).(\lambda (t2: T).(\lambda (H1: (pr0 t0 t2)).(\lambda (H2: (((eq T t0 -(THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: -T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 -t2))))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u t0) (THead -(Flat Cast) u1 t1))).(let H4 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) -\Rightarrow u | (THead _ t _) \Rightarrow t])) (THead (Flat Cast) u t0) -(THead (Flat Cast) u1 t1) H3) in ((let H5 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | -(TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Flat Cast) -u t0) (THead (Flat Cast) u1 t1) H3) in (\lambda (_: (eq T u u1)).(let H7 \def -(eq_ind T t0 (\lambda (t: T).((eq T t (THead (Flat Cast) u1 t1)) \to (or -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2)))) H2 t1 H5) in (let H8 \def -(eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H1 t1 H5) in (or_intror (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3)))) (pr0 t1 t2) H8))))) H4)))))))) y x H0))) H)))). -(* COMMENTS -Initial nodes: 2911 -END *) - -theorem pr0_gen_abbr: - \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Bind Abbr) u1 -t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead -(Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda -(u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y: T).(pr0 t1 y)) -(\lambda (y: T).(subst0 O u2 y t2))))))) (pr0 t1 (lift (S O) O x)))))) -\def - \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead -(Bind Abbr) u1 t1) x)).(insert_eq T (THead (Bind Abbr) u1 t1) (\lambda (t: -T).(pr0 t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 -u1 u2))) (\lambda (u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda -(y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y t2))))))) (pr0 t1 (lift (S -O) O x)))) (\lambda (y: T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: -T).(\lambda (t0: T).((eq T t (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T -(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: -T).(or (pr0 t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: -T).(subst0 O u2 y0 t2))))))) (pr0 t1 (lift (S O) O t0)))))) (\lambda (t: -T).(\lambda (H1: (eq T t (THead (Bind Abbr) u1 t1))).(let H2 \def (f_equal T -T (\lambda (e: T).e) t (THead (Bind Abbr) u1 t1) H1) in (eq_ind_r T (THead -(Bind Abbr) u1 t1) (\lambda (t0: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda -(t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T -(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 t2))))))) (pr0 -t1 (lift (S O) O t0)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T (THead (Bind Abbr) u1 t1) (THead (Bind Abbr) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: T).(or (pr0 -t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 -t2))))))) (pr0 t1 (lift (S O) O (THead (Bind Abbr) u1 t1))) (ex3_2_intro T T -(\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Bind Abbr) u1 t1) (THead -(Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda -(u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) -(\lambda (y0: T).(subst0 O u2 y0 t2)))))) u1 t1 (refl_equal T (THead (Bind -Abbr) u1 t1)) (pr0_refl u1) (or_introl (pr0 t1 t1) (ex2 T (\lambda (y0: -T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u1 y0 t1))) (pr0_refl t1)))) t -H2)))) (\lambda (u0: T).(\lambda (u2: T).(\lambda (H1: (pr0 u0 u2)).(\lambda -(H2: (((eq T u0 (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: -T).(\lambda (t2: T).(eq T u2 (THead (Bind Abbr) u3 t2)))) (\lambda (u3: -T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t2: T).(or (pr0 -t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 -t2))))))) (pr0 t1 (lift (S O) O u2)))))).(\lambda (t0: T).(\lambda (t2: -T).(\lambda (H3: (pr0 t0 t2)).(\lambda (H4: (((eq T t0 (THead (Bind Abbr) u1 -t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead -(Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda -(u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) -(\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O -t2)))))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u0 t0) (THead (Bind -Abbr) u1 t1))).(let H6 \def (f_equal T K (\lambda (e: T).(match e in T return -(\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | -(THead k0 _ _) \Rightarrow k0])) (THead k u0 t0) (THead (Bind Abbr) u1 t1) -H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 -| (THead _ t _) \Rightarrow t])) (THead k u0 t0) (THead (Bind Abbr) u1 t1) -H5) in ((let H8 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 -| (THead _ _ t) \Rightarrow t])) (THead k u0 t0) (THead (Bind Abbr) u1 t1) -H5) in (\lambda (H9: (eq T u0 u1)).(\lambda (H10: (eq K k (Bind -Abbr))).(eq_ind_r K (Bind Abbr) (\lambda (k0: K).(or (ex3_2 T T (\lambda (u3: -T).(\lambda (t3: T).(eq T (THead k0 u2 t2) (THead (Bind Abbr) u3 t3)))) -(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: -T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: -T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O (THead k0 u2 t2))))) (let -H11 \def (eq_ind T t0 (\lambda (t: T).((eq T t (THead (Bind Abbr) u1 t1)) \to -(or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) -u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: -T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) -(\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O t2))))) H4 -t1 H8) in (let H12 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H3 t1 H8) in -(let H13 \def (eq_ind T u0 (\lambda (t: T).((eq T t (THead (Bind Abbr) u1 -t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead -(Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda -(u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) -(\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O u2))))) H2 -u1 H9) in (let H14 \def (eq_ind T u0 (\lambda (t: T).(pr0 t u2)) H1 u1 H9) in -(or_introl (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind -Abbr) u2 t2) (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: -T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T -(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 -t1 (lift (S O) O (THead (Bind Abbr) u2 t2))) (ex3_2_intro T T (\lambda (u3: -T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 t2) (THead (Bind Abbr) u3 -t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: -T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) -(\lambda (y0: T).(subst0 O u3 y0 t3)))))) u2 t2 (refl_equal T (THead (Bind -Abbr) u2 t2)) H14 (or_introl (pr0 t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) -(\lambda (y0: T).(subst0 O u2 y0 t2))) H12))))))) k H10)))) H7)) -H6)))))))))))) (\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: -(pr0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind Abbr) u1 t1)) \to (or -(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind Abbr) u2 -t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: -T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) -(\lambda (y0: T).(subst0 O u2 y0 t2))))))) (pr0 t1 (lift (S O) O -v2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda -(_: (((eq T t0 (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0 -t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 -t3))))))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (H5: (eq T (THead (Flat -Appl) v1 (THead (Bind Abst) u t0)) (THead (Bind Abbr) u1 t1))).(let H6 \def -(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K -return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) -\Rightarrow True])])) I (THead (Bind Abbr) u1 t1) H5) in (False_ind (or -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) -(THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) -(\lambda (u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: -T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S -O) O (THead (Bind Abbr) v2 t2)))) H6)))))))))))) (\lambda (b: B).(\lambda (_: -(not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 -v2)).(\lambda (_: (((eq T v1 (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T -(\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind Abbr) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: -T).(or (pr0 t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: -T).(subst0 O u2 y0 t2))))))) (pr0 t1 (lift (S O) O v2)))))).(\lambda (u0: -T).(\lambda (u2: T).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (((eq T u0 (THead -(Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq -T u2 (THead (Bind Abbr) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 -u3))) (\lambda (u3: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y0: -T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 t2))))))) (pr0 t1 (lift (S -O) O u2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 -t2)).(\lambda (_: (((eq T t0 (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T -(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u3 t3)))) -(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: -T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: -T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (H8: (eq -T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead (Bind Abbr) u1 -t1))).(let H9 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) -(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow -(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | -(Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u1 t1) H8) in (False_ind -(or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Bind Abbr) u3 t3)))) -(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: -T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: -T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t2))))) H9))))))))))))))))) (\lambda (u0: -T).(\lambda (u2: T).(\lambda (H1: (pr0 u0 u2)).(\lambda (H2: (((eq T u0 -(THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: -T).(eq T u2 (THead (Bind Abbr) u3 t2)))) (\lambda (u3: T).(\lambda (_: -T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T -(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 t2))))))) (pr0 -t1 (lift (S O) O u2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H3: -(pr0 t0 t2)).(\lambda (H4: (((eq T t0 (THead (Bind Abbr) u1 t1)) \to (or -(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u3 -t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: -T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) -(\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O -t2)))))).(\lambda (w: T).(\lambda (H5: (subst0 O u2 t2 w)).(\lambda (H6: (eq -T (THead (Bind Abbr) u0 t0) (THead (Bind Abbr) u1 t1))).(let H7 \def (f_equal -T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) -(THead (Bind Abbr) u0 t0) (THead (Bind Abbr) u1 t1) H6) in ((let H8 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) -\Rightarrow t])) (THead (Bind Abbr) u0 t0) (THead (Bind Abbr) u1 t1) H6) in -(\lambda (H9: (eq T u0 u1)).(let H10 \def (eq_ind T t0 (\lambda (t: T).((eq T -t (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda -(t3: T).(eq T t2 (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: -T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T -(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 -t1 (lift (S O) O t2))))) H4 t1 H8) in (let H11 \def (eq_ind T t0 (\lambda (t: -T).(pr0 t t2)) H3 t1 H8) in (let H12 \def (eq_ind T u0 (\lambda (t: T).((eq T -t (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda -(t3: T).(eq T u2 (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: -T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T -(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 -t1 (lift (S O) O u2))))) H2 u1 H9) in (let H13 \def (eq_ind T u0 (\lambda (t: -T).(pr0 t u2)) H1 u1 H9) in (or_introl (ex3_2 T T (\lambda (u3: T).(\lambda -(t3: T).(eq T (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u3 t3)))) (\lambda -(u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or -(pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O -u3 y0 t3))))))) (pr0 t1 (lift (S O) O (THead (Bind Abbr) u2 w))) (ex3_2_intro -T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 w) (THead -(Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda -(u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) -(\lambda (y0: T).(subst0 O u3 y0 t3)))))) u2 w (refl_equal T (THead (Bind -Abbr) u2 w)) H13 (or_intror (pr0 t1 w) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) -(\lambda (y0: T).(subst0 O u2 y0 w))) (ex_intro2 T (\lambda (y0: T).(pr0 t1 -y0)) (\lambda (y0: T).(subst0 O u2 y0 w)) t2 H11 H5)))))))))) H7))))))))))))) -(\lambda (b: B).(\lambda (H1: (not (eq B b Abst))).(\lambda (t0: T).(\lambda -(t2: T).(\lambda (H2: (pr0 t0 t2)).(\lambda (H3: (((eq T t0 (THead (Bind -Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 -(THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) -(\lambda (u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: -T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S -O) O t2)))))).(\lambda (u: T).(\lambda (H4: (eq T (THead (Bind b) u (lift (S -O) O t0)) (THead (Bind Abbr) u1 t1))).(let H5 \def (f_equal T B (\lambda (e: -T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | -(TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k in K return -(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow -b])])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abbr) u1 t1) H4) in -((let H6 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: -T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) -\Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abbr) u1 -t1) H4) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat -\to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow -(TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true -\Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t3) \Rightarrow -(THead k (lref_map f d u0) (lref_map f (s k d) t3))]) in lref_map) (\lambda -(x0: nat).(plus x0 (S O))) O t0) | (TLRef _) \Rightarrow ((let rec lref_map -(f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) -\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with -[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t3) -\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t3))]) in -lref_map) (\lambda (x0: nat).(plus x0 (S O))) O t0) | (THead _ _ t) -\Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abbr) u1 -t1) H4) in (\lambda (_: (eq T u u1)).(\lambda (H9: (eq B b Abbr)).(let H10 -\def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H1 Abbr H9) in (let -H11 \def (eq_ind_r T t1 (\lambda (t: T).((eq T t0 (THead (Bind Abbr) u1 t)) -\to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: -T).(\lambda (t3: T).(or (pr0 t t3) (ex2 T (\lambda (y0: T).(pr0 t y0)) -(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t (lift (S O) O t2))))) H3 -(lift (S O) O t0) H7) in (eq_ind T (lift (S O) O t0) (\lambda (t: T).(or -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: -T).(\lambda (t3: T).(or (pr0 t t3) (ex2 T (\lambda (y0: T).(pr0 t y0)) -(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t (lift (S O) O t2)))) -(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: -T).(\lambda (t3: T).(or (pr0 (lift (S O) O t0) t3) (ex2 T (\lambda (y0: -T).(pr0 (lift (S O) O t0) y0)) (\lambda (y0: T).(subst0 O u2 y0 t3))))))) -(pr0 (lift (S O) O t0) (lift (S O) O t2)) (pr0_lift t0 t2 H2 (S O) O)) t1 -H7)))))) H6)) H5)))))))))) (\lambda (t0: T).(\lambda (t2: T).(\lambda (_: -(pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Abbr) u1 t1)) \to (or -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: -T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) -(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S O) O -t2)))))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u t0) (THead -(Bind Abbr) u1 t1))).(let H4 \def (eq_ind T (THead (Flat Cast) u t0) (\lambda -(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow -(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | -(Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u1 t1) H3) in (False_ind -(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: -T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) -(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S O) O t2))) -H4)))))))) y x H0))) H)))). -(* COMMENTS -Initial nodes: 4711 -END *) - -theorem pr0_gen_void: - \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Bind Void) u1 -t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead -(Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda -(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O x)))))) -\def - \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead -(Bind Void) u1 t1) x)).(insert_eq T (THead (Bind Void) u1 t1) (\lambda (t: -T).(pr0 t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 -u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) -O x)))) (\lambda (y: T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: -T).(\lambda (t0: T).((eq T t (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T -(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Void) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: -T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O t0)))))) (\lambda (t: T).(\lambda -(H1: (eq T t (THead (Bind Void) u1 t1))).(let H2 \def (f_equal T T (\lambda -(e: T).e) t (THead (Bind Void) u1 t1) H1) in (eq_ind_r T (THead (Bind Void) -u1 t1) (\lambda (t0: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq -T t0 (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 -u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O -t0)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead -(Bind Void) u1 t1) (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 -(lift (S O) O (THead (Bind Void) u1 t1))) (ex3_2_intro T T (\lambda (u2: -T).(\lambda (t2: T).(eq T (THead (Bind Void) u1 t1) (THead (Bind Void) u2 -t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: -T).(\lambda (t2: T).(pr0 t1 t2))) u1 t1 (refl_equal T (THead (Bind Void) u1 -t1)) (pr0_refl u1) (pr0_refl t1))) t H2)))) (\lambda (u0: T).(\lambda (u2: -T).(\lambda (H1: (pr0 u0 u2)).(\lambda (H2: (((eq T u0 (THead (Bind Void) u1 -t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead -(Bind Void) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda -(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O -u2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H3: (pr0 t0 -t2)).(\lambda (H4: (((eq T t0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T -(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u3 t3)))) -(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (k: K).(\lambda -(H5: (eq T (THead k u0 t0) (THead (Bind Void) u1 t1))).(let H6 \def (f_equal -T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) -\Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) -(THead k u0 t0) (THead (Bind Void) u1 t1) H5) in ((let H7 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) -(THead k u0 t0) (THead (Bind Void) u1 t1) H5) in ((let H8 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) -(THead k u0 t0) (THead (Bind Void) u1 t1) H5) in (\lambda (H9: (eq T u0 -u1)).(\lambda (H10: (eq K k (Bind Void))).(eq_ind_r K (Bind Void) (\lambda -(k0: K).(or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead k0 u2 -t2) (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 -u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O -(THead k0 u2 t2))))) (let H11 \def (eq_ind T t0 (\lambda (t: T).((eq T t -(THead (Bind Void) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: -T).(eq T t2 (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: -T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 -(lift (S O) O t2))))) H4 t1 H8) in (let H12 \def (eq_ind T t0 (\lambda (t: -T).(pr0 t t2)) H3 t1 H8) in (let H13 \def (eq_ind T u0 (\lambda (t: T).((eq T -t (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda -(t3: T).(eq T u2 (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: -T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 -(lift (S O) O u2))))) H2 u1 H9) in (let H14 \def (eq_ind T u0 (\lambda (t: -T).(pr0 t u2)) H1 u1 H9) in (or_introl (ex3_2 T T (\lambda (u3: T).(\lambda -(t3: T).(eq T (THead (Bind Void) u2 t2) (THead (Bind Void) u3 t3)))) (\lambda -(u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 -t1 t3)))) (pr0 t1 (lift (S O) O (THead (Bind Void) u2 t2))) (ex3_2_intro T T -(\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Void) u2 t2) (THead -(Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda -(_: T).(\lambda (t3: T).(pr0 t1 t3))) u2 t2 (refl_equal T (THead (Bind Void) -u2 t2)) H14 H12)))))) k H10)))) H7)) H6)))))))))))) (\lambda (u: T).(\lambda -(v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 -(THead (Bind Void) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T v2 (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 -(lift (S O) O v2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 -t2)).(\lambda (_: (((eq T t0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (H5: (eq T (THead -(Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Bind Void) u1 t1))).(let H6 -\def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K -return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) -\Rightarrow True])])) I (THead (Bind Void) u1 t1) H5) in (False_ind (or -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) -(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) -(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O (THead -(Bind Abbr) v2 t2)))) H6)))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B -b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 -v2)).(\lambda (_: (((eq T v1 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T -(\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind Void) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: -T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O v2)))))).(\lambda (u0: T).(\lambda -(u2: T).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (((eq T u0 (THead (Bind Void) -u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead -(Bind Void) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda -(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O -u2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda -(_: (((eq T t0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: -T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u3 t3)))) (\lambda (u3: -T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 -t3)))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (H8: (eq T (THead (Flat Appl) -v1 (THead (Bind b) u0 t0)) (THead (Bind Void) u1 t1))).(let H9 \def (eq_ind T -(THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (\lambda (ee: T).(match ee in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind Void) u1 t1) H8) in (False_ind (or (ex3_2 T T -(\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t2)) (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda -(_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 -(lift (S O) O (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2))))) -H9))))))))))))))))) (\lambda (u0: T).(\lambda (u2: T).(\lambda (_: (pr0 u0 -u2)).(\lambda (_: (((eq T u0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T -(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Bind Void) u3 t2)))) -(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: -T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O u2)))))).(\lambda (t0: T).(\lambda -(t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Void) -u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead -(Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda -(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O -t2)))))).(\lambda (w: T).(\lambda (_: (subst0 O u2 t2 w)).(\lambda (H6: (eq T -(THead (Bind Abbr) u0 t0) (THead (Bind Void) u1 t1))).(let H7 \def (eq_ind T -(THead (Bind Abbr) u0 t0) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr -\Rightarrow True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat -_) \Rightarrow False])])) I (THead (Bind Void) u1 t1) H6) in (False_ind (or -(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 w) -(THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) -(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O (THead -(Bind Abbr) u2 w)))) H7))))))))))))) (\lambda (b: B).(\lambda (H1: (not (eq B -b Abst))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H2: (pr0 t0 -t2)).(\lambda (H3: (((eq T t0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (u: T).(\lambda -(H4: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Void) u1 -t1))).(let H5 \def (f_equal T B (\lambda (e: T).(match e in T return (\lambda -(_: T).B) with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead k -_ _) \Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b0) -\Rightarrow b0 | (Flat _) \Rightarrow b])])) (THead (Bind b) u (lift (S O) O -t0)) (THead (Bind Void) u1 t1) H4) in ((let H6 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | -(TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t])) (THead (Bind b) u -(lift (S O) O t0)) (THead (Bind Void) u1 t1) H4) in ((let H7 \def (f_equal T -T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T -\def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow -(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) -| (THead k u0 t3) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) -t3))]) in lref_map) (\lambda (x0: nat).(plus x0 (S O))) O t0) | (TLRef _) -\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T -\def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow -(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) -| (THead k u0 t3) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) -t3))]) in lref_map) (\lambda (x0: nat).(plus x0 (S O))) O t0) | (THead _ _ t) -\Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Void) u1 -t1) H4) in (\lambda (_: (eq T u u1)).(\lambda (H9: (eq B b Void)).(let H10 -\def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H1 Void H9) in (let -H11 \def (eq_ind_r T t1 (\lambda (t: T).((eq T t0 (THead (Bind Void) u1 t)) -\to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t t3)))) (pr0 t (lift (S O) O t2))))) H3 (lift (S O) -O t0) H7) in (eq_ind T (lift (S O) O t0) (\lambda (t: T).(or (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t t3)))) (pr0 t (lift (S O) O t2)))) (or_intror (ex3_2 T T (\lambda -(u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 (lift -(S O) O t0) t3)))) (pr0 (lift (S O) O t0) (lift (S O) O t2)) (pr0_lift t0 t2 -H2 (S O) O)) t1 H7)))))) H6)) H5)))))))))) (\lambda (t0: T).(\lambda (t2: -T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Void) u1 -t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead -(Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda -(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O -t2)))))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u t0) (THead -(Bind Void) u1 t1))).(let H4 \def (eq_ind T (THead (Flat Cast) u t0) (\lambda -(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow -(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | -(Flat _) \Rightarrow True])])) I (THead (Bind Void) u1 t1) H3) in (False_ind -(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2))) H4)))))))) y x -H0))) H)))). -(* COMMENTS -Initial nodes: 3436 -END *) - -theorem pr0_gen_lift: - \forall (t1: T).(\forall (x: T).(\forall (h: nat).(\forall (d: nat).((pr0 -(lift h d t1) x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda -(t2: T).(pr0 t1 t2))))))) -\def - \lambda (t1: T).(\lambda (x: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda -(H: (pr0 (lift h d t1) x)).(insert_eq T (lift h d t1) (\lambda (t: T).(pr0 t -x)) (\lambda (_: T).(ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda -(t2: T).(pr0 t1 t2)))) (\lambda (y: T).(\lambda (H0: (pr0 y x)).(unintro nat -d (\lambda (n: nat).((eq T y (lift h n t1)) \to (ex2 T (\lambda (t2: T).(eq T -x (lift h n t2))) (\lambda (t2: T).(pr0 t1 t2))))) (unintro T t1 (\lambda (t: -T).(\forall (x0: nat).((eq T y (lift h x0 t)) \to (ex2 T (\lambda (t2: T).(eq -T x (lift h x0 t2))) (\lambda (t2: T).(pr0 t t2)))))) (pr0_ind (\lambda (t: -T).(\lambda (t0: T).(\forall (x0: T).(\forall (x1: nat).((eq T t (lift h x1 -x0)) \to (ex2 T (\lambda (t2: T).(eq T t0 (lift h x1 t2))) (\lambda (t2: -T).(pr0 x0 t2)))))))) (\lambda (t: T).(\lambda (x0: T).(\lambda (x1: -nat).(\lambda (H1: (eq T t (lift h x1 x0))).(ex_intro2 T (\lambda (t2: T).(eq -T t (lift h x1 t2))) (\lambda (t2: T).(pr0 x0 t2)) x0 H1 (pr0_refl x0)))))) -(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda (H2: -((\forall (x0: T).(\forall (x1: nat).((eq T u1 (lift h x1 x0)) \to (ex2 T -(\lambda (t2: T).(eq T u2 (lift h x1 t2))) (\lambda (t2: T).(pr0 x0 -t2)))))))).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2 -t3)).(\lambda (H4: ((\forall (x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 -x0)) \to (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: -T).(pr0 x0 t4)))))))).(\lambda (k: K).(\lambda (x0: T).(\lambda (x1: -nat).(\lambda (H5: (eq T (THead k u1 t2) (lift h x1 x0))).(K_ind (\lambda -(k0: K).((eq T (THead k0 u1 t2) (lift h x1 x0)) \to (ex2 T (\lambda (t4: -T).(eq T (THead k0 u2 t3) (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4))))) -(\lambda 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Abbr) u2 w) (lift h x1 t4))) (\lambda (t4: T).(pr0 -(THead (Bind Abbr) x2 x3) t4))) (\lambda (x4: T).(\lambda (H_x: (eq T t3 -(lift h (S x1) x4))).(\lambda (H10: (pr0 x3 x4)).(let H11 \def (eq_ind T t3 -(\lambda (t: T).(subst0 O u2 t w)) H5 (lift h (S x1) x4) H_x) in (ex2_ind T -(\lambda (t4: T).(eq T u2 (lift h x1 t4))) (\lambda (t4: T).(pr0 x2 t4)) (ex2 -T (\lambda (t4: T).(eq T (THead (Bind Abbr) u2 w) (lift h x1 t4))) (\lambda -(t4: T).(pr0 (THead (Bind Abbr) x2 x3) t4))) (\lambda (x5: T).(\lambda (H_x0: -(eq T u2 (lift h x1 x5))).(\lambda (H12: (pr0 x2 x5)).(eq_ind_r T (lift h x1 -x5) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) t w) -(lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind Abbr) x2 x3) t4)))) (let -H13 \def (eq_ind T u2 (\lambda (t: T).(subst0 O t (lift h (S x1) x4) w)) H11 -(lift h x1 x5) H_x0) in (let H14 \def (refl_equal nat (S (plus O x1))) in -(let H15 \def (eq_ind nat (S x1) (\lambda (n: nat).(subst0 O (lift h x1 x5) -(lift h n x4) w)) H13 (S (plus O x1)) H14) in (ex2_ind T (\lambda (t4: T).(eq -T w (lift h (S (plus O x1)) t4))) (\lambda (t4: T).(subst0 O x5 x4 t4)) (ex2 -T (\lambda (t4: T).(eq T (THead (Bind Abbr) (lift h x1 x5) w) (lift h x1 -t4))) (\lambda (t4: T).(pr0 (THead (Bind Abbr) x2 x3) t4))) (\lambda (x6: -T).(\lambda (H16: (eq T w (lift h (S (plus O x1)) x6))).(\lambda (H17: -(subst0 O x5 x4 x6)).(eq_ind_r T (lift h (S (plus O x1)) x6) (\lambda (t: -T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) (lift h x1 x5) t) (lift h -x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind Abbr) x2 x3) t4)))) (ex_intro2 T -(\lambda (t4: T).(eq T (THead (Bind Abbr) (lift h x1 x5) (lift h (S (plus O -x1)) x6)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind Abbr) x2 x3) -t4)) (THead (Bind Abbr) x5 x6) (sym_eq T (lift h x1 (THead (Bind Abbr) x5 -x6)) (THead (Bind Abbr) (lift h x1 x5) (lift h (S (plus O x1)) x6)) -(lift_bind Abbr x5 x6 h (plus O x1))) (pr0_delta x2 x5 H12 x3 x4 H10 x6 H17)) -w H16)))) (subst0_gen_lift_lt x5 x4 w O h x1 H15))))) u2 H_x0)))) (H2 x2 x1 -H8)))))) (H4 x3 (S x1) H9)) x0 H7)))))) (lift_gen_bind Abbr u1 t2 x0 h x1 -H6))))))))))))))) (\lambda (b: B).(\lambda (H1: (not (eq B b Abst))).(\lambda -(t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2 t3)).(\lambda (H3: ((\forall -(x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to (ex2 T (\lambda (t4: -T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4)))))))).(\lambda (u: -T).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H4: (eq T (THead (Bind b) u -(lift (S O) O t2)) (lift h x1 x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda -(z: T).(eq T x0 (THead (Bind b) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq -T u (lift h x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift (S O) O t2) -(lift h (S x1) z)))) (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) -(\lambda (t4: T).(pr0 x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda -(H5: (eq T x0 (THead (Bind b) x2 x3))).(\lambda (_: (eq T u (lift h x1 -x2))).(\lambda (H7: (eq T (lift (S O) O t2) (lift h (S x1) x3))).(eq_ind_r T -(THead (Bind b) x2 x3) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T t3 (lift -h x1 t4))) (\lambda (t4: T).(pr0 t t4)))) (let H8 \def (eq_ind_r nat (plus (S -O) x1) (\lambda (n: nat).(eq nat (S x1) n)) (refl_equal nat (plus (S O) x1)) -(plus x1 (S O)) (plus_sym x1 (S O))) in (let H9 \def (eq_ind nat (S x1) -(\lambda (n: nat).(eq T (lift (S O) O t2) (lift h n x3))) H7 (plus x1 (S O)) -H8) in (ex2_ind T (\lambda (t4: T).(eq T x3 (lift (S O) O t4))) (\lambda (t4: -T).(eq T t2 (lift h x1 t4))) (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 -t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 x3) t4))) (\lambda (x4: -T).(\lambda (H10: (eq T x3 (lift (S O) O x4))).(\lambda (H11: (eq T t2 (lift -h x1 x4))).(eq_ind_r T (lift (S O) O x4) (\lambda (t: T).(ex2 T (\lambda (t4: -T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 t) -t4)))) (ex2_ind T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: -T).(pr0 x4 t4)) (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda -(t4: T).(pr0 (THead (Bind b) x2 (lift (S O) O x4)) t4))) (\lambda (x5: -T).(\lambda (H_x: (eq T t3 (lift h x1 x5))).(\lambda (H12: (pr0 x4 -x5)).(eq_ind_r T (lift h x1 x5) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T -t (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 (lift (S O) O -x4)) t4)))) (ex_intro2 T (\lambda (t4: T).(eq T (lift h x1 x5) (lift h x1 -t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 (lift (S O) O x4)) t4)) x5 -(refl_equal T (lift h x1 x5)) (pr0_zeta b H1 x4 x5 H12 x2)) t3 H_x)))) (H3 x4 -x1 H11)) x3 H10)))) (lift_gen_lift t2 x3 (S O) h O x1 (le_O_n x1) H9)))) x0 -H5)))))) (lift_gen_bind b u (lift (S O) O t2) x0 h x1 H4)))))))))))) (\lambda -(t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2 t3)).(\lambda (H2: ((\forall -(x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to (ex2 T (\lambda (t4: -T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4)))))))).(\lambda (u: -T).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H3: (eq T (THead (Flat Cast) -u t2) (lift h x1 x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T -x0 (THead (Flat Cast) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u (lift -h x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h x1 z)))) (ex2 T -(\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4))) -(\lambda (x2: T).(\lambda (x3: T).(\lambda (H4: (eq T x0 (THead (Flat Cast) -x2 x3))).(\lambda (_: (eq T u (lift h x1 x2))).(\lambda (H6: (eq T t2 (lift h -x1 x3))).(eq_ind_r T (THead (Flat Cast) x2 x3) (\lambda (t: T).(ex2 T -(\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 t t4)))) -(ex2_ind T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 -x3 t4)) (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: -T).(pr0 (THead (Flat Cast) x2 x3) t4))) (\lambda (x4: T).(\lambda (H_x: (eq T -t3 (lift h x1 x4))).(\lambda (H7: (pr0 x3 x4)).(eq_ind_r T (lift h x1 x4) -(\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T t (lift h x1 t4))) (\lambda -(t4: T).(pr0 (THead (Flat Cast) x2 x3) t4)))) (ex_intro2 T (\lambda (t4: -T).(eq T (lift h x1 x4) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat -Cast) x2 x3) t4)) x4 (refl_equal T (lift h x1 x4)) (pr0_tau x3 x4 H7 x2)) t3 -H_x)))) (H2 x3 x1 H6)) x0 H4)))))) (lift_gen_flat Cast u t2 x0 h x1 -H3)))))))))) y x H0))))) H))))). -(* COMMENTS -Initial nodes: 7569 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr0/pr0.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr0/pr0.ma deleted file mode 100644 index 9a3b397fe..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr0/pr0.ma +++ /dev/null @@ -1,2507 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr0/fwd.ma". - -include "Basic-1/lift/tlt.ma". - -theorem pr0_confluence__pr0_cong_upsilon_refl: - \forall (b: B).((not (eq B b Abst)) \to (\forall (u0: T).(\forall (u3: -T).((pr0 u0 u3) \to (\forall (t4: T).(\forall (t5: T).((pr0 t4 t5) \to -(\forall (u2: T).(\forall (v2: T).(\forall (x: T).((pr0 u2 x) \to ((pr0 v2 x) -\to (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u0 t4)) -t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O -v2) t5)) t))))))))))))))) -\def - \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (u0: T).(\lambda -(u3: T).(\lambda (H0: (pr0 u0 u3)).(\lambda (t4: T).(\lambda (t5: T).(\lambda -(H1: (pr0 t4 t5)).(\lambda (u2: T).(\lambda (v2: T).(\lambda (x: T).(\lambda -(H2: (pr0 u2 x)).(\lambda (H3: (pr0 v2 x)).(ex_intro2 T (\lambda (t: T).(pr0 -(THead (Flat Appl) u2 (THead (Bind b) u0 t4)) t)) (\lambda (t: T).(pr0 (THead -(Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t)) (THead (Bind b) u3 -(THead (Flat Appl) (lift (S O) O x) t5)) (pr0_upsilon b H u2 x H2 u0 u3 H0 t4 -t5 H1) (pr0_comp u3 u3 (pr0_refl u3) (THead (Flat Appl) (lift (S O) O v2) t5) -(THead (Flat Appl) (lift (S O) O x) t5) (pr0_comp (lift (S O) O v2) (lift (S -O) O x) (pr0_lift v2 x H3 (S O) O) t5 t5 (pr0_refl t5) (Flat Appl)) (Bind -b))))))))))))))). -(* COMMENTS -Initial nodes: 257 -END *) - -theorem pr0_confluence__pr0_cong_upsilon_cong: - \forall (b: B).((not (eq B b Abst)) \to (\forall (u2: T).(\forall (v2: -T).(\forall (x: T).((pr0 u2 x) \to ((pr0 v2 x) \to (\forall (t2: T).(\forall -(t5: T).(\forall (x0: T).((pr0 t2 x0) \to ((pr0 t5 x0) \to (\forall (u5: -T).(\forall (u3: T).(\forall (x1: T).((pr0 u5 x1) \to ((pr0 u3 x1) \to (ex2 T -(\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 t2)) t)) -(\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) -t5)) t))))))))))))))))))) -\def - \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (u2: T).(\lambda -(v2: T).(\lambda (x: T).(\lambda (H0: (pr0 u2 x)).(\lambda (H1: (pr0 v2 -x)).(\lambda (t2: T).(\lambda (t5: T).(\lambda (x0: T).(\lambda (H2: (pr0 t2 -x0)).(\lambda (H3: (pr0 t5 x0)).(\lambda (u5: T).(\lambda (u3: T).(\lambda -(x1: T).(\lambda (H4: (pr0 u5 x1)).(\lambda (H5: (pr0 u3 x1)).(ex_intro2 T -(\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 t2)) t)) -(\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) -t5)) t)) (THead (Bind b) x1 (THead (Flat Appl) (lift (S O) O x) x0)) -(pr0_upsilon b H u2 x H0 u5 x1 H4 t2 x0 H2) (pr0_comp u3 x1 H5 (THead (Flat -Appl) (lift (S O) O v2) t5) (THead (Flat Appl) (lift (S O) O x) x0) (pr0_comp -(lift (S O) O v2) (lift (S O) O x) (pr0_lift v2 x H1 (S O) O) t5 x0 H3 (Flat -Appl)) (Bind b))))))))))))))))))). -(* COMMENTS -Initial nodes: 269 -END *) - -theorem pr0_confluence__pr0_cong_upsilon_delta: - (not (eq B Abbr Abst)) \to (\forall (u5: T).(\forall (t2: T).(\forall (w: -T).((subst0 O u5 t2 w) \to (\forall (u2: T).(\forall (v2: T).(\forall (x: -T).((pr0 u2 x) \to ((pr0 v2 x) \to (\forall (t5: T).(\forall (x0: T).((pr0 t2 -x0) \to ((pr0 t5 x0) \to (\forall (u3: T).(\forall (x1: T).((pr0 u5 x1) \to -((pr0 u3 x1) \to (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead -(Bind Abbr) u5 w)) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 (THead -(Flat Appl) (lift (S O) O v2) t5)) t)))))))))))))))))))) -\def - \lambda (H: (not (eq B Abbr Abst))).(\lambda (u5: T).(\lambda (t2: -T).(\lambda (w: T).(\lambda (H0: (subst0 O u5 t2 w)).(\lambda (u2: -T).(\lambda (v2: T).(\lambda (x: T).(\lambda (H1: (pr0 u2 x)).(\lambda (H2: -(pr0 v2 x)).(\lambda (t5: T).(\lambda (x0: T).(\lambda (H3: (pr0 t2 -x0)).(\lambda (H4: (pr0 t5 x0)).(\lambda (u3: T).(\lambda (x1: T).(\lambda -(H5: (pr0 u5 x1)).(\lambda (H6: (pr0 u3 x1)).(or_ind (pr0 w x0) (ex2 T -(\lambda (w2: T).(pr0 w w2)) (\lambda (w2: T).(subst0 O x1 x0 w2))) (ex2 T -(\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abbr) u5 w)) t)) -(\lambda (t: T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O -v2) t5)) t))) (\lambda (H7: (pr0 w x0)).(ex_intro2 T (\lambda (t: T).(pr0 -(THead (Flat Appl) u2 (THead (Bind Abbr) u5 w)) t)) (\lambda (t: T).(pr0 -(THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t)) (THead -(Bind Abbr) x1 (THead (Flat Appl) (lift (S O) O x) x0)) (pr0_upsilon Abbr H -u2 x H1 u5 x1 H5 w x0 H7) (pr0_comp u3 x1 H6 (THead (Flat Appl) (lift (S O) O -v2) t5) (THead (Flat Appl) (lift (S O) O x) x0) (pr0_comp (lift (S O) O v2) -(lift (S O) O x) (pr0_lift v2 x H2 (S O) O) t5 x0 H4 (Flat Appl)) (Bind -Abbr)))) (\lambda (H7: (ex2 T (\lambda (w2: T).(pr0 w w2)) (\lambda (w2: -T).(subst0 O x1 x0 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 w w2)) (\lambda -(w2: T).(subst0 O x1 x0 w2)) (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) -u2 (THead (Bind Abbr) u5 w)) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 -(THead (Flat Appl) (lift (S O) O v2) t5)) t))) (\lambda (x2: T).(\lambda (H8: -(pr0 w x2)).(\lambda (H9: (subst0 O x1 x0 x2)).(ex_intro2 T (\lambda (t: -T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abbr) u5 w)) t)) (\lambda (t: -T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t)) -(THead (Bind Abbr) x1 (THead (Flat Appl) (lift (S O) O x) x2)) (pr0_upsilon -Abbr H u2 x H1 u5 x1 H5 w x2 H8) (pr0_delta u3 x1 H6 (THead (Flat Appl) (lift -(S O) O v2) t5) (THead (Flat Appl) (lift (S O) O x) x0) (pr0_comp (lift (S O) -O v2) (lift (S O) O x) (pr0_lift v2 x H2 (S O) O) t5 x0 H4 (Flat Appl)) -(THead (Flat Appl) (lift (S O) O x) x2) (subst0_snd (Flat Appl) x1 x2 x0 O H9 -(lift (S O) O x))))))) H7)) (pr0_subst0 t2 x0 H3 u5 w O H0 x1 -H5))))))))))))))))))). -(* COMMENTS -Initial nodes: 769 -END *) - -theorem pr0_confluence__pr0_cong_upsilon_zeta: - \forall (b: B).((not (eq B b Abst)) \to (\forall (u0: T).(\forall (u3: -T).((pr0 u0 u3) \to (\forall (u2: T).(\forall (v2: T).(\forall (x0: T).((pr0 -u2 x0) \to ((pr0 v2 x0) \to (\forall (x: T).(\forall (t3: T).(\forall (x1: -T).((pr0 x x1) \to ((pr0 t3 x1) \to (ex2 T (\lambda (t: T).(pr0 (THead (Flat -Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) -(lift (S O) O v2) (lift (S O) O x))) t))))))))))))))))) -\def - \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (u0: T).(\lambda -(u3: T).(\lambda (_: (pr0 u0 u3)).(\lambda (u2: T).(\lambda (v2: T).(\lambda -(x0: T).(\lambda (H1: (pr0 u2 x0)).(\lambda (H2: (pr0 v2 x0)).(\lambda (x: -T).(\lambda (t3: T).(\lambda (x1: T).(\lambda (H3: (pr0 x x1)).(\lambda (H4: -(pr0 t3 x1)).(eq_ind T (lift (S O) O (THead (Flat Appl) v2 x)) (\lambda (t: -T).(ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3) t0)) (\lambda (t0: -T).(pr0 (THead (Bind b) u3 t) t0)))) (ex_intro2 T (\lambda (t: T).(pr0 (THead -(Flat Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (lift (S O) O -(THead (Flat Appl) v2 x))) t)) (THead (Flat Appl) x0 x1) (pr0_comp u2 x0 H1 -t3 x1 H4 (Flat Appl)) (pr0_zeta b H (THead (Flat Appl) v2 x) (THead (Flat -Appl) x0 x1) (pr0_comp v2 x0 H2 x x1 H3 (Flat Appl)) u3)) (THead (Flat Appl) -(lift (S O) O v2) (lift (S O) O x)) (lift_flat Appl v2 x (S O) -O)))))))))))))))). -(* COMMENTS -Initial nodes: 283 -END *) - -theorem pr0_confluence__pr0_cong_delta: - \forall (u3: T).(\forall (t5: T).(\forall (w: T).((subst0 O u3 t5 w) \to -(\forall (u2: T).(\forall (x: T).((pr0 u2 x) \to ((pr0 u3 x) \to (\forall -(t3: T).(\forall (x0: T).((pr0 t3 x0) \to ((pr0 t5 x0) \to (ex2 T (\lambda -(t: T).(pr0 (THead (Bind Abbr) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind -Abbr) u3 w) t)))))))))))))) -\def - \lambda (u3: T).(\lambda (t5: T).(\lambda (w: T).(\lambda (H: (subst0 O u3 -t5 w)).(\lambda (u2: T).(\lambda (x: T).(\lambda (H0: (pr0 u2 x)).(\lambda -(H1: (pr0 u3 x)).(\lambda (t3: T).(\lambda (x0: T).(\lambda (H2: (pr0 t3 -x0)).(\lambda (H3: (pr0 t5 x0)).(or_ind (pr0 w x0) (ex2 T (\lambda (w2: -T).(pr0 w w2)) (\lambda (w2: T).(subst0 O x x0 w2))) (ex2 T (\lambda (t: -T).(pr0 (THead (Bind Abbr) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) -u3 w) t))) (\lambda (H4: (pr0 w x0)).(ex_intro2 T (\lambda (t: T).(pr0 (THead -(Bind Abbr) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w) t)) -(THead (Bind Abbr) x x0) (pr0_comp u2 x H0 t3 x0 H2 (Bind Abbr)) (pr0_comp u3 -x H1 w x0 H4 (Bind Abbr)))) (\lambda (H4: (ex2 T (\lambda (w2: T).(pr0 w w2)) -(\lambda (w2: T).(subst0 O x x0 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 w -w2)) (\lambda (w2: T).(subst0 O x x0 w2)) (ex2 T (\lambda (t: T).(pr0 (THead -(Bind Abbr) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w) t))) -(\lambda (x1: T).(\lambda (H5: (pr0 w x1)).(\lambda (H6: (subst0 O x x0 -x1)).(ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 t3) t)) (\lambda -(t: T).(pr0 (THead (Bind Abbr) u3 w) t)) (THead (Bind Abbr) x x1) (pr0_delta -u2 x H0 t3 x0 H2 x1 H6) (pr0_comp u3 x H1 w x1 H5 (Bind Abbr)))))) H4)) -(pr0_subst0 t5 x0 H3 u3 w O H x H1))))))))))))). -(* COMMENTS -Initial nodes: 409 -END *) - -theorem pr0_confluence__pr0_upsilon_upsilon: - \forall (b: B).((not (eq B b Abst)) \to (\forall (v1: T).(\forall (v2: -T).(\forall (x0: T).((pr0 v1 x0) \to ((pr0 v2 x0) \to (\forall (u1: -T).(\forall (u2: T).(\forall (x1: T).((pr0 u1 x1) \to ((pr0 u2 x1) \to -(\forall (t1: T).(\forall (t2: T).(\forall (x2: T).((pr0 t1 x2) \to ((pr0 t2 -x2) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u1 (THead (Flat Appl) -(lift (S O) O v1) t1)) t)) (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t2)) t))))))))))))))))))) -\def - \lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda -(v2: T).(\lambda (x0: T).(\lambda (H0: (pr0 v1 x0)).(\lambda (H1: (pr0 v2 -x0)).(\lambda (u1: T).(\lambda (u2: T).(\lambda (x1: T).(\lambda (H2: (pr0 u1 -x1)).(\lambda (H3: (pr0 u2 x1)).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(x2: T).(\lambda (H4: (pr0 t1 x2)).(\lambda (H5: (pr0 t2 x2)).(ex_intro2 T -(\lambda (t: T).(pr0 (THead (Bind b) u1 (THead (Flat Appl) (lift (S O) O v1) -t1)) t)) (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t2)) t)) (THead (Bind b) x1 (THead (Flat Appl) (lift (S O) O x0) -x2)) (pr0_comp u1 x1 H2 (THead (Flat Appl) (lift (S O) O v1) t1) (THead (Flat -Appl) (lift (S O) O x0) x2) (pr0_comp (lift (S O) O v1) (lift (S O) O x0) -(pr0_lift v1 x0 H0 (S O) O) t1 x2 H4 (Flat Appl)) (Bind b)) (pr0_comp u2 x1 -H3 (THead (Flat Appl) (lift (S O) O v2) t2) (THead (Flat Appl) (lift (S O) O -x0) x2) (pr0_comp (lift (S O) O v2) (lift (S O) O x0) (pr0_lift v2 x0 H1 (S -O) O) t2 x2 H5 (Flat Appl)) (Bind b))))))))))))))))))). -(* COMMENTS -Initial nodes: 347 -END *) - -theorem pr0_confluence__pr0_delta_delta: - \forall (u2: T).(\forall (t3: T).(\forall (w: T).((subst0 O u2 t3 w) \to -(\forall (u3: T).(\forall (t5: T).(\forall (w0: T).((subst0 O u3 t5 w0) \to -(\forall (x: T).((pr0 u2 x) \to ((pr0 u3 x) \to (\forall (x0: T).((pr0 t3 x0) -\to ((pr0 t5 x0) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) -(\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t)))))))))))))))) -\def - \lambda (u2: T).(\lambda (t3: T).(\lambda (w: T).(\lambda (H: (subst0 O u2 -t3 w)).(\lambda (u3: T).(\lambda (t5: T).(\lambda (w0: T).(\lambda (H0: -(subst0 O u3 t5 w0)).(\lambda (x: T).(\lambda (H1: (pr0 u2 x)).(\lambda (H2: -(pr0 u3 x)).(\lambda (x0: T).(\lambda (H3: (pr0 t3 x0)).(\lambda (H4: (pr0 t5 -x0)).(or_ind (pr0 w0 x0) (ex2 T (\lambda (w2: T).(pr0 w0 w2)) (\lambda (w2: -T).(subst0 O x x0 w2))) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) -t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (H5: (pr0 w0 -x0)).(or_ind (pr0 w x0) (ex2 T (\lambda (w2: T).(pr0 w w2)) (\lambda (w2: -T).(subst0 O x x0 w2))) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) -t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (H6: (pr0 w -x0)).(ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda -(t: T).(pr0 (THead (Bind Abbr) u3 w0) t)) (THead (Bind Abbr) x x0) (pr0_comp -u2 x H1 w x0 H6 (Bind Abbr)) (pr0_comp u3 x H2 w0 x0 H5 (Bind Abbr)))) -(\lambda (H6: (ex2 T (\lambda (w2: T).(pr0 w w2)) (\lambda (w2: T).(subst0 O -x x0 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 w w2)) (\lambda (w2: T).(subst0 -O x x0 w2)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda -(t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (x1: T).(\lambda (H7: -(pr0 w x1)).(\lambda (H8: (subst0 O x x0 x1)).(ex_intro2 T (\lambda (t: -T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) -u3 w0) t)) (THead (Bind Abbr) x x1) (pr0_comp u2 x H1 w x1 H7 (Bind Abbr)) -(pr0_delta u3 x H2 w0 x0 H5 x1 H8))))) H6)) (pr0_subst0 t3 x0 H3 u2 w O H x -H1))) (\lambda (H5: (ex2 T (\lambda (w2: T).(pr0 w0 w2)) (\lambda (w2: -T).(subst0 O x x0 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 w0 w2)) (\lambda -(w2: T).(subst0 O x x0 w2)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 -w) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (x1: -T).(\lambda (H6: (pr0 w0 x1)).(\lambda (H7: (subst0 O x x0 x1)).(or_ind (pr0 -w x0) (ex2 T (\lambda (w2: T).(pr0 w w2)) (\lambda (w2: T).(subst0 O x x0 -w2))) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: -T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (H8: (pr0 w x0)).(ex_intro2 T -(\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead -(Bind Abbr) u3 w0) t)) (THead (Bind Abbr) x x1) (pr0_delta u2 x H1 w x0 H8 x1 -H7) (pr0_comp u3 x H2 w0 x1 H6 (Bind Abbr)))) (\lambda (H8: (ex2 T (\lambda -(w2: T).(pr0 w w2)) (\lambda (w2: T).(subst0 O x x0 w2)))).(ex2_ind T -(\lambda (w2: T).(pr0 w w2)) (\lambda (w2: T).(subst0 O x x0 w2)) (ex2 T -(\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead -(Bind Abbr) u3 w0) t))) (\lambda (x2: T).(\lambda (H9: (pr0 w x2)).(\lambda -(H10: (subst0 O x x0 x2)).(or4_ind (eq T x2 x1) (ex2 T (\lambda (t: -T).(subst0 O x x2 t)) (\lambda (t: T).(subst0 O x x1 t))) (subst0 O x x2 x1) -(subst0 O x x1 x2) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) -(\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (H11: (eq T x2 -x1)).(let H12 \def (eq_ind T x2 (\lambda (t: T).(pr0 w t)) H9 x1 H11) in -(ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: -T).(pr0 (THead (Bind Abbr) u3 w0) t)) (THead (Bind Abbr) x x1) (pr0_comp u2 x -H1 w x1 H12 (Bind Abbr)) (pr0_comp u3 x H2 w0 x1 H6 (Bind Abbr))))) (\lambda -(H11: (ex2 T (\lambda (t: T).(subst0 O x x2 t)) (\lambda (t: T).(subst0 O x -x1 t)))).(ex2_ind T (\lambda (t: T).(subst0 O x x2 t)) (\lambda (t: -T).(subst0 O x x1 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) -t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (x3: -T).(\lambda (H12: (subst0 O x x2 x3)).(\lambda (H13: (subst0 O x x1 -x3)).(ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda -(t: T).(pr0 (THead (Bind Abbr) u3 w0) t)) (THead (Bind Abbr) x x3) (pr0_delta -u2 x H1 w x2 H9 x3 H12) (pr0_delta u3 x H2 w0 x1 H6 x3 H13))))) H11)) -(\lambda (H11: (subst0 O x x2 x1)).(ex_intro2 T (\lambda (t: T).(pr0 (THead -(Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t)) -(THead (Bind Abbr) x x1) (pr0_delta u2 x H1 w x2 H9 x1 H11) (pr0_comp u3 x H2 -w0 x1 H6 (Bind Abbr)))) (\lambda (H11: (subst0 O x x1 x2)).(ex_intro2 T -(\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead -(Bind Abbr) u3 w0) t)) (THead (Bind Abbr) x x2) (pr0_comp u2 x H1 w x2 H9 -(Bind Abbr)) (pr0_delta u3 x H2 w0 x1 H6 x2 H11))) (subst0_confluence_eq x0 -x2 x O H10 x1 H7))))) H8)) (pr0_subst0 t3 x0 H3 u2 w O H x H1))))) H5)) -(pr0_subst0 t5 x0 H4 u3 w0 O H0 x H2))))))))))))))). -(* COMMENTS -Initial nodes: 1501 -END *) - -theorem pr0_confluence__pr0_delta_tau: - \forall (u2: T).(\forall (t3: T).(\forall (w: T).((subst0 O u2 t3 w) \to -(\forall (t4: T).((pr0 (lift (S O) O t4) t3) \to (\forall (t2: T).(ex2 T -(\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 t2 -t))))))))) -\def - \lambda (u2: T).(\lambda (t3: T).(\lambda (w: T).(\lambda (H: (subst0 O u2 -t3 w)).(\lambda (t4: T).(\lambda (H0: (pr0 (lift (S O) O t4) t3)).(\lambda -(t2: T).(ex2_ind T (\lambda (t5: T).(eq T t3 (lift (S O) O t5))) (\lambda -(t5: T).(pr0 t4 t5)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) -(\lambda (t: T).(pr0 t2 t))) (\lambda (x: T).(\lambda (H1: (eq T t3 (lift (S -O) O x))).(\lambda (_: (pr0 t4 x)).(let H3 \def (eq_ind T t3 (\lambda (t: -T).(subst0 O u2 t w)) H (lift (S O) O x) H1) in (subst0_gen_lift_false x u2 w -(S O) O O (le_n O) (eq_ind_r nat (plus (S O) O) (\lambda (n: nat).(lt O n)) -(le_n (plus (S O) O)) (plus O (S O)) (plus_sym O (S O))) H3 (ex2 T (\lambda -(t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 t2 t)))))))) -(pr0_gen_lift t4 t3 (S O) O H0)))))))). -(* COMMENTS -Initial nodes: 257 -END *) - -theorem pr0_confluence: - \forall (t0: T).(\forall (t1: T).((pr0 t0 t1) \to (\forall (t2: T).((pr0 t0 -t2) \to (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t))))))) -\def - \lambda (t0: T).(tlt_wf_ind (\lambda (t: T).(\forall (t1: T).((pr0 t t1) \to -(\forall (t2: T).((pr0 t t2) \to (ex2 T (\lambda (t3: T).(pr0 t1 t3)) -(\lambda (t3: T).(pr0 t2 t3)))))))) (\lambda (t: T).(\lambda (H: ((\forall -(v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 -v t2) \to (ex2 T (\lambda (t3: T).(pr0 t1 t3)) (\lambda (t3: T).(pr0 t2 -t3))))))))))).(\lambda (t1: T).(\lambda (H0: (pr0 t t1)).(\lambda (t2: -T).(\lambda (H1: (pr0 t t2)).(let H2 \def (match H0 in pr0 return (\lambda -(t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).((eq T t3 t) \to ((eq T t4 -t1) \to (ex2 T (\lambda (t5: T).(pr0 t1 t5)) (\lambda (t5: T).(pr0 t2 -t5)))))))) with [(pr0_refl t3) \Rightarrow (\lambda (H2: (eq T t3 -t)).(\lambda (H3: (eq T t3 t1)).(eq_ind T t (\lambda (t4: T).((eq T t4 t1) -\to (ex2 T (\lambda (t5: T).(pr0 t1 t5)) (\lambda (t5: T).(pr0 t2 t5))))) -(\lambda (H4: (eq T t t1)).(eq_ind T t1 (\lambda (_: T).(ex2 T (\lambda (t5: -T).(pr0 t1 t5)) (\lambda (t5: T).(pr0 t2 t5)))) (let H5 \def (match H1 in pr0 -return (\lambda (t4: T).(\lambda (t5: T).(\lambda (_: (pr0 t4 t5)).((eq T t4 -t) \to ((eq T t5 t2) \to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: -T).(pr0 t2 t6)))))))) with [(pr0_refl t4) \Rightarrow (\lambda (H5: (eq T t4 -t)).(\lambda (H6: (eq T t4 t2)).(eq_ind T t (\lambda (t5: T).((eq T t5 t2) -\to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6))))) -(\lambda (H7: (eq T t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t6: -T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6)))) (let H8 \def (eq_ind T t -(\lambda (t5: T).(eq T t4 t5)) H5 t2 H7) in (let H9 \def (eq_ind T t (\lambda -(t5: T).(eq T t5 t1)) H4 t2 H7) in (let H10 \def (eq_ind T t (\lambda (t5: -T).(eq T t3 t5)) H2 t2 H7) in (let H11 \def (eq_ind T t (\lambda (t5: -T).(\forall (v: T).((tlt v t5) \to (\forall (t6: T).((pr0 v t6) \to (\forall -(t7: T).((pr0 v t7) \to (ex2 T (\lambda (t8: T).(pr0 t6 t8)) (\lambda (t8: -T).(pr0 t7 t8)))))))))) H t2 H7) in (let H12 \def (eq_ind T t2 (\lambda (t5: -T).(\forall (v: T).((tlt v t5) \to (\forall (t6: T).((pr0 v t6) \to (\forall -(t7: T).((pr0 v t7) \to (ex2 T (\lambda (t8: T).(pr0 t6 t8)) (\lambda (t8: -T).(pr0 t7 t8)))))))))) H11 t1 H9) in (eq_ind_r T t1 (\lambda (t5: T).(ex2 T -(\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t5 t6)))) (let H13 \def -(eq_ind T t2 (\lambda (t5: T).(eq T t3 t5)) H10 t1 H9) in (ex_intro2 T -(\lambda (t5: T).(pr0 t1 t5)) (\lambda (t5: T).(pr0 t1 t5)) t1 (pr0_refl t1) -(pr0_refl t1))) t2 H9)))))) t (sym_eq T t t2 H7))) t4 (sym_eq T t4 t H5) -H6))) | (pr0_comp u1 u2 H5 t4 t5 H6 k) \Rightarrow (\lambda (H7: (eq T (THead -k u1 t4) t)).(\lambda (H8: (eq T (THead k u2 t5) t2)).(eq_ind T (THead k u1 -t4) (\lambda (_: T).((eq T (THead k u2 t5) t2) \to ((pr0 u1 u2) \to ((pr0 t4 -t5) \to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 -t7))))))) (\lambda (H9: (eq T (THead k u2 t5) t2)).(eq_ind T (THead k u2 t5) -(\lambda (t6: T).((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t7: -T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t6 t7)))))) (\lambda (H10: (pr0 u1 -u2)).(\lambda (H11: (pr0 t4 t5)).(let H12 \def (eq_ind_r T t (\lambda (t6: -T).(eq T t6 t1)) H4 (THead k u1 t4) H7) in (eq_ind T (THead k u1 t4) (\lambda -(t6: T).(ex2 T (\lambda (t7: T).(pr0 t6 t7)) (\lambda (t7: T).(pr0 (THead k -u2 t5) t7)))) (let H13 \def (eq_ind_r T t (\lambda (t6: T).(eq T t3 t6)) H2 -(THead k u1 t4) H7) in (let H14 \def (eq_ind_r T t (\lambda (t6: T).(\forall -(v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall (t8: -T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 -t8 t9)))))))))) H (THead k u1 t4) H7) in (ex_intro2 T (\lambda (t6: T).(pr0 -(THead k u1 t4) t6)) (\lambda (t6: T).(pr0 (THead k u2 t5) t6)) (THead k u2 -t5) (pr0_comp u1 u2 H10 t4 t5 H11 k) (pr0_refl (THead k u2 t5))))) t1 H12)))) -t2 H9)) t H7 H8 H5 H6))) | (pr0_beta u v1 v2 H5 t4 t5 H6) \Rightarrow -(\lambda (H7: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t4)) -t)).(\lambda (H8: (eq T (THead (Bind Abbr) v2 t5) t2)).(eq_ind T (THead (Flat -Appl) v1 (THead (Bind Abst) u t4)) (\lambda (_: T).((eq T (THead (Bind Abbr) -v2 t5) t2) \to ((pr0 v1 v2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t7: T).(pr0 -t1 t7)) (\lambda (t7: T).(pr0 t2 t7))))))) (\lambda (H9: (eq T (THead (Bind -Abbr) v2 t5) t2)).(eq_ind T (THead (Bind Abbr) v2 t5) (\lambda (t6: T).((pr0 -v1 v2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda -(t7: T).(pr0 t6 t7)))))) (\lambda (H10: (pr0 v1 v2)).(\lambda (H11: (pr0 t4 -t5)).(let H12 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 t1)) H4 (THead -(Flat Appl) v1 (THead (Bind Abst) u t4)) H7) in (eq_ind T (THead (Flat Appl) -v1 (THead (Bind Abst) u t4)) (\lambda (t6: T).(ex2 T (\lambda (t7: T).(pr0 t6 -t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t5) t7)))) (let H13 \def -(eq_ind_r T t (\lambda (t6: T).(eq T t3 t6)) H2 (THead (Flat Appl) v1 (THead -(Bind Abst) u t4)) H7) in (let H14 \def (eq_ind_r T t (\lambda (t6: -T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall -(t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: -T).(pr0 t8 t9)))))))))) H (THead (Flat Appl) v1 (THead (Bind Abst) u t4)) H7) -in (ex_intro2 T (\lambda (t6: T).(pr0 (THead (Flat Appl) v1 (THead (Bind -Abst) u t4)) t6)) (\lambda (t6: T).(pr0 (THead (Bind Abbr) v2 t5) t6)) (THead -(Bind Abbr) v2 t5) (pr0_beta u v1 v2 H10 t4 t5 H11) (pr0_refl (THead (Bind -Abbr) v2 t5))))) t1 H12)))) t2 H9)) t H7 H8 H5 H6))) | (pr0_upsilon b H5 v1 -v2 H6 u1 u2 H7 t4 t5 H8) \Rightarrow (\lambda (H9: (eq T (THead (Flat Appl) -v1 (THead (Bind b) u1 t4)) t)).(\lambda (H10: (eq T (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t5)) t2)).(eq_ind T (THead (Flat Appl) v1 -(THead (Bind b) u1 t4)) (\lambda (_: T).((eq T (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t5)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1 -v2) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t7: T).(pr0 t1 -t7)) (\lambda (t7: T).(pr0 t2 t7))))))))) (\lambda (H11: (eq T (THead (Bind -b) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) t2)).(eq_ind T (THead (Bind -b) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) (\lambda (t6: T).((not (eq B -b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T -(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t6 t7)))))))) (\lambda -(H12: (not (eq B b Abst))).(\lambda (H13: (pr0 v1 v2)).(\lambda (H14: (pr0 u1 -u2)).(\lambda (H15: (pr0 t4 t5)).(let H16 \def (eq_ind_r T t (\lambda (t6: -T).(eq T t6 t1)) H4 (THead (Flat Appl) v1 (THead (Bind b) u1 t4)) H9) in -(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t4)) (\lambda (t6: T).(ex2 -T (\lambda (t7: T).(pr0 t6 t7)) (\lambda (t7: T).(pr0 (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t5)) t7)))) (let H17 \def (eq_ind_r T t -(\lambda (t6: T).(eq T t3 t6)) H2 (THead (Flat Appl) v1 (THead (Bind b) u1 -t4)) H9) in (let H18 \def (eq_ind_r T t (\lambda (t6: T).(\forall (v: -T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall (t8: T).((pr0 v -t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8 -t9)))))))))) H (THead (Flat Appl) v1 (THead (Bind b) u1 t4)) H9) in -(pr0_confluence__pr0_cong_upsilon_refl b H12 u1 u2 H14 t4 t5 H15 v1 v2 v2 H13 -(pr0_refl v2)))) t1 H16)))))) t2 H11)) t H9 H10 H5 H6 H7 H8))) | (pr0_delta -u1 u2 H5 t4 t5 H6 w H7) \Rightarrow (\lambda (H8: (eq T (THead (Bind Abbr) u1 -t4) t)).(\lambda (H9: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead -(Bind Abbr) u1 t4) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t2) \to -((pr0 u1 u2) \to ((pr0 t4 t5) \to ((subst0 O u2 t5 w) \to (ex2 T (\lambda -(t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)))))))) (\lambda (H10: (eq T -(THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda -(t6: T).((pr0 u1 u2) \to ((pr0 t4 t5) \to ((subst0 O u2 t5 w) \to (ex2 T -(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t6 t7))))))) (\lambda -(H11: (pr0 u1 u2)).(\lambda (H12: (pr0 t4 t5)).(\lambda (H13: (subst0 O u2 t5 -w)).(let H14 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 t1)) H4 (THead -(Bind Abbr) u1 t4) H8) in (eq_ind T (THead (Bind Abbr) u1 t4) (\lambda (t6: -T).(ex2 T (\lambda (t7: T).(pr0 t6 t7)) (\lambda (t7: T).(pr0 (THead (Bind -Abbr) u2 w) t7)))) (let H15 \def (eq_ind_r T t (\lambda (t6: T).(eq T t3 t6)) -H2 (THead (Bind Abbr) u1 t4) H8) in (let H16 \def (eq_ind_r T t (\lambda (t6: -T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall -(t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: -T).(pr0 t8 t9)))))))))) H (THead (Bind Abbr) u1 t4) H8) in (ex_intro2 T -(\lambda (t6: T).(pr0 (THead (Bind Abbr) u1 t4) t6)) (\lambda (t6: T).(pr0 -(THead (Bind Abbr) u2 w) t6)) (THead (Bind Abbr) u2 w) (pr0_delta u1 u2 H11 -t4 t5 H12 w H13) (pr0_refl (THead (Bind Abbr) u2 w))))) t1 H14))))) t2 H10)) -t H8 H9 H5 H6 H7))) | (pr0_zeta b H5 t4 t5 H6 u) \Rightarrow (\lambda (H7: -(eq T (THead (Bind b) u (lift (S O) O t4)) t)).(\lambda (H8: (eq T t5 -t2)).(eq_ind T (THead (Bind b) u (lift (S O) O t4)) (\lambda (_: T).((eq T t5 -t2) \to ((not (eq B b Abst)) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t7: -T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))))))) (\lambda (H9: (eq T t5 -t2)).(eq_ind T t2 (\lambda (t6: T).((not (eq B b Abst)) \to ((pr0 t4 t6) \to -(ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)))))) -(\lambda (H10: (not (eq B b Abst))).(\lambda (H11: (pr0 t4 t2)).(let H12 \def -(eq_ind_r T t (\lambda (t6: T).(eq T t6 t1)) H4 (THead (Bind b) u (lift (S O) -O t4)) H7) in (eq_ind T (THead (Bind b) u (lift (S O) O t4)) (\lambda (t6: -T).(ex2 T (\lambda (t7: T).(pr0 t6 t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let -H13 \def (eq_ind_r T t (\lambda (t6: T).(eq T t3 t6)) H2 (THead (Bind b) u -(lift (S O) O t4)) H7) in (let H14 \def (eq_ind_r T t (\lambda (t6: -T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall -(t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: -T).(pr0 t8 t9)))))))))) H (THead (Bind b) u (lift (S O) O t4)) H7) in -(ex_intro2 T (\lambda (t6: T).(pr0 (THead (Bind b) u (lift (S O) O t4)) t6)) -(\lambda (t6: T).(pr0 t2 t6)) t2 (pr0_zeta b H10 t4 t2 H11 u) (pr0_refl -t2)))) t1 H12)))) t5 (sym_eq T t5 t2 H9))) t H7 H8 H5 H6))) | (pr0_tau t4 t5 -H5 u) \Rightarrow (\lambda (H6: (eq T (THead (Flat Cast) u t4) t)).(\lambda -(H7: (eq T t5 t2)).(eq_ind T (THead (Flat Cast) u t4) (\lambda (_: T).((eq T -t5 t2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda -(t7: T).(pr0 t2 t7)))))) (\lambda (H8: (eq T t5 t2)).(eq_ind T t2 (\lambda -(t6: T).((pr0 t4 t6) \to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: -T).(pr0 t2 t7))))) (\lambda (H9: (pr0 t4 t2)).(let H10 \def (eq_ind_r T t -(\lambda (t6: T).(eq T t6 t1)) H4 (THead (Flat Cast) u t4) H6) in (eq_ind T -(THead (Flat Cast) u t4) (\lambda (t6: T).(ex2 T (\lambda (t7: T).(pr0 t6 -t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let H11 \def (eq_ind_r T t (\lambda -(t6: T).(eq T t3 t6)) H2 (THead (Flat Cast) u t4) H6) in (let H12 \def -(eq_ind_r T t (\lambda (t6: T).(\forall (v: T).((tlt v t6) \to (\forall (t7: -T).((pr0 v t7) \to (\forall (t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: -T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead (Flat Cast) u -t4) H6) in (ex_intro2 T (\lambda (t6: T).(pr0 (THead (Flat Cast) u t4) t6)) -(\lambda (t6: T).(pr0 t2 t6)) t2 (pr0_tau t4 t2 H9 u) (pr0_refl t2)))) t1 -H10))) t5 (sym_eq T t5 t2 H8))) t H6 H7 H5)))]) in (H5 (refl_equal T t) -(refl_equal T t2))) t (sym_eq T t t1 H4))) t3 (sym_eq T t3 t H2) H3))) | -(pr0_comp u1 u2 H2 t3 t4 H3 k) \Rightarrow (\lambda (H4: (eq T (THead k u1 -t3) t)).(\lambda (H5: (eq T (THead k u2 t4) t1)).(eq_ind T (THead k u1 t3) -(\lambda (_: T).((eq T (THead k u2 t4) t1) \to ((pr0 u1 u2) \to ((pr0 t3 t4) -\to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6))))))) -(\lambda (H6: (eq T (THead k u2 t4) t1)).(eq_ind T (THead k u2 t4) (\lambda -(t5: T).((pr0 u1 u2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t5 -t6)) (\lambda (t6: T).(pr0 t2 t6)))))) (\lambda (H7: (pr0 u1 u2)).(\lambda -(H8: (pr0 t3 t4)).(let H9 \def (match H1 in pr0 return (\lambda (t5: -T).(\lambda (t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 t) \to ((eq T t6 t2) -\to (ex2 T (\lambda (t7: T).(pr0 (THead k u2 t4) t7)) (\lambda (t7: T).(pr0 -t2 t7)))))))) with [(pr0_refl t5) \Rightarrow (\lambda (H9: (eq T t5 -t)).(\lambda (H10: (eq T t5 t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2) -\to (ex2 T (\lambda (t7: T).(pr0 (THead k u2 t4) t7)) (\lambda (t7: T).(pr0 -t2 t7))))) (\lambda (H11: (eq T t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T -(\lambda (t7: T).(pr0 (THead k u2 t4) t7)) (\lambda (t7: T).(pr0 t2 t7)))) -(let H12 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 t2)) H11 (THead k u1 -t3) H4) in (eq_ind T (THead k u1 t3) (\lambda (t6: T).(ex2 T (\lambda (t7: -T).(pr0 (THead k u2 t4) t7)) (\lambda (t7: T).(pr0 t6 t7)))) (let H13 \def -(eq_ind_r T t (\lambda (t6: T).(eq T t5 t6)) H9 (THead k u1 t3) H4) in (let -H14 \def (eq_ind_r T t (\lambda (t6: T).(\forall (v: T).((tlt v t6) \to -(\forall (t7: T).((pr0 v t7) \to (\forall (t8: T).((pr0 v t8) \to (ex2 T -(\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead -k u1 t3) H4) in (ex_intro2 T (\lambda (t6: T).(pr0 (THead k u2 t4) t6)) -(\lambda (t6: T).(pr0 (THead k u1 t3) t6)) (THead k u2 t4) (pr0_refl (THead k -u2 t4)) (pr0_comp u1 u2 H7 t3 t4 H8 k)))) t2 H12)) t (sym_eq T t t2 H11))) t5 -(sym_eq T t5 t H9) H10))) | (pr0_comp u0 u3 H9 t5 t6 H10 k0) \Rightarrow -(\lambda (H11: (eq T (THead k0 u0 t5) t)).(\lambda (H12: (eq T (THead k0 u3 -t6) t2)).(eq_ind T (THead k0 u0 t5) (\lambda (_: T).((eq T (THead k0 u3 t6) -t2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead -k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H13: (eq T (THead -k0 u3 t6) t2)).(eq_ind T (THead k0 u3 t6) (\lambda (t7: T).((pr0 u0 u3) \to -((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda -(t8: T).(pr0 t7 t8)))))) (\lambda (H14: (pr0 u0 u3)).(\lambda (H15: (pr0 t5 -t6)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead k u1 t3) t7)) -H4 (THead k0 u0 t5) H11) in (let H17 \def (f_equal T K (\lambda (e: T).(match -e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) -\Rightarrow k | (THead k1 _ _) \Rightarrow k1])) (THead k u1 t3) (THead k0 u0 -t5) H16) in ((let H18 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 -| (THead _ t7 _) \Rightarrow t7])) (THead k u1 t3) (THead k0 u0 t5) H16) in -((let H19 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: -T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ -t7) \Rightarrow t7])) (THead k u1 t3) (THead k0 u0 t5) H16) in (\lambda (H20: -(eq T u1 u0)).(\lambda (H21: (eq K k k0)).(let H22 \def (eq_ind_r T t -(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) -\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) -(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead k0 u0 t5) H11) in (eq_ind_r -K k0 (\lambda (k1: K).(ex2 T (\lambda (t7: T).(pr0 (THead k1 u2 t4) t7)) -(\lambda (t7: T).(pr0 (THead k0 u3 t6) t7)))) (let H23 \def (eq_ind T u1 -(\lambda (t7: T).(pr0 t7 u2)) H7 u0 H20) in (let H24 \def (eq_ind T t3 -(\lambda (t7: T).(pr0 t7 t4)) H8 t5 H19) in (ex2_ind T (\lambda (t7: T).(pr0 -t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead k0 -u2 t4) t7)) (\lambda (t7: T).(pr0 (THead k0 u3 t6) t7))) (\lambda (x: -T).(\lambda (H25: (pr0 t4 x)).(\lambda (H26: (pr0 t6 x)).(ex2_ind T (\lambda -(t7: T).(pr0 u2 t7)) (\lambda (t7: T).(pr0 u3 t7)) (ex2 T (\lambda (t7: -T).(pr0 (THead k0 u2 t4) t7)) (\lambda (t7: T).(pr0 (THead k0 u3 t6) t7))) -(\lambda (x0: T).(\lambda (H27: (pr0 u2 x0)).(\lambda (H28: (pr0 u3 -x0)).(ex_intro2 T (\lambda (t7: T).(pr0 (THead k0 u2 t4) t7)) (\lambda (t7: -T).(pr0 (THead k0 u3 t6) t7)) (THead k0 x0 x) (pr0_comp u2 x0 H27 t4 x H25 -k0) (pr0_comp u3 x0 H28 t6 x H26 k0))))) (H22 u0 (tlt_head_sx k0 u0 t5) u2 -H23 u3 H14))))) (H22 t5 (tlt_head_dx k0 u0 t5) t4 H24 t6 H15)))) k H21))))) -H18)) H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_beta u v1 v2 H9 t5 t6 H10) -\Rightarrow (\lambda (H11: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u -t5)) t)).(\lambda (H12: (eq T (THead (Bind Abbr) v2 t6) t2)).(eq_ind T (THead -(Flat Appl) v1 (THead (Bind Abst) u t5)) (\lambda (_: T).((eq T (THead (Bind -Abbr) v2 t6) t2) \to ((pr0 v1 v2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: -T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda -(H13: (eq T (THead (Bind Abbr) v2 t6) t2)).(eq_ind T (THead (Bind Abbr) v2 -t6) (\lambda (t7: T).((pr0 v1 v2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: -T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (H14: -(pr0 v1 v2)).(\lambda (H15: (pr0 t5 t6)).(let H16 \def (eq_ind_r T t (\lambda -(t7: T).(eq T (THead k u1 t3) t7)) H4 (THead (Flat Appl) v1 (THead (Bind -Abst) u t5)) H11) in (let H17 \def (f_equal T K (\lambda (e: T).(match e in T -return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) -\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) (THead (Flat -Appl) v1 (THead (Bind Abst) u t5)) H16) in ((let H18 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) -(THead k u1 t3) (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) H16) in ((let -H19 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7) -\Rightarrow t7])) (THead k u1 t3) (THead (Flat Appl) v1 (THead (Bind Abst) u -t5)) H16) in (\lambda (H20: (eq T u1 v1)).(\lambda (H21: (eq K k (Flat -Appl))).(let H22 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v -t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to -(ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 -t10)))))))))) H (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) H11) in -(eq_ind_r K (Flat Appl) (\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 (THead -k0 u2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t6) t7)))) (let -H23 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H7 v1 H20) in (let H24 -\def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H8 (THead (Bind Abst) u t5) -H19) in (let H25 \def (match H24 in pr0 return (\lambda (t7: T).(\lambda (t8: -T).(\lambda (_: (pr0 t7 t8)).((eq T t7 (THead (Bind Abst) u t5)) \to ((eq T -t8 t4) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) -(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9)))))))) with [(pr0_refl -t7) \Rightarrow (\lambda (H25: (eq T t7 (THead (Bind Abst) u t5))).(\lambda -(H26: (eq T t7 t4)).(eq_ind T (THead (Bind Abst) u t5) (\lambda (t8: T).((eq -T t8 t4) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) -(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9))))) (\lambda (H27: (eq T -(THead (Bind Abst) u t5) t4)).(eq_ind T (THead (Bind Abst) u t5) (\lambda -(t8: T).(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t8) t9)) (\lambda -(t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9)))) (ex2_ind T (\lambda (t8: -T).(pr0 u2 t8)) (\lambda (t8: T).(pr0 v2 t8)) (ex2 T (\lambda (t8: T).(pr0 -(THead (Flat Appl) u2 (THead (Bind Abst) u t5)) t8)) (\lambda (t8: T).(pr0 -(THead (Bind Abbr) v2 t6) t8))) (\lambda (x: T).(\lambda (H28: (pr0 u2 -x)).(\lambda (H29: (pr0 v2 x)).(ex_intro2 T (\lambda (t8: T).(pr0 (THead -(Flat Appl) u2 (THead (Bind Abst) u t5)) t8)) (\lambda (t8: T).(pr0 (THead -(Bind Abbr) v2 t6) t8)) (THead (Bind Abbr) x t6) (pr0_beta u u2 x H28 t5 t6 -H15) (pr0_comp v2 x H29 t6 t6 (pr0_refl t6) (Bind Abbr)))))) (H22 v1 -(tlt_head_sx (Flat Appl) v1 (THead (Bind Abst) u t5)) u2 H23 v2 H14)) t4 -H27)) t7 (sym_eq T t7 (THead (Bind Abst) u t5) H25) H26))) | (pr0_comp u0 u3 -H25 t7 t8 H26 k0) \Rightarrow (\lambda (H27: (eq T (THead k0 u0 t7) (THead -(Bind Abst) u t5))).(\lambda (H28: (eq T (THead k0 u3 t8) t4)).((let H29 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow t7 | (TLRef _) \Rightarrow t7 | (THead _ _ t9) -\Rightarrow t9])) (THead k0 u0 t7) (THead (Bind Abst) u t5) H27) in ((let H30 -\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t9 _) -\Rightarrow t9])) (THead k0 u0 t7) (THead (Bind Abst) u t5) H27) in ((let H31 -\def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) -with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _ _) -\Rightarrow k1])) (THead k0 u0 t7) (THead (Bind Abst) u t5) H27) in (eq_ind K -(Bind Abst) (\lambda (k1: K).((eq T u0 u) \to ((eq T t7 t5) \to ((eq T (THead -k1 u3 t8) t4) \to ((pr0 u0 u3) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: -T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind -Abbr) v2 t6) t9))))))))) (\lambda (H32: (eq T u0 u)).(eq_ind T u (\lambda -(t9: T).((eq T t7 t5) \to ((eq T (THead (Bind Abst) u3 t8) t4) \to ((pr0 t9 -u3) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 -t4) t10)) (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t6) t10)))))))) -(\lambda (H33: (eq T t7 t5)).(eq_ind T t5 (\lambda (t9: T).((eq T (THead -(Bind Abst) u3 t8) t4) \to ((pr0 u u3) \to ((pr0 t9 t8) \to (ex2 T (\lambda -(t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda (t10: T).(pr0 (THead -(Bind Abbr) v2 t6) t10))))))) (\lambda (H34: (eq T (THead (Bind Abst) u3 t8) -t4)).(eq_ind T (THead (Bind Abst) u3 t8) (\lambda (t9: T).((pr0 u u3) \to -((pr0 t5 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t9) -t10)) (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t6) t10)))))) (\lambda (_: -(pr0 u u3)).(\lambda (H36: (pr0 t5 t8)).(ex2_ind T (\lambda (t9: T).(pr0 t8 -t9)) (\lambda (t9: T).(pr0 t6 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat -Appl) u2 (THead (Bind Abst) u3 t8)) t9)) (\lambda (t9: T).(pr0 (THead (Bind -Abbr) v2 t6) t9))) (\lambda (x: T).(\lambda (H37: (pr0 t8 x)).(\lambda (H38: -(pr0 t6 x)).(ex2_ind T (\lambda (t9: T).(pr0 u2 t9)) (\lambda (t9: T).(pr0 v2 -t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u3 -t8)) t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9))) (\lambda (x0: -T).(\lambda (H39: (pr0 u2 x0)).(\lambda (H40: (pr0 v2 x0)).(ex_intro2 T -(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u3 t8)) t9)) -(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9)) (THead (Bind Abbr) x0 x) -(pr0_beta u3 u2 x0 H39 t8 x H37) (pr0_comp v2 x0 H40 t6 x H38 (Bind -Abbr)))))) (H22 v1 (tlt_head_sx (Flat Appl) v1 (THead (Bind Abst) u t5)) u2 -H23 v2 H14))))) (H22 t5 (tlt_trans (THead (Bind Abst) u t5) t5 (THead (Flat -Appl) v1 (THead (Bind Abst) u t5)) (tlt_head_dx (Bind Abst) u t5) -(tlt_head_dx (Flat Appl) v1 (THead (Bind Abst) u t5))) t8 H36 t6 H15)))) t4 -H34)) t7 (sym_eq T t7 t5 H33))) u0 (sym_eq T u0 u H32))) k0 (sym_eq K k0 -(Bind Abst) H31))) H30)) H29)) H28 H25 H26))) | (pr0_beta u0 v0 v3 H25 t7 t8 -H26) \Rightarrow (\lambda (H27: (eq T (THead (Flat Appl) v0 (THead (Bind -Abst) u0 t7)) (THead (Bind Abst) u t5))).(\lambda (H28: (eq T (THead (Bind -Abbr) v3 t8) t4)).((let H29 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind -Abst) u0 t7)) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _) -\Rightarrow (match k0 in K return (\lambda (_: K).Prop) with [(Bind _) -\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u t5) -H27) in (False_ind ((eq T (THead (Bind Abbr) v3 t8) t4) \to ((pr0 v0 v3) \to -((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) -(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9)))))) H29)) H28 H25 H26))) -| (pr0_upsilon b H25 v0 v3 H26 u0 u3 H27 t7 t8 H28) \Rightarrow (\lambda -(H29: (eq T (THead (Flat Appl) v0 (THead (Bind b) u0 t7)) (THead (Bind Abst) -u t5))).(\lambda (H30: (eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S -O) O v3) t8)) t4)).((let H31 \def (eq_ind T (THead (Flat Appl) v0 (THead -(Bind b) u0 t7)) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) -with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ -_) \Rightarrow (match k0 in K return (\lambda (_: K).Prop) with [(Bind _) -\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u t5) -H29) in (False_ind ((eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O -v3) t8)) t4) \to ((not (eq B b Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3) \to -((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) -(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9)))))))) H31)) H30 H25 H26 -H27 H28))) | (pr0_delta u0 u3 H25 t7 t8 H26 w H27) \Rightarrow (\lambda (H28: -(eq T (THead (Bind Abbr) u0 t7) (THead (Bind Abst) u t5))).(\lambda (H29: (eq -T (THead (Bind Abbr) u3 w) t4)).((let H30 \def (eq_ind T (THead (Bind Abbr) -u0 t7) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _) -\Rightarrow (match k0 in K return (\lambda (_: K).Prop) with [(Bind b) -\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow -True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _) -\Rightarrow False])])) I (THead (Bind Abst) u t5) H28) in (False_ind ((eq T -(THead (Bind Abbr) u3 w) t4) \to ((pr0 u0 u3) \to ((pr0 t7 t8) \to ((subst0 O -u3 t8 w) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) -(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9))))))) H30)) H29 H25 H26 -H27))) | (pr0_zeta b H25 t7 t8 H26 u0) \Rightarrow (\lambda (H27: (eq T -(THead (Bind b) u0 (lift (S O) O t7)) (THead (Bind Abst) u t5))).(\lambda -(H28: (eq T t8 t4)).((let H29 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: -((nat \to nat))) (d: nat) (t9: T) on t9: T \def (match t9 with [(TSort n) -\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with -[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k0 u3 t10) -\Rightarrow (THead k0 (lref_map f d u3) (lref_map f (s k0 d) t10))]) in -lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (TLRef _) \Rightarrow -((let rec lref_map (f: ((nat \to nat))) (d: nat) (t9: T) on t9: T \def (match -t9 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef -(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | -(THead k0 u3 t10) \Rightarrow (THead k0 (lref_map f d u3) (lref_map f (s k0 -d) t10))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (THead _ _ -t9) \Rightarrow t9])) (THead (Bind b) u0 (lift (S O) O t7)) (THead (Bind -Abst) u t5) H27) in ((let H30 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) -\Rightarrow u0 | (THead _ t9 _) \Rightarrow t9])) (THead (Bind b) u0 (lift (S -O) O t7)) (THead (Bind Abst) u t5) H27) in ((let H31 \def (f_equal T B -(\lambda (e: T).(match e in T return (\lambda (_: T).B) with [(TSort _) -\Rightarrow b | (TLRef _) \Rightarrow b | (THead k0 _ _) \Rightarrow (match -k0 in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) -\Rightarrow b])])) (THead (Bind b) u0 (lift (S O) O t7)) (THead (Bind Abst) u -t5) H27) in (eq_ind B Abst (\lambda (b0: B).((eq T u0 u) \to ((eq T (lift (S -O) O t7) t5) \to ((eq T t8 t4) \to ((not (eq B b0 Abst)) \to ((pr0 t7 t8) \to -(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: -T).(pr0 (THead (Bind Abbr) v2 t6) t9))))))))) (\lambda (H32: (eq T u0 -u)).(eq_ind T u (\lambda (_: T).((eq T (lift (S O) O t7) t5) \to ((eq T t8 -t4) \to ((not (eq B Abst Abst)) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: -T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda (t10: T).(pr0 (THead (Bind -Abbr) v2 t6) t10)))))))) (\lambda (H33: (eq T (lift (S O) O t7) t5)).(eq_ind -T (lift (S O) O t7) (\lambda (_: T).((eq T t8 t4) \to ((not (eq B Abst Abst)) -\to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) -t10)) (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t6) t10))))))) (\lambda -(H34: (eq T t8 t4)).(eq_ind T t4 (\lambda (t9: T).((not (eq B Abst Abst)) \to -((pr0 t7 t9) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) -t10)) (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t6) t10)))))) (\lambda -(H35: (not (eq B Abst Abst))).(\lambda (_: (pr0 t7 t4)).(let H37 \def (match -(H35 (refl_equal B Abst)) in False return (\lambda (_: False).(ex2 T (\lambda -(t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead -(Bind Abbr) v2 t6) t9)))) with []) in H37))) t8 (sym_eq T t8 t4 H34))) t5 -H33)) u0 (sym_eq T u0 u H32))) b (sym_eq B b Abst H31))) H30)) H29)) H28 H25 -H26))) | (pr0_tau t7 t8 H25 u0) \Rightarrow (\lambda (H26: (eq T (THead (Flat -Cast) u0 t7) (THead (Bind Abst) u t5))).(\lambda (H27: (eq T t8 t4)).((let -H28 \def (eq_ind T (THead (Flat Cast) u0 t7) (\lambda (e: T).(match e in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind Abst) u t5) H26) in (False_ind ((eq T t8 t4) \to -((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) -(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9))))) H28)) H27 H25)))]) in -(H25 (refl_equal T (THead (Bind Abst) u t5)) (refl_equal T t4))))) k H21))))) -H18)) H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_upsilon b H9 v1 v2 H10 u0 -u3 H11 t5 t6 H12) \Rightarrow (\lambda (H13: (eq T (THead (Flat Appl) v1 -(THead (Bind b) u0 t5)) t)).(\lambda (H14: (eq T (THead (Bind b) u3 (THead -(Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Flat Appl) v1 -(THead (Bind b) u0 t5)) (\lambda (_: T).((eq T (THead (Bind b) u3 (THead -(Flat Appl) (lift (S O) O v2) t6)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1 -v2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead -k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H15: (eq T -(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T -(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) (\lambda (t7: -T).((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) -\to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 -t7 t8)))))))) (\lambda (H16: (not (eq B b Abst))).(\lambda (H17: (pr0 v1 -v2)).(\lambda (H18: (pr0 u0 u3)).(\lambda (H19: (pr0 t5 t6)).(let H20 \def -(eq_ind_r T t (\lambda (t7: T).(eq T (THead k u1 t3) t7)) H4 (THead (Flat -Appl) v1 (THead (Bind b) u0 t5)) H13) in (let H21 \def (f_equal T K (\lambda -(e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k -| (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) -(THead (Flat Appl) v1 (THead (Bind b) u0 t5)) H20) in ((let H22 \def (f_equal -T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) -(THead k u1 t3) (THead (Flat Appl) v1 (THead (Bind b) u0 t5)) H20) in ((let -H23 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7) -\Rightarrow t7])) (THead k u1 t3) (THead (Flat Appl) v1 (THead (Bind b) u0 -t5)) H20) in (\lambda (H24: (eq T u1 v1)).(\lambda (H25: (eq K k (Flat -Appl))).(let H26 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v -t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to -(ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 -t10)))))))))) H (THead (Flat Appl) v1 (THead (Bind b) u0 t5)) H13) in -(eq_ind_r K (Flat Appl) (\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 (THead -k0 u2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) -(lift (S O) O v2) t6)) t7)))) (let H27 \def (eq_ind T u1 (\lambda (t7: -T).(pr0 t7 u2)) H7 v1 H24) in (let H28 \def (eq_ind T t3 (\lambda (t7: -T).(pr0 t7 t4)) H8 (THead (Bind b) u0 t5) H23) in (let H29 \def (match H28 in -pr0 return (\lambda (t7: T).(\lambda (t8: T).(\lambda (_: (pr0 t7 t8)).((eq T -t7 (THead (Bind b) u0 t5)) \to ((eq T t8 t4) \to (ex2 T (\lambda (t9: T).(pr0 -(THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 -(THead (Flat Appl) (lift (S O) O v2) t6)) t9)))))))) with [(pr0_refl t7) -\Rightarrow (\lambda (H29: (eq T t7 (THead (Bind b) u0 t5))).(\lambda (H30: -(eq T t7 t4)).(eq_ind T (THead (Bind b) u0 t5) (\lambda (t8: T).((eq T t8 t4) -\to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: -T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9))))) -(\lambda (H31: (eq T (THead (Bind b) u0 t5) t4)).(eq_ind T (THead (Bind b) u0 -t5) (\lambda (t8: T).(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t8) -t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) -O v2) t6)) t9)))) (ex2_ind T (\lambda (t8: T).(pr0 u2 t8)) (\lambda (t8: -T).(pr0 v2 t8)) (ex2 T (\lambda (t8: T).(pr0 (THead (Flat Appl) u2 (THead -(Bind b) u0 t5)) t8)) (\lambda (t8: T).(pr0 (THead (Bind b) u3 (THead (Flat -Appl) (lift (S O) O v2) t6)) t8))) (\lambda (x: T).(\lambda (H32: (pr0 u2 -x)).(\lambda (H33: (pr0 v2 x)).(pr0_confluence__pr0_cong_upsilon_refl b H16 -u0 u3 H18 t5 t6 H19 u2 v2 x H32 H33)))) (H26 v1 (tlt_head_sx (Flat Appl) v1 -(THead (Bind b) u0 t5)) u2 H27 v2 H17)) t4 H31)) t7 (sym_eq T t7 (THead (Bind -b) u0 t5) H29) H30))) | (pr0_comp u4 u5 H29 t7 t8 H30 k0) \Rightarrow -(\lambda (H31: (eq T (THead k0 u4 t7) (THead (Bind b) u0 t5))).(\lambda (H32: -(eq T (THead k0 u5 t8) t4)).((let H33 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t7 | -(TLRef _) \Rightarrow t7 | (THead _ _ t9) \Rightarrow t9])) (THead k0 u4 t7) -(THead (Bind b) u0 t5) H31) in ((let H34 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u4 | -(TLRef _) \Rightarrow u4 | (THead _ t9 _) \Rightarrow t9])) (THead k0 u4 t7) -(THead (Bind b) u0 t5) H31) in ((let H35 \def (f_equal T K (\lambda (e: -T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k0 | -(TLRef _) \Rightarrow k0 | (THead k1 _ _) \Rightarrow k1])) (THead k0 u4 t7) -(THead (Bind b) u0 t5) H31) in (eq_ind K (Bind b) (\lambda (k1: K).((eq T u4 -u0) \to ((eq T t7 t5) \to ((eq T (THead k1 u5 t8) t4) \to ((pr0 u4 u5) \to -((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) -(\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) -t6)) t9))))))))) (\lambda (H36: (eq T u4 u0)).(eq_ind T u0 (\lambda (t9: -T).((eq T t7 t5) \to ((eq T (THead (Bind b) u5 t8) t4) \to ((pr0 t9 u5) \to -((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) -t10)) (\lambda (t10: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S -O) O v2) t6)) t10)))))))) (\lambda (H37: (eq T t7 t5)).(eq_ind T t5 (\lambda -(t9: T).((eq T (THead (Bind b) u5 t8) t4) \to ((pr0 u0 u5) \to ((pr0 t9 t8) -\to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda -(t10: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) -t10))))))) (\lambda (H38: (eq T (THead (Bind b) u5 t8) t4)).(eq_ind T (THead -(Bind b) u5 t8) (\lambda (t9: T).((pr0 u0 u5) \to ((pr0 t5 t8) \to (ex2 T -(\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t9) t10)) (\lambda (t10: T).(pr0 -(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t10)))))) -(\lambda (H39: (pr0 u0 u5)).(\lambda (H40: (pr0 t5 t8)).(ex2_ind T (\lambda -(t9: T).(pr0 t8 t9)) (\lambda (t9: T).(pr0 t6 t9)) (ex2 T (\lambda (t9: -T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 t8)) t9)) (\lambda (t9: -T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9))) -(\lambda (x: T).(\lambda (H41: (pr0 t8 x)).(\lambda (H42: (pr0 t6 -x)).(ex2_ind T (\lambda (t9: T).(pr0 u5 t9)) (\lambda (t9: T).(pr0 u3 t9)) -(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 t8)) -t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) -O v2) t6)) t9))) (\lambda (x0: T).(\lambda (H43: (pr0 u5 x0)).(\lambda (H44: -(pr0 u3 x0)).(ex2_ind T (\lambda (t9: T).(pr0 u2 t9)) (\lambda (t9: T).(pr0 -v2 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 -t8)) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift -(S O) O v2) t6)) t9))) (\lambda (x1: T).(\lambda (H45: (pr0 u2 x1)).(\lambda -(H46: (pr0 v2 x1)).(pr0_confluence__pr0_cong_upsilon_cong b H16 u2 v2 x1 H45 -H46 t8 t6 x H41 H42 u5 u3 x0 H43 H44)))) (H26 v1 (tlt_head_sx (Flat Appl) v1 -(THead (Bind b) u0 t5)) u2 H27 v2 H17))))) (H26 u0 (tlt_trans (THead (Bind b) -u0 t5) u0 (THead (Flat Appl) v1 (THead (Bind b) u0 t5)) (tlt_head_sx (Bind b) -u0 t5) (tlt_head_dx (Flat Appl) v1 (THead (Bind b) u0 t5))) u5 H39 u3 -H18))))) (H26 t5 (tlt_trans (THead (Bind b) u0 t5) t5 (THead (Flat Appl) v1 -(THead (Bind b) u0 t5)) (tlt_head_dx (Bind b) u0 t5) (tlt_head_dx (Flat Appl) -v1 (THead (Bind b) u0 t5))) t8 H40 t6 H19)))) t4 H38)) t7 (sym_eq T t7 t5 -H37))) u4 (sym_eq T u4 u0 H36))) k0 (sym_eq K k0 (Bind b) H35))) H34)) H33)) -H32 H29 H30))) | (pr0_beta u v0 v3 H29 t7 t8 H30) \Rightarrow (\lambda (H31: -(eq T (THead (Flat Appl) v0 (THead (Bind Abst) u t7)) (THead (Bind b) u0 -t5))).(\lambda (H32: (eq T (THead (Bind Abbr) v3 t8) t4)).((let H33 \def -(eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u t7)) (\lambda (e: -T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in -K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) -\Rightarrow True])])) I (THead (Bind b) u0 t5) H31) in (False_ind ((eq T -(THead (Bind Abbr) v3 t8) t4) \to ((pr0 v0 v3) \to ((pr0 t7 t8) \to (ex2 T -(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 -(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9)))))) H33)) -H32 H29 H30))) | (pr0_upsilon b0 H29 v0 v3 H30 u4 u5 H31 t7 t8 H32) -\Rightarrow (\lambda (H33: (eq T (THead (Flat Appl) v0 (THead (Bind b0) u4 -t7)) (THead (Bind b) u0 t5))).(\lambda (H34: (eq T (THead (Bind b0) u5 (THead -(Flat Appl) (lift (S O) O v3) t8)) t4)).((let H35 \def (eq_ind T (THead (Flat -Appl) v0 (THead (Bind b0) u4 t7)) (\lambda (e: T).(match e in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind b) u0 t5) H33) in (False_ind ((eq T (THead (Bind b0) -u5 (THead (Flat Appl) (lift (S O) O v3) t8)) t4) \to ((not (eq B b0 Abst)) -\to ((pr0 v0 v3) \to ((pr0 u4 u5) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: -T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) -u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9)))))))) H35)) H34 H29 H30 H31 -H32))) | (pr0_delta u4 u5 H29 t7 t8 H30 w H31) \Rightarrow (\lambda (H32: (eq -T (THead (Bind Abbr) u4 t7) (THead (Bind b) u0 t5))).(\lambda (H33: (eq T -(THead (Bind Abbr) u5 w) t4)).((let H34 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t7 | -(TLRef _) \Rightarrow t7 | (THead _ _ t9) \Rightarrow t9])) (THead (Bind -Abbr) u4 t7) (THead (Bind b) u0 t5) H32) in ((let H35 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow u4 | (TLRef _) \Rightarrow u4 | (THead _ t9 _) \Rightarrow t9])) -(THead (Bind Abbr) u4 t7) (THead (Bind b) u0 t5) H32) in ((let H36 \def -(f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with -[(TSort _) \Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | (THead k0 _ _) -\Rightarrow (match k0 in K return (\lambda (_: K).B) with [(Bind b0) -\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (THead (Bind Abbr) u4 t7) -(THead (Bind b) u0 t5) H32) in (eq_ind B Abbr (\lambda (b0: B).((eq T u4 u0) -\to ((eq T t7 t5) \to ((eq T (THead (Bind Abbr) u5 w) t4) \to ((pr0 u4 u5) -\to ((pr0 t7 t8) \to ((subst0 O u5 t8 w) \to (ex2 T (\lambda (t9: T).(pr0 -(THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind b0) u3 -(THead (Flat Appl) (lift (S O) O v2) t6)) t9)))))))))) (\lambda (H37: (eq T -u4 u0)).(eq_ind T u0 (\lambda (t9: T).((eq T t7 t5) \to ((eq T (THead (Bind -Abbr) u5 w) t4) \to ((pr0 t9 u5) \to ((pr0 t7 t8) \to ((subst0 O u5 t8 w) \to -(ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda (t10: -T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) -t10))))))))) (\lambda (H38: (eq T t7 t5)).(eq_ind T t5 (\lambda (t9: T).((eq -T (THead (Bind Abbr) u5 w) t4) \to ((pr0 u0 u5) \to ((pr0 t9 t8) \to ((subst0 -O u5 t8 w) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) -(\lambda (t10: T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O -v2) t6)) t10)))))))) (\lambda (H39: (eq T (THead (Bind Abbr) u5 w) -t4)).(eq_ind T (THead (Bind Abbr) u5 w) (\lambda (t9: T).((pr0 u0 u5) \to -((pr0 t5 t8) \to ((subst0 O u5 t8 w) \to (ex2 T (\lambda (t10: T).(pr0 (THead -(Flat Appl) u2 t9) t10)) (\lambda (t10: T).(pr0 (THead (Bind Abbr) u3 (THead -(Flat Appl) (lift (S O) O v2) t6)) t10))))))) (\lambda (H40: (pr0 u0 -u5)).(\lambda (H41: (pr0 t5 t8)).(\lambda (H42: (subst0 O u5 t8 w)).(let H43 -\def (eq_ind_r B b (\lambda (b0: B).(\forall (v: T).((tlt v (THead (Flat -Appl) v1 (THead (Bind b0) u0 t5))) \to (\forall (t9: T).((pr0 v t9) \to -(\forall (t10: T).((pr0 v t10) \to (ex2 T (\lambda (t11: T).(pr0 t9 t11)) -(\lambda (t11: T).(pr0 t10 t11)))))))))) H26 Abbr H36) in (let H44 \def -(eq_ind_r B b (\lambda (b0: B).(eq T t3 (THead (Bind b0) u0 t5))) H23 Abbr -H36) in (let H45 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 Abst))) -H16 Abbr H36) in (ex2_ind T (\lambda (t9: T).(pr0 t8 t9)) (\lambda (t9: -T).(pr0 t6 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead -(Bind Abbr) u5 w)) t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) u3 (THead -(Flat Appl) (lift (S O) O v2) t6)) t9))) (\lambda (x: T).(\lambda (H46: (pr0 -t8 x)).(\lambda (H47: (pr0 t6 x)).(ex2_ind T (\lambda (t9: T).(pr0 u5 t9)) -(\lambda (t9: T).(pr0 u3 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) -u2 (THead (Bind Abbr) u5 w)) t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) u3 -(THead (Flat Appl) (lift (S O) O v2) t6)) t9))) (\lambda (x0: T).(\lambda -(H48: (pr0 u5 x0)).(\lambda (H49: (pr0 u3 x0)).(ex2_ind T (\lambda (t9: -T).(pr0 u2 t9)) (\lambda (t9: T).(pr0 v2 t9)) (ex2 T (\lambda (t9: T).(pr0 -(THead (Flat Appl) u2 (THead (Bind Abbr) u5 w)) t9)) (\lambda (t9: T).(pr0 -(THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9))) -(\lambda (x1: T).(\lambda (H50: (pr0 u2 x1)).(\lambda (H51: (pr0 v2 -x1)).(pr0_confluence__pr0_cong_upsilon_delta H45 u5 t8 w H42 u2 v2 x1 H50 H51 -t6 x H46 H47 u3 x0 H48 H49)))) (H43 v1 (tlt_head_sx (Flat Appl) v1 (THead -(Bind Abbr) u0 t5)) u2 H27 v2 H17))))) (H43 u0 (tlt_trans (THead (Bind Abbr) -u0 t5) u0 (THead (Flat Appl) v1 (THead (Bind Abbr) u0 t5)) (tlt_head_sx (Bind -Abbr) u0 t5) (tlt_head_dx (Flat Appl) v1 (THead (Bind Abbr) u0 t5))) u5 H40 -u3 H18))))) (H43 t5 (tlt_trans (THead (Bind Abbr) u0 t5) t5 (THead (Flat -Appl) v1 (THead (Bind Abbr) u0 t5)) (tlt_head_dx (Bind Abbr) u0 t5) -(tlt_head_dx (Flat Appl) v1 (THead (Bind Abbr) u0 t5))) t8 H41 t6 H19)))))))) -t4 H39)) t7 (sym_eq T t7 t5 H38))) u4 (sym_eq T u4 u0 H37))) b H36)) H35)) -H34)) H33 H29 H30 H31))) | (pr0_zeta b0 H29 t7 t8 H30 u) \Rightarrow (\lambda -(H31: (eq T (THead (Bind b0) u (lift (S O) O t7)) (THead (Bind b) u0 -t5))).(\lambda (H32: (eq T t8 t4)).((let H33 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let -rec lref_map (f: ((nat \to nat))) (d: nat) (t9: T) on t9: T \def (match t9 -with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match -(blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k0 -u4 t10) \Rightarrow (THead k0 (lref_map f d u4) (lref_map f (s k0 d) t10))]) -in lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (TLRef _) \Rightarrow -((let rec lref_map (f: ((nat \to nat))) (d: nat) (t9: T) on t9: T \def (match -t9 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef -(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | -(THead k0 u4 t10) \Rightarrow (THead k0 (lref_map f d u4) (lref_map f (s k0 -d) t10))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (THead _ _ -t9) \Rightarrow t9])) (THead (Bind b0) u (lift (S O) O t7)) (THead (Bind b) -u0 t5) H31) in ((let H34 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) -\Rightarrow u | (THead _ t9 _) \Rightarrow t9])) (THead (Bind b0) u (lift (S -O) O t7)) (THead (Bind b) u0 t5) H31) in ((let H35 \def (f_equal T B (\lambda -(e: T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b0 -| (TLRef _) \Rightarrow b0 | (THead k0 _ _) \Rightarrow (match k0 in K return -(\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow -b0])])) (THead (Bind b0) u (lift (S O) O t7)) (THead (Bind b) u0 t5) H31) in -(eq_ind B b (\lambda (b1: B).((eq T u u0) \to ((eq T (lift (S O) O t7) t5) -\to ((eq T t8 t4) \to ((not (eq B b1 Abst)) \to ((pr0 t7 t8) \to (ex2 T -(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 -(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9))))))))) -(\lambda (H36: (eq T u u0)).(eq_ind T u0 (\lambda (_: T).((eq T (lift (S O) O -t7) t5) \to ((eq T t8 t4) \to ((not (eq B b Abst)) \to ((pr0 t7 t8) \to (ex2 -T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda (t10: -T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) -t10)))))))) (\lambda (H37: (eq T (lift (S O) O t7) t5)).(eq_ind T (lift (S O) -O t7) (\lambda (_: T).((eq T t8 t4) \to ((not (eq B b Abst)) \to ((pr0 t7 t8) -\to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda -(t10: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) -t10))))))) (\lambda (H38: (eq T t8 t4)).(eq_ind T t4 (\lambda (t9: T).((not -(eq B b Abst)) \to ((pr0 t7 t9) \to (ex2 T (\lambda (t10: T).(pr0 (THead -(Flat Appl) u2 t4) t10)) (\lambda (t10: T).(pr0 (THead (Bind b) u3 (THead -(Flat Appl) (lift (S O) O v2) t6)) t10)))))) (\lambda (H39: (not (eq B b -Abst))).(\lambda (H40: (pr0 t7 t4)).(let H41 \def (eq_ind_r T t5 (\lambda -(t9: T).(\forall (v: T).((tlt v (THead (Flat Appl) v1 (THead (Bind b) u0 -t9))) \to (\forall (t10: T).((pr0 v t10) \to (\forall (t11: T).((pr0 v t11) -\to (ex2 T (\lambda (t12: T).(pr0 t10 t12)) (\lambda (t12: T).(pr0 t11 -t12)))))))))) H26 (lift (S O) O t7) H37) in (let H42 \def (eq_ind_r T t5 -(\lambda (t9: T).(eq T t3 (THead (Bind b) u0 t9))) H23 (lift (S O) O t7) H37) -in (let H43 \def (eq_ind_r T t5 (\lambda (t9: T).(pr0 t9 t6)) H19 (lift (S O) -O t7) H37) in (ex2_ind T (\lambda (t9: T).(eq T t6 (lift (S O) O t9))) -(\lambda (t9: T).(pr0 t7 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) -u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift -(S O) O v2) t6)) t9))) (\lambda (x: T).(\lambda (H44: (eq T t6 (lift (S O) O -x))).(\lambda (H45: (pr0 t7 x)).(eq_ind_r T (lift (S O) O x) (\lambda (t9: -T).(ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda -(t10: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t9)) -t10)))) (ex2_ind T (\lambda (t9: T).(pr0 x t9)) (\lambda (t9: T).(pr0 t4 t9)) -(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: -T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) (lift (S O) O -x))) t9))) (\lambda (x0: T).(\lambda (H46: (pr0 x x0)).(\lambda (H47: (pr0 t4 -x0)).(ex2_ind T (\lambda (t9: T).(pr0 u2 t9)) (\lambda (t9: T).(pr0 v2 t9)) -(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: -T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) (lift (S O) O -x))) t9))) (\lambda (x1: T).(\lambda (H48: (pr0 u2 x1)).(\lambda (H49: (pr0 -v2 x1)).(pr0_confluence__pr0_cong_upsilon_zeta b H39 u0 u3 H18 u2 v2 x1 H48 -H49 x t4 x0 H46 H47)))) (H41 v1 (tlt_head_sx (Flat Appl) v1 (THead (Bind b) -u0 (lift (S O) O t7))) u2 H27 v2 H17))))) (H41 t7 (tlt_trans (THead (Bind b) -u0 (lift (S O) O t7)) t7 (THead (Flat Appl) v1 (THead (Bind b) u0 (lift (S O) -O t7))) (lift_tlt_dx (Bind b) u0 t7 (S O) O) (tlt_head_dx (Flat Appl) v1 -(THead (Bind b) u0 (lift (S O) O t7)))) x H45 t4 H40)) t6 H44)))) -(pr0_gen_lift t7 t6 (S O) O H43))))))) t8 (sym_eq T t8 t4 H38))) t5 H37)) u -(sym_eq T u u0 H36))) b0 (sym_eq B b0 b H35))) H34)) H33)) H32 H29 H30))) | -(pr0_tau t7 t8 H29 u) \Rightarrow (\lambda (H30: (eq T (THead (Flat Cast) u -t7) (THead (Bind b) u0 t5))).(\lambda (H31: (eq T t8 t4)).((let H32 \def -(eq_ind T (THead (Flat Cast) u t7) (\lambda (e: T).(match e in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind b) u0 t5) H30) in (False_ind ((eq T t8 t4) \to ((pr0 -t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) -(\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) -t6)) t9))))) H32)) H31 H29)))]) in (H29 (refl_equal T (THead (Bind b) u0 t5)) -(refl_equal T t4))))) k H25))))) H22)) H21))))))) t2 H15)) t H13 H14 H9 H10 -H11 H12))) | (pr0_delta u0 u3 H9 t5 t6 H10 w H11) \Rightarrow (\lambda (H12: -(eq T (THead (Bind Abbr) u0 t5) t)).(\lambda (H13: (eq T (THead (Bind Abbr) -u3 w) t2)).(eq_ind T (THead (Bind Abbr) u0 t5) (\lambda (_: T).((eq T (THead -(Bind Abbr) u3 w) t2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to ((subst0 O u3 t6 -w) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: -T).(pr0 t2 t8)))))))) (\lambda (H14: (eq T (THead (Bind Abbr) u3 w) -t2)).(eq_ind T (THead (Bind Abbr) u3 w) (\lambda (t7: T).((pr0 u0 u3) \to -((pr0 t5 t6) \to ((subst0 O u3 t6 w) \to (ex2 T (\lambda (t8: T).(pr0 (THead -k u2 t4) t8)) (\lambda (t8: T).(pr0 t7 t8))))))) (\lambda (H15: (pr0 u0 -u3)).(\lambda (H16: (pr0 t5 t6)).(\lambda (H17: (subst0 O u3 t6 w)).(let H18 -\def (eq_ind_r T t (\lambda (t7: T).(eq T (THead k u1 t3) t7)) H4 (THead -(Bind Abbr) u0 t5) H12) in (let H19 \def (f_equal T K (\lambda (e: T).(match -e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) -\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) (THead (Bind -Abbr) u0 t5) H18) in ((let H20 \def (f_equal T T (\lambda (e: T).(match e in -T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) -\Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t3) (THead -(Bind Abbr) u0 t5) H18) in ((let H21 \def (f_equal T T (\lambda (e: T).(match -e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) -\Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t3) (THead -(Bind Abbr) u0 t5) H18) in (\lambda (H22: (eq T u1 u0)).(\lambda (H23: (eq K -k (Bind Abbr))).(let H24 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: -T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v -t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 -t10)))))))))) H (THead (Bind Abbr) u0 t5) H12) in (eq_ind_r K (Bind Abbr) -(\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 (THead k0 u2 t4) t7)) (\lambda -(t7: T).(pr0 (THead (Bind Abbr) u3 w) t7)))) (let H25 \def (eq_ind T u1 -(\lambda (t7: T).(pr0 t7 u2)) H7 u0 H22) in (let H26 \def (eq_ind T t3 -(\lambda (t7: T).(pr0 t7 t4)) H8 t5 H21) in (ex2_ind T (\lambda (t7: T).(pr0 -t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead -(Bind Abbr) u2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u3 w) t7))) -(\lambda (x: T).(\lambda (H27: (pr0 t4 x)).(\lambda (H28: (pr0 t6 -x)).(ex2_ind T (\lambda (t7: T).(pr0 u2 t7)) (\lambda (t7: T).(pr0 u3 t7)) -(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 t4) t7)) (\lambda (t7: -T).(pr0 (THead (Bind Abbr) u3 w) t7))) (\lambda (x0: T).(\lambda (H29: (pr0 -u2 x0)).(\lambda (H30: (pr0 u3 x0)).(pr0_confluence__pr0_cong_delta u3 t6 w -H17 u2 x0 H29 H30 t4 x H27 H28)))) (H24 u0 (tlt_head_sx (Bind Abbr) u0 t5) u2 -H25 u3 H15))))) (H24 t5 (tlt_head_dx (Bind Abbr) u0 t5) t4 H26 t6 H16)))) k -H23))))) H20)) H19)))))) t2 H14)) t H12 H13 H9 H10 H11))) | (pr0_zeta b H9 t5 -t6 H10 u) \Rightarrow (\lambda (H11: (eq T (THead (Bind b) u (lift (S O) O -t5)) t)).(\lambda (H12: (eq T t6 t2)).(eq_ind T (THead (Bind b) u (lift (S O) -O t5)) (\lambda (_: T).((eq T t6 t2) \to ((not (eq B b Abst)) \to ((pr0 t5 -t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: -T).(pr0 t2 t8))))))) (\lambda (H13: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: -T).((not (eq B b Abst)) \to ((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 -(THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H14: (not -(eq B b Abst))).(\lambda (H15: (pr0 t5 t2)).(let H16 \def (eq_ind_r T t -(\lambda (t7: T).(eq T (THead k u1 t3) t7)) H4 (THead (Bind b) u (lift (S O) -O t5)) H11) in (let H17 \def (f_equal T K (\lambda (e: T).(match e in T -return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) -\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) (THead (Bind -b) u (lift (S O) O t5)) H16) in ((let H18 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | -(TLRef _) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t3) -(THead (Bind b) u (lift (S O) O t5)) H16) in ((let H19 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) -(THead k u1 t3) (THead (Bind b) u (lift (S O) O t5)) H16) in (\lambda (H20: -(eq T u1 u)).(\lambda (H21: (eq K k (Bind b))).(let H22 \def (eq_ind_r T t -(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) -\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) -(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Bind b) u (lift (S O) O -t5)) H11) in (eq_ind_r K (Bind b) (\lambda (k0: K).(ex2 T (\lambda (t7: -T).(pr0 (THead k0 u2 t4) t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let H23 \def -(eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H7 u H20) in (let H24 \def (eq_ind -T t3 (\lambda (t7: T).(pr0 t7 t4)) H8 (lift (S O) O t5) H19) in (ex2_ind T -(\lambda (t7: T).(eq T t4 (lift (S O) O t7))) (\lambda (t7: T).(pr0 t5 t7)) -(ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 t4) t7)) (\lambda (t7: -T).(pr0 t2 t7))) (\lambda (x: T).(\lambda (H25: (eq T t4 (lift (S O) O -x))).(\lambda (H26: (pr0 t5 x)).(eq_ind_r T (lift (S O) O x) (\lambda (t7: -T).(ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 t7) t8)) (\lambda (t8: -T).(pr0 t2 t8)))) (ex2_ind T (\lambda (t7: T).(pr0 x t7)) (\lambda (t7: -T).(pr0 t2 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (lift (S O) O -x)) t7)) (\lambda (t7: T).(pr0 t2 t7))) (\lambda (x0: T).(\lambda (H27: (pr0 -x x0)).(\lambda (H28: (pr0 t2 x0)).(ex_intro2 T (\lambda (t7: T).(pr0 (THead -(Bind b) u2 (lift (S O) O x)) t7)) (\lambda (t7: T).(pr0 t2 t7)) x0 (pr0_zeta -b H14 x x0 H27 u2) H28)))) (H22 t5 (lift_tlt_dx (Bind b) u t5 (S O) O) x H26 -t2 H15)) t4 H25)))) (pr0_gen_lift t5 t4 (S O) O H24)))) k H21))))) H18)) -H17))))) t6 (sym_eq T t6 t2 H13))) t H11 H12 H9 H10))) | (pr0_tau t5 t6 H9 u) -\Rightarrow (\lambda (H10: (eq T (THead (Flat Cast) u t5) t)).(\lambda (H11: -(eq T t6 t2)).(eq_ind T (THead (Flat Cast) u t5) (\lambda (_: T).((eq T t6 -t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) -(\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H12: (eq T t6 t2)).(eq_ind T t2 -(\lambda (t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 -t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))) (\lambda (H13: (pr0 t5 t2)).(let -H14 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead k u1 t3) t7)) H4 (THead -(Flat Cast) u t5) H10) in (let H15 \def (f_equal T K (\lambda (e: T).(match e -in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) -\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) (THead (Flat -Cast) u t5) H14) in ((let H16 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) -\Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t3) (THead -(Flat Cast) u t5) H14) in ((let H17 \def (f_equal T T (\lambda (e: T).(match -e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) -\Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t3) (THead -(Flat Cast) u t5) H14) in (\lambda (H18: (eq T u1 u)).(\lambda (H19: (eq K k -(Flat Cast))).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: -T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v -t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 -t10)))))))))) H (THead (Flat Cast) u t5) H10) in (eq_ind_r K (Flat Cast) -(\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 (THead k0 u2 t4) t7)) (\lambda -(t7: T).(pr0 t2 t7)))) (let H21 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 -u2)) H7 u H18) in (let H22 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H8 -t5 H17) in (ex2_ind T (\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t2 -t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Flat Cast) u2 t4) t7)) (\lambda -(t7: T).(pr0 t2 t7))) (\lambda (x: T).(\lambda (H23: (pr0 t4 x)).(\lambda -(H24: (pr0 t2 x)).(ex_intro2 T (\lambda (t7: T).(pr0 (THead (Flat Cast) u2 -t4) t7)) (\lambda (t7: T).(pr0 t2 t7)) x (pr0_tau t4 x H23 u2) H24)))) (H20 -t5 (tlt_head_dx (Flat Cast) u t5) t4 H22 t2 H13)))) k H19))))) H16)) H15)))) -t6 (sym_eq T t6 t2 H12))) t H10 H11 H9)))]) in (H9 (refl_equal T t) -(refl_equal T t2))))) t1 H6)) t H4 H5 H2 H3))) | (pr0_beta u v1 v2 H2 t3 t4 -H3) \Rightarrow (\lambda (H4: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) -u t3)) t)).(\lambda (H5: (eq T (THead (Bind Abbr) v2 t4) t1)).(eq_ind T -(THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (\lambda (_: T).((eq T (THead -(Bind Abbr) v2 t4) t1) \to ((pr0 v1 v2) \to ((pr0 t3 t4) \to (ex2 T (\lambda -(t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6))))))) (\lambda (H6: (eq T -(THead (Bind Abbr) v2 t4) t1)).(eq_ind T (THead (Bind Abbr) v2 t4) (\lambda -(t5: T).((pr0 v1 v2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t5 -t6)) (\lambda (t6: T).(pr0 t2 t6)))))) (\lambda (H7: (pr0 v1 v2)).(\lambda -(H8: (pr0 t3 t4)).(let H9 \def (match H1 in pr0 return (\lambda (t5: -T).(\lambda (t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 t) \to ((eq T t6 t2) -\to (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: -T).(pr0 t2 t7)))))))) with [(pr0_refl t5) \Rightarrow (\lambda (H9: (eq T t5 -t)).(\lambda (H10: (eq T t5 t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2) -\to (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: -T).(pr0 t2 t7))))) (\lambda (H11: (eq T t t2)).(eq_ind T t2 (\lambda (_: -T).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: -T).(pr0 t2 t7)))) (let H12 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 t2)) -H11 (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) H4) in (eq_ind T (THead -(Flat Appl) v1 (THead (Bind Abst) u t3)) (\lambda (t6: T).(ex2 T (\lambda -(t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 t6 t7)))) -(let H13 \def (eq_ind_r T t (\lambda (t6: T).(eq T t5 t6)) H9 (THead (Flat -Appl) v1 (THead (Bind Abst) u t3)) H4) in (let H14 \def (eq_ind_r T t -(\lambda (t6: T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) -\to (\forall (t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) -(\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead (Flat Appl) v1 (THead (Bind -Abst) u t3)) H4) in (ex_intro2 T (\lambda (t6: T).(pr0 (THead (Bind Abbr) v2 -t4) t6)) (\lambda (t6: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u -t3)) t6)) (THead (Bind Abbr) v2 t4) (pr0_refl (THead (Bind Abbr) v2 t4)) -(pr0_beta u v1 v2 H7 t3 t4 H8)))) t2 H12)) t (sym_eq T t t2 H11))) t5 (sym_eq -T t5 t H9) H10))) | (pr0_comp u1 u2 H9 t5 t6 H10 k) \Rightarrow (\lambda -(H11: (eq T (THead k u1 t5) t)).(\lambda (H12: (eq T (THead k u2 t6) -t2)).(eq_ind T (THead k u1 t5) (\lambda (_: T).((eq T (THead k u2 t6) t2) \to -((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind -Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H13: (eq T -(THead k u2 t6) t2)).(eq_ind T (THead k u2 t6) (\lambda (t7: T).((pr0 u1 u2) -\to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) -t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (H14: (pr0 u1 u2)).(\lambda -(H15: (pr0 t5 t6)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead -(Flat Appl) v1 (THead (Bind Abst) u t3)) t7)) H4 (THead k u1 t5) H11) in (let -H17 \def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) -with [(TSort _) \Rightarrow (Flat Appl) | (TLRef _) \Rightarrow (Flat Appl) | -(THead k0 _ _) \Rightarrow k0])) (THead (Flat Appl) v1 (THead (Bind Abst) u -t3)) (THead k u1 t5) H16) in ((let H18 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | -(TLRef _) \Rightarrow v1 | (THead _ t7 _) \Rightarrow t7])) (THead (Flat -Appl) v1 (THead (Bind Abst) u t3)) (THead k u1 t5) H16) in ((let H19 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow (THead (Bind Abst) u t3) | (TLRef _) \Rightarrow -(THead (Bind Abst) u t3) | (THead _ _ t7) \Rightarrow t7])) (THead (Flat -Appl) v1 (THead (Bind Abst) u t3)) (THead k u1 t5) H16) in (\lambda (H20: (eq -T v1 u1)).(\lambda (H21: (eq K (Flat Appl) k)).(eq_ind K (Flat Appl) (\lambda -(k0: K).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda -(t7: T).(pr0 (THead k0 u2 t6) t7)))) (let H22 \def (eq_ind_r K k (\lambda -(k0: K).(eq T (THead k0 u1 t5) t)) H11 (Flat Appl) H21) in (let H23 \def -(eq_ind_r T t5 (\lambda (t7: T).(pr0 t7 t6)) H15 (THead (Bind Abst) u t3) -H19) in (let H24 \def (match H23 in pr0 return (\lambda (t7: T).(\lambda (t8: -T).(\lambda (_: (pr0 t7 t8)).((eq T t7 (THead (Bind Abst) u t3)) \to ((eq T -t8 t6) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) -(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t6) t9)))))))) with [(pr0_refl -t7) \Rightarrow (\lambda (H24: (eq T t7 (THead (Bind Abst) u t3))).(\lambda -(H25: (eq T t7 t6)).(eq_ind T (THead (Bind Abst) u t3) (\lambda (t8: T).((eq -T t8 t6) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) -(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t6) t9))))) (\lambda (H26: (eq T -(THead (Bind Abst) u t3) t6)).(eq_ind T (THead (Bind Abst) u t3) (\lambda -(t8: T).(ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda -(t9: T).(pr0 (THead (Flat Appl) u2 t8) t9)))) (let H27 \def (eq_ind_r T t5 -(\lambda (t8: T).(eq T (THead (Flat Appl) u1 t8) t)) H22 (THead (Bind Abst) u -t3) H19) in (let H28 \def (eq_ind_r T t (\lambda (t8: T).(\forall (v: -T).((tlt v t8) \to (\forall (t9: T).((pr0 v t9) \to (\forall (t10: T).((pr0 v -t10) \to (ex2 T (\lambda (t11: T).(pr0 t9 t11)) (\lambda (t11: T).(pr0 t10 -t11)))))))))) H (THead (Flat Appl) u1 (THead (Bind Abst) u t3)) H27) in (let -H29 \def (eq_ind T v1 (\lambda (t8: T).(pr0 t8 v2)) H7 u1 H20) in (ex2_ind T -(\lambda (t8: T).(pr0 v2 t8)) (\lambda (t8: T).(pr0 u2 t8)) (ex2 T (\lambda -(t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 (THead -(Flat Appl) u2 (THead (Bind Abst) u t3)) t8))) (\lambda (x: T).(\lambda (H30: -(pr0 v2 x)).(\lambda (H31: (pr0 u2 x)).(ex_intro2 T (\lambda (t8: T).(pr0 -(THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 (THead (Flat Appl) u2 -(THead (Bind Abst) u t3)) t8)) (THead (Bind Abbr) x t4) (pr0_comp v2 x H30 t4 -t4 (pr0_refl t4) (Bind Abbr)) (pr0_beta u u2 x H31 t3 t4 H8))))) (H28 u1 -(tlt_head_sx (Flat Appl) u1 (THead (Bind Abst) u t3)) v2 H29 u2 H14))))) t6 -H26)) t7 (sym_eq T t7 (THead (Bind Abst) u t3) H24) H25))) | (pr0_comp u0 u3 -H24 t7 t8 H25 k0) \Rightarrow (\lambda (H26: (eq T (THead k0 u0 t7) (THead -(Bind Abst) u t3))).(\lambda (H27: (eq T (THead k0 u3 t8) t6)).((let H28 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow t7 | (TLRef _) \Rightarrow t7 | (THead _ _ t9) -\Rightarrow t9])) (THead k0 u0 t7) (THead (Bind Abst) u t3) H26) in ((let H29 -\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t9 _) -\Rightarrow t9])) (THead k0 u0 t7) (THead (Bind Abst) u t3) H26) in ((let H30 -\def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) -with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _ _) -\Rightarrow k1])) (THead k0 u0 t7) (THead (Bind Abst) u t3) H26) in (eq_ind K -(Bind Abst) (\lambda (k1: K).((eq T u0 u) \to ((eq T t7 t3) \to ((eq T (THead -k1 u3 t8) t6) \to ((pr0 u0 u3) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: -T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Flat -Appl) u2 t6) t9))))))))) (\lambda (H31: (eq T u0 u)).(eq_ind T u (\lambda -(t9: T).((eq T t7 t3) \to ((eq T (THead (Bind Abst) u3 t8) t6) \to ((pr0 t9 -u3) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 -t4) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t6) t10)))))))) -(\lambda (H32: (eq T t7 t3)).(eq_ind T t3 (\lambda (t9: T).((eq T (THead -(Bind Abst) u3 t8) t6) \to ((pr0 u u3) \to ((pr0 t9 t8) \to (ex2 T (\lambda -(t10: T).(pr0 (THead (Bind Abbr) v2 t4) t10)) (\lambda (t10: T).(pr0 (THead -(Flat Appl) u2 t6) t10))))))) (\lambda (H33: (eq T (THead (Bind Abst) u3 t8) -t6)).(eq_ind T (THead (Bind Abst) u3 t8) (\lambda (t9: T).((pr0 u u3) \to -((pr0 t3 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t4) -t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t9) t10)))))) (\lambda (_: -(pr0 u u3)).(\lambda (H35: (pr0 t3 t8)).(let H36 \def (eq_ind_r T t5 (\lambda -(t9: T).(eq T (THead (Flat Appl) u1 t9) t)) H22 (THead (Bind Abst) u t3) H19) -in (let H37 \def (eq_ind_r T t (\lambda (t9: T).(\forall (v: T).((tlt v t9) -\to (\forall (t10: T).((pr0 v t10) \to (\forall (t11: T).((pr0 v t11) \to -(ex2 T (\lambda (t12: T).(pr0 t10 t12)) (\lambda (t12: T).(pr0 t11 -t12)))))))))) H (THead (Flat Appl) u1 (THead (Bind Abst) u t3)) H36) in (let -H38 \def (eq_ind T v1 (\lambda (t9: T).(pr0 t9 v2)) H7 u1 H20) in (ex2_ind T -(\lambda (t9: T).(pr0 v2 t9)) (\lambda (t9: T).(pr0 u2 t9)) (ex2 T (\lambda -(t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead -(Flat Appl) u2 (THead (Bind Abst) u3 t8)) t9))) (\lambda (x: T).(\lambda -(H39: (pr0 v2 x)).(\lambda (H40: (pr0 u2 x)).(ex2_ind T (\lambda (t9: T).(pr0 -t8 t9)) (\lambda (t9: T).(pr0 t4 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead -(Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead -(Bind Abst) u3 t8)) t9))) (\lambda (x0: T).(\lambda (H41: (pr0 t8 -x0)).(\lambda (H42: (pr0 t4 x0)).(ex_intro2 T (\lambda (t9: T).(pr0 (THead -(Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead -(Bind Abst) u3 t8)) t9)) (THead (Bind Abbr) x x0) (pr0_comp v2 x H39 t4 x0 -H42 (Bind Abbr)) (pr0_beta u3 u2 x H40 t8 x0 H41))))) (H37 t3 (tlt_trans -(THead (Bind Abst) u t3) t3 (THead (Flat Appl) u1 (THead (Bind Abst) u t3)) -(tlt_head_dx (Bind Abst) u t3) (tlt_head_dx (Flat Appl) u1 (THead (Bind Abst) -u t3))) t8 H35 t4 H8))))) (H37 u1 (tlt_head_sx (Flat Appl) u1 (THead (Bind -Abst) u t3)) v2 H38 u2 H14))))))) t6 H33)) t7 (sym_eq T t7 t3 H32))) u0 -(sym_eq T u0 u H31))) k0 (sym_eq K k0 (Bind Abst) H30))) H29)) H28)) H27 H24 -H25))) | (pr0_beta u0 v0 v3 H24 t7 t8 H25) \Rightarrow (\lambda (H26: (eq T -(THead (Flat Appl) v0 (THead (Bind Abst) u0 t7)) (THead (Bind Abst) u -t3))).(\lambda (H27: (eq T (THead (Bind Abbr) v3 t8) t6)).((let H28 \def -(eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u0 t7)) (\lambda (e: -T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in -K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) -\Rightarrow True])])) I (THead (Bind Abst) u t3) H26) in (False_ind ((eq T -(THead (Bind Abbr) v3 t8) t6) \to ((pr0 v0 v3) \to ((pr0 t7 t8) \to (ex2 T -(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 -(THead (Flat Appl) u2 t6) t9)))))) H28)) H27 H24 H25))) | (pr0_upsilon b H24 -v0 v3 H25 u0 u3 H26 t7 t8 H27) \Rightarrow (\lambda (H28: (eq T (THead (Flat -Appl) v0 (THead (Bind b) u0 t7)) (THead (Bind Abst) u t3))).(\lambda (H29: -(eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v3) t8)) t6)).((let -H30 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind b) u0 t7)) (\lambda (e: -T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in -K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) -\Rightarrow True])])) I (THead (Bind Abst) u t3) H28) in (False_ind ((eq T -(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v3) t8)) t6) \to ((not -(eq B b Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3) \to ((pr0 t7 t8) \to (ex2 T -(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 -(THead (Flat Appl) u2 t6) t9)))))))) H30)) H29 H24 H25 H26 H27))) | -(pr0_delta u0 u3 H24 t7 t8 H25 w H26) \Rightarrow (\lambda (H27: (eq T (THead -(Bind Abbr) u0 t7) (THead (Bind Abst) u t3))).(\lambda (H28: (eq T (THead -(Bind Abbr) u3 w) t6)).((let H29 \def (eq_ind T (THead (Bind Abbr) u0 t7) -(\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow -(match k0 in K return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match -b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst -\Rightarrow False | Void \Rightarrow False]) | (Flat _) \Rightarrow -False])])) I (THead (Bind Abst) u t3) H27) in (False_ind ((eq T (THead (Bind -Abbr) u3 w) t6) \to ((pr0 u0 u3) \to ((pr0 t7 t8) \to ((subst0 O u3 t8 w) \to -(ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: -T).(pr0 (THead (Flat Appl) u2 t6) t9))))))) H29)) H28 H24 H25 H26))) | -(pr0_zeta b H24 t7 t8 H25 u0) \Rightarrow (\lambda (H26: (eq T (THead (Bind -b) u0 (lift (S O) O t7)) (THead (Bind Abst) u t3))).(\lambda (H27: (eq T t8 -t6)).((let H28 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat -\to nat))) (d: nat) (t9: T) on t9: T \def (match t9 with [(TSort n) -\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with -[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k0 u3 t10) -\Rightarrow (THead k0 (lref_map f d u3) (lref_map f (s k0 d) t10))]) in -lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (TLRef _) \Rightarrow -((let rec lref_map (f: ((nat \to nat))) (d: nat) (t9: T) on t9: T \def (match -t9 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef -(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | -(THead k0 u3 t10) \Rightarrow (THead k0 (lref_map f d u3) (lref_map f (s k0 -d) t10))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (THead _ _ -t9) \Rightarrow t9])) (THead (Bind b) u0 (lift (S O) O t7)) (THead (Bind -Abst) u t3) H26) in ((let H29 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) -\Rightarrow u0 | (THead _ t9 _) \Rightarrow t9])) (THead (Bind b) u0 (lift (S -O) O t7)) (THead (Bind Abst) u t3) H26) in ((let H30 \def (f_equal T B -(\lambda (e: T).(match e in T return (\lambda (_: T).B) with [(TSort _) -\Rightarrow b | (TLRef _) \Rightarrow b | (THead k0 _ _) \Rightarrow (match -k0 in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) -\Rightarrow b])])) (THead (Bind b) u0 (lift (S O) O t7)) (THead (Bind Abst) u -t3) H26) in (eq_ind B Abst (\lambda (b0: B).((eq T u0 u) \to ((eq T (lift (S -O) O t7) t3) \to ((eq T t8 t6) \to ((not (eq B b0 Abst)) \to ((pr0 t7 t8) \to -(ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: -T).(pr0 (THead (Flat Appl) u2 t6) t9))))))))) (\lambda (H31: (eq T u0 -u)).(eq_ind T u (\lambda (_: T).((eq T (lift (S O) O t7) t3) \to ((eq T t8 -t6) \to ((not (eq B Abst Abst)) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: -T).(pr0 (THead (Bind Abbr) v2 t4) t10)) (\lambda (t10: T).(pr0 (THead (Flat -Appl) u2 t6) t10)))))))) (\lambda (H32: (eq T (lift (S O) O t7) t3)).(eq_ind -T (lift (S O) O t7) (\lambda (_: T).((eq T t8 t6) \to ((not (eq B Abst Abst)) -\to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t4) -t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t6) t10))))))) (\lambda -(H33: (eq T t8 t6)).(eq_ind T t6 (\lambda (t9: T).((not (eq B Abst Abst)) \to -((pr0 t7 t9) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t4) -t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t6) t10)))))) (\lambda -(H34: (not (eq B Abst Abst))).(\lambda (_: (pr0 t7 t6)).(let H36 \def (match -(H34 (refl_equal B Abst)) in False return (\lambda (_: False).(ex2 T (\lambda -(t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead -(Flat Appl) u2 t6) t9)))) with []) in H36))) t8 (sym_eq T t8 t6 H33))) t3 -H32)) u0 (sym_eq T u0 u H31))) b (sym_eq B b Abst H30))) H29)) H28)) H27 H24 -H25))) | (pr0_tau t7 t8 H24 u0) \Rightarrow (\lambda (H25: (eq T (THead (Flat -Cast) u0 t7) (THead (Bind Abst) u t3))).(\lambda (H26: (eq T t8 t6)).((let -H27 \def (eq_ind T (THead (Flat Cast) u0 t7) (\lambda (e: T).(match e in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind Abst) u t3) H25) in (False_ind ((eq T t8 t6) \to -((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) -(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t6) t9))))) H27)) H26 H24)))]) in -(H24 (refl_equal T (THead (Bind Abst) u t3)) (refl_equal T t6))))) k H21)))) -H18)) H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_beta u0 v0 v3 H9 t5 t6 -H10) \Rightarrow (\lambda (H11: (eq T (THead (Flat Appl) v0 (THead (Bind -Abst) u0 t5)) t)).(\lambda (H12: (eq T (THead (Bind Abbr) v3 t6) t2)).(eq_ind -T (THead (Flat Appl) v0 (THead (Bind Abst) u0 t5)) (\lambda (_: T).((eq T -(THead (Bind Abbr) v3 t6) t2) \to ((pr0 v0 v3) \to ((pr0 t5 t6) \to (ex2 T -(\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 -t8))))))) (\lambda (H13: (eq T (THead (Bind Abbr) v3 t6) t2)).(eq_ind T -(THead (Bind Abbr) v3 t6) (\lambda (t7: T).((pr0 v0 v3) \to ((pr0 t5 t6) \to -(ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: -T).(pr0 t7 t8)))))) (\lambda (H14: (pr0 v0 v3)).(\lambda (H15: (pr0 t5 -t6)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1 -(THead (Bind Abst) u t3)) t7)) H4 (THead (Flat Appl) v0 (THead (Bind Abst) u0 -t5)) H11) in (let H17 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 -| (THead _ t7 _) \Rightarrow t7])) (THead (Flat Appl) v1 (THead (Bind Abst) u -t3)) (THead (Flat Appl) v0 (THead (Bind Abst) u0 t5)) H16) in ((let H18 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ _ t7) -\Rightarrow (match t7 in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t8 _) \Rightarrow t8])])) -(THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (THead (Flat Appl) v0 (THead -(Bind Abst) u0 t5)) H16) in ((let H19 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | -(TLRef _) \Rightarrow t3 | (THead _ _ t7) \Rightarrow (match t7 in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 -| (THead _ _ t8) \Rightarrow t8])])) (THead (Flat Appl) v1 (THead (Bind Abst) -u t3)) (THead (Flat Appl) v0 (THead (Bind Abst) u0 t5)) H16) in (\lambda (_: -(eq T u u0)).(\lambda (H21: (eq T v1 v0)).(let H22 \def (eq_ind_r T t -(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) -\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) -(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Appl) v0 (THead (Bind -Abst) u0 t5)) H11) in (let H23 \def (eq_ind T v1 (\lambda (t7: T).(pr0 t7 -v2)) H7 v0 H21) in (let H24 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) -H8 t5 H19) in (ex2_ind T (\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 -t6 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda -(t7: T).(pr0 (THead (Bind Abbr) v3 t6) t7))) (\lambda (x: T).(\lambda (H25: -(pr0 t4 x)).(\lambda (H26: (pr0 t6 x)).(ex2_ind T (\lambda (t7: T).(pr0 v2 -t7)) (\lambda (t7: T).(pr0 v3 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind -Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v3 t6) t7))) -(\lambda (x0: T).(\lambda (H27: (pr0 v2 x0)).(\lambda (H28: (pr0 v3 -x0)).(ex_intro2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) -(\lambda (t7: T).(pr0 (THead (Bind Abbr) v3 t6) t7)) (THead (Bind Abbr) x0 x) -(pr0_comp v2 x0 H27 t4 x H25 (Bind Abbr)) (pr0_comp v3 x0 H28 t6 x H26 (Bind -Abbr)))))) (H22 v0 (tlt_head_sx (Flat Appl) v0 (THead (Bind Abst) u0 t5)) v2 -H23 v3 H14))))) (H22 t5 (tlt_trans (THead (Bind Abst) u0 t5) t5 (THead (Flat -Appl) v0 (THead (Bind Abst) u0 t5)) (tlt_head_dx (Bind Abst) u0 t5) -(tlt_head_dx (Flat Appl) v0 (THead (Bind Abst) u0 t5))) t4 H24 t6 H15)))))))) -H18)) H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_upsilon b H9 v0 v3 H10 u1 -u2 H11 t5 t6 H12) \Rightarrow (\lambda (H13: (eq T (THead (Flat Appl) v0 -(THead (Bind b) u1 t5)) t)).(\lambda (H14: (eq T (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v3) t6)) t2)).(eq_ind T (THead (Flat Appl) v0 -(THead (Bind b) u1 t5)) (\lambda (_: T).((eq T (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v3) t6)) t2) \to ((not (eq B b Abst)) \to ((pr0 v0 -v3) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead -(Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H15: -(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v3) t6)) -t2)).(eq_ind T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v3) t6)) -(\lambda (t7: T).((not (eq B b Abst)) \to ((pr0 v0 v3) \to ((pr0 u1 u2) \to -((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) -(\lambda (t8: T).(pr0 t7 t8)))))))) (\lambda (H16: (not (eq B b -Abst))).(\lambda (_: (pr0 v0 v3)).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (pr0 -t5 t6)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) -v1 (THead (Bind Abst) u t3)) t7)) H4 (THead (Flat Appl) v0 (THead (Bind b) u1 -t5)) H13) in (let H21 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 -| (THead _ t7 _) \Rightarrow t7])) (THead (Flat Appl) v1 (THead (Bind Abst) u -t3)) (THead (Flat Appl) v0 (THead (Bind b) u1 t5)) H20) in ((let H22 \def -(f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with -[(TSort _) \Rightarrow Abst | (TLRef _) \Rightarrow Abst | (THead _ _ t7) -\Rightarrow (match t7 in T return (\lambda (_: T).B) with [(TSort _) -\Rightarrow Abst | (TLRef _) \Rightarrow Abst | (THead k _ _) \Rightarrow -(match k in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | -(Flat _) \Rightarrow Abst])])])) (THead (Flat Appl) v1 (THead (Bind Abst) u -t3)) (THead (Flat Appl) v0 (THead (Bind b) u1 t5)) H20) in ((let H23 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ _ t7) -\Rightarrow (match t7 in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t8 _) \Rightarrow t8])])) -(THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (THead (Flat Appl) v0 (THead -(Bind b) u1 t5)) H20) in ((let H24 \def (f_equal T T (\lambda (e: T).(match e -in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) -\Rightarrow t3 | (THead _ _ t7) \Rightarrow (match t7 in T return (\lambda -(_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead -_ _ t8) \Rightarrow t8])])) (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) -(THead (Flat Appl) v0 (THead (Bind b) u1 t5)) H20) in (\lambda (_: (eq T u -u1)).(\lambda (H26: (eq B Abst b)).(\lambda (_: (eq T v1 v0)).(eq_ind B Abst -(\lambda (b0: B).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) -(\lambda (t7: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O -v3) t6)) t7)))) (let H28 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 -Abst))) H16 Abst H26) in (let H29 \def (match (H28 (refl_equal B Abst)) in -False return (\lambda (_: False).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind -Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abst) u2 (THead (Flat -Appl) (lift (S O) O v3) t6)) t7)))) with []) in H29)) b H26))))) H23)) H22)) -H21))))))) t2 H15)) t H13 H14 H9 H10 H11 H12))) | (pr0_delta u1 u2 H9 t5 t6 -H10 w H11) \Rightarrow (\lambda (H12: (eq T (THead (Bind Abbr) u1 t5) -t)).(\lambda (H13: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind -Abbr) u1 t5) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t2) \to ((pr0 u1 -u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T (\lambda (t8: T).(pr0 -(THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8)))))))) (\lambda -(H14: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u2 w) -(\lambda (t7: T).((pr0 u1 u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to -(ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: -T).(pr0 t7 t8))))))) (\lambda (_: (pr0 u1 u2)).(\lambda (_: (pr0 t5 -t6)).(\lambda (_: (subst0 O u2 t6 w)).(let H18 \def (eq_ind_r T t (\lambda -(t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) t7)) H4 (THead -(Bind Abbr) u1 t5) H12) in (let H19 \def (eq_ind T (THead (Flat Appl) v1 -(THead (Bind Abst) u t3)) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind -Abbr) u1 t5) H18) in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind -Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7))) -H19)))))) t2 H14)) t H12 H13 H9 H10 H11))) | (pr0_zeta b H9 t5 t6 H10 u0) -\Rightarrow (\lambda (H11: (eq T (THead (Bind b) u0 (lift (S O) O t5)) -t)).(\lambda (H12: (eq T t6 t2)).(eq_ind T (THead (Bind b) u0 (lift (S O) O -t5)) (\lambda (_: T).((eq T t6 t2) \to ((not (eq B b Abst)) \to ((pr0 t5 t6) -\to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: -T).(pr0 t2 t8))))))) (\lambda (H13: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: -T).((not (eq B b Abst)) \to ((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 -(THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (_: -(not (eq B b Abst))).(\lambda (_: (pr0 t5 t2)).(let H16 \def (eq_ind_r T t -(\lambda (t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) t7)) -H4 (THead (Bind b) u0 (lift (S O) O t5)) H11) in (let H17 \def (eq_ind T -(THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (\lambda (ee: T).(match ee in -T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind b) u0 (lift (S O) O t5)) H16) in (False_ind (ex2 T -(\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 t2 -t7))) H17))))) t6 (sym_eq T t6 t2 H13))) t H11 H12 H9 H10))) | (pr0_tau t5 t6 -H9 u0) \Rightarrow (\lambda (H10: (eq T (THead (Flat Cast) u0 t5) -t)).(\lambda (H11: (eq T t6 t2)).(eq_ind T (THead (Flat Cast) u0 t5) (\lambda -(_: T).((eq T t6 t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead -(Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H12: (eq -T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8: -T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))) -(\lambda (_: (pr0 t5 t2)).(let H14 \def (eq_ind_r T t (\lambda (t7: T).(eq T -(THead (Flat Appl) v1 (THead (Bind Abst) u t3)) t7)) H4 (THead (Flat Cast) u0 -t5) H10) in (let H15 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) -u t3)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) -\Rightarrow False | (Flat f) \Rightarrow (match f in F return (\lambda (_: -F).Prop) with [Appl \Rightarrow True | Cast \Rightarrow False])])])) I (THead -(Flat Cast) u0 t5) H14) in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead -(Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 t2 t7))) H15)))) t6 (sym_eq T -t6 t2 H12))) t H10 H11 H9)))]) in (H9 (refl_equal T t) (refl_equal T t2))))) -t1 H6)) t H4 H5 H2 H3))) | (pr0_upsilon b H2 v1 v2 H3 u1 u2 H4 t3 t4 H5) -\Rightarrow (\lambda (H6: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) -t)).(\lambda (H7: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O -v2) t4)) t1)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) -(\lambda (_: T).((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O -v2) t4)) t1) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to -((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 -t2 t6))))))))) (\lambda (H8: (eq T (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4)) t1)).(eq_ind T (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4)) (\lambda (t5: T).((not (eq B b Abst)) \to ((pr0 v1 v2) -\to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t5 t6)) -(\lambda (t6: T).(pr0 t2 t6)))))))) (\lambda (H9: (not (eq B b -Abst))).(\lambda (H10: (pr0 v1 v2)).(\lambda (H11: (pr0 u1 u2)).(\lambda -(H12: (pr0 t3 t4)).(let H13 \def (match H1 in pr0 return (\lambda (t5: -T).(\lambda (t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 t) \to ((eq T t6 t2) -\to (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift -(S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 t2 t7)))))))) with [(pr0_refl t5) -\Rightarrow (\lambda (H13: (eq T t5 t)).(\lambda (H14: (eq T t5 t2)).(eq_ind -T t (\lambda (t6: T).((eq T t6 t2) \to (ex2 T (\lambda (t7: T).(pr0 (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: -T).(pr0 t2 t7))))) (\lambda (H15: (eq T t t2)).(eq_ind T t2 (\lambda (_: -T).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift -(S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let H16 \def (eq_ind_r -T t (\lambda (t6: T).(eq T t6 t2)) H15 (THead (Flat Appl) v1 (THead (Bind b) -u1 t3)) H6) in (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) -(\lambda (t6: T).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 t6 t7)))) (let H17 -\def (eq_ind_r T t (\lambda (t6: T).(eq T t5 t6)) H13 (THead (Flat Appl) v1 -(THead (Bind b) u1 t3)) H6) in (let H18 \def (eq_ind_r T t (\lambda (t6: -T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall -(t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: -T).(pr0 t8 t9)))))))))) H (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) H6) -in (ex2_sym T (pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 t3))) (pr0 (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) -(pr0_confluence__pr0_cong_upsilon_refl b H9 u1 u2 H11 t3 t4 H12 v1 v2 v2 H10 -(pr0_refl v2))))) t2 H16)) t (sym_eq T t t2 H15))) t5 (sym_eq T t5 t H13) -H14))) | (pr0_comp u0 u3 H13 t5 t6 H14 k) \Rightarrow (\lambda (H15: (eq T -(THead k u0 t5) t)).(\lambda (H16: (eq T (THead k u3 t6) t2)).(eq_ind T -(THead k u0 t5) (\lambda (_: T).((eq T (THead k u3 t6) t2) \to ((pr0 u0 u3) -\to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) -(\lambda (H17: (eq T (THead k u3 t6) t2)).(eq_ind T (THead k u3 t6) (\lambda -(t7: T).((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: -T).(pr0 t7 t8)))))) (\lambda (H18: (pr0 u0 u3)).(\lambda (H19: (pr0 t5 -t6)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1 -(THead (Bind b) u1 t3)) t7)) H6 (THead k u0 t5) H15) in (let H21 \def -(f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with -[(TSort _) \Rightarrow (Flat Appl) | (TLRef _) \Rightarrow (Flat Appl) | -(THead k0 _ _) \Rightarrow k0])) (THead (Flat Appl) v1 (THead (Bind b) u1 -t3)) (THead k u0 t5) H20) in ((let H22 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | -(TLRef _) \Rightarrow v1 | (THead _ t7 _) \Rightarrow t7])) (THead (Flat -Appl) v1 (THead (Bind b) u1 t3)) (THead k u0 t5) H20) in ((let H23 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow (THead (Bind b) u1 t3) | (TLRef _) \Rightarrow (THead -(Bind b) u1 t3) | (THead _ _ t7) \Rightarrow t7])) (THead (Flat Appl) v1 -(THead (Bind b) u1 t3)) (THead k u0 t5) H20) in (\lambda (H24: (eq T v1 -u0)).(\lambda (H25: (eq K (Flat Appl) k)).(eq_ind K (Flat Appl) (\lambda (k0: -K).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift -(S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 (THead k0 u3 t6) t7)))) (let H26 -\def (eq_ind_r K k (\lambda (k0: K).(eq T (THead k0 u0 t5) t)) H15 (Flat -Appl) H25) in (let H27 \def (eq_ind_r T t5 (\lambda (t7: T).(pr0 t7 t6)) H19 -(THead (Bind b) u1 t3) H23) in (let H28 \def (match H27 in pr0 return -(\lambda (t7: T).(\lambda (t8: T).(\lambda (_: (pr0 t7 t8)).((eq T t7 (THead -(Bind b) u1 t3)) \to ((eq T t8 t6) \to (ex2 T (\lambda (t9: T).(pr0 (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: -T).(pr0 (THead (Flat Appl) u3 t6) t9)))))))) with [(pr0_refl t7) \Rightarrow -(\lambda (H28: (eq T t7 (THead (Bind b) u1 t3))).(\lambda (H29: (eq T t7 -t6)).(eq_ind T (THead (Bind b) u1 t3) (\lambda (t8: T).((eq T t8 t6) \to (ex2 -T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O -v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9))))) -(\lambda (H30: (eq T (THead (Bind b) u1 t3) t6)).(eq_ind T (THead (Bind b) u1 -t3) (\lambda (t8: T).(ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat -Appl) u3 t8) t9)))) (let H31 \def (eq_ind_r T t5 (\lambda (t8: T).(eq T -(THead (Flat Appl) u0 t8) t)) H26 (THead (Bind b) u1 t3) H23) in (let H32 -\def (eq_ind_r T t (\lambda (t8: T).(\forall (v: T).((tlt v t8) \to (\forall -(t9: T).((pr0 v t9) \to (\forall (t10: T).((pr0 v t10) \to (ex2 T (\lambda -(t11: T).(pr0 t9 t11)) (\lambda (t11: T).(pr0 t10 t11)))))))))) H (THead -(Flat Appl) u0 (THead (Bind b) u1 t3)) H31) in (let H33 \def (eq_ind T v1 -(\lambda (t8: T).(pr0 t8 v2)) H10 u0 H24) in (ex2_ind T (\lambda (t8: T).(pr0 -v2 t8)) (\lambda (t8: T).(pr0 u3 t8)) (ex2 T (\lambda (t8: T).(pr0 (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: -T).(pr0 (THead (Flat Appl) u3 (THead (Bind b) u1 t3)) t8))) (\lambda (x: -T).(\lambda (H34: (pr0 v2 x)).(\lambda (H35: (pr0 u3 x)).(ex2_sym T (pr0 -(THead (Flat Appl) u3 (THead (Bind b) u1 t3))) (pr0 (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t4))) (pr0_confluence__pr0_cong_upsilon_refl b -H9 u1 u2 H11 t3 t4 H12 u3 v2 x H35 H34))))) (H32 u0 (tlt_head_sx (Flat Appl) -u0 (THead (Bind b) u1 t3)) v2 H33 u3 H18))))) t6 H30)) t7 (sym_eq T t7 (THead -(Bind b) u1 t3) H28) H29))) | (pr0_comp u4 u5 H28 t7 t8 H29 k0) \Rightarrow -(\lambda (H30: (eq T (THead k0 u4 t7) (THead (Bind b) u1 t3))).(\lambda (H31: -(eq T (THead k0 u5 t8) t6)).((let H32 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t7 | -(TLRef _) \Rightarrow t7 | (THead _ _ t9) \Rightarrow t9])) (THead k0 u4 t7) -(THead (Bind b) u1 t3) H30) in ((let H33 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u4 | -(TLRef _) \Rightarrow u4 | (THead _ t9 _) \Rightarrow t9])) (THead k0 u4 t7) -(THead (Bind b) u1 t3) H30) in ((let H34 \def (f_equal T K (\lambda (e: -T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k0 | -(TLRef _) \Rightarrow k0 | (THead k1 _ _) \Rightarrow k1])) (THead k0 u4 t7) -(THead (Bind b) u1 t3) H30) in (eq_ind K (Bind b) (\lambda (k1: K).((eq T u4 -u1) \to ((eq T t7 t3) \to ((eq T (THead k1 u5 t8) t6) \to ((pr0 u4 u5) \to -((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 -t6) t9))))))))) (\lambda (H35: (eq T u4 u1)).(eq_ind T u1 (\lambda (t9: -T).((eq T t7 t3) \to ((eq T (THead (Bind b) u5 t8) t6) \to ((pr0 t9 u5) \to -((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat -Appl) u3 t6) t10)))))))) (\lambda (H36: (eq T t7 t3)).(eq_ind T t3 (\lambda -(t9: T).((eq T (THead (Bind b) u5 t8) t6) \to ((pr0 u1 u5) \to ((pr0 t9 t8) -\to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift -(S O) O v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6) -t10))))))) (\lambda (H37: (eq T (THead (Bind b) u5 t8) t6)).(eq_ind T (THead -(Bind b) u5 t8) (\lambda (t9: T).((pr0 u1 u5) \to ((pr0 t3 t8) \to (ex2 T -(\lambda (t10: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O -v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t9) t10)))))) -(\lambda (H38: (pr0 u1 u5)).(\lambda (H39: (pr0 t3 t8)).(let H40 \def -(eq_ind_r T t5 (\lambda (t9: T).(eq T (THead (Flat Appl) u0 t9) t)) H26 -(THead (Bind b) u1 t3) H23) in (let H41 \def (eq_ind_r T t (\lambda (t9: -T).(\forall (v: T).((tlt v t9) \to (\forall (t10: T).((pr0 v t10) \to -(\forall (t11: T).((pr0 v t11) \to (ex2 T (\lambda (t12: T).(pr0 t10 t12)) -(\lambda (t12: T).(pr0 t11 t12)))))))))) H (THead (Flat Appl) u0 (THead (Bind -b) u1 t3)) H40) in (let H42 \def (eq_ind T v1 (\lambda (t9: T).(pr0 t9 v2)) -H10 u0 H24) in (ex2_ind T (\lambda (t9: T).(pr0 v2 t9)) (\lambda (t9: T).(pr0 -u3 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 -(THead (Bind b) u5 t8)) t9))) (\lambda (x: T).(\lambda (H43: (pr0 v2 -x)).(\lambda (H44: (pr0 u3 x)).(ex2_ind T (\lambda (t9: T).(pr0 t8 t9)) -(\lambda (t9: T).(pr0 t4 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead -(Flat Appl) u3 (THead (Bind b) u5 t8)) t9))) (\lambda (x0: T).(\lambda (H45: -(pr0 t8 x0)).(\lambda (H46: (pr0 t4 x0)).(ex2_ind T (\lambda (t9: T).(pr0 u5 -t9)) (\lambda (t9: T).(pr0 u2 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Bind -b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 -(THead (Flat Appl) u3 (THead (Bind b) u5 t8)) t9))) (\lambda (x1: T).(\lambda -(H47: (pr0 u5 x1)).(\lambda (H48: (pr0 u2 x1)).(ex2_sym T (pr0 (THead (Flat -Appl) u3 (THead (Bind b) u5 t8))) (pr0 (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4))) (pr0_confluence__pr0_cong_upsilon_cong b H9 u3 v2 x -H44 H43 t8 t4 x0 H45 H46 u5 u2 x1 H47 H48))))) (H41 u1 (tlt_trans (THead -(Bind b) u1 t3) u1 (THead (Flat Appl) u0 (THead (Bind b) u1 t3)) (tlt_head_sx -(Bind b) u1 t3) (tlt_head_dx (Flat Appl) u0 (THead (Bind b) u1 t3))) u5 H38 -u2 H11))))) (H41 t3 (tlt_trans (THead (Bind b) u1 t3) t3 (THead (Flat Appl) -u0 (THead (Bind b) u1 t3)) (tlt_head_dx (Bind b) u1 t3) (tlt_head_dx (Flat -Appl) u0 (THead (Bind b) u1 t3))) t8 H39 t4 H12))))) (H41 u0 (tlt_head_sx -(Flat Appl) u0 (THead (Bind b) u1 t3)) v2 H42 u3 H18))))))) t6 H37)) t7 -(sym_eq T t7 t3 H36))) u4 (sym_eq T u4 u1 H35))) k0 (sym_eq K k0 (Bind b) -H34))) H33)) H32)) H31 H28 H29))) | (pr0_beta u v0 v3 H28 t7 t8 H29) -\Rightarrow (\lambda (H30: (eq T (THead (Flat Appl) v0 (THead (Bind Abst) u -t7)) (THead (Bind b) u1 t3))).(\lambda (H31: (eq T (THead (Bind Abbr) v3 t8) -t6)).((let H32 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u t7)) -(\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow -(match k0 in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False -| (Flat _) \Rightarrow True])])) I (THead (Bind b) u1 t3) H30) in (False_ind -((eq T (THead (Bind Abbr) v3 t8) t6) \to ((pr0 v0 v3) \to ((pr0 t7 t8) \to -(ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9)))))) -H32)) H31 H28 H29))) | (pr0_upsilon b0 H28 v0 v3 H29 u4 u5 H30 t7 t8 H31) -\Rightarrow (\lambda (H32: (eq T (THead (Flat Appl) v0 (THead (Bind b0) u4 -t7)) (THead (Bind b) u1 t3))).(\lambda (H33: (eq T (THead (Bind b0) u5 (THead -(Flat Appl) (lift (S O) O v3) t8)) t6)).((let H34 \def (eq_ind T (THead (Flat -Appl) v0 (THead (Bind b0) u4 t7)) (\lambda (e: T).(match e in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind b) u1 t3) H32) in (False_ind ((eq T (THead (Bind b0) -u5 (THead (Flat Appl) (lift (S O) O v3) t8)) t6) \to ((not (eq B b0 Abst)) -\to ((pr0 v0 v3) \to ((pr0 u4 u5) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: -T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) -(\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9)))))))) H34)) H33 H28 H29 -H30 H31))) | (pr0_delta u4 u5 H28 t7 t8 H29 w H30) \Rightarrow (\lambda (H31: -(eq T (THead (Bind Abbr) u4 t7) (THead (Bind b) u1 t3))).(\lambda (H32: (eq T -(THead (Bind Abbr) u5 w) t6)).((let H33 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t7 | -(TLRef _) \Rightarrow t7 | (THead _ _ t9) \Rightarrow t9])) (THead (Bind -Abbr) u4 t7) (THead (Bind b) u1 t3) H31) in ((let H34 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow u4 | (TLRef _) \Rightarrow u4 | (THead _ t9 _) \Rightarrow t9])) -(THead (Bind Abbr) u4 t7) (THead (Bind b) u1 t3) H31) in ((let H35 \def -(f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with -[(TSort _) \Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | (THead k0 _ _) -\Rightarrow (match k0 in K return (\lambda (_: K).B) with [(Bind b0) -\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (THead (Bind Abbr) u4 t7) -(THead (Bind b) u1 t3) H31) in (eq_ind B Abbr (\lambda (b0: B).((eq T u4 u1) -\to ((eq T t7 t3) \to ((eq T (THead (Bind Abbr) u5 w) t6) \to ((pr0 u4 u5) -\to ((pr0 t7 t8) \to ((subst0 O u5 t8 w) \to (ex2 T (\lambda (t9: T).(pr0 -(THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda -(t9: T).(pr0 (THead (Flat Appl) u3 t6) t9)))))))))) (\lambda (H36: (eq T u4 -u1)).(eq_ind T u1 (\lambda (t9: T).((eq T t7 t3) \to ((eq T (THead (Bind -Abbr) u5 w) t6) \to ((pr0 t9 u5) \to ((pr0 t7 t8) \to ((subst0 O u5 t8 w) \to -(ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift -(S O) O v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6) -t10))))))))) (\lambda (H37: (eq T t7 t3)).(eq_ind T t3 (\lambda (t9: T).((eq -T (THead (Bind Abbr) u5 w) t6) \to ((pr0 u1 u5) \to ((pr0 t9 t8) \to ((subst0 -O u5 t8 w) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) u2 (THead -(Flat Appl) (lift (S O) O v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat -Appl) u3 t6) t10)))))))) (\lambda (H38: (eq T (THead (Bind Abbr) u5 w) -t6)).(eq_ind T (THead (Bind Abbr) u5 w) (\lambda (t9: T).((pr0 u1 u5) \to -((pr0 t3 t8) \to ((subst0 O u5 t8 w) \to (ex2 T (\lambda (t10: T).(pr0 (THead -(Bind Abbr) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t10)) (\lambda (t10: -T).(pr0 (THead (Flat Appl) u3 t9) t10))))))) (\lambda (H39: (pr0 u1 -u5)).(\lambda (H40: (pr0 t3 t8)).(\lambda (H41: (subst0 O u5 t8 w)).(let H42 -\def (eq_ind_r B b (\lambda (b0: B).(eq T (THead (Bind b0) u1 t3) t5)) H23 -Abbr H35) in (let H43 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 -Abst))) H9 Abbr H35) in (let H44 \def (eq_ind_r B b (\lambda (b0: B).(eq T -(THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t1)) H8 Abbr -H35) in (let H45 \def (eq_ind_r T t5 (\lambda (t9: T).(eq T (THead (Flat -Appl) u0 t9) t)) H26 (THead (Bind Abbr) u1 t3) H42) in (let H46 \def -(eq_ind_r T t (\lambda (t9: T).(\forall (v: T).((tlt v t9) \to (\forall (t10: -T).((pr0 v t10) \to (\forall (t11: T).((pr0 v t11) \to (ex2 T (\lambda (t12: -T).(pr0 t10 t12)) (\lambda (t12: T).(pr0 t11 t12)))))))))) H (THead (Flat -Appl) u0 (THead (Bind Abbr) u1 t3)) H45) in (let H47 \def (eq_ind T v1 -(\lambda (t9: T).(pr0 t9 v2)) H10 u0 H24) in (ex2_ind T (\lambda (t9: T).(pr0 -v2 t9)) (\lambda (t9: T).(pr0 u3 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead -(Bind Abbr) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: -T).(pr0 (THead (Flat Appl) u3 (THead (Bind Abbr) u5 w)) t9))) (\lambda (x: -T).(\lambda (H48: (pr0 v2 x)).(\lambda (H49: (pr0 u3 x)).(ex2_ind T (\lambda -(t9: T).(pr0 t8 t9)) (\lambda (t9: T).(pr0 t4 t9)) (ex2 T (\lambda (t9: -T).(pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) -(\lambda (t9: T).(pr0 (THead (Flat Appl) u3 (THead (Bind Abbr) u5 w)) t9))) -(\lambda (x0: T).(\lambda (H50: (pr0 t8 x0)).(\lambda (H51: (pr0 t4 -x0)).(ex2_ind T (\lambda (t9: T).(pr0 u5 t9)) (\lambda (t9: T).(pr0 u2 t9)) -(ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift -(S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 (THead -(Bind Abbr) u5 w)) t9))) (\lambda (x1: T).(\lambda (H52: (pr0 u5 -x1)).(\lambda (H53: (pr0 u2 x1)).(ex2_sym T (pr0 (THead (Flat Appl) u3 (THead -(Bind Abbr) u5 w))) (pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift (S O) -O v2) t4))) (pr0_confluence__pr0_cong_upsilon_delta H43 u5 t8 w H41 u3 v2 x -H49 H48 t4 x0 H50 H51 u2 x1 H52 H53))))) (H46 u1 (tlt_trans (THead (Bind -Abbr) u1 t3) u1 (THead (Flat Appl) u0 (THead (Bind Abbr) u1 t3)) (tlt_head_sx -(Bind Abbr) u1 t3) (tlt_head_dx (Flat Appl) u0 (THead (Bind Abbr) u1 t3))) u5 -H39 u2 H11))))) (H46 t3 (tlt_trans (THead (Bind Abbr) u1 t3) t3 (THead (Flat -Appl) u0 (THead (Bind Abbr) u1 t3)) (tlt_head_dx (Bind Abbr) u1 t3) -(tlt_head_dx (Flat Appl) u0 (THead (Bind Abbr) u1 t3))) t8 H40 t4 H12))))) -(H46 u0 (tlt_head_sx (Flat Appl) u0 (THead (Bind Abbr) u1 t3)) v2 H47 u3 -H18))))))))))) t6 H38)) t7 (sym_eq T t7 t3 H37))) u4 (sym_eq T u4 u1 H36))) b -H35)) H34)) H33)) H32 H28 H29 H30))) | (pr0_zeta b0 H28 t7 t8 H29 u) -\Rightarrow (\lambda (H30: (eq T (THead (Bind b0) u (lift (S O) O t7)) (THead -(Bind b) u1 t3))).(\lambda (H31: (eq T t8 t6)).((let H32 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t9: T) on t9: T -\def (match t9 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow -(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) -| (THead k0 u4 t10) \Rightarrow (THead k0 (lref_map f d u4) (lref_map f (s k0 -d) t10))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (TLRef _) -\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t9: T) on t9: T -\def (match t9 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow -(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) -| (THead k0 u4 t10) \Rightarrow (THead k0 (lref_map f d u4) (lref_map f (s k0 -d) t10))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (THead _ _ -t9) \Rightarrow t9])) (THead (Bind b0) u (lift (S O) O t7)) (THead (Bind b) -u1 t3) H30) in ((let H33 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) -\Rightarrow u | (THead _ t9 _) \Rightarrow t9])) (THead (Bind b0) u (lift (S -O) O t7)) (THead (Bind b) u1 t3) H30) in ((let H34 \def (f_equal T B (\lambda -(e: T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b0 -| (TLRef _) \Rightarrow b0 | (THead k0 _ _) \Rightarrow (match k0 in K return -(\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow -b0])])) (THead (Bind b0) u (lift (S O) O t7)) (THead (Bind b) u1 t3) H30) in -(eq_ind B b (\lambda (b1: B).((eq T u u1) \to ((eq T (lift (S O) O t7) t3) -\to ((eq T t8 t6) \to ((not (eq B b1 Abst)) \to ((pr0 t7 t8) \to (ex2 T -(\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) -t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9))))))))) -(\lambda (H35: (eq T u u1)).(eq_ind T u1 (\lambda (_: T).((eq T (lift (S O) O -t7) t3) \to ((eq T t8 t6) \to ((not (eq B b Abst)) \to ((pr0 t7 t8) \to (ex2 -T (\lambda (t10: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O -v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6) t10)))))))) -(\lambda (H36: (eq T (lift (S O) O t7) t3)).(eq_ind T (lift (S O) O t7) -(\lambda (_: T).((eq T t8 t6) \to ((not (eq B b Abst)) \to ((pr0 t7 t8) \to -(ex2 T (\lambda (t10: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6) -t10))))))) (\lambda (H37: (eq T t8 t6)).(eq_ind T t6 (\lambda (t9: T).((not -(eq B b Abst)) \to ((pr0 t7 t9) \to (ex2 T (\lambda (t10: T).(pr0 (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t10)) (\lambda (t10: -T).(pr0 (THead (Flat Appl) u3 t6) t10)))))) (\lambda (H38: (not (eq B b -Abst))).(\lambda (H39: (pr0 t7 t6)).(let H40 \def (eq_ind_r T t3 (\lambda -(t9: T).(eq T (THead (Bind b) u1 t9) t5)) H23 (lift (S O) O t7) H36) in (let -H41 \def (eq_ind_r T t5 (\lambda (t9: T).(eq T (THead (Flat Appl) u0 t9) t)) -H26 (THead (Bind b) u1 (lift (S O) O t7)) H40) in (let H42 \def (eq_ind_r T t -(\lambda (t9: T).(\forall (v: T).((tlt v t9) \to (\forall (t10: T).((pr0 v -t10) \to (\forall (t11: T).((pr0 v t11) \to (ex2 T (\lambda (t12: T).(pr0 t10 -t12)) (\lambda (t12: T).(pr0 t11 t12)))))))))) H (THead (Flat Appl) u0 (THead -(Bind b) u1 (lift (S O) O t7))) H41) in (let H43 \def (eq_ind_r T t3 (\lambda -(t9: T).(pr0 t9 t4)) H12 (lift (S O) O t7) H36) in (ex2_ind T (\lambda (t9: -T).(eq T t4 (lift (S O) O t9))) (\lambda (t9: T).(pr0 t7 t9)) (ex2 T (\lambda -(t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9))) (\lambda (x: -T).(\lambda (H44: (eq T t4 (lift (S O) O x))).(\lambda (H45: (pr0 t7 -x)).(eq_ind_r T (lift (S O) O x) (\lambda (t9: T).(ex2 T (\lambda (t10: -T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t9)) t10)) -(\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6) t10)))) (let H46 \def -(eq_ind T v1 (\lambda (t9: T).(pr0 t9 v2)) H10 u0 H24) in (ex2_ind T (\lambda -(t9: T).(pr0 v2 t9)) (\lambda (t9: T).(pr0 u3 t9)) (ex2 T (\lambda (t9: -T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (lift (S O) O -x))) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9))) (\lambda (x0: -T).(\lambda (H47: (pr0 v2 x0)).(\lambda (H48: (pr0 u3 x0)).(ex2_ind T -(\lambda (t9: T).(pr0 x t9)) (\lambda (t9: T).(pr0 t6 t9)) (ex2 T (\lambda -(t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (lift (S -O) O x))) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9))) (\lambda -(x1: T).(\lambda (H49: (pr0 x x1)).(\lambda (H50: (pr0 t6 x1)).(ex2_sym T -(pr0 (THead (Flat Appl) u3 t6)) (pr0 (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) (lift (S O) O x)))) (pr0_confluence__pr0_cong_upsilon_zeta -b H38 u1 u2 H11 u3 v2 x0 H48 H47 x t6 x1 H49 H50))))) (H42 t7 (tlt_trans -(THead (Bind b) u1 (lift (S O) O t7)) t7 (THead (Flat Appl) u0 (THead (Bind -b) u1 (lift (S O) O t7))) (lift_tlt_dx (Bind b) u1 t7 (S O) O) (tlt_head_dx -(Flat Appl) u0 (THead (Bind b) u1 (lift (S O) O t7)))) x H45 t6 H39))))) (H42 -u0 (tlt_head_sx (Flat Appl) u0 (THead (Bind b) u1 (lift (S O) O t7))) v2 H46 -u3 H18))) t4 H44)))) (pr0_gen_lift t7 t4 (S O) O H43)))))))) t8 (sym_eq T t8 -t6 H37))) t3 H36)) u (sym_eq T u u1 H35))) b0 (sym_eq B b0 b H34))) H33)) -H32)) H31 H28 H29))) | (pr0_tau t7 t8 H28 u) \Rightarrow (\lambda (H29: (eq T -(THead (Flat Cast) u t7) (THead (Bind b) u1 t3))).(\lambda (H30: (eq T t8 -t6)).((let H31 \def (eq_ind T (THead (Flat Cast) u t7) (\lambda (e: T).(match -e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | -(TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K -return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) -\Rightarrow True])])) I (THead (Bind b) u1 t3) H29) in (False_ind ((eq T t8 -t6) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead -(Flat Appl) u3 t6) t9))))) H31)) H30 H28)))]) in (H28 (refl_equal T (THead -(Bind b) u1 t3)) (refl_equal T t6))))) k H25)))) H22)) H21))))) t2 H17)) t -H15 H16 H13 H14))) | (pr0_beta u v0 v3 H13 t5 t6 H14) \Rightarrow (\lambda -(H15: (eq T (THead (Flat Appl) v0 (THead (Bind Abst) u t5)) t)).(\lambda -(H16: (eq T (THead (Bind Abbr) v3 t6) t2)).(eq_ind T (THead (Flat Appl) v0 -(THead (Bind Abst) u t5)) (\lambda (_: T).((eq T (THead (Bind Abbr) v3 t6) -t2) \to ((pr0 v0 v3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: -T).(pr0 t2 t8))))))) (\lambda (H17: (eq T (THead (Bind Abbr) v3 t6) -t2)).(eq_ind T (THead (Bind Abbr) v3 t6) (\lambda (t7: T).((pr0 v0 v3) \to -((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda -(_: (pr0 v0 v3)).(\lambda (_: (pr0 t5 t6)).(let H20 \def (eq_ind_r T t -(\lambda (t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) t7)) H6 -(THead (Flat Appl) v0 (THead (Bind Abst) u t5)) H15) in (let H21 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t7 _) -\Rightarrow t7])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat -Appl) v0 (THead (Bind Abst) u t5)) H20) in ((let H22 \def (f_equal T B -(\lambda (e: T).(match e in T return (\lambda (_: T).B) with [(TSort _) -\Rightarrow b | (TLRef _) \Rightarrow b | (THead _ _ t7) \Rightarrow (match -t7 in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | (TLRef _) -\Rightarrow b | (THead k _ _) \Rightarrow (match k in K return (\lambda (_: -K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow b])])])) (THead -(Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) v0 (THead (Bind -Abst) u t5)) H20) in ((let H23 \def (f_equal T T (\lambda (e: T).(match e in -T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) -\Rightarrow u1 | (THead _ _ t7) \Rightarrow (match t7 in T return (\lambda -(_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead -_ t8 _) \Rightarrow t8])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) -(THead (Flat Appl) v0 (THead (Bind Abst) u t5)) H20) in ((let H24 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7) -\Rightarrow (match t7 in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t8) \Rightarrow -t8])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) v0 -(THead (Bind Abst) u t5)) H20) in (\lambda (_: (eq T u1 u)).(\lambda (H26: -(eq B b Abst)).(\lambda (H27: (eq T v1 v0)).(let H28 \def (eq_ind_r T t -(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) -\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) -(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Appl) v0 (THead (Bind -Abst) u t5)) H15) in (let H29 \def (eq_ind T v1 (\lambda (t7: T).(pr0 t7 v2)) -H10 v0 H27) in (eq_ind_r B Abst (\lambda (b0: B).(ex2 T (\lambda (t7: T).(pr0 -(THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda -(t7: T).(pr0 (THead (Bind Abbr) v3 t6) t7)))) (let H30 \def (eq_ind B b -(\lambda (b0: B).(not (eq B b0 Abst))) H9 Abst H26) in (let H31 \def (match -(H30 (refl_equal B Abst)) in False return (\lambda (_: False).(ex2 T (\lambda -(t7: T).(pr0 (THead (Bind Abst) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v3 t6) t7)))) with []) in H31)) -b H26))))))) H23)) H22)) H21))))) t2 H17)) t H15 H16 H13 H14))) | -(pr0_upsilon b0 H13 v0 v3 H14 u0 u3 H15 t5 t6 H16) \Rightarrow (\lambda (H17: -(eq T (THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) t)).(\lambda (H18: (eq T -(THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) t2)).(eq_ind T -(THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) (\lambda (_: T).((eq T (THead -(Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) t2) \to ((not (eq B b0 -Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda -(t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -t8)) (\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H19: (eq T (THead (Bind -b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) t2)).(eq_ind T (THead (Bind -b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) (\lambda (t7: T).((not (eq B -b0 Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T -(\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) -t4)) t8)) (\lambda (t8: T).(pr0 t7 t8)))))))) (\lambda (_: (not (eq B b0 -Abst))).(\lambda (H21: (pr0 v0 v3)).(\lambda (H22: (pr0 u0 u3)).(\lambda -(H23: (pr0 t5 t6)).(let H24 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead -(Flat Appl) v1 (THead (Bind b) u1 t3)) t7)) H6 (THead (Flat Appl) v0 (THead -(Bind b0) u0 t5)) H17) in (let H25 \def (f_equal T T (\lambda (e: T).(match e -in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) -\Rightarrow v1 | (THead _ t7 _) \Rightarrow t7])) (THead (Flat Appl) v1 -(THead (Bind b) u1 t3)) (THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) H24) -in ((let H26 \def (f_equal T B (\lambda (e: T).(match e in T return (\lambda -(_: T).B) with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead _ -_ t7) \Rightarrow (match t7 in T return (\lambda (_: T).B) with [(TSort _) -\Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k -in K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) -\Rightarrow b])])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead -(Flat Appl) v0 (THead (Bind b0) u0 t5)) H24) in ((let H27 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ _ t7) \Rightarrow (match -t7 in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) -\Rightarrow u1 | (THead _ t8 _) \Rightarrow t8])])) (THead (Flat Appl) v1 -(THead (Bind b) u1 t3)) (THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) H24) -in ((let H28 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda -(_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead -_ _ t7) \Rightarrow (match t7 in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t8) \Rightarrow -t8])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) v0 -(THead (Bind b0) u0 t5)) H24) in (\lambda (H29: (eq T u1 u0)).(\lambda (H30: -(eq B b b0)).(\lambda (H31: (eq T v1 v0)).(let H32 \def (eq_ind_r T t -(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) -\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) -(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Appl) v0 (THead (Bind -b0) u0 t5)) H17) in (let H33 \def (eq_ind T v1 (\lambda (t7: T).(pr0 t7 v2)) -H10 v0 H31) in (eq_ind_r B b0 (\lambda (b1: B).(ex2 T (\lambda (t7: T).(pr0 -(THead (Bind b1) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda -(t7: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) -t7)))) (let H34 \def (eq_ind B b (\lambda (b1: B).(not (eq B b1 Abst))) H9 b0 -H30) in (let H35 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H11 u0 H29) -in (let H36 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H12 t5 H28) in -(ex2_ind T (\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T -(\lambda (t7: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O -v2) t4)) t7)) (\lambda (t7: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) -(lift (S O) O v3) t6)) t7))) (\lambda (x: T).(\lambda (H37: (pr0 t4 -x)).(\lambda (H38: (pr0 t6 x)).(ex2_ind T (\lambda (t7: T).(pr0 u2 t7)) -(\lambda (t7: T).(pr0 u3 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b0) -u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 -(THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) t7))) (\lambda -(x0: T).(\lambda (H39: (pr0 u2 x0)).(\lambda (H40: (pr0 u3 x0)).(ex2_ind T -(\lambda (t7: T).(pr0 v2 t7)) (\lambda (t7: T).(pr0 v3 t7)) (ex2 T (\lambda -(t7: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -t7)) (\lambda (t7: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) -O v3) t6)) t7))) (\lambda (x1: T).(\lambda (H41: (pr0 v2 x1)).(\lambda (H42: -(pr0 v3 x1)).(pr0_confluence__pr0_upsilon_upsilon b0 H34 v2 v3 x1 H41 H42 u2 -u3 x0 H39 H40 t4 t6 x H37 H38)))) (H32 v0 (tlt_head_sx (Flat Appl) v0 (THead -(Bind b0) u0 t5)) v2 H33 v3 H21))))) (H32 u0 (tlt_trans (THead (Bind b0) u0 -t5) u0 (THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) (tlt_head_sx (Bind b0) -u0 t5) (tlt_head_dx (Flat Appl) v0 (THead (Bind b0) u0 t5))) u2 H35 u3 -H22))))) (H32 t5 (tlt_trans (THead (Bind b0) u0 t5) t5 (THead (Flat Appl) v0 -(THead (Bind b0) u0 t5)) (tlt_head_dx (Bind b0) u0 t5) (tlt_head_dx (Flat -Appl) v0 (THead (Bind b0) u0 t5))) t4 H36 t6 H23))))) b H30))))))) H27)) -H26)) H25))))))) t2 H19)) t H17 H18 H13 H14 H15 H16))) | (pr0_delta u0 u3 H13 -t5 t6 H14 w H15) \Rightarrow (\lambda (H16: (eq T (THead (Bind Abbr) u0 t5) -t)).(\lambda (H17: (eq T (THead (Bind Abbr) u3 w) t2)).(eq_ind T (THead (Bind -Abbr) u0 t5) (\lambda (_: T).((eq T (THead (Bind Abbr) u3 w) t2) \to ((pr0 u0 -u3) \to ((pr0 t5 t6) \to ((subst0 O u3 t6 w) \to (ex2 T (\lambda (t8: T).(pr0 -(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda -(t8: T).(pr0 t2 t8)))))))) (\lambda (H18: (eq T (THead (Bind Abbr) u3 w) -t2)).(eq_ind T (THead (Bind Abbr) u3 w) (\lambda (t7: T).((pr0 u0 u3) \to -((pr0 t5 t6) \to ((subst0 O u3 t6 w) \to (ex2 T (\lambda (t8: T).(pr0 (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: -T).(pr0 t7 t8))))))) (\lambda (_: (pr0 u0 u3)).(\lambda (_: (pr0 t5 -t6)).(\lambda (_: (subst0 O u3 t6 w)).(let H22 \def (eq_ind_r T t (\lambda -(t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) t7)) H6 (THead -(Bind Abbr) u0 t5) H16) in (let H23 \def (eq_ind T (THead (Flat Appl) v1 -(THead (Bind b) u1 t3)) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind -Abbr) u0 t5) H22) in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 -(THead (Bind Abbr) u3 w) t7))) H23)))))) t2 H18)) t H16 H17 H13 H14 H15))) | -(pr0_zeta b0 H13 t5 t6 H14 u) \Rightarrow (\lambda (H15: (eq T (THead (Bind -b0) u (lift (S O) O t5)) t)).(\lambda (H16: (eq T t6 t2)).(eq_ind T (THead -(Bind b0) u (lift (S O) O t5)) (\lambda (_: T).((eq T t6 t2) \to ((not (eq B -b0 Abst)) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t2 -t8))))))) (\lambda (H17: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((not -(eq B b0 Abst)) \to ((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: -T).(pr0 t2 t8)))))) (\lambda (_: (not (eq B b0 Abst))).(\lambda (_: (pr0 t5 -t2)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1 -(THead (Bind b) u1 t3)) t7)) H6 (THead (Bind b0) u (lift (S O) O t5)) H15) in -(let H21 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) -(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow -(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | -(Flat _) \Rightarrow True])])) I (THead (Bind b0) u (lift (S O) O t5)) H20) -in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 t2 t7))) H21))))) t6 -(sym_eq T t6 t2 H17))) t H15 H16 H13 H14))) | (pr0_tau t5 t6 H13 u) -\Rightarrow (\lambda (H14: (eq T (THead (Flat Cast) u t5) t)).(\lambda (H15: -(eq T t6 t2)).(eq_ind T (THead (Flat Cast) u t5) (\lambda (_: T).((eq T t6 -t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t2 -t8)))))) (\lambda (H16: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5 -t7) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t2 t8))))) (\lambda (_: -(pr0 t5 t2)).(let H18 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat -Appl) v1 (THead (Bind b) u1 t3)) t7)) H6 (THead (Flat Cast) u t5) H14) in -(let H19 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) -(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow -(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | -(Flat f) \Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl -\Rightarrow True | Cast \Rightarrow False])])])) I (THead (Flat Cast) u t5) -H18) in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 t2 t7))) -H19)))) t6 (sym_eq T t6 t2 H16))) t H14 H15 H13)))]) in (H13 (refl_equal T t) -(refl_equal T t2))))))) t1 H8)) t H6 H7 H2 H3 H4 H5))) | (pr0_delta u1 u2 H2 -t3 t4 H3 w H4) \Rightarrow (\lambda (H5: (eq T (THead (Bind Abbr) u1 t3) -t)).(\lambda (H6: (eq T (THead (Bind Abbr) u2 w) t1)).(eq_ind T (THead (Bind -Abbr) u1 t3) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t1) \to ((pr0 u1 -u2) \to ((pr0 t3 t4) \to ((subst0 O u2 t4 w) \to (ex2 T (\lambda (t6: T).(pr0 -t1 t6)) (\lambda (t6: T).(pr0 t2 t6)))))))) (\lambda (H7: (eq T (THead (Bind -Abbr) u2 w) t1)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t5: T).((pr0 u1 -u2) \to ((pr0 t3 t4) \to ((subst0 O u2 t4 w) \to (ex2 T (\lambda (t6: T).(pr0 -t5 t6)) (\lambda (t6: T).(pr0 t2 t6))))))) (\lambda (H8: (pr0 u1 -u2)).(\lambda (H9: (pr0 t3 t4)).(\lambda (H10: (subst0 O u2 t4 w)).(let H11 -\def (match H1 in pr0 return (\lambda (t5: T).(\lambda (t6: T).(\lambda (_: -(pr0 t5 t6)).((eq T t5 t) \to ((eq T t6 t2) \to (ex2 T (\lambda (t7: T).(pr0 -(THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7)))))))) with -[(pr0_refl t5) \Rightarrow (\lambda (H11: (eq T t5 t)).(\lambda (H12: (eq T -t5 t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2) \to (ex2 T (\lambda (t7: -T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7))))) -(\lambda (H13: (eq T t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t7: -T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let -H14 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 t2)) H13 (THead (Bind Abbr) -u1 t3) H5) in (eq_ind T (THead (Bind Abbr) u1 t3) (\lambda (t6: T).(ex2 T -(\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t6 -t7)))) (let H15 \def (eq_ind_r T t (\lambda (t6: T).(eq T t5 t6)) H11 (THead -(Bind Abbr) u1 t3) H5) in (let H16 \def (eq_ind_r T t (\lambda (t6: -T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall -(t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: -T).(pr0 t8 t9)))))))))) H (THead (Bind Abbr) u1 t3) H5) in (ex_intro2 T -(\lambda (t6: T).(pr0 (THead (Bind Abbr) u2 w) t6)) (\lambda (t6: T).(pr0 -(THead (Bind Abbr) u1 t3) t6)) (THead (Bind Abbr) u2 w) (pr0_refl (THead -(Bind Abbr) u2 w)) (pr0_delta u1 u2 H8 t3 t4 H9 w H10)))) t2 H14)) t (sym_eq -T t t2 H13))) t5 (sym_eq T t5 t H11) H12))) | (pr0_comp u0 u3 H11 t5 t6 H12 -k) \Rightarrow (\lambda (H13: (eq T (THead k u0 t5) t)).(\lambda (H14: (eq T -(THead k u3 t6) t2)).(eq_ind T (THead k u0 t5) (\lambda (_: T).((eq T (THead -k u3 t6) t2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: -T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) -(\lambda (H15: (eq T (THead k u3 t6) t2)).(eq_ind T (THead k u3 t6) (\lambda -(t7: T).((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead -(Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (H16: (pr0 -u0 u3)).(\lambda (H17: (pr0 t5 t6)).(let H18 \def (eq_ind_r T t (\lambda (t7: -T).(eq T (THead (Bind Abbr) u1 t3) t7)) H5 (THead k u0 t5) H13) in (let H19 -\def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) -with [(TSort _) \Rightarrow (Bind Abbr) | (TLRef _) \Rightarrow (Bind Abbr) | -(THead k0 _ _) \Rightarrow k0])) (THead (Bind Abbr) u1 t3) (THead k u0 t5) -H18) in ((let H20 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 -| (THead _ t7 _) \Rightarrow t7])) (THead (Bind Abbr) u1 t3) (THead k u0 t5) -H18) in ((let H21 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 -| (THead _ _ t7) \Rightarrow t7])) (THead (Bind Abbr) u1 t3) (THead k u0 t5) -H18) in (\lambda (H22: (eq T u1 u0)).(\lambda (H23: (eq K (Bind Abbr) -k)).(eq_ind K (Bind Abbr) (\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 -(THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 (THead k0 u3 t6) t7)))) -(let H24 \def (eq_ind_r K k (\lambda (k0: K).(eq T (THead k0 u0 t5) t)) H13 -(Bind Abbr) H23) in (let H25 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: -T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v -t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 -t10)))))))))) H (THead (Bind Abbr) u0 t5) H24) in (let H26 \def (eq_ind T u1 -(\lambda (t7: T).(pr0 t7 u2)) H8 u0 H22) in (let H27 \def (eq_ind T t3 -(\lambda (t7: T).(pr0 t7 t4)) H9 t5 H21) in (ex2_ind T (\lambda (t7: T).(pr0 -t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead -(Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u3 t6) t7))) -(\lambda (x: T).(\lambda (H28: (pr0 t4 x)).(\lambda (H29: (pr0 t6 -x)).(ex2_ind T (\lambda (t7: T).(pr0 u2 t7)) (\lambda (t7: T).(pr0 u3 t7)) -(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: -T).(pr0 (THead (Bind Abbr) u3 t6) t7))) (\lambda (x0: T).(\lambda (H30: (pr0 -u2 x0)).(\lambda (H31: (pr0 u3 x0)).(ex2_sym T (pr0 (THead (Bind Abbr) u3 -t6)) (pr0 (THead (Bind Abbr) u2 w)) (pr0_confluence__pr0_cong_delta u2 t4 w -H10 u3 x0 H31 H30 t6 x H29 H28))))) (H25 u0 (tlt_head_sx (Bind Abbr) u0 t5) -u2 H26 u3 H16))))) (H25 t5 (tlt_head_dx (Bind Abbr) u0 t5) t4 H27 t6 -H17)))))) k H23)))) H20)) H19))))) t2 H15)) t H13 H14 H11 H12))) | (pr0_beta -u v1 v2 H11 t5 t6 H12) \Rightarrow (\lambda (H13: (eq T (THead (Flat Appl) v1 -(THead (Bind Abst) u t5)) t)).(\lambda (H14: (eq T (THead (Bind Abbr) v2 t6) -t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) (\lambda (_: -T).((eq T (THead (Bind Abbr) v2 t6) t2) \to ((pr0 v1 v2) \to ((pr0 t5 t6) \to -(ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: -T).(pr0 t2 t8))))))) (\lambda (H15: (eq T (THead (Bind Abbr) v2 t6) -t2)).(eq_ind T (THead (Bind Abbr) v2 t6) (\lambda (t7: T).((pr0 v1 v2) \to -((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) -(\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 -t5 t6)).(let H18 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind Abbr) -u1 t3) t7)) H5 (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) H13) in (let -H19 \def (eq_ind T (THead (Bind Abbr) u1 t3) (\lambda (ee: T).(match ee in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow -False])])) I (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) H18) in -(False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) -(\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t6) t7))) H19))))) t2 H15)) t H13 -H14 H11 H12))) | (pr0_upsilon b H11 v1 v2 H12 u0 u3 H13 t5 t6 H14) -\Rightarrow (\lambda (H15: (eq T (THead (Flat Appl) v1 (THead (Bind b) u0 -t5)) t)).(\lambda (H16: (eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S -O) O v2) t6)) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u0 t5)) -(\lambda (_: T).((eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O -v2) t6)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u3) \to -((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) -(\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H17: (eq T (THead (Bind b) u3 -(THead (Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Bind b) u3 -(THead (Flat Appl) (lift (S O) O v2) t6)) (\lambda (t7: T).((not (eq B b -Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda -(t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t7 -t8)))))))) (\lambda (_: (not (eq B b Abst))).(\lambda (_: (pr0 v1 -v2)).(\lambda (_: (pr0 u0 u3)).(\lambda (_: (pr0 t5 t6)).(let H22 \def -(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind Abbr) u1 t3) t7)) H5 (THead -(Flat Appl) v1 (THead (Bind b) u0 t5)) H15) in (let H23 \def (eq_ind T (THead -(Bind Abbr) u1 t3) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat -Appl) v1 (THead (Bind b) u0 t5)) H22) in (False_ind (ex2 T (\lambda (t7: -T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 (THead (Bind b) -u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t7))) H23))))))) t2 H17)) t H15 -H16 H11 H12 H13 H14))) | (pr0_delta u0 u3 H11 t5 t6 H12 w0 H13) \Rightarrow -(\lambda (H14: (eq T (THead (Bind Abbr) u0 t5) t)).(\lambda (H15: (eq T -(THead (Bind Abbr) u3 w0) t2)).(eq_ind T (THead (Bind Abbr) u0 t5) (\lambda -(_: T).((eq T (THead (Bind Abbr) u3 w0) t2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) -\to ((subst0 O u3 t6 w0) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) -u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8)))))))) (\lambda (H16: (eq T (THead -(Bind Abbr) u3 w0) t2)).(eq_ind T (THead (Bind Abbr) u3 w0) (\lambda (t7: -T).((pr0 u0 u3) \to ((pr0 t5 t6) \to ((subst0 O u3 t6 w0) \to (ex2 T (\lambda -(t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t7 t8))))))) -(\lambda (H17: (pr0 u0 u3)).(\lambda (H18: (pr0 t5 t6)).(\lambda (H19: -(subst0 O u3 t6 w0)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T -(THead (Bind Abbr) u1 t3) t7)) H5 (THead (Bind Abbr) u0 t5) H14) in (let H21 -\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t7 _) -\Rightarrow t7])) (THead (Bind Abbr) u1 t3) (THead (Bind Abbr) u0 t5) H20) in -((let H22 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: -T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ -t7) \Rightarrow t7])) (THead (Bind Abbr) u1 t3) (THead (Bind Abbr) u0 t5) -H20) in (\lambda (H23: (eq T u1 u0)).(let H24 \def (eq_ind_r T t (\lambda -(t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to -(\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) -(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Bind Abbr) u0 t5) H14) in -(let H25 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H8 u0 H23) in (let -H26 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H9 t5 H22) in (ex2_ind T -(\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda -(t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 (THead (Bind -Abbr) u3 w0) t7))) (\lambda (x: T).(\lambda (H27: (pr0 t4 x)).(\lambda (H28: -(pr0 t6 x)).(ex2_ind T (\lambda (t7: T).(pr0 u2 t7)) (\lambda (t7: T).(pr0 u3 -t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: -T).(pr0 (THead (Bind Abbr) u3 w0) t7))) (\lambda (x0: T).(\lambda (H29: (pr0 -u2 x0)).(\lambda (H30: (pr0 u3 x0)).(pr0_confluence__pr0_delta_delta u2 t4 w -H10 u3 t6 w0 H19 x0 H29 H30 x H27 H28)))) (H24 u0 (tlt_head_sx (Bind Abbr) u0 -t5) u2 H25 u3 H17))))) (H24 t5 (tlt_head_dx (Bind Abbr) u0 t5) t4 H26 t6 -H18))))))) H21)))))) t2 H16)) t H14 H15 H11 H12 H13))) | (pr0_zeta b H11 t5 -t6 H12 u) \Rightarrow (\lambda (H13: (eq T (THead (Bind b) u (lift (S O) O -t5)) t)).(\lambda (H14: (eq T t6 t2)).(eq_ind T (THead (Bind b) u (lift (S O) -O t5)) (\lambda (_: T).((eq T t6 t2) \to ((not (eq B b Abst)) \to ((pr0 t5 -t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda -(t8: T).(pr0 t2 t8))))))) (\lambda (H15: (eq T t6 t2)).(eq_ind T t2 (\lambda -(t7: T).((not (eq B b Abst)) \to ((pr0 t5 t7) \to (ex2 T (\lambda (t8: -T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) -(\lambda (H16: (not (eq B b Abst))).(\lambda (H17: (pr0 t5 t2)).(let H18 \def -(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind Abbr) u1 t3) t7)) H5 (THead -(Bind b) u (lift (S O) O t5)) H13) in (let H19 \def (f_equal T B (\lambda (e: -T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow Abbr | -(TLRef _) \Rightarrow Abbr | (THead k _ _) \Rightarrow (match k in K return -(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow -Abbr])])) (THead (Bind Abbr) u1 t3) (THead (Bind b) u (lift (S O) O t5)) H18) -in ((let H20 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda -(_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead -_ t7 _) \Rightarrow t7])) (THead (Bind Abbr) u1 t3) (THead (Bind b) u (lift -(S O) O t5)) H18) in ((let H21 \def (f_equal T T (\lambda (e: T).(match e in -T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) -\Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead (Bind Abbr) u1 t3) -(THead (Bind b) u (lift (S O) O t5)) H18) in (\lambda (H22: (eq T u1 -u)).(\lambda (H23: (eq B Abbr b)).(let H24 \def (eq_ind_r B b (\lambda (b0: -B).(not (eq B b0 Abst))) H16 Abbr H23) in (let H25 \def (eq_ind_r B b -(\lambda (b0: B).(eq T (THead (Bind b0) u (lift (S O) O t5)) t)) H13 Abbr -H23) in (let H26 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v -t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to -(ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 -t10)))))))))) H (THead (Bind Abbr) u (lift (S O) O t5)) H25) in (let H27 \def -(eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H8 u H22) in (let H28 \def (eq_ind -T t3 (\lambda (t7: T).(pr0 t7 t4)) H9 (lift (S O) O t5) H21) in (ex2_ind T -(\lambda (t7: T).(eq T t4 (lift (S O) O t7))) (\lambda (t7: T).(pr0 t5 t7)) -(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: -T).(pr0 t2 t7))) (\lambda (x: T).(\lambda (H29: (eq T t4 (lift (S O) O -x))).(\lambda (H30: (pr0 t5 x)).(let H31 \def (eq_ind T t4 (\lambda (t7: -T).(subst0 O u2 t7 w)) H10 (lift (S O) O x) H29) in (ex2_ind T (\lambda (t7: -T).(pr0 x t7)) (\lambda (t7: T).(pr0 t2 t7)) (ex2 T (\lambda (t7: T).(pr0 -(THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7))) (\lambda (x0: -T).(\lambda (_: (pr0 x x0)).(\lambda (_: (pr0 t2 -x0)).(pr0_confluence__pr0_delta_tau u2 (lift (S O) O x) w H31 x (pr0_refl -(lift (S O) O x)) t2)))) (H26 t5 (lift_tlt_dx (Bind Abbr) u t5 (S O) O) x H30 -t2 H17)))))) (pr0_gen_lift t5 t4 (S O) O H28)))))))))) H20)) H19))))) t6 -(sym_eq T t6 t2 H15))) t H13 H14 H11 H12))) | (pr0_tau t5 t6 H11 u) -\Rightarrow (\lambda (H12: (eq T (THead (Flat Cast) u t5) t)).(\lambda (H13: -(eq T t6 t2)).(eq_ind T (THead (Flat Cast) u t5) (\lambda (_: T).((eq T t6 -t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 -w) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H14: (eq T t6 -t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8: -T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8))))) -(\lambda (_: (pr0 t5 t2)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T -(THead (Bind Abbr) u1 t3) t7)) H5 (THead (Flat Cast) u t5) H12) in (let H17 -\def (eq_ind T (THead (Bind Abbr) u1 t3) (\lambda (ee: T).(match ee in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow -False])])) I (THead (Flat Cast) u t5) H16) in (False_ind (ex2 T (\lambda (t7: -T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7))) H17)))) -t6 (sym_eq T t6 t2 H14))) t H12 H13 H11)))]) in (H11 (refl_equal T t) -(refl_equal T t2)))))) t1 H7)) t H5 H6 H2 H3 H4))) | (pr0_zeta b H2 t3 t4 H3 -u) \Rightarrow (\lambda (H4: (eq T (THead (Bind b) u (lift (S O) O t3)) -t)).(\lambda (H5: (eq T t4 t1)).(eq_ind T (THead (Bind b) u (lift (S O) O -t3)) (\lambda (_: T).((eq T t4 t1) \to ((not (eq B b Abst)) \to ((pr0 t3 t4) -\to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6))))))) -(\lambda (H6: (eq T t4 t1)).(eq_ind T t1 (\lambda (t5: T).((not (eq B b -Abst)) \to ((pr0 t3 t5) \to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda -(t6: T).(pr0 t2 t6)))))) (\lambda (H7: (not (eq B b Abst))).(\lambda (H8: -(pr0 t3 t1)).(let H9 \def (match H1 in pr0 return (\lambda (t5: T).(\lambda -(t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 t) \to ((eq T t6 t2) \to (ex2 T -(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)))))))) with -[(pr0_refl t5) \Rightarrow (\lambda (H9: (eq T t5 t)).(\lambda (H10: (eq T t5 -t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2) \to (ex2 T (\lambda (t7: -T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))))) (\lambda (H11: (eq T t -t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t7: T).(pr0 t1 t7)) -(\lambda (t7: T).(pr0 t2 t7)))) (let H12 \def (eq_ind_r T t (\lambda (t6: -T).(eq T t6 t2)) H11 (THead (Bind b) u (lift (S O) O t3)) H4) in (eq_ind T -(THead (Bind b) u (lift (S O) O t3)) (\lambda (t6: T).(ex2 T (\lambda (t7: -T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t6 t7)))) (let H13 \def (eq_ind_r T t -(\lambda (t6: T).(eq T t5 t6)) H9 (THead (Bind b) u (lift (S O) O t3)) H4) in -(let H14 \def (eq_ind_r T t (\lambda (t6: T).(\forall (v: T).((tlt v t6) \to -(\forall (t7: T).((pr0 v t7) \to (\forall (t8: T).((pr0 v t8) \to (ex2 T -(\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead -(Bind b) u (lift (S O) O t3)) H4) in (ex_intro2 T (\lambda (t6: T).(pr0 t1 -t6)) (\lambda (t6: T).(pr0 (THead (Bind b) u (lift (S O) O t3)) t6)) t1 -(pr0_refl t1) (pr0_zeta b H7 t3 t1 H8 u)))) t2 H12)) t (sym_eq T t t2 H11))) -t5 (sym_eq T t5 t H9) H10))) | (pr0_comp u1 u2 H9 t5 t6 H10 k) \Rightarrow -(\lambda (H11: (eq T (THead k u1 t5) t)).(\lambda (H12: (eq T (THead k u2 t6) -t2)).(eq_ind T (THead k u1 t5) (\lambda (_: T).((eq T (THead k u2 t6) t2) \to -((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) -(\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H13: (eq T (THead k u2 t6) -t2)).(eq_ind T (THead k u2 t6) (\lambda (t7: T).((pr0 u1 u2) \to ((pr0 t5 t6) -\to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 t8)))))) -(\lambda (_: (pr0 u1 u2)).(\lambda (H15: (pr0 t5 t6)).(let H16 \def (eq_ind_r -T t (\lambda (t7: T).(eq T (THead (Bind b) u (lift (S O) O t3)) t7)) H4 -(THead k u1 t5) H11) in (let H17 \def (f_equal T K (\lambda (e: T).(match e -in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow (Bind b) | (TLRef -_) \Rightarrow (Bind b) | (THead k0 _ _) \Rightarrow k0])) (THead (Bind b) u -(lift (S O) O t3)) (THead k u1 t5) H16) in ((let H18 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t7 _) \Rightarrow t7])) -(THead (Bind b) u (lift (S O) O t3)) (THead k u1 t5) H16) in ((let H19 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t7: -T) on t7: T \def (match t7 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) -\Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false -\Rightarrow (f i)])) | (THead k0 u0 t8) \Rightarrow (THead k0 (lref_map f d -u0) (lref_map f (s k0 d) t8))]) in lref_map) (\lambda (x: nat).(plus x (S -O))) O t3) | (TLRef _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) -(d: nat) (t7: T) on t7: T \def (match t7 with [(TSort n) \Rightarrow (TSort -n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | -false \Rightarrow (f i)])) | (THead k0 u0 t8) \Rightarrow (THead k0 (lref_map -f d u0) (lref_map f (s k0 d) t8))]) in lref_map) (\lambda (x: nat).(plus x (S -O))) O t3) | (THead _ _ t7) \Rightarrow t7])) (THead (Bind b) u (lift (S O) O -t3)) (THead k u1 t5) H16) in (\lambda (_: (eq T u u1)).(\lambda (H21: (eq K -(Bind b) k)).(eq_ind K (Bind b) (\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 -t1 t7)) (\lambda (t7: T).(pr0 (THead k0 u2 t6) t7)))) (let H22 \def (eq_ind_r -K k (\lambda (k0: K).(eq T (THead k0 u1 t5) t)) H11 (Bind b) H21) in (let H23 -\def (eq_ind_r T t5 (\lambda (t7: T).(pr0 t7 t6)) H15 (lift (S O) O t3) H19) -in (ex2_ind T (\lambda (t7: T).(eq T t6 (lift (S O) O t7))) (\lambda (t7: -T).(pr0 t3 t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 -(THead (Bind b) u2 t6) t7))) (\lambda (x: T).(\lambda (H24: (eq T t6 (lift (S -O) O x))).(\lambda (H25: (pr0 t3 x)).(let H26 \def (eq_ind_r T t5 (\lambda -(t7: T).(eq T (THead (Bind b) u1 t7) t)) H22 (lift (S O) O t3) H19) in (let -H27 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to -(\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T -(\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H -(THead (Bind b) u1 (lift (S O) O t3)) H26) in (eq_ind_r T (lift (S O) O x) -(\lambda (t7: T).(ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 -(THead (Bind b) u2 t7) t8)))) (ex2_ind T (\lambda (t7: T).(pr0 x t7)) -(\lambda (t7: T).(pr0 t1 t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda -(t7: T).(pr0 (THead (Bind b) u2 (lift (S O) O x)) t7))) (\lambda (x0: -T).(\lambda (H28: (pr0 x x0)).(\lambda (H29: (pr0 t1 x0)).(ex_intro2 T -(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Bind b) u2 (lift -(S O) O x)) t7)) x0 H29 (pr0_zeta b H7 x x0 H28 u2))))) (H27 t3 (lift_tlt_dx -(Bind b) u1 t3 (S O) O) x H25 t1 H8)) t6 H24)))))) (pr0_gen_lift t3 t6 (S O) -O H23)))) k H21)))) H18)) H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_beta -u0 v1 v2 H9 t5 t6 H10) \Rightarrow (\lambda (H11: (eq T (THead (Flat Appl) v1 -(THead (Bind Abst) u0 t5)) t)).(\lambda (H12: (eq T (THead (Bind Abbr) v2 t6) -t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t5)) (\lambda (_: -T).((eq T (THead (Bind Abbr) v2 t6) t2) \to ((pr0 v1 v2) \to ((pr0 t5 t6) \to -(ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))))) -(\lambda (H13: (eq T (THead (Bind Abbr) v2 t6) t2)).(eq_ind T (THead (Bind -Abbr) v2 t6) (\lambda (t7: T).((pr0 v1 v2) \to ((pr0 t5 t6) \to (ex2 T -(\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (_: -(pr0 v1 v2)).(\lambda (_: (pr0 t5 t6)).(let H16 \def (eq_ind_r T t (\lambda -(t7: T).(eq T (THead (Bind b) u (lift (S O) O t3)) t7)) H4 (THead (Flat Appl) -v1 (THead (Bind Abst) u0 t5)) H11) in (let H17 \def (eq_ind T (THead (Bind b) -u (lift (S O) O t3)) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat -Appl) v1 (THead (Bind Abst) u0 t5)) H16) in (False_ind (ex2 T (\lambda (t7: -T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t6) t7))) -H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_upsilon b0 H9 v1 v2 H10 u1 u2 -H11 t5 t6 H12) \Rightarrow (\lambda (H13: (eq T (THead (Flat Appl) v1 (THead -(Bind b0) u1 t5)) t)).(\lambda (H14: (eq T (THead (Bind b0) u2 (THead (Flat -Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Flat Appl) v1 (THead -(Bind b0) u1 t5)) (\lambda (_: T).((eq T (THead (Bind b0) u2 (THead (Flat -Appl) (lift (S O) O v2) t6)) t2) \to ((not (eq B b0 Abst)) \to ((pr0 v1 v2) -\to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) -(\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H15: (eq T (THead (Bind b0) u2 -(THead (Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Bind b0) u2 -(THead (Flat Appl) (lift (S O) O v2) t6)) (\lambda (t7: T).((not (eq B b0 -Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda -(t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 t8)))))))) (\lambda (_: (not -(eq B b0 Abst))).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 u1 u2)).(\lambda -(_: (pr0 t5 t6)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead -(Bind b) u (lift (S O) O t3)) t7)) H4 (THead (Flat Appl) v1 (THead (Bind b0) -u1 t5)) H13) in (let H21 \def (eq_ind T (THead (Bind b) u (lift (S O) O t3)) -(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow -(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | -(Flat _) \Rightarrow False])])) I (THead (Flat Appl) v1 (THead (Bind b0) u1 -t5)) H20) in (False_ind (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: -T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t6)) t7))) -H21))))))) t2 H15)) t H13 H14 H9 H10 H11 H12))) | (pr0_delta u1 u2 H9 t5 t6 -H10 w H11) \Rightarrow (\lambda (H12: (eq T (THead (Bind Abbr) u1 t5) -t)).(\lambda (H13: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind -Abbr) u1 t5) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t2) \to ((pr0 u1 -u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T (\lambda (t8: T).(pr0 -t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))))) (\lambda (H14: (eq T (THead (Bind -Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t7: T).((pr0 u1 -u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T (\lambda (t8: T).(pr0 -t1 t8)) (\lambda (t8: T).(pr0 t7 t8))))))) (\lambda (_: (pr0 u1 u2)).(\lambda -(H16: (pr0 t5 t6)).(\lambda (H17: (subst0 O u2 t6 w)).(let H18 \def (eq_ind_r -T t (\lambda (t7: T).(eq T (THead (Bind b) u (lift (S O) O t3)) t7)) H4 -(THead (Bind Abbr) u1 t5) H12) in (let H19 \def (f_equal T B (\lambda (e: -T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | -(TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k in K return -(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow -b])])) (THead (Bind b) u (lift (S O) O t3)) (THead (Bind Abbr) u1 t5) H18) in -((let H20 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: -T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t7 -_) \Rightarrow t7])) (THead (Bind b) u (lift (S O) O t3)) (THead (Bind Abbr) -u1 t5) H18) in ((let H21 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: -((nat \to nat))) (d: nat) (t7: T) on t7: T \def (match t7 with [(TSort n) -\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with -[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t8) -\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t8))]) in -lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (TLRef _) \Rightarrow -((let rec lref_map (f: ((nat \to nat))) (d: nat) (t7: T) on t7: T \def (match -t7 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef -(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | -(THead k u0 t8) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) -t8))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (THead _ _ t7) -\Rightarrow t7])) (THead (Bind b) u (lift (S O) O t3)) (THead (Bind Abbr) u1 -t5) H18) in (\lambda (_: (eq T u u1)).(\lambda (H23: (eq B b Abbr)).(let H24 -\def (eq_ind_r T t5 (\lambda (t7: T).(pr0 t7 t6)) H16 (lift (S O) O t3) H21) -in (ex2_ind T (\lambda (t7: T).(eq T t6 (lift (S O) O t7))) (\lambda (t7: -T).(pr0 t3 t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 -(THead (Bind Abbr) u2 w) t7))) (\lambda (x: T).(\lambda (H25: (eq T t6 (lift -(S O) O x))).(\lambda (H26: (pr0 t3 x)).(let H27 \def (eq_ind_r T t5 (\lambda -(t7: T).(eq T (THead (Bind Abbr) u1 t7) t)) H12 (lift (S O) O t3) H21) in -(let H28 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to -(\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T -(\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H -(THead (Bind Abbr) u1 (lift (S O) O t3)) H27) in (let H29 \def (eq_ind T t6 -(\lambda (t7: T).(subst0 O u2 t7 w)) H17 (lift (S O) O x) H25) in (let H30 -\def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H7 Abbr H23) in -(ex2_ind T (\lambda (t7: T).(pr0 x t7)) (\lambda (t7: T).(pr0 t1 t7)) (ex2 T -(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) -t7))) (\lambda (x0: T).(\lambda (_: (pr0 x x0)).(\lambda (_: (pr0 t1 -x0)).(ex2_sym T (pr0 (THead (Bind Abbr) u2 w)) (pr0 t1) -(pr0_confluence__pr0_delta_tau u2 (lift (S O) O x) w H29 x (pr0_refl (lift (S -O) O x)) t1))))) (H28 t3 (lift_tlt_dx (Bind Abbr) u1 t3 (S O) O) x H26 t1 -H8))))))))) (pr0_gen_lift t3 t6 (S O) O H24)))))) H20)) H19)))))) t2 H14)) t -H12 H13 H9 H10 H11))) | (pr0_zeta b0 H9 t5 t6 H10 u0) \Rightarrow (\lambda -(H11: (eq T (THead (Bind b0) u0 (lift (S O) O t5)) t)).(\lambda (H12: (eq T -t6 t2)).(eq_ind T (THead (Bind b0) u0 (lift (S O) O t5)) (\lambda (_: T).((eq -T t6 t2) \to ((not (eq B b0 Abst)) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: -T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H13: (eq T t6 -t2)).(eq_ind T t2 (\lambda (t7: T).((not (eq B b0 Abst)) \to ((pr0 t5 t7) \to -(ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))) -(\lambda (_: (not (eq B b0 Abst))).(\lambda (H15: (pr0 t5 t2)).(let H16 \def -(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind b) u (lift (S O) O t3)) -t7)) H4 (THead (Bind b0) u0 (lift (S O) O t5)) H11) in (let H17 \def (f_equal -T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with [(TSort _) -\Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k -in K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) -\Rightarrow b])])) (THead (Bind b) u (lift (S O) O t3)) (THead (Bind b0) u0 -(lift (S O) O t5)) H16) in ((let H18 \def (f_equal T T (\lambda (e: T).(match -e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) -\Rightarrow u | (THead _ t7 _) \Rightarrow t7])) (THead (Bind b) u (lift (S -O) O t3)) (THead (Bind b0) u0 (lift (S O) O t5)) H16) in ((let H19 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t7: -T) on t7: T \def (match t7 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) -\Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false -\Rightarrow (f i)])) | (THead k u1 t8) \Rightarrow (THead k (lref_map f d u1) -(lref_map f (s k d) t8))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O -t3) | (TLRef _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) -(t7: T) on t7: T \def (match t7 with [(TSort n) \Rightarrow (TSort n) | -(TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | -false \Rightarrow (f i)])) | (THead k u1 t8) \Rightarrow (THead k (lref_map f -d u1) (lref_map f (s k d) t8))]) in lref_map) (\lambda (x: nat).(plus x (S -O))) O t3) | (THead _ _ t7) \Rightarrow t7])) (THead (Bind b) u (lift (S O) O -t3)) (THead (Bind b0) u0 (lift (S O) O t5)) H16) in (\lambda (_: (eq T u -u0)).(\lambda (H21: (eq B b b0)).(let H22 \def (eq_ind_r T t (\lambda (t7: -T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall -(t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: -T).(pr0 t9 t10)))))))))) H (THead (Bind b0) u0 (lift (S O) O t5)) H11) in -(let H23 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t1)) H8 t5 (lift_inj t3 -t5 (S O) O H19)) in (let H24 \def (eq_ind B b (\lambda (b1: B).(not (eq B b1 -Abst))) H7 b0 H21) in (ex2_ind T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: -T).(pr0 t2 t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 -t7))) (\lambda (x: T).(\lambda (H25: (pr0 t1 x)).(\lambda (H26: (pr0 t2 -x)).(ex_intro2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)) -x H25 H26)))) (H22 t5 (lift_tlt_dx (Bind b0) u0 t5 (S O) O) t1 H23 t2 -H15)))))))) H18)) H17))))) t6 (sym_eq T t6 t2 H13))) t H11 H12 H9 H10))) | -(pr0_tau t5 t6 H9 u0) \Rightarrow (\lambda (H10: (eq T (THead (Flat Cast) u0 -t5) t)).(\lambda (H11: (eq T t6 t2)).(eq_ind T (THead (Flat Cast) u0 t5) -(\lambda (_: T).((eq T t6 t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: -T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H12: (eq T t6 -t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8: -T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))) (\lambda (_: (pr0 t5 -t2)).(let H14 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind b) u -(lift (S O) O t3)) t7)) H4 (THead (Flat Cast) u0 t5) H10) in (let H15 \def -(eq_ind T (THead (Bind b) u (lift (S O) O t3)) (\lambda (ee: T).(match ee in -T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow -False])])) I (THead (Flat Cast) u0 t5) H14) in (False_ind (ex2 T (\lambda -(t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))) H15)))) t6 (sym_eq T t6 -t2 H12))) t H10 H11 H9)))]) in (H9 (refl_equal T t) (refl_equal T t2))))) t4 -(sym_eq T t4 t1 H6))) t H4 H5 H2 H3))) | (pr0_tau t3 t4 H2 u) \Rightarrow -(\lambda (H3: (eq T (THead (Flat Cast) u t3) t)).(\lambda (H4: (eq T t4 -t1)).(eq_ind T (THead (Flat Cast) u t3) (\lambda (_: T).((eq T t4 t1) \to -((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 -t2 t6)))))) (\lambda (H5: (eq T t4 t1)).(eq_ind T t1 (\lambda (t5: T).((pr0 -t3 t5) \to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 -t6))))) (\lambda (H6: (pr0 t3 t1)).(let H7 \def (match H1 in pr0 return -(\lambda (t5: T).(\lambda (t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 t) \to -((eq T t6 t2) \to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 -t2 t7)))))))) with [(pr0_refl t5) \Rightarrow (\lambda (H7: (eq T t5 -t)).(\lambda (H8: (eq T t5 t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2) -\to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))))) -(\lambda (H9: (eq T t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t7: -T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let H10 \def (eq_ind_r T t -(\lambda (t6: T).(eq T t6 t2)) H9 (THead (Flat Cast) u t3) H3) in (eq_ind T -(THead (Flat Cast) u t3) (\lambda (t6: T).(ex2 T (\lambda (t7: T).(pr0 t1 -t7)) (\lambda (t7: T).(pr0 t6 t7)))) (let H11 \def (eq_ind_r T t (\lambda -(t6: T).(eq T t5 t6)) H7 (THead (Flat Cast) u t3) H3) in (let H12 \def -(eq_ind_r T t (\lambda (t6: T).(\forall (v: T).((tlt v t6) \to (\forall (t7: -T).((pr0 v t7) \to (\forall (t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: -T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead (Flat Cast) u -t3) H3) in (ex_intro2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 -(THead (Flat Cast) u t3) t6)) t1 (pr0_refl t1) (pr0_tau t3 t1 H6 u)))) t2 -H10)) t (sym_eq T t t2 H9))) t5 (sym_eq T t5 t H7) H8))) | (pr0_comp u1 u2 H7 -t5 t6 H8 k) \Rightarrow (\lambda (H9: (eq T (THead k u1 t5) t)).(\lambda -(H10: (eq T (THead k u2 t6) t2)).(eq_ind T (THead k u1 t5) (\lambda (_: -T).((eq T (THead k u2 t6) t2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T -(\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda -(H11: (eq T (THead k u2 t6) t2)).(eq_ind T (THead k u2 t6) (\lambda (t7: -T).((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) -(\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (_: (pr0 u1 u2)).(\lambda (H13: -(pr0 t5 t6)).(let H14 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat -Cast) u t3) t7)) H3 (THead k u1 t5) H9) in (let H15 \def (f_equal T K -(\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) -\Rightarrow (Flat Cast) | (TLRef _) \Rightarrow (Flat Cast) | (THead k0 _ _) -\Rightarrow k0])) (THead (Flat Cast) u t3) (THead k u1 t5) H14) in ((let H16 -\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t7 _) -\Rightarrow t7])) (THead (Flat Cast) u t3) (THead k u1 t5) H14) in ((let H17 -\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7) -\Rightarrow t7])) (THead (Flat Cast) u t3) (THead k u1 t5) H14) in (\lambda -(_: (eq T u u1)).(\lambda (H19: (eq K (Flat Cast) k)).(eq_ind K (Flat Cast) -(\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 -(THead k0 u2 t6) t7)))) (let H20 \def (eq_ind_r K k (\lambda (k0: K).(eq T -(THead k0 u1 t5) t)) H9 (Flat Cast) H19) in (let H21 \def (eq_ind_r T t -(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) -\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) -(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Cast) u1 t5) H20) in -(let H22 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t1)) H6 t5 H17) in -(ex2_ind T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T -(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Flat Cast) u2 t6) -t7))) (\lambda (x: T).(\lambda (H23: (pr0 t1 x)).(\lambda (H24: (pr0 t6 -x)).(ex_intro2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead -(Flat Cast) u2 t6) t7)) x H23 (pr0_tau t6 x H24 u2))))) (H21 t5 (tlt_head_dx -(Flat Cast) u1 t5) t1 H22 t6 H13))))) k H19)))) H16)) H15))))) t2 H11)) t H9 -H10 H7 H8))) | (pr0_beta u0 v1 v2 H7 t5 t6 H8) \Rightarrow (\lambda (H9: (eq -T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t5)) t)).(\lambda (H10: (eq T -(THead (Bind Abbr) v2 t6) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind -Abst) u0 t5)) (\lambda (_: T).((eq T (THead (Bind Abbr) v2 t6) t2) \to ((pr0 -v1 v2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda -(t8: T).(pr0 t2 t8))))))) (\lambda (H11: (eq T (THead (Bind Abbr) v2 t6) -t2)).(eq_ind T (THead (Bind Abbr) v2 t6) (\lambda (t7: T).((pr0 v1 v2) \to -((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 -t7 t8)))))) (\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 t5 t6)).(let H14 \def -(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Cast) u t3) t7)) H3 (THead -(Flat Appl) v1 (THead (Bind Abst) u0 t5)) H9) in (let H15 \def (eq_ind T -(THead (Flat Cast) u t3) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f in F return -(\lambda (_: F).Prop) with [Appl \Rightarrow False | Cast \Rightarrow -True])])])) I (THead (Flat Appl) v1 (THead (Bind Abst) u0 t5)) H14) in -(False_ind (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead -(Bind Abbr) v2 t6) t7))) H15))))) t2 H11)) t H9 H10 H7 H8))) | (pr0_upsilon b -H7 v1 v2 H8 u1 u2 H9 t5 t6 H10) \Rightarrow (\lambda (H11: (eq T (THead (Flat -Appl) v1 (THead (Bind b) u1 t5)) t)).(\lambda (H12: (eq T (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Flat Appl) -v1 (THead (Bind b) u1 t5)) (\lambda (_: T).((eq T (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t6)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1 -v2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 -t8)) (\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H13: (eq T (THead (Bind -b) u2 (THead (Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Bind -b) u2 (THead (Flat Appl) (lift (S O) O v2) t6)) (\lambda (t7: T).((not (eq B -b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T -(\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 t8)))))))) (\lambda -(_: (not (eq B b Abst))).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 u1 -u2)).(\lambda (_: (pr0 t5 t6)).(let H18 \def (eq_ind_r T t (\lambda (t7: -T).(eq T (THead (Flat Cast) u t3) t7)) H3 (THead (Flat Appl) v1 (THead (Bind -b) u1 t5)) H11) in (let H19 \def (eq_ind T (THead (Flat Cast) u t3) (\lambda -(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow -(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | -(Flat f) \Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl -\Rightarrow False | Cast \Rightarrow True])])])) I (THead (Flat Appl) v1 -(THead (Bind b) u1 t5)) H18) in (False_ind (ex2 T (\lambda (t7: T).(pr0 t1 -t7)) (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) -O v2) t6)) t7))) H19))))))) t2 H13)) t H11 H12 H7 H8 H9 H10))) | (pr0_delta -u1 u2 H7 t5 t6 H8 w H9) \Rightarrow (\lambda (H10: (eq T (THead (Bind Abbr) -u1 t5) t)).(\lambda (H11: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T -(THead (Bind Abbr) u1 t5) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t2) -\to ((pr0 u1 u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T (\lambda -(t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))))) (\lambda (H12: (eq T -(THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda -(t7: T).((pr0 u1 u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T -(\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 t8))))))) (\lambda (_: -(pr0 u1 u2)).(\lambda (_: (pr0 t5 t6)).(\lambda (_: (subst0 O u2 t6 w)).(let -H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Cast) u t3) t7)) -H3 (THead (Bind Abbr) u1 t5) H10) in (let H17 \def (eq_ind T (THead (Flat -Cast) u t3) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) -\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u1 -t5) H16) in (False_ind (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: -T).(pr0 (THead (Bind Abbr) u2 w) t7))) H17)))))) t2 H12)) t H10 H11 H7 H8 -H9))) | (pr0_zeta b H7 t5 t6 H8 u0) \Rightarrow (\lambda (H9: (eq T (THead -(Bind b) u0 (lift (S O) O t5)) t)).(\lambda (H10: (eq T t6 t2)).(eq_ind T -(THead (Bind b) u0 (lift (S O) O t5)) (\lambda (_: T).((eq T t6 t2) \to ((not -(eq B b Abst)) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) -(\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H11: (eq T t6 t2)).(eq_ind T t2 -(\lambda (t7: T).((not (eq B b Abst)) \to ((pr0 t5 t7) \to (ex2 T (\lambda -(t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (_: (not (eq -B b Abst))).(\lambda (_: (pr0 t5 t2)).(let H14 \def (eq_ind_r T t (\lambda -(t7: T).(eq T (THead (Flat Cast) u t3) t7)) H3 (THead (Bind b) u0 (lift (S O) -O t5)) H9) in (let H15 \def (eq_ind T (THead (Flat Cast) u t3) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K -return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) -\Rightarrow True])])) I (THead (Bind b) u0 (lift (S O) O t5)) H14) in -(False_ind (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 -t7))) H15))))) t6 (sym_eq T t6 t2 H11))) t H9 H10 H7 H8))) | (pr0_tau t5 t6 -H7 u0) \Rightarrow (\lambda (H8: (eq T (THead (Flat Cast) u0 t5) t)).(\lambda -(H9: (eq T t6 t2)).(eq_ind T (THead (Flat Cast) u0 t5) (\lambda (_: T).((eq T -t6 t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda -(t8: T).(pr0 t2 t8)))))) (\lambda (H10: (eq T t6 t2)).(eq_ind T t2 (\lambda -(t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: -T).(pr0 t2 t8))))) (\lambda (H11: (pr0 t5 t2)).(let H12 \def (eq_ind_r T t -(\lambda (t7: T).(eq T (THead (Flat Cast) u t3) t7)) H3 (THead (Flat Cast) u0 -t5) H8) in (let H13 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | -(THead _ t7 _) \Rightarrow t7])) (THead (Flat Cast) u t3) (THead (Flat Cast) -u0 t5) H12) in ((let H14 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) -\Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead (Flat Cast) u t3) -(THead (Flat Cast) u0 t5) H12) in (\lambda (_: (eq T u u0)).(let H16 \def -(eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: -T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: -T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Cast) -u0 t5) H8) in (let H17 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t1)) H6 t5 -H14) in (ex2_ind T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 -t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))) -(\lambda (x: T).(\lambda (H18: (pr0 t1 x)).(\lambda (H19: (pr0 t2 -x)).(ex_intro2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)) -x H18 H19)))) (H16 t5 (tlt_head_dx (Flat Cast) u0 t5) t1 H17 t2 H11)))))) -H13)))) t6 (sym_eq T t6 t2 H10))) t H8 H9 H7)))]) in (H7 (refl_equal T t) -(refl_equal T t2)))) t4 (sym_eq T t4 t1 H5))) t H3 H4 H2)))]) in (H2 -(refl_equal T t) (refl_equal T t1))))))))) t0). -(* COMMENTS -Initial nodes: 46103 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr0/props.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr0/props.ma deleted file mode 100644 index d7f69d691..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr0/props.ma +++ /dev/null @@ -1,1758 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr0/defs.ma". - -include "Basic-1/subst0/subst0.ma". - -theorem pr0_lift: - \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (h: nat).(\forall -(d: nat).(pr0 (lift h d t1) (lift h d t2)))))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(pr0_ind (\lambda -(t: T).(\lambda (t0: T).(\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t) -(lift h d t0)))))) (\lambda (t: T).(\lambda (h: nat).(\lambda (d: -nat).(pr0_refl (lift h d t))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda -(_: (pr0 u1 u2)).(\lambda (H1: ((\forall (h: nat).(\forall (d: nat).(pr0 -(lift h d u1) (lift h d u2)))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda -(_: (pr0 t3 t4)).(\lambda (H3: ((\forall (h: nat).(\forall (d: nat).(pr0 -(lift h d t3) (lift h d t4)))))).(\lambda (k: K).(\lambda (h: nat).(\lambda -(d: nat).(eq_ind_r T (THead k (lift h d u1) (lift h (s k d) t3)) (\lambda (t: -T).(pr0 t (lift h d (THead k u2 t4)))) (eq_ind_r T (THead k (lift h d u2) -(lift h (s k d) t4)) (\lambda (t: T).(pr0 (THead k (lift h d u1) (lift h (s k -d) t3)) t)) (pr0_comp (lift h d u1) (lift h d u2) (H1 h d) (lift h (s k d) -t3) (lift h (s k d) t4) (H3 h (s k d)) k) (lift h d (THead k u2 t4)) -(lift_head k u2 t4 h d)) (lift h d (THead k u1 t3)) (lift_head k u1 t3 h -d))))))))))))) (\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: -(pr0 v1 v2)).(\lambda (H1: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h -d v1) (lift h d v2)))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 -t3 t4)).(\lambda (H3: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t3) -(lift h d t4)))))).(\lambda (h: nat).(\lambda (d: nat).(eq_ind_r T (THead -(Flat Appl) (lift h d v1) (lift h (s (Flat Appl) d) (THead (Bind Abst) u -t3))) (\lambda (t: T).(pr0 t (lift h d (THead (Bind Abbr) v2 t4)))) (eq_ind_r -T (THead (Bind Abst) (lift h (s (Flat Appl) d) u) (lift h (s (Bind Abst) (s -(Flat Appl) d)) t3)) (\lambda (t: T).(pr0 (THead (Flat Appl) (lift h d v1) t) -(lift h d (THead (Bind Abbr) v2 t4)))) (eq_ind_r T (THead (Bind Abbr) (lift h -d v2) (lift h (s (Bind Abbr) d) t4)) (\lambda (t: T).(pr0 (THead (Flat Appl) -(lift h d v1) (THead (Bind Abst) (lift h (s (Flat Appl) d) u) (lift h (s -(Bind Abst) (s (Flat Appl) d)) t3))) t)) (pr0_beta (lift h (s (Flat Appl) d) -u) (lift h d v1) (lift h d v2) (H1 h d) (lift h (s (Bind Abst) (s (Flat Appl) -d)) t3) (lift h (s (Bind Abbr) d) t4) (H3 h (s (Bind Abbr) d))) (lift h d -(THead (Bind Abbr) v2 t4)) (lift_head (Bind Abbr) v2 t4 h d)) (lift h (s -(Flat Appl) d) (THead (Bind Abst) u t3)) (lift_head (Bind Abst) u t3 h (s -(Flat Appl) d))) (lift h d (THead (Flat Appl) v1 (THead (Bind Abst) u t3))) -(lift_head (Flat Appl) v1 (THead (Bind Abst) u t3) h d))))))))))))) (\lambda -(b: B).(\lambda (H0: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: -T).(\lambda (_: (pr0 v1 v2)).(\lambda (H2: ((\forall (h: nat).(\forall (d: -nat).(pr0 (lift h d v1) (lift h d v2)))))).(\lambda (u1: T).(\lambda (u2: -T).(\lambda (_: (pr0 u1 u2)).(\lambda (H4: ((\forall (h: nat).(\forall (d: -nat).(pr0 (lift h d u1) (lift h d u2)))))).(\lambda (t3: T).(\lambda (t4: -T).(\lambda (_: (pr0 t3 t4)).(\lambda (H6: ((\forall (h: nat).(\forall (d: -nat).(pr0 (lift h d t3) (lift h d t4)))))).(\lambda (h: nat).(\lambda (d: -nat).(eq_ind_r T (THead (Flat Appl) (lift h d v1) (lift h (s (Flat Appl) d) -(THead (Bind b) u1 t3))) (\lambda (t: T).(pr0 t (lift h d (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t4))))) (eq_ind_r T (THead (Bind b) -(lift h (s (Flat Appl) d) u1) (lift h (s (Bind b) (s (Flat Appl) d)) t3)) -(\lambda (t: T).(pr0 (THead (Flat Appl) (lift h d v1) t) (lift h d (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))))) (eq_ind_r T (THead -(Bind b) (lift h d u2) (lift h (s (Bind b) d) (THead (Flat Appl) (lift (S O) -O v2) t4))) (\lambda (t: T).(pr0 (THead (Flat Appl) (lift h d v1) (THead -(Bind b) (lift h (s (Flat Appl) d) u1) (lift h (s (Bind b) (s (Flat Appl) d)) -t3))) t)) (eq_ind_r T (THead (Flat Appl) (lift h (s (Bind b) d) (lift (S O) O -v2)) (lift h (s (Flat Appl) (s (Bind b) d)) t4)) (\lambda (t: T).(pr0 (THead -(Flat Appl) (lift h d v1) (THead (Bind b) (lift h (s (Flat Appl) d) u1) (lift -h (s (Bind b) (s (Flat Appl) d)) t3))) (THead (Bind b) (lift h d u2) t))) -(eq_ind nat (plus (S O) d) (\lambda (n: nat).(pr0 (THead (Flat Appl) (lift h -d v1) (THead (Bind b) (lift h d u1) (lift h n t3))) (THead (Bind b) (lift h d -u2) (THead (Flat Appl) (lift h n (lift (S O) O v2)) (lift h n t4))))) -(eq_ind_r T (lift (S O) O (lift h d v2)) (\lambda (t: T).(pr0 (THead (Flat -Appl) (lift h d v1) (THead (Bind b) (lift h d u1) (lift h (plus (S O) d) -t3))) (THead (Bind b) (lift h d u2) (THead (Flat Appl) t (lift h (plus (S O) -d) t4))))) (pr0_upsilon b H0 (lift h d v1) (lift h d v2) (H2 h d) (lift h d -u1) (lift h d u2) (H4 h d) (lift h (plus (S O) d) t3) (lift h (plus (S O) d) -t4) (H6 h (plus (S O) d))) (lift h (plus (S O) d) (lift (S O) O v2)) (lift_d -v2 h (S O) d O (le_O_n d))) (S d) (refl_equal nat (S d))) (lift h (s (Bind b) -d) (THead (Flat Appl) (lift (S O) O v2) t4)) (lift_head (Flat Appl) (lift (S -O) O v2) t4 h (s (Bind b) d))) (lift h d (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4))) (lift_head (Bind b) u2 (THead (Flat Appl) (lift -(S O) O v2) t4) h d)) (lift h (s (Flat Appl) d) (THead (Bind b) u1 t3)) -(lift_head (Bind b) u1 t3 h (s (Flat Appl) d))) (lift h d (THead (Flat Appl) -v1 (THead (Bind b) u1 t3))) (lift_head (Flat Appl) v1 (THead (Bind b) u1 t3) -h d)))))))))))))))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 -u2)).(\lambda (H1: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d u1) -(lift h d u2)))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 -t4)).(\lambda (H3: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t3) -(lift h d t4)))))).(\lambda (w: T).(\lambda (H4: (subst0 O u2 t4 w)).(\lambda -(h: nat).(\lambda (d: nat).(eq_ind_r T (THead (Bind Abbr) (lift h d u1) (lift -h (s (Bind Abbr) d) t3)) (\lambda (t: T).(pr0 t (lift h d (THead (Bind Abbr) -u2 w)))) (eq_ind_r T (THead (Bind Abbr) (lift h d u2) (lift h (s (Bind Abbr) -d) w)) (\lambda (t: T).(pr0 (THead (Bind Abbr) (lift h d u1) (lift h (s (Bind -Abbr) d) t3)) t)) (pr0_delta (lift h d u1) (lift h d u2) (H1 h d) (lift h (S -d) t3) (lift h (S d) t4) (H3 h (S d)) (lift h (S d) w) (let d' \def (S d) in -(eq_ind nat (minus (S d) (S O)) (\lambda (n: nat).(subst0 O (lift h n u2) -(lift h d' t4) (lift h d' w))) (subst0_lift_lt t4 w u2 O H4 (S d) (le_n_S O d -(le_O_n d)) h) d (eq_ind nat d (\lambda (n: nat).(eq nat n d)) (refl_equal -nat d) (minus d O) (minus_n_O d))))) (lift h d (THead (Bind Abbr) u2 w)) -(lift_head (Bind Abbr) u2 w h d)) (lift h d (THead (Bind Abbr) u1 t3)) -(lift_head (Bind Abbr) u1 t3 h d)))))))))))))) (\lambda (b: B).(\lambda (H0: -(not (eq B b Abst))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 -t4)).(\lambda (H2: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t3) -(lift h d t4)))))).(\lambda (u: T).(\lambda (h: nat).(\lambda (d: -nat).(eq_ind_r T (THead (Bind b) (lift h d u) (lift h (s (Bind b) d) (lift (S -O) O t3))) (\lambda (t: T).(pr0 t (lift h d t4))) (eq_ind nat (plus (S O) d) -(\lambda (n: nat).(pr0 (THead (Bind b) (lift h d u) (lift h n (lift (S O) O -t3))) (lift h d t4))) (eq_ind_r T (lift (S O) O (lift h d t3)) (\lambda (t: -T).(pr0 (THead (Bind b) (lift h d u) t) (lift h d t4))) (pr0_zeta b H0 (lift -h d t3) (lift h d t4) (H2 h d) (lift h d u)) (lift h (plus (S O) d) (lift (S -O) O t3)) (lift_d t3 h (S O) d O (le_O_n d))) (S d) (refl_equal nat (S d))) -(lift h d (THead (Bind b) u (lift (S O) O t3))) (lift_head (Bind b) u (lift -(S O) O t3) h d))))))))))) (\lambda (t3: T).(\lambda (t4: T).(\lambda (_: -(pr0 t3 t4)).(\lambda (H1: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h -d t3) (lift h d t4)))))).(\lambda (u: T).(\lambda (h: nat).(\lambda (d: -nat).(eq_ind_r T (THead (Flat Cast) (lift h d u) (lift h (s (Flat Cast) d) -t3)) (\lambda (t: T).(pr0 t (lift h d t4))) (pr0_tau (lift h (s (Flat Cast) -d) t3) (lift h d t4) (H1 h d) (lift h d u)) (lift h d (THead (Flat Cast) u -t3)) (lift_head (Flat Cast) u t3 h d))))))))) t1 t2 H))). -(* COMMENTS -Initial nodes: 2845 -END *) - -theorem pr0_subst0_back: - \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst0 -i u2 t1 t2) \to (\forall (u1: T).((pr0 u1 u2) \to (ex2 T (\lambda (t: -T).(subst0 i u1 t1 t)) (\lambda (t: T).(pr0 t t2))))))))) -\def - \lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (i: nat).(\lambda -(H: (subst0 i u2 t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t: -T).(\lambda (t0: T).(\lambda (t3: T).(\forall (u1: T).((pr0 u1 t) \to (ex2 T -(\lambda (t4: T).(subst0 n u1 t0 t4)) (\lambda (t4: T).(pr0 t4 t3))))))))) -(\lambda (v: T).(\lambda (i0: nat).(\lambda (u1: T).(\lambda (H0: (pr0 u1 -v)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1 (TLRef i0) t)) (\lambda (t: -T).(pr0 t (lift (S i0) O v))) (lift (S i0) O u1) (subst0_lref u1 i0) -(pr0_lift u1 v H0 (S i0) O)))))) (\lambda (v: T).(\lambda (u3: T).(\lambda -(u1: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v u1 u3)).(\lambda (H1: -((\forall (u4: T).((pr0 u4 v) \to (ex2 T (\lambda (t: T).(subst0 i0 u4 u1 t)) -(\lambda (t: T).(pr0 t u3))))))).(\lambda (t: T).(\lambda (k: K).(\lambda -(u0: T).(\lambda (H2: (pr0 u0 v)).(ex2_ind T (\lambda (t0: T).(subst0 i0 u0 -u1 t0)) (\lambda (t0: T).(pr0 t0 u3)) (ex2 T (\lambda (t0: T).(subst0 i0 u0 -(THead k u1 t) t0)) (\lambda (t0: T).(pr0 t0 (THead k u3 t)))) (\lambda (x: -T).(\lambda (H3: (subst0 i0 u0 u1 x)).(\lambda (H4: (pr0 x u3)).(ex_intro2 T -(\lambda (t0: T).(subst0 i0 u0 (THead k u1 t) t0)) (\lambda (t0: T).(pr0 t0 -(THead k u3 t))) (THead k x t) (subst0_fst u0 x u1 i0 H3 t k) (pr0_comp x u3 -H4 t t (pr0_refl t) k))))) (H1 u0 H2)))))))))))) (\lambda (k: K).(\lambda (v: -T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (i0: nat).(\lambda (_: (subst0 -(s k i0) v t4 t3)).(\lambda (H1: ((\forall (u1: T).((pr0 u1 v) \to (ex2 T -(\lambda (t: T).(subst0 (s k i0) u1 t4 t)) (\lambda (t: T).(pr0 t -t3))))))).(\lambda (u: T).(\lambda (u1: T).(\lambda (H2: (pr0 u1 v)).(ex2_ind -T (\lambda (t: T).(subst0 (s k i0) u1 t4 t)) (\lambda (t: T).(pr0 t t3)) (ex2 -T (\lambda (t: T).(subst0 i0 u1 (THead k u t4) t)) (\lambda (t: T).(pr0 t -(THead k u t3)))) (\lambda (x: T).(\lambda (H3: (subst0 (s k i0) u1 t4 -x)).(\lambda (H4: (pr0 x t3)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1 -(THead k u t4) t)) (\lambda (t: T).(pr0 t (THead k u t3))) (THead k u x) -(subst0_snd k u1 x t4 i0 H3 u) (pr0_comp u u (pr0_refl u) x t3 H4 k))))) (H1 -u1 H2)))))))))))) (\lambda (v: T).(\lambda (u1: T).(\lambda (u3: T).(\lambda -(i0: nat).(\lambda (_: (subst0 i0 v u1 u3)).(\lambda (H1: ((\forall (u4: -T).((pr0 u4 v) \to (ex2 T (\lambda (t: T).(subst0 i0 u4 u1 t)) (\lambda (t: -T).(pr0 t u3))))))).(\lambda (k: K).(\lambda (t3: T).(\lambda (t4: -T).(\lambda (_: (subst0 (s k i0) v t3 t4)).(\lambda (H3: ((\forall (u4: -T).((pr0 u4 v) \to (ex2 T (\lambda (t: T).(subst0 (s k i0) u4 t3 t)) (\lambda -(t: T).(pr0 t t4))))))).(\lambda (u0: T).(\lambda (H4: (pr0 u0 v)).(ex2_ind T -(\lambda (t: T).(subst0 (s k i0) u0 t3 t)) (\lambda (t: T).(pr0 t t4)) (ex2 T -(\lambda (t: T).(subst0 i0 u0 (THead k u1 t3) t)) (\lambda (t: T).(pr0 t -(THead k u3 t4)))) (\lambda (x: T).(\lambda (H5: (subst0 (s k i0) u0 t3 -x)).(\lambda (H6: (pr0 x t4)).(ex2_ind T (\lambda (t: T).(subst0 i0 u0 u1 t)) -(\lambda (t: T).(pr0 t u3)) (ex2 T (\lambda (t: T).(subst0 i0 u0 (THead k u1 -t3) t)) (\lambda (t: T).(pr0 t (THead k u3 t4)))) (\lambda (x0: T).(\lambda -(H7: (subst0 i0 u0 u1 x0)).(\lambda (H8: (pr0 x0 u3)).(ex_intro2 T (\lambda -(t: T).(subst0 i0 u0 (THead k u1 t3) t)) (\lambda (t: T).(pr0 t (THead k u3 -t4))) (THead k x0 x) (subst0_both u0 u1 x0 i0 H7 k t3 x H5) (pr0_comp x0 u3 -H8 x t4 H6 k))))) (H1 u0 H4))))) (H3 u0 H4))))))))))))))) i u2 t1 t2 H))))). -(* COMMENTS -Initial nodes: 979 -END *) - -theorem pr0_subst0_fwd: - \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst0 -i u2 t1 t2) \to (\forall (u1: T).((pr0 u2 u1) \to (ex2 T (\lambda (t: -T).(subst0 i u1 t1 t)) (\lambda (t: T).(pr0 t2 t))))))))) -\def - \lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (i: nat).(\lambda -(H: (subst0 i u2 t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t: -T).(\lambda (t0: T).(\lambda (t3: T).(\forall (u1: T).((pr0 t u1) \to (ex2 T -(\lambda (t4: T).(subst0 n u1 t0 t4)) (\lambda (t4: T).(pr0 t3 t4))))))))) -(\lambda (v: T).(\lambda (i0: nat).(\lambda (u1: T).(\lambda (H0: (pr0 v -u1)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1 (TLRef i0) t)) (\lambda (t: -T).(pr0 (lift (S i0) O v) t)) (lift (S i0) O u1) (subst0_lref u1 i0) -(pr0_lift v u1 H0 (S i0) O)))))) (\lambda (v: T).(\lambda (u3: T).(\lambda -(u1: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v u1 u3)).(\lambda (H1: -((\forall (u4: T).((pr0 v u4) \to (ex2 T (\lambda (t: T).(subst0 i0 u4 u1 t)) -(\lambda (t: T).(pr0 u3 t))))))).(\lambda (t: T).(\lambda (k: K).(\lambda -(u0: T).(\lambda (H2: (pr0 v u0)).(ex2_ind T (\lambda (t0: T).(subst0 i0 u0 -u1 t0)) (\lambda (t0: T).(pr0 u3 t0)) (ex2 T (\lambda (t0: T).(subst0 i0 u0 -(THead k u1 t) t0)) (\lambda (t0: T).(pr0 (THead k u3 t) t0))) (\lambda (x: -T).(\lambda (H3: (subst0 i0 u0 u1 x)).(\lambda (H4: (pr0 u3 x)).(ex_intro2 T -(\lambda (t0: T).(subst0 i0 u0 (THead k u1 t) t0)) (\lambda (t0: T).(pr0 -(THead k u3 t) t0)) (THead k x t) (subst0_fst u0 x u1 i0 H3 t k) (pr0_comp u3 -x H4 t t (pr0_refl t) k))))) (H1 u0 H2)))))))))))) (\lambda (k: K).(\lambda -(v: T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (i0: nat).(\lambda (_: -(subst0 (s k i0) v t4 t3)).(\lambda (H1: ((\forall (u1: T).((pr0 v u1) \to -(ex2 T (\lambda (t: T).(subst0 (s k i0) u1 t4 t)) (\lambda (t: T).(pr0 t3 -t))))))).(\lambda (u: T).(\lambda (u1: T).(\lambda (H2: (pr0 v u1)).(ex2_ind -T (\lambda (t: T).(subst0 (s k i0) u1 t4 t)) (\lambda (t: T).(pr0 t3 t)) (ex2 -T (\lambda (t: T).(subst0 i0 u1 (THead k u t4) t)) (\lambda (t: T).(pr0 -(THead k u t3) t))) (\lambda (x: T).(\lambda (H3: (subst0 (s k i0) u1 t4 -x)).(\lambda (H4: (pr0 t3 x)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1 -(THead k u t4) t)) (\lambda (t: T).(pr0 (THead k u t3) t)) (THead k u x) -(subst0_snd k u1 x t4 i0 H3 u) (pr0_comp u u (pr0_refl u) t3 x H4 k))))) (H1 -u1 H2)))))))))))) (\lambda (v: T).(\lambda (u1: T).(\lambda (u3: T).(\lambda -(i0: nat).(\lambda (_: (subst0 i0 v u1 u3)).(\lambda (H1: ((\forall (u4: -T).((pr0 v u4) \to (ex2 T (\lambda (t: T).(subst0 i0 u4 u1 t)) (\lambda (t: -T).(pr0 u3 t))))))).(\lambda (k: K).(\lambda (t3: T).(\lambda (t4: -T).(\lambda (_: (subst0 (s k i0) v t3 t4)).(\lambda (H3: ((\forall (u4: -T).((pr0 v u4) \to (ex2 T (\lambda (t: T).(subst0 (s k i0) u4 t3 t)) (\lambda -(t: T).(pr0 t4 t))))))).(\lambda (u0: T).(\lambda (H4: (pr0 v u0)).(ex2_ind T -(\lambda (t: T).(subst0 (s k i0) u0 t3 t)) (\lambda (t: T).(pr0 t4 t)) (ex2 T -(\lambda (t: T).(subst0 i0 u0 (THead k u1 t3) t)) (\lambda (t: T).(pr0 (THead -k u3 t4) t))) (\lambda (x: T).(\lambda (H5: (subst0 (s k i0) u0 t3 -x)).(\lambda (H6: (pr0 t4 x)).(ex2_ind T (\lambda (t: T).(subst0 i0 u0 u1 t)) -(\lambda (t: T).(pr0 u3 t)) (ex2 T (\lambda (t: T).(subst0 i0 u0 (THead k u1 -t3) t)) (\lambda (t: T).(pr0 (THead k u3 t4) t))) (\lambda (x0: T).(\lambda -(H7: (subst0 i0 u0 u1 x0)).(\lambda (H8: (pr0 u3 x0)).(ex_intro2 T (\lambda -(t: T).(subst0 i0 u0 (THead k u1 t3) t)) (\lambda (t: T).(pr0 (THead k u3 t4) -t)) (THead k x0 x) (subst0_both u0 u1 x0 i0 H7 k t3 x H5) (pr0_comp u3 x0 H8 -t4 x H6 k))))) (H1 u0 H4))))) (H3 u0 H4))))))))))))))) i u2 t1 t2 H))))). -(* COMMENTS -Initial nodes: 979 -END *) - -theorem pr0_subst0: - \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (v1: T).(\forall -(w1: T).(\forall (i: nat).((subst0 i v1 t1 w1) \to (\forall (v2: T).((pr0 v1 -v2) \to (or (pr0 w1 t2) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: -T).(subst0 i v2 t2 w2)))))))))))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(pr0_ind (\lambda -(t: T).(\lambda (t0: T).(\forall (v1: T).(\forall (w1: T).(\forall (i: -nat).((subst0 i v1 t w1) \to (\forall (v2: T).((pr0 v1 v2) \to (or (pr0 w1 -t0) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t0 -w2)))))))))))) (\lambda (t: T).(\lambda (v1: T).(\lambda (w1: T).(\lambda (i: -nat).(\lambda (H0: (subst0 i v1 t w1)).(\lambda (v2: T).(\lambda (H1: (pr0 v1 -v2)).(or_intror (pr0 w1 t) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: -T).(subst0 i v2 t w2))) (ex2_sym T (subst0 i v2 t) (pr0 w1) (pr0_subst0_fwd -v1 t w1 i H0 v2 H1)))))))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (H0: -(pr0 u1 u2)).(\lambda (H1: ((\forall (v1: T).(\forall (w1: T).(\forall (i: -nat).((subst0 i v1 u1 w1) \to (\forall (v2: T).((pr0 v1 v2) \to (or (pr0 w1 -u2) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 u2 -w2)))))))))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 -t4)).(\lambda (H3: ((\forall (v1: T).(\forall (w1: T).(\forall (i: -nat).((subst0 i v1 t3 w1) \to (\forall (v2: T).((pr0 v1 v2) \to (or (pr0 w1 -t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 -w2)))))))))))).(\lambda (k: K).(\lambda (v1: T).(\lambda (w1: T).(\lambda (i: -nat).(\lambda (H4: (subst0 i v1 (THead k u1 t3) w1)).(\lambda (v2: -T).(\lambda (H5: (pr0 v1 v2)).(or3_ind (ex2 T (\lambda (u3: T).(eq T w1 -(THead k u3 t3))) (\lambda (u3: T).(subst0 i v1 u1 u3))) (ex2 T (\lambda (t5: -T).(eq T w1 (THead k u1 t5))) (\lambda (t5: T).(subst0 (s k i) v1 t3 t5))) -(ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T w1 (THead k u3 t5)))) -(\lambda (u3: T).(\lambda (_: T).(subst0 i v1 u1 u3))) (\lambda (_: -T).(\lambda (t5: T).(subst0 (s k i) v1 t3 t5)))) (or (pr0 w1 (THead k u2 t4)) -(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 (THead k -u2 t4) w2)))) (\lambda (H6: (ex2 T (\lambda (u3: T).(eq T w1 (THead k u3 -t3))) (\lambda (u3: T).(subst0 i v1 u1 u3)))).(ex2_ind T (\lambda (u3: T).(eq -T w1 (THead k u3 t3))) (\lambda (u3: T).(subst0 i v1 u1 u3)) (or (pr0 w1 -(THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: -T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x: T).(\lambda (H7: (eq T w1 -(THead k x t3))).(\lambda (H8: (subst0 i v1 u1 x)).(eq_ind_r T (THead k x t3) -(\lambda (t: T).(or (pr0 t (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 t -w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))))) (or_ind (pr0 x u2) -(ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 u2 w2))) -(or (pr0 (THead k x t3) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead -k x t3) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda -(H9: (pr0 x u2)).(or_introl (pr0 (THead k x t3) (THead k u2 t4)) (ex2 T -(\lambda (w2: T).(pr0 (THead k x t3) w2)) (\lambda (w2: T).(subst0 i v2 -(THead k u2 t4) w2))) (pr0_comp x u2 H9 t3 t4 H2 k))) (\lambda (H9: (ex2 T -(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 u2 w2)))).(ex2_ind -T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 u2 w2)) (or (pr0 -(THead k x t3) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x t3) -w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x0: -T).(\lambda (H10: (pr0 x x0)).(\lambda (H11: (subst0 i v2 u2 x0)).(or_intror -(pr0 (THead k x t3) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x -t3) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))) (ex_intro2 T -(\lambda (w2: T).(pr0 (THead k x t3) w2)) (\lambda (w2: T).(subst0 i v2 -(THead k u2 t4) w2)) (THead k x0 t4) (pr0_comp x x0 H10 t3 t4 H2 k) -(subst0_fst v2 x0 u2 i H11 t4 k)))))) H9)) (H1 v1 x i H8 v2 H5)) w1 H7)))) -H6)) (\lambda (H6: (ex2 T (\lambda (t5: T).(eq T w1 (THead k u1 t5))) -(\lambda (t5: T).(subst0 (s k i) v1 t3 t5)))).(ex2_ind T (\lambda (t5: T).(eq -T w1 (THead k u1 t5))) (\lambda (t5: T).(subst0 (s k i) v1 t3 t5)) (or (pr0 -w1 (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: -T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x: T).(\lambda (H7: (eq T w1 -(THead k u1 x))).(\lambda (H8: (subst0 (s k i) v1 t3 x)).(eq_ind_r T (THead k -u1 x) (\lambda (t: T).(or (pr0 t (THead k u2 t4)) (ex2 T (\lambda (w2: -T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))))) (or_ind -(pr0 x t4) (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s k -i) v2 t4 w2))) (or (pr0 (THead k u1 x) (THead k u2 t4)) (ex2 T (\lambda (w2: -T).(pr0 (THead k u1 x) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) -w2)))) (\lambda (H9: (pr0 x t4)).(or_introl (pr0 (THead k u1 x) (THead k u2 -t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k u1 x) w2)) (\lambda (w2: -T).(subst0 i v2 (THead k u2 t4) w2))) (pr0_comp u1 u2 H0 x t4 H9 k))) -(\lambda (H9: (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s -k i) v2 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: -T).(subst0 (s k i) v2 t4 w2)) (or (pr0 (THead k u1 x) (THead k u2 t4)) (ex2 T -(\lambda (w2: T).(pr0 (THead k u1 x) w2)) (\lambda (w2: T).(subst0 i v2 -(THead k u2 t4) w2)))) (\lambda (x0: T).(\lambda (H10: (pr0 x x0)).(\lambda -(H11: (subst0 (s k i) v2 t4 x0)).(or_intror (pr0 (THead k u1 x) (THead k u2 -t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k u1 x) w2)) (\lambda (w2: -T).(subst0 i v2 (THead k u2 t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 -(THead k u1 x) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)) (THead -k u2 x0) (pr0_comp u1 u2 H0 x x0 H10 k) (subst0_snd k v2 x0 t4 i H11 u2)))))) -H9)) (H3 v1 x (s k i) H8 v2 H5)) w1 H7)))) H6)) (\lambda (H6: (ex3_2 T T -(\lambda (u3: T).(\lambda (t5: T).(eq T w1 (THead k u3 t5)))) (\lambda (u3: -T).(\lambda (_: T).(subst0 i v1 u1 u3))) (\lambda (_: T).(\lambda (t5: -T).(subst0 (s k i) v1 t3 t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda -(t5: T).(eq T w1 (THead k u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 -i v1 u1 u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s k i) v1 t3 t5))) -(or (pr0 w1 (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda -(w2: T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H7: (eq T w1 (THead k x0 x1))).(\lambda (H8: (subst0 i v1 u1 -x0)).(\lambda (H9: (subst0 (s k i) v1 t3 x1)).(eq_ind_r T (THead k x0 x1) -(\lambda (t: T).(or (pr0 t (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 t -w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))))) (or_ind (pr0 x1 -t4) (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s k i) v2 -t4 w2))) (or (pr0 (THead k x0 x1) (THead k u2 t4)) (ex2 T (\lambda (w2: -T).(pr0 (THead k x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) -w2)))) (\lambda (H10: (pr0 x1 t4)).(or_ind (pr0 x0 u2) (ex2 T (\lambda (w2: -T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 u2 w2))) (or (pr0 (THead k x0 -x1) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x0 x1) w2)) -(\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (H11: (pr0 x0 -u2)).(or_introl (pr0 (THead k x0 x1) (THead k u2 t4)) (ex2 T (\lambda (w2: -T).(pr0 (THead k x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) -w2))) (pr0_comp x0 u2 H11 x1 t4 H10 k))) (\lambda (H11: (ex2 T (\lambda (w2: -T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 u2 w2)))).(ex2_ind T (\lambda -(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 u2 w2)) (or (pr0 (THead k -x0 x1) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x0 x1) w2)) -(\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x: T).(\lambda -(H12: (pr0 x0 x)).(\lambda (H13: (subst0 i v2 u2 x)).(or_intror (pr0 (THead k -x0 x1) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x0 x1) w2)) -(\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))) (ex_intro2 T (\lambda -(w2: T).(pr0 (THead k x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 -t4) w2)) (THead k x t4) (pr0_comp x0 x H12 x1 t4 H10 k) (subst0_fst v2 x u2 i -H13 t4 k)))))) H11)) (H1 v1 x0 i H8 v2 H5))) (\lambda (H10: (ex2 T (\lambda -(w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s k i) v2 t4 w2)))).(ex2_ind T -(\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s k i) v2 t4 w2)) (or -(pr0 (THead k x0 x1) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k -x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x: -T).(\lambda (H11: (pr0 x1 x)).(\lambda (H12: (subst0 (s k i) v2 t4 -x)).(or_ind (pr0 x0 u2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: -T).(subst0 i v2 u2 w2))) (or (pr0 (THead k x0 x1) (THead k u2 t4)) (ex2 T -(\lambda (w2: T).(pr0 (THead k x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 -(THead k u2 t4) w2)))) (\lambda (H13: (pr0 x0 u2)).(or_intror (pr0 (THead k -x0 x1) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x0 x1) w2)) -(\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))) (ex_intro2 T (\lambda -(w2: T).(pr0 (THead k x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 -t4) w2)) (THead k u2 x) (pr0_comp x0 u2 H13 x1 x H11 k) (subst0_snd k v2 x t4 -i H12 u2)))) (\lambda (H13: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda -(w2: T).(subst0 i v2 u2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) -(\lambda (w2: T).(subst0 i v2 u2 w2)) (or (pr0 (THead k x0 x1) (THead k u2 -t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x0 x1) w2)) (\lambda (w2: -T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x2: T).(\lambda (H14: (pr0 -x0 x2)).(\lambda (H15: (subst0 i v2 u2 x2)).(or_intror (pr0 (THead k x0 x1) -(THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x0 x1) w2)) (\lambda -(w2: T).(subst0 i v2 (THead k u2 t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 -(THead k x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)) -(THead k x2 x) (pr0_comp x0 x2 H14 x1 x H11 k) (subst0_both v2 u2 x2 i H15 k -t4 x H12)))))) H13)) (H1 v1 x0 i H8 v2 H5))))) H10)) (H3 v1 x1 (s k i) H9 v2 -H5)) w1 H7)))))) H6)) (subst0_gen_head k v1 u1 t3 w1 i H4))))))))))))))))) -(\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H0: (pr0 v1 -v2)).(\lambda (H1: ((\forall (v3: T).(\forall (w1: T).(\forall (i: -nat).((subst0 i v3 v1 w1) \to (\forall (v4: T).((pr0 v3 v4) \to (or (pr0 w1 -v2) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v4 v2 -w2)))))))))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 -t4)).(\lambda (H3: ((\forall (v3: T).(\forall (w1: T).(\forall (i: -nat).((subst0 i v3 t3 w1) \to (\forall (v4: T).((pr0 v3 v4) \to (or (pr0 w1 -t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v4 t4 -w2)))))))))))).(\lambda (v0: T).(\lambda (w1: T).(\lambda (i: nat).(\lambda -(H4: (subst0 i v0 (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) -w1)).(\lambda (v3: T).(\lambda (H5: (pr0 v0 v3)).(or3_ind (ex2 T (\lambda -(u2: T).(eq T w1 (THead (Flat Appl) u2 (THead (Bind Abst) u t3)))) (\lambda -(u2: T).(subst0 i v0 v1 u2))) (ex2 T (\lambda (t5: T).(eq T w1 (THead (Flat -Appl) v1 t5))) (\lambda (t5: T).(subst0 (s (Flat Appl) i) v0 (THead (Bind -Abst) u t3) t5))) (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T w1 -(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 v1 -u2))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Flat Appl) i) v0 (THead -(Bind Abst) u t3) t5)))) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T -(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind -Abbr) v2 t4) w2)))) (\lambda (H6: (ex2 T (\lambda (u2: T).(eq T w1 (THead -(Flat Appl) u2 (THead (Bind Abst) u t3)))) (\lambda (u2: T).(subst0 i v0 v1 -u2)))).(ex2_ind T (\lambda (u2: T).(eq T w1 (THead (Flat Appl) u2 (THead -(Bind Abst) u t3)))) (\lambda (u2: T).(subst0 i v0 v1 u2)) (or (pr0 w1 (THead -(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (x: T).(\lambda -(H7: (eq T w1 (THead (Flat Appl) x (THead (Bind Abst) u t3)))).(\lambda (H8: -(subst0 i v0 v1 x)).(eq_ind_r T (THead (Flat Appl) x (THead (Bind Abst) u -t3)) (\lambda (t: T).(or (pr0 t (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda -(w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) -w2))))) (or_ind (pr0 x v2) (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: -T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat Appl) x (THead (Bind Abst) u -t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat -Appl) x (THead (Bind Abst) u t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead -(Bind Abbr) v2 t4) w2)))) (\lambda (H9: (pr0 x v2)).(or_introl (pr0 (THead -(Flat Appl) x (THead (Bind Abst) u t3)) (THead (Bind Abbr) v2 t4)) (ex2 T -(\lambda (w2: T).(pr0 (THead (Flat Appl) x (THead (Bind Abst) u t3)) w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (pr0_beta u x -v2 H9 t3 t4 H2))) (\lambda (H9: (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda -(w2: T).(subst0 i v3 v2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) -(\lambda (w2: T).(subst0 i v3 v2 w2)) (or (pr0 (THead (Flat Appl) x (THead -(Bind Abst) u t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 -(THead (Flat Appl) x (THead (Bind Abst) u t3)) w2)) (\lambda (w2: T).(subst0 -i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (x0: T).(\lambda (H10: (pr0 x -x0)).(\lambda (H11: (subst0 i v3 v2 x0)).(or_intror (pr0 (THead (Flat Appl) x -(THead (Bind Abst) u t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x (THead (Bind Abst) u t3)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x (THead (Bind Abst) u t3)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)) (THead (Bind Abbr) x0 t4) -(pr0_beta u x x0 H10 t3 t4 H2) (subst0_fst v3 x0 v2 i H11 t4 (Bind -Abbr))))))) H9)) (H1 v0 x i H8 v3 H5)) w1 H7)))) H6)) (\lambda (H6: (ex2 T -(\lambda (t5: T).(eq T w1 (THead (Flat Appl) v1 t5))) (\lambda (t5: -T).(subst0 (s (Flat Appl) i) v0 (THead (Bind Abst) u t3) t5)))).(ex2_ind T -(\lambda (t5: T).(eq T w1 (THead (Flat Appl) v1 t5))) (\lambda (t5: -T).(subst0 (s (Flat Appl) i) v0 (THead (Bind Abst) u t3) t5)) (or (pr0 w1 -(THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (x: T).(\lambda -(H7: (eq T w1 (THead (Flat Appl) v1 x))).(\lambda (H8: (subst0 (s (Flat Appl) -i) v0 (THead (Bind Abst) u t3) x)).(or3_ind (ex2 T (\lambda (u2: T).(eq T x -(THead (Bind Abst) u2 t3))) (\lambda (u2: T).(subst0 (s (Flat Appl) i) v0 u -u2))) (ex2 T (\lambda (t5: T).(eq T x (THead (Bind Abst) u t5))) (\lambda -(t5: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5))) (ex3_2 T T -(\lambda (u2: T).(\lambda (t5: T).(eq T x (THead (Bind Abst) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u u2))) -(\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 -t3 t5)))) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 -w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) -(\lambda (H9: (ex2 T (\lambda (u2: T).(eq T x (THead (Bind Abst) u2 t3))) -(\lambda (u2: T).(subst0 (s (Flat Appl) i) v0 u u2)))).(ex2_ind T (\lambda -(u2: T).(eq T x (THead (Bind Abst) u2 t3))) (\lambda (u2: T).(subst0 (s (Flat -Appl) i) v0 u u2)) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda -(w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) -w2)))) (\lambda (x0: T).(\lambda (H10: (eq T x (THead (Bind Abst) x0 -t3))).(\lambda (_: (subst0 (s (Flat Appl) i) v0 u x0)).(let H12 \def (eq_ind -T x (\lambda (t: T).(eq T w1 (THead (Flat Appl) v1 t))) H7 (THead (Bind Abst) -x0 t3) H10) in (eq_ind_r T (THead (Flat Appl) v1 (THead (Bind Abst) x0 t3)) -(\lambda (t: T).(or (pr0 t (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: -T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) -w2))))) (or_introl (pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 t3)) -(THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 -(THead (Bind Abst) x0 t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind -Abbr) v2 t4) w2))) (pr0_beta x0 v1 v2 H0 t3 t4 H2)) w1 H12))))) H9)) (\lambda -(H9: (ex2 T (\lambda (t5: T).(eq T x (THead (Bind Abst) u t5))) (\lambda (t5: -T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5)))).(ex2_ind T (\lambda -(t5: T).(eq T x (THead (Bind Abst) u t5))) (\lambda (t5: T).(subst0 (s (Bind -Abst) (s (Flat Appl) i)) v0 t3 t5)) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) -(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead -(Bind Abbr) v2 t4) w2)))) (\lambda (x0: T).(\lambda (H10: (eq T x (THead -(Bind Abst) u x0))).(\lambda (H11: (subst0 (s (Bind Abst) (s (Flat Appl) i)) -v0 t3 x0)).(let H12 \def (eq_ind T x (\lambda (t: T).(eq T w1 (THead (Flat -Appl) v1 t))) H7 (THead (Bind Abst) u x0) H10) in (eq_ind_r T (THead (Flat -Appl) v1 (THead (Bind Abst) u x0)) (\lambda (t: T).(or (pr0 t (THead (Bind -Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i -v3 (THead (Bind Abbr) v2 t4) w2))))) (or_ind (pr0 x0 t4) (ex2 T (\lambda (w2: -T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 -t4 w2))) (or (pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u x0)) (THead -(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead -(Bind Abst) u x0)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 -t4) w2)))) (\lambda (H13: (pr0 x0 t4)).(or_introl (pr0 (THead (Flat Appl) v1 -(THead (Bind Abst) u x0)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u x0)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (pr0_beta u v1 v2 H0 x0 t4 -H13))) (\lambda (H13: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: -T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda -(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) -i)) v3 t4 w2)) (or (pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u x0)) -(THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 -(THead (Bind Abst) u x0)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind -Abbr) v2 t4) w2)))) (\lambda (x1: T).(\lambda (H14: (pr0 x0 x1)).(\lambda -(H15: (subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 x1)).(or_intror (pr0 -(THead (Flat Appl) v1 (THead (Bind Abst) u x0)) (THead (Bind Abbr) v2 t4)) -(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u x0)) -w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (ex_intro2 -T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u x0)) w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)) (THead (Bind -Abbr) v2 x1) (pr0_beta u v1 v2 H0 x0 x1 H14) (subst0_snd (Bind Abbr) v3 x1 t4 -i H15 v2)))))) H13)) (H3 v0 x0 (s (Bind Abst) (s (Flat Appl) i)) H11 v3 H5)) -w1 H12))))) H9)) (\lambda (H9: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: -T).(eq T x (THead (Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 (s (Flat Appl) i) v0 u u2))) (\lambda (_: T).(\lambda (t5: -T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5))))).(ex3_2_ind T T -(\lambda (u2: T).(\lambda (t5: T).(eq T x (THead (Bind Abst) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u u2))) -(\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 -t3 t5))) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 -w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) -(\lambda (x0: T).(\lambda (x1: T).(\lambda (H10: (eq T x (THead (Bind Abst) -x0 x1))).(\lambda (_: (subst0 (s (Flat Appl) i) v0 u x0)).(\lambda (H12: -(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 x1)).(let H13 \def (eq_ind T -x (\lambda (t: T).(eq T w1 (THead (Flat Appl) v1 t))) H7 (THead (Bind Abst) -x0 x1) H10) in (eq_ind_r T (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) -(\lambda (t: T).(or (pr0 t (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: -T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) -w2))))) (or_ind (pr0 x1 t4) (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda -(w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2))) (or (pr0 (THead -(Flat Appl) v1 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) v2 t4)) (ex2 T -(\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H14: -(pr0 x1 t4)).(or_introl (pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) -(THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 -(THead (Bind Abst) x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind -Abbr) v2 t4) w2))) (pr0_beta x0 v1 v2 H0 x1 t4 H14))) (\lambda (H14: (ex2 T -(\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s -(Flat Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x1 w2)) -(\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2)) (or -(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) v2 -t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 -x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) -(\lambda (x2: T).(\lambda (H15: (pr0 x1 x2)).(\lambda (H16: (subst0 (s (Bind -Abst) (s (Flat Appl) i)) v3 t4 x2)).(or_intror (pr0 (THead (Flat Appl) v1 -(THead (Bind Abst) x0 x1)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)) (THead (Bind Abbr) v2 x2) -(pr0_beta x0 v1 v2 H0 x1 x2 H15) (subst0_snd (Bind Abbr) v3 x2 t4 i H16 -v2)))))) H14)) (H3 v0 x1 (s (Bind Abst) (s (Flat Appl) i)) H12 v3 H5)) w1 -H13))))))) H9)) (subst0_gen_head (Bind Abst) v0 u t3 x (s (Flat Appl) i) -H8))))) H6)) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T -w1 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 -v1 u2))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Flat Appl) i) v0 (THead -(Bind Abst) u t3) t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: -T).(eq T w1 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 i v0 v1 u2))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Flat -Appl) i) v0 (THead (Bind Abst) u t3) t5))) (or (pr0 w1 (THead (Bind Abbr) v2 -t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 -(THead (Bind Abbr) v2 t4) w2)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda -(H7: (eq T w1 (THead (Flat Appl) x0 x1))).(\lambda (H8: (subst0 i v0 v1 -x0)).(\lambda (H9: (subst0 (s (Flat Appl) i) v0 (THead (Bind Abst) u t3) -x1)).(or3_ind (ex2 T (\lambda (u2: T).(eq T x1 (THead (Bind Abst) u2 t3))) -(\lambda (u2: T).(subst0 (s (Flat Appl) i) v0 u u2))) (ex2 T (\lambda (t5: -T).(eq T x1 (THead (Bind Abst) u t5))) (\lambda (t5: T).(subst0 (s (Bind -Abst) (s (Flat Appl) i)) v0 t3 t5))) (ex3_2 T T (\lambda (u2: T).(\lambda -(t5: T).(eq T x1 (THead (Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 (s (Flat Appl) i) v0 u u2))) (\lambda (_: T).(\lambda (t5: -T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5)))) (or (pr0 w1 (THead -(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H10: (ex2 T -(\lambda (u2: T).(eq T x1 (THead (Bind Abst) u2 t3))) (\lambda (u2: -T).(subst0 (s (Flat Appl) i) v0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T x1 -(THead (Bind Abst) u2 t3))) (\lambda (u2: T).(subst0 (s (Flat Appl) i) v0 u -u2)) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 -w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda -(x: T).(\lambda (H11: (eq T x1 (THead (Bind Abst) x t3))).(\lambda (_: -(subst0 (s (Flat Appl) i) v0 u x)).(let H13 \def (eq_ind T x1 (\lambda (t: -T).(eq T w1 (THead (Flat Appl) x0 t))) H7 (THead (Bind Abst) x t3) H11) in -(eq_ind_r T (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) (\lambda (t: -T).(or (pr0 t (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 t w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))))) (or_ind (pr0 -x0 v2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 -w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) (THead (Bind -Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind -Abst) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) -w2)))) (\lambda (H14: (pr0 x0 v2)).(or_introl (pr0 (THead (Flat Appl) x0 -(THead (Bind Abst) x t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (pr0_beta x x0 v2 H14 t3 t4 -H2))) (\lambda (H14: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: -T).(subst0 i v3 v2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda -(w2: T).(subst0 i v3 v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind -Abst) x t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead -(Flat Appl) x0 (THead (Bind Abst) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 -(THead (Bind Abbr) v2 t4) w2)))) (\lambda (x2: T).(\lambda (H15: (pr0 x0 -x2)).(\lambda (H16: (subst0 i v3 v2 x2)).(or_intror (pr0 (THead (Flat Appl) -x0 (THead (Bind Abst) x t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)) (THead (Bind Abbr) x2 t4) -(pr0_beta x x0 x2 H15 t3 t4 H2) (subst0_fst v3 x2 v2 i H16 t4 (Bind -Abbr))))))) H14)) (H1 v0 x0 i H8 v3 H5)) w1 H13))))) H10)) (\lambda (H10: -(ex2 T (\lambda (t5: T).(eq T x1 (THead (Bind Abst) u t5))) (\lambda (t5: -T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5)))).(ex2_ind T (\lambda -(t5: T).(eq T x1 (THead (Bind Abst) u t5))) (\lambda (t5: T).(subst0 (s (Bind -Abst) (s (Flat Appl) i)) v0 t3 t5)) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) -(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead -(Bind Abbr) v2 t4) w2)))) (\lambda (x: T).(\lambda (H11: (eq T x1 (THead -(Bind Abst) u x))).(\lambda (H12: (subst0 (s (Bind Abst) (s (Flat Appl) i)) -v0 t3 x)).(let H13 \def (eq_ind T x1 (\lambda (t: T).(eq T w1 (THead (Flat -Appl) x0 t))) H7 (THead (Bind Abst) u x) H11) in (eq_ind_r T (THead (Flat -Appl) x0 (THead (Bind Abst) u x)) (\lambda (t: T).(or (pr0 t (THead (Bind -Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i -v3 (THead (Bind Abbr) v2 t4) w2))))) (or_ind (pr0 x t4) (ex2 T (\lambda (w2: -T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 -t4 w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind -Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind -Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) -w2)))) (\lambda (H14: (pr0 x t4)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2: -T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat -Appl) x0 (THead (Bind Abst) u x)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda -(w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) w2)) (\lambda -(w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H15: (pr0 x0 -v2)).(or_introl (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead -(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead -(Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 -t4) w2))) (pr0_beta u x0 v2 H15 x t4 H14))) (\lambda (H15: (ex2 T (\lambda -(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)))).(ex2_ind T -(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)) (or (pr0 -(THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind Abbr) v2 t4)) -(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) -w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda -(x2: T).(\lambda (H16: (pr0 x0 x2)).(\lambda (H17: (subst0 i v3 v2 -x2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead -(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead -(Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 -t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead -(Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 -t4) w2)) (THead (Bind Abbr) x2 t4) (pr0_beta u x0 x2 H16 x t4 H14) -(subst0_fst v3 x2 v2 i H17 t4 (Bind Abbr))))))) H15)) (H1 v0 x0 i H8 v3 H5))) -(\lambda (H14: (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 -(s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda (w2: -T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 -t4 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind -Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind -Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) -w2)))) (\lambda (x2: T).(\lambda (H15: (pr0 x x2)).(\lambda (H16: (subst0 (s -(Bind Abst) (s (Flat Appl) i)) v3 t4 x2)).(or_ind (pr0 x0 v2) (ex2 T (\lambda -(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead -(Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind Abbr) v2 t4)) (ex2 T -(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H17: -(pr0 x0 v2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) -(THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 -(THead (Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind -Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 -(THead (Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind -Abbr) v2 t4) w2)) (THead (Bind Abbr) v2 x2) (pr0_beta u x0 v2 H17 x x2 H15) -(subst0_snd (Bind Abbr) v3 x2 t4 i H16 v2)))) (\lambda (H17: (ex2 T (\lambda -(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)))).(ex2_ind T -(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)) (or (pr0 -(THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind Abbr) v2 t4)) -(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) -w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda -(x3: T).(\lambda (H18: (pr0 x0 x3)).(\lambda (H19: (subst0 i v3 v2 -x3)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead -(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead -(Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 -t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead -(Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 -t4) w2)) (THead (Bind Abbr) x3 x2) (pr0_beta u x0 x3 H18 x x2 H15) -(subst0_both v3 v2 x3 i H19 (Bind Abbr) t4 x2 H16)))))) H17)) (H1 v0 x0 i H8 -v3 H5))))) H14)) (H3 v0 x (s (Bind Abst) (s (Flat Appl) i)) H12 v3 H5)) w1 -H13))))) H10)) (\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: -T).(eq T x1 (THead (Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 (s (Flat Appl) i) v0 u u2))) (\lambda (_: T).(\lambda (t5: -T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5))))).(ex3_2_ind T T -(\lambda (u2: T).(\lambda (t5: T).(eq T x1 (THead (Bind Abst) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u u2))) -(\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 -t3 t5))) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 -w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) -(\lambda (x2: T).(\lambda (x3: T).(\lambda (H11: (eq T x1 (THead (Bind Abst) -x2 x3))).(\lambda (_: (subst0 (s (Flat Appl) i) v0 u x2)).(\lambda (H13: -(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 x3)).(let H14 \def (eq_ind T -x1 (\lambda (t: T).(eq T w1 (THead (Flat Appl) x0 t))) H7 (THead (Bind Abst) -x2 x3) H11) in (eq_ind_r T (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) -(\lambda (t: T).(or (pr0 t (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: -T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) -w2))))) (or_ind (pr0 x3 t4) (ex2 T (\lambda (w2: T).(pr0 x3 w2)) (\lambda -(w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2))) (or (pr0 (THead -(Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead (Bind Abbr) v2 t4)) (ex2 T -(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H15: -(pr0 x3 t4)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) -(\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat Appl) x0 (THead -(Bind Abst) x2 x3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 -(THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H16: (pr0 x0 -v2)).(or_introl (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead -(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead -(Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 -t4) w2))) (pr0_beta x2 x0 v2 H16 x3 t4 H15))) (\lambda (H16: (ex2 T (\lambda -(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)))).(ex2_ind T -(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)) (or (pr0 -(THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead (Bind Abbr) v2 t4)) -(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) -w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda -(x: T).(\lambda (H17: (pr0 x0 x)).(\lambda (H18: (subst0 i v3 v2 -x)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead -(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead -(Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 -t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead -(Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 -t4) w2)) (THead (Bind Abbr) x t4) (pr0_beta x2 x0 x H17 x3 t4 H15) -(subst0_fst v3 x v2 i H18 t4 (Bind Abbr))))))) H16)) (H1 v0 x0 i H8 v3 H5))) -(\lambda (H15: (ex2 T (\lambda (w2: T).(pr0 x3 w2)) (\lambda (w2: T).(subst0 -(s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda (w2: -T).(pr0 x3 w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 -t4 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead -(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead -(Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 -t4) w2)))) (\lambda (x: T).(\lambda (H16: (pr0 x3 x)).(\lambda (H17: (subst0 -(s (Bind Abst) (s (Flat Appl) i)) v3 t4 x)).(or_ind (pr0 x0 v2) (ex2 T -(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 -(THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead (Bind Abbr) v2 t4)) -(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) -w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda -(H18: (pr0 x0 v2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) -x2 x3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat -Appl) x0 (THead (Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead -(Bind Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat -Appl) x0 (THead (Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead -(Bind Abbr) v2 t4) w2)) (THead (Bind Abbr) v2 x) (pr0_beta x2 x0 v2 H18 x3 x -H16) (subst0_snd (Bind Abbr) v3 x t4 i H17 v2)))) (\lambda (H18: (ex2 T -(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 -w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 -v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead -(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead -(Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 -t4) w2)))) (\lambda (x4: T).(\lambda (H19: (pr0 x0 x4)).(\lambda (H20: -(subst0 i v3 v2 x4)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) -x2 x3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat -Appl) x0 (THead (Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead -(Bind Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat -Appl) x0 (THead (Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead -(Bind Abbr) v2 t4) w2)) (THead (Bind Abbr) x4 x) (pr0_beta x2 x0 x4 H19 x3 x -H16) (subst0_both v3 v2 x4 i H20 (Bind Abbr) t4 x H17)))))) H18)) (H1 v0 x0 i -H8 v3 H5))))) H15)) (H3 v0 x3 (s (Bind Abst) (s (Flat Appl) i)) H13 v3 H5)) -w1 H14))))))) H10)) (subst0_gen_head (Bind Abst) v0 u t3 x1 (s (Flat Appl) i) -H9))))))) H6)) (subst0_gen_head (Flat Appl) v0 v1 (THead (Bind Abst) u t3) w1 -i H4))))))))))))))))) (\lambda (b: B).(\lambda (H0: (not (eq B b -Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H1: (pr0 v1 v2)).(\lambda -(H2: ((\forall (v3: T).(\forall (w1: T).(\forall (i: nat).((subst0 i v3 v1 -w1) \to (\forall (v4: T).((pr0 v3 v4) \to (or (pr0 w1 v2) (ex2 T (\lambda -(w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v4 v2 w2)))))))))))).(\lambda -(u1: T).(\lambda (u2: T).(\lambda (H3: (pr0 u1 u2)).(\lambda (H4: ((\forall -(v3: T).(\forall (w1: T).(\forall (i: nat).((subst0 i v3 u1 w1) \to (\forall -(v4: T).((pr0 v3 v4) \to (or (pr0 w1 u2) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) -(\lambda (w2: T).(subst0 i v4 u2 w2)))))))))))).(\lambda (t3: T).(\lambda -(t4: T).(\lambda (H5: (pr0 t3 t4)).(\lambda (H6: ((\forall (v3: T).(\forall -(w1: T).(\forall (i: nat).((subst0 i v3 t3 w1) \to (\forall (v4: T).((pr0 v3 -v4) \to (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: -T).(subst0 i v4 t4 w2)))))))))))).(\lambda (v0: T).(\lambda (w1: T).(\lambda -(i: nat).(\lambda (H7: (subst0 i v0 (THead (Flat Appl) v1 (THead (Bind b) u1 -t3)) w1)).(\lambda (v3: T).(\lambda (H8: (pr0 v0 v3)).(or3_ind (ex2 T -(\lambda (u3: T).(eq T w1 (THead (Flat Appl) u3 (THead (Bind b) u1 t3)))) -(\lambda (u3: T).(subst0 i v0 v1 u3))) (ex2 T (\lambda (t5: T).(eq T w1 -(THead (Flat Appl) v1 t5))) (\lambda (t5: T).(subst0 (s (Flat Appl) i) v0 -(THead (Bind b) u1 t3) t5))) (ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq -T w1 (THead (Flat Appl) u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i -v0 v1 u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Flat Appl) i) v0 -(THead (Bind b) u1 t3) t5)))) (or (pr0 w1 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda -(w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) -t4)) w2)))) (\lambda (H9: (ex2 T (\lambda (u3: T).(eq T w1 (THead (Flat Appl) -u3 (THead (Bind b) u1 t3)))) (\lambda (u3: T).(subst0 i v0 v1 u3)))).(ex2_ind -T (\lambda (u3: T).(eq T w1 (THead (Flat Appl) u3 (THead (Bind b) u1 t3)))) -(\lambda (u3: T).(subst0 i v0 v1 u3)) (or (pr0 w1 (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t4)) w2)))) (\lambda (x: T).(\lambda (H10: (eq T w1 (THead (Flat -Appl) x (THead (Bind b) u1 t3)))).(\lambda (H11: (subst0 i v0 v1 -x)).(eq_ind_r T (THead (Flat Appl) x (THead (Bind b) u1 t3)) (\lambda (t: -T).(or (pr0 t (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) -(ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2))))) (or_ind (pr0 x -v2) (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v3 v2 -w2))) (or (pr0 (THead (Flat Appl) x (THead (Bind b) u1 t3)) (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 -(THead (Flat Appl) x (THead (Bind b) u1 t3)) w2)) (\lambda (w2: T).(subst0 i -v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) -(\lambda (H12: (pr0 x v2)).(or_introl (pr0 (THead (Flat Appl) x (THead (Bind -b) u1 t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 -T (\lambda (w2: T).(pr0 (THead (Flat Appl) x (THead (Bind b) u1 t3)) w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t4)) w2))) (pr0_upsilon b H0 x v2 H12 u1 u2 H3 t3 t4 H5))) (\lambda -(H12: (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v3 v2 -w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v3 -v2 w2)) (or (pr0 (THead (Flat Appl) x (THead (Bind b) u1 t3)) (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 -(THead (Flat Appl) x (THead (Bind b) u1 t3)) w2)) (\lambda (w2: T).(subst0 i -v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) -(\lambda (x0: T).(\lambda (H13: (pr0 x x0)).(\lambda (H14: (subst0 i v3 v2 -x0)).(or_intror (pr0 (THead (Flat Appl) x (THead (Bind b) u1 t3)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x (THead (Bind b) u1 t3)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x (THead (Bind b) -u1 t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 (THead (Flat Appl) (lift -(S O) O x0) t4)) (pr0_upsilon b H0 x x0 H13 u1 u2 H3 t3 t4 H5) (subst0_snd -(Bind b) v3 (THead (Flat Appl) (lift (S O) O x0) t4) (THead (Flat Appl) (lift -(S O) O v2) t4) i (subst0_fst v3 (lift (S O) O x0) (lift (S O) O v2) (s (Bind -b) i) (subst0_lift_ge_s v2 x0 v3 i H14 O (le_O_n i) b) t4 (Flat Appl)) -u2)))))) H12)) (H2 v0 x i H11 v3 H8)) w1 H10)))) H9)) (\lambda (H9: (ex2 T -(\lambda (t5: T).(eq T w1 (THead (Flat Appl) v1 t5))) (\lambda (t5: -T).(subst0 (s (Flat Appl) i) v0 (THead (Bind b) u1 t3) t5)))).(ex2_ind T -(\lambda (t5: T).(eq T w1 (THead (Flat Appl) v1 t5))) (\lambda (t5: -T).(subst0 (s (Flat Appl) i) v0 (THead (Bind b) u1 t3) t5)) (or (pr0 w1 -(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda -(w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x: T).(\lambda (H10: (eq -T w1 (THead (Flat Appl) v1 x))).(\lambda (H11: (subst0 (s (Flat Appl) i) v0 -(THead (Bind b) u1 t3) x)).(or3_ind (ex2 T (\lambda (u3: T).(eq T x (THead -(Bind b) u3 t3))) (\lambda (u3: T).(subst0 (s (Flat Appl) i) v0 u1 u3))) (ex2 -T (\lambda (t5: T).(eq T x (THead (Bind b) u1 t5))) (\lambda (t5: T).(subst0 -(s (Bind b) (s (Flat Appl) i)) v0 t3 t5))) (ex3_2 T T (\lambda (u3: -T).(\lambda (t5: T).(eq T x (THead (Bind b) u3 t5)))) (\lambda (u3: -T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u1 u3))) (\lambda (_: -T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 t5)))) (or -(pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T -(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H12: (ex2 T -(\lambda (u3: T).(eq T x (THead (Bind b) u3 t3))) (\lambda (u3: T).(subst0 (s -(Flat Appl) i) v0 u1 u3)))).(ex2_ind T (\lambda (u3: T).(eq T x (THead (Bind -b) u3 t3))) (\lambda (u3: T).(subst0 (s (Flat Appl) i) v0 u1 u3)) (or (pr0 w1 -(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda -(w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x0: T).(\lambda (H13: (eq -T x (THead (Bind b) x0 t3))).(\lambda (H14: (subst0 (s (Flat Appl) i) v0 u1 -x0)).(let H15 \def (eq_ind T x (\lambda (t: T).(eq T w1 (THead (Flat Appl) v1 -t))) H10 (THead (Bind b) x0 t3) H13) in (eq_ind_r T (THead (Flat Appl) v1 -(THead (Bind b) x0 t3)) (\lambda (t: T).(or (pr0 t (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 t w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t4)) w2))))) (or_ind (pr0 x0 u2) (ex2 T (\lambda (w2: T).(pr0 x0 -w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 u2 w2))) (or (pr0 (THead -(Flat Appl) v1 (THead (Bind b) x0 t3)) (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 -(THead (Bind b) x0 t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H16: (pr0 x0 -u2)).(or_introl (pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 t3)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 t3)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2))) (pr0_upsilon b H0 v1 v2 H1 x0 u2 H16 t3 t4 H5))) (\lambda (H16: (ex2 T -(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 -u2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 -(s (Flat Appl) i) v3 u2 w2)) (or (pr0 (THead (Flat Appl) v1 (THead (Bind b) -x0 t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T -(\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 t3)) w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t4)) w2)))) (\lambda (x1: T).(\lambda (H17: (pr0 x0 x1)).(\lambda -(H18: (subst0 (s (Flat Appl) i) v3 u2 x1)).(or_intror (pr0 (THead (Flat Appl) -v1 (THead (Bind b) x0 t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) -O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind -b) x0 t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead -(Flat Appl) v1 (THead (Bind b) x0 t3)) w2)) (\lambda (w2: T).(subst0 i v3 -(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)) (THead -(Bind b) x1 (THead (Flat Appl) (lift (S O) O v2) t4)) (pr0_upsilon b H0 v1 v2 -H1 x0 x1 H17 t3 t4 H5) (subst0_fst v3 x1 u2 i H18 (THead (Flat Appl) (lift (S -O) O v2) t4) (Bind b))))))) H16)) (H4 v0 x0 (s (Flat Appl) i) H14 v3 H8)) w1 -H15))))) H12)) (\lambda (H12: (ex2 T (\lambda (t5: T).(eq T x (THead (Bind b) -u1 t5))) (\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 -t5)))).(ex2_ind T (\lambda (t5: T).(eq T x (THead (Bind b) u1 t5))) (\lambda -(t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 t5)) (or (pr0 w1 (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x0: T).(\lambda (H13: (eq T x -(THead (Bind b) u1 x0))).(\lambda (H14: (subst0 (s (Bind b) (s (Flat Appl) -i)) v0 t3 x0)).(let H15 \def (eq_ind T x (\lambda (t: T).(eq T w1 (THead -(Flat Appl) v1 t))) H10 (THead (Bind b) u1 x0) H13) in (eq_ind_r T (THead -(Flat Appl) v1 (THead (Bind b) u1 x0)) (\lambda (t: T).(or (pr0 t (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) w2))))) (or_ind (pr0 x0 t4) (ex2 T (\lambda (w2: -T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 -w2))) (or (pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 x0)) (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 -(THead (Flat Appl) v1 (THead (Bind b) u1 x0)) w2)) (\lambda (w2: T).(subst0 i -v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) -(\lambda (H16: (pr0 x0 t4)).(or_introl (pr0 (THead (Flat Appl) v1 (THead -(Bind b) u1 x0)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) -t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 -x0)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4)) w2))) (pr0_upsilon b H0 v1 v2 H1 u1 u2 H3 x0 t4 H16))) -(\lambda (H16: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 -(s (Bind b) (s (Flat Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 -x0 w2)) (\lambda (w2: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 w2)) -(or (pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 x0)) (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 -(THead (Flat Appl) v1 (THead (Bind b) u1 x0)) w2)) (\lambda (w2: T).(subst0 i -v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) -(\lambda (x1: T).(\lambda (H17: (pr0 x0 x1)).(\lambda (H18: (subst0 (s (Bind -b) (s (Flat Appl) i)) v3 t4 x1)).(or_intror (pr0 (THead (Flat Appl) v1 (THead -(Bind b) u1 x0)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) -t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 -x0)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat -Appl) v1 (THead (Bind b) u1 x0)) w2)) (\lambda (w2: T).(subst0 i v3 (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) x1)) (pr0_upsilon b H0 v1 v2 H1 u1 u2 H3 -x0 x1 H17) (subst0_snd (Bind b) v3 (THead (Flat Appl) (lift (S O) O v2) x1) -(THead (Flat Appl) (lift (S O) O v2) t4) i (subst0_snd (Flat Appl) v3 x1 t4 -(s (Bind b) i) H18 (lift (S O) O v2)) u2)))))) H16)) (H6 v0 x0 (s (Bind b) (s -(Flat Appl) i)) H14 v3 H8)) w1 H15))))) H12)) (\lambda (H12: (ex3_2 T T -(\lambda (u3: T).(\lambda (t5: T).(eq T x (THead (Bind b) u3 t5)))) (\lambda -(u3: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u1 u3))) (\lambda (_: -T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 -t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t5: T).(eq T x (THead (Bind -b) u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u1 -u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) -v0 t3 t5))) (or (pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O -v2) t4))) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 -(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda -(x0: T).(\lambda (x1: T).(\lambda (H13: (eq T x (THead (Bind b) x0 -x1))).(\lambda (H14: (subst0 (s (Flat Appl) i) v0 u1 x0)).(\lambda (H15: -(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 x1)).(let H16 \def (eq_ind T x -(\lambda (t: T).(eq T w1 (THead (Flat Appl) v1 t))) H10 (THead (Bind b) x0 -x1) H13) in (eq_ind_r T (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) -(\lambda (t: T).(or (pr0 t (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) -O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 -(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2))))) (or_ind -(pr0 x1 t4) (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s -(Bind b) (s (Flat Appl) i)) v3 t4 w2))) (or (pr0 (THead (Flat Appl) v1 (THead -(Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) -t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 -x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4)) w2)))) (\lambda (H17: (pr0 x1 t4)).(or_ind (pr0 x0 u2) -(ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) -i) v3 u2 w2))) (or (pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2)))) (\lambda (H18: (pr0 x0 u2)).(or_introl (pr0 (THead (Flat Appl) v1 -(THead (Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O -v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) -x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) w2))) (pr0_upsilon b H0 v1 v2 H1 x0 u2 H18 x1 t4 -H17))) (\lambda (H18: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: -T).(subst0 (s (Flat Appl) i) v3 u2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 -w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 u2 w2)) (or (pr0 (THead -(Flat Appl) v1 (THead (Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 -(THead (Bind b) x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x2: T).(\lambda -(H19: (pr0 x0 x2)).(\lambda (H20: (subst0 (s (Flat Appl) i) v3 u2 -x2)).(or_intror (pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind -b) x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) x2 (THead (Flat Appl) (lift -(S O) O v2) t4)) (pr0_upsilon b H0 v1 v2 H1 x0 x2 H19 x1 t4 H17) (subst0_fst -v3 x2 u2 i H20 (THead (Flat Appl) (lift (S O) O v2) t4) (Bind b))))))) H18)) -(H4 v0 x0 (s (Flat Appl) i) H14 v3 H8))) (\lambda (H17: (ex2 T (\lambda (w2: -T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 -w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s -(Bind b) (s (Flat Appl) i)) v3 t4 w2)) (or (pr0 (THead (Flat Appl) v1 (THead -(Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) -t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 -x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4)) w2)))) (\lambda (x2: T).(\lambda (H18: (pr0 x1 -x2)).(\lambda (H19: (subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 x2)).(or_ind -(pr0 x0 u2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s -(Flat Appl) i) v3 u2 w2))) (or (pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 -x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T -(\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t4)) w2)))) (\lambda (H20: (pr0 x0 u2)).(or_intror (pr0 (THead (Flat -Appl) v1 (THead (Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift -(S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead -(Bind b) x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 -(THead (Flat Appl) v1 (THead (Bind b) x0 x1)) w2)) (\lambda (w2: T).(subst0 i -v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) x2)) (pr0_upsilon b H0 v1 v2 -H1 x0 u2 H20 x1 x2 H18) (subst0_snd (Bind b) v3 (THead (Flat Appl) (lift (S -O) O v2) x2) (THead (Flat Appl) (lift (S O) O v2) t4) i (subst0_snd (Flat -Appl) v3 x2 t4 (s (Bind b) i) H19 (lift (S O) O v2)) u2)))) (\lambda (H20: -(ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) -i) v3 u2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: -T).(subst0 (s (Flat Appl) i) v3 u2 w2)) (or (pr0 (THead (Flat Appl) v1 (THead -(Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) -t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 -x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4)) w2)))) (\lambda (x3: T).(\lambda (H21: (pr0 x0 -x3)).(\lambda (H22: (subst0 (s (Flat Appl) i) v3 u2 x3)).(or_intror (pr0 -(THead (Flat Appl) v1 (THead (Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) -v1 (THead (Bind b) x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2)) (THead (Bind b) x3 (THead (Flat Appl) (lift (S O) O v2) x2)) -(pr0_upsilon b H0 v1 v2 H1 x0 x3 H21 x1 x2 H18) (subst0_both v3 u2 x3 i H22 -(Bind b) (THead (Flat Appl) (lift (S O) O v2) t4) (THead (Flat Appl) (lift (S -O) O v2) x2) (subst0_snd (Flat Appl) v3 x2 t4 (s (Bind b) i) H19 (lift (S O) -O v2)))))))) H20)) (H4 v0 x0 (s (Flat Appl) i) H14 v3 H8))))) H17)) (H6 v0 x1 -(s (Bind b) (s (Flat Appl) i)) H15 v3 H8)) w1 H16))))))) H12)) -(subst0_gen_head (Bind b) v0 u1 t3 x (s (Flat Appl) i) H11))))) H9)) (\lambda -(H9: (ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T w1 (THead (Flat Appl) -u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v0 v1 u3))) (\lambda (_: -T).(\lambda (t5: T).(subst0 (s (Flat Appl) i) v0 (THead (Bind b) u1 t3) -t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t5: T).(eq T w1 (THead -(Flat Appl) u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v0 v1 u3))) -(\lambda (_: T).(\lambda (t5: T).(subst0 (s (Flat Appl) i) v0 (THead (Bind b) -u1 t3) t5))) (or (pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O -v2) t4))) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 -(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda -(x0: T).(\lambda (x1: T).(\lambda (H10: (eq T w1 (THead (Flat Appl) x0 -x1))).(\lambda (H11: (subst0 i v0 v1 x0)).(\lambda (H12: (subst0 (s (Flat -Appl) i) v0 (THead (Bind b) u1 t3) x1)).(or3_ind (ex2 T (\lambda (u3: T).(eq -T x1 (THead (Bind b) u3 t3))) (\lambda (u3: T).(subst0 (s (Flat Appl) i) v0 -u1 u3))) (ex2 T (\lambda (t5: T).(eq T x1 (THead (Bind b) u1 t5))) (\lambda -(t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 t5))) (ex3_2 T T -(\lambda (u3: T).(\lambda (t5: T).(eq T x1 (THead (Bind b) u3 t5)))) (\lambda -(u3: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u1 u3))) (\lambda (_: -T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 t5)))) (or -(pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T -(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H13: (ex2 T -(\lambda (u3: T).(eq T x1 (THead (Bind b) u3 t3))) (\lambda (u3: T).(subst0 -(s (Flat Appl) i) v0 u1 u3)))).(ex2_ind T (\lambda (u3: T).(eq T x1 (THead -(Bind b) u3 t3))) (\lambda (u3: T).(subst0 (s (Flat Appl) i) v0 u1 u3)) (or -(pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T -(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x: T).(\lambda -(H14: (eq T x1 (THead (Bind b) x t3))).(\lambda (H15: (subst0 (s (Flat Appl) -i) v0 u1 x)).(let H16 \def (eq_ind T x1 (\lambda (t: T).(eq T w1 (THead (Flat -Appl) x0 t))) H10 (THead (Bind b) x t3) H14) in (eq_ind_r T (THead (Flat -Appl) x0 (THead (Bind b) x t3)) (\lambda (t: T).(or (pr0 t (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 t -w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4)) w2))))) (or_ind (pr0 x u2) (ex2 T (\lambda (w2: -T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 u2 w2))) (or -(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat -Appl) x0 (THead (Bind b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H17: -(pr0 x u2)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda -(w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) -x t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T -(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t4)) w2)))) (\lambda (H18: (pr0 x0 v2)).(or_introl (pr0 (THead (Flat -Appl) x0 (THead (Bind b) x t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift -(S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead -(Bind b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t4)) w2))) (pr0_upsilon b H0 x0 v2 H18 x u2 H17 -t3 t4 H5))) (\lambda (H18: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: -T).(subst0 i v3 v2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda -(w2: T).(subst0 i v3 v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x -t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T -(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t4)) w2)))) (\lambda (x2: T).(\lambda (H19: (pr0 x0 x2)).(\lambda -(H20: (subst0 i v3 v2 x2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind -b) x t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 -T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 -(THead (Bind b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O x2) t4)) (pr0_upsilon b H0 x0 x2 H19 x u2 H17 t3 t4 -H5) (subst0_snd (Bind b) v3 (THead (Flat Appl) (lift (S O) O x2) t4) (THead -(Flat Appl) (lift (S O) O v2) t4) i (subst0_fst v3 (lift (S O) O x2) (lift (S -O) O v2) (s (Bind b) i) (subst0_lift_ge_s v2 x2 v3 i H20 O (le_O_n i) b) t4 -(Flat Appl)) u2)))))) H18)) (H2 v0 x0 i H11 v3 H8))) (\lambda (H17: (ex2 T -(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 u2 -w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s -(Flat Appl) i) v3 u2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x -t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T -(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t4)) w2)))) (\lambda (x2: T).(\lambda (H18: (pr0 x x2)).(\lambda -(H19: (subst0 (s (Flat Appl) i) v3 u2 x2)).(or_ind (pr0 x0 v2) (ex2 T -(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 -(THead (Flat Appl) x0 (THead (Bind b) x t3)) (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) -x0 (THead (Bind b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H20: (pr0 x0 -v2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind -b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) x2 (THead (Flat Appl) (lift -(S O) O v2) t4)) (pr0_upsilon b H0 x0 v2 H20 x x2 H18 t3 t4 H5) (subst0_fst -v3 x2 u2 i H19 (THead (Flat Appl) (lift (S O) O v2) t4) (Bind b))))) (\lambda -(H20: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 -w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 -v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 -(THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) (\lambda (w2: T).(subst0 i -v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) -(\lambda (x3: T).(\lambda (H21: (pr0 x0 x3)).(\lambda (H22: (subst0 i v3 v2 -x3)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind -b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) x2 (THead (Flat Appl) (lift -(S O) O x3) t4)) (pr0_upsilon b H0 x0 x3 H21 x x2 H18 t3 t4 H5) (subst0_both -v3 u2 x2 i H19 (Bind b) (THead (Flat Appl) (lift (S O) O v2) t4) (THead (Flat -Appl) (lift (S O) O x3) t4) (subst0_fst v3 (lift (S O) O x3) (lift (S O) O -v2) (s (Bind b) i) (subst0_lift_ge_s v2 x3 v3 i H22 O (le_O_n i) b) t4 (Flat -Appl)))))))) H20)) (H2 v0 x0 i H11 v3 H8))))) H17)) (H4 v0 x (s (Flat Appl) -i) H15 v3 H8)) w1 H16))))) H13)) (\lambda (H13: (ex2 T (\lambda (t5: T).(eq T -x1 (THead (Bind b) u1 t5))) (\lambda (t5: T).(subst0 (s (Bind b) (s (Flat -Appl) i)) v0 t3 t5)))).(ex2_ind T (\lambda (t5: T).(eq T x1 (THead (Bind b) -u1 t5))) (\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 t5)) -(or (pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) -(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x: -T).(\lambda (H14: (eq T x1 (THead (Bind b) u1 x))).(\lambda (H15: (subst0 (s -(Bind b) (s (Flat Appl) i)) v0 t3 x)).(let H16 \def (eq_ind T x1 (\lambda (t: -T).(eq T w1 (THead (Flat Appl) x0 t))) H10 (THead (Bind b) u1 x) H14) in -(eq_ind_r T (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (\lambda (t: T).(or -(pr0 t (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T -(\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t4)) w2))))) (or_ind (pr0 x t4) (ex2 T -(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind b) (s (Flat -Appl) i)) v3 t4 w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) -(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda -(w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2)))) (\lambda (H17: (pr0 x t4)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2: -T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat -Appl) x0 (THead (Bind b) u1 x)) (THead (Bind b) u2 (THead (Flat Appl) (lift -(S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead -(Bind b) u1 x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H18: (pr0 x0 -v2)).(or_introl (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2))) (pr0_upsilon b H0 x0 v2 H18 u1 u2 H3 x t4 H17))) (\lambda (H18: (ex2 T -(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 -w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 -v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 -(THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: T).(subst0 i -v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) -(\lambda (x2: T).(\lambda (H19: (pr0 x0 x2)).(\lambda (H20: (subst0 i v3 v2 -x2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind -b) u1 x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 (THead (Flat Appl) (lift -(S O) O x2) t4)) (pr0_upsilon b H0 x0 x2 H19 u1 u2 H3 x t4 H17) (subst0_snd -(Bind b) v3 (THead (Flat Appl) (lift (S O) O x2) t4) (THead (Flat Appl) (lift -(S O) O v2) t4) i (subst0_fst v3 (lift (S O) O x2) (lift (S O) O v2) (s (Bind -b) i) (subst0_lift_ge_s v2 x2 v3 i H20 O (le_O_n i) b) t4 (Flat Appl)) -u2)))))) H18)) (H2 v0 x0 i H11 v3 H8))) (\lambda (H17: (ex2 T (\lambda (w2: -T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 -w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s -(Bind b) (s (Flat Appl) i)) v3 t4 w2)) (or (pr0 (THead (Flat Appl) x0 (THead -(Bind b) u1 x)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) -(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) -w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4)) w2)))) (\lambda (x2: T).(\lambda (H18: (pr0 x -x2)).(\lambda (H19: (subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 x2)).(or_ind -(pr0 x0 v2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i -v3 v2 w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2)))) (\lambda (H20: (pr0 x0 v2)).(or_intror (pr0 (THead (Flat Appl) x0 -(THead (Bind b) u1 x)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O -v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) -u1 x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead -(Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: T).(subst0 i v3 -(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) x2)) (pr0_upsilon b H0 x0 v2 -H20 u1 u2 H3 x x2 H18) (subst0_snd (Bind b) v3 (THead (Flat Appl) (lift (S O) -O v2) x2) (THead (Flat Appl) (lift (S O) O v2) t4) i (subst0_snd (Flat Appl) -v3 x2 t4 (s (Bind b) i) H19 (lift (S O) O v2)) u2)))) (\lambda (H20: (ex2 T -(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 -w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 -v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 -(THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: T).(subst0 i -v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) -(\lambda (x3: T).(\lambda (H21: (pr0 x0 x3)).(\lambda (H22: (subst0 i v3 v2 -x3)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind -b) u1 x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 (THead (Flat Appl) (lift -(S O) O x3) x2)) (pr0_upsilon b H0 x0 x3 H21 u1 u2 H3 x x2 H18) (subst0_snd -(Bind b) v3 (THead (Flat Appl) (lift (S O) O x3) x2) (THead (Flat Appl) (lift -(S O) O v2) t4) i (subst0_both v3 (lift (S O) O v2) (lift (S O) O x3) (s -(Bind b) i) (subst0_lift_ge_s v2 x3 v3 i H22 O (le_O_n i) b) (Flat Appl) t4 -x2 H19) u2)))))) H20)) (H2 v0 x0 i H11 v3 H8))))) H17)) (H6 v0 x (s (Bind b) -(s (Flat Appl) i)) H15 v3 H8)) w1 H16))))) H13)) (\lambda (H13: (ex3_2 T T -(\lambda (u3: T).(\lambda (t5: T).(eq T x1 (THead (Bind b) u3 t5)))) (\lambda -(u3: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u1 u3))) (\lambda (_: -T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 -t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t5: T).(eq T x1 (THead -(Bind b) u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) -v0 u1 u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat -Appl) i)) v0 t3 t5))) (or (pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift -(S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H14: (eq T x1 (THead (Bind -b) x2 x3))).(\lambda (H15: (subst0 (s (Flat Appl) i) v0 u1 x2)).(\lambda -(H16: (subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 x3)).(let H17 \def (eq_ind -T x1 (\lambda (t: T).(eq T w1 (THead (Flat Appl) x0 t))) H10 (THead (Bind b) -x2 x3) H14) in (eq_ind_r T (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) -(\lambda (t: T).(or (pr0 t (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) -O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 -(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2))))) (or_ind -(pr0 x3 t4) (ex2 T (\lambda (w2: T).(pr0 x3 w2)) (\lambda (w2: T).(subst0 (s -(Bind b) (s (Flat Appl) i)) v3 t4 w2))) (or (pr0 (THead (Flat Appl) x0 (THead -(Bind b) x2 x3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) -t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 -x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4)) w2)))) (\lambda (H18: (pr0 x3 t4)).(or_ind (pr0 x2 u2) -(ex2 T (\lambda (w2: T).(pr0 x2 w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) -i) v3 u2 w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2)))) (\lambda (H19: (pr0 x2 u2)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2: -T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat -Appl) x0 (THead (Bind b) x2 x3)) (THead (Bind b) u2 (THead (Flat Appl) (lift -(S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead -(Bind b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H20: (pr0 x0 -v2)).(or_introl (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2))) (pr0_upsilon b H0 x0 v2 H20 x2 u2 H19 x3 t4 H18))) (\lambda (H20: (ex2 -T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 -w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 -v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead (Bind -b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 -(THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i -v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) -(\lambda (x: T).(\lambda (H21: (pr0 x0 x)).(\lambda (H22: (subst0 i v3 v2 -x)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind -b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 (THead (Flat Appl) (lift -(S O) O x) t4)) (pr0_upsilon b H0 x0 x H21 x2 u2 H19 x3 t4 H18) (subst0_snd -(Bind b) v3 (THead (Flat Appl) (lift (S O) O x) t4) (THead (Flat Appl) (lift -(S O) O v2) t4) i (subst0_fst v3 (lift (S O) O x) (lift (S O) O v2) (s (Bind -b) i) (subst0_lift_ge_s v2 x v3 i H22 O (le_O_n i) b) t4 (Flat Appl)) -u2)))))) H20)) (H2 v0 x0 i H11 v3 H8))) (\lambda (H19: (ex2 T (\lambda (w2: -T).(pr0 x2 w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 u2 -w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x2 w2)) (\lambda (w2: T).(subst0 (s -(Flat Appl) i) v3 u2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 -x3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T -(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t4)) w2)))) (\lambda (x: T).(\lambda (H20: (pr0 x2 x)).(\lambda -(H21: (subst0 (s (Flat Appl) i) v3 u2 x)).(or_ind (pr0 x0 v2) (ex2 T (\lambda -(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead -(Flat Appl) x0 (THead (Bind b) x2 x3)) (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 -(THead (Bind b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H22: (pr0 x0 -v2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S 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(Flat Appl) (lift (S O) O v2) t4)) w2)))) -(\lambda (x4: T).(\lambda (H23: (pr0 x0 x4)).(\lambda (H24: (subst0 i v3 v2 -x4)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind -b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) x (THead (Flat Appl) (lift -(S O) O x4) t4)) (pr0_upsilon b H0 x0 x4 H23 x2 x H20 x3 t4 H18) (subst0_both -v3 u2 x i H21 (Bind b) (THead (Flat Appl) (lift (S O) O v2) t4) (THead (Flat -Appl) (lift (S O) O x4) t4) (subst0_fst v3 (lift (S O) O x4) (lift (S O) O -v2) (s (Bind b) i) (subst0_lift_ge_s v2 x4 v3 i H24 O (le_O_n i) b) t4 (Flat -Appl)))))))) H22)) (H2 v0 x0 i 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(THead (Bind Abbr) u1 x) (\lambda (t: T).(or (pr0 t -(THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: -T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))))) (or_ind (pr0 x t4) (ex2 T -(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind Abbr) i) v2 t4 -w2))) (or (pr0 (THead (Bind Abbr) u1 x) (THead (Bind Abbr) u2 w)) (ex2 T -(\lambda (w2: T).(pr0 (THead (Bind Abbr) u1 x) w2)) (\lambda (w2: T).(subst0 -i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (H10: (pr0 x t4)).(or_introl -(pr0 (THead (Bind Abbr) u1 x) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: -T).(pr0 (THead (Bind Abbr) u1 x) w2)) (\lambda (w2: T).(subst0 i v2 (THead -(Bind Abbr) u2 w) w2))) (pr0_delta u1 u2 H0 x t4 H10 w H4))) (\lambda (H10: -(ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind Abbr) -i) v2 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: -T).(subst0 (s (Bind Abbr) i) v2 t4 w2)) (or (pr0 (THead (Bind Abbr) u1 x) -(THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) u1 -x) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) -(\lambda (x0: T).(\lambda (H11: (pr0 x x0)).(\lambda (H12: (subst0 (s (Bind -Abbr) i) v2 t4 x0)).(ex2_ind T (\lambda (t: T).(subst0 O u2 x0 t)) (\lambda -(t: T).(subst0 (s (Bind Abbr) i) v2 w t)) (or (pr0 (THead (Bind Abbr) u1 x) -(THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) u1 -x) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) -(\lambda (x1: T).(\lambda (H13: (subst0 O u2 x0 x1)).(\lambda (H14: (subst0 -(s (Bind Abbr) i) v2 w x1)).(or_intror (pr0 (THead (Bind Abbr) u1 x) (THead -(Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) u1 x) w2)) -(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))) (ex_intro2 T -(\lambda (w2: T).(pr0 (THead (Bind Abbr) u1 x) w2)) (\lambda (w2: T).(subst0 -i v2 (THead (Bind Abbr) u2 w) w2)) (THead (Bind Abbr) u2 x1) (pr0_delta u1 u2 -H0 x x0 H11 x1 H13) (subst0_snd (Bind Abbr) v2 x1 w i H14 u2)))))) -(subst0_confluence_neq t4 x0 v2 (s (Bind Abbr) i) H12 w u2 O H4 (sym_not_eq -nat O (S i) (O_S i))))))) H10)) (H3 v1 x (s (Bind Abbr) i) H9 v2 H6)) w1 -H8)))) H7)) (\lambda (H7: (ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T -w1 (THead (Bind Abbr) u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v1 -u1 u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind Abbr) i) v1 t3 -t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t5: T).(eq T w1 (THead -(Bind Abbr) u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v1 u1 u3))) -(\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind Abbr) i) v1 t3 t5))) (or -(pr0 w1 (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) -(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H8: (eq T w1 (THead (Bind Abbr) x0 -x1))).(\lambda (H9: (subst0 i v1 u1 x0)).(\lambda (H10: (subst0 (s (Bind -Abbr) i) v1 t3 x1)).(eq_ind_r T (THead (Bind Abbr) x0 x1) (\lambda (t: T).(or -(pr0 t (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda -(w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))))) (or_ind (pr0 x1 t4) -(ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s (Bind Abbr) -i) v2 t4 w2))) (or (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) -(ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: -T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (H11: (pr0 x1 -t4)).(or_ind (pr0 x0 u2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: -T).(subst0 i v2 u2 w2))) (or (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind -Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) -(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (H12: -(pr0 x0 u2)).(or_introl (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 -w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: -T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))) (pr0_delta x0 u2 H12 x1 t4 H11 -w H4))) (\lambda (H12: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: -T).(subst0 i v2 u2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda -(w2: T).(subst0 i v2 u2 w2)) (or (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind -Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) -(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x: -T).(\lambda (H13: (pr0 x0 x)).(\lambda (H14: (subst0 i v2 u2 x)).(ex2_ind T -(\lambda (t: T).(subst0 O x t4 t)) (\lambda (t: T).(subst0 (S (plus i O)) v2 -w t)) (or (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T -(\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 -i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x2: T).(\lambda (H15: (subst0 -O x t4 x2)).(\lambda (H16: (subst0 (S (plus i O)) v2 w x2)).(let H17 \def -(f_equal nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O i))) in -(let H18 \def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n v2 w -x2)) H16 (S i) H17) in (or_intror (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind -Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) -(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))) (ex_intro2 T -(\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 -i v2 (THead (Bind Abbr) u2 w) w2)) (THead (Bind Abbr) x x2) (pr0_delta x0 x -H13 x1 t4 H11 x2 H15) (subst0_both v2 u2 x i H14 (Bind Abbr) w x2 H18)))))))) -(subst0_subst0_back t4 w u2 O H4 x v2 i H14))))) H12)) (H1 v1 x0 i H9 v2 -H6))) (\lambda (H11: (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: -T).(subst0 (s (Bind Abbr) i) v2 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x1 -w2)) (\lambda (w2: T).(subst0 (s (Bind Abbr) i) v2 t4 w2)) (or (pr0 (THead -(Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 -(THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind -Abbr) u2 w) w2)))) (\lambda (x: T).(\lambda (H12: (pr0 x1 x)).(\lambda (H13: -(subst0 (s (Bind Abbr) i) v2 t4 x)).(or_ind (pr0 x0 u2) (ex2 T (\lambda (w2: -T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 u2 w2))) (or (pr0 (THead (Bind -Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead -(Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 -w) w2)))) (\lambda (H14: (pr0 x0 u2)).(ex2_ind T (\lambda (t: T).(subst0 O u2 -x t)) (\lambda (t: T).(subst0 (s (Bind Abbr) i) v2 w t)) (or (pr0 (THead -(Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 -(THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind -Abbr) u2 w) w2)))) (\lambda (x2: T).(\lambda (H15: (subst0 O u2 x -x2)).(\lambda (H16: (subst0 (s (Bind Abbr) i) v2 w x2)).(or_intror (pr0 -(THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: -T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead -(Bind Abbr) u2 w) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) -x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)) -(THead (Bind Abbr) u2 x2) (pr0_delta x0 u2 H14 x1 x H12 x2 H15) (subst0_snd -(Bind Abbr) v2 x2 w i H16 u2)))))) (subst0_confluence_neq t4 x v2 (s (Bind -Abbr) i) H13 w u2 O H4 (sym_not_eq nat O (S i) (O_S i))))) (\lambda (H14: -(ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 u2 -w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 -u2 w2)) (or (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T -(\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 -i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x2: T).(\lambda (H15: (pr0 x0 -x2)).(\lambda (H16: (subst0 i v2 u2 x2)).(ex2_ind T (\lambda (t: T).(subst0 O -x2 t4 t)) (\lambda (t: T).(subst0 (S (plus i O)) v2 w t)) (or (pr0 (THead -(Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 -(THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind -Abbr) u2 w) w2)))) (\lambda (x3: T).(\lambda (H17: (subst0 O x2 t4 -x3)).(\lambda (H18: (subst0 (S (plus i O)) v2 w x3)).(let H19 \def (f_equal -nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O i))) in (let H20 -\def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n v2 w x3)) H18 (S -i) H19) in (ex2_ind T (\lambda (t: T).(subst0 (s (Bind Abbr) i) v2 x3 t)) -(\lambda (t: T).(subst0 O x2 x t)) (or (pr0 (THead (Bind Abbr) x0 x1) (THead -(Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) -w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda -(x4: T).(\lambda (H21: (subst0 (s (Bind Abbr) i) v2 x3 x4)).(\lambda (H22: -(subst0 O x2 x x4)).(or_intror (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind -Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) -(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))) (ex_intro2 T -(\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 -i v2 (THead (Bind Abbr) u2 w) w2)) (THead (Bind Abbr) x2 x4) (pr0_delta x0 x2 -H15 x1 x H12 x4 H22) (subst0_both v2 u2 x2 i H16 (Bind Abbr) w x4 -(subst0_trans x3 w v2 (s (Bind Abbr) i) H20 x4 H21))))))) -(subst0_confluence_neq t4 x3 x2 O H17 x v2 (s (Bind Abbr) i) H13 (O_S -i)))))))) (subst0_subst0_back t4 w u2 O H4 x2 v2 i H16))))) H14)) (H1 v1 x0 i -H9 v2 H6))))) H11)) (H3 v1 x1 (s (Bind Abbr) i) H10 v2 H6)) w1 H8)))))) H7)) -(subst0_gen_head (Bind Abbr) v1 u1 t3 w1 i H5)))))))))))))))))) (\lambda (b: -B).(\lambda (H0: (not (eq B b Abst))).(\lambda (t3: T).(\lambda (t4: -T).(\lambda (H1: (pr0 t3 t4)).(\lambda (H2: ((\forall (v1: T).(\forall (w1: -T).(\forall (i: nat).((subst0 i v1 t3 w1) \to (\forall (v2: T).((pr0 v1 v2) -\to (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: -T).(subst0 i v2 t4 w2)))))))))))).(\lambda (u: T).(\lambda (v1: T).(\lambda -(w1: T).(\lambda (i: nat).(\lambda (H3: (subst0 i v1 (THead (Bind b) u (lift -(S O) O t3)) w1)).(\lambda (v2: T).(\lambda (H4: (pr0 v1 v2)).(or3_ind (ex2 T -(\lambda (u2: T).(eq T w1 (THead (Bind b) u2 (lift (S O) O t3)))) (\lambda -(u2: T).(subst0 i v1 u u2))) (ex2 T (\lambda (t5: T).(eq T w1 (THead (Bind b) -u t5))) (\lambda (t5: T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5))) -(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T w1 (THead (Bind b) u2 -t5)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: -T).(\lambda (t5: T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5)))) (or -(pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i -v2 t4 w2)))) (\lambda (H5: (ex2 T (\lambda (u2: T).(eq T w1 (THead (Bind b) -u2 (lift (S O) O t3)))) (\lambda (u2: T).(subst0 i v1 u u2)))).(ex2_ind T -(\lambda (u2: T).(eq T w1 (THead (Bind b) u2 (lift (S O) O t3)))) (\lambda -(u2: T).(subst0 i v1 u u2)) (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 -w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H6: -(eq T w1 (THead (Bind b) x (lift (S O) O t3)))).(\lambda (_: (subst0 i v1 u -x)).(eq_ind_r T (THead (Bind b) x (lift (S O) O t3)) (\lambda (t: T).(or (pr0 -t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4 -w2))))) (or_introl (pr0 (THead (Bind b) x (lift (S O) O t3)) t4) (ex2 T -(\lambda (w2: T).(pr0 (THead (Bind b) x (lift (S O) O t3)) w2)) (\lambda (w2: -T).(subst0 i v2 t4 w2))) (pr0_zeta b H0 t3 t4 H1 x)) w1 H6)))) H5)) (\lambda -(H5: (ex2 T (\lambda (t5: T).(eq T w1 (THead (Bind b) u t5))) (\lambda (t5: -T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5)))).(ex2_ind T (\lambda -(t5: T).(eq T w1 (THead (Bind b) u t5))) (\lambda (t5: T).(subst0 (s (Bind b) -i) v1 (lift (S O) O t3) t5)) (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 -w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H6: -(eq T w1 (THead (Bind b) u x))).(\lambda (H7: (subst0 (s (Bind b) i) v1 (lift -(S O) O t3) x)).(ex2_ind T (\lambda (t5: T).(eq T x (lift (S O) O t5))) -(\lambda (t5: T).(subst0 (minus (s (Bind b) i) (S O)) v1 t3 t5)) (or (pr0 w1 -t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 -w2)))) (\lambda (x0: T).(\lambda (H8: (eq T x (lift (S O) O x0))).(\lambda -(H9: (subst0 (minus (s (Bind b) i) (S O)) v1 t3 x0)).(let H10 \def (eq_ind T -x (\lambda (t: T).(eq T w1 (THead (Bind b) u t))) H6 (lift (S O) O x0) H8) in -(eq_ind_r T (THead (Bind b) u (lift (S O) O x0)) (\lambda (t: T).(or (pr0 t -t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4 -w2))))) (let H11 \def (eq_ind_r nat (minus i O) (\lambda (n: nat).(subst0 n -v1 t3 x0)) H9 i (minus_n_O i)) in (or_ind (pr0 x0 t4) (ex2 T (\lambda (w2: -T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (or (pr0 (THead (Bind -b) u (lift (S O) O x0)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) u -(lift (S O) O x0)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda -(H12: (pr0 x0 t4)).(or_introl (pr0 (THead (Bind b) u (lift (S O) O x0)) t4) -(ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) u (lift (S O) O x0)) w2)) -(\lambda (w2: T).(subst0 i v2 t4 w2))) (pr0_zeta b H0 x0 t4 H12 u))) (\lambda -(H12: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 t4 -w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 -t4 w2)) (or (pr0 (THead (Bind b) u (lift (S O) O x0)) t4) (ex2 T (\lambda -(w2: T).(pr0 (THead (Bind b) u (lift (S O) O x0)) w2)) (\lambda (w2: -T).(subst0 i v2 t4 w2)))) (\lambda (x1: T).(\lambda (H13: (pr0 x0 -x1)).(\lambda (H14: (subst0 i v2 t4 x1)).(or_intror (pr0 (THead (Bind b) u -(lift (S O) O x0)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) u (lift -(S O) O x0)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (ex_intro2 T -(\lambda (w2: T).(pr0 (THead (Bind b) u (lift (S O) O x0)) w2)) (\lambda (w2: -T).(subst0 i v2 t4 w2)) x1 (pr0_zeta b H0 x0 x1 H13 u) H14))))) H12)) (H2 v1 -x0 i H11 v2 H4))) w1 H10))))) (subst0_gen_lift_ge v1 t3 x (s (Bind b) i) (S -O) O H7 (le_n_S O i (le_O_n i))))))) H5)) (\lambda (H5: (ex3_2 T T (\lambda -(u2: T).(\lambda (t5: T).(eq T w1 (THead (Bind b) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: -T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5))))).(ex3_2_ind T T -(\lambda (u2: T).(\lambda (t5: T).(eq T w1 (THead (Bind b) u2 t5)))) (\lambda -(u2: T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: -T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5))) (or (pr0 w1 t4) (ex2 T -(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) -(\lambda (x0: T).(\lambda (x1: T).(\lambda (H6: (eq T w1 (THead (Bind b) x0 -x1))).(\lambda (_: (subst0 i v1 u x0)).(\lambda (H8: (subst0 (s (Bind b) i) -v1 (lift (S O) O t3) x1)).(ex2_ind T (\lambda (t5: T).(eq T x1 (lift (S O) O -t5))) (\lambda (t5: T).(subst0 (minus (s (Bind b) i) (S O)) v1 t3 t5)) (or -(pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i -v2 t4 w2)))) (\lambda (x: T).(\lambda (H9: (eq T x1 (lift (S O) O -x))).(\lambda (H10: (subst0 (minus (s (Bind b) i) (S O)) v1 t3 x)).(let H11 -\def (eq_ind T x1 (\lambda (t: T).(eq T w1 (THead (Bind b) x0 t))) H6 (lift -(S O) O x) H9) in (eq_ind_r T (THead (Bind b) x0 (lift (S O) O x)) (\lambda -(t: T).(or (pr0 t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: -T).(subst0 i v2 t4 w2))))) (let H12 \def (eq_ind_r nat (minus i O) (\lambda -(n: nat).(subst0 n v1 t3 x)) H10 i (minus_n_O i)) in (or_ind (pr0 x t4) (ex2 -T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (or -(pr0 (THead (Bind b) x0 (lift (S O) O x)) t4) (ex2 T (\lambda (w2: T).(pr0 -(THead (Bind b) x0 (lift (S O) O x)) w2)) (\lambda (w2: T).(subst0 i v2 t4 -w2)))) (\lambda (H13: (pr0 x t4)).(or_introl (pr0 (THead (Bind b) x0 (lift (S -O) O x)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) x0 (lift (S O) O -x)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (pr0_zeta b H0 x t4 H13 x0))) -(\lambda (H13: (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i -v2 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 -i v2 t4 w2)) (or (pr0 (THead (Bind b) x0 (lift (S O) O x)) t4) (ex2 T -(\lambda (w2: T).(pr0 (THead (Bind b) x0 (lift (S O) O x)) w2)) (\lambda (w2: -T).(subst0 i v2 t4 w2)))) (\lambda (x2: T).(\lambda (H14: (pr0 x -x2)).(\lambda (H15: (subst0 i v2 t4 x2)).(or_intror (pr0 (THead (Bind b) x0 -(lift (S O) O x)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) x0 (lift -(S O) O x)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (ex_intro2 T (\lambda -(w2: T).(pr0 (THead (Bind b) x0 (lift (S O) O x)) w2)) (\lambda (w2: -T).(subst0 i v2 t4 w2)) x2 (pr0_zeta b H0 x x2 H14 x0) H15))))) H13)) (H2 v1 -x i H12 v2 H4))) w1 H11))))) (subst0_gen_lift_ge v1 t3 x1 (s (Bind b) i) (S -O) O H8 (le_n_S O i (le_O_n i))))))))) H5)) (subst0_gen_head (Bind b) v1 u -(lift (S O) O t3) w1 i H3))))))))))))))) (\lambda (t3: T).(\lambda (t4: -T).(\lambda (H0: (pr0 t3 t4)).(\lambda (H1: ((\forall (v1: T).(\forall (w1: -T).(\forall (i: nat).((subst0 i v1 t3 w1) \to (\forall (v2: T).((pr0 v1 v2) -\to (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: -T).(subst0 i v2 t4 w2)))))))))))).(\lambda (u: T).(\lambda (v1: T).(\lambda -(w1: T).(\lambda (i: nat).(\lambda (H2: (subst0 i v1 (THead (Flat Cast) u t3) -w1)).(\lambda (v2: T).(\lambda (H3: (pr0 v1 v2)).(or3_ind (ex2 T (\lambda -(u2: T).(eq T w1 (THead (Flat Cast) u2 t3))) (\lambda (u2: T).(subst0 i v1 u -u2))) (ex2 T (\lambda (t5: T).(eq T w1 (THead (Flat Cast) u t5))) (\lambda -(t5: T).(subst0 (s (Flat Cast) i) v1 t3 t5))) (ex3_2 T T (\lambda (u2: -T).(\lambda (t5: T).(eq T w1 (THead (Flat Cast) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: -T).(subst0 (s (Flat Cast) i) v1 t3 t5)))) (or (pr0 w1 t4) (ex2 T (\lambda -(w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (H4: -(ex2 T (\lambda (u2: T).(eq T w1 (THead (Flat Cast) u2 t3))) (\lambda (u2: -T).(subst0 i v1 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T w1 (THead (Flat -Cast) u2 t3))) (\lambda (u2: T).(subst0 i v1 u u2)) (or (pr0 w1 t4) (ex2 T -(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) -(\lambda (x: T).(\lambda (H5: (eq T w1 (THead (Flat Cast) x t3))).(\lambda -(_: (subst0 i v1 u x)).(eq_ind_r T (THead (Flat Cast) x t3) (\lambda (t: -T).(or (pr0 t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: -T).(subst0 i v2 t4 w2))))) (or_introl (pr0 (THead (Flat Cast) x t3) t4) (ex2 -T (\lambda (w2: T).(pr0 (THead (Flat Cast) x t3) w2)) (\lambda (w2: -T).(subst0 i v2 t4 w2))) (pr0_tau t3 t4 H0 x)) w1 H5)))) H4)) (\lambda (H4: -(ex2 T (\lambda (t5: T).(eq T w1 (THead (Flat Cast) u t5))) (\lambda (t5: -T).(subst0 (s (Flat Cast) i) v1 t3 t5)))).(ex2_ind T (\lambda (t5: T).(eq T -w1 (THead (Flat Cast) u t5))) (\lambda (t5: T).(subst0 (s (Flat Cast) i) v1 -t3 t5)) (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: -T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H5: (eq T w1 (THead (Flat -Cast) u x))).(\lambda (H6: (subst0 (s (Flat Cast) i) v1 t3 x)).(eq_ind_r T -(THead (Flat Cast) u x) (\lambda (t: T).(or (pr0 t t4) (ex2 T (\lambda (w2: -T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))))) (or_ind (pr0 x t4) -(ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Flat Cast) -i) v2 t4 w2))) (or (pr0 (THead (Flat Cast) u x) t4) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) -(\lambda (H7: (pr0 x t4)).(or_introl (pr0 (THead (Flat Cast) u x) t4) (ex2 T -(\lambda (w2: T).(pr0 (THead (Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i -v2 t4 w2))) (pr0_tau x t4 H7 u))) (\lambda (H7: (ex2 T (\lambda (w2: T).(pr0 -x w2)) (\lambda (w2: T).(subst0 (s (Flat Cast) i) v2 t4 w2)))).(ex2_ind T -(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Flat Cast) i) v2 t4 -w2)) (or (pr0 (THead (Flat Cast) u x) t4) (ex2 T (\lambda (w2: T).(pr0 (THead -(Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x0: -T).(\lambda (H8: (pr0 x x0)).(\lambda (H9: (subst0 (s (Flat Cast) i) v2 t4 -x0)).(or_intror (pr0 (THead (Flat Cast) u x) t4) (ex2 T (\lambda (w2: T).(pr0 -(THead (Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) -(ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) u x) w2)) (\lambda (w2: -T).(subst0 i v2 t4 w2)) x0 (pr0_tau x x0 H8 u) H9))))) H7)) (H1 v1 x (s (Flat -Cast) i) H6 v2 H3)) w1 H5)))) H4)) (\lambda (H4: (ex3_2 T T (\lambda (u2: -T).(\lambda (t5: T).(eq T w1 (THead (Flat Cast) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: -T).(subst0 (s (Flat Cast) i) v1 t3 t5))))).(ex3_2_ind T T (\lambda (u2: -T).(\lambda (t5: T).(eq T w1 (THead (Flat Cast) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: -T).(subst0 (s (Flat Cast) i) v1 t3 t5))) (or (pr0 w1 t4) (ex2 T (\lambda (w2: -T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H5: (eq T w1 (THead (Flat Cast) x0 -x1))).(\lambda (_: (subst0 i v1 u x0)).(\lambda (H7: (subst0 (s (Flat Cast) -i) v1 t3 x1)).(eq_ind_r T (THead (Flat Cast) x0 x1) (\lambda (t: T).(or (pr0 -t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4 -w2))))) (or_ind (pr0 x1 t4) (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda -(w2: T).(subst0 (s (Flat Cast) i) v2 t4 w2))) (or (pr0 (THead (Flat Cast) x0 -x1) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) x0 x1) w2)) (\lambda -(w2: T).(subst0 i v2 t4 w2)))) (\lambda (H8: (pr0 x1 t4)).(or_introl (pr0 -(THead (Flat Cast) x0 x1) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) -x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (pr0_tau x1 t4 H8 x0))) -(\lambda (H8: (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 -(s (Flat Cast) i) v2 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x1 w2)) -(\lambda (w2: T).(subst0 (s (Flat Cast) i) v2 t4 w2)) (or (pr0 (THead (Flat -Cast) x0 x1) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) x0 x1) w2)) -(\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H9: (pr0 x1 -x)).(\lambda (H10: (subst0 (s (Flat Cast) i) v2 t4 x)).(or_intror (pr0 (THead -(Flat Cast) x0 x1) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) x0 x1) -w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (ex_intro2 T (\lambda (w2: -T).(pr0 (THead (Flat Cast) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)) -x (pr0_tau x1 x H9 x0) H10))))) H8)) (H1 v1 x1 (s (Flat Cast) i) H7 v2 H3)) -w1 H5)))))) H4)) (subst0_gen_head (Flat Cast) v1 u t3 w1 i H2))))))))))))) t1 -t2 H))). -(* COMMENTS -Initial nodes: 38857 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr0/subst1.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr0/subst1.ma deleted file mode 100644 index 877f87f01..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr0/subst1.ma +++ /dev/null @@ -1,105 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr0/props.ma". - -include "Basic-1/subst1/defs.ma". - -theorem pr0_delta1: - \forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (t1: T).(\forall -(t2: T).((pr0 t1 t2) \to (\forall (w: T).((subst1 O u2 t2 w) \to (pr0 (THead -(Bind Abbr) u1 t1) (THead (Bind Abbr) u2 w))))))))) -\def - \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr0 u1 u2)).(\lambda (t1: -T).(\lambda (t2: T).(\lambda (H0: (pr0 t1 t2)).(\lambda (w: T).(\lambda (H1: -(subst1 O u2 t2 w)).(subst1_ind O u2 t2 (\lambda (t: T).(pr0 (THead (Bind -Abbr) u1 t1) (THead (Bind Abbr) u2 t))) (pr0_comp u1 u2 H t1 t2 H0 (Bind -Abbr)) (\lambda (t0: T).(\lambda (H2: (subst0 O u2 t2 t0)).(pr0_delta u1 u2 H -t1 t2 H0 t0 H2))) w H1)))))))). -(* COMMENTS -Initial nodes: 115 -END *) - -theorem pr0_subst1_back: - \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst1 -i u2 t1 t2) \to (\forall (u1: T).((pr0 u1 u2) \to (ex2 T (\lambda (t: -T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t t2))))))))) -\def - \lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (i: nat).(\lambda -(H: (subst1 i u2 t1 t2)).(subst1_ind i u2 t1 (\lambda (t: T).(\forall (u1: -T).((pr0 u1 u2) \to (ex2 T (\lambda (t0: T).(subst1 i u1 t1 t0)) (\lambda -(t0: T).(pr0 t0 t)))))) (\lambda (u1: T).(\lambda (_: (pr0 u1 u2)).(ex_intro2 -T (\lambda (t: T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t t1)) t1 -(subst1_refl i u1 t1) (pr0_refl t1)))) (\lambda (t0: T).(\lambda (H0: (subst0 -i u2 t1 t0)).(\lambda (u1: T).(\lambda (H1: (pr0 u1 u2)).(ex2_ind T (\lambda -(t: T).(subst0 i u1 t1 t)) (\lambda (t: T).(pr0 t t0)) (ex2 T (\lambda (t: -T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t t0))) (\lambda (x: T).(\lambda -(H2: (subst0 i u1 t1 x)).(\lambda (H3: (pr0 x t0)).(ex_intro2 T (\lambda (t: -T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t t0)) x (subst1_single i u1 t1 x -H2) H3)))) (pr0_subst0_back u2 t1 t0 i H0 u1 H1)))))) t2 H))))). -(* COMMENTS -Initial nodes: 251 -END *) - -theorem pr0_subst1_fwd: - \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst1 -i u2 t1 t2) \to (\forall (u1: T).((pr0 u2 u1) \to (ex2 T (\lambda (t: -T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t2 t))))))))) -\def - \lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (i: nat).(\lambda -(H: (subst1 i u2 t1 t2)).(subst1_ind i u2 t1 (\lambda (t: T).(\forall (u1: -T).((pr0 u2 u1) \to (ex2 T (\lambda (t0: T).(subst1 i u1 t1 t0)) (\lambda -(t0: T).(pr0 t t0)))))) (\lambda (u1: T).(\lambda (_: (pr0 u2 u1)).(ex_intro2 -T (\lambda (t: T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t1 t)) t1 -(subst1_refl i u1 t1) (pr0_refl t1)))) (\lambda (t0: T).(\lambda (H0: (subst0 -i u2 t1 t0)).(\lambda (u1: T).(\lambda (H1: (pr0 u2 u1)).(ex2_ind T (\lambda -(t: T).(subst0 i u1 t1 t)) (\lambda (t: T).(pr0 t0 t)) (ex2 T (\lambda (t: -T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t0 t))) (\lambda (x: T).(\lambda -(H2: (subst0 i u1 t1 x)).(\lambda (H3: (pr0 t0 x)).(ex_intro2 T (\lambda (t: -T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t0 t)) x (subst1_single i u1 t1 x -H2) H3)))) (pr0_subst0_fwd u2 t1 t0 i H0 u1 H1)))))) t2 H))))). -(* COMMENTS -Initial nodes: 251 -END *) - -theorem pr0_subst1: - \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (v1: T).(\forall -(w1: T).(\forall (i: nat).((subst1 i v1 t1 w1) \to (\forall (v2: T).((pr0 v1 -v2) \to (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst1 i v2 t2 -w2))))))))))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(\lambda (v1: -T).(\lambda (w1: T).(\lambda (i: nat).(\lambda (H0: (subst1 i v1 t1 -w1)).(subst1_ind i v1 t1 (\lambda (t: T).(\forall (v2: T).((pr0 v1 v2) \to -(ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst1 i v2 t2 w2)))))) -(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(ex_intro2 T (\lambda (w2: T).(pr0 -t1 w2)) (\lambda (w2: T).(subst1 i v2 t2 w2)) t2 H (subst1_refl i v2 t2)))) -(\lambda (t0: T).(\lambda (H1: (subst0 i v1 t1 t0)).(\lambda (v2: T).(\lambda -(H2: (pr0 v1 v2)).(or_ind (pr0 t0 t2) (ex2 T (\lambda (w2: T).(pr0 t0 w2)) -(\lambda (w2: T).(subst0 i v2 t2 w2))) (ex2 T (\lambda (w2: T).(pr0 t0 w2)) -(\lambda (w2: T).(subst1 i v2 t2 w2))) (\lambda (H3: (pr0 t0 t2)).(ex_intro2 -T (\lambda (w2: T).(pr0 t0 w2)) (\lambda (w2: T).(subst1 i v2 t2 w2)) t2 H3 -(subst1_refl i v2 t2))) (\lambda (H3: (ex2 T (\lambda (w2: T).(pr0 t0 w2)) -(\lambda (w2: T).(subst0 i v2 t2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t0 -w2)) (\lambda (w2: T).(subst0 i v2 t2 w2)) (ex2 T (\lambda (w2: T).(pr0 t0 -w2)) (\lambda (w2: T).(subst1 i v2 t2 w2))) (\lambda (x: T).(\lambda (H4: -(pr0 t0 x)).(\lambda (H5: (subst0 i v2 t2 x)).(ex_intro2 T (\lambda (w2: -T).(pr0 t0 w2)) (\lambda (w2: T).(subst1 i v2 t2 w2)) x H4 (subst1_single i -v2 t2 x H5))))) H3)) (pr0_subst0 t1 t2 H v1 t0 i H1 v2 H2)))))) w1 H0))))))). -(* COMMENTS -Initial nodes: 385 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr1/defs.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr1/defs.ma deleted file mode 100644 index 84db71acb..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr1/defs.ma +++ /dev/null @@ -1,23 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr0/defs.ma". - -inductive pr1: T \to (T \to Prop) \def -| pr1_refl: \forall (t: T).(pr1 t t) -| pr1_sing: \forall (t2: T).(\forall (t1: T).((pr0 t1 t2) \to (\forall (t3: -T).((pr1 t2 t3) \to (pr1 t1 t3))))). - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr1/pr1.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr1/pr1.ma deleted file mode 100644 index ec469da50..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr1/pr1.ma +++ /dev/null @@ -1,70 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr1/props.ma". - -include "Basic-1/pr0/pr0.ma". - -theorem pr1_strip: - \forall (t0: T).(\forall (t1: T).((pr1 t0 t1) \to (\forall (t2: T).((pr0 t0 -t2) \to (ex2 T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t))))))) -\def - \lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr1 t0 t1)).(pr1_ind (\lambda -(t: T).(\lambda (t2: T).(\forall (t3: T).((pr0 t t3) \to (ex2 T (\lambda (t4: -T).(pr1 t2 t4)) (\lambda (t4: T).(pr1 t3 t4))))))) (\lambda (t: T).(\lambda -(t2: T).(\lambda (H0: (pr0 t t2)).(ex_intro2 T (\lambda (t3: T).(pr1 t t3)) -(\lambda (t3: T).(pr1 t2 t3)) t2 (pr1_pr0 t t2 H0) (pr1_refl t2))))) (\lambda -(t2: T).(\lambda (t3: T).(\lambda (H0: (pr0 t3 t2)).(\lambda (t4: T).(\lambda -(_: (pr1 t2 t4)).(\lambda (H2: ((\forall (t5: T).((pr0 t2 t5) \to (ex2 T -(\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t))))))).(\lambda (t5: -T).(\lambda (H3: (pr0 t3 t5)).(ex2_ind T (\lambda (t: T).(pr0 t5 t)) (\lambda -(t: T).(pr0 t2 t)) (ex2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 -t))) (\lambda (x: T).(\lambda (H4: (pr0 t5 x)).(\lambda (H5: (pr0 t2 x)).(let -H6 \def (H2 x H5) in (ex2_ind T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: -T).(pr1 x t)) (ex2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t))) -(\lambda (x0: T).(\lambda (H7: (pr1 t4 x0)).(\lambda (H8: (pr1 x -x0)).(ex_intro2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t)) x0 -H7 (pr1_t x t5 (pr1_pr0 t5 x H4) x0 H8))))) H6))))) (pr0_confluence t3 t5 H3 -t2 H0)))))))))) t0 t1 H))). -(* COMMENTS -Initial nodes: 317 -END *) - -theorem pr1_confluence: - \forall (t0: T).(\forall (t1: T).((pr1 t0 t1) \to (\forall (t2: T).((pr1 t0 -t2) \to (ex2 T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t))))))) -\def - \lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr1 t0 t1)).(pr1_ind (\lambda -(t: T).(\lambda (t2: T).(\forall (t3: T).((pr1 t t3) \to (ex2 T (\lambda (t4: -T).(pr1 t2 t4)) (\lambda (t4: T).(pr1 t3 t4))))))) (\lambda (t: T).(\lambda -(t2: T).(\lambda (H0: (pr1 t t2)).(ex_intro2 T (\lambda (t3: T).(pr1 t t3)) -(\lambda (t3: T).(pr1 t2 t3)) t2 H0 (pr1_refl t2))))) (\lambda (t2: -T).(\lambda (t3: T).(\lambda (H0: (pr0 t3 t2)).(\lambda (t4: T).(\lambda (_: -(pr1 t2 t4)).(\lambda (H2: ((\forall (t5: T).((pr1 t2 t5) \to (ex2 T (\lambda -(t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t))))))).(\lambda (t5: T).(\lambda -(H3: (pr1 t3 t5)).(let H_x \def (pr1_strip t3 t5 H3 t2 H0) in (let H4 \def -H_x in (ex2_ind T (\lambda (t: T).(pr1 t5 t)) (\lambda (t: T).(pr1 t2 t)) -(ex2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t))) (\lambda (x: -T).(\lambda (H5: (pr1 t5 x)).(\lambda (H6: (pr1 t2 x)).(let H_x0 \def (H2 x -H6) in (let H7 \def H_x0 in (ex2_ind T (\lambda (t: T).(pr1 t4 t)) (\lambda -(t: T).(pr1 x t)) (ex2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 -t))) (\lambda (x0: T).(\lambda (H8: (pr1 t4 x0)).(\lambda (H9: (pr1 x -x0)).(ex_intro2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t)) x0 -H8 (pr1_t x t5 H5 x0 H9))))) H7)))))) H4))))))))))) t0 t1 H))). -(* COMMENTS -Initial nodes: 311 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr1/props.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr1/props.ma deleted file mode 100644 index 0615bed96..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr1/props.ma +++ /dev/null @@ -1,126 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr1/defs.ma". - -include "Basic-1/pr0/subst1.ma". - -include "Basic-1/subst1/props.ma". - -include "Basic-1/T/props.ma". - -theorem pr1_pr0: - \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (pr1 t1 t2))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(pr1_sing t2 t1 H -t2 (pr1_refl t2)))). -(* COMMENTS -Initial nodes: 23 -END *) - -theorem pr1_t: - \forall (t2: T).(\forall (t1: T).((pr1 t1 t2) \to (\forall (t3: T).((pr1 t2 -t3) \to (pr1 t1 t3))))) -\def - \lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pr1 t1 t2)).(pr1_ind (\lambda -(t: T).(\lambda (t0: T).(\forall (t3: T).((pr1 t0 t3) \to (pr1 t t3))))) -(\lambda (t: T).(\lambda (t3: T).(\lambda (H0: (pr1 t t3)).H0))) (\lambda -(t0: T).(\lambda (t3: T).(\lambda (H0: (pr0 t3 t0)).(\lambda (t4: T).(\lambda -(_: (pr1 t0 t4)).(\lambda (H2: ((\forall (t5: T).((pr1 t4 t5) \to (pr1 t0 -t5))))).(\lambda (t5: T).(\lambda (H3: (pr1 t4 t5)).(pr1_sing t0 t3 H0 t5 (H2 -t5 H3)))))))))) t1 t2 H))). -(* COMMENTS -Initial nodes: 103 -END *) - -theorem pr1_head_1: - \forall (u1: T).(\forall (u2: T).((pr1 u1 u2) \to (\forall (t: T).(\forall -(k: K).(pr1 (THead k u1 t) (THead k u2 t)))))) -\def - \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr1 u1 u2)).(\lambda (t: -T).(\lambda (k: K).(pr1_ind (\lambda (t0: T).(\lambda (t1: T).(pr1 (THead k -t0 t) (THead k t1 t)))) (\lambda (t0: T).(pr1_refl (THead k t0 t))) (\lambda -(t2: T).(\lambda (t1: T).(\lambda (H0: (pr0 t1 t2)).(\lambda (t3: T).(\lambda -(_: (pr1 t2 t3)).(\lambda (H2: (pr1 (THead k t2 t) (THead k t3 t))).(pr1_sing -(THead k t2 t) (THead k t1 t) (pr0_comp t1 t2 H0 t t (pr0_refl t) k) (THead k -t3 t) H2))))))) u1 u2 H))))). -(* COMMENTS -Initial nodes: 137 -END *) - -theorem pr1_head_2: - \forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall (u: T).(\forall -(k: K).(pr1 (THead k u t1) (THead k u t2)))))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr1 t1 t2)).(\lambda (u: -T).(\lambda (k: K).(pr1_ind (\lambda (t: T).(\lambda (t0: T).(pr1 (THead k u -t) (THead k u t0)))) (\lambda (t: T).(pr1_refl (THead k u t))) (\lambda (t0: -T).(\lambda (t3: T).(\lambda (H0: (pr0 t3 t0)).(\lambda (t4: T).(\lambda (_: -(pr1 t0 t4)).(\lambda (H2: (pr1 (THead k u t0) (THead k u t4))).(pr1_sing -(THead k u t0) (THead k u t3) (pr0_comp u u (pr0_refl u) t3 t0 H0 k) (THead k -u t4) H2))))))) t1 t2 H))))). -(* COMMENTS -Initial nodes: 137 -END *) - -theorem pr1_comp: - \forall (v: T).(\forall (w: T).((pr1 v w) \to (\forall (t: T).(\forall (u: -T).((pr1 t u) \to (\forall (k: K).(pr1 (THead k v t) (THead k w u)))))))) -\def - \lambda (v: T).(\lambda (w: T).(\lambda (H: (pr1 v w)).(pr1_ind (\lambda (t: -T).(\lambda (t0: T).(\forall (t1: T).(\forall (u: T).((pr1 t1 u) \to (\forall -(k: K).(pr1 (THead k t t1) (THead k t0 u)))))))) (\lambda (t: T).(\lambda -(t0: T).(\lambda (u: T).(\lambda (H0: (pr1 t0 u)).(\lambda (k: K).(pr1_head_2 -t0 u H0 t k)))))) (\lambda (t2: T).(\lambda (t1: T).(\lambda (H0: (pr0 t1 -t2)).(\lambda (t3: T).(\lambda (H1: (pr1 t2 t3)).(\lambda (_: ((\forall (t: -T).(\forall (u: T).((pr1 t u) \to (\forall (k: K).(pr1 (THead k t2 t) (THead -k t3 u)))))))).(\lambda (t: T).(\lambda (u: T).(\lambda (H3: (pr1 t -u)).(\lambda (k: K).(pr1_ind (\lambda (t0: T).(\lambda (t4: T).(pr1 (THead k -t1 t0) (THead k t3 t4)))) (\lambda (t0: T).(pr1_head_1 t1 t3 (pr1_sing t2 t1 -H0 t3 H1) t0 k)) (\lambda (t0: T).(\lambda (t4: T).(\lambda (H4: (pr0 t4 -t0)).(\lambda (t5: T).(\lambda (_: (pr1 t0 t5)).(\lambda (H6: (pr1 (THead k -t1 t0) (THead k t3 t5))).(pr1_sing (THead k t1 t0) (THead k t1 t4) (pr0_comp -t1 t1 (pr0_refl t1) t4 t0 H4 k) (THead k t3 t5) H6))))))) t u H3))))))))))) v -w H))). -(* COMMENTS -Initial nodes: 273 -END *) - -theorem pr1_eta: - \forall (w: T).(\forall (u: T).(let t \def (THead (Bind Abst) w u) in -(\forall (v: T).((pr1 v w) \to (pr1 (THead (Bind Abst) v (THead (Flat Appl) -(TLRef O) (lift (S O) O t))) t))))) -\def - \lambda (w: T).(\lambda (u: T).(let t \def (THead (Bind Abst) w u) in -(\lambda (v: T).(\lambda (H: (pr1 v w)).(eq_ind_r T (THead (Bind Abst) (lift -(S O) O w) (lift (S O) (S O) u)) (\lambda (t0: T).(pr1 (THead (Bind Abst) v -(THead (Flat Appl) (TLRef O) t0)) (THead (Bind Abst) w u))) (pr1_comp v w H -(THead (Flat Appl) (TLRef O) (THead (Bind Abst) (lift (S O) O w) (lift (S O) -(S O) u))) u (pr1_sing (THead (Bind Abbr) (TLRef O) (lift (S O) (S O) u)) -(THead (Flat Appl) (TLRef O) (THead (Bind Abst) (lift (S O) O w) (lift (S O) -(S O) u))) (pr0_beta (lift (S O) O w) (TLRef O) (TLRef O) (pr0_refl (TLRef -O)) (lift (S O) (S O) u) (lift (S O) (S O) u) (pr0_refl (lift (S O) (S O) -u))) u (pr1_sing (THead (Bind Abbr) (TLRef O) (lift (S O) O u)) (THead (Bind -Abbr) (TLRef O) (lift (S O) (S O) u)) (pr0_delta1 (TLRef O) (TLRef O) -(pr0_refl (TLRef O)) (lift (S O) (S O) u) (lift (S O) (S O) u) (pr0_refl -(lift (S O) (S O) u)) (lift (S O) O u) (subst1_lift_S u O O (le_n O))) u -(pr1_pr0 (THead (Bind Abbr) (TLRef O) (lift (S O) O u)) u (pr0_zeta Abbr -not_abbr_abst u u (pr0_refl u) (TLRef O))))) (Bind Abst)) (lift (S O) O -(THead (Bind Abst) w u)) (lift_bind Abst w u (S O) O)))))). -(* COMMENTS -Initial nodes: 463 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr2/clen.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr2/clen.ma deleted file mode 100644 index df9ea1fd9..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr2/clen.ma +++ /dev/null @@ -1,161 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr2/props.ma". - -include "Basic-1/clen/getl.ma". - -theorem pr2_gen_ctail: - \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall -(t2: T).((pr2 (CTail k u c) t1 t2) \to (or (pr2 c t1 t2) (ex3 T (\lambda (_: -T).(eq K k (Bind Abbr))) (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(subst0 -(clen c) u t t2))))))))) -\def - \lambda (k: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda -(t2: T).(\lambda (H: (pr2 (CTail k u c) t1 t2)).(insert_eq C (CTail k u c) -(\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(or (pr2 c t1 t2) (ex3 T -(\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t: T).(pr0 t1 t)) (\lambda -(t: T).(subst0 (clen c) u t t2))))) (\lambda (y: C).(\lambda (H0: (pr2 y t1 -t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 -(CTail k u c)) \to (or (pr2 c t t0) (ex3 T (\lambda (_: T).(eq K k (Bind -Abbr))) (\lambda (t3: T).(pr0 t t3)) (\lambda (t3: T).(subst0 (clen c) u t3 -t0)))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: -(pr0 t3 t4)).(\lambda (_: (eq C c0 (CTail k u c))).(or_introl (pr2 c t3 t4) -(ex3 T (\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t: T).(pr0 t3 t)) -(\lambda (t: T).(subst0 (clen c) u t t4))) (pr2_free c t3 t4 H1))))))) -(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda -(H1: (getl i c0 (CHead d (Bind Abbr) u0))).(\lambda (t3: T).(\lambda (t4: -T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: (subst0 i u0 t4 -t)).(\lambda (H4: (eq C c0 (CTail k u c))).(let H5 \def (eq_ind C c0 (\lambda -(c1: C).(getl i c1 (CHead d (Bind Abbr) u0))) H1 (CTail k u c) H4) in (let -H_x \def (getl_gen_tail k Abbr u u0 d c i H5) in (let H6 \def H_x in (or_ind -(ex2 C (\lambda (e: C).(eq C d (CTail k u e))) (\lambda (e: C).(getl i c -(CHead e (Bind Abbr) u0)))) (ex4 nat (\lambda (_: nat).(eq nat i (clen c))) -(\lambda (_: nat).(eq K k (Bind Abbr))) (\lambda (_: nat).(eq T u u0)) -(\lambda (n: nat).(eq C d (CSort n)))) (or (pr2 c t3 t) (ex3 T (\lambda (_: -T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0: -T).(subst0 (clen c) u t0 t)))) (\lambda (H7: (ex2 C (\lambda (e: C).(eq C d -(CTail k u e))) (\lambda (e: C).(getl i c (CHead e (Bind Abbr) -u0))))).(ex2_ind C (\lambda (e: C).(eq C d (CTail k u e))) (\lambda (e: -C).(getl i c (CHead e (Bind Abbr) u0))) (or (pr2 c t3 t) (ex3 T (\lambda (_: -T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0: -T).(subst0 (clen c) u t0 t)))) (\lambda (x: C).(\lambda (_: (eq C d (CTail k -u x))).(\lambda (H9: (getl i c (CHead x (Bind Abbr) u0))).(or_introl (pr2 c -t3 t) (ex3 T (\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3 -t0)) (\lambda (t0: T).(subst0 (clen c) u t0 t))) (pr2_delta c x u0 i H9 t3 t4 -H2 t H3))))) H7)) (\lambda (H7: (ex4 nat (\lambda (_: nat).(eq nat i (clen -c))) (\lambda (_: nat).(eq K k (Bind Abbr))) (\lambda (_: nat).(eq T u u0)) -(\lambda (n: nat).(eq C d (CSort n))))).(ex4_ind nat (\lambda (_: nat).(eq -nat i (clen c))) (\lambda (_: nat).(eq K k (Bind Abbr))) (\lambda (_: -nat).(eq T u u0)) (\lambda (n: nat).(eq C d (CSort n))) (or (pr2 c t3 t) (ex3 -T (\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3 t0)) -(\lambda (t0: T).(subst0 (clen c) u t0 t)))) (\lambda (x0: nat).(\lambda (H8: -(eq nat i (clen c))).(\lambda (H9: (eq K k (Bind Abbr))).(\lambda (H10: (eq T -u u0)).(\lambda (_: (eq C d (CSort x0))).(let H12 \def (eq_ind nat i (\lambda -(n: nat).(subst0 n u0 t4 t)) H3 (clen c) H8) in (let H13 \def (eq_ind_r T u0 -(\lambda (t0: T).(subst0 (clen c) t0 t4 t)) H12 u H10) in (eq_ind_r K (Bind -Abbr) (\lambda (k0: K).(or (pr2 c t3 t) (ex3 T (\lambda (_: T).(eq K k0 (Bind -Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0 -t))))) (or_intror (pr2 c t3 t) (ex3 T (\lambda (_: T).(eq K (Bind Abbr) (Bind -Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0 -t))) (ex3_intro T (\lambda (_: T).(eq K (Bind Abbr) (Bind Abbr))) (\lambda -(t0: T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0 t)) t4 -(refl_equal K (Bind Abbr)) H2 H13)) k H9)))))))) H7)) H6))))))))))))))) y t1 -t2 H0))) H)))))). -(* COMMENTS -Initial nodes: 1161 -END *) - -theorem pr2_gen_cbind: - \forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall -(t2: T).((pr2 (CHead c (Bind b) v) t1 t2) \to (pr2 c (THead (Bind b) v t1) -(THead (Bind b) v t2))))))) -\def - \lambda (b: B).(\lambda (c: C).(\lambda (v: T).(\lambda (t1: T).(\lambda -(t2: T).(\lambda (H: (pr2 (CHead c (Bind b) v) t1 t2)).(insert_eq C (CHead c -(Bind b) v) (\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 c (THead -(Bind b) v t1) (THead (Bind b) v t2))) (\lambda (y: C).(\lambda (H0: (pr2 y -t1 t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 -(CHead c (Bind b) v)) \to (pr2 c (THead (Bind b) v t) (THead (Bind b) v -t0)))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: -(pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c (Bind b) v))).(pr2_free c (THead -(Bind b) v t3) (THead (Bind b) v t4) (pr0_comp v v (pr0_refl v) t3 t4 H1 -(Bind b)))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: -nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t3: -T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: -(subst0 i u t4 t)).(\lambda (H4: (eq C c0 (CHead c (Bind b) v))).(let H5 \def -(eq_ind C c0 (\lambda (c1: C).(getl i c1 (CHead d (Bind Abbr) u))) H1 (CHead -c (Bind b) v) H4) in (let H_x \def (getl_gen_bind b c (CHead d (Bind Abbr) u) -v i H5) in (let H6 \def H_x in (or_ind (land (eq nat i O) (eq C (CHead d -(Bind Abbr) u) (CHead c (Bind b) v))) (ex2 nat (\lambda (j: nat).(eq nat i (S -j))) (\lambda (j: nat).(getl j c (CHead d (Bind Abbr) u)))) (pr2 c (THead -(Bind b) v t3) (THead (Bind b) v t)) (\lambda (H7: (land (eq nat i O) (eq C -(CHead d (Bind Abbr) u) (CHead c (Bind b) v)))).(land_ind (eq nat i O) (eq C -(CHead d (Bind Abbr) u) (CHead c (Bind b) v)) (pr2 c (THead (Bind b) v t3) -(THead (Bind b) v t)) (\lambda (H8: (eq nat i O)).(\lambda (H9: (eq C (CHead -d (Bind Abbr) u) (CHead c (Bind b) v))).(let H10 \def (f_equal C C (\lambda -(e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d -| (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u) (CHead c (Bind b) -v) H9) in ((let H11 \def (f_equal C B (\lambda (e: C).(match e in C return -(\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b0) -\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) -(CHead c (Bind b) v) H9) in ((let H12 \def (f_equal C T (\lambda (e: -C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | -(CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind Abbr) u) (CHead c (Bind b) v) -H9) in (\lambda (H13: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H15 \def -(eq_ind nat i (\lambda (n: nat).(subst0 n u t4 t)) H3 O H8) in (let H16 \def -(eq_ind T u (\lambda (t0: T).(subst0 O t0 t4 t)) H15 v H12) in (eq_ind B Abbr -(\lambda (b0: B).(pr2 c (THead (Bind b0) v t3) (THead (Bind b0) v t))) -(pr2_free c (THead (Bind Abbr) v t3) (THead (Bind Abbr) v t) (pr0_delta v v -(pr0_refl v) t3 t4 H2 t H16)) b H13)))))) H11)) H10)))) H7)) (\lambda (H7: -(ex2 nat (\lambda (j: nat).(eq nat i (S j))) (\lambda (j: nat).(getl j c -(CHead d (Bind Abbr) u))))).(ex2_ind nat (\lambda (j: nat).(eq nat i (S j))) -(\lambda (j: nat).(getl j c (CHead d (Bind Abbr) u))) (pr2 c (THead (Bind b) -v t3) (THead (Bind b) v t)) (\lambda (x: nat).(\lambda (H8: (eq nat i (S -x))).(\lambda (H9: (getl x c (CHead d (Bind Abbr) u))).(let H10 \def (f_equal -nat nat (\lambda (e: nat).e) i (S x) H8) in (let H11 \def (eq_ind nat i -(\lambda (n: nat).(subst0 n u t4 t)) H3 (S x) H10) in (pr2_head_2 c v t3 t -(Bind b) (pr2_delta (CHead c (Bind b) v) d u (S x) (getl_clear_bind b (CHead -c (Bind b) v) c v (clear_bind b c v) (CHead d (Bind Abbr) u) x H9) t3 t4 H2 t -H11))))))) H7)) H6))))))))))))))) y t1 t2 H0))) H)))))). -(* COMMENTS -Initial nodes: 1085 -END *) - -theorem pr2_gen_cflat: - \forall (f: F).(\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall -(t2: T).((pr2 (CHead c (Flat f) v) t1 t2) \to (pr2 c t1 t2)))))) -\def - \lambda (f: F).(\lambda (c: C).(\lambda (v: T).(\lambda (t1: T).(\lambda -(t2: T).(\lambda (H: (pr2 (CHead c (Flat f) v) t1 t2)).(insert_eq C (CHead c -(Flat f) v) (\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 c t1 t2)) -(\lambda (y: C).(\lambda (H0: (pr2 y t1 t2)).(pr2_ind (\lambda (c0: -C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CHead c (Flat f) v)) \to (pr2 -c t t0))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: -(pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c (Flat f) v))).(pr2_free c t3 t4 -H1)))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: -nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t3: -T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: -(subst0 i u t4 t)).(\lambda (H4: (eq C c0 (CHead c (Flat f) v))).(let H5 \def -(eq_ind C c0 (\lambda (c1: C).(getl i c1 (CHead d (Bind Abbr) u))) H1 (CHead -c (Flat f) v) H4) in (let H_y \def (getl_gen_flat f c (CHead d (Bind Abbr) u) -v i H5) in (pr2_delta c d u i H_y t3 t4 H2 t H3)))))))))))))) y t1 t2 H0))) -H)))))). -(* COMMENTS -Initial nodes: 293 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr2/defs.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr2/defs.ma deleted file mode 100644 index e21568232..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr2/defs.ma +++ /dev/null @@ -1,28 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr0/defs.ma". - -include "Basic-1/getl/defs.ma". - -inductive pr2: C \to (T \to (T \to Prop)) \def -| pr2_free: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to -(pr2 c t1 t2)))) -| pr2_delta: \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: -nat).((getl i c (CHead d (Bind Abbr) u)) \to (\forall (t1: T).(\forall (t2: -T).((pr0 t1 t2) \to (\forall (t: T).((subst0 i u t2 t) \to (pr2 c t1 -t)))))))))). - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr2/fwd.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr2/fwd.ma deleted file mode 100644 index 1a18f7dbc..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr2/fwd.ma +++ /dev/null @@ -1,3343 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr2/defs.ma". - -include "Basic-1/pr0/fwd.ma". - -include "Basic-1/getl/drop.ma". - -include "Basic-1/getl/clear.ma". - -theorem pr2_gen_sort: - \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr2 c (TSort n) x) \to -(eq T x (TSort n))))) -\def - \lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr2 c (TSort -n) x)).(insert_eq T (TSort n) (\lambda (t: T).(pr2 c t x)) (\lambda (t: -T).(eq T x t)) (\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda -(_: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (TSort n)) \to (eq T t0 -t))))) (\lambda (_: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H1: (pr0 -t1 t2)).(\lambda (H2: (eq T t1 (TSort n))).(let H3 \def (eq_ind T t1 (\lambda -(t: T).(pr0 t t2)) H1 (TSort n) H2) in (eq_ind_r T (TSort n) (\lambda (t: -T).(eq T t2 t)) (eq_ind_r T (TSort n) (\lambda (t: T).(eq T t (TSort n))) -(refl_equal T (TSort n)) t2 (pr0_gen_sort t2 n H3)) t1 H2))))))) (\lambda -(c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (_: (getl -i c0 (CHead d (Bind Abbr) u))).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H2: (pr0 t1 t2)).(\lambda (t: T).(\lambda (H3: (subst0 i u t2 t)).(\lambda -(H4: (eq T t1 (TSort n))).(let H5 \def (eq_ind T t1 (\lambda (t0: T).(pr0 t0 -t2)) H2 (TSort n) H4) in (eq_ind_r T (TSort n) (\lambda (t0: T).(eq T t t0)) -(let H6 \def (eq_ind T t2 (\lambda (t0: T).(subst0 i u t0 t)) H3 (TSort n) -(pr0_gen_sort t2 n H5)) in (subst0_gen_sort u t i n H6 (eq T t (TSort n)))) -t1 H4))))))))))))) c y x H0))) H)))). -(* COMMENTS -Initial nodes: 347 -END *) - -theorem pr2_gen_lref: - \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr2 c (TLRef n) x) \to -(or (eq T x (TLRef n)) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c -(CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T x (lift (S -n) O u))))))))) -\def - \lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr2 c (TLRef -n) x)).(insert_eq T (TLRef n) (\lambda (t: T).(pr2 c t x)) (\lambda (t: -T).(or (eq T x t) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c (CHead -d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T x (lift (S n) O -u))))))) (\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0: -C).(\lambda (t: T).(\lambda (t0: T).((eq T t (TLRef n)) \to (or (eq T t0 t) -(ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c0 (CHead d (Bind Abbr) -u)))) (\lambda (_: C).(\lambda (u: T).(eq T t0 (lift (S n) O u)))))))))) -(\lambda (c0: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H1: (pr0 t1 -t2)).(\lambda (H2: (eq T t1 (TLRef n))).(let H3 \def (eq_ind T t1 (\lambda -(t: T).(pr0 t t2)) H1 (TLRef n) H2) in (eq_ind_r T (TLRef n) (\lambda (t: -T).(or (eq T t2 t) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c0 -(CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T t2 (lift (S -n) O u))))))) (eq_ind_r T (TLRef n) (\lambda (t: T).(or (eq T t (TLRef n)) -(ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c0 (CHead d (Bind Abbr) -u)))) (\lambda (_: C).(\lambda (u: T).(eq T t (lift (S n) O u))))))) -(or_introl (eq T (TLRef n) (TLRef n)) (ex2_2 C T (\lambda (d: C).(\lambda (u: -T).(getl n c0 (CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq -T (TLRef n) (lift (S n) O u))))) (refl_equal T (TLRef n))) t2 (pr0_gen_lref -t2 n H3)) t1 H2))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: -T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr) -u))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H2: (pr0 t1 t2)).(\lambda -(t: T).(\lambda (H3: (subst0 i u t2 t)).(\lambda (H4: (eq T t1 (TLRef -n))).(let H5 \def (eq_ind T t1 (\lambda (t0: T).(pr0 t0 t2)) H2 (TLRef n) H4) -in (eq_ind_r T (TLRef n) (\lambda (t0: T).(or (eq T t t0) (ex2_2 C T (\lambda -(d0: C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind Abbr) u0)))) (\lambda (_: -C).(\lambda (u0: T).(eq T t (lift (S n) O u0))))))) (let H6 \def (eq_ind T t2 -(\lambda (t0: T).(subst0 i u t0 t)) H3 (TLRef n) (pr0_gen_lref t2 n H5)) in -(land_ind (eq nat n i) (eq T t (lift (S n) O u)) (or (eq T t (TLRef n)) -(ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind Abbr) -u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T t (lift (S n) O u0)))))) -(\lambda (H7: (eq nat n i)).(\lambda (H8: (eq T t (lift (S n) O -u))).(eq_ind_r T (lift (S n) O u) (\lambda (t0: T).(or (eq T t0 (TLRef n)) -(ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind Abbr) -u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T t0 (lift (S n) O u0))))))) (let -H9 \def (eq_ind_r nat i (\lambda (n0: nat).(getl n0 c0 (CHead d (Bind Abbr) -u))) H1 n H7) in (or_intror (eq T (lift (S n) O u) (TLRef n)) (ex2_2 C T -(\lambda (d0: C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind Abbr) u0)))) -(\lambda (_: C).(\lambda (u0: T).(eq T (lift (S n) O u) (lift (S n) O u0))))) -(ex2_2_intro C T (\lambda (d0: C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind -Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T (lift (S n) O u) (lift (S -n) O u0)))) d u H9 (refl_equal T (lift (S n) O u))))) t H8))) -(subst0_gen_lref u t i n H6))) t1 H4))))))))))))) c y x H0))) H)))). -(* COMMENTS -Initial nodes: 1003 -END *) - -theorem pr2_gen_abst: - \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c -(THead (Bind Abst) u1 t1) x) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 -c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: -T).(pr2 (CHead c (Bind b) u) t1 t2)))))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda -(H: (pr2 c (THead (Bind Abst) u1 t1) x)).(insert_eq T (THead (Bind Abst) u1 -t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall -(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2))))))) (\lambda (y: -T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0: C).(\lambda (t: -T).(\lambda (t0: T).((eq T t (THead (Bind Abst) u1 t1)) \to (ex3_2 T T -(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abst) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 -t2)))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H1: -(pr0 t0 t2)).(\lambda (H2: (eq T t0 (THead (Bind Abst) u1 t1))).(let H3 \def -(eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Bind Abst) u1 t1) H2) in -(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: -T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall -(u: T).(pr2 (CHead c0 (Bind b) u) t1 t3)))))) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H4: (eq T t2 (THead (Bind Abst) x0 x1))).(\lambda (H5: (pr0 u1 -x0)).(\lambda (H6: (pr0 t1 x1)).(eq_ind_r T (THead (Bind Abst) x0 x1) -(\lambda (t: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead -(Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) -(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead -c0 (Bind b) u) t1 t3))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: -T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall -(b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 t3))))) x0 x1 -(refl_equal T (THead (Bind Abst) x0 x1)) (pr2_free c0 u1 x0 H5) (\lambda (b: -B).(\lambda (u: T).(pr2_free (CHead c0 (Bind b) u) t1 x1 H6)))) t2 H4)))))) -(pr0_gen_abst u1 t1 t2 H3)))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda -(u: T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr) -u))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H2: (pr0 t0 t2)).(\lambda -(t: T).(\lambda (H3: (subst0 i u t2 t)).(\lambda (H4: (eq T t0 (THead (Bind -Abst) u1 t1))).(let H5 \def (eq_ind T t0 (\lambda (t3: T).(pr0 t3 t2)) H2 -(THead (Bind Abst) u1 t1) H4) in (ex3_2_ind T T (\lambda (u2: T).(\lambda -(t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 -t3)))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H6: (eq T t2 (THead -(Bind Abst) x0 x1))).(\lambda (H7: (pr0 u1 x0)).(\lambda (H8: (pr0 t1 -x1)).(let H9 \def (eq_ind T t2 (\lambda (t3: T).(subst0 i u t3 t)) H3 (THead -(Bind Abst) x0 x1) H6) in (or3_ind (ex2 T (\lambda (u2: T).(eq T t (THead -(Bind Abst) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2))) (ex2 T (\lambda -(t3: T).(eq T t (THead (Bind Abst) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind -Abst) i) u x1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t -(THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 -u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u x1 t3)))) -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) t1 t3)))))) (\lambda (H10: (ex2 T (\lambda (u2: T).(eq T t (THead (Bind -Abst) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2)))).(ex2_ind T (\lambda -(u2: T).(eq T t (THead (Bind Abst) u2 x1))) (\lambda (u2: T).(subst0 i u x0 -u2)) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst) -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) t1 t3)))))) (\lambda (x2: T).(\lambda (H11: (eq T t (THead (Bind Abst) x2 -x1))).(\lambda (H12: (subst0 i u x0 x2)).(eq_ind_r T (THead (Bind Abst) x2 -x1) (\lambda (t3: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 -(THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 -(CHead c0 (Bind b) u0) t1 t4))))))) (ex3_2_intro T T (\lambda (u2: -T).(\lambda (t3: T).(eq T (THead (Bind Abst) x2 x1) (THead (Bind Abst) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) t1 t3))))) x2 x1 (refl_equal T (THead (Bind Abst) x2 x1)) (pr2_delta c0 d -u i H1 u1 x0 H7 x2 H12) (\lambda (b: B).(\lambda (u0: T).(pr2_free (CHead c0 -(Bind b) u0) t1 x1 H8)))) t H11)))) H10)) (\lambda (H10: (ex2 T (\lambda (t3: -T).(eq T t (THead (Bind Abst) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind -Abst) i) u x1 t3)))).(ex2_ind T (\lambda (t3: T).(eq T t (THead (Bind Abst) -x0 t3))) (\lambda (t3: T).(subst0 (s (Bind Abst) i) u x1 t3)) (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 -t3)))))) (\lambda (x2: T).(\lambda (H11: (eq T t (THead (Bind Abst) x0 -x2))).(\lambda (H12: (subst0 (s (Bind Abst) i) u x1 x2)).(eq_ind_r T (THead -(Bind Abst) x0 x2) (\lambda (t3: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t4: -T).(eq T t3 (THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall -(u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t4))))))) (ex3_2_intro T T (\lambda -(u2: T).(\lambda (t3: T).(eq T (THead (Bind Abst) x0 x2) (THead (Bind Abst) -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) t1 t3))))) x0 x2 (refl_equal T (THead (Bind Abst) x0 x2)) (pr2_free c0 u1 -x0 H7) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u -(S i) (getl_head (Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 x1 H8 x2 -H12)))) t H11)))) H10)) (\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda -(t3: T).(eq T t (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind -Abst) i) u x1 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T -t (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u -x0 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u x1 -t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst) -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) t1 t3)))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H11: (eq T t -(THead (Bind Abst) x2 x3))).(\lambda (H12: (subst0 i u x0 x2)).(\lambda (H13: -(subst0 (s (Bind Abst) i) u x1 x3)).(eq_ind_r T (THead (Bind Abst) x2 x3) -(\lambda (t3: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead -(Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) -(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead -c0 (Bind b) u0) t1 t4))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: -T).(eq T (THead (Bind Abst) x2 x3) (THead (Bind Abst) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall -(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))))) x2 x3 -(refl_equal T (THead (Bind Abst) x2 x3)) (pr2_delta c0 d u i H1 u1 x0 H7 x2 -H12) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u -(S i) (getl_head (Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 x1 H8 x3 -H13)))) t H11)))))) H10)) (subst0_gen_head (Bind Abst) u x0 x1 t i H9)))))))) -(pr0_gen_abst u1 t1 t2 H5)))))))))))))) c y x H0))) H))))). -(* COMMENTS -Initial nodes: 2383 -END *) - -theorem pr2_gen_cast: - \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c -(THead (Flat Cast) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda -(t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c t1 t2)))) (pr2 c -t1 x)))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda -(H: (pr2 c (THead (Flat Cast) u1 t1) x)).(insert_eq T (THead (Flat Cast) u1 -t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c t1 -t2)))) (pr2 c t1 x))) (\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind -(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Flat Cast) -u1 t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead -(Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) -(\lambda (_: T).(\lambda (t2: T).(pr2 c0 t1 t2)))) (pr2 c0 t1 t0)))))) -(\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H1: (pr0 t0 -t2)).(\lambda (H2: (eq T t0 (THead (Flat Cast) u1 t1))).(let H3 \def (eq_ind -T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Flat Cast) u1 t1) H2) in (or_ind -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2) (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -t1 t3)))) (pr2 c0 t1 t2)) (\lambda (H4: (ex3_2 T T (\lambda (u2: T).(\lambda -(t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind -T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 -(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t2)) -(\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T t2 (THead (Flat Cast) -x0 x1))).(\lambda (H6: (pr0 u1 x0)).(\lambda (H7: (pr0 t1 x1)).(eq_ind_r T -(THead (Flat Cast) x0 x1) (\lambda (t: T).(or (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -t1 t3)))) (pr2 c0 t1 t))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda -(t3: T).(eq T (THead (Flat Cast) x0 x1) (THead (Flat Cast) u2 t3)))) (\lambda -(u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: -T).(pr2 c0 t1 t3)))) (pr2 c0 t1 (THead (Flat Cast) x0 x1)) (ex3_2_intro T T -(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Cast) x0 x1) (THead -(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) -(\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3))) x0 x1 (refl_equal T (THead -(Flat Cast) x0 x1)) (pr2_free c0 u1 x0 H6) (pr2_free c0 t1 x1 H7))) t2 -H5)))))) H4)) (\lambda (H4: (pr0 t1 t2)).(or_intror (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -t1 t3)))) (pr2 c0 t1 t2) (pr2_free c0 t1 t2 H4))) (pr0_gen_cast u1 t1 t2 -H3)))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: -nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t0: -T).(\lambda (t2: T).(\lambda (H2: (pr0 t0 t2)).(\lambda (t: T).(\lambda (H3: -(subst0 i u t2 t)).(\lambda (H4: (eq T t0 (THead (Flat Cast) u1 t1))).(let H5 -\def (eq_ind T t0 (\lambda (t3: T).(pr0 t3 t2)) H2 (THead (Flat Cast) u1 t1) -H4) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead -(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda -(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2) (or (ex3_2 T T (\lambda -(u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -t1 t3)))) (pr2 c0 t1 t)) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda -(t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind -T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t -(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t)) -(\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: (eq T t2 (THead (Flat Cast) -x0 x1))).(\lambda (H8: (pr0 u1 x0)).(\lambda (H9: (pr0 t1 x1)).(let H10 \def -(eq_ind T t2 (\lambda (t3: T).(subst0 i u t3 t)) H3 (THead (Flat Cast) x0 x1) -H7) in (or3_ind (ex2 T (\lambda (u2: T).(eq T t (THead (Flat Cast) u2 x1))) -(\lambda (u2: T).(subst0 i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T t (THead -(Flat Cast) x0 t3))) (\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3))) -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: -T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3)))) (or (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t)) (\lambda (H11: (ex2 T (\lambda (u2: -T).(eq T t (THead (Flat Cast) u2 x1))) (\lambda (u2: T).(subst0 i u x0 -u2)))).(ex2_ind T (\lambda (u2: T).(eq T t (THead (Flat Cast) u2 x1))) -(\lambda (u2: T).(subst0 i u x0 u2)) (or (ex3_2 T T (\lambda (u2: T).(\lambda -(t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 -c0 t1 t)) (\lambda (x2: T).(\lambda (H12: (eq T t (THead (Flat Cast) x2 -x1))).(\lambda (H13: (subst0 i u x0 x2)).(or_introl (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -t1 t3)))) (pr2 c0 t1 t) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: -T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 -c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3))) x2 x1 H12 -(pr2_delta c0 d u i H1 u1 x0 H8 x2 H13) (pr2_free c0 t1 x1 H9)))))) H11)) -(\lambda (H11: (ex2 T (\lambda (t3: T).(eq T t (THead (Flat Cast) x0 t3))) -(\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3)))).(ex2_ind T (\lambda -(t3: T).(eq T t (THead (Flat Cast) x0 t3))) (\lambda (t3: T).(subst0 (s (Flat -Cast) i) u x1 t3)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t -(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t)) -(\lambda (x2: T).(\lambda (H12: (eq T t (THead (Flat Cast) x0 x2))).(\lambda -(H13: (subst0 (s (Flat Cast) i) u x1 x2)).(or_introl (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -t1 t3)))) (pr2 c0 t1 t) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: -T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 -c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3))) x0 x2 H12 -(pr2_free c0 u1 x0 H8) (pr2_delta c0 d u i H1 t1 x1 H9 x2 H13)))))) H11)) -(\lambda (H11: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead -(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) -(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 -t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat -Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) -(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3))) (or -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t)) (\lambda (x2: -T).(\lambda (x3: T).(\lambda (H12: (eq T t (THead (Flat Cast) x2 -x3))).(\lambda (H13: (subst0 i u x0 x2)).(\lambda (H14: (subst0 (s (Flat -Cast) i) u x1 x3)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: -T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 -c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t) -(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast) -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr2 c0 t1 t3))) x2 x3 H12 (pr2_delta c0 d u i H1 u1 x0 -H8 x2 H13) (pr2_delta c0 d u i H1 t1 x1 H9 x3 H14)))))))) H11)) -(subst0_gen_head (Flat Cast) u x0 x1 t i H10)))))))) H6)) (\lambda (H6: (pr0 -t1 t2)).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t -(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t) -(pr2_delta c0 d u i H1 t1 t2 H6 t H3))) (pr0_gen_cast u1 t1 t2 -H5)))))))))))))) c y x H0))) H))))). -(* COMMENTS -Initial nodes: 2659 -END *) - -theorem pr2_gen_csort: - \forall (t1: T).(\forall (t2: T).(\forall (n: nat).((pr2 (CSort n) t1 t2) -\to (pr0 t1 t2)))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (n: nat).(\lambda (H: (pr2 (CSort -n) t1 t2)).(insert_eq C (CSort n) (\lambda (c: C).(pr2 c t1 t2)) (\lambda (_: -C).(pr0 t1 t2)) (\lambda (y: C).(\lambda (H0: (pr2 y t1 t2)).(pr2_ind -(\lambda (c: C).(\lambda (t: T).(\lambda (t0: T).((eq C c (CSort n)) \to (pr0 -t t0))))) (\lambda (c: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: -(pr0 t3 t4)).(\lambda (_: (eq C c (CSort n))).H1))))) (\lambda (c: -C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H1: (getl i c -(CHead d (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 -t3 t4)).(\lambda (t: T).(\lambda (_: (subst0 i u t4 t)).(\lambda (H4: (eq C c -(CSort n))).(let H5 \def (eq_ind C c (\lambda (c0: C).(getl i c0 (CHead d -(Bind Abbr) u))) H1 (CSort n) H4) in (getl_gen_sort n i (CHead d (Bind Abbr) -u) H5 (pr0 t3 t)))))))))))))) y t1 t2 H0))) H)))). -(* COMMENTS -Initial nodes: 221 -END *) - -theorem pr2_gen_appl: - \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c -(THead (Flat Appl) u1 t1) x) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda -(t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c t1 t2)))) (ex4_4 T -T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T -t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall -(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T -T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq -T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead -(Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda -(H: (pr2 c (THead (Flat Appl) u1 t1) x)).(insert_eq T (THead (Flat Appl) u1 -t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(or3 (ex3_2 T T (\lambda -(u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c t1 -t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind -Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) -z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) -(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) -z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) -y2) z1 z2)))))))))) (\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind -(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Flat Appl) -u1 t1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead -(Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) -(\lambda (_: T).(\lambda (t2: T).(pr2 c0 t1 t2)))) (ex4_4 T T T T (\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead -(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall -(b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) z1 t2)))))))) (ex6_6 B T T -T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T -t0 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 -y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 -z2))))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: T).(\lambda -(H1: (pr0 t0 t2)).(\lambda (H2: (eq T t0 (THead (Flat Appl) u1 t1))).(let H3 -\def (eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Flat Appl) u1 t1) -H2) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead -(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda -(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda -(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3)))))))) (or3 (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) -u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda -(u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift -(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 -(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (H4: (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 -t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead -(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda -(_: T).(\lambda (t3: T).(pr0 t1 t3))) (or3 (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) -u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda -(u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift -(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 -(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H5: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H6: (pr0 u1 -x0)).(\lambda (H7: (pr0 t1 x1)).(eq_ind_r T (THead (Flat Appl) x0 x1) -(\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t -(THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall -(b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T -T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t -(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda -(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 -y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 -z2)))))))))) (or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T -(THead (Flat Appl) x0 x1) (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) -x0 x1) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall -(u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Flat -Appl) x0 x1) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) -z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2)))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T -(THead (Flat Appl) x0 x1) (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -t1 t3))) x0 x1 (refl_equal T (THead (Flat Appl) x0 x1)) (pr2_free c0 u1 x0 -H6) (pr2_free c0 t1 x1 H7))) t2 H5)))))) H4)) (\lambda (H4: (ex4_4 T T T T -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda -(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 -t3))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda -(t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 -T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq -T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall -(b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T -T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T -t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 -y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 -z2))))))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: -T).(\lambda (H5: (eq T t1 (THead (Bind Abst) x0 x1))).(\lambda (H6: (eq T t2 -(THead (Bind Abbr) x2 x3))).(\lambda (H7: (pr0 u1 x2)).(\lambda (H8: (pr0 x1 -x3)).(eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda (t: T).(or3 (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t -(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 -(CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead -(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 -u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (eq_ind_r T (THead (Bind -Abst) x0 x1) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: -T).(eq T (THead (Bind Abbr) x2 x3) (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) -x2 x3) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall -(u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind -Abbr) x2 x3) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) -z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: -T).(eq T (THead (Bind Abbr) x2 x3) (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) u2 -t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: -T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) -u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead -(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind -Abbr) x2 x3) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) -z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead -(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) u2 -t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: -T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) -u) z1 t3))))))) x0 x1 x2 x3 (refl_equal T (THead (Bind Abst) x0 x1)) -(refl_equal T (THead (Bind Abbr) x2 x3)) (pr2_free c0 u1 x2 H7) (\lambda (b: -B).(\lambda (u: T).(pr2_free (CHead c0 (Bind b) u) x1 x3 H8))))) t1 H5) t2 -H6))))))))) H4)) (\lambda (H4: (ex6_6 B T T T T T (\lambda (b: B).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not -(eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) -y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead (Flat -Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 -t3))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead (Flat Appl) (lift -(S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 -t3))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead -(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) -(\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead -(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall -(b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T -T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T -t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 -y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 -z2))))))))) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: -T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H5: (not (eq B x0 -Abst))).(\lambda (H6: (eq T t1 (THead (Bind x0) x1 x2))).(\lambda (H7: (eq T -t2 (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)))).(\lambda -(H8: (pr0 u1 x3)).(\lambda (H9: (pr0 x1 x4)).(\lambda (H10: (pr0 x2 -x5)).(eq_ind_r T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) -x5)) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T -t (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall -(b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T -T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t -(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda -(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 -y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 -z2)))))))))) (eq_ind_r T (THead (Bind x0) x1 x2) (\lambda (t: T).(or3 (ex3_2 -T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat -Appl) (lift (S O) O x3) x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) -x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind Abbr) u2 t3)))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: -T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) -(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T t (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T -(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind b) -y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 -c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat -Appl) (lift (S O) O x3) x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) -(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 -(CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind -x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T -(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind b) -y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 -c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) -(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda -(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 -y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))) -x0 x1 x2 x5 x3 x4 H5 (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T -(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5))) (pr2_free c0 -u1 x3 H8) (pr2_free c0 x1 x4 H9) (pr2_free (CHead c0 (Bind x0) x4) x2 x5 -H10))) t1 H6) t2 H7))))))))))))) H4)) (pr0_gen_appl u1 t1 t2 H3)))))))) -(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t0: T).(\lambda (t2: -T).(\lambda (H2: (pr0 t0 t2)).(\lambda (t: T).(\lambda (H3: (subst0 i u t2 -t)).(\lambda (H4: (eq T t0 (THead (Flat Appl) u1 t1))).(let H5 \def (eq_ind T -t0 (\lambda (t3: T).(pr0 t3 t2)) H2 (THead (Flat Appl) u1 t1) H4) in (or3_ind -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 -z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: -T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda -(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3)))))))) (or3 (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind -Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda -(u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift -(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 -(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (H6: (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 -t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead -(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda -(_: T).(\lambda (t3: T).(pr0 t1 t3))) (or3 (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind -Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda -(u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift -(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 -(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H7: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H8: (pr0 u1 -x0)).(\lambda (H9: (pr0 t1 x1)).(let H10 \def (eq_ind T t2 (\lambda (t3: -T).(subst0 i u t3 t)) H3 (THead (Flat Appl) x0 x1) H7) in (or3_ind (ex2 T -(\lambda (u2: T).(eq T t (THead (Flat Appl) u2 x1))) (\lambda (u2: T).(subst0 -i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T t (THead (Flat Appl) x0 t3))) -(\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 t3))) (ex3_2 T T (\lambda -(u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3: -T).(subst0 (s (Flat Appl) i) u x1 t3)))) (or3 (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind -Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda -(u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift -(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 -(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (H11: (ex2 T (\lambda (u2: -T).(eq T t (THead (Flat Appl) u2 x1))) (\lambda (u2: T).(subst0 i u x0 -u2)))).(ex2_ind T (\lambda (u2: T).(eq T t (THead (Flat Appl) u2 x1))) -(\lambda (u2: T).(subst0 i u x0 u2)) (or3 (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind -Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda -(u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift -(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 -(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x2: T).(\lambda (H12: (eq T t -(THead (Flat Appl) x2 x1))).(\lambda (H13: (subst0 i u x0 x2)).(eq_ind_r T -(THead (Flat Appl) x2 x1) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: -T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 -t1 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind -Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda -(u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift -(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 -(CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro0 (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x1) (THead (Flat Appl) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t3: T).(eq T (THead (Flat Appl) x2 x1) (THead (Bind Abbr) u2 t3)))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: -T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) -(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T -(THead (Flat Appl) x2 x1) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O -u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2)))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T -(THead (Flat Appl) x2 x1) (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -t1 t3))) x2 x1 (refl_equal T (THead (Flat Appl) x2 x1)) (pr2_delta c0 d u i -H1 u1 x0 H8 x2 H13) (pr2_free c0 t1 x1 H9))) t H12)))) H11)) (\lambda (H11: -(ex2 T (\lambda (t3: T).(eq T t (THead (Flat Appl) x0 t3))) (\lambda (t3: -T).(subst0 (s (Flat Appl) i) u x1 t3)))).(ex2_ind T (\lambda (t3: T).(eq T t -(THead (Flat Appl) x0 t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 -t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat -Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda -(_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall -(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) -y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 -c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x2: T).(\lambda -(H12: (eq T t (THead (Flat Appl) x0 x2))).(\lambda (H13: (subst0 (s (Flat -Appl) i) u x1 x2)).(eq_ind_r T (THead (Flat Appl) x0 x2) (\lambda (t3: -T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat -Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda -(_: T).(\lambda (t4: T).(pr2 c0 t1 t4)))) (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall -(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) -(or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat -Appl) x0 x2) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 -T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq -T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x0 x2) (THead (Bind Abbr) -u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: -T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda -(u2: T).(\lambda (y2: T).(eq T (THead (Flat Appl) x0 x2) (THead (Bind b) y2 -(THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 -u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) (ex3_2_intro T T (\lambda -(u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x0 x2) (THead (Flat Appl) -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr2 c0 t1 t3))) x0 x2 (refl_equal T (THead (Flat Appl) -x0 x2)) (pr2_free c0 u1 x0 H8) (pr2_delta c0 d u i H1 t1 x1 H9 x2 H13))) t -H12)))) H11)) (\lambda (H11: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq -T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u -x0 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 -t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat -Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) -(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 t3))) (or3 -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall -(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) -y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 -c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x2: T).(\lambda -(x3: T).(\lambda (H12: (eq T t (THead (Flat Appl) x2 x3))).(\lambda (H13: -(subst0 i u x0 x2)).(\lambda (H14: (subst0 (s (Flat Appl) i) u x1 -x3)).(eq_ind_r T (THead (Flat Appl) x2 x3) (\lambda (t3: T).(or3 (ex3_2 T T -(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t4: T).(pr2 c0 t1 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 -(THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 -(CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead -(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 -u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro0 (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x3) (THead -(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) -(\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead -(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x3) (THead (Bind Abbr) u2 -t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: -T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda -(u2: T).(\lambda (y2: T).(eq T (THead (Flat Appl) x2 x3) (THead (Bind b) y2 -(THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 -u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) (ex3_2_intro T T (\lambda -(u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x3) (THead (Flat Appl) -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr2 c0 t1 t3))) x2 x3 (refl_equal T (THead (Flat Appl) -x2 x3)) (pr2_delta c0 d u i H1 u1 x0 H8 x2 H13) (pr2_delta c0 d u i H1 t1 x1 -H9 x3 H14))) t H12)))))) H11)) (subst0_gen_head (Flat Appl) u x0 x1 t i -H10)))))))) H6)) (\lambda (H6: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 -(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (t3: T).(pr0 z1 t3))))))).(ex4_4_ind T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))) (or3 (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t -(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 -(CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead -(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 -u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x0: T).(\lambda -(x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq T t1 (THead (Bind -Abst) x0 x1))).(\lambda (H8: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda -(H9: (pr0 u1 x2)).(\lambda (H10: (pr0 x1 x3)).(let H11 \def (eq_ind T t2 -(\lambda (t3: T).(subst0 i u t3 t)) H3 (THead (Bind Abbr) x2 x3) H8) in -(eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t3: T).(or3 (ex3_2 T T -(\lambda (u2: T).(\lambda (t4: T).(eq T t (THead (Flat Appl) u2 t4)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t4: T).(pr2 c0 t3 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T t3 (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t -(THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 -(CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t3 (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead -(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 -u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_ind (ex2 T (\lambda -(u2: T).(eq T t (THead (Bind Abbr) u2 x3))) (\lambda (u2: T).(subst0 i u x2 -u2))) (ex2 T (\lambda (t3: T).(eq T t (THead (Bind Abbr) x2 t3))) (\lambda -(t3: T).(subst0 (s (Bind Abbr) i) u x3 t3))) (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i u x2 u2))) (\lambda (_: T).(\lambda (t3: -T).(subst0 (s (Bind Abbr) i) u x3 t3)))) (or3 (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall -(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T -T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: -T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) -O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2))))))))) (\lambda (H12: (ex2 T (\lambda (u2: T).(eq T t (THead -(Bind Abbr) u2 x3))) (\lambda (u2: T).(subst0 i u x2 u2)))).(ex2_ind T -(\lambda (u2: T).(eq T t (THead (Bind Abbr) u2 x3))) (\lambda (u2: T).(subst0 -i u x2 u2)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead -(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) -(\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind Abst) x0 x1) t3)))) -(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda -(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind -Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead -(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) -y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 -c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x4: T).(\lambda -(H13: (eq T t (THead (Bind Abbr) x4 x3))).(\lambda (H14: (subst0 i u x2 -x4)).(eq_ind_r T (THead (Bind Abbr) x4 x3) (\lambda (t3: T).(or3 (ex3_2 T T -(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t4: T).(pr2 c0 (THead (Bind Abst) x0 x1) t4)))) (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) -x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: -T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) -(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: -T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S -O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: -T).(eq T (THead (Bind Abbr) x4 x3) (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x3) (THead (Bind Abbr) u2 -t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: -T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead -(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind -Abbr) x4 x3) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) -z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead -(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x3) (THead (Bind Abbr) u2 -t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: -T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) z1 t3))))))) x0 x1 x4 x3 (refl_equal T (THead (Bind Abst) x0 x1)) -(refl_equal T (THead (Bind Abbr) x4 x3)) (pr2_delta c0 d u i H1 u1 x2 H9 x4 -H14) (\lambda (b: B).(\lambda (u0: T).(pr2_free (CHead c0 (Bind b) u0) x1 x3 -H10))))) t H13)))) H12)) (\lambda (H12: (ex2 T (\lambda (t3: T).(eq T t -(THead (Bind Abbr) x2 t3))) (\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x3 -t3)))).(ex2_ind T (\lambda (t3: T).(eq T t (THead (Bind Abbr) x2 t3))) -(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x3 t3)) (or3 (ex3_2 T T (\lambda -(u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall -(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T -T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: -T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) -O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2))))))))) (\lambda (x4: T).(\lambda (H13: (eq T t (THead (Bind Abbr) -x2 x4))).(\lambda (H14: (subst0 (s (Bind Abbr) i) u x3 x4)).(eq_ind_r T -(THead (Bind Abbr) x2 x4) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: -T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 -(THead (Bind Abst) x0 x1) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall -(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T -T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: -T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S -O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: -T).(eq T (THead (Bind Abbr) x2 x4) (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x4) (THead (Bind Abbr) u2 -t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: -T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead -(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind -Abbr) x2 x4) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) -z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead -(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x4) (THead (Bind Abbr) u2 -t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: -T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) z1 t3))))))) x0 x1 x2 x4 (refl_equal T (THead (Bind Abst) x0 x1)) -(refl_equal T (THead (Bind Abbr) x2 x4)) (pr2_free c0 u1 x2 H9) (\lambda (b: -B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) -(getl_clear_bind b (CHead c0 (Bind b) u0) c0 u0 (clear_bind b c0 u0) (CHead d -(Bind Abbr) u) i H1) x1 x3 H10 x4 H14))))) t H13)))) H12)) (\lambda (H12: -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x2 u2))) (\lambda (_: -T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x3 t3))))).(ex3_2_ind T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 i u x2 u2))) (\lambda (_: -T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x3 t3))) (or3 (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t3: T).(pr2 c0 (THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) -x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: -T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) -(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: -T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) -O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2))))))))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H13: (eq T t -(THead (Bind Abbr) x4 x5))).(\lambda (H14: (subst0 i u x2 x4)).(\lambda (H15: -(subst0 (s (Bind Abbr) i) u x3 x5)).(eq_ind_r T (THead (Bind Abbr) x4 x5) -(\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 -(THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind Abst) x0 x1) -t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1 -z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: -T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall -(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) -z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: -T).(eq T (THead (Bind Abbr) x4 x5) (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2 -t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: -T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead -(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind -Abbr) x4 x5) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) -z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead -(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2 -t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: -T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) z1 t3))))))) x0 x1 x4 x5 (refl_equal T (THead (Bind Abst) x0 x1)) -(refl_equal T (THead (Bind Abbr) x4 x5)) (pr2_delta c0 d u i H1 u1 x2 H9 x4 -H14) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u -(S i) (getl_clear_bind b (CHead c0 (Bind b) u0) c0 u0 (clear_bind b c0 u0) -(CHead d (Bind Abbr) u) i H1) x1 x3 H10 x5 H15))))) t H13)))))) H12)) -(subst0_gen_head (Bind Abbr) u x2 x3 t i H11)) t1 H7)))))))))) H6)) (\lambda -(H6: (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) -(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda -(t3: T).(eq T t2 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) -t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda -(y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 -y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))))).(ex6_6_ind B T T T T -T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq -T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead -(Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda -(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3))))))) (or3 (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind -Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda -(u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift -(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 -(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x0: B).(\lambda (x1: -T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: -T).(\lambda (H7: (not (eq B x0 Abst))).(\lambda (H8: (eq T t1 (THead (Bind -x0) x1 x2))).(\lambda (H9: (eq T t2 (THead (Bind x0) x4 (THead (Flat Appl) -(lift (S O) O x3) x5)))).(\lambda (H10: (pr0 u1 x3)).(\lambda (H11: (pr0 x1 -x4)).(\lambda (H12: (pr0 x2 x5)).(let H13 \def (eq_ind T t2 (\lambda (t3: -T).(subst0 i u t3 t)) H3 (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O -x3) x5)) H9) in (eq_ind_r T (THead (Bind x0) x1 x2) (\lambda (t3: T).(or3 -(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t (THead (Flat Appl) u2 -t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t4: T).(pr2 c0 t3 t4)))) (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t3 (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t4: T).(eq T t (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall -(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t3 -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) -y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 -c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_ind (ex2 T (\lambda -(u2: T).(eq T t (THead (Bind x0) u2 (THead (Flat Appl) (lift (S O) O x3) -x5)))) (\lambda (u2: T).(subst0 i u x4 u2))) (ex2 T (\lambda (t3: T).(eq T t -(THead (Bind x0) x4 t3))) (\lambda (t3: T).(subst0 (s (Bind x0) i) u (THead -(Flat Appl) (lift (S O) O x3) x5) t3))) (ex3_2 T T (\lambda (u2: T).(\lambda -(t3: T).(eq T t (THead (Bind x0) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 i u x4 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind x0) -i) u (THead (Flat Appl) (lift (S O) O x3) x5) t3)))) (or3 (ex3_2 T T (\lambda -(u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall -(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: 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-(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind -Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead -(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 -u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x6: T).(\lambda -(H15: (eq T t (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) -x5)))).(\lambda (H16: (subst0 i u x4 x6)).(eq_ind_r T (THead (Bind x0) x6 -(THead (Flat Appl) (lift (S O) O x3) x5)) (\lambda (t3: T).(or3 (ex3_2 T T -(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 -x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (t4: 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T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) -z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2)))))))))) (or3_ind (ex2 T (\lambda (u2: T).(eq T x6 (THead (Flat -Appl) u2 x5))) (\lambda (u2: T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) -u2))) (ex2 T (\lambda (t3: T).(eq T x6 (THead (Flat Appl) (lift (S O) O x3) -t3))) (\lambda (t3: T).(subst0 (s 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c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T -T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: -T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind b) y2 (THead -(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 -u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (H17: (ex2 T -(\lambda (u2: T).(eq T x6 (THead (Flat Appl) u2 x5))) (\lambda (u2: -T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) u2)))).(ex2_ind T (\lambda -(u2: T).(eq T x6 (THead (Flat Appl) u2 x5))) (\lambda (u2: T).(subst0 (s -(Bind x0) i) u (lift (S O) O x3) u2)) (or3 (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 x6) (THead (Flat Appl) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) -x4 x6) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall -(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind b) y2 (THead (Flat Appl) -(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 -(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x7: T).(\lambda (H18: (eq T -x6 (THead (Flat Appl) x7 x5))).(\lambda (H19: (subst0 (s (Bind x0) i) u (lift -(S O) O x3) x7)).(eq_ind_r T (THead (Flat Appl) x7 x5) 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(Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) x7 x5)) (THead (Bind -Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead -(Flat Appl) x7 x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) -z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2))))))))) (\lambda (x8: T).(\lambda (H20: (eq T x7 (lift (S O) O -x8))).(\lambda (H21: (subst0 (minus (s (Bind x0) i) (S O)) u x3 x8)).(let H22 -\def (eq_ind nat (minus (s (Bind x0) i) (S O)) (\lambda (n: nat).(subst0 n u -x3 x8)) H21 i (s_arith1 x0 i)) in (eq_ind_r T (lift (S O) O x8) (\lambda (t3: -T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) -x4 (THead (Flat Appl) t3 x5)) (THead (Flat Appl) u2 t4)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 -(THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t4: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) t3 x5)) (THead (Bind -Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead -(Flat Appl) t3 x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) -z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: -T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x8) x5)) (THead -(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) -(\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) -(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) -x4 (THead (Flat Appl) (lift (S O) O x8) x5)) (THead (Bind Abbr) u2 t3)))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: -T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) -(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: -T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) -O x8) x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) -z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 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(_: -T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 -(CHead c0 (Bind b) y2) z1 z2))))))) x0 x1 x2 x5 x8 x4 H7 (refl_equal T (THead -(Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x4 (THead (Flat Appl) (lift -(S O) O x8) x5))) (pr2_delta c0 d u i H1 u1 x3 H10 x8 H22) (pr2_free c0 x1 x4 -H11) (pr2_free (CHead c0 (Bind x0) x4) x2 x5 H12))) x7 H20))))) -(subst0_gen_lift_ge u x3 x7 (s (Bind x0) i) (S O) O H19 (le_n_S O i (le_O_n -i)))) x6 H18)))) H17)) (\lambda (H17: (ex2 T (\lambda (t3: T).(eq T x6 (THead -(Flat Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) -(s (Bind x0) i)) u x5 t3)))).(ex2_ind T (\lambda (t3: T).(eq T x6 (THead -(Flat Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) -(s (Bind x0) i)) u x5 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: -T).(eq T (THead (Bind x0) x4 x6) (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t3: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind Abbr) u2 t3)))))) (\lambda -(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall -(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T -T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: -T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind b) y2 (THead -(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 -u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x7: T).(\lambda -(H18: (eq T x6 (THead (Flat Appl) (lift (S O) O x3) x7))).(\lambda (H19: -(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 x7)).(eq_ind_r T (THead (Flat -Appl) (lift (S O) O x3) x7) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: -T).(\lambda (t4: T).(eq T (THead (Bind x0) x4 t3) (THead (Flat Appl) u2 -t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) -x4 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall -(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(eq T (THead (Bind x0) x4 t3) (THead (Bind b) y2 (THead (Flat Appl) -(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 -(CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) -O x3) x7)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 -c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) -t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 -z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: -T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x7)) (THead -(Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 -(CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind -x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T -(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x7)) (THead (Bind b) -y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 -c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x7)) -(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda -(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 -y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))) -x0 x1 x2 x7 x3 x4 H7 (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T -(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x7))) (pr2_free c0 -u1 x3 H10) (pr2_free c0 x1 x4 H11) (pr2_delta (CHead c0 (Bind x0) x4) d u (S -i) (getl_clear_bind x0 (CHead c0 (Bind x0) x4) c0 x4 (clear_bind x0 c0 x4) -(CHead d (Bind Abbr) u) i H1) x2 x5 H12 x7 H19))) x6 H18)))) H17)) (\lambda -(H17: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x6 (THead (Flat -Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0) i) u -(lift (S O) O x3) u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat -Appl) (s (Bind x0) i)) u x5 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda -(t3: T).(eq T x6 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) u2))) (\lambda (_: T).(\lambda -(t3: T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3))) (or3 (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 x6) (THead (Flat -Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda -(_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) -x4 x6) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall -(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind b) y2 (THead (Flat Appl) -(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 -(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x7: T).(\lambda (x8: -T).(\lambda (H18: (eq T x6 (THead (Flat Appl) x7 x8))).(\lambda (H19: (subst0 -(s (Bind x0) i) u (lift (S O) O x3) x7)).(\lambda (H20: (subst0 (s (Flat -Appl) (s (Bind x0) i)) u x5 x8)).(eq_ind_r T (THead (Flat Appl) x7 x8) -(\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T -(THead (Bind x0) x4 t3) (THead (Flat Appl) u2 t4)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 -(THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t4: T).(eq T (THead (Bind x0) x4 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda -(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall -(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T -T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: -T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 t3) (THead (Bind b) y2 (THead -(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 -u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (ex2_ind T (\lambda (t3: -T).(eq T x7 (lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus (s (Bind x0) -i) (S O)) u x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T -(THead (Bind x0) x4 (THead (Flat Appl) x7 x8)) (THead (Flat Appl) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 -x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) x7 x8)) -(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 -(CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind -x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T -(THead (Bind x0) x4 (THead (Flat Appl) x7 x8)) (THead (Bind b) y2 (THead -(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 -u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x9: T).(\lambda -(H21: (eq T x7 (lift (S O) O x9))).(\lambda (H22: (subst0 (minus (s (Bind x0) -i) (S O)) u x3 x9)).(let H23 \def (eq_ind nat (minus (s (Bind x0) i) (S O)) -(\lambda (n: nat).(subst0 n u x3 x9)) H22 i (s_arith1 x0 i)) in (eq_ind_r T -(lift (S O) O x9) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda -(t4: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) t3 x8)) (THead (Flat -Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda -(_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) -x4 (THead (Flat Appl) t3 x8)) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall -(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T -T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: -T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) t3 x8)) -(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda -(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 -y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 -z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T -(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x9) x8)) (THead (Flat -Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda -(_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) -x4 (THead (Flat Appl) (lift (S O) O x9) x8)) (THead (Bind Abbr) u2 t3)))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: -T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) -(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: -T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) -O x9) x8)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) -z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B -b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead -(Flat Appl) (lift (S O) O x9) x8)) (THead (Bind b) y2 (THead (Flat Appl) -(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 -(CHead c0 (Bind b) y2) z1 z2))))))) x0 x1 x2 x8 x9 x4 H7 (refl_equal T (THead -(Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x4 (THead (Flat Appl) (lift -(S 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T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall -(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) -z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2))))))))) (\lambda (x6: T).(\lambda (x7: T).(\lambda (H15: (eq T t -(THead (Bind x0) x6 x7))).(\lambda (H16: (subst0 i u x4 x6)).(\lambda (H17: -(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) O x3) x5) -x7)).(eq_ind_r T (THead (Bind x0) x6 x7) (\lambda (t3: T).(or3 (ex3_2 T T -(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 -x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall -(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T -T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: -T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S -O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2)))))))))) (or3_ind (ex2 T (\lambda (u2: T).(eq T x7 (THead (Flat -Appl) u2 x5))) (\lambda (u2: T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) -u2))) (ex2 T (\lambda (t3: T).(eq T x7 (THead (Flat Appl) (lift (S O) O x3) -t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3))) -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x7 (THead (Flat Appl) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0) i) u (lift (S O) -O x3) u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind -x0) i)) u x5 t3)))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T -(THead (Bind x0) x6 x7) (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind Abbr) u2 t3)))))) (\lambda -(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall -(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T -T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: 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O) O x3) x8)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8)) -(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 -(CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind -x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T -(THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8)) (THead (Bind b) -y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 -c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8)) -(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda -(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 -y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))) -x0 x1 x2 x8 x3 x6 H7 (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T -(THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8))) (pr2_free c0 -u1 x3 H10) (pr2_delta c0 d u i H1 x1 x4 H11 x6 H16) (pr2_delta (CHead c0 -(Bind x0) x6) d u (S i) (getl_clear_bind x0 (CHead c0 (Bind x0) x6) c0 x6 -(clear_bind x0 c0 x6) (CHead d (Bind Abbr) u) i H1) x2 x5 H12 x8 H20))) x7 -H19)))) H18)) (\lambda (H18: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq -T x7 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s -(Bind x0) i) u (lift (S O) O x3) u2))) (\lambda (_: T).(\lambda (t3: -T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3))))).(ex3_2_ind T T -(\lambda (u2: T).(\lambda (t3: T).(eq T x7 (THead (Flat Appl) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) -u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind x0) -i)) u x5 t3))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead -(Bind x0) x6 x7) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) -x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 -z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: -T).(eq T (THead (Bind x0) x6 x7) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall -(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T -T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: -T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind b) y2 (THead -(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 -u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x8: T).(\lambda -(x9: T).(\lambda (H19: (eq T x7 (THead (Flat Appl) x8 x9))).(\lambda (H20: -(subst0 (s (Bind x0) i) u (lift (S O) O x3) x8)).(\lambda (H21: (subst0 (s -(Flat Appl) (s (Bind x0) i)) u x5 x9)).(eq_ind_r T (THead (Flat Appl) x8 x9) -(\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T -(THead (Bind x0) x6 t3) (THead (Flat Appl) u2 t4)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 -(THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t4: T).(eq T (THead (Bind x0) x6 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda -(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall -(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T -T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: -T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 t3) (THead (Bind b) y2 (THead -(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 -u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (ex2_ind T (\lambda (t3: -T).(eq T x8 (lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus (s (Bind x0) -i) (S O)) u x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T -(THead (Bind x0) x6 (THead (Flat Appl) x8 x9)) (THead (Flat Appl) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 -x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x8 x9)) -(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 -(CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind -x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T -(THead (Bind x0) x6 (THead (Flat Appl) x8 x9)) (THead (Bind b) y2 (THead -(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 -u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x10: T).(\lambda -(H22: (eq T x8 (lift (S O) O x10))).(\lambda (H23: (subst0 (minus (s (Bind -x0) i) (S O)) u x3 x10)).(let H24 \def (eq_ind nat (minus (s (Bind x0) i) (S -O)) (\lambda (n: nat).(subst0 n u x3 x10)) H23 i (s_arith1 x0 i)) in -(eq_ind_r T (lift (S O) O x10) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: -T).(\lambda (t4: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t3 x9)) -(THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) -(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) -x6 (THead (Flat Appl) t3 x9)) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall -(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T -T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: -T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t3 x9)) -(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda -(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 -y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 -z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T -(THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x10) x9)) (THead (Flat -Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda -(_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) -x6 (THead (Flat Appl) (lift (S O) O x10) x9)) (THead (Bind Abbr) u2 t3)))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: -T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) -(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: -T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) -O x10) x9)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) -z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B -b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead -(Flat Appl) (lift (S O) O x10) x9)) (THead (Bind b) y2 (THead (Flat Appl) -(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 -(CHead c0 (Bind b) y2) z1 z2))))))) x0 x1 x2 x9 x10 x6 H7 (refl_equal T -(THead (Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x6 (THead (Flat Appl) -(lift (S O) O x10) x9))) (pr2_delta c0 d u i H1 u1 x3 H10 x10 H24) (pr2_delta -c0 d u i H1 x1 x4 H11 x6 H16) (pr2_delta (CHead c0 (Bind x0) x6) d u (S i) -(getl_clear_bind x0 (CHead c0 (Bind x0) x6) c0 x6 (clear_bind x0 c0 x6) -(CHead d (Bind Abbr) u) i H1) x2 x5 H12 x9 H21))) x8 H22))))) -(subst0_gen_lift_ge u x3 x8 (s (Bind x0) i) (S O) O H20 (le_n_S O i (le_O_n -i)))) x7 H19)))))) H18)) (subst0_gen_head (Flat Appl) u (lift (S O) O x3) x5 -x7 (s (Bind x0) i) H17)) t H15)))))) H14)) (subst0_gen_head (Bind x0) u x4 -(THead (Flat Appl) (lift (S O) O x3) x5) t i H13)) t1 H8)))))))))))))) H6)) -(pr0_gen_appl u1 t1 t2 H5)))))))))))))) c y x H0))) H))))). -(* COMMENTS -Initial nodes: 38859 -END *) - -theorem pr2_gen_abbr: - \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c -(THead (Bind Abbr) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda -(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(or3 (\forall (b: -B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2))) (ex2 T (\lambda (u: -T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t2))) (ex3_2 T -T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) -(\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: -T).(pr2 (CHead c (Bind Abbr) u1) z t2)))))))) (\forall (b: B).(\forall (u: -T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda -(H: (pr2 c (THead (Bind Abbr) u1 t1) x)).(insert_eq T (THead (Bind Abbr) u1 -t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(or3 -(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2))) (ex2 T -(\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 -t2))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) -u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: -T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t2)))))))) (\forall (b: -B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))) -(\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0: C).(\lambda -(t: T).(\lambda (t0: T).((eq T t (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T -T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t2: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 -t2))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind -Abbr) u) t1 t2))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 -(Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) -(\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t2)))))))) -(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O -t0))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H1: -(pr0 t0 t2)).(\lambda (H2: (eq T t0 (THead (Bind Abbr) u1 t1))).(let H3 \def -(eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Bind Abbr) u1 t1) H2) in -(or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 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c0 (Bind b) u0) t1 t3))) (ex2 T -(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) -t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind -Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: -T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: -B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))) -(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 -(Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: -T).(pr2 (CHead c0 (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y0: -T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: -T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead -c0 (Bind Abbr) u1) z t3))))))) x3 x1 H13 (pr2_delta c0 d u i H1 u1 x0 H8 x3 -H14) (or3_intro1 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) -t1 x1))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 -(Bind Abbr) u0) t1 x1))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 -(CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 -z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z x1)))) -(ex_intro2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 -(Bind Abbr) u0) t1 x1)) x0 H8 (pr2_delta (CHead c0 (Bind Abbr) x0) c0 x0 O -(getl_refl Abbr c0 x0) t1 x2 H9 x1 H10)))))))) (pr0_subst0_back x0 x2 x1 O -H10 u1 H8))))) H12)) (\lambda (H12: (ex2 T (\lambda (t3: T).(eq T t (THead -(Bind Abbr) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x1 -t3)))).(ex2_ind T (\lambda (t3: T).(eq T t (THead (Bind Abbr) x0 t3))) -(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x1 t3)) (or (ex3_2 T T (\lambda -(u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 -(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T -(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) -t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind -Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: -T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: -B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))) -(\lambda (x3: T).(\lambda (H13: (eq T t (THead (Bind Abbr) x0 x3))).(\lambda -(H14: (subst0 (s (Bind Abbr) i) u x1 x3)).(ex2_ind T (\lambda (t3: T).(subst0 -O u1 x2 t3)) (\lambda (t3: T).(pr0 t3 x1)) (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 -(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T -(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) -t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind -Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: -T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: -B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))) -(\lambda (x4: T).(\lambda (H15: (subst0 O u1 x2 x4)).(\lambda (H16: (pr0 x4 -x1)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead -(Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) -(\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 -(CHead c0 (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda -(u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y0: -T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: -T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead -c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead -c0 (Bind b) u0) t1 (lift (S O) O t)))) (ex3_2_intro T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 -(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T -(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) -t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind -Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: -T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3))))))) x0 x3 H13 -(pr2_free c0 u1 x0 H8) (or3_intro2 (\forall (b: B).(\forall (u0: T).(pr2 -(CHead c0 (Bind b) u0) t1 x3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda -(u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 x3))) (ex3_2 T T (\lambda (y0: -T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: -T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead -c0 (Bind Abbr) u1) z x3)))) (ex3_2_intro T T (\lambda (y0: T).(\lambda (_: -T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: -T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) -u1) z x3))) x4 x1 (pr2_delta (CHead c0 (Bind Abbr) u1) c0 u1 O (getl_refl -Abbr c0 u1) t1 x2 H9 x4 H15) H16 (pr2_delta (CHead c0 (Bind Abbr) u1) d u (S -i) (getl_head (Bind Abbr) i c0 (CHead d (Bind Abbr) u) H1 u1) x1 x1 (pr0_refl -x1) x3 H14)))))))) (pr0_subst0_back x0 x2 x1 O H10 u1 H8))))) H12)) (\lambda -(H12: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: -T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x1 t3))))).(ex3_2_ind T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: -T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x1 t3))) (or (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 -t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 -(Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 -(CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 -z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z -t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 -(lift (S O) O t))))) (\lambda (x3: T).(\lambda (x4: T).(\lambda (H13: (eq T t -(THead (Bind Abbr) x3 x4))).(\lambda (H14: (subst0 i u x0 x3)).(\lambda (H15: -(subst0 (s (Bind Abbr) i) u x1 x4)).(ex2_ind T (\lambda (t3: T).(subst0 O u1 -x2 t3)) (\lambda (t3: T).(pr0 t3 x1)) (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 -(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T -(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) -t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind -Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: -T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: -B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))) -(\lambda (x5: T).(\lambda (H16: (subst0 O u1 x2 x5)).(\lambda (H17: (pr0 x5 -x1)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead -(Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) -(\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 -(CHead c0 (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda -(u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y0: -T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: -T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead -c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead -c0 (Bind b) u0) t1 (lift (S O) O t)))) (ex3_2_intro T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 -(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T -(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) -t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind -Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: -T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3))))))) x3 x4 H13 -(pr2_delta c0 d u i H1 u1 x0 H8 x3 H14) (or3_intro2 (\forall (b: B).(\forall -(u0: T).(pr2 (CHead c0 (Bind b) u0) t1 x4))) (ex2 T (\lambda (u0: T).(pr0 u1 -u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 x4))) (ex3_2 T T -(\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) -(\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: -T).(pr2 (CHead c0 (Bind Abbr) u1) z x4)))) (ex3_2_intro T T (\lambda (y0: -T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: -T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead -c0 (Bind Abbr) u1) z x4))) x5 x1 (pr2_delta (CHead c0 (Bind Abbr) u1) c0 u1 O -(getl_refl Abbr c0 u1) t1 x2 H9 x5 H16) H17 (pr2_delta (CHead c0 (Bind Abbr) -u1) d u (S i) (getl_head (Bind Abbr) i c0 (CHead d (Bind Abbr) u) H1 u1) x1 -x1 (pr0_refl x1) x4 H15)))))))) (pr0_subst0_back x0 x2 x1 O H10 u1 H8))))))) -H12)) (subst0_gen_head (Bind Abbr) u x0 x1 t i H11)))))) H_x0)) H_x)))))) -H6)) (\lambda (H6: (pr0 t1 (lift (S O) O t2))).(or_intror (ex3_2 T T (\lambda -(u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 -(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T -(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) -t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind -Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: -T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: -B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))) -(\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) -(getl_head (Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 (lift (S O) O t2) -H6 (lift (S O) O t) (subst0_lift_ge_S t2 t u i H3 O (le_O_n i))))))) -(pr0_gen_abbr u1 t1 t2 H5)))))))))))))) c y x H0))) H))))). -(* COMMENTS -Initial nodes: 11671 -END *) - -theorem pr2_gen_void: - \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c -(THead (Bind Void) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda -(t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall -(u: T).(pr2 (CHead c (Bind b) u) t1 t2)))))) (\forall (b: B).(\forall (u: -T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda -(H: (pr2 c (THead (Bind Void) u1 t1) x)).(insert_eq T (THead (Bind Void) u1 -t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall -(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2)))))) (\forall (b: -B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))) -(\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0: C).(\lambda -(t: T).(\lambda (t0: T).((eq T t (THead (Bind Void) u1 t1)) \to (or (ex3_2 T -T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Void) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 -t2)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 (lift -(S O) O t0))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: -T).(\lambda (H1: (pr0 t0 t2)).(\lambda (H2: (eq T t0 (THead (Bind Void) u1 -t1))).(let H3 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Bind -Void) u1 t1) H2) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq -T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 -u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O -t2)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda -(_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind -b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) -t1 (lift (S O) O t2))))) (\lambda (H4: (ex3_2 T T (\lambda (u2: T).(\lambda -(t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind -T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 -(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 -(CHead c0 (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead -c0 (Bind b) u) t1 (lift (S O) O t2))))) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H5: (eq T t2 (THead (Bind Void) x0 x1))).(\lambda (H6: (pr0 u1 -x0)).(\lambda (H7: (pr0 t1 x1)).(eq_ind_r T (THead (Bind Void) x0 x1) -(\lambda (t: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t -(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 -(CHead c0 (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead -c0 (Bind b) u) t1 (lift (S O) O t)))))) (or_introl (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T (THead (Bind Void) x0 x1) (THead (Bind Void) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) -u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 -(lift (S O) O (THead (Bind Void) x0 x1))))) (ex3_2_intro T T (\lambda (u2: -T).(\lambda (t3: T).(eq T (THead (Bind Void) x0 x1) (THead (Bind Void) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) -u) t1 t3))))) x0 x1 (refl_equal T (THead (Bind Void) x0 x1)) (pr2_free c0 u1 -x0 H6) (\lambda (b: B).(\lambda (u: T).(pr2_free (CHead c0 (Bind b) u) t1 x1 -H7))))) t2 H5)))))) H4)) (\lambda (H4: (pr0 t1 (lift (S O) O t2))).(or_intror -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) -u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 -(lift (S O) O t2)))) (\lambda (b: B).(\lambda (u: T).(pr2_free (CHead c0 -(Bind b) u) t1 (lift (S O) O t2) H4))))) (pr0_gen_void u1 t1 t2 H3)))))))) -(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t0: T).(\lambda (t2: -T).(\lambda (H2: (pr0 t0 t2)).(\lambda (t: T).(\lambda (H3: (subst0 i u t2 -t)).(\lambda (H4: (eq T t0 (THead (Bind Void) u1 t1))).(let H5 \def (eq_ind T -t0 (\lambda (t3: T).(pr0 t3 t2)) H2 (THead (Bind Void) u1 t1) H4) in (or_ind -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2)) (or (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 -t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 -(lift (S O) O t))))) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: -T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind -T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t -(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 -(CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 -(CHead c0 (Bind b) u0) t1 (lift (S O) O t))))) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H7: (eq T t2 (THead (Bind Void) x0 x1))).(\lambda (H8: (pr0 u1 -x0)).(\lambda (H9: (pr0 t1 x1)).(let H10 \def (eq_ind T t2 (\lambda (t3: -T).(subst0 i u t3 t)) H3 (THead (Bind Void) x0 x1) H7) in (or3_ind (ex2 T -(\lambda (u2: T).(eq T t (THead (Bind Void) u2 x1))) (\lambda (u2: T).(subst0 -i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T t (THead (Bind Void) x0 t3))) -(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3))) (ex3_2 T T (\lambda -(u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3: -T).(subst0 (s (Bind Void) i) u x1 t3)))) (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall -(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: -B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))) -(\lambda (H11: (ex2 T (\lambda (u2: T).(eq T t (THead (Bind Void) u2 x1))) -(\lambda (u2: T).(subst0 i u x0 u2)))).(ex2_ind T (\lambda (u2: T).(eq T t -(THead (Bind Void) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2)) (or (ex3_2 T -T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 -t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 -(lift (S O) O t))))) (\lambda (x2: T).(\lambda (H12: (eq T t (THead (Bind -Void) x2 x1))).(\lambda (H13: (subst0 i u x0 x2)).(or_introl (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 -t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 -(lift (S O) O t)))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T -t (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 -(CHead c0 (Bind b) u0) t1 t3))))) x2 x1 H12 (pr2_delta c0 d u i H1 u1 x0 H8 -x2 H13) (\lambda (b: B).(\lambda (u0: T).(pr2_free (CHead c0 (Bind b) u0) t1 -x1 H9)))))))) H11)) (\lambda (H11: (ex2 T (\lambda (t3: T).(eq T t (THead -(Bind Void) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 -t3)))).(ex2_ind T (\lambda (t3: T).(eq T t (THead (Bind Void) x0 t3))) -(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3)) (or (ex3_2 T T (\lambda -(u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall -(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: -B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))) -(\lambda (x2: T).(\lambda (H12: (eq T t (THead (Bind Void) x0 x2))).(\lambda -(H13: (subst0 (s (Bind Void) i) u x1 x2)).(or_introl (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall -(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: -B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))) -(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) t1 t3))))) x0 x2 H12 (pr2_free c0 u1 x0 H8) (\lambda (b: B).(\lambda (u0: -T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) (getl_head (Bind b) i c0 -(CHead d (Bind Abbr) u) H1 u0) t1 x1 H9 x2 H13)))))))) H11)) (\lambda (H11: -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: -T).(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3))))).(ex3_2_ind T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: -T).(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3))) (or (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 -t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 -(lift (S O) O t))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (eq T t -(THead (Bind Void) x2 x3))).(\lambda (H13: (subst0 i u x0 x2)).(\lambda (H14: -(subst0 (s (Bind Void) i) u x1 x3)).(or_introl (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall -(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: -B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))) -(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) t1 t3))))) x2 x3 H12 (pr2_delta c0 d u i H1 u1 x0 H8 x2 H13) (\lambda (b: -B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) (getl_head -(Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 x1 H9 x3 H14)))))))))) H11)) -(subst0_gen_head (Bind Void) u x0 x1 t i H10)))))))) H6)) (\lambda (H6: (pr0 -t1 (lift (S O) O t2))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: -T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 -c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: -T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0: -T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))) (\lambda (b: -B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) (getl_head -(Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 (lift (S O) O t2) H6 (lift (S -O) O t) (subst0_lift_ge_S t2 t u i H3 O (le_O_n i))))))) (pr0_gen_void u1 t1 -t2 H5)))))))))))))) c y x H0))) H))))). -(* COMMENTS -Initial nodes: 3467 -END *) - -theorem pr2_gen_lift: - \forall (c: C).(\forall (t1: T).(\forall (x: T).(\forall (h: nat).(\forall -(d: nat).((pr2 c (lift h d t1) x) \to (\forall (e: C).((drop h d c e) \to -(ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr2 e t1 -t2)))))))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (x: T).(\lambda (h: nat).(\lambda -(d: nat).(\lambda (H: (pr2 c (lift h d t1) x)).(insert_eq T (lift h d t1) -(\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(\forall (e: C).((drop h d c e) -\to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr2 e -t1 t2)))))) (\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0: -C).(\lambda (t: T).(\lambda (t0: T).((eq T t (lift h d t1)) \to (\forall (e: -C).((drop h d c0 e) \to (ex2 T (\lambda (t2: T).(eq T t0 (lift h d t2))) -(\lambda (t2: T).(pr2 e t1 t2))))))))) (\lambda (c0: C).(\lambda (t0: -T).(\lambda (t2: T).(\lambda (H1: (pr0 t0 t2)).(\lambda (H2: (eq T t0 (lift h -d t1))).(\lambda (e: C).(\lambda (_: (drop h d c0 e)).(let H4 \def (eq_ind T -t0 (\lambda (t: T).(pr0 t t2)) H1 (lift h d t1) H2) in (ex2_ind T (\lambda -(t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(pr0 t1 t3)) (ex2 T -(\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))) -(\lambda (x0: T).(\lambda (H5: (eq T t2 (lift h d x0))).(\lambda (H6: (pr0 t1 -x0)).(eq_ind_r T (lift h d x0) (\lambda (t: T).(ex2 T (\lambda (t3: T).(eq T -t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)))) (ex_intro2 T (\lambda -(t3: T).(eq T (lift h d x0) (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)) -x0 (refl_equal T (lift h d x0)) (pr2_free e t1 x0 H6)) t2 H5)))) -(pr0_gen_lift t1 t2 h d H4)))))))))) (\lambda (c0: C).(\lambda (d0: -C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d0 (Bind -Abbr) u))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H2: (pr0 t0 -t2)).(\lambda (t: T).(\lambda (H3: (subst0 i u t2 t)).(\lambda (H4: (eq T t0 -(lift h d t1))).(\lambda (e: C).(\lambda (H5: (drop h d c0 e)).(let H6 \def -(eq_ind T t0 (\lambda (t3: T).(pr0 t3 t2)) H2 (lift h d t1) H4) in (ex2_ind T -(\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(pr0 t1 t3)) (ex2 -T (\lambda (t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))) -(\lambda (x0: T).(\lambda (H7: (eq T t2 (lift h d x0))).(\lambda (H8: (pr0 t1 -x0)).(let H9 \def (eq_ind T t2 (\lambda (t3: T).(subst0 i u t3 t)) H3 (lift h -d x0) H7) in (lt_le_e i d (ex2 T (\lambda (t3: T).(eq T t (lift h d t3))) -(\lambda (t3: T).(pr2 e t1 t3))) (\lambda (H10: (lt i d)).(let H11 \def -(eq_ind nat d (\lambda (n: nat).(subst0 i u (lift h n x0) t)) H9 (S (plus i -(minus d (S i)))) (lt_plus_minus i d H10)) in (let H12 \def (eq_ind nat d -(\lambda (n: nat).(drop h n c0 e)) H5 (S (plus i (minus d (S i)))) -(lt_plus_minus i d H10)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: -C).(eq T u (lift h (minus d (S i)) v)))) (\lambda (v: T).(\lambda (e0: -C).(getl i e (CHead e0 (Bind Abbr) v)))) (\lambda (_: T).(\lambda (e0: -C).(drop h (minus d (S i)) d0 e0))) (ex2 T (\lambda (t3: T).(eq T t (lift h d -t3))) (\lambda (t3: T).(pr2 e t1 t3))) (\lambda (x1: T).(\lambda (x2: -C).(\lambda (H13: (eq T u (lift h (minus d (S i)) x1))).(\lambda (H14: (getl -i e (CHead x2 (Bind Abbr) x1))).(\lambda (_: (drop h (minus d (S i)) d0 -x2)).(let H16 \def (eq_ind T u (\lambda (t3: T).(subst0 i t3 (lift h (S (plus -i (minus d (S i)))) x0) t)) H11 (lift h (minus d (S i)) x1) H13) in (ex2_ind -T (\lambda (t3: T).(eq T t (lift h (S (plus i (minus d (S i)))) t3))) -(\lambda (t3: T).(subst0 i x1 x0 t3)) (ex2 T (\lambda (t3: T).(eq T t (lift h -d t3))) (\lambda (t3: T).(pr2 e t1 t3))) (\lambda (x3: T).(\lambda (H17: (eq -T t (lift h (S (plus i (minus d (S i)))) x3))).(\lambda (H18: (subst0 i x1 x0 -x3)).(let H19 \def (eq_ind_r nat (S (plus i (minus d (S i)))) (\lambda (n: -nat).(eq T t (lift h n x3))) H17 d (lt_plus_minus i d H10)) in (ex_intro2 T -(\lambda (t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)) x3 -H19 (pr2_delta e x2 x1 i H14 t1 x0 H8 x3 H18)))))) (subst0_gen_lift_lt x1 x0 -t i h (minus d (S i)) H16)))))))) (getl_drop_conf_lt Abbr c0 d0 u i H1 e h -(minus d (S i)) H12))))) (\lambda (H10: (le d i)).(lt_le_e i (plus d h) (ex2 -T (\lambda (t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))) -(\lambda (H11: (lt i (plus d h))).(subst0_gen_lift_false x0 u t h d i H10 H11 -H9 (ex2 T (\lambda (t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 -t3))))) (\lambda (H11: (le (plus d h) i)).(ex2_ind T (\lambda (t3: T).(eq T t -(lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u x0 t3)) (ex2 T -(\lambda (t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))) -(\lambda (x1: T).(\lambda (H12: (eq T t (lift h d x1))).(\lambda (H13: -(subst0 (minus i h) u x0 x1)).(ex_intro2 T (\lambda (t3: T).(eq T t (lift h d -t3))) (\lambda (t3: T).(pr2 e t1 t3)) x1 H12 (pr2_delta e d0 u (minus i h) -(getl_drop_conf_ge i (CHead d0 (Bind Abbr) u) c0 H1 e h d H5 H11) t1 x0 H8 x1 -H13))))) (subst0_gen_lift_ge u x0 t i h d H9 H11)))))))))) (pr0_gen_lift t1 -t2 h d H6)))))))))))))))) c y x H0))) H)))))). -(* COMMENTS -Initial nodes: 1579 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr2/pr2.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr2/pr2.ma deleted file mode 100644 index f8df4e9e2..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr2/pr2.ma +++ /dev/null @@ -1,258 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr2/defs.ma". - -include "Basic-1/pr0/pr0.ma". - -include "Basic-1/getl/props.ma". - -theorem pr2_confluence__pr2_free_free: - \forall (c: C).(\forall (t0: T).(\forall (t1: T).(\forall (t2: T).((pr0 t0 -t1) \to ((pr0 t0 t2) \to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: -T).(pr2 c t2 t)))))))) -\def - \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H: (pr0 t0 t1)).(\lambda (H0: (pr0 t0 t2)).(ex2_ind T (\lambda (t: T).(pr0 -t2 t)) (\lambda (t: T).(pr0 t1 t)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) -(\lambda (t: T).(pr2 c t2 t))) (\lambda (x: T).(\lambda (H1: (pr0 t2 -x)).(\lambda (H2: (pr0 t1 x)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) -(\lambda (t: T).(pr2 c t2 t)) x (pr2_free c t1 x H2) (pr2_free c t2 x H1))))) -(pr0_confluence t0 t2 H0 t1 H))))))). -(* COMMENTS -Initial nodes: 135 -END *) - -theorem pr2_confluence__pr2_free_delta: - \forall (c: C).(\forall (d: C).(\forall (t0: T).(\forall (t1: T).(\forall -(t2: T).(\forall (t4: T).(\forall (u: T).(\forall (i: nat).((pr0 t0 t1) \to -((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t4) \to ((subst0 i u t4 t2) -\to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 -t)))))))))))))) -\def - \lambda (c: C).(\lambda (d: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda -(t2: T).(\lambda (t4: T).(\lambda (u: T).(\lambda (i: nat).(\lambda (H: (pr0 -t0 t1)).(\lambda (H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H1: (pr0 -t0 t4)).(\lambda (H2: (subst0 i u t4 t2)).(ex2_ind T (\lambda (t: T).(pr0 t4 -t)) (\lambda (t: T).(pr0 t1 t)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda -(t: T).(pr2 c t2 t))) (\lambda (x: T).(\lambda (H3: (pr0 t4 x)).(\lambda (H4: -(pr0 t1 x)).(or_ind (pr0 t2 x) (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda -(w2: T).(subst0 i u x w2))) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: -T).(pr2 c t2 t))) (\lambda (H5: (pr0 t2 x)).(ex_intro2 T (\lambda (t: T).(pr2 -c t1 t)) (\lambda (t: T).(pr2 c t2 t)) x (pr2_free c t1 x H4) (pr2_free c t2 -x H5))) (\lambda (H5: (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: -T).(subst0 i u x w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t2 w2)) (\lambda -(w2: T).(subst0 i u x w2)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: -T).(pr2 c t2 t))) (\lambda (x0: T).(\lambda (H6: (pr0 t2 x0)).(\lambda (H7: -(subst0 i u x x0)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: -T).(pr2 c t2 t)) x0 (pr2_delta c d u i H0 t1 x H4 x0 H7) (pr2_free c t2 x0 -H6))))) H5)) (pr0_subst0 t4 x H3 u t2 i H2 u (pr0_refl u)))))) -(pr0_confluence t0 t4 H1 t1 H))))))))))))). -(* COMMENTS -Initial nodes: 403 -END *) - -theorem pr2_confluence__pr2_delta_delta: - \forall (c: C).(\forall (d: C).(\forall (d0: C).(\forall (t0: T).(\forall -(t1: T).(\forall (t2: T).(\forall (t3: T).(\forall (t4: T).(\forall (u: -T).(\forall (u0: T).(\forall (i: nat).(\forall (i0: nat).((getl i c (CHead d -(Bind Abbr) u)) \to ((pr0 t0 t3) \to ((subst0 i u t3 t1) \to ((getl i0 c -(CHead d0 (Bind Abbr) u0)) \to ((pr0 t0 t4) \to ((subst0 i0 u0 t4 t2) \to -(ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 -t)))))))))))))))))))) -\def - \lambda (c: C).(\lambda (d: C).(\lambda (d0: C).(\lambda (t0: T).(\lambda -(t1: T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (u: -T).(\lambda (u0: T).(\lambda (i: nat).(\lambda (i0: nat).(\lambda (H: (getl i -c (CHead d (Bind Abbr) u))).(\lambda (H0: (pr0 t0 t3)).(\lambda (H1: (subst0 -i u t3 t1)).(\lambda (H2: (getl i0 c (CHead d0 (Bind Abbr) u0))).(\lambda -(H3: (pr0 t0 t4)).(\lambda (H4: (subst0 i0 u0 t4 t2)).(ex2_ind T (\lambda (t: -T).(pr0 t4 t)) (\lambda (t: T).(pr0 t3 t)) (ex2 T (\lambda (t: T).(pr2 c t1 -t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (x: T).(\lambda (H5: (pr0 t4 -x)).(\lambda (H6: (pr0 t3 x)).(or_ind (pr0 t1 x) (ex2 T (\lambda (w2: T).(pr0 -t1 w2)) (\lambda (w2: T).(subst0 i u x w2))) (ex2 T (\lambda (t: T).(pr2 c t1 -t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (H7: (pr0 t1 x)).(or_ind (pr0 t2 -x) (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: T).(subst0 i0 u0 x -w2))) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) -(\lambda (H8: (pr0 t2 x)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda -(t: T).(pr2 c t2 t)) x (pr2_free c t1 x H7) (pr2_free c t2 x H8))) (\lambda -(H8: (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: T).(subst0 i0 u0 x -w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: T).(subst0 i0 -u0 x w2)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) -(\lambda (x0: T).(\lambda (H9: (pr0 t2 x0)).(\lambda (H10: (subst0 i0 u0 x -x0)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t)) -x0 (pr2_delta c d0 u0 i0 H2 t1 x H7 x0 H10) (pr2_free c t2 x0 H9))))) H8)) -(pr0_subst0 t4 x H5 u0 t2 i0 H4 u0 (pr0_refl u0)))) (\lambda (H7: (ex2 T -(\lambda (w2: T).(pr0 t1 w2)) (\lambda (w2: T).(subst0 i u x w2)))).(ex2_ind -T (\lambda (w2: T).(pr0 t1 w2)) (\lambda (w2: T).(subst0 i u x w2)) (ex2 T -(\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (x0: -T).(\lambda (H8: (pr0 t1 x0)).(\lambda (H9: (subst0 i u x x0)).(or_ind (pr0 -t2 x) (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: T).(subst0 i0 u0 x -w2))) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) -(\lambda (H10: (pr0 t2 x)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) -(\lambda (t: T).(pr2 c t2 t)) x0 (pr2_free c t1 x0 H8) (pr2_delta c d u i H -t2 x H10 x0 H9))) (\lambda (H10: (ex2 T (\lambda (w2: T).(pr0 t2 w2)) -(\lambda (w2: T).(subst0 i0 u0 x w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t2 -w2)) (\lambda (w2: T).(subst0 i0 u0 x w2)) (ex2 T (\lambda (t: T).(pr2 c t1 -t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (x1: T).(\lambda (H11: (pr0 t2 -x1)).(\lambda (H12: (subst0 i0 u0 x x1)).(neq_eq_e i i0 (ex2 T (\lambda (t: -T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (H13: (not (eq nat i -i0))).(ex2_ind T (\lambda (t: T).(subst0 i u x1 t)) (\lambda (t: T).(subst0 -i0 u0 x0 t)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 -t))) (\lambda (x2: T).(\lambda (H14: (subst0 i u x1 x2)).(\lambda (H15: -(subst0 i0 u0 x0 x2)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: -T).(pr2 c t2 t)) x2 (pr2_delta c d0 u0 i0 H2 t1 x0 H8 x2 H15) (pr2_delta c d -u i H t2 x1 H11 x2 H14))))) (subst0_confluence_neq x x1 u0 i0 H12 x0 u i H9 -(sym_not_eq nat i i0 H13)))) (\lambda (H13: (eq nat i i0)).(let H14 \def -(eq_ind_r nat i0 (\lambda (n: nat).(subst0 n u0 x x1)) H12 i H13) in (let H15 -\def (eq_ind_r nat i0 (\lambda (n: nat).(getl n c (CHead d0 (Bind Abbr) u0))) -H2 i H13) in (let H16 \def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c0: -C).(getl i c c0)) H (CHead d0 (Bind Abbr) u0) (getl_mono c (CHead d (Bind -Abbr) u) i H (CHead d0 (Bind Abbr) u0) H15)) in (let H17 \def (f_equal C C -(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abbr) u) -(CHead d0 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) i H (CHead d0 -(Bind Abbr) u0) H15)) in ((let H18 \def (f_equal C T (\lambda (e: C).(match e -in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) -\Rightarrow t])) (CHead d (Bind Abbr) u) (CHead d0 (Bind Abbr) u0) (getl_mono -c (CHead d (Bind Abbr) u) i H (CHead d0 (Bind Abbr) u0) H15)) in (\lambda -(H19: (eq C d d0)).(let H20 \def (eq_ind_r T u0 (\lambda (t: T).(subst0 i t x -x1)) H14 u H18) in (let H21 \def (eq_ind_r T u0 (\lambda (t: T).(getl i c -(CHead d0 (Bind Abbr) t))) H16 u H18) in (let H22 \def (eq_ind_r C d0 -(\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr) u))) H21 d H19) in (or4_ind -(eq T x1 x0) (ex2 T (\lambda (t: T).(subst0 i u x1 t)) (\lambda (t: -T).(subst0 i u x0 t))) (subst0 i u x1 x0) (subst0 i u x0 x1) (ex2 T (\lambda -(t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (H23: (eq T x1 -x0)).(let H24 \def (eq_ind T x1 (\lambda (t: T).(pr0 t2 t)) H11 x0 H23) in -(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t)) x0 -(pr2_free c t1 x0 H8) (pr2_free c t2 x0 H24)))) (\lambda (H23: (ex2 T -(\lambda (t: T).(subst0 i u x1 t)) (\lambda (t: T).(subst0 i u x0 -t)))).(ex2_ind T (\lambda (t: T).(subst0 i u x1 t)) (\lambda (t: T).(subst0 i -u x0 t)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) -(\lambda (x2: T).(\lambda (H24: (subst0 i u x1 x2)).(\lambda (H25: (subst0 i -u x0 x2)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c -t2 t)) x2 (pr2_delta c d u i H22 t1 x0 H8 x2 H25) (pr2_delta c d u i H22 t2 -x1 H11 x2 H24))))) H23)) (\lambda (H23: (subst0 i u x1 x0)).(ex_intro2 T -(\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t)) x0 (pr2_free c t1 -x0 H8) (pr2_delta c d u i H22 t2 x1 H11 x0 H23))) (\lambda (H23: (subst0 i u -x0 x1)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 -t)) x1 (pr2_delta c d u i H22 t1 x0 H8 x1 H23) (pr2_free c t2 x1 H11))) -(subst0_confluence_eq x x1 u i H20 x0 H9))))))) H17)))))))))) H10)) -(pr0_subst0 t4 x H5 u0 t2 i0 H4 u0 (pr0_refl u0)))))) H7)) (pr0_subst0 t3 x -H6 u t1 i H1 u (pr0_refl u)))))) (pr0_confluence t0 t4 H3 t3 -H0))))))))))))))))))). -(* COMMENTS -Initial nodes: 1901 -END *) - -theorem pr2_confluence: - \forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr2 c t0 t1) \to (\forall -(t2: T).((pr2 c t0 t2) \to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: -T).(pr2 c t2 t)))))))) -\def - \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr2 c t0 -t1)).(\lambda (t2: T).(\lambda (H0: (pr2 c t0 t2)).(let H1 \def (match H in -pr2 return (\lambda (c0: C).(\lambda (t: T).(\lambda (t3: T).(\lambda (_: -(pr2 c0 t t3)).((eq C c0 c) \to ((eq T t t0) \to ((eq T t3 t1) \to (ex2 T -(\lambda (t4: T).(pr2 c t1 t4)) (\lambda (t4: T).(pr2 c t2 t4)))))))))) with -[(pr2_free c0 t3 t4 H1) \Rightarrow (\lambda (H2: (eq C c0 c)).(\lambda (H3: -(eq T t3 t0)).(\lambda (H4: (eq T t4 t1)).(eq_ind C c (\lambda (_: C).((eq T -t3 t0) \to ((eq T t4 t1) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t: T).(pr2 c -t1 t)) (\lambda (t: T).(pr2 c t2 t))))))) (\lambda (H5: (eq T t3 t0)).(eq_ind -T t0 (\lambda (t: T).((eq T t4 t1) \to ((pr0 t t4) \to (ex2 T (\lambda (t5: -T).(pr2 c t1 t5)) (\lambda (t5: T).(pr2 c t2 t5)))))) (\lambda (H6: (eq T t4 -t1)).(eq_ind T t1 (\lambda (t: T).((pr0 t0 t) \to (ex2 T (\lambda (t5: -T).(pr2 c t1 t5)) (\lambda (t5: T).(pr2 c t2 t5))))) (\lambda (H7: (pr0 t0 -t1)).(let H8 \def (match H0 in pr2 return (\lambda (c1: C).(\lambda (t: -T).(\lambda (t5: T).(\lambda (_: (pr2 c1 t t5)).((eq C c1 c) \to ((eq T t t0) -\to ((eq T t5 t2) \to (ex2 T (\lambda (t6: T).(pr2 c t1 t6)) (\lambda (t6: -T).(pr2 c t2 t6)))))))))) with [(pr2_free c1 t5 t6 H8) \Rightarrow (\lambda -(H9: (eq C c1 c)).(\lambda (H10: (eq T t5 t0)).(\lambda (H11: (eq T t6 -t2)).(eq_ind C c (\lambda (_: C).((eq T t5 t0) \to ((eq T t6 t2) \to ((pr0 t5 -t6) \to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 -t))))))) (\lambda (H12: (eq T t5 t0)).(eq_ind T t0 (\lambda (t: T).((eq T t6 -t2) \to ((pr0 t t6) \to (ex2 T (\lambda (t7: T).(pr2 c t1 t7)) (\lambda (t7: -T).(pr2 c t2 t7)))))) (\lambda (H13: (eq T t6 t2)).(eq_ind T t2 (\lambda (t: -T).((pr0 t0 t) \to (ex2 T (\lambda (t7: T).(pr2 c t1 t7)) (\lambda (t7: -T).(pr2 c t2 t7))))) (\lambda (H14: (pr0 t0 -t2)).(pr2_confluence__pr2_free_free c t0 t1 t2 H7 H14)) t6 (sym_eq T t6 t2 -H13))) t5 (sym_eq T t5 t0 H12))) c1 (sym_eq C c1 c H9) H10 H11 H8)))) | -(pr2_delta c1 d u i H8 t5 t6 H9 t H10) \Rightarrow (\lambda (H11: (eq C c1 -c)).(\lambda (H12: (eq T t5 t0)).(\lambda (H13: (eq T t t2)).(eq_ind C c -(\lambda (c2: C).((eq T t5 t0) \to ((eq T t t2) \to ((getl i c2 (CHead d -(Bind Abbr) u)) \to ((pr0 t5 t6) \to ((subst0 i u t6 t) \to (ex2 T (\lambda -(t7: T).(pr2 c t1 t7)) (\lambda (t7: T).(pr2 c t2 t7))))))))) (\lambda (H14: -(eq T t5 t0)).(eq_ind T t0 (\lambda (t7: T).((eq T t t2) \to ((getl i c -(CHead d (Bind Abbr) u)) \to ((pr0 t7 t6) \to ((subst0 i u t6 t) \to (ex2 T -(\lambda (t8: T).(pr2 c t1 t8)) (\lambda (t8: T).(pr2 c t2 t8)))))))) -(\lambda (H15: (eq T t t2)).(eq_ind T t2 (\lambda (t7: T).((getl i c (CHead d -(Bind Abbr) u)) \to ((pr0 t0 t6) \to ((subst0 i u t6 t7) \to (ex2 T (\lambda -(t8: T).(pr2 c t1 t8)) (\lambda (t8: T).(pr2 c t2 t8))))))) (\lambda (H16: -(getl i c (CHead d (Bind Abbr) u))).(\lambda (H17: (pr0 t0 t6)).(\lambda -(H18: (subst0 i u t6 t2)).(pr2_confluence__pr2_free_delta c d t0 t1 t2 t6 u i -H7 H16 H17 H18)))) t (sym_eq T t t2 H15))) t5 (sym_eq T t5 t0 H14))) c1 -(sym_eq C c1 c H11) H12 H13 H8 H9 H10))))]) in (H8 (refl_equal C c) -(refl_equal T t0) (refl_equal T t2)))) t4 (sym_eq T t4 t1 H6))) t3 (sym_eq T -t3 t0 H5))) c0 (sym_eq C c0 c H2) H3 H4 H1)))) | (pr2_delta c0 d u i H1 t3 t4 -H2 t H3) \Rightarrow (\lambda (H4: (eq C c0 c)).(\lambda (H5: (eq T t3 -t0)).(\lambda (H6: (eq T t t1)).(eq_ind C c (\lambda (c1: C).((eq T t3 t0) -\to ((eq T t t1) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t3 t4) -\to ((subst0 i u t4 t) \to (ex2 T (\lambda (t5: T).(pr2 c t1 t5)) (\lambda -(t5: T).(pr2 c t2 t5))))))))) (\lambda (H7: (eq T t3 t0)).(eq_ind T t0 -(\lambda (t5: T).((eq T t t1) \to ((getl i c (CHead d (Bind Abbr) u)) \to -((pr0 t5 t4) \to ((subst0 i u t4 t) \to (ex2 T (\lambda (t6: T).(pr2 c t1 -t6)) (\lambda (t6: T).(pr2 c t2 t6)))))))) (\lambda (H8: (eq T t t1)).(eq_ind -T t1 (\lambda (t5: T).((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t4) -\to ((subst0 i u t4 t5) \to (ex2 T (\lambda (t6: T).(pr2 c t1 t6)) (\lambda -(t6: T).(pr2 c t2 t6))))))) (\lambda (H9: (getl i c (CHead d (Bind Abbr) -u))).(\lambda (H10: (pr0 t0 t4)).(\lambda (H11: (subst0 i u t4 t1)).(let H12 -\def (match H0 in pr2 return (\lambda (c1: C).(\lambda (t5: T).(\lambda (t6: -T).(\lambda (_: (pr2 c1 t5 t6)).((eq C c1 c) \to ((eq T t5 t0) \to ((eq T t6 -t2) \to (ex2 T (\lambda (t7: T).(pr2 c t1 t7)) (\lambda (t7: T).(pr2 c t2 -t7)))))))))) with [(pr2_free c1 t5 t6 H12) \Rightarrow (\lambda (H13: (eq C -c1 c)).(\lambda (H14: (eq T t5 t0)).(\lambda (H15: (eq T t6 t2)).(eq_ind C c -(\lambda (_: C).((eq T t5 t0) \to ((eq T t6 t2) \to ((pr0 t5 t6) \to (ex2 T -(\lambda (t7: T).(pr2 c t1 t7)) (\lambda (t7: T).(pr2 c t2 t7))))))) (\lambda -(H16: (eq T t5 t0)).(eq_ind T t0 (\lambda (t7: T).((eq T t6 t2) \to ((pr0 t7 -t6) \to (ex2 T (\lambda (t8: T).(pr2 c t1 t8)) (\lambda (t8: T).(pr2 c t2 -t8)))))) (\lambda (H17: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t0 -t7) \to (ex2 T (\lambda (t8: T).(pr2 c t1 t8)) (\lambda (t8: T).(pr2 c t2 -t8))))) (\lambda (H18: (pr0 t0 t2)).(ex2_sym T (pr2 c t2) (pr2 c t1) -(pr2_confluence__pr2_free_delta c d t0 t2 t1 t4 u i H18 H9 H10 H11))) t6 -(sym_eq T t6 t2 H17))) t5 (sym_eq T t5 t0 H16))) c1 (sym_eq C c1 c H13) H14 -H15 H12)))) | (pr2_delta c1 d0 u0 i0 H12 t5 t6 H13 t7 H14) \Rightarrow -(\lambda (H15: (eq C c1 c)).(\lambda (H16: (eq T t5 t0)).(\lambda (H17: (eq T -t7 t2)).(eq_ind C c (\lambda (c2: C).((eq T t5 t0) \to ((eq T t7 t2) \to -((getl i0 c2 (CHead d0 (Bind Abbr) u0)) \to ((pr0 t5 t6) \to ((subst0 i0 u0 -t6 t7) \to (ex2 T (\lambda (t8: T).(pr2 c t1 t8)) (\lambda (t8: T).(pr2 c t2 -t8))))))))) (\lambda (H18: (eq T t5 t0)).(eq_ind T t0 (\lambda (t8: T).((eq T -t7 t2) \to ((getl i0 c (CHead d0 (Bind Abbr) u0)) \to ((pr0 t8 t6) \to -((subst0 i0 u0 t6 t7) \to (ex2 T (\lambda (t9: T).(pr2 c t1 t9)) (\lambda -(t9: T).(pr2 c t2 t9)))))))) (\lambda (H19: (eq T t7 t2)).(eq_ind T t2 -(\lambda (t8: T).((getl i0 c (CHead d0 (Bind Abbr) u0)) \to ((pr0 t0 t6) \to -((subst0 i0 u0 t6 t8) \to (ex2 T (\lambda (t9: T).(pr2 c t1 t9)) (\lambda -(t9: T).(pr2 c t2 t9))))))) (\lambda (H20: (getl i0 c (CHead d0 (Bind Abbr) -u0))).(\lambda (H21: (pr0 t0 t6)).(\lambda (H22: (subst0 i0 u0 t6 -t2)).(pr2_confluence__pr2_delta_delta c d d0 t0 t1 t2 t4 t6 u u0 i i0 H9 H10 -H11 H20 H21 H22)))) t7 (sym_eq T t7 t2 H19))) t5 (sym_eq T t5 t0 H18))) c1 -(sym_eq C c1 c H15) H16 H17 H12 H13 H14))))]) in (H12 (refl_equal C c) -(refl_equal T t0) (refl_equal T t2)))))) t (sym_eq T t t1 H8))) t3 (sym_eq T -t3 t0 H7))) c0 (sym_eq C c0 c H4) H5 H6 H1 H2 H3))))]) in (H1 (refl_equal C -c) (refl_equal T t0) (refl_equal T t1)))))))). -(* COMMENTS -Initial nodes: 2087 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr2/props.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr2/props.ma deleted file mode 100644 index 2faeb6ebd..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr2/props.ma +++ /dev/null @@ -1,307 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr2/defs.ma". - -include "Basic-1/pr0/props.ma". - -include "Basic-1/getl/drop.ma". - -include "Basic-1/getl/clear.ma". - -theorem pr2_thin_dx: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall -(u: T).(\forall (f: F).(pr2 c (THead (Flat f) u t1) (THead (Flat f) u -t2))))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 -t2)).(\lambda (u: T).(\lambda (f: F).(pr2_ind (\lambda (c0: C).(\lambda (t: -T).(\lambda (t0: T).(pr2 c0 (THead (Flat f) u t) (THead (Flat f) u t0))))) -(\lambda (c0: C).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr0 t0 -t3)).(pr2_free c0 (THead (Flat f) u t0) (THead (Flat f) u t3) (pr0_comp u u -(pr0_refl u) t0 t3 H0 (Flat f))))))) (\lambda (c0: C).(\lambda (d: -C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind -Abbr) u0))).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H1: (pr0 t0 -t3)).(\lambda (t: T).(\lambda (H2: (subst0 i u0 t3 t)).(pr2_delta c0 d u0 i -H0 (THead (Flat f) u t0) (THead (Flat f) u t3) (pr0_comp u u (pr0_refl u) t0 -t3 H1 (Flat f)) (THead (Flat f) u t) (subst0_snd (Flat f) u0 t t3 i H2 -u)))))))))))) c t1 t2 H)))))). -(* COMMENTS -Initial nodes: 239 -END *) - -theorem pr2_head_1: - \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr2 c u1 u2) \to (\forall -(k: K).(\forall (t: T).(pr2 c (THead k u1 t) (THead k u2 t))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr2 c u1 -u2)).(\lambda (k: K).(\lambda (t: T).(pr2_ind (\lambda (c0: C).(\lambda (t0: -T).(\lambda (t1: T).(pr2 c0 (THead k t0 t) (THead k t1 t))))) (\lambda (c0: -C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr0 t1 t2)).(pr2_free c0 -(THead k t1 t) (THead k t2 t) (pr0_comp t1 t2 H0 t t (pr0_refl t) k)))))) -(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H0: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (H1: (pr0 t1 t2)).(\lambda (t0: T).(\lambda (H2: (subst0 i u t2 -t0)).(pr2_delta c0 d u i H0 (THead k t1 t) (THead k t2 t) (pr0_comp t1 t2 H1 -t t (pr0_refl t) k) (THead k t0 t) (subst0_fst u t0 t2 i H2 t k)))))))))))) c -u1 u2 H)))))). -(* COMMENTS -Initial nodes: 219 -END *) - -theorem pr2_head_2: - \forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).(\forall -(k: K).((pr2 (CHead c k u) t1 t2) \to (pr2 c (THead k u t1) (THead k u -t2))))))) -\def - \lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(k: K).(\lambda (H: (pr2 (CHead c k u) t1 t2)).(insert_eq C (CHead c k u) -(\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 c (THead k u t1) (THead -k u t2))) (\lambda (y: C).(\lambda (H0: (pr2 y t1 t2)).(pr2_ind (\lambda (c0: -C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CHead c k u)) \to (pr2 c -(THead k u t) (THead k u t0)))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda -(t4: T).(\lambda (H1: (pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c k -u))).(pr2_free c (THead k u t3) (THead k u t4) (pr0_comp u u (pr0_refl u) t3 -t4 H1 k))))))) (K_ind (\lambda (k0: K).(\forall (c0: C).(\forall (d: -C).(\forall (u0: T).(\forall (i: nat).((getl i c0 (CHead d (Bind Abbr) u0)) -\to (\forall (t3: T).(\forall (t4: T).((pr0 t3 t4) \to (\forall (t: -T).((subst0 i u0 t4 t) \to ((eq C c0 (CHead c k0 u)) \to (pr2 c (THead k0 u -t3) (THead k0 u t)))))))))))))) (\lambda (b: B).(\lambda (c0: C).(\lambda (d: -C).(\lambda (u0: T).(\lambda (i: nat).(nat_ind (\lambda (n: nat).((getl n c0 -(CHead d (Bind Abbr) u0)) \to (\forall (t3: T).(\forall (t4: T).((pr0 t3 t4) -\to (\forall (t: T).((subst0 n u0 t4 t) \to ((eq C c0 (CHead c (Bind b) u)) -\to (pr2 c (THead (Bind b) u t3) (THead (Bind b) u t)))))))))) (\lambda (H1: -(getl O c0 (CHead d (Bind Abbr) u0))).(\lambda (t3: T).(\lambda (t4: -T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: (subst0 O u0 t4 -t)).(\lambda (H4: (eq C c0 (CHead c (Bind b) u))).(let H5 \def (eq_ind C c0 -(\lambda (c1: C).(getl O c1 (CHead d (Bind Abbr) u0))) H1 (CHead c (Bind b) -u) H4) in (let H6 \def (f_equal C C (\lambda (e: C).(match e in C return -(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow -c1])) (CHead d (Bind Abbr) u0) (CHead c (Bind b) u) (clear_gen_bind b c -(CHead d (Bind Abbr) u0) u (getl_gen_O (CHead c (Bind b) u) (CHead d (Bind -Abbr) u0) H5))) in ((let H7 \def (f_equal C B (\lambda (e: C).(match e in C -return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _) -\Rightarrow (match k0 in K return (\lambda (_: K).B) with [(Bind b0) -\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u0) -(CHead c (Bind b) u) (clear_gen_bind b c (CHead d (Bind Abbr) u0) u -(getl_gen_O (CHead c (Bind b) u) (CHead d (Bind Abbr) u0) H5))) in ((let H8 -\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow u0 | (CHead _ _ t0) \Rightarrow t0])) (CHead d -(Bind Abbr) u0) (CHead c (Bind b) u) (clear_gen_bind b c (CHead d (Bind Abbr) -u0) u (getl_gen_O (CHead c (Bind b) u) (CHead d (Bind Abbr) u0) H5))) in -(\lambda (H9: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H11 \def (eq_ind T -u0 (\lambda (t0: T).(subst0 O t0 t4 t)) H3 u H8) in (eq_ind B Abbr (\lambda -(b0: B).(pr2 c (THead (Bind b0) u t3) (THead (Bind b0) u t))) (pr2_free c -(THead (Bind Abbr) u t3) (THead (Bind Abbr) u t) (pr0_delta u u (pr0_refl u) -t3 t4 H2 t H11)) b H9))))) H7)) H6)))))))))) (\lambda (n: nat).(\lambda (H1: -(((getl n c0 (CHead d (Bind Abbr) u0)) \to (\forall (t3: T).(\forall (t4: -T).((pr0 t3 t4) \to (\forall (t: T).((subst0 n u0 t4 t) \to ((eq C c0 (CHead -c (Bind b) u)) \to (pr2 c (THead (Bind b) u t3) (THead (Bind b) u -t))))))))))).(\lambda (H2: (getl (S n) c0 (CHead d (Bind Abbr) u0))).(\lambda -(t3: T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda (t: T).(\lambda -(H4: (subst0 (S n) u0 t4 t)).(\lambda (H5: (eq C c0 (CHead c (Bind b) -u))).(let H6 \def (eq_ind C c0 (\lambda (c1: C).(getl (S n) c1 (CHead d (Bind -Abbr) u0))) H2 (CHead c (Bind b) u) H5) in (let H7 \def (eq_ind C c0 (\lambda -(c1: C).((getl n c1 (CHead d (Bind Abbr) u0)) \to (\forall (t5: T).(\forall -(t6: T).((pr0 t5 t6) \to (\forall (t0: T).((subst0 n u0 t6 t0) \to ((eq C c1 -(CHead c (Bind b) u)) \to (pr2 c (THead (Bind b) u t5) (THead (Bind b) u -t0)))))))))) H1 (CHead c (Bind b) u) H5) in (pr2_delta c d u0 (r (Bind b) n) -(getl_gen_S (Bind b) c (CHead d (Bind Abbr) u0) u n H6) (THead (Bind b) u t3) -(THead (Bind b) u t4) (pr0_comp u u (pr0_refl u) t3 t4 H3 (Bind b)) (THead -(Bind b) u t) (subst0_snd (Bind b) u0 t t4 (r (Bind b) n) H4 u))))))))))))) -i)))))) (\lambda (f: F).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: -T).(\lambda (i: nat).(nat_ind (\lambda (n: nat).((getl n c0 (CHead d (Bind -Abbr) u0)) \to (\forall (t3: T).(\forall (t4: T).((pr0 t3 t4) \to (\forall -(t: T).((subst0 n u0 t4 t) \to ((eq C c0 (CHead c (Flat f) u)) \to (pr2 c -(THead (Flat f) u t3) (THead (Flat f) u t)))))))))) (\lambda (H1: (getl O c0 -(CHead d (Bind Abbr) u0))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: -(pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: (subst0 O u0 t4 t)).(\lambda (H4: -(eq C c0 (CHead c (Flat f) u))).(let H5 \def (eq_ind C c0 (\lambda (c1: -C).(getl O c1 (CHead d (Bind Abbr) u0))) H1 (CHead c (Flat f) u) H4) in -(pr2_delta c d u0 O (getl_intro O c (CHead d (Bind Abbr) u0) c (drop_refl c) -(clear_gen_flat f c (CHead d (Bind Abbr) u0) u (getl_gen_O (CHead c (Flat f) -u) (CHead d (Bind Abbr) u0) H5))) (THead (Flat f) u t3) (THead (Flat f) u t4) -(pr0_comp u u (pr0_refl u) t3 t4 H2 (Flat f)) (THead (Flat f) u t) -(subst0_snd (Flat f) u0 t t4 O H3 u)))))))))) (\lambda (n: nat).(\lambda (H1: -(((getl n c0 (CHead d (Bind Abbr) u0)) \to (\forall (t3: T).(\forall (t4: -T).((pr0 t3 t4) \to (\forall (t: T).((subst0 n u0 t4 t) \to ((eq C c0 (CHead -c (Flat f) u)) \to (pr2 c (THead (Flat f) u t3) (THead (Flat f) u -t))))))))))).(\lambda (H2: (getl (S n) c0 (CHead d (Bind Abbr) u0))).(\lambda -(t3: T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda (t: T).(\lambda -(H4: (subst0 (S n) u0 t4 t)).(\lambda (H5: (eq C c0 (CHead c (Flat f) -u))).(let H6 \def (eq_ind C c0 (\lambda (c1: C).(getl (S n) c1 (CHead d (Bind -Abbr) u0))) H2 (CHead c (Flat f) u) H5) in (let H7 \def (eq_ind C c0 (\lambda -(c1: C).((getl n c1 (CHead d (Bind Abbr) u0)) \to (\forall (t5: T).(\forall -(t6: T).((pr0 t5 t6) \to (\forall (t0: T).((subst0 n u0 t6 t0) \to ((eq C c1 -(CHead c (Flat f) u)) \to (pr2 c (THead (Flat f) u t5) (THead (Flat f) u -t0)))))))))) H1 (CHead c (Flat f) u) H5) in (pr2_delta c d u0 (r (Flat f) n) -(getl_gen_S (Flat f) c (CHead d (Bind Abbr) u0) u n H6) (THead (Flat f) u t3) -(THead (Flat f) u t4) (pr0_comp u u (pr0_refl u) t3 t4 H3 (Flat f)) (THead -(Flat f) u t) (subst0_snd (Flat f) u0 t t4 (r (Flat f) n) H4 u))))))))))))) -i)))))) k) y t1 t2 H0))) H)))))). -(* COMMENTS -Initial nodes: 1947 -END *) - -theorem clear_pr2_trans: - \forall (c2: C).(\forall (t1: T).(\forall (t2: T).((pr2 c2 t1 t2) \to -(\forall (c1: C).((clear c1 c2) \to (pr2 c1 t1 t2)))))) -\def - \lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c2 t1 -t2)).(pr2_ind (\lambda (c: C).(\lambda (t: T).(\lambda (t0: T).(\forall (c1: -C).((clear c1 c) \to (pr2 c1 t t0)))))) (\lambda (c: C).(\lambda (t3: -T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(\lambda (c1: C).(\lambda (_: -(clear c1 c)).(pr2_free c1 t3 t4 H0))))))) (\lambda (c: C).(\lambda (d: -C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d (Bind -Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 -t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (c1: -C).(\lambda (H3: (clear c1 c)).(pr2_delta c1 d u i (clear_getl_trans i c -(CHead d (Bind Abbr) u) H0 c1 H3) t3 t4 H1 t H2))))))))))))) c2 t1 t2 H)))). -(* COMMENTS -Initial nodes: 171 -END *) - -theorem pr2_cflat: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall -(f: F).(\forall (v: T).(pr2 (CHead c (Flat f) v) t1 t2)))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 -t2)).(\lambda (f: F).(\lambda (v: T).(pr2_ind (\lambda (c0: C).(\lambda (t: -T).(\lambda (t0: T).(pr2 (CHead c0 (Flat f) v) t t0)))) (\lambda (c0: -C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(pr2_free -(CHead c0 (Flat f) v) t3 t4 H0))))) (\lambda (c0: C).(\lambda (d: C).(\lambda -(u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr) -u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 t4)).(\lambda -(t: T).(\lambda (H2: (subst0 i u t4 t)).(pr2_delta (CHead c0 (Flat f) v) d u -i (getl_flat c0 (CHead d (Bind Abbr) u) i H0 f v) t3 t4 H1 t H2))))))))))) c -t1 t2 H)))))). -(* COMMENTS -Initial nodes: 175 -END *) - -theorem pr2_ctail: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall -(k: K).(\forall (u: T).(pr2 (CTail k u c) t1 t2)))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 -t2)).(\lambda (k: K).(\lambda (u: T).(pr2_ind (\lambda (c0: C).(\lambda (t: -T).(\lambda (t0: T).(pr2 (CTail k u c0) t t0)))) (\lambda (c0: C).(\lambda -(t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(pr2_free (CTail k u c0) -t3 t4 H0))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: -nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr) u0))).(\lambda (t3: -T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H2: -(subst0 i u0 t4 t)).(pr2_delta (CTail k u c0) (CTail k u d) u0 i (getl_ctail -Abbr c0 d u0 i H0 k u) t3 t4 H1 t H2))))))))))) c t1 t2 H)))))). -(* COMMENTS -Initial nodes: 171 -END *) - -theorem pr2_change: - \forall (b: B).((not (eq B b Abbr)) \to (\forall (c: C).(\forall (v1: -T).(\forall (t1: T).(\forall (t2: T).((pr2 (CHead c (Bind b) v1) t1 t2) \to -(\forall (v2: T).(pr2 (CHead c (Bind b) v2) t1 t2)))))))) -\def - \lambda (b: B).(\lambda (H: (not (eq B b Abbr))).(\lambda (c: C).(\lambda -(v1: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr2 (CHead c (Bind -b) v1) t1 t2)).(\lambda (v2: T).(insert_eq C (CHead c (Bind b) v1) (\lambda -(c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 (CHead c (Bind b) v2) t1 t2)) -(\lambda (y: C).(\lambda (H1: (pr2 y t1 t2)).(pr2_ind (\lambda (c0: -C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CHead c (Bind b) v1)) \to (pr2 -(CHead c (Bind b) v2) t t0))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda -(t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c (Bind b) -v1))).(pr2_free (CHead c (Bind b) v2) t3 t4 H2)))))) (\lambda (c0: -C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H2: (getl i c0 -(CHead d (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H3: -(pr0 t3 t4)).(\lambda (t: T).(\lambda (H4: (subst0 i u t4 t)).(\lambda (H5: -(eq C c0 (CHead c (Bind b) v1))).(let H6 \def (eq_ind C c0 (\lambda (c1: -C).(getl i c1 (CHead d (Bind Abbr) u))) H2 (CHead c (Bind b) v1) H5) in -(nat_ind (\lambda (n: nat).((getl n (CHead c (Bind b) v1) (CHead d (Bind -Abbr) u)) \to ((subst0 n u t4 t) \to (pr2 (CHead c (Bind b) v2) t3 t)))) -(\lambda (H7: (getl O (CHead c (Bind b) v1) (CHead d (Bind Abbr) -u))).(\lambda (H8: (subst0 O u t4 t)).(let H9 \def (f_equal C C (\lambda (e: -C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | -(CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u) (CHead c (Bind b) -v1) (clear_gen_bind b c (CHead d (Bind Abbr) u) v1 (getl_gen_O (CHead c (Bind -b) v1) (CHead d (Bind Abbr) u) H7))) in ((let H10 \def (f_equal C B (\lambda -(e: C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow -Abbr | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with -[(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind -Abbr) u) (CHead c (Bind b) v1) (clear_gen_bind b c (CHead d (Bind Abbr) u) v1 -(getl_gen_O (CHead c (Bind b) v1) (CHead d (Bind Abbr) u) H7))) in ((let H11 -\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d -(Bind Abbr) u) (CHead c (Bind b) v1) (clear_gen_bind b c (CHead d (Bind Abbr) -u) v1 (getl_gen_O (CHead c (Bind b) v1) (CHead d (Bind Abbr) u) H7))) in -(\lambda (H12: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H14 \def (eq_ind -T u (\lambda (t0: T).(subst0 O t0 t4 t)) H8 v1 H11) in (let H15 \def -(eq_ind_r B b (\lambda (b0: B).(not (eq B b0 Abbr))) H Abbr H12) in (eq_ind B -Abbr (\lambda (b0: B).(pr2 (CHead c (Bind b0) v2) t3 t)) (let H16 \def (match -(H15 (refl_equal B Abbr)) in False return (\lambda (_: False).(pr2 (CHead c -(Bind Abbr) v2) t3 t)) with []) in H16) b H12)))))) H10)) H9)))) (\lambda -(i0: nat).(\lambda (_: (((getl i0 (CHead c (Bind b) v1) (CHead d (Bind Abbr) -u)) \to ((subst0 i0 u t4 t) \to (pr2 (CHead c (Bind b) v2) t3 t))))).(\lambda -(H7: (getl (S i0) (CHead c (Bind b) v1) (CHead d (Bind Abbr) u))).(\lambda -(H8: (subst0 (S i0) u t4 t)).(pr2_delta (CHead c (Bind b) v2) d u (S i0) -(getl_head (Bind b) i0 c (CHead d (Bind Abbr) u) (getl_gen_S (Bind b) c -(CHead d (Bind Abbr) u) v1 i0 H7) v2) t3 t4 H3 t H8))))) i H6 H4))))))))))))) -y t1 t2 H1))) H0)))))))). -(* COMMENTS -Initial nodes: 913 -END *) - -theorem pr2_lift: - \forall (c: C).(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h -d c e) \to (\forall (t1: T).(\forall (t2: T).((pr2 e t1 t2) \to (pr2 c (lift -h d t1) (lift h d t2))))))))) -\def - \lambda (c: C).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda -(H: (drop h d c e)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr2 e t1 -t2)).(insert_eq C e (\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 c -(lift h d t1) (lift h d t2))) (\lambda (y: C).(\lambda (H1: (pr2 y t1 -t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 e) -\to (pr2 c (lift h d t) (lift h d t0)))))) (\lambda (c0: C).(\lambda (t3: -T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (_: (eq C c0 -e)).(pr2_free c (lift h d t3) (lift h d t4) (pr0_lift t3 t4 H2 h d))))))) -(\lambda (c0: C).(\lambda (d0: C).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H2: (getl i c0 (CHead d0 (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: -T).(\lambda (H3: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H4: (subst0 i u t4 -t)).(\lambda (H5: (eq C c0 e)).(let H6 \def (eq_ind C c0 (\lambda (c1: -C).(getl i c1 (CHead d0 (Bind Abbr) u))) H2 e H5) in (lt_le_e i d (pr2 c -(lift h d t3) (lift h d t)) (\lambda (H7: (lt i d)).(let H8 \def -(drop_getl_trans_le i d (le_S_n i d (le_S (S i) d H7)) c e h H (CHead d0 -(Bind Abbr) u) H6) in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop i -O c e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d i) e0 e1))) -(\lambda (_: C).(\lambda (e1: C).(clear e1 (CHead d0 (Bind Abbr) u)))) (pr2 c -(lift h d t3) (lift h d t)) (\lambda (x0: C).(\lambda (x1: C).(\lambda (H9: -(drop i O c x0)).(\lambda (H10: (drop h (minus d i) x0 x1)).(\lambda (H11: -(clear x1 (CHead d0 (Bind Abbr) u))).(let H12 \def (eq_ind nat (minus d i) -(\lambda (n: nat).(drop h n x0 x1)) H10 (S (minus d (S i))) (minus_x_Sy d i -H7)) in (let H13 \def (drop_clear_S x1 x0 h (minus d (S i)) H12 Abbr d0 u -H11) in (ex2_ind C (\lambda (c1: C).(clear x0 (CHead c1 (Bind Abbr) (lift h -(minus d (S i)) u)))) (\lambda (c1: C).(drop h (minus d (S i)) c1 d0)) (pr2 c -(lift h d t3) (lift h d t)) (\lambda (x: C).(\lambda (H14: (clear x0 (CHead x -(Bind Abbr) (lift h (minus d (S i)) u)))).(\lambda (_: (drop h (minus d (S -i)) x d0)).(pr2_delta c x (lift h (minus d (S i)) u) i (getl_intro i c (CHead -x (Bind Abbr) (lift h (minus d (S i)) u)) x0 H9 H14) (lift h d t3) (lift h d -t4) (pr0_lift t3 t4 H3 h d) (lift h d t) (subst0_lift_lt t4 t u i H4 d H7 -h))))) H13)))))))) H8))) (\lambda (H7: (le d i)).(pr2_delta c d0 u (plus i h) -(drop_getl_trans_ge i c e d h H (CHead d0 (Bind Abbr) u) H6 H7) (lift h d t3) -(lift h d t4) (pr0_lift t3 t4 H3 h d) (lift h d t) (subst0_lift_ge t4 t u i h -H4 d H7)))))))))))))))) y t1 t2 H1))) H0)))))))). -(* COMMENTS -Initial nodes: 849 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr2/subst1.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr2/subst1.ma deleted file mode 100644 index 47704dd52..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr2/subst1.ma +++ /dev/null @@ -1,281 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr2/defs.ma". - -include "Basic-1/pr0/subst1.ma". - -include "Basic-1/pr0/fwd.ma". - -include "Basic-1/csubst1/getl.ma". - -include "Basic-1/csubst1/fwd.ma". - -include "Basic-1/subst1/subst1.ma". - -include "Basic-1/getl/drop.ma". - -theorem pr2_delta1: - \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c -(CHead d (Bind Abbr) u)) \to (\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) -\to (\forall (t: T).((subst1 i u t2 t) \to (pr2 c t1 t)))))))))) -\def - \lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H: (getl i c (CHead d (Bind Abbr) u))).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (H0: (pr0 t1 t2)).(\lambda (t: T).(\lambda (H1: (subst1 i u t2 -t)).(subst1_ind i u t2 (\lambda (t0: T).(pr2 c t1 t0)) (pr2_free c t1 t2 H0) -(\lambda (t0: T).(\lambda (H2: (subst0 i u t2 t0)).(pr2_delta c d u i H t1 t2 -H0 t0 H2))) t H1)))))))))). -(* COMMENTS -Initial nodes: 111 -END *) - -theorem pr2_subst1: - \forall (c: C).(\forall (e: C).(\forall (v: T).(\forall (i: nat).((getl i c -(CHead e (Bind Abbr) v)) \to (\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) -\to (\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T (\lambda (w2: T).(pr2 c -w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2)))))))))))) -\def - \lambda (c: C).(\lambda (e: C).(\lambda (v: T).(\lambda (i: nat).(\lambda -(H: (getl i c (CHead e (Bind Abbr) v))).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (H0: (pr2 c t1 t2)).(insert_eq C c (\lambda (c0: C).(pr2 c0 t1 -t2)) (\lambda (c0: C).(\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T -(\lambda (w2: T).(pr2 c0 w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2)))))) -(\lambda (y: C).(\lambda (H1: (pr2 y t1 t2)).(pr2_ind (\lambda (c0: -C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 c) \to (\forall (w1: -T).((subst1 i v t w1) \to (ex2 T (\lambda (w2: T).(pr2 c0 w1 w2)) (\lambda -(w2: T).(subst1 i v t0 w2))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda -(t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (H3: (eq C c0 c)).(\lambda (w1: -T).(\lambda (H4: (subst1 i v t3 w1)).(eq_ind_r C c (\lambda (c1: C).(ex2 T -(\lambda (w2: T).(pr2 c1 w1 w2)) (\lambda (w2: T).(subst1 i v t4 w2)))) -(ex2_ind T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst1 i v t4 w2)) -(ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t4 w2))) -(\lambda (x: T).(\lambda (H5: (pr0 w1 x)).(\lambda (H6: (subst1 i v t4 -x)).(ex_intro2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v -t4 w2)) x (pr2_free c w1 x H5) H6)))) (pr0_subst1 t3 t4 H2 v w1 i H4 v -(pr0_refl v))) c0 H3)))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: -T).(\lambda (i0: nat).(\lambda (H2: (getl i0 c0 (CHead d (Bind Abbr) -u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda -(t: T).(\lambda (H4: (subst0 i0 u t4 t)).(\lambda (H5: (eq C c0 c)).(\lambda -(w1: T).(\lambda (H6: (subst1 i v t3 w1)).(let H7 \def (eq_ind C c0 (\lambda -(c1: C).(getl i0 c1 (CHead d (Bind Abbr) u))) H2 c H5) in (eq_ind_r C c -(\lambda (c1: C).(ex2 T (\lambda (w2: T).(pr2 c1 w1 w2)) (\lambda (w2: -T).(subst1 i v t w2)))) (ex2_ind T (\lambda (w2: T).(pr0 w1 w2)) (\lambda -(w2: T).(subst1 i v t4 w2)) (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda -(w2: T).(subst1 i v t w2))) (\lambda (x: T).(\lambda (H8: (pr0 w1 -x)).(\lambda (H9: (subst1 i v t4 x)).(neq_eq_e i i0 (ex2 T (\lambda (w2: -T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t w2))) (\lambda (H10: (not -(eq nat i i0))).(ex2_ind T (\lambda (t0: T).(subst1 i v t t0)) (\lambda (t0: -T).(subst1 i0 u x t0)) (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: -T).(subst1 i v t w2))) (\lambda (x0: T).(\lambda (H11: (subst1 i v t -x0)).(\lambda (H12: (subst1 i0 u x x0)).(ex_intro2 T (\lambda (w2: T).(pr2 c -w1 w2)) (\lambda (w2: T).(subst1 i v t w2)) x0 (pr2_delta1 c d u i0 H7 w1 x -H8 x0 H12) H11)))) (subst1_confluence_neq t4 t u i0 (subst1_single i0 u t4 t -H4) x v i H9 (sym_not_eq nat i i0 H10)))) (\lambda (H10: (eq nat i i0)).(let -H11 \def (eq_ind_r nat i0 (\lambda (n: nat).(subst0 n u t4 t)) H4 i H10) in -(let H12 \def (eq_ind_r nat i0 (\lambda (n: nat).(getl n c (CHead d (Bind -Abbr) u))) H7 i H10) in (let H13 \def (eq_ind C (CHead e (Bind Abbr) v) -(\lambda (c1: C).(getl i c c1)) H (CHead d (Bind Abbr) u) (getl_mono c (CHead -e (Bind Abbr) v) i H (CHead d (Bind Abbr) u) H12)) in (let H14 \def (f_equal -C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow e | (CHead c1 _ _) \Rightarrow c1])) (CHead e (Bind Abbr) v) -(CHead d (Bind Abbr) u) (getl_mono c (CHead e (Bind Abbr) v) i H (CHead d -(Bind Abbr) u) H12)) in ((let H15 \def (f_equal C T (\lambda (e0: C).(match -e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow v | (CHead _ _ -t0) \Rightarrow t0])) (CHead e (Bind Abbr) v) (CHead d (Bind Abbr) u) -(getl_mono c (CHead e (Bind Abbr) v) i H (CHead d (Bind Abbr) u) H12)) in -(\lambda (H16: (eq C e d)).(let H17 \def (eq_ind_r T u (\lambda (t0: T).(getl -i c (CHead d (Bind Abbr) t0))) H13 v H15) in (let H18 \def (eq_ind_r T u -(\lambda (t0: T).(subst0 i t0 t4 t)) H11 v H15) in (let H19 \def (eq_ind_r C -d (\lambda (c1: C).(getl i c (CHead c1 (Bind Abbr) v))) H17 e H16) in -(ex2_ind T (\lambda (t0: T).(subst1 i v t t0)) (\lambda (t0: T).(subst1 i v x -t0)) (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t -w2))) (\lambda (x0: T).(\lambda (H20: (subst1 i v t x0)).(\lambda (H21: -(subst1 i v x x0)).(ex_intro2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: -T).(subst1 i v t w2)) x0 (pr2_delta1 c e v i H19 w1 x H8 x0 H21) H20)))) -(subst1_confluence_eq t4 t v i (subst1_single i v t4 t H18) x H9))))))) -H14)))))))))) (pr0_subst1 t3 t4 H3 v w1 i H6 v (pr0_refl v))) c0 -H5))))))))))))))) y t1 t2 H1))) H0)))))))). -(* COMMENTS -Initial nodes: 1311 -END *) - -theorem pr2_gen_cabbr: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall -(e: C).(\forall (u: T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) -\to (\forall (a0: C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d -a0 a) \to (\forall (x1: T).((subst1 d u t1 (lift (S O) d x1)) \to (ex2 T -(\lambda (x2: T).(subst1 d u t2 (lift (S O) d x2))) (\lambda (x2: T).(pr2 a -x1 x2)))))))))))))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 -t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (e: -C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u)) \to -(\forall (a0: C).((csubst1 d u c0 a0) \to (\forall (a: C).((drop (S O) d a0 -a) \to (\forall (x1: T).((subst1 d u t (lift (S O) d x1)) \to (ex2 T (\lambda -(x2: T).(subst1 d u t0 (lift (S O) d x2))) (\lambda (x2: T).(pr2 a x1 -x2)))))))))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: -T).(\lambda (H0: (pr0 t3 t4)).(\lambda (e: C).(\lambda (u: T).(\lambda (d: -nat).(\lambda (_: (getl d c0 (CHead e (Bind Abbr) u))).(\lambda (a0: -C).(\lambda (_: (csubst1 d u c0 a0)).(\lambda (a: C).(\lambda (_: (drop (S O) -d a0 a)).(\lambda (x1: T).(\lambda (H4: (subst1 d u t3 (lift (S O) d -x1))).(ex2_ind T (\lambda (w2: T).(pr0 (lift (S O) d x1) w2)) (\lambda (w2: -T).(subst1 d u t4 w2)) (ex2 T (\lambda (x2: T).(subst1 d u t4 (lift (S O) d -x2))) (\lambda (x2: T).(pr2 a x1 x2))) (\lambda (x: T).(\lambda (H5: (pr0 -(lift (S O) d x1) x)).(\lambda (H6: (subst1 d u t4 x)).(ex2_ind T (\lambda -(t5: T).(eq T x (lift (S O) d t5))) (\lambda (t5: T).(pr0 x1 t5)) (ex2 T -(\lambda (x2: T).(subst1 d u t4 (lift (S O) d x2))) (\lambda (x2: T).(pr2 a -x1 x2))) (\lambda (x0: T).(\lambda (H7: (eq T x (lift (S O) d x0))).(\lambda -(H8: (pr0 x1 x0)).(let H9 \def (eq_ind T x (\lambda (t: T).(subst1 d u t4 t)) -H6 (lift (S O) d x0) H7) in (ex_intro2 T (\lambda (x2: T).(subst1 d u t4 -(lift (S O) d x2))) (\lambda (x2: T).(pr2 a x1 x2)) x0 H9 (pr2_free a x1 x0 -H8)))))) (pr0_gen_lift x1 x (S O) d H5))))) (pr0_subst1 t3 t4 H0 u (lift (S -O) d x1) d H4 u (pr0_refl u))))))))))))))))) (\lambda (c0: C).(\lambda (d: -C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind -Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 -t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (e: -C).(\lambda (u0: T).(\lambda (d0: nat).(\lambda (H3: (getl d0 c0 (CHead e -(Bind Abbr) u0))).(\lambda (a0: C).(\lambda (H4: (csubst1 d0 u0 c0 -a0)).(\lambda (a: C).(\lambda (H5: (drop (S O) d0 a0 a)).(\lambda (x1: -T).(\lambda (H6: (subst1 d0 u0 t3 (lift (S O) d0 x1))).(ex2_ind T (\lambda -(w2: T).(pr0 (lift (S O) d0 x1) w2)) (\lambda (w2: T).(subst1 d0 u0 t4 w2)) -(ex2 T (\lambda (x2: T).(subst1 d0 u0 t (lift (S O) d0 x2))) (\lambda (x2: -T).(pr2 a x1 x2))) (\lambda (x: T).(\lambda (H7: (pr0 (lift (S O) d0 x1) -x)).(\lambda (H8: (subst1 d0 u0 t4 x)).(ex2_ind T (\lambda (t5: T).(eq T x -(lift (S O) d0 t5))) (\lambda (t5: T).(pr0 x1 t5)) (ex2 T (\lambda (x2: -T).(subst1 d0 u0 t (lift (S O) d0 x2))) (\lambda (x2: T).(pr2 a x1 x2))) -(\lambda (x0: T).(\lambda (H9: (eq T x (lift (S O) d0 x0))).(\lambda (H10: -(pr0 x1 x0)).(let H11 \def (eq_ind T x (\lambda (t0: T).(subst1 d0 u0 t4 t0)) -H8 (lift (S O) d0 x0) H9) in (lt_eq_gt_e i d0 (ex2 T (\lambda (x2: T).(subst1 -d0 u0 t (lift (S O) d0 x2))) (\lambda (x2: T).(pr2 a x1 x2))) (\lambda (H12: -(lt i d0)).(ex2_ind T (\lambda (t0: T).(subst1 d0 u0 t t0)) (\lambda (t0: -T).(subst1 i u (lift (S O) d0 x0) t0)) (ex2 T (\lambda (x2: T).(subst1 d0 u0 -t (lift (S O) d0 x2))) (\lambda (x2: T).(pr2 a x1 x2))) (\lambda (x2: -T).(\lambda (H13: (subst1 d0 u0 t x2)).(\lambda (H14: (subst1 i u (lift (S O) -d0 x0) x2)).(ex2_ind C (\lambda (e2: C).(csubst1 (minus d0 i) u0 (CHead d -(Bind Abbr) u) e2)) (\lambda (e2: C).(getl i a0 e2)) (ex2 T (\lambda (x3: -T).(subst1 d0 u0 t (lift (S O) d0 x3))) (\lambda (x3: T).(pr2 a x1 x3))) -(\lambda (x3: C).(\lambda (H15: (csubst1 (minus d0 i) u0 (CHead d (Bind Abbr) -u) x3)).(\lambda (H16: (getl i a0 x3)).(let H17 \def (eq_ind nat (minus d0 i) -(\lambda (n: nat).(csubst1 n u0 (CHead d (Bind Abbr) u) x3)) H15 (S (minus d0 -(S i))) (minus_x_Sy d0 i H12)) in (let H18 \def (csubst1_gen_head (Bind Abbr) -d x3 u u0 (minus d0 (S i)) H17) in (ex3_2_ind T C (\lambda (u2: T).(\lambda -(c2: C).(eq C x3 (CHead c2 (Bind Abbr) u2)))) (\lambda (u2: T).(\lambda (_: -C).(subst1 (minus d0 (S i)) u0 u u2))) (\lambda (_: T).(\lambda (c2: -C).(csubst1 (minus d0 (S i)) u0 d c2))) (ex2 T (\lambda (x4: T).(subst1 d0 u0 -t (lift (S O) d0 x4))) (\lambda (x4: T).(pr2 a x1 x4))) (\lambda (x4: -T).(\lambda (x5: C).(\lambda (H19: (eq C x3 (CHead x5 (Bind Abbr) -x4))).(\lambda (H20: (subst1 (minus d0 (S i)) u0 u x4)).(\lambda (_: (csubst1 -(minus d0 (S i)) u0 d x5)).(let H22 \def (eq_ind C x3 (\lambda (c1: C).(getl -i a0 c1)) H16 (CHead x5 (Bind Abbr) x4) H19) in (let H23 \def (eq_ind nat d0 -(\lambda (n: nat).(drop (S O) n a0 a)) H5 (S (plus i (minus d0 (S i)))) -(lt_plus_minus i d0 H12)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: -C).(eq T x4 (lift (S O) (minus d0 (S i)) v)))) (\lambda (v: T).(\lambda (e0: -C).(getl i a (CHead e0 (Bind Abbr) v)))) (\lambda (_: T).(\lambda (e0: -C).(drop (S O) (minus d0 (S i)) x5 e0))) (ex2 T (\lambda (x6: T).(subst1 d0 -u0 t (lift (S O) d0 x6))) (\lambda (x6: T).(pr2 a x1 x6))) (\lambda (x6: -T).(\lambda (x7: C).(\lambda (H24: (eq T x4 (lift (S O) (minus d0 (S i)) -x6))).(\lambda (H25: (getl i a (CHead x7 (Bind Abbr) x6))).(\lambda (_: (drop -(S O) (minus d0 (S i)) x5 x7)).(let H27 \def (eq_ind T x4 (\lambda (t0: -T).(subst1 (minus d0 (S i)) u0 u t0)) H20 (lift (S O) (minus d0 (S i)) x6) -H24) in (ex2_ind T (\lambda (t0: T).(subst1 i (lift (S O) (minus d0 (S i)) -x6) (lift (S O) d0 x0) t0)) (\lambda (t0: T).(subst1 (S (plus (minus d0 (S -i)) i)) u0 x2 t0)) (ex2 T (\lambda (x8: T).(subst1 d0 u0 t (lift (S O) d0 -x8))) (\lambda (x8: T).(pr2 a x1 x8))) (\lambda (x8: T).(\lambda (H28: -(subst1 i (lift (S O) (minus d0 (S i)) x6) (lift (S O) d0 x0) x8)).(\lambda -(H29: (subst1 (S (plus (minus d0 (S i)) i)) u0 x2 x8)).(let H30 \def (eq_ind -nat d0 (\lambda (n: nat).(subst1 i (lift (S O) (minus d0 (S i)) x6) (lift (S -O) n x0) x8)) H28 (S (plus i (minus d0 (S i)))) (lt_plus_minus i d0 H12)) in -(ex2_ind T (\lambda (t5: T).(eq T x8 (lift (S O) (S (plus i (minus d0 (S -i)))) t5))) (\lambda (t5: T).(subst1 i x6 x0 t5)) (ex2 T (\lambda (x9: -T).(subst1 d0 u0 t (lift (S O) d0 x9))) (\lambda (x9: T).(pr2 a x1 x9))) -(\lambda (x9: T).(\lambda (H31: (eq T x8 (lift (S O) (S (plus i (minus d0 (S -i)))) x9))).(\lambda (H32: (subst1 i x6 x0 x9)).(let H33 \def (eq_ind T x8 -(\lambda (t0: T).(subst1 (S (plus (minus d0 (S i)) i)) u0 x2 t0)) H29 (lift -(S O) (S (plus i (minus d0 (S i)))) x9) H31) in (let H34 \def (eq_ind_r nat -(S (plus i (minus d0 (S i)))) (\lambda (n: nat).(subst1 (S (plus (minus d0 (S -i)) i)) u0 x2 (lift (S O) n x9))) H33 d0 (lt_plus_minus i d0 H12)) in (let -H35 \def (eq_ind_r nat (S (plus (minus d0 (S i)) i)) (\lambda (n: -nat).(subst1 n u0 x2 (lift (S O) d0 x9))) H34 d0 (lt_plus_minus_r i d0 H12)) -in (ex_intro2 T (\lambda (x10: T).(subst1 d0 u0 t (lift (S O) d0 x10))) -(\lambda (x10: T).(pr2 a x1 x10)) x9 (subst1_trans x2 t u0 d0 H13 (lift (S O) -d0 x9) H35) (pr2_delta1 a x7 x6 i H25 x1 x0 H10 x9 H32)))))))) -(subst1_gen_lift_lt x6 x0 x8 i (S O) (minus d0 (S i)) H30)))))) -(subst1_subst1_back (lift (S O) d0 x0) x2 u i H14 (lift (S O) (minus d0 (S -i)) x6) u0 (minus d0 (S i)) H27)))))))) (getl_drop_conf_lt Abbr a0 x5 x4 i -H22 a (S O) (minus d0 (S i)) H23))))))))) H18)))))) (csubst1_getl_lt d0 i H12 -c0 a0 u0 H4 (CHead d (Bind Abbr) u) H0))))) (subst1_confluence_neq t4 t u i -(subst1_single i u t4 t H2) (lift (S O) d0 x0) u0 d0 H11 (lt_neq i d0 H12)))) -(\lambda (H12: (eq nat i d0)).(let H13 \def (eq_ind_r nat d0 (\lambda (n: -nat).(subst1 n u0 t4 (lift (S O) n x0))) H11 i H12) in (let H14 \def -(eq_ind_r nat d0 (\lambda (n: nat).(drop (S O) n a0 a)) H5 i H12) in (let H15 -\def (eq_ind_r nat d0 (\lambda (n: nat).(csubst1 n u0 c0 a0)) H4 i H12) in -(let H16 \def (eq_ind_r nat d0 (\lambda (n: nat).(getl n c0 (CHead e (Bind -Abbr) u0))) H3 i H12) in (eq_ind nat i (\lambda (n: nat).(ex2 T (\lambda (x2: -T).(subst1 n u0 t (lift (S O) n x2))) (\lambda (x2: T).(pr2 a x1 x2)))) (let -H17 \def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c1: C).(getl i c0 c1)) -H0 (CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead -e (Bind Abbr) u0) H16)) in (let H18 \def (f_equal C C (\lambda (e0: C).(match -e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ -_) \Rightarrow c1])) (CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0) -(getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead e (Bind Abbr) u0) H16)) in -((let H19 \def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda -(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) -(CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d (Bind -Abbr) u) i H0 (CHead e (Bind Abbr) u0) H16)) in (\lambda (H20: (eq C d -e)).(let H21 \def (eq_ind_r T u0 (\lambda (t0: T).(getl i c0 (CHead e (Bind -Abbr) t0))) H17 u H19) in (let H22 \def (eq_ind_r T u0 (\lambda (t0: -T).(subst1 i t0 t4 (lift (S O) i x0))) H13 u H19) in (let H23 \def (eq_ind_r -T u0 (\lambda (t0: T).(csubst1 i t0 c0 a0)) H15 u H19) in (eq_ind T u -(\lambda (t0: T).(ex2 T (\lambda (x2: T).(subst1 i t0 t (lift (S O) i x2))) -(\lambda (x2: T).(pr2 a x1 x2)))) (let H24 \def (eq_ind_r C e (\lambda (c1: -C).(getl i c0 (CHead c1 (Bind Abbr) u))) H21 d H20) in (ex2_ind T (\lambda -(t0: T).(subst1 i u t t0)) (\lambda (t0: T).(subst1 i u (lift (S O) i x0) -t0)) (ex2 T (\lambda (x2: T).(subst1 i u t (lift (S O) i x2))) (\lambda (x2: -T).(pr2 a x1 x2))) (\lambda (x2: T).(\lambda (H25: (subst1 i u t -x2)).(\lambda (H26: (subst1 i u (lift (S O) i x0) x2)).(let H27 \def (eq_ind -T x2 (\lambda (t0: T).(subst1 i u t t0)) H25 (lift (S O) i x0) -(subst1_gen_lift_eq x0 u x2 (S O) i i (le_n i) (eq_ind_r nat (plus (S O) i) -(\lambda (n: nat).(lt i n)) (le_n (plus (S O) i)) (plus i (S O)) (plus_sym i -(S O))) H26)) in (ex_intro2 T (\lambda (x3: T).(subst1 i u t (lift (S O) i -x3))) (\lambda (x3: T).(pr2 a x1 x3)) x0 H27 (pr2_free a x1 x0 H10)))))) -(subst1_confluence_eq t4 t u i (subst1_single i u t4 t H2) (lift (S O) i x0) -H22))) u0 H19)))))) H18))) d0 H12)))))) (\lambda (H12: (lt d0 i)).(ex2_ind T -(\lambda (t0: T).(subst1 d0 u0 t t0)) (\lambda (t0: T).(subst1 i u (lift (S -O) d0 x0) t0)) (ex2 T (\lambda (x2: T).(subst1 d0 u0 t (lift (S O) d0 x2))) -(\lambda (x2: T).(pr2 a x1 x2))) (\lambda (x2: T).(\lambda (H13: (subst1 d0 -u0 t x2)).(\lambda (H14: (subst1 i u (lift (S O) d0 x0) x2)).(ex2_ind T -(\lambda (t5: T).(eq T x2 (lift (S O) d0 t5))) (\lambda (t5: T).(subst1 -(minus i (S O)) u x0 t5)) (ex2 T (\lambda (x3: T).(subst1 d0 u0 t (lift (S O) -d0 x3))) (\lambda (x3: T).(pr2 a x1 x3))) (\lambda (x3: T).(\lambda (H15: (eq -T x2 (lift (S O) d0 x3))).(\lambda (H16: (subst1 (minus i (S O)) u x0 -x3)).(let H17 \def (eq_ind T x2 (\lambda (t0: T).(subst1 d0 u0 t t0)) H13 -(lift (S O) d0 x3) H15) in (ex_intro2 T (\lambda (x4: T).(subst1 d0 u0 t -(lift (S O) d0 x4))) (\lambda (x4: T).(pr2 a x1 x4)) x3 H17 (pr2_delta1 a d u -(minus i (S O)) (getl_drop_conf_ge i (CHead d (Bind Abbr) u) a0 -(csubst1_getl_ge d0 i (le_S_n d0 i (le_S (S d0) i H12)) c0 a0 u0 H4 (CHead d -(Bind Abbr) u) H0) a (S O) d0 H5 (eq_ind_r nat (plus (S O) d0) (\lambda (n: -nat).(le n i)) H12 (plus d0 (S O)) (plus_sym d0 (S O)))) x1 x0 H10 x3 -H16)))))) (subst1_gen_lift_ge u x0 x2 i (S O) d0 H14 (eq_ind_r nat (plus (S -O) d0) (\lambda (n: nat).(le n i)) H12 (plus d0 (S O)) (plus_sym d0 (S -O)))))))) (subst1_confluence_neq t4 t u i (subst1_single i u t4 t H2) (lift -(S O) d0 x0) u0 d0 H11 (sym_not_eq nat d0 i (lt_neq d0 i H12)))))))))) -(pr0_gen_lift x1 x (S O) d0 H7))))) (pr0_subst1 t3 t4 H1 u0 (lift (S O) d0 -x1) d0 H6 u0 (pr0_refl u0))))))))))))))))))))))) c t1 t2 H)))). -(* COMMENTS -Initial nodes: 3757 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/defs.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/defs.ma deleted file mode 100644 index 9619ee1d7..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/defs.ma +++ /dev/null @@ -1,23 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr2/defs.ma". - -inductive pr3 (c: C): T \to (T \to Prop) \def -| pr3_refl: \forall (t: T).(pr3 c t t) -| pr3_sing: \forall (t2: T).(\forall (t1: T).((pr2 c t1 t2) \to (\forall (t3: -T).((pr3 c t2 t3) \to (pr3 c t1 t3))))). - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/fwd.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/fwd.ma deleted file mode 100644 index 726c420f0..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/fwd.ma +++ /dev/null @@ -1,1604 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr3/props.ma". - -include "Basic-1/pr2/fwd.ma". - -theorem pr3_gen_sort: - \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr3 c (TSort n) x) \to -(eq T x (TSort n))))) -\def - \lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr3 c (TSort -n) x)).(insert_eq T (TSort n) (\lambda (t: T).(pr3 c t x)) (\lambda (t: -T).(eq T x t)) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(pr3_ind c (\lambda -(t: T).(\lambda (t0: T).((eq T t (TSort n)) \to (eq T t0 t)))) (\lambda (t: -T).(\lambda (_: (eq T t (TSort n))).(refl_equal T t))) (\lambda (t2: -T).(\lambda (t1: T).(\lambda (H1: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda -(_: (pr3 c t2 t3)).(\lambda (H3: (((eq T t2 (TSort n)) \to (eq T t3 -t2)))).(\lambda (H4: (eq T t1 (TSort n))).(let H5 \def (eq_ind T t1 (\lambda -(t: T).(pr2 c t t2)) H1 (TSort n) H4) in (eq_ind_r T (TSort n) (\lambda (t: -T).(eq T t3 t)) (let H6 \def (eq_ind T t2 (\lambda (t: T).((eq T t (TSort n)) -\to (eq T t3 t))) H3 (TSort n) (pr2_gen_sort c t2 n H5)) in (H6 (refl_equal T -(TSort n)))) t1 H4))))))))) y x H0))) H)))). -(* COMMENTS -Initial nodes: 253 -END *) - -theorem pr3_gen_abst: - \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c -(THead (Bind Abst) u1 t1) x) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 -c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: -T).(pr3 (CHead c (Bind b) u) t1 t2)))))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda -(H: (pr3 c (THead (Bind Abst) u1 t1) x)).(insert_eq T (THead (Bind Abst) u1 -t1) (\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t1 t2))))))) (\lambda (y: -T).(\lambda (H0: (pr3 c y x)).(unintro T t1 (\lambda (t: T).((eq T y (THead -(Bind Abst) u1 t)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x -(THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) -(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead -c (Bind b) u) t t2)))))))) (unintro T u1 (\lambda (t: T).(\forall (x0: -T).((eq T y (THead (Bind Abst) t x0)) \to (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_: T).(\lambda (t2: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x0 t2))))))))) (pr3_ind c -(\lambda (t: T).(\lambda (t0: T).(\forall (x0: T).(\forall (x1: T).((eq T t -(THead (Bind Abst) x0 x1)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T t0 (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall -(u: T).(pr3 (CHead c (Bind b) u) x1 t2))))))))))) (\lambda (t: T).(\lambda -(x0: T).(\lambda (x1: T).(\lambda (H1: (eq T t (THead (Bind Abst) x0 -x1))).(ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T t (THead (Bind -Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: -T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -x1 t2))))) x0 x1 H1 (pr3_refl c x0) (\lambda (b: B).(\lambda (u: T).(pr3_refl -(CHead c (Bind b) u) x1)))))))) (\lambda (t2: T).(\lambda (t3: T).(\lambda -(H1: (pr2 c t3 t2)).(\lambda (t4: T).(\lambda (_: (pr3 c t2 t4)).(\lambda -(H3: ((\forall (x0: T).(\forall (x1: T).((eq T t2 (THead (Bind Abst) x0 x1)) -\to (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst) -u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: -T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -x1 t5))))))))))).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t3 -(THead (Bind Abst) x0 x1))).(let H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c -t t2)) H1 (THead (Bind Abst) x0 x1) H4) in (let H6 \def (pr2_gen_abst c x0 x1 -t2 H5) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead -(Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead -c (Bind b) u) x1 t5))))) (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T -t4 (THead (Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 -u2))) (\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 -(CHead c (Bind b) u) x1 t5)))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda -(H7: (eq T t2 (THead (Bind Abst) x2 x3))).(\lambda (H8: (pr2 c x0 -x2)).(\lambda (H9: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) -x1 x3))))).(let H10 \def (eq_ind T t2 (\lambda (t: T).(\forall (x4: -T).(\forall (x5: T).((eq T t (THead (Bind Abst) x4 x5)) \to (ex3_2 T T -(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x4 u2))) (\lambda (_: T).(\lambda -(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x5 -t5)))))))))) H3 (THead (Bind Abst) x2 x3) H7) in (let H11 \def (H10 x2 x3 -(refl_equal T (THead (Bind Abst) x2 x3))) in (ex3_2_ind T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 t5))))) (ex3_2 T T -(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) -(\lambda (x4: T).(\lambda (x5: T).(\lambda (H12: (eq T t4 (THead (Bind Abst) -x4 x5))).(\lambda (H13: (pr3 c x2 x4)).(\lambda (H14: ((\forall (b: -B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 x5))))).(ex3_2_intro T T -(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5))))) -x4 x5 H12 (pr3_sing c x2 x0 H8 x4 H13) (\lambda (b: B).(\lambda (u: -T).(pr3_sing (CHead c (Bind b) u) x3 x1 (H9 b u) x5 (H14 b u)))))))))) -H11)))))))) H6)))))))))))) y x H0))))) H))))). -(* COMMENTS -Initial nodes: 1261 -END *) - -theorem pr3_gen_cast: - \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c -(THead (Flat Cast) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda -(t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t1 t2)))) (pr3 c -t1 x)))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda -(H: (pr3 c (THead (Flat Cast) u1 t1) x)).(insert_eq T (THead (Flat Cast) u1 -t1) (\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t1 -t2)))) (pr3 c t1 x))) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(unintro T -t1 (\lambda (t: T).((eq T y (THead (Flat Cast) u1 t)) \to (or (ex3_2 T T -(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda -(t2: T).(pr3 c t t2)))) (pr3 c t x)))) (unintro T u1 (\lambda (t: T).(\forall -(x0: T).((eq T y (THead (Flat Cast) t x0)) \to (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c x0 -t2)))) (pr3 c x0 x))))) (pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall -(x0: T).(\forall (x1: T).((eq T t (THead (Flat Cast) x0 x1)) \to (or (ex3_2 T -T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Flat Cast) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t2: T).(pr3 c x1 t2)))) (pr3 c x1 t0))))))) (\lambda (t: T).(\lambda (x0: -T).(\lambda (x1: T).(\lambda (H1: (eq T t (THead (Flat Cast) x0 -x1))).(eq_ind_r T (THead (Flat Cast) x0 x1) (\lambda (t0: T).(or (ex3_2 T T -(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Flat Cast) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t2: T).(pr3 c x1 t2)))) (pr3 c x1 t0))) (or_introl (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T (THead (Flat Cast) x0 x1) (THead (Flat Cast) u2 -t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: -T).(\lambda (t2: T).(pr3 c x1 t2)))) (pr3 c x1 (THead (Flat Cast) x0 x1)) -(ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Cast) -x0 x1) (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c -x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c x1 t2))) x0 x1 (refl_equal T -(THead (Flat Cast) x0 x1)) (pr3_refl c x0) (pr3_refl c x1))) t H1))))) -(\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c t3 t2)).(\lambda (t4: -T).(\lambda (H2: (pr3 c t2 t4)).(\lambda (H3: ((\forall (x0: T).(\forall (x1: -T).((eq T t2 (THead (Flat Cast) x0 x1)) \to (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 -t5)))) (pr3 c x1 t4))))))).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: -(eq T t3 (THead (Flat Cast) x0 x1))).(let H5 \def (eq_ind T t3 (\lambda (t: -T).(pr2 c t t2)) H1 (THead (Flat Cast) x0 x1) H4) in (let H6 \def -(pr2_gen_cast c x0 x1 t2 H5) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda -(t5: T).(eq T t2 (THead (Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr2 c x1 t5)))) (pr2 c -x1 t2) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat -Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: -T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4)) (\lambda (H7: (ex3_2 T T -(\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Flat Cast) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda -(t5: T).(pr2 c x1 t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: -T).(eq T t2 (THead (Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr2 c x1 t5))) (or -(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 -t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: -T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4)) (\lambda (x2: T).(\lambda -(x3: T).(\lambda (H8: (eq T t2 (THead (Flat Cast) x2 x3))).(\lambda (H9: (pr2 -c x0 x2)).(\lambda (H10: (pr2 c x1 x3)).(let H11 \def (eq_ind T t2 (\lambda -(t: T).(\forall (x4: T).(\forall (x5: T).((eq T t (THead (Flat Cast) x4 x5)) -\to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat -Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x4 u2))) (\lambda (_: -T).(\lambda (t5: T).(pr3 c x5 t5)))) (pr3 c x5 t4)))))) H3 (THead (Flat Cast) -x2 x3) H8) in (let H12 \def (H11 x2 x3 (refl_equal T (THead (Flat Cast) x2 -x3))) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead -(Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr3 c x3 t5)))) (pr3 c x3 t4) (or (ex3_2 T -T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4)) (\lambda (H13: (ex3_2 T T (\lambda -(u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x3 -t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead -(Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr3 c x3 t5))) (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 -t5)))) (pr3 c x1 t4)) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H14: (eq T -t4 (THead (Flat Cast) x4 x5))).(\lambda (H15: (pr3 c x2 x4)).(\lambda (H16: -(pr3 c x3 x5)).(eq_ind_r T (THead (Flat Cast) x4 x5) (\lambda (t: T).(or -(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t (THead (Flat Cast) u2 -t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: -T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t))) (or_introl (ex3_2 T T -(\lambda (u2: T).(\lambda (t5: T).(eq T (THead (Flat Cast) x4 x5) (THead -(Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 (THead (Flat -Cast) x4 x5)) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t5: T).(eq T (THead -(Flat Cast) x4 x5) (THead (Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5))) x4 x5 -(refl_equal T (THead (Flat Cast) x4 x5)) (pr3_sing c x2 x0 H9 x4 H15) -(pr3_sing c x3 x1 H10 x5 H16))) t4 H14)))))) H13)) (\lambda (H13: (pr3 c x3 -t4)).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead -(Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4) (pr3_sing c -x3 x1 H10 t4 H13))) H12)))))))) H7)) (\lambda (H7: (pr2 c x1 t2)).(or_intror -(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 -t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: -T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4) (pr3_sing c t2 x1 H7 t4 -H2))) H6)))))))))))) y x H0))))) H))))). -(* COMMENTS -Initial nodes: 2001 -END *) - -theorem pr3_gen_lift: - \forall (c: C).(\forall (t1: T).(\forall (x: T).(\forall (h: nat).(\forall -(d: nat).((pr3 c (lift h d t1) x) \to (\forall (e: C).((drop h d c e) \to -(ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr3 e t1 -t2)))))))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (x: T).(\lambda (h: nat).(\lambda -(d: nat).(\lambda (H: (pr3 c (lift h d t1) x)).(insert_eq T (lift h d t1) -(\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(\forall (e: C).((drop h d c e) -\to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr3 e -t1 t2)))))) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(unintro T t1 (\lambda -(t: T).((eq T y (lift h d t)) \to (\forall (e: C).((drop h d c e) \to (ex2 T -(\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr3 e t t2))))))) -(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (x0: T).((eq T t (lift h -d x0)) \to (\forall (e: C).((drop h d c e) \to (ex2 T (\lambda (t2: T).(eq T -t0 (lift h d t2))) (\lambda (t2: T).(pr3 e x0 t2))))))))) (\lambda (t: -T).(\lambda (x0: T).(\lambda (H1: (eq T t (lift h d x0))).(\lambda (e: -C).(\lambda (_: (drop h d c e)).(ex_intro2 T (\lambda (t2: T).(eq T t (lift h -d t2))) (\lambda (t2: T).(pr3 e x0 t2)) x0 H1 (pr3_refl e x0))))))) (\lambda -(t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c t3 t2)).(\lambda (t4: -T).(\lambda (_: (pr3 c t2 t4)).(\lambda (H3: ((\forall (x0: T).((eq T t2 -(lift h d x0)) \to (\forall (e: C).((drop h d c e) \to (ex2 T (\lambda (t5: -T).(eq T t4 (lift h d t5))) (\lambda (t5: T).(pr3 e x0 t5))))))))).(\lambda -(x0: T).(\lambda (H4: (eq T t3 (lift h d x0))).(\lambda (e: C).(\lambda (H5: -(drop h d c e)).(let H6 \def (eq_ind T t3 (\lambda (t: T).(pr2 c t t2)) H1 -(lift h d x0) H4) in (let H7 \def (pr2_gen_lift c x0 t2 h d H6 e H5) in -(ex2_ind T (\lambda (t5: T).(eq T t2 (lift h d t5))) (\lambda (t5: T).(pr2 e -x0 t5)) (ex2 T (\lambda (t5: T).(eq T t4 (lift h d t5))) (\lambda (t5: -T).(pr3 e x0 t5))) (\lambda (x1: T).(\lambda (H8: (eq T t2 (lift h d -x1))).(\lambda (H9: (pr2 e x0 x1)).(ex2_ind T (\lambda (t5: T).(eq T t4 (lift -h d t5))) (\lambda (t5: T).(pr3 e x1 t5)) (ex2 T (\lambda (t5: T).(eq T t4 -(lift h d t5))) (\lambda (t5: T).(pr3 e x0 t5))) (\lambda (x2: T).(\lambda -(H10: (eq T t4 (lift h d x2))).(\lambda (H11: (pr3 e x1 x2)).(ex_intro2 T -(\lambda (t5: T).(eq T t4 (lift h d t5))) (\lambda (t5: T).(pr3 e x0 t5)) x2 -H10 (pr3_sing e x1 x0 H9 x2 H11))))) (H3 x1 H8 e H5))))) H7))))))))))))) y x -H0)))) H)))))). -(* COMMENTS -Initial nodes: 689 -END *) - -theorem pr3_gen_lref: - \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr3 c (TLRef n) x) \to -(or (eq T x (TLRef n)) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda -(_: T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: -T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(v: T).(eq T x (lift (S n) O v)))))))))) -\def - \lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr3 c (TLRef -n) x)).(insert_eq T (TLRef n) (\lambda (t: T).(pr3 c t x)) (\lambda (t: -T).(or (eq T x t) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: -T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: -T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(v: T).(eq T x (lift (S n) O v)))))))) (\lambda (y: T).(\lambda (H0: (pr3 c y -x)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).((eq T t (TLRef n)) \to (or -(eq T t0 t) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: -T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: -T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(v: T).(eq T t0 (lift (S n) O v)))))))))) (\lambda (t: T).(\lambda (_: (eq T -t (TLRef n))).(or_introl (eq T t t) (ex3_3 C T T (\lambda (d: C).(\lambda (u: -T).(\lambda (_: T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: -C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda -(_: T).(\lambda (v: T).(eq T t (lift (S n) O v)))))) (refl_equal T t)))) -(\lambda (t2: T).(\lambda (t1: T).(\lambda (H1: (pr2 c t1 t2)).(\lambda (t3: -T).(\lambda (H2: (pr3 c t2 t3)).(\lambda (H3: (((eq T t2 (TLRef n)) \to (or -(eq T t3 t2) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: -T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: -T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(v: T).(eq T t3 (lift (S n) O v)))))))))).(\lambda (H4: (eq T t1 (TLRef -n))).(let H5 \def (eq_ind T t1 (\lambda (t: T).(pr2 c t t2)) H1 (TLRef n) H4) -in (eq_ind_r T (TLRef n) (\lambda (t: T).(or (eq T t3 t) (ex3_3 C T T -(\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead d (Bind -Abbr) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T t3 (lift (S n) O -v)))))))) (let H6 \def (pr2_gen_lref c t2 n H5) in (or_ind (eq T t2 (TLRef -n)) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) -u)))) (\lambda (_: C).(\lambda (u: T).(eq T t2 (lift (S n) O u))))) (or (eq T -t3 (TLRef n)) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: -T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: -T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(v: T).(eq T t3 (lift (S n) O v))))))) (\lambda (H7: (eq T t2 (TLRef -n))).(let H8 \def (eq_ind T t2 (\lambda (t: T).((eq T t (TLRef n)) \to (or -(eq T t3 t) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: -T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: -T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(v: T).(eq T t3 (lift (S n) O v))))))))) H3 (TLRef n) H7) in (let H9 \def -(eq_ind T t2 (\lambda (t: T).(pr3 c t t3)) H2 (TLRef n) H7) in (H8 -(refl_equal T (TLRef n)))))) (\lambda (H7: (ex2_2 C T (\lambda (d: -C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda (_: -C).(\lambda (u: T).(eq T t2 (lift (S n) O u)))))).(ex2_2_ind C T (\lambda (d: -C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda (_: -C).(\lambda (u: T).(eq T t2 (lift (S n) O u)))) (or (eq T t3 (TLRef n)) -(ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead -d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (v: T).(pr3 d u -v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T t3 (lift (S n) O -v))))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H8: (getl n c (CHead x0 -(Bind Abbr) x1))).(\lambda (H9: (eq T t2 (lift (S n) O x1))).(let H10 \def -(eq_ind T t2 (\lambda (t: T).((eq T t (TLRef n)) \to (or (eq T t3 t) (ex3_3 C -T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead d (Bind -Abbr) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T t3 (lift (S n) O -v))))))))) H3 (lift (S n) O x1) H9) in (let H11 \def (eq_ind T t2 (\lambda -(t: T).(pr3 c t t3)) H2 (lift (S n) O x1) H9) in (let H12 \def (pr3_gen_lift -c x1 t3 (S n) O H11 x0 (getl_drop Abbr c x0 x1 n H8)) in (ex2_ind T (\lambda -(t4: T).(eq T t3 (lift (S n) O t4))) (\lambda (t4: T).(pr3 x0 x1 t4)) (or (eq -T t3 (TLRef n)) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: -T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: -T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(v: T).(eq T t3 (lift (S n) O v))))))) (\lambda (x2: T).(\lambda (H13: (eq T -t3 (lift (S n) O x2))).(\lambda (H14: (pr3 x0 x1 x2)).(or_intror (eq T t3 -(TLRef n)) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(getl -n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (v: -T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T t3 -(lift (S n) O v)))))) (ex3_3_intro C T T (\lambda (d: C).(\lambda (u: -T).(\lambda (_: T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: -C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda -(_: T).(\lambda (v: T).(eq T t3 (lift (S n) O v))))) x0 x1 x2 H8 H14 H13))))) -H12)))))))) H7)) H6)) t1 H4))))))))) y x H0))) H)))). -(* COMMENTS -Initial nodes: 1515 -END *) - -theorem pr3_gen_void: - \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c -(THead (Bind Void) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda -(t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall -(u: T).(pr3 (CHead c (Bind b) u) t1 t2)))))) (pr3 (CHead c (Bind Void) u1) t1 -(lift (S O) O x))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda -(H: (pr3 c (THead (Bind Void) u1 t1) x)).(insert_eq T (THead (Bind Void) u1 -t1) (\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t1 t2)))))) (pr3 (CHead c -(Bind Void) u1) t1 (lift (S O) O x)))) (\lambda (y: T).(\lambda (H0: (pr3 c y -x)).(unintro T t1 (\lambda (t: T).((eq T y (THead (Bind Void) u1 t)) \to (or -(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 -t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: -T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -t t2)))))) (pr3 (CHead c (Bind Void) u1) t (lift (S O) O x))))) (unintro T u1 -(\lambda (t: T).(\forall (x0: T).((eq T y (THead (Bind Void) t x0)) \to (or -(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 -t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_: -T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -x0 t2)))))) (pr3 (CHead c (Bind Void) t) x0 (lift (S O) O x)))))) (pr3_ind c -(\lambda (t: T).(\lambda (t0: T).(\forall (x0: T).(\forall (x1: T).((eq T t -(THead (Bind Void) x0 x1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T t0 (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall -(u: T).(pr3 (CHead c (Bind b) u) x1 t2)))))) (pr3 (CHead c (Bind Void) x0) x1 -(lift (S O) O t0)))))))) (\lambda (t: T).(\lambda (x0: T).(\lambda (x1: -T).(\lambda (H1: (eq T t (THead (Bind Void) x0 x1))).(eq_ind_r T (THead (Bind -Void) x0 x1) (\lambda (t0: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T t0 (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall -(u: T).(pr3 (CHead c (Bind b) u) x1 t2)))))) (pr3 (CHead c (Bind Void) x0) x1 -(lift (S O) O t0)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T (THead (Bind Void) x0 x1) (THead (Bind Void) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t2)))))) (pr3 (CHead c -(Bind Void) x0) x1 (lift (S O) O (THead (Bind Void) x0 x1))) (ex3_2_intro T T -(\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Bind Void) x0 x1) (THead -(Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) -(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead -c (Bind b) u) x1 t2))))) x0 x1 (refl_equal T (THead (Bind Void) x0 x1)) -(pr3_refl c x0) (\lambda (b: B).(\lambda (u: T).(pr3_refl (CHead c (Bind b) -u) x1))))) t H1))))) (\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c -t3 t2)).(\lambda (t4: T).(\lambda (H2: (pr3 c t2 t4)).(\lambda (H3: ((\forall -(x0: T).(\forall (x1: T).((eq T t2 (THead (Bind Void) x0 x1)) \to (or (ex3_2 -T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) -(pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4)))))))).(\lambda (x0: -T).(\lambda (x1: T).(\lambda (H4: (eq T t3 (THead (Bind Void) x0 x1))).(let -H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c t t2)) H1 (THead (Bind Void) x0 -x1) H4) in (let H6 \def (pr2_gen_void c x0 x1 t2 H5) in (or_ind (ex3_2 T T -(\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind Void) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda -(t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 t5)))))) -(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 (lift (S O) O -t2)))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind -Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: -T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4))) (\lambda -(H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind Void) -u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: -T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) -x1 t5))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead -(Bind Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead -c (Bind b) u) x1 t5))))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq -T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 -u2))) (\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 -(CHead c (Bind b) u) x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) -O t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H8: (eq T t2 (THead (Bind -Void) x2 x3))).(\lambda (H9: (pr2 c x0 x2)).(\lambda (H10: ((\forall (b: -B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 x3))))).(let H11 \def (eq_ind -T t2 (\lambda (t: T).(\forall (x4: T).(\forall (x5: T).((eq T t (THead (Bind -Void) x4 x5)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 -(THead (Bind Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x4 u2))) -(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead -c (Bind b) u) x5 t5)))))) (pr3 (CHead c (Bind Void) x4) x5 (lift (S O) O -t4))))))) H3 (THead (Bind Void) x2 x3) H8) in (let H12 \def (H11 x2 x3 -(refl_equal T (THead (Bind Void) x2 x3))) in (or_ind (ex3_2 T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 t5)))))) (pr3 (CHead c -(Bind Void) x2) x3 (lift (S O) O t4)) (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) (pr3 (CHead c -(Bind Void) x0) x1 (lift (S O) O t4))) (\lambda (H13: (ex3_2 T T (\lambda -(u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 t5))))))).(ex3_2_ind T T -(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda -(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 t5))))) -(or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) -u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: -T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4))) (\lambda -(x4: T).(\lambda (x5: T).(\lambda (H14: (eq T t4 (THead (Bind Void) x4 -x5))).(\lambda (H15: (pr3 c x2 x4)).(\lambda (H16: ((\forall (b: B).(\forall -(u: T).(pr3 (CHead c (Bind b) u) x3 x5))))).(or_introl (ex3_2 T T (\lambda -(u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) (pr3 (CHead c -(Bind Void) x0) x1 (lift (S O) O t4)) (ex3_2_intro T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5))))) x4 x5 H14 -(pr3_sing c x2 x0 H9 x4 H15) (\lambda (b: B).(\lambda (u: T).(pr3_sing (CHead -c (Bind b) u) x3 x1 (H10 b u) x5 (H16 b u))))))))))) H13)) (\lambda (H13: -(pr3 (CHead c (Bind Void) x2) x3 (lift (S O) O t4))).(or_intror (ex3_2 T T -(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) -(pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4)) (pr3_sing (CHead c (Bind -Void) x0) x3 x1 (H10 Void x0) (lift (S O) O t4) (pr3_pr2_pr3_t c x2 x3 (lift -(S O) O t4) (Bind Void) H13 x0 H9)))) H12)))))))) H7)) (\lambda (H7: -((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 (lift (S O) O -t2)))))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 -(THead (Bind Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead -c (Bind b) u) x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4)) -(pr3_sing (CHead c (Bind Void) x0) (lift (S O) O t2) x1 (H7 Void x0) (lift (S -O) O t4) (pr3_lift (CHead c (Bind Void) x0) c (S O) O (drop_drop (Bind Void) -O c c (drop_refl c) x0) t2 t4 H2)))) H6)))))))))))) y x H0))))) H))))). -(* COMMENTS -Initial nodes: 2645 -END *) - -theorem pr3_gen_abbr: - \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c -(THead (Bind Abbr) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda -(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) -u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O x))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda -(H: (pr3 c (THead (Bind Abbr) u1 t1) x)).(insert_eq T (THead (Bind Abbr) u1 -t1) (\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 -(CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S -O) O x)))) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(unintro T t1 (\lambda -(t: T).((eq T y (THead (Bind Abbr) u1 t)) \to (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 -(CHead c (Bind Abbr) u1) t t2)))) (pr3 (CHead c (Bind Abbr) u1) t (lift (S O) -O x))))) (unintro T u1 (\lambda (t: T).(\forall (x0: T).((eq T y (THead (Bind -Abbr) t x0)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x -(THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) -(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) t) x0 t2)))) (pr3 -(CHead c (Bind Abbr) t) x0 (lift (S O) O x)))))) (pr3_ind c (\lambda (t: -T).(\lambda (t0: T).(\forall (x0: T).(\forall (x1: T).((eq T t (THead (Bind -Abbr) x0 x1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 -(THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) -(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) x0) x1 t2)))) (pr3 -(CHead c (Bind Abbr) x0) x1 (lift (S O) O t0)))))))) (\lambda (t: T).(\lambda -(x0: T).(\lambda (x1: T).(\lambda (H1: (eq T t (THead (Bind Abbr) x0 -x1))).(eq_ind_r T (THead (Bind Abbr) x0 x1) (\lambda (t0: T).(or (ex3_2 T T -(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t2: T).(pr3 (CHead c (Bind Abbr) x0) x1 t2)))) (pr3 (CHead c (Bind Abbr) x0) -x1 (lift (S O) O t0)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 -(CHead c (Bind Abbr) x0) x1 t2)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S -O) O (THead (Bind Abbr) x0 x1))) (ex3_2_intro T T (\lambda (u2: T).(\lambda -(t2: T).(eq T (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 t2)))) (\lambda -(u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 -(CHead c (Bind Abbr) x0) x1 t2))) x0 x1 (refl_equal T (THead (Bind Abbr) x0 -x1)) (pr3_refl c x0) (pr3_refl (CHead c (Bind Abbr) x0) x1))) t H1))))) -(\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c t3 t2)).(\lambda (t4: -T).(\lambda (H2: (pr3 c t2 t4)).(\lambda (H3: ((\forall (x0: T).(\forall (x1: -T).((eq T t2 (THead (Bind Abbr) x0 x1)) \to (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S -O) O t4)))))))).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t3 -(THead (Bind Abbr) x0 x1))).(let H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c -t t2)) H1 (THead (Bind Abbr) x0 x1) H4) in (let H6 \def (pr2_gen_abbr c x0 x1 -t2 H5) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 -(THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 -(CHead c (Bind b) u) x1 t5))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: -T).(pr2 (CHead c (Bind Abbr) u) x1 t5))) (ex3_2 T T (\lambda (y0: T).(\lambda -(_: T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: -T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) -z t5)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 -(lift (S O) O t2)))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T -t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 -u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 -t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (H7: -(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind Abbr) u2 -t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: -T).(\lambda (t5: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind -b) u) x1 t5))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead -c (Bind Abbr) u) x1 t5))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 -(CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 -z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) z -t5))))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead -(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 -(CHead c (Bind b) u) x1 t5))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: -T).(pr2 (CHead c (Bind Abbr) u) x1 t5))) (ex3_2 T T (\lambda (y0: T).(\lambda -(_: T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: -T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) -z t5))))))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead -(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 -(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (x2: T).(\lambda -(x3: T).(\lambda (H8: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (H9: (pr2 -c x0 x2)).(\lambda (H10: (or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c -(Bind b) u) x1 x3))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 -(CHead c (Bind Abbr) u) x1 x3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: -T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: -T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) -z x3)))))).(or3_ind (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) -x1 x3))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead c -(Bind Abbr) u) x1 x3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 -(CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 -z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) z x3)))) -(or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) -u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: -T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c -(Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (H11: ((\forall (b: -B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 x3))))).(let H12 \def (eq_ind -T t2 (\lambda (t: T).(\forall (x4: T).(\forall (x5: T).((eq T t (THead (Bind -Abbr) x4 x5)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 -(THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x4 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x4) x5 t5)))) (pr3 -(CHead c (Bind Abbr) x4) x5 (lift (S O) O t4))))))) H3 (THead (Bind Abbr) x2 -x3) H8) in (let H13 \def (H12 x2 x3 (refl_equal T (THead (Bind Abbr) x2 x3))) -in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind -Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: -T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 t5)))) (pr3 (CHead c -(Bind Abbr) x2) x3 (lift (S O) O t4)) (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S -O) O t4))) (\lambda (H14: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T -t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 -u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 -t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead -(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 t5))) (or -(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 -t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: -T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c -(Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (x4: T).(\lambda (x5: -T).(\lambda (H15: (eq T t4 (THead (Bind Abbr) x4 x5))).(\lambda (H16: (pr3 c -x2 x4)).(\lambda (H17: (pr3 (CHead c (Bind Abbr) x2) x3 x5)).(eq_ind_r T -(THead (Bind Abbr) x4 x5) (\lambda (t: T).(or (ex3_2 T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t (THead (Bind Abbr) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S -O) O t)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T -(THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S -O) O (THead (Bind Abbr) x4 x5))) (ex3_2_intro T T (\lambda (u2: T).(\lambda -(t5: T).(eq T (THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2 t5)))) (\lambda -(u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x0) x1 t5))) x4 x5 (refl_equal T (THead (Bind Abbr) x4 -x5)) (pr3_sing c x2 x0 H9 x4 H16) (pr3_sing (CHead c (Bind Abbr) x0) x3 x1 -(H11 Abbr x0) x5 (pr3_pr2_pr3_t c x2 x3 x5 (Bind Abbr) H17 x0 H9)))) t4 -H15)))))) H14)) (\lambda (H14: (pr3 (CHead c (Bind Abbr) x2) x3 (lift (S O) O -t4))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead -(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 -(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4)) (pr3_sing (CHead c (Bind Abbr) -x0) x3 x1 (H11 Abbr x0) (lift (S O) O t4) (pr3_pr2_pr3_t c x2 x3 (lift (S O) -O t4) (Bind Abbr) H14 x0 H9)))) H13)))) (\lambda (H11: (ex2 T (\lambda (u: -T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) x1 -x3)))).(ex2_ind T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead c -(Bind Abbr) u) x1 x3)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T -t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 -u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 -t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (x4: -T).(\lambda (H12: (pr0 x0 x4)).(\lambda (H13: (pr2 (CHead c (Bind Abbr) x4) -x1 x3)).(let H14 \def (eq_ind T t2 (\lambda (t: T).(\forall (x5: T).(\forall -(x6: T).((eq T t (THead (Bind Abbr) x5 x6)) \to (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x5 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x5) x6 t5)))) (pr3 (CHead c (Bind Abbr) x5) x6 (lift (S -O) O t4))))))) H3 (THead (Bind Abbr) x2 x3) H8) in (let H15 \def (H14 x2 x3 -(refl_equal T (THead (Bind Abbr) x2 x3))) in (or_ind (ex3_2 T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x2) x3 t5)))) (pr3 (CHead c (Bind Abbr) x2) x3 (lift (S -O) O t4)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead -(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 -(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (H16: (ex3_2 T T -(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda -(t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 t5))))).(ex3_2_ind T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x2) x3 t5))) (or (ex3_2 T T (\lambda (u2: T).(\lambda -(t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) -x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda -(x5: T).(\lambda (x6: T).(\lambda (H17: (eq T t4 (THead (Bind Abbr) x5 -x6))).(\lambda (H18: (pr3 c x2 x5)).(\lambda (H19: (pr3 (CHead c (Bind Abbr) -x2) x3 x6)).(eq_ind_r T (THead (Bind Abbr) x5 x6) (\lambda (t: T).(or (ex3_2 -T T (\lambda (u2: T).(\lambda (t5: T).(eq T t (THead (Bind Abbr) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) -x1 (lift (S O) O t)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t5: -T).(eq T (THead (Bind Abbr) x5 x6) (THead (Bind Abbr) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S -O) O (THead (Bind Abbr) x5 x6))) (ex3_2_intro T T (\lambda (u2: T).(\lambda -(t5: T).(eq T (THead (Bind Abbr) x5 x6) (THead (Bind Abbr) u2 t5)))) (\lambda -(u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x0) x1 t5))) x5 x6 (refl_equal T (THead (Bind Abbr) x5 -x6)) (pr3_sing c x2 x0 H9 x5 H18) (pr3_t x3 x1 (CHead c (Bind Abbr) x0) -(pr3_pr0_pr2_t x0 x4 H12 c x1 x3 (Bind Abbr) H13) x6 (pr3_pr2_pr3_t c x2 x3 -x6 (Bind Abbr) H19 x0 H9)))) t4 H17)))))) H16)) (\lambda (H16: (pr3 (CHead c -(Bind Abbr) x2) x3 (lift (S O) O t4))).(or_intror (ex3_2 T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S -O) O t4)) (pr3_t x3 x1 (CHead c (Bind Abbr) x0) (pr3_pr0_pr2_t x0 x4 H12 c x1 -x3 (Bind Abbr) H13) (lift (S O) O t4) (pr3_pr2_pr3_t c x2 x3 (lift (S O) O -t4) (Bind Abbr) H16 x0 H9)))) H15)))))) H11)) (\lambda (H11: (ex3_2 T T -(\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) -(\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: -T).(pr2 (CHead c (Bind Abbr) x0) z x3))))).(ex3_2_ind T T (\lambda (y0: -T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: -T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c -(Bind Abbr) x0) z x3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq -T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 -u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 -t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (x4: -T).(\lambda (x5: T).(\lambda (H12: (pr2 (CHead c (Bind Abbr) x0) x1 -x4)).(\lambda (H13: (pr0 x4 x5)).(\lambda (H14: (pr2 (CHead c (Bind Abbr) x0) -x5 x3)).(let H15 \def (eq_ind T t2 (\lambda (t: T).(\forall (x6: T).(\forall -(x7: T).((eq T t (THead (Bind Abbr) x6 x7)) \to (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x6 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x6) x7 t5)))) (pr3 (CHead c (Bind Abbr) x6) x7 (lift (S -O) O t4))))))) H3 (THead (Bind Abbr) x2 x3) H8) in (let H16 \def (H15 x2 x3 -(refl_equal T (THead (Bind Abbr) x2 x3))) in (or_ind (ex3_2 T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x2) x3 t5)))) (pr3 (CHead c (Bind Abbr) x2) x3 (lift (S -O) O t4)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead -(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 -(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (H17: (ex3_2 T T -(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda -(t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 t5))))).(ex3_2_ind T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x2) x3 t5))) (or (ex3_2 T T (\lambda (u2: T).(\lambda -(t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) -x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda -(x6: T).(\lambda (x7: T).(\lambda (H18: (eq T t4 (THead (Bind Abbr) x6 -x7))).(\lambda (H19: (pr3 c x2 x6)).(\lambda (H20: (pr3 (CHead c (Bind Abbr) -x2) x3 x7)).(eq_ind_r T (THead (Bind Abbr) x6 x7) (\lambda (t: T).(or (ex3_2 -T T (\lambda (u2: T).(\lambda (t5: T).(eq T t (THead (Bind Abbr) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) -x1 (lift (S O) O t)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t5: -T).(eq T (THead (Bind Abbr) x6 x7) (THead (Bind Abbr) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S -O) O (THead (Bind Abbr) x6 x7))) (ex3_2_intro T T (\lambda (u2: T).(\lambda -(t5: T).(eq T (THead (Bind Abbr) x6 x7) (THead (Bind Abbr) u2 t5)))) (\lambda -(u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x0) x1 t5))) x6 x7 (refl_equal T (THead (Bind Abbr) x6 -x7)) (pr3_sing c x2 x0 H9 x6 H19) (pr3_sing (CHead c (Bind Abbr) x0) x4 x1 -H12 x7 (pr3_sing (CHead c (Bind Abbr) x0) x5 x4 (pr2_free (CHead c (Bind -Abbr) x0) x4 x5 H13) x7 (pr3_sing (CHead c (Bind Abbr) x0) x3 x5 H14 x7 -(pr3_pr2_pr3_t c x2 x3 x7 (Bind Abbr) H20 x0 H9)))))) t4 H18)))))) H17)) -(\lambda (H17: (pr3 (CHead c (Bind Abbr) x2) x3 (lift (S O) O -t4))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead -(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 -(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4)) (pr3_sing (CHead c (Bind Abbr) -x0) x4 x1 H12 (lift (S O) O t4) (pr3_sing (CHead c (Bind Abbr) x0) x5 x4 -(pr2_free (CHead c (Bind Abbr) x0) x4 x5 H13) (lift (S O) O t4) (pr3_sing -(CHead c (Bind Abbr) x0) x3 x5 H14 (lift (S O) O t4) (pr3_pr2_pr3_t c x2 x3 -(lift (S O) O t4) (Bind Abbr) H17 x0 H9)))))) H16)))))))) H11)) H10)))))) -H7)) (\lambda (H7: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) -x1 (lift (S O) O t2)))))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda -(t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) -x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4)) (pr3_sing -(CHead c (Bind Abbr) x0) (lift (S O) O t2) x1 (H7 Abbr x0) (lift (S O) O t4) -(pr3_lift (CHead c (Bind Abbr) x0) c (S O) O (drop_drop (Bind Abbr) O c c -(drop_refl c) x0) t2 t4 H2)))) H6)))))))))))) y x H0))))) H))))). -(* COMMENTS -Initial nodes: 5983 -END *) - -theorem pr3_gen_appl: - \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c -(THead (Flat Appl) u1 t1) x) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda -(t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t1 t2)))) (ex4_4 T -T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 -c (THead (Bind Abbr) u2 t2) x))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (_: T).(pr3 c u1 u2))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c t1 (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T -T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 -c t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) x))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda -(H: (pr3 c (THead (Flat Appl) u1 t1) x)).(insert_eq T (THead (Flat Appl) u1 -t1) (\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(or3 (ex3_2 T T (\lambda -(u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t1 -t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) x))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))))) -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c t1 -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) -(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(pr3 c t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) -x))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) -y2) z1 z2)))))))))) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(unintro T t1 -(\lambda (t: T).((eq T y (THead (Flat Appl) u1 t)) \to (or3 (ex3_2 T T -(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda -(t2: T).(pr3 c t t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) x))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 -u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(pr3 c t (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 -(CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c t (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead -(Flat Appl) (lift (S O) O u2) z2)) x))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 -u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))) (unintro T u1 (\lambda -(t: T).(\forall (x0: T).((eq T y (THead (Flat Appl) t x0)) \to (or3 (ex3_2 T -T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_: T).(\lambda (t2: -T).(pr3 c x0 t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) x))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))) (\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x0 (THead -(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) -(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(pr3 c x0 (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) -x))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) -y2) z1 z2)))))))))))) (pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall -(x0: T).(\forall (x1: T).((eq T t (THead (Flat Appl) x0 x1)) \to (or3 (ex3_2 -T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Flat Appl) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t2: T).(pr3 c x1 t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) t0))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 -u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 -(CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead -(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) -y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t0))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))))))) -(\lambda (t: T).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H1: (eq T t -(THead (Flat Appl) x0 x1))).(eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda -(t0: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead -(Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) -(\lambda (_: T).(\lambda (t2: T).(pr3 c x1 t2)))) (ex4_4 T T T T (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind -Abbr) u2 t2) t0))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T -T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 -c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t0))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))) -(or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat -Appl) x0 x1) (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c x1 t2)))) (ex4_4 T -T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 -c (THead (Bind Abbr) u2 t2) (THead (Flat Appl) x0 x1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))) -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) -(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) -(THead (Flat Appl) x0 x1)))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 -(CHead c (Bind b) y2) z1 z2)))))))) (ex3_2_intro T T (\lambda (u2: -T).(\lambda (t2: T).(eq T (THead (Flat Appl) x0 x1) (THead (Flat Appl) u2 -t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: -T).(\lambda (t2: T).(pr3 c x1 t2))) x0 x1 (refl_equal T (THead (Flat Appl) x0 -x1)) (pr3_refl c x0) (pr3_refl c x1))) t H1))))) (\lambda (t2: T).(\lambda -(t3: T).(\lambda (H1: (pr2 c t3 t2)).(\lambda (t4: T).(\lambda (H2: (pr3 c t2 -t4)).(\lambda (H3: ((\forall (x0: T).(\forall (x1: T).((eq T t2 (THead (Flat -Appl) x0 x1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 -(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind -Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T -T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 -c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 -z2)))))))))))))).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t3 -(THead (Flat Appl) x0 x1))).(let H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c -t t2)) H1 (THead (Flat Appl) x0 x1) H4) in (let H6 \def (pr2_gen_appl c x0 x1 -t2 H5) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 -(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr2 c x1 t5)))) (ex4_4 T T T T (\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x1 (THead -(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t5: T).(eq T t2 (THead (Bind Abbr) u2 t5)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T -T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq -T x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead -(Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (or3 (ex3_2 -T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 -u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 -(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead -(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) -y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda -(H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Flat Appl) -u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: -T).(\lambda (t5: T).(pr2 c x1 t5))))).(ex3_2_ind T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t2 (THead (Flat Appl) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr2 c x1 -t5))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat -Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: -T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda -(_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) -t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 -c x0 u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 -(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead -(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) -y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda -(x2: T).(\lambda (x3: T).(\lambda (H8: (eq T t2 (THead (Flat Appl) x2 -x3))).(\lambda (H9: (pr2 c x0 x2)).(\lambda (H10: (pr2 c x1 x3)).(let H11 -\def (eq_ind T t2 (\lambda (t: T).(\forall (x4: T).(\forall (x5: T).((eq T t -(THead (Flat Appl) x4 x5)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: -T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c x4 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x5 t5)))) (ex4_4 T -T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 -c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (_: T).(pr3 c x4 u2))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c x5 (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T -T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 -c x5 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c x4 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))))) H3 -(THead (Flat Appl) x2 x3) H8) in (let H12 \def (eq_ind T t2 (\lambda (t: -T).(pr3 c t t4)) H2 (THead (Flat Appl) x2 x3) H8) in (let H13 \def (H11 x2 x3 -(refl_equal T (THead (Flat Appl) x2 x3))) in (or3_ind (ex3_2 T T (\lambda -(u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x3 -t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))))) -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x3 -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) -(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(pr3 c x3 (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) -t4))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) -y2) z1 z2)))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 -(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind -Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T -T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 -c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda -(H14: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat -Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: -T).(\lambda (t5: T).(pr3 c x3 t5))))).(ex3_2_ind T T (\lambda (u2: -T).(\lambda (t5: T).(eq 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H18)))))))))) H14)) (\lambda (H14: (ex6_6 B T -T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(pr3 c x3 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c -(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr3 c x2 u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 -y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 -z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 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T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (ex6_6_intro -B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(pr3 c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c -(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 -y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))) -x4 x5 x6 x7 x8 x9 H15 (pr3_sing c x3 x1 H10 (THead (Bind x4) x5 x6) H16) H17 -(pr3_sing c x2 x0 H9 x8 H18) H19 H20)))))))))))))) H14)) H13))))))))) H7)) -(\lambda (H7: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind -Abbr) u2 t5)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c x0 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) -z1 t5))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (_: T).(eq T x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind -Abbr) u2 t5)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c x0 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) -z1 t5))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 -(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind -Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T -T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 -c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda -(x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H8: (eq -T x1 (THead (Bind Abst) x2 x3))).(\lambda (H9: (eq T t2 (THead (Bind Abbr) x4 -x5))).(\lambda (H10: (pr2 c x0 x4)).(\lambda (H11: ((\forall (b: B).(\forall -(u: T).(pr2 (CHead c (Bind b) u) x3 x5))))).(eq_ind_r T (THead (Bind Abst) x2 -x3) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T -t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 -u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c t t5)))) (ex4_4 T T T T -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c -(THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c t (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T -T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 -c t (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))) (let H12 -\def (eq_ind T t2 (\lambda (t: T).(\forall (x6: T).(\forall (x7: T).((eq T t -(THead (Flat Appl) x6 x7)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: -T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c x6 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x7 t5)))) (ex4_4 T -T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 -c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (_: T).(pr3 c x6 u2))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c x7 (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T -T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 -c x7 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c x6 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))))) H3 -(THead (Bind Abbr) x4 x5) H9) in (let H13 \def (eq_ind T t2 (\lambda (t: -T).(pr3 c t t4)) H2 (THead (Bind Abbr) x4 x5) H9) in (or3_intro1 (ex3_2 T T -(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t5: T).(pr3 c (THead (Bind Abst) x2 x3) t5)))) (ex4_4 T T T T (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind -Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) x2 x3) (THead -(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) -(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) x2 x3) (THead (Bind b) y1 -z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: -T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat -Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 -u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (ex4_4_intro T T T T -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c -(THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) x2 x3) (THead -(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -z1 t5))))))) x2 x3 x4 x5 H13 (pr3_pr2 c x0 x4 H10) (pr3_refl c (THead (Bind -Abst) x2 x3)) (\lambda (b: B).(\lambda (u: T).(pr3_pr2 (CHead c (Bind b) u) -x3 x5 (H11 b u)))))))) x1 H8))))))))) H7)) (\lambda (H7: (ex6_6 B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T x1 -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind -B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T -t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 -y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) -(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) -u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: -T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda -(_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) -t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 -c x0 u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 -(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead -(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) -y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda -(x2: B).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (x6: -T).(\lambda (x7: T).(\lambda (H8: (not (eq B x2 Abst))).(\lambda (H9: (eq T -x1 (THead (Bind x2) x3 x4))).(\lambda (H10: (eq T t2 (THead (Bind x2) x7 -(THead (Flat Appl) (lift (S O) O x6) x5)))).(\lambda (H11: (pr2 c x0 -x6)).(\lambda (H12: (pr2 c x3 x7)).(\lambda (H13: (pr2 (CHead c (Bind x2) x7) -x4 x5)).(eq_ind_r T (THead (Bind x2) x3 x4) (\lambda (t: T).(or3 (ex3_2 T T -(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t5: T).(pr3 c t t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 -u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(pr3 c t (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 -(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c t (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead -(Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 -u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))) (let H14 \def (eq_ind T t2 -(\lambda (t: T).(\forall (x8: T).(\forall (x9: T).((eq T t (THead (Flat Appl) -x8 x9)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead -(Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x8 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr3 c x9 t5)))) (ex4_4 T T T T (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind -Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr3 c x8 u2))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c x9 (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T -T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 -c x9 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c x8 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))))) H3 -(THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) O x6) x5)) H10) in (let -H15 \def (eq_ind T t2 (\lambda (t: T).(pr3 c t t4)) H2 (THead (Bind x2) x7 -(THead (Flat Appl) (lift (S O) O x6) x5)) H10) in (or3_intro2 (ex3_2 T T -(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t5: T).(pr3 c (THead (Bind x2) x3 x4) t5)))) (ex4_4 T T T T (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind -Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind x2) x3 x4) (THead -(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) -(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(pr3 c (THead (Bind x2) x3 x4) (THead (Bind b) y1 -z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: -T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat -Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 -u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c -(THead (Bind x2) x3 x4) (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) -t4))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) -y2) z1 z2))))))) x2 x3 x4 x5 x6 x7 H8 (pr3_refl c (THead (Bind x2) x3 x4)) -H15 (pr3_pr2 c x0 x6 H11) (pr3_pr2 c x3 x7 H12) (pr3_pr2 (CHead c (Bind x2) -x7) x4 x5 H13))))) x1 H9))))))))))))) H7)) H6)))))))))))) y x H0))))) H))))). -(* COMMENTS -Initial nodes: 12691 -END *) - -theorem pr3_gen_bind: - \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (u1: -T).(\forall (t1: T).(\forall (x: T).((pr3 c (THead (Bind b) u1 t1) x) \to (or -(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind b) u2 -t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: -T).(\lambda (t2: T).(pr3 (CHead c (Bind b) u1) t1 t2)))) (pr3 (CHead c (Bind -b) u1) t1 (lift (S O) O x))))))))) -\def - \lambda (b: B).(B_ind (\lambda (b0: B).((not (eq B b0 Abst)) \to (\forall -(c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c (THead (Bind -b0) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x -(THead (Bind b0) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) -(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind b0) u1) t1 t2)))) (pr3 -(CHead c (Bind b0) u1) t1 (lift (S O) O x)))))))))) (\lambda (_: (not (eq B -Abbr Abst))).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: -T).(\lambda (H0: (pr3 c (THead (Bind Abbr) u1 t1) x)).(let H1 \def -(pr3_gen_abbr c u1 t1 x H0) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda -(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) -u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O x)) (or (ex3_2 T -T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda -(t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) -t1 (lift (S O) O x))) (\lambda (H2: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 -c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 -t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind -Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: -T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2))) (or (ex3_2 T T -(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda -(t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) -t1 (lift (S O) O x))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H3: (eq T x -(THead (Bind Abbr) x0 x1))).(\lambda (H4: (pr3 c u1 x0)).(\lambda (H5: (pr3 -(CHead c (Bind Abbr) u1) t1 x1)).(or_introl (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 -(CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S -O) O x)) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead -(Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) -(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2))) x0 x1 -H3 H4 H5))))))) H2)) (\lambda (H2: (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S -O) O x))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x -(THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) -(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2)))) (pr3 -(CHead c (Bind Abbr) u1) t1 (lift (S O) O x)) H2)) H1)))))))) (\lambda (H: -(not (eq B Abst Abst))).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: -T).(\lambda (x: T).(\lambda (_: (pr3 c (THead (Bind Abst) u1 t1) x)).(let H1 -\def (match (H (refl_equal B Abst)) in False return (\lambda (_: False).(or -(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 -t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: -T).(\lambda (t2: T).(pr3 (CHead c (Bind Abst) u1) t1 t2)))) (pr3 (CHead c -(Bind Abst) u1) t1 (lift (S O) O x)))) with []) in H1))))))) (\lambda (_: -(not (eq B Void Abst))).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: -T).(\lambda (x: T).(\lambda (H0: (pr3 c (THead (Bind Void) u1 t1) x)).(let H1 -\def (pr3_gen_void c u1 t1 x H0) in (or_ind (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall -(b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t1 t2)))))) (pr3 (CHead c -(Bind Void) u1) t1 (lift (S O) O x)) (or (ex3_2 T T (\lambda (u2: T).(\lambda -(t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) -u1) t1 t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O x))) (\lambda -(H2: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) -u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: -T).(\lambda (t2: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) -u) t1 t2))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T x -(THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) -(\lambda (_: T).(\lambda (t2: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead -c (Bind b0) u) t1 t2))))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 -c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) u1) t1 -t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O x))) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H3: (eq T x (THead (Bind Void) x0 -x1))).(\lambda (H4: (pr3 c u1 x0)).(\lambda (H5: ((\forall (b0: B).(\forall -(u: T).(pr3 (CHead c (Bind b0) u) t1 x1))))).(or_introl (ex3_2 T T (\lambda -(u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 -(CHead c (Bind Void) u1) t1 t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S -O) O x)) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead -(Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) -(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) u1) t1 t2))) x0 x1 -H3 H4 (H5 Void u1)))))))) H2)) (\lambda (H2: (pr3 (CHead c (Bind Void) u1) t1 -(lift (S O) O x))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 -c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) u1) t1 -t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O x)) H2)) H1)))))))) b). -(* COMMENTS -Initial nodes: 1721 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/iso.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/iso.ma deleted file mode 100644 index c7b2fb7aa..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/iso.ma +++ /dev/null @@ -1,1155 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr3/fwd.ma". - -include "Basic-1/iso/props.ma". - -include "Basic-1/tlist/props.ma". - -theorem pr3_iso_appls_abbr: - \forall (c: C).(\forall (d: C).(\forall (w: T).(\forall (i: nat).((getl i c -(CHead d (Bind Abbr) w)) \to (\forall (vs: TList).(let u1 \def (THeads (Flat -Appl) vs (TLRef i)) in (\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to -(\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) vs (lift (S i) O w)) -u2)))))))))) -\def - \lambda (c: C).(\lambda (d: C).(\lambda (w: T).(\lambda (i: nat).(\lambda -(H: (getl i c (CHead d (Bind Abbr) w))).(\lambda (vs: TList).(TList_ind -(\lambda (t: TList).(let u1 \def (THeads (Flat Appl) t (TLRef i)) in (\forall -(u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to -(pr3 c (THeads (Flat Appl) t (lift (S i) O w)) u2)))))) (\lambda (u2: -T).(\lambda (H0: (pr3 c (TLRef i) u2)).(\lambda (H1: (((iso (TLRef i) u2) \to -(\forall (P: Prop).P)))).(let H2 \def (pr3_gen_lref c u2 i H0) in (or_ind (eq -T u2 (TLRef i)) (ex3_3 C T T (\lambda (d0: C).(\lambda (u: T).(\lambda (_: -T).(getl i c (CHead d0 (Bind Abbr) u))))) (\lambda (d0: C).(\lambda (u: -T).(\lambda (v: T).(pr3 d0 u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(v: T).(eq T u2 (lift (S i) O v)))))) (pr3 c (lift (S i) O w) u2) (\lambda -(H3: (eq T u2 (TLRef i))).(let H4 \def (eq_ind T u2 (\lambda (t: T).((iso -(TLRef i) t) \to (\forall (P: Prop).P))) H1 (TLRef i) H3) in (eq_ind_r T -(TLRef i) (\lambda (t: T).(pr3 c (lift (S i) O w) t)) (H4 (iso_refl (TLRef -i)) (pr3 c (lift (S i) O w) (TLRef i))) u2 H3))) (\lambda (H3: (ex3_3 C T T -(\lambda (d0: C).(\lambda (u: T).(\lambda (_: T).(getl i c (CHead d0 (Bind -Abbr) u))))) (\lambda (d0: C).(\lambda (u: T).(\lambda (v: T).(pr3 d0 u v)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T u2 (lift (S i) O -v))))))).(ex3_3_ind C T T (\lambda (d0: C).(\lambda (u: T).(\lambda (_: -T).(getl i c (CHead d0 (Bind Abbr) u))))) (\lambda (d0: C).(\lambda (u: -T).(\lambda (v: T).(pr3 d0 u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(v: T).(eq T u2 (lift (S i) O v))))) (pr3 c (lift (S i) O w) u2) (\lambda -(x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H4: (getl i c (CHead x0 -(Bind Abbr) x1))).(\lambda (H5: (pr3 x0 x1 x2)).(\lambda (H6: (eq T u2 (lift -(S i) O x2))).(let H7 \def (eq_ind T u2 (\lambda (t: T).((iso (TLRef i) t) -\to (\forall (P: Prop).P))) H1 (lift (S i) O x2) H6) in (eq_ind_r T (lift (S -i) O x2) (\lambda (t: T).(pr3 c (lift (S i) O w) t)) (let H8 \def (eq_ind C -(CHead d (Bind Abbr) w) (\lambda (c0: C).(getl i c c0)) H (CHead x0 (Bind -Abbr) x1) (getl_mono c (CHead d (Bind Abbr) w) i H (CHead x0 (Bind Abbr) x1) -H4)) in (let H9 \def (f_equal C C (\lambda (e: C).(match e in C return -(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow -c0])) (CHead d (Bind Abbr) w) (CHead x0 (Bind Abbr) x1) (getl_mono c (CHead d -(Bind Abbr) w) i H (CHead x0 (Bind Abbr) x1) H4)) in ((let H10 \def (f_equal -C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) -\Rightarrow w | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abbr) w) (CHead -x0 (Bind Abbr) x1) (getl_mono c (CHead d (Bind Abbr) w) i H (CHead x0 (Bind -Abbr) x1) H4)) in (\lambda (H11: (eq C d x0)).(let H12 \def (eq_ind_r T x1 -(\lambda (t: T).(getl i c (CHead x0 (Bind Abbr) t))) H8 w H10) in (let H13 -\def (eq_ind_r T x1 (\lambda (t: T).(pr3 x0 t x2)) H5 w H10) in (let H14 \def -(eq_ind_r C x0 (\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr) w))) H12 d -H11) in (let H15 \def (eq_ind_r C x0 (\lambda (c0: C).(pr3 c0 w x2)) H13 d -H11) in (pr3_lift c d (S i) O (getl_drop Abbr c d w i H14) w x2 H15))))))) -H9))) u2 H6)))))))) H3)) H2))))) (\lambda (t: T).(\lambda (t0: -TList).(\lambda (H0: ((\forall (u2: T).((pr3 c (THeads (Flat Appl) t0 (TLRef -i)) u2) \to ((((iso (THeads (Flat Appl) t0 (TLRef i)) u2) \to (\forall (P: -Prop).P))) \to (pr3 c (THeads (Flat Appl) t0 (lift (S i) O w)) -u2)))))).(\lambda (u2: T).(\lambda (H1: (pr3 c (THead (Flat Appl) t (THeads -(Flat Appl) t0 (TLRef i))) u2)).(\lambda (H2: (((iso (THead (Flat Appl) t -(THeads (Flat Appl) t0 (TLRef i))) u2) \to (\forall (P: Prop).P)))).(let H3 -\def (pr3_gen_appl c t (THeads (Flat Appl) t0 (TLRef i)) u2 H1) in (or3_ind -(ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 -t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t u3))) (\lambda (_: -T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) t2)))) (ex4_4 T -T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 -c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u3: T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) -(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead -(Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t -u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (pr3 c (THead (Flat Appl) t -(THeads (Flat Appl) t0 (lift (S i) O w))) u2) (\lambda (H4: (ex3_2 T T -(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) -(\lambda (u3: T).(\lambda (_: T).(pr3 c t u3))) (\lambda (_: T).(\lambda (t2: -T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) t2))))).(ex3_2_ind T T (\lambda -(u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: -T).(\lambda (_: T).(pr3 c t u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c -(THeads (Flat Appl) t0 (TLRef i)) t2))) (pr3 c (THead (Flat Appl) t (THeads -(Flat Appl) t0 (lift (S i) O w))) u2) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H5: (eq T u2 (THead (Flat Appl) x0 x1))).(\lambda (_: (pr3 c t -x0)).(\lambda (_: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) x1)).(let H8 \def -(eq_ind T u2 (\lambda (t1: T).((iso (THead (Flat Appl) t (THeads (Flat Appl) -t0 (TLRef i))) t1) \to (\forall (P: Prop).P))) H2 (THead (Flat Appl) x0 x1) -H5) in (eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda (t1: T).(pr3 c (THead -(Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O w))) t1)) (H8 (iso_head t -x0 (THeads (Flat Appl) t0 (TLRef i)) x1 (Flat Appl)) (pr3 c (THead (Flat -Appl) t (THeads (Flat Appl) t0 (lift (S i) O w))) (THead (Flat Appl) x0 x1))) -u2 H5))))))) H4)) (\lambda (H4: (ex4_4 T T T T (\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t -u3))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2))))))))).(ex4_4_ind T -T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 -c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u3: T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -z1 t2))))))) (pr3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O -w))) u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: -T).(\lambda (H5: (pr3 c (THead (Bind Abbr) x2 x3) u2)).(\lambda (H6: (pr3 c t -x2)).(\lambda (H7: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind -Abst) x0 x1))).(\lambda (H8: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c -(Bind b) u) x1 x3))))).(pr3_t (THead (Bind Abbr) t x1) (THead (Flat Appl) t -(THeads (Flat Appl) t0 (lift (S i) O w))) c (pr3_t (THead (Flat Appl) t -(THead (Bind Abst) x0 x1)) (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift -(S i) O w))) c (pr3_thin_dx c (THeads (Flat Appl) t0 (lift (S i) O w)) (THead -(Bind Abst) x0 x1) (H0 (THead (Bind Abst) x0 x1) H7 (\lambda (H9: (iso -(THeads (Flat Appl) t0 (TLRef i)) (THead (Bind Abst) x0 x1))).(\lambda (P: -Prop).(iso_flats_lref_bind_false Appl Abst i x0 x1 t0 H9 P)))) t Appl) (THead -(Bind Abbr) t x1) (pr3_pr2 c (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) -(THead (Bind Abbr) t x1) (pr2_free c (THead (Flat Appl) t (THead (Bind Abst) -x0 x1)) (THead (Bind Abbr) t x1) (pr0_beta x0 t t (pr0_refl t) x1 x1 -(pr0_refl x1))))) u2 (pr3_t (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) t -x1) c (pr3_head_12 c t x2 H6 (Bind Abbr) x1 x3 (H8 Abbr x2)) u2 H5)))))))))) -H4)) (\lambda (H4: (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B -b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) -y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda -(_: T).(pr3 c t u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind -B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind b) y1 z1)))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda -(u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift -(S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t u3))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 -(CHead c (Bind b) y2) z1 z2))))))) (pr3 c (THead (Flat Appl) t (THeads (Flat -Appl) t0 (lift (S i) O w))) u2) (\lambda (x0: B).(\lambda (x1: T).(\lambda -(x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H5: (not -(eq B x0 Abst))).(\lambda (H6: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) -(THead (Bind x0) x1 x2))).(\lambda (H7: (pr3 c (THead (Bind x0) x5 (THead -(Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda (H8: (pr3 c t x4)).(\lambda -(H9: (pr3 c x1 x5)).(\lambda (H10: (pr3 (CHead c (Bind x0) x5) x2 x3)).(pr3_t -(THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) (THead (Flat -Appl) t (THeads (Flat Appl) t0 (lift (S i) O w))) c (pr3_t (THead (Bind x0) -x1 (THead (Flat Appl) (lift (S O) O t) x2)) (THead (Flat Appl) t (THeads -(Flat Appl) t0 (lift (S i) O w))) c (pr3_t (THead (Flat Appl) t (THead (Bind -x0) x1 x2)) (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O w))) c -(pr3_thin_dx c (THeads (Flat Appl) t0 (lift (S i) O w)) (THead (Bind x0) x1 -x2) (H0 (THead (Bind x0) x1 x2) H6 (\lambda (H11: (iso (THeads (Flat Appl) t0 -(TLRef i)) (THead (Bind x0) x1 x2))).(\lambda (P: -Prop).(iso_flats_lref_bind_false Appl x0 i x1 x2 t0 H11 P)))) t Appl) (THead -(Bind x0) x1 (THead (Flat Appl) (lift (S O) O t) x2)) (pr3_pr2 c (THead (Flat -Appl) t (THead (Bind x0) x1 x2)) (THead (Bind x0) x1 (THead (Flat Appl) (lift -(S O) O t) x2)) (pr2_free c (THead (Flat Appl) t (THead (Bind x0) x1 x2)) -(THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O t) x2)) (pr0_upsilon x0 -H5 t t (pr0_refl t) x1 x1 (pr0_refl x1) x2 x2 (pr0_refl x2))))) (THead (Bind -x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) (pr3_head_12 c x1 x1 -(pr3_refl c x1) (Bind x0) (THead (Flat Appl) (lift (S O) O t) x2) (THead -(Flat Appl) (lift (S O) O x4) x2) (pr3_head_12 (CHead c (Bind x0) x1) (lift -(S O) O t) (lift (S O) O x4) (pr3_lift (CHead c (Bind x0) x1) c (S O) O -(drop_drop (Bind x0) O c c (drop_refl c) x1) t x4 H8) (Flat Appl) x2 x2 -(pr3_refl (CHead (CHead c (Bind x0) x1) (Flat Appl) (lift (S O) O x4)) x2)))) -u2 (pr3_t (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) -(THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) c (pr3_head_12 -c x1 x5 H9 (Bind x0) (THead (Flat Appl) (lift (S O) O x4) x2) (THead (Flat -Appl) (lift (S O) O x4) x3) (pr3_thin_dx (CHead c (Bind x0) x5) x2 x3 H10 -(lift (S O) O x4) Appl)) u2 H7)))))))))))))) H4)) H3)))))))) vs)))))). -(* COMMENTS -Initial nodes: 3759 -END *) - -theorem pr3_iso_appls_cast: - \forall (c: C).(\forall (v: T).(\forall (t: T).(\forall (vs: TList).(let u1 -\def (THeads (Flat Appl) vs (THead (Flat Cast) v t)) in (\forall (u2: -T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to (pr3 c -(THeads (Flat Appl) vs t) u2)))))))) -\def - \lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (vs: -TList).(TList_ind (\lambda (t0: TList).(let u1 \def (THeads (Flat Appl) t0 -(THead (Flat Cast) v t)) in (\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 -u2) \to (\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) t0 t) u2)))))) -(\lambda (u2: T).(\lambda (H: (pr3 c (THead (Flat Cast) v t) u2)).(\lambda -(H0: (((iso (THead (Flat Cast) v t) u2) \to (\forall (P: Prop).P)))).(let H1 -\def (pr3_gen_cast c v t u2 H) in (or_ind (ex3_2 T T (\lambda (u3: -T).(\lambda (t2: T).(eq T u2 (THead (Flat Cast) u3 t2)))) (\lambda (u3: -T).(\lambda (_: T).(pr3 c v u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t -t2)))) (pr3 c t u2) (pr3 c t u2) (\lambda (H2: (ex3_2 T T (\lambda (u3: -T).(\lambda (t2: T).(eq T u2 (THead (Flat Cast) u3 t2)))) (\lambda (u3: -T).(\lambda (_: T).(pr3 c v u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t -t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead -(Flat Cast) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c v u3))) -(\lambda (_: T).(\lambda (t2: T).(pr3 c t t2))) (pr3 c t u2) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H3: (eq T u2 (THead (Flat Cast) x0 -x1))).(\lambda (_: (pr3 c v x0)).(\lambda (_: (pr3 c t x1)).(let H6 \def -(eq_ind T u2 (\lambda (t0: T).((iso (THead (Flat Cast) v t) t0) \to (\forall -(P: Prop).P))) H0 (THead (Flat Cast) x0 x1) H3) in (eq_ind_r T (THead (Flat -Cast) x0 x1) (\lambda (t0: T).(pr3 c t t0)) (H6 (iso_head v x0 t x1 (Flat -Cast)) (pr3 c t (THead (Flat Cast) x0 x1))) u2 H3))))))) H2)) (\lambda (H2: -(pr3 c t u2)).H2) H1))))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H: -((\forall (u2: T).((pr3 c (THeads (Flat Appl) t1 (THead (Flat Cast) v t)) u2) -\to ((((iso (THeads (Flat Appl) t1 (THead (Flat Cast) v t)) u2) \to (\forall -(P: Prop).P))) \to (pr3 c (THeads (Flat Appl) t1 t) u2)))))).(\lambda (u2: -T).(\lambda (H0: (pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead -(Flat Cast) v t))) u2)).(\lambda (H1: (((iso (THead (Flat Appl) t0 (THeads -(Flat Appl) t1 (THead (Flat Cast) v t))) u2) \to (\forall (P: -Prop).P)))).(let H2 \def (pr3_gen_appl c t0 (THeads (Flat Appl) t1 (THead -(Flat Cast) v t)) u2 H0) in (or3_ind (ex3_2 T T (\lambda (u3: T).(\lambda -(t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: -T).(pr3 c t0 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat -Appl) t1 (THead (Flat Cast) v t)) t2)))) (ex4_4 T T T T (\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind -Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: -T).(\lambda (_: T).(pr3 c t0 u3))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat -Cast) v t)) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 -(CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat -Appl) t1 (THead (Flat Cast) v t)) (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u3: T).(\lambda -(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u3) z2)) -u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) -y2) z1 z2)))))))) (pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 t)) u2) -(\lambda (H3: (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead -(Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3))) -(\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat -Cast) v t)) t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 -(THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3))) -(\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat -Cast) v t)) t2))) (pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 t)) u2) -(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T u2 (THead (Flat Appl) -x0 x1))).(\lambda (_: (pr3 c t0 x0)).(\lambda (_: (pr3 c (THeads (Flat Appl) -t1 (THead (Flat Cast) v t)) x1)).(let H7 \def (eq_ind T u2 (\lambda (t2: -T).((iso (THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Flat Cast) v -t))) t2) \to (\forall (P: Prop).P))) H1 (THead (Flat Appl) x0 x1) H4) in -(eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda (t2: T).(pr3 c (THead (Flat -Appl) t0 (THeads (Flat Appl) t1 t)) t2)) (H7 (iso_head t0 x0 (THeads (Flat -Appl) t1 (THead (Flat Cast) v t)) x1 (Flat Appl)) (pr3 c (THead (Flat Appl) -t0 (THeads (Flat Appl) t1 t)) (THead (Flat Appl) x0 x1))) u2 H4))))))) H3)) -(\lambda (H3: (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: -T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3))))) -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c -(THeads (Flat Appl) t1 (THead (Flat Cast) v t)) (THead (Bind Abst) y1 -z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: -T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 -t2))))))))).(ex4_4_ind T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: -T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3))))) -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c -(THeads (Flat Appl) t1 (THead (Flat Cast) v t)) (THead (Bind Abst) y1 -z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: -T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2))))))) -(pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 t)) u2) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H4: (pr3 c -(THead (Bind Abbr) x2 x3) u2)).(\lambda (H5: (pr3 c t0 x2)).(\lambda (H6: -(pr3 c (THeads (Flat Appl) t1 (THead (Flat Cast) v t)) (THead (Bind Abst) x0 -x1))).(\lambda (H7: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) -u) x1 x3))))).(pr3_t (THead (Bind Abbr) t0 x1) (THead (Flat Appl) t0 (THeads -(Flat Appl) t1 t)) c (pr3_t (THead (Flat Appl) t0 (THead (Bind Abst) x0 x1)) -(THead (Flat Appl) t0 (THeads (Flat Appl) t1 t)) c (pr3_thin_dx c (THeads -(Flat Appl) t1 t) (THead (Bind Abst) x0 x1) (H (THead (Bind Abst) x0 x1) H6 -(\lambda (H8: (iso (THeads (Flat Appl) t1 (THead (Flat Cast) v t)) (THead -(Bind Abst) x0 x1))).(\lambda (P: Prop).(iso_flats_flat_bind_false Appl Cast -Abst x0 v x1 t t1 H8 P)))) t0 Appl) (THead (Bind Abbr) t0 x1) (pr3_pr2 c -(THead (Flat Appl) t0 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) t0 x1) -(pr2_free c (THead (Flat Appl) t0 (THead (Bind Abst) x0 x1)) (THead (Bind -Abbr) t0 x1) (pr0_beta x0 t0 t0 (pr0_refl t0) x1 x1 (pr0_refl x1))))) u2 -(pr3_t (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) t0 x1) c (pr3_head_12 c -t0 x2 H5 (Bind Abbr) x1 x3 (H7 Abbr x2)) u2 H4)))))))))) H3)) (\lambda (H3: -(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat Cast) v t)) (THead (Bind b) -y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: -T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat -Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t0 -u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c -(THeads (Flat Appl) t1 (THead (Flat Cast) v t)) (THead (Bind b) y1 z1)))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda -(u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift -(S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 -(CHead c (Bind b) y2) z1 z2))))))) (pr3 c (THead (Flat Appl) t0 (THeads (Flat -Appl) t1 t)) u2) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda -(x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H4: (not (eq B x0 -Abst))).(\lambda (H5: (pr3 c (THeads (Flat Appl) t1 (THead (Flat Cast) v t)) -(THead (Bind x0) x1 x2))).(\lambda (H6: (pr3 c (THead (Bind x0) x5 (THead -(Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda (H7: (pr3 c t0 x4)).(\lambda -(H8: (pr3 c x1 x5)).(\lambda (H9: (pr3 (CHead c (Bind x0) x5) x2 x3)).(pr3_t -(THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) (THead (Flat -Appl) t0 (THeads (Flat Appl) t1 t)) c (pr3_t (THead (Bind x0) x1 (THead (Flat -Appl) (lift (S O) O t0) x2)) (THead (Flat Appl) t0 (THeads (Flat Appl) t1 t)) -c (pr3_t (THead (Flat Appl) t0 (THead (Bind x0) x1 x2)) (THead (Flat Appl) t0 -(THeads (Flat Appl) t1 t)) c (pr3_thin_dx c (THeads (Flat Appl) t1 t) (THead -(Bind x0) x1 x2) (H (THead (Bind x0) x1 x2) H5 (\lambda (H10: (iso (THeads -(Flat Appl) t1 (THead (Flat Cast) v t)) (THead (Bind x0) x1 x2))).(\lambda -(P: Prop).(iso_flats_flat_bind_false Appl Cast x0 x1 v x2 t t1 H10 P)))) t0 -Appl) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O t0) x2)) (pr3_pr2 -c (THead (Flat Appl) t0 (THead (Bind x0) x1 x2)) (THead (Bind x0) x1 (THead -(Flat Appl) (lift (S O) O t0) x2)) (pr2_free c (THead (Flat Appl) t0 (THead -(Bind x0) x1 x2)) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O t0) -x2)) (pr0_upsilon x0 H4 t0 t0 (pr0_refl t0) x1 x1 (pr0_refl x1) x2 x2 -(pr0_refl x2))))) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) -x2)) (pr3_head_12 c x1 x1 (pr3_refl c x1) (Bind x0) (THead (Flat Appl) (lift -(S O) O t0) x2) (THead (Flat Appl) (lift (S O) O x4) x2) (pr3_head_12 (CHead -c (Bind x0) x1) (lift (S O) O t0) (lift (S O) O x4) (pr3_lift (CHead c (Bind -x0) x1) c (S O) O (drop_drop (Bind x0) O c c (drop_refl c) x1) t0 x4 H7) -(Flat Appl) x2 x2 (pr3_refl (CHead (CHead c (Bind x0) x1) (Flat Appl) (lift -(S O) O x4)) x2)))) u2 (pr3_t (THead (Bind x0) x5 (THead (Flat Appl) (lift (S -O) O x4) x3)) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) c -(pr3_head_12 c x1 x5 H8 (Bind x0) (THead (Flat Appl) (lift (S O) O x4) x2) -(THead (Flat Appl) (lift (S O) O x4) x3) (pr3_thin_dx (CHead c (Bind x0) x5) -x2 x3 H9 (lift (S O) O x4) Appl)) u2 H6)))))))))))))) H3)) H2)))))))) vs)))). -(* COMMENTS -Initial nodes: 3297 -END *) - -theorem pr3_iso_appl_bind: - \forall (b: B).((not (eq B b Abst)) \to (\forall (v1: T).(\forall (v2: -T).(\forall (t: T).(let u1 \def (THead (Flat Appl) v1 (THead (Bind b) v2 t)) -in (\forall (c: C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to -(\forall (P: Prop).P))) \to (pr3 c (THead (Bind b) v2 (THead (Flat Appl) -(lift (S O) O v1) t)) u2)))))))))) -\def - \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (v1: T).(\lambda -(v2: T).(\lambda (t: T).(\lambda (c: C).(\lambda (u2: T).(\lambda (H0: (pr3 c -(THead (Flat Appl) v1 (THead (Bind b) v2 t)) u2)).(\lambda (H1: (((iso (THead -(Flat Appl) v1 (THead (Bind b) v2 t)) u2) \to (\forall (P: Prop).P)))).(let -H2 \def (pr3_gen_appl c v1 (THead (Bind b) v2 t) u2 H0) in (or3_ind (ex3_2 T -T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) -(\lambda (u3: T).(\lambda (_: T).(pr3 c v1 u3))) (\lambda (_: T).(\lambda -(t2: T).(pr3 c (THead (Bind b) v2 t) t2)))) (ex4_4 T T T T (\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind -Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: -T).(\lambda (_: T).(pr3 c v1 u3))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind b) v2 t) (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(t2: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) z1 -t2)))))))) (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) -(\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(pr3 c (THead (Bind b) v2 t) (THead (Bind b0) y1 -z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: -T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b0) y2 (THead (Flat -Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v1 -u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b0: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2)))))))) (pr3 c (THead (Bind b) v2 -(THead (Flat Appl) (lift (S O) O v1) t)) u2) (\lambda (H3: (ex3_2 T T -(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) -(\lambda (u3: T).(\lambda (_: T).(pr3 c v1 u3))) (\lambda (_: T).(\lambda -(t2: T).(pr3 c (THead (Bind b) v2 t) t2))))).(ex3_2_ind T T (\lambda (u3: -T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: -T).(\lambda (_: T).(pr3 c v1 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c -(THead (Bind b) v2 t) t2))) (pr3 c (THead (Bind b) v2 (THead (Flat Appl) -(lift (S O) O v1) t)) u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq -T u2 (THead (Flat Appl) x0 x1))).(\lambda (_: (pr3 c v1 x0)).(\lambda (_: -(pr3 c (THead (Bind b) v2 t) x1)).(let H7 \def (eq_ind T u2 (\lambda (t0: -T).((iso (THead (Flat Appl) v1 (THead (Bind b) v2 t)) t0) \to (\forall (P: -Prop).P))) H1 (THead (Flat Appl) x0 x1) H4) in (eq_ind_r T (THead (Flat Appl) -x0 x1) (\lambda (t0: T).(pr3 c (THead (Bind b) v2 (THead (Flat Appl) (lift (S -O) O v1) t)) t0)) (H7 (iso_head v1 x0 (THead (Bind b) v2 t) x1 (Flat Appl)) -(pr3 c (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) t)) (THead -(Flat Appl) x0 x1))) u2 H4))))))) H3)) (\lambda (H3: (ex4_4 T T T T (\lambda -(_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind -Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: -T).(\lambda (_: T).(pr3 c v1 u3))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind b) v2 t) (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(t2: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) z1 -t2))))))))).(ex4_4_ind T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: -T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v1 u3))))) -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c -(THead (Bind b) v2 t) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b0: B).(\forall (u: -T).(pr3 (CHead c (Bind b0) u) z1 t2))))))) (pr3 c (THead (Bind b) v2 (THead -(Flat Appl) (lift (S O) O v1) t)) u2) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H4: (pr3 c (THead (Bind Abbr) -x2 x3) u2)).(\lambda (H5: (pr3 c v1 x2)).(\lambda (H6: (pr3 c (THead (Bind b) -v2 t) (THead (Bind Abst) x0 x1))).(\lambda (H7: ((\forall (b0: B).(\forall -(u: T).(pr3 (CHead c (Bind b0) u) x1 x3))))).(pr3_t (THead (Bind Abbr) x2 x3) -(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) t)) c (let H_x \def -(pr3_gen_bind b H c v2 t (THead (Bind Abst) x0 x1) H6) in (let H8 \def H_x in -(or_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind Abst) -x0 x1) (THead (Bind b) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c v2 -u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind b) v2) t t2)))) -(pr3 (CHead c (Bind b) v2) t (lift (S O) O (THead (Bind Abst) x0 x1))) (pr3 c -(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) t)) (THead (Bind -Abbr) x2 x3)) (\lambda (H9: (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq -T (THead (Bind Abst) x0 x1) (THead (Bind b) u3 t2)))) (\lambda (u3: -T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 -(CHead c (Bind b) v2) t t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: -T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) u3 t2)))) (\lambda (u3: -T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 -(CHead c (Bind b) v2) t t2))) (pr3 c (THead (Bind b) v2 (THead (Flat Appl) -(lift (S O) O v1) t)) (THead (Bind Abbr) x2 x3)) (\lambda (x4: T).(\lambda -(x5: T).(\lambda (H10: (eq T (THead (Bind Abst) x0 x1) (THead (Bind b) x4 -x5))).(\lambda (H11: (pr3 c v2 x4)).(\lambda (H12: (pr3 (CHead c (Bind b) v2) -t x5)).(let H13 \def (f_equal T B (\lambda (e: T).(match e in T return -(\lambda (_: T).B) with [(TSort _) \Rightarrow Abst | (TLRef _) \Rightarrow -Abst | (THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).B) with -[(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abst])])) (THead (Bind Abst) -x0 x1) (THead (Bind b) x4 x5) H10) in ((let H14 \def (f_equal T T (\lambda -(e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 -| (TLRef _) \Rightarrow x0 | (THead _ t0 _) \Rightarrow t0])) (THead (Bind -Abst) x0 x1) (THead (Bind b) x4 x5) H10) in ((let H15 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow x1 | (TLRef _) \Rightarrow x1 | (THead _ _ t0) \Rightarrow t0])) -(THead (Bind Abst) x0 x1) (THead (Bind b) x4 x5) H10) in (\lambda (H16: (eq T -x0 x4)).(\lambda (H17: (eq B Abst b)).(let H18 \def (eq_ind_r T x5 (\lambda -(t0: T).(pr3 (CHead c (Bind b) v2) t t0)) H12 x1 H15) in (let H19 \def -(eq_ind_r T x4 (\lambda (t0: T).(pr3 c v2 t0)) H11 x0 H16) in (let H20 \def -(eq_ind_r B b (\lambda (b0: B).(pr3 (CHead c (Bind b0) v2) t x1)) H18 Abst -H17) in (let H21 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 Abst))) H -Abst H17) in (eq_ind B Abst (\lambda (b0: B).(pr3 c (THead (Bind b0) v2 -(THead (Flat Appl) (lift (S O) O v1) t)) (THead (Bind Abbr) x2 x3))) (let H22 -\def (match (H21 (refl_equal B Abst)) in False return (\lambda (_: -False).(pr3 c (THead (Bind Abst) v2 (THead (Flat Appl) (lift (S O) O v1) t)) -(THead (Bind Abbr) x2 x3))) with []) in H22) b H17)))))))) H14)) H13))))))) -H9)) (\lambda (H9: (pr3 (CHead c (Bind b) v2) t (lift (S O) O (THead (Bind -Abst) x0 x1)))).(pr3_t (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O -x2) (lift (S O) O (THead (Bind Abst) x0 x1)))) (THead (Bind b) v2 (THead -(Flat Appl) (lift (S O) O v1) t)) c (pr3_head_2 c v2 (THead (Flat Appl) (lift -(S O) O v1) t) (THead (Flat Appl) (lift (S O) O x2) (lift (S O) O (THead -(Bind Abst) x0 x1))) (Bind b) (pr3_flat (CHead c (Bind b) v2) (lift (S O) O -v1) (lift (S O) O x2) (pr3_lift (CHead c (Bind b) v2) c (S O) O (drop_drop -(Bind b) O c c (drop_refl c) v2) v1 x2 H5) t (lift (S O) O (THead (Bind Abst) -x0 x1)) H9 Appl)) (THead (Bind Abbr) x2 x3) (eq_ind T (lift (S O) O (THead -(Flat Appl) x2 (THead (Bind Abst) x0 x1))) (\lambda (t0: T).(pr3 c (THead -(Bind b) v2 t0) (THead (Bind Abbr) x2 x3))) (pr3_sing c (THead (Bind Abbr) x2 -x1) (THead (Bind b) v2 (lift (S O) O (THead (Flat Appl) x2 (THead (Bind Abst) -x0 x1)))) (pr2_free c (THead (Bind b) v2 (lift (S O) O (THead (Flat Appl) x2 -(THead (Bind Abst) x0 x1)))) (THead (Bind Abbr) x2 x1) (pr0_zeta b H (THead -(Flat Appl) x2 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) x2 x1) (pr0_beta -x0 x2 x2 (pr0_refl x2) x1 x1 (pr0_refl x1)) v2)) (THead (Bind Abbr) x2 x3) -(pr3_head_12 c x2 x2 (pr3_refl c x2) (Bind Abbr) x1 x3 (H7 Abbr x2))) (THead -(Flat Appl) (lift (S O) O x2) (lift (S O) O (THead (Bind Abst) x0 x1))) -(lift_flat Appl x2 (THead (Bind Abst) x0 x1) (S O) O)))) H8))) u2 H4))))))))) -H3)) (\lambda (H3: (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B -b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind b) v2 t) (THead -(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b0) -y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda -(_: T).(pr3 c v1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2))))))))).(ex6_6_ind -B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(pr3 c (THead (Bind b) v2 t) (THead (Bind b0) y1 z1)))))))) (\lambda -(b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u3: -T).(\lambda (y2: T).(pr3 c (THead (Bind b0) y2 (THead (Flat Appl) (lift (S O) -O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v1 u3))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 c y1 y2))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) -y2) z1 z2))))))) (pr3 c (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O -v1) t)) u2) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: -T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H4: (not (eq B x0 -Abst))).(\lambda (H5: (pr3 c (THead (Bind b) v2 t) (THead (Bind x0) x1 -x2))).(\lambda (H6: (pr3 c (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) -O x4) x3)) u2)).(\lambda (H7: (pr3 c v1 x4)).(\lambda (H8: (pr3 c x1 -x5)).(\lambda (H9: (pr3 (CHead c (Bind x0) x5) x2 x3)).(pr3_t (THead (Bind -x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) (THead (Bind b) v2 (THead -(Flat Appl) (lift (S O) O v1) t)) c (let H_x \def (pr3_gen_bind b H c v2 t -(THead (Bind x0) x1 x2) H5) in (let H10 \def H_x in (or_ind (ex3_2 T T -(\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind -b) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: -T).(\lambda (t2: T).(pr3 (CHead c (Bind b) v2) t t2)))) (pr3 (CHead c (Bind -b) v2) t (lift (S O) O (THead (Bind x0) x1 x2))) (pr3 c (THead (Bind b) v2 -(THead (Flat Appl) (lift (S O) O v1) t)) (THead (Bind x0) x5 (THead (Flat -Appl) (lift (S O) O x4) x3))) (\lambda (H11: (ex3_2 T T (\lambda (u3: -T).(\lambda (t2: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) u3 t2)))) -(\lambda (u3: T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: T).(\lambda -(t2: T).(pr3 (CHead c (Bind b) v2) t t2))))).(ex3_2_ind T T (\lambda (u3: -T).(\lambda (t2: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) u3 t2)))) -(\lambda (u3: T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: T).(\lambda -(t2: T).(pr3 (CHead c (Bind b) v2) t t2))) (pr3 c (THead (Bind b) v2 (THead -(Flat Appl) (lift (S O) O v1) t)) (THead (Bind x0) x5 (THead (Flat Appl) -(lift (S O) O x4) x3))) (\lambda (x6: T).(\lambda (x7: T).(\lambda (H12: (eq -T (THead (Bind x0) x1 x2) (THead (Bind b) x6 x7))).(\lambda (H13: (pr3 c v2 -x6)).(\lambda (H14: (pr3 (CHead c (Bind b) v2) t x7)).(let H15 \def (f_equal -T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with [(TSort _) -\Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead k _ _) \Rightarrow (match -k in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) -\Rightarrow x0])])) (THead (Bind x0) x1 x2) (THead (Bind b) x6 x7) H12) in -((let H16 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: -T).T) with [(TSort _) \Rightarrow x1 | (TLRef _) \Rightarrow x1 | (THead _ t0 -_) \Rightarrow t0])) (THead (Bind x0) x1 x2) (THead (Bind b) x6 x7) H12) in -((let H17 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: -T).T) with [(TSort _) \Rightarrow x2 | (TLRef _) \Rightarrow x2 | (THead _ _ -t0) \Rightarrow t0])) (THead (Bind x0) x1 x2) (THead (Bind b) x6 x7) H12) in -(\lambda (H18: (eq T x1 x6)).(\lambda (H19: (eq B x0 b)).(let H20 \def -(eq_ind_r T x7 (\lambda (t0: T).(pr3 (CHead c (Bind b) v2) t t0)) H14 x2 H17) -in (let H21 \def (eq_ind_r T x6 (\lambda (t0: T).(pr3 c v2 t0)) H13 x1 H18) -in (let H22 \def (eq_ind B x0 (\lambda (b0: B).(pr3 (CHead c (Bind b0) x5) x2 -x3)) H9 b H19) in (let H23 \def (eq_ind B x0 (\lambda (b0: B).(not (eq B b0 -Abst))) H4 b H19) in (eq_ind_r B b (\lambda (b0: B).(pr3 c (THead (Bind b) v2 -(THead (Flat Appl) (lift (S O) O v1) t)) (THead (Bind b0) x5 (THead (Flat -Appl) (lift (S O) O x4) x3)))) (pr3_head_21 c v2 x5 (pr3_t x1 v2 c H21 x5 H8) -(Bind b) (THead (Flat Appl) (lift (S O) O v1) t) (THead (Flat Appl) (lift (S -O) O x4) x3) (pr3_flat (CHead c (Bind b) v2) (lift (S O) O v1) (lift (S O) O -x4) (pr3_lift (CHead c (Bind b) v2) c (S O) O (drop_drop (Bind b) O c c -(drop_refl c) v2) v1 x4 H7) t x3 (pr3_t x2 t (CHead c (Bind b) v2) H20 x3 -(pr3_pr3_pr3_t c v2 x1 H21 x2 x3 (Bind b) (pr3_pr3_pr3_t c x1 x5 H8 x2 x3 -(Bind b) H22))) Appl)) x0 H19)))))))) H16)) H15))))))) H11)) (\lambda (H11: -(pr3 (CHead c (Bind b) v2) t (lift (S O) O (THead (Bind x0) x1 x2)))).(pr3_t -(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O x4) (lift (S O) O (THead -(Bind x0) x1 x2)))) (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) -t)) c (pr3_head_2 c v2 (THead (Flat Appl) (lift (S O) O v1) t) (THead (Flat -Appl) (lift (S O) O x4) (lift (S O) O (THead (Bind x0) x1 x2))) (Bind b) -(pr3_flat (CHead c (Bind b) v2) (lift (S O) O v1) (lift (S O) O x4) (pr3_lift -(CHead c (Bind b) v2) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) v2) -v1 x4 H7) t (lift (S O) O (THead (Bind x0) x1 x2)) H11 Appl)) (THead (Bind -x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) (eq_ind T (lift (S O) O -(THead (Flat Appl) x4 (THead (Bind x0) x1 x2))) (\lambda (t0: T).(pr3 c -(THead (Bind b) v2 t0) (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O -x4) x3)))) (pr3_sing c (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O -x4) x2)) (THead (Bind b) v2 (lift (S O) O (THead (Flat Appl) x4 (THead (Bind -x0) x1 x2)))) (pr2_free c (THead (Bind b) v2 (lift (S O) O (THead (Flat Appl) -x4 (THead (Bind x0) x1 x2)))) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S -O) O x4) x2)) (pr0_zeta b H (THead (Flat Appl) x4 (THead (Bind x0) x1 x2)) -(THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) (pr0_upsilon x0 -H4 x4 x4 (pr0_refl x4) x1 x1 (pr0_refl x1) x2 x2 (pr0_refl x2)) v2)) (THead -(Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) (pr3_head_12 c x1 x5 -H8 (Bind x0) (THead (Flat Appl) (lift (S O) O x4) x2) (THead (Flat Appl) -(lift (S O) O x4) x3) (pr3_thin_dx (CHead c (Bind x0) x5) x2 x3 H9 (lift (S -O) O x4) Appl))) (THead (Flat Appl) (lift (S O) O x4) (lift (S O) O (THead -(Bind x0) x1 x2))) (lift_flat Appl x4 (THead (Bind x0) x1 x2) (S O) O)))) -H10))) u2 H6))))))))))))) H3)) H2)))))))))). -(* COMMENTS -Initial nodes: 4805 -END *) - -theorem pr3_iso_appls_appl_bind: - \forall (b: B).((not (eq B b Abst)) \to (\forall (v: T).(\forall (u: -T).(\forall (t: T).(\forall (vs: TList).(let u1 \def (THeads (Flat Appl) vs -(THead (Flat Appl) v (THead (Bind b) u t))) in (\forall (c: C).(\forall (u2: -T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to (pr3 c -(THeads (Flat Appl) vs (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) -t))) u2))))))))))) -\def - \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (v: T).(\lambda -(u: T).(\lambda (t: T).(\lambda (vs: TList).(TList_ind (\lambda (t0: -TList).(let u1 \def (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind -b) u t))) in (\forall (c: C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 -u2) \to (\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) t0 (THead -(Bind b) u (THead (Flat Appl) (lift (S O) O v) t))) u2))))))) (\lambda (c: -C).(\lambda (u2: T).(\lambda (H0: (pr3 c (THead (Flat Appl) v (THead (Bind b) -u t)) u2)).(\lambda (H1: (((iso (THead (Flat Appl) v (THead (Bind b) u t)) -u2) \to (\forall (P: Prop).P)))).(pr3_iso_appl_bind b H v u t c u2 H0 H1))))) -(\lambda (t0: T).(\lambda (t1: TList).(\lambda (H0: ((\forall (c: C).(\forall -(u2: T).((pr3 c (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u -t))) u2) \to ((((iso (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind -b) u t))) u2) \to (\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) t1 -(THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t))) u2))))))).(\lambda -(c: C).(\lambda (u2: T).(\lambda (H1: (pr3 c (THead (Flat Appl) t0 (THeads -(Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t)))) u2)).(\lambda -(H2: (((iso (THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Flat Appl) v -(THead (Bind b) u t)))) u2) \to (\forall (P: Prop).P)))).(let H3 \def -(pr3_gen_appl c t0 (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind -b) u t))) u2 H1) in (or3_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq -T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t0 -u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t1 (THead -(Flat Appl) v (THead (Bind b) u t))) t2)))) (ex4_4 T T T T (\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind -Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: -T).(\lambda (_: T).(pr3 c t0 u3))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat -Appl) v (THead (Bind b) u t))) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b0: -B).(\forall (u0: T).(pr3 (CHead c (Bind b0) u0) z1 t2)))))))) (ex6_6 B T T T -T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 -c (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t))) (THead -(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b0) -y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda -(_: T).(pr3 c t0 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2)))))))) (pr3 c -(THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat -Appl) (lift (S O) O v) t)))) u2) (\lambda (H4: (ex3_2 T T (\lambda (u3: -T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: -T).(\lambda (_: T).(pr3 c t0 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c -(THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t))) -t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead -(Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3))) -(\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat -Appl) v (THead (Bind b) u t))) t2))) (pr3 c (THead (Flat Appl) t0 (THeads -(Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) -u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T u2 (THead (Flat -Appl) x0 x1))).(\lambda (_: (pr3 c t0 x0)).(\lambda (_: (pr3 c (THeads (Flat -Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t))) x1)).(let H8 \def -(eq_ind T u2 (\lambda (t2: T).((iso (THead (Flat Appl) t0 (THeads (Flat Appl) -t1 (THead (Flat Appl) v (THead (Bind b) u t)))) t2) \to (\forall (P: -Prop).P))) H2 (THead (Flat Appl) x0 x1) H5) in (eq_ind_r T (THead (Flat Appl) -x0 x1) (\lambda (t2: T).(pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 -(THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) t2)) (H8 -(iso_head t0 x0 (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u -t))) x1 (Flat Appl)) (pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 -(THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) (THead (Flat -Appl) x0 x1))) u2 H5))))))) H4)) (\lambda (H4: (ex4_4 T T T T (\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind -Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: -T).(\lambda (_: T).(pr3 c t0 u3))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat -Appl) v (THead (Bind b) u t))) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b0: -B).(\forall (u0: T).(pr3 (CHead c (Bind b0) u0) z1 t2))))))))).(ex4_4_ind T T -T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c -(THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u3: T).(\lambda (_: T).(pr3 c t0 u3))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat -Appl) v (THead (Bind b) u t))) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b0: -B).(\forall (u0: T).(pr3 (CHead c (Bind b0) u0) z1 t2))))))) (pr3 c (THead -(Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) -(lift (S O) O v) t)))) u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: -T).(\lambda (x3: T).(\lambda (H5: (pr3 c (THead (Bind Abbr) x2 x3) -u2)).(\lambda (H6: (pr3 c t0 x2)).(\lambda (H7: (pr3 c (THeads (Flat Appl) t1 -(THead (Flat Appl) v (THead (Bind b) u t))) (THead (Bind Abst) x0 -x1))).(\lambda (H8: ((\forall (b0: B).(\forall (u0: T).(pr3 (CHead c (Bind -b0) u0) x1 x3))))).(pr3_t (THead (Bind Abbr) t0 x1) (THead (Flat Appl) t0 -(THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) -t)))) c (pr3_t (THead (Flat Appl) t0 (THead (Bind Abst) x0 x1)) (THead (Flat -Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) (lift (S -O) O v) t)))) c (pr3_thin_dx c (THeads (Flat Appl) t1 (THead (Bind b) u -(THead (Flat Appl) (lift (S O) O v) t))) (THead (Bind Abst) x0 x1) (H0 c -(THead (Bind Abst) x0 x1) H7 (\lambda (H9: (iso (THeads (Flat Appl) t1 (THead -(Flat Appl) v (THead (Bind b) u t))) (THead (Bind Abst) x0 x1))).(\lambda (P: -Prop).(iso_flats_flat_bind_false Appl Appl Abst x0 v x1 (THead (Bind b) u t) -t1 H9 P)))) t0 Appl) (THead (Bind Abbr) t0 x1) (pr3_pr2 c (THead (Flat Appl) -t0 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) t0 x1) (pr2_free c (THead -(Flat Appl) t0 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) t0 x1) (pr0_beta -x0 t0 t0 (pr0_refl t0) x1 x1 (pr0_refl x1))))) u2 (pr3_t (THead (Bind Abbr) -x2 x3) (THead (Bind Abbr) t0 x1) c (pr3_head_12 c t0 x2 H6 (Bind Abbr) x1 x3 -(H8 Abbr x2)) u2 H5)))))))))) H4)) (\lambda (H4: (ex6_6 B T T T T T (\lambda -(b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c -(THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t))) (THead -(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b0) -y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda -(_: T).(pr3 c t0 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2))))))))).(ex6_6_ind -B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u -t))) (THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind -b0) y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda -(_: T).(pr3 c t0 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2))))))) (pr3 c -(THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat -Appl) (lift (S O) O v) t)))) u2) (\lambda (x0: B).(\lambda (x1: T).(\lambda -(x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H5: (not -(eq B x0 Abst))).(\lambda (H6: (pr3 c (THeads (Flat Appl) t1 (THead (Flat -Appl) v (THead (Bind b) u t))) (THead (Bind x0) x1 x2))).(\lambda (H7: (pr3 c -(THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda -(H8: (pr3 c t0 x4)).(\lambda (H9: (pr3 c x1 x5)).(\lambda (H10: (pr3 (CHead c -(Bind x0) x5) x2 x3)).(pr3_t (THead (Bind x0) x1 (THead (Flat Appl) (lift (S -O) O x4) x2)) (THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u -(THead (Flat Appl) (lift (S O) O v) t)))) c (pr3_t (THead (Bind x0) x1 (THead -(Flat Appl) (lift (S O) O t0) x2)) (THead (Flat Appl) t0 (THeads (Flat Appl) -t1 (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) c (pr3_t -(THead (Flat Appl) t0 (THead (Bind x0) x1 x2)) (THead (Flat Appl) t0 (THeads -(Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) c -(pr3_thin_dx c (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) -(lift (S O) O v) t))) (THead (Bind x0) x1 x2) (H0 c (THead (Bind x0) x1 x2) -H6 (\lambda (H11: (iso (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead -(Bind b) u t))) (THead (Bind x0) x1 x2))).(\lambda (P: -Prop).(iso_flats_flat_bind_false Appl Appl x0 x1 v x2 (THead (Bind b) u t) t1 -H11 P)))) t0 Appl) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O t0) -x2)) (pr3_pr2 c (THead (Flat Appl) t0 (THead (Bind x0) x1 x2)) (THead (Bind -x0) x1 (THead (Flat Appl) (lift (S O) O t0) x2)) (pr2_free c (THead (Flat -Appl) t0 (THead (Bind x0) x1 x2)) (THead (Bind x0) x1 (THead (Flat Appl) -(lift (S O) O t0) x2)) (pr0_upsilon x0 H5 t0 t0 (pr0_refl t0) x1 x1 (pr0_refl -x1) x2 x2 (pr0_refl x2))))) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S -O) O x4) x2)) (pr3_head_12 c x1 x1 (pr3_refl c x1) (Bind x0) (THead (Flat -Appl) (lift (S O) O t0) x2) (THead (Flat Appl) (lift (S O) O x4) x2) -(pr3_head_12 (CHead c (Bind x0) x1) (lift (S O) O t0) (lift (S O) O x4) -(pr3_lift (CHead c (Bind x0) x1) c (S O) O (drop_drop (Bind x0) O c c -(drop_refl c) x1) t0 x4 H8) (Flat Appl) x2 x2 (pr3_refl (CHead (CHead c (Bind -x0) x1) (Flat Appl) (lift (S O) O x4)) x2)))) u2 (pr3_t (THead (Bind x0) x5 -(THead (Flat Appl) (lift (S O) O x4) x3)) (THead (Bind x0) x1 (THead (Flat -Appl) (lift (S O) O x4) x2)) c (pr3_head_12 c x1 x5 H9 (Bind x0) (THead (Flat -Appl) (lift (S O) O x4) x2) (THead (Flat Appl) (lift (S O) O x4) x3) -(pr3_thin_dx (CHead c (Bind x0) x5) x2 x3 H10 (lift (S O) O x4) Appl)) u2 -H7)))))))))))))) H4)) H3))))))))) vs)))))). -(* COMMENTS -Initial nodes: 3571 -END *) - -theorem pr3_iso_appls_bind: - \forall (b: B).((not (eq B b Abst)) \to (\forall (vs: TList).(\forall (u: -T).(\forall (t: T).(let u1 \def (THeads (Flat Appl) vs (THead (Bind b) u t)) -in (\forall (c: C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to -(\forall (P: Prop).P))) \to (pr3 c (THead (Bind b) u (THeads (Flat Appl) -(lifts (S O) O vs) t)) u2)))))))))) -\def - \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (vs: -TList).(tlist_ind_rev (\lambda (t: TList).(\forall (u: T).(\forall (t0: -T).(let u1 \def (THeads (Flat Appl) t (THead (Bind b) u t0)) in (\forall (c: -C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: -Prop).P))) \to (pr3 c (THead (Bind b) u (THeads (Flat Appl) (lifts (S O) O t) -t0)) u2))))))))) (\lambda (u: T).(\lambda (t: T).(\lambda (c: C).(\lambda -(u2: T).(\lambda (H0: (pr3 c (THead (Bind b) u t) u2)).(\lambda (_: (((iso -(THead (Bind b) u t) u2) \to (\forall (P: Prop).P)))).H0)))))) (\lambda (ts: -TList).(\lambda (t: T).(\lambda (H0: ((\forall (u: T).(\forall (t0: -T).(\forall (c: C).(\forall (u2: T).((pr3 c (THeads (Flat Appl) ts (THead -(Bind b) u t0)) u2) \to ((((iso (THeads (Flat Appl) ts (THead (Bind b) u t0)) -u2) \to (\forall (P: Prop).P))) \to (pr3 c (THead (Bind b) u (THeads (Flat -Appl) (lifts (S O) O ts) t0)) u2))))))))).(\lambda (u: T).(\lambda (t0: -T).(\lambda (c: C).(\lambda (u2: T).(\lambda (H1: (pr3 c (THeads (Flat Appl) -(TApp ts t) (THead (Bind b) u t0)) u2)).(\lambda (H2: (((iso (THeads (Flat -Appl) (TApp ts t) (THead (Bind b) u t0)) u2) \to (\forall (P: -Prop).P)))).(eq_ind_r TList (TApp (lifts (S O) O ts) (lift (S O) O t)) -(\lambda (t1: TList).(pr3 c (THead (Bind b) u (THeads (Flat Appl) t1 t0)) -u2)) (eq_ind_r T (THeads (Flat Appl) (lifts (S O) O ts) (THead (Flat Appl) -(lift (S O) O t) t0)) (\lambda (t1: T).(pr3 c (THead (Bind b) u t1) u2)) (let -H3 \def (eq_ind T (THeads (Flat Appl) (TApp ts t) (THead (Bind b) u t0)) -(\lambda (t1: T).(pr3 c t1 u2)) H1 (THeads (Flat Appl) ts (THead (Flat Appl) -t (THead (Bind b) u t0))) (theads_tapp (Flat Appl) t (THead (Bind b) u t0) -ts)) in (let H4 \def (eq_ind T (THeads (Flat Appl) (TApp ts t) (THead (Bind -b) u t0)) (\lambda (t1: T).((iso t1 u2) \to (\forall (P: Prop).P))) H2 -(THeads (Flat Appl) ts (THead (Flat Appl) t (THead (Bind b) u t0))) -(theads_tapp (Flat Appl) t (THead (Bind b) u t0) ts)) in (TList_ind (\lambda -(t1: TList).(((\forall (u0: T).(\forall (t2: T).(\forall (c0: C).(\forall -(u3: T).((pr3 c0 (THeads (Flat Appl) t1 (THead (Bind b) u0 t2)) u3) \to -((((iso (THeads (Flat Appl) t1 (THead (Bind b) u0 t2)) u3) \to (\forall (P: -Prop).P))) \to (pr3 c0 (THead (Bind b) u0 (THeads (Flat Appl) (lifts (S O) O -t1) t2)) u3)))))))) \to ((pr3 c (THeads (Flat Appl) t1 (THead (Flat Appl) t -(THead (Bind b) u t0))) u2) \to ((((iso (THeads (Flat Appl) t1 (THead (Flat -Appl) t (THead (Bind b) u t0))) u2) \to (\forall (P: Prop).P))) \to (pr3 c -(THead (Bind b) u (THeads (Flat Appl) (lifts (S O) O t1) (THead (Flat Appl) -(lift (S O) O t) t0))) u2))))) (\lambda (_: ((\forall (u0: T).(\forall (t1: -T).(\forall (c0: C).(\forall (u3: T).((pr3 c0 (THeads (Flat Appl) TNil (THead -(Bind b) u0 t1)) u3) \to ((((iso (THeads (Flat Appl) TNil (THead (Bind b) u0 -t1)) u3) \to (\forall (P: Prop).P))) \to (pr3 c0 (THead (Bind b) u0 (THeads -(Flat Appl) (lifts (S O) O TNil) t1)) u3))))))))).(\lambda (H6: (pr3 c -(THeads (Flat Appl) TNil (THead (Flat Appl) t (THead (Bind b) u t0))) -u2)).(\lambda (H7: (((iso (THeads (Flat Appl) TNil (THead (Flat Appl) t -(THead (Bind b) u t0))) u2) \to (\forall (P: Prop).P)))).(pr3_iso_appl_bind b -H t u t0 c u2 H6 H7)))) (\lambda (t1: T).(\lambda (ts0: TList).(\lambda (_: -((((\forall (u0: T).(\forall (t2: T).(\forall (c0: C).(\forall (u3: T).((pr3 -c0 (THeads (Flat Appl) ts0 (THead (Bind b) u0 t2)) u3) \to ((((iso (THeads -(Flat Appl) ts0 (THead (Bind b) u0 t2)) u3) \to (\forall (P: Prop).P))) \to -(pr3 c0 (THead (Bind b) u0 (THeads (Flat Appl) (lifts (S O) O ts0) t2)) -u3)))))))) \to ((pr3 c (THeads (Flat Appl) ts0 (THead (Flat Appl) t (THead -(Bind b) u t0))) u2) \to ((((iso (THeads (Flat Appl) ts0 (THead (Flat Appl) t -(THead (Bind b) u t0))) u2) \to (\forall (P: Prop).P))) \to (pr3 c (THead -(Bind b) u (THeads (Flat Appl) (lifts (S O) O ts0) (THead (Flat Appl) (lift -(S O) O t) t0))) u2)))))).(\lambda (H5: ((\forall (u0: T).(\forall (t2: -T).(\forall (c0: C).(\forall (u3: T).((pr3 c0 (THeads (Flat Appl) (TCons t1 -ts0) (THead (Bind b) u0 t2)) u3) \to ((((iso (THeads (Flat Appl) (TCons t1 -ts0) (THead (Bind b) u0 t2)) u3) \to (\forall (P: Prop).P))) \to (pr3 c0 -(THead (Bind b) u0 (THeads (Flat Appl) (lifts (S O) O (TCons t1 ts0)) t2)) -u3))))))))).(\lambda (H6: (pr3 c (THeads (Flat Appl) (TCons t1 ts0) (THead -(Flat Appl) t (THead (Bind b) u t0))) u2)).(\lambda (H7: (((iso (THeads (Flat -Appl) (TCons t1 ts0) (THead (Flat Appl) t (THead (Bind b) u t0))) u2) \to -(\forall (P: Prop).P)))).(H5 u (THead (Flat Appl) (lift (S O) O t) t0) c u2 -(pr3_iso_appls_appl_bind b H t u t0 (TCons t1 ts0) c u2 H6 H7) (\lambda (H8: -(iso (THeads (Flat Appl) (TCons t1 ts0) (THead (Bind b) u (THead (Flat Appl) -(lift (S O) O t) t0))) u2)).(\lambda (P: Prop).(H7 (iso_trans (THeads (Flat -Appl) (TCons t1 ts0) (THead (Flat Appl) t (THead (Bind b) u t0))) (THeads -(Flat Appl) (TCons t1 ts0) (THead (Bind b) u (THead (Flat Appl) (lift (S O) O -t) t0))) (iso_head t1 t1 (THeads (Flat Appl) ts0 (THead (Flat Appl) t (THead -(Bind b) u t0))) (THeads (Flat Appl) ts0 (THead (Bind b) u (THead (Flat Appl) -(lift (S O) O t) t0))) (Flat Appl)) u2 H8) P)))))))))) ts H0 H3 H4))) (THeads -(Flat Appl) (TApp (lifts (S O) O ts) (lift (S O) O t)) t0) (theads_tapp (Flat -Appl) (lift (S O) O t) t0 (lifts (S O) O ts))) (lifts (S O) O (TApp ts t)) -(lifts_tapp (S O) O t ts))))))))))) vs))). -(* COMMENTS -Initial nodes: 1681 -END *) - -theorem pr3_iso_beta: - \forall (v: T).(\forall (w: T).(\forall (t: T).(let u1 \def (THead (Flat -Appl) v (THead (Bind Abst) w t)) in (\forall (c: C).(\forall (u2: T).((pr3 c -u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to (pr3 c (THead (Bind -Abbr) v t) u2)))))))) -\def - \lambda (v: T).(\lambda (w: T).(\lambda (t: T).(\lambda (c: C).(\lambda (u2: -T).(\lambda (H: (pr3 c (THead (Flat Appl) v (THead (Bind Abst) w t)) -u2)).(\lambda (H0: (((iso (THead (Flat Appl) v (THead (Bind Abst) w t)) u2) -\to (\forall (P: Prop).P)))).(let H1 \def (pr3_gen_appl c v (THead (Bind -Abst) w t) u2 H) in (or3_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq -T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c v -u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THead (Bind Abst) w t) t2)))) -(ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: -T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v u3))))) (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) -w t) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind -b) u) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B -b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) w t) (THead -(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) -y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda -(_: T).(pr3 c v u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (pr3 c -(THead (Bind Abbr) v t) u2) (\lambda (H2: (ex3_2 T T (\lambda (u3: -T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: -T).(\lambda (_: T).(pr3 c v u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c -(THead (Bind Abst) w t) t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: -T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: -T).(pr3 c v u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THead (Bind Abst) -w t) t2))) (pr3 c (THead (Bind Abbr) v t) u2) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H3: (eq T u2 (THead (Flat Appl) x0 x1))).(\lambda (_: (pr3 c v -x0)).(\lambda (_: (pr3 c (THead (Bind Abst) w t) x1)).(let H6 \def (eq_ind T -u2 (\lambda (t0: T).((iso (THead (Flat Appl) v (THead (Bind Abst) w t)) t0) -\to (\forall (P: Prop).P))) H0 (THead (Flat Appl) x0 x1) H3) in (eq_ind_r T -(THead (Flat Appl) x0 x1) (\lambda (t0: T).(pr3 c (THead (Bind Abbr) v t) -t0)) (H6 (iso_head v x0 (THead (Bind Abst) w t) x1 (Flat Appl)) (pr3 c (THead -(Bind Abbr) v t) (THead (Flat Appl) x0 x1))) u2 H3))))))) H2)) (\lambda (H2: -(ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: -T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v u3))))) (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) -w t) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind -b) u) z1 t2))))))))).(ex4_4_ind T T T T (\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v -u3))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(pr3 c (THead (Bind Abst) w t) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall -(u: T).(pr3 (CHead c (Bind b) u) z1 t2))))))) (pr3 c (THead (Bind Abbr) v t) -u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: -T).(\lambda (H3: (pr3 c (THead (Bind Abbr) x2 x3) u2)).(\lambda (H4: (pr3 c v -x2)).(\lambda (H5: (pr3 c (THead (Bind Abst) w t) (THead (Bind Abst) x0 -x1))).(\lambda (H6: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) -u) x1 x3))))).(let H7 \def (pr3_gen_abst c w t (THead (Bind Abst) x0 x1) H5) -in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind Abst) -x0 x1) (THead (Bind Abst) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c w -u3))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 -(CHead c (Bind b) u) t t2))))) (pr3 c (THead (Bind Abbr) v t) u2) (\lambda -(x4: T).(\lambda (x5: T).(\lambda (H8: (eq T (THead (Bind Abst) x0 x1) (THead -(Bind Abst) x4 x5))).(\lambda (H9: (pr3 c w x4)).(\lambda (H10: ((\forall (b: -B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t x5))))).(let H11 \def (f_equal -T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ t0 _) \Rightarrow t0])) -(THead (Bind Abst) x0 x1) (THead (Bind Abst) x4 x5) H8) in ((let H12 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow x1 | (TLRef _) \Rightarrow x1 | (THead _ _ t0) -\Rightarrow t0])) (THead (Bind Abst) x0 x1) (THead (Bind Abst) x4 x5) H8) in -(\lambda (H13: (eq T x0 x4)).(let H14 \def (eq_ind_r T x5 (\lambda (t0: -T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t t0)))) H10 x1 -H12) in (let H15 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 c w t0)) H9 x0 -H13) in (pr3_t (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) v t) c -(pr3_head_12 c v x2 H4 (Bind Abbr) t x3 (pr3_t x1 t (CHead c (Bind Abbr) x2) -(H14 Abbr x2) x3 (H6 Abbr x2))) u2 H3))))) H11))))))) H7)))))))))) H2)) -(\lambda (H2: (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) -(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) w t) (THead (Bind b) y1 -z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: -T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat -Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v -u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c -(THead (Bind Abst) w t) (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u3: T).(\lambda -(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u3) z2)) -u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v u3))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) -y2) z1 z2))))))) (pr3 c (THead (Bind Abbr) v t) u2) (\lambda (x0: B).(\lambda -(x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: -T).(\lambda (H3: (not (eq B x0 Abst))).(\lambda (H4: (pr3 c (THead (Bind -Abst) w t) (THead (Bind x0) x1 x2))).(\lambda (H5: (pr3 c (THead (Bind x0) x5 -(THead (Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda (_: (pr3 c v -x4)).(\lambda (_: (pr3 c x1 x5)).(\lambda (H8: (pr3 (CHead c (Bind x0) x5) x2 -x3)).(let H9 \def (pr3_gen_abst c w t (THead (Bind x0) x1 x2) H4) in -(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind x0) x1 -x2) (THead (Bind Abst) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c w -u3))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 -(CHead c (Bind b) u) t t2))))) (pr3 c (THead (Bind Abbr) v t) u2) (\lambda -(x6: T).(\lambda (x7: T).(\lambda (H10: (eq T (THead (Bind x0) x1 x2) (THead -(Bind Abst) x6 x7))).(\lambda (H11: (pr3 c w x6)).(\lambda (H12: ((\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t x7))))).(let H13 \def -(f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with -[(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead k _ _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) -\Rightarrow b | (Flat _) \Rightarrow x0])])) (THead (Bind x0) x1 x2) (THead -(Bind Abst) x6 x7) H10) in ((let H14 \def (f_equal T T (\lambda (e: T).(match -e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x1 | (TLRef _) -\Rightarrow x1 | (THead _ t0 _) \Rightarrow t0])) (THead (Bind x0) x1 x2) -(THead (Bind Abst) x6 x7) H10) in ((let H15 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x2 | -(TLRef _) \Rightarrow x2 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind x0) -x1 x2) (THead (Bind Abst) x6 x7) H10) in (\lambda (H16: (eq T x1 -x6)).(\lambda (H17: (eq B x0 Abst)).(let H18 \def (eq_ind_r T x7 (\lambda -(t0: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t t0)))) -H12 x2 H15) in (let H19 \def (eq_ind_r T x6 (\lambda (t0: T).(pr3 c w t0)) -H11 x1 H16) in (let H20 \def (eq_ind B x0 (\lambda (b: B).(pr3 (CHead c (Bind -b) x5) x2 x3)) H8 Abst H17) in (let H21 \def (eq_ind B x0 (\lambda (b: -B).(pr3 c (THead (Bind b) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) u2)) -H5 Abst H17) in (let H22 \def (eq_ind B x0 (\lambda (b: B).(not (eq B b -Abst))) H3 Abst H17) in (let H23 \def (match (H22 (refl_equal B Abst)) in -False return (\lambda (_: False).(pr3 c (THead (Bind Abbr) v t) u2)) with []) -in H23))))))))) H14)) H13))))))) H9)))))))))))))) H2)) H1)))))))). -(* COMMENTS -Initial nodes: 2459 -END *) - -theorem pr3_iso_appls_beta: - \forall (us: TList).(\forall (v: T).(\forall (w: T).(\forall (t: T).(let u1 -\def (THeads (Flat Appl) us (THead (Flat Appl) v (THead (Bind Abst) w t))) in -(\forall (c: C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to -(\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) us (THead (Bind Abbr) -v t)) u2))))))))) -\def - \lambda (us: TList).(TList_ind (\lambda (t: TList).(\forall (v: T).(\forall -(w: T).(\forall (t0: T).(let u1 \def (THeads (Flat Appl) t (THead (Flat Appl) -v (THead (Bind Abst) w t0))) in (\forall (c: C).(\forall (u2: T).((pr3 c u1 -u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to (pr3 c (THeads (Flat -Appl) t (THead (Bind Abbr) v t0)) u2)))))))))) (\lambda (v: T).(\lambda (w: -T).(\lambda (t: T).(\lambda (c: C).(\lambda (u2: T).(\lambda (H: (pr3 c -(THead (Flat Appl) v (THead (Bind Abst) w t)) u2)).(\lambda (H0: (((iso -(THead (Flat Appl) v (THead (Bind Abst) w t)) u2) \to (\forall (P: -Prop).P)))).(pr3_iso_beta v w t c u2 H H0)))))))) (\lambda (t: T).(\lambda -(t0: TList).(\lambda (H: ((\forall (v: T).(\forall (w: T).(\forall (t1: -T).(\forall (c: C).(\forall (u2: T).((pr3 c (THeads (Flat Appl) t0 (THead -(Flat Appl) v (THead (Bind Abst) w t1))) u2) \to ((((iso (THeads (Flat Appl) -t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) u2) \to (\forall (P: -Prop).P))) \to (pr3 c (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1)) -u2)))))))))).(\lambda (v: T).(\lambda (w: T).(\lambda (t1: T).(\lambda (c: -C).(\lambda (u2: T).(\lambda (H0: (pr3 c (THead (Flat Appl) t (THeads (Flat -Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1)))) u2)).(\lambda (H1: -(((iso (THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Flat Appl) v -(THead (Bind Abst) w t1)))) u2) \to (\forall (P: Prop).P)))).(let H2 \def -(pr3_gen_appl c t (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind -Abst) w t1))) u2 H0) in (or3_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t2: -T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: -T).(pr3 c t u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) -t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) t2)))) (ex4_4 T T T T -(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c -(THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u3: T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat -Appl) v (THead (Bind Abst) w t1))) (THead (Bind Abst) y1 z1)))))) (\lambda -(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: -B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c -(THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) (THead -(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) -y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda -(_: T).(pr3 c t u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (pr3 c -(THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) u2) -(\lambda (H3: (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead -(Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t u3))) -(\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat -Appl) v (THead (Bind Abst) w t1))) t2))))).(ex3_2_ind T T (\lambda (u3: -T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: -T).(\lambda (_: T).(pr3 c t u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c -(THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) t2))) -(pr3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) -u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T u2 (THead (Flat -Appl) x0 x1))).(\lambda (_: (pr3 c t x0)).(\lambda (_: (pr3 c (THeads (Flat -Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) x1)).(let H7 \def -(eq_ind T u2 (\lambda (t2: T).((iso (THead (Flat Appl) t (THeads (Flat Appl) -t0 (THead (Flat Appl) v (THead (Bind Abst) w t1)))) t2) \to (\forall (P: -Prop).P))) H1 (THead (Flat Appl) x0 x1) H4) in (eq_ind_r T (THead (Flat Appl) -x0 x1) (\lambda (t2: T).(pr3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 -(THead (Bind Abbr) v t1))) t2)) (H7 (iso_head t x0 (THeads (Flat Appl) t0 -(THead (Flat Appl) v (THead (Bind Abst) w t1))) x1 (Flat Appl)) (pr3 c (THead -(Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) (THead (Flat -Appl) x0 x1))) u2 H4))))))) H3)) (\lambda (H3: (ex4_4 T T T T (\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind -Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: -T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat -Appl) v (THead (Bind Abst) w t1))) (THead (Bind Abst) y1 z1)))))) (\lambda -(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: -B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2))))))))).(ex4_4_ind T T T -T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c -(THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u3: T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat -Appl) v (THead (Bind Abst) w t1))) (THead (Bind Abst) y1 z1)))))) (\lambda -(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: -B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2))))))) (pr3 c (THead (Flat -Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) u2) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H4: (pr3 c -(THead (Bind Abbr) x2 x3) u2)).(\lambda (H5: (pr3 c t x2)).(\lambda (H6: (pr3 -c (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) -(THead (Bind Abst) x0 x1))).(\lambda (H7: ((\forall (b: B).(\forall (u: -T).(pr3 (CHead c (Bind b) u) x1 x3))))).(pr3_t (THead (Bind Abbr) t x1) -(THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) c -(pr3_t (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) (THead (Flat Appl) t -(THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) c (pr3_thin_dx c (THeads -(Flat Appl) t0 (THead (Bind Abbr) v t1)) (THead (Bind Abst) x0 x1) (H v w t1 -c (THead (Bind Abst) x0 x1) H6 (\lambda (H8: (iso (THeads (Flat Appl) t0 -(THead (Flat Appl) v (THead (Bind Abst) w t1))) (THead (Bind Abst) x0 -x1))).(\lambda (P: Prop).(iso_flats_flat_bind_false Appl Appl Abst x0 v x1 -(THead (Bind Abst) w t1) t0 H8 P)))) t Appl) (THead (Bind Abbr) t x1) -(pr3_pr2 c (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) -t x1) (pr2_free c (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) (THead -(Bind Abbr) t x1) (pr0_beta x0 t t (pr0_refl t) x1 x1 (pr0_refl x1))))) u2 -(pr3_t (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) t x1) c (pr3_head_12 c t -x2 H5 (Bind Abbr) x1 x3 (H7 Abbr x2)) u2 H4)))))))))) H3)) (\lambda (H3: -(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) -w t1))) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c -(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: -T).(\lambda (_: T).(pr3 c t u3))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 -y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 -z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat -Appl) v (THead (Bind Abst) w t1))) (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u3: T).(\lambda -(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u3) z2)) -u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t u3))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) -y2) z1 z2))))))) (pr3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (THead -(Bind Abbr) v t1))) u2) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: -T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H4: (not (eq -B x0 Abst))).(\lambda (H5: (pr3 c (THeads (Flat Appl) t0 (THead (Flat Appl) v -(THead (Bind Abst) w t1))) (THead (Bind x0) x1 x2))).(\lambda (H6: (pr3 c -(THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda -(H7: (pr3 c t x4)).(\lambda (H8: (pr3 c x1 x5)).(\lambda (H9: (pr3 (CHead c -(Bind x0) x5) x2 x3)).(pr3_t (THead (Bind x0) x1 (THead (Flat Appl) (lift (S -O) O x4) x2)) (THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) -v t1))) c (pr3_t (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O t) x2)) -(THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) c -(pr3_t (THead (Flat Appl) t (THead (Bind x0) x1 x2)) (THead (Flat Appl) t -(THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) c (pr3_thin_dx c (THeads -(Flat Appl) t0 (THead (Bind Abbr) v t1)) (THead (Bind x0) x1 x2) (H v w t1 c -(THead (Bind x0) x1 x2) H5 (\lambda (H10: (iso (THeads (Flat Appl) t0 (THead -(Flat Appl) v (THead (Bind Abst) w t1))) (THead (Bind x0) x1 x2))).(\lambda -(P: Prop).(iso_flats_flat_bind_false Appl Appl x0 x1 v x2 (THead (Bind Abst) -w t1) t0 H10 P)))) t Appl) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) -O t) x2)) (pr3_pr2 c (THead (Flat Appl) t (THead (Bind x0) x1 x2)) (THead -(Bind x0) x1 (THead (Flat Appl) (lift (S O) O t) x2)) (pr2_free c (THead -(Flat Appl) t (THead (Bind x0) x1 x2)) (THead (Bind x0) x1 (THead (Flat Appl) -(lift (S O) O t) x2)) (pr0_upsilon x0 H4 t t (pr0_refl t) x1 x1 (pr0_refl x1) -x2 x2 (pr0_refl x2))))) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O -x4) x2)) (pr3_head_12 c x1 x1 (pr3_refl c x1) (Bind x0) (THead (Flat Appl) -(lift (S O) O t) x2) (THead (Flat Appl) (lift (S O) O x4) x2) (pr3_head_12 -(CHead c (Bind x0) x1) (lift (S O) O t) (lift (S O) O x4) (pr3_lift (CHead c -(Bind x0) x1) c (S O) O (drop_drop (Bind x0) O c c (drop_refl c) x1) t x4 H7) -(Flat Appl) x2 x2 (pr3_refl (CHead (CHead c (Bind x0) x1) (Flat Appl) (lift -(S O) O x4)) x2)))) u2 (pr3_t (THead (Bind x0) x5 (THead (Flat Appl) (lift (S -O) O x4) x3)) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) c -(pr3_head_12 c x1 x5 H8 (Bind x0) (THead (Flat Appl) (lift (S O) O x4) x2) -(THead (Flat Appl) (lift (S O) O x4) x3) (pr3_thin_dx (CHead c (Bind x0) x5) -x2 x3 H9 (lift (S O) O x4) Appl)) u2 H6)))))))))))))) H3)) H2)))))))))))) us). -(* COMMENTS -Initial nodes: 3345 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/pr1.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/pr1.ma deleted file mode 100644 index e428daf37..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/pr1.ma +++ /dev/null @@ -1,34 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr3/defs.ma". - -include "Basic-1/pr1/defs.ma". - -theorem pr3_pr1: - \forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall (c: C).(pr3 c t1 -t2)))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr1 t1 t2)).(pr1_ind (\lambda -(t: T).(\lambda (t0: T).(\forall (c: C).(pr3 c t t0)))) (\lambda (t: -T).(\lambda (c: C).(pr3_refl c t))) (\lambda (t0: T).(\lambda (t3: -T).(\lambda (H0: (pr0 t3 t0)).(\lambda (t4: T).(\lambda (_: (pr1 t0 -t4)).(\lambda (H2: ((\forall (c: C).(pr3 c t0 t4)))).(\lambda (c: -C).(pr3_sing c t0 t3 (pr2_free c t3 t0 H0) t4 (H2 c))))))))) t1 t2 H))). -(* COMMENTS -Initial nodes: 95 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/pr3.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/pr3.ma deleted file mode 100644 index 935850b1c..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/pr3.ma +++ /dev/null @@ -1,74 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr3/props.ma". - -include "Basic-1/pr2/pr2.ma". - -theorem pr3_strip: - \forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr3 c t0 t1) \to (\forall -(t2: T).((pr2 c t0 t2) \to (ex2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: -T).(pr3 c t2 t)))))))) -\def - \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr3 c t0 -t1)).(pr3_ind c (\lambda (t: T).(\lambda (t2: T).(\forall (t3: T).((pr2 c t -t3) \to (ex2 T (\lambda (t4: T).(pr3 c t2 t4)) (\lambda (t4: T).(pr3 c t3 -t4))))))) (\lambda (t: T).(\lambda (t2: T).(\lambda (H0: (pr2 c t -t2)).(ex_intro2 T (\lambda (t3: T).(pr3 c t t3)) (\lambda (t3: T).(pr3 c t2 -t3)) t2 (pr3_pr2 c t t2 H0) (pr3_refl c t2))))) (\lambda (t2: T).(\lambda -(t3: T).(\lambda (H0: (pr2 c t3 t2)).(\lambda (t4: T).(\lambda (_: (pr3 c t2 -t4)).(\lambda (H2: ((\forall (t5: T).((pr2 c t2 t5) \to (ex2 T (\lambda (t: -T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t))))))).(\lambda (t5: T).(\lambda -(H3: (pr2 c t3 t5)).(ex2_ind T (\lambda (t: T).(pr2 c t5 t)) (\lambda (t: -T).(pr2 c t2 t)) (ex2 T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c -t5 t))) (\lambda (x: T).(\lambda (H4: (pr2 c t5 x)).(\lambda (H5: (pr2 c t2 -x)).(ex2_ind T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c x t)) -(ex2 T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t))) (\lambda -(x0: T).(\lambda (H6: (pr3 c t4 x0)).(\lambda (H7: (pr3 c x x0)).(ex_intro2 T -(\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t)) x0 H6 (pr3_sing c -x t5 H4 x0 H7))))) (H2 x H5))))) (pr2_confluence c t3 t5 H3 t2 H0)))))))))) -t0 t1 H)))). -(* COMMENTS -Initial nodes: 375 -END *) - -theorem pr3_confluence: - \forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr3 c t0 t1) \to (\forall -(t2: T).((pr3 c t0 t2) \to (ex2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: -T).(pr3 c t2 t)))))))) -\def - \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr3 c t0 -t1)).(pr3_ind c (\lambda (t: T).(\lambda (t2: T).(\forall (t3: T).((pr3 c t -t3) \to (ex2 T (\lambda (t4: T).(pr3 c t2 t4)) (\lambda (t4: T).(pr3 c t3 -t4))))))) (\lambda (t: T).(\lambda (t2: T).(\lambda (H0: (pr3 c t -t2)).(ex_intro2 T (\lambda (t3: T).(pr3 c t t3)) (\lambda (t3: T).(pr3 c t2 -t3)) t2 H0 (pr3_refl c t2))))) (\lambda (t2: T).(\lambda (t3: T).(\lambda -(H0: (pr2 c t3 t2)).(\lambda (t4: T).(\lambda (_: (pr3 c t2 t4)).(\lambda -(H2: ((\forall (t5: T).((pr3 c t2 t5) \to (ex2 T (\lambda (t: T).(pr3 c t4 -t)) (\lambda (t: T).(pr3 c t5 t))))))).(\lambda (t5: T).(\lambda (H3: (pr3 c -t3 t5)).(ex2_ind T (\lambda (t: T).(pr3 c t5 t)) (\lambda (t: T).(pr3 c t2 -t)) (ex2 T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t))) -(\lambda (x: T).(\lambda (H4: (pr3 c t5 x)).(\lambda (H5: (pr3 c t2 -x)).(ex2_ind T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c x t)) -(ex2 T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t))) (\lambda -(x0: T).(\lambda (H6: (pr3 c t4 x0)).(\lambda (H7: (pr3 c x x0)).(ex_intro2 T -(\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t)) x0 H6 (pr3_t x t5 -c H4 x0 H7))))) (H2 x H5))))) (pr3_strip c t3 t5 H3 t2 H0)))))))))) t0 t1 -H)))). -(* COMMENTS -Initial nodes: 367 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/props.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/props.ma deleted file mode 100644 index 777f5d85d..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/props.ma +++ /dev/null @@ -1,407 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr3/pr1.ma". - -include "Basic-1/pr2/props.ma". - -include "Basic-1/pr1/props.ma". - -theorem clear_pr3_trans: - \forall (c2: C).(\forall (t1: T).(\forall (t2: T).((pr3 c2 t1 t2) \to -(\forall (c1: C).((clear c1 c2) \to (pr3 c1 t1 t2)))))) -\def - \lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c2 t1 -t2)).(\lambda (c1: C).(\lambda (H0: (clear c1 c2)).(pr3_ind c2 (\lambda (t: -T).(\lambda (t0: T).(pr3 c1 t t0))) (\lambda (t: T).(pr3_refl c1 t)) (\lambda -(t3: T).(\lambda (t4: T).(\lambda (H1: (pr2 c2 t4 t3)).(\lambda (t5: -T).(\lambda (_: (pr3 c2 t3 t5)).(\lambda (H3: (pr3 c1 t3 t5)).(pr3_sing c1 t3 -t4 (clear_pr2_trans c2 t4 t3 H1 c1 H0) t5 H3))))))) t1 t2 H)))))). -(* COMMENTS -Initial nodes: 107 -END *) - -theorem pr3_pr2: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (pr3 c -t1 t2)))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 -t2)).(pr3_sing c t2 t1 H t2 (pr3_refl c t2))))). -(* COMMENTS -Initial nodes: 31 -END *) - -theorem pr3_t: - \forall (t2: T).(\forall (t1: T).(\forall (c: C).((pr3 c t1 t2) \to (\forall -(t3: T).((pr3 c t2 t3) \to (pr3 c t1 t3)))))) -\def - \lambda (t2: T).(\lambda (t1: T).(\lambda (c: C).(\lambda (H: (pr3 c t1 -t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (t3: T).((pr3 c t0 -t3) \to (pr3 c t t3))))) (\lambda (t: T).(\lambda (t3: T).(\lambda (H0: (pr3 -c t t3)).H0))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c t3 -t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2: ((\forall -(t5: T).((pr3 c t4 t5) \to (pr3 c t0 t5))))).(\lambda (t5: T).(\lambda (H3: -(pr3 c t4 t5)).(pr3_sing c t0 t3 H0 t5 (H2 t5 H3)))))))))) t1 t2 H)))). -(* COMMENTS -Initial nodes: 127 -END *) - -theorem pr3_thin_dx: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall -(u: T).(\forall (f: F).(pr3 c (THead (Flat f) u t1) (THead (Flat f) u -t2))))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 -t2)).(\lambda (u: T).(\lambda (f: F).(pr3_ind c (\lambda (t: T).(\lambda (t0: -T).(pr3 c (THead (Flat f) u t) (THead (Flat f) u t0)))) (\lambda (t: -T).(pr3_refl c (THead (Flat f) u t))) (\lambda (t0: T).(\lambda (t3: -T).(\lambda (H0: (pr2 c t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0 -t4)).(\lambda (H2: (pr3 c (THead (Flat f) u t0) (THead (Flat f) u -t4))).(pr3_sing c (THead (Flat f) u t0) (THead (Flat f) u t3) (pr2_thin_dx c -t3 t0 H0 u f) (THead (Flat f) u t4) H2))))))) t1 t2 H)))))). -(* COMMENTS -Initial nodes: 167 -END *) - -theorem pr3_head_1: - \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall -(k: K).(\forall (t: T).(pr3 c (THead k u1 t) (THead k u2 t))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1 -u2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (k: K).(\forall -(t1: T).(pr3 c (THead k t t1) (THead k t0 t1)))))) (\lambda (t: T).(\lambda -(k: K).(\lambda (t0: T).(pr3_refl c (THead k t t0))))) (\lambda (t2: -T).(\lambda (t1: T).(\lambda (H0: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda -(_: (pr3 c t2 t3)).(\lambda (H2: ((\forall (k: K).(\forall (t: T).(pr3 c -(THead k t2 t) (THead k t3 t)))))).(\lambda (k: K).(\lambda (t: T).(pr3_sing -c (THead k t2 t) (THead k t1 t) (pr2_head_1 c t1 t2 H0 k t) (THead k t3 t) -(H2 k t)))))))))) u1 u2 H)))). -(* COMMENTS -Initial nodes: 167 -END *) - -theorem pr3_head_2: - \forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).(\forall -(k: K).((pr3 (CHead c k u) t1 t2) \to (pr3 c (THead k u t1) (THead k u -t2))))))) -\def - \lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(k: K).(\lambda (H: (pr3 (CHead c k u) t1 t2)).(pr3_ind (CHead c k u) -(\lambda (t: T).(\lambda (t0: T).(pr3 c (THead k u t) (THead k u t0)))) -(\lambda (t: T).(pr3_refl c (THead k u t))) (\lambda (t0: T).(\lambda (t3: -T).(\lambda (H0: (pr2 (CHead c k u) t3 t0)).(\lambda (t4: T).(\lambda (_: -(pr3 (CHead c k u) t0 t4)).(\lambda (H2: (pr3 c (THead k u t0) (THead k u -t4))).(pr3_sing c (THead k u t0) (THead k u t3) (pr2_head_2 c u t3 t0 k H0) -(THead k u t4) H2))))))) t1 t2 H)))))). -(* COMMENTS -Initial nodes: 175 -END *) - -theorem pr3_head_21: - \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall -(k: K).(\forall (t1: T).(\forall (t2: T).((pr3 (CHead c k u1) t1 t2) \to (pr3 -c (THead k u1 t1) (THead k u2 t2))))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1 -u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 -(CHead c k u1) t1 t2)).(pr3_t (THead k u1 t2) (THead k u1 t1) c (pr3_head_2 c -u1 t1 t2 k H0) (THead k u2 t2) (pr3_head_1 c u1 u2 H k t2))))))))). -(* COMMENTS -Initial nodes: 89 -END *) - -theorem pr3_head_12: - \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall -(k: K).(\forall (t1: T).(\forall (t2: T).((pr3 (CHead c k u2) t1 t2) \to (pr3 -c (THead k u1 t1) (THead k u2 t2))))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1 -u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 -(CHead c k u2) t1 t2)).(pr3_t (THead k u2 t1) (THead k u1 t1) c (pr3_head_1 c -u1 u2 H k t1) (THead k u2 t2) (pr3_head_2 c u2 t1 t2 k H0))))))))). -(* COMMENTS -Initial nodes: 89 -END *) - -theorem pr3_cflat: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall -(f: F).(\forall (v: T).(pr3 (CHead c (Flat f) v) t1 t2)))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 -t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (f: F).(\forall (v: -T).(pr3 (CHead c (Flat f) v) t t0))))) (\lambda (t: T).(\lambda (f: -F).(\lambda (v: T).(pr3_refl (CHead c (Flat f) v) t)))) (\lambda (t3: -T).(\lambda (t4: T).(\lambda (H0: (pr2 c t4 t3)).(\lambda (t5: T).(\lambda -(_: (pr3 c t3 t5)).(\lambda (H2: ((\forall (f: F).(\forall (v: T).(pr3 (CHead -c (Flat f) v) t3 t5))))).(\lambda (f: F).(\lambda (v: T).(pr3_sing (CHead c -(Flat f) v) t3 t4 (pr2_cflat c t4 t3 H0 f v) t5 (H2 f v)))))))))) t1 t2 H)))). -(* COMMENTS -Initial nodes: 151 -END *) - -theorem pr3_flat: - \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall -(t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall (f: F).(pr3 c (THead -(Flat f) u1 t1) (THead (Flat f) u2 t2))))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1 -u2)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 c t1 t2)).(\lambda -(f: F).(pr3_head_12 c u1 u2 H (Flat f) t1 t2 (pr3_cflat c t1 t2 H0 f -u2))))))))). -(* COMMENTS -Initial nodes: 59 -END *) - -theorem pr3_pr0_pr2_t: - \forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (c: C).(\forall -(t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pr3 -(CHead c k u1) t1 t2)))))))) -\def - \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr0 u1 u2)).(\lambda (c: -C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k: K).(\lambda (H0: (pr2 -(CHead c k u2) t1 t2)).(insert_eq C (CHead c k u2) (\lambda (c0: C).(pr2 c0 -t1 t2)) (\lambda (_: C).(pr3 (CHead c k u1) t1 t2)) (\lambda (y: C).(\lambda -(H1: (pr2 y t1 t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: -T).((eq C c0 (CHead c k u2)) \to (pr3 (CHead c k u1) t t0))))) (\lambda (c0: -C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (_: -(eq C c0 (CHead c k u2))).(pr3_pr2 (CHead c k u1) t3 t4 (pr2_free (CHead c k -u1) t3 t4 H2))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda -(i: nat).(\lambda (H2: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t3: -T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H4: -(subst0 i u t4 t)).(\lambda (H5: (eq C c0 (CHead c k u2))).(let H6 \def -(eq_ind C c0 (\lambda (c1: C).(getl i c1 (CHead d (Bind Abbr) u))) H2 (CHead -c k u2) H5) in (nat_ind (\lambda (n: nat).((getl n (CHead c k u2) (CHead d -(Bind Abbr) u)) \to ((subst0 n u t4 t) \to (pr3 (CHead c k u1) t3 t)))) -(\lambda (H7: (getl O (CHead c k u2) (CHead d (Bind Abbr) u))).(\lambda (H8: -(subst0 O u t4 t)).(K_ind (\lambda (k0: K).((getl O (CHead c k0 u2) (CHead d -(Bind Abbr) u)) \to (pr3 (CHead c k0 u1) t3 t))) (\lambda (b: B).(\lambda -(H9: (getl O (CHead c (Bind b) u2) (CHead d (Bind Abbr) u))).(let H10 \def -(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with -[(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind -Abbr) u) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 -(getl_gen_O (CHead c (Bind b) u2) (CHead d (Bind Abbr) u) H9))) in ((let H11 -\def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) -with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow (match k0 in K -return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) -\Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead c (Bind b) u2) -(clear_gen_bind b c (CHead d (Bind Abbr) u) u2 (getl_gen_O (CHead c (Bind b) -u2) (CHead d (Bind Abbr) u) H9))) in ((let H12 \def (f_equal C T (\lambda (e: -C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | -(CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind Abbr) u) (CHead c (Bind b) -u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 (getl_gen_O (CHead c (Bind -b) u2) (CHead d (Bind Abbr) u) H9))) in (\lambda (H13: (eq B Abbr -b)).(\lambda (_: (eq C d c)).(let H15 \def (eq_ind T u (\lambda (t0: -T).(subst0 O t0 t4 t)) H8 u2 H12) in (eq_ind B Abbr (\lambda (b0: B).(pr3 -(CHead c (Bind b0) u1) t3 t)) (ex2_ind T (\lambda (t0: T).(subst0 O u1 t4 -t0)) (\lambda (t0: T).(pr0 t0 t)) (pr3 (CHead c (Bind Abbr) u1) t3 t) -(\lambda (x: T).(\lambda (H16: (subst0 O u1 t4 x)).(\lambda (H17: (pr0 x -t)).(pr3_sing (CHead c (Bind Abbr) u1) x t3 (pr2_delta (CHead c (Bind Abbr) -u1) c u1 O (getl_refl Abbr c u1) t3 t4 H3 x H16) t (pr3_pr2 (CHead c (Bind -Abbr) u1) x t (pr2_free (CHead c (Bind Abbr) u1) x t H17)))))) -(pr0_subst0_back u2 t4 t O H15 u1 H)) b H13))))) H11)) H10)))) (\lambda (f: -F).(\lambda (H9: (getl O (CHead c (Flat f) u2) (CHead d (Bind Abbr) -u))).(pr3_pr2 (CHead c (Flat f) u1) t3 t (pr2_cflat c t3 t (pr2_delta c d u O -(getl_intro O c (CHead d (Bind Abbr) u) c (drop_refl c) (clear_gen_flat f c -(CHead d (Bind Abbr) u) u2 (getl_gen_O (CHead c (Flat f) u2) (CHead d (Bind -Abbr) u) H9))) t3 t4 H3 t H8) f u1)))) k H7))) (\lambda (i0: nat).(\lambda -(IHi: (((getl i0 (CHead c k u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t4 -t) \to (pr3 (CHead c k u1) t3 t))))).(\lambda (H7: (getl (S i0) (CHead c k -u2) (CHead d (Bind Abbr) u))).(\lambda (H8: (subst0 (S i0) u t4 t)).(K_ind -(\lambda (k0: K).((getl (S i0) (CHead c k0 u2) (CHead d (Bind Abbr) u)) \to -((((getl i0 (CHead c k0 u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t4 t) -\to (pr3 (CHead c k0 u1) t3 t)))) \to (pr3 (CHead c k0 u1) t3 t)))) (\lambda -(b: B).(\lambda (H9: (getl (S i0) (CHead c (Bind b) u2) (CHead d (Bind Abbr) -u))).(\lambda (_: (((getl i0 (CHead c (Bind b) u2) (CHead d (Bind Abbr) u)) -\to ((subst0 i0 u t4 t) \to (pr3 (CHead c (Bind b) u1) t3 t))))).(pr3_pr2 -(CHead c (Bind b) u1) t3 t (pr2_delta (CHead c (Bind b) u1) d u (S i0) -(getl_head (Bind b) i0 c (CHead d (Bind Abbr) u) (getl_gen_S (Bind b) c -(CHead d (Bind Abbr) u) u2 i0 H9) u1) t3 t4 H3 t H8))))) (\lambda (f: -F).(\lambda (H9: (getl (S i0) (CHead c (Flat f) u2) (CHead d (Bind Abbr) -u))).(\lambda (_: (((getl i0 (CHead c (Flat f) u2) (CHead d (Bind Abbr) u)) -\to ((subst0 i0 u t4 t) \to (pr3 (CHead c (Flat f) u1) t3 t))))).(pr3_pr2 -(CHead c (Flat f) u1) t3 t (pr2_cflat c t3 t (pr2_delta c d u (r (Flat f) i0) -(getl_gen_S (Flat f) c (CHead d (Bind Abbr) u) u2 i0 H9) t3 t4 H3 t H8) f -u1))))) k H7 IHi))))) i H6 H4))))))))))))) y t1 t2 H1))) H0)))))))). -(* COMMENTS -Initial nodes: 1557 -END *) - -theorem pr3_pr2_pr2_t: - \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr2 c u1 u2) \to (\forall -(t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pr3 -(CHead c k u1) t1 t2)))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr2 c u1 -u2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (t1: -T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c0 k t0) t1 t2) \to (pr3 -(CHead c0 k t) t1 t2)))))))) (\lambda (c0: C).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (H0: (pr0 t1 t2)).(\lambda (t0: T).(\lambda (t3: T).(\lambda (k: -K).(\lambda (H1: (pr2 (CHead c0 k t2) t0 t3)).(pr3_pr0_pr2_t t1 t2 H0 c0 t0 -t3 k H1))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda -(i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t1: -T).(\lambda (t2: T).(\lambda (H1: (pr0 t1 t2)).(\lambda (t: T).(\lambda (H2: -(subst0 i u t2 t)).(\lambda (t0: T).(\lambda (t3: T).(\lambda (k: K).(\lambda -(H3: (pr2 (CHead c0 k t) t0 t3)).(insert_eq C (CHead c0 k t) (\lambda (c1: -C).(pr2 c1 t0 t3)) (\lambda (_: C).(pr3 (CHead c0 k t1) t0 t3)) (\lambda (y: -C).(\lambda (H4: (pr2 y t0 t3)).(pr2_ind (\lambda (c1: C).(\lambda (t4: -T).(\lambda (t5: T).((eq C c1 (CHead c0 k t)) \to (pr3 (CHead c0 k t1) t4 -t5))))) (\lambda (c1: C).(\lambda (t4: T).(\lambda (t5: T).(\lambda (H5: (pr0 -t4 t5)).(\lambda (_: (eq C c1 (CHead c0 k t))).(pr3_pr2 (CHead c0 k t1) t4 t5 -(pr2_free (CHead c0 k t1) t4 t5 H5))))))) (\lambda (c1: C).(\lambda (d0: -C).(\lambda (u0: T).(\lambda (i0: nat).(\lambda (H5: (getl i0 c1 (CHead d0 -(Bind Abbr) u0))).(\lambda (t4: T).(\lambda (t5: T).(\lambda (H6: (pr0 t4 -t5)).(\lambda (t6: T).(\lambda (H7: (subst0 i0 u0 t5 t6)).(\lambda (H8: (eq C -c1 (CHead c0 k t))).(let H9 \def (eq_ind C c1 (\lambda (c2: C).(getl i0 c2 -(CHead d0 (Bind Abbr) u0))) H5 (CHead c0 k t) H8) in (nat_ind (\lambda (n: -nat).((getl n (CHead c0 k t) (CHead d0 (Bind Abbr) u0)) \to ((subst0 n u0 t5 -t6) \to (pr3 (CHead c0 k t1) t4 t6)))) (\lambda (H10: (getl O (CHead c0 k t) -(CHead d0 (Bind Abbr) u0))).(\lambda (H11: (subst0 O u0 t5 t6)).(K_ind -(\lambda (k0: K).((clear (CHead c0 k0 t) (CHead d0 (Bind Abbr) u0)) \to (pr3 -(CHead c0 k0 t1) t4 t6))) (\lambda (b: B).(\lambda (H12: (clear (CHead c0 -(Bind b) t) (CHead d0 (Bind Abbr) u0))).(let H13 \def (f_equal C C (\lambda -(e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d0 -| (CHead c2 _ _) \Rightarrow c2])) (CHead d0 (Bind Abbr) u0) (CHead c0 (Bind -b) t) (clear_gen_bind b c0 (CHead d0 (Bind Abbr) u0) t H12)) in ((let H14 -\def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) -with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow (match k0 in K -return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) -\Rightarrow Abbr])])) (CHead d0 (Bind Abbr) u0) (CHead c0 (Bind b) t) -(clear_gen_bind b c0 (CHead d0 (Bind Abbr) u0) t H12)) in ((let H15 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow u0 | (CHead _ _ t7) \Rightarrow t7])) (CHead d0 (Bind -Abbr) u0) (CHead c0 (Bind b) t) (clear_gen_bind b c0 (CHead d0 (Bind Abbr) -u0) t H12)) in (\lambda (H16: (eq B Abbr b)).(\lambda (_: (eq C d0 c0)).(let -H18 \def (eq_ind T u0 (\lambda (t7: T).(subst0 O t7 t5 t6)) H11 t H15) in -(eq_ind B Abbr (\lambda (b0: B).(pr3 (CHead c0 (Bind b0) t1) t4 t6)) (ex2_ind -T (\lambda (t7: T).(subst0 O t2 t5 t7)) (\lambda (t7: T).(subst0 (S (plus i -O)) u t7 t6)) (pr3 (CHead c0 (Bind Abbr) t1) t4 t6) (\lambda (x: T).(\lambda -(H19: (subst0 O t2 t5 x)).(\lambda (H20: (subst0 (S (plus i O)) u x t6)).(let -H21 \def (f_equal nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O -i))) in (let H22 \def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n -u x t6)) H20 (S i) H21) in (ex2_ind T (\lambda (t7: T).(subst0 O t1 t5 t7)) -(\lambda (t7: T).(pr0 t7 x)) (pr3 (CHead c0 (Bind Abbr) t1) t4 t6) (\lambda -(x0: T).(\lambda (H23: (subst0 O t1 t5 x0)).(\lambda (H24: (pr0 x0 -x)).(pr3_sing (CHead c0 (Bind Abbr) t1) x0 t4 (pr2_delta (CHead c0 (Bind -Abbr) t1) c0 t1 O (getl_refl Abbr c0 t1) t4 t5 H6 x0 H23) t6 (pr3_pr2 (CHead -c0 (Bind Abbr) t1) x0 t6 (pr2_delta (CHead c0 (Bind Abbr) t1) d u (S i) -(getl_clear_bind Abbr (CHead c0 (Bind Abbr) t1) c0 t1 (clear_bind Abbr c0 t1) -(CHead d (Bind Abbr) u) i H0) x0 x H24 t6 H22)))))) (pr0_subst0_back t2 t5 x -O H19 t1 H1))))))) (subst0_subst0 t5 t6 t O H18 t2 u i H2)) b H16))))) H14)) -H13)))) (\lambda (f: F).(\lambda (H12: (clear (CHead c0 (Flat f) t) (CHead d0 -(Bind Abbr) u0))).(pr3_pr2 (CHead c0 (Flat f) t1) t4 t6 (pr2_cflat c0 t4 t6 -(pr2_delta c0 d0 u0 O (getl_intro O c0 (CHead d0 (Bind Abbr) u0) c0 -(drop_refl c0) (clear_gen_flat f c0 (CHead d0 (Bind Abbr) u0) t H12)) t4 t5 -H6 t6 H11) f t1)))) k (getl_gen_O (CHead c0 k t) (CHead d0 (Bind Abbr) u0) -H10)))) (\lambda (i1: nat).(\lambda (_: (((getl i1 (CHead c0 k t) (CHead d0 -(Bind Abbr) u0)) \to ((subst0 i1 u0 t5 t6) \to (pr3 (CHead c0 k t1) t4 -t6))))).(\lambda (H10: (getl (S i1) (CHead c0 k t) (CHead d0 (Bind Abbr) -u0))).(\lambda (H11: (subst0 (S i1) u0 t5 t6)).(K_ind (\lambda (k0: K).((getl -(S i1) (CHead c0 k0 t) (CHead d0 (Bind Abbr) u0)) \to (pr3 (CHead c0 k0 t1) -t4 t6))) (\lambda (b: B).(\lambda (H12: (getl (S i1) (CHead c0 (Bind b) t) -(CHead d0 (Bind Abbr) u0))).(pr3_pr2 (CHead c0 (Bind b) t1) t4 t6 (pr2_delta -(CHead c0 (Bind b) t1) d0 u0 (S i1) (getl_head (Bind b) i1 c0 (CHead d0 (Bind -Abbr) u0) (getl_gen_S (Bind b) c0 (CHead d0 (Bind Abbr) u0) t i1 H12) t1) t4 -t5 H6 t6 H11)))) (\lambda (f: F).(\lambda (H12: (getl (S i1) (CHead c0 (Flat -f) t) (CHead d0 (Bind Abbr) u0))).(pr3_pr2 (CHead c0 (Flat f) t1) t4 t6 -(pr2_cflat c0 t4 t6 (pr2_delta c0 d0 u0 (r (Flat f) i1) (getl_gen_S (Flat f) -c0 (CHead d0 (Bind Abbr) u0) t i1 H12) t4 t5 H6 t6 H11) f t1)))) k H10))))) -i0 H9 H7))))))))))))) y t0 t3 H4))) H3))))))))))))))) c u1 u2 H)))). -(* COMMENTS -Initial nodes: 1697 -END *) - -theorem pr3_pr2_pr3_t: - \forall (c: C).(\forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall -(k: K).((pr3 (CHead c k u2) t1 t2) \to (\forall (u1: T).((pr2 c u1 u2) \to -(pr3 (CHead c k u1) t1 t2)))))))) -\def - \lambda (c: C).(\lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(k: K).(\lambda (H: (pr3 (CHead c k u2) t1 t2)).(pr3_ind (CHead c k u2) -(\lambda (t: T).(\lambda (t0: T).(\forall (u1: T).((pr2 c u1 u2) \to (pr3 -(CHead c k u1) t t0))))) (\lambda (t: T).(\lambda (u1: T).(\lambda (_: (pr2 c -u1 u2)).(pr3_refl (CHead c k u1) t)))) (\lambda (t0: T).(\lambda (t3: -T).(\lambda (H0: (pr2 (CHead c k u2) t3 t0)).(\lambda (t4: T).(\lambda (_: -(pr3 (CHead c k u2) t0 t4)).(\lambda (H2: ((\forall (u1: T).((pr2 c u1 u2) -\to (pr3 (CHead c k u1) t0 t4))))).(\lambda (u1: T).(\lambda (H3: (pr2 c u1 -u2)).(pr3_t t0 t3 (CHead c k u1) (pr3_pr2_pr2_t c u1 u2 H3 t3 t0 k H0) t4 (H2 -u1 H3)))))))))) t1 t2 H)))))). -(* COMMENTS -Initial nodes: 199 -END *) - -theorem pr3_pr3_pr3_t: - \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall -(t1: T).(\forall (t2: T).(\forall (k: K).((pr3 (CHead c k u2) t1 t2) \to (pr3 -(CHead c k u1) t1 t2)))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1 -u2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (t1: T).(\forall -(t2: T).(\forall (k: K).((pr3 (CHead c k t0) t1 t2) \to (pr3 (CHead c k t) t1 -t2))))))) (\lambda (t: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k: -K).(\lambda (H0: (pr3 (CHead c k t) t1 t2)).H0))))) (\lambda (t2: T).(\lambda -(t1: T).(\lambda (H0: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda (_: (pr3 c t2 -t3)).(\lambda (H2: ((\forall (t4: T).(\forall (t5: T).(\forall (k: K).((pr3 -(CHead c k t3) t4 t5) \to (pr3 (CHead c k t2) t4 t5))))))).(\lambda (t0: -T).(\lambda (t4: T).(\lambda (k: K).(\lambda (H3: (pr3 (CHead c k t3) t0 -t4)).(pr3_pr2_pr3_t c t2 t0 t4 k (H2 t0 t4 k H3) t1 H0))))))))))) u1 u2 H)))). -(* COMMENTS -Initial nodes: 187 -END *) - -theorem pr3_lift: - \forall (c: C).(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h -d c e) \to (\forall (t1: T).(\forall (t2: T).((pr3 e t1 t2) \to (pr3 c (lift -h d t1) (lift h d t2))))))))) -\def - \lambda (c: C).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda -(H: (drop h d c e)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 e t1 -t2)).(pr3_ind e (\lambda (t: T).(\lambda (t0: T).(pr3 c (lift h d t) (lift h -d t0)))) (\lambda (t: T).(pr3_refl c (lift h d t))) (\lambda (t0: T).(\lambda -(t3: T).(\lambda (H1: (pr2 e t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 e t0 -t4)).(\lambda (H3: (pr3 c (lift h d t0) (lift h d t4))).(pr3_sing c (lift h d -t0) (lift h d t3) (pr2_lift c e h d H t3 t0 H1) (lift h d t4) H3))))))) t1 t2 -H0)))))))). -(* COMMENTS -Initial nodes: 167 -END *) - -theorem pr3_eta: - \forall (c: C).(\forall (w: T).(\forall (u: T).(let t \def (THead (Bind -Abst) w u) in (\forall (v: T).((pr3 c v w) \to (pr3 c (THead (Bind Abst) v -(THead (Flat Appl) (TLRef O) (lift (S O) O t))) t)))))) -\def - \lambda (c: C).(\lambda (w: T).(\lambda (u: T).(let t \def (THead (Bind -Abst) w u) in (\lambda (v: T).(\lambda (H: (pr3 c v w)).(eq_ind_r T (THead -(Bind Abst) (lift (S O) O w) (lift (S O) (S O) u)) (\lambda (t0: T).(pr3 c -(THead (Bind Abst) v (THead (Flat Appl) (TLRef O) t0)) (THead (Bind Abst) w -u))) (pr3_head_12 c v w H (Bind Abst) (THead (Flat Appl) (TLRef O) (THead -(Bind Abst) (lift (S O) O w) (lift (S O) (S O) u))) u (pr3_pr1 (THead (Flat -Appl) (TLRef O) (THead (Bind Abst) (lift (S O) O w) (lift (S O) (S O) u))) u -(pr1_sing (THead (Bind Abbr) (TLRef O) (lift (S O) (S O) u)) (THead (Flat -Appl) (TLRef O) (THead (Bind Abst) (lift (S O) O w) (lift (S O) (S O) u))) -(pr0_beta (lift (S O) O w) (TLRef O) (TLRef O) (pr0_refl (TLRef O)) (lift (S -O) (S O) u) (lift (S O) (S O) u) (pr0_refl (lift (S O) (S O) u))) u (pr1_sing -(THead (Bind Abbr) (TLRef O) (lift (S O) O u)) (THead (Bind Abbr) (TLRef O) -(lift (S O) (S O) u)) (pr0_delta1 (TLRef O) (TLRef O) (pr0_refl (TLRef O)) -(lift (S O) (S O) u) (lift (S O) (S O) u) (pr0_refl (lift (S O) (S O) u)) -(lift (S O) O u) (subst1_lift_S u O O (le_n O))) u (pr1_pr0 (THead (Bind -Abbr) (TLRef O) (lift (S O) O u)) u (pr0_zeta Abbr not_abbr_abst u u -(pr0_refl u) (TLRef O))))) (CHead c (Bind Abst) w))) (lift (S O) O (THead -(Bind Abst) w u)) (lift_bind Abst w u (S O) O))))))). -(* COMMENTS -Initial nodes: 523 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/subst1.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/subst1.ma deleted file mode 100644 index 3bcbd4965..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/subst1.ma +++ /dev/null @@ -1,95 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr3/defs.ma". - -include "Basic-1/pr2/subst1.ma". - -theorem pr3_subst1: - \forall (c: C).(\forall (e: C).(\forall (v: T).(\forall (i: nat).((getl i c -(CHead e (Bind Abbr) v)) \to (\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) -\to (\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T (\lambda (w2: T).(pr3 c -w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2)))))))))))) -\def - \lambda (c: C).(\lambda (e: C).(\lambda (v: T).(\lambda (i: nat).(\lambda -(H: (getl i c (CHead e (Bind Abbr) v))).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (H0: (pr3 c t1 t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: -T).(\forall (w1: T).((subst1 i v t w1) \to (ex2 T (\lambda (w2: T).(pr3 c w1 -w2)) (\lambda (w2: T).(subst1 i v t0 w2))))))) (\lambda (t: T).(\lambda (w1: -T).(\lambda (H1: (subst1 i v t w1)).(ex_intro2 T (\lambda (w2: T).(pr3 c w1 -w2)) (\lambda (w2: T).(subst1 i v t w2)) w1 (pr3_refl c w1) H1)))) (\lambda -(t3: T).(\lambda (t4: T).(\lambda (H1: (pr2 c t4 t3)).(\lambda (t5: -T).(\lambda (_: (pr3 c t3 t5)).(\lambda (H3: ((\forall (w1: T).((subst1 i v -t3 w1) \to (ex2 T (\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1 i -v t5 w2))))))).(\lambda (w1: T).(\lambda (H4: (subst1 i v t4 w1)).(ex2_ind T -(\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t3 w2)) (ex2 T -(\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1 i v t5 w2))) -(\lambda (x: T).(\lambda (H5: (pr2 c w1 x)).(\lambda (H6: (subst1 i v t3 -x)).(ex2_ind T (\lambda (w2: T).(pr3 c x w2)) (\lambda (w2: T).(subst1 i v t5 -w2)) (ex2 T (\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1 i v t5 -w2))) (\lambda (x0: T).(\lambda (H7: (pr3 c x x0)).(\lambda (H8: (subst1 i v -t5 x0)).(ex_intro2 T (\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1 -i v t5 w2)) x0 (pr3_sing c x w1 H5 x0 H7) H8)))) (H3 x H6))))) (pr2_subst1 c -e v i H t4 t3 H1 w1 H4)))))))))) t1 t2 H0)))))))). -(* COMMENTS -Initial nodes: 425 -END *) - -theorem pr3_gen_cabbr: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall -(e: C).(\forall (u: T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) -\to (\forall (a0: C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d -a0 a) \to (\forall (x1: T).((subst1 d u t1 (lift (S O) d x1)) \to (ex2 T -(\lambda (x2: T).(subst1 d u t2 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a -x1 x2)))))))))))))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 -t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (e: C).(\forall (u: -T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) \to (\forall (a0: -C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (\forall -(x1: T).((subst1 d u t (lift (S O) d x1)) \to (ex2 T (\lambda (x2: T).(subst1 -d u t0 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x1 x2))))))))))))))) -(\lambda (t: T).(\lambda (e: C).(\lambda (u: T).(\lambda (d: nat).(\lambda -(_: (getl d c (CHead e (Bind Abbr) u))).(\lambda (a0: C).(\lambda (_: -(csubst1 d u c a0)).(\lambda (a: C).(\lambda (_: (drop (S O) d a0 -a)).(\lambda (x1: T).(\lambda (H3: (subst1 d u t (lift (S O) d -x1))).(ex_intro2 T (\lambda (x2: T).(subst1 d u t (lift (S O) d x2))) -(\lambda (x2: T).(pr3 a x1 x2)) x1 H3 (pr3_refl a x1))))))))))))) (\lambda -(t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c t3 t0)).(\lambda (t4: -T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2: ((\forall (e: C).(\forall (u: -T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) \to (\forall (a0: -C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (\forall -(x1: T).((subst1 d u t0 (lift (S O) d x1)) \to (ex2 T (\lambda (x2: -T).(subst1 d u t4 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x1 -x2))))))))))))))).(\lambda (e: C).(\lambda (u: T).(\lambda (d: nat).(\lambda -(H3: (getl d c (CHead e (Bind Abbr) u))).(\lambda (a0: C).(\lambda (H4: -(csubst1 d u c a0)).(\lambda (a: C).(\lambda (H5: (drop (S O) d a0 -a)).(\lambda (x1: T).(\lambda (H6: (subst1 d u t3 (lift (S O) d -x1))).(ex2_ind T (\lambda (x2: T).(subst1 d u t0 (lift (S O) d x2))) (\lambda -(x2: T).(pr2 a x1 x2)) (ex2 T (\lambda (x2: T).(subst1 d u t4 (lift (S O) d -x2))) (\lambda (x2: T).(pr3 a x1 x2))) (\lambda (x: T).(\lambda (H7: (subst1 -d u t0 (lift (S O) d x))).(\lambda (H8: (pr2 a x1 x)).(ex2_ind T (\lambda -(x2: T).(subst1 d u t4 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x x2)) -(ex2 T (\lambda (x2: T).(subst1 d u t4 (lift (S O) d x2))) (\lambda (x2: -T).(pr3 a x1 x2))) (\lambda (x0: T).(\lambda (H9: (subst1 d u t4 (lift (S O) -d x0))).(\lambda (H10: (pr3 a x x0)).(ex_intro2 T (\lambda (x2: T).(subst1 d -u t4 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x1 x2)) x0 H9 (pr3_sing a x -x1 H8 x0 H10))))) (H2 e u d H3 a0 H4 a H5 x H7))))) (pr2_gen_cabbr c t3 t0 H0 -e u d H3 a0 H4 a H5 x1 H6)))))))))))))))))) t1 t2 H)))). -(* COMMENTS -Initial nodes: 731 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/wcpr0.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/wcpr0.ma deleted file mode 100644 index 052f4e5b2..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/wcpr0.ma +++ /dev/null @@ -1,66 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr3/props.ma". - -include "Basic-1/wcpr0/getl.ma". - -theorem pr3_wcpr0_t: - \forall (c1: C).(\forall (c2: C).((wcpr0 c2 c1) \to (\forall (t1: -T).(\forall (t2: T).((pr3 c1 t1 t2) \to (pr3 c2 t1 t2)))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c2 c1)).(wcpr0_ind -(\lambda (c: C).(\lambda (c0: C).(\forall (t1: T).(\forall (t2: T).((pr3 c0 -t1 t2) \to (pr3 c t1 t2)))))) (\lambda (c: C).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (H0: (pr3 c t1 t2)).H0)))) (\lambda (c0: C).(\lambda (c3: -C).(\lambda (H0: (wcpr0 c0 c3)).(\lambda (_: ((\forall (t1: T).(\forall (t2: -T).((pr3 c3 t1 t2) \to (pr3 c0 t1 t2)))))).(\lambda (u1: T).(\lambda (u2: -T).(\lambda (H2: (pr0 u1 u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (H3: (pr3 (CHead c3 k u2) t1 t2)).(pr3_ind (CHead c3 k u1) -(\lambda (t: T).(\lambda (t0: T).(pr3 (CHead c0 k u1) t t0))) (\lambda (t: -T).(pr3_refl (CHead c0 k u1) t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda -(H4: (pr2 (CHead c3 k u1) t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 (CHead -c3 k u1) t0 t4)).(\lambda (H6: (pr3 (CHead c0 k u1) t0 t4)).(pr3_t t0 t3 -(CHead c0 k u1) (insert_eq C (CHead c3 k u1) (\lambda (c: C).(pr2 c t3 t0)) -(\lambda (_: C).(pr3 (CHead c0 k u1) t3 t0)) (\lambda (y: C).(\lambda (H7: -(pr2 y t3 t0)).(pr2_ind (\lambda (c: C).(\lambda (t: T).(\lambda (t5: T).((eq -C c (CHead c3 k u1)) \to (pr3 (CHead c0 k u1) t t5))))) (\lambda (c: -C).(\lambda (t5: T).(\lambda (t6: T).(\lambda (H8: (pr0 t5 t6)).(\lambda (_: -(eq C c (CHead c3 k u1))).(pr3_pr2 (CHead c0 k u1) t5 t6 (pr2_free (CHead c0 -k u1) t5 t6 H8))))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: -T).(\lambda (i: nat).(\lambda (H8: (getl i c (CHead d (Bind Abbr) -u))).(\lambda (t5: T).(\lambda (t6: T).(\lambda (H9: (pr0 t5 t6)).(\lambda -(t: T).(\lambda (H10: (subst0 i u t6 t)).(\lambda (H11: (eq C c (CHead c3 k -u1))).(let H12 \def (eq_ind C c (\lambda (c4: C).(getl i c4 (CHead d (Bind -Abbr) u))) H8 (CHead c3 k u1) H11) in (ex3_2_ind C T (\lambda (e2: -C).(\lambda (u3: T).(getl i (CHead c0 k u1) (CHead e2 (Bind Abbr) u3)))) -(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 d))) (\lambda (_: C).(\lambda (u3: -T).(pr0 u3 u))) (pr3 (CHead c0 k u1) t5 t) (\lambda (x0: C).(\lambda (x1: -T).(\lambda (H13: (getl i (CHead c0 k u1) (CHead x0 (Bind Abbr) -x1))).(\lambda (_: (wcpr0 x0 d)).(\lambda (H15: (pr0 x1 u)).(ex2_ind T -(\lambda (t7: T).(subst0 i x1 t6 t7)) (\lambda (t7: T).(pr0 t7 t)) (pr3 -(CHead c0 k u1) t5 t) (\lambda (x: T).(\lambda (H16: (subst0 i x1 t6 -x)).(\lambda (H17: (pr0 x t)).(pr3_sing (CHead c0 k u1) x t5 (pr2_delta -(CHead c0 k u1) x0 x1 i H13 t5 t6 H9 x H16) t (pr3_pr2 (CHead c0 k u1) x t -(pr2_free (CHead c0 k u1) x t H17)))))) (pr0_subst0_back u t6 t i H10 x1 -H15))))))) (wcpr0_getl_back (CHead c3 k u1) (CHead c0 k u1) (wcpr0_comp c0 c3 -H0 u1 u1 (pr0_refl u1) k) i d u (Bind Abbr) H12)))))))))))))) y t3 t0 H7))) -H4) t4 H6))))))) t1 t2 (pr3_pr2_pr3_t c3 u2 t1 t2 k H3 u1 (pr2_free c3 u1 u2 -H2)))))))))))))) c2 c1 H))). -(* COMMENTS -Initial nodes: 799 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/preamble.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/preamble.ma deleted file mode 100644 index c9d6a172a..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/preamble.ma +++ /dev/null @@ -1,15 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "Ground-1/theory.ma". diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/r/defs.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/r/defs.ma deleted file mode 100644 index cde40fdf8..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/r/defs.ma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/T/defs.ma". - -definition r: - K \to (nat \to nat) -\def - \lambda (k: K).(\lambda (i: nat).(match k with [(Bind _) \Rightarrow i | -(Flat _) \Rightarrow (S i)])). - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/r/props.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/r/props.ma deleted file mode 100644 index 0815aaf5e..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/r/props.ma +++ /dev/null @@ -1,117 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/r/defs.ma". - -include "Basic-1/s/defs.ma". - -theorem r_S: - \forall (k: K).(\forall (i: nat).(eq nat (r k (S i)) (S (r k i)))) -\def - \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(eq nat (r k0 (S -i)) (S (r k0 i))))) (\lambda (b: B).(\lambda (i: nat).(refl_equal nat (S (r -(Bind b) i))))) (\lambda (f: F).(\lambda (i: nat).(refl_equal nat (S (r (Flat -f) i))))) k). -(* COMMENTS -Initial nodes: 65 -END *) - -theorem r_plus: - \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (r k (plus i j)) -(plus (r k i) j)))) -\def - \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: -nat).(eq nat (r k0 (plus i j)) (plus (r k0 i) j))))) (\lambda (b: B).(\lambda -(i: nat).(\lambda (j: nat).(refl_equal nat (plus (r (Bind b) i) j))))) -(\lambda (f: F).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (plus (r -(Flat f) i) j))))) k). -(* COMMENTS -Initial nodes: 79 -END *) - -theorem r_plus_sym: - \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (r k (plus i j)) -(plus i (r k j))))) -\def - \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: -nat).(eq nat (r k0 (plus i j)) (plus i (r k0 j)))))) (\lambda (_: B).(\lambda -(i: nat).(\lambda (j: nat).(refl_equal nat (plus i j))))) (\lambda (_: -F).(\lambda (i: nat).(\lambda (j: nat).(plus_n_Sm i j)))) k). -(* COMMENTS -Initial nodes: 63 -END *) - -theorem r_minus: - \forall (i: nat).(\forall (n: nat).((lt n i) \to (\forall (k: K).(eq nat -(minus (r k i) (S n)) (r k (minus i (S n))))))) -\def - \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (lt n i)).(\lambda (k: -K).(K_ind (\lambda (k0: K).(eq nat (minus (r k0 i) (S n)) (r k0 (minus i (S -n))))) (\lambda (_: B).(refl_equal nat (minus i (S n)))) (\lambda (_: -F).(minus_x_Sy i n H)) k)))). -(* COMMENTS -Initial nodes: 69 -END *) - -theorem r_dis: - \forall (k: K).(\forall (P: Prop).(((((\forall (i: nat).(eq nat (r k i) i))) -\to P)) \to (((((\forall (i: nat).(eq nat (r k i) (S i)))) \to P)) \to P))) -\def - \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (P: Prop).(((((\forall (i: -nat).(eq nat (r k0 i) i))) \to P)) \to (((((\forall (i: nat).(eq nat (r k0 i) -(S i)))) \to P)) \to P)))) (\lambda (b: B).(\lambda (P: Prop).(\lambda (H: -((((\forall (i: nat).(eq nat (r (Bind b) i) i))) \to P))).(\lambda (_: -((((\forall (i: nat).(eq nat (r (Bind b) i) (S i)))) \to P))).(H (\lambda (i: -nat).(refl_equal nat i))))))) (\lambda (f: F).(\lambda (P: Prop).(\lambda (_: -((((\forall (i: nat).(eq nat (r (Flat f) i) i))) \to P))).(\lambda (H0: -((((\forall (i: nat).(eq nat (r (Flat f) i) (S i)))) \to P))).(H0 (\lambda -(i: nat).(refl_equal nat (S i)))))))) k). -(* COMMENTS -Initial nodes: 151 -END *) - -theorem s_r: - \forall (k: K).(\forall (i: nat).(eq nat (s k (r k i)) (S i))) -\def - \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(eq nat (s k0 (r k0 -i)) (S i)))) (\lambda (_: B).(\lambda (i: nat).(refl_equal nat (S i)))) -(\lambda (_: F).(\lambda (i: nat).(refl_equal nat (S i)))) k). -(* COMMENTS -Initial nodes: 51 -END *) - -theorem r_arith0: - \forall (k: K).(\forall (i: nat).(eq nat (minus (r k (S i)) (S O)) (r k i))) -\def - \lambda (k: K).(\lambda (i: nat).(eq_ind_r nat (S (r k i)) (\lambda (n: -nat).(eq nat (minus n (S O)) (r k i))) (eq_ind_r nat (r k i) (\lambda (n: -nat).(eq nat n (r k i))) (refl_equal nat (r k i)) (minus (S (r k i)) (S O)) -(minus_Sx_SO (r k i))) (r k (S i)) (r_S k i))). -(* COMMENTS -Initial nodes: 105 -END *) - -theorem r_arith1: - \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (minus (r k (S -i)) (S j)) (minus (r k i) j)))) -\def - \lambda (k: K).(\lambda (i: nat).(\lambda (j: nat).(eq_ind_r nat (S (r k i)) -(\lambda (n: nat).(eq nat (minus n (S j)) (minus (r k i) j))) (refl_equal nat -(minus (r k i) j)) (r k (S i)) (r_S k i)))). -(* COMMENTS -Initial nodes: 69 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/s/defs.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/s/defs.ma deleted file mode 100644 index 0d1fb7914..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/s/defs.ma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/T/defs.ma". - -definition s: - K \to (nat \to nat) -\def - \lambda (k: K).(\lambda (i: nat).(match k with [(Bind _) \Rightarrow (S i) | -(Flat _) \Rightarrow i])). - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/s/props.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/s/props.ma deleted file mode 100644 index 3cb4fbd74..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/s/props.ma +++ /dev/null @@ -1,151 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/s/defs.ma". - -theorem s_S: - \forall (k: K).(\forall (i: nat).(eq nat (s k (S i)) (S (s k i)))) -\def - \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(eq nat (s k0 (S -i)) (S (s k0 i))))) (\lambda (b: B).(\lambda (i: nat).(refl_equal nat (S (s -(Bind b) i))))) (\lambda (f: F).(\lambda (i: nat).(refl_equal nat (S (s (Flat -f) i))))) k). -(* COMMENTS -Initial nodes: 65 -END *) - -theorem s_plus: - \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (s k (plus i j)) -(plus (s k i) j)))) -\def - \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: -nat).(eq nat (s k0 (plus i j)) (plus (s k0 i) j))))) (\lambda (b: B).(\lambda -(i: nat).(\lambda (j: nat).(refl_equal nat (plus (s (Bind b) i) j))))) -(\lambda (f: F).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (plus (s -(Flat f) i) j))))) k). -(* COMMENTS -Initial nodes: 79 -END *) - -theorem s_plus_sym: - \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (s k (plus i j)) -(plus i (s k j))))) -\def - \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: -nat).(eq nat (s k0 (plus i j)) (plus i (s k0 j)))))) (\lambda (_: B).(\lambda -(i: nat).(\lambda (j: nat).(eq_ind_r nat (plus i (S j)) (\lambda (n: nat).(eq -nat n (plus i (S j)))) (refl_equal nat (plus i (S j))) (S (plus i j)) -(plus_n_Sm i j))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (j: -nat).(refl_equal nat (plus i (s (Flat f) j)))))) k). -(* COMMENTS -Initial nodes: 117 -END *) - -theorem s_minus: - \forall (k: K).(\forall (i: nat).(\forall (j: nat).((le j i) \to (eq nat (s -k (minus i j)) (minus (s k i) j))))) -\def - \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: -nat).((le j i) \to (eq nat (s k0 (minus i j)) (minus (s k0 i) j)))))) -(\lambda (_: B).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (le j -i)).(eq_ind_r nat (minus (S i) j) (\lambda (n: nat).(eq nat n (minus (S i) -j))) (refl_equal nat (minus (S i) j)) (S (minus i j)) (minus_Sn_m i j H)))))) -(\lambda (f: F).(\lambda (i: nat).(\lambda (j: nat).(\lambda (_: (le j -i)).(refl_equal nat (minus (s (Flat f) i) j)))))) k). -(* COMMENTS -Initial nodes: 137 -END *) - -theorem minus_s_s: - \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (minus (s k i) (s -k j)) (minus i j)))) -\def - \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: -nat).(eq nat (minus (s k0 i) (s k0 j)) (minus i j))))) (\lambda (_: -B).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (minus i j))))) -(\lambda (_: F).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (minus i -j))))) k). -(* COMMENTS -Initial nodes: 67 -END *) - -theorem s_le: - \forall (k: K).(\forall (i: nat).(\forall (j: nat).((le i j) \to (le (s k i) -(s k j))))) -\def - \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: -nat).((le i j) \to (le (s k0 i) (s k0 j)))))) (\lambda (_: B).(\lambda (i: -nat).(\lambda (j: nat).(\lambda (H: (le i j)).(le_n_S i j H))))) (\lambda (_: -F).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (le i j)).H)))) k). -(* COMMENTS -Initial nodes: 65 -END *) - -theorem s_lt: - \forall (k: K).(\forall (i: nat).(\forall (j: nat).((lt i j) \to (lt (s k i) -(s k j))))) -\def - \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: -nat).((lt i j) \to (lt (s k0 i) (s k0 j)))))) (\lambda (_: B).(\lambda (i: -nat).(\lambda (j: nat).(\lambda (H: (lt i j)).(le_n_S (S i) j H))))) (\lambda -(_: F).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (lt i j)).H)))) k). -(* COMMENTS -Initial nodes: 67 -END *) - -theorem s_inj: - \forall (k: K).(\forall (i: nat).(\forall (j: nat).((eq nat (s k i) (s k j)) -\to (eq nat i j)))) -\def - \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: -nat).((eq nat (s k0 i) (s k0 j)) \to (eq nat i j))))) (\lambda (b: -B).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (eq nat (s (Bind b) i) (s -(Bind b) j))).(eq_add_S i j H))))) (\lambda (f: F).(\lambda (i: nat).(\lambda -(j: nat).(\lambda (H: (eq nat (s (Flat f) i) (s (Flat f) j))).H)))) k). -(* COMMENTS -Initial nodes: 97 -END *) - -theorem s_inc: - \forall (k: K).(\forall (i: nat).(le i (s k i))) -\def - \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(le i (s k0 i)))) -(\lambda (b: B).(\lambda (i: nat).(le_S_n i (s (Bind b) i) (le_S (S i) (s -(Bind b) i) (le_n (s (Bind b) i)))))) (\lambda (f: F).(\lambda (i: nat).(le_n -(s (Flat f) i)))) k). -(* COMMENTS -Initial nodes: 73 -END *) - -theorem s_arith0: - \forall (k: K).(\forall (i: nat).(eq nat (minus (s k i) (s k O)) i)) -\def - \lambda (k: K).(\lambda (i: nat).(eq_ind_r nat (minus i O) (\lambda (n: -nat).(eq nat n i)) (eq_ind nat i (\lambda (n: nat).(eq nat n i)) (refl_equal -nat i) (minus i O) (minus_n_O i)) (minus (s k i) (s k O)) (minus_s_s k i O))). -(* COMMENTS -Initial nodes: 77 -END *) - -theorem s_arith1: - \forall (b: B).(\forall (i: nat).(eq nat (minus (s (Bind b) i) (S O)) i)) -\def - \lambda (_: B).(\lambda (i: nat).(eq_ind nat i (\lambda (n: nat).(eq nat n -i)) (refl_equal nat i) (minus i O) (minus_n_O i))). -(* COMMENTS -Initial nodes: 35 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/sc3/arity.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/sc3/arity.ma deleted file mode 100644 index 651321580..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/sc3/arity.ma +++ /dev/null @@ -1,322 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubc/arity.ma". - -include "Basic-1/csubc/getl.ma". - -include "Basic-1/csubc/drop1.ma". - -include "Basic-1/csubc/props.ma". - -theorem sc3_arity_csubc: - \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1 -t a) \to (\forall (d1: C).(\forall (is: PList).((drop1 is d1 c1) \to (\forall -(c2: C).((csubc g d1 c2) \to (sc3 g a c2 (lift1 is t))))))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: -(arity g c1 t a)).(arity_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda (a0: -A).(\forall (d1: C).(\forall (is: PList).((drop1 is d1 c) \to (\forall (c2: -C).((csubc g d1 c2) \to (sc3 g a0 c2 (lift1 is t0)))))))))) (\lambda (c: -C).(\lambda (n: nat).(\lambda (d1: C).(\lambda (is: PList).(\lambda (_: -(drop1 is d1 c)).(\lambda (c2: C).(\lambda (_: (csubc g d1 c2)).(eq_ind_r T -(TSort n) (\lambda (t0: T).(land (arity g c2 t0 (ASort O n)) (sn3 c2 t0))) -(conj (arity g c2 (TSort n) (ASort O n)) (sn3 c2 (TSort n)) (arity_sort g c2 -n) (sn3_nf2 c2 (TSort n) (nf2_sort c2 n))) (lift1 is (TSort n)) (lift1_sort n -is))))))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: -nat).(\lambda (H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (a0: -A).(\lambda (_: (arity g d u a0)).(\lambda (H2: ((\forall (d1: C).(\forall -(is: PList).((drop1 is d1 d) \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g -a0 c2 (lift1 is u))))))))).(\lambda (d1: C).(\lambda (is: PList).(\lambda -(H3: (drop1 is d1 c)).(\lambda (c2: C).(\lambda (H4: (csubc g d1 c2)).(let -H_x \def (drop1_getl_trans is c d1 H3 Abbr d u i H0) in (let H5 \def H_x in -(ex2_ind C (\lambda (e2: C).(drop1 (ptrans is i) e2 d)) (\lambda (e2: -C).(getl (trans is i) d1 (CHead e2 (Bind Abbr) (lift1 (ptrans is i) u)))) -(sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x: C).(\lambda (_: (drop1 -(ptrans is i) x d)).(\lambda (H7: (getl (trans is i) d1 (CHead x (Bind Abbr) -(lift1 (ptrans is i) u)))).(let H_x0 \def (csubc_getl_conf g d1 (CHead x -(Bind Abbr) (lift1 (ptrans is i) u)) (trans is i) H7 c2 H4) in (let H8 \def -H_x0 in (ex2_ind C (\lambda (e2: C).(getl (trans is i) c2 e2)) (\lambda (e2: -C).(csubc g (CHead x (Bind Abbr) (lift1 (ptrans is i) u)) e2)) (sc3 g a0 c2 -(lift1 is (TLRef i))) (\lambda (x0: C).(\lambda (H9: (getl (trans is i) c2 -x0)).(\lambda (H10: (csubc g (CHead x (Bind Abbr) (lift1 (ptrans is i) u)) -x0)).(let H_x1 \def (csubc_gen_head_l g x x0 (lift1 (ptrans is i) u) (Bind -Abbr) H10) in (let H11 \def H_x1 in (or3_ind (ex2 C (\lambda (c3: C).(eq C x0 -(CHead c3 (Bind Abbr) (lift1 (ptrans is i) u)))) (\lambda (c3: C).(csubc g x -c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K -(Bind Abbr) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: -A).(eq C x0 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: -T).(\lambda (_: A).(csubc g x c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans is i) u))))) (\lambda (c3: -C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3 w))))) (ex4_3 B C T (\lambda -(b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C x0 (CHead c3 (Bind b) v2))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K (Bind Abbr) (Bind -Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b -Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g x c3))))) -(sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (H12: (ex2 C (\lambda (c3: C).(eq -C x0 (CHead c3 (Bind Abbr) (lift1 (ptrans is i) u)))) (\lambda (c3: C).(csubc -g x c3)))).(ex2_ind C (\lambda (c3: C).(eq C x0 (CHead c3 (Bind Abbr) (lift1 -(ptrans is i) u)))) (\lambda (c3: C).(csubc g x c3)) (sc3 g a0 c2 (lift1 is -(TLRef i))) (\lambda (x1: C).(\lambda (H13: (eq C x0 (CHead x1 (Bind Abbr) -(lift1 (ptrans is i) u)))).(\lambda (_: (csubc g x x1)).(let H15 \def (eq_ind -C x0 (\lambda (c0: C).(getl (trans is i) c2 c0)) H9 (CHead x1 (Bind Abbr) -(lift1 (ptrans is i) u)) H13) in (let H_y \def (sc3_abbr g a0 TNil) in -(eq_ind_r T (TLRef (trans is i)) (\lambda (t0: T).(sc3 g a0 c2 t0)) (H_y -(trans is i) x1 (lift1 (ptrans is i) u) c2 (eq_ind T (lift1 is (lift (S i) O -u)) (\lambda (t0: T).(sc3 g a0 c2 t0)) (eq_ind T (lift1 (PConsTail is (S i) -O) u) (\lambda (t0: T).(sc3 g a0 c2 t0)) (H2 d1 (PConsTail is (S i) O) -(drop1_cons_tail c d (S i) O (getl_drop Abbr c d u i H0) is d1 H3) c2 H4) -(lift1 is (lift (S i) O u)) (lift1_cons_tail u (S i) O is)) (lift (S (trans -is i)) O (lift1 (ptrans is i) u)) (lift1_free is i u)) H15) (lift1 is (TLRef -i)) (lift1_lref is i))))))) H12)) (\lambda (H12: (ex5_3 C T A (\lambda (_: -C).(\lambda (_: T).(\lambda (_: A).(eq K (Bind Abbr) (Bind Abst))))) (\lambda -(c3: C).(\lambda (w: T).(\lambda (_: A).(eq C x0 (CHead c3 (Bind Abbr) w))))) -(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g x c3)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans -is i) u))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3 -w)))))).(ex5_3_ind C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq -K (Bind Abbr) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: -A).(eq C x0 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: -T).(\lambda (_: A).(csubc g x c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans is i) u))))) (\lambda (c3: -C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3 w)))) (sc3 g a0 c2 (lift1 is -(TLRef i))) (\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: A).(\lambda (H13: -(eq K (Bind Abbr) (Bind Abst))).(\lambda (H14: (eq C x0 (CHead x1 (Bind Abbr) -x2))).(\lambda (_: (csubc g x x1)).(\lambda (_: (sc3 g (asucc g x3) x (lift1 -(ptrans is i) u))).(\lambda (_: (sc3 g x3 x1 x2)).(let H18 \def (eq_ind C x0 -(\lambda (c0: C).(getl (trans is i) c2 c0)) H9 (CHead x1 (Bind Abbr) x2) H14) -in (let H19 \def (eq_ind K (Bind Abbr) (\lambda (ee: K).(match ee in K return -(\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B return -(\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | -Void \Rightarrow False]) | (Flat _) \Rightarrow False])) I (Bind Abst) H13) -in (False_ind (sc3 g a0 c2 (lift1 is (TLRef i))) H19))))))))))) H12)) -(\lambda (H12: (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: -T).(eq C x0 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: T).(eq K (Bind Abbr) (Bind Void))))) (\lambda (b: B).(\lambda -(_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: -C).(\lambda (_: T).(csubc g x c3)))))).(ex4_3_ind B C T (\lambda (b: -B).(\lambda (c3: C).(\lambda (v2: T).(eq C x0 (CHead c3 (Bind b) v2))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K (Bind Abbr) (Bind -Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b -Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g x c3)))) -(sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x1: B).(\lambda (x2: C).(\lambda -(x3: T).(\lambda (H13: (eq C x0 (CHead x2 (Bind x1) x3))).(\lambda (H14: (eq -K (Bind Abbr) (Bind Void))).(\lambda (_: (not (eq B x1 Void))).(\lambda (_: -(csubc g x x2)).(let H17 \def (eq_ind C x0 (\lambda (c0: C).(getl (trans is -i) c2 c0)) H9 (CHead x2 (Bind x1) x3) H13) in (let H18 \def (eq_ind K (Bind -Abbr) (\lambda (ee: K).(match ee in K return (\lambda (_: K).Prop) with -[(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr -\Rightarrow True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat -_) \Rightarrow False])) I (Bind Void) H14) in (False_ind (sc3 g a0 c2 (lift1 -is (TLRef i))) H18)))))))))) H12)) H11)))))) H8)))))) H5)))))))))))))))) -(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H0: (getl i c (CHead d (Bind Abst) u))).(\lambda (a0: A).(\lambda (H1: -(arity g d u (asucc g a0))).(\lambda (_: ((\forall (d1: C).(\forall (is: -PList).((drop1 is d1 d) \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g -(asucc g a0) c2 (lift1 is u))))))))).(\lambda (d1: C).(\lambda (is: -PList).(\lambda (H3: (drop1 is d1 c)).(\lambda (c2: C).(\lambda (H4: (csubc g -d1 c2)).(let H5 \def H0 in (let H_x \def (drop1_getl_trans is c d1 H3 Abst d -u i H5) in (let H6 \def H_x in (ex2_ind C (\lambda (e2: C).(drop1 (ptrans is -i) e2 d)) (\lambda (e2: C).(getl (trans is i) d1 (CHead e2 (Bind Abst) (lift1 -(ptrans is i) u)))) (sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x: -C).(\lambda (H7: (drop1 (ptrans is i) x d)).(\lambda (H8: (getl (trans is i) -d1 (CHead x (Bind Abst) (lift1 (ptrans is i) u)))).(let H_x0 \def -(csubc_getl_conf g d1 (CHead x (Bind Abst) (lift1 (ptrans is i) u)) (trans is -i) H8 c2 H4) in (let H9 \def H_x0 in (ex2_ind C (\lambda (e2: C).(getl (trans -is i) c2 e2)) (\lambda (e2: C).(csubc g (CHead x (Bind Abst) (lift1 (ptrans -is i) u)) e2)) (sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x0: C).(\lambda -(H10: (getl (trans is i) c2 x0)).(\lambda (H11: (csubc g (CHead x (Bind Abst) -(lift1 (ptrans is i) u)) x0)).(let H_x1 \def (csubc_gen_head_l g x x0 (lift1 -(ptrans is i) u) (Bind Abst) H11) in (let H12 \def H_x1 in (or3_ind (ex2 C -(\lambda (c3: C).(eq C x0 (CHead c3 (Bind Abst) (lift1 (ptrans is i) u)))) -(\lambda (c3: C).(csubc g x c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: -T).(\lambda (_: A).(eq K (Bind Abst) (Bind Abst))))) (\lambda (c3: -C).(\lambda (w: T).(\lambda (_: A).(eq C x0 (CHead c3 (Bind Abbr) w))))) -(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g x c3)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans -is i) u))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3 -w))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C -x0 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -T).(eq K (Bind Abst) (Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda -(_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: -T).(csubc g x c3))))) (sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (H13: (ex2 -C (\lambda (c3: C).(eq C x0 (CHead c3 (Bind Abst) (lift1 (ptrans is i) u)))) -(\lambda (c3: C).(csubc g x c3)))).(ex2_ind C (\lambda (c3: C).(eq C x0 -(CHead c3 (Bind Abst) (lift1 (ptrans is i) u)))) (\lambda (c3: C).(csubc g x -c3)) (sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x1: C).(\lambda (H14: (eq C -x0 (CHead x1 (Bind Abst) (lift1 (ptrans is i) u)))).(\lambda (_: (csubc g x -x1)).(let H16 \def (eq_ind C x0 (\lambda (c0: C).(getl (trans is i) c2 c0)) -H10 (CHead x1 (Bind Abst) (lift1 (ptrans is i) u)) H14) in (let H_y \def -(sc3_abst g a0 TNil) in (eq_ind_r T (TLRef (trans is i)) (\lambda (t0: -T).(sc3 g a0 c2 t0)) (H_y c2 (trans is i) (csubc_arity_conf g d1 c2 H4 (TLRef -(trans is i)) a0 (eq_ind T (lift1 is (TLRef i)) (\lambda (t0: T).(arity g d1 -t0 a0)) (arity_lift1 g a0 c is d1 (TLRef i) H3 (arity_abst g c d u i H0 a0 -H1)) (TLRef (trans is i)) (lift1_lref is i))) (nf2_lref_abst c2 x1 (lift1 -(ptrans is i) u) (trans is i) H16) I) (lift1 is (TLRef i)) (lift1_lref is -i))))))) H13)) (\lambda (H13: (ex5_3 C T A (\lambda (_: C).(\lambda (_: -T).(\lambda (_: A).(eq K (Bind Abst) (Bind Abst))))) (\lambda (c3: -C).(\lambda (w: T).(\lambda (_: A).(eq C x0 (CHead c3 (Bind Abbr) w))))) -(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g x c3)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans -is i) u))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3 -w)))))).(ex5_3_ind C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq -K (Bind Abst) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: -A).(eq C x0 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: -T).(\lambda (_: A).(csubc g x c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans is i) u))))) (\lambda (c3: -C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3 w)))) (sc3 g a0 c2 (lift1 is -(TLRef i))) (\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: A).(\lambda (_: -(eq K (Bind Abst) (Bind Abst))).(\lambda (H15: (eq C x0 (CHead x1 (Bind Abbr) -x2))).(\lambda (_: (csubc g x x1)).(\lambda (H17: (sc3 g (asucc g x3) x -(lift1 (ptrans is i) u))).(\lambda (H18: (sc3 g x3 x1 x2)).(let H19 \def -(eq_ind C x0 (\lambda (c0: C).(getl (trans is i) c2 c0)) H10 (CHead x1 (Bind -Abbr) x2) H15) in (let H_y \def (sc3_abbr g a0 TNil) in (eq_ind_r T (TLRef -(trans is i)) (\lambda (t0: T).(sc3 g a0 c2 t0)) (H_y (trans is i) x1 x2 c2 -(let H_y0 \def (arity_lift1 g (asucc g a0) d (ptrans is i) x u H7 H1) in (let -H_y1 \def (sc3_arity_gen g x (lift1 (ptrans is i) u) (asucc g x3) H17) in -(sc3_repl g x3 c2 (lift (S (trans is i)) O x2) (sc3_lift g x3 x1 x2 H18 c2 (S -(trans is i)) O (getl_drop Abbr c2 x1 x2 (trans is i) H19)) a0 (asucc_inj g -x3 a0 (arity_mono g x (lift1 (ptrans is i) u) (asucc g x3) H_y1 (asucc g a0) -H_y0))))) H19) (lift1 is (TLRef i)) (lift1_lref is i)))))))))))) H13)) -(\lambda (H13: (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: -T).(eq C x0 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: T).(eq K (Bind Abst) (Bind Void))))) (\lambda (b: B).(\lambda -(_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: -C).(\lambda (_: T).(csubc g x c3)))))).(ex4_3_ind B C T (\lambda (b: -B).(\lambda (c3: C).(\lambda (v2: T).(eq C x0 (CHead c3 (Bind b) v2))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K (Bind Abst) (Bind -Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b -Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g x c3)))) -(sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x1: B).(\lambda (x2: C).(\lambda -(x3: T).(\lambda (H14: (eq C x0 (CHead x2 (Bind x1) x3))).(\lambda (H15: (eq -K (Bind Abst) (Bind Void))).(\lambda (_: (not (eq B x1 Void))).(\lambda (_: -(csubc g x x2)).(let H18 \def (eq_ind C x0 (\lambda (c0: C).(getl (trans is -i) c2 c0)) H10 (CHead x2 (Bind x1) x3) H14) in (let H19 \def (eq_ind K (Bind -Abst) (\lambda (ee: K).(match ee in K return (\lambda (_: K).Prop) with -[(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr -\Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat -_) \Rightarrow False])) I (Bind Void) H15) in (False_ind (sc3 g a0 c2 (lift1 -is (TLRef i))) H19)))))))))) H13)) H12)))))) H9)))))) H6))))))))))))))))) -(\lambda (b: B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c: C).(\lambda -(u: T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H2: -((\forall (d1: C).(\forall (is: PList).((drop1 is d1 c) \to (\forall (c2: -C).((csubc g d1 c2) \to (sc3 g a1 c2 (lift1 is u))))))))).(\lambda (t0: -T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c (Bind b) u) t0 -a2)).(\lambda (H4: ((\forall (d1: C).(\forall (is: PList).((drop1 is d1 -(CHead c (Bind b) u)) \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g a2 c2 -(lift1 is t0))))))))).(\lambda (d1: C).(\lambda (is: PList).(\lambda (H5: -(drop1 is d1 c)).(\lambda (c2: C).(\lambda (H6: (csubc g d1 c2)).(let H_y -\def (sc3_bind g b H0 a1 a2 TNil) in (eq_ind_r T (THead (Bind b) (lift1 is u) -(lift1 (Ss is) t0)) (\lambda (t1: T).(sc3 g a2 c2 t1)) (H_y c2 (lift1 is u) -(lift1 (Ss is) t0) (H4 (CHead d1 (Bind b) (lift1 is u)) (Ss is) -(drop1_skip_bind b c is d1 u H5) (CHead c2 (Bind b) (lift1 is u)) (csubc_head -g d1 c2 H6 (Bind b) (lift1 is u))) (H2 d1 is H5 c2 H6)) (lift1 is (THead -(Bind b) u t0)) (lift1_bind b is u t0))))))))))))))))))) (\lambda (c: -C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H0: (arity g c u (asucc g -a1))).(\lambda (H1: ((\forall (d1: C).(\forall (is: PList).((drop1 is d1 c) -\to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g (asucc g a1) c2 (lift1 is -u))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (H2: (arity g (CHead c -(Bind Abst) u) t0 a2)).(\lambda (H3: ((\forall (d1: C).(\forall (is: -PList).((drop1 is d1 (CHead c (Bind Abst) u)) \to (\forall (c2: C).((csubc g -d1 c2) \to (sc3 g a2 c2 (lift1 is t0))))))))).(\lambda (d1: C).(\lambda (is: -PList).(\lambda (H4: (drop1 is d1 c)).(\lambda (c2: C).(\lambda (H5: (csubc g -d1 c2)).(eq_ind_r T (THead (Bind Abst) (lift1 is u) (lift1 (Ss is) t0)) -(\lambda (t1: T).(land (arity g c2 t1 (AHead a1 a2)) (\forall (d: C).(\forall -(w: T).((sc3 g a1 d w) \to (\forall (is0: PList).((drop1 is0 d c2) \to (sc3 g -a2 d (THead (Flat Appl) w (lift1 is0 t1)))))))))) (conj (arity g c2 (THead -(Bind Abst) (lift1 is u) (lift1 (Ss is) t0)) (AHead a1 a2)) (\forall (d: -C).(\forall (w: T).((sc3 g a1 d w) \to (\forall (is0: PList).((drop1 is0 d -c2) \to (sc3 g a2 d (THead (Flat Appl) w (lift1 is0 (THead (Bind Abst) (lift1 -is u) (lift1 (Ss is) t0)))))))))) (csubc_arity_conf g d1 c2 H5 (THead (Bind -Abst) (lift1 is u) (lift1 (Ss is) t0)) (AHead a1 a2) (arity_head g d1 (lift1 -is u) a1 (arity_lift1 g (asucc g a1) c is d1 u H4 H0) (lift1 (Ss is) t0) a2 -(arity_lift1 g a2 (CHead c (Bind Abst) u) (Ss is) (CHead d1 (Bind Abst) -(lift1 is u)) t0 (drop1_skip_bind Abst c is d1 u H4) H2))) (\lambda (d: -C).(\lambda (w: T).(\lambda (H6: (sc3 g a1 d w)).(\lambda (is0: -PList).(\lambda (H7: (drop1 is0 d c2)).(eq_ind_r T (THead (Bind Abst) (lift1 -is0 (lift1 is u)) (lift1 (Ss is0) (lift1 (Ss is) t0))) (\lambda (t1: T).(sc3 -g a2 d (THead (Flat Appl) w t1))) (let H8 \def (sc3_appl g a1 a2 TNil) in (H8 -d w (lift1 (Ss is0) (lift1 (Ss is) t0)) (let H_y \def (sc3_bind g Abbr -(\lambda (H9: (eq B Abbr Abst)).(not_abbr_abst H9)) a1 a2 TNil) in (H_y d w -(lift1 (Ss is0) (lift1 (Ss is) t0)) (let H_x \def (csubc_drop1_conf_rev g is0 -d c2 H7 d1 H5) in (let H9 \def H_x in (ex2_ind C (\lambda (c3: C).(drop1 is0 -c3 d1)) (\lambda (c3: C).(csubc g c3 d)) (sc3 g a2 (CHead d (Bind Abbr) w) -(lift1 (Ss is0) (lift1 (Ss is) t0))) (\lambda (x: C).(\lambda (H10: (drop1 -is0 x d1)).(\lambda (H11: (csubc g x d)).(eq_ind_r T (lift1 (papp (Ss is0) -(Ss is)) t0) (\lambda (t1: T).(sc3 g a2 (CHead d (Bind Abbr) w) t1)) -(eq_ind_r PList (Ss (papp is0 is)) (\lambda (p: PList).(sc3 g a2 (CHead d -(Bind Abbr) w) (lift1 p t0))) (H3 (CHead x (Bind Abst) (lift1 (papp is0 is) -u)) (Ss (papp is0 is)) (drop1_skip_bind Abst c (papp is0 is) x u (drop1_trans -is0 x d1 H10 is c H4)) (CHead d (Bind Abbr) w) (csubc_abst g x d H11 (lift1 -(papp is0 is) u) a1 (H1 x (papp is0 is) (drop1_trans is0 x d1 H10 is c H4) x -(csubc_refl g x)) w H6)) (papp (Ss is0) (Ss is)) (papp_ss is0 is)) (lift1 (Ss -is0) (lift1 (Ss is) t0)) (lift1_lift1 (Ss is0) (Ss is) t0))))) H9))) H6)) H6 -(lift1 is0 (lift1 is u)) (sc3_lift1 g c2 (asucc g a1) is0 d (lift1 is u) (H1 -d1 is H4 c2 H5) H7))) (lift1 is0 (THead (Bind Abst) (lift1 is u) (lift1 (Ss -is) t0))) (lift1_bind Abst is0 (lift1 is u) (lift1 (Ss is) t0))))))))) (lift1 -is (THead (Bind Abst) u t0)) (lift1_bind Abst is u t0)))))))))))))))) -(\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c u -a1)).(\lambda (H1: ((\forall (d1: C).(\forall (is: PList).((drop1 is d1 c) -\to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g a1 c2 (lift1 is -u))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g c t0 -(AHead a1 a2))).(\lambda (H3: ((\forall (d1: C).(\forall (is: PList).((drop1 -is d1 c) \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g (AHead a1 a2) c2 -(lift1 is t0))))))))).(\lambda (d1: C).(\lambda (is: PList).(\lambda (H4: -(drop1 is d1 c)).(\lambda (c2: C).(\lambda (H5: (csubc g d1 c2)).(let H_y -\def (H1 d1 is H4 c2 H5) in (let H_y0 \def (H3 d1 is H4 c2 H5) in (let H6 -\def H_y0 in (land_ind (arity g c2 (lift1 is t0) (AHead a1 a2)) (\forall (d: -C).(\forall (w: T).((sc3 g a1 d w) \to (\forall (is0: PList).((drop1 is0 d -c2) \to (sc3 g a2 d (THead (Flat Appl) w (lift1 is0 (lift1 is t0))))))))) -(sc3 g a2 c2 (lift1 is (THead (Flat Appl) u t0))) (\lambda (_: (arity g c2 -(lift1 is t0) (AHead a1 a2))).(\lambda (H8: ((\forall (d: C).(\forall (w: -T).((sc3 g a1 d w) \to (\forall (is0: PList).((drop1 is0 d c2) \to (sc3 g a2 -d (THead (Flat Appl) w (lift1 is0 (lift1 is t0))))))))))).(let H_y1 \def (H8 -c2 (lift1 is u) H_y PNil) in (eq_ind_r T (THead (Flat Appl) (lift1 is u) -(lift1 is t0)) (\lambda (t1: T).(sc3 g a2 c2 t1)) (H_y1 (drop1_nil c2)) -(lift1 is (THead (Flat Appl) u t0)) (lift1_flat Appl is u t0))))) -H6)))))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a0: -A).(\lambda (_: (arity g c u (asucc g a0))).(\lambda (H1: ((\forall (d1: -C).(\forall (is: PList).((drop1 is d1 c) \to (\forall (c2: C).((csubc g d1 -c2) \to (sc3 g (asucc g a0) c2 (lift1 is u))))))))).(\lambda (t0: T).(\lambda -(_: (arity g c t0 a0)).(\lambda (H3: ((\forall (d1: C).(\forall (is: -PList).((drop1 is d1 c) \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g a0 -c2 (lift1 is t0))))))))).(\lambda (d1: C).(\lambda (is: PList).(\lambda (H4: -(drop1 is d1 c)).(\lambda (c2: C).(\lambda (H5: (csubc g d1 c2)).(let H_y -\def (sc3_cast g a0 TNil) in (eq_ind_r T (THead (Flat Cast) (lift1 is u) -(lift1 is t0)) (\lambda (t1: T).(sc3 g a0 c2 t1)) (H_y c2 (lift1 is u) (H1 d1 -is H4 c2 H5) (lift1 is t0) (H3 d1 is H4 c2 H5)) (lift1 is (THead (Flat Cast) -u t0)) (lift1_flat Cast is u t0)))))))))))))))) (\lambda (c: C).(\lambda (t0: -T).(\lambda (a1: A).(\lambda (_: (arity g c t0 a1)).(\lambda (H1: ((\forall -(d1: C).(\forall (is: PList).((drop1 is d1 c) \to (\forall (c2: C).((csubc g -d1 c2) \to (sc3 g a1 c2 (lift1 is t0))))))))).(\lambda (a2: A).(\lambda (H2: -(leq g a1 a2)).(\lambda (d1: C).(\lambda (is: PList).(\lambda (H3: (drop1 is -d1 c)).(\lambda (c2: C).(\lambda (H4: (csubc g d1 c2)).(sc3_repl g a1 c2 -(lift1 is t0) (H1 d1 is H3 c2 H4) a2 H2))))))))))))) c1 t a H))))). -(* COMMENTS -Initial nodes: 5940 -END *) - -theorem sc3_arity: - \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t -a) \to (sc3 g a c t))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: -(arity g c t a)).(let H_y \def (sc3_arity_csubc g c t a H c PNil) in (H_y -(drop1_nil c) c (csubc_refl g c))))))). -(* COMMENTS -Initial nodes: 47 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/sc3/defs.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/sc3/defs.ma deleted file mode 100644 index 32fdfda84..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/sc3/defs.ma +++ /dev/null @@ -1,32 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/sn3/defs.ma". - -include "Basic-1/arity/defs.ma". - -include "Basic-1/drop1/defs.ma". - -definition sc3: - G \to (A \to (C \to (T \to Prop))) -\def - let rec sc3 (g: G) (a: A) on a: (C \to (T \to Prop)) \def (\lambda (c: -C).(\lambda (t: T).(match a with [(ASort h n) \Rightarrow (land (arity g c t -(ASort h n)) (sn3 c t)) | (AHead a1 a2) \Rightarrow (land (arity g c t (AHead -a1 a2)) (\forall (d: C).(\forall (w: T).((sc3 g a1 d w) \to (\forall (is: -PList).((drop1 is d c) \to (sc3 g a2 d (THead (Flat Appl) w (lift1 is -t)))))))))]))) in sc3. - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/sc3/props.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/sc3/props.ma deleted file mode 100644 index e1d909251..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/sc3/props.ma +++ /dev/null @@ -1,728 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/sc3/defs.ma". - -include "Basic-1/sn3/lift1.ma". - -include "Basic-1/nf2/lift1.ma". - -include "Basic-1/csuba/arity.ma". - -include "Basic-1/arity/lift1.ma". - -include "Basic-1/arity/aprem.ma". - -include "Basic-1/llt/props.ma". - -include "Basic-1/drop1/getl.ma". - -include "Basic-1/drop1/props.ma". - -include "Basic-1/lift1/props.ma". - -theorem sc3_arity_gen: - \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((sc3 g a c -t) \to (arity g c t a))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(A_ind -(\lambda (a0: A).((sc3 g a0 c t) \to (arity g c t a0))) (\lambda (n: -nat).(\lambda (n0: nat).(\lambda (H: (land (arity g c t (ASort n n0)) (sn3 c -t))).(let H0 \def H in (land_ind (arity g c t (ASort n n0)) (sn3 c t) (arity -g c t (ASort n n0)) (\lambda (H1: (arity g c t (ASort n n0))).(\lambda (_: -(sn3 c t)).H1)) H0))))) (\lambda (a0: A).(\lambda (_: (((sc3 g a0 c t) \to -(arity g c t a0)))).(\lambda (a1: A).(\lambda (_: (((sc3 g a1 c t) \to (arity -g c t a1)))).(\lambda (H1: (land (arity g c t (AHead a0 a1)) (\forall (d: -C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) -\to (sc3 g a1 d (THead (Flat Appl) w (lift1 is t)))))))))).(let H2 \def H1 in -(land_ind (arity g c t (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g -a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat -Appl) w (lift1 is t)))))))) (arity g c t (AHead a0 a1)) (\lambda (H3: (arity -g c t (AHead a0 a1))).(\lambda (_: ((\forall (d: C).(\forall (w: T).((sc3 g -a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat -Appl) w (lift1 is t)))))))))).H3)) H2))))))) a)))). -(* COMMENTS -Initial nodes: 369 -END *) - -theorem sc3_repl: - \forall (g: G).(\forall (a1: A).(\forall (c: C).(\forall (t: T).((sc3 g a1 c -t) \to (\forall (a2: A).((leq g a1 a2) \to (sc3 g a2 c t))))))) -\def - \lambda (g: G).(\lambda (a1: A).(llt_wf_ind (\lambda (a: A).(\forall (c: -C).(\forall (t: T).((sc3 g a c t) \to (\forall (a2: A).((leq g a a2) \to (sc3 -g a2 c t))))))) (\lambda (a2: A).(A_ind (\lambda (a: A).(((\forall (a3: -A).((llt a3 a) \to (\forall (c: C).(\forall (t: T).((sc3 g a3 c t) \to -(\forall (a4: A).((leq g a3 a4) \to (sc3 g a4 c t))))))))) \to (\forall (c: -C).(\forall (t: T).((sc3 g a c t) \to (\forall (a3: A).((leq g a a3) \to (sc3 -g a3 c t)))))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (_: ((\forall -(a3: A).((llt a3 (ASort n n0)) \to (\forall (c: C).(\forall (t: T).((sc3 g a3 -c t) \to (\forall (a4: A).((leq g a3 a4) \to (sc3 g a4 c t)))))))))).(\lambda -(c: C).(\lambda (t: T).(\lambda (H0: (land (arity g c t (ASort n n0)) (sn3 c -t))).(\lambda (a3: A).(\lambda (H1: (leq g (ASort n n0) a3)).(let H2 \def H0 -in (land_ind (arity g c t (ASort n n0)) (sn3 c t) (sc3 g a3 c t) (\lambda -(H3: (arity g c t (ASort n n0))).(\lambda (H4: (sn3 c t)).(let H_y \def -(arity_repl g c t (ASort n n0) H3 a3 H1) in (let H_x \def (leq_gen_sort1 g n -n0 a3 H1) in (let H5 \def H_x in (ex2_3_ind nat nat nat (\lambda (n2: -nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort n n0) k) -(aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda -(_: nat).(eq A a3 (ASort h2 n2))))) (sc3 g a3 c t) (\lambda (x0: -nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (_: (eq A (aplus g (ASort -n n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H7: (eq A a3 (ASort x1 -x0))).(let H8 \def (f_equal A A (\lambda (e: A).e) a3 (ASort x1 x0) H7) in -(let H9 \def (eq_ind A a3 (\lambda (a: A).(arity g c t a)) H_y (ASort x1 x0) -H8) in (eq_ind_r A (ASort x1 x0) (\lambda (a: A).(sc3 g a c t)) (conj (arity -g c t (ASort x1 x0)) (sn3 c t) H9 H4) a3 H8)))))))) H5)))))) H2)))))))))) -(\lambda (a: A).(\lambda (_: ((((\forall (a3: A).((llt a3 a) \to (\forall (c: -C).(\forall (t: T).((sc3 g a3 c t) \to (\forall (a4: A).((leq g a3 a4) \to -(sc3 g a4 c t))))))))) \to (\forall (c: C).(\forall (t: T).((sc3 g a c t) \to -(\forall (a3: A).((leq g a a3) \to (sc3 g a3 c t))))))))).(\lambda (a0: -A).(\lambda (H0: ((((\forall (a3: A).((llt a3 a0) \to (\forall (c: -C).(\forall (t: T).((sc3 g a3 c t) \to (\forall (a4: A).((leq g a3 a4) \to -(sc3 g a4 c t))))))))) \to (\forall (c: C).(\forall (t: T).((sc3 g a0 c t) -\to (\forall (a3: A).((leq g a0 a3) \to (sc3 g a3 c t))))))))).(\lambda (H1: -((\forall (a3: A).((llt a3 (AHead a a0)) \to (\forall (c: C).(\forall (t: -T).((sc3 g a3 c t) \to (\forall (a4: A).((leq g a3 a4) \to (sc3 g a4 c -t)))))))))).(\lambda (c: C).(\lambda (t: T).(\lambda (H2: (land (arity g c t -(AHead a a0)) (\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall -(is: PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w (lift1 is -t)))))))))).(\lambda (a3: A).(\lambda (H3: (leq g (AHead a a0) a3)).(let H4 -\def H2 in (land_ind (arity g c t (AHead a a0)) (\forall (d: C).(\forall (w: -T).((sc3 g a d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a0 d -(THead (Flat Appl) w (lift1 is t)))))))) (sc3 g a3 c t) (\lambda (H5: (arity -g c t (AHead a a0))).(\lambda (H6: ((\forall (d: C).(\forall (w: T).((sc3 g a -d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat -Appl) w (lift1 is t)))))))))).(let H_x \def (leq_gen_head1 g a a0 a3 H3) in -(let H7 \def H_x in (ex3_2_ind A A (\lambda (a4: A).(\lambda (_: A).(leq g a -a4))) (\lambda (_: A).(\lambda (a5: A).(leq g a0 a5))) (\lambda (a4: -A).(\lambda (a5: A).(eq A a3 (AHead a4 a5)))) (sc3 g a3 c t) (\lambda (x0: -A).(\lambda (x1: A).(\lambda (H8: (leq g a x0)).(\lambda (H9: (leq g a0 -x1)).(\lambda (H10: (eq A a3 (AHead x0 x1))).(let H11 \def (f_equal A A -(\lambda (e: A).e) a3 (AHead x0 x1) H10) in (eq_ind_r A (AHead x0 x1) -(\lambda (a4: A).(sc3 g a4 c t)) (conj (arity g c t (AHead x0 x1)) (\forall -(d: C).(\forall (w: T).((sc3 g x0 d w) \to (\forall (is: PList).((drop1 is d -c) \to (sc3 g x1 d (THead (Flat Appl) w (lift1 is t)))))))) (arity_repl g c t -(AHead a a0) H5 (AHead x0 x1) (leq_head g a x0 H8 a0 x1 H9)) (\lambda (d: -C).(\lambda (w: T).(\lambda (H12: (sc3 g x0 d w)).(\lambda (is: -PList).(\lambda (H13: (drop1 is d c)).(H0 (\lambda (a4: A).(\lambda (H14: -(llt a4 a0)).(\lambda (c0: C).(\lambda (t0: T).(\lambda (H15: (sc3 g a4 c0 -t0)).(\lambda (a5: A).(\lambda (H16: (leq g a4 a5)).(H1 a4 (llt_trans a4 a0 -(AHead a a0) H14 (llt_head_dx a a0)) c0 t0 H15 a5 H16)))))))) d (THead (Flat -Appl) w (lift1 is t)) (H6 d w (H1 x0 (llt_repl g a x0 H8 (AHead a a0) -(llt_head_sx a a0)) d w H12 a (leq_sym g a x0 H8)) is H13) x1 H9))))))) a3 -H11))))))) H7))))) H4)))))))))))) a2)) a1)). -(* COMMENTS -Initial nodes: 1359 -END *) - -theorem sc3_lift: - \forall (g: G).(\forall (a: A).(\forall (e: C).(\forall (t: T).((sc3 g a e -t) \to (\forall (c: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) -\to (sc3 g a c (lift h d t)))))))))) -\def - \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(\forall (e: -C).(\forall (t: T).((sc3 g a0 e t) \to (\forall (c: C).(\forall (h: -nat).(\forall (d: nat).((drop h d c e) \to (sc3 g a0 c (lift h d t)))))))))) -(\lambda (n: nat).(\lambda (n0: nat).(\lambda (e: C).(\lambda (t: T).(\lambda -(H: (land (arity g e t (ASort n n0)) (sn3 e t))).(\lambda (c: C).(\lambda (h: -nat).(\lambda (d: nat).(\lambda (H0: (drop h d c e)).(let H1 \def H in -(land_ind (arity g e t (ASort n n0)) (sn3 e t) (land (arity g c (lift h d t) -(ASort n n0)) (sn3 c (lift h d t))) (\lambda (H2: (arity g e t (ASort n -n0))).(\lambda (H3: (sn3 e t)).(conj (arity g c (lift h d t) (ASort n n0)) -(sn3 c (lift h d t)) (arity_lift g e t (ASort n n0) H2 c h d H0) (sn3_lift e -t H3 c h d H0)))) H1))))))))))) (\lambda (a0: A).(\lambda (_: ((\forall (e: -C).(\forall (t: T).((sc3 g a0 e t) \to (\forall (c: C).(\forall (h: -nat).(\forall (d: nat).((drop h d c e) \to (sc3 g a0 c (lift h d -t))))))))))).(\lambda (a1: A).(\lambda (_: ((\forall (e: C).(\forall (t: -T).((sc3 g a1 e t) \to (\forall (c: C).(\forall (h: nat).(\forall (d: -nat).((drop h d c e) \to (sc3 g a1 c (lift h d t))))))))))).(\lambda (e: -C).(\lambda (t: T).(\lambda (H1: (land (arity g e t (AHead a0 a1)) (\forall -(d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d -e) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is t)))))))))).(\lambda (c: -C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2: (drop h d c e)).(let H3 -\def H1 in (land_ind (arity g e t (AHead a0 a1)) (\forall (d0: C).(\forall -(w: T).((sc3 g a0 d0 w) \to (\forall (is: PList).((drop1 is d0 e) \to (sc3 g -a1 d0 (THead (Flat Appl) w (lift1 is t)))))))) (land (arity g c (lift h d t) -(AHead a0 a1)) (\forall (d0: C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall -(is: PList).((drop1 is d0 c) \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is -(lift h d t)))))))))) (\lambda (H4: (arity g e t (AHead a0 a1))).(\lambda -(H5: ((\forall (d0: C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: -PList).((drop1 is d0 e) \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is -t)))))))))).(conj (arity g c (lift h d t) (AHead a0 a1)) (\forall (d0: -C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: PList).((drop1 is d0 c) -\to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is (lift h d t))))))))) -(arity_lift g e t (AHead a0 a1) H4 c h d H2) (\lambda (d0: C).(\lambda (w: -T).(\lambda (H6: (sc3 g a0 d0 w)).(\lambda (is: PList).(\lambda (H7: (drop1 -is d0 c)).(let H_y \def (H5 d0 w H6 (PConsTail is h d)) in (eq_ind T (lift1 -(PConsTail is h d) t) (\lambda (t0: T).(sc3 g a1 d0 (THead (Flat Appl) w -t0))) (H_y (drop1_cons_tail c e h d H2 is d0 H7)) (lift1 is (lift h d t)) -(lift1_cons_tail t h d is))))))))))) H3))))))))))))) a)). -(* COMMENTS -Initial nodes: 849 -END *) - -theorem sc3_lift1: - \forall (g: G).(\forall (e: C).(\forall (a: A).(\forall (hds: -PList).(\forall (c: C).(\forall (t: T).((sc3 g a e t) \to ((drop1 hds c e) -\to (sc3 g a c (lift1 hds t))))))))) -\def - \lambda (g: G).(\lambda (e: C).(\lambda (a: A).(\lambda (hds: -PList).(PList_ind (\lambda (p: PList).(\forall (c: C).(\forall (t: T).((sc3 g -a e t) \to ((drop1 p c e) \to (sc3 g a c (lift1 p t))))))) (\lambda (c: -C).(\lambda (t: T).(\lambda (H: (sc3 g a e t)).(\lambda (H0: (drop1 PNil c -e)).(let H_y \def (drop1_gen_pnil c e H0) in (eq_ind_r C e (\lambda (c0: -C).(sc3 g a c0 t)) H c H_y)))))) (\lambda (n: nat).(\lambda (n0: -nat).(\lambda (p: PList).(\lambda (H: ((\forall (c: C).(\forall (t: T).((sc3 -g a e t) \to ((drop1 p c e) \to (sc3 g a c (lift1 p t)))))))).(\lambda (c: -C).(\lambda (t: T).(\lambda (H0: (sc3 g a e t)).(\lambda (H1: (drop1 (PCons n -n0 p) c e)).(let H_x \def (drop1_gen_pcons c e p n n0 H1) in (let H2 \def H_x -in (ex2_ind C (\lambda (c2: C).(drop n n0 c c2)) (\lambda (c2: C).(drop1 p c2 -e)) (sc3 g a c (lift n n0 (lift1 p t))) (\lambda (x: C).(\lambda (H3: (drop n -n0 c x)).(\lambda (H4: (drop1 p x e)).(sc3_lift g a x (lift1 p t) (H x t H0 -H4) c n n0 H3)))) H2))))))))))) hds)))). -(* COMMENTS -Initial nodes: 289 -END *) - -theorem sc3_abbr: - \forall (g: G).(\forall (a: A).(\forall (vs: TList).(\forall (i: -nat).(\forall (d: C).(\forall (v: T).(\forall (c: C).((sc3 g a c (THeads -(Flat Appl) vs (lift (S i) O v))) \to ((getl i c (CHead d (Bind Abbr) v)) \to -(sc3 g a c (THeads (Flat Appl) vs (TLRef i))))))))))) -\def - \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(\forall (vs: -TList).(\forall (i: nat).(\forall (d: C).(\forall (v: T).(\forall (c: -C).((sc3 g a0 c (THeads (Flat Appl) vs (lift (S i) O v))) \to ((getl i c -(CHead d (Bind Abbr) v)) \to (sc3 g a0 c (THeads (Flat Appl) vs (TLRef -i))))))))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (vs: -TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(\lambda (c: -C).(\lambda (H: (land (arity g c (THeads (Flat Appl) vs (lift (S i) O v)) -(ASort n n0)) (sn3 c (THeads (Flat Appl) vs (lift (S i) O v))))).(\lambda -(H0: (getl i c (CHead d (Bind Abbr) v))).(let H1 \def H in (land_ind (arity g -c (THeads (Flat Appl) vs (lift (S i) O v)) (ASort n n0)) (sn3 c (THeads (Flat -Appl) vs (lift (S i) O v))) (land (arity g c (THeads (Flat Appl) vs (TLRef -i)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (TLRef i)))) (\lambda (H2: -(arity g c (THeads (Flat Appl) vs (lift (S i) O v)) (ASort n n0))).(\lambda -(H3: (sn3 c (THeads (Flat Appl) vs (lift (S i) O v)))).(conj (arity g c -(THeads (Flat Appl) vs (TLRef i)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs -(TLRef i))) (arity_appls_abbr g c d v i H0 vs (ASort n n0) H2) -(sn3_appls_abbr c d v i H0 vs H3)))) H1))))))))))) (\lambda (a0: A).(\lambda -(_: ((\forall (vs: TList).(\forall (i: nat).(\forall (d: C).(\forall (v: -T).(\forall (c: C).((sc3 g a0 c (THeads (Flat Appl) vs (lift (S i) O v))) \to -((getl i c (CHead d (Bind Abbr) v)) \to (sc3 g a0 c (THeads (Flat Appl) vs -(TLRef i)))))))))))).(\lambda (a1: A).(\lambda (H0: ((\forall (vs: -TList).(\forall (i: nat).(\forall (d: C).(\forall (v: T).(\forall (c: -C).((sc3 g a1 c (THeads (Flat Appl) vs (lift (S i) O v))) \to ((getl i c -(CHead d (Bind Abbr) v)) \to (sc3 g a1 c (THeads (Flat Appl) vs (TLRef -i)))))))))))).(\lambda (vs: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda -(v: T).(\lambda (c: C).(\lambda (H1: (land (arity g c (THeads (Flat Appl) vs -(lift (S i) O v)) (AHead a0 a1)) (\forall (d0: C).(\forall (w: T).((sc3 g a0 -d0 w) \to (\forall (is: PList).((drop1 is d0 c) \to (sc3 g a1 d0 (THead (Flat -Appl) w (lift1 is (THeads (Flat Appl) vs (lift (S i) O v)))))))))))).(\lambda -(H2: (getl i c (CHead d (Bind Abbr) v))).(let H3 \def H1 in (land_ind (arity -g c (THeads (Flat Appl) vs (lift (S i) O v)) (AHead a0 a1)) (\forall (d0: -C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: PList).((drop1 is d0 c) -\to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs (lift -(S i) O v)))))))))) (land (arity g c (THeads (Flat Appl) vs (TLRef i)) (AHead -a0 a1)) (\forall (d0: C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: -PList).((drop1 is d0 c) \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is -(THeads (Flat Appl) vs (TLRef i))))))))))) (\lambda (H4: (arity g c (THeads -(Flat Appl) vs (lift (S i) O v)) (AHead a0 a1))).(\lambda (H5: ((\forall (d0: -C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: PList).((drop1 is d0 c) -\to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs (lift -(S i) O v)))))))))))).(conj (arity g c (THeads (Flat Appl) vs (TLRef i)) -(AHead a0 a1)) (\forall (d0: C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall -(is: PList).((drop1 is d0 c) \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is -(THeads (Flat Appl) vs (TLRef i)))))))))) (arity_appls_abbr g c d v i H2 vs -(AHead a0 a1) H4) (\lambda (d0: C).(\lambda (w: T).(\lambda (H6: (sc3 g a0 d0 -w)).(\lambda (is: PList).(\lambda (H7: (drop1 is d0 c)).(let H_x \def -(drop1_getl_trans is c d0 H7 Abbr d v i H2) in (let H8 \def H_x in (ex2_ind C -(\lambda (e2: C).(drop1 (ptrans is i) e2 d)) (\lambda (e2: C).(getl (trans is -i) d0 (CHead e2 (Bind Abbr) (lift1 (ptrans is i) v)))) (sc3 g a1 d0 (THead -(Flat Appl) w (lift1 is (THeads (Flat Appl) vs (TLRef i))))) (\lambda (x: -C).(\lambda (_: (drop1 (ptrans is i) x d)).(\lambda (H10: (getl (trans is i) -d0 (CHead x (Bind Abbr) (lift1 (ptrans is i) v)))).(let H_y \def (H0 (TCons w -(lifts1 is vs))) in (eq_ind_r T (THeads (Flat Appl) (lifts1 is vs) (lift1 is -(TLRef i))) (\lambda (t: T).(sc3 g a1 d0 (THead (Flat Appl) w t))) (eq_ind_r -T (TLRef (trans is i)) (\lambda (t: T).(sc3 g a1 d0 (THead (Flat Appl) w -(THeads (Flat Appl) (lifts1 is vs) t)))) (H_y (trans is i) x (lift1 (ptrans -is i) v) d0 (eq_ind T (lift1 is (lift (S i) O v)) (\lambda (t: T).(sc3 g a1 -d0 (THead (Flat Appl) w (THeads (Flat Appl) (lifts1 is vs) t)))) (eq_ind T -(lift1 is (THeads (Flat Appl) vs (lift (S i) O v))) (\lambda (t: T).(sc3 g a1 -d0 (THead (Flat Appl) w t))) (H5 d0 w H6 is H7) (THeads (Flat Appl) (lifts1 -is vs) (lift1 is (lift (S i) O v))) (lifts1_flat Appl is (lift (S i) O v) -vs)) (lift (S (trans is i)) O (lift1 (ptrans is i) v)) (lift1_free is i v)) -H10) (lift1 is (TLRef i)) (lift1_lref is i)) (lift1 is (THeads (Flat Appl) vs -(TLRef i))) (lifts1_flat Appl is (TLRef i) vs)))))) H8))))))))))) -H3))))))))))))) a)). -(* COMMENTS -Initial nodes: 1563 -END *) - -theorem sc3_cast: - \forall (g: G).(\forall (a: A).(\forall (vs: TList).(\forall (c: C).(\forall -(u: T).((sc3 g (asucc g a) c (THeads (Flat Appl) vs u)) \to (\forall (t: -T).((sc3 g a c (THeads (Flat Appl) vs t)) \to (sc3 g a c (THeads (Flat Appl) -vs (THead (Flat Cast) u t)))))))))) -\def - \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(\forall (vs: -TList).(\forall (c: C).(\forall (u: T).((sc3 g (asucc g a0) c (THeads (Flat -Appl) vs u)) \to (\forall (t: T).((sc3 g a0 c (THeads (Flat Appl) vs t)) \to -(sc3 g a0 c (THeads (Flat Appl) vs (THead (Flat Cast) u t)))))))))) (\lambda -(n: nat).(\lambda (n0: nat).(\lambda (vs: TList).(\lambda (c: C).(\lambda (u: -T).(\lambda (H: (sc3 g (match n with [O \Rightarrow (ASort O (next g n0)) | -(S h) \Rightarrow (ASort h n0)]) c (THeads (Flat Appl) vs u))).(\lambda (t: -T).(\lambda (H0: (land (arity g c (THeads (Flat Appl) vs t) (ASort n n0)) -(sn3 c (THeads (Flat Appl) vs t)))).(nat_ind (\lambda (n1: nat).((sc3 g -(match n1 with [O \Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow -(ASort h n0)]) c (THeads (Flat Appl) vs u)) \to ((land (arity g c (THeads -(Flat Appl) vs t) (ASort n1 n0)) (sn3 c (THeads (Flat Appl) vs t))) \to (land -(arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort n1 n0)) -(sn3 c (THeads (Flat Appl) vs (THead (Flat Cast) u t))))))) (\lambda (H1: -(sc3 g (ASort O (next g n0)) c (THeads (Flat Appl) vs u))).(\lambda (H2: -(land (arity g c (THeads (Flat Appl) vs t) (ASort O n0)) (sn3 c (THeads (Flat -Appl) vs t)))).(let H3 \def H1 in (land_ind (arity g c (THeads (Flat Appl) vs -u) (ASort O (next g n0))) (sn3 c (THeads (Flat Appl) vs u)) (land (arity g c -(THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort O n0)) (sn3 c (THeads -(Flat Appl) vs (THead (Flat Cast) u t)))) (\lambda (H4: (arity g c (THeads -(Flat Appl) vs u) (ASort O (next g n0)))).(\lambda (H5: (sn3 c (THeads (Flat -Appl) vs u))).(let H6 \def H2 in (land_ind (arity g c (THeads (Flat Appl) vs -t) (ASort O n0)) (sn3 c (THeads (Flat Appl) vs t)) (land (arity g c (THeads -(Flat Appl) vs (THead (Flat Cast) u t)) (ASort O n0)) (sn3 c (THeads (Flat -Appl) vs (THead (Flat Cast) u t)))) (\lambda (H7: (arity g c (THeads (Flat -Appl) vs t) (ASort O n0))).(\lambda (H8: (sn3 c (THeads (Flat Appl) vs -t))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort -O n0)) (sn3 c (THeads (Flat Appl) vs (THead (Flat Cast) u t))) -(arity_appls_cast g c u t vs (ASort O n0) H4 H7) (sn3_appls_cast c vs u H5 t -H8)))) H6)))) H3)))) (\lambda (n1: nat).(\lambda (_: (((sc3 g (match n1 with -[O \Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)]) c -(THeads (Flat Appl) vs u)) \to ((land (arity g c (THeads (Flat Appl) vs t) -(ASort n1 n0)) (sn3 c (THeads (Flat Appl) vs t))) \to (land (arity g c -(THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort n1 n0)) (sn3 c (THeads -(Flat Appl) vs (THead (Flat Cast) u t)))))))).(\lambda (H1: (sc3 g (ASort n1 -n0) c (THeads (Flat Appl) vs u))).(\lambda (H2: (land (arity g c (THeads -(Flat Appl) vs t) (ASort (S n1) n0)) (sn3 c (THeads (Flat Appl) vs t)))).(let -H3 \def H1 in (land_ind (arity g c (THeads (Flat Appl) vs u) (ASort n1 n0)) -(sn3 c (THeads (Flat Appl) vs u)) (land (arity g c (THeads (Flat Appl) vs -(THead (Flat Cast) u t)) (ASort (S n1) n0)) (sn3 c (THeads (Flat Appl) vs -(THead (Flat Cast) u t)))) (\lambda (H4: (arity g c (THeads (Flat Appl) vs u) -(ASort n1 n0))).(\lambda (H5: (sn3 c (THeads (Flat Appl) vs u))).(let H6 \def -H2 in (land_ind (arity g c (THeads (Flat Appl) vs t) (ASort (S n1) n0)) (sn3 -c (THeads (Flat Appl) vs t)) (land (arity g c (THeads (Flat Appl) vs (THead -(Flat Cast) u t)) (ASort (S n1) n0)) (sn3 c (THeads (Flat Appl) vs (THead -(Flat Cast) u t)))) (\lambda (H7: (arity g c (THeads (Flat Appl) vs t) (ASort -(S n1) n0))).(\lambda (H8: (sn3 c (THeads (Flat Appl) vs t))).(conj (arity g -c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort (S n1) n0)) (sn3 c -(THeads (Flat Appl) vs (THead (Flat Cast) u t))) (arity_appls_cast g c u t vs -(ASort (S n1) n0) H4 H7) (sn3_appls_cast c vs u H5 t H8)))) H6)))) H3)))))) n -H H0))))))))) (\lambda (a0: A).(\lambda (_: ((\forall (vs: TList).(\forall -(c: C).(\forall (u: T).((sc3 g (asucc g a0) c (THeads (Flat Appl) vs u)) \to -(\forall (t: T).((sc3 g a0 c (THeads (Flat Appl) vs t)) \to (sc3 g a0 c -(THeads (Flat Appl) vs (THead (Flat Cast) u t))))))))))).(\lambda (a1: -A).(\lambda (H0: ((\forall (vs: TList).(\forall (c: C).(\forall (u: T).((sc3 -g (asucc g a1) c (THeads (Flat Appl) vs u)) \to (\forall (t: T).((sc3 g a1 c -(THeads (Flat Appl) vs t)) \to (sc3 g a1 c (THeads (Flat Appl) vs (THead -(Flat Cast) u t))))))))))).(\lambda (vs: TList).(\lambda (c: C).(\lambda (u: -T).(\lambda (H1: (land (arity g c (THeads (Flat Appl) vs u) (AHead a0 (asucc -g a1))) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: -PList).((drop1 is d c) \to (sc3 g (asucc g a1) d (THead (Flat Appl) w (lift1 -is (THeads (Flat Appl) vs u))))))))))).(\lambda (t: T).(\lambda (H2: (land -(arity g c (THeads (Flat Appl) vs t) (AHead a0 a1)) (\forall (d: C).(\forall -(w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 -d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs t))))))))))).(let H3 -\def H1 in (land_ind (arity g c (THeads (Flat Appl) vs u) (AHead a0 (asucc g -a1))) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: -PList).((drop1 is d c) \to (sc3 g (asucc g a1) d (THead (Flat Appl) w (lift1 -is (THeads (Flat Appl) vs u))))))))) (land (arity g c (THeads (Flat Appl) vs -(THead (Flat Cast) u t)) (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 -g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead -(Flat Appl) w (lift1 is (THeads (Flat Appl) vs (THead (Flat Cast) u -t))))))))))) (\lambda (H4: (arity g c (THeads (Flat Appl) vs u) (AHead a0 -(asucc g a1)))).(\lambda (H5: ((\forall (d: C).(\forall (w: T).((sc3 g a0 d -w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g (asucc g a1) d (THead -(Flat Appl) w (lift1 is (THeads (Flat Appl) vs u))))))))))).(let H6 \def H2 -in (land_ind (arity g c (THeads (Flat Appl) vs t) (AHead a0 a1)) (\forall (d: -C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) -\to (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs -t))))))))) (land (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) -(AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall -(is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is -(THeads (Flat Appl) vs (THead (Flat Cast) u t))))))))))) (\lambda (H7: (arity -g c (THeads (Flat Appl) vs t) (AHead a0 a1))).(\lambda (H8: ((\forall (d: -C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) -\to (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs -t))))))))))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) -(AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall -(is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is -(THeads (Flat Appl) vs (THead (Flat Cast) u t)))))))))) (arity_appls_cast g c -u t vs (AHead a0 a1) H4 H7) (\lambda (d: C).(\lambda (w: T).(\lambda (H9: -(sc3 g a0 d w)).(\lambda (is: PList).(\lambda (H10: (drop1 is d c)).(let H_y -\def (H0 (TCons w (lifts1 is vs))) in (eq_ind_r T (THeads (Flat Appl) (lifts1 -is vs) (lift1 is (THead (Flat Cast) u t))) (\lambda (t0: T).(sc3 g a1 d -(THead (Flat Appl) w t0))) (eq_ind_r T (THead (Flat Cast) (lift1 is u) (lift1 -is t)) (\lambda (t0: T).(sc3 g a1 d (THead (Flat Appl) w (THeads (Flat Appl) -(lifts1 is vs) t0)))) (H_y d (lift1 is u) (eq_ind T (lift1 is (THeads (Flat -Appl) vs u)) (\lambda (t0: T).(sc3 g (asucc g a1) d (THead (Flat Appl) w -t0))) (H5 d w H9 is H10) (THeads (Flat Appl) (lifts1 is vs) (lift1 is u)) -(lifts1_flat Appl is u vs)) (lift1 is t) (eq_ind T (lift1 is (THeads (Flat -Appl) vs t)) (\lambda (t0: T).(sc3 g a1 d (THead (Flat Appl) w t0))) (H8 d w -H9 is H10) (THeads (Flat Appl) (lifts1 is vs) (lift1 is t)) (lifts1_flat Appl -is t vs))) (lift1 is (THead (Flat Cast) u t)) (lift1_flat Cast is u t)) -(lift1 is (THeads (Flat Appl) vs (THead (Flat Cast) u t))) (lifts1_flat Appl -is (THead (Flat Cast) u t) vs))))))))))) H6)))) H3)))))))))))) a)). -(* COMMENTS -Initial nodes: 2625 -END *) - -theorem sc3_props__sc3_sn3_abst: - \forall (g: G).(\forall (a: A).(land (\forall (c: C).(\forall (t: T).((sc3 g -a c t) \to (sn3 c t)))) (\forall (vs: TList).(\forall (i: nat).(let t \def -(THeads (Flat Appl) vs (TLRef i)) in (\forall (c: C).((arity g c t a) \to -((nf2 c (TLRef i)) \to ((sns3 c vs) \to (sc3 g a c t)))))))))) -\def - \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(land (\forall (c: -C).(\forall (t: T).((sc3 g a0 c t) \to (sn3 c t)))) (\forall (vs: -TList).(\forall (i: nat).(let t \def (THeads (Flat Appl) vs (TLRef i)) in -(\forall (c: C).((arity g c t a0) \to ((nf2 c (TLRef i)) \to ((sns3 c vs) \to -(sc3 g a0 c t)))))))))) (\lambda (n: nat).(\lambda (n0: nat).(conj (\forall -(c: C).(\forall (t: T).((land (arity g c t (ASort n n0)) (sn3 c t)) \to (sn3 -c t)))) (\forall (vs: TList).(\forall (i: nat).(\forall (c: C).((arity g c -(THeads (Flat Appl) vs (TLRef i)) (ASort n n0)) \to ((nf2 c (TLRef i)) \to -((sns3 c vs) \to (land (arity g c (THeads (Flat Appl) vs (TLRef i)) (ASort n -n0)) (sn3 c (THeads (Flat Appl) vs (TLRef i)))))))))) (\lambda (c: -C).(\lambda (t: T).(\lambda (H: (land (arity g c t (ASort n n0)) (sn3 c -t))).(let H0 \def H in (land_ind (arity g c t (ASort n n0)) (sn3 c t) (sn3 c -t) (\lambda (_: (arity g c t (ASort n n0))).(\lambda (H2: (sn3 c t)).H2)) -H0))))) (\lambda (vs: TList).(\lambda (i: nat).(\lambda (c: C).(\lambda (H: -(arity g c (THeads (Flat Appl) vs (TLRef i)) (ASort n n0))).(\lambda (H0: -(nf2 c (TLRef i))).(\lambda (H1: (sns3 c vs)).(conj (arity g c (THeads (Flat -Appl) vs (TLRef i)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (TLRef i))) H -(sn3_appls_lref c i H0 vs H1))))))))))) (\lambda (a0: A).(\lambda (H: (land -(\forall (c: C).(\forall (t: T).((sc3 g a0 c t) \to (sn3 c t)))) (\forall -(vs: TList).(\forall (i: nat).(\forall (c: C).((arity g c (THeads (Flat Appl) -vs (TLRef i)) a0) \to ((nf2 c (TLRef i)) \to ((sns3 c vs) \to (sc3 g a0 c -(THeads (Flat Appl) vs (TLRef i))))))))))).(\lambda (a1: A).(\lambda (H0: -(land (\forall (c: C).(\forall (t: T).((sc3 g a1 c t) \to (sn3 c t)))) -(\forall (vs: TList).(\forall (i: nat).(\forall (c: C).((arity g c (THeads -(Flat Appl) vs (TLRef i)) a1) \to ((nf2 c (TLRef i)) \to ((sns3 c vs) \to -(sc3 g a1 c (THeads (Flat Appl) vs (TLRef i))))))))))).(conj (\forall (c: -C).(\forall (t: T).((land (arity g c t (AHead a0 a1)) (\forall (d: -C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) -\to (sc3 g a1 d (THead (Flat Appl) w (lift1 is t))))))))) \to (sn3 c t)))) -(\forall (vs: TList).(\forall (i: nat).(\forall (c: C).((arity g c (THeads -(Flat Appl) vs (TLRef i)) (AHead a0 a1)) \to ((nf2 c (TLRef i)) \to ((sns3 c -vs) \to (land (arity g c (THeads (Flat Appl) vs (TLRef i)) (AHead a0 a1)) -(\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: -PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads -(Flat Appl) vs (TLRef i))))))))))))))))) (\lambda (c: C).(\lambda (t: -T).(\lambda (H1: (land (arity g c t (AHead a0 a1)) (\forall (d: C).(\forall -(w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 -d (THead (Flat Appl) w (lift1 is t)))))))))).(let H2 \def H in (land_ind -(\forall (c0: C).(\forall (t0: T).((sc3 g a0 c0 t0) \to (sn3 c0 t0)))) -(\forall (vs: TList).(\forall (i: nat).(\forall (c0: C).((arity g c0 (THeads -(Flat Appl) vs (TLRef i)) a0) \to ((nf2 c0 (TLRef i)) \to ((sns3 c0 vs) \to -(sc3 g a0 c0 (THeads (Flat Appl) vs (TLRef i))))))))) (sn3 c t) (\lambda (_: -((\forall (c0: C).(\forall (t0: T).((sc3 g a0 c0 t0) \to (sn3 c0 -t0)))))).(\lambda (H4: ((\forall (vs: TList).(\forall (i: nat).(\forall (c0: -C).((arity g c0 (THeads (Flat Appl) vs (TLRef i)) a0) \to ((nf2 c0 (TLRef i)) -\to ((sns3 c0 vs) \to (sc3 g a0 c0 (THeads (Flat Appl) vs (TLRef -i))))))))))).(let H5 \def H0 in (land_ind (\forall (c0: C).(\forall (t0: -T).((sc3 g a1 c0 t0) \to (sn3 c0 t0)))) (\forall (vs: TList).(\forall (i: -nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) vs (TLRef i)) a1) \to -((nf2 c0 (TLRef i)) \to ((sns3 c0 vs) \to (sc3 g a1 c0 (THeads (Flat Appl) vs -(TLRef i))))))))) (sn3 c t) (\lambda (H6: ((\forall (c0: C).(\forall (t0: -T).((sc3 g a1 c0 t0) \to (sn3 c0 t0)))))).(\lambda (_: ((\forall (vs: -TList).(\forall (i: nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) vs -(TLRef i)) a1) \to ((nf2 c0 (TLRef i)) \to ((sns3 c0 vs) \to (sc3 g a1 c0 -(THeads (Flat Appl) vs (TLRef i))))))))))).(let H8 \def H1 in (land_ind -(arity g c t (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) -\to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w -(lift1 is t)))))))) (sn3 c t) (\lambda (H9: (arity g c t (AHead a0 -a1))).(\lambda (H10: ((\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to -(\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w -(lift1 is t)))))))))).(let H_y \def (arity_aprem g c t (AHead a0 a1) H9 O a0) -in (let H11 \def (H_y (aprem_zero a0 a1)) in (ex2_3_ind C T nat (\lambda (d: -C).(\lambda (_: T).(\lambda (j: nat).(drop j O d c)))) (\lambda (d: -C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g a0))))) (sn3 c t) -(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H12: (drop x2 -O x0 c)).(\lambda (H13: (arity g x0 x1 (asucc g a0))).(let H_y0 \def (H10 -(CHead x0 (Bind Abst) x1) (TLRef O) (H4 TNil O (CHead x0 (Bind Abst) x1) -(arity_abst g (CHead x0 (Bind Abst) x1) x0 x1 O (getl_refl Abst x0 x1) a0 -H13) (nf2_lref_abst (CHead x0 (Bind Abst) x1) x0 x1 O (getl_refl Abst x0 x1)) -I) (PCons (S x2) O PNil)) in (let H_y1 \def (H6 (CHead x0 (Bind Abst) x1) -(THead (Flat Appl) (TLRef O) (lift (S x2) O t)) (H_y0 (drop1_cons (CHead x0 -(Bind Abst) x1) c (S x2) O (drop_drop (Bind Abst) x2 x0 c H12 x1) c PNil -(drop1_nil c)))) in (let H_x \def (sn3_gen_flat Appl (CHead x0 (Bind Abst) -x1) (TLRef O) (lift (S x2) O t) H_y1) in (let H14 \def H_x in (land_ind (sn3 -(CHead x0 (Bind Abst) x1) (TLRef O)) (sn3 (CHead x0 (Bind Abst) x1) (lift (S -x2) O t)) (sn3 c t) (\lambda (_: (sn3 (CHead x0 (Bind Abst) x1) (TLRef -O))).(\lambda (H16: (sn3 (CHead x0 (Bind Abst) x1) (lift (S x2) O -t))).(sn3_gen_lift (CHead x0 (Bind Abst) x1) t (S x2) O H16 c (drop_drop -(Bind Abst) x2 x0 c H12 x1)))) H14)))))))))) H11))))) H8)))) H5)))) H2))))) -(\lambda (vs: TList).(\lambda (i: nat).(\lambda (c: C).(\lambda (H1: (arity g -c (THeads (Flat Appl) vs (TLRef i)) (AHead a0 a1))).(\lambda (H2: (nf2 c -(TLRef i))).(\lambda (H3: (sns3 c vs)).(conj (arity g c (THeads (Flat Appl) -vs (TLRef i)) (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) -\to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w -(lift1 is (THeads (Flat Appl) vs (TLRef i)))))))))) H1 (\lambda (d: -C).(\lambda (w: T).(\lambda (H4: (sc3 g a0 d w)).(\lambda (is: -PList).(\lambda (H5: (drop1 is d c)).(let H6 \def H in (land_ind (\forall -(c0: C).(\forall (t: T).((sc3 g a0 c0 t) \to (sn3 c0 t)))) (\forall (vs0: -TList).(\forall (i0: nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) -vs0 (TLRef i0)) a0) \to ((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a0 -c0 (THeads (Flat Appl) vs0 (TLRef i0))))))))) (sc3 g a1 d (THead (Flat Appl) -w (lift1 is (THeads (Flat Appl) vs (TLRef i))))) (\lambda (H7: ((\forall (c0: -C).(\forall (t: T).((sc3 g a0 c0 t) \to (sn3 c0 t)))))).(\lambda (_: -((\forall (vs0: TList).(\forall (i0: nat).(\forall (c0: C).((arity g c0 -(THeads (Flat Appl) vs0 (TLRef i0)) a0) \to ((nf2 c0 (TLRef i0)) \to ((sns3 -c0 vs0) \to (sc3 g a0 c0 (THeads (Flat Appl) vs0 (TLRef i0))))))))))).(let H9 -\def H0 in (land_ind (\forall (c0: C).(\forall (t: T).((sc3 g a1 c0 t) \to -(sn3 c0 t)))) (\forall (vs0: TList).(\forall (i0: nat).(\forall (c0: -C).((arity g c0 (THeads (Flat Appl) vs0 (TLRef i0)) a1) \to ((nf2 c0 (TLRef -i0)) \to ((sns3 c0 vs0) \to (sc3 g a1 c0 (THeads (Flat Appl) vs0 (TLRef -i0))))))))) (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs -(TLRef i))))) (\lambda (_: ((\forall (c0: C).(\forall (t: T).((sc3 g a1 c0 t) -\to (sn3 c0 t)))))).(\lambda (H11: ((\forall (vs0: TList).(\forall (i0: -nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) vs0 (TLRef i0)) a1) -\to ((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a1 c0 (THeads (Flat -Appl) vs0 (TLRef i0))))))))))).(let H_y \def (H11 (TCons w (lifts1 is vs))) -in (eq_ind_r T (THeads (Flat Appl) (lifts1 is vs) (lift1 is (TLRef i))) -(\lambda (t: T).(sc3 g a1 d (THead (Flat Appl) w t))) (eq_ind_r T (TLRef -(trans is i)) (\lambda (t: T).(sc3 g a1 d (THead (Flat Appl) w (THeads (Flat -Appl) (lifts1 is vs) t)))) (H_y (trans is i) d (eq_ind T (lift1 is (TLRef i)) -(\lambda (t: T).(arity g d (THead (Flat Appl) w (THeads (Flat Appl) (lifts1 -is vs) t)) a1)) (eq_ind T (lift1 is (THeads (Flat Appl) vs (TLRef i))) -(\lambda (t: T).(arity g d (THead (Flat Appl) w t) a1)) (arity_appl g d w a0 -(sc3_arity_gen g d w a0 H4) (lift1 is (THeads (Flat Appl) vs (TLRef i))) a1 -(arity_lift1 g (AHead a0 a1) c is d (THeads (Flat Appl) vs (TLRef i)) H5 H1)) -(THeads (Flat Appl) (lifts1 is vs) (lift1 is (TLRef i))) (lifts1_flat Appl is -(TLRef i) vs)) (TLRef (trans is i)) (lift1_lref is i)) (eq_ind T (lift1 is -(TLRef i)) (\lambda (t: T).(nf2 d t)) (nf2_lift1 c is d (TLRef i) H5 H2) -(TLRef (trans is i)) (lift1_lref is i)) (conj (sn3 d w) (sns3 d (lifts1 is -vs)) (H7 d w H4) (sns3_lifts1 c is d H5 vs H3))) (lift1 is (TLRef i)) -(lift1_lref is i)) (lift1 is (THeads (Flat Appl) vs (TLRef i))) (lifts1_flat -Appl is (TLRef i) vs))))) H9)))) H6))))))))))))))))))) a)). -(* COMMENTS -Initial nodes: 2737 -END *) - -theorem sc3_sn3: - \forall (g: G).(\forall (a: A).(\forall (c: C).(\forall (t: T).((sc3 g a c -t) \to (sn3 c t))))) -\def - \lambda (g: G).(\lambda (a: A).(\lambda (c: C).(\lambda (t: T).(\lambda (H: -(sc3 g a c t)).(let H_x \def (sc3_props__sc3_sn3_abst g a) in (let H0 \def -H_x in (land_ind (\forall (c0: C).(\forall (t0: T).((sc3 g a c0 t0) \to (sn3 -c0 t0)))) (\forall (vs: TList).(\forall (i: nat).(\forall (c0: C).((arity g -c0 (THeads (Flat Appl) vs (TLRef i)) a) \to ((nf2 c0 (TLRef i)) \to ((sns3 c0 -vs) \to (sc3 g a c0 (THeads (Flat Appl) vs (TLRef i))))))))) (sn3 c t) -(\lambda (H1: ((\forall (c0: C).(\forall (t0: T).((sc3 g a c0 t0) \to (sn3 c0 -t0)))))).(\lambda (_: ((\forall (vs: TList).(\forall (i: nat).(\forall (c0: -C).((arity g c0 (THeads (Flat Appl) vs (TLRef i)) a) \to ((nf2 c0 (TLRef i)) -\to ((sns3 c0 vs) \to (sc3 g a c0 (THeads (Flat Appl) vs (TLRef -i))))))))))).(H1 c t H))) H0))))))). -(* COMMENTS -Initial nodes: 203 -END *) - -theorem sc3_abst: - \forall (g: G).(\forall (a: A).(\forall (vs: TList).(\forall (c: C).(\forall -(i: nat).((arity g c (THeads (Flat Appl) vs (TLRef i)) a) \to ((nf2 c (TLRef -i)) \to ((sns3 c vs) \to (sc3 g a c (THeads (Flat Appl) vs (TLRef i)))))))))) -\def - \lambda (g: G).(\lambda (a: A).(\lambda (vs: TList).(\lambda (c: C).(\lambda -(i: nat).(\lambda (H: (arity g c (THeads (Flat Appl) vs (TLRef i)) -a)).(\lambda (H0: (nf2 c (TLRef i))).(\lambda (H1: (sns3 c vs)).(let H_x \def -(sc3_props__sc3_sn3_abst g a) in (let H2 \def H_x in (land_ind (\forall (c0: -C).(\forall (t: T).((sc3 g a c0 t) \to (sn3 c0 t)))) (\forall (vs0: -TList).(\forall (i0: nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) -vs0 (TLRef i0)) a) \to ((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a -c0 (THeads (Flat Appl) vs0 (TLRef i0))))))))) (sc3 g a c (THeads (Flat Appl) -vs (TLRef i))) (\lambda (_: ((\forall (c0: C).(\forall (t: T).((sc3 g a c0 t) -\to (sn3 c0 t)))))).(\lambda (H4: ((\forall (vs0: TList).(\forall (i0: -nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) vs0 (TLRef i0)) a) \to -((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a c0 (THeads (Flat Appl) -vs0 (TLRef i0))))))))))).(H4 vs i c H H0 H1))) H2)))))))))). -(* COMMENTS -Initial nodes: 249 -END *) - -theorem sc3_bind: - \forall (g: G).(\forall (b: B).((not (eq B b Abst)) \to (\forall (a1: -A).(\forall (a2: A).(\forall (vs: TList).(\forall (c: C).(\forall (v: -T).(\forall (t: T).((sc3 g a2 (CHead c (Bind b) v) (THeads (Flat Appl) (lifts -(S O) O vs) t)) \to ((sc3 g a1 c v) \to (sc3 g a2 c (THeads (Flat Appl) vs -(THead (Bind b) v t))))))))))))) -\def - \lambda (g: G).(\lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda -(a1: A).(\lambda (a2: A).(A_ind (\lambda (a: A).(\forall (vs: TList).(\forall -(c: C).(\forall (v: T).(\forall (t: T).((sc3 g a (CHead c (Bind b) v) (THeads -(Flat Appl) (lifts (S O) O vs) t)) \to ((sc3 g a1 c v) \to (sc3 g a c (THeads -(Flat Appl) vs (THead (Bind b) v t)))))))))) (\lambda (n: nat).(\lambda (n0: -nat).(\lambda (vs: TList).(\lambda (c: C).(\lambda (v: T).(\lambda (t: -T).(\lambda (H0: (land (arity g (CHead c (Bind b) v) (THeads (Flat Appl) -(lifts (S O) O vs) t) (ASort n n0)) (sn3 (CHead c (Bind b) v) (THeads (Flat -Appl) (lifts (S O) O vs) t)))).(\lambda (H1: (sc3 g a1 c v)).(let H2 \def H0 -in (land_ind (arity g (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O -vs) t) (ASort n n0)) (sn3 (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S -O) O vs) t)) (land (arity g c (THeads (Flat Appl) vs (THead (Bind b) v t)) -(ASort n n0)) (sn3 c (THeads (Flat Appl) vs (THead (Bind b) v t)))) (\lambda -(H3: (arity g (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O vs) t) -(ASort n n0))).(\lambda (H4: (sn3 (CHead c (Bind b) v) (THeads (Flat Appl) -(lifts (S O) O vs) t))).(conj (arity g c (THeads (Flat Appl) vs (THead (Bind -b) v t)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (THead (Bind b) v t))) -(arity_appls_bind g b H c v a1 (sc3_arity_gen g c v a1 H1) t vs (ASort n n0) -H3) (sn3_appls_bind b H c v (sc3_sn3 g a1 c v H1) vs t H4)))) H2)))))))))) -(\lambda (a: A).(\lambda (_: ((\forall (vs: TList).(\forall (c: C).(\forall -(v: T).(\forall (t: T).((sc3 g a (CHead c (Bind b) v) (THeads (Flat Appl) -(lifts (S O) O vs) t)) \to ((sc3 g a1 c v) \to (sc3 g a c (THeads (Flat Appl) -vs (THead (Bind b) v t))))))))))).(\lambda (a0: A).(\lambda (H1: ((\forall -(vs: TList).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sc3 g a0 (CHead -c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O vs) t)) \to ((sc3 g a1 c v) -\to (sc3 g a0 c (THeads (Flat Appl) vs (THead (Bind b) v -t))))))))))).(\lambda (vs: TList).(\lambda (c: C).(\lambda (v: T).(\lambda -(t: T).(\lambda (H2: (land (arity g (CHead c (Bind b) v) (THeads (Flat Appl) -(lifts (S O) O vs) t) (AHead a a0)) (\forall (d: C).(\forall (w: T).((sc3 g a -d w) \to (\forall (is: PList).((drop1 is d (CHead c (Bind b) v)) \to (sc3 g -a0 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) (lifts (S O) O vs) -t))))))))))).(\lambda (H3: (sc3 g a1 c v)).(let H4 \def H2 in (land_ind -(arity g (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O vs) t) -(AHead a a0)) (\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall -(is: PList).((drop1 is d (CHead c (Bind b) v)) \to (sc3 g a0 d (THead (Flat -Appl) w (lift1 is (THeads (Flat Appl) (lifts (S O) O vs) t))))))))) (land -(arity g c (THeads (Flat Appl) vs (THead (Bind b) v t)) (AHead a a0)) -(\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall (is: -PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w (lift1 is (THeads -(Flat Appl) vs (THead (Bind b) v t))))))))))) (\lambda (H5: (arity g (CHead c -(Bind b) v) (THeads (Flat Appl) (lifts (S O) O vs) t) (AHead a a0))).(\lambda -(H6: ((\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall (is: -PList).((drop1 is d (CHead c (Bind b) v)) \to (sc3 g a0 d (THead (Flat Appl) -w (lift1 is (THeads (Flat Appl) (lifts (S O) O vs) t))))))))))).(conj (arity -g c (THeads (Flat Appl) vs (THead (Bind b) v t)) (AHead a a0)) (\forall (d: -C).(\forall (w: T).((sc3 g a d w) \to (\forall (is: PList).((drop1 is d c) -\to (sc3 g a0 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs (THead -(Bind b) v t)))))))))) (arity_appls_bind g b H c v a1 (sc3_arity_gen g c v a1 -H3) t vs (AHead a a0) H5) (\lambda (d: C).(\lambda (w: T).(\lambda (H7: (sc3 -g a d w)).(\lambda (is: PList).(\lambda (H8: (drop1 is d c)).(let H_y \def -(H1 (TCons w (lifts1 is vs))) in (eq_ind_r T (THeads (Flat Appl) (lifts1 is -vs) (lift1 is (THead (Bind b) v t))) (\lambda (t0: T).(sc3 g a0 d (THead -(Flat Appl) w t0))) (eq_ind_r T (THead (Bind b) (lift1 is v) (lift1 (Ss is) -t)) (\lambda (t0: T).(sc3 g a0 d (THead (Flat Appl) w (THeads (Flat Appl) -(lifts1 is vs) t0)))) (H_y d (lift1 is v) (lift1 (Ss is) t) (eq_ind TList -(lifts1 (Ss is) (lifts (S O) O vs)) (\lambda (t0: TList).(sc3 g a0 (CHead d -(Bind b) (lift1 is v)) (THead (Flat Appl) (lift (S O) O w) (THeads (Flat -Appl) t0 (lift1 (Ss is) t))))) (eq_ind T (lift1 (Ss is) (THeads (Flat Appl) -(lifts (S O) O vs) t)) (\lambda (t0: T).(sc3 g a0 (CHead d (Bind b) (lift1 is -v)) (THead (Flat Appl) (lift (S O) O w) t0))) (H6 (CHead d (Bind b) (lift1 is -v)) (lift (S O) O w) (sc3_lift g a d w H7 (CHead d (Bind b) (lift1 is v)) (S -O) O (drop_drop (Bind b) O d d (drop_refl d) (lift1 is v))) (Ss is) -(drop1_skip_bind b c is d v H8)) (THeads (Flat Appl) (lifts1 (Ss is) (lifts -(S O) O vs)) (lift1 (Ss is) t)) (lifts1_flat Appl (Ss is) t (lifts (S O) O -vs))) (lifts (S O) O (lifts1 is vs)) (lifts1_xhg is vs)) (sc3_lift1 g c a1 is -d v H3 H8)) (lift1 is (THead (Bind b) v t)) (lift1_bind b is v t)) (lift1 is -(THeads (Flat Appl) vs (THead (Bind b) v t))) (lifts1_flat Appl is (THead -(Bind b) v t) vs))))))))))) H4)))))))))))) a2))))). -(* COMMENTS -Initial nodes: 1797 -END *) - -theorem sc3_appl: - \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (vs: -TList).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sc3 g a2 c (THeads -(Flat Appl) vs (THead (Bind Abbr) v t))) \to ((sc3 g a1 c v) \to (\forall (w: -T).((sc3 g (asucc g a1) c w) \to (sc3 g a2 c (THeads (Flat Appl) vs (THead -(Flat Appl) v (THead (Bind Abst) w t)))))))))))))) -\def - \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(A_ind (\lambda (a: -A).(\forall (vs: TList).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sc3 -g a c (THeads (Flat Appl) vs (THead (Bind Abbr) v t))) \to ((sc3 g a1 c v) -\to (\forall (w: T).((sc3 g (asucc g a1) c w) \to (sc3 g a c (THeads (Flat -Appl) vs (THead (Flat Appl) v (THead (Bind Abst) w t))))))))))))) (\lambda -(n: nat).(\lambda (n0: nat).(\lambda (vs: TList).(\lambda (c: C).(\lambda (v: -T).(\lambda (t: T).(\lambda (H: (land (arity g c (THeads (Flat Appl) vs -(THead (Bind Abbr) v t)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (THead -(Bind Abbr) v t))))).(\lambda (H0: (sc3 g a1 c v)).(\lambda (w: T).(\lambda -(H1: (sc3 g (asucc g a1) c w)).(let H2 \def H in (land_ind (arity g c (THeads -(Flat Appl) vs (THead (Bind Abbr) v t)) (ASort n n0)) (sn3 c (THeads (Flat -Appl) vs (THead (Bind Abbr) v t))) (land (arity g c (THeads (Flat Appl) vs -(THead (Flat Appl) v (THead (Bind Abst) w t))) (ASort n n0)) (sn3 c (THeads -(Flat Appl) vs (THead (Flat Appl) v (THead (Bind Abst) w t))))) (\lambda (H3: -(arity g c (THeads (Flat Appl) vs (THead (Bind Abbr) v t)) (ASort n -n0))).(\lambda (H4: (sn3 c (THeads (Flat Appl) vs (THead (Bind Abbr) v -t)))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat Appl) v (THead -(Bind Abst) w t))) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (THead (Flat -Appl) v (THead (Bind Abst) w t)))) (arity_appls_appl g c v a1 (sc3_arity_gen -g c v a1 H0) w (sc3_arity_gen g c w (asucc g a1) H1) t vs (ASort n n0) H3) -(sn3_appls_beta c v t vs H4 w (sc3_sn3 g (asucc g a1) c w H1))))) -H2)))))))))))) (\lambda (a: A).(\lambda (_: ((\forall (vs: TList).(\forall -(c: C).(\forall (v: T).(\forall (t: T).((sc3 g a c (THeads (Flat Appl) vs -(THead (Bind Abbr) v t))) \to ((sc3 g a1 c v) \to (\forall (w: T).((sc3 g -(asucc g a1) c w) \to (sc3 g a c (THeads (Flat Appl) vs (THead (Flat Appl) v -(THead (Bind Abst) w t)))))))))))))).(\lambda (a0: A).(\lambda (H0: ((\forall -(vs: TList).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sc3 g a0 c -(THeads (Flat Appl) vs (THead (Bind Abbr) v t))) \to ((sc3 g a1 c v) \to -(\forall (w: T).((sc3 g (asucc g a1) c w) \to (sc3 g a0 c (THeads (Flat Appl) -vs (THead (Flat Appl) v (THead (Bind Abst) w t)))))))))))))).(\lambda (vs: -TList).(\lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (H1: (land -(arity g c (THeads (Flat Appl) vs (THead (Bind Abbr) v t)) (AHead a a0)) -(\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall (is: -PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w (lift1 is (THeads -(Flat Appl) vs (THead (Bind Abbr) v t)))))))))))).(\lambda (H2: (sc3 g a1 c -v)).(\lambda (w: T).(\lambda (H3: (sc3 g (asucc g a1) c w)).(let H4 \def H1 -in (land_ind (arity g c (THeads (Flat Appl) vs (THead (Bind Abbr) v t)) -(AHead a a0)) (\forall (d: C).(\forall (w0: T).((sc3 g a d w0) \to (\forall -(is: PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w0 (lift1 is -(THeads (Flat Appl) vs (THead (Bind Abbr) v t)))))))))) (land (arity g c -(THeads (Flat Appl) vs (THead (Flat Appl) v (THead (Bind Abst) w t))) (AHead -a a0)) (\forall (d: C).(\forall (w0: T).((sc3 g a d w0) \to (\forall (is: -PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w0 (lift1 is -(THeads (Flat Appl) vs (THead (Flat Appl) v (THead (Bind Abst) w -t)))))))))))) (\lambda (H5: (arity g c (THeads (Flat Appl) vs (THead (Bind -Abbr) v t)) (AHead a a0))).(\lambda (H6: ((\forall (d: C).(\forall (w0: -T).((sc3 g a d w0) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a0 d -(THead (Flat Appl) w0 (lift1 is (THeads (Flat Appl) vs (THead (Bind Abbr) v -t)))))))))))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat Appl) v -(THead (Bind Abst) w t))) (AHead a a0)) (\forall (d: C).(\forall (w0: -T).((sc3 g a d w0) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a0 d -(THead (Flat Appl) w0 (lift1 is (THeads (Flat Appl) vs (THead (Flat Appl) v -(THead (Bind Abst) w t))))))))))) (arity_appls_appl g c v a1 (sc3_arity_gen g -c v a1 H2) w (sc3_arity_gen g c w (asucc g a1) H3) t vs (AHead a a0) H5) -(\lambda (d: C).(\lambda (w0: T).(\lambda (H7: (sc3 g a d w0)).(\lambda (is: -PList).(\lambda (H8: (drop1 is d c)).(eq_ind_r T (THeads (Flat Appl) (lifts1 -is vs) (lift1 is (THead (Flat Appl) v (THead (Bind Abst) w t)))) (\lambda -(t0: T).(sc3 g a0 d (THead (Flat Appl) w0 t0))) (eq_ind_r T (THead (Flat -Appl) (lift1 is v) (lift1 is (THead (Bind Abst) w t))) (\lambda (t0: T).(sc3 -g a0 d (THead (Flat Appl) w0 (THeads (Flat Appl) (lifts1 is vs) t0)))) -(eq_ind_r T (THead (Bind Abst) (lift1 is w) (lift1 (Ss is) t)) (\lambda (t0: -T).(sc3 g a0 d (THead (Flat Appl) w0 (THeads (Flat Appl) (lifts1 is vs) -(THead (Flat Appl) (lift1 is v) t0))))) (let H_y \def (H0 (TCons w0 (lifts1 -is vs))) in (H_y d (lift1 is v) (lift1 (Ss is) t) (eq_ind T (lift1 is (THead -(Bind Abbr) v t)) (\lambda (t0: T).(sc3 g a0 d (THead (Flat Appl) w0 (THeads -(Flat Appl) (lifts1 is vs) t0)))) (eq_ind T (lift1 is (THeads (Flat Appl) vs -(THead (Bind Abbr) v t))) (\lambda (t0: T).(sc3 g a0 d (THead (Flat Appl) w0 -t0))) (H6 d w0 H7 is H8) (THeads (Flat Appl) (lifts1 is vs) (lift1 is (THead -(Bind Abbr) v t))) (lifts1_flat Appl is (THead (Bind Abbr) v t) vs)) (THead -(Bind Abbr) (lift1 is v) (lift1 (Ss is) t)) (lift1_bind Abbr is v t)) -(sc3_lift1 g c a1 is d v H2 H8) (lift1 is w) (sc3_lift1 g c (asucc g a1) is d -w H3 H8))) (lift1 is (THead (Bind Abst) w t)) (lift1_bind Abst is w t)) -(lift1 is (THead (Flat Appl) v (THead (Bind Abst) w t))) (lift1_flat Appl is -v (THead (Bind Abst) w t))) (lift1 is (THeads (Flat Appl) vs (THead (Flat -Appl) v (THead (Bind Abst) w t)))) (lifts1_flat Appl is (THead (Flat Appl) v -(THead (Bind Abst) w t)) vs)))))))))) H4)))))))))))))) a2))). -(* COMMENTS -Initial nodes: 1901 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/sn3/defs.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/sn3/defs.ma deleted file mode 100644 index 686e6c673..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/sn3/defs.ma +++ /dev/null @@ -1,29 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr3/defs.ma". - -inductive sn3 (c: C): T \to Prop \def -| sn3_sing: \forall (t1: T).(((\forall (t2: T).((((eq T t1 t2) \to (\forall -(P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2))))) \to (sn3 c t1)). - -definition sns3: - C \to (TList \to Prop) -\def - let rec sns3 (c: C) (ts: TList) on ts: Prop \def (match ts with [TNil -\Rightarrow True | (TCons t ts0) \Rightarrow (land (sn3 c t) (sns3 c ts0))]) -in sns3. - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/sn3/fwd.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/sn3/fwd.ma deleted file mode 100644 index 68276fe9f..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/sn3/fwd.ma +++ /dev/null @@ -1,197 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/sn3/defs.ma". - -include "Basic-1/pr3/props.ma". - -theorem sn3_gen_bind: - \forall (b: B).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 c -(THead (Bind b) u t)) \to (land (sn3 c u) (sn3 (CHead c (Bind b) u) t)))))) -\def - \lambda (b: B).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: -(sn3 c (THead (Bind b) u t))).(insert_eq T (THead (Bind b) u t) (\lambda (t0: -T).(sn3 c t0)) (\lambda (_: T).(land (sn3 c u) (sn3 (CHead c (Bind b) u) t))) -(\lambda (y: T).(\lambda (H0: (sn3 c y)).(unintro T t (\lambda (t0: T).((eq T -y (THead (Bind b) u t0)) \to (land (sn3 c u) (sn3 (CHead c (Bind b) u) t0)))) -(unintro T u (\lambda (t0: T).(\forall (x: T).((eq T y (THead (Bind b) t0 x)) -\to (land (sn3 c t0) (sn3 (CHead c (Bind b) t0) x))))) (sn3_ind c (\lambda -(t0: T).(\forall (x: T).(\forall (x0: T).((eq T t0 (THead (Bind b) x x0)) \to -(land (sn3 c x) (sn3 (CHead c (Bind b) x) x0)))))) (\lambda (t1: T).(\lambda -(H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 -c t1 t2) \to (sn3 c t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2) -\to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (\forall (x: T).(\forall -(x0: T).((eq T t2 (THead (Bind b) x x0)) \to (land (sn3 c x) (sn3 (CHead c -(Bind b) x) x0)))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: (eq T -t1 (THead (Bind b) x x0))).(let H4 \def (eq_ind T t1 (\lambda (t0: -T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c -t0 t2) \to (\forall (x1: T).(\forall (x2: T).((eq T t2 (THead (Bind b) x1 -x2)) \to (land (sn3 c x1) (sn3 (CHead c (Bind b) x1) x2))))))))) H2 (THead -(Bind b) x x0) H3) in (let H5 \def (eq_ind T t1 (\lambda (t0: T).(\forall -(t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to -(sn3 c t2))))) H1 (THead (Bind b) x x0) H3) in (conj (sn3 c x) (sn3 (CHead c -(Bind b) x) x0) (sn3_sing c x (\lambda (t2: T).(\lambda (H6: (((eq T x t2) -\to (\forall (P: Prop).P)))).(\lambda (H7: (pr3 c x t2)).(let H8 \def (H4 -(THead (Bind b) t2 x0) (\lambda (H8: (eq T (THead (Bind b) x x0) (THead (Bind -b) t2 x0))).(\lambda (P: Prop).(let H9 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x | -(TLRef _) \Rightarrow x | (THead _ t0 _) \Rightarrow t0])) (THead (Bind b) x -x0) (THead (Bind b) t2 x0) H8) in (let H10 \def (eq_ind_r T t2 (\lambda (t0: -T).(pr3 c x t0)) H7 x H9) in (let H11 \def (eq_ind_r T t2 (\lambda (t0: -T).((eq T x t0) \to (\forall (P0: Prop).P0))) H6 x H9) in (H11 (refl_equal T -x) P)))))) (pr3_head_12 c x t2 H7 (Bind b) x0 x0 (pr3_refl (CHead c (Bind b) -t2) x0)) t2 x0 (refl_equal T (THead (Bind b) t2 x0))) in (land_ind (sn3 c t2) -(sn3 (CHead c (Bind b) t2) x0) (sn3 c t2) (\lambda (H9: (sn3 c t2)).(\lambda -(_: (sn3 (CHead c (Bind b) t2) x0)).H9)) H8)))))) (sn3_sing (CHead c (Bind b) -x) x0 (\lambda (t2: T).(\lambda (H6: (((eq T x0 t2) \to (\forall (P: -Prop).P)))).(\lambda (H7: (pr3 (CHead c (Bind b) x) x0 t2)).(let H8 \def (H4 -(THead (Bind b) x t2) (\lambda (H8: (eq T (THead (Bind b) x x0) (THead (Bind -b) x t2))).(\lambda (P: Prop).(let H9 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | -(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind b) x -x0) (THead (Bind b) x t2) H8) in (let H10 \def (eq_ind_r T t2 (\lambda (t0: -T).(pr3 (CHead c (Bind b) x) x0 t0)) H7 x0 H9) in (let H11 \def (eq_ind_r T -t2 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H6 x0 H9) in -(H11 (refl_equal T x0) P)))))) (pr3_head_12 c x x (pr3_refl c x) (Bind b) x0 -t2 H7) x t2 (refl_equal T (THead (Bind b) x t2))) in (land_ind (sn3 c x) (sn3 -(CHead c (Bind b) x) t2) (sn3 (CHead c (Bind b) x) t2) (\lambda (_: (sn3 c -x)).(\lambda (H10: (sn3 (CHead c (Bind b) x) t2)).H10)) H8))))))))))))))) y -H0))))) H))))). -(* COMMENTS -Initial nodes: 1055 -END *) - -theorem sn3_gen_flat: - \forall (f: F).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 c -(THead (Flat f) u t)) \to (land (sn3 c u) (sn3 c t)))))) -\def - \lambda (f: F).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: -(sn3 c (THead (Flat f) u t))).(insert_eq T (THead (Flat f) u t) (\lambda (t0: -T).(sn3 c t0)) (\lambda (_: T).(land (sn3 c u) (sn3 c t))) (\lambda (y: -T).(\lambda (H0: (sn3 c y)).(unintro T t (\lambda (t0: T).((eq T y (THead -(Flat f) u t0)) \to (land (sn3 c u) (sn3 c t0)))) (unintro T u (\lambda (t0: -T).(\forall (x: T).((eq T y (THead (Flat f) t0 x)) \to (land (sn3 c t0) (sn3 -c x))))) (sn3_ind c (\lambda (t0: T).(\forall (x: T).(\forall (x0: T).((eq T -t0 (THead (Flat f) x x0)) \to (land (sn3 c x) (sn3 c x0)))))) (\lambda (t1: -T).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: -Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H2: ((\forall -(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to -(\forall (x: T).(\forall (x0: T).((eq T t2 (THead (Flat f) x x0)) \to (land -(sn3 c x) (sn3 c x0)))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: -(eq T t1 (THead (Flat f) x x0))).(let H4 \def (eq_ind T t1 (\lambda (t0: -T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c -t0 t2) \to (\forall (x1: T).(\forall (x2: T).((eq T t2 (THead (Flat f) x1 -x2)) \to (land (sn3 c x1) (sn3 c x2))))))))) H2 (THead (Flat f) x x0) H3) in -(let H5 \def (eq_ind T t1 (\lambda (t0: T).(\forall (t2: T).((((eq T t0 t2) -\to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (sn3 c t2))))) H1 (THead -(Flat f) x x0) H3) in (conj (sn3 c x) (sn3 c x0) (sn3_sing c x (\lambda (t2: -T).(\lambda (H6: (((eq T x t2) \to (\forall (P: Prop).P)))).(\lambda (H7: -(pr3 c x t2)).(let H8 \def (H4 (THead (Flat f) t2 x0) (\lambda (H8: (eq T -(THead (Flat f) x x0) (THead (Flat f) t2 x0))).(\lambda (P: Prop).(let H9 -\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t0 _) -\Rightarrow t0])) (THead (Flat f) x x0) (THead (Flat f) t2 x0) H8) in (let -H10 \def (eq_ind_r T t2 (\lambda (t0: T).(pr3 c x t0)) H7 x H9) in (let H11 -\def (eq_ind_r T t2 (\lambda (t0: T).((eq T x t0) \to (\forall (P0: -Prop).P0))) H6 x H9) in (H11 (refl_equal T x) P)))))) (pr3_head_12 c x t2 H7 -(Flat f) x0 x0 (pr3_refl (CHead c (Flat f) t2) x0)) t2 x0 (refl_equal T -(THead (Flat f) t2 x0))) in (land_ind (sn3 c t2) (sn3 c x0) (sn3 c t2) -(\lambda (H9: (sn3 c t2)).(\lambda (_: (sn3 c x0)).H9)) H8)))))) (sn3_sing c -x0 (\lambda (t2: T).(\lambda (H6: (((eq T x0 t2) \to (\forall (P: -Prop).P)))).(\lambda (H7: (pr3 c x0 t2)).(let H8 \def (H4 (THead (Flat f) x -t2) (\lambda (H8: (eq T (THead (Flat f) x x0) (THead (Flat f) x -t2))).(\lambda (P: Prop).(let H9 \def (f_equal T T (\lambda (e: T).(match e -in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) -\Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat f) x x0) -(THead (Flat f) x t2) H8) in (let H10 \def (eq_ind_r T t2 (\lambda (t0: -T).(pr3 c x0 t0)) H7 x0 H9) in (let H11 \def (eq_ind_r T t2 (\lambda (t0: -T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H6 x0 H9) in (H11 (refl_equal -T x0) P)))))) (pr3_thin_dx c x0 t2 H7 x f) x t2 (refl_equal T (THead (Flat f) -x t2))) in (land_ind (sn3 c x) (sn3 c t2) (sn3 c t2) (\lambda (_: (sn3 c -x)).(\lambda (H10: (sn3 c t2)).H10)) H8))))))))))))))) y H0))))) H))))). -(* COMMENTS -Initial nodes: 925 -END *) - -theorem sn3_gen_head: - \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 c -(THead k u t)) \to (sn3 c u))))) -\def - \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (c: C).(\forall (u: -T).(\forall (t: T).((sn3 c (THead k0 u t)) \to (sn3 c u)))))) (\lambda (b: -B).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (sn3 c (THead -(Bind b) u t))).(let H_x \def (sn3_gen_bind b c u t H) in (let H0 \def H_x in -(land_ind (sn3 c u) (sn3 (CHead c (Bind b) u) t) (sn3 c u) (\lambda (H1: (sn3 -c u)).(\lambda (_: (sn3 (CHead c (Bind b) u) t)).H1)) H0)))))))) (\lambda (f: -F).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (sn3 c (THead -(Flat f) u t))).(let H_x \def (sn3_gen_flat f c u t H) in (let H0 \def H_x in -(land_ind (sn3 c u) (sn3 c t) (sn3 c u) (\lambda (H1: (sn3 c u)).(\lambda (_: -(sn3 c t)).H1)) H0)))))))) k). -(* COMMENTS -Initial nodes: 191 -END *) - -theorem sn3_gen_cflat: - \forall (f: F).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 (CHead -c (Flat f) u) t) \to (sn3 c t))))) -\def - \lambda (f: F).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: -(sn3 (CHead c (Flat f) u) t)).(sn3_ind (CHead c (Flat f) u) (\lambda (t0: -T).(sn3 c t0)) (\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 -t2) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Flat f) u) t1 t2) \to -(sn3 (CHead c (Flat f) u) t2)))))).(\lambda (H1: ((\forall (t2: T).((((eq T -t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Flat f) u) t1 t2) \to -(sn3 c t2)))))).(sn3_sing c t1 (\lambda (t2: T).(\lambda (H2: (((eq T t1 t2) -\to (\forall (P: Prop).P)))).(\lambda (H3: (pr3 c t1 t2)).(H1 t2 H2 -(pr3_cflat c t1 t2 H3 f u))))))))) t H))))). -(* COMMENTS -Initial nodes: 175 -END *) - -theorem sn3_gen_lift: - \forall (c1: C).(\forall (t: T).(\forall (h: nat).(\forall (d: nat).((sn3 c1 -(lift h d t)) \to (\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 t))))))) -\def - \lambda (c1: C).(\lambda (t: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda -(H: (sn3 c1 (lift h d t))).(insert_eq T (lift h d t) (\lambda (t0: T).(sn3 c1 -t0)) (\lambda (_: T).(\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 t)))) -(\lambda (y: T).(\lambda (H0: (sn3 c1 y)).(unintro T t (\lambda (t0: T).((eq -T y (lift h d t0)) \to (\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 t0))))) -(sn3_ind c1 (\lambda (t0: T).(\forall (x: T).((eq T t0 (lift h d x)) \to -(\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 x)))))) (\lambda (t1: -T).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: -Prop).P))) \to ((pr3 c1 t1 t2) \to (sn3 c1 t2)))))).(\lambda (H2: ((\forall -(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c1 t1 t2) \to -(\forall (x: T).((eq T t2 (lift h d x)) \to (\forall (c2: C).((drop h d c1 -c2) \to (sn3 c2 x)))))))))).(\lambda (x: T).(\lambda (H3: (eq T t1 (lift h d -x))).(\lambda (c2: C).(\lambda (H4: (drop h d c1 c2)).(let H5 \def (eq_ind T -t1 (\lambda (t0: T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P: -Prop).P))) \to ((pr3 c1 t0 t2) \to (\forall (x0: T).((eq T t2 (lift h d x0)) -\to (\forall (c3: C).((drop h d c1 c3) \to (sn3 c3 x0))))))))) H2 (lift h d -x) H3) in (let H6 \def (eq_ind T t1 (\lambda (t0: T).(\forall (t2: T).((((eq -T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c1 t0 t2) \to (sn3 c1 t2))))) -H1 (lift h d x) H3) in (sn3_sing c2 x (\lambda (t2: T).(\lambda (H7: (((eq T -x t2) \to (\forall (P: Prop).P)))).(\lambda (H8: (pr3 c2 x t2)).(H5 (lift h d -t2) (\lambda (H9: (eq T (lift h d x) (lift h d t2))).(\lambda (P: Prop).(let -H10 \def (eq_ind_r T t2 (\lambda (t0: T).(pr3 c2 x t0)) H8 x (lift_inj x t2 h -d H9)) in (let H11 \def (eq_ind_r T t2 (\lambda (t0: T).((eq T x t0) \to -(\forall (P0: Prop).P0))) H7 x (lift_inj x t2 h d H9)) in (H11 (refl_equal T -x) P))))) (pr3_lift c1 c2 h d H4 x t2 H8) t2 (refl_equal T (lift h d t2)) c2 -H4)))))))))))))) y H0)))) H))))). -(* COMMENTS -Initial nodes: 565 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/sn3/lift1.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/sn3/lift1.ma deleted file mode 100644 index 1b64c22bb..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/sn3/lift1.ma +++ /dev/null @@ -1,46 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/sn3/props.ma". - -include "Basic-1/drop1/fwd.ma". - -include "Basic-1/lift1/fwd.ma". - -theorem sns3_lifts1: - \forall (e: C).(\forall (hds: PList).(\forall (c: C).((drop1 hds c e) \to -(\forall (ts: TList).((sns3 e ts) \to (sns3 c (lifts1 hds ts))))))) -\def - \lambda (e: C).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall -(c: C).((drop1 p c e) \to (\forall (ts: TList).((sns3 e ts) \to (sns3 c -(lifts1 p ts))))))) (\lambda (c: C).(\lambda (H: (drop1 PNil c e)).(\lambda -(ts: TList).(\lambda (H0: (sns3 e ts)).(let H_y \def (drop1_gen_pnil c e H) -in (eq_ind_r C e (\lambda (c0: C).(sns3 c0 (lifts1 PNil ts))) (eq_ind_r TList -ts (\lambda (t: TList).(sns3 e t)) H0 (lifts1 PNil ts) (lifts1_nil ts)) c -H_y)))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda -(H: ((\forall (c: C).((drop1 p c e) \to (\forall (ts: TList).((sns3 e ts) \to -(sns3 c (lifts1 p ts)))))))).(\lambda (c: C).(\lambda (H0: (drop1 (PCons n n0 -p) c e)).(\lambda (ts: TList).(\lambda (H1: (sns3 e ts)).(let H_x \def -(drop1_gen_pcons c e p n n0 H0) in (let H2 \def H_x in (ex2_ind C (\lambda -(c2: C).(drop n n0 c c2)) (\lambda (c2: C).(drop1 p c2 e)) (sns3 c (lifts1 -(PCons n n0 p) ts)) (\lambda (x: C).(\lambda (H3: (drop n n0 c x)).(\lambda -(H4: (drop1 p x e)).(eq_ind_r TList (lifts n n0 (lifts1 p ts)) (\lambda (t: -TList).(sns3 c t)) (sns3_lifts c x n n0 H3 (lifts1 p ts) (H x H4 ts H1)) -(lifts1 (PCons n n0 p) ts) (lifts1_cons n n0 p ts))))) H2))))))))))) hds)). -(* COMMENTS -Initial nodes: 323 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/sn3/nf2.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/sn3/nf2.ma deleted file mode 100644 index 824428084..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/sn3/nf2.ma +++ /dev/null @@ -1,66 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/sn3/defs.ma". - -include "Basic-1/nf2/dec.ma". - -include "Basic-1/nf2/pr3.ma". - -theorem sn3_nf2: - \forall (c: C).(\forall (t: T).((nf2 c t) \to (sn3 c t))) -\def - \lambda (c: C).(\lambda (t: T).(\lambda (H: (nf2 c t)).(sn3_sing c t -(\lambda (t2: T).(\lambda (H0: (((eq T t t2) \to (\forall (P: -Prop).P)))).(\lambda (H1: (pr3 c t t2)).(let H_y \def (nf2_pr3_unfold c t t2 -H1 H) in (let H2 \def (eq_ind_r T t2 (\lambda (t0: T).(pr3 c t t0)) H1 t H_y) -in (let H3 \def (eq_ind_r T t2 (\lambda (t0: T).((eq T t t0) \to (\forall (P: -Prop).P))) H0 t H_y) in (eq_ind T t (\lambda (t0: T).(sn3 c t0)) (H3 -(refl_equal T t) (sn3 c t)) t2 H_y)))))))))). -(* COMMENTS -Initial nodes: 129 -END *) - -theorem nf2_sn3: - \forall (c: C).(\forall (t: T).((sn3 c t) \to (ex2 T (\lambda (u: T).(pr3 c -t u)) (\lambda (u: T).(nf2 c u))))) -\def - \lambda (c: C).(\lambda (t: T).(\lambda (H: (sn3 c t)).(sn3_ind c (\lambda -(t0: T).(ex2 T (\lambda (u: T).(pr3 c t0 u)) (\lambda (u: T).(nf2 c u)))) -(\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) \to (\forall -(P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H1: ((\forall -(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to -(ex2 T (\lambda (u: T).(pr3 c t2 u)) (\lambda (u: T).(nf2 c u)))))))).(let -H_x \def (nf2_dec c t1) in (let H2 \def H_x in (or_ind (nf2 c t1) (ex2 T -(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: -T).(pr2 c t1 t2))) (ex2 T (\lambda (u: T).(pr3 c t1 u)) (\lambda (u: T).(nf2 -c u))) (\lambda (H3: (nf2 c t1)).(ex_intro2 T (\lambda (u: T).(pr3 c t1 u)) -(\lambda (u: T).(nf2 c u)) t1 (pr3_refl c t1) H3)) (\lambda (H3: (ex2 T -(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: -T).(pr2 c t1 t2)))).(ex2_ind T (\lambda (t2: T).((eq T t1 t2) \to (\forall -(P: Prop).P))) (\lambda (t2: T).(pr2 c t1 t2)) (ex2 T (\lambda (u: T).(pr3 c -t1 u)) (\lambda (u: T).(nf2 c u))) (\lambda (x: T).(\lambda (H4: (((eq T t1 -x) \to (\forall (P: Prop).P)))).(\lambda (H5: (pr2 c t1 x)).(let H_y \def (H1 -x H4) in (let H6 \def (H_y (pr3_pr2 c t1 x H5)) in (ex2_ind T (\lambda (u: -T).(pr3 c x u)) (\lambda (u: T).(nf2 c u)) (ex2 T (\lambda (u: T).(pr3 c t1 -u)) (\lambda (u: T).(nf2 c u))) (\lambda (x0: T).(\lambda (H7: (pr3 c x -x0)).(\lambda (H8: (nf2 c x0)).(ex_intro2 T (\lambda (u: T).(pr3 c t1 u)) -(\lambda (u: T).(nf2 c u)) x0 (pr3_sing c x t1 H5 x0 H7) H8)))) H6)))))) H3)) -H2)))))) t H))). -(* COMMENTS -Initial nodes: 443 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/sn3/props.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/sn3/props.ma deleted file mode 100644 index ea72c8869..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/sn3/props.ma +++ /dev/null @@ -1,2575 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/sn3/nf2.ma". - -include "Basic-1/sn3/fwd.ma". - -include "Basic-1/nf2/iso.ma". - -include "Basic-1/pr3/iso.ma". - -theorem sn3_pr3_trans: - \forall (c: C).(\forall (t1: T).((sn3 c t1) \to (\forall (t2: T).((pr3 c t1 -t2) \to (sn3 c t2))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (H: (sn3 c t1)).(sn3_ind c (\lambda -(t: T).(\forall (t2: T).((pr3 c t t2) \to (sn3 c t2)))) (\lambda (t2: -T).(\lambda (H0: ((\forall (t3: T).((((eq T t2 t3) \to (\forall (P: -Prop).P))) \to ((pr3 c t2 t3) \to (sn3 c t3)))))).(\lambda (H1: ((\forall -(t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to -(\forall (t4: T).((pr3 c t3 t4) \to (sn3 c t4)))))))).(\lambda (t3: -T).(\lambda (H2: (pr3 c t2 t3)).(sn3_sing c t3 (\lambda (t0: T).(\lambda (H3: -(((eq T t3 t0) \to (\forall (P: Prop).P)))).(\lambda (H4: (pr3 c t3 t0)).(let -H_x \def (term_dec t2 t3) in (let H5 \def H_x in (or_ind (eq T t2 t3) ((eq T -t2 t3) \to (\forall (P: Prop).P)) (sn3 c t0) (\lambda (H6: (eq T t2 t3)).(let -H7 \def (eq_ind_r T t3 (\lambda (t: T).(pr3 c t t0)) H4 t2 H6) in (let H8 -\def (eq_ind_r T t3 (\lambda (t: T).((eq T t t0) \to (\forall (P: Prop).P))) -H3 t2 H6) in (let H9 \def (eq_ind_r T t3 (\lambda (t: T).(pr3 c t2 t)) H2 t2 -H6) in (H0 t0 H8 H7))))) (\lambda (H6: (((eq T t2 t3) \to (\forall (P: -Prop).P)))).(H1 t3 H6 H2 t0 H4)) H5)))))))))))) t1 H))). -(* COMMENTS -Initial nodes: 289 -END *) - -theorem sn3_pr2_intro: - \forall (c: C).(\forall (t1: T).(((\forall (t2: T).((((eq T t1 t2) \to -(\forall (P: Prop).P))) \to ((pr2 c t1 t2) \to (sn3 c t2))))) \to (sn3 c t1))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (H: ((\forall (t2: T).((((eq T t1 -t2) \to (\forall (P: Prop).P))) \to ((pr2 c t1 t2) \to (sn3 c -t2)))))).(sn3_sing c t1 (\lambda (t2: T).(\lambda (H0: (((eq T t1 t2) \to -(\forall (P: Prop).P)))).(\lambda (H1: (pr3 c t1 t2)).(let H2 \def H0 in -((let H3 \def H in (pr3_ind c (\lambda (t: T).(\lambda (t0: T).(((\forall -(t3: T).((((eq T t t3) \to (\forall (P: Prop).P))) \to ((pr2 c t t3) \to (sn3 -c t3))))) \to ((((eq T t t0) \to (\forall (P: Prop).P))) \to (sn3 c t0))))) -(\lambda (t: T).(\lambda (H4: ((\forall (t3: T).((((eq T t t3) \to (\forall -(P: Prop).P))) \to ((pr2 c t t3) \to (sn3 c t3)))))).(\lambda (H5: (((eq T t -t) \to (\forall (P: Prop).P)))).(H4 t H5 (pr2_free c t t (pr0_refl t)))))) -(\lambda (t3: T).(\lambda (t4: T).(\lambda (H4: (pr2 c t4 t3)).(\lambda (t5: -T).(\lambda (H5: (pr3 c t3 t5)).(\lambda (H6: ((((\forall (t6: T).((((eq T t3 -t6) \to (\forall (P: Prop).P))) \to ((pr2 c t3 t6) \to (sn3 c t6))))) \to -((((eq T t3 t5) \to (\forall (P: Prop).P))) \to (sn3 c t5))))).(\lambda (H7: -((\forall (t6: T).((((eq T t4 t6) \to (\forall (P: Prop).P))) \to ((pr2 c t4 -t6) \to (sn3 c t6)))))).(\lambda (H8: (((eq T t4 t5) \to (\forall (P: -Prop).P)))).(let H_x \def (term_dec t4 t3) in (let H9 \def H_x in (or_ind (eq -T t4 t3) ((eq T t4 t3) \to (\forall (P: Prop).P)) (sn3 c t5) (\lambda (H10: -(eq T t4 t3)).(let H11 \def (eq_ind T t4 (\lambda (t: T).((eq T t t5) \to -(\forall (P: Prop).P))) H8 t3 H10) in (let H12 \def (eq_ind T t4 (\lambda (t: -T).(\forall (t6: T).((((eq T t t6) \to (\forall (P: Prop).P))) \to ((pr2 c t -t6) \to (sn3 c t6))))) H7 t3 H10) in (let H13 \def (eq_ind T t4 (\lambda (t: -T).(pr2 c t t3)) H4 t3 H10) in (H6 H12 H11))))) (\lambda (H10: (((eq T t4 t3) -\to (\forall (P: Prop).P)))).(sn3_pr3_trans c t3 (H7 t3 H10 H4) t5 H5)) -H9))))))))))) t1 t2 H1 H3)) H2)))))))). -(* COMMENTS -Initial nodes: 467 -END *) - -theorem sn3_cast: - \forall (c: C).(\forall (u: T).((sn3 c u) \to (\forall (t: T).((sn3 c t) \to -(sn3 c (THead (Flat Cast) u t)))))) -\def - \lambda (c: C).(\lambda (u: T).(\lambda (H: (sn3 c u)).(sn3_ind c (\lambda -(t: T).(\forall (t0: T).((sn3 c t0) \to (sn3 c (THead (Flat Cast) t t0))))) -(\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) \to (\forall -(P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H1: ((\forall -(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to -(\forall (t: T).((sn3 c t) \to (sn3 c (THead (Flat Cast) t2 -t))))))))).(\lambda (t: T).(\lambda (H2: (sn3 c t)).(sn3_ind c (\lambda (t0: -T).(sn3 c (THead (Flat Cast) t1 t0))) (\lambda (t0: T).(\lambda (H3: -((\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t0 -t2) \to (sn3 c t2)))))).(\lambda (H4: ((\forall (t2: T).((((eq T t0 t2) \to -(\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (sn3 c (THead (Flat Cast) t1 -t2))))))).(sn3_pr2_intro c (THead (Flat Cast) t1 t0) (\lambda (t2: -T).(\lambda (H5: (((eq T (THead (Flat Cast) t1 t0) t2) \to (\forall (P: -Prop).P)))).(\lambda (H6: (pr2 c (THead (Flat Cast) t1 t0) t2)).(let H7 \def -(pr2_gen_cast c t1 t0 t2 H6) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda -(t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c t1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t0 t3)))) (pr2 c -t0 t2) (sn3 c t2) (\lambda (H8: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: -T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c t1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t0 -t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead -(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2))) -(\lambda (_: T).(\lambda (t3: T).(pr2 c t0 t3))) (sn3 c t2) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H9: (eq T t2 (THead (Flat Cast) x0 -x1))).(\lambda (H10: (pr2 c t1 x0)).(\lambda (H11: (pr2 c t0 x1)).(let H12 -\def (eq_ind T t2 (\lambda (t3: T).((eq T (THead (Flat Cast) t1 t0) t3) \to -(\forall (P: Prop).P))) H5 (THead (Flat Cast) x0 x1) H9) in (eq_ind_r T -(THead (Flat Cast) x0 x1) (\lambda (t3: T).(sn3 c t3)) (let H_x \def -(term_dec x0 t1) in (let H13 \def H_x in (or_ind (eq T x0 t1) ((eq T x0 t1) -\to (\forall (P: Prop).P)) (sn3 c (THead (Flat Cast) x0 x1)) (\lambda (H14: -(eq T x0 t1)).(let H15 \def (eq_ind T x0 (\lambda (t3: T).((eq T (THead (Flat -Cast) t1 t0) (THead (Flat Cast) t3 x1)) \to (\forall (P: Prop).P))) H12 t1 -H14) in (let H16 \def (eq_ind T x0 (\lambda (t3: T).(pr2 c t1 t3)) H10 t1 -H14) in (eq_ind_r T t1 (\lambda (t3: T).(sn3 c (THead (Flat Cast) t3 x1))) -(let H_x0 \def (term_dec t0 x1) in (let H17 \def H_x0 in (or_ind (eq T t0 x1) -((eq T t0 x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Cast) t1 x1)) -(\lambda (H18: (eq T t0 x1)).(let H19 \def (eq_ind_r T x1 (\lambda (t3: -T).((eq T (THead (Flat Cast) t1 t0) (THead (Flat Cast) t1 t3)) \to (\forall -(P: Prop).P))) H15 t0 H18) in (let H20 \def (eq_ind_r T x1 (\lambda (t3: -T).(pr2 c t0 t3)) H11 t0 H18) in (eq_ind T t0 (\lambda (t3: T).(sn3 c (THead -(Flat Cast) t1 t3))) (H19 (refl_equal T (THead (Flat Cast) t1 t0)) (sn3 c -(THead (Flat Cast) t1 t0))) x1 H18)))) (\lambda (H18: (((eq T t0 x1) \to -(\forall (P: Prop).P)))).(H4 x1 H18 (pr3_pr2 c t0 x1 H11))) H17))) x0 H14)))) -(\lambda (H14: (((eq T x0 t1) \to (\forall (P: Prop).P)))).(H1 x0 (\lambda -(H15: (eq T t1 x0)).(\lambda (P: Prop).(let H16 \def (eq_ind_r T x0 (\lambda -(t3: T).((eq T t3 t1) \to (\forall (P0: Prop).P0))) H14 t1 H15) in (let H17 -\def (eq_ind_r T x0 (\lambda (t3: T).((eq T (THead (Flat Cast) t1 t0) (THead -(Flat Cast) t3 x1)) \to (\forall (P0: Prop).P0))) H12 t1 H15) in (let H18 -\def (eq_ind_r T x0 (\lambda (t3: T).(pr2 c t1 t3)) H10 t1 H15) in (H16 -(refl_equal T t1) P)))))) (pr3_pr2 c t1 x0 H10) x1 (let H_x0 \def (term_dec -t0 x1) in (let H15 \def H_x0 in (or_ind (eq T t0 x1) ((eq T t0 x1) \to -(\forall (P: Prop).P)) (sn3 c x1) (\lambda (H16: (eq T t0 x1)).(let H17 \def -(eq_ind_r T x1 (\lambda (t3: T).((eq T (THead (Flat Cast) t1 t0) (THead (Flat -Cast) x0 t3)) \to (\forall (P: Prop).P))) H12 t0 H16) in (let H18 \def -(eq_ind_r T x1 (\lambda (t3: T).(pr2 c t0 t3)) H11 t0 H16) in (eq_ind T t0 -(\lambda (t3: T).(sn3 c t3)) (sn3_sing c t0 H3) x1 H16)))) (\lambda (H16: -(((eq T t0 x1) \to (\forall (P: Prop).P)))).(H3 x1 H16 (pr3_pr2 c t0 x1 -H11))) H15))))) H13))) t2 H9))))))) H8)) (\lambda (H8: (pr2 c t0 -t2)).(sn3_pr3_trans c t0 (sn3_sing c t0 H3) t2 (pr3_pr2 c t0 t2 H8))) -H7))))))))) t H2)))))) u H))). -(* COMMENTS -Initial nodes: 1239 -END *) - -theorem sn3_cflat: - \forall (c: C).(\forall (t: T).((sn3 c t) \to (\forall (f: F).(\forall (u: -T).(sn3 (CHead c (Flat f) u) t))))) -\def - \lambda (c: C).(\lambda (t: T).(\lambda (H: (sn3 c t)).(\lambda (f: -F).(\lambda (u: T).(sn3_ind c (\lambda (t0: T).(sn3 (CHead c (Flat f) u) t0)) -(\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) \to (\forall -(P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H1: ((\forall -(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to -(sn3 (CHead c (Flat f) u) t2)))))).(sn3_pr2_intro (CHead c (Flat f) u) t1 -(\lambda (t2: T).(\lambda (H2: (((eq T t1 t2) \to (\forall (P: -Prop).P)))).(\lambda (H3: (pr2 (CHead c (Flat f) u) t1 t2)).(H1 t2 H2 -(pr3_pr2 c t1 t2 (pr2_gen_cflat f c u t1 t2 H3)))))))))) t H))))). -(* COMMENTS -Initial nodes: 175 -END *) - -theorem sn3_shift: - \forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sn3 c -(THead (Bind b) v t)) \to (sn3 (CHead c (Bind b) v) t))))) -\def - \lambda (b: B).(\lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (H: -(sn3 c (THead (Bind b) v t))).(let H_x \def (sn3_gen_bind b c v t H) in (let -H0 \def H_x in (land_ind (sn3 c v) (sn3 (CHead c (Bind b) v) t) (sn3 (CHead c -(Bind b) v) t) (\lambda (_: (sn3 c v)).(\lambda (H2: (sn3 (CHead c (Bind b) -v) t)).H2)) H0))))))). -(* COMMENTS -Initial nodes: 95 -END *) - -theorem sn3_change: - \forall (b: B).((not (eq B b Abbr)) \to (\forall (c: C).(\forall (v1: -T).(\forall (t: T).((sn3 (CHead c (Bind b) v1) t) \to (\forall (v2: T).(sn3 -(CHead c (Bind b) v2) t))))))) -\def - \lambda (b: B).(\lambda (H: (not (eq B b Abbr))).(\lambda (c: C).(\lambda -(v1: T).(\lambda (t: T).(\lambda (H0: (sn3 (CHead c (Bind b) v1) t)).(\lambda -(v2: T).(sn3_ind (CHead c (Bind b) v1) (\lambda (t0: T).(sn3 (CHead c (Bind -b) v2) t0)) (\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) -\to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) v1) t1 t2) \to (sn3 -(CHead c (Bind b) v1) t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 -t2) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) v1) t1 t2) \to -(sn3 (CHead c (Bind b) v2) t2)))))).(sn3_pr2_intro (CHead c (Bind b) v2) t1 -(\lambda (t2: T).(\lambda (H3: (((eq T t1 t2) \to (\forall (P: -Prop).P)))).(\lambda (H4: (pr2 (CHead c (Bind b) v2) t1 t2)).(H2 t2 H3 -(pr3_pr2 (CHead c (Bind b) v1) t1 t2 (pr2_change b H c v2 t1 t2 H4 -v1)))))))))) t H0))))))). -(* COMMENTS -Initial nodes: 239 -END *) - -theorem sn3_gen_def: - \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c -(CHead d (Bind Abbr) v)) \to ((sn3 c (TLRef i)) \to (sn3 d v)))))) -\def - \lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda -(H: (getl i c (CHead d (Bind Abbr) v))).(\lambda (H0: (sn3 c (TLRef -i))).(sn3_gen_lift c v (S i) O (sn3_pr3_trans c (TLRef i) H0 (lift (S i) O v) -(pr3_pr2 c (TLRef i) (lift (S i) O v) (pr2_delta c d v i H (TLRef i) (TLRef -i) (pr0_refl (TLRef i)) (lift (S i) O v) (subst0_lref v i)))) d (getl_drop -Abbr c d v i H))))))). -(* COMMENTS -Initial nodes: 139 -END *) - -theorem sn3_cdelta: - \forall (v: T).(\forall (t: T).(\forall (i: nat).(((\forall (w: T).(ex T -(\lambda (u: T).(subst0 i w t u))))) \to (\forall (c: C).(\forall (d: -C).((getl i c (CHead d (Bind Abbr) v)) \to ((sn3 c t) \to (sn3 d v)))))))) -\def - \lambda (v: T).(\lambda (t: T).(\lambda (i: nat).(\lambda (H: ((\forall (w: -T).(ex T (\lambda (u: T).(subst0 i w t u)))))).(let H_x \def (H v) in (let H0 -\def H_x in (ex_ind T (\lambda (u: T).(subst0 i v t u)) (\forall (c: -C).(\forall (d: C).((getl i c (CHead d (Bind Abbr) v)) \to ((sn3 c t) \to -(sn3 d v))))) (\lambda (x: T).(\lambda (H1: (subst0 i v t x)).(subst0_ind -(\lambda (n: nat).(\lambda (t0: T).(\lambda (t1: T).(\lambda (_: T).(\forall -(c: C).(\forall (d: C).((getl n c (CHead d (Bind Abbr) t0)) \to ((sn3 c t1) -\to (sn3 d t0))))))))) (\lambda (v0: T).(\lambda (i0: nat).(\lambda (c: -C).(\lambda (d: C).(\lambda (H2: (getl i0 c (CHead d (Bind Abbr) -v0))).(\lambda (H3: (sn3 c (TLRef i0))).(sn3_gen_def c d v0 i0 H2 H3))))))) -(\lambda (v0: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i0: -nat).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (H3: ((\forall (c: -C).(\forall (d: C).((getl i0 c (CHead d (Bind Abbr) v0)) \to ((sn3 c u1) \to -(sn3 d v0))))))).(\lambda (t0: T).(\lambda (k: K).(\lambda (c: C).(\lambda -(d: C).(\lambda (H4: (getl i0 c (CHead d (Bind Abbr) v0))).(\lambda (H5: (sn3 -c (THead k u1 t0))).(let H_y \def (sn3_gen_head k c u1 t0 H5) in (H3 c d H4 -H_y)))))))))))))) (\lambda (k: K).(\lambda (v0: T).(\lambda (t2: T).(\lambda -(t1: T).(\lambda (i0: nat).(\lambda (H2: (subst0 (s k i0) v0 t1 t2)).(\lambda -(H3: ((\forall (c: C).(\forall (d: C).((getl (s k i0) c (CHead d (Bind Abbr) -v0)) \to ((sn3 c t1) \to (sn3 d v0))))))).(\lambda (u: T).(\lambda (c: -C).(\lambda (d: C).(\lambda (H4: (getl i0 c (CHead d (Bind Abbr) -v0))).(\lambda (H5: (sn3 c (THead k u t1))).(K_ind (\lambda (k0: K).((subst0 -(s k0 i0) v0 t1 t2) \to (((\forall (c0: C).(\forall (d0: C).((getl (s k0 i0) -c0 (CHead d0 (Bind Abbr) v0)) \to ((sn3 c0 t1) \to (sn3 d0 v0)))))) \to ((sn3 -c (THead k0 u t1)) \to (sn3 d v0))))) (\lambda (b: B).(\lambda (_: (subst0 (s -(Bind b) i0) v0 t1 t2)).(\lambda (H7: ((\forall (c0: C).(\forall (d0: -C).((getl (s (Bind b) i0) c0 (CHead d0 (Bind Abbr) v0)) \to ((sn3 c0 t1) \to -(sn3 d0 v0))))))).(\lambda (H8: (sn3 c (THead (Bind b) u t1))).(let H_x0 \def -(sn3_gen_bind b c u t1 H8) in (let H9 \def H_x0 in (land_ind (sn3 c u) (sn3 -(CHead c (Bind b) u) t1) (sn3 d v0) (\lambda (_: (sn3 c u)).(\lambda (H11: -(sn3 (CHead c (Bind b) u) t1)).(H7 (CHead c (Bind b) u) d (getl_clear_bind b -(CHead c (Bind b) u) c u (clear_bind b c u) (CHead d (Bind Abbr) v0) i0 H4) -H11))) H9))))))) (\lambda (f: F).(\lambda (_: (subst0 (s (Flat f) i0) v0 t1 -t2)).(\lambda (H7: ((\forall (c0: C).(\forall (d0: C).((getl (s (Flat f) i0) -c0 (CHead d0 (Bind Abbr) v0)) \to ((sn3 c0 t1) \to (sn3 d0 v0))))))).(\lambda -(H8: (sn3 c (THead (Flat f) u t1))).(let H_x0 \def (sn3_gen_flat f c u t1 H8) -in (let H9 \def H_x0 in (land_ind (sn3 c u) (sn3 c t1) (sn3 d v0) (\lambda -(_: (sn3 c u)).(\lambda (H11: (sn3 c t1)).(H7 c d H4 H11))) H9))))))) k H2 H3 -H5))))))))))))) (\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda -(i0: nat).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (H3: ((\forall (c: -C).(\forall (d: C).((getl i0 c (CHead d (Bind Abbr) v0)) \to ((sn3 c u1) \to -(sn3 d v0))))))).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(_: (subst0 (s k i0) v0 t1 t2)).(\lambda (_: ((\forall (c: C).(\forall (d: -C).((getl (s k i0) c (CHead d (Bind Abbr) v0)) \to ((sn3 c t1) \to (sn3 d -v0))))))).(\lambda (c: C).(\lambda (d: C).(\lambda (H6: (getl i0 c (CHead d -(Bind Abbr) v0))).(\lambda (H7: (sn3 c (THead k u1 t1))).(let H_y \def -(sn3_gen_head k c u1 t1 H7) in (H3 c d H6 H_y))))))))))))))))) i v t x H1))) -H0)))))). -(* COMMENTS -Initial nodes: 949 -END *) - -theorem sn3_cpr3_trans: - \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall -(k: K).(\forall (t: T).((sn3 (CHead c k u1) t) \to (sn3 (CHead c k u2) -t))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1 -u2)).(\lambda (k: K).(\lambda (t: T).(\lambda (H0: (sn3 (CHead c k u1) -t)).(sn3_ind (CHead c k u1) (\lambda (t0: T).(sn3 (CHead c k u2) t0)) -(\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) \to (\forall -(P: Prop).P))) \to ((pr3 (CHead c k u1) t1 t2) \to (sn3 (CHead c k u1) -t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: -Prop).P))) \to ((pr3 (CHead c k u1) t1 t2) \to (sn3 (CHead c k u2) -t2)))))).(sn3_sing (CHead c k u2) t1 (\lambda (t2: T).(\lambda (H3: (((eq T -t1 t2) \to (\forall (P: Prop).P)))).(\lambda (H4: (pr3 (CHead c k u2) t1 -t2)).(H2 t2 H3 (pr3_pr3_pr3_t c u1 u2 H t1 t2 k H4))))))))) t H0))))))). -(* COMMENTS -Initial nodes: 203 -END *) - -theorem sn3_bind: - \forall (b: B).(\forall (c: C).(\forall (u: T).((sn3 c u) \to (\forall (t: -T).((sn3 (CHead c (Bind b) u) t) \to (sn3 c (THead (Bind b) u t))))))) -\def - \lambda (b: B).(\lambda (c: C).(\lambda (u: T).(\lambda (H: (sn3 c -u)).(sn3_ind c (\lambda (t: T).(\forall (t0: T).((sn3 (CHead c (Bind b) t) -t0) \to (sn3 c (THead (Bind b) t t0))))) (\lambda (t1: T).(\lambda (_: -((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 -t2) \to (sn3 c t2)))))).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to -(\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (\forall (t: T).((sn3 (CHead c -(Bind b) t2) t) \to (sn3 c (THead (Bind b) t2 t))))))))).(\lambda (t: -T).(\lambda (H2: (sn3 (CHead c (Bind b) t1) t)).(sn3_ind (CHead c (Bind b) -t1) (\lambda (t0: T).(sn3 c (THead (Bind b) t1 t0))) (\lambda (t2: -T).(\lambda (H3: ((\forall (t3: T).((((eq T t2 t3) \to (\forall (P: -Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to (sn3 (CHead c (Bind b) -t1) t3)))))).(\lambda (H4: ((\forall (t3: T).((((eq T t2 t3) \to (\forall (P: -Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to (sn3 c (THead (Bind b) -t1 t3))))))).(sn3_sing c (THead (Bind b) t1 t2) (\lambda (t3: T).(\lambda -(H5: (((eq T (THead (Bind b) t1 t2) t3) \to (\forall (P: Prop).P)))).(\lambda -(H6: (pr3 c (THead (Bind b) t1 t2) t3)).(let H_x \def (bind_dec_not b Abst) -in (let H7 \def H_x in (or_ind (eq B b Abst) (not (eq B b Abst)) (sn3 c t3) -(\lambda (H8: (eq B b Abst)).(let H9 \def (eq_ind B b (\lambda (b0: B).(pr3 c -(THead (Bind b0) t1 t2) t3)) H6 Abst H8) in (let H10 \def (eq_ind B b -(\lambda (b0: B).((eq T (THead (Bind b0) t1 t2) t3) \to (\forall (P: -Prop).P))) H5 Abst H8) in (let H11 \def (eq_ind B b (\lambda (b0: B).(\forall -(t4: T).((((eq T t2 t4) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind -b0) t1) t2 t4) \to (sn3 c (THead (Bind b0) t1 t4)))))) H4 Abst H8) in (let -H12 \def (eq_ind B b (\lambda (b0: B).(\forall (t4: T).((((eq T t2 t4) \to -(\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b0) t1) t2 t4) \to (sn3 -(CHead c (Bind b0) t1) t4))))) H3 Abst H8) in (let H13 \def (eq_ind B b -(\lambda (b0: B).(\forall (t4: T).((((eq T t1 t4) \to (\forall (P: Prop).P))) -\to ((pr3 c t1 t4) \to (\forall (t0: T).((sn3 (CHead c (Bind b0) t4) t0) \to -(sn3 c (THead (Bind b0) t4 t0)))))))) H1 Abst H8) in (let H14 \def -(pr3_gen_abst c t1 t2 t3 H9) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: -T).(eq T t3 (THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b0: B).(\forall -(u0: T).(pr3 (CHead c (Bind b0) u0) t2 t4))))) (sn3 c t3) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H15: (eq T t3 (THead (Bind Abst) x0 -x1))).(\lambda (H16: (pr3 c t1 x0)).(\lambda (H17: ((\forall (b0: B).(\forall -(u0: T).(pr3 (CHead c (Bind b0) u0) t2 x1))))).(let H18 \def (eq_ind T t3 -(\lambda (t0: T).((eq T (THead (Bind Abst) t1 t2) t0) \to (\forall (P: -Prop).P))) H10 (THead (Bind Abst) x0 x1) H15) in (eq_ind_r T (THead (Bind -Abst) x0 x1) (\lambda (t0: T).(sn3 c t0)) (let H_x0 \def (term_dec t1 x0) in -(let H19 \def H_x0 in (or_ind (eq T t1 x0) ((eq T t1 x0) \to (\forall (P: -Prop).P)) (sn3 c (THead (Bind Abst) x0 x1)) (\lambda (H20: (eq T t1 x0)).(let -H21 \def (eq_ind_r T x0 (\lambda (t0: T).((eq T (THead (Bind Abst) t1 t2) -(THead (Bind Abst) t0 x1)) \to (\forall (P: Prop).P))) H18 t1 H20) in (let -H22 \def (eq_ind_r T x0 (\lambda (t0: T).(pr3 c t1 t0)) H16 t1 H20) in -(eq_ind T t1 (\lambda (t0: T).(sn3 c (THead (Bind Abst) t0 x1))) (let H_x1 -\def (term_dec t2 x1) in (let H23 \def H_x1 in (or_ind (eq T t2 x1) ((eq T t2 -x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Bind Abst) t1 x1)) (\lambda -(H24: (eq T t2 x1)).(let H25 \def (eq_ind_r T x1 (\lambda (t0: T).((eq T -(THead (Bind Abst) t1 t2) (THead (Bind Abst) t1 t0)) \to (\forall (P: -Prop).P))) H21 t2 H24) in (let H26 \def (eq_ind_r T x1 (\lambda (t0: -T).(\forall (b0: B).(\forall (u0: T).(pr3 (CHead c (Bind b0) u0) t2 t0)))) -H17 t2 H24) in (eq_ind T t2 (\lambda (t0: T).(sn3 c (THead (Bind Abst) t1 -t0))) (H25 (refl_equal T (THead (Bind Abst) t1 t2)) (sn3 c (THead (Bind Abst) -t1 t2))) x1 H24)))) (\lambda (H24: (((eq T t2 x1) \to (\forall (P: -Prop).P)))).(H11 x1 H24 (H17 Abst t1))) H23))) x0 H20)))) (\lambda (H20: -(((eq T t1 x0) \to (\forall (P: Prop).P)))).(let H_x1 \def (term_dec t2 x1) -in (let H21 \def H_x1 in (or_ind (eq T t2 x1) ((eq T t2 x1) \to (\forall (P: -Prop).P)) (sn3 c (THead (Bind Abst) x0 x1)) (\lambda (H22: (eq T t2 x1)).(let -H23 \def (eq_ind_r T x1 (\lambda (t0: T).(\forall (b0: B).(\forall (u0: -T).(pr3 (CHead c (Bind b0) u0) t2 t0)))) H17 t2 H22) in (eq_ind T t2 (\lambda -(t0: T).(sn3 c (THead (Bind Abst) x0 t0))) (H13 x0 H20 H16 t2 (sn3_cpr3_trans -c t1 x0 H16 (Bind Abst) t2 (sn3_sing (CHead c (Bind Abst) t1) t2 H12))) x1 -H22))) (\lambda (H22: (((eq T t2 x1) \to (\forall (P: Prop).P)))).(H13 x0 H20 -H16 x1 (sn3_cpr3_trans c t1 x0 H16 (Bind Abst) x1 (H12 x1 H22 (H17 Abst -t1))))) H21)))) H19))) t3 H15))))))) H14)))))))) (\lambda (H8: (not (eq B b -Abst))).(let H_x0 \def (pr3_gen_bind b H8 c t1 t2 t3 H6) in (let H9 \def H_x0 -in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind -b) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_: -T).(\lambda (t4: T).(pr3 (CHead c (Bind b) t1) t2 t4)))) (pr3 (CHead c (Bind -b) t1) t2 (lift (S O) O t3)) (sn3 c t3) (\lambda (H10: (ex3_2 T T (\lambda -(u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr3 -(CHead c (Bind b) t1) t2 t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda -(t4: T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr3 (CHead c (Bind b) -t1) t2 t4))) (sn3 c t3) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq -T t3 (THead (Bind b) x0 x1))).(\lambda (H12: (pr3 c t1 x0)).(\lambda (H13: -(pr3 (CHead c (Bind b) t1) t2 x1)).(let H14 \def (eq_ind T t3 (\lambda (t0: -T).((eq T (THead (Bind b) t1 t2) t0) \to (\forall (P: Prop).P))) H5 (THead -(Bind b) x0 x1) H11) in (eq_ind_r T (THead (Bind b) x0 x1) (\lambda (t0: -T).(sn3 c t0)) (let H_x1 \def (term_dec t1 x0) in (let H15 \def H_x1 in -(or_ind (eq T t1 x0) ((eq T t1 x0) \to (\forall (P: Prop).P)) (sn3 c (THead -(Bind b) x0 x1)) (\lambda (H16: (eq T t1 x0)).(let H17 \def (eq_ind_r T x0 -(\lambda (t0: T).((eq T (THead (Bind b) t1 t2) (THead (Bind b) t0 x1)) \to -(\forall (P: Prop).P))) H14 t1 H16) in (let H18 \def (eq_ind_r T x0 (\lambda -(t0: T).(pr3 c t1 t0)) H12 t1 H16) in (eq_ind T t1 (\lambda (t0: T).(sn3 c -(THead (Bind b) t0 x1))) (let H_x2 \def (term_dec t2 x1) in (let H19 \def -H_x2 in (or_ind (eq T t2 x1) ((eq T t2 x1) \to (\forall (P: Prop).P)) (sn3 c -(THead (Bind b) t1 x1)) (\lambda (H20: (eq T t2 x1)).(let H21 \def (eq_ind_r -T x1 (\lambda (t0: T).((eq T (THead (Bind b) t1 t2) (THead (Bind b) t1 t0)) -\to (\forall (P: Prop).P))) H17 t2 H20) in (let H22 \def (eq_ind_r T x1 -(\lambda (t0: T).(pr3 (CHead c (Bind b) t1) t2 t0)) H13 t2 H20) in (eq_ind T -t2 (\lambda (t0: T).(sn3 c (THead (Bind b) t1 t0))) (H21 (refl_equal T (THead -(Bind b) t1 t2)) (sn3 c (THead (Bind b) t1 t2))) x1 H20)))) (\lambda (H20: -(((eq T t2 x1) \to (\forall (P: Prop).P)))).(H4 x1 H20 H13)) H19))) x0 -H16)))) (\lambda (H16: (((eq T t1 x0) \to (\forall (P: Prop).P)))).(let H_x2 -\def (term_dec t2 x1) in (let H17 \def H_x2 in (or_ind (eq T t2 x1) ((eq T t2 -x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Bind b) x0 x1)) (\lambda (H18: -(eq T t2 x1)).(let H19 \def (eq_ind_r T x1 (\lambda (t0: T).(pr3 (CHead c -(Bind b) t1) t2 t0)) H13 t2 H18) in (eq_ind T t2 (\lambda (t0: T).(sn3 c -(THead (Bind b) x0 t0))) (H1 x0 H16 H12 t2 (sn3_cpr3_trans c t1 x0 H12 (Bind -b) t2 (sn3_sing (CHead c (Bind b) t1) t2 H3))) x1 H18))) (\lambda (H18: (((eq -T t2 x1) \to (\forall (P: Prop).P)))).(H1 x0 H16 H12 x1 (sn3_cpr3_trans c t1 -x0 H12 (Bind b) x1 (H3 x1 H18 H13)))) H17)))) H15))) t3 H11))))))) H10)) -(\lambda (H10: (pr3 (CHead c (Bind b) t1) t2 (lift (S O) O -t3))).(sn3_gen_lift (CHead c (Bind b) t1) t3 (S O) O (sn3_pr3_trans (CHead c -(Bind b) t1) t2 (sn3_sing (CHead c (Bind b) t1) t2 H3) (lift (S O) O t3) H10) -c (drop_drop (Bind b) O c c (drop_refl c) t1))) H9)))) H7)))))))))) t -H2)))))) u H)))). -(* COMMENTS -Initial nodes: 2401 -END *) - -theorem sn3_beta: - \forall (c: C).(\forall (v: T).(\forall (t: T).((sn3 c (THead (Bind Abbr) v -t)) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) v (THead -(Bind Abst) w t)))))))) -\def - \lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (H: (sn3 c (THead -(Bind Abbr) v t))).(insert_eq T (THead (Bind Abbr) v t) (\lambda (t0: T).(sn3 -c t0)) (\lambda (_: T).(\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat -Appl) v (THead (Bind Abst) w t)))))) (\lambda (y: T).(\lambda (H0: (sn3 c -y)).(unintro T t (\lambda (t0: T).((eq T y (THead (Bind Abbr) v t0)) \to -(\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) v (THead (Bind Abst) -w t0))))))) (unintro T v (\lambda (t0: T).(\forall (x: T).((eq T y (THead -(Bind Abbr) t0 x)) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat -Appl) t0 (THead (Bind Abst) w x)))))))) (sn3_ind c (\lambda (t0: T).(\forall -(x: T).(\forall (x0: T).((eq T t0 (THead (Bind Abbr) x x0)) \to (\forall (w: -T).((sn3 c w) \to (sn3 c (THead (Flat Appl) x (THead (Bind Abst) w -x0))))))))) (\lambda (t1: T).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) -\to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda -(H2: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 -c t1 t2) \to (\forall (x: T).(\forall (x0: T).((eq T t2 (THead (Bind Abbr) x -x0)) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) x (THead -(Bind Abst) w x0))))))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: -(eq T t1 (THead (Bind Abbr) x x0))).(\lambda (w: T).(\lambda (H4: (sn3 c -w)).(let H5 \def (eq_ind T t1 (\lambda (t0: T).(\forall (t2: T).((((eq T t0 -t2) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (\forall (x1: -T).(\forall (x2: T).((eq T t2 (THead (Bind Abbr) x1 x2)) \to (\forall (w0: -T).((sn3 c w0) \to (sn3 c (THead (Flat Appl) x1 (THead (Bind Abst) w0 -x2)))))))))))) H2 (THead (Bind Abbr) x x0) H3) in (let H6 \def (eq_ind T t1 -(\lambda (t0: T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) -\to ((pr3 c t0 t2) \to (sn3 c t2))))) H1 (THead (Bind Abbr) x x0) H3) in -(sn3_ind c (\lambda (t0: T).(sn3 c (THead (Flat Appl) x (THead (Bind Abst) t0 -x0)))) (\lambda (t2: T).(\lambda (H7: ((\forall (t3: T).((((eq T t2 t3) \to -(\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to (sn3 c t3)))))).(\lambda (H8: -((\forall (t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t2 -t3) \to (sn3 c (THead (Flat Appl) x (THead (Bind Abst) t3 -x0)))))))).(sn3_pr2_intro c (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) -(\lambda (t3: T).(\lambda (H9: (((eq T (THead (Flat Appl) x (THead (Bind -Abst) t2 x0)) t3) \to (\forall (P: Prop).P)))).(\lambda (H10: (pr2 c (THead -(Flat Appl) x (THead (Bind Abst) t2 x0)) t3)).(let H11 \def (pr2_gen_appl c x -(THead (Bind Abst) t2 x0) t3 H10) in (or3_ind (ex3_2 T T (\lambda (u2: -T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c -(THead (Bind Abst) t2 x0) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) t2 x0) -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda -(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: -B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4)))))))) (ex6_6 B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind Abst) t2 x0) (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) -z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) -y2) z1 z2)))))))) (sn3 c t3) (\lambda (H12: (ex3_2 T T (\lambda (u2: -T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c -(THead (Bind Abst) t2 x0) t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda -(t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind Abst) -t2 x0) t4))) (sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (H13: (eq -T t3 (THead (Flat Appl) x1 x2))).(\lambda (H14: (pr2 c x x1)).(\lambda (H15: -(pr2 c (THead (Bind Abst) t2 x0) x2)).(let H16 \def (eq_ind T t3 (\lambda -(t0: T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) t0) \to -(\forall (P: Prop).P))) H9 (THead (Flat Appl) x1 x2) H13) in (eq_ind_r T -(THead (Flat Appl) x1 x2) (\lambda (t0: T).(sn3 c t0)) (let H17 \def -(pr2_gen_abst c t2 x0 x2 H15) in (ex3_2_ind T T (\lambda (u2: T).(\lambda -(t4: T).(eq T x2 (THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c t2 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall -(u: T).(pr2 (CHead c (Bind b) u) x0 t4))))) (sn3 c (THead (Flat Appl) x1 x2)) -(\lambda (x3: T).(\lambda (x4: T).(\lambda (H18: (eq T x2 (THead (Bind Abst) -x3 x4))).(\lambda (H19: (pr2 c t2 x3)).(\lambda (H20: ((\forall (b: -B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x0 x4))))).(let H21 \def (eq_ind -T x2 (\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) -(THead (Flat Appl) x1 t0)) \to (\forall (P: Prop).P))) H16 (THead (Bind Abst) -x3 x4) H18) in (eq_ind_r T (THead (Bind Abst) x3 x4) (\lambda (t0: T).(sn3 c -(THead (Flat Appl) x1 t0))) (let H_x \def (term_dec t2 x3) in (let H22 \def -H_x in (or_ind (eq T t2 x3) ((eq T t2 x3) \to (\forall (P: Prop).P)) (sn3 c -(THead (Flat Appl) x1 (THead (Bind Abst) x3 x4))) (\lambda (H23: (eq T t2 -x3)).(let H24 \def (eq_ind_r T x3 (\lambda (t0: T).((eq T (THead (Flat Appl) -x (THead (Bind Abst) t2 x0)) (THead (Flat Appl) x1 (THead (Bind Abst) t0 -x4))) \to (\forall (P: Prop).P))) H21 t2 H23) in (let H25 \def (eq_ind_r T x3 -(\lambda (t0: T).(pr2 c t2 t0)) H19 t2 H23) in (eq_ind T t2 (\lambda (t0: -T).(sn3 c (THead (Flat Appl) x1 (THead (Bind Abst) t0 x4)))) (let H_x0 \def -(term_dec x x1) in (let H26 \def H_x0 in (or_ind (eq T x x1) ((eq T x x1) \to -(\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Bind Abst) t2 -x4))) (\lambda (H27: (eq T x x1)).(let H28 \def (eq_ind_r T x1 (\lambda (t0: -T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) (THead (Flat Appl) -t0 (THead (Bind Abst) t2 x4))) \to (\forall (P: Prop).P))) H24 x H27) in (let -H29 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14 x H27) in (eq_ind -T x (\lambda (t0: T).(sn3 c (THead (Flat Appl) t0 (THead (Bind Abst) t2 -x4)))) (let H_x1 \def (term_dec x0 x4) in (let H30 \def H_x1 in (or_ind (eq T -x0 x4) ((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x -(THead (Bind Abst) t2 x4))) (\lambda (H31: (eq T x0 x4)).(let H32 \def -(eq_ind_r T x4 (\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind -Abst) t2 x0)) (THead (Flat Appl) x (THead (Bind Abst) t2 t0))) \to (\forall -(P: Prop).P))) H28 x0 H31) in (let H33 \def (eq_ind_r T x4 (\lambda (t0: -T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 -H31) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Flat Appl) x (THead -(Bind Abst) t2 t0)))) (H32 (refl_equal T (THead (Flat Appl) x (THead (Bind -Abst) t2 x0))) (sn3 c (THead (Flat Appl) x (THead (Bind Abst) t2 x0)))) x4 -H31)))) (\lambda (H31: (((eq T x0 x4) \to (\forall (P: Prop).P)))).(H5 (THead -(Bind Abbr) x x4) (\lambda (H32: (eq T (THead (Bind Abbr) x x0) (THead (Bind -Abbr) x x4))).(\lambda (P: Prop).(let H33 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | -(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind -Abbr) x x0) (THead (Bind Abbr) x x4) H32) in (let H34 \def (eq_ind_r T x4 -(\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H31 x0 H33) in -(let H35 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: -T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H33) in (H34 (refl_equal T x0) -P)))))) (pr3_pr2 c (THead (Bind Abbr) x x0) (THead (Bind Abbr) x x4) -(pr2_head_2 c x x0 x4 (Bind Abbr) (H20 Abbr x))) x x4 (refl_equal T (THead -(Bind Abbr) x x4)) t2 (sn3_sing c t2 H7))) H30))) x1 H27)))) (\lambda (H27: -(((eq T x x1) \to (\forall (P: Prop).P)))).(H5 (THead (Bind Abbr) x1 x4) -(\lambda (H28: (eq T (THead (Bind Abbr) x x0) (THead (Bind Abbr) x1 -x4))).(\lambda (P: Prop).(let H29 \def (f_equal T T (\lambda (e: T).(match e -in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) -\Rightarrow x | (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abbr) x x0) -(THead (Bind Abbr) x1 x4) H28) in ((let H30 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | -(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind -Abbr) x x0) (THead (Bind Abbr) x1 x4) H28) in (\lambda (H31: (eq T x -x1)).(let H32 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall -(u: T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H30) in (let H33 \def -(eq_ind_r T x1 (\lambda (t0: T).((eq T x t0) \to (\forall (P0: Prop).P0))) -H27 x H31) in (let H34 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14 -x H31) in (H33 (refl_equal T x) P)))))) H29)))) (pr3_head_12 c x x1 (pr3_pr2 -c x x1 H14) (Bind Abbr) x0 x4 (pr3_pr2 (CHead c (Bind Abbr) x1) x0 x4 (H20 -Abbr x1))) x1 x4 (refl_equal T (THead (Bind Abbr) x1 x4)) t2 (sn3_sing c t2 -H7))) H26))) x3 H23)))) (\lambda (H23: (((eq T t2 x3) \to (\forall (P: -Prop).P)))).(let H_x0 \def (term_dec x x1) in (let H24 \def H_x0 in (or_ind -(eq T x x1) ((eq T x x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) -x1 (THead (Bind Abst) x3 x4))) (\lambda (H25: (eq T x x1)).(let H26 \def -(eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14 x H25) in (eq_ind T x -(\lambda (t0: T).(sn3 c (THead (Flat Appl) t0 (THead (Bind Abst) x3 x4)))) -(let H_x1 \def (term_dec x0 x4) in (let H27 \def H_x1 in (or_ind (eq T x0 x4) -((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x (THead -(Bind Abst) x3 x4))) (\lambda (H28: (eq T x0 x4)).(let H29 \def (eq_ind_r T -x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) -x0 t0)))) H20 x0 H28) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Flat -Appl) x (THead (Bind Abst) x3 t0)))) (H8 x3 H23 (pr3_pr2 c t2 x3 H19)) x4 -H28))) (\lambda (H28: (((eq T x0 x4) \to (\forall (P: Prop).P)))).(H5 (THead -(Bind Abbr) x x4) (\lambda (H29: (eq T (THead (Bind Abbr) x x0) (THead (Bind -Abbr) x x4))).(\lambda (P: Prop).(let H30 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | -(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind -Abbr) x x0) (THead (Bind Abbr) x x4) H29) in (let H31 \def (eq_ind_r T x4 -(\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H28 x0 H30) in -(let H32 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: -T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H30) in (H31 (refl_equal T x0) -P)))))) (pr3_pr2 c (THead (Bind Abbr) x x0) (THead (Bind Abbr) x x4) -(pr2_head_2 c x x0 x4 (Bind Abbr) (H20 Abbr x))) x x4 (refl_equal T (THead -(Bind Abbr) x x4)) x3 (H7 x3 H23 (pr3_pr2 c t2 x3 H19)))) H27))) x1 H25))) -(\lambda (H25: (((eq T x x1) \to (\forall (P: Prop).P)))).(H5 (THead (Bind -Abbr) x1 x4) (\lambda (H26: (eq T (THead (Bind Abbr) x x0) (THead (Bind Abbr) -x1 x4))).(\lambda (P: Prop).(let H27 \def (f_equal T T (\lambda (e: T).(match -e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) -\Rightarrow x | (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abbr) x x0) -(THead (Bind Abbr) x1 x4) H26) in ((let H28 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | -(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind -Abbr) x x0) (THead (Bind Abbr) x1 x4) H26) in (\lambda (H29: (eq T x -x1)).(let H30 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall -(u: T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H28) in (let H31 \def -(eq_ind_r T x1 (\lambda (t0: T).((eq T x t0) \to (\forall (P0: Prop).P0))) -H25 x H29) in (let H32 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14 -x H29) in (H31 (refl_equal T x) P)))))) H27)))) (pr3_head_12 c x x1 (pr3_pr2 -c x x1 H14) (Bind Abbr) x0 x4 (pr3_pr2 (CHead c (Bind Abbr) x1) x0 x4 (H20 -Abbr x1))) x1 x4 (refl_equal T (THead (Bind Abbr) x1 x4)) x3 (H7 x3 H23 -(pr3_pr2 c t2 x3 H19)))) H24)))) H22))) x2 H18))))))) H17)) t3 H13))))))) -H12)) (\lambda (H12: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) t2 x0) (THead -(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda -(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: -B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4))))))))).(ex4_4_ind T T T -T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind Abst) t2 x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind -Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t4: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) -z1 t4))))))) (sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: -T).(\lambda (x4: T).(\lambda (H13: (eq T (THead (Bind Abst) t2 x0) (THead -(Bind Abst) x1 x2))).(\lambda (H14: (eq T t3 (THead (Bind Abbr) x3 -x4))).(\lambda (H15: (pr2 c x x3)).(\lambda (H16: ((\forall (b: B).(\forall -(u: T).(pr2 (CHead c (Bind b) u) x2 x4))))).(let H17 \def (eq_ind T t3 -(\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) t0) -\to (\forall (P: Prop).P))) H9 (THead (Bind Abbr) x3 x4) H14) in (eq_ind_r T -(THead (Bind Abbr) x3 x4) (\lambda (t0: T).(sn3 c t0)) (let H18 \def (f_equal -T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow t2 | (TLRef _) \Rightarrow t2 | (THead _ t0 _) \Rightarrow t0])) -(THead (Bind Abst) t2 x0) (THead (Bind Abst) x1 x2) H13) in ((let H19 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ t0) -\Rightarrow t0])) (THead (Bind Abst) t2 x0) (THead (Bind Abst) x1 x2) H13) in -(\lambda (_: (eq T t2 x1)).(let H21 \def (eq_ind_r T x2 (\lambda (t0: -T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t0 x4)))) H16 x0 -H19) in (let H_x \def (term_dec x x3) in (let H22 \def H_x in (or_ind (eq T x -x3) ((eq T x x3) \to (\forall (P: Prop).P)) (sn3 c (THead (Bind Abbr) x3 x4)) -(\lambda (H23: (eq T x x3)).(let H24 \def (eq_ind_r T x3 (\lambda (t0: -T).(pr2 c x t0)) H15 x H23) in (eq_ind T x (\lambda (t0: T).(sn3 c (THead -(Bind Abbr) t0 x4))) (let H_x0 \def (term_dec x0 x4) in (let H25 \def H_x0 in -(or_ind (eq T x0 x4) ((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead -(Bind Abbr) x x4)) (\lambda (H26: (eq T x0 x4)).(let H27 \def (eq_ind_r T x4 -(\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x0 -t0)))) H21 x0 H26) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Bind Abbr) -x t0))) (sn3_sing c (THead (Bind Abbr) x x0) H6) x4 H26))) (\lambda (H26: -(((eq T x0 x4) \to (\forall (P: Prop).P)))).(H6 (THead (Bind Abbr) x x4) -(\lambda (H27: (eq T (THead (Bind Abbr) x x0) (THead (Bind Abbr) x -x4))).(\lambda (P: Prop).(let H28 \def (f_equal T T (\lambda (e: T).(match e -in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) -\Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind Abbr) x x0) -(THead (Bind Abbr) x x4) H27) in (let H29 \def (eq_ind_r T x4 (\lambda (t0: -T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H26 x0 H28) in (let H30 \def -(eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c -(Bind b) u) x0 t0)))) H21 x0 H28) in (H29 (refl_equal T x0) P)))))) (pr3_pr2 -c (THead (Bind Abbr) x x0) (THead (Bind Abbr) x x4) (pr2_head_2 c x x0 x4 -(Bind Abbr) (H21 Abbr x))))) H25))) x3 H23))) (\lambda (H23: (((eq T x x3) -\to (\forall (P: Prop).P)))).(H6 (THead (Bind Abbr) x3 x4) (\lambda (H24: (eq -T (THead (Bind Abbr) x x0) (THead (Bind Abbr) x3 x4))).(\lambda (P: -Prop).(let H25 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | -(THead _ t0 _) \Rightarrow t0])) (THead (Bind Abbr) x x0) (THead (Bind Abbr) -x3 x4) H24) in ((let H26 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) -\Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind Abbr) x x0) -(THead (Bind Abbr) x3 x4) H24) in (\lambda (H27: (eq T x x3)).(let H28 \def -(eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c -(Bind b) u) x0 t0)))) H21 x0 H26) in (let H29 \def (eq_ind_r T x3 (\lambda -(t0: T).((eq T x t0) \to (\forall (P0: Prop).P0))) H23 x H27) in (let H30 -\def (eq_ind_r T x3 (\lambda (t0: T).(pr2 c x t0)) H15 x H27) in (H29 -(refl_equal T x) P)))))) H25)))) (pr3_head_12 c x x3 (pr3_pr2 c x x3 H15) -(Bind Abbr) x0 x4 (pr3_pr2 (CHead c (Bind Abbr) x3) x0 x4 (H21 Abbr x3))))) -H22)))))) H18)) t3 H14)))))))))) H12)) (\lambda (H12: (ex6_6 B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind Abst) t2 x0) (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) -z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) -y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B -b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) t2 x0) (THead -(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) (sn3 c t3) -(\lambda (x1: B).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda -(x5: T).(\lambda (x6: T).(\lambda (H13: (not (eq B x1 Abst))).(\lambda (H14: -(eq T (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3))).(\lambda (H15: (eq -T t3 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))).(\lambda -(_: (pr2 c x x5)).(\lambda (H17: (pr2 c x2 x6)).(\lambda (H18: (pr2 (CHead c -(Bind x1) x6) x3 x4)).(let H19 \def (eq_ind T t3 (\lambda (t0: T).((eq T -(THead (Flat Appl) x (THead (Bind Abst) t2 x0)) t0) \to (\forall (P: -Prop).P))) H9 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)) -H15) in (eq_ind_r T (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) -x4)) (\lambda (t0: T).(sn3 c t0)) (let H20 \def (f_equal T B (\lambda (e: -T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow Abst | -(TLRef _) \Rightarrow Abst | (THead k _ _) \Rightarrow (match k in K return -(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow -Abst])])) (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3) H14) in ((let H21 -\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow t2 | (TLRef _) \Rightarrow t2 | (THead _ t0 _) -\Rightarrow t0])) (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3) H14) in -((let H22 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: -T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ -t0) \Rightarrow t0])) (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3) H14) -in (\lambda (H23: (eq T t2 x2)).(\lambda (H24: (eq B Abst x1)).(let H25 \def -(eq_ind_r T x3 (\lambda (t0: T).(pr2 (CHead c (Bind x1) x6) t0 x4)) H18 x0 -H22) in (let H26 \def (eq_ind_r T x2 (\lambda (t0: T).(pr2 c t0 x6)) H17 t2 -H23) in (let H27 \def (eq_ind_r B x1 (\lambda (b: B).(pr2 (CHead c (Bind b) -x6) x0 x4)) H25 Abst H24) in (let H28 \def (eq_ind_r B x1 (\lambda (b: -B).(not (eq B b Abst))) H13 Abst H24) in (eq_ind B Abst (\lambda (b: B).(sn3 -c (THead (Bind b) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))) (let H29 -\def (match (H28 (refl_equal B Abst)) in False return (\lambda (_: -False).(sn3 c (THead (Bind Abst) x6 (THead (Flat Appl) (lift (S O) O x5) -x4)))) with []) in H29) x1 H24)))))))) H21)) H20)) t3 H15)))))))))))))) H12)) -H11))))))))) w H4))))))))))) y H0))))) H)))). -(* COMMENTS -Initial nodes: 5699 -END *) - -theorem sn3_appl_lref: - \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (v: -T).((sn3 c v) \to (sn3 c (THead (Flat Appl) v (TLRef i))))))) -\def - \lambda (c: C).(\lambda (i: nat).(\lambda (H: (nf2 c (TLRef i))).(\lambda -(v: T).(\lambda (H0: (sn3 c v)).(sn3_ind c (\lambda (t: T).(sn3 c (THead -(Flat Appl) t (TLRef i)))) (\lambda (t1: T).(\lambda (_: ((\forall (t2: -T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c -t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: -Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c (THead (Flat Appl) t2 (TLRef -i)))))))).(sn3_pr2_intro c (THead (Flat Appl) t1 (TLRef i)) (\lambda (t2: -T).(\lambda (H3: (((eq T (THead (Flat Appl) t1 (TLRef i)) t2) \to (\forall -(P: Prop).P)))).(\lambda (H4: (pr2 c (THead (Flat Appl) t1 (TLRef i)) -t2)).(let H5 \def (pr2_gen_appl c t1 (TLRef i) t2 H4) in (or3_ind (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2))) (\lambda (_: T).(\lambda -(t3: T).(pr2 c (TLRef i) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i) (THead (Bind Abst) y1 -z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: -T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2))))) (\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall -(u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T -(TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T -t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c t1 u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 -y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) -(sn3 c t2) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T -t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c t1 -u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (TLRef i) t3))))).(ex3_2_ind T -T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2))) (\lambda (_: T).(\lambda -(t3: T).(pr2 c (TLRef i) t3))) (sn3 c t2) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H7: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H8: (pr2 c t1 -x0)).(\lambda (H9: (pr2 c (TLRef i) x1)).(let H10 \def (eq_ind T t2 (\lambda -(t: T).((eq T (THead (Flat Appl) t1 (TLRef i)) t) \to (\forall (P: Prop).P))) -H3 (THead (Flat Appl) x0 x1) H7) in (eq_ind_r T (THead (Flat Appl) x0 x1) -(\lambda (t: T).(sn3 c t)) (let H11 \def (eq_ind_r T x1 (\lambda (t: T).((eq -T (THead (Flat Appl) t1 (TLRef i)) (THead (Flat Appl) x0 t)) \to (\forall (P: -Prop).P))) H10 (TLRef i) (H x1 H9)) in (let H12 \def (eq_ind_r T x1 (\lambda -(t: T).(pr2 c (TLRef i) t)) H9 (TLRef i) (H x1 H9)) in (eq_ind T (TLRef i) -(\lambda (t: T).(sn3 c (THead (Flat Appl) x0 t))) (let H_x \def (term_dec t1 -x0) in (let H13 \def H_x in (or_ind (eq T t1 x0) ((eq T t1 x0) \to (\forall -(P: Prop).P)) (sn3 c (THead (Flat Appl) x0 (TLRef i))) (\lambda (H14: (eq T -t1 x0)).(let H15 \def (eq_ind_r T x0 (\lambda (t: T).((eq T (THead (Flat -Appl) t1 (TLRef i)) (THead (Flat Appl) t (TLRef i))) \to (\forall (P: -Prop).P))) H11 t1 H14) in (let H16 \def (eq_ind_r T x0 (\lambda (t: T).(pr2 c -t1 t)) H8 t1 H14) in (eq_ind T t1 (\lambda (t: T).(sn3 c (THead (Flat Appl) t -(TLRef i)))) (H15 (refl_equal T (THead (Flat Appl) t1 (TLRef i))) (sn3 c -(THead (Flat Appl) t1 (TLRef i)))) x0 H14)))) (\lambda (H14: (((eq T t1 x0) -\to (\forall (P: Prop).P)))).(H2 x0 H14 (pr3_pr2 c t1 x0 H8))) H13))) x1 (H -x1 H9)))) t2 H7))))))) H6)) (\lambda (H6: (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i) (THead -(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall -(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3))))))))).(ex4_4_ind T -T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T -(TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 -u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: -T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3))))))) -(sn3 c t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: -T).(\lambda (H7: (eq T (TLRef i) (THead (Bind Abst) x0 x1))).(\lambda (H8: -(eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (_: (pr2 c t1 x2)).(\lambda (_: -((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 x3))))).(let -H11 \def (eq_ind T t2 (\lambda (t: T).((eq T (THead (Flat Appl) t1 (TLRef i)) -t) \to (\forall (P: Prop).P))) H3 (THead (Bind Abbr) x2 x3) H8) in (eq_ind_r -T (THead (Bind Abbr) x2 x3) (\lambda (t: T).(sn3 c t)) (let H12 \def (eq_ind -T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) -\Rightarrow False])) I (THead (Bind Abst) x0 x1) H7) in (False_ind (sn3 c -(THead (Bind Abbr) x2 x3)) H12)) t2 H8)))))))))) H6)) (\lambda (H6: (ex6_6 B -T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T (TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda -(_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq -T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c t1 u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 -y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 -z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i) (THead (Bind b) y1 -z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: -T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat -Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 -(CHead c (Bind b) y2) z1 z2))))))) (sn3 c t2) (\lambda (x0: B).(\lambda (x1: -T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: -T).(\lambda (_: (not (eq B x0 Abst))).(\lambda (H8: (eq T (TLRef i) (THead -(Bind x0) x1 x2))).(\lambda (H9: (eq T t2 (THead (Bind x0) x5 (THead (Flat -Appl) (lift (S O) O x4) x3)))).(\lambda (_: (pr2 c t1 x4)).(\lambda (_: (pr2 -c x1 x5)).(\lambda (_: (pr2 (CHead c (Bind x0) x5) x2 x3)).(let H13 \def -(eq_ind T t2 (\lambda (t: T).((eq T (THead (Flat Appl) t1 (TLRef i)) t) \to -(\forall (P: Prop).P))) H3 (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) -O x4) x3)) H9) in (eq_ind_r T (THead (Bind x0) x5 (THead (Flat Appl) (lift (S -O) O x4) x3)) (\lambda (t: T).(sn3 c t)) (let H14 \def (eq_ind T (TLRef i) -(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow -False])) I (THead (Bind x0) x1 x2) H8) in (False_ind (sn3 c (THead (Bind x0) -x5 (THead (Flat Appl) (lift (S O) O x4) x3))) H14)) t2 H9)))))))))))))) H6)) -H5))))))))) v H0))))). -(* COMMENTS -Initial nodes: 2125 -END *) - -theorem sn3_appl_abbr: - \forall (c: C).(\forall (d: C).(\forall (w: T).(\forall (i: nat).((getl i c -(CHead d (Bind Abbr) w)) \to (\forall (v: T).((sn3 c (THead (Flat Appl) v -(lift (S i) O w))) \to (sn3 c (THead (Flat Appl) v (TLRef i))))))))) -\def - \lambda (c: C).(\lambda (d: C).(\lambda (w: T).(\lambda (i: nat).(\lambda -(H: (getl i c (CHead d (Bind Abbr) w))).(\lambda (v: T).(\lambda (H0: (sn3 c -(THead (Flat Appl) v (lift (S i) O w)))).(insert_eq T (THead (Flat Appl) v -(lift (S i) O w)) (\lambda (t: T).(sn3 c t)) (\lambda (_: T).(sn3 c (THead -(Flat Appl) v (TLRef i)))) (\lambda (y: T).(\lambda (H1: (sn3 c y)).(unintro -T v (\lambda (t: T).((eq T y (THead (Flat Appl) t (lift (S i) O w))) \to (sn3 -c (THead (Flat Appl) t (TLRef i))))) (sn3_ind c (\lambda (t: T).(\forall (x: -T).((eq T t (THead (Flat Appl) x (lift (S i) O w))) \to (sn3 c (THead (Flat -Appl) x (TLRef i)))))) (\lambda (t1: T).(\lambda (H2: ((\forall (t2: -T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c -t2)))))).(\lambda (H3: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: -Prop).P))) \to ((pr3 c t1 t2) \to (\forall (x: T).((eq T t2 (THead (Flat -Appl) x (lift (S i) O w))) \to (sn3 c (THead (Flat Appl) x (TLRef -i)))))))))).(\lambda (x: T).(\lambda (H4: (eq T t1 (THead (Flat Appl) x (lift -(S i) O w)))).(let H5 \def (eq_ind T t1 (\lambda (t: T).(\forall (t2: -T).((((eq T t t2) \to (\forall (P: Prop).P))) \to ((pr3 c t t2) \to (\forall -(x0: T).((eq T t2 (THead (Flat Appl) x0 (lift (S i) O w))) \to (sn3 c (THead -(Flat Appl) x0 (TLRef i))))))))) H3 (THead (Flat Appl) x (lift (S i) O w)) -H4) in (let H6 \def (eq_ind T t1 (\lambda (t: T).(\forall (t2: T).((((eq T t -t2) \to (\forall (P: Prop).P))) \to ((pr3 c t t2) \to (sn3 c t2))))) H2 -(THead (Flat Appl) x (lift (S i) O w)) H4) in (sn3_pr2_intro c (THead (Flat -Appl) x (TLRef i)) (\lambda (t2: T).(\lambda (H7: (((eq T (THead (Flat Appl) -x (TLRef i)) t2) \to (\forall (P: Prop).P)))).(\lambda (H8: (pr2 c (THead -(Flat Appl) x (TLRef i)) t2)).(let H9 \def (pr2_gen_appl c x (TLRef i) t2 H8) -in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead -(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) -(\lambda (_: T).(\lambda (t3: T).(pr2 c (TLRef i) t3)))) (ex4_4 T T T T -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T -(TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x -u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: -T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) -(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T (TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda -(_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq -T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 -y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) -(sn3 c t2) (\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T -t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x -u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (TLRef i) t3))))).(ex3_2_ind T -T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t3: -T).(pr2 c (TLRef i) t3))) (sn3 c t2) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H11: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H12: (pr2 c -x x0)).(\lambda (H13: (pr2 c (TLRef i) x1)).(let H14 \def (eq_ind T t2 -(\lambda (t: T).((eq T (THead (Flat Appl) x (TLRef i)) t) \to (\forall (P: -Prop).P))) H7 (THead (Flat Appl) x0 x1) H11) in (eq_ind_r T (THead (Flat -Appl) x0 x1) (\lambda (t: T).(sn3 c t)) (let H15 \def (pr2_gen_lref c x1 i -H13) in (or_ind (eq T x1 (TLRef i)) (ex2_2 C T (\lambda (d0: C).(\lambda (u: -T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq -T x1 (lift (S i) O u))))) (sn3 c (THead (Flat Appl) x0 x1)) (\lambda (H16: -(eq T x1 (TLRef i))).(let H17 \def (eq_ind T x1 (\lambda (t: T).((eq T (THead -(Flat Appl) x (TLRef i)) (THead (Flat Appl) x0 t)) \to (\forall (P: -Prop).P))) H14 (TLRef i) H16) in (eq_ind_r T (TLRef i) (\lambda (t: T).(sn3 c -(THead (Flat Appl) x0 t))) (let H_x \def (term_dec x x0) in (let H18 \def H_x -in (or_ind (eq T x x0) ((eq T x x0) \to (\forall (P: Prop).P)) (sn3 c (THead -(Flat Appl) x0 (TLRef i))) (\lambda (H19: (eq T x x0)).(let H20 \def -(eq_ind_r T x0 (\lambda (t: T).((eq T (THead (Flat Appl) x (TLRef i)) (THead -(Flat Appl) t (TLRef i))) \to (\forall (P: Prop).P))) H17 x H19) in (let H21 -\def (eq_ind_r T x0 (\lambda (t: T).(pr2 c x t)) H12 x H19) in (eq_ind T x -(\lambda (t: T).(sn3 c (THead (Flat Appl) t (TLRef i)))) (H20 (refl_equal T -(THead (Flat Appl) x (TLRef i))) (sn3 c (THead (Flat Appl) x (TLRef i)))) x0 -H19)))) (\lambda (H19: (((eq T x x0) \to (\forall (P: Prop).P)))).(H5 (THead -(Flat Appl) x0 (lift (S i) O w)) (\lambda (H20: (eq T (THead (Flat Appl) x -(lift (S i) O w)) (THead (Flat Appl) x0 (lift (S i) O w)))).(\lambda (P: -Prop).(let H21 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | -(THead _ t _) \Rightarrow t])) (THead (Flat Appl) x (lift (S i) O w)) (THead -(Flat Appl) x0 (lift (S i) O w)) H20) in (let H22 \def (eq_ind_r T x0 -(\lambda (t: T).((eq T x t) \to (\forall (P0: Prop).P0))) H19 x H21) in (let -H23 \def (eq_ind_r T x0 (\lambda (t: T).(pr2 c x t)) H12 x H21) in (H22 -(refl_equal T x) P)))))) (pr3_pr2 c (THead (Flat Appl) x (lift (S i) O w)) -(THead (Flat Appl) x0 (lift (S i) O w)) (pr2_head_1 c x x0 H12 (Flat Appl) -(lift (S i) O w))) x0 (refl_equal T (THead (Flat Appl) x0 (lift (S i) O -w))))) H18))) x1 H16))) (\lambda (H16: (ex2_2 C T (\lambda (d0: C).(\lambda -(u: T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: -T).(eq T x1 (lift (S i) O u)))))).(ex2_2_ind C T (\lambda (d0: C).(\lambda -(u: T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: -T).(eq T x1 (lift (S i) O u)))) (sn3 c (THead (Flat Appl) x0 x1)) (\lambda -(x2: C).(\lambda (x3: T).(\lambda (H17: (getl i c (CHead x2 (Bind Abbr) -x3))).(\lambda (H18: (eq T x1 (lift (S i) O x3))).(let H19 \def (eq_ind T x1 -(\lambda (t: T).((eq T (THead (Flat Appl) x (TLRef i)) (THead (Flat Appl) x0 -t)) \to (\forall (P: Prop).P))) H14 (lift (S i) O x3) H18) in (eq_ind_r T -(lift (S i) O x3) (\lambda (t: T).(sn3 c (THead (Flat Appl) x0 t))) (let H20 -\def (eq_ind C (CHead d (Bind Abbr) w) (\lambda (c0: C).(getl i c c0)) H -(CHead x2 (Bind Abbr) x3) (getl_mono c (CHead d (Bind Abbr) w) i H (CHead x2 -(Bind Abbr) x3) H17)) in (let H21 \def (f_equal C C (\lambda (e: C).(match e -in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) -\Rightarrow c0])) (CHead d (Bind Abbr) w) (CHead x2 (Bind Abbr) x3) -(getl_mono c (CHead d (Bind Abbr) w) i H (CHead x2 (Bind Abbr) x3) H17)) in -((let H22 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: -C).T) with [(CSort _) \Rightarrow w | (CHead _ _ t) \Rightarrow t])) (CHead d -(Bind Abbr) w) (CHead x2 (Bind Abbr) x3) (getl_mono c (CHead d (Bind Abbr) w) -i H (CHead x2 (Bind Abbr) x3) H17)) in (\lambda (H23: (eq C d x2)).(let H24 -\def (eq_ind_r T x3 (\lambda (t: T).(getl i c (CHead x2 (Bind Abbr) t))) H20 -w H22) in (eq_ind T w (\lambda (t: T).(sn3 c (THead (Flat Appl) x0 (lift (S -i) O t)))) (let H25 \def (eq_ind_r C x2 (\lambda (c0: C).(getl i c (CHead c0 -(Bind Abbr) w))) H24 d H23) in (let H_x \def (term_dec x x0) in (let H26 \def -H_x in (or_ind (eq T x x0) ((eq T x x0) \to (\forall (P: Prop).P)) (sn3 c -(THead (Flat Appl) x0 (lift (S i) O w))) (\lambda (H27: (eq T x x0)).(let H28 -\def (eq_ind_r T x0 (\lambda (t: T).(pr2 c x t)) H12 x H27) in (eq_ind T x -(\lambda (t: T).(sn3 c (THead (Flat Appl) t (lift (S i) O w)))) (sn3_sing c -(THead (Flat Appl) x (lift (S i) O w)) H6) x0 H27))) (\lambda (H27: (((eq T x -x0) \to (\forall (P: Prop).P)))).(H6 (THead (Flat Appl) x0 (lift (S i) O w)) -(\lambda (H28: (eq T (THead (Flat Appl) x (lift (S i) O w)) (THead (Flat -Appl) x0 (lift (S i) O w)))).(\lambda (P: Prop).(let H29 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t _) \Rightarrow t])) -(THead (Flat Appl) x (lift (S i) O w)) (THead (Flat Appl) x0 (lift (S i) O -w)) H28) in (let H30 \def (eq_ind_r T x0 (\lambda (t: T).((eq T x t) \to -(\forall (P0: Prop).P0))) H27 x H29) in (let H31 \def (eq_ind_r T x0 (\lambda -(t: T).(pr2 c x t)) H12 x H29) in (H30 (refl_equal T x) P)))))) (pr3_pr2 c -(THead (Flat Appl) x (lift (S i) O w)) (THead (Flat Appl) x0 (lift (S i) O -w)) (pr2_head_1 c x x0 H12 (Flat Appl) (lift (S i) O w))))) H26)))) x3 -H22)))) H21))) x1 H18)))))) H16)) H15)) t2 H11))))))) H10)) (\lambda (H10: -(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(eq T (TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda -(_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 -t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: -T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 -t3))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T (TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda -(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead -(Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind -b) u) z1 t3))))))) (sn3 c t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: -T).(\lambda (x3: T).(\lambda (H11: (eq T (TLRef i) (THead (Bind Abst) x0 -x1))).(\lambda (H12: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (_: (pr2 c -x x2)).(\lambda (_: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) -u) x1 x3))))).(let H15 \def (eq_ind T t2 (\lambda (t: T).((eq T (THead (Flat -Appl) x (TLRef i)) t) \to (\forall (P: Prop).P))) H7 (THead (Bind Abbr) x2 -x3) H12) in (eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda (t: T).(sn3 c t)) -(let H16 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) x0 -x1) H11) in (False_ind (sn3 c (THead (Bind Abbr) x2 x3)) H16)) t2 -H12)))))))))) H10)) (\lambda (H10: (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i) -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind -B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T (TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda -(_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq -T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 -y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) -(sn3 c t2) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: -T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (not (eq B x0 -Abst))).(\lambda (H12: (eq T (TLRef i) (THead (Bind x0) x1 x2))).(\lambda -(H13: (eq T t2 (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) -x3)))).(\lambda (_: (pr2 c x x4)).(\lambda (_: (pr2 c x1 x5)).(\lambda (_: -(pr2 (CHead c (Bind x0) x5) x2 x3)).(let H17 \def (eq_ind T t2 (\lambda (t: -T).((eq T (THead (Flat Appl) x (TLRef i)) t) \to (\forall (P: Prop).P))) H7 -(THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) H13) in -(eq_ind_r T (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) -(\lambda (t: T).(sn3 c t)) (let H18 \def (eq_ind T (TLRef i) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I -(THead (Bind x0) x1 x2) H12) in (False_ind (sn3 c (THead (Bind x0) x5 (THead -(Flat Appl) (lift (S O) O x4) x3))) H18)) t2 H13)))))))))))))) H10)) -H9))))))))))))) y H1)))) H0))))))). -(* COMMENTS -Initial nodes: 3727 -END *) - -theorem sn3_appl_cast: - \forall (c: C).(\forall (v: T).(\forall (u: T).((sn3 c (THead (Flat Appl) v -u)) \to (\forall (t: T).((sn3 c (THead (Flat Appl) v t)) \to (sn3 c (THead -(Flat Appl) v (THead (Flat Cast) u t)))))))) -\def - \lambda (c: C).(\lambda (v: T).(\lambda (u: T).(\lambda (H: (sn3 c (THead -(Flat Appl) v u))).(insert_eq T (THead (Flat Appl) v u) (\lambda (t: T).(sn3 -c t)) (\lambda (_: T).(\forall (t0: T).((sn3 c (THead (Flat Appl) v t0)) \to -(sn3 c (THead (Flat Appl) v (THead (Flat Cast) u t0)))))) (\lambda (y: -T).(\lambda (H0: (sn3 c y)).(unintro T u (\lambda (t: T).((eq T y (THead -(Flat Appl) v t)) \to (\forall (t0: T).((sn3 c (THead (Flat Appl) v t0)) \to -(sn3 c (THead (Flat Appl) v (THead (Flat Cast) t t0))))))) (unintro T v -(\lambda (t: T).(\forall (x: T).((eq T y (THead (Flat Appl) t x)) \to -(\forall (t0: T).((sn3 c (THead (Flat Appl) t t0)) \to (sn3 c (THead (Flat -Appl) t (THead (Flat Cast) x t0)))))))) (sn3_ind c (\lambda (t: T).(\forall -(x: T).(\forall (x0: T).((eq T t (THead (Flat Appl) x x0)) \to (\forall (t0: -T).((sn3 c (THead (Flat Appl) x t0)) \to (sn3 c (THead (Flat Appl) x (THead -(Flat Cast) x0 t0))))))))) (\lambda (t1: T).(\lambda (H1: ((\forall (t2: -T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c -t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: -Prop).P))) \to ((pr3 c t1 t2) \to (\forall (x: T).(\forall (x0: T).((eq T t2 -(THead (Flat Appl) x x0)) \to (\forall (t: T).((sn3 c (THead (Flat Appl) x -t)) \to (sn3 c (THead (Flat Appl) x (THead (Flat Cast) x0 -t))))))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: (eq T t1 (THead -(Flat Appl) x x0))).(\lambda (t: T).(\lambda (H4: (sn3 c (THead (Flat Appl) x -t))).(insert_eq T (THead (Flat Appl) x t) (\lambda (t0: T).(sn3 c t0)) -(\lambda (_: T).(sn3 c (THead (Flat Appl) x (THead (Flat Cast) x0 t)))) -(\lambda (y0: T).(\lambda (H5: (sn3 c y0)).(unintro T t (\lambda (t0: T).((eq -T y0 (THead (Flat Appl) x t0)) \to (sn3 c (THead (Flat Appl) x (THead (Flat -Cast) x0 t0))))) (sn3_ind c (\lambda (t0: T).(\forall (x1: T).((eq T t0 -(THead (Flat Appl) x x1)) \to (sn3 c (THead (Flat Appl) x (THead (Flat Cast) -x0 x1)))))) (\lambda (t0: T).(\lambda (H6: ((\forall (t2: T).((((eq T t0 t2) -\to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (sn3 c t2)))))).(\lambda -(H7: ((\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 -c t0 t2) \to (\forall (x1: T).((eq T t2 (THead (Flat Appl) x x1)) \to (sn3 c -(THead (Flat Appl) x (THead (Flat Cast) x0 x1)))))))))).(\lambda (x1: -T).(\lambda (H8: (eq T t0 (THead (Flat Appl) x x1))).(let H9 \def (eq_ind T -t0 (\lambda (t2: T).(\forall (t3: T).((((eq T t2 t3) \to (\forall (P: -Prop).P))) \to ((pr3 c t2 t3) \to (\forall (x2: T).((eq T t3 (THead (Flat -Appl) x x2)) \to (sn3 c (THead (Flat Appl) x (THead (Flat Cast) x0 -x2))))))))) H7 (THead (Flat Appl) x x1) H8) in (let H10 \def (eq_ind T t0 -(\lambda (t2: T).(\forall (t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) -\to ((pr3 c t2 t3) \to (sn3 c t3))))) H6 (THead (Flat Appl) x x1) H8) in (let -H11 \def (eq_ind T t1 (\lambda (t2: T).(\forall (t3: T).((((eq T t2 t3) \to -(\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to (\forall (x2: T).(\forall (x3: -T).((eq T t3 (THead (Flat Appl) x2 x3)) \to (\forall (t4: T).((sn3 c (THead -(Flat Appl) x2 t4)) \to (sn3 c (THead (Flat Appl) x2 (THead (Flat Cast) x3 -t4)))))))))))) H2 (THead (Flat Appl) x x0) H3) in (let H12 \def (eq_ind T t1 -(\lambda (t2: T).(\forall (t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) -\to ((pr3 c t2 t3) \to (sn3 c t3))))) H1 (THead (Flat Appl) x x0) H3) in -(sn3_pr2_intro c (THead (Flat Appl) x (THead (Flat Cast) x0 x1)) (\lambda -(t2: T).(\lambda (H13: (((eq T (THead (Flat Appl) x (THead (Flat Cast) x0 -x1)) t2) \to (\forall (P: Prop).P)))).(\lambda (H14: (pr2 c (THead (Flat -Appl) x (THead (Flat Cast) x0 x1)) t2)).(let H15 \def (pr2_gen_appl c x -(THead (Flat Cast) x0 x1) t2 H14) in (or3_ind (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c -(THead (Flat Cast) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Cast) x0 x1) -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda -(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: -B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T -T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq -T (THead (Flat Cast) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) -z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) -y2) z1 z2)))))))) (sn3 c t2) (\lambda (H16: (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c -(THead (Flat Cast) x0 x1) t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda -(t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Flat Cast) -x0 x1) t3))) (sn3 c t2) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H17: (eq -T t2 (THead (Flat Appl) x2 x3))).(\lambda (H18: (pr2 c x x2)).(\lambda (H19: -(pr2 c (THead (Flat Cast) x0 x1) x3)).(let H20 \def (eq_ind T t2 (\lambda -(t3: T).((eq T (THead (Flat Appl) x (THead (Flat Cast) x0 x1)) t3) \to -(\forall (P: Prop).P))) H13 (THead (Flat Appl) x2 x3) H17) in (eq_ind_r T -(THead (Flat Appl) x2 x3) (\lambda (t3: T).(sn3 c t3)) (let H21 \def -(pr2_gen_cast c x0 x1 x3 H19) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda -(t3: T).(eq T x3 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c x1 t3)))) (pr2 c -x1 x3) (sn3 c (THead (Flat Appl) x2 x3)) (\lambda (H22: (ex3_2 T T (\lambda -(u2: T).(\lambda (t3: T).(eq T x3 (THead (Flat Cast) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c x1 -t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x3 (THead -(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) -(\lambda (_: T).(\lambda (t3: T).(pr2 c x1 t3))) (sn3 c (THead (Flat Appl) x2 -x3)) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H23: (eq T x3 (THead (Flat -Cast) x4 x5))).(\lambda (H24: (pr2 c x0 x4)).(\lambda (H25: (pr2 c x1 -x5)).(let H26 \def (eq_ind T x3 (\lambda (t3: T).((eq T (THead (Flat Appl) x -(THead (Flat Cast) x0 x1)) (THead (Flat Appl) x2 t3)) \to (\forall (P: -Prop).P))) H20 (THead (Flat Cast) x4 x5) H23) in (eq_ind_r T (THead (Flat -Cast) x4 x5) (\lambda (t3: T).(sn3 c (THead (Flat Appl) x2 t3))) (let H_x -\def (term_dec (THead (Flat Appl) x x0) (THead (Flat Appl) x2 x4)) in (let -H27 \def H_x in (or_ind (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x2 -x4)) ((eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x2 x4)) \to (\forall -(P: Prop).P)) (sn3 c (THead (Flat Appl) x2 (THead (Flat Cast) x4 x5))) -(\lambda (H28: (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x2 -x4))).(let H29 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | -(THead _ t3 _) \Rightarrow t3])) (THead (Flat Appl) x x0) (THead (Flat Appl) -x2 x4) H28) in ((let H30 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) -\Rightarrow x0 | (THead _ _ t3) \Rightarrow t3])) (THead (Flat Appl) x x0) -(THead (Flat Appl) x2 x4) H28) in (\lambda (H31: (eq T x x2)).(let H32 \def -(eq_ind_r T x4 (\lambda (t3: T).((eq T (THead (Flat Appl) x (THead (Flat -Cast) x0 x1)) (THead (Flat Appl) x2 (THead (Flat Cast) t3 x5))) \to (\forall -(P: Prop).P))) H26 x0 H30) in (let H33 \def (eq_ind_r T x4 (\lambda (t3: -T).(pr2 c x0 t3)) H24 x0 H30) in (eq_ind T x0 (\lambda (t3: T).(sn3 c (THead -(Flat Appl) x2 (THead (Flat Cast) t3 x5)))) (let H34 \def (eq_ind_r T x2 -(\lambda (t3: T).((eq T (THead (Flat Appl) x (THead (Flat Cast) x0 x1)) -(THead (Flat Appl) t3 (THead (Flat Cast) x0 x5))) \to (\forall (P: Prop).P))) -H32 x H31) in (let H35 \def (eq_ind_r T x2 (\lambda (t3: T).(pr2 c x t3)) H18 -x H31) in (eq_ind T x (\lambda (t3: T).(sn3 c (THead (Flat Appl) t3 (THead -(Flat Cast) x0 x5)))) (let H_x0 \def (term_dec (THead (Flat Appl) x x1) -(THead (Flat Appl) x x5)) in (let H36 \def H_x0 in (or_ind (eq T (THead (Flat -Appl) x x1) (THead (Flat Appl) x x5)) ((eq T (THead (Flat Appl) x x1) (THead -(Flat Appl) x x5)) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x -(THead (Flat Cast) x0 x5))) (\lambda (H37: (eq T (THead (Flat Appl) x x1) -(THead (Flat Appl) x x5))).(let H38 \def (f_equal T T (\lambda (e: T).(match -e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x1 | (TLRef _) -\Rightarrow x1 | (THead _ _ t3) \Rightarrow t3])) (THead (Flat Appl) x x1) -(THead (Flat Appl) x x5) H37) in (let H39 \def (eq_ind_r T x5 (\lambda (t3: -T).((eq T (THead (Flat Appl) x (THead (Flat Cast) x0 x1)) (THead (Flat Appl) -x (THead (Flat Cast) x0 t3))) \to (\forall (P: Prop).P))) H34 x1 H38) in (let -H40 \def (eq_ind_r T x5 (\lambda (t3: T).(pr2 c x1 t3)) H25 x1 H38) in -(eq_ind T x1 (\lambda (t3: T).(sn3 c (THead (Flat Appl) x (THead (Flat Cast) -x0 t3)))) (H39 (refl_equal T (THead (Flat Appl) x (THead (Flat Cast) x0 x1))) -(sn3 c (THead (Flat Appl) x (THead (Flat Cast) x0 x1)))) x5 H38))))) (\lambda -(H37: (((eq T (THead (Flat Appl) x x1) (THead (Flat Appl) x x5)) \to (\forall -(P: Prop).P)))).(H9 (THead (Flat Appl) x x5) H37 (pr3_pr2 c (THead (Flat -Appl) x x1) (THead (Flat Appl) x x5) (pr2_thin_dx c x1 x5 H25 x Appl)) x5 -(refl_equal T (THead (Flat Appl) x x5)))) H36))) x2 H31))) x4 H30))))) H29))) -(\lambda (H28: (((eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x2 x4)) -\to (\forall (P: Prop).P)))).(let H_x0 \def (term_dec (THead (Flat Appl) x -x1) (THead (Flat Appl) x2 x5)) in (let H29 \def H_x0 in (or_ind (eq T (THead -(Flat Appl) x x1) (THead (Flat Appl) x2 x5)) ((eq T (THead (Flat Appl) x x1) -(THead (Flat Appl) x2 x5)) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat -Appl) x2 (THead (Flat Cast) x4 x5))) (\lambda (H30: (eq T (THead (Flat Appl) -x x1) (THead (Flat Appl) x2 x5))).(let H31 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x | -(TLRef _) \Rightarrow x | (THead _ t3 _) \Rightarrow t3])) (THead (Flat Appl) -x x1) (THead (Flat Appl) x2 x5) H30) in ((let H32 \def (f_equal T T (\lambda -(e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x1 -| (TLRef _) \Rightarrow x1 | (THead _ _ t3) \Rightarrow t3])) (THead (Flat -Appl) x x1) (THead (Flat Appl) x2 x5) H30) in (\lambda (H33: (eq T x -x2)).(let H34 \def (eq_ind_r T x5 (\lambda (t3: T).(pr2 c x1 t3)) H25 x1 H32) -in (eq_ind T x1 (\lambda (t3: T).(sn3 c (THead (Flat Appl) x2 (THead (Flat -Cast) x4 t3)))) (let H35 \def (eq_ind_r T x2 (\lambda (t3: T).((eq T (THead -(Flat Appl) x x0) (THead (Flat Appl) t3 x4)) \to (\forall (P: Prop).P))) H28 -x H33) in (let H36 \def (eq_ind_r T x2 (\lambda (t3: T).(pr2 c x t3)) H18 x -H33) in (eq_ind T x (\lambda (t3: T).(sn3 c (THead (Flat Appl) t3 (THead -(Flat Cast) x4 x1)))) (H11 (THead (Flat Appl) x x4) H35 (pr3_pr2 c (THead -(Flat Appl) x x0) (THead (Flat Appl) x x4) (pr2_thin_dx c x0 x4 H24 x Appl)) -x x4 (refl_equal T (THead (Flat Appl) x x4)) x1 (sn3_sing c (THead (Flat -Appl) x x1) H10)) x2 H33))) x5 H32)))) H31))) (\lambda (H30: (((eq T (THead -(Flat Appl) x x1) (THead (Flat Appl) x2 x5)) \to (\forall (P: -Prop).P)))).(H11 (THead (Flat Appl) x2 x4) H28 (pr3_flat c x x2 (pr3_pr2 c x -x2 H18) x0 x4 (pr3_pr2 c x0 x4 H24) Appl) x2 x4 (refl_equal T (THead (Flat -Appl) x2 x4)) x5 (H10 (THead (Flat Appl) x2 x5) H30 (pr3_flat c x x2 (pr3_pr2 -c x x2 H18) x1 x5 (pr3_pr2 c x1 x5 H25) Appl)))) H29)))) H27))) x3 H23))))))) -H22)) (\lambda (H22: (pr2 c x1 x3)).(let H_x \def (term_dec (THead (Flat -Appl) x x1) (THead (Flat Appl) x2 x3)) in (let H23 \def H_x in (or_ind (eq T -(THead (Flat Appl) x x1) (THead (Flat Appl) x2 x3)) ((eq T (THead (Flat Appl) -x x1) (THead (Flat Appl) x2 x3)) \to (\forall (P: Prop).P)) (sn3 c (THead -(Flat Appl) x2 x3)) (\lambda (H24: (eq T (THead (Flat Appl) x x1) (THead -(Flat Appl) x2 x3))).(let H25 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) -\Rightarrow x | (THead _ t3 _) \Rightarrow t3])) (THead (Flat Appl) x x1) -(THead (Flat Appl) x2 x3) H24) in ((let H26 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x1 | -(TLRef _) \Rightarrow x1 | (THead _ _ t3) \Rightarrow t3])) (THead (Flat -Appl) x x1) (THead (Flat Appl) x2 x3) H24) in (\lambda (H27: (eq T x -x2)).(let H28 \def (eq_ind_r T x3 (\lambda (t3: T).(pr2 c x1 t3)) H22 x1 H26) -in (let H29 \def (eq_ind_r T x3 (\lambda (t3: T).((eq T (THead (Flat Appl) x -(THead (Flat Cast) x0 x1)) (THead (Flat Appl) x2 t3)) \to (\forall (P: -Prop).P))) H20 x1 H26) in (eq_ind T x1 (\lambda (t3: T).(sn3 c (THead (Flat -Appl) x2 t3))) (let H30 \def (eq_ind_r T x2 (\lambda (t3: T).((eq T (THead -(Flat Appl) x (THead (Flat Cast) x0 x1)) (THead (Flat Appl) t3 x1)) \to -(\forall (P: Prop).P))) H29 x H27) in (let H31 \def (eq_ind_r T x2 (\lambda -(t3: T).(pr2 c x t3)) H18 x H27) in (eq_ind T x (\lambda (t3: T).(sn3 c -(THead (Flat Appl) t3 x1))) (sn3_sing c (THead (Flat Appl) x x1) H10) x2 -H27))) x3 H26))))) H25))) (\lambda (H24: (((eq T (THead (Flat Appl) x x1) -(THead (Flat Appl) x2 x3)) \to (\forall (P: Prop).P)))).(H10 (THead (Flat -Appl) x2 x3) H24 (pr3_flat c x x2 (pr3_pr2 c x x2 H18) x1 x3 (pr3_pr2 c x1 x3 -H22) Appl))) H23)))) H21)) t2 H17))))))) H16)) (\lambda (H16: (ex4_4 T T T T -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Flat Cast) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) -u0) z1 t3))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Cast) x0 x1) (THead -(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda -(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: -B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3))))))) (sn3 c t2) -(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda -(H17: (eq T (THead (Flat Cast) x0 x1) (THead (Bind Abst) x2 x3))).(\lambda -(H18: (eq T t2 (THead (Bind Abbr) x4 x5))).(\lambda (_: (pr2 c x -x4)).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) -u0) x3 x5))))).(let H21 \def (eq_ind T t2 (\lambda (t3: T).((eq T (THead -(Flat Appl) x (THead (Flat Cast) x0 x1)) t3) \to (\forall (P: Prop).P))) H13 -(THead (Bind Abbr) x4 x5) H18) in (eq_ind_r T (THead (Bind Abbr) x4 x5) -(\lambda (t3: T).(sn3 c t3)) (let H22 \def (eq_ind T (THead (Flat Cast) x0 -x1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort -_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) -\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) x2 -x3) H17) in (False_ind (sn3 c (THead (Bind Abbr) x4 x5)) H22)) t2 -H18)))))))))) H16)) (\lambda (H16: (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat -Cast) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T -t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 -y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 -z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Cast) x0 x1) (THead -(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) (sn3 c t2) -(\lambda (x2: B).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda -(x6: T).(\lambda (x7: T).(\lambda (_: (not (eq B x2 Abst))).(\lambda (H18: -(eq T (THead (Flat Cast) x0 x1) (THead (Bind x2) x3 x4))).(\lambda (H19: (eq -T t2 (THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) O x6) x5)))).(\lambda -(_: (pr2 c x x6)).(\lambda (_: (pr2 c x3 x7)).(\lambda (_: (pr2 (CHead c -(Bind x2) x7) x4 x5)).(let H23 \def (eq_ind T t2 (\lambda (t3: T).((eq T -(THead (Flat Appl) x (THead (Flat Cast) x0 x1)) t3) \to (\forall (P: -Prop).P))) H13 (THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) O x6) x5)) -H19) in (eq_ind_r T (THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) O x6) -x5)) (\lambda (t3: T).(sn3 c t3)) (let H24 \def (eq_ind T (THead (Flat Cast) -x0 x1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) -\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind x2) x3 x4) -H18) in (False_ind (sn3 c (THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) -O x6) x5))) H24)) t2 H19)))))))))))))) H16)) H15))))))))))))))) y0 H5)))) -H4))))))))) y H0))))) H)))). -(* COMMENTS -Initial nodes: 5149 -END *) - -theorem sn3_appl_bind: - \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (u: -T).((sn3 c u) \to (\forall (t: T).(\forall (v: T).((sn3 (CHead c (Bind b) u) -(THead (Flat Appl) (lift (S O) O v) t)) \to (sn3 c (THead (Flat Appl) v -(THead (Bind b) u t)))))))))) -\def - \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (c: C).(\lambda -(u: T).(\lambda (H0: (sn3 c u)).(sn3_ind c (\lambda (t: T).(\forall (t0: -T).(\forall (v: T).((sn3 (CHead c (Bind b) t) (THead (Flat Appl) (lift (S O) -O v) t0)) \to (sn3 c (THead (Flat Appl) v (THead (Bind b) t t0))))))) -(\lambda (t1: T).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall -(P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H2: ((\forall -(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to -(\forall (t: T).(\forall (v: T).((sn3 (CHead c (Bind b) t2) (THead (Flat -Appl) (lift (S O) O v) t)) \to (sn3 c (THead (Flat Appl) v (THead (Bind b) t2 -t))))))))))).(\lambda (t: T).(\lambda (v: T).(\lambda (H3: (sn3 (CHead c -(Bind b) t1) (THead (Flat Appl) (lift (S O) O v) t))).(insert_eq T (THead -(Flat Appl) (lift (S O) O v) t) (\lambda (t0: T).(sn3 (CHead c (Bind b) t1) -t0)) (\lambda (_: T).(sn3 c (THead (Flat Appl) v (THead (Bind b) t1 t)))) -(\lambda (y: T).(\lambda (H4: (sn3 (CHead c (Bind b) t1) y)).(unintro T t -(\lambda (t0: T).((eq T y (THead (Flat Appl) (lift (S O) O v) t0)) \to (sn3 c -(THead (Flat Appl) v (THead (Bind b) t1 t0))))) (unintro T v (\lambda (t0: -T).(\forall (x: T).((eq T y (THead (Flat Appl) (lift (S O) O t0) x)) \to (sn3 -c (THead (Flat Appl) t0 (THead (Bind b) t1 x)))))) (sn3_ind (CHead c (Bind b) -t1) (\lambda (t0: T).(\forall (x: T).(\forall (x0: T).((eq T t0 (THead (Flat -Appl) (lift (S O) O x) x0)) \to (sn3 c (THead (Flat Appl) x (THead (Bind b) -t1 x0))))))) (\lambda (t2: T).(\lambda (H5: ((\forall (t3: T).((((eq T t2 t3) -\to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to (sn3 -(CHead c (Bind b) t1) t3)))))).(\lambda (H6: ((\forall (t3: T).((((eq T t2 -t3) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to -(\forall (x: T).(\forall (x0: T).((eq T t3 (THead (Flat Appl) (lift (S O) O -x) x0)) \to (sn3 c (THead (Flat Appl) x (THead (Bind b) t1 -x0))))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H7: (eq T t2 (THead -(Flat Appl) (lift (S O) O x) x0))).(let H8 \def (eq_ind T t2 (\lambda (t0: -T).(\forall (t3: T).((((eq T t0 t3) \to (\forall (P: Prop).P))) \to ((pr3 -(CHead c (Bind b) t1) t0 t3) \to (\forall (x1: T).(\forall (x2: T).((eq T t3 -(THead (Flat Appl) (lift (S O) O x1) x2)) \to (sn3 c (THead (Flat Appl) x1 -(THead (Bind b) t1 x2)))))))))) H6 (THead (Flat Appl) (lift (S O) O x) x0) -H7) in (let H9 \def (eq_ind T t2 (\lambda (t0: T).(\forall (t3: T).((((eq T -t0 t3) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t0 t3) \to -(sn3 (CHead c (Bind b) t1) t3))))) H5 (THead (Flat Appl) (lift (S O) O x) x0) -H7) in (sn3_pr2_intro c (THead (Flat Appl) x (THead (Bind b) t1 x0)) (\lambda -(t3: T).(\lambda (H10: (((eq T (THead (Flat Appl) x (THead (Bind b) t1 x0)) -t3) \to (\forall (P: Prop).P)))).(\lambda (H11: (pr2 c (THead (Flat Appl) x -(THead (Bind b) t1 x0)) t3)).(let H12 \def (pr2_gen_appl c x (THead (Bind b) -t1 x0) t3 H11) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T -t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x -u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind b) t1 x0) t4)))) -(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(eq T (THead (Bind b) t1 x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind -Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t4: T).(\forall (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind b0) -u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B -b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) t1 x0) (THead -(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind -b0) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) -(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b0) y2) z1 z2)))))))) (sn3 c t3) -(\lambda (H13: (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead -(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) -(\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind b) t1 x0) -t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead -(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) -(\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind b) t1 x0) t4))) (sn3 c -t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (H14: (eq T t3 (THead (Flat -Appl) x1 x2))).(\lambda (H15: (pr2 c x x1)).(\lambda (H16: (pr2 c (THead -(Bind b) t1 x0) x2)).(let H17 \def (eq_ind T t3 (\lambda (t0: T).((eq T -(THead (Flat Appl) x (THead (Bind b) t1 x0)) t0) \to (\forall (P: Prop).P))) -H10 (THead (Flat Appl) x1 x2) H14) in (eq_ind_r T (THead (Flat Appl) x1 x2) -(\lambda (t0: T).(sn3 c t0)) (let H_x \def (pr3_gen_bind b H c t1 x0 x2) in -(let H18 \def (H_x (pr3_pr2 c (THead (Bind b) t1 x0) x2 H16)) in (or_ind -(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead (Bind b) u2 -t4)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_: -T).(\lambda (t4: T).(pr3 (CHead c (Bind b) t1) x0 t4)))) (pr3 (CHead c (Bind -b) t1) x0 (lift (S O) O x2)) (sn3 c (THead (Flat Appl) x1 x2)) (\lambda (H19: -(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead (Bind b) u2 -t4)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_: -T).(\lambda (t4: T).(pr3 (CHead c (Bind b) t1) x0 t4))))).(ex3_2_ind T T -(\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead (Bind b) u2 t4)))) (\lambda -(u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr3 -(CHead c (Bind b) t1) x0 t4))) (sn3 c (THead (Flat Appl) x1 x2)) (\lambda -(x3: T).(\lambda (x4: T).(\lambda (H20: (eq T x2 (THead (Bind b) x3 -x4))).(\lambda (H21: (pr3 c t1 x3)).(\lambda (H22: (pr3 (CHead c (Bind b) t1) -x0 x4)).(let H23 \def (eq_ind T x2 (\lambda (t0: T).((eq T (THead (Flat Appl) -x (THead (Bind b) t1 x0)) (THead (Flat Appl) x1 t0)) \to (\forall (P: -Prop).P))) H17 (THead (Bind b) x3 x4) H20) in (eq_ind_r T (THead (Bind b) x3 -x4) (\lambda (t0: T).(sn3 c (THead (Flat Appl) x1 t0))) (let H_x0 \def -(term_dec t1 x3) in (let H24 \def H_x0 in (or_ind (eq T t1 x3) ((eq T t1 x3) -\to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Bind b) x3 -x4))) (\lambda (H25: (eq T t1 x3)).(let H26 \def (eq_ind_r T x3 (\lambda (t0: -T).((eq T (THead (Flat Appl) x (THead (Bind b) t1 x0)) (THead (Flat Appl) x1 -(THead (Bind b) t0 x4))) \to (\forall (P: Prop).P))) H23 t1 H25) in (let H27 -\def (eq_ind_r T x3 (\lambda (t0: T).(pr3 c t1 t0)) H21 t1 H25) in (eq_ind T -t1 (\lambda (t0: T).(sn3 c (THead (Flat Appl) x1 (THead (Bind b) t0 x4)))) -(let H_x1 \def (term_dec x0 x4) in (let H28 \def H_x1 in (or_ind (eq T x0 x4) -((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead -(Bind b) t1 x4))) (\lambda (H29: (eq T x0 x4)).(let H30 \def (eq_ind_r T x4 -(\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind b) t1 x0)) (THead -(Flat Appl) x1 (THead (Bind b) t1 t0))) \to (\forall (P: Prop).P))) H26 x0 -H29) in (let H31 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 (CHead c (Bind b) -t1) x0 t0)) H22 x0 H29) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Flat -Appl) x1 (THead (Bind b) t1 t0)))) (let H_x2 \def (term_dec x x1) in (let H32 -\def H_x2 in (or_ind (eq T x x1) ((eq T x x1) \to (\forall (P: Prop).P)) (sn3 -c (THead (Flat Appl) x1 (THead (Bind b) t1 x0))) (\lambda (H33: (eq T x -x1)).(let H34 \def (eq_ind_r T x1 (\lambda (t0: T).((eq T (THead (Flat Appl) -x (THead (Bind b) t1 x0)) (THead (Flat Appl) t0 (THead (Bind b) t1 x0))) \to -(\forall (P: Prop).P))) H30 x H33) in (let H35 \def (eq_ind_r T x1 (\lambda -(t0: T).(pr2 c x t0)) H15 x H33) in (eq_ind T x (\lambda (t0: T).(sn3 c -(THead (Flat Appl) t0 (THead (Bind b) t1 x0)))) (H34 (refl_equal T (THead -(Flat Appl) x (THead (Bind b) t1 x0))) (sn3 c (THead (Flat Appl) x (THead -(Bind b) t1 x0)))) x1 H33)))) (\lambda (H33: (((eq T x x1) \to (\forall (P: -Prop).P)))).(H8 (THead (Flat Appl) (lift (S O) O x1) x0) (\lambda (H34: (eq T -(THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) -x0))).(\lambda (P: Prop).(let H35 \def (f_equal T T (\lambda (e: T).(match e -in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map -(f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n) -\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with -[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4) -\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in -lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow -((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match -t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef -(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | -(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) -t4))]) in lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _) -\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) -(lift (S O) O x1) x0) H34) in (let H36 \def (eq_ind_r T x1 (\lambda (t0: -T).((eq T x t0) \to (\forall (P0: Prop).P0))) H33 x (lift_inj x x1 (S O) O -H35)) in (let H37 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x -(lift_inj x x1 (S O) O H35)) in (H36 (refl_equal T x) P)))))) (pr3_flat -(CHead c (Bind b) t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c -(Bind b) t1) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1 -(pr3_pr2 c x x1 H15)) x0 x0 (pr3_refl (CHead c (Bind b) t1) x0) Appl) x1 x0 -(refl_equal T (THead (Flat Appl) (lift (S O) O x1) x0)))) H32))) x4 H29)))) -(\lambda (H29: (((eq T x0 x4) \to (\forall (P: Prop).P)))).(H8 (THead (Flat -Appl) (lift (S O) O x1) x4) (\lambda (H30: (eq T (THead (Flat Appl) (lift (S -O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) x4))).(\lambda (P: -Prop).(let H31 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat -\to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n) -\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with -[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4) -\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in -lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow -((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match -t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef -(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | -(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) -t4))]) in lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _) -\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) -(lift (S O) O x1) x4) H30) in ((let H32 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | -(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat -Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) x4) H30) in -(\lambda (H33: (eq T (lift (S O) O x) (lift (S O) O x1))).(let H34 \def -(eq_ind_r T x4 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) -H29 x0 H32) in (let H35 \def (eq_ind_r T x4 (\lambda (t0: T).((eq T (THead -(Flat Appl) x (THead (Bind b) t1 x0)) (THead (Flat Appl) x1 (THead (Bind b) -t1 t0))) \to (\forall (P0: Prop).P0))) H26 x0 H32) in (let H36 \def (eq_ind_r -T x4 (\lambda (t0: T).(pr3 (CHead c (Bind b) t1) x0 t0)) H22 x0 H32) in (let -H37 \def (eq_ind_r T x1 (\lambda (t0: T).((eq T (THead (Flat Appl) x (THead -(Bind b) t1 x0)) (THead (Flat Appl) t0 (THead (Bind b) t1 x0))) \to (\forall -(P0: Prop).P0))) H35 x (lift_inj x x1 (S O) O H33)) in (let H38 \def -(eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x (lift_inj x x1 (S O) O -H33)) in (H34 (refl_equal T x0) P)))))))) H31)))) (pr3_flat (CHead c (Bind b) -t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c (Bind b) t1) c (S -O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1 (pr3_pr2 c x x1 H15)) -x0 x4 H22 Appl) x1 x4 (refl_equal T (THead (Flat Appl) (lift (S O) O x1) -x4)))) H28))) x3 H25)))) (\lambda (H25: (((eq T t1 x3) \to (\forall (P: -Prop).P)))).(H2 x3 H25 H21 x4 x1 (sn3_cpr3_trans c t1 x3 H21 (Bind b) (THead -(Flat Appl) (lift (S O) O x1) x4) (let H_x1 \def (term_dec x0 x4) in (let H26 -\def H_x1 in (or_ind (eq T x0 x4) ((eq T x0 x4) \to (\forall (P: Prop).P)) -(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x1) x4)) (\lambda -(H27: (eq T x0 x4)).(let H28 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 (CHead -c (Bind b) t1) x0 t0)) H22 x0 H27) in (eq_ind T x0 (\lambda (t0: T).(sn3 -(CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x1) t0))) (let H_x2 -\def (term_dec x x1) in (let H29 \def H_x2 in (or_ind (eq T x x1) ((eq T x -x1) \to (\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat Appl) -(lift (S O) O x1) x0)) (\lambda (H30: (eq T x x1)).(let H31 \def (eq_ind_r T -x1 (\lambda (t0: T).(pr2 c x t0)) H15 x H30) in (eq_ind T x (\lambda (t0: -T).(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O t0) x0))) -(sn3_sing (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x) x0) H9) -x1 H30))) (\lambda (H30: (((eq T x x1) \to (\forall (P: Prop).P)))).(H9 -(THead (Flat Appl) (lift (S O) O x1) x0) (\lambda (H31: (eq T (THead (Flat -Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) -x0))).(\lambda (P: Prop).(let H32 \def (f_equal T T (\lambda (e: T).(match e -in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map -(f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n) -\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with -[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4) -\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in -lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow -((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match -t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef -(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | -(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) -t4))]) in lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _) -\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) -(lift (S O) O x1) x0) H31) in (let H33 \def (eq_ind_r T x1 (\lambda (t0: -T).((eq T x t0) \to (\forall (P0: Prop).P0))) H30 x (lift_inj x x1 (S O) O -H32)) in (let H34 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x -(lift_inj x x1 (S O) O H32)) in (H33 (refl_equal T x) P)))))) (pr3_flat -(CHead c (Bind b) t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c -(Bind b) t1) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1 -(pr3_pr2 c x x1 H15)) x0 x0 (pr3_refl (CHead c (Bind b) t1) x0) Appl))) -H29))) x4 H27))) (\lambda (H27: (((eq T x0 x4) \to (\forall (P: -Prop).P)))).(H9 (THead (Flat Appl) (lift (S O) O x1) x4) (\lambda (H28: (eq T -(THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) -x4))).(\lambda (P: Prop).(let H29 \def (f_equal T T (\lambda (e: T).(match e -in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map -(f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n) -\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with -[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4) -\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in -lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow -((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match -t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef -(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | -(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) -t4))]) in lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _) -\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) -(lift (S O) O x1) x4) H28) in ((let H30 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | -(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat -Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) x4) H28) in -(\lambda (H31: (eq T (lift (S O) O x) (lift (S O) O x1))).(let H32 \def -(eq_ind_r T x4 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) -H27 x0 H30) in (let H33 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 (CHead c -(Bind b) t1) x0 t0)) H22 x0 H30) in (let H34 \def (eq_ind_r T x1 (\lambda -(t0: T).(pr2 c x t0)) H15 x (lift_inj x x1 (S O) O H31)) in (H32 (refl_equal -T x0) P)))))) H29)))) (pr3_flat (CHead c (Bind b) t1) (lift (S O) O x) (lift -(S O) O x1) (pr3_lift (CHead c (Bind b) t1) c (S O) O (drop_drop (Bind b) O c -c (drop_refl c) t1) x x1 (pr3_pr2 c x x1 H15)) x0 x4 H22 Appl))) H26)))))) -H24))) x2 H20))))))) H19)) (\lambda (H19: (pr3 (CHead c (Bind b) t1) x0 (lift -(S O) O x2))).(sn3_gen_lift (CHead c (Bind b) t1) (THead (Flat Appl) x1 x2) -(S O) O (eq_ind_r T (THead (Flat Appl) (lift (S O) O x1) (lift (S O) (s (Flat -Appl) O) x2)) (\lambda (t0: T).(sn3 (CHead c (Bind b) t1) t0)) (sn3_pr3_trans -(CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x1) x0) (let H_x0 \def -(term_dec x x1) in (let H20 \def H_x0 in (or_ind (eq T x x1) ((eq T x x1) \to -(\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S -O) O x1) x0)) (\lambda (H21: (eq T x x1)).(let H22 \def (eq_ind_r T x1 -(\lambda (t0: T).(pr2 c x t0)) H15 x H21) in (eq_ind T x (\lambda (t0: -T).(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O t0) x0))) -(sn3_sing (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x) x0) H9) -x1 H21))) (\lambda (H21: (((eq T x x1) \to (\forall (P: Prop).P)))).(H9 -(THead (Flat Appl) (lift (S O) O x1) x0) (\lambda (H22: (eq T (THead (Flat -Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) -x0))).(\lambda (P: Prop).(let H23 \def (f_equal T T (\lambda (e: T).(match e -in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map -(f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n) -\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with -[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4) -\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in -lref_map) (\lambda (x3: nat).(plus x3 (S O))) O x) | (TLRef _) \Rightarrow -((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match -t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef -(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | -(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) -t4))]) in lref_map) (\lambda (x3: nat).(plus x3 (S O))) O x) | (THead _ t0 _) -\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) -(lift (S O) O x1) x0) H22) in (let H24 \def (eq_ind_r T x1 (\lambda (t0: -T).((eq T x t0) \to (\forall (P0: Prop).P0))) H21 x (lift_inj x x1 (S O) O -H23)) in (let H25 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x -(lift_inj x x1 (S O) O H23)) in (H24 (refl_equal T x) P)))))) (pr3_flat -(CHead c (Bind b) t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c -(Bind b) t1) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1 -(pr3_pr2 c x x1 H15)) x0 x0 (pr3_refl (CHead c (Bind b) t1) x0) Appl))) -H20))) (THead (Flat Appl) (lift (S O) O x1) (lift (S O) O x2)) (pr3_thin_dx -(CHead c (Bind b) t1) x0 (lift (S O) O x2) H19 (lift (S O) O x1) Appl)) (lift -(S O) O (THead (Flat Appl) x1 x2)) (lift_head (Flat Appl) x1 x2 (S O) O)) c -(drop_drop (Bind b) O c c (drop_refl c) t1))) H18))) t3 H14))))))) H13)) -(\lambda (H13: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T (THead (Bind b) t1 x0) (THead (Bind Abst) y1 -z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: -T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b0: B).(\forall (u0: -T).(pr2 (CHead c (Bind b0) u0) z1 t4))))))))).(ex4_4_ind T T T T (\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind -b) t1 x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x -u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: -T).(\forall (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind b0) u0) z1 t4))))))) -(sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: -T).(\lambda (H14: (eq T (THead (Bind b) t1 x0) (THead (Bind Abst) x1 -x2))).(\lambda (H15: (eq T t3 (THead (Bind Abbr) x3 x4))).(\lambda (_: (pr2 c -x x3)).(\lambda (H17: ((\forall (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind -b0) u0) x2 x4))))).(let H18 \def (eq_ind T t3 (\lambda (t0: T).((eq T (THead -(Flat Appl) x (THead (Bind b) t1 x0)) t0) \to (\forall (P: Prop).P))) H10 -(THead (Bind Abbr) x3 x4) H15) in (eq_ind_r T (THead (Bind Abbr) x3 x4) -(\lambda (t0: T).(sn3 c t0)) (let H19 \def (f_equal T B (\lambda (e: -T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | -(TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k in K return -(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow -b])])) (THead (Bind b) t1 x0) (THead (Bind Abst) x1 x2) H14) in ((let H20 -\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ t0 _) -\Rightarrow t0])) (THead (Bind b) t1 x0) (THead (Bind Abst) x1 x2) H14) in -((let H21 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: -T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ -t0) \Rightarrow t0])) (THead (Bind b) t1 x0) (THead (Bind Abst) x1 x2) H14) -in (\lambda (_: (eq T t1 x1)).(\lambda (H23: (eq B b Abst)).(let H24 \def -(eq_ind_r T x2 (\lambda (t0: T).(\forall (b0: B).(\forall (u0: T).(pr2 (CHead -c (Bind b0) u0) t0 x4)))) H17 x0 H21) in (let H25 \def (eq_ind B b (\lambda -(b0: B).((eq T (THead (Flat Appl) x (THead (Bind b0) t1 x0)) (THead (Bind -Abbr) x3 x4)) \to (\forall (P: Prop).P))) H18 Abst H23) in (let H26 \def -(eq_ind B b (\lambda (b0: B).(\forall (t4: T).((((eq T (THead (Flat Appl) -(lift (S O) O x) x0) t4) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind -b0) t1) (THead (Flat Appl) (lift (S O) O x) x0) t4) \to (sn3 (CHead c (Bind -b0) t1) t4))))) H9 Abst H23) in (let H27 \def (eq_ind B b (\lambda (b0: -B).(\forall (t4: T).((((eq T (THead (Flat Appl) (lift (S O) O x) x0) t4) \to -(\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b0) t1) (THead (Flat Appl) -(lift (S O) O x) x0) t4) \to (\forall (x5: T).(\forall (x6: T).((eq T t4 -(THead (Flat Appl) (lift (S O) O x5) x6)) \to (sn3 c (THead (Flat Appl) x5 -(THead (Bind b0) t1 x6)))))))))) H8 Abst H23) in (let H28 \def (eq_ind B b -(\lambda (b0: B).(\forall (t4: T).((((eq T t1 t4) \to (\forall (P: Prop).P))) -\to ((pr3 c t1 t4) \to (\forall (t0: T).(\forall (v0: T).((sn3 (CHead c (Bind -b0) t4) (THead (Flat Appl) (lift (S O) O v0) t0)) \to (sn3 c (THead (Flat -Appl) v0 (THead (Bind b0) t4 t0)))))))))) H2 Abst H23) in (let H29 \def -(eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H Abst H23) in (let H30 -\def (match (H29 (refl_equal B Abst)) in False return (\lambda (_: -False).(sn3 c (THead (Bind Abbr) x3 x4))) with []) in H30)))))))))) H20)) -H19)) t3 H15)))))))))) H13)) (\lambda (H13: (ex6_6 B T T T T T (\lambda (b0: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) -t1 x0) (THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T -t3 (THead (Bind b0) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 -y2))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b0) y2) z1 -z2))))))))).(ex6_6_ind B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 -Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) t1 x0) (THead (Bind -b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind b0) y2 (THead -(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x -u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b0: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c (Bind b0) y2) z1 z2))))))) (sn3 c t3) (\lambda (x1: -B).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: -T).(\lambda (x6: T).(\lambda (_: (not (eq B x1 Abst))).(\lambda (H15: (eq T -(THead (Bind b) t1 x0) (THead (Bind x1) x2 x3))).(\lambda (H16: (eq T t3 -(THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))).(\lambda -(H17: (pr2 c x x5)).(\lambda (H18: (pr2 c x2 x6)).(\lambda (H19: (pr2 (CHead -c (Bind x1) x6) x3 x4)).(let H20 \def (eq_ind T t3 (\lambda (t0: T).((eq T -(THead (Flat Appl) x (THead (Bind b) t1 x0)) t0) \to (\forall (P: Prop).P))) -H10 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)) H16) in -(eq_ind_r T (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)) -(\lambda (t0: T).(sn3 c t0)) (let H21 \def (f_equal T B (\lambda (e: -T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | -(TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k in K return -(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow -b])])) (THead (Bind b) t1 x0) (THead (Bind x1) x2 x3) H15) in ((let H22 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ t0 _) -\Rightarrow t0])) (THead (Bind b) t1 x0) (THead (Bind x1) x2 x3) H15) in -((let H23 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: -T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ -t0) \Rightarrow t0])) (THead (Bind b) t1 x0) (THead (Bind x1) x2 x3) H15) in -(\lambda (H24: (eq T t1 x2)).(\lambda (H25: (eq B b x1)).(let H26 \def -(eq_ind_r T x3 (\lambda (t0: T).(pr2 (CHead c (Bind x1) x6) t0 x4)) H19 x0 -H23) in (let H27 \def (eq_ind_r T x2 (\lambda (t0: T).(pr2 c t0 x6)) H18 t1 -H24) in (let H28 \def (eq_ind_r B x1 (\lambda (b0: B).(pr2 (CHead c (Bind b0) -x6) x0 x4)) H26 b H25) in (eq_ind B b (\lambda (b0: B).(sn3 c (THead (Bind -b0) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))) (sn3_pr3_trans c (THead -(Bind b) t1 (THead (Flat Appl) (lift (S O) O x5) x4)) (sn3_bind b c t1 -(sn3_sing c t1 H1) (THead (Flat Appl) (lift (S O) O x5) x4) (let H_x \def -(term_dec x x5) in (let H29 \def H_x in (or_ind (eq T x x5) ((eq T x x5) \to -(\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S -O) O x5) x4)) (\lambda (H30: (eq T x x5)).(let H31 \def (eq_ind_r T x5 -(\lambda (t0: T).(pr2 c x t0)) H17 x H30) in (eq_ind T x (\lambda (t0: -T).(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O t0) x4))) (let -H_x0 \def (term_dec x0 x4) in (let H32 \def H_x0 in (or_ind (eq T x0 x4) ((eq -T x0 x4) \to (\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat -Appl) (lift (S O) O x) x4)) (\lambda (H33: (eq T x0 x4)).(let H34 \def -(eq_ind_r T x4 (\lambda (t0: T).(pr2 (CHead c (Bind b) x6) x0 t0)) H28 x0 -H33) in (eq_ind T x0 (\lambda (t0: T).(sn3 (CHead c (Bind b) t1) (THead (Flat -Appl) (lift (S O) O x) t0))) (sn3_sing (CHead c (Bind b) t1) (THead (Flat -Appl) (lift (S O) O x) x0) H9) x4 H33))) (\lambda (H33: (((eq T x0 x4) \to -(\forall (P: Prop).P)))).(H9 (THead (Flat Appl) (lift (S O) O x) x4) (\lambda -(H34: (eq T (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift -(S O) O x) x4))).(\lambda (P: Prop).(let H35 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | -(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat -Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x) x4) H34) in -(let H36 \def (eq_ind_r T x4 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: -Prop).P0))) H33 x0 H35) in (let H37 \def (eq_ind_r T x4 (\lambda (t0: T).(pr2 -(CHead c (Bind b) x6) x0 t0)) H28 x0 H35) in (H36 (refl_equal T x0) P)))))) -(pr3_pr3_pr3_t c t1 x6 (pr3_pr2 c t1 x6 H27) (THead (Flat Appl) (lift (S O) O -x) x0) (THead (Flat Appl) (lift (S O) O x) x4) (Bind b) (pr3_pr2 (CHead c -(Bind b) x6) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift -(S O) O x) x4) (pr2_thin_dx (CHead c (Bind b) x6) x0 x4 H28 (lift (S O) O x) -Appl))))) H32))) x5 H30))) (\lambda (H30: (((eq T x x5) \to (\forall (P: -Prop).P)))).(H9 (THead (Flat Appl) (lift (S O) O x5) x4) (\lambda (H31: (eq T -(THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x5) -x4))).(\lambda (P: Prop).(let H32 \def (f_equal T T (\lambda (e: T).(match e -in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map -(f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n) -\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with -[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4) -\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in -lref_map) (\lambda (x7: nat).(plus x7 (S O))) O x) | (TLRef _) \Rightarrow -((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match -t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef -(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | -(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) -t4))]) in lref_map) (\lambda (x7: nat).(plus x7 (S O))) O x) | (THead _ t0 _) -\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) -(lift (S O) O x5) x4) H31) in ((let H33 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | -(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat -Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x5) x4) H31) in -(\lambda (H34: (eq T (lift (S O) O x) (lift (S O) O x5))).(let H35 \def -(eq_ind_r T x5 (\lambda (t0: T).((eq T x t0) \to (\forall (P0: Prop).P0))) -H30 x (lift_inj x x5 (S O) O H34)) in (let H36 \def (eq_ind_r T x5 (\lambda -(t0: T).(pr2 c x t0)) H17 x (lift_inj x x5 (S O) O H34)) in (let H37 \def -(eq_ind_r T x4 (\lambda (t0: T).(pr2 (CHead c (Bind b) x6) x0 t0)) H28 x0 -H33) in (H35 (refl_equal T x) P)))))) H32)))) (pr3_pr3_pr3_t c t1 x6 (pr3_pr2 -c t1 x6 H27) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift -(S O) O x5) x4) (Bind b) (pr3_flat (CHead c (Bind b) x6) (lift (S O) O x) -(lift (S O) O x5) (pr3_lift (CHead c (Bind b) x6) c (S O) O (drop_drop (Bind -b) O c c (drop_refl c) x6) x x5 (pr3_pr2 c x x5 H17)) x0 x4 (pr3_pr2 (CHead c -(Bind b) x6) x0 x4 H28) Appl)))) H29)))) (THead (Bind b) x6 (THead (Flat -Appl) (lift (S O) O x5) x4)) (pr3_pr2 c (THead (Bind b) t1 (THead (Flat Appl) -(lift (S O) O x5) x4)) (THead (Bind b) x6 (THead (Flat Appl) (lift (S O) O -x5) x4)) (pr2_head_1 c t1 x6 H27 (Bind b) (THead (Flat Appl) (lift (S O) O -x5) x4)))) x1 H25))))))) H22)) H21)) t3 H16)))))))))))))) H13)) -H12)))))))))))))) y H4))))) H3))))))) u H0))))). -(* COMMENTS -Initial nodes: 9191 -END *) - -theorem sn3_appl_appl: - \forall (v1: T).(\forall (t1: T).(let u1 \def (THead (Flat Appl) v1 t1) in -(\forall (c: C).((sn3 c u1) \to (\forall (v2: T).((sn3 c v2) \to (((\forall -(u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to -(sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c (THead (Flat Appl) v2 -u1))))))))) -\def - \lambda (v1: T).(\lambda (t1: T).(let u1 \def (THead (Flat Appl) v1 t1) in -(\lambda (c: C).(\lambda (H: (sn3 c (THead (Flat Appl) v1 t1))).(insert_eq T -(THead (Flat Appl) v1 t1) (\lambda (t: T).(sn3 c t)) (\lambda (t: T).(\forall -(v2: T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c t u2) \to ((((iso t u2) -\to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to -(sn3 c (THead (Flat Appl) v2 t)))))) (\lambda (y: T).(\lambda (H0: (sn3 c -y)).(unintro T t1 (\lambda (t: T).((eq T y (THead (Flat Appl) v1 t)) \to -(\forall (v2: T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c y u2) \to ((((iso -y u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) -\to (sn3 c (THead (Flat Appl) v2 y))))))) (unintro T v1 (\lambda (t: -T).(\forall (x: T).((eq T y (THead (Flat Appl) t x)) \to (\forall (v2: -T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c y u2) \to ((((iso y u2) \to -(\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c -(THead (Flat Appl) v2 y)))))))) (sn3_ind c (\lambda (t: T).(\forall (x: -T).(\forall (x0: T).((eq T t (THead (Flat Appl) x x0)) \to (\forall (v2: -T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c t u2) \to ((((iso t u2) \to -(\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c -(THead (Flat Appl) v2 t))))))))) (\lambda (t2: T).(\lambda (H1: ((\forall -(t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to -(sn3 c t3)))))).(\lambda (H2: ((\forall (t3: T).((((eq T t2 t3) \to (\forall -(P: Prop).P))) \to ((pr3 c t2 t3) \to (\forall (x: T).(\forall (x0: T).((eq T -t3 (THead (Flat Appl) x x0)) \to (\forall (v2: T).((sn3 c v2) \to (((\forall -(u2: T).((pr3 c t3 u2) \to ((((iso t3 u2) \to (\forall (P: Prop).P))) \to -(sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c (THead (Flat Appl) v2 -t3))))))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: (eq T t2 -(THead (Flat Appl) x x0))).(\lambda (v2: T).(\lambda (H4: (sn3 c -v2)).(sn3_ind c (\lambda (t: T).(((\forall (u2: T).((pr3 c t2 u2) \to ((((iso -t2 u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) t u2)))))) -\to (sn3 c (THead (Flat Appl) t t2)))) (\lambda (t0: T).(\lambda (H5: -((\forall (t3: T).((((eq T t0 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t0 -t3) \to (sn3 c t3)))))).(\lambda (H6: ((\forall (t3: T).((((eq T t0 t3) \to -(\forall (P: Prop).P))) \to ((pr3 c t0 t3) \to (((\forall (u2: T).((pr3 c t2 -u2) \to ((((iso t2 u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat -Appl) t3 u2)))))) \to (sn3 c (THead (Flat Appl) t3 t2)))))))).(\lambda (H7: -((\forall (u2: T).((pr3 c t2 u2) \to ((((iso t2 u2) \to (\forall (P: -Prop).P))) \to (sn3 c (THead (Flat Appl) t0 u2))))))).(let H8 \def (eq_ind T -t2 (\lambda (t: T).(\forall (u2: T).((pr3 c t u2) \to ((((iso t u2) \to -(\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) t0 u2)))))) H7 (THead -(Flat Appl) x x0) H3) in (let H9 \def (eq_ind T t2 (\lambda (t: T).(\forall -(t3: T).((((eq T t0 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t3) \to -(((\forall (u2: T).((pr3 c t u2) \to ((((iso t u2) \to (\forall (P: -Prop).P))) \to (sn3 c (THead (Flat Appl) t3 u2)))))) \to (sn3 c (THead (Flat -Appl) t3 t))))))) H6 (THead (Flat Appl) x x0) H3) in (let H10 \def (eq_ind T -t2 (\lambda (t: T).(\forall (t3: T).((((eq T t t3) \to (\forall (P: -Prop).P))) \to ((pr3 c t t3) \to (\forall (x1: T).(\forall (x2: T).((eq T t3 -(THead (Flat Appl) x1 x2)) \to (\forall (v3: T).((sn3 c v3) \to (((\forall -(u2: T).((pr3 c t3 u2) \to ((((iso t3 u2) \to (\forall (P: Prop).P))) \to -(sn3 c (THead (Flat Appl) v3 u2)))))) \to (sn3 c (THead (Flat Appl) v3 -t3)))))))))))) H2 (THead (Flat Appl) x x0) H3) in (let H11 \def (eq_ind T t2 -(\lambda (t: T).(\forall (t3: T).((((eq T t t3) \to (\forall (P: Prop).P))) -\to ((pr3 c t t3) \to (sn3 c t3))))) H1 (THead (Flat Appl) x x0) H3) in -(eq_ind_r T (THead (Flat Appl) x x0) (\lambda (t: T).(sn3 c (THead (Flat -Appl) t0 t))) (sn3_pr2_intro c (THead (Flat Appl) t0 (THead (Flat Appl) x -x0)) (\lambda (t3: T).(\lambda (H12: (((eq T (THead (Flat Appl) t0 (THead -(Flat Appl) x x0)) t3) \to (\forall (P: Prop).P)))).(\lambda (H13: (pr2 c -(THead (Flat Appl) t0 (THead (Flat Appl) x x0)) t3)).(let H14 \def -(pr2_gen_appl c t0 (THead (Flat Appl) x x0) t3 H13) in (or3_ind (ex3_2 T T -(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))) (\lambda (_: T).(\lambda -(t4: T).(pr2 c (THead (Flat Appl) x x0) t4)))) (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Appl) -x x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall -(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4)))))))) (ex6_6 B T T T -T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq -T (THead (Flat Appl) x x0) (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) -z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) -y2) z1 z2)))))))) (sn3 c t3) (\lambda (H15: (ex3_2 T T (\lambda (u2: -T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c t0 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c -(THead (Flat Appl) x x0) t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda -(t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c t0 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Flat Appl) -x x0) t4))) (sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (H16: (eq T -t3 (THead (Flat Appl) x1 x2))).(\lambda (H17: (pr2 c t0 x1)).(\lambda (H18: -(pr2 c (THead (Flat Appl) x x0) x2)).(let H19 \def (eq_ind T t3 (\lambda (t: -T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) t) \to (\forall (P: -Prop).P))) H12 (THead (Flat Appl) x1 x2) H16) in (eq_ind_r T (THead (Flat -Appl) x1 x2) (\lambda (t: T).(sn3 c t)) (let H20 \def (pr2_gen_appl c x x0 x2 -H18) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead -(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) -(\lambda (_: T).(\lambda (t4: T).(pr2 c x0 t4)))) (ex4_4 T T T T (\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x0 (THead -(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t4: T).(eq T x2 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda -(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: -B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4)))))))) (ex6_6 B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T x0 -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x2 (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (sn3 c -(THead (Flat Appl) x1 x2)) (\lambda (H21: (ex3_2 T T (\lambda (u2: -T).(\lambda (t4: T).(eq T x2 (THead (Flat Appl) u2 t4)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c x0 -t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead -(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) -(\lambda (_: T).(\lambda (t4: T).(pr2 c x0 t4))) (sn3 c (THead (Flat Appl) x1 -x2)) (\lambda (x3: T).(\lambda (x4: T).(\lambda (H22: (eq T x2 (THead (Flat -Appl) x3 x4))).(\lambda (H23: (pr2 c x x3)).(\lambda (H24: (pr2 c x0 -x4)).(let H25 \def (eq_ind T x2 (\lambda (t: T).((eq T (THead (Flat Appl) t0 -(THead (Flat Appl) x x0)) (THead (Flat Appl) x1 t)) \to (\forall (P: -Prop).P))) H19 (THead (Flat Appl) x3 x4) H22) in (eq_ind_r T (THead (Flat -Appl) x3 x4) (\lambda (t: T).(sn3 c (THead (Flat Appl) x1 t))) (let H_x \def -(term_dec (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4)) in (let H26 -\def H_x in (or_ind (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4)) -((eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4)) \to (\forall (P: -Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Flat Appl) x3 x4))) (\lambda -(H27: (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4))).(let H28 -\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t _) -\Rightarrow t])) (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4) H27) in -((let H29 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: -T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ -t) \Rightarrow t])) (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4) H27) -in (\lambda (H30: (eq T x x3)).(let H31 \def (eq_ind_r T x4 (\lambda (t: -T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) (THead (Flat Appl) -x1 (THead (Flat Appl) x3 t))) \to (\forall (P: Prop).P))) H25 x0 H29) in (let -H32 \def (eq_ind_r T x4 (\lambda (t: T).(pr2 c x0 t)) H24 x0 H29) in (eq_ind -T x0 (\lambda (t: T).(sn3 c (THead (Flat Appl) x1 (THead (Flat Appl) x3 t)))) -(let H33 \def (eq_ind_r T x3 (\lambda (t: T).((eq T (THead (Flat Appl) t0 -(THead (Flat Appl) x x0)) (THead (Flat Appl) x1 (THead (Flat Appl) t x0))) -\to (\forall (P: Prop).P))) H31 x H30) in (let H34 \def (eq_ind_r T x3 -(\lambda (t: T).(pr2 c x t)) H23 x H30) in (eq_ind T x (\lambda (t: T).(sn3 c -(THead (Flat Appl) x1 (THead (Flat Appl) t x0)))) (let H_x0 \def (term_dec t0 -x1) in (let H35 \def H_x0 in (or_ind (eq T t0 x1) ((eq T t0 x1) \to (\forall -(P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Flat Appl) x x0))) -(\lambda (H36: (eq T t0 x1)).(let H37 \def (eq_ind_r T x1 (\lambda (t: -T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) (THead (Flat Appl) -t (THead (Flat Appl) x x0))) \to (\forall (P: Prop).P))) H33 t0 H36) in (let -H38 \def (eq_ind_r T x1 (\lambda (t: T).(pr2 c t0 t)) H17 t0 H36) in (eq_ind -T t0 (\lambda (t: T).(sn3 c (THead (Flat Appl) t (THead (Flat Appl) x x0)))) -(H37 (refl_equal T (THead (Flat Appl) t0 (THead (Flat Appl) x x0))) (sn3 c -(THead (Flat Appl) t0 (THead (Flat Appl) x x0)))) x1 H36)))) (\lambda (H36: -(((eq T t0 x1) \to (\forall (P: Prop).P)))).(H9 x1 H36 (pr3_pr2 c t0 x1 H17) -(\lambda (u2: T).(\lambda (H37: (pr3 c (THead (Flat Appl) x x0) u2)).(\lambda -(H38: (((iso (THead (Flat Appl) x x0) u2) \to (\forall (P: -Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) t0 u2) (H8 u2 H37 H38) (THead -(Flat Appl) x1 u2) (pr3_pr2 c (THead (Flat Appl) t0 u2) (THead (Flat Appl) x1 -u2) (pr2_head_1 c t0 x1 H17 (Flat Appl) u2)))))))) H35))) x3 H30))) x4 -H29))))) H28))) (\lambda (H27: (((eq T (THead (Flat Appl) x x0) (THead (Flat -Appl) x3 x4)) \to (\forall (P: Prop).P)))).(H10 (THead (Flat Appl) x3 x4) H27 -(pr3_flat c x x3 (pr3_pr2 c x x3 H23) x0 x4 (pr3_pr2 c x0 x4 H24) Appl) x3 x4 -(refl_equal T (THead (Flat Appl) x3 x4)) x1 (sn3_pr3_trans c t0 (sn3_sing c -t0 H5) x1 (pr3_pr2 c t0 x1 H17)) (\lambda (u2: T).(\lambda (H28: (pr3 c -(THead (Flat Appl) x3 x4) u2)).(\lambda (H29: (((iso (THead (Flat Appl) x3 -x4) u2) \to (\forall (P: Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) t0 -u2) (H8 u2 (pr3_sing c (THead (Flat Appl) x x4) (THead (Flat Appl) x x0) -(pr2_thin_dx c x0 x4 H24 x Appl) u2 (pr3_sing c (THead (Flat Appl) x3 x4) -(THead (Flat Appl) x x4) (pr2_head_1 c x x3 H23 (Flat Appl) x4) u2 H28)) -(\lambda (H30: (iso (THead (Flat Appl) x x0) u2)).(\lambda (P: Prop).(H29 -(iso_trans (THead (Flat Appl) x3 x4) (THead (Flat Appl) x x0) (iso_head x3 x -x4 x0 (Flat Appl)) u2 H30) P)))) (THead (Flat Appl) x1 u2) (pr3_pr2 c (THead -(Flat Appl) t0 u2) (THead (Flat Appl) x1 u2) (pr2_head_1 c t0 x1 H17 (Flat -Appl) u2)))))))) H26))) x2 H22))))))) H21)) (\lambda (H21: (ex4_4 T T T T -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x0 -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t4: T).(eq T x2 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda -(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: -B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4))))))))).(ex4_4_ind T T T -T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x0 -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t4: T).(eq T x2 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda -(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: -B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4))))))) (sn3 c (THead (Flat -Appl) x1 x2)) (\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda -(x6: T).(\lambda (H22: (eq T x0 (THead (Bind Abst) x3 x4))).(\lambda (H23: -(eq T x2 (THead (Bind Abbr) x5 x6))).(\lambda (H24: (pr2 c x x5)).(\lambda -(H25: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x4 -x6))))).(let H26 \def (eq_ind T x2 (\lambda (t: T).((eq T (THead (Flat Appl) -t0 (THead (Flat Appl) x x0)) (THead (Flat Appl) x1 t)) \to (\forall (P: -Prop).P))) H19 (THead (Bind Abbr) x5 x6) H23) in (eq_ind_r T (THead (Bind -Abbr) x5 x6) (\lambda (t: T).(sn3 c (THead (Flat Appl) x1 t))) (let H27 \def -(eq_ind T x0 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) -x t)) (THead (Flat Appl) x1 (THead (Bind Abbr) x5 x6))) \to (\forall (P: -Prop).P))) H26 (THead (Bind Abst) x3 x4) H22) in (let H28 \def (eq_ind T x0 -(\lambda (t: T).(\forall (t4: T).((((eq T (THead (Flat Appl) x t) t4) \to -(\forall (P: Prop).P))) \to ((pr3 c (THead (Flat Appl) x t) t4) \to (sn3 c -t4))))) H11 (THead (Bind Abst) x3 x4) H22) in (let H29 \def (eq_ind T x0 -(\lambda (t: T).(\forall (t4: T).((((eq T (THead (Flat Appl) x t) t4) \to -(\forall (P: Prop).P))) \to ((pr3 c (THead (Flat Appl) x t) t4) \to (\forall -(x7: T).(\forall (x8: T).((eq T t4 (THead (Flat Appl) x7 x8)) \to (\forall -(v3: T).((sn3 c v3) \to (((\forall (u2: T).((pr3 c t4 u2) \to ((((iso t4 u2) -\to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v3 u2)))))) \to -(sn3 c (THead (Flat Appl) v3 t4)))))))))))) H10 (THead (Bind Abst) x3 x4) -H22) in (let H30 \def (eq_ind T x0 (\lambda (t: T).(\forall (u2: T).((pr3 c -(THead (Flat Appl) x t) u2) \to ((((iso (THead (Flat Appl) x t) u2) \to -(\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) t0 u2)))))) H8 (THead -(Bind Abst) x3 x4) H22) in (let H31 \def (eq_ind T x0 (\lambda (t: -T).(\forall (t4: T).((((eq T t0 t4) \to (\forall (P: Prop).P))) \to ((pr3 c -t0 t4) \to (((\forall (u2: T).((pr3 c (THead (Flat Appl) x t) u2) \to ((((iso -(THead (Flat Appl) x t) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead -(Flat Appl) t4 u2)))))) \to (sn3 c (THead (Flat Appl) t4 (THead (Flat Appl) x -t)))))))) H9 (THead (Bind Abst) x3 x4) H22) in (sn3_pr3_trans c (THead (Flat -Appl) t0 (THead (Bind Abbr) x5 x6)) (H30 (THead (Bind Abbr) x5 x6) (pr3_sing -c (THead (Bind Abbr) x x4) (THead (Flat Appl) x (THead (Bind Abst) x3 x4)) -(pr2_free c (THead (Flat Appl) x (THead (Bind Abst) x3 x4)) (THead (Bind -Abbr) x x4) (pr0_beta x3 x x (pr0_refl x) x4 x4 (pr0_refl x4))) (THead (Bind -Abbr) x5 x6) (pr3_head_12 c x x5 (pr3_pr2 c x x5 H24) (Bind Abbr) x4 x6 -(pr3_pr2 (CHead c (Bind Abbr) x5) x4 x6 (H25 Abbr x5)))) (\lambda (H32: (iso -(THead (Flat Appl) x (THead (Bind Abst) x3 x4)) (THead (Bind Abbr) x5 -x6))).(\lambda (P: Prop).(let H33 \def (match H32 in iso return (\lambda (t: -T).(\lambda (t4: T).(\lambda (_: (iso t t4)).((eq T t (THead (Flat Appl) x -(THead (Bind Abst) x3 x4))) \to ((eq T t4 (THead (Bind Abbr) x5 x6)) \to -P))))) with [(iso_sort n1 n2) \Rightarrow (\lambda (H33: (eq T (TSort n1) -(THead (Flat Appl) x (THead (Bind Abst) x3 x4)))).(\lambda (H34: (eq T (TSort -n2) (THead (Bind Abbr) x5 x6))).((let H35 \def (eq_ind T (TSort n1) (\lambda -(e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I -(THead (Flat Appl) x (THead (Bind Abst) x3 x4)) H33) in (False_ind ((eq T -(TSort n2) (THead (Bind Abbr) x5 x6)) \to P) H35)) H34))) | (iso_lref i1 i2) -\Rightarrow (\lambda (H33: (eq T (TLRef i1) (THead (Flat Appl) x (THead (Bind -Abst) x3 x4)))).(\lambda (H34: (eq T (TLRef i2) (THead (Bind Abbr) x5 -x6))).((let H35 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) x -(THead (Bind Abst) x3 x4)) H33) in (False_ind ((eq T (TLRef i2) (THead (Bind -Abbr) x5 x6)) \to P) H35)) H34))) | (iso_head v4 v5 t4 t5 k) \Rightarrow -(\lambda (H33: (eq T (THead k v4 t4) (THead (Flat Appl) x (THead (Bind Abst) -x3 x4)))).(\lambda (H34: (eq T (THead k v5 t5) (THead (Bind Abbr) x5 -x6))).((let H35 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow t4 | (TLRef _) \Rightarrow t4 -| (THead _ _ t) \Rightarrow t])) (THead k v4 t4) (THead (Flat Appl) x (THead -(Bind Abst) x3 x4)) H33) in ((let H36 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v4 | -(TLRef _) \Rightarrow v4 | (THead _ t _) \Rightarrow t])) (THead k v4 t4) -(THead (Flat Appl) x (THead (Bind Abst) x3 x4)) H33) in ((let H37 \def -(f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with -[(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) -\Rightarrow k0])) (THead k v4 t4) (THead (Flat Appl) x (THead (Bind Abst) x3 -x4)) H33) in (eq_ind K (Flat Appl) (\lambda (k0: K).((eq T v4 x) \to ((eq T -t4 (THead (Bind Abst) x3 x4)) \to ((eq T (THead k0 v5 t5) (THead (Bind Abbr) -x5 x6)) \to P)))) (\lambda (H38: (eq T v4 x)).(eq_ind T x (\lambda (_: -T).((eq T t4 (THead (Bind Abst) x3 x4)) \to ((eq T (THead (Flat Appl) v5 t5) -(THead (Bind Abbr) x5 x6)) \to P))) (\lambda (H39: (eq T t4 (THead (Bind -Abst) x3 x4))).(eq_ind T (THead (Bind Abst) x3 x4) (\lambda (_: T).((eq T -(THead (Flat Appl) v5 t5) (THead (Bind Abbr) x5 x6)) \to P)) (\lambda (H40: -(eq T (THead (Flat Appl) v5 t5) (THead (Bind Abbr) x5 x6))).(let H41 \def -(eq_ind T (THead (Flat Appl) v5 t5) (\lambda (e: T).(match e in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind Abbr) x5 x6) H40) in (False_ind P H41))) t4 (sym_eq -T t4 (THead (Bind Abst) x3 x4) H39))) v4 (sym_eq T v4 x H38))) k (sym_eq K k -(Flat Appl) H37))) H36)) H35)) H34)))]) in (H33 (refl_equal T (THead (Flat -Appl) x (THead (Bind Abst) x3 x4))) (refl_equal T (THead (Bind Abbr) x5 -x6))))))) (THead (Flat Appl) x1 (THead (Bind Abbr) x5 x6)) (pr3_pr2 c (THead -(Flat Appl) t0 (THead (Bind Abbr) x5 x6)) (THead (Flat Appl) x1 (THead (Bind -Abbr) x5 x6)) (pr2_head_1 c t0 x1 H17 (Flat Appl) (THead (Bind Abbr) x5 -x6))))))))) x2 H23)))))))))) H21)) (\lambda (H21: (ex6_6 B T T T T T (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T x0 -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x2 (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind -B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T x0 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T -x2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 -y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) -(sn3 c (THead (Flat Appl) x1 x2)) (\lambda (x3: B).(\lambda (x4: T).(\lambda -(x5: T).(\lambda (x6: T).(\lambda (x7: T).(\lambda (x8: T).(\lambda (H22: -(not (eq B x3 Abst))).(\lambda (H23: (eq T x0 (THead (Bind x3) x4 -x5))).(\lambda (H24: (eq T x2 (THead (Bind x3) x8 (THead (Flat Appl) (lift (S -O) O x7) x6)))).(\lambda (H25: (pr2 c x x7)).(\lambda (H26: (pr2 c x4 -x8)).(\lambda (H27: (pr2 (CHead c (Bind x3) x8) x5 x6)).(let H28 \def (eq_ind -T x2 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) -(THead (Flat Appl) x1 t)) \to (\forall (P: Prop).P))) H19 (THead (Bind x3) x8 -(THead (Flat Appl) (lift (S O) O x7) x6)) H24) in (eq_ind_r T (THead (Bind -x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)) (\lambda (t: T).(sn3 c -(THead (Flat Appl) x1 t))) (let H29 \def (eq_ind T x0 (\lambda (t: T).((eq T -(THead (Flat Appl) t0 (THead (Flat Appl) x t)) (THead (Flat Appl) x1 (THead -(Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))) \to (\forall (P: -Prop).P))) H28 (THead (Bind x3) x4 x5) H23) in (let H30 \def (eq_ind T x0 -(\lambda (t: T).(\forall (t4: T).((((eq T (THead (Flat Appl) x t) t4) \to -(\forall (P: Prop).P))) \to ((pr3 c (THead (Flat Appl) x t) t4) \to (sn3 c -t4))))) H11 (THead (Bind x3) x4 x5) H23) in (let H31 \def (eq_ind T x0 -(\lambda (t: T).(\forall (t4: T).((((eq T (THead (Flat Appl) x t) t4) \to -(\forall (P: Prop).P))) \to ((pr3 c (THead (Flat Appl) x t) t4) \to (\forall -(x9: T).(\forall (x10: T).((eq T t4 (THead (Flat Appl) x9 x10)) \to (\forall -(v3: T).((sn3 c v3) \to (((\forall (u2: T).((pr3 c t4 u2) \to ((((iso t4 u2) -\to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v3 u2)))))) \to -(sn3 c (THead (Flat Appl) v3 t4)))))))))))) H10 (THead (Bind x3) x4 x5) H23) -in (let H32 \def (eq_ind T x0 (\lambda (t: T).(\forall (u2: T).((pr3 c (THead -(Flat Appl) x t) u2) \to ((((iso (THead (Flat Appl) x t) u2) \to (\forall (P: -Prop).P))) \to (sn3 c (THead (Flat Appl) t0 u2)))))) H8 (THead (Bind x3) x4 -x5) H23) in (let H33 \def (eq_ind T x0 (\lambda (t: T).(\forall (t4: -T).((((eq T t0 t4) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t4) \to -(((\forall (u2: T).((pr3 c (THead (Flat Appl) x t) u2) \to ((((iso (THead -(Flat Appl) x t) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat -Appl) t4 u2)))))) \to (sn3 c (THead (Flat Appl) t4 (THead (Flat Appl) x -t)))))))) H9 (THead (Bind x3) x4 x5) H23) in (sn3_pr3_trans c (THead (Flat -Appl) t0 (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) (H32 -(THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)) (pr3_sing c -(THead (Bind x3) x4 (THead (Flat Appl) (lift (S O) O x) x5)) (THead (Flat -Appl) x (THead (Bind x3) x4 x5)) (pr2_free c (THead (Flat Appl) x (THead -(Bind x3) x4 x5)) (THead (Bind x3) x4 (THead (Flat Appl) (lift (S O) O x) -x5)) (pr0_upsilon x3 H22 x x (pr0_refl x) x4 x4 (pr0_refl x4) x5 x5 (pr0_refl -x5))) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)) -(pr3_head_12 c x4 x8 (pr3_pr2 c x4 x8 H26) (Bind x3) (THead (Flat Appl) (lift -(S O) O x) x5) (THead (Flat Appl) (lift (S O) O x7) x6) (pr3_head_12 (CHead c -(Bind x3) x8) (lift (S O) O x) (lift (S O) O x7) (pr3_lift (CHead c (Bind x3) -x8) c (S O) O (drop_drop (Bind x3) O c c (drop_refl c) x8) x x7 (pr3_pr2 c x -x7 H25)) (Flat Appl) x5 x6 (pr3_pr2 (CHead (CHead c (Bind x3) x8) (Flat Appl) -(lift (S O) O x7)) x5 x6 (pr2_cflat (CHead c (Bind x3) x8) x5 x6 H27 Appl -(lift (S O) O x7)))))) (\lambda (H34: (iso (THead (Flat Appl) x (THead (Bind -x3) x4 x5)) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) -x6)))).(\lambda (P: Prop).(let H35 \def (match H34 in iso return (\lambda (t: -T).(\lambda (t4: T).(\lambda (_: (iso t t4)).((eq T t (THead (Flat Appl) x -(THead (Bind x3) x4 x5))) \to ((eq T t4 (THead (Bind x3) x8 (THead (Flat -Appl) (lift (S O) O x7) x6))) \to P))))) with [(iso_sort n1 n2) \Rightarrow -(\lambda (H35: (eq T (TSort n1) (THead (Flat Appl) x (THead (Bind x3) x4 -x5)))).(\lambda (H36: (eq T (TSort n2) (THead (Bind x3) x8 (THead (Flat Appl) -(lift (S O) O x7) x6)))).((let H37 \def (eq_ind T (TSort n1) (\lambda (e: -T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I -(THead (Flat Appl) x (THead (Bind x3) x4 x5)) H35) in (False_ind ((eq T -(TSort n2) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to -P) H37)) H36))) | (iso_lref i1 i2) \Rightarrow (\lambda (H35: (eq T (TLRef -i1) (THead (Flat Appl) x (THead (Bind x3) x4 x5)))).(\lambda (H36: (eq T -(TLRef i2) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) -x6)))).((let H37 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) x -(THead (Bind x3) x4 x5)) H35) in (False_ind ((eq T (TLRef i2) (THead (Bind -x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to P) H37)) H36))) | -(iso_head v4 v5 t4 t5 k) \Rightarrow (\lambda (H35: (eq T (THead k v4 t4) -(THead (Flat Appl) x (THead (Bind x3) x4 x5)))).(\lambda (H36: (eq T (THead k -v5 t5) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))).((let -H37 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow t4 | (TLRef _) \Rightarrow t4 | (THead _ _ t) -\Rightarrow t])) (THead k v4 t4) (THead (Flat Appl) x (THead (Bind x3) x4 -x5)) H35) in ((let H38 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow v4 | (TLRef _) \Rightarrow v4 -| (THead _ t _) \Rightarrow t])) (THead k v4 t4) (THead (Flat Appl) x (THead -(Bind x3) x4 x5)) H35) in ((let H39 \def (f_equal T K (\lambda (e: T).(match -e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) -\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k v4 t4) (THead (Flat -Appl) x (THead (Bind x3) x4 x5)) H35) in (eq_ind K (Flat Appl) (\lambda (k0: -K).((eq T v4 x) \to ((eq T t4 (THead (Bind x3) x4 x5)) \to ((eq T (THead k0 -v5 t5) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to -P)))) (\lambda (H40: (eq T v4 x)).(eq_ind T x (\lambda (_: T).((eq T t4 -(THead (Bind x3) x4 x5)) \to ((eq T (THead (Flat Appl) v5 t5) (THead (Bind -x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to P))) (\lambda (H41: (eq -T t4 (THead (Bind x3) x4 x5))).(eq_ind T (THead (Bind x3) x4 x5) (\lambda (_: -T).((eq T (THead (Flat Appl) v5 t5) (THead (Bind x3) x8 (THead (Flat Appl) -(lift (S O) O x7) x6))) \to P)) (\lambda (H42: (eq T (THead (Flat Appl) v5 -t5) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))).(let H43 -\def (eq_ind T (THead (Flat Appl) v5 t5) (\lambda (e: T).(match e in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)) -H42) in (False_ind P H43))) t4 (sym_eq T t4 (THead (Bind x3) x4 x5) H41))) v4 -(sym_eq T v4 x H40))) k (sym_eq K k (Flat Appl) H39))) H38)) H37)) H36)))]) -in (H35 (refl_equal T (THead (Flat Appl) x (THead (Bind x3) x4 x5))) -(refl_equal T (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) -x6)))))))) (THead (Flat Appl) x1 (THead (Bind x3) x8 (THead (Flat Appl) (lift -(S O) O x7) x6))) (pr3_pr2 c (THead (Flat Appl) t0 (THead (Bind x3) x8 (THead -(Flat Appl) (lift (S O) O x7) x6))) (THead (Flat Appl) x1 (THead (Bind x3) x8 -(THead (Flat Appl) (lift (S O) O x7) x6))) (pr2_head_1 c t0 x1 H17 (Flat -Appl) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))))))))) -x2 H24)))))))))))))) H21)) H20)) t3 H16))))))) H15)) (\lambda (H15: (ex4_4 T -T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Flat Appl) x x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind -Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c t0 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t4: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) -z1 t4))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (_: T).(eq T (THead (Flat Appl) x x0) (THead (Bind Abst) y1 -z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: -T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))))) (\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall -(u: T).(pr2 (CHead c (Bind b) u) z1 t4))))))) (sn3 c t3) (\lambda (x1: -T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H16: (eq T -(THead (Flat Appl) x x0) (THead (Bind Abst) x1 x2))).(\lambda (H17: (eq T t3 -(THead (Bind Abbr) x3 x4))).(\lambda (_: (pr2 c t0 x3)).(\lambda (_: -((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x2 x4))))).(let -H20 \def (eq_ind T t3 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead -(Flat Appl) x x0)) t) \to (\forall (P: Prop).P))) H12 (THead (Bind Abbr) x3 -x4) H17) in (eq_ind_r T (THead (Bind Abbr) x3 x4) (\lambda (t: T).(sn3 c t)) -(let H21 \def (eq_ind T (THead (Flat Appl) x x0) (\lambda (ee: T).(match ee -in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef -_) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return -(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind Abst) x1 x2) H16) in (False_ind (sn3 c (THead (Bind -Abbr) x3 x4)) H21)) t3 H17)))))))))) H15)) (\lambda (H15: (ex6_6 B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Flat Appl) x x0) (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) -z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) -y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B -b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Appl) x x0) (THead -(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c t0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) (sn3 c t3) -(\lambda (x1: B).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda -(x5: T).(\lambda (x6: T).(\lambda (_: (not (eq B x1 Abst))).(\lambda (H17: -(eq T (THead (Flat Appl) x x0) (THead (Bind x1) x2 x3))).(\lambda (H18: (eq T -t3 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))).(\lambda -(_: (pr2 c t0 x5)).(\lambda (_: (pr2 c x2 x6)).(\lambda (_: (pr2 (CHead c -(Bind x1) x6) x3 x4)).(let H22 \def (eq_ind T t3 (\lambda (t: T).((eq T -(THead (Flat Appl) t0 (THead (Flat Appl) x x0)) t) \to (\forall (P: -Prop).P))) H12 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)) -H18) in (eq_ind_r T (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) -x4)) (\lambda (t: T).(sn3 c t)) (let H23 \def (eq_ind T (THead (Flat Appl) x -x0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort -_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) -\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind x1) x2 x3) -H17) in (False_ind (sn3 c (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) -O x5) x4))) H23)) t3 H18)))))))))))))) H15)) H14)))))) t2 H3))))))))) v2 -H4))))))))) y H0))))) H))))). -(* COMMENTS -Initial nodes: 9317 -END *) - -theorem sn3_appl_beta: - \forall (c: C).(\forall (u: T).(\forall (v: T).(\forall (t: T).((sn3 c -(THead (Flat Appl) u (THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w) -\to (sn3 c (THead (Flat Appl) u (THead (Flat Appl) v (THead (Bind Abst) w -t)))))))))) -\def - \lambda (c: C).(\lambda (u: T).(\lambda (v: T).(\lambda (t: T).(\lambda (H: -(sn3 c (THead (Flat Appl) u (THead (Bind Abbr) v t)))).(\lambda (w: -T).(\lambda (H0: (sn3 c w)).(let H_x \def (sn3_gen_flat Appl c u (THead (Bind -Abbr) v t) H) in (let H1 \def H_x in (land_ind (sn3 c u) (sn3 c (THead (Bind -Abbr) v t)) (sn3 c (THead (Flat Appl) u (THead (Flat Appl) v (THead (Bind -Abst) w t)))) (\lambda (H2: (sn3 c u)).(\lambda (H3: (sn3 c (THead (Bind -Abbr) v t))).(sn3_appl_appl v (THead (Bind Abst) w t) c (sn3_beta c v t H3 w -H0) u H2 (\lambda (u2: T).(\lambda (H4: (pr3 c (THead (Flat Appl) v (THead -(Bind Abst) w t)) u2)).(\lambda (H5: (((iso (THead (Flat Appl) v (THead (Bind -Abst) w t)) u2) \to (\forall (P: Prop).P)))).(sn3_pr3_trans c (THead (Flat -Appl) u (THead (Bind Abbr) v t)) H (THead (Flat Appl) u u2) (pr3_thin_dx c -(THead (Bind Abbr) v t) u2 (pr3_iso_beta v w t c u2 H4 H5) u Appl)))))))) -H1))))))))). -(* COMMENTS -Initial nodes: 289 -END *) - -theorem sn3_appl_appls: - \forall (v1: T).(\forall (t1: T).(\forall (vs: TList).(let u1 \def (THeads -(Flat Appl) (TCons v1 vs) t1) in (\forall (c: C).((sn3 c u1) \to (\forall -(v2: T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) -\to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to -(sn3 c (THead (Flat Appl) v2 u1)))))))))) -\def - \lambda (v1: T).(\lambda (t1: T).(\lambda (vs: TList).(let u1 \def (THeads -(Flat Appl) (TCons v1 vs) t1) in (\lambda (c: C).(\lambda (H: (sn3 c (THead -(Flat Appl) v1 (THeads (Flat Appl) vs t1)))).(\lambda (v2: T).(\lambda (H0: -(sn3 c v2)).(\lambda (H1: ((\forall (u2: T).((pr3 c (THead (Flat Appl) v1 -(THeads (Flat Appl) vs t1)) u2) \to ((((iso (THead (Flat Appl) v1 (THeads -(Flat Appl) vs t1)) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat -Appl) v2 u2))))))).(sn3_appl_appl v1 (THeads (Flat Appl) vs t1) c H v2 H0 -H1))))))))). -(* COMMENTS -Initial nodes: 141 -END *) - -theorem sn3_appls_lref: - \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (us: -TList).((sns3 c us) \to (sn3 c (THeads (Flat Appl) us (TLRef i))))))) -\def - \lambda (c: C).(\lambda (i: nat).(\lambda (H: (nf2 c (TLRef i))).(\lambda -(us: TList).(TList_ind (\lambda (t: TList).((sns3 c t) \to (sn3 c (THeads -(Flat Appl) t (TLRef i))))) (\lambda (_: True).(sn3_nf2 c (TLRef i) H)) -(\lambda (t: T).(\lambda (t0: TList).(TList_ind (\lambda (t1: TList).((((sns3 -c t1) \to (sn3 c (THeads (Flat Appl) t1 (TLRef i))))) \to ((land (sn3 c t) -(sns3 c t1)) \to (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t1 (TLRef -i))))))) (\lambda (_: (((sns3 c TNil) \to (sn3 c (THeads (Flat Appl) TNil -(TLRef i)))))).(\lambda (H1: (land (sn3 c t) (sns3 c TNil))).(let H2 \def H1 -in (land_ind (sn3 c t) True (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) -TNil (TLRef i)))) (\lambda (H3: (sn3 c t)).(\lambda (_: True).(sn3_appl_lref -c i H t H3))) H2)))) (\lambda (t1: T).(\lambda (t2: TList).(\lambda (_: -(((((sns3 c t2) \to (sn3 c (THeads (Flat Appl) t2 (TLRef i))))) \to ((land -(sn3 c t) (sns3 c t2)) \to (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t2 -(TLRef i)))))))).(\lambda (H1: (((sns3 c (TCons t1 t2)) \to (sn3 c (THeads -(Flat Appl) (TCons t1 t2) (TLRef i)))))).(\lambda (H2: (land (sn3 c t) (sns3 -c (TCons t1 t2)))).(let H3 \def H2 in (land_ind (sn3 c t) (land (sn3 c t1) -(sns3 c t2)) (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) (TCons t1 t2) -(TLRef i)))) (\lambda (H4: (sn3 c t)).(\lambda (H5: (land (sn3 c t1) (sns3 c -t2))).(land_ind (sn3 c t1) (sns3 c t2) (sn3 c (THead (Flat Appl) t (THeads -(Flat Appl) (TCons t1 t2) (TLRef i)))) (\lambda (H6: (sn3 c t1)).(\lambda -(H7: (sns3 c t2)).(sn3_appl_appls t1 (TLRef i) t2 c (H1 (conj (sn3 c t1) -(sns3 c t2) H6 H7)) t H4 (\lambda (u2: T).(\lambda (H8: (pr3 c (THeads (Flat -Appl) (TCons t1 t2) (TLRef i)) u2)).(\lambda (H9: (((iso (THeads (Flat Appl) -(TCons t1 t2) (TLRef i)) u2) \to (\forall (P: Prop).P)))).(H9 -(nf2_iso_appls_lref c i H (TCons t1 t2) u2 H8) (sn3 c (THead (Flat Appl) t -u2))))))))) H5))) H3))))))) t0))) us)))). -(* COMMENTS -Initial nodes: 577 -END *) - -theorem sn3_appls_cast: - \forall (c: C).(\forall (vs: TList).(\forall (u: T).((sn3 c (THeads (Flat -Appl) vs u)) \to (\forall (t: T).((sn3 c (THeads (Flat Appl) vs t)) \to (sn3 -c (THeads (Flat Appl) vs (THead (Flat Cast) u t)))))))) -\def - \lambda (c: C).(\lambda (vs: TList).(TList_ind (\lambda (t: TList).(\forall -(u: T).((sn3 c (THeads (Flat Appl) t u)) \to (\forall (t0: T).((sn3 c (THeads -(Flat Appl) t t0)) \to (sn3 c (THeads (Flat Appl) t (THead (Flat Cast) u -t0)))))))) (\lambda (u: T).(\lambda (H: (sn3 c u)).(\lambda (t: T).(\lambda -(H0: (sn3 c t)).(sn3_cast c u H t H0))))) (\lambda (t: T).(\lambda (t0: -TList).(TList_ind (\lambda (t1: TList).(((\forall (u: T).((sn3 c (THeads -(Flat Appl) t1 u)) \to (\forall (t2: T).((sn3 c (THeads (Flat Appl) t1 t2)) -\to (sn3 c (THeads (Flat Appl) t1 (THead (Flat Cast) u t2)))))))) \to -(\forall (u: T).((sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t1 u))) \to -(\forall (t2: T).((sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t1 t2))) -\to (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t1 (THead (Flat Cast) u -t2)))))))))) (\lambda (_: ((\forall (u: T).((sn3 c (THeads (Flat Appl) TNil -u)) \to (\forall (t1: T).((sn3 c (THeads (Flat Appl) TNil t1)) \to (sn3 c -(THeads (Flat Appl) TNil (THead (Flat Cast) u t1))))))))).(\lambda (u: -T).(\lambda (H0: (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) TNil -u)))).(\lambda (t1: T).(\lambda (H1: (sn3 c (THead (Flat Appl) t (THeads -(Flat Appl) TNil t1)))).(sn3_appl_cast c t u H0 t1 H1)))))) (\lambda (t1: -T).(\lambda (t2: TList).(\lambda (_: ((((\forall (u: T).((sn3 c (THeads (Flat -Appl) t2 u)) \to (\forall (t3: T).((sn3 c (THeads (Flat Appl) t2 t3)) \to -(sn3 c (THeads (Flat Appl) t2 (THead (Flat Cast) u t3)))))))) \to (\forall -(u: T).((sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t2 u))) \to (\forall -(t3: T).((sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t2 t3))) \to (sn3 c -(THead (Flat Appl) t (THeads (Flat Appl) t2 (THead (Flat Cast) u -t3))))))))))).(\lambda (H0: ((\forall (u: T).((sn3 c (THeads (Flat Appl) -(TCons t1 t2) u)) \to (\forall (t3: T).((sn3 c (THeads (Flat Appl) (TCons t1 -t2) t3)) \to (sn3 c (THeads (Flat Appl) (TCons t1 t2) (THead (Flat Cast) u -t3))))))))).(\lambda (u: T).(\lambda (H1: (sn3 c (THead (Flat Appl) t (THeads -(Flat Appl) (TCons t1 t2) u)))).(\lambda (t3: T).(\lambda (H2: (sn3 c (THead -(Flat Appl) t (THeads (Flat Appl) (TCons t1 t2) t3)))).(let H_x \def -(sn3_gen_flat Appl c t (THeads (Flat Appl) (TCons t1 t2) t3) H2) in (let H3 -\def H_x in (land_ind (sn3 c t) (sn3 c (THead (Flat Appl) t1 (THeads (Flat -Appl) t2 t3))) (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) (TCons t1 t2) -(THead (Flat Cast) u t3)))) (\lambda (_: (sn3 c t)).(\lambda (H5: (sn3 c -(THead (Flat Appl) t1 (THeads (Flat Appl) t2 t3)))).(let H6 \def H5 in (let -H_x0 \def (sn3_gen_flat Appl c t (THeads (Flat Appl) (TCons t1 t2) u) H1) in -(let H7 \def H_x0 in (land_ind (sn3 c t) (sn3 c (THead (Flat Appl) t1 (THeads -(Flat Appl) t2 u))) (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) (TCons t1 -t2) (THead (Flat Cast) u t3)))) (\lambda (H8: (sn3 c t)).(\lambda (H9: (sn3 c -(THead (Flat Appl) t1 (THeads (Flat Appl) t2 u)))).(let H10 \def H9 in -(sn3_appl_appls t1 (THead (Flat Cast) u t3) t2 c (H0 u H10 t3 H6) t H8 -(\lambda (u2: T).(\lambda (H11: (pr3 c (THeads (Flat Appl) (TCons t1 t2) -(THead (Flat Cast) u t3)) u2)).(\lambda (H12: (((iso (THeads (Flat Appl) -(TCons t1 t2) (THead (Flat Cast) u t3)) u2) \to (\forall (P: -Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) t (THeads (Flat Appl) (TCons -t1 t2) t3)) H2 (THead (Flat Appl) t u2) (pr3_thin_dx c (THeads (Flat Appl) -(TCons t1 t2) t3) u2 (pr3_iso_appls_cast c u t3 (TCons t1 t2) u2 H11 H12) t -Appl))))))))) H7)))))) H3))))))))))) t0))) vs)). -(* COMMENTS -Initial nodes: 1025 -END *) - -theorem sn3_appls_bind: - \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (u: -T).((sn3 c u) \to (\forall (vs: TList).(\forall (t: T).((sn3 (CHead c (Bind -b) u) (THeads (Flat Appl) (lifts (S O) O vs) t)) \to (sn3 c (THeads (Flat -Appl) vs (THead (Bind b) u t)))))))))) -\def - \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (c: C).(\lambda -(u: T).(\lambda (H0: (sn3 c u)).(\lambda (vs: TList).(TList_ind (\lambda (t: -TList).(\forall (t0: T).((sn3 (CHead c (Bind b) u) (THeads (Flat Appl) (lifts -(S O) O t) t0)) \to (sn3 c (THeads (Flat Appl) t (THead (Bind b) u t0)))))) -(\lambda (t: T).(\lambda (H1: (sn3 (CHead c (Bind b) u) t)).(sn3_bind b c u -H0 t H1))) (\lambda (v: T).(\lambda (vs0: TList).(TList_ind (\lambda (t: -TList).(((\forall (t0: T).((sn3 (CHead c (Bind b) u) (THeads (Flat Appl) -(lifts (S O) O t) t0)) \to (sn3 c (THeads (Flat Appl) t (THead (Bind b) u -t0)))))) \to (\forall (t0: T).((sn3 (CHead c (Bind b) u) (THead (Flat Appl) -(lift (S O) O v) (THeads (Flat Appl) (lifts (S O) O t) t0))) \to (sn3 c -(THead (Flat Appl) v (THeads (Flat Appl) t (THead (Bind b) u t0)))))))) -(\lambda (_: ((\forall (t: T).((sn3 (CHead c (Bind b) u) (THeads (Flat Appl) -(lifts (S O) O TNil) t)) \to (sn3 c (THeads (Flat Appl) TNil (THead (Bind b) -u t))))))).(\lambda (t: T).(\lambda (H2: (sn3 (CHead c (Bind b) u) (THead -(Flat Appl) (lift (S O) O v) (THeads (Flat Appl) (lifts (S O) O TNil) -t)))).(sn3_appl_bind b H c u H0 t v H2)))) (\lambda (t: T).(\lambda (t0: -TList).(\lambda (_: ((((\forall (t1: T).((sn3 (CHead c (Bind b) u) (THeads -(Flat Appl) (lifts (S O) O t0) t1)) \to (sn3 c (THeads (Flat Appl) t0 (THead -(Bind b) u t1)))))) \to (\forall (t1: T).((sn3 (CHead c (Bind b) u) (THead -(Flat Appl) (lift (S O) O v) (THeads (Flat Appl) (lifts (S O) O t0) t1))) \to -(sn3 c (THead (Flat Appl) v (THeads (Flat Appl) t0 (THead (Bind b) u -t1))))))))).(\lambda (H2: ((\forall (t1: T).((sn3 (CHead c (Bind b) u) -(THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1)) \to (sn3 c (THeads -(Flat Appl) (TCons t t0) (THead (Bind b) u t1))))))).(\lambda (t1: -T).(\lambda (H3: (sn3 (CHead c (Bind b) u) (THead (Flat Appl) (lift (S O) O -v) (THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1)))).(let H_x \def -(sn3_gen_flat Appl (CHead c (Bind b) u) (lift (S O) O v) (THeads (Flat Appl) -(lifts (S O) O (TCons t t0)) t1) H3) in (let H4 \def H_x in (land_ind (sn3 -(CHead c (Bind b) u) (lift (S O) O v)) (sn3 (CHead c (Bind b) u) (THead (Flat -Appl) (lift (S O) O t) (THeads (Flat Appl) (lifts (S O) O t0) t1))) (sn3 c -(THead (Flat Appl) v (THeads (Flat Appl) (TCons t t0) (THead (Bind b) u -t1)))) (\lambda (H5: (sn3 (CHead c (Bind b) u) (lift (S O) O v))).(\lambda -(H6: (sn3 (CHead c (Bind b) u) (THead (Flat Appl) (lift (S O) O t) (THeads -(Flat Appl) (lifts (S O) O t0) t1)))).(let H_y \def (sn3_gen_lift (CHead c -(Bind b) u) v (S O) O H5 c) in (sn3_appl_appls t (THead (Bind b) u t1) t0 c -(H2 t1 H6) v (H_y (drop_drop (Bind b) O c c (drop_refl c) u)) (\lambda (u2: -T).(\lambda (H7: (pr3 c (THeads (Flat Appl) (TCons t t0) (THead (Bind b) u -t1)) u2)).(\lambda (H8: (((iso (THeads (Flat Appl) (TCons t t0) (THead (Bind -b) u t1)) u2) \to (\forall (P: Prop).P)))).(let H9 \def (pr3_iso_appls_bind b -H (TCons t t0) u t1 c u2 H7 H8) in (sn3_pr3_trans c (THead (Flat Appl) v -(THead (Bind b) u (THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1))) -(sn3_appl_bind b H c u H0 (THeads (Flat Appl) (lifts (S O) O (TCons t t0)) -t1) v H3) (THead (Flat Appl) v u2) (pr3_flat c v v (pr3_refl c v) (THead -(Bind b) u (THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1)) u2 H9 -Appl)))))))))) H4))))))))) vs0))) vs)))))). -(* COMMENTS -Initial nodes: 1143 -END *) - -theorem sn3_appls_beta: - \forall (c: C).(\forall (v: T).(\forall (t: T).(\forall (us: TList).((sn3 c -(THeads (Flat Appl) us (THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c -w) \to (sn3 c (THeads (Flat Appl) us (THead (Flat Appl) v (THead (Bind Abst) -w t)))))))))) -\def - \lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (us: -TList).(TList_ind (\lambda (t0: TList).((sn3 c (THeads (Flat Appl) t0 (THead -(Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THeads (Flat -Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t)))))))) (\lambda (H: -(sn3 c (THead (Bind Abbr) v t))).(\lambda (w: T).(\lambda (H0: (sn3 c -w)).(sn3_beta c v t H w H0)))) (\lambda (u: T).(\lambda (us0: -TList).(TList_ind (\lambda (t0: TList).((((sn3 c (THeads (Flat Appl) t0 -(THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THeads -(Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t)))))))) \to ((sn3 -c (THead (Flat Appl) u (THeads (Flat Appl) t0 (THead (Bind Abbr) v t)))) \to -(\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) u (THeads (Flat -Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t)))))))))) (\lambda (_: -(((sn3 c (THeads (Flat Appl) TNil (THead (Bind Abbr) v t))) \to (\forall (w: -T).((sn3 c w) \to (sn3 c (THeads (Flat Appl) TNil (THead (Flat Appl) v (THead -(Bind Abst) w t))))))))).(\lambda (H0: (sn3 c (THead (Flat Appl) u (THeads -(Flat Appl) TNil (THead (Bind Abbr) v t))))).(\lambda (w: T).(\lambda (H1: -(sn3 c w)).(sn3_appl_beta c u v t H0 w H1))))) (\lambda (t0: T).(\lambda (t1: -TList).(\lambda (_: (((((sn3 c (THeads (Flat Appl) t1 (THead (Bind Abbr) v -t))) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THeads (Flat Appl) t1 (THead -(Flat Appl) v (THead (Bind Abst) w t)))))))) \to ((sn3 c (THead (Flat Appl) u -(THeads (Flat Appl) t1 (THead (Bind Abbr) v t)))) \to (\forall (w: T).((sn3 c -w) \to (sn3 c (THead (Flat Appl) u (THeads (Flat Appl) t1 (THead (Flat Appl) -v (THead (Bind Abst) w t))))))))))).(\lambda (H0: (((sn3 c (THeads (Flat -Appl) (TCons t0 t1) (THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w) -\to (sn3 c (THeads (Flat Appl) (TCons t0 t1) (THead (Flat Appl) v (THead -(Bind Abst) w t))))))))).(\lambda (H1: (sn3 c (THead (Flat Appl) u (THeads -(Flat Appl) (TCons t0 t1) (THead (Bind Abbr) v t))))).(\lambda (w: -T).(\lambda (H2: (sn3 c w)).(let H_x \def (sn3_gen_flat Appl c u (THeads -(Flat Appl) (TCons t0 t1) (THead (Bind Abbr) v t)) H1) in (let H3 \def H_x in -(land_ind (sn3 c u) (sn3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 -(THead (Bind Abbr) v t)))) (sn3 c (THead (Flat Appl) u (THeads (Flat Appl) -(TCons t0 t1) (THead (Flat Appl) v (THead (Bind Abst) w t))))) (\lambda (H4: -(sn3 c u)).(\lambda (H5: (sn3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 -(THead (Bind Abbr) v t))))).(sn3_appl_appls t0 (THead (Flat Appl) v (THead -(Bind Abst) w t)) t1 c (H0 H5 w H2) u H4 (\lambda (u2: T).(\lambda (H6: (pr3 -c (THeads (Flat Appl) (TCons t0 t1) (THead (Flat Appl) v (THead (Bind Abst) w -t))) u2)).(\lambda (H7: (((iso (THeads (Flat Appl) (TCons t0 t1) (THead (Flat -Appl) v (THead (Bind Abst) w t))) u2) \to (\forall (P: Prop).P)))).(let H8 -\def (pr3_iso_appls_beta (TCons t0 t1) v w t c u2 H6 H7) in (sn3_pr3_trans c -(THead (Flat Appl) u (THeads (Flat Appl) (TCons t0 t1) (THead (Bind Abbr) v -t))) H1 (THead (Flat Appl) u u2) (pr3_thin_dx c (THeads (Flat Appl) (TCons t0 -t1) (THead (Bind Abbr) v t)) u2 H8 u Appl))))))))) H3)))))))))) us0))) us)))). -(* COMMENTS -Initial nodes: 987 -END *) - -theorem sn3_lift: - \forall (d: C).(\forall (t: T).((sn3 d t) \to (\forall (c: C).(\forall (h: -nat).(\forall (i: nat).((drop h i c d) \to (sn3 c (lift h i t)))))))) -\def - \lambda (d: C).(\lambda (t: T).(\lambda (H: (sn3 d t)).(sn3_ind d (\lambda -(t0: T).(\forall (c: C).(\forall (h: nat).(\forall (i: nat).((drop h i c d) -\to (sn3 c (lift h i t0))))))) (\lambda (t1: T).(\lambda (_: ((\forall (t2: -T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 d t1 t2) \to (sn3 d -t2)))))).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: -Prop).P))) \to ((pr3 d t1 t2) \to (\forall (c: C).(\forall (h: nat).(\forall -(i: nat).((drop h i c d) \to (sn3 c (lift h i t2))))))))))).(\lambda (c: -C).(\lambda (h: nat).(\lambda (i: nat).(\lambda (H2: (drop h i c -d)).(sn3_pr2_intro c (lift h i t1) (\lambda (t2: T).(\lambda (H3: (((eq T -(lift h i t1) t2) \to (\forall (P: Prop).P)))).(\lambda (H4: (pr2 c (lift h i -t1) t2)).(let H5 \def (pr2_gen_lift c t1 t2 h i H4 d H2) in (ex2_ind T -(\lambda (t3: T).(eq T t2 (lift h i t3))) (\lambda (t3: T).(pr2 d t1 t3)) -(sn3 c t2) (\lambda (x: T).(\lambda (H6: (eq T t2 (lift h i x))).(\lambda -(H7: (pr2 d t1 x)).(let H8 \def (eq_ind T t2 (\lambda (t0: T).((eq T (lift h -i t1) t0) \to (\forall (P: Prop).P))) H3 (lift h i x) H6) in (eq_ind_r T -(lift h i x) (\lambda (t0: T).(sn3 c t0)) (H1 x (\lambda (H9: (eq T t1 -x)).(\lambda (P: Prop).(let H10 \def (eq_ind_r T x (\lambda (t0: T).((eq T -(lift h i t1) (lift h i t0)) \to (\forall (P0: Prop).P0))) H8 t1 H9) in (let -H11 \def (eq_ind_r T x (\lambda (t0: T).(pr2 d t1 t0)) H7 t1 H9) in (H10 -(refl_equal T (lift h i t1)) P))))) (pr3_pr2 d t1 x H7) c h i H2) t2 H6))))) -H5))))))))))))) t H))). -(* COMMENTS -Initial nodes: 439 -END *) - -theorem sn3_abbr: - \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c -(CHead d (Bind Abbr) v)) \to ((sn3 d v) \to (sn3 c (TLRef i))))))) -\def - \lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda -(H: (getl i c (CHead d (Bind Abbr) v))).(\lambda (H0: (sn3 d -v)).(sn3_pr2_intro c (TLRef i) (\lambda (t2: T).(\lambda (H1: (((eq T (TLRef -i) t2) \to (\forall (P: Prop).P)))).(\lambda (H2: (pr2 c (TLRef i) t2)).(let -H3 \def (pr2_gen_lref c t2 i H2) in (or_ind (eq T t2 (TLRef i)) (ex2_2 C T -(\lambda (d0: C).(\lambda (u: T).(getl i c (CHead d0 (Bind Abbr) u)))) -(\lambda (_: C).(\lambda (u: T).(eq T t2 (lift (S i) O u))))) (sn3 c t2) -(\lambda (H4: (eq T t2 (TLRef i))).(let H5 \def (eq_ind T t2 (\lambda (t: -T).((eq T (TLRef i) t) \to (\forall (P: Prop).P))) H1 (TLRef i) H4) in -(eq_ind_r T (TLRef i) (\lambda (t: T).(sn3 c t)) (H5 (refl_equal T (TLRef i)) -(sn3 c (TLRef i))) t2 H4))) (\lambda (H4: (ex2_2 C T (\lambda (d0: -C).(\lambda (u: T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_: -C).(\lambda (u: T).(eq T t2 (lift (S i) O u)))))).(ex2_2_ind C T (\lambda -(d0: C).(\lambda (u: T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_: -C).(\lambda (u: T).(eq T t2 (lift (S i) O u)))) (sn3 c t2) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (H5: (getl i c (CHead x0 (Bind Abbr) -x1))).(\lambda (H6: (eq T t2 (lift (S i) O x1))).(let H7 \def (eq_ind T t2 -(\lambda (t: T).((eq T (TLRef i) t) \to (\forall (P: Prop).P))) H1 (lift (S -i) O x1) H6) in (eq_ind_r T (lift (S i) O x1) (\lambda (t: T).(sn3 c t)) (let -H8 \def (eq_ind C (CHead d (Bind Abbr) v) (\lambda (c0: C).(getl i c c0)) H -(CHead x0 (Bind Abbr) x1) (getl_mono c (CHead d (Bind Abbr) v) i H (CHead x0 -(Bind Abbr) x1) H5)) in (let H9 \def (f_equal C C (\lambda (e: C).(match e in -C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) -\Rightarrow c0])) (CHead d (Bind Abbr) v) (CHead x0 (Bind Abbr) x1) -(getl_mono c (CHead d (Bind Abbr) v) i H (CHead x0 (Bind Abbr) x1) H5)) in -((let H10 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: -C).T) with [(CSort _) \Rightarrow v | (CHead _ _ t) \Rightarrow t])) (CHead d -(Bind Abbr) v) (CHead x0 (Bind Abbr) x1) (getl_mono c (CHead d (Bind Abbr) v) -i H (CHead x0 (Bind Abbr) x1) H5)) in (\lambda (H11: (eq C d x0)).(let H12 -\def (eq_ind_r T x1 (\lambda (t: T).(getl i c (CHead x0 (Bind Abbr) t))) H8 v -H10) in (eq_ind T v (\lambda (t: T).(sn3 c (lift (S i) O t))) (let H13 \def -(eq_ind_r C x0 (\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr) v))) H12 d -H11) in (sn3_lift d v H0 c (S i) O (getl_drop Abbr c d v i H13))) x1 H10)))) -H9))) t2 H6)))))) H4)) H3))))))))))). -(* COMMENTS -Initial nodes: 743 -END *) - -theorem sn3_appls_abbr: - \forall (c: C).(\forall (d: C).(\forall (w: T).(\forall (i: nat).((getl i c -(CHead d (Bind Abbr) w)) \to (\forall (vs: TList).((sn3 c (THeads (Flat Appl) -vs (lift (S i) O w))) \to (sn3 c (THeads (Flat Appl) vs (TLRef i))))))))) -\def - \lambda (c: C).(\lambda (d: C).(\lambda (w: T).(\lambda (i: nat).(\lambda -(H: (getl i c (CHead d (Bind Abbr) w))).(\lambda (vs: TList).(TList_ind -(\lambda (t: TList).((sn3 c (THeads (Flat Appl) t (lift (S i) O w))) \to (sn3 -c (THeads (Flat Appl) t (TLRef i))))) (\lambda (H0: (sn3 c (lift (S i) O -w))).(let H_y \def (sn3_gen_lift c w (S i) O H0 d (getl_drop Abbr c d w i H)) -in (sn3_abbr c d w i H H_y))) (\lambda (v: T).(\lambda (vs0: -TList).(TList_ind (\lambda (t: TList).((((sn3 c (THeads (Flat Appl) t (lift -(S i) O w))) \to (sn3 c (THeads (Flat Appl) t (TLRef i))))) \to ((sn3 c -(THead (Flat Appl) v (THeads (Flat Appl) t (lift (S i) O w)))) \to (sn3 c -(THead (Flat Appl) v (THeads (Flat Appl) t (TLRef i))))))) (\lambda (_: -(((sn3 c (THeads (Flat Appl) TNil (lift (S i) O w))) \to (sn3 c (THeads (Flat -Appl) TNil (TLRef i)))))).(\lambda (H1: (sn3 c (THead (Flat Appl) v (THeads -(Flat Appl) TNil (lift (S i) O w))))).(sn3_appl_abbr c d w i H v H1))) -(\lambda (t: T).(\lambda (t0: TList).(\lambda (_: (((((sn3 c (THeads (Flat -Appl) t0 (lift (S i) O w))) \to (sn3 c (THeads (Flat Appl) t0 (TLRef i))))) -\to ((sn3 c (THead (Flat Appl) v (THeads (Flat Appl) t0 (lift (S i) O w)))) -\to (sn3 c (THead (Flat Appl) v (THeads (Flat Appl) t0 (TLRef -i)))))))).(\lambda (H1: (((sn3 c (THeads (Flat Appl) (TCons t t0) (lift (S i) -O w))) \to (sn3 c (THeads (Flat Appl) (TCons t t0) (TLRef i)))))).(\lambda -(H2: (sn3 c (THead (Flat Appl) v (THeads (Flat Appl) (TCons t t0) (lift (S i) -O w))))).(let H_x \def (sn3_gen_flat Appl c v (THeads (Flat Appl) (TCons t -t0) (lift (S i) O w)) H2) in (let H3 \def H_x in (land_ind (sn3 c v) (sn3 c -(THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O w)))) (sn3 c (THead -(Flat Appl) v (THeads (Flat Appl) (TCons t t0) (TLRef i)))) (\lambda (H4: -(sn3 c v)).(\lambda (H5: (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 -(lift (S i) O w))))).(sn3_appl_appls t (TLRef i) t0 c (H1 H5) v H4 (\lambda -(u2: T).(\lambda (H6: (pr3 c (THeads (Flat Appl) (TCons t t0) (TLRef i)) -u2)).(\lambda (H7: (((iso (THeads (Flat Appl) (TCons t t0) (TLRef i)) u2) \to -(\forall (P: Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) v (THeads (Flat -Appl) (TCons t t0) (lift (S i) O w))) H2 (THead (Flat Appl) v u2) -(pr3_thin_dx c (THeads (Flat Appl) (TCons t t0) (lift (S i) O w)) u2 -(pr3_iso_appls_abbr c d w i H (TCons t t0) u2 H6 H7) v Appl)))))))) -H3)))))))) vs0))) vs)))))). -(* COMMENTS -Initial nodes: 797 -END *) - -theorem sns3_lifts: - \forall (c: C).(\forall (d: C).(\forall (h: nat).(\forall (i: nat).((drop h -i c d) \to (\forall (ts: TList).((sns3 d ts) \to (sns3 c (lifts h i ts)))))))) -\def - \lambda (c: C).(\lambda (d: C).(\lambda (h: nat).(\lambda (i: nat).(\lambda -(H: (drop h i c d)).(\lambda (ts: TList).(TList_ind (\lambda (t: -TList).((sns3 d t) \to (sns3 c (lifts h i t)))) (\lambda (H0: True).H0) -(\lambda (t: T).(\lambda (t0: TList).(\lambda (H0: (((sns3 d t0) \to (sns3 c -(lifts h i t0))))).(\lambda (H1: (land (sn3 d t) (sns3 d t0))).(let H2 \def -H1 in (land_ind (sn3 d t) (sns3 d t0) (land (sn3 c (lift h i t)) (sns3 c -(lifts h i t0))) (\lambda (H3: (sn3 d t)).(\lambda (H4: (sns3 d t0)).(conj -(sn3 c (lift h i t)) (sns3 c (lifts h i t0)) (sn3_lift d t H3 c h i H) (H0 -H4)))) H2)))))) ts)))))). -(* COMMENTS -Initial nodes: 185 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/spare.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/spare.ma deleted file mode 100644 index 95d2c24ee..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/spare.ma +++ /dev/null @@ -1,38 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/theory.ma". - -axiom pc3_gen_appls_sort_abst: - \forall (c: C).(\forall (vs: TList).(\forall (w: T).(\forall (u: T).(\forall -(n: nat).((pc3 c (THeads (Flat Appl) vs (TSort n)) (THead (Bind Abst) w u)) -\to False))))) -. - -axiom pc3_gen_appls_lref_abst: - \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c -(CHead d (Bind Abst) v)) \to (\forall (vs: TList).(\forall (w: T).(\forall -(u: T).((pc3 c (THeads (Flat Appl) vs (TLRef i)) (THead (Bind Abst) w u)) \to -False)))))))) -. - -axiom pc3_gen_appls_lref_sort: - \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c -(CHead d (Bind Abst) v)) \to (\forall (vs: TList).(\forall (ws: -TList).(\forall (n: nat).((pc3 c (THeads (Flat Appl) vs (TLRef i)) (THeads -(Flat Appl) ws (TSort n))) \to False)))))))) -. - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/sty0/defs.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/sty0/defs.ma deleted file mode 100644 index 6468d9dbb..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/sty0/defs.ma +++ /dev/null @@ -1,39 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/G/defs.ma". - -include "Basic-1/getl/defs.ma". - -inductive sty0 (g: G): C \to (T \to (T \to Prop)) \def -| sty0_sort: \forall (c: C).(\forall (n: nat).(sty0 g c (TSort n) (TSort -(next g n)))) -| sty0_abbr: \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: -nat).((getl i c (CHead d (Bind Abbr) v)) \to (\forall (w: T).((sty0 g d v w) -\to (sty0 g c (TLRef i) (lift (S i) O w)))))))) -| sty0_abst: \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: -nat).((getl i c (CHead d (Bind Abst) v)) \to (\forall (w: T).((sty0 g d v w) -\to (sty0 g c (TLRef i) (lift (S i) O v)))))))) -| sty0_bind: \forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t1: -T).(\forall (t2: T).((sty0 g (CHead c (Bind b) v) t1 t2) \to (sty0 g c (THead -(Bind b) v t1) (THead (Bind b) v t2))))))) -| sty0_appl: \forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall (t2: -T).((sty0 g c t1 t2) \to (sty0 g c (THead (Flat Appl) v t1) (THead (Flat -Appl) v t2)))))) -| sty0_cast: \forall (c: C).(\forall (v1: T).(\forall (v2: T).((sty0 g c v1 -v2) \to (\forall (t1: T).(\forall (t2: T).((sty0 g c t1 t2) \to (sty0 g c -(THead (Flat Cast) v1 t1) (THead (Flat Cast) v2 t2)))))))). - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/sty0/fwd.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/sty0/fwd.ma deleted file mode 100644 index 134ec3c10..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/sty0/fwd.ma +++ /dev/null @@ -1,562 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/sty0/defs.ma". - -theorem sty0_gen_sort: - \forall (g: G).(\forall (c: C).(\forall (x: T).(\forall (n: nat).((sty0 g c -(TSort n) x) \to (eq T x (TSort (next g n))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda -(H: (sty0 g c (TSort n) x)).(insert_eq T (TSort n) (\lambda (t: T).(sty0 g c -t x)) (\lambda (_: T).(eq T x (TSort (next g n)))) (\lambda (y: T).(\lambda -(H0: (sty0 g c y x)).(sty0_ind g (\lambda (_: C).(\lambda (t: T).(\lambda -(t0: T).((eq T t (TSort n)) \to (eq T t0 (TSort (next g n))))))) (\lambda (_: -C).(\lambda (n0: nat).(\lambda (H1: (eq T (TSort n0) (TSort n))).(let H2 \def -(f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with -[(TSort n1) \Rightarrow n1 | (TLRef _) \Rightarrow n0 | (THead _ _ _) -\Rightarrow n0])) (TSort n0) (TSort n) H1) in (eq_ind_r nat n (\lambda (n1: -nat).(eq T (TSort (next g n1)) (TSort (next g n)))) (refl_equal T (TSort -(next g n))) n0 H2))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (v: -T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind Abbr) -v))).(\lambda (w: T).(\lambda (_: (sty0 g d v w)).(\lambda (_: (((eq T v -(TSort n)) \to (eq T w (TSort (next g n)))))).(\lambda (H4: (eq T (TLRef i) -(TSort n))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (TSort n) H4) in -(False_ind (eq T (lift (S i) O w) (TSort (next g n))) H5))))))))))) (\lambda -(c0: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (_: (getl -i c0 (CHead d (Bind Abst) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v -w)).(\lambda (_: (((eq T v (TSort n)) \to (eq T w (TSort (next g -n)))))).(\lambda (H4: (eq T (TLRef i) (TSort n))).(let H5 \def (eq_ind T -(TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) -\Rightarrow False])) I (TSort n) H4) in (False_ind (eq T (lift (S i) O v) -(TSort (next g n))) H5))))))))))) (\lambda (b: B).(\lambda (c0: C).(\lambda -(v: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (sty0 g (CHead c0 (Bind -b) v) t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2 (TSort (next g -n)))))).(\lambda (H3: (eq T (THead (Bind b) v t1) (TSort n))).(let H4 \def -(eq_ind T (THead (Bind b) v t1) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H3) in -(False_ind (eq T (THead (Bind b) v t2) (TSort (next g n))) H4)))))))))) -(\lambda (c0: C).(\lambda (v: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(_: (sty0 g c0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2 (TSort -(next g n)))))).(\lambda (H3: (eq T (THead (Flat Appl) v t1) (TSort n))).(let -H4 \def (eq_ind T (THead (Flat Appl) v t1) (\lambda (ee: T).(match ee in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H3) in -(False_ind (eq T (THead (Flat Appl) v t2) (TSort (next g n))) H4))))))))) -(\lambda (c0: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (sty0 g c0 v1 -v2)).(\lambda (_: (((eq T v1 (TSort n)) \to (eq T v2 (TSort (next g -n)))))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (sty0 g c0 t1 -t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2 (TSort (next g -n)))))).(\lambda (H5: (eq T (THead (Flat Cast) v1 t1) (TSort n))).(let H6 -\def (eq_ind T (THead (Flat Cast) v1 t1) (\lambda (ee: T).(match ee in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H5) in -(False_ind (eq T (THead (Flat Cast) v2 t2) (TSort (next g n))) H6)))))))))))) -c y x H0))) H))))). -(* COMMENTS -Initial nodes: 869 -END *) - -theorem sty0_gen_lref: - \forall (g: G).(\forall (c: C).(\forall (x: T).(\forall (n: nat).((sty0 g c -(TLRef n) x) \to (or (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda -(_: T).(getl n c (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: -T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(t: T).(eq T x (lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda -(u: T).(\lambda (_: T).(getl n c (CHead e (Bind Abst) u))))) (\lambda (e: -C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq T x (lift (S n) O u))))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda -(H: (sty0 g c (TLRef n) x)).(insert_eq T (TLRef n) (\lambda (t: T).(sty0 g c -t x)) (\lambda (_: T).(or (ex3_3 C T T (\lambda (e: C).(\lambda (u: -T).(\lambda (_: T).(getl n c (CHead e (Bind Abbr) u))))) (\lambda (e: -C).(\lambda (u: T).(\lambda (t0: T).(sty0 g e u t0)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (t0: T).(eq T x (lift (S n) O t0)))))) (ex3_3 C T -T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead e (Bind -Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t0: T).(sty0 g e u -t0)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T x (lift (S n) O -u)))))))) (\lambda (y: T).(\lambda (H0: (sty0 g c y x)).(sty0_ind g (\lambda -(c0: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (TLRef n)) \to (or (ex3_3 C -T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind -Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t1: T).(sty0 g e u -t1)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t1: T).(eq T t0 (lift (S n) -O t1)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl -n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda -(t1: T).(sty0 g e u t1)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: -T).(eq T t0 (lift (S n) O u))))))))))) (\lambda (c0: C).(\lambda (n0: -nat).(\lambda (H1: (eq T (TSort n0) (TLRef n))).(let H2 \def (eq_ind T (TSort -n0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort -_) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow -False])) I (TLRef n) H1) in (False_ind (or (ex3_3 C T T (\lambda (e: -C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u))))) -(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (t: T).(eq T (TSort (next g n0)) (lift (S n) -O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl -n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: -T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T -(TSort (next g n0)) (lift (S n) O u))))))) H2))))) (\lambda (c0: C).(\lambda -(d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d -(Bind Abbr) v))).(\lambda (w: T).(\lambda (H2: (sty0 g d v w)).(\lambda (_: -(((eq T v (TLRef n)) \to (or (ex3_3 C T T (\lambda (e: C).(\lambda (u: -T).(\lambda (_: T).(getl n d (CHead e (Bind Abbr) u))))) (\lambda (e: -C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda -(_: T).(\lambda (t: T).(eq T w (lift (S n) O t)))))) (ex3_3 C T T (\lambda -(e: C).(\lambda (u: T).(\lambda (_: T).(getl n d (CHead e (Bind Abst) u))))) -(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda -(_: C).(\lambda (u: T).(\lambda (_: T).(eq T w (lift (S n) O -u)))))))))).(\lambda (H4: (eq T (TLRef i) (TLRef n))).(let H5 \def (f_equal T -nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort _) -\Rightarrow i | (TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow i])) -(TLRef i) (TLRef n) H4) in (let H6 \def (eq_ind nat i (\lambda (n0: -nat).(getl n0 c0 (CHead d (Bind Abbr) v))) H1 n H5) in (eq_ind_r nat n -(\lambda (n0: nat).(or (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda -(_: T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: -T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(t: T).(eq T (lift (S n0) O w) (lift (S n) O t)))))) (ex3_3 C T T (\lambda -(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) -(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda -(_: C).(\lambda (u: T).(\lambda (_: T).(eq T (lift (S n0) O w) (lift (S n) O -u)))))))) (or_introl (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda -(_: T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: -T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(t: T).(eq T (lift (S n) O w) (lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: -C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) -(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda -(_: C).(\lambda (u: T).(\lambda (_: T).(eq T (lift (S n) O w) (lift (S n) O -u)))))) (ex3_3_intro C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: -T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: -T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(t: T).(eq T (lift (S n) O w) (lift (S n) O t))))) d v w H6 H2 (refl_equal T -(lift (S n) O w)))) i H5)))))))))))) (\lambda (c0: C).(\lambda (d: -C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d (Bind -Abst) v))).(\lambda (w: T).(\lambda (H2: (sty0 g d v w)).(\lambda (_: (((eq T -v (TLRef n)) \to (or (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda -(_: T).(getl n d (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: -T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(t: T).(eq T w (lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda -(u: T).(\lambda (_: T).(getl n d (CHead e (Bind Abst) u))))) (\lambda (e: -C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq T w (lift (S n) O u)))))))))).(\lambda (H4: (eq T -(TLRef i) (TLRef n))).(let H5 \def (f_equal T nat (\lambda (e: T).(match e in -T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow i | (TLRef n0) -\Rightarrow n0 | (THead _ _ _) \Rightarrow i])) (TLRef i) (TLRef n) H4) in -(let H6 \def (eq_ind nat i (\lambda (n0: nat).(getl n0 c0 (CHead d (Bind -Abst) v))) H1 n H5) in (eq_ind_r nat n (\lambda (n0: nat).(or (ex3_3 C T T -(\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind -Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u -t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T (lift (S n0) O v) -(lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda -(_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: -T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda -(_: T).(eq T (lift (S n0) O v) (lift (S n) O u)))))))) (or_intror (ex3_3 C T -T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind -Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u -t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T (lift (S n) O v) -(lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda -(_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: -T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda -(_: T).(eq T (lift (S n) O v) (lift (S n) O u)))))) (ex3_3_intro C T T -(\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind -Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u -t)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T (lift (S n) O v) -(lift (S n) O u))))) d v w H6 H2 (refl_equal T (lift (S n) O v)))) i -H5)))))))))))) (\lambda (b: B).(\lambda (c0: C).(\lambda (v: T).(\lambda (t1: -T).(\lambda (t2: T).(\lambda (_: (sty0 g (CHead c0 (Bind b) v) t1 -t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (or (ex3_3 C T T (\lambda (e: -C).(\lambda (u: T).(\lambda (_: T).(getl n (CHead c0 (Bind b) v) (CHead e -(Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e -u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T t2 (lift (S n) -O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl -n (CHead c0 (Bind b) v) (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda -(u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(eq T t2 (lift (S n) O u)))))))))).(\lambda (H3: (eq T -(THead (Bind b) v t1) (TLRef n))).(let H4 \def (eq_ind T (THead (Bind b) v -t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort -_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) -\Rightarrow True])) I (TLRef n) H3) in (False_ind (or (ex3_3 C T T (\lambda -(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u))))) -(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (t: T).(eq T (THead (Bind b) v t2) (lift (S -n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: -T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: -T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda -(_: T).(eq T (THead (Bind b) v t2) (lift (S n) O u))))))) H4)))))))))) -(\lambda (c0: C).(\lambda (v: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(_: (sty0 g c0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (or (ex3_3 C T -T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind -Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u -t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T t2 (lift (S n) O -t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n -c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: -T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T t2 -(lift (S n) O u)))))))))).(\lambda (H3: (eq T (THead (Flat Appl) v t1) (TLRef -n))).(let H4 \def (eq_ind T (THead (Flat Appl) v t1) (\lambda (ee: T).(match -ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | -(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) -H3) in (False_ind (or (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda -(_: T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: -T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(t: T).(eq T (THead (Flat Appl) v t2) (lift (S n) O t)))))) (ex3_3 C T T -(\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind -Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u -t)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T (THead (Flat -Appl) v t2) (lift (S n) O u))))))) H4))))))))) (\lambda (c0: C).(\lambda (v1: -T).(\lambda (v2: T).(\lambda (_: (sty0 g c0 v1 v2)).(\lambda (_: (((eq T v1 -(TLRef n)) \to (or (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: -T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: -T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(t: T).(eq T v2 (lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda -(u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: -C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq T v2 (lift (S n) O u)))))))))).(\lambda (t1: -T).(\lambda (t2: T).(\lambda (_: (sty0 g c0 t1 t2)).(\lambda (_: (((eq T t1 -(TLRef n)) \to (or (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: -T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: -T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(t: T).(eq T t2 (lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda -(u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: -C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq T t2 (lift (S n) O u)))))))))).(\lambda (H5: (eq T -(THead (Flat Cast) v1 t1) (TLRef n))).(let H6 \def (eq_ind T (THead (Flat -Cast) v1 t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) -with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ -_) \Rightarrow True])) I (TLRef n) H5) in (False_ind (or (ex3_3 C T T -(\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind -Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u -t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T (THead (Flat -Cast) v2 t2) (lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: -T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: -C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq T (THead (Flat Cast) v2 t2) (lift (S n) O u))))))) -H6)))))))))))) c y x H0))) H))))). -(* COMMENTS -Initial nodes: 3231 -END *) - -theorem sty0_gen_bind: - \forall (g: G).(\forall (b: B).(\forall (c: C).(\forall (u: T).(\forall (t1: -T).(\forall (x: T).((sty0 g c (THead (Bind b) u t1) x) \to (ex2 T (\lambda -(t2: T).(sty0 g (CHead c (Bind b) u) t1 t2)) (\lambda (t2: T).(eq T x (THead -(Bind b) u t2)))))))))) -\def - \lambda (g: G).(\lambda (b: B).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: -T).(\lambda (x: T).(\lambda (H: (sty0 g c (THead (Bind b) u t1) -x)).(insert_eq T (THead (Bind b) u t1) (\lambda (t: T).(sty0 g c t x)) -(\lambda (_: T).(ex2 T (\lambda (t2: T).(sty0 g (CHead c (Bind b) u) t1 t2)) -(\lambda (t2: T).(eq T x (THead (Bind b) u t2))))) (\lambda (y: T).(\lambda -(H0: (sty0 g c y x)).(sty0_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda -(t0: T).((eq T t (THead (Bind b) u t1)) \to (ex2 T (\lambda (t2: T).(sty0 g -(CHead c0 (Bind b) u) t1 t2)) (\lambda (t2: T).(eq T t0 (THead (Bind b) u -t2)))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (H1: (eq T (TSort n) -(THead (Bind b) u t1))).(let H2 \def (eq_ind T (TSort n) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I -(THead (Bind b) u t1) H1) in (False_ind (ex2 T (\lambda (t2: T).(sty0 g -(CHead c0 (Bind b) u) t1 t2)) (\lambda (t2: T).(eq T (TSort (next g n)) -(THead (Bind b) u t2)))) H2))))) (\lambda (c0: C).(\lambda (d: C).(\lambda -(v: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind Abbr) -v))).(\lambda (w: T).(\lambda (_: (sty0 g d v w)).(\lambda (_: (((eq T v -(THead (Bind b) u t1)) \to (ex2 T (\lambda (t2: T).(sty0 g (CHead d (Bind b) -u) t1 t2)) (\lambda (t2: T).(eq T w (THead (Bind b) u t2))))))).(\lambda (H4: -(eq T (TLRef i) (THead (Bind b) u t1))).(let H5 \def (eq_ind T (TLRef i) -(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow -False])) I (THead (Bind b) u t1) H4) in (False_ind (ex2 T (\lambda (t2: -T).(sty0 g (CHead c0 (Bind b) u) t1 t2)) (\lambda (t2: T).(eq T (lift (S i) O -w) (THead (Bind b) u t2)))) H5))))))))))) (\lambda (c0: C).(\lambda (d: -C).(\lambda (v: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind -Abst) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v w)).(\lambda (_: (((eq T -v (THead (Bind b) u t1)) \to (ex2 T (\lambda (t2: T).(sty0 g (CHead d (Bind -b) u) t1 t2)) (\lambda (t2: T).(eq T w (THead (Bind b) u t2))))))).(\lambda -(H4: (eq T (TLRef i) (THead (Bind b) u t1))).(let H5 \def (eq_ind T (TLRef i) -(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow -False])) I (THead (Bind b) u t1) H4) in (False_ind (ex2 T (\lambda (t2: -T).(sty0 g (CHead c0 (Bind b) u) t1 t2)) (\lambda (t2: T).(eq T (lift (S i) O -v) (THead (Bind b) u t2)))) H5))))))))))) (\lambda (b0: B).(\lambda (c0: -C).(\lambda (v: T).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H1: (sty0 g -(CHead c0 (Bind b0) v) t0 t2)).(\lambda (H2: (((eq T t0 (THead (Bind b) u -t1)) \to (ex2 T (\lambda (t3: T).(sty0 g (CHead (CHead c0 (Bind b0) v) (Bind -b) u) t1 t3)) (\lambda (t3: T).(eq T t2 (THead (Bind b) u t3))))))).(\lambda -(H3: (eq T (THead (Bind b0) v t0) (THead (Bind b) u t1))).(let H4 \def -(f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with -[(TSort _) \Rightarrow b0 | (TLRef _) \Rightarrow b0 | (THead k _ _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b1) -\Rightarrow b1 | (Flat _) \Rightarrow b0])])) (THead (Bind b0) v t0) (THead -(Bind b) u t1) H3) in ((let H5 \def (f_equal T T (\lambda (e: T).(match e in -T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v | (TLRef _) -\Rightarrow v | (THead _ t _) \Rightarrow t])) (THead (Bind b0) v t0) (THead -(Bind b) u t1) H3) in ((let H6 \def (f_equal T T (\lambda (e: T).(match e in -T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) -\Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Bind b0) v t0) (THead -(Bind b) u t1) H3) in (\lambda (H7: (eq T v u)).(\lambda (H8: (eq B b0 -b)).(let H9 \def (eq_ind T t0 (\lambda (t: T).((eq T t (THead (Bind b) u t1)) -\to (ex2 T (\lambda (t3: T).(sty0 g (CHead (CHead c0 (Bind b0) v) (Bind b) u) -t1 t3)) (\lambda (t3: T).(eq T t2 (THead (Bind b) u t3)))))) H2 t1 H6) in -(let H10 \def (eq_ind T t0 (\lambda (t: T).(sty0 g (CHead c0 (Bind b0) v) t -t2)) H1 t1 H6) in (let H11 \def (eq_ind T v (\lambda (t: T).((eq T t1 (THead -(Bind b) u t1)) \to (ex2 T (\lambda (t3: T).(sty0 g (CHead (CHead c0 (Bind -b0) t) (Bind b) u) t1 t3)) (\lambda (t3: T).(eq T t2 (THead (Bind b) u -t3)))))) H9 u H7) in (let H12 \def (eq_ind T v (\lambda (t: T).(sty0 g (CHead -c0 (Bind b0) t) t1 t2)) H10 u H7) in (eq_ind_r T u (\lambda (t: T).(ex2 T -(\lambda (t3: T).(sty0 g (CHead c0 (Bind b) u) t1 t3)) (\lambda (t3: T).(eq T -(THead (Bind b0) t t2) (THead (Bind b) u t3))))) (let H13 \def (eq_ind B b0 -(\lambda (b1: B).((eq T t1 (THead (Bind b) u t1)) \to (ex2 T (\lambda (t3: -T).(sty0 g (CHead (CHead c0 (Bind b1) u) (Bind b) u) t1 t3)) (\lambda (t3: -T).(eq T t2 (THead (Bind b) u t3)))))) H11 b H8) in (let H14 \def (eq_ind B -b0 (\lambda (b1: B).(sty0 g (CHead c0 (Bind b1) u) t1 t2)) H12 b H8) in -(eq_ind_r B b (\lambda (b1: B).(ex2 T (\lambda (t3: T).(sty0 g (CHead c0 -(Bind b) u) t1 t3)) (\lambda (t3: T).(eq T (THead (Bind b1) u t2) (THead -(Bind b) u t3))))) (ex_intro2 T (\lambda (t3: T).(sty0 g (CHead c0 (Bind b) -u) t1 t3)) (\lambda (t3: T).(eq T (THead (Bind b) u t2) (THead (Bind b) u -t3))) t2 H14 (refl_equal T (THead (Bind b) u t2))) b0 H8))) v H7)))))))) H5)) -H4)))))))))) (\lambda (c0: C).(\lambda (v: T).(\lambda (t0: T).(\lambda (t2: -T).(\lambda (_: (sty0 g c0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind b) u -t1)) \to (ex2 T (\lambda (t3: T).(sty0 g (CHead c0 (Bind b) u) t1 t3)) -(\lambda (t3: T).(eq T t2 (THead (Bind b) u t3))))))).(\lambda (H3: (eq T -(THead (Flat Appl) v t0) (THead (Bind b) u t1))).(let H4 \def (eq_ind T -(THead (Flat Appl) v t0) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind -b) u t1) H3) in (False_ind (ex2 T (\lambda (t3: T).(sty0 g (CHead c0 (Bind b) -u) t1 t3)) (\lambda (t3: T).(eq T (THead (Flat Appl) v t2) (THead (Bind b) u -t3)))) H4))))))))) (\lambda (c0: C).(\lambda (v1: T).(\lambda (v2: -T).(\lambda (_: (sty0 g c0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind b) u -t1)) \to (ex2 T (\lambda (t2: T).(sty0 g (CHead c0 (Bind b) u) t1 t2)) -(\lambda (t2: T).(eq T v2 (THead (Bind b) u t2))))))).(\lambda (t0: -T).(\lambda (t2: T).(\lambda (_: (sty0 g c0 t0 t2)).(\lambda (_: (((eq T t0 -(THead (Bind b) u t1)) \to (ex2 T (\lambda (t3: T).(sty0 g (CHead c0 (Bind b) -u) t1 t3)) (\lambda (t3: T).(eq T t2 (THead (Bind b) u t3))))))).(\lambda -(H5: (eq T (THead (Flat Cast) v1 t0) (THead (Bind b) u t1))).(let H6 \def -(eq_ind T (THead (Flat Cast) v1 t0) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind b) u t1) H5) in (False_ind (ex2 T (\lambda (t3: -T).(sty0 g (CHead c0 (Bind b) u) t1 t3)) (\lambda (t3: T).(eq T (THead (Flat -Cast) v2 t2) (THead (Bind b) u t3)))) H6)))))))))))) c y x H0))) H))))))). -(* COMMENTS -Initial nodes: 1975 -END *) - -theorem sty0_gen_appl: - \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (x: -T).((sty0 g c (THead (Flat Appl) u t1) x) \to (ex2 T (\lambda (t2: T).(sty0 g -c t1 t2)) (\lambda (t2: T).(eq T x (THead (Flat Appl) u t2))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (x: -T).(\lambda (H: (sty0 g c (THead (Flat Appl) u t1) x)).(insert_eq T (THead -(Flat Appl) u t1) (\lambda (t: T).(sty0 g c t x)) (\lambda (_: T).(ex2 T -(\lambda (t2: T).(sty0 g c t1 t2)) (\lambda (t2: T).(eq T x (THead (Flat -Appl) u t2))))) (\lambda (y: T).(\lambda (H0: (sty0 g c y x)).(sty0_ind g -(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Flat Appl) -u t1)) \to (ex2 T (\lambda (t2: T).(sty0 g c0 t1 t2)) (\lambda (t2: T).(eq T -t0 (THead (Flat Appl) u t2)))))))) (\lambda (c0: C).(\lambda (n: -nat).(\lambda (H1: (eq T (TSort n) (THead (Flat Appl) u t1))).(let H2 \def -(eq_ind T (TSort n) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | -(THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) u t1) H1) in -(False_ind (ex2 T (\lambda (t2: T).(sty0 g c0 t1 t2)) (\lambda (t2: T).(eq T -(TSort (next g n)) (THead (Flat Appl) u t2)))) H2))))) (\lambda (c0: -C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (_: (getl i c0 -(CHead d (Bind Abbr) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v -w)).(\lambda (_: (((eq T v (THead (Flat Appl) u t1)) \to (ex2 T (\lambda (t2: -T).(sty0 g d t1 t2)) (\lambda (t2: T).(eq T w (THead (Flat Appl) u -t2))))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat Appl) u t1))).(let H5 -\def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | -(THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) u t1) H4) in -(False_ind (ex2 T (\lambda (t2: T).(sty0 g c0 t1 t2)) (\lambda (t2: T).(eq T -(lift (S i) O w) (THead (Flat Appl) u t2)))) H5))))))))))) (\lambda (c0: -C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (_: (getl i c0 -(CHead d (Bind Abst) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v -w)).(\lambda (_: (((eq T v (THead (Flat Appl) u t1)) \to (ex2 T (\lambda (t2: -T).(sty0 g d t1 t2)) (\lambda (t2: T).(eq T w (THead (Flat Appl) u -t2))))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat Appl) u t1))).(let H5 -\def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | -(THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) u t1) H4) in -(False_ind (ex2 T (\lambda (t2: T).(sty0 g c0 t1 t2)) (\lambda (t2: T).(eq T -(lift (S i) O v) (THead (Flat Appl) u t2)))) H5))))))))))) (\lambda (b: -B).(\lambda (c0: C).(\lambda (v: T).(\lambda (t0: T).(\lambda (t2: -T).(\lambda (_: (sty0 g (CHead c0 (Bind b) v) t0 t2)).(\lambda (_: (((eq T t0 -(THead (Flat Appl) u t1)) \to (ex2 T (\lambda (t3: T).(sty0 g (CHead c0 (Bind -b) v) t1 t3)) (\lambda (t3: T).(eq T t2 (THead (Flat Appl) u -t3))))))).(\lambda (H3: (eq T (THead (Bind b) v t0) (THead (Flat Appl) u -t1))).(let H4 \def (eq_ind T (THead (Bind b) v t0) (\lambda (ee: T).(match ee -in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef -_) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return -(\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow -False])])) I (THead (Flat Appl) u t1) H3) in (False_ind (ex2 T (\lambda (t3: -T).(sty0 g c0 t1 t3)) (\lambda (t3: T).(eq T (THead (Bind b) v t2) (THead -(Flat Appl) u t3)))) H4)))))))))) (\lambda (c0: C).(\lambda (v: T).(\lambda -(t0: T).(\lambda (t2: T).(\lambda (H1: (sty0 g c0 t0 t2)).(\lambda (H2: (((eq -T t0 (THead (Flat Appl) u t1)) \to (ex2 T (\lambda (t3: T).(sty0 g c0 t1 t3)) -(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u t3))))))).(\lambda (H3: (eq T -(THead (Flat Appl) v t0) (THead (Flat Appl) u t1))).(let H4 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow v | (TLRef _) \Rightarrow v | (THead _ t _) \Rightarrow t])) -(THead (Flat Appl) v t0) (THead (Flat Appl) u t1) H3) in ((let H5 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) -\Rightarrow t])) (THead (Flat Appl) v t0) (THead (Flat Appl) u t1) H3) in -(\lambda (H6: (eq T v u)).(let H7 \def (eq_ind T t0 (\lambda (t: T).((eq T t -(THead (Flat Appl) u t1)) \to (ex2 T (\lambda (t3: T).(sty0 g c0 t1 t3)) -(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u t3)))))) H2 t1 H5) in (let H8 -\def (eq_ind T t0 (\lambda (t: T).(sty0 g c0 t t2)) H1 t1 H5) in (eq_ind_r T -u (\lambda (t: T).(ex2 T (\lambda (t3: T).(sty0 g c0 t1 t3)) (\lambda (t3: -T).(eq T (THead (Flat Appl) t t2) (THead (Flat Appl) u t3))))) (ex_intro2 T -(\lambda (t3: T).(sty0 g c0 t1 t3)) (\lambda (t3: T).(eq T (THead (Flat Appl) -u t2) (THead (Flat Appl) u t3))) t2 H8 (refl_equal T (THead (Flat Appl) u -t2))) v H6))))) H4))))))))) (\lambda (c0: C).(\lambda (v1: T).(\lambda (v2: -T).(\lambda (_: (sty0 g c0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Flat Appl) -u t1)) \to (ex2 T (\lambda (t2: T).(sty0 g c0 t1 t2)) (\lambda (t2: T).(eq T -v2 (THead (Flat Appl) u t2))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda -(_: (sty0 g c0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u t1)) \to -(ex2 T (\lambda (t3: T).(sty0 g c0 t1 t3)) (\lambda (t3: T).(eq T t2 (THead -(Flat Appl) u t3))))))).(\lambda (H5: (eq T (THead (Flat Cast) v1 t0) (THead -(Flat Appl) u t1))).(let H6 \def (eq_ind T (THead (Flat Cast) v1 t0) (\lambda -(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow -(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | -(Flat f) \Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl -\Rightarrow False | Cast \Rightarrow True])])])) I (THead (Flat Appl) u t1) -H5) in (False_ind (ex2 T (\lambda (t3: T).(sty0 g c0 t1 t3)) (\lambda (t3: -T).(eq T (THead (Flat Cast) v2 t2) (THead (Flat Appl) u t3)))) H6)))))))))))) -c y x H0))) H)))))). -(* COMMENTS -Initial nodes: 1489 -END *) - -theorem sty0_gen_cast: - \forall (g: G).(\forall (c: C).(\forall (v1: T).(\forall (t1: T).(\forall -(x: T).((sty0 g c (THead (Flat Cast) v1 t1) x) \to (ex3_2 T T (\lambda (v2: -T).(\lambda (_: T).(sty0 g c v1 v2))) (\lambda (_: T).(\lambda (t2: T).(sty0 -g c t1 t2))) (\lambda (v2: T).(\lambda (t2: T).(eq T x (THead (Flat Cast) v2 -t2)))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (v1: T).(\lambda (t1: T).(\lambda -(x: T).(\lambda (H: (sty0 g c (THead (Flat Cast) v1 t1) x)).(insert_eq T -(THead (Flat Cast) v1 t1) (\lambda (t: T).(sty0 g c t x)) (\lambda (_: -T).(ex3_2 T T (\lambda (v2: T).(\lambda (_: T).(sty0 g c v1 v2))) (\lambda -(_: T).(\lambda (t2: T).(sty0 g c t1 t2))) (\lambda (v2: T).(\lambda (t2: -T).(eq T x (THead (Flat Cast) v2 t2)))))) (\lambda (y: T).(\lambda (H0: (sty0 -g c y x)).(sty0_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq -T t (THead (Flat Cast) v1 t1)) \to (ex3_2 T T (\lambda (v2: T).(\lambda (_: -T).(sty0 g c0 v1 v2))) (\lambda (_: T).(\lambda (t2: T).(sty0 g c0 t1 t2))) -(\lambda (v2: T).(\lambda (t2: T).(eq T t0 (THead (Flat Cast) v2 t2))))))))) -(\lambda (c0: C).(\lambda (n: nat).(\lambda (H1: (eq T (TSort n) (THead (Flat -Cast) v1 t1))).(let H2 \def (eq_ind T (TSort n) (\lambda (ee: T).(match ee in -T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) -v1 t1) H1) in (False_ind (ex3_2 T T (\lambda (v2: T).(\lambda (_: T).(sty0 g -c0 v1 v2))) (\lambda (_: T).(\lambda (t2: T).(sty0 g c0 t1 t2))) (\lambda -(v2: T).(\lambda (t2: T).(eq T (TSort (next g n)) (THead (Flat Cast) v2 -t2))))) H2))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: -nat).(\lambda (_: (getl i c0 (CHead d (Bind Abbr) v))).(\lambda (w: -T).(\lambda (_: (sty0 g d v w)).(\lambda (_: (((eq T v (THead (Flat Cast) v1 -t1)) \to (ex3_2 T T (\lambda (v2: T).(\lambda (_: T).(sty0 g d v1 v2))) -(\lambda (_: T).(\lambda (t2: T).(sty0 g d t1 t2))) (\lambda (v2: T).(\lambda -(t2: T).(eq T w (THead (Flat Cast) v2 t2)))))))).(\lambda (H4: (eq T (TLRef -i) (THead (Flat Cast) v1 t1))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I -(THead (Flat Cast) v1 t1) H4) in (False_ind (ex3_2 T T (\lambda (v2: -T).(\lambda (_: T).(sty0 g c0 v1 v2))) (\lambda (_: T).(\lambda (t2: T).(sty0 -g c0 t1 t2))) (\lambda (v2: T).(\lambda (t2: T).(eq T (lift (S i) O w) (THead -(Flat Cast) v2 t2))))) H5))))))))))) (\lambda (c0: C).(\lambda (d: -C).(\lambda (v: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind -Abst) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v w)).(\lambda (_: (((eq T -v (THead (Flat Cast) v1 t1)) \to (ex3_2 T T (\lambda (v2: T).(\lambda (_: -T).(sty0 g d v1 v2))) (\lambda (_: T).(\lambda (t2: T).(sty0 g d t1 t2))) -(\lambda (v2: T).(\lambda (t2: T).(eq T w (THead (Flat Cast) v2 -t2)))))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat Cast) v1 t1))).(let H5 -\def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | -(THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) v1 t1) H4) in -(False_ind (ex3_2 T T (\lambda (v2: T).(\lambda (_: T).(sty0 g c0 v1 v2))) -(\lambda (_: T).(\lambda (t2: T).(sty0 g c0 t1 t2))) (\lambda (v2: -T).(\lambda (t2: T).(eq T (lift (S i) O v) (THead (Flat Cast) v2 t2))))) -H5))))))))))) (\lambda (b: B).(\lambda (c0: C).(\lambda (v: T).(\lambda (t0: -T).(\lambda (t2: T).(\lambda (_: (sty0 g (CHead c0 (Bind b) v) t0 -t2)).(\lambda (_: (((eq T t0 (THead (Flat Cast) v1 t1)) \to (ex3_2 T T -(\lambda (v2: T).(\lambda (_: T).(sty0 g (CHead c0 (Bind b) v) v1 v2))) -(\lambda (_: T).(\lambda (t3: T).(sty0 g (CHead c0 (Bind b) v) t1 t3))) -(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) v2 -t3)))))))).(\lambda (H3: (eq T (THead (Bind b) v t0) (THead (Flat Cast) v1 -t1))).(let H4 \def (eq_ind T (THead (Bind b) v t0) (\lambda (ee: T).(match ee -in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef -_) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return -(\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow -False])])) I (THead (Flat Cast) v1 t1) H3) in (False_ind (ex3_2 T T (\lambda -(v2: T).(\lambda (_: T).(sty0 g c0 v1 v2))) (\lambda (_: T).(\lambda (t3: -T).(sty0 g c0 t1 t3))) (\lambda (v2: T).(\lambda (t3: T).(eq T (THead (Bind -b) v t2) (THead (Flat Cast) v2 t3))))) H4)))))))))) (\lambda (c0: C).(\lambda -(v: T).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (sty0 g c0 t0 -t2)).(\lambda (_: (((eq T t0 (THead (Flat Cast) v1 t1)) \to (ex3_2 T T -(\lambda (v2: T).(\lambda (_: T).(sty0 g c0 v1 v2))) (\lambda (_: T).(\lambda -(t3: T).(sty0 g c0 t1 t3))) (\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead -(Flat Cast) v2 t3)))))))).(\lambda (H3: (eq T (THead (Flat Appl) v t0) (THead -(Flat Cast) v1 t1))).(let H4 \def (eq_ind T (THead (Flat Appl) v t0) (\lambda -(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow -(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | -(Flat f) \Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl -\Rightarrow True | Cast \Rightarrow False])])])) I (THead (Flat Cast) v1 t1) -H3) in (False_ind (ex3_2 T T (\lambda (v2: T).(\lambda (_: T).(sty0 g c0 v1 -v2))) (\lambda (_: T).(\lambda (t3: T).(sty0 g c0 t1 t3))) (\lambda (v2: -T).(\lambda (t3: T).(eq T (THead (Flat Appl) v t2) (THead (Flat Cast) v2 -t3))))) H4))))))))) (\lambda (c0: C).(\lambda (v0: T).(\lambda (v2: -T).(\lambda (H1: (sty0 g c0 v0 v2)).(\lambda (H2: (((eq T v0 (THead (Flat -Cast) v1 t1)) \to (ex3_2 T T (\lambda (v3: T).(\lambda (_: T).(sty0 g c0 v1 -v3))) (\lambda (_: T).(\lambda (t2: T).(sty0 g c0 t1 t2))) (\lambda (v3: -T).(\lambda (t2: T).(eq T v2 (THead (Flat Cast) v3 t2)))))))).(\lambda (t0: -T).(\lambda (t2: T).(\lambda (H3: (sty0 g c0 t0 t2)).(\lambda (H4: (((eq T t0 -(THead (Flat Cast) v1 t1)) \to (ex3_2 T T (\lambda (v3: T).(\lambda (_: -T).(sty0 g c0 v1 v3))) (\lambda (_: T).(\lambda (t3: T).(sty0 g c0 t1 t3))) -(\lambda (v3: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) v3 -t3)))))))).(\lambda (H5: (eq T (THead (Flat Cast) v0 t0) (THead (Flat Cast) -v1 t1))).(let H6 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow v0 | (TLRef _) \Rightarrow v0 -| (THead _ t _) \Rightarrow t])) (THead (Flat Cast) v0 t0) (THead (Flat Cast) -v1 t1) H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 -| (THead _ _ t) \Rightarrow t])) (THead (Flat Cast) v0 t0) (THead (Flat Cast) -v1 t1) H5) in (\lambda (H8: (eq T v0 v1)).(let H9 \def (eq_ind T t0 (\lambda -(t: T).((eq T t (THead (Flat Cast) v1 t1)) \to (ex3_2 T T (\lambda (v3: -T).(\lambda (_: T).(sty0 g c0 v1 v3))) (\lambda (_: T).(\lambda (t3: T).(sty0 -g c0 t1 t3))) (\lambda (v3: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) -v3 t3))))))) H4 t1 H7) in (let H10 \def (eq_ind T t0 (\lambda (t: T).(sty0 g -c0 t t2)) H3 t1 H7) in (let H11 \def (eq_ind T v0 (\lambda (t: T).((eq T t -(THead (Flat Cast) v1 t1)) \to (ex3_2 T T (\lambda (v3: T).(\lambda (_: -T).(sty0 g c0 v1 v3))) (\lambda (_: T).(\lambda (t3: T).(sty0 g c0 t1 t3))) -(\lambda (v3: T).(\lambda (t3: T).(eq T v2 (THead (Flat Cast) v3 t3))))))) H2 -v1 H8) in (let H12 \def (eq_ind T v0 (\lambda (t: T).(sty0 g c0 t v2)) H1 v1 -H8) in (ex3_2_intro T T (\lambda (v3: T).(\lambda (_: T).(sty0 g c0 v1 v3))) -(\lambda (_: T).(\lambda (t3: T).(sty0 g c0 t1 t3))) (\lambda (v3: -T).(\lambda (t3: T).(eq T (THead (Flat Cast) v2 t2) (THead (Flat Cast) v3 -t3)))) v2 t2 H12 H10 (refl_equal T (THead (Flat Cast) v2 t2))))))))) -H6)))))))))))) c y x H0))) H)))))). -(* COMMENTS -Initial nodes: 1855 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/sty0/props.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/sty0/props.ma deleted file mode 100644 index 00c9f6818..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/sty0/props.ma +++ /dev/null @@ -1,217 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/sty0/defs.ma". - -include "Basic-1/getl/drop.ma". - -theorem sty0_lift: - \forall (g: G).(\forall (e: C).(\forall (t1: T).(\forall (t2: T).((sty0 g e -t1 t2) \to (\forall (c: C).(\forall (h: nat).(\forall (d: nat).((drop h d c -e) \to (sty0 g c (lift h d t1) (lift h d t2)))))))))) -\def - \lambda (g: G).(\lambda (e: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H: (sty0 g e t1 t2)).(sty0_ind g (\lambda (c: C).(\lambda (t: T).(\lambda -(t0: T).(\forall (c0: C).(\forall (h: nat).(\forall (d: nat).((drop h d c0 c) -\to (sty0 g c0 (lift h d t) (lift h d t0))))))))) (\lambda (c: C).(\lambda -(n: nat).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (_: -(drop h d c0 c)).(eq_ind_r T (TSort n) (\lambda (t: T).(sty0 g c0 t (lift h d -(TSort (next g n))))) (eq_ind_r T (TSort (next g n)) (\lambda (t: T).(sty0 g -c0 (TSort n) t)) (sty0_sort g c0 n) (lift h d (TSort (next g n))) (lift_sort -(next g n) h d)) (lift h d (TSort n)) (lift_sort n h d)))))))) (\lambda (c: -C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H0: (getl i c -(CHead d (Bind Abbr) v))).(\lambda (w: T).(\lambda (H1: (sty0 g d v -w)).(\lambda (H2: ((\forall (c0: C).(\forall (h: nat).(\forall (d0: -nat).((drop h d0 c0 d) \to (sty0 g c0 (lift h d0 v) (lift h d0 -w)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d0: nat).(\lambda (H3: -(drop h d0 c0 c)).(lt_le_e i d0 (sty0 g c0 (lift h d0 (TLRef i)) (lift h d0 -(lift (S i) O w))) (\lambda (H4: (lt i d0)).(let H5 \def (drop_getl_trans_le -i d0 (le_S_n i d0 (le_S (S i) d0 H4)) c0 c h H3 (CHead d (Bind Abbr) v) H0) -in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop i O c0 e0))) -(\lambda (e0: C).(\lambda (e1: C).(drop h (minus d0 i) e0 e1))) (\lambda (_: -C).(\lambda (e1: C).(clear e1 (CHead d (Bind Abbr) v)))) (sty0 g c0 (lift h -d0 (TLRef i)) (lift h d0 (lift (S i) O w))) (\lambda (x0: C).(\lambda (x1: -C).(\lambda (H6: (drop i O c0 x0)).(\lambda (H7: (drop h (minus d0 i) x0 -x1)).(\lambda (H8: (clear x1 (CHead d (Bind Abbr) v))).(let H9 \def (eq_ind -nat (minus d0 i) (\lambda (n: nat).(drop h n x0 x1)) H7 (S (minus d0 (S i))) -(minus_x_Sy d0 i H4)) in (let H10 \def (drop_clear_S x1 x0 h (minus d0 (S i)) -H9 Abbr d v H8) in (ex2_ind C (\lambda (c1: C).(clear x0 (CHead c1 (Bind -Abbr) (lift h (minus d0 (S i)) v)))) (\lambda (c1: C).(drop h (minus d0 (S -i)) c1 d)) (sty0 g c0 (lift h d0 (TLRef i)) (lift h d0 (lift (S i) O w))) -(\lambda (x: C).(\lambda (H11: (clear x0 (CHead x (Bind Abbr) (lift h (minus -d0 (S i)) v)))).(\lambda (H12: (drop h (minus d0 (S i)) x d)).(eq_ind_r T -(TLRef i) (\lambda (t: T).(sty0 g c0 t (lift h d0 (lift (S i) O w)))) (eq_ind -nat (plus (S i) (minus d0 (S i))) (\lambda (n: nat).(sty0 g c0 (TLRef i) -(lift h n (lift (S i) O w)))) (eq_ind_r T (lift (S i) O (lift h (minus d0 (S -i)) w)) (\lambda (t: T).(sty0 g c0 (TLRef i) t)) (eq_ind nat d0 (\lambda (_: -nat).(sty0 g c0 (TLRef i) (lift (S i) O (lift h (minus d0 (S i)) w)))) -(sty0_abbr g c0 x (lift h (minus d0 (S i)) v) i (getl_intro i c0 (CHead x -(Bind Abbr) (lift h (minus d0 (S i)) v)) x0 H6 H11) (lift h (minus d0 (S i)) -w) (H2 x h (minus d0 (S i)) H12)) (plus (S i) (minus d0 (S i))) -(le_plus_minus (S i) d0 H4)) (lift h (plus (S i) (minus d0 (S i))) (lift (S -i) O w)) (lift_d w h (S i) (minus d0 (S i)) O (le_O_n (minus d0 (S i))))) d0 -(le_plus_minus_r (S i) d0 H4)) (lift h d0 (TLRef i)) (lift_lref_lt i h d0 -H4))))) H10)))))))) H5))) (\lambda (H4: (le d0 i)).(eq_ind_r T (TLRef (plus i -h)) (\lambda (t: T).(sty0 g c0 t (lift h d0 (lift (S i) O w)))) (eq_ind nat -(S i) (\lambda (_: nat).(sty0 g c0 (TLRef (plus i h)) (lift h d0 (lift (S i) -O w)))) (eq_ind_r T (lift (plus h (S i)) O w) (\lambda (t: T).(sty0 g c0 -(TLRef (plus i h)) t)) (eq_ind_r nat (plus (S i) h) (\lambda (n: nat).(sty0 g -c0 (TLRef (plus i h)) (lift n O w))) (sty0_abbr g c0 d v (plus i h) -(drop_getl_trans_ge i c0 c d0 h H3 (CHead d (Bind Abbr) v) H0 H4) w H1) (plus -h (S i)) (plus_sym h (S i))) (lift h d0 (lift (S i) O w)) (lift_free w (S i) -h O d0 (le_S d0 i H4) (le_O_n d0))) (plus i (S O)) (eq_ind_r nat (plus (S O) -i) (\lambda (n: nat).(eq nat (S i) n)) (refl_equal nat (plus (S O) i)) (plus -i (S O)) (plus_sym i (S O)))) (lift h d0 (TLRef i)) (lift_lref_ge i h d0 -H4)))))))))))))))) (\lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda -(i: nat).(\lambda (H0: (getl i c (CHead d (Bind Abst) v))).(\lambda (w: -T).(\lambda (H1: (sty0 g d v w)).(\lambda (H2: ((\forall (c0: C).(\forall (h: -nat).(\forall (d0: nat).((drop h d0 c0 d) \to (sty0 g c0 (lift h d0 v) (lift -h d0 w)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d0: nat).(\lambda -(H3: (drop h d0 c0 c)).(lt_le_e i d0 (sty0 g c0 (lift h d0 (TLRef i)) (lift h -d0 (lift (S i) O v))) (\lambda (H4: (lt i d0)).(let H5 \def -(drop_getl_trans_le i d0 (le_S_n i d0 (le_S (S i) d0 H4)) c0 c h H3 (CHead d -(Bind Abst) v) H0) in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop i -O c0 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d0 i) e0 e1))) -(\lambda (_: C).(\lambda (e1: C).(clear e1 (CHead d (Bind Abst) v)))) (sty0 g -c0 (lift h d0 (TLRef i)) (lift h d0 (lift (S i) O v))) (\lambda (x0: -C).(\lambda (x1: C).(\lambda (H6: (drop i O c0 x0)).(\lambda (H7: (drop h -(minus d0 i) x0 x1)).(\lambda (H8: (clear x1 (CHead d (Bind Abst) v))).(let -H9 \def (eq_ind nat (minus d0 i) (\lambda (n: nat).(drop h n x0 x1)) H7 (S -(minus d0 (S i))) (minus_x_Sy d0 i H4)) in (let H10 \def (drop_clear_S x1 x0 -h (minus d0 (S i)) H9 Abst d v H8) in (ex2_ind C (\lambda (c1: C).(clear x0 -(CHead c1 (Bind Abst) (lift h (minus d0 (S i)) v)))) (\lambda (c1: C).(drop h -(minus d0 (S i)) c1 d)) (sty0 g c0 (lift h d0 (TLRef i)) (lift h d0 (lift (S -i) O v))) (\lambda (x: C).(\lambda (H11: (clear x0 (CHead x (Bind Abst) (lift -h (minus d0 (S i)) v)))).(\lambda (H12: (drop h (minus d0 (S i)) x -d)).(eq_ind_r T (TLRef i) (\lambda (t: T).(sty0 g c0 t (lift h d0 (lift (S i) -O v)))) (eq_ind nat (plus (S i) (minus d0 (S i))) (\lambda (n: nat).(sty0 g -c0 (TLRef i) (lift h n (lift (S i) O v)))) (eq_ind_r T (lift (S i) O (lift h -(minus d0 (S i)) v)) (\lambda (t: T).(sty0 g c0 (TLRef i) t)) (eq_ind nat d0 -(\lambda (_: nat).(sty0 g c0 (TLRef i) (lift (S i) O (lift h (minus d0 (S i)) -v)))) (sty0_abst g c0 x (lift h (minus d0 (S i)) v) i (getl_intro i c0 (CHead -x (Bind Abst) (lift h (minus d0 (S i)) v)) x0 H6 H11) (lift h (minus d0 (S -i)) w) (H2 x h (minus d0 (S i)) H12)) (plus (S i) (minus d0 (S i))) -(le_plus_minus (S i) d0 H4)) (lift h (plus (S i) (minus d0 (S i))) (lift (S -i) O v)) (lift_d v h (S i) (minus d0 (S i)) O (le_O_n (minus d0 (S i))))) d0 -(le_plus_minus_r (S i) d0 H4)) (lift h d0 (TLRef i)) (lift_lref_lt i h d0 -H4))))) H10)))))))) H5))) (\lambda (H4: (le d0 i)).(eq_ind_r T (TLRef (plus i -h)) (\lambda (t: T).(sty0 g c0 t (lift h d0 (lift (S i) O v)))) (eq_ind nat -(S i) (\lambda (_: nat).(sty0 g c0 (TLRef (plus i h)) (lift h d0 (lift (S i) -O v)))) (eq_ind_r T (lift (plus h (S i)) O v) (\lambda (t: T).(sty0 g c0 -(TLRef (plus i h)) t)) (eq_ind_r nat (plus (S i) h) (\lambda (n: nat).(sty0 g -c0 (TLRef (plus i h)) (lift n O v))) (sty0_abst g c0 d v (plus i h) -(drop_getl_trans_ge i c0 c d0 h H3 (CHead d (Bind Abst) v) H0 H4) w H1) (plus -h (S i)) (plus_sym h (S i))) (lift h d0 (lift (S i) O v)) (lift_free v (S i) -h O d0 (le_S d0 i H4) (le_O_n d0))) (plus i (S O)) (eq_ind_r nat (plus (S O) -i) (\lambda (n: nat).(eq nat (S i) n)) (refl_equal nat (plus (S O) i)) (plus -i (S O)) (plus_sym i (S O)))) (lift h d0 (TLRef i)) (lift_lref_ge i h d0 -H4)))))))))))))))) (\lambda (b: B).(\lambda (c: C).(\lambda (v: T).(\lambda -(t3: T).(\lambda (t4: T).(\lambda (_: (sty0 g (CHead c (Bind b) v) t3 -t4)).(\lambda (H1: ((\forall (c0: C).(\forall (h: nat).(\forall (d: -nat).((drop h d c0 (CHead c (Bind b) v)) \to (sty0 g c0 (lift h d t3) (lift h -d t4)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda -(H2: (drop h d c0 c)).(eq_ind_r T (THead (Bind b) (lift h d v) (lift h (s -(Bind b) d) t3)) (\lambda (t: T).(sty0 g c0 t (lift h d (THead (Bind b) v -t4)))) (eq_ind_r T (THead (Bind b) (lift h d v) (lift h (s (Bind b) d) t4)) -(\lambda (t: T).(sty0 g c0 (THead (Bind b) (lift h d v) (lift h (s (Bind b) -d) t3)) t)) (sty0_bind g b c0 (lift h d v) (lift h (S d) t3) (lift h (S d) -t4) (H1 (CHead c0 (Bind b) (lift h d v)) h (S d) (drop_skip_bind h d c0 c H2 -b v))) (lift h d (THead (Bind b) v t4)) (lift_head (Bind b) v t4 h d)) (lift -h d (THead (Bind b) v t3)) (lift_head (Bind b) v t3 h d))))))))))))) (\lambda -(c: C).(\lambda (v: T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (sty0 g -c t3 t4)).(\lambda (H1: ((\forall (c0: C).(\forall (h: nat).(\forall (d: -nat).((drop h d c0 c) \to (sty0 g c0 (lift h d t3) (lift h d -t4)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2: -(drop h d c0 c)).(eq_ind_r T (THead (Flat Appl) (lift h d v) (lift h (s (Flat -Appl) d) t3)) (\lambda (t: T).(sty0 g c0 t (lift h d (THead (Flat Appl) v -t4)))) (eq_ind_r T (THead (Flat Appl) (lift h d v) (lift h (s (Flat Appl) d) -t4)) (\lambda (t: T).(sty0 g c0 (THead (Flat Appl) (lift h d v) (lift h (s -(Flat Appl) d) t3)) t)) (sty0_appl g c0 (lift h d v) (lift h (s (Flat Appl) -d) t3) (lift h (s (Flat Appl) d) t4) (H1 c0 h (s (Flat Appl) d) H2)) (lift h -d (THead (Flat Appl) v t4)) (lift_head (Flat Appl) v t4 h d)) (lift h d -(THead (Flat Appl) v t3)) (lift_head (Flat Appl) v t3 h d)))))))))))) -(\lambda (c: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (sty0 g c v1 -v2)).(\lambda (H1: ((\forall (c0: C).(\forall (h: nat).(\forall (d: -nat).((drop h d c0 c) \to (sty0 g c0 (lift h d v1) (lift h d -v2)))))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (sty0 g c t3 -t4)).(\lambda (H3: ((\forall (c0: C).(\forall (h: nat).(\forall (d: -nat).((drop h d c0 c) \to (sty0 g c0 (lift h d t3) (lift h d -t4)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H4: -(drop h d c0 c)).(eq_ind_r T (THead (Flat Cast) (lift h d v1) (lift h (s -(Flat Cast) d) t3)) (\lambda (t: T).(sty0 g c0 t (lift h d (THead (Flat Cast) -v2 t4)))) (eq_ind_r T (THead (Flat Cast) (lift h d v2) (lift h (s (Flat Cast) -d) t4)) (\lambda (t: T).(sty0 g c0 (THead (Flat Cast) (lift h d v1) (lift h -(s (Flat Cast) d) t3)) t)) (sty0_cast g c0 (lift h d v1) (lift h d v2) (H1 c0 -h d H4) (lift h (s (Flat Cast) d) t3) (lift h (s (Flat Cast) d) t4) (H3 c0 h -(s (Flat Cast) d) H4)) (lift h d (THead (Flat Cast) v2 t4)) (lift_head (Flat -Cast) v2 t4 h d)) (lift h d (THead (Flat Cast) v1 t3)) (lift_head (Flat Cast) -v1 t3 h d))))))))))))))) e t1 t2 H))))). -(* COMMENTS -Initial nodes: 3677 -END *) - -theorem sty0_correct: - \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((sty0 g c -t1 t) \to (ex T (\lambda (t2: T).(sty0 g c t t2))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H: -(sty0 g c t1 t)).(sty0_ind g (\lambda (c0: C).(\lambda (_: T).(\lambda (t2: -T).(ex T (\lambda (t3: T).(sty0 g c0 t2 t3)))))) (\lambda (c0: C).(\lambda -(n: nat).(ex_intro T (\lambda (t2: T).(sty0 g c0 (TSort (next g n)) t2)) -(TSort (next g (next g n))) (sty0_sort g c0 (next g n))))) (\lambda (c0: -C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 -(CHead d (Bind Abbr) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v -w)).(\lambda (H2: (ex T (\lambda (t2: T).(sty0 g d w t2)))).(let H3 \def H2 -in (ex_ind T (\lambda (t2: T).(sty0 g d w t2)) (ex T (\lambda (t2: T).(sty0 g -c0 (lift (S i) O w) t2))) (\lambda (x: T).(\lambda (H4: (sty0 g d w -x)).(ex_intro T (\lambda (t2: T).(sty0 g c0 (lift (S i) O w) t2)) (lift (S i) -O x) (sty0_lift g d w x H4 c0 (S i) O (getl_drop Abbr c0 d v i H0))))) -H3)))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: -nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abst) v))).(\lambda (w: -T).(\lambda (H1: (sty0 g d v w)).(\lambda (H2: (ex T (\lambda (t2: T).(sty0 g -d w t2)))).(let H3 \def H2 in (ex_ind T (\lambda (t2: T).(sty0 g d w t2)) (ex -T (\lambda (t2: T).(sty0 g c0 (lift (S i) O v) t2))) (\lambda (x: T).(\lambda -(_: (sty0 g d w x)).(ex_intro T (\lambda (t2: T).(sty0 g c0 (lift (S i) O v) -t2)) (lift (S i) O w) (sty0_lift g d v w H1 c0 (S i) O (getl_drop Abst c0 d v -i H0))))) H3)))))))))) (\lambda (b: B).(\lambda (c0: C).(\lambda (v: -T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (sty0 g (CHead c0 (Bind b) -v) t2 t3)).(\lambda (H1: (ex T (\lambda (t4: T).(sty0 g (CHead c0 (Bind b) v) -t3 t4)))).(let H2 \def H1 in (ex_ind T (\lambda (t4: T).(sty0 g (CHead c0 -(Bind b) v) t3 t4)) (ex T (\lambda (t4: T).(sty0 g c0 (THead (Bind b) v t3) -t4))) (\lambda (x: T).(\lambda (H3: (sty0 g (CHead c0 (Bind b) v) t3 -x)).(ex_intro T (\lambda (t4: T).(sty0 g c0 (THead (Bind b) v t3) t4)) (THead -(Bind b) v x) (sty0_bind g b c0 v t3 x H3)))) H2))))))))) (\lambda (c0: -C).(\lambda (v: T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (sty0 g c0 -t2 t3)).(\lambda (H1: (ex T (\lambda (t4: T).(sty0 g c0 t3 t4)))).(let H2 -\def H1 in (ex_ind T (\lambda (t4: T).(sty0 g c0 t3 t4)) (ex T (\lambda (t4: -T).(sty0 g c0 (THead (Flat Appl) v t3) t4))) (\lambda (x: T).(\lambda (H3: -(sty0 g c0 t3 x)).(ex_intro T (\lambda (t4: T).(sty0 g c0 (THead (Flat Appl) -v t3) t4)) (THead (Flat Appl) v x) (sty0_appl g c0 v t3 x H3)))) H2)))))))) -(\lambda (c0: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (sty0 g c0 v1 -v2)).(\lambda (H1: (ex T (\lambda (t2: T).(sty0 g c0 v2 t2)))).(\lambda (t2: -T).(\lambda (t3: T).(\lambda (_: (sty0 g c0 t2 t3)).(\lambda (H3: (ex T -(\lambda (t4: T).(sty0 g c0 t3 t4)))).(let H4 \def H1 in (ex_ind T (\lambda -(t4: T).(sty0 g c0 v2 t4)) (ex T (\lambda (t4: T).(sty0 g c0 (THead (Flat -Cast) v2 t3) t4))) (\lambda (x: T).(\lambda (H5: (sty0 g c0 v2 x)).(let H6 -\def H3 in (ex_ind T (\lambda (t4: T).(sty0 g c0 t3 t4)) (ex T (\lambda (t4: -T).(sty0 g c0 (THead (Flat Cast) v2 t3) t4))) (\lambda (x0: T).(\lambda (H7: -(sty0 g c0 t3 x0)).(ex_intro T (\lambda (t4: T).(sty0 g c0 (THead (Flat Cast) -v2 t3) t4)) (THead (Flat Cast) x x0) (sty0_cast g c0 v2 x H5 t3 x0 H7)))) -H6)))) H4))))))))))) c t1 t H))))). -(* COMMENTS -Initial nodes: 991 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/sty1/cnt.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/sty1/cnt.ma deleted file mode 100644 index 0f39bdfd9..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/sty1/cnt.ma +++ /dev/null @@ -1,89 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/sty1/props.ma". - -include "Basic-1/cnt/props.ma". - -theorem sty1_cnt: - \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((sty0 g c -t1 t) \to (ex2 T (\lambda (t2: T).(sty1 g c t1 t2)) (\lambda (t2: T).(cnt -t2))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H: -(sty0 g c t1 t)).(sty0_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (_: -T).(ex2 T (\lambda (t3: T).(sty1 g c0 t0 t3)) (\lambda (t3: T).(cnt t3)))))) -(\lambda (c0: C).(\lambda (n: nat).(ex_intro2 T (\lambda (t2: T).(sty1 g c0 -(TSort n) t2)) (\lambda (t2: T).(cnt t2)) (TSort (next g n)) (sty1_sty0 g c0 -(TSort n) (TSort (next g n)) (sty0_sort g c0 n)) (cnt_sort (next g n))))) -(\lambda (c0: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda -(H0: (getl i c0 (CHead d (Bind Abbr) v))).(\lambda (w: T).(\lambda (_: (sty0 -g d v w)).(\lambda (H2: (ex2 T (\lambda (t2: T).(sty1 g d v t2)) (\lambda -(t2: T).(cnt t2)))).(let H3 \def H2 in (ex2_ind T (\lambda (t2: T).(sty1 g d -v t2)) (\lambda (t2: T).(cnt t2)) (ex2 T (\lambda (t2: T).(sty1 g c0 (TLRef -i) t2)) (\lambda (t2: T).(cnt t2))) (\lambda (x: T).(\lambda (H4: (sty1 g d v -x)).(\lambda (H5: (cnt x)).(ex_intro2 T (\lambda (t2: T).(sty1 g c0 (TLRef i) -t2)) (\lambda (t2: T).(cnt t2)) (lift (S i) O x) (sty1_abbr g c0 d v i H0 x -H4) (cnt_lift x H5 (S i) O))))) H3)))))))))) (\lambda (c0: C).(\lambda (d: -C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind -Abst) v))).(\lambda (w: T).(\lambda (H1: (sty0 g d v w)).(\lambda (H2: (ex2 T -(\lambda (t2: T).(sty1 g d v t2)) (\lambda (t2: T).(cnt t2)))).(let H3 \def -H2 in (ex2_ind T (\lambda (t2: T).(sty1 g d v t2)) (\lambda (t2: T).(cnt t2)) -(ex2 T (\lambda (t2: T).(sty1 g c0 (TLRef i) t2)) (\lambda (t2: T).(cnt t2))) -(\lambda (x: T).(\lambda (H4: (sty1 g d v x)).(\lambda (H5: (cnt -x)).(ex_intro2 T (\lambda (t2: T).(sty1 g c0 (TLRef i) t2)) (\lambda (t2: -T).(cnt t2)) (lift (S i) O x) (sty1_trans g c0 (TLRef i) (lift (S i) O v) -(sty1_sty0 g c0 (TLRef i) (lift (S i) O v) (sty0_abst g c0 d v i H0 w H1)) -(lift (S i) O x) (sty1_lift g d v x H4 c0 (S i) O (getl_drop Abst c0 d v i -H0))) (cnt_lift x H5 (S i) O))))) H3)))))))))) (\lambda (b: B).(\lambda (c0: -C).(\lambda (v: T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (sty0 g -(CHead c0 (Bind b) v) t2 t3)).(\lambda (H1: (ex2 T (\lambda (t4: T).(sty1 g -(CHead c0 (Bind b) v) t2 t4)) (\lambda (t4: T).(cnt t4)))).(let H2 \def H1 in -(ex2_ind T (\lambda (t4: T).(sty1 g (CHead c0 (Bind b) v) t2 t4)) (\lambda -(t4: T).(cnt t4)) (ex2 T (\lambda (t4: T).(sty1 g c0 (THead (Bind b) v t2) -t4)) (\lambda (t4: T).(cnt t4))) (\lambda (x: T).(\lambda (H3: (sty1 g (CHead -c0 (Bind b) v) t2 x)).(\lambda (H4: (cnt x)).(ex_intro2 T (\lambda (t4: -T).(sty1 g c0 (THead (Bind b) v t2) t4)) (\lambda (t4: T).(cnt t4)) (THead -(Bind b) v x) (sty1_bind g b c0 v t2 x H3) (cnt_head x H4 (Bind b) v))))) -H2))))))))) (\lambda (c0: C).(\lambda (v: T).(\lambda (t2: T).(\lambda (t3: -T).(\lambda (_: (sty0 g c0 t2 t3)).(\lambda (H1: (ex2 T (\lambda (t4: -T).(sty1 g c0 t2 t4)) (\lambda (t4: T).(cnt t4)))).(let H2 \def H1 in -(ex2_ind T (\lambda (t4: T).(sty1 g c0 t2 t4)) (\lambda (t4: T).(cnt t4)) -(ex2 T (\lambda (t4: T).(sty1 g c0 (THead (Flat Appl) v t2) t4)) (\lambda -(t4: T).(cnt t4))) (\lambda (x: T).(\lambda (H3: (sty1 g c0 t2 x)).(\lambda -(H4: (cnt x)).(ex_intro2 T (\lambda (t4: T).(sty1 g c0 (THead (Flat Appl) v -t2) t4)) (\lambda (t4: T).(cnt t4)) (THead (Flat Appl) v x) (sty1_appl g c0 v -t2 x H3) (cnt_head x H4 (Flat Appl) v))))) H2)))))))) (\lambda (c0: -C).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H0: (sty0 g c0 v1 -v2)).(\lambda (_: (ex2 T (\lambda (t2: T).(sty1 g c0 v1 t2)) (\lambda (t2: -T).(cnt t2)))).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (sty0 g c0 t2 -t3)).(\lambda (H3: (ex2 T (\lambda (t4: T).(sty1 g c0 t2 t4)) (\lambda (t4: -T).(cnt t4)))).(let H4 \def H3 in (ex2_ind T (\lambda (t4: T).(sty1 g c0 t2 -t4)) (\lambda (t4: T).(cnt t4)) (ex2 T (\lambda (t4: T).(sty1 g c0 (THead -(Flat Cast) v1 t2) t4)) (\lambda (t4: T).(cnt t4))) (\lambda (x: T).(\lambda -(H5: (sty1 g c0 t2 x)).(\lambda (H6: (cnt x)).(let H_x \def (sty1_cast2 g c0 -t2 x H5 v1 v2 H0) in (let H7 \def H_x in (ex2_ind T (\lambda (v3: T).(sty1 g -c0 v1 v3)) (\lambda (v3: T).(sty1 g c0 (THead (Flat Cast) v1 t2) (THead (Flat -Cast) v3 x))) (ex2 T (\lambda (t4: T).(sty1 g c0 (THead (Flat Cast) v1 t2) -t4)) (\lambda (t4: T).(cnt t4))) (\lambda (x0: T).(\lambda (_: (sty1 g c0 v1 -x0)).(\lambda (H9: (sty1 g c0 (THead (Flat Cast) v1 t2) (THead (Flat Cast) x0 -x))).(ex_intro2 T (\lambda (t4: T).(sty1 g c0 (THead (Flat Cast) v1 t2) t4)) -(\lambda (t4: T).(cnt t4)) (THead (Flat Cast) x0 x) H9 (cnt_head x H6 (Flat -Cast) x0))))) H7)))))) H4))))))))))) c t1 t H))))). -(* COMMENTS -Initial nodes: 1313 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/sty1/defs.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/sty1/defs.ma deleted file mode 100644 index 8fd219524..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/sty1/defs.ma +++ /dev/null @@ -1,23 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/sty0/defs.ma". - -inductive sty1 (g: G) (c: C) (t1: T): T \to Prop \def -| sty1_sty0: \forall (t2: T).((sty0 g c t1 t2) \to (sty1 g c t1 t2)) -| sty1_sing: \forall (t: T).((sty1 g c t1 t) \to (\forall (t2: T).((sty0 g c -t t2) \to (sty1 g c t1 t2)))). - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/sty1/props.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/sty1/props.ma deleted file mode 100644 index 53061952e..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/sty1/props.ma +++ /dev/null @@ -1,163 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/sty1/defs.ma". - -include "Basic-1/sty0/props.ma". - -theorem sty1_trans: - \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((sty1 g c -t1 t) \to (\forall (t2: T).((sty1 g c t t2) \to (sty1 g c t1 t2))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H: -(sty1 g c t1 t)).(\lambda (t2: T).(\lambda (H0: (sty1 g c t t2)).(sty1_ind g -c t (\lambda (t0: T).(sty1 g c t1 t0)) (\lambda (t3: T).(\lambda (H1: (sty0 g -c t t3)).(sty1_sing g c t1 t H t3 H1))) (\lambda (t0: T).(\lambda (_: (sty1 g -c t t0)).(\lambda (H2: (sty1 g c t1 t0)).(\lambda (t3: T).(\lambda (H3: (sty0 -g c t0 t3)).(sty1_sing g c t1 t0 H2 t3 H3)))))) t2 H0))))))). -(* COMMENTS -Initial nodes: 131 -END *) - -theorem sty1_bind: - \forall (g: G).(\forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t1: -T).(\forall (t2: T).((sty1 g (CHead c (Bind b) v) t1 t2) \to (sty1 g c (THead -(Bind b) v t1) (THead (Bind b) v t2)))))))) -\def - \lambda (g: G).(\lambda (b: B).(\lambda (c: C).(\lambda (v: T).(\lambda (t1: -T).(\lambda (t2: T).(\lambda (H: (sty1 g (CHead c (Bind b) v) t1 -t2)).(sty1_ind g (CHead c (Bind b) v) t1 (\lambda (t: T).(sty1 g c (THead -(Bind b) v t1) (THead (Bind b) v t))) (\lambda (t3: T).(\lambda (H0: (sty0 g -(CHead c (Bind b) v) t1 t3)).(sty1_sty0 g c (THead (Bind b) v t1) (THead -(Bind b) v t3) (sty0_bind g b c v t1 t3 H0)))) (\lambda (t: T).(\lambda (_: -(sty1 g (CHead c (Bind b) v) t1 t)).(\lambda (H1: (sty1 g c (THead (Bind b) v -t1) (THead (Bind b) v t))).(\lambda (t3: T).(\lambda (H2: (sty0 g (CHead c -(Bind b) v) t t3)).(sty1_sing g c (THead (Bind b) v t1) (THead (Bind b) v t) -H1 (THead (Bind b) v t3) (sty0_bind g b c v t t3 H2))))))) t2 H))))))). -(* COMMENTS -Initial nodes: 259 -END *) - -theorem sty1_appl: - \forall (g: G).(\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall -(t2: T).((sty1 g c t1 t2) \to (sty1 g c (THead (Flat Appl) v t1) (THead (Flat -Appl) v t2))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (v: T).(\lambda (t1: T).(\lambda -(t2: T).(\lambda (H: (sty1 g c t1 t2)).(sty1_ind g c t1 (\lambda (t: T).(sty1 -g c (THead (Flat Appl) v t1) (THead (Flat Appl) v t))) (\lambda (t3: -T).(\lambda (H0: (sty0 g c t1 t3)).(sty1_sty0 g c (THead (Flat Appl) v t1) -(THead (Flat Appl) v t3) (sty0_appl g c v t1 t3 H0)))) (\lambda (t: -T).(\lambda (_: (sty1 g c t1 t)).(\lambda (H1: (sty1 g c (THead (Flat Appl) v -t1) (THead (Flat Appl) v t))).(\lambda (t3: T).(\lambda (H2: (sty0 g c t -t3)).(sty1_sing g c (THead (Flat Appl) v t1) (THead (Flat Appl) v t) H1 -(THead (Flat Appl) v t3) (sty0_appl g c v t t3 H2))))))) t2 H)))))). -(* COMMENTS -Initial nodes: 213 -END *) - -theorem sty1_lift: - \forall (g: G).(\forall (e: C).(\forall (t1: T).(\forall (t2: T).((sty1 g e -t1 t2) \to (\forall (c: C).(\forall (h: nat).(\forall (d: nat).((drop h d c -e) \to (sty1 g c (lift h d t1) (lift h d t2)))))))))) -\def - \lambda (g: G).(\lambda (e: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H: (sty1 g e t1 t2)).(sty1_ind g e t1 (\lambda (t: T).(\forall (c: -C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to (sty1 g c (lift h -d t1) (lift h d t))))))) (\lambda (t3: T).(\lambda (H0: (sty0 g e t1 -t3)).(\lambda (c: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: (drop -h d c e)).(sty1_sty0 g c (lift h d t1) (lift h d t3) (sty0_lift g e t1 t3 H0 -c h d H1)))))))) (\lambda (t: T).(\lambda (_: (sty1 g e t1 t)).(\lambda (H1: -((\forall (c: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to -(sty1 g c (lift h d t1) (lift h d t)))))))).(\lambda (t3: T).(\lambda (H2: -(sty0 g e t t3)).(\lambda (c: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda -(H3: (drop h d c e)).(sty1_sing g c (lift h d t1) (lift h d t) (H1 c h d H3) -(lift h d t3) (sty0_lift g e t t3 H2 c h d H3))))))))))) t2 H))))). -(* COMMENTS -Initial nodes: 277 -END *) - -theorem sty1_correct: - \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((sty1 g c -t1 t) \to (ex T (\lambda (t2: T).(sty0 g c t t2))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H: -(sty1 g c t1 t)).(sty1_ind g c t1 (\lambda (t0: T).(ex T (\lambda (t2: -T).(sty0 g c t0 t2)))) (\lambda (t2: T).(\lambda (H0: (sty0 g c t1 -t2)).(sty0_correct g c t1 t2 H0))) (\lambda (t0: T).(\lambda (_: (sty1 g c t1 -t0)).(\lambda (_: (ex T (\lambda (t2: T).(sty0 g c t0 t2)))).(\lambda (t2: -T).(\lambda (H2: (sty0 g c t0 t2)).(sty0_correct g c t0 t2 H2)))))) t H))))). -(* COMMENTS -Initial nodes: 123 -END *) - -theorem sty1_abbr: - \forall (g: G).(\forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: -nat).((getl i c (CHead d (Bind Abbr) v)) \to (\forall (w: T).((sty1 g d v w) -\to (sty1 g c (TLRef i) (lift (S i) O w))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: -nat).(\lambda (H: (getl i c (CHead d (Bind Abbr) v))).(\lambda (w: -T).(\lambda (H0: (sty1 g d v w)).(sty1_ind g d v (\lambda (t: T).(sty1 g c -(TLRef i) (lift (S i) O t))) (\lambda (t2: T).(\lambda (H1: (sty0 g d v -t2)).(sty1_sty0 g c (TLRef i) (lift (S i) O t2) (sty0_abbr g c d v i H t2 -H1)))) (\lambda (t: T).(\lambda (_: (sty1 g d v t)).(\lambda (H2: (sty1 g c -(TLRef i) (lift (S i) O t))).(\lambda (t2: T).(\lambda (H3: (sty0 g d t -t2)).(sty1_sing g c (TLRef i) (lift (S i) O t) H2 (lift (S i) O t2) -(sty0_lift g d t t2 H3 c (S i) O (getl_drop Abbr c d v i H)))))))) w -H0)))))))). -(* COMMENTS -Initial nodes: 231 -END *) - -theorem sty1_cast2: - \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((sty1 g c -t1 t2) \to (\forall (v1: T).(\forall (v2: T).((sty0 g c v1 v2) \to (ex2 T -(\lambda (v3: T).(sty1 g c v1 v3)) (\lambda (v3: T).(sty1 g c (THead (Flat -Cast) v1 t1) (THead (Flat Cast) v3 t2))))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H: (sty1 g c t1 t2)).(sty1_ind g c t1 (\lambda (t: T).(\forall (v1: -T).(\forall (v2: T).((sty0 g c v1 v2) \to (ex2 T (\lambda (v3: T).(sty1 g c -v1 v3)) (\lambda (v3: T).(sty1 g c (THead (Flat Cast) v1 t1) (THead (Flat -Cast) v3 t)))))))) (\lambda (t3: T).(\lambda (H0: (sty0 g c t1 t3)).(\lambda -(v1: T).(\lambda (v2: T).(\lambda (H1: (sty0 g c v1 v2)).(ex_intro2 T -(\lambda (v3: T).(sty1 g c v1 v3)) (\lambda (v3: T).(sty1 g c (THead (Flat -Cast) v1 t1) (THead (Flat Cast) v3 t3))) v2 (sty1_sty0 g c v1 v2 H1) -(sty1_sty0 g c (THead (Flat Cast) v1 t1) (THead (Flat Cast) v2 t3) (sty0_cast -g c v1 v2 H1 t1 t3 H0)))))))) (\lambda (t: T).(\lambda (_: (sty1 g c t1 -t)).(\lambda (H1: ((\forall (v1: T).(\forall (v2: T).((sty0 g c v1 v2) \to -(ex2 T (\lambda (v3: T).(sty1 g c v1 v3)) (\lambda (v3: T).(sty1 g c (THead -(Flat Cast) v1 t1) (THead (Flat Cast) v3 t))))))))).(\lambda (t3: T).(\lambda -(H2: (sty0 g c t t3)).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H3: (sty0 g -c v1 v2)).(let H_x \def (H1 v1 v2 H3) in (let H4 \def H_x in (ex2_ind T -(\lambda (v3: T).(sty1 g c v1 v3)) (\lambda (v3: T).(sty1 g c (THead (Flat -Cast) v1 t1) (THead (Flat Cast) v3 t))) (ex2 T (\lambda (v3: T).(sty1 g c v1 -v3)) (\lambda (v3: T).(sty1 g c (THead (Flat Cast) v1 t1) (THead (Flat Cast) -v3 t3)))) (\lambda (x: T).(\lambda (H5: (sty1 g c v1 x)).(\lambda (H6: (sty1 -g c (THead (Flat Cast) v1 t1) (THead (Flat Cast) x t))).(let H_x0 \def -(sty1_correct g c v1 x H5) in (let H7 \def H_x0 in (ex_ind T (\lambda (t4: -T).(sty0 g c x t4)) (ex2 T (\lambda (v3: T).(sty1 g c v1 v3)) (\lambda (v3: -T).(sty1 g c (THead (Flat Cast) v1 t1) (THead (Flat Cast) v3 t3)))) (\lambda -(x0: T).(\lambda (H8: (sty0 g c x x0)).(ex_intro2 T (\lambda (v3: T).(sty1 g -c v1 v3)) (\lambda (v3: T).(sty1 g c (THead (Flat Cast) v1 t1) (THead (Flat -Cast) v3 t3))) x0 (sty1_sing g c v1 x H5 x0 H8) (sty1_sing g c (THead (Flat -Cast) v1 t1) (THead (Flat Cast) x t) H6 (THead (Flat Cast) x0 t3) (sty0_cast -g c x x0 H8 t t3 H2))))) H7)))))) H4))))))))))) t2 H))))). -(* COMMENTS -Initial nodes: 657 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/subst/defs.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/subst/defs.ma deleted file mode 100644 index e0ed86b52..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/subst/defs.ma +++ /dev/null @@ -1,27 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/lift/defs.ma". - -definition subst: - nat \to (T \to (T \to T)) -\def - let rec subst (d: nat) (v: T) (t: T) on t: T \def (match t with [(TSort n) -\Rightarrow (TSort n) | (TLRef i) \Rightarrow (match (blt i d) with [true -\Rightarrow (TLRef i) | false \Rightarrow (match (blt d i) with [true -\Rightarrow (TLRef (pred i)) | false \Rightarrow (lift d O v)])]) | (THead k -u t0) \Rightarrow (THead k (subst d v u) (subst (s k d) v t0))]) in subst. - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/subst/fwd.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/subst/fwd.ma deleted file mode 100644 index a0678e5fb..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/subst/fwd.ma +++ /dev/null @@ -1,79 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/subst/defs.ma". - -theorem subst_sort: - \forall (v: T).(\forall (d: nat).(\forall (k: nat).(eq T (subst d v (TSort -k)) (TSort k)))) -\def - \lambda (_: T).(\lambda (_: nat).(\lambda (k: nat).(refl_equal T (TSort -k)))). -(* COMMENTS -Initial nodes: 13 -END *) - -theorem subst_lref_lt: - \forall (v: T).(\forall (d: nat).(\forall (i: nat).((lt i d) \to (eq T -(subst d v (TLRef i)) (TLRef i))))) -\def - \lambda (v: T).(\lambda (d: nat).(\lambda (i: nat).(\lambda (H: (lt i -d)).(eq_ind_r bool true (\lambda (b: bool).(eq T (match b with [true -\Rightarrow (TLRef i) | false \Rightarrow (match (blt d i) with [true -\Rightarrow (TLRef (pred i)) | false \Rightarrow (lift d O v)])]) (TLRef i))) -(refl_equal T (TLRef i)) (blt i d) (lt_blt d i H))))). -(* COMMENTS -Initial nodes: 73 -END *) - -theorem subst_lref_eq: - \forall (v: T).(\forall (i: nat).(eq T (subst i v (TLRef i)) (lift i O v))) -\def - \lambda (v: T).(\lambda (i: nat).(eq_ind_r bool false (\lambda (b: bool).(eq -T (match b with [true \Rightarrow (TLRef i) | false \Rightarrow (match b with -[true \Rightarrow (TLRef (pred i)) | false \Rightarrow (lift i O v)])]) (lift -i O v))) (refl_equal T (lift i O v)) (blt i i) (le_bge i i (le_n i)))). -(* COMMENTS -Initial nodes: 71 -END *) - -theorem subst_lref_gt: - \forall (v: T).(\forall (d: nat).(\forall (i: nat).((lt d i) \to (eq T -(subst d v (TLRef i)) (TLRef (pred i)))))) -\def - \lambda (v: T).(\lambda (d: nat).(\lambda (i: nat).(\lambda (H: (lt d -i)).(eq_ind_r bool false (\lambda (b: bool).(eq T (match b with [true -\Rightarrow (TLRef i) | false \Rightarrow (match (blt d i) with [true -\Rightarrow (TLRef (pred i)) | false \Rightarrow (lift d O v)])]) (TLRef -(pred i)))) (eq_ind_r bool true (\lambda (b: bool).(eq T (match b with [true -\Rightarrow (TLRef (pred i)) | false \Rightarrow (lift d O v)]) (TLRef (pred -i)))) (refl_equal T (TLRef (pred i))) (blt d i) (lt_blt i d H)) (blt i d) -(le_bge d i (lt_le_weak d i H)))))). -(* COMMENTS -Initial nodes: 130 -END *) - -theorem subst_head: - \forall (k: K).(\forall (w: T).(\forall (u: T).(\forall (t: T).(\forall (d: -nat).(eq T (subst d w (THead k u t)) (THead k (subst d w u) (subst (s k d) w -t))))))) -\def - \lambda (k: K).(\lambda (w: T).(\lambda (u: T).(\lambda (t: T).(\lambda (d: -nat).(refl_equal T (THead k (subst d w u) (subst (s k d) w t))))))). -(* COMMENTS -Initial nodes: 37 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/subst/props.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/subst/props.ma deleted file mode 100644 index 3bad044e3..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/subst/props.ma +++ /dev/null @@ -1,116 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/subst/fwd.ma". - -include "Basic-1/subst0/defs.ma". - -include "Basic-1/lift/props.ma". - -theorem subst_lift_SO: - \forall (v: T).(\forall (t: T).(\forall (d: nat).(eq T (subst d v (lift (S -O) d t)) t))) -\def - \lambda (v: T).(\lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d: nat).(eq -T (subst d v (lift (S O) d t0)) t0))) (\lambda (n: nat).(\lambda (d: -nat).(eq_ind_r T (TSort n) (\lambda (t0: T).(eq T (subst d v t0) (TSort n))) -(eq_ind_r T (TSort n) (\lambda (t0: T).(eq T t0 (TSort n))) (refl_equal T -(TSort n)) (subst d v (TSort n)) (subst_sort v d n)) (lift (S O) d (TSort n)) -(lift_sort n (S O) d)))) (\lambda (n: nat).(\lambda (d: nat).(lt_le_e n d (eq -T (subst d v (lift (S O) d (TLRef n))) (TLRef n)) (\lambda (H: (lt n -d)).(eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (subst d v t0) (TLRef n))) -(eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 (TLRef n))) (refl_equal T -(TLRef n)) (subst d v (TLRef n)) (subst_lref_lt v d n H)) (lift (S O) d -(TLRef n)) (lift_lref_lt n (S O) d H))) (\lambda (H: (le d n)).(eq_ind_r T -(TLRef (plus n (S O))) (\lambda (t0: T).(eq T (subst d v t0) (TLRef n))) -(eq_ind nat (S (plus n O)) (\lambda (n0: nat).(eq T (subst d v (TLRef n0)) -(TLRef n))) (eq_ind_r T (TLRef (pred (S (plus n O)))) (\lambda (t0: T).(eq T -t0 (TLRef n))) (eq_ind nat (plus n O) (\lambda (n0: nat).(eq T (TLRef n0) -(TLRef n))) (f_equal nat T TLRef (plus n O) n (sym_eq nat n (plus n O) -(plus_n_O n))) (pred (S (plus n O))) (pred_Sn (plus n O))) (subst d v (TLRef -(S (plus n O)))) (subst_lref_gt v d (S (plus n O)) (le_n_S d (plus n O) -(le_plus_trans d n O H)))) (plus n (S O)) (plus_n_Sm n O)) (lift (S O) d -(TLRef n)) (lift_lref_ge n (S O) d H)))))) (\lambda (k: K).(\lambda (t0: -T).(\lambda (H: ((\forall (d: nat).(eq T (subst d v (lift (S O) d t0)) -t0)))).(\lambda (t1: T).(\lambda (H0: ((\forall (d: nat).(eq T (subst d v -(lift (S O) d t1)) t1)))).(\lambda (d: nat).(eq_ind_r T (THead k (lift (S O) -d t0) (lift (S O) (s k d) t1)) (\lambda (t2: T).(eq T (subst d v t2) (THead k -t0 t1))) (eq_ind_r T (THead k (subst d v (lift (S O) d t0)) (subst (s k d) v -(lift (S O) (s k d) t1))) (\lambda (t2: T).(eq T t2 (THead k t0 t1))) -(f_equal3 K T T T THead k k (subst d v (lift (S O) d t0)) t0 (subst (s k d) v -(lift (S O) (s k d) t1)) t1 (refl_equal K k) (H d) (H0 (s k d))) (subst d v -(THead k (lift (S O) d t0) (lift (S O) (s k d) t1))) (subst_head k v (lift (S -O) d t0) (lift (S O) (s k d) t1) d)) (lift (S O) d (THead k t0 t1)) -(lift_head k t0 t1 (S O) d)))))))) t)). -(* COMMENTS -Initial nodes: 879 -END *) - -theorem subst_subst0: - \forall (v: T).(\forall (t1: T).(\forall (t2: T).(\forall (d: nat).((subst0 -d v t1 t2) \to (eq T (subst d v t1) (subst d v t2)))))) -\def - \lambda (v: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (d: nat).(\lambda -(H: (subst0 d v t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t: -T).(\lambda (t0: T).(\lambda (t3: T).(eq T (subst n t t0) (subst n t t3)))))) -(\lambda (v0: T).(\lambda (i: nat).(eq_ind_r T (lift i O v0) (\lambda (t: -T).(eq T t (subst i v0 (lift (S i) O v0)))) (eq_ind nat (plus (S O) i) -(\lambda (n: nat).(eq T (lift i O v0) (subst i v0 (lift n O v0)))) (eq_ind T -(lift (S O) i (lift i O v0)) (\lambda (t: T).(eq T (lift i O v0) (subst i v0 -t))) (eq_ind_r T (lift i O v0) (\lambda (t: T).(eq T (lift i O v0) t)) -(refl_equal T (lift i O v0)) (subst i v0 (lift (S O) i (lift i O v0))) -(subst_lift_SO v0 (lift i O v0) i)) (lift (plus (S O) i) O v0) (lift_free v0 -i (S O) O i (le_n (plus O i)) (le_O_n i))) (S i) (refl_equal nat (S i))) -(subst i v0 (TLRef i)) (subst_lref_eq v0 i)))) (\lambda (v0: T).(\lambda (u2: -T).(\lambda (u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v0 u1 -u2)).(\lambda (H1: (eq T (subst i v0 u1) (subst i v0 u2))).(\lambda (t: -T).(\lambda (k: K).(eq_ind_r T (THead k (subst i v0 u1) (subst (s k i) v0 t)) -(\lambda (t0: T).(eq T t0 (subst i v0 (THead k u2 t)))) (eq_ind_r T (THead k -(subst i v0 u2) (subst (s k i) v0 t)) (\lambda (t0: T).(eq T (THead k (subst -i v0 u1) (subst (s k i) v0 t)) t0)) (eq_ind_r T (subst i v0 u2) (\lambda (t0: -T).(eq T (THead k t0 (subst (s k i) v0 t)) (THead k (subst i v0 u2) (subst (s -k i) v0 t)))) (refl_equal T (THead k (subst i v0 u2) (subst (s k i) v0 t))) -(subst i v0 u1) H1) (subst i v0 (THead k u2 t)) (subst_head k v0 u2 t i)) -(subst i v0 (THead k u1 t)) (subst_head k v0 u1 t i)))))))))) (\lambda (k: -K).(\lambda (v0: T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (i: -nat).(\lambda (_: (subst0 (s k i) v0 t4 t3)).(\lambda (H1: (eq T (subst (s k -i) v0 t4) (subst (s k i) v0 t3))).(\lambda (u: T).(eq_ind_r T (THead k (subst -i v0 u) (subst (s k i) v0 t4)) (\lambda (t: T).(eq T t (subst i v0 (THead k u -t3)))) (eq_ind_r T (THead k (subst i v0 u) (subst (s k i) v0 t3)) (\lambda -(t: T).(eq T (THead k (subst i v0 u) (subst (s k i) v0 t4)) t)) (eq_ind_r T -(subst (s k i) v0 t3) (\lambda (t: T).(eq T (THead k (subst i v0 u) t) (THead -k (subst i v0 u) (subst (s k i) v0 t3)))) (refl_equal T (THead k (subst i v0 -u) (subst (s k i) v0 t3))) (subst (s k i) v0 t4) H1) (subst i v0 (THead k u -t3)) (subst_head k v0 u t3 i)) (subst i v0 (THead k u t4)) (subst_head k v0 u -t4 i)))))))))) (\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda -(i: nat).(\lambda (_: (subst0 i v0 u1 u2)).(\lambda (H1: (eq T (subst i v0 -u1) (subst i v0 u2))).(\lambda (k: K).(\lambda (t3: T).(\lambda (t4: -T).(\lambda (_: (subst0 (s k i) v0 t3 t4)).(\lambda (H3: (eq T (subst (s k i) -v0 t3) (subst (s k i) v0 t4))).(eq_ind_r T (THead k (subst i v0 u1) (subst (s -k i) v0 t3)) (\lambda (t: T).(eq T t (subst i v0 (THead k u2 t4)))) (eq_ind_r -T (THead k (subst i v0 u2) (subst (s k i) v0 t4)) (\lambda (t: T).(eq T -(THead k (subst i v0 u1) (subst (s k i) v0 t3)) t)) (eq_ind_r T (subst i v0 -u2) (\lambda (t: T).(eq T (THead k t (subst (s k i) v0 t3)) (THead k (subst i -v0 u2) (subst (s k i) v0 t4)))) (eq_ind_r T (subst (s k i) v0 t4) (\lambda -(t: T).(eq T (THead k (subst i v0 u2) t) (THead k (subst i v0 u2) (subst (s k -i) v0 t4)))) (refl_equal T (THead k (subst i v0 u2) (subst (s k i) v0 t4))) -(subst (s k i) v0 t3) H3) (subst i v0 u1) H1) (subst i v0 (THead k u2 t4)) -(subst_head k v0 u2 t4 i)) (subst i v0 (THead k u1 t3)) (subst_head k v0 u1 -t3 i))))))))))))) d v t1 t2 H))))). -(* COMMENTS -Initial nodes: 1363 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/subst0/dec.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/subst0/dec.ma deleted file mode 100644 index 0234ff06c..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/subst0/dec.ma +++ /dev/null @@ -1,182 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/subst0/defs.ma". - -include "Basic-1/lift/props.ma". - -theorem dnf_dec2: - \forall (t: T).(\forall (d: nat).(or (\forall (w: T).(ex T (\lambda (v: -T).(subst0 d w t (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T t (lift (S -O) d v)))))) -\def - \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d: nat).(or (\forall (w: -T).(ex T (\lambda (v: T).(subst0 d w t0 (lift (S O) d v))))) (ex T (\lambda -(v: T).(eq T t0 (lift (S O) d v))))))) (\lambda (n: nat).(\lambda (d: -nat).(or_intror (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (TSort n) -(lift (S O) d v))))) (ex T (\lambda (v: T).(eq T (TSort n) (lift (S O) d -v)))) (ex_intro T (\lambda (v: T).(eq T (TSort n) (lift (S O) d v))) (TSort -n) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq T (TSort n) t0)) (refl_equal T -(TSort n)) (lift (S O) d (TSort n)) (lift_sort n (S O) d)))))) (\lambda (n: -nat).(\lambda (d: nat).(lt_eq_gt_e n d (or (\forall (w: T).(ex T (\lambda (v: -T).(subst0 d w (TLRef n) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T -(TLRef n) (lift (S O) d v))))) (\lambda (H: (lt n d)).(or_intror (\forall (w: -T).(ex T (\lambda (v: T).(subst0 d w (TLRef n) (lift (S O) d v))))) (ex T -(\lambda (v: T).(eq T (TLRef n) (lift (S O) d v)))) (ex_intro T (\lambda (v: -T).(eq T (TLRef n) (lift (S O) d v))) (TLRef n) (eq_ind_r T (TLRef n) -(\lambda (t0: T).(eq T (TLRef n) t0)) (refl_equal T (TLRef n)) (lift (S O) d -(TLRef n)) (lift_lref_lt n (S O) d H))))) (\lambda (H: (eq nat n d)).(eq_ind -nat n (\lambda (n0: nat).(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 n0 -w (TLRef n) (lift (S O) n0 v))))) (ex T (\lambda (v: T).(eq T (TLRef n) (lift -(S O) n0 v)))))) (or_introl (\forall (w: T).(ex T (\lambda (v: T).(subst0 n w -(TLRef n) (lift (S O) n v))))) (ex T (\lambda (v: T).(eq T (TLRef n) (lift (S -O) n v)))) (\lambda (w: T).(ex_intro T (\lambda (v: T).(subst0 n w (TLRef n) -(lift (S O) n v))) (lift n O w) (eq_ind_r T (lift (plus (S O) n) O w) -(\lambda (t0: T).(subst0 n w (TLRef n) t0)) (subst0_lref w n) (lift (S O) n -(lift n O w)) (lift_free w n (S O) O n (le_n (plus O n)) (le_O_n n)))))) d -H)) (\lambda (H: (lt d n)).(or_intror (\forall (w: T).(ex T (\lambda (v: -T).(subst0 d w (TLRef n) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T -(TLRef n) (lift (S O) d v)))) (ex_intro T (\lambda (v: T).(eq T (TLRef n) -(lift (S O) d v))) (TLRef (pred n)) (eq_ind_r T (TLRef n) (\lambda (t0: -T).(eq T (TLRef n) t0)) (refl_equal T (TLRef n)) (lift (S O) d (TLRef (pred -n))) (lift_lref_gt d n H)))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda -(H: ((\forall (d: nat).(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w -t0 (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T t0 (lift (S O) d -v)))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (d: nat).(or (\forall (w: -T).(ex T (\lambda (v: T).(subst0 d w t1 (lift (S O) d v))))) (ex T (\lambda -(v: T).(eq T t1 (lift (S O) d v)))))))).(\lambda (d: nat).(let H_x \def (H d) -in (let H1 \def H_x in (or_ind (\forall (w: T).(ex T (\lambda (v: T).(subst0 -d w t0 (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T t0 (lift (S O) d -v)))) (or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) -(lift (S O) d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 t1) (lift (S O) -d v))))) (\lambda (H2: ((\forall (w: T).(ex T (\lambda (v: T).(subst0 d w t0 -(lift (S O) d v))))))).(let H_x0 \def (H0 (s k d)) in (let H3 \def H_x0 in -(or_ind (\forall (w: T).(ex T (\lambda (v: T).(subst0 (s k d) w t1 (lift (S -O) (s k d) v))))) (ex T (\lambda (v: T).(eq T t1 (lift (S O) (s k d) v)))) -(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) (lift -(S O) d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 t1) (lift (S O) d -v))))) (\lambda (H4: ((\forall (w: T).(ex T (\lambda (v: T).(subst0 (s k d) w -t1 (lift (S O) (s k d) v))))))).(or_introl (\forall (w: T).(ex T (\lambda (v: -T).(subst0 d w (THead k t0 t1) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq -T (THead k t0 t1) (lift (S O) d v)))) (\lambda (w: T).(let H_x1 \def (H4 w) -in (let H5 \def H_x1 in (ex_ind T (\lambda (v: T).(subst0 (s k d) w t1 (lift -(S O) (s k d) v))) (ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) (lift (S -O) d v)))) (\lambda (x: T).(\lambda (H6: (subst0 (s k d) w t1 (lift (S O) (s -k d) x))).(let H_x2 \def (H2 w) in (let H7 \def H_x2 in (ex_ind T (\lambda -(v: T).(subst0 d w t0 (lift (S O) d v))) (ex T (\lambda (v: T).(subst0 d w -(THead k t0 t1) (lift (S O) d v)))) (\lambda (x0: T).(\lambda (H8: (subst0 d -w t0 (lift (S O) d x0))).(ex_intro T (\lambda (v: T).(subst0 d w (THead k t0 -t1) (lift (S O) d v))) (THead k x0 x) (eq_ind_r T (THead k (lift (S O) d x0) -(lift (S O) (s k d) x)) (\lambda (t2: T).(subst0 d w (THead k t0 t1) t2)) -(subst0_both w t0 (lift (S O) d x0) d H8 k t1 (lift (S O) (s k d) x) H6) -(lift (S O) d (THead k x0 x)) (lift_head k x0 x (S O) d))))) H7))))) H5)))))) -(\lambda (H4: (ex T (\lambda (v: T).(eq T t1 (lift (S O) (s k d) -v))))).(ex_ind T (\lambda (v: T).(eq T t1 (lift (S O) (s k d) v))) (or -(\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) (lift (S O) -d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 t1) (lift (S O) d v))))) -(\lambda (x: T).(\lambda (H5: (eq T t1 (lift (S O) (s k d) x))).(eq_ind_r T -(lift (S O) (s k d) x) (\lambda (t2: T).(or (\forall (w: T).(ex T (\lambda -(v: T).(subst0 d w (THead k t0 t2) (lift (S O) d v))))) (ex T (\lambda (v: -T).(eq T (THead k t0 t2) (lift (S O) d v)))))) (or_introl (\forall (w: T).(ex -T (\lambda (v: T).(subst0 d w (THead k t0 (lift (S O) (s k d) x)) (lift (S O) -d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 (lift (S O) (s k d) x)) -(lift (S O) d v)))) (\lambda (w: T).(let H_x1 \def (H2 w) in (let H6 \def -H_x1 in (ex_ind T (\lambda (v: T).(subst0 d w t0 (lift (S O) d v))) (ex T -(\lambda (v: T).(subst0 d w (THead k t0 (lift (S O) (s k d) x)) (lift (S O) d -v)))) (\lambda (x0: T).(\lambda (H7: (subst0 d w t0 (lift (S O) d -x0))).(ex_intro T (\lambda (v: T).(subst0 d w (THead k t0 (lift (S O) (s k d) -x)) (lift (S O) d v))) (THead k x0 x) (eq_ind_r T (THead k (lift (S O) d x0) -(lift (S O) (s k d) x)) (\lambda (t2: T).(subst0 d w (THead k t0 (lift (S O) -(s k d) x)) t2)) (subst0_fst w (lift (S O) d x0) t0 d H7 (lift (S O) (s k d) -x) k) (lift (S O) d (THead k x0 x)) (lift_head k x0 x (S O) d))))) H6))))) t1 -H5))) H4)) H3)))) (\lambda (H2: (ex T (\lambda (v: T).(eq T t0 (lift (S O) d -v))))).(ex_ind T (\lambda (v: T).(eq T t0 (lift (S O) d v))) (or (\forall (w: -T).(ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) (lift (S O) d v))))) (ex -T (\lambda (v: T).(eq T (THead k t0 t1) (lift (S O) d v))))) (\lambda (x: -T).(\lambda (H3: (eq T t0 (lift (S O) d x))).(let H_x0 \def (H0 (s k d)) in -(let H4 \def H_x0 in (or_ind (\forall (w: T).(ex T (\lambda (v: T).(subst0 (s -k d) w t1 (lift (S O) (s k d) v))))) (ex T (\lambda (v: T).(eq T t1 (lift (S -O) (s k d) v)))) (or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead -k t0 t1) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 t1) -(lift (S O) d v))))) (\lambda (H5: ((\forall (w: T).(ex T (\lambda (v: -T).(subst0 (s k d) w t1 (lift (S O) (s k d) v))))))).(eq_ind_r T (lift (S O) -d x) (\lambda (t2: T).(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w -(THead k t2 t1) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T (THead k t2 -t1) (lift (S O) d v)))))) (or_introl (\forall (w: T).(ex T (\lambda (v: -T).(subst0 d w (THead k (lift (S O) d x) t1) (lift (S O) d v))))) (ex T -(\lambda (v: T).(eq T (THead k (lift (S O) d x) t1) (lift (S O) d v)))) -(\lambda (w: T).(let H_x1 \def (H5 w) in (let H6 \def H_x1 in (ex_ind T -(\lambda (v: T).(subst0 (s k d) w t1 (lift (S O) (s k d) v))) (ex T (\lambda -(v: T).(subst0 d w (THead k (lift (S O) d x) t1) (lift (S O) d v)))) (\lambda -(x0: T).(\lambda (H7: (subst0 (s k d) w t1 (lift (S O) (s k d) -x0))).(ex_intro T (\lambda (v: T).(subst0 d w (THead k (lift (S O) d x) t1) -(lift (S O) d v))) (THead k x x0) (eq_ind_r T (THead k (lift (S O) d x) (lift -(S O) (s k d) x0)) (\lambda (t2: T).(subst0 d w (THead k (lift (S O) d x) t1) -t2)) (subst0_snd k w (lift (S O) (s k d) x0) t1 d H7 (lift (S O) d x)) (lift -(S O) d (THead k x x0)) (lift_head k x x0 (S O) d))))) H6))))) t0 H3)) -(\lambda (H5: (ex T (\lambda (v: T).(eq T t1 (lift (S O) (s k d) -v))))).(ex_ind T (\lambda (v: T).(eq T t1 (lift (S O) (s k d) v))) (or -(\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) (lift (S O) -d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 t1) (lift (S O) d v))))) -(\lambda (x0: T).(\lambda (H6: (eq T t1 (lift (S O) (s k d) x0))).(eq_ind_r T -(lift (S O) (s k d) x0) (\lambda (t2: T).(or (\forall (w: T).(ex T (\lambda -(v: T).(subst0 d w (THead k t0 t2) (lift (S O) d v))))) (ex T (\lambda (v: -T).(eq T (THead k t0 t2) (lift (S O) d v)))))) (eq_ind_r T (lift (S O) d x) -(\lambda (t2: T).(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead -k t2 (lift (S O) (s k d) x0)) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq -T (THead k t2 (lift (S O) (s k d) x0)) (lift (S O) d v)))))) (or_intror -(\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead k (lift (S O) d x) -(lift (S O) (s k d) x0)) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T -(THead k (lift (S O) d x) (lift (S O) (s k d) x0)) (lift (S O) d v)))) -(ex_intro T (\lambda (v: T).(eq T (THead k (lift (S O) d x) (lift (S O) (s k -d) x0)) (lift (S O) d v))) (THead k x x0) (eq_ind_r T (THead k (lift (S O) d -x) (lift (S O) (s k d) x0)) (\lambda (t2: T).(eq T (THead k (lift (S O) d x) -(lift (S O) (s k d) x0)) t2)) (refl_equal T (THead k (lift (S O) d x) (lift -(S O) (s k d) x0))) (lift (S O) d (THead k x x0)) (lift_head k x x0 (S O) -d)))) t0 H3) t1 H6))) H5)) H4))))) H2)) H1))))))))) t). -(* COMMENTS -Initial nodes: 3549 -END *) - -theorem dnf_dec: - \forall (w: T).(\forall (t: T).(\forall (d: nat).(ex T (\lambda (v: T).(or -(subst0 d w t (lift (S O) d v)) (eq T t (lift (S O) d v))))))) -\def - \lambda (w: T).(\lambda (t: T).(\lambda (d: nat).(let H_x \def (dnf_dec2 t -d) in (let H \def H_x in (or_ind (\forall (w0: T).(ex T (\lambda (v: -T).(subst0 d w0 t (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T t (lift (S -O) d v)))) (ex T (\lambda (v: T).(or (subst0 d w t (lift (S O) d v)) (eq T t -(lift (S O) d v))))) (\lambda (H0: ((\forall (w0: T).(ex T (\lambda (v: -T).(subst0 d w0 t (lift (S O) d v))))))).(let H_x0 \def (H0 w) in (let H1 -\def H_x0 in (ex_ind T (\lambda (v: T).(subst0 d w t (lift (S O) d v))) (ex T -(\lambda (v: T).(or (subst0 d w t (lift (S O) d v)) (eq T t (lift (S O) d -v))))) (\lambda (x: T).(\lambda (H2: (subst0 d w t (lift (S O) d -x))).(ex_intro T (\lambda (v: T).(or (subst0 d w t (lift (S O) d v)) (eq T t -(lift (S O) d v)))) x (or_introl (subst0 d w t (lift (S O) d x)) (eq T t -(lift (S O) d x)) H2)))) H1)))) (\lambda (H0: (ex T (\lambda (v: T).(eq T t -(lift (S O) d v))))).(ex_ind T (\lambda (v: T).(eq T t (lift (S O) d v))) (ex -T (\lambda (v: T).(or (subst0 d w t (lift (S O) d v)) (eq T t (lift (S O) d -v))))) (\lambda (x: T).(\lambda (H1: (eq T t (lift (S O) d x))).(eq_ind_r T -(lift (S O) d x) (\lambda (t0: T).(ex T (\lambda (v: T).(or (subst0 d w t0 -(lift (S O) d v)) (eq T t0 (lift (S O) d v)))))) (ex_intro T (\lambda (v: -T).(or (subst0 d w (lift (S O) d x) (lift (S O) d v)) (eq T (lift (S O) d x) -(lift (S O) d v)))) x (or_intror (subst0 d w (lift (S O) d x) (lift (S O) d -x)) (eq T (lift (S O) d x) (lift (S O) d x)) (refl_equal T (lift (S O) d -x)))) t H1))) H0)) H))))). -(* COMMENTS -Initial nodes: 603 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/subst0/defs.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/subst0/defs.ma deleted file mode 100644 index a493a7ac2..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/subst0/defs.ma +++ /dev/null @@ -1,32 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/lift/defs.ma". - -inductive subst0: nat \to (T \to (T \to (T \to Prop))) \def -| subst0_lref: \forall (v: T).(\forall (i: nat).(subst0 i v (TLRef i) (lift -(S i) O v))) -| subst0_fst: \forall (v: T).(\forall (u2: T).(\forall (u1: T).(\forall (i: -nat).((subst0 i v u1 u2) \to (\forall (t: T).(\forall (k: K).(subst0 i v -(THead k u1 t) (THead k u2 t)))))))) -| subst0_snd: \forall (k: K).(\forall (v: T).(\forall (t2: T).(\forall (t1: -T).(\forall (i: nat).((subst0 (s k i) v t1 t2) \to (\forall (u: T).(subst0 i -v (THead k u t1) (THead k u t2)))))))) -| subst0_both: \forall (v: T).(\forall (u1: T).(\forall (u2: T).(\forall (i: -nat).((subst0 i v u1 u2) \to (\forall (k: K).(\forall (t1: T).(\forall (t2: -T).((subst0 (s k i) v t1 t2) \to (subst0 i v (THead k u1 t1) (THead k u2 -t2)))))))))). - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/subst0/fwd.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/subst0/fwd.ma deleted file mode 100644 index 165555fe2..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/subst0/fwd.ma +++ /dev/null @@ -1,728 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/subst0/defs.ma". - -include "Basic-1/lift/props.ma". - -theorem subst0_gen_sort: - \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst0 -i v (TSort n) x) \to (\forall (P: Prop).P))))) -\def - \lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda -(H: (subst0 i v (TSort n) x)).(\lambda (P: Prop).(insert_eq T (TSort n) -(\lambda (t: T).(subst0 i v t x)) (\lambda (_: T).P) (\lambda (y: T).(\lambda -(H0: (subst0 i v y x)).(subst0_ind (\lambda (_: nat).(\lambda (_: T).(\lambda -(t0: T).(\lambda (_: T).((eq T t0 (TSort n)) \to P))))) (\lambda (_: -T).(\lambda (i0: nat).(\lambda (H1: (eq T (TLRef i0) (TSort n))).(let H2 \def -(eq_ind T (TLRef i0) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | -(THead _ _ _) \Rightarrow False])) I (TSort n) H1) in (False_ind P H2))))) -(\lambda (v0: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i0: -nat).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (_: (((eq T u1 (TSort n)) -\to P))).(\lambda (t: T).(\lambda (k: K).(\lambda (H3: (eq T (THead k u1 t) -(TSort n))).(let H4 \def (eq_ind T (THead k u1 t) (\lambda (ee: T).(match ee -in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef -_) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H3) in -(False_ind P H4))))))))))) (\lambda (k: K).(\lambda (v0: T).(\lambda (t2: -T).(\lambda (t1: T).(\lambda (i0: nat).(\lambda (_: (subst0 (s k i0) v0 t1 -t2)).(\lambda (_: (((eq T t1 (TSort n)) \to P))).(\lambda (u: T).(\lambda -(H3: (eq T (THead k u t1) (TSort n))).(let H4 \def (eq_ind T (THead k u t1) -(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow -True])) I (TSort n) H3) in (False_ind P H4))))))))))) (\lambda (v0: -T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i0: nat).(\lambda (_: (subst0 -i0 v0 u1 u2)).(\lambda (_: (((eq T u1 (TSort n)) \to P))).(\lambda (k: -K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (subst0 (s k i0) v0 t1 -t2)).(\lambda (_: (((eq T t1 (TSort n)) \to P))).(\lambda (H5: (eq T (THead k -u1 t1) (TSort n))).(let H6 \def (eq_ind T (THead k u1 t1) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I -(TSort n) H5) in (False_ind P H6)))))))))))))) i v y x H0))) H)))))). -(* COMMENTS -Initial nodes: 445 -END *) - -theorem subst0_gen_lref: - \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst0 -i v (TLRef n) x) \to (land (eq nat n i) (eq T x (lift (S n) O v))))))) -\def - \lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda -(H: (subst0 i v (TLRef n) x)).(insert_eq T (TLRef n) (\lambda (t: T).(subst0 -i v t x)) (\lambda (_: T).(land (eq nat n i) (eq T x (lift (S n) O v)))) -(\lambda (y: T).(\lambda (H0: (subst0 i v y x)).(subst0_ind (\lambda (n0: -nat).(\lambda (t: T).(\lambda (t0: T).(\lambda (t1: T).((eq T t0 (TLRef n)) -\to (land (eq nat n n0) (eq T t1 (lift (S n) O t)))))))) (\lambda (v0: -T).(\lambda (i0: nat).(\lambda (H1: (eq T (TLRef i0) (TLRef n))).(let H2 \def -(f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with -[(TSort _) \Rightarrow i0 | (TLRef n0) \Rightarrow n0 | (THead _ _ _) -\Rightarrow i0])) (TLRef i0) (TLRef n) H1) in (eq_ind_r nat n (\lambda (n0: -nat).(land (eq nat n n0) (eq T (lift (S n0) O v0) (lift (S n) O v0)))) (conj -(eq nat n n) (eq T (lift (S n) O v0) (lift (S n) O v0)) (refl_equal nat n) -(refl_equal T (lift (S n) O v0))) i0 H2))))) (\lambda (v0: T).(\lambda (u2: -T).(\lambda (u1: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v0 u1 -u2)).(\lambda (_: (((eq T u1 (TLRef n)) \to (land (eq nat n i0) (eq T u2 -(lift (S n) O v0)))))).(\lambda (t: T).(\lambda (k: K).(\lambda (H3: (eq T -(THead k u1 t) (TLRef n))).(let H4 \def (eq_ind T (THead k u1 t) (\lambda -(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow -True])) I (TLRef n) H3) in (False_ind (land (eq nat n i0) (eq T (THead k u2 -t) (lift (S n) O v0))) H4))))))))))) (\lambda (k: K).(\lambda (v0: -T).(\lambda (t2: T).(\lambda (t1: T).(\lambda (i0: nat).(\lambda (_: (subst0 -(s k i0) v0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (land (eq nat n (s -k i0)) (eq T t2 (lift (S n) O v0)))))).(\lambda (u: T).(\lambda (H3: (eq T -(THead k u t1) (TLRef n))).(let H4 \def (eq_ind T (THead k u t1) (\lambda -(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow -True])) I (TLRef n) H3) in (False_ind (land (eq nat n i0) (eq T (THead k u -t2) (lift (S n) O v0))) H4))))))))))) (\lambda (v0: T).(\lambda (u1: -T).(\lambda (u2: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v0 u1 -u2)).(\lambda (_: (((eq T u1 (TLRef n)) \to (land (eq nat n i0) (eq T u2 -(lift (S n) O v0)))))).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (_: (subst0 (s k i0) v0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef -n)) \to (land (eq nat n (s k i0)) (eq T t2 (lift (S n) O v0)))))).(\lambda -(H5: (eq T (THead k u1 t1) (TLRef n))).(let H6 \def (eq_ind T (THead k u1 t1) -(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow -True])) I (TLRef n) H5) in (False_ind (land (eq nat n i0) (eq T (THead k u2 -t2) (lift (S n) O v0))) H6)))))))))))))) i v y x H0))) H))))). -(* COMMENTS -Initial nodes: 779 -END *) - -theorem subst0_gen_head: - \forall (k: K).(\forall (v: T).(\forall (u1: T).(\forall (t1: T).(\forall -(x: T).(\forall (i: nat).((subst0 i v (THead k u1 t1) x) \to (or3 (ex2 T -(\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 -u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k u1 t2))) (\lambda (t2: -T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1 -u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1 t2))))))))))) -\def - \lambda (k: K).(\lambda (v: T).(\lambda (u1: T).(\lambda (t1: T).(\lambda -(x: T).(\lambda (i: nat).(\lambda (H: (subst0 i v (THead k u1 t1) -x)).(insert_eq T (THead k u1 t1) (\lambda (t: T).(subst0 i v t x)) (\lambda -(_: T).(or3 (ex2 T (\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2: -T).(subst0 i v u1 u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k u1 t2))) -(\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1 -t2)))))) (\lambda (y: T).(\lambda (H0: (subst0 i v y x)).(subst0_ind (\lambda -(n: nat).(\lambda (t: T).(\lambda (t0: T).(\lambda (t2: T).((eq T t0 (THead k -u1 t1)) \to (or3 (ex2 T (\lambda (u2: T).(eq T t2 (THead k u2 t1))) (\lambda -(u2: T).(subst0 n t u1 u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead k u1 -t3))) (\lambda (t3: T).(subst0 (s k n) t t1 t3))) (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 n t u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k n) t t1 -t3)))))))))) (\lambda (v0: T).(\lambda (i0: nat).(\lambda (H1: (eq T (TLRef -i0) (THead k u1 t1))).(let H2 \def (eq_ind T (TLRef i0) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I -(THead k u1 t1) H1) in (False_ind (or3 (ex2 T (\lambda (u2: T).(eq T (lift (S -i0) O v0) (THead k u2 t1))) (\lambda (u2: T).(subst0 i0 v0 u1 u2))) (ex2 T -(\lambda (t2: T).(eq T (lift (S i0) O v0) (THead k u1 t2))) (\lambda (t2: -T).(subst0 (s k i0) v0 t1 t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T (lift (S i0) O v0) (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 i0 v0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i0) -v0 t1 t2))))) H2))))) (\lambda (v0: T).(\lambda (u2: T).(\lambda (u0: -T).(\lambda (i0: nat).(\lambda (H1: (subst0 i0 v0 u0 u2)).(\lambda (H2: (((eq -T u0 (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T u2 (THead k u3 -t1))) (\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t2: T).(eq T -u2 (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))) (ex3_2 T T -(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead k u3 t2)))) (\lambda (u3: -T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t2: -T).(subst0 (s k i0) v0 t1 t2)))))))).(\lambda (t: T).(\lambda (k0: -K).(\lambda (H3: (eq T (THead k0 u0 t) (THead k u1 t1))).(let H4 \def -(f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with -[(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _ _) -\Rightarrow k1])) (THead k0 u0 t) (THead k u1 t1) H3) in ((let H5 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t0 _) -\Rightarrow t0])) (THead k0 u0 t) (THead k u1 t1) H3) in ((let H6 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t0) -\Rightarrow t0])) (THead k0 u0 t) (THead k u1 t1) H3) in (\lambda (H7: (eq T -u0 u1)).(\lambda (H8: (eq K k0 k)).(eq_ind_r K k (\lambda (k1: K).(or3 (ex2 T -(\lambda (u3: T).(eq T (THead k1 u2 t) (THead k u3 t1))) (\lambda (u3: -T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t2: T).(eq T (THead k1 u2 t) -(THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))) (ex3_2 T T -(\lambda (u3: T).(\lambda (t2: T).(eq T (THead k1 u2 t) (THead k u3 t2)))) -(\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: -T).(\lambda (t2: T).(subst0 (s k i0) v0 t1 t2)))))) (eq_ind_r T t1 (\lambda -(t0: T).(or3 (ex2 T (\lambda (u3: T).(eq T (THead k u2 t0) (THead k u3 t1))) -(\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t2: T).(eq T (THead -k u2 t0) (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))) -(ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead k u2 t0) (THead k -u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda -(_: T).(\lambda (t2: T).(subst0 (s k i0) v0 t1 t2)))))) (let H9 \def (eq_ind -T u0 (\lambda (t0: T).((eq T t0 (THead k u1 t1)) \to (or3 (ex2 T (\lambda -(u3: T).(eq T u2 (THead k u3 t1))) (\lambda (u3: T).(subst0 i0 v0 u1 u3))) -(ex2 T (\lambda (t2: T).(eq T u2 (THead k u1 t2))) (\lambda (t2: T).(subst0 -(s k i0) v0 t1 t2))) (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 -(THead k u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) -(\lambda (_: T).(\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))))))) H2 u1 H7) -in (let H10 \def (eq_ind T u0 (\lambda (t0: T).(subst0 i0 v0 t0 u2)) H1 u1 -H7) in (or3_intro0 (ex2 T (\lambda (u3: T).(eq T (THead k u2 t1) (THead k u3 -t1))) (\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t2: T).(eq T -(THead k u2 t1) (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1 -t2))) (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead k u2 t1) -(THead k u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) -(\lambda (_: T).(\lambda (t2: T).(subst0 (s k i0) v0 t1 t2)))) (ex_intro2 T -(\lambda (u3: T).(eq T (THead k u2 t1) (THead k u3 t1))) (\lambda (u3: -T).(subst0 i0 v0 u1 u3)) u2 (refl_equal T (THead k u2 t1)) H10)))) t H6) k0 -H8)))) H5)) H4))))))))))) (\lambda (k0: K).(\lambda (v0: T).(\lambda (t2: -T).(\lambda (t0: T).(\lambda (i0: nat).(\lambda (H1: (subst0 (s k0 i0) v0 t0 -t2)).(\lambda (H2: (((eq T t0 (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u2: -T).(eq T t2 (THead k u2 t1))) (\lambda (u2: T).(subst0 (s k0 i0) v0 u1 u2))) -(ex2 T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 -(s k (s k0 i0)) v0 t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq -T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s k0 i0) v0 -u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k0 i0)) v0 t1 -t3)))))))).(\lambda (u: T).(\lambda (H3: (eq T (THead k0 u t0) (THead k u1 -t1))).(let H4 \def (f_equal T K (\lambda (e: T).(match e in T return (\lambda -(_: T).K) with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead -k1 _ _) \Rightarrow k1])) (THead k0 u t0) (THead k u1 t1) H3) in ((let H5 -\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) -\Rightarrow t])) (THead k0 u t0) (THead k u1 t1) H3) in ((let H6 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) -\Rightarrow t])) (THead k0 u t0) (THead k u1 t1) H3) in (\lambda (H7: (eq T u -u1)).(\lambda (H8: (eq K k0 k)).(eq_ind_r T u1 (\lambda (t: T).(or3 (ex2 T -(\lambda (u2: T).(eq T (THead k0 t t2) (THead k u2 t1))) (\lambda (u2: -T).(subst0 i0 v0 u1 u2))) (ex2 T (\lambda (t3: T).(eq T (THead k0 t t2) -(THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T (THead k0 t t2) (THead k u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(subst0 (s k i0) v0 t1 t3)))))) (let H9 \def (eq_ind T t0 -(\lambda (t: T).((eq T t (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u2: -T).(eq T t2 (THead k u2 t1))) (\lambda (u2: T).(subst0 (s k0 i0) v0 u1 u2))) -(ex2 T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 -(s k (s k0 i0)) v0 t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq -T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s k0 i0) v0 -u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k0 i0)) v0 t1 -t3))))))) H2 t1 H6) in (let H10 \def (eq_ind T t0 (\lambda (t: T).(subst0 (s -k0 i0) v0 t t2)) H1 t1 H6) in (let H11 \def (eq_ind K k0 (\lambda (k1: -K).((eq T t1 (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u2: T).(eq T t2 -(THead k u2 t1))) (\lambda (u2: T).(subst0 (s k1 i0) v0 u1 u2))) (ex2 T -(\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k (s -k1 i0)) v0 t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 -(THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s k1 i0) v0 u1 -u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k1 i0)) v0 t1 -t3))))))) H9 k H8) in (let H12 \def (eq_ind K k0 (\lambda (k1: K).(subst0 (s -k1 i0) v0 t1 t2)) H10 k H8) in (eq_ind_r K k (\lambda (k1: K).(or3 (ex2 T -(\lambda (u2: T).(eq T (THead k1 u1 t2) (THead k u2 t1))) (\lambda (u2: -T).(subst0 i0 v0 u1 u2))) (ex2 T (\lambda (t3: T).(eq T (THead k1 u1 t2) -(THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T (THead k1 u1 t2) (THead k u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(subst0 (s k i0) v0 t1 t3)))))) (or3_intro1 (ex2 T -(\lambda (u2: T).(eq T (THead k u1 t2) (THead k u2 t1))) (\lambda (u2: -T).(subst0 i0 v0 u1 u2))) (ex2 T (\lambda (t3: T).(eq T (THead k u1 t2) -(THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T (THead k u1 t2) (THead k u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(subst0 (s k i0) v0 t1 t3)))) (ex_intro2 T (\lambda (t3: -T).(eq T (THead k u1 t2) (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) -v0 t1 t3)) t2 (refl_equal T (THead k u1 t2)) H12)) k0 H8))))) u H7)))) H5)) -H4))))))))))) (\lambda (v0: T).(\lambda (u0: T).(\lambda (u2: T).(\lambda -(i0: nat).(\lambda (H1: (subst0 i0 v0 u0 u2)).(\lambda (H2: (((eq T u0 (THead -k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T u2 (THead k u3 t1))) -(\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t2: T).(eq T u2 -(THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))) (ex3_2 T T -(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead k u3 t2)))) (\lambda (u3: -T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t2: -T).(subst0 (s k i0) v0 t1 t2)))))))).(\lambda (k0: K).(\lambda (t0: -T).(\lambda (t2: T).(\lambda (H3: (subst0 (s k0 i0) v0 t0 t2)).(\lambda (H4: -(((eq T t0 (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T t2 (THead -k u3 t1))) (\lambda (u3: T).(subst0 (s k0 i0) v0 u1 u3))) (ex2 T (\lambda -(t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k (s k0 i0)) -v0 t1 t3))) (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead k u3 -t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 (s k0 i0) v0 u1 u3))) -(\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k0 i0)) v0 t1 -t3)))))))).(\lambda (H5: (eq T (THead k0 u0 t0) (THead k u1 t1))).(let H6 -\def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) -with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _ _) -\Rightarrow k1])) (THead k0 u0 t0) (THead k u1 t1) H5) in ((let H7 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) -\Rightarrow t])) (THead k0 u0 t0) (THead k u1 t1) H5) in ((let H8 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) -\Rightarrow t])) (THead k0 u0 t0) (THead k u1 t1) H5) in (\lambda (H9: (eq T -u0 u1)).(\lambda (H10: (eq K k0 k)).(let H11 \def (eq_ind T t0 (\lambda (t: -T).((eq T t (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T t2 (THead -k u3 t1))) (\lambda (u3: T).(subst0 (s k0 i0) v0 u1 u3))) (ex2 T (\lambda -(t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k (s k0 i0)) -v0 t1 t3))) (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead k u3 -t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 (s k0 i0) v0 u1 u3))) -(\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k0 i0)) v0 t1 t3))))))) H4 -t1 H8) in (let H12 \def (eq_ind T t0 (\lambda (t: T).(subst0 (s k0 i0) v0 t -t2)) H3 t1 H8) in (let H13 \def (eq_ind K k0 (\lambda (k1: K).((eq T t1 -(THead k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T t2 (THead k u3 t1))) -(\lambda (u3: T).(subst0 (s k1 i0) v0 u1 u3))) (ex2 T (\lambda (t3: T).(eq T -t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k (s k1 i0)) v0 t1 t3))) -(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead k u3 t3)))) -(\lambda (u3: T).(\lambda (_: T).(subst0 (s k1 i0) v0 u1 u3))) (\lambda (_: -T).(\lambda (t3: T).(subst0 (s k (s k1 i0)) v0 t1 t3))))))) H11 k H10) in -(let H14 \def (eq_ind K k0 (\lambda (k1: K).(subst0 (s k1 i0) v0 t1 t2)) H12 -k H10) in (eq_ind_r K k (\lambda (k1: K).(or3 (ex2 T (\lambda (u3: T).(eq T -(THead k1 u2 t2) (THead k u3 t1))) (\lambda (u3: T).(subst0 i0 v0 u1 u3))) -(ex2 T (\lambda (t3: T).(eq T (THead k1 u2 t2) (THead k u1 t3))) (\lambda -(t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T (\lambda (u3: T).(\lambda -(t3: T).(eq T (THead k1 u2 t2) (THead k u3 t3)))) (\lambda (u3: T).(\lambda -(_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k -i0) v0 t1 t3)))))) (let H15 \def (eq_ind T u0 (\lambda (t: T).((eq T t (THead -k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T u2 (THead k u3 t1))) -(\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t3: T).(eq T u2 -(THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T -(\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead k u3 t3)))) (\lambda (u3: -T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t3: -T).(subst0 (s k i0) v0 t1 t3))))))) H2 u1 H9) in (let H16 \def (eq_ind T u0 -(\lambda (t: T).(subst0 i0 v0 t u2)) H1 u1 H9) in (or3_intro2 (ex2 T (\lambda -(u3: T).(eq T (THead k u2 t2) (THead k u3 t1))) (\lambda (u3: T).(subst0 i0 -v0 u1 u3))) (ex2 T (\lambda (t3: T).(eq T (THead k u2 t2) (THead k u1 t3))) -(\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T (\lambda (u3: -T).(\lambda (t3: T).(eq T (THead k u2 t2) (THead k u3 t3)))) (\lambda (u3: -T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t3: -T).(subst0 (s k i0) v0 t1 t3)))) (ex3_2_intro T T (\lambda (u3: T).(\lambda -(t3: T).(eq T (THead k u2 t2) (THead k u3 t3)))) (\lambda (u3: T).(\lambda -(_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k -i0) v0 t1 t3))) u2 t2 (refl_equal T (THead k u2 t2)) H16 H14)))) k0 -H10)))))))) H7)) H6)))))))))))))) i v y x H0))) H))))))). -(* COMMENTS -Initial nodes: 4255 -END *) - -theorem subst0_gen_lift_lt: - \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall -(h: nat).(\forall (d: nat).((subst0 i (lift h d u) (lift h (S (plus i d)) t1) -x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda -(t2: T).(subst0 i u t1 t2))))))))) -\def - \lambda (u: T).(\lambda (t1: T).(T_ind (\lambda (t: T).(\forall (x: -T).(\forall (i: nat).(\forall (h: nat).(\forall (d: nat).((subst0 i (lift h d -u) (lift h (S (plus i d)) t) x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h -(S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u t t2))))))))) (\lambda (n: -nat).(\lambda (x: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda (d: -nat).(\lambda (H: (subst0 i (lift h d u) (lift h (S (plus i d)) (TSort n)) -x)).(let H0 \def (eq_ind T (lift h (S (plus i d)) (TSort n)) (\lambda (t: -T).(subst0 i (lift h d u) t x)) H (TSort n) (lift_sort n h (S (plus i d)))) -in (subst0_gen_sort (lift h d u) x i n H0 (ex2 T (\lambda (t2: T).(eq T x -(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TSort n) -t2))))))))))) (\lambda (n: nat).(\lambda (x: T).(\lambda (i: nat).(\lambda -(h: nat).(\lambda (d: nat).(\lambda (H: (subst0 i (lift h d u) (lift h (S -(plus i d)) (TLRef n)) x)).(lt_le_e n (S (plus i d)) (ex2 T (\lambda (t2: -T).(eq T x (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef -n) t2))) (\lambda (H0: (lt n (S (plus i d)))).(let H1 \def (eq_ind T (lift h -(S (plus i d)) (TLRef n)) (\lambda (t: T).(subst0 i (lift h d u) t x)) H -(TLRef n) (lift_lref_lt n h (S (plus i d)) H0)) in (land_ind (eq nat n i) (eq -T x (lift (S n) O (lift h d u))) (ex2 T (\lambda (t2: T).(eq T x (lift h (S -(plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef n) t2))) (\lambda (H2: -(eq nat n i)).(\lambda (H3: (eq T x (lift (S n) O (lift h d u)))).(eq_ind_r T -(lift (S n) O (lift h d u)) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t -(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef n) t2)))) -(eq_ind_r nat i (\lambda (n0: nat).(ex2 T (\lambda (t2: T).(eq T (lift (S n0) -O (lift h d u)) (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u -(TLRef n0) t2)))) (eq_ind T (lift h (plus (S i) d) (lift (S i) O u)) (\lambda -(t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h (S (plus i d)) t2))) (\lambda -(t2: T).(subst0 i u (TLRef i) t2)))) (ex_intro2 T (\lambda (t2: T).(eq T -(lift h (S (plus i d)) (lift (S i) O u)) (lift h (S (plus i d)) t2))) -(\lambda (t2: T).(subst0 i u (TLRef i) t2)) (lift (S i) O u) (refl_equal T -(lift h (S (plus i d)) (lift (S i) O u))) (subst0_lref u i)) (lift (S i) O -(lift h d u)) (lift_d u h (S i) d O (le_O_n d))) n H2) x H3))) -(subst0_gen_lref (lift h d u) x i n H1)))) (\lambda (H0: (le (S (plus i d)) -n)).(let H1 \def (eq_ind T (lift h (S (plus i d)) (TLRef n)) (\lambda (t: -T).(subst0 i (lift h d u) t x)) H (TLRef (plus n h)) (lift_lref_ge n h (S -(plus i d)) H0)) in (land_ind (eq nat (plus n h) i) (eq T x (lift (S (plus n -h)) O (lift h d u))) (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) -t2))) (\lambda (t2: T).(subst0 i u (TLRef n) t2))) (\lambda (H2: (eq nat -(plus n h) i)).(\lambda (_: (eq T x (lift (S (plus n h)) O (lift h d -u)))).(let H4 \def (eq_ind_r nat i (\lambda (n0: nat).(le (S (plus n0 d)) n)) -H0 (plus n h) H2) in (le_false n (plus (plus n h) d) (ex2 T (\lambda (t2: -T).(eq T x (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef -n) t2))) (le_plus_trans n (plus n h) d (le_plus_l n h)) H4)))) -(subst0_gen_lref (lift h d u) x i (plus n h) H1))))))))))) (\lambda (k: -K).(\lambda (t: T).(\lambda (H: ((\forall (x: T).(\forall (i: nat).(\forall -(h: nat).(\forall (d: nat).((subst0 i (lift h d u) (lift h (S (plus i d)) t) -x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda -(t2: T).(subst0 i u t t2)))))))))).(\lambda (t0: T).(\lambda (H0: ((\forall -(x: T).(\forall (i: nat).(\forall (h: nat).(\forall (d: nat).((subst0 i (lift -h d u) (lift h (S (plus i d)) t0) x) \to (ex2 T (\lambda (t2: T).(eq T x -(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u t0 -t2)))))))))).(\lambda (x: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda (d: -nat).(\lambda (H1: (subst0 i (lift h d u) (lift h (S (plus i d)) (THead k t -t0)) x)).(let H2 \def (eq_ind T (lift h (S (plus i d)) (THead k t t0)) -(\lambda (t2: T).(subst0 i (lift h d u) t2 x)) H1 (THead k (lift h (S (plus i -d)) t) (lift h (s k (S (plus i d))) t0)) (lift_head k t t0 h (S (plus i d)))) -in (or3_ind (ex2 T (\lambda (u2: T).(eq T x (THead k u2 (lift h (s k (S (plus -i d))) t0)))) (\lambda (u2: T).(subst0 i (lift h d u) (lift h (S (plus i d)) -t) u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h (S (plus i d)) t) -t2))) (\lambda (t2: T).(subst0 (s k i) (lift h d u) (lift h (s k (S (plus i -d))) t0) t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k -u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i (lift h d u) (lift h (S -(plus i d)) t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) (lift h -d u) (lift h (s k (S (plus i d))) t0) t2)))) (ex2 T (\lambda (t2: T).(eq T x -(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) -t2))) (\lambda (H3: (ex2 T (\lambda (u2: T).(eq T x (THead k u2 (lift h (s k -(S (plus i d))) t0)))) (\lambda (u2: T).(subst0 i (lift h d u) (lift h (S -(plus i d)) t) u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead k u2 (lift h -(s k (S (plus i d))) t0)))) (\lambda (u2: T).(subst0 i (lift h d u) (lift h -(S (plus i d)) t) u2)) (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) -t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2))) (\lambda (x0: -T).(\lambda (H4: (eq T x (THead k x0 (lift h (s k (S (plus i d))) -t0)))).(\lambda (H5: (subst0 i (lift h d u) (lift h (S (plus i d)) t) -x0)).(eq_ind_r T (THead k x0 (lift h (s k (S (plus i d))) t0)) (\lambda (t2: -T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda -(t3: T).(subst0 i u (THead k t t0) t3)))) (ex2_ind T (\lambda (t2: T).(eq T -x0 (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u t t2)) (ex2 T -(\lambda (t2: T).(eq T (THead k x0 (lift h (s k (S (plus i d))) t0)) (lift h -(S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2))) -(\lambda (x1: T).(\lambda (H6: (eq T x0 (lift h (S (plus i d)) x1))).(\lambda -(H7: (subst0 i u t x1)).(eq_ind_r T (lift h (S (plus i d)) x1) (\lambda (t2: -T).(ex2 T (\lambda (t3: T).(eq T (THead k t2 (lift h (s k (S (plus i d))) -t0)) (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u (THead k t t0) -t3)))) (eq_ind T (lift h (S (plus i d)) (THead k x1 t0)) (\lambda (t2: -T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda -(t3: T).(subst0 i u (THead k t t0) t3)))) (ex_intro2 T (\lambda (t2: T).(eq T -(lift h (S (plus i d)) (THead k x1 t0)) (lift h (S (plus i d)) t2))) (\lambda -(t2: T).(subst0 i u (THead k t t0) t2)) (THead k x1 t0) (refl_equal T (lift h -(S (plus i d)) (THead k x1 t0))) (subst0_fst u x1 t i H7 t0 k)) (THead k -(lift h (S (plus i d)) x1) (lift h (s k (S (plus i d))) t0)) (lift_head k x1 -t0 h (S (plus i d)))) x0 H6)))) (H x0 i h d H5)) x H4)))) H3)) (\lambda (H3: -(ex2 T (\lambda (t2: T).(eq T x (THead k (lift h (S (plus i d)) t) t2))) -(\lambda (t2: T).(subst0 (s k i) (lift h d u) (lift h (s k (S (plus i d))) -t0) t2)))).(ex2_ind T (\lambda (t2: T).(eq T x (THead k (lift h (S (plus i -d)) t) t2))) (\lambda (t2: T).(subst0 (s k i) (lift h d u) (lift h (s k (S -(plus i d))) t0) t2)) (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) -t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2))) (\lambda (x0: -T).(\lambda (H4: (eq T x (THead k (lift h (S (plus i d)) t) x0))).(\lambda -(H5: (subst0 (s k i) (lift h d u) (lift h (s k (S (plus i d))) t0) -x0)).(eq_ind_r T (THead k (lift h (S (plus i d)) t) x0) (\lambda (t2: T).(ex2 -T (\lambda (t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda (t3: -T).(subst0 i u (THead k t t0) t3)))) (let H6 \def (eq_ind nat (s k (S (plus i -d))) (\lambda (n: nat).(subst0 (s k i) (lift h d u) (lift h n t0) x0)) H5 (S -(s k (plus i d))) (s_S k (plus i d))) in (let H7 \def (eq_ind nat (s k (plus -i d)) (\lambda (n: nat).(subst0 (s k i) (lift h d u) (lift h (S n) t0) x0)) -H6 (plus (s k i) d) (s_plus k i d)) in (ex2_ind T (\lambda (t2: T).(eq T x0 -(lift h (S (plus (s k i) d)) t2))) (\lambda (t2: T).(subst0 (s k i) u t0 t2)) -(ex2 T (\lambda (t2: T).(eq T (THead k (lift h (S (plus i d)) t) x0) (lift h -(S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2))) -(\lambda (x1: T).(\lambda (H8: (eq T x0 (lift h (S (plus (s k i) d)) -x1))).(\lambda (H9: (subst0 (s k i) u t0 x1)).(eq_ind_r T (lift h (S (plus (s -k i) d)) x1) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k (lift h -(S (plus i d)) t) t2) (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i -u (THead k t t0) t3)))) (eq_ind nat (s k (plus i d)) (\lambda (n: nat).(ex2 T -(\lambda (t2: T).(eq T (THead k (lift h (S (plus i d)) t) (lift h (S n) x1)) -(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) -t2)))) (eq_ind nat (s k (S (plus i d))) (\lambda (n: nat).(ex2 T (\lambda -(t2: T).(eq T (THead k (lift h (S (plus i d)) t) (lift h n x1)) (lift h (S -(plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2)))) (eq_ind -T (lift h (S (plus i d)) (THead k t x1)) (\lambda (t2: T).(ex2 T (\lambda -(t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u -(THead k t t0) t3)))) (ex_intro2 T (\lambda (t2: T).(eq T (lift h (S (plus i -d)) (THead k t x1)) (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u -(THead k t t0) t2)) (THead k t x1) (refl_equal T (lift h (S (plus i d)) -(THead k t x1))) (subst0_snd k u x1 t0 i H9 t)) (THead k (lift h (S (plus i -d)) t) (lift h (s k (S (plus i d))) x1)) (lift_head k t x1 h (S (plus i d)))) -(S (s k (plus i d))) (s_S k (plus i d))) (plus (s k i) d) (s_plus k i d)) x0 -H8)))) (H0 x0 (s k i) h d H7)))) x H4)))) H3)) (\lambda (H3: (ex3_2 T T -(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i (lift h d u) (lift h (S (plus i d)) t) u2))) -(\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) (lift h d u) (lift h (s k (S -(plus i d))) t0) t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq -T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i (lift h d -u) (lift h (S (plus i d)) t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 -(s k i) (lift h d u) (lift h (s k (S (plus i d))) t0) t2))) (ex2 T (\lambda -(t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u -(THead k t t0) t2))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T x -(THead k x0 x1))).(\lambda (H5: (subst0 i (lift h d u) (lift h (S (plus i d)) -t) x0)).(\lambda (H6: (subst0 (s k i) (lift h d u) (lift h (s k (S (plus i -d))) t0) x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t2: T).(ex2 T (\lambda -(t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u -(THead k t t0) t3)))) (let H7 \def (eq_ind nat (s k (S (plus i d))) (\lambda -(n: nat).(subst0 (s k i) (lift h d u) (lift h n t0) x1)) H6 (S (s k (plus i -d))) (s_S k (plus i d))) in (let H8 \def (eq_ind nat (s k (plus i d)) -(\lambda (n: nat).(subst0 (s k i) (lift h d u) (lift h (S n) t0) x1)) H7 -(plus (s k i) d) (s_plus k i d)) in (ex2_ind T (\lambda (t2: T).(eq T x1 -(lift h (S (plus (s k i) d)) t2))) (\lambda (t2: T).(subst0 (s k i) u t0 t2)) -(ex2 T (\lambda (t2: T).(eq T (THead k x0 x1) (lift h (S (plus i d)) t2))) -(\lambda (t2: T).(subst0 i u (THead k t t0) t2))) (\lambda (x2: T).(\lambda -(H9: (eq T x1 (lift h (S (plus (s k i) d)) x2))).(\lambda (H10: (subst0 (s k -i) u t0 x2)).(ex2_ind T (\lambda (t2: T).(eq T x0 (lift h (S (plus i d)) -t2))) (\lambda (t2: T).(subst0 i u t t2)) (ex2 T (\lambda (t2: T).(eq T -(THead k x0 x1) (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u -(THead k t t0) t2))) (\lambda (x3: T).(\lambda (H11: (eq T x0 (lift h (S -(plus i d)) x3))).(\lambda (H12: (subst0 i u t x3)).(eq_ind_r T (lift h (S -(plus i d)) x3) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k t2 -x1) (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u (THead k t t0) -t3)))) (eq_ind_r T (lift h (S (plus (s k i) d)) x2) (\lambda (t2: T).(ex2 T -(\lambda (t3: T).(eq T (THead k (lift h (S (plus i d)) x3) t2) (lift h (S -(plus i d)) t3))) (\lambda (t3: T).(subst0 i u (THead k t t0) t3)))) (eq_ind -nat (s k (plus i d)) (\lambda (n: nat).(ex2 T (\lambda (t2: T).(eq T (THead k -(lift h (S (plus i d)) x3) (lift h (S n) x2)) (lift h (S (plus i d)) t2))) -(\lambda (t2: T).(subst0 i u (THead k t t0) t2)))) (eq_ind nat (s k (S (plus -i d))) (\lambda (n: nat).(ex2 T (\lambda (t2: T).(eq T (THead k (lift h (S -(plus i d)) x3) (lift h n x2)) (lift h (S (plus i d)) t2))) (\lambda (t2: -T).(subst0 i u (THead k t t0) t2)))) (eq_ind T (lift h (S (plus i d)) (THead -k x3 x2)) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h (S (plus -i d)) t3))) (\lambda (t3: T).(subst0 i u (THead k t t0) t3)))) (ex_intro2 T -(\lambda (t2: T).(eq T (lift h (S (plus i d)) (THead k x3 x2)) (lift h (S -(plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2)) (THead k -x3 x2) (refl_equal T (lift h (S (plus i d)) (THead k x3 x2))) (subst0_both u -t x3 i H12 k t0 x2 H10)) (THead k (lift h (S (plus i d)) x3) (lift h (s k (S -(plus i d))) x2)) (lift_head k x3 x2 h (S (plus i d)))) (S (s k (plus i d))) -(s_S k (plus i d))) (plus (s k i) d) (s_plus k i d)) x1 H9) x0 H11)))) (H x0 -i h d H5))))) (H0 x1 (s k i) h d H8)))) x H4)))))) H3)) (subst0_gen_head k -(lift h d u) (lift h (S (plus i d)) t) (lift h (s k (S (plus i d))) t0) x i -H2))))))))))))) t1)). -(* COMMENTS -Initial nodes: 5157 -END *) - -theorem subst0_gen_lift_false: - \forall (t: T).(\forall (u: T).(\forall (x: T).(\forall (h: nat).(\forall -(d: nat).(\forall (i: nat).((le d i) \to ((lt i (plus d h)) \to ((subst0 i u -(lift h d t) x) \to (\forall (P: Prop).P))))))))) -\def - \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (u: T).(\forall (x: -T).(\forall (h: nat).(\forall (d: nat).(\forall (i: nat).((le d i) \to ((lt i -(plus d h)) \to ((subst0 i u (lift h d t0) x) \to (\forall (P: -Prop).P)))))))))) (\lambda (n: nat).(\lambda (u: T).(\lambda (x: T).(\lambda -(h: nat).(\lambda (d: nat).(\lambda (i: nat).(\lambda (_: (le d i)).(\lambda -(_: (lt i (plus d h))).(\lambda (H1: (subst0 i u (lift h d (TSort n)) -x)).(\lambda (P: Prop).(let H2 \def (eq_ind T (lift h d (TSort n)) (\lambda -(t0: T).(subst0 i u t0 x)) H1 (TSort n) (lift_sort n h d)) in -(subst0_gen_sort u x i n H2 P)))))))))))) (\lambda (n: nat).(\lambda (u: -T).(\lambda (x: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (i: -nat).(\lambda (H: (le d i)).(\lambda (H0: (lt i (plus d h))).(\lambda (H1: -(subst0 i u (lift h d (TLRef n)) x)).(\lambda (P: Prop).(lt_le_e n d P -(\lambda (H2: (lt n d)).(let H3 \def (eq_ind T (lift h d (TLRef n)) (\lambda -(t0: T).(subst0 i u t0 x)) H1 (TLRef n) (lift_lref_lt n h d H2)) in (land_ind -(eq nat n i) (eq T x (lift (S n) O u)) P (\lambda (H4: (eq nat n i)).(\lambda -(_: (eq T x (lift (S n) O u))).(let H6 \def (eq_ind nat n (\lambda (n0: -nat).(lt n0 d)) H2 i H4) in (le_false d i P H H6)))) (subst0_gen_lref u x i n -H3)))) (\lambda (H2: (le d n)).(let H3 \def (eq_ind T (lift h d (TLRef n)) -(\lambda (t0: T).(subst0 i u t0 x)) H1 (TLRef (plus n h)) (lift_lref_ge n h d -H2)) in (land_ind (eq nat (plus n h) i) (eq T x (lift (S (plus n h)) O u)) P -(\lambda (H4: (eq nat (plus n h) i)).(\lambda (_: (eq T x (lift (S (plus n -h)) O u))).(let H6 \def (eq_ind_r nat i (\lambda (n0: nat).(lt n0 (plus d -h))) H0 (plus n h) H4) in (le_false d n P H2 (lt_le_S n d (simpl_lt_plus_r h -n d H6)))))) (subst0_gen_lref u x i (plus n h) H3))))))))))))))) (\lambda (k: -K).(\lambda (t0: T).(\lambda (H: ((\forall (u: T).(\forall (x: T).(\forall -(h: nat).(\forall (d: nat).(\forall (i: nat).((le d i) \to ((lt i (plus d h)) -\to ((subst0 i u (lift h d t0) x) \to (\forall (P: -Prop).P))))))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (u: T).(\forall -(x: T).(\forall (h: nat).(\forall (d: nat).(\forall (i: nat).((le d i) \to -((lt i (plus d h)) \to ((subst0 i u (lift h d t1) x) \to (\forall (P: -Prop).P))))))))))).(\lambda (u: T).(\lambda (x: T).(\lambda (h: nat).(\lambda -(d: nat).(\lambda (i: nat).(\lambda (H1: (le d i)).(\lambda (H2: (lt i (plus -d h))).(\lambda (H3: (subst0 i u (lift h d (THead k t0 t1)) x)).(\lambda (P: -Prop).(let H4 \def (eq_ind T (lift h d (THead k t0 t1)) (\lambda (t2: -T).(subst0 i u t2 x)) H3 (THead k (lift h d t0) (lift h (s k d) t1)) -(lift_head k t0 t1 h d)) in (or3_ind (ex2 T (\lambda (u2: T).(eq T x (THead k -u2 (lift h (s k d) t1)))) (\lambda (u2: T).(subst0 i u (lift h d t0) u2))) -(ex2 T (\lambda (t2: T).(eq T x (THead k (lift h d t0) t2))) (\lambda (t2: -T).(subst0 (s k i) u (lift h (s k d) t1) t2))) (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 i u (lift h d t0) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 -(s k i) u (lift h (s k d) t1) t2)))) P (\lambda (H5: (ex2 T (\lambda (u2: -T).(eq T x (THead k u2 (lift h (s k d) t1)))) (\lambda (u2: T).(subst0 i u -(lift h d t0) u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead k u2 (lift h -(s k d) t1)))) (\lambda (u2: T).(subst0 i u (lift h d t0) u2)) P (\lambda -(x0: T).(\lambda (_: (eq T x (THead k x0 (lift h (s k d) t1)))).(\lambda (H7: -(subst0 i u (lift h d t0) x0)).(H u x0 h d i H1 H2 H7 P)))) H5)) (\lambda -(H5: (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h d t0) t2))) (\lambda -(t2: T).(subst0 (s k i) u (lift h (s k d) t1) t2)))).(ex2_ind T (\lambda (t2: -T).(eq T x (THead k (lift h d t0) t2))) (\lambda (t2: T).(subst0 (s k i) u -(lift h (s k d) t1) t2)) P (\lambda (x0: T).(\lambda (_: (eq T x (THead k -(lift h d t0) x0))).(\lambda (H7: (subst0 (s k i) u (lift h (s k d) t1) -x0)).(H0 u x0 h (s k d) (s k i) (s_le k d i H1) (eq_ind nat (s k (plus d h)) -(\lambda (n: nat).(lt (s k i) n)) (lt_le_S (s k i) (s k (plus d h)) (s_lt k i -(plus d h) H2)) (plus (s k d) h) (s_plus k d h)) H7 P)))) H5)) (\lambda (H5: -(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 i u (lift h d t0) u2))) (\lambda (_: -T).(\lambda (t2: T).(subst0 (s k i) u (lift h (s k d) t1) t2))))).(ex3_2_ind -T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda -(u2: T).(\lambda (_: T).(subst0 i u (lift h d t0) u2))) (\lambda (_: -T).(\lambda (t2: T).(subst0 (s k i) u (lift h (s k d) t1) t2))) P (\lambda -(x0: T).(\lambda (x1: T).(\lambda (_: (eq T x (THead k x0 x1))).(\lambda (H7: -(subst0 i u (lift h d t0) x0)).(\lambda (_: (subst0 (s k i) u (lift h (s k d) -t1) x1)).(H u x0 h d i H1 H2 H7 P)))))) H5)) (subst0_gen_head k u (lift h d -t0) (lift h (s k d) t1) x i H4))))))))))))))))) t). -(* COMMENTS -Initial nodes: 1621 -END *) - -theorem subst0_gen_lift_ge: - \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall -(h: nat).(\forall (d: nat).((subst0 i u (lift h d t1) x) \to ((le (plus d h) -i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: -T).(subst0 (minus i h) u t1 t2)))))))))) -\def - \lambda (u: T).(\lambda (t1: T).(T_ind (\lambda (t: T).(\forall (x: -T).(\forall (i: nat).(\forall (h: nat).(\forall (d: nat).((subst0 i u (lift h -d t) x) \to ((le (plus d h) i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d -t2))) (\lambda (t2: T).(subst0 (minus i h) u t t2)))))))))) (\lambda (n: -nat).(\lambda (x: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda (d: -nat).(\lambda (H: (subst0 i u (lift h d (TSort n)) x)).(\lambda (_: (le (plus -d h) i)).(let H1 \def (eq_ind T (lift h d (TSort n)) (\lambda (t: T).(subst0 -i u t x)) H (TSort n) (lift_sort n h d)) in (subst0_gen_sort u x i n H1 (ex2 -T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i -h) u (TSort n) t2)))))))))))) (\lambda (n: nat).(\lambda (x: T).(\lambda (i: -nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (subst0 i u (lift h d -(TLRef n)) x)).(\lambda (H0: (le (plus d h) i)).(lt_le_e n d (ex2 T (\lambda -(t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (TLRef -n) t2))) (\lambda (H1: (lt n d)).(let H2 \def (eq_ind T (lift h d (TLRef n)) -(\lambda (t: T).(subst0 i u t x)) H (TLRef n) (lift_lref_lt n h d H1)) in -(land_ind (eq nat n i) (eq T x (lift (S n) O u)) (ex2 T (\lambda (t2: T).(eq -T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (TLRef n) t2))) -(\lambda (H3: (eq nat n i)).(\lambda (_: (eq T x (lift (S n) O u))).(let H5 -\def (eq_ind nat n (\lambda (n0: nat).(lt n0 d)) H1 i H3) in (le_false (plus -d h) i (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: -T).(subst0 (minus i h) u (TLRef n) t2))) H0 (le_plus_trans (S i) d h H5))))) -(subst0_gen_lref u x i n H2)))) (\lambda (H1: (le d n)).(let H2 \def (eq_ind -T (lift h d (TLRef n)) (\lambda (t: T).(subst0 i u t x)) H (TLRef (plus n h)) -(lift_lref_ge n h d H1)) in (land_ind (eq nat (plus n h) i) (eq T x (lift (S -(plus n h)) O u)) (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda -(t2: T).(subst0 (minus i h) u (TLRef n) t2))) (\lambda (H3: (eq nat (plus n -h) i)).(\lambda (H4: (eq T x (lift (S (plus n h)) O u))).(eq_ind nat (plus n -h) (\lambda (n0: nat).(ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) -(\lambda (t2: T).(subst0 (minus n0 h) u (TLRef n) t2)))) (eq_ind_r T (lift (S -(plus n h)) O u) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h d -t2))) (\lambda (t2: T).(subst0 (minus (plus n h) h) u (TLRef n) t2)))) -(eq_ind_r nat n (\lambda (n0: nat).(ex2 T (\lambda (t2: T).(eq T (lift (S -(plus n h)) O u) (lift h d t2))) (\lambda (t2: T).(subst0 n0 u (TLRef n) -t2)))) (ex_intro2 T (\lambda (t2: T).(eq T (lift (S (plus n h)) O u) (lift h -d t2))) (\lambda (t2: T).(subst0 n u (TLRef n) t2)) (lift (S n) O u) -(eq_ind_r T (lift (plus h (S n)) O u) (\lambda (t: T).(eq T (lift (S (plus n -h)) O u) t)) (eq_ind_r nat (plus h n) (\lambda (n0: nat).(eq T (lift (S n0) O -u) (lift (plus h (S n)) O u))) (eq_ind_r nat (plus h (S n)) (\lambda (n0: -nat).(eq T (lift n0 O u) (lift (plus h (S n)) O u))) (refl_equal T (lift -(plus h (S n)) O u)) (S (plus h n)) (plus_n_Sm h n)) (plus n h) (plus_sym n -h)) (lift h d (lift (S n) O u)) (lift_free u (S n) h O d (le_trans_plus_r O d -(plus O (S n)) (le_plus_plus O O d (S n) (le_n O) (le_S d n H1))) (le_O_n -d))) (subst0_lref u n)) (minus (plus n h) h) (minus_plus_r n h)) x H4) i -H3))) (subst0_gen_lref u x i (plus n h) H2)))))))))))) (\lambda (k: -K).(\lambda (t: T).(\lambda (H: ((\forall (x: T).(\forall (i: nat).(\forall -(h: nat).(\forall (d: nat).((subst0 i u (lift h d t) x) \to ((le (plus d h) -i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: -T).(subst0 (minus i h) u t t2))))))))))).(\lambda (t0: T).(\lambda (H0: -((\forall (x: T).(\forall (i: nat).(\forall (h: nat).(\forall (d: -nat).((subst0 i u (lift h d t0) x) \to ((le (plus d h) i) \to (ex2 T (\lambda -(t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u t0 -t2))))))))))).(\lambda (x: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda -(d: nat).(\lambda (H1: (subst0 i u (lift h d (THead k t t0)) x)).(\lambda -(H2: (le (plus d h) i)).(let H3 \def (eq_ind T (lift h d (THead k t t0)) -(\lambda (t2: T).(subst0 i u t2 x)) H1 (THead k (lift h d t) (lift h (s k d) -t0)) (lift_head k t t0 h d)) in (or3_ind (ex2 T (\lambda (u2: T).(eq T x -(THead k u2 (lift h (s k d) t0)))) (\lambda (u2: T).(subst0 i u (lift h d t) -u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h d t) t2))) (\lambda -(t2: T).(subst0 (s k i) u (lift h (s k d) t0) t2))) (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 i u (lift h d t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s -k i) u (lift h (s k d) t0) t2)))) (ex2 T (\lambda (t2: T).(eq T x (lift h d -t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda -(H4: (ex2 T (\lambda (u2: T).(eq T x (THead k u2 (lift h (s k d) t0)))) -(\lambda (u2: T).(subst0 i u (lift h d t) u2)))).(ex2_ind T (\lambda (u2: -T).(eq T x (THead k u2 (lift h (s k d) t0)))) (\lambda (u2: T).(subst0 i u -(lift h d t) u2)) (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda -(t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (x0: T).(\lambda -(H5: (eq T x (THead k x0 (lift h (s k d) t0)))).(\lambda (H6: (subst0 i u -(lift h d t) x0)).(eq_ind_r T (THead k x0 (lift h (s k d) t0)) (\lambda (t2: -T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst0 -(minus i h) u (THead k t t0) t3)))) (ex2_ind T (\lambda (t2: T).(eq T x0 -(lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u t t2)) (ex2 T (\lambda -(t2: T).(eq T (THead k x0 (lift h (s k d) t0)) (lift h d t2))) (\lambda (t2: -T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (x1: T).(\lambda (H7: -(eq T x0 (lift h d x1))).(\lambda (H8: (subst0 (minus i h) u t x1)).(eq_ind_r -T (lift h d x1) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k t2 -(lift h (s k d) t0)) (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u -(THead k t t0) t3)))) (eq_ind T (lift h d (THead k x1 t0)) (\lambda (t2: -T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst0 -(minus i h) u (THead k t t0) t3)))) (ex_intro2 T (\lambda (t2: T).(eq T (lift -h d (THead k x1 t0)) (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u -(THead k t t0) t2)) (THead k x1 t0) (refl_equal T (lift h d (THead k x1 t0))) -(subst0_fst u x1 t (minus i h) H8 t0 k)) (THead k (lift h d x1) (lift h (s k -d) t0)) (lift_head k x1 t0 h d)) x0 H7)))) (H x0 i h d H6 H2)) x H5)))) H4)) -(\lambda (H4: (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h d t) t2))) -(\lambda (t2: T).(subst0 (s k i) u (lift h (s k d) t0) t2)))).(ex2_ind T -(\lambda (t2: T).(eq T x (THead k (lift h d t) t2))) (\lambda (t2: T).(subst0 -(s k i) u (lift h (s k d) t0) t2)) (ex2 T (\lambda (t2: T).(eq T x (lift h d -t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda -(x0: T).(\lambda (H5: (eq T x (THead k (lift h d t) x0))).(\lambda (H6: -(subst0 (s k i) u (lift h (s k d) t0) x0)).(eq_ind_r T (THead k (lift h d t) -x0) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3))) -(\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) (ex2_ind T -(\lambda (t2: T).(eq T x0 (lift h (s k d) t2))) (\lambda (t2: T).(subst0 -(minus (s k i) h) u t0 t2)) (ex2 T (\lambda (t2: T).(eq T (THead k (lift h d -t) x0) (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead k t t0) -t2))) (\lambda (x1: T).(\lambda (H7: (eq T x0 (lift h (s k d) x1))).(\lambda -(H8: (subst0 (minus (s k i) h) u t0 x1)).(eq_ind_r T (lift h (s k d) x1) -(\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k (lift h d t) t2) -(lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) -(eq_ind T (lift h d (THead k t x1)) (\lambda (t2: T).(ex2 T (\lambda (t3: -T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t -t0) t3)))) (let H9 \def (eq_ind_r nat (minus (s k i) h) (\lambda (n: -nat).(subst0 n u t0 x1)) H8 (s k (minus i h)) (s_minus k i h (le_trans_plus_r -d h i H2))) in (ex_intro2 T (\lambda (t2: T).(eq T (lift h d (THead k t x1)) -(lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead k t t0) t2)) -(THead k t x1) (refl_equal T (lift h d (THead k t x1))) (subst0_snd k u x1 t0 -(minus i h) H9 t))) (THead k (lift h d t) (lift h (s k d) x1)) (lift_head k t -x1 h d)) x0 H7)))) (H0 x0 (s k i) h (s k d) H6 (eq_ind nat (s k (plus d h)) -(\lambda (n: nat).(le n (s k i))) (s_le k (plus d h) i H2) (plus (s k d) h) -(s_plus k d h)))) x H5)))) H4)) (\lambda (H4: (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 i u (lift h d t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s -k i) u (lift h (s k d) t0) t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda -(t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i -u (lift h d t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) u (lift -h (s k d) t0) t2))) (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda -(t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (x0: T).(\lambda -(x1: T).(\lambda (H5: (eq T x (THead k x0 x1))).(\lambda (H6: (subst0 i u -(lift h d t) x0)).(\lambda (H7: (subst0 (s k i) u (lift h (s k d) t0) -x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq -T t2 (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) -t3)))) (ex2_ind T (\lambda (t2: T).(eq T x1 (lift h (s k d) t2))) (\lambda -(t2: T).(subst0 (minus (s k i) h) u t0 t2)) (ex2 T (\lambda (t2: T).(eq T -(THead k x0 x1) (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead -k t t0) t2))) (\lambda (x2: T).(\lambda (H8: (eq T x1 (lift h (s k d) -x2))).(\lambda (H9: (subst0 (minus (s k i) h) u t0 x2)).(ex2_ind T (\lambda -(t2: T).(eq T x0 (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u t -t2)) (ex2 T (\lambda (t2: T).(eq T (THead k x0 x1) (lift h d t2))) (\lambda -(t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (x3: T).(\lambda -(H10: (eq T x0 (lift h d x3))).(\lambda (H11: (subst0 (minus i h) u t -x3)).(eq_ind_r T (lift h d x3) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T -(THead k t2 x1) (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead -k t t0) t3)))) (eq_ind_r T (lift h (s k d) x2) (\lambda (t2: T).(ex2 T -(\lambda (t3: T).(eq T (THead k (lift h d x3) t2) (lift h d t3))) (\lambda -(t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) (eq_ind T (lift h d -(THead k x3 x2)) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d -t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) (let H12 -\def (eq_ind_r nat (minus (s k i) h) (\lambda (n: nat).(subst0 n u t0 x2)) H9 -(s k (minus i h)) (s_minus k i h (le_trans_plus_r d h i H2))) in (ex_intro2 T -(\lambda (t2: T).(eq T (lift h d (THead k x3 x2)) (lift h d t2))) (\lambda -(t2: T).(subst0 (minus i h) u (THead k t t0) t2)) (THead k x3 x2) (refl_equal -T (lift h d (THead k x3 x2))) (subst0_both u t x3 (minus i h) H11 k t0 x2 -H12))) (THead k (lift h d x3) (lift h (s k d) x2)) (lift_head k x3 x2 h d)) -x1 H8) x0 H10)))) (H x0 i h d H6 H2))))) (H0 x1 (s k i) h (s k d) H7 (eq_ind -nat (s k (plus d h)) (\lambda (n: nat).(le n (s k i))) (s_le k (plus d h) i -H2) (plus (s k d) h) (s_plus k d h)))) x H5)))))) H4)) (subst0_gen_head k u -(lift h d t) (lift h (s k d) t0) x i H3)))))))))))))) t1)). -(* COMMENTS -Initial nodes: 4191 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/subst0/props.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/subst0/props.ma deleted file mode 100644 index 5da05fa2a..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/subst0/props.ma +++ /dev/null @@ -1,241 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/subst0/fwd.ma". - -theorem subst0_refl: - \forall (u: T).(\forall (t: T).(\forall (d: nat).((subst0 d u t t) \to -(\forall (P: Prop).P)))) -\def - \lambda (u: T).(\lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d: -nat).((subst0 d u t0 t0) \to (\forall (P: Prop).P)))) (\lambda (n: -nat).(\lambda (d: nat).(\lambda (H: (subst0 d u (TSort n) (TSort -n))).(\lambda (P: Prop).(subst0_gen_sort u (TSort n) d n H P))))) (\lambda -(n: nat).(\lambda (d: nat).(\lambda (H: (subst0 d u (TLRef n) (TLRef -n))).(\lambda (P: Prop).(land_ind (eq nat n d) (eq T (TLRef n) (lift (S n) O -u)) P (\lambda (_: (eq nat n d)).(\lambda (H1: (eq T (TLRef n) (lift (S n) O -u))).(lift_gen_lref_false (S n) O n (le_O_n n) (le_n (plus O (S n))) u H1 -P))) (subst0_gen_lref u (TLRef n) d n H)))))) (\lambda (k: K).(\lambda (t0: -T).(\lambda (H: ((\forall (d: nat).((subst0 d u t0 t0) \to (\forall (P: -Prop).P))))).(\lambda (t1: T).(\lambda (H0: ((\forall (d: nat).((subst0 d u -t1 t1) \to (\forall (P: Prop).P))))).(\lambda (d: nat).(\lambda (H1: (subst0 -d u (THead k t0 t1) (THead k t0 t1))).(\lambda (P: Prop).(or3_ind (ex2 T -(\lambda (u2: T).(eq T (THead k t0 t1) (THead k u2 t1))) (\lambda (u2: -T).(subst0 d u t0 u2))) (ex2 T (\lambda (t2: T).(eq T (THead k t0 t1) (THead -k t0 t2))) (\lambda (t2: T).(subst0 (s k d) u t1 t2))) (ex3_2 T T (\lambda -(u2: T).(\lambda (t2: T).(eq T (THead k t0 t1) (THead k u2 t2)))) (\lambda -(u2: T).(\lambda (_: T).(subst0 d u t0 u2))) (\lambda (_: T).(\lambda (t2: -T).(subst0 (s k d) u t1 t2)))) P (\lambda (H2: (ex2 T (\lambda (u2: T).(eq T -(THead k t0 t1) (THead k u2 t1))) (\lambda (u2: T).(subst0 d u t0 -u2)))).(ex2_ind T (\lambda (u2: T).(eq T (THead k t0 t1) (THead k u2 t1))) -(\lambda (u2: T).(subst0 d u t0 u2)) P (\lambda (x: T).(\lambda (H3: (eq T -(THead k t0 t1) (THead k x t1))).(\lambda (H4: (subst0 d u t0 x)).(let H5 -\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ t2 _) -\Rightarrow t2])) (THead k t0 t1) (THead k x t1) H3) in (let H6 \def -(eq_ind_r T x (\lambda (t2: T).(subst0 d u t0 t2)) H4 t0 H5) in (H d H6 -P)))))) H2)) (\lambda (H2: (ex2 T (\lambda (t2: T).(eq T (THead k t0 t1) -(THead k t0 t2))) (\lambda (t2: T).(subst0 (s k d) u t1 t2)))).(ex2_ind T -(\lambda (t2: T).(eq T (THead k t0 t1) (THead k t0 t2))) (\lambda (t2: -T).(subst0 (s k d) u t1 t2)) P (\lambda (x: T).(\lambda (H3: (eq T (THead k -t0 t1) (THead k t0 x))).(\lambda (H4: (subst0 (s k d) u t1 x)).(let H5 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ _ t2) -\Rightarrow t2])) (THead k t0 t1) (THead k t0 x) H3) in (let H6 \def -(eq_ind_r T x (\lambda (t2: T).(subst0 (s k d) u t1 t2)) H4 t1 H5) in (H0 (s -k d) H6 P)))))) H2)) (\lambda (H2: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T (THead k t0 t1) (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 d u t0 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k d) u t1 -t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead k t0 -t1) (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 d u t0 u2))) -(\lambda (_: T).(\lambda (t2: T).(subst0 (s k d) u t1 t2))) P (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H3: (eq T (THead k t0 t1) (THead k x0 -x1))).(\lambda (H4: (subst0 d u t0 x0)).(\lambda (H5: (subst0 (s k d) u t1 -x1)).(let H6 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda -(_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead -_ t2 _) \Rightarrow t2])) (THead k t0 t1) (THead k x0 x1) H3) in ((let H7 -\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ _ t2) -\Rightarrow t2])) (THead k t0 t1) (THead k x0 x1) H3) in (\lambda (H8: (eq T -t0 x0)).(let H9 \def (eq_ind_r T x1 (\lambda (t2: T).(subst0 (s k d) u t1 -t2)) H5 t1 H7) in (let H10 \def (eq_ind_r T x0 (\lambda (t2: T).(subst0 d u -t0 t2)) H4 t0 H8) in (H d H10 P))))) H6))))))) H2)) (subst0_gen_head k u t0 -t1 (THead k t0 t1) d H1)))))))))) t)). -(* COMMENTS -Initial nodes: 1119 -END *) - -theorem subst0_lift_lt: - \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).((subst0 -i u t1 t2) \to (\forall (d: nat).((lt i d) \to (\forall (h: nat).(subst0 i -(lift h (minus d (S i)) u) (lift h d t1) (lift h d t2))))))))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H: (subst0 i u t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t: -T).(\lambda (t0: T).(\lambda (t3: T).(\forall (d: nat).((lt n d) \to (\forall -(h: nat).(subst0 n (lift h (minus d (S n)) t) (lift h d t0) (lift h d -t3))))))))) (\lambda (v: T).(\lambda (i0: nat).(\lambda (d: nat).(\lambda -(H0: (lt i0 d)).(\lambda (h: nat).(eq_ind_r T (TLRef i0) (\lambda (t: -T).(subst0 i0 (lift h (minus d (S i0)) v) t (lift h d (lift (S i0) O v)))) -(let w \def (minus d (S i0)) in (eq_ind nat (plus (S i0) (minus d (S i0))) -(\lambda (n: nat).(subst0 i0 (lift h w v) (TLRef i0) (lift h n (lift (S i0) O -v)))) (eq_ind_r T (lift (S i0) O (lift h (minus d (S i0)) v)) (\lambda (t: -T).(subst0 i0 (lift h w v) (TLRef i0) t)) (subst0_lref (lift h (minus d (S -i0)) v) i0) (lift h (plus (S i0) (minus d (S i0))) (lift (S i0) O v)) (lift_d -v h (S i0) (minus d (S i0)) O (le_O_n (minus d (S i0))))) d (le_plus_minus_r -(S i0) d H0))) (lift h d (TLRef i0)) (lift_lref_lt i0 h d H0))))))) (\lambda -(v: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i0: nat).(\lambda (_: -(subst0 i0 v u1 u2)).(\lambda (H1: ((\forall (d: nat).((lt i0 d) \to (\forall -(h: nat).(subst0 i0 (lift h (minus d (S i0)) v) (lift h d u1) (lift h d -u2))))))).(\lambda (t: T).(\lambda (k: K).(\lambda (d: nat).(\lambda (H2: (lt -i0 d)).(\lambda (h: nat).(eq_ind_r T (THead k (lift h d u1) (lift h (s k d) -t)) (\lambda (t0: T).(subst0 i0 (lift h (minus d (S i0)) v) t0 (lift h d -(THead k u2 t)))) (eq_ind_r T (THead k (lift h d u2) (lift h (s k d) t)) -(\lambda (t0: T).(subst0 i0 (lift h (minus d (S i0)) v) (THead k (lift h d -u1) (lift h (s k d) t)) t0)) (subst0_fst (lift h (minus d (S i0)) v) (lift h -d u2) (lift h d u1) i0 (H1 d H2 h) (lift h (s k d) t) k) (lift h d (THead k -u2 t)) (lift_head k u2 t h d)) (lift h d (THead k u1 t)) (lift_head k u1 t h -d))))))))))))) (\lambda (k: K).(\lambda (v: T).(\lambda (t0: T).(\lambda (t3: -T).(\lambda (i0: nat).(\lambda (_: (subst0 (s k i0) v t3 t0)).(\lambda (H1: -((\forall (d: nat).((lt (s k i0) d) \to (\forall (h: nat).(subst0 (s k i0) -(lift h (minus d (S (s k i0))) v) (lift h d t3) (lift h d t0))))))).(\lambda -(u0: T).(\lambda (d: nat).(\lambda (H2: (lt i0 d)).(\lambda (h: nat).(let H3 -\def (eq_ind_r nat (S (s k i0)) (\lambda (n: nat).(\forall (d0: nat).((lt (s -k i0) d0) \to (\forall (h0: nat).(subst0 (s k i0) (lift h0 (minus d0 n) v) -(lift h0 d0 t3) (lift h0 d0 t0)))))) H1 (s k (S i0)) (s_S k i0)) in (eq_ind_r -T (THead k (lift h d u0) (lift h (s k d) t3)) (\lambda (t: T).(subst0 i0 -(lift h (minus d (S i0)) v) t (lift h d (THead k u0 t0)))) (eq_ind_r T (THead -k (lift h d u0) (lift h (s k d) t0)) (\lambda (t: T).(subst0 i0 (lift h -(minus d (S i0)) v) (THead k (lift h d u0) (lift h (s k d) t3)) t)) (eq_ind -nat (minus (s k d) (s k (S i0))) (\lambda (n: nat).(subst0 i0 (lift h n v) -(THead k (lift h d u0) (lift h (s k d) t3)) (THead k (lift h d u0) (lift h (s -k d) t0)))) (subst0_snd k (lift h (minus (s k d) (s k (S i0))) v) (lift h (s -k d) t0) (lift h (s k d) t3) i0 (H3 (s k d) (s_lt k i0 d H2) h) (lift h d -u0)) (minus d (S i0)) (minus_s_s k d (S i0))) (lift h d (THead k u0 t0)) -(lift_head k u0 t0 h d)) (lift h d (THead k u0 t3)) (lift_head k u0 t3 h -d)))))))))))))) (\lambda (v: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda -(i0: nat).(\lambda (_: (subst0 i0 v u1 u2)).(\lambda (H1: ((\forall (d: -nat).((lt i0 d) \to (\forall (h: nat).(subst0 i0 (lift h (minus d (S i0)) v) -(lift h d u1) (lift h d u2))))))).(\lambda (k: K).(\lambda (t0: T).(\lambda -(t3: T).(\lambda (_: (subst0 (s k i0) v t0 t3)).(\lambda (H3: ((\forall (d: -nat).((lt (s k i0) d) \to (\forall (h: nat).(subst0 (s k i0) (lift h (minus d -(S (s k i0))) v) (lift h d t0) (lift h d t3))))))).(\lambda (d: nat).(\lambda -(H4: (lt i0 d)).(\lambda (h: nat).(let H5 \def (eq_ind_r nat (S (s k i0)) -(\lambda (n: nat).(\forall (d0: nat).((lt (s k i0) d0) \to (\forall (h0: -nat).(subst0 (s k i0) (lift h0 (minus d0 n) v) (lift h0 d0 t0) (lift h0 d0 -t3)))))) H3 (s k (S i0)) (s_S k i0)) in (eq_ind_r T (THead k (lift h d u1) -(lift h (s k d) t0)) (\lambda (t: T).(subst0 i0 (lift h (minus d (S i0)) v) t -(lift h d (THead k u2 t3)))) (eq_ind_r T (THead k (lift h d u2) (lift h (s k -d) t3)) (\lambda (t: T).(subst0 i0 (lift h (minus d (S i0)) v) (THead k (lift -h d u1) (lift h (s k d) t0)) t)) (subst0_both (lift h (minus d (S i0)) v) -(lift h d u1) (lift h d u2) i0 (H1 d H4 h) k (lift h (s k d) t0) (lift h (s k -d) t3) (eq_ind nat (minus (s k d) (s k (S i0))) (\lambda (n: nat).(subst0 (s -k i0) (lift h n v) (lift h (s k d) t0) (lift h (s k d) t3))) (H5 (s k d) -(s_lt k i0 d H4) h) (minus d (S i0)) (minus_s_s k d (S i0)))) (lift h d -(THead k u2 t3)) (lift_head k u2 t3 h d)) (lift h d (THead k u1 t0)) -(lift_head k u1 t0 h d))))))))))))))))) i u t1 t2 H))))). -(* COMMENTS -Initial nodes: 1805 -END *) - -theorem subst0_lift_ge: - \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).(\forall -(h: nat).((subst0 i u t1 t2) \to (\forall (d: nat).((le d i) \to (subst0 -(plus i h) u (lift h d t1) (lift h d t2))))))))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda -(h: nat).(\lambda (H: (subst0 i u t1 t2)).(subst0_ind (\lambda (n: -nat).(\lambda (t: T).(\lambda (t0: T).(\lambda (t3: T).(\forall (d: nat).((le -d n) \to (subst0 (plus n h) t (lift h d t0) (lift h d t3)))))))) (\lambda (v: -T).(\lambda (i0: nat).(\lambda (d: nat).(\lambda (H0: (le d i0)).(eq_ind_r T -(TLRef (plus i0 h)) (\lambda (t: T).(subst0 (plus i0 h) v t (lift h d (lift -(S i0) O v)))) (eq_ind_r T (lift (plus h (S i0)) O v) (\lambda (t: T).(subst0 -(plus i0 h) v (TLRef (plus i0 h)) t)) (eq_ind nat (S (plus h i0)) (\lambda -(n: nat).(subst0 (plus i0 h) v (TLRef (plus i0 h)) (lift n O v))) (eq_ind_r -nat (plus h i0) (\lambda (n: nat).(subst0 n v (TLRef n) (lift (S (plus h i0)) -O v))) (subst0_lref v (plus h i0)) (plus i0 h) (plus_sym i0 h)) (plus h (S -i0)) (plus_n_Sm h i0)) (lift h d (lift (S i0) O v)) (lift_free v (S i0) h O d -(le_S d i0 H0) (le_O_n d))) (lift h d (TLRef i0)) (lift_lref_ge i0 h d -H0)))))) (\lambda (v: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i0: -nat).(\lambda (_: (subst0 i0 v u1 u2)).(\lambda (H1: ((\forall (d: nat).((le -d i0) \to (subst0 (plus i0 h) v (lift h d u1) (lift h d u2)))))).(\lambda (t: -T).(\lambda (k: K).(\lambda (d: nat).(\lambda (H2: (le d i0)).(eq_ind_r T -(THead k (lift h d u1) (lift h (s k d) t)) (\lambda (t0: T).(subst0 (plus i0 -h) v t0 (lift h d (THead k u2 t)))) (eq_ind_r T (THead k (lift h d u2) (lift -h (s k d) t)) (\lambda (t0: T).(subst0 (plus i0 h) v (THead k (lift h d u1) -(lift h (s k d) t)) t0)) (subst0_fst v (lift h d u2) (lift h d u1) (plus i0 -h) (H1 d H2) (lift h (s k d) t) k) (lift h d (THead k u2 t)) (lift_head k u2 -t h d)) (lift h d (THead k u1 t)) (lift_head k u1 t h d)))))))))))) (\lambda -(k: K).(\lambda (v: T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (i0: -nat).(\lambda (_: (subst0 (s k i0) v t3 t0)).(\lambda (H1: ((\forall (d: -nat).((le d (s k i0)) \to (subst0 (plus (s k i0) h) v (lift h d t3) (lift h d -t0)))))).(\lambda (u0: T).(\lambda (d: nat).(\lambda (H2: (le d i0)).(let H3 -\def (eq_ind_r nat (plus (s k i0) h) (\lambda (n: nat).(\forall (d0: -nat).((le d0 (s k i0)) \to (subst0 n v (lift h d0 t3) (lift h d0 t0))))) H1 -(s k (plus i0 h)) (s_plus k i0 h)) in (eq_ind_r T (THead k (lift h d u0) -(lift h (s k d) t3)) (\lambda (t: T).(subst0 (plus i0 h) v t (lift h d (THead -k u0 t0)))) (eq_ind_r T (THead k (lift h d u0) (lift h (s k d) t0)) (\lambda -(t: T).(subst0 (plus i0 h) v (THead k (lift h d u0) (lift h (s k d) t3)) t)) -(subst0_snd k v (lift h (s k d) t0) (lift h (s k d) t3) (plus i0 h) (H3 (s k -d) (s_le k d i0 H2)) (lift h d u0)) (lift h d (THead k u0 t0)) (lift_head k -u0 t0 h d)) (lift h d (THead k u0 t3)) (lift_head k u0 t3 h d))))))))))))) -(\lambda (v: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i0: nat).(\lambda -(_: (subst0 i0 v u1 u2)).(\lambda (H1: ((\forall (d: nat).((le d i0) \to -(subst0 (plus i0 h) v (lift h d u1) (lift h d u2)))))).(\lambda (k: -K).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (subst0 (s k i0) v t0 -t3)).(\lambda (H3: ((\forall (d: nat).((le d (s k i0)) \to (subst0 (plus (s k -i0) h) v (lift h d t0) (lift h d t3)))))).(\lambda (d: nat).(\lambda (H4: (le -d i0)).(let H5 \def (eq_ind_r nat (plus (s k i0) h) (\lambda (n: -nat).(\forall (d0: nat).((le d0 (s k i0)) \to (subst0 n v (lift h d0 t0) -(lift h d0 t3))))) H3 (s k (plus i0 h)) (s_plus k i0 h)) in (eq_ind_r T -(THead k (lift h d u1) (lift h (s k d) t0)) (\lambda (t: T).(subst0 (plus i0 -h) v t (lift h d (THead k u2 t3)))) (eq_ind_r T (THead k (lift h d u2) (lift -h (s k d) t3)) (\lambda (t: T).(subst0 (plus i0 h) v (THead k (lift h d u1) -(lift h (s k d) t0)) t)) (subst0_both v (lift h d u1) (lift h d u2) (plus i0 -h) (H1 d H4) k (lift h (s k d) t0) (lift h (s k d) t3) (H5 (s k d) (s_le k d -i0 H4))) (lift h d (THead k u2 t3)) (lift_head k u2 t3 h d)) (lift h d (THead -k u1 t0)) (lift_head k u1 t0 h d)))))))))))))))) i u t1 t2 H)))))). -(* COMMENTS -Initial nodes: 1449 -END *) - -theorem subst0_lift_ge_S: - \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).((subst0 -i u t1 t2) \to (\forall (d: nat).((le d i) \to (subst0 (S i) u (lift (S O) d -t1) (lift (S O) d t2)))))))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H: (subst0 i u t1 t2)).(\lambda (d: nat).(\lambda (H0: (le d i)).(eq_ind nat -(plus i (S O)) (\lambda (n: nat).(subst0 n u (lift (S O) d t1) (lift (S O) d -t2))) (subst0_lift_ge t1 t2 u i (S O) H d H0) (S i) (eq_ind_r nat (plus (S O) -i) (\lambda (n: nat).(eq nat n (S i))) (refl_equal nat (S i)) (plus i (S O)) -(plus_sym i (S O)))))))))). -(* COMMENTS -Initial nodes: 137 -END *) - -theorem subst0_lift_ge_s: - \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).((subst0 -i u t1 t2) \to (\forall (d: nat).((le d i) \to (\forall (b: B).(subst0 (s -(Bind b) i) u (lift (S O) d t1) (lift (S O) d t2))))))))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H: (subst0 i u t1 t2)).(\lambda (d: nat).(\lambda (H0: (le d i)).(\lambda -(_: B).(subst0_lift_ge_S t1 t2 u i H d H0)))))))). -(* COMMENTS -Initial nodes: 43 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/subst0/subst0.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/subst0/subst0.ma deleted file mode 100644 index 66c167d2b..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/subst0/subst0.ma +++ /dev/null @@ -1,1407 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/subst0/props.ma". - -theorem subst0_subst0: - \forall (t1: T).(\forall (t2: T).(\forall (u2: T).(\forall (j: nat).((subst0 -j u2 t1 t2) \to (\forall (u1: T).(\forall (u: T).(\forall (i: nat).((subst0 i -u u1 u2) \to (ex2 T (\lambda (t: T).(subst0 j u1 t1 t)) (\lambda (t: -T).(subst0 (S (plus i j)) u t t2))))))))))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (u2: T).(\lambda (j: nat).(\lambda -(H: (subst0 j u2 t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t: -T).(\lambda (t0: T).(\lambda (t3: T).(\forall (u1: T).(\forall (u: -T).(\forall (i: nat).((subst0 i u u1 t) \to (ex2 T (\lambda (t4: T).(subst0 n -u1 t0 t4)) (\lambda (t4: T).(subst0 (S (plus i n)) u t4 t3))))))))))) -(\lambda (v: T).(\lambda (i: nat).(\lambda (u1: T).(\lambda (u: T).(\lambda -(i0: nat).(\lambda (H0: (subst0 i0 u u1 v)).(eq_ind nat (plus i0 (S i)) -(\lambda (n: nat).(ex2 T (\lambda (t: T).(subst0 i u1 (TLRef i) t)) (\lambda -(t: T).(subst0 n u t (lift (S i) O v))))) (ex_intro2 T (\lambda (t: -T).(subst0 i u1 (TLRef i) t)) (\lambda (t: T).(subst0 (plus i0 (S i)) u t -(lift (S i) O v))) (lift (S i) O u1) (subst0_lref u1 i) (subst0_lift_ge u1 v -u i0 (S i) H0 O (le_O_n i0))) (S (plus i0 i)) (sym_eq nat (S (plus i0 i)) -(plus i0 (S i)) (plus_n_Sm i0 i))))))))) (\lambda (v: T).(\lambda (u0: -T).(\lambda (u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1 -u0)).(\lambda (H1: ((\forall (u3: T).(\forall (u: T).(\forall (i0: -nat).((subst0 i0 u u3 v) \to (ex2 T (\lambda (t: T).(subst0 i u3 u1 t)) -(\lambda (t: T).(subst0 (S (plus i0 i)) u t u0))))))))).(\lambda (t: -T).(\lambda (k: K).(\lambda (u3: T).(\lambda (u: T).(\lambda (i0: -nat).(\lambda (H2: (subst0 i0 u u3 v)).(ex2_ind T (\lambda (t0: T).(subst0 i -u3 u1 t0)) (\lambda (t0: T).(subst0 (S (plus i0 i)) u t0 u0)) (ex2 T (\lambda -(t0: T).(subst0 i u3 (THead k u1 t) t0)) (\lambda (t0: T).(subst0 (S (plus i0 -i)) u t0 (THead k u0 t)))) (\lambda (x: T).(\lambda (H3: (subst0 i u3 u1 -x)).(\lambda (H4: (subst0 (S (plus i0 i)) u x u0)).(ex_intro2 T (\lambda (t0: -T).(subst0 i u3 (THead k u1 t) t0)) (\lambda (t0: T).(subst0 (S (plus i0 i)) -u t0 (THead k u0 t))) (THead k x t) (subst0_fst u3 x u1 i H3 t k) (subst0_fst -u u0 x (S (plus i0 i)) H4 t k))))) (H1 u3 u i0 H2)))))))))))))) (\lambda (k: -K).(\lambda (v: T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (i: -nat).(\lambda (_: (subst0 (s k i) v t3 t0)).(\lambda (H1: ((\forall (u1: -T).(\forall (u: T).(\forall (i0: nat).((subst0 i0 u u1 v) \to (ex2 T (\lambda -(t: T).(subst0 (s k i) u1 t3 t)) (\lambda (t: T).(subst0 (S (plus i0 (s k -i))) u t t0))))))))).(\lambda (u: T).(\lambda (u1: T).(\lambda (u0: -T).(\lambda (i0: nat).(\lambda (H2: (subst0 i0 u0 u1 v)).(ex2_ind T (\lambda -(t: T).(subst0 (s k i) u1 t3 t)) (\lambda (t: T).(subst0 (S (plus i0 (s k -i))) u0 t t0)) (ex2 T (\lambda (t: T).(subst0 i u1 (THead k u t3) t)) -(\lambda (t: T).(subst0 (S (plus i0 i)) u0 t (THead k u t0)))) (\lambda (x: -T).(\lambda (H3: (subst0 (s k i) u1 t3 x)).(\lambda (H4: (subst0 (S (plus i0 -(s k i))) u0 x t0)).(let H5 \def (eq_ind_r nat (plus i0 (s k i)) (\lambda (n: -nat).(subst0 (S n) u0 x t0)) H4 (s k (plus i0 i)) (s_plus_sym k i0 i)) in -(let H6 \def (eq_ind_r nat (S (s k (plus i0 i))) (\lambda (n: nat).(subst0 n -u0 x t0)) H5 (s k (S (plus i0 i))) (s_S k (plus i0 i))) in (ex_intro2 T -(\lambda (t: T).(subst0 i u1 (THead k u t3) t)) (\lambda (t: T).(subst0 (S -(plus i0 i)) u0 t (THead k u t0))) (THead k u x) (subst0_snd k u1 x t3 i H3 -u) (subst0_snd k u0 t0 x (S (plus i0 i)) H6 u))))))) (H1 u1 u0 i0 -H2)))))))))))))) (\lambda (v: T).(\lambda (u1: T).(\lambda (u0: T).(\lambda -(i: nat).(\lambda (_: (subst0 i v u1 u0)).(\lambda (H1: ((\forall (u3: -T).(\forall (u: T).(\forall (i0: nat).((subst0 i0 u u3 v) \to (ex2 T (\lambda -(t: T).(subst0 i u3 u1 t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u t -u0))))))))).(\lambda (k: K).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: -(subst0 (s k i) v t0 t3)).(\lambda (H3: ((\forall (u3: T).(\forall (u: -T).(\forall (i0: nat).((subst0 i0 u u3 v) \to (ex2 T (\lambda (t: T).(subst0 -(s k i) u3 t0 t)) (\lambda (t: T).(subst0 (S (plus i0 (s k i))) u t -t3))))))))).(\lambda (u3: T).(\lambda (u: T).(\lambda (i0: nat).(\lambda (H4: -(subst0 i0 u u3 v)).(ex2_ind T (\lambda (t: T).(subst0 (s k i) u3 t0 t)) -(\lambda (t: T).(subst0 (S (plus i0 (s k i))) u t t3)) (ex2 T (\lambda (t: -T).(subst0 i u3 (THead k u1 t0) t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u -t (THead k u0 t3)))) (\lambda (x: T).(\lambda (H5: (subst0 (s k i) u3 t0 -x)).(\lambda (H6: (subst0 (S (plus i0 (s k i))) u x t3)).(ex2_ind T (\lambda -(t: T).(subst0 i u3 u1 t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u t u0)) -(ex2 T (\lambda (t: T).(subst0 i u3 (THead k u1 t0) t)) (\lambda (t: -T).(subst0 (S (plus i0 i)) u t (THead k u0 t3)))) (\lambda (x0: T).(\lambda -(H7: (subst0 i u3 u1 x0)).(\lambda (H8: (subst0 (S (plus i0 i)) u x0 -u0)).(let H9 \def (eq_ind_r nat (plus i0 (s k i)) (\lambda (n: nat).(subst0 -(S n) u x t3)) H6 (s k (plus i0 i)) (s_plus_sym k i0 i)) in (let H10 \def -(eq_ind_r nat (S (s k (plus i0 i))) (\lambda (n: nat).(subst0 n u x t3)) H9 -(s k (S (plus i0 i))) (s_S k (plus i0 i))) in (ex_intro2 T (\lambda (t: -T).(subst0 i u3 (THead k u1 t0) t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u -t (THead k u0 t3))) (THead k x0 x) (subst0_both u3 u1 x0 i H7 k t0 x H5) -(subst0_both u x0 u0 (S (plus i0 i)) H8 k x t3 H10))))))) (H1 u3 u i0 H4))))) -(H3 u3 u i0 H4))))))))))))))))) j u2 t1 t2 H))))). -(* COMMENTS -Initial nodes: 1613 -END *) - -theorem subst0_subst0_back: - \forall (t1: T).(\forall (t2: T).(\forall (u2: T).(\forall (j: nat).((subst0 -j u2 t1 t2) \to (\forall (u1: T).(\forall (u: T).(\forall (i: nat).((subst0 i -u u2 u1) \to (ex2 T (\lambda (t: T).(subst0 j u1 t1 t)) (\lambda (t: -T).(subst0 (S (plus i j)) u t2 t))))))))))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (u2: T).(\lambda (j: nat).(\lambda -(H: (subst0 j u2 t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t: -T).(\lambda (t0: T).(\lambda (t3: T).(\forall (u1: T).(\forall (u: -T).(\forall (i: nat).((subst0 i u t u1) \to (ex2 T (\lambda (t4: T).(subst0 n -u1 t0 t4)) (\lambda (t4: T).(subst0 (S (plus i n)) u t3 t4))))))))))) -(\lambda (v: T).(\lambda (i: nat).(\lambda (u1: T).(\lambda (u: T).(\lambda -(i0: nat).(\lambda (H0: (subst0 i0 u v u1)).(eq_ind nat (plus i0 (S i)) -(\lambda (n: nat).(ex2 T (\lambda (t: T).(subst0 i u1 (TLRef i) t)) (\lambda -(t: T).(subst0 n u (lift (S i) O v) t)))) (ex_intro2 T (\lambda (t: -T).(subst0 i u1 (TLRef i) t)) (\lambda (t: T).(subst0 (plus i0 (S i)) u (lift -(S i) O v) t)) (lift (S i) O u1) (subst0_lref u1 i) (subst0_lift_ge v u1 u i0 -(S i) H0 O (le_O_n i0))) (S (plus i0 i)) (sym_eq nat (S (plus i0 i)) (plus i0 -(S i)) (plus_n_Sm i0 i))))))))) (\lambda (v: T).(\lambda (u0: T).(\lambda -(u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1 u0)).(\lambda (H1: -((\forall (u3: T).(\forall (u: T).(\forall (i0: nat).((subst0 i0 u v u3) \to -(ex2 T (\lambda (t: T).(subst0 i u3 u1 t)) (\lambda (t: T).(subst0 (S (plus -i0 i)) u u0 t))))))))).(\lambda (t: T).(\lambda (k: K).(\lambda (u3: -T).(\lambda (u: T).(\lambda (i0: nat).(\lambda (H2: (subst0 i0 u v -u3)).(ex2_ind T (\lambda (t0: T).(subst0 i u3 u1 t0)) (\lambda (t0: -T).(subst0 (S (plus i0 i)) u u0 t0)) (ex2 T (\lambda (t0: T).(subst0 i u3 -(THead k u1 t) t0)) (\lambda (t0: T).(subst0 (S (plus i0 i)) u (THead k u0 t) -t0))) (\lambda (x: T).(\lambda (H3: (subst0 i u3 u1 x)).(\lambda (H4: (subst0 -(S (plus i0 i)) u u0 x)).(ex_intro2 T (\lambda (t0: T).(subst0 i u3 (THead k -u1 t) t0)) (\lambda (t0: T).(subst0 (S (plus i0 i)) u (THead k u0 t) t0)) -(THead k x t) (subst0_fst u3 x u1 i H3 t k) (subst0_fst u x u0 (S (plus i0 -i)) H4 t k))))) (H1 u3 u i0 H2)))))))))))))) (\lambda (k: K).(\lambda (v: -T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (i: nat).(\lambda (_: (subst0 -(s k i) v t3 t0)).(\lambda (H1: ((\forall (u1: T).(\forall (u: T).(\forall -(i0: nat).((subst0 i0 u v u1) \to (ex2 T (\lambda (t: T).(subst0 (s k i) u1 -t3 t)) (\lambda (t: T).(subst0 (S (plus i0 (s k i))) u t0 t))))))))).(\lambda -(u: T).(\lambda (u1: T).(\lambda (u0: T).(\lambda (i0: nat).(\lambda (H2: -(subst0 i0 u0 v u1)).(ex2_ind T (\lambda (t: T).(subst0 (s k i) u1 t3 t)) -(\lambda (t: T).(subst0 (S (plus i0 (s k i))) u0 t0 t)) (ex2 T (\lambda (t: -T).(subst0 i u1 (THead k u t3) t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u0 -(THead k u t0) t))) (\lambda (x: T).(\lambda (H3: (subst0 (s k i) u1 t3 -x)).(\lambda (H4: (subst0 (S (plus i0 (s k i))) u0 t0 x)).(let H5 \def -(eq_ind_r nat (plus i0 (s k i)) (\lambda (n: nat).(subst0 (S n) u0 t0 x)) H4 -(s k (plus i0 i)) (s_plus_sym k i0 i)) in (let H6 \def (eq_ind_r nat (S (s k -(plus i0 i))) (\lambda (n: nat).(subst0 n u0 t0 x)) H5 (s k (S (plus i0 i))) -(s_S k (plus i0 i))) in (ex_intro2 T (\lambda (t: T).(subst0 i u1 (THead k u -t3) t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u0 (THead k u t0) t)) (THead -k u x) (subst0_snd k u1 x t3 i H3 u) (subst0_snd k u0 x t0 (S (plus i0 i)) H6 -u))))))) (H1 u1 u0 i0 H2)))))))))))))) (\lambda (v: T).(\lambda (u1: -T).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1 -u0)).(\lambda (H1: ((\forall (u3: T).(\forall (u: T).(\forall (i0: -nat).((subst0 i0 u v u3) \to (ex2 T (\lambda (t: T).(subst0 i u3 u1 t)) -(\lambda (t: T).(subst0 (S (plus i0 i)) u u0 t))))))))).(\lambda (k: -K).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (subst0 (s k i) v t0 -t3)).(\lambda (H3: ((\forall (u3: T).(\forall (u: T).(\forall (i0: -nat).((subst0 i0 u v u3) \to (ex2 T (\lambda (t: T).(subst0 (s k i) u3 t0 t)) -(\lambda (t: T).(subst0 (S (plus i0 (s k i))) u t3 t))))))))).(\lambda (u3: -T).(\lambda (u: T).(\lambda (i0: nat).(\lambda (H4: (subst0 i0 u v -u3)).(ex2_ind T (\lambda (t: T).(subst0 (s k i) u3 t0 t)) (\lambda (t: -T).(subst0 (S (plus i0 (s k i))) u t3 t)) (ex2 T (\lambda (t: T).(subst0 i u3 -(THead k u1 t0) t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u (THead k u0 t3) -t))) (\lambda (x: T).(\lambda (H5: (subst0 (s k i) u3 t0 x)).(\lambda (H6: -(subst0 (S (plus i0 (s k i))) u t3 x)).(ex2_ind T (\lambda (t: T).(subst0 i -u3 u1 t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u u0 t)) (ex2 T (\lambda -(t: T).(subst0 i u3 (THead k u1 t0) t)) (\lambda (t: T).(subst0 (S (plus i0 -i)) u (THead k u0 t3) t))) (\lambda (x0: T).(\lambda (H7: (subst0 i u3 u1 -x0)).(\lambda (H8: (subst0 (S (plus i0 i)) u u0 x0)).(let H9 \def (eq_ind_r -nat (plus i0 (s k i)) (\lambda (n: nat).(subst0 (S n) u t3 x)) H6 (s k (plus -i0 i)) (s_plus_sym k i0 i)) in (let H10 \def (eq_ind_r nat (S (s k (plus i0 -i))) (\lambda (n: nat).(subst0 n u t3 x)) H9 (s k (S (plus i0 i))) (s_S k -(plus i0 i))) in (ex_intro2 T (\lambda (t: T).(subst0 i u3 (THead k u1 t0) -t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u (THead k u0 t3) t)) (THead k x0 -x) (subst0_both u3 u1 x0 i H7 k t0 x H5) (subst0_both u u0 x0 (S (plus i0 i)) -H8 k t3 x H10))))))) (H1 u3 u i0 H4))))) (H3 u3 u i0 H4))))))))))))))))) j u2 -t1 t2 H))))). -(* COMMENTS -Initial nodes: 1613 -END *) - -theorem subst0_trans: - \forall (t2: T).(\forall (t1: T).(\forall (v: T).(\forall (i: nat).((subst0 -i v t1 t2) \to (\forall (t3: T).((subst0 i v t2 t3) \to (subst0 i v t1 -t3))))))) -\def - \lambda (t2: T).(\lambda (t1: T).(\lambda (v: T).(\lambda (i: nat).(\lambda -(H: (subst0 i v t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t: -T).(\lambda (t0: T).(\lambda (t3: T).(\forall (t4: T).((subst0 n t t3 t4) \to -(subst0 n t t0 t4))))))) (\lambda (v0: T).(\lambda (i0: nat).(\lambda (t3: -T).(\lambda (H0: (subst0 i0 v0 (lift (S i0) O v0) t3)).(subst0_gen_lift_false -v0 v0 t3 (S i0) O i0 (le_O_n i0) (le_n (plus O (S i0))) H0 (subst0 i0 v0 -(TLRef i0) t3)))))) (\lambda (v0: T).(\lambda (u2: T).(\lambda (u1: -T).(\lambda (i0: nat).(\lambda (H0: (subst0 i0 v0 u1 u2)).(\lambda (H1: -((\forall (t3: T).((subst0 i0 v0 u2 t3) \to (subst0 i0 v0 u1 t3))))).(\lambda -(t: T).(\lambda (k: K).(\lambda (t3: T).(\lambda (H2: (subst0 i0 v0 (THead k -u2 t) t3)).(or3_ind (ex2 T (\lambda (u3: T).(eq T t3 (THead k u3 t))) -(\lambda (u3: T).(subst0 i0 v0 u2 u3))) (ex2 T (\lambda (t4: T).(eq T t3 -(THead k u2 t4))) (\lambda (t4: T).(subst0 (s k i0) v0 t t4))) (ex3_2 T T -(\lambda (u3: T).(\lambda (t4: T).(eq T t3 (THead k u3 t4)))) (\lambda (u3: -T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) (\lambda (_: T).(\lambda (t4: -T).(subst0 (s k i0) v0 t t4)))) (subst0 i0 v0 (THead k u1 t) t3) (\lambda -(H3: (ex2 T (\lambda (u3: T).(eq T t3 (THead k u3 t))) (\lambda (u3: -T).(subst0 i0 v0 u2 u3)))).(ex2_ind T (\lambda (u3: T).(eq T t3 (THead k u3 -t))) (\lambda (u3: T).(subst0 i0 v0 u2 u3)) (subst0 i0 v0 (THead k u1 t) t3) -(\lambda (x: T).(\lambda (H4: (eq T t3 (THead k x t))).(\lambda (H5: (subst0 -i0 v0 u2 x)).(eq_ind_r T (THead k x t) (\lambda (t0: T).(subst0 i0 v0 (THead -k u1 t) t0)) (subst0_fst v0 x u1 i0 (H1 x H5) t k) t3 H4)))) H3)) (\lambda -(H3: (ex2 T (\lambda (t4: T).(eq T t3 (THead k u2 t4))) (\lambda (t4: -T).(subst0 (s k i0) v0 t t4)))).(ex2_ind T (\lambda (t4: T).(eq T t3 (THead k -u2 t4))) (\lambda (t4: T).(subst0 (s k i0) v0 t t4)) (subst0 i0 v0 (THead k -u1 t) t3) (\lambda (x: T).(\lambda (H4: (eq T t3 (THead k u2 x))).(\lambda -(H5: (subst0 (s k i0) v0 t x)).(eq_ind_r T (THead k u2 x) (\lambda (t0: -T).(subst0 i0 v0 (THead k u1 t) t0)) (subst0_both v0 u1 u2 i0 H0 k t x H5) t3 -H4)))) H3)) (\lambda (H3: (ex3_2 T T (\lambda (u3: T).(\lambda (t4: T).(eq T -t3 (THead k u3 t4)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) -(\lambda (_: T).(\lambda (t4: T).(subst0 (s k i0) v0 t t4))))).(ex3_2_ind T T -(\lambda (u3: T).(\lambda (t4: T).(eq T t3 (THead k u3 t4)))) (\lambda (u3: -T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) (\lambda (_: T).(\lambda (t4: -T).(subst0 (s k i0) v0 t t4))) (subst0 i0 v0 (THead k u1 t) t3) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H4: (eq T t3 (THead k x0 x1))).(\lambda (H5: -(subst0 i0 v0 u2 x0)).(\lambda (H6: (subst0 (s k i0) v0 t x1)).(eq_ind_r T -(THead k x0 x1) (\lambda (t0: T).(subst0 i0 v0 (THead k u1 t) t0)) -(subst0_both v0 u1 x0 i0 (H1 x0 H5) k t x1 H6) t3 H4)))))) H3)) -(subst0_gen_head k v0 u2 t t3 i0 H2)))))))))))) (\lambda (k: K).(\lambda (v0: -T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (i0: nat).(\lambda (H0: (subst0 -(s k i0) v0 t3 t0)).(\lambda (H1: ((\forall (t4: T).((subst0 (s k i0) v0 t0 -t4) \to (subst0 (s k i0) v0 t3 t4))))).(\lambda (u: T).(\lambda (t4: -T).(\lambda (H2: (subst0 i0 v0 (THead k u t0) t4)).(or3_ind (ex2 T (\lambda -(u2: T).(eq T t4 (THead k u2 t0))) (\lambda (u2: T).(subst0 i0 v0 u u2))) -(ex2 T (\lambda (t5: T).(eq T t4 (THead k u t5))) (\lambda (t5: T).(subst0 (s -k i0) v0 t0 t5))) (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 -(THead k u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u u2))) -(\lambda (_: T).(\lambda (t5: T).(subst0 (s k i0) v0 t0 t5)))) (subst0 i0 v0 -(THead k u t3) t4) (\lambda (H3: (ex2 T (\lambda (u2: T).(eq T t4 (THead k u2 -t0))) (\lambda (u2: T).(subst0 i0 v0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq -T t4 (THead k u2 t0))) (\lambda (u2: T).(subst0 i0 v0 u u2)) (subst0 i0 v0 -(THead k u t3) t4) (\lambda (x: T).(\lambda (H4: (eq T t4 (THead k x -t0))).(\lambda (H5: (subst0 i0 v0 u x)).(eq_ind_r T (THead k x t0) (\lambda -(t: T).(subst0 i0 v0 (THead k u t3) t)) (subst0_both v0 u x i0 H5 k t3 t0 H0) -t4 H4)))) H3)) (\lambda (H3: (ex2 T (\lambda (t5: T).(eq T t4 (THead k u -t5))) (\lambda (t5: T).(subst0 (s k i0) v0 t0 t5)))).(ex2_ind T (\lambda (t5: -T).(eq T t4 (THead k u t5))) (\lambda (t5: T).(subst0 (s k i0) v0 t0 t5)) -(subst0 i0 v0 (THead k u t3) t4) (\lambda (x: T).(\lambda (H4: (eq T t4 -(THead k u x))).(\lambda (H5: (subst0 (s k i0) v0 t0 x)).(eq_ind_r T (THead k -u x) (\lambda (t: T).(subst0 i0 v0 (THead k u t3) t)) (subst0_snd k v0 x t3 -i0 (H1 x H5) u) t4 H4)))) H3)) (\lambda (H3: (ex3_2 T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead k u2 t5)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 i0 v0 u u2))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s k i0) v0 -t0 t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead k -u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u u2))) (\lambda (_: -T).(\lambda (t5: T).(subst0 (s k i0) v0 t0 t5))) (subst0 i0 v0 (THead k u t3) -t4) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t4 (THead k x0 -x1))).(\lambda (H5: (subst0 i0 v0 u x0)).(\lambda (H6: (subst0 (s k i0) v0 t0 -x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t: T).(subst0 i0 v0 (THead k u t3) -t)) (subst0_both v0 u x0 i0 H5 k t3 x1 (H1 x1 H6)) t4 H4)))))) H3)) -(subst0_gen_head k v0 u t0 t4 i0 H2)))))))))))) (\lambda (v0: T).(\lambda -(u1: T).(\lambda (u2: T).(\lambda (i0: nat).(\lambda (H0: (subst0 i0 v0 u1 -u2)).(\lambda (H1: ((\forall (t3: T).((subst0 i0 v0 u2 t3) \to (subst0 i0 v0 -u1 t3))))).(\lambda (k: K).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H2: -(subst0 (s k i0) v0 t0 t3)).(\lambda (H3: ((\forall (t4: T).((subst0 (s k i0) -v0 t3 t4) \to (subst0 (s k i0) v0 t0 t4))))).(\lambda (t4: T).(\lambda (H4: -(subst0 i0 v0 (THead k u2 t3) t4)).(or3_ind (ex2 T (\lambda (u3: T).(eq T t4 -(THead k u3 t3))) (\lambda (u3: T).(subst0 i0 v0 u2 u3))) (ex2 T (\lambda -(t5: T).(eq T t4 (THead k u2 t5))) (\lambda (t5: T).(subst0 (s k i0) v0 t3 -t5))) (ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T t4 (THead k u3 -t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) (\lambda (_: -T).(\lambda (t5: T).(subst0 (s k i0) v0 t3 t5)))) (subst0 i0 v0 (THead k u1 -t0) t4) (\lambda (H5: (ex2 T (\lambda (u3: T).(eq T t4 (THead k u3 t3))) -(\lambda (u3: T).(subst0 i0 v0 u2 u3)))).(ex2_ind T (\lambda (u3: T).(eq T t4 -(THead k u3 t3))) (\lambda (u3: T).(subst0 i0 v0 u2 u3)) (subst0 i0 v0 (THead -k u1 t0) t4) (\lambda (x: T).(\lambda (H6: (eq T t4 (THead k x t3))).(\lambda -(H7: (subst0 i0 v0 u2 x)).(eq_ind_r T (THead k x t3) (\lambda (t: T).(subst0 -i0 v0 (THead k u1 t0) t)) (subst0_both v0 u1 x i0 (H1 x H7) k t0 t3 H2) t4 -H6)))) H5)) (\lambda (H5: (ex2 T (\lambda (t5: T).(eq T t4 (THead k u2 t5))) -(\lambda (t5: T).(subst0 (s k i0) v0 t3 t5)))).(ex2_ind T (\lambda (t5: -T).(eq T t4 (THead k u2 t5))) (\lambda (t5: T).(subst0 (s k i0) v0 t3 t5)) -(subst0 i0 v0 (THead k u1 t0) t4) (\lambda (x: T).(\lambda (H6: (eq T t4 -(THead k u2 x))).(\lambda (H7: (subst0 (s k i0) v0 t3 x)).(eq_ind_r T (THead -k u2 x) (\lambda (t: T).(subst0 i0 v0 (THead k u1 t0) t)) (subst0_both v0 u1 -u2 i0 H0 k t0 x (H3 x H7)) t4 H6)))) H5)) (\lambda (H5: (ex3_2 T T (\lambda -(u3: T).(\lambda (t5: T).(eq T t4 (THead k u3 t5)))) (\lambda (u3: -T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) (\lambda (_: T).(\lambda (t5: -T).(subst0 (s k i0) v0 t3 t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda -(t5: T).(eq T t4 (THead k u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 -i0 v0 u2 u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s k i0) v0 t3 t5))) -(subst0 i0 v0 (THead k u1 t0) t4) (\lambda (x0: T).(\lambda (x1: T).(\lambda -(H6: (eq T t4 (THead k x0 x1))).(\lambda (H7: (subst0 i0 v0 u2 x0)).(\lambda -(H8: (subst0 (s k i0) v0 t3 x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t: -T).(subst0 i0 v0 (THead k u1 t0) t)) (subst0_both v0 u1 x0 i0 (H1 x0 H7) k t0 -x1 (H3 x1 H8)) t4 H6)))))) H5)) (subst0_gen_head k v0 u2 t3 t4 i0 -H4))))))))))))))) i v t1 t2 H))))). -(* COMMENTS -Initial nodes: 2555 -END *) - -theorem subst0_confluence_neq: - \forall (t0: T).(\forall (t1: T).(\forall (u1: T).(\forall (i1: -nat).((subst0 i1 u1 t0 t1) \to (\forall (t2: T).(\forall (u2: T).(\forall -(i2: nat).((subst0 i2 u2 t0 t2) \to ((not (eq nat i1 i2)) \to (ex2 T (\lambda -(t: T).(subst0 i2 u2 t1 t)) (\lambda (t: T).(subst0 i1 u1 t2 t)))))))))))) -\def - \lambda (t0: T).(\lambda (t1: T).(\lambda (u1: T).(\lambda (i1: -nat).(\lambda (H: (subst0 i1 u1 t0 t1)).(subst0_ind (\lambda (n: -nat).(\lambda (t: T).(\lambda (t2: T).(\lambda (t3: T).(\forall (t4: -T).(\forall (u2: T).(\forall (i2: nat).((subst0 i2 u2 t2 t4) \to ((not (eq -nat n i2)) \to (ex2 T (\lambda (t5: T).(subst0 i2 u2 t3 t5)) (\lambda (t5: -T).(subst0 n t t4 t5)))))))))))) (\lambda (v: T).(\lambda (i: nat).(\lambda -(t2: T).(\lambda (u2: T).(\lambda (i2: nat).(\lambda (H0: (subst0 i2 u2 -(TLRef i) t2)).(\lambda (H1: (not (eq nat i i2))).(land_ind (eq nat i i2) (eq -T t2 (lift (S i) O u2)) (ex2 T (\lambda (t: T).(subst0 i2 u2 (lift (S i) O v) -t)) (\lambda (t: T).(subst0 i v t2 t))) (\lambda (H2: (eq nat i i2)).(\lambda -(H3: (eq T t2 (lift (S i) O u2))).(let H4 \def (eq_ind nat i (\lambda (n: -nat).(not (eq nat n i2))) H1 i2 H2) in (eq_ind_r T (lift (S i) O u2) (\lambda -(t: T).(ex2 T (\lambda (t3: T).(subst0 i2 u2 (lift (S i) O v) t3)) (\lambda -(t3: T).(subst0 i v t t3)))) (let H5 \def (match (H4 (refl_equal nat i2)) in -False return (\lambda (_: False).(ex2 T (\lambda (t: T).(subst0 i2 u2 (lift -(S i) O v) t)) (\lambda (t: T).(subst0 i v (lift (S i) O u2) t)))) with []) -in H5) t2 H3)))) (subst0_gen_lref u2 t2 i2 i H0))))))))) (\lambda (v: -T).(\lambda (u2: T).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H0: (subst0 -i v u0 u2)).(\lambda (H1: ((\forall (t2: T).(\forall (u3: T).(\forall (i2: -nat).((subst0 i2 u3 u0 t2) \to ((not (eq nat i i2)) \to (ex2 T (\lambda (t: -T).(subst0 i2 u3 u2 t)) (\lambda (t: T).(subst0 i v t2 t)))))))))).(\lambda -(t: T).(\lambda (k: K).(\lambda (t2: T).(\lambda (u3: T).(\lambda (i2: -nat).(\lambda (H2: (subst0 i2 u3 (THead k u0 t) t2)).(\lambda (H3: (not (eq -nat i i2))).(or3_ind (ex2 T (\lambda (u4: T).(eq T t2 (THead k u4 t))) -(\lambda (u4: T).(subst0 i2 u3 u0 u4))) (ex2 T (\lambda (t3: T).(eq T t2 -(THead k u0 t3))) (\lambda (t3: T).(subst0 (s k i2) u3 t t3))) (ex3_2 T T -(\lambda (u4: T).(\lambda (t3: T).(eq T t2 (THead k u4 t3)))) (\lambda (u4: -T).(\lambda (_: T).(subst0 i2 u3 u0 u4))) (\lambda (_: T).(\lambda (t3: -T).(subst0 (s k i2) u3 t t3)))) (ex2 T (\lambda (t3: T).(subst0 i2 u3 (THead -k u2 t) t3)) (\lambda (t3: T).(subst0 i v t2 t3))) (\lambda (H4: (ex2 T -(\lambda (u4: T).(eq T t2 (THead k u4 t))) (\lambda (u4: T).(subst0 i2 u3 u0 -u4)))).(ex2_ind T (\lambda (u4: T).(eq T t2 (THead k u4 t))) (\lambda (u4: -T).(subst0 i2 u3 u0 u4)) (ex2 T (\lambda (t3: T).(subst0 i2 u3 (THead k u2 t) -t3)) (\lambda (t3: T).(subst0 i v t2 t3))) (\lambda (x: T).(\lambda (H5: (eq -T t2 (THead k x t))).(\lambda (H6: (subst0 i2 u3 u0 x)).(eq_ind_r T (THead k -x t) (\lambda (t3: T).(ex2 T (\lambda (t4: T).(subst0 i2 u3 (THead k u2 t) -t4)) (\lambda (t4: T).(subst0 i v t3 t4)))) (ex2_ind T (\lambda (t3: -T).(subst0 i2 u3 u2 t3)) (\lambda (t3: T).(subst0 i v x t3)) (ex2 T (\lambda -(t3: T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i v (THead -k x t) t3))) (\lambda (x0: T).(\lambda (H7: (subst0 i2 u3 u2 x0)).(\lambda -(H8: (subst0 i v x x0)).(ex_intro2 T (\lambda (t3: T).(subst0 i2 u3 (THead k -u2 t) t3)) (\lambda (t3: T).(subst0 i v (THead k x t) t3)) (THead k x0 t) -(subst0_fst u3 x0 u2 i2 H7 t k) (subst0_fst v x0 x i H8 t k))))) (H1 x u3 i2 -H6 H3)) t2 H5)))) H4)) (\lambda (H4: (ex2 T (\lambda (t3: T).(eq T t2 (THead -k u0 t3))) (\lambda (t3: T).(subst0 (s k i2) u3 t t3)))).(ex2_ind T (\lambda -(t3: T).(eq T t2 (THead k u0 t3))) (\lambda (t3: T).(subst0 (s k i2) u3 t -t3)) (ex2 T (\lambda (t3: T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda (t3: -T).(subst0 i v t2 t3))) (\lambda (x: T).(\lambda (H5: (eq T t2 (THead k u0 -x))).(\lambda (H6: (subst0 (s k i2) u3 t x)).(eq_ind_r T (THead k u0 x) -(\lambda (t3: T).(ex2 T (\lambda (t4: T).(subst0 i2 u3 (THead k u2 t) t4)) -(\lambda (t4: T).(subst0 i v t3 t4)))) (ex_intro2 T (\lambda (t3: T).(subst0 -i2 u3 (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i v (THead k u0 x) t3)) -(THead k u2 x) (subst0_snd k u3 x t i2 H6 u2) (subst0_fst v u2 u0 i H0 x k)) -t2 H5)))) H4)) (\lambda (H4: (ex3_2 T T (\lambda (u4: T).(\lambda (t3: T).(eq -T t2 (THead k u4 t3)))) (\lambda (u4: T).(\lambda (_: T).(subst0 i2 u3 u0 -u4))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i2) u3 t -t3))))).(ex3_2_ind T T (\lambda (u4: T).(\lambda (t3: T).(eq T t2 (THead k u4 -t3)))) (\lambda (u4: T).(\lambda (_: T).(subst0 i2 u3 u0 u4))) (\lambda (_: -T).(\lambda (t3: T).(subst0 (s k i2) u3 t t3))) (ex2 T (\lambda (t3: -T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i v t2 t3))) -(\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T t2 (THead k x0 -x1))).(\lambda (H6: (subst0 i2 u3 u0 x0)).(\lambda (H7: (subst0 (s k i2) u3 t -x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t3: T).(ex2 T (\lambda (t4: -T).(subst0 i2 u3 (THead k u2 t) t4)) (\lambda (t4: T).(subst0 i v t3 t4)))) -(ex2_ind T (\lambda (t3: T).(subst0 i2 u3 u2 t3)) (\lambda (t3: T).(subst0 i -v x0 t3)) (ex2 T (\lambda (t3: T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda -(t3: T).(subst0 i v (THead k x0 x1) t3))) (\lambda (x: T).(\lambda (H8: -(subst0 i2 u3 u2 x)).(\lambda (H9: (subst0 i v x0 x)).(ex_intro2 T (\lambda -(t3: T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i v (THead -k x0 x1) t3)) (THead k x x1) (subst0_both u3 u2 x i2 H8 k t x1 H7) -(subst0_fst v x x0 i H9 x1 k))))) (H1 x0 u3 i2 H6 H3)) t2 H5)))))) H4)) -(subst0_gen_head k u3 u0 t t2 i2 H2))))))))))))))) (\lambda (k: K).(\lambda -(v: T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (i: nat).(\lambda (H0: -(subst0 (s k i) v t3 t2)).(\lambda (H1: ((\forall (t4: T).(\forall (u2: -T).(\forall (i2: nat).((subst0 i2 u2 t3 t4) \to ((not (eq nat (s k i) i2)) -\to (ex2 T (\lambda (t: T).(subst0 i2 u2 t2 t)) (\lambda (t: T).(subst0 (s k -i) v t4 t)))))))))).(\lambda (u: T).(\lambda (t4: T).(\lambda (u2: -T).(\lambda (i2: nat).(\lambda (H2: (subst0 i2 u2 (THead k u t3) -t4)).(\lambda (H3: (not (eq nat i i2))).(or3_ind (ex2 T (\lambda (u3: T).(eq -T t4 (THead k u3 t3))) (\lambda (u3: T).(subst0 i2 u2 u u3))) (ex2 T (\lambda -(t5: T).(eq T t4 (THead k u t5))) (\lambda (t5: T).(subst0 (s k i2) u2 t3 -t5))) (ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T t4 (THead k u3 -t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i2 u2 u u3))) (\lambda (_: -T).(\lambda (t5: T).(subst0 (s k i2) u2 t3 t5)))) (ex2 T (\lambda (t: -T).(subst0 i2 u2 (THead k u t2) t)) (\lambda (t: T).(subst0 i v t4 t))) -(\lambda (H4: (ex2 T (\lambda (u3: T).(eq T t4 (THead k u3 t3))) (\lambda -(u3: T).(subst0 i2 u2 u u3)))).(ex2_ind T (\lambda (u3: T).(eq T t4 (THead k -u3 t3))) (\lambda (u3: T).(subst0 i2 u2 u u3)) (ex2 T (\lambda (t: T).(subst0 -i2 u2 (THead k u t2) t)) (\lambda (t: T).(subst0 i v t4 t))) (\lambda (x: -T).(\lambda (H5: (eq T t4 (THead k x t3))).(\lambda (H6: (subst0 i2 u2 u -x)).(eq_ind_r T (THead k x t3) (\lambda (t: T).(ex2 T (\lambda (t5: -T).(subst0 i2 u2 (THead k u t2) t5)) (\lambda (t5: T).(subst0 i v t t5)))) -(ex_intro2 T (\lambda (t: T).(subst0 i2 u2 (THead k u t2) t)) (\lambda (t: -T).(subst0 i v (THead k x t3) t)) (THead k x t2) (subst0_fst u2 x u i2 H6 t2 -k) (subst0_snd k v t2 t3 i H0 x)) t4 H5)))) H4)) (\lambda (H4: (ex2 T -(\lambda (t5: T).(eq T t4 (THead k u t5))) (\lambda (t5: T).(subst0 (s k i2) -u2 t3 t5)))).(ex2_ind T (\lambda (t5: T).(eq T t4 (THead k u t5))) (\lambda -(t5: T).(subst0 (s k i2) u2 t3 t5)) (ex2 T (\lambda (t: T).(subst0 i2 u2 -(THead k u t2) t)) (\lambda (t: T).(subst0 i v t4 t))) (\lambda (x: -T).(\lambda (H5: (eq T t4 (THead k u x))).(\lambda (H6: (subst0 (s k i2) u2 -t3 x)).(eq_ind_r T (THead k u x) (\lambda (t: T).(ex2 T (\lambda (t5: -T).(subst0 i2 u2 (THead k u t2) t5)) (\lambda (t5: T).(subst0 i v t t5)))) -(ex2_ind T (\lambda (t: T).(subst0 (s k i2) u2 t2 t)) (\lambda (t: T).(subst0 -(s k i) v x t)) (ex2 T (\lambda (t: T).(subst0 i2 u2 (THead k u t2) t)) -(\lambda (t: T).(subst0 i v (THead k u x) t))) (\lambda (x0: T).(\lambda (H7: -(subst0 (s k i2) u2 t2 x0)).(\lambda (H8: (subst0 (s k i) v x x0)).(ex_intro2 -T (\lambda (t: T).(subst0 i2 u2 (THead k u t2) t)) (\lambda (t: T).(subst0 i -v (THead k u x) t)) (THead k u x0) (subst0_snd k u2 x0 t2 i2 H7 u) -(subst0_snd k v x0 x i H8 u))))) (H1 x u2 (s k i2) H6 (ex2_ind T (\lambda (t: -T).(subst0 (s k i2) u2 t2 t)) (\lambda (t: T).(subst0 (s k i) v x t)) ((eq -nat (s k i) (s k i2)) \to False) (\lambda (x0: T).(\lambda (_: (subst0 (s k -i2) u2 t2 x0)).(\lambda (_: (subst0 (s k i) v x x0)).(\lambda (H9: (eq nat (s -k i) (s k i2))).(H3 (s_inj k i i2 H9)))))) (H1 x u2 (s k i2) H6 (\lambda (H7: -(eq nat (s k i) (s k i2))).(H3 (s_inj k i i2 H7))))))) t4 H5)))) H4)) -(\lambda (H4: (ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T t4 (THead k -u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i2 u2 u u3))) (\lambda (_: -T).(\lambda (t5: T).(subst0 (s k i2) u2 t3 t5))))).(ex3_2_ind T T (\lambda -(u3: T).(\lambda (t5: T).(eq T t4 (THead k u3 t5)))) (\lambda (u3: -T).(\lambda (_: T).(subst0 i2 u2 u u3))) (\lambda (_: T).(\lambda (t5: -T).(subst0 (s k i2) u2 t3 t5))) (ex2 T (\lambda (t: T).(subst0 i2 u2 (THead k -u t2) t)) (\lambda (t: T).(subst0 i v t4 t))) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H5: (eq T t4 (THead k x0 x1))).(\lambda (H6: (subst0 i2 u2 u -x0)).(\lambda (H7: (subst0 (s k i2) u2 t3 x1)).(eq_ind_r T (THead k x0 x1) -(\lambda (t: T).(ex2 T (\lambda (t5: T).(subst0 i2 u2 (THead k u t2) t5)) -(\lambda (t5: T).(subst0 i v t t5)))) (ex2_ind T (\lambda (t: T).(subst0 (s k -i2) u2 t2 t)) (\lambda (t: T).(subst0 (s k i) v x1 t)) (ex2 T (\lambda (t: -T).(subst0 i2 u2 (THead k u t2) t)) (\lambda (t: T).(subst0 i v (THead k x0 -x1) t))) (\lambda (x: T).(\lambda (H8: (subst0 (s k i2) u2 t2 x)).(\lambda -(H9: (subst0 (s k i) v x1 x)).(ex_intro2 T (\lambda (t: T).(subst0 i2 u2 -(THead k u t2) t)) (\lambda (t: T).(subst0 i v (THead k x0 x1) t)) (THead k -x0 x) (subst0_both u2 u x0 i2 H6 k t2 x H8) (subst0_snd k v x x1 i H9 x0))))) -(H1 x1 u2 (s k i2) H7 (ex2_ind T (\lambda (t: T).(subst0 (s k i2) u2 t2 t)) -(\lambda (t: T).(subst0 (s k i) v x1 t)) ((eq nat (s k i) (s k i2)) \to -False) (\lambda (x: T).(\lambda (_: (subst0 (s k i2) u2 t2 x)).(\lambda (_: -(subst0 (s k i) v x1 x)).(\lambda (H10: (eq nat (s k i) (s k i2))).(H3 (s_inj -k i i2 H10)))))) (H1 x1 u2 (s k i2) H7 (\lambda (H8: (eq nat (s k i) (s k -i2))).(H3 (s_inj k i i2 H8))))))) t4 H5)))))) H4)) (subst0_gen_head k u2 u t3 -t4 i2 H2))))))))))))))) (\lambda (v: T).(\lambda (u0: T).(\lambda (u2: -T).(\lambda (i: nat).(\lambda (H0: (subst0 i v u0 u2)).(\lambda (H1: -((\forall (t2: T).(\forall (u3: T).(\forall (i2: nat).((subst0 i2 u3 u0 t2) -\to ((not (eq nat i i2)) \to (ex2 T (\lambda (t: T).(subst0 i2 u3 u2 t)) -(\lambda (t: T).(subst0 i v t2 t)))))))))).(\lambda (k: K).(\lambda (t2: -T).(\lambda (t3: T).(\lambda (H2: (subst0 (s k i) v t2 t3)).(\lambda (H3: -((\forall (t4: T).(\forall (u3: T).(\forall (i2: nat).((subst0 i2 u3 t2 t4) -\to ((not (eq nat (s k i) i2)) \to (ex2 T (\lambda (t: T).(subst0 i2 u3 t3 -t)) (\lambda (t: T).(subst0 (s k i) v t4 t)))))))))).(\lambda (t4: -T).(\lambda (u3: T).(\lambda (i2: nat).(\lambda (H4: (subst0 i2 u3 (THead k -u0 t2) t4)).(\lambda (H5: (not (eq nat i i2))).(or3_ind (ex2 T (\lambda (u4: -T).(eq T t4 (THead k u4 t2))) (\lambda (u4: T).(subst0 i2 u3 u0 u4))) (ex2 T -(\lambda (t5: T).(eq T t4 (THead k u0 t5))) (\lambda (t5: T).(subst0 (s k i2) -u3 t2 t5))) (ex3_2 T T (\lambda (u4: T).(\lambda (t5: T).(eq T t4 (THead k u4 -t5)))) (\lambda (u4: T).(\lambda (_: T).(subst0 i2 u3 u0 u4))) (\lambda (_: -T).(\lambda (t5: T).(subst0 (s k i2) u3 t2 t5)))) (ex2 T (\lambda (t: -T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v t4 t))) -(\lambda (H6: (ex2 T (\lambda (u4: T).(eq T t4 (THead k u4 t2))) (\lambda -(u4: T).(subst0 i2 u3 u0 u4)))).(ex2_ind T (\lambda (u4: T).(eq T t4 (THead k -u4 t2))) (\lambda (u4: T).(subst0 i2 u3 u0 u4)) (ex2 T (\lambda (t: -T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v t4 t))) -(\lambda (x: T).(\lambda (H7: (eq T t4 (THead k x t2))).(\lambda (H8: (subst0 -i2 u3 u0 x)).(eq_ind_r T (THead k x t2) (\lambda (t: T).(ex2 T (\lambda (t5: -T).(subst0 i2 u3 (THead k u2 t3) t5)) (\lambda (t5: T).(subst0 i v t t5)))) -(ex2_ind T (\lambda (t: T).(subst0 i2 u3 u2 t)) (\lambda (t: T).(subst0 i v x -t)) (ex2 T (\lambda (t: T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: -T).(subst0 i v (THead k x t2) t))) (\lambda (x0: T).(\lambda (H9: (subst0 i2 -u3 u2 x0)).(\lambda (H10: (subst0 i v x x0)).(ex_intro2 T (\lambda (t: -T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v (THead k x -t2) t)) (THead k x0 t3) (subst0_fst u3 x0 u2 i2 H9 t3 k) (subst0_both v x x0 -i H10 k t2 t3 H2))))) (H1 x u3 i2 H8 H5)) t4 H7)))) H6)) (\lambda (H6: (ex2 T -(\lambda (t5: T).(eq T t4 (THead k u0 t5))) (\lambda (t5: T).(subst0 (s k i2) -u3 t2 t5)))).(ex2_ind T (\lambda (t5: T).(eq T t4 (THead k u0 t5))) (\lambda -(t5: T).(subst0 (s k i2) u3 t2 t5)) (ex2 T (\lambda (t: T).(subst0 i2 u3 -(THead k u2 t3) t)) (\lambda (t: T).(subst0 i v t4 t))) (\lambda (x: -T).(\lambda (H7: (eq T t4 (THead k u0 x))).(\lambda (H8: (subst0 (s k i2) u3 -t2 x)).(eq_ind_r T (THead k u0 x) (\lambda (t: T).(ex2 T (\lambda (t5: -T).(subst0 i2 u3 (THead k u2 t3) t5)) (\lambda (t5: T).(subst0 i v t t5)))) -(ex2_ind T (\lambda (t: T).(subst0 (s k i2) u3 t3 t)) (\lambda (t: T).(subst0 -(s k i) v x t)) (ex2 T (\lambda (t: T).(subst0 i2 u3 (THead k u2 t3) t)) -(\lambda (t: T).(subst0 i v (THead k u0 x) t))) (\lambda (x0: T).(\lambda -(H9: (subst0 (s k i2) u3 t3 x0)).(\lambda (H10: (subst0 (s k i) v x -x0)).(ex_intro2 T (\lambda (t: T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda -(t: T).(subst0 i v (THead k u0 x) t)) (THead k u2 x0) (subst0_snd k u3 x0 t3 -i2 H9 u2) (subst0_both v u0 u2 i H0 k x x0 H10))))) (H3 x u3 (s k i2) H8 -(ex2_ind T (\lambda (t: T).(subst0 (s k i2) u3 t3 t)) (\lambda (t: T).(subst0 -(s k i) v x t)) ((eq nat (s k i) (s k i2)) \to False) (\lambda (x0: -T).(\lambda (_: (subst0 (s k i2) u3 t3 x0)).(\lambda (_: (subst0 (s k i) v x -x0)).(\lambda (H11: (eq nat (s k i) (s k i2))).(H5 (s_inj k i i2 H11)))))) -(H3 x u3 (s k i2) H8 (\lambda (H9: (eq nat (s k i) (s k i2))).(H5 (s_inj k i -i2 H9))))))) t4 H7)))) H6)) (\lambda (H6: (ex3_2 T T (\lambda (u4: -T).(\lambda (t5: T).(eq T t4 (THead k u4 t5)))) (\lambda (u4: T).(\lambda (_: -T).(subst0 i2 u3 u0 u4))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s k i2) -u3 t2 t5))))).(ex3_2_ind T T (\lambda (u4: T).(\lambda (t5: T).(eq T t4 -(THead k u4 t5)))) (\lambda (u4: T).(\lambda (_: T).(subst0 i2 u3 u0 u4))) -(\lambda (_: T).(\lambda (t5: T).(subst0 (s k i2) u3 t2 t5))) (ex2 T (\lambda -(t: T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v t4 t))) -(\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: (eq T t4 (THead k x0 -x1))).(\lambda (H8: (subst0 i2 u3 u0 x0)).(\lambda (H9: (subst0 (s k i2) u3 -t2 x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t: T).(ex2 T (\lambda (t5: -T).(subst0 i2 u3 (THead k u2 t3) t5)) (\lambda (t5: T).(subst0 i v t t5)))) -(ex2_ind T (\lambda (t: T).(subst0 i2 u3 u2 t)) (\lambda (t: T).(subst0 i v -x0 t)) (ex2 T (\lambda (t: T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: -T).(subst0 i v (THead k x0 x1) t))) (\lambda (x: T).(\lambda (H10: (subst0 i2 -u3 u2 x)).(\lambda (H11: (subst0 i v x0 x)).(ex2_ind T (\lambda (t: -T).(subst0 (s k i2) u3 t3 t)) (\lambda (t: T).(subst0 (s k i) v x1 t)) (ex2 T -(\lambda (t: T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v -(THead k x0 x1) t))) (\lambda (x2: T).(\lambda (H12: (subst0 (s k i2) u3 t3 -x2)).(\lambda (H13: (subst0 (s k i) v x1 x2)).(ex_intro2 T (\lambda (t: -T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v (THead k x0 -x1) t)) (THead k x x2) (subst0_both u3 u2 x i2 H10 k t3 x2 H12) (subst0_both -v x0 x i H11 k x1 x2 H13))))) (H3 x1 u3 (s k i2) H9 (ex2_ind T (\lambda (t: -T).(subst0 (s k i2) u3 t3 t)) (\lambda (t: T).(subst0 (s k i) v x1 t)) ((eq -nat (s k i) (s k i2)) \to False) (\lambda (x2: T).(\lambda (_: (subst0 (s k -i2) u3 t3 x2)).(\lambda (_: (subst0 (s k i) v x1 x2)).(\lambda (H14: (eq nat -(s k i) (s k i2))).(H5 (s_inj k i i2 H14)))))) (H3 x1 u3 (s k i2) H9 (\lambda -(H12: (eq nat (s k i) (s k i2))).(H5 (s_inj k i i2 H12)))))))))) (H1 x0 u3 i2 -H8 H5)) t4 H7)))))) H6)) (subst0_gen_head k u3 u0 t2 t4 i2 -H4)))))))))))))))))) i1 u1 t0 t1 H))))). -(* COMMENTS -Initial nodes: 5375 -END *) - -theorem subst0_confluence_eq: - \forall (t0: T).(\forall (t1: T).(\forall (u: T).(\forall (i: nat).((subst0 -i u t0 t1) \to (\forall (t2: T).((subst0 i u t0 t2) \to (or4 (eq T t1 t2) -(ex2 T (\lambda (t: T).(subst0 i u t1 t)) (\lambda (t: T).(subst0 i u t2 t))) -(subst0 i u t1 t2) (subst0 i u t2 t1)))))))) -\def - \lambda (t0: T).(\lambda (t1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H: (subst0 i u t0 t1)).(subst0_ind (\lambda (n: nat).(\lambda (t: -T).(\lambda (t2: T).(\lambda (t3: T).(\forall (t4: T).((subst0 n t t2 t4) \to -(or4 (eq T t3 t4) (ex2 T (\lambda (t5: T).(subst0 n t t3 t5)) (\lambda (t5: -T).(subst0 n t t4 t5))) (subst0 n t t3 t4) (subst0 n t t4 t3)))))))) (\lambda -(v: T).(\lambda (i0: nat).(\lambda (t2: T).(\lambda (H0: (subst0 i0 v (TLRef -i0) t2)).(land_ind (eq nat i0 i0) (eq T t2 (lift (S i0) O v)) (or4 (eq T -(lift (S i0) O v) t2) (ex2 T (\lambda (t: T).(subst0 i0 v (lift (S i0) O v) -t)) (\lambda (t: T).(subst0 i0 v t2 t))) (subst0 i0 v (lift (S i0) O v) t2) -(subst0 i0 v t2 (lift (S i0) O v))) (\lambda (_: (eq nat i0 i0)).(\lambda -(H2: (eq T t2 (lift (S i0) O v))).(or4_intro0 (eq T (lift (S i0) O v) t2) -(ex2 T (\lambda (t: T).(subst0 i0 v (lift (S i0) O v) t)) (\lambda (t: -T).(subst0 i0 v t2 t))) (subst0 i0 v (lift (S i0) O v) t2) (subst0 i0 v t2 -(lift (S i0) O v)) (sym_eq T t2 (lift (S i0) O v) H2)))) (subst0_gen_lref v -t2 i0 i0 H0)))))) (\lambda (v: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda -(i0: nat).(\lambda (H0: (subst0 i0 v u1 u2)).(\lambda (H1: ((\forall (t2: -T).((subst0 i0 v u1 t2) \to (or4 (eq T u2 t2) (ex2 T (\lambda (t: T).(subst0 -i0 v u2 t)) (\lambda (t: T).(subst0 i0 v t2 t))) (subst0 i0 v u2 t2) (subst0 -i0 v t2 u2)))))).(\lambda (t: T).(\lambda (k: K).(\lambda (t2: T).(\lambda -(H2: (subst0 i0 v (THead k u1 t) t2)).(or3_ind (ex2 T (\lambda (u3: T).(eq T -t2 (THead k u3 t))) (\lambda (u3: T).(subst0 i0 v u1 u3))) (ex2 T (\lambda -(t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v t -t3))) (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead k u3 -t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v u1 u3))) (\lambda (_: -T).(\lambda (t3: T).(subst0 (s k i0) v t t3)))) (or4 (eq T (THead k u2 t) t2) -(ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: -T).(subst0 i0 v t2 t3))) (subst0 i0 v (THead k u2 t) t2) (subst0 i0 v t2 -(THead k u2 t))) (\lambda (H3: (ex2 T (\lambda (u3: T).(eq T t2 (THead k u3 -t))) (\lambda (u3: T).(subst0 i0 v u1 u3)))).(ex2_ind T (\lambda (u3: T).(eq -T t2 (THead k u3 t))) (\lambda (u3: T).(subst0 i0 v u1 u3)) (or4 (eq T (THead -k u2 t) t2) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda -(t3: T).(subst0 i0 v t2 t3))) (subst0 i0 v (THead k u2 t) t2) (subst0 i0 v t2 -(THead k u2 t))) (\lambda (x: T).(\lambda (H4: (eq T t2 (THead k x -t))).(\lambda (H5: (subst0 i0 v u1 x)).(eq_ind_r T (THead k x t) (\lambda -(t3: T).(or4 (eq T (THead k u2 t) t3) (ex2 T (\lambda (t4: T).(subst0 i0 v -(THead k u2 t) t4)) (\lambda (t4: T).(subst0 i0 v t3 t4))) (subst0 i0 v -(THead k u2 t) t3) (subst0 i0 v t3 (THead k u2 t)))) (or4_ind (eq T u2 x) -(ex2 T (\lambda (t3: T).(subst0 i0 v u2 t3)) (\lambda (t3: T).(subst0 i0 v x -t3))) (subst0 i0 v u2 x) (subst0 i0 v x u2) (or4 (eq T (THead k u2 t) (THead -k x t)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda -(t3: T).(subst0 i0 v (THead k x t) t3))) (subst0 i0 v (THead k u2 t) (THead k -x t)) (subst0 i0 v (THead k x t) (THead k u2 t))) (\lambda (H6: (eq T u2 -x)).(eq_ind_r T x (\lambda (t3: T).(or4 (eq T (THead k t3 t) (THead k x t)) -(ex2 T (\lambda (t4: T).(subst0 i0 v (THead k t3 t) t4)) (\lambda (t4: -T).(subst0 i0 v (THead k x t) t4))) (subst0 i0 v (THead k t3 t) (THead k x -t)) (subst0 i0 v (THead k x t) (THead k t3 t)))) (or4_intro0 (eq T (THead k x -t) (THead k x t)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k x t) t3)) -(\lambda (t3: T).(subst0 i0 v (THead k x t) t3))) (subst0 i0 v (THead k x t) -(THead k x t)) (subst0 i0 v (THead k x t) (THead k x t)) (refl_equal T (THead -k x t))) u2 H6)) (\lambda (H6: (ex2 T (\lambda (t3: T).(subst0 i0 v u2 t3)) -(\lambda (t3: T).(subst0 i0 v x t3)))).(ex2_ind T (\lambda (t3: T).(subst0 i0 -v u2 t3)) (\lambda (t3: T).(subst0 i0 v x t3)) (or4 (eq T (THead k u2 t) -(THead k x t)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) -(\lambda (t3: T).(subst0 i0 v (THead k x t) t3))) (subst0 i0 v (THead k u2 t) -(THead k x t)) (subst0 i0 v (THead k x t) (THead k u2 t))) (\lambda (x0: -T).(\lambda (H7: (subst0 i0 v u2 x0)).(\lambda (H8: (subst0 i0 v x -x0)).(or4_intro1 (eq T (THead k u2 t) (THead k x t)) (ex2 T (\lambda (t3: -T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k x -t) t3))) (subst0 i0 v (THead k u2 t) (THead k x t)) (subst0 i0 v (THead k x -t) (THead k u2 t)) (ex_intro2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) -t3)) (\lambda (t3: T).(subst0 i0 v (THead k x t) t3)) (THead k x0 t) -(subst0_fst v x0 u2 i0 H7 t k) (subst0_fst v x0 x i0 H8 t k)))))) H6)) -(\lambda (H6: (subst0 i0 v u2 x)).(or4_intro2 (eq T (THead k u2 t) (THead k x -t)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: -T).(subst0 i0 v (THead k x t) t3))) (subst0 i0 v (THead k u2 t) (THead k x -t)) (subst0 i0 v (THead k x t) (THead k u2 t)) (subst0_fst v x u2 i0 H6 t -k))) (\lambda (H6: (subst0 i0 v x u2)).(or4_intro3 (eq T (THead k u2 t) -(THead k x t)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) -(\lambda (t3: T).(subst0 i0 v (THead k x t) t3))) (subst0 i0 v (THead k u2 t) -(THead k x t)) (subst0 i0 v (THead k x t) (THead k u2 t)) (subst0_fst v u2 x -i0 H6 t k))) (H1 x H5)) t2 H4)))) H3)) (\lambda (H3: (ex2 T (\lambda (t3: -T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v t -t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: -T).(subst0 (s k i0) v t t3)) (or4 (eq T (THead k u2 t) t2) (ex2 T (\lambda -(t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0 v t2 -t3))) (subst0 i0 v (THead k u2 t) t2) (subst0 i0 v t2 (THead k u2 t))) -(\lambda (x: T).(\lambda (H4: (eq T t2 (THead k u1 x))).(\lambda (H5: (subst0 -(s k i0) v t x)).(eq_ind_r T (THead k u1 x) (\lambda (t3: T).(or4 (eq T -(THead k u2 t) t3) (ex2 T (\lambda (t4: T).(subst0 i0 v (THead k u2 t) t4)) -(\lambda (t4: T).(subst0 i0 v t3 t4))) (subst0 i0 v (THead k u2 t) t3) -(subst0 i0 v t3 (THead k u2 t)))) (or4_ind (eq T u2 u2) (ex2 T (\lambda (t3: -T).(subst0 i0 v u2 t3)) (\lambda (t3: T).(subst0 i0 v u2 t3))) (subst0 i0 v -u2 u2) (subst0 i0 v u2 u2) (or4 (eq T (THead k u2 t) (THead k u1 x)) (ex2 T -(\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0 -v (THead k u1 x) t3))) (subst0 i0 v (THead k u2 t) (THead k u1 x)) (subst0 i0 -v (THead k u1 x) (THead k u2 t))) (\lambda (_: (eq T u2 u2)).(or4_intro1 (eq -T (THead k u2 t) (THead k u1 x)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead -k u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k u1 x) t3))) (subst0 i0 v -(THead k u2 t) (THead k u1 x)) (subst0 i0 v (THead k u1 x) (THead k u2 t)) -(ex_intro2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: -T).(subst0 i0 v (THead k u1 x) t3)) (THead k u2 x) (subst0_snd k v x t i0 H5 -u2) (subst0_fst v u2 u1 i0 H0 x k)))) (\lambda (H6: (ex2 T (\lambda (t3: -T).(subst0 i0 v u2 t3)) (\lambda (t3: T).(subst0 i0 v u2 t3)))).(ex2_ind T -(\lambda (t3: T).(subst0 i0 v u2 t3)) (\lambda (t3: T).(subst0 i0 v u2 t3)) -(or4 (eq T (THead k u2 t) (THead k u1 x)) (ex2 T (\lambda (t3: T).(subst0 i0 -v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k u1 x) t3))) -(subst0 i0 v (THead k u2 t) (THead k u1 x)) (subst0 i0 v (THead k u1 x) -(THead k u2 t))) (\lambda (x0: T).(\lambda (_: (subst0 i0 v u2 x0)).(\lambda -(_: (subst0 i0 v u2 x0)).(or4_intro1 (eq T (THead k u2 t) (THead k u1 x)) -(ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: -T).(subst0 i0 v (THead k u1 x) t3))) (subst0 i0 v (THead k u2 t) (THead k u1 -x)) (subst0 i0 v (THead k u1 x) (THead k u2 t)) (ex_intro2 T (\lambda (t3: -T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k u1 -x) t3)) (THead k u2 x) (subst0_snd k v x t i0 H5 u2) (subst0_fst v u2 u1 i0 -H0 x k)))))) H6)) (\lambda (_: (subst0 i0 v u2 u2)).(or4_intro1 (eq T (THead -k u2 t) (THead k u1 x)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) -t3)) (\lambda (t3: T).(subst0 i0 v (THead k u1 x) t3))) (subst0 i0 v (THead k -u2 t) (THead k u1 x)) (subst0 i0 v (THead k u1 x) (THead k u2 t)) (ex_intro2 -T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 -i0 v (THead k u1 x) t3)) (THead k u2 x) (subst0_snd k v x t i0 H5 u2) -(subst0_fst v u2 u1 i0 H0 x k)))) (\lambda (_: (subst0 i0 v u2 -u2)).(or4_intro1 (eq T (THead k u2 t) (THead k u1 x)) (ex2 T (\lambda (t3: -T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k u1 -x) t3))) (subst0 i0 v (THead k u2 t) (THead k u1 x)) (subst0 i0 v (THead k u1 -x) (THead k u2 t)) (ex_intro2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) -t3)) (\lambda (t3: T).(subst0 i0 v (THead k u1 x) t3)) (THead k u2 x) -(subst0_snd k v x t i0 H5 u2) (subst0_fst v u2 u1 i0 H0 x k)))) (H1 u2 H0)) -t2 H4)))) H3)) (\lambda (H3: (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq -T t2 (THead k u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v u1 -u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i0) v t -t3))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead k u3 -t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v u1 u3))) (\lambda (_: -T).(\lambda (t3: T).(subst0 (s k i0) v t t3))) (or4 (eq T (THead k u2 t) t2) -(ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: -T).(subst0 i0 v t2 t3))) (subst0 i0 v (THead k u2 t) t2) (subst0 i0 v t2 -(THead k u2 t))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t2 -(THead k x0 x1))).(\lambda (H5: (subst0 i0 v u1 x0)).(\lambda (H6: (subst0 (s -k i0) v t x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t3: T).(or4 (eq T (THead -k u2 t) t3) (ex2 T (\lambda (t4: T).(subst0 i0 v (THead k u2 t) t4)) (\lambda -(t4: T).(subst0 i0 v t3 t4))) (subst0 i0 v (THead k u2 t) t3) (subst0 i0 v t3 -(THead k u2 t)))) (or4_ind (eq T u2 x0) (ex2 T (\lambda (t3: T).(subst0 i0 v -u2 t3)) (\lambda (t3: T).(subst0 i0 v x0 t3))) (subst0 i0 v u2 x0) (subst0 i0 -v x0 u2) (or4 (eq T (THead k u2 t) (THead k x0 x1)) (ex2 T (\lambda (t3: -T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k x0 -x1) t3))) (subst0 i0 v (THead k u2 t) (THead k x0 x1)) (subst0 i0 v (THead k -x0 x1) (THead k u2 t))) (\lambda (H7: (eq T u2 x0)).(eq_ind_r T x0 (\lambda -(t3: T).(or4 (eq T (THead k t3 t) (THead k x0 x1)) (ex2 T (\lambda (t4: -T).(subst0 i0 v (THead k t3 t) t4)) (\lambda (t4: T).(subst0 i0 v (THead k x0 -x1) t4))) (subst0 i0 v (THead k t3 t) (THead k x0 x1)) (subst0 i0 v (THead k -x0 x1) (THead k t3 t)))) (or4_intro2 (eq T (THead k x0 t) (THead k x0 x1)) -(ex2 T (\lambda (t3: T).(subst0 i0 v (THead k x0 t) t3)) (\lambda (t3: -T).(subst0 i0 v (THead k x0 x1) t3))) (subst0 i0 v (THead k x0 t) (THead k x0 -x1)) (subst0 i0 v (THead k x0 x1) (THead k x0 t)) (subst0_snd k v x1 t i0 H6 -x0)) u2 H7)) (\lambda (H7: (ex2 T (\lambda (t3: T).(subst0 i0 v u2 t3)) -(\lambda (t3: T).(subst0 i0 v x0 t3)))).(ex2_ind T (\lambda (t3: T).(subst0 -i0 v u2 t3)) (\lambda (t3: T).(subst0 i0 v x0 t3)) (or4 (eq T (THead k u2 t) -(THead k x0 x1)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) -(\lambda (t3: T).(subst0 i0 v (THead k x0 x1) t3))) (subst0 i0 v (THead k u2 -t) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t))) (\lambda -(x: T).(\lambda (H8: (subst0 i0 v u2 x)).(\lambda (H9: (subst0 i0 v x0 -x)).(or4_intro1 (eq T (THead k u2 t) (THead k x0 x1)) (ex2 T (\lambda (t3: -T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k x0 -x1) t3))) (subst0 i0 v (THead k u2 t) (THead k x0 x1)) (subst0 i0 v (THead k -x0 x1) (THead k u2 t)) (ex_intro2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 -t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k x0 x1) t3)) (THead k x x1) -(subst0_both v u2 x i0 H8 k t x1 H6) (subst0_fst v x x0 i0 H9 x1 k)))))) H7)) -(\lambda (H7: (subst0 i0 v u2 x0)).(or4_intro2 (eq T (THead k u2 t) (THead k -x0 x1)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda -(t3: T).(subst0 i0 v (THead k x0 x1) t3))) (subst0 i0 v (THead k u2 t) (THead -k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t)) (subst0_both v u2 x0 -i0 H7 k t x1 H6))) (\lambda (H7: (subst0 i0 v x0 u2)).(or4_intro1 (eq T -(THead k u2 t) (THead k x0 x1)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k -u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k x0 x1) t3))) (subst0 i0 v -(THead k u2 t) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t)) -(ex_intro2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: -T).(subst0 i0 v (THead k x0 x1) t3)) (THead k u2 x1) (subst0_snd k v x1 t i0 -H6 u2) (subst0_fst v u2 x0 i0 H7 x1 k)))) (H1 x0 H5)) t2 H4)))))) H3)) -(subst0_gen_head k v u1 t t2 i0 H2)))))))))))) (\lambda (k: K).(\lambda (v: -T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (i0: nat).(\lambda (H0: (subst0 -(s k i0) v t3 t2)).(\lambda (H1: ((\forall (t4: T).((subst0 (s k i0) v t3 t4) -\to (or4 (eq T t2 t4) (ex2 T (\lambda (t: T).(subst0 (s k i0) v t2 t)) -(\lambda (t: T).(subst0 (s k i0) v t4 t))) (subst0 (s k i0) v t2 t4) (subst0 -(s k i0) v t4 t2)))))).(\lambda (u0: T).(\lambda (t4: T).(\lambda (H2: -(subst0 i0 v (THead k u0 t3) t4)).(or3_ind (ex2 T (\lambda (u2: T).(eq T t4 -(THead k u2 t3))) (\lambda (u2: T).(subst0 i0 v u0 u2))) (ex2 T (\lambda (t5: -T).(eq T t4 (THead k u0 t5))) (\lambda (t5: T).(subst0 (s k i0) v t3 t5))) -(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead k u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 i0 v u0 u2))) (\lambda (_: -T).(\lambda (t5: T).(subst0 (s k i0) v t3 t5)))) (or4 (eq T (THead k u0 t2) -t4) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u0 t2) t)) (\lambda (t: -T).(subst0 i0 v t4 t))) (subst0 i0 v (THead k u0 t2) t4) (subst0 i0 v t4 -(THead k u0 t2))) (\lambda (H3: (ex2 T (\lambda (u2: T).(eq T t4 (THead k u2 -t3))) (\lambda (u2: T).(subst0 i0 v u0 u2)))).(ex2_ind T (\lambda (u2: T).(eq -T t4 (THead k u2 t3))) (\lambda (u2: T).(subst0 i0 v u0 u2)) (or4 (eq T -(THead k u0 t2) t4) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u0 t2) t)) -(\lambda (t: T).(subst0 i0 v t4 t))) (subst0 i0 v (THead k u0 t2) t4) (subst0 -i0 v t4 (THead k u0 t2))) (\lambda (x: T).(\lambda (H4: (eq T t4 (THead k x -t3))).(\lambda (H5: (subst0 i0 v u0 x)).(eq_ind_r T (THead k x t3) (\lambda -(t: T).(or4 (eq T (THead k u0 t2) t) (ex2 T (\lambda (t5: T).(subst0 i0 v -(THead k u0 t2) t5)) (\lambda (t5: T).(subst0 i0 v t t5))) (subst0 i0 v -(THead k u0 t2) t) (subst0 i0 v t (THead k u0 t2)))) (or4_ind (eq T t2 t2) -(ex2 T (\lambda (t: T).(subst0 (s k i0) v t2 t)) (\lambda (t: T).(subst0 (s k -i0) v t2 t))) (subst0 (s k i0) v t2 t2) (subst0 (s k i0) v t2 t2) (or4 (eq T -(THead k u0 t2) (THead k x t3)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k -u0 t2) t)) (\lambda (t: T).(subst0 i0 v (THead k x t3) t))) (subst0 i0 v -(THead k u0 t2) (THead k x t3)) (subst0 i0 v (THead k x t3) (THead k u0 t2))) -(\lambda (_: (eq T t2 t2)).(or4_intro1 (eq T (THead k u0 t2) (THead k x t3)) -(ex2 T (\lambda (t: T).(subst0 i0 v (THead k u0 t2) t)) (\lambda (t: -T).(subst0 i0 v (THead k x t3) t))) (subst0 i0 v (THead k u0 t2) (THead k x -t3)) (subst0 i0 v (THead k x t3) (THead k u0 t2)) (ex_intro2 T (\lambda (t: -T).(subst0 i0 v (THead k u0 t2) t)) (\lambda (t: T).(subst0 i0 v (THead k x -t3) t)) (THead k x t2) (subst0_fst v x u0 i0 H5 t2 k) (subst0_snd k v t2 t3 -i0 H0 x)))) (\lambda (H6: (ex2 T (\lambda (t: T).(subst0 (s k i0) v t2 t)) -(\lambda (t: T).(subst0 (s k i0) v t2 t)))).(ex2_ind T (\lambda (t: -T).(subst0 (s k i0) v t2 t)) (\lambda (t: T).(subst0 (s k i0) v t2 t)) (or4 -(eq T (THead k u0 t2) (THead k x t3)) (ex2 T (\lambda (t: T).(subst0 i0 v -(THead k u0 t2) t)) (\lambda (t: T).(subst0 i0 v (THead k x t3) t))) (subst0 -i0 v (THead k u0 t2) (THead k x t3)) (subst0 i0 v (THead k x t3) (THead k u0 -t2))) (\lambda (x0: T).(\lambda (_: (subst0 (s k i0) v t2 x0)).(\lambda (_: -(subst0 (s k i0) v t2 x0)).(or4_intro1 (eq T (THead k u0 t2) (THead k x t3)) -(ex2 T (\lambda (t: T).(subst0 i0 v (THead k u0 t2) t)) (\lambda (t: -T).(subst0 i0 v (THead k x t3) t))) (subst0 i0 v (THead k u0 t2) (THead k x -t3)) (subst0 i0 v (THead k x t3) (THead k u0 t2)) (ex_intro2 T (\lambda (t: -T).(subst0 i0 v (THead k u0 t2) t)) (\lambda (t: T).(subst0 i0 v (THead k x -t3) t)) (THead k x t2) (subst0_fst v x u0 i0 H5 t2 k) (subst0_snd k v t2 t3 -i0 H0 x)))))) H6)) (\lambda (_: (subst0 (s k i0) v t2 t2)).(or4_intro1 (eq T -(THead k u0 t2) (THead k x t3)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k -u0 t2) t)) (\lambda (t: T).(subst0 i0 v (THead k x t3) t))) (subst0 i0 v -(THead k u0 t2) (THead k x t3)) (subst0 i0 v (THead k x t3) (THead k u0 t2)) -(ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k u0 t2) t)) (\lambda (t: -T).(subst0 i0 v (THead k x t3) t)) (THead k x t2) (subst0_fst v x u0 i0 H5 t2 -k) (subst0_snd k v t2 t3 i0 H0 x)))) (\lambda (_: (subst0 (s k i0) v t2 -t2)).(or4_intro1 (eq T (THead k u0 t2) (THead k x t3)) (ex2 T (\lambda (t: -T).(subst0 i0 v (THead k u0 t2) t)) (\lambda (t: T).(subst0 i0 v (THead k x -t3) t))) (subst0 i0 v (THead k u0 t2) (THead k x t3)) (subst0 i0 v (THead k x -t3) (THead k u0 t2)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k u0 -t2) t)) (\lambda (t: T).(subst0 i0 v (THead k x t3) t)) (THead k x t2) -(subst0_fst v x u0 i0 H5 t2 k) (subst0_snd k v t2 t3 i0 H0 x)))) (H1 t2 H0)) -t4 H4)))) H3)) (\lambda (H3: (ex2 T (\lambda (t5: T).(eq T t4 (THead k u0 -t5))) (\lambda (t5: T).(subst0 (s k i0) v t3 t5)))).(ex2_ind T (\lambda (t5: -T).(eq T t4 (THead k u0 t5))) (\lambda (t5: T).(subst0 (s k i0) v t3 t5)) -(or4 (eq T (THead k u0 t2) t4) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k -u0 t2) t)) (\lambda (t: T).(subst0 i0 v t4 t))) (subst0 i0 v (THead k u0 t2) -t4) (subst0 i0 v t4 (THead k u0 t2))) (\lambda (x: T).(\lambda (H4: (eq T t4 -(THead k u0 x))).(\lambda (H5: (subst0 (s k i0) v t3 x)).(eq_ind_r T (THead k -u0 x) (\lambda (t: T).(or4 (eq T (THead k u0 t2) t) (ex2 T (\lambda (t5: -T).(subst0 i0 v (THead k u0 t2) t5)) (\lambda (t5: T).(subst0 i0 v t t5))) -(subst0 i0 v (THead k u0 t2) t) (subst0 i0 v t (THead k u0 t2)))) (or4_ind -(eq T t2 x) (ex2 T (\lambda (t: T).(subst0 (s k i0) v t2 t)) (\lambda (t: -T).(subst0 (s k i0) v x t))) (subst0 (s k i0) v t2 x) (subst0 (s k i0) v x -t2) (or4 (eq T (THead k u0 t2) (THead k u0 x)) (ex2 T (\lambda (t: T).(subst0 -i0 v (THead k u0 t2) t)) (\lambda (t: T).(subst0 i0 v (THead k u0 x) t))) -(subst0 i0 v (THead k u0 t2) (THead k u0 x)) (subst0 i0 v (THead k u0 x) -(THead k u0 t2))) (\lambda (H6: (eq T t2 x)).(eq_ind_r T x (\lambda (t: -T).(or4 (eq T (THead k u0 t) (THead k u0 x)) (ex2 T (\lambda (t5: T).(subst0 -i0 v (THead k u0 t) t5)) (\lambda (t5: T).(subst0 i0 v (THead k u0 x) t5))) -(subst0 i0 v (THead k u0 t) (THead k u0 x)) (subst0 i0 v (THead k u0 x) -(THead k u0 t)))) (or4_intro0 (eq T (THead k u0 x) (THead k u0 x)) (ex2 T -(\lambda (t: T).(subst0 i0 v (THead k u0 x) t)) (\lambda (t: T).(subst0 i0 v -(THead k u0 x) t))) (subst0 i0 v (THead k u0 x) (THead k u0 x)) (subst0 i0 v -(THead k u0 x) (THead k u0 x)) (refl_equal T (THead k u0 x))) t2 H6)) -(\lambda (H6: (ex2 T (\lambda (t: T).(subst0 (s k i0) v t2 t)) (\lambda (t: -T).(subst0 (s k i0) v x t)))).(ex2_ind T (\lambda (t: T).(subst0 (s k i0) v -t2 t)) (\lambda (t: T).(subst0 (s k i0) v x t)) (or4 (eq T (THead k u0 t2) -(THead k u0 x)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u0 t2) t)) -(\lambda (t: T).(subst0 i0 v (THead k u0 x) t))) (subst0 i0 v (THead k u0 t2) -(THead k u0 x)) (subst0 i0 v (THead k u0 x) (THead k u0 t2))) (\lambda (x0: -T).(\lambda (H7: (subst0 (s k i0) v t2 x0)).(\lambda (H8: (subst0 (s k i0) v -x x0)).(or4_intro1 (eq T (THead k u0 t2) (THead k u0 x)) (ex2 T (\lambda (t: -T).(subst0 i0 v (THead k u0 t2) t)) (\lambda (t: T).(subst0 i0 v (THead k u0 -x) t))) (subst0 i0 v (THead k u0 t2) (THead k u0 x)) (subst0 i0 v (THead k u0 -x) (THead k u0 t2)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k u0 t2) -t)) (\lambda (t: T).(subst0 i0 v (THead k u0 x) t)) (THead k u0 x0) -(subst0_snd k v x0 t2 i0 H7 u0) (subst0_snd k v x0 x i0 H8 u0)))))) H6)) -(\lambda (H6: (subst0 (s k i0) v t2 x)).(or4_intro2 (eq T (THead k u0 t2) -(THead k u0 x)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u0 t2) t)) -(\lambda (t: T).(subst0 i0 v (THead k u0 x) t))) (subst0 i0 v (THead k u0 t2) -(THead k u0 x)) (subst0 i0 v (THead k u0 x) (THead k u0 t2)) (subst0_snd k v -x t2 i0 H6 u0))) (\lambda (H6: (subst0 (s k i0) v x t2)).(or4_intro3 (eq T -(THead k u0 t2) (THead k u0 x)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k -u0 t2) t)) (\lambda (t: T).(subst0 i0 v (THead k u0 x) t))) (subst0 i0 v -(THead k u0 t2) (THead k u0 x)) (subst0 i0 v (THead k u0 x) (THead k u0 t2)) -(subst0_snd k v t2 x i0 H6 u0))) (H1 x H5)) t4 H4)))) H3)) (\lambda (H3: -(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead k u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 i0 v u0 u2))) (\lambda (_: -T).(\lambda (t5: T).(subst0 (s k i0) v t3 t5))))).(ex3_2_ind T T (\lambda -(u2: T).(\lambda (t5: T).(eq T t4 (THead k u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i0 v u0 u2))) (\lambda (_: T).(\lambda (t5: -T).(subst0 (s k i0) v t3 t5))) (or4 (eq T (THead k u0 t2) t4) (ex2 T (\lambda -(t: T).(subst0 i0 v (THead k u0 t2) t)) (\lambda (t: T).(subst0 i0 v t4 t))) -(subst0 i0 v (THead k u0 t2) t4) (subst0 i0 v t4 (THead k u0 t2))) (\lambda -(x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t4 (THead k x0 x1))).(\lambda -(H5: (subst0 i0 v u0 x0)).(\lambda (H6: (subst0 (s k i0) v t3 x1)).(eq_ind_r -T (THead k x0 x1) (\lambda (t: T).(or4 (eq T (THead k u0 t2) t) (ex2 T -(\lambda (t5: T).(subst0 i0 v (THead k u0 t2) t5)) (\lambda (t5: T).(subst0 -i0 v t t5))) (subst0 i0 v (THead k u0 t2) t) (subst0 i0 v t (THead k u0 -t2)))) (or4_ind (eq T t2 x1) (ex2 T (\lambda (t: T).(subst0 (s k i0) v t2 t)) -(\lambda (t: T).(subst0 (s k i0) v x1 t))) (subst0 (s k i0) v t2 x1) (subst0 -(s k i0) v x1 t2) (or4 (eq T (THead k u0 t2) (THead k x0 x1)) (ex2 T (\lambda -(t: T).(subst0 i0 v (THead k u0 t2) t)) (\lambda (t: T).(subst0 i0 v (THead k -x0 x1) t))) (subst0 i0 v (THead k u0 t2) (THead k x0 x1)) (subst0 i0 v (THead -k x0 x1) (THead k u0 t2))) (\lambda (H7: (eq T t2 x1)).(eq_ind_r T x1 -(\lambda (t: T).(or4 (eq T (THead k u0 t) (THead k x0 x1)) (ex2 T (\lambda -(t5: T).(subst0 i0 v (THead k u0 t) t5)) (\lambda (t5: T).(subst0 i0 v (THead -k x0 x1) t5))) (subst0 i0 v (THead k u0 t) (THead k x0 x1)) (subst0 i0 v -(THead k x0 x1) (THead k u0 t)))) (or4_intro2 (eq T (THead k u0 x1) (THead k -x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u0 x1) t)) (\lambda (t: -T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u0 x1) (THead k x0 -x1)) (subst0 i0 v (THead k x0 x1) (THead k u0 x1)) (subst0_fst v x0 u0 i0 H5 -x1 k)) t2 H7)) (\lambda (H7: (ex2 T (\lambda (t: T).(subst0 (s k i0) v t2 t)) -(\lambda (t: T).(subst0 (s k i0) v x1 t)))).(ex2_ind T (\lambda (t: -T).(subst0 (s k i0) v t2 t)) (\lambda (t: T).(subst0 (s k i0) v x1 t)) (or4 -(eq T (THead k u0 t2) (THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v -(THead k u0 t2) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 -i0 v (THead k u0 t2) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k -u0 t2))) (\lambda (x: T).(\lambda (H8: (subst0 (s k i0) v t2 x)).(\lambda -(H9: (subst0 (s k i0) v x1 x)).(or4_intro1 (eq T (THead k u0 t2) (THead k x0 -x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u0 t2) t)) (\lambda (t: -T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u0 t2) (THead k x0 -x1)) (subst0 i0 v (THead k x0 x1) (THead k u0 t2)) (ex_intro2 T (\lambda (t: -T).(subst0 i0 v (THead k u0 t2) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 -x1) t)) (THead k x0 x) (subst0_both v u0 x0 i0 H5 k t2 x H8) (subst0_snd k v -x x1 i0 H9 x0)))))) H7)) (\lambda (H7: (subst0 (s k i0) v t2 x1)).(or4_intro2 -(eq T (THead k u0 t2) (THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v -(THead k u0 t2) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 -i0 v (THead k u0 t2) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k -u0 t2)) (subst0_both v u0 x0 i0 H5 k t2 x1 H7))) (\lambda (H7: (subst0 (s k -i0) v x1 t2)).(or4_intro1 (eq T (THead k u0 t2) (THead k x0 x1)) (ex2 T -(\lambda (t: T).(subst0 i0 v (THead k u0 t2) t)) (\lambda (t: T).(subst0 i0 v -(THead k x0 x1) t))) (subst0 i0 v (THead k u0 t2) (THead k x0 x1)) (subst0 i0 -v (THead k x0 x1) (THead k u0 t2)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v -(THead k u0 t2) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t)) (THead k -x0 t2) (subst0_fst v x0 u0 i0 H5 t2 k) (subst0_snd k v t2 x1 i0 H7 x0)))) (H1 -x1 H6)) t4 H4)))))) H3)) (subst0_gen_head k v u0 t3 t4 i0 H2)))))))))))) -(\lambda (v: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i0: nat).(\lambda -(H0: (subst0 i0 v u1 u2)).(\lambda (H1: ((\forall (t2: T).((subst0 i0 v u1 -t2) \to (or4 (eq T u2 t2) (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda -(t: T).(subst0 i0 v t2 t))) (subst0 i0 v u2 t2) (subst0 i0 v t2 -u2)))))).(\lambda (k: K).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H2: -(subst0 (s k i0) v t2 t3)).(\lambda (H3: ((\forall (t4: T).((subst0 (s k i0) -v t2 t4) \to (or4 (eq T t3 t4) (ex2 T (\lambda (t: T).(subst0 (s k i0) v t3 -t)) (\lambda (t: T).(subst0 (s k i0) v t4 t))) (subst0 (s k i0) v t3 t4) -(subst0 (s k i0) v t4 t3)))))).(\lambda (t4: T).(\lambda (H4: (subst0 i0 v -(THead k u1 t2) t4)).(or3_ind (ex2 T (\lambda (u3: T).(eq T t4 (THead k u3 -t2))) (\lambda (u3: T).(subst0 i0 v u1 u3))) (ex2 T (\lambda (t5: T).(eq T t4 -(THead k u1 t5))) (\lambda (t5: T).(subst0 (s k i0) v t2 t5))) (ex3_2 T T -(\lambda (u3: T).(\lambda (t5: T).(eq T t4 (THead k u3 t5)))) (\lambda (u3: -T).(\lambda (_: T).(subst0 i0 v u1 u3))) (\lambda (_: T).(\lambda (t5: -T).(subst0 (s k i0) v t2 t5)))) (or4 (eq T (THead k u2 t3) t4) (ex2 T -(\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v -t4 t))) (subst0 i0 v (THead k u2 t3) t4) (subst0 i0 v t4 (THead k u2 t3))) -(\lambda (H5: (ex2 T (\lambda (u3: T).(eq T t4 (THead k u3 t2))) (\lambda -(u3: T).(subst0 i0 v u1 u3)))).(ex2_ind T (\lambda (u3: T).(eq T t4 (THead k -u3 t2))) (\lambda (u3: T).(subst0 i0 v u1 u3)) (or4 (eq T (THead k u2 t3) t4) -(ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: -T).(subst0 i0 v t4 t))) (subst0 i0 v (THead k u2 t3) t4) (subst0 i0 v t4 -(THead k u2 t3))) (\lambda (x: T).(\lambda (H6: (eq T t4 (THead k x -t2))).(\lambda (H7: (subst0 i0 v u1 x)).(eq_ind_r T (THead k x t2) (\lambda -(t: T).(or4 (eq T (THead k u2 t3) t) (ex2 T (\lambda (t5: T).(subst0 i0 v -(THead k u2 t3) t5)) (\lambda (t5: T).(subst0 i0 v t t5))) (subst0 i0 v -(THead k u2 t3) t) (subst0 i0 v t (THead k u2 t3)))) (or4_ind (eq T t3 t3) -(ex2 T (\lambda (t: T).(subst0 (s k i0) v t3 t)) (\lambda (t: T).(subst0 (s k -i0) v t3 t))) (subst0 (s k i0) v t3 t3) (subst0 (s k i0) v t3 t3) (or4 (eq T -(THead k u2 t3) (THead k x t2)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k -u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x t2) t))) (subst0 i0 v -(THead k u2 t3) (THead k x t2)) (subst0 i0 v (THead k x t2) (THead k u2 t3))) -(\lambda (_: (eq T t3 t3)).(or4_ind (eq T u2 x) (ex2 T (\lambda (t: -T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0 i0 v x t))) (subst0 i0 v u2 x) -(subst0 i0 v x u2) (or4 (eq T (THead k u2 t3) (THead k x t2)) (ex2 T (\lambda -(t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k -x t2) t))) (subst0 i0 v (THead k u2 t3) (THead k x t2)) (subst0 i0 v (THead k -x t2) (THead k u2 t3))) (\lambda (H9: (eq T u2 x)).(eq_ind_r T x (\lambda (t: -T).(or4 (eq T (THead k t t3) (THead k x t2)) (ex2 T (\lambda (t5: T).(subst0 -i0 v (THead k t t3) t5)) (\lambda (t5: T).(subst0 i0 v (THead k x t2) t5))) -(subst0 i0 v (THead k t t3) (THead k x t2)) (subst0 i0 v (THead k x t2) -(THead k t t3)))) (or4_intro3 (eq T (THead k x t3) (THead k x t2)) (ex2 T -(\lambda (t: T).(subst0 i0 v 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(\lambda (t: T).(subst0 (s k i0) -v t3 t)) (\lambda (t: T).(subst0 (s k i0) v t3 t)))).(ex2_ind T (\lambda (t: -T).(subst0 (s k i0) v t3 t)) (\lambda (t: T).(subst0 (s k i0) v t3 t)) (or4 -(eq T (THead k u2 t3) (THead k x t2)) (ex2 T (\lambda (t: T).(subst0 i0 v -(THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x t2) t))) (subst0 -i0 v (THead k u2 t3) (THead k x t2)) (subst0 i0 v (THead k x t2) (THead k u2 -t3))) (\lambda (x0: T).(\lambda (_: (subst0 (s k i0) v t3 x0)).(\lambda (_: -(subst0 (s k i0) v t3 x0)).(or4_ind (eq T u2 x) (ex2 T (\lambda (t: -T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0 i0 v x t))) (subst0 i0 v u2 x) -(subst0 i0 v x u2) (or4 (eq T (THead k u2 t3) (THead k x t2)) (ex2 T (\lambda -(t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k -x t2) t))) (subst0 i0 v (THead k u2 t3) (THead k x t2)) (subst0 i0 v (THead k -x t2) (THead k u2 t3))) (\lambda (H11: (eq T u2 x)).(eq_ind_r T x (\lambda -(t: T).(or4 (eq T (THead k t t3) (THead k x t2)) (ex2 T (\lambda (t5: -T).(subst0 i0 v (THead k t t3) t5)) (\lambda (t5: T).(subst0 i0 v (THead k x -t2) t5))) (subst0 i0 v (THead k t t3) (THead k x t2)) (subst0 i0 v (THead k x -t2) (THead k t t3)))) (or4_intro3 (eq T (THead k x t3) (THead k x t2)) (ex2 T -(\lambda (t: T).(subst0 i0 v (THead k x t3) t)) (\lambda (t: T).(subst0 i0 v -(THead k x t2) t))) (subst0 i0 v (THead k x t3) (THead k x t2)) (subst0 i0 v -(THead k x t2) (THead k x t3)) (subst0_snd k v t3 t2 i0 H2 x)) u2 H11)) -(\lambda (H11: (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: -T).(subst0 i0 v x t)))).(ex2_ind T (\lambda (t: T).(subst0 i0 v u2 t)) -(\lambda (t: T).(subst0 i0 v x t)) (or4 (eq T (THead k u2 t3) (THead k x t2)) -(ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: -T).(subst0 i0 v (THead k x t2) t))) (subst0 i0 v (THead k u2 t3) (THead k x -t2)) (subst0 i0 v (THead k x t2) (THead k u2 t3))) (\lambda (x1: T).(\lambda -(H12: (subst0 i0 v u2 x1)).(\lambda (H13: (subst0 i0 v x x1)).(or4_intro1 (eq -T (THead k u2 t3) (THead k x t2)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead -k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x t2) t))) (subst0 i0 v -(THead k u2 t3) (THead k x t2)) (subst0 i0 v (THead k x t2) (THead k u2 t3)) -(ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: -T).(subst0 i0 v (THead k x t2) t)) (THead k x1 t3) (subst0_fst v x1 u2 i0 H12 -t3 k) (subst0_both v x x1 i0 H13 k t2 t3 H2)))))) H11)) (\lambda (H11: -(subst0 i0 v u2 x)).(or4_intro1 (eq T (THead k u2 t3) (THead k x t2)) (ex2 T -(\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v -(THead k x t2) t))) (subst0 i0 v (THead k u2 t3) (THead k x t2)) (subst0 i0 v -(THead k x t2) (THead k u2 t3)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v -(THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x t2) t)) (THead k -x t3) (subst0_fst v x u2 i0 H11 t3 k) (subst0_snd k v t3 t2 i0 H2 x)))) -(\lambda (H11: (subst0 i0 v x u2)).(or4_intro3 (eq T (THead k u2 t3) (THead k -x t2)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: -T).(subst0 i0 v (THead k x t2) t))) (subst0 i0 v (THead k u2 t3) (THead k x -t2)) (subst0 i0 v (THead k x t2) (THead k u2 t3)) (subst0_both v x u2 i0 H11 -k t2 t3 H2))) (H1 x H7))))) H8)) (\lambda (_: (subst0 (s k i0) v t3 -t3)).(or4_ind (eq T u2 x) (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda -(t: T).(subst0 i0 v x t))) (subst0 i0 v u2 x) (subst0 i0 v x u2) (or4 (eq T -(THead k u2 t3) (THead k x t2)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k -u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x t2) t))) (subst0 i0 v -(THead k u2 t3) (THead k x t2)) (subst0 i0 v (THead k x t2) (THead k u2 t3))) -(\lambda (H9: (eq T u2 x)).(eq_ind_r T x (\lambda (t: T).(or4 (eq T (THead k -t t3) (THead k x t2)) (ex2 T (\lambda (t5: T).(subst0 i0 v (THead k t t3) -t5)) (\lambda (t5: T).(subst0 i0 v (THead k x t2) t5))) (subst0 i0 v (THead k -t t3) (THead k x t2)) (subst0 i0 v (THead k x t2) (THead k t t3)))) -(or4_intro3 (eq T (THead k x t3) (THead k x t2)) (ex2 T (\lambda (t: -T).(subst0 i0 v (THead k x t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x -t2) t))) (subst0 i0 v (THead k x t3) (THead k x t2)) (subst0 i0 v (THead k x -t2) (THead k x t3)) (subst0_snd k v t3 t2 i0 H2 x)) u2 H9)) (\lambda (H9: -(ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0 i0 v x -t)))).(ex2_ind T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0 -i0 v x t)) (or4 (eq T (THead k u2 t3) (THead k x t2)) (ex2 T (\lambda (t: -T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x -t2) t))) (subst0 i0 v (THead k u2 t3) (THead k x t2)) (subst0 i0 v (THead k x -t2) (THead k u2 t3))) (\lambda (x0: T).(\lambda (H10: (subst0 i0 v u2 -x0)).(\lambda (H11: (subst0 i0 v x x0)).(or4_intro1 (eq T (THead k u2 t3) -(THead k x t2)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) -(\lambda (t: T).(subst0 i0 v (THead k x t2) t))) (subst0 i0 v (THead k u2 t3) -(THead k x t2)) (subst0 i0 v (THead k x t2) (THead k u2 t3)) (ex_intro2 T -(\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v -(THead k x t2) t)) (THead k x0 t3) (subst0_fst v x0 u2 i0 H10 t3 k) -(subst0_both v x x0 i0 H11 k t2 t3 H2)))))) H9)) (\lambda (H9: (subst0 i0 v -u2 x)).(or4_intro1 (eq T (THead k u2 t3) (THead k x t2)) (ex2 T (\lambda (t: -T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x -t2) t))) (subst0 i0 v (THead k u2 t3) (THead k x t2)) (subst0 i0 v (THead k x -t2) (THead k u2 t3)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k u2 -t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x t2) t)) (THead k x t3) -(subst0_fst v x u2 i0 H9 t3 k) (subst0_snd k v t3 t2 i0 H2 x)))) (\lambda -(H9: (subst0 i0 v x u2)).(or4_intro3 (eq T (THead k u2 t3) (THead k x t2)) -(ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: -T).(subst0 i0 v (THead k x t2) t))) (subst0 i0 v (THead k u2 t3) (THead k x -t2)) (subst0 i0 v 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x t2)) (subst0 i0 v (THead k x t2) (THead k x t3)) (subst0_snd k v -t3 t2 i0 H2 x)) u2 H9)) (\lambda (H9: (ex2 T (\lambda (t: T).(subst0 i0 v u2 -t)) (\lambda (t: T).(subst0 i0 v x t)))).(ex2_ind T (\lambda (t: T).(subst0 -i0 v u2 t)) (\lambda (t: T).(subst0 i0 v x t)) (or4 (eq T (THead k u2 t3) -(THead k x t2)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) -(\lambda (t: T).(subst0 i0 v (THead k x t2) t))) (subst0 i0 v (THead k u2 t3) -(THead k x t2)) (subst0 i0 v (THead k x t2) (THead k u2 t3))) (\lambda (x0: -T).(\lambda (H10: (subst0 i0 v u2 x0)).(\lambda (H11: (subst0 i0 v x -x0)).(or4_intro1 (eq T (THead k u2 t3) (THead k x t2)) (ex2 T (\lambda (t: -T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x -t2) t))) (subst0 i0 v (THead k u2 t3) (THead k x t2)) (subst0 i0 v (THead k x -t2) (THead k u2 t3)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k u2 -t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x t2) t)) (THead k x0 t3) -(subst0_fst v x0 u2 i0 H10 t3 k) (subst0_both v x x0 i0 H11 k t2 t3 H2)))))) -H9)) (\lambda (H9: (subst0 i0 v u2 x)).(or4_intro1 (eq T (THead k u2 t3) -(THead k x t2)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) -(\lambda (t: T).(subst0 i0 v (THead k x t2) t))) (subst0 i0 v (THead k u2 t3) -(THead k x t2)) (subst0 i0 v (THead k x t2) (THead k u2 t3)) (ex_intro2 T -(\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v -(THead k x t2) t)) (THead k x t3) (subst0_fst v x u2 i0 H9 t3 k) (subst0_snd -k v t3 t2 i0 H2 x)))) (\lambda (H9: (subst0 i0 v x u2)).(or4_intro3 (eq T -(THead k u2 t3) (THead k x t2)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k -u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x t2) t))) (subst0 i0 v -(THead k u2 t3) (THead k x t2)) (subst0 i0 v (THead k x t2) (THead k u2 t3)) -(subst0_both v x u2 i0 H9 k t2 t3 H2))) (H1 x H7))) (H3 t3 H2)) t4 H6)))) -H5)) (\lambda (H5: (ex2 T (\lambda (t5: T).(eq T t4 (THead k u1 t5))) -(\lambda (t5: T).(subst0 (s k i0) v t2 t5)))).(ex2_ind T (\lambda (t5: T).(eq -T t4 (THead k u1 t5))) (\lambda (t5: T).(subst0 (s k i0) v t2 t5)) (or4 (eq T -(THead k u2 t3) t4) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) -(\lambda (t: T).(subst0 i0 v t4 t))) (subst0 i0 v (THead k u2 t3) t4) (subst0 -i0 v t4 (THead k u2 t3))) (\lambda (x: T).(\lambda (H6: (eq T t4 (THead k u1 -x))).(\lambda (H7: (subst0 (s k i0) v t2 x)).(eq_ind_r T (THead k u1 x) -(\lambda (t: T).(or4 (eq T (THead k u2 t3) t) (ex2 T (\lambda (t5: T).(subst0 -i0 v (THead k u2 t3) t5)) (\lambda (t5: T).(subst0 i0 v t t5))) (subst0 i0 v -(THead k u2 t3) t) (subst0 i0 v t (THead k u2 t3)))) (or4_ind (eq T t3 x) -(ex2 T (\lambda (t: T).(subst0 (s k i0) v t3 t)) (\lambda (t: T).(subst0 (s k -i0) v x t))) (subst0 (s k i0) v t3 x) (subst0 (s k i0) v x t3) (or4 (eq T -(THead k u2 t3) (THead k u1 x)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k -u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k u1 x) t))) (subst0 i0 v -(THead k u2 t3) (THead k u1 x)) 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(subst0_fst v u2 u1 i0 H0 x -k))) (\lambda (H9: (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: -T).(subst0 i0 v u2 t)))).(ex2_ind T (\lambda (t: T).(subst0 i0 v u2 t)) -(\lambda (t: T).(subst0 i0 v u2 t)) (or4 (eq T (THead k u2 x) (THead k u1 x)) -(ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 x) t)) (\lambda (t: -T).(subst0 i0 v (THead k u1 x) t))) (subst0 i0 v (THead k u2 x) (THead k u1 -x)) (subst0 i0 v (THead k u1 x) (THead k u2 x))) (\lambda (x0: T).(\lambda -(_: (subst0 i0 v u2 x0)).(\lambda (_: (subst0 i0 v u2 x0)).(or4_intro3 (eq T -(THead k u2 x) (THead k u1 x)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k -u2 x) t)) (\lambda (t: T).(subst0 i0 v (THead k u1 x) t))) (subst0 i0 v -(THead k u2 x) (THead k u1 x)) (subst0 i0 v (THead k u1 x) (THead k u2 x)) -(subst0_fst v u2 u1 i0 H0 x k))))) H9)) (\lambda (_: (subst0 i0 v u2 -u2)).(or4_intro3 (eq T (THead k u2 x) (THead k u1 x)) (ex2 T (\lambda (t: -T).(subst0 i0 v (THead k u2 x) t)) (\lambda (t: T).(subst0 i0 v (THead k 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T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0 i0 v u2 t)) (or4 -(eq T (THead k u2 t3) (THead k u1 x)) (ex2 T (\lambda (t: T).(subst0 i0 v -(THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k u1 x) t))) (subst0 -i0 v (THead k u2 t3) (THead k u1 x)) (subst0 i0 v (THead k u1 x) (THead k u2 -t3))) (\lambda (x1: T).(\lambda (_: (subst0 i0 v u2 x1)).(\lambda (_: (subst0 -i0 v u2 x1)).(or4_intro1 (eq T (THead k u2 t3) (THead k u1 x)) (ex2 T -(\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v -(THead k u1 x) t))) (subst0 i0 v (THead k u2 t3) (THead k u1 x)) (subst0 i0 v -(THead k u1 x) (THead k u2 t3)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v -(THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k u1 x) t)) (THead k -u2 x0) (subst0_snd k v x0 t3 i0 H9 u2) (subst0_both v u1 u2 i0 H0 k x x0 -H10)))))) H11)) (\lambda (_: (subst0 i0 v u2 u2)).(or4_intro1 (eq T (THead k -u2 t3) (THead k u1 x)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) -t)) (\lambda (t: T).(subst0 i0 v (THead k u1 x) t))) (subst0 i0 v (THead k u2 -t3) (THead k u1 x)) (subst0 i0 v (THead k u1 x) (THead k u2 t3)) (ex_intro2 T -(\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v -(THead k u1 x) t)) (THead k u2 x0) (subst0_snd k v x0 t3 i0 H9 u2) -(subst0_both v u1 u2 i0 H0 k x x0 H10)))) (\lambda (_: (subst0 i0 v u2 -u2)).(or4_intro1 (eq T (THead k u2 t3) (THead k u1 x)) (ex2 T (\lambda (t: -T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k u1 -x) t))) (subst0 i0 v (THead k u2 t3) (THead k u1 x)) (subst0 i0 v (THead k u1 -x) (THead k u2 t3)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) -t)) (\lambda (t: T).(subst0 i0 v (THead k u1 x) t)) (THead k u2 x0) -(subst0_snd k v x0 t3 i0 H9 u2) (subst0_both v u1 u2 i0 H0 k x x0 H10)))) (H1 -u2 H0))))) H8)) (\lambda (H8: (subst0 (s k i0) v t3 x)).(or4_ind (eq T u2 u2) -(ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0 i0 v u2 -t))) (subst0 i0 v u2 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(t: T).(subst0 i0 v (THead k u2 t3) -t)) (\lambda (t: T).(subst0 i0 v (THead k u1 x) t)) (THead k u2 x) -(subst0_snd k v x t3 i0 H8 u2) (subst0_fst v u2 u1 i0 H0 x k)))) (\lambda (_: -(subst0 i0 v u2 u2)).(or4_intro1 (eq T (THead k u2 t3) (THead k u1 x)) (ex2 T -(\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v -(THead k u1 x) t))) (subst0 i0 v (THead k u2 t3) (THead k u1 x)) (subst0 i0 v -(THead k u1 x) (THead k u2 t3)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v -(THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k u1 x) t)) (THead k -u2 x) (subst0_snd k v x t3 i0 H8 u2) (subst0_fst v u2 u1 i0 H0 x k)))) (H1 u2 -H0))) (\lambda (H8: (subst0 (s k i0) v x t3)).(or4_ind (eq T u2 u2) (ex2 T -(\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0 i0 v u2 t))) -(subst0 i0 v u2 u2) (subst0 i0 v u2 u2) (or4 (eq T (THead k u2 t3) (THead k -u1 x)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: -T).(subst0 i0 v (THead k u1 x) t))) (subst0 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T).(\lambda (_: T).(subst0 i0 v -u1 u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s k i0) v t2 -t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t5: T).(eq T t4 (THead k u3 -t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v u1 u3))) (\lambda (_: -T).(\lambda (t5: T).(subst0 (s k i0) v t2 t5))) (or4 (eq T (THead k u2 t3) -t4) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: -T).(subst0 i0 v t4 t))) (subst0 i0 v (THead k u2 t3) t4) (subst0 i0 v t4 -(THead k u2 t3))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H6: (eq T t4 -(THead k x0 x1))).(\lambda (H7: (subst0 i0 v u1 x0)).(\lambda (H8: (subst0 (s -k i0) v t2 x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t: T).(or4 (eq T (THead -k u2 t3) t) (ex2 T (\lambda (t5: T).(subst0 i0 v (THead k u2 t3) t5)) -(\lambda (t5: T).(subst0 i0 v t t5))) (subst0 i0 v (THead k u2 t3) t) (subst0 -i0 v t (THead k u2 t3)))) (or4_ind (eq T t3 x1) (ex2 T (\lambda (t: -T).(subst0 (s k i0) v t3 t)) (\lambda (t: T).(subst0 (s k i0) v x1 t))) -(subst0 (s k i0) v t3 x1) (subst0 (s k i0) v x1 t3) (or4 (eq T (THead k u2 -t3) (THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) -(\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 -t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t3))) (\lambda -(H9: (eq T t3 x1)).(eq_ind_r T x1 (\lambda (t: T).(or4 (eq T (THead k u2 t) -(THead k x0 x1)) (ex2 T (\lambda (t5: T).(subst0 i0 v (THead k u2 t) t5)) -(\lambda (t5: T).(subst0 i0 v (THead k x0 x1) t5))) (subst0 i0 v (THead k u2 -t) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t)))) (or4_ind -(eq T u2 x0) (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: -T).(subst0 i0 v x0 t))) (subst0 i0 v u2 x0) (subst0 i0 v x0 u2) (or4 (eq T -(THead k u2 x1) (THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k -u2 x1) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v -(THead k u2 x1) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 -x1))) (\lambda (H10: (eq T u2 x0)).(eq_ind_r T x0 (\lambda (t: T).(or4 (eq T -(THead k t x1) (THead k x0 x1)) (ex2 T (\lambda (t5: T).(subst0 i0 v (THead k -t x1) t5)) (\lambda (t5: T).(subst0 i0 v (THead k x0 x1) t5))) (subst0 i0 v -(THead k t x1) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k t -x1)))) (or4_intro0 (eq T (THead k x0 x1) (THead k x0 x1)) (ex2 T (\lambda (t: -T).(subst0 i0 v (THead k x0 x1) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 -x1) t))) (subst0 i0 v (THead k x0 x1) (THead k x0 x1)) (subst0 i0 v (THead k -x0 x1) (THead k x0 x1)) (refl_equal T (THead k x0 x1))) u2 H10)) (\lambda -(H10: (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0 i0 v -x0 t)))).(ex2_ind T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: -T).(subst0 i0 v x0 t)) (or4 (eq T (THead k u2 x1) (THead k x0 x1)) (ex2 T -(\lambda (t: T).(subst0 i0 v (THead k u2 x1) t)) (\lambda (t: T).(subst0 i0 v -(THead k x0 x1) t))) (subst0 i0 v (THead k u2 x1) (THead k x0 x1)) (subst0 i0 -v (THead k x0 x1) (THead k u2 x1))) (\lambda (x: T).(\lambda (H11: (subst0 i0 -v u2 x)).(\lambda (H12: (subst0 i0 v x0 x)).(or4_intro1 (eq T (THead k u2 x1) -(THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 x1) t)) -(\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 -x1) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 x1)) (ex_intro2 -T (\lambda (t: T).(subst0 i0 v (THead k u2 x1) t)) (\lambda (t: T).(subst0 i0 -v (THead k x0 x1) t)) (THead k x x1) (subst0_fst v x u2 i0 H11 x1 k) -(subst0_fst v x x0 i0 H12 x1 k)))))) H10)) (\lambda (H10: (subst0 i0 v u2 -x0)).(or4_intro2 (eq T (THead k u2 x1) (THead k x0 x1)) (ex2 T (\lambda (t: -T).(subst0 i0 v (THead k u2 x1) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 -x1) t))) (subst0 i0 v (THead k u2 x1) (THead k x0 x1)) (subst0 i0 v (THead k -x0 x1) (THead k u2 x1)) (subst0_fst v x0 u2 i0 H10 x1 k))) (\lambda (H10: -(subst0 i0 v x0 u2)).(or4_intro3 (eq T (THead k u2 x1) (THead k x0 x1)) (ex2 -T (\lambda (t: T).(subst0 i0 v (THead k u2 x1) t)) (\lambda (t: T).(subst0 i0 -v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 x1) (THead k x0 x1)) (subst0 -i0 v (THead k x0 x1) (THead k u2 x1)) (subst0_fst v u2 x0 i0 H10 x1 k))) (H1 -x0 H7)) t3 H9)) (\lambda (H9: (ex2 T (\lambda (t: T).(subst0 (s k i0) v t3 -t)) (\lambda (t: T).(subst0 (s k i0) v x1 t)))).(ex2_ind T (\lambda (t: -T).(subst0 (s k i0) v t3 t)) (\lambda (t: T).(subst0 (s k i0) v x1 t)) (or4 -(eq T (THead k u2 t3) (THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v -(THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 -i0 v (THead k u2 t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k -u2 t3))) (\lambda (x: T).(\lambda (H10: (subst0 (s k i0) v t3 x)).(\lambda -(H11: (subst0 (s k i0) v x1 x)).(or4_ind (eq T u2 x0) (ex2 T (\lambda (t: -T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0 i0 v x0 t))) (subst0 i0 v u2 -x0) (subst0 i0 v x0 u2) (or4 (eq T (THead k u2 t3) (THead k x0 x1)) (ex2 T -(\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v -(THead k x0 x1) t))) (subst0 i0 v (THead k u2 t3) (THead k x0 x1)) (subst0 i0 -v (THead k x0 x1) (THead k u2 t3))) (\lambda (H12: (eq T u2 x0)).(eq_ind_r T -x0 (\lambda (t: T).(or4 (eq T (THead k t t3) (THead k x0 x1)) (ex2 T (\lambda -(t5: T).(subst0 i0 v (THead k t t3) t5)) (\lambda (t5: T).(subst0 i0 v (THead -k x0 x1) t5))) (subst0 i0 v (THead k t t3) (THead k x0 x1)) (subst0 i0 v -(THead k x0 x1) (THead k t t3)))) (or4_intro1 (eq T (THead k x0 t3) (THead k -x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k x0 t3) t)) (\lambda (t: -T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k x0 t3) (THead k x0 -x1)) (subst0 i0 v (THead k x0 x1) (THead k x0 t3)) (ex_intro2 T (\lambda (t: -T).(subst0 i0 v (THead k x0 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 -x1) t)) (THead k x0 x) (subst0_snd k v x t3 i0 H10 x0) (subst0_snd k v x x1 -i0 H11 x0))) u2 H12)) (\lambda (H12: (ex2 T (\lambda (t: T).(subst0 i0 v u2 -t)) (\lambda (t: T).(subst0 i0 v x0 t)))).(ex2_ind T (\lambda (t: T).(subst0 -i0 v u2 t)) (\lambda (t: T).(subst0 i0 v x0 t)) (or4 (eq T (THead k u2 t3) -(THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) -(\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 -t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t3))) (\lambda -(x2: T).(\lambda (H13: (subst0 i0 v u2 x2)).(\lambda (H14: (subst0 i0 v x0 -x2)).(or4_intro1 (eq T (THead k u2 t3) (THead k x0 x1)) (ex2 T (\lambda (t: -T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 -x1) t))) (subst0 i0 v (THead k u2 t3) (THead k x0 x1)) (subst0 i0 v (THead k -x0 x1) (THead k u2 t3)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k u2 -t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t)) (THead k x2 x) -(subst0_both v u2 x2 i0 H13 k t3 x H10) (subst0_both v x0 x2 i0 H14 k x1 x -H11)))))) H12)) (\lambda (H12: (subst0 i0 v u2 x0)).(or4_intro1 (eq T (THead -k u2 t3) (THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) -t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k -u2 t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t3)) -(ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: -T).(subst0 i0 v (THead k x0 x1) t)) (THead k x0 x) (subst0_both v u2 x0 i0 -H12 k t3 x H10) (subst0_snd k v x x1 i0 H11 x0)))) (\lambda (H12: (subst0 i0 -v x0 u2)).(or4_intro1 (eq T (THead k u2 t3) (THead k x0 x1)) (ex2 T (\lambda -(t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k -x0 x1) t))) (subst0 i0 v (THead k u2 t3) (THead k x0 x1)) (subst0 i0 v (THead -k x0 x1) (THead k u2 t3)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k -u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t)) (THead k u2 x) -(subst0_snd k v x t3 i0 H10 u2) (subst0_both v x0 u2 i0 H12 k x1 x H11)))) -(H1 x0 H7))))) H9)) (\lambda (H9: (subst0 (s k i0) v t3 x1)).(or4_ind (eq T -u2 x0) (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0 i0 -v x0 t))) (subst0 i0 v u2 x0) (subst0 i0 v x0 u2) (or4 (eq T (THead k u2 t3) -(THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) -(\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 -t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t3))) (\lambda -(H10: (eq T u2 x0)).(eq_ind_r T x0 (\lambda (t: T).(or4 (eq T (THead k t t3) -(THead k x0 x1)) (ex2 T (\lambda (t5: T).(subst0 i0 v (THead k t t3) t5)) -(\lambda (t5: T).(subst0 i0 v (THead k x0 x1) t5))) (subst0 i0 v (THead k t -t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k t t3)))) -(or4_intro2 (eq T (THead k x0 t3) (THead k x0 x1)) (ex2 T (\lambda (t: -T).(subst0 i0 v (THead k x0 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 -x1) t))) (subst0 i0 v (THead k x0 t3) (THead k x0 x1)) (subst0 i0 v (THead k -x0 x1) (THead k x0 t3)) (subst0_snd k v x1 t3 i0 H9 x0)) u2 H10)) (\lambda -(H10: (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0 i0 v -x0 t)))).(ex2_ind T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: -T).(subst0 i0 v x0 t)) (or4 (eq T (THead k u2 t3) (THead k x0 x1)) (ex2 T -(\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v -(THead k x0 x1) t))) (subst0 i0 v (THead k u2 t3) (THead k x0 x1)) (subst0 i0 -v (THead k x0 x1) (THead k u2 t3))) (\lambda (x: T).(\lambda (H11: (subst0 i0 -v u2 x)).(\lambda (H12: (subst0 i0 v x0 x)).(or4_intro1 (eq T (THead k u2 t3) -(THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) -(\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 -t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t3)) (ex_intro2 -T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 -v (THead k x0 x1) t)) (THead k x x1) (subst0_both v u2 x i0 H11 k t3 x1 H9) -(subst0_fst v x x0 i0 H12 x1 k)))))) H10)) (\lambda (H10: (subst0 i0 v u2 -x0)).(or4_intro2 (eq T (THead k u2 t3) (THead k x0 x1)) (ex2 T (\lambda (t: -T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 -x1) t))) (subst0 i0 v (THead k u2 t3) (THead k x0 x1)) (subst0 i0 v (THead k -x0 x1) (THead k u2 t3)) (subst0_both v u2 x0 i0 H10 k t3 x1 H9))) (\lambda -(H10: (subst0 i0 v x0 u2)).(or4_intro1 (eq T (THead k u2 t3) (THead k x0 x1)) -(ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: -T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 t3) (THead k x0 -x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t3)) (ex_intro2 T (\lambda (t: -T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 -x1) t)) (THead k u2 x1) (subst0_snd k v x1 t3 i0 H9 u2) (subst0_fst v u2 x0 -i0 H10 x1 k)))) (H1 x0 H7))) (\lambda (H9: (subst0 (s k i0) v x1 -t3)).(or4_ind (eq T u2 x0) (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) -(\lambda (t: T).(subst0 i0 v x0 t))) (subst0 i0 v u2 x0) (subst0 i0 v x0 u2) -(or4 (eq T (THead k u2 t3) (THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 -v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) -(subst0 i0 v (THead k u2 t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) -(THead k u2 t3))) (\lambda (H10: (eq T u2 x0)).(eq_ind_r T x0 (\lambda (t: -T).(or4 (eq T (THead k t t3) (THead k x0 x1)) (ex2 T (\lambda (t5: T).(subst0 -i0 v (THead k t t3) t5)) (\lambda (t5: T).(subst0 i0 v (THead k x0 x1) t5))) -(subst0 i0 v (THead k t t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) -(THead k t t3)))) (or4_intro3 (eq T (THead k x0 t3) (THead k x0 x1)) (ex2 T -(\lambda (t: T).(subst0 i0 v (THead k x0 t3) t)) (\lambda (t: T).(subst0 i0 v -(THead k x0 x1) t))) (subst0 i0 v (THead k x0 t3) (THead k x0 x1)) (subst0 i0 -v (THead k x0 x1) (THead k x0 t3)) (subst0_snd k v t3 x1 i0 H9 x0)) u2 H10)) -(\lambda (H10: (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: -T).(subst0 i0 v x0 t)))).(ex2_ind T (\lambda (t: T).(subst0 i0 v u2 t)) -(\lambda (t: T).(subst0 i0 v x0 t)) (or4 (eq T (THead k u2 t3) (THead k x0 -x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: -T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 t3) (THead k x0 -x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t3))) (\lambda (x: T).(\lambda -(H11: (subst0 i0 v u2 x)).(\lambda (H12: (subst0 i0 v x0 x)).(or4_intro1 (eq -T (THead k u2 t3) (THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead -k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v -(THead k u2 t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 -t3)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda -(t: T).(subst0 i0 v (THead k x0 x1) t)) (THead k x t3) (subst0_fst v x u2 i0 -H11 t3 k) (subst0_both v x0 x i0 H12 k x1 t3 H9)))))) H10)) (\lambda (H10: -(subst0 i0 v u2 x0)).(or4_intro1 (eq T (THead k u2 t3) (THead k x0 x1)) (ex2 -T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 -v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 t3) (THead k x0 x1)) (subst0 -i0 v (THead k x0 x1) (THead k u2 t3)) (ex_intro2 T (\lambda (t: T).(subst0 i0 -v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t)) (THead -k x0 t3) (subst0_fst v x0 u2 i0 H10 t3 k) (subst0_snd k v t3 x1 i0 H9 x0)))) -(\lambda (H10: (subst0 i0 v x0 u2)).(or4_intro3 (eq T (THead k u2 t3) (THead -k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda -(t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 t3) (THead -k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t3)) (subst0_both v x0 u2 -i0 H10 k x1 t3 H9))) (H1 x0 H7))) (H3 x1 H8)) t4 H6)))))) H5)) -(subst0_gen_head k v u1 t2 t4 i0 H4))))))))))))))) i u t0 t1 H))))). -(* COMMENTS -Initial nodes: 25595 -END *) - -theorem subst0_confluence_lift: - \forall (t0: T).(\forall (t1: T).(\forall (u: T).(\forall (i: nat).((subst0 -i u t0 (lift (S O) i t1)) \to (\forall (t2: T).((subst0 i u t0 (lift (S O) i -t2)) \to (eq T t1 t2))))))) -\def - \lambda (t0: T).(\lambda (t1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H: (subst0 i u t0 (lift (S O) i t1))).(\lambda (t2: T).(\lambda (H0: (subst0 -i u t0 (lift (S O) i t2))).(or4_ind (eq T (lift (S O) i t2) (lift (S O) i -t1)) (ex2 T (\lambda (t: T).(subst0 i u (lift (S O) i t2) t)) (\lambda (t: -T).(subst0 i u (lift (S O) i t1) t))) (subst0 i u (lift (S O) i t2) (lift (S -O) i t1)) (subst0 i u (lift (S O) i t1) (lift (S O) i t2)) (eq T t1 t2) -(\lambda (H1: (eq T (lift (S O) i t2) (lift (S O) i t1))).(let H2 \def -(sym_eq T (lift (S O) i t2) (lift (S O) i t1) H1) in (lift_inj t1 t2 (S O) i -H2))) (\lambda (H1: (ex2 T (\lambda (t: T).(subst0 i u (lift (S O) i t2) t)) -(\lambda (t: T).(subst0 i u (lift (S O) i t1) t)))).(ex2_ind T (\lambda (t: -T).(subst0 i u (lift (S O) i t2) t)) (\lambda (t: T).(subst0 i u (lift (S O) -i t1) t)) (eq T t1 t2) (\lambda (x: T).(\lambda (_: (subst0 i u (lift (S O) i -t2) x)).(\lambda (H3: (subst0 i u (lift (S O) i t1) -x)).(subst0_gen_lift_false t1 u x (S O) i i (le_n i) (eq_ind_r nat (plus (S -O) i) (\lambda (n: nat).(lt i n)) (le_n (plus (S O) i)) (plus i (S O)) -(plus_sym i (S O))) H3 (eq T t1 t2))))) H1)) (\lambda (H1: (subst0 i u (lift -(S O) i t2) (lift (S O) i t1))).(subst0_gen_lift_false t2 u (lift (S O) i t1) -(S O) i i (le_n i) (eq_ind_r nat (plus (S O) i) (\lambda (n: nat).(lt i n)) -(le_n (plus (S O) i)) (plus i (S O)) (plus_sym i (S O))) H1 (eq T t1 t2))) -(\lambda (H1: (subst0 i u (lift (S O) i t1) (lift (S O) i -t2))).(subst0_gen_lift_false t1 u (lift (S O) i t2) (S O) i i (le_n i) -(eq_ind_r nat (plus (S O) i) (\lambda (n: nat).(lt i n)) (le_n (plus (S O) -i)) (plus i (S O)) (plus_sym i (S O))) H1 (eq T t1 t2))) -(subst0_confluence_eq t0 (lift (S O) i t2) u i H0 (lift (S O) i t1) H)))))))). -(* COMMENTS -Initial nodes: 703 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/subst0/tlt.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/subst0/tlt.ma deleted file mode 100644 index c8e8420bb..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/subst0/tlt.ma +++ /dev/null @@ -1,468 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/subst0/defs.ma". - -include "Basic-1/lift/props.ma". - -include "Basic-1/lift/tlt.ma". - -theorem subst0_weight_le: - \forall (u: T).(\forall (t: T).(\forall (z: T).(\forall (d: nat).((subst0 d -u t z) \to (\forall (f: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S -d) O u)) (g d)) \to (le (weight_map f z) (weight_map g t)))))))))) -\def - \lambda (u: T).(\lambda (t: T).(\lambda (z: T).(\lambda (d: nat).(\lambda -(H: (subst0 d u t z)).(subst0_ind (\lambda (n: nat).(\lambda (t0: T).(\lambda -(t1: T).(\lambda (t2: T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S -n) O t0)) (g n)) \to (le (weight_map f t2) (weight_map g t1)))))))))) -(\lambda (v: T).(\lambda (i: nat).(\lambda (f: ((nat \to nat))).(\lambda (g: -((nat \to nat))).(\lambda (_: ((\forall (m: nat).(le (f m) (g m))))).(\lambda -(H1: (lt (weight_map f (lift (S i) O v)) (g i))).(le_S_n (weight_map f (lift -(S i) O v)) (weight_map g (TLRef i)) (le_S (S (weight_map f (lift (S i) O -v))) (weight_map g (TLRef i)) H1)))))))) (\lambda (v: T).(\lambda (u2: -T).(\lambda (u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1 -u2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S -i) O v)) (g i)) \to (le (weight_map f u2) (weight_map g u1)))))))).(\lambda -(t0: T).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (f: ((nat \to -nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) -\to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (le (weight_map f (THead -k0 u2 t0)) (weight_map g (THead k0 u1 t0)))))))) (\lambda (b: B).(B_ind -(\lambda (b0: B).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S -i) O v)) (g i)) \to (le (weight_map f (THead (Bind b0) u2 t0)) (weight_map g -(THead (Bind b0) u1 t0)))))))) (\lambda (f: ((nat \to nat))).(\lambda (g: -((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g -m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g i))).(le_n_S -(plus (weight_map f u2) (weight_map (wadd f (S (weight_map f u2))) t0)) (plus -(weight_map g u1) (weight_map (wadd g (S (weight_map g u1))) t0)) -(le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f (S -(weight_map f u2))) t0) (weight_map (wadd g (S (weight_map g u1))) t0) (H1 f -g H2 H3) (weight_le t0 (wadd f (S (weight_map f u2))) (wadd g (S (weight_map -g u1))) (\lambda (n: nat).(wadd_le f g H2 (S (weight_map f u2)) (S -(weight_map g u1)) (le_n_S (weight_map f u2) (weight_map g u1) (H1 f g H2 -H3)) n))))))))) (\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to -nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3: (lt -(weight_map f (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map f u2) -(weight_map (wadd f O) t0)) (plus (weight_map g u1) (weight_map (wadd g O) -t0)) (le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f O) -t0) (weight_map (wadd g O) t0) (H1 f g H2 H3) (weight_le t0 (wadd f O) (wadd -g O) (\lambda (n: nat).(wadd_le f g H2 O O (le_n O) n))))))))) (\lambda (f: -((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: -nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g -i))).(le_n_S (plus (weight_map f u2) (weight_map (wadd f O) t0)) (plus -(weight_map g u1) (weight_map (wadd g O) t0)) (le_plus_plus (weight_map f u2) -(weight_map g u1) (weight_map (wadd f O) t0) (weight_map (wadd g O) t0) (H1 f -g H2 H3) (weight_le t0 (wadd f O) (wadd g O) (\lambda (n: nat).(wadd_le f g -H2 O O (le_n O) n))))))))) b)) (\lambda (_: F).(\lambda (f0: ((nat \to -nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f0 -m) (g m))))).(\lambda (H3: (lt (weight_map f0 (lift (S i) O v)) (g -i))).(le_n_S (plus (weight_map f0 u2) (weight_map f0 t0)) (plus (weight_map g -u1) (weight_map g t0)) (le_plus_plus (weight_map f0 u2) (weight_map g u1) -(weight_map f0 t0) (weight_map g t0) (H1 f0 g H2 H3) (weight_le t0 f0 g -H2)))))))) k))))))))) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (v: -T).(\forall (t2: T).(\forall (t1: T).(\forall (i: nat).((subst0 (s k0 i) v t1 -t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S -(s k0 i)) O v)) (g (s k0 i))) \to (le (weight_map f t2) (weight_map g -t1))))))) \to (\forall (u0: T).(\forall (f: ((nat \to nat))).(\forall (g: -((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map -f (lift (S i) O v)) (g i)) \to (le (weight_map f (THead k0 u0 t2)) -(weight_map g (THead k0 u0 t1))))))))))))))) (\lambda (b: B).(B_ind (\lambda -(b0: B).(\forall (v: T).(\forall (t2: T).(\forall (t1: T).(\forall (i: -nat).((subst0 (s (Bind b0) i) v t1 t2) \to (((\forall (f: ((nat \to -nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) -\to ((lt (weight_map f (lift (S (s (Bind b0) i)) O v)) (g (s (Bind b0) i))) -\to (le (weight_map f t2) (weight_map g t1))))))) \to (\forall (u0: -T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: -nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to -(le (weight_map f (THead (Bind b0) u0 t2)) (weight_map g (THead (Bind b0) u0 -t1))))))))))))))) (\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda -(i: nat).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H1: ((\forall (f: -((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) -(g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (le -(weight_map f t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f: -((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: -nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g -i))).(le_n_S (plus (weight_map f u0) (weight_map (wadd f (S (weight_map f -u0))) t2)) (plus (weight_map g u0) (weight_map (wadd g (S (weight_map g u0))) -t1)) (le_plus_plus (weight_map f u0) (weight_map g u0) (weight_map (wadd f (S -(weight_map f u0))) t2) (weight_map (wadd g (S (weight_map g u0))) t1) -(weight_le u0 f g H2) (H1 (wadd f (S (weight_map f u0))) (wadd g (S -(weight_map g u0))) (\lambda (m: nat).(wadd_le f g H2 (S (weight_map f u0)) -(S (weight_map g u0)) (le_n_S (weight_map f u0) (weight_map g u0) (weight_le -u0 f g H2)) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda (n: -nat).(lt n (g i))) H3 (weight_map (wadd f (S (weight_map f u0))) (lift (S (S -i)) O v)) (lift_weight_add_O (S (weight_map f u0)) v (S i) f)))))))))))))))) -(\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda (i: nat).(\lambda -(_: (subst0 (S i) v t1 t2)).(\lambda (H1: ((\forall (f: ((nat \to -nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) -\to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (le (weight_map f -t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f: ((nat \to -nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f -m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g -i))).(le_n_S (plus (weight_map f u0) (weight_map (wadd f O) t2)) (plus -(weight_map g u0) (weight_map (wadd g O) t1)) (le_plus_plus (weight_map f u0) -(weight_map g u0) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1) -(weight_le u0 f g H2) (H1 (wadd f O) (wadd g O) (\lambda (m: nat).(wadd_le f -g H2 O O (le_n O) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda -(n: nat).(lt n (g i))) H3 (weight_map (wadd f O) (lift (S (S i)) O v)) -(lift_weight_add_O O v (S i) f)))))))))))))))) (\lambda (v: T).(\lambda (t2: -T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (_: (subst0 (S i) v t1 -t2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S -(S i)) O v)) (g (S i))) \to (le (weight_map f t2) (weight_map g -t1)))))))).(\lambda (u0: T).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat -\to nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3: -(lt (weight_map f (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map f u0) -(weight_map (wadd f O) t2)) (plus (weight_map g u0) (weight_map (wadd g O) -t1)) (le_plus_plus (weight_map f u0) (weight_map g u0) (weight_map (wadd f O) -t2) (weight_map (wadd g O) t1) (weight_le u0 f g H2) (H1 (wadd f O) (wadd g -O) (\lambda (m: nat).(wadd_le f g H2 O O (le_n O) m)) (eq_ind nat (weight_map -f (lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H3 (weight_map (wadd f O) -(lift (S (S i)) O v)) (lift_weight_add_O O v (S i) f)))))))))))))))) b)) -(\lambda (_: F).(\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda -(i: nat).(\lambda (_: (subst0 i v t1 t2)).(\lambda (H1: ((\forall (f0: ((nat -\to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f0 m) (g -m)))) \to ((lt (weight_map f0 (lift (S i) O v)) (g i)) \to (le (weight_map f0 -t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f0: ((nat \to -nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f0 -m) (g m))))).(\lambda (H3: (lt (weight_map f0 (lift (S i) O v)) (g -i))).(le_n_S (plus (weight_map f0 u0) (weight_map f0 t2)) (plus (weight_map g -u0) (weight_map g t1)) (le_plus_plus (weight_map f0 u0) (weight_map g u0) -(weight_map f0 t2) (weight_map g t1) (weight_le u0 f0 g H2) (H1 f0 g H2 -H3))))))))))))))) k)) (\lambda (v: T).(\lambda (u1: T).(\lambda (u2: -T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1 u2)).(\lambda (H1: ((\forall -(f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f -m) (g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (le -(weight_map f u2) (weight_map g u1)))))))).(\lambda (k: K).(K_ind (\lambda -(k0: K).(\forall (t1: T).(\forall (t2: T).((subst0 (s k0 i) v t1 t2) \to -(((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: -nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S (s k0 i)) O v)) (g (s -k0 i))) \to (le (weight_map f t2) (weight_map g t1))))))) \to (\forall (f: -((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) -(g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (le (weight_map -f (THead k0 u2 t2)) (weight_map g (THead k0 u1 t1)))))))))))) (\lambda (b: -B).(B_ind (\lambda (b0: B).(\forall (t1: T).(\forall (t2: T).((subst0 (s -(Bind b0) i) v t1 t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat -\to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f -(lift (S (s (Bind b0) i)) O v)) (g (s (Bind b0) i))) \to (le (weight_map f -t2) (weight_map g t1))))))) \to (\forall (f: ((nat \to nat))).(\forall (g: -((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map -f (lift (S i) O v)) (g i)) \to (le (weight_map f (THead (Bind b0) u2 t2)) -(weight_map g (THead (Bind b0) u1 t1)))))))))))) (\lambda (t1: T).(\lambda -(t2: T).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H3: ((\forall (f: -((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) -(g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (le -(weight_map f t2) (weight_map g t1)))))))).(\lambda (f: ((nat \to -nat))).(\lambda (g: ((nat \to nat))).(\lambda (H4: ((\forall (m: nat).(le (f -m) (g m))))).(\lambda (H5: (lt (weight_map f (lift (S i) O v)) (g -i))).(le_n_S (plus (weight_map f u2) (weight_map (wadd f (S (weight_map f -u2))) t2)) (plus (weight_map g u1) (weight_map (wadd g (S (weight_map g u1))) -t1)) (le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f (S -(weight_map f u2))) t2) (weight_map (wadd g (S (weight_map g u1))) t1) (H1 f -g H4 H5) (H3 (wadd f (S (weight_map f u2))) (wadd g (S (weight_map g u1))) -(\lambda (m: nat).(wadd_le f g H4 (S (weight_map f u2)) (S (weight_map g u1)) -(le_n_S (weight_map f u2) (weight_map g u1) (H1 f g H4 H5)) m)) (eq_ind nat -(weight_map f (lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H5 -(weight_map (wadd f (S (weight_map f u2))) (lift (S (S i)) O v)) -(lift_weight_add_O (S (weight_map f u2)) v (S i) f))))))))))))) (\lambda (t1: -T).(\lambda (t2: T).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H3: -((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: -nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S -i))) \to (le (weight_map f t2) (weight_map g t1)))))))).(\lambda (f: ((nat -\to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H4: ((\forall (m: nat).(le -(f m) (g m))))).(\lambda (H5: (lt (weight_map f (lift (S i) O v)) (g -i))).(le_n_S (plus (weight_map f u2) (weight_map (wadd f O) t2)) (plus -(weight_map g u1) (weight_map (wadd g O) t1)) (le_plus_plus (weight_map f u2) -(weight_map g u1) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1) (H1 f -g H4 H5) (H3 (wadd f O) (wadd g O) (\lambda (m: nat).(wadd_le f g H4 O O -(le_n O) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda (n: -nat).(lt n (g i))) H5 (weight_map (wadd f O) (lift (S (S i)) O v)) -(lift_weight_add_O O v (S i) f))))))))))))) (\lambda (t1: T).(\lambda (t2: -T).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H3: ((\forall (f: ((nat \to -nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) -\to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (le (weight_map f -t2) (weight_map g t1)))))))).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat -\to nat))).(\lambda (H4: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H5: -(lt (weight_map f (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map f u2) -(weight_map (wadd f O) t2)) (plus (weight_map g u1) (weight_map (wadd g O) -t1)) (le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f O) -t2) (weight_map (wadd g O) t1) (H1 f g H4 H5) (H3 (wadd f O) (wadd g O) -(\lambda (m: nat).(wadd_le f g H4 O O (le_n O) m)) (eq_ind nat (weight_map f -(lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H5 (weight_map (wadd f O) -(lift (S (S i)) O v)) (lift_weight_add_O O v (S i) f))))))))))))) b)) -(\lambda (_: F).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (subst0 i v t1 -t2)).(\lambda (H3: ((\forall (f0: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (m: nat).(le (f0 m) (g m)))) \to ((lt (weight_map f0 (lift -(S i) O v)) (g i)) \to (le (weight_map f0 t2) (weight_map g -t1)))))))).(\lambda (f0: ((nat \to nat))).(\lambda (g: ((nat \to -nat))).(\lambda (H4: ((\forall (m: nat).(le (f0 m) (g m))))).(\lambda (H5: -(lt (weight_map f0 (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map f0 u2) -(weight_map f0 t2)) (plus (weight_map g u1) (weight_map g t1)) (le_plus_plus -(weight_map f0 u2) (weight_map g u1) (weight_map f0 t2) (weight_map g t1) (H1 -f0 g H4 H5) (H3 f0 g H4 H5)))))))))))) k)))))))) d u t z H))))). -(* COMMENTS -Initial nodes: 4101 -END *) - -theorem subst0_weight_lt: - \forall (u: T).(\forall (t: T).(\forall (z: T).(\forall (d: nat).((subst0 d -u t z) \to (\forall (f: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S -d) O u)) (g d)) \to (lt (weight_map f z) (weight_map g t)))))))))) -\def - \lambda (u: T).(\lambda (t: T).(\lambda (z: T).(\lambda (d: nat).(\lambda -(H: (subst0 d u t z)).(subst0_ind (\lambda (n: nat).(\lambda (t0: T).(\lambda -(t1: T).(\lambda (t2: T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S -n) O t0)) (g n)) \to (lt (weight_map f t2) (weight_map g t1)))))))))) -(\lambda (v: T).(\lambda (i: nat).(\lambda (f: ((nat \to nat))).(\lambda (g: -((nat \to nat))).(\lambda (_: ((\forall (m: nat).(le (f m) (g m))))).(\lambda -(H1: (lt (weight_map f (lift (S i) O v)) (g i))).H1)))))) (\lambda (v: -T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i -v u1 u2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S -i) O v)) (g i)) \to (lt (weight_map f u2) (weight_map g u1)))))))).(\lambda -(t0: T).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (f: ((nat \to -nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) -\to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (lt (weight_map f (THead -k0 u2 t0)) (weight_map g (THead k0 u1 t0)))))))) (\lambda (b: B).(B_ind -(\lambda (b0: B).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S -i) O v)) (g i)) \to (lt (weight_map f (THead (Bind b0) u2 t0)) (weight_map g -(THead (Bind b0) u1 t0)))))))) (\lambda (f: ((nat \to nat))).(\lambda (g: -((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g -m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g i))).(lt_n_S -(plus (weight_map f u2) (weight_map (wadd f (S (weight_map f u2))) t0)) (plus -(weight_map g u1) (weight_map (wadd g (S (weight_map g u1))) t0)) -(lt_le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f (S -(weight_map f u2))) t0) (weight_map (wadd g (S (weight_map g u1))) t0) (H1 f -g H2 H3) (weight_le t0 (wadd f (S (weight_map f u2))) (wadd g (S (weight_map -g u1))) (\lambda (n: nat).(wadd_lt f g H2 (S (weight_map f u2)) (S -(weight_map g u1)) (lt_n_S (weight_map f u2) (weight_map g u1) (H1 f g H2 -H3)) n))))))))) (\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to -nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3: (lt -(weight_map f (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map f u2) -(weight_map (wadd f O) t0)) (plus (weight_map g u1) (weight_map (wadd g O) -t0)) (lt_le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f -O) t0) (weight_map (wadd g O) t0) (H1 f g H2 H3) (weight_le t0 (wadd f O) -(wadd g O) (\lambda (n: nat).(le_S_n (wadd f O n) (wadd g O n) (le_n_S (wadd -f O n) (wadd g O n) (wadd_le f g H2 O O (le_n O) n))))))))))) (\lambda (f: -((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: -nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g -i))).(lt_n_S (plus (weight_map f u2) (weight_map (wadd f O) t0)) (plus -(weight_map g u1) (weight_map (wadd g O) t0)) (lt_le_plus_plus (weight_map f -u2) (weight_map g u1) (weight_map (wadd f O) t0) (weight_map (wadd g O) t0) -(H1 f g H2 H3) (weight_le t0 (wadd f O) (wadd g O) (\lambda (n: nat).(le_S_n -(wadd f O n) (wadd g O n) (le_n_S (wadd f O n) (wadd g O n) (wadd_le f g H2 O -O (le_n O) n))))))))))) b)) (\lambda (_: F).(\lambda (f0: ((nat \to -nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f0 -m) (g m))))).(\lambda (H3: (lt (weight_map f0 (lift (S i) O v)) (g -i))).(lt_n_S (plus (weight_map f0 u2) (weight_map f0 t0)) (plus (weight_map g -u1) (weight_map g t0)) (lt_le_plus_plus (weight_map f0 u2) (weight_map g u1) -(weight_map f0 t0) (weight_map g t0) (H1 f0 g H2 H3) (weight_le t0 f0 g -H2)))))))) k))))))))) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (v: -T).(\forall (t2: T).(\forall (t1: T).(\forall (i: nat).((subst0 (s k0 i) v t1 -t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S -(s k0 i)) O v)) (g (s k0 i))) \to (lt (weight_map f t2) (weight_map g -t1))))))) \to (\forall (u0: T).(\forall (f: ((nat \to nat))).(\forall (g: -((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map -f (lift (S i) O v)) (g i)) \to (lt (weight_map f (THead k0 u0 t2)) -(weight_map g (THead k0 u0 t1))))))))))))))) (\lambda (b: B).(B_ind (\lambda -(b0: B).(\forall (v: T).(\forall (t2: T).(\forall (t1: T).(\forall (i: -nat).((subst0 (s (Bind b0) i) v t1 t2) \to (((\forall (f: ((nat \to -nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) -\to ((lt (weight_map f (lift (S (s (Bind b0) i)) O v)) (g (s (Bind b0) i))) -\to (lt (weight_map f t2) (weight_map g t1))))))) \to (\forall (u0: -T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: -nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to -(lt (weight_map f (THead (Bind b0) u0 t2)) (weight_map g (THead (Bind b0) u0 -t1))))))))))))))) (\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda -(i: nat).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H1: ((\forall (f: -((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) -(g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (lt -(weight_map f t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f: -((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: -nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g -i))).(lt_n_S (plus (weight_map f u0) (weight_map (wadd f (S (weight_map f -u0))) t2)) (plus (weight_map g u0) (weight_map (wadd g (S (weight_map g u0))) -t1)) (le_lt_plus_plus (weight_map f u0) (weight_map g u0) (weight_map (wadd f -(S (weight_map f u0))) t2) (weight_map (wadd g (S (weight_map g u0))) t1) -(weight_le u0 f g H2) (H1 (wadd f (S (weight_map f u0))) (wadd g (S -(weight_map g u0))) (\lambda (m: nat).(wadd_le f g H2 (S (weight_map f u0)) -(S (weight_map g u0)) (le_n_S (weight_map f u0) (weight_map g u0) (weight_le -u0 f g H2)) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda (n: -nat).(lt n (g i))) H3 (weight_map (wadd f (S (weight_map f u0))) (lift (S (S -i)) O v)) (lift_weight_add_O (S (weight_map f u0)) v (S i) f)))))))))))))))) -(\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda (i: nat).(\lambda -(_: (subst0 (S i) v t1 t2)).(\lambda (H1: ((\forall (f: ((nat \to -nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) -\to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (lt (weight_map f -t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f: ((nat \to -nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f -m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g -i))).(lt_n_S (plus (weight_map f u0) (weight_map (wadd f O) t2)) (plus -(weight_map g u0) (weight_map (wadd g O) t1)) (le_lt_plus_plus (weight_map f -u0) (weight_map g u0) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1) -(weight_le u0 f g H2) (H1 (wadd f O) (wadd g O) (\lambda (m: nat).(wadd_le f -g H2 O O (le_n O) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda -(n: nat).(lt n (g i))) H3 (weight_map (wadd f O) (lift (S (S i)) O v)) -(lift_weight_add_O O v (S i) f)))))))))))))))) (\lambda (v: T).(\lambda (t2: -T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (_: (subst0 (S i) v t1 -t2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S -(S i)) O v)) (g (S i))) \to (lt (weight_map f t2) (weight_map g -t1)))))))).(\lambda (u0: T).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat -\to nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3: -(lt (weight_map f (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map f u0) -(weight_map (wadd f O) t2)) (plus (weight_map g u0) (weight_map (wadd g O) -t1)) (le_lt_plus_plus (weight_map f u0) (weight_map g u0) (weight_map (wadd f -O) t2) (weight_map (wadd g O) t1) (weight_le u0 f g H2) (H1 (wadd f O) (wadd -g O) (\lambda (m: nat).(wadd_le f g H2 O O (le_n O) m)) (eq_ind nat -(weight_map f (lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H3 -(weight_map (wadd f O) (lift (S (S i)) O v)) (lift_weight_add_O O v (S i) -f)))))))))))))))) b)) (\lambda (_: F).(\lambda (v: T).(\lambda (t2: -T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v t1 -t2)).(\lambda (H1: ((\forall (f0: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (m: nat).(le (f0 m) (g m)))) \to ((lt (weight_map f0 (lift -(S i) O v)) (g i)) \to (lt (weight_map f0 t2) (weight_map g -t1)))))))).(\lambda (u0: T).(\lambda (f0: ((nat \to nat))).(\lambda (g: ((nat -\to nat))).(\lambda (H2: ((\forall (m: nat).(le (f0 m) (g m))))).(\lambda -(H3: (lt (weight_map f0 (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map -f0 u0) (weight_map f0 t2)) (plus (weight_map g u0) (weight_map g t1)) -(le_lt_plus_plus (weight_map f0 u0) (weight_map g u0) (weight_map f0 t2) -(weight_map g t1) (weight_le u0 f0 g H2) (H1 f0 g H2 H3))))))))))))))) k)) -(\lambda (v: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i: nat).(\lambda -(_: (subst0 i v u1 u2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall -(g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt -(weight_map f (lift (S i) O v)) (g i)) \to (lt (weight_map f u2) (weight_map -g u1)))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t1: -T).(\forall (t2: T).((subst0 (s k0 i) v t1 t2) \to (((\forall (f: ((nat \to -nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) -\to ((lt (weight_map f (lift (S (s k0 i)) O v)) (g (s k0 i))) \to (lt -(weight_map f t2) (weight_map g t1))))))) \to (\forall (f: ((nat \to -nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) -\to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (lt (weight_map f (THead -k0 u2 t2)) (weight_map g (THead k0 u1 t1)))))))))))) (\lambda (b: B).(B_ind -(\lambda (b0: B).(\forall (t1: T).(\forall (t2: T).((subst0 (s (Bind b0) i) v -t1 t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S -(s (Bind b0) i)) O v)) (g (s (Bind b0) i))) \to (lt (weight_map f t2) -(weight_map g t1))))))) \to (\forall (f: ((nat \to nat))).(\forall (g: ((nat -\to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f -(lift (S i) O v)) (g i)) \to (lt (weight_map f (THead (Bind b0) u2 t2)) -(weight_map g (THead (Bind b0) u1 t1)))))))))))) (\lambda (t1: T).(\lambda -(t2: T).(\lambda (H2: (subst0 (S i) v t1 t2)).(\lambda (_: ((\forall (f: -((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) -(g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (lt -(weight_map f t2) (weight_map g t1)))))))).(\lambda (f: ((nat \to -nat))).(\lambda (g: ((nat \to nat))).(\lambda (H4: ((\forall (m: nat).(le (f -m) (g m))))).(\lambda (H5: (lt (weight_map f (lift (S i) O v)) (g -i))).(lt_n_S (plus (weight_map f u2) (weight_map (wadd f (S (weight_map f -u2))) t2)) (plus (weight_map g u1) (weight_map (wadd g (S (weight_map g u1))) -t1)) (lt_le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f -(S (weight_map f u2))) t2) (weight_map (wadd g (S (weight_map g u1))) t1) (H1 -f g H4 H5) (subst0_weight_le v t1 t2 (S i) H2 (wadd f (S (weight_map f u2))) -(wadd g (S (weight_map g u1))) (\lambda (m: nat).(wadd_lt f g H4 (S -(weight_map f u2)) (S (weight_map g u1)) (lt_n_S (weight_map f u2) -(weight_map g u1) (H1 f g H4 H5)) m)) (eq_ind nat (weight_map f (lift (S i) O -v)) (\lambda (n: nat).(lt n (g i))) H5 (weight_map (wadd f (S (weight_map f -u2))) (lift (S (S i)) O v)) (lift_weight_add_O (S (weight_map f u2)) v (S i) -f))))))))))))) (\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (subst0 (S i) v -t1 t2)).(\lambda (H3: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S -(S i)) O v)) (g (S i))) \to (lt (weight_map f t2) (weight_map g -t1)))))))).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to -nat))).(\lambda (H4: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H5: (lt -(weight_map f (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map f u2) -(weight_map (wadd f O) t2)) (plus (weight_map g u1) (weight_map (wadd g O) -t1)) (lt_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f O) -t2) (weight_map (wadd g O) t1) (H1 f g H4 H5) (H3 (wadd f O) (wadd g O) -(\lambda (m: nat).(le_S_n (wadd f O m) (wadd g O m) (le_n_S (wadd f O m) -(wadd g O m) (wadd_le f g H4 O O (le_n O) m)))) (eq_ind nat (weight_map f -(lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H5 (weight_map (wadd f O) -(lift (S (S i)) O v)) (lift_weight_add_O O v (S i) f))))))))))))) (\lambda -(t1: T).(\lambda (t2: T).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H3: -((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: -nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S -i))) \to (lt (weight_map f t2) (weight_map g t1)))))))).(\lambda (f: ((nat -\to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H4: ((\forall (m: nat).(le -(f m) (g m))))).(\lambda (H5: (lt (weight_map f (lift (S i) O v)) (g -i))).(lt_n_S (plus (weight_map f u2) (weight_map (wadd f O) t2)) (plus -(weight_map g u1) (weight_map (wadd g O) t1)) (lt_plus_plus (weight_map f u2) -(weight_map g u1) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1) (H1 f -g H4 H5) (H3 (wadd f O) (wadd g O) (\lambda (m: nat).(le_S_n (wadd f O m) -(wadd g O m) (le_n_S (wadd f O m) (wadd g O m) (wadd_le f g H4 O O (le_n O) -m)))) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda (n: nat).(lt n (g -i))) H5 (weight_map (wadd f O) (lift (S (S i)) O v)) (lift_weight_add_O O v -(S i) f))))))))))))) b)) (\lambda (_: F).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (_: (subst0 i v t1 t2)).(\lambda (H3: ((\forall (f0: ((nat \to -nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f0 m) (g m)))) -\to ((lt (weight_map f0 (lift (S i) O v)) (g i)) \to (lt (weight_map f0 t2) -(weight_map g t1)))))))).(\lambda (f0: ((nat \to nat))).(\lambda (g: ((nat -\to nat))).(\lambda (H4: ((\forall (m: nat).(le (f0 m) (g m))))).(\lambda -(H5: (lt (weight_map f0 (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map -f0 u2) (weight_map f0 t2)) (plus (weight_map g u1) (weight_map g t1)) -(lt_plus_plus (weight_map f0 u2) (weight_map g u1) (weight_map f0 t2) -(weight_map g t1) (H1 f0 g H4 H5) (H3 f0 g H4 H5)))))))))))) k)))))))) d u t -z H))))). -(* COMMENTS -Initial nodes: 4207 -END *) - -theorem subst0_tlt_head: - \forall (u: T).(\forall (t: T).(\forall (z: T).((subst0 O u t z) \to (tlt -(THead (Bind Abbr) u z) (THead (Bind Abbr) u t))))) -\def - \lambda (u: T).(\lambda (t: T).(\lambda (z: T).(\lambda (H: (subst0 O u t -z)).(lt_n_S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd -(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) z)) (plus -(weight_map (\lambda (_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) (S -(weight_map (\lambda (_: nat).O) u))) t)) (le_lt_plus_plus (weight_map -(\lambda (_: nat).O) u) (weight_map (\lambda (_: nat).O) u) (weight_map (wadd -(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) z) (weight_map -(wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t) (le_n -(weight_map (\lambda (_: nat).O) u)) (subst0_weight_lt u t z O H (wadd -(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) (wadd (\lambda -(_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) (\lambda (m: nat).(le_n -(wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u)) m))) -(eq_ind nat (weight_map (\lambda (_: nat).O) (lift O O u)) (\lambda (n: -nat).(lt n (S (weight_map (\lambda (_: nat).O) u)))) (eq_ind_r T u (\lambda -(t0: T).(lt (weight_map (\lambda (_: nat).O) t0) (S (weight_map (\lambda (_: -nat).O) u)))) (le_n (S (weight_map (\lambda (_: nat).O) u))) (lift O O u) -(lift_r u O)) (weight_map (wadd (\lambda (_: nat).O) (S (weight_map (\lambda -(_: nat).O) u))) (lift (S O) O u)) (lift_weight_add_O (S (weight_map (\lambda -(_: nat).O) u)) u O (\lambda (_: nat).O))))))))). -(* COMMENTS -Initial nodes: 347 -END *) - -theorem subst0_tlt: - \forall (u: T).(\forall (t: T).(\forall (z: T).((subst0 O u t z) \to (tlt z -(THead (Bind Abbr) u t))))) -\def - \lambda (u: T).(\lambda (t: T).(\lambda (z: T).(\lambda (H: (subst0 O u t -z)).(tlt_trans (THead (Bind Abbr) u z) z (THead (Bind Abbr) u t) (tlt_head_dx -(Bind Abbr) u z) (subst0_tlt_head u t z H))))). -(* COMMENTS -Initial nodes: 59 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/subst1/defs.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/subst1/defs.ma deleted file mode 100644 index 6a51bcfb5..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/subst1/defs.ma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/subst0/defs.ma". - -inductive subst1 (i: nat) (v: T) (t1: T): T \to Prop \def -| subst1_refl: subst1 i v t1 t1 -| subst1_single: \forall (t2: T).((subst0 i v t1 t2) \to (subst1 i v t1 t2)). - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/subst1/fwd.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/subst1/fwd.ma deleted file mode 100644 index a2bc1edd6..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/subst1/fwd.ma +++ /dev/null @@ -1,182 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/subst1/defs.ma". - -include "Basic-1/subst0/props.ma". - -theorem subst1_gen_sort: - \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst1 -i v (TSort n) x) \to (eq T x (TSort n)))))) -\def - \lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda -(H: (subst1 i v (TSort n) x)).(subst1_ind i v (TSort n) (\lambda (t: T).(eq T -t (TSort n))) (refl_equal T (TSort n)) (\lambda (t2: T).(\lambda (H0: (subst0 -i v (TSort n) t2)).(subst0_gen_sort v t2 i n H0 (eq T t2 (TSort n))))) x -H))))). -(* COMMENTS -Initial nodes: 89 -END *) - -theorem subst1_gen_lref: - \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst1 -i v (TLRef n) x) \to (or (eq T x (TLRef n)) (land (eq nat n i) (eq T x (lift -(S n) O v)))))))) -\def - \lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda -(H: (subst1 i v (TLRef n) x)).(subst1_ind i v (TLRef n) (\lambda (t: T).(or -(eq T t (TLRef n)) (land (eq nat n i) (eq T t (lift (S n) O v))))) (or_introl -(eq T (TLRef n) (TLRef n)) (land (eq nat n i) (eq T (TLRef n) (lift (S n) O -v))) (refl_equal T (TLRef n))) (\lambda (t2: T).(\lambda (H0: (subst0 i v -(TLRef n) t2)).(land_ind (eq nat n i) (eq T t2 (lift (S n) O v)) (or (eq T t2 -(TLRef n)) (land (eq nat n i) (eq T t2 (lift (S n) O v)))) (\lambda (H1: (eq -nat n i)).(\lambda (H2: (eq T t2 (lift (S n) O v))).(or_intror (eq T t2 -(TLRef n)) (land (eq nat n i) (eq T t2 (lift (S n) O v))) (conj (eq nat n i) -(eq T t2 (lift (S n) O v)) H1 H2)))) (subst0_gen_lref v t2 i n H0)))) x -H))))). -(* COMMENTS -Initial nodes: 305 -END *) - -theorem subst1_gen_head: - \forall (k: K).(\forall (v: T).(\forall (u1: T).(\forall (t1: T).(\forall -(x: T).(\forall (i: nat).((subst1 i v (THead k u1 t1) x) \to (ex3_2 T T -(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (t2: -T).(subst1 (s k i) v t1 t2)))))))))) -\def - \lambda (k: K).(\lambda (v: T).(\lambda (u1: T).(\lambda (t1: T).(\lambda -(x: T).(\lambda (i: nat).(\lambda (H: (subst1 i v (THead k u1 t1) -x)).(subst1_ind i v (THead k u1 t1) (\lambda (t: T).(ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T t (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (t2: T).(subst1 (s k i) v t1 -t2))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead k u1 -t1) (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2))) -(\lambda (_: T).(\lambda (t2: T).(subst1 (s k i) v t1 t2))) u1 t1 (refl_equal -T (THead k u1 t1)) (subst1_refl i v u1) (subst1_refl (s k i) v t1)) (\lambda -(t2: T).(\lambda (H0: (subst0 i v (THead k u1 t1) t2)).(or3_ind (ex2 T -(\lambda (u2: T).(eq T t2 (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 -u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: -T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: -T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v -u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3)))) (ex3_2 -T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) (\lambda -(u2: T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (t3: -T).(subst1 (s k i) v t1 t3)))) (\lambda (H1: (ex2 T (\lambda (u2: T).(eq T t2 -(THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2)))).(ex2_ind T (\lambda -(u2: T).(eq T t2 (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2)) -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(subst1 (s k i) v t1 t3)))) (\lambda (x0: T).(\lambda -(H2: (eq T t2 (THead k x0 t1))).(\lambda (H3: (subst0 i v u1 -x0)).(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(subst1 (s k i) v t1 t3))) x0 t1 H2 (subst1_single i v u1 -x0 H3) (subst1_refl (s k i) v t1))))) H1)) (\lambda (H1: (ex2 T (\lambda (t3: -T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 -t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: -T).(subst0 (s k i) v t1 t3)) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq -T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2))) -(\lambda (_: T).(\lambda (t3: T).(subst1 (s k i) v t1 t3)))) (\lambda (x0: -T).(\lambda (H2: (eq T t2 (THead k u1 x0))).(\lambda (H3: (subst0 (s k i) v -t1 x0)).(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(subst1 (s k i) v t1 t3))) u1 x0 H2 (subst1_refl i v u1) -(subst1_single (s k i) v t1 x0 H3))))) H1)) (\lambda (H1: (ex3_2 T T (\lambda -(u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t3: -T).(subst0 (s k i) v t1 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: -T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v -u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T -T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (t3: -T).(subst1 (s k i) v t1 t3)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda -(H2: (eq T t2 (THead k x0 x1))).(\lambda (H3: (subst0 i v u1 x0)).(\lambda -(H4: (subst0 (s k i) v t1 x1)).(ex3_2_intro T T (\lambda (u2: T).(\lambda -(t3: T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1 -i v u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst1 (s k i) v t1 t3))) x0 -x1 H2 (subst1_single i v u1 x0 H3) (subst1_single (s k i) v t1 x1 H4))))))) -H1)) (subst0_gen_head k v u1 t1 t2 i H0)))) x H))))))). -(* COMMENTS -Initial nodes: 1199 -END *) - -theorem subst1_gen_lift_lt: - \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall -(h: nat).(\forall (d: nat).((subst1 i (lift h d u) (lift h (S (plus i d)) t1) -x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda -(t2: T).(subst1 i u t1 t2))))))))) -\def - \lambda (u: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (i: nat).(\lambda -(h: nat).(\lambda (d: nat).(\lambda (H: (subst1 i (lift h d u) (lift h (S -(plus i d)) t1) x)).(subst1_ind i (lift h d u) (lift h (S (plus i d)) t1) -(\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h (S (plus i d)) t2))) -(\lambda (t2: T).(subst1 i u t1 t2)))) (ex_intro2 T (\lambda (t2: T).(eq T -(lift h (S (plus i d)) t1) (lift h (S (plus i d)) t2))) (\lambda (t2: -T).(subst1 i u t1 t2)) t1 (refl_equal T (lift h (S (plus i d)) t1)) -(subst1_refl i u t1)) (\lambda (t2: T).(\lambda (H0: (subst0 i (lift h d u) -(lift h (S (plus i d)) t1) t2)).(ex2_ind T (\lambda (t3: T).(eq T t2 (lift h -(S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u t1 t3)) (ex2 T (\lambda -(t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst1 i u t1 -t3))) (\lambda (x0: T).(\lambda (H1: (eq T t2 (lift h (S (plus i d)) -x0))).(\lambda (H2: (subst0 i u t1 x0)).(ex_intro2 T (\lambda (t3: T).(eq T -t2 (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst1 i u t1 t3)) x0 H1 -(subst1_single i u t1 x0 H2))))) (subst0_gen_lift_lt u t1 t2 i h d H0)))) x -H))))))). -(* COMMENTS -Initial nodes: 395 -END *) - -theorem subst1_gen_lift_eq: - \forall (t: T).(\forall (u: T).(\forall (x: T).(\forall (h: nat).(\forall -(d: nat).(\forall (i: nat).((le d i) \to ((lt i (plus d h)) \to ((subst1 i u -(lift h d t) x) \to (eq T x (lift h d t)))))))))) -\def - \lambda (t: T).(\lambda (u: T).(\lambda (x: T).(\lambda (h: nat).(\lambda -(d: nat).(\lambda (i: nat).(\lambda (H: (le d i)).(\lambda (H0: (lt i (plus d -h))).(\lambda (H1: (subst1 i u (lift h d t) x)).(subst1_ind i u (lift h d t) -(\lambda (t0: T).(eq T t0 (lift h d t))) (refl_equal T (lift h d t)) (\lambda -(t2: T).(\lambda (H2: (subst0 i u (lift h d t) t2)).(subst0_gen_lift_false t -u t2 h d i H H0 H2 (eq T t2 (lift h d t))))) x H1))))))))). -(* COMMENTS -Initial nodes: 141 -END *) - -theorem subst1_gen_lift_ge: - \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall -(h: nat).(\forall (d: nat).((subst1 i u (lift h d t1) x) \to ((le (plus d h) -i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: -T).(subst1 (minus i h) u t1 t2)))))))))) -\def - \lambda (u: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (i: nat).(\lambda -(h: nat).(\lambda (d: nat).(\lambda (H: (subst1 i u (lift h d t1) -x)).(\lambda (H0: (le (plus d h) i)).(subst1_ind i u (lift h d t1) (\lambda -(t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h d t2))) (\lambda (t2: -T).(subst1 (minus i h) u t1 t2)))) (ex_intro2 T (\lambda (t2: T).(eq T (lift -h d t1) (lift h d t2))) (\lambda (t2: T).(subst1 (minus i h) u t1 t2)) t1 -(refl_equal T (lift h d t1)) (subst1_refl (minus i h) u t1)) (\lambda (t2: -T).(\lambda (H1: (subst0 i u (lift h d t1) t2)).(ex2_ind T (\lambda (t3: -T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u t1 t3)) -(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst1 -(minus i h) u t1 t3))) (\lambda (x0: T).(\lambda (H2: (eq T t2 (lift h d -x0))).(\lambda (H3: (subst0 (minus i h) u t1 x0)).(ex_intro2 T (\lambda (t3: -T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst1 (minus i h) u t1 t3)) x0 -H2 (subst1_single (minus i h) u t1 x0 H3))))) (subst0_gen_lift_ge u t1 t2 i h -d H1 H0)))) x H)))))))). -(* COMMENTS -Initial nodes: 355 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/subst1/props.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/subst1/props.ma deleted file mode 100644 index ac8dea954..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/subst1/props.ma +++ /dev/null @@ -1,179 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/subst1/defs.ma". - -include "Basic-1/subst0/props.ma". - -theorem subst1_head: - \forall (v: T).(\forall (u1: T).(\forall (u2: T).(\forall (i: nat).((subst1 -i v u1 u2) \to (\forall (k: K).(\forall (t1: T).(\forall (t2: T).((subst1 (s -k i) v t1 t2) \to (subst1 i v (THead k u1 t1) (THead k u2 t2)))))))))) -\def - \lambda (v: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i: nat).(\lambda -(H: (subst1 i v u1 u2)).(subst1_ind i v u1 (\lambda (t: T).(\forall (k: -K).(\forall (t1: T).(\forall (t2: T).((subst1 (s k i) v t1 t2) \to (subst1 i -v (THead k u1 t1) (THead k t t2))))))) (\lambda (k: K).(\lambda (t1: -T).(\lambda (t2: T).(\lambda (H0: (subst1 (s k i) v t1 t2)).(subst1_ind (s k -i) v t1 (\lambda (t: T).(subst1 i v (THead k u1 t1) (THead k u1 t))) -(subst1_refl i v (THead k u1 t1)) (\lambda (t3: T).(\lambda (H1: (subst0 (s k -i) v t1 t3)).(subst1_single i v (THead k u1 t1) (THead k u1 t3) (subst0_snd k -v t3 t1 i H1 u1)))) t2 H0))))) (\lambda (t2: T).(\lambda (H0: (subst0 i v u1 -t2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t0: T).(\lambda (H1: (subst1 -(s k i) v t1 t0)).(subst1_ind (s k i) v t1 (\lambda (t: T).(subst1 i v (THead -k u1 t1) (THead k t2 t))) (subst1_single i v (THead k u1 t1) (THead k t2 t1) -(subst0_fst v t2 u1 i H0 t1 k)) (\lambda (t3: T).(\lambda (H2: (subst0 (s k -i) v t1 t3)).(subst1_single i v (THead k u1 t1) (THead k t2 t3) (subst0_both -v u1 t2 i H0 k t1 t3 H2)))) t0 H1))))))) u2 H))))). -(* COMMENTS -Initial nodes: 369 -END *) - -theorem subst1_lift_lt: - \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).((subst1 -i u t1 t2) \to (\forall (d: nat).((lt i d) \to (\forall (h: nat).(subst1 i -(lift h (minus d (S i)) u) (lift h d t1) (lift h d t2))))))))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H: (subst1 i u t1 t2)).(subst1_ind i u t1 (\lambda (t: T).(\forall (d: -nat).((lt i d) \to (\forall (h: nat).(subst1 i (lift h (minus d (S i)) u) -(lift h d t1) (lift h d t)))))) (\lambda (d: nat).(\lambda (_: (lt i -d)).(\lambda (h: nat).(subst1_refl i (lift h (minus d (S i)) u) (lift h d -t1))))) (\lambda (t3: T).(\lambda (H0: (subst0 i u t1 t3)).(\lambda (d: -nat).(\lambda (H1: (lt i d)).(\lambda (h: nat).(subst1_single i (lift h -(minus d (S i)) u) (lift h d t1) (lift h d t3) (subst0_lift_lt t1 t3 u i H0 d -H1 h))))))) t2 H))))). -(* COMMENTS -Initial nodes: 185 -END *) - -theorem subst1_lift_ge: - \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).(\forall -(h: nat).((subst1 i u t1 t2) \to (\forall (d: nat).((le d i) \to (subst1 -(plus i h) u (lift h d t1) (lift h d t2))))))))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda -(h: nat).(\lambda (H: (subst1 i u t1 t2)).(subst1_ind i u t1 (\lambda (t: -T).(\forall (d: nat).((le d i) \to (subst1 (plus i h) u (lift h d t1) (lift h -d t))))) (\lambda (d: nat).(\lambda (_: (le d i)).(subst1_refl (plus i h) u -(lift h d t1)))) (\lambda (t3: T).(\lambda (H0: (subst0 i u t1 t3)).(\lambda -(d: nat).(\lambda (H1: (le d i)).(subst1_single (plus i h) u (lift h d t1) -(lift h d t3) (subst0_lift_ge t1 t3 u i h H0 d H1)))))) t2 H)))))). -(* COMMENTS -Initial nodes: 157 -END *) - -theorem subst1_ex: - \forall (u: T).(\forall (t1: T).(\forall (d: nat).(ex T (\lambda (t2: -T).(subst1 d u t1 (lift (S O) d t2)))))) -\def - \lambda (u: T).(\lambda (t1: T).(T_ind (\lambda (t: T).(\forall (d: nat).(ex -T (\lambda (t2: T).(subst1 d u t (lift (S O) d t2)))))) (\lambda (n: -nat).(\lambda (d: nat).(ex_intro T (\lambda (t2: T).(subst1 d u (TSort n) -(lift (S O) d t2))) (TSort n) (eq_ind_r T (TSort n) (\lambda (t: T).(subst1 d -u (TSort n) t)) (subst1_refl d u (TSort n)) (lift (S O) d (TSort n)) -(lift_sort n (S O) d))))) (\lambda (n: nat).(\lambda (d: nat).(lt_eq_gt_e n d -(ex T (\lambda (t2: T).(subst1 d u (TLRef n) (lift (S O) d t2)))) (\lambda -(H: (lt n d)).(ex_intro T (\lambda (t2: T).(subst1 d u (TLRef n) (lift (S O) -d t2))) (TLRef n) (eq_ind_r T (TLRef n) (\lambda (t: T).(subst1 d u (TLRef n) -t)) (subst1_refl d u (TLRef n)) (lift (S O) d (TLRef n)) (lift_lref_lt n (S -O) d H)))) (\lambda (H: (eq nat n d)).(eq_ind nat n (\lambda (n0: nat).(ex T -(\lambda (t2: T).(subst1 n0 u (TLRef n) (lift (S O) n0 t2))))) (ex_intro T -(\lambda (t2: T).(subst1 n u (TLRef n) (lift (S O) n t2))) (lift n O u) -(eq_ind_r T (lift (plus (S O) n) O u) (\lambda (t: T).(subst1 n u (TLRef n) -t)) (subst1_single n u (TLRef n) (lift (S n) O u) (subst0_lref u n)) (lift (S -O) n (lift n O u)) (lift_free u n (S O) O n (le_n (plus O n)) (le_O_n n)))) d -H)) (\lambda (H: (lt d n)).(ex_intro T (\lambda (t2: T).(subst1 d u (TLRef n) -(lift (S O) d t2))) (TLRef (pred n)) (eq_ind_r T (TLRef n) (\lambda (t: -T).(subst1 d u (TLRef n) t)) (subst1_refl d u (TLRef n)) (lift (S O) d (TLRef -(pred n))) (lift_lref_gt d n H))))))) (\lambda (k: K).(\lambda (t: -T).(\lambda (H: ((\forall (d: nat).(ex T (\lambda (t2: T).(subst1 d u t (lift -(S O) d t2))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (d: nat).(ex T -(\lambda (t2: T).(subst1 d u t0 (lift (S O) d t2))))))).(\lambda (d: -nat).(let H_x \def (H d) in (let H1 \def H_x in (ex_ind T (\lambda (t2: -T).(subst1 d u t (lift (S O) d t2))) (ex T (\lambda (t2: T).(subst1 d u -(THead k t t0) (lift (S O) d t2)))) (\lambda (x: T).(\lambda (H2: (subst1 d u -t (lift (S O) d x))).(let H_x0 \def (H0 (s k d)) in (let H3 \def H_x0 in -(ex_ind T (\lambda (t2: T).(subst1 (s k d) u t0 (lift (S O) (s k d) t2))) (ex -T (\lambda (t2: T).(subst1 d u (THead k t t0) (lift (S O) d t2)))) (\lambda -(x0: T).(\lambda (H4: (subst1 (s k d) u t0 (lift (S O) (s k d) -x0))).(ex_intro T (\lambda (t2: T).(subst1 d u (THead k t t0) (lift (S O) d -t2))) (THead k x x0) (eq_ind_r T (THead k (lift (S O) d x) (lift (S O) (s k -d) x0)) (\lambda (t2: T).(subst1 d u (THead k t t0) t2)) (subst1_head u t -(lift (S O) d x) d H2 k t0 (lift (S O) (s k d) x0) H4) (lift (S O) d (THead k -x x0)) (lift_head k x x0 (S O) d))))) H3))))) H1))))))))) t1)). -(* COMMENTS -Initial nodes: 925 -END *) - -theorem subst1_lift_S: - \forall (u: T).(\forall (i: nat).(\forall (h: nat).((le h i) \to (subst1 i -(TLRef h) (lift (S h) (S i) u) (lift (S h) i u))))) -\def - \lambda (u: T).(T_ind (\lambda (t: T).(\forall (i: nat).(\forall (h: -nat).((le h i) \to (subst1 i (TLRef h) (lift (S h) (S i) t) (lift (S h) i -t)))))) (\lambda (n: nat).(\lambda (i: nat).(\lambda (h: nat).(\lambda (_: -(le h i)).(eq_ind_r T (TSort n) (\lambda (t: T).(subst1 i (TLRef h) t (lift -(S h) i (TSort n)))) (eq_ind_r T (TSort n) (\lambda (t: T).(subst1 i (TLRef -h) (TSort n) t)) (subst1_refl i (TLRef h) (TSort n)) (lift (S h) i (TSort n)) -(lift_sort n (S h) i)) (lift (S h) (S i) (TSort n)) (lift_sort n (S h) (S -i))))))) (\lambda (n: nat).(\lambda (i: nat).(\lambda (h: nat).(\lambda (H: -(le h i)).(lt_eq_gt_e n i (subst1 i (TLRef h) (lift (S h) (S i) (TLRef n)) -(lift (S h) i (TLRef n))) (\lambda (H0: (lt n i)).(eq_ind_r T (TLRef n) -(\lambda (t: T).(subst1 i (TLRef h) t (lift (S h) i (TLRef n)))) (eq_ind_r T -(TLRef n) (\lambda (t: T).(subst1 i (TLRef h) (TLRef n) t)) (subst1_refl i -(TLRef h) (TLRef n)) (lift (S h) i (TLRef n)) (lift_lref_lt n (S h) i H0)) -(lift (S h) (S i) (TLRef n)) (lift_lref_lt n (S h) (S i) (le_S (S n) i H0)))) -(\lambda (H0: (eq nat n i)).(let H1 \def (eq_ind_r nat i (\lambda (n0: -nat).(le h n0)) H n H0) in (eq_ind nat n (\lambda (n0: nat).(subst1 n0 (TLRef -h) (lift (S h) (S n0) (TLRef n)) (lift (S h) n0 (TLRef n)))) (eq_ind_r T -(TLRef n) (\lambda (t: T).(subst1 n (TLRef h) t (lift (S h) n (TLRef n)))) -(eq_ind_r T (TLRef (plus n (S h))) (\lambda (t: T).(subst1 n (TLRef h) (TLRef -n) t)) (eq_ind nat (S (plus n h)) (\lambda (n0: nat).(subst1 n (TLRef h) -(TLRef n) (TLRef n0))) (eq_ind_r nat (plus h n) (\lambda (n0: nat).(subst1 n -(TLRef h) (TLRef n) (TLRef (S n0)))) (eq_ind nat (plus h (S n)) (\lambda (n0: -nat).(subst1 n (TLRef h) (TLRef n) (TLRef n0))) (eq_ind T (lift (S n) O -(TLRef h)) (\lambda (t: T).(subst1 n (TLRef h) (TLRef n) t)) (subst1_single n -(TLRef h) (TLRef n) (lift (S n) O (TLRef h)) (subst0_lref (TLRef h) n)) -(TLRef (plus h (S n))) (lift_lref_ge h (S n) O (le_O_n h))) (S (plus h n)) -(sym_eq nat (S (plus h n)) (plus h (S n)) (plus_n_Sm h n))) (plus n h) -(plus_sym n h)) (plus n (S h)) (plus_n_Sm n h)) (lift (S h) n (TLRef n)) -(lift_lref_ge n (S h) n (le_n n))) (lift (S h) (S n) (TLRef n)) (lift_lref_lt -n (S h) (S n) (le_n (S n)))) i H0))) (\lambda (H0: (lt i n)).(eq_ind_r T -(TLRef (plus n (S h))) (\lambda (t: T).(subst1 i (TLRef h) t (lift (S h) i -(TLRef n)))) (eq_ind_r T (TLRef (plus n (S h))) (\lambda (t: T).(subst1 i -(TLRef h) (TLRef (plus n (S h))) t)) (subst1_refl i (TLRef h) (TLRef (plus n -(S h)))) (lift (S h) i (TLRef n)) (lift_lref_ge n (S h) i (le_S_n i n (le_S -(S i) n H0)))) (lift (S h) (S i) (TLRef n)) (lift_lref_ge n (S h) (S i) -H0)))))))) (\lambda (k: K).(\lambda (t: T).(\lambda (H: ((\forall (i: -nat).(\forall (h: nat).((le h i) \to (subst1 i (TLRef h) (lift (S h) (S i) t) -(lift (S h) i t))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (i: -nat).(\forall (h: nat).((le h i) \to (subst1 i (TLRef h) (lift (S h) (S i) -t0) (lift (S h) i t0))))))).(\lambda (i: nat).(\lambda (h: nat).(\lambda (H1: -(le h i)).(eq_ind_r T (THead k (lift (S h) (S i) t) (lift (S h) (s k (S i)) -t0)) (\lambda (t1: T).(subst1 i (TLRef h) t1 (lift (S h) i (THead k t t0)))) -(eq_ind_r T (THead k (lift (S h) i t) (lift (S h) (s k i) t0)) (\lambda (t1: -T).(subst1 i (TLRef h) (THead k (lift (S h) (S i) t) (lift (S h) (s k (S i)) -t0)) t1)) (subst1_head (TLRef h) (lift (S h) (S i) t) (lift (S h) i t) i (H i -h H1) k (lift (S h) (s k (S i)) t0) (lift (S h) (s k i) t0) (eq_ind_r nat (S -(s k i)) (\lambda (n: nat).(subst1 (s k i) (TLRef h) (lift (S h) n t0) (lift -(S h) (s k i) t0))) (H0 (s k i) h (le_trans h i (s k i) H1 (s_inc k i))) (s k -(S i)) (s_S k i))) (lift (S h) i (THead k t t0)) (lift_head k t t0 (S h) i)) -(lift (S h) (S i) (THead k t t0)) (lift_head k t t0 (S h) (S i))))))))))) u). -(* COMMENTS -Initial nodes: 1421 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/subst1/subst1.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/subst1/subst1.ma deleted file mode 100644 index e6dae0dc9..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/subst1/subst1.ma +++ /dev/null @@ -1,214 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/subst1/fwd.ma". - -include "Basic-1/subst0/subst0.ma". - -theorem subst1_subst1: - \forall (t1: T).(\forall (t2: T).(\forall (u2: T).(\forall (j: nat).((subst1 -j u2 t1 t2) \to (\forall (u1: T).(\forall (u: T).(\forall (i: nat).((subst1 i -u u1 u2) \to (ex2 T (\lambda (t: T).(subst1 j u1 t1 t)) (\lambda (t: -T).(subst1 (S (plus i j)) u t t2))))))))))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (u2: T).(\lambda (j: nat).(\lambda -(H: (subst1 j u2 t1 t2)).(subst1_ind j u2 t1 (\lambda (t: T).(\forall (u1: -T).(\forall (u: T).(\forall (i: nat).((subst1 i u u1 u2) \to (ex2 T (\lambda -(t0: T).(subst1 j u1 t1 t0)) (\lambda (t0: T).(subst1 (S (plus i j)) u t0 -t)))))))) (\lambda (u1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda (_: -(subst1 i u u1 u2)).(ex_intro2 T (\lambda (t: T).(subst1 j u1 t1 t)) (\lambda -(t: T).(subst1 (S (plus i j)) u t t1)) t1 (subst1_refl j u1 t1) (subst1_refl -(S (plus i j)) u t1)))))) (\lambda (t3: T).(\lambda (H0: (subst0 j u2 t1 -t3)).(\lambda (u1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda (H1: (subst1 -i u u1 u2)).(insert_eq T u2 (\lambda (t: T).(subst1 i u u1 t)) (\lambda (_: -T).(ex2 T (\lambda (t0: T).(subst1 j u1 t1 t0)) (\lambda (t0: T).(subst1 (S -(plus i j)) u t0 t3)))) (\lambda (y: T).(\lambda (H2: (subst1 i u u1 -y)).(subst1_ind i u u1 (\lambda (t: T).((eq T t u2) \to (ex2 T (\lambda (t0: -T).(subst1 j u1 t1 t0)) (\lambda (t0: T).(subst1 (S (plus i j)) u t0 t3))))) -(\lambda (H3: (eq T u1 u2)).(eq_ind_r T u2 (\lambda (t: T).(ex2 T (\lambda -(t0: T).(subst1 j t t1 t0)) (\lambda (t0: T).(subst1 (S (plus i j)) u t0 -t3)))) (ex_intro2 T (\lambda (t: T).(subst1 j u2 t1 t)) (\lambda (t: -T).(subst1 (S (plus i j)) u t t3)) t3 (subst1_single j u2 t1 t3 H0) -(subst1_refl (S (plus i j)) u t3)) u1 H3)) (\lambda (t0: T).(\lambda (H3: -(subst0 i u u1 t0)).(\lambda (H4: (eq T t0 u2)).(let H5 \def (eq_ind T t0 -(\lambda (t: T).(subst0 i u u1 t)) H3 u2 H4) in (ex2_ind T (\lambda (t: -T).(subst0 j u1 t1 t)) (\lambda (t: T).(subst0 (S (plus i j)) u t t3)) (ex2 T -(\lambda (t: T).(subst1 j u1 t1 t)) (\lambda (t: T).(subst1 (S (plus i j)) u -t t3))) (\lambda (x: T).(\lambda (H6: (subst0 j u1 t1 x)).(\lambda (H7: -(subst0 (S (plus i j)) u x t3)).(ex_intro2 T (\lambda (t: T).(subst1 j u1 t1 -t)) (\lambda (t: T).(subst1 (S (plus i j)) u t t3)) x (subst1_single j u1 t1 -x H6) (subst1_single (S (plus i j)) u x t3 H7))))) (subst0_subst0 t1 t3 u2 j -H0 u1 u i H5)))))) y H2))) H1))))))) t2 H))))). -(* COMMENTS -Initial nodes: 649 -END *) - -theorem subst1_subst1_back: - \forall (t1: T).(\forall (t2: T).(\forall (u2: T).(\forall (j: nat).((subst1 -j u2 t1 t2) \to (\forall (u1: T).(\forall (u: T).(\forall (i: nat).((subst1 i -u u2 u1) \to (ex2 T (\lambda (t: T).(subst1 j u1 t1 t)) (\lambda (t: -T).(subst1 (S (plus i j)) u t2 t))))))))))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (u2: T).(\lambda (j: nat).(\lambda -(H: (subst1 j u2 t1 t2)).(subst1_ind j u2 t1 (\lambda (t: T).(\forall (u1: -T).(\forall (u: T).(\forall (i: nat).((subst1 i u u2 u1) \to (ex2 T (\lambda -(t0: T).(subst1 j u1 t1 t0)) (\lambda (t0: T).(subst1 (S (plus i j)) u t -t0)))))))) (\lambda (u1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda (_: -(subst1 i u u2 u1)).(ex_intro2 T (\lambda (t: T).(subst1 j u1 t1 t)) (\lambda -(t: T).(subst1 (S (plus i j)) u t1 t)) t1 (subst1_refl j u1 t1) (subst1_refl -(S (plus i j)) u t1)))))) (\lambda (t3: T).(\lambda (H0: (subst0 j u2 t1 -t3)).(\lambda (u1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda (H1: (subst1 -i u u2 u1)).(subst1_ind i u u2 (\lambda (t: T).(ex2 T (\lambda (t0: -T).(subst1 j t t1 t0)) (\lambda (t0: T).(subst1 (S (plus i j)) u t3 t0)))) -(ex_intro2 T (\lambda (t: T).(subst1 j u2 t1 t)) (\lambda (t: T).(subst1 (S -(plus i j)) u t3 t)) t3 (subst1_single j u2 t1 t3 H0) (subst1_refl (S (plus i -j)) u t3)) (\lambda (t0: T).(\lambda (H2: (subst0 i u u2 t0)).(ex2_ind T -(\lambda (t: T).(subst0 j t0 t1 t)) (\lambda (t: T).(subst0 (S (plus i j)) u -t3 t)) (ex2 T (\lambda (t: T).(subst1 j t0 t1 t)) (\lambda (t: T).(subst1 (S -(plus i j)) u t3 t))) (\lambda (x: T).(\lambda (H3: (subst0 j t0 t1 -x)).(\lambda (H4: (subst0 (S (plus i j)) u t3 x)).(ex_intro2 T (\lambda (t: -T).(subst1 j t0 t1 t)) (\lambda (t: T).(subst1 (S (plus i j)) u t3 t)) x -(subst1_single j t0 t1 x H3) (subst1_single (S (plus i j)) u t3 x H4))))) -(subst0_subst0_back t1 t3 u2 j H0 t0 u i H2)))) u1 H1))))))) t2 H))))). -(* COMMENTS -Initial nodes: 487 -END *) - -theorem subst1_trans: - \forall (t2: T).(\forall (t1: T).(\forall (v: T).(\forall (i: nat).((subst1 -i v t1 t2) \to (\forall (t3: T).((subst1 i v t2 t3) \to (subst1 i v t1 -t3))))))) -\def - \lambda (t2: T).(\lambda (t1: T).(\lambda (v: T).(\lambda (i: nat).(\lambda -(H: (subst1 i v t1 t2)).(subst1_ind i v t1 (\lambda (t: T).(\forall (t3: -T).((subst1 i v t t3) \to (subst1 i v t1 t3)))) (\lambda (t3: T).(\lambda -(H0: (subst1 i v t1 t3)).H0)) (\lambda (t3: T).(\lambda (H0: (subst0 i v t1 -t3)).(\lambda (t4: T).(\lambda (H1: (subst1 i v t3 t4)).(subst1_ind i v t3 -(\lambda (t: T).(subst1 i v t1 t)) (subst1_single i v t1 t3 H0) (\lambda (t0: -T).(\lambda (H2: (subst0 i v t3 t0)).(subst1_single i v t1 t0 (subst0_trans -t3 t1 v i H0 t0 H2)))) t4 H1))))) t2 H))))). -(* COMMENTS -Initial nodes: 165 -END *) - -theorem subst1_confluence_neq: - \forall (t0: T).(\forall (t1: T).(\forall (u1: T).(\forall (i1: -nat).((subst1 i1 u1 t0 t1) \to (\forall (t2: T).(\forall (u2: T).(\forall -(i2: nat).((subst1 i2 u2 t0 t2) \to ((not (eq nat i1 i2)) \to (ex2 T (\lambda -(t: T).(subst1 i2 u2 t1 t)) (\lambda (t: T).(subst1 i1 u1 t2 t)))))))))))) -\def - \lambda (t0: T).(\lambda (t1: T).(\lambda (u1: T).(\lambda (i1: -nat).(\lambda (H: (subst1 i1 u1 t0 t1)).(subst1_ind i1 u1 t0 (\lambda (t: -T).(\forall (t2: T).(\forall (u2: T).(\forall (i2: nat).((subst1 i2 u2 t0 t2) -\to ((not (eq nat i1 i2)) \to (ex2 T (\lambda (t3: T).(subst1 i2 u2 t t3)) -(\lambda (t3: T).(subst1 i1 u1 t2 t3))))))))) (\lambda (t2: T).(\lambda (u2: -T).(\lambda (i2: nat).(\lambda (H0: (subst1 i2 u2 t0 t2)).(\lambda (_: (not -(eq nat i1 i2))).(ex_intro2 T (\lambda (t: T).(subst1 i2 u2 t0 t)) (\lambda -(t: T).(subst1 i1 u1 t2 t)) t2 H0 (subst1_refl i1 u1 t2))))))) (\lambda (t2: -T).(\lambda (H0: (subst0 i1 u1 t0 t2)).(\lambda (t3: T).(\lambda (u2: -T).(\lambda (i2: nat).(\lambda (H1: (subst1 i2 u2 t0 t3)).(\lambda (H2: (not -(eq nat i1 i2))).(subst1_ind i2 u2 t0 (\lambda (t: T).(ex2 T (\lambda (t4: -T).(subst1 i2 u2 t2 t4)) (\lambda (t4: T).(subst1 i1 u1 t t4)))) (ex_intro2 T -(\lambda (t: T).(subst1 i2 u2 t2 t)) (\lambda (t: T).(subst1 i1 u1 t0 t)) t2 -(subst1_refl i2 u2 t2) (subst1_single i1 u1 t0 t2 H0)) (\lambda (t4: -T).(\lambda (H3: (subst0 i2 u2 t0 t4)).(ex2_ind T (\lambda (t: T).(subst0 i1 -u1 t4 t)) (\lambda (t: T).(subst0 i2 u2 t2 t)) (ex2 T (\lambda (t: T).(subst1 -i2 u2 t2 t)) (\lambda (t: T).(subst1 i1 u1 t4 t))) (\lambda (x: T).(\lambda -(H4: (subst0 i1 u1 t4 x)).(\lambda (H5: (subst0 i2 u2 t2 x)).(ex_intro2 T -(\lambda (t: T).(subst1 i2 u2 t2 t)) (\lambda (t: T).(subst1 i1 u1 t4 t)) x -(subst1_single i2 u2 t2 x H5) (subst1_single i1 u1 t4 x H4))))) -(subst0_confluence_neq t0 t4 u2 i2 H3 t2 u1 i1 H0 (sym_not_eq nat i1 i2 -H2))))) t3 H1)))))))) t1 H))))). -(* COMMENTS -Initial nodes: 455 -END *) - -theorem subst1_confluence_eq: - \forall (t0: T).(\forall (t1: T).(\forall (u: T).(\forall (i: nat).((subst1 -i u t0 t1) \to (\forall (t2: T).((subst1 i u t0 t2) \to (ex2 T (\lambda (t: -T).(subst1 i u t1 t)) (\lambda (t: T).(subst1 i u t2 t))))))))) -\def - \lambda (t0: T).(\lambda (t1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H: (subst1 i u t0 t1)).(subst1_ind i u t0 (\lambda (t: T).(\forall (t2: -T).((subst1 i u t0 t2) \to (ex2 T (\lambda (t3: T).(subst1 i u t t3)) -(\lambda (t3: T).(subst1 i u t2 t3)))))) (\lambda (t2: T).(\lambda (H0: -(subst1 i u t0 t2)).(ex_intro2 T (\lambda (t: T).(subst1 i u t0 t)) (\lambda -(t: T).(subst1 i u t2 t)) t2 H0 (subst1_refl i u t2)))) (\lambda (t2: -T).(\lambda (H0: (subst0 i u t0 t2)).(\lambda (t3: T).(\lambda (H1: (subst1 i -u t0 t3)).(subst1_ind i u t0 (\lambda (t: T).(ex2 T (\lambda (t4: T).(subst1 -i u t2 t4)) (\lambda (t4: T).(subst1 i u t t4)))) (ex_intro2 T (\lambda (t: -T).(subst1 i u t2 t)) (\lambda (t: T).(subst1 i u t0 t)) t2 (subst1_refl i u -t2) (subst1_single i u t0 t2 H0)) (\lambda (t4: T).(\lambda (H2: (subst0 i u -t0 t4)).(or4_ind (eq T t4 t2) (ex2 T (\lambda (t: T).(subst0 i u t4 t)) -(\lambda (t: T).(subst0 i u t2 t))) (subst0 i u t4 t2) (subst0 i u t2 t4) -(ex2 T (\lambda (t: T).(subst1 i u t2 t)) (\lambda (t: T).(subst1 i u t4 t))) -(\lambda (H3: (eq T t4 t2)).(eq_ind_r T t2 (\lambda (t: T).(ex2 T (\lambda -(t5: T).(subst1 i u t2 t5)) (\lambda (t5: T).(subst1 i u t t5)))) (ex_intro2 -T (\lambda (t: T).(subst1 i u t2 t)) (\lambda (t: T).(subst1 i u t2 t)) t2 -(subst1_refl i u t2) (subst1_refl i u t2)) t4 H3)) (\lambda (H3: (ex2 T -(\lambda (t: T).(subst0 i u t4 t)) (\lambda (t: T).(subst0 i u t2 -t)))).(ex2_ind T (\lambda (t: T).(subst0 i u t4 t)) (\lambda (t: T).(subst0 i -u t2 t)) (ex2 T (\lambda (t: T).(subst1 i u t2 t)) (\lambda (t: T).(subst1 i -u t4 t))) (\lambda (x: T).(\lambda (H4: (subst0 i u t4 x)).(\lambda (H5: -(subst0 i u t2 x)).(ex_intro2 T (\lambda (t: T).(subst1 i u t2 t)) (\lambda -(t: T).(subst1 i u t4 t)) x (subst1_single i u t2 x H5) (subst1_single i u t4 -x H4))))) H3)) (\lambda (H3: (subst0 i u t4 t2)).(ex_intro2 T (\lambda (t: -T).(subst1 i u t2 t)) (\lambda (t: T).(subst1 i u t4 t)) t2 (subst1_refl i u -t2) (subst1_single i u t4 t2 H3))) (\lambda (H3: (subst0 i u t2 -t4)).(ex_intro2 T (\lambda (t: T).(subst1 i u t2 t)) (\lambda (t: T).(subst1 -i u t4 t)) t4 (subst1_single i u t2 t4 H3) (subst1_refl i u t4))) -(subst0_confluence_eq t0 t4 u i H2 t2 H0)))) t3 H1))))) t1 H))))). -(* COMMENTS -Initial nodes: 729 -END *) - -theorem subst1_confluence_lift: - \forall (t0: T).(\forall (t1: T).(\forall (u: T).(\forall (i: nat).((subst1 -i u t0 (lift (S O) i t1)) \to (\forall (t2: T).((subst1 i u t0 (lift (S O) i -t2)) \to (eq T t1 t2))))))) -\def - \lambda (t0: T).(\lambda (t1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H: (subst1 i u t0 (lift (S O) i t1))).(insert_eq T (lift (S O) i t1) -(\lambda (t: T).(subst1 i u t0 t)) (\lambda (_: T).(\forall (t2: T).((subst1 -i u t0 (lift (S O) i t2)) \to (eq T t1 t2)))) (\lambda (y: T).(\lambda (H0: -(subst1 i u t0 y)).(subst1_ind i u t0 (\lambda (t: T).((eq T t (lift (S O) i -t1)) \to (\forall (t2: T).((subst1 i u t0 (lift (S O) i t2)) \to (eq T t1 -t2))))) (\lambda (H1: (eq T t0 (lift (S O) i t1))).(\lambda (t2: T).(\lambda -(H2: (subst1 i u t0 (lift (S O) i t2))).(let H3 \def (eq_ind T t0 (\lambda -(t: T).(subst1 i u t (lift (S O) i t2))) H2 (lift (S O) i t1) H1) in (let H4 -\def (sym_eq T (lift (S O) i t2) (lift (S O) i t1) (subst1_gen_lift_eq t1 u -(lift (S O) i t2) (S O) i i (le_n i) (eq_ind_r nat (plus (S O) i) (\lambda -(n: nat).(lt i n)) (le_n (plus (S O) i)) (plus i (S O)) (plus_sym i (S O))) -H3)) in (lift_inj t1 t2 (S O) i H4)))))) (\lambda (t2: T).(\lambda (H1: -(subst0 i u t0 t2)).(\lambda (H2: (eq T t2 (lift (S O) i t1))).(\lambda (t3: -T).(\lambda (H3: (subst1 i u t0 (lift (S O) i t3))).(let H4 \def (eq_ind T t2 -(\lambda (t: T).(subst0 i u t0 t)) H1 (lift (S O) i t1) H2) in (insert_eq T -(lift (S O) i t3) (\lambda (t: T).(subst1 i u t0 t)) (\lambda (_: T).(eq T t1 -t3)) (\lambda (y0: T).(\lambda (H5: (subst1 i u t0 y0)).(subst1_ind i u t0 -(\lambda (t: T).((eq T t (lift (S O) i t3)) \to (eq T t1 t3))) (\lambda (H6: -(eq T t0 (lift (S O) i t3))).(let H7 \def (eq_ind T t0 (\lambda (t: -T).(subst0 i u t (lift (S O) i t1))) H4 (lift (S O) i t3) H6) in -(subst0_gen_lift_false t3 u (lift (S O) i t1) (S O) i i (le_n i) (eq_ind_r -nat (plus (S O) i) (\lambda (n: nat).(lt i n)) (le_n (plus (S O) i)) (plus i -(S O)) (plus_sym i (S O))) H7 (eq T t1 t3)))) (\lambda (t4: T).(\lambda (H6: -(subst0 i u t0 t4)).(\lambda (H7: (eq T t4 (lift (S O) i t3))).(let H8 \def -(eq_ind T t4 (\lambda (t: T).(subst0 i u t0 t)) H6 (lift (S O) i t3) H7) in -(sym_eq T t3 t1 (subst0_confluence_lift t0 t3 u i H8 t1 H4)))))) y0 H5))) -H3))))))) y H0))) H))))). -(* COMMENTS -Initial nodes: 735 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/theory.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/theory.ma deleted file mode 100644 index f405be588..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/theory.ma +++ /dev/null @@ -1,42 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/subst0/tlt.ma". - -include "Basic-1/subst/props.ma". - -include "Basic-1/sty1/cnt.ma". - -include "Basic-1/ex0/props.ma". - -include "Basic-1/wcpr0/fwd.ma". - -include "Basic-1/pr3/wcpr0.ma". - -include "Basic-1/ex2/props.ma". - -include "Basic-1/ex1/props.ma". - -include "Basic-1/ty3/sty0.ma". - -include "Basic-1/csubt/csuba.ma". - -include "Basic-1/ty3/fwd_nf2.ma". - -include "Basic-1/ty3/nf2.ma". - -include "Basic-1/wf3/props.ma". - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/tlist/defs.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/tlist/defs.ma deleted file mode 100644 index 9445013d4..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/tlist/defs.ma +++ /dev/null @@ -1,47 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/T/defs.ma". - -inductive TList: Set \def -| TNil: TList -| TCons: T \to (TList \to TList). - -definition THeads: - K \to (TList \to (T \to T)) -\def - let rec THeads (k: K) (us: TList) on us: (T \to T) \def (\lambda (t: -T).(match us with [TNil \Rightarrow t | (TCons u ul) \Rightarrow (THead k u -(THeads k ul t))])) in THeads. - -definition TApp: - TList \to (T \to TList) -\def - let rec TApp (ts: TList) on ts: (T \to TList) \def (\lambda (v: T).(match ts -with [TNil \Rightarrow (TCons v TNil) | (TCons t ts0) \Rightarrow (TCons t -(TApp ts0 v))])) in TApp. - -definition tslen: - TList \to nat -\def - let rec tslen (ts: TList) on ts: nat \def (match ts with [TNil \Rightarrow O -| (TCons _ ts0) \Rightarrow (S (tslen ts0))]) in tslen. - -definition tslt: - TList \to (TList \to Prop) -\def - \lambda (ts1: TList).(\lambda (ts2: TList).(lt (tslen ts1) (tslen ts2))). - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/tlist/props.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/tlist/props.ma deleted file mode 100644 index 92b0d05ff..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/tlist/props.ma +++ /dev/null @@ -1,131 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/tlist/defs.ma". - -theorem tslt_wf__q_ind: - \forall (P: ((TList \to Prop))).(((\forall (n: nat).((\lambda (P0: ((TList -\to Prop))).(\lambda (n0: nat).(\forall (ts: TList).((eq nat (tslen ts) n0) -\to (P0 ts))))) P n))) \to (\forall (ts: TList).(P ts))) -\def - let Q \def (\lambda (P: ((TList \to Prop))).(\lambda (n: nat).(\forall (ts: -TList).((eq nat (tslen ts) n) \to (P ts))))) in (\lambda (P: ((TList \to -Prop))).(\lambda (H: ((\forall (n: nat).(\forall (ts: TList).((eq nat (tslen -ts) n) \to (P ts)))))).(\lambda (ts: TList).(H (tslen ts) ts (refl_equal nat -(tslen ts)))))). -(* COMMENTS -Initial nodes: 61 -END *) - -theorem tslt_wf_ind: - \forall (P: ((TList \to Prop))).(((\forall (ts2: TList).(((\forall (ts1: -TList).((tslt ts1 ts2) \to (P ts1)))) \to (P ts2)))) \to (\forall (ts: -TList).(P ts))) -\def - let Q \def (\lambda (P: ((TList \to Prop))).(\lambda (n: nat).(\forall (ts: -TList).((eq nat (tslen ts) n) \to (P ts))))) in (\lambda (P: ((TList \to -Prop))).(\lambda (H: ((\forall (ts2: TList).(((\forall (ts1: TList).((lt -(tslen ts1) (tslen ts2)) \to (P ts1)))) \to (P ts2))))).(\lambda (ts: -TList).(tslt_wf__q_ind (\lambda (t: TList).(P t)) (\lambda (n: -nat).(lt_wf_ind n (Q (\lambda (t: TList).(P t))) (\lambda (n0: nat).(\lambda -(H0: ((\forall (m: nat).((lt m n0) \to (Q (\lambda (t: TList).(P t)) -m))))).(\lambda (ts0: TList).(\lambda (H1: (eq nat (tslen ts0) n0)).(let H2 -\def (eq_ind_r nat n0 (\lambda (n1: nat).(\forall (m: nat).((lt m n1) \to -(\forall (ts1: TList).((eq nat (tslen ts1) m) \to (P ts1)))))) H0 (tslen ts0) -H1) in (H ts0 (\lambda (ts1: TList).(\lambda (H3: (lt (tslen ts1) (tslen -ts0))).(H2 (tslen ts1) H3 ts1 (refl_equal nat (tslen ts1))))))))))))) ts)))). -(* COMMENTS -Initial nodes: 179 -END *) - -theorem theads_tapp: - \forall (k: K).(\forall (v: T).(\forall (t: T).(\forall (vs: TList).(eq T -(THeads k (TApp vs v) t) (THeads k vs (THead k v t)))))) -\def - \lambda (k: K).(\lambda (v: T).(\lambda (t: T).(\lambda (vs: -TList).(TList_ind (\lambda (t0: TList).(eq T (THeads k (TApp t0 v) t) (THeads -k t0 (THead k v t)))) (refl_equal T (THead k v t)) (\lambda (t0: T).(\lambda -(t1: TList).(\lambda (H: (eq T (THeads k (TApp t1 v) t) (THeads k t1 (THead k -v t)))).(eq_ind T (THeads k (TApp t1 v) t) (\lambda (t2: T).(eq T (THead k t0 -(THeads k (TApp t1 v) t)) (THead k t0 t2))) (refl_equal T (THead k t0 (THeads -k (TApp t1 v) t))) (THeads k t1 (THead k v t)) H)))) vs)))). -(* COMMENTS -Initial nodes: 175 -END *) - -theorem tcons_tapp_ex: - \forall (ts1: TList).(\forall (t1: T).(ex2_2 TList T (\lambda (ts2: -TList).(\lambda (t2: T).(eq TList (TCons t1 ts1) (TApp ts2 t2)))) (\lambda -(ts2: TList).(\lambda (_: T).(eq nat (tslen ts1) (tslen ts2)))))) -\def - \lambda (ts1: TList).(TList_ind (\lambda (t: TList).(\forall (t1: T).(ex2_2 -TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons t1 t) (TApp -ts2 t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (tslen t) (tslen -ts2))))))) (\lambda (t1: T).(ex2_2_intro TList T (\lambda (ts2: -TList).(\lambda (t2: T).(eq TList (TCons t1 TNil) (TApp ts2 t2)))) (\lambda -(ts2: TList).(\lambda (_: T).(eq nat O (tslen ts2)))) TNil t1 (refl_equal -TList (TApp TNil t1)) (refl_equal nat (tslen TNil)))) (\lambda (t: -T).(\lambda (t0: TList).(\lambda (H: ((\forall (t1: T).(ex2_2 TList T -(\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons t1 t0) (TApp ts2 -t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (tslen t0) (tslen -ts2)))))))).(\lambda (t1: T).(let H_x \def (H t) in (let H0 \def H_x in -(ex2_2_ind TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons t -t0) (TApp ts2 t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (tslen t0) -(tslen ts2)))) (ex2_2 TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq -TList (TCons t1 (TCons t t0)) (TApp ts2 t2)))) (\lambda (ts2: TList).(\lambda -(_: T).(eq nat (S (tslen t0)) (tslen ts2))))) (\lambda (x0: TList).(\lambda -(x1: T).(\lambda (H1: (eq TList (TCons t t0) (TApp x0 x1))).(\lambda (H2: (eq -nat (tslen t0) (tslen x0))).(eq_ind_r TList (TApp x0 x1) (\lambda (t2: -TList).(ex2_2 TList T (\lambda (ts2: TList).(\lambda (t3: T).(eq TList (TCons -t1 t2) (TApp ts2 t3)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (S -(tslen t0)) (tslen ts2)))))) (eq_ind_r nat (tslen x0) (\lambda (n: -nat).(ex2_2 TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons -t1 (TApp x0 x1)) (TApp ts2 t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq -nat (S n) (tslen ts2)))))) (ex2_2_intro TList T (\lambda (ts2: -TList).(\lambda (t2: T).(eq TList (TCons t1 (TApp x0 x1)) (TApp ts2 t2)))) -(\lambda (ts2: TList).(\lambda (_: T).(eq nat (S (tslen x0)) (tslen ts2)))) -(TCons t1 x0) x1 (refl_equal TList (TApp (TCons t1 x0) x1)) (refl_equal nat -(tslen (TCons t1 x0)))) (tslen t0) H2) (TCons t t0) H1))))) H0))))))) ts1). -(* COMMENTS -Initial nodes: 503 -END *) - -theorem tlist_ind_rev: - \forall (P: ((TList \to Prop))).((P TNil) \to (((\forall (ts: -TList).(\forall (t: T).((P ts) \to (P (TApp ts t)))))) \to (\forall (ts: -TList).(P ts)))) -\def - \lambda (P: ((TList \to Prop))).(\lambda (H: (P TNil)).(\lambda (H0: -((\forall (ts: TList).(\forall (t: T).((P ts) \to (P (TApp ts -t))))))).(\lambda (ts: TList).(tslt_wf_ind (\lambda (t: TList).(P t)) -(\lambda (ts2: TList).(TList_ind (\lambda (t: TList).(((\forall (ts1: -TList).((tslt ts1 t) \to (P ts1)))) \to (P t))) (\lambda (_: ((\forall (ts1: -TList).((tslt ts1 TNil) \to (P ts1))))).H) (\lambda (t: T).(\lambda (t0: -TList).(\lambda (_: ((((\forall (ts1: TList).((tslt ts1 t0) \to (P ts1)))) -\to (P t0)))).(\lambda (H2: ((\forall (ts1: TList).((tslt ts1 (TCons t t0)) -\to (P ts1))))).(let H_x \def (tcons_tapp_ex t0 t) in (let H3 \def H_x in -(ex2_2_ind TList T (\lambda (ts3: TList).(\lambda (t2: T).(eq TList (TCons t -t0) (TApp ts3 t2)))) (\lambda (ts3: TList).(\lambda (_: T).(eq nat (tslen t0) -(tslen ts3)))) (P (TCons t t0)) (\lambda (x0: TList).(\lambda (x1: -T).(\lambda (H4: (eq TList (TCons t t0) (TApp x0 x1))).(\lambda (H5: (eq nat -(tslen t0) (tslen x0))).(eq_ind_r TList (TApp x0 x1) (\lambda (t1: TList).(P -t1)) (H0 x0 x1 (H2 x0 (eq_ind nat (tslen t0) (\lambda (n: nat).(lt n (tslen -(TCons t t0)))) (le_n (tslen (TCons t t0))) (tslen x0) H5))) (TCons t t0) -H4))))) H3))))))) ts2)) ts)))). -(* COMMENTS -Initial nodes: 273 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/tlt/defs.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/tlt/defs.ma deleted file mode 100644 index 546bdb78d..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/tlt/defs.ma +++ /dev/null @@ -1,46 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/T/defs.ma". - -definition wadd: - ((nat \to nat)) \to (nat \to (nat \to nat)) -\def - \lambda (f: ((nat \to nat))).(\lambda (w: nat).(\lambda (n: nat).(match n -with [O \Rightarrow w | (S m) \Rightarrow (f m)]))). - -definition weight_map: - ((nat \to nat)) \to (T \to nat) -\def - let rec weight_map (f: ((nat \to nat))) (t: T) on t: nat \def (match t with -[(TSort _) \Rightarrow O | (TLRef n) \Rightarrow (f n) | (THead k u t0) -\Rightarrow (match k with [(Bind b) \Rightarrow (match b with [Abbr -\Rightarrow (S (plus (weight_map f u) (weight_map (wadd f (S (weight_map f -u))) t0))) | Abst \Rightarrow (S (plus (weight_map f u) (weight_map (wadd f -O) t0))) | Void \Rightarrow (S (plus (weight_map f u) (weight_map (wadd f O) -t0)))]) | (Flat _) \Rightarrow (S (plus (weight_map f u) (weight_map f -t0)))])]) in weight_map. - -definition weight: - T \to nat -\def - weight_map (\lambda (_: nat).O). - -definition tlt: - T \to (T \to Prop) -\def - \lambda (t1: T).(\lambda (t2: T).(lt (weight t1) (weight t2))). - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/tlt/props.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/tlt/props.ma deleted file mode 100644 index 0b4f16d41..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/tlt/props.ma +++ /dev/null @@ -1,300 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/tlt/defs.ma". - -theorem wadd_le: - \forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (n: -nat).(le (f n) (g n)))) \to (\forall (v: nat).(\forall (w: nat).((le v w) \to -(\forall (n: nat).(le (wadd f v n) (wadd g w n)))))))) -\def - \lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H: -((\forall (n: nat).(le (f n) (g n))))).(\lambda (v: nat).(\lambda (w: -nat).(\lambda (H0: (le v w)).(\lambda (n: nat).(nat_ind (\lambda (n0: -nat).(le (wadd f v n0) (wadd g w n0))) H0 (\lambda (n0: nat).(\lambda (_: (le -(wadd f v n0) (wadd g w n0))).(H n0))) n))))))). -(* COMMENTS -Initial nodes: 81 -END *) - -theorem wadd_lt: - \forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (n: -nat).(le (f n) (g n)))) \to (\forall (v: nat).(\forall (w: nat).((lt v w) \to -(\forall (n: nat).(le (wadd f v n) (wadd g w n)))))))) -\def - \lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H: -((\forall (n: nat).(le (f n) (g n))))).(\lambda (v: nat).(\lambda (w: -nat).(\lambda (H0: (lt v w)).(\lambda (n: nat).(nat_ind (\lambda (n0: -nat).(le (wadd f v n0) (wadd g w n0))) (le_S_n v w (le_S (S v) w H0)) -(\lambda (n0: nat).(\lambda (_: (le (wadd f v n0) (wadd g w n0))).(H n0))) -n))))))). -(* COMMENTS -Initial nodes: 95 -END *) - -theorem wadd_O: - \forall (n: nat).(eq nat (wadd (\lambda (_: nat).O) O n) O) -\def - \lambda (n: nat).(nat_ind (\lambda (n0: nat).(eq nat (wadd (\lambda (_: -nat).O) O n0) O)) (refl_equal nat O) (\lambda (n0: nat).(\lambda (_: (eq nat -(wadd (\lambda (_: nat).O) O n0) O)).(refl_equal nat O))) n). -(* COMMENTS -Initial nodes: 53 -END *) - -theorem weight_le: - \forall (t: T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (n: nat).(le (f n) (g n)))) \to (le (weight_map f t) -(weight_map g t))))) -\def - \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (f: ((nat \to -nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g n)))) -\to (le (weight_map f t0) (weight_map g t0)))))) (\lambda (n: nat).(\lambda -(f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (_: ((\forall -(n0: nat).(le (f n0) (g n0))))).(le_n (weight_map g (TSort n))))))) (\lambda -(n: nat).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda -(H: ((\forall (n0: nat).(le (f n0) (g n0))))).(H n))))) (\lambda (k: -K).(K_ind (\lambda (k0: K).(\forall (t0: T).(((\forall (f: ((nat \to -nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g n)))) -\to (le (weight_map f t0) (weight_map g t0)))))) \to (\forall (t1: -T).(((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall -(n: nat).(le (f n) (g n)))) \to (le (weight_map f t1) (weight_map g t1)))))) -\to (\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall -(n: nat).(le (f n) (g n)))) \to (le (weight_map f (THead k0 t0 t1)) -(weight_map g (THead k0 t0 t1))))))))))) (\lambda (b: B).(B_ind (\lambda (b0: -B).(\forall (t0: T).(((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (n: nat).(le (f n) (g n)))) \to (le (weight_map f t0) -(weight_map g t0)))))) \to (\forall (t1: T).(((\forall (f: ((nat \to -nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g n)))) -\to (le (weight_map f t1) (weight_map g t1)))))) \to (\forall (f: ((nat \to -nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g n)))) -\to (le (match b0 with [Abbr \Rightarrow (S (plus (weight_map f t0) -(weight_map (wadd f (S (weight_map f t0))) t1))) | Abst \Rightarrow (S (plus -(weight_map f t0) (weight_map (wadd f O) t1))) | Void \Rightarrow (S (plus -(weight_map f t0) (weight_map (wadd f O) t1)))]) (match b0 with [Abbr -\Rightarrow (S (plus (weight_map g t0) (weight_map (wadd g (S (weight_map g -t0))) t1))) | Abst \Rightarrow (S (plus (weight_map g t0) (weight_map (wadd g -O) t1))) | Void \Rightarrow (S (plus (weight_map g t0) (weight_map (wadd g O) -t1)))])))))))))) (\lambda (t0: T).(\lambda (H: ((\forall (f: ((nat \to -nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g n)))) -\to (le (weight_map f t0) (weight_map g t0))))))).(\lambda (t1: T).(\lambda -(H0: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall -(n: nat).(le (f n) (g n)))) \to (le (weight_map f t1) (weight_map g -t1))))))).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to -nat))).(\lambda (H1: ((\forall (n: nat).(le (f n) (g n))))).(le_n_S (plus -(weight_map f t0) (weight_map (wadd f (S (weight_map f t0))) t1)) (plus -(weight_map g t0) (weight_map (wadd g (S (weight_map g t0))) t1)) -(le_plus_plus (weight_map f t0) (weight_map g t0) (weight_map (wadd f (S -(weight_map f t0))) t1) (weight_map (wadd g (S (weight_map g t0))) t1) (H f g -H1) (H0 (wadd f (S (weight_map f t0))) (wadd g (S (weight_map g t0))) -(\lambda (n: nat).(wadd_le f g H1 (S (weight_map f t0)) (S (weight_map g t0)) -(le_n_S (weight_map f t0) (weight_map g t0) (H f g H1)) n)))))))))))) -(\lambda (t0: T).(\lambda (H: ((\forall (f: ((nat \to nat))).(\forall (g: -((nat \to nat))).(((\forall (n: nat).(le (f n) (g n)))) \to (le (weight_map f -t0) (weight_map g t0))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (f: -((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) -(g n)))) \to (le (weight_map f t1) (weight_map g t1))))))).(\lambda (f: ((nat -\to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H1: ((\forall (n: nat).(le -(f n) (g n))))).(le_n_S (plus (weight_map f t0) (weight_map (wadd f O) t1)) -(plus (weight_map g t0) (weight_map (wadd g O) t1)) (le_plus_plus (weight_map -f t0) (weight_map g t0) (weight_map (wadd f O) t1) (weight_map (wadd g O) t1) -(H f g H1) (H0 (wadd f O) (wadd g O) (\lambda (n: nat).(wadd_le f g H1 O O -(le_n O) n)))))))))))) (\lambda (t0: T).(\lambda (H: ((\forall (f: ((nat \to -nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g n)))) -\to (le (weight_map f t0) (weight_map g t0))))))).(\lambda (t1: T).(\lambda -(H0: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall -(n: nat).(le (f n) (g n)))) \to (le (weight_map f t1) (weight_map g -t1))))))).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to -nat))).(\lambda (H1: ((\forall (n: nat).(le (f n) (g n))))).(le_n_S (plus -(weight_map f t0) (weight_map (wadd f O) t1)) (plus (weight_map g t0) -(weight_map (wadd g O) t1)) (le_plus_plus (weight_map f t0) (weight_map g t0) -(weight_map (wadd f O) t1) (weight_map (wadd g O) t1) (H f g H1) (H0 (wadd f -O) (wadd g O) (\lambda (n: nat).(wadd_le f g H1 O O (le_n O) n)))))))))))) -b)) (\lambda (_: F).(\lambda (t0: T).(\lambda (H: ((\forall (f0: ((nat \to -nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f0 n) (g n)))) -\to (le (weight_map f0 t0) (weight_map g t0))))))).(\lambda (t1: T).(\lambda -(H0: ((\forall (f0: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall -(n: nat).(le (f0 n) (g n)))) \to (le (weight_map f0 t1) (weight_map g -t1))))))).(\lambda (f0: ((nat \to nat))).(\lambda (g: ((nat \to -nat))).(\lambda (H1: ((\forall (n: nat).(le (f0 n) (g n))))).(le_n_S (plus -(weight_map f0 t0) (weight_map f0 t1)) (plus (weight_map g t0) (weight_map g -t1)) (le_plus_plus (weight_map f0 t0) (weight_map g t0) (weight_map f0 t1) -(weight_map g t1) (H f0 g H1) (H0 f0 g H1))))))))))) k)) t). -(* COMMENTS -Initial nodes: 1309 -END *) - -theorem weight_eq: - \forall (t: T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (n: nat).(eq nat (f n) (g n)))) \to (eq nat (weight_map f -t) (weight_map g t))))) -\def - \lambda (t: T).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to -nat))).(\lambda (H: ((\forall (n: nat).(eq nat (f n) (g n))))).(le_antisym -(weight_map f t) (weight_map g t) (weight_le t f g (\lambda (n: -nat).(eq_ind_r nat (g n) (\lambda (n0: nat).(le n0 (g n))) (le_n (g n)) (f n) -(H n)))) (weight_le t g f (\lambda (n: nat).(eq_ind_r nat (g n) (\lambda (n0: -nat).(le (g n) n0)) (le_n (g n)) (f n) (H n)))))))). -(* COMMENTS -Initial nodes: 121 -END *) - -theorem weight_add_O: - \forall (t: T).(eq nat (weight_map (wadd (\lambda (_: nat).O) O) t) -(weight_map (\lambda (_: nat).O) t)) -\def - \lambda (t: T).(weight_eq t (wadd (\lambda (_: nat).O) O) (\lambda (_: -nat).O) (\lambda (n: nat).(wadd_O n))). -(* COMMENTS -Initial nodes: 23 -END *) - -theorem weight_add_S: - \forall (t: T).(\forall (m: nat).(le (weight_map (wadd (\lambda (_: nat).O) -O) t) (weight_map (wadd (\lambda (_: nat).O) (S m)) t))) -\def - \lambda (t: T).(\lambda (m: nat).(weight_le t (wadd (\lambda (_: nat).O) O) -(wadd (\lambda (_: nat).O) (S m)) (\lambda (n: nat).(wadd_le (\lambda (_: -nat).O) (\lambda (_: nat).O) (\lambda (_: nat).(le_n O)) O (S m) (le_S O m -(le_O_n m)) n)))). -(* COMMENTS -Initial nodes: 61 -END *) - -theorem tlt_trans: - \forall (v: T).(\forall (u: T).(\forall (t: T).((tlt u v) \to ((tlt v t) \to -(tlt u t))))) -\def - \lambda (v: T).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (lt (weight u) -(weight v))).(\lambda (H0: (lt (weight v) (weight t))).(lt_trans (weight u) -(weight v) (weight t) H H0))))). -(* COMMENTS -Initial nodes: 43 -END *) - -theorem tlt_head_sx: - \forall (k: K).(\forall (u: T).(\forall (t: T).(tlt u (THead k u t)))) -\def - \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (u: T).(\forall (t: T).(lt -(weight_map (\lambda (_: nat).O) u) (weight_map (\lambda (_: nat).O) (THead -k0 u t)))))) (\lambda (b: B).(B_ind (\lambda (b0: B).(\forall (u: T).(\forall -(t: T).(lt (weight_map (\lambda (_: nat).O) u) (match b0 with [Abbr -\Rightarrow (S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd -(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t))) | Abst -\Rightarrow (S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd -(\lambda (_: nat).O) O) t))) | Void \Rightarrow (S (plus (weight_map (\lambda -(_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) O) t)))]))))) (\lambda -(u: T).(\lambda (t: T).(le_n_S (weight_map (\lambda (_: nat).O) u) (plus -(weight_map (\lambda (_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) (S -(weight_map (\lambda (_: nat).O) u))) t)) (le_plus_l (weight_map (\lambda (_: -nat).O) u) (weight_map (wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: -nat).O) u))) t))))) (\lambda (u: T).(\lambda (t: T).(le_n_S (weight_map -(\lambda (_: nat).O) u) (plus (weight_map (\lambda (_: nat).O) u) (weight_map -(wadd (\lambda (_: nat).O) O) t)) (le_plus_l (weight_map (\lambda (_: nat).O) -u) (weight_map (wadd (\lambda (_: nat).O) O) t))))) (\lambda (u: T).(\lambda -(t: T).(le_n_S (weight_map (\lambda (_: nat).O) u) (plus (weight_map (\lambda -(_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) O) t)) (le_plus_l -(weight_map (\lambda (_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) O) -t))))) b)) (\lambda (_: F).(\lambda (u: T).(\lambda (t: T).(le_n_S -(weight_map (\lambda (_: nat).O) u) (plus (weight_map (\lambda (_: nat).O) u) -(weight_map (\lambda (_: nat).O) t)) (le_plus_l (weight_map (\lambda (_: -nat).O) u) (weight_map (\lambda (_: nat).O) t)))))) k). -(* COMMENTS -Initial nodes: 379 -END *) - -theorem tlt_head_dx: - \forall (k: K).(\forall (u: T).(\forall (t: T).(tlt t (THead k u t)))) -\def - \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (u: T).(\forall (t: T).(lt -(weight_map (\lambda (_: nat).O) t) (weight_map (\lambda (_: nat).O) (THead -k0 u t)))))) (\lambda (b: B).(B_ind (\lambda (b0: B).(\forall (u: T).(\forall -(t: T).(lt (weight_map (\lambda (_: nat).O) t) (match b0 with [Abbr -\Rightarrow (S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd -(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t))) | Abst -\Rightarrow (S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd -(\lambda (_: nat).O) O) t))) | Void \Rightarrow (S (plus (weight_map (\lambda -(_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) O) t)))]))))) (\lambda -(u: T).(\lambda (t: T).(lt_le_trans (weight_map (\lambda (_: nat).O) t) (S -(weight_map (\lambda (_: nat).O) t)) (S (plus (weight_map (\lambda (_: -nat).O) u) (weight_map (wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: -nat).O) u))) t))) (lt_n_Sn (weight_map (\lambda (_: nat).O) t)) (le_n_S -(weight_map (\lambda (_: nat).O) t) (plus (weight_map (\lambda (_: nat).O) u) -(weight_map (wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) -u))) t)) (le_trans (weight_map (\lambda (_: nat).O) t) (weight_map (wadd -(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t) (plus -(weight_map (\lambda (_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) (S -(weight_map (\lambda (_: nat).O) u))) t)) (eq_ind nat (weight_map (wadd -(\lambda (_: nat).O) O) t) (\lambda (n: nat).(le n (weight_map (wadd (\lambda -(_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t))) (weight_add_S t -(weight_map (\lambda (_: nat).O) u)) (weight_map (\lambda (_: nat).O) t) -(weight_add_O t)) (le_plus_r (weight_map (\lambda (_: nat).O) u) (weight_map -(wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t))))))) -(\lambda (u: T).(\lambda (t: T).(eq_ind_r nat (weight_map (\lambda (_: -nat).O) t) (\lambda (n: nat).(lt (weight_map (\lambda (_: nat).O) t) (S (plus -(weight_map (\lambda (_: nat).O) u) n)))) (le_n_S (weight_map (\lambda (_: -nat).O) t) (plus (weight_map (\lambda (_: nat).O) u) (weight_map (\lambda (_: -nat).O) t)) (le_plus_r (weight_map (\lambda (_: nat).O) u) (weight_map -(\lambda (_: nat).O) t))) (weight_map (wadd (\lambda (_: nat).O) O) t) -(weight_add_O t)))) (\lambda (u: T).(\lambda (t: T).(eq_ind_r nat (weight_map -(\lambda (_: nat).O) t) (\lambda (n: nat).(lt (weight_map (\lambda (_: -nat).O) t) (S (plus (weight_map (\lambda (_: nat).O) u) n)))) (le_n_S -(weight_map (\lambda (_: nat).O) t) (plus (weight_map (\lambda (_: nat).O) u) -(weight_map (\lambda (_: nat).O) t)) (le_plus_r (weight_map (\lambda (_: -nat).O) u) (weight_map (\lambda (_: nat).O) t))) (weight_map (wadd (\lambda -(_: nat).O) O) t) (weight_add_O t)))) b)) (\lambda (_: F).(\lambda (u: -T).(\lambda (t: T).(le_n_S (weight_map (\lambda (_: nat).O) t) (plus -(weight_map (\lambda (_: nat).O) u) (weight_map (\lambda (_: nat).O) t)) -(le_plus_r (weight_map (\lambda (_: nat).O) u) (weight_map (\lambda (_: -nat).O) t)))))) k). -(* COMMENTS -Initial nodes: 659 -END *) - -theorem tlt_wf__q_ind: - \forall (P: ((T \to Prop))).(((\forall (n: nat).((\lambda (P0: ((T \to -Prop))).(\lambda (n0: nat).(\forall (t: T).((eq nat (weight t) n0) \to (P0 -t))))) P n))) \to (\forall (t: T).(P t))) -\def - let Q \def (\lambda (P: ((T \to Prop))).(\lambda (n: nat).(\forall (t: -T).((eq nat (weight t) n) \to (P t))))) in (\lambda (P: ((T \to -Prop))).(\lambda (H: ((\forall (n: nat).(\forall (t: T).((eq nat (weight t) -n) \to (P t)))))).(\lambda (t: T).(H (weight t) t (refl_equal nat (weight -t)))))). -(* COMMENTS -Initial nodes: 61 -END *) - -theorem tlt_wf_ind: - \forall (P: ((T \to Prop))).(((\forall (t: T).(((\forall (v: T).((tlt v t) -\to (P v)))) \to (P t)))) \to (\forall (t: T).(P t))) -\def - let Q \def (\lambda (P: ((T \to Prop))).(\lambda (n: nat).(\forall (t: -T).((eq nat (weight t) n) \to (P t))))) in (\lambda (P: ((T \to -Prop))).(\lambda (H: ((\forall (t: T).(((\forall (v: T).((lt (weight v) -(weight t)) \to (P v)))) \to (P t))))).(\lambda (t: T).(tlt_wf__q_ind -(\lambda (t0: T).(P t0)) (\lambda (n: nat).(lt_wf_ind n (Q (\lambda (t0: -T).(P t0))) (\lambda (n0: nat).(\lambda (H0: ((\forall (m: nat).((lt m n0) -\to (Q (\lambda (t0: T).(P t0)) m))))).(\lambda (t0: T).(\lambda (H1: (eq nat -(weight t0) n0)).(let H2 \def (eq_ind_r nat n0 (\lambda (n1: nat).(\forall -(m: nat).((lt m n1) \to (\forall (t1: T).((eq nat (weight t1) m) \to (P -t1)))))) H0 (weight t0) H1) in (H t0 (\lambda (v: T).(\lambda (H3: (lt -(weight v) (weight t0))).(H2 (weight v) H3 v (refl_equal nat (weight -v))))))))))))) t)))). -(* COMMENTS -Initial nodes: 179 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/arity.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/arity.ma deleted file mode 100644 index 4400e1540..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/arity.ma +++ /dev/null @@ -1,186 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/ty3/pr3_props.ma". - -include "Basic-1/arity/pr3.ma". - -include "Basic-1/asucc/fwd.ma". - -theorem ty3_arity: - \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c -t1 t2) \to (ex2 A (\lambda (a1: A).(arity g c t1 a1)) (\lambda (a1: A).(arity -g c t2 (asucc g a1)))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H: (ty3 g c t1 t2)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda -(t0: T).(ex2 A (\lambda (a1: A).(arity g c0 t a1)) (\lambda (a1: A).(arity g -c0 t0 (asucc g a1))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t: -T).(\lambda (_: (ty3 g c0 t3 t)).(\lambda (H1: (ex2 A (\lambda (a1: A).(arity -g c0 t3 a1)) (\lambda (a1: A).(arity g c0 t (asucc g a1))))).(\lambda (u: -T).(\lambda (t4: T).(\lambda (_: (ty3 g c0 u t4)).(\lambda (H3: (ex2 A -(\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t4 (asucc g -a1))))).(\lambda (H4: (pc3 c0 t4 t3)).(let H5 \def H1 in (ex2_ind A (\lambda -(a1: A).(arity g c0 t3 a1)) (\lambda (a1: A).(arity g c0 t (asucc g a1))) -(ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t3 -(asucc g a1)))) (\lambda (x: A).(\lambda (H6: (arity g c0 t3 x)).(\lambda (_: -(arity g c0 t (asucc g x))).(let H8 \def H3 in (ex2_ind A (\lambda (a1: -A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t4 (asucc g a1))) (ex2 A -(\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t3 (asucc g -a1)))) (\lambda (x0: A).(\lambda (H9: (arity g c0 u x0)).(\lambda (H10: -(arity g c0 t4 (asucc g x0))).(let H11 \def H4 in (ex2_ind T (\lambda (t0: -T).(pr3 c0 t4 t0)) (\lambda (t0: T).(pr3 c0 t3 t0)) (ex2 A (\lambda (a1: -A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t3 (asucc g a1)))) -(\lambda (x1: T).(\lambda (H12: (pr3 c0 t4 x1)).(\lambda (H13: (pr3 c0 t3 -x1)).(ex_intro2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity -g c0 t3 (asucc g a1))) x0 H9 (arity_repl g c0 t3 x H6 (asucc g x0) (leq_sym g -(asucc g x0) x (arity_mono g c0 x1 (asucc g x0) (arity_sred_pr3 c0 t4 x1 H12 -g (asucc g x0) H10) x (arity_sred_pr3 c0 t3 x1 H13 g x H6)))))))) H11))))) -H8))))) H5)))))))))))) (\lambda (c0: C).(\lambda (m: nat).(ex_intro2 A -(\lambda (a1: A).(arity g c0 (TSort m) a1)) (\lambda (a1: A).(arity g c0 -(TSort (next g m)) (asucc g a1))) (ASort O m) (arity_sort g c0 m) (arity_sort -g c0 (next g m))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: -C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abbr) -u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2: (ex2 A -(\lambda (a1: A).(arity g d u a1)) (\lambda (a1: A).(arity g d t (asucc g -a1))))).(let H3 \def H2 in (ex2_ind A (\lambda (a1: A).(arity g d u a1)) -(\lambda (a1: A).(arity g d t (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g -c0 (TLRef n) a1)) (\lambda (a1: A).(arity g c0 (lift (S n) O t) (asucc g -a1)))) (\lambda (x: A).(\lambda (H4: (arity g d u x)).(\lambda (H5: (arity g -d t (asucc g x))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (TLRef n) a1)) -(\lambda (a1: A).(arity g c0 (lift (S n) O t) (asucc g a1))) x (arity_abbr g -c0 d u n H0 x H4) (arity_lift g d t (asucc g x) H5 c0 (S n) O (getl_drop Abbr -c0 d u n H0)))))) H3)))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda -(d: C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abst) -u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2: (ex2 A -(\lambda (a1: A).(arity g d u a1)) (\lambda (a1: A).(arity g d t (asucc g -a1))))).(let H3 \def H2 in (ex2_ind A (\lambda (a1: A).(arity g d u a1)) -(\lambda (a1: A).(arity g d t (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g -c0 (TLRef n) a1)) (\lambda (a1: A).(arity g c0 (lift (S n) O u) (asucc g -a1)))) (\lambda (x: A).(\lambda (H4: (arity g d u x)).(\lambda (_: (arity g d -t (asucc g x))).(let H_x \def (leq_asucc g x) in (let H6 \def H_x in (ex_ind -A (\lambda (a0: A).(leq g x (asucc g a0))) (ex2 A (\lambda (a1: A).(arity g -c0 (TLRef n) a1)) (\lambda (a1: A).(arity g c0 (lift (S n) O u) (asucc g -a1)))) (\lambda (x0: A).(\lambda (H7: (leq g x (asucc g x0))).(ex_intro2 A -(\lambda (a1: A).(arity g c0 (TLRef n) a1)) (\lambda (a1: A).(arity g c0 -(lift (S n) O u) (asucc g a1))) x0 (arity_abst g c0 d u n H0 x0 (arity_repl g -d u x H4 (asucc g x0) H7)) (arity_repl g c0 (lift (S n) O u) x (arity_lift g -d u x H4 c0 (S n) O (getl_drop Abst c0 d u n H0)) (asucc g x0) H7)))) -H6)))))) H3)))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (t: -T).(\lambda (_: (ty3 g c0 u t)).(\lambda (H1: (ex2 A (\lambda (a1: A).(arity -g c0 u a1)) (\lambda (a1: A).(arity g c0 t (asucc g a1))))).(\lambda (b: -B).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) -u) t3 t4)).(\lambda (H3: (ex2 A (\lambda (a1: A).(arity g (CHead c0 (Bind b) -u) t3 a1)) (\lambda (a1: A).(arity g (CHead c0 (Bind b) u) t4 (asucc g -a1))))).(let H4 \def H1 in (ex2_ind A (\lambda (a1: A).(arity g c0 u a1)) -(\lambda (a1: A).(arity g c0 t (asucc g a1))) (ex2 A (\lambda (a1: A).(arity -g c0 (THead (Bind b) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind b) -u t4) (asucc g a1)))) (\lambda (x: A).(\lambda (H5: (arity g c0 u -x)).(\lambda (_: (arity g c0 t (asucc g x))).(let H7 \def H3 in (ex2_ind A -(\lambda (a1: A).(arity g (CHead c0 (Bind b) u) t3 a1)) (\lambda (a1: -A).(arity g (CHead c0 (Bind b) u) t4 (asucc g a1))) (ex2 A (\lambda (a1: -A).(arity g c0 (THead (Bind b) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead -(Bind b) u t4) (asucc g a1)))) (\lambda (x0: A).(\lambda (H8: (arity g (CHead -c0 (Bind b) u) t3 x0)).(\lambda (H9: (arity g (CHead c0 (Bind b) u) t4 (asucc -g x0))).(let H_x \def (leq_asucc g x) in (let H10 \def H_x in (ex_ind A -(\lambda (a0: A).(leq g x (asucc g a0))) (ex2 A (\lambda (a1: A).(arity g c0 -(THead (Bind b) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind b) u t4) -(asucc g a1)))) (\lambda (x1: A).(\lambda (H11: (leq g x (asucc g -x1))).(B_ind (\lambda (b0: B).((arity g (CHead c0 (Bind b0) u) t3 x0) \to -((arity g (CHead c0 (Bind b0) u) t4 (asucc g x0)) \to (ex2 A (\lambda (a1: -A).(arity g c0 (THead (Bind b0) u t3) a1)) (\lambda (a1: A).(arity g c0 -(THead (Bind b0) u t4) (asucc g a1))))))) (\lambda (H12: (arity g (CHead c0 -(Bind Abbr) u) t3 x0)).(\lambda (H13: (arity g (CHead c0 (Bind Abbr) u) t4 -(asucc g x0))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Bind Abbr) u -t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind Abbr) u t4) (asucc g a1))) -x0 (arity_bind g Abbr not_abbr_abst c0 u x H5 t3 x0 H12) (arity_bind g Abbr -not_abbr_abst c0 u x H5 t4 (asucc g x0) H13)))) (\lambda (H12: (arity g -(CHead c0 (Bind Abst) u) t3 x0)).(\lambda (H13: (arity g (CHead c0 (Bind -Abst) u) t4 (asucc g x0))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (THead -(Bind Abst) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind Abst) u t4) -(asucc g a1))) (AHead x1 x0) (arity_head g c0 u x1 (arity_repl g c0 u x H5 -(asucc g x1) H11) t3 x0 H12) (arity_repl g c0 (THead (Bind Abst) u t4) (AHead -x1 (asucc g x0)) (arity_head g c0 u x1 (arity_repl g c0 u x H5 (asucc g x1) -H11) t4 (asucc g x0) H13) (asucc g (AHead x1 x0)) (leq_refl g (asucc g (AHead -x1 x0))))))) (\lambda (H12: (arity g (CHead c0 (Bind Void) u) t3 -x0)).(\lambda (H13: (arity g (CHead c0 (Bind Void) u) t4 (asucc g -x0))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Bind Void) u t3) a1)) -(\lambda (a1: A).(arity g c0 (THead (Bind Void) u t4) (asucc g a1))) x0 -(arity_bind g Void (sym_not_eq B Abst Void not_abst_void) c0 u x H5 t3 x0 -H12) (arity_bind g Void (sym_not_eq B Abst Void not_abst_void) c0 u x H5 t4 -(asucc g x0) H13)))) b H8 H9))) H10)))))) H7))))) H4)))))))))))) (\lambda -(c0: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda -(H1: (ex2 A (\lambda (a1: A).(arity g c0 w a1)) (\lambda (a1: A).(arity g c0 -u (asucc g a1))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v -(THead (Bind Abst) u t))).(\lambda (H3: (ex2 A (\lambda (a1: A).(arity g c0 v -a1)) (\lambda (a1: A).(arity g c0 (THead (Bind Abst) u t) (asucc g -a1))))).(let H4 \def H1 in (ex2_ind A (\lambda (a1: A).(arity g c0 w a1)) -(\lambda (a1: A).(arity g c0 u (asucc g a1))) (ex2 A (\lambda (a1: A).(arity -g c0 (THead (Flat Appl) w v) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat -Appl) w (THead (Bind Abst) u t)) (asucc g a1)))) (\lambda (x: A).(\lambda -(H5: (arity g c0 w x)).(\lambda (H6: (arity g c0 u (asucc g x))).(let H7 \def -H3 in (ex2_ind A (\lambda (a1: A).(arity g c0 v a1)) (\lambda (a1: A).(arity -g c0 (THead (Bind Abst) u t) (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g -c0 (THead (Flat Appl) w v) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat -Appl) w (THead (Bind Abst) u t)) (asucc g a1)))) (\lambda (x0: A).(\lambda -(H8: (arity g c0 v x0)).(\lambda (H9: (arity g c0 (THead (Bind Abst) u t) -(asucc g x0))).(let H10 \def (arity_gen_abst g c0 u t (asucc g x0) H9) in -(ex3_2_ind A A (\lambda (a1: A).(\lambda (a2: A).(eq A (asucc g x0) (AHead a1 -a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g c0 u (asucc g a1)))) -(\lambda (_: A).(\lambda (a2: A).(arity g (CHead c0 (Bind Abst) u) t a2))) -(ex2 A (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w v) a1)) (\lambda -(a1: A).(arity g c0 (THead (Flat Appl) w (THead (Bind Abst) u t)) (asucc g -a1)))) (\lambda (x1: A).(\lambda (x2: A).(\lambda (H11: (eq A (asucc g x0) -(AHead x1 x2))).(\lambda (H12: (arity g c0 u (asucc g x1))).(\lambda (H13: -(arity g (CHead c0 (Bind Abst) u) t x2)).(let H14 \def (sym_eq A (asucc g x0) -(AHead x1 x2) H11) in (let H15 \def (asucc_gen_head g x1 x2 x0 H14) in -(ex2_ind A (\lambda (a0: A).(eq A x0 (AHead x1 a0))) (\lambda (a0: A).(eq A -x2 (asucc g a0))) (ex2 A (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w v) -a1)) (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w (THead (Bind Abst) u -t)) (asucc g a1)))) (\lambda (x3: A).(\lambda (H16: (eq A x0 (AHead x1 -x3))).(\lambda (H17: (eq A x2 (asucc g x3))).(let H18 \def (eq_ind A x2 -(\lambda (a: A).(arity g (CHead c0 (Bind Abst) u) t a)) H13 (asucc g x3) H17) -in (let H19 \def (eq_ind A x0 (\lambda (a: A).(arity g c0 v a)) H8 (AHead x1 -x3) H16) in (ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w v) -a1)) (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w (THead (Bind Abst) u -t)) (asucc g a1))) x3 (arity_appl g c0 w x1 (arity_repl g c0 w x H5 x1 -(leq_sym g x1 x (asucc_inj g x1 x (arity_mono g c0 u (asucc g x1) H12 (asucc -g x) H6)))) v x3 H19) (arity_appl g c0 w x H5 (THead (Bind Abst) u t) (asucc -g x3) (arity_head g c0 u x H6 t (asucc g x3) H18)))))))) H15)))))))) H10))))) -H7))))) H4))))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: -T).(\lambda (_: (ty3 g c0 t3 t4)).(\lambda (H1: (ex2 A (\lambda (a1: -A).(arity g c0 t3 a1)) (\lambda (a1: A).(arity g c0 t4 (asucc g -a1))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t4 t0)).(\lambda (H3: (ex2 A -(\lambda (a1: A).(arity g c0 t4 a1)) (\lambda (a1: A).(arity g c0 t0 (asucc g -a1))))).(let H4 \def H1 in (ex2_ind A (\lambda (a1: A).(arity g c0 t3 a1)) -(\lambda (a1: A).(arity g c0 t4 (asucc g a1))) (ex2 A (\lambda (a1: A).(arity -g c0 (THead (Flat Cast) t4 t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat -Cast) t0 t4) (asucc g a1)))) (\lambda (x: A).(\lambda (H5: (arity g c0 t3 -x)).(\lambda (H6: (arity g c0 t4 (asucc g x))).(let H7 \def H3 in (ex2_ind A -(\lambda (a1: A).(arity g c0 t4 a1)) (\lambda (a1: A).(arity g c0 t0 (asucc g -a1))) (ex2 A (\lambda (a1: A).(arity g c0 (THead (Flat Cast) t4 t3) a1)) -(\lambda (a1: A).(arity g c0 (THead (Flat Cast) t0 t4) (asucc g a1)))) -(\lambda (x0: A).(\lambda (H8: (arity g c0 t4 x0)).(\lambda (H9: (arity g c0 -t0 (asucc g x0))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Flat -Cast) t4 t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat Cast) t0 t4) -(asucc g a1))) x (arity_cast g c0 t4 x H6 t3 H5) (arity_cast g c0 t0 (asucc g -x) (arity_repl g c0 t0 (asucc g x0) H9 (asucc g (asucc g x)) (asucc_repl g x0 -(asucc g x) (arity_mono g c0 t4 x0 H8 (asucc g x) H6))) t4 H6))))) H7))))) -H4)))))))))) c t1 t2 H))))). -(* COMMENTS -Initial nodes: 3761 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/arity_props.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/arity_props.ma deleted file mode 100644 index 49d6c0572..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/arity_props.ma +++ /dev/null @@ -1,117 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/ty3/arity.ma". - -include "Basic-1/sc3/arity.ma". - -theorem ty3_predicative: - \forall (g: G).(\forall (c: C).(\forall (v: T).(\forall (t: T).(\forall (u: -T).((ty3 g c (THead (Bind Abst) v t) u) \to ((pc3 c u v) \to (\forall (P: -Prop).P))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (u: -T).(\lambda (H: (ty3 g c (THead (Bind Abst) v t) u)).(\lambda (H0: (pc3 c u -v)).(\lambda (P: Prop).(let H1 \def H in (ex3_2_ind T T (\lambda (t2: -T).(\lambda (_: T).(pc3 c (THead (Bind Abst) v t2) u))) (\lambda (_: -T).(\lambda (t0: T).(ty3 g c v t0))) (\lambda (t2: T).(\lambda (_: T).(ty3 g -(CHead c (Bind Abst) v) t t2))) P (\lambda (x0: T).(\lambda (x1: T).(\lambda -(_: (pc3 c (THead (Bind Abst) v x0) u)).(\lambda (H3: (ty3 g c v -x1)).(\lambda (_: (ty3 g (CHead c (Bind Abst) v) t x0)).(let H_y \def -(ty3_conv g c v x1 H3 (THead (Bind Abst) v t) u H H0) in (let H_x \def -(ty3_arity g c (THead (Bind Abst) v t) v H_y) in (let H5 \def H_x in (ex2_ind -A (\lambda (a1: A).(arity g c (THead (Bind Abst) v t) a1)) (\lambda (a1: -A).(arity g c v (asucc g a1))) P (\lambda (x: A).(\lambda (H6: (arity g c -(THead (Bind Abst) v t) x)).(\lambda (H7: (arity g c v (asucc g x))).(let H8 -\def (arity_gen_abst g c v t x H6) in (ex3_2_ind A A (\lambda (a1: -A).(\lambda (a2: A).(eq A x (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: -A).(arity g c v (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g -(CHead c (Bind Abst) v) t a2))) P (\lambda (x2: A).(\lambda (x3: A).(\lambda -(H9: (eq A x (AHead x2 x3))).(\lambda (H10: (arity g c v (asucc g -x2))).(\lambda (_: (arity g (CHead c (Bind Abst) v) t x3)).(let H12 \def -(eq_ind A x (\lambda (a: A).(arity g c v (asucc g a))) H7 (AHead x2 x3) H9) -in (leq_ahead_asucc_false g x2 (asucc g x3) (arity_mono g c v (asucc g (AHead -x2 x3)) H12 (asucc g x2) H10) P))))))) H8))))) H5))))))))) (ty3_gen_bind g -Abst c v t u H1)))))))))). -(* COMMENTS -Initial nodes: 497 -END *) - -theorem ty3_repellent: - \forall (g: G).(\forall (c: C).(\forall (w: T).(\forall (t: T).(\forall (u1: -T).((ty3 g c (THead (Bind Abst) w t) u1) \to (\forall (u2: T).((ty3 g (CHead -c (Bind Abst) w) t (lift (S O) O u2)) \to ((pc3 c u1 u2) \to (\forall (P: -Prop).P))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (w: T).(\lambda (t: T).(\lambda (u1: -T).(\lambda (H: (ty3 g c (THead (Bind Abst) w t) u1)).(\lambda (u2: -T).(\lambda (H0: (ty3 g (CHead c (Bind Abst) w) t (lift (S O) O -u2))).(\lambda (H1: (pc3 c u1 u2)).(\lambda (P: Prop).(ex_ind T (\lambda (t0: -T).(ty3 g (CHead c (Bind Abst) w) (lift (S O) O u2) t0)) P (\lambda (x: -T).(\lambda (H2: (ty3 g (CHead c (Bind Abst) w) (lift (S O) O u2) x)).(let H3 -\def (ty3_gen_lift g (CHead c (Bind Abst) w) u2 x (S O) O H2 c (drop_drop -(Bind Abst) O c c (drop_refl c) w)) in (ex2_ind T (\lambda (t2: T).(pc3 -(CHead c (Bind Abst) w) (lift (S O) O t2) x)) (\lambda (t2: T).(ty3 g c u2 -t2)) P (\lambda (x0: T).(\lambda (_: (pc3 (CHead c (Bind Abst) w) (lift (S O) -O x0) x)).(\lambda (H5: (ty3 g c u2 x0)).(let H_y \def (ty3_conv g c u2 x0 H5 -(THead (Bind Abst) w t) u1 H H1) in (let H_x \def (ty3_arity g (CHead c (Bind -Abst) w) t (lift (S O) O u2) H0) in (let H6 \def H_x in (ex2_ind A (\lambda -(a1: A).(arity g (CHead c (Bind Abst) w) t a1)) (\lambda (a1: A).(arity g -(CHead c (Bind Abst) w) (lift (S O) O u2) (asucc g a1))) P (\lambda (x1: -A).(\lambda (H7: (arity g (CHead c (Bind Abst) w) t x1)).(\lambda (H8: (arity -g (CHead c (Bind Abst) w) (lift (S O) O u2) (asucc g x1))).(let H_x0 \def -(ty3_arity g c (THead (Bind Abst) w t) u2 H_y) in (let H9 \def H_x0 in -(ex2_ind A (\lambda (a1: A).(arity g c (THead (Bind Abst) w t) a1)) (\lambda -(a1: A).(arity g c u2 (asucc g a1))) P (\lambda (x2: A).(\lambda (H10: (arity -g c (THead (Bind Abst) w t) x2)).(\lambda (H11: (arity g c u2 (asucc g -x2))).(arity_repellent g c w t x1 H7 x2 H10 (asucc_inj g x1 x2 (arity_mono g -c u2 (asucc g x1) (arity_gen_lift g (CHead c (Bind Abst) w) u2 (asucc g x1) -(S O) O H8 c (drop_drop (Bind Abst) O c c (drop_refl c) w)) (asucc g x2) -H11)) P)))) H9)))))) H6))))))) H3)))) (ty3_correct g (CHead c (Bind Abst) w) -t (lift (S O) O u2) H0))))))))))). -(* COMMENTS -Initial nodes: 651 -END *) - -theorem ty3_acyclic: - \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t -u) \to ((pc3 c u t) \to (\forall (P: Prop).P)))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H: -(ty3 g c t u)).(\lambda (H0: (pc3 c u t)).(\lambda (P: Prop).(let H_y \def -(ty3_conv g c t u H t u H H0) in (let H_x \def (ty3_arity g c t t H_y) in -(let H1 \def H_x in (ex2_ind A (\lambda (a1: A).(arity g c t a1)) (\lambda -(a1: A).(arity g c t (asucc g a1))) P (\lambda (x: A).(\lambda (H2: (arity g -c t x)).(\lambda (H3: (arity g c t (asucc g x))).(leq_asucc_false g x -(arity_mono g c t (asucc g x) H3 x H2) P)))) H1)))))))))). -(* COMMENTS -Initial nodes: 151 -END *) - -theorem ty3_sn3: - \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t -u) \to (sn3 c t))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H: -(ty3 g c t u)).(let H_x \def (ty3_arity g c t u H) in (let H0 \def H_x in -(ex2_ind A (\lambda (a1: A).(arity g c t a1)) (\lambda (a1: A).(arity g c u -(asucc g a1))) (sn3 c t) (\lambda (x: A).(\lambda (H1: (arity g c t -x)).(\lambda (_: (arity g c u (asucc g x))).(sc3_sn3 g x c t (sc3_arity g c t -x H1))))) H0))))))). -(* COMMENTS -Initial nodes: 119 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/dec.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/dec.ma deleted file mode 100644 index c176d635e..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/dec.ma +++ /dev/null @@ -1,438 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pc3/dec.ma". - -include "Basic-1/getl/flt.ma". - -include "Basic-1/getl/dec.ma". - -theorem ty3_inference: - \forall (g: G).(\forall (c: C).(\forall (t1: T).(or (ex T (\lambda (t2: -T).(ty3 g c t1 t2))) (\forall (t2: T).((ty3 g c t1 t2) \to False))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(flt_wf_ind (\lambda (c0: -C).(\lambda (t: T).(or (ex T (\lambda (t2: T).(ty3 g c0 t t2))) (\forall (t2: -T).((ty3 g c0 t t2) \to False))))) (\lambda (c2: C).(\lambda (t2: T).(T_ind -(\lambda (t: T).(((\forall (c1: C).(\forall (t3: T).((flt c1 t3 c2 t) \to (or -(ex T (\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4) -\to False))))))) \to (or (ex T (\lambda (t3: T).(ty3 g c2 t t3))) (\forall -(t3: T).((ty3 g c2 t t3) \to False))))) (\lambda (n: nat).(\lambda (_: -((\forall (c1: C).(\forall (t3: T).((flt c1 t3 c2 (TSort n)) \to (or (ex T -(\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to -False)))))))).(or_introl (ex T (\lambda (t3: T).(ty3 g c2 (TSort n) t3))) -(\forall (t3: T).((ty3 g c2 (TSort n) t3) \to False)) (ex_intro T (\lambda -(t3: T).(ty3 g c2 (TSort n) t3)) (TSort (next g n)) (ty3_sort g c2 n))))) -(\lambda (n: nat).(\lambda (H: ((\forall (c1: C).(\forall (t3: T).((flt c1 t3 -c2 (TLRef n)) \to (or (ex T (\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: -T).((ty3 g c1 t3 t4) \to False)))))))).(let H_x \def (getl_dec c2 n) in (let -H0 \def H_x in (or_ind (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda -(v: T).(getl n c2 (CHead e (Bind b) v)))))) (\forall (d: C).((getl n c2 d) -\to (\forall (P: Prop).P))) (or (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) -t3))) (\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to False))) (\lambda (H1: -(ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl n c2 (CHead -e (Bind b) v))))))).(ex_3_ind C B T (\lambda (e: C).(\lambda (b: B).(\lambda -(v: T).(getl n c2 (CHead e (Bind b) v))))) (or (ex T (\lambda (t3: T).(ty3 g -c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to False))) -(\lambda (x0: C).(\lambda (x1: B).(\lambda (x2: T).(\lambda (H2: (getl n c2 -(CHead x0 (Bind x1) x2))).(let H3 \def (H x0 x2 (getl_flt x1 c2 x0 x2 n H2)) -in (or_ind (ex T (\lambda (t3: T).(ty3 g x0 x2 t3))) (\forall (t3: T).((ty3 g -x0 x2 t3) \to False)) (or (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) -(\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to False))) (\lambda (H4: (ex T -(\lambda (t3: T).(ty3 g x0 x2 t3)))).(ex_ind T (\lambda (t3: T).(ty3 g x0 x2 -t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: -T).((ty3 g c2 (TLRef n) t3) \to False))) (\lambda (x: T).(\lambda (H5: (ty3 g -x0 x2 x)).(B_ind (\lambda (b: B).((getl n c2 (CHead x0 (Bind b) x2)) \to (or -(ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 -(TLRef n) t3) \to False))))) (\lambda (H6: (getl n c2 (CHead x0 (Bind Abbr) -x2))).(or_introl (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall -(t3: T).((ty3 g c2 (TLRef n) t3) \to False)) (ex_intro T (\lambda (t3: -T).(ty3 g c2 (TLRef n) t3)) (lift (S n) O x) (ty3_abbr g n c2 x0 x2 H6 x -H5)))) (\lambda (H6: (getl n c2 (CHead x0 (Bind Abst) x2))).(or_introl (ex T -(\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef -n) t3) \to False)) (ex_intro T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3)) -(lift (S n) O x2) (ty3_abst g n c2 x0 x2 H6 x H5)))) (\lambda (H6: (getl n c2 -(CHead x0 (Bind Void) x2))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 -(TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to False)) -(\lambda (t3: T).(\lambda (H7: (ty3 g c2 (TLRef n) t3)).(or_ind (ex3_3 C T T -(\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) -t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e -(Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u -t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 -(lift (S n) O u) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl -n c2 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: -T).(ty3 g e u t))))) False (\lambda (H8: (ex3_3 C T T (\lambda (_: -C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda -(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u))))) -(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind -C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O -t) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e -(Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u -t)))) False (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: -(pc3 c2 (lift (S n) O x5) t3)).(\lambda (H10: (getl n c2 (CHead x3 (Bind -Abbr) x4))).(\lambda (_: (ty3 g x3 x4 x5)).(let H12 \def (eq_ind C (CHead x0 -(Bind Void) x2) (\lambda (c0: C).(getl n c2 c0)) H6 (CHead x3 (Bind Abbr) x4) -(getl_mono c2 (CHead x0 (Bind Void) x2) n H6 (CHead x3 (Bind Abbr) x4) H10)) -in (let H13 \def (eq_ind C (CHead x0 (Bind Void) x2) (\lambda (ee: C).(match -ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | -(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr -\Rightarrow False | Abst \Rightarrow False | Void \Rightarrow True]) | (Flat -_) \Rightarrow False])])) I (CHead x3 (Bind Abbr) x4) (getl_mono c2 (CHead x0 -(Bind Void) x2) n H6 (CHead x3 (Bind Abbr) x4) H10)) in (False_ind False -H13))))))))) H8)) (\lambda (H8: (ex3_3 C T T (\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e: C).(\lambda -(u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u))))) (\lambda (e: -C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T -(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) -t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e -(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u -t)))) False (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: -(pc3 c2 (lift (S n) O x4) t3)).(\lambda (H10: (getl n c2 (CHead x3 (Bind -Abst) x4))).(\lambda (_: (ty3 g x3 x4 x5)).(let H12 \def (eq_ind C (CHead x0 -(Bind Void) x2) (\lambda (c0: C).(getl n c2 c0)) H6 (CHead x3 (Bind Abst) x4) -(getl_mono c2 (CHead x0 (Bind Void) x2) n H6 (CHead x3 (Bind Abst) x4) H10)) -in (let H13 \def (eq_ind C (CHead x0 (Bind Void) x2) (\lambda (ee: C).(match -ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | -(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr -\Rightarrow False | Abst \Rightarrow False | Void \Rightarrow True]) | (Flat -_) \Rightarrow False])])) I (CHead x3 (Bind Abst) x4) (getl_mono c2 (CHead x0 -(Bind Void) x2) n H6 (CHead x3 (Bind Abst) x4) H10)) in (False_ind False -H13))))))))) H8)) (ty3_gen_lref g c2 t3 n H7)))))) x1 H2))) H4)) (\lambda -(H4: ((\forall (t3: T).((ty3 g x0 x2 t3) \to False)))).(or_intror (ex T -(\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef -n) t3) \to False)) (\lambda (t3: T).(\lambda (H5: (ty3 g c2 (TLRef n) -t3)).(or_ind (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: -T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda -(_: T).(getl n c2 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: -T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e: -C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u))))) -(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))) False -(\lambda (H6: (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: -T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda -(_: T).(getl n c2 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: -T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T (\lambda (_: -C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda -(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u))))) -(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))) False -(\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (pc3 c2 (lift -(S n) O x5) t3)).(\lambda (H8: (getl n c2 (CHead x3 (Bind Abbr) -x4))).(\lambda (H9: (ty3 g x3 x4 x5)).(let H10 \def (eq_ind C (CHead x0 (Bind -x1) x2) (\lambda (c0: C).(getl n c2 c0)) H2 (CHead x3 (Bind Abbr) x4) -(getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind Abbr) x4) H8)) in -(let H11 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: -C).C) with [(CSort _) \Rightarrow x0 | (CHead c0 _ _) \Rightarrow c0])) -(CHead x0 (Bind x1) x2) (CHead x3 (Bind Abbr) x4) (getl_mono c2 (CHead x0 -(Bind x1) x2) n H2 (CHead x3 (Bind Abbr) x4) H8)) in ((let H12 \def (f_equal -C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with [(CSort _) -\Rightarrow x1 | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: -K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow x1])])) (CHead x0 -(Bind x1) x2) (CHead x3 (Bind Abbr) x4) (getl_mono c2 (CHead x0 (Bind x1) x2) -n H2 (CHead x3 (Bind Abbr) x4) H8)) in ((let H13 \def (f_equal C T (\lambda -(e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow x2 -| (CHead _ _ t) \Rightarrow t])) (CHead x0 (Bind x1) x2) (CHead x3 (Bind -Abbr) x4) (getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind Abbr) -x4) H8)) in (\lambda (_: (eq B x1 Abbr)).(\lambda (H15: (eq C x0 x3)).(let -H16 \def (eq_ind_r T x4 (\lambda (t: T).(getl n c2 (CHead x3 (Bind Abbr) t))) -H10 x2 H13) in (let H17 \def (eq_ind_r T x4 (\lambda (t: T).(ty3 g x3 t x5)) -H9 x2 H13) in (let H18 \def (eq_ind_r C x3 (\lambda (c0: C).(getl n c2 (CHead -c0 (Bind Abbr) x2))) H16 x0 H15) in (let H19 \def (eq_ind_r C x3 (\lambda -(c0: C).(ty3 g c0 x2 x5)) H17 x0 H15) in (H4 x5 H19)))))))) H12)) -H11))))))))) H6)) (\lambda (H6: (ex3_3 C T T (\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e: C).(\lambda -(u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u))))) (\lambda (e: -C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T -(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) -t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e -(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u -t)))) False (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H7: -(pc3 c2 (lift (S n) O x4) t3)).(\lambda (H8: (getl n c2 (CHead x3 (Bind Abst) -x4))).(\lambda (H9: (ty3 g x3 x4 x5)).(let H10 \def (eq_ind C (CHead x0 (Bind -x1) x2) (\lambda (c0: C).(getl n c2 c0)) H2 (CHead x3 (Bind Abst) x4) -(getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind Abst) x4) H8)) in -(let H11 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: -C).C) with [(CSort _) \Rightarrow x0 | (CHead c0 _ _) \Rightarrow c0])) -(CHead x0 (Bind x1) x2) (CHead x3 (Bind Abst) x4) (getl_mono c2 (CHead x0 -(Bind x1) x2) n H2 (CHead x3 (Bind Abst) x4) H8)) in ((let H12 \def (f_equal -C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with [(CSort _) -\Rightarrow x1 | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: -K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow x1])])) (CHead x0 -(Bind x1) x2) (CHead x3 (Bind Abst) x4) (getl_mono c2 (CHead x0 (Bind x1) x2) -n H2 (CHead x3 (Bind Abst) x4) H8)) in ((let H13 \def (f_equal C T (\lambda -(e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow x2 -| (CHead _ _ t) \Rightarrow t])) (CHead x0 (Bind x1) x2) (CHead x3 (Bind -Abst) x4) (getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind Abst) -x4) H8)) in (\lambda (_: (eq B x1 Abst)).(\lambda (H15: (eq C x0 x3)).(let -H16 \def (eq_ind_r T x4 (\lambda (t: T).(getl n c2 (CHead x3 (Bind Abst) t))) -H10 x2 H13) in (let H17 \def (eq_ind_r T x4 (\lambda (t: T).(ty3 g x3 t x5)) -H9 x2 H13) in (let H18 \def (eq_ind_r T x4 (\lambda (t: T).(pc3 c2 (lift (S -n) O t) t3)) H7 x2 H13) in (let H19 \def (eq_ind_r C x3 (\lambda (c0: -C).(getl n c2 (CHead c0 (Bind Abst) x2))) H16 x0 H15) in (let H20 \def -(eq_ind_r C x3 (\lambda (c0: C).(ty3 g c0 x2 x5)) H17 x0 H15) in (H4 x5 -H20))))))))) H12)) H11))))))))) H6)) (ty3_gen_lref g c2 t3 n H5)))))) -H3)))))) H1)) (\lambda (H1: ((\forall (d: C).((getl n c2 d) \to (\forall (P: -Prop).P))))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) -(\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to False)) (\lambda (t3: -T).(\lambda (H2: (ty3 g c2 (TLRef n) t3)).(or_ind (ex3_3 C T T (\lambda (_: -C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda -(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u))))) -(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T -T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) -t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e -(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u -t))))) False (\lambda (H3: (ex3_3 C T T (\lambda (_: C).(\lambda (_: -T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda (e: C).(\lambda -(u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u))))) (\lambda (e: -C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T -(\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) -t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e -(Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u -t)))) False (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: -(pc3 c2 (lift (S n) O x2) t3)).(\lambda (H5: (getl n c2 (CHead x0 (Bind Abbr) -x1))).(\lambda (_: (ty3 g x0 x1 x2)).(H1 (CHead x0 (Bind Abbr) x1) H5 -False))))))) H3)) (\lambda (H3: (ex3_3 C T T (\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e: C).(\lambda -(u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u))))) (\lambda (e: -C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T -(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) -t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e -(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u -t)))) False (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: -(pc3 c2 (lift (S n) O x1) t3)).(\lambda (H5: (getl n c2 (CHead x0 (Bind Abst) -x1))).(\lambda (_: (ty3 g x0 x1 x2)).(H1 (CHead x0 (Bind Abst) x1) H5 -False))))))) H3)) (ty3_gen_lref g c2 t3 n H2)))))) H0))))) (\lambda (k: -K).(\lambda (t: T).(\lambda (_: ((((\forall (c1: C).(\forall (t3: T).((flt c1 -t3 c2 t) \to (or (ex T (\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: -T).((ty3 g c1 t3 t4) \to False))))))) \to (or (ex T (\lambda (t3: T).(ty3 g -c2 t t3))) (\forall (t3: T).((ty3 g c2 t t3) \to False)))))).(\lambda (t0: -T).(\lambda (_: ((((\forall (c1: C).(\forall (t3: T).((flt c1 t3 c2 t0) \to -(or (ex T (\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3 g c1 t3 -t4) \to False))))))) \to (or (ex T (\lambda (t3: T).(ty3 g c2 t0 t3))) -(\forall (t3: T).((ty3 g c2 t0 t3) \to False)))))).(\lambda (H1: ((\forall -(c1: C).(\forall (t3: T).((flt c1 t3 c2 (THead k t t0)) \to (or (ex T -(\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to -False)))))))).(K_ind (\lambda (k0: K).(((\forall (c1: C).(\forall (t3: -T).((flt c1 t3 c2 (THead k0 t t0)) \to (or (ex T (\lambda (t4: T).(ty3 g c1 -t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to False))))))) \to (or (ex T -(\lambda (t3: T).(ty3 g c2 (THead k0 t t0) t3))) (\forall (t3: T).((ty3 g c2 -(THead k0 t t0) t3) \to False))))) (\lambda (b: B).(\lambda (H2: ((\forall -(c1: C).(\forall (t3: T).((flt c1 t3 c2 (THead (Bind b) t t0)) \to (or (ex T -(\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to -False)))))))).(let H3 \def (H2 c2 t (flt_thead_sx (Bind b) c2 t t0)) in -(or_ind (ex T (\lambda (t3: T).(ty3 g c2 t t3))) (\forall (t3: T).((ty3 g c2 -t t3) \to False)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Bind b) t t0) -t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0) t3) \to False))) -(\lambda (H4: (ex T (\lambda (t3: T).(ty3 g c2 t t3)))).(ex_ind T (\lambda -(t3: T).(ty3 g c2 t t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Bind b) -t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0) t3) \to -False))) (\lambda (x: T).(\lambda (H5: (ty3 g c2 t x)).(let H6 \def (H2 -(CHead c2 (Bind b) t) t0 (flt_shift (Bind b) c2 t t0)) in (or_ind (ex T -(\lambda (t3: T).(ty3 g (CHead c2 (Bind b) t) t0 t3))) (\forall (t3: T).((ty3 -g (CHead c2 (Bind b) t) t0 t3) \to False)) (or (ex T (\lambda (t3: T).(ty3 g -c2 (THead (Bind b) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t -t0) t3) \to False))) (\lambda (H7: (ex T (\lambda (t3: T).(ty3 g (CHead c2 -(Bind b) t) t0 t3)))).(ex_ind T (\lambda (t3: T).(ty3 g (CHead c2 (Bind b) t) -t0 t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Bind b) t t0) t3))) -(\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0) t3) \to False))) (\lambda -(x0: T).(\lambda (H8: (ty3 g (CHead c2 (Bind b) t) t0 x0)).(or_introl (ex T -(\lambda (t3: T).(ty3 g c2 (THead (Bind b) t t0) t3))) (\forall (t3: T).((ty3 -g c2 (THead (Bind b) t t0) t3) \to False)) (ex_intro T (\lambda (t3: T).(ty3 -g c2 (THead (Bind b) t t0) t3)) (THead (Bind b) t x0) (ty3_bind g c2 t x H5 b -t0 x0 H8))))) H7)) (\lambda (H7: ((\forall (t3: T).((ty3 g (CHead c2 (Bind b) -t) t0 t3) \to False)))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead -(Bind b) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0) t3) -\to False)) (\lambda (t3: T).(\lambda (H8: (ty3 g c2 (THead (Bind b) t t0) -t3)).(ex3_2_ind T T (\lambda (t4: T).(\lambda (_: T).(pc3 c2 (THead (Bind b) -t t4) t3))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c2 t t5))) (\lambda (t4: -T).(\lambda (_: T).(ty3 g (CHead c2 (Bind b) t) t0 t4))) False (\lambda (x0: -T).(\lambda (x1: T).(\lambda (_: (pc3 c2 (THead (Bind b) t x0) t3)).(\lambda -(_: (ty3 g c2 t x1)).(\lambda (H11: (ty3 g (CHead c2 (Bind b) t) t0 x0)).(H7 -x0 H11)))))) (ty3_gen_bind g b c2 t t0 t3 H8)))))) H6)))) H4)) (\lambda (H4: -((\forall (t3: T).((ty3 g c2 t t3) \to False)))).(or_intror (ex T (\lambda -(t3: T).(ty3 g c2 (THead (Bind b) t t0) t3))) (\forall (t3: T).((ty3 g c2 -(THead (Bind b) t t0) t3) \to False)) (\lambda (t3: T).(\lambda (H5: (ty3 g -c2 (THead (Bind b) t t0) t3)).(ex3_2_ind T T (\lambda (t4: T).(\lambda (_: -T).(pc3 c2 (THead (Bind b) t t4) t3))) (\lambda (_: T).(\lambda (t5: T).(ty3 -g c2 t t5))) (\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead c2 (Bind b) t) t0 -t4))) False (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 c2 (THead -(Bind b) t x0) t3)).(\lambda (H7: (ty3 g c2 t x1)).(\lambda (_: (ty3 g (CHead -c2 (Bind b) t) t0 x0)).(H4 x1 H7)))))) (ty3_gen_bind g b c2 t t0 t3 H5)))))) -H3)))) (\lambda (f: F).(\lambda (H2: ((\forall (c1: C).(\forall (t3: T).((flt -c1 t3 c2 (THead (Flat f) t t0)) \to (or (ex T (\lambda (t4: T).(ty3 g c1 t3 -t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to False)))))))).(F_ind (\lambda -(f0: F).(((\forall (c1: C).(\forall (t3: T).((flt c1 t3 c2 (THead (Flat f0) t -t0)) \to (or (ex T (\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3 -g c1 t3 t4) \to False))))))) \to (or (ex T (\lambda (t3: T).(ty3 g c2 (THead -(Flat f0) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat f0) t t0) t3) -\to False))))) (\lambda (H3: ((\forall (c1: C).(\forall (t3: T).((flt c1 t3 -c2 (THead (Flat Appl) t t0)) \to (or (ex T (\lambda (t4: T).(ty3 g c1 t3 -t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to False)))))))).(let H4 \def (H3 -c2 t (flt_thead_sx (Flat Appl) c2 t t0)) in (or_ind (ex T (\lambda (t3: -T).(ty3 g c2 t t3))) (\forall (t3: T).((ty3 g c2 t t3) \to False)) (or (ex T -(\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: -T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False))) (\lambda (H5: (ex T -(\lambda (t3: T).(ty3 g c2 t t3)))).(ex_ind T (\lambda (t3: T).(ty3 g c2 t -t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) -(\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False))) -(\lambda (x: T).(\lambda (H6: (ty3 g c2 t x)).(let H7 \def (H3 c2 t0 -(flt_thead_dx (Flat Appl) c2 t t0)) in (or_ind (ex T (\lambda (t3: T).(ty3 g -c2 t0 t3))) (\forall (t3: T).((ty3 g c2 t0 t3) \to False)) (or (ex T (\lambda -(t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 -(THead (Flat Appl) t t0) t3) \to False))) (\lambda (H8: (ex T (\lambda (t3: -T).(ty3 g c2 t0 t3)))).(ex_ind T (\lambda (t3: T).(ty3 g c2 t0 t3)) (or (ex T -(\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: -T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False))) (\lambda (x0: -T).(\lambda (H9: (ty3 g c2 t0 x0)).(ex_ind T (\lambda (t3: T).(ty3 g c2 x0 -t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) -(\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False))) -(\lambda (x1: T).(\lambda (H10: (ty3 g c2 x0 x1)).(ex_ind T (\lambda (t3: -T).(ty3 g c2 x t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t -t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to -False))) (\lambda (x2: T).(\lambda (H11: (ty3 g c2 x x2)).(let H12 \def -(ty3_sn3 g c2 x x2 H11) in (let H_x \def (nf2_sn3 c2 x H12) in (let H13 \def -H_x in (ex2_ind T (\lambda (u: T).(pr3 c2 x u)) (\lambda (u: T).(nf2 c2 u)) -(or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall -(t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False))) (\lambda (x3: -T).(\lambda (H14: (pr3 c2 x x3)).(\lambda (H15: (nf2 c2 x3)).(let H16 \def -(ty3_sred_pr3 c2 x x3 H14 g x2 H11) in (let H_x0 \def (pc3_abst_dec g c2 x0 -x1 H10 x3 x2 H16) in (let H17 \def H_x0 in (or_ind (ex4_2 T T (\lambda (u: -T).(\lambda (_: T).(pc3 c2 x0 (THead (Bind Abst) x3 u)))) (\lambda (u: -T).(\lambda (v2: T).(ty3 g c2 (THead (Bind Abst) v2 u) x1))) (\lambda (_: -T).(\lambda (v2: T).(pr3 c2 x3 v2))) (\lambda (_: T).(\lambda (v2: T).(nf2 c2 -v2)))) (\forall (u: T).((pc3 c2 x0 (THead (Bind Abst) x3 u)) \to False)) (or -(ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: -T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False))) (\lambda (H18: (ex4_2 -T T (\lambda (u: T).(\lambda (_: T).(pc3 c2 x0 (THead (Bind Abst) x3 u)))) -(\lambda (u: T).(\lambda (v2: T).(ty3 g c2 (THead (Bind Abst) v2 u) x1))) -(\lambda (_: T).(\lambda (v2: T).(pr3 c2 x3 v2))) (\lambda (_: T).(\lambda -(v2: T).(nf2 c2 v2))))).(ex4_2_ind T T (\lambda (u: T).(\lambda (_: T).(pc3 -c2 x0 (THead (Bind Abst) x3 u)))) (\lambda (u: T).(\lambda (v2: T).(ty3 g c2 -(THead (Bind Abst) v2 u) x1))) (\lambda (_: T).(\lambda (v2: T).(pr3 c2 x3 -v2))) (\lambda (_: T).(\lambda (v2: T).(nf2 c2 v2))) (or (ex T (\lambda (t3: -T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 -(THead (Flat Appl) t t0) t3) \to False))) (\lambda (x4: T).(\lambda (x5: -T).(\lambda (H19: (pc3 c2 x0 (THead (Bind Abst) x3 x4))).(\lambda (H20: (ty3 -g c2 (THead (Bind Abst) x5 x4) x1)).(\lambda (H21: (pr3 c2 x3 x5)).(\lambda -(_: (nf2 c2 x5)).(let H_y \def (nf2_pr3_unfold c2 x3 x5 H21 H15) in (let H23 -\def (eq_ind_r T x5 (\lambda (t3: T).(pr3 c2 x3 t3)) H21 x3 H_y) in (let H24 -\def (eq_ind_r T x5 (\lambda (t3: T).(ty3 g c2 (THead (Bind Abst) t3 x4) x1)) -H20 x3 H_y) in (or_introl (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) -t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to -False)) (ex_intro T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3)) -(THead (Flat Appl) t (THead (Bind Abst) x3 x4)) (ty3_appl g c2 t x3 (ty3_tred -g c2 t x H6 x3 H14) t0 x4 (ty3_conv g c2 (THead (Bind Abst) x3 x4) x1 H24 t0 -x0 H9 H19))))))))))))) H18)) (\lambda (H18: ((\forall (u: T).((pc3 c2 x0 -(THead (Bind Abst) x3 u)) \to False)))).(or_intror (ex T (\lambda (t3: -T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 -(THead (Flat Appl) t t0) t3) \to False)) (\lambda (t3: T).(\lambda (H19: (ty3 -g c2 (THead (Flat Appl) t t0) t3)).(ex3_2_ind T T (\lambda (u: T).(\lambda -(t4: T).(pc3 c2 (THead (Flat Appl) t (THead (Bind Abst) u t4)) t3))) (\lambda -(u: T).(\lambda (t4: T).(ty3 g c2 t0 (THead (Bind Abst) u t4)))) (\lambda (u: -T).(\lambda (_: T).(ty3 g c2 t u))) False (\lambda (x4: T).(\lambda (x5: -T).(\lambda (_: (pc3 c2 (THead (Flat Appl) t (THead (Bind Abst) x4 x5)) -t3)).(\lambda (H21: (ty3 g c2 t0 (THead (Bind Abst) x4 x5))).(\lambda (H22: -(ty3 g c2 t x4)).(let H_y \def (ty3_unique g c2 t x4 H22 x H6) in (let H_y0 -\def (ty3_unique g c2 t0 (THead (Bind Abst) x4 x5) H21 x0 H9) in (H18 x5 -(pc3_t (THead (Bind Abst) x4 x5) c2 x0 (pc3_s c2 x0 (THead (Bind Abst) x4 x5) -H_y0) (THead (Bind Abst) x3 x5) (pc3_head_1 c2 x4 x3 (pc3_t x c2 x4 H_y x3 -(pc3_pr3_r c2 x x3 H14)) (Bind Abst) x5)))))))))) (ty3_gen_appl g c2 t t0 t3 -H19)))))) H17))))))) H13)))))) (ty3_correct g c2 t x H6)))) (ty3_correct g c2 -t0 x0 H9)))) H8)) (\lambda (H8: ((\forall (t3: T).((ty3 g c2 t0 t3) \to -False)))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t -t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to -False)) (\lambda (t3: T).(\lambda (H9: (ty3 g c2 (THead (Flat Appl) t t0) -t3)).(ex3_2_ind T T (\lambda (u: T).(\lambda (t4: T).(pc3 c2 (THead (Flat -Appl) t (THead (Bind Abst) u t4)) t3))) (\lambda (u: T).(\lambda (t4: T).(ty3 -g c2 t0 (THead (Bind Abst) u t4)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c2 -t u))) False (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 c2 (THead -(Flat Appl) t (THead (Bind Abst) x0 x1)) t3)).(\lambda (H11: (ty3 g c2 t0 -(THead (Bind Abst) x0 x1))).(\lambda (_: (ty3 g c2 t x0)).(H8 (THead (Bind -Abst) x0 x1) H11)))))) (ty3_gen_appl g c2 t t0 t3 H9)))))) H7)))) H5)) -(\lambda (H5: ((\forall (t3: T).((ty3 g c2 t t3) \to False)))).(or_intror (ex -T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: -T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False)) (\lambda (t3: -T).(\lambda (H6: (ty3 g c2 (THead (Flat Appl) t t0) t3)).(ex3_2_ind T T -(\lambda (u: T).(\lambda (t4: T).(pc3 c2 (THead (Flat Appl) t (THead (Bind -Abst) u t4)) t3))) (\lambda (u: T).(\lambda (t4: T).(ty3 g c2 t0 (THead (Bind -Abst) u t4)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c2 t u))) False -(\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 c2 (THead (Flat Appl) t -(THead (Bind Abst) x0 x1)) t3)).(\lambda (_: (ty3 g c2 t0 (THead (Bind Abst) -x0 x1))).(\lambda (H9: (ty3 g c2 t x0)).(H5 x0 H9)))))) (ty3_gen_appl g c2 t -t0 t3 H6)))))) H4))) (\lambda (H3: ((\forall (c1: C).(\forall (t3: T).((flt -c1 t3 c2 (THead (Flat Cast) t t0)) \to (or (ex T (\lambda (t4: T).(ty3 g c1 -t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to False)))))))).(let H4 \def -(H3 c2 t (flt_thead_sx (Flat Cast) c2 t t0)) in (or_ind (ex T (\lambda (t3: -T).(ty3 g c2 t t3))) (\forall (t3: T).((ty3 g c2 t t3) \to False)) (or (ex T -(\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3: -T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False))) (\lambda (H5: (ex T -(\lambda (t3: T).(ty3 g c2 t t3)))).(ex_ind T (\lambda (t3: T).(ty3 g c2 t -t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) -(\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False))) -(\lambda (x: T).(\lambda (H6: (ty3 g c2 t x)).(let H7 \def (H3 c2 t0 -(flt_thead_dx (Flat Cast) c2 t t0)) in (or_ind (ex T (\lambda (t3: T).(ty3 g -c2 t0 t3))) (\forall (t3: T).((ty3 g c2 t0 t3) \to False)) (or (ex T (\lambda -(t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3: T).((ty3 g c2 -(THead (Flat Cast) t t0) t3) \to False))) (\lambda (H8: (ex T (\lambda (t3: -T).(ty3 g c2 t0 t3)))).(ex_ind T (\lambda (t3: T).(ty3 g c2 t0 t3)) (or (ex T -(\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3: -T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False))) (\lambda (x0: -T).(\lambda (H9: (ty3 g c2 t0 x0)).(ex_ind T (\lambda (t3: T).(ty3 g c2 x0 -t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) -(\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False))) -(\lambda (x1: T).(\lambda (H10: (ty3 g c2 x0 x1)).(let H_x \def (pc3_dec g c2 -x0 x1 H10 t x H6) in (let H11 \def H_x in (or_ind (pc3 c2 x0 t) ((pc3 c2 x0 -t) \to False) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) -t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False))) -(\lambda (H12: (pc3 c2 x0 t)).(or_introl (ex T (\lambda (t3: T).(ty3 g c2 -(THead (Flat Cast) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) -t t0) t3) \to False)) (ex_intro T (\lambda (t3: T).(ty3 g c2 (THead (Flat -Cast) t t0) t3)) (THead (Flat Cast) x t) (ty3_cast g c2 t0 t (ty3_conv g c2 t -x H6 t0 x0 H9 H12) x H6)))) (\lambda (H12: (((pc3 c2 x0 t) \to -False))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) -t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False)) -(\lambda (t3: T).(\lambda (H13: (ty3 g c2 (THead (Flat Cast) t t0) -t3)).(ex3_ind T (\lambda (t4: T).(pc3 c2 (THead (Flat Cast) t4 t) t3)) -(\lambda (_: T).(ty3 g c2 t0 t)) (\lambda (t4: T).(ty3 g c2 t t4)) False -(\lambda (x2: T).(\lambda (_: (pc3 c2 (THead (Flat Cast) x2 t) t3)).(\lambda -(H15: (ty3 g c2 t0 t)).(\lambda (H16: (ty3 g c2 t x2)).(let H_y \def -(ty3_unique g c2 t x2 H16 x H6) in (let H_y0 \def (ty3_unique g c2 t0 t H15 -x0 H9) in (H12 (ex2_sym T (pr3 c2 t) (pr3 c2 x0) H_y0)))))))) (ty3_gen_cast g -c2 t0 t t3 H13)))))) H11))))) (ty3_correct g c2 t0 x0 H9)))) H8)) (\lambda -(H8: ((\forall (t3: T).((ty3 g c2 t0 t3) \to False)))).(or_intror (ex T -(\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3: -T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False)) (\lambda (t3: -T).(\lambda (H9: (ty3 g c2 (THead (Flat Cast) t t0) t3)).(ex3_ind T (\lambda -(t4: T).(pc3 c2 (THead (Flat Cast) t4 t) t3)) (\lambda (_: T).(ty3 g c2 t0 -t)) (\lambda (t4: T).(ty3 g c2 t t4)) False (\lambda (x0: T).(\lambda (_: -(pc3 c2 (THead (Flat Cast) x0 t) t3)).(\lambda (H11: (ty3 g c2 t0 -t)).(\lambda (_: (ty3 g c2 t x0)).(H8 t H11))))) (ty3_gen_cast g c2 t0 t t3 -H9)))))) H7)))) H5)) (\lambda (H5: ((\forall (t3: T).((ty3 g c2 t t3) \to -False)))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t -t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to -False)) (\lambda (t3: T).(\lambda (H6: (ty3 g c2 (THead (Flat Cast) t t0) -t3)).(ex3_ind T (\lambda (t4: T).(pc3 c2 (THead (Flat Cast) t4 t) t3)) -(\lambda (_: T).(ty3 g c2 t0 t)) (\lambda (t4: T).(ty3 g c2 t t4)) False -(\lambda (x0: T).(\lambda (_: (pc3 c2 (THead (Flat Cast) x0 t) t3)).(\lambda -(_: (ty3 g c2 t0 t)).(\lambda (H9: (ty3 g c2 t x0)).(ex_ind T (\lambda (t4: -T).(ty3 g c2 x0 t4)) False (\lambda (x: T).(\lambda (_: (ty3 g c2 x0 x)).(H5 -x0 H9))) (ty3_correct g c2 t x0 H9)))))) (ty3_gen_cast g c2 t0 t t3 H6)))))) -H4))) f H2))) k H1))))))) t2))) c t1))). -(* COMMENTS -Initial nodes: 9001 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/defs.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/defs.ma deleted file mode 100644 index d1f500cdf..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/defs.ma +++ /dev/null @@ -1,49 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/G/defs.ma". - -include "Basic-1/pc3/defs.ma". - -inductive ty3 (g: G): C \to (T \to (T \to Prop)) \def -| ty3_conv: \forall (c: C).(\forall (t2: T).(\forall (t: T).((ty3 g c t2 t) -\to (\forall (u: T).(\forall (t1: T).((ty3 g c u t1) \to ((pc3 c t1 t2) \to -(ty3 g c u t2)))))))) -| ty3_sort: \forall (c: C).(\forall (m: nat).(ty3 g c (TSort m) (TSort (next -g m)))) -| ty3_abbr: \forall (n: nat).(\forall (c: C).(\forall (d: C).(\forall (u: -T).((getl n c (CHead d (Bind Abbr) u)) \to (\forall (t: T).((ty3 g d u t) \to -(ty3 g c (TLRef n) (lift (S n) O t)))))))) -| ty3_abst: \forall (n: nat).(\forall (c: C).(\forall (d: C).(\forall (u: -T).((getl n c (CHead d (Bind Abst) u)) \to (\forall (t: T).((ty3 g d u t) \to -(ty3 g c (TLRef n) (lift (S n) O u)))))))) -| ty3_bind: \forall (c: C).(\forall (u: T).(\forall (t: T).((ty3 g c u t) \to -(\forall (b: B).(\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c (Bind b) -u) t1 t2) \to (ty3 g c (THead (Bind b) u t1) (THead (Bind b) u t2))))))))) -| ty3_appl: \forall (c: C).(\forall (w: T).(\forall (u: T).((ty3 g c w u) \to -(\forall (v: T).(\forall (t: T).((ty3 g c v (THead (Bind Abst) u t)) \to (ty3 -g c (THead (Flat Appl) w v) (THead (Flat Appl) w (THead (Bind Abst) u -t))))))))) -| ty3_cast: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c t1 t2) -\to (\forall (t0: T).((ty3 g c t2 t0) \to (ty3 g c (THead (Flat Cast) t2 t1) -(THead (Flat Cast) t0 t2))))))). - -inductive tys3 (g: G) (c: C): TList \to (T \to Prop) \def -| tys3_nil: \forall (u: T).(\forall (u0: T).((ty3 g c u u0) \to (tys3 g c -TNil u))) -| tys3_cons: \forall (t: T).(\forall (u: T).((ty3 g c t u) \to (\forall (ts: -TList).((tys3 g c ts u) \to (tys3 g c (TCons t ts) u))))). - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/fsubst0.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/fsubst0.ma deleted file mode 100644 index f92366a26..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/fsubst0.ma +++ /dev/null @@ -1,995 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/ty3/props.ma". - -include "Basic-1/pc3/fsubst0.ma". - -include "Basic-1/getl/getl.ma". - -theorem ty3_fsubst0: - \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t: T).((ty3 g c1 -t1 t) \to (\forall (i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t2: -T).((fsubst0 i u c1 t1 c2 t2) \to (\forall (e: C).((getl i c1 (CHead e (Bind -Abbr) u)) \to (ty3 g c2 t2 t)))))))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t: T).(\lambda -(H: (ty3 g c1 t1 t)).(ty3_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda -(t2: T).(\forall (i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t3: -T).((fsubst0 i u c t0 c2 t3) \to (\forall (e: C).((getl i c (CHead e (Bind -Abbr) u)) \to (ty3 g c2 t3 t2))))))))))) (\lambda (c: C).(\lambda (t2: -T).(\lambda (t0: T).(\lambda (H0: (ty3 g c t2 t0)).(\lambda (H1: ((\forall -(i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t3: T).((fsubst0 i u c t2 -c2 t3) \to (\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) \to (ty3 g c2 -t3 t0)))))))))).(\lambda (u: T).(\lambda (t3: T).(\lambda (_: (ty3 g c u -t3)).(\lambda (H3: ((\forall (i: nat).(\forall (u0: T).(\forall (c2: -C).(\forall (t4: T).((fsubst0 i u0 c u c2 t4) \to (\forall (e: C).((getl i c -(CHead e (Bind Abbr) u0)) \to (ty3 g c2 t4 t3)))))))))).(\lambda (H4: (pc3 c -t3 t2)).(\lambda (i: nat).(\lambda (u0: T).(\lambda (c2: C).(\lambda (t4: -T).(\lambda (H5: (fsubst0 i u0 c u c2 t4)).(fsubst0_ind i u0 c u (\lambda -(c0: C).(\lambda (t5: T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u0)) -\to (ty3 g c0 t5 t2))))) (\lambda (t5: T).(\lambda (H6: (subst0 i u0 u -t5)).(\lambda (e: C).(\lambda (H7: (getl i c (CHead e (Bind Abbr) -u0))).(ty3_conv g c t2 t0 H0 t5 t3 (H3 i u0 c t5 (fsubst0_snd i u0 c u t5 H6) -e H7) H4))))) (\lambda (c3: C).(\lambda (H6: (csubst0 i u0 c c3)).(\lambda -(e: C).(\lambda (H7: (getl i c (CHead e (Bind Abbr) u0))).(ty3_conv g c3 t2 -t0 (H1 i u0 c3 t2 (fsubst0_fst i u0 c t2 c3 H6) e H7) u t3 (H3 i u0 c3 u -(fsubst0_fst i u0 c u c3 H6) e H7) (pc3_fsubst0 c t3 t2 H4 i u0 c3 t3 -(fsubst0_fst i u0 c t3 c3 H6) e H7)))))) (\lambda (t5: T).(\lambda (H6: -(subst0 i u0 u t5)).(\lambda (c3: C).(\lambda (H7: (csubst0 i u0 c -c3)).(\lambda (e: C).(\lambda (H8: (getl i c (CHead e (Bind Abbr) -u0))).(ty3_conv g c3 t2 t0 (H1 i u0 c3 t2 (fsubst0_fst i u0 c t2 c3 H7) e H8) -t5 t3 (H3 i u0 c3 t5 (fsubst0_both i u0 c u t5 H6 c3 H7) e H8) (pc3_fsubst0 c -t3 t2 H4 i u0 c3 t3 (fsubst0_fst i u0 c t3 c3 H7) e H8)))))))) c2 t4 -H5)))))))))))))))) (\lambda (c: C).(\lambda (m: nat).(\lambda (i: -nat).(\lambda (u: T).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H0: (fsubst0 -i u c (TSort m) c2 t2)).(fsubst0_ind i u c (TSort m) (\lambda (c0: -C).(\lambda (t0: T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) \to -(ty3 g c0 t0 (TSort (next g m))))))) (\lambda (t3: T).(\lambda (H1: (subst0 i -u (TSort m) t3)).(\lambda (e: C).(\lambda (_: (getl i c (CHead e (Bind Abbr) -u))).(subst0_gen_sort u t3 i m H1 (ty3 g c t3 (TSort (next g m)))))))) -(\lambda (c3: C).(\lambda (_: (csubst0 i u c c3)).(\lambda (e: C).(\lambda -(_: (getl i c (CHead e (Bind Abbr) u))).(ty3_sort g c3 m))))) (\lambda (t3: -T).(\lambda (H1: (subst0 i u (TSort m) t3)).(\lambda (c3: C).(\lambda (_: -(csubst0 i u c c3)).(\lambda (e: C).(\lambda (_: (getl i c (CHead e (Bind -Abbr) u))).(subst0_gen_sort u t3 i m H1 (ty3 g c3 t3 (TSort (next g -m)))))))))) c2 t2 H0)))))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d: -C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind Abbr) u))).(\lambda -(t0: T).(\lambda (H1: (ty3 g d u t0)).(\lambda (H2: ((\forall (i: -nat).(\forall (u0: T).(\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 d u c2 -t2) \to (\forall (e: C).((getl i d (CHead e (Bind Abbr) u0)) \to (ty3 g c2 t2 -t0)))))))))).(\lambda (i: nat).(\lambda (u0: T).(\lambda (c2: C).(\lambda -(t2: T).(\lambda (H3: (fsubst0 i u0 c (TLRef n) c2 t2)).(fsubst0_ind i u0 c -(TLRef n) (\lambda (c0: C).(\lambda (t3: T).(\forall (e: C).((getl i c (CHead -e (Bind Abbr) u0)) \to (ty3 g c0 t3 (lift (S n) O t0)))))) (\lambda (t3: -T).(\lambda (H4: (subst0 i u0 (TLRef n) t3)).(\lambda (e: C).(\lambda (H5: -(getl i c (CHead e (Bind Abbr) u0))).(land_ind (eq nat n i) (eq T t3 (lift (S -n) O u0)) (ty3 g c t3 (lift (S n) O t0)) (\lambda (H6: (eq nat n i)).(\lambda -(H7: (eq T t3 (lift (S n) O u0))).(eq_ind_r T (lift (S n) O u0) (\lambda (t4: -T).(ty3 g c t4 (lift (S n) O t0))) (let H8 \def (eq_ind_r nat i (\lambda (n0: -nat).(getl n0 c (CHead e (Bind Abbr) u0))) H5 n H6) in (let H9 \def (eq_ind C -(CHead d (Bind Abbr) u) (\lambda (c0: C).(getl n c c0)) H0 (CHead e (Bind -Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) n H0 (CHead e (Bind Abbr) u0) -H8)) in (let H10 \def (f_equal C C (\lambda (e0: C).(match e0 in C return -(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow -c0])) (CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono c (CHead d -(Bind Abbr) u) n H0 (CHead e (Bind Abbr) u0) H8)) in ((let H11 \def (f_equal -C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) -\Rightarrow u | (CHead _ _ t4) \Rightarrow t4])) (CHead d (Bind Abbr) u) -(CHead e (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) n H0 (CHead e -(Bind Abbr) u0) H8)) in (\lambda (H12: (eq C d e)).(let H13 \def (eq_ind_r C -e (\lambda (c0: C).(getl n c (CHead c0 (Bind Abbr) u0))) H9 d H12) in (let -H14 \def (eq_ind_r T u0 (\lambda (t4: T).(getl n c (CHead d (Bind Abbr) t4))) -H13 u H11) in (eq_ind T u (\lambda (t4: T).(ty3 g c (lift (S n) O t4) (lift -(S n) O t0))) (ty3_lift g d u t0 H1 c O (S n) (getl_drop Abbr c d u n H14)) -u0 H11))))) H10)))) t3 H7))) (subst0_gen_lref u0 t3 i n H4)))))) (\lambda -(c3: C).(\lambda (H4: (csubst0 i u0 c c3)).(\lambda (e: C).(\lambda (H5: -(getl i c (CHead e (Bind Abbr) u0))).(lt_le_e n i (ty3 g c3 (TLRef n) (lift -(S n) O t0)) (\lambda (H6: (lt n i)).(let H7 \def (csubst0_getl_lt i n H6 c -c3 u0 H4 (CHead d (Bind Abbr) u) H0) in (or4_ind (getl n c3 (CHead d (Bind -Abbr) u)) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b) u1)))))) -(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 -(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: -T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w)))))) (ex3_4 B C C T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C -(CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda -(_: C).(\lambda (e2: C).(\lambda (u1: T).(getl n c3 (CHead e2 (Bind b) -u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus i (S n)) u0 e1 e2)))))) (ex4_5 B C C T T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C -(CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e2 -(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -(minus i (S n)) u0 e1 e2))))))) (ty3 g c3 (TLRef n) (lift (S n) O t0)) -(\lambda (H8: (getl n c3 (CHead d (Bind Abbr) u))).(ty3_abbr g n c3 d u H8 t0 -H1)) (\lambda (H8: (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda -(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b) -u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(getl n c3 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 -w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b) u1)))))) -(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 -(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: -T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w))))) (ty3 g c3 (TLRef n) -(lift (S n) O t0)) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: -T).(\lambda (x3: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead x1 -(Bind x0) x2))).(\lambda (H10: (getl n c3 (CHead x1 (Bind x0) x3))).(\lambda -(H11: (subst0 (minus i (S n)) u0 x2 x3)).(let H12 \def (f_equal C C (\lambda -(e0: C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow -d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abbr) u) (CHead x1 (Bind -x0) x2) H9) in ((let H13 \def (f_equal C B (\lambda (e0: C).(match e0 in C -return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) -\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead -x1 (Bind x0) x2) H9) in ((let H14 \def (f_equal C T (\lambda (e0: C).(match -e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ -t3) \Rightarrow t3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H9) in -(\lambda (H15: (eq B Abbr x0)).(\lambda (H16: (eq C d x1)).(let H17 \def -(eq_ind_r T x2 (\lambda (t3: T).(subst0 (minus i (S n)) u0 t3 x3)) H11 u H14) -in (let H18 \def (eq_ind_r C x1 (\lambda (c0: C).(getl n c3 (CHead c0 (Bind -x0) x3))) H10 d H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b: B).(getl n -c3 (CHead d (Bind b) x3))) H18 Abbr H15) in (let H20 \def (eq_ind nat (minus -i n) (\lambda (n0: nat).(getl n0 (CHead d (Bind Abbr) x3) (CHead e (Bind -Abbr) u0))) (getl_conf_le i (CHead e (Bind Abbr) u0) c3 (csubst0_getl_ge i i -(le_n i) c c3 u0 H4 (CHead e (Bind Abbr) u0) H5) (CHead d (Bind Abbr) x3) n -H19 (le_S_n n i (le_S (S n) i H6))) (S (minus i (S n))) (minus_x_Sy i n H6)) -in (ty3_abbr g n c3 d x3 H19 t0 (H2 (minus i (S n)) u0 d x3 (fsubst0_snd -(minus i (S n)) u0 d u x3 H17) e (getl_gen_S (Bind Abbr) d (CHead e (Bind -Abbr) u0) x3 (minus i (S n)) H20)))))))))) H13)) H12))))))))) H8)) (\lambda -(H8: (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda -(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl n c3 (CHead e2 -(Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 (minus i (S n)) u0 e1 e2))))))).(ex3_4_ind B C C T (\lambda -(b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind -Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u1: T).(getl n c3 (CHead e2 (Bind b) u1)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) -u0 e1 e2))))) (ty3 g c3 (TLRef n) (lift (S n) O t0)) (\lambda (x0: -B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H9: (eq C -(CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3))).(\lambda (H10: (getl n c3 -(CHead x2 (Bind x0) x3))).(\lambda (H11: (csubst0 (minus i (S n)) u0 x1 -x2)).(let H12 \def (f_equal C C (\lambda (e0: C).(match e0 in C return -(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow -c0])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H13 \def -(f_equal C B (\lambda (e0: C).(match e0 in C return (\lambda (_: C).B) with -[(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k in K return -(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow -Abbr])])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H14 -\def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow u | (CHead _ _ t3) \Rightarrow t3])) (CHead d -(Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in (\lambda (H15: (eq B Abbr -x0)).(\lambda (H16: (eq C d x1)).(let H17 \def (eq_ind_r T x3 (\lambda (t3: -T).(getl n c3 (CHead x2 (Bind x0) t3))) H10 u H14) in (let H18 \def (eq_ind_r -C x1 (\lambda (c0: C).(csubst0 (minus i (S n)) u0 c0 x2)) H11 d H16) in (let -H19 \def (eq_ind_r B x0 (\lambda (b: B).(getl n c3 (CHead x2 (Bind b) u))) -H17 Abbr H15) in (let H20 \def (eq_ind nat (minus i n) (\lambda (n0: -nat).(getl n0 (CHead x2 (Bind Abbr) u) (CHead e (Bind Abbr) u0))) -(getl_conf_le i (CHead e (Bind Abbr) u0) c3 (csubst0_getl_ge i i (le_n i) c -c3 u0 H4 (CHead e (Bind Abbr) u0) H5) (CHead x2 (Bind Abbr) u) n H19 (le_S_n -n i (le_S (S n) i H6))) (S (minus i (S n))) (minus_x_Sy i n H6)) in (ty3_abbr -g n c3 x2 u H19 t0 (H2 (minus i (S n)) u0 x2 u (fsubst0_fst (minus i (S n)) -u0 d u x2 H18) e (csubst0_getl_ge_back (minus i (S n)) (minus i (S n)) (le_n -(minus i (S n))) d x2 u0 H18 (CHead e (Bind Abbr) u0) (getl_gen_S (Bind Abbr) -x2 (CHead e (Bind Abbr) u0) u (minus i (S n)) H20))))))))))) H13)) -H12))))))))) H8)) (\lambda (H8: (ex4_5 B C C T T (\lambda (b: B).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind -Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e2 -(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -(minus i (S n)) u0 e1 e2)))))))).(ex4_5_ind B C C T T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C -(CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e2 -(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -(minus i (S n)) u0 e1 e2)))))) (ty3 g c3 (TLRef n) (lift (S n) O t0)) -(\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda -(x4: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) -x3))).(\lambda (H10: (getl n c3 (CHead x2 (Bind x0) x4))).(\lambda (H11: -(subst0 (minus i (S n)) u0 x3 x4)).(\lambda (H12: (csubst0 (minus i (S n)) u0 -x1 x2)).(let H13 \def (f_equal C C (\lambda (e0: C).(match e0 in C return -(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow -c0])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H14 \def -(f_equal C B (\lambda (e0: C).(match e0 in C return (\lambda (_: C).B) with -[(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k in K return -(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow -Abbr])])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H15 -\def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow u | (CHead _ _ t3) \Rightarrow t3])) (CHead d -(Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in (\lambda (H16: (eq B Abbr -x0)).(\lambda (H17: (eq C d x1)).(let H18 \def (eq_ind_r T x3 (\lambda (t3: -T).(subst0 (minus i (S n)) u0 t3 x4)) H11 u H15) in (let H19 \def (eq_ind_r C -x1 (\lambda (c0: C).(csubst0 (minus i (S n)) u0 c0 x2)) H12 d H17) in (let -H20 \def (eq_ind_r B x0 (\lambda (b: B).(getl n c3 (CHead x2 (Bind b) x4))) -H10 Abbr H16) in (let H21 \def (eq_ind nat (minus i n) (\lambda (n0: -nat).(getl n0 (CHead x2 (Bind Abbr) x4) (CHead e (Bind Abbr) u0))) -(getl_conf_le i (CHead e (Bind Abbr) u0) c3 (csubst0_getl_ge i i (le_n i) c -c3 u0 H4 (CHead e (Bind Abbr) u0) H5) (CHead x2 (Bind Abbr) x4) n H20 (le_S_n -n i (le_S (S n) i H6))) (S (minus i (S n))) (minus_x_Sy i n H6)) in (ty3_abbr -g n c3 x2 x4 H20 t0 (H2 (minus i (S n)) u0 x2 x4 (fsubst0_both (minus i (S -n)) u0 d u x4 H18 x2 H19) e (csubst0_getl_ge_back (minus i (S n)) (minus i (S -n)) (le_n (minus i (S n))) d x2 u0 H19 (CHead e (Bind Abbr) u0) (getl_gen_S -(Bind Abbr) x2 (CHead e (Bind Abbr) u0) x4 (minus i (S n)) H21))))))))))) -H14)) H13))))))))))) H8)) H7))) (\lambda (H6: (le i n)).(ty3_abbr g n c3 d u -(csubst0_getl_ge i n H6 c c3 u0 H4 (CHead d (Bind Abbr) u) H0) t0 H1))))))) -(\lambda (t3: T).(\lambda (H4: (subst0 i u0 (TLRef n) t3)).(\lambda (c3: -C).(\lambda (H5: (csubst0 i u0 c c3)).(\lambda (e: C).(\lambda (H6: (getl i c -(CHead e (Bind Abbr) u0))).(land_ind (eq nat n i) (eq T t3 (lift (S n) O u0)) -(ty3 g c3 t3 (lift (S n) O t0)) (\lambda (H7: (eq nat n i)).(\lambda (H8: (eq -T t3 (lift (S n) O u0))).(eq_ind_r T (lift (S n) O u0) (\lambda (t4: T).(ty3 -g c3 t4 (lift (S n) O t0))) (let H9 \def (eq_ind_r nat i (\lambda (n0: -nat).(getl n0 c (CHead e (Bind Abbr) u0))) H6 n H7) in (let H10 \def -(eq_ind_r nat i (\lambda (n0: nat).(csubst0 n0 u0 c c3)) H5 n H7) in (let H11 -\def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c0: C).(getl n c c0)) H0 -(CHead e (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) n H0 (CHead e -(Bind Abbr) u0) H9)) in (let H12 \def (f_equal C C (\lambda (e0: C).(match e0 -in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) -\Rightarrow c0])) (CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono -c (CHead d (Bind Abbr) u) n H0 (CHead e (Bind Abbr) u0) H9)) in ((let H13 -\def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow u | (CHead _ _ t4) \Rightarrow t4])) (CHead d -(Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) -n H0 (CHead e (Bind Abbr) u0) H9)) in (\lambda (H14: (eq C d e)).(let H15 -\def (eq_ind_r C e (\lambda (c0: C).(getl n c (CHead c0 (Bind Abbr) u0))) H11 -d H14) in (let H16 \def (eq_ind_r T u0 (\lambda (t4: T).(getl n c (CHead d -(Bind Abbr) t4))) H15 u H13) in (let H17 \def (eq_ind_r T u0 (\lambda (t4: -T).(csubst0 n t4 c c3)) H10 u H13) in (eq_ind T u (\lambda (t4: T).(ty3 g c3 -(lift (S n) O t4) (lift (S n) O t0))) (ty3_lift g d u t0 H1 c3 O (S n) -(getl_drop Abbr c3 d u n (csubst0_getl_ge n n (le_n n) c c3 u H17 (CHead d -(Bind Abbr) u) H16))) u0 H13)))))) H12))))) t3 H8))) (subst0_gen_lref u0 t3 i -n H4)))))))) c2 t2 H3)))))))))))))) (\lambda (n: nat).(\lambda (c: -C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind -Abst) u))).(\lambda (t0: T).(\lambda (H1: (ty3 g d u t0)).(\lambda (H2: -((\forall (i: nat).(\forall (u0: T).(\forall (c2: C).(\forall (t2: -T).((fsubst0 i u0 d u c2 t2) \to (\forall (e: C).((getl i d (CHead e (Bind -Abbr) u0)) \to (ty3 g c2 t2 t0)))))))))).(\lambda (i: nat).(\lambda (u0: -T).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H3: (fsubst0 i u0 c (TLRef n) -c2 t2)).(fsubst0_ind i u0 c (TLRef n) (\lambda (c0: C).(\lambda (t3: -T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u0)) \to (ty3 g c0 t3 -(lift (S n) O u)))))) (\lambda (t3: T).(\lambda (H4: (subst0 i u0 (TLRef n) -t3)).(\lambda (e: C).(\lambda (H5: (getl i c (CHead e (Bind Abbr) -u0))).(land_ind (eq nat n i) (eq T t3 (lift (S n) O u0)) (ty3 g c t3 (lift (S -n) O u)) (\lambda (H6: (eq nat n i)).(\lambda (H7: (eq T t3 (lift (S n) O -u0))).(eq_ind_r T (lift (S n) O u0) (\lambda (t4: T).(ty3 g c t4 (lift (S n) -O u))) (let H8 \def (eq_ind_r nat i (\lambda (n0: nat).(getl n0 c (CHead e -(Bind Abbr) u0))) H5 n H6) in (let H9 \def (eq_ind C (CHead d (Bind Abst) u) -(\lambda (c0: C).(getl n c c0)) H0 (CHead e (Bind Abbr) u0) (getl_mono c -(CHead d (Bind Abst) u) n H0 (CHead e (Bind Abbr) u0) H8)) in (let H10 \def -(eq_ind C (CHead d (Bind Abst) u) (\lambda (ee: C).(match ee in C return -(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b) -\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow -False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) -\Rightarrow False])])) I (CHead e (Bind Abbr) u0) (getl_mono c (CHead d (Bind -Abst) u) n H0 (CHead e (Bind Abbr) u0) H8)) in (False_ind (ty3 g c (lift (S -n) O u0) (lift (S n) O u)) H10)))) t3 H7))) (subst0_gen_lref u0 t3 i n -H4)))))) (\lambda (c3: C).(\lambda (H4: (csubst0 i u0 c c3)).(\lambda (e: -C).(\lambda (H5: (getl i c (CHead e (Bind Abbr) u0))).(lt_le_e n i (ty3 g c3 -(TLRef n) (lift (S n) O u)) (\lambda (H6: (lt n i)).(let H7 \def -(csubst0_getl_lt i n H6 c c3 u0 H4 (CHead d (Bind Abst) u) H0) in (or4_ind -(getl n c3 (CHead d (Bind Abst) u)) (ex3_4 B C T T (\lambda (b: B).(\lambda -(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead -e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(getl n c3 (CHead e0 (Bind b) w)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i (S n)) -u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1)))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl n c3 -(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus i (S n)) u0 e1 e2)))))) (ex4_5 B C C T T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1))))))) (\lambda -(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl -n c3 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus i (S n)) u0 e1 e2))))))) (ty3 g c3 (TLRef n) (lift (S -n) O u)) (\lambda (H8: (getl n c3 (CHead d (Bind Abst) u))).(ty3_abst g n c3 -d u H8 t0 H1)) (\lambda (H8: (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead e0 -(Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda -(w: T).(getl n c3 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 -w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead e0 (Bind b) u1)))))) -(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 -(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: -T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w))))) (ty3 g c3 (TLRef n) -(lift (S n) O u)) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda -(x3: T).(\lambda (H9: (eq C (CHead d (Bind Abst) u) (CHead x1 (Bind x0) -x2))).(\lambda (H10: (getl n c3 (CHead x1 (Bind x0) x3))).(\lambda (H11: -(subst0 (minus i (S n)) u0 x2 x3)).(let H12 \def (f_equal C C (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | -(CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) -x2) H9) in ((let H13 \def (f_equal C B (\lambda (e0: C).(match e0 in C return -(\lambda (_: C).B) with [(CSort _) \Rightarrow Abst | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) -\Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead d (Bind Abst) u) (CHead -x1 (Bind x0) x2) H9) in ((let H14 \def (f_equal C T (\lambda (e0: C).(match -e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ -t3) \Rightarrow t3])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x2) H9) in -(\lambda (H15: (eq B Abst x0)).(\lambda (H16: (eq C d x1)).(let H17 \def -(eq_ind_r T x2 (\lambda (t3: T).(subst0 (minus i (S n)) u0 t3 x3)) H11 u H14) -in (let H18 \def (eq_ind_r C x1 (\lambda (c0: C).(getl n c3 (CHead c0 (Bind -x0) x3))) H10 d H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b: B).(getl n -c3 (CHead d (Bind b) x3))) H18 Abst H15) in (let H20 \def (eq_ind nat (minus -i n) (\lambda (n0: nat).(getl n0 (CHead d (Bind Abst) x3) (CHead e (Bind -Abbr) u0))) (getl_conf_le i (CHead e (Bind Abbr) u0) c3 (csubst0_getl_ge i i -(le_n i) c c3 u0 H4 (CHead e (Bind Abbr) u0) H5) (CHead d (Bind Abst) x3) n -H19 (le_S_n n i (le_S (S n) i H6))) (S (minus i (S n))) (minus_x_Sy i n H6)) -in (ty3_conv g c3 (lift (S n) O u) (lift (S n) O t0) (ty3_lift g d u t0 H1 c3 -O (S n) (getl_drop Abst c3 d x3 n H19)) (TLRef n) (lift (S n) O x3) (ty3_abst -g n c3 d x3 H19 t0 (H2 (minus i (S n)) u0 d x3 (fsubst0_snd (minus i (S n)) -u0 d u x3 H17) e (getl_gen_S (Bind Abst) d (CHead e (Bind Abbr) u0) x3 (minus -i (S n)) H20))) (pc3_lift c3 d (S n) O (getl_drop Abst c3 d x3 n H19) x3 u -(pc3_pr2_x d x3 u (pr2_delta d e u0 (r (Bind Abst) (minus i (S n))) -(getl_gen_S (Bind Abst) d (CHead e (Bind Abbr) u0) x3 (minus i (S n)) H20) u -u (pr0_refl u) x3 H17))))))))))) H13)) H12))))))))) H8)) (\lambda (H8: (ex3_4 -B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq -C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl n c3 (CHead e2 -(Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 (minus i (S n)) u0 e1 e2))))))).(ex3_4_ind B C C T (\lambda -(b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind -Abst) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u1: T).(getl n c3 (CHead e2 (Bind b) u1)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) -u0 e1 e2))))) (ty3 g c3 (TLRef n) (lift (S n) O u)) (\lambda (x0: B).(\lambda -(x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H9: (eq C (CHead d (Bind -Abst) u) (CHead x1 (Bind x0) x3))).(\lambda (H10: (getl n c3 (CHead x2 (Bind -x0) x3))).(\lambda (H11: (csubst0 (minus i (S n)) u0 x1 x2)).(let H12 \def -(f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with -[(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind -Abst) u) (CHead x1 (Bind x0) x3) H9) in ((let H13 \def (f_equal C B (\lambda -(e0: C).(match e0 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow -Abst | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with -[(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead d (Bind Abst) -u) (CHead x1 (Bind x0) x3) H9) in ((let H14 \def (f_equal C T (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | -(CHead _ _ t3) \Rightarrow t3])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) -x3) H9) in (\lambda (H15: (eq B Abst x0)).(\lambda (H16: (eq C d x1)).(let -H17 \def (eq_ind_r T x3 (\lambda (t3: T).(getl n c3 (CHead x2 (Bind x0) t3))) -H10 u H14) in (let H18 \def (eq_ind_r C x1 (\lambda (c0: C).(csubst0 (minus i -(S n)) u0 c0 x2)) H11 d H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b: -B).(getl n c3 (CHead x2 (Bind b) u))) H17 Abst H15) in (let H20 \def (eq_ind -nat (minus i n) (\lambda (n0: nat).(getl n0 (CHead x2 (Bind Abst) u) (CHead e -(Bind Abbr) u0))) (getl_conf_le i (CHead e (Bind Abbr) u0) c3 -(csubst0_getl_ge i i (le_n i) c c3 u0 H4 (CHead e (Bind Abbr) u0) H5) (CHead -x2 (Bind Abst) u) n H19 (le_S_n n i (le_S (S n) i H6))) (S (minus i (S n))) -(minus_x_Sy i n H6)) in (ty3_abst g n c3 x2 u H19 t0 (H2 (minus i (S n)) u0 -x2 u (fsubst0_fst (minus i (S n)) u0 d u x2 H18) e (csubst0_getl_ge_back -(minus i (S n)) (minus i (S n)) (le_n (minus i (S n))) d x2 u0 H18 (CHead e -(Bind Abbr) u0) (getl_gen_S (Bind Abst) x2 (CHead e (Bind Abbr) u0) u (minus -i (S n)) H20))))))))))) H13)) H12))))))))) H8)) (\lambda (H8: (ex4_5 B C C T -T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1))))))) (\lambda -(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl -n c3 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus i (S n)) u0 e1 e2)))))))).(ex4_5_ind B C C T T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1))))))) (\lambda -(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl -n c3 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus i (S n)) u0 e1 e2)))))) (ty3 g c3 (TLRef n) (lift (S -n) O u)) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: -T).(\lambda (x4: T).(\lambda (H9: (eq C (CHead d (Bind Abst) u) (CHead x1 -(Bind x0) x3))).(\lambda (H10: (getl n c3 (CHead x2 (Bind x0) x4))).(\lambda -(H11: (subst0 (minus i (S n)) u0 x3 x4)).(\lambda (H12: (csubst0 (minus i (S -n)) u0 x1 x2)).(let H13 \def (f_equal C C (\lambda (e0: C).(match e0 in C -return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) -\Rightarrow c0])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3) H9) in -((let H14 \def (f_equal C B (\lambda (e0: C).(match e0 in C return (\lambda -(_: C).B) with [(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match -k in K return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) -\Rightarrow Abst])])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3) H9) in -((let H15 \def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda -(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t3) \Rightarrow t3])) -(CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3) H9) in (\lambda (H16: (eq B -Abst x0)).(\lambda (H17: (eq C d x1)).(let H18 \def (eq_ind_r T x3 (\lambda -(t3: T).(subst0 (minus i (S n)) u0 t3 x4)) H11 u H15) in (let H19 \def -(eq_ind_r C x1 (\lambda (c0: C).(csubst0 (minus i (S n)) u0 c0 x2)) H12 d -H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b: B).(getl n c3 (CHead x2 -(Bind b) x4))) H10 Abst H16) in (let H21 \def (eq_ind nat (minus i n) -(\lambda (n0: nat).(getl n0 (CHead x2 (Bind Abst) x4) (CHead e (Bind Abbr) -u0))) (getl_conf_le i (CHead e (Bind Abbr) u0) c3 (csubst0_getl_ge i i (le_n -i) c c3 u0 H4 (CHead e (Bind Abbr) u0) H5) (CHead x2 (Bind Abst) x4) n H20 -(le_S_n n i (le_S (S n) i H6))) (S (minus i (S n))) (minus_x_Sy i n H6)) in -(ty3_conv g c3 (lift (S n) O u) (lift (S n) O t0) (ty3_lift g x2 u t0 (H2 -(minus i (S n)) u0 x2 u (fsubst0_fst (minus i (S n)) u0 d u x2 H19) e -(csubst0_getl_ge_back (minus i (S n)) (minus i (S n)) (le_n (minus i (S n))) -d x2 u0 H19 (CHead e (Bind Abbr) u0) (getl_gen_S (Bind Abst) x2 (CHead e -(Bind Abbr) u0) x4 (minus i (S n)) H21))) c3 O (S n) (getl_drop Abst c3 x2 x4 -n H20)) (TLRef n) (lift (S n) O x4) (ty3_abst g n c3 x2 x4 H20 t0 (H2 (minus -i (S n)) u0 x2 x4 (fsubst0_both (minus i (S n)) u0 d u x4 H18 x2 H19) e -(csubst0_getl_ge_back (minus i (S n)) (minus i (S n)) (le_n (minus i (S n))) -d x2 u0 H19 (CHead e (Bind Abbr) u0) (getl_gen_S (Bind Abst) x2 (CHead e -(Bind Abbr) u0) x4 (minus i (S n)) H21)))) (pc3_lift c3 x2 (S n) O (getl_drop -Abst c3 x2 x4 n H20) x4 u (pc3_fsubst0 d u u (pc3_refl d u) (minus i (S n)) -u0 x2 x4 (fsubst0_both (minus i (S n)) u0 d u x4 H18 x2 H19) e -(csubst0_getl_ge_back (minus i (S n)) (minus i (S n)) (le_n (minus i (S n))) -d x2 u0 H19 (CHead e (Bind Abbr) u0) (getl_gen_S (Bind Abst) x2 (CHead e -(Bind Abbr) u0) x4 (minus i (S n)) H21)))))))))))) H14)) H13))))))))))) H8)) -H7))) (\lambda (H6: (le i n)).(ty3_abst g n c3 d u (csubst0_getl_ge i n H6 c -c3 u0 H4 (CHead d (Bind Abst) u) H0) t0 H1))))))) (\lambda (t3: T).(\lambda -(H4: (subst0 i u0 (TLRef n) t3)).(\lambda (c3: C).(\lambda (H5: (csubst0 i u0 -c c3)).(\lambda (e: C).(\lambda (H6: (getl i c (CHead e (Bind Abbr) -u0))).(land_ind (eq nat n i) (eq T t3 (lift (S n) O u0)) (ty3 g c3 t3 (lift -(S n) O u)) (\lambda (H7: (eq nat n i)).(\lambda (H8: (eq T t3 (lift (S n) O -u0))).(eq_ind_r T (lift (S n) O u0) (\lambda (t4: T).(ty3 g c3 t4 (lift (S n) -O u))) (let H9 \def (eq_ind_r nat i (\lambda (n0: nat).(getl n0 c (CHead e -(Bind Abbr) u0))) H6 n H7) in (let H10 \def (eq_ind_r nat i (\lambda (n0: -nat).(csubst0 n0 u0 c c3)) H5 n H7) in (let H11 \def (eq_ind C (CHead d (Bind -Abst) u) (\lambda (c0: C).(getl n c c0)) H0 (CHead e (Bind Abbr) u0) -(getl_mono c (CHead d (Bind Abst) u) n H0 (CHead e (Bind Abbr) u0) H9)) in -(let H12 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (ee: C).(match ee in -C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k -_) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b) -\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow -False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) -\Rightarrow False])])) I (CHead e (Bind Abbr) u0) (getl_mono c (CHead d (Bind -Abst) u) n H0 (CHead e (Bind Abbr) u0) H9)) in (False_ind (ty3 g c3 (lift (S -n) O u0) (lift (S n) O u)) H12))))) t3 H8))) (subst0_gen_lref u0 t3 i n -H4)))))))) c2 t2 H3)))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda -(t0: T).(\lambda (H0: (ty3 g c u t0)).(\lambda (H1: ((\forall (i: -nat).(\forall (u0: T).(\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 c u c2 -t2) \to (\forall (e: C).((getl i c (CHead e (Bind Abbr) u0)) \to (ty3 g c2 t2 -t0)))))))))).(\lambda (b: B).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H2: -(ty3 g (CHead c (Bind b) u) t2 t3)).(\lambda (H3: ((\forall (i: nat).(\forall -(u0: T).(\forall (c2: C).(\forall (t4: T).((fsubst0 i u0 (CHead c (Bind b) u) -t2 c2 t4) \to (\forall (e: C).((getl i (CHead c (Bind b) u) (CHead e (Bind -Abbr) u0)) \to (ty3 g c2 t4 t3)))))))))).(\lambda (i: nat).(\lambda (u0: -T).(\lambda (c2: C).(\lambda (t4: T).(\lambda (H4: (fsubst0 i u0 c (THead -(Bind b) u t2) c2 t4)).(fsubst0_ind i u0 c (THead (Bind b) u t2) (\lambda -(c0: C).(\lambda (t5: T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u0)) -\to (ty3 g c0 t5 (THead (Bind b) u t3)))))) (\lambda (t5: T).(\lambda (H5: -(subst0 i u0 (THead (Bind b) u t2) t5)).(\lambda (e: C).(\lambda (H6: (getl i -c (CHead e (Bind Abbr) u0))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t5 (THead -(Bind b) u2 t2))) (\lambda (u2: T).(subst0 i u0 u u2))) (ex2 T (\lambda (t6: -T).(eq T t5 (THead (Bind b) u t6))) (\lambda (t6: T).(subst0 (s (Bind b) i) -u0 t2 t6))) (ex3_2 T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead -(Bind b) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) -(\lambda (_: T).(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 t6)))) (ty3 g c -t5 (THead (Bind b) u t3)) (\lambda (H7: (ex2 T (\lambda (u2: T).(eq T t5 -(THead (Bind b) u2 t2))) (\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T -(\lambda (u2: T).(eq T t5 (THead (Bind b) u2 t2))) (\lambda (u2: T).(subst0 i -u0 u u2)) (ty3 g c t5 (THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H8: -(eq T t5 (THead (Bind b) x t2))).(\lambda (H9: (subst0 i u0 u x)).(eq_ind_r T -(THead (Bind b) x t2) (\lambda (t6: T).(ty3 g c t6 (THead (Bind b) u t3))) -(ex_ind T (\lambda (t6: T).(ty3 g (CHead c (Bind b) u) t3 t6)) (ty3 g c -(THead (Bind b) x t2) (THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (H10: -(ty3 g (CHead c (Bind b) u) t3 x0)).(ex_ind T (\lambda (t6: T).(ty3 g (CHead -c (Bind b) x) t3 t6)) (ty3 g c (THead (Bind b) x t2) (THead (Bind b) u t3)) -(\lambda (x1: T).(\lambda (_: (ty3 g (CHead c (Bind b) x) t3 x1)).(ty3_conv g -c (THead (Bind b) u t3) (THead (Bind b) u x0) (ty3_bind g c u t0 H0 b t3 x0 -H10) (THead (Bind b) x t2) (THead (Bind b) x t3) (ty3_bind g c x t0 (H1 i u0 -c x (fsubst0_snd i u0 c u x H9) e H6) b t2 t3 (H3 (S i) u0 (CHead c (Bind b) -x) t2 (fsubst0_fst (S i) u0 (CHead c (Bind b) u) t2 (CHead c (Bind b) x) -(csubst0_snd_bind b i u0 u x H9 c)) e (getl_head (Bind b) i c (CHead e (Bind -Abbr) u0) H6 u))) (pc3_fsubst0 c (THead (Bind b) u t3) (THead (Bind b) u t3) -(pc3_refl c (THead (Bind b) u t3)) i u0 c (THead (Bind b) x t3) (fsubst0_snd -i u0 c (THead (Bind b) u t3) (THead (Bind b) x t3) (subst0_fst u0 x u i H9 t3 -(Bind b))) e H6)))) (ty3_correct g (CHead c (Bind b) x) t2 t3 (H3 (S i) u0 -(CHead c (Bind b) x) t2 (fsubst0_fst (S i) u0 (CHead c (Bind b) u) t2 (CHead -c (Bind b) x) (csubst0_snd_bind b i u0 u x H9 c)) e (getl_head (Bind b) i c -(CHead e (Bind Abbr) u0) H6 u)))))) (ty3_correct g (CHead c (Bind b) u) t2 t3 -H2)) t5 H8)))) H7)) (\lambda (H7: (ex2 T (\lambda (t6: T).(eq T t5 (THead -(Bind b) u t6))) (\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 -t6)))).(ex2_ind T (\lambda (t6: T).(eq T t5 (THead (Bind b) u t6))) (\lambda -(t6: T).(subst0 (s (Bind b) i) u0 t2 t6)) (ty3 g c t5 (THead (Bind b) u t3)) -(\lambda (x: T).(\lambda (H8: (eq T t5 (THead (Bind b) u x))).(\lambda (H9: -(subst0 (s (Bind b) i) u0 t2 x)).(eq_ind_r T (THead (Bind b) u x) (\lambda -(t6: T).(ty3 g c t6 (THead (Bind b) u t3))) (ex_ind T (\lambda (t6: T).(ty3 g -(CHead c (Bind b) u) t3 t6)) (ty3 g c (THead (Bind b) u x) (THead (Bind b) u -t3)) (\lambda (x0: T).(\lambda (_: (ty3 g (CHead c (Bind b) u) t3 -x0)).(ty3_bind g c u t0 H0 b x t3 (H3 (S i) u0 (CHead c (Bind b) u) x -(fsubst0_snd (S i) u0 (CHead c (Bind b) u) t2 x H9) e (getl_head (Bind b) i c -(CHead e (Bind Abbr) u0) H6 u))))) (ty3_correct g (CHead c (Bind b) u) x t3 -(H3 (S i) u0 (CHead c (Bind b) u) x (fsubst0_snd (S i) u0 (CHead c (Bind b) -u) t2 x H9) e (getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H6 u)))) t5 -H8)))) H7)) (\lambda (H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t6: T).(eq T -t5 (THead (Bind b) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u -u2))) (\lambda (_: T).(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 -t6))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead -(Bind b) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) -(\lambda (_: T).(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 t6))) (ty3 g c -t5 (THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H8: (eq -T t5 (THead (Bind b) x0 x1))).(\lambda (H9: (subst0 i u0 u x0)).(\lambda -(H10: (subst0 (s (Bind b) i) u0 t2 x1)).(eq_ind_r T (THead (Bind b) x0 x1) -(\lambda (t6: T).(ty3 g c t6 (THead (Bind b) u t3))) (ex_ind T (\lambda (t6: -T).(ty3 g (CHead c (Bind b) u) t3 t6)) (ty3 g c (THead (Bind b) x0 x1) (THead -(Bind b) u t3)) (\lambda (x: T).(\lambda (H11: (ty3 g (CHead c (Bind b) u) t3 -x)).(ex_ind T (\lambda (t6: T).(ty3 g (CHead c (Bind b) x0) t3 t6)) (ty3 g c -(THead (Bind b) x0 x1) (THead (Bind b) u t3)) (\lambda (x2: T).(\lambda (_: -(ty3 g (CHead c (Bind b) x0) t3 x2)).(ty3_conv g c (THead (Bind b) u t3) -(THead (Bind b) u x) (ty3_bind g c u t0 H0 b t3 x H11) (THead (Bind b) x0 x1) -(THead (Bind b) x0 t3) (ty3_bind g c x0 t0 (H1 i u0 c x0 (fsubst0_snd i u0 c -u x0 H9) e H6) b x1 t3 (H3 (S i) u0 (CHead c (Bind b) x0) x1 (fsubst0_both (S -i) u0 (CHead c (Bind b) u) t2 x1 H10 (CHead c (Bind b) x0) (csubst0_snd_bind -b i u0 u x0 H9 c)) e (getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H6 u))) -(pc3_fsubst0 c (THead (Bind b) u t3) (THead (Bind b) u t3) (pc3_refl c (THead -(Bind b) u t3)) i u0 c (THead (Bind b) x0 t3) (fsubst0_snd i u0 c (THead -(Bind b) u t3) (THead (Bind b) x0 t3) (subst0_fst u0 x0 u i H9 t3 (Bind b))) -e H6)))) (ty3_correct g (CHead c (Bind b) x0) x1 t3 (H3 (S i) u0 (CHead c -(Bind b) x0) x1 (fsubst0_both (S i) u0 (CHead c (Bind b) u) t2 x1 H10 (CHead -c (Bind b) x0) (csubst0_snd_bind b i u0 u x0 H9 c)) e (getl_head (Bind b) i c -(CHead e (Bind Abbr) u0) H6 u)))))) (ty3_correct g (CHead c (Bind b) u) t2 t3 -H2)) t5 H8)))))) H7)) (subst0_gen_head (Bind b) u0 u t2 t5 i H5)))))) -(\lambda (c3: C).(\lambda (H5: (csubst0 i u0 c c3)).(\lambda (e: C).(\lambda -(H6: (getl i c (CHead e (Bind Abbr) u0))).(ex_ind T (\lambda (t5: T).(ty3 g -(CHead c3 (Bind b) u) t3 t5)) (ty3 g c3 (THead (Bind b) u t2) (THead (Bind b) -u t3)) (\lambda (x: T).(\lambda (_: (ty3 g (CHead c3 (Bind b) u) t3 -x)).(ty3_bind g c3 u t0 (H1 i u0 c3 u (fsubst0_fst i u0 c u c3 H5) e H6) b t2 -t3 (H3 (S i) u0 (CHead c3 (Bind b) u) t2 (fsubst0_fst (S i) u0 (CHead c (Bind -b) u) t2 (CHead c3 (Bind b) u) (csubst0_fst_bind b i c c3 u0 H5 u)) e -(getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H6 u))))) (ty3_correct g -(CHead c3 (Bind b) u) t2 t3 (H3 (S i) u0 (CHead c3 (Bind b) u) t2 -(fsubst0_fst (S i) u0 (CHead c (Bind b) u) t2 (CHead c3 (Bind b) u) -(csubst0_fst_bind b i c c3 u0 H5 u)) e (getl_head (Bind b) i c (CHead e (Bind -Abbr) u0) H6 u)))))))) (\lambda (t5: T).(\lambda (H5: (subst0 i u0 (THead -(Bind b) u t2) t5)).(\lambda (c3: C).(\lambda (H6: (csubst0 i u0 c -c3)).(\lambda (e: C).(\lambda (H7: (getl i c (CHead e (Bind Abbr) -u0))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t5 (THead (Bind b) u2 t2))) -(\lambda (u2: T).(subst0 i u0 u u2))) (ex2 T (\lambda (t6: T).(eq T t5 (THead -(Bind b) u t6))) (\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 t6))) (ex3_2 T -T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead (Bind b) u2 t6)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: -T).(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 t6)))) (ty3 g c3 t5 (THead -(Bind b) u t3)) (\lambda (H8: (ex2 T (\lambda (u2: T).(eq T t5 (THead (Bind -b) u2 t2))) (\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T (\lambda (u2: -T).(eq T t5 (THead (Bind b) u2 t2))) (\lambda (u2: T).(subst0 i u0 u u2)) -(ty3 g c3 t5 (THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H9: (eq T t5 -(THead (Bind b) x t2))).(\lambda (H10: (subst0 i u0 u x)).(eq_ind_r T (THead -(Bind b) x t2) (\lambda (t6: T).(ty3 g c3 t6 (THead (Bind b) u t3))) (ex_ind -T (\lambda (t6: T).(ty3 g (CHead c3 (Bind b) u) t3 t6)) (ty3 g c3 (THead -(Bind b) x t2) (THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (H11: (ty3 g -(CHead c3 (Bind b) u) t3 x0)).(ex_ind T (\lambda (t6: T).(ty3 g (CHead c3 -(Bind b) u) x0 t6)) (ty3 g c3 (THead (Bind b) x t2) (THead (Bind b) u t3)) -(\lambda (x1: T).(\lambda (_: (ty3 g (CHead c3 (Bind b) u) x0 x1)).(ex_ind T -(\lambda (t6: T).(ty3 g (CHead c3 (Bind b) x) t3 t6)) (ty3 g c3 (THead (Bind -b) x t2) (THead (Bind b) u t3)) (\lambda (x2: T).(\lambda (_: (ty3 g (CHead -c3 (Bind b) x) t3 x2)).(ty3_conv g c3 (THead (Bind b) u t3) (THead (Bind b) u -x0) (ty3_bind g c3 u t0 (H1 i u0 c3 u (fsubst0_fst i u0 c u c3 H6) e H7) b t3 -x0 H11) (THead (Bind b) x t2) (THead (Bind b) x t3) (ty3_bind g c3 x t0 (H1 i -u0 c3 x (fsubst0_both i u0 c u x H10 c3 H6) e H7) b t2 t3 (H3 (S i) u0 (CHead -c3 (Bind b) x) t2 (fsubst0_fst (S i) u0 (CHead c (Bind b) u) t2 (CHead c3 -(Bind b) x) (csubst0_both_bind b i u0 u x H10 c c3 H6)) e (getl_head (Bind b) -i c (CHead e (Bind Abbr) u0) H7 u))) (pc3_fsubst0 c (THead (Bind b) u t3) -(THead (Bind b) u t3) (pc3_refl c (THead (Bind b) u t3)) i u0 c3 (THead (Bind -b) x t3) (fsubst0_both i u0 c (THead (Bind b) u t3) (THead (Bind b) x t3) -(subst0_fst u0 x u i H10 t3 (Bind b)) c3 H6) e H7)))) (ty3_correct g (CHead -c3 (Bind b) x) t2 t3 (H3 (S i) u0 (CHead c3 (Bind b) x) t2 (fsubst0_fst (S i) -u0 (CHead c (Bind b) u) t2 (CHead c3 (Bind b) x) (csubst0_both_bind b i u0 u -x H10 c c3 H6)) e (getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H7 u)))))) -(ty3_correct g (CHead c3 (Bind b) u) t3 x0 H11)))) (ty3_correct g (CHead c3 -(Bind b) u) t2 t3 (H3 (S i) u0 (CHead c3 (Bind b) u) t2 (fsubst0_fst (S i) u0 -(CHead c (Bind b) u) t2 (CHead c3 (Bind b) u) (csubst0_fst_bind b i c c3 u0 -H6 u)) e (getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H7 u)))) t5 H9)))) -H8)) (\lambda (H8: (ex2 T (\lambda (t6: T).(eq T t5 (THead (Bind b) u t6))) -(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 t6)))).(ex2_ind T (\lambda (t6: -T).(eq T t5 (THead (Bind b) u t6))) (\lambda (t6: T).(subst0 (s (Bind b) i) -u0 t2 t6)) (ty3 g c3 t5 (THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H9: -(eq T t5 (THead (Bind b) u x))).(\lambda (H10: (subst0 (s (Bind b) i) u0 t2 -x)).(eq_ind_r T (THead (Bind b) u x) (\lambda (t6: T).(ty3 g c3 t6 (THead -(Bind b) u t3))) (ex_ind T (\lambda (t6: T).(ty3 g (CHead c3 (Bind b) u) t3 -t6)) (ty3 g c3 (THead (Bind b) u x) (THead (Bind b) u t3)) (\lambda (x0: -T).(\lambda (_: (ty3 g (CHead c3 (Bind b) u) t3 x0)).(ty3_bind g c3 u t0 (H1 -i u0 c3 u (fsubst0_fst i u0 c u c3 H6) e H7) b x t3 (H3 (S i) u0 (CHead c3 -(Bind b) u) x (fsubst0_both (S i) u0 (CHead c (Bind b) u) t2 x H10 (CHead c3 -(Bind b) u) (csubst0_fst_bind b i c c3 u0 H6 u)) e (getl_head (Bind b) i c -(CHead e (Bind Abbr) u0) H7 u))))) (ty3_correct g (CHead c3 (Bind b) u) x t3 -(H3 (S i) u0 (CHead c3 (Bind b) u) x (fsubst0_both (S i) u0 (CHead c (Bind b) -u) t2 x H10 (CHead c3 (Bind b) u) (csubst0_fst_bind b i c c3 u0 H6 u)) e -(getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H7 u)))) t5 H9)))) H8)) -(\lambda (H8: (ex3_2 T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead -(Bind b) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) -(\lambda (_: T).(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 -t6))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead -(Bind b) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) -(\lambda (_: T).(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 t6))) (ty3 g c3 -t5 (THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H9: (eq -T t5 (THead (Bind b) x0 x1))).(\lambda (H10: (subst0 i u0 u x0)).(\lambda -(H11: (subst0 (s (Bind b) i) u0 t2 x1)).(eq_ind_r T (THead (Bind b) x0 x1) -(\lambda (t6: T).(ty3 g c3 t6 (THead (Bind b) u t3))) (ex_ind T (\lambda (t6: -T).(ty3 g (CHead c3 (Bind b) u) t3 t6)) (ty3 g c3 (THead (Bind b) x0 x1) -(THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H12: (ty3 g (CHead c3 (Bind -b) u) t3 x)).(ex_ind T (\lambda (t6: T).(ty3 g (CHead c3 (Bind b) u) x t6)) -(ty3 g c3 (THead (Bind b) x0 x1) (THead (Bind b) u t3)) (\lambda (x2: -T).(\lambda (_: (ty3 g (CHead c3 (Bind b) u) x x2)).(ex_ind T (\lambda (t6: -T).(ty3 g (CHead c3 (Bind b) x0) t3 t6)) (ty3 g c3 (THead (Bind b) x0 x1) -(THead (Bind b) u t3)) (\lambda (x3: T).(\lambda (_: (ty3 g (CHead c3 (Bind -b) x0) t3 x3)).(ty3_conv g c3 (THead (Bind b) u t3) (THead (Bind b) u x) -(ty3_bind g c3 u t0 (H1 i u0 c3 u (fsubst0_fst i u0 c u c3 H6) e H7) b t3 x -H12) (THead (Bind b) x0 x1) (THead (Bind b) x0 t3) (ty3_bind g c3 x0 t0 (H1 i -u0 c3 x0 (fsubst0_both i u0 c u x0 H10 c3 H6) e H7) b x1 t3 (H3 (S i) u0 -(CHead c3 (Bind b) x0) x1 (fsubst0_both (S i) u0 (CHead c (Bind b) u) t2 x1 -H11 (CHead c3 (Bind b) x0) (csubst0_both_bind b i u0 u x0 H10 c c3 H6)) e -(getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H7 u))) (pc3_fsubst0 c -(THead (Bind b) u t3) (THead (Bind b) u t3) (pc3_refl c (THead (Bind b) u -t3)) i u0 c3 (THead (Bind b) x0 t3) (fsubst0_both i u0 c (THead (Bind b) u -t3) (THead (Bind b) x0 t3) (subst0_fst u0 x0 u i H10 t3 (Bind b)) c3 H6) e -H7)))) (ty3_correct g (CHead c3 (Bind b) x0) x1 t3 (H3 (S i) u0 (CHead c3 -(Bind b) x0) x1 (fsubst0_both (S i) u0 (CHead c (Bind b) u) t2 x1 H11 (CHead -c3 (Bind b) x0) (csubst0_both_bind b i u0 u x0 H10 c c3 H6)) e (getl_head -(Bind b) i c (CHead e (Bind Abbr) u0) H7 u)))))) (ty3_correct g (CHead c3 -(Bind b) u) t3 x H12)))) (ty3_correct g (CHead c3 (Bind b) u) t2 t3 (H3 (S i) -u0 (CHead c3 (Bind b) u) t2 (fsubst0_fst (S i) u0 (CHead c (Bind b) u) t2 -(CHead c3 (Bind b) u) (csubst0_fst_bind b i c c3 u0 H6 u)) e (getl_head (Bind -b) i c (CHead e (Bind Abbr) u0) H7 u)))) t5 H9)))))) H8)) (subst0_gen_head -(Bind b) u0 u t2 t5 i H5)))))))) c2 t4 H4)))))))))))))))) (\lambda (c: -C).(\lambda (w: T).(\lambda (u: T).(\lambda (H0: (ty3 g c w u)).(\lambda (H1: -((\forall (i: nat).(\forall (u0: T).(\forall (c2: C).(\forall (t2: -T).((fsubst0 i u0 c w c2 t2) \to (\forall (e: C).((getl i c (CHead e (Bind -Abbr) u0)) \to (ty3 g c2 t2 u)))))))))).(\lambda (v: T).(\lambda (t0: -T).(\lambda (H2: (ty3 g c v (THead (Bind Abst) u t0))).(\lambda (H3: -((\forall (i: nat).(\forall (u0: T).(\forall (c2: C).(\forall (t2: -T).((fsubst0 i u0 c v c2 t2) \to (\forall (e: C).((getl i c (CHead e (Bind -Abbr) u0)) \to (ty3 g c2 t2 (THead (Bind Abst) u t0))))))))))).(\lambda (i: -nat).(\lambda (u0: T).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H4: -(fsubst0 i u0 c (THead (Flat Appl) w v) c2 t2)).(fsubst0_ind i u0 c (THead -(Flat Appl) w v) (\lambda (c0: C).(\lambda (t3: T).(\forall (e: C).((getl i c -(CHead e (Bind Abbr) u0)) \to (ty3 g c0 t3 (THead (Flat Appl) w (THead (Bind -Abst) u t0))))))) (\lambda (t3: T).(\lambda (H5: (subst0 i u0 (THead (Flat -Appl) w v) t3)).(\lambda (e: C).(\lambda (H6: (getl i c (CHead 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t0)))) -(ex_ind T (\lambda (t4: T).(ty3 g c (THead (Bind Abst) u t0) t4)) (ty3 g c -(THead (Flat Appl) x v) (THead (Flat Appl) w (THead (Bind Abst) u t0))) -(\lambda (x0: T).(\lambda (H10: (ty3 g c (THead (Bind Abst) u t0) -x0)).(ex3_2_ind T T (\lambda (t4: T).(\lambda (_: T).(pc3 c (THead (Bind -Abst) u t4) x0))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c u t5))) (\lambda -(t4: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t0 t4))) (ty3 g c -(THead (Flat Appl) x v) (THead (Flat Appl) w (THead (Bind Abst) u t0))) -(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 c (THead (Bind Abst) u -x1) x0)).(\lambda (_: (ty3 g c u x2)).(\lambda (H13: (ty3 g (CHead c (Bind -Abst) u) t0 x1)).(ex_ind T (\lambda (t4: T).(ty3 g c u t4)) (ty3 g c (THead -(Flat Appl) x v) (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda -(x3: T).(\lambda (H14: (ty3 g c u x3)).(ty3_conv g c (THead (Flat Appl) w -(THead (Bind Abst) u t0)) (THead (Flat Appl) w (THead (Bind Abst) u x1)) -(ty3_appl g c w u H0 (THead (Bind Abst) u t0) x1 (ty3_bind g c u x3 H14 Abst -t0 x1 H13)) (THead (Flat Appl) x v) (THead (Flat Appl) x (THead (Bind Abst) u -t0)) (ty3_appl g c x u (H1 i u0 c x (fsubst0_snd i u0 c w x H9) e H6) v t0 -H2) (pc3_fsubst0 c (THead (Flat Appl) w (THead (Bind Abst) u t0)) (THead -(Flat Appl) w (THead (Bind Abst) u t0)) (pc3_refl c (THead (Flat Appl) w -(THead (Bind Abst) u t0))) i u0 c (THead (Flat Appl) x (THead (Bind Abst) u -t0)) (fsubst0_snd i u0 c (THead (Flat Appl) w (THead (Bind Abst) u t0)) -(THead (Flat Appl) x (THead (Bind Abst) u t0)) (subst0_fst u0 x w i H9 (THead -(Bind Abst) u t0) (Flat Appl))) e H6)))) (ty3_correct g c x u (H1 i u0 c x -(fsubst0_snd i u0 c w x H9) e H6)))))))) (ty3_gen_bind g Abst c u t0 x0 -H10)))) (ty3_correct g c v (THead (Bind Abst) u t0) H2)) t3 H8)))) H7)) -(\lambda (H7: (ex2 T (\lambda (t4: T).(eq T t3 (THead (Flat Appl) w t4))) -(\lambda (t4: T).(subst0 (s (Flat Appl) i) u0 v t4)))).(ex2_ind T (\lambda -(t4: T).(eq T t3 (THead (Flat Appl) w 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(x1: -T).(\lambda (H8: (eq T t3 (THead (Flat Appl) x0 x1))).(\lambda (H9: (subst0 i -u0 w x0)).(\lambda (H10: (subst0 (s (Flat Appl) i) u0 v x1)).(eq_ind_r T -(THead (Flat Appl) x0 x1) (\lambda (t4: T).(ty3 g c t4 (THead (Flat Appl) w -(THead (Bind Abst) u t0)))) (ex_ind T (\lambda (t4: T).(ty3 g c (THead (Bind -Abst) u t0) t4)) (ty3 g c (THead (Flat Appl) x0 x1) (THead (Flat Appl) w -(THead (Bind Abst) u t0))) (\lambda (x: T).(\lambda (H11: (ty3 g c (THead -(Bind Abst) u t0) x)).(ex3_2_ind T T (\lambda (t4: T).(\lambda (_: T).(pc3 c -(THead (Bind Abst) u t4) x))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c u -t5))) (\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t0 -t4))) (ty3 g c (THead (Flat Appl) x0 x1) (THead (Flat Appl) w (THead (Bind -Abst) u t0))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (_: (pc3 c (THead -(Bind Abst) u x2) x)).(\lambda (_: (ty3 g c u x3)).(\lambda (H14: (ty3 g -(CHead c (Bind Abst) u) t0 x2)).(ex_ind T (\lambda (t4: T).(ty3 g c u t4)) -(ty3 g c 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H0))))))) (ty3_gen_bind g Abst c u t0 x H11)))) -(ty3_correct g c v (THead (Bind Abst) u t0) H2)) t3 H8)))))) H7)) -(subst0_gen_head (Flat Appl) u0 w v t3 i H5)))))) (\lambda (c3: C).(\lambda -(H5: (csubst0 i u0 c c3)).(\lambda (e: C).(\lambda (H6: (getl i c (CHead e -(Bind Abbr) u0))).(ty3_appl g c3 w u (H1 i u0 c3 w (fsubst0_fst i u0 c w c3 -H5) e H6) v t0 (H3 i u0 c3 v (fsubst0_fst i u0 c v c3 H5) e H6)))))) (\lambda -(t3: T).(\lambda (H5: (subst0 i u0 (THead (Flat Appl) w v) t3)).(\lambda (c3: -C).(\lambda (H6: (csubst0 i u0 c c3)).(\lambda (e: C).(\lambda (H7: (getl i c -(CHead e (Bind Abbr) u0))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t3 (THead -(Flat Appl) u2 v))) (\lambda (u2: T).(subst0 i u0 w u2))) (ex2 T (\lambda -(t4: T).(eq T t3 (THead (Flat Appl) w t4))) (\lambda (t4: T).(subst0 (s (Flat -Appl) i) u0 v t4))) (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 -(THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 w -u2))) (\lambda (_: T).(\lambda 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c (THead (Flat Appl) w (THead (Bind Abst) u -t0)) (THead (Flat Appl) x (THead (Bind Abst) u t0)) (subst0_fst u0 x w i H10 -(THead (Bind Abst) u t0) (Flat Appl)) c3 H6) e H7))))))) (ty3_gen_bind g Abst -c3 u t0 x0 H11)))) (ty3_correct g c3 v (THead (Bind Abst) u t0) (H3 i u0 c3 v -(fsubst0_fst i u0 c v c3 H6) e H7))) t3 H9)))) H8)) (\lambda (H8: (ex2 T -(\lambda (t4: T).(eq T t3 (THead (Flat Appl) w t4))) (\lambda (t4: T).(subst0 -(s (Flat Appl) i) u0 v t4)))).(ex2_ind T (\lambda (t4: T).(eq T t3 (THead -(Flat Appl) w t4))) (\lambda (t4: T).(subst0 (s (Flat Appl) i) u0 v t4)) (ty3 -g c3 t3 (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x: -T).(\lambda (H9: (eq T t3 (THead (Flat Appl) w x))).(\lambda (H10: (subst0 (s -(Flat Appl) i) u0 v x)).(eq_ind_r T (THead (Flat Appl) w x) (\lambda (t4: -T).(ty3 g c3 t4 (THead (Flat Appl) w (THead (Bind Abst) u t0)))) (ty3_appl g -c3 w u (H1 i u0 c3 w (fsubst0_fst i u0 c w c3 H6) e H7) x t0 (H3 i u0 c3 x -(fsubst0_both i u0 c v x H10 c3 H6) e H7)) t3 H9)))) H8)) (\lambda (H8: -(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 -t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 w u2))) (\lambda (_: -T).(\lambda (t4: T).(subst0 (s (Flat Appl) i) u0 v t4))))).(ex3_2_ind T T -(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 i u0 w u2))) (\lambda (_: -T).(\lambda (t4: T).(subst0 (s (Flat Appl) i) u0 v t4))) (ty3 g c3 t3 (THead -(Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H9: (eq T t3 (THead (Flat Appl) x0 x1))).(\lambda (H10: (subst0 -i u0 w x0)).(\lambda (H11: (subst0 (s (Flat Appl) i) u0 v x1)).(eq_ind_r T -(THead (Flat Appl) x0 x1) (\lambda (t4: T).(ty3 g c3 t4 (THead (Flat Appl) w -(THead (Bind Abst) u t0)))) (ex_ind T (\lambda (t4: T).(ty3 g c3 (THead (Bind -Abst) u t0) t4)) (ty3 g c3 (THead (Flat Appl) x0 x1) (THead (Flat Appl) w -(THead (Bind Abst) u t0))) (\lambda (x: T).(\lambda (H12: (ty3 g c3 (THead -(Bind Abst) u t0) x)).(ex3_2_ind T T (\lambda (t4: T).(\lambda (_: T).(pc3 c3 -(THead (Bind Abst) u t4) x))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c3 u -t5))) (\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead c3 (Bind Abst) u) t0 -t4))) (ty3 g c3 (THead (Flat Appl) x0 x1) (THead (Flat Appl) w (THead (Bind -Abst) u t0))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (_: (pc3 c3 (THead -(Bind Abst) u x2) x)).(\lambda (_: (ty3 g c3 u x3)).(\lambda (H15: (ty3 g -(CHead c3 (Bind Abst) u) t0 x2)).(ex_ind T (\lambda (t4: T).(ty3 g c3 u t4)) -(ty3 g c3 (THead (Flat Appl) x0 x1) (THead (Flat Appl) w (THead (Bind Abst) u -t0))) (\lambda (x4: T).(\lambda (H16: (ty3 g c3 u x4)).(ty3_conv g c3 (THead -(Flat Appl) w (THead (Bind Abst) u t0)) (THead (Flat Appl) w (THead (Bind -Abst) u x2)) (ty3_appl g c3 w u (H1 i u0 c3 w (fsubst0_fst i u0 c w c3 H6) e -H7) (THead (Bind Abst) u t0) x2 (ty3_bind g c3 u x4 H16 Abst t0 x2 H15)) -(THead (Flat Appl) x0 x1) (THead (Flat Appl) x0 (THead (Bind Abst) u t0)) -(ty3_appl g c3 x0 u (H1 i u0 c3 x0 (fsubst0_both i u0 c w x0 H10 c3 H6) e H7) -x1 t0 (H3 i u0 c3 x1 (fsubst0_both i u0 c v x1 H11 c3 H6) e H7)) (pc3_fsubst0 -c (THead (Flat Appl) w (THead (Bind Abst) u t0)) (THead (Flat Appl) w (THead -(Bind Abst) u t0)) (pc3_refl c (THead (Flat Appl) w (THead (Bind Abst) u -t0))) i u0 c3 (THead (Flat Appl) x0 (THead (Bind Abst) u t0)) (fsubst0_both i -u0 c (THead (Flat Appl) w (THead (Bind Abst) u t0)) (THead (Flat Appl) x0 -(THead (Bind Abst) u t0)) (subst0_fst u0 x0 w i H10 (THead (Bind Abst) u t0) -(Flat Appl)) c3 H6) e H7)))) (ty3_correct g c3 w u (H1 i u0 c3 w (fsubst0_fst -i u0 c w c3 H6) e H7)))))))) (ty3_gen_bind g Abst c3 u t0 x H12)))) -(ty3_correct g c3 v (THead (Bind Abst) u t0) (H3 i u0 c3 v (fsubst0_fst i u0 -c v c3 H6) e H7))) t3 H9)))))) H8)) (subst0_gen_head (Flat Appl) u0 w v t3 i -H5)))))))) c2 t2 H4))))))))))))))) (\lambda (c: C).(\lambda (t2: T).(\lambda -(t3: T).(\lambda (H0: (ty3 g c t2 t3)).(\lambda (H1: ((\forall (i: -nat).(\forall (u: T).(\forall (c2: C).(\forall (t4: T).((fsubst0 i u c t2 c2 -t4) \to (\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) \to (ty3 g c2 t4 -t3)))))))))).(\lambda (t0: T).(\lambda (H2: (ty3 g c t3 t0)).(\lambda (H3: -((\forall (i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t4: -T).((fsubst0 i u c t3 c2 t4) \to (\forall (e: C).((getl i c (CHead e (Bind -Abbr) u)) \to (ty3 g c2 t4 t0)))))))))).(\lambda (i: nat).(\lambda (u: -T).(\lambda (c2: C).(\lambda (t4: T).(\lambda (H4: (fsubst0 i u c (THead -(Flat Cast) t3 t2) c2 t4)).(fsubst0_ind i u c (THead (Flat Cast) t3 t2) -(\lambda (c0: C).(\lambda (t5: T).(\forall (e: C).((getl i c (CHead e (Bind -Abbr) u)) \to (ty3 g c0 t5 (THead (Flat Cast) t0 t3)))))) (\lambda (t5: -T).(\lambda (H5: (subst0 i u (THead (Flat Cast) t3 t2) t5)).(\lambda (e: -C).(\lambda (H6: (getl i c (CHead e (Bind Abbr) u))).(or3_ind (ex2 T (\lambda -(u2: T).(eq T t5 (THead (Flat Cast) u2 t2))) (\lambda (u2: T).(subst0 i u t3 -u2))) (ex2 T (\lambda (t6: T).(eq T t5 (THead (Flat Cast) t3 t6))) (\lambda -(t6: T).(subst0 (s (Flat Cast) i) u t2 t6))) (ex3_2 T T (\lambda (u2: -T).(\lambda (t6: T).(eq T t5 (THead (Flat Cast) u2 t6)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i u t3 u2))) (\lambda (_: T).(\lambda (t6: -T).(subst0 (s (Flat Cast) i) u t2 t6)))) (ty3 g c t5 (THead (Flat Cast) t0 -t3)) (\lambda (H7: (ex2 T (\lambda (u2: T).(eq T t5 (THead (Flat Cast) u2 -t2))) (\lambda (u2: T).(subst0 i u t3 u2)))).(ex2_ind T (\lambda (u2: T).(eq -T t5 (THead (Flat Cast) u2 t2))) (\lambda (u2: T).(subst0 i u t3 u2)) (ty3 g -c t5 (THead (Flat Cast) t0 t3)) (\lambda (x: T).(\lambda (H8: (eq T t5 (THead -(Flat Cast) x t2))).(\lambda (H9: (subst0 i u t3 x)).(eq_ind_r T (THead (Flat -Cast) x t2) (\lambda (t6: T).(ty3 g c t6 (THead (Flat Cast) t0 t3))) (ex_ind -T (\lambda (t6: T).(ty3 g c t0 t6)) (ty3 g c (THead (Flat Cast) x t2) (THead -(Flat Cast) t0 t3)) (\lambda (x0: T).(\lambda (H10: (ty3 g c t0 -x0)).(ty3_conv g c (THead (Flat Cast) t0 t3) (THead (Flat Cast) x0 t0) -(ty3_cast g c t3 t0 H2 x0 H10) (THead (Flat Cast) x t2) (THead (Flat Cast) t0 -x) (ty3_cast g c t2 x (ty3_conv g c x t0 (H3 i u c x (fsubst0_snd i u c t3 x -H9) e H6) t2 t3 H0 (pc3_s c t3 x (pc3_fsubst0 c t3 t3 (pc3_refl c t3) i u c x -(fsubst0_snd i u c t3 x H9) e H6))) t0 (H3 i u c x (fsubst0_snd i u c t3 x -H9) e H6)) (pc3_fsubst0 c (THead (Flat Cast) t0 t3) (THead (Flat Cast) t0 t3) -(pc3_refl c (THead (Flat Cast) t0 t3)) i u c (THead (Flat Cast) t0 x) -(fsubst0_snd i u c (THead (Flat Cast) t0 t3) (THead (Flat Cast) t0 x) -(subst0_snd (Flat Cast) u x t3 i H9 t0)) e H6)))) (ty3_correct g c x t0 (H3 i -u c x (fsubst0_snd i u c t3 x H9) e H6))) t5 H8)))) H7)) (\lambda (H7: (ex2 T -(\lambda (t6: T).(eq T t5 (THead (Flat Cast) t3 t6))) (\lambda (t6: -T).(subst0 (s (Flat Cast) i) u t2 t6)))).(ex2_ind T (\lambda (t6: T).(eq T t5 -(THead (Flat Cast) t3 t6))) (\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 -t6)) (ty3 g c t5 (THead (Flat Cast) t0 t3)) (\lambda (x: T).(\lambda (H8: (eq -T t5 (THead (Flat Cast) t3 x))).(\lambda (H9: (subst0 (s (Flat Cast) i) u t2 -x)).(eq_ind_r T (THead (Flat Cast) t3 x) (\lambda (t6: T).(ty3 g c t6 (THead -(Flat Cast) t0 t3))) (ty3_cast g c x t3 (H1 (s (Flat Cast) i) u c x -(fsubst0_snd (s (Flat Cast) i) u c t2 x H9) e H6) t0 H2) t5 H8)))) H7)) -(\lambda (H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead -(Flat Cast) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u t3 u2))) -(\lambda (_: T).(\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 -t6))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead -(Flat Cast) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u t3 u2))) -(\lambda (_: T).(\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 t6))) (ty3 g -c t5 (THead (Flat Cast) t0 t3)) (\lambda (x0: T).(\lambda (x1: T).(\lambda -(H8: (eq T t5 (THead (Flat Cast) x0 x1))).(\lambda (H9: (subst0 i u t3 -x0)).(\lambda (H10: (subst0 (s (Flat Cast) i) u t2 x1)).(eq_ind_r T (THead -(Flat Cast) x0 x1) (\lambda (t6: T).(ty3 g c t6 (THead (Flat Cast) t0 t3))) -(ex_ind T (\lambda (t6: T).(ty3 g c t0 t6)) (ty3 g c (THead (Flat Cast) x0 -x1) (THead (Flat Cast) t0 t3)) (\lambda (x: T).(\lambda (H11: (ty3 g c t0 -x)).(ty3_conv g c (THead (Flat Cast) t0 t3) (THead (Flat Cast) x t0) -(ty3_cast g c t3 t0 H2 x H11) (THead (Flat Cast) x0 x1) (THead (Flat Cast) t0 -x0) (ty3_cast g c x1 x0 (ty3_conv g c x0 t0 (H3 i u c x0 (fsubst0_snd i u c -t3 x0 H9) e H6) x1 t3 (H1 (s (Flat Cast) i) u c x1 (fsubst0_snd (s (Flat -Cast) i) u c t2 x1 H10) e H6) (pc3_s c t3 x0 (pc3_fsubst0 c t3 t3 (pc3_refl c -t3) i u c x0 (fsubst0_snd i u c t3 x0 H9) e H6))) t0 (H3 i u c x0 -(fsubst0_snd i u c t3 x0 H9) e H6)) (pc3_fsubst0 c (THead (Flat Cast) t0 t3) -(THead (Flat Cast) t0 t3) (pc3_refl c (THead (Flat Cast) t0 t3)) i u c (THead -(Flat Cast) t0 x0) (fsubst0_snd i u c (THead (Flat Cast) t0 t3) (THead (Flat -Cast) t0 x0) (subst0_snd (Flat Cast) u x0 t3 i H9 t0)) e H6)))) (ty3_correct -g c x0 t0 (H3 i u c x0 (fsubst0_snd i u c t3 x0 H9) e H6))) t5 H8)))))) H7)) -(subst0_gen_head (Flat Cast) u t3 t2 t5 i H5)))))) (\lambda (c3: C).(\lambda -(H5: (csubst0 i u c c3)).(\lambda (e: C).(\lambda (H6: (getl i c (CHead e -(Bind Abbr) u))).(ty3_cast g c3 t2 t3 (H1 i u c3 t2 (fsubst0_fst i u c t2 c3 -H5) e H6) t0 (H3 i u c3 t3 (fsubst0_fst i u c t3 c3 H5) e H6)))))) (\lambda -(t5: T).(\lambda (H5: (subst0 i u (THead (Flat Cast) t3 t2) t5)).(\lambda -(c3: C).(\lambda (H6: (csubst0 i u c c3)).(\lambda (e: C).(\lambda (H7: (getl -i c (CHead e (Bind Abbr) u))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t5 -(THead (Flat Cast) u2 t2))) (\lambda (u2: T).(subst0 i u t3 u2))) (ex2 T -(\lambda (t6: T).(eq T t5 (THead (Flat Cast) t3 t6))) (\lambda (t6: -T).(subst0 (s (Flat Cast) i) u t2 t6))) (ex3_2 T T (\lambda (u2: T).(\lambda -(t6: T).(eq T t5 (THead (Flat Cast) u2 t6)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 i u t3 u2))) (\lambda (_: T).(\lambda (t6: T).(subst0 (s (Flat -Cast) i) u t2 t6)))) (ty3 g c3 t5 (THead (Flat Cast) t0 t3)) (\lambda (H8: -(ex2 T (\lambda (u2: T).(eq T t5 (THead (Flat Cast) u2 t2))) (\lambda (u2: -T).(subst0 i u t3 u2)))).(ex2_ind T (\lambda (u2: T).(eq T t5 (THead (Flat -Cast) u2 t2))) (\lambda (u2: T).(subst0 i u t3 u2)) (ty3 g c3 t5 (THead (Flat -Cast) t0 t3)) (\lambda (x: T).(\lambda (H9: (eq T t5 (THead (Flat Cast) x -t2))).(\lambda (H10: (subst0 i u t3 x)).(eq_ind_r T (THead (Flat Cast) x t2) -(\lambda (t6: T).(ty3 g c3 t6 (THead (Flat Cast) t0 t3))) (ex_ind T (\lambda -(t6: T).(ty3 g c3 t0 t6)) (ty3 g c3 (THead (Flat Cast) x t2) (THead (Flat -Cast) t0 t3)) (\lambda (x0: T).(\lambda (H11: (ty3 g c3 t0 x0)).(ty3_conv g -c3 (THead (Flat Cast) t0 t3) (THead (Flat Cast) x0 t0) (ty3_cast g c3 t3 t0 -(H3 i u c3 t3 (fsubst0_fst i u c t3 c3 H6) e H7) x0 H11) (THead (Flat Cast) x -t2) (THead (Flat Cast) t0 x) (ty3_cast g c3 t2 x (ty3_conv g c3 x t0 (H3 i u -c3 x (fsubst0_both i u c t3 x H10 c3 H6) e H7) t2 t3 (H1 i u c3 t2 -(fsubst0_fst i u c t2 c3 H6) e H7) (pc3_s c3 t3 x (pc3_fsubst0 c t3 t3 -(pc3_refl c t3) i u c3 x (fsubst0_both i u c t3 x H10 c3 H6) e H7))) t0 (H3 i -u c3 x (fsubst0_both i u c t3 x H10 c3 H6) e H7)) (pc3_fsubst0 c (THead (Flat -Cast) t0 t3) (THead (Flat Cast) t0 t3) (pc3_refl c (THead (Flat Cast) t0 t3)) -i u c3 (THead (Flat Cast) t0 x) (fsubst0_both i u c (THead (Flat Cast) t0 t3) -(THead (Flat Cast) t0 x) (subst0_snd (Flat Cast) u x t3 i H10 t0) c3 H6) e -H7)))) (ty3_correct g c3 t3 t0 (H3 i u c3 t3 (fsubst0_fst i u c t3 c3 H6) e -H7))) t5 H9)))) H8)) (\lambda (H8: (ex2 T (\lambda (t6: T).(eq T t5 (THead -(Flat Cast) t3 t6))) (\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 -t6)))).(ex2_ind T (\lambda (t6: T).(eq T t5 (THead (Flat Cast) t3 t6))) -(\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 t6)) (ty3 g c3 t5 (THead -(Flat Cast) t0 t3)) (\lambda (x: T).(\lambda (H9: (eq T t5 (THead (Flat Cast) -t3 x))).(\lambda (H10: (subst0 (s (Flat Cast) i) u t2 x)).(eq_ind_r T (THead -(Flat Cast) t3 x) (\lambda (t6: T).(ty3 g c3 t6 (THead (Flat Cast) t0 t3))) -(ty3_cast g c3 x t3 (H1 i u c3 x (fsubst0_both i u c t2 x H10 c3 H6) e H7) t0 -(H3 i u c3 t3 (fsubst0_fst i u c t3 c3 H6) e H7)) t5 H9)))) H8)) (\lambda -(H8: (ex3_2 T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead (Flat Cast) -u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u t3 u2))) (\lambda (_: -T).(\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 t6))))).(ex3_2_ind T T -(\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead (Flat Cast) u2 t6)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 i u t3 u2))) (\lambda (_: -T).(\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 t6))) (ty3 g c3 t5 (THead -(Flat Cast) t0 t3)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H9: (eq T t5 -(THead (Flat Cast) x0 x1))).(\lambda (H10: (subst0 i u t3 x0)).(\lambda (H11: -(subst0 (s (Flat Cast) i) u t2 x1)).(eq_ind_r T (THead (Flat Cast) x0 x1) -(\lambda (t6: T).(ty3 g c3 t6 (THead (Flat Cast) t0 t3))) (ex_ind T (\lambda -(t6: T).(ty3 g c3 t0 t6)) (ty3 g c3 (THead (Flat Cast) x0 x1) (THead (Flat -Cast) t0 t3)) (\lambda (x: T).(\lambda (H12: (ty3 g c3 t0 x)).(ty3_conv g c3 -(THead (Flat Cast) t0 t3) (THead (Flat Cast) x t0) (ty3_cast g c3 t3 t0 (H3 i -u c3 t3 (fsubst0_fst i u c t3 c3 H6) e H7) x H12) (THead (Flat Cast) x0 x1) -(THead (Flat Cast) t0 x0) (ty3_cast g c3 x1 x0 (ty3_conv g c3 x0 t0 (H3 i u -c3 x0 (fsubst0_both i u c t3 x0 H10 c3 H6) e H7) x1 t3 (H1 i u c3 x1 -(fsubst0_both i u c t2 x1 H11 c3 H6) e H7) (pc3_s c3 t3 x0 (pc3_fsubst0 c t3 -t3 (pc3_refl c t3) i u c3 x0 (fsubst0_both i u c t3 x0 H10 c3 H6) e H7))) t0 -(H3 i u c3 x0 (fsubst0_both i u c t3 x0 H10 c3 H6) e H7)) (pc3_fsubst0 c -(THead (Flat Cast) t0 t3) (THead (Flat Cast) t0 t3) (pc3_refl c (THead (Flat -Cast) t0 t3)) i u c3 (THead (Flat Cast) t0 x0) (fsubst0_both i u c (THead -(Flat Cast) t0 t3) (THead (Flat Cast) t0 x0) (subst0_snd (Flat Cast) u x0 t3 -i H10 t0) c3 H6) e H7)))) (ty3_correct g c3 t3 t0 (H3 i u c3 t3 (fsubst0_fst -i u c t3 c3 H6) e H7))) t5 H9)))))) H8)) (subst0_gen_head (Flat Cast) u t3 t2 -t5 i H5)))))))) c2 t4 H4)))))))))))))) c1 t1 t H))))). -(* COMMENTS -Initial nodes: 23439 -END *) - -theorem ty3_csubst0: - \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c1 -t1 t2) \to (\forall (e: C).(\forall (u: T).(\forall (i: nat).((getl i c1 -(CHead e (Bind Abbr) u)) \to (\forall (c2: C).((csubst0 i u c1 c2) \to (ty3 g -c2 t1 t2))))))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H: (ty3 g c1 t1 t2)).(\lambda (e: C).(\lambda (u: T).(\lambda (i: -nat).(\lambda (H0: (getl i c1 (CHead e (Bind Abbr) u))).(\lambda (c2: -C).(\lambda (H1: (csubst0 i u c1 c2)).(ty3_fsubst0 g c1 t1 t2 H i u c2 t1 -(fsubst0_fst i u c1 t1 c2 H1) e H0))))))))))). -(* COMMENTS -Initial nodes: 89 -END *) - -theorem ty3_subst0: - \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((ty3 g c t1 -t) \to (\forall (e: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead e -(Bind Abbr) u)) \to (\forall (t2: T).((subst0 i u t1 t2) \to (ty3 g c t2 -t))))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H: -(ty3 g c t1 t)).(\lambda (e: C).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H0: (getl i c (CHead e (Bind Abbr) u))).(\lambda (t2: T).(\lambda (H1: -(subst0 i u t1 t2)).(ty3_fsubst0 g c t1 t H i u c t2 (fsubst0_snd i u c t1 t2 -H1) e H0))))))))))). -(* COMMENTS -Initial nodes: 89 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/fwd.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/fwd.ma deleted file mode 100644 index bf6634e45..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/fwd.ma +++ /dev/null @@ -1,922 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/ty3/defs.ma". - -include "Basic-1/pc3/props.ma". - -theorem ty3_gen_sort: - \forall (g: G).(\forall (c: C).(\forall (x: T).(\forall (n: nat).((ty3 g c -(TSort n) x) \to (pc3 c (TSort (next g n)) x))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda -(H: (ty3 g c (TSort n) x)).(insert_eq T (TSort n) (\lambda (t: T).(ty3 g c t -x)) (\lambda (_: T).(pc3 c (TSort (next g n)) x)) (\lambda (y: T).(\lambda -(H0: (ty3 g c y x)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: -T).((eq T t (TSort n)) \to (pc3 c0 (TSort (next g n)) t0))))) (\lambda (c0: -C).(\lambda (t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda -(_: (((eq T t2 (TSort n)) \to (pc3 c0 (TSort (next g n)) t)))).(\lambda (u: -T).(\lambda (t1: T).(\lambda (H3: (ty3 g c0 u t1)).(\lambda (H4: (((eq T u -(TSort n)) \to (pc3 c0 (TSort (next g n)) t1)))).(\lambda (H5: (pc3 c0 t1 -t2)).(\lambda (H6: (eq T u (TSort n))).(let H7 \def (f_equal T T (\lambda (e: -T).e) u (TSort n) H6) in (let H8 \def (eq_ind T u (\lambda (t0: T).((eq T t0 -(TSort n)) \to (pc3 c0 (TSort (next g n)) t1))) H4 (TSort n) H7) in (let H9 -\def (eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 t1)) H3 (TSort n) H7) in -(pc3_t t1 c0 (TSort (next g n)) (H8 (refl_equal T (TSort n))) t2 -H5))))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq T -(TSort m) (TSort n))).(let H2 \def (f_equal T nat (\lambda (e: T).(match e in -T return (\lambda (_: T).nat) with [(TSort n0) \Rightarrow n0 | (TLRef _) -\Rightarrow m | (THead _ _ _) \Rightarrow m])) (TSort m) (TSort n) H1) in -(eq_ind_r nat n (\lambda (n0: nat).(pc3 c0 (TSort (next g n)) (TSort (next g -n0)))) (pc3_refl c0 (TSort (next g n))) m H2))))) (\lambda (n0: nat).(\lambda -(c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (_: (getl n0 c0 (CHead d -(Bind Abbr) u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (_: -(((eq T u (TSort n)) \to (pc3 d (TSort (next g n)) t)))).(\lambda (H4: (eq T -(TLRef n0) (TSort n))).(let H5 \def (eq_ind T (TLRef n0) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I -(TSort n) H4) in (False_ind (pc3 c0 (TSort (next g n)) (lift (S n0) O t)) -H5))))))))))) (\lambda (n0: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda -(u: T).(\lambda (_: (getl n0 c0 (CHead d (Bind Abst) u))).(\lambda (t: -T).(\lambda (_: (ty3 g d u t)).(\lambda (_: (((eq T u (TSort n)) \to (pc3 d -(TSort (next g n)) t)))).(\lambda (H4: (eq T (TLRef n0) (TSort n))).(let H5 -\def (eq_ind T (TLRef n0) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | -(THead _ _ _) \Rightarrow False])) I (TSort n) H4) in (False_ind (pc3 c0 -(TSort (next g n)) (lift (S n0) O u)) H5))))))))))) (\lambda (c0: C).(\lambda -(u: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((eq T u -(TSort n)) \to (pc3 c0 (TSort (next g n)) t)))).(\lambda (b: B).(\lambda (t1: -T).(\lambda (t2: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t1 -t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (pc3 (CHead c0 (Bind b) u) (TSort -(next g n)) t2)))).(\lambda (H5: (eq T (THead (Bind b) u t1) (TSort n))).(let -H6 \def (eq_ind T (THead (Bind b) u t1) (\lambda (ee: T).(match ee in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H5) in -(False_ind (pc3 c0 (TSort (next g n)) (THead (Bind b) u t2)) H6))))))))))))) -(\lambda (c0: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w -u)).(\lambda (_: (((eq T w (TSort n)) \to (pc3 c0 (TSort (next g n)) -u)))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead (Bind -Abst) u t))).(\lambda (_: (((eq T v (TSort n)) \to (pc3 c0 (TSort (next g n)) -(THead (Bind Abst) u t))))).(\lambda (H5: (eq T (THead (Flat Appl) w v) -(TSort n))).(let H6 \def (eq_ind T (THead (Flat Appl) w v) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I -(TSort n) H5) in (False_ind (pc3 c0 (TSort (next g n)) (THead (Flat Appl) w -(THead (Bind Abst) u t))) H6)))))))))))) (\lambda (c0: C).(\lambda (t1: -T).(\lambda (t2: T).(\lambda (_: (ty3 g c0 t1 t2)).(\lambda (_: (((eq T t1 -(TSort n)) \to (pc3 c0 (TSort (next g n)) t2)))).(\lambda (t0: T).(\lambda -(_: (ty3 g c0 t2 t0)).(\lambda (_: (((eq T t2 (TSort n)) \to (pc3 c0 (TSort -(next g n)) t0)))).(\lambda (H5: (eq T (THead (Flat Cast) t2 t1) (TSort -n))).(let H6 \def (eq_ind T (THead (Flat Cast) t2 t1) (\lambda (ee: T).(match -ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | -(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) -H5) in (False_ind (pc3 c0 (TSort (next g n)) (THead (Flat Cast) t0 t2)) -H6))))))))))) c y x H0))) H))))). -(* COMMENTS -Initial nodes: 1179 -END *) - -theorem ty3_gen_lref: - \forall (g: G).(\forall (c: C).(\forall (x: T).(\forall (n: nat).((ty3 g c -(TLRef n) x) \to (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda -(t: T).(pc3 c (lift (S n) O t) x)))) (\lambda (e: C).(\lambda (u: T).(\lambda -(_: T).(getl n c (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: -T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(pc3 c (lift (S n) O u) x)))) (\lambda (e: C).(\lambda -(u: T).(\lambda (_: T).(getl n c (CHead e (Bind Abst) u))))) (\lambda (e: -C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda -(H: (ty3 g c (TLRef n) x)).(insert_eq T (TLRef n) (\lambda (t: T).(ty3 g c t -x)) (\lambda (_: T).(or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda -(t0: T).(pc3 c (lift (S n) O t0) x)))) (\lambda (e: C).(\lambda (u: -T).(\lambda (_: T).(getl n c (CHead e (Bind Abbr) u))))) (\lambda (e: -C).(\lambda (u: T).(\lambda (t0: T).(ty3 g e u t0))))) (ex3_3 C T T (\lambda -(_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c (lift (S n) O u) x)))) (\lambda -(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead e (Bind Abst) u))))) -(\lambda (e: C).(\lambda (u: T).(\lambda (t0: T).(ty3 g e u t0))))))) -(\lambda (y: T).(\lambda (H0: (ty3 g c y x)).(ty3_ind g (\lambda (c0: -C).(\lambda (t: T).(\lambda (t0: T).((eq T t (TLRef n)) \to (or (ex3_3 C T T -(\lambda (_: C).(\lambda (_: T).(\lambda (t1: T).(pc3 c0 (lift (S n) O t1) -t0)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e -(Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t1: T).(ty3 g e -u t1))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 -c0 (lift (S n) O u) t0)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: -T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: -T).(\lambda (t1: T).(ty3 g e u t1)))))))))) (\lambda (c0: C).(\lambda (t2: -T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda (_: (((eq T t2 -(TLRef n)) \to (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: -T).(pc3 c0 (lift (S n) O t0) t)))) (\lambda (e: C).(\lambda (u: T).(\lambda -(_: T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: -T).(\lambda (t0: T).(ty3 g e u t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u) t)))) (\lambda (e: -C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) -(\lambda (e: C).(\lambda (u: T).(\lambda (t0: T).(ty3 g e u -t0))))))))).(\lambda (u: T).(\lambda (t1: T).(\lambda (H3: (ty3 g c0 u -t1)).(\lambda (H4: (((eq T u (TLRef n)) \to (or (ex3_3 C T T (\lambda (_: -C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t1)))) (\lambda -(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) -u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) -(ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift -(S n) O u0) t1)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n -c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(t0: T).(ty3 g e u0 t0))))))))).(\lambda (H5: (pc3 c0 t1 t2)).(\lambda (H6: -(eq T u (TLRef n))).(let H7 \def (f_equal T T (\lambda (e: T).e) u (TLRef n) -H6) in (let H8 \def (eq_ind T u (\lambda (t0: T).((eq T t0 (TLRef n)) \to (or -(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t3: T).(pc3 c0 (lift -(S n) O t3) t1)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n -c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(t3: T).(ty3 g e u0 t3))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: -T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) t1)))) (\lambda (e: C).(\lambda -(u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: -C).(\lambda (u0: T).(\lambda (t3: T).(ty3 g e u0 t3)))))))) H4 (TLRef n) H7) -in (let H9 \def (eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 t1)) H3 (TLRef n) -H7) in (let H10 \def (H8 (refl_equal T (TLRef n))) in (or_ind (ex3_3 C T T -(\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) -t1)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e -(Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g -e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: -T).(pc3 c0 (lift (S n) O u0) t1)))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (t0: T).(ty3 g e u0 t0))))) (or (ex3_3 C T T (\lambda (_: -C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda -(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) -u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) -(ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift -(S n) O u0) t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n -c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(t0: T).(ty3 g e u0 t0)))))) (\lambda (H11: (ex3_3 C T T (\lambda (_: -C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t1)))) (\lambda -(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) -u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 -t0)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: -T).(pc3 c0 (lift (S n) O t0) t1)))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(_: T).(getl n c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (t0: T).(ty3 g e u0 t0)))) (or (ex3_3 C T T (\lambda (_: -C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda -(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) -u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) -(ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift -(S n) O u0) t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n -c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(t0: T).(ty3 g e u0 t0)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: -T).(\lambda (H12: (pc3 c0 (lift (S n) O x2) t1)).(\lambda (H13: (getl n c0 -(CHead x0 (Bind Abbr) x1))).(\lambda (H14: (ty3 g x0 x1 x2)).(or_introl -(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift -(S n) O t0) t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n -c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: -T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) t2)))) (\lambda (e: C).(\lambda -(u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: -C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3_intro C T T -(\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) -t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e -(Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g -e u0 t0)))) x0 x1 x2 (pc3_t t1 c0 (lift (S n) O x2) H12 t2 H5) H13 -H14)))))))) H11)) (\lambda (H11: (ex3_3 C T T (\lambda (_: C).(\lambda (u0: -T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) t1)))) (\lambda (e: C).(\lambda -(u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: -C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))))).(ex3_3_ind C T T -(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) -t1)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e -(Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g -e u0 t0)))) (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: -T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(_: T).(getl n c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda -(u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) t2)))) (\lambda (e: -C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) -(\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))))) -(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H12: (pc3 c0 -(lift (S n) O x1) t1)).(\lambda (H13: (getl n c0 (CHead x0 (Bind Abst) -x1))).(\lambda (H14: (ty3 g x0 x1 x2)).(or_intror (ex3_3 C T T (\lambda (_: -C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda -(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) -u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) -(ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift -(S n) O u0) t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n -c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(t0: T).(ty3 g e u0 t0))))) (ex3_3_intro C T T (\lambda (_: C).(\lambda (u0: -T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) t2)))) (\lambda (e: C).(\lambda -(u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: -C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))) x0 x1 x2 (pc3_t t1 c0 -(lift (S n) O x1) H12 t2 H5) H13 H14)))))))) H11)) H10)))))))))))))))) -(\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq T (TSort m) (TLRef -n))).(let H2 \def (eq_ind T (TSort m) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef n) H1) in -(False_ind (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: -T).(pc3 c0 (lift (S n) O t) (TSort (next g m)))))) (\lambda (e: C).(\lambda -(u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: -C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T T (\lambda -(_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u) (TSort (next -g m)))))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e -(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u -t)))))) H2))))) (\lambda (n0: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda -(u: T).(\lambda (H1: (getl n0 c0 (CHead d (Bind Abbr) u))).(\lambda (t: -T).(\lambda (H2: (ty3 g d u t)).(\lambda (_: (((eq T u (TLRef n)) \to (or -(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 d (lift (S -n) O t0) t)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n d -(CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: -T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda -(_: T).(pc3 d (lift (S n) O u0) t)))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (_: T).(getl n d (CHead e (Bind Abst) u0))))) (\lambda (e: -C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))))))).(\lambda (H4: -(eq T (TLRef n0) (TLRef n))).(let H5 \def (f_equal T nat (\lambda (e: -T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow n0 | -(TLRef n1) \Rightarrow n1 | (THead _ _ _) \Rightarrow n0])) (TLRef n0) (TLRef -n) H4) in (let H6 \def (eq_ind nat n0 (\lambda (n1: nat).(getl n1 c0 (CHead d -(Bind Abbr) u))) H1 n H5) in (eq_ind_r nat n (\lambda (n1: nat).(or (ex3_3 C -T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O -t0) (lift (S n1) O t))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: -T).(getl n c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda -(u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) (lift (S n1) O t))))) -(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind -Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 -t0))))))) (or_introl (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda -(t0: T).(pc3 c0 (lift (S n) O t0) (lift (S n) O t))))) (\lambda (e: -C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u0))))) -(\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3 -C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O -u0) (lift (S n) O t))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: -T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3_intro C T T (\lambda (_: -C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) (lift (S n) O -t))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e -(Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g -e u0 t0)))) d u t (pc3_refl c0 (lift (S n) O t)) H6 H2)) n0 H5)))))))))))) -(\lambda (n0: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda -(H1: (getl n0 c0 (CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (H2: (ty3 -g d u t)).(\lambda (_: (((eq T u (TLRef n)) \to (or (ex3_3 C T T (\lambda (_: -C).(\lambda (_: T).(\lambda (t0: T).(pc3 d (lift (S n) O t0) t)))) (\lambda -(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n d (CHead e (Bind Abbr) -u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) -(ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 d (lift (S -n) O u0) t)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n d -(CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: -T).(ty3 g e u0 t0))))))))).(\lambda (H4: (eq T (TLRef n0) (TLRef n))).(let H5 -\def (f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) -with [(TSort _) \Rightarrow n0 | (TLRef n1) \Rightarrow n1 | (THead _ _ _) -\Rightarrow n0])) (TLRef n0) (TLRef n) H4) in (let H6 \def (eq_ind nat n0 -(\lambda (n1: nat).(getl n1 c0 (CHead d (Bind Abst) u))) H1 n H5) in -(eq_ind_r nat n (\lambda (n1: nat).(or (ex3_3 C T T (\lambda (_: C).(\lambda -(_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) (lift (S n1) O u))))) -(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind -Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 -t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 -(lift (S n) O u0) (lift (S n1) O u))))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: -C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))))) (or_intror (ex3_3 -C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O -t0) (lift (S n) O u))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: -T).(getl n c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda -(u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) (lift (S n) O u))))) -(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind -Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 -t0))))) (ex3_3_intro C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: -T).(pc3 c0 (lift (S n) O u0) (lift (S n) O u))))) (\lambda (e: C).(\lambda -(u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: -C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))) d u t (pc3_refl c0 -(lift (S n) O u)) H6 H2)) n0 H5)))))))))))) (\lambda (c0: C).(\lambda (u: -T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((eq T u (TLRef -n)) \to (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: -T).(pc3 c0 (lift (S n) O t0) t)))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(_: T).(getl n c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda -(u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) t)))) (\lambda (e: -C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) -(\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 -t0))))))))).(\lambda (b: B).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: -(ty3 g (CHead c0 (Bind b) u) t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to -(or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 (CHead -c0 (Bind b) u) (lift (S n) O t0) t2)))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (_: T).(getl n (CHead c0 (Bind b) u) (CHead e (Bind Abbr) u0))))) -(\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3 -C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 (CHead c0 (Bind -b) u) (lift (S n) O u0) t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: -T).(getl n (CHead c0 (Bind b) u) (CHead e (Bind Abst) u0))))) (\lambda (e: -C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))))))).(\lambda (H5: -(eq T (THead (Bind b) u t1) (TLRef n))).(let H6 \def (eq_ind T (THead (Bind -b) u t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) -\Rightarrow True])) I (TLRef n) H5) in (False_ind (or (ex3_3 C T T (\lambda -(_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) (THead -(Bind b) u t2))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n -c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: -T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) (THead (Bind b) u t2))))) -(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind -Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 -t0)))))) H6))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda (u: -T).(\lambda (_: (ty3 g c0 w u)).(\lambda (_: (((eq T w (TLRef n)) \to (or -(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c0 (lift (S -n) O t) u)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 -(CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t: -T).(ty3 g e u0 t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda -(_: T).(pc3 c0 (lift (S n) O u0) u)))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: -C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t))))))))).(\lambda (v: -T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead (Bind Abst) u -t))).(\lambda (_: (((eq T v (TLRef n)) \to (or (ex3_3 C T T (\lambda (_: -C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) (THead (Bind -Abst) u t))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 -(CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: -T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda -(_: T).(pc3 c0 (lift (S n) O u0) (THead (Bind Abst) u t))))) (\lambda (e: -C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) -(\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 -t0))))))))).(\lambda (H5: (eq T (THead (Flat Appl) w v) (TLRef n))).(let H6 -\def (eq_ind T (THead (Flat Appl) w v) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H5) in -(False_ind (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: -T).(pc3 c0 (lift (S n) O t0) (THead (Flat Appl) w (THead (Bind Abst) u -t)))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e -(Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g -e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: -T).(pc3 c0 (lift (S n) O u0) (THead (Flat Appl) w (THead (Bind Abst) u -t)))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e -(Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g -e u0 t0)))))) H6)))))))))))) (\lambda (c0: C).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (_: (ty3 g c0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (or -(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c0 (lift (S -n) O t) t2)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 -(CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: -T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda -(_: T).(pc3 c0 (lift (S n) O u) t2)))) (\lambda (e: C).(\lambda (u: -T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: -C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))))))).(\lambda (t0: -T).(\lambda (_: (ty3 g c0 t2 t0)).(\lambda (_: (((eq T t2 (TLRef n)) \to (or -(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c0 (lift (S -n) O t) t0)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 -(CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: -T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda -(_: T).(pc3 c0 (lift (S n) O u) t0)))) (\lambda (e: C).(\lambda (u: -T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: -C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))))))).(\lambda (H5: (eq T -(THead (Flat Cast) t2 t1) (TLRef n))).(let H6 \def (eq_ind T (THead (Flat -Cast) t2 t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) -with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ -_) \Rightarrow True])) I (TLRef n) H5) in (False_ind (or (ex3_3 C T T -(\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c0 (lift (S n) O t) -(THead (Flat Cast) t0 t2))))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: -T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: -T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u) (THead (Flat Cast) t0 t2))))) -(\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind -Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u -t)))))) H6))))))))))) c y x H0))) H))))). -(* COMMENTS -Initial nodes: 5569 -END *) - -theorem ty3_gen_bind: - \forall (g: G).(\forall (b: B).(\forall (c: C).(\forall (u: T).(\forall (t1: -T).(\forall (x: T).((ty3 g c (THead (Bind b) u t1) x) \to (ex3_2 T T (\lambda -(t2: T).(\lambda (_: T).(pc3 c (THead (Bind b) u t2) x))) (\lambda (_: -T).(\lambda (t: T).(ty3 g c u t))) (\lambda (t2: T).(\lambda (_: T).(ty3 g -(CHead c (Bind b) u) t1 t2)))))))))) -\def - \lambda (g: G).(\lambda (b: B).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: -T).(\lambda (x: T).(\lambda (H: (ty3 g c (THead (Bind b) u t1) x)).(insert_eq -T (THead (Bind b) u t1) (\lambda (t: T).(ty3 g c t x)) (\lambda (_: T).(ex3_2 -T T (\lambda (t2: T).(\lambda (_: T).(pc3 c (THead (Bind b) u t2) x))) -(\lambda (_: T).(\lambda (t0: T).(ty3 g c u t0))) (\lambda (t2: T).(\lambda -(_: T).(ty3 g (CHead c (Bind b) u) t1 t2))))) (\lambda (y: T).(\lambda (H0: -(ty3 g c y x)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: -T).((eq T t (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda (t2: T).(\lambda -(_: T).(pc3 c0 (THead (Bind b) u t2) t0))) (\lambda (_: T).(\lambda (t3: -T).(ty3 g c0 u t3))) (\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind -b) u) t1 t2)))))))) (\lambda (c0: C).(\lambda (t2: T).(\lambda (t: -T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda (_: (((eq T t2 (THead (Bind b) u -t1)) \to (ex3_2 T T (\lambda (t3: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) -u t3) t))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u t0))) (\lambda (t3: -T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3))))))).(\lambda (u0: -T).(\lambda (t0: T).(\lambda (H3: (ty3 g c0 u0 t0)).(\lambda (H4: (((eq T u0 -(THead (Bind b) u t1)) \to (ex3_2 T T (\lambda (t3: T).(\lambda (_: T).(pc3 -c0 (THead (Bind b) u t3) t0))) (\lambda (_: T).(\lambda (t4: T).(ty3 g c0 u -t4))) (\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 -t3))))))).(\lambda (H5: (pc3 c0 t0 t2)).(\lambda (H6: (eq T u0 (THead (Bind -b) u t1))).(let H7 \def (f_equal T T (\lambda (e: T).e) u0 (THead (Bind b) u -t1) H6) in (let H8 \def (eq_ind T u0 (\lambda (t3: T).((eq T t3 (THead (Bind -b) u t1)) \to (ex3_2 T T (\lambda (t4: T).(\lambda (_: T).(pc3 c0 (THead -(Bind b) u t4) t0))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c0 u t5))) -(\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t4)))))) H4 -(THead (Bind b) u t1) H7) in (let H9 \def (eq_ind T u0 (\lambda (t3: T).(ty3 -g c0 t3 t0)) H3 (THead (Bind b) u t1) H7) in (let H10 \def (H8 (refl_equal T -(THead (Bind b) u t1))) in (ex3_2_ind T T (\lambda (t3: T).(\lambda (_: -T).(pc3 c0 (THead (Bind b) u t3) t0))) (\lambda (_: T).(\lambda (t4: T).(ty3 -g c0 u t4))) (\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 -t3))) (ex3_2 T T (\lambda (t3: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u -t3) t2))) (\lambda (_: T).(\lambda (t4: T).(ty3 g c0 u t4))) (\lambda (t3: -T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3)))) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H11: (pc3 c0 (THead (Bind b) u x0) -t0)).(\lambda (H12: (ty3 g c0 u x1)).(\lambda (H13: (ty3 g (CHead c0 (Bind b) -u) t1 x0)).(ex3_2_intro T T (\lambda (t3: T).(\lambda (_: T).(pc3 c0 (THead -(Bind b) u t3) t2))) (\lambda (_: T).(\lambda (t4: T).(ty3 g c0 u t4))) -(\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3))) x0 x1 -(pc3_t t0 c0 (THead (Bind b) u x0) H11 t2 H5) H12 H13)))))) -H10)))))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq T -(TSort m) (THead (Bind b) u t1))).(let H2 \def (eq_ind T (TSort m) (\lambda -(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow -False])) I (THead (Bind b) u t1) H1) in (False_ind (ex3_2 T T (\lambda (t2: -T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t2) (TSort (next g m))))) -(\lambda (_: T).(\lambda (t: T).(ty3 g c0 u t))) (\lambda (t2: T).(\lambda -(_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2)))) H2))))) (\lambda (n: -nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (_: (getl n -c0 (CHead d (Bind Abbr) u0))).(\lambda (t: T).(\lambda (_: (ty3 g d u0 -t)).(\lambda (_: (((eq T u0 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda -(t2: T).(\lambda (_: T).(pc3 d (THead (Bind b) u t2) t))) (\lambda (_: -T).(\lambda (t0: T).(ty3 g d u t0))) (\lambda (t2: T).(\lambda (_: T).(ty3 g -(CHead d (Bind b) u) t1 t2))))))).(\lambda (H4: (eq T (TLRef n) (THead (Bind -b) u t1))).(let H5 \def (eq_ind T (TLRef n) (\lambda (ee: T).(match ee in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t1) -H4) in (False_ind (ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3 c0 (THead -(Bind b) u t2) (lift (S n) O t)))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c0 -u t0))) (\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 -t2)))) H5))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: -C).(\lambda (u0: T).(\lambda (_: (getl n c0 (CHead d (Bind Abst) -u0))).(\lambda (t: T).(\lambda (_: (ty3 g d u0 t)).(\lambda (_: (((eq T u0 -(THead (Bind b) u t1)) \to (ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3 d -(THead (Bind b) u t2) t))) (\lambda (_: T).(\lambda (t0: T).(ty3 g d u t0))) -(\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead d (Bind b) u) t1 -t2))))))).(\lambda (H4: (eq T (TLRef n) (THead (Bind b) u t1))).(let H5 \def -(eq_ind T (TLRef n) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | -(THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t1) H4) in (False_ind -(ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t2) -(lift (S n) O u0)))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u t0))) -(\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2)))) -H5))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (t: T).(\lambda (H1: -(ty3 g c0 u0 t)).(\lambda (H2: (((eq T u0 (THead (Bind b) u t1)) \to (ex3_2 T -T (\lambda (t2: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t2) t))) -(\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u t0))) (\lambda (t2: T).(\lambda -(_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2))))))).(\lambda (b0: B).(\lambda -(t0: T).(\lambda (t2: T).(\lambda (H3: (ty3 g (CHead c0 (Bind b0) u0) t0 -t2)).(\lambda (H4: (((eq T t0 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda -(t3: T).(\lambda (_: T).(pc3 (CHead c0 (Bind b0) u0) (THead (Bind b) u t3) -t2))) (\lambda (_: T).(\lambda (t4: T).(ty3 g (CHead c0 (Bind b0) u0) u t4))) -(\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead (CHead c0 (Bind b0) u0) (Bind -b) u) t1 t3))))))).(\lambda (H5: (eq T (THead (Bind b0) u0 t0) (THead (Bind -b) u t1))).(let H6 \def (f_equal T B (\lambda (e: T).(match e in T return -(\lambda (_: T).B) with [(TSort _) \Rightarrow b0 | (TLRef _) \Rightarrow b0 -| (THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).B) with -[(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow b0])])) (THead (Bind b0) u0 -t0) (THead (Bind b) u t1) H5) in ((let H7 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | -(TLRef _) \Rightarrow u0 | (THead _ t3 _) \Rightarrow t3])) (THead (Bind b0) -u0 t0) (THead (Bind b) u t1) H5) in ((let H8 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | -(TLRef _) \Rightarrow t0 | (THead _ _ t3) \Rightarrow t3])) (THead (Bind b0) -u0 t0) (THead (Bind b) u t1) H5) in (\lambda (H9: (eq T u0 u)).(\lambda (H10: -(eq B b0 b)).(let H11 \def (eq_ind T t0 (\lambda (t3: T).((eq T t3 (THead -(Bind b) u t1)) \to (ex3_2 T T (\lambda (t4: T).(\lambda (_: T).(pc3 (CHead -c0 (Bind b0) u0) (THead (Bind b) u t4) t2))) (\lambda (_: T).(\lambda (t5: -T).(ty3 g (CHead c0 (Bind b0) u0) u t5))) (\lambda (t4: T).(\lambda (_: -T).(ty3 g (CHead (CHead c0 (Bind b0) u0) (Bind b) u) t1 t4)))))) H4 t1 H8) in -(let H12 \def (eq_ind T t0 (\lambda (t3: T).(ty3 g (CHead c0 (Bind b0) u0) t3 -t2)) H3 t1 H8) in (let H13 \def (eq_ind B b0 (\lambda (b1: B).((eq T t1 -(THead (Bind b) u t1)) \to (ex3_2 T T (\lambda (t3: T).(\lambda (_: T).(pc3 -(CHead c0 (Bind b1) u0) (THead (Bind b) u t3) t2))) (\lambda (_: T).(\lambda -(t4: T).(ty3 g (CHead c0 (Bind b1) u0) u t4))) (\lambda (t3: T).(\lambda (_: -T).(ty3 g (CHead (CHead c0 (Bind b1) u0) (Bind b) u) t1 t3)))))) H11 b H10) -in (let H14 \def (eq_ind B b0 (\lambda (b1: B).(ty3 g (CHead c0 (Bind b1) u0) -t1 t2)) H12 b H10) in (eq_ind_r B b (\lambda (b1: B).(ex3_2 T T (\lambda (t3: -T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t3) (THead (Bind b1) u0 t2)))) -(\lambda (_: T).(\lambda (t4: T).(ty3 g c0 u t4))) (\lambda (t3: T).(\lambda -(_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3))))) (let H15 \def (eq_ind T u0 -(\lambda (t3: T).((eq T t1 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda -(t4: T).(\lambda (_: T).(pc3 (CHead c0 (Bind b) t3) (THead (Bind b) u t4) -t2))) (\lambda (_: T).(\lambda (t5: T).(ty3 g (CHead c0 (Bind b) t3) u t5))) -(\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead (CHead c0 (Bind b) t3) (Bind -b) u) t1 t4)))))) H13 u H9) in (let H16 \def (eq_ind T u0 (\lambda (t3: -T).(ty3 g (CHead c0 (Bind b) t3) t1 t2)) H14 u H9) in (let H17 \def (eq_ind T -u0 (\lambda (t3: T).((eq T t3 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda -(t4: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t4) t))) (\lambda (_: -T).(\lambda (t5: T).(ty3 g c0 u t5))) (\lambda (t4: T).(\lambda (_: T).(ty3 g -(CHead c0 (Bind b) u) t1 t4)))))) H2 u H9) in (let H18 \def (eq_ind T u0 -(\lambda (t3: T).(ty3 g c0 t3 t)) H1 u H9) in (eq_ind_r T u (\lambda (t3: -T).(ex3_2 T T (\lambda (t4: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t4) -(THead (Bind b) t3 t2)))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c0 u t5))) -(\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t4))))) -(ex3_2_intro T T (\lambda (t3: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u -t3) (THead (Bind b) u t2)))) (\lambda (_: T).(\lambda (t4: T).(ty3 g c0 u -t4))) (\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3))) -t2 t (pc3_refl c0 (THead (Bind b) u t2)) H18 H16) u0 H9))))) b0 H10)))))))) -H7)) H6))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda (u0: -T).(\lambda (_: (ty3 g c0 w u0)).(\lambda (_: (((eq T w (THead (Bind b) u -t1)) \to (ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) -u t2) u0))) (\lambda (_: T).(\lambda (t: T).(ty3 g c0 u t))) (\lambda (t2: -T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2))))))).(\lambda (v: -T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead (Bind Abst) u0 -t))).(\lambda (_: (((eq T v (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda -(t2: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t2) (THead (Bind Abst) u0 -t)))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u t0))) (\lambda (t2: -T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2))))))).(\lambda (H5: -(eq T (THead (Flat Appl) w v) (THead (Bind b) u t1))).(let H6 \def (eq_ind T -(THead (Flat Appl) w v) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind -b) u t1) H5) in (False_ind (ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3 -c0 (THead (Bind b) u t2) (THead (Flat Appl) w (THead (Bind Abst) u0 t))))) -(\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u t0))) (\lambda (t2: T).(\lambda -(_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2)))) H6)))))))))))) (\lambda (c0: -C).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (ty3 g c0 t0 t2)).(\lambda -(_: (((eq T t0 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda (t3: -T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t3) t2))) (\lambda (_: -T).(\lambda (t: T).(ty3 g c0 u t))) (\lambda (t3: T).(\lambda (_: T).(ty3 g -(CHead c0 (Bind b) u) t1 t3))))))).(\lambda (t3: T).(\lambda (_: (ty3 g c0 t2 -t3)).(\lambda (_: (((eq T t2 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda -(t4: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t4) t3))) (\lambda (_: -T).(\lambda (t: T).(ty3 g c0 u t))) (\lambda (t4: T).(\lambda (_: T).(ty3 g -(CHead c0 (Bind b) u) t1 t4))))))).(\lambda (H5: (eq T (THead (Flat Cast) t2 -t0) (THead (Bind b) u t1))).(let H6 \def (eq_ind T (THead (Flat Cast) t2 t0) -(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow -(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | -(Flat _) \Rightarrow True])])) I (THead (Bind b) u t1) H5) in (False_ind -(ex3_2 T T (\lambda (t4: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t4) -(THead (Flat Cast) t3 t2)))) (\lambda (_: T).(\lambda (t: T).(ty3 g c0 u t))) -(\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t4)))) -H6))))))))))) c y x H0))) H))))))). -(* COMMENTS -Initial nodes: 3389 -END *) - -theorem ty3_gen_appl: - \forall (g: G).(\forall (c: C).(\forall (w: T).(\forall (v: T).(\forall (x: -T).((ty3 g c (THead (Flat Appl) w v) x) \to (ex3_2 T T (\lambda (u: -T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x))) -(\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) -(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (w: T).(\lambda (v: T).(\lambda (x: -T).(\lambda (H: (ty3 g c (THead (Flat Appl) w v) x)).(insert_eq T (THead -(Flat Appl) w v) (\lambda (t: T).(ty3 g c t x)) (\lambda (_: T).(ex3_2 T T -(\lambda (u: T).(\lambda (t0: T).(pc3 c (THead (Flat Appl) w (THead (Bind -Abst) u t0)) x))) (\lambda (u: T).(\lambda (t0: T).(ty3 g c v (THead (Bind -Abst) u t0)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c w u))))) (\lambda (y: -T).(\lambda (H0: (ty3 g c y x)).(ty3_ind g (\lambda (c0: C).(\lambda (t: -T).(\lambda (t0: T).((eq T t (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda -(u: T).(\lambda (t1: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u -t1)) t0))) (\lambda (u: T).(\lambda (t1: T).(ty3 g c0 v (THead (Bind Abst) u -t1)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c0 w u)))))))) (\lambda (c0: -C).(\lambda (t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda -(_: (((eq T t2 (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda (u: -T).(\lambda (t0: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u t0)) -t))) (\lambda (u: T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u t0)))) -(\lambda (u: T).(\lambda (_: T).(ty3 g c0 w u))))))).(\lambda (u: T).(\lambda -(t1: T).(\lambda (H3: (ty3 g c0 u t1)).(\lambda (H4: (((eq T u (THead (Flat -Appl) w v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0 (THead -(Flat Appl) w (THead (Bind Abst) u0 t0)) t1))) (\lambda (u0: T).(\lambda (t0: -T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_: -T).(ty3 g c0 w u0))))))).(\lambda (H5: (pc3 c0 t1 t2)).(\lambda (H6: (eq T u -(THead (Flat Appl) w v))).(let H7 \def (f_equal T T (\lambda (e: T).e) u -(THead (Flat Appl) w v) H6) in (let H8 \def (eq_ind T u (\lambda (t0: T).((eq -T t0 (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t3: -T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t3)) t1))) (\lambda -(u0: T).(\lambda (t3: T).(ty3 g c0 v (THead (Bind Abst) u0 t3)))) (\lambda -(u0: T).(\lambda (_: T).(ty3 g c0 w u0)))))) H4 (THead (Flat Appl) w v) H7) -in (let H9 \def (eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 t1)) H3 (THead -(Flat Appl) w v) H7) in (let H10 \def (H8 (refl_equal T (THead (Flat Appl) w -v))) in (ex3_2_ind T T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0 (THead (Flat -Appl) w (THead (Bind Abst) u0 t0)) t1))) (\lambda (u0: T).(\lambda (t0: -T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_: -T).(ty3 g c0 w u0))) (ex3_2 T T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0 -(THead (Flat Appl) w (THead (Bind Abst) u0 t0)) t2))) (\lambda (u0: -T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: -T).(\lambda (_: T).(ty3 g c0 w u0)))) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H11: (pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) x0 x1)) -t1)).(\lambda (H12: (ty3 g c0 v (THead (Bind Abst) x0 x1))).(\lambda (H13: -(ty3 g c0 w x0)).(ex3_2_intro T T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0 -(THead (Flat Appl) w (THead (Bind Abst) u0 t0)) t2))) (\lambda (u0: -T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: -T).(\lambda (_: T).(ty3 g c0 w u0))) x0 x1 (pc3_t t1 c0 (THead (Flat Appl) w -(THead (Bind Abst) x0 x1)) H11 t2 H5) H12 H13)))))) H10)))))))))))))))) -(\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq T (TSort m) (THead (Flat -Appl) w v))).(let H2 \def (eq_ind T (TSort m) (\lambda (ee: T).(match ee in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) w -v) H1) in (False_ind (ex3_2 T T (\lambda (u: T).(\lambda (t: T).(pc3 c0 -(THead (Flat Appl) w (THead (Bind Abst) u t)) (TSort (next g m))))) (\lambda -(u: T).(\lambda (t: T).(ty3 g c0 v (THead (Bind Abst) u t)))) (\lambda (u: -T).(\lambda (_: T).(ty3 g c0 w u)))) H2))))) (\lambda (n: nat).(\lambda (c0: -C).(\lambda (d: C).(\lambda (u: T).(\lambda (_: (getl n c0 (CHead d (Bind -Abbr) u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (_: (((eq T u -(THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t0: -T).(pc3 d (THead (Flat Appl) w (THead (Bind Abst) u0 t0)) t))) (\lambda (u0: -T).(\lambda (t0: T).(ty3 g d v (THead (Bind Abst) u0 t0)))) (\lambda (u0: -T).(\lambda (_: T).(ty3 g d w u0))))))).(\lambda (H4: (eq T (TLRef n) (THead -(Flat Appl) w v))).(let H5 \def (eq_ind T (TLRef n) (\lambda (ee: T).(match -ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | -(TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead -(Flat Appl) w v) H4) in (False_ind (ex3_2 T T (\lambda (u0: T).(\lambda (t0: -T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t0)) (lift (S n) O -t)))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u0 -t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w u0)))) H5))))))))))) -(\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda -(_: (getl n c0 (CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (_: (ty3 g -d u t)).(\lambda (_: (((eq T u (THead (Flat Appl) w v)) \to (ex3_2 T T -(\lambda (u0: T).(\lambda (t0: T).(pc3 d (THead (Flat Appl) w (THead (Bind -Abst) u0 t0)) t))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g d v (THead (Bind -Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g d w -u0))))))).(\lambda (H4: (eq T (TLRef n) (THead (Flat Appl) w v))).(let H5 -\def (eq_ind T (TLRef n) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | -(THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) w v) H4) in -(False_ind (ex3_2 T T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0 (THead (Flat -Appl) w (THead (Bind Abst) u0 t0)) (lift (S n) O u)))) (\lambda (u0: -T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: -T).(\lambda (_: T).(ty3 g c0 w u0)))) H5))))))))))) (\lambda (c0: C).(\lambda -(u: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((eq T u -(THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t0: -T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t0)) t))) (\lambda (u0: -T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: -T).(\lambda (_: T).(ty3 g c0 w u0))))))).(\lambda (b: B).(\lambda (t1: -T).(\lambda (t2: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t1 -t2)).(\lambda (_: (((eq T t1 (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda -(u0: T).(\lambda (t0: T).(pc3 (CHead c0 (Bind b) u) (THead (Flat Appl) w -(THead (Bind Abst) u0 t0)) t2))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g -(CHead c0 (Bind b) u) v (THead (Bind Abst) u0 t0)))) (\lambda (u0: -T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) w u0))))))).(\lambda (H5: (eq -T (THead (Bind b) u t1) (THead (Flat Appl) w v))).(let H6 \def (eq_ind T -(THead (Bind b) u t1) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat -Appl) w v) H5) in (False_ind (ex3_2 T T (\lambda (u0: T).(\lambda (t0: -T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t0)) (THead (Bind b) u -t2)))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u0 -t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w u0)))) H6))))))))))))) -(\lambda (c0: C).(\lambda (w0: T).(\lambda (u: T).(\lambda (H1: (ty3 g c0 w0 -u)).(\lambda (H2: (((eq T w0 (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda -(u0: T).(\lambda (t: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 -t)) u))) (\lambda (u0: T).(\lambda (t: T).(ty3 g c0 v (THead (Bind Abst) u0 -t)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w u0))))))).(\lambda (v0: -T).(\lambda (t: T).(\lambda (H3: (ty3 g c0 v0 (THead (Bind Abst) u -t))).(\lambda (H4: (((eq T v0 (THead (Flat Appl) w v)) \to (ex3_2 T T -(\lambda (u0: T).(\lambda (t0: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind -Abst) u0 t0)) (THead (Bind Abst) u t)))) (\lambda (u0: T).(\lambda (t0: -T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_: -T).(ty3 g c0 w u0))))))).(\lambda (H5: (eq T (THead (Flat Appl) w0 v0) (THead -(Flat Appl) w v))).(let H6 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow w0 | (TLRef _) -\Rightarrow w0 | (THead _ t0 _) \Rightarrow t0])) (THead (Flat Appl) w0 v0) -(THead (Flat Appl) w v) H5) in ((let H7 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v0 | -(TLRef _) \Rightarrow v0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat -Appl) w0 v0) (THead (Flat Appl) w v) H5) in (\lambda (H8: (eq T w0 w)).(let -H9 \def (eq_ind T v0 (\lambda (t0: T).((eq T t0 (THead (Flat Appl) w v)) \to -(ex3_2 T T (\lambda (u0: T).(\lambda (t1: T).(pc3 c0 (THead (Flat Appl) w -(THead (Bind Abst) u0 t1)) (THead (Bind Abst) u t)))) (\lambda (u0: -T).(\lambda (t1: T).(ty3 g c0 v (THead (Bind Abst) u0 t1)))) (\lambda (u0: -T).(\lambda (_: T).(ty3 g c0 w u0)))))) H4 v H7) in (let H10 \def (eq_ind T -v0 (\lambda (t0: T).(ty3 g c0 t0 (THead (Bind Abst) u t))) H3 v H7) in (let -H11 \def (eq_ind T w0 (\lambda (t0: T).((eq T t0 (THead (Flat Appl) w v)) \to -(ex3_2 T T (\lambda (u0: T).(\lambda (t1: T).(pc3 c0 (THead (Flat Appl) w -(THead (Bind Abst) u0 t1)) u))) (\lambda (u0: T).(\lambda (t1: T).(ty3 g c0 v -(THead (Bind Abst) u0 t1)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w -u0)))))) H2 w H8) in (let H12 \def (eq_ind T w0 (\lambda (t0: T).(ty3 g c0 t0 -u)) H1 w H8) in (eq_ind_r T w (\lambda (t0: T).(ex3_2 T T (\lambda (u0: -T).(\lambda (t1: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t1)) -(THead (Flat Appl) t0 (THead (Bind Abst) u t))))) (\lambda (u0: T).(\lambda -(t1: T).(ty3 g c0 v (THead (Bind Abst) u0 t1)))) (\lambda (u0: T).(\lambda -(_: T).(ty3 g c0 w u0))))) (ex3_2_intro T T (\lambda (u0: T).(\lambda (t0: -T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t0)) (THead (Flat Appl) -w (THead (Bind Abst) u t))))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g c0 v -(THead (Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w -u0))) u t (pc3_refl c0 (THead (Flat Appl) w (THead (Bind Abst) u t))) H10 -H12) w0 H8))))))) H6)))))))))))) (\lambda (c0: C).(\lambda (t1: T).(\lambda -(t2: T).(\lambda (_: (ty3 g c0 t1 t2)).(\lambda (_: (((eq T t1 (THead (Flat -Appl) w v)) \to (ex3_2 T T (\lambda (u: T).(\lambda (t: T).(pc3 c0 (THead -(Flat Appl) w (THead (Bind Abst) u t)) t2))) (\lambda (u: T).(\lambda (t: -T).(ty3 g c0 v (THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (_: -T).(ty3 g c0 w u))))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t2 -t0)).(\lambda (_: (((eq T t2 (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda -(u: T).(\lambda (t: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u t)) -t0))) (\lambda (u: T).(\lambda (t: T).(ty3 g c0 v (THead (Bind Abst) u t)))) -(\lambda (u: T).(\lambda (_: T).(ty3 g c0 w u))))))).(\lambda (H5: (eq T -(THead (Flat Cast) t2 t1) (THead (Flat Appl) w v))).(let H6 \def (eq_ind T -(THead (Flat Cast) t2 t1) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f in F return -(\lambda (_: F).Prop) with [Appl \Rightarrow False | Cast \Rightarrow -True])])])) I (THead (Flat Appl) w v) H5) in (False_ind (ex3_2 T T (\lambda -(u: T).(\lambda (t: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u t)) -(THead (Flat Cast) t0 t2)))) (\lambda (u: T).(\lambda (t: T).(ty3 g c0 v -(THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c0 w u)))) -H6))))))))))) c y x H0))) H)))))). -(* COMMENTS -Initial nodes: 3171 -END *) - -theorem ty3_gen_cast: - \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).(\forall -(x: T).((ty3 g c (THead (Flat Cast) t2 t1) x) \to (ex3 T (\lambda (t0: -T).(pc3 c (THead (Flat Cast) t0 t2) x)) (\lambda (_: T).(ty3 g c t1 t2)) -(\lambda (t0: T).(ty3 g c t2 t0)))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(x: T).(\lambda (H: (ty3 g c (THead (Flat Cast) t2 t1) x)).(insert_eq T -(THead (Flat Cast) t2 t1) (\lambda (t: T).(ty3 g c t x)) (\lambda (_: T).(ex3 -T (\lambda (t0: T).(pc3 c (THead (Flat Cast) t0 t2) x)) (\lambda (_: T).(ty3 -g c t1 t2)) (\lambda (t0: T).(ty3 g c t2 t0)))) (\lambda (y: T).(\lambda (H0: -(ty3 g c y x)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: -T).((eq T t (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t3: T).(pc3 c0 -(THead (Flat Cast) t3 t2) t0)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda -(t3: T).(ty3 g c0 t2 t3))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t: -T).(\lambda (_: (ty3 g c0 t0 t)).(\lambda (_: (((eq T t0 (THead (Flat Cast) -t2 t1)) \to (ex3 T (\lambda (t3: T).(pc3 c0 (THead (Flat Cast) t3 t2) t)) -(\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t3: T).(ty3 g c0 t2 -t3)))))).(\lambda (u: T).(\lambda (t3: T).(\lambda (H3: (ty3 g c0 u -t3)).(\lambda (H4: (((eq T u (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda -(t4: T).(pc3 c0 (THead (Flat Cast) t4 t2) t3)) (\lambda (_: T).(ty3 g c0 t1 -t2)) (\lambda (t4: T).(ty3 g c0 t2 t4)))))).(\lambda (H5: (pc3 c0 t3 -t0)).(\lambda (H6: (eq T u (THead (Flat Cast) t2 t1))).(let H7 \def (f_equal -T T (\lambda (e: T).e) u (THead (Flat Cast) t2 t1) H6) in (let H8 \def -(eq_ind T u (\lambda (t4: T).((eq T t4 (THead (Flat Cast) t2 t1)) \to (ex3 T -(\lambda (t5: T).(pc3 c0 (THead (Flat Cast) t5 t2) t3)) (\lambda (_: T).(ty3 -g c0 t1 t2)) (\lambda (t5: T).(ty3 g c0 t2 t5))))) H4 (THead (Flat Cast) t2 -t1) H7) in (let H9 \def (eq_ind T u (\lambda (t4: T).(ty3 g c0 t4 t3)) H3 -(THead (Flat Cast) t2 t1) H7) in (let H10 \def (H8 (refl_equal T (THead (Flat -Cast) t2 t1))) in (ex3_ind T (\lambda (t4: T).(pc3 c0 (THead (Flat Cast) t4 -t2) t3)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t4: T).(ty3 g c0 t2 t4)) -(ex3 T (\lambda (t4: T).(pc3 c0 (THead (Flat Cast) t4 t2) t0)) (\lambda (_: -T).(ty3 g c0 t1 t2)) (\lambda (t4: T).(ty3 g c0 t2 t4))) (\lambda (x0: -T).(\lambda (H11: (pc3 c0 (THead (Flat Cast) x0 t2) t3)).(\lambda (H12: (ty3 -g c0 t1 t2)).(\lambda (H13: (ty3 g c0 t2 x0)).(ex3_intro T (\lambda (t4: -T).(pc3 c0 (THead (Flat Cast) t4 t2) t0)) (\lambda (_: T).(ty3 g c0 t1 t2)) -(\lambda (t4: T).(ty3 g c0 t2 t4)) x0 (pc3_t t3 c0 (THead (Flat Cast) x0 t2) -H11 t0 H5) H12 H13))))) H10)))))))))))))))) (\lambda (c0: C).(\lambda (m: -nat).(\lambda (H1: (eq T (TSort m) (THead (Flat Cast) t2 t1))).(let H2 \def -(eq_ind T (TSort m) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | -(THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) t2 t1) H1) in -(False_ind (ex3 T (\lambda (t0: T).(pc3 c0 (THead (Flat Cast) t0 t2) (TSort -(next g m)))) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0 t2 -t0))) H2))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: -T).(\lambda (_: (getl n c0 (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda -(_: (ty3 g d u t)).(\lambda (_: (((eq T u (THead (Flat Cast) t2 t1)) \to (ex3 -T (\lambda (t0: T).(pc3 d (THead (Flat Cast) t0 t2) t)) (\lambda (_: T).(ty3 -g d t1 t2)) (\lambda (t0: T).(ty3 g d t2 t0)))))).(\lambda (H4: (eq T (TLRef -n) (THead (Flat Cast) t2 t1))).(let H5 \def (eq_ind T (TLRef n) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I -(THead (Flat Cast) t2 t1) H4) in (False_ind (ex3 T (\lambda (t0: T).(pc3 c0 -(THead (Flat Cast) t0 t2) (lift (S n) O t))) (\lambda (_: T).(ty3 g c0 t1 -t2)) (\lambda (t0: T).(ty3 g c0 t2 t0))) H5))))))))))) (\lambda (n: -nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (_: (getl n c0 -(CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (_: (ty3 g d u -t)).(\lambda (_: (((eq T u (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda -(t0: T).(pc3 d (THead (Flat Cast) t0 t2) t)) (\lambda (_: T).(ty3 g d t1 t2)) -(\lambda (t0: T).(ty3 g d t2 t0)))))).(\lambda (H4: (eq T (TLRef n) (THead -(Flat Cast) t2 t1))).(let H5 \def (eq_ind T (TLRef n) (\lambda (ee: T).(match -ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | -(TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead -(Flat Cast) t2 t1) H4) in (False_ind (ex3 T (\lambda (t0: T).(pc3 c0 (THead -(Flat Cast) t0 t2) (lift (S n) O u))) (\lambda (_: T).(ty3 g c0 t1 t2)) -(\lambda (t0: T).(ty3 g c0 t2 t0))) H5))))))))))) (\lambda (c0: C).(\lambda -(u: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((eq T u -(THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t0: T).(pc3 c0 (THead (Flat -Cast) t0 t2) t)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0 -t2 t0)))))).(\lambda (b: B).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: -(ty3 g (CHead c0 (Bind b) u) t0 t3)).(\lambda (_: (((eq T t0 (THead (Flat -Cast) t2 t1)) \to (ex3 T (\lambda (t4: T).(pc3 (CHead c0 (Bind b) u) (THead -(Flat Cast) t4 t2) t3)) (\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2)) -(\lambda (t4: T).(ty3 g (CHead c0 (Bind b) u) t2 t4)))))).(\lambda (H5: (eq T -(THead (Bind b) u t0) (THead (Flat Cast) t2 t1))).(let H6 \def (eq_ind T -(THead (Bind b) u t0) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat -Cast) t2 t1) H5) in (False_ind (ex3 T (\lambda (t4: T).(pc3 c0 (THead (Flat -Cast) t4 t2) (THead (Bind b) u t3))) (\lambda (_: T).(ty3 g c0 t1 t2)) -(\lambda (t4: T).(ty3 g c0 t2 t4))) H6))))))))))))) (\lambda (c0: C).(\lambda -(w: T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda (_: (((eq T w -(THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t0: T).(pc3 c0 (THead (Flat -Cast) t0 t2) u)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0 -t2 t0)))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead -(Bind Abst) u t))).(\lambda (_: (((eq T v (THead (Flat Cast) t2 t1)) \to (ex3 -T (\lambda (t0: T).(pc3 c0 (THead (Flat Cast) t0 t2) (THead (Bind Abst) u -t))) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0 t2 -t0)))))).(\lambda (H5: (eq T (THead (Flat Appl) w v) (THead (Flat Cast) t2 -t1))).(let H6 \def (eq_ind T (THead (Flat Appl) w v) (\lambda (ee: T).(match -ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | -(TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return -(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow -(match f in F return (\lambda (_: F).Prop) with [Appl \Rightarrow True | Cast -\Rightarrow False])])])) I (THead (Flat Cast) t2 t1) H5) in (False_ind (ex3 T -(\lambda (t0: T).(pc3 c0 (THead (Flat Cast) t0 t2) (THead (Flat Appl) w -(THead (Bind Abst) u t)))) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0: -T).(ty3 g c0 t2 t0))) H6)))))))))))) (\lambda (c0: C).(\lambda (t0: -T).(\lambda (t3: T).(\lambda (H1: (ty3 g c0 t0 t3)).(\lambda (H2: (((eq T t0 -(THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t4: T).(pc3 c0 (THead (Flat -Cast) t4 t2) t3)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t4: T).(ty3 g -c0 t2 t4)))))).(\lambda (t4: T).(\lambda (H3: (ty3 g c0 t3 t4)).(\lambda (H4: -(((eq T t3 (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t5: T).(pc3 c0 -(THead (Flat Cast) t5 t2) t4)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda -(t5: T).(ty3 g c0 t2 t5)))))).(\lambda (H5: (eq T (THead (Flat Cast) t3 t0) -(THead (Flat Cast) t2 t1))).(let H6 \def (f_equal T T (\lambda (e: T).(match -e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) -\Rightarrow t3 | (THead _ t _) \Rightarrow t])) (THead (Flat Cast) t3 t0) -(THead (Flat Cast) t2 t1) H5) in ((let H7 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | -(TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Flat Cast) -t3 t0) (THead (Flat Cast) t2 t1) H5) in (\lambda (H8: (eq T t3 t2)).(let H9 -\def (eq_ind T t3 (\lambda (t: T).((eq T t (THead (Flat Cast) t2 t1)) \to -(ex3 T (\lambda (t5: T).(pc3 c0 (THead (Flat Cast) t5 t2) t4)) (\lambda (_: -T).(ty3 g c0 t1 t2)) (\lambda (t5: T).(ty3 g c0 t2 t5))))) H4 t2 H8) in (let -H10 \def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 t t4)) H3 t2 H8) in (let H11 -\def (eq_ind T t3 (\lambda (t: T).((eq T t0 (THead (Flat Cast) t2 t1)) \to -(ex3 T (\lambda (t5: T).(pc3 c0 (THead (Flat Cast) t5 t2) t)) (\lambda (_: -T).(ty3 g c0 t1 t2)) (\lambda (t5: T).(ty3 g c0 t2 t5))))) H2 t2 H8) in (let -H12 \def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 t0 t)) H1 t2 H8) in (eq_ind_r -T t2 (\lambda (t: T).(ex3 T (\lambda (t5: T).(pc3 c0 (THead (Flat Cast) t5 -t2) (THead (Flat Cast) t4 t))) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda -(t5: T).(ty3 g c0 t2 t5)))) (let H13 \def (eq_ind T t0 (\lambda (t: T).((eq T -t (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t5: T).(pc3 c0 (THead (Flat -Cast) t5 t2) t2)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t5: T).(ty3 g -c0 t2 t5))))) H11 t1 H7) in (let H14 \def (eq_ind T t0 (\lambda (t: T).(ty3 g -c0 t t2)) H12 t1 H7) in (ex3_intro T (\lambda (t5: T).(pc3 c0 (THead (Flat -Cast) t5 t2) (THead (Flat Cast) t4 t2))) (\lambda (_: T).(ty3 g c0 t1 t2)) -(\lambda (t5: T).(ty3 g c0 t2 t5)) t4 (pc3_refl c0 (THead (Flat Cast) t4 t2)) -H14 H10))) t3 H8))))))) H6))))))))))) c y x H0))) H)))))). -(* COMMENTS -Initial nodes: 2609 -END *) - -theorem tys3_gen_nil: - \forall (g: G).(\forall (c: C).(\forall (u: T).((tys3 g c TNil u) \to (ex T -(\lambda (u0: T).(ty3 g c u u0)))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (H: (tys3 g c TNil -u)).(insert_eq TList TNil (\lambda (t: TList).(tys3 g c t u)) (\lambda (_: -TList).(ex T (\lambda (u0: T).(ty3 g c u u0)))) (\lambda (y: TList).(\lambda -(H0: (tys3 g c y u)).(tys3_ind g c (\lambda (t: TList).(\lambda (t0: T).((eq -TList t TNil) \to (ex T (\lambda (u0: T).(ty3 g c t0 u0)))))) (\lambda (u0: -T).(\lambda (u1: T).(\lambda (H1: (ty3 g c u0 u1)).(\lambda (_: (eq TList -TNil TNil)).(ex_intro T (\lambda (u2: T).(ty3 g c u0 u2)) u1 H1))))) (\lambda -(t: T).(\lambda (u0: T).(\lambda (_: (ty3 g c t u0)).(\lambda (ts: -TList).(\lambda (_: (tys3 g c ts u0)).(\lambda (_: (((eq TList ts TNil) \to -(ex T (\lambda (u1: T).(ty3 g c u0 u1)))))).(\lambda (H4: (eq TList (TCons t -ts) TNil)).(let H5 \def (eq_ind TList (TCons t ts) (\lambda (ee: -TList).(match ee in TList return (\lambda (_: TList).Prop) with [TNil -\Rightarrow False | (TCons _ _) \Rightarrow True])) I TNil H4) in (False_ind -(ex T (\lambda (u1: T).(ty3 g c u0 u1))) H5))))))))) y u H0))) H)))). -(* COMMENTS -Initial nodes: 255 -END *) - -theorem tys3_gen_cons: - \forall (g: G).(\forall (c: C).(\forall (ts: TList).(\forall (t: T).(\forall -(u: T).((tys3 g c (TCons t ts) u) \to (land (ty3 g c t u) (tys3 g c ts -u))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (ts: TList).(\lambda (t: T).(\lambda -(u: T).(\lambda (H: (tys3 g c (TCons t ts) u)).(insert_eq TList (TCons t ts) -(\lambda (t0: TList).(tys3 g c t0 u)) (\lambda (_: TList).(land (ty3 g c t u) -(tys3 g c ts u))) (\lambda (y: TList).(\lambda (H0: (tys3 g c y u)).(tys3_ind -g c (\lambda (t0: TList).(\lambda (t1: T).((eq TList t0 (TCons t ts)) \to -(land (ty3 g c t t1) (tys3 g c ts t1))))) (\lambda (u0: T).(\lambda (u1: -T).(\lambda (_: (ty3 g c u0 u1)).(\lambda (H2: (eq TList TNil (TCons t -ts))).(let H3 \def (eq_ind TList TNil (\lambda (ee: TList).(match ee in TList -return (\lambda (_: TList).Prop) with [TNil \Rightarrow True | (TCons _ _) -\Rightarrow False])) I (TCons t ts) H2) in (False_ind (land (ty3 g c t u0) -(tys3 g c ts u0)) H3)))))) (\lambda (t0: T).(\lambda (u0: T).(\lambda (H1: -(ty3 g c t0 u0)).(\lambda (ts0: TList).(\lambda (H2: (tys3 g c ts0 -u0)).(\lambda (H3: (((eq TList ts0 (TCons t ts)) \to (land (ty3 g c t u0) -(tys3 g c ts u0))))).(\lambda (H4: (eq TList (TCons t0 ts0) (TCons t -ts))).(let H5 \def (f_equal TList T (\lambda (e: TList).(match e in TList -return (\lambda (_: TList).T) with [TNil \Rightarrow t0 | (TCons t1 _) -\Rightarrow t1])) (TCons t0 ts0) (TCons t ts) H4) in ((let H6 \def (f_equal -TList TList (\lambda (e: TList).(match e in TList return (\lambda (_: -TList).TList) with [TNil \Rightarrow ts0 | (TCons _ t1) \Rightarrow t1])) -(TCons t0 ts0) (TCons t ts) H4) in (\lambda (H7: (eq T t0 t)).(let H8 \def -(eq_ind TList ts0 (\lambda (t1: TList).((eq TList t1 (TCons t ts)) \to (land -(ty3 g c t u0) (tys3 g c ts u0)))) H3 ts H6) in (let H9 \def (eq_ind TList -ts0 (\lambda (t1: TList).(tys3 g c t1 u0)) H2 ts H6) in (let H10 \def (eq_ind -T t0 (\lambda (t1: T).(ty3 g c t1 u0)) H1 t H7) in (conj (ty3 g c t u0) (tys3 -g c ts u0) H10 H9)))))) H5))))))))) y u H0))) H)))))). -(* COMMENTS -Initial nodes: 479 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/fwd_nf2.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/fwd_nf2.ma deleted file mode 100644 index ca4e40c7b..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/fwd_nf2.ma +++ /dev/null @@ -1,301 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/ty3/arity_props.ma". - -include "Basic-1/pc3/nf2.ma". - -include "Basic-1/nf2/fwd.ma". - -theorem ty3_gen_appl_nf2: - \forall (g: G).(\forall (c: C).(\forall (w: T).(\forall (v: T).(\forall (x: -T).((ty3 g c (THead (Flat Appl) w v) x) \to (ex4_2 T T (\lambda (u: -T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x))) -(\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) -(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t: -T).(nf2 c (THead (Bind Abst) u t)))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (w: T).(\lambda (v: T).(\lambda (x: -T).(\lambda (H: (ty3 g c (THead (Flat Appl) w v) x)).(ex3_2_ind T T (\lambda -(u: T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) -x))) (\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) -(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (ex4_2 T T (\lambda (u: -T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x))) -(\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) -(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t: -T).(nf2 c (THead (Bind Abst) u t))))) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H0: (pc3 c (THead (Flat Appl) w (THead (Bind Abst) x0 x1)) -x)).(\lambda (H1: (ty3 g c v (THead (Bind Abst) x0 x1))).(\lambda (H2: (ty3 g -c w x0)).(let H_x \def (ty3_correct g c v (THead (Bind Abst) x0 x1) H1) in -(let H3 \def H_x in (ex_ind T (\lambda (t: T).(ty3 g c (THead (Bind Abst) x0 -x1) t)) (ex4_2 T T (\lambda (u: T).(\lambda (t: T).(pc3 c (THead (Flat Appl) -w (THead (Bind Abst) u t)) x))) (\lambda (u: T).(\lambda (t: T).(ty3 g c v -(THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) -(\lambda (u: T).(\lambda (t: T).(nf2 c (THead (Bind Abst) u t))))) (\lambda -(x2: T).(\lambda (H4: (ty3 g c (THead (Bind Abst) x0 x1) x2)).(let H_x0 \def -(ty3_correct g c w x0 H2) in (let H5 \def H_x0 in (ex_ind T (\lambda (t: -T).(ty3 g c x0 t)) (ex4_2 T T (\lambda (u: T).(\lambda (t: T).(pc3 c (THead -(Flat Appl) w (THead (Bind Abst) u t)) x))) (\lambda (u: T).(\lambda (t: -T).(ty3 g c v (THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3 -g c w u))) (\lambda (u: T).(\lambda (t: T).(nf2 c (THead (Bind Abst) u t))))) -(\lambda (x3: T).(\lambda (H6: (ty3 g c x0 x3)).(let H7 \def (ty3_sn3 g c -(THead (Bind Abst) x0 x1) x2 H4) in (let H_x1 \def (nf2_sn3 c (THead (Bind -Abst) x0 x1) H7) in (let H8 \def H_x1 in (ex2_ind T (\lambda (u: T).(pr3 c -(THead (Bind Abst) x0 x1) u)) (\lambda (u: T).(nf2 c u)) (ex4_2 T T (\lambda -(u: T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) -x))) (\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) -(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t: -T).(nf2 c (THead (Bind Abst) u t))))) (\lambda (x4: T).(\lambda (H9: (pr3 c -(THead (Bind Abst) x0 x1) x4)).(\lambda (H10: (nf2 c x4)).(let H11 \def -(pr3_gen_abst c x0 x1 x4 H9) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: -T).(eq T x4 (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall -(u: T).(pr3 (CHead c (Bind b) u) x1 t2))))) (ex4_2 T T (\lambda (u: -T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x))) -(\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) -(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t: -T).(nf2 c (THead (Bind Abst) u t))))) (\lambda (x5: T).(\lambda (x6: -T).(\lambda (H12: (eq T x4 (THead (Bind Abst) x5 x6))).(\lambda (H13: (pr3 c -x0 x5)).(\lambda (H14: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind -b) u) x1 x6))))).(let H15 \def (eq_ind T x4 (\lambda (t: T).(nf2 c t)) H10 -(THead (Bind Abst) x5 x6) H12) in (let H16 \def (pr3_head_12 c x0 x5 H13 -(Bind Abst) x1 x6 (H14 Abst x5)) in (ex4_2_intro T T (\lambda (u: T).(\lambda -(t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x))) (\lambda (u: -T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) (\lambda (u: -T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t: T).(nf2 c -(THead (Bind Abst) u t)))) x5 x6 (pc3_pr3_conf c (THead (Flat Appl) w (THead -(Bind Abst) x0 x1)) x H0 (THead (Flat Appl) w (THead (Bind Abst) x5 x6)) -(pr3_thin_dx c (THead (Bind Abst) x0 x1) (THead (Bind Abst) x5 x6) H16 w -Appl)) (ty3_conv g c (THead (Bind Abst) x5 x6) x2 (ty3_sred_pr3 c (THead -(Bind Abst) x0 x1) (THead (Bind Abst) x5 x6) H16 g x2 H4) v (THead (Bind -Abst) x0 x1) H1 (pc3_pr3_r c (THead (Bind Abst) x0 x1) (THead (Bind Abst) x5 -x6) H16)) (ty3_conv g c x5 x3 (ty3_sred_pr3 c x0 x5 H13 g x3 H6) w x0 H2 -(pc3_pr3_r c x0 x5 H13)) H15)))))))) H11))))) H8)))))) H5))))) H3)))))))) -(ty3_gen_appl g c w v x H))))))). -(* COMMENTS -Initial nodes: 1289 -END *) - -theorem ty3_inv_lref_nf2_pc3: - \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (i: nat).((ty3 g c -(TLRef i) u1) \to ((nf2 c (TLRef i)) \to (\forall (u2: T).((nf2 c u2) \to -((pc3 c u1 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift (S i) O u)))))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (u1: T).(\lambda (i: nat).(\lambda -(H: (ty3 g c (TLRef i) u1)).(insert_eq T (TLRef i) (\lambda (t: T).(ty3 g c t -u1)) (\lambda (t: T).((nf2 c t) \to (\forall (u2: T).((nf2 c u2) \to ((pc3 c -u1 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift (S i) O u))))))))) (\lambda -(y: T).(\lambda (H0: (ty3 g c y u1)).(ty3_ind g (\lambda (c0: C).(\lambda (t: -T).(\lambda (t0: T).((eq T t (TLRef i)) \to ((nf2 c0 t) \to (\forall (u2: -T).((nf2 c0 u2) \to ((pc3 c0 t0 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift -(S i) O u)))))))))))) (\lambda (c0: C).(\lambda (t2: T).(\lambda (t: -T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda (_: (((eq T t2 (TLRef i)) \to ((nf2 -c0 t2) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 t u2) \to (ex T -(\lambda (u: T).(eq T u2 (lift (S i) O u))))))))))).(\lambda (u: T).(\lambda -(t1: T).(\lambda (H3: (ty3 g c0 u t1)).(\lambda (H4: (((eq T u (TLRef i)) \to -((nf2 c0 u) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 t1 u2) \to (ex T -(\lambda (u0: T).(eq T u2 (lift (S i) O u0))))))))))).(\lambda (H5: (pc3 c0 -t1 t2)).(\lambda (H6: (eq T u (TLRef i))).(\lambda (H7: (nf2 c0 u)).(\lambda -(u2: T).(\lambda (H8: (nf2 c0 u2)).(\lambda (H9: (pc3 c0 t2 u2)).(let H10 -\def (eq_ind T u (\lambda (t0: T).(nf2 c0 t0)) H7 (TLRef i) H6) in (let H11 -\def (eq_ind T u (\lambda (t0: T).((eq T t0 (TLRef i)) \to ((nf2 c0 t0) \to -(\forall (u3: T).((nf2 c0 u3) \to ((pc3 c0 t1 u3) \to (ex T (\lambda (u0: -T).(eq T u3 (lift (S i) O u0)))))))))) H4 (TLRef i) H6) in (let H12 \def -(eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 t1)) H3 (TLRef i) H6) in (let H_y -\def (H11 (refl_equal T (TLRef i)) H10 u2 H8) in (H_y (pc3_t t2 c0 t1 H5 u2 -H9))))))))))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq -T (TSort m) (TLRef i))).(\lambda (_: (nf2 c0 (TSort m))).(\lambda (u2: -T).(\lambda (_: (nf2 c0 u2)).(\lambda (_: (pc3 c0 (TSort (next g m)) -u2)).(let H5 \def (eq_ind T (TSort m) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef i) H1) in -(False_ind (ex T (\lambda (u: T).(eq T u2 (lift (S i) O u)))) H5))))))))) -(\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda -(H1: (getl n c0 (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda (_: (ty3 g -d u t)).(\lambda (_: (((eq T u (TLRef i)) \to ((nf2 d u) \to (\forall (u2: -T).((nf2 d u2) \to ((pc3 d t u2) \to (ex T (\lambda (u0: T).(eq T u2 (lift (S -i) O u0))))))))))).(\lambda (H4: (eq T (TLRef n) (TLRef i))).(\lambda (H5: -(nf2 c0 (TLRef n))).(\lambda (u2: T).(\lambda (_: (nf2 c0 u2)).(\lambda (H7: -(pc3 c0 (lift (S n) O t) u2)).(let H8 \def (f_equal T nat (\lambda (e: -T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow n | -(TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef -i) H4) in (let H9 \def (eq_ind nat n (\lambda (n0: nat).(pc3 c0 (lift (S n0) -O t) u2)) H7 i H8) in (let H10 \def (eq_ind nat n (\lambda (n0: nat).(nf2 c0 -(TLRef n0))) H5 i H8) in (let H11 \def (eq_ind nat n (\lambda (n0: nat).(getl -n0 c0 (CHead d (Bind Abbr) u))) H1 i H8) in (nf2_gen_lref c0 d u i H11 H10 -(ex T (\lambda (u0: T).(eq T u2 (lift (S i) O u0)))))))))))))))))))))) -(\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda -(H1: (getl n c0 (CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (_: (ty3 g -d u t)).(\lambda (_: (((eq T u (TLRef i)) \to ((nf2 d u) \to (\forall (u2: -T).((nf2 d u2) \to ((pc3 d t u2) \to (ex T (\lambda (u0: T).(eq T u2 (lift (S -i) O u0))))))))))).(\lambda (H4: (eq T (TLRef n) (TLRef i))).(\lambda (H5: -(nf2 c0 (TLRef n))).(\lambda (u2: T).(\lambda (H6: (nf2 c0 u2)).(\lambda (H7: -(pc3 c0 (lift (S n) O u) u2)).(let H8 \def (f_equal T nat (\lambda (e: -T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow n | -(TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef -i) H4) in (let H9 \def (eq_ind nat n (\lambda (n0: nat).(pc3 c0 (lift (S n0) -O u) u2)) H7 i H8) in (let H10 \def (eq_ind nat n (\lambda (n0: nat).(nf2 c0 -(TLRef n0))) H5 i H8) in (let H11 \def (eq_ind nat n (\lambda (n0: nat).(getl -n0 c0 (CHead d (Bind Abst) u))) H1 i H8) in (let H_y \def (pc3_nf2_unfold c0 -(lift (S i) O u) u2 H9 H6) in (let H12 \def (pr3_gen_lift c0 u u2 (S i) O H_y -d (getl_drop Abst c0 d u i H11)) in (ex2_ind T (\lambda (t2: T).(eq T u2 -(lift (S i) O t2))) (\lambda (t2: T).(pr3 d u t2)) (ex T (\lambda (u0: T).(eq -T u2 (lift (S i) O u0)))) (\lambda (x: T).(\lambda (H13: (eq T u2 (lift (S i) -O x))).(\lambda (_: (pr3 d u x)).(eq_ind_r T (lift (S i) O x) (\lambda (t0: -T).(ex T (\lambda (u0: T).(eq T t0 (lift (S i) O u0))))) (ex_intro T (\lambda -(u0: T).(eq T (lift (S i) O x) (lift (S i) O u0))) x (refl_equal T (lift (S -i) O x))) u2 H13)))) H12)))))))))))))))))))) (\lambda (c0: C).(\lambda (u: -T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((eq T u (TLRef -i)) \to ((nf2 c0 u) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 t u2) \to -(ex T (\lambda (u0: T).(eq T u2 (lift (S i) O u0))))))))))).(\lambda (b: -B).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) -u) t1 t2)).(\lambda (_: (((eq T t1 (TLRef i)) \to ((nf2 (CHead c0 (Bind b) u) -t1) \to (\forall (u2: T).((nf2 (CHead c0 (Bind b) u) u2) \to ((pc3 (CHead c0 -(Bind b) u) t2 u2) \to (ex T (\lambda (u0: T).(eq T u2 (lift (S i) O -u0))))))))))).(\lambda (H5: (eq T (THead (Bind b) u t1) (TLRef i))).(\lambda -(_: (nf2 c0 (THead (Bind b) u t1))).(\lambda (u2: T).(\lambda (_: (nf2 c0 -u2)).(\lambda (_: (pc3 c0 (THead (Bind b) u t2) u2)).(let H9 \def (eq_ind T -(THead (Bind b) u t1) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead _ _ _) \Rightarrow True])) I (TLRef i) H5) in (False_ind (ex T -(\lambda (u0: T).(eq T u2 (lift (S i) O u0)))) H9))))))))))))))))) (\lambda -(c0: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda -(_: (((eq T w (TLRef i)) \to ((nf2 c0 w) \to (\forall (u2: T).((nf2 c0 u2) -\to ((pc3 c0 u u2) \to (ex T (\lambda (u0: T).(eq T u2 (lift (S i) O -u0))))))))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead -(Bind Abst) u t))).(\lambda (_: (((eq T v (TLRef i)) \to ((nf2 c0 v) \to -(\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 (THead (Bind Abst) u t) u2) \to -(ex T (\lambda (u0: T).(eq T u2 (lift (S i) O u0))))))))))).(\lambda (H5: (eq -T (THead (Flat Appl) w v) (TLRef i))).(\lambda (_: (nf2 c0 (THead (Flat Appl) -w v))).(\lambda (u2: T).(\lambda (_: (nf2 c0 u2)).(\lambda (_: (pc3 c0 (THead -(Flat Appl) w (THead (Bind Abst) u t)) u2)).(let H9 \def (eq_ind T (THead -(Flat Appl) w v) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) -with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ -_) \Rightarrow True])) I (TLRef i) H5) in (False_ind (ex T (\lambda (u0: -T).(eq T u2 (lift (S i) O u0)))) H9)))))))))))))))) (\lambda (c0: C).(\lambda -(t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g c0 t1 t2)).(\lambda (_: (((eq T -t1 (TLRef i)) \to ((nf2 c0 t1) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 -t2 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift (S i) O u))))))))))).(\lambda -(t0: T).(\lambda (_: (ty3 g c0 t2 t0)).(\lambda (_: (((eq T t2 (TLRef i)) \to -((nf2 c0 t2) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 t0 u2) \to (ex T -(\lambda (u: T).(eq T u2 (lift (S i) O u))))))))))).(\lambda (H5: (eq T -(THead (Flat Cast) t2 t1) (TLRef i))).(\lambda (_: (nf2 c0 (THead (Flat Cast) -t2 t1))).(\lambda (u2: T).(\lambda (_: (nf2 c0 u2)).(\lambda (_: (pc3 c0 -(THead (Flat Cast) t0 t2) u2)).(let H9 \def (eq_ind T (THead (Flat Cast) t2 -t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort -_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) -\Rightarrow True])) I (TLRef i) H5) in (False_ind (ex T (\lambda (u: T).(eq T -u2 (lift (S i) O u)))) H9))))))))))))))) c y u1 H0))) H))))). -(* COMMENTS -Initial nodes: 2175 -END *) - -theorem ty3_inv_lref_nf2: - \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (i: nat).((ty3 g c -(TLRef i) u) \to ((nf2 c (TLRef i)) \to ((nf2 c u) \to (ex T (\lambda (u0: -T).(eq T u (lift (S i) O u0)))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H: (ty3 g c (TLRef i) u)).(\lambda (H0: (nf2 c (TLRef i))).(\lambda (H1: -(nf2 c u)).(ty3_inv_lref_nf2_pc3 g c u i H H0 u H1 (pc3_refl c u)))))))). -(* COMMENTS -Initial nodes: 57 -END *) - -theorem ty3_inv_appls_lref_nf2: - \forall (g: G).(\forall (c: C).(\forall (vs: TList).(\forall (u1: -T).(\forall (i: nat).((ty3 g c (THeads (Flat Appl) vs (TLRef i)) u1) \to -((nf2 c (TLRef i)) \to ((nf2 c u1) \to (ex2 T (\lambda (u: T).(nf2 c (lift (S -i) O u))) (\lambda (u: T).(pc3 c (THeads (Flat Appl) vs (lift (S i) O u)) -u1)))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (vs: TList).(TList_ind (\lambda (t: -TList).(\forall (u1: T).(\forall (i: nat).((ty3 g c (THeads (Flat Appl) t -(TLRef i)) u1) \to ((nf2 c (TLRef i)) \to ((nf2 c u1) \to (ex2 T (\lambda (u: -T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3 c (THeads (Flat Appl) t -(lift (S i) O u)) u1))))))))) (\lambda (u1: T).(\lambda (i: nat).(\lambda (H: -(ty3 g c (TLRef i) u1)).(\lambda (H0: (nf2 c (TLRef i))).(\lambda (H1: (nf2 c -u1)).(let H_x \def (ty3_inv_lref_nf2 g c u1 i H H0 H1) in (let H2 \def H_x in -(ex_ind T (\lambda (u0: T).(eq T u1 (lift (S i) O u0))) (ex2 T (\lambda (u: -T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3 c (lift (S i) O u) u1))) -(\lambda (x: T).(\lambda (H3: (eq T u1 (lift (S i) O x))).(let H4 \def -(eq_ind T u1 (\lambda (t: T).(nf2 c t)) H1 (lift (S i) O x) H3) in (eq_ind_r -T (lift (S i) O x) (\lambda (t: T).(ex2 T (\lambda (u: T).(nf2 c (lift (S i) -O u))) (\lambda (u: T).(pc3 c (lift (S i) O u) t)))) (ex_intro2 T (\lambda -(u: T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3 c (lift (S i) O u) -(lift (S i) O x))) x H4 (pc3_refl c (lift (S i) O x))) u1 H3)))) H2)))))))) -(\lambda (t: T).(\lambda (t0: TList).(\lambda (H: ((\forall (u1: T).(\forall -(i: nat).((ty3 g c (THeads (Flat Appl) t0 (TLRef i)) u1) \to ((nf2 c (TLRef -i)) \to ((nf2 c u1) \to (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O u))) -(\lambda (u: T).(pc3 c (THeads (Flat Appl) t0 (lift (S i) O u)) -u1)))))))))).(\lambda (u1: T).(\lambda (i: nat).(\lambda (H0: (ty3 g c (THead -(Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) u1)).(\lambda (H1: (nf2 c -(TLRef i))).(\lambda (_: (nf2 c u1)).(let H_x \def (ty3_gen_appl_nf2 g c t -(THeads (Flat Appl) t0 (TLRef i)) u1 H0) in (let H3 \def H_x in (ex4_2_ind T -T (\lambda (u: T).(\lambda (t1: T).(pc3 c (THead (Flat Appl) t (THead (Bind -Abst) u t1)) u1))) (\lambda (u: T).(\lambda (t1: T).(ty3 g c (THeads (Flat -Appl) t0 (TLRef i)) (THead (Bind Abst) u t1)))) (\lambda (u: T).(\lambda (_: -T).(ty3 g c t u))) (\lambda (u: T).(\lambda (t1: T).(nf2 c (THead (Bind Abst) -u t1)))) (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O u))) (\lambda (u: -T).(pc3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O u))) -u1))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (pc3 c (THead (Flat -Appl) t (THead (Bind Abst) x0 x1)) u1)).(\lambda (H5: (ty3 g c (THeads (Flat -Appl) t0 (TLRef i)) (THead (Bind Abst) x0 x1))).(\lambda (_: (ty3 g c t -x0)).(\lambda (H7: (nf2 c (THead (Bind Abst) x0 x1))).(let H8 \def -(nf2_gen_abst c x0 x1 H7) in (land_ind (nf2 c x0) (nf2 (CHead c (Bind Abst) -x0) x1) (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3 -c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O u))) u1))) -(\lambda (H9: (nf2 c x0)).(\lambda (H10: (nf2 (CHead c (Bind Abst) x0) -x1)).(let H_y \def (H (THead (Bind Abst) x0 x1) i H5 H1) in (let H11 \def -(H_y (nf2_abst_shift c x0 H9 x1 H10)) in (ex2_ind T (\lambda (u: T).(nf2 c -(lift (S i) O u))) (\lambda (u: T).(pc3 c (THeads (Flat Appl) t0 (lift (S i) -O u)) (THead (Bind Abst) x0 x1))) (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O -u))) (\lambda (u: T).(pc3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift -(S i) O u))) u1))) (\lambda (x: T).(\lambda (H12: (nf2 c (lift (S i) O -x))).(\lambda (H13: (pc3 c (THeads (Flat Appl) t0 (lift (S i) O x)) (THead -(Bind Abst) x0 x1))).(ex_intro2 T (\lambda (u: T).(nf2 c (lift (S i) O u))) -(\lambda (u: T).(pc3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S -i) O u))) u1)) x H12 (pc3_t (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) c -(THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O x))) (pc3_thin_dx c -(THeads (Flat Appl) t0 (lift (S i) O x)) (THead (Bind Abst) x0 x1) H13 t -Appl) u1 H4))))) H11))))) H8)))))))) H3))))))))))) vs))). -(* COMMENTS -Initial nodes: 1213 -END *) - -theorem ty3_inv_lref_lref_nf2: - \forall (g: G).(\forall (c: C).(\forall (i: nat).(\forall (j: nat).((ty3 g c -(TLRef i) (TLRef j)) \to ((nf2 c (TLRef i)) \to ((nf2 c (TLRef j)) \to (lt i -j))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (i: nat).(\lambda (j: nat).(\lambda -(H: (ty3 g c (TLRef i) (TLRef j))).(\lambda (H0: (nf2 c (TLRef i))).(\lambda -(H1: (nf2 c (TLRef j))).(let H_x \def (ty3_inv_lref_nf2 g c (TLRef j) i H H0 -H1) in (let H2 \def H_x in (ex_ind T (\lambda (u0: T).(eq T (TLRef j) (lift -(S i) O u0))) (lt i j) (\lambda (x: T).(\lambda (H3: (eq T (TLRef j) (lift (S -i) O x))).(let H_x0 \def (lift_gen_lref x O (S i) j H3) in (let H4 \def H_x0 -in (or_ind (land (lt j O) (eq T x (TLRef j))) (land (le (S i) j) (eq T x -(TLRef (minus j (S i))))) (lt i j) (\lambda (H5: (land (lt j O) (eq T x -(TLRef j)))).(land_ind (lt j O) (eq T x (TLRef j)) (lt i j) (\lambda (H6: (lt -j O)).(\lambda (_: (eq T x (TLRef j))).(lt_x_O j H6 (lt i j)))) H5)) (\lambda -(H5: (land (le (S i) j) (eq T x (TLRef (minus j (S i)))))).(land_ind (le (S -i) j) (eq T x (TLRef (minus j (S i)))) (lt i j) (\lambda (H6: (le (S i) -j)).(\lambda (_: (eq T x (TLRef (minus j (S i))))).H6)) H5)) H4))))) -H2))))))))). -(* COMMENTS -Initial nodes: 337 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/nf2.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/nf2.ma deleted file mode 100644 index 47b675663..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/nf2.ma +++ /dev/null @@ -1,472 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/ty3/arity.ma". - -include "Basic-1/pc3/nf2.ma". - -include "Basic-1/nf2/arity.ma". - -definition ty3_nf2_inv_abst_premise: - C \to (T \to (T \to Prop)) -\def - \lambda (c: C).(\lambda (w: T).(\lambda (u: T).(\forall (d: C).(\forall (wi: -T).(\forall (i: nat).((getl i c (CHead d (Bind Abst) wi)) \to (\forall (vs: -TList).((pc3 c (THeads (Flat Appl) vs (lift (S i) O wi)) (THead (Bind Abst) w -u)) \to False)))))))). - -theorem ty3_nf2_inv_abst_premise_csort: - \forall (w: T).(\forall (u: T).(\forall (m: nat).(ty3_nf2_inv_abst_premise -(CSort m) w u))) -\def - \lambda (w: T).(\lambda (u: T).(\lambda (m: nat).(\lambda (d: C).(\lambda -(wi: T).(\lambda (i: nat).(\lambda (H: (getl i (CSort m) (CHead d (Bind Abst) -wi))).(\lambda (vs: TList).(\lambda (_: (pc3 (CSort m) (THeads (Flat Appl) vs -(lift (S i) O wi)) (THead (Bind Abst) w u))).(getl_gen_sort m i (CHead d -(Bind Abst) wi) H False))))))))). -(* COMMENTS -Initial nodes: 85 -END *) - -theorem ty3_nf2_inv_all: - \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t -u) \to ((nf2 c t) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T -t (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c w))) -(\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c (Bind Abst) w) u0)))) (ex nat -(\lambda (n: nat).(eq T t (TSort n)))) (ex3_2 TList nat (\lambda (ws: -TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i))))) -(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c (TLRef i))))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H: -(ty3 g c t u)).(\lambda (H0: (nf2 c t)).(let H_x \def (ty3_arity g c t u H) -in (let H1 \def H_x in (ex2_ind A (\lambda (a1: A).(arity g c t a1)) (\lambda -(a1: A).(arity g c u (asucc g a1))) (or3 (ex3_2 T T (\lambda (w: T).(\lambda -(u0: T).(eq T t (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: -T).(nf2 c w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c (Bind Abst) w) -u0)))) (ex nat (\lambda (n: nat).(eq T t (TSort n)))) (ex3_2 TList nat -(\lambda (ws: TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef -i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c (TLRef i)))))) (\lambda (x: A).(\lambda (H2: -(arity g c t x)).(\lambda (_: (arity g c u (asucc g x))).(arity_nf2_inv_all g -c t x H2 H0)))) H1)))))))). -(* COMMENTS -Initial nodes: 233 -END *) - -theorem ty3_nf2_inv_sort: - \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (m: nat).((ty3 g c t -(TSort m)) \to ((nf2 c t) \to (or (ex2 nat (\lambda (n: nat).(eq T t (TSort -n))) (\lambda (n: nat).(eq nat m (next g n)))) (ex3_2 TList nat (\lambda (ws: -TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i))))) -(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c (TLRef i))))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (m: nat).(\lambda -(H: (ty3 g c t (TSort m))).(\lambda (H0: (nf2 c t)).(let H_x \def -(ty3_nf2_inv_all g c t (TSort m) H H0) in (let H1 \def H_x in (or3_ind (ex3_2 -T T (\lambda (w: T).(\lambda (u: T).(eq T t (THead (Bind Abst) w u)))) -(\lambda (w: T).(\lambda (_: T).(nf2 c w))) (\lambda (w: T).(\lambda (u: -T).(nf2 (CHead c (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t -(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t -(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef i))))) -(or (ex2 nat (\lambda (n: nat).(eq T t (TSort n))) (\lambda (n: nat).(eq nat -m (next g n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T -t (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef -i)))))) (\lambda (H2: (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t -(THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c w))) -(\lambda (w: T).(\lambda (u: T).(nf2 (CHead c (Bind Abst) w) -u))))).(ex3_2_ind T T (\lambda (w: T).(\lambda (u: T).(eq T t (THead (Bind -Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c w))) (\lambda (w: -T).(\lambda (u: T).(nf2 (CHead c (Bind Abst) w) u))) (or (ex2 nat (\lambda -(n: nat).(eq T t (TSort n))) (\lambda (n: nat).(eq nat m (next g n)))) (ex3_2 -TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) -ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) -(\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef i)))))) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H3: (eq T t (THead (Bind Abst) x0 -x1))).(\lambda (_: (nf2 c x0)).(\lambda (_: (nf2 (CHead c (Bind Abst) x0) -x1)).(let H6 \def (eq_ind T t (\lambda (t0: T).(ty3 g c t0 (TSort m))) H -(THead (Bind Abst) x0 x1) H3) in (eq_ind_r T (THead (Bind Abst) x0 x1) -(\lambda (t0: T).(or (ex2 nat (\lambda (n: nat).(eq T t0 (TSort n))) (\lambda -(n: nat).(eq nat m (next g n)))) (ex3_2 TList nat (\lambda (ws: -TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i))))) -(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c (TLRef i))))))) (ex3_2_ind T T (\lambda (t2: -T).(\lambda (_: T).(pc3 c (THead (Bind Abst) x0 t2) (TSort m)))) (\lambda (_: -T).(\lambda (t0: T).(ty3 g c x0 t0))) (\lambda (t2: T).(\lambda (_: T).(ty3 g -(CHead c (Bind Abst) x0) x1 t2))) (or (ex2 nat (\lambda (n: nat).(eq T (THead -(Bind Abst) x0 x1) (TSort n))) (\lambda (n: nat).(eq nat m (next g n)))) -(ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Bind -Abst) x0 x1) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: -TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: -nat).(nf2 c (TLRef i)))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: -(pc3 c (THead (Bind Abst) x0 x2) (TSort m))).(\lambda (_: (ty3 g c x0 -x3)).(\lambda (_: (ty3 g (CHead c (Bind Abst) x0) x1 x2)).(pc3_gen_sort_abst -c x0 x2 m (pc3_s c (TSort m) (THead (Bind Abst) x0 x2) H7) (or (ex2 nat -(\lambda (n: nat).(eq T (THead (Bind Abst) x0 x1) (TSort n))) (\lambda (n: -nat).(eq nat m (next g n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda -(i: nat).(eq T (THead (Bind Abst) x0 x1) (THeads (Flat Appl) ws (TLRef i))))) -(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c (TLRef i)))))))))))) (ty3_gen_bind g Abst c -x0 x1 (TSort m) H6)) t H3))))))) H2)) (\lambda (H2: (ex nat (\lambda (n: -nat).(eq T t (TSort n))))).(ex_ind nat (\lambda (n: nat).(eq T t (TSort n))) -(or (ex2 nat (\lambda (n: nat).(eq T t (TSort n))) (\lambda (n: nat).(eq nat -m (next g n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T -t (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef -i)))))) (\lambda (x: nat).(\lambda (H3: (eq T t (TSort x))).(let H4 \def -(eq_ind T t (\lambda (t0: T).(ty3 g c t0 (TSort m))) H (TSort x) H3) in -(eq_ind_r T (TSort x) (\lambda (t0: T).(or (ex2 nat (\lambda (n: nat).(eq T -t0 (TSort n))) (\lambda (n: nat).(eq nat m (next g n)))) (ex3_2 TList nat -(\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef -i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c (TLRef i))))))) (eq_ind nat (next g x) -(\lambda (n: nat).(or (ex2 nat (\lambda (n0: nat).(eq T (TSort x) (TSort -n0))) (\lambda (n0: nat).(eq nat n (next g n0)))) (ex3_2 TList nat (\lambda -(ws: TList).(\lambda (i: nat).(eq T (TSort x) (THeads (Flat Appl) ws (TLRef -i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c (TLRef i))))))) (or_introl (ex2 nat (\lambda -(n: nat).(eq T (TSort x) (TSort n))) (\lambda (n: nat).(eq nat (next g x) -(next g n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T -(TSort x) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda -(_: nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef -i))))) (ex_intro2 nat (\lambda (n: nat).(eq T (TSort x) (TSort n))) (\lambda -(n: nat).(eq nat (next g x) (next g n))) x (refl_equal T (TSort x)) -(refl_equal nat (next g x)))) m (pc3_gen_sort c (next g x) m (ty3_gen_sort g -c (TSort m) x H4))) t H3)))) H2)) (\lambda (H2: (ex3_2 TList nat (\lambda -(ws: TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i))))) -(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c (TLRef i)))))).(ex3_2_ind TList nat (\lambda -(ws: TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i))))) -(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c (TLRef i)))) (or (ex2 nat (\lambda (n: -nat).(eq T t (TSort n))) (\lambda (n: nat).(eq nat m (next g n)))) (ex3_2 -TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) -ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) -(\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef i)))))) (\lambda (x0: -TList).(\lambda (x1: nat).(\lambda (H3: (eq T t (THeads (Flat Appl) x0 (TLRef -x1)))).(\lambda (H4: (nfs2 c x0)).(\lambda (H5: (nf2 c (TLRef x1))).(let H6 -\def (eq_ind T t (\lambda (t0: T).(ty3 g c t0 (TSort m))) H (THeads (Flat -Appl) x0 (TLRef x1)) H3) in (eq_ind_r T (THeads (Flat Appl) x0 (TLRef x1)) -(\lambda (t0: T).(or (ex2 nat (\lambda (n: nat).(eq T t0 (TSort n))) (\lambda -(n: nat).(eq nat m (next g n)))) (ex3_2 TList nat (\lambda (ws: -TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i))))) -(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c (TLRef i))))))) (or_intror (ex2 nat (\lambda -(n: nat).(eq T (THeads (Flat Appl) x0 (TLRef x1)) (TSort n))) (\lambda (n: -nat).(eq nat m (next g n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda -(i: nat).(eq T (THeads (Flat Appl) x0 (TLRef x1)) (THeads (Flat Appl) ws -(TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda -(_: TList).(\lambda (i: nat).(nf2 c (TLRef i))))) (ex3_2_intro TList nat -(\lambda (ws: TList).(\lambda (i: nat).(eq T (THeads (Flat Appl) x0 (TLRef -x1)) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef i)))) -x0 x1 (refl_equal T (THeads (Flat Appl) x0 (TLRef x1))) H4 H5)) t H3))))))) -H2)) H1)))))))). -(* COMMENTS -Initial nodes: 2045 -END *) - -theorem ty3_nf2_gen__ty3_nf2_inv_abst_aux: - \forall (c: C).(\forall (w1: T).(\forall (u1: T).((ty3_nf2_inv_abst_premise -c w1 u1) \to (\forall (t: T).(\forall (w2: T).(\forall (u2: T).((pc3 c (THead -(Flat Appl) t (THead (Bind Abst) w2 u2)) (THead (Bind Abst) w1 u1)) \to -(ty3_nf2_inv_abst_premise c w2 u2)))))))) -\def - \lambda (c: C).(\lambda (w1: T).(\lambda (u1: T).(\lambda (H: ((\forall (d: -C).(\forall (wi: T).(\forall (i: nat).((getl i c (CHead d (Bind Abst) wi)) -\to (\forall (vs: TList).((pc3 c (THeads (Flat Appl) vs (lift (S i) O wi)) -(THead (Bind Abst) w1 u1)) \to False)))))))).(\lambda (t: T).(\lambda (w2: -T).(\lambda (u2: T).(\lambda (H0: (pc3 c (THead (Flat Appl) t (THead (Bind -Abst) w2 u2)) (THead (Bind Abst) w1 u1))).(\lambda (d: C).(\lambda (wi: -T).(\lambda (i: nat).(\lambda (H1: (getl i c (CHead d (Bind Abst) -wi))).(\lambda (vs: TList).(\lambda (H2: (pc3 c (THeads (Flat Appl) vs (lift -(S i) O wi)) (THead (Bind Abst) w2 u2))).(H d wi i H1 (TCons t vs) (pc3_t -(THead (Flat Appl) t (THead (Bind Abst) w2 u2)) c (THead (Flat Appl) t -(THeads (Flat Appl) vs (lift (S i) O wi))) (pc3_thin_dx c (THeads (Flat Appl) -vs (lift (S i) O wi)) (THead (Bind Abst) w2 u2) H2 t Appl) (THead (Bind Abst) -w1 u1) H0))))))))))))))). -(* COMMENTS -Initial nodes: 271 -END *) - -theorem ty3_nf2_inv_abst: - \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (w: T).(\forall (u: -T).((ty3 g c t (THead (Bind Abst) w u)) \to ((nf2 c t) \to ((nf2 c w) \to -((ty3_nf2_inv_abst_premise c w u) \to (ex4_2 T T (\lambda (v: T).(\lambda (_: -T).(eq T t (THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g -c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v -u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) -v)))))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (w: T).(\lambda (u: -T).(\lambda (H: (ty3 g c t (THead (Bind Abst) w u))).(\lambda (H0: (nf2 c -t)).(\lambda (H1: (nf2 c w)).(\lambda (H2: (ty3_nf2_inv_abst_premise c w -u)).(let H_x \def (ty3_nf2_inv_all g c t (THead (Bind Abst) w u) H H0) in -(let H3 \def H_x in (or3_ind (ex3_2 T T (\lambda (w0: T).(\lambda (u0: T).(eq -T t (THead (Bind Abst) w0 u0)))) (\lambda (w0: T).(\lambda (_: T).(nf2 c -w0))) (\lambda (w0: T).(\lambda (u0: T).(nf2 (CHead c (Bind Abst) w0) u0)))) -(ex nat (\lambda (n: nat).(eq T t (TSort n)))) (ex3_2 TList nat (\lambda (ws: -TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i))))) -(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c (TLRef i))))) (ex4_2 T T (\lambda (v: -T).(\lambda (_: T).(eq T t (THead (Bind Abst) w v)))) (\lambda (_: -T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g -(CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c -(Bind Abst) w) v)))) (\lambda (H4: (ex3_2 T T (\lambda (w0: T).(\lambda (u0: -T).(eq T t (THead (Bind Abst) w0 u0)))) (\lambda (w0: T).(\lambda (_: T).(nf2 -c w0))) (\lambda (w0: T).(\lambda (u0: T).(nf2 (CHead c (Bind Abst) w0) -u0))))).(ex3_2_ind T T (\lambda (w0: T).(\lambda (u0: T).(eq T t (THead (Bind -Abst) w0 u0)))) (\lambda (w0: T).(\lambda (_: T).(nf2 c w0))) (\lambda (w0: -T).(\lambda (u0: T).(nf2 (CHead c (Bind Abst) w0) u0))) (ex4_2 T T (\lambda -(v: T).(\lambda (_: T).(eq T t (THead (Bind Abst) w v)))) (\lambda (_: -T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g -(CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c -(Bind Abst) w) v)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T t -(THead (Bind Abst) x0 x1))).(\lambda (H6: (nf2 c x0)).(\lambda (H7: (nf2 -(CHead c (Bind Abst) x0) x1)).(let H8 \def (eq_ind T t (\lambda (t0: T).(ty3 -g c t0 (THead (Bind Abst) w u))) H (THead (Bind Abst) x0 x1) H5) in (eq_ind_r -T (THead (Bind Abst) x0 x1) (\lambda (t0: T).(ex4_2 T T (\lambda (v: -T).(\lambda (_: T).(eq T t0 (THead (Bind Abst) w v)))) (\lambda (_: -T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g -(CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c -(Bind Abst) w) v))))) (ex_ind T (\lambda (t0: T).(ty3 g c (THead (Bind Abst) -w u) t0)) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (THead (Bind Abst) -x0 x1) (THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w -w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u))) -(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v)))) (\lambda -(x: T).(\lambda (H9: (ty3 g c (THead (Bind Abst) w u) x)).(ex3_2_ind T T -(\lambda (t2: T).(\lambda (_: T).(pc3 c (THead (Bind Abst) w t2) x))) -(\lambda (_: T).(\lambda (t0: T).(ty3 g c w t0))) (\lambda (t2: T).(\lambda -(_: T).(ty3 g (CHead c (Bind Abst) w) u t2))) (ex4_2 T T (\lambda (v: -T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) w v)))) -(\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda -(_: T).(ty3 g (CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: -T).(nf2 (CHead c (Bind Abst) w) v)))) (\lambda (x2: T).(\lambda (x3: -T).(\lambda (_: (pc3 c (THead (Bind Abst) w x2) x)).(\lambda (H11: (ty3 g c w -x3)).(\lambda (H12: (ty3 g (CHead c (Bind Abst) w) u x2)).(ex3_2_ind T T -(\lambda (t2: T).(\lambda (_: T).(pc3 c (THead (Bind Abst) x0 t2) (THead -(Bind Abst) w u)))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c x0 t0))) -(\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x0) x1 t2))) -(ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) -(THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) -(\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u))) -(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v)))) (\lambda -(x4: T).(\lambda (x5: T).(\lambda (H13: (pc3 c (THead (Bind Abst) x0 x4) -(THead (Bind Abst) w u))).(\lambda (_: (ty3 g c x0 x5)).(\lambda (H15: (ty3 g -(CHead c (Bind Abst) x0) x1 x4)).(land_ind (pc3 c x0 w) (\forall (b: -B).(\forall (u0: T).(pc3 (CHead c (Bind b) u0) x4 u))) (ex4_2 T T (\lambda -(v: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) w -v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: -T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u))) (\lambda (v: -T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v)))) (\lambda (H16: (pc3 c -x0 w)).(\lambda (H17: ((\forall (b: B).(\forall (u0: T).(pc3 (CHead c (Bind -b) u0) x4 u))))).(let H_y \def (pc3_nf2 c x0 w H16 H6 H1) in (let H18 \def -(eq_ind T x0 (\lambda (t0: T).(ty3 g (CHead c (Bind Abst) t0) x1 x4)) H15 w -H_y) in (let H19 \def (eq_ind T x0 (\lambda (t0: T).(nf2 (CHead c (Bind Abst) -t0) x1)) H7 w H_y) in (eq_ind_r T w (\lambda (t0: T).(ex4_2 T T (\lambda (v: -T).(\lambda (_: T).(eq T (THead (Bind Abst) t0 x1) (THead (Bind Abst) w v)))) -(\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda -(_: T).(ty3 g (CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: -T).(nf2 (CHead c (Bind Abst) w) v))))) (ex4_2_intro T T (\lambda (v: -T).(\lambda (_: T).(eq T (THead (Bind Abst) w x1) (THead (Bind Abst) w v)))) -(\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda -(_: T).(ty3 g (CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: -T).(nf2 (CHead c (Bind Abst) w) v))) x1 x3 (refl_equal T (THead (Bind Abst) w -x1)) H11 (ty3_conv g (CHead c (Bind Abst) w) u x2 H12 x1 x4 H18 (H17 Abst w)) -H19) x0 H_y)))))) (pc3_gen_abst c x0 w x4 u H13))))))) (ty3_gen_bind g Abst c -x0 x1 (THead (Bind Abst) w u) H8))))))) (ty3_gen_bind g Abst c w u x H9)))) -(ty3_correct g c (THead (Bind Abst) x0 x1) (THead (Bind Abst) w u) H8)) t -H5))))))) H4)) (\lambda (H4: (ex nat (\lambda (n: nat).(eq T t (TSort -n))))).(ex_ind nat (\lambda (n: nat).(eq T t (TSort n))) (ex4_2 T T (\lambda -(v: T).(\lambda (_: T).(eq T t (THead (Bind Abst) w v)))) (\lambda (_: -T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g -(CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c -(Bind Abst) w) v)))) (\lambda (x: nat).(\lambda (H5: (eq T t (TSort x))).(let -H6 \def (eq_ind T t (\lambda (t0: T).(ty3 g c t0 (THead (Bind Abst) w u))) H -(TSort x) H5) in (eq_ind_r T (TSort x) (\lambda (t0: T).(ex4_2 T T (\lambda -(v: T).(\lambda (_: T).(eq T t0 (THead (Bind Abst) w v)))) (\lambda (_: -T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g -(CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c -(Bind Abst) w) v))))) (pc3_gen_sort_abst c w u (next g x) (ty3_gen_sort g c -(THead (Bind Abst) w u) x H6) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq -T (TSort x) (THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 -g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v -u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v))))) t -H5)))) H4)) (\lambda (H4: (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: -nat).(eq T t (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: -TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: -nat).(nf2 c (TLRef i)))))).(ex3_2_ind TList nat (\lambda (ws: TList).(\lambda -(i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: -TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: -nat).(nf2 c (TLRef i)))) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T t -(THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) -(\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u))) -(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v)))) (\lambda -(x0: TList).(\lambda (x1: nat).(\lambda (H5: (eq T t (THeads (Flat Appl) x0 -(TLRef x1)))).(\lambda (_: (nfs2 c x0)).(\lambda (H7: (nf2 c (TLRef -x1))).(let H8 \def (eq_ind T t (\lambda (t0: T).(ty3 g c t0 (THead (Bind -Abst) w u))) H (THeads (Flat Appl) x0 (TLRef x1)) H5) in (eq_ind_r T (THeads -(Flat Appl) x0 (TLRef x1)) (\lambda (t0: T).(ex4_2 T T (\lambda (v: -T).(\lambda (_: T).(eq T t0 (THead (Bind Abst) w v)))) (\lambda (_: -T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g -(CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c -(Bind Abst) w) v))))) (let H9 \def H2 in ((let H10 \def H8 in (unintro T u -(\lambda (t0: T).((ty3 g c (THeads (Flat Appl) x0 (TLRef x1)) (THead (Bind -Abst) w t0)) \to ((ty3_nf2_inv_abst_premise c w t0) \to (ex4_2 T T (\lambda -(v: T).(\lambda (_: T).(eq T (THeads (Flat Appl) x0 (TLRef x1)) (THead (Bind -Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: -T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v t0))) (\lambda (v: -T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v))))))) (unintro T w -(\lambda (t0: T).(\forall (x: T).((ty3 g c (THeads (Flat Appl) x0 (TLRef x1)) -(THead (Bind Abst) t0 x)) \to ((ty3_nf2_inv_abst_premise c t0 x) \to (ex4_2 T -T (\lambda (v: T).(\lambda (_: T).(eq T (THeads (Flat Appl) x0 (TLRef x1)) -(THead (Bind Abst) t0 v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c t0 -w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) t0) v x))) -(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) t0) v)))))))) -(TList_ind (\lambda (t0: TList).(\forall (x: T).(\forall (x2: T).((ty3 g c -(THeads (Flat Appl) t0 (TLRef x1)) (THead (Bind Abst) x x2)) \to -((ty3_nf2_inv_abst_premise c x x2) \to (ex4_2 T T (\lambda (v: T).(\lambda -(_: T).(eq T (THeads (Flat Appl) t0 (TLRef x1)) (THead (Bind Abst) x v)))) -(\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda -(_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_: -T).(nf2 (CHead c (Bind Abst) x) v))))))))) (\lambda (x: T).(\lambda (x2: -T).(\lambda (H11: (ty3 g c (TLRef x1) (THead (Bind Abst) x x2))).(\lambda -(H12: (ty3_nf2_inv_abst_premise c x x2)).(or_ind (ex3_3 C T T (\lambda (_: -C).(\lambda (_: T).(\lambda (t0: T).(pc3 c (lift (S x1) O t0) (THead (Bind -Abst) x x2))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl x1 c -(CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: -T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda -(_: T).(pc3 c (lift (S x1) O u0) (THead (Bind Abst) x x2))))) (\lambda (e: -C).(\lambda (u0: T).(\lambda (_: T).(getl x1 c (CHead e (Bind Abst) u0))))) -(\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex4_2 -T T (\lambda (v: T).(\lambda (_: T).(eq T (TLRef x1) (THead (Bind Abst) x -v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: -T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v: -T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) v)))) (\lambda (H13: (ex3_3 C -T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c (lift (S x1) O -t0) (THead (Bind Abst) x x2))))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(_: T).(getl x1 c (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (t0: T).(ty3 g e u0 t0)))))).(ex3_3_ind C T T (\lambda (_: -C).(\lambda (_: T).(\lambda (t0: T).(pc3 c (lift (S x1) O t0) (THead (Bind -Abst) x x2))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl x1 c -(CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: -T).(ty3 g e u0 t0)))) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (TLRef -x1) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c x -w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) -(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) v)))) (\lambda -(x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (pc3 c (lift (S x1) O -x5) (THead (Bind Abst) x x2))).(\lambda (H15: (getl x1 c (CHead x3 (Bind -Abbr) x4))).(\lambda (_: (ty3 g x3 x4 x5)).(nf2_gen_lref c x3 x4 x1 H15 H7 -(ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (TLRef x1) (THead (Bind -Abst) x v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: -T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v: -T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) v))))))))))) H13)) (\lambda -(H13: (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c -(lift (S x1) O u0) (THead (Bind Abst) x x2))))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (_: T).(getl x1 c (CHead e (Bind Abst) u0))))) (\lambda (e: -C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))))).(ex3_3_ind C T T -(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c (lift (S x1) O u0) -(THead (Bind Abst) x x2))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: -T).(getl x1 c (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (t0: T).(ty3 g e u0 t0)))) (ex4_2 T T (\lambda (v: T).(\lambda -(_: T).(eq T (TLRef x1) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda -(w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c -(Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind -Abst) x) v)))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda -(H14: (pc3 c (lift (S x1) O x4) (THead (Bind Abst) x x2))).(\lambda (H15: -(getl x1 c (CHead x3 (Bind Abst) x4))).(\lambda (_: (ty3 g x3 x4 x5)).(let -H_x0 \def (H12 x3 x4 x1 H15 TNil H14) in (let H17 \def H_x0 in (False_ind -(ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (TLRef x1) (THead (Bind -Abst) x v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: -T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v: -T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) v)))) H17))))))))) H13)) -(ty3_gen_lref g c (THead (Bind Abst) x x2) x1 H11)))))) (\lambda (t0: -T).(\lambda (t1: TList).(\lambda (H11: ((\forall (x: T).(\forall (x2: -T).((ty3 g c (THeads (Flat Appl) t1 (TLRef x1)) (THead (Bind Abst) x x2)) \to -((ty3_nf2_inv_abst_premise c x x2) \to (ex4_2 T T (\lambda (v: T).(\lambda -(_: T).(eq T (THeads (Flat Appl) t1 (TLRef x1)) (THead (Bind Abst) x v)))) -(\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda -(_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_: -T).(nf2 (CHead c (Bind Abst) x) v)))))))))).(\lambda (x: T).(\lambda (x2: -T).(\lambda (H12: (ty3 g c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 -(TLRef x1))) (THead (Bind Abst) x x2))).(\lambda (H13: -(ty3_nf2_inv_abst_premise c x x2)).(ex3_2_ind T T (\lambda (u0: T).(\lambda -(t2: T).(pc3 c (THead (Flat Appl) t0 (THead (Bind Abst) u0 t2)) (THead (Bind -Abst) x x2)))) (\lambda (u0: T).(\lambda (t2: T).(ty3 g c (THeads (Flat Appl) -t1 (TLRef x1)) (THead (Bind Abst) u0 t2)))) (\lambda (u0: T).(\lambda (_: -T).(ty3 g c t0 u0))) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (THead -(Flat Appl) t0 (THeads (Flat Appl) t1 (TLRef x1))) (THead (Bind Abst) x v)))) -(\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda -(_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_: -T).(nf2 (CHead c (Bind Abst) x) v)))) (\lambda (x3: T).(\lambda (x4: -T).(\lambda (H14: (pc3 c (THead (Flat Appl) t0 (THead (Bind Abst) x3 x4)) -(THead (Bind Abst) x x2))).(\lambda (H15: (ty3 g c (THeads (Flat Appl) t1 -(TLRef x1)) (THead (Bind Abst) x3 x4))).(\lambda (_: (ty3 g c t0 x3)).(let -H_y \def (ty3_nf2_gen__ty3_nf2_inv_abst_aux c x x2 H13 t0 x3 x4 H14) in (let -H_x0 \def (H11 x3 x4 H15 H_y) in (let H17 \def H_x0 in (ex4_2_ind T T -(\lambda (v: T).(\lambda (_: T).(eq T (THeads (Flat Appl) t1 (TLRef x1)) -(THead (Bind Abst) x3 v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c x3 -w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x3) v x4))) -(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x3) v))) (ex4_2 T T -(\lambda (v: T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (THeads (Flat -Appl) t1 (TLRef x1))) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda -(w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c -(Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind -Abst) x) v)))) (\lambda (x5: T).(\lambda (x6: T).(\lambda (H18: (eq T (THeads -(Flat Appl) t1 (TLRef x1)) (THead (Bind Abst) x3 x5))).(\lambda (_: (ty3 g c -x3 x6)).(\lambda (_: (ty3 g (CHead c (Bind Abst) x3) x5 x4)).(\lambda (_: -(nf2 (CHead c (Bind Abst) x3) x5)).(TList_ind (\lambda (t2: TList).((eq T -(THeads (Flat Appl) t2 (TLRef x1)) (THead (Bind Abst) x3 x5)) \to (ex4_2 T T -(\lambda (v: T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (THeads (Flat -Appl) t2 (TLRef x1))) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda -(w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c -(Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind -Abst) x) v)))))) (\lambda (H22: (eq T (THeads (Flat Appl) TNil (TLRef x1)) -(THead (Bind Abst) x3 x5))).(let H23 \def (eq_ind T (TLRef x1) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I -(THead (Bind Abst) x3 x5) H22) in (False_ind (ex4_2 T T (\lambda (v: -T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (THeads (Flat Appl) TNil -(TLRef x1))) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 -g c x w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v -x2))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) v)))) -H23))) (\lambda (t2: T).(\lambda (t3: TList).(\lambda (_: (((eq T (THeads -(Flat Appl) t3 (TLRef x1)) (THead (Bind Abst) x3 x5)) \to (ex4_2 T T (\lambda -(v: T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (THeads (Flat Appl) t3 -(TLRef x1))) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 -g c x w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v -x2))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) -v))))))).(\lambda (H22: (eq T (THeads (Flat Appl) (TCons t2 t3) (TLRef x1)) -(THead (Bind Abst) x3 x5))).(let H23 \def (eq_ind T (THead (Flat Appl) t2 -(THeads (Flat Appl) t3 (TLRef x1))) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind Abst) x3 x5) H22) in (False_ind (ex4_2 T T (\lambda -(v: T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (THeads (Flat Appl) (TCons -t2 t3) (TLRef x1))) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda (w0: -T).(ty3 g c x w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind -Abst) x) v x2))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) -v)))) H23)))))) t1 H18))))))) H17))))))))) (ty3_gen_appl g c t0 (THeads (Flat -Appl) t1 (TLRef x1)) (THead (Bind Abst) x x2) H12))))))))) x0)) H10)) H9)) t -H5))))))) H4)) H3))))))))))). -(* COMMENTS -Initial nodes: 5333 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/pr3.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/pr3.ma deleted file mode 100644 index 8d184aa58..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/pr3.ma +++ /dev/null @@ -1,728 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubt/ty3.ma". - -include "Basic-1/ty3/subst1.ma". - -include "Basic-1/ty3/fsubst0.ma". - -include "Basic-1/pc3/pc1.ma". - -include "Basic-1/pc3/wcpr0.ma". - -include "Basic-1/pc1/props.ma". - -theorem ty3_sred_wcpr0_pr0: - \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t: T).((ty3 g c1 -t1 t) \to (\forall (c2: C).((wcpr0 c1 c2) \to (\forall (t2: T).((pr0 t1 t2) -\to (ty3 g c2 t2 t))))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t: T).(\lambda -(H: (ty3 g c1 t1 t)).(ty3_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda -(t2: T).(\forall (c2: C).((wcpr0 c c2) \to (\forall (t3: T).((pr0 t0 t3) \to -(ty3 g c2 t3 t2)))))))) (\lambda (c: C).(\lambda (t2: T).(\lambda (t0: -T).(\lambda (_: (ty3 g c t2 t0)).(\lambda (H1: ((\forall (c2: C).((wcpr0 c -c2) \to (\forall (t3: T).((pr0 t2 t3) \to (ty3 g c2 t3 t0))))))).(\lambda (u: -T).(\lambda (t3: T).(\lambda (_: (ty3 g c u t3)).(\lambda (H3: ((\forall (c2: -C).((wcpr0 c c2) \to (\forall (t4: T).((pr0 u t4) \to (ty3 g c2 t4 -t3))))))).(\lambda (H4: (pc3 c t3 t2)).(\lambda (c2: C).(\lambda (H5: (wcpr0 -c c2)).(\lambda (t4: T).(\lambda (H6: (pr0 u t4)).(ty3_conv g c2 t2 t0 (H1 c2 -H5 t2 (pr0_refl t2)) t4 t3 (H3 c2 H5 t4 H6) (pc3_wcpr0 c c2 H5 t3 t2 -H4)))))))))))))))) (\lambda (c: C).(\lambda (m: nat).(\lambda (c2: -C).(\lambda (_: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H1: (pr0 (TSort m) -t2)).(eq_ind_r T (TSort m) (\lambda (t0: T).(ty3 g c2 t0 (TSort (next g m)))) -(ty3_sort g c2 m) t2 (pr0_gen_sort t2 m H1)))))))) (\lambda (n: nat).(\lambda -(c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind -Abbr) u))).(\lambda (t0: T).(\lambda (_: (ty3 g d u t0)).(\lambda (H2: -((\forall (c2: C).((wcpr0 d c2) \to (\forall (t2: T).((pr0 u t2) \to (ty3 g -c2 t2 t0))))))).(\lambda (c2: C).(\lambda (H3: (wcpr0 c c2)).(\lambda (t2: -T).(\lambda (H4: (pr0 (TLRef n) t2)).(eq_ind_r T (TLRef n) (\lambda (t3: -T).(ty3 g c2 t3 (lift (S n) O t0))) (ex3_2_ind C T (\lambda (e2: C).(\lambda -(u2: T).(getl n c2 (CHead e2 (Bind Abbr) u2)))) (\lambda (e2: C).(\lambda (_: -T).(wcpr0 d e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u u2))) (ty3 g c2 -(TLRef n) (lift (S n) O t0)) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: -(getl n c2 (CHead x0 (Bind Abbr) x1))).(\lambda (H6: (wcpr0 d x0)).(\lambda -(H7: (pr0 u x1)).(ty3_abbr g n c2 x0 x1 H5 t0 (H2 x0 H6 x1 H7))))))) -(wcpr0_getl c c2 H3 n d u (Bind Abbr) H0)) t2 (pr0_gen_lref t2 n -H4)))))))))))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d: C).(\lambda -(u: T).(\lambda (H0: (getl n c (CHead d (Bind Abst) u))).(\lambda (t0: -T).(\lambda (_: (ty3 g d u t0)).(\lambda (H2: ((\forall (c2: C).((wcpr0 d c2) -\to (\forall (t2: T).((pr0 u t2) \to (ty3 g c2 t2 t0))))))).(\lambda (c2: -C).(\lambda (H3: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H4: (pr0 (TLRef n) -t2)).(eq_ind_r T (TLRef n) (\lambda (t3: T).(ty3 g c2 t3 (lift (S n) O u))) -(ex3_2_ind C T (\lambda (e2: C).(\lambda (u2: T).(getl n c2 (CHead e2 (Bind -Abst) u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 d e2))) (\lambda (_: -C).(\lambda (u2: T).(pr0 u u2))) (ty3 g c2 (TLRef n) (lift (S n) O u)) -(\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: (getl n c2 (CHead x0 (Bind -Abst) x1))).(\lambda (H6: (wcpr0 d x0)).(\lambda (H7: (pr0 u x1)).(ty3_conv g -c2 (lift (S n) O u) (lift (S n) O t0) (ty3_lift g x0 u t0 (H2 x0 H6 u -(pr0_refl u)) c2 O (S n) (getl_drop Abst c2 x0 x1 n H5)) (TLRef n) (lift (S -n) O x1) (ty3_abst g n c2 x0 x1 H5 t0 (H2 x0 H6 x1 H7)) (pc3_lift c2 x0 (S n) -O (getl_drop Abst c2 x0 x1 n H5) x1 u (pc3_pr2_x x0 x1 u (pr2_free x0 u x1 -H7))))))))) (wcpr0_getl c c2 H3 n d u (Bind Abst) H0)) t2 (pr0_gen_lref t2 n -H4)))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (t0: T).(\lambda -(_: (ty3 g c u t0)).(\lambda (H1: ((\forall (c2: C).((wcpr0 c c2) \to -(\forall (t2: T).((pr0 u t2) \to (ty3 g c2 t2 t0))))))).(\lambda (b: -B).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H2: (ty3 g (CHead c (Bind b) -u) t2 t3)).(\lambda (H3: ((\forall (c2: C).((wcpr0 (CHead c (Bind b) u) c2) -\to (\forall (t4: T).((pr0 t2 t4) \to (ty3 g c2 t4 t3))))))).(\lambda (c2: -C).(\lambda (H4: (wcpr0 c c2)).(\lambda (t4: T).(\lambda (H5: (pr0 (THead -(Bind b) u t2) t4)).(let H6 \def (match H5 in pr0 return (\lambda (t5: -T).(\lambda (t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 (THead (Bind b) u -t2)) \to ((eq T t6 t4) \to (ty3 g c2 t4 (THead (Bind b) u t3))))))) with -[(pr0_refl t5) \Rightarrow (\lambda (H6: (eq T t5 (THead (Bind b) u -t2))).(\lambda (H7: (eq T t5 t4)).(eq_ind T (THead (Bind b) u t2) (\lambda -(t6: T).((eq T t6 t4) \to (ty3 g c2 t4 (THead (Bind b) u t3)))) (\lambda (H8: -(eq T (THead (Bind b) u t2) t4)).(eq_ind T (THead (Bind b) u t2) (\lambda -(t6: T).(ty3 g c2 t6 (THead (Bind b) u t3))) (ty3_bind g c2 u t0 (H1 c2 H4 u -(pr0_refl u)) b t2 t3 (H3 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H4 u u -(pr0_refl u) (Bind b)) t2 (pr0_refl t2))) t4 H8)) t5 (sym_eq T t5 (THead -(Bind b) u t2) H6) H7))) | (pr0_comp u1 u2 H6 t5 t6 H7 k) \Rightarrow -(\lambda (H8: (eq T (THead k u1 t5) (THead (Bind b) u t2))).(\lambda (H9: (eq -T (THead k u2 t6) t4)).((let H10 \def (f_equal T T (\lambda (e: T).(match e -in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t5 | (TLRef _) -\Rightarrow t5 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t5) (THead -(Bind b) u t2) H8) in ((let H11 \def (f_equal T T (\lambda (e: T).(match e in -T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) -\Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t5) (THead -(Bind b) u t2) H8) in ((let H12 \def (f_equal T K (\lambda (e: T).(match e in -T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) -\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t5) (THead (Bind -b) u t2) H8) in (eq_ind K (Bind b) (\lambda (k0: K).((eq T u1 u) \to ((eq T -t5 t2) \to ((eq T (THead k0 u2 t6) t4) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to -(ty3 g c2 t4 (THead (Bind b) u t3)))))))) (\lambda (H13: (eq T u1 u)).(eq_ind -T u (\lambda (t7: T).((eq T t5 t2) \to ((eq T (THead (Bind b) u2 t6) t4) \to -((pr0 t7 u2) \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3))))))) -(\lambda (H14: (eq T t5 t2)).(eq_ind T t2 (\lambda (t7: T).((eq T (THead -(Bind b) u2 t6) t4) \to ((pr0 u u2) \to ((pr0 t7 t6) \to (ty3 g c2 t4 (THead -(Bind b) u t3)))))) (\lambda (H15: (eq T (THead (Bind b) u2 t6) t4)).(eq_ind -T (THead (Bind b) u2 t6) (\lambda (t7: T).((pr0 u u2) \to ((pr0 t2 t6) \to -(ty3 g c2 t7 (THead (Bind b) u t3))))) (\lambda (H16: (pr0 u u2)).(\lambda -(H17: (pr0 t2 t6)).(ex_ind T (\lambda (t7: T).(ty3 g (CHead c2 (Bind b) u) t3 -t7)) (ty3 g c2 (THead (Bind b) u2 t6) (THead (Bind b) u t3)) (\lambda (x: -T).(\lambda (H18: (ty3 g (CHead c2 (Bind b) u) t3 x)).(ex_ind T (\lambda (t7: -T).(ty3 g (CHead c2 (Bind b) u2) t3 t7)) (ty3 g c2 (THead (Bind b) u2 t6) -(THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (_: (ty3 g (CHead c2 (Bind -b) u2) t3 x0)).(ty3_conv g c2 (THead (Bind b) u t3) (THead (Bind b) u x) -(ty3_bind g c2 u t0 (H1 c2 H4 u (pr0_refl u)) b t3 x H18) (THead (Bind b) u2 -t6) (THead (Bind b) u2 t3) (ty3_bind g c2 u2 t0 (H1 c2 H4 u2 H16) b t6 t3 (H3 -(CHead c2 (Bind b) u2) (wcpr0_comp c c2 H4 u u2 H16 (Bind b)) t6 H17)) -(pc3_pr2_x c2 (THead (Bind b) u2 t3) (THead (Bind b) u t3) (pr2_head_1 c2 u -u2 (pr2_free c2 u u2 H16) (Bind b) t3))))) (ty3_correct g (CHead c2 (Bind b) -u2) t6 t3 (H3 (CHead c2 (Bind b) u2) (wcpr0_comp c c2 H4 u u2 H16 (Bind b)) -t6 H17))))) (ty3_correct g (CHead c2 (Bind b) u) t2 t3 (H3 (CHead c2 (Bind b) -u) (wcpr0_comp c c2 H4 u u (pr0_refl u) (Bind b)) t2 (pr0_refl t2)))))) t4 -H15)) t5 (sym_eq T t5 t2 H14))) u1 (sym_eq T u1 u H13))) k (sym_eq K k (Bind -b) H12))) H11)) H10)) H9 H6 H7))) | (pr0_beta u0 v1 v2 H6 t5 t6 H7) -\Rightarrow (\lambda (H8: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u0 -t5)) (THead (Bind b) u t2))).(\lambda (H9: (eq T (THead (Bind Abbr) v2 t6) -t4)).((let H10 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u0 -t5)) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort -_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) -\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u t2) -H8) in (False_ind ((eq T (THead (Bind Abbr) v2 t6) t4) \to ((pr0 v1 v2) \to -((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3))))) H10)) H9 H6 H7))) | -(pr0_upsilon b0 H6 v1 v2 H7 u1 u2 H8 t5 t6 H9) \Rightarrow (\lambda (H10: (eq -T (THead (Flat Appl) v1 (THead (Bind b0) u1 t5)) (THead (Bind b) u -t2))).(\lambda (H11: (eq T (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) -O v2) t6)) t4)).((let H12 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind -b0) u1 t5)) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) -\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u t2) -H10) in (False_ind ((eq T (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) -O v2) t6)) t4) \to ((not (eq B b0 Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) -\to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3))))))) H12)) H11 H6 H7 -H8 H9))) | (pr0_delta u1 u2 H6 t5 t6 H7 w H8) \Rightarrow (\lambda (H9: (eq T -(THead (Bind Abbr) u1 t5) (THead (Bind b) u t2))).(\lambda (H10: (eq T (THead -(Bind Abbr) u2 w) t4)).((let H11 \def (f_equal T T (\lambda (e: T).(match e -in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t5 | (TLRef _) -\Rightarrow t5 | (THead _ _ t7) \Rightarrow t7])) (THead (Bind Abbr) u1 t5) -(THead (Bind b) u t2) H9) in ((let H12 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | -(TLRef _) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead (Bind -Abbr) u1 t5) (THead (Bind b) u t2) H9) in ((let H13 \def (f_equal T B -(\lambda (e: T).(match e in T return (\lambda (_: T).B) with [(TSort _) -\Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | (THead k _ _) \Rightarrow -(match k in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | -(Flat _) \Rightarrow Abbr])])) (THead (Bind Abbr) u1 t5) (THead (Bind b) u -t2) H9) in (eq_ind B Abbr (\lambda (b0: B).((eq T u1 u) \to ((eq T t5 t2) \to -((eq T (THead (Bind Abbr) u2 w) t4) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to -((subst0 O u2 t6 w) \to (ty3 g c2 t4 (THead (Bind b0) u t3))))))))) (\lambda -(H14: (eq T u1 u)).(eq_ind T u (\lambda (t7: T).((eq T t5 t2) \to ((eq T -(THead (Bind Abbr) u2 w) t4) \to ((pr0 t7 u2) \to ((pr0 t5 t6) \to ((subst0 O -u2 t6 w) \to (ty3 g c2 t4 (THead (Bind Abbr) u t3)))))))) (\lambda (H15: (eq -T t5 t2)).(eq_ind T t2 (\lambda (t7: T).((eq T (THead (Bind Abbr) u2 w) t4) -\to ((pr0 u u2) \to ((pr0 t7 t6) \to ((subst0 O u2 t6 w) \to (ty3 g c2 t4 -(THead (Bind Abbr) u t3))))))) (\lambda (H16: (eq T (THead (Bind Abbr) u2 w) -t4)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t7: T).((pr0 u u2) \to -((pr0 t2 t6) \to ((subst0 O u2 t6 w) \to (ty3 g c2 t7 (THead (Bind Abbr) u -t3)))))) (\lambda (H17: (pr0 u u2)).(\lambda (H18: (pr0 t2 t6)).(\lambda -(H19: (subst0 O u2 t6 w)).(let H20 \def (eq_ind_r B b (\lambda (b0: -B).(\forall (c3: C).((wcpr0 (CHead c (Bind b0) u) c3) \to (\forall (t7: -T).((pr0 t2 t7) \to (ty3 g c3 t7 t3)))))) H3 Abbr H13) in (let H21 \def -(eq_ind_r B b (\lambda (b0: B).(ty3 g (CHead c (Bind b0) u) t2 t3)) H2 Abbr -H13) in (ex_ind T (\lambda (t7: T).(ty3 g (CHead c2 (Bind Abbr) u) t3 t7)) -(ty3 g c2 (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u t3)) (\lambda (x: -T).(\lambda (H22: (ty3 g (CHead c2 (Bind Abbr) u) t3 x)).(ex_ind T (\lambda -(t7: T).(ty3 g (CHead c2 (Bind Abbr) u2) t3 t7)) (ty3 g c2 (THead (Bind Abbr) -u2 w) (THead (Bind Abbr) u t3)) (\lambda (x0: T).(\lambda (_: (ty3 g (CHead -c2 (Bind Abbr) u2) t3 x0)).(ty3_conv g c2 (THead (Bind Abbr) u t3) (THead -(Bind Abbr) u x) (ty3_bind g c2 u t0 (H1 c2 H4 u (pr0_refl u)) Abbr t3 x H22) -(THead (Bind Abbr) u2 w) (THead (Bind Abbr) u2 t3) (ty3_bind g c2 u2 t0 (H1 -c2 H4 u2 H17) Abbr w t3 (ty3_subst0 g (CHead c2 (Bind Abbr) u2) t6 t3 (H20 -(CHead c2 (Bind Abbr) u2) (wcpr0_comp c c2 H4 u u2 H17 (Bind Abbr)) t6 H18) -c2 u2 O (getl_refl Abbr c2 u2) w H19)) (pc3_pr2_x c2 (THead (Bind Abbr) u2 -t3) (THead (Bind Abbr) u t3) (pr2_head_1 c2 u u2 (pr2_free c2 u u2 H17) (Bind -Abbr) t3))))) (ty3_correct g (CHead c2 (Bind Abbr) u2) t6 t3 (H20 (CHead c2 -(Bind Abbr) u2) (wcpr0_comp c c2 H4 u u2 H17 (Bind Abbr)) t6 H18))))) -(ty3_correct g (CHead c2 (Bind Abbr) u) t2 t3 (H20 (CHead c2 (Bind Abbr) u) -(wcpr0_comp c c2 H4 u u (pr0_refl u) (Bind Abbr)) t2 (pr0_refl t2))))))))) t4 -H16)) t5 (sym_eq T t5 t2 H15))) u1 (sym_eq T u1 u H14))) b H13)) H12)) H11)) -H10 H6 H7 H8))) | (pr0_zeta b0 H6 t5 t6 H7 u0) \Rightarrow (\lambda (H8: (eq -T (THead (Bind b0) u0 (lift (S O) O t5)) (THead (Bind b) u t2))).(\lambda -(H9: (eq T t6 t4)).((let H10 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: -((nat \to nat))) (d: nat) (t7: T) on t7: T \def (match t7 with [(TSort n) -\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with -[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u1 t8) -\Rightarrow (THead k (lref_map f d u1) (lref_map f (s k d) t8))]) in -lref_map) (\lambda (x: nat).(plus x (S O))) O t5) | (TLRef _) \Rightarrow -((let rec lref_map (f: ((nat \to nat))) (d: nat) (t7: T) on t7: T \def (match -t7 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef -(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | -(THead k u1 t8) \Rightarrow (THead k (lref_map f d u1) (lref_map f (s k d) -t8))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t5) | (THead _ _ t7) -\Rightarrow t7])) (THead (Bind b0) u0 (lift (S O) O t5)) (THead (Bind b) u -t2) H8) in ((let H11 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 -| (THead _ t7 _) \Rightarrow t7])) (THead (Bind b0) u0 (lift (S O) O t5)) -(THead (Bind b) u t2) H8) in ((let H12 \def (f_equal T B (\lambda (e: -T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b0 | -(TLRef _) \Rightarrow b0 | (THead k _ _) \Rightarrow (match k in K return -(\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow -b0])])) (THead (Bind b0) u0 (lift (S O) O t5)) (THead (Bind b) u t2) H8) in -(eq_ind B b (\lambda (b1: B).((eq T u0 u) \to ((eq T (lift (S O) O t5) t2) -\to ((eq T t6 t4) \to ((not (eq B b1 Abst)) \to ((pr0 t5 t6) \to (ty3 g c2 t4 -(THead (Bind b) u t3)))))))) (\lambda (H13: (eq T u0 u)).(eq_ind T u (\lambda -(_: T).((eq T (lift (S O) O t5) t2) \to ((eq T t6 t4) \to ((not (eq B b -Abst)) \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3))))))) (\lambda -(H14: (eq T (lift (S O) O t5) t2)).(eq_ind T (lift (S O) O t5) (\lambda (_: -T).((eq T t6 t4) \to ((not (eq B b Abst)) \to ((pr0 t5 t6) \to (ty3 g c2 t4 -(THead (Bind b) u t3)))))) (\lambda (H15: (eq T t6 t4)).(eq_ind T t4 (\lambda -(t7: T).((not (eq B b Abst)) \to ((pr0 t5 t7) \to (ty3 g c2 t4 (THead (Bind -b) u t3))))) (\lambda (H16: (not (eq B b Abst))).(\lambda (H17: (pr0 t5 -t4)).(let H18 \def (eq_ind_r T t2 (\lambda (t7: T).(\forall (c3: C).((wcpr0 -(CHead c (Bind b) u) c3) \to (\forall (t8: T).((pr0 t7 t8) \to (ty3 g c3 t8 -t3)))))) H3 (lift (S O) O t5) H14) in (let H19 \def (eq_ind_r T t2 (\lambda -(t7: T).(ty3 g (CHead c (Bind b) u) t7 t3)) H2 (lift (S O) O t5) H14) in -(ex_ind T (\lambda (t7: T).(ty3 g (CHead c2 (Bind b) u) t3 t7)) (ty3 g c2 t4 -(THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H20: (ty3 g (CHead c2 (Bind -b) u) t3 x)).(B_ind (\lambda (b1: B).((not (eq B b1 Abst)) \to ((ty3 g (CHead -c2 (Bind b1) u) t3 x) \to ((ty3 g (CHead c2 (Bind b1) u) (lift (S O) O t4) -t3) \to (ty3 g c2 t4 (THead (Bind b1) u t3)))))) (\lambda (H21: (not (eq B -Abbr Abst))).(\lambda (H22: (ty3 g (CHead c2 (Bind Abbr) u) t3 x)).(\lambda -(H23: (ty3 g (CHead c2 (Bind Abbr) u) (lift (S O) O t4) t3)).(let H24 \def -(ty3_gen_cabbr g (CHead c2 (Bind Abbr) u) (lift (S O) O t4) t3 H23 c2 u O -(getl_refl Abbr c2 u) (CHead c2 (Bind Abbr) u) (csubst1_refl O u (CHead c2 -(Bind Abbr) u)) c2 (drop_drop (Bind Abbr) O c2 c2 (drop_refl c2) u)) in -(ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(subst1 O u (lift (S O) O t4) -(lift (S O) O y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 O u t3 (lift (S -O) O y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g c2 y1 y2))) (ty3 g c2 t4 -(THead (Bind Abbr) u t3)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H25: -(subst1 O u (lift (S O) O t4) (lift (S O) O x0))).(\lambda (H26: (subst1 O u -t3 (lift (S O) O x1))).(\lambda (H27: (ty3 g c2 x0 x1)).(let H28 \def (eq_ind -T x0 (\lambda (t7: T).(ty3 g c2 t7 x1)) H27 t4 (lift_inj x0 t4 (S O) O -(subst1_gen_lift_eq t4 u (lift (S O) O x0) (S O) O O (le_n O) (eq_ind_r nat -(plus (S O) O) (\lambda (n: nat).(lt O n)) (le_n (plus (S O) O)) (plus O (S -O)) (plus_sym O (S O))) H25))) in (ty3_conv g c2 (THead (Bind Abbr) u t3) -(THead (Bind Abbr) u x) (ty3_bind g c2 u t0 (H1 c2 H4 u (pr0_refl u)) Abbr t3 -x H22) t4 x1 H28 (pc3_pr3_x c2 x1 (THead (Bind Abbr) u t3) (pr3_t (THead -(Bind Abbr) u (lift (S O) O x1)) (THead (Bind Abbr) u t3) c2 (pr3_pr2 c2 -(THead (Bind Abbr) u t3) (THead (Bind Abbr) u (lift (S O) O x1)) (pr2_free c2 -(THead (Bind Abbr) u t3) (THead (Bind Abbr) u (lift (S O) O x1)) (pr0_delta1 -u u (pr0_refl u) t3 t3 (pr0_refl t3) (lift (S O) O x1) H26))) x1 (pr3_pr2 c2 -(THead (Bind Abbr) u (lift (S O) O x1)) x1 (pr2_free c2 (THead (Bind Abbr) u -(lift (S O) O x1)) x1 (pr0_zeta Abbr H21 x1 x1 (pr0_refl x1) u)))))))))))) -H24))))) (\lambda (H21: (not (eq B Abst Abst))).(\lambda (_: (ty3 g (CHead c2 -(Bind Abst) u) t3 x)).(\lambda (_: (ty3 g (CHead c2 (Bind Abst) u) (lift (S -O) O t4) t3)).(let H24 \def (match (H21 (refl_equal B Abst)) in False return -(\lambda (_: False).(ty3 g c2 t4 (THead (Bind Abst) u t3))) with []) in -H24)))) (\lambda (H21: (not (eq B Void Abst))).(\lambda (H22: (ty3 g (CHead -c2 (Bind Void) u) t3 x)).(\lambda (H23: (ty3 g (CHead c2 (Bind Void) u) (lift -(S O) O t4) t3)).(let H24 \def (ty3_gen_cvoid g (CHead c2 (Bind Void) u) -(lift (S O) O t4) t3 H23 c2 u O (getl_refl Void c2 u) c2 (drop_drop (Bind -Void) O c2 c2 (drop_refl c2) u)) in (ex3_2_ind T T (\lambda (y1: T).(\lambda -(_: T).(eq T (lift (S O) O t4) (lift (S O) O y1)))) (\lambda (_: T).(\lambda -(y2: T).(eq T t3 (lift (S O) O y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 -g c2 y1 y2))) (ty3 g c2 t4 (THead (Bind Void) u t3)) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H25: (eq T (lift (S O) O t4) (lift (S O) O -x0))).(\lambda (H26: (eq T t3 (lift (S O) O x1))).(\lambda (H27: (ty3 g c2 x0 -x1)).(let H28 \def (eq_ind T t3 (\lambda (t7: T).(ty3 g (CHead c2 (Bind Void) -u) t7 x)) H22 (lift (S O) O x1) H26) in (eq_ind_r T (lift (S O) O x1) -(\lambda (t7: T).(ty3 g c2 t4 (THead (Bind Void) u t7))) (let H29 \def -(eq_ind_r T x0 (\lambda (t7: T).(ty3 g c2 t7 x1)) H27 t4 (lift_inj t4 x0 (S -O) O H25)) in (ty3_conv g c2 (THead (Bind Void) u (lift (S O) O x1)) (THead -(Bind Void) u x) (ty3_bind g c2 u t0 (H1 c2 H4 u (pr0_refl u)) Void (lift (S -O) O x1) x H28) t4 x1 H29 (pc3_s c2 x1 (THead (Bind Void) u (lift (S O) O -x1)) (pc3_pr2_r c2 (THead (Bind Void) u (lift (S O) O x1)) x1 (pr2_free c2 -(THead (Bind Void) u (lift (S O) O x1)) x1 (pr0_zeta Void H21 x1 x1 (pr0_refl -x1) u)))))) t3 H26))))))) H24))))) b H16 H20 (H18 (CHead c2 (Bind b) u) -(wcpr0_comp c c2 H4 u u (pr0_refl u) (Bind b)) (lift (S O) O t4) (pr0_lift t5 -t4 H17 (S O) O))))) (ty3_correct g (CHead c2 (Bind b) u) (lift (S O) O t4) t3 -(H18 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H4 u u (pr0_refl u) (Bind b)) -(lift (S O) O t4) (pr0_lift t5 t4 H17 (S O) O)))))))) t6 (sym_eq T t6 t4 -H15))) t2 H14)) u0 (sym_eq T u0 u H13))) b0 (sym_eq B b0 b H12))) H11)) H10)) -H9 H6 H7))) | (pr0_tau t5 t6 H6 u0) \Rightarrow (\lambda (H7: (eq T (THead -(Flat Cast) u0 t5) (THead (Bind b) u t2))).(\lambda (H8: (eq T t6 t4)).((let -H9 \def (eq_ind T (THead (Flat Cast) u0 t5) (\lambda (e: T).(match e in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind b) u t2) H7) in (False_ind ((eq T t6 t4) \to ((pr0 -t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3)))) H9)) H8 H6)))]) in (H6 -(refl_equal T (THead (Bind b) u t2)) (refl_equal T t4))))))))))))))))) -(\lambda (c: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c w -u)).(\lambda (H1: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 -w t2) \to (ty3 g c2 t2 u))))))).(\lambda (v: T).(\lambda (t0: T).(\lambda -(H2: (ty3 g c v (THead (Bind Abst) u t0))).(\lambda (H3: ((\forall (c2: -C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 v t2) \to (ty3 g c2 t2 (THead -(Bind Abst) u t0)))))))).(\lambda (c2: C).(\lambda (H4: (wcpr0 c -c2)).(\lambda (t2: T).(\lambda (H5: (pr0 (THead (Flat Appl) w v) t2)).(let H6 -\def (match H5 in pr0 return (\lambda (t3: T).(\lambda (t4: T).(\lambda (_: -(pr0 t3 t4)).((eq T t3 (THead (Flat Appl) w v)) \to ((eq T t4 t2) \to (ty3 g -c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u t0)))))))) with [(pr0_refl -t3) \Rightarrow (\lambda (H6: (eq T t3 (THead (Flat Appl) w v))).(\lambda -(H7: (eq T t3 t2)).(eq_ind T (THead (Flat Appl) w v) (\lambda (t4: T).((eq T -t4 t2) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u t0))))) -(\lambda (H8: (eq T (THead (Flat Appl) w v) t2)).(eq_ind T (THead (Flat Appl) -w v) (\lambda (t4: T).(ty3 g c2 t4 (THead (Flat Appl) w (THead (Bind Abst) u -t0)))) (ty3_appl g c2 w u (H1 c2 H4 w (pr0_refl w)) v t0 (H3 c2 H4 v -(pr0_refl v))) t2 H8)) t3 (sym_eq T t3 (THead (Flat Appl) w v) H6) H7))) | -(pr0_comp u1 u2 H6 t3 t4 H7 k) \Rightarrow (\lambda (H8: (eq T (THead k u1 -t3) (THead (Flat Appl) w v))).(\lambda (H9: (eq T (THead k u2 t4) t2)).((let -H10 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t5) -\Rightarrow t5])) (THead k u1 t3) (THead (Flat Appl) w v) H8) in ((let H11 -\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t5 _) -\Rightarrow t5])) (THead k u1 t3) (THead (Flat Appl) w v) H8) in ((let H12 -\def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) -with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) -\Rightarrow k0])) (THead k u1 t3) (THead (Flat Appl) w v) H8) in (eq_ind K -(Flat Appl) (\lambda (k0: K).((eq T u1 w) \to ((eq T t3 v) \to ((eq T (THead -k0 u2 t4) t2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat -Appl) w (THead (Bind Abst) u t0))))))))) (\lambda (H13: (eq T u1 w)).(eq_ind -T w (\lambda (t5: T).((eq T t3 v) \to ((eq T (THead (Flat Appl) u2 t4) t2) -\to ((pr0 t5 u2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w -(THead (Bind Abst) u t0)))))))) (\lambda (H14: (eq T t3 v)).(eq_ind T v -(\lambda (t5: T).((eq T (THead (Flat Appl) u2 t4) t2) \to ((pr0 w u2) \to -((pr0 t5 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u -t0))))))) (\lambda (H15: (eq T (THead (Flat Appl) u2 t4) t2)).(eq_ind T -(THead (Flat Appl) u2 t4) (\lambda (t5: T).((pr0 w u2) \to ((pr0 v t4) \to -(ty3 g c2 t5 (THead (Flat Appl) w (THead (Bind Abst) u t0)))))) (\lambda -(H16: (pr0 w u2)).(\lambda (H17: (pr0 v t4)).(ex_ind T (\lambda (t5: T).(ty3 -g c2 (THead (Bind Abst) u t0) t5)) (ty3 g c2 (THead (Flat Appl) u2 t4) (THead -(Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x: T).(\lambda (H18: (ty3 -g c2 (THead (Bind Abst) u t0) x)).(ex3_2_ind T T (\lambda (t5: T).(\lambda -(_: T).(pc3 c2 (THead (Bind Abst) u t5) x))) (\lambda (_: T).(\lambda (t6: -T).(ty3 g c2 u t6))) (\lambda (t5: T).(\lambda (_: T).(ty3 g (CHead c2 (Bind -Abst) u) t0 t5))) (ty3 g c2 (THead (Flat Appl) u2 t4) (THead (Flat Appl) w -(THead (Bind Abst) u t0))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: -(pc3 c2 (THead (Bind Abst) u x0) x)).(\lambda (H20: (ty3 g c2 u x1)).(\lambda -(H21: (ty3 g (CHead c2 (Bind Abst) u) t0 x0)).(ty3_conv g c2 (THead (Flat -Appl) w (THead (Bind Abst) u t0)) (THead (Flat Appl) w (THead (Bind Abst) u -x0)) (ty3_appl g c2 w u (H1 c2 H4 w (pr0_refl w)) (THead (Bind Abst) u t0) x0 -(ty3_bind g c2 u x1 H20 Abst t0 x0 H21)) (THead (Flat Appl) u2 t4) (THead -(Flat Appl) u2 (THead (Bind Abst) u t0)) (ty3_appl g c2 u2 u (H1 c2 H4 u2 -H16) t4 t0 (H3 c2 H4 t4 H17)) (pc3_pr2_x c2 (THead (Flat Appl) u2 (THead -(Bind Abst) u t0)) (THead (Flat Appl) w (THead (Bind Abst) u t0)) (pr2_head_1 -c2 w u2 (pr2_free c2 w u2 H16) (Flat Appl) (THead (Bind Abst) u t0))))))))) -(ty3_gen_bind g Abst c2 u t0 x H18)))) (ty3_correct g c2 v (THead (Bind Abst) -u t0) (H3 c2 H4 v (pr0_refl v)))))) t2 H15)) t3 (sym_eq T t3 v H14))) u1 -(sym_eq T u1 w H13))) k (sym_eq K k (Flat Appl) H12))) H11)) H10)) H9 H6 -H7))) | (pr0_beta u0 v1 v2 H6 t3 t4 H7) \Rightarrow (\lambda (H8: (eq T -(THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) (THead (Flat Appl) w -v))).(\lambda (H9: (eq T (THead (Bind Abbr) v2 t4) t2)).((let H10 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow (THead (Bind Abst) u0 t3) | (TLRef _) \Rightarrow -(THead (Bind Abst) u0 t3) | (THead _ _ t5) \Rightarrow t5])) (THead (Flat -Appl) v1 (THead (Bind Abst) u0 t3)) (THead (Flat Appl) w v) H8) in ((let H11 -\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t5 _) -\Rightarrow t5])) (THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) (THead -(Flat Appl) w v) H8) in (eq_ind T w (\lambda (t5: T).((eq T (THead (Bind -Abst) u0 t3) v) \to ((eq T (THead (Bind Abbr) v2 t4) t2) \to ((pr0 t5 v2) \to -((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u -t0)))))))) (\lambda (H12: (eq T (THead (Bind Abst) u0 t3) v)).(eq_ind T -(THead (Bind Abst) u0 t3) (\lambda (_: T).((eq T (THead (Bind Abbr) v2 t4) -t2) \to ((pr0 w v2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w -(THead (Bind Abst) u t0))))))) (\lambda (H13: (eq T (THead (Bind Abbr) v2 t4) -t2)).(eq_ind T (THead (Bind Abbr) v2 t4) (\lambda (t5: T).((pr0 w v2) \to -((pr0 t3 t4) \to (ty3 g c2 t5 (THead (Flat Appl) w (THead (Bind Abst) u -t0)))))) (\lambda (H14: (pr0 w v2)).(\lambda (H15: (pr0 t3 t4)).(let H16 \def -(eq_ind_r T v (\lambda (t5: T).(\forall (c3: C).((wcpr0 c c3) \to (\forall -(t6: T).((pr0 t5 t6) \to (ty3 g c3 t6 (THead (Bind Abst) u t0))))))) H3 -(THead (Bind Abst) u0 t3) H12) in (let H17 \def (eq_ind_r T v (\lambda (t5: -T).(ty3 g c t5 (THead (Bind Abst) u t0))) H2 (THead (Bind Abst) u0 t3) H12) -in (ex_ind T (\lambda (t5: T).(ty3 g c2 (THead (Bind Abst) u t0) t5)) (ty3 g -c2 (THead (Bind Abbr) v2 t4) (THead (Flat Appl) w (THead (Bind Abst) u t0))) -(\lambda (x: T).(\lambda (H18: (ty3 g c2 (THead (Bind Abst) u t0) -x)).(ex3_2_ind T T (\lambda (t5: T).(\lambda (_: T).(pc3 c2 (THead (Bind -Abst) u t5) x))) (\lambda (_: T).(\lambda (t6: T).(ty3 g c2 u t6))) (\lambda -(t5: T).(\lambda (_: T).(ty3 g (CHead c2 (Bind Abst) u) t0 t5))) (ty3 g c2 -(THead (Bind Abbr) v2 t4) (THead (Flat Appl) w (THead (Bind Abst) u t0))) -(\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 c2 (THead (Bind Abst) u -x0) x)).(\lambda (H20: (ty3 g c2 u x1)).(\lambda (H21: (ty3 g (CHead c2 (Bind -Abst) u) t0 x0)).(ex3_2_ind T T (\lambda (t5: T).(\lambda (_: T).(pc3 c2 -(THead (Bind Abst) u0 t5) (THead (Bind Abst) u t0)))) (\lambda (_: -T).(\lambda (t6: T).(ty3 g c2 u0 t6))) (\lambda (t5: T).(\lambda (_: T).(ty3 -g (CHead c2 (Bind Abst) u0) t4 t5))) (ty3 g c2 (THead (Bind Abbr) v2 t4) -(THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x2: T).(\lambda -(x3: T).(\lambda (H22: (pc3 c2 (THead (Bind Abst) u0 x2) (THead (Bind Abst) u -t0))).(\lambda (H23: (ty3 g c2 u0 x3)).(\lambda (H24: (ty3 g (CHead c2 (Bind -Abst) u0) t4 x2)).(land_ind (pc3 c2 u0 u) (\forall (b: B).(\forall (u1: -T).(pc3 (CHead c2 (Bind b) u1) x2 t0))) (ty3 g c2 (THead (Bind Abbr) v2 t4) -(THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (H25: (pc3 c2 u0 -u)).(\lambda (H26: ((\forall (b: B).(\forall (u1: T).(pc3 (CHead c2 (Bind b) -u1) x2 t0))))).(ty3_conv g c2 (THead (Flat Appl) w (THead (Bind Abst) u t0)) -(THead (Flat Appl) w (THead (Bind Abst) u x0)) (ty3_appl g c2 w u (H1 c2 H4 w -(pr0_refl w)) (THead (Bind Abst) u t0) x0 (ty3_bind g c2 u x1 H20 Abst t0 x0 -H21)) (THead (Bind Abbr) v2 t4) (THead (Bind Abbr) v2 x2) (ty3_bind g c2 v2 u -(H1 c2 H4 v2 H14) Abbr t4 x2 (csubt_ty3_ld g c2 v2 u0 (ty3_conv g c2 u0 x3 -H23 v2 u (H1 c2 H4 v2 H14) (pc3_s c2 u u0 H25)) t4 x2 H24)) (pc3_t (THead -(Bind Abbr) v2 t0) c2 (THead (Bind Abbr) v2 x2) (pc3_head_2 c2 v2 x2 t0 (Bind -Abbr) (H26 Abbr v2)) (THead (Flat Appl) w (THead (Bind Abst) u t0)) -(pc3_pr2_x c2 (THead (Bind Abbr) v2 t0) (THead (Flat Appl) w (THead (Bind -Abst) u t0)) (pr2_free c2 (THead (Flat Appl) w (THead (Bind Abst) u t0)) -(THead (Bind Abbr) v2 t0) (pr0_beta u w v2 H14 t0 t0 (pr0_refl t0)))))))) -(pc3_gen_abst c2 u0 u x2 t0 H22))))))) (ty3_gen_bind g Abst c2 u0 t4 (THead -(Bind Abst) u t0) (H16 c2 H4 (THead (Bind Abst) u0 t4) (pr0_comp u0 u0 -(pr0_refl u0) t3 t4 H15 (Bind Abst)))))))))) (ty3_gen_bind g Abst c2 u t0 x -H18)))) (ty3_correct g c2 (THead (Bind Abst) u0 t3) (THead (Bind Abst) u t0) -(H16 c2 H4 (THead (Bind Abst) u0 t3) (pr0_refl (THead (Bind Abst) u0 -t3))))))))) t2 H13)) v H12)) v1 (sym_eq T v1 w H11))) H10)) H9 H6 H7))) | -(pr0_upsilon b H6 v1 v2 H7 u1 u2 H8 t3 t4 H9) \Rightarrow (\lambda (H10: (eq -T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) w -v))).(\lambda (H11: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O -v2) t4)) t2)).((let H12 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow (THead (Bind b) u1 t3) -| (TLRef _) \Rightarrow (THead (Bind b) u1 t3) | (THead _ _ t5) \Rightarrow -t5])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) w v) -H10) in ((let H13 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 -| (THead _ t5 _) \Rightarrow t5])) (THead (Flat Appl) v1 (THead (Bind b) u1 -t3)) (THead (Flat Appl) w v) H10) in (eq_ind T w (\lambda (t5: T).((eq T -(THead (Bind b) u1 t3) v) \to ((eq T (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4)) t2) \to ((not (eq B b Abst)) \to ((pr0 t5 v2) \to -((pr0 u1 u2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead -(Bind Abst) u t0)))))))))) (\lambda (H14: (eq T (THead (Bind b) u1 t3) -v)).(eq_ind T (THead (Bind b) u1 t3) (\lambda (_: T).((eq T (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t2) \to ((not (eq B b Abst)) \to -((pr0 w v2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat -Appl) w (THead (Bind Abst) u t0))))))))) (\lambda (H15: (eq T (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t2)).(eq_ind T (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t4)) (\lambda (t5: T).((not (eq B b -Abst)) \to ((pr0 w v2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ty3 g c2 t5 -(THead (Flat Appl) w (THead (Bind Abst) u t0)))))))) (\lambda (H16: (not (eq -B b Abst))).(\lambda (H17: (pr0 w v2)).(\lambda (H18: (pr0 u1 u2)).(\lambda -(H19: (pr0 t3 t4)).(let H20 \def (eq_ind_r T v (\lambda (t5: T).(\forall (c3: -C).((wcpr0 c c3) \to (\forall (t6: T).((pr0 t5 t6) \to (ty3 g c3 t6 (THead -(Bind Abst) u t0))))))) H3 (THead (Bind b) u1 t3) H14) in (let H21 \def -(eq_ind_r T v (\lambda (t5: T).(ty3 g c t5 (THead (Bind Abst) u t0))) H2 -(THead (Bind b) u1 t3) H14) in (ex_ind T (\lambda (t5: T).(ty3 g c2 (THead -(Bind Abst) u t0) t5)) (ty3 g c2 (THead (Bind b) u2 (THead (Flat Appl) (lift -(S O) O v2) t4)) (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x: -T).(\lambda (H22: (ty3 g c2 (THead (Bind Abst) u t0) x)).(let H23 \def H22 in -(ex3_2_ind T T (\lambda (t5: T).(\lambda (_: T).(pc3 c2 (THead (Bind Abst) u -t5) x))) (\lambda (_: T).(\lambda (t6: T).(ty3 g c2 u t6))) (\lambda (t5: -T).(\lambda (_: T).(ty3 g (CHead c2 (Bind Abst) u) t0 t5))) (ty3 g c2 (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) (THead (Flat Appl) w -(THead (Bind Abst) u t0))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: -(pc3 c2 (THead (Bind Abst) u x0) x)).(\lambda (H25: (ty3 g c2 u x1)).(\lambda -(H26: (ty3 g (CHead c2 (Bind Abst) u) t0 x0)).(ex3_2_ind T T (\lambda (t5: -T).(\lambda (_: T).(pc3 c2 (THead (Bind b) u2 t5) (THead (Bind Abst) u t0)))) -(\lambda (_: T).(\lambda (t6: T).(ty3 g c2 u2 t6))) (\lambda (t5: T).(\lambda -(_: T).(ty3 g (CHead c2 (Bind b) u2) t4 t5))) (ty3 g c2 (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t4)) (THead (Flat Appl) w (THead (Bind -Abst) u t0))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H27: (pc3 c2 (THead -(Bind b) u2 x2) (THead (Bind Abst) u t0))).(\lambda (H28: (ty3 g c2 u2 -x3)).(\lambda (H29: (ty3 g (CHead c2 (Bind b) u2) t4 x2)).(let H30 \def -(eq_ind T (lift (S O) O (THead (Bind Abst) u t0)) (\lambda (t5: T).(pc3 -(CHead c2 (Bind b) u2) x2 t5)) (pc3_gen_not_abst b H16 c2 x2 t0 u2 u H27) -(THead (Bind Abst) (lift (S O) O u) (lift (S O) (S O) t0)) (lift_bind Abst u -t0 (S O) O)) in (let H31 \def (eq_ind T (lift (S O) O (THead (Bind Abst) u -t0)) (\lambda (t5: T).(ty3 g (CHead c2 (Bind b) u2) t5 (lift (S O) O x))) -(ty3_lift g c2 (THead (Bind Abst) u t0) x H22 (CHead c2 (Bind b) u2) O (S O) -(drop_drop (Bind b) O c2 c2 (drop_refl c2) u2)) (THead (Bind Abst) (lift (S -O) O u) (lift (S O) (S O) t0)) (lift_bind Abst u t0 (S O) O)) in (ex3_2_ind T -T (\lambda (t5: T).(\lambda (_: T).(pc3 (CHead c2 (Bind b) u2) (THead (Bind -Abst) (lift (S O) O u) t5) (lift (S O) O x)))) (\lambda (_: T).(\lambda (t6: -T).(ty3 g (CHead c2 (Bind b) u2) (lift (S O) O u) t6))) (\lambda (t5: -T).(\lambda (_: T).(ty3 g (CHead (CHead c2 (Bind b) u2) (Bind Abst) (lift (S -O) O u)) (lift (S O) (S O) t0) t5))) (ty3 g c2 (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t4)) (THead (Flat Appl) w (THead (Bind Abst) u -t0))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (pc3 (CHead c2 (Bind b) -u2) (THead (Bind Abst) (lift (S O) O u) x4) (lift (S O) O x))).(\lambda (H33: -(ty3 g (CHead c2 (Bind b) u2) (lift (S O) O u) x5)).(\lambda (H34: (ty3 g -(CHead (CHead c2 (Bind b) u2) (Bind Abst) (lift (S O) O u)) (lift (S O) (S O) -t0) x4)).(ty3_conv g c2 (THead (Flat Appl) w (THead (Bind Abst) u t0)) (THead -(Flat Appl) w (THead (Bind Abst) u x0)) (ty3_appl g c2 w u (H1 c2 H4 w -(pr0_refl w)) (THead (Bind Abst) u t0) x0 (ty3_bind g c2 u x1 H25 Abst t0 x0 -H26)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (THead (Bind Abst) (lift (S -O) O u) (lift (S O) (S O) t0)))) (ty3_bind g c2 u2 x3 H28 b (THead (Flat -Appl) (lift (S O) O v2) t4) (THead (Flat Appl) (lift (S O) O v2) (THead (Bind -Abst) (lift (S O) O u) (lift (S O) (S O) t0))) (ty3_appl g (CHead c2 (Bind b) -u2) (lift (S O) O v2) (lift (S O) O u) (ty3_lift g c2 v2 u (H1 c2 H4 v2 H17) -(CHead c2 (Bind b) u2) O (S O) (drop_drop (Bind b) O c2 c2 (drop_refl c2) -u2)) t4 (lift (S O) (S O) t0) (ty3_conv g (CHead c2 (Bind b) u2) (THead (Bind -Abst) (lift (S O) O u) (lift (S O) (S O) t0)) (THead (Bind Abst) (lift (S O) -O u) x4) (ty3_bind g (CHead c2 (Bind b) u2) (lift (S O) O u) x5 H33 Abst -(lift (S O) (S O) t0) x4 H34) t4 x2 H29 H30))) (eq_ind T (lift (S O) O (THead -(Bind Abst) u t0)) (\lambda (t5: T).(pc3 c2 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t5)) (THead (Flat Appl) w (THead (Bind Abst) u t0)))) -(pc3_pc1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (lift (S O) -O (THead (Bind Abst) u t0)))) (THead (Flat Appl) w (THead (Bind Abst) u t0)) -(pc1_pr0_u2 (THead (Flat Appl) v2 (THead (Bind b) u2 (lift (S O) O (THead -(Bind Abst) u t0)))) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) -(lift (S O) O (THead (Bind Abst) u t0)))) (pr0_upsilon b H16 v2 v2 (pr0_refl -v2) u2 u2 (pr0_refl u2) (lift (S O) O (THead (Bind Abst) u t0)) (lift (S O) O -(THead (Bind Abst) u t0)) (pr0_refl (lift (S O) O (THead (Bind Abst) u t0)))) -(THead (Flat Appl) w (THead (Bind Abst) u t0)) (pc1_head v2 w (pc1_pr0_x v2 w -H17) (THead (Bind b) u2 (lift (S O) O (THead (Bind Abst) u t0))) (THead (Bind -Abst) u t0) (pc1_pr0_r (THead (Bind b) u2 (lift (S O) O (THead (Bind Abst) u -t0))) (THead (Bind Abst) u t0) (pr0_zeta b H16 (THead (Bind Abst) u t0) -(THead (Bind Abst) u t0) (pr0_refl (THead (Bind Abst) u t0)) u2)) (Flat -Appl))) c2) (THead (Bind Abst) (lift (S O) O u) (lift (S O) (S O) t0)) -(lift_bind Abst u t0 (S O) O)))))))) (ty3_gen_bind g Abst (CHead c2 (Bind b) -u2) (lift (S O) O u) (lift (S O) (S O) t0) (lift (S O) O x) H31))))))))) -(ty3_gen_bind g b c2 u2 t4 (THead (Bind Abst) u t0) (H20 c2 H4 (THead (Bind -b) u2 t4) (pr0_comp u1 u2 H18 t3 t4 H19 (Bind b)))))))))) (ty3_gen_bind g -Abst c2 u t0 x H23))))) (ty3_correct g c2 (THead (Bind b) u2 t4) (THead (Bind -Abst) u t0) (H20 c2 H4 (THead (Bind b) u2 t4) (pr0_comp u1 u2 H18 t3 t4 H19 -(Bind b))))))))))) t2 H15)) v H14)) v1 (sym_eq T v1 w H13))) H12)) H11 H6 H7 -H8 H9))) | (pr0_delta u1 u2 H6 t3 t4 H7 w0 H8) \Rightarrow (\lambda (H9: (eq -T (THead (Bind Abbr) u1 t3) (THead (Flat Appl) w v))).(\lambda (H10: (eq T -(THead (Bind Abbr) u2 w0) t2)).((let H11 \def (eq_ind T (THead (Bind Abbr) u1 -t3) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort -_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) -\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) w v) -H9) in (False_ind ((eq T (THead (Bind Abbr) u2 w0) t2) \to ((pr0 u1 u2) \to -((pr0 t3 t4) \to ((subst0 O u2 t4 w0) \to (ty3 g c2 t2 (THead (Flat Appl) w -(THead (Bind Abst) u t0))))))) H11)) H10 H6 H7 H8))) | (pr0_zeta b H6 t3 t4 -H7 u0) \Rightarrow (\lambda (H8: (eq T (THead (Bind b) u0 (lift (S O) O t3)) -(THead (Flat Appl) w v))).(\lambda (H9: (eq T t4 t2)).((let H10 \def (eq_ind -T (THead (Bind b) u0 (lift (S O) O t3)) (\lambda (e: T).(match e in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow -False])])) I (THead (Flat Appl) w v) H8) in (False_ind ((eq T t4 t2) \to -((not (eq B b Abst)) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w -(THead (Bind Abst) u t0)))))) H10)) H9 H6 H7))) | (pr0_tau t3 t4 H6 u0) -\Rightarrow (\lambda (H7: (eq T (THead (Flat Cast) u0 t3) (THead (Flat Appl) -w v))).(\lambda (H8: (eq T t4 t2)).((let H9 \def (eq_ind T (THead (Flat Cast) -u0 t3) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) -\Rightarrow False | (Flat f) \Rightarrow (match f in F return (\lambda (_: -F).Prop) with [Appl \Rightarrow False | Cast \Rightarrow True])])])) I (THead -(Flat Appl) w v) H7) in (False_ind ((eq T t4 t2) \to ((pr0 t3 t4) \to (ty3 g -c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u t0))))) H9)) H8 H6)))]) in -(H6 (refl_equal T (THead (Flat Appl) w v)) (refl_equal T t2)))))))))))))))) -(\lambda (c: C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (ty3 g c t2 -t3)).(\lambda (H1: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t4: T).((pr0 -t2 t4) \to (ty3 g c2 t4 t3))))))).(\lambda (t0: T).(\lambda (_: (ty3 g c t3 -t0)).(\lambda (H3: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t4: T).((pr0 -t3 t4) \to (ty3 g c2 t4 t0))))))).(\lambda (c2: C).(\lambda (H4: (wcpr0 c -c2)).(\lambda (t4: T).(\lambda (H5: (pr0 (THead (Flat Cast) t3 t2) t4)).(let -H6 \def (match H5 in pr0 return (\lambda (t5: T).(\lambda (t6: T).(\lambda -(_: (pr0 t5 t6)).((eq T t5 (THead (Flat Cast) t3 t2)) \to ((eq T t6 t4) \to -(ty3 g c2 t4 (THead (Flat Cast) t0 t3))))))) with [(pr0_refl t5) \Rightarrow -(\lambda (H6: (eq T t5 (THead (Flat Cast) t3 t2))).(\lambda (H7: (eq T t5 -t4)).(eq_ind T (THead (Flat Cast) t3 t2) (\lambda (t6: T).((eq T t6 t4) \to -(ty3 g c2 t4 (THead (Flat Cast) t0 t3)))) (\lambda (H8: (eq T (THead (Flat -Cast) t3 t2) t4)).(eq_ind T (THead (Flat Cast) t3 t2) (\lambda (t6: T).(ty3 g -c2 t6 (THead (Flat Cast) t0 t3))) (ty3_cast g c2 t2 t3 (H1 c2 H4 t2 (pr0_refl -t2)) t0 (H3 c2 H4 t3 (pr0_refl t3))) t4 H8)) t5 (sym_eq T t5 (THead (Flat -Cast) t3 t2) H6) H7))) | (pr0_comp u1 u2 H6 t5 t6 H7 k) \Rightarrow (\lambda -(H8: (eq T (THead k u1 t5) (THead (Flat Cast) t3 t2))).(\lambda (H9: (eq T -(THead k u2 t6) t4)).((let H10 \def (f_equal T T (\lambda (e: T).(match e in -T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t5 | (TLRef _) -\Rightarrow t5 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t5) (THead -(Flat Cast) t3 t2) H8) in ((let H11 \def (f_equal T T (\lambda (e: T).(match -e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) -\Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t5) (THead -(Flat Cast) t3 t2) H8) in ((let H12 \def (f_equal T K (\lambda (e: T).(match -e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) -\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t5) (THead (Flat -Cast) t3 t2) H8) in (eq_ind K (Flat Cast) (\lambda (k0: K).((eq T u1 t3) \to -((eq T t5 t2) \to ((eq T (THead k0 u2 t6) t4) \to ((pr0 u1 u2) \to ((pr0 t5 -t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3)))))))) (\lambda (H13: (eq T u1 -t3)).(eq_ind T t3 (\lambda (t7: T).((eq T t5 t2) \to ((eq T (THead (Flat -Cast) u2 t6) t4) \to ((pr0 t7 u2) \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead -(Flat Cast) t0 t3))))))) (\lambda (H14: (eq T t5 t2)).(eq_ind T t2 (\lambda -(t7: T).((eq T (THead (Flat Cast) u2 t6) t4) \to ((pr0 t3 u2) \to ((pr0 t7 -t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3)))))) (\lambda (H15: (eq T -(THead (Flat Cast) u2 t6) t4)).(eq_ind T (THead (Flat Cast) u2 t6) (\lambda -(t7: T).((pr0 t3 u2) \to ((pr0 t2 t6) \to (ty3 g c2 t7 (THead (Flat Cast) t0 -t3))))) (\lambda (H16: (pr0 t3 u2)).(\lambda (H17: (pr0 t2 t6)).(ex_ind T -(\lambda (t7: T).(ty3 g c2 t0 t7)) (ty3 g c2 (THead (Flat Cast) u2 t6) (THead -(Flat Cast) t0 t3)) (\lambda (x: T).(\lambda (H18: (ty3 g c2 t0 x)).(ty3_conv -g c2 (THead (Flat Cast) t0 t3) (THead (Flat Cast) x t0) (ty3_cast g c2 t3 t0 -(H3 c2 H4 t3 (pr0_refl t3)) x H18) (THead (Flat Cast) u2 t6) (THead (Flat -Cast) t0 u2) (ty3_cast g c2 t6 u2 (ty3_conv g c2 u2 t0 (H3 c2 H4 u2 H16) t6 -t3 (H1 c2 H4 t6 H17) (pc3_pr2_r c2 t3 u2 (pr2_free c2 t3 u2 H16))) t0 (H3 c2 -H4 u2 H16)) (pc3_s c2 (THead (Flat Cast) t0 u2) (THead (Flat Cast) t0 t3) -(pc3_pr2_r c2 (THead (Flat Cast) t0 t3) (THead (Flat Cast) t0 u2) -(pr2_thin_dx c2 t3 u2 (pr2_free c2 t3 u2 H16) t0 Cast)))))) (ty3_correct g c2 -t3 t0 (H3 c2 H4 t3 (pr0_refl t3)))))) t4 H15)) t5 (sym_eq T t5 t2 H14))) u1 -(sym_eq T u1 t3 H13))) k (sym_eq K k (Flat Cast) H12))) H11)) H10)) H9 H6 -H7))) | (pr0_beta u v1 v2 H6 t5 t6 H7) \Rightarrow (\lambda (H8: (eq T (THead -(Flat Appl) v1 (THead (Bind Abst) u t5)) (THead (Flat Cast) t3 t2))).(\lambda -(H9: (eq T (THead (Bind Abbr) v2 t6) t4)).((let H10 \def (eq_ind T (THead -(Flat Appl) v1 (THead (Bind Abst) u t5)) (\lambda (e: T).(match e in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f -in F return (\lambda (_: F).Prop) with [Appl \Rightarrow True | Cast -\Rightarrow False])])])) I (THead (Flat Cast) t3 t2) H8) in (False_ind ((eq T -(THead (Bind Abbr) v2 t6) t4) \to ((pr0 v1 v2) \to ((pr0 t5 t6) \to (ty3 g c2 -t4 (THead (Flat Cast) t0 t3))))) H10)) H9 H6 H7))) | (pr0_upsilon b H6 v1 v2 -H7 u1 u2 H8 t5 t6 H9) \Rightarrow (\lambda (H10: (eq T (THead (Flat Appl) v1 -(THead (Bind b) u1 t5)) (THead (Flat Cast) t3 t2))).(\lambda (H11: (eq T -(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t6)) t4)).((let H12 -\def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t5)) (\lambda (e: -T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K -return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) -\Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl \Rightarrow -True | Cast \Rightarrow False])])])) I (THead (Flat Cast) t3 t2) H10) in -(False_ind ((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) -t6)) t4) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 -t5 t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3))))))) H12)) H11 H6 H7 H8 -H9))) | (pr0_delta u1 u2 H6 t5 t6 H7 w H8) \Rightarrow (\lambda (H9: (eq T -(THead (Bind Abbr) u1 t5) (THead (Flat Cast) t3 t2))).(\lambda (H10: (eq T -(THead (Bind Abbr) u2 w) t4)).((let H11 \def (eq_ind T (THead (Bind Abbr) u1 -t5) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort -_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) -\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Cast) t3 -t2) H9) in (False_ind ((eq T (THead (Bind Abbr) u2 w) t4) \to ((pr0 u1 u2) -\to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ty3 g c2 t4 (THead (Flat Cast) -t0 t3)))))) H11)) H10 H6 H7 H8))) | (pr0_zeta b H6 t5 t6 H7 u) \Rightarrow -(\lambda (H8: (eq T (THead (Bind b) u (lift (S O) O t5)) (THead (Flat Cast) -t3 t2))).(\lambda (H9: (eq T t6 t4)).((let H10 \def (eq_ind T (THead (Bind b) -u (lift (S O) O t5)) (\lambda (e: T).(match e in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat -Cast) t3 t2) H8) in (False_ind ((eq T t6 t4) \to ((not (eq B b Abst)) \to -((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3))))) H10)) H9 H6 H7))) -| (pr0_tau t5 t6 H6 u) \Rightarrow (\lambda (H7: (eq T (THead (Flat Cast) u -t5) (THead (Flat Cast) t3 t2))).(\lambda (H8: (eq T t6 t4)).((let H9 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow t5 | (TLRef _) \Rightarrow t5 | (THead _ _ t7) -\Rightarrow t7])) (THead (Flat Cast) u t5) (THead (Flat Cast) t3 t2) H7) in -((let H10 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: -T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t7 -_) \Rightarrow t7])) (THead (Flat Cast) u t5) (THead (Flat Cast) t3 t2) H7) -in (eq_ind T t3 (\lambda (_: T).((eq T t5 t2) \to ((eq T t6 t4) \to ((pr0 t5 -t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3)))))) (\lambda (H11: (eq T t5 -t2)).(eq_ind T t2 (\lambda (t7: T).((eq T t6 t4) \to ((pr0 t7 t6) \to (ty3 g -c2 t4 (THead (Flat Cast) t0 t3))))) (\lambda (H12: (eq T t6 t4)).(eq_ind T t4 -(\lambda (t7: T).((pr0 t2 t7) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3)))) -(\lambda (H13: (pr0 t2 t4)).(ex_ind T (\lambda (t7: T).(ty3 g c2 t0 t7)) (ty3 -g c2 t4 (THead (Flat Cast) t0 t3)) (\lambda (x: T).(\lambda (H14: (ty3 g c2 -t0 x)).(ty3_conv g c2 (THead (Flat Cast) t0 t3) (THead (Flat Cast) x t0) -(ty3_cast g c2 t3 t0 (H3 c2 H4 t3 (pr0_refl t3)) x H14) t4 t3 (H1 c2 H4 t4 -H13) (pc3_pr2_x c2 t3 (THead (Flat Cast) t0 t3) (pr2_free c2 (THead (Flat -Cast) t0 t3) t3 (pr0_tau t3 t3 (pr0_refl t3) t0)))))) (ty3_correct g c2 t3 t0 -(H3 c2 H4 t3 (pr0_refl t3))))) t6 (sym_eq T t6 t4 H12))) t5 (sym_eq T t5 t2 -H11))) u (sym_eq T u t3 H10))) H9)) H8 H6)))]) in (H6 (refl_equal T (THead -(Flat Cast) t3 t2)) (refl_equal T t4))))))))))))))) c1 t1 t H))))). -(* COMMENTS -Initial nodes: 14710 -END *) - -theorem ty3_sred_pr0: - \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (g: G).(\forall -(c: C).(\forall (t: T).((ty3 g c t1 t) \to (ty3 g c t2 t))))))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(\lambda (g: -G).(\lambda (c: C).(\lambda (t: T).(\lambda (H0: (ty3 g c t1 -t)).(ty3_sred_wcpr0_pr0 g c t1 t H0 c (wcpr0_refl c) t2 H))))))). -(* COMMENTS -Initial nodes: 47 -END *) - -theorem ty3_sred_pr1: - \forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall (g: G).(\forall -(c: C).(\forall (t: T).((ty3 g c t1 t) \to (ty3 g c t2 t))))))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr1 t1 t2)).(pr1_ind (\lambda -(t: T).(\lambda (t0: T).(\forall (g: G).(\forall (c: C).(\forall (t3: -T).((ty3 g c t t3) \to (ty3 g c t0 t3))))))) (\lambda (t: T).(\lambda (g: -G).(\lambda (c: C).(\lambda (t0: T).(\lambda (H0: (ty3 g c t t0)).H0))))) -(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t4 t3)).(\lambda (t5: -T).(\lambda (_: (pr1 t3 t5)).(\lambda (H2: ((\forall (g: G).(\forall (c: -C).(\forall (t: T).((ty3 g c t3 t) \to (ty3 g c t5 t))))))).(\lambda (g: -G).(\lambda (c: C).(\lambda (t: T).(\lambda (H3: (ty3 g c t4 t)).(H2 g c t -(ty3_sred_pr0 t4 t3 H0 g c t H3)))))))))))) t1 t2 H))). -(* COMMENTS -Initial nodes: 151 -END *) - -theorem ty3_sred_pr2: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall -(g: G).(\forall (t: T).((ty3 g c t1 t) \to (ty3 g c t2 t))))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 -t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (g: -G).(\forall (t3: T).((ty3 g c0 t t3) \to (ty3 g c0 t0 t3))))))) (\lambda (c0: -C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(\lambda (g: -G).(\lambda (t: T).(\lambda (H1: (ty3 g c0 t3 t)).(ty3_sred_wcpr0_pr0 g c0 t3 -t H1 c0 (wcpr0_refl c0) t4 H0)))))))) (\lambda (c0: C).(\lambda (d: -C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind -Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 -t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (g: -G).(\lambda (t0: T).(\lambda (H3: (ty3 g c0 t3 t0)).(ty3_subst0 g c0 t4 t0 -(ty3_sred_wcpr0_pr0 g c0 t3 t0 H3 c0 (wcpr0_refl c0) t4 H1) d u i H0 t -H2)))))))))))))) c t1 t2 H)))). -(* COMMENTS -Initial nodes: 205 -END *) - -theorem ty3_sred_pr3: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall -(g: G).(\forall (t: T).((ty3 g c t1 t) \to (ty3 g c t2 t))))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 -t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (g: G).(\forall -(t3: T).((ty3 g c t t3) \to (ty3 g c t0 t3)))))) (\lambda (t: T).(\lambda (g: -G).(\lambda (t0: T).(\lambda (H0: (ty3 g c t t0)).H0)))) (\lambda (t3: -T).(\lambda (t4: T).(\lambda (H0: (pr2 c t4 t3)).(\lambda (t5: T).(\lambda -(_: (pr3 c t3 t5)).(\lambda (H2: ((\forall (g: G).(\forall (t: T).((ty3 g c -t3 t) \to (ty3 g c t5 t)))))).(\lambda (g: G).(\lambda (t: T).(\lambda (H3: -(ty3 g c t4 t)).(H2 g t (ty3_sred_pr2 c t4 t3 H0 g t H3))))))))))) t1 t2 -H)))). -(* COMMENTS -Initial nodes: 151 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/pr3_props.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/pr3_props.ma deleted file mode 100644 index b5bb1fb69..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/pr3_props.ma +++ /dev/null @@ -1,513 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/ty3/pr3.ma". - -theorem ty3_cred_pr2: - \forall (g: G).(\forall (c: C).(\forall (v1: T).(\forall (v2: T).((pr2 c v1 -v2) \to (\forall (b: B).(\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c -(Bind b) v1) t1 t2) \to (ty3 g (CHead c (Bind b) v2) t1 t2))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda -(H: (pr2 c v1 v2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: -T).(\forall (b: B).(\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c0 (Bind -b) t) t1 t2) \to (ty3 g (CHead c0 (Bind b) t0) t1 t2)))))))) (\lambda (c0: -C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr0 t1 t2)).(\lambda (b: -B).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H1: (ty3 g (CHead c0 (Bind b) -t1) t0 t3)).(ty3_sred_wcpr0_pr0 g (CHead c0 (Bind b) t1) t0 t3 H1 (CHead c0 -(Bind b) t2) (wcpr0_comp c0 c0 (wcpr0_refl c0) t1 t2 H0 (Bind b)) t0 -(pr0_refl t0)))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: -T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr) -u))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H1: (pr0 t1 t2)).(\lambda -(t: T).(\lambda (H2: (subst0 i u t2 t)).(\lambda (b: B).(\lambda (t0: -T).(\lambda (t3: T).(\lambda (H3: (ty3 g (CHead c0 (Bind b) t1) t0 -t3)).(ty3_csubst0 g (CHead c0 (Bind b) t2) t0 t3 (ty3_sred_wcpr0_pr0 g (CHead -c0 (Bind b) t1) t0 t3 H3 (CHead c0 (Bind b) t2) (wcpr0_comp c0 c0 (wcpr0_refl -c0) t1 t2 H1 (Bind b)) t0 (pr0_refl t0)) d u (S i) (getl_clear_bind b (CHead -c0 (Bind b) t2) c0 t2 (clear_bind b c0 t2) (CHead d (Bind Abbr) u) i H0) -(CHead c0 (Bind b) t) (csubst0_snd_bind b i u t2 t H2 c0)))))))))))))))) c v1 -v2 H))))). -(* COMMENTS -Initial nodes: 383 -END *) - -theorem ty3_cred_pr3: - \forall (g: G).(\forall (c: C).(\forall (v1: T).(\forall (v2: T).((pr3 c v1 -v2) \to (\forall (b: B).(\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c -(Bind b) v1) t1 t2) \to (ty3 g (CHead c (Bind b) v2) t1 t2))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda -(H: (pr3 c v1 v2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (b: -B).(\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c (Bind b) t) t1 t2) \to -(ty3 g (CHead c (Bind b) t0) t1 t2))))))) (\lambda (t: T).(\lambda (b: -B).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (ty3 g (CHead c (Bind b) -t) t1 t2)).H0))))) (\lambda (t2: T).(\lambda (t1: T).(\lambda (H0: (pr2 c t1 -t2)).(\lambda (t3: T).(\lambda (_: (pr3 c t2 t3)).(\lambda (H2: ((\forall (b: -B).(\forall (t4: T).(\forall (t5: T).((ty3 g (CHead c (Bind b) t2) t4 t5) \to -(ty3 g (CHead c (Bind b) t3) t4 t5))))))).(\lambda (b: B).(\lambda (t0: -T).(\lambda (t4: T).(\lambda (H3: (ty3 g (CHead c (Bind b) t1) t0 t4)).(H2 b -t0 t4 (ty3_cred_pr2 g c t1 t2 H0 b t0 t4 H3)))))))))))) v1 v2 H))))). -(* COMMENTS -Initial nodes: 215 -END *) - -theorem ty3_gen_lift: - \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (x: T).(\forall (h: -nat).(\forall (d: nat).((ty3 g c (lift h d t1) x) \to (\forall (e: C).((drop -h d c e) \to (ex2 T (\lambda (t2: T).(pc3 c (lift h d t2) x)) (\lambda (t2: -T).(ty3 g e t1 t2))))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (x: T).(\lambda (h: -nat).(\lambda (d: nat).(\lambda (H: (ty3 g c (lift h d t1) x)).(insert_eq T -(lift h d t1) (\lambda (t: T).(ty3 g c t x)) (\lambda (_: T).(\forall (e: -C).((drop h d c e) \to (ex2 T (\lambda (t2: T).(pc3 c (lift h d t2) x)) -(\lambda (t2: T).(ty3 g e t1 t2)))))) (\lambda (y: T).(\lambda (H0: (ty3 g c -y x)).(unintro nat d (\lambda (n: nat).((eq T y (lift h n t1)) \to (\forall -(e: C).((drop h n c e) \to (ex2 T (\lambda (t2: T).(pc3 c (lift h n t2) x)) -(\lambda (t2: T).(ty3 g e t1 t2))))))) (unintro T t1 (\lambda (t: T).(\forall -(x0: nat).((eq T y (lift h x0 t)) \to (\forall (e: C).((drop h x0 c e) \to -(ex2 T (\lambda (t2: T).(pc3 c (lift h x0 t2) x)) (\lambda (t2: T).(ty3 g e t -t2)))))))) (ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: -T).(\forall (x0: T).(\forall (x1: nat).((eq T t (lift h x1 x0)) \to (\forall -(e: C).((drop h x1 c0 e) \to (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) -t0)) (\lambda (t2: T).(ty3 g e x0 t2))))))))))) (\lambda (c0: C).(\lambda -(t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda (_: ((\forall -(x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to (\forall (e: -C).((drop h x1 c0 e) \to (ex2 T (\lambda (t3: T).(pc3 c0 (lift h x1 t3) t)) -(\lambda (t3: T).(ty3 g e x0 t3)))))))))).(\lambda (u: T).(\lambda (t3: -T).(\lambda (H3: (ty3 g c0 u t3)).(\lambda (H4: ((\forall (x0: T).(\forall -(x1: nat).((eq T u (lift h x1 x0)) \to (\forall (e: C).((drop h x1 c0 e) \to -(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) t3)) (\lambda (t4: T).(ty3 g e -x0 t4)))))))))).(\lambda (H5: (pc3 c0 t3 t2)).(\lambda (x0: T).(\lambda (x1: -nat).(\lambda (H6: (eq T u (lift h x1 x0))).(\lambda (e: C).(\lambda (H7: -(drop h x1 c0 e)).(let H8 \def (eq_ind T u (\lambda (t0: T).(\forall (x2: -T).(\forall (x3: nat).((eq T t0 (lift h x3 x2)) \to (\forall (e0: C).((drop h -x3 c0 e0) \to (ex2 T (\lambda (t4: T).(pc3 c0 (lift h x3 t4) t3)) (\lambda -(t4: T).(ty3 g e0 x2 t4))))))))) H4 (lift h x1 x0) H6) in (let H9 \def -(eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 t3)) H3 (lift h x1 x0) H6) in (let -H10 \def (H8 x0 x1 (refl_equal T (lift h x1 x0)) e H7) in (ex2_ind T (\lambda -(t4: T).(pc3 c0 (lift h x1 t4) t3)) (\lambda (t4: T).(ty3 g e x0 t4)) (ex2 T -(\lambda (t4: T).(pc3 c0 (lift h x1 t4) t2)) (\lambda (t4: T).(ty3 g e x0 -t4))) (\lambda (x2: T).(\lambda (H11: (pc3 c0 (lift h x1 x2) t3)).(\lambda -(H12: (ty3 g e x0 x2)).(ex_intro2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) -t2)) (\lambda (t4: T).(ty3 g e x0 t4)) x2 (pc3_t t3 c0 (lift h x1 x2) H11 t2 -H5) H12)))) H10))))))))))))))))))) (\lambda (c0: C).(\lambda (m: -nat).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H1: (eq T (TSort m) (lift -h x1 x0))).(\lambda (e: C).(\lambda (_: (drop h x1 c0 e)).(eq_ind_r T (TSort -m) (\lambda (t: T).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (TSort -(next g m)))) (\lambda (t2: T).(ty3 g e t t2)))) (ex_intro2 T (\lambda (t2: -T).(pc3 c0 (lift h x1 t2) (TSort (next g m)))) (\lambda (t2: T).(ty3 g e -(TSort m) t2)) (TSort (next g m)) (eq_ind_r T (TSort (next g m)) (\lambda (t: -T).(pc3 c0 t (TSort (next g m)))) (pc3_refl c0 (TSort (next g m))) (lift h x1 -(TSort (next g m))) (lift_sort (next g m) h x1)) (ty3_sort g e m)) x0 -(lift_gen_sort h x1 m x0 H1))))))))) (\lambda (n: nat).(\lambda (c0: -C).(\lambda (d0: C).(\lambda (u: T).(\lambda (H1: (getl n c0 (CHead d0 (Bind -Abbr) u))).(\lambda (t: T).(\lambda (H2: (ty3 g d0 u t)).(\lambda (H3: -((\forall (x0: T).(\forall (x1: nat).((eq T u (lift h x1 x0)) \to (\forall -(e: C).((drop h x1 d0 e) \to (ex2 T (\lambda (t2: T).(pc3 d0 (lift h x1 t2) -t)) (\lambda (t2: T).(ty3 g e x0 t2)))))))))).(\lambda (x0: T).(\lambda (x1: -nat).(\lambda (H4: (eq T (TLRef n) (lift h x1 x0))).(\lambda (e: C).(\lambda -(H5: (drop h x1 c0 e)).(let H_x \def (lift_gen_lref x0 x1 h n H4) in (let H6 -\def H_x in (or_ind (land (lt n x1) (eq T x0 (TLRef n))) (land (le (plus x1 -h) n) (eq T x0 (TLRef (minus n h)))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h -x1 t2) (lift (S n) O t))) (\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (H7: -(land (lt n x1) (eq T x0 (TLRef n)))).(land_ind (lt n x1) (eq T x0 (TLRef n)) -(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t))) (\lambda -(t2: T).(ty3 g e x0 t2))) (\lambda (H8: (lt n x1)).(\lambda (H9: (eq T x0 -(TLRef n))).(eq_ind_r T (TLRef n) (\lambda (t0: T).(ex2 T (\lambda (t2: -T).(pc3 c0 (lift h x1 t2) (lift (S n) O t))) (\lambda (t2: T).(ty3 g e t0 -t2)))) (let H10 \def (eq_ind nat x1 (\lambda (n0: nat).(drop h n0 c0 e)) H5 -(S (plus n (minus x1 (S n)))) (lt_plus_minus n x1 H8)) in (ex3_2_ind T C -(\lambda (v: T).(\lambda (_: C).(eq T u (lift h (minus x1 (S n)) v)))) -(\lambda (v: T).(\lambda (e0: C).(getl n e (CHead e0 (Bind Abbr) v)))) -(\lambda (_: T).(\lambda (e0: C).(drop h (minus x1 (S n)) d0 e0))) (ex2 T -(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t))) (\lambda (t2: -T).(ty3 g e (TLRef n) t2))) (\lambda (x2: T).(\lambda (x3: C).(\lambda (H11: -(eq T u (lift h (minus x1 (S n)) x2))).(\lambda (H12: (getl n e (CHead x3 -(Bind Abbr) x2))).(\lambda (H13: (drop h (minus x1 (S n)) d0 x3)).(let H14 -\def (eq_ind T u (\lambda (t0: T).(\forall (x4: T).(\forall (x5: nat).((eq T -t0 (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 d0 e0) \to (ex2 T -(\lambda (t2: T).(pc3 d0 (lift h x5 t2) t)) (\lambda (t2: T).(ty3 g e0 x4 -t2))))))))) H3 (lift h (minus x1 (S n)) x2) H11) in (let H15 \def (eq_ind T u -(\lambda (t0: T).(ty3 g d0 t0 t)) H2 (lift h (minus x1 (S n)) x2) H11) in -(let H16 \def (H14 x2 (minus x1 (S n)) (refl_equal T (lift h (minus x1 (S n)) -x2)) x3 H13) in (ex2_ind T (\lambda (t2: T).(pc3 d0 (lift h (minus x1 (S n)) -t2) t)) (\lambda (t2: T).(ty3 g x3 x2 t2)) (ex2 T (\lambda (t2: T).(pc3 c0 -(lift h x1 t2) (lift (S n) O t))) (\lambda (t2: T).(ty3 g e (TLRef n) t2))) -(\lambda (x4: T).(\lambda (H17: (pc3 d0 (lift h (minus x1 (S n)) x4) -t)).(\lambda (H18: (ty3 g x3 x2 x4)).(eq_ind_r nat (plus (S n) (minus x1 (S -n))) (\lambda (n0: nat).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h n0 t2) (lift -(S n) O t))) (\lambda (t2: T).(ty3 g e (TLRef n) t2)))) (ex_intro2 T (\lambda -(t2: T).(pc3 c0 (lift h (plus (S n) (minus x1 (S n))) t2) (lift (S n) O t))) -(\lambda (t2: T).(ty3 g e (TLRef n) t2)) (lift (S n) O x4) (eq_ind_r T (lift -(S n) O (lift h (minus x1 (S n)) x4)) (\lambda (t0: T).(pc3 c0 t0 (lift (S n) -O t))) (pc3_lift c0 d0 (S n) O (getl_drop Abbr c0 d0 u n H1) (lift h (minus -x1 (S n)) x4) t H17) (lift h (plus (S n) (minus x1 (S n))) (lift (S n) O x4)) -(lift_d x4 h (S n) (minus x1 (S n)) O (le_O_n (minus x1 (S n))))) (ty3_abbr g -n e x3 x2 H12 x4 H18)) x1 (le_plus_minus (S n) x1 H8))))) H16))))))))) -(getl_drop_conf_lt Abbr c0 d0 u n H1 e h (minus x1 (S n)) H10))) x0 H9))) -H7)) (\lambda (H7: (land (le (plus x1 h) n) (eq T x0 (TLRef (minus n -h))))).(land_ind (le (plus x1 h) n) (eq T x0 (TLRef (minus n h))) (ex2 T -(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t))) (\lambda (t2: -T).(ty3 g e x0 t2))) (\lambda (H8: (le (plus x1 h) n)).(\lambda (H9: (eq T x0 -(TLRef (minus n h)))).(eq_ind_r T (TLRef (minus n h)) (\lambda (t0: T).(ex2 T -(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t))) (\lambda (t2: -T).(ty3 g e t0 t2)))) (ex_intro2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) -(lift (S n) O t))) (\lambda (t2: T).(ty3 g e (TLRef (minus n h)) t2)) (lift -(S (minus n h)) O t) (eq_ind_r T (lift (plus h (S (minus n h))) O t) (\lambda -(t0: T).(pc3 c0 t0 (lift (S n) O t))) (eq_ind nat (S (plus h (minus n h))) -(\lambda (n0: nat).(pc3 c0 (lift n0 O t) (lift (S n) O t))) (eq_ind nat n -(\lambda (n0: nat).(pc3 c0 (lift (S n0) O t) (lift (S n) O t))) (pc3_refl c0 -(lift (S n) O t)) (plus h (minus n h)) (le_plus_minus h n (le_trans h (plus -x1 h) n (le_plus_r x1 h) H8))) (plus h (S (minus n h))) (plus_n_Sm h (minus n -h))) (lift h x1 (lift (S (minus n h)) O t)) (lift_free t (S (minus n h)) h O -x1 (le_trans x1 (S (minus n h)) (plus O (S (minus n h))) (le_S_minus x1 h n -H8) (le_n (plus O (S (minus n h))))) (le_O_n x1))) (ty3_abbr g (minus n h) e -d0 u (getl_drop_conf_ge n (CHead d0 (Bind Abbr) u) c0 H1 e h x1 H5 H8) t H2)) -x0 H9))) H7)) H6)))))))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda -(d0: C).(\lambda (u: T).(\lambda (H1: (getl n c0 (CHead d0 (Bind Abst) -u))).(\lambda (t: T).(\lambda (H2: (ty3 g d0 u t)).(\lambda (H3: ((\forall -(x0: T).(\forall (x1: nat).((eq T u (lift h x1 x0)) \to (\forall (e: -C).((drop h x1 d0 e) \to (ex2 T (\lambda (t2: T).(pc3 d0 (lift h x1 t2) t)) -(\lambda (t2: T).(ty3 g e x0 t2)))))))))).(\lambda (x0: T).(\lambda (x1: -nat).(\lambda (H4: (eq T (TLRef n) (lift h x1 x0))).(\lambda (e: C).(\lambda -(H5: (drop h x1 c0 e)).(let H_x \def (lift_gen_lref x0 x1 h n H4) in (let H6 -\def H_x in (or_ind (land (lt n x1) (eq T x0 (TLRef n))) (land (le (plus x1 -h) n) (eq T x0 (TLRef (minus n h)))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h -x1 t2) (lift (S n) O u))) (\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (H7: -(land (lt n x1) (eq T x0 (TLRef n)))).(land_ind (lt n x1) (eq T x0 (TLRef n)) -(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O u))) (\lambda -(t2: T).(ty3 g e x0 t2))) (\lambda (H8: (lt n x1)).(\lambda (H9: (eq T x0 -(TLRef n))).(eq_ind_r T (TLRef n) (\lambda (t0: T).(ex2 T (\lambda (t2: -T).(pc3 c0 (lift h x1 t2) (lift (S n) O u))) (\lambda (t2: T).(ty3 g e t0 -t2)))) (let H10 \def (eq_ind nat x1 (\lambda (n0: nat).(drop h n0 c0 e)) H5 -(S (plus n (minus x1 (S n)))) (lt_plus_minus n x1 H8)) in (ex3_2_ind T C -(\lambda (v: T).(\lambda (_: C).(eq T u (lift h (minus x1 (S n)) v)))) -(\lambda (v: T).(\lambda (e0: C).(getl n e (CHead e0 (Bind Abst) v)))) -(\lambda (_: T).(\lambda (e0: C).(drop h (minus x1 (S n)) d0 e0))) (ex2 T -(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O u))) (\lambda (t2: -T).(ty3 g e (TLRef n) t2))) (\lambda (x2: T).(\lambda (x3: C).(\lambda (H11: -(eq T u (lift h (minus x1 (S n)) x2))).(\lambda (H12: (getl n e (CHead x3 -(Bind Abst) x2))).(\lambda (H13: (drop h (minus x1 (S n)) d0 x3)).(let H14 -\def (eq_ind T u (\lambda (t0: T).(\forall (x4: T).(\forall (x5: nat).((eq T -t0 (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 d0 e0) \to (ex2 T -(\lambda (t2: T).(pc3 d0 (lift h x5 t2) t)) (\lambda (t2: T).(ty3 g e0 x4 -t2))))))))) H3 (lift h (minus x1 (S n)) x2) H11) in (let H15 \def (eq_ind T u -(\lambda (t0: T).(ty3 g d0 t0 t)) H2 (lift h (minus x1 (S n)) x2) H11) in -(eq_ind_r T (lift h (minus x1 (S n)) x2) (\lambda (t0: T).(ex2 T (\lambda -(t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t0))) (\lambda (t2: T).(ty3 g e -(TLRef n) t2)))) (let H16 \def (H14 x2 (minus x1 (S n)) (refl_equal T (lift h -(minus x1 (S n)) x2)) x3 H13) in (ex2_ind T (\lambda (t2: T).(pc3 d0 (lift h -(minus x1 (S n)) t2) t)) (\lambda (t2: T).(ty3 g x3 x2 t2)) (ex2 T (\lambda -(t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O (lift h (minus x1 (S n)) x2)))) -(\lambda (t2: T).(ty3 g e (TLRef n) t2))) (\lambda (x4: T).(\lambda (_: (pc3 -d0 (lift h (minus x1 (S n)) x4) t)).(\lambda (H18: (ty3 g x3 x2 -x4)).(eq_ind_r nat (plus (S n) (minus x1 (S n))) (\lambda (n0: nat).(ex2 T -(\lambda (t2: T).(pc3 c0 (lift h n0 t2) (lift (S n) O (lift h (minus n0 (S -n)) x2)))) (\lambda (t2: T).(ty3 g e (TLRef n) t2)))) (ex_intro2 T (\lambda -(t2: T).(pc3 c0 (lift h (plus (S n) (minus x1 (S n))) t2) (lift (S n) O (lift -h (minus (plus (S n) (minus x1 (S n))) (S n)) x2)))) (\lambda (t2: T).(ty3 g -e (TLRef n) t2)) (lift (S n) O x2) (eq_ind_r T (lift (S n) O (lift h (minus -x1 (S n)) x2)) (\lambda (t0: T).(pc3 c0 t0 (lift (S n) O (lift h (minus (plus -(S n) (minus x1 (S n))) (S n)) x2)))) (eq_ind nat x1 (\lambda (n0: nat).(pc3 -c0 (lift (S n) O (lift h (minus x1 (S n)) x2)) (lift (S n) O (lift h (minus -n0 (S n)) x2)))) (pc3_refl c0 (lift (S n) O (lift h (minus x1 (S n)) x2))) -(plus (S n) (minus x1 (S n))) (le_plus_minus (S n) x1 H8)) (lift h (plus (S -n) (minus x1 (S n))) (lift (S n) O x2)) (lift_d x2 h (S n) (minus x1 (S n)) O -(le_O_n (minus x1 (S n))))) (ty3_abst g n e x3 x2 H12 x4 H18)) x1 -(le_plus_minus (S n) x1 H8))))) H16)) u H11)))))))) (getl_drop_conf_lt Abst -c0 d0 u n H1 e h (minus x1 (S n)) H10))) x0 H9))) H7)) (\lambda (H7: (land -(le (plus x1 h) n) (eq T x0 (TLRef (minus n h))))).(land_ind (le (plus x1 h) -n) (eq T x0 (TLRef (minus n h))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 -t2) (lift (S n) O u))) (\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (H8: (le -(plus x1 h) n)).(\lambda (H9: (eq T x0 (TLRef (minus n h)))).(eq_ind_r T -(TLRef (minus n h)) (\lambda (t0: T).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h -x1 t2) (lift (S n) O u))) (\lambda (t2: T).(ty3 g e t0 t2)))) (ex_intro2 T -(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O u))) (\lambda (t2: -T).(ty3 g e (TLRef (minus n h)) t2)) (lift (S (minus n h)) O u) (eq_ind_r T -(lift (plus h (S (minus n h))) O u) (\lambda (t0: T).(pc3 c0 t0 (lift (S n) O -u))) (eq_ind nat (S (plus h (minus n h))) (\lambda (n0: nat).(pc3 c0 (lift n0 -O u) (lift (S n) O u))) (eq_ind nat n (\lambda (n0: nat).(pc3 c0 (lift (S n0) -O u) (lift (S n) O u))) (pc3_refl c0 (lift (S n) O u)) (plus h (minus n h)) -(le_plus_minus h n (le_trans h (plus x1 h) n (le_plus_r x1 h) H8))) (plus h -(S (minus n h))) (plus_n_Sm h (minus n h))) (lift h x1 (lift (S (minus n h)) -O u)) (lift_free u (S (minus n h)) h O x1 (le_trans x1 (S (minus n h)) (plus -O (S (minus n h))) (le_S_minus x1 h n H8) (le_n (plus O (S (minus n h))))) -(le_O_n x1))) (ty3_abst g (minus n h) e d0 u (getl_drop_conf_ge n (CHead d0 -(Bind Abst) u) c0 H1 e h x1 H5 H8) t H2)) x0 H9))) H7)) H6)))))))))))))))) -(\lambda (c0: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H1: (ty3 g c0 u -t)).(\lambda (H2: ((\forall (x0: T).(\forall (x1: nat).((eq T u (lift h x1 -x0)) \to (\forall (e: C).((drop h x1 c0 e) \to (ex2 T (\lambda (t2: T).(pc3 -c0 (lift h x1 t2) t)) (\lambda (t2: T).(ty3 g e x0 t2)))))))))).(\lambda (b: -B).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H3: (ty3 g (CHead c0 (Bind b) -u) t2 t3)).(\lambda (H4: ((\forall (x0: T).(\forall (x1: nat).((eq T t2 (lift -h x1 x0)) \to (\forall (e: C).((drop h x1 (CHead c0 (Bind b) u) e) \to (ex2 T -(\lambda (t4: T).(pc3 (CHead c0 (Bind b) u) (lift h x1 t4) t3)) (\lambda (t4: -T).(ty3 g e x0 t4)))))))))).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H5: -(eq T (THead (Bind b) u t2) (lift h x1 x0))).(\lambda (e: C).(\lambda (H6: -(drop h x1 c0 e)).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 -(THead (Bind b) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u (lift h x1 -y0)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h (S x1) z)))) (ex2 T -(\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b) u t3))) (\lambda (t4: -T).(ty3 g e x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq T x0 -(THead (Bind b) x2 x3))).(\lambda (H8: (eq T u (lift h x1 x2))).(\lambda (H9: -(eq T t2 (lift h (S x1) x3))).(eq_ind_r T (THead (Bind b) x2 x3) (\lambda -(t0: T).(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b) u -t3))) (\lambda (t4: T).(ty3 g e t0 t4)))) (let H10 \def (eq_ind T t2 (\lambda -(t0: T).(\forall (x4: T).(\forall (x5: nat).((eq T t0 (lift h x5 x4)) \to -(\forall (e0: C).((drop h x5 (CHead c0 (Bind b) u) e0) \to (ex2 T (\lambda -(t4: T).(pc3 (CHead c0 (Bind b) u) (lift h x5 t4) t3)) (\lambda (t4: T).(ty3 -g e0 x4 t4))))))))) H4 (lift h (S x1) x3) H9) in (let H11 \def (eq_ind T t2 -(\lambda (t0: T).(ty3 g (CHead c0 (Bind b) u) t0 t3)) H3 (lift h (S x1) x3) -H9) in (let H12 \def (eq_ind T u (\lambda (t0: T).(ty3 g (CHead c0 (Bind b) -t0) (lift h (S x1) x3) t3)) H11 (lift h x1 x2) H8) in (let H13 \def (eq_ind T -u (\lambda (t0: T).(\forall (x4: T).(\forall (x5: nat).((eq T (lift h (S x1) -x3) (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 (CHead c0 (Bind b) t0) -e0) \to (ex2 T (\lambda (t4: T).(pc3 (CHead c0 (Bind b) t0) (lift h x5 t4) -t3)) (\lambda (t4: T).(ty3 g e0 x4 t4))))))))) H10 (lift h x1 x2) H8) in (let -H14 \def (eq_ind T u (\lambda (t0: T).(\forall (x4: T).(\forall (x5: -nat).((eq T t0 (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 c0 e0) \to -(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x5 t4) t)) (\lambda (t4: T).(ty3 g e0 -x4 t4))))))))) H2 (lift h x1 x2) H8) in (let H15 \def (eq_ind T u (\lambda -(t0: T).(ty3 g c0 t0 t)) H1 (lift h x1 x2) H8) in (eq_ind_r T (lift h x1 x2) -(\lambda (t0: T).(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind -b) t0 t3))) (\lambda (t4: T).(ty3 g e (THead (Bind b) x2 x3) t4)))) (let H16 -\def (H14 x2 x1 (refl_equal T (lift h x1 x2)) e H6) in (ex2_ind T (\lambda -(t4: T).(pc3 c0 (lift h x1 t4) t)) (\lambda (t4: T).(ty3 g e x2 t4)) (ex2 T -(\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b) (lift h x1 x2) t3))) -(\lambda (t4: T).(ty3 g e (THead (Bind b) x2 x3) t4))) (\lambda (x4: -T).(\lambda (_: (pc3 c0 (lift h x1 x4) t)).(\lambda (H18: (ty3 g e x2 -x4)).(let H19 \def (H13 x3 (S x1) (refl_equal T (lift h (S x1) x3)) (CHead e -(Bind b) x2) (drop_skip_bind h x1 c0 e H6 b x2)) in (ex2_ind T (\lambda (t4: -T).(pc3 (CHead c0 (Bind b) (lift h x1 x2)) (lift h (S x1) t4) t3)) (\lambda -(t4: T).(ty3 g (CHead e (Bind b) x2) x3 t4)) (ex2 T (\lambda (t4: T).(pc3 c0 -(lift h x1 t4) (THead (Bind b) (lift h x1 x2) t3))) (\lambda (t4: T).(ty3 g e -(THead (Bind b) x2 x3) t4))) (\lambda (x5: T).(\lambda (H20: (pc3 (CHead c0 -(Bind b) (lift h x1 x2)) (lift h (S x1) x5) t3)).(\lambda (H21: (ty3 g (CHead -e (Bind b) x2) x3 x5)).(ex_ind T (\lambda (t0: T).(ty3 g (CHead e (Bind b) -x2) x5 t0)) (ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b) -(lift h x1 x2) t3))) (\lambda (t4: T).(ty3 g e (THead (Bind b) x2 x3) t4))) -(\lambda (x6: T).(\lambda (_: (ty3 g (CHead e (Bind b) x2) x5 x6)).(ex_intro2 -T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b) (lift h x1 x2) -t3))) (\lambda (t4: T).(ty3 g e (THead (Bind b) x2 x3) t4)) (THead (Bind b) -x2 x5) (eq_ind_r T (THead (Bind b) (lift h x1 x2) (lift h (S x1) x5)) -(\lambda (t0: T).(pc3 c0 t0 (THead (Bind b) (lift h x1 x2) t3))) (pc3_head_2 -c0 (lift h x1 x2) (lift h (S x1) x5) t3 (Bind b) H20) (lift h x1 (THead (Bind -b) x2 x5)) (lift_bind b x2 x5 h x1)) (ty3_bind g e x2 x4 H18 b x3 x5 H21)))) -(ty3_correct g (CHead e (Bind b) x2) x3 x5 H21))))) H19))))) H16)) u -H8))))))) x0 H7)))))) (lift_gen_bind b u t2 x0 h x1 H5))))))))))))))))) -(\lambda (c0: C).(\lambda (w: T).(\lambda (u: T).(\lambda (H1: (ty3 g c0 w -u)).(\lambda (H2: ((\forall (x0: T).(\forall (x1: nat).((eq T w (lift h x1 -x0)) \to (\forall (e: C).((drop h x1 c0 e) \to (ex2 T (\lambda (t2: T).(pc3 -c0 (lift h x1 t2) u)) (\lambda (t2: T).(ty3 g e x0 t2)))))))))).(\lambda (v: -T).(\lambda (t: T).(\lambda (H3: (ty3 g c0 v (THead (Bind Abst) u -t))).(\lambda (H4: ((\forall (x0: T).(\forall (x1: nat).((eq T v (lift h x1 -x0)) \to (\forall (e: C).((drop h x1 c0 e) \to (ex2 T (\lambda (t2: T).(pc3 -c0 (lift h x1 t2) (THead (Bind Abst) u t))) (\lambda (t2: T).(ty3 g e x0 -t2)))))))))).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H5: (eq T (THead -(Flat Appl) w v) (lift h x1 x0))).(\lambda (e: C).(\lambda (H6: (drop h x1 c0 -e)).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Flat -Appl) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T w (lift h x1 y0)))) -(\lambda (_: T).(\lambda (z: T).(eq T v (lift h x1 z)))) (ex2 T (\lambda (t2: -T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) w (THead (Bind Abst) u t)))) -(\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (x2: T).(\lambda (x3: T).(\lambda -(H7: (eq T x0 (THead (Flat Appl) x2 x3))).(\lambda (H8: (eq T w (lift h x1 -x2))).(\lambda (H9: (eq T v (lift h x1 x3))).(eq_ind_r T (THead (Flat Appl) -x2 x3) (\lambda (t0: T).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead -(Flat Appl) w (THead (Bind Abst) u t)))) (\lambda (t2: T).(ty3 g e t0 t2)))) -(let H10 \def (eq_ind T v (\lambda (t0: T).(\forall (x4: T).(\forall (x5: -nat).((eq T t0 (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 c0 e0) \to -(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x5 t2) (THead (Bind Abst) u t))) -(\lambda (t2: T).(ty3 g e0 x4 t2))))))))) H4 (lift h x1 x3) H9) in (let H11 -\def (eq_ind T v (\lambda (t0: T).(ty3 g c0 t0 (THead (Bind Abst) u t))) H3 -(lift h x1 x3) H9) in (let H12 \def (eq_ind T w (\lambda (t0: T).(\forall -(x4: T).(\forall (x5: nat).((eq T t0 (lift h x5 x4)) \to (\forall (e0: -C).((drop h x5 c0 e0) \to (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x5 t2) u)) -(\lambda (t2: T).(ty3 g e0 x4 t2))))))))) H2 (lift h x1 x2) H8) in (let H13 -\def (eq_ind T w (\lambda (t0: T).(ty3 g c0 t0 u)) H1 (lift h x1 x2) H8) in -(eq_ind_r T (lift h x1 x2) (\lambda (t0: T).(ex2 T (\lambda (t2: T).(pc3 c0 -(lift h x1 t2) (THead (Flat Appl) t0 (THead (Bind Abst) u t)))) (\lambda (t2: -T).(ty3 g e (THead (Flat Appl) x2 x3) t2)))) (let H14 \def (H12 x2 x1 -(refl_equal T (lift h x1 x2)) e H6) in (ex2_ind T (\lambda (t2: T).(pc3 c0 -(lift h x1 t2) u)) (\lambda (t2: T).(ty3 g e x2 t2)) (ex2 T (\lambda (t2: -T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) (lift h x1 x2) (THead (Bind -Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead (Flat Appl) x2 x3) t2))) -(\lambda (x4: T).(\lambda (H15: (pc3 c0 (lift h x1 x4) u)).(\lambda (H16: -(ty3 g e x2 x4)).(let H17 \def (H10 x3 x1 (refl_equal T (lift h x1 x3)) e H6) -in (ex2_ind T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead (Bind Abst) u -t))) (\lambda (t2: T).(ty3 g e x3 t2)) (ex2 T (\lambda (t2: T).(pc3 c0 (lift -h x1 t2) (THead (Flat Appl) (lift h x1 x2) (THead (Bind Abst) u t)))) -(\lambda (t2: T).(ty3 g e (THead (Flat Appl) x2 x3) t2))) (\lambda (x5: -T).(\lambda (H18: (pc3 c0 (lift h x1 x5) (THead (Bind Abst) u t))).(\lambda -(H19: (ty3 g e x3 x5)).(ex3_2_ind T T (\lambda (u1: T).(\lambda (t2: T).(pr3 -e x5 (THead (Bind Abst) u1 t2)))) (\lambda (u1: T).(\lambda (_: T).(pr3 c0 u -(lift h x1 u1)))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall -(u0: T).(pr3 (CHead c0 (Bind b) u0) t (lift h (S x1) t2)))))) (ex2 T (\lambda -(t2: T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) (lift h x1 x2) (THead (Bind -Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead (Flat Appl) x2 x3) t2))) -(\lambda (x6: T).(\lambda (x7: T).(\lambda (H20: (pr3 e x5 (THead (Bind Abst) -x6 x7))).(\lambda (H21: (pr3 c0 u (lift h x1 x6))).(\lambda (H22: ((\forall -(b: B).(\forall (u0: T).(pr3 (CHead c0 (Bind b) u0) t (lift h (S x1) -x7)))))).(ex_ind T (\lambda (t0: T).(ty3 g e x5 t0)) (ex2 T (\lambda (t2: -T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) (lift h x1 x2) (THead (Bind -Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead (Flat Appl) x2 x3) t2))) -(\lambda (x8: T).(\lambda (H23: (ty3 g e x5 x8)).(let H_y \def (ty3_sred_pr3 -e x5 (THead (Bind Abst) x6 x7) H20 g x8 H23) in (ex3_2_ind T T (\lambda (t2: -T).(\lambda (_: T).(pc3 e (THead (Bind Abst) x6 t2) x8))) (\lambda (_: -T).(\lambda (t0: T).(ty3 g e x6 t0))) (\lambda (t2: T).(\lambda (_: T).(ty3 g -(CHead e (Bind Abst) x6) x7 t2))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 -t2) (THead (Flat Appl) (lift h x1 x2) (THead (Bind Abst) u t)))) (\lambda -(t2: T).(ty3 g e (THead (Flat Appl) x2 x3) t2))) (\lambda (x9: T).(\lambda -(x10: T).(\lambda (_: (pc3 e (THead (Bind Abst) x6 x9) x8)).(\lambda (H25: -(ty3 g e x6 x10)).(\lambda (H26: (ty3 g (CHead e (Bind Abst) x6) x7 -x9)).(ex_intro2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) -(lift h x1 x2) (THead (Bind Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead -(Flat Appl) x2 x3) t2)) (THead (Flat Appl) x2 (THead (Bind Abst) x6 x7)) -(eq_ind_r T (THead (Flat Appl) (lift h x1 x2) (lift h x1 (THead (Bind Abst) -x6 x7))) (\lambda (t0: T).(pc3 c0 t0 (THead (Flat Appl) (lift h x1 x2) (THead -(Bind Abst) u t)))) (pc3_thin_dx c0 (lift h x1 (THead (Bind Abst) x6 x7)) -(THead (Bind Abst) u t) (eq_ind_r T (THead (Bind Abst) (lift h x1 x6) (lift h -(S x1) x7)) (\lambda (t0: T).(pc3 c0 t0 (THead (Bind Abst) u t))) -(pc3_head_21 c0 (lift h x1 x6) u (pc3_pr3_x c0 (lift h x1 x6) u H21) (Bind -Abst) (lift h (S x1) x7) t (pc3_pr3_x (CHead c0 (Bind Abst) (lift h x1 x6)) -(lift h (S x1) x7) t (H22 Abst (lift h x1 x6)))) (lift h x1 (THead (Bind -Abst) x6 x7)) (lift_bind Abst x6 x7 h x1)) (lift h x1 x2) Appl) (lift h x1 -(THead (Flat Appl) x2 (THead (Bind Abst) x6 x7))) (lift_flat Appl x2 (THead -(Bind Abst) x6 x7) h x1)) (ty3_appl g e x2 x6 (ty3_conv g e x6 x10 H25 x2 x4 -H16 (pc3_gen_lift c0 x4 x6 h x1 (pc3_t u c0 (lift h x1 x4) H15 (lift h x1 x6) -(pc3_pr3_r c0 u (lift h x1 x6) H21)) e H6)) x3 x7 (ty3_conv g e (THead (Bind -Abst) x6 x7) (THead (Bind Abst) x6 x9) (ty3_bind g e x6 x10 H25 Abst x7 x9 -H26) x3 x5 H19 (pc3_pr3_r e x5 (THead (Bind Abst) x6 x7) H20))))))))) -(ty3_gen_bind g Abst e x6 x7 x8 H_y))))) (ty3_correct g e x3 x5 H19))))))) -(pc3_gen_lift_abst c0 x5 t u h x1 H18 e H6))))) H17))))) H14)) w H8))))) x0 -H7)))))) (lift_gen_flat Appl w v x0 h x1 H5)))))))))))))))) (\lambda (c0: -C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (ty3 g c0 t2 t3)).(\lambda -(H2: ((\forall (x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to -(\forall (e: C).((drop h x1 c0 e) \to (ex2 T (\lambda (t4: T).(pc3 c0 (lift h -x1 t4) t3)) (\lambda (t4: T).(ty3 g e x0 t4)))))))))).(\lambda (t0: -T).(\lambda (H3: (ty3 g c0 t3 t0)).(\lambda (H4: ((\forall (x0: T).(\forall -(x1: nat).((eq T t3 (lift h x1 x0)) \to (\forall (e: C).((drop h x1 c0 e) \to -(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) t0)) (\lambda (t4: T).(ty3 g e -x0 t4)))))))))).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H5: (eq T -(THead (Flat Cast) t3 t2) (lift h x1 x0))).(\lambda (e: C).(\lambda (H6: -(drop h x1 c0 e)).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 -(THead (Flat Cast) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T t3 (lift h -x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h x1 z)))) (ex2 T -(\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Flat Cast) t0 t3))) (\lambda -(t4: T).(ty3 g e x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq -T x0 (THead (Flat Cast) x2 x3))).(\lambda (H8: (eq T t3 (lift h x1 -x2))).(\lambda (H9: (eq T t2 (lift h x1 x3))).(eq_ind_r T (THead (Flat Cast) -x2 x3) (\lambda (t: T).(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead -(Flat Cast) t0 t3))) (\lambda (t4: T).(ty3 g e t t4)))) (let H10 \def (eq_ind -T t3 (\lambda (t: T).(\forall (x4: T).(\forall (x5: nat).((eq T t (lift h x5 -x4)) \to (\forall (e0: C).((drop h x5 c0 e0) \to (ex2 T (\lambda (t4: T).(pc3 -c0 (lift h x5 t4) t0)) (\lambda (t4: T).(ty3 g e0 x4 t4))))))))) H4 (lift h -x1 x2) H8) in (let H11 \def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 t t0)) H3 -(lift h x1 x2) H8) in (let H12 \def (eq_ind T t3 (\lambda (t: T).(\forall -(x4: T).(\forall (x5: nat).((eq T t2 (lift h x5 x4)) \to (\forall (e0: -C).((drop h x5 c0 e0) \to (ex2 T (\lambda (t4: T).(pc3 c0 (lift h x5 t4) t)) -(\lambda (t4: T).(ty3 g e0 x4 t4))))))))) H2 (lift h x1 x2) H8) in (let H13 -\def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 t2 t)) H1 (lift h x1 x2) H8) in -(eq_ind_r T (lift h x1 x2) (\lambda (t: T).(ex2 T (\lambda (t4: T).(pc3 c0 -(lift h x1 t4) (THead (Flat Cast) t0 t))) (\lambda (t4: T).(ty3 g e (THead -(Flat Cast) x2 x3) t4)))) (let H14 \def (eq_ind T t2 (\lambda (t: T).(ty3 g -c0 t (lift h x1 x2))) H13 (lift h x1 x3) H9) in (let H15 \def (eq_ind T t2 -(\lambda (t: T).(\forall (x4: T).(\forall (x5: nat).((eq T t (lift h x5 x4)) -\to (\forall (e0: C).((drop h x5 c0 e0) \to (ex2 T (\lambda (t4: T).(pc3 c0 -(lift h x5 t4) (lift h x1 x2))) (\lambda (t4: T).(ty3 g e0 x4 t4))))))))) H12 -(lift h x1 x3) H9) in (let H16 \def (H15 x3 x1 (refl_equal T (lift h x1 x3)) -e H6) in (ex2_ind T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (lift h x1 x2))) -(\lambda (t4: T).(ty3 g e x3 t4)) (ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 -t4) (THead (Flat Cast) t0 (lift h x1 x2)))) (\lambda (t4: T).(ty3 g e (THead -(Flat Cast) x2 x3) t4))) (\lambda (x4: T).(\lambda (H17: (pc3 c0 (lift h x1 -x4) (lift h x1 x2))).(\lambda (H18: (ty3 g e x3 x4)).(let H19 \def (H10 x2 x1 -(refl_equal T (lift h x1 x2)) e H6) in (ex2_ind T (\lambda (t4: T).(pc3 c0 -(lift h x1 t4) t0)) (\lambda (t4: T).(ty3 g e x2 t4)) (ex2 T (\lambda (t4: -T).(pc3 c0 (lift h x1 t4) (THead (Flat Cast) t0 (lift h x1 x2)))) (\lambda -(t4: T).(ty3 g e (THead (Flat Cast) x2 x3) t4))) (\lambda (x5: T).(\lambda -(H20: (pc3 c0 (lift h x1 x5) t0)).(\lambda (H21: (ty3 g e x2 x5)).(ex_intro2 -T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Flat Cast) t0 (lift h x1 -x2)))) (\lambda (t4: T).(ty3 g e (THead (Flat Cast) x2 x3) t4)) (THead (Flat -Cast) x5 x2) (eq_ind_r T (THead (Flat Cast) (lift h x1 x5) (lift h x1 x2)) -(\lambda (t: T).(pc3 c0 t (THead (Flat Cast) t0 (lift h x1 x2)))) (pc3_head_1 -c0 (lift h x1 x5) t0 H20 (Flat Cast) (lift h x1 x2)) (lift h x1 (THead (Flat -Cast) x5 x2)) (lift_flat Cast x5 x2 h x1)) (ty3_cast g e x3 x2 (ty3_conv g e -x2 x5 H21 x3 x4 H18 (pc3_gen_lift c0 x4 x2 h x1 H17 e H6)) x5 H21))))) -H19))))) H16)))) t3 H8))))) x0 H7)))))) (lift_gen_flat Cast t3 t2 x0 h x1 -H5))))))))))))))) c y x H0))))) H))))))). -(* COMMENTS -Initial nodes: 9781 -END *) - -theorem ty3_tred: - \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).((ty3 g c u -t1) \to (\forall (t2: T).((pr3 c t1 t2) \to (ty3 g c u t2))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (H: -(ty3 g c u t1)).(\lambda (t2: T).(\lambda (H0: (pr3 c t1 t2)).(ex_ind T -(\lambda (t: T).(ty3 g c t1 t)) (ty3 g c u t2) (\lambda (x: T).(\lambda (H1: -(ty3 g c t1 x)).(let H_y \def (ty3_sred_pr3 c t1 t2 H0 g x H1) in (ty3_conv g -c t2 x H_y u t1 H (pc3_pr3_r c t1 t2 H0))))) (ty3_correct g c u t1 H)))))))). -(* COMMENTS -Initial nodes: 121 -END *) - -theorem ty3_sconv_pc3: - \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (t1: T).((ty3 g c -u1 t1) \to (\forall (u2: T).(\forall (t2: T).((ty3 g c u2 t2) \to ((pc3 c u1 -u2) \to (pc3 c t1 t2))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda -(H: (ty3 g c u1 t1)).(\lambda (u2: T).(\lambda (t2: T).(\lambda (H0: (ty3 g c -u2 t2)).(\lambda (H1: (pc3 c u1 u2)).(let H2 \def H1 in (ex2_ind T (\lambda -(t: T).(pr3 c u1 t)) (\lambda (t: T).(pr3 c u2 t)) (pc3 c t1 t2) (\lambda (x: -T).(\lambda (H3: (pr3 c u1 x)).(\lambda (H4: (pr3 c u2 x)).(let H_y \def -(ty3_sred_pr3 c u2 x H4 g t2 H0) in (let H_y0 \def (ty3_sred_pr3 c u1 x H3 g -t1 H) in (ty3_unique g c x t1 H_y0 t2 H_y)))))) H2)))))))))). -(* COMMENTS -Initial nodes: 141 -END *) - -theorem ty3_sred_back: - \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t0: T).((ty3 g c -t1 t0) \to (\forall (t2: T).((pr3 c t1 t2) \to (\forall (t: T).((ty3 g c t2 -t) \to (ty3 g c t1 t))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t0: T).(\lambda -(H: (ty3 g c t1 t0)).(\lambda (t2: T).(\lambda (H0: (pr3 c t1 t2)).(\lambda -(t: T).(\lambda (H1: (ty3 g c t2 t)).(ex_ind T (\lambda (t3: T).(ty3 g c t -t3)) (ty3 g c t1 t) (\lambda (x: T).(\lambda (H2: (ty3 g c t x)).(ty3_conv g -c t x H2 t1 t0 H (ty3_unique g c t2 t0 (ty3_sred_pr3 c t1 t2 H0 g t0 H) t -H1)))) (ty3_correct g c t2 t H1)))))))))). -(* COMMENTS -Initial nodes: 137 -END *) - -theorem ty3_sconv: - \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (t1: T).((ty3 g c -u1 t1) \to (\forall (u2: T).(\forall (t2: T).((ty3 g c u2 t2) \to ((pc3 c u1 -u2) \to (ty3 g c u1 t2))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda -(H: (ty3 g c u1 t1)).(\lambda (u2: T).(\lambda (t2: T).(\lambda (H0: (ty3 g c -u2 t2)).(\lambda (H1: (pc3 c u1 u2)).(let H2 \def H1 in (ex2_ind T (\lambda -(t: T).(pr3 c u1 t)) (\lambda (t: T).(pr3 c u2 t)) (ty3 g c u1 t2) (\lambda -(x: T).(\lambda (H3: (pr3 c u1 x)).(\lambda (H4: (pr3 c u2 x)).(ty3_sred_back -g c u1 t1 H x H3 t2 (ty3_sred_pr3 c u2 x H4 g t2 H0))))) H2)))))))))). -(* COMMENTS -Initial nodes: 129 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/props.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/props.ma deleted file mode 100644 index 579a8dd69..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/props.ma +++ /dev/null @@ -1,691 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/ty3/fwd.ma". - -include "Basic-1/pc3/fwd.ma". - -theorem ty3_lift: - \forall (g: G).(\forall (e: C).(\forall (t1: T).(\forall (t2: T).((ty3 g e -t1 t2) \to (\forall (c: C).(\forall (d: nat).(\forall (h: nat).((drop h d c -e) \to (ty3 g c (lift h d t1) (lift h d t2)))))))))) -\def - \lambda (g: G).(\lambda (e: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H: (ty3 g e t1 t2)).(ty3_ind g (\lambda (c: C).(\lambda (t: T).(\lambda (t0: -T).(\forall (c0: C).(\forall (d: nat).(\forall (h: nat).((drop h d c0 c) \to -(ty3 g c0 (lift h d t) (lift h d t0))))))))) (\lambda (c: C).(\lambda (t0: -T).(\lambda (t: T).(\lambda (_: (ty3 g c t0 t)).(\lambda (H1: ((\forall (c0: -C).(\forall (d: nat).(\forall (h: nat).((drop h d c0 c) \to (ty3 g c0 (lift h -d t0) (lift h d t)))))))).(\lambda (u: T).(\lambda (t3: T).(\lambda (_: (ty3 -g c u t3)).(\lambda (H3: ((\forall (c0: C).(\forall (d: nat).(\forall (h: -nat).((drop h d c0 c) \to (ty3 g c0 (lift h d u) (lift h d -t3)))))))).(\lambda (H4: (pc3 c t3 t0)).(\lambda (c0: C).(\lambda (d: -nat).(\lambda (h: nat).(\lambda (H5: (drop h d c0 c)).(ty3_conv g c0 (lift h -d t0) (lift h d t) (H1 c0 d h H5) (lift h d u) (lift h d t3) (H3 c0 d h H5) -(pc3_lift c0 c h d H5 t3 t0 H4)))))))))))))))) (\lambda (c: C).(\lambda (m: -nat).(\lambda (c0: C).(\lambda (d: nat).(\lambda (h: nat).(\lambda (_: (drop -h d c0 c)).(eq_ind_r T (TSort m) (\lambda (t: T).(ty3 g c0 t (lift h d (TSort -(next g m))))) (eq_ind_r T (TSort (next g m)) (\lambda (t: T).(ty3 g c0 -(TSort m) t)) (ty3_sort g c0 m) (lift h d (TSort (next g m))) (lift_sort -(next g m) h d)) (lift h d (TSort m)) (lift_sort m h d)))))))) (\lambda (n: -nat).(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c -(CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u -t)).(\lambda (H2: ((\forall (c0: C).(\forall (d0: nat).(\forall (h: -nat).((drop h d0 c0 d) \to (ty3 g c0 (lift h d0 u) (lift h d0 -t)))))))).(\lambda (c0: C).(\lambda (d0: nat).(\lambda (h: nat).(\lambda (H3: -(drop h d0 c0 c)).(lt_le_e n d0 (ty3 g c0 (lift h d0 (TLRef n)) (lift h d0 -(lift (S n) O t))) (\lambda (H4: (lt n d0)).(let H5 \def (drop_getl_trans_le -n d0 (le_S_n n d0 (le_S (S n) d0 H4)) c0 c h H3 (CHead d (Bind Abbr) u) H0) -in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop n O c0 e0))) -(\lambda (e0: C).(\lambda (e1: C).(drop h (minus d0 n) e0 e1))) (\lambda (_: -C).(\lambda (e1: C).(clear e1 (CHead d (Bind Abbr) u)))) (ty3 g c0 (lift h d0 -(TLRef n)) (lift h d0 (lift (S n) O t))) (\lambda (x0: C).(\lambda (x1: -C).(\lambda (H6: (drop n O c0 x0)).(\lambda (H7: (drop h (minus d0 n) x0 -x1)).(\lambda (H8: (clear x1 (CHead d (Bind Abbr) u))).(let H9 \def (eq_ind -nat (minus d0 n) (\lambda (n0: nat).(drop h n0 x0 x1)) H7 (S (minus d0 (S -n))) (minus_x_Sy d0 n H4)) in (let H10 \def (drop_clear_S x1 x0 h (minus d0 -(S n)) H9 Abbr d u H8) in (ex2_ind C (\lambda (c1: C).(clear x0 (CHead c1 -(Bind Abbr) (lift h (minus d0 (S n)) u)))) (\lambda (c1: C).(drop h (minus d0 -(S n)) c1 d)) (ty3 g c0 (lift h d0 (TLRef n)) (lift h d0 (lift (S n) O t))) -(\lambda (x: C).(\lambda (H11: (clear x0 (CHead x (Bind Abbr) (lift h (minus -d0 (S n)) u)))).(\lambda (H12: (drop h (minus d0 (S n)) x d)).(eq_ind_r T -(TLRef n) (\lambda (t0: T).(ty3 g c0 t0 (lift h d0 (lift (S n) O t)))) -(eq_ind nat (plus (S n) (minus d0 (S n))) (\lambda (n0: nat).(ty3 g c0 (TLRef -n) (lift h n0 (lift (S n) O t)))) (eq_ind_r T (lift (S n) O (lift h (minus d0 -(S n)) t)) (\lambda (t0: T).(ty3 g c0 (TLRef n) t0)) (eq_ind nat d0 (\lambda -(_: nat).(ty3 g c0 (TLRef n) (lift (S n) O (lift h (minus d0 (S n)) t)))) -(ty3_abbr g n c0 x (lift h (minus d0 (S n)) u) (getl_intro n c0 (CHead x -(Bind Abbr) (lift h (minus d0 (S n)) u)) x0 H6 H11) (lift h (minus d0 (S n)) -t) (H2 x (minus d0 (S n)) h H12)) (plus (S n) (minus d0 (S n))) -(le_plus_minus (S n) d0 H4)) (lift h (plus (S n) (minus d0 (S n))) (lift (S -n) O t)) (lift_d t h (S n) (minus d0 (S n)) O (le_O_n (minus d0 (S n))))) d0 -(le_plus_minus_r (S n) d0 H4)) (lift h d0 (TLRef n)) (lift_lref_lt n h d0 -H4))))) H10)))))))) H5))) (\lambda (H4: (le d0 n)).(eq_ind_r T (TLRef (plus n -h)) (\lambda (t0: T).(ty3 g c0 t0 (lift h d0 (lift (S n) O t)))) (eq_ind nat -(S n) (\lambda (_: nat).(ty3 g c0 (TLRef (plus n h)) (lift h d0 (lift (S n) O -t)))) (eq_ind_r T (lift (plus h (S n)) O t) (\lambda (t0: T).(ty3 g c0 (TLRef -(plus n h)) t0)) (eq_ind_r nat (plus (S n) h) (\lambda (n0: nat).(ty3 g c0 -(TLRef (plus n h)) (lift n0 O t))) (ty3_abbr g (plus n h) c0 d u -(drop_getl_trans_ge n c0 c d0 h H3 (CHead d (Bind Abbr) u) H0 H4) t H1) (plus -h (S n)) (plus_sym h (S n))) (lift h d0 (lift (S n) O t)) (lift_free t (S n) -h O d0 (le_S d0 n H4) (le_O_n d0))) (plus n (S O)) (eq_ind_r nat (plus (S O) -n) (\lambda (n0: nat).(eq nat (S n) n0)) (refl_equal nat (plus (S O) n)) -(plus n (S O)) (plus_sym n (S O)))) (lift h d0 (TLRef n)) (lift_lref_ge n h -d0 H4)))))))))))))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d: -C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind Abst) u))).(\lambda -(t: T).(\lambda (H1: (ty3 g d u t)).(\lambda (H2: ((\forall (c0: C).(\forall -(d0: nat).(\forall (h: nat).((drop h d0 c0 d) \to (ty3 g c0 (lift h d0 u) -(lift h d0 t)))))))).(\lambda (c0: C).(\lambda (d0: nat).(\lambda (h: -nat).(\lambda (H3: (drop h d0 c0 c)).(lt_le_e n d0 (ty3 g c0 (lift h d0 -(TLRef n)) (lift h d0 (lift (S n) O u))) (\lambda (H4: (lt n d0)).(let H5 -\def (drop_getl_trans_le n d0 (le_S_n n d0 (le_S (S n) d0 H4)) c0 c h H3 -(CHead d (Bind Abst) u) H0) in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: -C).(drop n O c0 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d0 n) -e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear e1 (CHead d (Bind Abst) -u)))) (ty3 g c0 (lift h d0 (TLRef n)) (lift h d0 (lift (S n) O u))) (\lambda -(x0: C).(\lambda (x1: C).(\lambda (H6: (drop n O c0 x0)).(\lambda (H7: (drop -h (minus d0 n) x0 x1)).(\lambda (H8: (clear x1 (CHead d (Bind Abst) u))).(let -H9 \def (eq_ind nat (minus d0 n) (\lambda (n0: nat).(drop h n0 x0 x1)) H7 (S -(minus d0 (S n))) (minus_x_Sy d0 n H4)) in (let H10 \def (drop_clear_S x1 x0 -h (minus d0 (S n)) H9 Abst d u H8) in (ex2_ind C (\lambda (c1: C).(clear x0 -(CHead c1 (Bind Abst) (lift h (minus d0 (S n)) u)))) (\lambda (c1: C).(drop h -(minus d0 (S n)) c1 d)) (ty3 g c0 (lift h d0 (TLRef n)) (lift h d0 (lift (S -n) O u))) (\lambda (x: C).(\lambda (H11: (clear x0 (CHead x (Bind Abst) (lift -h (minus d0 (S n)) u)))).(\lambda (H12: (drop h (minus d0 (S n)) x -d)).(eq_ind_r T (TLRef n) (\lambda (t0: T).(ty3 g c0 t0 (lift h d0 (lift (S -n) O u)))) (eq_ind nat (plus (S n) (minus d0 (S n))) (\lambda (n0: nat).(ty3 -g c0 (TLRef n) (lift h n0 (lift (S n) O u)))) (eq_ind_r T (lift (S n) O (lift -h (minus d0 (S n)) u)) (\lambda (t0: T).(ty3 g c0 (TLRef n) t0)) (eq_ind nat -d0 (\lambda (_: nat).(ty3 g c0 (TLRef n) (lift (S n) O (lift h (minus d0 (S -n)) u)))) (ty3_abst g n c0 x (lift h (minus d0 (S n)) u) (getl_intro n c0 -(CHead x (Bind Abst) (lift h (minus d0 (S n)) u)) x0 H6 H11) (lift h (minus -d0 (S n)) t) (H2 x (minus d0 (S n)) h H12)) (plus (S n) (minus d0 (S n))) -(le_plus_minus (S n) d0 H4)) (lift h (plus (S n) (minus d0 (S n))) (lift (S -n) O u)) (lift_d u h (S n) (minus d0 (S n)) O (le_O_n (minus d0 (S n))))) d0 -(le_plus_minus_r (S n) d0 H4)) (lift h d0 (TLRef n)) (lift_lref_lt n h d0 -H4))))) H10)))))))) H5))) (\lambda (H4: (le d0 n)).(eq_ind_r T (TLRef (plus n -h)) (\lambda (t0: T).(ty3 g c0 t0 (lift h d0 (lift (S n) O u)))) (eq_ind nat -(S n) (\lambda (_: nat).(ty3 g c0 (TLRef (plus n h)) (lift h d0 (lift (S n) O -u)))) (eq_ind_r T (lift (plus h (S n)) O u) (\lambda (t0: T).(ty3 g c0 (TLRef -(plus n h)) t0)) (eq_ind_r nat (plus (S n) h) (\lambda (n0: nat).(ty3 g c0 -(TLRef (plus n h)) (lift n0 O u))) (ty3_abst g (plus n h) c0 d u -(drop_getl_trans_ge n c0 c d0 h H3 (CHead d (Bind Abst) u) H0 H4) t H1) (plus -h (S n)) (plus_sym h (S n))) (lift h d0 (lift (S n) O u)) (lift_free u (S n) -h O d0 (le_S d0 n H4) (le_O_n d0))) (plus n (S O)) (eq_ind_r nat (plus (S O) -n) (\lambda (n0: nat).(eq nat (S n) n0)) (refl_equal nat (plus (S O) n)) -(plus n (S O)) (plus_sym n (S O)))) (lift h d0 (TLRef n)) (lift_lref_ge n h -d0 H4)))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (t: -T).(\lambda (_: (ty3 g c u t)).(\lambda (H1: ((\forall (c0: C).(\forall (d: -nat).(\forall (h: nat).((drop h d c0 c) \to (ty3 g c0 (lift h d u) (lift h d -t)))))))).(\lambda (b: B).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (ty3 -g (CHead c (Bind b) u) t0 t3)).(\lambda (H3: ((\forall (c0: C).(\forall (d: -nat).(\forall (h: nat).((drop h d c0 (CHead c (Bind b) u)) \to (ty3 g c0 -(lift h d t0) (lift h d t3)))))))).(\lambda (c0: C).(\lambda (d: -nat).(\lambda (h: nat).(\lambda (H4: (drop h d c0 c)).(eq_ind_r T (THead -(Bind b) (lift h d u) (lift h (s (Bind b) d) t0)) (\lambda (t4: T).(ty3 g c0 -t4 (lift h d (THead (Bind b) u t3)))) (eq_ind_r T (THead (Bind b) (lift h d -u) (lift h (s (Bind b) d) t3)) (\lambda (t4: T).(ty3 g c0 (THead (Bind b) -(lift h d u) (lift h (s (Bind b) d) t0)) t4)) (ty3_bind g c0 (lift h d u) -(lift h d t) (H1 c0 d h H4) b (lift h (S d) t0) (lift h (S d) t3) (H3 (CHead -c0 (Bind b) (lift h d u)) (S d) h (drop_skip_bind h d c0 c H4 b u))) (lift h -d (THead (Bind b) u t3)) (lift_head (Bind b) u t3 h d)) (lift h d (THead -(Bind b) u t0)) (lift_head (Bind b) u t0 h d)))))))))))))))) (\lambda (c: -C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c w u)).(\lambda (H1: -((\forall (c0: C).(\forall (d: nat).(\forall (h: nat).((drop h d c0 c) \to -(ty3 g c0 (lift h d w) (lift h d u)))))))).(\lambda (v: T).(\lambda (t: -T).(\lambda (_: (ty3 g c v (THead (Bind Abst) u t))).(\lambda (H3: ((\forall -(c0: C).(\forall (d: nat).(\forall (h: nat).((drop h d c0 c) \to (ty3 g c0 -(lift h d v) (lift h d (THead (Bind Abst) u t))))))))).(\lambda (c0: -C).(\lambda (d: nat).(\lambda (h: nat).(\lambda (H4: (drop h d c0 -c)).(eq_ind_r T (THead (Flat Appl) (lift h d w) (lift h (s (Flat Appl) d) v)) -(\lambda (t0: T).(ty3 g c0 t0 (lift h d (THead (Flat Appl) w (THead (Bind -Abst) u t))))) (eq_ind_r T (THead (Flat Appl) (lift h d w) (lift h (s (Flat -Appl) d) (THead (Bind Abst) u t))) (\lambda (t0: T).(ty3 g c0 (THead (Flat -Appl) (lift h d w) (lift h (s (Flat Appl) d) v)) t0)) (eq_ind_r T (THead -(Bind Abst) (lift h (s (Flat Appl) d) u) (lift h (s (Bind Abst) (s (Flat -Appl) d)) t)) (\lambda (t0: T).(ty3 g c0 (THead (Flat Appl) (lift h d w) -(lift h (s (Flat Appl) d) v)) (THead (Flat Appl) (lift h d w) t0))) (ty3_appl -g c0 (lift h d w) (lift h d u) (H1 c0 d h H4) (lift h d v) (lift h (S d) t) -(eq_ind T (lift h d (THead (Bind Abst) u t)) (\lambda (t0: T).(ty3 g c0 (lift -h d v) t0)) (H3 c0 d h H4) (THead (Bind Abst) (lift h d u) (lift h (S d) t)) -(lift_bind Abst u t h d))) (lift h (s (Flat Appl) d) (THead (Bind Abst) u t)) -(lift_head (Bind Abst) u t h (s (Flat Appl) d))) (lift h d (THead (Flat Appl) -w (THead (Bind Abst) u t))) (lift_head (Flat Appl) w (THead (Bind Abst) u t) -h d)) (lift h d (THead (Flat Appl) w v)) (lift_head (Flat Appl) w v h -d))))))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda (t3: T).(\lambda -(_: (ty3 g c t0 t3)).(\lambda (H1: ((\forall (c0: C).(\forall (d: -nat).(\forall (h: nat).((drop h d c0 c) \to (ty3 g c0 (lift h d t0) (lift h d -t3)))))))).(\lambda (t4: T).(\lambda (_: (ty3 g c t3 t4)).(\lambda (H3: -((\forall (c0: C).(\forall (d: nat).(\forall (h: nat).((drop h d c0 c) \to -(ty3 g c0 (lift h d t3) (lift h d t4)))))))).(\lambda (c0: C).(\lambda (d: -nat).(\lambda (h: nat).(\lambda (H4: (drop h d c0 c)).(eq_ind_r T (THead -(Flat Cast) (lift h d t3) (lift h (s (Flat Cast) d) t0)) (\lambda (t: T).(ty3 -g c0 t (lift h d (THead (Flat Cast) t4 t3)))) (eq_ind_r T (THead (Flat Cast) -(lift h d t4) (lift h (s (Flat Cast) d) t3)) (\lambda (t: T).(ty3 g c0 (THead -(Flat Cast) (lift h d t3) (lift h (s (Flat Cast) d) t0)) t)) (ty3_cast g c0 -(lift h (s (Flat Cast) d) t0) (lift h (s (Flat Cast) d) t3) (H1 c0 (s (Flat -Cast) d) h H4) (lift h d t4) (H3 c0 d h H4)) (lift h d (THead (Flat Cast) t4 -t3)) (lift_head (Flat Cast) t4 t3 h d)) (lift h d (THead (Flat Cast) t3 t0)) -(lift_head (Flat Cast) t3 t0 h d)))))))))))))) e t1 t2 H))))). -(* COMMENTS -Initial nodes: 4253 -END *) - -theorem ty3_correct: - \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c -t1 t2) \to (ex T (\lambda (t: T).(ty3 g c t2 t))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H: (ty3 g c t1 t2)).(ty3_ind g (\lambda (c0: C).(\lambda (_: T).(\lambda -(t0: T).(ex T (\lambda (t3: T).(ty3 g c0 t0 t3)))))) (\lambda (c0: -C).(\lambda (t0: T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 t0 t)).(\lambda -(_: (ex T (\lambda (t3: T).(ty3 g c0 t t3)))).(\lambda (u: T).(\lambda (t3: -T).(\lambda (_: (ty3 g c0 u t3)).(\lambda (_: (ex T (\lambda (t4: T).(ty3 g -c0 t3 t4)))).(\lambda (_: (pc3 c0 t3 t0)).(ex_intro T (\lambda (t4: T).(ty3 g -c0 t0 t4)) t H0))))))))))) (\lambda (c0: C).(\lambda (m: nat).(ex_intro T -(\lambda (t: T).(ty3 g c0 (TSort (next g m)) t)) (TSort (next g (next g m))) -(ty3_sort g c0 (next g m))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: -C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abbr) -u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2: (ex T (\lambda -(t0: T).(ty3 g d t t0)))).(let H3 \def H2 in (ex_ind T (\lambda (t0: T).(ty3 -g d t t0)) (ex T (\lambda (t0: T).(ty3 g c0 (lift (S n) O t) t0))) (\lambda -(x: T).(\lambda (H4: (ty3 g d t x)).(ex_intro T (\lambda (t0: T).(ty3 g c0 -(lift (S n) O t) t0)) (lift (S n) O x) (ty3_lift g d t x H4 c0 O (S n) -(getl_drop Abbr c0 d u n H0))))) H3)))))))))) (\lambda (n: nat).(\lambda (c0: -C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind -Abst) u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u t)).(\lambda (_: (ex T -(\lambda (t0: T).(ty3 g d t t0)))).(ex_intro T (\lambda (t0: T).(ty3 g c0 -(lift (S n) O u) t0)) (lift (S n) O t) (ty3_lift g d u t H1 c0 O (S n) -(getl_drop Abst c0 d u n H0))))))))))) (\lambda (c0: C).(\lambda (u: -T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 u t)).(\lambda (_: (ex T (\lambda -(t0: T).(ty3 g c0 t t0)))).(\lambda (b: B).(\lambda (t0: T).(\lambda (t3: -T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t0 t3)).(\lambda (H3: (ex T -(\lambda (t4: T).(ty3 g (CHead c0 (Bind b) u) t3 t4)))).(let H4 \def H3 in -(ex_ind T (\lambda (t4: T).(ty3 g (CHead c0 (Bind b) u) t3 t4)) (ex T -(\lambda (t4: T).(ty3 g c0 (THead (Bind b) u t3) t4))) (\lambda (x: -T).(\lambda (H5: (ty3 g (CHead c0 (Bind b) u) t3 x)).(ex_intro T (\lambda -(t4: T).(ty3 g c0 (THead (Bind b) u t3) t4)) (THead (Bind b) u x) (ty3_bind g -c0 u t H0 b t3 x H5)))) H4)))))))))))) (\lambda (c0: C).(\lambda (w: -T).(\lambda (u: T).(\lambda (H0: (ty3 g c0 w u)).(\lambda (H1: (ex T (\lambda -(t: T).(ty3 g c0 u t)))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g -c0 v (THead (Bind Abst) u t))).(\lambda (H3: (ex T (\lambda (t0: T).(ty3 g c0 -(THead (Bind Abst) u t) t0)))).(let H4 \def H1 in (ex_ind T (\lambda (t0: -T).(ty3 g c0 u t0)) (ex T (\lambda (t0: T).(ty3 g c0 (THead (Flat Appl) w -(THead (Bind Abst) u t)) t0))) (\lambda (x: T).(\lambda (_: (ty3 g c0 u -x)).(let H6 \def H3 in (ex_ind T (\lambda (t0: T).(ty3 g c0 (THead (Bind -Abst) u t) t0)) (ex T (\lambda (t0: T).(ty3 g c0 (THead (Flat Appl) w (THead -(Bind Abst) u t)) t0))) (\lambda (x0: T).(\lambda (H7: (ty3 g c0 (THead (Bind -Abst) u t) x0)).(ex3_2_ind T T (\lambda (t3: T).(\lambda (_: T).(pc3 c0 -(THead (Bind Abst) u t3) x0))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u -t0))) (\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind Abst) u) t -t3))) (ex T (\lambda (t0: T).(ty3 g c0 (THead (Flat Appl) w (THead (Bind -Abst) u t)) t0))) (\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 c0 -(THead (Bind Abst) u x1) x0)).(\lambda (H9: (ty3 g c0 u x2)).(\lambda (H10: -(ty3 g (CHead c0 (Bind Abst) u) t x1)).(ex_intro T (\lambda (t0: T).(ty3 g c0 -(THead (Flat Appl) w (THead (Bind Abst) u t)) t0)) (THead (Flat Appl) w -(THead (Bind Abst) u x1)) (ty3_appl g c0 w u H0 (THead (Bind Abst) u t) x1 -(ty3_bind g c0 u x2 H9 Abst t x1 H10)))))))) (ty3_gen_bind g Abst c0 u t x0 -H7)))) H6)))) H4))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t3: -T).(\lambda (_: (ty3 g c0 t0 t3)).(\lambda (_: (ex T (\lambda (t: T).(ty3 g -c0 t3 t)))).(\lambda (t4: T).(\lambda (H2: (ty3 g c0 t3 t4)).(\lambda (H3: -(ex T (\lambda (t: T).(ty3 g c0 t4 t)))).(let H4 \def H3 in (ex_ind T -(\lambda (t: T).(ty3 g c0 t4 t)) (ex T (\lambda (t: T).(ty3 g c0 (THead (Flat -Cast) t4 t3) t))) (\lambda (x: T).(\lambda (H5: (ty3 g c0 t4 x)).(ex_intro T -(\lambda (t: T).(ty3 g c0 (THead (Flat Cast) t4 t3) t)) (THead (Flat Cast) x -t4) (ty3_cast g c0 t3 t4 H2 x H5)))) H4)))))))))) c t1 t2 H))))). -(* COMMENTS -Initial nodes: 1333 -END *) - -theorem ty3_unique: - \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).((ty3 g c u -t1) \to (\forall (t2: T).((ty3 g c u t2) \to (pc3 c t1 t2))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (H: -(ty3 g c u t1)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: -T).(\forall (t2: T).((ty3 g c0 t t2) \to (pc3 c0 t0 t2)))))) (\lambda (c0: -C).(\lambda (t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda -(_: ((\forall (t3: T).((ty3 g c0 t2 t3) \to (pc3 c0 t t3))))).(\lambda (u0: -T).(\lambda (t0: T).(\lambda (_: (ty3 g c0 u0 t0)).(\lambda (H3: ((\forall -(t3: T).((ty3 g c0 u0 t3) \to (pc3 c0 t0 t3))))).(\lambda (H4: (pc3 c0 t0 -t2)).(\lambda (t3: T).(\lambda (H5: (ty3 g c0 u0 t3)).(pc3_t t0 c0 t2 (pc3_s -c0 t2 t0 H4) t3 (H3 t3 H5)))))))))))))) (\lambda (c0: C).(\lambda (m: -nat).(\lambda (t2: T).(\lambda (H0: (ty3 g c0 (TSort m) t2)).(ty3_gen_sort g -c0 t2 m H0))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda -(u0: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abbr) u0))).(\lambda (t: -T).(\lambda (_: (ty3 g d u0 t)).(\lambda (H2: ((\forall (t2: T).((ty3 g d u0 -t2) \to (pc3 d t t2))))).(\lambda (t2: T).(\lambda (H3: (ty3 g c0 (TLRef n) -t2)).(or_ind (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: -T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda -(_: T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: C).(\lambda (u1: -T).(\lambda (t0: T).(ty3 g e u1 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda -(u1: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u1) t2)))) (\lambda (e: -C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u1))))) -(\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 t0))))) (pc3 c0 -(lift (S n) O t) t2) (\lambda (H4: (ex3_3 C T T (\lambda (_: C).(\lambda (_: -T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda (e: C).(\lambda -(u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: -C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 t0)))))).(ex3_3_ind C T T -(\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) -t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e -(Bind Abbr) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g -e u1 t0)))) (pc3 c0 (lift (S n) O t) t2) (\lambda (x0: C).(\lambda (x1: -T).(\lambda (x2: T).(\lambda (H5: (pc3 c0 (lift (S n) O x2) t2)).(\lambda -(H6: (getl n c0 (CHead x0 (Bind Abbr) x1))).(\lambda (H7: (ty3 g x0 x1 -x2)).(let H8 \def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda (c1: C).(getl n -c0 c1)) H0 (CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind Abbr) u0) n -H0 (CHead x0 (Bind Abbr) x1) H6)) in (let H9 \def (f_equal C C (\lambda (e: -C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | -(CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u0) (CHead x0 (Bind -Abbr) x1) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead x0 (Bind Abbr) -x1) H6)) in ((let H10 \def (f_equal C T (\lambda (e: C).(match e in C return -(\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t0) -\Rightarrow t0])) (CHead d (Bind Abbr) u0) (CHead x0 (Bind Abbr) x1) -(getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in -(\lambda (H11: (eq C d x0)).(let H12 \def (eq_ind_r T x1 (\lambda (t0: -T).(getl n c0 (CHead x0 (Bind Abbr) t0))) H8 u0 H10) in (let H13 \def -(eq_ind_r T x1 (\lambda (t0: T).(ty3 g x0 t0 x2)) H7 u0 H10) in (let H14 \def -(eq_ind_r C x0 (\lambda (c1: C).(getl n c0 (CHead c1 (Bind Abbr) u0))) H12 d -H11) in (let H15 \def (eq_ind_r C x0 (\lambda (c1: C).(ty3 g c1 u0 x2)) H13 d -H11) in (pc3_t (lift (S n) O x2) c0 (lift (S n) O t) (pc3_lift c0 d (S n) O -(getl_drop Abbr c0 d u0 n H14) t x2 (H2 x2 H15)) t2 H5))))))) H9))))))))) -H4)) (\lambda (H4: (ex3_3 C T T (\lambda (_: C).(\lambda (u1: T).(\lambda (_: -T).(pc3 c0 (lift (S n) O u1) t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda -(_: T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1: -T).(\lambda (t0: T).(ty3 g e u1 t0)))))).(ex3_3_ind C T T (\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u1) t2)))) (\lambda -(e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) -u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 t0)))) -(pc3 c0 (lift (S n) O t) t2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: -T).(\lambda (_: (pc3 c0 (lift (S n) O x1) t2)).(\lambda (H6: (getl n c0 -(CHead x0 (Bind Abst) x1))).(\lambda (_: (ty3 g x0 x1 x2)).(let H8 \def -(eq_ind C (CHead d (Bind Abbr) u0) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead -x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead x0 -(Bind Abst) x1) H6)) in (let H9 \def (eq_ind C (CHead d (Bind Abbr) u0) -(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind b) \Rightarrow (match b in B return (\lambda (_: -B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | Void -\Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead x0 (Bind Abst) -x1) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead x0 (Bind Abst) x1) -H6)) in (False_ind (pc3 c0 (lift (S n) O t) t2) H9))))))))) H4)) -(ty3_gen_lref g c0 t2 n H3)))))))))))) (\lambda (n: nat).(\lambda (c0: -C).(\lambda (d: C).(\lambda (u0: T).(\lambda (H0: (getl n c0 (CHead d (Bind -Abst) u0))).(\lambda (t: T).(\lambda (_: (ty3 g d u0 t)).(\lambda (_: -((\forall (t2: T).((ty3 g d u0 t2) \to (pc3 d t t2))))).(\lambda (t2: -T).(\lambda (H3: (ty3 g c0 (TLRef n) t2)).(or_ind (ex3_3 C T T (\lambda (_: -C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda -(e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) -u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 t0))))) -(ex3_3 C T T (\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(pc3 c0 (lift -(S n) O u1) t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n -c0 (CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda -(t0: T).(ty3 g e u1 t0))))) (pc3 c0 (lift (S n) O u0) t2) (\lambda (H4: -(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift -(S n) O t0) t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n -c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda -(t0: T).(ty3 g e u1 t0)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (_: -T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda (e: C).(\lambda -(u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: -C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 t0)))) (pc3 c0 (lift (S n) O -u0) t2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 -c0 (lift (S n) O x2) t2)).(\lambda (H6: (getl n c0 (CHead x0 (Bind Abbr) -x1))).(\lambda (_: (ty3 g x0 x1 x2)).(let H8 \def (eq_ind C (CHead d (Bind -Abst) u0) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead x0 (Bind Abbr) x1) -(getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in -(let H9 \def (eq_ind C (CHead d (Bind Abst) u0) (\lambda (ee: C).(match ee in -C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k -_) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b) -\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow -False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) -\Rightarrow False])])) I (CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d -(Bind Abst) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in (False_ind (pc3 c0 -(lift (S n) O u0) t2) H9))))))))) H4)) (\lambda (H4: (ex3_3 C T T (\lambda -(_: C).(\lambda (u1: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u1) t2)))) -(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind -Abst) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 -t0)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (u1: T).(\lambda (_: -T).(pc3 c0 (lift (S n) O u1) t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda -(_: T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1: -T).(\lambda (t0: T).(ty3 g e u1 t0)))) (pc3 c0 (lift (S n) O u0) t2) (\lambda -(x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H5: (pc3 c0 (lift (S n) O -x1) t2)).(\lambda (H6: (getl n c0 (CHead x0 (Bind Abst) x1))).(\lambda (H7: -(ty3 g x0 x1 x2)).(let H8 \def (eq_ind C (CHead d (Bind Abst) u0) (\lambda -(c1: C).(getl n c0 c1)) H0 (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d -(Bind Abst) u0) n H0 (CHead x0 (Bind Abst) x1) H6)) in (let H9 \def (f_equal -C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abst) u0) -(CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead -x0 (Bind Abst) x1) H6)) in ((let H10 \def (f_equal C T (\lambda (e: C).(match -e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ -t0) \Rightarrow t0])) (CHead d (Bind Abst) u0) (CHead x0 (Bind Abst) x1) -(getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead x0 (Bind Abst) x1) H6)) in -(\lambda (H11: (eq C d x0)).(let H12 \def (eq_ind_r T x1 (\lambda (t0: -T).(getl n c0 (CHead x0 (Bind Abst) t0))) H8 u0 H10) in (let H13 \def -(eq_ind_r T x1 (\lambda (t0: T).(ty3 g x0 t0 x2)) H7 u0 H10) in (let H14 \def -(eq_ind_r T x1 (\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)) H5 u0 H10) in -(let H15 \def (eq_ind_r C x0 (\lambda (c1: C).(getl n c0 (CHead c1 (Bind -Abst) u0))) H12 d H11) in (let H16 \def (eq_ind_r C x0 (\lambda (c1: C).(ty3 -g c1 u0 x2)) H13 d H11) in H14))))))) H9))))))))) H4)) (ty3_gen_lref g c0 t2 -n H3)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (t: T).(\lambda -(_: (ty3 g c0 u0 t)).(\lambda (_: ((\forall (t2: T).((ty3 g c0 u0 t2) \to -(pc3 c0 t t2))))).(\lambda (b: B).(\lambda (t0: T).(\lambda (t2: T).(\lambda -(_: (ty3 g (CHead c0 (Bind b) u0) t0 t2)).(\lambda (H3: ((\forall (t3: -T).((ty3 g (CHead c0 (Bind b) u0) t0 t3) \to (pc3 (CHead c0 (Bind b) u0) t2 -t3))))).(\lambda (t3: T).(\lambda (H4: (ty3 g c0 (THead (Bind b) u0 t0) -t3)).(ex3_2_ind T T (\lambda (t4: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) -u0 t4) t3))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c0 u0 t5))) (\lambda -(t4: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u0) t0 t4))) (pc3 c0 (THead -(Bind b) u0 t2) t3) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (pc3 c0 -(THead (Bind b) u0 x0) t3)).(\lambda (_: (ty3 g c0 u0 x1)).(\lambda (H7: (ty3 -g (CHead c0 (Bind b) u0) t0 x0)).(pc3_t (THead (Bind b) u0 x0) c0 (THead -(Bind b) u0 t2) (pc3_head_2 c0 u0 t2 x0 (Bind b) (H3 x0 H7)) t3 H5)))))) -(ty3_gen_bind g b c0 u0 t0 t3 H4)))))))))))))) (\lambda (c0: C).(\lambda (w: -T).(\lambda (u0: T).(\lambda (_: (ty3 g c0 w u0)).(\lambda (_: ((\forall (t2: -T).((ty3 g c0 w t2) \to (pc3 c0 u0 t2))))).(\lambda (v: T).(\lambda (t: -T).(\lambda (_: (ty3 g c0 v (THead (Bind Abst) u0 t))).(\lambda (H3: -((\forall (t2: T).((ty3 g c0 v t2) \to (pc3 c0 (THead (Bind Abst) u0 t) -t2))))).(\lambda (t2: T).(\lambda (H4: (ty3 g c0 (THead (Flat Appl) w v) -t2)).(ex3_2_ind T T (\lambda (u1: T).(\lambda (t0: T).(pc3 c0 (THead (Flat -Appl) w (THead (Bind Abst) u1 t0)) t2))) (\lambda (u1: T).(\lambda (t0: -T).(ty3 g c0 v (THead (Bind Abst) u1 t0)))) (\lambda (u1: T).(\lambda (_: -T).(ty3 g c0 w u1))) (pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t)) -t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (pc3 c0 (THead (Flat -Appl) w (THead (Bind Abst) x0 x1)) t2)).(\lambda (H6: (ty3 g c0 v (THead -(Bind Abst) x0 x1))).(\lambda (_: (ty3 g c0 w x0)).(pc3_t (THead (Flat Appl) -w (THead (Bind Abst) x0 x1)) c0 (THead (Flat Appl) w (THead (Bind Abst) u0 -t)) (pc3_thin_dx c0 (THead (Bind Abst) u0 t) (THead (Bind Abst) x0 x1) (H3 -(THead (Bind Abst) x0 x1) H6) w Appl) t2 H5)))))) (ty3_gen_appl g c0 w v t2 -H4))))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: T).(\lambda -(_: (ty3 g c0 t0 t2)).(\lambda (_: ((\forall (t3: T).((ty3 g c0 t0 t3) \to -(pc3 c0 t2 t3))))).(\lambda (t3: T).(\lambda (_: (ty3 g c0 t2 t3)).(\lambda -(H3: ((\forall (t4: T).((ty3 g c0 t2 t4) \to (pc3 c0 t3 t4))))).(\lambda (t4: -T).(\lambda (H4: (ty3 g c0 (THead (Flat Cast) t2 t0) t4)).(ex3_ind T (\lambda -(t5: T).(pc3 c0 (THead (Flat Cast) t5 t2) t4)) (\lambda (_: T).(ty3 g c0 t0 -t2)) (\lambda (t5: T).(ty3 g c0 t2 t5)) (pc3 c0 (THead (Flat Cast) t3 t2) t4) -(\lambda (x0: T).(\lambda (H5: (pc3 c0 (THead (Flat Cast) x0 t2) -t4)).(\lambda (_: (ty3 g c0 t0 t2)).(\lambda (H7: (ty3 g c0 t2 x0)).(pc3_t -(THead (Flat Cast) x0 t2) c0 (THead (Flat Cast) t3 t2) (pc3_head_1 c0 t3 x0 -(H3 x0 H7) (Flat Cast) t2) t4 H5))))) (ty3_gen_cast g c0 t0 t2 t4 -H4)))))))))))) c u t1 H))))). -(* COMMENTS -Initial nodes: 3459 -END *) - -theorem ty3_gen_abst_abst: - \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall -(t2: T).((ty3 g c (THead (Bind Abst) u t1) (THead (Bind Abst) u t2)) \to (ex2 -T (\lambda (w: T).(ty3 g c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst) -u) t1 t2)))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda -(t2: T).(\lambda (H: (ty3 g c (THead (Bind Abst) u t1) (THead (Bind Abst) u -t2))).(ex_ind T (\lambda (t: T).(ty3 g c (THead (Bind Abst) u t2) t)) (ex2 T -(\lambda (w: T).(ty3 g c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) -t1 t2))) (\lambda (x: T).(\lambda (H0: (ty3 g c (THead (Bind Abst) u t2) -x)).(ex3_2_ind T T (\lambda (t3: T).(\lambda (_: T).(pc3 c (THead (Bind Abst) -u t3) x))) (\lambda (_: T).(\lambda (t: T).(ty3 g c u t))) (\lambda (t3: -T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t2 t3))) (ex2 T (\lambda -(w: T).(ty3 g c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t1 t2))) -(\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 c (THead (Bind Abst) u -x0) x)).(\lambda (_: (ty3 g c u x1)).(\lambda (H3: (ty3 g (CHead c (Bind -Abst) u) t2 x0)).(ex3_2_ind T T (\lambda (t3: T).(\lambda (_: T).(pc3 c -(THead (Bind Abst) u t3) (THead (Bind Abst) u t2)))) (\lambda (_: T).(\lambda -(t: T).(ty3 g c u t))) (\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c (Bind -Abst) u) t1 t3))) (ex2 T (\lambda (w: T).(ty3 g c u w)) (\lambda (_: T).(ty3 -g (CHead c (Bind Abst) u) t1 t2))) (\lambda (x2: T).(\lambda (x3: T).(\lambda -(H4: (pc3 c (THead (Bind Abst) u x2) (THead (Bind Abst) u t2))).(\lambda (H5: -(ty3 g c u x3)).(\lambda (H6: (ty3 g (CHead c (Bind Abst) u) t1 x2)).(let H_y -\def (pc3_gen_abst_shift c u x2 t2 H4) in (ex_intro2 T (\lambda (w: T).(ty3 g -c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t1 t2)) x3 H5 -(ty3_conv g (CHead c (Bind Abst) u) t2 x0 H3 t1 x2 H6 H_y)))))))) -(ty3_gen_bind g Abst c u t1 (THead (Bind Abst) u t2) H))))))) (ty3_gen_bind g -Abst c u t2 x H0)))) (ty3_correct g c (THead (Bind Abst) u t1) (THead (Bind -Abst) u t2) H))))))). -(* COMMENTS -Initial nodes: 571 -END *) - -theorem ty3_typecheck: - \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (v: T).((ty3 g c t -v) \to (ex T (\lambda (u: T).(ty3 g c (THead (Flat Cast) v t) u))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (v: T).(\lambda (H: -(ty3 g c t v)).(ex_ind T (\lambda (t0: T).(ty3 g c v t0)) (ex T (\lambda (u: -T).(ty3 g c (THead (Flat Cast) v t) u))) (\lambda (x: T).(\lambda (H0: (ty3 g -c v x)).(ex_intro T (\lambda (u: T).(ty3 g c (THead (Flat Cast) v t) u)) -(THead (Flat Cast) x v) (ty3_cast g c t v H x H0)))) (ty3_correct g c t v -H)))))). -(* COMMENTS -Initial nodes: 131 -END *) - -theorem ty3_getl_subst0: - \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t -u) \to (\forall (v0: T).(\forall (t0: T).(\forall (i: nat).((subst0 i v0 t -t0) \to (\forall (b: B).(\forall (d: C).(\forall (v: T).((getl i c (CHead d -(Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v w))))))))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H: -(ty3 g c t u)).(ty3_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (_: -T).(\forall (v0: T).(\forall (t2: T).(\forall (i: nat).((subst0 i v0 t0 t2) -\to (\forall (b: B).(\forall (d: C).(\forall (v: T).((getl i c0 (CHead d -(Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v w)))))))))))))) (\lambda -(c0: C).(\lambda (t2: T).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t2 -t0)).(\lambda (_: ((\forall (v0: T).(\forall (t1: T).(\forall (i: -nat).((subst0 i v0 t2 t1) \to (\forall (b: B).(\forall (d: C).(\forall (v: -T).((getl i c0 (CHead d (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v -w))))))))))))).(\lambda (u0: T).(\lambda (t1: T).(\lambda (_: (ty3 g c0 u0 -t1)).(\lambda (H3: ((\forall (v0: T).(\forall (t3: T).(\forall (i: -nat).((subst0 i v0 u0 t3) \to (\forall (b: B).(\forall (d: C).(\forall (v: -T).((getl i c0 (CHead d (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v -w))))))))))))).(\lambda (_: (pc3 c0 t1 t2)).(\lambda (v0: T).(\lambda (t3: -T).(\lambda (i: nat).(\lambda (H5: (subst0 i v0 u0 t3)).(\lambda (b: -B).(\lambda (d: C).(\lambda (v: T).(\lambda (H6: (getl i c0 (CHead d (Bind b) -v))).(H3 v0 t3 i H5 b d v H6))))))))))))))))))) (\lambda (c0: C).(\lambda (m: -nat).(\lambda (v0: T).(\lambda (t0: T).(\lambda (i: nat).(\lambda (H0: -(subst0 i v0 (TSort m) t0)).(\lambda (b: B).(\lambda (d: C).(\lambda (v: -T).(\lambda (_: (getl i c0 (CHead d (Bind b) v))).(subst0_gen_sort v0 t0 i m -H0 (ex T (\lambda (w: T).(ty3 g d v w)))))))))))))) (\lambda (n: -nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (H0: (getl n -c0 (CHead d (Bind Abbr) u0))).(\lambda (t0: T).(\lambda (H1: (ty3 g d u0 -t0)).(\lambda (_: ((\forall (v0: T).(\forall (t1: T).(\forall (i: -nat).((subst0 i v0 u0 t1) \to (\forall (b: B).(\forall (d0: C).(\forall (v: -T).((getl i d (CHead d0 (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d0 v -w))))))))))))).(\lambda (v0: T).(\lambda (t1: T).(\lambda (i: nat).(\lambda -(H3: (subst0 i v0 (TLRef n) t1)).(\lambda (b: B).(\lambda (d0: C).(\lambda -(v: T).(\lambda (H4: (getl i c0 (CHead d0 (Bind b) v))).(land_ind (eq nat n -i) (eq T t1 (lift (S n) O v0)) (ex T (\lambda (w: T).(ty3 g d0 v w))) -(\lambda (H5: (eq nat n i)).(\lambda (_: (eq T t1 (lift (S n) O v0))).(let H7 -\def (eq_ind_r nat i (\lambda (n0: nat).(getl n0 c0 (CHead d0 (Bind b) v))) -H4 n H5) in (let H8 \def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda (c1: -C).(getl n c0 c1)) H0 (CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind -Abbr) u0) n H0 (CHead d0 (Bind b) v) H7)) in (let H9 \def (f_equal C C -(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u0) -(CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead d0 -(Bind b) v) H7)) in ((let H10 \def (f_equal C B (\lambda (e: C).(match e in C -return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b0) -\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u0) -(CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead d0 -(Bind b) v) H7)) in ((let H11 \def (f_equal C T (\lambda (e: C).(match e in C -return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t2) -\Rightarrow t2])) (CHead d (Bind Abbr) u0) (CHead d0 (Bind b) v) (getl_mono -c0 (CHead d (Bind Abbr) u0) n H0 (CHead d0 (Bind b) v) H7)) in (\lambda (H12: -(eq B Abbr b)).(\lambda (H13: (eq C d d0)).(let H14 \def (eq_ind_r T v -(\lambda (t2: T).(getl n c0 (CHead d0 (Bind b) t2))) H8 u0 H11) in (eq_ind T -u0 (\lambda (t2: T).(ex T (\lambda (w: T).(ty3 g d0 t2 w)))) (let H15 \def -(eq_ind_r C d0 (\lambda (c1: C).(getl n c0 (CHead c1 (Bind b) u0))) H14 d -H13) in (eq_ind C d (\lambda (c1: C).(ex T (\lambda (w: T).(ty3 g c1 u0 w)))) -(let H16 \def (eq_ind_r B b (\lambda (b0: B).(getl n c0 (CHead d (Bind b0) -u0))) H15 Abbr H12) in (ex_intro T (\lambda (w: T).(ty3 g d u0 w)) t0 H1)) d0 -H13)) v H11))))) H10)) H9)))))) (subst0_gen_lref v0 t1 i n -H3)))))))))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: -C).(\lambda (u0: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abst) -u0))).(\lambda (t0: T).(\lambda (H1: (ty3 g d u0 t0)).(\lambda (_: ((\forall -(v0: T).(\forall (t1: T).(\forall (i: nat).((subst0 i v0 u0 t1) \to (\forall -(b: B).(\forall (d0: C).(\forall (v: T).((getl i d (CHead d0 (Bind b) v)) \to -(ex T (\lambda (w: T).(ty3 g d0 v w))))))))))))).(\lambda (v0: T).(\lambda -(t1: T).(\lambda (i: nat).(\lambda (H3: (subst0 i v0 (TLRef n) t1)).(\lambda -(b: B).(\lambda (d0: C).(\lambda (v: T).(\lambda (H4: (getl i c0 (CHead d0 -(Bind b) v))).(land_ind (eq nat n i) (eq T t1 (lift (S n) O v0)) (ex T -(\lambda (w: T).(ty3 g d0 v w))) (\lambda (H5: (eq nat n i)).(\lambda (_: (eq -T t1 (lift (S n) O v0))).(let H7 \def (eq_ind_r nat i (\lambda (n0: -nat).(getl n0 c0 (CHead d0 (Bind b) v))) H4 n H5) in (let H8 \def (eq_ind C -(CHead d (Bind Abst) u0) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead d0 (Bind -b) v) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead d0 (Bind b) v) H7)) -in (let H9 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda -(_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) -(CHead d (Bind Abst) u0) (CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind -Abst) u0) n H0 (CHead d0 (Bind b) v) H7)) in ((let H10 \def (f_equal C B -(\lambda (e: C).(match e in C return (\lambda (_: C).B) with [(CSort _) -\Rightarrow Abst | (CHead _ k _) \Rightarrow (match k in K return (\lambda -(_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abst])])) -(CHead d (Bind Abst) u0) (CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind -Abst) u0) n H0 (CHead d0 (Bind b) v) H7)) in ((let H11 \def (f_equal C T -(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) -\Rightarrow u0 | (CHead _ _ t2) \Rightarrow t2])) (CHead d (Bind Abst) u0) -(CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead d0 -(Bind b) v) H7)) in (\lambda (H12: (eq B Abst b)).(\lambda (H13: (eq C d -d0)).(let H14 \def (eq_ind_r T v (\lambda (t2: T).(getl n c0 (CHead d0 (Bind -b) t2))) H8 u0 H11) in (eq_ind T u0 (\lambda (t2: T).(ex T (\lambda (w: -T).(ty3 g d0 t2 w)))) (let H15 \def (eq_ind_r C d0 (\lambda (c1: C).(getl n -c0 (CHead c1 (Bind b) u0))) H14 d H13) in (eq_ind C d (\lambda (c1: C).(ex T -(\lambda (w: T).(ty3 g c1 u0 w)))) (let H16 \def (eq_ind_r B b (\lambda (b0: -B).(getl n c0 (CHead d (Bind b0) u0))) H15 Abst H12) in (ex_intro T (\lambda -(w: T).(ty3 g d u0 w)) t0 H1)) d0 H13)) v H11))))) H10)) H9)))))) -(subst0_gen_lref v0 t1 i n H3)))))))))))))))))) (\lambda (c0: C).(\lambda -(u0: T).(\lambda (t0: T).(\lambda (_: (ty3 g c0 u0 t0)).(\lambda (H1: -((\forall (v0: T).(\forall (t1: T).(\forall (i: nat).((subst0 i v0 u0 t1) \to -(\forall (b: B).(\forall (d: C).(\forall (v: T).((getl i c0 (CHead d (Bind b) -v)) \to (ex T (\lambda (w: T).(ty3 g d v w))))))))))))).(\lambda (b: -B).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) -u0) t1 t2)).(\lambda (H3: ((\forall (v0: T).(\forall (t3: T).(\forall (i: -nat).((subst0 i v0 t1 t3) \to (\forall (b0: B).(\forall (d: C).(\forall (v: -T).((getl i (CHead c0 (Bind b) u0) (CHead d (Bind b0) v)) \to (ex T (\lambda -(w: T).(ty3 g d v w))))))))))))).(\lambda (v0: T).(\lambda (t3: T).(\lambda -(i: nat).(\lambda (H4: (subst0 i v0 (THead (Bind b) u0 t1) t3)).(\lambda (b0: -B).(\lambda (d: C).(\lambda (v: T).(\lambda (H5: (getl i c0 (CHead d (Bind -b0) v))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t3 (THead (Bind b) u2 t1))) -(\lambda (u2: T).(subst0 i v0 u0 u2))) (ex2 T (\lambda (t4: T).(eq T t3 -(THead (Bind b) u0 t4))) (\lambda (t4: T).(subst0 (s (Bind b) i) v0 t1 t4))) -(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind b) u2 -t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 u0 u2))) (\lambda (_: -T).(\lambda (t4: T).(subst0 (s (Bind b) i) v0 t1 t4)))) (ex T (\lambda (w: -T).(ty3 g d v w))) (\lambda (H6: (ex2 T (\lambda (u2: T).(eq T t3 (THead -(Bind b) u2 t1))) (\lambda (u2: T).(subst0 i v0 u0 u2)))).(ex2_ind T (\lambda -(u2: T).(eq T t3 (THead (Bind b) u2 t1))) (\lambda (u2: T).(subst0 i v0 u0 -u2)) (ex T (\lambda (w: T).(ty3 g d v w))) (\lambda (x: T).(\lambda (_: (eq T -t3 (THead (Bind b) x t1))).(\lambda (H8: (subst0 i v0 u0 x)).(H1 v0 x i H8 b0 -d v H5)))) H6)) (\lambda (H6: (ex2 T (\lambda (t4: T).(eq T t3 (THead (Bind -b) u0 t4))) (\lambda (t4: T).(subst0 (s (Bind b) i) v0 t1 t4)))).(ex2_ind T -(\lambda (t4: T).(eq T t3 (THead (Bind b) u0 t4))) (\lambda (t4: T).(subst0 -(s (Bind b) i) v0 t1 t4)) (ex T (\lambda (w: T).(ty3 g d v w))) (\lambda (x: -T).(\lambda (_: (eq T t3 (THead (Bind b) u0 x))).(\lambda (H8: (subst0 (s -(Bind b) i) v0 t1 x)).(H3 v0 x (S i) H8 b0 d v (getl_head (Bind b) i c0 -(CHead d (Bind b0) v) H5 u0))))) H6)) (\lambda (H6: (ex3_2 T T (\lambda (u2: -T).(\lambda (t4: T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i v0 u0 u2))) (\lambda (_: T).(\lambda (t4: -T).(subst0 (s (Bind b) i) v0 t1 t4))))).(ex3_2_ind T T (\lambda (u2: -T).(\lambda (t4: T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i v0 u0 u2))) (\lambda (_: T).(\lambda (t4: -T).(subst0 (s (Bind b) i) v0 t1 t4))) (ex T (\lambda (w: T).(ty3 g d v w))) -(\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (eq T t3 (THead (Bind b) x0 -x1))).(\lambda (H8: (subst0 i v0 u0 x0)).(\lambda (_: (subst0 (s (Bind b) i) -v0 t1 x1)).(H1 v0 x0 i H8 b0 d v H5)))))) H6)) (subst0_gen_head (Bind b) v0 -u0 t1 t3 i H4)))))))))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda -(u0: T).(\lambda (_: (ty3 g c0 w u0)).(\lambda (H1: ((\forall (v0: -T).(\forall (t0: T).(\forall (i: nat).((subst0 i v0 w t0) \to (\forall (b: -B).(\forall (d: C).(\forall (v: T).((getl i c0 (CHead d (Bind b) v)) \to (ex -T (\lambda (w0: T).(ty3 g d v w0))))))))))))).(\lambda (v: T).(\lambda (t0: -T).(\lambda (_: (ty3 g c0 v (THead (Bind Abst) u0 t0))).(\lambda (H3: -((\forall (v0: T).(\forall (t1: T).(\forall (i: nat).((subst0 i v0 v t1) \to -(\forall (b: B).(\forall (d: C).(\forall (v1: T).((getl i c0 (CHead d (Bind -b) v1)) \to (ex T (\lambda (w0: T).(ty3 g d v1 w0))))))))))))).(\lambda (v0: -T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (H4: (subst0 i v0 (THead (Flat -Appl) w v) t1)).(\lambda (b: B).(\lambda (d: C).(\lambda (v1: T).(\lambda -(H5: (getl i c0 (CHead d (Bind b) v1))).(or3_ind (ex2 T (\lambda (u2: T).(eq -T t1 (THead (Flat Appl) u2 v))) (\lambda (u2: T).(subst0 i v0 w u2))) (ex2 T -(\lambda (t2: T).(eq T t1 (THead (Flat Appl) w t2))) (\lambda (t2: T).(subst0 -(s (Flat Appl) i) v0 v t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq -T t1 (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i -v0 w u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s (Flat Appl) i) v0 v -t2)))) (ex T (\lambda (w0: T).(ty3 g d v1 w0))) (\lambda (H6: (ex2 T (\lambda -(u2: T).(eq T t1 (THead (Flat Appl) u2 v))) (\lambda (u2: T).(subst0 i v0 w -u2)))).(ex2_ind T (\lambda (u2: T).(eq T t1 (THead (Flat Appl) u2 v))) -(\lambda (u2: T).(subst0 i v0 w u2)) (ex T (\lambda (w0: T).(ty3 g d v1 w0))) -(\lambda (x: T).(\lambda (_: (eq T t1 (THead (Flat Appl) x v))).(\lambda (H8: -(subst0 i v0 w x)).(H1 v0 x i H8 b d v1 H5)))) H6)) (\lambda (H6: (ex2 T -(\lambda (t2: T).(eq T t1 (THead (Flat Appl) w t2))) (\lambda (t2: T).(subst0 -(s (Flat Appl) i) v0 v t2)))).(ex2_ind T (\lambda (t2: T).(eq T t1 (THead -(Flat Appl) w t2))) (\lambda (t2: T).(subst0 (s (Flat Appl) i) v0 v t2)) (ex -T (\lambda (w0: T).(ty3 g d v1 w0))) (\lambda (x: T).(\lambda (_: (eq T t1 -(THead (Flat Appl) w x))).(\lambda (H8: (subst0 (s (Flat Appl) i) v0 v -x)).(H3 v0 x (s (Flat Appl) i) H8 b d v1 H5)))) H6)) (\lambda (H6: (ex3_2 T T -(\lambda (u2: T).(\lambda (t2: T).(eq T t1 (THead (Flat Appl) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 i v0 w u2))) (\lambda (_: -T).(\lambda (t2: T).(subst0 (s (Flat Appl) i) v0 v t2))))).(ex3_2_ind T T -(\lambda (u2: T).(\lambda (t2: T).(eq T t1 (THead (Flat Appl) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 i v0 w u2))) (\lambda (_: -T).(\lambda (t2: T).(subst0 (s (Flat Appl) i) v0 v t2))) (ex T (\lambda (w0: -T).(ty3 g d v1 w0))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (eq T t1 -(THead (Flat Appl) x0 x1))).(\lambda (_: (subst0 i v0 w x0)).(\lambda (H9: -(subst0 (s (Flat Appl) i) v0 v x1)).(H3 v0 x1 (s (Flat Appl) i) H9 b d v1 -H5)))))) H6)) (subst0_gen_head (Flat Appl) v0 w v t1 i H4))))))))))))))))))) -(\lambda (c0: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g c0 t1 -t2)).(\lambda (H1: ((\forall (v0: T).(\forall (t0: T).(\forall (i: -nat).((subst0 i v0 t1 t0) \to (\forall (b: B).(\forall (d: C).(\forall (v: -T).((getl i c0 (CHead d (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v -w))))))))))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t2 t0)).(\lambda (H3: -((\forall (v0: T).(\forall (t3: T).(\forall (i: nat).((subst0 i v0 t2 t3) \to -(\forall (b: B).(\forall (d: C).(\forall (v: T).((getl i c0 (CHead d (Bind b) -v)) \to (ex T (\lambda (w: T).(ty3 g d v w))))))))))))).(\lambda (v0: -T).(\lambda (t3: T).(\lambda (i: nat).(\lambda (H4: (subst0 i v0 (THead (Flat -Cast) t2 t1) t3)).(\lambda (b: B).(\lambda (d: C).(\lambda (v: T).(\lambda -(H5: (getl i c0 (CHead d (Bind b) v))).(or3_ind (ex2 T (\lambda (u2: T).(eq T -t3 (THead (Flat Cast) u2 t1))) (\lambda (u2: T).(subst0 i v0 t2 u2))) (ex2 T -(\lambda (t4: T).(eq T t3 (THead (Flat Cast) t2 t4))) (\lambda (t4: -T).(subst0 (s (Flat Cast) i) v0 t1 t4))) (ex3_2 T T (\lambda (u2: T).(\lambda -(t4: T).(eq T t3 (THead (Flat Cast) u2 t4)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 i v0 t2 u2))) (\lambda (_: T).(\lambda (t4: T).(subst0 (s (Flat -Cast) i) v0 t1 t4)))) (ex T (\lambda (w: T).(ty3 g d v w))) (\lambda (H6: -(ex2 T (\lambda (u2: T).(eq T t3 (THead (Flat Cast) u2 t1))) (\lambda (u2: -T).(subst0 i v0 t2 u2)))).(ex2_ind T (\lambda (u2: T).(eq T t3 (THead (Flat -Cast) u2 t1))) (\lambda (u2: T).(subst0 i v0 t2 u2)) (ex T (\lambda (w: -T).(ty3 g d v w))) (\lambda (x: T).(\lambda (_: (eq T t3 (THead (Flat Cast) x -t1))).(\lambda (H8: (subst0 i v0 t2 x)).(H3 v0 x i H8 b d v H5)))) H6)) -(\lambda (H6: (ex2 T (\lambda (t4: T).(eq T t3 (THead (Flat Cast) t2 t4))) -(\lambda (t4: T).(subst0 (s (Flat Cast) i) v0 t1 t4)))).(ex2_ind T (\lambda -(t4: T).(eq T t3 (THead (Flat Cast) t2 t4))) (\lambda (t4: T).(subst0 (s -(Flat Cast) i) v0 t1 t4)) (ex T (\lambda (w: T).(ty3 g d v w))) (\lambda (x: -T).(\lambda (_: (eq T t3 (THead (Flat Cast) t2 x))).(\lambda (H8: (subst0 (s -(Flat Cast) i) v0 t1 x)).(H1 v0 x (s (Flat Cast) i) H8 b d v H5)))) H6)) -(\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead -(Flat Cast) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 t2 u2))) -(\lambda (_: T).(\lambda (t4: T).(subst0 (s (Flat Cast) i) v0 t1 -t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead -(Flat Cast) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 t2 u2))) -(\lambda (_: T).(\lambda (t4: T).(subst0 (s (Flat Cast) i) v0 t1 t4))) (ex T -(\lambda (w: T).(ty3 g d v w))) (\lambda (x0: T).(\lambda (x1: T).(\lambda -(_: (eq T t3 (THead (Flat Cast) x0 x1))).(\lambda (H8: (subst0 i v0 t2 -x0)).(\lambda (_: (subst0 (s (Flat Cast) i) v0 t1 x1)).(H3 v0 x0 i H8 b d v -H5)))))) H6)) (subst0_gen_head (Flat Cast) v0 t2 t1 t3 i H4)))))))))))))))))) -c t u H))))). -(* COMMENTS -Initial nodes: 4343 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/sty0.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/sty0.ma deleted file mode 100644 index 3c630662c..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/sty0.ma +++ /dev/null @@ -1,237 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/ty3/pr3_props.ma". - -include "Basic-1/sty0/fwd.ma". - -theorem ty3_sty0: - \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).((ty3 g c u -t1) \to (\forall (t2: T).((sty0 g c u t2) \to (ty3 g c u t2))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (H: -(ty3 g c u t1)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (_: -T).(\forall (t2: T).((sty0 g c0 t t2) \to (ty3 g c0 t t2)))))) (\lambda (c0: -C).(\lambda (t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda -(_: ((\forall (t3: T).((sty0 g c0 t2 t3) \to (ty3 g c0 t2 t3))))).(\lambda -(u0: T).(\lambda (t3: T).(\lambda (_: (ty3 g c0 u0 t3)).(\lambda (H3: -((\forall (t4: T).((sty0 g c0 u0 t4) \to (ty3 g c0 u0 t4))))).(\lambda (_: -(pc3 c0 t3 t2)).(\lambda (t0: T).(\lambda (H5: (sty0 g c0 u0 t0)).(H3 t0 -H5))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (t2: T).(\lambda -(H0: (sty0 g c0 (TSort m) t2)).(let H_y \def (sty0_gen_sort g c0 t2 m H0) in -(let H1 \def (f_equal T T (\lambda (e: T).e) t2 (TSort (next g m)) H_y) in -(eq_ind_r T (TSort (next g m)) (\lambda (t: T).(ty3 g c0 (TSort m) t)) -(ty3_sort g c0 m) t2 H1))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda -(d: C).(\lambda (u0: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abbr) -u0))).(\lambda (t: T).(\lambda (_: (ty3 g d u0 t)).(\lambda (H2: ((\forall -(t2: T).((sty0 g d u0 t2) \to (ty3 g d u0 t2))))).(\lambda (t2: T).(\lambda -(H3: (sty0 g c0 (TLRef n) t2)).(let H_x \def (sty0_gen_lref g c0 t2 n H3) in -(let H4 \def H_x in (or_ind (ex3_3 C T T (\lambda (e: C).(\lambda (u1: -T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: -C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (t0: T).(eq T t2 (lift (S n) O t0)))))) (ex3_3 C -T T (\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e -(Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g -e u1 t0)))) (\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq T t2 (lift -(S n) O u1)))))) (ty3 g c0 (TLRef n) t2) (\lambda (H5: (ex3_3 C T T (\lambda -(e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) -u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g e u1 t0)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(eq T t2 (lift (S n) O -t0))))))).(ex3_3_ind C T T (\lambda (e: C).(\lambda (u1: T).(\lambda (_: -T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: C).(\lambda (u1: -T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (t0: T).(eq T t2 (lift (S n) O t0))))) (ty3 g c0 (TLRef n) t2) -(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H6: (getl n c0 -(CHead x0 (Bind Abbr) x1))).(\lambda (H7: (sty0 g x0 x1 x2)).(\lambda (H8: -(eq T t2 (lift (S n) O x2))).(let H9 \def (f_equal T T (\lambda (e: T).e) t2 -(lift (S n) O x2) H8) in (eq_ind_r T (lift (S n) O x2) (\lambda (t0: T).(ty3 -g c0 (TLRef n) t0)) (let H10 \def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda -(c1: C).(getl n c0 c1)) H0 (CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d -(Bind Abbr) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in (let H11 \def (f_equal -C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u0) -(CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead -x0 (Bind Abbr) x1) H6)) in ((let H12 \def (f_equal C T (\lambda (e: C).(match -e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ -t0) \Rightarrow t0])) (CHead d (Bind Abbr) u0) (CHead x0 (Bind Abbr) x1) -(getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in -(\lambda (H13: (eq C d x0)).(let H14 \def (eq_ind_r T x1 (\lambda (t0: -T).(getl n c0 (CHead x0 (Bind Abbr) t0))) H10 u0 H12) in (let H15 \def -(eq_ind_r T x1 (\lambda (t0: T).(sty0 g x0 t0 x2)) H7 u0 H12) in (let H16 -\def (eq_ind_r C x0 (\lambda (c1: C).(getl n c0 (CHead c1 (Bind Abbr) u0))) -H14 d H13) in (let H17 \def (eq_ind_r C x0 (\lambda (c1: C).(sty0 g c1 u0 -x2)) H15 d H13) in (ty3_abbr g n c0 d u0 H16 x2 (H2 x2 H17)))))))) H11))) t2 -H9)))))))) H5)) (\lambda (H5: (ex3_3 C T T (\lambda (e: C).(\lambda (u1: -T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e: -C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq T t2 (lift (S n) O -u1))))))).(ex3_3_ind C T T (\lambda (e: C).(\lambda (u1: T).(\lambda (_: -T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1: -T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_: C).(\lambda (u1: -T).(\lambda (_: T).(eq T t2 (lift (S n) O u1))))) (ty3 g c0 (TLRef n) t2) -(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H6: (getl n c0 -(CHead x0 (Bind Abst) x1))).(\lambda (_: (sty0 g x0 x1 x2)).(\lambda (H8: (eq -T t2 (lift (S n) O x1))).(let H9 \def (f_equal T T (\lambda (e: T).e) t2 -(lift (S n) O x1) H8) in (eq_ind_r T (lift (S n) O x1) (\lambda (t0: T).(ty3 -g c0 (TLRef n) t0)) (let H10 \def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda -(c1: C).(getl n c0 c1)) H0 (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d -(Bind Abbr) u0) n H0 (CHead x0 (Bind Abst) x1) H6)) in (let H11 \def (eq_ind -C (CHead d (Bind Abbr) u0) (\lambda (ee: C).(match ee in C return (\lambda -(_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow -(match k in K return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match -b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst -\Rightarrow False | Void \Rightarrow False]) | (Flat _) \Rightarrow -False])])) I (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abbr) u0) -n H0 (CHead x0 (Bind Abst) x1) H6)) in (False_ind (ty3 g c0 (TLRef n) (lift -(S n) O x1)) H11))) t2 H9)))))))) H5)) H4))))))))))))) (\lambda (n: -nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (H0: (getl n -c0 (CHead d (Bind Abst) u0))).(\lambda (t: T).(\lambda (H1: (ty3 g d u0 -t)).(\lambda (_: ((\forall (t2: T).((sty0 g d u0 t2) \to (ty3 g d u0 -t2))))).(\lambda (t2: T).(\lambda (H3: (sty0 g c0 (TLRef n) t2)).(let H_x -\def (sty0_gen_lref g c0 t2 n H3) in (let H4 \def H_x in (or_ind (ex3_3 C T T -(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind -Abbr) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g e u1 -t0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(eq T t2 (lift (S n) -O t0)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u1: T).(\lambda (_: -T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1: -T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_: C).(\lambda (u1: -T).(\lambda (_: T).(eq T t2 (lift (S n) O u1)))))) (ty3 g c0 (TLRef n) t2) -(\lambda (H5: (ex3_3 C T T (\lambda (e: C).(\lambda (u1: T).(\lambda (_: -T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: C).(\lambda (u1: -T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (t0: T).(eq T t2 (lift (S n) O t0))))))).(ex3_3_ind C T T -(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind -Abbr) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g e u1 -t0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(eq T t2 (lift (S n) -O t0))))) (ty3 g c0 (TLRef n) t2) (\lambda (x0: C).(\lambda (x1: T).(\lambda -(x2: T).(\lambda (H6: (getl n c0 (CHead x0 (Bind Abbr) x1))).(\lambda (_: -(sty0 g x0 x1 x2)).(\lambda (H8: (eq T t2 (lift (S n) O x2))).(let H9 \def -(f_equal T T (\lambda (e: T).e) t2 (lift (S n) O x2) H8) in (eq_ind_r T (lift -(S n) O x2) (\lambda (t0: T).(ty3 g c0 (TLRef n) t0)) (let H10 \def (eq_ind C -(CHead d (Bind Abst) u0) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead x0 (Bind -Abbr) x1) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead x0 (Bind Abbr) -x1) H6)) in (let H11 \def (eq_ind C (CHead d (Bind Abst) u0) (\lambda (ee: -C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow -False | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) -with [(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with -[Abbr \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | -(Flat _) \Rightarrow False])])) I (CHead x0 (Bind Abbr) x1) (getl_mono c0 -(CHead d (Bind Abst) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in (False_ind -(ty3 g c0 (TLRef n) (lift (S n) O x2)) H11))) t2 H9)))))))) H5)) (\lambda -(H5: (ex3_3 C T T (\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 -(CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: -T).(sty0 g e u1 t0)))) (\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq T -t2 (lift (S n) O u1))))))).(ex3_3_ind C T T (\lambda (e: C).(\lambda (u1: -T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e: -C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq T t2 (lift (S n) O u1))))) (ty3 g c0 -(TLRef n) t2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda -(H6: (getl n c0 (CHead x0 (Bind Abst) x1))).(\lambda (H7: (sty0 g x0 x1 -x2)).(\lambda (H8: (eq T t2 (lift (S n) O x1))).(let H9 \def (f_equal T T -(\lambda (e: T).e) t2 (lift (S n) O x1) H8) in (eq_ind_r T (lift (S n) O x1) -(\lambda (t0: T).(ty3 g c0 (TLRef n) t0)) (let H10 \def (eq_ind C (CHead d -(Bind Abst) u0) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead x0 (Bind Abst) x1) -(getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead x0 (Bind Abst) x1) H6)) in -(let H11 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: -C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead -d (Bind Abst) u0) (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind -Abst) u0) n H0 (CHead x0 (Bind Abst) x1) H6)) in ((let H12 \def (f_equal C T -(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) -\Rightarrow u0 | (CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind Abst) u0) -(CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead -x0 (Bind Abst) x1) H6)) in (\lambda (H13: (eq C d x0)).(let H14 \def -(eq_ind_r T x1 (\lambda (t0: T).(getl n c0 (CHead x0 (Bind Abst) t0))) H10 u0 -H12) in (let H15 \def (eq_ind_r T x1 (\lambda (t0: T).(sty0 g x0 t0 x2)) H7 -u0 H12) in (eq_ind T u0 (\lambda (t0: T).(ty3 g c0 (TLRef n) (lift (S n) O -t0))) (let H16 \def (eq_ind_r C x0 (\lambda (c1: C).(getl n c0 (CHead c1 -(Bind Abst) u0))) H14 d H13) in (let H17 \def (eq_ind_r C x0 (\lambda (c1: -C).(sty0 g c1 u0 x2)) H15 d H13) in (ty3_abst g n c0 d u0 H16 t H1))) x1 -H12))))) H11))) t2 H9)))))))) H5)) H4))))))))))))) (\lambda (c0: C).(\lambda -(u0: T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 u0 t)).(\lambda (_: ((\forall -(t2: T).((sty0 g c0 u0 t2) \to (ty3 g c0 u0 t2))))).(\lambda (b: B).(\lambda -(t2: T).(\lambda (t3: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u0) t2 -t3)).(\lambda (H3: ((\forall (t4: T).((sty0 g (CHead c0 (Bind b) u0) t2 t4) -\to (ty3 g (CHead c0 (Bind b) u0) t2 t4))))).(\lambda (t0: T).(\lambda (H4: -(sty0 g c0 (THead (Bind b) u0 t2) t0)).(let H_x \def (sty0_gen_bind g b c0 u0 -t2 t0 H4) in (let H5 \def H_x in (ex2_ind T (\lambda (t4: T).(sty0 g (CHead -c0 (Bind b) u0) t2 t4)) (\lambda (t4: T).(eq T t0 (THead (Bind b) u0 t4))) -(ty3 g c0 (THead (Bind b) u0 t2) t0) (\lambda (x: T).(\lambda (H6: (sty0 g -(CHead c0 (Bind b) u0) t2 x)).(\lambda (H7: (eq T t0 (THead (Bind b) u0 -x))).(let H8 \def (f_equal T T (\lambda (e: T).e) t0 (THead (Bind b) u0 x) -H7) in (eq_ind_r T (THead (Bind b) u0 x) (\lambda (t4: T).(ty3 g c0 (THead -(Bind b) u0 t2) t4)) (ty3_bind g c0 u0 t H0 b t2 x (H3 x H6)) t0 H8))))) -H5))))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda (u0: T).(\lambda -(H0: (ty3 g c0 w u0)).(\lambda (_: ((\forall (t2: T).((sty0 g c0 w t2) \to -(ty3 g c0 w t2))))).(\lambda (v: T).(\lambda (t: T).(\lambda (H2: (ty3 g c0 v -(THead (Bind Abst) u0 t))).(\lambda (H3: ((\forall (t2: T).((sty0 g c0 v t2) -\to (ty3 g c0 v t2))))).(\lambda (t2: T).(\lambda (H4: (sty0 g c0 (THead -(Flat Appl) w v) t2)).(let H_x \def (sty0_gen_appl g c0 w v t2 H4) in (let H5 -\def H_x in (ex2_ind T (\lambda (t3: T).(sty0 g c0 v t3)) (\lambda (t3: -T).(eq T t2 (THead (Flat Appl) w t3))) (ty3 g c0 (THead (Flat Appl) w v) t2) -(\lambda (x: T).(\lambda (H6: (sty0 g c0 v x)).(\lambda (H7: (eq T t2 (THead -(Flat Appl) w x))).(let H8 \def (f_equal T T (\lambda (e: T).e) t2 (THead -(Flat Appl) w x) H7) in (eq_ind_r T (THead (Flat Appl) w x) (\lambda (t0: -T).(ty3 g c0 (THead (Flat Appl) w v) t0)) (let H_y \def (H3 x H6) in (let H9 -\def (ty3_unique g c0 v x H_y (THead (Bind Abst) u0 t) H2) in (ex_ind T -(\lambda (t0: T).(ty3 g c0 x t0)) (ty3 g c0 (THead (Flat Appl) w v) (THead -(Flat Appl) w x)) (\lambda (x0: T).(\lambda (H10: (ty3 g c0 x x0)).(ex_ind T -(\lambda (t0: T).(ty3 g c0 u0 t0)) (ty3 g c0 (THead (Flat Appl) w v) (THead -(Flat Appl) w x)) (\lambda (x1: T).(\lambda (_: (ty3 g c0 u0 x1)).(ex_ind T -(\lambda (t0: T).(ty3 g c0 (THead (Bind Abst) u0 t) t0)) (ty3 g c0 (THead -(Flat Appl) w v) (THead (Flat Appl) w x)) (\lambda (x2: T).(\lambda (H12: -(ty3 g c0 (THead (Bind Abst) u0 t) x2)).(ex3_2_ind T T (\lambda (t3: -T).(\lambda (_: T).(pc3 c0 (THead (Bind Abst) u0 t3) x2))) (\lambda (_: -T).(\lambda (t0: T).(ty3 g c0 u0 t0))) (\lambda (t3: T).(\lambda (_: T).(ty3 -g (CHead c0 (Bind Abst) u0) t t3))) (ty3 g c0 (THead (Flat Appl) w v) (THead -(Flat Appl) w x)) (\lambda (x3: T).(\lambda (x4: T).(\lambda (_: (pc3 c0 -(THead (Bind Abst) u0 x3) x2)).(\lambda (H14: (ty3 g c0 u0 x4)).(\lambda -(H15: (ty3 g (CHead c0 (Bind Abst) u0) t x3)).(ty3_conv g c0 (THead (Flat -Appl) w x) (THead (Flat Appl) w (THead (Bind Abst) u0 x3)) (ty3_appl g c0 w -u0 H0 x x3 (ty3_sconv g c0 x x0 H10 (THead (Bind Abst) u0 t) (THead (Bind -Abst) u0 x3) (ty3_bind g c0 u0 x4 H14 Abst t x3 H15) H9)) (THead (Flat Appl) -w v) (THead (Flat Appl) w (THead (Bind Abst) u0 t)) (ty3_appl g c0 w u0 H0 v -t H2) (pc3_thin_dx c0 (THead (Bind Abst) u0 t) x (ty3_unique g c0 v (THead -(Bind Abst) u0 t) H2 x H_y) w Appl))))))) (ty3_gen_bind g Abst c0 u0 t x2 -H12)))) (ty3_correct g c0 v (THead (Bind Abst) u0 t) H2)))) (ty3_correct g c0 -w u0 H0)))) (ty3_correct g c0 v x H_y)))) t2 H8))))) H5)))))))))))))) -(\lambda (c0: C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H0: (ty3 g c0 t2 -t3)).(\lambda (H1: ((\forall (t4: T).((sty0 g c0 t2 t4) \to (ty3 g c0 t2 -t4))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t3 t0)).(\lambda (H3: -((\forall (t4: T).((sty0 g c0 t3 t4) \to (ty3 g c0 t3 t4))))).(\lambda (t4: -T).(\lambda (H4: (sty0 g c0 (THead (Flat Cast) t3 t2) t4)).(let H_x \def -(sty0_gen_cast g c0 t3 t2 t4 H4) in (let H5 \def H_x in (ex3_2_ind T T -(\lambda (v2: T).(\lambda (_: T).(sty0 g c0 t3 v2))) (\lambda (_: T).(\lambda -(t5: T).(sty0 g c0 t2 t5))) (\lambda (v2: T).(\lambda (t5: T).(eq T t4 (THead -(Flat Cast) v2 t5)))) (ty3 g c0 (THead (Flat Cast) t3 t2) t4) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H6: (sty0 g c0 t3 x0)).(\lambda (H7: (sty0 g c0 -t2 x1)).(\lambda (H8: (eq T t4 (THead (Flat Cast) x0 x1))).(let H9 \def -(f_equal T T (\lambda (e: T).e) t4 (THead (Flat Cast) x0 x1) H8) in (eq_ind_r -T (THead (Flat Cast) x0 x1) (\lambda (t: T).(ty3 g c0 (THead (Flat Cast) t3 -t2) t)) (let H_y \def (H1 x1 H7) in (let H_y0 \def (H3 x0 H6) in (let H10 -\def (ty3_unique g c0 t2 x1 H_y t3 H0) in (ex_ind T (\lambda (t: T).(ty3 g c0 -x0 t)) (ty3 g c0 (THead (Flat Cast) t3 t2) (THead (Flat Cast) x0 x1)) -(\lambda (x: T).(\lambda (H11: (ty3 g c0 x0 x)).(ex_ind T (\lambda (t: -T).(ty3 g c0 x1 t)) (ty3 g c0 (THead (Flat Cast) t3 t2) (THead (Flat Cast) x0 -x1)) (\lambda (x2: T).(\lambda (H12: (ty3 g c0 x1 x2)).(ty3_conv g c0 (THead -(Flat Cast) x0 x1) (THead (Flat Cast) x x0) (ty3_cast g c0 x1 x0 (ty3_sconv g -c0 x1 x2 H12 t3 x0 H_y0 H10) x H11) (THead (Flat Cast) t3 t2) (THead (Flat -Cast) x0 t3) (ty3_cast g c0 t2 t3 H0 x0 H_y0) (pc3_thin_dx c0 t3 x1 -(ty3_unique g c0 t2 t3 H0 x1 H_y) x0 Cast)))) (ty3_correct g c0 t2 x1 H_y)))) -(ty3_correct g c0 t3 x0 H_y0))))) t4 H9))))))) H5))))))))))))) c u t1 H))))). -(* COMMENTS -Initial nodes: 4539 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/subst1.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/subst1.ma deleted file mode 100644 index ca9516e58..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/subst1.ma +++ /dev/null @@ -1,1102 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/ty3/props.ma". - -include "Basic-1/pc3/subst1.ma". - -include "Basic-1/getl/getl.ma". - -theorem ty3_gen_cabbr: - \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c -t1 t2) \to (\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c -(CHead e (Bind Abbr) u)) \to (\forall (a0: C).((csubst1 d u c a0) \to -(\forall (a: C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda -(_: T).(subst1 d u t1 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: -T).(subst1 d u t2 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 -g a y1 y2)))))))))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H: (ty3 g c t1 t2)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda -(t0: T).(\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead -e (Bind Abbr) u)) \to (\forall (a0: C).((csubst1 d u c0 a0) \to (\forall (a: -C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: -T).(subst1 d u t (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: -T).(subst1 d u t0 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 -g a y1 y2))))))))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t: -T).(\lambda (_: (ty3 g c0 t3 t)).(\lambda (H1: ((\forall (e: C).(\forall (u: -T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u)) \to (\forall (a0: -C).((csubst1 d u c0 a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (ex3_2 T -T (\lambda (y1: T).(\lambda (_: T).(subst1 d u t3 (lift (S O) d y1)))) -(\lambda (_: T).(\lambda (y2: T).(subst1 d u t (lift (S O) d y2)))) (\lambda -(y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))))).(\lambda (u: -T).(\lambda (t4: T).(\lambda (_: (ty3 g c0 u t4)).(\lambda (H3: ((\forall (e: -C).(\forall (u0: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u0)) -\to (\forall (a0: C).((csubst1 d u0 c0 a0) \to (\forall (a: C).((drop (S O) d -a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u (lift (S -O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t4 (lift (S O) d -y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 -y2)))))))))))))).(\lambda (H4: (pc3 c0 t4 t3)).(\lambda (e: C).(\lambda (u0: -T).(\lambda (d: nat).(\lambda (H5: (getl d c0 (CHead e (Bind Abbr) -u0))).(\lambda (a0: C).(\lambda (H6: (csubst1 d u0 c0 a0)).(\lambda (a: -C).(\lambda (H7: (drop (S O) d a0 a)).(let H8 \def (H3 e u0 d H5 a0 H6 a H7) -in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u (lift (S O) -d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t4 (lift (S O) d -y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T -(\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u (lift (S O) d y1)))) (\lambda -(_: T).(\lambda (y2: T).(subst1 d u0 t3 (lift (S O) d y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H9: (subst1 d u0 u (lift (S O) d x0))).(\lambda (H10: (subst1 d -u0 t4 (lift (S O) d x1))).(\lambda (H11: (ty3 g a x0 x1)).(let H12 \def (H1 e -u0 d H5 a0 H6 a H7) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: -T).(subst1 d u0 t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: -T).(subst1 d u0 t (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 -g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u (lift -(S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t3 (lift (S O) d -y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: -T).(\lambda (x3: T).(\lambda (H13: (subst1 d u0 t3 (lift (S O) d -x2))).(\lambda (_: (subst1 d u0 t (lift (S O) d x3))).(\lambda (H15: (ty3 g a -x2 x3)).(ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u -(lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t3 (lift -(S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) x0 x2 H9 -H13 (ty3_conv g a x2 x3 H15 x0 x1 H11 (pc3_gen_cabbr c0 t4 t3 H4 e u0 d H5 a0 -H6 a H7 x1 H10 x2 H13)))))))) H12))))))) H8)))))))))))))))))))) (\lambda (c0: -C).(\lambda (m: nat).(\lambda (e: C).(\lambda (u: T).(\lambda (d: -nat).(\lambda (_: (getl d c0 (CHead e (Bind Abbr) u))).(\lambda (a0: -C).(\lambda (_: (csubst1 d u c0 a0)).(\lambda (a: C).(\lambda (_: (drop (S O) -d a0 a)).(ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u (TSort -m) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u (TSort -(next g m)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a -y1 y2))) (TSort m) (TSort (next g m)) (eq_ind_r T (TSort m) (\lambda (t: -T).(subst1 d u (TSort m) t)) (subst1_refl d u (TSort m)) (lift (S O) d (TSort -m)) (lift_sort m (S O) d)) (eq_ind_r T (TSort (next g m)) (\lambda (t: -T).(subst1 d u (TSort (next g m)) t)) (subst1_refl d u (TSort (next g m))) -(lift (S O) d (TSort (next g m))) (lift_sort (next g m) (S O) d)) (ty3_sort g -a m)))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda -(u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abbr) u))).(\lambda (t: -T).(\lambda (H1: (ty3 g d u t)).(\lambda (H2: ((\forall (e: C).(\forall (u0: -T).(\forall (d0: nat).((getl d0 d (CHead e (Bind Abbr) u0)) \to (\forall (a0: -C).((csubst1 d0 u0 d a0) \to (\forall (a: C).((drop (S O) d0 a0 a) \to (ex3_2 -T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 u (lift (S O) d0 y1)))) -(\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 t (lift (S O) d0 y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))))).(\lambda (e: -C).(\lambda (u0: T).(\lambda (d0: nat).(\lambda (H3: (getl d0 c0 (CHead e -(Bind Abbr) u0))).(\lambda (a0: C).(\lambda (H4: (csubst1 d0 u0 c0 -a0)).(\lambda (a: C).(\lambda (H5: (drop (S O) d0 a0 a)).(lt_eq_gt_e n d0 -(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S -O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O t) -(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) -(\lambda (H6: (lt n d0)).(let H7 \def (eq_ind nat (minus d0 n) (\lambda (n0: -nat).(getl n0 (CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0))) -(getl_conf_le d0 (CHead e (Bind Abbr) u0) c0 H3 (CHead d (Bind Abbr) u) n H0 -(le_S_n n d0 (le_S (S n) d0 H6))) (S (minus d0 (S n))) (minus_x_Sy d0 n H6)) -in (ex2_ind C (\lambda (e2: C).(csubst1 (minus d0 n) u0 (CHead d (Bind Abbr) -u) e2)) (\lambda (e2: C).(getl n a0 e2)) (ex3_2 T T (\lambda (y1: T).(\lambda -(_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda -(y2: T).(subst1 d0 u0 (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x: C).(\lambda (H8: (csubst1 -(minus d0 n) u0 (CHead d (Bind Abbr) u) x)).(\lambda (H9: (getl n a0 x)).(let -H10 \def (eq_ind nat (minus d0 n) (\lambda (n0: nat).(csubst1 n0 u0 (CHead d -(Bind Abbr) u) x)) H8 (S (minus d0 (S n))) (minus_x_Sy d0 n H6)) in (let H11 -\def (csubst1_gen_head (Bind Abbr) d x u u0 (minus d0 (S n)) H10) in -(ex3_2_ind T C (\lambda (u2: T).(\lambda (c2: C).(eq C x (CHead c2 (Bind -Abbr) u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 (minus d0 (S n)) u0 u -u2))) (\lambda (_: T).(\lambda (c2: C).(csubst1 (minus d0 (S n)) u0 d c2))) -(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S -O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O t) -(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) -(\lambda (x0: T).(\lambda (x1: C).(\lambda (H12: (eq C x (CHead x1 (Bind -Abbr) x0))).(\lambda (H13: (subst1 (minus d0 (S n)) u0 u x0)).(\lambda (H14: -(csubst1 (minus d0 (S n)) u0 d x1)).(let H15 \def (eq_ind C x (\lambda (c1: -C).(getl n a0 c1)) H9 (CHead x1 (Bind Abbr) x0) H12) in (let H16 \def (eq_ind -nat d0 (\lambda (n0: nat).(drop (S O) n0 a0 a)) H5 (S (plus n (minus d0 (S -n)))) (lt_plus_minus n d0 H6)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: -C).(eq T x0 (lift (S O) (minus d0 (S n)) v)))) (\lambda (v: T).(\lambda (e0: -C).(getl n a (CHead e0 (Bind Abbr) v)))) (\lambda (_: T).(\lambda (e0: -C).(drop (S O) (minus d0 (S n)) x1 e0))) (ex3_2 T T (\lambda (y1: T).(\lambda -(_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda -(y2: T).(subst1 d0 u0 (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: T).(\lambda (x3: -C).(\lambda (H17: (eq T x0 (lift (S O) (minus d0 (S n)) x2))).(\lambda (H18: -(getl n a (CHead x3 (Bind Abbr) x2))).(\lambda (H19: (drop (S O) (minus d0 (S -n)) x1 x3)).(let H20 \def (eq_ind T x0 (\lambda (t0: T).(subst1 (minus d0 (S -n)) u0 u t0)) H13 (lift (S O) (minus d0 (S n)) x2) H17) in (let H21 \def (H2 -e u0 (minus d0 (S n)) (getl_gen_S (Bind Abbr) d (CHead e (Bind Abbr) u0) u -(minus d0 (S n)) H7) x1 H14 x3 H19) in (ex3_2_ind T T (\lambda (y1: -T).(\lambda (_: T).(subst1 (minus d0 (S n)) u0 u (lift (S O) (minus d0 (S n)) -y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 (minus d0 (S n)) u0 t (lift -(S O) (minus d0 (S n)) y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g x3 y1 -y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) -(lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S -n) O t) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 -y2)))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H22: (subst1 (minus d0 (S -n)) u0 u (lift (S O) (minus d0 (S n)) x4))).(\lambda (H23: (subst1 (minus d0 -(S n)) u0 t (lift (S O) (minus d0 (S n)) x5))).(\lambda (H24: (ty3 g x3 x4 -x5)).(let H25 \def (eq_ind T x4 (\lambda (t0: T).(ty3 g x3 t0 x5)) H24 x2 -(subst1_confluence_lift u x4 u0 (minus d0 (S n)) H22 x2 H20)) in (eq_ind_r -nat (plus (minus d0 (S n)) (S n)) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: -T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: -T).(\lambda (y2: T).(subst1 n0 u0 (lift (S n) O t) (lift (S O) d0 y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r nat (plus (S -n) (minus d0 (S n))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda -(_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda -(y2: T).(subst1 (plus (minus d0 (S n)) (S n)) u0 (lift (S n) O t) (lift (S O) -n0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro -T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 -y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 (plus (minus d0 (S n)) (S n)) -u0 (lift (S n) O t) (lift (S O) (plus (S n) (minus d0 (S n))) y2)))) (\lambda -(y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (TLRef n) (lift (S n) O x5) -(eq_ind_r T (TLRef n) (\lambda (t0: T).(subst1 d0 u0 (TLRef n) t0)) -(subst1_refl d0 u0 (TLRef n)) (lift (S O) d0 (TLRef n)) (lift_lref_lt n (S O) -d0 H6)) (eq_ind_r T (lift (S n) O (lift (S O) (minus d0 (S n)) x5)) (\lambda -(t0: T).(subst1 (plus (minus d0 (S n)) (S n)) u0 (lift (S n) O t) t0)) -(subst1_lift_ge t (lift (S O) (minus d0 (S n)) x5) u0 (minus d0 (S n)) (S n) -H23 O (le_O_n (minus d0 (S n)))) (lift (S O) (plus (S n) (minus d0 (S n))) -(lift (S n) O x5)) (lift_d x5 (S O) (S n) (minus d0 (S n)) O (le_O_n (minus -d0 (S n))))) (ty3_abbr g n a x3 x2 H18 x5 H25)) d0 (le_plus_minus (S n) d0 -H6)) d0 (le_plus_minus_sym (S n) d0 H6)))))))) H21)))))))) (getl_drop_conf_lt -Abbr a0 x1 x0 n H15 a (S O) (minus d0 (S n)) H16))))))))) H11)))))) -(csubst1_getl_lt d0 n H6 c0 a0 u0 H4 (CHead d (Bind Abbr) u) H0)))) (\lambda -(H6: (eq nat n d0)).(let H7 \def (eq_ind_r nat d0 (\lambda (n0: nat).(drop (S -O) n0 a0 a)) H5 n H6) in (let H8 \def (eq_ind_r nat d0 (\lambda (n0: -nat).(csubst1 n0 u0 c0 a0)) H4 n H6) in (let H9 \def (eq_ind_r nat d0 -(\lambda (n0: nat).(getl n0 c0 (CHead e (Bind Abbr) u0))) H3 n H6) in (eq_ind -nat n (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 -n0 u0 (TLRef n) (lift (S O) n0 y1)))) (\lambda (_: T).(\lambda (y2: -T).(subst1 n0 u0 (lift (S n) O t) (lift (S O) n0 y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H10 \def (eq_ind C (CHead d -(Bind Abbr) u) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead e (Bind Abbr) u0) -(getl_mono c0 (CHead d (Bind Abbr) u) n H0 (CHead e (Bind Abbr) u0) H9)) in -(let H11 \def (f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda -(_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) -(CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d (Bind -Abbr) u) n H0 (CHead e (Bind Abbr) u0) H9)) in ((let H12 \def (f_equal C T -(\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) -\Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind Abbr) u) -(CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d (Bind Abbr) u) n H0 (CHead e -(Bind Abbr) u0) H9)) in (\lambda (H13: (eq C d e)).(let H14 \def (eq_ind_r T -u0 (\lambda (t0: T).(getl n c0 (CHead e (Bind Abbr) t0))) H10 u H12) in (let -H15 \def (eq_ind_r T u0 (\lambda (t0: T).(csubst1 n t0 c0 a0)) H8 u H12) in -(eq_ind T u (\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: -T).(subst1 n t0 (TLRef n) (lift (S O) n y1)))) (\lambda (_: T).(\lambda (y2: -T).(subst1 n t0 (lift (S n) O t) (lift (S O) n y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H16 \def (eq_ind_r C e (\lambda -(c1: C).(getl n c0 (CHead c1 (Bind Abbr) u))) H14 d H13) in (ex3_2_intro T T -(\lambda (y1: T).(\lambda (_: T).(subst1 n u (TLRef n) (lift (S O) n y1)))) -(\lambda (_: T).(\lambda (y2: T).(subst1 n u (lift (S n) O t) (lift (S O) n -y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (lift n O u) (lift -n O t) (subst1_single n u (TLRef n) (lift (S O) n (lift n O u)) (eq_ind_r T -(lift (plus (S O) n) O u) (\lambda (t0: T).(subst0 n u (TLRef n) t0)) -(subst0_lref u n) (lift (S O) n (lift n O u)) (lift_free u n (S O) O n (le_n -(plus O n)) (le_O_n n)))) (eq_ind_r T (lift (plus (S O) n) O t) (\lambda (t0: -T).(subst1 n u (lift (S n) O t) t0)) (subst1_refl n u (lift (S n) O t)) (lift -(S O) n (lift n O t)) (lift_free t n (S O) O n (le_n (plus O n)) (le_O_n n))) -(ty3_lift g d u t H1 a O n (getl_conf_ge_drop Abbr a0 d u n (csubst1_getl_ge -n n (le_n n) c0 a0 u H15 (CHead d (Bind Abbr) u) H16) a H7)))) u0 H12))))) -H11))) d0 H6))))) (\lambda (H6: (lt d0 n)).(eq_ind_r nat (S (plus O (minus n -(S O)))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: -T).(subst1 d0 u0 (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda -(y2: T).(subst1 d0 u0 (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind nat (plus (S O) (minus n (S -O))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 -d0 u0 (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: -T).(subst1 d0 u0 (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r nat (plus (minus n (S O)) -(S O)) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 -d0 u0 (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: -T).(subst1 d0 u0 (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1: -T).(\lambda (_: T).(subst1 d0 u0 (TLRef (plus (minus n (S O)) (S O))) (lift -(S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O -t) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) -(TLRef (minus n (S O))) (lift n O t) (eq_ind_r T (TLRef (plus (minus n (S O)) -(S O))) (\lambda (t0: T).(subst1 d0 u0 (TLRef (plus (minus n (S O)) (S O))) -t0)) (subst1_refl d0 u0 (TLRef (plus (minus n (S O)) (S O)))) (lift (S O) d0 -(TLRef (minus n (S O)))) (lift_lref_ge (minus n (S O)) (S O) d0 (lt_le_minus -d0 n H6))) (eq_ind_r T (lift (plus (S O) n) O t) (\lambda (t0: T).(subst1 d0 -u0 (lift (S n) O t) t0)) (subst1_refl d0 u0 (lift (S n) O t)) (lift (S O) d0 -(lift n O t)) (lift_free t n (S O) O d0 (le_S_n d0 (plus O n) (le_S (S d0) -(plus O n) H6)) (le_O_n d0))) (eq_ind_r nat (S (minus n (S O))) (\lambda (n0: -nat).(ty3 g a (TLRef (minus n (S O))) (lift n0 O t))) (ty3_abbr g (minus n (S -O)) a d u (getl_drop_conf_ge n (CHead d (Bind Abbr) u) a0 (csubst1_getl_ge d0 -n (le_S_n d0 n (le_S (S d0) n H6)) c0 a0 u0 H4 (CHead d (Bind Abbr) u) H0) a -(S O) d0 H5 (eq_ind_r nat (plus (S O) d0) (\lambda (n0: nat).(le n0 n)) H6 -(plus d0 (S O)) (plus_sym d0 (S O)))) t H1) n (minus_x_SO n (le_lt_trans O d0 -n (le_O_n d0) H6)))) (plus (S O) (minus n (S O))) (plus_sym (S O) (minus n (S -O)))) (S (plus O (minus n (S O)))) (refl_equal nat (S (plus O (minus n (S -O)))))) n (lt_plus_minus O n (le_lt_trans O d0 n (le_O_n d0) -H6))))))))))))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: -C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abst) -u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u t)).(\lambda (H2: ((\forall (e: -C).(\forall (u0: T).(\forall (d0: nat).((getl d0 d (CHead e (Bind Abbr) u0)) -\to (\forall (a0: C).((csubst1 d0 u0 d a0) \to (\forall (a: C).((drop (S O) -d0 a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 u -(lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 t (lift -(S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 -y2)))))))))))))).(\lambda (e: C).(\lambda (u0: T).(\lambda (d0: nat).(\lambda -(H3: (getl d0 c0 (CHead e (Bind Abbr) u0))).(\lambda (a0: C).(\lambda (H4: -(csubst1 d0 u0 c0 a0)).(\lambda (a: C).(\lambda (H5: (drop (S O) d0 a0 -a)).(lt_eq_gt_e n d0 (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 -u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 -d0 u0 (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: -T).(ty3 g a y1 y2)))) (\lambda (H6: (lt n d0)).(let H7 \def (eq_ind nat -(minus d0 n) (\lambda (n0: nat).(getl n0 (CHead d (Bind Abst) u) (CHead e -(Bind Abbr) u0))) (getl_conf_le d0 (CHead e (Bind Abbr) u0) c0 H3 (CHead d -(Bind Abst) u) n H0 (le_S_n n d0 (le_S (S n) d0 H6))) (S (minus d0 (S n))) -(minus_x_Sy d0 n H6)) in (ex2_ind C (\lambda (e2: C).(csubst1 (minus d0 n) u0 -(CHead d (Bind Abst) u) e2)) (\lambda (e2: C).(getl n a0 e2)) (ex3_2 T T -(\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 -y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O u) (lift -(S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda -(x: C).(\lambda (H8: (csubst1 (minus d0 n) u0 (CHead d (Bind Abst) u) -x)).(\lambda (H9: (getl n a0 x)).(let H10 \def (eq_ind nat (minus d0 n) -(\lambda (n0: nat).(csubst1 n0 u0 (CHead d (Bind Abst) u) x)) H8 (S (minus d0 -(S n))) (minus_x_Sy d0 n H6)) in (let H11 \def (csubst1_gen_head (Bind Abst) -d x u u0 (minus d0 (S n)) H10) in (ex3_2_ind T C (\lambda (u2: T).(\lambda -(c2: C).(eq C x (CHead c2 (Bind Abst) u2)))) (\lambda (u2: T).(\lambda (_: -C).(subst1 (minus d0 (S n)) u0 u u2))) (\lambda (_: T).(\lambda (c2: -C).(csubst1 (minus d0 (S n)) u0 d c2))) (ex3_2 T T (\lambda (y1: T).(\lambda -(_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda -(y2: T).(subst1 d0 u0 (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1: -C).(\lambda (H12: (eq C x (CHead x1 (Bind Abst) x0))).(\lambda (H13: (subst1 -(minus d0 (S n)) u0 u x0)).(\lambda (H14: (csubst1 (minus d0 (S n)) u0 d -x1)).(let H15 \def (eq_ind C x (\lambda (c1: C).(getl n a0 c1)) H9 (CHead x1 -(Bind Abst) x0) H12) in (let H16 \def (eq_ind nat d0 (\lambda (n0: nat).(drop -(S O) n0 a0 a)) H5 (S (plus n (minus d0 (S n)))) (lt_plus_minus n d0 H6)) in -(ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T x0 (lift (S O) (minus d0 -(S n)) v)))) (\lambda (v: T).(\lambda (e0: C).(getl n a (CHead e0 (Bind Abst) -v)))) (\lambda (_: T).(\lambda (e0: C).(drop (S O) (minus d0 (S n)) x1 e0))) -(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S -O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O u) -(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) -(\lambda (x2: T).(\lambda (x3: C).(\lambda (H17: (eq T x0 (lift (S O) (minus -d0 (S n)) x2))).(\lambda (H18: (getl n a (CHead x3 (Bind Abst) x2))).(\lambda -(H19: (drop (S O) (minus d0 (S n)) x1 x3)).(let H20 \def (eq_ind T x0 -(\lambda (t0: T).(subst1 (minus d0 (S n)) u0 u t0)) H13 (lift (S O) (minus d0 -(S n)) x2) H17) in (let H21 \def (H2 e u0 (minus d0 (S n)) (getl_gen_S (Bind -Abst) d (CHead e (Bind Abbr) u0) u (minus d0 (S n)) H7) x1 H14 x3 H19) in -(ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(subst1 (minus d0 (S n)) u0 u -(lift (S O) (minus d0 (S n)) y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 -(minus d0 (S n)) u0 t (lift (S O) (minus d0 (S n)) y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g x3 y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda -(_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda -(y2: T).(subst1 d0 u0 (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x4: T).(\lambda (x5: -T).(\lambda (H22: (subst1 (minus d0 (S n)) u0 u (lift (S O) (minus d0 (S n)) -x4))).(\lambda (_: (subst1 (minus d0 (S n)) u0 t (lift (S O) (minus d0 (S n)) -x5))).(\lambda (H24: (ty3 g x3 x4 x5)).(let H25 \def (eq_ind T x4 (\lambda -(t0: T).(ty3 g x3 t0 x5)) H24 x2 (subst1_confluence_lift u x4 u0 (minus d0 (S -n)) H22 x2 H20)) in (eq_ind_r nat (plus (minus d0 (S n)) (S n)) (\lambda (n0: -nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) -(lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 n0 u0 (lift (S -n) O u) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 -y2))))) (eq_ind_r nat (plus (S n) (minus d0 (S n))) (\lambda (n0: nat).(ex3_2 -T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 -y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 (plus (minus d0 (S n)) (S n)) -u0 (lift (S n) O u) (lift (S O) n0 y2)))) (\lambda (y1: T).(\lambda (y2: -T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: -T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda -(y2: T).(subst1 (plus (minus d0 (S n)) (S n)) u0 (lift (S n) O u) (lift (S O) -(plus (S n) (minus d0 (S n))) y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g -a y1 y2))) (TLRef n) (lift (S n) O x2) (eq_ind_r T (TLRef n) (\lambda (t0: -T).(subst1 d0 u0 (TLRef n) t0)) (subst1_refl d0 u0 (TLRef n)) (lift (S O) d0 -(TLRef n)) (lift_lref_lt n (S O) d0 H6)) (eq_ind_r T (lift (S n) O (lift (S -O) (minus d0 (S n)) x2)) (\lambda (t0: T).(subst1 (plus (minus d0 (S n)) (S -n)) u0 (lift (S n) O u) t0)) (subst1_lift_ge u (lift (S O) (minus d0 (S n)) -x2) u0 (minus d0 (S n)) (S n) H20 O (le_O_n (minus d0 (S n)))) (lift (S O) -(plus (S n) (minus d0 (S n))) (lift (S n) O x2)) (lift_d x2 (S O) (S n) -(minus d0 (S n)) O (le_O_n (minus d0 (S n))))) (ty3_abst g n a x3 x2 H18 x5 -H25)) d0 (le_plus_minus (S n) d0 H6)) d0 (le_plus_minus_sym (S n) d0 -H6)))))))) H21)))))))) (getl_drop_conf_lt Abst a0 x1 x0 n H15 a (S O) (minus -d0 (S n)) H16))))))))) H11)))))) (csubst1_getl_lt d0 n H6 c0 a0 u0 H4 (CHead -d (Bind Abst) u) H0)))) (\lambda (H6: (eq nat n d0)).(let H7 \def (eq_ind_r -nat d0 (\lambda (n0: nat).(drop (S O) n0 a0 a)) H5 n H6) in (let H8 \def -(eq_ind_r nat d0 (\lambda (n0: nat).(csubst1 n0 u0 c0 a0)) H4 n H6) in (let -H9 \def (eq_ind_r nat d0 (\lambda (n0: nat).(getl n0 c0 (CHead e (Bind Abbr) -u0))) H3 n H6) in (eq_ind nat n (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: -T).(\lambda (_: T).(subst1 n0 u0 (TLRef n) (lift (S O) n0 y1)))) (\lambda (_: -T).(\lambda (y2: T).(subst1 n0 u0 (lift (S n) O u) (lift (S O) n0 y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H10 \def (eq_ind C -(CHead d (Bind Abst) u) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead e (Bind -Abbr) u0) (getl_mono c0 (CHead d (Bind Abst) u) n H0 (CHead e (Bind Abbr) u0) -H9)) in (let H11 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (ee: -C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow -False | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) -with [(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with -[Abbr \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | -(Flat _) \Rightarrow False])])) I (CHead e (Bind Abbr) u0) (getl_mono c0 -(CHead d (Bind Abst) u) n H0 (CHead e (Bind Abbr) u0) H9)) in (False_ind -(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 n u0 (TLRef n) (lift (S -O) n y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 n u0 (lift (S n) O u) -(lift (S O) n y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) -H11))) d0 H6))))) (\lambda (H6: (lt d0 n)).(eq_ind_r nat (S (plus O (minus n -(S O)))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: -T).(subst1 d0 u0 (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda -(y2: T).(subst1 d0 u0 (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind nat (plus (S O) (minus n (S -O))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 -d0 u0 (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: -T).(subst1 d0 u0 (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r nat (plus (minus n (S O)) -(S O)) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 -d0 u0 (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: -T).(subst1 d0 u0 (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1: -T).(\lambda (_: T).(subst1 d0 u0 (TLRef (plus (minus n (S O)) (S O))) (lift -(S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O -u) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) -(TLRef (minus n (S O))) (lift n O u) (eq_ind_r T (TLRef (plus (minus n (S O)) -(S O))) (\lambda (t0: T).(subst1 d0 u0 (TLRef (plus (minus n (S O)) (S O))) -t0)) (subst1_refl d0 u0 (TLRef (plus (minus n (S O)) (S O)))) (lift (S O) d0 -(TLRef (minus n (S O)))) (lift_lref_ge (minus n (S O)) (S O) d0 (lt_le_minus -d0 n H6))) (eq_ind_r T (lift (plus (S O) n) O u) (\lambda (t0: T).(subst1 d0 -u0 (lift (S n) O u) t0)) (subst1_refl d0 u0 (lift (S n) O u)) (lift (S O) d0 -(lift n O u)) (lift_free u n (S O) O d0 (le_S_n d0 (plus O n) (le_S (S d0) -(plus O n) H6)) (le_O_n d0))) (eq_ind_r nat (S (minus n (S O))) (\lambda (n0: -nat).(ty3 g a (TLRef (minus n (S O))) (lift n0 O u))) (ty3_abst g (minus n (S -O)) a d u (getl_drop_conf_ge n (CHead d (Bind Abst) u) a0 (csubst1_getl_ge d0 -n (le_S_n d0 n (le_S (S d0) n H6)) c0 a0 u0 H4 (CHead d (Bind Abst) u) H0) a -(S O) d0 H5 (eq_ind_r nat (plus (S O) d0) (\lambda (n0: nat).(le n0 n)) H6 -(plus d0 (S O)) (plus_sym d0 (S O)))) t H1) n (minus_x_SO n (le_lt_trans O d0 -n (le_O_n d0) H6)))) (plus (S O) (minus n (S O))) (plus_sym (S O) (minus n (S -O)))) (S (plus O (minus n (S O)))) (refl_equal nat (S (plus O (minus n (S -O)))))) n (lt_plus_minus O n (le_lt_trans O d0 n (le_O_n d0) -H6))))))))))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (t: -T).(\lambda (_: (ty3 g c0 u t)).(\lambda (H1: ((\forall (e: C).(\forall (u0: -T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u0)) \to (\forall (a0: -C).((csubst1 d u0 c0 a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (ex3_2 -T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u (lift (S O) d y1)))) -(\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t (lift (S O) d y2)))) (\lambda -(y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))))).(\lambda (b: -B).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) -u) t3 t4)).(\lambda (H3: ((\forall (e: C).(\forall (u0: T).(\forall (d: -nat).((getl d (CHead c0 (Bind b) u) (CHead e (Bind Abbr) u0)) \to (\forall -(a0: C).((csubst1 d u0 (CHead c0 (Bind b) u) a0) \to (\forall (a: C).((drop -(S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 t3 -(lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t4 (lift -(S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 -y2)))))))))))))).(\lambda (e: C).(\lambda (u0: T).(\lambda (d: nat).(\lambda -(H4: (getl d c0 (CHead e (Bind Abbr) u0))).(\lambda (a0: C).(\lambda (H5: -(csubst1 d u0 c0 a0)).(\lambda (a: C).(\lambda (H6: (drop (S O) d a0 a)).(let -H7 \def (H1 e u0 d H4 a0 H5 a H6) in (ex3_2_ind T T (\lambda (y1: T).(\lambda -(_: T).(subst1 d u0 u (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: -T).(subst1 d u0 t (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 -g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 (THead -(Bind b) u t3) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 -d u0 (THead (Bind b) u t4) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda -(y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H8: -(subst1 d u0 u (lift (S O) d x0))).(\lambda (_: (subst1 d u0 t (lift (S O) d -x1))).(\lambda (H10: (ty3 g a x0 x1)).(let H11 \def (H3 e u0 (S d) (getl_head -(Bind b) d c0 (CHead e (Bind Abbr) u0) H4 u) (CHead a0 (Bind b) (lift (S O) d -x0)) (csubst1_bind b d u0 u (lift (S O) d x0) H8 c0 a0 H5) (CHead a (Bind b) -x0) (drop_skip_bind (S O) d a0 a H6 b x0)) in (ex3_2_ind T T (\lambda (y1: -T).(\lambda (_: T).(subst1 (S d) u0 t3 (lift (S O) (S d) y1)))) (\lambda (_: -T).(\lambda (y2: T).(subst1 (S d) u0 t4 (lift (S O) (S d) y2)))) (\lambda -(y1: T).(\lambda (y2: T).(ty3 g (CHead a (Bind b) x0) y1 y2))) (ex3_2 T T -(\lambda (y1: T).(\lambda (_: T).(subst1 d u0 (THead (Bind b) u t3) (lift (S -O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Bind b) u -t4) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) -(\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (subst1 (S d) u0 t3 (lift (S -O) (S d) x2))).(\lambda (H13: (subst1 (S d) u0 t4 (lift (S O) (S d) -x3))).(\lambda (H14: (ty3 g (CHead a (Bind b) x0) x2 x3)).(ex3_2_intro T T -(\lambda (y1: T).(\lambda (_: T).(subst1 d u0 (THead (Bind b) u t3) (lift (S -O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Bind b) u -t4) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) -(THead (Bind b) x0 x2) (THead (Bind b) x0 x3) (eq_ind_r T (THead (Bind b) -(lift (S O) d x0) (lift (S O) (S d) x2)) (\lambda (t0: T).(subst1 d u0 (THead -(Bind b) u t3) t0)) (subst1_head u0 u (lift (S O) d x0) d H8 (Bind b) t3 -(lift (S O) (S d) x2) H12) (lift (S O) d (THead (Bind b) x0 x2)) (lift_bind b -x0 x2 (S O) d)) (eq_ind_r T (THead (Bind b) (lift (S O) d x0) (lift (S O) (S -d) x3)) (\lambda (t0: T).(subst1 d u0 (THead (Bind b) u t4) t0)) (subst1_head -u0 u (lift (S O) d x0) d H8 (Bind b) t4 (lift (S O) (S d) x3) H13) (lift (S -O) d (THead (Bind b) x0 x3)) (lift_bind b x0 x3 (S O) d)) (ty3_bind g a x0 x1 -H10 b x2 x3 H14))))))) H11))))))) H7)))))))))))))))))))) (\lambda (c0: -C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda (H1: -((\forall (e: C).(\forall (u0: T).(\forall (d: nat).((getl d c0 (CHead e -(Bind Abbr) u0)) \to (\forall (a0: C).((csubst1 d u0 c0 a0) \to (\forall (a: -C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: -T).(subst1 d u0 w (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: -T).(subst1 d u0 u (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 -g a y1 y2)))))))))))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g -c0 v (THead (Bind Abst) u t))).(\lambda (H3: ((\forall (e: C).(\forall (u0: -T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u0)) \to (\forall (a0: -C).((csubst1 d u0 c0 a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (ex3_2 -T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 v (lift (S O) d y1)))) -(\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Bind Abst) u t) (lift -(S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 -y2)))))))))))))).(\lambda (e: C).(\lambda (u0: T).(\lambda (d: nat).(\lambda -(H4: (getl d c0 (CHead e (Bind Abbr) u0))).(\lambda (a0: C).(\lambda (H5: -(csubst1 d u0 c0 a0)).(\lambda (a: C).(\lambda (H6: (drop (S O) d a0 a)).(let -H7 \def (H3 e u0 d H4 a0 H5 a H6) in (ex3_2_ind T T (\lambda (y1: T).(\lambda -(_: T).(subst1 d u0 v (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: -T).(subst1 d u0 (THead (Bind Abst) u t) (lift (S O) d y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda -(_: T).(subst1 d u0 (THead (Flat Appl) w v) (lift (S O) d y1)))) (\lambda (_: -T).(\lambda (y2: T).(subst1 d u0 (THead (Flat Appl) w (THead (Bind Abst) u -t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) -(\lambda (x0: T).(\lambda (x1: T).(\lambda (H8: (subst1 d u0 v (lift (S O) d -x0))).(\lambda (H9: (subst1 d u0 (THead (Bind Abst) u t) (lift (S O) d -x1))).(\lambda (H10: (ty3 g a x0 x1)).(let H11 \def (H1 e u0 d H4 a0 H5 a H6) -in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 w (lift (S O) -d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 u (lift (S O) d y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: -T).(\lambda (_: T).(subst1 d u0 (THead (Flat Appl) w v) (lift (S O) d y1)))) -(\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Flat Appl) w (THead -(Bind Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 -g a y1 y2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (subst1 d u0 w -(lift (S O) d x2))).(\lambda (H13: (subst1 d u0 u (lift (S O) d -x3))).(\lambda (H14: (ty3 g a x2 x3)).(let H_x \def (subst1_gen_head (Bind -Abst) u0 u t (lift (S O) d x1) d H9) in (let H15 \def H_x in (ex3_2_ind T T -(\lambda (u2: T).(\lambda (t3: T).(eq T (lift (S O) d x1) (THead (Bind Abst) -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1 d u0 u u2))) (\lambda (_: -T).(\lambda (t3: T).(subst1 (S d) u0 t t3))) (ex3_2 T T (\lambda (y1: -T).(\lambda (_: T).(subst1 d u0 (THead (Flat Appl) w v) (lift (S O) d y1)))) -(\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Flat Appl) w (THead -(Bind Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 -g a y1 y2)))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H16: (eq T (lift (S -O) d x1) (THead (Bind Abst) x4 x5))).(\lambda (H17: (subst1 d u0 u -x4)).(\lambda (H18: (subst1 (S d) u0 t x5)).(let H19 \def (sym_eq T (lift (S -O) d x1) (THead (Bind Abst) x4 x5) H16) in (ex3_2_ind T T (\lambda (y: -T).(\lambda (z: T).(eq T x1 (THead (Bind Abst) y z)))) (\lambda (y: -T).(\lambda (_: T).(eq T x4 (lift (S O) d y)))) (\lambda (_: T).(\lambda (z: -T).(eq T x5 (lift (S O) (S d) z)))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: -T).(subst1 d u0 (THead (Flat Appl) w v) (lift (S O) d y1)))) (\lambda (_: -T).(\lambda (y2: T).(subst1 d u0 (THead (Flat Appl) w (THead (Bind Abst) u -t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) -(\lambda (x6: T).(\lambda (x7: T).(\lambda (H20: (eq T x1 (THead (Bind Abst) -x6 x7))).(\lambda (H21: (eq T x4 (lift (S O) d x6))).(\lambda (H22: (eq T x5 -(lift (S O) (S d) x7))).(let H23 \def (eq_ind T x5 (\lambda (t0: T).(subst1 -(S d) u0 t t0)) H18 (lift (S O) (S d) x7) H22) in (let H24 \def (eq_ind T x4 -(\lambda (t0: T).(subst1 d u0 u t0)) H17 (lift (S O) d x6) H21) in (let H25 -\def (eq_ind T x1 (\lambda (t0: T).(ty3 g a x0 t0)) H10 (THead (Bind Abst) x6 -x7) H20) in (let H26 \def (eq_ind T x6 (\lambda (t0: T).(ty3 g a x0 (THead -(Bind Abst) t0 x7))) H25 x3 (subst1_confluence_lift u x6 u0 d H24 x3 H13)) in -(ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 (THead (Flat -Appl) w v) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 -(THead (Flat Appl) w (THead (Bind Abst) u t)) (lift (S O) d y2)))) (\lambda -(y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (THead (Flat Appl) x2 x0) (THead -(Flat Appl) x2 (THead (Bind Abst) x3 x7)) (eq_ind_r T (THead (Flat Appl) -(lift (S O) d x2) (lift (S O) d x0)) (\lambda (t0: T).(subst1 d u0 (THead -(Flat Appl) w v) t0)) (subst1_head u0 w (lift (S O) d x2) d H12 (Flat Appl) v -(lift (S O) d x0) H8) (lift (S O) d (THead (Flat Appl) x2 x0)) (lift_flat -Appl x2 x0 (S O) d)) (eq_ind_r T (THead (Flat Appl) (lift (S O) d x2) (lift -(S O) d (THead (Bind Abst) x3 x7))) (\lambda (t0: T).(subst1 d u0 (THead -(Flat Appl) w (THead (Bind Abst) u t)) t0)) (subst1_head u0 w (lift (S O) d -x2) d H12 (Flat Appl) (THead (Bind Abst) u t) (lift (S O) d (THead (Bind -Abst) x3 x7)) (eq_ind_r T (THead (Bind Abst) (lift (S O) d x3) (lift (S O) (S -d) x7)) (\lambda (t0: T).(subst1 (s (Flat Appl) d) u0 (THead (Bind Abst) u t) -t0)) (subst1_head u0 u (lift (S O) d x3) (s (Flat Appl) d) H13 (Bind Abst) t -(lift (S O) (S d) x7) H23) (lift (S O) d (THead (Bind Abst) x3 x7)) -(lift_bind Abst x3 x7 (S O) d))) (lift (S O) d (THead (Flat Appl) x2 (THead -(Bind Abst) x3 x7))) (lift_flat Appl x2 (THead (Bind Abst) x3 x7) (S O) d)) -(ty3_appl g a x2 x3 H14 x0 x7 H26))))))))))) (lift_gen_bind Abst x4 x5 x1 (S -O) d H19)))))))) H15)))))))) H11))))))) H7))))))))))))))))))) (\lambda (c0: -C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (ty3 g c0 t3 t4)).(\lambda -(H1: ((\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead e -(Bind Abbr) u)) \to (\forall (a0: C).((csubst1 d u c0 a0) \to (\forall (a: -C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: -T).(subst1 d u t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: -T).(subst1 d u t4 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 -g a y1 y2)))))))))))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t4 -t0)).(\lambda (H3: ((\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl -d c0 (CHead e (Bind Abbr) u)) \to (\forall (a0: C).((csubst1 d u c0 a0) \to -(\forall (a: C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda -(_: T).(subst1 d u t4 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: -T).(subst1 d u t0 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 -g a y1 y2)))))))))))))).(\lambda (e: C).(\lambda (u: T).(\lambda (d: -nat).(\lambda (H4: (getl d c0 (CHead e (Bind Abbr) u))).(\lambda (a0: -C).(\lambda (H5: (csubst1 d u c0 a0)).(\lambda (a: C).(\lambda (H6: (drop (S -O) d a0 a)).(let H7 \def (H3 e u d H4 a0 H5 a H6) in (ex3_2_ind T T (\lambda -(y1: T).(\lambda (_: T).(subst1 d u t4 (lift (S O) d y1)))) (\lambda (_: -T).(\lambda (y2: T).(subst1 d u t0 (lift (S O) d y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda -(_: T).(subst1 d u (THead (Flat Cast) t4 t3) (lift (S O) d y1)))) (\lambda -(_: T).(\lambda (y2: T).(subst1 d u (THead (Flat Cast) t0 t4) (lift (S O) d -y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H8: (subst1 d u t4 (lift (S O) d x0))).(\lambda -(H9: (subst1 d u t0 (lift (S O) d x1))).(\lambda (H10: (ty3 g a x0 x1)).(let -H11 \def (H1 e u d H4 a0 H5 a H6) in (ex3_2_ind T T (\lambda (y1: T).(\lambda -(_: T).(subst1 d u t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: -T).(subst1 d u t4 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 -g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u (THead -(Flat Cast) t4 t3) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: -T).(subst1 d u (THead (Flat Cast) t0 t4) (lift (S O) d y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: T).(\lambda (x3: -T).(\lambda (H12: (subst1 d u t3 (lift (S O) d x2))).(\lambda (H13: (subst1 d -u t4 (lift (S O) d x3))).(\lambda (H14: (ty3 g a x2 x3)).(let H15 \def -(eq_ind T x3 (\lambda (t: T).(ty3 g a x2 t)) H14 x0 (subst1_confluence_lift -t4 x3 u d H13 x0 H8)) in (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: -T).(subst1 d u (THead (Flat Cast) t4 t3) (lift (S O) d y1)))) (\lambda (_: -T).(\lambda (y2: T).(subst1 d u (THead (Flat Cast) t0 t4) (lift (S O) d -y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (THead (Flat Cast) -x0 x2) (THead (Flat Cast) x1 x0) (eq_ind_r T (THead (Flat Cast) (lift (S O) d -x0) (lift (S O) d x2)) (\lambda (t: T).(subst1 d u (THead (Flat Cast) t4 t3) -t)) (subst1_head u t4 (lift (S O) d x0) d H8 (Flat Cast) t3 (lift (S O) d x2) -H12) (lift (S O) d (THead (Flat Cast) x0 x2)) (lift_flat Cast x0 x2 (S O) d)) -(eq_ind_r T (THead (Flat Cast) (lift (S O) d x1) (lift (S O) d x0)) (\lambda -(t: T).(subst1 d u (THead (Flat Cast) t0 t4) t)) (subst1_head u t0 (lift (S -O) d x1) d H9 (Flat Cast) t4 (lift (S O) d x0) H8) (lift (S O) d (THead (Flat -Cast) x1 x0)) (lift_flat Cast x1 x0 (S O) d)) (ty3_cast g a x2 x0 H15 x1 -H10)))))))) H11))))))) H7)))))))))))))))))) c t1 t2 H))))). -(* COMMENTS -Initial nodes: 12848 -END *) - -theorem ty3_gen_cvoid: - \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c -t1 t2) \to (\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c -(CHead e (Bind Void) u)) \to (\forall (a: C).((drop (S O) d c a) \to (ex3_2 T -T (\lambda (y1: T).(\lambda (_: T).(eq T t1 (lift (S O) d y1)))) (\lambda (_: -T).(\lambda (y2: T).(eq T t2 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda -(y2: T).(ty3 g a y1 y2)))))))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H: (ty3 g c t1 t2)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda -(t0: T).(\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead -e (Bind Void) u)) \to (\forall (a: C).((drop (S O) d c0 a) \to (ex3_2 T T -(\lambda (y1: T).(\lambda (_: T).(eq T t (lift (S O) d y1)))) (\lambda (_: -T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda -(y2: T).(ty3 g a y1 y2))))))))))))) (\lambda (c0: C).(\lambda (t3: -T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 t3 t)).(\lambda (H1: ((\forall (e: -C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Void) u)) \to -(\forall (a: C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda -(_: T).(eq T t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t -(lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 -y2)))))))))))).(\lambda (u: T).(\lambda (t4: T).(\lambda (H2: (ty3 g c0 u -t4)).(\lambda (H3: ((\forall (e: C).(\forall (u0: T).(\forall (d: nat).((getl -d c0 (CHead e (Bind Void) u0)) \to (\forall (a: C).((drop (S O) d c0 a) \to -(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T u (lift (S O) d y1)))) -(\lambda (_: T).(\lambda (y2: T).(eq T t4 (lift (S O) d y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (H4: (pc3 c0 t4 -t3)).(\lambda (e: C).(\lambda (u0: T).(\lambda (d: nat).(\lambda (H5: (getl d -c0 (CHead e (Bind Void) u0))).(\lambda (a: C).(\lambda (H6: (drop (S O) d c0 -a)).(let H7 \def (H3 e u0 d H5 a H6) in (ex3_2_ind T T (\lambda (y1: -T).(\lambda (_: T).(eq T u (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: -T).(eq T t4 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a -y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T u (lift (S O) d -y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t3 (lift (S O) d y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H8: (eq T u (lift (S O) d x0))).(\lambda (H9: -(eq T t4 (lift (S O) d x1))).(\lambda (H10: (ty3 g a x0 x1)).(let H11 \def -(eq_ind T t4 (\lambda (t0: T).(pc3 c0 t0 t3)) H4 (lift (S O) d x1) H9) in -(let H12 \def (eq_ind T t4 (\lambda (t0: T).(ty3 g c0 u t0)) H2 (lift (S O) d -x1) H9) in (let H13 \def (eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 (lift (S -O) d x1))) H12 (lift (S O) d x0) H8) in (eq_ind_r T (lift (S O) d x0) -(\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t0 (lift -(S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t3 (lift (S O) d y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H14 \def (H1 e u0 -d H5 a H6) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T t3 (lift -(S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: -T).(\lambda (_: T).(eq T (lift (S O) d x0) (lift (S O) d y1)))) (\lambda (_: -T).(\lambda (y2: T).(eq T t3 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda -(y2: T).(ty3 g a y1 y2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H15: -(eq T t3 (lift (S O) d x2))).(\lambda (H16: (eq T t (lift (S O) d -x3))).(\lambda (H17: (ty3 g a x2 x3)).(let H18 \def (eq_ind T t (\lambda (t0: -T).(ty3 g c0 t3 t0)) H0 (lift (S O) d x3) H16) in (let H19 \def (eq_ind T t3 -(\lambda (t0: T).(ty3 g c0 t0 (lift (S O) d x3))) H18 (lift (S O) d x2) H15) -in (let H20 \def (eq_ind T t3 (\lambda (t0: T).(pc3 c0 (lift (S O) d x1) t0)) -H11 (lift (S O) d x2) H15) in (eq_ind_r T (lift (S O) d x2) (\lambda (t0: -T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (lift (S O) d x0) (lift -(S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T -(\lambda (y1: T).(\lambda (_: T).(eq T (lift (S O) d x0) (lift (S O) d y1)))) -(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d x2) (lift (S O) d y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) x0 x2 (refl_equal T (lift -(S O) d x0)) (refl_equal T (lift (S O) d x2)) (ty3_conv g a x2 x3 H17 x0 x1 -H10 (pc3_gen_lift c0 x1 x2 (S O) d H20 a H6))) t3 H15))))))))) H14)) u -H8))))))))) H7)))))))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda -(e: C).(\lambda (u: T).(\lambda (d: nat).(\lambda (_: (getl d c0 (CHead e -(Bind Void) u))).(\lambda (a: C).(\lambda (_: (drop (S O) d c0 -a)).(ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(eq T (TSort m) (lift -(S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (TSort (next g m)) -(lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) -(TSort m) (TSort (next g m)) (eq_ind_r T (TSort m) (\lambda (t: T).(eq T -(TSort m) t)) (refl_equal T (TSort m)) (lift (S O) d (TSort m)) (lift_sort m -(S O) d)) (eq_ind_r T (TSort (next g m)) (\lambda (t: T).(eq T (TSort (next g -m)) t)) (refl_equal T (TSort (next g m))) (lift (S O) d (TSort (next g m))) -(lift_sort (next g m) (S O) d)) (ty3_sort g a m)))))))))) (\lambda (n: -nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n -c0 (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u -t)).(\lambda (H2: ((\forall (e: C).(\forall (u0: T).(\forall (d0: nat).((getl -d0 d (CHead e (Bind Void) u0)) \to (\forall (a: C).((drop (S O) d0 d a) \to -(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T u (lift (S O) d0 y1)))) -(\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d0 y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (e: C).(\lambda (u0: -T).(\lambda (d0: nat).(\lambda (H3: (getl d0 c0 (CHead e (Bind Void) -u0))).(\lambda (a: C).(\lambda (H4: (drop (S O) d0 c0 a)).(lt_eq_gt_e n d0 -(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 -y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O t) (lift (S O) d0 -y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (H5: (lt -n d0)).(let H6 \def (eq_ind nat (minus d0 n) (\lambda (n0: nat).(getl n0 -(CHead d (Bind Abbr) u) (CHead e (Bind Void) u0))) (getl_conf_le d0 (CHead e -(Bind Void) u0) c0 H3 (CHead d (Bind Abbr) u) n H0 (le_S_n n d0 (le_S (S n) -d0 H5))) (S (minus d0 (S n))) (minus_x_Sy d0 n H5)) in (let H7 \def (eq_ind -nat d0 (\lambda (n0: nat).(drop (S O) n0 c0 a)) H4 (S (plus n (minus d0 (S -n)))) (lt_plus_minus n d0 H5)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: -C).(eq T u (lift (S O) (minus d0 (S n)) v)))) (\lambda (v: T).(\lambda (e0: -C).(getl n a (CHead e0 (Bind Abbr) v)))) (\lambda (_: T).(\lambda (e0: -C).(drop (S O) (minus d0 (S n)) d e0))) (ex3_2 T T (\lambda (y1: T).(\lambda -(_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: -T).(eq T (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda -(y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1: C).(\lambda (H8: -(eq T u (lift (S O) (minus d0 (S n)) x0))).(\lambda (H9: (getl n a (CHead x1 -(Bind Abbr) x0))).(\lambda (H10: (drop (S O) (minus d0 (S n)) d x1)).(let H11 -\def (eq_ind T u (\lambda (t0: T).(\forall (e0: C).(\forall (u1: T).(\forall -(d1: nat).((getl d1 d (CHead e0 (Bind Void) u1)) \to (\forall (a0: C).((drop -(S O) d1 d a0) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t0 (lift -(S O) d1 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d1 -y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a0 y1 y2))))))))))) H2 (lift -(S O) (minus d0 (S n)) x0) H8) in (let H12 \def (eq_ind T u (\lambda (t0: -T).(ty3 g d t0 t)) H1 (lift (S O) (minus d0 (S n)) x0) H8) in (let H13 \def -(H11 e u0 (minus d0 (S n)) (getl_gen_S (Bind Abbr) d (CHead e (Bind Void) u0) -u (minus d0 (S n)) H6) x1 H10) in (ex3_2_ind T T (\lambda (y1: T).(\lambda -(_: T).(eq T (lift (S O) (minus d0 (S n)) x0) (lift (S O) (minus d0 (S n)) -y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) (minus d0 (S n)) -y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g x1 y1 y2))) (ex3_2 T T -(\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) -(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O t) (lift (S O) d0 y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: -T).(\lambda (x3: T).(\lambda (H14: (eq T (lift (S O) (minus d0 (S n)) x0) -(lift (S O) (minus d0 (S n)) x2))).(\lambda (H15: (eq T t (lift (S O) (minus -d0 (S n)) x3))).(\lambda (H16: (ty3 g x1 x2 x3)).(let H17 \def (eq_ind T t -(\lambda (t0: T).(ty3 g d (lift (S O) (minus d0 (S n)) x0) t0)) H12 (lift (S -O) (minus d0 (S n)) x3) H15) in (eq_ind_r T (lift (S O) (minus d0 (S n)) x3) -(\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) -(lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O -t0) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 -y2))))) (let H18 \def (eq_ind_r T x2 (\lambda (t0: T).(ty3 g x1 t0 x3)) H16 -x0 (lift_inj x0 x2 (S O) (minus d0 (S n)) H14)) in (eq_ind T (lift (S O) -(plus (S n) (minus d0 (S n))) (lift (S n) O x3)) (\lambda (t0: T).(ex3_2 T T -(\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) -(\lambda (_: T).(\lambda (y2: T).(eq T t0 (lift (S O) d0 y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind nat d0 (\lambda (n0: -nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) -d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) n0 (lift (S n) O -x3)) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 -y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) -(lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d0 -(lift (S n) O x3)) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: -T).(ty3 g a y1 y2))) (TLRef n) (lift (S n) O x3) (eq_ind_r T (TLRef n) -(\lambda (t0: T).(eq T (TLRef n) t0)) (refl_equal T (TLRef n)) (lift (S O) d0 -(TLRef n)) (lift_lref_lt n (S O) d0 H5)) (refl_equal T (lift (S O) d0 (lift -(S n) O x3))) (ty3_abbr g n a x1 x0 H9 x3 H18)) (plus (S n) (minus d0 (S n))) -(le_plus_minus (S n) d0 H5)) (lift (S n) O (lift (S O) (minus d0 (S n)) x3)) -(lift_d x3 (S O) (S n) (minus d0 (S n)) O (le_O_n (minus d0 (S n)))))) t -H15))))))) H13))))))))) (getl_drop_conf_lt Abbr c0 d u n H0 a (S O) (minus d0 -(S n)) H7))))) (\lambda (H5: (eq nat n d0)).(let H6 \def (eq_ind_r nat d0 -(\lambda (n0: nat).(drop (S O) n0 c0 a)) H4 n H5) in (let H7 \def (eq_ind_r -nat d0 (\lambda (n0: nat).(getl n0 c0 (CHead e (Bind Void) u0))) H3 n H5) in -(eq_ind nat n (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: -T).(eq T (TLRef n) (lift (S O) n0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq -T (lift (S n) O t) (lift (S O) n0 y2)))) (\lambda (y1: T).(\lambda (y2: -T).(ty3 g a y1 y2))))) (let H8 \def (eq_ind C (CHead d (Bind Abbr) u) -(\lambda (c1: C).(getl n c0 c1)) H0 (CHead e (Bind Void) u0) (getl_mono c0 -(CHead d (Bind Abbr) u) n H0 (CHead e (Bind Void) u0) H7)) in (let H9 \def -(eq_ind C (CHead d (Bind Abbr) u) (\lambda (ee: C).(match ee in C return -(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b) -\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow -True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _) -\Rightarrow False])])) I (CHead e (Bind Void) u0) (getl_mono c0 (CHead d -(Bind Abbr) u) n H0 (CHead e (Bind Void) u0) H7)) in (False_ind (ex3_2 T T -(\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) n y1)))) -(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O t) (lift (S O) n y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) H9))) d0 H5)))) (\lambda -(H5: (lt d0 n)).(eq_ind_r nat (S (plus O (minus n (S O)))) (\lambda (n0: -nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n0) (lift (S O) -d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O t) (lift (S O) -d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind nat -(plus (S O) (minus n (S O))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: -T).(\lambda (_: T).(eq T (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: -T).(\lambda (y2: T).(eq T (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda -(y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r nat (plus (minus n (S -O)) (S O)) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq -T (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T -(lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 -g a y1 y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef -(plus (minus n (S O)) (S O))) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda -(y2: T).(eq T (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2))) (TLRef (minus n (S O))) (lift n O t) -(eq_ind_r T (TLRef (plus (minus n (S O)) (S O))) (\lambda (t0: T).(eq T -(TLRef (plus (minus n (S O)) (S O))) t0)) (refl_equal T (TLRef (plus (minus n -(S O)) (S O)))) (lift (S O) d0 (TLRef (minus n (S O)))) (lift_lref_ge (minus -n (S O)) (S O) d0 (lt_le_minus d0 n H5))) (eq_ind_r T (lift (plus (S O) n) O -t) (\lambda (t0: T).(eq T (lift (S n) O t) t0)) (refl_equal T (lift (S n) O -t)) (lift (S O) d0 (lift n O t)) (lift_free t n (S O) O d0 (le_S_n d0 (plus O -n) (le_S (S d0) (plus O n) H5)) (le_O_n d0))) (eq_ind_r nat (S (minus n (S -O))) (\lambda (n0: nat).(ty3 g a (TLRef (minus n (S O))) (lift n0 O t))) -(ty3_abbr g (minus n (S O)) a d u (getl_drop_conf_ge n (CHead d (Bind Abbr) -u) c0 H0 a (S O) d0 H4 (eq_ind_r nat (plus (S O) d0) (\lambda (n0: nat).(le -n0 n)) H5 (plus d0 (S O)) (plus_sym d0 (S O)))) t H1) n (minus_x_SO n -(le_lt_trans O d0 n (le_O_n d0) H5)))) (plus (S O) (minus n (S O))) (plus_sym -(S O) (minus n (S O)))) (S (plus O (minus n (S O)))) (refl_equal nat (S (plus -O (minus n (S O)))))) n (lt_plus_minus O n (le_lt_trans O d0 n (le_O_n d0) -H5))))))))))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: -C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abst) -u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u t)).(\lambda (H2: ((\forall (e: -C).(\forall (u0: T).(\forall (d0: nat).((getl d0 d (CHead e (Bind Void) u0)) -\to (\forall (a: C).((drop (S O) d0 d a) \to (ex3_2 T T (\lambda (y1: -T).(\lambda (_: T).(eq T u (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda -(y2: T).(eq T t (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 -g a y1 y2)))))))))))).(\lambda (e: C).(\lambda (u0: T).(\lambda (d0: -nat).(\lambda (H3: (getl d0 c0 (CHead e (Bind Void) u0))).(\lambda (a: -C).(\lambda (H4: (drop (S O) d0 c0 a)).(lt_eq_gt_e n d0 (ex3_2 T T (\lambda -(y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: -T).(\lambda (y2: T).(eq T (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda -(y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (H5: (lt n d0)).(let H6 -\def (eq_ind nat (minus d0 n) (\lambda (n0: nat).(getl n0 (CHead d (Bind -Abst) u) (CHead e (Bind Void) u0))) (getl_conf_le d0 (CHead e (Bind Void) u0) -c0 H3 (CHead d (Bind Abst) u) n H0 (le_S_n n d0 (le_S (S n) d0 H5))) (S -(minus d0 (S n))) (minus_x_Sy d0 n H5)) in (let H7 \def (eq_ind nat d0 -(\lambda (n0: nat).(drop (S O) n0 c0 a)) H4 (S (plus n (minus d0 (S n)))) -(lt_plus_minus n d0 H5)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: -C).(eq T u (lift (S O) (minus d0 (S n)) v)))) (\lambda (v: T).(\lambda (e0: -C).(getl n a (CHead e0 (Bind Abst) v)))) (\lambda (_: T).(\lambda (e0: -C).(drop (S O) (minus d0 (S n)) d e0))) (ex3_2 T T (\lambda (y1: T).(\lambda -(_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: -T).(eq T (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda -(y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1: C).(\lambda (H8: -(eq T u (lift (S O) (minus d0 (S n)) x0))).(\lambda (H9: (getl n a (CHead x1 -(Bind Abst) x0))).(\lambda (H10: (drop (S O) (minus d0 (S n)) d x1)).(let H11 -\def (eq_ind T u (\lambda (t0: T).(\forall (e0: C).(\forall (u1: T).(\forall -(d1: nat).((getl d1 d (CHead e0 (Bind Void) u1)) \to (\forall (a0: C).((drop -(S O) d1 d a0) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t0 (lift -(S O) d1 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d1 -y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a0 y1 y2))))))))))) H2 (lift -(S O) (minus d0 (S n)) x0) H8) in (let H12 \def (eq_ind T u (\lambda (t0: -T).(ty3 g d t0 t)) H1 (lift (S O) (minus d0 (S n)) x0) H8) in (eq_ind_r T -(lift (S O) (minus d0 (S n)) x0) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: -T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: -T).(\lambda (y2: T).(eq T (lift (S n) O t0) (lift (S O) d0 y2)))) (\lambda -(y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H13 \def (H11 e u0 (minus -d0 (S n)) (getl_gen_S (Bind Abst) d (CHead e (Bind Void) u0) u (minus d0 (S -n)) H6) x1 H10) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T -(lift (S O) (minus d0 (S n)) x0) (lift (S O) (minus d0 (S n)) y1)))) (\lambda -(_: T).(\lambda (y2: T).(eq T t (lift (S O) (minus d0 (S n)) y2)))) (\lambda -(y1: T).(\lambda (y2: T).(ty3 g x1 y1 y2))) (ex3_2 T T (\lambda (y1: -T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: -T).(\lambda (y2: T).(eq T (lift (S n) O (lift (S O) (minus d0 (S n)) x0)) -(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) -(\lambda (x2: T).(\lambda (x3: T).(\lambda (H14: (eq T (lift (S O) (minus d0 -(S n)) x0) (lift (S O) (minus d0 (S n)) x2))).(\lambda (H15: (eq T t (lift (S -O) (minus d0 (S n)) x3))).(\lambda (H16: (ty3 g x1 x2 x3)).(let H17 \def -(eq_ind T t (\lambda (t0: T).(ty3 g d (lift (S O) (minus d0 (S n)) x0) t0)) -H12 (lift (S O) (minus d0 (S n)) x3) H15) in (let H18 \def (eq_ind_r T x2 -(\lambda (t0: T).(ty3 g x1 t0 x3)) H16 x0 (lift_inj x0 x2 (S O) (minus d0 (S -n)) H14)) in (eq_ind T (lift (S O) (plus (S n) (minus d0 (S n))) (lift (S n) -O x0)) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T -(TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t0 -(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) -(eq_ind nat d0 (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: -T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq -T (lift (S O) n0 (lift (S n) O x0)) (lift (S O) d0 y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1: -T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: -T).(\lambda (y2: T).(eq T (lift (S O) d0 (lift (S n) O x0)) (lift (S O) d0 -y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (TLRef n) (lift (S -n) O x0) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (TLRef n) t0)) -(refl_equal T (TLRef n)) (lift (S O) d0 (TLRef n)) (lift_lref_lt n (S O) d0 -H5)) (refl_equal T (lift (S O) d0 (lift (S n) O x0))) (ty3_abst g n a x1 x0 -H9 x3 H18)) (plus (S n) (minus d0 (S n))) (le_plus_minus (S n) d0 H5)) (lift -(S n) O (lift (S O) (minus d0 (S n)) x0)) (lift_d x0 (S O) (S n) (minus d0 (S -n)) O (le_O_n (minus d0 (S n)))))))))))) H13)) u H8)))))))) -(getl_drop_conf_lt Abst c0 d u n H0 a (S O) (minus d0 (S n)) H7))))) (\lambda -(H5: (eq nat n d0)).(let H6 \def (eq_ind_r nat d0 (\lambda (n0: nat).(drop (S -O) n0 c0 a)) H4 n H5) in (let H7 \def (eq_ind_r nat d0 (\lambda (n0: -nat).(getl n0 c0 (CHead e (Bind Void) u0))) H3 n H5) in (eq_ind nat n -(\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef -n) (lift (S O) n0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O -u) (lift (S O) n0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 -y2))))) (let H8 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (c1: C).(getl -n c0 c1)) H0 (CHead e (Bind Void) u0) (getl_mono c0 (CHead d (Bind Abst) u) n -H0 (CHead e (Bind Void) u0) H7)) in (let H9 \def (eq_ind C (CHead d (Bind -Abst) u) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with -[(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return -(\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B return -(\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow True | -Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead e (Bind -Void) u0) (getl_mono c0 (CHead d (Bind Abst) u) n H0 (CHead e (Bind Void) u0) -H7)) in (False_ind (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef -n) (lift (S O) n y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O -u) (lift (S O) n y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) -H9))) d0 H5)))) (\lambda (H5: (lt d0 n)).(eq_ind_r nat (S (plus O (minus n (S -O)))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T -(TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift -(S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a -y1 y2))))) (eq_ind nat (plus (S O) (minus n (S O))) (\lambda (n0: nat).(ex3_2 -T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n0) (lift (S O) d0 y1)))) -(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O u) (lift (S O) d0 y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r nat (plus -(minus n (S O)) (S O)) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: -T).(\lambda (_: T).(eq T (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: -T).(\lambda (y2: T).(eq T (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda -(y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1: -T).(\lambda (_: T).(eq T (TLRef (plus (minus n (S O)) (S O))) (lift (S O) d0 -y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O u) (lift (S O) d0 -y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (TLRef (minus n (S -O))) (lift n O u) (eq_ind_r T (TLRef (plus (minus n (S O)) (S O))) (\lambda -(t0: T).(eq T (TLRef (plus (minus n (S O)) (S O))) t0)) (refl_equal T (TLRef -(plus (minus n (S O)) (S O)))) (lift (S O) d0 (TLRef (minus n (S O)))) -(lift_lref_ge (minus n (S O)) (S O) d0 (lt_le_minus d0 n H5))) (eq_ind_r T -(lift (plus (S O) n) O u) (\lambda (t0: T).(eq T (lift (S n) O u) t0)) -(refl_equal T (lift (S n) O u)) (lift (S O) d0 (lift n O u)) (lift_free u n -(S O) O d0 (le_S_n d0 (plus O n) (le_S (S d0) (plus O n) H5)) (le_O_n d0))) -(eq_ind_r nat (S (minus n (S O))) (\lambda (n0: nat).(ty3 g a (TLRef (minus n -(S O))) (lift n0 O u))) (ty3_abst g (minus n (S O)) a d u (getl_drop_conf_ge -n (CHead d (Bind Abst) u) c0 H0 a (S O) d0 H4 (eq_ind_r nat (plus (S O) d0) -(\lambda (n0: nat).(le n0 n)) H5 (plus d0 (S O)) (plus_sym d0 (S O)))) t H1) -n (minus_x_SO n (le_lt_trans O d0 n (le_O_n d0) H5)))) (plus (S O) (minus n -(S O))) (plus_sym (S O) (minus n (S O)))) (S (plus O (minus n (S O)))) -(refl_equal nat (S (plus O (minus n (S O)))))) n (lt_plus_minus O n -(le_lt_trans O d0 n (le_O_n d0) H5))))))))))))))))))) (\lambda (c0: -C).(\lambda (u: T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 u t)).(\lambda -(H1: ((\forall (e: C).(\forall (u0: T).(\forall (d: nat).((getl d c0 (CHead e -(Bind Void) u0)) \to (\forall (a: C).((drop (S O) d c0 a) \to (ex3_2 T T -(\lambda (y1: T).(\lambda (_: T).(eq T u (lift (S O) d y1)))) (\lambda (_: -T).(\lambda (y2: T).(eq T t (lift (S O) d y2)))) (\lambda (y1: T).(\lambda -(y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (b: B).(\lambda (t3: T).(\lambda -(t4: T).(\lambda (H2: (ty3 g (CHead c0 (Bind b) u) t3 t4)).(\lambda (H3: -((\forall (e: C).(\forall (u0: T).(\forall (d: nat).((getl d (CHead c0 (Bind -b) u) (CHead e (Bind Void) u0)) \to (\forall (a: C).((drop (S O) d (CHead c0 -(Bind b) u) a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t3 (lift -(S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t4 (lift (S O) d y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (e: -C).(\lambda (u0: T).(\lambda (d: nat).(\lambda (H4: (getl d c0 (CHead e (Bind -Void) u0))).(\lambda (a: C).(\lambda (H5: (drop (S O) d c0 a)).(let H6 \def -(H1 e u0 d H4 a H5) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T -u (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d -y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T -(\lambda (y1: T).(\lambda (_: T).(eq T (THead (Bind b) u t3) (lift (S O) d -y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Bind b) u t4) (lift (S -O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda -(x0: T).(\lambda (x1: T).(\lambda (H7: (eq T u (lift (S O) d x0))).(\lambda -(H8: (eq T t (lift (S O) d x1))).(\lambda (H9: (ty3 g a x0 x1)).(let H10 \def -(eq_ind T t (\lambda (t0: T).(ty3 g c0 u t0)) H0 (lift (S O) d x1) H8) in -(let H11 \def (eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 (lift (S O) d x1))) -H10 (lift (S O) d x0) H7) in (let H12 \def (eq_ind T u (\lambda (t0: -T).(\forall (e0: C).(\forall (u1: T).(\forall (d0: nat).((getl d0 (CHead c0 -(Bind b) t0) (CHead e0 (Bind Void) u1)) \to (\forall (a0: C).((drop (S O) d0 -(CHead c0 (Bind b) t0) a0) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: -T).(eq T t3 (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t4 -(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a0 y1 -y2))))))))))) H3 (lift (S O) d x0) H7) in (let H13 \def (eq_ind T u (\lambda -(t0: T).(ty3 g (CHead c0 (Bind b) t0) t3 t4)) H2 (lift (S O) d x0) H7) in -(eq_ind_r T (lift (S O) d x0) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: -T).(\lambda (_: T).(eq T (THead (Bind b) t0 t3) (lift (S O) d y1)))) (\lambda -(_: T).(\lambda (y2: T).(eq T (THead (Bind b) t0 t4) (lift (S O) d y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H14 \def (H12 e u0 -(S d) (getl_head (Bind b) d c0 (CHead e (Bind Void) u0) H4 (lift (S O) d x0)) -(CHead a (Bind b) x0) (drop_skip_bind (S O) d c0 a H5 b x0)) in (ex3_2_ind T -T (\lambda (y1: T).(\lambda (_: T).(eq T t3 (lift (S O) (S d) y1)))) (\lambda -(_: T).(\lambda (y2: T).(eq T t4 (lift (S O) (S d) y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g (CHead a (Bind b) x0) y1 y2))) (ex3_2 T T (\lambda -(y1: T).(\lambda (_: T).(eq T (THead (Bind b) (lift (S O) d x0) t3) (lift (S -O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Bind b) (lift (S -O) d x0) t4) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a -y1 y2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H15: (eq T t3 (lift (S -O) (S d) x2))).(\lambda (H16: (eq T t4 (lift (S O) (S d) x3))).(\lambda (H17: -(ty3 g (CHead a (Bind b) x0) x2 x3)).(eq_ind_r T (lift (S O) (S d) x3) -(\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead -(Bind b) (lift (S O) d x0) t3) (lift (S O) d y1)))) (\lambda (_: T).(\lambda -(y2: T).(eq T (THead (Bind b) (lift (S O) d x0) t0) (lift (S O) d y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r T (lift (S O) -(S d) x2) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T -(THead (Bind b) (lift (S O) d x0) t0) (lift (S O) d y1)))) (\lambda (_: -T).(\lambda (y2: T).(eq T (THead (Bind b) (lift (S O) d x0) (lift (S O) (S d) -x3)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 -y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Bind -b) (lift (S O) d x0) (lift (S O) (S d) x2)) (lift (S O) d y1)))) (\lambda (_: -T).(\lambda (y2: T).(eq T (THead (Bind b) (lift (S O) d x0) (lift (S O) (S d) -x3)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) -(THead (Bind b) x0 x2) (THead (Bind b) x0 x3) (sym_eq T (lift (S O) d (THead -(Bind b) x0 x2)) (THead (Bind b) (lift (S O) d x0) (lift (S O) (S d) x2)) -(lift_bind b x0 x2 (S O) d)) (sym_eq T (lift (S O) d (THead (Bind b) x0 x3)) -(THead (Bind b) (lift (S O) d x0) (lift (S O) (S d) x3)) (lift_bind b x0 x3 -(S O) d)) (ty3_bind g a x0 x1 H9 b x2 x3 H17)) t3 H15) t4 H16)))))) H14)) u -H7)))))))))) H6)))))))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda -(u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda (H1: ((\forall (e: C).(\forall -(u0: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Void) u0)) \to (\forall -(a: C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: -T).(eq T w (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T u -(lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 -y2)))))))))))).(\lambda (v: T).(\lambda (t: T).(\lambda (H2: (ty3 g c0 v -(THead (Bind Abst) u t))).(\lambda (H3: ((\forall (e: C).(\forall (u0: -T).(\forall (d: nat).((getl d c0 (CHead e (Bind Void) u0)) \to (\forall (a: -C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T -v (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Bind -Abst) u t) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 -y2)))))))))))).(\lambda (e: C).(\lambda (u0: T).(\lambda (d: nat).(\lambda -(H4: (getl d c0 (CHead e (Bind Void) u0))).(\lambda (a: C).(\lambda (H5: -(drop (S O) d c0 a)).(let H6 \def (H3 e u0 d H4 a H5) in (ex3_2_ind T T -(\lambda (y1: T).(\lambda (_: T).(eq T v (lift (S O) d y1)))) (\lambda (_: -T).(\lambda (y2: T).(eq T (THead (Bind Abst) u t) (lift (S O) d y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: -T).(\lambda (_: T).(eq T (THead (Flat Appl) w v) (lift (S O) d y1)))) -(\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) w (THead (Bind -Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a -y1 y2)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: (eq T v (lift (S O) -d x0))).(\lambda (H8: (eq T (THead (Bind Abst) u t) (lift (S O) d -x1))).(\lambda (H9: (ty3 g a x0 x1)).(let H10 \def (eq_ind T v (\lambda (t0: -T).(ty3 g c0 t0 (THead (Bind Abst) u t))) H2 (lift (S O) d x0) H7) in -(eq_ind_r T (lift (S O) d x0) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: -T).(\lambda (_: T).(eq T (THead (Flat Appl) w t0) (lift (S O) d y1)))) -(\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) w (THead (Bind -Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a -y1 y2))))) (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x1 (THead -(Bind Abst) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift (S O) d -y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift (S O) (S d) z)))) (ex3_2 -T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Appl) w (lift (S O) d -x0)) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat -Appl) w (THead (Bind Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: T).(\lambda (x3: -T).(\lambda (H11: (eq T x1 (THead (Bind Abst) x2 x3))).(\lambda (H12: (eq T u -(lift (S O) d x2))).(\lambda (H13: (eq T t (lift (S O) (S d) x3))).(let H14 -\def (eq_ind T x1 (\lambda (t0: T).(ty3 g a x0 t0)) H9 (THead (Bind Abst) x2 -x3) H11) in (eq_ind_r T (lift (S O) (S d) x3) (\lambda (t0: T).(ex3_2 T T -(\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Appl) w (lift (S O) d -x0)) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat -Appl) w (THead (Bind Abst) u t0)) (lift (S O) d y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H15 \def (eq_ind T u (\lambda -(t0: T).(\forall (e0: C).(\forall (u1: T).(\forall (d0: nat).((getl d0 c0 -(CHead e0 (Bind Void) u1)) \to (\forall (a0: C).((drop (S O) d0 c0 a0) \to -(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T w (lift (S O) d0 y1)))) -(\lambda (_: T).(\lambda (y2: T).(eq T t0 (lift (S O) d0 y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a0 y1 y2))))))))))) H1 (lift (S O) d x2) H12) in -(eq_ind_r T (lift (S O) d x2) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: -T).(\lambda (_: T).(eq T (THead (Flat Appl) w (lift (S O) d x0)) (lift (S O) -d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) w (THead -(Bind Abst) t0 (lift (S O) (S d) x3))) (lift (S O) d y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H16 \def (H15 e u0 d H4 a H5) in -(ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T w (lift (S O) d y1)))) -(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d x2) (lift (S O) d y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: -T).(\lambda (_: T).(eq T (THead (Flat Appl) w (lift (S O) d x0)) (lift (S O) -d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) w (THead -(Bind Abst) (lift (S O) d x2) (lift (S O) (S d) x3))) (lift (S O) d y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x4: -T).(\lambda (x5: T).(\lambda (H17: (eq T w (lift (S O) d x4))).(\lambda (H18: -(eq T (lift (S O) d x2) (lift (S O) d x5))).(\lambda (H19: (ty3 g a x4 -x5)).(eq_ind_r T (lift (S O) d x4) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: -T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (lift (S O) d x0)) (lift (S O) -d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) t0 (THead -(Bind Abst) (lift (S O) d x2) (lift (S O) (S d) x3))) (lift (S O) d y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H20 \def (eq_ind_r -T x5 (\lambda (t0: T).(ty3 g a x4 t0)) H19 x2 (lift_inj x2 x5 (S O) d H18)) -in (eq_ind T (lift (S O) d (THead (Bind Abst) x2 x3)) (\lambda (t0: T).(ex3_2 -T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Appl) (lift (S O) d -x4) (lift (S O) d x0)) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: -T).(eq T (THead (Flat Appl) (lift (S O) d x4) t0) (lift (S O) d y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind T (lift (S O) d -(THead (Flat Appl) x4 x0)) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: -T).(\lambda (_: T).(eq T t0 (lift (S O) d y1)))) (\lambda (_: T).(\lambda -(y2: T).(eq T (THead (Flat Appl) (lift (S O) d x4) (lift (S O) d (THead (Bind -Abst) x2 x3))) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g -a y1 y2))))) (eq_ind T (lift (S O) d (THead (Flat Appl) x4 (THead (Bind Abst) -x2 x3))) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T -(lift (S O) d (THead (Flat Appl) x4 x0)) (lift (S O) d y1)))) (\lambda (_: -T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda -(y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: -T).(eq T (lift (S O) d (THead (Flat Appl) x4 x0)) (lift (S O) d y1)))) -(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d (THead (Flat Appl) x4 -(THead (Bind Abst) x2 x3))) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda -(y2: T).(ty3 g a y1 y2))) (THead (Flat Appl) x4 x0) (THead (Flat Appl) x4 -(THead (Bind Abst) x2 x3)) (refl_equal T (lift (S O) d (THead (Flat Appl) x4 -x0))) (refl_equal T (lift (S O) d (THead (Flat Appl) x4 (THead (Bind Abst) x2 -x3)))) (ty3_appl g a x4 x2 H20 x0 x3 H14)) (THead (Flat Appl) (lift (S O) d -x4) (lift (S O) d (THead (Bind Abst) x2 x3))) (lift_flat Appl x4 (THead (Bind -Abst) x2 x3) (S O) d)) (THead (Flat Appl) (lift (S O) d x4) (lift (S O) d -x0)) (lift_flat Appl x4 x0 (S O) d)) (THead (Bind Abst) (lift (S O) d x2) -(lift (S O) (S d) x3)) (lift_bind Abst x2 x3 (S O) d))) w H17)))))) H16)) u -H12)) t H13))))))) (lift_gen_bind Abst u t x1 (S O) d H8)) v H7))))))) -H6))))))))))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: -T).(\lambda (H0: (ty3 g c0 t3 t4)).(\lambda (H1: ((\forall (e: C).(\forall -(u: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Void) u)) \to (\forall -(a: C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: -T).(eq T t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t4 -(lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 -y2)))))))))))).(\lambda (t0: T).(\lambda (H2: (ty3 g c0 t4 t0)).(\lambda (H3: -((\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead e (Bind -Void) u)) \to (\forall (a: C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda -(y1: T).(\lambda (_: T).(eq T t4 (lift (S O) d y1)))) (\lambda (_: -T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda -(y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (e: C).(\lambda (u: T).(\lambda -(d: nat).(\lambda (H4: (getl d c0 (CHead e (Bind Void) u))).(\lambda (a: -C).(\lambda (H5: (drop (S O) d c0 a)).(let H6 \def (H3 e u d H4 a H5) in -(ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T t4 (lift (S O) d y1)))) -(\lambda (_: T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda -(_: T).(eq T (THead (Flat Cast) t4 t3) (lift (S O) d y1)))) (\lambda (_: -T).(\lambda (y2: T).(eq T (THead (Flat Cast) t0 t4) (lift (S O) d y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H7: (eq T t4 (lift (S O) d x0))).(\lambda (H8: -(eq T t0 (lift (S O) d x1))).(\lambda (H9: (ty3 g a x0 x1)).(let H10 \def -(eq_ind T t0 (\lambda (t: T).(ty3 g c0 t4 t)) H2 (lift (S O) d x1) H8) in -(eq_ind_r T (lift (S O) d x1) (\lambda (t: T).(ex3_2 T T (\lambda (y1: -T).(\lambda (_: T).(eq T (THead (Flat Cast) t4 t3) (lift (S O) d y1)))) -(\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Cast) t t4) (lift (S O) d -y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H11 \def -(eq_ind T t4 (\lambda (t: T).(ty3 g c0 t (lift (S O) d x1))) H10 (lift (S O) -d x0) H7) in (let H12 \def (eq_ind T t4 (\lambda (t: T).(\forall (e0: -C).(\forall (u0: T).(\forall (d0: nat).((getl d0 c0 (CHead e0 (Bind Void) -u0)) \to (\forall (a0: C).((drop (S O) d0 c0 a0) \to (ex3_2 T T (\lambda (y1: -T).(\lambda (_: T).(eq T t3 (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda -(y2: T).(eq T t (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 -g a0 y1 y2))))))))))) H1 (lift (S O) d x0) H7) in (let H13 \def (eq_ind T t4 -(\lambda (t: T).(ty3 g c0 t3 t)) H0 (lift (S O) d x0) H7) in (eq_ind_r T -(lift (S O) d x0) (\lambda (t: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: -T).(eq T (THead (Flat Cast) t t3) (lift (S O) d y1)))) (\lambda (_: -T).(\lambda (y2: T).(eq T (THead (Flat Cast) (lift (S O) d x1) t) (lift (S O) -d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H14 \def -(H12 e u d H4 a H5) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T -t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d -x0) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) -(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Cast) (lift (S -O) d x0) t3) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T -(THead (Flat Cast) (lift (S O) d x1) (lift (S O) d x0)) (lift (S O) d y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: -T).(\lambda (x3: T).(\lambda (H15: (eq T t3 (lift (S O) d x2))).(\lambda -(H16: (eq T (lift (S O) d x0) (lift (S O) d x3))).(\lambda (H17: (ty3 g a x2 -x3)).(let H18 \def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 t (lift (S O) d -x0))) H13 (lift (S O) d x2) H15) in (eq_ind_r T (lift (S O) d x2) (\lambda -(t: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Cast) -(lift (S O) d x0) t) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: -T).(eq T (THead (Flat Cast) (lift (S O) d x1) (lift (S O) d x0)) (lift (S O) -d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H19 \def -(eq_ind_r T x3 (\lambda (t: T).(ty3 g a x2 t)) H17 x0 (lift_inj x0 x3 (S O) d -H16)) in (eq_ind T (lift (S O) d (THead (Flat Cast) x0 x2)) (\lambda (t: -T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t (lift (S O) d y1)))) -(\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Cast) (lift (S O) d x1) -(lift (S O) d x0)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: -T).(ty3 g a y1 y2))))) (eq_ind T (lift (S O) d (THead (Flat Cast) x1 x0)) -(\lambda (t: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (lift (S O) -d (THead (Flat Cast) x0 x2)) (lift (S O) d y1)))) (\lambda (_: T).(\lambda -(y2: T).(eq T t (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g -a y1 y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(eq T (lift (S -O) d (THead (Flat Cast) x0 x2)) (lift (S O) d y1)))) (\lambda (_: T).(\lambda -(y2: T).(eq T (lift (S O) d (THead (Flat Cast) x1 x0)) (lift (S O) d y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (THead (Flat Cast) x0 x2) -(THead (Flat Cast) x1 x0) (refl_equal T (lift (S O) d (THead (Flat Cast) x0 -x2))) (refl_equal T (lift (S O) d (THead (Flat Cast) x1 x0))) (ty3_cast g a -x2 x0 H19 x1 H9)) (THead (Flat Cast) (lift (S O) d x1) (lift (S O) d x0)) -(lift_flat Cast x1 x0 (S O) d)) (THead (Flat Cast) (lift (S O) d x0) (lift (S -O) d x2)) (lift_flat Cast x0 x2 (S O) d))) t3 H15))))))) H14)) t4 H7)))) t0 -H8))))))) H6)))))))))))))))) c t1 t2 H))))). -(* COMMENTS -Initial nodes: 13105 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/wcpr0/defs.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/wcpr0/defs.ma deleted file mode 100644 index cdc9bf3a4..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/wcpr0/defs.ma +++ /dev/null @@ -1,26 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr0/defs.ma". - -include "Basic-1/C/defs.ma". - -inductive wcpr0: C \to (C \to Prop) \def -| wcpr0_refl: \forall (c: C).(wcpr0 c c) -| wcpr0_comp: \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall -(u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (k: K).(wcpr0 (CHead c1 k -u1) (CHead c2 k u2)))))))). - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/wcpr0/fwd.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/wcpr0/fwd.ma deleted file mode 100644 index 2b0531a8a..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/wcpr0/fwd.ma +++ /dev/null @@ -1,105 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/wcpr0/defs.ma". - -theorem wcpr0_gen_sort: - \forall (x: C).(\forall (n: nat).((wcpr0 (CSort n) x) \to (eq C x (CSort -n)))) -\def - \lambda (x: C).(\lambda (n: nat).(\lambda (H: (wcpr0 (CSort n) -x)).(insert_eq C (CSort n) (\lambda (c: C).(wcpr0 c x)) (\lambda (c: C).(eq C -x c)) (\lambda (y: C).(\lambda (H0: (wcpr0 y x)).(wcpr0_ind (\lambda (c: -C).(\lambda (c0: C).((eq C c (CSort n)) \to (eq C c0 c)))) (\lambda (c: -C).(\lambda (H1: (eq C c (CSort n))).(let H2 \def (f_equal C C (\lambda (e: -C).e) c (CSort n) H1) in (eq_ind_r C (CSort n) (\lambda (c0: C).(eq C c0 c0)) -(refl_equal C (CSort n)) c H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda -(_: (wcpr0 c1 c2)).(\lambda (_: (((eq C c1 (CSort n)) \to (eq C c2 -c1)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda -(k: K).(\lambda (H4: (eq C (CHead c1 k u1) (CSort n))).(let H5 \def (eq_ind C -(CHead c1 k u1) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) -with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I -(CSort n) H4) in (False_ind (eq C (CHead c2 k u2) (CHead c1 k u1)) -H5))))))))))) y x H0))) H))). -(* COMMENTS -Initial nodes: 249 -END *) - -theorem wcpr0_gen_head: - \forall (k: K).(\forall (c1: C).(\forall (x: C).(\forall (u1: T).((wcpr0 -(CHead c1 k u1) x) \to (or (eq C x (CHead c1 k u1)) (ex3_2 C T (\lambda (c2: -C).(\lambda (u2: T).(eq C x (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: -T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2))))))))) -\def - \lambda (k: K).(\lambda (c1: C).(\lambda (x: C).(\lambda (u1: T).(\lambda -(H: (wcpr0 (CHead c1 k u1) x)).(insert_eq C (CHead c1 k u1) (\lambda (c: -C).(wcpr0 c x)) (\lambda (c: C).(or (eq C x c) (ex3_2 C T (\lambda (c2: -C).(\lambda (u2: T).(eq C x (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: -T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))) (\lambda -(y: C).(\lambda (H0: (wcpr0 y x)).(wcpr0_ind (\lambda (c: C).(\lambda (c0: -C).((eq C c (CHead c1 k u1)) \to (or (eq C c0 c) (ex3_2 C T (\lambda (c2: -C).(\lambda (u2: T).(eq C c0 (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: -T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))))) -(\lambda (c: C).(\lambda (H1: (eq C c (CHead c1 k u1))).(let H2 \def (f_equal -C C (\lambda (e: C).e) c (CHead c1 k u1) H1) in (eq_ind_r C (CHead c1 k u1) -(\lambda (c0: C).(or (eq C c0 c0) (ex3_2 C T (\lambda (c2: C).(\lambda (u2: -T).(eq C c0 (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: T).(wcpr0 c1 -c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))) (or_introl (eq C -(CHead c1 k u1) (CHead c1 k u1)) (ex3_2 C T (\lambda (c2: C).(\lambda (u2: -T).(eq C (CHead c1 k u1) (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: -T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))) -(refl_equal C (CHead c1 k u1))) c H2)))) (\lambda (c0: C).(\lambda (c2: -C).(\lambda (H1: (wcpr0 c0 c2)).(\lambda (H2: (((eq C c0 (CHead c1 k u1)) \to -(or (eq C c2 c0) (ex3_2 C T (\lambda (c3: C).(\lambda (u2: T).(eq C c2 (CHead -c3 k u2)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: -C).(\lambda (u2: T).(pr0 u1 u2)))))))).(\lambda (u0: T).(\lambda (u2: -T).(\lambda (H3: (pr0 u0 u2)).(\lambda (k0: K).(\lambda (H4: (eq C (CHead c0 -k0 u0) (CHead c1 k u1))).(let H5 \def (f_equal C C (\lambda (e: C).(match e -in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) -\Rightarrow c])) (CHead c0 k0 u0) (CHead c1 k u1) H4) in ((let H6 \def -(f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with -[(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead c0 k0 u0) -(CHead c1 k u1) H4) in ((let H7 \def (f_equal C T (\lambda (e: C).(match e in -C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) -\Rightarrow t])) (CHead c0 k0 u0) (CHead c1 k u1) H4) in (\lambda (H8: (eq K -k0 k)).(\lambda (H9: (eq C c0 c1)).(eq_ind_r K k (\lambda (k1: K).(or (eq C -(CHead c2 k1 u2) (CHead c0 k1 u0)) (ex3_2 C T (\lambda (c3: C).(\lambda (u3: -T).(eq C (CHead c2 k1 u2) (CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_: -T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3)))))) (let H10 -\def (eq_ind T u0 (\lambda (t: T).(pr0 t u2)) H3 u1 H7) in (eq_ind_r T u1 -(\lambda (t: T).(or (eq C (CHead c2 k u2) (CHead c0 k t)) (ex3_2 C T (\lambda -(c3: C).(\lambda (u3: T).(eq C (CHead c2 k u2) (CHead c3 k u3)))) (\lambda -(c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 -u1 u3)))))) (let H11 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 k -u1)) \to (or (eq C c2 c) (ex3_2 C T (\lambda (c3: C).(\lambda (u3: T).(eq C -c2 (CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) -(\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))))))) H2 c1 H9) in (let H12 \def -(eq_ind C c0 (\lambda (c: C).(wcpr0 c c2)) H1 c1 H9) in (eq_ind_r C c1 -(\lambda (c: C).(or (eq C (CHead c2 k u2) (CHead c k u1)) (ex3_2 C T (\lambda -(c3: C).(\lambda (u3: T).(eq C (CHead c2 k u2) (CHead c3 k u3)))) (\lambda -(c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 -u1 u3)))))) (or_intror (eq C (CHead c2 k u2) (CHead c1 k u1)) (ex3_2 C T -(\lambda (c3: C).(\lambda (u3: T).(eq C (CHead c2 k u2) (CHead c3 k u3)))) -(\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda -(u3: T).(pr0 u1 u3)))) (ex3_2_intro C T (\lambda (c3: C).(\lambda (u3: T).(eq -C (CHead c2 k u2) (CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 -c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))) c2 u2 (refl_equal C -(CHead c2 k u2)) H12 H10)) c0 H9))) u0 H7)) k0 H8)))) H6)) H5))))))))))) y x -H0))) H))))). -(* COMMENTS -Initial nodes: 1133 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/wcpr0/getl.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/wcpr0/getl.ma deleted file mode 100644 index d3a109e0d..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/wcpr0/getl.ma +++ /dev/null @@ -1,464 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/wcpr0/defs.ma". - -include "Basic-1/getl/props.ma". - -theorem wcpr0_drop: - \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (h: -nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((drop h O c1 (CHead -e1 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c2 -(CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda -(_: C).(\lambda (u2: T).(pr0 u1 u2))))))))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(wcpr0_ind -(\lambda (c: C).(\lambda (c0: C).(\forall (h: nat).(\forall (e1: C).(\forall -(u1: T).(\forall (k: K).((drop h O c (CHead e1 k u1)) \to (ex3_2 C T (\lambda -(e2: C).(\lambda (u2: T).(drop h O c0 (CHead e2 k u2)))) (\lambda (e2: -C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 -u2))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda -(u1: T).(\lambda (k: K).(\lambda (H0: (drop h O c (CHead e1 k -u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c (CHead -e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: -C).(\lambda (u2: T).(pr0 u1 u2))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl -u1)))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wcpr0 c3 -c4)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1: -T).(\forall (k: K).((drop h O c3 (CHead e1 k u1)) \to (ex3_2 C T (\lambda -(e2: C).(\lambda (u2: T).(drop h O c4 (CHead e2 k u2)))) (\lambda (e2: -C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 -u2))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 -u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall -(e1: C).(\forall (u3: T).(\forall (k0: K).((drop n O (CHead c3 k u1) (CHead -e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead -c4 k u2) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) -(\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))) (\lambda (e1: -C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (drop O O (CHead c3 k u1) -(CHead e1 k0 u0))).(let H4 \def (f_equal C C (\lambda (e: C).(match e in C -return (\lambda (_: C).C) with [(CSort _) \Rightarrow c3 | (CHead c _ _) -\Rightarrow c])) (CHead c3 k u1) (CHead e1 k0 u0) (drop_gen_refl (CHead c3 k -u1) (CHead e1 k0 u0) H3)) in ((let H5 \def (f_equal C K (\lambda (e: -C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | -(CHead _ k1 _) \Rightarrow k1])) (CHead c3 k u1) (CHead e1 k0 u0) -(drop_gen_refl (CHead c3 k u1) (CHead e1 k0 u0) H3)) in ((let H6 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead c3 k u1) -(CHead e1 k0 u0) (drop_gen_refl (CHead c3 k u1) (CHead e1 k0 u0) H3)) in -(\lambda (H7: (eq K k k0)).(\lambda (H8: (eq C c3 e1)).(eq_ind K k (\lambda -(k1: K).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c4 k -u2) (CHead e2 k1 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) -(\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))))) (eq_ind T u1 (\lambda (t: -T).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c4 k u2) -(CHead e2 k u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda -(_: C).(\lambda (u3: T).(pr0 t u3))))) (eq_ind C c3 (\lambda (c: C).(ex3_2 C -T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c4 k u2) (CHead e2 k -u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 c e2))) (\lambda (_: -C).(\lambda (u3: T).(pr0 u1 u3))))) (ex3_2_intro C T (\lambda (e2: -C).(\lambda (u3: T).(drop O O (CHead c4 k u2) (CHead e2 k u3)))) (\lambda -(e2: C).(\lambda (_: T).(wcpr0 c3 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 -u1 u3))) c4 u2 (drop_refl (CHead c4 k u2)) H0 H2) e1 H8) u0 H6) k0 H7)))) -H5)) H4)))))) (K_ind (\lambda (k0: K).(\forall (n: nat).(((\forall (e1: -C).(\forall (u3: T).(\forall (k1: K).((drop n O (CHead c3 k0 u1) (CHead e1 k1 -u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c4 k0 -u2) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) -(\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))) \to (\forall (e1: -C).(\forall (u3: T).(\forall (k1: K).((drop (S n) O (CHead c3 k0 u1) (CHead -e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop (S n) O -(CHead c4 k0 u2) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 -e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))))) (\lambda (b: -B).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall -(k0: K).((drop n O (CHead c3 (Bind b) u1) (CHead e1 k0 u3)) \to (ex3_2 C T -(\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c4 (Bind b) u2) (CHead e2 -k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: -C).(\lambda (u4: T).(pr0 u3 u4)))))))))).(\lambda (e1: C).(\lambda (u0: -T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c3 (Bind b) u1) (CHead -e1 k0 u0))).(let H5 \def (H1 n e1 u0 k0 (drop_gen_drop (Bind b) c3 (CHead e1 -k0 u0) u1 n H4)) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop n -O c4 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) -(\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) (ex3_2 C T (\lambda (e2: -C).(\lambda (u3: T).(drop (S n) O (CHead c4 (Bind b) u2) (CHead e2 k0 u3)))) -(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda -(u3: T).(pr0 u0 u3)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (drop -n O c4 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 e1 x0)).(\lambda (H8: (pr0 u0 -x1)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead -c4 (Bind b) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 -e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) x0 x1 (drop_drop -(Bind b) n c4 (CHead x0 k0 x1) H6 u2) H7 H8)))))) H5))))))))) (\lambda (f: -F).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall -(k0: K).((drop n O (CHead c3 (Flat f) u1) (CHead e1 k0 u3)) \to (ex3_2 C T -(\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c4 (Flat f) u2) (CHead e2 -k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: -C).(\lambda (u4: T).(pr0 u3 u4)))))))))).(\lambda (e1: C).(\lambda (u0: -T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c3 (Flat f) u1) (CHead -e1 k0 u0))).(let H5 \def (H1 (S n) e1 u0 k0 (drop_gen_drop (Flat f) c3 (CHead -e1 k0 u0) u1 n H4)) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop -(S n) O c4 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 -e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) (ex3_2 C T (\lambda (e2: -C).(\lambda (u3: T).(drop (S n) O (CHead c4 (Flat f) u2) (CHead e2 k0 u3)))) -(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda -(u3: T).(pr0 u0 u3)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (drop -(S n) O c4 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 e1 x0)).(\lambda (H8: (pr0 -u0 x1)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O -(CHead c4 (Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: -T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) x0 x1 -(drop_drop (Flat f) n c4 (CHead x0 k0 x1) H6 u2) H7 H8)))))) H5))))))))) k) -h)))))))))) c1 c2 H))). -(* COMMENTS -Initial nodes: 1755 -END *) - -theorem wcpr0_drop_back: - \forall (c1: C).(\forall (c2: C).((wcpr0 c2 c1) \to (\forall (h: -nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((drop h O c1 (CHead -e1 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c2 -(CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda -(_: C).(\lambda (u2: T).(pr0 u2 u1))))))))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c2 c1)).(wcpr0_ind -(\lambda (c: C).(\lambda (c0: C).(\forall (h: nat).(\forall (e1: C).(\forall -(u1: T).(\forall (k: K).((drop h O c0 (CHead e1 k u1)) \to (ex3_2 C T -(\lambda (e2: C).(\lambda (u2: T).(drop h O c (CHead e2 k u2)))) (\lambda -(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 -u2 u1))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda -(u1: T).(\lambda (k: K).(\lambda (H0: (drop h O c (CHead e1 k -u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c (CHead -e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: -C).(\lambda (u2: T).(pr0 u2 u1))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl -u1)))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wcpr0 c3 -c4)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1: -T).(\forall (k: K).((drop h O c4 (CHead e1 k u1)) \to (ex3_2 C T (\lambda -(e2: C).(\lambda (u2: T).(drop h O c3 (CHead e2 k u2)))) (\lambda (e2: -C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 -u1))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 -u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall -(e1: C).(\forall (u3: T).(\forall (k0: K).((drop n O (CHead c4 k u2) (CHead -e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead -c3 k u1) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) -(\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))) (\lambda (e1: -C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (drop O O (CHead c4 k u2) -(CHead e1 k0 u0))).(let H4 \def (f_equal C C (\lambda (e: C).(match e in C -return (\lambda (_: C).C) with [(CSort _) \Rightarrow c4 | (CHead c _ _) -\Rightarrow c])) (CHead c4 k u2) (CHead e1 k0 u0) (drop_gen_refl (CHead c4 k -u2) (CHead e1 k0 u0) H3)) in ((let H5 \def (f_equal C K (\lambda (e: -C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | -(CHead _ k1 _) \Rightarrow k1])) (CHead c4 k u2) (CHead e1 k0 u0) -(drop_gen_refl (CHead c4 k u2) (CHead e1 k0 u0) H3)) in ((let H6 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c4 k u2) -(CHead e1 k0 u0) (drop_gen_refl (CHead c4 k u2) (CHead e1 k0 u0) H3)) in -(\lambda (H7: (eq K k k0)).(\lambda (H8: (eq C c4 e1)).(eq_ind K k (\lambda -(k1: K).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c3 k -u1) (CHead e2 k1 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) -(\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))))) (eq_ind T u2 (\lambda (t: -T).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c3 k u1) -(CHead e2 k u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda -(_: C).(\lambda (u3: T).(pr0 u3 t))))) (eq_ind C c4 (\lambda (c: C).(ex3_2 C -T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c3 k u1) (CHead e2 k -u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 c))) (\lambda (_: -C).(\lambda (u3: T).(pr0 u3 u2))))) (ex3_2_intro C T (\lambda (e2: -C).(\lambda (u3: T).(drop O O (CHead c3 k u1) (CHead e2 k u3)))) (\lambda -(e2: C).(\lambda (_: T).(wcpr0 e2 c4))) (\lambda (_: C).(\lambda (u3: T).(pr0 -u3 u2))) c3 u1 (drop_refl (CHead c3 k u1)) H0 H2) e1 H8) u0 H6) k0 H7)))) -H5)) H4)))))) (K_ind (\lambda (k0: K).(\forall (n: nat).(((\forall (e1: -C).(\forall (u3: T).(\forall (k1: K).((drop n O (CHead c4 k0 u2) (CHead e1 k1 -u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c3 k0 -u1) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) -(\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))) \to (\forall (e1: -C).(\forall (u3: T).(\forall (k1: K).((drop (S n) O (CHead c4 k0 u2) (CHead -e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop (S n) O -(CHead c3 k0 u1) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 -e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))))) (\lambda (b: -B).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall -(k0: K).((drop n O (CHead c4 (Bind b) u2) (CHead e1 k0 u3)) \to (ex3_2 C T -(\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c3 (Bind b) u1) (CHead e2 -k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: -C).(\lambda (u4: T).(pr0 u4 u3)))))))))).(\lambda (e1: C).(\lambda (u0: -T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c4 (Bind b) u2) (CHead -e1 k0 u0))).(let H5 \def (H1 n e1 u0 k0 (drop_gen_drop (Bind b) c4 (CHead e1 -k0 u0) u2 n H4)) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop n -O c3 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) -(\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) (ex3_2 C T (\lambda (e2: -C).(\lambda (u3: T).(drop (S n) O (CHead c3 (Bind b) u1) (CHead e2 k0 u3)))) -(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda -(u3: T).(pr0 u3 u0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (drop -n O c3 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 x0 e1)).(\lambda (H8: (pr0 x1 -u0)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead -c3 (Bind b) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 -e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) x0 x1 (drop_drop -(Bind b) n c3 (CHead x0 k0 x1) H6 u1) H7 H8)))))) H5))))))))) (\lambda (f: -F).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall -(k0: K).((drop n O (CHead c4 (Flat f) u2) (CHead e1 k0 u3)) \to (ex3_2 C T -(\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c3 (Flat f) u1) (CHead e2 -k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: -C).(\lambda (u4: T).(pr0 u4 u3)))))))))).(\lambda (e1: C).(\lambda (u0: -T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c4 (Flat f) u2) (CHead -e1 k0 u0))).(let H5 \def (H1 (S n) e1 u0 k0 (drop_gen_drop (Flat f) c4 (CHead -e1 k0 u0) u2 n H4)) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop -(S n) O c3 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 -e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) (ex3_2 C T (\lambda (e2: -C).(\lambda (u3: T).(drop (S n) O (CHead c3 (Flat f) u1) (CHead e2 k0 u3)))) -(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda -(u3: T).(pr0 u3 u0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (drop -(S n) O c3 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 x0 e1)).(\lambda (H8: (pr0 -x1 u0)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O -(CHead c3 (Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: -T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) x0 x1 -(drop_drop (Flat f) n c3 (CHead x0 k0 x1) H6 u1) H7 H8)))))) H5))))))))) k) -h)))))))))) c2 c1 H))). -(* COMMENTS -Initial nodes: 1755 -END *) - -theorem wcpr0_getl: - \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (h: -nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((getl h c1 (CHead e1 -k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(getl h c2 (CHead e2 -k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: -C).(\lambda (u2: T).(pr0 u1 u2))))))))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(wcpr0_ind -(\lambda (c: C).(\lambda (c0: C).(\forall (h: nat).(\forall (e1: C).(\forall -(u1: T).(\forall (k: K).((getl h c (CHead e1 k u1)) \to (ex3_2 C T (\lambda -(e2: C).(\lambda (u2: T).(getl h c0 (CHead e2 k u2)))) (\lambda (e2: -C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 -u2))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda -(u1: T).(\lambda (k: K).(\lambda (H0: (getl h c (CHead e1 k -u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(getl h c (CHead e2 -k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: -C).(\lambda (u2: T).(pr0 u1 u2))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl -u1)))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wcpr0 c3 -c4)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1: -T).(\forall (k: K).((getl h c3 (CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2: -C).(\lambda (u2: T).(getl h c4 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda -(_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 -u2))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 -u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall -(e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c3 k u1) (CHead e1 -k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c4 k -u2) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) -(\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))) (\lambda (e1: -C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (getl O (CHead c3 k u1) -(CHead e1 k0 u0))).(K_ind (\lambda (k1: K).((clear (CHead c3 k1 u1) (CHead e1 -k0 u0)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c4 k1 -u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) -(\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))))) (\lambda (b: B).(\lambda -(H4: (clear (CHead c3 (Bind b) u1) (CHead e1 k0 u0))).(let H5 \def (f_equal C -C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow e1 | (CHead c _ _) \Rightarrow c])) (CHead e1 k0 u0) (CHead c3 -(Bind b) u1) (clear_gen_bind b c3 (CHead e1 k0 u0) u1 H4)) in ((let H6 \def -(f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with -[(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead e1 k0 u0) -(CHead c3 (Bind b) u1) (clear_gen_bind b c3 (CHead e1 k0 u0) u1 H4)) in ((let -H7 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead e1 k0 -u0) (CHead c3 (Bind b) u1) (clear_gen_bind b c3 (CHead e1 k0 u0) u1 H4)) in -(\lambda (H8: (eq K k0 (Bind b))).(\lambda (H9: (eq C e1 c3)).(eq_ind_r K -(Bind b) (\lambda (k1: K).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl -O (CHead c4 (Bind b) u2) (CHead e2 k1 u3)))) (\lambda (e2: C).(\lambda (_: -T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))))) (eq_ind_r -T u1 (\lambda (t: T).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O -(CHead c4 (Bind b) u2) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda -(_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 t u3))))) -(eq_ind_r C c3 (\lambda (c: C).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: -T).(getl O (CHead c4 (Bind b) u2) (CHead e2 (Bind b) u3)))) (\lambda (e2: -C).(\lambda (_: T).(wcpr0 c e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 -u3))))) (ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c4 -(Bind b) u2) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda (_: -T).(wcpr0 c3 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))) c4 u2 -(getl_refl b c4 u2) H0 H2) e1 H9) u0 H7) k0 H8)))) H6)) H5)))) (\lambda (f: -F).(\lambda (H4: (clear (CHead c3 (Flat f) u1) (CHead e1 k0 u0))).(let H5 -\def (H1 O e1 u0 k0 (getl_intro O c3 (CHead e1 k0 u0) c3 (drop_refl c3) -(clear_gen_flat f c3 (CHead e1 k0 u0) u1 H4))) in (ex3_2_ind C T (\lambda -(e2: C).(\lambda (u3: T).(getl O c4 (CHead e2 k0 u3)))) (\lambda (e2: -C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 -u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c4 (Flat f) -u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) -(\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))) (\lambda (x0: C).(\lambda -(x1: T).(\lambda (H6: (getl O c4 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 e1 -x0)).(\lambda (H8: (pr0 u0 x1)).(ex3_2_intro C T (\lambda (e2: C).(\lambda -(u3: T).(getl O (CHead c4 (Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2: -C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 -u3))) x0 x1 (getl_flat c4 (CHead x0 k0 x1) O H6 f u2) H7 H8)))))) H5)))) k -(getl_gen_O (CHead c3 k u1) (CHead e1 k0 u0) H3)))))) (K_ind (\lambda (k0: -K).(\forall (n: nat).(((\forall (e1: C).(\forall (u3: T).(\forall (k1: -K).((getl n (CHead c3 k0 u1) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: -C).(\lambda (u4: T).(getl n (CHead c4 k0 u2) (CHead e2 k1 u4)))) (\lambda -(e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 -u3 u4))))))))) \to (\forall (e1: C).(\forall (u3: T).(\forall (k1: K).((getl -(S n) (CHead c3 k0 u1) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: -C).(\lambda (u4: T).(getl (S n) (CHead c4 k0 u2) (CHead e2 k1 u4)))) (\lambda -(e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 -u3 u4))))))))))) (\lambda (b: B).(\lambda (n: nat).(\lambda (_: ((\forall -(e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c3 (Bind b) u1) -(CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n -(CHead c4 (Bind b) u2) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: -T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 u3 -u4)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4: -(getl (S n) (CHead c3 (Bind b) u1) (CHead e1 k0 u0))).(let H5 \def (H1 n e1 -u0 k0 (getl_gen_S (Bind b) c3 (CHead e1 k0 u0) u1 n H4)) in (ex3_2_ind C T -(\lambda (e2: C).(\lambda (u3: T).(getl n c4 (CHead e2 k0 u3)))) (\lambda -(e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 -u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c4 -(Bind b) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 -e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (H6: (getl n c4 (CHead x0 k0 x1))).(\lambda (H7: -(wcpr0 e1 x0)).(\lambda (H8: (pr0 u0 x1)).(ex3_2_intro C T (\lambda (e2: -C).(\lambda (u3: T).(getl (S n) (CHead c4 (Bind b) u2) (CHead e2 k0 u3)))) -(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda -(u3: T).(pr0 u0 u3))) x0 x1 (getl_head (Bind b) n c4 (CHead x0 k0 x1) H6 u2) -H7 H8)))))) H5))))))))) (\lambda (f: F).(\lambda (n: nat).(\lambda (_: -((\forall (e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c3 (Flat -f) u1) (CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: -T).(getl n (CHead c4 (Flat f) u2) (CHead e2 k0 u4)))) (\lambda (e2: -C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 u3 -u4)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4: -(getl (S n) (CHead c3 (Flat f) u1) (CHead e1 k0 u0))).(let H5 \def (H1 (S n) -e1 u0 k0 (getl_gen_S (Flat f) c3 (CHead e1 k0 u0) u1 n H4)) in (ex3_2_ind C T -(\lambda (e2: C).(\lambda (u3: T).(getl (S n) c4 (CHead e2 k0 u3)))) (\lambda -(e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 -u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c4 -(Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 -e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (H6: (getl (S n) c4 (CHead x0 k0 x1))).(\lambda -(H7: (wcpr0 e1 x0)).(\lambda (H8: (pr0 u0 x1)).(ex3_2_intro C T (\lambda (e2: -C).(\lambda (u3: T).(getl (S n) (CHead c4 (Flat f) u2) (CHead e2 k0 u3)))) -(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda -(u3: T).(pr0 u0 u3))) x0 x1 (getl_head (Flat f) n c4 (CHead x0 k0 x1) H6 u2) -H7 H8)))))) H5))))))))) k) h)))))))))) c1 c2 H))). -(* COMMENTS -Initial nodes: 2103 -END *) - -theorem wcpr0_getl_back: - \forall (c1: C).(\forall (c2: C).((wcpr0 c2 c1) \to (\forall (h: -nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((getl h c1 (CHead e1 -k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(getl h c2 (CHead e2 -k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: -C).(\lambda (u2: T).(pr0 u2 u1))))))))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c2 c1)).(wcpr0_ind -(\lambda (c: C).(\lambda (c0: C).(\forall (h: nat).(\forall (e1: C).(\forall -(u1: T).(\forall (k: K).((getl h c0 (CHead e1 k u1)) \to (ex3_2 C T (\lambda -(e2: C).(\lambda (u2: T).(getl h c (CHead e2 k u2)))) (\lambda (e2: -C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 -u1))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda -(u1: T).(\lambda (k: K).(\lambda (H0: (getl h c (CHead e1 k -u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(getl h c (CHead e2 -k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: -C).(\lambda (u2: T).(pr0 u2 u1))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl -u1)))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wcpr0 c3 -c4)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1: -T).(\forall (k: K).((getl h c4 (CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2: -C).(\lambda (u2: T).(getl h c3 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda -(_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 -u1))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 -u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall -(e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c4 k u2) (CHead e1 -k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c3 k -u1) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) -(\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))) (\lambda (e1: -C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (getl O (CHead c4 k u2) -(CHead e1 k0 u0))).(K_ind (\lambda (k1: K).((clear (CHead c4 k1 u2) (CHead e1 -k0 u0)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 k1 -u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) -(\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))))) (\lambda (b: B).(\lambda -(H4: (clear (CHead c4 (Bind b) u2) (CHead e1 k0 u0))).(let H5 \def (f_equal C -C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow e1 | (CHead c _ _) \Rightarrow c])) (CHead e1 k0 u0) (CHead c4 -(Bind b) u2) (clear_gen_bind b c4 (CHead e1 k0 u0) u2 H4)) in ((let H6 \def -(f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with -[(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead e1 k0 u0) -(CHead c4 (Bind b) u2) (clear_gen_bind b c4 (CHead e1 k0 u0) u2 H4)) in ((let -H7 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead e1 k0 -u0) (CHead c4 (Bind b) u2) (clear_gen_bind b c4 (CHead e1 k0 u0) u2 H4)) in -(\lambda (H8: (eq K k0 (Bind b))).(\lambda (H9: (eq C e1 c4)).(eq_ind_r K -(Bind b) (\lambda (k1: K).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl -O (CHead c3 (Bind b) u1) (CHead e2 k1 u3)))) (\lambda (e2: C).(\lambda (_: -T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))))) (eq_ind_r -T u2 (\lambda (t: T).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O -(CHead c3 (Bind b) u1) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda -(_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 t))))) -(eq_ind_r C c4 (\lambda (c: C).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: -T).(getl O (CHead c3 (Bind b) u1) (CHead e2 (Bind b) u3)))) (\lambda (e2: -C).(\lambda (_: T).(wcpr0 e2 c))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 -u2))))) (ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 -(Bind b) u1) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda (_: -T).(wcpr0 e2 c4))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u2))) c3 u1 -(getl_refl b c3 u1) H0 H2) e1 H9) u0 H7) k0 H8)))) H6)) H5)))) (\lambda (f: -F).(\lambda (H4: (clear (CHead c4 (Flat f) u2) (CHead e1 k0 u0))).(let H5 -\def (H1 O e1 u0 k0 (getl_intro O c4 (CHead e1 k0 u0) c4 (drop_refl c4) -(clear_gen_flat f c4 (CHead e1 k0 u0) u2 H4))) in (ex3_2_ind C T (\lambda -(e2: C).(\lambda (u3: T).(getl O c3 (CHead e2 k0 u3)))) (\lambda (e2: -C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 -u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 (Flat f) -u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) -(\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))) (\lambda (x0: C).(\lambda -(x1: T).(\lambda (H6: (getl O c3 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 x0 -e1)).(\lambda (H8: (pr0 x1 u0)).(ex3_2_intro C T (\lambda (e2: C).(\lambda -(u3: T).(getl O (CHead c3 (Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2: -C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 -u0))) x0 x1 (getl_flat c3 (CHead x0 k0 x1) O H6 f u1) H7 H8)))))) H5)))) k -(getl_gen_O (CHead c4 k u2) (CHead e1 k0 u0) H3)))))) (K_ind (\lambda (k0: -K).(\forall (n: nat).(((\forall (e1: C).(\forall (u3: T).(\forall (k1: -K).((getl n (CHead c4 k0 u2) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: -C).(\lambda (u4: T).(getl n (CHead c3 k0 u1) (CHead e2 k1 u4)))) (\lambda -(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 -u4 u3))))))))) \to (\forall (e1: C).(\forall (u3: T).(\forall (k1: K).((getl -(S n) (CHead c4 k0 u2) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: -C).(\lambda (u4: T).(getl (S n) (CHead c3 k0 u1) (CHead e2 k1 u4)))) (\lambda -(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 -u4 u3))))))))))) (\lambda (b: B).(\lambda (n: nat).(\lambda (_: ((\forall -(e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c4 (Bind b) u2) -(CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n -(CHead c3 (Bind b) u1) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: -T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 u4 -u3)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4: -(getl (S n) (CHead c4 (Bind b) u2) (CHead e1 k0 u0))).(let H5 \def (H1 n e1 -u0 k0 (getl_gen_S (Bind b) c4 (CHead e1 k0 u0) u2 n H4)) in (ex3_2_ind C T -(\lambda (e2: C).(\lambda (u3: T).(getl n c3 (CHead e2 k0 u3)))) (\lambda -(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 -u3 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c3 -(Bind b) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 -e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (H6: (getl n c3 (CHead x0 k0 x1))).(\lambda (H7: -(wcpr0 x0 e1)).(\lambda (H8: (pr0 x1 u0)).(ex3_2_intro C T (\lambda (e2: -C).(\lambda (u3: T).(getl (S n) (CHead c3 (Bind b) u1) (CHead e2 k0 u3)))) -(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda -(u3: T).(pr0 u3 u0))) x0 x1 (getl_head (Bind b) n c3 (CHead x0 k0 x1) H6 u1) -H7 H8)))))) H5))))))))) (\lambda (f: F).(\lambda (n: nat).(\lambda (_: -((\forall (e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c4 (Flat -f) u2) (CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: -T).(getl n (CHead c3 (Flat f) u1) (CHead e2 k0 u4)))) (\lambda (e2: -C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 u4 -u3)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4: -(getl (S n) (CHead c4 (Flat f) u2) (CHead e1 k0 u0))).(let H5 \def (H1 (S n) -e1 u0 k0 (getl_gen_S (Flat f) c4 (CHead e1 k0 u0) u2 n H4)) in (ex3_2_ind C T -(\lambda (e2: C).(\lambda (u3: T).(getl (S n) c3 (CHead e2 k0 u3)))) (\lambda -(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 -u3 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c3 -(Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 -e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (H6: (getl (S n) c3 (CHead x0 k0 x1))).(\lambda -(H7: (wcpr0 x0 e1)).(\lambda (H8: (pr0 x1 u0)).(ex3_2_intro C T (\lambda (e2: -C).(\lambda (u3: T).(getl (S n) (CHead c3 (Flat f) u1) (CHead e2 k0 u3)))) -(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda -(u3: T).(pr0 u3 u0))) x0 x1 (getl_head (Flat f) n c3 (CHead x0 k0 x1) H6 u1) -H7 H8)))))) H5))))))))) k) h)))))))))) c2 c1 H))). -(* COMMENTS -Initial nodes: 2103 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/wf3/clear.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/wf3/clear.ma deleted file mode 100644 index b2ddd47af..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/wf3/clear.ma +++ /dev/null @@ -1,91 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/wf3/fwd.ma". - -theorem wf3_clear_conf: - \forall (c1: C).(\forall (c: C).((clear c1 c) \to (\forall (g: G).(\forall -(c2: C).((wf3 g c1 c2) \to (wf3 g c c2)))))) -\def - \lambda (c1: C).(\lambda (c: C).(\lambda (H: (clear c1 c)).(clear_ind -(\lambda (c0: C).(\lambda (c2: C).(\forall (g: G).(\forall (c3: C).((wf3 g c0 -c3) \to (wf3 g c2 c3)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (u: -T).(\lambda (g: G).(\lambda (c2: C).(\lambda (H0: (wf3 g (CHead e (Bind b) u) -c2)).H0)))))) (\lambda (e: C).(\lambda (c0: C).(\lambda (_: (clear e -c0)).(\lambda (H1: ((\forall (g: G).(\forall (c2: C).((wf3 g e c2) \to (wf3 g -c0 c2)))))).(\lambda (f: F).(\lambda (u: T).(\lambda (g: G).(\lambda (c2: -C).(\lambda (H2: (wf3 g (CHead e (Flat f) u) c2)).(let H_y \def -(wf3_gen_flat1 g e c2 u f H2) in (H1 g c2 H_y))))))))))) c1 c H))). -(* COMMENTS -Initial nodes: 145 -END *) - -theorem clear_wf3_trans: - \forall (c1: C).(\forall (d1: C).((clear c1 d1) \to (\forall (g: G).(\forall -(d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda -(c2: C).(clear c2 d2)))))))) -\def - \lambda (c1: C).(\lambda (d1: C).(\lambda (H: (clear c1 d1)).(clear_ind -(\lambda (c: C).(\lambda (c0: C).(\forall (g: G).(\forall (d2: C).((wf3 g c0 -d2) \to (ex2 C (\lambda (c2: C).(wf3 g c c2)) (\lambda (c2: C).(clear c2 -d2)))))))) (\lambda (b: B).(\lambda (e: C).(\lambda (u: T).(\lambda (g: -G).(\lambda (d2: C).(\lambda (H0: (wf3 g (CHead e (Bind b) u) d2)).(let H_x -\def (wf3_gen_bind1 g e d2 u b H0) in (let H1 \def H_x in (or_ind (ex3_2 C T -(\lambda (c2: C).(\lambda (_: T).(eq C d2 (CHead c2 (Bind b) u)))) (\lambda -(c2: C).(\lambda (_: T).(wf3 g e c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g -e u w)))) (ex3 C (\lambda (c2: C).(eq C d2 (CHead c2 (Bind Void) (TSort O)))) -(\lambda (c2: C).(wf3 g e c2)) (\lambda (_: C).(\forall (w: T).((ty3 g e u w) -\to False)))) (ex2 C (\lambda (c2: C).(wf3 g (CHead e (Bind b) u) c2)) -(\lambda (c2: C).(clear c2 d2))) (\lambda (H2: (ex3_2 C T (\lambda (c2: -C).(\lambda (_: T).(eq C d2 (CHead c2 (Bind b) u)))) (\lambda (c2: -C).(\lambda (_: T).(wf3 g e c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g e u -w))))).(ex3_2_ind C T (\lambda (c2: C).(\lambda (_: T).(eq C d2 (CHead c2 -(Bind b) u)))) (\lambda (c2: C).(\lambda (_: T).(wf3 g e c2))) (\lambda (_: -C).(\lambda (w: T).(ty3 g e u w))) (ex2 C (\lambda (c2: C).(wf3 g (CHead e -(Bind b) u) c2)) (\lambda (c2: C).(clear c2 d2))) (\lambda (x0: C).(\lambda -(x1: T).(\lambda (H3: (eq C d2 (CHead x0 (Bind b) u))).(\lambda (H4: (wf3 g e -x0)).(\lambda (H5: (ty3 g e u x1)).(eq_ind_r C (CHead x0 (Bind b) u) (\lambda -(c: C).(ex2 C (\lambda (c2: C).(wf3 g (CHead e (Bind b) u) c2)) (\lambda (c2: -C).(clear c2 c)))) (ex_intro2 C (\lambda (c2: C).(wf3 g (CHead e (Bind b) u) -c2)) (\lambda (c2: C).(clear c2 (CHead x0 (Bind b) u))) (CHead x0 (Bind b) u) -(wf3_bind g e x0 H4 u x1 H5 b) (clear_bind b x0 u)) d2 H3)))))) H2)) (\lambda -(H2: (ex3 C (\lambda (c2: C).(eq C d2 (CHead c2 (Bind Void) (TSort O)))) -(\lambda (c2: C).(wf3 g e c2)) (\lambda (_: C).(\forall (w: T).((ty3 g e u w) -\to False))))).(ex3_ind C (\lambda (c2: C).(eq C d2 (CHead c2 (Bind Void) -(TSort O)))) (\lambda (c2: C).(wf3 g e c2)) (\lambda (_: C).(\forall (w: -T).((ty3 g e u w) \to False))) (ex2 C (\lambda (c2: C).(wf3 g (CHead e (Bind -b) u) c2)) (\lambda (c2: C).(clear c2 d2))) (\lambda (x0: C).(\lambda (H3: -(eq C d2 (CHead x0 (Bind Void) (TSort O)))).(\lambda (H4: (wf3 g e -x0)).(\lambda (H5: ((\forall (w: T).((ty3 g e u w) \to False)))).(eq_ind_r C -(CHead x0 (Bind Void) (TSort O)) (\lambda (c: C).(ex2 C (\lambda (c2: C).(wf3 -g (CHead e (Bind b) u) c2)) (\lambda (c2: C).(clear c2 c)))) (ex_intro2 C -(\lambda (c2: C).(wf3 g (CHead e (Bind b) u) c2)) (\lambda (c2: C).(clear c2 -(CHead x0 (Bind Void) (TSort O)))) (CHead x0 (Bind Void) (TSort O)) (wf3_void -g e x0 H4 u H5 b) (clear_bind Void x0 (TSort O))) d2 H3))))) H2)) H1))))))))) -(\lambda (e: C).(\lambda (c: C).(\lambda (_: (clear e c)).(\lambda (H1: -((\forall (g: G).(\forall (d2: C).((wf3 g c d2) \to (ex2 C (\lambda (c2: -C).(wf3 g e c2)) (\lambda (c2: C).(clear c2 d2)))))))).(\lambda (f: -F).(\lambda (u: T).(\lambda (g: G).(\lambda (d2: C).(\lambda (H2: (wf3 g c -d2)).(let H_x \def (H1 g d2 H2) in (let H3 \def H_x in (ex2_ind C (\lambda -(c2: C).(wf3 g e c2)) (\lambda (c2: C).(clear c2 d2)) (ex2 C (\lambda (c2: -C).(wf3 g (CHead e (Flat f) u) c2)) (\lambda (c2: C).(clear c2 d2))) (\lambda -(x: C).(\lambda (H4: (wf3 g e x)).(\lambda (H5: (clear x d2)).(ex_intro2 C -(\lambda (c2: C).(wf3 g (CHead e (Flat f) u) c2)) (\lambda (c2: C).(clear c2 -d2)) x (wf3_flat g e x H4 u f) H5)))) H3)))))))))))) c1 d1 H))). -(* COMMENTS -Initial nodes: 1023 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/wf3/defs.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/wf3/defs.ma deleted file mode 100644 index a99661941..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/wf3/defs.ma +++ /dev/null @@ -1,29 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/ty3/defs.ma". - -inductive wf3 (g: G): C \to (C \to Prop) \def -| wf3_sort: \forall (m: nat).(wf3 g (CSort m) (CSort m)) -| wf3_bind: \forall (c1: C).(\forall (c2: C).((wf3 g c1 c2) \to (\forall (u: -T).(\forall (t: T).((ty3 g c1 u t) \to (\forall (b: B).(wf3 g (CHead c1 (Bind -b) u) (CHead c2 (Bind b) u)))))))) -| wf3_void: \forall (c1: C).(\forall (c2: C).((wf3 g c1 c2) \to (\forall (u: -T).(((\forall (t: T).((ty3 g c1 u t) \to False))) \to (\forall (b: B).(wf3 g -(CHead c1 (Bind b) u) (CHead c2 (Bind Void) (TSort O)))))))) -| wf3_flat: \forall (c1: C).(\forall (c2: C).((wf3 g c1 c2) \to (\forall (u: -T).(\forall (f: F).(wf3 g (CHead c1 (Flat f) u) c2))))). - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/wf3/fwd.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/wf3/fwd.ma deleted file mode 100644 index 71c903f06..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/wf3/fwd.ma +++ /dev/null @@ -1,311 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/wf3/defs.ma". - -theorem wf3_gen_sort1: - \forall (g: G).(\forall (x: C).(\forall (m: nat).((wf3 g (CSort m) x) \to -(eq C x (CSort m))))) -\def - \lambda (g: G).(\lambda (x: C).(\lambda (m: nat).(\lambda (H: (wf3 g (CSort -m) x)).(insert_eq C (CSort m) (\lambda (c: C).(wf3 g c x)) (\lambda (c: -C).(eq C x c)) (\lambda (y: C).(\lambda (H0: (wf3 g y x)).(wf3_ind g (\lambda -(c: C).(\lambda (c0: C).((eq C c (CSort m)) \to (eq C c0 c)))) (\lambda (m0: -nat).(\lambda (H1: (eq C (CSort m0) (CSort m))).(let H2 \def (f_equal C nat -(\lambda (e: C).(match e in C return (\lambda (_: C).nat) with [(CSort n) -\Rightarrow n | (CHead _ _ _) \Rightarrow m0])) (CSort m0) (CSort m) H1) in -(eq_ind_r nat m (\lambda (n: nat).(eq C (CSort n) (CSort n))) (refl_equal C -(CSort m)) m0 H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (wf3 g c1 -c2)).(\lambda (_: (((eq C c1 (CSort m)) \to (eq C c2 c1)))).(\lambda (u: -T).(\lambda (t: T).(\lambda (_: (ty3 g c1 u t)).(\lambda (b: B).(\lambda (H4: -(eq C (CHead c1 (Bind b) u) (CSort m))).(let H5 \def (eq_ind C (CHead c1 -(Bind b) u) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with -[(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort m) -H4) in (False_ind (eq C (CHead c2 (Bind b) u) (CHead c1 (Bind b) u)) -H5))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (wf3 g c1 -c2)).(\lambda (_: (((eq C c1 (CSort m)) \to (eq C c2 c1)))).(\lambda (u: -T).(\lambda (_: ((\forall (t: T).((ty3 g c1 u t) \to False)))).(\lambda (b: -B).(\lambda (H4: (eq C (CHead c1 (Bind b) u) (CSort m))).(let H5 \def (eq_ind -C (CHead c1 (Bind b) u) (\lambda (ee: C).(match ee in C return (\lambda (_: -C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow -True])) I (CSort m) H4) in (False_ind (eq C (CHead c2 (Bind Void) (TSort O)) -(CHead c1 (Bind b) u)) H5)))))))))) (\lambda (c1: C).(\lambda (c2: -C).(\lambda (_: (wf3 g c1 c2)).(\lambda (_: (((eq C c1 (CSort m)) \to (eq C -c2 c1)))).(\lambda (u: T).(\lambda (f: F).(\lambda (H3: (eq C (CHead c1 (Flat -f) u) (CSort m))).(let H4 \def (eq_ind C (CHead c1 (Flat f) u) (\lambda (ee: -C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow -False | (CHead _ _ _) \Rightarrow True])) I (CSort m) H3) in (False_ind (eq C -c2 (CHead c1 (Flat f) u)) H4))))))))) y x H0))) H)))). -(* COMMENTS -Initial nodes: 523 -END *) - -theorem wf3_gen_bind1: - \forall (g: G).(\forall (c1: C).(\forall (x: C).(\forall (v: T).(\forall (b: -B).((wf3 g (CHead c1 (Bind b) v) x) \to (or (ex3_2 C T (\lambda (c2: -C).(\lambda (_: T).(eq C x (CHead c2 (Bind b) v)))) (\lambda (c2: C).(\lambda -(_: T).(wf3 g c1 c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 -C (\lambda (c2: C).(eq C x (CHead c2 (Bind Void) (TSort O)))) (\lambda (c2: -C).(wf3 g c1 c2)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to -False)))))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (x: C).(\lambda (v: T).(\lambda (b: -B).(\lambda (H: (wf3 g (CHead c1 (Bind b) v) x)).(insert_eq C (CHead c1 (Bind -b) v) (\lambda (c: C).(wf3 g c x)) (\lambda (_: C).(or (ex3_2 C T (\lambda -(c2: C).(\lambda (_: T).(eq C x (CHead c2 (Bind b) v)))) (\lambda (c2: -C).(\lambda (_: T).(wf3 g c1 c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 -v w)))) (ex3 C (\lambda (c2: C).(eq C x (CHead c2 (Bind Void) (TSort O)))) -(\lambda (c2: C).(wf3 g c1 c2)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v -w) \to False)))))) (\lambda (y: C).(\lambda (H0: (wf3 g y x)).(wf3_ind g -(\lambda (c: C).(\lambda (c0: C).((eq C c (CHead c1 (Bind b) v)) \to (or -(ex3_2 C T (\lambda (c2: C).(\lambda (_: T).(eq C c0 (CHead c2 (Bind b) v)))) -(\lambda (c2: C).(\lambda (_: T).(wf3 g c1 c2))) (\lambda (_: C).(\lambda (w: -T).(ty3 g c1 v w)))) (ex3 C (\lambda (c2: C).(eq C c0 (CHead c2 (Bind Void) -(TSort O)))) (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (_: C).(\forall (w: -T).((ty3 g c1 v w) \to False)))))))) (\lambda (m: nat).(\lambda (H1: (eq C -(CSort m) (CHead c1 (Bind b) v))).(let H2 \def (eq_ind C (CSort m) (\lambda -(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead c1 (Bind b) v) -H1) in (False_ind (or (ex3_2 C T (\lambda (c2: C).(\lambda (_: T).(eq C -(CSort m) (CHead c2 (Bind b) v)))) (\lambda (c2: C).(\lambda (_: T).(wf3 g c1 -c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c2: -C).(eq C (CSort m) (CHead c2 (Bind Void) (TSort O)))) (\lambda (c2: C).(wf3 g -c1 c2)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to False))))) H2)))) -(\lambda (c0: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c0 c2)).(\lambda (H2: -(((eq C c0 (CHead c1 (Bind b) v)) \to (or (ex3_2 C T (\lambda (c3: -C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda (c3: -C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 -v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort O)))) -(\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v -w) \to False)))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H3: (ty3 g c0 -u t)).(\lambda (b0: B).(\lambda (H4: (eq C (CHead c0 (Bind b0) u) (CHead c1 -(Bind b) v))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return -(\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow -c])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) H4) in ((let H6 \def -(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with -[(CSort _) \Rightarrow b0 | (CHead _ k _) \Rightarrow (match k in K return -(\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow -b0])])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) H4) in ((let H7 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead c0 (Bind -b0) u) (CHead c1 (Bind b) v) H4) in (\lambda (H8: (eq B b0 b)).(\lambda (H9: -(eq C c0 c1)).(eq_ind_r B b (\lambda (b1: B).(or (ex3_2 C T (\lambda (c3: -C).(\lambda (_: T).(eq C (CHead c2 (Bind b1) u) (CHead c3 (Bind b) v)))) -(\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: -T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C (CHead c2 (Bind b1) u) -(CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda -(_: C).(\forall (w: T).((ty3 g c1 v w) \to False)))))) (let H10 \def (eq_ind -T u (\lambda (t0: T).(ty3 g c0 t0 t)) H3 v H7) in (eq_ind_r T v (\lambda (t0: -T).(or (ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C (CHead c2 (Bind b) -t0) (CHead c3 (Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) -(\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq -C (CHead c2 (Bind b) t0) (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: -C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to -False)))))) (let H11 \def (eq_ind C c0 (\lambda (c: C).(ty3 g c v t)) H10 c1 -H9) in (let H12 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 (Bind b) -v)) \to (or (ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 -(Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: -C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead -c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: -C).(\forall (w: T).((ty3 g c1 v w) \to False))))))) H2 c1 H9) in (let H13 -\def (eq_ind C c0 (\lambda (c: C).(wf3 g c c2)) H1 c1 H9) in (or_introl -(ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C (CHead c2 (Bind b) v) -(CHead c3 (Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) -(\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq -C (CHead c2 (Bind b) v) (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: -C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to -False)))) (ex3_2_intro C T (\lambda (c3: C).(\lambda (_: T).(eq C (CHead c2 -(Bind b) v) (CHead c3 (Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g -c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w))) c2 t (refl_equal C -(CHead c2 (Bind b) v)) H13 H11))))) u H7)) b0 H8)))) H6)) H5))))))))))) -(\lambda (c0: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c0 c2)).(\lambda (H2: -(((eq C c0 (CHead c1 (Bind b) v)) \to (or (ex3_2 C T (\lambda (c3: -C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda (c3: -C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 -v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort O)))) -(\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v -w) \to False)))))))).(\lambda (u: T).(\lambda (H3: ((\forall (t: T).((ty3 g -c0 u t) \to False)))).(\lambda (b0: B).(\lambda (H4: (eq C (CHead c0 (Bind -b0) u) (CHead c1 (Bind b) v))).(let H5 \def (f_equal C C (\lambda (e: -C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | -(CHead c _ _) \Rightarrow c])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) -H4) in ((let H6 \def (f_equal C B (\lambda (e: C).(match e in C return -(\lambda (_: C).B) with [(CSort _) \Rightarrow b0 | (CHead _ k _) \Rightarrow -(match k in K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | -(Flat _) \Rightarrow b0])])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) H4) -in ((let H7 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda -(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) -(CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) H4) in (\lambda (_: (eq B b0 -b)).(\lambda (H9: (eq C c0 c1)).(let H10 \def (eq_ind T u (\lambda (t: -T).(\forall (t0: T).((ty3 g c0 t t0) \to False))) H3 v H7) in (let H11 \def -(eq_ind C c0 (\lambda (c: C).(\forall (t: T).((ty3 g c v t) \to False))) H10 -c1 H9) in (let H12 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 (Bind -b) v)) \to (or (ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 -(Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: -C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead -c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: -C).(\forall (w: T).((ty3 g c1 v w) \to False))))))) H2 c1 H9) in (let H13 -\def (eq_ind C c0 (\lambda (c: C).(wf3 g c c2)) H1 c1 H9) in (or_intror -(ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C (CHead c2 (Bind Void) -(TSort O)) (CHead c3 (Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g -c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda -(c3: C).(eq C (CHead c2 (Bind Void) (TSort O)) (CHead c3 (Bind Void) (TSort -O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g -c1 v w) \to False)))) (ex3_intro C (\lambda (c3: C).(eq C (CHead c2 (Bind -Void) (TSort O)) (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g -c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to False))) c2 -(refl_equal C (CHead c2 (Bind Void) (TSort O))) H13 H11))))))))) H6)) -H5)))))))))) (\lambda (c0: C).(\lambda (c2: C).(\lambda (_: (wf3 g c0 -c2)).(\lambda (_: (((eq C c0 (CHead c1 (Bind b) v)) \to (or (ex3_2 C T -(\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda -(c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 -g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort -O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g -c1 v w) \to False)))))))).(\lambda (u: T).(\lambda (f: F).(\lambda (H3: (eq C -(CHead c0 (Flat f) u) (CHead c1 (Bind b) v))).(let H4 \def (eq_ind C (CHead -c0 (Flat f) u) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) -with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K -return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) -\Rightarrow True])])) I (CHead c1 (Bind b) v) H3) in (False_ind (or (ex3_2 C -T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda -(c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 -g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort -O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g -c1 v w) \to False))))) H4))))))))) y x H0))) H)))))). -(* COMMENTS -Initial nodes: 2507 -END *) - -theorem wf3_gen_flat1: - \forall (g: G).(\forall (c1: C).(\forall (x: C).(\forall (v: T).(\forall (f: -F).((wf3 g (CHead c1 (Flat f) v) x) \to (wf3 g c1 x)))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (x: C).(\lambda (v: T).(\lambda (f: -F).(\lambda (H: (wf3 g (CHead c1 (Flat f) v) x)).(insert_eq C (CHead c1 (Flat -f) v) (\lambda (c: C).(wf3 g c x)) (\lambda (_: C).(wf3 g c1 x)) (\lambda (y: -C).(\lambda (H0: (wf3 g y x)).(wf3_ind g (\lambda (c: C).(\lambda (c0: -C).((eq C c (CHead c1 (Flat f) v)) \to (wf3 g c1 c0)))) (\lambda (m: -nat).(\lambda (H1: (eq C (CSort m) (CHead c1 (Flat f) v))).(let H2 \def -(eq_ind C (CSort m) (\lambda (ee: C).(match ee in C return (\lambda (_: -C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow -False])) I (CHead c1 (Flat f) v) H1) in (False_ind (wf3 g c1 (CSort m)) -H2)))) (\lambda (c0: C).(\lambda (c2: C).(\lambda (_: (wf3 g c0 c2)).(\lambda -(_: (((eq C c0 (CHead c1 (Flat f) v)) \to (wf3 g c1 c2)))).(\lambda (u: -T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (b: B).(\lambda (H4: -(eq C (CHead c0 (Bind b) u) (CHead c1 (Flat f) v))).(let H5 \def (eq_ind C -(CHead c0 (Bind b) u) (\lambda (ee: C).(match ee in C return (\lambda (_: -C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match -k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat -_) \Rightarrow False])])) I (CHead c1 (Flat f) v) H4) in (False_ind (wf3 g c1 -(CHead c2 (Bind b) u)) H5))))))))))) (\lambda (c0: C).(\lambda (c2: -C).(\lambda (_: (wf3 g c0 c2)).(\lambda (_: (((eq C c0 (CHead c1 (Flat f) v)) -\to (wf3 g c1 c2)))).(\lambda (u: T).(\lambda (_: ((\forall (t: T).((ty3 g c0 -u t) \to False)))).(\lambda (b: B).(\lambda (H4: (eq C (CHead c0 (Bind b) u) -(CHead c1 (Flat f) v))).(let H5 \def (eq_ind C (CHead c0 (Bind b) u) (\lambda -(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow -False])])) I (CHead c1 (Flat f) v) H4) in (False_ind (wf3 g c1 (CHead c2 -(Bind Void) (TSort O))) H5)))))))))) (\lambda (c0: C).(\lambda (c2: -C).(\lambda (H1: (wf3 g c0 c2)).(\lambda (H2: (((eq C c0 (CHead c1 (Flat f) -v)) \to (wf3 g c1 c2)))).(\lambda (u: T).(\lambda (f0: F).(\lambda (H3: (eq C -(CHead c0 (Flat f0) u) (CHead c1 (Flat f) v))).(let H4 \def (f_equal C C -(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 (Flat f0) u) (CHead -c1 (Flat f) v) H3) in ((let H5 \def (f_equal C F (\lambda (e: C).(match e in -C return (\lambda (_: C).F) with [(CSort _) \Rightarrow f0 | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).F) with [(Bind _) -\Rightarrow f0 | (Flat f1) \Rightarrow f1])])) (CHead c0 (Flat f0) u) (CHead -c1 (Flat f) v) H3) in ((let H6 \def (f_equal C T (\lambda (e: C).(match e in -C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) -\Rightarrow t])) (CHead c0 (Flat f0) u) (CHead c1 (Flat f) v) H3) in (\lambda -(_: (eq F f0 f)).(\lambda (H8: (eq C c0 c1)).(let H9 \def (eq_ind C c0 -(\lambda (c: C).((eq C c (CHead c1 (Flat f) v)) \to (wf3 g c1 c2))) H2 c1 H8) -in (let H10 \def (eq_ind C c0 (\lambda (c: C).(wf3 g c c2)) H1 c1 H8) in -H10))))) H5)) H4))))))))) y x H0))) H)))))). -(* COMMENTS -Initial nodes: 737 -END *) - -theorem wf3_gen_head2: - \forall (g: G).(\forall (x: C).(\forall (c: C).(\forall (v: T).(\forall (k: -K).((wf3 g x (CHead c k v)) \to (ex B (\lambda (b: B).(eq K k (Bind b))))))))) -\def - \lambda (g: G).(\lambda (x: C).(\lambda (c: C).(\lambda (v: T).(\lambda (k: -K).(\lambda (H: (wf3 g x (CHead c k v))).(insert_eq C (CHead c k v) (\lambda -(c0: C).(wf3 g x c0)) (\lambda (_: C).(ex B (\lambda (b: B).(eq K k (Bind -b))))) (\lambda (y: C).(\lambda (H0: (wf3 g x y)).(wf3_ind g (\lambda (_: -C).(\lambda (c1: C).((eq C c1 (CHead c k v)) \to (ex B (\lambda (b: B).(eq K -k (Bind b))))))) (\lambda (m: nat).(\lambda (H1: (eq C (CSort m) (CHead c k -v))).(let H2 \def (eq_ind C (CSort m) (\lambda (ee: C).(match ee in C return -(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) -\Rightarrow False])) I (CHead c k v) H1) in (False_ind (ex B (\lambda (b: -B).(eq K k (Bind b)))) H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: -(wf3 g c1 c2)).(\lambda (H2: (((eq C c2 (CHead c k v)) \to (ex B (\lambda (b: -B).(eq K k (Bind b))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H3: (ty3 -g c1 u t)).(\lambda (b: B).(\lambda (H4: (eq C (CHead c2 (Bind b) u) (CHead c -k v))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return -(\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ _) -\Rightarrow c0])) (CHead c2 (Bind b) u) (CHead c k v) H4) in ((let H6 \def -(f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with -[(CSort _) \Rightarrow (Bind b) | (CHead _ k0 _) \Rightarrow k0])) (CHead c2 -(Bind b) u) (CHead c k v) H4) in ((let H7 \def (f_equal C T (\lambda (e: -C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | -(CHead _ _ t0) \Rightarrow t0])) (CHead c2 (Bind b) u) (CHead c k v) H4) in -(\lambda (H8: (eq K (Bind b) k)).(\lambda (H9: (eq C c2 c)).(let H10 \def -(eq_ind T u (\lambda (t0: T).(ty3 g c1 t0 t)) H3 v H7) in (let H11 \def -(eq_ind C c2 (\lambda (c0: C).((eq C c0 (CHead c k v)) \to (ex B (\lambda -(b0: B).(eq K k (Bind b0)))))) H2 c H9) in (let H12 \def (eq_ind C c2 -(\lambda (c0: C).(wf3 g c1 c0)) H1 c H9) in (let H13 \def (eq_ind_r K k -(\lambda (k0: K).((eq C c (CHead c k0 v)) \to (ex B (\lambda (b0: B).(eq K k0 -(Bind b0)))))) H11 (Bind b) H8) in (eq_ind K (Bind b) (\lambda (k0: K).(ex B -(\lambda (b0: B).(eq K k0 (Bind b0))))) (ex_intro B (\lambda (b0: B).(eq K -(Bind b) (Bind b0))) b (refl_equal K (Bind b))) k H8)))))))) H6)) -H5))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c1 -c2)).(\lambda (H2: (((eq C c2 (CHead c k v)) \to (ex B (\lambda (b: B).(eq K -k (Bind b))))))).(\lambda (u: T).(\lambda (_: ((\forall (t: T).((ty3 g c1 u -t) \to False)))).(\lambda (_: B).(\lambda (H4: (eq C (CHead c2 (Bind Void) -(TSort O)) (CHead c k v))).(let H5 \def (f_equal C C (\lambda (e: C).(match e -in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ -_) \Rightarrow c0])) (CHead c2 (Bind Void) (TSort O)) (CHead c k v) H4) in -((let H6 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: -C).K) with [(CSort _) \Rightarrow (Bind Void) | (CHead _ k0 _) \Rightarrow -k0])) (CHead c2 (Bind Void) (TSort O)) (CHead c k v) H4) in ((let H7 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow (TSort O) | (CHead _ _ t) \Rightarrow t])) (CHead c2 -(Bind Void) (TSort O)) (CHead c k v) H4) in (\lambda (H8: (eq K (Bind Void) -k)).(\lambda (H9: (eq C c2 c)).(let H10 \def (eq_ind C c2 (\lambda (c0: -C).((eq C c0 (CHead c k v)) \to (ex B (\lambda (b0: B).(eq K k (Bind b0)))))) -H2 c H9) in (let H11 \def (eq_ind C c2 (\lambda (c0: C).(wf3 g c1 c0)) H1 c -H9) in (let H12 \def (eq_ind_r K k (\lambda (k0: K).((eq C c (CHead c k0 v)) -\to (ex B (\lambda (b0: B).(eq K k0 (Bind b0)))))) H10 (Bind Void) H8) in -(eq_ind K (Bind Void) (\lambda (k0: K).(ex B (\lambda (b0: B).(eq K k0 (Bind -b0))))) (let H13 \def (eq_ind_r T v (\lambda (t: T).((eq C c (CHead c (Bind -Void) t)) \to (ex B (\lambda (b0: B).(eq K (Bind Void) (Bind b0)))))) H12 -(TSort O) H7) in (ex_intro B (\lambda (b0: B).(eq K (Bind Void) (Bind b0))) -Void (refl_equal K (Bind Void)))) k H8))))))) H6)) H5)))))))))) (\lambda (c1: -C).(\lambda (c2: C).(\lambda (H1: (wf3 g c1 c2)).(\lambda (H2: (((eq C c2 -(CHead c k v)) \to (ex B (\lambda (b: B).(eq K k (Bind b))))))).(\lambda (_: -T).(\lambda (_: F).(\lambda (H3: (eq C c2 (CHead c k v))).(let H4 \def -(f_equal C C (\lambda (e: C).e) c2 (CHead c k v) H3) in (let H5 \def (eq_ind -C c2 (\lambda (c0: C).((eq C c0 (CHead c k v)) \to (ex B (\lambda (b: B).(eq -K k (Bind b)))))) H2 (CHead c k v) H4) in (let H6 \def (eq_ind C c2 (\lambda -(c0: C).(wf3 g c1 c0)) H1 (CHead c k v) H4) in (H5 (refl_equal C (CHead c k -v))))))))))))) x y H0))) H)))))). -(* COMMENTS -Initial nodes: 1225 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/wf3/getl.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/wf3/getl.ma deleted file mode 100644 index 8a8f13780..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/wf3/getl.ma +++ /dev/null @@ -1,205 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/wf3/clear.ma". - -include "Basic-1/ty3/dec.ma". - -theorem wf3_getl_conf: - \forall (b: B).(\forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall -(v: T).((getl i c1 (CHead d1 (Bind b) v)) \to (\forall (g: G).(\forall (c2: -C).((wf3 g c1 c2) \to (\forall (w: T).((ty3 g d1 v w) \to (ex2 C (\lambda -(d2: C).(getl i c2 (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 -d2))))))))))))) -\def - \lambda (b: B).(\lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: -C).(\forall (d1: C).(\forall (v: T).((getl n c1 (CHead d1 (Bind b) v)) \to -(\forall (g: G).(\forall (c2: C).((wf3 g c1 c2) \to (\forall (w: T).((ty3 g -d1 v w) \to (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind b) v))) -(\lambda (d2: C).(wf3 g d1 d2))))))))))))) (\lambda (c1: C).(\lambda (d1: -C).(\lambda (v: T).(\lambda (H: (getl O c1 (CHead d1 (Bind b) v))).(\lambda -(g: G).(\lambda (c2: C).(\lambda (H0: (wf3 g c1 c2)).(\lambda (w: T).(\lambda -(H1: (ty3 g d1 v w)).(let H_y \def (wf3_clear_conf c1 (CHead d1 (Bind b) v) -(getl_gen_O c1 (CHead d1 (Bind b) v) H) g c2 H0) in (let H_x \def -(wf3_gen_bind1 g d1 c2 v b H_y) in (let H2 \def H_x in (or_ind (ex3_2 C T -(\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda -(c3: C).(\lambda (_: T).(wf3 g d1 c3))) (\lambda (_: C).(\lambda (w0: T).(ty3 -g d1 v w0)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort -O)))) (\lambda (c3: C).(wf3 g d1 c3)) (\lambda (_: C).(\forall (w0: T).((ty3 -g d1 v w0) \to False)))) (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind -b) v))) (\lambda (d2: C).(wf3 g d1 d2))) (\lambda (H3: (ex3_2 C T (\lambda -(c3: C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda (c3: -C).(\lambda (_: T).(wf3 g d1 c3))) (\lambda (_: C).(\lambda (w0: T).(ty3 g d1 -v w0))))).(ex3_2_ind C T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 -(Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g d1 c3))) (\lambda (_: -C).(\lambda (w0: T).(ty3 g d1 v w0))) (ex2 C (\lambda (d2: C).(getl O c2 -(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (H4: (eq C c2 (CHead x0 (Bind b) v))).(\lambda -(H5: (wf3 g d1 x0)).(\lambda (_: (ty3 g d1 v x1)).(eq_ind_r C (CHead x0 (Bind -b) v) (\lambda (c: C).(ex2 C (\lambda (d2: C).(getl O c (CHead d2 (Bind b) -v))) (\lambda (d2: C).(wf3 g d1 d2)))) (ex_intro2 C (\lambda (d2: C).(getl O -(CHead x0 (Bind b) v) (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)) -x0 (getl_refl b x0 v) H5) c2 H4)))))) H3)) (\lambda (H3: (ex3 C (\lambda (c3: -C).(eq C c2 (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g d1 -c3)) (\lambda (_: C).(\forall (w0: T).((ty3 g d1 v w0) \to -False))))).(ex3_ind C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort -O)))) (\lambda (c3: C).(wf3 g d1 c3)) (\lambda (_: C).(\forall (w0: T).((ty3 -g d1 v w0) \to False))) (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind b) -v))) (\lambda (d2: C).(wf3 g d1 d2))) (\lambda (x0: C).(\lambda (H4: (eq C c2 -(CHead x0 (Bind Void) (TSort O)))).(\lambda (_: (wf3 g d1 x0)).(\lambda (H6: -((\forall (w0: T).((ty3 g d1 v w0) \to False)))).(eq_ind_r C (CHead x0 (Bind -Void) (TSort O)) (\lambda (c: C).(ex2 C (\lambda (d2: C).(getl O c (CHead d2 -(Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)))) (let H_x0 \def (H6 w H1) in -(let H7 \def H_x0 in (False_ind (ex2 C (\lambda (d2: C).(getl O (CHead x0 -(Bind Void) (TSort O)) (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 -d2))) H7))) c2 H4))))) H3)) H2))))))))))))) (\lambda (n: nat).(\lambda (H: -((\forall (c1: C).(\forall (d1: C).(\forall (v: T).((getl n c1 (CHead d1 -(Bind b) v)) \to (\forall (g: G).(\forall (c2: C).((wf3 g c1 c2) \to (\forall -(w: T).((ty3 g d1 v w) \to (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind -b) v))) (\lambda (d2: C).(wf3 g d1 d2)))))))))))))).(\lambda (c1: C).(C_ind -(\lambda (c: C).(\forall (d1: C).(\forall (v: T).((getl (S n) c (CHead d1 -(Bind b) v)) \to (\forall (g: G).(\forall (c2: C).((wf3 g c c2) \to (\forall -(w: T).((ty3 g d1 v w) \to (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 -(Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)))))))))))) (\lambda (n0: -nat).(\lambda (d1: C).(\lambda (v: T).(\lambda (H0: (getl (S n) (CSort n0) -(CHead d1 (Bind b) v))).(\lambda (g: G).(\lambda (c2: C).(\lambda (_: (wf3 g -(CSort n0) c2)).(\lambda (w: T).(\lambda (_: (ty3 g d1 v w)).(getl_gen_sort -n0 (S n) (CHead d1 (Bind b) v) H0 (ex2 C (\lambda (d2: C).(getl (S n) c2 -(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2))))))))))))) (\lambda -(c: C).(\lambda (H0: ((\forall (d1: C).(\forall (v: T).((getl (S n) c (CHead -d1 (Bind b) v)) \to (\forall (g: G).(\forall (c2: C).((wf3 g c c2) \to -(\forall (w: T).((ty3 g d1 v w) \to (ex2 C (\lambda (d2: C).(getl (S n) c2 -(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2))))))))))))).(\lambda -(k: K).(\lambda (t: T).(\lambda (d1: C).(\lambda (v: T).(\lambda (H1: (getl -(S n) (CHead c k t) (CHead d1 (Bind b) v))).(\lambda (g: G).(\lambda (c2: -C).(\lambda (H2: (wf3 g (CHead c k t) c2)).(\lambda (w: T).(\lambda (H3: (ty3 -g d1 v w)).(K_ind (\lambda (k0: K).((wf3 g (CHead c k0 t) c2) \to ((getl (r -k0 n) c (CHead d1 (Bind b) v)) \to (ex2 C (\lambda (d2: C).(getl (S n) c2 -(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)))))) (\lambda (b0: -B).(\lambda (H4: (wf3 g (CHead c (Bind b0) t) c2)).(\lambda (H5: (getl (r -(Bind b0) n) c (CHead d1 (Bind b) v))).(let H_x \def (wf3_gen_bind1 g c c2 t -b0 H4) in (let H6 \def H_x in (or_ind (ex3_2 C T (\lambda (c3: C).(\lambda -(_: T).(eq C c2 (CHead c3 (Bind b0) t)))) (\lambda (c3: C).(\lambda (_: -T).(wf3 g c c3))) (\lambda (_: C).(\lambda (w0: T).(ty3 g c t w0)))) (ex3 C -(\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: -C).(wf3 g c c3)) (\lambda (_: C).(\forall (w0: T).((ty3 g c t w0) \to -False)))) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind b) v))) -(\lambda (d2: C).(wf3 g d1 d2))) (\lambda (H7: (ex3_2 C T (\lambda (c3: -C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b0) t)))) (\lambda (c3: -C).(\lambda (_: T).(wf3 g c c3))) (\lambda (_: C).(\lambda (w0: T).(ty3 g c t -w0))))).(ex3_2_ind C T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 -(Bind b0) t)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c c3))) (\lambda (_: -C).(\lambda (w0: T).(ty3 g c t w0))) (ex2 C (\lambda (d2: C).(getl (S n) c2 -(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (H8: (eq C c2 (CHead x0 (Bind b0) t))).(\lambda -(H9: (wf3 g c x0)).(\lambda (_: (ty3 g c t x1)).(eq_ind_r C (CHead x0 (Bind -b0) t) (\lambda (c0: C).(ex2 C (\lambda (d2: C).(getl (S n) c0 (CHead d2 -(Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)))) (let H_x0 \def (H c d1 v H5 g -x0 H9 w H3) in (let H11 \def H_x0 in (ex2_ind C (\lambda (d2: C).(getl n x0 -(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)) (ex2 C (\lambda (d2: -C).(getl (S n) (CHead x0 (Bind b0) t) (CHead d2 (Bind b) v))) (\lambda (d2: -C).(wf3 g d1 d2))) (\lambda (x: C).(\lambda (H12: (getl n x0 (CHead x (Bind -b) v))).(\lambda (H13: (wf3 g d1 x)).(ex_intro2 C (\lambda (d2: C).(getl (S -n) (CHead x0 (Bind b0) t) (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 -d2)) x (getl_head (Bind b0) n x0 (CHead x (Bind b) v) H12 t) H13)))) H11))) -c2 H8)))))) H7)) (\lambda (H7: (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 -(Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c c3)) (\lambda (_: -C).(\forall (w0: T).((ty3 g c t w0) \to False))))).(ex3_ind C (\lambda (c3: -C).(eq C c2 (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c c3)) -(\lambda (_: C).(\forall (w0: T).((ty3 g c t w0) \to False))) (ex2 C (\lambda -(d2: C).(getl (S n) c2 (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 -d2))) (\lambda (x0: C).(\lambda (H8: (eq C c2 (CHead x0 (Bind Void) (TSort -O)))).(\lambda (H9: (wf3 g c x0)).(\lambda (_: ((\forall (w0: T).((ty3 g c t -w0) \to False)))).(eq_ind_r C (CHead x0 (Bind Void) (TSort O)) (\lambda (c0: -C).(ex2 C (\lambda (d2: C).(getl (S n) c0 (CHead d2 (Bind b) v))) (\lambda -(d2: C).(wf3 g d1 d2)))) (let H_x0 \def (H c d1 v H5 g x0 H9 w H3) in (let -H11 \def H_x0 in (ex2_ind C (\lambda (d2: C).(getl n x0 (CHead d2 (Bind b) -v))) (\lambda (d2: C).(wf3 g d1 d2)) (ex2 C (\lambda (d2: C).(getl (S n) -(CHead x0 (Bind Void) (TSort O)) (CHead d2 (Bind b) v))) (\lambda (d2: -C).(wf3 g d1 d2))) (\lambda (x: C).(\lambda (H12: (getl n x0 (CHead x (Bind -b) v))).(\lambda (H13: (wf3 g d1 x)).(ex_intro2 C (\lambda (d2: C).(getl (S -n) (CHead x0 (Bind Void) (TSort O)) (CHead d2 (Bind b) v))) (\lambda (d2: -C).(wf3 g d1 d2)) x (getl_head (Bind Void) n x0 (CHead x (Bind b) v) H12 -(TSort O)) H13)))) H11))) c2 H8))))) H7)) H6)))))) (\lambda (f: F).(\lambda -(H4: (wf3 g (CHead c (Flat f) t) c2)).(\lambda (H5: (getl (r (Flat f) n) c -(CHead d1 (Bind b) v))).(let H_y \def (wf3_gen_flat1 g c c2 t f H4) in (H0 d1 -v H5 g c2 H_y w H3))))) k H2 (getl_gen_S k c (CHead d1 (Bind b) v) t n -H1)))))))))))))) c1)))) i)). -(* COMMENTS -Initial nodes: 2531 -END *) - -theorem getl_wf3_trans: - \forall (i: nat).(\forall (c1: C).(\forall (d1: C).((getl i c1 d1) \to -(\forall (g: G).(\forall (d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda (c2: -C).(wf3 g c1 c2)) (\lambda (c2: C).(getl i c2 d2))))))))) -\def - \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (d1: -C).((getl n c1 d1) \to (\forall (g: G).(\forall (d2: C).((wf3 g d1 d2) \to -(ex2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(getl n c2 -d2)))))))))) (\lambda (c1: C).(\lambda (d1: C).(\lambda (H: (getl O c1 -d1)).(\lambda (g: G).(\lambda (d2: C).(\lambda (H0: (wf3 g d1 d2)).(let H_x -\def (clear_wf3_trans c1 d1 (getl_gen_O c1 d1 H) g d2 H0) in (let H1 \def H_x -in (ex2_ind C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(clear c2 d2)) -(ex2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(getl O c2 d2))) -(\lambda (x: C).(\lambda (H2: (wf3 g c1 x)).(\lambda (H3: (clear x -d2)).(ex_intro2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(getl O c2 -d2)) x H2 (getl_intro O x d2 x (drop_refl x) H3))))) H1))))))))) (\lambda (n: -nat).(\lambda (H: ((\forall (c1: C).(\forall (d1: C).((getl n c1 d1) \to -(\forall (g: G).(\forall (d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda (c2: -C).(wf3 g c1 c2)) (\lambda (c2: C).(getl n c2 d2))))))))))).(\lambda (c1: -C).(C_ind (\lambda (c: C).(\forall (d1: C).((getl (S n) c d1) \to (\forall -(g: G).(\forall (d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda (c2: C).(wf3 g c -c2)) (\lambda (c2: C).(getl (S n) c2 d2))))))))) (\lambda (n0: nat).(\lambda -(d1: C).(\lambda (H0: (getl (S n) (CSort n0) d1)).(\lambda (g: G).(\lambda -(d2: C).(\lambda (_: (wf3 g d1 d2)).(getl_gen_sort n0 (S n) d1 H0 (ex2 C -(\lambda (c2: C).(wf3 g (CSort n0) c2)) (\lambda (c2: C).(getl (S n) c2 -d2)))))))))) (\lambda (c: C).(\lambda (H0: ((\forall (d1: C).((getl (S n) c -d1) \to (\forall (g: G).(\forall (d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda -(c2: C).(wf3 g c c2)) (\lambda (c2: C).(getl (S n) c2 d2)))))))))).(\lambda -(k: K).(\lambda (t: T).(\lambda (d1: C).(\lambda (H1: (getl (S n) (CHead c k -t) d1)).(\lambda (g: G).(\lambda (d2: C).(\lambda (H2: (wf3 g d1 d2)).(K_ind -(\lambda (k0: K).((getl (r k0 n) c d1) \to (ex2 C (\lambda (c2: C).(wf3 g -(CHead c k0 t) c2)) (\lambda (c2: C).(getl (S n) c2 d2))))) (\lambda (b: -B).(\lambda (H3: (getl (r (Bind b) n) c d1)).(let H_x \def (H c d1 H3 g d2 -H2) in (let H4 \def H_x in (ex2_ind C (\lambda (c2: C).(wf3 g c c2)) (\lambda -(c2: C).(getl n c2 d2)) (ex2 C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) -c2)) (\lambda (c2: C).(getl (S n) c2 d2))) (\lambda (x: C).(\lambda (H5: (wf3 -g c x)).(\lambda (H6: (getl n x d2)).(let H_x0 \def (ty3_inference g c t) in -(let H7 \def H_x0 in (or_ind (ex T (\lambda (t2: T).(ty3 g c t t2))) (\forall -(t2: T).((ty3 g c t t2) \to False)) (ex2 C (\lambda (c2: C).(wf3 g (CHead c -(Bind b) t) c2)) (\lambda (c2: C).(getl (S n) c2 d2))) (\lambda (H8: (ex T -(\lambda (t2: T).(ty3 g c t t2)))).(ex_ind T (\lambda (t2: T).(ty3 g c t t2)) -(ex2 C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2)) (\lambda (c2: -C).(getl (S n) c2 d2))) (\lambda (x0: T).(\lambda (H9: (ty3 g c t -x0)).(ex_intro2 C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2)) (\lambda -(c2: C).(getl (S n) c2 d2)) (CHead x (Bind b) t) (wf3_bind g c x H5 t x0 H9 -b) (getl_head (Bind b) n x d2 H6 t)))) H8)) (\lambda (H8: ((\forall (t2: -T).((ty3 g c t t2) \to False)))).(ex_intro2 C (\lambda (c2: C).(wf3 g (CHead -c (Bind b) t) c2)) (\lambda (c2: C).(getl (S n) c2 d2)) (CHead x (Bind Void) -(TSort O)) (wf3_void g c x H5 t H8 b) (getl_head (Bind Void) n x d2 H6 (TSort -O)))) H7)))))) H4))))) (\lambda (f: F).(\lambda (H3: (getl (r (Flat f) n) c -d1)).(let H_x \def (H0 d1 H3 g d2 H2) in (let H4 \def H_x in (ex2_ind C -(\lambda (c2: C).(wf3 g c c2)) (\lambda (c2: C).(getl (S n) c2 d2)) (ex2 C -(\lambda (c2: C).(wf3 g (CHead c (Flat f) t) c2)) (\lambda (c2: C).(getl (S -n) c2 d2))) (\lambda (x: C).(\lambda (H5: (wf3 g c x)).(\lambda (H6: (getl (S -n) x d2)).(ex_intro2 C (\lambda (c2: C).(wf3 g (CHead c (Flat f) t) c2)) -(\lambda (c2: C).(getl (S n) c2 d2)) x (wf3_flat g c x H5 t f) H6)))) H4))))) -k (getl_gen_S k c d1 t n H1))))))))))) c1)))) i). -(* COMMENTS -Initial nodes: 1139 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/wf3/props.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/wf3/props.ma deleted file mode 100644 index 98a05c637..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/wf3/props.ma +++ /dev/null @@ -1,248 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/wf3/ty3.ma". - -include "Basic-1/app/defs.ma". - -theorem wf3_mono: - \forall (g: G).(\forall (c: C).(\forall (c1: C).((wf3 g c c1) \to (\forall -(c2: C).((wf3 g c c2) \to (eq C c1 c2)))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (c1: C).(\lambda (H: (wf3 g c -c1)).(wf3_ind g (\lambda (c0: C).(\lambda (c2: C).(\forall (c3: C).((wf3 g c0 -c3) \to (eq C c2 c3))))) (\lambda (m: nat).(\lambda (c2: C).(\lambda (H0: -(wf3 g (CSort m) c2)).(let H_y \def (wf3_gen_sort1 g c2 m H0) in (eq_ind_r C -(CSort m) (\lambda (c0: C).(eq C (CSort m) c0)) (refl_equal C (CSort m)) c2 -H_y))))) (\lambda (c2: C).(\lambda (c3: C).(\lambda (_: (wf3 g c2 -c3)).(\lambda (H1: ((\forall (c4: C).((wf3 g c2 c4) \to (eq C c3 -c4))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H2: (ty3 g c2 u -t)).(\lambda (b: B).(\lambda (c0: C).(\lambda (H3: (wf3 g (CHead c2 (Bind b) -u) c0)).(let H_x \def (wf3_gen_bind1 g c2 c0 u b H3) in (let H4 \def H_x in -(or_ind (ex3_2 C T (\lambda (c4: C).(\lambda (_: T).(eq C c0 (CHead c4 (Bind -b) u)))) (\lambda (c4: C).(\lambda (_: T).(wf3 g c2 c4))) (\lambda (_: -C).(\lambda (w: T).(ty3 g c2 u w)))) (ex3 C (\lambda (c4: C).(eq C c0 (CHead -c4 (Bind Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda (_: -C).(\forall (w: T).((ty3 g c2 u w) \to False)))) (eq C (CHead c3 (Bind b) u) -c0) (\lambda (H5: (ex3_2 C T (\lambda (c4: C).(\lambda (_: T).(eq C c0 (CHead -c4 (Bind b) u)))) (\lambda (c4: C).(\lambda (_: T).(wf3 g c2 c4))) (\lambda -(_: C).(\lambda (w: T).(ty3 g c2 u w))))).(ex3_2_ind C T (\lambda (c4: -C).(\lambda (_: T).(eq C c0 (CHead c4 (Bind b) u)))) (\lambda (c4: -C).(\lambda (_: T).(wf3 g c2 c4))) (\lambda (_: C).(\lambda (w: T).(ty3 g c2 -u w))) (eq C (CHead c3 (Bind b) u) c0) (\lambda (x0: C).(\lambda (x1: -T).(\lambda (H6: (eq C c0 (CHead x0 (Bind b) u))).(\lambda (H7: (wf3 g c2 -x0)).(\lambda (_: (ty3 g c2 u x1)).(eq_ind_r C (CHead x0 (Bind b) u) (\lambda -(c4: C).(eq C (CHead c3 (Bind b) u) c4)) (f_equal3 C K T C CHead c3 x0 (Bind -b) (Bind b) u u (H1 x0 H7) (refl_equal K (Bind b)) (refl_equal T u)) c0 -H6)))))) H5)) (\lambda (H5: (ex3 C (\lambda (c4: C).(eq C c0 (CHead c4 (Bind -Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda (_: C).(\forall -(w: T).((ty3 g c2 u w) \to False))))).(ex3_ind C (\lambda (c4: C).(eq C c0 -(CHead c4 (Bind Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda -(_: C).(\forall (w: T).((ty3 g c2 u w) \to False))) (eq C (CHead c3 (Bind b) -u) c0) (\lambda (x0: C).(\lambda (H6: (eq C c0 (CHead x0 (Bind Void) (TSort -O)))).(\lambda (_: (wf3 g c2 x0)).(\lambda (H8: ((\forall (w: T).((ty3 g c2 u -w) \to False)))).(eq_ind_r C (CHead x0 (Bind Void) (TSort O)) (\lambda (c4: -C).(eq C (CHead c3 (Bind b) u) c4)) (let H_x0 \def (H8 t H2) in (let H9 \def -H_x0 in (False_ind (eq C (CHead c3 (Bind b) u) (CHead x0 (Bind Void) (TSort -O))) H9))) c0 H6))))) H5)) H4))))))))))))) (\lambda (c2: C).(\lambda (c3: -C).(\lambda (_: (wf3 g c2 c3)).(\lambda (H1: ((\forall (c4: C).((wf3 g c2 c4) -\to (eq C c3 c4))))).(\lambda (u: T).(\lambda (H2: ((\forall (t: T).((ty3 g -c2 u t) \to False)))).(\lambda (b: B).(\lambda (c0: C).(\lambda (H3: (wf3 g -(CHead c2 (Bind b) u) c0)).(let H_x \def (wf3_gen_bind1 g c2 c0 u b H3) in -(let H4 \def H_x in (or_ind (ex3_2 C T (\lambda (c4: C).(\lambda (_: T).(eq C -c0 (CHead c4 (Bind b) u)))) (\lambda (c4: C).(\lambda (_: T).(wf3 g c2 c4))) -(\lambda (_: C).(\lambda (w: T).(ty3 g c2 u w)))) (ex3 C (\lambda (c4: C).(eq -C c0 (CHead c4 (Bind Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) -(\lambda (_: C).(\forall (w: T).((ty3 g c2 u w) \to False)))) (eq C (CHead c3 -(Bind Void) (TSort O)) c0) (\lambda (H5: (ex3_2 C T (\lambda (c4: C).(\lambda -(_: T).(eq C c0 (CHead c4 (Bind b) u)))) (\lambda (c4: C).(\lambda (_: -T).(wf3 g c2 c4))) (\lambda (_: C).(\lambda (w: T).(ty3 g c2 u -w))))).(ex3_2_ind C T (\lambda (c4: C).(\lambda (_: T).(eq C c0 (CHead c4 -(Bind b) u)))) (\lambda (c4: C).(\lambda (_: T).(wf3 g c2 c4))) (\lambda (_: -C).(\lambda (w: T).(ty3 g c2 u w))) (eq C (CHead c3 (Bind Void) (TSort O)) -c0) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C c0 (CHead x0 (Bind -b) u))).(\lambda (_: (wf3 g c2 x0)).(\lambda (H8: (ty3 g c2 u x1)).(eq_ind_r -C (CHead x0 (Bind b) u) (\lambda (c4: C).(eq C (CHead c3 (Bind Void) (TSort -O)) c4)) (let H_x0 \def (H2 x1 H8) in (let H9 \def H_x0 in (False_ind (eq C -(CHead c3 (Bind Void) (TSort O)) (CHead x0 (Bind b) u)) H9))) c0 H6)))))) -H5)) (\lambda (H5: (ex3 C (\lambda (c4: C).(eq C c0 (CHead c4 (Bind Void) -(TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda (_: C).(\forall (w: -T).((ty3 g c2 u w) \to False))))).(ex3_ind C (\lambda (c4: C).(eq C c0 (CHead -c4 (Bind Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda (_: -C).(\forall (w: T).((ty3 g c2 u w) \to False))) (eq C (CHead c3 (Bind Void) -(TSort O)) c0) (\lambda (x0: C).(\lambda (H6: (eq C c0 (CHead x0 (Bind Void) -(TSort O)))).(\lambda (H7: (wf3 g c2 x0)).(\lambda (_: ((\forall (w: T).((ty3 -g c2 u w) \to False)))).(eq_ind_r C (CHead x0 (Bind Void) (TSort O)) (\lambda -(c4: C).(eq C (CHead c3 (Bind Void) (TSort O)) c4)) (f_equal3 C K T C CHead -c3 x0 (Bind Void) (Bind Void) (TSort O) (TSort O) (H1 x0 H7) (refl_equal K -(Bind Void)) (refl_equal T (TSort O))) c0 H6))))) H5)) H4)))))))))))) -(\lambda (c2: C).(\lambda (c3: C).(\lambda (_: (wf3 g c2 c3)).(\lambda (H1: -((\forall (c4: C).((wf3 g c2 c4) \to (eq C c3 c4))))).(\lambda (u: -T).(\lambda (f: F).(\lambda (c0: C).(\lambda (H2: (wf3 g (CHead c2 (Flat f) -u) c0)).(let H_y \def (wf3_gen_flat1 g c2 c0 u f H2) in (H1 c0 H_y)))))))))) -c c1 H)))). -(* COMMENTS -Initial nodes: 1555 -END *) - -theorem wf3_total: - \forall (g: G).(\forall (c1: C).(ex C (\lambda (c2: C).(wf3 g c1 c2)))) -\def - \lambda (g: G).(\lambda (c1: C).(C_ind (\lambda (c: C).(ex C (\lambda (c2: -C).(wf3 g c c2)))) (\lambda (n: nat).(ex_intro C (\lambda (c2: C).(wf3 g -(CSort n) c2)) (CSort n) (wf3_sort g n))) (\lambda (c: C).(\lambda (H: (ex C -(\lambda (c2: C).(wf3 g c c2)))).(\lambda (k: K).(\lambda (t: T).(let H0 \def -H in (ex_ind C (\lambda (c2: C).(wf3 g c c2)) (ex C (\lambda (c2: C).(wf3 g -(CHead c k t) c2))) (\lambda (x: C).(\lambda (H1: (wf3 g c x)).(K_ind -(\lambda (k0: K).(ex C (\lambda (c2: C).(wf3 g (CHead c k0 t) c2)))) (\lambda -(b: B).(let H_x \def (ty3_inference g c t) in (let H2 \def H_x in (or_ind (ex -T (\lambda (t2: T).(ty3 g c t t2))) (\forall (t2: T).((ty3 g c t t2) \to -False)) (ex C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2))) (\lambda -(H3: (ex T (\lambda (t2: T).(ty3 g c t t2)))).(ex_ind T (\lambda (t2: T).(ty3 -g c t t2)) (ex C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2))) (\lambda -(x0: T).(\lambda (H4: (ty3 g c t x0)).(ex_intro C (\lambda (c2: C).(wf3 g -(CHead c (Bind b) t) c2)) (CHead x (Bind b) t) (wf3_bind g c x H1 t x0 H4 -b)))) H3)) (\lambda (H3: ((\forall (t2: T).((ty3 g c t t2) \to -False)))).(ex_intro C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2)) -(CHead x (Bind Void) (TSort O)) (wf3_void g c x H1 t H3 b))) H2)))) (\lambda -(f: F).(ex_intro C (\lambda (c2: C).(wf3 g (CHead c (Flat f) t) c2)) x -(wf3_flat g c x H1 t f))) k))) H0)))))) c1)). -(* COMMENTS -Initial nodes: 435 -END *) - -theorem ty3_shift1: - \forall (g: G).(\forall (c: C).((wf3 g c c) \to (\forall (t1: T).(\forall -(t2: T).((ty3 g c t1 t2) \to (ty3 g (CSort (cbk c)) (app1 c t1) (app1 c -t2))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (H: (wf3 g c c)).(insert_eq C c -(\lambda (c0: C).(wf3 g c0 c)) (\lambda (c0: C).(\forall (t1: T).(\forall -(t2: T).((ty3 g c0 t1 t2) \to (ty3 g (CSort (cbk c0)) (app1 c0 t1) (app1 c0 -t2)))))) (\lambda (y: C).(\lambda (H0: (wf3 g y c)).(wf3_ind g (\lambda (c0: -C).(\lambda (c1: C).((eq C c0 c1) \to (\forall (t1: T).(\forall (t2: T).((ty3 -g c0 t1 t2) \to (ty3 g (CSort (cbk c0)) (app1 c0 t1) (app1 c0 t2)))))))) -(\lambda (m: nat).(\lambda (_: (eq C (CSort m) (CSort m))).(\lambda (t1: -T).(\lambda (t2: T).(\lambda (H2: (ty3 g (CSort m) t1 t2)).H2))))) (\lambda -(c1: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c1 c2)).(\lambda (H2: (((eq C -c1 c2) \to (\forall (t1: T).(\forall (t2: T).((ty3 g c1 t1 t2) \to (ty3 g -(CSort (cbk c1)) (app1 c1 t1) (app1 c1 t2)))))))).(\lambda (u: T).(\lambda -(t: T).(\lambda (H3: (ty3 g c1 u t)).(\lambda (b: B).(\lambda (H4: (eq C -(CHead c1 (Bind b) u) (CHead c2 (Bind b) u))).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (H5: (ty3 g (CHead c1 (Bind b) u) t1 t2)).(let H6 \def (f_equal C -C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow c1 | (CHead c0 _ _) \Rightarrow c0])) (CHead c1 (Bind b) u) -(CHead c2 (Bind b) u) H4) in (let H7 \def (eq_ind_r C c2 (\lambda (c0: -C).((eq C c1 c0) \to (\forall (t3: T).(\forall (t4: T).((ty3 g c1 t3 t4) \to -(ty3 g (CSort (cbk c1)) (app1 c1 t3) (app1 c1 t4))))))) H2 c1 H6) in (let H8 -\def (eq_ind_r C c2 (\lambda (c0: C).(wf3 g c1 c0)) H1 c1 H6) in (ex_ind T -(\lambda (t0: T).(ty3 g (CHead c1 (Bind b) u) t2 t0)) (ty3 g (CSort (cbk c1)) -(app1 c1 (THead (Bind b) u t1)) (app1 c1 (THead (Bind b) u t2))) (\lambda (x: -T).(\lambda (_: (ty3 g (CHead c1 (Bind b) u) t2 x)).(H7 (refl_equal C c1) -(THead (Bind b) u t1) (THead (Bind b) u t2) (ty3_bind g c1 u t H3 b t1 t2 -H5)))) (ty3_correct g (CHead c1 (Bind b) u) t1 t2 H5))))))))))))))))) -(\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c1 c2)).(\lambda (H2: -(((eq C c1 c2) \to (\forall (t1: T).(\forall (t2: T).((ty3 g c1 t1 t2) \to -(ty3 g (CSort (cbk c1)) (app1 c1 t1) (app1 c1 t2)))))))).(\lambda (u: -T).(\lambda (H3: ((\forall (t: T).((ty3 g c1 u t) \to False)))).(\lambda (b: -B).(\lambda (H4: (eq C (CHead c1 (Bind b) u) (CHead c2 (Bind Void) (TSort -O)))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H5: (ty3 g (CHead c1 (Bind -b) u) t1 t2)).(let H6 \def (f_equal C C (\lambda (e: C).(match e in C return -(\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _ _) -\Rightarrow c0])) (CHead c1 (Bind b) u) (CHead c2 (Bind Void) (TSort O)) H4) -in ((let H7 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda -(_: C).B) with [(CSort _) \Rightarrow b | (CHead _ k _) \Rightarrow (match k -in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) -\Rightarrow b])])) (CHead c1 (Bind b) u) (CHead c2 (Bind Void) (TSort O)) H4) -in ((let H8 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda -(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) -(CHead c1 (Bind b) u) (CHead c2 (Bind Void) (TSort O)) H4) in (\lambda (H9: -(eq B b Void)).(\lambda (H10: (eq C c1 c2)).(let H11 \def (eq_ind B b -(\lambda (b0: B).(ty3 g (CHead c1 (Bind b0) u) t1 t2)) H5 Void H9) in -(eq_ind_r B Void (\lambda (b0: B).(ty3 g (CSort (cbk (CHead c1 (Bind b0) u))) -(app1 (CHead c1 (Bind b0) u) t1) (app1 (CHead c1 (Bind b0) u) t2))) (let H12 -\def (eq_ind T u (\lambda (t: T).(ty3 g (CHead c1 (Bind Void) t) t1 t2)) H11 -(TSort O) H8) in (let H13 \def (eq_ind T u (\lambda (t: T).(\forall (t0: -T).((ty3 g c1 t t0) \to False))) H3 (TSort O) H8) in (eq_ind_r T (TSort O) -(\lambda (t: T).(ty3 g (CSort (cbk (CHead c1 (Bind Void) t))) (app1 (CHead c1 -(Bind Void) t) t1) (app1 (CHead c1 (Bind Void) t) t2))) (let H14 \def -(eq_ind_r C c2 (\lambda (c0: C).((eq C c1 c0) \to (\forall (t3: T).(\forall -(t4: T).((ty3 g c1 t3 t4) \to (ty3 g (CSort (cbk c1)) (app1 c1 t3) (app1 c1 -t4))))))) H2 c1 H10) in (let H15 \def (eq_ind_r C c2 (\lambda (c0: C).(wf3 g -c1 c0)) H1 c1 H10) in (ex_ind T (\lambda (t: T).(ty3 g (CHead c1 (Bind Void) -(TSort O)) t2 t)) (ty3 g (CSort (cbk c1)) (app1 c1 (THead (Bind Void) (TSort -O) t1)) (app1 c1 (THead (Bind Void) (TSort O) t2))) (\lambda (x: T).(\lambda -(_: (ty3 g (CHead c1 (Bind Void) (TSort O)) t2 x)).(H14 (refl_equal C c1) -(THead (Bind Void) (TSort O) t1) (THead (Bind Void) (TSort O) t2) (ty3_bind g -c1 (TSort O) (TSort (next g O)) (ty3_sort g c1 O) Void t1 t2 H12)))) -(ty3_correct g (CHead c1 (Bind Void) (TSort O)) t1 t2 H12)))) u H8))) b -H9))))) H7)) H6))))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: -(wf3 g c1 c2)).(\lambda (H2: (((eq C c1 c2) \to (\forall (t1: T).(\forall -(t2: T).((ty3 g c1 t1 t2) \to (ty3 g (CSort (cbk c1)) (app1 c1 t1) (app1 c1 -t2)))))))).(\lambda (u: T).(\lambda (f: F).(\lambda (H3: (eq C (CHead c1 -(Flat f) u) c2)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g (CHead -c1 (Flat f) u) t1 t2)).(let H5 \def (f_equal C C (\lambda (e: C).e) (CHead c1 -(Flat f) u) c2 H3) in (let H6 \def (eq_ind_r C c2 (\lambda (c0: C).((eq C c1 -c0) \to (\forall (t3: T).(\forall (t4: T).((ty3 g c1 t3 t4) \to (ty3 g (CSort -(cbk c1)) (app1 c1 t3) (app1 c1 t4))))))) H2 (CHead c1 (Flat f) u) H5) in -(let H7 \def (eq_ind_r C c2 (\lambda (c0: C).(wf3 g c1 c0)) H1 (CHead c1 -(Flat f) u) H5) in (let H_x \def (wf3_gen_head2 g c1 c1 u (Flat f) H7) in -(let H8 \def H_x in (ex_ind B (\lambda (b: B).(eq K (Flat f) (Bind b))) (ty3 -g (CSort (cbk c1)) (app1 c1 (THead (Flat f) u t1)) (app1 c1 (THead (Flat f) u -t2))) (\lambda (x: B).(\lambda (H9: (eq K (Flat f) (Bind x))).(let H10 \def -(eq_ind K (Flat f) (\lambda (ee: K).(match ee in K return (\lambda (_: -K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])) I -(Bind x) H9) in (False_ind (ty3 g (CSort (cbk c1)) (app1 c1 (THead (Flat f) u -t1)) (app1 c1 (THead (Flat f) u t2))) H10)))) H8)))))))))))))))) y c H0))) -H))). -(* COMMENTS -Initial nodes: 1677 -END *) - -theorem wf3_idem: - \forall (g: G).(\forall (c1: C).(\forall (c2: C).((wf3 g c1 c2) \to (wf3 g -c2 c2)))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wf3 g c1 -c2)).(wf3_ind g (\lambda (_: C).(\lambda (c0: C).(wf3 g c0 c0))) (\lambda (m: -nat).(wf3_sort g m)) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wf3 g -c3 c4)).(\lambda (H1: (wf3 g c4 c4)).(\lambda (u: T).(\lambda (t: T).(\lambda -(H2: (ty3 g c3 u t)).(\lambda (b: B).(wf3_bind g c4 c4 H1 u t (wf3_ty3_conf g -c3 u t H2 c4 H0) b))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (_: -(wf3 g c3 c4)).(\lambda (H1: (wf3 g c4 c4)).(\lambda (u: T).(\lambda (_: -((\forall (t: T).((ty3 g c3 u t) \to False)))).(\lambda (_: B).(wf3_bind g c4 -c4 H1 (TSort O) (TSort (next g O)) (ty3_sort g c4 O) Void)))))))) (\lambda -(c3: C).(\lambda (c4: C).(\lambda (_: (wf3 g c3 c4)).(\lambda (H1: (wf3 g c4 -c4)).(\lambda (_: T).(\lambda (_: F).H1)))))) c1 c2 H)))). -(* COMMENTS -Initial nodes: 207 -END *) - -theorem wf3_ty3: - \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (u: T).((ty3 g c1 t -u) \to (ex2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(ty3 g c2 t -u))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H: -(ty3 g c1 t u)).(let H_x \def (wf3_total g c1) in (let H0 \def H_x in (ex_ind -C (\lambda (c2: C).(wf3 g c1 c2)) (ex2 C (\lambda (c2: C).(wf3 g c1 c2)) -(\lambda (c2: C).(ty3 g c2 t u))) (\lambda (x: C).(\lambda (H1: (wf3 g c1 -x)).(ex_intro2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(ty3 g c2 t -u)) x H1 (wf3_ty3_conf g c1 t u H x H1)))) H0))))))). -(* COMMENTS -Initial nodes: 123 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/wf3/ty3.ma b/matita/matita/contribs/LAMBDA-TYPES/Basic-1/wf3/ty3.ma deleted file mode 100644 index e47d4a2e0..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Basic-1/wf3/ty3.ma +++ /dev/null @@ -1,138 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/wf3/getl.ma". - -theorem wf3_pr2_conf: - \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pr2 c1 -t1 t2) \to (\forall (c2: C).((wf3 g c1 c2) \to (\forall (u: T).((ty3 g c1 t1 -u) \to (pr2 c2 t1 t2))))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H: (pr2 c1 t1 t2)).(pr2_ind (\lambda (c: C).(\lambda (t: T).(\lambda (t0: -T).(\forall (c2: C).((wf3 g c c2) \to (\forall (u: T).((ty3 g c t u) \to (pr2 -c2 t t0)))))))) (\lambda (c: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda -(H0: (pr0 t3 t4)).(\lambda (c2: C).(\lambda (_: (wf3 g c c2)).(\lambda (u: -T).(\lambda (_: (ty3 g c t3 u)).(pr2_free c2 t3 t4 H0))))))))) (\lambda (c: -C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c -(CHead d (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: -(pr0 t3 t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (c2: -C).(\lambda (H3: (wf3 g c c2)).(\lambda (u0: T).(\lambda (H4: (ty3 g c t3 -u0)).(let H_y \def (ty3_sred_pr0 t3 t4 H1 g c u0 H4) in (let H_x \def -(ty3_getl_subst0 g c t4 u0 H_y u t i H2 Abbr d u H0) in (let H5 \def H_x in -(ex_ind T (\lambda (w: T).(ty3 g d u w)) (pr2 c2 t3 t) (\lambda (x: -T).(\lambda (H6: (ty3 g d u x)).(let H_x0 \def (wf3_getl_conf Abbr i c d u H0 -g c2 H3 x H6) in (let H7 \def H_x0 in (ex2_ind C (\lambda (d2: C).(getl i c2 -(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(wf3 g d d2)) (pr2 c2 t3 t) -(\lambda (x0: C).(\lambda (H8: (getl i c2 (CHead x0 (Bind Abbr) u))).(\lambda -(_: (wf3 g d x0)).(pr2_delta c2 x0 u i H8 t3 t4 H1 t H2)))) H7))))) -H5)))))))))))))))))) c1 t1 t2 H))))). -(* COMMENTS -Initial nodes: 373 -END *) - -theorem wf3_pr3_conf: - \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pr3 c1 -t1 t2) \to (\forall (c2: C).((wf3 g c1 c2) \to (\forall (u: T).((ty3 g c1 t1 -u) \to (pr3 c2 t1 t2))))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H: (pr3 c1 t1 t2)).(pr3_ind c1 (\lambda (t: T).(\lambda (t0: T).(\forall -(c2: C).((wf3 g c1 c2) \to (\forall (u: T).((ty3 g c1 t u) \to (pr3 c2 t -t0))))))) (\lambda (t: T).(\lambda (c2: C).(\lambda (_: (wf3 g c1 -c2)).(\lambda (u: T).(\lambda (_: (ty3 g c1 t u)).(pr3_refl c2 t)))))) -(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr2 c1 t4 t3)).(\lambda (t5: -T).(\lambda (_: (pr3 c1 t3 t5)).(\lambda (H2: ((\forall (c2: C).((wf3 g c1 -c2) \to (\forall (u: T).((ty3 g c1 t3 u) \to (pr3 c2 t3 t5))))))).(\lambda -(c2: C).(\lambda (H3: (wf3 g c1 c2)).(\lambda (u: T).(\lambda (H4: (ty3 g c1 -t4 u)).(pr3_sing c2 t3 t4 (wf3_pr2_conf g c1 t4 t3 H0 c2 H3 u H4) t5 (H2 c2 -H3 u (ty3_sred_pr2 c1 t4 t3 H0 g u H4))))))))))))) t1 t2 H))))). -(* COMMENTS -Initial nodes: 217 -END *) - -theorem wf3_pc3_conf: - \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pc3 c1 -t1 t2) \to (\forall (c2: C).((wf3 g c1 c2) \to (\forall (u1: T).((ty3 g c1 t1 -u1) \to (\forall (u2: T).((ty3 g c1 t2 u2) \to (pc3 c2 t1 t2))))))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H: (pc3 c1 t1 t2)).(\lambda (c2: C).(\lambda (H0: (wf3 g c1 c2)).(\lambda -(u1: T).(\lambda (H1: (ty3 g c1 t1 u1)).(\lambda (u2: T).(\lambda (H2: (ty3 g -c1 t2 u2)).(let H3 \def H in (ex2_ind T (\lambda (t: T).(pr3 c1 t1 t)) -(\lambda (t: T).(pr3 c1 t2 t)) (pc3 c2 t1 t2) (\lambda (x: T).(\lambda (H4: -(pr3 c1 t1 x)).(\lambda (H5: (pr3 c1 t2 x)).(pc3_pr3_t c2 t1 x (wf3_pr3_conf -g c1 t1 x H4 c2 H0 u1 H1) t2 (wf3_pr3_conf g c1 t2 x H5 c2 H0 u2 H2))))) -H3)))))))))))). -(* COMMENTS -Initial nodes: 153 -END *) - -theorem wf3_ty3_conf: - \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c1 -t1 t2) \to (\forall (c2: C).((wf3 g c1 c2) \to (ty3 g c2 t1 t2))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H: (ty3 g c1 t1 t2)).(ty3_ind g (\lambda (c: C).(\lambda (t: T).(\lambda -(t0: T).(\forall (c2: C).((wf3 g c c2) \to (ty3 g c2 t t0)))))) (\lambda (c: -C).(\lambda (t3: T).(\lambda (t: T).(\lambda (H0: (ty3 g c t3 t)).(\lambda -(H1: ((\forall (c2: C).((wf3 g c c2) \to (ty3 g c2 t3 t))))).(\lambda (u: -T).(\lambda (t4: T).(\lambda (H2: (ty3 g c u t4)).(\lambda (H3: ((\forall -(c2: C).((wf3 g c c2) \to (ty3 g c2 u t4))))).(\lambda (H4: (pc3 c t4 -t3)).(\lambda (c2: C).(\lambda (H5: (wf3 g c c2)).(ex_ind T (\lambda (t0: -T).(ty3 g c t4 t0)) (ty3 g c2 u t3) (\lambda (x: T).(\lambda (H6: (ty3 g c t4 -x)).(ty3_conv g c2 t3 t (H1 c2 H5) u t4 (H3 c2 H5) (wf3_pc3_conf g c t4 t3 H4 -c2 H5 x H6 t H0)))) (ty3_correct g c u t4 H2)))))))))))))) (\lambda (c: -C).(\lambda (m: nat).(\lambda (c2: C).(\lambda (_: (wf3 g c c2)).(ty3_sort g -c2 m))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d: C).(\lambda (u: -T).(\lambda (H0: (getl n c (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda -(H1: (ty3 g d u t)).(\lambda (H2: ((\forall (c2: C).((wf3 g d c2) \to (ty3 g -c2 u t))))).(\lambda (c2: C).(\lambda (H3: (wf3 g c c2)).(let H_x \def -(wf3_getl_conf Abbr n c d u H0 g c2 H3 t H1) in (let H4 \def H_x in (ex2_ind -C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: -C).(wf3 g d d2)) (ty3 g c2 (TLRef n) (lift (S n) O t)) (\lambda (x: -C).(\lambda (H5: (getl n c2 (CHead x (Bind Abbr) u))).(\lambda (H6: (wf3 g d -x)).(ty3_abbr g n c2 x u H5 t (H2 x H6))))) H4))))))))))))) (\lambda (n: -nat).(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c -(CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u -t)).(\lambda (H2: ((\forall (c2: C).((wf3 g d c2) \to (ty3 g c2 u -t))))).(\lambda (c2: C).(\lambda (H3: (wf3 g c c2)).(let H_x \def -(wf3_getl_conf Abst n c d u H0 g c2 H3 t H1) in (let H4 \def H_x in (ex2_ind -C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(wf3 g d d2)) (ty3 g c2 (TLRef n) (lift (S n) O u)) (\lambda (x: -C).(\lambda (H5: (getl n c2 (CHead x (Bind Abst) u))).(\lambda (H6: (wf3 g d -x)).(ty3_abst g n c2 x u H5 t (H2 x H6))))) H4))))))))))))) (\lambda (c: -C).(\lambda (u: T).(\lambda (t: T).(\lambda (H0: (ty3 g c u t)).(\lambda (H1: -((\forall (c2: C).((wf3 g c c2) \to (ty3 g c2 u t))))).(\lambda (b: -B).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (ty3 g (CHead c (Bind b) u) -t3 t4)).(\lambda (H3: ((\forall (c2: C).((wf3 g (CHead c (Bind b) u) c2) \to -(ty3 g c2 t3 t4))))).(\lambda (c2: C).(\lambda (H4: (wf3 g c c2)).(ty3_bind g -c2 u t (H1 c2 H4) b t3 t4 (H3 (CHead c2 (Bind b) u) (wf3_bind g c c2 H4 u t -H0 b))))))))))))))) (\lambda (c: C).(\lambda (w: T).(\lambda (u: T).(\lambda -(_: (ty3 g c w u)).(\lambda (H1: ((\forall (c2: C).((wf3 g c c2) \to (ty3 g -c2 w u))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c v (THead -(Bind Abst) u t))).(\lambda (H3: ((\forall (c2: C).((wf3 g c c2) \to (ty3 g -c2 v (THead (Bind Abst) u t)))))).(\lambda (c2: C).(\lambda (H4: (wf3 g c -c2)).(ty3_appl g c2 w u (H1 c2 H4) v t (H3 c2 H4))))))))))))) (\lambda (c: -C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (ty3 g c t3 t4)).(\lambda -(H1: ((\forall (c2: C).((wf3 g c c2) \to (ty3 g c2 t3 t4))))).(\lambda (t0: -T).(\lambda (_: (ty3 g c t4 t0)).(\lambda (H3: ((\forall (c2: C).((wf3 g c -c2) \to (ty3 g c2 t4 t0))))).(\lambda (c2: C).(\lambda (H4: (wf3 g c -c2)).(ty3_cast g c2 t3 t4 (H1 c2 H4) t0 (H3 c2 H4)))))))))))) c1 t1 t2 H))))). -(* COMMENTS -Initial nodes: 1027 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Ground-1/blt/defs.ma b/matita/matita/contribs/LAMBDA-TYPES/Ground-1/blt/defs.ma deleted file mode 100644 index 009627a3b..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Ground-1/blt/defs.ma +++ /dev/null @@ -1,25 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Ground-1/preamble.ma". - -definition blt: - nat \to (nat \to bool) -\def - let rec blt (m: nat) (n: nat) on n: bool \def (match n with [O \Rightarrow -false | (S n0) \Rightarrow (match m with [O \Rightarrow true | (S m0) -\Rightarrow (blt m0 n0)])]) in blt. - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Ground-1/blt/props.ma b/matita/matita/contribs/LAMBDA-TYPES/Ground-1/blt/props.ma deleted file mode 100644 index e5b569925..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Ground-1/blt/props.ma +++ /dev/null @@ -1,112 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Ground-1/blt/defs.ma". - -theorem lt_blt: - \forall (x: nat).(\forall (y: nat).((lt y x) \to (eq bool (blt y x) true))) -\def - \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((lt y n) \to -(eq bool (blt y n) true)))) (\lambda (y: nat).(\lambda (H: (lt y O)).(let H0 -\def (match H in le return (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat -n O) \to (eq bool (blt y O) true)))) with [le_n \Rightarrow (\lambda (H0: (eq -nat (S y) O)).(let H1 \def (eq_ind nat (S y) (\lambda (e: nat).(match e in -nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) -\Rightarrow True])) I O H0) in (False_ind (eq bool (blt y O) true) H1))) | -(le_S m H0) \Rightarrow (\lambda (H1: (eq nat (S m) O)).((let H2 \def (eq_ind -nat (S m) (\lambda (e: nat).(match e in nat return (\lambda (_: nat).Prop) -with [O \Rightarrow False | (S _) \Rightarrow True])) I O H1) in (False_ind -((le (S y) m) \to (eq bool (blt y O) true)) H2)) H0))]) in (H0 (refl_equal -nat O))))) (\lambda (n: nat).(\lambda (H: ((\forall (y: nat).((lt y n) \to -(eq bool (blt y n) true))))).(\lambda (y: nat).(nat_ind (\lambda (n0: -nat).((lt n0 (S n)) \to (eq bool (blt n0 (S n)) true))) (\lambda (_: (lt O (S -n))).(refl_equal bool true)) (\lambda (n0: nat).(\lambda (_: (((lt n0 (S n)) -\to (eq bool (match n0 with [O \Rightarrow true | (S m) \Rightarrow (blt m -n)]) true)))).(\lambda (H1: (lt (S n0) (S n))).(H n0 (le_S_n (S n0) n H1))))) -y)))) x). -(* COMMENTS -Initial nodes: 291 -END *) - -theorem le_bge: - \forall (x: nat).(\forall (y: nat).((le x y) \to (eq bool (blt y x) false))) -\def - \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((le n y) \to -(eq bool (blt y n) false)))) (\lambda (y: nat).(\lambda (_: (le O -y)).(refl_equal bool false))) (\lambda (n: nat).(\lambda (H: ((\forall (y: -nat).((le n y) \to (eq bool (blt y n) false))))).(\lambda (y: nat).(nat_ind -(\lambda (n0: nat).((le (S n) n0) \to (eq bool (blt n0 (S n)) false))) -(\lambda (H0: (le (S n) O)).(let H1 \def (match H0 in le return (\lambda (n0: -nat).(\lambda (_: (le ? n0)).((eq nat n0 O) \to (eq bool (blt O (S n)) -false)))) with [le_n \Rightarrow (\lambda (H1: (eq nat (S n) O)).(let H2 \def -(eq_ind nat (S n) (\lambda (e: nat).(match e in nat return (\lambda (_: -nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H1) in -(False_ind (eq bool (blt O (S n)) false) H2))) | (le_S m H1) \Rightarrow -(\lambda (H2: (eq nat (S m) O)).((let H3 \def (eq_ind nat (S m) (\lambda (e: -nat).(match e in nat return (\lambda (_: nat).Prop) with [O \Rightarrow False -| (S _) \Rightarrow True])) I O H2) in (False_ind ((le (S n) m) \to (eq bool -(blt O (S n)) false)) H3)) H1))]) in (H1 (refl_equal nat O)))) (\lambda (n0: -nat).(\lambda (_: (((le (S n) n0) \to (eq bool (blt n0 (S n)) -false)))).(\lambda (H1: (le (S n) (S n0))).(H n0 (le_S_n n n0 H1))))) y)))) -x). -(* COMMENTS -Initial nodes: 293 -END *) - -theorem blt_lt: - \forall (x: nat).(\forall (y: nat).((eq bool (blt y x) true) \to (lt y x))) -\def - \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((eq bool (blt -y n) true) \to (lt y n)))) (\lambda (y: nat).(\lambda (H: (eq bool (blt y O) -true)).(let H0 \def (match H in eq return (\lambda (b: bool).(\lambda (_: (eq -? ? b)).((eq bool b true) \to (lt y O)))) with [refl_equal \Rightarrow -(\lambda (H0: (eq bool (blt y O) true)).(let H1 \def (eq_ind bool (blt y O) -(\lambda (e: bool).(match e in bool return (\lambda (_: bool).Prop) with -[true \Rightarrow False | false \Rightarrow True])) I true H0) in (False_ind -(lt y O) H1)))]) in (H0 (refl_equal bool true))))) (\lambda (n: nat).(\lambda -(H: ((\forall (y: nat).((eq bool (blt y n) true) \to (lt y n))))).(\lambda -(y: nat).(nat_ind (\lambda (n0: nat).((eq bool (blt n0 (S n)) true) \to (lt -n0 (S n)))) (\lambda (_: (eq bool true true)).(le_S_n (S O) (S n) (le_n_S (S -O) (S n) (le_n_S O n (le_O_n n))))) (\lambda (n0: nat).(\lambda (_: (((eq -bool (match n0 with [O \Rightarrow true | (S m) \Rightarrow (blt m n)]) true) -\to (lt n0 (S n))))).(\lambda (H1: (eq bool (blt n0 n) true)).(lt_n_S n0 n (H -n0 H1))))) y)))) x). -(* COMMENTS -Initial nodes: 252 -END *) - -theorem bge_le: - \forall (x: nat).(\forall (y: nat).((eq bool (blt y x) false) \to (le x y))) -\def - \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((eq bool (blt -y n) false) \to (le n y)))) (\lambda (y: nat).(\lambda (_: (eq bool (blt y O) -false)).(le_O_n y))) (\lambda (n: nat).(\lambda (H: ((\forall (y: nat).((eq -bool (blt y n) false) \to (le n y))))).(\lambda (y: nat).(nat_ind (\lambda -(n0: nat).((eq bool (blt n0 (S n)) false) \to (le (S n) n0))) (\lambda (H0: -(eq bool (blt O (S n)) false)).(let H1 \def (match H0 in eq return (\lambda -(b: bool).(\lambda (_: (eq ? ? b)).((eq bool b false) \to (le (S n) O)))) -with [refl_equal \Rightarrow (\lambda (H1: (eq bool (blt O (S n)) -false)).(let H2 \def (eq_ind bool (blt O (S n)) (\lambda (e: bool).(match e -in bool return (\lambda (_: bool).Prop) with [true \Rightarrow True | false -\Rightarrow False])) I false H1) in (False_ind (le (S n) O) H2)))]) in (H1 -(refl_equal bool false)))) (\lambda (n0: nat).(\lambda (_: (((eq bool (blt n0 -(S n)) false) \to (le (S n) n0)))).(\lambda (H1: (eq bool (blt (S n0) (S n)) -false)).(le_S_n (S n) (S n0) (le_n_S (S n) (S n0) (le_n_S n n0 (H n0 -H1))))))) y)))) x). -(* COMMENTS -Initial nodes: 262 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Ground-1/definitions.ma b/matita/matita/contribs/LAMBDA-TYPES/Ground-1/definitions.ma deleted file mode 100644 index df31468aa..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Ground-1/definitions.ma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Ground-1/types/defs.ma". - -include "Ground-1/blt/defs.ma". - -include "Ground-1/plist/defs.ma". - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Ground-1/ext/arith.ma b/matita/matita/contribs/LAMBDA-TYPES/Ground-1/ext/arith.ma deleted file mode 100644 index f9796e7fd..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Ground-1/ext/arith.ma +++ /dev/null @@ -1,737 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Ground-1/preamble.ma". - -theorem nat_dec: - \forall (n1: nat).(\forall (n2: nat).(or (eq nat n1 n2) ((eq nat n1 n2) \to -(\forall (P: Prop).P)))) -\def - \lambda (n1: nat).(nat_ind (\lambda (n: nat).(\forall (n2: nat).(or (eq nat -n n2) ((eq nat n n2) \to (\forall (P: Prop).P))))) (\lambda (n2: -nat).(nat_ind (\lambda (n: nat).(or (eq nat O n) ((eq nat O n) \to (\forall -(P: Prop).P)))) (or_introl (eq nat O O) ((eq nat O O) \to (\forall (P: -Prop).P)) (refl_equal nat O)) (\lambda (n: nat).(\lambda (_: (or (eq nat O n) -((eq nat O n) \to (\forall (P: Prop).P)))).(or_intror (eq nat O (S n)) ((eq -nat O (S n)) \to (\forall (P: Prop).P)) (\lambda (H0: (eq nat O (S -n))).(\lambda (P: Prop).(let H1 \def (eq_ind nat O (\lambda (ee: nat).(match -ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) -\Rightarrow False])) I (S n) H0) in (False_ind P H1))))))) n2)) (\lambda (n: -nat).(\lambda (H: ((\forall (n2: nat).(or (eq nat n n2) ((eq nat n n2) \to -(\forall (P: Prop).P)))))).(\lambda (n2: nat).(nat_ind (\lambda (n0: nat).(or -(eq nat (S n) n0) ((eq nat (S n) n0) \to (\forall (P: Prop).P)))) (or_intror -(eq nat (S n) O) ((eq nat (S n) O) \to (\forall (P: Prop).P)) (\lambda (H0: -(eq nat (S n) O)).(\lambda (P: Prop).(let H1 \def (eq_ind nat (S n) (\lambda -(ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow -False | (S _) \Rightarrow True])) I O H0) in (False_ind P H1))))) (\lambda -(n0: nat).(\lambda (H0: (or (eq nat (S n) n0) ((eq nat (S n) n0) \to (\forall -(P: Prop).P)))).(or_ind (eq nat n n0) ((eq nat n n0) \to (\forall (P: -Prop).P)) (or (eq nat (S n) (S n0)) ((eq nat (S n) (S n0)) \to (\forall (P: -Prop).P))) (\lambda (H1: (eq nat n n0)).(let H2 \def (eq_ind_r nat n0 -(\lambda (n3: nat).(or (eq nat (S n) n3) ((eq nat (S n) n3) \to (\forall (P: -Prop).P)))) H0 n H1) in (eq_ind nat n (\lambda (n3: nat).(or (eq nat (S n) (S -n3)) ((eq nat (S n) (S n3)) \to (\forall (P: Prop).P)))) (or_introl (eq nat -(S n) (S n)) ((eq nat (S n) (S n)) \to (\forall (P: Prop).P)) (refl_equal nat -(S n))) n0 H1))) (\lambda (H1: (((eq nat n n0) \to (\forall (P: -Prop).P)))).(or_intror (eq nat (S n) (S n0)) ((eq nat (S n) (S n0)) \to -(\forall (P: Prop).P)) (\lambda (H2: (eq nat (S n) (S n0))).(\lambda (P: -Prop).(let H3 \def (f_equal nat nat (\lambda (e: nat).(match e in nat return -(\lambda (_: nat).nat) with [O \Rightarrow n | (S n3) \Rightarrow n3])) (S n) -(S n0) H2) in (let H4 \def (eq_ind_r nat n0 (\lambda (n3: nat).((eq nat n n3) -\to (\forall (P0: Prop).P0))) H1 n H3) in (let H5 \def (eq_ind_r nat n0 -(\lambda (n3: nat).(or (eq nat (S n) n3) ((eq nat (S n) n3) \to (\forall (P0: -Prop).P0)))) H0 n H3) in (H4 (refl_equal nat n) P)))))))) (H n0)))) n2)))) -n1). -(* COMMENTS -Initial nodes: 676 -END *) - -theorem simpl_plus_r: - \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((eq nat (plus m n) -(plus p n)) \to (eq nat m p)))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (eq nat -(plus m n) (plus p n))).(simpl_plus_l n m p (eq_ind_r nat (plus m n) (\lambda -(n0: nat).(eq nat n0 (plus n p))) (eq_ind_r nat (plus p n) (\lambda (n0: -nat).(eq nat n0 (plus n p))) (sym_eq nat (plus n p) (plus p n) (plus_sym n -p)) (plus m n) H) (plus n m) (plus_sym n m)))))). -(* COMMENTS -Initial nodes: 119 -END *) - -theorem minus_Sx_Sy: - \forall (x: nat).(\forall (y: nat).(eq nat (minus (S x) (S y)) (minus x y))) -\def - \lambda (x: nat).(\lambda (y: nat).(refl_equal nat (minus x y))). -(* COMMENTS -Initial nodes: 13 -END *) - -theorem minus_plus_r: - \forall (m: nat).(\forall (n: nat).(eq nat (minus (plus m n) n) m)) -\def - \lambda (m: nat).(\lambda (n: nat).(eq_ind_r nat (plus n m) (\lambda (n0: -nat).(eq nat (minus n0 n) m)) (minus_plus n m) (plus m n) (plus_sym m n))). -(* COMMENTS -Initial nodes: 45 -END *) - -theorem plus_permute_2_in_3: - \forall (x: nat).(\forall (y: nat).(\forall (z: nat).(eq nat (plus (plus x -y) z) (plus (plus x z) y)))) -\def - \lambda (x: nat).(\lambda (y: nat).(\lambda (z: nat).(eq_ind_r nat (plus x -(plus y z)) (\lambda (n: nat).(eq nat n (plus (plus x z) y))) (eq_ind_r nat -(plus z y) (\lambda (n: nat).(eq nat (plus x n) (plus (plus x z) y))) (eq_ind -nat (plus (plus x z) y) (\lambda (n: nat).(eq nat n (plus (plus x z) y))) -(refl_equal nat (plus (plus x z) y)) (plus x (plus z y)) (plus_assoc_r x z -y)) (plus y z) (plus_sym y z)) (plus (plus x y) z) (plus_assoc_r x y z)))). -(* COMMENTS -Initial nodes: 163 -END *) - -theorem plus_permute_2_in_3_assoc: - \forall (n: nat).(\forall (h: nat).(\forall (k: nat).(eq nat (plus (plus n -h) k) (plus n (plus k h))))) -\def - \lambda (n: nat).(\lambda (h: nat).(\lambda (k: nat).(eq_ind_r nat (plus -(plus n k) h) (\lambda (n0: nat).(eq nat n0 (plus n (plus k h)))) (eq_ind_r -nat (plus (plus n k) h) (\lambda (n0: nat).(eq nat (plus (plus n k) h) n0)) -(refl_equal nat (plus (plus n k) h)) (plus n (plus k h)) (plus_assoc_l n k -h)) (plus (plus n h) k) (plus_permute_2_in_3 n h k)))). -(* COMMENTS -Initial nodes: 119 -END *) - -theorem plus_O: - \forall (x: nat).(\forall (y: nat).((eq nat (plus x y) O) \to (land (eq nat -x O) (eq nat y O)))) -\def - \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((eq nat (plus -n y) O) \to (land (eq nat n O) (eq nat y O))))) (\lambda (y: nat).(\lambda -(H: (eq nat (plus O y) O)).(conj (eq nat O O) (eq nat y O) (refl_equal nat O) -H))) (\lambda (n: nat).(\lambda (_: ((\forall (y: nat).((eq nat (plus n y) O) -\to (land (eq nat n O) (eq nat y O)))))).(\lambda (y: nat).(\lambda (H0: (eq -nat (plus (S n) y) O)).(let H1 \def (match H0 in eq return (\lambda (n0: -nat).(\lambda (_: (eq ? ? n0)).((eq nat n0 O) \to (land (eq nat (S n) O) (eq -nat y O))))) with [refl_equal \Rightarrow (\lambda (H1: (eq nat (plus (S n) -y) O)).(let H2 \def (eq_ind nat (plus (S n) y) (\lambda (e: nat).(match e in -nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) -\Rightarrow True])) I O H1) in (False_ind (land (eq nat (S n) O) (eq nat y -O)) H2)))]) in (H1 (refl_equal nat O))))))) x). -(* COMMENTS -Initial nodes: 233 -END *) - -theorem minus_Sx_SO: - \forall (x: nat).(eq nat (minus (S x) (S O)) x) -\def - \lambda (x: nat).(eq_ind nat x (\lambda (n: nat).(eq nat n x)) (refl_equal -nat x) (minus x O) (minus_n_O x)). -(* COMMENTS -Initial nodes: 33 -END *) - -theorem eq_nat_dec: - \forall (i: nat).(\forall (j: nat).(or (not (eq nat i j)) (eq nat i j))) -\def - \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (j: nat).(or (not (eq -nat n j)) (eq nat n j)))) (\lambda (j: nat).(nat_ind (\lambda (n: nat).(or -(not (eq nat O n)) (eq nat O n))) (or_intror (not (eq nat O O)) (eq nat O O) -(refl_equal nat O)) (\lambda (n: nat).(\lambda (_: (or (not (eq nat O n)) (eq -nat O n))).(or_introl (not (eq nat O (S n))) (eq nat O (S n)) (O_S n)))) j)) -(\lambda (n: nat).(\lambda (H: ((\forall (j: nat).(or (not (eq nat n j)) (eq -nat n j))))).(\lambda (j: nat).(nat_ind (\lambda (n0: nat).(or (not (eq nat -(S n) n0)) (eq nat (S n) n0))) (or_introl (not (eq nat (S n) O)) (eq nat (S -n) O) (sym_not_eq nat O (S n) (O_S n))) (\lambda (n0: nat).(\lambda (_: (or -(not (eq nat (S n) n0)) (eq nat (S n) n0))).(or_ind (not (eq nat n n0)) (eq -nat n n0) (or (not (eq nat (S n) (S n0))) (eq nat (S n) (S n0))) (\lambda -(H1: (not (eq nat n n0))).(or_introl (not (eq nat (S n) (S n0))) (eq nat (S -n) (S n0)) (not_eq_S n n0 H1))) (\lambda (H1: (eq nat n n0)).(or_intror (not -(eq nat (S n) (S n0))) (eq nat (S n) (S n0)) (f_equal nat nat S n n0 H1))) (H -n0)))) j)))) i). -(* COMMENTS -Initial nodes: 401 -END *) - -theorem neq_eq_e: - \forall (i: nat).(\forall (j: nat).(\forall (P: Prop).((((not (eq nat i j)) -\to P)) \to ((((eq nat i j) \to P)) \to P)))) -\def - \lambda (i: nat).(\lambda (j: nat).(\lambda (P: Prop).(\lambda (H: (((not -(eq nat i j)) \to P))).(\lambda (H0: (((eq nat i j) \to P))).(let o \def -(eq_nat_dec i j) in (or_ind (not (eq nat i j)) (eq nat i j) P H H0 o)))))). -(* COMMENTS -Initial nodes: 61 -END *) - -theorem le_false: - \forall (m: nat).(\forall (n: nat).(\forall (P: Prop).((le m n) \to ((le (S -n) m) \to P)))) -\def - \lambda (m: nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).(\forall (P: -Prop).((le n n0) \to ((le (S n0) n) \to P))))) (\lambda (n: nat).(\lambda (P: -Prop).(\lambda (_: (le O n)).(\lambda (H0: (le (S n) O)).(let H1 \def (match -H0 in le return (\lambda (n0: nat).(\lambda (_: (le ? n0)).((eq nat n0 O) \to -P))) with [le_n \Rightarrow (\lambda (H1: (eq nat (S n) O)).(let H2 \def -(eq_ind nat (S n) (\lambda (e: nat).(match e in nat return (\lambda (_: -nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H1) in -(False_ind P H2))) | (le_S m0 H1) \Rightarrow (\lambda (H2: (eq nat (S m0) -O)).((let H3 \def (eq_ind nat (S m0) (\lambda (e: nat).(match e in nat return -(\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) -I O H2) in (False_ind ((le (S n) m0) \to P) H3)) H1))]) in (H1 (refl_equal -nat O))))))) (\lambda (n: nat).(\lambda (H: ((\forall (n0: nat).(\forall (P: -Prop).((le n n0) \to ((le (S n0) n) \to P)))))).(\lambda (n0: nat).(nat_ind -(\lambda (n1: nat).(\forall (P: Prop).((le (S n) n1) \to ((le (S n1) (S n)) -\to P)))) (\lambda (P: Prop).(\lambda (H0: (le (S n) O)).(\lambda (_: (le (S -O) (S n))).(let H2 \def (match H0 in le return (\lambda (n1: nat).(\lambda -(_: (le ? n1)).((eq nat n1 O) \to P))) with [le_n \Rightarrow (\lambda (H2: -(eq nat (S n) O)).(let H3 \def (eq_ind nat (S n) (\lambda (e: nat).(match e -in nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) -\Rightarrow True])) I O H2) in (False_ind P H3))) | (le_S m0 H2) \Rightarrow -(\lambda (H3: (eq nat (S m0) O)).((let H4 \def (eq_ind nat (S m0) (\lambda -(e: nat).(match e in nat return (\lambda (_: nat).Prop) with [O \Rightarrow -False | (S _) \Rightarrow True])) I O H3) in (False_ind ((le (S n) m0) \to P) -H4)) H2))]) in (H2 (refl_equal nat O)))))) (\lambda (n1: nat).(\lambda (_: -((\forall (P: Prop).((le (S n) n1) \to ((le (S n1) (S n)) \to P))))).(\lambda -(P: Prop).(\lambda (H1: (le (S n) (S n1))).(\lambda (H2: (le (S (S n1)) (S -n))).(H n1 P (le_S_n n n1 H1) (le_S_n (S n1) n H2))))))) n0)))) m). -(* COMMENTS -Initial nodes: 409 -END *) - -theorem le_Sx_x: - \forall (x: nat).((le (S x) x) \to (\forall (P: Prop).P)) -\def - \lambda (x: nat).(\lambda (H: (le (S x) x)).(\lambda (P: Prop).(let H0 \def -le_Sn_n in (False_ind P (H0 x H))))). -(* COMMENTS -Initial nodes: 23 -END *) - -theorem le_n_pred: - \forall (n: nat).(\forall (m: nat).((le n m) \to (le (pred n) (pred m)))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(le_ind n (\lambda -(n0: nat).(le (pred n) (pred n0))) (le_n (pred n)) (\lambda (m0: -nat).(\lambda (_: (le n m0)).(\lambda (H1: (le (pred n) (pred m0))).(le_trans -(pred n) (pred m0) m0 H1 (le_pred_n m0))))) m H))). -(* COMMENTS -Initial nodes: 71 -END *) - -theorem minus_le: - \forall (x: nat).(\forall (y: nat).(le (minus x y) x)) -\def - \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).(le (minus n -y) n))) (\lambda (_: nat).(le_n O)) (\lambda (n: nat).(\lambda (H: ((\forall -(y: nat).(le (minus n y) n)))).(\lambda (y: nat).(nat_ind (\lambda (n0: -nat).(le (minus (S n) n0) (S n))) (le_n (S n)) (\lambda (n0: nat).(\lambda -(_: (le (match n0 with [O \Rightarrow (S n) | (S l) \Rightarrow (minus n l)]) -(S n))).(le_S (minus n n0) n (H n0)))) y)))) x). -(* COMMENTS -Initial nodes: 101 -END *) - -theorem le_plus_minus_sym: - \forall (n: nat).(\forall (m: nat).((le n m) \to (eq nat m (plus (minus m n) -n)))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(eq_ind_r nat -(plus n (minus m n)) (\lambda (n0: nat).(eq nat m n0)) (le_plus_minus n m H) -(plus (minus m n) n) (plus_sym (minus m n) n)))). -(* COMMENTS -Initial nodes: 61 -END *) - -theorem le_minus_minus: - \forall (x: nat).(\forall (y: nat).((le x y) \to (\forall (z: nat).((le y z) -\to (le (minus y x) (minus z x)))))) -\def - \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (le x y)).(\lambda (z: -nat).(\lambda (H0: (le y z)).(simpl_le_plus_l x (minus y x) (minus z x) -(eq_ind_r nat y (\lambda (n: nat).(le n (plus x (minus z x)))) (eq_ind_r nat -z (\lambda (n: nat).(le y n)) H0 (plus x (minus z x)) (le_plus_minus_r x z -(le_trans x y z H H0))) (plus x (minus y x)) (le_plus_minus_r x y H))))))). -(* COMMENTS -Initial nodes: 117 -END *) - -theorem le_minus_plus: - \forall (z: nat).(\forall (x: nat).((le z x) \to (\forall (y: nat).(eq nat -(minus (plus x y) z) (plus (minus x z) y))))) -\def - \lambda (z: nat).(nat_ind (\lambda (n: nat).(\forall (x: nat).((le n x) \to -(\forall (y: nat).(eq nat (minus (plus x y) n) (plus (minus x n) y)))))) -(\lambda (x: nat).(\lambda (H: (le O x)).(let H0 \def (match H in le return -(\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat n x) \to (\forall (y: -nat).(eq nat (minus (plus x y) O) (plus (minus x O) y)))))) with [le_n -\Rightarrow (\lambda (H0: (eq nat O x)).(eq_ind nat O (\lambda (n: -nat).(\forall (y: nat).(eq nat (minus (plus n y) O) (plus (minus n O) y)))) -(\lambda (y: nat).(sym_eq nat (plus (minus O O) y) (minus (plus O y) O) -(minus_n_O (plus O y)))) x H0)) | (le_S m H0) \Rightarrow (\lambda (H1: (eq -nat (S m) x)).(eq_ind nat (S m) (\lambda (n: nat).((le O m) \to (\forall (y: -nat).(eq nat (minus (plus n y) O) (plus (minus n O) y))))) (\lambda (_: (le O -m)).(\lambda (y: nat).(refl_equal nat (plus (minus (S m) O) y)))) x H1 H0))]) -in (H0 (refl_equal nat x))))) (\lambda (z0: nat).(\lambda (H: ((\forall (x: -nat).((le z0 x) \to (\forall (y: nat).(eq nat (minus (plus x y) z0) (plus -(minus x z0) y))))))).(\lambda (x: nat).(nat_ind (\lambda (n: nat).((le (S -z0) n) \to (\forall (y: nat).(eq nat (minus (plus n y) (S z0)) (plus (minus n -(S z0)) y))))) (\lambda (H0: (le (S z0) O)).(\lambda (y: nat).(let H1 \def -(match H0 in le return (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat n O) -\to (eq nat (minus (plus O y) (S z0)) (plus (minus O (S z0)) y))))) with -[le_n \Rightarrow (\lambda (H1: (eq nat (S z0) O)).(let H2 \def (eq_ind nat -(S z0) (\lambda (e: nat).(match e in nat return (\lambda (_: nat).Prop) with -[O \Rightarrow False | (S _) \Rightarrow True])) I O H1) in (False_ind (eq -nat (minus (plus O y) (S z0)) (plus (minus O (S z0)) y)) H2))) | (le_S m H1) -\Rightarrow (\lambda (H2: (eq nat (S m) O)).((let H3 \def (eq_ind nat (S m) -(\lambda (e: nat).(match e in nat return (\lambda (_: nat).Prop) with [O -\Rightarrow False | (S _) \Rightarrow True])) I O H2) in (False_ind ((le (S -z0) m) \to (eq nat (minus (plus O y) (S z0)) (plus (minus O (S z0)) y))) H3)) -H1))]) in (H1 (refl_equal nat O))))) (\lambda (n: nat).(\lambda (_: (((le (S -z0) n) \to (\forall (y: nat).(eq nat (minus (plus n y) (S z0)) (plus (minus n -(S z0)) y)))))).(\lambda (H1: (le (S z0) (S n))).(\lambda (y: nat).(H n -(le_S_n z0 n H1) y))))) x)))) z). -(* COMMENTS -Initial nodes: 603 -END *) - -theorem le_minus: - \forall (x: nat).(\forall (z: nat).(\forall (y: nat).((le (plus x y) z) \to -(le x (minus z y))))) -\def - \lambda (x: nat).(\lambda (z: nat).(\lambda (y: nat).(\lambda (H: (le (plus -x y) z)).(eq_ind nat (minus (plus x y) y) (\lambda (n: nat).(le n (minus z -y))) (le_minus_minus y (plus x y) (le_plus_r x y) z H) x (minus_plus_r x -y))))). -(* COMMENTS -Initial nodes: 69 -END *) - -theorem le_trans_plus_r: - \forall (x: nat).(\forall (y: nat).(\forall (z: nat).((le (plus x y) z) \to -(le y z)))) -\def - \lambda (x: nat).(\lambda (y: nat).(\lambda (z: nat).(\lambda (H: (le (plus -x y) z)).(le_trans y (plus x y) z (le_plus_r x y) H)))). -(* COMMENTS -Initial nodes: 35 -END *) - -theorem lt_x_O: - \forall (x: nat).((lt x O) \to (\forall (P: Prop).P)) -\def - \lambda (x: nat).(\lambda (H: (le (S x) O)).(\lambda (P: Prop).(let H_y \def -(le_n_O_eq (S x) H) in (let H0 \def (eq_ind nat O (\lambda (ee: nat).(match -ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) -\Rightarrow False])) I (S x) H_y) in (False_ind P H0))))). -(* COMMENTS -Initial nodes: 48 -END *) - -theorem le_gen_S: - \forall (m: nat).(\forall (x: nat).((le (S m) x) \to (ex2 nat (\lambda (n: -nat).(eq nat x (S n))) (\lambda (n: nat).(le m n))))) -\def - \lambda (m: nat).(\lambda (x: nat).(\lambda (H: (le (S m) x)).(let H0 \def -(match H in le return (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat n x) -\to (ex2 nat (\lambda (n0: nat).(eq nat x (S n0))) (\lambda (n0: nat).(le m -n0)))))) with [le_n \Rightarrow (\lambda (H0: (eq nat (S m) x)).(eq_ind nat -(S m) (\lambda (n: nat).(ex2 nat (\lambda (n0: nat).(eq nat n (S n0))) -(\lambda (n0: nat).(le m n0)))) (ex_intro2 nat (\lambda (n: nat).(eq nat (S -m) (S n))) (\lambda (n: nat).(le m n)) m (refl_equal nat (S m)) (le_n m)) x -H0)) | (le_S m0 H0) \Rightarrow (\lambda (H1: (eq nat (S m0) x)).(eq_ind nat -(S m0) (\lambda (n: nat).((le (S m) m0) \to (ex2 nat (\lambda (n0: nat).(eq -nat n (S n0))) (\lambda (n0: nat).(le m n0))))) (\lambda (H2: (le (S m) -m0)).(ex_intro2 nat (\lambda (n: nat).(eq nat (S m0) (S n))) (\lambda (n: -nat).(le m n)) m0 (refl_equal nat (S m0)) (le_S_n m m0 (le_S (S m) m0 H2)))) -x H1 H0))]) in (H0 (refl_equal nat x))))). -(* COMMENTS -Initial nodes: 261 -END *) - -theorem lt_x_plus_x_Sy: - \forall (x: nat).(\forall (y: nat).(lt x (plus x (S y)))) -\def - \lambda (x: nat).(\lambda (y: nat).(eq_ind_r nat (plus (S y) x) (\lambda (n: -nat).(lt x n)) (le_S_n (S x) (S (plus y x)) (le_n_S (S x) (S (plus y x)) -(le_n_S x (plus y x) (le_plus_r y x)))) (plus x (S y)) (plus_sym x (S y)))). -(* COMMENTS -Initial nodes: 83 -END *) - -theorem simpl_lt_plus_r: - \forall (p: nat).(\forall (n: nat).(\forall (m: nat).((lt (plus n p) (plus m -p)) \to (lt n m)))) -\def - \lambda (p: nat).(\lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt (plus -n p) (plus m p))).(simpl_lt_plus_l n m p (let H0 \def (eq_ind nat (plus n p) -(\lambda (n0: nat).(lt n0 (plus m p))) H (plus p n) (plus_sym n p)) in (let -H1 \def (eq_ind nat (plus m p) (\lambda (n0: nat).(lt (plus p n) n0)) H0 -(plus p m) (plus_sym m p)) in H1)))))). -(* COMMENTS -Initial nodes: 101 -END *) - -theorem minus_x_Sy: - \forall (x: nat).(\forall (y: nat).((lt y x) \to (eq nat (minus x y) (S -(minus x (S y)))))) -\def - \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((lt y n) \to -(eq nat (minus n y) (S (minus n (S y))))))) (\lambda (y: nat).(\lambda (H: -(lt y O)).(let H0 \def (match H in le return (\lambda (n: nat).(\lambda (_: -(le ? n)).((eq nat n O) \to (eq nat (minus O y) (S (minus O (S y))))))) with -[le_n \Rightarrow (\lambda (H0: (eq nat (S y) O)).(let H1 \def (eq_ind nat (S -y) (\lambda (e: nat).(match e in nat return (\lambda (_: nat).Prop) with [O -\Rightarrow False | (S _) \Rightarrow True])) I O H0) in (False_ind (eq nat -(minus O y) (S (minus O (S y)))) H1))) | (le_S m H0) \Rightarrow (\lambda -(H1: (eq nat (S m) O)).((let H2 \def (eq_ind nat (S m) (\lambda (e: -nat).(match e in nat return (\lambda (_: nat).Prop) with [O \Rightarrow False -| (S _) \Rightarrow True])) I O H1) in (False_ind ((le (S y) m) \to (eq nat -(minus O y) (S (minus O (S y))))) H2)) H0))]) in (H0 (refl_equal nat O))))) -(\lambda (n: nat).(\lambda (H: ((\forall (y: nat).((lt y n) \to (eq nat -(minus n y) (S (minus n (S y)))))))).(\lambda (y: nat).(nat_ind (\lambda (n0: -nat).((lt n0 (S n)) \to (eq nat (minus (S n) n0) (S (minus (S n) (S n0)))))) -(\lambda (_: (lt O (S n))).(eq_ind nat n (\lambda (n0: nat).(eq nat (S n) (S -n0))) (refl_equal nat (S n)) (minus n O) (minus_n_O n))) (\lambda (n0: -nat).(\lambda (_: (((lt n0 (S n)) \to (eq nat (minus (S n) n0) (S (minus (S -n) (S n0))))))).(\lambda (H1: (lt (S n0) (S n))).(let H2 \def (le_S_n (S n0) -n H1) in (H n0 H2))))) y)))) x). -(* COMMENTS -Initial nodes: 383 -END *) - -theorem lt_plus_minus: - \forall (x: nat).(\forall (y: nat).((lt x y) \to (eq nat y (S (plus x (minus -y (S x))))))) -\def - \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(le_plus_minus (S -x) y H))). -(* COMMENTS -Initial nodes: 19 -END *) - -theorem lt_plus_minus_r: - \forall (x: nat).(\forall (y: nat).((lt x y) \to (eq nat y (S (plus (minus y -(S x)) x))))) -\def - \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(eq_ind_r nat -(plus x (minus y (S x))) (\lambda (n: nat).(eq nat y (S n))) (lt_plus_minus x -y H) (plus (minus y (S x)) x) (plus_sym (minus y (S x)) x)))). -(* COMMENTS -Initial nodes: 69 -END *) - -theorem minus_x_SO: - \forall (x: nat).((lt O x) \to (eq nat x (S (minus x (S O))))) -\def - \lambda (x: nat).(\lambda (H: (lt O x)).(eq_ind nat (minus x O) (\lambda (n: -nat).(eq nat x n)) (eq_ind nat x (\lambda (n: nat).(eq nat x n)) (refl_equal -nat x) (minus x O) (minus_n_O x)) (S (minus x (S O))) (minus_x_Sy x O H))). -(* COMMENTS -Initial nodes: 77 -END *) - -theorem le_x_pred_y: - \forall (y: nat).(\forall (x: nat).((lt x y) \to (le x (pred y)))) -\def - \lambda (y: nat).(nat_ind (\lambda (n: nat).(\forall (x: nat).((lt x n) \to -(le x (pred n))))) (\lambda (x: nat).(\lambda (H: (lt x O)).(let H0 \def -(match H in le return (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat n O) -\to (le x O)))) with [le_n \Rightarrow (\lambda (H0: (eq nat (S x) O)).(let -H1 \def (eq_ind nat (S x) (\lambda (e: nat).(match e in nat return (\lambda -(_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H0) -in (False_ind (le x O) H1))) | (le_S m H0) \Rightarrow (\lambda (H1: (eq nat -(S m) O)).((let H2 \def (eq_ind nat (S m) (\lambda (e: nat).(match e in nat -return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow -True])) I O H1) in (False_ind ((le (S x) m) \to (le x O)) H2)) H0))]) in (H0 -(refl_equal nat O))))) (\lambda (n: nat).(\lambda (_: ((\forall (x: nat).((lt -x n) \to (le x (pred n)))))).(\lambda (x: nat).(\lambda (H0: (lt x (S -n))).(le_S_n x n H0))))) y). -(* COMMENTS -Initial nodes: 189 -END *) - -theorem lt_le_minus: - \forall (x: nat).(\forall (y: nat).((lt x y) \to (le x (minus y (S O))))) -\def - \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(le_minus x y (S -O) (eq_ind_r nat (plus (S O) x) (\lambda (n: nat).(le n y)) H (plus x (S O)) -(plus_sym x (S O)))))). -(* COMMENTS -Initial nodes: 57 -END *) - -theorem lt_le_e: - \forall (n: nat).(\forall (d: nat).(\forall (P: Prop).((((lt n d) \to P)) -\to ((((le d n) \to P)) \to P)))) -\def - \lambda (n: nat).(\lambda (d: nat).(\lambda (P: Prop).(\lambda (H: (((lt n -d) \to P))).(\lambda (H0: (((le d n) \to P))).(let H1 \def (le_or_lt d n) in -(or_ind (le d n) (lt n d) P H0 H H1)))))). -(* COMMENTS -Initial nodes: 49 -END *) - -theorem lt_eq_e: - \forall (x: nat).(\forall (y: nat).(\forall (P: Prop).((((lt x y) \to P)) -\to ((((eq nat x y) \to P)) \to ((le x y) \to P))))) -\def - \lambda (x: nat).(\lambda (y: nat).(\lambda (P: Prop).(\lambda (H: (((lt x -y) \to P))).(\lambda (H0: (((eq nat x y) \to P))).(\lambda (H1: (le x -y)).(or_ind (lt x y) (eq nat x y) P H H0 (le_lt_or_eq x y H1))))))). -(* COMMENTS -Initial nodes: 59 -END *) - -theorem lt_eq_gt_e: - \forall (x: nat).(\forall (y: nat).(\forall (P: Prop).((((lt x y) \to P)) -\to ((((eq nat x y) \to P)) \to ((((lt y x) \to P)) \to P))))) -\def - \lambda (x: nat).(\lambda (y: nat).(\lambda (P: Prop).(\lambda (H: (((lt x -y) \to P))).(\lambda (H0: (((eq nat x y) \to P))).(\lambda (H1: (((lt y x) -\to P))).(lt_le_e x y P H (\lambda (H2: (le y x)).(lt_eq_e y x P H1 (\lambda -(H3: (eq nat y x)).(H0 (sym_eq nat y x H3))) H2)))))))). -(* COMMENTS -Initial nodes: 79 -END *) - -theorem lt_gen_xS: - \forall (x: nat).(\forall (n: nat).((lt x (S n)) \to (or (eq nat x O) (ex2 -nat (\lambda (m: nat).(eq nat x (S m))) (\lambda (m: nat).(lt m n)))))) -\def - \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).((lt n (S -n0)) \to (or (eq nat n O) (ex2 nat (\lambda (m: nat).(eq nat n (S m))) -(\lambda (m: nat).(lt m n0))))))) (\lambda (n: nat).(\lambda (_: (lt O (S -n))).(or_introl (eq nat O O) (ex2 nat (\lambda (m: nat).(eq nat O (S m))) -(\lambda (m: nat).(lt m n))) (refl_equal nat O)))) (\lambda (n: nat).(\lambda -(_: ((\forall (n0: nat).((lt n (S n0)) \to (or (eq nat n O) (ex2 nat (\lambda -(m: nat).(eq nat n (S m))) (\lambda (m: nat).(lt m n0)))))))).(\lambda (n0: -nat).(\lambda (H0: (lt (S n) (S n0))).(or_intror (eq nat (S n) O) (ex2 nat -(\lambda (m: nat).(eq nat (S n) (S m))) (\lambda (m: nat).(lt m n0))) -(ex_intro2 nat (\lambda (m: nat).(eq nat (S n) (S m))) (\lambda (m: nat).(lt -m n0)) n (refl_equal nat (S n)) (le_S_n (S n) n0 H0))))))) x). -(* COMMENTS -Initial nodes: 243 -END *) - -theorem le_lt_false: - \forall (x: nat).(\forall (y: nat).((le x y) \to ((lt y x) \to (\forall (P: -Prop).P)))) -\def - \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (le x y)).(\lambda (H0: (lt -y x)).(\lambda (P: Prop).(False_ind P (le_not_lt x y H H0)))))). -(* COMMENTS -Initial nodes: 31 -END *) - -theorem lt_neq: - \forall (x: nat).(\forall (y: nat).((lt x y) \to (not (eq nat x y)))) -\def - \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(\lambda (H0: (eq -nat x y)).(let H1 \def (eq_ind nat x (\lambda (n: nat).(lt n y)) H y H0) in -(lt_n_n y H1))))). -(* COMMENTS -Initial nodes: 43 -END *) - -theorem arith0: - \forall (h2: nat).(\forall (d2: nat).(\forall (n: nat).((le (plus d2 h2) n) -\to (\forall (h1: nat).(le (plus d2 h1) (minus (plus n h1) h2)))))) -\def - \lambda (h2: nat).(\lambda (d2: nat).(\lambda (n: nat).(\lambda (H: (le -(plus d2 h2) n)).(\lambda (h1: nat).(eq_ind nat (minus (plus h2 (plus d2 h1)) -h2) (\lambda (n0: nat).(le n0 (minus (plus n h1) h2))) (le_minus_minus h2 -(plus h2 (plus d2 h1)) (le_plus_l h2 (plus d2 h1)) (plus n h1) (eq_ind_r nat -(plus (plus h2 d2) h1) (\lambda (n0: nat).(le n0 (plus n h1))) (eq_ind_r nat -(plus d2 h2) (\lambda (n0: nat).(le (plus n0 h1) (plus n h1))) (le_S_n (plus -(plus d2 h2) h1) (plus n h1) (le_n_S (plus (plus d2 h2) h1) (plus n h1) -(le_plus_plus (plus d2 h2) n h1 h1 H (le_n h1)))) (plus h2 d2) (plus_sym h2 -d2)) (plus h2 (plus d2 h1)) (plus_assoc_l h2 d2 h1))) (plus d2 h1) -(minus_plus h2 (plus d2 h1))))))). -(* COMMENTS -Initial nodes: 235 -END *) - -theorem O_minus: - \forall (x: nat).(\forall (y: nat).((le x y) \to (eq nat (minus x y) O))) -\def - \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((le n y) \to -(eq nat (minus n y) O)))) (\lambda (y: nat).(\lambda (_: (le O -y)).(refl_equal nat O))) (\lambda (x0: nat).(\lambda (H: ((\forall (y: -nat).((le x0 y) \to (eq nat (minus x0 y) O))))).(\lambda (y: nat).(nat_ind -(\lambda (n: nat).((le (S x0) n) \to (eq nat (match n with [O \Rightarrow (S -x0) | (S l) \Rightarrow (minus x0 l)]) O))) (\lambda (H0: (le (S x0) -O)).(ex2_ind nat (\lambda (n: nat).(eq nat O (S n))) (\lambda (n: nat).(le x0 -n)) (eq nat (S x0) O) (\lambda (x1: nat).(\lambda (H1: (eq nat O (S -x1))).(\lambda (_: (le x0 x1)).(let H3 \def (eq_ind nat O (\lambda (ee: -nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True -| (S _) \Rightarrow False])) I (S x1) H1) in (False_ind (eq nat (S x0) O) -H3))))) (le_gen_S x0 O H0))) (\lambda (n: nat).(\lambda (_: (((le (S x0) n) -\to (eq nat (match n with [O \Rightarrow (S x0) | (S l) \Rightarrow (minus x0 -l)]) O)))).(\lambda (H1: (le (S x0) (S n))).(H n (le_S_n x0 n H1))))) y)))) -x). -(* COMMENTS -Initial nodes: 252 -END *) - -theorem minus_minus: - \forall (z: nat).(\forall (x: nat).(\forall (y: nat).((le z x) \to ((le z y) -\to ((eq nat (minus x z) (minus y z)) \to (eq nat x y)))))) -\def - \lambda (z: nat).(nat_ind (\lambda (n: nat).(\forall (x: nat).(\forall (y: -nat).((le n x) \to ((le n y) \to ((eq nat (minus x n) (minus y n)) \to (eq -nat x y))))))) (\lambda (x: nat).(\lambda (y: nat).(\lambda (_: (le O -x)).(\lambda (_: (le O y)).(\lambda (H1: (eq nat (minus x O) (minus y -O))).(let H2 \def (eq_ind_r nat (minus x O) (\lambda (n: nat).(eq nat n -(minus y O))) H1 x (minus_n_O x)) in (let H3 \def (eq_ind_r nat (minus y O) -(\lambda (n: nat).(eq nat x n)) H2 y (minus_n_O y)) in H3))))))) (\lambda -(z0: nat).(\lambda (IH: ((\forall (x: nat).(\forall (y: nat).((le z0 x) \to -((le z0 y) \to ((eq nat (minus x z0) (minus y z0)) \to (eq nat x -y)))))))).(\lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((le -(S z0) n) \to ((le (S z0) y) \to ((eq nat (minus n (S z0)) (minus y (S z0))) -\to (eq nat n y)))))) (\lambda (y: nat).(\lambda (H: (le (S z0) O)).(\lambda -(_: (le (S z0) y)).(\lambda (_: (eq nat (minus O (S z0)) (minus y (S -z0)))).(ex2_ind nat (\lambda (n: nat).(eq nat O (S n))) (\lambda (n: nat).(le -z0 n)) (eq nat O y) (\lambda (x0: nat).(\lambda (H2: (eq nat O (S -x0))).(\lambda (_: (le z0 x0)).(let H4 \def (eq_ind nat O (\lambda (ee: -nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True -| (S _) \Rightarrow False])) I (S x0) H2) in (False_ind (eq nat O y) H4))))) -(le_gen_S z0 O H)))))) (\lambda (x0: nat).(\lambda (_: ((\forall (y: -nat).((le (S z0) x0) \to ((le (S z0) y) \to ((eq nat (minus x0 (S z0)) (minus -y (S z0))) \to (eq nat x0 y))))))).(\lambda (y: nat).(nat_ind (\lambda (n: -nat).((le (S z0) (S x0)) \to ((le (S z0) n) \to ((eq nat (minus (S x0) (S -z0)) (minus n (S z0))) \to (eq nat (S x0) n))))) (\lambda (H: (le (S z0) (S -x0))).(\lambda (H0: (le (S z0) O)).(\lambda (_: (eq nat (minus (S x0) (S z0)) -(minus O (S z0)))).(let H_y \def (le_S_n z0 x0 H) in (ex2_ind nat (\lambda -(n: nat).(eq nat O (S n))) (\lambda (n: nat).(le z0 n)) (eq nat (S x0) O) -(\lambda (x1: nat).(\lambda (H2: (eq nat O (S x1))).(\lambda (_: (le z0 -x1)).(let H4 \def (eq_ind nat O (\lambda (ee: nat).(match ee in nat return -(\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) -I (S x1) H2) in (False_ind (eq nat (S x0) O) H4))))) (le_gen_S z0 O H0)))))) -(\lambda (y0: nat).(\lambda (_: (((le (S z0) (S x0)) \to ((le (S z0) y0) \to -((eq nat (minus (S x0) (S z0)) (minus y0 (S z0))) \to (eq nat (S x0) -y0)))))).(\lambda (H: (le (S z0) (S x0))).(\lambda (H0: (le (S z0) (S -y0))).(\lambda (H1: (eq nat (minus (S x0) (S z0)) (minus (S y0) (S -z0)))).(f_equal nat nat S x0 y0 (IH x0 y0 (le_S_n z0 x0 H) (le_S_n z0 y0 H0) -H1))))))) y)))) x)))) z). -(* COMMENTS -Initial nodes: 751 -END *) - -theorem plus_plus: - \forall (z: nat).(\forall (x1: nat).(\forall (x2: nat).(\forall (y1: -nat).(\forall (y2: nat).((le x1 z) \to ((le x2 z) \to ((eq nat (plus (minus z -x1) y1) (plus (minus z x2) y2)) \to (eq nat (plus x1 y2) (plus x2 y1))))))))) -\def - \lambda (z: nat).(nat_ind (\lambda (n: nat).(\forall (x1: nat).(\forall (x2: -nat).(\forall (y1: nat).(\forall (y2: nat).((le x1 n) \to ((le x2 n) \to ((eq -nat (plus (minus n x1) y1) (plus (minus n x2) y2)) \to (eq nat (plus x1 y2) -(plus x2 y1)))))))))) (\lambda (x1: nat).(\lambda (x2: nat).(\lambda (y1: -nat).(\lambda (y2: nat).(\lambda (H: (le x1 O)).(\lambda (H0: (le x2 -O)).(\lambda (H1: (eq nat y1 y2)).(eq_ind nat y1 (\lambda (n: nat).(eq nat -(plus x1 n) (plus x2 y1))) (let H_y \def (le_n_O_eq x2 H0) in (eq_ind nat O -(\lambda (n: nat).(eq nat (plus x1 y1) (plus n y1))) (let H_y0 \def -(le_n_O_eq x1 H) in (eq_ind nat O (\lambda (n: nat).(eq nat (plus n y1) (plus -O y1))) (refl_equal nat (plus O y1)) x1 H_y0)) x2 H_y)) y2 H1)))))))) -(\lambda (z0: nat).(\lambda (IH: ((\forall (x1: nat).(\forall (x2: -nat).(\forall (y1: nat).(\forall (y2: nat).((le x1 z0) \to ((le x2 z0) \to -((eq nat (plus (minus z0 x1) y1) (plus (minus z0 x2) y2)) \to (eq nat (plus -x1 y2) (plus x2 y1))))))))))).(\lambda (x1: nat).(nat_ind (\lambda (n: -nat).(\forall (x2: nat).(\forall (y1: nat).(\forall (y2: nat).((le n (S z0)) -\to ((le x2 (S z0)) \to ((eq nat (plus (minus (S z0) n) y1) (plus (minus (S -z0) x2) y2)) \to (eq nat (plus n y2) (plus x2 y1))))))))) (\lambda (x2: -nat).(nat_ind (\lambda (n: nat).(\forall (y1: nat).(\forall (y2: nat).((le O -(S z0)) \to ((le n (S z0)) \to ((eq nat (plus (minus (S z0) O) y1) (plus -(minus (S z0) n) y2)) \to (eq nat (plus O y2) (plus n y1)))))))) (\lambda -(y1: nat).(\lambda (y2: nat).(\lambda (_: (le O (S z0))).(\lambda (_: (le O -(S z0))).(\lambda (H1: (eq nat (S (plus z0 y1)) (S (plus z0 y2)))).(let H_y -\def (IH O O) in (let H2 \def (eq_ind_r nat (minus z0 O) (\lambda (n: -nat).(\forall (y3: nat).(\forall (y4: nat).((le O z0) \to ((le O z0) \to ((eq -nat (plus n y3) (plus n y4)) \to (eq nat y4 y3))))))) H_y z0 (minus_n_O z0)) -in (H2 y1 y2 (le_O_n z0) (le_O_n z0) (eq_add_S (plus z0 y1) (plus z0 y2) -H1))))))))) (\lambda (x3: nat).(\lambda (_: ((\forall (y1: nat).(\forall (y2: -nat).((le O (S z0)) \to ((le x3 (S z0)) \to ((eq nat (S (plus z0 y1)) (plus -(match x3 with [O \Rightarrow (S z0) | (S l) \Rightarrow (minus z0 l)]) y2)) -\to (eq nat y2 (plus x3 y1))))))))).(\lambda (y1: nat).(\lambda (y2: -nat).(\lambda (_: (le O (S z0))).(\lambda (H0: (le (S x3) (S z0))).(\lambda -(H1: (eq nat (S (plus z0 y1)) (plus (minus z0 x3) y2))).(let H_y \def (IH O -x3 (S y1)) in (let H2 \def (eq_ind_r nat (minus z0 O) (\lambda (n: -nat).(\forall (y3: nat).((le O z0) \to ((le x3 z0) \to ((eq nat (plus n (S -y1)) (plus (minus z0 x3) y3)) \to (eq nat y3 (plus x3 (S y1)))))))) H_y z0 -(minus_n_O z0)) in (let H3 \def (eq_ind_r nat (plus z0 (S y1)) (\lambda (n: -nat).(\forall (y3: nat).((le O z0) \to ((le x3 z0) \to ((eq nat n (plus -(minus z0 x3) y3)) \to (eq nat y3 (plus x3 (S y1)))))))) H2 (S (plus z0 y1)) -(plus_n_Sm z0 y1)) in (let H4 \def (eq_ind_r nat (plus x3 (S y1)) (\lambda -(n: nat).(\forall (y3: nat).((le O z0) \to ((le x3 z0) \to ((eq nat (S (plus -z0 y1)) (plus (minus z0 x3) y3)) \to (eq nat y3 n)))))) H3 (S (plus x3 y1)) -(plus_n_Sm x3 y1)) in (H4 y2 (le_O_n z0) (le_S_n x3 z0 H0) H1)))))))))))) -x2)) (\lambda (x2: nat).(\lambda (_: ((\forall (x3: nat).(\forall (y1: -nat).(\forall (y2: nat).((le x2 (S z0)) \to ((le x3 (S z0)) \to ((eq nat -(plus (minus (S z0) x2) y1) (plus (minus (S z0) x3) y2)) \to (eq nat (plus x2 -y2) (plus x3 y1)))))))))).(\lambda (x3: nat).(nat_ind (\lambda (n: -nat).(\forall (y1: nat).(\forall (y2: nat).((le (S x2) (S z0)) \to ((le n (S -z0)) \to ((eq nat (plus (minus (S z0) (S x2)) y1) (plus (minus (S z0) n) y2)) -\to (eq nat (plus (S x2) y2) (plus n y1)))))))) (\lambda (y1: nat).(\lambda -(y2: nat).(\lambda (H: (le (S x2) (S z0))).(\lambda (_: (le O (S -z0))).(\lambda (H1: (eq nat (plus (minus z0 x2) y1) (S (plus z0 y2)))).(let -H_y \def (IH x2 O y1 (S y2)) in (let H2 \def (eq_ind_r nat (minus z0 O) -(\lambda (n: nat).((le x2 z0) \to ((le O z0) \to ((eq nat (plus (minus z0 x2) -y1) (plus n (S y2))) \to (eq nat (plus x2 (S y2)) y1))))) H_y z0 (minus_n_O -z0)) in (let H3 \def (eq_ind_r nat (plus z0 (S y2)) (\lambda (n: nat).((le x2 -z0) \to ((le O z0) \to ((eq nat (plus (minus z0 x2) y1) n) \to (eq nat (plus -x2 (S y2)) y1))))) H2 (S (plus z0 y2)) (plus_n_Sm z0 y2)) in (let H4 \def -(eq_ind_r nat (plus x2 (S y2)) (\lambda (n: nat).((le x2 z0) \to ((le O z0) -\to ((eq nat (plus (minus z0 x2) y1) (S (plus z0 y2))) \to (eq nat n y1))))) -H3 (S (plus x2 y2)) (plus_n_Sm x2 y2)) in (H4 (le_S_n x2 z0 H) (le_O_n z0) -H1)))))))))) (\lambda (x4: nat).(\lambda (_: ((\forall (y1: nat).(\forall -(y2: nat).((le (S x2) (S z0)) \to ((le x4 (S z0)) \to ((eq nat (plus (minus -z0 x2) y1) (plus (match x4 with [O \Rightarrow (S z0) | (S l) \Rightarrow -(minus z0 l)]) y2)) \to (eq nat (S (plus x2 y2)) (plus x4 -y1))))))))).(\lambda (y1: nat).(\lambda (y2: nat).(\lambda (H: (le (S x2) (S -z0))).(\lambda (H0: (le (S x4) (S z0))).(\lambda (H1: (eq nat (plus (minus z0 -x2) y1) (plus (minus z0 x4) y2))).(f_equal nat nat S (plus x2 y2) (plus x4 -y1) (IH x2 x4 y1 y2 (le_S_n x2 z0 H) (le_S_n x4 z0 H0) H1))))))))) x3)))) -x1)))) z). -(* COMMENTS -Initial nodes: 1495 -END *) - -theorem le_S_minus: - \forall (d: nat).(\forall (h: nat).(\forall (n: nat).((le (plus d h) n) \to -(le d (S (minus n h)))))) -\def - \lambda (d: nat).(\lambda (h: nat).(\lambda (n: nat).(\lambda (H: (le (plus -d h) n)).(let H0 \def (le_trans d (plus d h) n (le_plus_l d h) H) in (let H1 -\def (eq_ind nat n (\lambda (n0: nat).(le d n0)) H0 (plus (minus n h) h) -(le_plus_minus_sym h n (le_trans h (plus d h) n (le_plus_r d h) H))) in (le_S -d (minus n h) (le_minus d n h H))))))). -(* COMMENTS -Initial nodes: 107 -END *) - -theorem lt_x_pred_y: - \forall (x: nat).(\forall (y: nat).((lt x (pred y)) \to (lt (S x) y))) -\def - \lambda (x: nat).(\lambda (y: nat).(nat_ind (\lambda (n: nat).((lt x (pred -n)) \to (lt (S x) n))) (\lambda (H: (lt x O)).(lt_x_O x H (lt (S x) O))) -(\lambda (n: nat).(\lambda (_: (((lt x (pred n)) \to (lt (S x) n)))).(\lambda -(H0: (lt x n)).(le_S_n (S (S x)) (S n) (le_n_S (S (S x)) (S n) (le_n_S (S x) -n H0)))))) y)). -(* COMMENTS -Initial nodes: 103 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Ground-1/ext/tactics.ma b/matita/matita/contribs/LAMBDA-TYPES/Ground-1/ext/tactics.ma deleted file mode 100644 index 766db9f1f..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Ground-1/ext/tactics.ma +++ /dev/null @@ -1,50 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Ground-1/preamble.ma". - -theorem insert_eq: - \forall (S: Set).(\forall (x: S).(\forall (P: ((S \to Prop))).(\forall (G: -((S \to Prop))).(((\forall (y: S).((P y) \to ((eq S y x) \to (G y))))) \to -((P x) \to (G x)))))) -\def - \lambda (S: Set).(\lambda (x: S).(\lambda (P: ((S \to Prop))).(\lambda (G: -((S \to Prop))).(\lambda (H: ((\forall (y: S).((P y) \to ((eq S y x) \to (G -y)))))).(\lambda (H0: (P x)).(H x H0 (refl_equal S x))))))). -(* COMMENTS -Initial nodes: 45 -END *) - -theorem unintro: - \forall (A: Set).(\forall (a: A).(\forall (P: ((A \to Prop))).(((\forall (x: -A).(P x))) \to (P a)))) -\def - \lambda (A: Set).(\lambda (a: A).(\lambda (P: ((A \to Prop))).(\lambda (H: -((\forall (x: A).(P x)))).(H a)))). -(* COMMENTS -Initial nodes: 17 -END *) - -theorem xinduction: - \forall (A: Set).(\forall (t: A).(\forall (P: ((A \to Prop))).(((\forall (x: -A).((eq A t x) \to (P x)))) \to (P t)))) -\def - \lambda (A: Set).(\lambda (t: A).(\lambda (P: ((A \to Prop))).(\lambda (H: -((\forall (x: A).((eq A t x) \to (P x))))).(H t (refl_equal A t))))). -(* COMMENTS -Initial nodes: 31 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Ground-1/plist/defs.ma b/matita/matita/contribs/LAMBDA-TYPES/Ground-1/plist/defs.ma deleted file mode 100644 index 8f9c1d3cc..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Ground-1/plist/defs.ma +++ /dev/null @@ -1,43 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Ground-1/preamble.ma". - -inductive PList: Set \def -| PNil: PList -| PCons: nat \to (nat \to (PList \to PList)). - -definition PConsTail: - PList \to (nat \to (nat \to PList)) -\def - let rec PConsTail (hds: PList) on hds: (nat \to (nat \to PList)) \def -(\lambda (h0: nat).(\lambda (d0: nat).(match hds with [PNil \Rightarrow -(PCons h0 d0 PNil) | (PCons h d hds0) \Rightarrow (PCons h d (PConsTail hds0 -h0 d0))]))) in PConsTail. - -definition Ss: - PList \to PList -\def - let rec Ss (hds: PList) on hds: PList \def (match hds with [PNil \Rightarrow -PNil | (PCons h d hds0) \Rightarrow (PCons h (S d) (Ss hds0))]) in Ss. - -definition papp: - PList \to (PList \to PList) -\def - let rec papp (a: PList) on a: (PList \to PList) \def (\lambda (b: -PList).(match a with [PNil \Rightarrow b | (PCons h d a0) \Rightarrow (PCons -h d (papp a0 b))])) in papp. - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Ground-1/plist/props.ma b/matita/matita/contribs/LAMBDA-TYPES/Ground-1/plist/props.ma deleted file mode 100644 index 990397229..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Ground-1/plist/props.ma +++ /dev/null @@ -1,34 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Ground-1/plist/defs.ma". - -theorem papp_ss: - \forall (is1: PList).(\forall (is2: PList).(eq PList (papp (Ss is1) (Ss -is2)) (Ss (papp is1 is2)))) -\def - \lambda (is1: PList).(PList_ind (\lambda (p: PList).(\forall (is2: -PList).(eq PList (papp (Ss p) (Ss is2)) (Ss (papp p is2))))) (\lambda (is2: -PList).(refl_equal PList (Ss is2))) (\lambda (n: nat).(\lambda (n0: -nat).(\lambda (p: PList).(\lambda (H: ((\forall (is2: PList).(eq PList (papp -(Ss p) (Ss is2)) (Ss (papp p is2)))))).(\lambda (is2: PList).(eq_ind_r PList -(Ss (papp p is2)) (\lambda (p0: PList).(eq PList (PCons n (S n0) p0) (PCons n -(S n0) (Ss (papp p is2))))) (refl_equal PList (PCons n (S n0) (Ss (papp p -is2)))) (papp (Ss p) (Ss is2)) (H is2))))))) is1). -(* COMMENTS -Initial nodes: 151 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Ground-1/preamble.ma b/matita/matita/contribs/LAMBDA-TYPES/Ground-1/preamble.ma deleted file mode 100644 index 16ff2dc44..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Ground-1/preamble.ma +++ /dev/null @@ -1,15 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "Legacy-1/theory.ma". diff --git a/matita/matita/contribs/LAMBDA-TYPES/Ground-1/spare.ma b/matita/matita/contribs/LAMBDA-TYPES/Ground-1/spare.ma deleted file mode 100644 index a966c2ae2..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Ground-1/spare.ma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Ground-1/theory.ma". - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Ground-1/theory.ma b/matita/matita/contribs/LAMBDA-TYPES/Ground-1/theory.ma deleted file mode 100644 index 3b59dc6e8..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Ground-1/theory.ma +++ /dev/null @@ -1,26 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Ground-1/ext/tactics.ma". - -include "Ground-1/ext/arith.ma". - -include "Ground-1/types/props.ma". - -include "Ground-1/blt/props.ma". - -include "Ground-1/plist/props.ma". - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Ground-1/types/defs.ma b/matita/matita/contribs/LAMBDA-TYPES/Ground-1/types/defs.ma deleted file mode 100644 index f94969d88..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Ground-1/types/defs.ma +++ /dev/null @@ -1,172 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Ground-1/preamble.ma". - -inductive and3 (P0: Prop) (P1: Prop) (P2: Prop): Prop \def -| and3_intro: P0 \to (P1 \to (P2 \to (and3 P0 P1 P2))). - -inductive and4 (P0: Prop) (P1: Prop) (P2: Prop) (P3: Prop): Prop \def -| and4_intro: P0 \to (P1 \to (P2 \to (P3 \to (and4 P0 P1 P2 P3)))). - -inductive and5 (P0: Prop) (P1: Prop) (P2: Prop) (P3: Prop) (P4: Prop): Prop -\def -| and5_intro: P0 \to (P1 \to (P2 \to (P3 \to (P4 \to (and5 P0 P1 P2 P3 -P4))))). - -inductive or3 (P0: Prop) (P1: Prop) (P2: Prop): Prop \def -| or3_intro0: P0 \to (or3 P0 P1 P2) -| or3_intro1: P1 \to (or3 P0 P1 P2) -| or3_intro2: P2 \to (or3 P0 P1 P2). - -inductive or4 (P0: Prop) (P1: Prop) (P2: Prop) (P3: Prop): Prop \def -| or4_intro0: P0 \to (or4 P0 P1 P2 P3) -| or4_intro1: P1 \to (or4 P0 P1 P2 P3) -| or4_intro2: P2 \to (or4 P0 P1 P2 P3) -| or4_intro3: P3 \to (or4 P0 P1 P2 P3). - -inductive or5 (P0: Prop) (P1: Prop) (P2: Prop) (P3: Prop) (P4: Prop): Prop -\def -| or5_intro0: P0 \to (or5 P0 P1 P2 P3 P4) -| or5_intro1: P1 \to (or5 P0 P1 P2 P3 P4) -| or5_intro2: P2 \to (or5 P0 P1 P2 P3 P4) -| or5_intro3: P3 \to (or5 P0 P1 P2 P3 P4) -| or5_intro4: P4 \to (or5 P0 P1 P2 P3 P4). - -inductive ex3 (A0: Set) (P0: A0 \to Prop) (P1: A0 \to Prop) (P2: A0 \to -Prop): Prop \def -| ex3_intro: \forall (x0: A0).((P0 x0) \to ((P1 x0) \to ((P2 x0) \to (ex3 A0 -P0 P1 P2)))). - -inductive ex4 (A0: Set) (P0: A0 \to Prop) (P1: A0 \to Prop) (P2: A0 \to Prop) -(P3: A0 \to Prop): Prop \def -| ex4_intro: \forall (x0: A0).((P0 x0) \to ((P1 x0) \to ((P2 x0) \to ((P3 x0) -\to (ex4 A0 P0 P1 P2 P3))))). - -inductive ex_2 (A0: Set) (A1: Set) (P0: A0 \to (A1 \to Prop)): Prop \def -| ex_2_intro: \forall (x0: A0).(\forall (x1: A1).((P0 x0 x1) \to (ex_2 A0 A1 -P0))). - -inductive ex2_2 (A0: Set) (A1: Set) (P0: A0 \to (A1 \to Prop)) (P1: A0 \to -(A1 \to Prop)): Prop \def -| ex2_2_intro: \forall (x0: A0).(\forall (x1: A1).((P0 x0 x1) \to ((P1 x0 x1) -\to (ex2_2 A0 A1 P0 P1)))). - -inductive ex3_2 (A0: Set) (A1: Set) (P0: A0 \to (A1 \to Prop)) (P1: A0 \to -(A1 \to Prop)) (P2: A0 \to (A1 \to Prop)): Prop \def -| ex3_2_intro: \forall (x0: A0).(\forall (x1: A1).((P0 x0 x1) \to ((P1 x0 x1) -\to ((P2 x0 x1) \to (ex3_2 A0 A1 P0 P1 P2))))). - -inductive ex4_2 (A0: Set) (A1: Set) (P0: A0 \to (A1 \to Prop)) (P1: A0 \to -(A1 \to Prop)) (P2: A0 \to (A1 \to Prop)) (P3: A0 \to (A1 \to Prop)): Prop -\def -| ex4_2_intro: \forall (x0: A0).(\forall (x1: A1).((P0 x0 x1) \to ((P1 x0 x1) -\to ((P2 x0 x1) \to ((P3 x0 x1) \to (ex4_2 A0 A1 P0 P1 P2 P3)))))). - -inductive ex_3 (A0: Set) (A1: Set) (A2: Set) (P0: A0 \to (A1 \to (A2 \to -Prop))): Prop \def -| ex_3_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).((P0 x0 x1 -x2) \to (ex_3 A0 A1 A2 P0)))). - -inductive ex2_3 (A0: Set) (A1: Set) (A2: Set) (P0: A0 \to (A1 \to (A2 \to -Prop))) (P1: A0 \to (A1 \to (A2 \to Prop))): Prop \def -| ex2_3_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).((P0 x0 -x1 x2) \to ((P1 x0 x1 x2) \to (ex2_3 A0 A1 A2 P0 P1))))). - -inductive ex3_3 (A0: Set) (A1: Set) (A2: Set) (P0: A0 \to (A1 \to (A2 \to -Prop))) (P1: A0 \to (A1 \to (A2 \to Prop))) (P2: A0 \to (A1 \to (A2 \to -Prop))): Prop \def -| ex3_3_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).((P0 x0 -x1 x2) \to ((P1 x0 x1 x2) \to ((P2 x0 x1 x2) \to (ex3_3 A0 A1 A2 P0 P1 -P2)))))). - -inductive ex4_3 (A0: Set) (A1: Set) (A2: Set) (P0: A0 \to (A1 \to (A2 \to -Prop))) (P1: A0 \to (A1 \to (A2 \to Prop))) (P2: A0 \to (A1 \to (A2 \to -Prop))) (P3: A0 \to (A1 \to (A2 \to Prop))): Prop \def -| ex4_3_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).((P0 x0 -x1 x2) \to ((P1 x0 x1 x2) \to ((P2 x0 x1 x2) \to ((P3 x0 x1 x2) \to (ex4_3 A0 -A1 A2 P0 P1 P2 P3))))))). - -inductive ex5_3 (A0: Set) (A1: Set) (A2: Set) (P0: A0 \to (A1 \to (A2 \to -Prop))) (P1: A0 \to (A1 \to (A2 \to Prop))) (P2: A0 \to (A1 \to (A2 \to -Prop))) (P3: A0 \to (A1 \to (A2 \to Prop))) (P4: A0 \to (A1 \to (A2 \to -Prop))): Prop \def -| ex5_3_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).((P0 x0 -x1 x2) \to ((P1 x0 x1 x2) \to ((P2 x0 x1 x2) \to ((P3 x0 x1 x2) \to ((P4 x0 -x1 x2) \to (ex5_3 A0 A1 A2 P0 P1 P2 P3 P4)))))))). - -inductive ex3_4 (A0: Set) (A1: Set) (A2: Set) (A3: Set) (P0: A0 \to (A1 \to -(A2 \to (A3 \to Prop)))) (P1: A0 \to (A1 \to (A2 \to (A3 \to Prop)))) (P2: A0 -\to (A1 \to (A2 \to (A3 \to Prop)))): Prop \def -| ex3_4_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall -(x3: A3).((P0 x0 x1 x2 x3) \to ((P1 x0 x1 x2 x3) \to ((P2 x0 x1 x2 x3) \to -(ex3_4 A0 A1 A2 A3 P0 P1 P2))))))). - -inductive ex4_4 (A0: Set) (A1: Set) (A2: Set) (A3: Set) (P0: A0 \to (A1 \to -(A2 \to (A3 \to Prop)))) (P1: A0 \to (A1 \to (A2 \to (A3 \to Prop)))) (P2: A0 -\to (A1 \to (A2 \to (A3 \to Prop)))) (P3: A0 \to (A1 \to (A2 \to (A3 \to -Prop)))): Prop \def -| ex4_4_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall -(x3: A3).((P0 x0 x1 x2 x3) \to ((P1 x0 x1 x2 x3) \to ((P2 x0 x1 x2 x3) \to -((P3 x0 x1 x2 x3) \to (ex4_4 A0 A1 A2 A3 P0 P1 P2 P3)))))))). - -inductive ex4_5 (A0: Set) (A1: Set) (A2: Set) (A3: Set) (A4: Set) (P0: A0 \to -(A1 \to (A2 \to (A3 \to (A4 \to Prop))))) (P1: A0 \to (A1 \to (A2 \to (A3 \to -(A4 \to Prop))))) (P2: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to Prop))))) (P3: -A0 \to (A1 \to (A2 \to (A3 \to (A4 \to Prop))))): Prop \def -| ex4_5_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall -(x3: A3).(\forall (x4: A4).((P0 x0 x1 x2 x3 x4) \to ((P1 x0 x1 x2 x3 x4) \to -((P2 x0 x1 x2 x3 x4) \to ((P3 x0 x1 x2 x3 x4) \to (ex4_5 A0 A1 A2 A3 A4 P0 P1 -P2 P3))))))))). - -inductive ex5_5 (A0: Set) (A1: Set) (A2: Set) (A3: Set) (A4: Set) (P0: A0 \to -(A1 \to (A2 \to (A3 \to (A4 \to Prop))))) (P1: A0 \to (A1 \to (A2 \to (A3 \to -(A4 \to Prop))))) (P2: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to Prop))))) (P3: -A0 \to (A1 \to (A2 \to (A3 \to (A4 \to Prop))))) (P4: A0 \to (A1 \to (A2 \to -(A3 \to (A4 \to Prop))))): Prop \def -| ex5_5_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall -(x3: A3).(\forall (x4: A4).((P0 x0 x1 x2 x3 x4) \to ((P1 x0 x1 x2 x3 x4) \to -((P2 x0 x1 x2 x3 x4) \to ((P3 x0 x1 x2 x3 x4) \to ((P4 x0 x1 x2 x3 x4) \to -(ex5_5 A0 A1 A2 A3 A4 P0 P1 P2 P3 P4)))))))))). - -inductive ex6_6 (A0: Set) (A1: Set) (A2: Set) (A3: Set) (A4: Set) (A5: Set) -(P0: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to Prop)))))) (P1: A0 \to -(A1 \to (A2 \to (A3 \to (A4 \to (A5 \to Prop)))))) (P2: A0 \to (A1 \to (A2 -\to (A3 \to (A4 \to (A5 \to Prop)))))) (P3: A0 \to (A1 \to (A2 \to (A3 \to -(A4 \to (A5 \to Prop)))))) (P4: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 -\to Prop)))))) (P5: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to -Prop)))))): Prop \def -| ex6_6_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall -(x3: A3).(\forall (x4: A4).(\forall (x5: A5).((P0 x0 x1 x2 x3 x4 x5) \to ((P1 -x0 x1 x2 x3 x4 x5) \to ((P2 x0 x1 x2 x3 x4 x5) \to ((P3 x0 x1 x2 x3 x4 x5) -\to ((P4 x0 x1 x2 x3 x4 x5) \to ((P5 x0 x1 x2 x3 x4 x5) \to (ex6_6 A0 A1 A2 -A3 A4 A5 P0 P1 P2 P3 P4 P5)))))))))))). - -inductive ex6_7 (A0: Set) (A1: Set) (A2: Set) (A3: Set) (A4: Set) (A5: Set) -(A6: Set) (P0: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 \to -Prop))))))) (P1: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 \to -Prop))))))) (P2: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 \to -Prop))))))) (P3: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 \to -Prop))))))) (P4: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 \to -Prop))))))) (P5: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 \to -Prop))))))): Prop \def -| ex6_7_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall -(x3: A3).(\forall (x4: A4).(\forall (x5: A5).(\forall (x6: A6).((P0 x0 x1 x2 -x3 x4 x5 x6) \to ((P1 x0 x1 x2 x3 x4 x5 x6) \to ((P2 x0 x1 x2 x3 x4 x5 x6) -\to ((P3 x0 x1 x2 x3 x4 x5 x6) \to ((P4 x0 x1 x2 x3 x4 x5 x6) \to ((P5 x0 x1 -x2 x3 x4 x5 x6) \to (ex6_7 A0 A1 A2 A3 A4 A5 A6 P0 P1 P2 P3 P4 -P5))))))))))))). - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Ground-1/types/props.ma b/matita/matita/contribs/LAMBDA-TYPES/Ground-1/types/props.ma deleted file mode 100644 index 9c326f44c..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Ground-1/types/props.ma +++ /dev/null @@ -1,33 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Ground-1/types/defs.ma". - -theorem ex2_sym: - \forall (A: Set).(\forall (P: ((A \to Prop))).(\forall (Q: ((A \to -Prop))).((ex2 A (\lambda (x: A).(P x)) (\lambda (x: A).(Q x))) \to (ex2 A -(\lambda (x: A).(Q x)) (\lambda (x: A).(P x)))))) -\def - \lambda (A: Set).(\lambda (P: ((A \to Prop))).(\lambda (Q: ((A \to -Prop))).(\lambda (H: (ex2 A (\lambda (x: A).(P x)) (\lambda (x: A).(Q -x)))).(ex2_ind A (\lambda (x: A).(P x)) (\lambda (x: A).(Q x)) (ex2 A -(\lambda (x: A).(Q x)) (\lambda (x: A).(P x))) (\lambda (x: A).(\lambda (H0: -(P x)).(\lambda (H1: (Q x)).(ex_intro2 A (\lambda (x0: A).(Q x0)) (\lambda -(x0: A).(P x0)) x H1 H0)))) H)))). -(* COMMENTS -Initial nodes: 91 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Legacy-1/coq/defs.ma b/matita/matita/contribs/LAMBDA-TYPES/Legacy-1/coq/defs.ma deleted file mode 100644 index 7d4696229..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Legacy-1/coq/defs.ma +++ /dev/null @@ -1,99 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Legacy-1/preamble.ma". - -inductive eq (A: Set) (x: A): A \to Prop \def -| refl_equal: eq A x x. - -inductive True: Prop \def -| I: True. - -inductive land (A: Prop) (B: Prop): Prop \def -| conj: A \to (B \to (land A B)). - -inductive or (A: Prop) (B: Prop): Prop \def -| or_introl: A \to (or A B) -| or_intror: B \to (or A B). - -inductive ex (A: Set) (P: A \to Prop): Prop \def -| ex_intro: \forall (x: A).((P x) \to (ex A P)). - -inductive ex2 (A: Set) (P: A \to Prop) (Q: A \to Prop): Prop \def -| ex_intro2: \forall (x: A).((P x) \to ((Q x) \to (ex2 A P Q))). - -definition not: - Prop \to Prop -\def - \lambda (A: Prop).(A \to False). - -inductive bool: Set \def -| true: bool -| false: bool. - -inductive nat: Set \def -| O: nat -| S: nat \to nat. - -inductive le (n: nat): nat \to Prop \def -| le_n: le n n -| le_S: \forall (m: nat).((le n m) \to (le n (S m))). - -definition lt: - nat \to (nat \to Prop) -\def - \lambda (n: nat).(\lambda (m: nat).(le (S n) m)). - -definition IsSucc: - nat \to Prop -\def - \lambda (n: nat).(match n with [O \Rightarrow False | (S _) \Rightarrow -True]). - -definition pred: - nat \to nat -\def - \lambda (n: nat).(match n with [O \Rightarrow O | (S u) \Rightarrow u]). - -definition plus: - nat \to (nat \to nat) -\def - let rec plus (n: nat) on n: (nat \to nat) \def (\lambda (m: nat).(match n -with [O \Rightarrow m | (S p) \Rightarrow (S (plus p m))])) in plus. - -definition minus: - nat \to (nat \to nat) -\def - let rec minus (n: nat) on n: (nat \to nat) \def (\lambda (m: nat).(match n -with [O \Rightarrow O | (S k) \Rightarrow (match m with [O \Rightarrow (S k) -| (S l) \Rightarrow (minus k l)])])) in minus. - -inductive Acc (A: Set) (R: A \to (A \to Prop)): A \to Prop \def -| Acc_intro: \forall (x: A).(((\forall (y: A).((R y x) \to (Acc A R y)))) \to -(Acc A R x)). - -definition well_founded: - \forall (A: Set).(((A \to (A \to Prop))) \to Prop) -\def - \lambda (A: Set).(\lambda (R: ((A \to (A \to Prop)))).(\forall (a: A).(Acc A -R a))). - -definition ltof: - \forall (A: Set).(((A \to nat)) \to (A \to (A \to Prop))) -\def - \lambda (A: Set).(\lambda (f: ((A \to nat))).(\lambda (a: A).(\lambda (b: -A).(lt (f a) (f b))))). - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Legacy-1/coq/props.ma b/matita/matita/contribs/LAMBDA-TYPES/Legacy-1/coq/props.ma deleted file mode 100644 index 0b9d97b42..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Legacy-1/coq/props.ma +++ /dev/null @@ -1,805 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Legacy-1/coq/defs.ma". - -theorem f_equal: - \forall (A: Set).(\forall (B: Set).(\forall (f: ((A \to B))).(\forall (x: -A).(\forall (y: A).((eq A x y) \to (eq B (f x) (f y))))))) -\def - \lambda (A: Set).(\lambda (B: Set).(\lambda (f: ((A \to B))).(\lambda (x: -A).(\lambda (y: A).(\lambda (H: (eq A x y)).(eq_ind A x (\lambda (a: A).(eq B -(f x) (f a))) (refl_equal B (f x)) y H)))))). -(* COMMENTS -Initial nodes: 51 -END *) - -theorem f_equal2: - \forall (A1: Set).(\forall (A2: Set).(\forall (B: Set).(\forall (f: ((A1 \to -(A2 \to B)))).(\forall (x1: A1).(\forall (y1: A1).(\forall (x2: A2).(\forall -(y2: A2).((eq A1 x1 y1) \to ((eq A2 x2 y2) \to (eq B (f x1 x2) (f y1 -y2))))))))))) -\def - \lambda (A1: Set).(\lambda (A2: Set).(\lambda (B: Set).(\lambda (f: ((A1 \to -(A2 \to B)))).(\lambda (x1: A1).(\lambda (y1: A1).(\lambda (x2: A2).(\lambda -(y2: A2).(\lambda (H: (eq A1 x1 y1)).(eq_ind A1 x1 (\lambda (a: A1).((eq A2 -x2 y2) \to (eq B (f x1 x2) (f a y2)))) (\lambda (H0: (eq A2 x2 y2)).(eq_ind -A2 x2 (\lambda (a: A2).(eq B (f x1 x2) (f x1 a))) (refl_equal B (f x1 x2)) y2 -H0)) y1 H))))))))). -(* COMMENTS -Initial nodes: 109 -END *) - -theorem f_equal3: - \forall (A1: Set).(\forall (A2: Set).(\forall (A3: Set).(\forall (B: -Set).(\forall (f: ((A1 \to (A2 \to (A3 \to B))))).(\forall (x1: A1).(\forall -(y1: A1).(\forall (x2: A2).(\forall (y2: A2).(\forall (x3: A3).(\forall (y3: -A3).((eq A1 x1 y1) \to ((eq A2 x2 y2) \to ((eq A3 x3 y3) \to (eq B (f x1 x2 -x3) (f y1 y2 y3))))))))))))))) -\def - \lambda (A1: Set).(\lambda (A2: Set).(\lambda (A3: Set).(\lambda (B: -Set).(\lambda (f: ((A1 \to (A2 \to (A3 \to B))))).(\lambda (x1: A1).(\lambda -(y1: A1).(\lambda (x2: A2).(\lambda (y2: A2).(\lambda (x3: A3).(\lambda (y3: -A3).(\lambda (H: (eq A1 x1 y1)).(eq_ind A1 x1 (\lambda (a: A1).((eq A2 x2 y2) -\to ((eq A3 x3 y3) \to (eq B (f x1 x2 x3) (f a y2 y3))))) (\lambda (H0: (eq -A2 x2 y2)).(eq_ind A2 x2 (\lambda (a: A2).((eq A3 x3 y3) \to (eq B (f x1 x2 -x3) (f x1 a y3)))) (\lambda (H1: (eq A3 x3 y3)).(eq_ind A3 x3 (\lambda (a: -A3).(eq B (f x1 x2 x3) (f x1 x2 a))) (refl_equal B (f x1 x2 x3)) y3 H1)) y2 -H0)) y1 H)))))))))))). -(* COMMENTS -Initial nodes: 183 -END *) - -theorem sym_eq: - \forall (A: Set).(\forall (x: A).(\forall (y: A).((eq A x y) \to (eq A y -x)))) -\def - \lambda (A: Set).(\lambda (x: A).(\lambda (y: A).(\lambda (H: (eq A x -y)).(eq_ind A x (\lambda (a: A).(eq A a x)) (refl_equal A x) y H)))). -(* COMMENTS -Initial nodes: 39 -END *) - -theorem eq_ind_r: - \forall (A: Set).(\forall (x: A).(\forall (P: ((A \to Prop))).((P x) \to -(\forall (y: A).((eq A y x) \to (P y)))))) -\def - \lambda (A: Set).(\lambda (x: A).(\lambda (P: ((A \to Prop))).(\lambda (H: -(P x)).(\lambda (y: A).(\lambda (H0: (eq A y x)).(match (sym_eq A y x H0) in -eq return (\lambda (a: A).(\lambda (_: (eq ? ? a)).(P a))) with [refl_equal -\Rightarrow H])))))). -(* COMMENTS -Initial nodes: 38 -END *) - -theorem trans_eq: - \forall (A: Set).(\forall (x: A).(\forall (y: A).(\forall (z: A).((eq A x y) -\to ((eq A y z) \to (eq A x z)))))) -\def - \lambda (A: Set).(\lambda (x: A).(\lambda (y: A).(\lambda (z: A).(\lambda -(H: (eq A x y)).(\lambda (H0: (eq A y z)).(eq_ind A y (\lambda (a: A).(eq A x -a)) H z H0)))))). -(* COMMENTS -Initial nodes: 45 -END *) - -theorem sym_not_eq: - \forall (A: Set).(\forall (x: A).(\forall (y: A).((not (eq A x y)) \to (not -(eq A y x))))) -\def - \lambda (A: Set).(\lambda (x: A).(\lambda (y: A).(\lambda (h1: (not (eq A x -y))).(\lambda (h2: (eq A y x)).(h1 (eq_ind A y (\lambda (a: A).(eq A a y)) -(refl_equal A y) x h2)))))). -(* COMMENTS -Initial nodes: 51 -END *) - -theorem nat_double_ind: - \forall (R: ((nat \to (nat \to Prop)))).(((\forall (n: nat).(R O n))) \to -(((\forall (n: nat).(R (S n) O))) \to (((\forall (n: nat).(\forall (m: -nat).((R n m) \to (R (S n) (S m)))))) \to (\forall (n: nat).(\forall (m: -nat).(R n m)))))) -\def - \lambda (R: ((nat \to (nat \to Prop)))).(\lambda (H: ((\forall (n: nat).(R O -n)))).(\lambda (H0: ((\forall (n: nat).(R (S n) O)))).(\lambda (H1: ((\forall -(n: nat).(\forall (m: nat).((R n m) \to (R (S n) (S m))))))).(\lambda (n: -nat).(nat_ind (\lambda (n0: nat).(\forall (m: nat).(R n0 m))) H (\lambda (n0: -nat).(\lambda (H2: ((\forall (m: nat).(R n0 m)))).(\lambda (m: nat).(nat_ind -(\lambda (n1: nat).(R (S n0) n1)) (H0 n0) (\lambda (n1: nat).(\lambda (_: (R -(S n0) n1)).(H1 n0 n1 (H2 n1)))) m)))) n))))). -(* COMMENTS -Initial nodes: 111 -END *) - -theorem eq_add_S: - \forall (n: nat).(\forall (m: nat).((eq nat (S n) (S m)) \to (eq nat n m))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (eq nat (S n) (S -m))).(f_equal nat nat pred (S n) (S m) H))). -(* COMMENTS -Initial nodes: 33 -END *) - -theorem O_S: - \forall (n: nat).(not (eq nat O (S n))) -\def - \lambda (n: nat).(\lambda (H: (eq nat O (S n))).(eq_ind nat (S n) (\lambda -(n0: nat).(IsSucc n0)) I O (sym_eq nat O (S n) H))). -(* COMMENTS -Initial nodes: 41 -END *) - -theorem not_eq_S: - \forall (n: nat).(\forall (m: nat).((not (eq nat n m)) \to (not (eq nat (S -n) (S m))))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (not (eq nat n m))).(\lambda -(H0: (eq nat (S n) (S m))).(H (eq_add_S n m H0))))). -(* COMMENTS -Initial nodes: 35 -END *) - -theorem pred_Sn: - \forall (m: nat).(eq nat m (pred (S m))) -\def - \lambda (m: nat).(refl_equal nat (pred (S m))). -(* COMMENTS -Initial nodes: 11 -END *) - -theorem S_pred: - \forall (n: nat).(\forall (m: nat).((lt m n) \to (eq nat n (S (pred n))))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt m n)).(le_ind (S m) -(\lambda (n0: nat).(eq nat n0 (S (pred n0)))) (refl_equal nat (S (pred (S -m)))) (\lambda (m0: nat).(\lambda (_: (le (S m) m0)).(\lambda (_: (eq nat m0 -(S (pred m0)))).(refl_equal nat (S (pred (S m0))))))) n H))). -(* COMMENTS -Initial nodes: 79 -END *) - -theorem le_trans: - \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((le n m) \to ((le m p) -\to (le n p))))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (le n -m)).(\lambda (H0: (le m p)).(le_ind m (\lambda (n0: nat).(le n n0)) H -(\lambda (m0: nat).(\lambda (_: (le m m0)).(\lambda (IHle: (le n m0)).(le_S n -m0 IHle)))) p H0))))). -(* COMMENTS -Initial nodes: 57 -END *) - -theorem le_trans_S: - \forall (n: nat).(\forall (m: nat).((le (S n) m) \to (le n m))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le (S n) m)).(le_trans n (S -n) m (le_S n n (le_n n)) H))). -(* COMMENTS -Initial nodes: 33 -END *) - -theorem le_n_S: - \forall (n: nat).(\forall (m: nat).((le n m) \to (le (S n) (S m)))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(le_ind n (\lambda -(n0: nat).(le (S n) (S n0))) (le_n (S n)) (\lambda (m0: nat).(\lambda (_: (le -n m0)).(\lambda (IHle: (le (S n) (S m0))).(le_S (S n) (S m0) IHle)))) m H))). -(* COMMENTS -Initial nodes: 65 -END *) - -theorem le_O_n: - \forall (n: nat).(le O n) -\def - \lambda (n: nat).(nat_ind (\lambda (n0: nat).(le O n0)) (le_n O) (\lambda -(n0: nat).(\lambda (IHn: (le O n0)).(le_S O n0 IHn))) n). -(* COMMENTS -Initial nodes: 33 -END *) - -theorem le_S_n: - \forall (n: nat).(\forall (m: nat).((le (S n) (S m)) \to (le n m))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le (S n) (S m))).(le_ind (S -n) (\lambda (n0: nat).(le (pred (S n)) (pred n0))) (le_n n) (\lambda (m0: -nat).(\lambda (H0: (le (S n) m0)).(\lambda (_: (le n (pred m0))).(le_trans_S -n m0 H0)))) (S m) H))). -(* COMMENTS -Initial nodes: 69 -END *) - -theorem le_Sn_O: - \forall (n: nat).(not (le (S n) O)) -\def - \lambda (n: nat).(\lambda (H: (le (S n) O)).(le_ind (S n) (\lambda (n0: -nat).(IsSucc n0)) I (\lambda (m: nat).(\lambda (_: (le (S n) m)).(\lambda (_: -(IsSucc m)).I))) O H)). -(* COMMENTS -Initial nodes: 43 -END *) - -theorem le_Sn_n: - \forall (n: nat).(not (le (S n) n)) -\def - \lambda (n: nat).(nat_ind (\lambda (n0: nat).(not (le (S n0) n0))) (le_Sn_O -O) (\lambda (n0: nat).(\lambda (IHn: (not (le (S n0) n0))).(\lambda (H: (le -(S (S n0)) (S n0))).(IHn (le_S_n (S n0) n0 H))))) n). -(* COMMENTS -Initial nodes: 57 -END *) - -theorem le_antisym: - \forall (n: nat).(\forall (m: nat).((le n m) \to ((le m n) \to (eq nat n -m)))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (h: (le n m)).(le_ind n (\lambda -(n0: nat).((le n0 n) \to (eq nat n n0))) (\lambda (_: (le n n)).(refl_equal -nat n)) (\lambda (m0: nat).(\lambda (H: (le n m0)).(\lambda (_: (((le m0 n) -\to (eq nat n m0)))).(\lambda (H1: (le (S m0) n)).(False_ind (eq nat n (S -m0)) (let H2 \def (le_trans (S m0) n m0 H1 H) in ((let H3 \def (le_Sn_n m0) -in (\lambda (H4: (le (S m0) m0)).(H3 H4))) H2))))))) m h))). -(* COMMENTS -Initial nodes: 119 -END *) - -theorem le_n_O_eq: - \forall (n: nat).((le n O) \to (eq nat O n)) -\def - \lambda (n: nat).(\lambda (H: (le n O)).(le_antisym O n (le_O_n n) H)). -(* COMMENTS -Initial nodes: 19 -END *) - -theorem le_elim_rel: - \forall (P: ((nat \to (nat \to Prop)))).(((\forall (p: nat).(P O p))) \to -(((\forall (p: nat).(\forall (q: nat).((le p q) \to ((P p q) \to (P (S p) (S -q))))))) \to (\forall (n: nat).(\forall (m: nat).((le n m) \to (P n m)))))) -\def - \lambda (P: ((nat \to (nat \to Prop)))).(\lambda (H: ((\forall (p: nat).(P O -p)))).(\lambda (H0: ((\forall (p: nat).(\forall (q: nat).((le p q) \to ((P p -q) \to (P (S p) (S q)))))))).(\lambda (n: nat).(nat_ind (\lambda (n0: -nat).(\forall (m: nat).((le n0 m) \to (P n0 m)))) (\lambda (m: nat).(\lambda -(_: (le O m)).(H m))) (\lambda (n0: nat).(\lambda (IHn: ((\forall (m: -nat).((le n0 m) \to (P n0 m))))).(\lambda (m: nat).(\lambda (Le: (le (S n0) -m)).(le_ind (S n0) (\lambda (n1: nat).(P (S n0) n1)) (H0 n0 n0 (le_n n0) (IHn -n0 (le_n n0))) (\lambda (m0: nat).(\lambda (H1: (le (S n0) m0)).(\lambda (_: -(P (S n0) m0)).(H0 n0 m0 (le_trans_S n0 m0 H1) (IHn m0 (le_trans_S n0 m0 -H1)))))) m Le))))) n)))). -(* COMMENTS -Initial nodes: 181 -END *) - -theorem lt_n_n: - \forall (n: nat).(not (lt n n)) -\def - le_Sn_n. -(* COMMENTS -Initial nodes: 1 -END *) - -theorem lt_n_S: - \forall (n: nat).(\forall (m: nat).((lt n m) \to (lt (S n) (S m)))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt n m)).(le_n_S (S n) m -H))). -(* COMMENTS -Initial nodes: 19 -END *) - -theorem lt_n_Sn: - \forall (n: nat).(lt n (S n)) -\def - \lambda (n: nat).(le_n (S n)). -(* COMMENTS -Initial nodes: 7 -END *) - -theorem lt_S_n: - \forall (n: nat).(\forall (m: nat).((lt (S n) (S m)) \to (lt n m))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt (S n) (S m))).(le_S_n (S -n) m H))). -(* COMMENTS -Initial nodes: 23 -END *) - -theorem lt_n_O: - \forall (n: nat).(not (lt n O)) -\def - le_Sn_O. -(* COMMENTS -Initial nodes: 1 -END *) - -theorem lt_trans: - \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((lt n m) \to ((lt m p) -\to (lt n p))))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (lt n -m)).(\lambda (H0: (lt m p)).(le_ind (S m) (\lambda (n0: nat).(lt n n0)) (le_S -(S n) m H) (\lambda (m0: nat).(\lambda (_: (le (S m) m0)).(\lambda (IHle: (lt -n m0)).(le_S (S n) m0 IHle)))) p H0))))). -(* COMMENTS -Initial nodes: 71 -END *) - -theorem lt_O_Sn: - \forall (n: nat).(lt O (S n)) -\def - \lambda (n: nat).(le_n_S O n (le_O_n n)). -(* COMMENTS -Initial nodes: 11 -END *) - -theorem lt_le_S: - \forall (n: nat).(\forall (p: nat).((lt n p) \to (le (S n) p))) -\def - \lambda (n: nat).(\lambda (p: nat).(\lambda (H: (lt n p)).H)). -(* COMMENTS -Initial nodes: 11 -END *) - -theorem le_not_lt: - \forall (n: nat).(\forall (m: nat).((le n m) \to (not (lt m n)))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(le_ind n (\lambda -(n0: nat).(not (lt n0 n))) (lt_n_n n) (\lambda (m0: nat).(\lambda (_: (le n -m0)).(\lambda (IHle: (not (lt m0 n))).(\lambda (H1: (lt (S m0) n)).(IHle -(le_trans_S (S m0) n H1)))))) m H))). -(* COMMENTS -Initial nodes: 67 -END *) - -theorem le_lt_n_Sm: - \forall (n: nat).(\forall (m: nat).((le n m) \to (lt n (S m)))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(le_n_S n m H))). -(* COMMENTS -Initial nodes: 17 -END *) - -theorem le_lt_trans: - \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((le n m) \to ((lt m p) -\to (lt n p))))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (le n -m)).(\lambda (H0: (lt m p)).(le_ind (S m) (\lambda (n0: nat).(lt n n0)) -(le_n_S n m H) (\lambda (m0: nat).(\lambda (_: (le (S m) m0)).(\lambda (IHle: -(lt n m0)).(le_S (S n) m0 IHle)))) p H0))))). -(* COMMENTS -Initial nodes: 69 -END *) - -theorem lt_le_trans: - \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((lt n m) \to ((le m p) -\to (lt n p))))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (lt n -m)).(\lambda (H0: (le m p)).(le_ind m (\lambda (n0: nat).(lt n n0)) H -(\lambda (m0: nat).(\lambda (_: (le m m0)).(\lambda (IHle: (lt n m0)).(le_S -(S n) m0 IHle)))) p H0))))). -(* COMMENTS -Initial nodes: 59 -END *) - -theorem lt_le_weak: - \forall (n: nat).(\forall (m: nat).((lt n m) \to (le n m))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt n m)).(le_trans_S n m -H))). -(* COMMENTS -Initial nodes: 17 -END *) - -theorem lt_n_Sm_le: - \forall (n: nat).(\forall (m: nat).((lt n (S m)) \to (le n m))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt n (S m))).(le_S_n n m -H))). -(* COMMENTS -Initial nodes: 19 -END *) - -theorem le_lt_or_eq: - \forall (n: nat).(\forall (m: nat).((le n m) \to (or (lt n m) (eq nat n m)))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(le_ind n (\lambda -(n0: nat).(or (lt n n0) (eq nat n n0))) (or_intror (lt n n) (eq nat n n) -(refl_equal nat n)) (\lambda (m0: nat).(\lambda (H0: (le n m0)).(\lambda (_: -(or (lt n m0) (eq nat n m0))).(or_introl (lt n (S m0)) (eq nat n (S m0)) -(le_n_S n m0 H0))))) m H))). -(* COMMENTS -Initial nodes: 109 -END *) - -theorem le_or_lt: - \forall (n: nat).(\forall (m: nat).(or (le n m) (lt m n))) -\def - \lambda (n: nat).(\lambda (m: nat).(nat_double_ind (\lambda (n0: -nat).(\lambda (n1: nat).(or (le n0 n1) (lt n1 n0)))) (\lambda (n0: -nat).(or_introl (le O n0) (lt n0 O) (le_O_n n0))) (\lambda (n0: -nat).(or_intror (le (S n0) O) (lt O (S n0)) (lt_le_S O (S n0) (lt_O_Sn n0)))) -(\lambda (n0: nat).(\lambda (m0: nat).(\lambda (H: (or (le n0 m0) (lt m0 -n0))).(or_ind (le n0 m0) (lt m0 n0) (or (le (S n0) (S m0)) (lt (S m0) (S -n0))) (\lambda (H0: (le n0 m0)).(or_introl (le (S n0) (S m0)) (lt (S m0) (S -n0)) (le_n_S n0 m0 H0))) (\lambda (H0: (lt m0 n0)).(or_intror (le (S n0) (S -m0)) (lt (S m0) (S n0)) (le_n_S (S m0) n0 H0))) H)))) n m)). -(* COMMENTS -Initial nodes: 209 -END *) - -theorem plus_n_O: - \forall (n: nat).(eq nat n (plus n O)) -\def - \lambda (n: nat).(nat_ind (\lambda (n0: nat).(eq nat n0 (plus n0 O))) -(refl_equal nat O) (\lambda (n0: nat).(\lambda (H: (eq nat n0 (plus n0 -O))).(f_equal nat nat S n0 (plus n0 O) H))) n). -(* COMMENTS -Initial nodes: 57 -END *) - -theorem plus_n_Sm: - \forall (n: nat).(\forall (m: nat).(eq nat (S (plus n m)) (plus n (S m)))) -\def - \lambda (m: nat).(\lambda (n: nat).(nat_ind (\lambda (n0: nat).(eq nat (S -(plus n0 n)) (plus n0 (S n)))) (refl_equal nat (S n)) (\lambda (n0: -nat).(\lambda (H: (eq nat (S (plus n0 n)) (plus n0 (S n)))).(f_equal nat nat -S (S (plus n0 n)) (plus n0 (S n)) H))) m)). -(* COMMENTS -Initial nodes: 85 -END *) - -theorem plus_sym: - \forall (n: nat).(\forall (m: nat).(eq nat (plus n m) (plus m n))) -\def - \lambda (n: nat).(\lambda (m: nat).(nat_ind (\lambda (n0: nat).(eq nat (plus -n0 m) (plus m n0))) (plus_n_O m) (\lambda (y: nat).(\lambda (H: (eq nat (plus -y m) (plus m y))).(eq_ind nat (S (plus m y)) (\lambda (n0: nat).(eq nat (S -(plus y m)) n0)) (f_equal nat nat S (plus y m) (plus m y) H) (plus m (S y)) -(plus_n_Sm m y)))) n)). -(* COMMENTS -Initial nodes: 111 -END *) - -theorem plus_Snm_nSm: - \forall (n: nat).(\forall (m: nat).(eq nat (plus (S n) m) (plus n (S m)))) -\def - \lambda (n: nat).(\lambda (m: nat).(eq_ind_r nat (plus m n) (\lambda (n0: -nat).(eq nat (S n0) (plus n (S m)))) (eq_ind_r nat (plus (S m) n) (\lambda -(n0: nat).(eq nat (S (plus m n)) n0)) (refl_equal nat (plus (S m) n)) (plus n -(S m)) (plus_sym n (S m))) (plus n m) (plus_sym n m))). -(* COMMENTS -Initial nodes: 99 -END *) - -theorem plus_assoc_l: - \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(eq nat (plus n (plus m -p)) (plus (plus n m) p)))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(nat_ind (\lambda (n0: -nat).(eq nat (plus n0 (plus m p)) (plus (plus n0 m) p))) (refl_equal nat -(plus m p)) (\lambda (n0: nat).(\lambda (H: (eq nat (plus n0 (plus m p)) -(plus (plus n0 m) p))).(f_equal nat nat S (plus n0 (plus m p)) (plus (plus n0 -m) p) H))) n))). -(* COMMENTS -Initial nodes: 101 -END *) - -theorem plus_assoc_r: - \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(eq nat (plus (plus n -m) p) (plus n (plus m p))))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(sym_eq nat (plus n -(plus m p)) (plus (plus n m) p) (plus_assoc_l n m p)))). -(* COMMENTS -Initial nodes: 37 -END *) - -theorem simpl_plus_l: - \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((eq nat (plus n m) -(plus n p)) \to (eq nat m p)))) -\def - \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (m: nat).(\forall (p: -nat).((eq nat (plus n0 m) (plus n0 p)) \to (eq nat m p))))) (\lambda (m: -nat).(\lambda (p: nat).(\lambda (H: (eq nat m p)).H))) (\lambda (n0: -nat).(\lambda (IHn: ((\forall (m: nat).(\forall (p: nat).((eq nat (plus n0 m) -(plus n0 p)) \to (eq nat m p)))))).(\lambda (m: nat).(\lambda (p: -nat).(\lambda (H: (eq nat (S (plus n0 m)) (S (plus n0 p)))).(IHn m p (IHn -(plus n0 m) (plus n0 p) (f_equal nat nat (plus n0) (plus n0 m) (plus n0 p) -(eq_add_S (plus n0 m) (plus n0 p) H))))))))) n). -(* COMMENTS -Initial nodes: 161 -END *) - -theorem minus_n_O: - \forall (n: nat).(eq nat n (minus n O)) -\def - \lambda (n: nat).(nat_ind (\lambda (n0: nat).(eq nat n0 (minus n0 O))) -(refl_equal nat O) (\lambda (n0: nat).(\lambda (_: (eq nat n0 (minus n0 -O))).(refl_equal nat (S n0)))) n). -(* COMMENTS -Initial nodes: 47 -END *) - -theorem minus_n_n: - \forall (n: nat).(eq nat O (minus n n)) -\def - \lambda (n: nat).(nat_ind (\lambda (n0: nat).(eq nat O (minus n0 n0))) -(refl_equal nat O) (\lambda (n0: nat).(\lambda (IHn: (eq nat O (minus n0 -n0))).IHn)) n). -(* COMMENTS -Initial nodes: 41 -END *) - -theorem minus_Sn_m: - \forall (n: nat).(\forall (m: nat).((le m n) \to (eq nat (S (minus n m)) -(minus (S n) m)))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (Le: (le m n)).(le_elim_rel -(\lambda (n0: nat).(\lambda (n1: nat).(eq nat (S (minus n1 n0)) (minus (S n1) -n0)))) (\lambda (p: nat).(f_equal nat nat S (minus p O) p (sym_eq nat p -(minus p O) (minus_n_O p)))) (\lambda (p: nat).(\lambda (q: nat).(\lambda (_: -(le p q)).(\lambda (H0: (eq nat (S (minus q p)) (match p with [O \Rightarrow -(S q) | (S l) \Rightarrow (minus q l)]))).H0)))) m n Le))). -(* COMMENTS -Initial nodes: 111 -END *) - -theorem plus_minus: - \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((eq nat n (plus m p)) -\to (eq nat p (minus n m))))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(nat_double_ind -(\lambda (n0: nat).(\lambda (n1: nat).((eq nat n1 (plus n0 p)) \to (eq nat p -(minus n1 n0))))) (\lambda (n0: nat).(\lambda (H: (eq nat n0 p)).(eq_ind nat -n0 (\lambda (n1: nat).(eq nat p n1)) (sym_eq nat n0 p H) (minus n0 O) -(minus_n_O n0)))) (\lambda (n0: nat).(\lambda (H: (eq nat O (S (plus n0 -p)))).(False_ind (eq nat p O) (let H0 \def H in ((let H1 \def (O_S (plus n0 -p)) in (\lambda (H2: (eq nat O (S (plus n0 p)))).(H1 H2))) H0))))) (\lambda -(n0: nat).(\lambda (m0: nat).(\lambda (H: (((eq nat m0 (plus n0 p)) \to (eq -nat p (minus m0 n0))))).(\lambda (H0: (eq nat (S m0) (S (plus n0 p)))).(H -(eq_add_S m0 (plus n0 p) H0)))))) m n))). -(* COMMENTS -Initial nodes: 199 -END *) - -theorem minus_plus: - \forall (n: nat).(\forall (m: nat).(eq nat (minus (plus n m) n) m)) -\def - \lambda (n: nat).(\lambda (m: nat).(sym_eq nat m (minus (plus n m) n) -(plus_minus (plus n m) n m (refl_equal nat (plus n m))))). -(* COMMENTS -Initial nodes: 41 -END *) - -theorem le_pred_n: - \forall (n: nat).(le (pred n) n) -\def - \lambda (n: nat).(nat_ind (\lambda (n0: nat).(le (pred n0) n0)) (le_n O) -(\lambda (n0: nat).(\lambda (_: (le (pred n0) n0)).(le_S (pred (S n0)) n0 -(le_n n0)))) n). -(* COMMENTS -Initial nodes: 43 -END *) - -theorem le_plus_l: - \forall (n: nat).(\forall (m: nat).(le n (plus n m))) -\def - \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (m: nat).(le n0 (plus -n0 m)))) (\lambda (m: nat).(le_O_n m)) (\lambda (n0: nat).(\lambda (IHn: -((\forall (m: nat).(le n0 (plus n0 m))))).(\lambda (m: nat).(le_n_S n0 (plus -n0 m) (IHn m))))) n). -(* COMMENTS -Initial nodes: 55 -END *) - -theorem le_plus_r: - \forall (n: nat).(\forall (m: nat).(le m (plus n m))) -\def - \lambda (n: nat).(\lambda (m: nat).(nat_ind (\lambda (n0: nat).(le m (plus -n0 m))) (le_n m) (\lambda (n0: nat).(\lambda (H: (le m (plus n0 m))).(le_S m -(plus n0 m) H))) n)). -(* COMMENTS -Initial nodes: 47 -END *) - -theorem simpl_le_plus_l: - \forall (p: nat).(\forall (n: nat).(\forall (m: nat).((le (plus p n) (plus p -m)) \to (le n m)))) -\def - \lambda (p: nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).(\forall (m: -nat).((le (plus n n0) (plus n m)) \to (le n0 m))))) (\lambda (n: -nat).(\lambda (m: nat).(\lambda (H: (le n m)).H))) (\lambda (p0: -nat).(\lambda (IHp: ((\forall (n: nat).(\forall (m: nat).((le (plus p0 n) -(plus p0 m)) \to (le n m)))))).(\lambda (n: nat).(\lambda (m: nat).(\lambda -(H: (le (S (plus p0 n)) (S (plus p0 m)))).(IHp n m (le_S_n (plus p0 n) (plus -p0 m) H))))))) p). -(* COMMENTS -Initial nodes: 113 -END *) - -theorem le_plus_trans: - \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((le n m) \to (le n -(plus m p))))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (le n -m)).(le_trans n m (plus m p) H (le_plus_l m p))))). -(* COMMENTS -Initial nodes: 31 -END *) - -theorem le_reg_l: - \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((le n m) \to (le (plus -p n) (plus p m))))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(nat_ind (\lambda (n0: -nat).((le n m) \to (le (plus n0 n) (plus n0 m)))) (\lambda (H: (le n m)).H) -(\lambda (p0: nat).(\lambda (IHp: (((le n m) \to (le (plus p0 n) (plus p0 -m))))).(\lambda (H: (le n m)).(le_n_S (plus p0 n) (plus p0 m) (IHp H))))) -p))). -(* COMMENTS -Initial nodes: 85 -END *) - -theorem le_plus_plus: - \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(\forall (q: nat).((le -n m) \to ((le p q) \to (le (plus n p) (plus m q))))))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (q: -nat).(\lambda (H: (le n m)).(\lambda (H0: (le p q)).(le_ind n (\lambda (n0: -nat).(le (plus n p) (plus n0 q))) (le_reg_l p q n H0) (\lambda (m0: -nat).(\lambda (_: (le n m0)).(\lambda (H2: (le (plus n p) (plus m0 q))).(le_S -(plus n p) (plus m0 q) H2)))) m H)))))). -(* COMMENTS -Initial nodes: 91 -END *) - -theorem le_plus_minus: - \forall (n: nat).(\forall (m: nat).((le n m) \to (eq nat m (plus n (minus m -n))))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (Le: (le n m)).(le_elim_rel -(\lambda (n0: nat).(\lambda (n1: nat).(eq nat n1 (plus n0 (minus n1 n0))))) -(\lambda (p: nat).(minus_n_O p)) (\lambda (p: nat).(\lambda (q: nat).(\lambda -(_: (le p q)).(\lambda (H0: (eq nat q (plus p (minus q p)))).(f_equal nat nat -S q (plus p (minus q p)) H0))))) n m Le))). -(* COMMENTS -Initial nodes: 91 -END *) - -theorem le_plus_minus_r: - \forall (n: nat).(\forall (m: nat).((le n m) \to (eq nat (plus n (minus m -n)) m))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(sym_eq nat m -(plus n (minus m n)) (le_plus_minus n m H)))). -(* COMMENTS -Initial nodes: 33 -END *) - -theorem simpl_lt_plus_l: - \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((lt (plus p n) (plus p -m)) \to (lt n m)))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(nat_ind (\lambda (n0: -nat).((lt (plus n0 n) (plus n0 m)) \to (lt n m))) (\lambda (H: (lt n m)).H) -(\lambda (p0: nat).(\lambda (IHp: (((lt (plus p0 n) (plus p0 m)) \to (lt n -m)))).(\lambda (H: (lt (S (plus p0 n)) (S (plus p0 m)))).(IHp (le_S_n (S -(plus p0 n)) (plus p0 m) H))))) p))). -(* COMMENTS -Initial nodes: 99 -END *) - -theorem lt_reg_l: - \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((lt n m) \to (lt (plus -p n) (plus p m))))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(nat_ind (\lambda (n0: -nat).((lt n m) \to (lt (plus n0 n) (plus n0 m)))) (\lambda (H: (lt n m)).H) -(\lambda (p0: nat).(\lambda (IHp: (((lt n m) \to (lt (plus p0 n) (plus p0 -m))))).(\lambda (H: (lt n m)).(lt_n_S (plus p0 n) (plus p0 m) (IHp H))))) -p))). -(* COMMENTS -Initial nodes: 85 -END *) - -theorem lt_reg_r: - \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((lt n m) \to (lt (plus -n p) (plus m p))))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (lt n -m)).(eq_ind_r nat (plus p n) (\lambda (n0: nat).(lt n0 (plus m p))) (eq_ind_r -nat (plus p m) (\lambda (n0: nat).(lt (plus p n) n0)) (nat_ind (\lambda (n0: -nat).(lt (plus n0 n) (plus n0 m))) H (\lambda (n0: nat).(\lambda (_: (lt -(plus n0 n) (plus n0 m))).(lt_reg_l n m (S n0) H))) p) (plus m p) (plus_sym m -p)) (plus n p) (plus_sym n p))))). -(* COMMENTS -Initial nodes: 129 -END *) - -theorem le_lt_plus_plus: - \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(\forall (q: nat).((le -n m) \to ((lt p q) \to (lt (plus n p) (plus m q))))))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (q: -nat).(\lambda (H: (le n m)).(\lambda (H0: (le (S p) q)).(eq_ind_r nat (plus n -(S p)) (\lambda (n0: nat).(le n0 (plus m q))) (le_plus_plus n m (S p) q H H0) -(plus (S n) p) (plus_Snm_nSm n p))))))). -(* COMMENTS -Initial nodes: 75 -END *) - -theorem lt_le_plus_plus: - \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(\forall (q: nat).((lt -n m) \to ((le p q) \to (lt (plus n p) (plus m q))))))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (q: -nat).(\lambda (H: (le (S n) m)).(\lambda (H0: (le p q)).(le_plus_plus (S n) m -p q H H0)))))). -(* COMMENTS -Initial nodes: 37 -END *) - -theorem lt_plus_plus: - \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(\forall (q: nat).((lt -n m) \to ((lt p q) \to (lt (plus n p) (plus m q))))))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (q: -nat).(\lambda (H: (lt n m)).(\lambda (H0: (lt p q)).(lt_le_plus_plus n m p q -H (lt_le_weak p q H0))))))). -(* COMMENTS -Initial nodes: 39 -END *) - -theorem well_founded_ltof: - \forall (A: Set).(\forall (f: ((A \to nat))).(well_founded A (ltof A f))) -\def - \lambda (A: Set).(\lambda (f: ((A \to nat))).(let H \def (\lambda (n: -nat).(nat_ind (\lambda (n0: nat).(\forall (a: A).((lt (f a) n0) \to (Acc A -(ltof A f) a)))) (\lambda (a: A).(\lambda (H: (lt (f a) O)).(False_ind (Acc A -(ltof A f) a) (let H0 \def H in ((let H1 \def (lt_n_O (f a)) in (\lambda (H2: -(lt (f a) O)).(H1 H2))) H0))))) (\lambda (n0: nat).(\lambda (IHn: ((\forall -(a: A).((lt (f a) n0) \to (Acc A (ltof A f) a))))).(\lambda (a: A).(\lambda -(ltSma: (lt (f a) (S n0))).(Acc_intro A (ltof A f) a (\lambda (b: A).(\lambda -(ltfafb: (lt (f b) (f a))).(IHn b (lt_le_trans (f b) (f a) n0 ltfafb -(lt_n_Sm_le (f a) n0 ltSma)))))))))) n)) in (\lambda (a: A).(H (S (f a)) a -(le_n (S (f a))))))). -(* COMMENTS -Initial nodes: 189 -END *) - -theorem lt_wf: - well_founded nat lt -\def - well_founded_ltof nat (\lambda (m: nat).m). -(* COMMENTS -Initial nodes: 7 -END *) - -theorem lt_wf_ind: - \forall (p: nat).(\forall (P: ((nat \to Prop))).(((\forall (n: -nat).(((\forall (m: nat).((lt m n) \to (P m)))) \to (P n)))) \to (P p))) -\def - \lambda (p: nat).(\lambda (P: ((nat \to Prop))).(\lambda (H: ((\forall (n: -nat).(((\forall (m: nat).((lt m n) \to (P m)))) \to (P n))))).(Acc_ind nat lt -(\lambda (n: nat).(P n)) (\lambda (x: nat).(\lambda (_: ((\forall (y: -nat).((lt y x) \to (Acc nat lt y))))).(\lambda (H1: ((\forall (y: nat).((lt y -x) \to (P y))))).(H x H1)))) p (lt_wf p)))). -(* COMMENTS -Initial nodes: 77 -END *) - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Legacy-1/definitions.ma b/matita/matita/contribs/LAMBDA-TYPES/Legacy-1/definitions.ma deleted file mode 100644 index 63fc85890..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Legacy-1/definitions.ma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Legacy-1/coq/defs.ma". - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Legacy-1/preamble.ma b/matita/matita/contribs/LAMBDA-TYPES/Legacy-1/preamble.ma deleted file mode 100644 index 96c1bc1fa..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Legacy-1/preamble.ma +++ /dev/null @@ -1,15 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -inductive False: Prop \def . diff --git a/matita/matita/contribs/LAMBDA-TYPES/Legacy-1/spare.ma b/matita/matita/contribs/LAMBDA-TYPES/Legacy-1/spare.ma deleted file mode 100644 index 77939a1b6..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Legacy-1/spare.ma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Legacy-1/theory.ma". - diff --git a/matita/matita/contribs/LAMBDA-TYPES/Legacy-1/theory.ma b/matita/matita/contribs/LAMBDA-TYPES/Legacy-1/theory.ma deleted file mode 100644 index 4ee597e09..000000000 --- a/matita/matita/contribs/LAMBDA-TYPES/Legacy-1/theory.ma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Legacy-1/coq/props.ma". - diff --git a/matita/matita/contribs/lambdadelta/basic_1/A/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/A/defs.ma new file mode 100644 index 000000000..2290c3de4 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/A/defs.ma @@ -0,0 +1,22 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/preamble.ma". + +inductive A: Set \def +| ASort: nat \to (nat \to A) +| AHead: A \to (A \to A). + diff --git a/matita/matita/contribs/lambdadelta/basic_1/C/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/C/defs.ma new file mode 100644 index 000000000..cecc08267 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/C/defs.ma @@ -0,0 +1,45 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/T/defs.ma". + +inductive C: Set \def +| CSort: nat \to C +| CHead: C \to (K \to (T \to C)). + +definition cweight: + C \to nat +\def + let rec cweight (c: C) on c: nat \def (match c with [(CSort _) \Rightarrow O +| (CHead c0 _ t) \Rightarrow (plus (cweight c0) (tweight t))]) in cweight. + +definition clt: + C \to (C \to Prop) +\def + \lambda (c1: C).(\lambda (c2: C).(lt (cweight c1) (cweight c2))). + +definition cle: + C \to (C \to Prop) +\def + \lambda (c1: C).(\lambda (c2: C).(le (cweight c1) (cweight c2))). + +definition CTail: + K \to (T \to (C \to C)) +\def + let rec CTail (k: K) (t: T) (c: C) on c: C \def (match c with [(CSort n) +\Rightarrow (CHead (CSort n) k t) | (CHead d h u) \Rightarrow (CHead (CTail k +t d) h u)]) in CTail. + diff --git a/matita/matita/contribs/lambdadelta/basic_1/C/props.ma b/matita/matita/contribs/lambdadelta/basic_1/C/props.ma new file mode 100644 index 000000000..878abb3bc --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/C/props.ma @@ -0,0 +1,139 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/C/defs.ma". + +include "Basic-1/T/props.ma". + +theorem clt_cong: + \forall (c: C).(\forall (d: C).((clt c d) \to (\forall (k: K).(\forall (t: +T).(clt (CHead c k t) (CHead d k t)))))) +\def + \lambda (c: C).(\lambda (d: C).(\lambda (H: (lt (cweight c) (cweight +d))).(\lambda (_: K).(\lambda (t: T).(lt_reg_r (cweight c) (cweight d) +(tweight t) H))))). +(* COMMENTS +Initial nodes: 33 +END *) + +theorem clt_head: + \forall (k: K).(\forall (c: C).(\forall (u: T).(clt c (CHead c k u)))) +\def + \lambda (_: K).(\lambda (c: C).(\lambda (u: T).(eq_ind_r nat (plus (cweight +c) O) (\lambda (n: nat).(lt n (plus (cweight c) (tweight u)))) +(le_lt_plus_plus (cweight c) (cweight c) O (tweight u) (le_n (cweight c)) +(tweight_lt u)) (cweight c) (plus_n_O (cweight c))))). +(* COMMENTS +Initial nodes: 69 +END *) + +theorem clt_wf__q_ind: + \forall (P: ((C \to Prop))).(((\forall (n: nat).((\lambda (P0: ((C \to +Prop))).(\lambda (n0: nat).(\forall (c: C).((eq nat (cweight c) n0) \to (P0 +c))))) P n))) \to (\forall (c: C).(P c))) +\def + let Q \def (\lambda (P: ((C \to Prop))).(\lambda (n: nat).(\forall (c: +C).((eq nat (cweight c) n) \to (P c))))) in (\lambda (P: ((C \to +Prop))).(\lambda (H: ((\forall (n: nat).(\forall (c: C).((eq nat (cweight c) +n) \to (P c)))))).(\lambda (c: C).(H (cweight c) c (refl_equal nat (cweight +c)))))). +(* COMMENTS +Initial nodes: 61 +END *) + +theorem clt_wf_ind: + \forall (P: ((C \to Prop))).(((\forall (c: C).(((\forall (d: C).((clt d c) +\to (P d)))) \to (P c)))) \to (\forall (c: C).(P c))) +\def + let Q \def (\lambda (P: ((C \to Prop))).(\lambda (n: nat).(\forall (c: +C).((eq nat (cweight c) n) \to (P c))))) in (\lambda (P: ((C \to +Prop))).(\lambda (H: ((\forall (c: C).(((\forall (d: C).((lt (cweight d) +(cweight c)) \to (P d)))) \to (P c))))).(\lambda (c: C).(clt_wf__q_ind +(\lambda (c0: C).(P c0)) (\lambda (n: nat).(lt_wf_ind n (Q (\lambda (c0: +C).(P c0))) (\lambda (n0: nat).(\lambda (H0: ((\forall (m: nat).((lt m n0) +\to (Q (\lambda (c0: C).(P c0)) m))))).(\lambda (c0: C).(\lambda (H1: (eq nat +(cweight c0) n0)).(let H2 \def (eq_ind_r nat n0 (\lambda (n1: nat).(\forall +(m: nat).((lt m n1) \to (\forall (c1: C).((eq nat (cweight c1) m) \to (P +c1)))))) H0 (cweight c0) H1) in (H c0 (\lambda (d: C).(\lambda (H3: (lt +(cweight d) (cweight c0))).(H2 (cweight d) H3 d (refl_equal nat (cweight +d))))))))))))) c)))). +(* COMMENTS +Initial nodes: 179 +END *) + +theorem chead_ctail: + \forall (c: C).(\forall (t: T).(\forall (k: K).(ex_3 K C T (\lambda (h: +K).(\lambda (d: C).(\lambda (u: T).(eq C (CHead c k t) (CTail h u d)))))))) +\def + \lambda (c: C).(C_ind (\lambda (c0: C).(\forall (t: T).(\forall (k: K).(ex_3 +K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C (CHead c0 k t) +(CTail h u d))))))))) (\lambda (n: nat).(\lambda (t: T).(\lambda (k: +K).(ex_3_intro K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C +(CHead (CSort n) k t) (CTail h u d))))) k (CSort n) t (refl_equal C (CHead +(CSort n) k t)))))) (\lambda (c0: C).(\lambda (H: ((\forall (t: T).(\forall +(k: K).(ex_3 K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C +(CHead c0 k t) (CTail h u d)))))))))).(\lambda (k: K).(\lambda (t: +T).(\lambda (t0: T).(\lambda (k0: K).(let H_x \def (H t k) in (let H0 \def +H_x in (ex_3_ind K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C +(CHead c0 k t) (CTail h u d))))) (ex_3 K C T (\lambda (h: K).(\lambda (d: +C).(\lambda (u: T).(eq C (CHead (CHead c0 k t) k0 t0) (CTail h u d)))))) +(\lambda (x0: K).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H1: (eq C (CHead +c0 k t) (CTail x0 x2 x1))).(eq_ind_r C (CTail x0 x2 x1) (\lambda (c1: +C).(ex_3 K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C (CHead +c1 k0 t0) (CTail h u d))))))) (ex_3_intro K C T (\lambda (h: K).(\lambda (d: +C).(\lambda (u: T).(eq C (CHead (CTail x0 x2 x1) k0 t0) (CTail h u d))))) x0 +(CHead x1 k0 t0) x2 (refl_equal C (CHead (CTail x0 x2 x1) k0 t0))) (CHead c0 +k t) H1))))) H0))))))))) c). +(* COMMENTS +Initial nodes: 395 +END *) + +theorem clt_thead: + \forall (k: K).(\forall (u: T).(\forall (c: C).(clt c (CTail k u c)))) +\def + \lambda (k: K).(\lambda (u: T).(\lambda (c: C).(C_ind (\lambda (c0: C).(clt +c0 (CTail k u c0))) (\lambda (n: nat).(clt_head k (CSort n) u)) (\lambda (c0: +C).(\lambda (H: (clt c0 (CTail k u c0))).(\lambda (k0: K).(\lambda (t: +T).(clt_cong c0 (CTail k u c0) H k0 t))))) c))). +(* COMMENTS +Initial nodes: 71 +END *) + +theorem c_tail_ind: + \forall (P: ((C \to Prop))).(((\forall (n: nat).(P (CSort n)))) \to +(((\forall (c: C).((P c) \to (\forall (k: K).(\forall (t: T).(P (CTail k t +c))))))) \to (\forall (c: C).(P c)))) +\def + \lambda (P: ((C \to Prop))).(\lambda (H: ((\forall (n: nat).(P (CSort +n))))).(\lambda (H0: ((\forall (c: C).((P c) \to (\forall (k: K).(\forall (t: +T).(P (CTail k t c)))))))).(\lambda (c: C).(clt_wf_ind (\lambda (c0: C).(P +c0)) (\lambda (c0: C).(C_ind (\lambda (c1: C).(((\forall (d: C).((clt d c1) +\to (P d)))) \to (P c1))) (\lambda (n: nat).(\lambda (_: ((\forall (d: +C).((clt d (CSort n)) \to (P d))))).(H n))) (\lambda (c1: C).(\lambda (_: +((((\forall (d: C).((clt d c1) \to (P d)))) \to (P c1)))).(\lambda (k: +K).(\lambda (t: T).(\lambda (H2: ((\forall (d: C).((clt d (CHead c1 k t)) \to +(P d))))).(let H_x \def (chead_ctail c1 t k) in (let H3 \def H_x in (ex_3_ind +K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C (CHead c1 k t) +(CTail h u d))))) (P (CHead c1 k t)) (\lambda (x0: K).(\lambda (x1: +C).(\lambda (x2: T).(\lambda (H4: (eq C (CHead c1 k t) (CTail x0 x2 +x1))).(eq_ind_r C (CTail x0 x2 x1) (\lambda (c2: C).(P c2)) (let H5 \def +(eq_ind C (CHead c1 k t) (\lambda (c2: C).(\forall (d: C).((clt d c2) \to (P +d)))) H2 (CTail x0 x2 x1) H4) in (H0 x1 (H5 x1 (clt_thead x0 x2 x1)) x0 x2)) +(CHead c1 k t) H4))))) H3)))))))) c0)) c)))). +(* COMMENTS +Initial nodes: 295 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/G/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/G/defs.ma new file mode 100644 index 000000000..23edf5789 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/G/defs.ma @@ -0,0 +1,23 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/preamble.ma". + +record G : Set \def { + next: (nat \to nat); + next_lt: (\forall (n: nat).(lt n (next n))) +}. + diff --git a/matita/matita/contribs/lambdadelta/basic_1/T/dec.ma b/matita/matita/contribs/lambdadelta/basic_1/T/dec.ma new file mode 100644 index 000000000..a088c40e3 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/T/dec.ma @@ -0,0 +1,446 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/T/defs.ma". + +theorem terms_props__bind_dec: + \forall (b1: B).(\forall (b2: B).(or (eq B b1 b2) ((eq B b1 b2) \to (\forall +(P: Prop).P)))) +\def + \lambda (b1: B).(B_ind (\lambda (b: B).(\forall (b2: B).(or (eq B b b2) ((eq +B b b2) \to (\forall (P: Prop).P))))) (\lambda (b2: B).(B_ind (\lambda (b: +B).(or (eq B Abbr b) ((eq B Abbr b) \to (\forall (P: Prop).P)))) (or_introl +(eq B Abbr Abbr) ((eq B Abbr Abbr) \to (\forall (P: Prop).P)) (refl_equal B +Abbr)) (or_intror (eq B Abbr Abst) ((eq B Abbr Abst) \to (\forall (P: +Prop).P)) (\lambda (H: (eq B Abbr Abst)).(\lambda (P: Prop).(let H0 \def +(eq_ind B Abbr (\lambda (ee: B).(match ee in B return (\lambda (_: B).Prop) +with [Abbr \Rightarrow True | Abst \Rightarrow False | Void \Rightarrow +False])) I Abst H) in (False_ind P H0))))) (or_intror (eq B Abbr Void) ((eq B +Abbr Void) \to (\forall (P: Prop).P)) (\lambda (H: (eq B Abbr Void)).(\lambda +(P: Prop).(let H0 \def (eq_ind B Abbr (\lambda (ee: B).(match ee in B return +(\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | +Void \Rightarrow False])) I Void H) in (False_ind P H0))))) b2)) (\lambda +(b2: B).(B_ind (\lambda (b: B).(or (eq B Abst b) ((eq B Abst b) \to (\forall +(P: Prop).P)))) (or_intror (eq B Abst Abbr) ((eq B Abst Abbr) \to (\forall +(P: Prop).P)) (\lambda (H: (eq B Abst Abbr)).(\lambda (P: Prop).(let H0 \def +(eq_ind B Abst (\lambda (ee: B).(match ee in B return (\lambda (_: B).Prop) +with [Abbr \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow +False])) I Abbr H) in (False_ind P H0))))) (or_introl (eq B Abst Abst) ((eq B +Abst Abst) \to (\forall (P: Prop).P)) (refl_equal B Abst)) (or_intror (eq B +Abst Void) ((eq B Abst Void) \to (\forall (P: Prop).P)) (\lambda (H: (eq B +Abst Void)).(\lambda (P: Prop).(let H0 \def (eq_ind B Abst (\lambda (ee: +B).(match ee in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | +Abst \Rightarrow True | Void \Rightarrow False])) I Void H) in (False_ind P +H0))))) b2)) (\lambda (b2: B).(B_ind (\lambda (b: B).(or (eq B Void b) ((eq B +Void b) \to (\forall (P: Prop).P)))) (or_intror (eq B Void Abbr) ((eq B Void +Abbr) \to (\forall (P: Prop).P)) (\lambda (H: (eq B Void Abbr)).(\lambda (P: +Prop).(let H0 \def (eq_ind B Void (\lambda (ee: B).(match ee in B return +(\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False | +Void \Rightarrow True])) I Abbr H) in (False_ind P H0))))) (or_intror (eq B +Void Abst) ((eq B Void Abst) \to (\forall (P: Prop).P)) (\lambda (H: (eq B +Void Abst)).(\lambda (P: Prop).(let H0 \def (eq_ind B Void (\lambda (ee: +B).(match ee in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | +Abst \Rightarrow False | Void \Rightarrow True])) I Abst H) in (False_ind P +H0))))) (or_introl (eq B Void Void) ((eq B Void Void) \to (\forall (P: +Prop).P)) (refl_equal B Void)) b2)) b1). +(* COMMENTS +Initial nodes: 559 +END *) + +theorem bind_dec_not: + \forall (b1: B).(\forall (b2: B).(or (eq B b1 b2) (not (eq B b1 b2)))) +\def + \lambda (b1: B).(\lambda (b2: B).(let H_x \def (terms_props__bind_dec b1 b2) +in (let H \def H_x in (or_ind (eq B b1 b2) ((eq B b1 b2) \to (\forall (P: +Prop).P)) (or (eq B b1 b2) ((eq B b1 b2) \to False)) (\lambda (H0: (eq B b1 +b2)).(or_introl (eq B b1 b2) ((eq B b1 b2) \to False) H0)) (\lambda (H0: +(((eq B b1 b2) \to (\forall (P: Prop).P)))).(or_intror (eq B b1 b2) ((eq B b1 +b2) \to False) (\lambda (H1: (eq B b1 b2)).(H0 H1 False)))) H)))). +(* COMMENTS +Initial nodes: 131 +END *) + +theorem terms_props__flat_dec: + \forall (f1: F).(\forall (f2: F).(or (eq F f1 f2) ((eq F f1 f2) \to (\forall +(P: Prop).P)))) +\def + \lambda (f1: F).(F_ind (\lambda (f: F).(\forall (f2: F).(or (eq F f f2) ((eq +F f f2) \to (\forall (P: Prop).P))))) (\lambda (f2: F).(F_ind (\lambda (f: +F).(or (eq F Appl f) ((eq F Appl f) \to (\forall (P: Prop).P)))) (or_introl +(eq F Appl Appl) ((eq F Appl Appl) \to (\forall (P: Prop).P)) (refl_equal F +Appl)) (or_intror (eq F Appl Cast) ((eq F Appl Cast) \to (\forall (P: +Prop).P)) (\lambda (H: (eq F Appl Cast)).(\lambda (P: Prop).(let H0 \def +(eq_ind F Appl (\lambda (ee: F).(match ee in F return (\lambda (_: F).Prop) +with [Appl \Rightarrow True | Cast \Rightarrow False])) I Cast H) in +(False_ind P H0))))) f2)) (\lambda (f2: F).(F_ind (\lambda (f: F).(or (eq F +Cast f) ((eq F Cast f) \to (\forall (P: Prop).P)))) (or_intror (eq F Cast +Appl) ((eq F Cast Appl) \to (\forall (P: Prop).P)) (\lambda (H: (eq F Cast +Appl)).(\lambda (P: Prop).(let H0 \def (eq_ind F Cast (\lambda (ee: F).(match +ee in F return (\lambda (_: F).Prop) with [Appl \Rightarrow False | Cast +\Rightarrow True])) I Appl H) in (False_ind P H0))))) (or_introl (eq F Cast +Cast) ((eq F Cast Cast) \to (\forall (P: Prop).P)) (refl_equal F Cast)) f2)) +f1). +(* COMMENTS +Initial nodes: 263 +END *) + +theorem terms_props__kind_dec: + \forall (k1: K).(\forall (k2: K).(or (eq K k1 k2) ((eq K k1 k2) \to (\forall +(P: Prop).P)))) +\def + \lambda (k1: K).(K_ind (\lambda (k: K).(\forall (k2: K).(or (eq K k k2) ((eq +K k k2) \to (\forall (P: Prop).P))))) (\lambda (b: B).(\lambda (k2: K).(K_ind +(\lambda (k: K).(or (eq K (Bind b) k) ((eq K (Bind b) k) \to (\forall (P: +Prop).P)))) (\lambda (b0: B).(let H_x \def (terms_props__bind_dec b b0) in +(let H \def H_x in (or_ind (eq B b b0) ((eq B b b0) \to (\forall (P: +Prop).P)) (or (eq K (Bind b) (Bind b0)) ((eq K (Bind b) (Bind b0)) \to +(\forall (P: Prop).P))) (\lambda (H0: (eq B b b0)).(eq_ind B b (\lambda (b1: +B).(or (eq K (Bind b) (Bind b1)) ((eq K (Bind b) (Bind b1)) \to (\forall (P: +Prop).P)))) (or_introl (eq K (Bind b) (Bind b)) ((eq K (Bind b) (Bind b)) \to +(\forall (P: Prop).P)) (refl_equal K (Bind b))) b0 H0)) (\lambda (H0: (((eq B +b b0) \to (\forall (P: Prop).P)))).(or_intror (eq K (Bind b) (Bind b0)) ((eq +K (Bind b) (Bind b0)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq K (Bind b) +(Bind b0))).(\lambda (P: Prop).(let H2 \def (f_equal K B (\lambda (e: +K).(match e in K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | +(Flat _) \Rightarrow b])) (Bind b) (Bind b0) H1) in (let H3 \def (eq_ind_r B +b0 (\lambda (b1: B).((eq B b b1) \to (\forall (P0: Prop).P0))) H0 b H2) in +(H3 (refl_equal B b) P))))))) H)))) (\lambda (f: F).(or_intror (eq K (Bind b) +(Flat f)) ((eq K (Bind b) (Flat f)) \to (\forall (P: Prop).P)) (\lambda (H: +(eq K (Bind b) (Flat f))).(\lambda (P: Prop).(let H0 \def (eq_ind K (Bind b) +(\lambda (ee: K).(match ee in K return (\lambda (_: K).Prop) with [(Bind _) +\Rightarrow True | (Flat _) \Rightarrow False])) I (Flat f) H) in (False_ind +P H0)))))) k2))) (\lambda (f: F).(\lambda (k2: K).(K_ind (\lambda (k: K).(or +(eq K (Flat f) k) ((eq K (Flat f) k) \to (\forall (P: Prop).P)))) (\lambda +(b: B).(or_intror (eq K (Flat f) (Bind b)) ((eq K (Flat f) (Bind b)) \to +(\forall (P: Prop).P)) (\lambda (H: (eq K (Flat f) (Bind b))).(\lambda (P: +Prop).(let H0 \def (eq_ind K (Flat f) (\lambda (ee: K).(match ee in K return +(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow +True])) I (Bind b) H) in (False_ind P H0)))))) (\lambda (f0: F).(let H_x \def +(terms_props__flat_dec f f0) in (let H \def H_x in (or_ind (eq F f f0) ((eq F +f f0) \to (\forall (P: Prop).P)) (or (eq K (Flat f) (Flat f0)) ((eq K (Flat +f) (Flat f0)) \to (\forall (P: Prop).P))) (\lambda (H0: (eq F f f0)).(eq_ind +F f (\lambda (f1: F).(or (eq K (Flat f) (Flat f1)) ((eq K (Flat f) (Flat f1)) +\to (\forall (P: Prop).P)))) (or_introl (eq K (Flat f) (Flat f)) ((eq K (Flat +f) (Flat f)) \to (\forall (P: Prop).P)) (refl_equal K (Flat f))) f0 H0)) +(\lambda (H0: (((eq F f f0) \to (\forall (P: Prop).P)))).(or_intror (eq K +(Flat f) (Flat f0)) ((eq K (Flat f) (Flat f0)) \to (\forall (P: Prop).P)) +(\lambda (H1: (eq K (Flat f) (Flat f0))).(\lambda (P: Prop).(let H2 \def +(f_equal K F (\lambda (e: K).(match e in K return (\lambda (_: K).F) with +[(Bind _) \Rightarrow f | (Flat f1) \Rightarrow f1])) (Flat f) (Flat f0) H1) +in (let H3 \def (eq_ind_r F f0 (\lambda (f1: F).((eq F f f1) \to (\forall +(P0: Prop).P0))) H0 f H2) in (H3 (refl_equal F f) P))))))) H)))) k2))) k1). +(* COMMENTS +Initial nodes: 799 +END *) + +theorem term_dec: + \forall (t1: T).(\forall (t2: T).(or (eq T t1 t2) ((eq T t1 t2) \to (\forall +(P: Prop).P)))) +\def + \lambda (t1: T).(T_ind (\lambda (t: T).(\forall (t2: T).(or (eq T t t2) ((eq +T t t2) \to (\forall (P: Prop).P))))) (\lambda (n: nat).(\lambda (t2: +T).(T_ind (\lambda (t: T).(or (eq T (TSort n) t) ((eq T (TSort n) t) \to +(\forall (P: Prop).P)))) (\lambda (n0: nat).(let H_x \def (nat_dec n n0) in +(let H \def H_x in (or_ind (eq nat n n0) ((eq nat n n0) \to (\forall (P: +Prop).P)) (or (eq T (TSort n) (TSort n0)) ((eq T (TSort n) (TSort n0)) \to +(\forall (P: Prop).P))) (\lambda (H0: (eq nat n n0)).(eq_ind nat n (\lambda +(n1: nat).(or (eq T (TSort n) (TSort n1)) ((eq T (TSort n) (TSort n1)) \to +(\forall (P: Prop).P)))) (or_introl (eq T (TSort n) (TSort n)) ((eq T (TSort +n) (TSort n)) \to (\forall (P: Prop).P)) (refl_equal T (TSort n))) n0 H0)) +(\lambda (H0: (((eq nat n n0) \to (\forall (P: Prop).P)))).(or_intror (eq T +(TSort n) (TSort n0)) ((eq T (TSort n) (TSort n0)) \to (\forall (P: Prop).P)) +(\lambda (H1: (eq T (TSort n) (TSort n0))).(\lambda (P: Prop).(let H2 \def +(f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with +[(TSort n1) \Rightarrow n1 | (TLRef _) \Rightarrow n | (THead _ _ _) +\Rightarrow n])) (TSort n) (TSort n0) H1) in (let H3 \def (eq_ind_r nat n0 +(\lambda (n1: nat).((eq nat n n1) \to (\forall (P0: Prop).P0))) H0 n H2) in +(H3 (refl_equal nat n) P))))))) H)))) (\lambda (n0: nat).(or_intror (eq T +(TSort n) (TLRef n0)) ((eq T (TSort n) (TLRef n0)) \to (\forall (P: Prop).P)) +(\lambda (H: (eq T (TSort n) (TLRef n0))).(\lambda (P: Prop).(let H0 \def +(eq_ind T (TSort n) (\lambda (ee: T).(match ee in T return (\lambda (_: +T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | +(THead _ _ _) \Rightarrow False])) I (TLRef n0) H) in (False_ind P H0)))))) +(\lambda (k: K).(\lambda (t: T).(\lambda (_: (or (eq T (TSort n) t) ((eq T +(TSort n) t) \to (\forall (P: Prop).P)))).(\lambda (t0: T).(\lambda (_: (or +(eq T (TSort n) t0) ((eq T (TSort n) t0) \to (\forall (P: +Prop).P)))).(or_intror (eq T (TSort n) (THead k t t0)) ((eq T (TSort n) +(THead k t t0)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq T (TSort n) +(THead k t t0))).(\lambda (P: Prop).(let H2 \def (eq_ind T (TSort n) (\lambda +(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) +\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow +False])) I (THead k t t0) H1) in (False_ind P H2)))))))))) t2))) (\lambda (n: +nat).(\lambda (t2: T).(T_ind (\lambda (t: T).(or (eq T (TLRef n) t) ((eq T +(TLRef n) t) \to (\forall (P: Prop).P)))) (\lambda (n0: nat).(or_intror (eq T +(TLRef n) (TSort n0)) ((eq T (TLRef n) (TSort n0)) \to (\forall (P: Prop).P)) +(\lambda (H: (eq T (TLRef n) (TSort n0))).(\lambda (P: Prop).(let H0 \def +(eq_ind T (TLRef n) (\lambda (ee: T).(match ee in T return (\lambda (_: +T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | +(THead _ _ _) \Rightarrow False])) I (TSort n0) H) in (False_ind P H0)))))) +(\lambda (n0: nat).(let H_x \def (nat_dec n n0) in (let H \def H_x in (or_ind +(eq nat n n0) ((eq nat n n0) \to (\forall (P: Prop).P)) (or (eq T (TLRef n) +(TLRef n0)) ((eq T (TLRef n) (TLRef n0)) \to (\forall (P: Prop).P))) (\lambda +(H0: (eq nat n n0)).(eq_ind nat n (\lambda (n1: nat).(or (eq T (TLRef n) +(TLRef n1)) ((eq T (TLRef n) (TLRef n1)) \to (\forall (P: Prop).P)))) +(or_introl (eq T (TLRef n) (TLRef n)) ((eq T (TLRef n) (TLRef n)) \to +(\forall (P: Prop).P)) (refl_equal T (TLRef n))) n0 H0)) (\lambda (H0: (((eq +nat n n0) \to (\forall (P: Prop).P)))).(or_intror (eq T (TLRef n) (TLRef n0)) +((eq T (TLRef n) (TLRef n0)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq T +(TLRef n) (TLRef n0))).(\lambda (P: Prop).(let H2 \def (f_equal T nat +(\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort _) +\Rightarrow n | (TLRef n1) \Rightarrow n1 | (THead _ _ _) \Rightarrow n])) +(TLRef n) (TLRef n0) H1) in (let H3 \def (eq_ind_r nat n0 (\lambda (n1: +nat).((eq nat n n1) \to (\forall (P0: Prop).P0))) H0 n H2) in (H3 (refl_equal +nat n) P))))))) H)))) (\lambda (k: K).(\lambda (t: T).(\lambda (_: (or (eq T +(TLRef n) t) ((eq T (TLRef n) t) \to (\forall (P: Prop).P)))).(\lambda (t0: +T).(\lambda (_: (or (eq T (TLRef n) t0) ((eq T (TLRef n) t0) \to (\forall (P: +Prop).P)))).(or_intror (eq T (TLRef n) (THead k t t0)) ((eq T (TLRef n) +(THead k t t0)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq T (TLRef n) +(THead k t t0))).(\lambda (P: Prop).(let H2 \def (eq_ind T (TLRef n) (\lambda +(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) +\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow +False])) I (THead k t t0) H1) in (False_ind P H2)))))))))) t2))) (\lambda (k: +K).(\lambda (t: T).(\lambda (H: ((\forall (t2: T).(or (eq T t t2) ((eq T t +t2) \to (\forall (P: Prop).P)))))).(\lambda (t0: T).(\lambda (H0: ((\forall +(t2: T).(or (eq T t0 t2) ((eq T t0 t2) \to (\forall (P: +Prop).P)))))).(\lambda (t2: T).(T_ind (\lambda (t3: T).(or (eq T (THead k t +t0) t3) ((eq T (THead k t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (n: +nat).(or_intror (eq T (THead k t t0) (TSort n)) ((eq T (THead k t t0) (TSort +n)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq T (THead k t t0) (TSort +n))).(\lambda (P: Prop).(let H2 \def (eq_ind T (THead k t t0) (\lambda (ee: +T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow +False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I +(TSort n) H1) in (False_ind P H2)))))) (\lambda (n: nat).(or_intror (eq T +(THead k t t0) (TLRef n)) ((eq T (THead k t t0) (TLRef n)) \to (\forall (P: +Prop).P)) (\lambda (H1: (eq T (THead k t t0) (TLRef n))).(\lambda (P: +Prop).(let H2 \def (eq_ind T (THead k t t0) (\lambda (ee: T).(match ee in T +return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H1) in +(False_ind P H2)))))) (\lambda (k0: K).(\lambda (t3: T).(\lambda (H1: (or (eq +T (THead k t t0) t3) ((eq T (THead k t t0) t3) \to (\forall (P: +Prop).P)))).(\lambda (t4: T).(\lambda (H2: (or (eq T (THead k t t0) t4) ((eq +T (THead k t t0) t4) \to (\forall (P: Prop).P)))).(let H_x \def (H t3) in +(let H3 \def H_x in (or_ind (eq T t t3) ((eq T t t3) \to (\forall (P: +Prop).P)) (or (eq T (THead k t t0) (THead k0 t3 t4)) ((eq T (THead k t t0) +(THead k0 t3 t4)) \to (\forall (P: Prop).P))) (\lambda (H4: (eq T t t3)).(let +H5 \def (eq_ind_r T t3 (\lambda (t5: T).(or (eq T (THead k t t0) t5) ((eq T +(THead k t t0) t5) \to (\forall (P: Prop).P)))) H1 t H4) in (eq_ind T t +(\lambda (t5: T).(or (eq T (THead k t t0) (THead k0 t5 t4)) ((eq T (THead k t +t0) (THead k0 t5 t4)) \to (\forall (P: Prop).P)))) (let H_x0 \def (H0 t4) in +(let H6 \def H_x0 in (or_ind (eq T t0 t4) ((eq T t0 t4) \to (\forall (P: +Prop).P)) (or (eq T (THead k t t0) (THead k0 t t4)) ((eq T (THead k t t0) +(THead k0 t t4)) \to (\forall (P: Prop).P))) (\lambda (H7: (eq T t0 t4)).(let +H8 \def (eq_ind_r T t4 (\lambda (t5: T).(or (eq T (THead k t t0) t5) ((eq T +(THead k t t0) t5) \to (\forall (P: Prop).P)))) H2 t0 H7) in (eq_ind T t0 +(\lambda (t5: T).(or (eq T (THead k t t0) (THead k0 t t5)) ((eq T (THead k t +t0) (THead k0 t t5)) \to (\forall (P: Prop).P)))) (let H_x1 \def +(terms_props__kind_dec k k0) in (let H9 \def H_x1 in (or_ind (eq K k k0) ((eq +K k k0) \to (\forall (P: Prop).P)) (or (eq T (THead k t t0) (THead k0 t t0)) +((eq T (THead k t t0) (THead k0 t t0)) \to (\forall (P: Prop).P))) (\lambda +(H10: (eq K k k0)).(eq_ind K k (\lambda (k1: K).(or (eq T (THead k t t0) +(THead k1 t t0)) ((eq T (THead k t t0) (THead k1 t t0)) \to (\forall (P: +Prop).P)))) (or_introl (eq T (THead k t t0) (THead k t t0)) ((eq T (THead k t +t0) (THead k t t0)) \to (\forall (P: Prop).P)) (refl_equal T (THead k t t0))) +k0 H10)) (\lambda (H10: (((eq K k k0) \to (\forall (P: Prop).P)))).(or_intror +(eq T (THead k t t0) (THead k0 t t0)) ((eq T (THead k t t0) (THead k0 t t0)) +\to (\forall (P: Prop).P)) (\lambda (H11: (eq T (THead k t t0) (THead k0 t +t0))).(\lambda (P: Prop).(let H12 \def (f_equal T K (\lambda (e: T).(match e +in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) +\Rightarrow k | (THead k1 _ _) \Rightarrow k1])) (THead k t t0) (THead k0 t +t0) H11) in (let H13 \def (eq_ind_r K k0 (\lambda (k1: K).((eq K k k1) \to +(\forall (P0: Prop).P0))) H10 k H12) in (H13 (refl_equal K k) P))))))) H9))) +t4 H7))) (\lambda (H7: (((eq T t0 t4) \to (\forall (P: Prop).P)))).(or_intror +(eq T (THead k t t0) (THead k0 t t4)) ((eq T (THead k t t0) (THead k0 t t4)) +\to (\forall (P: Prop).P)) (\lambda (H8: (eq T (THead k t t0) (THead k0 t +t4))).(\lambda (P: Prop).(let H9 \def (f_equal T K (\lambda (e: T).(match e +in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) +\Rightarrow k | (THead k1 _ _) \Rightarrow k1])) (THead k t t0) (THead k0 t +t4) H8) in ((let H10 \def (f_equal T T (\lambda (e: T).(match e in T return +(\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 +| (THead _ _ t5) \Rightarrow t5])) (THead k t t0) (THead k0 t t4) H8) in +(\lambda (_: (eq K k k0)).(let H12 \def (eq_ind_r T t4 (\lambda (t5: T).((eq +T t0 t5) \to (\forall (P0: Prop).P0))) H7 t0 H10) in (let H13 \def (eq_ind_r +T t4 (\lambda (t5: T).(or (eq T (THead k t t0) t5) ((eq T (THead k t t0) t5) +\to (\forall (P0: Prop).P0)))) H2 t0 H10) in (H12 (refl_equal T t0) P))))) +H9)))))) H6))) t3 H4))) (\lambda (H4: (((eq T t t3) \to (\forall (P: +Prop).P)))).(or_intror (eq T (THead k t t0) (THead k0 t3 t4)) ((eq T (THead k +t t0) (THead k0 t3 t4)) \to (\forall (P: Prop).P)) (\lambda (H5: (eq T (THead +k t t0) (THead k0 t3 t4))).(\lambda (P: Prop).(let H6 \def (f_equal T K +(\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) +\Rightarrow k | (TLRef _) \Rightarrow k | (THead k1 _ _) \Rightarrow k1])) +(THead k t t0) (THead k0 t3 t4) H5) in ((let H7 \def (f_equal T T (\lambda +(e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t +| (TLRef _) \Rightarrow t | (THead _ t5 _) \Rightarrow t5])) (THead k t t0) +(THead k0 t3 t4) H5) in ((let H8 \def (f_equal T T (\lambda (e: T).(match e +in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) +\Rightarrow t0 | (THead _ _ t5) \Rightarrow t5])) (THead k t t0) (THead k0 t3 +t4) H5) in (\lambda (H9: (eq T t t3)).(\lambda (_: (eq K k k0)).(let H11 \def +(eq_ind_r T t4 (\lambda (t5: T).(or (eq T (THead k t t0) t5) ((eq T (THead k +t t0) t5) \to (\forall (P0: Prop).P0)))) H2 t0 H8) in (let H12 \def (eq_ind_r +T t3 (\lambda (t5: T).((eq T t t5) \to (\forall (P0: Prop).P0))) H4 t H9) in +(let H13 \def (eq_ind_r T t3 (\lambda (t5: T).(or (eq T (THead k t t0) t5) +((eq T (THead k t t0) t5) \to (\forall (P0: Prop).P0)))) H1 t H9) in (H12 +(refl_equal T t) P))))))) H7)) H6)))))) H3)))))))) t2))))))) t1). +(* COMMENTS +Initial nodes: 2821 +END *) + +theorem binder_dec: + \forall (t: T).(or (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: +T).(eq T t (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall +(u: T).((eq T t (THead (Bind b) w u)) \to (\forall (P: Prop).P)))))) +\def + \lambda (t: T).(T_ind (\lambda (t0: T).(or (ex_3 B T T (\lambda (b: +B).(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w u)))))) +(\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t0 (THead (Bind b) w +u)) \to (\forall (P: Prop).P))))))) (\lambda (n: nat).(or_intror (ex_3 B T T +(\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T (TSort n) (THead (Bind +b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T (TSort n) +(THead (Bind b) w u)) \to (\forall (P: Prop).P))))) (\lambda (b: B).(\lambda +(w: T).(\lambda (u: T).(\lambda (H: (eq T (TSort n) (THead (Bind b) w +u))).(\lambda (P: Prop).(let H0 \def (eq_ind T (TSort n) (\lambda (ee: +T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow +True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I +(THead (Bind b) w u) H) in (False_ind P H0))))))))) (\lambda (n: +nat).(or_intror (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: +T).(eq T (TLRef n) (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: +T).(\forall (u: T).((eq T (TLRef n) (THead (Bind b) w u)) \to (\forall (P: +Prop).P))))) (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(\lambda (H: (eq +T (TLRef n) (THead (Bind b) w u))).(\lambda (P: Prop).(let H0 \def (eq_ind T +(TLRef n) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with +[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) +\Rightarrow False])) I (THead (Bind b) w u) H) in (False_ind P H0))))))))) +(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t0: T).((or (ex_3 B T T +(\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w +u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t0 (THead +(Bind b) w u)) \to (\forall (P: Prop).P)))))) \to (\forall (t1: T).((or (ex_3 +B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t1 (THead (Bind +b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t1 (THead +(Bind b) w u)) \to (\forall (P: Prop).P)))))) \to (or (ex_3 B T T (\lambda +(b: B).(\lambda (w: T).(\lambda (u: T).(eq T (THead k0 t0 t1) (THead (Bind b) +w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T (THead k0 t0 +t1) (THead (Bind b) w u)) \to (\forall (P: Prop).P))))))))))) (\lambda (b: +B).(\lambda (t0: T).(\lambda (_: (or (ex_3 B T T (\lambda (b0: B).(\lambda +(w: T).(\lambda (u: T).(eq T t0 (THead (Bind b0) w u)))))) (\forall (b0: +B).(\forall (w: T).(\forall (u: T).((eq T t0 (THead (Bind b0) w u)) \to +(\forall (P: Prop).P))))))).(\lambda (t1: T).(\lambda (_: (or (ex_3 B T T +(\lambda (b0: B).(\lambda (w: T).(\lambda (u: T).(eq T t1 (THead (Bind b0) w +u)))))) (\forall (b0: B).(\forall (w: T).(\forall (u: T).((eq T t1 (THead +(Bind b0) w u)) \to (\forall (P: Prop).P))))))).(or_introl (ex_3 B T T +(\lambda (b0: B).(\lambda (w: T).(\lambda (u: T).(eq T (THead (Bind b) t0 t1) +(THead (Bind b0) w u)))))) (\forall (b0: B).(\forall (w: T).(\forall (u: +T).((eq T (THead (Bind b) t0 t1) (THead (Bind b0) w u)) \to (\forall (P: +Prop).P))))) (ex_3_intro B T T (\lambda (b0: B).(\lambda (w: T).(\lambda (u: +T).(eq T (THead (Bind b) t0 t1) (THead (Bind b0) w u))))) b t0 t1 (refl_equal +T (THead (Bind b) t0 t1))))))))) (\lambda (f: F).(\lambda (t0: T).(\lambda +(_: (or (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t0 +(THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: +T).((eq T t0 (THead (Bind b) w u)) \to (\forall (P: Prop).P))))))).(\lambda +(t1: T).(\lambda (_: (or (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda +(u: T).(eq T t1 (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: +T).(\forall (u: T).((eq T t1 (THead (Bind b) w u)) \to (\forall (P: +Prop).P))))))).(or_intror (ex_3 B T T (\lambda (b: B).(\lambda (w: +T).(\lambda (u: T).(eq T (THead (Flat f) t0 t1) (THead (Bind b) w u)))))) +(\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T (THead (Flat f) t0 t1) +(THead (Bind b) w u)) \to (\forall (P: Prop).P))))) (\lambda (b: B).(\lambda +(w: T).(\lambda (u: T).(\lambda (H1: (eq T (THead (Flat f) t0 t1) (THead +(Bind b) w u))).(\lambda (P: Prop).(let H2 \def (eq_ind T (THead (Flat f) t0 +t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort +_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _) +\Rightarrow (match k0 in K return (\lambda (_: K).Prop) with [(Bind _) +\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) w u) H1) +in (False_ind P H2))))))))))))) k)) t). +(* COMMENTS +Initial nodes: 1063 +END *) + +theorem abst_dec: + \forall (u: T).(\forall (v: T).(or (ex T (\lambda (t: T).(eq T u (THead +(Bind Abst) v t)))) (\forall (t: T).((eq T u (THead (Bind Abst) v t)) \to +(\forall (P: Prop).P))))) +\def + \lambda (u: T).(T_ind (\lambda (t: T).(\forall (v: T).(or (ex T (\lambda +(t0: T).(eq T t (THead (Bind Abst) v t0)))) (\forall (t0: T).((eq T t (THead +(Bind Abst) v t0)) \to (\forall (P: Prop).P)))))) (\lambda (n: nat).(\lambda +(v: T).(or_intror (ex T (\lambda (t: T).(eq T (TSort n) (THead (Bind Abst) v +t)))) (\forall (t: T).((eq T (TSort n) (THead (Bind Abst) v t)) \to (\forall +(P: Prop).P))) (\lambda (t: T).(\lambda (H: (eq T (TSort n) (THead (Bind +Abst) v t))).(\lambda (P: Prop).(let H0 \def (eq_ind T (TSort n) (\lambda +(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) +\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow +False])) I (THead (Bind Abst) v t) H) in (False_ind P H0)))))))) (\lambda (n: +nat).(\lambda (v: T).(or_intror (ex T (\lambda (t: T).(eq T (TLRef n) (THead +(Bind Abst) v t)))) (\forall (t: T).((eq T (TLRef n) (THead (Bind Abst) v t)) +\to (\forall (P: Prop).P))) (\lambda (t: T).(\lambda (H: (eq T (TLRef n) +(THead (Bind Abst) v t))).(\lambda (P: Prop).(let H0 \def (eq_ind T (TLRef n) +(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) +\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow +False])) I (THead (Bind Abst) v t) H) in (False_ind P H0)))))))) (\lambda (k: +K).(\lambda (t: T).(\lambda (_: ((\forall (v: T).(or (ex T (\lambda (t0: +T).(eq T t (THead (Bind Abst) v t0)))) (\forall (t0: T).((eq T t (THead (Bind +Abst) v t0)) \to (\forall (P: Prop).P))))))).(\lambda (t0: T).(\lambda (_: +((\forall (v: T).(or (ex T (\lambda (t1: T).(eq T t0 (THead (Bind Abst) v +t1)))) (\forall (t1: T).((eq T t0 (THead (Bind Abst) v t1)) \to (\forall (P: +Prop).P))))))).(\lambda (v: T).(let H_x \def (terms_props__kind_dec k (Bind +Abst)) in (let H1 \def H_x in (or_ind (eq K k (Bind Abst)) ((eq K k (Bind +Abst)) \to (\forall (P: Prop).P)) (or (ex T (\lambda (t1: T).(eq T (THead k t +t0) (THead (Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead k t t0) (THead +(Bind Abst) v t1)) \to (\forall (P: Prop).P)))) (\lambda (H2: (eq K k (Bind +Abst))).(eq_ind_r K (Bind Abst) (\lambda (k0: K).(or (ex T (\lambda (t1: +T).(eq T (THead k0 t t0) (THead (Bind Abst) v t1)))) (\forall (t1: T).((eq T +(THead k0 t t0) (THead (Bind Abst) v t1)) \to (\forall (P: Prop).P))))) (let +H_x0 \def (term_dec t v) in (let H3 \def H_x0 in (or_ind (eq T t v) ((eq T t +v) \to (\forall (P: Prop).P)) (or (ex T (\lambda (t1: T).(eq T (THead (Bind +Abst) t t0) (THead (Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead (Bind +Abst) t t0) (THead (Bind Abst) v t1)) \to (\forall (P: Prop).P)))) (\lambda +(H4: (eq T t v)).(eq_ind T t (\lambda (t1: T).(or (ex T (\lambda (t2: T).(eq +T (THead (Bind Abst) t t0) (THead (Bind Abst) t1 t2)))) (\forall (t2: T).((eq +T (THead (Bind Abst) t t0) (THead (Bind Abst) t1 t2)) \to (\forall (P: +Prop).P))))) (or_introl (ex T (\lambda (t1: T).(eq T (THead (Bind Abst) t t0) +(THead (Bind Abst) t t1)))) (\forall (t1: T).((eq T (THead (Bind Abst) t t0) +(THead (Bind Abst) t t1)) \to (\forall (P: Prop).P))) (ex_intro T (\lambda +(t1: T).(eq T (THead (Bind Abst) t t0) (THead (Bind Abst) t t1))) t0 +(refl_equal T (THead (Bind Abst) t t0)))) v H4)) (\lambda (H4: (((eq T t v) +\to (\forall (P: Prop).P)))).(or_intror (ex T (\lambda (t1: T).(eq T (THead +(Bind Abst) t t0) (THead (Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead +(Bind Abst) t t0) (THead (Bind Abst) v t1)) \to (\forall (P: Prop).P))) +(\lambda (t1: T).(\lambda (H5: (eq T (THead (Bind Abst) t t0) (THead (Bind +Abst) v t1))).(\lambda (P: Prop).(let H6 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t | +(TLRef _) \Rightarrow t | (THead _ t2 _) \Rightarrow t2])) (THead (Bind Abst) +t t0) (THead (Bind Abst) v t1) H5) in ((let H7 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | +(TLRef _) \Rightarrow t0 | (THead _ _ t2) \Rightarrow t2])) (THead (Bind +Abst) t t0) (THead (Bind Abst) v t1) H5) in (\lambda (H8: (eq T t v)).(H4 H8 +P))) H6))))))) H3))) k H2)) (\lambda (H2: (((eq K k (Bind Abst)) \to (\forall +(P: Prop).P)))).(or_intror (ex T (\lambda (t1: T).(eq T (THead k t t0) (THead +(Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead k t t0) (THead (Bind +Abst) v t1)) \to (\forall (P: Prop).P))) (\lambda (t1: T).(\lambda (H3: (eq T +(THead k t t0) (THead (Bind Abst) v t1))).(\lambda (P: Prop).(let H4 \def +(f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with +[(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) +\Rightarrow k0])) (THead k t t0) (THead (Bind Abst) v t1) H3) in ((let H5 +\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) +with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ t2 _) +\Rightarrow t2])) (THead k t t0) (THead (Bind Abst) v t1) H3) in ((let H6 +\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) +with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t2) +\Rightarrow t2])) (THead k t t0) (THead (Bind Abst) v t1) H3) in (\lambda (_: +(eq T t v)).(\lambda (H8: (eq K k (Bind Abst))).(H2 H8 P)))) H5)) H4))))))) +H1))))))))) u). +(* COMMENTS +Initial nodes: 1305 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/T/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/T/defs.ma new file mode 100644 index 000000000..6ddbe6d0d --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/T/defs.ma @@ -0,0 +1,43 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/preamble.ma". + +inductive B: Set \def +| Abbr: B +| Abst: B +| Void: B. + +inductive F: Set \def +| Appl: F +| Cast: F. + +inductive K: Set \def +| Bind: B \to K +| Flat: F \to K. + +inductive T: Set \def +| TSort: nat \to T +| TLRef: nat \to T +| THead: K \to (T \to (T \to T)). + +definition tweight: + T \to nat +\def + let rec tweight (t: T) on t: nat \def (match t with [(TSort _) \Rightarrow +(S O) | (TLRef _) \Rightarrow (S O) | (THead _ u t0) \Rightarrow (S (plus +(tweight u) (tweight t0)))]) in tweight. + diff --git a/matita/matita/contribs/lambdadelta/basic_1/T/props.ma b/matita/matita/contribs/lambdadelta/basic_1/T/props.ma new file mode 100644 index 000000000..faa9ed95d --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/T/props.ma @@ -0,0 +1,111 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/T/defs.ma". + +theorem not_abbr_abst: + not (eq B Abbr Abst) +\def + \lambda (H: (eq B Abbr Abst)).(let H0 \def (eq_ind B Abbr (\lambda (ee: +B).(match ee in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow True | +Abst \Rightarrow False | Void \Rightarrow False])) I Abst H) in (False_ind +False H0)). +(* COMMENTS +Initial nodes: 34 +END *) + +theorem not_void_abst: + not (eq B Void Abst) +\def + \lambda (H: (eq B Void Abst)).(let H0 \def (eq_ind B Void (\lambda (ee: +B).(match ee in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | +Abst \Rightarrow False | Void \Rightarrow True])) I Abst H) in (False_ind +False H0)). +(* COMMENTS +Initial nodes: 34 +END *) + +theorem not_abbr_void: + not (eq B Abbr Void) +\def + \lambda (H: (eq B Abbr Void)).(let H0 \def (eq_ind B Abbr (\lambda (ee: +B).(match ee in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow True | +Abst \Rightarrow False | Void \Rightarrow False])) I Void H) in (False_ind +False H0)). +(* COMMENTS +Initial nodes: 34 +END *) + +theorem not_abst_void: + not (eq B Abst Void) +\def + \lambda (H: (eq B Abst Void)).(let H0 \def (eq_ind B Abst (\lambda (ee: +B).(match ee in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | +Abst \Rightarrow True | Void \Rightarrow False])) I Void H) in (False_ind +False H0)). +(* COMMENTS +Initial nodes: 34 +END *) + +theorem thead_x_y_y: + \forall (k: K).(\forall (v: T).(\forall (t: T).((eq T (THead k v t) t) \to +(\forall (P: Prop).P)))) +\def + \lambda (k: K).(\lambda (v: T).(\lambda (t: T).(T_ind (\lambda (t0: T).((eq +T (THead k v t0) t0) \to (\forall (P: Prop).P))) (\lambda (n: nat).(\lambda +(H: (eq T (THead k v (TSort n)) (TSort n))).(\lambda (P: Prop).(let H0 \def +(eq_ind T (THead k v (TSort n)) (\lambda (ee: T).(match ee in T return +(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H) in +(False_ind P H0))))) (\lambda (n: nat).(\lambda (H: (eq T (THead k v (TLRef +n)) (TLRef n))).(\lambda (P: Prop).(let H0 \def (eq_ind T (THead k v (TLRef +n)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort +_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) +\Rightarrow True])) I (TLRef n) H) in (False_ind P H0))))) (\lambda (k0: +K).(\lambda (t0: T).(\lambda (_: (((eq T (THead k v t0) t0) \to (\forall (P: +Prop).P)))).(\lambda (t1: T).(\lambda (H0: (((eq T (THead k v t1) t1) \to +(\forall (P: Prop).P)))).(\lambda (H1: (eq T (THead k v (THead k0 t0 t1)) +(THead k0 t0 t1))).(\lambda (P: Prop).(let H2 \def (f_equal T K (\lambda (e: +T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | +(TLRef _) \Rightarrow k | (THead k1 _ _) \Rightarrow k1])) (THead k v (THead +k0 t0 t1)) (THead k0 t0 t1) H1) in ((let H3 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v | +(TLRef _) \Rightarrow v | (THead _ t2 _) \Rightarrow t2])) (THead k v (THead +k0 t0 t1)) (THead k0 t0 t1) H1) in ((let H4 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow (THead +k0 t0 t1) | (TLRef _) \Rightarrow (THead k0 t0 t1) | (THead _ _ t2) +\Rightarrow t2])) (THead k v (THead k0 t0 t1)) (THead k0 t0 t1) H1) in +(\lambda (H5: (eq T v t0)).(\lambda (H6: (eq K k k0)).(let H7 \def (eq_ind T +v (\lambda (t2: T).((eq T (THead k t2 t1) t1) \to (\forall (P0: Prop).P0))) +H0 t0 H5) in (let H8 \def (eq_ind K k (\lambda (k1: K).((eq T (THead k1 t0 +t1) t1) \to (\forall (P0: Prop).P0))) H7 k0 H6) in (H8 H4 P)))))) H3)) +H2))))))))) t))). +(* COMMENTS +Initial nodes: 461 +END *) + +theorem tweight_lt: + \forall (t: T).(lt O (tweight t)) +\def + \lambda (t: T).(T_ind (\lambda (t0: T).(lt O (tweight t0))) (\lambda (_: +nat).(le_n (S O))) (\lambda (_: nat).(le_n (S O))) (\lambda (_: K).(\lambda +(t0: T).(\lambda (H: (lt O (tweight t0))).(\lambda (t1: T).(\lambda (_: (lt O +(tweight t1))).(le_S (S O) (plus (tweight t0) (tweight t1)) (le_plus_trans (S +O) (tweight t0) (tweight t1) H))))))) t). +(* COMMENTS +Initial nodes: 85 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/aplus/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/aplus/defs.ma new file mode 100644 index 000000000..4095b163b --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/aplus/defs.ma @@ -0,0 +1,24 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/asucc/defs.ma". + +definition aplus: + G \to (A \to (nat \to A)) +\def + let rec aplus (g: G) (a: A) (n: nat) on n: A \def (match n with [O +\Rightarrow a | (S n0) \Rightarrow (asucc g (aplus g a n0))]) in aplus. + diff --git a/matita/matita/contribs/lambdadelta/basic_1/aplus/props.ma b/matita/matita/contribs/lambdadelta/basic_1/aplus/props.ma new file mode 100644 index 000000000..94bb9a069 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/aplus/props.ma @@ -0,0 +1,282 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/aplus/defs.ma". + +include "Basic-1/next_plus/props.ma". + +theorem aplus_reg_r: + \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (h1: nat).(\forall +(h2: nat).((eq A (aplus g a1 h1) (aplus g a2 h2)) \to (\forall (h: nat).(eq A +(aplus g a1 (plus h h1)) (aplus g a2 (plus h h2))))))))) +\def + \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (h1: nat).(\lambda +(h2: nat).(\lambda (H: (eq A (aplus g a1 h1) (aplus g a2 h2))).(\lambda (h: +nat).(nat_ind (\lambda (n: nat).(eq A (aplus g a1 (plus n h1)) (aplus g a2 +(plus n h2)))) H (\lambda (n: nat).(\lambda (H0: (eq A (aplus g a1 (plus n +h1)) (aplus g a2 (plus n h2)))).(f_equal2 G A A asucc g g (aplus g a1 (plus n +h1)) (aplus g a2 (plus n h2)) (refl_equal G g) H0))) h))))))). +(* COMMENTS +Initial nodes: 143 +END *) + +theorem aplus_assoc: + \forall (g: G).(\forall (a: A).(\forall (h1: nat).(\forall (h2: nat).(eq A +(aplus g (aplus g a h1) h2) (aplus g a (plus h1 h2)))))) +\def + \lambda (g: G).(\lambda (a: A).(\lambda (h1: nat).(nat_ind (\lambda (n: +nat).(\forall (h2: nat).(eq A (aplus g (aplus g a n) h2) (aplus g a (plus n +h2))))) (\lambda (h2: nat).(refl_equal A (aplus g a h2))) (\lambda (n: +nat).(\lambda (_: ((\forall (h2: nat).(eq A (aplus g (aplus g a n) h2) (aplus +g a (plus n h2)))))).(\lambda (h2: nat).(nat_ind (\lambda (n0: nat).(eq A +(aplus g (asucc g (aplus g a n)) n0) (asucc g (aplus g a (plus n n0))))) +(eq_ind nat n (\lambda (n0: nat).(eq A (asucc g (aplus g a n)) (asucc g +(aplus g a n0)))) (refl_equal A (asucc g (aplus g a n))) (plus n O) (plus_n_O +n)) (\lambda (n0: nat).(\lambda (H0: (eq A (aplus g (asucc g (aplus g a n)) +n0) (asucc g (aplus g a (plus n n0))))).(eq_ind nat (S (plus n n0)) (\lambda +(n1: nat).(eq A (asucc g (aplus g (asucc g (aplus g a n)) n0)) (asucc g +(aplus g a n1)))) (f_equal2 G A A asucc g g (aplus g (asucc g (aplus g a n)) +n0) (asucc g (aplus g a (plus n n0))) (refl_equal G g) H0) (plus n (S n0)) +(plus_n_Sm n n0)))) h2)))) h1))). +(* COMMENTS +Initial nodes: 361 +END *) + +theorem aplus_asucc: + \forall (g: G).(\forall (h: nat).(\forall (a: A).(eq A (aplus g (asucc g a) +h) (asucc g (aplus g a h))))) +\def + \lambda (g: G).(\lambda (h: nat).(\lambda (a: A).(eq_ind_r A (aplus g a +(plus (S O) h)) (\lambda (a0: A).(eq A a0 (asucc g (aplus g a h)))) +(refl_equal A (asucc g (aplus g a h))) (aplus g (aplus g a (S O)) h) +(aplus_assoc g a (S O) h)))). +(* COMMENTS +Initial nodes: 87 +END *) + +theorem aplus_sort_O_S_simpl: + \forall (g: G).(\forall (n: nat).(\forall (k: nat).(eq A (aplus g (ASort O +n) (S k)) (aplus g (ASort O (next g n)) k)))) +\def + \lambda (g: G).(\lambda (n: nat).(\lambda (k: nat).(eq_ind A (aplus g (asucc +g (ASort O n)) k) (\lambda (a: A).(eq A a (aplus g (ASort O (next g n)) k))) +(refl_equal A (aplus g (ASort O (next g n)) k)) (asucc g (aplus g (ASort O n) +k)) (aplus_asucc g k (ASort O n))))). +(* COMMENTS +Initial nodes: 97 +END *) + +theorem aplus_sort_S_S_simpl: + \forall (g: G).(\forall (n: nat).(\forall (h: nat).(\forall (k: nat).(eq A +(aplus g (ASort (S h) n) (S k)) (aplus g (ASort h n) k))))) +\def + \lambda (g: G).(\lambda (n: nat).(\lambda (h: nat).(\lambda (k: nat).(eq_ind +A (aplus g (asucc g (ASort (S h) n)) k) (\lambda (a: A).(eq A a (aplus g +(ASort h n) k))) (refl_equal A (aplus g (ASort h n) k)) (asucc g (aplus g +(ASort (S h) n) k)) (aplus_asucc g k (ASort (S h) n)))))). +(* COMMENTS +Initial nodes: 97 +END *) + +theorem aplus_asort_O_simpl: + \forall (g: G).(\forall (h: nat).(\forall (n: nat).(eq A (aplus g (ASort O +n) h) (ASort O (next_plus g n h))))) +\def + \lambda (g: G).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall (n0: +nat).(eq A (aplus g (ASort O n0) n) (ASort O (next_plus g n0 n))))) (\lambda +(n: nat).(refl_equal A (ASort O n))) (\lambda (n: nat).(\lambda (H: ((\forall +(n0: nat).(eq A (aplus g (ASort O n0) n) (ASort O (next_plus g n0 +n)))))).(\lambda (n0: nat).(eq_ind A (aplus g (asucc g (ASort O n0)) n) +(\lambda (a: A).(eq A a (ASort O (next g (next_plus g n0 n))))) (eq_ind nat +(next_plus g (next g n0) n) (\lambda (n1: nat).(eq A (aplus g (ASort O (next +g n0)) n) (ASort O n1))) (H (next g n0)) (next g (next_plus g n0 n)) +(next_plus_next g n0 n)) (asucc g (aplus g (ASort O n0) n)) (aplus_asucc g n +(ASort O n0)))))) h)). +(* COMMENTS +Initial nodes: 229 +END *) + +theorem aplus_asort_le_simpl: + \forall (g: G).(\forall (h: nat).(\forall (k: nat).(\forall (n: nat).((le h +k) \to (eq A (aplus g (ASort k n) h) (ASort (minus k h) n)))))) +\def + \lambda (g: G).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall (k: +nat).(\forall (n0: nat).((le n k) \to (eq A (aplus g (ASort k n0) n) (ASort +(minus k n) n0)))))) (\lambda (k: nat).(\lambda (n: nat).(\lambda (_: (le O +k)).(eq_ind nat k (\lambda (n0: nat).(eq A (ASort k n) (ASort n0 n))) +(refl_equal A (ASort k n)) (minus k O) (minus_n_O k))))) (\lambda (h0: +nat).(\lambda (H: ((\forall (k: nat).(\forall (n: nat).((le h0 k) \to (eq A +(aplus g (ASort k n) h0) (ASort (minus k h0) n))))))).(\lambda (k: +nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).((le (S h0) n) \to (eq A +(asucc g (aplus g (ASort n n0) h0)) (ASort (minus n (S h0)) n0))))) (\lambda +(n: nat).(\lambda (H0: (le (S h0) O)).(ex2_ind nat (\lambda (n0: nat).(eq nat +O (S n0))) (\lambda (n0: nat).(le h0 n0)) (eq A (asucc g (aplus g (ASort O n) +h0)) (ASort (minus O (S h0)) n)) (\lambda (x: nat).(\lambda (H1: (eq nat O (S +x))).(\lambda (_: (le h0 x)).(let H3 \def (eq_ind nat O (\lambda (ee: +nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True +| (S _) \Rightarrow False])) I (S x) H1) in (False_ind (eq A (asucc g (aplus +g (ASort O n) h0)) (ASort (minus O (S h0)) n)) H3))))) (le_gen_S h0 O H0)))) +(\lambda (n: nat).(\lambda (_: ((\forall (n0: nat).((le (S h0) n) \to (eq A +(asucc g (aplus g (ASort n n0) h0)) (ASort (minus n (S h0)) n0)))))).(\lambda +(n0: nat).(\lambda (H1: (le (S h0) (S n))).(eq_ind A (aplus g (asucc g (ASort +(S n) n0)) h0) (\lambda (a: A).(eq A a (ASort (minus (S n) (S h0)) n0))) (H n +n0 (le_S_n h0 n H1)) (asucc g (aplus g (ASort (S n) n0) h0)) (aplus_asucc g +h0 (ASort (S n) n0))))))) k)))) h)). +(* COMMENTS +Initial nodes: 484 +END *) + +theorem aplus_asort_simpl: + \forall (g: G).(\forall (h: nat).(\forall (k: nat).(\forall (n: nat).(eq A +(aplus g (ASort k n) h) (ASort (minus k h) (next_plus g n (minus h k))))))) +\def + \lambda (g: G).(\lambda (h: nat).(\lambda (k: nat).(\lambda (n: +nat).(lt_le_e k h (eq A (aplus g (ASort k n) h) (ASort (minus k h) (next_plus +g n (minus h k)))) (\lambda (H: (lt k h)).(eq_ind_r nat (plus k (minus h k)) +(\lambda (n0: nat).(eq A (aplus g (ASort k n) n0) (ASort (minus k h) +(next_plus g n (minus h k))))) (eq_ind A (aplus g (aplus g (ASort k n) k) +(minus h k)) (\lambda (a: A).(eq A a (ASort (minus k h) (next_plus g n (minus +h k))))) (eq_ind_r A (ASort (minus k k) n) (\lambda (a: A).(eq A (aplus g a +(minus h k)) (ASort (minus k h) (next_plus g n (minus h k))))) (eq_ind nat O +(\lambda (n0: nat).(eq A (aplus g (ASort n0 n) (minus h k)) (ASort (minus k +h) (next_plus g n (minus h k))))) (eq_ind_r nat O (\lambda (n0: nat).(eq A +(aplus g (ASort O n) (minus h k)) (ASort n0 (next_plus g n (minus h k))))) +(aplus_asort_O_simpl g (minus h k) n) (minus k h) (O_minus k h (le_S_n k h +(le_S (S k) h H)))) (minus k k) (minus_n_n k)) (aplus g (ASort k n) k) +(aplus_asort_le_simpl g k k n (le_n k))) (aplus g (ASort k n) (plus k (minus +h k))) (aplus_assoc g (ASort k n) k (minus h k))) h (le_plus_minus k h +(le_S_n k h (le_S (S k) h H))))) (\lambda (H: (le h k)).(eq_ind_r A (ASort +(minus k h) n) (\lambda (a: A).(eq A a (ASort (minus k h) (next_plus g n +(minus h k))))) (eq_ind_r nat O (\lambda (n0: nat).(eq A (ASort (minus k h) +n) (ASort (minus k h) (next_plus g n n0)))) (refl_equal A (ASort (minus k h) +(next_plus g n O))) (minus h k) (O_minus h k H)) (aplus g (ASort k n) h) +(aplus_asort_le_simpl g h k n H))))))). +(* COMMENTS +Initial nodes: 587 +END *) + +theorem aplus_ahead_simpl: + \forall (g: G).(\forall (h: nat).(\forall (a1: A).(\forall (a2: A).(eq A +(aplus g (AHead a1 a2) h) (AHead a1 (aplus g a2 h)))))) +\def + \lambda (g: G).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall (a1: +A).(\forall (a2: A).(eq A (aplus g (AHead a1 a2) n) (AHead a1 (aplus g a2 +n)))))) (\lambda (a1: A).(\lambda (a2: A).(refl_equal A (AHead a1 a2)))) +(\lambda (n: nat).(\lambda (H: ((\forall (a1: A).(\forall (a2: A).(eq A +(aplus g (AHead a1 a2) n) (AHead a1 (aplus g a2 n))))))).(\lambda (a1: +A).(\lambda (a2: A).(eq_ind A (aplus g (asucc g (AHead a1 a2)) n) (\lambda +(a: A).(eq A a (AHead a1 (asucc g (aplus g a2 n))))) (eq_ind A (aplus g +(asucc g a2) n) (\lambda (a: A).(eq A (aplus g (asucc g (AHead a1 a2)) n) +(AHead a1 a))) (H a1 (asucc g a2)) (asucc g (aplus g a2 n)) (aplus_asucc g n +a2)) (asucc g (aplus g (AHead a1 a2) n)) (aplus_asucc g n (AHead a1 a2))))))) +h)). +(* COMMENTS +Initial nodes: 239 +END *) + +theorem aplus_asucc_false: + \forall (g: G).(\forall (a: A).(\forall (h: nat).((eq A (aplus g (asucc g a) +h) a) \to (\forall (P: Prop).P)))) +\def + \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(\forall (h: +nat).((eq A (aplus g (asucc g a0) h) a0) \to (\forall (P: Prop).P)))) +(\lambda (n: nat).(\lambda (n0: nat).(\lambda (h: nat).(\lambda (H: (eq A +(aplus g (match n with [O \Rightarrow (ASort O (next g n0)) | (S h0) +\Rightarrow (ASort h0 n0)]) h) (ASort n n0))).(\lambda (P: Prop).(nat_ind +(\lambda (n1: nat).((eq A (aplus g (match n1 with [O \Rightarrow (ASort O +(next g n0)) | (S h0) \Rightarrow (ASort h0 n0)]) h) (ASort n1 n0)) \to P)) +(\lambda (H0: (eq A (aplus g (ASort O (next g n0)) h) (ASort O n0))).(let H1 +\def (eq_ind A (aplus g (ASort O (next g n0)) h) (\lambda (a0: A).(eq A a0 +(ASort O n0))) H0 (ASort (minus O h) (next_plus g (next g n0) (minus h O))) +(aplus_asort_simpl g h O (next g n0))) in (let H2 \def (f_equal A nat +(\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort _ n1) +\Rightarrow n1 | (AHead _ _) \Rightarrow ((let rec next_plus (g0: G) (n1: +nat) (i: nat) on i: nat \def (match i with [O \Rightarrow n1 | (S i0) +\Rightarrow (next g0 (next_plus g0 n1 i0))]) in next_plus) g (next g n0) +(minus h O))])) (ASort (minus O h) (next_plus g (next g n0) (minus h O))) +(ASort O n0) H1) in (let H3 \def (eq_ind_r nat (minus h O) (\lambda (n1: +nat).(eq nat (next_plus g (next g n0) n1) n0)) H2 h (minus_n_O h)) in +(le_lt_false (next_plus g (next g n0) h) n0 (eq_ind nat (next_plus g (next g +n0) h) (\lambda (n1: nat).(le (next_plus g (next g n0) h) n1)) (le_n +(next_plus g (next g n0) h)) n0 H3) (next_plus_lt g h n0) P))))) (\lambda +(n1: nat).(\lambda (_: (((eq A (aplus g (match n1 with [O \Rightarrow (ASort +O (next g n0)) | (S h0) \Rightarrow (ASort h0 n0)]) h) (ASort n1 n0)) \to +P))).(\lambda (H0: (eq A (aplus g (ASort n1 n0) h) (ASort (S n1) n0))).(let +H1 \def (eq_ind A (aplus g (ASort n1 n0) h) (\lambda (a0: A).(eq A a0 (ASort +(S n1) n0))) H0 (ASort (minus n1 h) (next_plus g n0 (minus h n1))) +(aplus_asort_simpl g h n1 n0)) in (let H2 \def (f_equal A nat (\lambda (e: +A).(match e in A return (\lambda (_: A).nat) with [(ASort n2 _) \Rightarrow +n2 | (AHead _ _) \Rightarrow ((let rec minus (n2: nat) on n2: (nat \to nat) +\def (\lambda (m: nat).(match n2 with [O \Rightarrow O | (S k) \Rightarrow +(match m with [O \Rightarrow (S k) | (S l) \Rightarrow (minus k l)])])) in +minus) n1 h)])) (ASort (minus n1 h) (next_plus g n0 (minus h n1))) (ASort (S +n1) n0) H1) in ((let H3 \def (f_equal A nat (\lambda (e: A).(match e in A +return (\lambda (_: A).nat) with [(ASort _ n2) \Rightarrow n2 | (AHead _ _) +\Rightarrow ((let rec next_plus (g0: G) (n2: nat) (i: nat) on i: nat \def +(match i with [O \Rightarrow n2 | (S i0) \Rightarrow (next g0 (next_plus g0 +n2 i0))]) in next_plus) g n0 (minus h n1))])) (ASort (minus n1 h) (next_plus +g n0 (minus h n1))) (ASort (S n1) n0) H1) in (\lambda (H4: (eq nat (minus n1 +h) (S n1))).(le_Sx_x n1 (eq_ind nat (minus n1 h) (\lambda (n2: nat).(le n2 +n1)) (minus_le n1 h) (S n1) H4) P))) H2)))))) n H)))))) (\lambda (a0: +A).(\lambda (_: ((\forall (h: nat).((eq A (aplus g (asucc g a0) h) a0) \to +(\forall (P: Prop).P))))).(\lambda (a1: A).(\lambda (H0: ((\forall (h: +nat).((eq A (aplus g (asucc g a1) h) a1) \to (\forall (P: +Prop).P))))).(\lambda (h: nat).(\lambda (H1: (eq A (aplus g (AHead a0 (asucc +g a1)) h) (AHead a0 a1))).(\lambda (P: Prop).(let H2 \def (eq_ind A (aplus g +(AHead a0 (asucc g a1)) h) (\lambda (a2: A).(eq A a2 (AHead a0 a1))) H1 +(AHead a0 (aplus g (asucc g a1) h)) (aplus_ahead_simpl g h a0 (asucc g a1))) +in (let H3 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda +(_: A).A) with [(ASort _ _) \Rightarrow ((let rec aplus (g0: G) (a2: A) (n: +nat) on n: A \def (match n with [O \Rightarrow a2 | (S n0) \Rightarrow (asucc +g0 (aplus g0 a2 n0))]) in aplus) g (asucc g a1) h) | (AHead _ a2) \Rightarrow +a2])) (AHead a0 (aplus g (asucc g a1) h)) (AHead a0 a1) H2) in (H0 h H3 +P)))))))))) a)). +(* COMMENTS +Initial nodes: 977 +END *) + +theorem aplus_inj: + \forall (g: G).(\forall (h1: nat).(\forall (h2: nat).(\forall (a: A).((eq A +(aplus g a h1) (aplus g a h2)) \to (eq nat h1 h2))))) +\def + \lambda (g: G).(\lambda (h1: nat).(nat_ind (\lambda (n: nat).(\forall (h2: +nat).(\forall (a: A).((eq A (aplus g a n) (aplus g a h2)) \to (eq nat n +h2))))) (\lambda (h2: nat).(nat_ind (\lambda (n: nat).(\forall (a: A).((eq A +(aplus g a O) (aplus g a n)) \to (eq nat O n)))) (\lambda (a: A).(\lambda (_: +(eq A a a)).(refl_equal nat O))) (\lambda (n: nat).(\lambda (_: ((\forall (a: +A).((eq A a (aplus g a n)) \to (eq nat O n))))).(\lambda (a: A).(\lambda (H0: +(eq A a (asucc g (aplus g a n)))).(let H1 \def (eq_ind_r A (asucc g (aplus g +a n)) (\lambda (a0: A).(eq A a a0)) H0 (aplus g (asucc g a) n) (aplus_asucc g +n a)) in (aplus_asucc_false g a n (sym_eq A a (aplus g (asucc g a) n) H1) (eq +nat O (S n)))))))) h2)) (\lambda (n: nat).(\lambda (H: ((\forall (h2: +nat).(\forall (a: A).((eq A (aplus g a n) (aplus g a h2)) \to (eq nat n +h2)))))).(\lambda (h2: nat).(nat_ind (\lambda (n0: nat).(\forall (a: A).((eq +A (aplus g a (S n)) (aplus g a n0)) \to (eq nat (S n) n0)))) (\lambda (a: +A).(\lambda (H0: (eq A (asucc g (aplus g a n)) a)).(let H1 \def (eq_ind_r A +(asucc g (aplus g a n)) (\lambda (a0: A).(eq A a0 a)) H0 (aplus g (asucc g a) +n) (aplus_asucc g n a)) in (aplus_asucc_false g a n H1 (eq nat (S n) O))))) +(\lambda (n0: nat).(\lambda (_: ((\forall (a: A).((eq A (asucc g (aplus g a +n)) (aplus g a n0)) \to (eq nat (S n) n0))))).(\lambda (a: A).(\lambda (H1: +(eq A (asucc g (aplus g a n)) (asucc g (aplus g a n0)))).(let H2 \def +(eq_ind_r A (asucc g (aplus g a n)) (\lambda (a0: A).(eq A a0 (asucc g (aplus +g a n0)))) H1 (aplus g (asucc g a) n) (aplus_asucc g n a)) in (let H3 \def +(eq_ind_r A (asucc g (aplus g a n0)) (\lambda (a0: A).(eq A (aplus g (asucc g +a) n) a0)) H2 (aplus g (asucc g a) n0) (aplus_asucc g n0 a)) in (f_equal nat +nat S n n0 (H n0 (asucc g a) H3)))))))) h2)))) h1)). +(* COMMENTS +Initial nodes: 599 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/app/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/app/defs.ma new file mode 100644 index 000000000..9cf1e37d1 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/app/defs.ma @@ -0,0 +1,31 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/C/defs.ma". + +definition cbk: + C \to nat +\def + let rec cbk (c: C) on c: nat \def (match c with [(CSort m) \Rightarrow m | +(CHead c0 _ _) \Rightarrow (cbk c0)]) in cbk. + +definition app1: + C \to (T \to T) +\def + let rec app1 (c: C) on c: (T \to T) \def (\lambda (t: T).(match c with +[(CSort _) \Rightarrow t | (CHead c0 k u) \Rightarrow (app1 c0 (THead k u +t))])) in app1. + diff --git a/matita/matita/contribs/lambdadelta/basic_1/aprem/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/aprem/defs.ma new file mode 100644 index 000000000..78b49e920 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/aprem/defs.ma @@ -0,0 +1,23 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/A/defs.ma". + +inductive aprem: nat \to (A \to (A \to Prop)) \def +| aprem_zero: \forall (a1: A).(\forall (a2: A).(aprem O (AHead a1 a2) a1)) +| aprem_succ: \forall (a2: A).(\forall (a: A).(\forall (i: nat).((aprem i a2 +a) \to (\forall (a1: A).(aprem (S i) (AHead a1 a2) a))))). + diff --git a/matita/matita/contribs/lambdadelta/basic_1/aprem/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/aprem/fwd.ma new file mode 100644 index 000000000..ed48846a8 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/aprem/fwd.ma @@ -0,0 +1,120 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/aprem/defs.ma". + +theorem aprem_gen_sort: + \forall (x: A).(\forall (i: nat).(\forall (h: nat).(\forall (n: nat).((aprem +i (ASort h n) x) \to False)))) +\def + \lambda (x: A).(\lambda (i: nat).(\lambda (h: nat).(\lambda (n: +nat).(\lambda (H: (aprem i (ASort h n) x)).(insert_eq A (ASort h n) (\lambda +(a: A).(aprem i a x)) (\lambda (_: A).False) (\lambda (y: A).(\lambda (H0: +(aprem i y x)).(aprem_ind (\lambda (_: nat).(\lambda (a: A).(\lambda (_: +A).((eq A a (ASort h n)) \to False)))) (\lambda (a1: A).(\lambda (a2: +A).(\lambda (H1: (eq A (AHead a1 a2) (ASort h n))).(let H2 \def (eq_ind A +(AHead a1 a2) (\lambda (ee: A).(match ee in A return (\lambda (_: A).Prop) +with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I +(ASort h n) H1) in (False_ind False H2))))) (\lambda (a2: A).(\lambda (a: +A).(\lambda (i0: nat).(\lambda (_: (aprem i0 a2 a)).(\lambda (_: (((eq A a2 +(ASort h n)) \to False))).(\lambda (a1: A).(\lambda (H3: (eq A (AHead a1 a2) +(ASort h n))).(let H4 \def (eq_ind A (AHead a1 a2) (\lambda (ee: A).(match ee +in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | +(AHead _ _) \Rightarrow True])) I (ASort h n) H3) in (False_ind False +H4))))))))) i y x H0))) H))))). +(* COMMENTS +Initial nodes: 227 +END *) + +theorem aprem_gen_head_O: + \forall (a1: A).(\forall (a2: A).(\forall (x: A).((aprem O (AHead a1 a2) x) +\to (eq A x a1)))) +\def + \lambda (a1: A).(\lambda (a2: A).(\lambda (x: A).(\lambda (H: (aprem O +(AHead a1 a2) x)).(insert_eq A (AHead a1 a2) (\lambda (a: A).(aprem O a x)) +(\lambda (_: A).(eq A x a1)) (\lambda (y: A).(\lambda (H0: (aprem O y +x)).(insert_eq nat O (\lambda (n: nat).(aprem n y x)) (\lambda (_: nat).((eq +A y (AHead a1 a2)) \to (eq A x a1))) (\lambda (y0: nat).(\lambda (H1: (aprem +y0 y x)).(aprem_ind (\lambda (n: nat).(\lambda (a: A).(\lambda (a0: A).((eq +nat n O) \to ((eq A a (AHead a1 a2)) \to (eq A a0 a1)))))) (\lambda (a0: +A).(\lambda (a3: A).(\lambda (_: (eq nat O O)).(\lambda (H3: (eq A (AHead a0 +a3) (AHead a1 a2))).(let H4 \def (f_equal A A (\lambda (e: A).(match e in A +return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead a _) +\Rightarrow a])) (AHead a0 a3) (AHead a1 a2) H3) in ((let H5 \def (f_equal A +A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) +\Rightarrow a3 | (AHead _ a) \Rightarrow a])) (AHead a0 a3) (AHead a1 a2) H3) +in (\lambda (H6: (eq A a0 a1)).H6)) H4)))))) (\lambda (a0: A).(\lambda (a: +A).(\lambda (i: nat).(\lambda (H2: (aprem i a0 a)).(\lambda (H3: (((eq nat i +O) \to ((eq A a0 (AHead a1 a2)) \to (eq A a a1))))).(\lambda (a3: A).(\lambda +(H4: (eq nat (S i) O)).(\lambda (H5: (eq A (AHead a3 a0) (AHead a1 a2))).(let +H6 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) +with [(ASort _ _) \Rightarrow a3 | (AHead a4 _) \Rightarrow a4])) (AHead a3 +a0) (AHead a1 a2) H5) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e +in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead _ +a4) \Rightarrow a4])) (AHead a3 a0) (AHead a1 a2) H5) in (\lambda (_: (eq A +a3 a1)).(let H9 \def (eq_ind A a0 (\lambda (a4: A).((eq nat i O) \to ((eq A +a4 (AHead a1 a2)) \to (eq A a a1)))) H3 a2 H7) in (let H10 \def (eq_ind A a0 +(\lambda (a4: A).(aprem i a4 a)) H2 a2 H7) in (let H11 \def (eq_ind nat (S i) +(\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O +\Rightarrow False | (S _) \Rightarrow True])) I O H4) in (False_ind (eq A a +a1) H11)))))) H6)))))))))) y0 y x H1))) H0))) H)))). +(* COMMENTS +Initial nodes: 500 +END *) + +theorem aprem_gen_head_S: + \forall (a1: A).(\forall (a2: A).(\forall (x: A).(\forall (i: nat).((aprem +(S i) (AHead a1 a2) x) \to (aprem i a2 x))))) +\def + \lambda (a1: A).(\lambda (a2: A).(\lambda (x: A).(\lambda (i: nat).(\lambda +(H: (aprem (S i) (AHead a1 a2) x)).(insert_eq A (AHead a1 a2) (\lambda (a: +A).(aprem (S i) a x)) (\lambda (_: A).(aprem i a2 x)) (\lambda (y: +A).(\lambda (H0: (aprem (S i) y x)).(insert_eq nat (S i) (\lambda (n: +nat).(aprem n y x)) (\lambda (_: nat).((eq A y (AHead a1 a2)) \to (aprem i a2 +x))) (\lambda (y0: nat).(\lambda (H1: (aprem y0 y x)).(aprem_ind (\lambda (n: +nat).(\lambda (a: A).(\lambda (a0: A).((eq nat n (S i)) \to ((eq A a (AHead +a1 a2)) \to (aprem i a2 a0)))))) (\lambda (a0: A).(\lambda (a3: A).(\lambda +(H2: (eq nat O (S i))).(\lambda (H3: (eq A (AHead a0 a3) (AHead a1 a2))).(let +H4 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) +with [(ASort _ _) \Rightarrow a0 | (AHead a _) \Rightarrow a])) (AHead a0 a3) +(AHead a1 a2) H3) in ((let H5 \def (f_equal A A (\lambda (e: A).(match e in A +return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a3 | (AHead _ a) +\Rightarrow a])) (AHead a0 a3) (AHead a1 a2) H3) in (\lambda (H6: (eq A a0 +a1)).(eq_ind_r A a1 (\lambda (a: A).(aprem i a2 a)) (let H7 \def (eq_ind nat +O (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O +\Rightarrow True | (S _) \Rightarrow False])) I (S i) H2) in (False_ind +(aprem i a2 a1) H7)) a0 H6))) H4)))))) (\lambda (a0: A).(\lambda (a: +A).(\lambda (i0: nat).(\lambda (H2: (aprem i0 a0 a)).(\lambda (H3: (((eq nat +i0 (S i)) \to ((eq A a0 (AHead a1 a2)) \to (aprem i a2 a))))).(\lambda (a3: +A).(\lambda (H4: (eq nat (S i0) (S i))).(\lambda (H5: (eq A (AHead a3 a0) +(AHead a1 a2))).(let H6 \def (f_equal A A (\lambda (e: A).(match e in A +return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a3 | (AHead a4 _) +\Rightarrow a4])) (AHead a3 a0) (AHead a1 a2) H5) in ((let H7 \def (f_equal A +A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) +\Rightarrow a0 | (AHead _ a4) \Rightarrow a4])) (AHead a3 a0) (AHead a1 a2) +H5) in (\lambda (_: (eq A a3 a1)).(let H9 \def (eq_ind A a0 (\lambda (a4: +A).((eq nat i0 (S i)) \to ((eq A a4 (AHead a1 a2)) \to (aprem i a2 a)))) H3 +a2 H7) in (let H10 \def (eq_ind A a0 (\lambda (a4: A).(aprem i0 a4 a)) H2 a2 +H7) in (let H11 \def (f_equal nat nat (\lambda (e: nat).(match e in nat +return (\lambda (_: nat).nat) with [O \Rightarrow i0 | (S n) \Rightarrow n])) +(S i0) (S i) H4) in (let H12 \def (eq_ind nat i0 (\lambda (n: nat).((eq nat n +(S i)) \to ((eq A a2 (AHead a1 a2)) \to (aprem i a2 a)))) H9 i H11) in (let +H13 \def (eq_ind nat i0 (\lambda (n: nat).(aprem n a2 a)) H10 i H11) in +H13))))))) H6)))))))))) y0 y x H1))) H0))) H))))). +(* COMMENTS +Initial nodes: 631 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/aprem/props.ma b/matita/matita/contribs/lambdadelta/basic_1/aprem/props.ma new file mode 100644 index 000000000..fb8321062 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/aprem/props.ma @@ -0,0 +1,76 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/aprem/fwd.ma". + +include "Basic-1/leq/defs.ma". + +theorem aprem_repl: + \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall +(i: nat).(\forall (b2: A).((aprem i a2 b2) \to (ex2 A (\lambda (b1: A).(leq g +b1 b2)) (\lambda (b1: A).(aprem i a1 b1))))))))) +\def + \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1 +a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(\forall (i: nat).(\forall +(b2: A).((aprem i a0 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda +(b1: A).(aprem i a b1)))))))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda +(n1: nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (_: (eq A (aplus g +(ASort h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (i: nat).(\lambda (b2: +A).(\lambda (H1: (aprem i (ASort h2 n2) b2)).(let H_x \def (aprem_gen_sort b2 +i h2 n2 H1) in (let H2 \def H_x in (False_ind (ex2 A (\lambda (b1: A).(leq g +b1 b2)) (\lambda (b1: A).(aprem i (ASort h1 n1) b1))) H2)))))))))))) (\lambda +(a0: A).(\lambda (a3: A).(\lambda (H0: (leq g a0 a3)).(\lambda (_: ((\forall +(i: nat).(\forall (b2: A).((aprem i a3 b2) \to (ex2 A (\lambda (b1: A).(leq g +b1 b2)) (\lambda (b1: A).(aprem i a0 b1)))))))).(\lambda (a4: A).(\lambda +(a5: A).(\lambda (_: (leq g a4 a5)).(\lambda (H3: ((\forall (i: nat).(\forall +(b2: A).((aprem i a5 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda +(b1: A).(aprem i a4 b1)))))))).(\lambda (i: nat).(\lambda (b2: A).(\lambda +(H4: (aprem i (AHead a3 a5) b2)).(nat_ind (\lambda (n: nat).((aprem n (AHead +a3 a5) b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem +n (AHead a0 a4) b1))))) (\lambda (H5: (aprem O (AHead a3 a5) b2)).(let H_y +\def (aprem_gen_head_O a3 a5 b2 H5) in (eq_ind_r A a3 (\lambda (a: A).(ex2 A +(\lambda (b1: A).(leq g b1 a)) (\lambda (b1: A).(aprem O (AHead a0 a4) b1)))) +(ex_intro2 A (\lambda (b1: A).(leq g b1 a3)) (\lambda (b1: A).(aprem O (AHead +a0 a4) b1)) a0 H0 (aprem_zero a0 a4)) b2 H_y))) (\lambda (i0: nat).(\lambda +(_: (((aprem i0 (AHead a3 a5) b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) +(\lambda (b1: A).(aprem i0 (AHead a0 a4) b1)))))).(\lambda (H5: (aprem (S i0) +(AHead a3 a5) b2)).(let H_y \def (aprem_gen_head_S a3 a5 b2 i0 H5) in (let +H_x \def (H3 i0 b2 H_y) in (let H6 \def H_x in (ex2_ind A (\lambda (b1: +A).(leq g b1 b2)) (\lambda (b1: A).(aprem i0 a4 b1)) (ex2 A (\lambda (b1: +A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S i0) (AHead a0 a4) b1))) (\lambda +(x: A).(\lambda (H7: (leq g x b2)).(\lambda (H8: (aprem i0 a4 x)).(ex_intro2 +A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S i0) (AHead a0 +a4) b1)) x H7 (aprem_succ a4 x i0 H8 a0))))) H6))))))) i H4)))))))))))) a1 a2 +H)))). +(* COMMENTS +Initial nodes: 621 +END *) + +theorem aprem_asucc: + \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (i: nat).((aprem i +a1 a2) \to (aprem i (asucc g a1) a2))))) +\def + \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (i: nat).(\lambda +(H: (aprem i a1 a2)).(aprem_ind (\lambda (n: nat).(\lambda (a: A).(\lambda +(a0: A).(aprem n (asucc g a) a0)))) (\lambda (a0: A).(\lambda (a3: +A).(aprem_zero a0 (asucc g a3)))) (\lambda (a0: A).(\lambda (a: A).(\lambda +(i0: nat).(\lambda (_: (aprem i0 a0 a)).(\lambda (H1: (aprem i0 (asucc g a0) +a)).(\lambda (a3: A).(aprem_succ (asucc g a0) a i0 H1 a3))))))) i a1 a2 +H))))). +(* COMMENTS +Initial nodes: 101 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/arity/aprem.ma b/matita/matita/contribs/lambdadelta/basic_1/arity/aprem.ma new file mode 100644 index 000000000..35e8f58e0 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/arity/aprem.ma @@ -0,0 +1,260 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/arity/props.ma". + +include "Basic-1/arity/cimp.ma". + +include "Basic-1/aprem/props.ma". + +theorem arity_aprem: + \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t +a) \to (\forall (i: nat).(\forall (b: A).((aprem i a b) \to (ex2_3 C T nat +(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c)))) +(\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g +b))))))))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: +(arity g c t a)).(arity_ind g (\lambda (c0: C).(\lambda (_: T).(\lambda (a0: +A).(\forall (i: nat).(\forall (b: A).((aprem i a0 b) \to (ex2_3 C T nat +(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) +(\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g +b)))))))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (i: nat).(\lambda +(b: A).(\lambda (H0: (aprem i (ASort O n) b)).(let H_x \def (aprem_gen_sort b +i O n H0) in (let H1 \def H_x in (False_ind (ex2_3 C T nat (\lambda (d: +C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: +C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b)))))) H1)))))))) +(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda +(H0: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (a0: A).(\lambda (_: +(arity g d u a0)).(\lambda (H2: ((\forall (i0: nat).(\forall (b: A).((aprem +i0 a0 b) \to (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: +nat).(drop (plus i0 j) O d0 d)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda +(_: nat).(arity g d0 u0 (asucc g b))))))))))).(\lambda (i0: nat).(\lambda (b: +A).(\lambda (H3: (aprem i0 a0 b)).(let H_x \def (H2 i0 b H3) in (let H4 \def +H_x in (ex2_3_ind C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: +nat).(drop (plus i0 j) O d0 d)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda +(_: nat).(arity g d0 u0 (asucc g b))))) (ex2_3 C T nat (\lambda (d0: +C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d0 c0)))) (\lambda +(d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b)))))) +(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H5: (drop +(plus i0 x2) O x0 d)).(\lambda (H6: (arity g x0 x1 (asucc g b))).(let H_x0 +\def (getl_drop_conf_rev (plus i0 x2) x0 d H5 Abbr c0 u i H0) in (let H7 \def +H_x0 in (ex2_ind C (\lambda (c1: C).(drop (plus i0 x2) O c1 c0)) (\lambda +(c1: C).(drop (S i) (plus i0 x2) c1 x0)) (ex2_3 C T nat (\lambda (d0: +C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d0 c0)))) (\lambda +(d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b)))))) +(\lambda (x: C).(\lambda (H8: (drop (plus i0 x2) O x c0)).(\lambda (H9: (drop +(S i) (plus i0 x2) x x0)).(ex2_3_intro C T nat (\lambda (d0: C).(\lambda (_: +T).(\lambda (j: nat).(drop (plus i0 j) O d0 c0)))) (\lambda (d0: C).(\lambda +(u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b))))) x (lift (S i) (plus +i0 x2) x1) x2 H8 (arity_lift g x0 x1 (asucc g b) H6 x (S i) (plus i0 x2) +H9))))) H7)))))))) H4)))))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda +(u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abst) +u))).(\lambda (a0: A).(\lambda (_: (arity g d u (asucc g a0))).(\lambda (H2: +((\forall (i0: nat).(\forall (b: A).((aprem i0 (asucc g a0) b) \to (ex2_3 C T +nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d0 +d)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 +(asucc g b))))))))))).(\lambda (i0: nat).(\lambda (b: A).(\lambda (H3: (aprem +i0 a0 b)).(let H4 \def (H2 i0 b (aprem_asucc g a0 b i0 H3)) in (ex2_3_ind C T +nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d0 +d)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 +(asucc g b))))) (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: +nat).(drop (plus i0 j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda +(_: nat).(arity g d0 u0 (asucc g b)))))) (\lambda (x0: C).(\lambda (x1: +T).(\lambda (x2: nat).(\lambda (H5: (drop (plus i0 x2) O x0 d)).(\lambda (H6: +(arity g x0 x1 (asucc g b))).(let H_x \def (getl_drop_conf_rev (plus i0 x2) +x0 d H5 Abst c0 u i H0) in (let H7 \def H_x in (ex2_ind C (\lambda (c1: +C).(drop (plus i0 x2) O c1 c0)) (\lambda (c1: C).(drop (S i) (plus i0 x2) c1 +x0)) (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop +(plus i0 j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: +nat).(arity g d0 u0 (asucc g b)))))) (\lambda (x: C).(\lambda (H8: (drop +(plus i0 x2) O x c0)).(\lambda (H9: (drop (S i) (plus i0 x2) x +x0)).(ex2_3_intro C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: +nat).(drop (plus i0 j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda +(_: nat).(arity g d0 u0 (asucc g b))))) x (lift (S i) (plus i0 x2) x1) x2 H8 +(arity_lift g x0 x1 (asucc g b) H6 x (S i) (plus i0 x2) H9))))) H7)))))))) +H4))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda +(c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u +a1)).(\lambda (_: ((\forall (i: nat).(\forall (b0: A).((aprem i a1 b0) \to +(ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus +i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d +u0 (asucc g b0))))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: +(arity g (CHead c0 (Bind b) u) t0 a2)).(\lambda (H4: ((\forall (i: +nat).(\forall (b0: A).((aprem i a2 b0) \to (ex2_3 C T nat (\lambda (d: +C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d (CHead c0 (Bind b) +u))))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 +(asucc g b0))))))))))).(\lambda (i: nat).(\lambda (b0: A).(\lambda (H5: +(aprem i a2 b0)).(let H_x \def (H4 i b0 H5) in (let H6 \def H_x in (ex2_3_ind +C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O +d (CHead c0 (Bind b) u))))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: +nat).(arity g d u0 (asucc g b0))))) (ex2_3 C T nat (\lambda (d: C).(\lambda +(_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda +(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b0)))))) (\lambda (x0: +C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H7: (drop (plus i x2) O x0 +(CHead c0 (Bind b) u))).(\lambda (H8: (arity g x0 x1 (asucc g b0))).(let H9 +\def (eq_ind nat (S (plus i x2)) (\lambda (n: nat).(drop n O x0 c0)) (drop_S +b x0 c0 u (plus i x2) H7) (plus i (S x2)) (plus_n_Sm i x2)) in (ex2_3_intro C +T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d +c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 +(asucc g b0))))) x0 x1 (S x2) H9 H8))))))) H6))))))))))))))))) (\lambda (c0: +C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H0: (arity g c0 u (asucc g +a1))).(\lambda (_: ((\forall (i: nat).(\forall (b: A).((aprem i (asucc g a1) +b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop +(plus i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: +nat).(arity g d u0 (asucc g b))))))))))).(\lambda (t0: T).(\lambda (a2: +A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u) t0 a2)).(\lambda (H3: +((\forall (i: nat).(\forall (b: A).((aprem i a2 b) \to (ex2_3 C T nat +(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d (CHead +c0 (Bind Abst) u))))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: +nat).(arity g d u0 (asucc g b))))))))))).(\lambda (i: nat).(\lambda (b: +A).(\lambda (H4: (aprem i (AHead a1 a2) b)).(nat_ind (\lambda (n: +nat).((aprem n (AHead a1 a2) b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda +(_: T).(\lambda (j: nat).(drop (plus n j) O d c0)))) (\lambda (d: C).(\lambda +(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b)))))))) (\lambda (H5: +(aprem O (AHead a1 a2) b)).(let H_y \def (aprem_gen_head_O a1 a2 b H5) in +(eq_ind_r A a1 (\lambda (a0: A).(ex2_3 C T nat (\lambda (d: C).(\lambda (_: +T).(\lambda (j: nat).(drop (plus O j) O d c0)))) (\lambda (d: C).(\lambda +(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g a0))))))) (ex2_3_intro C T +nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus O j) O d +c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 +(asucc g a1))))) c0 u O (drop_refl c0) H0) b H_y))) (\lambda (i0: +nat).(\lambda (_: (((aprem i0 (AHead a1 a2) b) \to (ex2_3 C T nat (\lambda +(d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d c0)))) +(\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g +b))))))))).(\lambda (H5: (aprem (S i0) (AHead a1 a2) b)).(let H_y \def +(aprem_gen_head_S a1 a2 b i0 H5) in (let H_x \def (H3 i0 b H_y) in (let H6 +\def H_x in (ex2_3_ind C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: +nat).(drop (plus i0 j) O d (CHead c0 (Bind Abst) u))))) (\lambda (d: +C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))) (ex2_3 C +T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus (S i0) j) +O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 +(asucc g b)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: +nat).(\lambda (H7: (drop (plus i0 x2) O x0 (CHead c0 (Bind Abst) +u))).(\lambda (H8: (arity g x0 x1 (asucc g b))).(ex2_3_intro C T nat (\lambda +(d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus (S i0) j) O d c0)))) +(\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g +b))))) x0 x1 x2 (drop_S Abst x0 c0 u (plus i0 x2) H7) H8)))))) H6))))))) i +H4))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda +(_: (arity g c0 u a1)).(\lambda (_: ((\forall (i: nat).(\forall (b: +A).((aprem i a1 b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: +T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda +(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))))))))).(\lambda (t0: +T).(\lambda (a2: A).(\lambda (_: (arity g c0 t0 (AHead a1 a2))).(\lambda (H3: +((\forall (i: nat).(\forall (b: A).((aprem i (AHead a1 a2) b) \to (ex2_3 C T +nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d +c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 +(asucc g b))))))))))).(\lambda (i: nat).(\lambda (b: A).(\lambda (H4: (aprem +i a2 b)).(let H5 \def (H3 (S i) b (aprem_succ a2 b i H4 a1)) in (ex2_3_ind C +T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (S (plus i j)) +O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 +(asucc g b))))) (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: +nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda +(_: nat).(arity g d u0 (asucc g b)))))) (\lambda (x0: C).(\lambda (x1: +T).(\lambda (x2: nat).(\lambda (H6: (drop (S (plus i x2)) O x0 c0)).(\lambda +(H7: (arity g x0 x1 (asucc g b))).(C_ind (\lambda (c1: C).((drop (S (plus i +x2)) O c1 c0) \to ((arity g c1 x1 (asucc g b)) \to (ex2_3 C T nat (\lambda +(d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda +(d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))))))) +(\lambda (n: nat).(\lambda (H8: (drop (S (plus i x2)) O (CSort n) +c0)).(\lambda (_: (arity g (CSort n) x1 (asucc g b))).(and3_ind (eq C c0 +(CSort n)) (eq nat (S (plus i x2)) O) (eq nat O O) (ex2_3 C T nat (\lambda +(d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda +(d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b)))))) +(\lambda (_: (eq C c0 (CSort n))).(\lambda (H11: (eq nat (S (plus i x2)) +O)).(\lambda (_: (eq nat O O)).(let H13 \def (eq_ind nat (S (plus i x2)) +(\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O +\Rightarrow False | (S _) \Rightarrow True])) I O H11) in (False_ind (ex2_3 C +T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d +c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 +(asucc g b)))))) H13))))) (drop_gen_sort n (S (plus i x2)) O c0 H8))))) +(\lambda (d: C).(\lambda (IHd: (((drop (S (plus i x2)) O d c0) \to ((arity g +d x1 (asucc g b)) \to (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: +T).(\lambda (j: nat).(drop (plus i j) O d0 c0)))) (\lambda (d0: C).(\lambda +(u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b)))))))))).(\lambda (k: +K).(\lambda (t1: T).(\lambda (H8: (drop (S (plus i x2)) O (CHead d k t1) +c0)).(\lambda (H9: (arity g (CHead d k t1) x1 (asucc g b))).(K_ind (\lambda +(k0: K).((arity g (CHead d k0 t1) x1 (asucc g b)) \to ((drop (r k0 (plus i +x2)) O d c0) \to (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: +nat).(drop (plus i j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda +(_: nat).(arity g d0 u0 (asucc g b))))))))) (\lambda (b0: B).(\lambda (H10: +(arity g (CHead d (Bind b0) t1) x1 (asucc g b))).(\lambda (H11: (drop (r +(Bind b0) (plus i x2)) O d c0)).(ex2_3_intro C T nat (\lambda (d0: +C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d0 c0)))) (\lambda +(d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b))))) +(CHead d (Bind b0) t1) x1 (S x2) (eq_ind nat (S (plus i x2)) (\lambda (n: +nat).(drop n O (CHead d (Bind b0) t1) c0)) (drop_drop (Bind b0) (plus i x2) d +c0 H11 t1) (plus i (S x2)) (plus_n_Sm i x2)) H10)))) (\lambda (f: F).(\lambda +(H10: (arity g (CHead d (Flat f) t1) x1 (asucc g b))).(\lambda (H11: (drop (r +(Flat f) (plus i x2)) O d c0)).(let H12 \def (IHd H11 (arity_cimp_conf g +(CHead d (Flat f) t1) x1 (asucc g b) H10 d (cimp_flat_sx f d t1))) in +(ex2_3_ind C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop +(plus i j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: +nat).(arity g d0 u0 (asucc g b))))) (ex2_3 C T nat (\lambda (d0: C).(\lambda +(_: T).(\lambda (j: nat).(drop (plus i j) O d0 c0)))) (\lambda (d0: +C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b)))))) +(\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: nat).(\lambda (H13: (drop +(plus i x5) O x3 c0)).(\lambda (H14: (arity g x3 x4 (asucc g +b))).(ex2_3_intro C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: +nat).(drop (plus i j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda +(_: nat).(arity g d0 u0 (asucc g b))))) x3 x4 x5 H13 H14)))))) H12))))) k H9 +(drop_gen_drop k d c0 t1 (plus i x2) H8)))))))) x0 H6 H7)))))) +H5)))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a0: A).(\lambda +(_: (arity g c0 u (asucc g a0))).(\lambda (_: ((\forall (i: nat).(\forall (b: +A).((aprem i (asucc g a0) b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: +T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda +(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))))))))).(\lambda (t0: +T).(\lambda (_: (arity g c0 t0 a0)).(\lambda (H3: ((\forall (i: nat).(\forall +(b: A).((aprem i a0 b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: +T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda +(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))))))))).(\lambda (i: +nat).(\lambda (b: A).(\lambda (H4: (aprem i a0 b)).(let H_x \def (H3 i b H4) +in (let H5 \def H_x in (ex2_3_ind C T nat (\lambda (d: C).(\lambda (_: +T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda +(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))) (ex2_3 C T nat +(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) +(\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g +b)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H6: +(drop (plus i x2) O x0 c0)).(\lambda (H7: (arity g x0 x1 (asucc g +b))).(ex2_3_intro C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: +nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda +(_: nat).(arity g d u0 (asucc g b))))) x0 x1 x2 H6 H7)))))) H5)))))))))))))) +(\lambda (c0: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda (_: (arity g c0 +t0 a1)).(\lambda (H1: ((\forall (i: nat).(\forall (b: A).((aprem i a1 b) \to +(ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus +i j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d +u (asucc g b))))))))))).(\lambda (a2: A).(\lambda (H2: (leq g a1 +a2)).(\lambda (i: nat).(\lambda (b: A).(\lambda (H3: (aprem i a2 b)).(let H_x +\def (aprem_repl g a1 a2 H2 i b H3) in (let H4 \def H_x in (ex2_ind A +(\lambda (b1: A).(leq g b1 b)) (\lambda (b1: A).(aprem i a1 b1)) (ex2_3 C T +nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d +c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc +g b)))))) (\lambda (x: A).(\lambda (H5: (leq g x b)).(\lambda (H6: (aprem i +a1 x)).(let H_x0 \def (H1 i x H6) in (let H7 \def H_x0 in (ex2_3_ind C T nat +(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) +(\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g +x))))) (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop +(plus i j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: +nat).(arity g d u (asucc g b)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda +(x2: nat).(\lambda (H8: (drop (plus i x2) O x0 c0)).(\lambda (H9: (arity g x0 +x1 (asucc g x))).(ex2_3_intro C T nat (\lambda (d: C).(\lambda (_: +T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u: +T).(\lambda (_: nat).(arity g d u (asucc g b))))) x0 x1 x2 H8 (arity_repl g +x0 x1 (asucc g x) H9 (asucc g b) (asucc_repl g x b H5)))))))) H7)))))) +H4))))))))))))) c t a H))))). +(* COMMENTS +Initial nodes: 4526 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/arity/cimp.ma b/matita/matita/contribs/lambdadelta/basic_1/arity/cimp.ma new file mode 100644 index 000000000..c6212ac15 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/arity/cimp.ma @@ -0,0 +1,102 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/arity/defs.ma". + +include "Basic-1/cimp/props.ma". + +theorem arity_cimp_conf: + \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1 +t a) \to (\forall (c2: C).((cimp c1 c2) \to (arity g c2 t a))))))) +\def + \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: +(arity g c1 t a)).(arity_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda (a0: +A).(\forall (c2: C).((cimp c c2) \to (arity g c2 t0 a0)))))) (\lambda (c: +C).(\lambda (n: nat).(\lambda (c2: C).(\lambda (_: (cimp c c2)).(arity_sort g +c2 n))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: +nat).(\lambda (H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (a0: +A).(\lambda (_: (arity g d u a0)).(\lambda (H2: ((\forall (c2: C).((cimp d +c2) \to (arity g c2 u a0))))).(\lambda (c2: C).(\lambda (H3: (cimp c +c2)).(let H_x \def (H3 Abbr d u i H0) in (let H4 \def H_x in (ex_ind C +(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (arity g c2 (TLRef i) +a0) (\lambda (x: C).(\lambda (H5: (getl i c2 (CHead x (Bind Abbr) u))).(let +H_x0 \def (cimp_getl_conf c c2 H3 Abbr d u i H0) in (let H6 \def H_x0 in +(ex2_ind C (\lambda (d2: C).(cimp d d2)) (\lambda (d2: C).(getl i c2 (CHead +d2 (Bind Abbr) u))) (arity g c2 (TLRef i) a0) (\lambda (x0: C).(\lambda (H7: +(cimp d x0)).(\lambda (H8: (getl i c2 (CHead x0 (Bind Abbr) u))).(let H9 \def +(eq_ind C (CHead x (Bind Abbr) u) (\lambda (c0: C).(getl i c2 c0)) H5 (CHead +x0 (Bind Abbr) u) (getl_mono c2 (CHead x (Bind Abbr) u) i H5 (CHead x0 (Bind +Abbr) u) H8)) in (let H10 \def (f_equal C C (\lambda (e: C).(match e in C +return (\lambda (_: C).C) with [(CSort _) \Rightarrow x | (CHead c0 _ _) +\Rightarrow c0])) (CHead x (Bind Abbr) u) (CHead x0 (Bind Abbr) u) (getl_mono +c2 (CHead x (Bind Abbr) u) i H5 (CHead x0 (Bind Abbr) u) H8)) in (let H11 +\def (eq_ind_r C x0 (\lambda (c0: C).(getl i c2 (CHead c0 (Bind Abbr) u))) H9 +x H10) in (let H12 \def (eq_ind_r C x0 (\lambda (c0: C).(cimp d c0)) H7 x +H10) in (arity_abbr g c2 x u i H11 a0 (H2 x H12))))))))) H6))))) +H4))))))))))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: +nat).(\lambda (H0: (getl i c (CHead d (Bind Abst) u))).(\lambda (a0: +A).(\lambda (_: (arity g d u (asucc g a0))).(\lambda (H2: ((\forall (c2: +C).((cimp d c2) \to (arity g c2 u (asucc g a0)))))).(\lambda (c2: C).(\lambda +(H3: (cimp c c2)).(let H_x \def (H3 Abst d u i H0) in (let H4 \def H_x in +(ex_ind C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) (arity g c2 +(TLRef i) a0) (\lambda (x: C).(\lambda (H5: (getl i c2 (CHead x (Bind Abst) +u))).(let H_x0 \def (cimp_getl_conf c c2 H3 Abst d u i H0) in (let H6 \def +H_x0 in (ex2_ind C (\lambda (d2: C).(cimp d d2)) (\lambda (d2: C).(getl i c2 +(CHead d2 (Bind Abst) u))) (arity g c2 (TLRef i) a0) (\lambda (x0: +C).(\lambda (H7: (cimp d x0)).(\lambda (H8: (getl i c2 (CHead x0 (Bind Abst) +u))).(let H9 \def (eq_ind C (CHead x (Bind Abst) u) (\lambda (c0: C).(getl i +c2 c0)) H5 (CHead x0 (Bind Abst) u) (getl_mono c2 (CHead x (Bind Abst) u) i +H5 (CHead x0 (Bind Abst) u) H8)) in (let H10 \def (f_equal C C (\lambda (e: +C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow x | +(CHead c0 _ _) \Rightarrow c0])) (CHead x (Bind Abst) u) (CHead x0 (Bind +Abst) u) (getl_mono c2 (CHead x (Bind Abst) u) i H5 (CHead x0 (Bind Abst) u) +H8)) in (let H11 \def (eq_ind_r C x0 (\lambda (c0: C).(getl i c2 (CHead c0 +(Bind Abst) u))) H9 x H10) in (let H12 \def (eq_ind_r C x0 (\lambda (c0: +C).(cimp d c0)) H7 x H10) in (arity_abst g c2 x u i H11 a0 (H2 x H12))))))))) +H6))))) H4))))))))))))) (\lambda (b: B).(\lambda (H0: (not (eq B b +Abst))).(\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity +g c u a1)).(\lambda (H2: ((\forall (c2: C).((cimp c c2) \to (arity g c2 u +a1))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c +(Bind b) u) t0 a2)).(\lambda (H4: ((\forall (c2: C).((cimp (CHead c (Bind b) +u) c2) \to (arity g c2 t0 a2))))).(\lambda (c2: C).(\lambda (H5: (cimp c +c2)).(arity_bind g b H0 c2 u a1 (H2 c2 H5) t0 a2 (H4 (CHead c2 (Bind b) u) +(cimp_bind c c2 H5 b u)))))))))))))))) (\lambda (c: C).(\lambda (u: +T).(\lambda (a1: A).(\lambda (_: (arity g c u (asucc g a1))).(\lambda (H1: +((\forall (c2: C).((cimp c c2) \to (arity g c2 u (asucc g a1)))))).(\lambda +(t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c (Bind Abst) u) t0 +a2)).(\lambda (H3: ((\forall (c2: C).((cimp (CHead c (Bind Abst) u) c2) \to +(arity g c2 t0 a2))))).(\lambda (c2: C).(\lambda (H4: (cimp c +c2)).(arity_head g c2 u a1 (H1 c2 H4) t0 a2 (H3 (CHead c2 (Bind Abst) u) +(cimp_bind c c2 H4 Abst u)))))))))))))) (\lambda (c: C).(\lambda (u: +T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H1: ((\forall +(c2: C).((cimp c c2) \to (arity g c2 u a1))))).(\lambda (t0: T).(\lambda (a2: +A).(\lambda (_: (arity g c t0 (AHead a1 a2))).(\lambda (H3: ((\forall (c2: +C).((cimp c c2) \to (arity g c2 t0 (AHead a1 a2)))))).(\lambda (c2: +C).(\lambda (H4: (cimp c c2)).(arity_appl g c2 u a1 (H1 c2 H4) t0 a2 (H3 c2 +H4))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a0: A).(\lambda (_: +(arity g c u (asucc g a0))).(\lambda (H1: ((\forall (c2: C).((cimp c c2) \to +(arity g c2 u (asucc g a0)))))).(\lambda (t0: T).(\lambda (_: (arity g c t0 +a0)).(\lambda (H3: ((\forall (c2: C).((cimp c c2) \to (arity g c2 t0 +a0))))).(\lambda (c2: C).(\lambda (H4: (cimp c c2)).(arity_cast g c2 u a0 (H1 +c2 H4) t0 (H3 c2 H4)))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda +(a1: A).(\lambda (_: (arity g c t0 a1)).(\lambda (H1: ((\forall (c2: +C).((cimp c c2) \to (arity g c2 t0 a1))))).(\lambda (a2: A).(\lambda (H2: +(leq g a1 a2)).(\lambda (c2: C).(\lambda (H3: (cimp c c2)).(arity_repl g c2 +t0 a1 (H1 c2 H3) a2 H2)))))))))) c1 t a H))))). +(* COMMENTS +Initial nodes: 1505 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/arity/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/arity/defs.ma new file mode 100644 index 000000000..99324dcbc --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/arity/defs.ma @@ -0,0 +1,45 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/leq/defs.ma". + +include "Basic-1/getl/defs.ma". + +inductive arity (g: G): C \to (T \to (A \to Prop)) \def +| arity_sort: \forall (c: C).(\forall (n: nat).(arity g c (TSort n) (ASort O +n))) +| arity_abbr: \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: +nat).((getl i c (CHead d (Bind Abbr) u)) \to (\forall (a: A).((arity g d u a) +\to (arity g c (TLRef i) a))))))) +| arity_abst: \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: +nat).((getl i c (CHead d (Bind Abst) u)) \to (\forall (a: A).((arity g d u +(asucc g a)) \to (arity g c (TLRef i) a))))))) +| arity_bind: \forall (b: B).((not (eq B b Abst)) \to (\forall (c: +C).(\forall (u: T).(\forall (a1: A).((arity g c u a1) \to (\forall (t: +T).(\forall (a2: A).((arity g (CHead c (Bind b) u) t a2) \to (arity g c +(THead (Bind b) u t) a2))))))))) +| arity_head: \forall (c: C).(\forall (u: T).(\forall (a1: A).((arity g c u +(asucc g a1)) \to (\forall (t: T).(\forall (a2: A).((arity g (CHead c (Bind +Abst) u) t a2) \to (arity g c (THead (Bind Abst) u t) (AHead a1 a2)))))))) +| arity_appl: \forall (c: C).(\forall (u: T).(\forall (a1: A).((arity g c u +a1) \to (\forall (t: T).(\forall (a2: A).((arity g c t (AHead a1 a2)) \to +(arity g c (THead (Flat Appl) u t) a2))))))) +| arity_cast: \forall (c: C).(\forall (u: T).(\forall (a: A).((arity g c u +(asucc g a)) \to (\forall (t: T).((arity g c t a) \to (arity g c (THead (Flat +Cast) u t) a)))))) +| arity_repl: \forall (c: C).(\forall (t: T).(\forall (a1: A).((arity g c t +a1) \to (\forall (a2: A).((leq g a1 a2) \to (arity g c t a2)))))). + diff --git a/matita/matita/contribs/lambdadelta/basic_1/arity/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/arity/fwd.ma new file mode 100644 index 000000000..31fa35f3c --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/arity/fwd.ma @@ -0,0 +1,1163 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/arity/defs.ma". + +include "Basic-1/leq/asucc.ma". + +include "Basic-1/getl/drop.ma". + +theorem arity_gen_sort: + \forall (g: G).(\forall (c: C).(\forall (n: nat).(\forall (a: A).((arity g c +(TSort n) a) \to (leq g a (ASort O n)))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (n: nat).(\lambda (a: A).(\lambda +(H: (arity g c (TSort n) a)).(insert_eq T (TSort n) (\lambda (t: T).(arity g +c t a)) (\lambda (_: T).(leq g a (ASort O n))) (\lambda (y: T).(\lambda (H0: +(arity g c y a)).(arity_ind g (\lambda (_: C).(\lambda (t: T).(\lambda (a0: +A).((eq T t (TSort n)) \to (leq g a0 (ASort O n)))))) (\lambda (_: +C).(\lambda (n0: nat).(\lambda (H1: (eq T (TSort n0) (TSort n))).(let H2 \def +(f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with +[(TSort n1) \Rightarrow n1 | (TLRef _) \Rightarrow n0 | (THead _ _ _) +\Rightarrow n0])) (TSort n0) (TSort n) H1) in (eq_ind_r nat n (\lambda (n1: +nat).(leq g (ASort O n1) (ASort O n))) (leq_refl g (ASort O n)) n0 H2))))) +(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda +(_: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (a0: A).(\lambda (_: (arity +g d u a0)).(\lambda (_: (((eq T u (TSort n)) \to (leq g a0 (ASort O +n))))).(\lambda (H4: (eq T (TLRef i) (TSort n))).(let H5 \def (eq_ind T +(TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with +[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) +\Rightarrow False])) I (TSort n) H4) in (False_ind (leq g a0 (ASort O n)) +H5))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: +nat).(\lambda (_: (getl i c0 (CHead d (Bind Abst) u))).(\lambda (a0: +A).(\lambda (_: (arity g d u (asucc g a0))).(\lambda (_: (((eq T u (TSort n)) +\to (leq g (asucc g a0) (ASort O n))))).(\lambda (H4: (eq T (TLRef i) (TSort +n))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return +(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (TSort n) H4) in +(False_ind (leq g a0 (ASort O n)) H5))))))))))) (\lambda (b: B).(\lambda (_: +(not (eq B b Abst))).(\lambda (c0: C).(\lambda (u: T).(\lambda (a1: +A).(\lambda (_: (arity g c0 u a1)).(\lambda (_: (((eq T u (TSort n)) \to (leq +g a1 (ASort O n))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (_: (arity g +(CHead c0 (Bind b) u) t a2)).(\lambda (_: (((eq T t (TSort n)) \to (leq g a2 +(ASort O n))))).(\lambda (H6: (eq T (THead (Bind b) u t) (TSort n))).(let H7 +\def (eq_ind T (THead (Bind b) u t) (\lambda (ee: T).(match ee in T return +(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H6) in +(False_ind (leq g a2 (ASort O n)) H7)))))))))))))) (\lambda (c0: C).(\lambda +(u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u (asucc g a1))).(\lambda +(_: (((eq T u (TSort n)) \to (leq g (asucc g a1) (ASort O n))))).(\lambda (t: +T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u) t +a2)).(\lambda (_: (((eq T t (TSort n)) \to (leq g a2 (ASort O n))))).(\lambda +(H5: (eq T (THead (Bind Abst) u t) (TSort n))).(let H6 \def (eq_ind T (THead +(Bind Abst) u t) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) +with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ +_) \Rightarrow True])) I (TSort n) H5) in (False_ind (leq g (AHead a1 a2) +(ASort O n)) H6)))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a1: +A).(\lambda (_: (arity g c0 u a1)).(\lambda (_: (((eq T u (TSort n)) \to (leq +g a1 (ASort O n))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (_: (arity g +c0 t (AHead a1 a2))).(\lambda (_: (((eq T t (TSort n)) \to (leq g (AHead a1 +a2) (ASort O n))))).(\lambda (H5: (eq T (THead (Flat Appl) u t) (TSort +n))).(let H6 \def (eq_ind T (THead (Flat Appl) u t) (\lambda (ee: T).(match +ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | +(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) +H5) in (False_ind (leq g a2 (ASort O n)) H6)))))))))))) (\lambda (c0: +C).(\lambda (u: T).(\lambda (a0: A).(\lambda (_: (arity g c0 u (asucc g +a0))).(\lambda (_: (((eq T u (TSort n)) \to (leq g (asucc g a0) (ASort O +n))))).(\lambda (t: T).(\lambda (_: (arity g c0 t a0)).(\lambda (_: (((eq T t +(TSort n)) \to (leq g a0 (ASort O n))))).(\lambda (H5: (eq T (THead (Flat +Cast) u t) (TSort n))).(let H6 \def (eq_ind T (THead (Flat Cast) u t) +(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) +\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow +True])) I (TSort n) H5) in (False_ind (leq g a0 (ASort O n)) H6))))))))))) +(\lambda (c0: C).(\lambda (t: T).(\lambda (a1: A).(\lambda (H1: (arity g c0 t +a1)).(\lambda (H2: (((eq T t (TSort n)) \to (leq g a1 (ASort O +n))))).(\lambda (a2: A).(\lambda (H3: (leq g a1 a2)).(\lambda (H4: (eq T t +(TSort n))).(let H5 \def (f_equal T T (\lambda (e: T).e) t (TSort n) H4) in +(let H6 \def (eq_ind T t (\lambda (t0: T).((eq T t0 (TSort n)) \to (leq g a1 +(ASort O n)))) H2 (TSort n) H5) in (let H7 \def (eq_ind T t (\lambda (t0: +T).(arity g c0 t0 a1)) H1 (TSort n) H5) in (leq_trans g a2 a1 (leq_sym g a1 +a2 H3) (ASort O n) (H6 (refl_equal T (TSort n))))))))))))))) c y a H0))) +H))))). +(* COMMENTS +Initial nodes: 1235 +END *) + +theorem arity_gen_lref: + \forall (g: G).(\forall (c: C).(\forall (i: nat).(\forall (a: A).((arity g c +(TLRef i) a) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c +(CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u a)))) +(ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c (CHead d (Bind Abst) +u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a)))))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (i: nat).(\lambda (a: A).(\lambda +(H: (arity g c (TLRef i) a)).(insert_eq T (TLRef i) (\lambda (t: T).(arity g +c t a)) (\lambda (_: T).(or (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl +i c (CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u +a)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c (CHead d (Bind +Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a))))))) +(\lambda (y: T).(\lambda (H0: (arity g c y a)).(arity_ind g (\lambda (c0: +C).(\lambda (t: T).(\lambda (a0: A).((eq T t (TLRef i)) \to (or (ex2_2 C T +(\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u)))) +(\lambda (d: C).(\lambda (u: T).(arity g d u a0)))) (ex2_2 C T (\lambda (d: +C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d: +C).(\lambda (u: T).(arity g d u (asucc g a0)))))))))) (\lambda (c0: +C).(\lambda (n: nat).(\lambda (H1: (eq T (TSort n) (TLRef i))).(let H2 \def +(eq_ind T (TSort n) (\lambda (ee: T).(match ee in T return (\lambda (_: +T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | +(THead _ _ _) \Rightarrow False])) I (TLRef i) H1) in (False_ind (or (ex2_2 C +T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u)))) +(\lambda (d: C).(\lambda (u: T).(arity g d u (ASort O n))))) (ex2_2 C T +(\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) +(\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g (ASort O n))))))) +H2))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i0: +nat).(\lambda (H1: (getl i0 c0 (CHead d (Bind Abbr) u))).(\lambda (a0: +A).(\lambda (H2: (arity g d u a0)).(\lambda (_: (((eq T u (TLRef i)) \to (or +(ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl i d (CHead d0 (Bind Abbr) +u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a0)))) (ex2_2 C T +(\lambda (d0: C).(\lambda (u0: T).(getl i d (CHead d0 (Bind Abst) u0)))) +(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g +a0))))))))).(\lambda (H4: (eq T (TLRef i0) (TLRef i))).(let H5 \def (f_equal +T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort +_) \Rightarrow i0 | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow i0])) +(TLRef i0) (TLRef i) H4) in (let H6 \def (eq_ind nat i0 (\lambda (n: +nat).(getl n c0 (CHead d (Bind Abbr) u))) H1 i H5) in (or_introl (ex2_2 C T +(\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abbr) u0)))) +(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a0)))) (ex2_2 C T (\lambda +(d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0)))) (\lambda +(d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a0))))) (ex2_2_intro C T +(\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abbr) u0)))) +(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a0))) d u H6 H2))))))))))))) +(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i0: nat).(\lambda +(H1: (getl i0 c0 (CHead d (Bind Abst) u))).(\lambda (a0: A).(\lambda (H2: +(arity g d u (asucc g a0))).(\lambda (_: (((eq T u (TLRef i)) \to (or (ex2_2 +C T (\lambda (d0: C).(\lambda (u0: T).(getl i d (CHead d0 (Bind Abbr) u0)))) +(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a0))))) (ex2_2 C T +(\lambda (d0: C).(\lambda (u0: T).(getl i d (CHead d0 (Bind Abst) u0)))) +(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g (asucc g +a0)))))))))).(\lambda (H4: (eq T (TLRef i0) (TLRef i))).(let H5 \def (f_equal +T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort +_) \Rightarrow i0 | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow i0])) +(TLRef i0) (TLRef i) H4) in (let H6 \def (eq_ind nat i0 (\lambda (n: +nat).(getl n c0 (CHead d (Bind Abst) u))) H1 i H5) in (or_intror (ex2_2 C T +(\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abbr) u0)))) +(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a0)))) (ex2_2 C T (\lambda +(d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0)))) (\lambda +(d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a0))))) (ex2_2_intro C T +(\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0)))) +(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a0)))) d u H6 +H2))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda +(c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u +a1)).(\lambda (_: (((eq T u (TLRef i)) \to (or (ex2_2 C T (\lambda (d: +C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d: +C).(\lambda (u0: T).(arity g d u0 a1)))) (ex2_2 C T (\lambda (d: C).(\lambda +(u0: T).(getl i c0 (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0: +T).(arity g d u0 (asucc g a1))))))))).(\lambda (t: T).(\lambda (a2: +A).(\lambda (_: (arity g (CHead c0 (Bind b) u) t a2)).(\lambda (_: (((eq T t +(TLRef i)) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i (CHead +c0 (Bind b) u) (CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: +T).(arity g d u0 a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i +(CHead c0 (Bind b) u) (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda +(u0: T).(arity g d u0 (asucc g a2))))))))).(\lambda (H6: (eq T (THead (Bind +b) u t) (TLRef i))).(let H7 \def (eq_ind T (THead (Bind b) u t) (\lambda (ee: +T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow +False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I +(TLRef i) H6) in (False_ind (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: +T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: +T).(arity g d u0 a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i +c0 (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 +(asucc g a2)))))) H7)))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda +(a1: A).(\lambda (_: (arity g c0 u (asucc g a1))).(\lambda (_: (((eq T u +(TLRef i)) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 +(CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 +(asucc g a1))))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 +(CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 +(asucc g (asucc g a1)))))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (_: +(arity g (CHead c0 (Bind Abst) u) t a2)).(\lambda (_: (((eq T t (TLRef i)) +\to (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i (CHead c0 (Bind +Abst) u) (CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity +g d u0 a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i (CHead c0 +(Bind Abst) u) (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0: +T).(arity g d u0 (asucc g a2))))))))).(\lambda (H5: (eq T (THead (Bind Abst) +u t) (TLRef i))).(let H6 \def (eq_ind T (THead (Bind Abst) u t) (\lambda (ee: +T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow +False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I +(TLRef i) H5) in (False_ind (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: +T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: +T).(arity g d u0 (AHead a1 a2))))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: +T).(getl i c0 (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0: +T).(arity g d u0 (asucc g (AHead a1 a2))))))) H6)))))))))))) (\lambda (c0: +C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u a1)).(\lambda +(_: (((eq T u (TLRef i)) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: +T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: +T).(arity g d u0 a1)))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i +c0 (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 +(asucc g a1))))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (_: (arity g +c0 t (AHead a1 a2))).(\lambda (_: (((eq T t (TLRef i)) \to (or (ex2_2 C T +(\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abbr) u0)))) +(\lambda (d: C).(\lambda (u0: T).(arity g d u0 (AHead a1 a2))))) (ex2_2 C T +(\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abst) u0)))) +(\lambda (d: C).(\lambda (u0: T).(arity g d u0 (asucc g (AHead a1 +a2)))))))))).(\lambda (H5: (eq T (THead (Flat Appl) u t) (TLRef i))).(let H6 +\def (eq_ind T (THead (Flat Appl) u t) (\lambda (ee: T).(match ee in T return +(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef i) H5) in +(False_ind (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead +d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 a2)))) +(ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abst) +u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 (asucc g a2)))))) +H6)))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a0: A).(\lambda (_: +(arity g c0 u (asucc g a0))).(\lambda (_: (((eq T u (TLRef i)) \to (or (ex2_2 +C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abbr) u0)))) +(\lambda (d: C).(\lambda (u0: T).(arity g d u0 (asucc g a0))))) (ex2_2 C T +(\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abst) u0)))) +(\lambda (d: C).(\lambda (u0: T).(arity g d u0 (asucc g (asucc g +a0)))))))))).(\lambda (t: T).(\lambda (_: (arity g c0 t a0)).(\lambda (_: +(((eq T t (TLRef i)) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: +T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: +T).(arity g d u0 a0)))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i +c0 (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 +(asucc g a0))))))))).(\lambda (H5: (eq T (THead (Flat Cast) u t) (TLRef +i))).(let H6 \def (eq_ind T (THead (Flat Cast) u t) (\lambda (ee: T).(match +ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | +(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef i) +H5) in (False_ind (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 +(CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 +a0)))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind +Abst) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 (asucc g a0)))))) +H6))))))))))) (\lambda (c0: C).(\lambda (t: T).(\lambda (a1: A).(\lambda (H1: +(arity g c0 t a1)).(\lambda (H2: (((eq T t (TLRef i)) \to (or (ex2_2 C T +(\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u)))) +(\lambda (d: C).(\lambda (u: T).(arity g d u a1)))) (ex2_2 C T (\lambda (d: +C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d: +C).(\lambda (u: T).(arity g d u (asucc g a1))))))))).(\lambda (a2: +A).(\lambda (H3: (leq g a1 a2)).(\lambda (H4: (eq T t (TLRef i))).(let H5 +\def (f_equal T T (\lambda (e: T).e) t (TLRef i) H4) in (let H6 \def (eq_ind +T t (\lambda (t0: T).((eq T t0 (TLRef i)) \to (or (ex2_2 C T (\lambda (d: +C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u)))) (\lambda (d: +C).(\lambda (u: T).(arity g d u a1)))) (ex2_2 C T (\lambda (d: C).(\lambda +(u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u: +T).(arity g d u (asucc g a1)))))))) H2 (TLRef i) H5) in (let H7 \def (eq_ind +T t (\lambda (t0: T).(arity g c0 t0 a1)) H1 (TLRef i) H5) in (let H8 \def (H6 +(refl_equal T (TLRef i))) in (or_ind (ex2_2 C T (\lambda (d: C).(\lambda (u: +T).(getl i c0 (CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: +T).(arity g d u a1)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 +(CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u +(asucc g a1))))) (or (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 +(CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u +a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind +Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a2)))))) +(\lambda (H9: (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d +(Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u +a1))))).(ex2_2_ind C T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d +(Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u a1))) (or +(ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) +u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u a2)))) (ex2_2 C T (\lambda +(d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d: +C).(\lambda (u: T).(arity g d u (asucc g a2)))))) (\lambda (x0: C).(\lambda +(x1: T).(\lambda (H10: (getl i c0 (CHead x0 (Bind Abbr) x1))).(\lambda (H11: +(arity g x0 x1 a1)).(or_introl (ex2_2 C T (\lambda (d: C).(\lambda (u: +T).(getl i c0 (CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: +T).(arity g d u a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 +(CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u +(asucc g a2))))) (ex2_2_intro C T (\lambda (d: C).(\lambda (u: T).(getl i c0 +(CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u a2))) +x0 x1 H10 (arity_repl g x0 x1 a1 H11 a2 H3))))))) H9)) (\lambda (H9: (ex2_2 C +T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) +(\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a1)))))).(ex2_2_ind C T +(\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) +(\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a1)))) (or (ex2_2 C T +(\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u)))) +(\lambda (d: C).(\lambda (u: T).(arity g d u a2)))) (ex2_2 C T (\lambda (d: +C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d: +C).(\lambda (u: T).(arity g d u (asucc g a2)))))) (\lambda (x0: C).(\lambda +(x1: T).(\lambda (H10: (getl i c0 (CHead x0 (Bind Abst) x1))).(\lambda (H11: +(arity g x0 x1 (asucc g a1))).(or_intror (ex2_2 C T (\lambda (d: C).(\lambda +(u: T).(getl i c0 (CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: +T).(arity g d u a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 +(CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u +(asucc g a2))))) (ex2_2_intro C T (\lambda (d: C).(\lambda (u: T).(getl i c0 +(CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u +(asucc g a2)))) x0 x1 H10 (arity_repl g x0 x1 (asucc g a1) H11 (asucc g a2) +(asucc_repl g a1 a2 H3)))))))) H9)) H8))))))))))))) c y a H0))) H))))). +(* COMMENTS +Initial nodes: 3853 +END *) + +theorem arity_gen_bind: + \forall (b: B).((not (eq B b Abst)) \to (\forall (g: G).(\forall (c: +C).(\forall (u: T).(\forall (t: T).(\forall (a2: A).((arity g c (THead (Bind +b) u t) a2) \to (ex2 A (\lambda (a1: A).(arity g c u a1)) (\lambda (_: +A).(arity g (CHead c (Bind b) u) t a2)))))))))) +\def + \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (g: G).(\lambda +(c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (a2: A).(\lambda (H0: (arity +g c (THead (Bind b) u t) a2)).(insert_eq T (THead (Bind b) u t) (\lambda (t0: +T).(arity g c t0 a2)) (\lambda (_: T).(ex2 A (\lambda (a1: A).(arity g c u +a1)) (\lambda (_: A).(arity g (CHead c (Bind b) u) t a2)))) (\lambda (y: +T).(\lambda (H1: (arity g c y a2)).(arity_ind g (\lambda (c0: C).(\lambda +(t0: T).(\lambda (a: A).((eq T t0 (THead (Bind b) u t)) \to (ex2 A (\lambda +(a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t +a))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (H2: (eq T (TSort n) +(THead (Bind b) u t))).(let H3 \def (eq_ind T (TSort n) (\lambda (ee: +T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow +True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I +(THead (Bind b) u t) H2) in (False_ind (ex2 A (\lambda (a1: A).(arity g c0 u +a1)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t (ASort O n)))) H3))))) +(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda +(_: (getl i c0 (CHead d (Bind Abbr) u0))).(\lambda (a: A).(\lambda (_: (arity +g d u0 a)).(\lambda (_: (((eq T u0 (THead (Bind b) u t)) \to (ex2 A (\lambda +(a1: A).(arity g d u a1)) (\lambda (_: A).(arity g (CHead d (Bind b) u) t +a)))))).(\lambda (H5: (eq T (TLRef i) (THead (Bind b) u t))).(let H6 \def +(eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_: +T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | +(THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t) H5) in (False_ind +(ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g (CHead c0 +(Bind b) u) t a))) H6))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda +(u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind Abst) +u0))).(\lambda (a: A).(\lambda (_: (arity g d u0 (asucc g a))).(\lambda (_: +(((eq T u0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a1: A).(arity g d u +a1)) (\lambda (_: A).(arity g (CHead d (Bind b) u) t (asucc g +a))))))).(\lambda (H5: (eq T (TLRef i) (THead (Bind b) u t))).(let H6 \def +(eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_: +T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | +(THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t) H5) in (False_ind +(ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g (CHead c0 +(Bind b) u) t a))) H6))))))))))) (\lambda (b0: B).(\lambda (H2: (not (eq B b0 +Abst))).(\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (H3: +(arity g c0 u0 a1)).(\lambda (H4: (((eq T u0 (THead (Bind b) u t)) \to (ex2 A +(\lambda (a3: A).(arity g c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind +b) u) t a1)))))).(\lambda (t0: T).(\lambda (a0: A).(\lambda (H5: (arity g +(CHead c0 (Bind b0) u0) t0 a0)).(\lambda (H6: (((eq T t0 (THead (Bind b) u +t)) \to (ex2 A (\lambda (a3: A).(arity g (CHead c0 (Bind b0) u0) u a3)) +(\lambda (_: A).(arity g (CHead (CHead c0 (Bind b0) u0) (Bind b) u) t +a0)))))).(\lambda (H7: (eq T (THead (Bind b0) u0 t0) (THead (Bind b) u +t))).(let H8 \def (f_equal T B (\lambda (e: T).(match e in T return (\lambda +(_: T).B) with [(TSort _) \Rightarrow b0 | (TLRef _) \Rightarrow b0 | (THead +k _ _) \Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b1) +\Rightarrow b1 | (Flat _) \Rightarrow b0])])) (THead (Bind b0) u0 t0) (THead +(Bind b) u t) H7) in ((let H9 \def (f_equal T T (\lambda (e: T).(match e in T +return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) +\Rightarrow u0 | (THead _ t1 _) \Rightarrow t1])) (THead (Bind b0) u0 t0) +(THead (Bind b) u t) H7) in ((let H10 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | +(TLRef _) \Rightarrow t0 | (THead _ _ t1) \Rightarrow t1])) (THead (Bind b0) +u0 t0) (THead (Bind b) u t) H7) in (\lambda (H11: (eq T u0 u)).(\lambda (H12: +(eq B b0 b)).(let H13 \def (eq_ind T t0 (\lambda (t1: T).((eq T t1 (THead +(Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g (CHead c0 (Bind b0) u0) u +a3)) (\lambda (_: A).(arity g (CHead (CHead c0 (Bind b0) u0) (Bind b) u) t +a0))))) H6 t H10) in (let H14 \def (eq_ind T t0 (\lambda (t1: T).(arity g +(CHead c0 (Bind b0) u0) t1 a0)) H5 t H10) in (let H15 \def (eq_ind T u0 +(\lambda (t1: T).((eq T t (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: +A).(arity g (CHead c0 (Bind b0) t1) u a3)) (\lambda (_: A).(arity g (CHead +(CHead c0 (Bind b0) t1) (Bind b) u) t a0))))) H13 u H11) in (let H16 \def +(eq_ind T u0 (\lambda (t1: T).(arity g (CHead c0 (Bind b0) t1) t a0)) H14 u +H11) in (let H17 \def (eq_ind T u0 (\lambda (t1: T).((eq T t1 (THead (Bind b) +u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (_: A).(arity g +(CHead c0 (Bind b) u) t a1))))) H4 u H11) in (let H18 \def (eq_ind T u0 +(\lambda (t1: T).(arity g c0 t1 a1)) H3 u H11) in (let H19 \def (eq_ind B b0 +(\lambda (b1: B).((eq T t (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: +A).(arity g (CHead c0 (Bind b1) u) u a3)) (\lambda (_: A).(arity g (CHead +(CHead c0 (Bind b1) u) (Bind b) u) t a0))))) H15 b H12) in (let H20 \def +(eq_ind B b0 (\lambda (b1: B).(arity g (CHead c0 (Bind b1) u) t a0)) H16 b +H12) in (let H21 \def (eq_ind B b0 (\lambda (b1: B).(not (eq B b1 Abst))) H2 +b H12) in (ex_intro2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (_: +A).(arity g (CHead c0 (Bind b) u) t a0)) a1 H18 H20))))))))))))) H9)) +H8)))))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: A).(\lambda +(H2: (arity g c0 u0 (asucc g a1))).(\lambda (H3: (((eq T u0 (THead (Bind b) u +t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (_: A).(arity g +(CHead c0 (Bind b) u) t (asucc g a1))))))).(\lambda (t0: T).(\lambda (a0: +A).(\lambda (H4: (arity g (CHead c0 (Bind Abst) u0) t0 a0)).(\lambda (H5: +(((eq T t0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g (CHead +c0 (Bind Abst) u0) u a3)) (\lambda (_: A).(arity g (CHead (CHead c0 (Bind +Abst) u0) (Bind b) u) t a0)))))).(\lambda (H6: (eq T (THead (Bind Abst) u0 +t0) (THead (Bind b) u t))).(let H7 \def (f_equal T B (\lambda (e: T).(match e +in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow Abst | (TLRef _) +\Rightarrow Abst | (THead k _ _) \Rightarrow (match k in K return (\lambda +(_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abst])])) +(THead (Bind Abst) u0 t0) (THead (Bind b) u t) H6) in ((let H8 \def (f_equal +T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t1 _) \Rightarrow t1])) +(THead (Bind Abst) u0 t0) (THead (Bind b) u t) H6) in ((let H9 \def (f_equal +T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t1) \Rightarrow t1])) +(THead (Bind Abst) u0 t0) (THead (Bind b) u t) H6) in (\lambda (H10: (eq T u0 +u)).(\lambda (H11: (eq B Abst b)).(let H12 \def (eq_ind T t0 (\lambda (t1: +T).((eq T t1 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g +(CHead c0 (Bind Abst) u0) u a3)) (\lambda (_: A).(arity g (CHead (CHead c0 +(Bind Abst) u0) (Bind b) u) t a0))))) H5 t H9) in (let H13 \def (eq_ind T t0 +(\lambda (t1: T).(arity g (CHead c0 (Bind Abst) u0) t1 a0)) H4 t H9) in (let +H14 \def (eq_ind T u0 (\lambda (t1: T).((eq T t (THead (Bind b) u t)) \to +(ex2 A (\lambda (a3: A).(arity g (CHead c0 (Bind Abst) t1) u a3)) (\lambda +(_: A).(arity g (CHead (CHead c0 (Bind Abst) t1) (Bind b) u) t a0))))) H12 u +H10) in (let H15 \def (eq_ind T u0 (\lambda (t1: T).(arity g (CHead c0 (Bind +Abst) t1) t a0)) H13 u H10) in (let H16 \def (eq_ind T u0 (\lambda (t1: +T).((eq T t1 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u +a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t (asucc g a1)))))) H3 u +H10) in (let H17 \def (eq_ind T u0 (\lambda (t1: T).(arity g c0 t1 (asucc g +a1))) H2 u H10) in (let H18 \def (eq_ind_r B b (\lambda (b0: B).((eq T t +(THead (Bind b0) u t)) \to (ex2 A (\lambda (a3: A).(arity g (CHead c0 (Bind +Abst) u) u a3)) (\lambda (_: A).(arity g (CHead (CHead c0 (Bind Abst) u) +(Bind b0) u) t a0))))) H14 Abst H11) in (let H19 \def (eq_ind_r B b (\lambda +(b0: B).((eq T u (THead (Bind b0) u t)) \to (ex2 A (\lambda (a3: A).(arity g +c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b0) u) t (asucc g a1)))))) +H16 Abst H11) in (let H20 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 +Abst))) H Abst H11) in (eq_ind B Abst (\lambda (b0: B).(ex2 A (\lambda (a3: +A).(arity g c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b0) u) t +(AHead a1 a0))))) (let H21 \def (match (H20 (refl_equal B Abst)) in False +return (\lambda (_: False).(ex2 A (\lambda (a3: A).(arity g c0 u a3)) +(\lambda (_: A).(arity g (CHead c0 (Bind Abst) u) t (AHead a1 a0))))) with +[]) in H21) b H11))))))))))))) H8)) H7)))))))))))) (\lambda (c0: C).(\lambda +(u0: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u0 a1)).(\lambda (_: (((eq +T u0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) +(\lambda (_: A).(arity g (CHead c0 (Bind b) u) t a1)))))).(\lambda (t0: +T).(\lambda (a0: A).(\lambda (_: (arity g c0 t0 (AHead a1 a0))).(\lambda (_: +(((eq T t0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u +a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t (AHead a1 +a0))))))).(\lambda (H6: (eq T (THead (Flat Appl) u0 t0) (THead (Bind b) u +t))).(let H7 \def (eq_ind T (THead (Flat Appl) u0 t0) (\lambda (ee: T).(match +ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | +(TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return +(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow +True])])) I (THead (Bind b) u t) H6) in (False_ind (ex2 A (\lambda (a3: +A).(arity g c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t a0))) +H7)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a: A).(\lambda (_: +(arity g c0 u0 (asucc g a))).(\lambda (_: (((eq T u0 (THead (Bind b) u t)) +\to (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g +(CHead c0 (Bind b) u) t (asucc g a))))))).(\lambda (t0: T).(\lambda (_: +(arity g c0 t0 a)).(\lambda (_: (((eq T t0 (THead (Bind b) u t)) \to (ex2 A +(\lambda (a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g (CHead c0 (Bind +b) u) t a)))))).(\lambda (H6: (eq T (THead (Flat Cast) u0 t0) (THead (Bind b) +u t))).(let H7 \def (eq_ind T (THead (Flat Cast) u0 t0) (\lambda (ee: +T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow +False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K +return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) +\Rightarrow True])])) I (THead (Bind b) u t) H6) in (False_ind (ex2 A +(\lambda (a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g (CHead c0 (Bind +b) u) t a))) H7))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (a1: +A).(\lambda (H2: (arity g c0 t0 a1)).(\lambda (H3: (((eq T t0 (THead (Bind b) +u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (_: A).(arity g +(CHead c0 (Bind b) u) t a1)))))).(\lambda (a0: A).(\lambda (H4: (leq g a1 +a0)).(\lambda (H5: (eq T t0 (THead (Bind b) u t))).(let H6 \def (f_equal T T +(\lambda (e: T).e) t0 (THead (Bind b) u t) H5) in (let H7 \def (eq_ind T t0 +(\lambda (t1: T).((eq T t1 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: +A).(arity g c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t +a1))))) H3 (THead (Bind b) u t) H6) in (let H8 \def (eq_ind T t0 (\lambda +(t1: T).(arity g c0 t1 a1)) H2 (THead (Bind b) u t) H6) in (let H9 \def (H7 +(refl_equal T (THead (Bind b) u t))) in (ex2_ind A (\lambda (a3: A).(arity g +c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t a1)) (ex2 A +(\lambda (a3: A).(arity g c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind +b) u) t a0))) (\lambda (x: A).(\lambda (H10: (arity g c0 u x)).(\lambda (H11: +(arity g (CHead c0 (Bind b) u) t a1)).(ex_intro2 A (\lambda (a3: A).(arity g +c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t a0)) x H10 +(arity_repl g (CHead c0 (Bind b) u) t a1 H11 a0 H4))))) H9))))))))))))) c y +a2 H1))) H0)))))))). +(* COMMENTS +Initial nodes: 3365 +END *) + +theorem arity_gen_abst: + \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (a: +A).((arity g c (THead (Bind Abst) u t) a) \to (ex3_2 A A (\lambda (a1: +A).(\lambda (a2: A).(eq A a (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: +A).(arity g c u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g +(CHead c (Bind Abst) u) t a2))))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (a: +A).(\lambda (H: (arity g c (THead (Bind Abst) u t) a)).(insert_eq T (THead +(Bind Abst) u t) (\lambda (t0: T).(arity g c t0 a)) (\lambda (_: T).(ex3_2 A +A (\lambda (a1: A).(\lambda (a2: A).(eq A a (AHead a1 a2)))) (\lambda (a1: +A).(\lambda (_: A).(arity g c u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: +A).(arity g (CHead c (Bind Abst) u) t a2))))) (\lambda (y: T).(\lambda (H0: +(arity g c y a)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (a0: +A).((eq T t0 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a1: +A).(\lambda (a2: A).(eq A a0 (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: +A).(arity g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g +(CHead c0 (Bind Abst) u) t a2)))))))) (\lambda (c0: C).(\lambda (n: +nat).(\lambda (H1: (eq T (TSort n) (THead (Bind Abst) u t))).(let H2 \def +(eq_ind T (TSort n) (\lambda (ee: T).(match ee in T return (\lambda (_: +T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | +(THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) u t) H1) in +(False_ind (ex3_2 A A (\lambda (a1: A).(\lambda (a2: A).(eq A (ASort O n) +(AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g c0 u (asucc g +a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g (CHead c0 (Bind Abst) u) t +a2)))) H2))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: +nat).(\lambda (_: (getl i c0 (CHead d (Bind Abbr) u0))).(\lambda (a0: +A).(\lambda (_: (arity g d u0 a0)).(\lambda (_: (((eq T u0 (THead (Bind Abst) +u t)) \to (ex3_2 A A (\lambda (a1: A).(\lambda (a2: A).(eq A a0 (AHead a1 +a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g d u (asucc g a1)))) (\lambda +(_: A).(\lambda (a2: A).(arity g (CHead d (Bind Abst) u) t a2))))))).(\lambda +(H4: (eq T (TLRef i) (THead (Bind Abst) u t))).(let H5 \def (eq_ind T (TLRef +i) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort +_) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow +False])) I (THead (Bind Abst) u t) H4) in (False_ind (ex3_2 A A (\lambda (a1: +A).(\lambda (a2: A).(eq A a0 (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: +A).(arity g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g +(CHead c0 (Bind Abst) u) t a2)))) H5))))))))))) (\lambda (c0: C).(\lambda (d: +C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind +Abst) u0))).(\lambda (a0: A).(\lambda (_: (arity g d u0 (asucc g +a0))).(\lambda (_: (((eq T u0 (THead (Bind Abst) u t)) \to (ex3_2 A A +(\lambda (a1: A).(\lambda (a2: A).(eq A (asucc g a0) (AHead a1 a2)))) +(\lambda (a1: A).(\lambda (_: A).(arity g d u (asucc g a1)))) (\lambda (_: +A).(\lambda (a2: A).(arity g (CHead d (Bind Abst) u) t a2))))))).(\lambda +(H4: (eq T (TLRef i) (THead (Bind Abst) u t))).(let H5 \def (eq_ind T (TLRef +i) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort +_) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow +False])) I (THead (Bind Abst) u t) H4) in (False_ind (ex3_2 A A (\lambda (a1: +A).(\lambda (a2: A).(eq A a0 (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: +A).(arity g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g +(CHead c0 (Bind Abst) u) t a2)))) H5))))))))))) (\lambda (b: B).(\lambda (H1: +(not (eq B b Abst))).(\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: +A).(\lambda (H2: (arity g c0 u0 a1)).(\lambda (H3: (((eq T u0 (THead (Bind +Abst) u t)) \to (ex3_2 A A (\lambda (a2: A).(\lambda (a3: A).(eq A a1 (AHead +a2 a3)))) (\lambda (a2: A).(\lambda (_: A).(arity g c0 u (asucc g a2)))) +(\lambda (_: A).(\lambda (a3: A).(arity g (CHead c0 (Bind Abst) u) t +a3))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (H4: (arity g (CHead c0 +(Bind b) u0) t0 a2)).(\lambda (H5: (((eq T t0 (THead (Bind Abst) u t)) \to +(ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) +(\lambda (a3: A).(\lambda (_: A).(arity g (CHead c0 (Bind b) u0) u (asucc g +a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead (CHead c0 (Bind b) +u0) (Bind Abst) u) t a4))))))).(\lambda (H6: (eq T (THead (Bind b) u0 t0) +(THead (Bind Abst) u t))).(let H7 \def (f_equal T B (\lambda (e: T).(match e +in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | (TLRef _) +\Rightarrow b | (THead k _ _) \Rightarrow (match k in K return (\lambda (_: +K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow b])])) (THead +(Bind b) u0 t0) (THead (Bind Abst) u t) H6) in ((let H8 \def (f_equal T T +(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t1 _) \Rightarrow t1])) +(THead (Bind b) u0 t0) (THead (Bind Abst) u t) H6) in ((let H9 \def (f_equal +T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t1) \Rightarrow t1])) +(THead (Bind b) u0 t0) (THead (Bind Abst) u t) H6) in (\lambda (H10: (eq T u0 +u)).(\lambda (H11: (eq B b Abst)).(let H12 \def (eq_ind T t0 (\lambda (t1: +T).((eq T t1 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3: +A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: +A).(arity g (CHead c0 (Bind b) u0) u (asucc g a3)))) (\lambda (_: A).(\lambda +(a4: A).(arity g (CHead (CHead c0 (Bind b) u0) (Bind Abst) u) t a4)))))) H5 t +H9) in (let H13 \def (eq_ind T t0 (\lambda (t1: T).(arity g (CHead c0 (Bind +b) u0) t1 a2)) H4 t H9) in (let H14 \def (eq_ind T u0 (\lambda (t1: T).((eq T +t (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: +A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g (CHead +c0 (Bind b) t1) u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g +(CHead (CHead c0 (Bind b) t1) (Bind Abst) u) t a4)))))) H12 u H10) in (let +H15 \def (eq_ind T u0 (\lambda (t1: T).(arity g (CHead c0 (Bind b) t1) t a2)) +H13 u H10) in (let H16 \def (eq_ind T u0 (\lambda (t1: T).((eq T t1 (THead +(Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a1 +(AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g +a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t +a4)))))) H3 u H10) in (let H17 \def (eq_ind T u0 (\lambda (t1: T).(arity g c0 +t1 a1)) H2 u H10) in (let H18 \def (eq_ind B b (\lambda (b0: B).((eq T t +(THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq +A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g (CHead c0 +(Bind b0) u) u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g +(CHead (CHead c0 (Bind b0) u) (Bind Abst) u) t a4)))))) H14 Abst H11) in (let +H19 \def (eq_ind B b (\lambda (b0: B).(arity g (CHead c0 (Bind b0) u) t a2)) +H15 Abst H11) in (let H20 \def (eq_ind B b (\lambda (b0: B).(not (eq B b0 +Abst))) H1 Abst H11) in (let H21 \def (match (H20 (refl_equal B Abst)) in +False return (\lambda (_: False).(ex3_2 A A (\lambda (a3: A).(\lambda (a4: +A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u +(asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind +Abst) u) t a4))))) with []) in H21))))))))))))) H8)) H7)))))))))))))) +(\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (H1: (arity g c0 +u0 (asucc g a1))).(\lambda (H2: (((eq T u0 (THead (Bind Abst) u t)) \to +(ex3_2 A A (\lambda (a2: A).(\lambda (a3: A).(eq A (asucc g a1) (AHead a2 +a3)))) (\lambda (a2: A).(\lambda (_: A).(arity g c0 u (asucc g a2)))) +(\lambda (_: A).(\lambda (a3: A).(arity g (CHead c0 (Bind Abst) u) t +a3))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (H3: (arity g (CHead c0 +(Bind Abst) u0) t0 a2)).(\lambda (H4: (((eq T t0 (THead (Bind Abst) u t)) \to +(ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) +(\lambda (a3: A).(\lambda (_: A).(arity g (CHead c0 (Bind Abst) u0) u (asucc +g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead (CHead c0 (Bind +Abst) u0) (Bind Abst) u) t a4))))))).(\lambda (H5: (eq T (THead (Bind Abst) +u0 t0) (THead (Bind Abst) u t))).(let H6 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | +(TLRef _) \Rightarrow u0 | (THead _ t1 _) \Rightarrow t1])) (THead (Bind +Abst) u0 t0) (THead (Bind Abst) u t) H5) in ((let H7 \def (f_equal T T +(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t1) \Rightarrow t1])) +(THead (Bind Abst) u0 t0) (THead (Bind Abst) u t) H5) in (\lambda (H8: (eq T +u0 u)).(let H9 \def (eq_ind T t0 (\lambda (t1: T).((eq T t1 (THead (Bind +Abst) u t)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead +a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g (CHead c0 (Bind Abst) u0) +u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead (CHead c0 +(Bind Abst) u0) (Bind Abst) u) t a4)))))) H4 t H7) in (let H10 \def (eq_ind T +t0 (\lambda (t1: T).(arity g (CHead c0 (Bind Abst) u0) t1 a2)) H3 t H7) in +(let H11 \def (eq_ind T u0 (\lambda (t1: T).((eq T t (THead (Bind Abst) u t)) +\to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) +(\lambda (a3: A).(\lambda (_: A).(arity g (CHead c0 (Bind Abst) t1) u (asucc +g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead (CHead c0 (Bind +Abst) t1) (Bind Abst) u) t a4)))))) H9 u H8) in (let H12 \def (eq_ind T u0 +(\lambda (t1: T).(arity g (CHead c0 (Bind Abst) t1) t a2)) H10 u H8) in (let +H13 \def (eq_ind T u0 (\lambda (t1: T).((eq T t1 (THead (Bind Abst) u t)) \to +(ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A (asucc g a1) (AHead a3 +a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g a3)))) +(\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t a4)))))) +H2 u H8) in (let H14 \def (eq_ind T u0 (\lambda (t1: T).(arity g c0 t1 (asucc +g a1))) H1 u H8) in (ex3_2_intro A A (\lambda (a3: A).(\lambda (a4: A).(eq A +(AHead a1 a2) (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u +(asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind +Abst) u) t a4))) a1 a2 (refl_equal A (AHead a1 a2)) H14 H12))))))))) +H6)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: A).(\lambda +(_: (arity g c0 u0 a1)).(\lambda (_: (((eq T u0 (THead (Bind Abst) u t)) \to +(ex3_2 A A (\lambda (a2: A).(\lambda (a3: A).(eq A a1 (AHead a2 a3)))) +(\lambda (a2: A).(\lambda (_: A).(arity g c0 u (asucc g a2)))) (\lambda (_: +A).(\lambda (a3: A).(arity g (CHead c0 (Bind Abst) u) t a3))))))).(\lambda +(t0: T).(\lambda (a2: A).(\lambda (_: (arity g c0 t0 (AHead a1 a2))).(\lambda +(_: (((eq T t0 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3: +A).(\lambda (a4: A).(eq A (AHead a1 a2) (AHead a3 a4)))) (\lambda (a3: +A).(\lambda (_: A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda +(a4: A).(arity g (CHead c0 (Bind Abst) u) t a4))))))).(\lambda (H5: (eq T +(THead (Flat Appl) u0 t0) (THead (Bind Abst) u t))).(let H6 \def (eq_ind T +(THead (Flat Appl) u0 t0) (\lambda (ee: T).(match ee in T return (\lambda (_: +T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | +(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with +[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind +Abst) u t) H5) in (False_ind (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq +A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g +a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t +a4)))) H6)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a0: +A).(\lambda (_: (arity g c0 u0 (asucc g a0))).(\lambda (_: (((eq T u0 (THead +(Bind Abst) u t)) \to (ex3_2 A A (\lambda (a1: A).(\lambda (a2: A).(eq A +(asucc g a0) (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g c0 u +(asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g (CHead c0 (Bind +Abst) u) t a2))))))).(\lambda (t0: T).(\lambda (_: (arity g c0 t0 +a0)).(\lambda (_: (((eq T t0 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda +(a1: A).(\lambda (a2: A).(eq A a0 (AHead a1 a2)))) (\lambda (a1: A).(\lambda +(_: A).(arity g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity +g (CHead c0 (Bind Abst) u) t a2))))))).(\lambda (H5: (eq T (THead (Flat Cast) +u0 t0) (THead (Bind Abst) u t))).(let H6 \def (eq_ind T (THead (Flat Cast) u0 +t0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort +_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) +\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) +\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u t) +H5) in (False_ind (ex3_2 A A (\lambda (a1: A).(\lambda (a2: A).(eq A a0 +(AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g c0 u (asucc g +a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g (CHead c0 (Bind Abst) u) t +a2)))) H6))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (a1: +A).(\lambda (H1: (arity g c0 t0 a1)).(\lambda (H2: (((eq T t0 (THead (Bind +Abst) u t)) \to (ex3_2 A A (\lambda (a2: A).(\lambda (a3: A).(eq A a1 (AHead +a2 a3)))) (\lambda (a2: A).(\lambda (_: A).(arity g c0 u (asucc g a2)))) +(\lambda (_: A).(\lambda (a3: A).(arity g (CHead c0 (Bind Abst) u) t +a3))))))).(\lambda (a2: A).(\lambda (H3: (leq g a1 a2)).(\lambda (H4: (eq T +t0 (THead (Bind Abst) u t))).(let H5 \def (f_equal T T (\lambda (e: T).e) t0 +(THead (Bind Abst) u t) H4) in (let H6 \def (eq_ind T t0 (\lambda (t1: +T).((eq T t1 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3: +A).(\lambda (a4: A).(eq A a1 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: +A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g +(CHead c0 (Bind Abst) u) t a4)))))) H2 (THead (Bind Abst) u t) H5) in (let H7 +\def (eq_ind T t0 (\lambda (t1: T).(arity g c0 t1 a1)) H1 (THead (Bind Abst) +u t) H5) in (let H8 \def (H6 (refl_equal T (THead (Bind Abst) u t))) in +(ex3_2_ind A A (\lambda (a3: A).(\lambda (a4: A).(eq A a1 (AHead a3 a4)))) +(\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g a3)))) (\lambda (_: +A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t a4))) (ex3_2 A A +(\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: +A).(\lambda (_: A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda +(a4: A).(arity g (CHead c0 (Bind Abst) u) t a4)))) (\lambda (x0: A).(\lambda +(x1: A).(\lambda (H9: (eq A a1 (AHead x0 x1))).(\lambda (H10: (arity g c0 u +(asucc g x0))).(\lambda (H11: (arity g (CHead c0 (Bind Abst) u) t x1)).(let +H12 \def (eq_ind A a1 (\lambda (a0: A).(leq g a0 a2)) H3 (AHead x0 x1) H9) in +(let H13 \def (eq_ind A a1 (\lambda (a0: A).(arity g c0 (THead (Bind Abst) u +t) a0)) H7 (AHead x0 x1) H9) in (let H_x \def (leq_gen_head1 g x0 x1 a2 H12) +in (let H14 \def H_x in (ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq +g x0 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g x1 a4))) (\lambda (a3: +A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) (ex3_2 A A (\lambda (a3: +A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: +A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g +(CHead c0 (Bind Abst) u) t a4)))) (\lambda (x2: A).(\lambda (x3: A).(\lambda +(H15: (leq g x0 x2)).(\lambda (H16: (leq g x1 x3)).(\lambda (H17: (eq A a2 +(AHead x2 x3))).(let H18 \def (f_equal A A (\lambda (e: A).e) a2 (AHead x2 +x3) H17) in (eq_ind_r A (AHead x2 x3) (\lambda (a0: A).(ex3_2 A A (\lambda +(a3: A).(\lambda (a4: A).(eq A a0 (AHead a3 a4)))) (\lambda (a3: A).(\lambda +(_: A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity +g (CHead c0 (Bind Abst) u) t a4))))) (ex3_2_intro A A (\lambda (a3: +A).(\lambda (a4: A).(eq A (AHead x2 x3) (AHead a3 a4)))) (\lambda (a3: +A).(\lambda (_: A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda +(a4: A).(arity g (CHead c0 (Bind Abst) u) t a4))) x2 x3 (refl_equal A (AHead +x2 x3)) (arity_repl g c0 u (asucc g x0) H10 (asucc g x2) (asucc_repl g x0 x2 +H15)) (arity_repl g (CHead c0 (Bind Abst) u) t x1 H11 x3 H16)) a2 H18))))))) +H14)))))))))) H8))))))))))))) c y a H0))) H)))))). +(* COMMENTS +Initial nodes: 4265 +END *) + +theorem arity_gen_appl: + \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (a2: +A).((arity g c (THead (Flat Appl) u t) a2) \to (ex2 A (\lambda (a1: A).(arity +g c u a1)) (\lambda (a1: A).(arity g c t (AHead a1 a2))))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (a2: +A).(\lambda (H: (arity g c (THead (Flat Appl) u t) a2)).(insert_eq T (THead +(Flat Appl) u t) (\lambda (t0: T).(arity g c t0 a2)) (\lambda (_: T).(ex2 A +(\lambda (a1: A).(arity g c u a1)) (\lambda (a1: A).(arity g c t (AHead a1 +a2))))) (\lambda (y: T).(\lambda (H0: (arity g c y a2)).(arity_ind g (\lambda +(c0: C).(\lambda (t0: T).(\lambda (a: A).((eq T t0 (THead (Flat Appl) u t)) +\to (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t +(AHead a1 a)))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (H1: (eq T +(TSort n) (THead (Flat Appl) u t))).(let H2 \def (eq_ind T (TSort n) (\lambda +(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) +\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow +False])) I (THead (Flat Appl) u t) H1) in (False_ind (ex2 A (\lambda (a1: +A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t (AHead a1 (ASort O +n))))) H2))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: +nat).(\lambda (_: (getl i c0 (CHead d (Bind Abbr) u0))).(\lambda (a: +A).(\lambda (_: (arity g d u0 a)).(\lambda (_: (((eq T u0 (THead (Flat Appl) +u t)) \to (ex2 A (\lambda (a1: A).(arity g d u a1)) (\lambda (a1: A).(arity g +d t (AHead a1 a))))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat Appl) u +t))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return +(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) u +t) H4) in (False_ind (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: +A).(arity g c0 t (AHead a1 a)))) H5))))))))))) (\lambda (c0: C).(\lambda (d: +C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind +Abst) u0))).(\lambda (a: A).(\lambda (_: (arity g d u0 (asucc g a))).(\lambda +(_: (((eq T u0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda (a1: A).(arity g +d u a1)) (\lambda (a1: A).(arity g d t (AHead a1 (asucc g a)))))))).(\lambda +(H4: (eq T (TLRef i) (THead (Flat Appl) u t))).(let H5 \def (eq_ind T (TLRef +i) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort +_) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow +False])) I (THead (Flat Appl) u t) H4) in (False_ind (ex2 A (\lambda (a1: +A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t (AHead a1 a)))) +H5))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (c0: +C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u0 +a1)).(\lambda (_: (((eq T u0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda +(a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3 +a1))))))).(\lambda (t0: T).(\lambda (a0: A).(\lambda (_: (arity g (CHead c0 +(Bind b) u0) t0 a0)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u t)) \to +(ex2 A (\lambda (a3: A).(arity g (CHead c0 (Bind b) u0) u a3)) (\lambda (a3: +A).(arity g (CHead c0 (Bind b) u0) t (AHead a3 a0))))))).(\lambda (H6: (eq T +(THead (Bind b) u0 t0) (THead (Flat Appl) u t))).(let H7 \def (eq_ind T +(THead (Bind b) u0 t0) (\lambda (ee: T).(match ee in T return (\lambda (_: +T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | +(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with +[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat +Appl) u t) H6) in (False_ind (ex2 A (\lambda (a3: A).(arity g c0 u a3)) +(\lambda (a3: A).(arity g c0 t (AHead a3 a0)))) H7)))))))))))))) (\lambda +(c0: C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u0 (asucc +g a1))).(\lambda (_: (((eq T u0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda +(a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3 (asucc g +a1)))))))).(\lambda (t0: T).(\lambda (a0: A).(\lambda (_: (arity g (CHead c0 +(Bind Abst) u0) t0 a0)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u t)) \to +(ex2 A (\lambda (a3: A).(arity g (CHead c0 (Bind Abst) u0) u a3)) (\lambda +(a3: A).(arity g (CHead c0 (Bind Abst) u0) t (AHead a3 a0))))))).(\lambda +(H5: (eq T (THead (Bind Abst) u0 t0) (THead (Flat Appl) u t))).(let H6 \def +(eq_ind T (THead (Bind Abst) u0 t0) (\lambda (ee: T).(match ee in T return +(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda +(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow +False])])) I (THead (Flat Appl) u t) H5) in (False_ind (ex2 A (\lambda (a3: +A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3 (AHead a1 +a0))))) H6)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: +A).(\lambda (H1: (arity g c0 u0 a1)).(\lambda (H2: (((eq T u0 (THead (Flat +Appl) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3: +A).(arity g c0 t (AHead a3 a1))))))).(\lambda (t0: T).(\lambda (a0: +A).(\lambda (H3: (arity g c0 t0 (AHead a1 a0))).(\lambda (H4: (((eq T t0 +(THead (Flat Appl) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) +(\lambda (a3: A).(arity g c0 t (AHead a3 (AHead a1 a0)))))))).(\lambda (H5: +(eq T (THead (Flat Appl) u0 t0) (THead (Flat Appl) u t))).(let H6 \def +(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with +[(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t1 _) +\Rightarrow t1])) (THead (Flat Appl) u0 t0) (THead (Flat Appl) u t) H5) in +((let H7 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: +T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ +t1) \Rightarrow t1])) (THead (Flat Appl) u0 t0) (THead (Flat Appl) u t) H5) +in (\lambda (H8: (eq T u0 u)).(let H9 \def (eq_ind T t0 (\lambda (t1: T).((eq +T t1 (THead (Flat Appl) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) +(\lambda (a3: A).(arity g c0 t (AHead a3 (AHead a1 a0))))))) H4 t H7) in (let +H10 \def (eq_ind T t0 (\lambda (t1: T).(arity g c0 t1 (AHead a1 a0))) H3 t +H7) in (let H11 \def (eq_ind T u0 (\lambda (t1: T).((eq T t1 (THead (Flat +Appl) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3: +A).(arity g c0 t (AHead a3 a1)))))) H2 u H8) in (let H12 \def (eq_ind T u0 +(\lambda (t1: T).(arity g c0 t1 a1)) H1 u H8) in (ex_intro2 A (\lambda (a3: +A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3 a0))) a1 H12 +H10))))))) H6)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a: +A).(\lambda (_: (arity g c0 u0 (asucc g a))).(\lambda (_: (((eq T u0 (THead +(Flat Appl) u t)) \to (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda +(a1: A).(arity g c0 t (AHead a1 (asucc g a)))))))).(\lambda (t0: T).(\lambda +(_: (arity g c0 t0 a)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u t)) \to +(ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t +(AHead a1 a))))))).(\lambda (H5: (eq T (THead (Flat Cast) u0 t0) (THead (Flat +Appl) u t))).(let H6 \def (eq_ind T (THead (Flat Cast) u0 t0) (\lambda (ee: +T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow +False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K +return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) +\Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl \Rightarrow +False | Cast \Rightarrow True])])])) I (THead (Flat Appl) u t) H5) in +(False_ind (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity +g c0 t (AHead a1 a)))) H6))))))))))) (\lambda (c0: C).(\lambda (t0: +T).(\lambda (a1: A).(\lambda (H1: (arity g c0 t0 a1)).(\lambda (H2: (((eq T +t0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) +(\lambda (a3: A).(arity g c0 t (AHead a3 a1))))))).(\lambda (a0: A).(\lambda +(H3: (leq g a1 a0)).(\lambda (H4: (eq T t0 (THead (Flat Appl) u t))).(let H5 +\def (f_equal T T (\lambda (e: T).e) t0 (THead (Flat Appl) u t) H4) in (let +H6 \def (eq_ind T t0 (\lambda (t1: T).((eq T t1 (THead (Flat Appl) u t)) \to +(ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t +(AHead a3 a1)))))) H2 (THead (Flat Appl) u t) H5) in (let H7 \def (eq_ind T +t0 (\lambda (t1: T).(arity g c0 t1 a1)) H1 (THead (Flat Appl) u t) H5) in +(let H8 \def (H6 (refl_equal T (THead (Flat Appl) u t))) in (ex2_ind A +(\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3 +a1))) (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 +t (AHead a3 a0)))) (\lambda (x: A).(\lambda (H9: (arity g c0 u x)).(\lambda +(H10: (arity g c0 t (AHead x a1))).(ex_intro2 A (\lambda (a3: A).(arity g c0 +u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3 a0))) x H9 (arity_repl g c0 t +(AHead x a1) H10 (AHead x a0) (leq_head g x x (leq_refl g x) a1 a0 H3)))))) +H8))))))))))))) c y a2 H0))) H)))))). +(* COMMENTS +Initial nodes: 2277 +END *) + +theorem arity_gen_cast: + \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (a: +A).((arity g c (THead (Flat Cast) u t) a) \to (land (arity g c u (asucc g a)) +(arity g c t a))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (a: +A).(\lambda (H: (arity g c (THead (Flat Cast) u t) a)).(insert_eq T (THead +(Flat Cast) u t) (\lambda (t0: T).(arity g c t0 a)) (\lambda (_: T).(land +(arity g c u (asucc g a)) (arity g c t a))) (\lambda (y: T).(\lambda (H0: +(arity g c y a)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (a0: +A).((eq T t0 (THead (Flat Cast) u t)) \to (land (arity g c0 u (asucc g a0)) +(arity g c0 t a0)))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (H1: (eq T +(TSort n) (THead (Flat Cast) u t))).(let H2 \def (eq_ind T (TSort n) (\lambda +(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) +\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow +False])) I (THead (Flat Cast) u t) H1) in (False_ind (land (arity g c0 u +(asucc g (ASort O n))) (arity g c0 t (ASort O n))) H2))))) (\lambda (c0: +C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0 +(CHead d (Bind Abbr) u0))).(\lambda (a0: A).(\lambda (_: (arity g d u0 +a0)).(\lambda (_: (((eq T u0 (THead (Flat Cast) u t)) \to (land (arity g d u +(asucc g a0)) (arity g d t a0))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat +Cast) u t))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T +return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) u +t) H4) in (False_ind (land (arity g c0 u (asucc g a0)) (arity g c0 t a0)) +H5))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: +nat).(\lambda (_: (getl i c0 (CHead d (Bind Abst) u0))).(\lambda (a0: +A).(\lambda (_: (arity g d u0 (asucc g a0))).(\lambda (_: (((eq T u0 (THead +(Flat Cast) u t)) \to (land (arity g d u (asucc g (asucc g a0))) (arity g d t +(asucc g a0)))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat Cast) u +t))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return +(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) u +t) H4) in (False_ind (land (arity g c0 u (asucc g a0)) (arity g c0 t a0)) +H5))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (c0: +C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u0 +a1)).(\lambda (_: (((eq T u0 (THead (Flat Cast) u t)) \to (land (arity g c0 u +(asucc g a1)) (arity g c0 t a1))))).(\lambda (t0: T).(\lambda (a2: +A).(\lambda (_: (arity g (CHead c0 (Bind b) u0) t0 a2)).(\lambda (_: (((eq T +t0 (THead (Flat Cast) u t)) \to (land (arity g (CHead c0 (Bind b) u0) u +(asucc g a2)) (arity g (CHead c0 (Bind b) u0) t a2))))).(\lambda (H6: (eq T +(THead (Bind b) u0 t0) (THead (Flat Cast) u t))).(let H7 \def (eq_ind T +(THead (Bind b) u0 t0) (\lambda (ee: T).(match ee in T return (\lambda (_: +T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | +(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with +[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat +Cast) u t) H6) in (False_ind (land (arity g c0 u (asucc g a2)) (arity g c0 t +a2)) H7)))))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: +A).(\lambda (_: (arity g c0 u0 (asucc g a1))).(\lambda (_: (((eq T u0 (THead +(Flat Cast) u t)) \to (land (arity g c0 u (asucc g (asucc g a1))) (arity g c0 +t (asucc g a1)))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g +(CHead c0 (Bind Abst) u0) t0 a2)).(\lambda (_: (((eq T t0 (THead (Flat Cast) +u t)) \to (land (arity g (CHead c0 (Bind Abst) u0) u (asucc g a2)) (arity g +(CHead c0 (Bind Abst) u0) t a2))))).(\lambda (H5: (eq T (THead (Bind Abst) u0 +t0) (THead (Flat Cast) u t))).(let H6 \def (eq_ind T (THead (Bind Abst) u0 +t0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort +_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) +\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) +\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Cast) u t) +H5) in (False_ind (land (arity g c0 u (asucc g (AHead a1 a2))) (arity g c0 t +(AHead a1 a2))) H6)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda +(a1: A).(\lambda (_: (arity g c0 u0 a1)).(\lambda (_: (((eq T u0 (THead (Flat +Cast) u t)) \to (land (arity g c0 u (asucc g a1)) (arity g c0 t +a1))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g c0 t0 (AHead +a1 a2))).(\lambda (_: (((eq T t0 (THead (Flat Cast) u t)) \to (land (arity g +c0 u (asucc g (AHead a1 a2))) (arity g c0 t (AHead a1 a2)))))).(\lambda (H5: +(eq T (THead (Flat Appl) u0 t0) (THead (Flat Cast) u t))).(let H6 \def +(eq_ind T (THead (Flat Appl) u0 t0) (\lambda (ee: T).(match ee in T return +(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda +(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f +in F return (\lambda (_: F).Prop) with [Appl \Rightarrow True | Cast +\Rightarrow False])])])) I (THead (Flat Cast) u t) H5) in (False_ind (land +(arity g c0 u (asucc g a2)) (arity g c0 t a2)) H6)))))))))))) (\lambda (c0: +C).(\lambda (u0: T).(\lambda (a0: A).(\lambda (H1: (arity g c0 u0 (asucc g +a0))).(\lambda (H2: (((eq T u0 (THead (Flat Cast) u t)) \to (land (arity g c0 +u (asucc g (asucc g a0))) (arity g c0 t (asucc g a0)))))).(\lambda (t0: +T).(\lambda (H3: (arity g c0 t0 a0)).(\lambda (H4: (((eq T t0 (THead (Flat +Cast) u t)) \to (land (arity g c0 u (asucc g a0)) (arity g c0 t +a0))))).(\lambda (H5: (eq T (THead (Flat Cast) u0 t0) (THead (Flat Cast) u +t))).(let H6 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda +(_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead +_ t1 _) \Rightarrow t1])) (THead (Flat Cast) u0 t0) (THead (Flat Cast) u t) +H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T return +(\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 +| (THead _ _ t1) \Rightarrow t1])) (THead (Flat Cast) u0 t0) (THead (Flat +Cast) u t) H5) in (\lambda (H8: (eq T u0 u)).(let H9 \def (eq_ind T t0 +(\lambda (t1: T).((eq T t1 (THead (Flat Cast) u t)) \to (land (arity g c0 u +(asucc g a0)) (arity g c0 t a0)))) H4 t H7) in (let H10 \def (eq_ind T t0 +(\lambda (t1: T).(arity g c0 t1 a0)) H3 t H7) in (let H11 \def (eq_ind T u0 +(\lambda (t1: T).((eq T t1 (THead (Flat Cast) u t)) \to (land (arity g c0 u +(asucc g (asucc g a0))) (arity g c0 t (asucc g a0))))) H2 u H8) in (let H12 +\def (eq_ind T u0 (\lambda (t1: T).(arity g c0 t1 (asucc g a0))) H1 u H8) in +(conj (arity g c0 u (asucc g a0)) (arity g c0 t a0) H12 H10))))))) +H6))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda +(H1: (arity g c0 t0 a1)).(\lambda (H2: (((eq T t0 (THead (Flat Cast) u t)) +\to (land (arity g c0 u (asucc g a1)) (arity g c0 t a1))))).(\lambda (a2: +A).(\lambda (H3: (leq g a1 a2)).(\lambda (H4: (eq T t0 (THead (Flat Cast) u +t))).(let H5 \def (f_equal T T (\lambda (e: T).e) t0 (THead (Flat Cast) u t) +H4) in (let H6 \def (eq_ind T t0 (\lambda (t1: T).((eq T t1 (THead (Flat +Cast) u t)) \to (land (arity g c0 u (asucc g a1)) (arity g c0 t a1)))) H2 +(THead (Flat Cast) u t) H5) in (let H7 \def (eq_ind T t0 (\lambda (t1: +T).(arity g c0 t1 a1)) H1 (THead (Flat Cast) u t) H5) in (let H8 \def (H6 +(refl_equal T (THead (Flat Cast) u t))) in (land_ind (arity g c0 u (asucc g +a1)) (arity g c0 t a1) (land (arity g c0 u (asucc g a2)) (arity g c0 t a2)) +(\lambda (H9: (arity g c0 u (asucc g a1))).(\lambda (H10: (arity g c0 t +a1)).(conj (arity g c0 u (asucc g a2)) (arity g c0 t a2) (arity_repl g c0 u +(asucc g a1) H9 (asucc g a2) (asucc_repl g a1 a2 H3)) (arity_repl g c0 t a1 +H10 a2 H3)))) H8))))))))))))) c y a H0))) H)))))). +(* COMMENTS +Initial nodes: 2147 +END *) + +theorem arity_gen_appls: + \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (vs: TList).(\forall +(a2: A).((arity g c (THeads (Flat Appl) vs t) a2) \to (ex A (\lambda (a: +A).(arity g c t a)))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (vs: +TList).(TList_ind (\lambda (t0: TList).(\forall (a2: A).((arity g c (THeads +(Flat Appl) t0 t) a2) \to (ex A (\lambda (a: A).(arity g c t a)))))) (\lambda +(a2: A).(\lambda (H: (arity g c t a2)).(ex_intro A (\lambda (a: A).(arity g c +t a)) a2 H))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H: ((\forall +(a2: A).((arity g c (THeads (Flat Appl) t1 t) a2) \to (ex A (\lambda (a: +A).(arity g c t a))))))).(\lambda (a2: A).(\lambda (H0: (arity g c (THead +(Flat Appl) t0 (THeads (Flat Appl) t1 t)) a2)).(let H1 \def (arity_gen_appl g +c t0 (THeads (Flat Appl) t1 t) a2 H0) in (ex2_ind A (\lambda (a1: A).(arity g +c t0 a1)) (\lambda (a1: A).(arity g c (THeads (Flat Appl) t1 t) (AHead a1 +a2))) (ex A (\lambda (a: A).(arity g c t a))) (\lambda (x: A).(\lambda (_: +(arity g c t0 x)).(\lambda (H3: (arity g c (THeads (Flat Appl) t1 t) (AHead x +a2))).(let H_x \def (H (AHead x a2) H3) in (let H4 \def H_x in (ex_ind A +(\lambda (a: A).(arity g c t a)) (ex A (\lambda (a: A).(arity g c t a))) +(\lambda (x0: A).(\lambda (H5: (arity g c t x0)).(ex_intro A (\lambda (a: +A).(arity g c t a)) x0 H5))) H4)))))) H1))))))) vs)))). +(* COMMENTS +Initial nodes: 341 +END *) + +theorem arity_gen_lift: + \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).(\forall (h: +nat).(\forall (d: nat).((arity g c1 (lift h d t) a) \to (\forall (c2: +C).((drop h d c1 c2) \to (arity g c2 t a))))))))) +\def + \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (a: A).(\lambda (h: +nat).(\lambda (d: nat).(\lambda (H: (arity g c1 (lift h d t) a)).(insert_eq T +(lift h d t) (\lambda (t0: T).(arity g c1 t0 a)) (\lambda (_: T).(\forall +(c2: C).((drop h d c1 c2) \to (arity g c2 t a)))) (\lambda (y: T).(\lambda +(H0: (arity g c1 y a)).(unintro T t (\lambda (t0: T).((eq T y (lift h d t0)) +\to (\forall (c2: C).((drop h d c1 c2) \to (arity g c2 t0 a))))) (unintro nat +d (\lambda (n: nat).(\forall (x: T).((eq T y (lift h n x)) \to (\forall (c2: +C).((drop h n c1 c2) \to (arity g c2 x a)))))) (arity_ind g (\lambda (c: +C).(\lambda (t0: T).(\lambda (a0: A).(\forall (x: nat).(\forall (x0: T).((eq +T t0 (lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to (arity g c2 x0 +a0))))))))) (\lambda (c: C).(\lambda (n: nat).(\lambda (x: nat).(\lambda (x0: +T).(\lambda (H1: (eq T (TSort n) (lift h x x0))).(\lambda (c2: C).(\lambda +(_: (drop h x c c2)).(eq_ind_r T (TSort n) (\lambda (t0: T).(arity g c2 t0 +(ASort O n))) (arity_sort g c2 n) x0 (lift_gen_sort h x n x0 H1))))))))) +(\lambda (c: C).(\lambda (d0: C).(\lambda (u: T).(\lambda (i: nat).(\lambda +(H1: (getl i c (CHead d0 (Bind Abbr) u))).(\lambda (a0: A).(\lambda (H2: +(arity g d0 u a0)).(\lambda (H3: ((\forall (x: nat).(\forall (x0: T).((eq T u +(lift h x x0)) \to (\forall (c2: C).((drop h x d0 c2) \to (arity g c2 x0 +a0)))))))).(\lambda (x: nat).(\lambda (x0: T).(\lambda (H4: (eq T (TLRef i) +(lift h x x0))).(\lambda (c2: C).(\lambda (H5: (drop h x c c2)).(let H_x \def +(lift_gen_lref x0 x h i H4) in (let H6 \def H_x in (or_ind (land (lt i x) (eq +T x0 (TLRef i))) (land (le (plus x h) i) (eq T x0 (TLRef (minus i h)))) +(arity g c2 x0 a0) (\lambda (H7: (land (lt i x) (eq T x0 (TLRef +i)))).(land_ind (lt i x) (eq T x0 (TLRef i)) (arity g c2 x0 a0) (\lambda (H8: +(lt i x)).(\lambda (H9: (eq T x0 (TLRef i))).(eq_ind_r T (TLRef i) (\lambda +(t0: T).(arity g c2 t0 a0)) (let H10 \def (eq_ind nat x (\lambda (n: +nat).(drop h n c c2)) H5 (S (plus i (minus x (S i)))) (lt_plus_minus i x H8)) +in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (minus x (S +i)) v)))) (\lambda (v: T).(\lambda (e0: C).(getl i c2 (CHead e0 (Bind Abbr) +v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (minus x (S i)) d0 e0))) +(arity g c2 (TLRef i) a0) (\lambda (x1: T).(\lambda (x2: C).(\lambda (H11: +(eq T u (lift h (minus x (S i)) x1))).(\lambda (H12: (getl i c2 (CHead x2 +(Bind Abbr) x1))).(\lambda (H13: (drop h (minus x (S i)) d0 x2)).(let H14 +\def (eq_ind T u (\lambda (t0: T).(\forall (x3: nat).(\forall (x4: T).((eq T +t0 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 d0 c3) \to (arity g c3 x4 +a0))))))) H3 (lift h (minus x (S i)) x1) H11) in (let H15 \def (eq_ind T u +(\lambda (t0: T).(arity g d0 t0 a0)) H2 (lift h (minus x (S i)) x1) H11) in +(arity_abbr g c2 x2 x1 i H12 a0 (H14 (minus x (S i)) x1 (refl_equal T (lift h +(minus x (S i)) x1)) x2 H13))))))))) (getl_drop_conf_lt Abbr c d0 u i H1 c2 h +(minus x (S i)) H10))) x0 H9))) H7)) (\lambda (H7: (land (le (plus x h) i) +(eq T x0 (TLRef (minus i h))))).(land_ind (le (plus x h) i) (eq T x0 (TLRef +(minus i h))) (arity g c2 x0 a0) (\lambda (H8: (le (plus x h) i)).(\lambda +(H9: (eq T x0 (TLRef (minus i h)))).(eq_ind_r T (TLRef (minus i h)) (\lambda +(t0: T).(arity g c2 t0 a0)) (arity_abbr g c2 d0 u (minus i h) +(getl_drop_conf_ge i (CHead d0 (Bind Abbr) u) c H1 c2 h x H5 H8) a0 H2) x0 +H9))) H7)) H6)))))))))))))))) (\lambda (c: C).(\lambda (d0: C).(\lambda (u: +T).(\lambda (i: nat).(\lambda (H1: (getl i c (CHead d0 (Bind Abst) +u))).(\lambda (a0: A).(\lambda (H2: (arity g d0 u (asucc g a0))).(\lambda +(H3: ((\forall (x: nat).(\forall (x0: T).((eq T u (lift h x x0)) \to (\forall +(c2: C).((drop h x d0 c2) \to (arity g c2 x0 (asucc g a0))))))))).(\lambda +(x: nat).(\lambda (x0: T).(\lambda (H4: (eq T (TLRef i) (lift h x +x0))).(\lambda (c2: C).(\lambda (H5: (drop h x c c2)).(let H_x \def +(lift_gen_lref x0 x h i H4) in (let H6 \def H_x in (or_ind (land (lt i x) (eq +T x0 (TLRef i))) (land (le (plus x h) i) (eq T x0 (TLRef (minus i h)))) +(arity g c2 x0 a0) (\lambda (H7: (land (lt i x) (eq T x0 (TLRef +i)))).(land_ind (lt i x) (eq T x0 (TLRef i)) (arity g c2 x0 a0) (\lambda (H8: +(lt i x)).(\lambda (H9: (eq T x0 (TLRef i))).(eq_ind_r T (TLRef i) (\lambda +(t0: T).(arity g c2 t0 a0)) (let H10 \def (eq_ind nat x (\lambda (n: +nat).(drop h n c c2)) H5 (S (plus i (minus x (S i)))) (lt_plus_minus i x H8)) +in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (minus x (S +i)) v)))) (\lambda (v: T).(\lambda (e0: C).(getl i c2 (CHead e0 (Bind Abst) +v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (minus x (S i)) d0 e0))) +(arity g c2 (TLRef i) a0) (\lambda (x1: T).(\lambda (x2: C).(\lambda (H11: +(eq T u (lift h (minus x (S i)) x1))).(\lambda (H12: (getl i c2 (CHead x2 +(Bind Abst) x1))).(\lambda (H13: (drop h (minus x (S i)) d0 x2)).(let H14 +\def (eq_ind T u (\lambda (t0: T).(\forall (x3: nat).(\forall (x4: T).((eq T +t0 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 d0 c3) \to (arity g c3 x4 +(asucc g a0)))))))) H3 (lift h (minus x (S i)) x1) H11) in (let H15 \def +(eq_ind T u (\lambda (t0: T).(arity g d0 t0 (asucc g a0))) H2 (lift h (minus +x (S i)) x1) H11) in (arity_abst g c2 x2 x1 i H12 a0 (H14 (minus x (S i)) x1 +(refl_equal T (lift h (minus x (S i)) x1)) x2 H13))))))))) (getl_drop_conf_lt +Abst c d0 u i H1 c2 h (minus x (S i)) H10))) x0 H9))) H7)) (\lambda (H7: +(land (le (plus x h) i) (eq T x0 (TLRef (minus i h))))).(land_ind (le (plus x +h) i) (eq T x0 (TLRef (minus i h))) (arity g c2 x0 a0) (\lambda (H8: (le +(plus x h) i)).(\lambda (H9: (eq T x0 (TLRef (minus i h)))).(eq_ind_r T +(TLRef (minus i h)) (\lambda (t0: T).(arity g c2 t0 a0)) (arity_abst g c2 d0 +u (minus i h) (getl_drop_conf_ge i (CHead d0 (Bind Abst) u) c H1 c2 h x H5 +H8) a0 H2) x0 H9))) H7)) H6)))))))))))))))) (\lambda (b: B).(\lambda (H1: +(not (eq B b Abst))).(\lambda (c: C).(\lambda (u: T).(\lambda (a1: +A).(\lambda (H2: (arity g c u a1)).(\lambda (H3: ((\forall (x: nat).(\forall +(x0: T).((eq T u (lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to +(arity g c2 x0 a1)))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (H4: +(arity g (CHead c (Bind b) u) t0 a2)).(\lambda (H5: ((\forall (x: +nat).(\forall (x0: T).((eq T t0 (lift h x x0)) \to (\forall (c2: C).((drop h +x (CHead c (Bind b) u) c2) \to (arity g c2 x0 a2)))))))).(\lambda (x: +nat).(\lambda (x0: T).(\lambda (H6: (eq T (THead (Bind b) u t0) (lift h x +x0))).(\lambda (c2: C).(\lambda (H7: (drop h x c c2)).(ex3_2_ind T T (\lambda +(y0: T).(\lambda (z: T).(eq T x0 (THead (Bind b) y0 z)))) (\lambda (y0: +T).(\lambda (_: T).(eq T u (lift h x y0)))) (\lambda (_: T).(\lambda (z: +T).(eq T t0 (lift h (S x) z)))) (arity g c2 x0 a2) (\lambda (x1: T).(\lambda +(x2: T).(\lambda (H8: (eq T x0 (THead (Bind b) x1 x2))).(\lambda (H9: (eq T u +(lift h x x1))).(\lambda (H10: (eq T t0 (lift h (S x) x2))).(eq_ind_r T +(THead (Bind b) x1 x2) (\lambda (t1: T).(arity g c2 t1 a2)) (let H11 \def +(eq_ind T t0 (\lambda (t1: T).(\forall (x3: nat).(\forall (x4: T).((eq T t1 +(lift h x3 x4)) \to (\forall (c3: C).((drop h x3 (CHead c (Bind b) u) c3) \to +(arity g c3 x4 a2))))))) H5 (lift h (S x) x2) H10) in (let H12 \def (eq_ind T +t0 (\lambda (t1: T).(arity g (CHead c (Bind b) u) t1 a2)) H4 (lift h (S x) +x2) H10) in (let H13 \def (eq_ind T u (\lambda (t1: T).(arity g (CHead c +(Bind b) t1) (lift h (S x) x2) a2)) H12 (lift h x x1) H9) in (let H14 \def +(eq_ind T u (\lambda (t1: T).(\forall (x3: nat).(\forall (x4: T).((eq T (lift +h (S x) x2) (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 (CHead c (Bind +b) t1) c3) \to (arity g c3 x4 a2))))))) H11 (lift h x x1) H9) in (let H15 +\def (eq_ind T u (\lambda (t1: T).(\forall (x3: nat).(\forall (x4: T).((eq T +t1 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 c c3) \to (arity g c3 x4 +a1))))))) H3 (lift h x x1) H9) in (let H16 \def (eq_ind T u (\lambda (t1: +T).(arity g c t1 a1)) H2 (lift h x x1) H9) in (arity_bind g b H1 c2 x1 a1 +(H15 x x1 (refl_equal T (lift h x x1)) c2 H7) x2 a2 (H14 (S x) x2 (refl_equal +T (lift h (S x) x2)) (CHead c2 (Bind b) x1) (drop_skip_bind h x c c2 H7 b +x1))))))))) x0 H8)))))) (lift_gen_bind b u t0 x0 h x H6)))))))))))))))))) +(\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H1: (arity g c u +(asucc g a1))).(\lambda (H2: ((\forall (x: nat).(\forall (x0: T).((eq T u +(lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to (arity g c2 x0 +(asucc g a1))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (H3: (arity g +(CHead c (Bind Abst) u) t0 a2)).(\lambda (H4: ((\forall (x: nat).(\forall +(x0: T).((eq T t0 (lift h x x0)) \to (\forall (c2: C).((drop h x (CHead c +(Bind Abst) u) c2) \to (arity g c2 x0 a2)))))))).(\lambda (x: nat).(\lambda +(x0: T).(\lambda (H5: (eq T (THead (Bind Abst) u t0) (lift h x x0))).(\lambda +(c2: C).(\lambda (H6: (drop h x c c2)).(ex3_2_ind T T (\lambda (y0: +T).(\lambda (z: T).(eq T x0 (THead (Bind Abst) y0 z)))) (\lambda (y0: +T).(\lambda (_: T).(eq T u (lift h x y0)))) (\lambda (_: T).(\lambda (z: +T).(eq T t0 (lift h (S x) z)))) (arity g c2 x0 (AHead a1 a2)) (\lambda (x1: +T).(\lambda (x2: T).(\lambda (H7: (eq T x0 (THead (Bind Abst) x1 +x2))).(\lambda (H8: (eq T u (lift h x x1))).(\lambda (H9: (eq T t0 (lift h (S +x) x2))).(eq_ind_r T (THead (Bind Abst) x1 x2) (\lambda (t1: T).(arity g c2 +t1 (AHead a1 a2))) (let H10 \def (eq_ind T t0 (\lambda (t1: T).(\forall (x3: +nat).(\forall (x4: T).((eq T t1 (lift h x3 x4)) \to (\forall (c3: C).((drop h +x3 (CHead c (Bind Abst) u) c3) \to (arity g c3 x4 a2))))))) H4 (lift h (S x) +x2) H9) in (let H11 \def (eq_ind T t0 (\lambda (t1: T).(arity g (CHead c +(Bind Abst) u) t1 a2)) H3 (lift h (S x) x2) H9) in (let H12 \def (eq_ind T u +(\lambda (t1: T).(arity g (CHead c (Bind Abst) t1) (lift h (S x) x2) a2)) H11 +(lift h x x1) H8) in (let H13 \def (eq_ind T u (\lambda (t1: T).(\forall (x3: +nat).(\forall (x4: T).((eq T (lift h (S x) x2) (lift h x3 x4)) \to (\forall +(c3: C).((drop h x3 (CHead c (Bind Abst) t1) c3) \to (arity g c3 x4 a2))))))) +H10 (lift h x x1) H8) in (let H14 \def (eq_ind T u (\lambda (t1: T).(\forall +(x3: nat).(\forall (x4: T).((eq T t1 (lift h x3 x4)) \to (\forall (c3: +C).((drop h x3 c c3) \to (arity g c3 x4 (asucc g a1)))))))) H2 (lift h x x1) +H8) in (let H15 \def (eq_ind T u (\lambda (t1: T).(arity g c t1 (asucc g +a1))) H1 (lift h x x1) H8) in (arity_head g c2 x1 a1 (H14 x x1 (refl_equal T +(lift h x x1)) c2 H6) x2 a2 (H13 (S x) x2 (refl_equal T (lift h (S x) x2)) +(CHead c2 (Bind Abst) x1) (drop_skip_bind h x c c2 H6 Abst x1))))))))) x0 +H7)))))) (lift_gen_bind Abst u t0 x0 h x H5)))))))))))))))) (\lambda (c: +C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H1: (arity g c u a1)).(\lambda +(H2: ((\forall (x: nat).(\forall (x0: T).((eq T u (lift h x x0)) \to (\forall +(c2: C).((drop h x c c2) \to (arity g c2 x0 a1)))))))).(\lambda (t0: +T).(\lambda (a2: A).(\lambda (H3: (arity g c t0 (AHead a1 a2))).(\lambda (H4: +((\forall (x: nat).(\forall (x0: T).((eq T t0 (lift h x x0)) \to (\forall +(c2: C).((drop h x c c2) \to (arity g c2 x0 (AHead a1 a2))))))))).(\lambda +(x: nat).(\lambda (x0: T).(\lambda (H5: (eq T (THead (Flat Appl) u t0) (lift +h x x0))).(\lambda (c2: C).(\lambda (H6: (drop h x c c2)).(ex3_2_ind T T +(\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Flat Appl) y0 z)))) +(\lambda (y0: T).(\lambda (_: T).(eq T u (lift h x y0)))) (\lambda (_: +T).(\lambda (z: T).(eq T t0 (lift h x z)))) (arity g c2 x0 a2) (\lambda (x1: +T).(\lambda (x2: T).(\lambda (H7: (eq T x0 (THead (Flat Appl) x1 +x2))).(\lambda (H8: (eq T u (lift h x x1))).(\lambda (H9: (eq T t0 (lift h x +x2))).(eq_ind_r T (THead (Flat Appl) x1 x2) (\lambda (t1: T).(arity g c2 t1 +a2)) (let H10 \def (eq_ind T t0 (\lambda (t1: T).(\forall (x3: nat).(\forall +(x4: T).((eq T t1 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 c c3) \to +(arity g c3 x4 (AHead a1 a2)))))))) H4 (lift h x x2) H9) in (let H11 \def +(eq_ind T t0 (\lambda (t1: T).(arity g c t1 (AHead a1 a2))) H3 (lift h x x2) +H9) in (let H12 \def (eq_ind T u (\lambda (t1: T).(\forall (x3: nat).(\forall +(x4: T).((eq T t1 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 c c3) \to +(arity g c3 x4 a1))))))) H2 (lift h x x1) H8) in (let H13 \def (eq_ind T u +(\lambda (t1: T).(arity g c t1 a1)) H1 (lift h x x1) H8) in (arity_appl g c2 +x1 a1 (H12 x x1 (refl_equal T (lift h x x1)) c2 H6) x2 a2 (H10 x x2 +(refl_equal T (lift h x x2)) c2 H6)))))) x0 H7)))))) (lift_gen_flat Appl u t0 +x0 h x H5)))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a0: +A).(\lambda (H1: (arity g c u (asucc g a0))).(\lambda (H2: ((\forall (x: +nat).(\forall (x0: T).((eq T u (lift h x x0)) \to (\forall (c2: C).((drop h x +c c2) \to (arity g c2 x0 (asucc g a0))))))))).(\lambda (t0: T).(\lambda (H3: +(arity g c t0 a0)).(\lambda (H4: ((\forall (x: nat).(\forall (x0: T).((eq T +t0 (lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to (arity g c2 x0 +a0)))))))).(\lambda (x: nat).(\lambda (x0: T).(\lambda (H5: (eq T (THead +(Flat Cast) u t0) (lift h x x0))).(\lambda (c2: C).(\lambda (H6: (drop h x c +c2)).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Flat +Cast) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u (lift h x y0)))) +(\lambda (_: T).(\lambda (z: T).(eq T t0 (lift h x z)))) (arity g c2 x0 a0) +(\lambda (x1: T).(\lambda (x2: T).(\lambda (H7: (eq T x0 (THead (Flat Cast) +x1 x2))).(\lambda (H8: (eq T u (lift h x x1))).(\lambda (H9: (eq T t0 (lift h +x x2))).(eq_ind_r T (THead (Flat Cast) x1 x2) (\lambda (t1: T).(arity g c2 t1 +a0)) (let H10 \def (eq_ind T t0 (\lambda (t1: T).(\forall (x3: nat).(\forall +(x4: T).((eq T t1 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 c c3) \to +(arity g c3 x4 a0))))))) H4 (lift h x x2) H9) in (let H11 \def (eq_ind T t0 +(\lambda (t1: T).(arity g c t1 a0)) H3 (lift h x x2) H9) in (let H12 \def +(eq_ind T u (\lambda (t1: T).(\forall (x3: nat).(\forall (x4: T).((eq T t1 +(lift h x3 x4)) \to (\forall (c3: C).((drop h x3 c c3) \to (arity g c3 x4 +(asucc g a0)))))))) H2 (lift h x x1) H8) in (let H13 \def (eq_ind T u +(\lambda (t1: T).(arity g c t1 (asucc g a0))) H1 (lift h x x1) H8) in +(arity_cast g c2 x1 a0 (H12 x x1 (refl_equal T (lift h x x1)) c2 H6) x2 (H10 +x x2 (refl_equal T (lift h x x2)) c2 H6)))))) x0 H7)))))) (lift_gen_flat Cast +u t0 x0 h x H5))))))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda (a1: +A).(\lambda (_: (arity g c t0 a1)).(\lambda (H2: ((\forall (x: nat).(\forall +(x0: T).((eq T t0 (lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to +(arity g c2 x0 a1)))))))).(\lambda (a2: A).(\lambda (H3: (leq g a1 +a2)).(\lambda (x: nat).(\lambda (x0: T).(\lambda (H4: (eq T t0 (lift h x +x0))).(\lambda (c2: C).(\lambda (H5: (drop h x c c2)).(arity_repl g c2 x0 a1 +(H2 x x0 H4 c2 H5) a2 H3))))))))))))) c1 y a H0))))) H))))))). +(* COMMENTS +Initial nodes: 4693 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/arity/lift1.ma b/matita/matita/contribs/lambdadelta/basic_1/arity/lift1.ma new file mode 100644 index 000000000..4077a8a1c --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/arity/lift1.ma @@ -0,0 +1,44 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/arity/props.ma". + +include "Basic-1/drop1/fwd.ma". + +theorem arity_lift1: + \forall (g: G).(\forall (a: A).(\forall (c2: C).(\forall (hds: +PList).(\forall (c1: C).(\forall (t: T).((drop1 hds c1 c2) \to ((arity g c2 t +a) \to (arity g c1 (lift1 hds t) a)))))))) +\def + \lambda (g: G).(\lambda (a: A).(\lambda (c2: C).(\lambda (hds: +PList).(PList_ind (\lambda (p: PList).(\forall (c1: C).(\forall (t: +T).((drop1 p c1 c2) \to ((arity g c2 t a) \to (arity g c1 (lift1 p t) a)))))) +(\lambda (c1: C).(\lambda (t: T).(\lambda (H: (drop1 PNil c1 c2)).(\lambda +(H0: (arity g c2 t a)).(let H_y \def (drop1_gen_pnil c1 c2 H) in (eq_ind_r C +c2 (\lambda (c: C).(arity g c t a)) H0 c1 H_y)))))) (\lambda (n: +nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c1: +C).(\forall (t: T).((drop1 p c1 c2) \to ((arity g c2 t a) \to (arity g c1 +(lift1 p t) a))))))).(\lambda (c1: C).(\lambda (t: T).(\lambda (H0: (drop1 +(PCons n n0 p) c1 c2)).(\lambda (H1: (arity g c2 t a)).(let H_x \def +(drop1_gen_pcons c1 c2 p n n0 H0) in (let H2 \def H_x in (ex2_ind C (\lambda +(c3: C).(drop n n0 c1 c3)) (\lambda (c3: C).(drop1 p c3 c2)) (arity g c1 +(lift n n0 (lift1 p t)) a) (\lambda (x: C).(\lambda (H3: (drop n n0 c1 +x)).(\lambda (H4: (drop1 p x c2)).(arity_lift g x (lift1 p t) a (H x t H4 H1) +c1 n n0 H3)))) H2))))))))))) hds)))). +(* COMMENTS +Initial nodes: 289 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/arity/pr3.ma b/matita/matita/contribs/lambdadelta/basic_1/arity/pr3.ma new file mode 100644 index 000000000..f8952088e --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/arity/pr3.ma @@ -0,0 +1,635 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/csuba/arity.ma". + +include "Basic-1/pr3/defs.ma". + +include "Basic-1/pr1/defs.ma". + +include "Basic-1/wcpr0/getl.ma". + +include "Basic-1/pr0/fwd.ma". + +include "Basic-1/arity/subst0.ma". + +theorem arity_sred_wcpr0_pr0: + \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (a: A).((arity g +c1 t1 a) \to (\forall (c2: C).((wcpr0 c1 c2) \to (\forall (t2: T).((pr0 t1 +t2) \to (arity g c2 t2 a))))))))) +\def + \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (a: A).(\lambda +(H: (arity g c1 t1 a)).(arity_ind g (\lambda (c: C).(\lambda (t: T).(\lambda +(a0: A).(\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 t t2) \to +(arity g c2 t2 a0)))))))) (\lambda (c: C).(\lambda (n: nat).(\lambda (c2: +C).(\lambda (_: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H1: (pr0 (TSort n) +t2)).(eq_ind_r T (TSort n) (\lambda (t: T).(arity g c2 t (ASort O n))) +(arity_sort g c2 n) t2 (pr0_gen_sort t2 n H1)))))))) (\lambda (c: C).(\lambda +(d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d +(Bind Abbr) u))).(\lambda (a0: A).(\lambda (_: (arity g d u a0)).(\lambda +(H2: ((\forall (c2: C).((wcpr0 d c2) \to (\forall (t2: T).((pr0 u t2) \to +(arity g c2 t2 a0))))))).(\lambda (c2: C).(\lambda (H3: (wcpr0 c +c2)).(\lambda (t2: T).(\lambda (H4: (pr0 (TLRef i) t2)).(eq_ind_r T (TLRef i) +(\lambda (t: T).(arity g c2 t a0)) (ex3_2_ind C T (\lambda (e2: C).(\lambda +(u2: T).(getl i c2 (CHead e2 (Bind Abbr) u2)))) (\lambda (e2: C).(\lambda (_: +T).(wcpr0 d e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u u2))) (arity g c2 +(TLRef i) a0) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: (getl i c2 +(CHead x0 (Bind Abbr) x1))).(\lambda (H6: (wcpr0 d x0)).(\lambda (H7: (pr0 u +x1)).(arity_abbr g c2 x0 x1 i H5 a0 (H2 x0 H6 x1 H7))))))) (wcpr0_getl c c2 +H3 i d u (Bind Abbr) H0)) t2 (pr0_gen_lref t2 i H4)))))))))))))) (\lambda (c: +C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c +(CHead d (Bind Abst) u))).(\lambda (a0: A).(\lambda (_: (arity g d u (asucc g +a0))).(\lambda (H2: ((\forall (c2: C).((wcpr0 d c2) \to (\forall (t2: +T).((pr0 u t2) \to (arity g c2 t2 (asucc g a0)))))))).(\lambda (c2: +C).(\lambda (H3: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H4: (pr0 (TLRef i) +t2)).(eq_ind_r T (TLRef i) (\lambda (t: T).(arity g c2 t a0)) (ex3_2_ind C T +(\lambda (e2: C).(\lambda (u2: T).(getl i c2 (CHead e2 (Bind Abst) u2)))) +(\lambda (e2: C).(\lambda (_: T).(wcpr0 d e2))) (\lambda (_: C).(\lambda (u2: +T).(pr0 u u2))) (arity g c2 (TLRef i) a0) (\lambda (x0: C).(\lambda (x1: +T).(\lambda (H5: (getl i c2 (CHead x0 (Bind Abst) x1))).(\lambda (H6: (wcpr0 +d x0)).(\lambda (H7: (pr0 u x1)).(arity_abst g c2 x0 x1 i H5 a0 (H2 x0 H6 x1 +H7))))))) (wcpr0_getl c c2 H3 i d u (Bind Abst) H0)) t2 (pr0_gen_lref t2 i +H4)))))))))))))) (\lambda (b: B).(\lambda (H0: (not (eq B b Abst))).(\lambda +(c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c u +a1)).(\lambda (H2: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 +u t2) \to (arity g c2 t2 a1))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda +(H3: (arity g (CHead c (Bind b) u) t a2)).(\lambda (H4: ((\forall (c2: +C).((wcpr0 (CHead c (Bind b) u) c2) \to (\forall (t2: T).((pr0 t t2) \to +(arity g c2 t2 a2))))))).(\lambda (c2: C).(\lambda (H5: (wcpr0 c +c2)).(\lambda (t2: T).(\lambda (H6: (pr0 (THead (Bind b) u t) t2)).(insert_eq +T (THead (Bind b) u t) (\lambda (t0: T).(pr0 t0 t2)) (\lambda (_: T).(arity g +c2 t2 a2)) (\lambda (y: T).(\lambda (H7: (pr0 y t2)).(pr0_ind (\lambda (t0: +T).(\lambda (t3: T).((eq T t0 (THead (Bind b) u t)) \to (arity g c2 t3 a2)))) +(\lambda (t0: T).(\lambda (H8: (eq T t0 (THead (Bind b) u t))).(let H9 \def +(f_equal T T (\lambda (e: T).e) t0 (THead (Bind b) u t) H8) in (eq_ind_r T +(THead (Bind b) u t) (\lambda (t3: T).(arity g c2 t3 a2)) (arity_bind g b H0 +c2 u a1 (H2 c2 H5 u (pr0_refl u)) t a2 (H4 (CHead c2 (Bind b) u) (wcpr0_comp +c c2 H5 u u (pr0_refl u) (Bind b)) t (pr0_refl t))) t0 H9)))) (\lambda (u1: +T).(\lambda (u2: T).(\lambda (H8: (pr0 u1 u2)).(\lambda (H9: (((eq T u1 +(THead (Bind b) u t)) \to (arity g c2 u2 a2)))).(\lambda (t3: T).(\lambda +(t4: T).(\lambda (H10: (pr0 t3 t4)).(\lambda (H11: (((eq T t3 (THead (Bind b) +u t)) \to (arity g c2 t4 a2)))).(\lambda (k: K).(\lambda (H12: (eq T (THead k +u1 t3) (THead (Bind b) u t))).(let H13 \def (f_equal T K (\lambda (e: +T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | +(TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) +(THead (Bind b) u t) H12) in ((let H14 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | +(TLRef _) \Rightarrow u1 | (THead _ t0 _) \Rightarrow t0])) (THead k u1 t3) +(THead (Bind b) u t) H12) in ((let H15 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | +(TLRef _) \Rightarrow t3 | (THead _ _ t0) \Rightarrow t0])) (THead k u1 t3) +(THead (Bind b) u t) H12) in (\lambda (H16: (eq T u1 u)).(\lambda (H17: (eq K +k (Bind b))).(eq_ind_r K (Bind b) (\lambda (k0: K).(arity g c2 (THead k0 u2 +t4) a2)) (let H18 \def (eq_ind T t3 (\lambda (t0: T).((eq T t0 (THead (Bind +b) u t)) \to (arity g c2 t4 a2))) H11 t H15) in (let H19 \def (eq_ind T t3 +(\lambda (t0: T).(pr0 t0 t4)) H10 t H15) in (let H20 \def (eq_ind T u1 +(\lambda (t0: T).((eq T t0 (THead (Bind b) u t)) \to (arity g c2 u2 a2))) H9 +u H16) in (let H21 \def (eq_ind T u1 (\lambda (t0: T).(pr0 t0 u2)) H8 u H16) +in (arity_bind g b H0 c2 u2 a1 (H2 c2 H5 u2 H21) t4 a2 (H4 (CHead c2 (Bind b) +u2) (wcpr0_comp c c2 H5 u u2 H21 (Bind b)) t4 H19)))))) k H17)))) H14)) +H13)))))))))))) (\lambda (u0: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda +(_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind b) u t)) \to (arity g +c2 v2 a2)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 +t4)).(\lambda (_: (((eq T t3 (THead (Bind b) u t)) \to (arity g c2 t4 +a2)))).(\lambda (H12: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) +(THead (Bind b) u t))).(let H13 \def (eq_ind T (THead (Flat Appl) v1 (THead +(Bind Abst) u0 t3)) (\lambda (ee: T).(match ee in T return (\lambda (_: +T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | +(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with +[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind +b) u t) H12) in (False_ind (arity g c2 (THead (Bind Abbr) v2 t4) a2) +H13)))))))))))) (\lambda (b0: B).(\lambda (_: (not (eq B b0 Abst))).(\lambda +(v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 +(THead (Bind b) u t)) \to (arity g c2 v2 a2)))).(\lambda (u1: T).(\lambda +(u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 (THead (Bind b) u +t)) \to (arity g c2 u2 a2)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: +(pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead (Bind b) u t)) \to (arity g c2 t4 +a2)))).(\lambda (H15: (eq T (THead (Flat Appl) v1 (THead (Bind b0) u1 t3)) +(THead (Bind b) u t))).(let H16 \def (eq_ind T (THead (Flat Appl) v1 (THead +(Bind b0) u1 t3)) (\lambda (ee: T).(match ee in T return (\lambda (_: +T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | +(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with +[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind +b) u t) H15) in (False_ind (arity g c2 (THead (Bind b0) u2 (THead (Flat Appl) +(lift (S O) O v2) t4)) a2) H16))))))))))))))))) (\lambda (u1: T).(\lambda +(u2: T).(\lambda (H8: (pr0 u1 u2)).(\lambda (H9: (((eq T u1 (THead (Bind b) u +t)) \to (arity g c2 u2 a2)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda +(H10: (pr0 t3 t4)).(\lambda (H11: (((eq T t3 (THead (Bind b) u t)) \to (arity +g c2 t4 a2)))).(\lambda (w: T).(\lambda (H12: (subst0 O u2 t4 w)).(\lambda +(H13: (eq T (THead (Bind Abbr) u1 t3) (THead (Bind b) u t))).(let H14 \def +(f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with +[(TSort _) \Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | (THead k _ _) +\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b0) +\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (THead (Bind Abbr) u1 t3) +(THead (Bind b) u t) H13) in ((let H15 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | +(TLRef _) \Rightarrow u1 | (THead _ t0 _) \Rightarrow t0])) (THead (Bind +Abbr) u1 t3) (THead (Bind b) u t) H13) in ((let H16 \def (f_equal T T +(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t0) \Rightarrow t0])) +(THead (Bind Abbr) u1 t3) (THead (Bind b) u t) H13) in (\lambda (H17: (eq T +u1 u)).(\lambda (H18: (eq B Abbr b)).(let H19 \def (eq_ind T t3 (\lambda (t0: +T).((eq T t0 (THead (Bind b) u t)) \to (arity g c2 t4 a2))) H11 t H16) in +(let H20 \def (eq_ind T t3 (\lambda (t0: T).(pr0 t0 t4)) H10 t H16) in (let +H21 \def (eq_ind T u1 (\lambda (t0: T).((eq T t0 (THead (Bind b) u t)) \to +(arity g c2 u2 a2))) H9 u H17) in (let H22 \def (eq_ind T u1 (\lambda (t0: +T).(pr0 t0 u2)) H8 u H17) in (let H23 \def (eq_ind_r B b (\lambda (b0: +B).((eq T t (THead (Bind b0) u t)) \to (arity g c2 t4 a2))) H19 Abbr H18) in +(let H24 \def (eq_ind_r B b (\lambda (b0: B).((eq T u (THead (Bind b0) u t)) +\to (arity g c2 u2 a2))) H21 Abbr H18) in (let H25 \def (eq_ind_r B b +(\lambda (b0: B).(\forall (c3: C).((wcpr0 (CHead c (Bind b0) u) c3) \to +(\forall (t5: T).((pr0 t t5) \to (arity g c3 t5 a2)))))) H4 Abbr H18) in (let +H26 \def (eq_ind_r B b (\lambda (b0: B).(arity g (CHead c (Bind b0) u) t a2)) +H3 Abbr H18) in (let H27 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 +Abst))) H0 Abbr H18) in (arity_bind g Abbr H27 c2 u2 a1 (H2 c2 H5 u2 H22) w +a2 (arity_subst0 g (CHead c2 (Bind Abbr) u2) t4 a2 (H25 (CHead c2 (Bind Abbr) +u2) (wcpr0_comp c c2 H5 u u2 H22 (Bind Abbr)) t4 H20) c2 u2 O (getl_refl Abbr +c2 u2) w H12)))))))))))))) H15)) H14))))))))))))) (\lambda (b0: B).(\lambda +(H8: (not (eq B b0 Abst))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H9: +(pr0 t3 t4)).(\lambda (H10: (((eq T t3 (THead (Bind b) u t)) \to (arity g c2 +t4 a2)))).(\lambda (u0: T).(\lambda (H11: (eq T (THead (Bind b0) u0 (lift (S +O) O t3)) (THead (Bind b) u t))).(let H12 \def (f_equal T B (\lambda (e: +T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b0 | +(TLRef _) \Rightarrow b0 | (THead k _ _) \Rightarrow (match k in K return +(\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow +b0])])) (THead (Bind b0) u0 (lift (S O) O t3)) (THead (Bind b) u t) H11) in +((let H13 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: +T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t0 +_) \Rightarrow t0])) (THead (Bind b0) u0 (lift (S O) O t3)) (THead (Bind b) u +t) H11) in ((let H14 \def (f_equal T T (\lambda (e: T).(match e in T return +(\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat +\to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n) +\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with +[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u1 t5) +\Rightarrow (THead k (lref_map f d u1) (lref_map f (s k d) t5))]) in +lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (TLRef _) \Rightarrow +((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match +t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef +(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | +(THead k u1 t5) \Rightarrow (THead k (lref_map f d u1) (lref_map f (s k d) +t5))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (THead _ _ t0) +\Rightarrow t0])) (THead (Bind b0) u0 (lift (S O) O t3)) (THead (Bind b) u t) +H11) in (\lambda (_: (eq T u0 u)).(\lambda (H16: (eq B b0 b)).(let H17 \def +(eq_ind B b0 (\lambda (b1: B).(not (eq B b1 Abst))) H8 b H16) in (let H18 +\def (eq_ind_r T t (\lambda (t0: T).((eq T t3 (THead (Bind b) u t0)) \to +(arity g c2 t4 a2))) H10 (lift (S O) O t3) H14) in (let H19 \def (eq_ind_r T +t (\lambda (t0: T).(\forall (c3: C).((wcpr0 (CHead c (Bind b) u) c3) \to +(\forall (t5: T).((pr0 t0 t5) \to (arity g c3 t5 a2)))))) H4 (lift (S O) O +t3) H14) in (let H20 \def (eq_ind_r T t (\lambda (t0: T).(arity g (CHead c +(Bind b) u) t0 a2)) H3 (lift (S O) O t3) H14) in (arity_gen_lift g (CHead c2 +(Bind b) u) t4 a2 (S O) O (H19 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H5 u u +(pr0_refl u) (Bind b)) (lift (S O) O t4) (pr0_lift t3 t4 H9 (S O) O)) c2 +(drop_drop (Bind b) O c2 c2 (drop_refl c2) u))))))))) H13)) H12)))))))))) +(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: +(((eq T t3 (THead (Bind b) u t)) \to (arity g c2 t4 a2)))).(\lambda (u0: +T).(\lambda (H10: (eq T (THead (Flat Cast) u0 t3) (THead (Bind b) u t))).(let +H11 \def (eq_ind T (THead (Flat Cast) u0 t3) (\lambda (ee: T).(match ee in T +return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda +(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow +True])])) I (THead (Bind b) u t) H10) in (False_ind (arity g c2 t4 a2) +H11)))))))) y t2 H7))) H6)))))))))))))))) (\lambda (c: C).(\lambda (u: +T).(\lambda (a1: A).(\lambda (_: (arity g c u (asucc g a1))).(\lambda (H1: +((\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 u t2) \to (arity g +c2 t2 (asucc g a1)))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (H2: +(arity g (CHead c (Bind Abst) u) t a2)).(\lambda (H3: ((\forall (c2: +C).((wcpr0 (CHead c (Bind Abst) u) c2) \to (\forall (t2: T).((pr0 t t2) \to +(arity g c2 t2 a2))))))).(\lambda (c2: C).(\lambda (H4: (wcpr0 c +c2)).(\lambda (t2: T).(\lambda (H5: (pr0 (THead (Bind Abst) u t) +t2)).(insert_eq T (THead (Bind Abst) u t) (\lambda (t0: T).(pr0 t0 t2)) +(\lambda (_: T).(arity g c2 t2 (AHead a1 a2))) (\lambda (y: T).(\lambda (H6: +(pr0 y t2)).(pr0_ind (\lambda (t0: T).(\lambda (t3: T).((eq T t0 (THead (Bind +Abst) u t)) \to (arity g c2 t3 (AHead a1 a2))))) (\lambda (t0: T).(\lambda +(H7: (eq T t0 (THead (Bind Abst) u t))).(let H8 \def (f_equal T T (\lambda +(e: T).e) t0 (THead (Bind Abst) u t) H7) in (eq_ind_r T (THead (Bind Abst) u +t) (\lambda (t3: T).(arity g c2 t3 (AHead a1 a2))) (arity_head g c2 u a1 (H1 +c2 H4 u (pr0_refl u)) t a2 (H3 (CHead c2 (Bind Abst) u) (wcpr0_comp c c2 H4 u +u (pr0_refl u) (Bind Abst)) t (pr0_refl t))) t0 H8)))) (\lambda (u1: +T).(\lambda (u2: T).(\lambda (H7: (pr0 u1 u2)).(\lambda (H8: (((eq T u1 +(THead (Bind Abst) u t)) \to (arity g c2 u2 (AHead a1 a2))))).(\lambda (t3: +T).(\lambda (t4: T).(\lambda (H9: (pr0 t3 t4)).(\lambda (H10: (((eq T t3 +(THead (Bind Abst) u t)) \to (arity g c2 t4 (AHead a1 a2))))).(\lambda (k: +K).(\lambda (H11: (eq T (THead k u1 t3) (THead (Bind Abst) u t))).(let H12 +\def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) +with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) +\Rightarrow k0])) (THead k u1 t3) (THead (Bind Abst) u t) H11) in ((let H13 +\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) +with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t0 _) +\Rightarrow t0])) (THead k u1 t3) (THead (Bind Abst) u t) H11) in ((let H14 +\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) +with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t0) +\Rightarrow t0])) (THead k u1 t3) (THead (Bind Abst) u t) H11) in (\lambda +(H15: (eq T u1 u)).(\lambda (H16: (eq K k (Bind Abst))).(eq_ind_r K (Bind +Abst) (\lambda (k0: K).(arity g c2 (THead k0 u2 t4) (AHead a1 a2))) (let H17 +\def (eq_ind T t3 (\lambda (t0: T).((eq T t0 (THead (Bind Abst) u t)) \to +(arity g c2 t4 (AHead a1 a2)))) H10 t H14) in (let H18 \def (eq_ind T t3 +(\lambda (t0: T).(pr0 t0 t4)) H9 t H14) in (let H19 \def (eq_ind T u1 +(\lambda (t0: T).((eq T t0 (THead (Bind Abst) u t)) \to (arity g c2 u2 (AHead +a1 a2)))) H8 u H15) in (let H20 \def (eq_ind T u1 (\lambda (t0: T).(pr0 t0 +u2)) H7 u H15) in (arity_head g c2 u2 a1 (H1 c2 H4 u2 H20) t4 a2 (H3 (CHead +c2 (Bind Abst) u2) (wcpr0_comp c c2 H4 u u2 H20 (Bind Abst)) t4 H18)))))) k +H16)))) H13)) H12)))))))))))) (\lambda (u0: T).(\lambda (v1: T).(\lambda (v2: +T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind Abst) u t)) +\to (arity g c2 v2 (AHead a1 a2))))).(\lambda (t3: T).(\lambda (t4: +T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead (Bind Abst) u t)) +\to (arity g c2 t4 (AHead a1 a2))))).(\lambda (H11: (eq T (THead (Flat Appl) +v1 (THead (Bind Abst) u0 t3)) (THead (Bind Abst) u t))).(let H12 \def (eq_ind +T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) (\lambda (ee: T).(match ee +in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef +_) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return +(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow +True])])) I (THead (Bind Abst) u t) H11) in (False_ind (arity g c2 (THead +(Bind Abbr) v2 t4) (AHead a1 a2)) H12)))))))))))) (\lambda (b: B).(\lambda +(_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 +v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind Abst) u t)) \to (arity g c2 v2 +(AHead a1 a2))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 +u2)).(\lambda (_: (((eq T u1 (THead (Bind Abst) u t)) \to (arity g c2 u2 +(AHead a1 a2))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 +t4)).(\lambda (_: (((eq T t3 (THead (Bind Abst) u t)) \to (arity g c2 t4 +(AHead a1 a2))))).(\lambda (H14: (eq T (THead (Flat Appl) v1 (THead (Bind b) +u1 t3)) (THead (Bind Abst) u t))).(let H15 \def (eq_ind T (THead (Flat Appl) +v1 (THead (Bind b) u1 t3)) (\lambda (ee: T).(match ee in T return (\lambda +(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False +| (THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with +[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind +Abst) u t) H14) in (False_ind (arity g c2 (THead (Bind b) u2 (THead (Flat +Appl) (lift (S O) O v2) t4)) (AHead a1 a2)) H15))))))))))))))))) (\lambda +(u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 +(THead (Bind Abst) u t)) \to (arity g c2 u2 (AHead a1 a2))))).(\lambda (t3: +T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead +(Bind Abst) u t)) \to (arity g c2 t4 (AHead a1 a2))))).(\lambda (w: +T).(\lambda (_: (subst0 O u2 t4 w)).(\lambda (H12: (eq T (THead (Bind Abbr) +u1 t3) (THead (Bind Abst) u t))).(let H13 \def (eq_ind T (THead (Bind Abbr) +u1 t3) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with +[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) +\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b) +\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow +True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _) +\Rightarrow False])])) I (THead (Bind Abst) u t) H12) in (False_ind (arity g +c2 (THead (Bind Abbr) u2 w) (AHead a1 a2)) H13))))))))))))) (\lambda (b: +B).(\lambda (H7: (not (eq B b Abst))).(\lambda (t3: T).(\lambda (t4: +T).(\lambda (_: (pr0 t3 t4)).(\lambda (H9: (((eq T t3 (THead (Bind Abst) u +t)) \to (arity g c2 t4 (AHead a1 a2))))).(\lambda (u0: T).(\lambda (H10: (eq +T (THead (Bind b) u0 (lift (S O) O t3)) (THead (Bind Abst) u t))).(let H11 +\def (f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) +with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _) +\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b0) +\Rightarrow b0 | (Flat _) \Rightarrow b])])) (THead (Bind b) u0 (lift (S O) O +t3)) (THead (Bind Abst) u t) H10) in ((let H12 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | +(TLRef _) \Rightarrow u0 | (THead _ t0 _) \Rightarrow t0])) (THead (Bind b) +u0 (lift (S O) O t3)) (THead (Bind Abst) u t) H10) in ((let H13 \def (f_equal +T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T +\def (match t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow +(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) +| (THead k u1 t5) \Rightarrow (THead k (lref_map f d u1) (lref_map f (s k d) +t5))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (TLRef _) +\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T +\def (match t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow +(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) +| (THead k u1 t5) \Rightarrow (THead k (lref_map f d u1) (lref_map f (s k d) +t5))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (THead _ _ t0) +\Rightarrow t0])) (THead (Bind b) u0 (lift (S O) O t3)) (THead (Bind Abst) u +t) H10) in (\lambda (_: (eq T u0 u)).(\lambda (H15: (eq B b Abst)).(let H16 +\def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H7 Abst H15) in (let +H17 \def (eq_ind_r T t (\lambda (t0: T).((eq T t3 (THead (Bind Abst) u t0)) +\to (arity g c2 t4 (AHead a1 a2)))) H9 (lift (S O) O t3) H13) in (let H18 +\def (eq_ind_r T t (\lambda (t0: T).(\forall (c3: C).((wcpr0 (CHead c (Bind +Abst) u) c3) \to (\forall (t5: T).((pr0 t0 t5) \to (arity g c3 t5 a2)))))) H3 +(lift (S O) O t3) H13) in (let H19 \def (eq_ind_r T t (\lambda (t0: T).(arity +g (CHead c (Bind Abst) u) t0 a2)) H2 (lift (S O) O t3) H13) in (let H20 \def +(match (H16 (refl_equal B Abst)) in False return (\lambda (_: False).(arity g +c2 t4 (AHead a1 a2))) with []) in H20)))))))) H12)) H11)))))))))) (\lambda +(t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 +(THead (Bind Abst) u t)) \to (arity g c2 t4 (AHead a1 a2))))).(\lambda (u0: +T).(\lambda (H9: (eq T (THead (Flat Cast) u0 t3) (THead (Bind Abst) u +t))).(let H10 \def (eq_ind T (THead (Flat Cast) u0 t3) (\lambda (ee: +T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow +False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K +return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) +\Rightarrow True])])) I (THead (Bind Abst) u t) H9) in (False_ind (arity g c2 +t4 (AHead a1 a2)) H10)))))))) y t2 H6))) H5)))))))))))))) (\lambda (c: +C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda +(H1: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 u t2) \to +(arity g c2 t2 a1))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (H2: +(arity g c t (AHead a1 a2))).(\lambda (H3: ((\forall (c2: C).((wcpr0 c c2) +\to (\forall (t2: T).((pr0 t t2) \to (arity g c2 t2 (AHead a1 +a2)))))))).(\lambda (c2: C).(\lambda (H4: (wcpr0 c c2)).(\lambda (t2: +T).(\lambda (H5: (pr0 (THead (Flat Appl) u t) t2)).(insert_eq T (THead (Flat +Appl) u t) (\lambda (t0: T).(pr0 t0 t2)) (\lambda (_: T).(arity g c2 t2 a2)) +(\lambda (y: T).(\lambda (H6: (pr0 y t2)).(pr0_ind (\lambda (t0: T).(\lambda +(t3: T).((eq T t0 (THead (Flat Appl) u t)) \to (arity g c2 t3 a2)))) (\lambda +(t0: T).(\lambda (H7: (eq T t0 (THead (Flat Appl) u t))).(let H8 \def +(f_equal T T (\lambda (e: T).e) t0 (THead (Flat Appl) u t) H7) in (eq_ind_r T +(THead (Flat Appl) u t) (\lambda (t3: T).(arity g c2 t3 a2)) (arity_appl g c2 +u a1 (H1 c2 H4 u (pr0_refl u)) t a2 (H3 c2 H4 t (pr0_refl t))) t0 H8)))) +(\lambda (u1: T).(\lambda (u2: T).(\lambda (H7: (pr0 u1 u2)).(\lambda (H8: +(((eq T u1 (THead (Flat Appl) u t)) \to (arity g c2 u2 a2)))).(\lambda (t3: +T).(\lambda (t4: T).(\lambda (H9: (pr0 t3 t4)).(\lambda (H10: (((eq T t3 +(THead (Flat Appl) u t)) \to (arity g c2 t4 a2)))).(\lambda (k: K).(\lambda +(H11: (eq T (THead k u1 t3) (THead (Flat Appl) u t))).(let H12 \def (f_equal +T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) +\Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) +(THead k u1 t3) (THead (Flat Appl) u t) H11) in ((let H13 \def (f_equal T T +(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t0 _) \Rightarrow t0])) +(THead k u1 t3) (THead (Flat Appl) u t) H11) in ((let H14 \def (f_equal T T +(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t0) \Rightarrow t0])) +(THead k u1 t3) (THead (Flat Appl) u t) H11) in (\lambda (H15: (eq T u1 +u)).(\lambda (H16: (eq K k (Flat Appl))).(eq_ind_r K (Flat Appl) (\lambda +(k0: K).(arity g c2 (THead k0 u2 t4) a2)) (let H17 \def (eq_ind T t3 (\lambda +(t0: T).((eq T t0 (THead (Flat Appl) u t)) \to (arity g c2 t4 a2))) H10 t +H14) in (let H18 \def (eq_ind T t3 (\lambda (t0: T).(pr0 t0 t4)) H9 t H14) in +(let H19 \def (eq_ind T u1 (\lambda (t0: T).((eq T t0 (THead (Flat Appl) u +t)) \to (arity g c2 u2 a2))) H8 u H15) in (let H20 \def (eq_ind T u1 (\lambda +(t0: T).(pr0 t0 u2)) H7 u H15) in (arity_appl g c2 u2 a1 (H1 c2 H4 u2 H20) t4 +a2 (H3 c2 H4 t4 H18)))))) k H16)))) H13)) H12)))))))))))) (\lambda (u0: +T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H7: (pr0 v1 v2)).(\lambda (H8: +(((eq T v1 (THead (Flat Appl) u t)) \to (arity g c2 v2 a2)))).(\lambda (t3: +T).(\lambda (t4: T).(\lambda (H9: (pr0 t3 t4)).(\lambda (H10: (((eq T t3 +(THead (Flat Appl) u t)) \to (arity g c2 t4 a2)))).(\lambda (H11: (eq T +(THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) (THead (Flat Appl) u +t))).(let H12 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda +(_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead +_ t0 _) \Rightarrow t0])) (THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) +(THead (Flat Appl) u t) H11) in ((let H13 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow (THead +(Bind Abst) u0 t3) | (TLRef _) \Rightarrow (THead (Bind Abst) u0 t3) | (THead +_ _ t0) \Rightarrow t0])) (THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) +(THead (Flat Appl) u t) H11) in (\lambda (H14: (eq T v1 u)).(let H15 \def +(eq_ind T v1 (\lambda (t0: T).((eq T t0 (THead (Flat Appl) u t)) \to (arity g +c2 v2 a2))) H8 u H14) in (let H16 \def (eq_ind T v1 (\lambda (t0: T).(pr0 t0 +v2)) H7 u H14) in (let H17 \def (eq_ind_r T t (\lambda (t0: T).((eq T t3 +(THead (Flat Appl) u t0)) \to (arity g c2 t4 a2))) H10 (THead (Bind Abst) u0 +t3) H13) in (let H18 \def (eq_ind_r T t (\lambda (t0: T).((eq T u (THead +(Flat Appl) u t0)) \to (arity g c2 v2 a2))) H15 (THead (Bind Abst) u0 t3) +H13) in (let H19 \def (eq_ind_r T t (\lambda (t0: T).(\forall (c3: C).((wcpr0 +c c3) \to (\forall (t5: T).((pr0 t0 t5) \to (arity g c3 t5 (AHead a1 +a2))))))) H3 (THead (Bind Abst) u0 t3) H13) in (let H20 \def (eq_ind_r T t +(\lambda (t0: T).(arity g c t0 (AHead a1 a2))) H2 (THead (Bind Abst) u0 t3) +H13) in (let H21 \def (H1 c2 H4 v2 H16) in (let H22 \def (H19 c2 H4 (THead +(Bind Abst) u0 t4) (pr0_comp u0 u0 (pr0_refl u0) t3 t4 H9 (Bind Abst))) in +(let H23 \def (arity_gen_abst g c2 u0 t4 (AHead a1 a2) H22) in (ex3_2_ind A A +(\lambda (a3: A).(\lambda (a4: A).(eq A (AHead a1 a2) (AHead a3 a4)))) +(\lambda (a3: A).(\lambda (_: A).(arity g c2 u0 (asucc g a3)))) (\lambda (_: +A).(\lambda (a4: A).(arity g (CHead c2 (Bind Abst) u0) t4 a4))) (arity g c2 +(THead (Bind Abbr) v2 t4) a2) (\lambda (x0: A).(\lambda (x1: A).(\lambda +(H24: (eq A (AHead a1 a2) (AHead x0 x1))).(\lambda (H25: (arity g c2 u0 +(asucc g x0))).(\lambda (H26: (arity g (CHead c2 (Bind Abst) u0) t4 x1)).(let +H27 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) +with [(ASort _ _) \Rightarrow a1 | (AHead a0 _) \Rightarrow a0])) (AHead a1 +a2) (AHead x0 x1) H24) in ((let H28 \def (f_equal A A (\lambda (e: A).(match +e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a2 | (AHead _ +a0) \Rightarrow a0])) (AHead a1 a2) (AHead x0 x1) H24) in (\lambda (H29: (eq +A a1 x0)).(let H30 \def (eq_ind_r A x1 (\lambda (a0: A).(arity g (CHead c2 +(Bind Abst) u0) t4 a0)) H26 a2 H28) in (let H31 \def (eq_ind_r A x0 (\lambda +(a0: A).(arity g c2 u0 (asucc g a0))) H25 a1 H29) in (arity_bind g Abbr +not_abbr_abst c2 v2 a1 H21 t4 a2 (csuba_arity g (CHead c2 (Bind Abst) u0) t4 +a2 H30 (CHead c2 (Bind Abbr) v2) (csuba_abst g c2 c2 (csuba_refl g c2) u0 a1 +H31 v2 H21))))))) H27))))))) H23)))))))))))) H12)))))))))))) (\lambda (b: +B).(\lambda (H7: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: +T).(\lambda (H8: (pr0 v1 v2)).(\lambda (H9: (((eq T v1 (THead (Flat Appl) u +t)) \to (arity g c2 v2 a2)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda +(H10: (pr0 u1 u2)).(\lambda (H11: (((eq T u1 (THead (Flat Appl) u t)) \to +(arity g c2 u2 a2)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H12: (pr0 +t3 t4)).(\lambda (H13: (((eq T t3 (THead (Flat Appl) u t)) \to (arity g c2 t4 +a2)))).(\lambda (H14: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) +(THead (Flat Appl) u t))).(let H15 \def (f_equal T T (\lambda (e: T).(match e +in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) +\Rightarrow v1 | (THead _ t0 _) \Rightarrow t0])) (THead (Flat Appl) v1 +(THead (Bind b) u1 t3)) (THead (Flat Appl) u t) H14) in ((let H16 \def +(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with +[(TSort _) \Rightarrow (THead (Bind b) u1 t3) | (TLRef _) \Rightarrow (THead +(Bind b) u1 t3) | (THead _ _ t0) \Rightarrow t0])) (THead (Flat Appl) v1 +(THead (Bind b) u1 t3)) (THead (Flat Appl) u t) H14) in (\lambda (H17: (eq T +v1 u)).(let H18 \def (eq_ind T v1 (\lambda (t0: T).((eq T t0 (THead (Flat +Appl) u t)) \to (arity g c2 v2 a2))) H9 u H17) in (let H19 \def (eq_ind T v1 +(\lambda (t0: T).(pr0 t0 v2)) H8 u H17) in (let H20 \def (eq_ind_r T t +(\lambda (t0: T).((eq T t3 (THead (Flat Appl) u t0)) \to (arity g c2 t4 a2))) +H13 (THead (Bind b) u1 t3) H16) in (let H21 \def (eq_ind_r T t (\lambda (t0: +T).((eq T u1 (THead (Flat Appl) u t0)) \to (arity g c2 u2 a2))) H11 (THead +(Bind b) u1 t3) H16) in (let H22 \def (eq_ind_r T t (\lambda (t0: T).((eq T u +(THead (Flat Appl) u t0)) \to (arity g c2 v2 a2))) H18 (THead (Bind b) u1 t3) +H16) in (let H23 \def (eq_ind_r T t (\lambda (t0: T).(\forall (c3: C).((wcpr0 +c c3) \to (\forall (t5: T).((pr0 t0 t5) \to (arity g c3 t5 (AHead a1 +a2))))))) H3 (THead (Bind b) u1 t3) H16) in (let H24 \def (eq_ind_r T t +(\lambda (t0: T).(arity g c t0 (AHead a1 a2))) H2 (THead (Bind b) u1 t3) H16) +in (let H25 \def (H1 c2 H4 v2 H19) in (let H26 \def (H23 c2 H4 (THead (Bind +b) u2 t4) (pr0_comp u1 u2 H10 t3 t4 H12 (Bind b))) in (let H27 \def +(arity_gen_bind b H7 g c2 u2 t4 (AHead a1 a2) H26) in (ex2_ind A (\lambda +(a3: A).(arity g c2 u2 a3)) (\lambda (_: A).(arity g (CHead c2 (Bind b) u2) +t4 (AHead a1 a2))) (arity g c2 (THead (Bind b) u2 (THead (Flat Appl) (lift (S +O) O v2) t4)) a2) (\lambda (x: A).(\lambda (H28: (arity g c2 u2 x)).(\lambda +(H29: (arity g (CHead c2 (Bind b) u2) t4 (AHead a1 a2))).(arity_bind g b H7 +c2 u2 x H28 (THead (Flat Appl) (lift (S O) O v2) t4) a2 (arity_appl g (CHead +c2 (Bind b) u2) (lift (S O) O v2) a1 (arity_lift g c2 v2 a1 H25 (CHead c2 +(Bind b) u2) (S O) O (drop_drop (Bind b) O c2 c2 (drop_refl c2) u2)) t4 a2 +H29))))) H27))))))))))))) H15))))))))))))))))) (\lambda (u1: T).(\lambda (u2: +T).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 (THead (Flat Appl) u t)) +\to (arity g c2 u2 a2)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 +t3 t4)).(\lambda (_: (((eq T t3 (THead (Flat Appl) u t)) \to (arity g c2 t4 +a2)))).(\lambda (w: T).(\lambda (_: (subst0 O u2 t4 w)).(\lambda (H12: (eq T +(THead (Bind Abbr) u1 t3) (THead (Flat Appl) u t))).(let H13 \def (eq_ind T +(THead (Bind Abbr) u1 t3) (\lambda (ee: T).(match ee in T return (\lambda (_: +T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | +(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with +[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat +Appl) u t) H12) in (False_ind (arity g c2 (THead (Bind Abbr) u2 w) a2) +H13))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda +(t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 +(THead (Flat Appl) u t)) \to (arity g c2 t4 a2)))).(\lambda (u0: T).(\lambda +(H10: (eq T (THead (Bind b) u0 (lift (S O) O t3)) (THead (Flat Appl) u +t))).(let H11 \def (eq_ind T (THead (Bind b) u0 (lift (S O) O t3)) (\lambda +(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) +\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow +(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | +(Flat _) \Rightarrow False])])) I (THead (Flat Appl) u t) H10) in (False_ind +(arity g c2 t4 a2) H11)))))))))) (\lambda (t3: T).(\lambda (t4: T).(\lambda +(_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead (Flat Appl) u t)) \to (arity +g c2 t4 a2)))).(\lambda (u0: T).(\lambda (H9: (eq T (THead (Flat Cast) u0 t3) +(THead (Flat Appl) u t))).(let H10 \def (eq_ind T (THead (Flat Cast) u0 t3) +(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) +\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow +(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | +(Flat f) \Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl +\Rightarrow False | Cast \Rightarrow True])])])) I (THead (Flat Appl) u t) +H9) in (False_ind (arity g c2 t4 a2) H10)))))))) y t2 H6))) H5)))))))))))))) +(\lambda (c: C).(\lambda (u: T).(\lambda (a0: A).(\lambda (_: (arity g c u +(asucc g a0))).(\lambda (H1: ((\forall (c2: C).((wcpr0 c c2) \to (\forall +(t2: T).((pr0 u t2) \to (arity g c2 t2 (asucc g a0)))))))).(\lambda (t: +T).(\lambda (_: (arity g c t a0)).(\lambda (H3: ((\forall (c2: C).((wcpr0 c +c2) \to (\forall (t2: T).((pr0 t t2) \to (arity g c2 t2 a0))))))).(\lambda +(c2: C).(\lambda (H4: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H5: (pr0 +(THead (Flat Cast) u t) t2)).(insert_eq T (THead (Flat Cast) u t) (\lambda +(t0: T).(pr0 t0 t2)) (\lambda (_: T).(arity g c2 t2 a0)) (\lambda (y: +T).(\lambda (H6: (pr0 y t2)).(pr0_ind (\lambda (t0: T).(\lambda (t3: T).((eq +T t0 (THead (Flat Cast) u t)) \to (arity g c2 t3 a0)))) (\lambda (t0: +T).(\lambda (H7: (eq T t0 (THead (Flat Cast) u t))).(let H8 \def (f_equal T T +(\lambda (e: T).e) t0 (THead (Flat Cast) u t) H7) in (eq_ind_r T (THead (Flat +Cast) u t) (\lambda (t3: T).(arity g c2 t3 a0)) (arity_cast g c2 u a0 (H1 c2 +H4 u (pr0_refl u)) t (H3 c2 H4 t (pr0_refl t))) t0 H8)))) (\lambda (u1: +T).(\lambda (u2: T).(\lambda (H7: (pr0 u1 u2)).(\lambda (H8: (((eq T u1 +(THead (Flat Cast) u t)) \to (arity g c2 u2 a0)))).(\lambda (t3: T).(\lambda +(t4: T).(\lambda (H9: (pr0 t3 t4)).(\lambda (H10: (((eq T t3 (THead (Flat +Cast) u t)) \to (arity g c2 t4 a0)))).(\lambda (k: K).(\lambda (H11: (eq T +(THead k u1 t3) (THead (Flat Cast) u t))).(let H12 \def (f_equal T K (\lambda +(e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k +| (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) +(THead (Flat Cast) u t) H11) in ((let H13 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | +(TLRef _) \Rightarrow u1 | (THead _ t0 _) \Rightarrow t0])) (THead k u1 t3) +(THead (Flat Cast) u t) H11) in ((let H14 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | +(TLRef _) \Rightarrow t3 | (THead _ _ t0) \Rightarrow t0])) (THead k u1 t3) +(THead (Flat Cast) u t) H11) in (\lambda (H15: (eq T u1 u)).(\lambda (H16: +(eq K k (Flat Cast))).(eq_ind_r K (Flat Cast) (\lambda (k0: K).(arity g c2 +(THead k0 u2 t4) a0)) (let H17 \def (eq_ind T t3 (\lambda (t0: T).((eq T t0 +(THead (Flat Cast) u t)) \to (arity g c2 t4 a0))) H10 t H14) in (let H18 \def +(eq_ind T t3 (\lambda (t0: T).(pr0 t0 t4)) H9 t H14) in (let H19 \def (eq_ind +T u1 (\lambda (t0: T).((eq T t0 (THead (Flat Cast) u t)) \to (arity g c2 u2 +a0))) H8 u H15) in (let H20 \def (eq_ind T u1 (\lambda (t0: T).(pr0 t0 u2)) +H7 u H15) in (arity_cast g c2 u2 a0 (H1 c2 H4 u2 H20) t4 (H3 c2 H4 t4 +H18)))))) k H16)))) H13)) H12)))))))))))) (\lambda (u0: T).(\lambda (v1: +T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (THead +(Flat Cast) u t)) \to (arity g c2 v2 a0)))).(\lambda (t3: T).(\lambda (t4: +T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead (Flat Cast) u t)) +\to (arity g c2 t4 a0)))).(\lambda (H11: (eq T (THead (Flat Appl) v1 (THead +(Bind Abst) u0 t3)) (THead (Flat Cast) u t))).(let H12 \def (eq_ind T (THead +(Flat Appl) v1 (THead (Bind Abst) u0 t3)) (\lambda (ee: T).(match ee in T +return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda +(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f +in F return (\lambda (_: F).Prop) with [Appl \Rightarrow True | Cast +\Rightarrow False])])])) I (THead (Flat Cast) u t) H11) in (False_ind (arity +g c2 (THead (Bind Abbr) v2 t4) a0) H12)))))))))))) (\lambda (b: B).(\lambda +(_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 +v1 v2)).(\lambda (_: (((eq T v1 (THead (Flat Cast) u t)) \to (arity g c2 v2 +a0)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda +(_: (((eq T u1 (THead (Flat Cast) u t)) \to (arity g c2 u2 a0)))).(\lambda +(t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 +(THead (Flat Cast) u t)) \to (arity g c2 t4 a0)))).(\lambda (H14: (eq T +(THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Cast) u t))).(let +H15 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (\lambda +(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) +\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow +(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | +(Flat f) \Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl +\Rightarrow True | Cast \Rightarrow False])])])) I (THead (Flat Cast) u t) +H14) in (False_ind (arity g c2 (THead (Bind b) u2 (THead (Flat Appl) (lift (S +O) O v2) t4)) a0) H15))))))))))))))))) (\lambda (u1: T).(\lambda (u2: +T).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 (THead (Flat Cast) u t)) +\to (arity g c2 u2 a0)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 +t3 t4)).(\lambda (_: (((eq T t3 (THead (Flat Cast) u t)) \to (arity g c2 t4 +a0)))).(\lambda (w: T).(\lambda (_: (subst0 O u2 t4 w)).(\lambda (H12: (eq T +(THead (Bind Abbr) u1 t3) (THead (Flat Cast) u t))).(let H13 \def (eq_ind T +(THead (Bind Abbr) u1 t3) (\lambda (ee: T).(match ee in T return (\lambda (_: +T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | +(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with +[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat +Cast) u t) H12) in (False_ind (arity g c2 (THead (Bind Abbr) u2 w) a0) +H13))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda +(t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 +(THead (Flat Cast) u t)) \to (arity g c2 t4 a0)))).(\lambda (u0: T).(\lambda +(H10: (eq T (THead (Bind b) u0 (lift (S O) O t3)) (THead (Flat Cast) u +t))).(let H11 \def (eq_ind T (THead (Bind b) u0 (lift (S O) O t3)) (\lambda +(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) +\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow +(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | +(Flat _) \Rightarrow False])])) I (THead (Flat Cast) u t) H10) in (False_ind +(arity g c2 t4 a0) H11)))))))))) (\lambda (t3: T).(\lambda (t4: T).(\lambda +(H7: (pr0 t3 t4)).(\lambda (H8: (((eq T t3 (THead (Flat Cast) u t)) \to +(arity g c2 t4 a0)))).(\lambda (u0: T).(\lambda (H9: (eq T (THead (Flat Cast) +u0 t3) (THead (Flat Cast) u t))).(let H10 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | +(TLRef _) \Rightarrow u0 | (THead _ t0 _) \Rightarrow t0])) (THead (Flat +Cast) u0 t3) (THead (Flat Cast) u t) H9) in ((let H11 \def (f_equal T T +(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t0) \Rightarrow t0])) +(THead (Flat Cast) u0 t3) (THead (Flat Cast) u t) H9) in (\lambda (_: (eq T +u0 u)).(let H13 \def (eq_ind T t3 (\lambda (t0: T).((eq T t0 (THead (Flat +Cast) u t)) \to (arity g c2 t4 a0))) H8 t H11) in (let H14 \def (eq_ind T t3 +(\lambda (t0: T).(pr0 t0 t4)) H7 t H11) in (H3 c2 H4 t4 H14))))) H10)))))))) +y t2 H6))) H5))))))))))))) (\lambda (c: C).(\lambda (t: T).(\lambda (a1: +A).(\lambda (_: (arity g c t a1)).(\lambda (H1: ((\forall (c2: C).((wcpr0 c +c2) \to (\forall (t2: T).((pr0 t t2) \to (arity g c2 t2 a1))))))).(\lambda +(a2: A).(\lambda (H2: (leq g a1 a2)).(\lambda (c2: C).(\lambda (H3: (wcpr0 c +c2)).(\lambda (t2: T).(\lambda (H4: (pr0 t t2)).(arity_repl g c2 t2 a1 (H1 c2 +H3 t2 H4) a2 H2)))))))))))) c1 t1 a H))))). +(* COMMENTS +Initial nodes: 10246 +END *) + +theorem arity_sred_wcpr0_pr1: + \forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall (g: G).(\forall +(c1: C).(\forall (a: A).((arity g c1 t1 a) \to (\forall (c2: C).((wcpr0 c1 +c2) \to (arity g c2 t2 a))))))))) +\def + \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr1 t1 t2)).(pr1_ind (\lambda +(t: T).(\lambda (t0: T).(\forall (g: G).(\forall (c1: C).(\forall (a: +A).((arity g c1 t a) \to (\forall (c2: C).((wcpr0 c1 c2) \to (arity g c2 t0 +a))))))))) (\lambda (t: T).(\lambda (g: G).(\lambda (c1: C).(\lambda (a: +A).(\lambda (H0: (arity g c1 t a)).(\lambda (c2: C).(\lambda (H1: (wcpr0 c1 +c2)).(arity_sred_wcpr0_pr0 g c1 t a H0 c2 H1 t (pr0_refl t))))))))) (\lambda +(t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t4 t3)).(\lambda (t5: T).(\lambda +(_: (pr1 t3 t5)).(\lambda (H2: ((\forall (g: G).(\forall (c1: C).(\forall (a: +A).((arity g c1 t3 a) \to (\forall (c2: C).((wcpr0 c1 c2) \to (arity g c2 t5 +a))))))))).(\lambda (g: G).(\lambda (c1: C).(\lambda (a: A).(\lambda (H3: +(arity g c1 t4 a)).(\lambda (c2: C).(\lambda (H4: (wcpr0 c1 c2)).(H2 g c2 a +(arity_sred_wcpr0_pr0 g c1 t4 a H3 c2 H4 t3 H0) c2 (wcpr0_refl +c2)))))))))))))) t1 t2 H))). +(* COMMENTS +Initial nodes: 213 +END *) + +theorem arity_sred_pr2: + \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall +(g: G).(\forall (a: A).((arity g c t1 a) \to (arity g c t2 a))))))) +\def + \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 +t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (g: +G).(\forall (a: A).((arity g c0 t a) \to (arity g c0 t0 a))))))) (\lambda +(c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(\lambda +(g: G).(\lambda (a: A).(\lambda (H1: (arity g c0 t3 a)).(arity_sred_wcpr0_pr0 +g c0 t3 a H1 c0 (wcpr0_refl c0) t4 H0)))))))) (\lambda (c0: C).(\lambda (d: +C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind +Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 +t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (g: +G).(\lambda (a: A).(\lambda (H3: (arity g c0 t3 a)).(arity_subst0 g c0 t4 a +(arity_sred_wcpr0_pr0 g c0 t3 a H3 c0 (wcpr0_refl c0) t4 H1) d u i H0 t +H2)))))))))))))) c t1 t2 H)))). +(* COMMENTS +Initial nodes: 205 +END *) + +theorem arity_sred_pr3: + \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall +(g: G).(\forall (a: A).((arity g c t1 a) \to (arity g c t2 a))))))) +\def + \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 +t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (g: G).(\forall (a: +A).((arity g c t a) \to (arity g c t0 a)))))) (\lambda (t: T).(\lambda (g: +G).(\lambda (a: A).(\lambda (H0: (arity g c t a)).H0)))) (\lambda (t3: +T).(\lambda (t4: T).(\lambda (H0: (pr2 c t4 t3)).(\lambda (t5: T).(\lambda +(_: (pr3 c t3 t5)).(\lambda (H2: ((\forall (g: G).(\forall (a: A).((arity g c +t3 a) \to (arity g c t5 a)))))).(\lambda (g: G).(\lambda (a: A).(\lambda (H3: +(arity g c t4 a)).(H2 g a (arity_sred_pr2 c t4 t3 H0 g a H3))))))))))) t1 t2 +H)))). +(* COMMENTS +Initial nodes: 151 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/arity/props.ma b/matita/matita/contribs/lambdadelta/basic_1/arity/props.ma new file mode 100644 index 000000000..6f131df09 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/arity/props.ma @@ -0,0 +1,434 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/arity/fwd.ma". + +theorem node_inh: + \forall (g: G).(\forall (n: nat).(\forall (k: nat).(ex_2 C T (\lambda (c: +C).(\lambda (t: T).(arity g c t (ASort k n))))))) +\def + \lambda (g: G).(\lambda (n: nat).(\lambda (k: nat).(nat_ind (\lambda (n0: +nat).(ex_2 C T (\lambda (c: C).(\lambda (t: T).(arity g c t (ASort n0 n)))))) +(ex_2_intro C T (\lambda (c: C).(\lambda (t: T).(arity g c t (ASort O n)))) +(CSort O) (TSort n) (arity_sort g (CSort O) n)) (\lambda (n0: nat).(\lambda +(H: (ex_2 C T (\lambda (c: C).(\lambda (t: T).(arity g c t (ASort n0 +n)))))).(let H0 \def H in (ex_2_ind C T (\lambda (c: C).(\lambda (t: +T).(arity g c t (ASort n0 n)))) (ex_2 C T (\lambda (c: C).(\lambda (t: +T).(arity g c t (ASort (S n0) n))))) (\lambda (x0: C).(\lambda (x1: +T).(\lambda (H1: (arity g x0 x1 (ASort n0 n))).(ex_2_intro C T (\lambda (c: +C).(\lambda (t: T).(arity g c t (ASort (S n0) n)))) (CHead x0 (Bind Abst) x1) +(TLRef O) (arity_abst g (CHead x0 (Bind Abst) x1) x0 x1 O (getl_refl Abst x0 +x1) (ASort (S n0) n) H1))))) H0)))) k))). +(* COMMENTS +Initial nodes: 253 +END *) + +theorem arity_lift: + \forall (g: G).(\forall (c2: C).(\forall (t: T).(\forall (a: A).((arity g c2 +t a) \to (\forall (c1: C).(\forall (h: nat).(\forall (d: nat).((drop h d c1 +c2) \to (arity g c1 (lift h d t) a))))))))) +\def + \lambda (g: G).(\lambda (c2: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: +(arity g c2 t a)).(arity_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda (a0: +A).(\forall (c1: C).(\forall (h: nat).(\forall (d: nat).((drop h d c1 c) \to +(arity g c1 (lift h d t0) a0)))))))) (\lambda (c: C).(\lambda (n: +nat).(\lambda (c1: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (_: (drop +h d c1 c)).(eq_ind_r T (TSort n) (\lambda (t0: T).(arity g c1 t0 (ASort O +n))) (arity_sort g c1 n) (lift h d (TSort n)) (lift_sort n h d)))))))) +(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda +(H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (a0: A).(\lambda (H1: +(arity g d u a0)).(\lambda (H2: ((\forall (c1: C).(\forall (h: nat).(\forall +(d0: nat).((drop h d0 c1 d) \to (arity g c1 (lift h d0 u) a0))))))).(\lambda +(c1: C).(\lambda (h: nat).(\lambda (d0: nat).(\lambda (H3: (drop h d0 c1 +c)).(lt_le_e i d0 (arity g c1 (lift h d0 (TLRef i)) a0) (\lambda (H4: (lt i +d0)).(eq_ind_r T (TLRef i) (\lambda (t0: T).(arity g c1 t0 a0)) (let H5 \def +(drop_getl_trans_le i d0 (le_S_n i d0 (le_S (S i) d0 H4)) c1 c h H3 (CHead d +(Bind Abbr) u) H0) in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop i +O c1 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d0 i) e0 e1))) +(\lambda (_: C).(\lambda (e1: C).(clear e1 (CHead d (Bind Abbr) u)))) (arity +g c1 (TLRef i) a0) (\lambda (x0: C).(\lambda (x1: C).(\lambda (H6: (drop i O +c1 x0)).(\lambda (H7: (drop h (minus d0 i) x0 x1)).(\lambda (H8: (clear x1 +(CHead d (Bind Abbr) u))).(let H9 \def (eq_ind nat (minus d0 i) (\lambda (n: +nat).(drop h n x0 x1)) H7 (S (minus d0 (S i))) (minus_x_Sy d0 i H4)) in (let +H10 \def (drop_clear_S x1 x0 h (minus d0 (S i)) H9 Abbr d u H8) in (ex2_ind C +(\lambda (c3: C).(clear x0 (CHead c3 (Bind Abbr) (lift h (minus d0 (S i)) +u)))) (\lambda (c3: C).(drop h (minus d0 (S i)) c3 d)) (arity g c1 (TLRef i) +a0) (\lambda (x: C).(\lambda (H11: (clear x0 (CHead x (Bind Abbr) (lift h +(minus d0 (S i)) u)))).(\lambda (H12: (drop h (minus d0 (S i)) x +d)).(arity_abbr g c1 x (lift h (minus d0 (S i)) u) i (getl_intro i c1 (CHead +x (Bind Abbr) (lift h (minus d0 (S i)) u)) x0 H6 H11) a0 (H2 x h (minus d0 (S +i)) H12))))) H10)))))))) H5)) (lift h d0 (TLRef i)) (lift_lref_lt i h d0 +H4))) (\lambda (H4: (le d0 i)).(eq_ind_r T (TLRef (plus i h)) (\lambda (t0: +T).(arity g c1 t0 a0)) (arity_abbr g c1 d u (plus i h) (drop_getl_trans_ge i +c1 c d0 h H3 (CHead d (Bind Abbr) u) H0 H4) a0 H1) (lift h d0 (TLRef i)) +(lift_lref_ge i h d0 H4)))))))))))))))) (\lambda (c: C).(\lambda (d: +C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d (Bind +Abst) u))).(\lambda (a0: A).(\lambda (H1: (arity g d u (asucc g +a0))).(\lambda (H2: ((\forall (c1: C).(\forall (h: nat).(\forall (d0: +nat).((drop h d0 c1 d) \to (arity g c1 (lift h d0 u) (asucc g +a0)))))))).(\lambda (c1: C).(\lambda (h: nat).(\lambda (d0: nat).(\lambda +(H3: (drop h d0 c1 c)).(lt_le_e i d0 (arity g c1 (lift h d0 (TLRef i)) a0) +(\lambda (H4: (lt i d0)).(eq_ind_r T (TLRef i) (\lambda (t0: T).(arity g c1 +t0 a0)) (let H5 \def (drop_getl_trans_le i d0 (le_S_n i d0 (le_S (S i) d0 +H4)) c1 c h H3 (CHead d (Bind Abst) u) H0) in (ex3_2_ind C C (\lambda (e0: +C).(\lambda (_: C).(drop i O c1 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop +h (minus d0 i) e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear e1 (CHead d +(Bind Abst) u)))) (arity g c1 (TLRef i) a0) (\lambda (x0: C).(\lambda (x1: +C).(\lambda (H6: (drop i O c1 x0)).(\lambda (H7: (drop h (minus d0 i) x0 +x1)).(\lambda (H8: (clear x1 (CHead d (Bind Abst) u))).(let H9 \def (eq_ind +nat (minus d0 i) (\lambda (n: nat).(drop h n x0 x1)) H7 (S (minus d0 (S i))) +(minus_x_Sy d0 i H4)) in (let H10 \def (drop_clear_S x1 x0 h (minus d0 (S i)) +H9 Abst d u H8) in (ex2_ind C (\lambda (c3: C).(clear x0 (CHead c3 (Bind +Abst) (lift h (minus d0 (S i)) u)))) (\lambda (c3: C).(drop h (minus d0 (S +i)) c3 d)) (arity g c1 (TLRef i) a0) (\lambda (x: C).(\lambda (H11: (clear x0 +(CHead x (Bind Abst) (lift h (minus d0 (S i)) u)))).(\lambda (H12: (drop h +(minus d0 (S i)) x d)).(arity_abst g c1 x (lift h (minus d0 (S i)) u) i +(getl_intro i c1 (CHead x (Bind Abst) (lift h (minus d0 (S i)) u)) x0 H6 H11) +a0 (H2 x h (minus d0 (S i)) H12))))) H10)))))))) H5)) (lift h d0 (TLRef i)) +(lift_lref_lt i h d0 H4))) (\lambda (H4: (le d0 i)).(eq_ind_r T (TLRef (plus +i h)) (\lambda (t0: T).(arity g c1 t0 a0)) (arity_abst g c1 d u (plus i h) +(drop_getl_trans_ge i c1 c d0 h H3 (CHead d (Bind Abst) u) H0 H4) a0 H1) +(lift h d0 (TLRef i)) (lift_lref_ge i h d0 H4)))))))))))))))) (\lambda (b: +B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c: C).(\lambda (u: +T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H2: ((\forall +(c1: C).(\forall (h: nat).(\forall (d: nat).((drop h d c1 c) \to (arity g c1 +(lift h d u) a1))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity +g (CHead c (Bind b) u) t0 a2)).(\lambda (H4: ((\forall (c1: C).(\forall (h: +nat).(\forall (d: nat).((drop h d c1 (CHead c (Bind b) u)) \to (arity g c1 +(lift h d t0) a2))))))).(\lambda (c1: C).(\lambda (h: nat).(\lambda (d: +nat).(\lambda (H5: (drop h d c1 c)).(eq_ind_r T (THead (Bind b) (lift h d u) +(lift h (s (Bind b) d) t0)) (\lambda (t1: T).(arity g c1 t1 a2)) (arity_bind +g b H0 c1 (lift h d u) a1 (H2 c1 h d H5) (lift h (s (Bind b) d) t0) a2 (H4 +(CHead c1 (Bind b) (lift h d u)) h (s (Bind b) d) (drop_skip_bind h d c1 c H5 +b u))) (lift h d (THead (Bind b) u t0)) (lift_head (Bind b) u t0 h +d))))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda +(_: (arity g c u (asucc g a1))).(\lambda (H1: ((\forall (c1: C).(\forall (h: +nat).(\forall (d: nat).((drop h d c1 c) \to (arity g c1 (lift h d u) (asucc g +a1)))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c +(Bind Abst) u) t0 a2)).(\lambda (H3: ((\forall (c1: C).(\forall (h: +nat).(\forall (d: nat).((drop h d c1 (CHead c (Bind Abst) u)) \to (arity g c1 +(lift h d t0) a2))))))).(\lambda (c1: C).(\lambda (h: nat).(\lambda (d: +nat).(\lambda (H4: (drop h d c1 c)).(eq_ind_r T (THead (Bind Abst) (lift h d +u) (lift h (s (Bind Abst) d) t0)) (\lambda (t1: T).(arity g c1 t1 (AHead a1 +a2))) (arity_head g c1 (lift h d u) a1 (H1 c1 h d H4) (lift h (s (Bind Abst) +d) t0) a2 (H3 (CHead c1 (Bind Abst) (lift h d u)) h (s (Bind Abst) d) +(drop_skip_bind h d c1 c H4 Abst u))) (lift h d (THead (Bind Abst) u t0)) +(lift_head (Bind Abst) u t0 h d))))))))))))))) (\lambda (c: C).(\lambda (u: +T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H1: ((\forall +(c1: C).(\forall (h: nat).(\forall (d: nat).((drop h d c1 c) \to (arity g c1 +(lift h d u) a1))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity +g c t0 (AHead a1 a2))).(\lambda (H3: ((\forall (c1: C).(\forall (h: +nat).(\forall (d: nat).((drop h d c1 c) \to (arity g c1 (lift h d t0) (AHead +a1 a2)))))))).(\lambda (c1: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda +(H4: (drop h d c1 c)).(eq_ind_r T (THead (Flat Appl) (lift h d u) (lift h (s +(Flat Appl) d) t0)) (\lambda (t1: T).(arity g c1 t1 a2)) (arity_appl g c1 +(lift h d u) a1 (H1 c1 h d H4) (lift h (s (Flat Appl) d) t0) a2 (H3 c1 h (s +(Flat Appl) d) H4)) (lift h d (THead (Flat Appl) u t0)) (lift_head (Flat +Appl) u t0 h d))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a0: +A).(\lambda (_: (arity g c u (asucc g a0))).(\lambda (H1: ((\forall (c1: +C).(\forall (h: nat).(\forall (d: nat).((drop h d c1 c) \to (arity g c1 (lift +h d u) (asucc g a0)))))))).(\lambda (t0: T).(\lambda (_: (arity g c t0 +a0)).(\lambda (H3: ((\forall (c1: C).(\forall (h: nat).(\forall (d: +nat).((drop h d c1 c) \to (arity g c1 (lift h d t0) a0))))))).(\lambda (c1: +C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H4: (drop h d c1 +c)).(eq_ind_r T (THead (Flat Cast) (lift h d u) (lift h (s (Flat Cast) d) +t0)) (\lambda (t1: T).(arity g c1 t1 a0)) (arity_cast g c1 (lift h d u) a0 +(H1 c1 h d H4) (lift h (s (Flat Cast) d) t0) (H3 c1 h (s (Flat Cast) d) H4)) +(lift h d (THead (Flat Cast) u t0)) (lift_head (Flat Cast) u t0 h +d)))))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda +(_: (arity g c t0 a1)).(\lambda (H1: ((\forall (c1: C).(\forall (h: +nat).(\forall (d: nat).((drop h d c1 c) \to (arity g c1 (lift h d t0) +a1))))))).(\lambda (a2: A).(\lambda (H2: (leq g a1 a2)).(\lambda (c1: +C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H3: (drop h d c1 +c)).(arity_repl g c1 (lift h d t0) a1 (H1 c1 h d H3) a2 H2)))))))))))) c2 t a +H))))). +(* COMMENTS +Initial nodes: 2661 +END *) + +theorem arity_mono: + \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a1: A).((arity g c +t a1) \to (\forall (a2: A).((arity g c t a2) \to (leq g a1 a2))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a1: A).(\lambda (H: +(arity g c t a1)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (a: +A).(\forall (a2: A).((arity g c0 t0 a2) \to (leq g a a2)))))) (\lambda (c0: +C).(\lambda (n: nat).(\lambda (a2: A).(\lambda (H0: (arity g c0 (TSort n) +a2)).(leq_sym g a2 (ASort O n) (arity_gen_sort g c0 n a2 H0)))))) (\lambda +(c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl +i c0 (CHead d (Bind Abbr) u))).(\lambda (a: A).(\lambda (_: (arity g d u +a)).(\lambda (H2: ((\forall (a2: A).((arity g d u a2) \to (leq g a +a2))))).(\lambda (a2: A).(\lambda (H3: (arity g c0 (TLRef i) a2)).(let H4 +\def (arity_gen_lref g c0 i a2 H3) in (or_ind (ex2_2 C T (\lambda (d0: +C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abbr) u0)))) (\lambda (d0: +C).(\lambda (u0: T).(arity g d0 u0 a2)))) (ex2_2 C T (\lambda (d0: +C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0)))) (\lambda (d0: +C).(\lambda (u0: T).(arity g d0 u0 (asucc g a2))))) (leq g a a2) (\lambda +(H5: (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind +Abbr) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 +a2))))).(ex2_2_ind C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 +(Bind Abbr) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a2))) +(leq g a a2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl i c0 +(CHead x0 (Bind Abbr) x1))).(\lambda (H7: (arity g x0 x1 a2)).(let H8 \def +(eq_ind C (CHead d (Bind Abbr) u) (\lambda (c1: C).(getl i c0 c1)) H0 (CHead +x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead x0 (Bind +Abbr) x1) H6)) in (let H9 \def (f_equal C C (\lambda (e: C).(match e in C +return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) +\Rightarrow c1])) (CHead d (Bind Abbr) u) (CHead x0 (Bind Abbr) x1) +(getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead x0 (Bind Abbr) x1) H6)) in +((let H10 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: +C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead +d (Bind Abbr) u) (CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind Abbr) +u) i H0 (CHead x0 (Bind Abbr) x1) H6)) in (\lambda (H11: (eq C d x0)).(let +H12 \def (eq_ind_r T x1 (\lambda (t0: T).(getl i c0 (CHead x0 (Bind Abbr) +t0))) H8 u H10) in (let H13 \def (eq_ind_r T x1 (\lambda (t0: T).(arity g x0 +t0 a2)) H7 u H10) in (let H14 \def (eq_ind_r C x0 (\lambda (c1: C).(getl i c0 +(CHead c1 (Bind Abbr) u))) H12 d H11) in (let H15 \def (eq_ind_r C x0 +(\lambda (c1: C).(arity g c1 u a2)) H13 d H11) in (H2 a2 H15))))))) H9))))))) +H5)) (\lambda (H5: (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 +(CHead d0 (Bind Abst) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 +(asucc g a2)))))).(ex2_2_ind C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 +(CHead d0 (Bind Abst) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 +(asucc g a2)))) (leq g a a2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: +(getl i c0 (CHead x0 (Bind Abst) x1))).(\lambda (_: (arity g x0 x1 (asucc g +a2))).(let H8 \def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c1: C).(getl i +c0 c1)) H0 (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abbr) u) i +H0 (CHead x0 (Bind Abst) x1) H6)) in (let H9 \def (eq_ind C (CHead d (Bind +Abbr) u) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with +[(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return +(\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B return +(\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | +Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead x0 (Bind +Abst) x1) (getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead x0 (Bind Abst) +x1) H6)) in (False_ind (leq g a a2) H9))))))) H5)) H4)))))))))))) (\lambda +(c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl +i c0 (CHead d (Bind Abst) u))).(\lambda (a: A).(\lambda (_: (arity g d u +(asucc g a))).(\lambda (H2: ((\forall (a2: A).((arity g d u a2) \to (leq g +(asucc g a) a2))))).(\lambda (a2: A).(\lambda (H3: (arity g c0 (TLRef i) +a2)).(let H4 \def (arity_gen_lref g c0 i a2 H3) in (or_ind (ex2_2 C T +(\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abbr) u0)))) +(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a2)))) (ex2_2 C T (\lambda +(d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0)))) (\lambda +(d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a2))))) (leq g a a2) +(\lambda (H5: (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead +d0 (Bind Abbr) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 +a2))))).(ex2_2_ind C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 +(Bind Abbr) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a2))) +(leq g a a2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl i c0 +(CHead x0 (Bind Abbr) x1))).(\lambda (_: (arity g x0 x1 a2)).(let H8 \def +(eq_ind C (CHead d (Bind Abst) u) (\lambda (c1: C).(getl i c0 c1)) H0 (CHead +x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind Abst) u) i H0 (CHead x0 (Bind +Abbr) x1) H6)) in (let H9 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda +(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) +\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda +(_: K).Prop) with [(Bind b) \Rightarrow (match b in B return (\lambda (_: +B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow True | Void +\Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead x0 (Bind Abbr) +x1) (getl_mono c0 (CHead d (Bind Abst) u) i H0 (CHead x0 (Bind Abbr) x1) H6)) +in (False_ind (leq g a a2) H9))))))) H5)) (\lambda (H5: (ex2_2 C T (\lambda +(d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0)))) (\lambda +(d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a2)))))).(ex2_2_ind C T +(\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0)))) +(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a2)))) (leq g a a2) +(\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl i c0 (CHead x0 (Bind +Abst) x1))).(\lambda (H7: (arity g x0 x1 (asucc g a2))).(let H8 \def (eq_ind +C (CHead d (Bind Abst) u) (\lambda (c1: C).(getl i c0 c1)) H0 (CHead x0 (Bind +Abst) x1) (getl_mono c0 (CHead d (Bind Abst) u) i H0 (CHead x0 (Bind Abst) +x1) H6)) in (let H9 \def (f_equal C C (\lambda (e: C).(match e in C return +(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow +c1])) (CHead d (Bind Abst) u) (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead +d (Bind Abst) u) i H0 (CHead x0 (Bind Abst) x1) H6)) in ((let H10 \def +(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with +[(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind +Abst) u) (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abst) u) i H0 +(CHead x0 (Bind Abst) x1) H6)) in (\lambda (H11: (eq C d x0)).(let H12 \def +(eq_ind_r T x1 (\lambda (t0: T).(getl i c0 (CHead x0 (Bind Abst) t0))) H8 u +H10) in (let H13 \def (eq_ind_r T x1 (\lambda (t0: T).(arity g x0 t0 (asucc g +a2))) H7 u H10) in (let H14 \def (eq_ind_r C x0 (\lambda (c1: C).(getl i c0 +(CHead c1 (Bind Abst) u))) H12 d H11) in (let H15 \def (eq_ind_r C x0 +(\lambda (c1: C).(arity g c1 u (asucc g a2))) H13 d H11) in (asucc_inj g a a2 +(H2 (asucc g a2) H15)))))))) H9))))))) H5)) H4)))))))))))) (\lambda (b: +B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c0: C).(\lambda (u: +T).(\lambda (a2: A).(\lambda (_: (arity g c0 u a2)).(\lambda (_: ((\forall +(a3: A).((arity g c0 u a3) \to (leq g a2 a3))))).(\lambda (t0: T).(\lambda +(a3: A).(\lambda (_: (arity g (CHead c0 (Bind b) u) t0 a3)).(\lambda (H4: +((\forall (a4: A).((arity g (CHead c0 (Bind b) u) t0 a4) \to (leq g a3 +a4))))).(\lambda (a0: A).(\lambda (H5: (arity g c0 (THead (Bind b) u t0) +a0)).(let H6 \def (arity_gen_bind b H0 g c0 u t0 a0 H5) in (ex2_ind A +(\lambda (a4: A).(arity g c0 u a4)) (\lambda (_: A).(arity g (CHead c0 (Bind +b) u) t0 a0)) (leq g a3 a0) (\lambda (x: A).(\lambda (_: (arity g c0 u +x)).(\lambda (H8: (arity g (CHead c0 (Bind b) u) t0 a0)).(H4 a0 H8)))) +H6))))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a2: A).(\lambda +(_: (arity g c0 u (asucc g a2))).(\lambda (H1: ((\forall (a3: A).((arity g c0 +u a3) \to (leq g (asucc g a2) a3))))).(\lambda (t0: T).(\lambda (a3: +A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u) t0 a3)).(\lambda (H3: +((\forall (a4: A).((arity g (CHead c0 (Bind Abst) u) t0 a4) \to (leq g a3 +a4))))).(\lambda (a0: A).(\lambda (H4: (arity g c0 (THead (Bind Abst) u t0) +a0)).(let H5 \def (arity_gen_abst g c0 u t0 a0 H4) in (ex3_2_ind A A (\lambda +(a4: A).(\lambda (a5: A).(eq A a0 (AHead a4 a5)))) (\lambda (a4: A).(\lambda +(_: A).(arity g c0 u (asucc g a4)))) (\lambda (_: A).(\lambda (a5: A).(arity +g (CHead c0 (Bind Abst) u) t0 a5))) (leq g (AHead a2 a3) a0) (\lambda (x0: +A).(\lambda (x1: A).(\lambda (H6: (eq A a0 (AHead x0 x1))).(\lambda (H7: +(arity g c0 u (asucc g x0))).(\lambda (H8: (arity g (CHead c0 (Bind Abst) u) +t0 x1)).(eq_ind_r A (AHead x0 x1) (\lambda (a: A).(leq g (AHead a2 a3) a)) +(leq_head g a2 x0 (asucc_inj g a2 x0 (H1 (asucc g x0) H7)) a3 x1 (H3 x1 H8)) +a0 H6)))))) H5))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a2: +A).(\lambda (_: (arity g c0 u a2)).(\lambda (_: ((\forall (a3: A).((arity g +c0 u a3) \to (leq g a2 a3))))).(\lambda (t0: T).(\lambda (a3: A).(\lambda (_: +(arity g c0 t0 (AHead a2 a3))).(\lambda (H3: ((\forall (a4: A).((arity g c0 +t0 a4) \to (leq g (AHead a2 a3) a4))))).(\lambda (a0: A).(\lambda (H4: (arity +g c0 (THead (Flat Appl) u t0) a0)).(let H5 \def (arity_gen_appl g c0 u t0 a0 +H4) in (ex2_ind A (\lambda (a4: A).(arity g c0 u a4)) (\lambda (a4: A).(arity +g c0 t0 (AHead a4 a0))) (leq g a3 a0) (\lambda (x: A).(\lambda (_: (arity g +c0 u x)).(\lambda (H7: (arity g c0 t0 (AHead x a0))).(ahead_inj_snd g a2 a3 x +a0 (H3 (AHead x a0) H7))))) H5))))))))))))) (\lambda (c0: C).(\lambda (u: +T).(\lambda (a: A).(\lambda (_: (arity g c0 u (asucc g a))).(\lambda (_: +((\forall (a2: A).((arity g c0 u a2) \to (leq g (asucc g a) a2))))).(\lambda +(t0: T).(\lambda (_: (arity g c0 t0 a)).(\lambda (H3: ((\forall (a2: +A).((arity g c0 t0 a2) \to (leq g a a2))))).(\lambda (a2: A).(\lambda (H4: +(arity g c0 (THead (Flat Cast) u t0) a2)).(let H5 \def (arity_gen_cast g c0 u +t0 a2 H4) in (land_ind (arity g c0 u (asucc g a2)) (arity g c0 t0 a2) (leq g +a a2) (\lambda (_: (arity g c0 u (asucc g a2))).(\lambda (H7: (arity g c0 t0 +a2)).(H3 a2 H7))) H5)))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda +(a2: A).(\lambda (_: (arity g c0 t0 a2)).(\lambda (H1: ((\forall (a3: +A).((arity g c0 t0 a3) \to (leq g a2 a3))))).(\lambda (a3: A).(\lambda (H2: +(leq g a2 a3)).(\lambda (a0: A).(\lambda (H3: (arity g c0 t0 a0)).(leq_trans +g a3 a2 (leq_sym g a2 a3 H2) a0 (H1 a0 H3))))))))))) c t a1 H))))). +(* COMMENTS +Initial nodes: 2947 +END *) + +theorem arity_repellent: + \forall (g: G).(\forall (c: C).(\forall (w: T).(\forall (t: T).(\forall (a1: +A).((arity g (CHead c (Bind Abst) w) t a1) \to (\forall (a2: A).((arity g c +(THead (Bind Abst) w t) a2) \to ((leq g a1 a2) \to (\forall (P: +Prop).P))))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (w: T).(\lambda (t: T).(\lambda (a1: +A).(\lambda (H: (arity g (CHead c (Bind Abst) w) t a1)).(\lambda (a2: +A).(\lambda (H0: (arity g c (THead (Bind Abst) w t) a2)).(\lambda (H1: (leq g +a1 a2)).(\lambda (P: Prop).(let H_y \def (arity_repl g (CHead c (Bind Abst) +w) t a1 H a2 H1) in (let H2 \def (arity_gen_abst g c w t a2 H0) in (ex3_2_ind +A A (\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: +A).(\lambda (_: A).(arity g c w (asucc g a3)))) (\lambda (_: A).(\lambda (a4: +A).(arity g (CHead c (Bind Abst) w) t a4))) P (\lambda (x0: A).(\lambda (x1: +A).(\lambda (H3: (eq A a2 (AHead x0 x1))).(\lambda (_: (arity g c w (asucc g +x0))).(\lambda (H5: (arity g (CHead c (Bind Abst) w) t x1)).(let H6 \def +(eq_ind A a2 (\lambda (a: A).(arity g (CHead c (Bind Abst) w) t a)) H_y +(AHead x0 x1) H3) in (leq_ahead_false_2 g x1 x0 (arity_mono g (CHead c (Bind +Abst) w) t (AHead x0 x1) H6 x1 H5) P))))))) H2)))))))))))). +(* COMMENTS +Initial nodes: 283 +END *) + +theorem arity_appls_cast: + \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (vs: +TList).(\forall (a: A).((arity g c (THeads (Flat Appl) vs u) (asucc g a)) \to +((arity g c (THeads (Flat Appl) vs t) a) \to (arity g c (THeads (Flat Appl) +vs (THead (Flat Cast) u t)) a)))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (vs: +TList).(TList_ind (\lambda (t0: TList).(\forall (a: A).((arity g c (THeads +(Flat Appl) t0 u) (asucc g a)) \to ((arity g c (THeads (Flat Appl) t0 t) a) +\to (arity g c (THeads (Flat Appl) t0 (THead (Flat Cast) u t)) a))))) +(\lambda (a: A).(\lambda (H: (arity g c u (asucc g a))).(\lambda (H0: (arity +g c t a)).(arity_cast g c u a H t H0)))) (\lambda (t0: T).(\lambda (t1: +TList).(\lambda (H: ((\forall (a: A).((arity g c (THeads (Flat Appl) t1 u) +(asucc g a)) \to ((arity g c (THeads (Flat Appl) t1 t) a) \to (arity g c +(THeads (Flat Appl) t1 (THead (Flat Cast) u t)) a)))))).(\lambda (a: +A).(\lambda (H0: (arity g c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 u)) +(asucc g a))).(\lambda (H1: (arity g c (THead (Flat Appl) t0 (THeads (Flat +Appl) t1 t)) a)).(let H2 \def (arity_gen_appl g c t0 (THeads (Flat Appl) t1 +t) a H1) in (ex2_ind A (\lambda (a1: A).(arity g c t0 a1)) (\lambda (a1: +A).(arity g c (THeads (Flat Appl) t1 t) (AHead a1 a))) (arity g c (THead +(Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Flat Cast) u t))) a) (\lambda +(x: A).(\lambda (H3: (arity g c t0 x)).(\lambda (H4: (arity g c (THeads (Flat +Appl) t1 t) (AHead x a))).(let H5 \def (arity_gen_appl g c t0 (THeads (Flat +Appl) t1 u) (asucc g a) H0) in (ex2_ind A (\lambda (a1: A).(arity g c t0 a1)) +(\lambda (a1: A).(arity g c (THeads (Flat Appl) t1 u) (AHead a1 (asucc g +a)))) (arity g c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Flat +Cast) u t))) a) (\lambda (x0: A).(\lambda (H6: (arity g c t0 x0)).(\lambda +(H7: (arity g c (THeads (Flat Appl) t1 u) (AHead x0 (asucc g +a)))).(arity_appl g c t0 x H3 (THeads (Flat Appl) t1 (THead (Flat Cast) u t)) +a (H (AHead x a) (arity_repl g c (THeads (Flat Appl) t1 u) (AHead x (asucc g +a)) (arity_repl g c (THeads (Flat Appl) t1 u) (AHead x0 (asucc g a)) H7 +(AHead x (asucc g a)) (leq_head g x0 x (arity_mono g c t0 x0 H6 x H3) (asucc +g a) (asucc g a) (leq_refl g (asucc g a)))) (asucc g (AHead x a)) (leq_refl g +(asucc g (AHead x a)))) H4))))) H5))))) H2)))))))) vs))))). +(* COMMENTS +Initial nodes: 707 +END *) + +theorem arity_appls_abbr: + \forall (g: G).(\forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: +nat).((getl i c (CHead d (Bind Abbr) v)) \to (\forall (vs: TList).(\forall +(a: A).((arity g c (THeads (Flat Appl) vs (lift (S i) O v)) a) \to (arity g c +(THeads (Flat Appl) vs (TLRef i)) a))))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: +nat).(\lambda (H: (getl i c (CHead d (Bind Abbr) v))).(\lambda (vs: +TList).(TList_ind (\lambda (t: TList).(\forall (a: A).((arity g c (THeads +(Flat Appl) t (lift (S i) O v)) a) \to (arity g c (THeads (Flat Appl) t +(TLRef i)) a)))) (\lambda (a: A).(\lambda (H0: (arity g c (lift (S i) O v) +a)).(arity_abbr g c d v i H a (arity_gen_lift g c v a (S i) O H0 d (getl_drop +Abbr c d v i H))))) (\lambda (t: T).(\lambda (t0: TList).(\lambda (H0: +((\forall (a: A).((arity g c (THeads (Flat Appl) t0 (lift (S i) O v)) a) \to +(arity g c (THeads (Flat Appl) t0 (TLRef i)) a))))).(\lambda (a: A).(\lambda +(H1: (arity g c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O +v))) a)).(let H2 \def (arity_gen_appl g c t (THeads (Flat Appl) t0 (lift (S +i) O v)) a H1) in (ex2_ind A (\lambda (a1: A).(arity g c t a1)) (\lambda (a1: +A).(arity g c (THeads (Flat Appl) t0 (lift (S i) O v)) (AHead a1 a))) (arity +g c (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) a) (\lambda (x: +A).(\lambda (H3: (arity g c t x)).(\lambda (H4: (arity g c (THeads (Flat +Appl) t0 (lift (S i) O v)) (AHead x a))).(arity_appl g c t x H3 (THeads (Flat +Appl) t0 (TLRef i)) a (H0 (AHead x a) H4))))) H2))))))) vs))))))). +(* COMMENTS +Initial nodes: 425 +END *) + +theorem arity_appls_bind: + \forall (g: G).(\forall (b: B).((not (eq B b Abst)) \to (\forall (c: +C).(\forall (v: T).(\forall (a1: A).((arity g c v a1) \to (\forall (t: +T).(\forall (vs: TList).(\forall (a2: A).((arity g (CHead c (Bind b) v) +(THeads (Flat Appl) (lifts (S O) O vs) t) a2) \to (arity g c (THeads (Flat +Appl) vs (THead (Bind b) v t)) a2))))))))))) +\def + \lambda (g: G).(\lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda +(c: C).(\lambda (v: T).(\lambda (a1: A).(\lambda (H0: (arity g c v +a1)).(\lambda (t: T).(\lambda (vs: TList).(TList_ind (\lambda (t0: +TList).(\forall (a2: A).((arity g (CHead c (Bind b) v) (THeads (Flat Appl) +(lifts (S O) O t0) t) a2) \to (arity g c (THeads (Flat Appl) t0 (THead (Bind +b) v t)) a2)))) (\lambda (a2: A).(\lambda (H1: (arity g (CHead c (Bind b) v) +t a2)).(arity_bind g b H c v a1 H0 t a2 H1))) (\lambda (t0: T).(\lambda (t1: +TList).(\lambda (H1: ((\forall (a2: A).((arity g (CHead c (Bind b) v) (THeads +(Flat Appl) (lifts (S O) O t1) t) a2) \to (arity g c (THeads (Flat Appl) t1 +(THead (Bind b) v t)) a2))))).(\lambda (a2: A).(\lambda (H2: (arity g (CHead +c (Bind b) v) (THead (Flat Appl) (lift (S O) O t0) (THeads (Flat Appl) (lifts +(S O) O t1) t)) a2)).(let H3 \def (arity_gen_appl g (CHead c (Bind b) v) +(lift (S O) O t0) (THeads (Flat Appl) (lifts (S O) O t1) t) a2 H2) in +(ex2_ind A (\lambda (a3: A).(arity g (CHead c (Bind b) v) (lift (S O) O t0) +a3)) (\lambda (a3: A).(arity g (CHead c (Bind b) v) (THeads (Flat Appl) +(lifts (S O) O t1) t) (AHead a3 a2))) (arity g c (THead (Flat Appl) t0 +(THeads (Flat Appl) t1 (THead (Bind b) v t))) a2) (\lambda (x: A).(\lambda +(H4: (arity g (CHead c (Bind b) v) (lift (S O) O t0) x)).(\lambda (H5: (arity +g (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O t1) t) (AHead x +a2))).(arity_appl g c t0 x (arity_gen_lift g (CHead c (Bind b) v) t0 x (S O) +O H4 c (drop_drop (Bind b) O c c (drop_refl c) v)) (THeads (Flat Appl) t1 +(THead (Bind b) v t)) a2 (H1 (AHead x a2) H5))))) H3))))))) vs))))))))). +(* COMMENTS +Initial nodes: 567 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/arity/subst0.ma b/matita/matita/contribs/lambdadelta/basic_1/arity/subst0.ma new file mode 100644 index 000000000..16046993b --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/arity/subst0.ma @@ -0,0 +1,1137 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/arity/props.ma". + +include "Basic-1/fsubst0/fwd.ma". + +include "Basic-1/csubst0/getl.ma". + +include "Basic-1/subst0/dec.ma". + +include "Basic-1/subst0/fwd.ma". + +include "Basic-1/getl/getl.ma". + +theorem arity_gen_cvoid_subst0: + \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t +a) \to (\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d +(Bind Void) u)) \to (\forall (w: T).(\forall (v: T).((subst0 i w t v) \to +(\forall (P: Prop).P)))))))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: +(arity g c t a)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (_: +A).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c0 (CHead d +(Bind Void) u)) \to (\forall (w: T).(\forall (v: T).((subst0 i w t0 v) \to +(\forall (P: Prop).P))))))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda +(d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d +(Bind Void) u))).(\lambda (w: T).(\lambda (v: T).(\lambda (H1: (subst0 i w +(TSort n) v)).(\lambda (P: Prop).(subst0_gen_sort w v i n H1 P))))))))))) +(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda +(H0: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (a0: A).(\lambda (_: +(arity g d u a0)).(\lambda (_: ((\forall (d0: C).(\forall (u0: T).(\forall +(i0: nat).((getl i0 d (CHead d0 (Bind Void) u0)) \to (\forall (w: T).(\forall +(v: T).((subst0 i0 w u v) \to (\forall (P: Prop).P)))))))))).(\lambda (d0: +C).(\lambda (u0: T).(\lambda (i0: nat).(\lambda (H3: (getl i0 c0 (CHead d0 +(Bind Void) u0))).(\lambda (w: T).(\lambda (v: T).(\lambda (H4: (subst0 i0 w +(TLRef i) v)).(\lambda (P: Prop).(land_ind (eq nat i i0) (eq T v (lift (S i) +O w)) P (\lambda (H5: (eq nat i i0)).(\lambda (_: (eq T v (lift (S i) O +w))).(let H7 \def (eq_ind_r nat i0 (\lambda (n: nat).(getl n c0 (CHead d0 +(Bind Void) u0))) H3 i H5) in (let H8 \def (eq_ind C (CHead d (Bind Abbr) u) +(\lambda (c1: C).(getl i c0 c1)) H0 (CHead d0 (Bind Void) u0) (getl_mono c0 +(CHead d (Bind Abbr) u) i H0 (CHead d0 (Bind Void) u0) H7)) in (let H9 \def +(eq_ind C (CHead d (Bind Abbr) u) (\lambda (ee: C).(match ee in C return +(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) +\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b) +\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow +True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _) +\Rightarrow False])])) I (CHead d0 (Bind Void) u0) (getl_mono c0 (CHead d +(Bind Abbr) u) i H0 (CHead d0 (Bind Void) u0) H7)) in (False_ind P H9)))))) +(subst0_gen_lref w v i0 i H4)))))))))))))))))) (\lambda (c0: C).(\lambda (d: +C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind +Abst) u))).(\lambda (a0: A).(\lambda (_: (arity g d u (asucc g a0))).(\lambda +(_: ((\forall (d0: C).(\forall (u0: T).(\forall (i0: nat).((getl i0 d (CHead +d0 (Bind Void) u0)) \to (\forall (w: T).(\forall (v: T).((subst0 i0 w u v) +\to (\forall (P: Prop).P)))))))))).(\lambda (d0: C).(\lambda (u0: T).(\lambda +(i0: nat).(\lambda (H3: (getl i0 c0 (CHead d0 (Bind Void) u0))).(\lambda (w: +T).(\lambda (v: T).(\lambda (H4: (subst0 i0 w (TLRef i) v)).(\lambda (P: +Prop).(land_ind (eq nat i i0) (eq T v (lift (S i) O w)) P (\lambda (H5: (eq +nat i i0)).(\lambda (_: (eq T v (lift (S i) O w))).(let H7 \def (eq_ind_r nat +i0 (\lambda (n: nat).(getl n c0 (CHead d0 (Bind Void) u0))) H3 i H5) in (let +H8 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (c1: C).(getl i c0 c1)) H0 +(CHead d0 (Bind Void) u0) (getl_mono c0 (CHead d (Bind Abst) u) i H0 (CHead +d0 (Bind Void) u0) H7)) in (let H9 \def (eq_ind C (CHead d (Bind Abst) u) +(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) +\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda +(_: K).Prop) with [(Bind b) \Rightarrow (match b in B return (\lambda (_: +B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow True | Void +\Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead d0 (Bind Void) +u0) (getl_mono c0 (CHead d (Bind Abst) u) i H0 (CHead d0 (Bind Void) u0) H7)) +in (False_ind P H9)))))) (subst0_gen_lref w v i0 i H4)))))))))))))))))) +(\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (c0: C).(\lambda +(u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u a1)).(\lambda (H2: +((\forall (d: C).(\forall (u0: T).(\forall (i: nat).((getl i c0 (CHead d +(Bind Void) u0)) \to (\forall (w: T).(\forall (v: T).((subst0 i w u v) \to +(\forall (P: Prop).P)))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: +(arity g (CHead c0 (Bind b) u) t0 a2)).(\lambda (H4: ((\forall (d: +C).(\forall (u0: T).(\forall (i: nat).((getl i (CHead c0 (Bind b) u) (CHead d +(Bind Void) u0)) \to (\forall (w: T).(\forall (v: T).((subst0 i w t0 v) \to +(\forall (P: Prop).P)))))))))).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: +nat).(\lambda (H5: (getl i c0 (CHead d (Bind Void) u0))).(\lambda (w: +T).(\lambda (v: T).(\lambda (H6: (subst0 i w (THead (Bind b) u t0) +v)).(\lambda (P: Prop).(or3_ind (ex2 T (\lambda (u2: T).(eq T v (THead (Bind +b) u2 t0))) (\lambda (u2: T).(subst0 i w u u2))) (ex2 T (\lambda (t2: T).(eq +T v (THead (Bind b) u t2))) (\lambda (t2: T).(subst0 (s (Bind b) i) w t0 +t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Bind b) u2 +t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: +T).(\lambda (t2: T).(subst0 (s (Bind b) i) w t0 t2)))) P (\lambda (H7: (ex2 T +(\lambda (u2: T).(eq T v (THead (Bind b) u2 t0))) (\lambda (u2: T).(subst0 i +w u u2)))).(ex2_ind T (\lambda (u2: T).(eq T v (THead (Bind b) u2 t0))) +(\lambda (u2: T).(subst0 i w u u2)) P (\lambda (x: T).(\lambda (_: (eq T v +(THead (Bind b) x t0))).(\lambda (H9: (subst0 i w u x)).(H2 d u0 i H5 w x H9 +P)))) H7)) (\lambda (H7: (ex2 T (\lambda (t2: T).(eq T v (THead (Bind b) u +t2))) (\lambda (t2: T).(subst0 (s (Bind b) i) w t0 t2)))).(ex2_ind T (\lambda +(t2: T).(eq T v (THead (Bind b) u t2))) (\lambda (t2: T).(subst0 (s (Bind b) +i) w t0 t2)) P (\lambda (x: T).(\lambda (_: (eq T v (THead (Bind b) u +x))).(\lambda (H9: (subst0 (s (Bind b) i) w t0 x)).(H4 d u0 (S i) +(getl_clear_bind b (CHead c0 (Bind b) u) c0 u (clear_bind b c0 u) (CHead d +(Bind Void) u0) i H5) w x H9 P)))) H7)) (\lambda (H7: (ex3_2 T T (\lambda +(u2: T).(\lambda (t2: T).(eq T v (THead (Bind b) u2 t2)))) (\lambda (u2: +T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda (t2: +T).(subst0 (s (Bind b) i) w t0 t2))))).(ex3_2_ind T T (\lambda (u2: +T).(\lambda (t2: T).(eq T v (THead (Bind b) u2 t2)))) (\lambda (u2: +T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda (t2: +T).(subst0 (s (Bind b) i) w t0 t2))) P (\lambda (x0: T).(\lambda (x1: +T).(\lambda (_: (eq T v (THead (Bind b) x0 x1))).(\lambda (H9: (subst0 i w u +x0)).(\lambda (_: (subst0 (s (Bind b) i) w t0 x1)).(H2 d u0 i H5 w x0 H9 +P)))))) H7)) (subst0_gen_head (Bind b) w u t0 v i H6))))))))))))))))))))) +(\lambda (c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u +(asucc g a1))).(\lambda (H1: ((\forall (d: C).(\forall (u0: T).(\forall (i: +nat).((getl i c0 (CHead d (Bind Void) u0)) \to (\forall (w: T).(\forall (v: +T).((subst0 i w u v) \to (\forall (P: Prop).P)))))))))).(\lambda (t0: +T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u) t0 +a2)).(\lambda (H3: ((\forall (d: C).(\forall (u0: T).(\forall (i: nat).((getl +i (CHead c0 (Bind Abst) u) (CHead d (Bind Void) u0)) \to (\forall (w: +T).(\forall (v: T).((subst0 i w t0 v) \to (\forall (P: +Prop).P)))))))))).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda +(H4: (getl i c0 (CHead d (Bind Void) u0))).(\lambda (w: T).(\lambda (v: +T).(\lambda (H5: (subst0 i w (THead (Bind Abst) u t0) v)).(\lambda (P: +Prop).(or3_ind (ex2 T (\lambda (u2: T).(eq T v (THead (Bind Abst) u2 t0))) +(\lambda (u2: T).(subst0 i w u u2))) (ex2 T (\lambda (t2: T).(eq T v (THead +(Bind Abst) u t2))) (\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0 t2))) +(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Bind Abst) u2 +t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: +T).(\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0 t2)))) P (\lambda (H6: +(ex2 T (\lambda (u2: T).(eq T v (THead (Bind Abst) u2 t0))) (\lambda (u2: +T).(subst0 i w u u2)))).(ex2_ind T (\lambda (u2: T).(eq T v (THead (Bind +Abst) u2 t0))) (\lambda (u2: T).(subst0 i w u u2)) P (\lambda (x: T).(\lambda +(_: (eq T v (THead (Bind Abst) x t0))).(\lambda (H8: (subst0 i w u x)).(H1 d +u0 i H4 w x H8 P)))) H6)) (\lambda (H6: (ex2 T (\lambda (t2: T).(eq T v +(THead (Bind Abst) u t2))) (\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0 +t2)))).(ex2_ind T (\lambda (t2: T).(eq T v (THead (Bind Abst) u t2))) +(\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0 t2)) P (\lambda (x: +T).(\lambda (_: (eq T v (THead (Bind Abst) u x))).(\lambda (H8: (subst0 (s +(Bind Abst) i) w t0 x)).(H3 d u0 (S i) (getl_clear_bind Abst (CHead c0 (Bind +Abst) u) c0 u (clear_bind Abst c0 u) (CHead d (Bind Void) u0) i H4) w x H8 +P)))) H6)) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v +(THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w u +u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0 +t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Bind +Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda +(_: T).(\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0 t2))) P (\lambda (x0: +T).(\lambda (x1: T).(\lambda (_: (eq T v (THead (Bind Abst) x0 x1))).(\lambda +(H8: (subst0 i w u x0)).(\lambda (_: (subst0 (s (Bind Abst) i) w t0 x1)).(H1 +d u0 i H4 w x0 H8 P)))))) H6)) (subst0_gen_head (Bind Abst) w u t0 v i +H5))))))))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a1: +A).(\lambda (_: (arity g c0 u a1)).(\lambda (H1: ((\forall (d: C).(\forall +(u0: T).(\forall (i: nat).((getl i c0 (CHead d (Bind Void) u0)) \to (\forall +(w: T).(\forall (v: T).((subst0 i w u v) \to (\forall (P: +Prop).P)))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g c0 +t0 (AHead a1 a2))).(\lambda (H3: ((\forall (d: C).(\forall (u0: T).(\forall +(i: nat).((getl i c0 (CHead d (Bind Void) u0)) \to (\forall (w: T).(\forall +(v: T).((subst0 i w t0 v) \to (\forall (P: Prop).P)))))))))).(\lambda (d: +C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H4: (getl i c0 (CHead d (Bind +Void) u0))).(\lambda (w: T).(\lambda (v: T).(\lambda (H5: (subst0 i w (THead +(Flat Appl) u t0) v)).(\lambda (P: Prop).(or3_ind (ex2 T (\lambda (u2: T).(eq +T v (THead (Flat Appl) u2 t0))) (\lambda (u2: T).(subst0 i w u u2))) (ex2 T +(\lambda (t2: T).(eq T v (THead (Flat Appl) u t2))) (\lambda (t2: T).(subst0 +(s (Flat Appl) i) w t0 t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq +T v (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w +u u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s (Flat Appl) i) w t0 +t2)))) P (\lambda (H6: (ex2 T (\lambda (u2: T).(eq T v (THead (Flat Appl) u2 +t0))) (\lambda (u2: T).(subst0 i w u u2)))).(ex2_ind T (\lambda (u2: T).(eq T +v (THead (Flat Appl) u2 t0))) (\lambda (u2: T).(subst0 i w u u2)) P (\lambda +(x: T).(\lambda (_: (eq T v (THead (Flat Appl) x t0))).(\lambda (H8: (subst0 +i w u x)).(H1 d u0 i H4 w x H8 P)))) H6)) (\lambda (H6: (ex2 T (\lambda (t2: +T).(eq T v (THead (Flat Appl) u t2))) (\lambda (t2: T).(subst0 (s (Flat Appl) +i) w t0 t2)))).(ex2_ind T (\lambda (t2: T).(eq T v (THead (Flat Appl) u t2))) +(\lambda (t2: T).(subst0 (s (Flat Appl) i) w t0 t2)) P (\lambda (x: +T).(\lambda (_: (eq T v (THead (Flat Appl) u x))).(\lambda (H8: (subst0 (s +(Flat Appl) i) w t0 x)).(H3 d u0 i H4 w x H8 P)))) H6)) (\lambda (H6: (ex3_2 +T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Flat Appl) u2 t2)))) +(\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda +(t2: T).(subst0 (s (Flat Appl) i) w t0 t2))))).(ex3_2_ind T T (\lambda (u2: +T).(\lambda (t2: T).(eq T v (THead (Flat Appl) u2 t2)))) (\lambda (u2: +T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda (t2: +T).(subst0 (s (Flat Appl) i) w t0 t2))) P (\lambda (x0: T).(\lambda (x1: +T).(\lambda (_: (eq T v (THead (Flat Appl) x0 x1))).(\lambda (H8: (subst0 i w +u x0)).(\lambda (_: (subst0 (s (Flat Appl) i) w t0 x1)).(H1 d u0 i H4 w x0 H8 +P)))))) H6)) (subst0_gen_head (Flat Appl) w u t0 v i H5))))))))))))))))))) +(\lambda (c0: C).(\lambda (u: T).(\lambda (a0: A).(\lambda (_: (arity g c0 u +(asucc g a0))).(\lambda (H1: ((\forall (d: C).(\forall (u0: T).(\forall (i: +nat).((getl i c0 (CHead d (Bind Void) u0)) \to (\forall (w: T).(\forall (v: +T).((subst0 i w u v) \to (\forall (P: Prop).P)))))))))).(\lambda (t0: +T).(\lambda (_: (arity g c0 t0 a0)).(\lambda (H3: ((\forall (d: C).(\forall +(u0: T).(\forall (i: nat).((getl i c0 (CHead d (Bind Void) u0)) \to (\forall +(w: T).(\forall (v: T).((subst0 i w t0 v) \to (\forall (P: +Prop).P)))))))))).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda +(H4: (getl i c0 (CHead d (Bind Void) u0))).(\lambda (w: T).(\lambda (v: +T).(\lambda (H5: (subst0 i w (THead (Flat Cast) u t0) v)).(\lambda (P: +Prop).(or3_ind (ex2 T (\lambda (u2: T).(eq T v (THead (Flat Cast) u2 t0))) +(\lambda (u2: T).(subst0 i w u u2))) (ex2 T (\lambda (t2: T).(eq T v (THead +(Flat Cast) u t2))) (\lambda (t2: T).(subst0 (s (Flat Cast) i) w t0 t2))) +(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Flat Cast) u2 +t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: +T).(\lambda (t2: T).(subst0 (s (Flat Cast) i) w t0 t2)))) P (\lambda (H6: +(ex2 T (\lambda (u2: T).(eq T v (THead (Flat Cast) u2 t0))) (\lambda (u2: +T).(subst0 i w u u2)))).(ex2_ind T (\lambda (u2: T).(eq T v (THead (Flat +Cast) u2 t0))) (\lambda (u2: T).(subst0 i w u u2)) P (\lambda (x: T).(\lambda +(_: (eq T v (THead (Flat Cast) x t0))).(\lambda (H8: (subst0 i w u x)).(H1 d +u0 i H4 w x H8 P)))) H6)) (\lambda (H6: (ex2 T (\lambda (t2: T).(eq T v +(THead (Flat Cast) u t2))) (\lambda (t2: T).(subst0 (s (Flat Cast) i) w t0 +t2)))).(ex2_ind T (\lambda (t2: T).(eq T v (THead (Flat Cast) u t2))) +(\lambda (t2: T).(subst0 (s (Flat Cast) i) w t0 t2)) P (\lambda (x: +T).(\lambda (_: (eq T v (THead (Flat Cast) u x))).(\lambda (H8: (subst0 (s +(Flat Cast) i) w t0 x)).(H3 d u0 i H4 w x H8 P)))) H6)) (\lambda (H6: (ex3_2 +T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Flat Cast) u2 t2)))) +(\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda +(t2: T).(subst0 (s (Flat Cast) i) w t0 t2))))).(ex3_2_ind T T (\lambda (u2: +T).(\lambda (t2: T).(eq T v (THead (Flat Cast) u2 t2)))) (\lambda (u2: +T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda (t2: +T).(subst0 (s (Flat Cast) i) w t0 t2))) P (\lambda (x0: T).(\lambda (x1: +T).(\lambda (_: (eq T v (THead (Flat Cast) x0 x1))).(\lambda (H8: (subst0 i w +u x0)).(\lambda (_: (subst0 (s (Flat Cast) i) w t0 x1)).(H1 d u0 i H4 w x0 H8 +P)))))) H6)) (subst0_gen_head (Flat Cast) w u t0 v i H5)))))))))))))))))) +(\lambda (c0: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda (_: (arity g c0 +t0 a1)).(\lambda (H1: ((\forall (d: C).(\forall (u: T).(\forall (i: +nat).((getl i c0 (CHead d (Bind Void) u)) \to (\forall (w: T).(\forall (v: +T).((subst0 i w t0 v) \to (\forall (P: Prop).P)))))))))).(\lambda (a2: +A).(\lambda (_: (leq g a1 a2)).(\lambda (d: C).(\lambda (u: T).(\lambda (i: +nat).(\lambda (H3: (getl i c0 (CHead d (Bind Void) u))).(\lambda (w: +T).(\lambda (v: T).(\lambda (H4: (subst0 i w t0 v)).(\lambda (P: Prop).(H1 d +u i H3 w v H4 P)))))))))))))))) c t a H))))). +(* COMMENTS +Initial nodes: 4131 +END *) + +theorem arity_gen_cvoid: + \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t +a) \to (\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d +(Bind Void) u)) \to (ex T (\lambda (v: T).(eq T t (lift (S O) i v)))))))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: +(arity g c t a)).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda +(H0: (getl i c (CHead d (Bind Void) u))).(let H_x \def (dnf_dec u t i) in +(let H1 \def H_x in (ex_ind T (\lambda (v: T).(or (subst0 i u t (lift (S O) i +v)) (eq T t (lift (S O) i v)))) (ex T (\lambda (v: T).(eq T t (lift (S O) i +v)))) (\lambda (x: T).(\lambda (H2: (or (subst0 i u t (lift (S O) i x)) (eq T +t (lift (S O) i x)))).(or_ind (subst0 i u t (lift (S O) i x)) (eq T t (lift +(S O) i x)) (ex T (\lambda (v: T).(eq T t (lift (S O) i v)))) (\lambda (H3: +(subst0 i u t (lift (S O) i x))).(arity_gen_cvoid_subst0 g c t a H d u i H0 u +(lift (S O) i x) H3 (ex T (\lambda (v: T).(eq T t (lift (S O) i v)))))) +(\lambda (H3: (eq T t (lift (S O) i x))).(let H4 \def (eq_ind T t (\lambda +(t0: T).(arity g c t0 a)) H (lift (S O) i x) H3) in (eq_ind_r T (lift (S O) i +x) (\lambda (t0: T).(ex T (\lambda (v: T).(eq T t0 (lift (S O) i v))))) +(ex_intro T (\lambda (v: T).(eq T (lift (S O) i x) (lift (S O) i v))) x +(refl_equal T (lift (S O) i x))) t H3))) H2))) H1))))))))))). +(* COMMENTS +Initial nodes: 423 +END *) + +theorem arity_fsubst0: + \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (a: A).((arity g +c1 t1 a) \to (\forall (d1: C).(\forall (u: T).(\forall (i: nat).((getl i c1 +(CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u +c1 t1 c2 t2) \to (arity g c2 t2 a)))))))))))) +\def + \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (a: A).(\lambda +(H: (arity g c1 t1 a)).(arity_ind g (\lambda (c: C).(\lambda (t: T).(\lambda +(a0: A).(\forall (d1: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead +d1 (Bind Abbr) u)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u c t c2 +t2) \to (arity g c2 t2 a0))))))))))) (\lambda (c: C).(\lambda (n: +nat).(\lambda (d1: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (_: (getl i +c (CHead d1 (Bind Abbr) u))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H1: +(fsubst0 i u c (TSort n) c2 t2)).(let H_x \def (fsubst0_gen_base c c2 (TSort +n) t2 u i H1) in (let H2 \def H_x in (or3_ind (land (eq C c c2) (subst0 i u +(TSort n) t2)) (land (eq T (TSort n) t2) (csubst0 i u c c2)) (land (subst0 i +u (TSort n) t2) (csubst0 i u c c2)) (arity g c2 t2 (ASort O n)) (\lambda (H3: +(land (eq C c c2) (subst0 i u (TSort n) t2))).(land_ind (eq C c c2) (subst0 i +u (TSort n) t2) (arity g c2 t2 (ASort O n)) (\lambda (H4: (eq C c +c2)).(\lambda (H5: (subst0 i u (TSort n) t2)).(eq_ind C c (\lambda (c0: +C).(arity g c0 t2 (ASort O n))) (subst0_gen_sort u t2 i n H5 (arity g c t2 +(ASort O n))) c2 H4))) H3)) (\lambda (H3: (land (eq T (TSort n) t2) (csubst0 +i u c c2))).(land_ind (eq T (TSort n) t2) (csubst0 i u c c2) (arity g c2 t2 +(ASort O n)) (\lambda (H4: (eq T (TSort n) t2)).(\lambda (_: (csubst0 i u c +c2)).(eq_ind T (TSort n) (\lambda (t: T).(arity g c2 t (ASort O n))) +(arity_sort g c2 n) t2 H4))) H3)) (\lambda (H3: (land (subst0 i u (TSort n) +t2) (csubst0 i u c c2))).(land_ind (subst0 i u (TSort n) t2) (csubst0 i u c +c2) (arity g c2 t2 (ASort O n)) (\lambda (H4: (subst0 i u (TSort n) +t2)).(\lambda (_: (csubst0 i u c c2)).(subst0_gen_sort u t2 i n H4 (arity g +c2 t2 (ASort O n))))) H3)) H2)))))))))))) (\lambda (c: C).(\lambda (d: +C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d (Bind +Abbr) u))).(\lambda (a0: A).(\lambda (H1: (arity g d u a0)).(\lambda (H2: +((\forall (d1: C).(\forall (u0: T).(\forall (i0: nat).((getl i0 d (CHead d1 +(Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i0 u0 d u c2 +t2) \to (arity g c2 t2 a0)))))))))).(\lambda (d1: C).(\lambda (u0: +T).(\lambda (i0: nat).(\lambda (H3: (getl i0 c (CHead d1 (Bind Abbr) +u0))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H4: (fsubst0 i0 u0 c (TLRef +i) c2 t2)).(let H_x \def (fsubst0_gen_base c c2 (TLRef i) t2 u0 i0 H4) in +(let H5 \def H_x in (or3_ind (land (eq C c c2) (subst0 i0 u0 (TLRef i) t2)) +(land (eq T (TLRef i) t2) (csubst0 i0 u0 c c2)) (land (subst0 i0 u0 (TLRef i) +t2) (csubst0 i0 u0 c c2)) (arity g c2 t2 a0) (\lambda (H6: (land (eq C c c2) +(subst0 i0 u0 (TLRef i) t2))).(land_ind (eq C c c2) (subst0 i0 u0 (TLRef i) +t2) (arity g c2 t2 a0) (\lambda (H7: (eq C c c2)).(\lambda (H8: (subst0 i0 u0 +(TLRef i) t2)).(eq_ind C c (\lambda (c0: C).(arity g c0 t2 a0)) (land_ind (eq +nat i i0) (eq T t2 (lift (S i) O u0)) (arity g c t2 a0) (\lambda (H9: (eq nat +i i0)).(\lambda (H10: (eq T t2 (lift (S i) O u0))).(eq_ind_r T (lift (S i) O +u0) (\lambda (t: T).(arity g c t a0)) (let H11 \def (eq_ind_r nat i0 (\lambda +(n: nat).(getl n c (CHead d1 (Bind Abbr) u0))) H3 i H9) in (let H12 \def +(eq_ind C (CHead d (Bind Abbr) u) (\lambda (c0: C).(getl i c c0)) H0 (CHead +d1 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) i H0 (CHead d1 (Bind +Abbr) u0) H11)) in (let H13 \def (f_equal C C (\lambda (e: C).(match e in C +return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) +\Rightarrow c0])) (CHead d (Bind Abbr) u) (CHead d1 (Bind Abbr) u0) +(getl_mono c (CHead d (Bind Abbr) u) i H0 (CHead d1 (Bind Abbr) u0) H11)) in +((let H14 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: +C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d +(Bind Abbr) u) (CHead d1 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) +i H0 (CHead d1 (Bind Abbr) u0) H11)) in (\lambda (H15: (eq C d d1)).(let H16 +\def (eq_ind_r T u0 (\lambda (t: T).(getl i c (CHead d1 (Bind Abbr) t))) H12 +u H14) in (eq_ind T u (\lambda (t: T).(arity g c (lift (S i) O t) a0)) (let +H17 \def (eq_ind_r C d1 (\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr) u))) +H16 d H15) in (arity_lift g d u a0 H1 c (S i) O (getl_drop Abbr c d u i +H17))) u0 H14)))) H13)))) t2 H10))) (subst0_gen_lref u0 t2 i0 i H8)) c2 H7))) +H6)) (\lambda (H6: (land (eq T (TLRef i) t2) (csubst0 i0 u0 c c2))).(land_ind +(eq T (TLRef i) t2) (csubst0 i0 u0 c c2) (arity g c2 t2 a0) (\lambda (H7: (eq +T (TLRef i) t2)).(\lambda (H8: (csubst0 i0 u0 c c2)).(eq_ind T (TLRef i) +(\lambda (t: T).(arity g c2 t a0)) (lt_le_e i i0 (arity g c2 (TLRef i) a0) +(\lambda (H9: (lt i i0)).(let H10 \def (csubst0_getl_lt i0 i H9 c c2 u0 H8 +(CHead d (Bind Abbr) u) H0) in (or4_ind (getl i c2 (CHead d (Bind Abbr) u)) +(ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: +T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: +B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl i c2 (CHead e0 +(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda +(w: T).(subst0 (minus i0 (S i)) u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: +B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind +Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda +(e2: C).(\lambda (u1: T).(getl i c2 (CHead e2 (Bind b) u1)))))) (\lambda (_: +B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S +i)) u0 e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda +(_: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead +e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (w: T).(getl i c2 (CHead e2 (Bind b) w))))))) +(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: +T).(subst0 (minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) +u0 e1 e2))))))) (arity g c2 (TLRef i) a0) (\lambda (H11: (getl i c2 (CHead d +(Bind Abbr) u))).(let H12 \def (eq_ind nat (minus i0 i) (\lambda (n: +nat).(getl n (CHead d (Bind Abbr) u) (CHead d1 (Bind Abbr) u0))) +(getl_conf_le i0 (CHead d1 (Bind Abbr) u0) c H3 (CHead d (Bind Abbr) u) i H0 +(le_S_n i i0 (le_S (S i) i0 H9))) (S (minus i0 (S i))) (minus_x_Sy i0 i H9)) +in (arity_abbr g c2 d u i H11 a0 H1))) (\lambda (H11: (ex3_4 B C T T (\lambda +(b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind +Abbr) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: +C).(\lambda (_: T).(\lambda (w: T).(getl i c2 (CHead e0 (Bind b) w)))))) +(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 +(minus i0 (S i)) u0 u1 w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda +(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead +e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: +T).(\lambda (w: T).(getl i c2 (CHead e0 (Bind b) w)))))) (\lambda (_: +B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) +u0 u1 w))))) (arity g c2 (TLRef i) a0) (\lambda (x0: B).(\lambda (x1: +C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (eq C (CHead d (Bind +Abbr) u) (CHead x1 (Bind x0) x2))).(\lambda (H13: (getl i c2 (CHead x1 (Bind +x0) x3))).(\lambda (H14: (subst0 (minus i0 (S i)) u0 x2 x3)).(let H15 \def +(eq_ind nat (minus i0 i) (\lambda (n: nat).(getl n (CHead d (Bind Abbr) u) +(CHead d1 (Bind Abbr) u0))) (getl_conf_le i0 (CHead d1 (Bind Abbr) u0) c H3 +(CHead d (Bind Abbr) u) i H0 (le_S_n i i0 (le_S (S i) i0 H9))) (S (minus i0 +(S i))) (minus_x_Sy i0 i H9)) in (let H16 \def (f_equal C C (\lambda (e: +C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | +(CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) +x2) H12) in ((let H17 \def (f_equal C B (\lambda (e: C).(match e in C return +(\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) +\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) +\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead +x1 (Bind x0) x2) H12) in ((let H18 \def (f_equal C T (\lambda (e: C).(match e +in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) +\Rightarrow t])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H12) in +(\lambda (H19: (eq B Abbr x0)).(\lambda (H20: (eq C d x1)).(let H21 \def +(eq_ind_r T x2 (\lambda (t: T).(subst0 (minus i0 (S i)) u0 t x3)) H14 u H18) +in (let H22 \def (eq_ind_r C x1 (\lambda (c0: C).(getl i c2 (CHead c0 (Bind +x0) x3))) H13 d H20) in (let H23 \def (eq_ind_r B x0 (\lambda (b: B).(getl i +c2 (CHead d (Bind b) x3))) H22 Abbr H19) in (arity_abbr g c2 d x3 i H23 a0 +(H2 d1 u0 (r (Bind Abbr) (minus i0 (S i))) (getl_gen_S (Bind Abbr) d (CHead +d1 (Bind Abbr) u0) u (minus i0 (S i)) H15) d x3 (fsubst0_snd (r (Bind Abbr) +(minus i0 (S i))) u0 d u x3 H21))))))))) H17)) H16)))))))))) H11)) (\lambda +(H11: (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) +(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c2 +(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))).(ex3_4_ind B C C +T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C +(CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda +(_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c2 (CHead e2 (Bind b) +u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(csubst0 (minus i0 (S i)) u0 e1 e2))))) (arity g c2 (TLRef i) a0) (\lambda +(x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H12: (eq +C (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3))).(\lambda (H13: (getl i c2 +(CHead x2 (Bind x0) x3))).(\lambda (H14: (csubst0 (minus i0 (S i)) u0 x1 +x2)).(let H15 \def (eq_ind nat (minus i0 i) (\lambda (n: nat).(getl n (CHead +d (Bind Abbr) u) (CHead d1 (Bind Abbr) u0))) (getl_conf_le i0 (CHead d1 (Bind +Abbr) u0) c H3 (CHead d (Bind Abbr) u) i H0 (le_S_n i i0 (le_S (S i) i0 H9))) +(S (minus i0 (S i))) (minus_x_Sy i0 i H9)) in (let H16 \def (f_equal C C +(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) +\Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abbr) u) +(CHead x1 (Bind x0) x3) H12) in ((let H17 \def (f_equal C B (\lambda (e: +C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | +(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind +b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) +(CHead x1 (Bind x0) x3) H12) in ((let H18 \def (f_equal C T (\lambda (e: +C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | +(CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) +x3) H12) in (\lambda (H19: (eq B Abbr x0)).(\lambda (H20: (eq C d x1)).(let +H21 \def (eq_ind_r T x3 (\lambda (t: T).(getl i c2 (CHead x2 (Bind x0) t))) +H13 u H18) in (let H22 \def (eq_ind_r C x1 (\lambda (c0: C).(csubst0 (minus +i0 (S i)) u0 c0 x2)) H14 d H20) in (let H23 \def (eq_ind_r B x0 (\lambda (b: +B).(getl i c2 (CHead x2 (Bind b) u))) H21 Abbr H19) in (arity_abbr g c2 x2 u +i H23 a0 (H2 d1 u0 (r (Bind Abbr) (minus i0 (S i))) (getl_gen_S (Bind Abbr) d +(CHead d1 (Bind Abbr) u0) u (minus i0 (S i)) H15) x2 u (fsubst0_fst (r (Bind +Abbr) (minus i0 (S i))) u0 d u x2 H22))))))))) H17)) H16)))))))))) H11)) +(\lambda (H11: (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 +(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (w: T).(getl i c2 (CHead e2 (Bind b) w))))))) (\lambda (_: +B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 +(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 +e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 +(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (w: T).(getl i c2 (CHead e2 (Bind b) w))))))) (\lambda (_: +B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 +(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) +(arity g c2 (TLRef i) a0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: +C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H12: (eq C (CHead d (Bind +Abbr) u) (CHead x1 (Bind x0) x3))).(\lambda (H13: (getl i c2 (CHead x2 (Bind +x0) x4))).(\lambda (H14: (subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H15: +(csubst0 (minus i0 (S i)) u0 x1 x2)).(let H16 \def (eq_ind nat (minus i0 i) +(\lambda (n: nat).(getl n (CHead d (Bind Abbr) u) (CHead d1 (Bind Abbr) u0))) +(getl_conf_le i0 (CHead d1 (Bind Abbr) u0) c H3 (CHead d (Bind Abbr) u) i H0 +(le_S_n i i0 (le_S (S i) i0 H9))) (S (minus i0 (S i))) (minus_x_Sy i0 i H9)) +in (let H17 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda +(_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) +(CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H12) in ((let H18 \def +(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with +[(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k in K return +(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow +Abbr])])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H12) in ((let H19 +\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) +with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind +Abbr) u) (CHead x1 (Bind x0) x3) H12) in (\lambda (H20: (eq B Abbr +x0)).(\lambda (H21: (eq C d x1)).(let H22 \def (eq_ind_r T x3 (\lambda (t: +T).(subst0 (minus i0 (S i)) u0 t x4)) H14 u H19) in (let H23 \def (eq_ind_r C +x1 (\lambda (c0: C).(csubst0 (minus i0 (S i)) u0 c0 x2)) H15 d H21) in (let +H24 \def (eq_ind_r B x0 (\lambda (b: B).(getl i c2 (CHead x2 (Bind b) x4))) +H13 Abbr H20) in (arity_abbr g c2 x2 x4 i H24 a0 (H2 d1 u0 (r (Bind Abbr) +(minus i0 (S i))) (getl_gen_S (Bind Abbr) d (CHead d1 (Bind Abbr) u0) u +(minus i0 (S i)) H16) x2 x4 (fsubst0_both (r (Bind Abbr) (minus i0 (S i))) u0 +d u x4 H22 x2 H23))))))))) H18)) H17)))))))))))) H11)) H10))) (\lambda (H9: +(le i0 i)).(arity_abbr g c2 d u i (csubst0_getl_ge i0 i H9 c c2 u0 H8 (CHead +d (Bind Abbr) u) H0) a0 H1))) t2 H7))) H6)) (\lambda (H6: (land (subst0 i0 u0 +(TLRef i) t2) (csubst0 i0 u0 c c2))).(land_ind (subst0 i0 u0 (TLRef i) t2) +(csubst0 i0 u0 c c2) (arity g c2 t2 a0) (\lambda (H7: (subst0 i0 u0 (TLRef i) +t2)).(\lambda (H8: (csubst0 i0 u0 c c2)).(land_ind (eq nat i i0) (eq T t2 +(lift (S i) O u0)) (arity g c2 t2 a0) (\lambda (H9: (eq nat i i0)).(\lambda +(H10: (eq T t2 (lift (S i) O u0))).(eq_ind_r T (lift (S i) O u0) (\lambda (t: +T).(arity g c2 t a0)) (let H11 \def (eq_ind_r nat i0 (\lambda (n: +nat).(csubst0 n u0 c c2)) H8 i H9) in (let H12 \def (eq_ind_r nat i0 (\lambda +(n: nat).(getl n c (CHead d1 (Bind Abbr) u0))) H3 i H9) in (let H13 \def +(eq_ind C (CHead d (Bind Abbr) u) (\lambda (c0: C).(getl i c c0)) H0 (CHead +d1 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) i H0 (CHead d1 (Bind +Abbr) u0) H12)) in (let H14 \def (f_equal C C (\lambda (e: C).(match e in C +return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) +\Rightarrow c0])) (CHead d (Bind Abbr) u) (CHead d1 (Bind Abbr) u0) +(getl_mono c (CHead d (Bind Abbr) u) i H0 (CHead d1 (Bind Abbr) u0) H12)) in +((let H15 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: +C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d +(Bind Abbr) u) (CHead d1 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) +i H0 (CHead d1 (Bind Abbr) u0) H12)) in (\lambda (H16: (eq C d d1)).(let H17 +\def (eq_ind_r T u0 (\lambda (t: T).(getl i c (CHead d1 (Bind Abbr) t))) H13 +u H15) in (let H18 \def (eq_ind_r T u0 (\lambda (t: T).(csubst0 i t c c2)) +H11 u H15) in (eq_ind T u (\lambda (t: T).(arity g c2 (lift (S i) O t) a0)) +(let H19 \def (eq_ind_r C d1 (\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr) +u))) H17 d H16) in (arity_lift g d u a0 H1 c2 (S i) O (getl_drop Abbr c2 d u +i (csubst0_getl_ge i i (le_n i) c c2 u H18 (CHead d (Bind Abbr) u) H19)))) u0 +H15))))) H14))))) t2 H10))) (subst0_gen_lref u0 t2 i0 i H7)))) H6)) +H5)))))))))))))))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda +(i: nat).(\lambda (H0: (getl i c (CHead d (Bind Abst) u))).(\lambda (a0: +A).(\lambda (H1: (arity g d u (asucc g a0))).(\lambda (H2: ((\forall (d1: +C).(\forall (u0: T).(\forall (i0: nat).((getl i0 d (CHead d1 (Bind Abbr) u0)) +\to (\forall (c2: C).(\forall (t2: T).((fsubst0 i0 u0 d u c2 t2) \to (arity g +c2 t2 (asucc g a0))))))))))).(\lambda (d1: C).(\lambda (u0: T).(\lambda (i0: +nat).(\lambda (H3: (getl i0 c (CHead d1 (Bind Abbr) u0))).(\lambda (c2: +C).(\lambda (t2: T).(\lambda (H4: (fsubst0 i0 u0 c (TLRef i) c2 t2)).(let H_x +\def (fsubst0_gen_base c c2 (TLRef i) t2 u0 i0 H4) in (let H5 \def H_x in +(or3_ind (land (eq C c c2) (subst0 i0 u0 (TLRef i) t2)) (land (eq T (TLRef i) +t2) (csubst0 i0 u0 c c2)) (land (subst0 i0 u0 (TLRef i) t2) (csubst0 i0 u0 c +c2)) (arity g c2 t2 a0) (\lambda (H6: (land (eq C c c2) (subst0 i0 u0 (TLRef +i) t2))).(land_ind (eq C c c2) (subst0 i0 u0 (TLRef i) t2) (arity g c2 t2 a0) +(\lambda (H7: (eq C c c2)).(\lambda (H8: (subst0 i0 u0 (TLRef i) t2)).(eq_ind +C c (\lambda (c0: C).(arity g c0 t2 a0)) (land_ind (eq nat i i0) (eq T t2 +(lift (S i) O u0)) (arity g c t2 a0) (\lambda (H9: (eq nat i i0)).(\lambda +(H10: (eq T t2 (lift (S i) O u0))).(eq_ind_r T (lift (S i) O u0) (\lambda (t: +T).(arity g c t a0)) (let H11 \def (eq_ind_r nat i0 (\lambda (n: nat).(getl n +c (CHead d1 (Bind Abbr) u0))) H3 i H9) in (let H12 \def (eq_ind C (CHead d +(Bind Abst) u) (\lambda (c0: C).(getl i c c0)) H0 (CHead d1 (Bind Abbr) u0) +(getl_mono c (CHead d (Bind Abst) u) i H0 (CHead d1 (Bind Abbr) u0) H11)) in +(let H13 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (ee: C).(match ee in +C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k +_) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b) +\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow +False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) +\Rightarrow False])])) I (CHead d1 (Bind Abbr) u0) (getl_mono c (CHead d +(Bind Abst) u) i H0 (CHead d1 (Bind Abbr) u0) H11)) in (False_ind (arity g c +(lift (S i) O u0) a0) H13)))) t2 H10))) (subst0_gen_lref u0 t2 i0 i H8)) c2 +H7))) H6)) (\lambda (H6: (land (eq T (TLRef i) t2) (csubst0 i0 u0 c +c2))).(land_ind (eq T (TLRef i) t2) (csubst0 i0 u0 c c2) (arity g c2 t2 a0) +(\lambda (H7: (eq T (TLRef i) t2)).(\lambda (H8: (csubst0 i0 u0 c +c2)).(eq_ind T (TLRef i) (\lambda (t: T).(arity g c2 t a0)) (lt_le_e i i0 +(arity g c2 (TLRef i) a0) (\lambda (H9: (lt i i0)).(let H10 \def +(csubst0_getl_lt i0 i H9 c c2 u0 H8 (CHead d (Bind Abst) u) H0) in (or4_ind +(getl i c2 (CHead d (Bind Abst) u)) (ex3_4 B C T T (\lambda (b: B).(\lambda +(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead +e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: +T).(\lambda (w: T).(getl i c2 (CHead e0 (Bind b) w)))))) (\lambda (_: +B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) +u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u1: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1)))))) +(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c2 +(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) (ex4_5 B C C T T +(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda +(_: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1))))))) (\lambda +(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl +i c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: +C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w)))))) +(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))) (arity g c2 (TLRef i) a0) +(\lambda (H11: (getl i c2 (CHead d (Bind Abst) u))).(let H12 \def (eq_ind nat +(minus i0 i) (\lambda (n: nat).(getl n (CHead d (Bind Abst) u) (CHead d1 +(Bind Abbr) u0))) (getl_conf_le i0 (CHead d1 (Bind Abbr) u0) c H3 (CHead d +(Bind Abst) u) i H0 (le_S_n i i0 (le_S (S i) i0 H9))) (S (minus i0 (S i))) +(minus_x_Sy i0 i H9)) in (arity_abst g c2 d u i H11 a0 H1))) (\lambda (H11: +(ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: +T).(eq C (CHead d (Bind Abst) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: +B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl i c2 (CHead e0 +(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda +(w: T).(subst0 (minus i0 (S i)) u0 u1 w))))))).(ex3_4_ind B C T T (\lambda +(b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind +Abst) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: +C).(\lambda (_: T).(\lambda (w: T).(getl i c2 (CHead e0 (Bind b) w)))))) +(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 +(minus i0 (S i)) u0 u1 w))))) (arity g c2 (TLRef i) a0) (\lambda (x0: +B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (eq C +(CHead d (Bind Abst) u) (CHead x1 (Bind x0) x2))).(\lambda (H13: (getl i c2 +(CHead x1 (Bind x0) x3))).(\lambda (H14: (subst0 (minus i0 (S i)) u0 x2 +x3)).(let H15 \def (eq_ind nat (minus i0 i) (\lambda (n: nat).(getl n (CHead +d (Bind Abst) u) (CHead d1 (Bind Abbr) u0))) (getl_conf_le i0 (CHead d1 (Bind +Abbr) u0) c H3 (CHead d (Bind Abst) u) i H0 (le_S_n i i0 (le_S (S i) i0 H9))) +(S (minus i0 (S i))) (minus_x_Sy i0 i H9)) in (let H16 \def (f_equal C C +(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) +\Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abst) u) +(CHead x1 (Bind x0) x2) H12) in ((let H17 \def (f_equal C B (\lambda (e: +C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abst | +(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind +b) \Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead d (Bind Abst) u) +(CHead x1 (Bind x0) x2) H12) in ((let H18 \def (f_equal C T (\lambda (e: +C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | +(CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) +x2) H12) in (\lambda (H19: (eq B Abst x0)).(\lambda (H20: (eq C d x1)).(let +H21 \def (eq_ind_r T x2 (\lambda (t: T).(subst0 (minus i0 (S i)) u0 t x3)) +H14 u H18) in (let H22 \def (eq_ind_r C x1 (\lambda (c0: C).(getl i c2 (CHead +c0 (Bind x0) x3))) H13 d H20) in (let H23 \def (eq_ind_r B x0 (\lambda (b: +B).(getl i c2 (CHead d (Bind b) x3))) H22 Abst H19) in (arity_abst g c2 d x3 +i H23 a0 (H2 d1 u0 (r (Bind Abst) (minus i0 (S i))) (getl_gen_S (Bind Abst) d +(CHead d1 (Bind Abbr) u0) u (minus i0 (S i)) H15) d x3 (fsubst0_snd (r (Bind +Abst) (minus i0 (S i))) u0 d u x3 H21))))))))) H17)) H16)))))))))) H11)) +(\lambda (H11: (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u1: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1)))))) +(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c2 +(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))).(ex3_4_ind B C C +T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C +(CHead d (Bind Abst) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda +(_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c2 (CHead e2 (Bind b) +u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(csubst0 (minus i0 (S i)) u0 e1 e2))))) (arity g c2 (TLRef i) a0) (\lambda +(x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H12: (eq +C (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3))).(\lambda (H13: (getl i c2 +(CHead x2 (Bind x0) x3))).(\lambda (H14: (csubst0 (minus i0 (S i)) u0 x1 +x2)).(let H15 \def (eq_ind nat (minus i0 i) (\lambda (n: nat).(getl n (CHead +d (Bind Abst) u) (CHead d1 (Bind Abbr) u0))) (getl_conf_le i0 (CHead d1 (Bind +Abbr) u0) c H3 (CHead d (Bind Abst) u) i H0 (le_S_n i i0 (le_S (S i) i0 H9))) +(S (minus i0 (S i))) (minus_x_Sy i0 i H9)) in (let H16 \def (f_equal C C +(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) +\Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abst) u) +(CHead x1 (Bind x0) x3) H12) in ((let H17 \def (f_equal C B (\lambda (e: +C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abst | +(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind +b) \Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead d (Bind Abst) u) +(CHead x1 (Bind x0) x3) H12) in ((let H18 \def (f_equal C T (\lambda (e: +C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | +(CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) +x3) H12) in (\lambda (H19: (eq B Abst x0)).(\lambda (H20: (eq C d x1)).(let +H21 \def (eq_ind_r T x3 (\lambda (t: T).(getl i c2 (CHead x2 (Bind x0) t))) +H13 u H18) in (let H22 \def (eq_ind_r C x1 (\lambda (c0: C).(csubst0 (minus +i0 (S i)) u0 c0 x2)) H14 d H20) in (let H23 \def (eq_ind_r B x0 (\lambda (b: +B).(getl i c2 (CHead x2 (Bind b) u))) H21 Abst H19) in (arity_abst g c2 x2 u +i H23 a0 (H2 d1 u0 (r (Bind Abst) (minus i0 (S i))) (getl_gen_S (Bind Abst) d +(CHead d1 (Bind Abbr) u0) u (minus i0 (S i)) H15) x2 u (fsubst0_fst (r (Bind +Abst) (minus i0 (S i))) u0 d u x2 H22))))))))) H17)) H16)))))))))) H11)) +(\lambda (H11: (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead e1 +(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (w: T).(getl i c2 (CHead e2 (Bind b) w))))))) (\lambda (_: +B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 +(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 +e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead e1 +(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (w: T).(getl i c2 (CHead e2 (Bind b) w))))))) (\lambda (_: +B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 +(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) +(arity g c2 (TLRef i) a0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: +C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H12: (eq C (CHead d (Bind +Abst) u) (CHead x1 (Bind x0) x3))).(\lambda (H13: (getl i c2 (CHead x2 (Bind +x0) x4))).(\lambda (H14: (subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H15: +(csubst0 (minus i0 (S i)) u0 x1 x2)).(let H16 \def (eq_ind nat (minus i0 i) +(\lambda (n: nat).(getl n (CHead d (Bind Abst) u) (CHead d1 (Bind Abbr) u0))) +(getl_conf_le i0 (CHead d1 (Bind Abbr) u0) c H3 (CHead d (Bind Abst) u) i H0 +(le_S_n i i0 (le_S (S i) i0 H9))) (S (minus i0 (S i))) (minus_x_Sy i0 i H9)) +in (let H17 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda +(_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) +(CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3) H12) in ((let H18 \def +(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with +[(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match k in K return +(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow +Abst])])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3) H12) in ((let H19 +\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) +with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind +Abst) u) (CHead x1 (Bind x0) x3) H12) in (\lambda (H20: (eq B Abst +x0)).(\lambda (H21: (eq C d x1)).(let H22 \def (eq_ind_r T x3 (\lambda (t: +T).(subst0 (minus i0 (S i)) u0 t x4)) H14 u H19) in (let H23 \def (eq_ind_r C +x1 (\lambda (c0: C).(csubst0 (minus i0 (S i)) u0 c0 x2)) H15 d H21) in (let +H24 \def (eq_ind_r B x0 (\lambda (b: B).(getl i c2 (CHead x2 (Bind b) x4))) +H13 Abst H20) in (arity_abst g c2 x2 x4 i H24 a0 (H2 d1 u0 (r (Bind Abst) +(minus i0 (S i))) (getl_gen_S (Bind Abst) d (CHead d1 (Bind Abbr) u0) u +(minus i0 (S i)) H16) x2 x4 (fsubst0_both (r (Bind Abst) (minus i0 (S i))) u0 +d u x4 H22 x2 H23))))))))) H18)) H17)))))))))))) H11)) H10))) (\lambda (H9: +(le i0 i)).(arity_abst g c2 d u i (csubst0_getl_ge i0 i H9 c c2 u0 H8 (CHead +d (Bind Abst) u) H0) a0 H1))) t2 H7))) H6)) (\lambda (H6: (land (subst0 i0 u0 +(TLRef i) t2) (csubst0 i0 u0 c c2))).(land_ind (subst0 i0 u0 (TLRef i) t2) +(csubst0 i0 u0 c c2) (arity g c2 t2 a0) (\lambda (H7: (subst0 i0 u0 (TLRef i) +t2)).(\lambda (H8: (csubst0 i0 u0 c c2)).(land_ind (eq nat i i0) (eq T t2 +(lift (S i) O u0)) (arity g c2 t2 a0) (\lambda (H9: (eq nat i i0)).(\lambda +(H10: (eq T t2 (lift (S i) O u0))).(eq_ind_r T (lift (S i) O u0) (\lambda (t: +T).(arity g c2 t a0)) (let H11 \def (eq_ind_r nat i0 (\lambda (n: +nat).(csubst0 n u0 c c2)) H8 i H9) in (let H12 \def (eq_ind_r nat i0 (\lambda +(n: nat).(getl n c (CHead d1 (Bind Abbr) u0))) H3 i H9) in (let H13 \def +(eq_ind C (CHead d (Bind Abst) u) (\lambda (c0: C).(getl i c c0)) H0 (CHead +d1 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abst) u) i H0 (CHead d1 (Bind +Abbr) u0) H12)) in (let H14 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda +(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) +\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda +(_: K).Prop) with [(Bind b) \Rightarrow (match b in B return (\lambda (_: +B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow True | Void +\Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead d1 (Bind Abbr) +u0) (getl_mono c (CHead d (Bind Abst) u) i H0 (CHead d1 (Bind Abbr) u0) H12)) +in (False_ind (arity g c2 (lift (S i) O u0) a0) H14))))) t2 H10))) +(subst0_gen_lref u0 t2 i0 i H7)))) H6)) H5)))))))))))))))))) (\lambda (b: +B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c: C).(\lambda (u: +T).(\lambda (a1: A).(\lambda (H1: (arity g c u a1)).(\lambda (H2: ((\forall +(d1: C).(\forall (u0: T).(\forall (i: nat).((getl i c (CHead d1 (Bind Abbr) +u0)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 c u c2 t2) \to +(arity g c2 t2 a1)))))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (_: +(arity g (CHead c (Bind b) u) t a2)).(\lambda (H4: ((\forall (d1: C).(\forall +(u0: T).(\forall (i: nat).((getl i (CHead c (Bind b) u) (CHead d1 (Bind Abbr) +u0)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 (CHead c (Bind b) +u) t c2 t2) \to (arity g c2 t2 a2)))))))))).(\lambda (d1: C).(\lambda (u0: +T).(\lambda (i: nat).(\lambda (H5: (getl i c (CHead d1 (Bind Abbr) +u0))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H6: (fsubst0 i u0 c (THead +(Bind b) u t) c2 t2)).(let H_x \def (fsubst0_gen_base c c2 (THead (Bind b) u +t) t2 u0 i H6) in (let H7 \def H_x in (or3_ind (land (eq C c c2) (subst0 i u0 +(THead (Bind b) u t) t2)) (land (eq T (THead (Bind b) u t) t2) (csubst0 i u0 +c c2)) (land (subst0 i u0 (THead (Bind b) u t) t2) (csubst0 i u0 c c2)) +(arity g c2 t2 a2) (\lambda (H8: (land (eq C c c2) (subst0 i u0 (THead (Bind +b) u t) t2))).(land_ind (eq C c c2) (subst0 i u0 (THead (Bind b) u t) t2) +(arity g c2 t2 a2) (\lambda (H9: (eq C c c2)).(\lambda (H10: (subst0 i u0 +(THead (Bind b) u t) t2)).(eq_ind C c (\lambda (c0: C).(arity g c0 t2 a2)) +(or3_ind (ex2 T (\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t))) (\lambda +(u2: T).(subst0 i u0 u u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead (Bind b) +u t3))) (\lambda (t3: T).(subst0 (s (Bind b) i) u0 t t3))) (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) u2 t3)))) (\lambda +(u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: +T).(subst0 (s (Bind b) i) u0 t t3)))) (arity g c t2 a2) (\lambda (H11: (ex2 T +(\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t))) (\lambda (u2: T).(subst0 i +u0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t))) +(\lambda (u2: T).(subst0 i u0 u u2)) (arity g c t2 a2) (\lambda (x: +T).(\lambda (H12: (eq T t2 (THead (Bind b) x t))).(\lambda (H13: (subst0 i u0 +u x)).(eq_ind_r T (THead (Bind b) x t) (\lambda (t0: T).(arity g c t0 a2)) +(arity_bind g b H0 c x a1 (H2 d1 u0 i H5 c x (fsubst0_snd i u0 c u x H13)) t +a2 (H4 d1 u0 (S i) (getl_clear_bind b (CHead c (Bind b) u) c u (clear_bind b +c u) (CHead d1 (Bind Abbr) u0) i H5) (CHead c (Bind b) x) t (fsubst0_fst (S +i) u0 (CHead c (Bind b) u) t (CHead c (Bind b) x) (csubst0_snd_bind b i u0 u +x H13 c)))) t2 H12)))) H11)) (\lambda (H11: (ex2 T (\lambda (t3: T).(eq T t2 +(THead (Bind b) u t3))) (\lambda (t3: T).(subst0 (s (Bind b) i) u0 t +t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead (Bind b) u t3))) (\lambda +(t3: T).(subst0 (s (Bind b) i) u0 t t3)) (arity g c t2 a2) (\lambda (x: +T).(\lambda (H12: (eq T t2 (THead (Bind b) u x))).(\lambda (H13: (subst0 (s +(Bind b) i) u0 t x)).(eq_ind_r T (THead (Bind b) u x) (\lambda (t0: T).(arity +g c t0 a2)) (arity_bind g b H0 c u a1 H1 x a2 (H4 d1 u0 (S i) +(getl_clear_bind b (CHead c (Bind b) u) c u (clear_bind b c u) (CHead d1 +(Bind Abbr) u0) i H5) (CHead c (Bind b) u) x (fsubst0_snd (S i) u0 (CHead c +(Bind b) u) t x H13))) t2 H12)))) H11)) (\lambda (H11: (ex3_2 T T (\lambda +(u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: +T).(subst0 (s (Bind b) i) u0 t t3))))).(ex3_2_ind T T (\lambda (u2: +T).(\lambda (t3: T).(eq T t2 (THead (Bind b) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: +T).(subst0 (s (Bind b) i) u0 t t3))) (arity g c t2 a2) (\lambda (x0: +T).(\lambda (x1: T).(\lambda (H12: (eq T t2 (THead (Bind b) x0 x1))).(\lambda +(H13: (subst0 i u0 u x0)).(\lambda (H14: (subst0 (s (Bind b) i) u0 t +x1)).(eq_ind_r T (THead (Bind b) x0 x1) (\lambda (t0: T).(arity g c t0 a2)) +(arity_bind g b H0 c x0 a1 (H2 d1 u0 i H5 c x0 (fsubst0_snd i u0 c u x0 H13)) +x1 a2 (H4 d1 u0 (S i) (getl_clear_bind b (CHead c (Bind b) u) c u (clear_bind +b c u) (CHead d1 (Bind Abbr) u0) i H5) (CHead c (Bind b) x0) x1 (fsubst0_both +(S i) u0 (CHead c (Bind b) u) t x1 H14 (CHead c (Bind b) x0) +(csubst0_snd_bind b i u0 u x0 H13 c)))) t2 H12)))))) H11)) (subst0_gen_head +(Bind b) u0 u t t2 i H10)) c2 H9))) H8)) (\lambda (H8: (land (eq T (THead +(Bind b) u t) t2) (csubst0 i u0 c c2))).(land_ind (eq T (THead (Bind b) u t) +t2) (csubst0 i u0 c c2) (arity g c2 t2 a2) (\lambda (H9: (eq T (THead (Bind +b) u t) t2)).(\lambda (H10: (csubst0 i u0 c c2)).(eq_ind T (THead (Bind b) u +t) (\lambda (t0: T).(arity g c2 t0 a2)) (arity_bind g b H0 c2 u a1 (H2 d1 u0 +i H5 c2 u (fsubst0_fst i u0 c u c2 H10)) t a2 (H4 d1 u0 (S i) +(getl_clear_bind b (CHead c (Bind b) u) c u (clear_bind b c u) (CHead d1 +(Bind Abbr) u0) i H5) (CHead c2 (Bind b) u) t (fsubst0_fst (S i) u0 (CHead c +(Bind b) u) t (CHead c2 (Bind b) u) (csubst0_fst_bind b i c c2 u0 H10 u)))) +t2 H9))) H8)) (\lambda (H8: (land (subst0 i u0 (THead (Bind b) u t) t2) +(csubst0 i u0 c c2))).(land_ind (subst0 i u0 (THead (Bind b) u t) t2) +(csubst0 i u0 c c2) (arity g c2 t2 a2) (\lambda (H9: (subst0 i u0 (THead +(Bind b) u t) t2)).(\lambda (H10: (csubst0 i u0 c c2)).(or3_ind (ex2 T +(\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t))) (\lambda (u2: T).(subst0 i +u0 u u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead (Bind b) u t3))) (\lambda +(t3: T).(subst0 (s (Bind b) i) u0 t t3))) (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T t2 (THead (Bind b) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: +T).(subst0 (s (Bind b) i) u0 t t3)))) (arity g c2 t2 a2) (\lambda (H11: (ex2 +T (\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t))) (\lambda (u2: T).(subst0 +i u0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t))) +(\lambda (u2: T).(subst0 i u0 u u2)) (arity g c2 t2 a2) (\lambda (x: +T).(\lambda (H12: (eq T t2 (THead (Bind b) x t))).(\lambda (H13: (subst0 i u0 +u x)).(eq_ind_r T (THead (Bind b) x t) (\lambda (t0: T).(arity g c2 t0 a2)) +(arity_bind g b H0 c2 x a1 (H2 d1 u0 i H5 c2 x (fsubst0_both i u0 c u x H13 +c2 H10)) t a2 (H4 d1 u0 (S i) (getl_clear_bind b (CHead c (Bind b) u) c u +(clear_bind b c u) (CHead d1 (Bind Abbr) u0) i H5) (CHead c2 (Bind b) x) t +(fsubst0_fst (S i) u0 (CHead c (Bind b) u) t (CHead c2 (Bind b) x) +(csubst0_both_bind b i u0 u x H13 c c2 H10)))) t2 H12)))) H11)) (\lambda +(H11: (ex2 T (\lambda (t3: T).(eq T t2 (THead (Bind b) u t3))) (\lambda (t3: +T).(subst0 (s (Bind b) i) u0 t t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 +(THead (Bind b) u t3))) (\lambda (t3: T).(subst0 (s (Bind b) i) u0 t t3)) +(arity g c2 t2 a2) (\lambda (x: T).(\lambda (H12: (eq T t2 (THead (Bind b) u +x))).(\lambda (H13: (subst0 (s (Bind b) i) u0 t x)).(eq_ind_r T (THead (Bind +b) u x) (\lambda (t0: T).(arity g c2 t0 a2)) (arity_bind g b H0 c2 u a1 (H2 +d1 u0 i H5 c2 u (fsubst0_fst i u0 c u c2 H10)) x a2 (H4 d1 u0 (S i) +(getl_clear_bind b (CHead c (Bind b) u) c u (clear_bind b c u) (CHead d1 +(Bind Abbr) u0) i H5) (CHead c2 (Bind b) u) x (fsubst0_both (S i) u0 (CHead c +(Bind b) u) t x H13 (CHead c2 (Bind b) u) (csubst0_fst_bind b i c c2 u0 H10 +u)))) t2 H12)))) H11)) (\lambda (H11: (ex3_2 T T (\lambda (u2: T).(\lambda +(t3: T).(eq T t2 (THead (Bind b) u2 t3)))) (\lambda (u2: T).(\lambda (_: +T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind b) +i) u0 t t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 +(THead (Bind b) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u +u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind b) i) u0 t t3))) +(arity g c2 t2 a2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H12: (eq T t2 +(THead (Bind b) x0 x1))).(\lambda (H13: (subst0 i u0 u x0)).(\lambda (H14: +(subst0 (s (Bind b) i) u0 t x1)).(eq_ind_r T (THead (Bind b) x0 x1) (\lambda +(t0: T).(arity g c2 t0 a2)) (arity_bind g b H0 c2 x0 a1 (H2 d1 u0 i H5 c2 x0 +(fsubst0_both i u0 c u x0 H13 c2 H10)) x1 a2 (H4 d1 u0 (S i) (getl_clear_bind +b (CHead c (Bind b) u) c u (clear_bind b c u) (CHead d1 (Bind Abbr) u0) i H5) +(CHead c2 (Bind b) x0) x1 (fsubst0_both (S i) u0 (CHead c (Bind b) u) t x1 +H14 (CHead c2 (Bind b) x0) (csubst0_both_bind b i u0 u x0 H13 c c2 H10)))) t2 +H12)))))) H11)) (subst0_gen_head (Bind b) u0 u t t2 i H9)))) H8)) +H7))))))))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a1: +A).(\lambda (H0: (arity g c u (asucc g a1))).(\lambda (H1: ((\forall (d1: +C).(\forall (u0: T).(\forall (i: nat).((getl i c (CHead d1 (Bind Abbr) u0)) +\to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 c u c2 t2) \to (arity g +c2 t2 (asucc g a1))))))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (_: +(arity g (CHead c (Bind Abst) u) t a2)).(\lambda (H3: ((\forall (d1: +C).(\forall (u0: T).(\forall (i: nat).((getl i (CHead c (Bind Abst) u) (CHead +d1 (Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 +(CHead c (Bind Abst) u) t c2 t2) \to (arity g c2 t2 a2)))))))))).(\lambda +(d1: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H4: (getl i c (CHead d1 +(Bind Abbr) u0))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H5: (fsubst0 i +u0 c (THead (Bind Abst) u t) c2 t2)).(let H_x \def (fsubst0_gen_base c c2 +(THead (Bind Abst) u t) t2 u0 i H5) in (let H6 \def H_x in (or3_ind (land (eq +C c c2) (subst0 i u0 (THead (Bind Abst) u t) t2)) (land (eq T (THead (Bind +Abst) u t) t2) (csubst0 i u0 c c2)) (land (subst0 i u0 (THead (Bind Abst) u +t) t2) (csubst0 i u0 c c2)) (arity g c2 t2 (AHead a1 a2)) (\lambda (H7: (land +(eq C c c2) (subst0 i u0 (THead (Bind Abst) u t) t2))).(land_ind (eq C c c2) +(subst0 i u0 (THead (Bind Abst) u t) t2) (arity g c2 t2 (AHead a1 a2)) +(\lambda (H8: (eq C c c2)).(\lambda (H9: (subst0 i u0 (THead (Bind Abst) u t) +t2)).(eq_ind C c (\lambda (c0: C).(arity g c0 t2 (AHead a1 a2))) (or3_ind +(ex2 T (\lambda (u2: T).(eq T t2 (THead (Bind Abst) u2 t))) (\lambda (u2: +T).(subst0 i u0 u u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead (Bind Abst) u +t3))) (\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3))) (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: +T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3)))) (arity g c t2 +(AHead a1 a2)) (\lambda (H10: (ex2 T (\lambda (u2: T).(eq T t2 (THead (Bind +Abst) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T (\lambda (u2: +T).(eq T t2 (THead (Bind Abst) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)) +(arity g c t2 (AHead a1 a2)) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead +(Bind Abst) x t))).(\lambda (H12: (subst0 i u0 u x)).(eq_ind_r T (THead (Bind +Abst) x t) (\lambda (t0: T).(arity g c t0 (AHead a1 a2))) (arity_head g c x +a1 (H1 d1 u0 i H4 c x (fsubst0_snd i u0 c u x H12)) t a2 (H3 d1 u0 (S i) +(getl_clear_bind Abst (CHead c (Bind Abst) u) c u (clear_bind Abst c u) +(CHead d1 (Bind Abbr) u0) i H4) (CHead c (Bind Abst) x) t (fsubst0_fst (S i) +u0 (CHead c (Bind Abst) u) t (CHead c (Bind Abst) x) (csubst0_snd_bind Abst i +u0 u x H12 c)))) t2 H11)))) H10)) (\lambda (H10: (ex2 T (\lambda (t3: T).(eq +T t2 (THead (Bind Abst) u t3))) (\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 +t t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead (Bind Abst) u t3))) +(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3)) (arity g c t2 (AHead a1 +a2)) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Bind Abst) u +x))).(\lambda (H12: (subst0 (s (Bind Abst) i) u0 t x)).(eq_ind_r T (THead +(Bind Abst) u x) (\lambda (t0: T).(arity g c t0 (AHead a1 a2))) (arity_head g +c u a1 H0 x a2 (H3 d1 u0 (S i) (getl_clear_bind Abst (CHead c (Bind Abst) u) +c u (clear_bind Abst c u) (CHead d1 (Bind Abbr) u0) i H4) (CHead c (Bind +Abst) u) x (fsubst0_snd (S i) u0 (CHead c (Bind Abst) u) t x H12))) t2 +H11)))) H10)) (\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq +T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i +u0 u u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t +t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead +(Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) +(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3))) (arity +g c t2 (AHead a1 a2)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq T +t2 (THead (Bind Abst) x0 x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda +(H13: (subst0 (s (Bind Abst) i) u0 t x1)).(eq_ind_r T (THead (Bind Abst) x0 +x1) (\lambda (t0: T).(arity g c t0 (AHead a1 a2))) (arity_head g c x0 a1 (H1 +d1 u0 i H4 c x0 (fsubst0_snd i u0 c u x0 H12)) x1 a2 (H3 d1 u0 (S i) +(getl_clear_bind Abst (CHead c (Bind Abst) u) c u (clear_bind Abst c u) +(CHead d1 (Bind Abbr) u0) i H4) (CHead c (Bind Abst) x0) x1 (fsubst0_both (S +i) u0 (CHead c (Bind Abst) u) t x1 H13 (CHead c (Bind Abst) x0) +(csubst0_snd_bind Abst i u0 u x0 H12 c)))) t2 H11)))))) H10)) +(subst0_gen_head (Bind Abst) u0 u t t2 i H9)) c2 H8))) H7)) (\lambda (H7: +(land (eq T (THead (Bind Abst) u t) t2) (csubst0 i u0 c c2))).(land_ind (eq T +(THead (Bind Abst) u t) t2) (csubst0 i u0 c c2) (arity g c2 t2 (AHead a1 a2)) +(\lambda (H8: (eq T (THead (Bind Abst) u t) t2)).(\lambda (H9: (csubst0 i u0 +c c2)).(eq_ind T (THead (Bind Abst) u t) (\lambda (t0: T).(arity g c2 t0 +(AHead a1 a2))) (arity_head g c2 u a1 (H1 d1 u0 i H4 c2 u (fsubst0_fst i u0 c +u c2 H9)) t a2 (H3 d1 u0 (S i) (getl_clear_bind Abst (CHead c (Bind Abst) u) +c u (clear_bind Abst c u) (CHead d1 (Bind Abbr) u0) i H4) (CHead c2 (Bind +Abst) u) t (fsubst0_fst (S i) u0 (CHead c (Bind Abst) u) t (CHead c2 (Bind +Abst) u) (csubst0_fst_bind Abst i c c2 u0 H9 u)))) t2 H8))) H7)) (\lambda +(H7: (land (subst0 i u0 (THead (Bind Abst) u t) t2) (csubst0 i u0 c +c2))).(land_ind (subst0 i u0 (THead (Bind Abst) u t) t2) (csubst0 i u0 c c2) +(arity g c2 t2 (AHead a1 a2)) (\lambda (H8: (subst0 i u0 (THead (Bind Abst) u +t) t2)).(\lambda (H9: (csubst0 i u0 c c2)).(or3_ind (ex2 T (\lambda (u2: +T).(eq T t2 (THead (Bind Abst) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2))) +(ex2 T (\lambda (t3: T).(eq T t2 (THead (Bind Abst) u t3))) (\lambda (t3: +T).(subst0 (s (Bind Abst) i) u0 t t3))) (ex3_2 T T (\lambda (u2: T).(\lambda +(t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: +T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind +Abst) i) u0 t t3)))) (arity g c2 t2 (AHead a1 a2)) (\lambda (H10: (ex2 T +(\lambda (u2: T).(eq T t2 (THead (Bind Abst) u2 t))) (\lambda (u2: T).(subst0 +i u0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T t2 (THead (Bind Abst) u2 t))) +(\lambda (u2: T).(subst0 i u0 u u2)) (arity g c2 t2 (AHead a1 a2)) (\lambda +(x: T).(\lambda (H11: (eq T t2 (THead (Bind Abst) x t))).(\lambda (H12: +(subst0 i u0 u x)).(eq_ind_r T (THead (Bind Abst) x t) (\lambda (t0: +T).(arity g c2 t0 (AHead a1 a2))) (arity_head g c2 x a1 (H1 d1 u0 i H4 c2 x +(fsubst0_both i u0 c u x H12 c2 H9)) t a2 (H3 d1 u0 (S i) (getl_clear_bind +Abst (CHead c (Bind Abst) u) c u (clear_bind Abst c u) (CHead d1 (Bind Abbr) +u0) i H4) (CHead c2 (Bind Abst) x) t (fsubst0_fst (S i) u0 (CHead c (Bind +Abst) u) t (CHead c2 (Bind Abst) x) (csubst0_both_bind Abst i u0 u x H12 c c2 +H9)))) t2 H11)))) H10)) (\lambda (H10: (ex2 T (\lambda (t3: T).(eq T t2 +(THead (Bind Abst) u t3))) (\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t +t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead (Bind Abst) u t3))) +(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3)) (arity g c2 t2 (AHead a1 +a2)) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Bind Abst) u +x))).(\lambda (H12: (subst0 (s (Bind Abst) i) u0 t x)).(eq_ind_r T (THead +(Bind Abst) u x) (\lambda (t0: T).(arity g c2 t0 (AHead a1 a2))) (arity_head +g c2 u a1 (H1 d1 u0 i H4 c2 u (fsubst0_fst i u0 c u c2 H9)) x a2 (H3 d1 u0 (S +i) (getl_clear_bind Abst (CHead c (Bind Abst) u) c u (clear_bind Abst c u) +(CHead d1 (Bind Abbr) u0) i H4) (CHead c2 (Bind Abst) u) x (fsubst0_both (S +i) u0 (CHead c (Bind Abst) u) t x H12 (CHead c2 (Bind Abst) u) +(csubst0_fst_bind Abst i c c2 u0 H9 u)))) t2 H11)))) H10)) (\lambda (H10: +(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 +t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: +T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3))))).(ex3_2_ind T T +(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: +T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3))) (arity g c2 t2 +(AHead a1 a2)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq T t2 +(THead (Bind Abst) x0 x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda (H13: +(subst0 (s (Bind Abst) i) u0 t x1)).(eq_ind_r T (THead (Bind Abst) x0 x1) +(\lambda (t0: T).(arity g c2 t0 (AHead a1 a2))) (arity_head g c2 x0 a1 (H1 d1 +u0 i H4 c2 x0 (fsubst0_both i u0 c u x0 H12 c2 H9)) x1 a2 (H3 d1 u0 (S i) +(getl_clear_bind Abst (CHead c (Bind Abst) u) c u (clear_bind Abst c u) +(CHead d1 (Bind Abbr) u0) i H4) (CHead c2 (Bind Abst) x0) x1 (fsubst0_both (S +i) u0 (CHead c (Bind Abst) u) t x1 H13 (CHead c2 (Bind Abst) x0) +(csubst0_both_bind Abst i u0 u x0 H12 c c2 H9)))) t2 H11)))))) H10)) +(subst0_gen_head (Bind Abst) u0 u t t2 i H8)))) H7)) H6))))))))))))))))))) +(\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H0: (arity g c u +a1)).(\lambda (H1: ((\forall (d1: C).(\forall (u0: T).(\forall (i: +nat).((getl i c (CHead d1 (Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2: +T).((fsubst0 i u0 c u c2 t2) \to (arity g c2 t2 a1)))))))))).(\lambda (t: +T).(\lambda (a2: A).(\lambda (H2: (arity g c t (AHead a1 a2))).(\lambda (H3: +((\forall (d1: C).(\forall (u0: T).(\forall (i: nat).((getl i c (CHead d1 +(Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 c t c2 +t2) \to (arity g c2 t2 (AHead a1 a2))))))))))).(\lambda (d1: C).(\lambda (u0: +T).(\lambda (i: nat).(\lambda (H4: (getl i c (CHead d1 (Bind Abbr) +u0))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H5: (fsubst0 i u0 c (THead +(Flat Appl) u t) c2 t2)).(let H_x \def (fsubst0_gen_base c c2 (THead (Flat +Appl) u t) t2 u0 i H5) in (let H6 \def H_x in (or3_ind (land (eq C c c2) +(subst0 i u0 (THead (Flat Appl) u t) t2)) (land (eq T (THead (Flat Appl) u t) +t2) (csubst0 i u0 c c2)) (land (subst0 i u0 (THead (Flat Appl) u t) t2) +(csubst0 i u0 c c2)) (arity g c2 t2 a2) (\lambda (H7: (land (eq C c c2) +(subst0 i u0 (THead (Flat Appl) u t) t2))).(land_ind (eq C c c2) (subst0 i u0 +(THead (Flat Appl) u t) t2) (arity g c2 t2 a2) (\lambda (H8: (eq C c +c2)).(\lambda (H9: (subst0 i u0 (THead (Flat Appl) u t) t2)).(eq_ind C c +(\lambda (c0: C).(arity g c0 t2 a2)) (or3_ind (ex2 T (\lambda (u2: T).(eq T +t2 (THead (Flat Appl) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2))) (ex2 T +(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u t3))) (\lambda (t3: T).(subst0 +(s (Flat Appl) i) u0 t t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq +T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i +u0 u u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 t +t3)))) (arity g c t2 a2) (\lambda (H10: (ex2 T (\lambda (u2: T).(eq T t2 +(THead (Flat Appl) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T +(\lambda (u2: T).(eq T t2 (THead (Flat Appl) u2 t))) (\lambda (u2: T).(subst0 +i u0 u u2)) (arity g c t2 a2) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead +(Flat Appl) x t))).(\lambda (H12: (subst0 i u0 u x)).(eq_ind_r T (THead (Flat +Appl) x t) (\lambda (t0: T).(arity g c t0 a2)) (arity_appl g c x a1 (H1 d1 u0 +i H4 c x (fsubst0_snd i u0 c u x H12)) t a2 H2) t2 H11)))) H10)) (\lambda +(H10: (ex2 T (\lambda (t3: T).(eq T t2 (THead (Flat Appl) u t3))) (\lambda +(t3: T).(subst0 (s (Flat Appl) i) u0 t t3)))).(ex2_ind T (\lambda (t3: T).(eq +T t2 (THead (Flat Appl) u t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 +t t3)) (arity g c t2 a2) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat +Appl) u x))).(\lambda (H12: (subst0 (s (Flat Appl) i) u0 t x)).(eq_ind_r T +(THead (Flat Appl) u x) (\lambda (t0: T).(arity g c t0 a2)) (arity_appl g c u +a1 H0 x a2 (H3 d1 u0 i H4 c x (fsubst0_snd i u0 c t x H12))) t2 H11)))) H10)) +(\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead +(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) +(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 t +t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead +(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) +(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 t t3))) (arity +g c t2 a2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq T t2 (THead +(Flat Appl) x0 x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda (H13: +(subst0 (s (Flat Appl) i) u0 t x1)).(eq_ind_r T (THead (Flat Appl) x0 x1) +(\lambda (t0: T).(arity g c t0 a2)) (arity_appl g c x0 a1 (H1 d1 u0 i H4 c x0 +(fsubst0_snd i u0 c u x0 H12)) x1 a2 (H3 d1 u0 i H4 c x1 (fsubst0_snd i u0 c +t x1 H13))) t2 H11)))))) H10)) (subst0_gen_head (Flat Appl) u0 u t t2 i H9)) +c2 H8))) H7)) (\lambda (H7: (land (eq T (THead (Flat Appl) u t) t2) (csubst0 +i u0 c c2))).(land_ind (eq T (THead (Flat Appl) u t) t2) (csubst0 i u0 c c2) +(arity g c2 t2 a2) (\lambda (H8: (eq T (THead (Flat Appl) u t) t2)).(\lambda +(H9: (csubst0 i u0 c c2)).(eq_ind T (THead (Flat Appl) u t) (\lambda (t0: +T).(arity g c2 t0 a2)) (arity_appl g c2 u a1 (H1 d1 u0 i H4 c2 u (fsubst0_fst +i u0 c u c2 H9)) t a2 (H3 d1 u0 i H4 c2 t (fsubst0_fst i u0 c t c2 H9))) t2 +H8))) H7)) (\lambda (H7: (land (subst0 i u0 (THead (Flat Appl) u t) t2) +(csubst0 i u0 c c2))).(land_ind (subst0 i u0 (THead (Flat Appl) u t) t2) +(csubst0 i u0 c c2) (arity g c2 t2 a2) (\lambda (H8: (subst0 i u0 (THead +(Flat Appl) u t) t2)).(\lambda (H9: (csubst0 i u0 c c2)).(or3_ind (ex2 T +(\lambda (u2: T).(eq T t2 (THead (Flat Appl) u2 t))) (\lambda (u2: T).(subst0 +i u0 u u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead (Flat Appl) u t3))) +(\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 t t3))) (ex3_2 T T (\lambda +(u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: +T).(subst0 (s (Flat Appl) i) u0 t t3)))) (arity g c2 t2 a2) (\lambda (H10: +(ex2 T (\lambda (u2: T).(eq T t2 (THead (Flat Appl) u2 t))) (\lambda (u2: +T).(subst0 i u0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T t2 (THead (Flat +Appl) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)) (arity g c2 t2 a2) +(\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat Appl) x t))).(\lambda +(H12: (subst0 i u0 u x)).(eq_ind_r T (THead (Flat Appl) x t) (\lambda (t0: +T).(arity g c2 t0 a2)) (arity_appl g c2 x a1 (H1 d1 u0 i H4 c2 x +(fsubst0_both i u0 c u x H12 c2 H9)) t a2 (H3 d1 u0 i H4 c2 t (fsubst0_fst i +u0 c t c2 H9))) t2 H11)))) H10)) (\lambda (H10: (ex2 T (\lambda (t3: T).(eq T +t2 (THead (Flat Appl) u t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 t +t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead (Flat Appl) u t3))) +(\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 t t3)) (arity g c2 t2 a2) +(\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat Appl) u x))).(\lambda +(H12: (subst0 (s (Flat Appl) i) u0 t x)).(eq_ind_r T (THead (Flat Appl) u x) +(\lambda (t0: T).(arity g c2 t0 a2)) (arity_appl g c2 u a1 (H1 d1 u0 i H4 c2 +u (fsubst0_fst i u0 c u c2 H9)) x a2 (H3 d1 u0 i H4 c2 x (fsubst0_both i u0 c +t x H12 c2 H9))) t2 H11)))) H10)) (\lambda (H10: (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: +T).(subst0 (s (Flat Appl) i) u0 t t3))))).(ex3_2_ind T T (\lambda (u2: +T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: +T).(subst0 (s (Flat Appl) i) u0 t t3))) (arity g c2 t2 a2) (\lambda (x0: +T).(\lambda (x1: T).(\lambda (H11: (eq T t2 (THead (Flat Appl) x0 +x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda (H13: (subst0 (s (Flat +Appl) i) u0 t x1)).(eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda (t0: +T).(arity g c2 t0 a2)) (arity_appl g c2 x0 a1 (H1 d1 u0 i H4 c2 x0 +(fsubst0_both i u0 c u x0 H12 c2 H9)) x1 a2 (H3 d1 u0 i H4 c2 x1 +(fsubst0_both i u0 c t x1 H13 c2 H9))) t2 H11)))))) H10)) (subst0_gen_head +(Flat Appl) u0 u t t2 i H8)))) H7)) H6))))))))))))))))))) (\lambda (c: +C).(\lambda (u: T).(\lambda (a0: A).(\lambda (H0: (arity g c u (asucc g +a0))).(\lambda (H1: ((\forall (d1: C).(\forall (u0: T).(\forall (i: +nat).((getl i c (CHead d1 (Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2: +T).((fsubst0 i u0 c u c2 t2) \to (arity g c2 t2 (asucc g +a0))))))))))).(\lambda (t: T).(\lambda (H2: (arity g c t a0)).(\lambda (H3: +((\forall (d1: C).(\forall (u0: T).(\forall (i: nat).((getl i c (CHead d1 +(Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 c t c2 +t2) \to (arity g c2 t2 a0)))))))))).(\lambda (d1: C).(\lambda (u0: +T).(\lambda (i: nat).(\lambda (H4: (getl i c (CHead d1 (Bind Abbr) +u0))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H5: (fsubst0 i u0 c (THead +(Flat Cast) u t) c2 t2)).(let H_x \def (fsubst0_gen_base c c2 (THead (Flat +Cast) u t) t2 u0 i H5) in (let H6 \def H_x in (or3_ind (land (eq C c c2) +(subst0 i u0 (THead (Flat Cast) u t) t2)) (land (eq T (THead (Flat Cast) u t) +t2) (csubst0 i u0 c c2)) (land (subst0 i u0 (THead (Flat Cast) u t) t2) +(csubst0 i u0 c c2)) (arity g c2 t2 a0) (\lambda (H7: (land (eq C c c2) +(subst0 i u0 (THead (Flat Cast) u t) t2))).(land_ind (eq C c c2) (subst0 i u0 +(THead (Flat Cast) u t) t2) (arity g c2 t2 a0) (\lambda (H8: (eq C c +c2)).(\lambda (H9: (subst0 i u0 (THead (Flat Cast) u t) t2)).(eq_ind C c +(\lambda (c0: C).(arity g c0 t2 a0)) (or3_ind (ex2 T (\lambda (u2: T).(eq T +t2 (THead (Flat Cast) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2))) (ex2 T +(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u t3))) (\lambda (t3: T).(subst0 +(s (Flat Cast) i) u0 t t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq +T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i +u0 u u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t +t3)))) (arity g c t2 a0) (\lambda (H10: (ex2 T (\lambda (u2: T).(eq T t2 +(THead (Flat Cast) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T +(\lambda (u2: T).(eq T t2 (THead (Flat Cast) u2 t))) (\lambda (u2: T).(subst0 +i u0 u u2)) (arity g c t2 a0) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead +(Flat Cast) x t))).(\lambda (H12: (subst0 i u0 u x)).(eq_ind_r T (THead (Flat +Cast) x t) (\lambda (t0: T).(arity g c t0 a0)) (arity_cast g c x a0 (H1 d1 u0 +i H4 c x (fsubst0_snd i u0 c u x H12)) t H2) t2 H11)))) H10)) (\lambda (H10: +(ex2 T (\lambda (t3: T).(eq T t2 (THead (Flat Cast) u t3))) (\lambda (t3: +T).(subst0 (s (Flat Cast) i) u0 t t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 +(THead (Flat Cast) u t3))) (\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t +t3)) (arity g c t2 a0) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat +Cast) u x))).(\lambda (H12: (subst0 (s (Flat Cast) i) u0 t x)).(eq_ind_r T +(THead (Flat Cast) u x) (\lambda (t0: T).(arity g c t0 a0)) (arity_cast g c u +a0 H0 x (H3 d1 u0 i H4 c x (fsubst0_snd i u0 c t x H12))) t2 H11)))) H10)) +(\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead +(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) +(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t +t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead +(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) +(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t t3))) (arity +g c t2 a0) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq T t2 (THead +(Flat Cast) x0 x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda (H13: +(subst0 (s (Flat Cast) i) u0 t x1)).(eq_ind_r T (THead (Flat Cast) x0 x1) +(\lambda (t0: T).(arity g c t0 a0)) (arity_cast g c x0 a0 (H1 d1 u0 i H4 c x0 +(fsubst0_snd i u0 c u x0 H12)) x1 (H3 d1 u0 i H4 c x1 (fsubst0_snd i u0 c t +x1 H13))) t2 H11)))))) H10)) (subst0_gen_head (Flat Cast) u0 u t t2 i H9)) c2 +H8))) H7)) (\lambda (H7: (land (eq T (THead (Flat Cast) u t) t2) (csubst0 i +u0 c c2))).(land_ind (eq T (THead (Flat Cast) u t) t2) (csubst0 i u0 c c2) +(arity g c2 t2 a0) (\lambda (H8: (eq T (THead (Flat Cast) u t) t2)).(\lambda +(H9: (csubst0 i u0 c c2)).(eq_ind T (THead (Flat Cast) u t) (\lambda (t0: +T).(arity g c2 t0 a0)) (arity_cast g c2 u a0 (H1 d1 u0 i H4 c2 u (fsubst0_fst +i u0 c u c2 H9)) t (H3 d1 u0 i H4 c2 t (fsubst0_fst i u0 c t c2 H9))) t2 +H8))) H7)) (\lambda (H7: (land (subst0 i u0 (THead (Flat Cast) u t) t2) +(csubst0 i u0 c c2))).(land_ind (subst0 i u0 (THead (Flat Cast) u t) t2) +(csubst0 i u0 c c2) (arity g c2 t2 a0) (\lambda (H8: (subst0 i u0 (THead +(Flat Cast) u t) t2)).(\lambda (H9: (csubst0 i u0 c c2)).(or3_ind (ex2 T +(\lambda (u2: T).(eq T t2 (THead (Flat Cast) u2 t))) (\lambda (u2: T).(subst0 +i u0 u u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead (Flat Cast) u t3))) +(\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t t3))) (ex3_2 T T (\lambda +(u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: +T).(subst0 (s (Flat Cast) i) u0 t t3)))) (arity g c2 t2 a0) (\lambda (H10: +(ex2 T (\lambda (u2: T).(eq T t2 (THead (Flat Cast) u2 t))) (\lambda (u2: +T).(subst0 i u0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T t2 (THead (Flat +Cast) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)) (arity g c2 t2 a0) +(\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat Cast) x t))).(\lambda +(H12: (subst0 i u0 u x)).(eq_ind_r T (THead (Flat Cast) x t) (\lambda (t0: +T).(arity g c2 t0 a0)) (arity_cast g c2 x a0 (H1 d1 u0 i H4 c2 x +(fsubst0_both i u0 c u x H12 c2 H9)) t (H3 d1 u0 i H4 c2 t (fsubst0_fst i u0 +c t c2 H9))) t2 H11)))) H10)) (\lambda (H10: (ex2 T (\lambda (t3: T).(eq T t2 +(THead (Flat Cast) u t3))) (\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t +t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead (Flat Cast) u t3))) +(\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t t3)) (arity g c2 t2 a0) +(\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat Cast) u x))).(\lambda +(H12: (subst0 (s (Flat Cast) i) u0 t x)).(eq_ind_r T (THead (Flat Cast) u x) +(\lambda (t0: T).(arity g c2 t0 a0)) (arity_cast g c2 u a0 (H1 d1 u0 i H4 c2 +u (fsubst0_fst i u0 c u c2 H9)) x (H3 d1 u0 i H4 c2 x (fsubst0_both i u0 c t +x H12 c2 H9))) t2 H11)))) H10)) (\lambda (H10: (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: +T).(subst0 (s (Flat Cast) i) u0 t t3))))).(ex3_2_ind T T (\lambda (u2: +T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: +T).(subst0 (s (Flat Cast) i) u0 t t3))) (arity g c2 t2 a0) (\lambda (x0: +T).(\lambda (x1: T).(\lambda (H11: (eq T t2 (THead (Flat Cast) x0 +x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda (H13: (subst0 (s (Flat +Cast) i) u0 t x1)).(eq_ind_r T (THead (Flat Cast) x0 x1) (\lambda (t0: +T).(arity g c2 t0 a0)) (arity_cast g c2 x0 a0 (H1 d1 u0 i H4 c2 x0 +(fsubst0_both i u0 c u x0 H12 c2 H9)) x1 (H3 d1 u0 i H4 c2 x1 (fsubst0_both i +u0 c t x1 H13 c2 H9))) t2 H11)))))) H10)) (subst0_gen_head (Flat Cast) u0 u t +t2 i H8)))) H7)) H6)))))))))))))))))) (\lambda (c: C).(\lambda (t: +T).(\lambda (a1: A).(\lambda (_: (arity g c t a1)).(\lambda (H1: ((\forall +(d1: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d1 (Bind Abbr) +u)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u c t c2 t2) \to (arity +g c2 t2 a1)))))))))).(\lambda (a2: A).(\lambda (H2: (leq g a1 a2)).(\lambda +(d1: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H3: (getl i c (CHead d1 +(Bind Abbr) u))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H4: (fsubst0 i u +c t c2 t2)).(let H_x \def (fsubst0_gen_base c c2 t t2 u i H4) in (let H5 \def +H_x in (or3_ind (land (eq C c c2) (subst0 i u t t2)) (land (eq T t t2) +(csubst0 i u c c2)) (land (subst0 i u t t2) (csubst0 i u c c2)) (arity g c2 +t2 a2) (\lambda (H6: (land (eq C c c2) (subst0 i u t t2))).(land_ind (eq C c +c2) (subst0 i u t t2) (arity g c2 t2 a2) (\lambda (H7: (eq C c c2)).(\lambda +(H8: (subst0 i u t t2)).(eq_ind C c (\lambda (c0: C).(arity g c0 t2 a2)) +(arity_repl g c t2 a1 (H1 d1 u i H3 c t2 (fsubst0_snd i u c t t2 H8)) a2 H2) +c2 H7))) H6)) (\lambda (H6: (land (eq T t t2) (csubst0 i u c c2))).(land_ind +(eq T t t2) (csubst0 i u c c2) (arity g c2 t2 a2) (\lambda (H7: (eq T t +t2)).(\lambda (H8: (csubst0 i u c c2)).(eq_ind T t (\lambda (t0: T).(arity g +c2 t0 a2)) (arity_repl g c2 t a1 (H1 d1 u i H3 c2 t (fsubst0_fst i u c t c2 +H8)) a2 H2) t2 H7))) H6)) (\lambda (H6: (land (subst0 i u t t2) (csubst0 i u +c c2))).(land_ind (subst0 i u t t2) (csubst0 i u c c2) (arity g c2 t2 a2) +(\lambda (H7: (subst0 i u t t2)).(\lambda (H8: (csubst0 i u c +c2)).(arity_repl g c2 t2 a1 (H1 d1 u i H3 c2 t2 (fsubst0_both i u c t t2 H7 +c2 H8)) a2 H2))) H6)) H5))))))))))))))))) c1 t1 a H))))). +(* COMMENTS +Initial nodes: 20387 +END *) + +theorem arity_subst0: + \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (a: A).((arity g c +t1 a) \to (\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead +d (Bind Abbr) u)) \to (\forall (t2: T).((subst0 i u t1 t2) \to (arity g c t2 +a))))))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (a: A).(\lambda (H: +(arity g c t1 a)).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda +(H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (t2: T).(\lambda (H1: +(subst0 i u t1 t2)).(arity_fsubst0 g c t1 a H d u i H0 c t2 (fsubst0_snd i u +c t1 t2 H1)))))))))))). +(* COMMENTS +Initial nodes: 89 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/asucc/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/asucc/defs.ma new file mode 100644 index 000000000..2f1af0279 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/asucc/defs.ma @@ -0,0 +1,28 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/A/defs.ma". + +include "Basic-1/G/defs.ma". + +definition asucc: + G \to (A \to A) +\def + let rec asucc (g: G) (l: A) on l: A \def (match l with [(ASort n0 n) +\Rightarrow (match n0 with [O \Rightarrow (ASort O (next g n)) | (S h) +\Rightarrow (ASort h n)]) | (AHead a1 a2) \Rightarrow (AHead a1 (asucc g +a2))]) in asucc. + diff --git a/matita/matita/contribs/lambdadelta/basic_1/asucc/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/asucc/fwd.ma new file mode 100644 index 000000000..61fcb5799 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/asucc/fwd.ma @@ -0,0 +1,99 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/asucc/defs.ma". + +theorem asucc_gen_sort: + \forall (g: G).(\forall (h: nat).(\forall (n: nat).(\forall (a: A).((eq A +(ASort h n) (asucc g a)) \to (ex_2 nat nat (\lambda (h0: nat).(\lambda (n0: +nat).(eq A a (ASort h0 n0))))))))) +\def + \lambda (g: G).(\lambda (h: nat).(\lambda (n: nat).(\lambda (a: A).(A_ind +(\lambda (a0: A).((eq A (ASort h n) (asucc g a0)) \to (ex_2 nat nat (\lambda +(h0: nat).(\lambda (n0: nat).(eq A a0 (ASort h0 n0))))))) (\lambda (n0: +nat).(\lambda (n1: nat).(\lambda (H: (eq A (ASort h n) (asucc g (ASort n0 +n1)))).(let H0 \def (f_equal A A (\lambda (e: A).e) (ASort h n) (match n0 +with [O \Rightarrow (ASort O (next g n1)) | (S h0) \Rightarrow (ASort h0 +n1)]) H) in (ex_2_intro nat nat (\lambda (h0: nat).(\lambda (n2: nat).(eq A +(ASort n0 n1) (ASort h0 n2)))) n0 n1 (refl_equal A (ASort n0 n1))))))) +(\lambda (a0: A).(\lambda (_: (((eq A (ASort h n) (asucc g a0)) \to (ex_2 nat +nat (\lambda (h0: nat).(\lambda (n0: nat).(eq A a0 (ASort h0 +n0)))))))).(\lambda (a1: A).(\lambda (_: (((eq A (ASort h n) (asucc g a1)) +\to (ex_2 nat nat (\lambda (h0: nat).(\lambda (n0: nat).(eq A a1 (ASort h0 +n0)))))))).(\lambda (H1: (eq A (ASort h n) (asucc g (AHead a0 a1)))).(let H2 +\def (eq_ind A (ASort h n) (\lambda (ee: A).(match ee in A return (\lambda +(_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow +False])) I (asucc g (AHead a0 a1)) H1) in (False_ind (ex_2 nat nat (\lambda +(h0: nat).(\lambda (n0: nat).(eq A (AHead a0 a1) (ASort h0 n0))))) H2))))))) +a)))). +(* COMMENTS +Initial nodes: 317 +END *) + +theorem asucc_gen_head: + \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a: A).((eq A +(AHead a1 a2) (asucc g a)) \to (ex2 A (\lambda (a0: A).(eq A a (AHead a1 +a0))) (\lambda (a0: A).(eq A a2 (asucc g a0)))))))) +\def + \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a: A).(A_ind +(\lambda (a0: A).((eq A (AHead a1 a2) (asucc g a0)) \to (ex2 A (\lambda (a3: +A).(eq A a0 (AHead a1 a3))) (\lambda (a3: A).(eq A a2 (asucc g a3)))))) +(\lambda (n: nat).(\lambda (n0: nat).(\lambda (H: (eq A (AHead a1 a2) (asucc +g (ASort n n0)))).(nat_ind (\lambda (n1: nat).((eq A (AHead a1 a2) (asucc g +(ASort n1 n0))) \to (ex2 A (\lambda (a0: A).(eq A (ASort n1 n0) (AHead a1 +a0))) (\lambda (a0: A).(eq A a2 (asucc g a0)))))) (\lambda (H0: (eq A (AHead +a1 a2) (asucc g (ASort O n0)))).(let H1 \def (eq_ind A (AHead a1 a2) (\lambda +(ee: A).(match ee in A return (\lambda (_: A).Prop) with [(ASort _ _) +\Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort O (next g n0)) +H0) in (False_ind (ex2 A (\lambda (a0: A).(eq A (ASort O n0) (AHead a1 a0))) +(\lambda (a0: A).(eq A a2 (asucc g a0)))) H1))) (\lambda (n1: nat).(\lambda +(_: (((eq A (AHead a1 a2) (asucc g (ASort n1 n0))) \to (ex2 A (\lambda (a0: +A).(eq A (ASort n1 n0) (AHead a1 a0))) (\lambda (a0: A).(eq A a2 (asucc g +a0))))))).(\lambda (H0: (eq A (AHead a1 a2) (asucc g (ASort (S n1) +n0)))).(let H1 \def (eq_ind A (AHead a1 a2) (\lambda (ee: A).(match ee in A +return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ +_) \Rightarrow True])) I (ASort n1 n0) H0) in (False_ind (ex2 A (\lambda (a0: +A).(eq A (ASort (S n1) n0) (AHead a1 a0))) (\lambda (a0: A).(eq A a2 (asucc g +a0)))) H1))))) n H)))) (\lambda (a0: A).(\lambda (H: (((eq A (AHead a1 a2) +(asucc g a0)) \to (ex2 A (\lambda (a3: A).(eq A a0 (AHead a1 a3))) (\lambda +(a3: A).(eq A a2 (asucc g a3))))))).(\lambda (a3: A).(\lambda (H0: (((eq A +(AHead a1 a2) (asucc g a3)) \to (ex2 A (\lambda (a4: A).(eq A a3 (AHead a1 +a4))) (\lambda (a4: A).(eq A a2 (asucc g a4))))))).(\lambda (H1: (eq A (AHead +a1 a2) (asucc g (AHead a0 a3)))).(let H2 \def (f_equal A A (\lambda (e: +A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a1 | +(AHead a4 _) \Rightarrow a4])) (AHead a1 a2) (AHead a0 (asucc g a3)) H1) in +((let H3 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: +A).A) with [(ASort _ _) \Rightarrow a2 | (AHead _ a4) \Rightarrow a4])) +(AHead a1 a2) (AHead a0 (asucc g a3)) H1) in (\lambda (H4: (eq A a1 a0)).(let +H5 \def (eq_ind_r A a0 (\lambda (a4: A).((eq A (AHead a1 a2) (asucc g a4)) +\to (ex2 A (\lambda (a5: A).(eq A a4 (AHead a1 a5))) (\lambda (a5: A).(eq A +a2 (asucc g a5)))))) H a1 H4) in (eq_ind A a1 (\lambda (a4: A).(ex2 A +(\lambda (a5: A).(eq A (AHead a4 a3) (AHead a1 a5))) (\lambda (a5: A).(eq A +a2 (asucc g a5))))) (let H6 \def (eq_ind A a2 (\lambda (a4: A).((eq A (AHead +a1 a4) (asucc g a3)) \to (ex2 A (\lambda (a5: A).(eq A a3 (AHead a1 a5))) +(\lambda (a5: A).(eq A a4 (asucc g a5)))))) H0 (asucc g a3) H3) in (let H7 +\def (eq_ind A a2 (\lambda (a4: A).((eq A (AHead a1 a4) (asucc g a1)) \to +(ex2 A (\lambda (a5: A).(eq A a1 (AHead a1 a5))) (\lambda (a5: A).(eq A a4 +(asucc g a5)))))) H5 (asucc g a3) H3) in (eq_ind_r A (asucc g a3) (\lambda +(a4: A).(ex2 A (\lambda (a5: A).(eq A (AHead a1 a3) (AHead a1 a5))) (\lambda +(a5: A).(eq A a4 (asucc g a5))))) (ex_intro2 A (\lambda (a4: A).(eq A (AHead +a1 a3) (AHead a1 a4))) (\lambda (a4: A).(eq A (asucc g a3) (asucc g a4))) a3 +(refl_equal A (AHead a1 a3)) (refl_equal A (asucc g a3))) a2 H3))) a0 H4)))) +H2))))))) a)))). +(* COMMENTS +Initial nodes: 957 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/cimp/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/cimp/defs.ma new file mode 100644 index 000000000..b45b64b59 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/cimp/defs.ma @@ -0,0 +1,25 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/getl/defs.ma". + +definition cimp: + C \to (C \to Prop) +\def + \lambda (c1: C).(\lambda (c2: C).(\forall (b: B).(\forall (d1: C).(\forall +(w: T).(\forall (h: nat).((getl h c1 (CHead d1 (Bind b) w)) \to (ex C +(\lambda (d2: C).(getl h c2 (CHead d2 (Bind b) w)))))))))). + diff --git a/matita/matita/contribs/lambdadelta/basic_1/cimp/props.ma b/matita/matita/contribs/lambdadelta/basic_1/cimp/props.ma new file mode 100644 index 000000000..0834a7afe --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/cimp/props.ma @@ -0,0 +1,139 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/cimp/defs.ma". + +include "Basic-1/getl/getl.ma". + +theorem cimp_flat_sx: + \forall (f: F).(\forall (c: C).(\forall (v: T).(cimp (CHead c (Flat f) v) +c))) +\def + \lambda (f: F).(\lambda (c: C).(\lambda (v: T).(\lambda (b: B).(\lambda (d1: +C).(\lambda (w: T).(\lambda (h: nat).(\lambda (H: (getl h (CHead c (Flat f) +v) (CHead d1 (Bind b) w))).(nat_ind (\lambda (n: nat).((getl n (CHead c (Flat +f) v) (CHead d1 (Bind b) w)) \to (ex C (\lambda (d2: C).(getl n c (CHead d2 +(Bind b) w)))))) (\lambda (H0: (getl O (CHead c (Flat f) v) (CHead d1 (Bind +b) w))).(ex_intro C (\lambda (d2: C).(getl O c (CHead d2 (Bind b) w))) d1 +(getl_intro O c (CHead d1 (Bind b) w) c (drop_refl c) (clear_gen_flat f c +(CHead d1 (Bind b) w) v (getl_gen_O (CHead c (Flat f) v) (CHead d1 (Bind b) +w) H0))))) (\lambda (h0: nat).(\lambda (_: (((getl h0 (CHead c (Flat f) v) +(CHead d1 (Bind b) w)) \to (ex C (\lambda (d2: C).(getl h0 c (CHead d2 (Bind +b) w))))))).(\lambda (H0: (getl (S h0) (CHead c (Flat f) v) (CHead d1 (Bind +b) w))).(ex_intro C (\lambda (d2: C).(getl (S h0) c (CHead d2 (Bind b) w))) +d1 (getl_gen_S (Flat f) c (CHead d1 (Bind b) w) v h0 H0))))) h H)))))))). +(* COMMENTS +Initial nodes: 327 +END *) + +theorem cimp_flat_dx: + \forall (f: F).(\forall (c: C).(\forall (v: T).(cimp c (CHead c (Flat f) +v)))) +\def + \lambda (f: F).(\lambda (c: C).(\lambda (v: T).(\lambda (b: B).(\lambda (d1: +C).(\lambda (w: T).(\lambda (h: nat).(\lambda (H: (getl h c (CHead d1 (Bind +b) w))).(ex_intro C (\lambda (d2: C).(getl h (CHead c (Flat f) v) (CHead d2 +(Bind b) w))) d1 (getl_flat c (CHead d1 (Bind b) w) h H f v))))))))). +(* COMMENTS +Initial nodes: 83 +END *) + +theorem cimp_bind: + \forall (c1: C).(\forall (c2: C).((cimp c1 c2) \to (\forall (b: B).(\forall +(v: T).(cimp (CHead c1 (Bind b) v) (CHead c2 (Bind b) v)))))) +\def + \lambda (c1: C).(\lambda (c2: C).(\lambda (H: ((\forall (b: B).(\forall (d1: +C).(\forall (w: T).(\forall (h: nat).((getl h c1 (CHead d1 (Bind b) w)) \to +(ex C (\lambda (d2: C).(getl h c2 (CHead d2 (Bind b) w))))))))))).(\lambda +(b: B).(\lambda (v: T).(\lambda (b0: B).(\lambda (d1: C).(\lambda (w: +T).(\lambda (h: nat).(\lambda (H0: (getl h (CHead c1 (Bind b) v) (CHead d1 +(Bind b0) w))).(nat_ind (\lambda (n: nat).((getl n (CHead c1 (Bind b) v) +(CHead d1 (Bind b0) w)) \to (ex C (\lambda (d2: C).(getl n (CHead c2 (Bind b) +v) (CHead d2 (Bind b0) w)))))) (\lambda (H1: (getl O (CHead c1 (Bind b) v) +(CHead d1 (Bind b0) w))).(let H2 \def (f_equal C C (\lambda (e: C).(match e +in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d1 | (CHead c _ _) +\Rightarrow c])) (CHead d1 (Bind b0) w) (CHead c1 (Bind b) v) (clear_gen_bind +b c1 (CHead d1 (Bind b0) w) v (getl_gen_O (CHead c1 (Bind b) v) (CHead d1 +(Bind b0) w) H1))) in ((let H3 \def (f_equal C B (\lambda (e: C).(match e in +C return (\lambda (_: C).B) with [(CSort _) \Rightarrow b0 | (CHead _ k _) +\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b1) +\Rightarrow b1 | (Flat _) \Rightarrow b0])])) (CHead d1 (Bind b0) w) (CHead +c1 (Bind b) v) (clear_gen_bind b c1 (CHead d1 (Bind b0) w) v (getl_gen_O +(CHead c1 (Bind b) v) (CHead d1 (Bind b0) w) H1))) in ((let H4 \def (f_equal +C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) +\Rightarrow w | (CHead _ _ t) \Rightarrow t])) (CHead d1 (Bind b0) w) (CHead +c1 (Bind b) v) (clear_gen_bind b c1 (CHead d1 (Bind b0) w) v (getl_gen_O +(CHead c1 (Bind b) v) (CHead d1 (Bind b0) w) H1))) in (\lambda (H5: (eq B b0 +b)).(\lambda (_: (eq C d1 c1)).(eq_ind_r T v (\lambda (t: T).(ex C (\lambda +(d2: C).(getl O (CHead c2 (Bind b) v) (CHead d2 (Bind b0) t))))) (eq_ind_r B +b (\lambda (b1: B).(ex C (\lambda (d2: C).(getl O (CHead c2 (Bind b) v) +(CHead d2 (Bind b1) v))))) (ex_intro C (\lambda (d2: C).(getl O (CHead c2 +(Bind b) v) (CHead d2 (Bind b) v))) c2 (getl_refl b c2 v)) b0 H5) w H4)))) +H3)) H2))) (\lambda (h0: nat).(\lambda (_: (((getl h0 (CHead c1 (Bind b) v) +(CHead d1 (Bind b0) w)) \to (ex C (\lambda (d2: C).(getl h0 (CHead c2 (Bind +b) v) (CHead d2 (Bind b0) w))))))).(\lambda (H1: (getl (S h0) (CHead c1 (Bind +b) v) (CHead d1 (Bind b0) w))).(let H_x \def (H b0 d1 w (r (Bind b) h0) +(getl_gen_S (Bind b) c1 (CHead d1 (Bind b0) w) v h0 H1)) in (let H2 \def H_x +in (ex_ind C (\lambda (d2: C).(getl h0 c2 (CHead d2 (Bind b0) w))) (ex C +(\lambda (d2: C).(getl (S h0) (CHead c2 (Bind b) v) (CHead d2 (Bind b0) w)))) +(\lambda (x: C).(\lambda (H3: (getl h0 c2 (CHead x (Bind b0) w))).(ex_intro C +(\lambda (d2: C).(getl (S h0) (CHead c2 (Bind b) v) (CHead d2 (Bind b0) w))) +x (getl_head (Bind b) h0 c2 (CHead x (Bind b0) w) H3 v)))) H2)))))) h +H0)))))))))). +(* COMMENTS +Initial nodes: 817 +END *) + +theorem cimp_getl_conf: + \forall (c1: C).(\forall (c2: C).((cimp c1 c2) \to (\forall (b: B).(\forall +(d1: C).(\forall (w: T).(\forall (i: nat).((getl i c1 (CHead d1 (Bind b) w)) +\to (ex2 C (\lambda (d2: C).(cimp d1 d2)) (\lambda (d2: C).(getl i c2 (CHead +d2 (Bind b) w))))))))))) +\def + \lambda (c1: C).(\lambda (c2: C).(\lambda (H: ((\forall (b: B).(\forall (d1: +C).(\forall (w: T).(\forall (h: nat).((getl h c1 (CHead d1 (Bind b) w)) \to +(ex C (\lambda (d2: C).(getl h c2 (CHead d2 (Bind b) w))))))))))).(\lambda +(b: B).(\lambda (d1: C).(\lambda (w: T).(\lambda (i: nat).(\lambda (H0: (getl +i c1 (CHead d1 (Bind b) w))).(let H_x \def (H b d1 w i H0) in (let H1 \def +H_x in (ex_ind C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind b) w))) (ex2 C +(\lambda (d2: C).(\forall (b0: B).(\forall (d3: C).(\forall (w0: T).(\forall +(h: nat).((getl h d1 (CHead d3 (Bind b0) w0)) \to (ex C (\lambda (d4: +C).(getl h d2 (CHead d4 (Bind b0) w0)))))))))) (\lambda (d2: C).(getl i c2 +(CHead d2 (Bind b) w)))) (\lambda (x: C).(\lambda (H2: (getl i c2 (CHead x +(Bind b) w))).(ex_intro2 C (\lambda (d2: C).(\forall (b0: B).(\forall (d3: +C).(\forall (w0: T).(\forall (h: nat).((getl h d1 (CHead d3 (Bind b0) w0)) +\to (ex C (\lambda (d4: C).(getl h d2 (CHead d4 (Bind b0) w0)))))))))) +(\lambda (d2: C).(getl i c2 (CHead d2 (Bind b) w))) x (\lambda (b0: +B).(\lambda (d0: C).(\lambda (w0: T).(\lambda (h: nat).(\lambda (H3: (getl h +d1 (CHead d0 (Bind b0) w0))).(let H_y \def (getl_trans (S h) c1 (CHead d1 +(Bind b) w) i H0) in (let H_x0 \def (H b0 d0 w0 (plus (S h) i) (H_y (CHead d0 +(Bind b0) w0) (getl_head (Bind b) h d1 (CHead d0 (Bind b0) w0) H3 w))) in +(let H4 \def H_x0 in (ex_ind C (\lambda (d2: C).(getl (S (plus h i)) c2 +(CHead d2 (Bind b0) w0))) (ex C (\lambda (d2: C).(getl h x (CHead d2 (Bind +b0) w0)))) (\lambda (x0: C).(\lambda (H5: (getl (S (plus h i)) c2 (CHead x0 +(Bind b0) w0))).(let H_y0 \def (getl_conf_le (S (plus h i)) (CHead x0 (Bind +b0) w0) c2 H5 (CHead x (Bind b) w) i H2) in (let H6 \def (refl_equal nat +(plus (S h) i)) in (let H7 \def (eq_ind nat (S (plus h i)) (\lambda (n: +nat).(getl (minus n i) (CHead x (Bind b) w) (CHead x0 (Bind b0) w0))) (H_y0 +(le_S i (plus h i) (le_plus_r h i))) (plus (S h) i) H6) in (let H8 \def +(eq_ind nat (minus (plus (S h) i) i) (\lambda (n: nat).(getl n (CHead x (Bind +b) w) (CHead x0 (Bind b0) w0))) H7 (S h) (minus_plus_r (S h) i)) in (ex_intro +C (\lambda (d2: C).(getl h x (CHead d2 (Bind b0) w0))) x0 (getl_gen_S (Bind +b) x (CHead x0 (Bind b0) w0) w h H8)))))))) H4))))))))) H2))) H1)))))))))). +(* COMMENTS +Initial nodes: 673 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/clear/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/clear/defs.ma new file mode 100644 index 000000000..30607a4fc --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/clear/defs.ma @@ -0,0 +1,24 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/C/defs.ma". + +inductive clear: C \to (C \to Prop) \def +| clear_bind: \forall (b: B).(\forall (e: C).(\forall (u: T).(clear (CHead e +(Bind b) u) (CHead e (Bind b) u)))) +| clear_flat: \forall (e: C).(\forall (c: C).((clear e c) \to (\forall (f: +F).(\forall (u: T).(clear (CHead e (Flat f) u) c))))). + diff --git a/matita/matita/contribs/lambdadelta/basic_1/clear/drop.ma b/matita/matita/contribs/lambdadelta/basic_1/clear/drop.ma new file mode 100644 index 000000000..aae3fedc0 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/clear/drop.ma @@ -0,0 +1,181 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/clear/fwd.ma". + +include "Basic-1/drop/fwd.ma". + +theorem drop_clear: + \forall (c1: C).(\forall (c2: C).(\forall (i: nat).((drop (S i) O c1 c2) \to +(ex2_3 B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear c1 (CHead +e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e +c2)))))))) +\def + \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (i: +nat).((drop (S i) O c c2) \to (ex2_3 B C T (\lambda (b: B).(\lambda (e: +C).(\lambda (v: T).(clear c (CHead e (Bind b) v))))) (\lambda (_: B).(\lambda +(e: C).(\lambda (_: T).(drop i O e c2))))))))) (\lambda (n: nat).(\lambda +(c2: C).(\lambda (i: nat).(\lambda (H: (drop (S i) O (CSort n) c2)).(and3_ind +(eq C c2 (CSort n)) (eq nat (S i) O) (eq nat O O) (ex2_3 B C T (\lambda (b: +B).(\lambda (e: C).(\lambda (v: T).(clear (CSort n) (CHead e (Bind b) v))))) +(\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2))))) (\lambda +(_: (eq C c2 (CSort n))).(\lambda (H1: (eq nat (S i) O)).(\lambda (_: (eq nat +O O)).(let H3 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee in nat +return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow +True])) I O H1) in (False_ind (ex2_3 B C T (\lambda (b: B).(\lambda (e: +C).(\lambda (v: T).(clear (CSort n) (CHead e (Bind b) v))))) (\lambda (_: +B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2))))) H3))))) (drop_gen_sort +n (S i) O c2 H)))))) (\lambda (c: C).(\lambda (H: ((\forall (c2: C).(\forall +(i: nat).((drop (S i) O c c2) \to (ex2_3 B C T (\lambda (b: B).(\lambda (e: +C).(\lambda (v: T).(clear c (CHead e (Bind b) v))))) (\lambda (_: B).(\lambda +(e: C).(\lambda (_: T).(drop i O e c2)))))))))).(\lambda (k: K).(\lambda (t: +T).(\lambda (c2: C).(\lambda (i: nat).(\lambda (H0: (drop (S i) O (CHead c k +t) c2)).(K_ind (\lambda (k0: K).((drop (r k0 i) O c c2) \to (ex2_3 B C T +(\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear (CHead c k0 t) (CHead +e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e +c2))))))) (\lambda (b: B).(\lambda (H1: (drop (r (Bind b) i) O c +c2)).(ex2_3_intro B C T (\lambda (b0: B).(\lambda (e: C).(\lambda (v: +T).(clear (CHead c (Bind b) t) (CHead e (Bind b0) v))))) (\lambda (_: +B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2)))) b c t (clear_bind b c +t) H1))) (\lambda (f: F).(\lambda (H1: (drop (r (Flat f) i) O c c2)).(let H2 +\def (H c2 i H1) in (ex2_3_ind B C T (\lambda (b: B).(\lambda (e: C).(\lambda +(v: T).(clear c (CHead e (Bind b) v))))) (\lambda (_: B).(\lambda (e: +C).(\lambda (_: T).(drop i O e c2)))) (ex2_3 B C T (\lambda (b: B).(\lambda +(e: C).(\lambda (v: T).(clear (CHead c (Flat f) t) (CHead e (Bind b) v))))) +(\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2))))) (\lambda +(x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H3: (clear c (CHead x1 +(Bind x0) x2))).(\lambda (H4: (drop i O x1 c2)).(ex2_3_intro B C T (\lambda +(b: B).(\lambda (e: C).(\lambda (v: T).(clear (CHead c (Flat f) t) (CHead e +(Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e +c2)))) x0 x1 x2 (clear_flat c (CHead x1 (Bind x0) x2) H3 f t) H4)))))) H2)))) +k (drop_gen_drop k c c2 t i H0))))))))) c1). +(* COMMENTS +Initial nodes: 770 +END *) + +theorem drop_clear_O: + \forall (b: B).(\forall (c: C).(\forall (e1: C).(\forall (u: T).((clear c +(CHead e1 (Bind b) u)) \to (\forall (e2: C).(\forall (i: nat).((drop i O e1 +e2) \to (drop (S i) O c e2)))))))) +\def + \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e1: +C).(\forall (u: T).((clear c0 (CHead e1 (Bind b) u)) \to (\forall (e2: +C).(\forall (i: nat).((drop i O e1 e2) \to (drop (S i) O c0 e2)))))))) +(\lambda (n: nat).(\lambda (e1: C).(\lambda (u: T).(\lambda (H: (clear (CSort +n) (CHead e1 (Bind b) u))).(\lambda (e2: C).(\lambda (i: nat).(\lambda (_: +(drop i O e1 e2)).(clear_gen_sort (CHead e1 (Bind b) u) n H (drop (S i) O +(CSort n) e2))))))))) (\lambda (c0: C).(\lambda (H: ((\forall (e1: +C).(\forall (u: T).((clear c0 (CHead e1 (Bind b) u)) \to (\forall (e2: +C).(\forall (i: nat).((drop i O e1 e2) \to (drop (S i) O c0 +e2))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e1: C).(\lambda (u: +T).(\lambda (H0: (clear (CHead c0 k t) (CHead e1 (Bind b) u))).(\lambda (e2: +C).(\lambda (i: nat).(\lambda (H1: (drop i O e1 e2)).(K_ind (\lambda (k0: +K).((clear (CHead c0 k0 t) (CHead e1 (Bind b) u)) \to (drop (S i) O (CHead c0 +k0 t) e2))) (\lambda (b0: B).(\lambda (H2: (clear (CHead c0 (Bind b0) t) +(CHead e1 (Bind b) u))).(let H3 \def (f_equal C C (\lambda (e: C).(match e in +C return (\lambda (_: C).C) with [(CSort _) \Rightarrow e1 | (CHead c1 _ _) +\Rightarrow c1])) (CHead e1 (Bind b) u) (CHead c0 (Bind b0) t) +(clear_gen_bind b0 c0 (CHead e1 (Bind b) u) t H2)) in ((let H4 \def (f_equal +C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with [(CSort _) +\Rightarrow b | (CHead _ k0 _) \Rightarrow (match k0 in K return (\lambda (_: +K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow b])])) (CHead e1 +(Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e1 (Bind b) +u) t H2)) in ((let H5 \def (f_equal C T (\lambda (e: C).(match e in C return +(\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow +t0])) (CHead e1 (Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 +(CHead e1 (Bind b) u) t H2)) in (\lambda (H6: (eq B b b0)).(\lambda (H7: (eq +C e1 c0)).(let H8 \def (eq_ind C e1 (\lambda (c1: C).(drop i O c1 e2)) H1 c0 +H7) in (eq_ind B b (\lambda (b1: B).(drop (S i) O (CHead c0 (Bind b1) t) e2)) +(drop_drop (Bind b) i c0 e2 H8 t) b0 H6))))) H4)) H3)))) (\lambda (f: +F).(\lambda (H2: (clear (CHead c0 (Flat f) t) (CHead e1 (Bind b) +u))).(drop_drop (Flat f) i c0 e2 (H e1 u (clear_gen_flat f c0 (CHead e1 (Bind +b) u) t H2) e2 i H1) t))) k H0))))))))))) c)). +(* COMMENTS +Initial nodes: 619 +END *) + +theorem drop_clear_S: + \forall (x2: C).(\forall (x1: C).(\forall (h: nat).(\forall (d: nat).((drop +h (S d) x1 x2) \to (\forall (b: B).(\forall (c2: C).(\forall (u: T).((clear +x2 (CHead c2 (Bind b) u)) \to (ex2 C (\lambda (c1: C).(clear x1 (CHead c1 +(Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2))))))))))) +\def + \lambda (x2: C).(C_ind (\lambda (c: C).(\forall (x1: C).(\forall (h: +nat).(\forall (d: nat).((drop h (S d) x1 c) \to (\forall (b: B).(\forall (c2: +C).(\forall (u: T).((clear c (CHead c2 (Bind b) u)) \to (ex2 C (\lambda (c1: +C).(clear x1 (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 +c2)))))))))))) (\lambda (n: nat).(\lambda (x1: C).(\lambda (h: nat).(\lambda +(d: nat).(\lambda (_: (drop h (S d) x1 (CSort n))).(\lambda (b: B).(\lambda +(c2: C).(\lambda (u: T).(\lambda (H0: (clear (CSort n) (CHead c2 (Bind b) +u))).(clear_gen_sort (CHead c2 (Bind b) u) n H0 (ex2 C (\lambda (c1: +C).(clear x1 (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 +c2))))))))))))) (\lambda (c: C).(\lambda (H: ((\forall (x1: C).(\forall (h: +nat).(\forall (d: nat).((drop h (S d) x1 c) \to (\forall (b: B).(\forall (c2: +C).(\forall (u: T).((clear c (CHead c2 (Bind b) u)) \to (ex2 C (\lambda (c1: +C).(clear x1 (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 +c2))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (x1: C).(\lambda (h: +nat).(\lambda (d: nat).(\lambda (H0: (drop h (S d) x1 (CHead c k +t))).(\lambda (b: B).(\lambda (c2: C).(\lambda (u: T).(\lambda (H1: (clear +(CHead c k t) (CHead c2 (Bind b) u))).(ex2_ind C (\lambda (e: C).(eq C x1 +(CHead e k (lift h (r k d) t)))) (\lambda (e: C).(drop h (r k d) e c)) (ex2 C +(\lambda (c1: C).(clear x1 (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: +C).(drop h d c1 c2))) (\lambda (x: C).(\lambda (H2: (eq C x1 (CHead x k (lift +h (r k d) t)))).(\lambda (H3: (drop h (r k d) x c)).(eq_ind_r C (CHead x k +(lift h (r k d) t)) (\lambda (c0: C).(ex2 C (\lambda (c1: C).(clear c0 (CHead +c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2)))) (K_ind +(\lambda (k0: K).((clear (CHead c k0 t) (CHead c2 (Bind b) u)) \to ((drop h +(r k0 d) x c) \to (ex2 C (\lambda (c1: C).(clear (CHead x k0 (lift h (r k0 d) +t)) (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2)))))) +(\lambda (b0: B).(\lambda (H4: (clear (CHead c (Bind b0) t) (CHead c2 (Bind +b) u))).(\lambda (H5: (drop h (r (Bind b0) d) x c)).(let H6 \def (f_equal C C +(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) +\Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) (CHead c2 (Bind b) u) +(CHead c (Bind b0) t) (clear_gen_bind b0 c (CHead c2 (Bind b) u) t H4)) in +((let H7 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: +C).B) with [(CSort _) \Rightarrow b | (CHead _ k0 _) \Rightarrow (match k0 in +K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) +\Rightarrow b])])) (CHead c2 (Bind b) u) (CHead c (Bind b0) t) +(clear_gen_bind b0 c (CHead c2 (Bind b) u) t H4)) in ((let H8 \def (f_equal C +T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) +\Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead c2 (Bind b) u) (CHead +c (Bind b0) t) (clear_gen_bind b0 c (CHead c2 (Bind b) u) t H4)) in (\lambda +(H9: (eq B b b0)).(\lambda (H10: (eq C c2 c)).(eq_ind_r T t (\lambda (t0: +T).(ex2 C (\lambda (c1: C).(clear (CHead x (Bind b0) (lift h (r (Bind b0) d) +t)) (CHead c1 (Bind b) (lift h d t0)))) (\lambda (c1: C).(drop h d c1 c2)))) +(eq_ind_r C c (\lambda (c0: C).(ex2 C (\lambda (c1: C).(clear (CHead x (Bind +b0) (lift h (r (Bind b0) d) t)) (CHead c1 (Bind b) (lift h d t)))) (\lambda +(c1: C).(drop h d c1 c0)))) (eq_ind_r B b0 (\lambda (b1: B).(ex2 C (\lambda +(c1: C).(clear (CHead x (Bind b0) (lift h (r (Bind b0) d) t)) (CHead c1 (Bind +b1) (lift h d t)))) (\lambda (c1: C).(drop h d c1 c)))) (ex_intro2 C (\lambda +(c1: C).(clear (CHead x (Bind b0) (lift h (r (Bind b0) d) t)) (CHead c1 (Bind +b0) (lift h d t)))) (\lambda (c1: C).(drop h d c1 c)) x (clear_bind b0 x +(lift h d t)) H5) b H9) c2 H10) u H8)))) H7)) H6))))) (\lambda (f: +F).(\lambda (H4: (clear (CHead c (Flat f) t) (CHead c2 (Bind b) u))).(\lambda +(H5: (drop h (r (Flat f) d) x c)).(let H6 \def (H x h d H5 b c2 u +(clear_gen_flat f c (CHead c2 (Bind b) u) t H4)) in (ex2_ind C (\lambda (c1: +C).(clear x (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 +c2)) (ex2 C (\lambda (c1: C).(clear (CHead x (Flat f) (lift h (r (Flat f) d) +t)) (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2))) +(\lambda (x0: C).(\lambda (H7: (clear x (CHead x0 (Bind b) (lift h d +u)))).(\lambda (H8: (drop h d x0 c2)).(ex_intro2 C (\lambda (c1: C).(clear +(CHead x (Flat f) (lift h (r (Flat f) d) t)) (CHead c1 (Bind b) (lift h d +u)))) (\lambda (c1: C).(drop h d c1 c2)) x0 (clear_flat x (CHead x0 (Bind b) +(lift h d u)) H7 f (lift h (r (Flat f) d) t)) H8)))) H6))))) k H1 H3) x1 +H2)))) (drop_gen_skip_r c x1 t h d k H0)))))))))))))) x2). +(* COMMENTS +Initial nodes: 1449 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/clear/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/clear/fwd.ma new file mode 100644 index 000000000..d64ff77be --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/clear/fwd.ma @@ -0,0 +1,164 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/clear/defs.ma". + +theorem clear_gen_sort: + \forall (x: C).(\forall (n: nat).((clear (CSort n) x) \to (\forall (P: +Prop).P))) +\def + \lambda (x: C).(\lambda (n: nat).(\lambda (H: (clear (CSort n) x)).(\lambda +(P: Prop).(insert_eq C (CSort n) (\lambda (c: C).(clear c x)) (\lambda (_: +C).P) (\lambda (y: C).(\lambda (H0: (clear y x)).(clear_ind (\lambda (c: +C).(\lambda (_: C).((eq C c (CSort n)) \to P))) (\lambda (b: B).(\lambda (e: +C).(\lambda (u: T).(\lambda (H1: (eq C (CHead e (Bind b) u) (CSort n))).(let +H2 \def (eq_ind C (CHead e (Bind b) u) (\lambda (ee: C).(match ee in C return +(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) +\Rightarrow True])) I (CSort n) H1) in (False_ind P H2)))))) (\lambda (e: +C).(\lambda (c: C).(\lambda (_: (clear e c)).(\lambda (_: (((eq C e (CSort +n)) \to P))).(\lambda (f: F).(\lambda (u: T).(\lambda (H3: (eq C (CHead e +(Flat f) u) (CSort n))).(let H4 \def (eq_ind C (CHead e (Flat f) u) (\lambda +(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) +\Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H3) in +(False_ind P H4))))))))) y x H0))) H)))). +(* COMMENTS +Initial nodes: 215 +END *) + +theorem clear_gen_bind: + \forall (b: B).(\forall (e: C).(\forall (x: C).(\forall (u: T).((clear +(CHead e (Bind b) u) x) \to (eq C x (CHead e (Bind b) u)))))) +\def + \lambda (b: B).(\lambda (e: C).(\lambda (x: C).(\lambda (u: T).(\lambda (H: +(clear (CHead e (Bind b) u) x)).(insert_eq C (CHead e (Bind b) u) (\lambda +(c: C).(clear c x)) (\lambda (c: C).(eq C x c)) (\lambda (y: C).(\lambda (H0: +(clear y x)).(clear_ind (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead e +(Bind b) u)) \to (eq C c0 c)))) (\lambda (b0: B).(\lambda (e0: C).(\lambda +(u0: T).(\lambda (H1: (eq C (CHead e0 (Bind b0) u0) (CHead e (Bind b) +u))).(let H2 \def (f_equal C C (\lambda (e1: C).(match e1 in C return +(\lambda (_: C).C) with [(CSort _) \Rightarrow e0 | (CHead c _ _) \Rightarrow +c])) (CHead e0 (Bind b0) u0) (CHead e (Bind b) u) H1) in ((let H3 \def +(f_equal C B (\lambda (e1: C).(match e1 in C return (\lambda (_: C).B) with +[(CSort _) \Rightarrow b0 | (CHead _ k _) \Rightarrow (match k in K return +(\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow +b0])])) (CHead e0 (Bind b0) u0) (CHead e (Bind b) u) H1) in ((let H4 \def +(f_equal C T (\lambda (e1: C).(match e1 in C return (\lambda (_: C).T) with +[(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead e0 (Bind +b0) u0) (CHead e (Bind b) u) H1) in (\lambda (H5: (eq B b0 b)).(\lambda (H6: +(eq C e0 e)).(eq_ind_r T u (\lambda (t: T).(eq C (CHead e0 (Bind b0) t) +(CHead e0 (Bind b0) t))) (eq_ind_r C e (\lambda (c: C).(eq C (CHead c (Bind +b0) u) (CHead c (Bind b0) u))) (eq_ind_r B b (\lambda (b1: B).(eq C (CHead e +(Bind b1) u) (CHead e (Bind b1) u))) (refl_equal C (CHead e (Bind b) u)) b0 +H5) e0 H6) u0 H4)))) H3)) H2)))))) (\lambda (e0: C).(\lambda (c: C).(\lambda +(_: (clear e0 c)).(\lambda (_: (((eq C e0 (CHead e (Bind b) u)) \to (eq C c +e0)))).(\lambda (f: F).(\lambda (u0: T).(\lambda (H3: (eq C (CHead e0 (Flat +f) u0) (CHead e (Bind b) u))).(let H4 \def (eq_ind C (CHead e0 (Flat f) u0) +(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) +\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda +(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow +True])])) I (CHead e (Bind b) u) H3) in (False_ind (eq C c (CHead e0 (Flat f) +u0)) H4))))))))) y x H0))) H))))). +(* COMMENTS +Initial nodes: 525 +END *) + +theorem clear_gen_flat: + \forall (f: F).(\forall (e: C).(\forall (x: C).(\forall (u: T).((clear +(CHead e (Flat f) u) x) \to (clear e x))))) +\def + \lambda (f: F).(\lambda (e: C).(\lambda (x: C).(\lambda (u: T).(\lambda (H: +(clear (CHead e (Flat f) u) x)).(insert_eq C (CHead e (Flat f) u) (\lambda +(c: C).(clear c x)) (\lambda (_: C).(clear e x)) (\lambda (y: C).(\lambda +(H0: (clear y x)).(clear_ind (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead +e (Flat f) u)) \to (clear e c0)))) (\lambda (b: B).(\lambda (e0: C).(\lambda +(u0: T).(\lambda (H1: (eq C (CHead e0 (Bind b) u0) (CHead e (Flat f) +u))).(let H2 \def (eq_ind C (CHead e0 (Bind b) u0) (\lambda (ee: C).(match ee +in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead +_ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) +\Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead e (Flat f) u) H1) +in (False_ind (clear e (CHead e0 (Bind b) u0)) H2)))))) (\lambda (e0: +C).(\lambda (c: C).(\lambda (H1: (clear e0 c)).(\lambda (H2: (((eq C e0 +(CHead e (Flat f) u)) \to (clear e c)))).(\lambda (f0: F).(\lambda (u0: +T).(\lambda (H3: (eq C (CHead e0 (Flat f0) u0) (CHead e (Flat f) u))).(let H4 +\def (f_equal C C (\lambda (e1: C).(match e1 in C return (\lambda (_: C).C) +with [(CSort _) \Rightarrow e0 | (CHead c0 _ _) \Rightarrow c0])) (CHead e0 +(Flat f0) u0) (CHead e (Flat f) u) H3) in ((let H5 \def (f_equal C F (\lambda +(e1: C).(match e1 in C return (\lambda (_: C).F) with [(CSort _) \Rightarrow +f0 | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).F) with +[(Bind _) \Rightarrow f0 | (Flat f1) \Rightarrow f1])])) (CHead e0 (Flat f0) +u0) (CHead e (Flat f) u) H3) in ((let H6 \def (f_equal C T (\lambda (e1: +C).(match e1 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | +(CHead _ _ t) \Rightarrow t])) (CHead e0 (Flat f0) u0) (CHead e (Flat f) u) +H3) in (\lambda (_: (eq F f0 f)).(\lambda (H8: (eq C e0 e)).(let H9 \def +(eq_ind C e0 (\lambda (c0: C).((eq C c0 (CHead e (Flat f) u)) \to (clear e +c))) H2 e H8) in (let H10 \def (eq_ind C e0 (\lambda (c0: C).(clear c0 c)) H1 +e H8) in H10))))) H5)) H4))))))))) y x H0))) H))))). +(* COMMENTS +Initial nodes: 453 +END *) + +theorem clear_gen_flat_r: + \forall (f: F).(\forall (x: C).(\forall (e: C).(\forall (u: T).((clear x +(CHead e (Flat f) u)) \to (\forall (P: Prop).P))))) +\def + \lambda (f: F).(\lambda (x: C).(\lambda (e: C).(\lambda (u: T).(\lambda (H: +(clear x (CHead e (Flat f) u))).(\lambda (P: Prop).(insert_eq C (CHead e +(Flat f) u) (\lambda (c: C).(clear x c)) (\lambda (_: C).P) (\lambda (y: +C).(\lambda (H0: (clear x y)).(clear_ind (\lambda (_: C).(\lambda (c0: +C).((eq C c0 (CHead e (Flat f) u)) \to P))) (\lambda (b: B).(\lambda (e0: +C).(\lambda (u0: T).(\lambda (H1: (eq C (CHead e0 (Bind b) u0) (CHead e (Flat +f) u))).(let H2 \def (eq_ind C (CHead e0 (Bind b) u0) (\lambda (ee: C).(match +ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | +(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with +[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead e (Flat +f) u) H1) in (False_ind P H2)))))) (\lambda (e0: C).(\lambda (c: C).(\lambda +(H1: (clear e0 c)).(\lambda (H2: (((eq C c (CHead e (Flat f) u)) \to +P))).(\lambda (_: F).(\lambda (_: T).(\lambda (H3: (eq C c (CHead e (Flat f) +u))).(let H4 \def (eq_ind C c (\lambda (c0: C).((eq C c0 (CHead e (Flat f) +u)) \to P)) H2 (CHead e (Flat f) u) H3) in (let H5 \def (eq_ind C c (\lambda +(c0: C).(clear e0 c0)) H1 (CHead e (Flat f) u) H3) in (H4 (refl_equal C +(CHead e (Flat f) u)))))))))))) x y H0))) H)))))). +(* COMMENTS +Initial nodes: 303 +END *) + +theorem clear_gen_all: + \forall (c1: C).(\forall (c2: C).((clear c1 c2) \to (ex_3 B C T (\lambda (b: +B).(\lambda (e: C).(\lambda (u: T).(eq C c2 (CHead e (Bind b) u)))))))) +\def + \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (clear c1 c2)).(clear_ind +(\lambda (_: C).(\lambda (c0: C).(ex_3 B C T (\lambda (b: B).(\lambda (e: +C).(\lambda (u: T).(eq C c0 (CHead e (Bind b) u)))))))) (\lambda (b: +B).(\lambda (e: C).(\lambda (u: T).(ex_3_intro B C T (\lambda (b0: +B).(\lambda (e0: C).(\lambda (u0: T).(eq C (CHead e (Bind b) u) (CHead e0 +(Bind b0) u0))))) b e u (refl_equal C (CHead e (Bind b) u)))))) (\lambda (e: +C).(\lambda (c: C).(\lambda (H0: (clear e c)).(\lambda (H1: (ex_3 B C T +(\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(eq C c (CHead e0 (Bind b) +u))))))).(\lambda (_: F).(\lambda (_: T).(let H2 \def H1 in (ex_3_ind B C T +(\lambda (b: B).(\lambda (e0: C).(\lambda (u0: T).(eq C c (CHead e0 (Bind b) +u0))))) (ex_3 B C T (\lambda (b: B).(\lambda (e0: C).(\lambda (u0: T).(eq C c +(CHead e0 (Bind b) u0)))))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: +T).(\lambda (H3: (eq C c (CHead x1 (Bind x0) x2))).(let H4 \def (eq_ind C c +(\lambda (c0: C).(clear e c0)) H0 (CHead x1 (Bind x0) x2) H3) in (eq_ind_r C +(CHead x1 (Bind x0) x2) (\lambda (c0: C).(ex_3 B C T (\lambda (b: B).(\lambda +(e0: C).(\lambda (u0: T).(eq C c0 (CHead e0 (Bind b) u0))))))) (ex_3_intro B +C T (\lambda (b: B).(\lambda (e0: C).(\lambda (u0: T).(eq C (CHead x1 (Bind +x0) x2) (CHead e0 (Bind b) u0))))) x0 x1 x2 (refl_equal C (CHead x1 (Bind x0) +x2))) c H3)))))) H2)))))))) c1 c2 H))). +(* COMMENTS +Initial nodes: 381 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/clear/props.ma b/matita/matita/contribs/lambdadelta/basic_1/clear/props.ma new file mode 100644 index 000000000..68e250d76 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/clear/props.ma @@ -0,0 +1,152 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/clear/fwd.ma". + +theorem clear_clear: + \forall (c1: C).(\forall (c2: C).((clear c1 c2) \to (clear c2 c2))) +\def + \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).((clear c c2) \to +(clear c2 c2)))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (H: (clear +(CSort n) c2)).(clear_gen_sort c2 n H (clear c2 c2))))) (\lambda (c: +C).(\lambda (H: ((\forall (c2: C).((clear c c2) \to (clear c2 +c2))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (H0: (clear +(CHead c k t) c2)).(K_ind (\lambda (k0: K).((clear (CHead c k0 t) c2) \to +(clear c2 c2))) (\lambda (b: B).(\lambda (H1: (clear (CHead c (Bind b) t) +c2)).(eq_ind_r C (CHead c (Bind b) t) (\lambda (c0: C).(clear c0 c0)) +(clear_bind b c t) c2 (clear_gen_bind b c c2 t H1)))) (\lambda (f: +F).(\lambda (H1: (clear (CHead c (Flat f) t) c2)).(H c2 (clear_gen_flat f c +c2 t H1)))) k H0))))))) c1). +(* COMMENTS +Initial nodes: 199 +END *) + +theorem clear_mono: + \forall (c: C).(\forall (c1: C).((clear c c1) \to (\forall (c2: C).((clear c +c2) \to (eq C c1 c2))))) +\def + \lambda (c: C).(C_ind (\lambda (c0: C).(\forall (c1: C).((clear c0 c1) \to +(\forall (c2: C).((clear c0 c2) \to (eq C c1 c2)))))) (\lambda (n: +nat).(\lambda (c1: C).(\lambda (_: (clear (CSort n) c1)).(\lambda (c2: +C).(\lambda (H0: (clear (CSort n) c2)).(clear_gen_sort c2 n H0 (eq C c1 +c2))))))) (\lambda (c0: C).(\lambda (H: ((\forall (c1: C).((clear c0 c1) \to +(\forall (c2: C).((clear c0 c2) \to (eq C c1 c2))))))).(\lambda (k: +K).(\lambda (t: T).(\lambda (c1: C).(\lambda (H0: (clear (CHead c0 k t) +c1)).(\lambda (c2: C).(\lambda (H1: (clear (CHead c0 k t) c2)).(K_ind +(\lambda (k0: K).((clear (CHead c0 k0 t) c1) \to ((clear (CHead c0 k0 t) c2) +\to (eq C c1 c2)))) (\lambda (b: B).(\lambda (H2: (clear (CHead c0 (Bind b) +t) c1)).(\lambda (H3: (clear (CHead c0 (Bind b) t) c2)).(eq_ind_r C (CHead c0 +(Bind b) t) (\lambda (c3: C).(eq C c1 c3)) (eq_ind_r C (CHead c0 (Bind b) t) +(\lambda (c3: C).(eq C c3 (CHead c0 (Bind b) t))) (refl_equal C (CHead c0 +(Bind b) t)) c1 (clear_gen_bind b c0 c1 t H2)) c2 (clear_gen_bind b c0 c2 t +H3))))) (\lambda (f: F).(\lambda (H2: (clear (CHead c0 (Flat f) t) +c1)).(\lambda (H3: (clear (CHead c0 (Flat f) t) c2)).(H c1 (clear_gen_flat f +c0 c1 t H2) c2 (clear_gen_flat f c0 c2 t H3))))) k H0 H1))))))))) c). +(* COMMENTS +Initial nodes: 357 +END *) + +theorem clear_trans: + \forall (c1: C).(\forall (c: C).((clear c1 c) \to (\forall (c2: C).((clear c +c2) \to (clear c1 c2))))) +\def + \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c0: C).((clear c c0) \to +(\forall (c2: C).((clear c0 c2) \to (clear c c2)))))) (\lambda (n: +nat).(\lambda (c: C).(\lambda (H: (clear (CSort n) c)).(\lambda (c2: +C).(\lambda (_: (clear c c2)).(clear_gen_sort c n H (clear (CSort n) +c2))))))) (\lambda (c: C).(\lambda (H: ((\forall (c0: C).((clear c c0) \to +(\forall (c2: C).((clear c0 c2) \to (clear c c2))))))).(\lambda (k: +K).(\lambda (t: T).(\lambda (c0: C).(\lambda (H0: (clear (CHead c k t) +c0)).(\lambda (c2: C).(\lambda (H1: (clear c0 c2)).(K_ind (\lambda (k0: +K).((clear (CHead c k0 t) c0) \to (clear (CHead c k0 t) c2))) (\lambda (b: +B).(\lambda (H2: (clear (CHead c (Bind b) t) c0)).(let H3 \def (eq_ind C c0 +(\lambda (c3: C).(clear c3 c2)) H1 (CHead c (Bind b) t) (clear_gen_bind b c +c0 t H2)) in (eq_ind_r C (CHead c (Bind b) t) (\lambda (c3: C).(clear (CHead +c (Bind b) t) c3)) (clear_bind b c t) c2 (clear_gen_bind b c c2 t H3))))) +(\lambda (f: F).(\lambda (H2: (clear (CHead c (Flat f) t) c0)).(clear_flat c +c2 (H c0 (clear_gen_flat f c c0 t H2) c2 H1) f t))) k H0))))))))) c1). +(* COMMENTS +Initial nodes: 299 +END *) + +theorem clear_ctail: + \forall (b: B).(\forall (c1: C).(\forall (c2: C).(\forall (u2: T).((clear c1 +(CHead c2 (Bind b) u2)) \to (\forall (k: K).(\forall (u1: T).(clear (CTail k +u1 c1) (CHead (CTail k u1 c2) (Bind b) u2)))))))) +\def + \lambda (b: B).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: +C).(\forall (u2: T).((clear c (CHead c2 (Bind b) u2)) \to (\forall (k: +K).(\forall (u1: T).(clear (CTail k u1 c) (CHead (CTail k u1 c2) (Bind b) +u2)))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (u2: T).(\lambda (H: +(clear (CSort n) (CHead c2 (Bind b) u2))).(\lambda (k: K).(\lambda (u1: +T).(K_ind (\lambda (k0: K).(clear (CHead (CSort n) k0 u1) (CHead (CTail k0 u1 +c2) (Bind b) u2))) (\lambda (b0: B).(clear_gen_sort (CHead c2 (Bind b) u2) n +H (clear (CHead (CSort n) (Bind b0) u1) (CHead (CTail (Bind b0) u1 c2) (Bind +b) u2)))) (\lambda (f: F).(clear_gen_sort (CHead c2 (Bind b) u2) n H (clear +(CHead (CSort n) (Flat f) u1) (CHead (CTail (Flat f) u1 c2) (Bind b) u2)))) +k))))))) (\lambda (c: C).(\lambda (H: ((\forall (c2: C).(\forall (u2: +T).((clear c (CHead c2 (Bind b) u2)) \to (\forall (k: K).(\forall (u1: +T).(clear (CTail k u1 c) (CHead (CTail k u1 c2) (Bind b) u2))))))))).(\lambda +(k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (u2: T).(\lambda (H0: (clear +(CHead c k t) (CHead c2 (Bind b) u2))).(\lambda (k0: K).(\lambda (u1: +T).(K_ind (\lambda (k1: K).((clear (CHead c k1 t) (CHead c2 (Bind b) u2)) \to +(clear (CHead (CTail k0 u1 c) k1 t) (CHead (CTail k0 u1 c2) (Bind b) u2)))) +(\lambda (b0: B).(\lambda (H1: (clear (CHead c (Bind b0) t) (CHead c2 (Bind +b) u2))).(let H2 \def (f_equal C C (\lambda (e: C).(match e in C return +(\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ _) +\Rightarrow c0])) (CHead c2 (Bind b) u2) (CHead c (Bind b0) t) +(clear_gen_bind b0 c (CHead c2 (Bind b) u2) t H1)) in ((let H3 \def (f_equal +C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with [(CSort _) +\Rightarrow b | (CHead _ k1 _) \Rightarrow (match k1 in K return (\lambda (_: +K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow b])])) (CHead c2 +(Bind b) u2) (CHead c (Bind b0) t) (clear_gen_bind b0 c (CHead c2 (Bind b) +u2) t H1)) in ((let H4 \def (f_equal C T (\lambda (e: C).(match e in C return +(\lambda (_: C).T) with [(CSort _) \Rightarrow u2 | (CHead _ _ t0) +\Rightarrow t0])) (CHead c2 (Bind b) u2) (CHead c (Bind b0) t) +(clear_gen_bind b0 c (CHead c2 (Bind b) u2) t H1)) in (\lambda (H5: (eq B b +b0)).(\lambda (H6: (eq C c2 c)).(eq_ind_r T t (\lambda (t0: T).(clear (CHead +(CTail k0 u1 c) (Bind b0) t) (CHead (CTail k0 u1 c2) (Bind b) t0))) (eq_ind_r +C c (\lambda (c0: C).(clear (CHead (CTail k0 u1 c) (Bind b0) t) (CHead (CTail +k0 u1 c0) (Bind b) t))) (eq_ind B b (\lambda (b1: B).(clear (CHead (CTail k0 +u1 c) (Bind b1) t) (CHead (CTail k0 u1 c) (Bind b) t))) (clear_bind b (CTail +k0 u1 c) t) b0 H5) c2 H6) u2 H4)))) H3)) H2)))) (\lambda (f: F).(\lambda (H1: +(clear (CHead c (Flat f) t) (CHead c2 (Bind b) u2))).(clear_flat (CTail k0 u1 +c) (CHead (CTail k0 u1 c2) (Bind b) u2) (H c2 u2 (clear_gen_flat f c (CHead +c2 (Bind b) u2) t H1) k0 u1) f t))) k H0)))))))))) c1)). +(* COMMENTS +Initial nodes: 819 +END *) + +theorem clear_cle: + \forall (c1: C).(\forall (c2: C).((clear c1 c2) \to (cle c2 c1))) +\def + \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).((clear c c2) \to +(le (cweight c2) (cweight c))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda +(H: (clear (CSort n) c2)).(clear_gen_sort c2 n H (le (cweight c2) O))))) +(\lambda (c: C).(\lambda (H: ((\forall (c2: C).((clear c c2) \to (le (cweight +c2) (cweight c)))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: +C).(\lambda (H0: (clear (CHead c k t) c2)).(K_ind (\lambda (k0: K).((clear +(CHead c k0 t) c2) \to (le (cweight c2) (plus (cweight c) (tweight t))))) +(\lambda (b: B).(\lambda (H1: (clear (CHead c (Bind b) t) c2)).(eq_ind_r C +(CHead c (Bind b) t) (\lambda (c0: C).(le (cweight c0) (plus (cweight c) +(tweight t)))) (le_n (plus (cweight c) (tweight t))) c2 (clear_gen_bind b c +c2 t H1)))) (\lambda (f: F).(\lambda (H1: (clear (CHead c (Flat f) t) +c2)).(le_plus_trans (cweight c2) (cweight c) (tweight t) (H c2 +(clear_gen_flat f c c2 t H1))))) k H0))))))) c1). +(* COMMENTS +Initial nodes: 247 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/clen/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/clen/defs.ma new file mode 100644 index 000000000..07539ffad --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/clen/defs.ma @@ -0,0 +1,26 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/C/defs.ma". + +include "Basic-1/s/defs.ma". + +definition clen: + C \to nat +\def + let rec clen (c: C) on c: nat \def (match c with [(CSort _) \Rightarrow O | +(CHead c0 k _) \Rightarrow (s k (clen c0))]) in clen. + diff --git a/matita/matita/contribs/lambdadelta/basic_1/clen/getl.ma b/matita/matita/contribs/lambdadelta/basic_1/clen/getl.ma new file mode 100644 index 000000000..af8a96cf5 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/clen/getl.ma @@ -0,0 +1,361 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/clen/defs.ma". + +include "Basic-1/getl/props.ma". + +theorem getl_ctail_clen: + \forall (b: B).(\forall (t: T).(\forall (c: C).(ex nat (\lambda (n: +nat).(getl (clen c) (CTail (Bind b) t c) (CHead (CSort n) (Bind b) t)))))) +\def + \lambda (b: B).(\lambda (t: T).(\lambda (c: C).(C_ind (\lambda (c0: C).(ex +nat (\lambda (n: nat).(getl (clen c0) (CTail (Bind b) t c0) (CHead (CSort n) +(Bind b) t))))) (\lambda (n: nat).(ex_intro nat (\lambda (n0: nat).(getl O +(CHead (CSort n) (Bind b) t) (CHead (CSort n0) (Bind b) t))) n (getl_refl b +(CSort n) t))) (\lambda (c0: C).(\lambda (H: (ex nat (\lambda (n: nat).(getl +(clen c0) (CTail (Bind b) t c0) (CHead (CSort n) (Bind b) t))))).(\lambda (k: +K).(\lambda (t0: T).(let H0 \def H in (ex_ind nat (\lambda (n: nat).(getl +(clen c0) (CTail (Bind b) t c0) (CHead (CSort n) (Bind b) t))) (ex nat +(\lambda (n: nat).(getl (s k (clen c0)) (CHead (CTail (Bind b) t c0) k t0) +(CHead (CSort n) (Bind b) t)))) (\lambda (x: nat).(\lambda (H1: (getl (clen +c0) (CTail (Bind b) t c0) (CHead (CSort x) (Bind b) t))).(K_ind (\lambda (k0: +K).(ex nat (\lambda (n: nat).(getl (s k0 (clen c0)) (CHead (CTail (Bind b) t +c0) k0 t0) (CHead (CSort n) (Bind b) t))))) (\lambda (b0: B).(ex_intro nat +(\lambda (n: nat).(getl (S (clen c0)) (CHead (CTail (Bind b) t c0) (Bind b0) +t0) (CHead (CSort n) (Bind b) t))) x (getl_head (Bind b0) (clen c0) (CTail +(Bind b) t c0) (CHead (CSort x) (Bind b) t) H1 t0))) (\lambda (f: +F).(ex_intro nat (\lambda (n: nat).(getl (clen c0) (CHead (CTail (Bind b) t +c0) (Flat f) t0) (CHead (CSort n) (Bind b) t))) x (getl_flat (CTail (Bind b) +t c0) (CHead (CSort x) (Bind b) t) (clen c0) H1 f t0))) k))) H0)))))) c))). +(* COMMENTS +Initial nodes: 459 +END *) + +theorem getl_gen_tail: + \forall (k: K).(\forall (b: B).(\forall (u1: T).(\forall (u2: T).(\forall +(c2: C).(\forall (c1: C).(\forall (i: nat).((getl i (CTail k u1 c1) (CHead c2 +(Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) +(\lambda (e: C).(getl i c1 (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: +nat).(eq nat i (clen c1))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: +nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n)))))))))))) +\def + \lambda (k: K).(\lambda (b: B).(\lambda (u1: T).(\lambda (u2: T).(\lambda +(c2: C).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (i: nat).((getl i +(CTail k u1 c) (CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C +c2 (CTail k u1 e))) (\lambda (e: C).(getl i c (CHead e (Bind b) u2)))) (ex4 +nat (\lambda (_: nat).(eq nat i (clen c))) (\lambda (_: nat).(eq K k (Bind +b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort +n)))))))) (\lambda (n: nat).(\lambda (i: nat).(nat_ind (\lambda (n0: +nat).((getl n0 (CTail k u1 (CSort n)) (CHead c2 (Bind b) u2)) \to (or (ex2 C +(\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl n0 (CSort n) +(CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat n0 (clen (CSort +n)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) +(\lambda (n1: nat).(eq C c2 (CSort n1))))))) (\lambda (H: (getl O (CHead +(CSort n) k u1) (CHead c2 (Bind b) u2))).(K_ind (\lambda (k0: K).((clear +(CHead (CSort n) k0 u1) (CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda (e: +C).(eq C c2 (CTail k0 u1 e))) (\lambda (e: C).(getl O (CSort n) (CHead e +(Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O O)) (\lambda (_: +nat).(eq K k0 (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: +nat).(eq C c2 (CSort n0))))))) (\lambda (b0: B).(\lambda (H0: (clear (CHead +(CSort n) (Bind b0) u1) (CHead c2 (Bind b) u2))).(let H1 \def (f_equal C C +(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) +\Rightarrow c2 | (CHead c _ _) \Rightarrow c])) (CHead c2 (Bind b) u2) (CHead +(CSort n) (Bind b0) u1) (clear_gen_bind b0 (CSort n) (CHead c2 (Bind b) u2) +u1 H0)) in ((let H2 \def (f_equal C B (\lambda (e: C).(match e in C return +(\lambda (_: C).B) with [(CSort _) \Rightarrow b | (CHead _ k0 _) \Rightarrow +(match k0 in K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | +(Flat _) \Rightarrow b])])) (CHead c2 (Bind b) u2) (CHead (CSort n) (Bind b0) +u1) (clear_gen_bind b0 (CSort n) (CHead c2 (Bind b) u2) u1 H0)) in ((let H3 +\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) +with [(CSort _) \Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c2 +(Bind b) u2) (CHead (CSort n) (Bind b0) u1) (clear_gen_bind b0 (CSort n) +(CHead c2 (Bind b) u2) u1 H0)) in (\lambda (H4: (eq B b b0)).(\lambda (H5: +(eq C c2 (CSort n))).(eq_ind_r C (CSort n) (\lambda (c: C).(or (ex2 C +(\lambda (e: C).(eq C c (CTail (Bind b0) u1 e))) (\lambda (e: C).(getl O +(CSort n) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O O)) +(\lambda (_: nat).(eq K (Bind b0) (Bind b))) (\lambda (_: nat).(eq T u1 u2)) +(\lambda (n0: nat).(eq C c (CSort n0)))))) (eq_ind_r T u1 (\lambda (t: T).(or +(ex2 C (\lambda (e: C).(eq C (CSort n) (CTail (Bind b0) u1 e))) (\lambda (e: +C).(getl O (CSort n) (CHead e (Bind b) t)))) (ex4 nat (\lambda (_: nat).(eq +nat O O)) (\lambda (_: nat).(eq K (Bind b0) (Bind b))) (\lambda (_: nat).(eq +T u1 t)) (\lambda (n0: nat).(eq C (CSort n) (CSort n0)))))) (eq_ind_r B b0 +(\lambda (b1: B).(or (ex2 C (\lambda (e: C).(eq C (CSort n) (CTail (Bind b0) +u1 e))) (\lambda (e: C).(getl O (CSort n) (CHead e (Bind b1) u1)))) (ex4 nat +(\lambda (_: nat).(eq nat O O)) (\lambda (_: nat).(eq K (Bind b0) (Bind b1))) +(\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: nat).(eq C (CSort n) (CSort +n0)))))) (or_intror (ex2 C (\lambda (e: C).(eq C (CSort n) (CTail (Bind b0) +u1 e))) (\lambda (e: C).(getl O (CSort n) (CHead e (Bind b0) u1)))) (ex4 nat +(\lambda (_: nat).(eq nat O O)) (\lambda (_: nat).(eq K (Bind b0) (Bind b0))) +(\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: nat).(eq C (CSort n) (CSort +n0)))) (ex4_intro nat (\lambda (_: nat).(eq nat O O)) (\lambda (_: nat).(eq K +(Bind b0) (Bind b0))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: nat).(eq +C (CSort n) (CSort n0))) n (refl_equal nat O) (refl_equal K (Bind b0)) +(refl_equal T u1) (refl_equal C (CSort n)))) b H4) u2 H3) c2 H5)))) H2)) +H1)))) (\lambda (f: F).(\lambda (H0: (clear (CHead (CSort n) (Flat f) u1) +(CHead c2 (Bind b) u2))).(clear_gen_sort (CHead c2 (Bind b) u2) n +(clear_gen_flat f (CSort n) (CHead c2 (Bind b) u2) u1 H0) (or (ex2 C (\lambda +(e: C).(eq C c2 (CTail (Flat f) u1 e))) (\lambda (e: C).(getl O (CSort n) +(CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O O)) (\lambda +(_: nat).(eq K (Flat f) (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda +(n0: nat).(eq C c2 (CSort n0)))))))) k (getl_gen_O (CHead (CSort n) k u1) +(CHead c2 (Bind b) u2) H))) (\lambda (n0: nat).(\lambda (_: (((getl n0 (CHead +(CSort n) k u1) (CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C +c2 (CTail k u1 e))) (\lambda (e: C).(getl n0 (CSort n) (CHead e (Bind b) +u2)))) (ex4 nat (\lambda (_: nat).(eq nat n0 O)) (\lambda (_: nat).(eq K k +(Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n1: nat).(eq C c2 (CSort +n1)))))))).(\lambda (H0: (getl (S n0) (CHead (CSort n) k u1) (CHead c2 (Bind +b) u2))).(getl_gen_sort n (r k n0) (CHead c2 (Bind b) u2) (getl_gen_S k +(CSort n) (CHead c2 (Bind b) u2) u1 n0 H0) (or (ex2 C (\lambda (e: C).(eq C +c2 (CTail k u1 e))) (\lambda (e: C).(getl (S n0) (CSort n) (CHead e (Bind b) +u2)))) (ex4 nat (\lambda (_: nat).(eq nat (S n0) O)) (\lambda (_: nat).(eq K +k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n1: nat).(eq C c2 +(CSort n1))))))))) i))) (\lambda (c: C).(\lambda (H: ((\forall (i: +nat).((getl i (CTail k u1 c) (CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda +(e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl i c (CHead e (Bind b) +u2)))) (ex4 nat (\lambda (_: nat).(eq nat i (clen c))) (\lambda (_: nat).(eq +K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 +(CSort n))))))))).(\lambda (k0: K).(\lambda (t: T).(\lambda (i: nat).(nat_ind +(\lambda (n: nat).((getl n (CTail k u1 (CHead c k0 t)) (CHead c2 (Bind b) +u2)) \to (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: +C).(getl n (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: +nat).(eq nat n (clen (CHead c k0 t)))) (\lambda (_: nat).(eq K k (Bind b))) +(\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0))))))) +(\lambda (H0: (getl O (CHead (CTail k u1 c) k0 t) (CHead c2 (Bind b) +u2))).(K_ind (\lambda (k1: K).((clear (CHead (CTail k u1 c) k1 t) (CHead c2 +(Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) +(\lambda (e: C).(getl O (CHead c k1 t) (CHead e (Bind b) u2)))) (ex4 nat +(\lambda (_: nat).(eq nat O (s k1 (clen c)))) (\lambda (_: nat).(eq K k (Bind +b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort +n))))))) (\lambda (b0: B).(\lambda (H1: (clear (CHead (CTail k u1 c) (Bind +b0) t) (CHead c2 (Bind b) u2))).(let H2 \def (f_equal C C (\lambda (e: +C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | +(CHead c0 _ _) \Rightarrow c0])) (CHead c2 (Bind b) u2) (CHead (CTail k u1 c) +(Bind b0) t) (clear_gen_bind b0 (CTail k u1 c) (CHead c2 (Bind b) u2) t H1)) +in ((let H3 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda +(_: C).B) with [(CSort _) \Rightarrow b | (CHead _ k1 _) \Rightarrow (match +k1 in K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) +\Rightarrow b])])) (CHead c2 (Bind b) u2) (CHead (CTail k u1 c) (Bind b0) t) +(clear_gen_bind b0 (CTail k u1 c) (CHead c2 (Bind b) u2) t H1)) in ((let H4 +\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) +with [(CSort _) \Rightarrow u2 | (CHead _ _ t0) \Rightarrow t0])) (CHead c2 +(Bind b) u2) (CHead (CTail k u1 c) (Bind b0) t) (clear_gen_bind b0 (CTail k +u1 c) (CHead c2 (Bind b) u2) t H1)) in (\lambda (H5: (eq B b b0)).(\lambda +(H6: (eq C c2 (CTail k u1 c))).(eq_ind T u2 (\lambda (t0: T).(or (ex2 C +(\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c +(Bind b0) t0) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O +(s (Bind b0) (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: +nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n)))))) (eq_ind B b +(\lambda (b1: B).(or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) +(\lambda (e: C).(getl O (CHead c (Bind b1) u2) (CHead e (Bind b) u2)))) (ex4 +nat (\lambda (_: nat).(eq nat O (s (Bind b1) (clen c)))) (\lambda (_: +nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq +C c2 (CSort n)))))) (let H7 \def (eq_ind C c2 (\lambda (c0: C).(\forall (i0: +nat).((getl i0 (CTail k u1 c) (CHead c0 (Bind b) u2)) \to (or (ex2 C (\lambda +(e: C).(eq C c0 (CTail k u1 e))) (\lambda (e: C).(getl i0 c (CHead e (Bind b) +u2)))) (ex4 nat (\lambda (_: nat).(eq nat i0 (clen c))) (\lambda (_: nat).(eq +K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c0 +(CSort n)))))))) H (CTail k u1 c) H6) in (eq_ind_r C (CTail k u1 c) (\lambda +(c0: C).(or (ex2 C (\lambda (e: C).(eq C c0 (CTail k u1 e))) (\lambda (e: +C).(getl O (CHead c (Bind b) u2) (CHead e (Bind b) u2)))) (ex4 nat (\lambda +(_: nat).(eq nat O (s (Bind b) (clen c)))) (\lambda (_: nat).(eq K k (Bind +b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c0 (CSort +n)))))) (or_introl (ex2 C (\lambda (e: C).(eq C (CTail k u1 c) (CTail k u1 +e))) (\lambda (e: C).(getl O (CHead c (Bind b) u2) (CHead e (Bind b) u2)))) +(ex4 nat (\lambda (_: nat).(eq nat O (s (Bind b) (clen c)))) (\lambda (_: +nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq +C (CTail k u1 c) (CSort n)))) (ex_intro2 C (\lambda (e: C).(eq C (CTail k u1 +c) (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Bind b) u2) (CHead e +(Bind b) u2))) c (refl_equal C (CTail k u1 c)) (getl_refl b c u2))) c2 H6)) +b0 H5) t H4)))) H3)) H2)))) (\lambda (f: F).(\lambda (H1: (clear (CHead +(CTail k u1 c) (Flat f) t) (CHead c2 (Bind b) u2))).(let H2 \def (H O +(getl_intro O (CTail k u1 c) (CHead c2 (Bind b) u2) (CTail k u1 c) (drop_refl +(CTail k u1 c)) (clear_gen_flat f (CTail k u1 c) (CHead c2 (Bind b) u2) t +H1))) in (or_ind (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda +(e: C).(getl O c (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat +O (clen c))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 +u2)) (\lambda (n: nat).(eq C c2 (CSort n)))) (or (ex2 C (\lambda (e: C).(eq C +c2 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e +(Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) +(\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda +(n: nat).(eq C c2 (CSort n))))) (\lambda (H3: (ex2 C (\lambda (e: C).(eq C c2 +(CTail k u1 e))) (\lambda (e: C).(getl O c (CHead e (Bind b) u2))))).(ex2_ind +C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl O c (CHead +e (Bind b) u2))) (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) +(\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind b) u2)))) (ex4 +nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq +K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 +(CSort n))))) (\lambda (x: C).(\lambda (H4: (eq C c2 (CTail k u1 +x))).(\lambda (H5: (getl O c (CHead x (Bind b) u2))).(eq_ind_r C (CTail k u1 +x) (\lambda (c0: C).(or (ex2 C (\lambda (e: C).(eq C c0 (CTail k u1 e))) +(\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind b) u2)))) (ex4 +nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq +K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c0 +(CSort n)))))) (or_introl (ex2 C (\lambda (e: C).(eq C (CTail k u1 x) (CTail +k u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind b) +u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) (\lambda +(_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: +nat).(eq C (CTail k u1 x) (CSort n)))) (ex_intro2 C (\lambda (e: C).(eq C +(CTail k u1 x) (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) +(CHead e (Bind b) u2))) x (refl_equal C (CTail k u1 x)) (getl_flat c (CHead x +(Bind b) u2) O H5 f t))) c2 H4)))) H3)) (\lambda (H3: (ex4 nat (\lambda (_: +nat).(eq nat O (clen c))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: +nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n))))).(ex4_ind nat +(\lambda (_: nat).(eq nat O (clen c))) (\lambda (_: nat).(eq K k (Bind b))) +(\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n))) (or +(ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl O +(CHead c (Flat f) t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq +nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda +(_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n))))) (\lambda (x0: +nat).(\lambda (H4: (eq nat O (clen c))).(\lambda (H5: (eq K k (Bind +b))).(\lambda (H6: (eq T u1 u2)).(\lambda (H7: (eq C c2 (CSort +x0))).(eq_ind_r C (CSort x0) (\lambda (c0: C).(or (ex2 C (\lambda (e: C).(eq +C c0 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e +(Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) +(\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda +(n: nat).(eq C c0 (CSort n)))))) (eq_ind T u1 (\lambda (t0: T).(or (ex2 C +(\lambda (e: C).(eq C (CSort x0) (CTail k u1 e))) (\lambda (e: C).(getl O +(CHead c (Flat f) t) (CHead e (Bind b) t0)))) (ex4 nat (\lambda (_: nat).(eq +nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda +(_: nat).(eq T u1 t0)) (\lambda (n: nat).(eq C (CSort x0) (CSort n)))))) +(eq_ind_r K (Bind b) (\lambda (k1: K).(or (ex2 C (\lambda (e: C).(eq C (CSort +x0) (CTail k1 u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e +(Bind b) u1)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) +(\lambda (_: nat).(eq K k1 (Bind b))) (\lambda (_: nat).(eq T u1 u1)) +(\lambda (n: nat).(eq C (CSort x0) (CSort n)))))) (or_intror (ex2 C (\lambda +(e: C).(eq C (CSort x0) (CTail (Bind b) u1 e))) (\lambda (e: C).(getl O +(CHead c (Flat f) t) (CHead e (Bind b) u1)))) (ex4 nat (\lambda (_: nat).(eq +nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq K (Bind b) (Bind b))) +(\lambda (_: nat).(eq T u1 u1)) (\lambda (n: nat).(eq C (CSort x0) (CSort +n)))) (ex4_intro nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) +(\lambda (_: nat).(eq K (Bind b) (Bind b))) (\lambda (_: nat).(eq T u1 u1)) +(\lambda (n: nat).(eq C (CSort x0) (CSort n))) x0 H4 (refl_equal K (Bind b)) +(refl_equal T u1) (refl_equal C (CSort x0)))) k H5) u2 H6) c2 H7)))))) H3)) +H2)))) k0 (getl_gen_O (CHead (CTail k u1 c) k0 t) (CHead c2 (Bind b) u2) +H0))) (\lambda (n: nat).(\lambda (H0: (((getl n (CHead (CTail k u1 c) k0 t) +(CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 +e))) (\lambda (e: C).(getl n (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat +(\lambda (_: nat).(eq nat n (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind +b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort +n0)))))))).(\lambda (H1: (getl (S n) (CHead (CTail k u1 c) k0 t) (CHead c2 +(Bind b) u2))).(let H_x \def (H (r k0 n) (getl_gen_S k0 (CTail k u1 c) (CHead +c2 (Bind b) u2) t n H1)) in (let H2 \def H_x in (or_ind (ex2 C (\lambda (e: +C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl (r k0 n) c (CHead e (Bind +b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat (r k0 n) (clen c))) (\lambda (_: +nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: +nat).(eq C c2 (CSort n0)))) (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 +e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 +nat (\lambda (_: nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K +k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 +(CSort n0))))) (\lambda (H3: (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) +(\lambda (e: C).(getl (r k0 n) c (CHead e (Bind b) u2))))).(ex2_ind C +(\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl (r k0 n) c +(CHead e (Bind b) u2))) (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) +(\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat +(\lambda (_: nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k +(Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort +n0))))) (\lambda (x: C).(\lambda (H4: (eq C c2 (CTail k u1 x))).(\lambda (H5: +(getl (r k0 n) c (CHead x (Bind b) u2))).(let H6 \def (eq_ind C c2 (\lambda +(c0: C).(getl (r k0 n) (CTail k u1 c) (CHead c0 (Bind b) u2))) (getl_gen_S k0 +(CTail k u1 c) (CHead c2 (Bind b) u2) t n H1) (CTail k u1 x) H4) in (let H7 +\def (eq_ind C c2 (\lambda (c0: C).((getl n (CHead (CTail k u1 c) k0 t) +(CHead c0 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c0 (CTail k u1 +e))) (\lambda (e: C).(getl n (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat +(\lambda (_: nat).(eq nat n (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind +b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c0 (CSort +n0))))))) H0 (CTail k u1 x) H4) in (eq_ind_r C (CTail k u1 x) (\lambda (c0: +C).(or (ex2 C (\lambda (e: C).(eq C c0 (CTail k u1 e))) (\lambda (e: C).(getl +(S n) (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq +nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: +nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c0 (CSort n0)))))) (or_introl +(ex2 C (\lambda (e: C).(eq C (CTail k u1 x) (CTail k u1 e))) (\lambda (e: +C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: +nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) +(\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C (CTail k u1 x) +(CSort n0)))) (ex_intro2 C (\lambda (e: C).(eq C (CTail k u1 x) (CTail k u1 +e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u2))) x +(refl_equal C (CTail k u1 x)) (getl_head k0 n c (CHead x (Bind b) u2) H5 t))) +c2 H4)))))) H3)) (\lambda (H3: (ex4 nat (\lambda (_: nat).(eq nat (r k0 n) +(clen c))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 +u2)) (\lambda (n0: nat).(eq C c2 (CSort n0))))).(ex4_ind nat (\lambda (_: +nat).(eq nat (r k0 n) (clen c))) (\lambda (_: nat).(eq K k (Bind b))) +(\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0))) (or +(ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl (S n) +(CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat (S +n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: +nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0))))) (\lambda (x0: +nat).(\lambda (H4: (eq nat (r k0 n) (clen c))).(\lambda (H5: (eq K k (Bind +b))).(\lambda (H6: (eq T u1 u2)).(\lambda (H7: (eq C c2 (CSort x0))).(let H8 +\def (eq_ind C c2 (\lambda (c0: C).(getl (r k0 n) (CTail k u1 c) (CHead c0 +(Bind b) u2))) (getl_gen_S k0 (CTail k u1 c) (CHead c2 (Bind b) u2) t n H1) +(CSort x0) H7) in (let H9 \def (eq_ind C c2 (\lambda (c0: C).((getl n (CHead +(CTail k u1 c) k0 t) (CHead c0 (Bind b) u2)) \to (or (ex2 C (\lambda (e: +C).(eq C c0 (CTail k u1 e))) (\lambda (e: C).(getl n (CHead c k0 t) (CHead e +(Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat n (s k0 (clen c)))) +(\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda +(n0: nat).(eq C c0 (CSort n0))))))) H0 (CSort x0) H7) in (eq_ind_r C (CSort +x0) (\lambda (c0: C).(or (ex2 C (\lambda (e: C).(eq C c0 (CTail k u1 e))) +(\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat +(\lambda (_: nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k +(Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c0 (CSort +n0)))))) (let H10 \def (eq_ind_r T u2 (\lambda (t0: T).((getl n (CHead (CTail +k u1 c) k0 t) (CHead (CSort x0) (Bind b) t0)) \to (or (ex2 C (\lambda (e: +C).(eq C (CSort x0) (CTail k u1 e))) (\lambda (e: C).(getl n (CHead c k0 t) +(CHead e (Bind b) t0)))) (ex4 nat (\lambda (_: nat).(eq nat n (s k0 (clen +c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 t0)) +(\lambda (n0: nat).(eq C (CSort x0) (CSort n0))))))) H9 u1 H6) in (let H11 +\def (eq_ind_r T u2 (\lambda (t0: T).(getl (r k0 n) (CTail k u1 c) (CHead +(CSort x0) (Bind b) t0))) H8 u1 H6) in (eq_ind T u1 (\lambda (t0: T).(or (ex2 +C (\lambda (e: C).(eq C (CSort x0) (CTail k u1 e))) (\lambda (e: C).(getl (S +n) (CHead c k0 t) (CHead e (Bind b) t0)))) (ex4 nat (\lambda (_: nat).(eq nat +(S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: +nat).(eq T u1 t0)) (\lambda (n0: nat).(eq C (CSort x0) (CSort n0)))))) (let +H12 \def (eq_ind K k (\lambda (k1: K).((getl n (CHead (CTail k1 u1 c) k0 t) +(CHead (CSort x0) (Bind b) u1)) \to (or (ex2 C (\lambda (e: C).(eq C (CSort +x0) (CTail k1 u1 e))) (\lambda (e: C).(getl n (CHead c k0 t) (CHead e (Bind +b) u1)))) (ex4 nat (\lambda (_: nat).(eq nat n (s k0 (clen c)))) (\lambda (_: +nat).(eq K k1 (Bind b))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: +nat).(eq C (CSort x0) (CSort n0))))))) H10 (Bind b) H5) in (let H13 \def +(eq_ind K k (\lambda (k1: K).(getl (r k0 n) (CTail k1 u1 c) (CHead (CSort x0) +(Bind b) u1))) H11 (Bind b) H5) in (eq_ind_r K (Bind b) (\lambda (k1: K).(or +(ex2 C (\lambda (e: C).(eq C (CSort x0) (CTail k1 u1 e))) (\lambda (e: +C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u1)))) (ex4 nat (\lambda (_: +nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k1 (Bind b))) +(\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: nat).(eq C (CSort x0) (CSort +n0)))))) (eq_ind nat (r k0 n) (\lambda (n0: nat).(or (ex2 C (\lambda (e: +C).(eq C (CSort x0) (CTail (Bind b) u1 e))) (\lambda (e: C).(getl (S n) +(CHead c k0 t) (CHead e (Bind b) u1)))) (ex4 nat (\lambda (_: nat).(eq nat (S +n) (s k0 n0))) (\lambda (_: nat).(eq K (Bind b) (Bind b))) (\lambda (_: +nat).(eq T u1 u1)) (\lambda (n1: nat).(eq C (CSort x0) (CSort n1)))))) +(eq_ind_r nat (S n) (\lambda (n0: nat).(or (ex2 C (\lambda (e: C).(eq C +(CSort x0) (CTail (Bind b) u1 e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) +(CHead e (Bind b) u1)))) (ex4 nat (\lambda (_: nat).(eq nat (S n) n0)) +(\lambda (_: nat).(eq K (Bind b) (Bind b))) (\lambda (_: nat).(eq T u1 u1)) +(\lambda (n1: nat).(eq C (CSort x0) (CSort n1)))))) (or_intror (ex2 C +(\lambda (e: C).(eq C (CSort x0) (CTail (Bind b) u1 e))) (\lambda (e: +C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u1)))) (ex4 nat (\lambda (_: +nat).(eq nat (S n) (S n))) (\lambda (_: nat).(eq K (Bind b) (Bind b))) +(\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: nat).(eq C (CSort x0) (CSort +n0)))) (ex4_intro nat (\lambda (_: nat).(eq nat (S n) (S n))) (\lambda (_: +nat).(eq K (Bind b) (Bind b))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: +nat).(eq C (CSort x0) (CSort n0))) x0 (refl_equal nat (S n)) (refl_equal K +(Bind b)) (refl_equal T u1) (refl_equal C (CSort x0)))) (s k0 (r k0 n)) (s_r +k0 n)) (clen c) H4) k H5))) u2 H6))) c2 H7)))))))) H3)) H2)))))) i)))))) +c1)))))). +(* COMMENTS +Initial nodes: 7489 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/cnt/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/cnt/defs.ma new file mode 100644 index 000000000..d6ab61d78 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/cnt/defs.ma @@ -0,0 +1,23 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/T/defs.ma". + +inductive cnt: T \to Prop \def +| cnt_sort: \forall (n: nat).(cnt (TSort n)) +| cnt_head: \forall (t: T).((cnt t) \to (\forall (k: K).(\forall (v: T).(cnt +(THead k v t))))). + diff --git a/matita/matita/contribs/lambdadelta/basic_1/cnt/props.ma b/matita/matita/contribs/lambdadelta/basic_1/cnt/props.ma new file mode 100644 index 000000000..7fcd315eb --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/cnt/props.ma @@ -0,0 +1,37 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/cnt/defs.ma". + +include "Basic-1/lift/fwd.ma". + +theorem cnt_lift: + \forall (t: T).((cnt t) \to (\forall (i: nat).(\forall (d: nat).(cnt (lift i +d t))))) +\def + \lambda (t: T).(\lambda (H: (cnt t)).(cnt_ind (\lambda (t0: T).(\forall (i: +nat).(\forall (d: nat).(cnt (lift i d t0))))) (\lambda (n: nat).(\lambda (i: +nat).(\lambda (d: nat).(eq_ind_r T (TSort n) (\lambda (t0: T).(cnt t0)) +(cnt_sort n) (lift i d (TSort n)) (lift_sort n i d))))) (\lambda (t0: +T).(\lambda (_: (cnt t0)).(\lambda (H1: ((\forall (i: nat).(\forall (d: +nat).(cnt (lift i d t0)))))).(\lambda (k: K).(\lambda (v: T).(\lambda (i: +nat).(\lambda (d: nat).(eq_ind_r T (THead k (lift i d v) (lift i (s k d) t0)) +(\lambda (t1: T).(cnt t1)) (cnt_head (lift i (s k d) t0) (H1 i (s k d)) k +(lift i d v)) (lift i d (THead k v t0)) (lift_head k v t0 i d))))))))) t H)). +(* COMMENTS +Initial nodes: 191 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/csuba/arity.ma b/matita/matita/contribs/lambdadelta/basic_1/csuba/arity.ma new file mode 100644 index 000000000..fe021cd0f --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/csuba/arity.ma @@ -0,0 +1,336 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/csuba/getl.ma". + +include "Basic-1/csuba/props.ma". + +include "Basic-1/arity/props.ma". + +include "Basic-1/csubv/getl.ma". + +theorem csuba_arity: + \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1 +t a) \to (\forall (c2: C).((csuba g c1 c2) \to (arity g c2 t a))))))) +\def + \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: +(arity g c1 t a)).(arity_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda (a0: +A).(\forall (c2: C).((csuba g c c2) \to (arity g c2 t0 a0)))))) (\lambda (c: +C).(\lambda (n: nat).(\lambda (c2: C).(\lambda (_: (csuba g c +c2)).(arity_sort g c2 n))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: +T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d (Bind Abbr) +u))).(\lambda (a0: A).(\lambda (_: (arity g d u a0)).(\lambda (H2: ((\forall +(c2: C).((csuba g d c2) \to (arity g c2 u a0))))).(\lambda (c2: C).(\lambda +(H3: (csuba g c c2)).(let H4 \def (csuba_getl_abbr g c d u i H0 c2 H3) in +(ex2_ind C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda +(d2: C).(csuba g d d2)) (arity g c2 (TLRef i) a0) (\lambda (x: C).(\lambda +(H5: (getl i c2 (CHead x (Bind Abbr) u))).(\lambda (H6: (csuba g d +x)).(arity_abbr g c2 x u i H5 a0 (H2 x H6))))) H4)))))))))))) (\lambda (c: +C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c +(CHead d (Bind Abst) u))).(\lambda (a0: A).(\lambda (H1: (arity g d u (asucc +g a0))).(\lambda (H2: ((\forall (c2: C).((csuba g d c2) \to (arity g c2 u +(asucc g a0)))))).(\lambda (c2: C).(\lambda (H3: (csuba g c c2)).(let H4 \def +(csuba_getl_abst g c d u i H0 c2 H3) in (or_ind (ex2 C (\lambda (d2: C).(getl +i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d d2))) (ex4_3 C T +A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 +(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g +d d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d u (asucc +g a1))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2 +a1))))) (arity g c2 (TLRef i) a0) (\lambda (H5: (ex2 C (\lambda (d2: C).(getl +i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d d2)))).(ex2_ind C +(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: +C).(csuba g d d2)) (arity g c2 (TLRef i) a0) (\lambda (x: C).(\lambda (H6: +(getl i c2 (CHead x (Bind Abst) u))).(\lambda (H7: (csuba g d x)).(arity_abst +g c2 x u i H6 a0 (H2 x H7))))) H5)) (\lambda (H5: (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) +(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d d2)))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d u (asucc g a1))))) +(\lambda (d2: C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2 +a1)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: +A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: +T).(\lambda (_: A).(csuba g d d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a1: A).(arity g d u (asucc g a1))))) (\lambda (d2: C).(\lambda (u2: +T).(\lambda (a1: A).(arity g d2 u2 a1)))) (arity g c2 (TLRef i) a0) (\lambda +(x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H6: (getl i c2 (CHead x0 +(Bind Abbr) x1))).(\lambda (_: (csuba g d x0)).(\lambda (H8: (arity g d u +(asucc g x2))).(\lambda (H9: (arity g x0 x1 x2)).(arity_repl g c2 (TLRef i) +x2 (arity_abbr g c2 x0 x1 i H6 x2 H9) a0 (asucc_inj g x2 a0 (arity_mono g d u +(asucc g x2) H8 (asucc g a0) H1)))))))))) H5)) H4)))))))))))) (\lambda (b: +B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c: C).(\lambda (u: +T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H2: ((\forall +(c2: C).((csuba g c c2) \to (arity g c2 u a1))))).(\lambda (t0: T).(\lambda +(a2: A).(\lambda (_: (arity g (CHead c (Bind b) u) t0 a2)).(\lambda (H4: +((\forall (c2: C).((csuba g (CHead c (Bind b) u) c2) \to (arity g c2 t0 +a2))))).(\lambda (c2: C).(\lambda (H5: (csuba g c c2)).(arity_bind g b H0 c2 +u a1 (H2 c2 H5) t0 a2 (H4 (CHead c2 (Bind b) u) (csuba_head g c c2 H5 (Bind +b) u)))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a1: +A).(\lambda (_: (arity g c u (asucc g a1))).(\lambda (H1: ((\forall (c2: +C).((csuba g c c2) \to (arity g c2 u (asucc g a1)))))).(\lambda (t0: +T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c (Bind Abst) u) t0 +a2)).(\lambda (H3: ((\forall (c2: C).((csuba g (CHead c (Bind Abst) u) c2) +\to (arity g c2 t0 a2))))).(\lambda (c2: C).(\lambda (H4: (csuba g c +c2)).(arity_head g c2 u a1 (H1 c2 H4) t0 a2 (H3 (CHead c2 (Bind Abst) u) +(csuba_head g c c2 H4 (Bind Abst) u)))))))))))))) (\lambda (c: C).(\lambda +(u: T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H1: +((\forall (c2: C).((csuba g c c2) \to (arity g c2 u a1))))).(\lambda (t0: +T).(\lambda (a2: A).(\lambda (_: (arity g c t0 (AHead a1 a2))).(\lambda (H3: +((\forall (c2: C).((csuba g c c2) \to (arity g c2 t0 (AHead a1 +a2)))))).(\lambda (c2: C).(\lambda (H4: (csuba g c c2)).(arity_appl g c2 u a1 +(H1 c2 H4) t0 a2 (H3 c2 H4))))))))))))) (\lambda (c: C).(\lambda (u: +T).(\lambda (a0: A).(\lambda (_: (arity g c u (asucc g a0))).(\lambda (H1: +((\forall (c2: C).((csuba g c c2) \to (arity g c2 u (asucc g +a0)))))).(\lambda (t0: T).(\lambda (_: (arity g c t0 a0)).(\lambda (H3: +((\forall (c2: C).((csuba g c c2) \to (arity g c2 t0 a0))))).(\lambda (c2: +C).(\lambda (H4: (csuba g c c2)).(arity_cast g c2 u a0 (H1 c2 H4) t0 (H3 c2 +H4)))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda (_: +(arity g c t0 a1)).(\lambda (H1: ((\forall (c2: C).((csuba g c c2) \to (arity +g c2 t0 a1))))).(\lambda (a2: A).(\lambda (H2: (leq g a1 a2)).(\lambda (c2: +C).(\lambda (H3: (csuba g c c2)).(arity_repl g c2 t0 a1 (H1 c2 H3) a2 +H2)))))))))) c1 t a H))))). +(* COMMENTS +Initial nodes: 1505 +END *) + +theorem csuba_arity_rev: + \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1 +t a) \to (\forall (c2: C).((csuba g c2 c1) \to ((csubv c2 c1) \to (arity g c2 +t a)))))))) +\def + \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: +(arity g c1 t a)).(arity_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda (a0: +A).(\forall (c2: C).((csuba g c2 c) \to ((csubv c2 c) \to (arity g c2 t0 +a0))))))) (\lambda (c: C).(\lambda (n: nat).(\lambda (c2: C).(\lambda (_: +(csuba g c2 c)).(\lambda (_: (csubv c2 c)).(arity_sort g c2 n)))))) (\lambda +(c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl +i c (CHead d (Bind Abbr) u))).(\lambda (a0: A).(\lambda (H1: (arity g d u +a0)).(\lambda (H2: ((\forall (c2: C).((csuba g c2 d) \to ((csubv c2 d) \to +(arity g c2 u a0)))))).(\lambda (c2: C).(\lambda (H3: (csuba g c2 +c)).(\lambda (H4: (csubv c2 c)).(let H_x \def (csuba_getl_abbr_rev g c d u i +H0 c2 H3) in (let H5 \def H_x in (or3_ind (ex2 C (\lambda (d2: C).(getl i c2 +(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d2 d))) (ex4_3 C T A +(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind +Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 +d)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2 +(asucc g a1))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d +u a1))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d)))) (arity +g c2 (TLRef i) a0) (\lambda (H6: (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 +(Bind Abbr) u))) (\lambda (d2: C).(csuba g d2 d)))).(ex2_ind C (\lambda (d2: +C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d2 d)) +(arity g c2 (TLRef i) a0) (\lambda (x: C).(\lambda (H7: (getl i c2 (CHead x +(Bind Abbr) u))).(\lambda (H8: (csuba g x d)).(let H_x0 \def (csubv_getl_conf +c2 c H4 Abbr x u i H7) in (let H9 \def H_x0 in (ex2_3_ind B C T (\lambda (_: +B).(\lambda (d2: C).(\lambda (_: T).(csubv x d2)))) (\lambda (b2: B).(\lambda +(d2: C).(\lambda (v2: T).(getl i c (CHead d2 (Bind b2) v2))))) (arity g c2 +(TLRef i) a0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda +(H10: (csubv x x1)).(\lambda (H11: (getl i c (CHead x1 (Bind x0) x2))).(let +H12 \def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c0: C).(getl i c c0)) H0 +(CHead x1 (Bind x0) x2) (getl_mono c (CHead d (Bind Abbr) u) i H0 (CHead x1 +(Bind x0) x2) H11)) in (let H13 \def (f_equal C C (\lambda (e: C).(match e in +C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) +\Rightarrow c0])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) (getl_mono +c (CHead d (Bind Abbr) u) i H0 (CHead x1 (Bind x0) x2) H11)) in ((let H14 +\def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) +with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k in K +return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow +Abbr])])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) (getl_mono c (CHead +d (Bind Abbr) u) i H0 (CHead x1 (Bind x0) x2) H11)) in ((let H15 \def +(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with +[(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind +Abbr) u) (CHead x1 (Bind x0) x2) (getl_mono c (CHead d (Bind Abbr) u) i H0 +(CHead x1 (Bind x0) x2) H11)) in (\lambda (H16: (eq B Abbr x0)).(\lambda +(H17: (eq C d x1)).(let H18 \def (eq_ind_r T x2 (\lambda (t0: T).(getl i c +(CHead x1 (Bind x0) t0))) H12 u H15) in (let H19 \def (eq_ind_r C x1 (\lambda +(c0: C).(getl i c (CHead c0 (Bind x0) u))) H18 d H17) in (let H20 \def +(eq_ind_r C x1 (\lambda (c0: C).(csubv x c0)) H10 d H17) in (let H21 \def +(eq_ind_r B x0 (\lambda (b: B).(getl i c (CHead d (Bind b) u))) H19 Abbr H16) +in (arity_abbr g c2 x u i H7 a0 (H2 x H8 H20))))))))) H14)) H13)))))))) +H9)))))) H6)) (\lambda (H6: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: +T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: +C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d)))) (\lambda (d2: +C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2 (asucc g a1))))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d u a1)))))).(ex4_3_ind C T +A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 +(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g +d2 d)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2 +(asucc g a1))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d +u a1)))) (arity g c2 (TLRef i) a0) (\lambda (x0: C).(\lambda (x1: T).(\lambda +(x2: A).(\lambda (H7: (getl i c2 (CHead x0 (Bind Abst) x1))).(\lambda (_: +(csuba g x0 d)).(\lambda (H9: (arity g x0 x1 (asucc g x2))).(\lambda (H10: +(arity g d u x2)).(arity_repl g c2 (TLRef i) x2 (arity_abst g c2 x0 x1 i H7 +x2 H9) a0 (arity_mono g d u x2 H10 a0 H1))))))))) H6)) (\lambda (H6: (ex2_2 C +T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d))))).(ex2_2_ind C T (\lambda +(d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda +(d2: C).(\lambda (_: T).(csuba g d2 d))) (arity g c2 (TLRef i) a0) (\lambda +(x0: C).(\lambda (x1: T).(\lambda (H7: (getl i c2 (CHead x0 (Bind Void) +x1))).(\lambda (_: (csuba g x0 d)).(let H_x0 \def (csubv_getl_conf_void c2 c +H4 x0 x1 i H7) in (let H9 \def H_x0 in (ex2_2_ind C T (\lambda (d2: +C).(\lambda (_: T).(csubv x0 d2))) (\lambda (d2: C).(\lambda (v2: T).(getl i +c (CHead d2 (Bind Void) v2)))) (arity g c2 (TLRef i) a0) (\lambda (x2: +C).(\lambda (x3: T).(\lambda (_: (csubv x0 x2)).(\lambda (H11: (getl i c +(CHead x2 (Bind Void) x3))).(let H12 \def (eq_ind C (CHead d (Bind Abbr) u) +(\lambda (c0: C).(getl i c c0)) H0 (CHead x2 (Bind Void) x3) (getl_mono c +(CHead d (Bind Abbr) u) i H0 (CHead x2 (Bind Void) x3) H11)) in (let H13 \def +(eq_ind C (CHead d (Bind Abbr) u) (\lambda (ee: C).(match ee in C return +(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) +\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b) +\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow +True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _) +\Rightarrow False])])) I (CHead x2 (Bind Void) x3) (getl_mono c (CHead d +(Bind Abbr) u) i H0 (CHead x2 (Bind Void) x3) H11)) in (False_ind (arity g c2 +(TLRef i) a0) H13))))))) H9))))))) H6)) H5)))))))))))))) (\lambda (c: +C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c +(CHead d (Bind Abst) u))).(\lambda (a0: A).(\lambda (_: (arity g d u (asucc g +a0))).(\lambda (H2: ((\forall (c2: C).((csuba g c2 d) \to ((csubv c2 d) \to +(arity g c2 u (asucc g a0))))))).(\lambda (c2: C).(\lambda (H3: (csuba g c2 +c)).(\lambda (H4: (csubv c2 c)).(let H_x \def (csuba_getl_abst_rev g c d u i +H0 c2 H3) in (let H5 \def H_x in (or_ind (ex2 C (\lambda (d2: C).(getl i c2 +(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d)))) (arity g c2 (TLRef i) a0) +(\lambda (H6: (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) +(\lambda (d2: C).(csuba g d2 d)))).(ex2_ind C (\lambda (d2: C).(getl i c2 +(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d)) (arity g c2 +(TLRef i) a0) (\lambda (x: C).(\lambda (H7: (getl i c2 (CHead x (Bind Abst) +u))).(\lambda (H8: (csuba g x d)).(let H_x0 \def (csubv_getl_conf c2 c H4 +Abst x u i H7) in (let H9 \def H_x0 in (ex2_3_ind B C T (\lambda (_: +B).(\lambda (d2: C).(\lambda (_: T).(csubv x d2)))) (\lambda (b2: B).(\lambda +(d2: C).(\lambda (v2: T).(getl i c (CHead d2 (Bind b2) v2))))) (arity g c2 +(TLRef i) a0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda +(H10: (csubv x x1)).(\lambda (H11: (getl i c (CHead x1 (Bind x0) x2))).(let +H12 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (c0: C).(getl i c c0)) H0 +(CHead x1 (Bind x0) x2) (getl_mono c (CHead d (Bind Abst) u) i H0 (CHead x1 +(Bind x0) x2) H11)) in (let H13 \def (f_equal C C (\lambda (e: C).(match e in +C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) +\Rightarrow c0])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x2) (getl_mono +c (CHead d (Bind Abst) u) i H0 (CHead x1 (Bind x0) x2) H11)) in ((let H14 +\def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) +with [(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match k in K +return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow +Abst])])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x2) (getl_mono c (CHead +d (Bind Abst) u) i H0 (CHead x1 (Bind x0) x2) H11)) in ((let H15 \def +(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with +[(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind +Abst) u) (CHead x1 (Bind x0) x2) (getl_mono c (CHead d (Bind Abst) u) i H0 +(CHead x1 (Bind x0) x2) H11)) in (\lambda (H16: (eq B Abst x0)).(\lambda +(H17: (eq C d x1)).(let H18 \def (eq_ind_r T x2 (\lambda (t0: T).(getl i c +(CHead x1 (Bind x0) t0))) H12 u H15) in (let H19 \def (eq_ind_r C x1 (\lambda +(c0: C).(getl i c (CHead c0 (Bind x0) u))) H18 d H17) in (let H20 \def +(eq_ind_r C x1 (\lambda (c0: C).(csubv x c0)) H10 d H17) in (let H21 \def +(eq_ind_r B x0 (\lambda (b: B).(getl i c (CHead d (Bind b) u))) H19 Abst H16) +in (arity_abst g c2 x u i H7 a0 (H2 x H8 H20))))))))) H14)) H13)))))))) +H9)))))) H6)) (\lambda (H6: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(getl +i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g +d2 d))) (arity g c2 (TLRef i) a0) (\lambda (x0: C).(\lambda (x1: T).(\lambda +(H7: (getl i c2 (CHead x0 (Bind Void) x1))).(\lambda (_: (csuba g x0 d)).(let +H_x0 \def (csubv_getl_conf_void c2 c H4 x0 x1 i H7) in (let H9 \def H_x0 in +(ex2_2_ind C T (\lambda (d2: C).(\lambda (_: T).(csubv x0 d2))) (\lambda (d2: +C).(\lambda (v2: T).(getl i c (CHead d2 (Bind Void) v2)))) (arity g c2 (TLRef +i) a0) (\lambda (x2: C).(\lambda (x3: T).(\lambda (_: (csubv x0 x2)).(\lambda +(H11: (getl i c (CHead x2 (Bind Void) x3))).(let H12 \def (eq_ind C (CHead d +(Bind Abst) u) (\lambda (c0: C).(getl i c c0)) H0 (CHead x2 (Bind Void) x3) +(getl_mono c (CHead d (Bind Abst) u) i H0 (CHead x2 (Bind Void) x3) H11)) in +(let H13 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (ee: C).(match ee in +C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k +_) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b) +\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow +False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) +\Rightarrow False])])) I (CHead x2 (Bind Void) x3) (getl_mono c (CHead d +(Bind Abst) u) i H0 (CHead x2 (Bind Void) x3) H11)) in (False_ind (arity g c2 +(TLRef i) a0) H13))))))) H9))))))) H6)) H5)))))))))))))) (\lambda (b: +B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c: C).(\lambda (u: +T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H2: ((\forall +(c2: C).((csuba g c2 c) \to ((csubv c2 c) \to (arity g c2 u a1)))))).(\lambda +(t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c (Bind b) u) t0 +a2)).(\lambda (H4: ((\forall (c2: C).((csuba g c2 (CHead c (Bind b) u)) \to +((csubv c2 (CHead c (Bind b) u)) \to (arity g c2 t0 a2)))))).(\lambda (c2: +C).(\lambda (H5: (csuba g c2 c)).(\lambda (H6: (csubv c2 c)).(arity_bind g b +H0 c2 u a1 (H2 c2 H5 H6) t0 a2 (H4 (CHead c2 (Bind b) u) (csuba_head g c2 c +H5 (Bind b) u) (csubv_bind_same c2 c H6 b u u))))))))))))))))) (\lambda (c: +C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c u (asucc g +a1))).(\lambda (H1: ((\forall (c2: C).((csuba g c2 c) \to ((csubv c2 c) \to +(arity g c2 u (asucc g a1))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda +(_: (arity g (CHead c (Bind Abst) u) t0 a2)).(\lambda (H3: ((\forall (c2: +C).((csuba g c2 (CHead c (Bind Abst) u)) \to ((csubv c2 (CHead c (Bind Abst) +u)) \to (arity g c2 t0 a2)))))).(\lambda (c2: C).(\lambda (H4: (csuba g c2 +c)).(\lambda (H5: (csubv c2 c)).(arity_head g c2 u a1 (H1 c2 H4 H5) t0 a2 (H3 +(CHead c2 (Bind Abst) u) (csuba_head g c2 c H4 (Bind Abst) u) +(csubv_bind_same c2 c H5 Abst u u))))))))))))))) (\lambda (c: C).(\lambda (u: +T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H1: ((\forall +(c2: C).((csuba g c2 c) \to ((csubv c2 c) \to (arity g c2 u a1)))))).(\lambda +(t0: T).(\lambda (a2: A).(\lambda (_: (arity g c t0 (AHead a1 a2))).(\lambda +(H3: ((\forall (c2: C).((csuba g c2 c) \to ((csubv c2 c) \to (arity g c2 t0 +(AHead a1 a2))))))).(\lambda (c2: C).(\lambda (H4: (csuba g c2 c)).(\lambda +(H5: (csubv c2 c)).(arity_appl g c2 u a1 (H1 c2 H4 H5) t0 a2 (H3 c2 H4 +H5)))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a0: A).(\lambda +(_: (arity g c u (asucc g a0))).(\lambda (H1: ((\forall (c2: C).((csuba g c2 +c) \to ((csubv c2 c) \to (arity g c2 u (asucc g a0))))))).(\lambda (t0: +T).(\lambda (_: (arity g c t0 a0)).(\lambda (H3: ((\forall (c2: C).((csuba g +c2 c) \to ((csubv c2 c) \to (arity g c2 t0 a0)))))).(\lambda (c2: C).(\lambda +(H4: (csuba g c2 c)).(\lambda (H5: (csubv c2 c)).(arity_cast g c2 u a0 (H1 c2 +H4 H5) t0 (H3 c2 H4 H5))))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda +(a1: A).(\lambda (_: (arity g c t0 a1)).(\lambda (H1: ((\forall (c2: +C).((csuba g c2 c) \to ((csubv c2 c) \to (arity g c2 t0 a1)))))).(\lambda +(a2: A).(\lambda (H2: (leq g a1 a2)).(\lambda (c2: C).(\lambda (H3: (csuba g +c2 c)).(\lambda (H4: (csubv c2 c)).(arity_repl g c2 t0 a1 (H1 c2 H3 H4) a2 +H2))))))))))) c1 t a H))))). +(* COMMENTS +Initial nodes: 3597 +END *) + +theorem arity_appls_appl: + \forall (g: G).(\forall (c: C).(\forall (v: T).(\forall (a1: A).((arity g c +v a1) \to (\forall (u: T).((arity g c u (asucc g a1)) \to (\forall (t: +T).(\forall (vs: TList).(\forall (a2: A).((arity g c (THeads (Flat Appl) vs +(THead (Bind Abbr) v t)) a2) \to (arity g c (THeads (Flat Appl) vs (THead +(Flat Appl) v (THead (Bind Abst) u t))) a2))))))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (v: T).(\lambda (a1: A).(\lambda (H: +(arity g c v a1)).(\lambda (u: T).(\lambda (H0: (arity g c u (asucc g +a1))).(\lambda (t: T).(\lambda (vs: TList).(TList_ind (\lambda (t0: +TList).(\forall (a2: A).((arity g c (THeads (Flat Appl) t0 (THead (Bind Abbr) +v t)) a2) \to (arity g c (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead +(Bind Abst) u t))) a2)))) (\lambda (a2: A).(\lambda (H1: (arity g c (THead +(Bind Abbr) v t) a2)).(let H_x \def (arity_gen_bind Abbr (\lambda (H2: (eq B +Abbr Abst)).(not_abbr_abst H2)) g c v t a2 H1) in (let H2 \def H_x in +(ex2_ind A (\lambda (a3: A).(arity g c v a3)) (\lambda (_: A).(arity g (CHead +c (Bind Abbr) v) t a2)) (arity g c (THead (Flat Appl) v (THead (Bind Abst) u +t)) a2) (\lambda (x: A).(\lambda (_: (arity g c v x)).(\lambda (H4: (arity g +(CHead c (Bind Abbr) v) t a2)).(arity_appl g c v a1 H (THead (Bind Abst) u t) +a2 (arity_head g c u a1 H0 t a2 (csuba_arity_rev g (CHead c (Bind Abbr) v) t +a2 H4 (CHead c (Bind Abst) u) (csuba_abst g c c (csuba_refl g c) u a1 H0 v H) +(csubv_bind c c (csubv_refl c) Abst (sym_not_eq B Void Abst not_void_abst) +Abbr u v))))))) H2))))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H1: +((\forall (a2: A).((arity g c (THeads (Flat Appl) t1 (THead (Bind Abbr) v t)) +a2) \to (arity g c (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind +Abst) u t))) a2))))).(\lambda (a2: A).(\lambda (H2: (arity g c (THead (Flat +Appl) t0 (THeads (Flat Appl) t1 (THead (Bind Abbr) v t))) a2)).(let H3 \def +(arity_gen_appl g c t0 (THeads (Flat Appl) t1 (THead (Bind Abbr) v t)) a2 H2) +in (ex2_ind A (\lambda (a3: A).(arity g c t0 a3)) (\lambda (a3: A).(arity g c +(THeads (Flat Appl) t1 (THead (Bind Abbr) v t)) (AHead a3 a2))) (arity g c +(THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead +(Bind Abst) u t)))) a2) (\lambda (x: A).(\lambda (H4: (arity g c t0 +x)).(\lambda (H5: (arity g c (THeads (Flat Appl) t1 (THead (Bind Abbr) v t)) +(AHead x a2))).(arity_appl g c t0 x H4 (THeads (Flat Appl) t1 (THead (Flat +Appl) v (THead (Bind Abst) u t))) a2 (H1 (AHead x a2) H5))))) H3))))))) +vs))))))))). +(* COMMENTS +Initial nodes: 687 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/csuba/clear.ma b/matita/matita/contribs/lambdadelta/basic_1/csuba/clear.ma new file mode 100644 index 000000000..2350fd1e8 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/csuba/clear.ma @@ -0,0 +1,128 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/csuba/defs.ma". + +include "Basic-1/clear/fwd.ma". + +theorem csuba_clear_conf: + \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csuba g c1 c2) \to +(\forall (e1: C).((clear c1 e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) +(\lambda (e2: C).(clear c2 e2)))))))) +\def + \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csuba g c1 +c2)).(csuba_ind g (\lambda (c: C).(\lambda (c0: C).(\forall (e1: C).((clear c +e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2: C).(clear c0 +e2))))))) (\lambda (n: nat).(\lambda (e1: C).(\lambda (H0: (clear (CSort n) +e1)).(clear_gen_sort e1 n H0 (ex2 C (\lambda (e2: C).(csuba g e1 e2)) +(\lambda (e2: C).(clear (CSort n) e2))))))) (\lambda (c3: C).(\lambda (c4: +C).(\lambda (H0: (csuba g c3 c4)).(\lambda (H1: ((\forall (e1: C).((clear c3 +e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2: C).(clear c4 +e2))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (e1: C).(\lambda (H2: +(clear (CHead c3 k u) e1)).(K_ind (\lambda (k0: K).((clear (CHead c3 k0 u) +e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2: C).(clear +(CHead c4 k0 u) e2))))) (\lambda (b: B).(\lambda (H3: (clear (CHead c3 (Bind +b) u) e1)).(eq_ind_r C (CHead c3 (Bind b) u) (\lambda (c: C).(ex2 C (\lambda +(e2: C).(csuba g c e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b) u) e2)))) +(ex_intro2 C (\lambda (e2: C).(csuba g (CHead c3 (Bind b) u) e2)) (\lambda +(e2: C).(clear (CHead c4 (Bind b) u) e2)) (CHead c4 (Bind b) u) (csuba_head g +c3 c4 H0 (Bind b) u) (clear_bind b c4 u)) e1 (clear_gen_bind b c3 e1 u H3)))) +(\lambda (f: F).(\lambda (H3: (clear (CHead c3 (Flat f) u) e1)).(let H4 \def +(H1 e1 (clear_gen_flat f c3 e1 u H3)) in (ex2_ind C (\lambda (e2: C).(csuba g +e1 e2)) (\lambda (e2: C).(clear c4 e2)) (ex2 C (\lambda (e2: C).(csuba g e1 +e2)) (\lambda (e2: C).(clear (CHead c4 (Flat f) u) e2))) (\lambda (x: +C).(\lambda (H5: (csuba g e1 x)).(\lambda (H6: (clear c4 x)).(ex_intro2 C +(\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2: C).(clear (CHead c4 (Flat f) +u) e2)) x H5 (clear_flat c4 x H6 f u))))) H4)))) k H2))))))))) (\lambda (c3: +C).(\lambda (c4: C).(\lambda (H0: (csuba g c3 c4)).(\lambda (_: ((\forall +(e1: C).((clear c3 e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) (\lambda +(e2: C).(clear c4 e2))))))).(\lambda (b: B).(\lambda (H2: (not (eq B b +Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (e1: C).(\lambda (H3: +(clear (CHead c3 (Bind Void) u1) e1)).(eq_ind_r C (CHead c3 (Bind Void) u1) +(\lambda (c: C).(ex2 C (\lambda (e2: C).(csuba g c e2)) (\lambda (e2: +C).(clear (CHead c4 (Bind b) u2) e2)))) (ex_intro2 C (\lambda (e2: C).(csuba +g (CHead c3 (Bind Void) u1) e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b) +u2) e2)) (CHead c4 (Bind b) u2) (csuba_void g c3 c4 H0 b H2 u1 u2) +(clear_bind b c4 u2)) e1 (clear_gen_bind Void c3 e1 u1 H3)))))))))))) +(\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csuba g c3 c4)).(\lambda (_: +((\forall (e1: C).((clear c3 e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) +(\lambda (e2: C).(clear c4 e2))))))).(\lambda (t: T).(\lambda (a: A).(\lambda +(H2: (arity g c3 t (asucc g a))).(\lambda (u: T).(\lambda (H3: (arity g c4 u +a)).(\lambda (e1: C).(\lambda (H4: (clear (CHead c3 (Bind Abst) t) +e1)).(eq_ind_r C (CHead c3 (Bind Abst) t) (\lambda (c: C).(ex2 C (\lambda +(e2: C).(csuba g c e2)) (\lambda (e2: C).(clear (CHead c4 (Bind Abbr) u) +e2)))) (ex_intro2 C (\lambda (e2: C).(csuba g (CHead c3 (Bind Abst) t) e2)) +(\lambda (e2: C).(clear (CHead c4 (Bind Abbr) u) e2)) (CHead c4 (Bind Abbr) +u) (csuba_abst g c3 c4 H0 t a H2 u H3) (clear_bind Abbr c4 u)) e1 +(clear_gen_bind Abst c3 e1 t H4))))))))))))) c1 c2 H)))). +(* COMMENTS +Initial nodes: 937 +END *) + +theorem csuba_clear_trans: + \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csuba g c2 c1) \to +(\forall (e1: C).((clear c1 e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) +(\lambda (e2: C).(clear c2 e2)))))))) +\def + \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csuba g c2 +c1)).(csuba_ind g (\lambda (c: C).(\lambda (c0: C).(\forall (e1: C).((clear +c0 e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) (\lambda (e2: C).(clear c +e2))))))) (\lambda (n: nat).(\lambda (e1: C).(\lambda (H0: (clear (CSort n) +e1)).(clear_gen_sort e1 n H0 (ex2 C (\lambda (e2: C).(csuba g e2 e1)) +(\lambda (e2: C).(clear (CSort n) e2))))))) (\lambda (c3: C).(\lambda (c4: +C).(\lambda (H0: (csuba g c3 c4)).(\lambda (H1: ((\forall (e1: C).((clear c4 +e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) (\lambda (e2: C).(clear c3 +e2))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (e1: C).(\lambda (H2: +(clear (CHead c4 k u) e1)).(K_ind (\lambda (k0: K).((clear (CHead c4 k0 u) +e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) (\lambda (e2: C).(clear +(CHead c3 k0 u) e2))))) (\lambda (b: B).(\lambda (H3: (clear (CHead c4 (Bind +b) u) e1)).(eq_ind_r C (CHead c4 (Bind b) u) (\lambda (c: C).(ex2 C (\lambda +(e2: C).(csuba g e2 c)) (\lambda (e2: C).(clear (CHead c3 (Bind b) u) e2)))) +(ex_intro2 C (\lambda (e2: C).(csuba g e2 (CHead c4 (Bind b) u))) (\lambda +(e2: C).(clear (CHead c3 (Bind b) u) e2)) (CHead c3 (Bind b) u) (csuba_head g +c3 c4 H0 (Bind b) u) (clear_bind b c3 u)) e1 (clear_gen_bind b c4 e1 u H3)))) +(\lambda (f: F).(\lambda (H3: (clear (CHead c4 (Flat f) u) e1)).(let H4 \def +(H1 e1 (clear_gen_flat f c4 e1 u H3)) in (ex2_ind C (\lambda (e2: C).(csuba g +e2 e1)) (\lambda (e2: C).(clear c3 e2)) (ex2 C (\lambda (e2: C).(csuba g e2 +e1)) (\lambda (e2: C).(clear (CHead c3 (Flat f) u) e2))) (\lambda (x: +C).(\lambda (H5: (csuba g x e1)).(\lambda (H6: (clear c3 x)).(ex_intro2 C +(\lambda (e2: C).(csuba g e2 e1)) (\lambda (e2: C).(clear (CHead c3 (Flat f) +u) e2)) x H5 (clear_flat c3 x H6 f u))))) H4)))) k H2))))))))) (\lambda (c3: +C).(\lambda (c4: C).(\lambda (H0: (csuba g c3 c4)).(\lambda (_: ((\forall +(e1: C).((clear c4 e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) (\lambda +(e2: C).(clear c3 e2))))))).(\lambda (b: B).(\lambda (H2: (not (eq B b +Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (e1: C).(\lambda (H3: +(clear (CHead c4 (Bind b) u2) e1)).(eq_ind_r C (CHead c4 (Bind b) u2) +(\lambda (c: C).(ex2 C (\lambda (e2: C).(csuba g e2 c)) (\lambda (e2: +C).(clear (CHead c3 (Bind Void) u1) e2)))) (ex_intro2 C (\lambda (e2: +C).(csuba g e2 (CHead c4 (Bind b) u2))) (\lambda (e2: C).(clear (CHead c3 +(Bind Void) u1) e2)) (CHead c3 (Bind Void) u1) (csuba_void g c3 c4 H0 b H2 u1 +u2) (clear_bind Void c3 u1)) e1 (clear_gen_bind b c4 e1 u2 H3)))))))))))) +(\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csuba g c3 c4)).(\lambda (_: +((\forall (e1: C).((clear c4 e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) +(\lambda (e2: C).(clear c3 e2))))))).(\lambda (t: T).(\lambda (a: A).(\lambda +(H2: (arity g c3 t (asucc g a))).(\lambda (u: T).(\lambda (H3: (arity g c4 u +a)).(\lambda (e1: C).(\lambda (H4: (clear (CHead c4 (Bind Abbr) u) +e1)).(eq_ind_r C (CHead c4 (Bind Abbr) u) (\lambda (c: C).(ex2 C (\lambda +(e2: C).(csuba g e2 c)) (\lambda (e2: C).(clear (CHead c3 (Bind Abst) t) +e2)))) (ex_intro2 C (\lambda (e2: C).(csuba g e2 (CHead c4 (Bind Abbr) u))) +(\lambda (e2: C).(clear (CHead c3 (Bind Abst) t) e2)) (CHead c3 (Bind Abst) +t) (csuba_abst g c3 c4 H0 t a H2 u H3) (clear_bind Abst c3 t)) e1 +(clear_gen_bind Abbr c4 e1 u H4))))))))))))) c2 c1 H)))). +(* COMMENTS +Initial nodes: 937 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/csuba/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/csuba/defs.ma new file mode 100644 index 000000000..cc6a46cd8 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/csuba/defs.ma @@ -0,0 +1,30 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/arity/defs.ma". + +inductive csuba (g: G): C \to (C \to Prop) \def +| csuba_sort: \forall (n: nat).(csuba g (CSort n) (CSort n)) +| csuba_head: \forall (c1: C).(\forall (c2: C).((csuba g c1 c2) \to (\forall +(k: K).(\forall (u: T).(csuba g (CHead c1 k u) (CHead c2 k u)))))) +| csuba_void: \forall (c1: C).(\forall (c2: C).((csuba g c1 c2) \to (\forall +(b: B).((not (eq B b Void)) \to (\forall (u1: T).(\forall (u2: T).(csuba g +(CHead c1 (Bind Void) u1) (CHead c2 (Bind b) u2)))))))) +| csuba_abst: \forall (c1: C).(\forall (c2: C).((csuba g c1 c2) \to (\forall +(t: T).(\forall (a: A).((arity g c1 t (asucc g a)) \to (\forall (u: +T).((arity g c2 u a) \to (csuba g (CHead c1 (Bind Abst) t) (CHead c2 (Bind +Abbr) u))))))))). + diff --git a/matita/matita/contribs/lambdadelta/basic_1/csuba/drop.ma b/matita/matita/contribs/lambdadelta/basic_1/csuba/drop.ma new file mode 100644 index 000000000..1047ac13d --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/csuba/drop.ma @@ -0,0 +1,2468 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/csuba/fwd.ma". + +include "Basic-1/drop/fwd.ma". + +theorem csuba_drop_abbr: + \forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).((drop i +O c1 (CHead d1 (Bind Abbr) u)) \to (\forall (g: G).(\forall (c2: C).((csuba g +c1 c2) \to (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr) u))) +(\lambda (d2: C).(csuba g d1 d2)))))))))) +\def + \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (d1: +C).(\forall (u: T).((drop n O c1 (CHead d1 (Bind Abbr) u)) \to (\forall (g: +G).(\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(drop n O c2 +(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))))))))) +(\lambda (c1: C).(\lambda (d1: C).(\lambda (u: T).(\lambda (H: (drop O O c1 +(CHead d1 (Bind Abbr) u))).(\lambda (g: G).(\lambda (c2: C).(\lambda (H0: +(csuba g c1 c2)).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csuba g c c2)) H0 +(CHead d1 (Bind Abbr) u) (drop_gen_refl c1 (CHead d1 (Bind Abbr) u) H)) in +(let H_x \def (csuba_gen_abbr g d1 c2 u H1) in (let H2 \def H_x in (ex2_ind C +(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba +g d1 d2)) (ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abbr) u))) +(\lambda (d2: C).(csuba g d1 d2))) (\lambda (x: C).(\lambda (H3: (eq C c2 +(CHead x (Bind Abbr) u))).(\lambda (H4: (csuba g d1 x)).(eq_ind_r C (CHead x +(Bind Abbr) u) (\lambda (c: C).(ex2 C (\lambda (d2: C).(drop O O c (CHead d2 +(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (ex_intro2 C (\lambda +(d2: C).(drop O O (CHead x (Bind Abbr) u) (CHead d2 (Bind Abbr) u))) (\lambda +(d2: C).(csuba g d1 d2)) x (drop_refl (CHead x (Bind Abbr) u)) H4) c2 H3)))) +H2))))))))))) (\lambda (n: nat).(\lambda (H: ((\forall (c1: C).(\forall (d1: +C).(\forall (u: T).((drop n O c1 (CHead d1 (Bind Abbr) u)) \to (\forall (g: +G).(\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(drop n O c2 +(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 +d2)))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (d1: +C).(\forall (u: T).((drop (S n) O c (CHead d1 (Bind Abbr) u)) \to (\forall +(g: G).(\forall (c2: C).((csuba g c c2) \to (ex2 C (\lambda (d2: C).(drop (S +n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))))))) +(\lambda (n0: nat).(\lambda (d1: C).(\lambda (u: T).(\lambda (H0: (drop (S n) +O (CSort n0) (CHead d1 (Bind Abbr) u))).(\lambda (g: G).(\lambda (c2: +C).(\lambda (_: (csuba g (CSort n0) c2)).(and3_ind (eq C (CHead d1 (Bind +Abbr) u) (CSort n0)) (eq nat (S n) O) (eq nat O O) (ex2 C (\lambda (d2: +C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 +d2))) (\lambda (_: (eq C (CHead d1 (Bind Abbr) u) (CSort n0))).(\lambda (H3: +(eq nat (S n) O)).(\lambda (_: (eq nat O O)).(let H5 \def (eq_ind nat (S n) +(\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O +\Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind (ex2 C +(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: +C).(csuba g d1 d2))) H5))))) (drop_gen_sort n0 (S n) O (CHead d1 (Bind Abbr) +u) H0))))))))) (\lambda (c: C).(\lambda (H0: ((\forall (d1: C).(\forall (u: +T).((drop (S n) O c (CHead d1 (Bind Abbr) u)) \to (\forall (g: G).(\forall +(c2: C).((csuba g c c2) \to (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead +d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))))))))).(\lambda (k: +K).(\lambda (t: T).(\lambda (d1: C).(\lambda (u: T).(\lambda (H1: (drop (S n) +O (CHead c k t) (CHead d1 (Bind Abbr) u))).(\lambda (g: G).(\lambda (c2: +C).(\lambda (H2: (csuba g (CHead c k t) c2)).(K_ind (\lambda (k0: K).((csuba +g (CHead c k0 t) c2) \to ((drop (r k0 n) O c (CHead d1 (Bind Abbr) u)) \to +(ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda +(d2: C).(csuba g d1 d2)))))) (\lambda (b: B).(\lambda (H3: (csuba g (CHead c +(Bind b) t) c2)).(\lambda (H4: (drop (r (Bind b) n) O c (CHead d1 (Bind Abbr) +u))).(B_ind (\lambda (b0: B).((csuba g (CHead c (Bind b0) t) c2) \to ((drop +(r (Bind b0) n) O c (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: +C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 +d2)))))) (\lambda (H5: (csuba g (CHead c (Bind Abbr) t) c2)).(\lambda (H6: +(drop (r (Bind Abbr) n) O c (CHead d1 (Bind Abbr) u))).(let H_x \def +(csuba_gen_abbr g c c2 t H5) in (let H7 \def H_x in (ex2_ind C (\lambda (d2: +C).(eq C c2 (CHead d2 (Bind Abbr) t))) (\lambda (d2: C).(csuba g c d2)) (ex2 +C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: +C).(csuba g d1 d2))) (\lambda (x: C).(\lambda (H8: (eq C c2 (CHead x (Bind +Abbr) t))).(\lambda (H9: (csuba g c x)).(eq_ind_r C (CHead x (Bind Abbr) t) +(\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind +Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H10 \def (H c d1 u H6 g x +H9) in (ex2_ind C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u))) +(\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(drop (S n) O +(CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g +d1 d2))) (\lambda (x0: C).(\lambda (H11: (drop n O x (CHead x0 (Bind Abbr) +u))).(\lambda (H12: (csuba g d1 x0)).(let H13 \def (refl_equal nat (r (Bind +Abbr) n)) in (let H14 \def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x +(CHead x0 (Bind Abbr) u))) H11 (r (Bind Abbr) n) H13) in (ex_intro2 C +(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) +u))) (\lambda (d2: C).(csuba g d1 d2)) x0 (drop_drop (Bind Abbr) n x (CHead +x0 (Bind Abbr) u) H14 t) H12)))))) H10)) c2 H8)))) H7))))) (\lambda (H5: +(csuba g (CHead c (Bind Abst) t) c2)).(\lambda (H6: (drop (r (Bind Abst) n) O +c (CHead d1 (Bind Abbr) u))).(let H_x \def (csuba_gen_abst g c c2 t H5) in +(let H7 \def H_x in (or_ind (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind +Abst) t))) (\lambda (d2: C).(csuba g c d2))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) +(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g c d2)))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t (asucc g a))))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex2 C +(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: +C).(csuba g d1 d2))) (\lambda (H8: (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 +(Bind Abst) t))) (\lambda (d2: C).(csuba g c d2)))).(ex2_ind C (\lambda (d2: +C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba g c d2)) (ex2 +C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: +C).(csuba g d1 d2))) (\lambda (x: C).(\lambda (H9: (eq C c2 (CHead x (Bind +Abst) t))).(\lambda (H10: (csuba g c x)).(eq_ind_r C (CHead x (Bind Abst) t) +(\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind +Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H11 \def (H c d1 u H6 g x +H10) in (ex2_ind C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u))) +(\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(drop (S n) O +(CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g +d1 d2))) (\lambda (x0: C).(\lambda (H12: (drop n O x (CHead x0 (Bind Abbr) +u))).(\lambda (H13: (csuba g d1 x0)).(let H14 \def (refl_equal nat (r (Bind +Abbr) n)) in (let H15 \def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x +(CHead x0 (Bind Abbr) u))) H12 (r (Bind Abbr) n) H14) in (ex_intro2 C +(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) +u))) (\lambda (d2: C).(csuba g d1 d2)) x0 (drop_drop (Bind Abst) n x (CHead +x0 (Bind Abbr) u) H15 t) H13)))))) H11)) c2 H9)))) H8)) (\lambda (H8: (ex4_3 +C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 +(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g +c d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t (asucc +g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: +A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: +T).(\lambda (_: A).(csuba g c d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g c t (asucc g a))))) (\lambda (d2: C).(\lambda (u2: +T).(\lambda (a: A).(arity g d2 u2 a)))) (ex2 C (\lambda (d2: C).(drop (S n) O +c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda +(x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H9: (eq C c2 (CHead x0 +(Bind Abbr) x1))).(\lambda (H10: (csuba g c x0)).(\lambda (_: (arity g c t +(asucc g x2))).(\lambda (_: (arity g x0 x1 x2)).(eq_ind_r C (CHead x0 (Bind +Abbr) x1) (\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 +(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H13 \def (H c d1 u +H6 g x0 H10) in (ex2_ind C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind +Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(drop (S +n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abbr) u))) (\lambda (d2: +C).(csuba g d1 d2))) (\lambda (x: C).(\lambda (H14: (drop n O x0 (CHead x +(Bind Abbr) u))).(\lambda (H15: (csuba g d1 x)).(let H16 \def (refl_equal nat +(r (Bind Abbr) n)) in (let H17 \def (eq_ind nat n (\lambda (n0: nat).(drop n0 +O x0 (CHead x (Bind Abbr) u))) H14 (r (Bind Abbr) n) H16) in (ex_intro2 C +(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind +Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x (drop_drop (Bind Abbr) n x0 +(CHead x (Bind Abbr) u) H17 x1) H15)))))) H13)) c2 H9)))))))) H8)) H7))))) +(\lambda (H5: (csuba g (CHead c (Bind Void) t) c2)).(\lambda (H6: (drop (r +(Bind Void) n) O c (CHead d1 (Bind Abbr) u))).(let H_x \def (csuba_gen_void g +c c2 t H5) in (let H7 \def H_x in (ex2_3_ind B C T (\lambda (b0: B).(\lambda +(d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind b0) u2))))) (\lambda (_: +B).(\lambda (d2: C).(\lambda (_: T).(csuba g c d2)))) (ex2 C (\lambda (d2: +C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 +d2))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H8: (eq C +c2 (CHead x1 (Bind x0) x2))).(\lambda (H9: (csuba g c x1)).(eq_ind_r C (CHead +x1 (Bind x0) x2) (\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0 +(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H10 \def +(H c d1 u H6 g x1 H9) in (ex2_ind C (\lambda (d2: C).(drop n O x1 (CHead d2 +(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: +C).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abbr) u))) (\lambda +(d2: C).(csuba g d1 d2))) (\lambda (x: C).(\lambda (H11: (drop n O x1 (CHead +x (Bind Abbr) u))).(\lambda (H12: (csuba g d1 x)).(let H13 \def (refl_equal +nat (r (Bind Abbr) n)) in (let H14 \def (eq_ind nat n (\lambda (n0: +nat).(drop n0 O x1 (CHead x (Bind Abbr) u))) H11 (r (Bind Abbr) n) H13) in +(ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2 +(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x (drop_drop (Bind x0) n +x1 (CHead x (Bind Abbr) u) H14 x2) H12)))))) H10)) c2 H8)))))) H7))))) b H3 +H4)))) (\lambda (f: F).(\lambda (H3: (csuba g (CHead c (Flat f) t) +c2)).(\lambda (H4: (drop (r (Flat f) n) O c (CHead d1 (Bind Abbr) u))).(let +H_x \def (csuba_gen_flat g c c2 t f H3) in (let H5 \def H_x in (ex2_2_ind C T +(\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Flat f) u2)))) (\lambda +(d2: C).(\lambda (_: T).(csuba g c d2))) (ex2 C (\lambda (d2: C).(drop (S n) +O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda +(x0: C).(\lambda (x1: T).(\lambda (H6: (eq C c2 (CHead x0 (Flat f) +x1))).(\lambda (H7: (csuba g c x0)).(eq_ind_r C (CHead x0 (Flat f) x1) +(\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind +Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H8 \def (H0 d1 u H4 g x0 +H7) in (ex2_ind C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind Abbr) +u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(drop (S n) O +(CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g +d1 d2))) (\lambda (x: C).(\lambda (H9: (drop (S n) O x0 (CHead x (Bind Abbr) +u))).(\lambda (H10: (csuba g d1 x)).(ex_intro2 C (\lambda (d2: C).(drop (S n) +O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g +d1 d2)) x (drop_drop (Flat f) n x0 (CHead x (Bind Abbr) u) H9 x1) H10)))) +H8)) c2 H6))))) H5)))))) k H2 (drop_gen_drop k c (CHead d1 (Bind Abbr) u) t n +H1)))))))))))) c1)))) i). +(* COMMENTS +Initial nodes: 3648 +END *) + +theorem csuba_drop_abst: + \forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u1: T).((drop i +O c1 (CHead d1 (Bind Abst) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba +g c1 c2) \to (or (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst) +u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abbr) +u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g +a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +a))))))))))))) +\def + \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (d1: +C).(\forall (u1: T).((drop n O c1 (CHead d1 (Bind Abst) u1)) \to (\forall (g: +G).(\forall (c2: C).((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(drop n +O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C +T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O c2 (CHead d2 +(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g +d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 +(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 +u2 a)))))))))))))) (\lambda (c1: C).(\lambda (d1: C).(\lambda (u1: +T).(\lambda (H: (drop O O c1 (CHead d1 (Bind Abst) u1))).(\lambda (g: +G).(\lambda (c2: C).(\lambda (H0: (csuba g c1 c2)).(let H1 \def (eq_ind C c1 +(\lambda (c: C).(csuba g c c2)) H0 (CHead d1 (Bind Abst) u1) (drop_gen_refl +c1 (CHead d1 (Bind Abst) u1) H)) in (let H_x \def (csuba_gen_abst g d1 c2 u1 +H1) in (let H2 \def H_x in (or_ind (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 +(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda +(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) +u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g +a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +a))))) (or (ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) u1))) +(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda +(u2: T).(\lambda (_: A).(drop O O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda +(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: +C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (H3: +(ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: +C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind +Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: +C).(drop O O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 +d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O +O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda +(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: +A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda +(a: A).(arity g d2 u2 a)))))) (\lambda (x: C).(\lambda (H4: (eq C c2 (CHead x +(Bind Abst) u1))).(\lambda (H5: (csuba g d1 x)).(eq_ind_r C (CHead x (Bind +Abst) u1) (\lambda (c: C).(or (ex2 C (\lambda (d2: C).(drop O O c (CHead d2 +(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda +(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O c (CHead d2 (Bind Abbr) +u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g +a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +a))))))) (or_introl (ex2 C (\lambda (d2: C).(drop O O (CHead x (Bind Abst) +u1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T +A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O (CHead x (Bind +Abst) u1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: +T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: +T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda +(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: +C).(drop O O (CHead x (Bind Abst) u1) (CHead d2 (Bind Abst) u1))) (\lambda +(d2: C).(csuba g d1 d2)) x (drop_refl (CHead x (Bind Abst) u1)) H5)) c2 +H4)))) H3)) (\lambda (H3: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: +T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: +C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: +C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))).(ex4_3_ind C +T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 +(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g +d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 +(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 +u2 a)))) (or (ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) u1))) +(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda +(u2: T).(\lambda (_: A).(drop O O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda +(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: +C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x0: +C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H4: (eq C c2 (CHead x0 (Bind +Abbr) x1))).(\lambda (H5: (csuba g d1 x0)).(\lambda (H6: (arity g d1 u1 +(asucc g x2))).(\lambda (H7: (arity g x0 x1 x2)).(eq_ind_r C (CHead x0 (Bind +Abbr) x1) (\lambda (c: C).(or (ex2 C (\lambda (d2: C).(drop O O c (CHead d2 +(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda +(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O c (CHead d2 (Bind Abbr) +u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g +a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +a))))))) (or_intror (ex2 C (\lambda (d2: C).(drop O O (CHead x0 (Bind Abbr) +x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T +A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O (CHead x0 (Bind +Abbr) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: +T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: +T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda +(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C T A (\lambda +(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O (CHead x0 (Bind Abbr) x1) +(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: +A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity +g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 a)))) x0 x1 x2 (drop_refl (CHead x0 (Bind Abbr) x1)) H5 H6 +H7)) c2 H4)))))))) H3)) H2))))))))))) (\lambda (n: nat).(\lambda (H: +((\forall (c1: C).(\forall (d1: C).(\forall (u1: T).((drop n O c1 (CHead d1 +(Bind Abst) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba g c1 c2) \to +(or (ex2 C (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abst) u1))) (\lambda +(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: +T).(\lambda (_: A).(drop n O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: +C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: +C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +a))))))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (d1: +C).(\forall (u1: T).((drop (S n) O c (CHead d1 (Bind Abst) u1)) \to (\forall +(g: G).(\forall (c2: C).((csuba g c c2) \to (or (ex2 C (\lambda (d2: C).(drop +(S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) +(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O +c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda +(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: +A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda +(a: A).(arity g d2 u2 a))))))))))))) (\lambda (n0: nat).(\lambda (d1: +C).(\lambda (u1: T).(\lambda (H0: (drop (S n) O (CSort n0) (CHead d1 (Bind +Abst) u1))).(\lambda (g: G).(\lambda (c2: C).(\lambda (_: (csuba g (CSort n0) +c2)).(and3_ind (eq C (CHead d1 (Bind Abst) u1) (CSort n0)) (eq nat (S n) O) +(eq nat O O) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind +Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) +u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g +a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +a)))))) (\lambda (_: (eq C (CHead d1 (Bind Abst) u1) (CSort n0))).(\lambda +(H3: (eq nat (S n) O)).(\lambda (_: (eq nat O O)).(let H5 \def (eq_ind nat (S +n) (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O +\Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind (or (ex2 +C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: +C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda +(_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: +C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: +C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) H5))))) +(drop_gen_sort n0 (S n) O (CHead d1 (Bind Abst) u1) H0))))))))) (\lambda (c: +C).(\lambda (H0: ((\forall (d1: C).(\forall (u1: T).((drop (S n) O c (CHead +d1 (Bind Abst) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba g c c2) \to +(or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) +(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda +(u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) u2))))) +(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) +(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +a)))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (d1: C).(\lambda +(u1: T).(\lambda (H1: (drop (S n) O (CHead c k t) (CHead d1 (Bind Abst) +u1))).(\lambda (g: G).(\lambda (c2: C).(\lambda (H2: (csuba g (CHead c k t) +c2)).(K_ind (\lambda (k0: K).((csuba g (CHead c k0 t) c2) \to ((drop (r k0 n) +O c (CHead d1 (Bind Abst) u1)) \to (or (ex2 C (\lambda (d2: C).(drop (S n) O +c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T +A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead +d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: +A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity +g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 a))))))))) (\lambda (b: B).(\lambda (H3: (csuba g (CHead c +(Bind b) t) c2)).(\lambda (H4: (drop (r (Bind b) n) O c (CHead d1 (Bind Abst) +u1))).(B_ind (\lambda (b0: B).((csuba g (CHead c (Bind b0) t) c2) \to ((drop +(r (Bind b0) n) O c (CHead d1 (Bind Abst) u1)) \to (or (ex2 C (\lambda (d2: +C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 +d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S +n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: +T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: +T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda +(u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))) (\lambda (H5: (csuba g +(CHead c (Bind Abbr) t) c2)).(\lambda (H6: (drop (r (Bind Abbr) n) O c (CHead +d1 (Bind Abst) u1))).(let H_x \def (csuba_gen_abbr g c c2 t H5) in (let H7 +\def H_x in (ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) +(\lambda (d2: C).(csuba g c d2)) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 +(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A +(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 +(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g +d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 +(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 +u2 a)))))) (\lambda (x: C).(\lambda (H8: (eq C c2 (CHead x (Bind Abbr) +t))).(\lambda (H9: (csuba g c x)).(eq_ind_r C (CHead x (Bind Abbr) t) +(\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind +Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind Abbr) +u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g +a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +a))))))) (let H10 \def (H c d1 u1 H6 g x H9) in (or_ind (ex2 C (\lambda (d2: +C).(drop n O x (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) +(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x +(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: +A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity +g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x +(Bind Abbr) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) +(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O +(CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: +C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: +C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda +(H11: (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) u1))) +(\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(drop n O x +(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C +(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) +u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t) +(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: +A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity +g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 a)))))) (\lambda (x0: C).(\lambda (H12: (drop n O x (CHead +x0 (Bind Abst) u1))).(\lambda (H13: (csuba g d1 x0)).(let H14 \def +(refl_equal nat (r (Bind Abbr) n)) in (let H15 \def (eq_ind nat n (\lambda +(n0: nat).(drop n0 O x (CHead x0 (Bind Abst) u1))) H12 (r (Bind Abbr) n) H14) +in (or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) +(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A +(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x +(Bind Abbr) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: +T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: +T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda +(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: +C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u1))) (\lambda +(d2: C).(csuba g d1 d2)) x0 (drop_drop (Bind Abbr) n x (CHead x0 (Bind Abst) +u1) H15 t) H13))))))) H11)) (\lambda (H11: (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abbr) u2))))) +(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) +(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: +A).(drop n O x (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: +T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: +T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda +(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2: +C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u1))) (\lambda +(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: +T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind +Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 +d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc +g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +a)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H12: +(drop n O x (CHead x0 (Bind Abbr) x1))).(\lambda (H13: (csuba g d1 +x0)).(\lambda (H14: (arity g d1 u1 (asucc g x2))).(\lambda (H15: (arity g x0 +x1 x2)).(let H16 \def (refl_equal nat (r (Bind Abbr) n)) in (let H17 \def +(eq_ind nat n (\lambda (n0: nat).(drop n0 O x (CHead x0 (Bind Abbr) x1))) H12 +(r (Bind Abbr) n) H16) in (or_intror (ex2 C (\lambda (d2: C).(drop (S n) O +(CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g +d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop +(S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: +C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: +C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C +T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x +(Bind Abbr) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: +T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: +T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda +(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x0 x1 x2 (drop_drop (Bind Abbr) +n x (CHead x0 (Bind Abbr) x1) H17 t) H13 H14 H15))))))))))) H11)) H10)) c2 +H8)))) H7))))) (\lambda (H5: (csuba g (CHead c (Bind Abst) t) c2)).(\lambda +(H6: (drop (r (Bind Abst) n) O c (CHead d1 (Bind Abst) u1))).(let H_x \def +(csuba_gen_abst g c c2 t H5) in (let H7 \def H_x in (or_ind (ex2 C (\lambda +(d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba g c d2))) +(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 +(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: +A).(csuba g c d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g +c t (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity +g d2 u2 a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind +Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) +u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g +a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +a)))))) (\lambda (H8: (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) +t))) (\lambda (d2: C).(csuba g c d2)))).(ex2_ind C (\lambda (d2: C).(eq C c2 +(CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba g c d2)) (or (ex2 C +(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: +C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda +(_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: +C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: +C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x: +C).(\lambda (H9: (eq C c2 (CHead x (Bind Abst) t))).(\lambda (H10: (csuba g c +x)).(eq_ind_r C (CHead x (Bind Abst) t) (\lambda (c0: C).(or (ex2 C (\lambda +(d2: C).(drop (S n) O c0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba +g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: +A).(drop (S n) O c0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda +(_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: +T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda +(u2: T).(\lambda (a: A).(arity g d2 u2 a))))))) (let H11 \def (H c d1 u1 H6 g +x H10) in (or_ind (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) +u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abbr) u2))))) +(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) +(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (or +(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind +Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) +(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: +A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity +g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 a)))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x +(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C +(\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) u1))) (\lambda (d2: +C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind +Abst) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) +(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O +(CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: +C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: +C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x0: +C).(\lambda (H13: (drop n O x (CHead x0 (Bind Abst) u1))).(\lambda (H14: +(csuba g d1 x0)).(let H15 \def (refl_equal nat (r (Bind Abbr) n)) in (let H16 +\def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x (CHead x0 (Bind Abst) +u1))) H13 (r (Bind Abbr) n) H15) in (or_introl (ex2 C (\lambda (d2: C).(drop +(S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: +C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda +(_: A).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u2))))) +(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) +(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) +(ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 +(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) x0 (drop_drop (Bind Abst) +n x (CHead x0 (Bind Abst) u1) H16 t) H14))))))) H12)) (\lambda (H12: (ex4_3 C +T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 +(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g +d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 +(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 +u2 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: +A).(drop n O x (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: +T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: +T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda +(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2: +C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u1))) (\lambda +(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: +T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind +Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 +d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc +g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +a)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H13: +(drop n O x (CHead x0 (Bind Abbr) x1))).(\lambda (H14: (csuba g d1 +x0)).(\lambda (H15: (arity g d1 u1 (asucc g x2))).(\lambda (H16: (arity g x0 +x1 x2)).(let H17 \def (refl_equal nat (r (Bind Abbr) n)) in (let H18 \def +(eq_ind nat n (\lambda (n0: nat).(drop n0 O x (CHead x0 (Bind Abbr) x1))) H13 +(r (Bind Abbr) n) H17) in (or_intror (ex2 C (\lambda (d2: C).(drop (S n) O +(CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g +d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop +(S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: +C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: +C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C +T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x +(Bind Abst) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: +T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: +T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda +(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x0 x1 x2 (drop_drop (Bind Abst) +n x (CHead x0 (Bind Abbr) x1) H18 t) H14 H15 H16))))))))))) H12)) H11)) c2 +H9)))) H8)) (\lambda (H8: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: +T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: +C).(\lambda (_: T).(\lambda (_: A).(csuba g c d2)))) (\lambda (_: C).(\lambda +(_: T).(\lambda (a: A).(arity g c t (asucc g a))))) (\lambda (d2: C).(\lambda +(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))).(ex4_3_ind C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) +(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g c d2)))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t (asucc g a))))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C +(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: +C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda +(_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: +C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: +C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x0: +C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H9: (eq C c2 (CHead x0 (Bind +Abbr) x1))).(\lambda (H10: (csuba g c x0)).(\lambda (_: (arity g c t (asucc g +x2))).(\lambda (_: (arity g x0 x1 x2)).(eq_ind_r C (CHead x0 (Bind Abbr) x1) +(\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind +Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind Abbr) +u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g +a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +a))))))) (let H13 \def (H c d1 u1 H6 g x0 H10) in (or_ind (ex2 C (\lambda +(d2: C).(drop n O x0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 +d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n +O x0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda +(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: +A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda +(a: A).(arity g d2 u2 a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead +x0 (Bind Abbr) x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 +d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S +n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: +C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: +C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda +(H14: (ex2 C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind Abst) u1))) +(\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(drop n O x0 +(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C +(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind +Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abbr) x1) +(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: +A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity +g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 a)))))) (\lambda (x: C).(\lambda (H15: (drop n O x0 (CHead +x (Bind Abst) u1))).(\lambda (H16: (csuba g d1 x)).(let H17 \def (refl_equal +nat (r (Bind Abbr) n)) in (let H18 \def (eq_ind nat n (\lambda (n0: +nat).(drop n0 O x0 (CHead x (Bind Abst) u1))) H15 (r (Bind Abbr) n) H17) in +(or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abbr) x1) +(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A +(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 +(Bind Abbr) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: +T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: +T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda +(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: +C).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abst) u1))) +(\lambda (d2: C).(csuba g d1 d2)) x (drop_drop (Bind Abbr) n x0 (CHead x +(Bind Abst) u1) H18 x1) H16))))))) H14)) (\lambda (H14: (ex4_3 C T A (\lambda +(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x0 (CHead d2 (Bind Abbr) +u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g +a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: +A).(drop n O x0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: +T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: +T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda +(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2: +C).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abst) u1))) +(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda +(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 +(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g +d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 +(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 +u2 a)))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: A).(\lambda (H15: +(drop n O x0 (CHead x3 (Bind Abbr) x4))).(\lambda (H16: (csuba g d1 +x3)).(\lambda (H17: (arity g d1 u1 (asucc g x5))).(\lambda (H18: (arity g x3 +x4 x5)).(let H19 \def (refl_equal nat (r (Bind Abbr) n)) in (let H20 \def +(eq_ind nat n (\lambda (n0: nat).(drop n0 O x0 (CHead x3 (Bind Abbr) x4))) +H15 (r (Bind Abbr) n) H19) in (or_intror (ex2 C (\lambda (d2: C).(drop (S n) +O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: +C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda +(_: A).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abbr) u2))))) +(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) +(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) +(ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S +n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: +C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: +C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x3 x4 x5 +(drop_drop (Bind Abbr) n x0 (CHead x3 (Bind Abbr) x4) H20 x1) H16 H17 +H18))))))))))) H14)) H13)) c2 H9)))))))) H8)) H7))))) (\lambda (H5: (csuba g +(CHead c (Bind Void) t) c2)).(\lambda (H6: (drop (r (Bind Void) n) O c (CHead +d1 (Bind Abst) u1))).(let H_x \def (csuba_gen_void g c c2 t H5) in (let H7 +\def H_x in (ex2_3_ind B C T (\lambda (b0: B).(\lambda (d2: C).(\lambda (u2: +T).(eq C c2 (CHead d2 (Bind b0) u2))))) (\lambda (_: B).(\lambda (d2: +C).(\lambda (_: T).(csuba g c d2)))) (or (ex2 C (\lambda (d2: C).(drop (S n) +O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C +T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead +d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: +A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity +g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 a)))))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: +T).(\lambda (H8: (eq C c2 (CHead x1 (Bind x0) x2))).(\lambda (H9: (csuba g c +x1)).(eq_ind_r C (CHead x1 (Bind x0) x2) (\lambda (c0: C).(or (ex2 C (\lambda +(d2: C).(drop (S n) O c0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba +g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: +A).(drop (S n) O c0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda +(_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: +T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda +(u2: T).(\lambda (a: A).(arity g d2 u2 a))))))) (let H10 \def (H c d1 u1 H6 g +x1 H9) in (or_ind (ex2 C (\lambda (d2: C).(drop n O x1 (CHead d2 (Bind Abst) +u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(drop n O x1 (CHead d2 (Bind Abbr) +u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g +a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x1 (Bind x0) x2) +(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A +(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x1 +(Bind x0) x2) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: +T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: +T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda +(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (H11: (ex2 C (\lambda +(d2: C).(drop n O x1 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 +d2)))).(ex2_ind C (\lambda (d2: C).(drop n O x1 (CHead d2 (Bind Abst) u1))) +(\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: C).(drop (S n) O +(CHead x1 (Bind x0) x2) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g +d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop +(S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: +C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: +C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x: +C).(\lambda (H12: (drop n O x1 (CHead x (Bind Abst) u1))).(\lambda (H13: +(csuba g d1 x)).(let H14 \def (refl_equal nat (r (Bind Abbr) n)) in (let H15 +\def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x1 (CHead x (Bind Abst) +u1))) H12 (r (Bind Abbr) n) H14) in (or_introl (ex2 C (\lambda (d2: C).(drop +(S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abst) u1))) (\lambda (d2: +C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda +(_: A).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abbr) u2))))) +(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) +(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) +(ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2 +(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) x (drop_drop (Bind x0) n +x1 (CHead x (Bind Abst) u1) H15 x2) H13))))))) H11)) (\lambda (H11: (ex4_3 C +T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x1 (CHead d2 +(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g +d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 +(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 +u2 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: +A).(drop n O x1 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: +T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: +T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda +(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2: +C).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abst) u1))) (\lambda +(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: +T).(\lambda (_: A).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind +Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 +d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc +g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +a)))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: A).(\lambda (H12: +(drop n O x1 (CHead x3 (Bind Abbr) x4))).(\lambda (H13: (csuba g d1 +x3)).(\lambda (H14: (arity g d1 u1 (asucc g x5))).(\lambda (H15: (arity g x3 +x4 x5)).(let H16 \def (refl_equal nat (r (Bind Abbr) n)) in (let H17 \def +(eq_ind nat n (\lambda (n0: nat).(drop n0 O x1 (CHead x3 (Bind Abbr) x4))) +H12 (r (Bind Abbr) n) H16) in (or_intror (ex2 C (\lambda (d2: C).(drop (S n) +O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba +g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: +A).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abbr) u2))))) +(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) +(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) +(ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S +n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: +C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: +C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x3 x4 x5 +(drop_drop (Bind x0) n x1 (CHead x3 (Bind Abbr) x4) H17 x2) H13 H14 +H15))))))))))) H11)) H10)) c2 H8)))))) H7))))) b H3 H4)))) (\lambda (f: +F).(\lambda (H3: (csuba g (CHead c (Flat f) t) c2)).(\lambda (H4: (drop (r +(Flat f) n) O c (CHead d1 (Bind Abst) u1))).(let H_x \def (csuba_gen_flat g c +c2 t f H3) in (let H5 \def H_x in (ex2_2_ind C T (\lambda (d2: C).(\lambda +(u2: T).(eq C c2 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g c d2))) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 +(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda +(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind +Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 +d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc +g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +a)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C c2 (CHead x0 +(Flat f) x1))).(\lambda (H7: (csuba g c x0)).(eq_ind_r C (CHead x0 (Flat f) +x1) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 +(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda +(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind +Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 +d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc +g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +a))))))) (let H8 \def (H0 d1 u1 H4 g x0 H7) in (or_ind (ex2 C (\lambda (d2: +C).(drop (S n) O x0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 +d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S +n) O x0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: +T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: +T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda +(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (or (ex2 C (\lambda (d2: +C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u1))) (\lambda +(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: +T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) +u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g +a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +a)))))) (\lambda (H9: (ex2 C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 +(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda +(d2: C).(drop (S n) O x0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba +g d1 d2)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) +(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A +(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 +(Flat f) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: +T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: +T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda +(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x: C).(\lambda (H10: +(drop (S n) O x0 (CHead x (Bind Abst) u1))).(\lambda (H11: (csuba g d1 +x)).(or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) +(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A +(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 +(Flat f) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: +T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: +T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda +(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: +C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u1))) (\lambda +(d2: C).(csuba g d1 d2)) x (drop_drop (Flat f) n x0 (CHead x (Bind Abst) u1) +H10 x1) H11))))) H9)) (\lambda (H9: (ex4_3 C T A (\lambda (d2: C).(\lambda +(u2: T).(\lambda (_: A).(drop (S n) O x0 (CHead d2 (Bind Abbr) u2))))) +(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) +(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: +A).(drop (S n) O x0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda +(_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: +T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda +(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2: +C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u1))) (\lambda +(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: +T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) +u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g +a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +a)))))) (\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: A).(\lambda (H10: +(drop (S n) O x0 (CHead x2 (Bind Abbr) x3))).(\lambda (H11: (csuba g d1 +x2)).(\lambda (H12: (arity g d1 u1 (asucc g x4))).(\lambda (H13: (arity g x2 +x3 x4)).(or_intror (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) +x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T +A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 +(Flat f) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: +T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: +T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda +(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C T A (\lambda +(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) +(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: +A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity +g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 a)))) x2 x3 x4 (drop_drop (Flat f) n x0 (CHead x2 (Bind +Abbr) x3) H10 x1) H11 H12 H13))))))))) H9)) H8)) c2 H6))))) H5)))))) k H2 +(drop_gen_drop k c (CHead d1 (Bind Abst) u1) t n H1)))))))))))) c1)))) i). +(* COMMENTS +Initial nodes: 12528 +END *) + +theorem csuba_drop_abst_rev: + \forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).((drop i +O c1 (CHead d1 (Bind Abst) u)) \to (\forall (g: G).(\forall (c2: C).((csuba g +c2 c1) \to (or (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst) +u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda +(u2: T).(drop i O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda +(_: T).(csuba g d2 d1)))))))))))) +\def + \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (d1: +C).(\forall (u: T).((drop n O c1 (CHead d1 (Bind Abst) u)) \to (\forall (g: +G).(\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: C).(drop n +O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(drop n O c2 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))))) (\lambda (c1: +C).(\lambda (d1: C).(\lambda (u: T).(\lambda (H: (drop O O c1 (CHead d1 (Bind +Abst) u))).(\lambda (g: G).(\lambda (c2: C).(\lambda (H0: (csuba g c2 +c1)).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csuba g c2 c)) H0 (CHead d1 +(Bind Abst) u) (drop_gen_refl c1 (CHead d1 (Bind Abst) u) H)) in (let H_x +\def (csuba_gen_abst_rev g d1 c2 u H1) in (let H2 \def H_x in (or_ind (ex2 C +(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba +g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or +(ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: +C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O +c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1))))) (\lambda (H3: (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) +u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(eq C c2 +(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C +(\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: +C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O +c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1))))) (\lambda (x: C).(\lambda (H4: (eq C c2 (CHead x (Bind Abst) +u))).(\lambda (H5: (csuba g x d1)).(eq_ind_r C (CHead x (Bind Abst) u) +(\lambda (c: C).(or (ex2 C (\lambda (d2: C).(drop O O c (CHead d2 (Bind Abst) +u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda +(u2: T).(drop O O c (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda +(_: T).(csuba g d2 d1)))))) (or_introl (ex2 C (\lambda (d2: C).(drop O O +(CHead x (Bind Abst) u) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g +d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O (CHead x +(Bind Abst) u) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop O O (CHead x (Bind +Abst) u) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x +(drop_refl (CHead x (Bind Abst) u)) H5)) c2 H4)))) H3)) (\lambda (H3: (ex2_2 +C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda +(d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1))) (or (ex2 C (\lambda (d2: C).(drop O O c2 +(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(drop O O c2 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: +C).(\lambda (x1: T).(\lambda (H4: (eq C c2 (CHead x0 (Bind Void) +x1))).(\lambda (H5: (csuba g x0 d1)).(eq_ind_r C (CHead x0 (Bind Void) x1) +(\lambda (c: C).(or (ex2 C (\lambda (d2: C).(drop O O c (CHead d2 (Bind Abst) +u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda +(u2: T).(drop O O c (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda +(_: T).(csuba g d2 d1)))))) (or_intror (ex2 C (\lambda (d2: C).(drop O O +(CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba +g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O (CHead x0 +(Bind Void) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: +T).(drop O O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) (\lambda +(d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1 (drop_refl (CHead x0 (Bind +Void) x1)) H5)) c2 H4))))) H3)) H2))))))))))) (\lambda (n: nat).(\lambda (H: +((\forall (c1: C).(\forall (d1: C).(\forall (u: T).((drop n O c1 (CHead d1 +(Bind Abst) u)) \to (\forall (g: G).(\forall (c2: C).((csuba g c2 c1) \to (or +(ex2 C (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: +C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O +c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1)))))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (d1: +C).(\forall (u: T).((drop (S n) O c (CHead d1 (Bind Abst) u)) \to (\forall +(g: G).(\forall (c2: C).((csuba g c2 c) \to (or (ex2 C (\lambda (d2: C).(drop +(S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))))))) +(\lambda (n0: nat).(\lambda (d1: C).(\lambda (u: T).(\lambda (H0: (drop (S n) +O (CSort n0) (CHead d1 (Bind Abst) u))).(\lambda (g: G).(\lambda (c2: +C).(\lambda (_: (csuba g c2 (CSort n0))).(and3_ind (eq C (CHead d1 (Bind +Abst) u) (CSort n0)) (eq nat (S n) O) (eq nat O O) (or (ex2 C (\lambda (d2: +C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 +d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) +(\lambda (_: (eq C (CHead d1 (Bind Abst) u) (CSort n0))).(\lambda (H3: (eq +nat (S n) O)).(\lambda (_: (eq nat O O)).(let H5 \def (eq_ind nat (S n) +(\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O +\Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind (or (ex2 +C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: +C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) +O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g +d2 d1))))) H5))))) (drop_gen_sort n0 (S n) O (CHead d1 (Bind Abst) u) +H0))))))))) (\lambda (c: C).(\lambda (H0: ((\forall (d1: C).(\forall (u: +T).((drop (S n) O c (CHead d1 (Bind Abst) u)) \to (\forall (g: G).(\forall +(c2: C).((csuba g c2 c) \to (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 +(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) +u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))))).(\lambda +(k: K).(\lambda (t: T).(\lambda (d1: C).(\lambda (u: T).(\lambda (H1: (drop +(S n) O (CHead c k t) (CHead d1 (Bind Abst) u))).(\lambda (g: G).(\lambda +(c2: C).(\lambda (H2: (csuba g c2 (CHead c k t))).(K_ind (\lambda (k0: +K).((csuba g c2 (CHead c k0 t)) \to ((drop (r k0 n) O c (CHead d1 (Bind Abst) +u)) \to (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) +u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda +(u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1)))))))) (\lambda (b: B).(\lambda (H3: +(csuba g c2 (CHead c (Bind b) t))).(\lambda (H4: (drop (r (Bind b) n) O c +(CHead d1 (Bind Abst) u))).(B_ind (\lambda (b0: B).((csuba g c2 (CHead c +(Bind b0) t)) \to ((drop (r (Bind b0) n) O c (CHead d1 (Bind Abst) u)) \to +(or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) +(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda +(_: T).(csuba g d2 d1)))))))) (\lambda (H5: (csuba g c2 (CHead c (Bind Abbr) +t))).(\lambda (H6: (drop (r (Bind Abbr) n) O c (CHead d1 (Bind Abst) +u))).(let H_x \def (csuba_gen_abbr_rev g c c2 t H5) in (let H7 \def H_x in +(or3_ind (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) (\lambda +(d2: C).(csuba g d2 c))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: +T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: +C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 c)))) (\lambda (d2: +C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t a))))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 c)))) (or (ex2 C (\lambda (d2: +C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 +d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) +(\lambda (H8: (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) +(\lambda (d2: C).(csuba g d2 c)))).(ex2_ind C (\lambda (d2: C).(eq C c2 +(CHead d2 (Bind Abbr) t))) (\lambda (d2: C).(csuba g d2 c)) (or (ex2 C +(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: +C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) +O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g +d2 d1))))) (\lambda (x: C).(\lambda (H9: (eq C c2 (CHead x (Bind Abbr) +t))).(\lambda (H10: (csuba g x c)).(eq_ind_r C (CHead x (Bind Abbr) t) +(\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind +Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 (Bind Void) u2)))) (\lambda +(d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H11 \def (H c d1 u H6 g x +H10) in (or_ind (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) +u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda +(u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda +(_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x +(Bind Abbr) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind +Abbr) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba +g d2 d1))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 +(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: +C).(drop n O x (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) +(or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 +(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda +(x0: C).(\lambda (H13: (drop n O x (CHead x0 (Bind Abst) u))).(\lambda (H14: +(csuba g x0 d1)).(or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x +(Bind Abbr) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind +Abbr) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba +g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) +t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x0 (drop_drop +(Bind Abbr) n x (CHead x0 (Bind Abst) u) H13 t) H14))))) H12)) (\lambda (H12: +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind +C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) +u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or (ex2 C (\lambda +(d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u))) +(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: +C).(\lambda (x1: T).(\lambda (H13: (drop n O x (CHead x0 (Bind Void) +x1))).(\lambda (H14: (csuba g x0 d1)).(or_intror (ex2 C (\lambda (d2: +C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u))) (\lambda +(d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop +(S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: +C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1 +(drop_drop (Bind Abbr) n x (CHead x0 (Bind Void) x1) H13 t) H14)))))) H12)) +H11)) c2 H9)))) H8)) (\lambda (H8: (ex4_3 C T A (\lambda (d2: C).(\lambda +(u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: +C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 c)))) (\lambda (d2: +C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t a)))))).(ex4_3_ind C T A +(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind +Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 +c)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc +g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t a)))) +(or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) +(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda +(_: T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: +A).(\lambda (H9: (eq C c2 (CHead x0 (Bind Abst) x1))).(\lambda (H10: (csuba g +x0 c)).(\lambda (_: (arity g x0 x1 (asucc g x2))).(\lambda (_: (arity g c t +x2)).(eq_ind_r C (CHead x0 (Bind Abst) x1) (\lambda (c0: C).(or (ex2 C +(\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: +C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) +O c0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g +d2 d1)))))) (let H13 \def (H c d1 u H6 g x0 H10) in (or_ind (ex2 C (\lambda +(d2: C).(drop n O x0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 +d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or +(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 +(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda +(H14: (ex2 C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind Abst) u))) +(\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop n O x0 +(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C +(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind +Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda +(x: C).(\lambda (H15: (drop n O x0 (CHead x (Bind Abst) u))).(\lambda (H16: +(csuba g x d1)).(or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 +(Bind Abst) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind +Abst) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x0 +(Bind Abst) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) +x (drop_drop (Bind Abst) n x0 (CHead x (Bind Abst) u) H15 x1) H16))))) H14)) +(\lambda (H14: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 +(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead +d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or +(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 +(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda +(x3: C).(\lambda (x4: T).(\lambda (H15: (drop n O x0 (CHead x3 (Bind Void) +x4))).(\lambda (H16: (csuba g x3 d1)).(or_intror (ex2 C (\lambda (d2: +C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u))) +(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda +(d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x3 x4 +(drop_drop (Bind Abst) n x0 (CHead x3 (Bind Void) x4) H15 x1) H16)))))) H14)) +H13)) c2 H9)))))))) H8)) (\lambda (H8: (ex2_2 C T (\lambda (d2: C).(\lambda +(u2: T).(eq C c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 c))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(eq C +c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +c))) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) +(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda +(_: T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H9: +(eq C c2 (CHead x0 (Bind Void) x1))).(\lambda (H10: (csuba g x0 c)).(eq_ind_r +C (CHead x0 (Bind Void) x1) (\lambda (c0: C).(or (ex2 C (\lambda (d2: +C).(drop (S n) O c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 +d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let +H11 \def (H c d1 u H6 g x0 H10) in (or_ind (ex2 C (\lambda (d2: C).(drop n O +x0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: +C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u))) +(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H12: (ex2 C +(\lambda (d2: C).(drop n O x0 (CHead d2 (Bind Abst) u))) (\lambda (d2: +C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop n O x0 (CHead d2 +(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: +C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u))) +(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda +(H13: (drop n O x0 (CHead x (Bind Abst) u))).(\lambda (H14: (csuba g x +d1)).(or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) +x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) +(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) +(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop +(Bind Void) n x0 (CHead x (Bind Abst) u) H13 x1) H14))))) H12)) (\lambda +(H12: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead +d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or +(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 +(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda +(x2: C).(\lambda (x3: T).(\lambda (H13: (drop n O x0 (CHead x2 (Bind Void) +x3))).(\lambda (H14: (csuba g x2 d1)).(or_intror (ex2 C (\lambda (d2: +C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u))) +(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda +(d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x2 x3 +(drop_drop (Bind Void) n x0 (CHead x2 (Bind Void) x3) H13 x1) H14)))))) H12)) +H11)) c2 H9))))) H8)) H7))))) (\lambda (H5: (csuba g c2 (CHead c (Bind Abst) +t))).(\lambda (H6: (drop (r (Bind Abst) n) O c (CHead d1 (Bind Abst) +u))).(let H_x \def (csuba_gen_abst_rev g c c2 t H5) in (let H7 \def H_x in +(or_ind (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda +(d2: C).(csuba g d2 c))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C +c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +c)))) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) +(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda +(_: T).(csuba g d2 d1))))) (\lambda (H8: (ex2 C (\lambda (d2: C).(eq C c2 +(CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba g d2 c)))).(ex2_ind C +(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba +g d2 c)) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) +u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda +(u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda (H9: (eq C c2 +(CHead x (Bind Abst) t))).(\lambda (H10: (csuba g x c)).(eq_ind_r C (CHead x +(Bind Abst) t) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 +(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 (Bind Void) +u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H11 \def (H +c d1 u H6 g x H10) in (or_ind (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 +(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: C).(drop (S n) +O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba +g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead +x (Bind Abst) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x +(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C +(\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) u))) (\lambda (d2: +C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind +Abst) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 +C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abst) t) +(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1))))) (\lambda (x0: C).(\lambda (H13: (drop n O x (CHead x0 (Bind Abst) +u))).(\lambda (H14: (csuba g x0 d1)).(or_introl (ex2 C (\lambda (d2: C).(drop +(S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: +C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) +O (CHead x (Bind Abst) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S +n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: +C).(csuba g d2 d1)) x0 (drop_drop (Bind Abst) n x (CHead x0 (Bind Abst) u) +H13 t) H14))))) H12)) (\lambda (H12: (ex2_2 C T (\lambda (d2: C).(\lambda +(u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda +(_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: +T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind +Abst) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 +C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abst) t) +(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H13: (drop n O x (CHead +x0 (Bind Void) x1))).(\lambda (H14: (csuba g x0 d1)).(or_intror (ex2 C +(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) +u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda +(u2: T).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda +(d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1 +(drop_drop (Bind Abst) n x (CHead x0 (Bind Void) x1) H13 t) H14)))))) H12)) +H11)) c2 H9)))) H8)) (\lambda (H8: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(eq C c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 c))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(eq C +c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +c))) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) +(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda +(_: T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H9: +(eq C c2 (CHead x0 (Bind Void) x1))).(\lambda (H10: (csuba g x0 c)).(eq_ind_r +C (CHead x0 (Bind Void) x1) (\lambda (c0: C).(or (ex2 C (\lambda (d2: +C).(drop (S n) O c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 +d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let +H11 \def (H c d1 u H6 g x0 H10) in (or_ind (ex2 C (\lambda (d2: C).(drop n O +x0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: +C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u))) +(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H12: (ex2 C +(\lambda (d2: C).(drop n O x0 (CHead d2 (Bind Abst) u))) (\lambda (d2: +C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop n O x0 (CHead d2 +(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: +C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u))) +(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda +(H13: (drop n O x0 (CHead x (Bind Abst) u))).(\lambda (H14: (csuba g x +d1)).(or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) +x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) +(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) +(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop +(Bind Void) n x0 (CHead x (Bind Abst) u) H13 x1) H14))))) H12)) (\lambda +(H12: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead +d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or +(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 +(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda +(x2: C).(\lambda (x3: T).(\lambda (H13: (drop n O x0 (CHead x2 (Bind Void) +x3))).(\lambda (H14: (csuba g x2 d1)).(or_intror (ex2 C (\lambda (d2: +C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u))) +(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda +(d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x2 x3 +(drop_drop (Bind Void) n x0 (CHead x2 (Bind Void) x3) H13 x1) H14)))))) H12)) +H11)) c2 H9))))) H8)) H7))))) (\lambda (H5: (csuba g c2 (CHead c (Bind Void) +t))).(\lambda (H6: (drop (r (Bind Void) n) O c (CHead d1 (Bind Abst) +u))).(let H_x \def (csuba_gen_void_rev g c c2 t H5) in (let H7 \def H_x in +(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Void) t))) (\lambda (d2: +C).(csuba g d2 c)) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 +(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda +(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda (H8: (eq +C c2 (CHead x (Bind Void) t))).(\lambda (H9: (csuba g x c)).(eq_ind_r C +(CHead x (Bind Void) t) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S +n) O c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 +C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 (Bind Void) +u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H10 \def (H +c d1 u H6 g x H9) in (or_ind (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 +(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: C).(drop (S n) +O (CHead x (Bind Void) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba +g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead +x (Bind Void) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1))))) (\lambda (H11: (ex2 C (\lambda (d2: C).(drop n O x +(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C +(\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) u))) (\lambda (d2: +C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind +Void) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 +C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Void) t) +(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1))))) (\lambda (x0: C).(\lambda (H12: (drop n O x (CHead x0 (Bind Abst) +u))).(\lambda (H13: (csuba g x0 d1)).(or_introl (ex2 C (\lambda (d2: C).(drop +(S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: +C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) +O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S +n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: +C).(csuba g d2 d1)) x0 (drop_drop (Bind Void) n x (CHead x0 (Bind Abst) u) +H12 t) H13))))) H11)) (\lambda (H11: (ex2_2 C T (\lambda (d2: C).(\lambda +(u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda +(_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: +T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind +Void) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 +C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Void) t) +(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H12: (drop n O x (CHead +x0 (Bind Void) x1))).(\lambda (H13: (csuba g x0 d1)).(or_intror (ex2 C +(\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abst) +u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda +(u2: T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda +(d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1 +(drop_drop (Bind Void) n x (CHead x0 (Bind Void) x1) H12 t) H13)))))) H11)) +H10)) c2 H8)))) H7))))) b H3 H4)))) (\lambda (f: F).(\lambda (H3: (csuba g c2 +(CHead c (Flat f) t))).(\lambda (H4: (drop (r (Flat f) n) O c (CHead d1 (Bind +Abst) u))).(let H_x \def (csuba_gen_flat_rev g c c2 t f H3) in (let H5 \def +H_x in (ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 +(Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 c))) (or (ex2 C +(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: +C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) +O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g +d2 d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C c2 (CHead x0 +(Flat f) x1))).(\lambda (H7: (csuba g x0 c)).(eq_ind_r C (CHead x0 (Flat f) +x1) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 +(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 (Bind Void) u2)))) (\lambda +(d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H8 \def (H0 d1 u H4 g x0 +H7) in (or_ind (ex2 C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind Abst) +u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda +(u2: T).(drop (S n) O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: C).(drop (S n) +O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g +d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 +(Flat f) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1))))) (\lambda (H9: (ex2 C (\lambda (d2: C).(drop (S n) O x0 +(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C +(\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind Abst) u))) (\lambda (d2: +C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat +f) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C +T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f) x1) +(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1))))) (\lambda (x: C).(\lambda (H10: (drop (S n) O x0 (CHead x (Bind Abst) +u))).(\lambda (H11: (csuba g x d1)).(or_introl (ex2 C (\lambda (d2: C).(drop +(S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: +C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) +O (CHead x0 (Flat f) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S +n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: +C).(csuba g d2 d1)) x (drop_drop (Flat f) n x0 (CHead x (Bind Abst) u) H10 +x1) H11))))) H9)) (\lambda (H9: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda +(_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda +(_: T).(csuba g d2 d1))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 +(Flat f) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f) +x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1))))) (\lambda (x2: C).(\lambda (x3: T).(\lambda (H10: (drop (S n) O x0 +(CHead x2 (Bind Void) x3))).(\lambda (H11: (csuba g x2 d1)).(or_intror (ex2 C +(\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) +u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda +(u2: T).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda +(d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x2 x3 +(drop_drop (Flat f) n x0 (CHead x2 (Bind Void) x3) H10 x1) H11)))))) H9)) +H8)) c2 H6))))) H5)))))) k H2 (drop_gen_drop k c (CHead d1 (Bind Abst) u) t n +H1)))))))))))) c1)))) i). +(* COMMENTS +Initial nodes: 11438 +END *) + +theorem csuba_drop_abbr_rev: + \forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u1: T).((drop i +O c1 (CHead d1 (Bind Abbr) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba +g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr) +u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abst) +u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) +(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g +a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop i O c2 (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))))))) +\def + \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (d1: +C).(\forall (u1: T).((drop n O c1 (CHead d1 (Bind Abbr) u1)) \to (\forall (g: +G).(\forall (c2: C).((csuba g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(drop n +O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C +T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O c2 (CHead d2 +(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g +d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 +u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O c2 (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1))))))))))))) (\lambda (c1: C).(\lambda (d1: C).(\lambda (u1: T).(\lambda +(H: (drop O O c1 (CHead d1 (Bind Abbr) u1))).(\lambda (g: G).(\lambda (c2: +C).(\lambda (H0: (csuba g c2 c1)).(let H1 \def (eq_ind C c1 (\lambda (c: +C).(csuba g c2 c)) H0 (CHead d1 (Bind Abbr) u1) (drop_gen_refl c1 (CHead d1 +(Bind Abbr) u1) H)) in (let H_x \def (csuba_gen_abbr_rev g d1 c2 u1 H1) in +(let H2 \def H_x in (or3_ind (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind +Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abst) u2))))) +(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 +C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: +C).(drop O O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 +d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O +O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda +(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop O O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1))))) (\lambda (H3: (ex2 C (\lambda (d2: C).(eq C c2 (CHead +d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda +(d2: C).(eq C c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 +d1)) (or3 (ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abbr) u1))) +(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda +(u2: T).(\lambda (_: A).(drop O O c2 (CHead d2 (Bind Abst) u2))))) (\lambda +(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: +C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(drop O O c2 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda +(H4: (eq C c2 (CHead x (Bind Abbr) u1))).(\lambda (H5: (csuba g x +d1)).(eq_ind_r C (CHead x (Bind Abbr) u1) (\lambda (c: C).(or3 (ex2 C +(\lambda (d2: C).(drop O O c (CHead d2 (Bind Abbr) u1))) (\lambda (d2: +C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda +(_: A).(drop O O c (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda +(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: +T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda +(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(drop O O c (CHead d2 (Bind Void) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1)))))) (or3_intro0 (ex2 C (\lambda (d2: +C).(drop O O (CHead x (Bind Abbr) u1) (CHead d2 (Bind Abbr) u1))) (\lambda +(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: +T).(\lambda (_: A).(drop O O (CHead x (Bind Abbr) u1) (CHead d2 (Bind Abst) +u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) +(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g +a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O (CHead x (Bind Abbr) +u1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1)))) (ex_intro2 C (\lambda (d2: C).(drop O O (CHead x (Bind Abbr) u1) +(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_refl +(CHead x (Bind Abbr) u1)) H5)) c2 H4)))) H3)) (\lambda (H3: (ex4_3 C T A +(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind +Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 +d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 +u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: +A).(eq C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: +T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: +T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda +(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C (\lambda (d2: +C).(drop O O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 +d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O +O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda +(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop O O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: +A).(\lambda (H4: (eq C c2 (CHead x0 (Bind Abst) x1))).(\lambda (H5: (csuba g +x0 d1)).(\lambda (H6: (arity g x0 x1 (asucc g x2))).(\lambda (H7: (arity g d1 +u1 x2)).(eq_ind_r C (CHead x0 (Bind Abst) x1) (\lambda (c: C).(or3 (ex2 C +(\lambda (d2: C).(drop O O c (CHead d2 (Bind Abbr) u1))) (\lambda (d2: +C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda +(_: A).(drop O O c (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda +(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: +T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda +(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(drop O O c (CHead d2 (Bind Void) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1)))))) (or3_intro1 (ex2 C (\lambda (d2: +C).(drop O O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abbr) u1))) (\lambda +(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: +T).(\lambda (_: A).(drop O O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) +u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) +(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g +a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O (CHead x0 (Bind Abst) +x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1)))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: +A).(drop O O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) (\lambda +(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: +C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x0 x1 x2 +(drop_refl (CHead x0 (Bind Abst) x1)) H5 H6 H7)) c2 H4)))))))) H3)) (\lambda +(H3: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind +C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C (\lambda (d2: +C).(drop O O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 +d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O +O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda +(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop O O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H4: (eq C +c2 (CHead x0 (Bind Void) x1))).(\lambda (H5: (csuba g x0 d1)).(eq_ind_r C +(CHead x0 (Bind Void) x1) (\lambda (c: C).(or3 (ex2 C (\lambda (d2: C).(drop +O O c (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C +T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O c (CHead d2 +(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g +d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 +u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O c (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) +(or3_intro2 (ex2 C (\lambda (d2: C).(drop O O (CHead x0 (Bind Void) x1) +(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A +(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O (CHead x0 (Bind +Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: +T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: +T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda +(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(drop O O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) +u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T +(\lambda (d2: C).(\lambda (u2: T).(drop O O (CHead x0 (Bind Void) x1) (CHead +d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 +x1 (drop_refl (CHead x0 (Bind Void) x1)) H5)) c2 H4))))) H3)) H2))))))))))) +(\lambda (n: nat).(\lambda (H: ((\forall (c1: C).(\forall (d1: C).(\forall +(u1: T).((drop n O c1 (CHead d1 (Bind Abbr) u1)) \to (\forall (g: G).(\forall +(c2: C).((csuba g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(drop n O c2 (CHead +d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda +(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O c2 (CHead d2 (Bind Abst) +u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) +(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g +a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O c2 (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1)))))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (d1: +C).(\forall (u1: T).((drop (S n) O c (CHead d1 (Bind Abbr) u1)) \to (\forall +(g: G).(\forall (c2: C).((csuba g c2 c) \to (or3 (ex2 C (\lambda (d2: +C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 +d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S +n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: +T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: +T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda +(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda +(d2: C).(\lambda (_: T).(csuba g d2 d1)))))))))))) (\lambda (n0: +nat).(\lambda (d1: C).(\lambda (u1: T).(\lambda (H0: (drop (S n) O (CSort n0) +(CHead d1 (Bind Abbr) u1))).(\lambda (g: G).(\lambda (c2: C).(\lambda (_: +(csuba g c2 (CSort n0))).(and3_ind (eq C (CHead d1 (Bind Abbr) u1) (CSort +n0)) (eq nat (S n) O) (eq nat O O) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O +c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T +A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead +d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: +A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda +(_: T).(csuba g d2 d1))))) (\lambda (_: (eq C (CHead d1 (Bind Abbr) u1) +(CSort n0))).(\lambda (H3: (eq nat (S n) O)).(\lambda (_: (eq nat O O)).(let +H5 \def (eq_ind nat (S n) (\lambda (ee: nat).(match ee in nat return (\lambda +(_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H3) +in (False_ind (or3 (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind +Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) +u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) +(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g +a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) H5))))) +(drop_gen_sort n0 (S n) O (CHead d1 (Bind Abbr) u1) H0))))))))) (\lambda (c: +C).(\lambda (H0: ((\forall (d1: C).(\forall (u1: T).((drop (S n) O c (CHead +d1 (Bind Abbr) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba g c2 c) \to +(or3 (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) +(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda +(u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) +(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 +C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) +u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))))).(\lambda +(k: K).(\lambda (t: T).(\lambda (d1: C).(\lambda (u1: T).(\lambda (H1: (drop +(S n) O (CHead c k t) (CHead d1 (Bind Abbr) u1))).(\lambda (g: G).(\lambda +(c2: C).(\lambda (H2: (csuba g c2 (CHead c k t))).(K_ind (\lambda (k0: +K).((csuba g c2 (CHead c k0 t)) \to ((drop (r k0 n) O c (CHead d1 (Bind Abbr) +u1)) \to (or3 (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) +u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) +u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) +(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g +a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))) (\lambda +(b: B).(\lambda (H3: (csuba g c2 (CHead c (Bind b) t))).(\lambda (H4: (drop +(r (Bind b) n) O c (CHead d1 (Bind Abbr) u1))).(B_ind (\lambda (b0: +B).((csuba g c2 (CHead c (Bind b0) t)) \to ((drop (r (Bind b0) n) O c (CHead +d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead +d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda +(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind +Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 +d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 +u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 +(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1)))))))) (\lambda (H5: (csuba g c2 (CHead c (Bind Abbr) t))).(\lambda (H6: +(drop (r (Bind Abbr) n) O c (CHead d1 (Bind Abbr) u1))).(let H_x \def +(csuba_gen_abbr_rev g c c2 t H5) in (let H7 \def H_x in (or3_ind (ex2 C +(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) (\lambda (d2: C).(csuba +g d2 c))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq +C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda +(_: A).(csuba g d2 c)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g c t a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C +c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +c)))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) +u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) +u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) +(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g +a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda +(H8: (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) (\lambda +(d2: C).(csuba g d2 c)))).(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 +(Bind Abbr) t))) (\lambda (d2: C).(csuba g d2 c)) (or3 (ex2 C (\lambda (d2: +C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 +d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S +n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: +T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: +T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda +(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda +(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda (H9: (eq +C c2 (CHead x (Bind Abbr) t))).(\lambda (H10: (csuba g x c)).(eq_ind_r C +(CHead x (Bind Abbr) t) (\lambda (c0: C).(or3 (ex2 C (\lambda (d2: C).(drop +(S n) O c0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) +(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O +c0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda +(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O c0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda +(_: T).(csuba g d2 d1)))))) (let H11 \def (H c d1 u1 H6 g x H10) in (or3_ind +(ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2: +C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda +(_: A).(drop n O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda +(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: +T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda +(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(drop (S +n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: +C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda +(_: A).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u2))))) +(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 +C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abbr) t) +(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind +Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: +C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) +(or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 +(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda +(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) +t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda +(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: +C).(\lambda 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T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abst) u2))))) +(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 c)))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t a)))) (or3 (ex2 +C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: +C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda +(_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: +C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: +C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) +u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: +C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H9: (eq C c2 (CHead x0 (Bind +Abst) 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(d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) +(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1)))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: +A).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) +(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x3 x4 x5 +(drop_drop (Bind Abst) n x0 (CHead x3 (Bind Abst) x4) H15 x1) H16 H17 +H18))))))))) H14)) (\lambda (H14: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(drop +n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g +d2 d1))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) +(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A +(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 +(Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: +T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: +T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda +(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda +(x3: C).(\lambda (x4: T).(\lambda (H15: (drop n O x0 (CHead x3 (Bind Void) +x4))).(\lambda (H16: (csuba g x3 d1)).(or3_intro2 (ex2 C (\lambda (d2: +C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abbr) u1))) +(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda +(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 +(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g +d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 +u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead +x0 (Bind Abst) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x3 x4 (drop_drop (Bind +Abst) n x0 (CHead x3 (Bind Void) x4) H15 x1) H16)))))) H14)) H13)) c2 +H9)))))))) H8)) (\lambda (H8: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(eq C c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 c))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(eq C +c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +c))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) +u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) +u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) +(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g +a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda +(x0: C).(\lambda (x1: T).(\lambda (H9: (eq C c2 (CHead x0 (Bind Void) +x1))).(\lambda (H10: (csuba g x0 c)).(eq_ind_r C (CHead x0 (Bind Void) x1) +(\lambda (c0: C).(or3 (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 +(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda +(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind +Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 +d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 +u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0 +(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1)))))) (let H11 \def (H c d1 u1 H6 g x0 H10) in (or3_ind (ex2 C (\lambda +(d2: C).(drop n O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 +d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n +O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda +(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 +(Bind Void) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 +d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S +n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: +C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: +C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) +(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind +Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: +C).(drop n O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) +(or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead +d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda +(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Void) +x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda +(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda +(H13: (drop n O x0 (CHead x (Bind Abbr) u1))).(\lambda (H14: (csuba g x +d1)).(or3_intro0 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) +x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T +A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 +(Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: +T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: +T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda +(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C +(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind +Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop (Bind Void) n x0 +(CHead x (Bind Abbr) u1) H13 x1) H14))))) H12)) (\lambda (H12: (ex4_3 C T A +(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x0 (CHead d2 +(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g +d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 +u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: +A).(drop n O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: +T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: +T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda +(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C (\lambda (d2: +C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abbr) u1))) +(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda +(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 +(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g +d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 +u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead +x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1))))) (\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: +A).(\lambda (H13: (drop n O x0 (CHead x2 (Bind Abst) x3))).(\lambda (H14: +(csuba g x2 d1)).(\lambda (H15: (arity g x2 x3 (asucc g x4))).(\lambda (H16: +(arity g d1 u1 x4)).(or3_intro1 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead +x0 (Bind Void) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 +d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S +n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: +C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: +C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) +(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1)))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: +A).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u2))))) +(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x2 x3 x4 +(drop_drop (Bind Void) n x0 (CHead x2 (Bind Abst) x3) H13 x1) H14 H15 +H16))))))))) H12)) (\lambda (H12: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(drop +n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g +d2 d1))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) +(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A +(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 +(Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: +T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: +T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda +(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda +(x2: C).(\lambda (x3: T).(\lambda (H13: (drop n O x0 (CHead x2 (Bind Void) +x3))).(\lambda (H14: (csuba g x2 d1)).(or3_intro2 (ex2 C (\lambda (d2: +C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abbr) u1))) +(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda +(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 +(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g +d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 +u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead +x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x2 x3 (drop_drop (Bind +Void) n x0 (CHead x2 (Bind Void) x3) H13 x1) H14)))))) H12)) H11)) c2 H9))))) +H8)) H7))))) (\lambda (H5: (csuba g c2 (CHead c (Bind Abst) t))).(\lambda +(H6: (drop (r (Bind Abst) n) O c (CHead d1 (Bind Abbr) u1))).(let H_x \def +(csuba_gen_abst_rev g c c2 t H5) in (let H7 \def H_x in (or_ind (ex2 C +(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba +g d2 c))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 c)))) (or3 +(ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda +(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: +T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda +(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: +C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) +u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H8: +(ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2: +C).(csuba g d2 c)))).(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind +Abst) t))) (\lambda (d2: C).(csuba g d2 c)) (or3 (ex2 C (\lambda (d2: +C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 +d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S +n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: +T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: +T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda +(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda +(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda (H9: (eq +C c2 (CHead x (Bind Abst) t))).(\lambda (H10: (csuba g x c)).(eq_ind_r C +(CHead x (Bind Abst) t) (\lambda (c0: C).(or3 (ex2 C (\lambda (d2: C).(drop +(S n) O c0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) +(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O +c0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda +(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O c0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda +(_: T).(csuba g d2 d1)))))) (let H11 \def (H c d1 u1 H6 g x H10) in (or3_ind +(ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2: +C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda +(_: A).(drop n O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda +(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: +T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda +(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(drop (S +n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: +C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda +(_: A).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u2))))) +(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 +C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abst) t) +(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind +Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: +C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) +(or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 +(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda +(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) +t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda +(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: +C).(\lambda (H13: (drop n O x (CHead x0 (Bind Abbr) u1))).(\lambda (H14: +(csuba g x0 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x0 (CHead x2 (Bind Abst) x3) H13 x1) H14 H15 +H16))))))))) H12)) (\lambda (H12: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(drop +n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g +d2 d1))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) +(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A +(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 +(Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: +T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: +T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda +(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda +(x2: C).(\lambda (x3: T).(\lambda (H13: (drop n O x0 (CHead x2 (Bind Void) +x3))).(\lambda (H14: (csuba g x2 d1)).(or3_intro2 (ex2 C (\lambda (d2: +C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abbr) u1))) +(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda +(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 +(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g +d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 +u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead +x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x2 x3 (drop_drop (Bind +Void) n x0 (CHead x2 (Bind Void) x3) H13 x1) H14)))))) H12)) H11)) c2 H9))))) +H8)) H7))))) (\lambda (H5: (csuba g c2 (CHead c (Bind Void) t))).(\lambda +(H6: (drop (r (Bind Void) n) O c (CHead d1 (Bind Abbr) u1))).(let H_x \def +(csuba_gen_void_rev g c c2 t H5) in (let H7 \def H_x in (ex2_ind C (\lambda +(d2: C).(eq C c2 (CHead d2 (Bind Void) t))) (\lambda (d2: C).(csuba g d2 c)) +(or3 (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) +(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda +(u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) +(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 +C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) +u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: +C).(\lambda (H8: (eq C c2 (CHead x (Bind Void) t))).(\lambda (H9: (csuba g x +c)).(eq_ind_r C (CHead x (Bind Void) t) (\lambda (c0: C).(or3 (ex2 C (\lambda +(d2: C).(drop (S n) O c0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba +g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: +A).(drop (S n) O c0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda +(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: +T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda +(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 (Bind Void) u2)))) (\lambda +(d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H10 \def (H c d1 u1 H6 g x +H9) in (or3_ind (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) +u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abst) u2))))) +(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 +C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) +u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C +(\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) +u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) +(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: +A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H11: (ex2 C +(\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2: +C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop n O x (CHead d2 (Bind +Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) (or3 (ex2 C (\lambda (d2: +C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) u1))) (\lambda +(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: +T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind +Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 +d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 +u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x +(Bind Void) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (H12: (drop n O x (CHead x0 +(Bind Abbr) u1))).(\lambda (H13: (csuba g x0 d1)).(or3_intro0 (ex2 C (\lambda +(d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) u1))) +(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda +(u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind +Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 +d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 +u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x +(Bind Void) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x +(Bind Void) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) +x0 (drop_drop (Bind Void) n x (CHead x0 (Bind Abbr) u1) H12 t) H13))))) H11)) +(\lambda (H11: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: +A).(drop n O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: +T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: +T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda +(_: T).(\lambda (a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abst) u2))))) +(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 +(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind +Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) +(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: +A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: +C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H12: (drop n O x (CHead x0 +(Bind Abst) x1))).(\lambda (H13: (csuba g x0 d1)).(\lambda (H14: (arity g x0 +x1 (asucc g x2))).(\lambda (H15: (arity g d1 u1 x2)).(or3_intro1 (ex2 C +(\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) +u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) +(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: +A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex4_3_intro C T A +(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x +(Bind Void) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: +T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: +T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda +(_: T).(\lambda (a: A).(arity g d1 u1 a)))) x0 x1 x2 (drop_drop (Bind Void) n +x (CHead x0 (Bind Abst) x1) H12 t) H13 H14 H15))))))))) H11)) (\lambda (H11: +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind +C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) +u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C +(\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) +u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) +(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: +A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: +C).(\lambda (x1: T).(\lambda (H12: (drop n O x (CHead x0 (Bind Void) +x1))).(\lambda (H13: (csuba g x0 d1)).(or3_intro2 (ex2 C (\lambda (d2: +C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) u1))) (\lambda +(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: +T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind +Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 +d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 +u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x +(Bind Void) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1 (drop_drop (Bind +Void) n x (CHead x0 (Bind Void) x1) H12 t) H13)))))) H11)) H10)) c2 H8)))) +H7))))) b H3 H4)))) (\lambda (f: F).(\lambda (H3: (csuba g c2 (CHead c (Flat +f) t))).(\lambda (H4: (drop (r (Flat f) n) O c (CHead d1 (Bind Abbr) +u1))).(let H_x \def (csuba_gen_flat_rev g c c2 t f H3) in (let H5 \def H_x in +(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Flat f) +u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 c))) (or3 (ex2 C (\lambda +(d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba +g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: +A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda +(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: +T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda +(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda +(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (x1: +T).(\lambda (H6: (eq C c2 (CHead x0 (Flat f) x1))).(\lambda (H7: (csuba g x0 +c)).(eq_ind_r C (CHead x0 (Flat f) x1) (\lambda (c0: C).(or3 (ex2 C (\lambda +(d2: C).(drop (S n) O c0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba +g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: +A).(drop (S n) O c0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda +(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: +T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda +(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 (Bind Void) u2)))) (\lambda +(d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H8 \def (H0 d1 u1 H4 g x0 +H7) in (or3_ind (ex2 C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind +Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O x0 (CHead d2 (Bind Abst) +u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) +(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g +a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O x0 (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C +(\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) +u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) +(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: +A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Void) u2)))) (\lambda +(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H9: (ex2 C (\lambda +(d2: C).(drop (S n) O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba +g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind +Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) (or3 (ex2 C (\lambda (d2: +C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u1))) (\lambda +(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: +T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) +u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) +(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g +a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f) +x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1))))) (\lambda (x: C).(\lambda (H10: (drop (S n) O x0 (CHead x (Bind Abbr) +u1))).(\lambda (H11: (csuba g x d1)).(or3_intro0 (ex2 C (\lambda (d2: +C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u1))) (\lambda +(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: +T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) +u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) +(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g +a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f) +x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) +(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop +(Flat f) n x0 (CHead x (Bind Abbr) u1) H10 x1) H11))))) H9)) (\lambda (H9: +(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O +x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda +(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: +T).(\lambda (_: A).(drop (S n) O x0 (CHead d2 (Bind Abst) u2))))) (\lambda +(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: +C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C +(\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) +u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) +(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: +A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Void) u2)))) (\lambda +(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x2: C).(\lambda (x3: +T).(\lambda (x4: A).(\lambda (H10: (drop (S n) O x0 (CHead x2 (Bind Abst) +x3))).(\lambda (H11: (csuba g x2 d1)).(\lambda (H12: (arity g x2 x3 (asucc g +x4))).(\lambda (H13: (arity g d1 u1 x4)).(or3_intro1 (ex2 C (\lambda (d2: +C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u1))) (\lambda +(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: +T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) +u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) +(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g +a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f) +x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1)))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: +A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u2))))) +(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x2 x3 x4 +(drop_drop (Flat f) n x0 (CHead x2 (Bind Abst) x3) H10 x1) H11 H12 +H13))))))))) H9)) (\lambda (H9: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda +(_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda +(_: T).(csuba g d2 d1))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 +(Flat f) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) +(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O +(CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: +C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: +C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f) x1) (CHead +d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) +(\lambda (x2: C).(\lambda (x3: T).(\lambda (H10: (drop (S n) O x0 (CHead x2 +(Bind Void) x3))).(\lambda (H11: (csuba g x2 d1)).(or3_intro2 (ex2 C (\lambda +(d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u1))) +(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda +(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind +Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 +d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 +u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead +x0 (Flat f) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: +T).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Void) u2)))) (\lambda +(d2: C).(\lambda (_: T).(csuba g d2 d1))) x2 x3 (drop_drop (Flat f) n x0 +(CHead x2 (Bind Void) x3) H10 x1) H11)))))) H9)) H8)) c2 H6))))) H5)))))) k +H2 (drop_gen_drop k c (CHead d1 (Bind Abbr) u1) t n H1)))))))))))) c1)))) i). +(* COMMENTS +Initial nodes: 23852 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/csuba/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/csuba/fwd.ma new file mode 100644 index 000000000..a618761fe --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/csuba/fwd.ma @@ -0,0 +1,1083 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/csuba/defs.ma". + +theorem csuba_gen_abbr: + \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g +(CHead d1 (Bind Abbr) u) c) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 +(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))))) +\def + \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u: T).(\lambda (H: +(csuba g (CHead d1 (Bind Abbr) u) c)).(insert_eq C (CHead d1 (Bind Abbr) u) +(\lambda (c0: C).(csuba g c0 c)) (\lambda (_: C).(ex2 C (\lambda (d2: C).(eq +C c (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (\lambda +(y: C).(\lambda (H0: (csuba g y c)).(csuba_ind g (\lambda (c0: C).(\lambda +(c1: C).((eq C c0 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(eq C +c1 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))) (\lambda +(n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Abbr) u))).(let H2 +\def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return (\lambda (_: +C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow +False])) I (CHead d1 (Bind Abbr) u) H1) in (False_ind (ex2 C (\lambda (d2: +C).(eq C (CSort n) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 +d2))) H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 +c2)).(\lambda (H2: (((eq C c1 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda +(d2: C).(eq C c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 +d2)))))).(\lambda (k: K).(\lambda (u0: T).(\lambda (H3: (eq C (CHead c1 k u0) +(CHead d1 (Bind Abbr) u))).(let H4 \def (f_equal C C (\lambda (e: C).(match e +in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _ +_) \Rightarrow c0])) (CHead c1 k u0) (CHead d1 (Bind Abbr) u) H3) in ((let H5 +\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) +with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c1 k +u0) (CHead d1 (Bind Abbr) u) H3) in ((let H6 \def (f_equal C T (\lambda (e: +C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | +(CHead _ _ t) \Rightarrow t])) (CHead c1 k u0) (CHead d1 (Bind Abbr) u) H3) +in (\lambda (H7: (eq K k (Bind Abbr))).(\lambda (H8: (eq C c1 d1)).(eq_ind_r +T u (\lambda (t: T).(ex2 C (\lambda (d2: C).(eq C (CHead c2 k t) (CHead d2 +(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (eq_ind_r K (Bind Abbr) +(\lambda (k0: K).(ex2 C (\lambda (d2: C).(eq C (CHead c2 k0 u) (CHead d2 +(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H9 \def (eq_ind C +c1 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda +(d2: C).(eq C c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 +d2))))) H2 d1 H8) in (let H10 \def (eq_ind C c1 (\lambda (c0: C).(csuba g c0 +c2)) H1 d1 H8) in (ex_intro2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Abbr) +u) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) c2 +(refl_equal C (CHead c2 (Bind Abbr) u)) H10))) k H7) u0 H6)))) H5)) +H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1 +c2)).(\lambda (_: (((eq C c1 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda +(d2: C).(eq C c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 +d2)))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: +T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u1) (CHead d1 +(Bind Abbr) u))).(let H5 \def (eq_ind C (CHead c1 (Bind Void) u1) (\lambda +(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) +\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda +(_: K).Prop) with [(Bind b0) \Rightarrow (match b0 in B return (\lambda (_: +B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False | Void +\Rightarrow True]) | (Flat _) \Rightarrow False])])) I (CHead d1 (Bind Abbr) +u) H4) in (False_ind (ex2 C (\lambda (d2: C).(eq C (CHead c2 (Bind b) u2) +(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) H5))))))))))) +(\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1 c2)).(\lambda (_: +(((eq C c1 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(eq C c2 +(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))).(\lambda (t: +T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g a))).(\lambda (u0: +T).(\lambda (_: (arity g c2 u0 a)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) +t) (CHead d1 (Bind Abbr) u))).(let H6 \def (eq_ind C (CHead c1 (Bind Abst) t) +(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) +\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda +(_: K).Prop) with [(Bind b) \Rightarrow (match b in B return (\lambda (_: +B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow True | Void +\Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead d1 (Bind Abbr) +u) H5) in (False_ind (ex2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Abbr) u0) +(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) H6)))))))))))) +y c H0))) H))))). +(* COMMENTS +Initial nodes: 1117 +END *) + +theorem csuba_gen_void: + \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).((csuba g +(CHead d1 (Bind Void) u1) c) \to (ex2_3 B C T (\lambda (b: B).(\lambda (d2: +C).(\lambda (u2: T).(eq C c (CHead d2 (Bind b) u2))))) (\lambda (_: +B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))))))))) +\def + \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda +(H: (csuba g (CHead d1 (Bind Void) u1) c)).(insert_eq C (CHead d1 (Bind Void) +u1) (\lambda (c0: C).(csuba g c0 c)) (\lambda (_: C).(ex2_3 B C T (\lambda +(b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Bind b) u2))))) +(\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))))) +(\lambda (y: C).(\lambda (H0: (csuba g y c)).(csuba_ind g (\lambda (c0: +C).(\lambda (c1: C).((eq C c0 (CHead d1 (Bind Void) u1)) \to (ex2_3 B C T +(\lambda (b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 (Bind b) +u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 +d2)))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind +Void) u1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C +return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) +\Rightarrow False])) I (CHead d1 (Bind Void) u1) H1) in (False_ind (ex2_3 B C +T (\lambda (b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C (CSort n) (CHead d2 +(Bind b) u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 +d2))))) H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 +c2)).(\lambda (H2: (((eq C c1 (CHead d1 (Bind Void) u1)) \to (ex2_3 B C T +(\lambda (b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind b) +u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 +d2)))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c1 k u) +(CHead d1 (Bind Void) u1))).(let H4 \def (f_equal C C (\lambda (e: C).(match +e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _ +_) \Rightarrow c0])) (CHead c1 k u) (CHead d1 (Bind Void) u1) H3) in ((let H5 +\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) +with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c1 k +u) (CHead d1 (Bind Void) u1) H3) in ((let H6 \def (f_equal C T (\lambda (e: +C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | +(CHead _ _ t) \Rightarrow t])) (CHead c1 k u) (CHead d1 (Bind Void) u1) H3) +in (\lambda (H7: (eq K k (Bind Void))).(\lambda (H8: (eq C c1 d1)).(eq_ind_r +T u1 (\lambda (t: T).(ex2_3 B C T (\lambda (b: B).(\lambda (d2: C).(\lambda +(u2: T).(eq C (CHead c2 k t) (CHead d2 (Bind b) u2))))) (\lambda (_: +B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))))) (eq_ind_r K (Bind +Void) (\lambda (k0: K).(ex2_3 B C T (\lambda (b: B).(\lambda (d2: C).(\lambda +(u2: T).(eq C (CHead c2 k0 u1) (CHead d2 (Bind b) u2))))) (\lambda (_: +B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))))) (let H9 \def (eq_ind +C c1 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Void) u1)) \to (ex2_3 B C T +(\lambda (b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind b) +u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 +d2))))))) H2 d1 H8) in (let H10 \def (eq_ind C c1 (\lambda (c0: C).(csuba g +c0 c2)) H1 d1 H8) in (ex2_3_intro B C T (\lambda (b: B).(\lambda (d2: +C).(\lambda (u2: T).(eq C (CHead c2 (Bind Void) u1) (CHead d2 (Bind b) +u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))) +Void c2 u1 (refl_equal C (CHead c2 (Bind Void) u1)) H10))) k H7) u H6)))) +H5)) H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 +c2)).(\lambda (H2: (((eq C c1 (CHead d1 (Bind Void) u1)) \to (ex2_3 B C T +(\lambda (b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind b) +u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 +d2)))))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u0: +T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u0) (CHead d1 +(Bind Void) u1))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C +return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _ _) +\Rightarrow c0])) (CHead c1 (Bind Void) u0) (CHead d1 (Bind Void) u1) H4) in +((let H6 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: +C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead +c1 (Bind Void) u0) (CHead d1 (Bind Void) u1) H4) in (\lambda (H7: (eq C c1 +d1)).(let H8 \def (eq_ind C c1 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind +Void) u1)) \to (ex2_3 B C T (\lambda (b0: B).(\lambda (d2: C).(\lambda (u3: +T).(eq C c2 (CHead d2 (Bind b0) u3))))) (\lambda (_: B).(\lambda (d2: +C).(\lambda (_: T).(csuba g d1 d2))))))) H2 d1 H7) in (let H9 \def (eq_ind C +c1 (\lambda (c0: C).(csuba g c0 c2)) H1 d1 H7) in (ex2_3_intro B C T (\lambda +(b0: B).(\lambda (d2: C).(\lambda (u3: T).(eq C (CHead c2 (Bind b) u2) (CHead +d2 (Bind b0) u3))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba +g d1 d2)))) b c2 u2 (refl_equal C (CHead c2 (Bind b) u2)) H9))))) +H5))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1 +c2)).(\lambda (_: (((eq C c1 (CHead d1 (Bind Void) u1)) \to (ex2_3 B C T +(\lambda (b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind b) +u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 +d2)))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc +g a))).(\lambda (u: T).(\lambda (_: (arity g c2 u a)).(\lambda (H5: (eq C +(CHead c1 (Bind Abst) t) (CHead d1 (Bind Void) u1))).(let H6 \def (eq_ind C +(CHead c1 (Bind Abst) t) (\lambda (ee: C).(match ee in C return (\lambda (_: +C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match +k in K return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B +return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow +True | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead d1 +(Bind Void) u1) H5) in (False_ind (ex2_3 B C T (\lambda (b: B).(\lambda (d2: +C).(\lambda (u2: T).(eq C (CHead c2 (Bind Abbr) u) (CHead d2 (Bind b) u2))))) +(\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))))) +H6)))))))))))) y c H0))) H))))). +(* COMMENTS +Initial nodes: 1418 +END *) + +theorem csuba_gen_abst: + \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).((csuba g +(CHead d1 (Bind Abst) u1) c) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead +d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda +(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abbr) +u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g +a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +a)))))))))) +\def + \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda +(H: (csuba g (CHead d1 (Bind Abst) u1) c)).(insert_eq C (CHead d1 (Bind Abst) +u1) (\lambda (c0: C).(csuba g c0 c)) (\lambda (_: C).(or (ex2 C (\lambda (d2: +C).(eq C c (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) +(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead +d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: +A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity +g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 a))))))) (\lambda (y: C).(\lambda (H0: (csuba g y +c)).(csuba_ind g (\lambda (c0: C).(\lambda (c1: C).((eq C c0 (CHead d1 (Bind +Abst) u1)) \to (or (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abst) +u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind Abbr) u2))))) +(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) +(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))) +(\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Abst) +u1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return +(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) +\Rightarrow False])) I (CHead d1 (Bind Abst) u1) H1) in (False_ind (or (ex2 C +(\lambda (d2: C).(eq C (CSort n) (CHead d2 (Bind Abst) u1))) (\lambda (d2: +C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda +(_: A).(eq C (CSort n) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: +C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: +C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) H2)))) +(\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda +(H2: (((eq C c1 (CHead d1 (Bind Abst) u1)) \to (or (ex2 C (\lambda (d2: +C).(eq C c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) +(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 +(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: +A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity +g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 a))))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: +(eq C (CHead c1 k u) (CHead d1 (Bind Abst) u1))).(let H4 \def (f_equal C C +(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) +\Rightarrow c1 | (CHead c0 _ _) \Rightarrow c0])) (CHead c1 k u) (CHead d1 +(Bind Abst) u1) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e in +C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) +\Rightarrow k0])) (CHead c1 k u) (CHead d1 (Bind Abst) u1) H3) in ((let H6 +\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) +with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u) +(CHead d1 (Bind Abst) u1) H3) in (\lambda (H7: (eq K k (Bind Abst))).(\lambda +(H8: (eq C c1 d1)).(eq_ind_r T u1 (\lambda (t: T).(or (ex2 C (\lambda (d2: +C).(eq C (CHead c2 k t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g +d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C +(CHead c2 k t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: +T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: +T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda +(u2: T).(\lambda (a: A).(arity g d2 u2 a))))))) (eq_ind_r K (Bind Abst) +(\lambda (k0: K).(or (ex2 C (\lambda (d2: C).(eq C (CHead c2 k0 u1) (CHead d2 +(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda +(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c2 k0 u1) (CHead d2 +(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g +d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 +(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 +u2 a))))))) (let H9 \def (eq_ind C c1 (\lambda (c0: C).((eq C c0 (CHead d1 +(Bind Abst) u1)) \to (or (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind +Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) +(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) +(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))))) H2 +d1 H8) in (let H10 \def (eq_ind C c1 (\lambda (c0: C).(csuba g c0 c2)) H1 d1 +H8) in (or_introl (ex2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Abst) u1) +(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A +(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abst) +u1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda +(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: +A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda +(a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: C).(eq C (CHead c2 +(Bind Abst) u1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) +c2 (refl_equal C (CHead c2 (Bind Abst) u1)) H10)))) k H7) u H6)))) H5)) +H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1 +c2)).(\lambda (_: (((eq C c1 (CHead d1 (Bind Abst) u1)) \to (or (ex2 C +(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba +g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq +C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda +(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: +A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda +(a: A).(arity g d2 u2 a))))))))).(\lambda (b: B).(\lambda (_: (not (eq B b +Void))).(\lambda (u0: T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind +Void) u0) (CHead d1 (Bind Abst) u1))).(let H5 \def (eq_ind C (CHead c1 (Bind +Void) u0) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with +[(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return +(\lambda (_: K).Prop) with [(Bind b0) \Rightarrow (match b0 in B return +(\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False | +Void \Rightarrow True]) | (Flat _) \Rightarrow False])])) I (CHead d1 (Bind +Abst) u1) H4) in (False_ind (or (ex2 C (\lambda (d2: C).(eq C (CHead c2 (Bind +b) u2) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 +C T A (\lambda (d2: C).(\lambda (u3: T).(\lambda (_: A).(eq C (CHead c2 (Bind +b) u2) (CHead d2 (Bind Abbr) u3))))) (\lambda (d2: C).(\lambda (_: +T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: +T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda +(u3: T).(\lambda (a: A).(arity g d2 u3 a)))))) H5))))))))))) (\lambda (c1: +C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c1 +(CHead d1 (Bind Abst) u1)) \to (or (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 +(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda +(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) +u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g +a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +a))))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (H3: (arity g c1 t (asucc +g a))).(\lambda (u: T).(\lambda (H4: (arity g c2 u a)).(\lambda (H5: (eq C +(CHead c1 (Bind Abst) t) (CHead d1 (Bind Abst) u1))).(let H6 \def (f_equal C +C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) +\Rightarrow c1 | (CHead c0 _ _) \Rightarrow c0])) (CHead c1 (Bind Abst) t) +(CHead d1 (Bind Abst) u1) H5) in ((let H7 \def (f_equal C T (\lambda (e: +C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow t | +(CHead _ _ t0) \Rightarrow t0])) (CHead c1 (Bind Abst) t) (CHead d1 (Bind +Abst) u1) H5) in (\lambda (H8: (eq C c1 d1)).(let H9 \def (eq_ind T t +(\lambda (t0: T).(arity g c1 t0 (asucc g a))) H3 u1 H7) in (let H10 \def +(eq_ind C c1 (\lambda (c0: C).(arity g c0 u1 (asucc g a))) H9 d1 H8) in (let +H11 \def (eq_ind C c1 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abst) u1)) +\to (or (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) u1))) (\lambda +(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: +T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: +C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: +C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 (asucc g a0))))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 a0)))))))) H2 d1 H8) +in (let H12 \def (eq_ind C c1 (\lambda (c0: C).(csuba g c0 c2)) H1 d1 H8) in +(or_intror (ex2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Abbr) u) (CHead d2 +(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda +(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) u) +(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: +A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity +g d1 u1 (asucc g a0))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: +A).(arity g d2 u2 a0))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: +T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) u) (CHead d2 (Bind Abbr) +u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 (asucc g +a0))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 +a0)))) c2 u a (refl_equal C (CHead c2 (Bind Abbr) u)) H12 H10 H4)))))))) +H6)))))))))))) y c H0))) H))))). +(* COMMENTS +Initial nodes: 2550 +END *) + +theorem csuba_gen_flat: + \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).(\forall +(f: F).((csuba g (CHead d1 (Flat f) u1) c) \to (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d1 d2))))))))) +\def + \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda +(f: F).(\lambda (H: (csuba g (CHead d1 (Flat f) u1) c)).(insert_eq C (CHead +d1 (Flat f) u1) (\lambda (c0: C).(csuba g c0 c)) (\lambda (_: C).(ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda +(d2: C).(\lambda (_: T).(csuba g d1 d2))))) (\lambda (y: C).(\lambda (H0: +(csuba g y c)).(csuba_ind g (\lambda (c0: C).(\lambda (c1: C).((eq C c0 +(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq +C c1 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 +d2))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Flat f) +u1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return +(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) +\Rightarrow False])) I (CHead d1 (Flat f) u1) H1) in (False_ind (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(eq C (CSort n) (CHead d2 (Flat f) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))) H2)))) (\lambda (c1: +C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c1 +(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq +C c2 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 +d2))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c1 k u) +(CHead d1 (Flat f) u1))).(let H4 \def (f_equal C C (\lambda (e: C).(match e +in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _ +_) \Rightarrow c0])) (CHead c1 k u) (CHead d1 (Flat f) u1) H3) in ((let H5 +\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) +with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c1 k +u) (CHead d1 (Flat f) u1) H3) in ((let H6 \def (f_equal C T (\lambda (e: +C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | +(CHead _ _ t) \Rightarrow t])) (CHead c1 k u) (CHead d1 (Flat f) u1) H3) in +(\lambda (H7: (eq K k (Flat f))).(\lambda (H8: (eq C c1 d1)).(eq_ind_r T u1 +(\lambda (t: T).(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c2 +k t) (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 +d2))))) (eq_ind_r K (Flat f) (\lambda (k0: K).(ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(eq C (CHead c2 k0 u1) (CHead d2 (Flat f) u2)))) (\lambda +(d2: C).(\lambda (_: T).(csuba g d1 d2))))) (let H9 \def (eq_ind C c1 +(\lambda (c0: C).((eq C c0 (CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda +(d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Flat f) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d1 d2)))))) H2 d1 H8) in (let H10 \def (eq_ind C +c1 (\lambda (c0: C).(csuba g c0 c2)) H1 d1 H8) in (ex2_2_intro C T (\lambda +(d2: C).(\lambda (u2: T).(eq C (CHead c2 (Flat f) u1) (CHead d2 (Flat f) +u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))) c2 u1 (refl_equal C +(CHead c2 (Flat f) u1)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: +C).(\lambda (c2: C).(\lambda (_: (csuba g c1 c2)).(\lambda (_: (((eq C c1 +(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq +C c2 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 +d2))))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u0: +T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u0) (CHead d1 +(Flat f) u1))).(let H5 \def (eq_ind C (CHead c1 (Bind Void) u0) (\lambda (ee: +C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow +False | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) +with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead d1 +(Flat f) u1) H4) in (False_ind (ex2_2 C T (\lambda (d2: C).(\lambda (u3: +T).(eq C (CHead c2 (Bind b) u2) (CHead d2 (Flat f) u3)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d1 d2)))) H5))))))))))) (\lambda (c1: C).(\lambda +(c2: C).(\lambda (_: (csuba g c1 c2)).(\lambda (_: (((eq C c1 (CHead d1 (Flat +f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 +(Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 +d2))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g +a))).(\lambda (u: T).(\lambda (_: (arity g c2 u a)).(\lambda (H5: (eq C +(CHead c1 (Bind Abst) t) (CHead d1 (Flat f) u1))).(let H6 \def (eq_ind C +(CHead c1 (Bind Abst) t) (\lambda (ee: C).(match ee in C return (\lambda (_: +C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match +k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat +_) \Rightarrow False])])) I (CHead d1 (Flat f) u1) H5) in (False_ind (ex2_2 C +T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c2 (Bind Abbr) u) (CHead d2 +(Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))) +H6)))))))))))) y c H0))) H)))))). +(* COMMENTS +Initial nodes: 1183 +END *) + +theorem csuba_gen_bind: + \forall (g: G).(\forall (b1: B).(\forall (e1: C).(\forall (c2: C).(\forall +(v1: T).((csuba g (CHead e1 (Bind b1) v1) c2) \to (ex2_3 B C T (\lambda (b2: +B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) +(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2)))))))))) +\def + \lambda (g: G).(\lambda (b1: B).(\lambda (e1: C).(\lambda (c2: C).(\lambda +(v1: T).(\lambda (H: (csuba g (CHead e1 (Bind b1) v1) c2)).(insert_eq C +(CHead e1 (Bind b1) v1) (\lambda (c: C).(csuba g c c2)) (\lambda (_: +C).(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 +(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: +T).(csuba g e1 e2)))))) (\lambda (y: C).(\lambda (H0: (csuba g y +c2)).(csuba_ind g (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead e1 (Bind +b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: +T).(eq C c0 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: +C).(\lambda (_: T).(csuba g e1 e2)))))))) (\lambda (n: nat).(\lambda (H1: (eq +C (CSort n) (CHead e1 (Bind b1) v1))).(let H2 \def (eq_ind C (CSort n) +(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) +\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 (Bind b1) +v1) H1) in (False_ind (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda +(v2: T).(eq C (CSort n) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda +(e2: C).(\lambda (_: T).(csuba g e1 e2))))) H2)))) (\lambda (c1: C).(\lambda +(c3: C).(\lambda (H1: (csuba g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 +(Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda +(v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: +C).(\lambda (_: T).(csuba g e1 e2)))))))).(\lambda (k: K).(\lambda (u: +T).(\lambda (H3: (eq C (CHead c1 k u) (CHead e1 (Bind b1) v1))).(let H4 \def +(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with +[(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u) +(CHead e1 (Bind b1) v1) H3) in ((let H5 \def (f_equal C K (\lambda (e: +C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | +(CHead _ k0 _) \Rightarrow k0])) (CHead c1 k u) (CHead e1 (Bind b1) v1) H3) +in ((let H6 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda +(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) +(CHead c1 k u) (CHead e1 (Bind b1) v1) H3) in (\lambda (H7: (eq K k (Bind +b1))).(\lambda (H8: (eq C c1 e1)).(eq_ind_r T v1 (\lambda (t: T).(ex2_3 B C T +(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 k t) +(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: +T).(csuba g e1 e2)))))) (eq_ind_r K (Bind b1) (\lambda (k0: K).(ex2_3 B C T +(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 k0 v1) +(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: +T).(csuba g e1 e2)))))) (let H9 \def (eq_ind C c1 (\lambda (c: C).((eq C c +(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: +C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: +B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2))))))) H2 e1 H8) in (let +H10 \def (eq_ind C c1 (\lambda (c: C).(csuba g c c3)) H1 e1 H8) in +(ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C +(CHead c3 (Bind b1) v1) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda +(e2: C).(\lambda (_: T).(csuba g e1 e2)))) b1 c3 v1 (refl_equal C (CHead c3 +(Bind b1) v1)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: +C).(\lambda (c3: C).(\lambda (H1: (csuba g c1 c3)).(\lambda (H2: (((eq C c1 +(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: +C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: +B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2)))))))).(\lambda (b: +B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: +T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) +v1))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda +(_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) +(CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) v1) H4) in ((let H6 \def +(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with +[(CSort _) \Rightarrow Void | (CHead _ k _) \Rightarrow (match k in K return +(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow +Void])])) (CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) v1) H4) in ((let H7 +\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) +with [(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead c1 +(Bind Void) u1) (CHead e1 (Bind b1) v1) H4) in (\lambda (H8: (eq B Void +b1)).(\lambda (H9: (eq C c1 e1)).(let H10 \def (eq_ind C c1 (\lambda (c: +C).((eq C c (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: +B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) +(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2))))))) H2 e1 +H9) in (let H11 \def (eq_ind C c1 (\lambda (c: C).(csuba g c c3)) H1 e1 H9) +in (let H12 \def (eq_ind_r B b1 (\lambda (b0: B).((eq C e1 (CHead e1 (Bind +b0) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: +T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: +C).(\lambda (_: T).(csuba g e1 e2))))))) H10 Void H8) in (ex2_3_intro B C T +(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind b) +u2) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: +T).(csuba g e1 e2)))) b c3 u2 (refl_equal C (CHead c3 (Bind b) u2)) +H11))))))) H6)) H5))))))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1: +(csuba g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind b1) v1)) \to (ex2_3 +B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 +(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g +e1 e2)))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (H3: (arity g c1 t +(asucc g a))).(\lambda (u: T).(\lambda (_: (arity g c3 u a)).(\lambda (H5: +(eq C (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1))).(let H6 \def +(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with +[(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 (Bind +Abst) t) (CHead e1 (Bind b1) v1) H5) in ((let H7 \def (f_equal C B (\lambda +(e: C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow +Abst | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with +[(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead c1 (Bind +Abst) t) (CHead e1 (Bind b1) v1) H5) in ((let H8 \def (f_equal C T (\lambda +(e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow t +| (CHead _ _ t0) \Rightarrow t0])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind +b1) v1) H5) in (\lambda (H9: (eq B Abst b1)).(\lambda (H10: (eq C c1 +e1)).(let H11 \def (eq_ind T t (\lambda (t0: T).(arity g c1 t0 (asucc g a))) +H3 v1 H8) in (let H12 \def (eq_ind C c1 (\lambda (c: C).(arity g c v1 (asucc +g a))) H11 e1 H10) in (let H13 \def (eq_ind C c1 (\lambda (c: C).((eq C c +(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: +C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: +B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2))))))) H2 e1 H10) in (let +H14 \def (eq_ind C c1 (\lambda (c: C).(csuba g c c3)) H1 e1 H10) in (let H15 +\def (eq_ind_r B b1 (\lambda (b: B).((eq C e1 (CHead e1 (Bind b) v1)) \to +(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 +(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: +T).(csuba g e1 e2))))))) H13 Abst H9) in (ex2_3_intro B C T (\lambda (b2: +B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 +(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g +e1 e2)))) Abbr c3 u (refl_equal C (CHead c3 (Bind Abbr) u)) H14))))))))) H7)) +H6)))))))))))) y c2 H0))) H)))))). +(* COMMENTS +Initial nodes: 1889 +END *) + +theorem csuba_gen_abst_rev: + \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g c +(CHead d1 (Bind Abst) u)) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead d2 +(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(eq C c (CHead d2 (Bind Void) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1))))))))) +\def + \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u: T).(\lambda (H: +(csuba g c (CHead d1 (Bind Abst) u))).(insert_eq C (CHead d1 (Bind Abst) u) +(\lambda (c0: C).(csuba g c c0)) (\lambda (_: C).(or (ex2 C (\lambda (d2: +C).(eq C c (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Bind Void) +u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (\lambda (y: +C).(\lambda (H0: (csuba g c y)).(csuba_ind g (\lambda (c0: C).(\lambda (c1: +C).((eq C c1 (CHead d1 (Bind Abst) u)) \to (or (ex2 C (\lambda (d2: C).(eq C +c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(eq C c0 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))) (\lambda (n: +nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Abst) u))).(let H2 \def +(eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return (\lambda (_: +C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow +False])) I (CHead d1 (Bind Abst) u) H1) in (False_ind (or (ex2 C (\lambda +(d2: C).(eq C (CSort n) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g +d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CSort n) (CHead +d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) +H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 +c2)).(\lambda (H2: (((eq C c2 (CHead d1 (Bind Abst) u)) \to (or (ex2 C +(\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba +g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1)))))))).(\lambda (k: K).(\lambda (u0: T).(\lambda (H3: (eq C (CHead c2 k +u0) (CHead d1 (Bind Abst) u))).(let H4 \def (f_equal C C (\lambda (e: +C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | +(CHead c0 _ _) \Rightarrow c0])) (CHead c2 k u0) (CHead d1 (Bind Abst) u) H3) +in ((let H5 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda +(_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) +(CHead c2 k u0) (CHead d1 (Bind Abst) u) H3) in ((let H6 \def (f_equal C T +(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) +\Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead c2 k u0) (CHead d1 +(Bind Abst) u) H3) in (\lambda (H7: (eq K k (Bind Abst))).(\lambda (H8: (eq C +c2 d1)).(eq_ind_r T u (\lambda (t: T).(or (ex2 C (\lambda (d2: C).(eq C +(CHead c1 k t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 k t) (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) +(eq_ind_r K (Bind Abst) (\lambda (k0: K).(or (ex2 C (\lambda (d2: C).(eq C +(CHead c1 k0 u) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 k0 u) (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let +H9 \def (eq_ind C c2 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abst) u)) \to +(or (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abst) u))) (\lambda (d2: +C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 +(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1))))))) H2 d1 H8) in (let H10 \def (eq_ind C c2 (\lambda (c0: C).(csuba g +c1 c0)) H1 d1 H8) in (or_introl (ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind +Abst) u) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 +C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 (Bind Abst) u) (CHead +d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) +(ex_intro2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abst) u) (CHead d2 (Bind +Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) c1 (refl_equal C (CHead c1 (Bind +Abst) u)) H10)))) k H7) u0 H6)))) H5)) H4))))))))) (\lambda (c1: C).(\lambda +(c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c2 (CHead d1 +(Bind Abst) u)) \to (or (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind +Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(eq C c1 (CHead d2 (Bind Void) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1)))))))).(\lambda (b: B).(\lambda (H3: (not +(eq B b Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead +c2 (Bind b) u2) (CHead d1 (Bind Abst) u))).(let H5 \def (f_equal C C (\lambda +(e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 +| (CHead c0 _ _) \Rightarrow c0])) (CHead c2 (Bind b) u2) (CHead d1 (Bind +Abst) u) H4) in ((let H6 \def (f_equal C B (\lambda (e: C).(match e in C +return (\lambda (_: C).B) with [(CSort _) \Rightarrow b | (CHead _ k _) +\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b0) +\Rightarrow b0 | (Flat _) \Rightarrow b])])) (CHead c2 (Bind b) u2) (CHead d1 +(Bind Abst) u) H4) in ((let H7 \def (f_equal C T (\lambda (e: C).(match e in +C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u2 | (CHead _ _ t) +\Rightarrow t])) (CHead c2 (Bind b) u2) (CHead d1 (Bind Abst) u) H4) in +(\lambda (H8: (eq B b Abst)).(\lambda (H9: (eq C c2 d1)).(let H10 \def +(eq_ind B b (\lambda (b0: B).(not (eq B b0 Void))) H3 Abst H8) in (let H11 +\def (eq_ind C c2 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abst) u)) \to +(or (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abst) u))) (\lambda (d2: +C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u3: T).(eq C c1 +(CHead d2 (Bind Void) u3)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1))))))) H2 d1 H9) in (let H12 \def (eq_ind C c2 (\lambda (c0: C).(csuba g +c1 c0)) H1 d1 H9) in (or_intror (ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind +Void) u1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u3: T).(eq C (CHead c1 (Bind Void) u1) +(CHead d2 (Bind Void) u3)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u3: T).(eq C (CHead c1 +(Bind Void) u1) (CHead d2 (Bind Void) u3)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1))) c1 u1 (refl_equal C (CHead c1 (Bind Void) u1)) +H12)))))))) H6)) H5))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: +(csuba g c1 c2)).(\lambda (_: (((eq C c2 (CHead d1 (Bind Abst) u)) \to (or +(ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abst) u))) (\lambda (d2: +C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 +(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1)))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc +g a))).(\lambda (u0: T).(\lambda (_: (arity g c2 u0 a)).(\lambda (H5: (eq C +(CHead c2 (Bind Abbr) u0) (CHead d1 (Bind Abst) u))).(let H6 \def (eq_ind C +(CHead c2 (Bind Abbr) u0) (\lambda (ee: C).(match ee in C return (\lambda (_: +C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match +k in K return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B +return (\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow +False | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead +d1 (Bind Abst) u) H5) in (False_ind (or (ex2 C (\lambda (d2: C).(eq C (CHead +c1 (Bind Abst) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 +d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 (Bind +Abst) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba +g d2 d1))))) H6)))))))))))) c y H0))) H))))). +(* COMMENTS +Initial nodes: 1980 +END *) + +theorem csuba_gen_void_rev: + \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g c +(CHead d1 (Bind Void) u)) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind +Void) u))) (\lambda (d2: C).(csuba g d2 d1))))))) +\def + \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u: T).(\lambda (H: +(csuba g c (CHead d1 (Bind Void) u))).(insert_eq C (CHead d1 (Bind Void) u) +(\lambda (c0: C).(csuba g c c0)) (\lambda (_: C).(ex2 C (\lambda (d2: C).(eq +C c (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))) (\lambda +(y: C).(\lambda (H0: (csuba g c y)).(csuba_ind g (\lambda (c0: C).(\lambda +(c1: C).((eq C c1 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda (d2: C).(eq C +c0 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))))) (\lambda +(n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Void) u))).(let H2 +\def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return (\lambda (_: +C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow +False])) I (CHead d1 (Bind Void) u) H1) in (False_ind (ex2 C (\lambda (d2: +C).(eq C (CSort n) (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 +d1))) H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 +c2)).(\lambda (H2: (((eq C c2 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda +(d2: C).(eq C c1 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 +d1)))))).(\lambda (k: K).(\lambda (u0: T).(\lambda (H3: (eq C (CHead c2 k u0) +(CHead d1 (Bind Void) u))).(let H4 \def (f_equal C C (\lambda (e: C).(match e +in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ +_) \Rightarrow c0])) (CHead c2 k u0) (CHead d1 (Bind Void) u) H3) in ((let H5 +\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) +with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c2 k +u0) (CHead d1 (Bind Void) u) H3) in ((let H6 \def (f_equal C T (\lambda (e: +C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | +(CHead _ _ t) \Rightarrow t])) (CHead c2 k u0) (CHead d1 (Bind Void) u) H3) +in (\lambda (H7: (eq K k (Bind Void))).(\lambda (H8: (eq C c2 d1)).(eq_ind_r +T u (\lambda (t: T).(ex2 C (\lambda (d2: C).(eq C (CHead c1 k t) (CHead d2 +(Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))) (eq_ind_r K (Bind Void) +(\lambda (k0: K).(ex2 C (\lambda (d2: C).(eq C (CHead c1 k0 u) (CHead d2 +(Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))) (let H9 \def (eq_ind C +c2 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda +(d2: C).(eq C c1 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 +d1))))) H2 d1 H8) in (let H10 \def (eq_ind C c2 (\lambda (c0: C).(csuba g c1 +c0)) H1 d1 H8) in (ex_intro2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Void) +u) (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)) c1 +(refl_equal C (CHead c1 (Bind Void) u)) H10))) k H7) u0 H6)))) H5)) +H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 +c2)).(\lambda (H2: (((eq C c2 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda +(d2: C).(eq C c1 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 +d1)))))).(\lambda (b: B).(\lambda (H3: (not (eq B b Void))).(\lambda (u1: +T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c2 (Bind b) u2) (CHead d1 +(Bind Void) u))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C +return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ _) +\Rightarrow c0])) (CHead c2 (Bind b) u2) (CHead d1 (Bind Void) u) H4) in +((let H6 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: +C).B) with [(CSort _) \Rightarrow b | (CHead _ k _) \Rightarrow (match k in K +return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) +\Rightarrow b])])) (CHead c2 (Bind b) u2) (CHead d1 (Bind Void) u) H4) in +((let H7 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: +C).T) with [(CSort _) \Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead +c2 (Bind b) u2) (CHead d1 (Bind Void) u) H4) in (\lambda (H8: (eq B b +Void)).(\lambda (H9: (eq C c2 d1)).(let H10 \def (eq_ind B b (\lambda (b0: +B).(not (eq B b0 Void))) H3 Void H8) in (let H11 \def (eq_ind C c2 (\lambda +(c0: C).((eq C c0 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda (d2: C).(eq C +c1 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1))))) H2 d1 H9) +in (let H12 \def (eq_ind C c2 (\lambda (c0: C).(csuba g c1 c0)) H1 d1 H9) in +(let H13 \def (match (H10 (refl_equal B Void)) in False return (\lambda (_: +False).(ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Void) u1) (CHead d2 +(Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))) with []) in H13))))))) +H6)) H5))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1 +c2)).(\lambda (_: (((eq C c2 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda +(d2: C).(eq C c1 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 +d1)))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g +a))).(\lambda (u0: T).(\lambda (_: (arity g c2 u0 a)).(\lambda (H5: (eq C +(CHead c2 (Bind Abbr) u0) (CHead d1 (Bind Void) u))).(let H6 \def (eq_ind C +(CHead c2 (Bind Abbr) u0) (\lambda (ee: C).(match ee in C return (\lambda (_: +C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match +k in K return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B +return (\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow +False | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead +d1 (Bind Void) u) H5) in (False_ind (ex2 C (\lambda (d2: C).(eq C (CHead c1 +(Bind Abst) t) (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1))) +H6)))))))))))) c y H0))) H))))). +(* COMMENTS +Initial nodes: 1326 +END *) + +theorem csuba_gen_abbr_rev: + \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).((csuba g c +(CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(eq C c (CHead d2 +(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda +(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abst) +u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) +(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g +a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Bind Void) +u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))) +\def + \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda +(H: (csuba g c (CHead d1 (Bind Abbr) u1))).(insert_eq C (CHead d1 (Bind Abbr) +u1) (\lambda (c0: C).(csuba g c c0)) (\lambda (_: C).(or3 (ex2 C (\lambda +(d2: C).(eq C c (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 +d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c +(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: +A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq +C c (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1)))))) (\lambda (y: C).(\lambda (H0: (csuba g c y)).(csuba_ind g (\lambda +(c0: C).(\lambda (c1: C).((eq C c1 (CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C +(\lambda (d2: C).(eq C c0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba +g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq +C c0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda +(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq +C c0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g +d2 d1)))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind +Abbr) u1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C +return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) +\Rightarrow False])) I (CHead d1 (Bind Abbr) u1) H1) in (False_ind (or3 (ex2 +C (\lambda (d2: C).(eq C (CSort n) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: +C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda +(_: A).(eq C (CSort n) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: +C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: +C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(eq C (CSort n) (CHead d2 (Bind Void) +u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) H2)))) (\lambda +(c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C +c2 (CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(eq C c1 +(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A +(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind +Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 +d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 +u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1)))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c2 k u) +(CHead d1 (Bind Abbr) u1))).(let H4 \def (f_equal C C (\lambda (e: C).(match +e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ +_) \Rightarrow c0])) (CHead c2 k u) (CHead d1 (Bind Abbr) u1) H3) in ((let H5 +\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) +with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c2 k +u) (CHead d1 (Bind Abbr) u1) H3) in ((let H6 \def (f_equal C T (\lambda (e: +C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | +(CHead _ _ t) \Rightarrow t])) (CHead c2 k u) (CHead d1 (Bind Abbr) u1) H3) +in (\lambda (H7: (eq K k (Bind Abbr))).(\lambda (H8: (eq C c2 d1)).(eq_ind_r +T u1 (\lambda (t: T).(or3 (ex2 C (\lambda (d2: C).(eq C (CHead c1 k t) (CHead +d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda +(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c1 k t) (CHead d2 (Bind +Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 +d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 +u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 k t) +(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1)))))) (eq_ind_r K (Bind Abbr) (\lambda (k0: K).(or3 (ex2 C (\lambda (d2: +C).(eq C (CHead c1 k0 u1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba +g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq +C (CHead c1 k0 u1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda +(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: +T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda +(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(eq C (CHead c1 k0 u1) (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H9 \def (eq_ind C +c2 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C +(\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba +g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq +C c1 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda +(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq +C c1 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g +d2 d1))))))) H2 d1 H8) in (let H10 \def (eq_ind C c2 (\lambda (c0: C).(csuba +g c1 c0)) H1 d1 H8) in (or3_intro0 (ex2 C (\lambda (d2: C).(eq C (CHead c1 +(Bind Abbr) u1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 +d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C +(CHead c1 (Bind Abbr) u1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: +C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: +C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 (Bind Abbr) u1) (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) +(ex_intro2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abbr) u1) (CHead d2 (Bind +Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) c1 (refl_equal C (CHead c1 +(Bind Abbr) u1)) H10)))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: +C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c2 +(CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(eq C c1 (CHead +d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda +(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind Abst) +u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) +(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g +a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 (Bind Void) +u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))).(\lambda (b: +B).(\lambda (H3: (not (eq B b Void))).(\lambda (u0: T).(\lambda (u2: +T).(\lambda (H4: (eq C (CHead c2 (Bind b) u2) (CHead d1 (Bind Abbr) +u1))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda +(_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) +(CHead c2 (Bind b) u2) (CHead d1 (Bind Abbr) u1) H4) in ((let H6 \def +(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with +[(CSort _) \Rightarrow b | (CHead _ k _) \Rightarrow (match k in K return +(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow +b])])) (CHead c2 (Bind b) u2) (CHead d1 (Bind Abbr) u1) H4) in ((let H7 \def +(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with +[(CSort _) \Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c2 (Bind b) +u2) (CHead d1 (Bind Abbr) u1) H4) in (\lambda (H8: (eq B b Abbr)).(\lambda +(H9: (eq C c2 d1)).(let H10 \def (eq_ind B b (\lambda (b0: B).(not (eq B b0 +Void))) H3 Abbr H8) in (let H11 \def (eq_ind C c2 (\lambda (c0: C).((eq C c0 +(CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(eq C c1 (CHead +d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda +(d2: C).(\lambda (u3: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind Abst) +u3))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) +(\lambda (d2: C).(\lambda (u3: T).(\lambda (a: A).(arity g d2 u3 (asucc g +a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u3: T).(eq C c1 (CHead d2 (Bind Void) +u3)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))) H2 d1 H9) in +(let H12 \def (eq_ind C c2 (\lambda (c0: C).(csuba g c1 c0)) H1 d1 H9) in +(or3_intro2 (ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Void) u0) (CHead d2 +(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda +(d2: C).(\lambda (u3: T).(\lambda (_: A).(eq C (CHead c1 (Bind Void) u0) +(CHead d2 (Bind Abst) u3))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: +A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u3: T).(\lambda (a: +A).(arity g d2 u3 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u3: T).(eq +C (CHead c1 (Bind Void) u0) (CHead d2 (Bind Void) u3)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: +C).(\lambda (u3: T).(eq C (CHead c1 (Bind Void) u0) (CHead d2 (Bind Void) +u3)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) c1 u0 (refl_equal C +(CHead c1 (Bind Void) u0)) H12)))))))) H6)) H5))))))))))) (\lambda (c1: +C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c2 +(CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(eq C c1 (CHead +d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda +(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind Abst) +u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) +(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g +a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 (Bind Void) +u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))).(\lambda (t: +T).(\lambda (a: A).(\lambda (H3: (arity g c1 t (asucc g a))).(\lambda (u: +T).(\lambda (H4: (arity g c2 u a)).(\lambda (H5: (eq C (CHead c2 (Bind Abbr) +u) (CHead d1 (Bind Abbr) u1))).(let H6 \def (f_equal C C (\lambda (e: +C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | +(CHead c0 _ _) \Rightarrow c0])) (CHead c2 (Bind Abbr) u) (CHead d1 (Bind +Abbr) u1) H5) in ((let H7 \def (f_equal C T (\lambda (e: C).(match e in C +return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) +\Rightarrow t0])) (CHead c2 (Bind Abbr) u) (CHead d1 (Bind Abbr) u1) H5) in +(\lambda (H8: (eq C c2 d1)).(let H9 \def (eq_ind T u (\lambda (t0: T).(arity +g c2 t0 a)) H4 u1 H7) in (let H10 \def (eq_ind C c2 (\lambda (c0: C).(arity g +c0 u1 a)) H9 d1 H8) in (let H11 \def (eq_ind C c2 (\lambda (c0: C).((eq C c0 +(CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(eq C c1 (CHead +d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda +(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind Abst) +u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) +(\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 (asucc g +a0))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 +a0))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))) H2 d1 H8) +in (let H12 \def (eq_ind C c2 (\lambda (c0: C).(csuba g c1 c0)) H1 d1 H8) in +(or3_intro1 (ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abst) t) (CHead d2 +(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda +(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) t) +(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: +A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: +A).(arity g d2 u2 (asucc g a0))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a0: A).(arity g d1 u1 a0))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(eq C (CHead c1 (Bind Abst) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1)))) (ex4_3_intro C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) t) (CHead d2 +(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g +d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 +(asucc g a0))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 +u1 a0)))) c1 t a (refl_equal C (CHead c1 (Bind Abst) t)) H12 H3 H10)))))))) +H6)))))))))))) c y H0))) H))))). +(* COMMENTS +Initial nodes: 3459 +END *) + +theorem csuba_gen_flat_rev: + \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).(\forall +(f: F).((csuba g c (CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1))))))))) +\def + \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda +(f: F).(\lambda (H: (csuba g c (CHead d1 (Flat f) u1))).(insert_eq C (CHead +d1 (Flat f) u1) (\lambda (c0: C).(csuba g c c0)) (\lambda (_: C).(ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda +(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (y: C).(\lambda (H0: +(csuba g c y)).(csuba_ind g (\lambda (c0: C).(\lambda (c1: C).((eq C c1 +(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq +C c0 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Flat f) +u1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return +(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) +\Rightarrow False])) I (CHead d1 (Flat f) u1) H1) in (False_ind (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(eq C (CSort n) (CHead d2 (Flat f) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) H2)))) (\lambda (c1: +C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c2 +(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq +C c1 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c2 k u) +(CHead d1 (Flat f) u1))).(let H4 \def (f_equal C C (\lambda (e: C).(match e +in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ +_) \Rightarrow c0])) (CHead c2 k u) (CHead d1 (Flat f) u1) H3) in ((let H5 +\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) +with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c2 k +u) (CHead d1 (Flat f) u1) H3) in ((let H6 \def (f_equal C T (\lambda (e: +C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | +(CHead _ _ t) \Rightarrow t])) (CHead c2 k u) (CHead d1 (Flat f) u1) H3) in +(\lambda (H7: (eq K k (Flat f))).(\lambda (H8: (eq C c2 d1)).(eq_ind_r T u1 +(\lambda (t: T).(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 +k t) (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1))))) (eq_ind_r K (Flat f) (\lambda (k0: K).(ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(eq C (CHead c1 k0 u1) (CHead d2 (Flat f) u2)))) (\lambda +(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (let H9 \def (eq_ind C c2 +(\lambda (c0: C).((eq C c0 (CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda +(d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 (Flat f) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1)))))) H2 d1 H8) in (let H10 \def (eq_ind C +c2 (\lambda (c0: C).(csuba g c1 c0)) H1 d1 H8) in (ex2_2_intro C T (\lambda +(d2: C).(\lambda (u2: T).(eq C (CHead c1 (Flat f) u1) (CHead d2 (Flat f) +u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) c1 u1 (refl_equal C +(CHead c1 (Flat f) u1)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: +C).(\lambda (c2: C).(\lambda (_: (csuba g c1 c2)).(\lambda (_: (((eq C c2 +(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq +C c1 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1))))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u0: +T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c2 (Bind b) u2) (CHead d1 +(Flat f) u1))).(let H5 \def (eq_ind C (CHead c2 (Bind b) u2) (\lambda (ee: +C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow +False | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) +with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead d1 +(Flat f) u1) H4) in (False_ind (ex2_2 C T (\lambda (d2: C).(\lambda (u3: +T).(eq C (CHead c1 (Bind Void) u0) (CHead d2 (Flat f) u3)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1)))) H5))))))))))) (\lambda (c1: C).(\lambda +(c2: C).(\lambda (_: (csuba g c1 c2)).(\lambda (_: (((eq C c2 (CHead d1 (Flat +f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 +(Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g +a))).(\lambda (u: T).(\lambda (_: (arity g c2 u a)).(\lambda (H5: (eq C +(CHead c2 (Bind Abbr) u) (CHead d1 (Flat f) u1))).(let H6 \def (eq_ind C +(CHead c2 (Bind Abbr) u) (\lambda (ee: C).(match ee in C return (\lambda (_: +C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match +k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat +_) \Rightarrow False])])) I (CHead d1 (Flat f) u1) H5) in (False_ind (ex2_2 C +T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 (Bind Abst) t) (CHead d2 +(Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) +H6)))))))))))) c y H0))) H)))))). +(* COMMENTS +Initial nodes: 1183 +END *) + +theorem csuba_gen_bind_rev: + \forall (g: G).(\forall (b1: B).(\forall (e1: C).(\forall (c2: C).(\forall +(v1: T).((csuba g c2 (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: +B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) +(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1)))))))))) +\def + \lambda (g: G).(\lambda (b1: B).(\lambda (e1: C).(\lambda (c2: C).(\lambda +(v1: T).(\lambda (H: (csuba g c2 (CHead e1 (Bind b1) v1))).(insert_eq C +(CHead e1 (Bind b1) v1) (\lambda (c: C).(csuba g c2 c)) (\lambda (_: +C).(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 +(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: +T).(csuba g e2 e1)))))) (\lambda (y: C).(\lambda (H0: (csuba g c2 +y)).(csuba_ind g (\lambda (c: C).(\lambda (c0: C).((eq C c0 (CHead e1 (Bind +b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: +T).(eq C c (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: +C).(\lambda (_: T).(csuba g e2 e1)))))))) (\lambda (n: nat).(\lambda (H1: (eq +C (CSort n) (CHead e1 (Bind b1) v1))).(let H2 \def (eq_ind C (CSort n) +(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) +\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 (Bind b1) +v1) H1) in (False_ind (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda +(v2: T).(eq C (CSort n) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda +(e2: C).(\lambda (_: T).(csuba g e2 e1))))) H2)))) (\lambda (c1: C).(\lambda +(c3: C).(\lambda (H1: (csuba g c1 c3)).(\lambda (H2: (((eq C c3 (CHead e1 +(Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda +(v2: T).(eq C c1 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: +C).(\lambda (_: T).(csuba g e2 e1)))))))).(\lambda (k: K).(\lambda (u: +T).(\lambda (H3: (eq C (CHead c3 k u) (CHead e1 (Bind b1) v1))).(let H4 \def +(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with +[(CSort _) \Rightarrow c3 | (CHead c _ _) \Rightarrow c])) (CHead c3 k u) +(CHead e1 (Bind b1) v1) H3) in ((let H5 \def (f_equal C K (\lambda (e: +C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | +(CHead _ k0 _) \Rightarrow k0])) (CHead c3 k u) (CHead e1 (Bind b1) v1) H3) +in ((let H6 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda +(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) +(CHead c3 k u) (CHead e1 (Bind b1) v1) H3) in (\lambda (H7: (eq K k (Bind +b1))).(\lambda (H8: (eq C c3 e1)).(eq_ind_r T v1 (\lambda (t: T).(ex2_3 B C T +(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c1 k t) +(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: +T).(csuba g e2 e1)))))) (eq_ind_r K (Bind b1) (\lambda (k0: K).(ex2_3 B C T +(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c1 k0 v1) +(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: +T).(csuba g e2 e1)))))) (let H9 \def (eq_ind C c3 (\lambda (c: C).((eq C c +(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: +C).(\lambda (v2: T).(eq C c1 (CHead e2 (Bind b2) v2))))) (\lambda (_: +B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1))))))) H2 e1 H8) in (let +H10 \def (eq_ind C c3 (\lambda (c: C).(csuba g c1 c)) H1 e1 H8) in +(ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C +(CHead c1 (Bind b1) v1) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda +(e2: C).(\lambda (_: T).(csuba g e2 e1)))) b1 c1 v1 (refl_equal C (CHead c1 +(Bind b1) v1)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: +C).(\lambda (c3: C).(\lambda (H1: (csuba g c1 c3)).(\lambda (H2: (((eq C c3 +(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: +C).(\lambda (v2: T).(eq C c1 (CHead e2 (Bind b2) v2))))) (\lambda (_: +B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1)))))))).(\lambda (b: +B).(\lambda (H3: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: +T).(\lambda (H4: (eq C (CHead c3 (Bind b) u2) (CHead e1 (Bind b1) v1))).(let +H5 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) +with [(CSort _) \Rightarrow c3 | (CHead c _ _) \Rightarrow c])) (CHead c3 +(Bind b) u2) (CHead e1 (Bind b1) v1) H4) in ((let H6 \def (f_equal C B +(\lambda (e: C).(match e in C return (\lambda (_: C).B) with [(CSort _) +\Rightarrow b | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: +K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow b])])) (CHead c3 +(Bind b) u2) (CHead e1 (Bind b1) v1) H4) in ((let H7 \def (f_equal C T +(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) +\Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c3 (Bind b) u2) (CHead +e1 (Bind b1) v1) H4) in (\lambda (H8: (eq B b b1)).(\lambda (H9: (eq C c3 +e1)).(let H10 \def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Void))) H3 b1 +H8) in (let H11 \def (eq_ind C c3 (\lambda (c: C).((eq C c (CHead e1 (Bind +b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: +T).(eq C c1 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: +C).(\lambda (_: T).(csuba g e2 e1))))))) H2 e1 H9) in (let H12 \def (eq_ind C +c3 (\lambda (c: C).(csuba g c1 c)) H1 e1 H9) in (ex2_3_intro B C T (\lambda +(b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c1 (Bind Void) u1) +(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: +T).(csuba g e2 e1)))) Void c1 u1 (refl_equal C (CHead c1 (Bind Void) u1)) +H12))))))) H6)) H5))))))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1: +(csuba g c1 c3)).(\lambda (H2: (((eq C c3 (CHead e1 (Bind b1) v1)) \to (ex2_3 +B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c1 (CHead e2 +(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g +e2 e1)))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t +(asucc g a))).(\lambda (u: T).(\lambda (H4: (arity g c3 u a)).(\lambda (H5: +(eq C (CHead c3 (Bind Abbr) u) (CHead e1 (Bind b1) v1))).(let H6 \def +(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with +[(CSort _) \Rightarrow c3 | (CHead c _ _) \Rightarrow c])) (CHead c3 (Bind +Abbr) u) (CHead e1 (Bind b1) v1) H5) in ((let H7 \def (f_equal C B (\lambda +(e: C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow +Abbr | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with +[(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead c3 (Bind +Abbr) u) (CHead e1 (Bind b1) v1) H5) in ((let H8 \def (f_equal C T (\lambda +(e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u +| (CHead _ _ t0) \Rightarrow t0])) (CHead c3 (Bind Abbr) u) (CHead e1 (Bind +b1) v1) H5) in (\lambda (H9: (eq B Abbr b1)).(\lambda (H10: (eq C c3 +e1)).(let H11 \def (eq_ind T u (\lambda (t0: T).(arity g c3 t0 a)) H4 v1 H8) +in (let H12 \def (eq_ind C c3 (\lambda (c: C).(arity g c v1 a)) H11 e1 H10) +in (let H13 \def (eq_ind C c3 (\lambda (c: C).((eq C c (CHead e1 (Bind b1) +v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq +C c1 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda +(_: T).(csuba g e2 e1))))))) H2 e1 H10) in (let H14 \def (eq_ind C c3 +(\lambda (c: C).(csuba g c1 c)) H1 e1 H10) in (let H15 \def (eq_ind_r B b1 +(\lambda (b: B).((eq C e1 (CHead e1 (Bind b) v1)) \to (ex2_3 B C T (\lambda +(b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c1 (CHead e2 (Bind b2) +v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 +e1))))))) H13 Abbr H9) in (ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: +C).(\lambda (v2: T).(eq C (CHead c1 (Bind Abst) t) (CHead e2 (Bind b2) +v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1)))) +Abst c1 t (refl_equal C (CHead c1 (Bind Abst) t)) H14))))))))) H7)) +H6)))))))))))) c2 y H0))) H)))))). +(* COMMENTS +Initial nodes: 1831 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/csuba/getl.ma b/matita/matita/contribs/lambdadelta/basic_1/csuba/getl.ma new file mode 100644 index 000000000..b49d9b7c2 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/csuba/getl.ma @@ -0,0 +1,1178 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/csuba/drop.ma". + +include "Basic-1/csuba/clear.ma". + +include "Basic-1/getl/clear.ma". + +theorem csuba_getl_abbr: + \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).(\forall +(i: nat).((getl i c1 (CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csuba g +c1 c2) \to (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) +(\lambda (d2: C).(csuba g d1 d2)))))))))) +\def + \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u: T).(\lambda +(i: nat).(\lambda (H: (getl i c1 (CHead d1 (Bind Abbr) u))).(let H0 \def +(getl_gen_all c1 (CHead d1 (Bind Abbr) u) i H) in (ex2_ind C (\lambda (e: +C).(drop i O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abbr) u))) +(\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(getl i c2 +(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))) (\lambda (x: +C).(\lambda (H1: (drop i O c1 x)).(\lambda (H2: (clear x (CHead d1 (Bind +Abbr) u))).(C_ind (\lambda (c: C).((drop i O c1 c) \to ((clear c (CHead d1 +(Bind Abbr) u)) \to (\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda +(d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 +d2)))))))) (\lambda (n: nat).(\lambda (_: (drop i O c1 (CSort n))).(\lambda +(H4: (clear (CSort n) (CHead d1 (Bind Abbr) u))).(clear_gen_sort (CHead d1 +(Bind Abbr) u) n H4 (\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda +(d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 +d2))))))))) (\lambda (x0: C).(\lambda (_: (((drop i O c1 x0) \to ((clear x0 +(CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csuba g c1 c2) \to (ex2 C +(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: +C).(csuba g d1 d2))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (H3: +(drop i O c1 (CHead x0 k t))).(\lambda (H4: (clear (CHead x0 k t) (CHead d1 +(Bind Abbr) u))).(K_ind (\lambda (k0: K).((drop i O c1 (CHead x0 k0 t)) \to +((clear (CHead x0 k0 t) (CHead d1 (Bind Abbr) u)) \to (\forall (c2: +C).((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind +Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))))) (\lambda (b: B).(\lambda +(H5: (drop i O c1 (CHead x0 (Bind b) t))).(\lambda (H6: (clear (CHead x0 +(Bind b) t) (CHead d1 (Bind Abbr) u))).(let H7 \def (f_equal C C (\lambda (e: +C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d1 | +(CHead c _ _) \Rightarrow c])) (CHead d1 (Bind Abbr) u) (CHead x0 (Bind b) t) +(clear_gen_bind b x0 (CHead d1 (Bind Abbr) u) t H6)) in ((let H8 \def +(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with +[(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow (match k0 in K +return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) +\Rightarrow Abbr])])) (CHead d1 (Bind Abbr) u) (CHead x0 (Bind b) t) +(clear_gen_bind b x0 (CHead d1 (Bind Abbr) u) t H6)) in ((let H9 \def +(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with +[(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d1 (Bind +Abbr) u) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abbr) u) +t H6)) in (\lambda (H10: (eq B Abbr b)).(\lambda (H11: (eq C d1 x0)).(\lambda +(c2: C).(\lambda (H12: (csuba g c1 c2)).(let H13 \def (eq_ind_r T t (\lambda +(t0: T).(drop i O c1 (CHead x0 (Bind b) t0))) H5 u H9) in (let H14 \def +(eq_ind_r B b (\lambda (b0: B).(drop i O c1 (CHead x0 (Bind b0) u))) H13 Abbr +H10) in (let H15 \def (eq_ind_r C x0 (\lambda (c: C).(drop i O c1 (CHead c +(Bind Abbr) u))) H14 d1 H11) in (let H16 \def (csuba_drop_abbr i c1 d1 u H15 +g c2 H12) in (ex2_ind C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr) +u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(getl i c2 +(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x1: +C).(\lambda (H17: (drop i O c2 (CHead x1 (Bind Abbr) u))).(\lambda (H18: +(csuba g d1 x1)).(ex_intro2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind +Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x1 (getl_intro i c2 (CHead x1 +(Bind Abbr) u) (CHead x1 (Bind Abbr) u) H17 (clear_bind Abbr x1 u)) H18)))) +H16)))))))))) H8)) H7))))) (\lambda (f: F).(\lambda (H5: (drop i O c1 (CHead +x0 (Flat f) t))).(\lambda (H6: (clear (CHead x0 (Flat f) t) (CHead d1 (Bind +Abbr) u))).(let H7 \def H5 in (unintro C c1 (\lambda (c: C).((drop i O c +(CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g c c2) \to (ex2 C +(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: +C).(csuba g d1 d2))))))) (nat_ind (\lambda (n: nat).(\forall (x1: C).((drop n +O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g x1 c2) \to (ex2 C +(\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: +C).(csuba g d1 d2)))))))) (\lambda (x1: C).(\lambda (H8: (drop O O x1 (CHead +x0 (Flat f) t))).(\lambda (c2: C).(\lambda (H9: (csuba g x1 c2)).(let H10 +\def (eq_ind C x1 (\lambda (c: C).(csuba g c c2)) H9 (CHead x0 (Flat f) t) +(drop_gen_refl x1 (CHead x0 (Flat f) t) H8)) in (let H_y \def (clear_flat x0 +(CHead d1 (Bind Abbr) u) (clear_gen_flat f x0 (CHead d1 (Bind Abbr) u) t H6) +f t) in (let H11 \def (csuba_clear_conf g (CHead x0 (Flat f) t) c2 H10 (CHead +d1 (Bind Abbr) u) H_y) in (ex2_ind C (\lambda (e2: C).(csuba g (CHead d1 +(Bind Abbr) u) e2)) (\lambda (e2: C).(clear c2 e2)) (ex2 C (\lambda (d2: +C).(getl O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) +(\lambda (x2: C).(\lambda (H12: (csuba g (CHead d1 (Bind Abbr) u) +x2)).(\lambda (H13: (clear c2 x2)).(let H_x \def (csuba_gen_abbr g d1 x2 u +H12) in (let H14 \def H_x in (ex2_ind C (\lambda (d2: C).(eq C x2 (CHead d2 +(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: +C).(getl O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) +(\lambda (x3: C).(\lambda (H15: (eq C x2 (CHead x3 (Bind Abbr) u))).(\lambda +(H16: (csuba g d1 x3)).(let H17 \def (eq_ind C x2 (\lambda (c: C).(clear c2 +c)) H13 (CHead x3 (Bind Abbr) u) H15) in (ex_intro2 C (\lambda (d2: C).(getl +O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x3 +(getl_intro O c2 (CHead x3 (Bind Abbr) u) c2 (drop_refl c2) H17) H16))))) +H14)))))) H11)))))))) (\lambda (n: nat).(\lambda (H8: ((\forall (x1: +C).((drop n O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g x1 c2) +\to (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))) (\lambda +(d2: C).(csuba g d1 d2))))))))).(\lambda (x1: C).(\lambda (H9: (drop (S n) O +x1 (CHead x0 (Flat f) t))).(\lambda (c2: C).(\lambda (H10: (csuba g x1 +c2)).(let H11 \def (drop_clear x1 (CHead x0 (Flat f) t) n H9) in (ex2_3_ind B +C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear x1 (CHead e (Bind +b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop n O e (CHead +x0 (Flat f) t))))) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind +Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x2: B).(\lambda (x3: +C).(\lambda (x4: T).(\lambda (H12: (clear x1 (CHead x3 (Bind x2) +x4))).(\lambda (H13: (drop n O x3 (CHead x0 (Flat f) t))).(let H14 \def +(csuba_clear_conf g x1 c2 H10 (CHead x3 (Bind x2) x4) H12) in (ex2_ind C +(\lambda (e2: C).(csuba g (CHead x3 (Bind x2) x4) e2)) (\lambda (e2: +C).(clear c2 e2)) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind +Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x5: C).(\lambda (H15: +(csuba g (CHead x3 (Bind x2) x4) x5)).(\lambda (H16: (clear c2 x5)).(let H_x +\def (csuba_gen_bind g x2 x3 x5 x4 H15) in (let H17 \def H_x in (ex2_3_ind B +C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C x5 (CHead e2 +(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g +x3 e2)))) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) u))) +(\lambda (d2: C).(csuba g d1 d2))) (\lambda (x6: B).(\lambda (x7: C).(\lambda +(x8: T).(\lambda (H18: (eq C x5 (CHead x7 (Bind x6) x8))).(\lambda (H19: +(csuba g x3 x7)).(let H20 \def (eq_ind C x5 (\lambda (c: C).(clear c2 c)) H16 +(CHead x7 (Bind x6) x8) H18) in (let H21 \def (H8 x3 H13 x7 H19) in (ex2_ind +C (\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abbr) u))) (\lambda (d2: +C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind +Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x9: C).(\lambda (H22: +(getl n x7 (CHead x9 (Bind Abbr) u))).(\lambda (H23: (csuba g d1 +x9)).(ex_intro2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) u))) +(\lambda (d2: C).(csuba g d1 d2)) x9 (getl_clear_bind x6 c2 x7 x8 H20 (CHead +x9 (Bind Abbr) u) n H22) H23)))) H21)))))))) H17)))))) H14))))))) H11)))))))) +i) H7))))) k H3 H4))))))) x H1 H2)))) H0))))))). +(* COMMENTS +Initial nodes: 2319 +END *) + +theorem csuba_getl_abst: + \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u1: T).(\forall +(i: nat).((getl i c1 (CHead d1 (Bind Abst) u1)) \to (\forall (c2: C).((csuba +g c1 c2) \to (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) +u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) +(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) +(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +a))))))))))))) +\def + \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u1: T).(\lambda +(i: nat).(\lambda (H: (getl i c1 (CHead d1 (Bind Abst) u1))).(let H0 \def +(getl_gen_all c1 (CHead d1 (Bind Abst) u1) i H) in (ex2_ind C (\lambda (e: +C).(drop i O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abst) u1))) +(\forall (c2: C).((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(getl i c2 +(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A +(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind +Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 +d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc +g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +a)))))))) (\lambda (x: C).(\lambda (H1: (drop i O c1 x)).(\lambda (H2: (clear +x (CHead d1 (Bind Abst) u1))).(C_ind (\lambda (c: C).((drop i O c1 c) \to +((clear c (CHead d1 (Bind Abst) u1)) \to (\forall (c2: C).((csuba g c1 c2) +\to (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) +(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda +(u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda +(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: +C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))))) (\lambda +(n: nat).(\lambda (_: (drop i O c1 (CSort n))).(\lambda (H4: (clear (CSort n) +(CHead d1 (Bind Abst) u1))).(clear_gen_sort (CHead d1 (Bind Abst) u1) n H4 +(\forall (c2: C).((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(getl i c2 +(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A +(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind +Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 +d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc +g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +a)))))))))))) (\lambda (x0: C).(\lambda (_: (((drop i O c1 x0) \to ((clear x0 +(CHead d1 (Bind Abst) u1)) \to (\forall (c2: C).((csuba g c1 c2) \to (or (ex2 +C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: +C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda +(_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: +T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: +T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda +(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))))))))).(\lambda (k: K).(\lambda +(t: T).(\lambda (H3: (drop i O c1 (CHead x0 k t))).(\lambda (H4: (clear +(CHead x0 k t) (CHead d1 (Bind Abst) u1))).(K_ind (\lambda (k0: K).((drop i O +c1 (CHead x0 k0 t)) \to ((clear (CHead x0 k0 t) (CHead d1 (Bind Abst) u1)) +\to (\forall (c2: C).((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(getl i +c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T +A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 +(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g +d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 +(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 +u2 a))))))))))) (\lambda (b: B).(\lambda (H5: (drop i O c1 (CHead x0 (Bind b) +t))).(\lambda (H6: (clear (CHead x0 (Bind b) t) (CHead d1 (Bind Abst) +u1))).(let H7 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda +(_: C).C) with [(CSort _) \Rightarrow d1 | (CHead c _ _) \Rightarrow c])) +(CHead d1 (Bind Abst) u1) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead +d1 (Bind Abst) u1) t H6)) in ((let H8 \def (f_equal C B (\lambda (e: +C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abst | +(CHead _ k0 _) \Rightarrow (match k0 in K return (\lambda (_: K).B) with +[(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abst])])) (CHead d1 (Bind +Abst) u1) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abst) +u1) t H6)) in ((let H9 \def (f_equal C T (\lambda (e: C).(match e in C return +(\lambda (_: C).T) with [(CSort _) \Rightarrow u1 | (CHead _ _ t0) +\Rightarrow t0])) (CHead d1 (Bind Abst) u1) (CHead x0 (Bind b) t) +(clear_gen_bind b x0 (CHead d1 (Bind Abst) u1) t H6)) in (\lambda (H10: (eq B +Abst b)).(\lambda (H11: (eq C d1 x0)).(\lambda (c2: C).(\lambda (H12: (csuba +g c1 c2)).(let H13 \def (eq_ind_r T t (\lambda (t0: T).(drop i O c1 (CHead x0 +(Bind b) t0))) H5 u1 H9) in (let H14 \def (eq_ind_r B b (\lambda (b0: +B).(drop i O c1 (CHead x0 (Bind b0) u1))) H13 Abst H10) in (let H15 \def +(eq_ind_r C x0 (\lambda (c: C).(drop i O c1 (CHead c (Bind Abst) u1))) H14 d1 +H11) in (let H16 \def (csuba_drop_abst i c1 d1 u1 H15 g c2 H12) in (or_ind +(ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst) u1))) (\lambda +(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: +T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: +C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: +C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (or (ex2 C +(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: +C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda +(_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: +T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: +T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda +(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (H17: (ex2 C (\lambda +(d2: C).(drop i O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 +d2)))).(ex2_ind C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst) u1))) +(\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: C).(getl i c2 +(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A +(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind +Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 +d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc +g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +a)))))) (\lambda (x1: C).(\lambda (H18: (drop i O c2 (CHead x1 (Bind Abst) +u1))).(\lambda (H19: (csuba g d1 x1)).(or_introl (ex2 C (\lambda (d2: +C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) +(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 +(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: +A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity +g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: C).(getl i c2 (CHead d2 +(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) x1 (getl_intro i c2 +(CHead x1 (Bind Abst) u1) (CHead x1 (Bind Abst) u1) H18 (clear_bind Abst x1 +u1)) H19))))) H17)) (\lambda (H17: (ex4_3 C T A (\lambda (d2: C).(\lambda +(u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda +(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: +C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))).(ex4_3_ind C +T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 +(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g +d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 +(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 +u2 a)))) (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) +(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda +(u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda +(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: +C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x1: +C).(\lambda (x2: T).(\lambda (x3: A).(\lambda (H18: (drop i O c2 (CHead x1 +(Bind Abbr) x2))).(\lambda (H19: (csuba g d1 x1)).(\lambda (H20: (arity g d1 +u1 (asucc g x3))).(\lambda (H21: (arity g x1 x2 x3)).(or_intror (ex2 C +(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: +C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda +(_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: +T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: +T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda +(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C T A (\lambda +(d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) +u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g +a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) +x1 x2 x3 (getl_intro i c2 (CHead x1 (Bind Abbr) x2) (CHead x1 (Bind Abbr) x2) +H18 (clear_bind Abbr x1 x2)) H19 H20 H21))))))))) H17)) H16)))))))))) H8)) +H7))))) (\lambda (f: F).(\lambda (H5: (drop i O c1 (CHead x0 (Flat f) +t))).(\lambda (H6: (clear (CHead x0 (Flat f) t) (CHead d1 (Bind Abst) +u1))).(let H7 \def H5 in (unintro C c1 (\lambda (c: C).((drop i O c (CHead x0 +(Flat f) t)) \to (\forall (c2: C).((csuba g c c2) \to (or (ex2 C (\lambda +(d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 +d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i +c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda +(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: +A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda +(a: A).(arity g d2 u2 a)))))))))) (nat_ind (\lambda (n: nat).(\forall (x1: +C).((drop n O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g x1 c2) +\to (or (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) u1))) +(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda +(u2: T).(\lambda (_: A).(getl n c2 (CHead d2 (Bind Abbr) u2))))) (\lambda +(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: +C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))))) (\lambda +(x1: C).(\lambda (H8: (drop O O x1 (CHead x0 (Flat f) t))).(\lambda (c2: +C).(\lambda (H9: (csuba g x1 c2)).(let H10 \def (eq_ind C x1 (\lambda (c: +C).(csuba g c c2)) H9 (CHead x0 (Flat f) t) (drop_gen_refl x1 (CHead x0 (Flat +f) t) H8)) in (let H_y \def (clear_flat x0 (CHead d1 (Bind Abst) u1) +(clear_gen_flat f x0 (CHead d1 (Bind Abst) u1) t H6) f t) in (let H11 \def +(csuba_clear_conf g (CHead x0 (Flat f) t) c2 H10 (CHead d1 (Bind Abst) u1) +H_y) in (ex2_ind C (\lambda (e2: C).(csuba g (CHead d1 (Bind Abst) u1) e2)) +(\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(getl O c2 (CHead +d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda +(d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abbr) +u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g +a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +a)))))) (\lambda (x2: C).(\lambda (H12: (csuba g (CHead d1 (Bind Abst) u1) +x2)).(\lambda (H13: (clear c2 x2)).(let H_x \def (csuba_gen_abst g d1 x2 u1 +H12) in (let H14 \def H_x in (or_ind (ex2 C (\lambda (d2: C).(eq C x2 (CHead +d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda +(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C x2 (CHead d2 (Bind Abbr) +u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g +a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +a))))) (or (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u1))) +(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda +(u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda +(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: +C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda +(H15: (ex2 C (\lambda (d2: C).(eq C x2 (CHead d2 (Bind Abst) u1))) (\lambda +(d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(eq C x2 (CHead d2 +(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: +C).(getl O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) +(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 +(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: +A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity +g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 a)))))) (\lambda (x3: C).(\lambda (H16: (eq C x2 (CHead x3 +(Bind Abst) u1))).(\lambda (H17: (csuba g d1 x3)).(let H18 \def (eq_ind C x2 +(\lambda (c: C).(clear c2 c)) H13 (CHead x3 (Bind Abst) u1) H16) in +(or_introl (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u1))) +(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda +(u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda +(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: +C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C +(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: +C).(csuba g d1 d2)) x3 (getl_intro O c2 (CHead x3 (Bind Abst) u1) c2 +(drop_refl c2) H18) H17)))))) H15)) (\lambda (H15: (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(eq C x2 (CHead d2 (Bind Abbr) u2))))) +(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) +(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: +A).(eq C x2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: +T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: +T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda +(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2: +C).(getl O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) +(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 +(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: +A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity +g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 a)))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: +A).(\lambda (H16: (eq C x2 (CHead x3 (Bind Abbr) x4))).(\lambda (H17: (csuba +g d1 x3)).(\lambda (H18: (arity g d1 u1 (asucc g x5))).(\lambda (H19: (arity +g x3 x4 x5)).(let H20 \def (eq_ind C x2 (\lambda (c: C).(clear c2 c)) H13 +(CHead x3 (Bind Abbr) x4) H16) in (or_intror (ex2 C (\lambda (d2: C).(getl O +c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T +A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 +(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g +d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 +(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 +u2 a))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: +A).(getl O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: +T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: +T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda +(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x3 x4 x5 (getl_intro O c2 (CHead +x3 (Bind Abbr) x4) c2 (drop_refl c2) H20) H17 H18 H19)))))))))) H15)) +H14)))))) H11)))))))) (\lambda (n: nat).(\lambda (H8: ((\forall (x1: +C).((drop n O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g x1 c2) +\to (or (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) u1))) +(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda +(u2: T).(\lambda (_: A).(getl n c2 (CHead d2 (Bind Abbr) u2))))) (\lambda +(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: +C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +a)))))))))))).(\lambda (x1: C).(\lambda (H9: (drop (S n) O x1 (CHead x0 (Flat +f) t))).(\lambda (c2: C).(\lambda (H10: (csuba g x1 c2)).(let H11 \def +(drop_clear x1 (CHead x0 (Flat f) t) n H9) in (ex2_3_ind B C T (\lambda (b: +B).(\lambda (e: C).(\lambda (v: T).(clear x1 (CHead e (Bind b) v))))) +(\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop n O e (CHead x0 (Flat +f) t))))) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) +u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abbr) +u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g +a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +a)))))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: T).(\lambda (H12: +(clear x1 (CHead x3 (Bind x2) x4))).(\lambda (H13: (drop n O x3 (CHead x0 +(Flat f) t))).(let H14 \def (csuba_clear_conf g x1 c2 H10 (CHead x3 (Bind x2) +x4) H12) in (ex2_ind C (\lambda (e2: C).(csuba g (CHead x3 (Bind x2) x4) e2)) +(\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 +(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A +(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 +(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g +d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 +(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 +u2 a)))))) (\lambda (x5: C).(\lambda (H15: (csuba g (CHead x3 (Bind x2) x4) +x5)).(\lambda (H16: (clear c2 x5)).(let H_x \def (csuba_gen_bind g x2 x3 x5 +x4 H15) in (let H17 \def H_x in (ex2_3_ind B C T (\lambda (b2: B).(\lambda +(e2: C).(\lambda (v2: T).(eq C x5 (CHead e2 (Bind b2) v2))))) (\lambda (_: +B).(\lambda (e2: C).(\lambda (_: T).(csuba g x3 e2)))) (or (ex2 C (\lambda +(d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g +d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl +(S n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: +T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: +T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda +(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x6: B).(\lambda (x7: +C).(\lambda (x8: T).(\lambda (H18: (eq C x5 (CHead x7 (Bind x6) +x8))).(\lambda (H19: (csuba g x3 x7)).(let H20 \def (eq_ind C x5 (\lambda (c: +C).(clear c2 c)) H16 (CHead x7 (Bind x6) x8) H18) in (let H21 \def (H8 x3 H13 +x7 H19) in (or_ind (ex2 C (\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abst) +u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(getl n x7 (CHead d2 (Bind Abbr) u2))))) +(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) +(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (or +(ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u1))) (\lambda +(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: +T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: +C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: +C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda +(H22: (ex2 C (\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abst) u1))) (\lambda +(d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(getl n x7 (CHead d2 +(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: +C).(getl (S n) c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 +d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S +n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda +(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: +A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda +(a: A).(arity g d2 u2 a)))))) (\lambda (x9: C).(\lambda (H23: (getl n x7 +(CHead x9 (Bind Abst) u1))).(\lambda (H24: (csuba g d1 x9)).(or_introl (ex2 C +(\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: +C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda +(_: A).(getl (S n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda +(_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: +T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda +(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: +C).(getl (S n) c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 +d2)) x9 (getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Abst) u1) n H23) +H24))))) H22)) (\lambda (H22: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: +T).(\lambda (_: A).(getl n x7 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: +C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: +C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))).(ex4_3_ind C +T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl n x7 (CHead d2 +(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g +d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 +(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 +u2 a)))) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) +u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abbr) +u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g +a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +a)))))) (\lambda (x9: C).(\lambda (x10: T).(\lambda (x11: A).(\lambda (H23: +(getl n x7 (CHead x9 (Bind Abbr) x10))).(\lambda (H24: (csuba g d1 +x9)).(\lambda (H25: (arity g d1 u1 (asucc g x11))).(\lambda (H26: (arity g x9 +x10 x11)).(or_intror (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind +Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abbr) +u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g +a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +a))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: +A).(getl (S n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: +T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: +T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda +(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x9 x10 x11 (getl_clear_bind x6 +c2 x7 x8 H20 (CHead x9 (Bind Abbr) x10) n H23) H24 H25 H26))))))))) H22)) +H21)))))))) H17)))))) H14))))))) H11)))))))) i) H7))))) k H3 H4))))))) x H1 +H2)))) H0))))))). +(* COMMENTS +Initial nodes: 6437 +END *) + +theorem csuba_getl_abst_rev: + \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).(\forall +(i: nat).((getl i c1 (CHead d1 (Bind Abst) u)) \to (\forall (c2: C).((csuba g +c2 c1) \to (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) +(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1)))))))))))) +\def + \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u: T).(\lambda +(i: nat).(\lambda (H: (getl i c1 (CHead d1 (Bind Abst) u))).(let H0 \def +(getl_gen_all c1 (CHead d1 (Bind Abst) u) i H) in (ex2_ind C (\lambda (e: +C).(drop i O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abst) u))) +(\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: C).(getl i c2 +(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))) (\lambda (x: +C).(\lambda (H1: (drop i O c1 x)).(\lambda (H2: (clear x (CHead d1 (Bind +Abst) u))).(C_ind (\lambda (c: C).((drop i O c1 c) \to ((clear c (CHead d1 +(Bind Abst) u)) \to (\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda +(d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 +d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))))) +(\lambda (n: nat).(\lambda (_: (drop i O c1 (CSort n))).(\lambda (H4: (clear +(CSort n) (CHead d1 (Bind Abst) u))).(clear_gen_sort (CHead d1 (Bind Abst) u) +n H4 (\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: C).(getl +i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))) (\lambda (x0: +C).(\lambda (_: (((drop i O c1 x0) \to ((clear x0 (CHead d1 (Bind Abst) u)) +\to (\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: C).(getl i +c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))).(\lambda (k: +K).(\lambda (t: T).(\lambda (H3: (drop i O c1 (CHead x0 k t))).(\lambda (H4: +(clear (CHead x0 k t) (CHead d1 (Bind Abst) u))).(K_ind (\lambda (k0: +K).((drop i O c1 (CHead x0 k0 t)) \to ((clear (CHead x0 k0 t) (CHead d1 (Bind +Abst) u)) \to (\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: +C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) +u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))))) (\lambda (b: +B).(\lambda (H5: (drop i O c1 (CHead x0 (Bind b) t))).(\lambda (H6: (clear +(CHead x0 (Bind b) t) (CHead d1 (Bind Abst) u))).(let H7 \def (f_equal C C +(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) +\Rightarrow d1 | (CHead c _ _) \Rightarrow c])) (CHead d1 (Bind Abst) u) +(CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abst) u) t H6)) in +((let H8 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: +C).B) with [(CSort _) \Rightarrow Abst | (CHead _ k0 _) \Rightarrow (match k0 +in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) +\Rightarrow Abst])])) (CHead d1 (Bind Abst) u) (CHead x0 (Bind b) t) +(clear_gen_bind b x0 (CHead d1 (Bind Abst) u) t H6)) in ((let H9 \def +(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with +[(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d1 (Bind +Abst) u) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abst) u) +t H6)) in (\lambda (H10: (eq B Abst b)).(\lambda (H11: (eq C d1 x0)).(\lambda +(c2: C).(\lambda (H12: (csuba g c2 c1)).(let H13 \def (eq_ind_r T t (\lambda +(t0: T).(drop i O c1 (CHead x0 (Bind b) t0))) H5 u H9) in (let H14 \def +(eq_ind_r B b (\lambda (b0: B).(drop i O c1 (CHead x0 (Bind b0) u))) H13 Abst +H10) in (let H15 \def (eq_ind_r C x0 (\lambda (c: C).(drop i O c1 (CHead c +(Bind Abst) u))) H14 d1 H11) in (let H16 \def (csuba_drop_abst_rev i c1 d1 u +H15 g c2 H12) in (or_ind (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind +Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(drop i O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: C).(getl i c2 +(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H17: (ex2 C +(\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: +C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop i O c2 (CHead d2 +(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: +C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) +u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x1: +C).(\lambda (H18: (drop i O c2 (CHead x1 (Bind Abst) u))).(\lambda (H19: +(csuba g x1 d1)).(or_introl (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 +(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(getl i +c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x1 +(getl_intro i c2 (CHead x1 (Bind Abst) u) (CHead x1 (Bind Abst) u) H18 +(clear_bind Abst x1 u)) H19))))) H17)) (\lambda (H17: (ex2_2 C T (\lambda +(d2: C).(\lambda (u2: T).(drop i O c2 (CHead d2 (Bind Void) u2)))) (\lambda +(d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: +C).(\lambda (u2: T).(drop i O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1))) (or (ex2 C (\lambda (d2: C).(getl i c2 +(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x1: +C).(\lambda (x2: T).(\lambda (H18: (drop i O c2 (CHead x1 (Bind Void) +x2))).(\lambda (H19: (csuba g x1 d1)).(or_intror (ex2 C (\lambda (d2: +C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) +u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T +(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x1 x2 (getl_intro i c2 +(CHead x1 (Bind Void) x2) (CHead x1 (Bind Void) x2) H18 (clear_bind Void x1 +x2)) H19)))))) H17)) H16)))))))))) H8)) H7))))) (\lambda (f: F).(\lambda (H5: +(drop i O c1 (CHead x0 (Flat f) t))).(\lambda (H6: (clear (CHead x0 (Flat f) +t) (CHead d1 (Bind Abst) u))).(let H7 \def H5 in (unintro C c1 (\lambda (c: +C).((drop i O c (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g c2 c) +\to (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) +(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1))))))))) (nat_ind (\lambda (n: nat).(\forall (x1: C).((drop +n O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g c2 x1) \to (or +(ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: +C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl n c2 +(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1)))))))))) (\lambda (x1: C).(\lambda (H8: (drop O O x1 (CHead x0 (Flat f) +t))).(\lambda (c2: C).(\lambda (H9: (csuba g c2 x1)).(let H10 \def (eq_ind C +x1 (\lambda (c: C).(csuba g c2 c)) H9 (CHead x0 (Flat f) t) (drop_gen_refl x1 +(CHead x0 (Flat f) t) H8)) in (let H_y \def (clear_flat x0 (CHead d1 (Bind +Abst) u) (clear_gen_flat f x0 (CHead d1 (Bind Abst) u) t H6) f t) in (let H11 +\def (csuba_clear_trans g (CHead x0 (Flat f) t) c2 H10 (CHead d1 (Bind Abst) +u) H_y) in (ex2_ind C (\lambda (e2: C).(csuba g e2 (CHead d1 (Bind Abst) u))) +(\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(getl O c2 (CHead +d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda +(d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 (Bind Void) u2)))) (\lambda +(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x2: C).(\lambda (H12: +(csuba g x2 (CHead d1 (Bind Abst) u))).(\lambda (H13: (clear c2 x2)).(let H_x +\def (csuba_gen_abst_rev g d1 x2 u H12) in (let H14 \def H_x in (or_ind (ex2 +C (\lambda (d2: C).(eq C x2 (CHead d2 (Bind Abst) u))) (\lambda (d2: +C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C x2 +(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1)))) (or (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u))) +(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(getl O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1))))) (\lambda (H15: (ex2 C (\lambda (d2: C).(eq C x2 (CHead +d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda +(d2: C).(eq C x2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) +(or (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u))) (\lambda +(d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl +O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g +d2 d1))))) (\lambda (x3: C).(\lambda (H16: (eq C x2 (CHead x3 (Bind Abst) +u))).(\lambda (H17: (csuba g x3 d1)).(let H18 \def (eq_ind C x2 (\lambda (c: +C).(clear c2 c)) H13 (CHead x3 (Bind Abst) u) H16) in (or_introl (ex2 C +(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: +C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl O c2 +(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1)))) (ex_intro2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u))) +(\lambda (d2: C).(csuba g d2 d1)) x3 (getl_intro O c2 (CHead x3 (Bind Abst) +u) c2 (drop_refl c2) H18) H17)))))) H15)) (\lambda (H15: (ex2_2 C T (\lambda +(d2: C).(\lambda (u2: T).(eq C x2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: +C).(\lambda (u2: T).(eq C x2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1))) (or (ex2 C (\lambda (d2: C).(getl O c2 +(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x3: +C).(\lambda (x4: T).(\lambda (H16: (eq C x2 (CHead x3 (Bind Void) +x4))).(\lambda (H17: (csuba g x3 d1)).(let H18 \def (eq_ind C x2 (\lambda (c: +C).(clear c2 c)) H13 (CHead x3 (Bind Void) x4) H16) in (or_intror (ex2 C +(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: +C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl O c2 +(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead +d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x3 +x4 (getl_intro O c2 (CHead x3 (Bind Void) x4) c2 (drop_refl c2) H18) +H17))))))) H15)) H14)))))) H11)))))))) (\lambda (n: nat).(\lambda (H8: +((\forall (x1: C).((drop n O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: +C).((csuba g c2 x1) \to (or (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 +(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(getl n c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1))))))))))).(\lambda (x1: C).(\lambda (H9: +(drop (S n) O x1 (CHead x0 (Flat f) t))).(\lambda (c2: C).(\lambda (H10: +(csuba g c2 x1)).(let H11 \def (drop_clear x1 (CHead x0 (Flat f) t) n H9) in +(ex2_3_ind B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear x1 +(CHead e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: +T).(drop n O e (CHead x0 (Flat f) t))))) (or (ex2 C (\lambda (d2: C).(getl (S +n) c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C +T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind Void) +u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x2: +B).(\lambda (x3: C).(\lambda (x4: T).(\lambda (H12: (clear x1 (CHead x3 (Bind +x2) x4))).(\lambda (H13: (drop n O x3 (CHead x0 (Flat f) t))).(let H14 \def +(csuba_clear_trans g x1 c2 H10 (CHead x3 (Bind x2) x4) H12) in (ex2_ind C +(\lambda (e2: C).(csuba g e2 (CHead x3 (Bind x2) x4))) (\lambda (e2: +C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind +Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x5: C).(\lambda (H15: (csuba +g x5 (CHead x3 (Bind x2) x4))).(\lambda (H16: (clear c2 x5)).(let H_x \def +(csuba_gen_bind_rev g x2 x3 x5 x4 H15) in (let H17 \def H_x in (ex2_3_ind B C +T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C x5 (CHead e2 (Bind +b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 +x3)))) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u))) +(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1))))) (\lambda (x6: B).(\lambda (x7: C).(\lambda (x8: +T).(\lambda (H18: (eq C x5 (CHead x7 (Bind x6) x8))).(\lambda (H19: (csuba g +x7 x3)).(let H20 \def (eq_ind C x5 (\lambda (c: C).(clear c2 c)) H16 (CHead +x7 (Bind x6) x8) H18) in (let H21 \def (H8 x3 H13 x7 H19) in (or_ind (ex2 C +(\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abst) u))) (\lambda (d2: +C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl n x7 +(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1)))) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u))) +(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1))))) (\lambda (H22: (ex2 C (\lambda (d2: C).(getl n x7 +(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C +(\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abst) u))) (\lambda (d2: +C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 +(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x9: C).(\lambda (H23: (getl +n x7 (CHead x9 (Bind Abst) u))).(\lambda (H24: (csuba g x9 d1)).(or_introl +(ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u))) (\lambda +(d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl +(S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba +g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind +Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x9 (getl_clear_bind x6 c2 x7 x8 +H20 (CHead x9 (Bind Abst) u) n H23) H24))))) H22)) (\lambda (H22: (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(getl n x7 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda +(d2: C).(\lambda (u2: T).(getl n x7 (CHead d2 (Bind Void) u2)))) (\lambda +(d2: C).(\lambda (_: T).(csuba g d2 d1))) (or (ex2 C (\lambda (d2: C).(getl +(S n) c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 +C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind Void) +u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x9: +C).(\lambda (x10: T).(\lambda (H23: (getl n x7 (CHead x9 (Bind Void) +x10))).(\lambda (H24: (csuba g x9 d1)).(or_intror (ex2 C (\lambda (d2: +C).(getl (S n) c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 +d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) +(ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x9 x10 +(getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Void) x10) n H23) H24)))))) +H22)) H21)))))))) H17)))))) H14))))))) H11)))))))) i) H7))))) k H3 H4))))))) +x H1 H2)))) H0))))))). +(* COMMENTS +Initial nodes: 4703 +END *) + +theorem csuba_getl_abbr_rev: + \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u1: T).(\forall +(i: nat).((getl i c1 (CHead d1 (Bind Abbr) u1)) \to (\forall (c2: C).((csuba +g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) +u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) +(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 +C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))))))) +\def + \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u1: T).(\lambda +(i: nat).(\lambda (H: (getl i c1 (CHead d1 (Bind Abbr) u1))).(let H0 \def +(getl_gen_all c1 (CHead d1 (Bind Abbr) u1) i H) in (ex2_ind C (\lambda (e: +C).(drop i O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abbr) u1))) +(\forall (c2: C).((csuba g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(getl i c2 +(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A +(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind +Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 +d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 +u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))) +(\lambda (x: C).(\lambda (H1: (drop i O c1 x)).(\lambda (H2: (clear x (CHead +d1 (Bind Abbr) u1))).(C_ind (\lambda (c: C).((drop i O c1 c) \to ((clear c +(CHead d1 (Bind Abbr) u1)) \to (\forall (c2: C).((csuba g c2 c1) \to (or3 +(ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: +C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda +(_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: +T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: +T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda +(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1)))))))))) (\lambda (n: nat).(\lambda (_: +(drop i O c1 (CSort n))).(\lambda (H4: (clear (CSort n) (CHead d1 (Bind Abbr) +u1))).(clear_gen_sort (CHead d1 (Bind Abbr) u1) n H4 (\forall (c2: C).((csuba +g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) +u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) +(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 +C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))) (\lambda (x0: +C).(\lambda (_: (((drop i O c1 x0) \to ((clear x0 (CHead d1 (Bind Abbr) u1)) +\to (\forall (c2: C).((csuba g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(getl +i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C +T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 +(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g +d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 +u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (H3: (drop i O c1 +(CHead x0 k t))).(\lambda (H4: (clear (CHead x0 k t) (CHead d1 (Bind Abbr) +u1))).(K_ind (\lambda (k0: K).((drop i O c1 (CHead x0 k0 t)) \to ((clear +(CHead x0 k0 t) (CHead d1 (Bind Abbr) u1)) \to (\forall (c2: C).((csuba g c2 +c1) \to (or3 (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) +(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda +(u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda +(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: +C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))))) (\lambda (b: +B).(\lambda (H5: (drop i O c1 (CHead x0 (Bind b) t))).(\lambda (H6: (clear +(CHead x0 (Bind b) t) (CHead d1 (Bind Abbr) u1))).(let H7 \def (f_equal C C +(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) +\Rightarrow d1 | (CHead c _ _) \Rightarrow c])) (CHead d1 (Bind Abbr) u1) +(CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abbr) u1) t H6)) +in ((let H8 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda +(_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow +(match k0 in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | +(Flat _) \Rightarrow Abbr])])) (CHead d1 (Bind Abbr) u1) (CHead x0 (Bind b) +t) (clear_gen_bind b x0 (CHead d1 (Bind Abbr) u1) t H6)) in ((let H9 \def +(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with +[(CSort _) \Rightarrow u1 | (CHead _ _ t0) \Rightarrow t0])) (CHead d1 (Bind +Abbr) u1) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abbr) +u1) t H6)) in (\lambda (H10: (eq B Abbr b)).(\lambda (H11: (eq C d1 +x0)).(\lambda (c2: C).(\lambda (H12: (csuba g c2 c1)).(let H13 \def (eq_ind_r +T t (\lambda (t0: T).(drop i O c1 (CHead x0 (Bind b) t0))) H5 u1 H9) in (let +H14 \def (eq_ind_r B b (\lambda (b0: B).(drop i O c1 (CHead x0 (Bind b0) +u1))) H13 Abbr H10) in (let H15 \def (eq_ind_r C x0 (\lambda (c: C).(drop i O +c1 (CHead c (Bind Abbr) u1))) H14 d1 H11) in (let H16 \def +(csuba_drop_abbr_rev i c1 d1 u1 H15 g c2 H12) in (or3_ind (ex2 C (\lambda +(d2: C).(drop i O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 +d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop i +O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda +(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(drop i O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind +Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) +(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 +C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H17: (ex2 C +(\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: +C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop i O c2 (CHead d2 +(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) (or3 (ex2 C (\lambda (d2: +C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) +(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 +(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: +A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1))))) (\lambda (x1: C).(\lambda (H18: (drop i O c2 (CHead x1 +(Bind Abbr) u1))).(\lambda (H19: (csuba g x1 d1)).(or3_intro0 (ex2 C (\lambda +(d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 +d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i +c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda +(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(getl i c2 (CHead d2 +(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x1 (getl_intro i c2 +(CHead x1 (Bind Abbr) u1) (CHead x1 (Bind Abbr) u1) H18 (clear_bind Abbr x1 +u1)) H19))))) H17)) (\lambda (H17: (ex4_3 C T A (\lambda (d2: C).(\lambda +(u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abst) u2))))) (\lambda +(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: +C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T +A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 +(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g +d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 +u1 a)))) (or3 (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) +(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda +(u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda +(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: +C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x1: +C).(\lambda (x2: T).(\lambda (x3: A).(\lambda (H18: (drop i O c2 (CHead x1 +(Bind Abst) x2))).(\lambda (H19: (csuba g x1 d1)).(\lambda (H20: (arity g x1 +x2 (asucc g x3))).(\lambda (H21: (arity g d1 u1 x3)).(or3_intro1 (ex2 C +(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: +C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda +(_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: +T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: +T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda +(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1)))) (ex4_3_intro C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) +(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x1 x2 x3 +(getl_intro i c2 (CHead x1 (Bind Abst) x2) (CHead x1 (Bind Abst) x2) H18 +(clear_bind Abst x1 x2)) H19 H20 H21))))))))) H17)) (\lambda (H17: (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(drop i O c2 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda +(d2: C).(\lambda (u2: T).(drop i O c2 (CHead d2 (Bind Void) u2)))) (\lambda +(d2: C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C (\lambda (d2: C).(getl +i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C +T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 +(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g +d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 +u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) +(\lambda (x1: C).(\lambda (x2: T).(\lambda (H18: (drop i O c2 (CHead x1 (Bind +Void) x2))).(\lambda (H19: (csuba g x1 d1)).(or3_intro2 (ex2 C (\lambda (d2: +C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) +(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 +(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: +A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: +T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1))) x1 x2 (getl_intro i c2 (CHead x1 (Bind Void) x2) (CHead +x1 (Bind Void) x2) H18 (clear_bind Void x1 x2)) H19)))))) H17)) H16)))))))))) +H8)) H7))))) (\lambda (f: F).(\lambda (H5: (drop i O c1 (CHead x0 (Flat f) +t))).(\lambda (H6: (clear (CHead x0 (Flat f) t) (CHead d1 (Bind Abbr) +u1))).(let H7 \def H5 in (unintro C c1 (\lambda (c: C).((drop i O c (CHead x0 +(Flat f) t)) \to (\forall (c2: C).((csuba g c2 c) \to (or3 (ex2 C (\lambda +(d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 +d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i +c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda +(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1))))))))) (nat_ind (\lambda (n: nat).(\forall (x1: C).((drop +n O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g c2 x1) \to (or3 +(ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: +C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda +(_: A).(getl n c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: +T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: +T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda +(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(getl n c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1)))))))))) (\lambda (x1: C).(\lambda (H8: +(drop O O x1 (CHead x0 (Flat f) t))).(\lambda (c2: C).(\lambda (H9: (csuba g +c2 x1)).(let H10 \def (eq_ind C x1 (\lambda (c: C).(csuba g c2 c)) H9 (CHead +x0 (Flat f) t) (drop_gen_refl x1 (CHead x0 (Flat f) t) H8)) in (let H_y \def +(clear_flat x0 (CHead d1 (Bind Abbr) u1) (clear_gen_flat f x0 (CHead d1 (Bind +Abbr) u1) t H6) f t) in (let H11 \def (csuba_clear_trans g (CHead x0 (Flat f) +t) c2 H10 (CHead d1 (Bind Abbr) u1) H_y) in (ex2_ind C (\lambda (e2: +C).(csuba g e2 (CHead d1 (Bind Abbr) u1))) (\lambda (e2: C).(clear c2 e2)) +(or3 (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u1))) (\lambda +(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: +T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: +C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: +C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x2: +C).(\lambda (H12: (csuba g x2 (CHead d1 (Bind Abbr) u1))).(\lambda (H13: +(clear c2 x2)).(let H_x \def (csuba_gen_abbr_rev g d1 x2 u1 H12) in (let H14 +\def H_x in (or3_ind (ex2 C (\lambda (d2: C).(eq C x2 (CHead d2 (Bind Abbr) +u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(eq C x2 (CHead d2 (Bind Abst) u2))))) +(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 +C T (\lambda (d2: C).(\lambda (u2: T).(eq C x2 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: +C).(getl O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) +(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 +(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: +A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(getl O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1))))) (\lambda (H15: (ex2 C (\lambda (d2: C).(eq C x2 (CHead +d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda +(d2: C).(eq C x2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 +d1)) (or3 (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u1))) +(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda +(u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abst) u2))))) (\lambda +(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: +C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x3: +C).(\lambda (H16: (eq C x2 (CHead x3 (Bind Abbr) u1))).(\lambda (H17: (csuba +g x3 d1)).(let H18 \def (eq_ind C x2 (\lambda (c: C).(clear c2 c)) H13 (CHead +x3 (Bind Abbr) u1) H16) in (or3_intro0 (ex2 C (\lambda (d2: C).(getl O c2 +(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A +(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind +Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 +d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 +u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) +(ex_intro2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u1))) (\lambda +(d2: C).(csuba g d2 d1)) x3 (getl_intro O c2 (CHead x3 (Bind Abbr) u1) c2 +(drop_refl c2) H18) H17)))))) H15)) (\lambda (H15: (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(eq C x2 (CHead d2 (Bind Abst) u2))))) +(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 +a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: +A).(eq C x2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: +T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: +T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda +(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C (\lambda (d2: +C).(getl O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) +(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 +(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: +A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(getl O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: +A).(\lambda (H16: (eq C x2 (CHead x3 (Bind Abst) x4))).(\lambda (H17: (csuba +g x3 d1)).(\lambda (H18: (arity g x3 x4 (asucc g x5))).(\lambda (H19: (arity +g d1 u1 x5)).(let H20 \def (eq_ind C x2 (\lambda (c: C).(clear c2 c)) H13 +(CHead x3 (Bind Abst) x4) H16) in (or3_intro1 (ex2 C (\lambda (d2: C).(getl O +c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T +A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 +(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g +d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 +u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) +(ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O +c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda +(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g d1 u1 a)))) x3 x4 x5 (getl_intro O c2 (CHead x3 (Bind Abst) +x4) c2 (drop_refl c2) H20) H17 H18 H19)))))))))) H15)) (\lambda (H15: (ex2_2 +C T (\lambda (d2: C).(\lambda (u2: T).(eq C x2 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda +(d2: C).(\lambda (u2: T).(eq C x2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C (\lambda (d2: C).(getl O c2 +(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A +(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind +Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 +d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 +u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 +(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) +(\lambda (x3: C).(\lambda (x4: T).(\lambda (H16: (eq C x2 (CHead x3 (Bind +Void) x4))).(\lambda (H17: (csuba g x3 d1)).(let H18 \def (eq_ind C x2 +(\lambda (c: C).(clear c2 c)) H13 (CHead x3 (Bind Void) x4) H16) in +(or3_intro2 (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u1))) +(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda +(u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abst) u2))))) (\lambda +(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: +C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda +(d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 (Bind Void) u2)))) (\lambda +(d2: C).(\lambda (_: T).(csuba g d2 d1))) x3 x4 (getl_intro O c2 (CHead x3 +(Bind Void) x4) c2 (drop_refl c2) H18) H17))))))) H15)) H14)))))) H11)))))))) +(\lambda (n: nat).(\lambda (H8: ((\forall (x1: C).((drop n O x1 (CHead x0 +(Flat f) t)) \to (\forall (c2: C).((csuba g c2 x1) \to (or3 (ex2 C (\lambda +(d2: C).(getl n c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 +d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl n +c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda +(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(getl n c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1))))))))))).(\lambda (x1: C).(\lambda (H9: (drop (S n) O x1 +(CHead x0 (Flat f) t))).(\lambda (c2: C).(\lambda (H10: (csuba g c2 x1)).(let +H11 \def (drop_clear x1 (CHead x0 (Flat f) t) n H9) in (ex2_3_ind B C T +(\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear x1 (CHead e (Bind b) +v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop n O e (CHead x0 +(Flat f) t))))) (or3 (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind +Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst) +u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) +(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g +a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda +(x2: B).(\lambda (x3: C).(\lambda (x4: T).(\lambda (H12: (clear x1 (CHead x3 +(Bind x2) x4))).(\lambda (H13: (drop n O x3 (CHead x0 (Flat f) t))).(let H14 +\def (csuba_clear_trans g x1 c2 H10 (CHead x3 (Bind x2) x4) H12) in (ex2_ind +C (\lambda (e2: C).(csuba g e2 (CHead x3 (Bind x2) x4))) (\lambda (e2: +C).(clear c2 e2)) (or3 (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind +Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst) +u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) +(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g +a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda +(x5: C).(\lambda (H15: (csuba g x5 (CHead x3 (Bind x2) x4))).(\lambda (H16: +(clear c2 x5)).(let H_x \def (csuba_gen_bind_rev g x2 x3 x5 x4 H15) in (let +H17 \def H_x in (ex2_3_ind B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda +(v2: T).(eq C x5 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: +C).(\lambda (_: T).(csuba g e2 x3)))) (or3 (ex2 C (\lambda (d2: C).(getl (S +n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C +T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead +d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: +A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1))))) (\lambda (x6: B).(\lambda (x7: C).(\lambda (x8: +T).(\lambda (H18: (eq C x5 (CHead x7 (Bind x6) x8))).(\lambda (H19: (csuba g +x7 x3)).(let H20 \def (eq_ind C x5 (\lambda (c: C).(clear c2 c)) H16 (CHead +x7 (Bind x6) x8) H18) in (let H21 \def (H8 x3 H13 x7 H19) in (or3_ind (ex2 C +(\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: +C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda +(_: A).(getl n x7 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: +T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: +T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda +(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(getl n x7 (CHead d2 (Bind Void) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(getl (S +n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C +T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead +d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: +A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1))))) (\lambda (H22: (ex2 C (\lambda (d2: C).(getl n x7 +(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C +(\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: +C).(csuba g d2 d1)) (or3 (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 +(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda +(d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst) +u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) +(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g +a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda +(x9: C).(\lambda (H23: (getl n x7 (CHead x9 (Bind Abbr) u1))).(\lambda (H24: +(csuba g x9 d1)).(or3_intro0 (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 +(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda +(d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst) +u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) +(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g +a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C +(\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: +C).(csuba g d2 d1)) x9 (getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Abbr) +u1) n H23) H24))))) H22)) (\lambda (H22: (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(getl n x7 (CHead d2 (Bind Abst) u2))))) +(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 +a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: +A).(getl n x7 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: +T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: +T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda +(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C (\lambda (d2: +C).(getl (S n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 +d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S +n) c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda +(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1))))) (\lambda (x9: C).(\lambda (x10: T).(\lambda (x11: +A).(\lambda (H23: (getl n x7 (CHead x9 (Bind Abst) x10))).(\lambda (H24: +(csuba g x9 d1)).(\lambda (H25: (arity g x9 x10 (asucc g x11))).(\lambda +(H26: (arity g d1 u1 x11)).(or3_intro1 (ex2 C (\lambda (d2: C).(getl (S n) c2 +(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A +(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 +(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g +d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 +u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead +d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) +(ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S +n) c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda +(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g d1 u1 a)))) x9 x10 x11 (getl_clear_bind x6 c2 x7 x8 H20 +(CHead x9 (Bind Abst) x10) n H23) H24 H25 H26))))))))) H22)) (\lambda (H22: +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl n x7 (CHead d2 (Bind Void) +u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind C T +(\lambda (d2: C).(\lambda (u2: T).(getl n x7 (CHead d2 (Bind Void) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C (\lambda (d2: +C).(getl (S n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 +d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S +n) c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda +(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: +T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: +T).(csuba g d2 d1))))) (\lambda (x9: C).(\lambda (x10: T).(\lambda (H23: +(getl n x7 (CHead x9 (Bind Void) x10))).(\lambda (H24: (csuba g x9 +d1)).(or3_intro2 (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) +u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst) +u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) +(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g +a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) +(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind +Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro +C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind Void) +u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x9 x10 +(getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Void) x10) n H23) H24)))))) +H22)) H21)))))))) H17)))))) H14))))))) H11)))))))) i) H7))))) k H3 H4))))))) +x H1 H2)))) H0))))))). +(* COMMENTS +Initial nodes: 9091 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/csuba/props.ma b/matita/matita/contribs/lambdadelta/basic_1/csuba/props.ma new file mode 100644 index 000000000..823e88825 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/csuba/props.ma @@ -0,0 +1,28 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/csuba/defs.ma". + +theorem csuba_refl: + \forall (g: G).(\forall (c: C).(csuba g c c)) +\def + \lambda (g: G).(\lambda (c: C).(C_ind (\lambda (c0: C).(csuba g c0 c0)) +(\lambda (n: nat).(csuba_sort g n)) (\lambda (c0: C).(\lambda (H: (csuba g c0 +c0)).(\lambda (k: K).(\lambda (t: T).(csuba_head g c0 c0 H k t))))) c)). +(* COMMENTS +Initial nodes: 53 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubc/arity.ma b/matita/matita/contribs/lambdadelta/basic_1/csubc/arity.ma new file mode 100644 index 000000000..62922d311 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/csubc/arity.ma @@ -0,0 +1,42 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/csubc/csuba.ma". + +theorem csubc_arity_conf: + \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to +(\forall (t: T).(\forall (a: A).((arity g c1 t a) \to (arity g c2 t a))))))) +\def + \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubc g c1 +c2)).(\lambda (t: T).(\lambda (a: A).(\lambda (H0: (arity g c1 t +a)).(csuba_arity g c1 t a H0 c2 (csubc_csuba g c1 c2 H)))))))). +(* COMMENTS +Initial nodes: 51 +END *) + +theorem csubc_arity_trans: + \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to +((csubv c1 c2) \to (\forall (t: T).(\forall (a: A).((arity g c2 t a) \to +(arity g c1 t a)))))))) +\def + \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubc g c1 +c2)).(\lambda (H0: (csubv c1 c2)).(\lambda (t: T).(\lambda (a: A).(\lambda +(H1: (arity g c2 t a)).(csuba_arity_rev g c2 t a H1 c1 (csubc_csuba g c1 c2 +H) H0)))))))). +(* COMMENTS +Initial nodes: 59 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubc/clear.ma b/matita/matita/contribs/lambdadelta/basic_1/csubc/clear.ma new file mode 100644 index 000000000..185bdff75 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/csubc/clear.ma @@ -0,0 +1,170 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/csubc/fwd.ma". + +theorem csubc_clear_conf: + \forall (g: G).(\forall (c1: C).(\forall (e1: C).((clear c1 e1) \to (\forall +(c2: C).((csubc g c1 c2) \to (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda +(e2: C).(csubc g e1 e2)))))))) +\def + \lambda (g: G).(\lambda (c1: C).(\lambda (e1: C).(\lambda (H: (clear c1 +e1)).(clear_ind (\lambda (c: C).(\lambda (c0: C).(\forall (c2: C).((csubc g c +c2) \to (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2: C).(csubc g c0 +e2))))))) (\lambda (b: B).(\lambda (e: C).(\lambda (u: T).(\lambda (c2: +C).(\lambda (H0: (csubc g (CHead e (Bind b) u) c2)).(let H_x \def +(csubc_gen_head_l g e c2 u (Bind b) H0) in (let H1 \def H_x in (or3_ind (ex2 +C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind b) u))) (\lambda (c3: C).(csubc g +e c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K +(Bind b) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq +C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda +(_: A).(csubc g e c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 +g (asucc g a) e u)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g +a c3 w))))) (ex4_3 B C T (\lambda (b0: B).(\lambda (c3: C).(\lambda (v2: +T).(eq C c2 (CHead c3 (Bind b0) v2))))) (\lambda (_: B).(\lambda (_: +C).(\lambda (_: T).(eq K (Bind b) (Bind Void))))) (\lambda (b0: B).(\lambda +(_: C).(\lambda (_: T).(not (eq B b0 Void))))) (\lambda (_: B).(\lambda (c3: +C).(\lambda (_: T).(csubc g e c3))))) (ex2 C (\lambda (e2: C).(clear c2 e2)) +(\lambda (e2: C).(csubc g (CHead e (Bind b) u) e2))) (\lambda (H2: (ex2 C +(\lambda (c3: C).(eq C c2 (CHead c3 (Bind b) u))) (\lambda (c3: C).(csubc g e +c3)))).(ex2_ind C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind b) u))) (\lambda +(c3: C).(csubc g e c3)) (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2: +C).(csubc g (CHead e (Bind b) u) e2))) (\lambda (x: C).(\lambda (H3: (eq C c2 +(CHead x (Bind b) u))).(\lambda (H4: (csubc g e x)).(eq_ind_r C (CHead x +(Bind b) u) (\lambda (c: C).(ex2 C (\lambda (e2: C).(clear c e2)) (\lambda +(e2: C).(csubc g (CHead e (Bind b) u) e2)))) (ex_intro2 C (\lambda (e2: +C).(clear (CHead x (Bind b) u) e2)) (\lambda (e2: C).(csubc g (CHead e (Bind +b) u) e2)) (CHead x (Bind b) u) (clear_bind b x u) (csubc_head g e x H4 (Bind +b) u)) c2 H3)))) H2)) (\lambda (H2: (ex5_3 C T A (\lambda (_: C).(\lambda (_: +T).(\lambda (_: A).(eq K (Bind b) (Bind Abst))))) (\lambda (c3: C).(\lambda +(w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: +C).(\lambda (_: T).(\lambda (_: A).(csubc g e c3)))) (\lambda (_: C).(\lambda +(_: T).(\lambda (a: A).(sc3 g (asucc g a) e u)))) (\lambda (c3: C).(\lambda +(w: T).(\lambda (a: A).(sc3 g a c3 w)))))).(ex5_3_ind C T A (\lambda (_: +C).(\lambda (_: T).(\lambda (_: A).(eq K (Bind b) (Bind Abst))))) (\lambda +(c3: C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) +(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g e c3)))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) e u)))) (\lambda +(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w)))) (ex2 C (\lambda +(e2: C).(clear c2 e2)) (\lambda (e2: C).(csubc g (CHead e (Bind b) u) e2))) +(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H3: (eq K (Bind +b) (Bind Abst))).(\lambda (H4: (eq C c2 (CHead x0 (Bind Abbr) x1))).(\lambda +(H5: (csubc g e x0)).(\lambda (H6: (sc3 g (asucc g x2) e u)).(\lambda (H7: +(sc3 g x2 x0 x1)).(eq_ind_r C (CHead x0 (Bind Abbr) x1) (\lambda (c: C).(ex2 +C (\lambda (e2: C).(clear c e2)) (\lambda (e2: C).(csubc g (CHead e (Bind b) +u) e2)))) (let H8 \def (f_equal K B (\lambda (e0: K).(match e0 in K return +(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow b])) +(Bind b) (Bind Abst) H3) in (eq_ind_r B Abst (\lambda (b0: B).(ex2 C (\lambda +(e2: C).(clear (CHead x0 (Bind Abbr) x1) e2)) (\lambda (e2: C).(csubc g +(CHead e (Bind b0) u) e2)))) (ex_intro2 C (\lambda (e2: C).(clear (CHead x0 +(Bind Abbr) x1) e2)) (\lambda (e2: C).(csubc g (CHead e (Bind Abst) u) e2)) +(CHead x0 (Bind Abbr) x1) (clear_bind Abbr x0 x1) (csubc_abst g e x0 H5 u x2 +H6 x1 H7)) b H8)) c2 H4))))))))) H2)) (\lambda (H2: (ex4_3 B C T (\lambda +(b0: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b0) +v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K (Bind b) (Bind +Void))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b0 +Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g e +c3)))))).(ex4_3_ind B C T (\lambda (b0: B).(\lambda (c3: C).(\lambda (v2: +T).(eq C c2 (CHead c3 (Bind b0) v2))))) (\lambda (_: B).(\lambda (_: +C).(\lambda (_: T).(eq K (Bind b) (Bind Void))))) (\lambda (b0: B).(\lambda +(_: C).(\lambda (_: T).(not (eq B b0 Void))))) (\lambda (_: B).(\lambda (c3: +C).(\lambda (_: T).(csubc g e c3)))) (ex2 C (\lambda (e2: C).(clear c2 e2)) +(\lambda (e2: C).(csubc g (CHead e (Bind b) u) e2))) (\lambda (x0: +B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H3: (eq C c2 (CHead x1 (Bind +x0) x2))).(\lambda (H4: (eq K (Bind b) (Bind Void))).(\lambda (H5: (not (eq B +x0 Void))).(\lambda (H6: (csubc g e x1)).(eq_ind_r C (CHead x1 (Bind x0) x2) +(\lambda (c: C).(ex2 C (\lambda (e2: C).(clear c e2)) (\lambda (e2: C).(csubc +g (CHead e (Bind b) u) e2)))) (let H7 \def (f_equal K B (\lambda (e0: +K).(match e0 in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | +(Flat _) \Rightarrow b])) (Bind b) (Bind Void) H4) in (eq_ind_r B Void +(\lambda (b0: B).(ex2 C (\lambda (e2: C).(clear (CHead x1 (Bind x0) x2) e2)) +(\lambda (e2: C).(csubc g (CHead e (Bind b0) u) e2)))) (ex_intro2 C (\lambda +(e2: C).(clear (CHead x1 (Bind x0) x2) e2)) (\lambda (e2: C).(csubc g (CHead +e (Bind Void) u) e2)) (CHead x1 (Bind x0) x2) (clear_bind x0 x1 x2) +(csubc_void g e x1 H6 x0 H5 u x2)) b H7)) c2 H3)))))))) H2)) H1)))))))) +(\lambda (e: C).(\lambda (c: C).(\lambda (_: (clear e c)).(\lambda (H1: +((\forall (c2: C).((csubc g e c2) \to (ex2 C (\lambda (e2: C).(clear c2 e2)) +(\lambda (e2: C).(csubc g c e2))))))).(\lambda (f: F).(\lambda (u: +T).(\lambda (c2: C).(\lambda (H2: (csubc g (CHead e (Flat f) u) c2)).(let H_x +\def (csubc_gen_head_l g e c2 u (Flat f) H2) in (let H3 \def H_x in (or3_ind +(ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 (Flat f) u))) (\lambda (c3: +C).(csubc g e c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: +A).(eq K (Flat f) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda +(_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: +T).(\lambda (_: A).(csubc g e c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(sc3 g (asucc g a) e u)))) (\lambda (c3: C).(\lambda (w: T).(\lambda +(a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: +C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_: +B).(\lambda (_: C).(\lambda (_: T).(eq K (Flat f) (Bind Void))))) (\lambda +(b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: +B).(\lambda (c3: C).(\lambda (_: T).(csubc g e c3))))) (ex2 C (\lambda (e2: +C).(clear c2 e2)) (\lambda (e2: C).(csubc g c e2))) (\lambda (H4: (ex2 C +(\lambda (c3: C).(eq C c2 (CHead c3 (Flat f) u))) (\lambda (c3: C).(csubc g e +c3)))).(ex2_ind C (\lambda (c3: C).(eq C c2 (CHead c3 (Flat f) u))) (\lambda +(c3: C).(csubc g e c3)) (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2: +C).(csubc g c e2))) (\lambda (x: C).(\lambda (H5: (eq C c2 (CHead x (Flat f) +u))).(\lambda (H6: (csubc g e x)).(eq_ind_r C (CHead x (Flat f) u) (\lambda +(c0: C).(ex2 C (\lambda (e2: C).(clear c0 e2)) (\lambda (e2: C).(csubc g c +e2)))) (let H_x0 \def (H1 x H6) in (let H7 \def H_x0 in (ex2_ind C (\lambda +(e2: C).(clear x e2)) (\lambda (e2: C).(csubc g c e2)) (ex2 C (\lambda (e2: +C).(clear (CHead x (Flat f) u) e2)) (\lambda (e2: C).(csubc g c e2))) +(\lambda (x0: C).(\lambda (H8: (clear x x0)).(\lambda (H9: (csubc g c +x0)).(ex_intro2 C (\lambda (e2: C).(clear (CHead x (Flat f) u) e2)) (\lambda +(e2: C).(csubc g c e2)) x0 (clear_flat x x0 H8 f u) H9)))) H7))) c2 H5)))) +H4)) (\lambda (H4: (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: +A).(eq K (Flat f) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda +(_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: +T).(\lambda (_: A).(csubc g e c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(sc3 g (asucc g a) e u)))) (\lambda (c3: C).(\lambda (w: T).(\lambda +(a: A).(sc3 g a c3 w)))))).(ex5_3_ind C T A (\lambda (_: C).(\lambda (_: +T).(\lambda (_: A).(eq K (Flat f) (Bind Abst))))) (\lambda (c3: C).(\lambda +(w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: +C).(\lambda (_: T).(\lambda (_: A).(csubc g e c3)))) (\lambda (_: C).(\lambda +(_: T).(\lambda (a: A).(sc3 g (asucc g a) e u)))) (\lambda (c3: C).(\lambda +(w: T).(\lambda (a: A).(sc3 g a c3 w)))) (ex2 C (\lambda (e2: C).(clear c2 +e2)) (\lambda (e2: C).(csubc g c e2))) (\lambda (x0: C).(\lambda (x1: +T).(\lambda (x2: A).(\lambda (H5: (eq K (Flat f) (Bind Abst))).(\lambda (H6: +(eq C c2 (CHead x0 (Bind Abbr) x1))).(\lambda (_: (csubc g e x0)).(\lambda +(_: (sc3 g (asucc g x2) e u)).(\lambda (_: (sc3 g x2 x0 x1)).(eq_ind_r C +(CHead x0 (Bind Abbr) x1) (\lambda (c0: C).(ex2 C (\lambda (e2: C).(clear c0 +e2)) (\lambda (e2: C).(csubc g c e2)))) (let H10 \def (eq_ind K (Flat f) +(\lambda (ee: K).(match ee in K return (\lambda (_: K).Prop) with [(Bind _) +\Rightarrow False | (Flat _) \Rightarrow True])) I (Bind Abst) H5) in +(False_ind (ex2 C (\lambda (e2: C).(clear (CHead x0 (Bind Abbr) x1) e2)) +(\lambda (e2: C).(csubc g c e2))) H10)) c2 H6))))))))) H4)) (\lambda (H4: +(ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 +(CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: +T).(eq K (Flat f) (Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda +(_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: +T).(csubc g e c3)))))).(ex4_3_ind B C T (\lambda (b: B).(\lambda (c3: +C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_: +B).(\lambda (_: C).(\lambda (_: T).(eq K (Flat f) (Bind Void))))) (\lambda +(b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: +B).(\lambda (c3: C).(\lambda (_: T).(csubc g e c3)))) (ex2 C (\lambda (e2: +C).(clear c2 e2)) (\lambda (e2: C).(csubc g c e2))) (\lambda (x0: B).(\lambda +(x1: C).(\lambda (x2: T).(\lambda (H5: (eq C c2 (CHead x1 (Bind x0) +x2))).(\lambda (H6: (eq K (Flat f) (Bind Void))).(\lambda (_: (not (eq B x0 +Void))).(\lambda (_: (csubc g e x1)).(eq_ind_r C (CHead x1 (Bind x0) x2) +(\lambda (c0: C).(ex2 C (\lambda (e2: C).(clear c0 e2)) (\lambda (e2: +C).(csubc g c e2)))) (let H9 \def (eq_ind K (Flat f) (\lambda (ee: K).(match +ee in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat +_) \Rightarrow True])) I (Bind Void) H6) in (False_ind (ex2 C (\lambda (e2: +C).(clear (CHead x1 (Bind x0) x2) e2)) (\lambda (e2: C).(csubc g c e2))) H9)) +c2 H5)))))))) H4)) H3))))))))))) c1 e1 H)))). +(* COMMENTS +Initial nodes: 2837 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubc/csuba.ma b/matita/matita/contribs/lambdadelta/basic_1/csubc/csuba.ma new file mode 100644 index 000000000..2abfe2a1f --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/csubc/csuba.ma @@ -0,0 +1,40 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/csubc/defs.ma". + +include "Basic-1/sc3/props.ma". + +theorem csubc_csuba: + \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to (csuba +g c1 c2)))) +\def + \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubc g c1 +c2)).(csubc_ind g (\lambda (c: C).(\lambda (c0: C).(csuba g c c0))) (\lambda +(n: nat).(csuba_refl g (CSort n))) (\lambda (c3: C).(\lambda (c4: C).(\lambda +(_: (csubc g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (k: K).(\lambda +(v: T).(csuba_head g c3 c4 H1 k v))))))) (\lambda (c3: C).(\lambda (c4: +C).(\lambda (_: (csubc g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (b: +B).(\lambda (H2: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: +T).(csuba_void g c3 c4 H1 b H2 u1 u2))))))))) (\lambda (c3: C).(\lambda (c4: +C).(\lambda (_: (csubc g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (v: +T).(\lambda (a: A).(\lambda (H2: (sc3 g (asucc g a) c3 v)).(\lambda (w: +T).(\lambda (H3: (sc3 g a c4 w)).(csuba_abst g c3 c4 H1 v a (sc3_arity_gen g +c3 v (asucc g a) H2) w (sc3_arity_gen g c4 w a H3))))))))))) c1 c2 H)))). +(* COMMENTS +Initial nodes: 231 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubc/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/csubc/defs.ma new file mode 100644 index 000000000..73a4c5625 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/csubc/defs.ma @@ -0,0 +1,30 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/sc3/defs.ma". + +inductive csubc (g: G): C \to (C \to Prop) \def +| csubc_sort: \forall (n: nat).(csubc g (CSort n) (CSort n)) +| csubc_head: \forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to (\forall +(k: K).(\forall (v: T).(csubc g (CHead c1 k v) (CHead c2 k v)))))) +| csubc_void: \forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to (\forall +(b: B).((not (eq B b Void)) \to (\forall (u1: T).(\forall (u2: T).(csubc g +(CHead c1 (Bind Void) u1) (CHead c2 (Bind b) u2)))))))) +| csubc_abst: \forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to (\forall +(v: T).(\forall (a: A).((sc3 g (asucc g a) c1 v) \to (\forall (w: T).((sc3 g +a c2 w) \to (csubc g (CHead c1 (Bind Abst) v) (CHead c2 (Bind Abbr) +w))))))))). + diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubc/drop.ma b/matita/matita/contribs/lambdadelta/basic_1/csubc/drop.ma new file mode 100644 index 000000000..195d36491 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/csubc/drop.ma @@ -0,0 +1,475 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/csubc/fwd.ma". + +include "Basic-1/sc3/props.ma". + +theorem csubc_drop_conf_O: + \forall (g: G).(\forall (c1: C).(\forall (e1: C).(\forall (h: nat).((drop h +O c1 e1) \to (\forall (c2: C).((csubc g c1 c2) \to (ex2 C (\lambda (e2: +C).(drop h O c2 e2)) (\lambda (e2: C).(csubc g e1 e2))))))))) +\def + \lambda (g: G).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (e1: +C).(\forall (h: nat).((drop h O c e1) \to (\forall (c2: C).((csubc g c c2) +\to (ex2 C (\lambda (e2: C).(drop h O c2 e2)) (\lambda (e2: C).(csubc g e1 +e2))))))))) (\lambda (n: nat).(\lambda (e1: C).(\lambda (h: nat).(\lambda (H: +(drop h O (CSort n) e1)).(\lambda (c2: C).(\lambda (H0: (csubc g (CSort n) +c2)).(and3_ind (eq C e1 (CSort n)) (eq nat h O) (eq nat O O) (ex2 C (\lambda +(e2: C).(drop h O c2 e2)) (\lambda (e2: C).(csubc g e1 e2))) (\lambda (H1: +(eq C e1 (CSort n))).(\lambda (H2: (eq nat h O)).(\lambda (_: (eq nat O +O)).(eq_ind_r nat O (\lambda (n0: nat).(ex2 C (\lambda (e2: C).(drop n0 O c2 +e2)) (\lambda (e2: C).(csubc g e1 e2)))) (eq_ind_r C (CSort n) (\lambda (c: +C).(ex2 C (\lambda (e2: C).(drop O O c2 e2)) (\lambda (e2: C).(csubc g c +e2)))) (ex_intro2 C (\lambda (e2: C).(drop O O c2 e2)) (\lambda (e2: +C).(csubc g (CSort n) e2)) c2 (drop_refl c2) H0) e1 H1) h H2)))) +(drop_gen_sort n h O e1 H)))))))) (\lambda (c: C).(\lambda (H: ((\forall (e1: +C).(\forall (h: nat).((drop h O c e1) \to (\forall (c2: C).((csubc g c c2) +\to (ex2 C (\lambda (e2: C).(drop h O c2 e2)) (\lambda (e2: C).(csubc g e1 +e2)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e1: C).(\lambda (h: +nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c k t) e1) \to (\forall +(c2: C).((csubc g (CHead c k t) c2) \to (ex2 C (\lambda (e2: C).(drop n O c2 +e2)) (\lambda (e2: C).(csubc g e1 e2))))))) (\lambda (H0: (drop O O (CHead c +k t) e1)).(\lambda (c2: C).(\lambda (H1: (csubc g (CHead c k t) c2)).(eq_ind +C (CHead c k t) (\lambda (c0: C).(ex2 C (\lambda (e2: C).(drop O O c2 e2)) +(\lambda (e2: C).(csubc g c0 e2)))) (ex_intro2 C (\lambda (e2: C).(drop O O +c2 e2)) (\lambda (e2: C).(csubc g (CHead c k t) e2)) c2 (drop_refl c2) H1) e1 +(drop_gen_refl (CHead c k t) e1 H0))))) (\lambda (n: nat).(\lambda (H0: +(((drop n O (CHead c k t) e1) \to (\forall (c2: C).((csubc g (CHead c k t) +c2) \to (ex2 C (\lambda (e2: C).(drop n O c2 e2)) (\lambda (e2: C).(csubc g +e1 e2)))))))).(\lambda (H1: (drop (S n) O (CHead c k t) e1)).(\lambda (c2: +C).(\lambda (H2: (csubc g (CHead c k t) c2)).(let H_x \def (csubc_gen_head_l +g c c2 t k H2) in (let H3 \def H_x in (or3_ind (ex2 C (\lambda (c3: C).(eq C +c2 (CHead c3 k t))) (\lambda (c3: C).(csubc g c c3))) (ex5_3 C T A (\lambda +(_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3: +C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) +(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c c3)))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c t)))) (\lambda +(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T +(\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b) +v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind +Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b +Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c c3))))) +(ex2 C (\lambda (e2: C).(drop (S n) O c2 e2)) (\lambda (e2: C).(csubc g e1 +e2))) (\lambda (H4: (ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 k t))) +(\lambda (c3: C).(csubc g c c3)))).(ex2_ind C (\lambda (c3: C).(eq C c2 +(CHead c3 k t))) (\lambda (c3: C).(csubc g c c3)) (ex2 C (\lambda (e2: +C).(drop (S n) O c2 e2)) (\lambda (e2: C).(csubc g e1 e2))) (\lambda (x: +C).(\lambda (H5: (eq C c2 (CHead x k t))).(\lambda (H6: (csubc g c +x)).(eq_ind_r C (CHead x k t) (\lambda (c0: C).(ex2 C (\lambda (e2: C).(drop +(S n) O c0 e2)) (\lambda (e2: C).(csubc g e1 e2)))) (let H_x0 \def (H e1 (r k +n) (drop_gen_drop k c e1 t n H1) x H6) in (let H7 \def H_x0 in (ex2_ind C +(\lambda (e2: C).(drop (r k n) O x e2)) (\lambda (e2: C).(csubc g e1 e2)) +(ex2 C (\lambda (e2: C).(drop (S n) O (CHead x k t) e2)) (\lambda (e2: +C).(csubc g e1 e2))) (\lambda (x0: C).(\lambda (H8: (drop (r k n) O x +x0)).(\lambda (H9: (csubc g e1 x0)).(ex_intro2 C (\lambda (e2: C).(drop (S n) +O (CHead x k t) e2)) (\lambda (e2: C).(csubc g e1 e2)) x0 (drop_drop k n x x0 +H8 t) H9)))) H7))) c2 H5)))) H4)) (\lambda (H4: (ex5_3 C T A (\lambda (_: +C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3: +C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) +(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c c3)))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c t)))) (\lambda +(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w)))))).(ex5_3_ind C T A +(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) +(\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind +Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c +c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c +t)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w)))) (ex2 +C (\lambda (e2: C).(drop (S n) O c2 e2)) (\lambda (e2: C).(csubc g e1 e2))) +(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H5: (eq K k +(Bind Abst))).(\lambda (H6: (eq C c2 (CHead x0 (Bind Abbr) x1))).(\lambda +(H7: (csubc g c x0)).(\lambda (_: (sc3 g (asucc g x2) c t)).(\lambda (_: (sc3 +g x2 x0 x1)).(eq_ind_r C (CHead x0 (Bind Abbr) x1) (\lambda (c0: C).(ex2 C +(\lambda (e2: C).(drop (S n) O c0 e2)) (\lambda (e2: C).(csubc g e1 e2)))) +(let H10 \def (eq_ind K k (\lambda (k0: K).(drop (r k0 n) O c e1)) +(drop_gen_drop k c e1 t n H1) (Bind Abst) H5) in (let H11 \def (eq_ind K k +(\lambda (k0: K).((drop n O (CHead c k0 t) e1) \to (\forall (c3: C).((csubc g +(CHead c k0 t) c3) \to (ex2 C (\lambda (e2: C).(drop n O c3 e2)) (\lambda +(e2: C).(csubc g e1 e2))))))) H0 (Bind Abst) H5) in (let H_x0 \def (H e1 (r +(Bind Abst) n) H10 x0 H7) in (let H12 \def H_x0 in (ex2_ind C (\lambda (e2: +C).(drop n O x0 e2)) (\lambda (e2: C).(csubc g e1 e2)) (ex2 C (\lambda (e2: +C).(drop (S n) O (CHead x0 (Bind Abbr) x1) e2)) (\lambda (e2: C).(csubc g e1 +e2))) (\lambda (x: C).(\lambda (H13: (drop n O x0 x)).(\lambda (H14: (csubc g +e1 x)).(ex_intro2 C (\lambda (e2: C).(drop (S n) O (CHead x0 (Bind Abbr) x1) +e2)) (\lambda (e2: C).(csubc g e1 e2)) x (drop_drop (Bind Abbr) n x0 x H13 +x1) H14)))) H12))))) c2 H6))))))))) H4)) (\lambda (H4: (ex4_3 B C T (\lambda +(b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b) v2))))) +(\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind Void))))) +(\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) +(\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c +c3)))))).(ex4_3_ind B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: +T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: +C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: +C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: +C).(\lambda (_: T).(csubc g c c3)))) (ex2 C (\lambda (e2: C).(drop (S n) O c2 +e2)) (\lambda (e2: C).(csubc g e1 e2))) (\lambda (x0: B).(\lambda (x1: +C).(\lambda (x2: T).(\lambda (H5: (eq C c2 (CHead x1 (Bind x0) x2))).(\lambda +(H6: (eq K k (Bind Void))).(\lambda (_: (not (eq B x0 Void))).(\lambda (H8: +(csubc g c x1)).(eq_ind_r C (CHead x1 (Bind x0) x2) (\lambda (c0: C).(ex2 C +(\lambda (e2: C).(drop (S n) O c0 e2)) (\lambda (e2: C).(csubc g e1 e2)))) +(let H9 \def (eq_ind K k (\lambda (k0: K).(drop (r k0 n) O c e1)) +(drop_gen_drop k c e1 t n H1) (Bind Void) H6) in (let H10 \def (eq_ind K k +(\lambda (k0: K).((drop n O (CHead c k0 t) e1) \to (\forall (c3: C).((csubc g +(CHead c k0 t) c3) \to (ex2 C (\lambda (e2: C).(drop n O c3 e2)) (\lambda +(e2: C).(csubc g e1 e2))))))) H0 (Bind Void) H6) in (let H_x0 \def (H e1 (r +(Bind Void) n) H9 x1 H8) in (let H11 \def H_x0 in (ex2_ind C (\lambda (e2: +C).(drop n O x1 e2)) (\lambda (e2: C).(csubc g e1 e2)) (ex2 C (\lambda (e2: +C).(drop (S n) O (CHead x1 (Bind x0) x2) e2)) (\lambda (e2: C).(csubc g e1 +e2))) (\lambda (x: C).(\lambda (H12: (drop n O x1 x)).(\lambda (H13: (csubc g +e1 x)).(ex_intro2 C (\lambda (e2: C).(drop (S n) O (CHead x1 (Bind x0) x2) +e2)) (\lambda (e2: C).(csubc g e1 e2)) x (drop_drop (Bind x0) n x1 x H12 x2) +H13)))) H11))))) c2 H5)))))))) H4)) H3)))))))) h))))))) c1)). +(* COMMENTS +Initial nodes: 2389 +END *) + +theorem drop_csubc_trans: + \forall (g: G).(\forall (c2: C).(\forall (e2: C).(\forall (d: nat).(\forall +(h: nat).((drop h d c2 e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C +(\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c2 c1)))))))))) +\def + \lambda (g: G).(\lambda (c2: C).(C_ind (\lambda (c: C).(\forall (e2: +C).(\forall (d: nat).(\forall (h: nat).((drop h d c e2) \to (\forall (e1: +C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda +(c1: C).(csubc g c c1)))))))))) (\lambda (n: nat).(\lambda (e2: C).(\lambda +(d: nat).(\lambda (h: nat).(\lambda (H: (drop h d (CSort n) e2)).(\lambda +(e1: C).(\lambda (H0: (csubc g e2 e1)).(and3_ind (eq C e2 (CSort n)) (eq nat +h O) (eq nat d O) (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: +C).(csubc g (CSort n) c1))) (\lambda (H1: (eq C e2 (CSort n))).(\lambda (H2: +(eq nat h O)).(\lambda (H3: (eq nat d O)).(eq_ind_r nat O (\lambda (n0: +nat).(ex2 C (\lambda (c1: C).(drop n0 d c1 e1)) (\lambda (c1: C).(csubc g +(CSort n) c1)))) (eq_ind_r nat O (\lambda (n0: nat).(ex2 C (\lambda (c1: +C).(drop O n0 c1 e1)) (\lambda (c1: C).(csubc g (CSort n) c1)))) (let H4 \def +(eq_ind C e2 (\lambda (c: C).(csubc g c e1)) H0 (CSort n) H1) in (ex_intro2 C +(\lambda (c1: C).(drop O O c1 e1)) (\lambda (c1: C).(csubc g (CSort n) c1)) +e1 (drop_refl e1) H4)) d H3) h H2)))) (drop_gen_sort n h d e2 H))))))))) +(\lambda (c: C).(\lambda (H: ((\forall (e2: C).(\forall (d: nat).(\forall (h: +nat).((drop h d c e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C +(\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c +c1))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e2: C).(\lambda (d: +nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n (CHead c k t) +e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop h +n c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1)))))))) (\lambda (h: +nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c k t) e2) \to (\forall +(e1: C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop n O c1 e1)) +(\lambda (c1: C).(csubc g (CHead c k t) c1))))))) (\lambda (H0: (drop O O +(CHead c k t) e2)).(\lambda (e1: C).(\lambda (H1: (csubc g e2 e1)).(let H2 +\def (eq_ind_r C e2 (\lambda (c0: C).(csubc g c0 e1)) H1 (CHead c k t) +(drop_gen_refl (CHead c k t) e2 H0)) in (ex_intro2 C (\lambda (c1: C).(drop O +O c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1)) e1 (drop_refl e1) +H2))))) (\lambda (n: nat).(\lambda (_: (((drop n O (CHead c k t) e2) \to +(\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop n O c1 +e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1)))))))).(\lambda (H1: (drop +(S n) O (CHead c k t) e2)).(\lambda (e1: C).(\lambda (H2: (csubc g e2 +e1)).(let H_x \def (H e2 O (r k n) (drop_gen_drop k c e2 t n H1) e1 H2) in +(let H3 \def H_x in (ex2_ind C (\lambda (c1: C).(drop (r k n) O c1 e1)) +(\lambda (c1: C).(csubc g c c1)) (ex2 C (\lambda (c1: C).(drop (S n) O c1 +e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1))) (\lambda (x: C).(\lambda +(H4: (drop (r k n) O x e1)).(\lambda (H5: (csubc g c x)).(ex_intro2 C +(\lambda (c1: C).(drop (S n) O c1 e1)) (\lambda (c1: C).(csubc g (CHead c k +t) c1)) (CHead x k t) (drop_drop k n x e1 H4 t) (csubc_head g c x H5 k t))))) +H3)))))))) h)) (\lambda (n: nat).(\lambda (H0: ((\forall (h: nat).((drop h n +(CHead c k t) e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda +(c1: C).(drop h n c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t) +c1))))))))).(\lambda (h: nat).(\lambda (H1: (drop h (S n) (CHead c k t) +e2)).(\lambda (e1: C).(\lambda (H2: (csubc g e2 e1)).(ex3_2_ind C T (\lambda +(e: C).(\lambda (v: T).(eq C e2 (CHead e k v)))) (\lambda (_: C).(\lambda (v: +T).(eq T t (lift h (r k n) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r k +n) c e))) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1: +C).(csubc g (CHead c k t) c1))) (\lambda (x0: C).(\lambda (x1: T).(\lambda +(H3: (eq C e2 (CHead x0 k x1))).(\lambda (H4: (eq T t (lift h (r k n) +x1))).(\lambda (H5: (drop h (r k n) c x0)).(let H6 \def (eq_ind C e2 (\lambda +(c0: C).(csubc g c0 e1)) H2 (CHead x0 k x1) H3) in (let H7 \def (eq_ind C e2 +(\lambda (c0: C).(\forall (h0: nat).((drop h0 n (CHead c k t) c0) \to +(\forall (e3: C).((csubc g c0 e3) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 +e3)) (\lambda (c1: C).(csubc g (CHead c k t) c1)))))))) H0 (CHead x0 k x1) +H3) in (let H8 \def (eq_ind T t (\lambda (t0: T).(\forall (h0: nat).((drop h0 +n (CHead c k t0) (CHead x0 k x1)) \to (\forall (e3: C).((csubc g (CHead x0 k +x1) e3) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 e3)) (\lambda (c1: +C).(csubc g (CHead c k t0) c1)))))))) H7 (lift h (r k n) x1) H4) in (eq_ind_r +T (lift h (r k n) x1) (\lambda (t0: T).(ex2 C (\lambda (c1: C).(drop h (S n) +c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t0) c1)))) (let H_x \def +(csubc_gen_head_l g x0 e1 x1 k H6) in (let H9 \def H_x in (or3_ind (ex2 C +(\lambda (c3: C).(eq C e1 (CHead c3 k x1))) (\lambda (c3: C).(csubc g x0 +c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k +(Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C e1 +(CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: +A).(csubc g x0 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g +(asucc g a) x0 x1)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g +a c3 w))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: +T).(eq C e1 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: +C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: +C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: +C).(\lambda (_: T).(csubc g x0 c3))))) (ex2 C (\lambda (c1: C).(drop h (S n) +c1 e1)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1))) +(\lambda (H10: (ex2 C (\lambda (c3: C).(eq C e1 (CHead c3 k x1))) (\lambda +(c3: C).(csubc g x0 c3)))).(ex2_ind C (\lambda (c3: C).(eq C e1 (CHead c3 k +x1))) (\lambda (c3: C).(csubc g x0 c3)) (ex2 C (\lambda (c1: C).(drop h (S n) +c1 e1)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1))) +(\lambda (x: C).(\lambda (H11: (eq C e1 (CHead x k x1))).(\lambda (H12: +(csubc g x0 x)).(eq_ind_r C (CHead x k x1) (\lambda (c0: C).(ex2 C (\lambda +(c1: C).(drop h (S n) c1 c0)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r +k n) x1)) c1)))) (let H_x0 \def (H x0 (r k n) h H5 x H12) in (let H13 \def +H_x0 in (ex2_ind C (\lambda (c1: C).(drop h (r k n) c1 x)) (\lambda (c1: +C).(csubc g c c1)) (ex2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x k x1))) +(\lambda (c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1))) (\lambda (x2: +C).(\lambda (H14: (drop h (r k n) x2 x)).(\lambda (H15: (csubc g c +x2)).(ex_intro2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x k x1))) (\lambda +(c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1)) (CHead x2 k (lift h (r +k n) x1)) (drop_skip k h n x2 x H14 x1) (csubc_head g c x2 H15 k (lift h (r k +n) x1)))))) H13))) e1 H11)))) H10)) (\lambda (H10: (ex5_3 C T A (\lambda (_: +C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3: +C).(\lambda (w: T).(\lambda (_: A).(eq C e1 (CHead c3 (Bind Abbr) w))))) +(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g x0 c3)))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) x0 x1)))) (\lambda +(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w)))))).(ex5_3_ind C T A +(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) +(\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C e1 (CHead c3 (Bind +Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g x0 +c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) x0 +x1)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w)))) +(ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1: C).(csubc g +(CHead c k (lift h (r k n) x1)) c1))) (\lambda (x2: C).(\lambda (x3: +T).(\lambda (x4: A).(\lambda (H11: (eq K k (Bind Abst))).(\lambda (H12: (eq C +e1 (CHead x2 (Bind Abbr) x3))).(\lambda (H13: (csubc g x0 x2)).(\lambda (H14: +(sc3 g (asucc g x4) x0 x1)).(\lambda (H15: (sc3 g x4 x2 x3)).(eq_ind_r C +(CHead x2 (Bind Abbr) x3) (\lambda (c0: C).(ex2 C (\lambda (c1: C).(drop h (S +n) c1 c0)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1)))) +(let H16 \def (eq_ind K k (\lambda (k0: K).(\forall (h0: nat).((drop h0 n +(CHead c k0 (lift h (r k0 n) x1)) (CHead x0 k0 x1)) \to (\forall (e3: +C).((csubc g (CHead x0 k0 x1) e3) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 +e3)) (\lambda (c1: C).(csubc g (CHead c k0 (lift h (r k0 n) x1)) c1)))))))) +H8 (Bind Abst) H11) in (let H17 \def (eq_ind K k (\lambda (k0: K).(drop h (r +k0 n) c x0)) H5 (Bind Abst) H11) in (eq_ind_r K (Bind Abst) (\lambda (k0: +K).(ex2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x2 (Bind Abbr) x3))) +(\lambda (c1: C).(csubc g (CHead c k0 (lift h (r k0 n) x1)) c1)))) (let H_x0 +\def (H x0 (r (Bind Abst) n) h H17 x2 H13) in (let H18 \def H_x0 in (ex2_ind +C (\lambda (c1: C).(drop h n c1 x2)) (\lambda (c1: C).(csubc g c c1)) (ex2 C +(\lambda (c1: C).(drop h (S n) c1 (CHead x2 (Bind Abbr) x3))) (\lambda (c1: +C).(csubc g (CHead c (Bind Abst) (lift h (r (Bind Abst) n) x1)) c1))) +(\lambda (x: C).(\lambda (H19: (drop h n x x2)).(\lambda (H20: (csubc g c +x)).(ex_intro2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x2 (Bind Abbr) +x3))) (\lambda (c1: C).(csubc g (CHead c (Bind Abst) (lift h (r (Bind Abst) +n) x1)) c1)) (CHead x (Bind Abbr) (lift h n x3)) (drop_skip_bind h n x x2 H19 +Abbr x3) (csubc_abst g c x H20 (lift h (r (Bind Abst) n) x1) x4 (sc3_lift g +(asucc g x4) x0 x1 H14 c h (r (Bind Abst) n) H17) (lift h n x3) (sc3_lift g +x4 x2 x3 H15 x h n H19)))))) H18))) k H11))) e1 H12))))))))) H10)) (\lambda +(H10: (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C e1 +(CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: +T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: +T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: +T).(csubc g x0 c3)))))).(ex4_3_ind B C T (\lambda (b: B).(\lambda (c3: +C).(\lambda (v2: T).(eq C e1 (CHead c3 (Bind b) v2))))) (\lambda (_: +B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: +B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: +B).(\lambda (c3: C).(\lambda (_: T).(csubc g x0 c3)))) (ex2 C (\lambda (c1: +C).(drop h (S n) c1 e1)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r k n) +x1)) c1))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: T).(\lambda (H11: +(eq C e1 (CHead x3 (Bind x2) x4))).(\lambda (H12: (eq K k (Bind +Void))).(\lambda (H13: (not (eq B x2 Void))).(\lambda (H14: (csubc g x0 +x3)).(eq_ind_r C (CHead x3 (Bind x2) x4) (\lambda (c0: C).(ex2 C (\lambda +(c1: C).(drop h (S n) c1 c0)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r +k n) x1)) c1)))) (let H15 \def (eq_ind K k (\lambda (k0: K).(\forall (h0: +nat).((drop h0 n (CHead c k0 (lift h (r k0 n) x1)) (CHead x0 k0 x1)) \to +(\forall (e3: C).((csubc g (CHead x0 k0 x1) e3) \to (ex2 C (\lambda (c1: +C).(drop h0 n c1 e3)) (\lambda (c1: C).(csubc g (CHead c k0 (lift h (r k0 n) +x1)) c1)))))))) H8 (Bind Void) H12) in (let H16 \def (eq_ind K k (\lambda +(k0: K).(drop h (r k0 n) c x0)) H5 (Bind Void) H12) in (eq_ind_r K (Bind +Void) (\lambda (k0: K).(ex2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x3 +(Bind x2) x4))) (\lambda (c1: C).(csubc g (CHead c k0 (lift h (r k0 n) x1)) +c1)))) (let H_x0 \def (H x0 (r (Bind Void) n) h H16 x3 H14) in (let H17 \def +H_x0 in (ex2_ind C (\lambda (c1: C).(drop h n c1 x3)) (\lambda (c1: C).(csubc +g c c1)) (ex2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x3 (Bind x2) x4))) +(\lambda (c1: C).(csubc g (CHead c (Bind Void) (lift h (r (Bind Void) n) x1)) +c1))) (\lambda (x: C).(\lambda (H18: (drop h n x x3)).(\lambda (H19: (csubc g +c x)).(ex_intro2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x3 (Bind x2) +x4))) (\lambda (c1: C).(csubc g (CHead c (Bind Void) (lift h (r (Bind Void) +n) x1)) c1)) (CHead x (Bind x2) (lift h n x4)) (drop_skip_bind h n x x3 H18 +x2 x4) (csubc_void g c x H19 x2 H13 (lift h (r (Bind Void) n) x1) (lift h n +x4)))))) H17))) k H12))) e1 H11)))))))) H10)) H9))) t H4))))))))) +(drop_gen_skip_l c e2 t h n k H1)))))))) d))))))) c2)). +(* COMMENTS +Initial nodes: 3747 +END *) + +theorem csubc_drop_conf_rev: + \forall (g: G).(\forall (c2: C).(\forall (e2: C).(\forall (d: nat).(\forall +(h: nat).((drop h d c2 e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C +(\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c1 c2)))))))))) +\def + \lambda (g: G).(\lambda (c2: C).(C_ind (\lambda (c: C).(\forall (e2: +C).(\forall (d: nat).(\forall (h: nat).((drop h d c e2) \to (\forall (e1: +C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda +(c1: C).(csubc g c1 c)))))))))) (\lambda (n: nat).(\lambda (e2: C).(\lambda +(d: nat).(\lambda (h: nat).(\lambda (H: (drop h d (CSort n) e2)).(\lambda +(e1: C).(\lambda (H0: (csubc g e1 e2)).(and3_ind (eq C e2 (CSort n)) (eq nat +h O) (eq nat d O) (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: +C).(csubc g c1 (CSort n)))) (\lambda (H1: (eq C e2 (CSort n))).(\lambda (H2: +(eq nat h O)).(\lambda (H3: (eq nat d O)).(eq_ind_r nat O (\lambda (n0: +nat).(ex2 C (\lambda (c1: C).(drop n0 d c1 e1)) (\lambda (c1: C).(csubc g c1 +(CSort n))))) (eq_ind_r nat O (\lambda (n0: nat).(ex2 C (\lambda (c1: +C).(drop O n0 c1 e1)) (\lambda (c1: C).(csubc g c1 (CSort n))))) (let H4 \def +(eq_ind C e2 (\lambda (c: C).(csubc g e1 c)) H0 (CSort n) H1) in (ex_intro2 C +(\lambda (c1: C).(drop O O c1 e1)) (\lambda (c1: C).(csubc g c1 (CSort n))) +e1 (drop_refl e1) H4)) d H3) h H2)))) (drop_gen_sort n h d e2 H))))))))) +(\lambda (c: C).(\lambda (H: ((\forall (e2: C).(\forall (d: nat).(\forall (h: +nat).((drop h d c e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C +(\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c1 +c))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e2: C).(\lambda (d: +nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n (CHead c k t) +e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop h +n c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t))))))))) (\lambda (h: +nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c k t) e2) \to (\forall +(e1: C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop n O c1 e1)) +(\lambda (c1: C).(csubc g c1 (CHead c k t)))))))) (\lambda (H0: (drop O O +(CHead c k t) e2)).(\lambda (e1: C).(\lambda (H1: (csubc g e1 e2)).(let H2 +\def (eq_ind_r C e2 (\lambda (c0: C).(csubc g e1 c0)) H1 (CHead c k t) +(drop_gen_refl (CHead c k t) e2 H0)) in (ex_intro2 C (\lambda (c1: C).(drop O +O c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t))) e1 (drop_refl e1) +H2))))) (\lambda (n: nat).(\lambda (_: (((drop n O (CHead c k t) e2) \to +(\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop n O c1 +e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t))))))))).(\lambda (H1: (drop +(S n) O (CHead c k t) e2)).(\lambda (e1: C).(\lambda (H2: (csubc g e1 +e2)).(let H_x \def (H e2 O (r k n) (drop_gen_drop k c e2 t n H1) e1 H2) in +(let H3 \def H_x in (ex2_ind C (\lambda (c1: C).(drop (r k n) O c1 e1)) +(\lambda (c1: C).(csubc g c1 c)) (ex2 C (\lambda (c1: C).(drop (S n) O c1 +e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t)))) (\lambda (x: C).(\lambda +(H4: (drop (r k n) O x e1)).(\lambda (H5: (csubc g x c)).(ex_intro2 C +(\lambda (c1: C).(drop (S n) O c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c +k t))) (CHead x k t) (drop_drop k n x e1 H4 t) (csubc_head g x c H5 k t))))) +H3)))))))) h)) (\lambda (n: nat).(\lambda (H0: ((\forall (h: nat).((drop h n +(CHead c k t) e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda +(c1: C).(drop h n c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k +t)))))))))).(\lambda (h: nat).(\lambda (H1: (drop h (S n) (CHead c k t) +e2)).(\lambda (e1: C).(\lambda (H2: (csubc g e1 e2)).(ex3_2_ind C T (\lambda +(e: C).(\lambda (v: T).(eq C e2 (CHead e k v)))) (\lambda (_: C).(\lambda (v: +T).(eq T t (lift h (r k n) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r k +n) c e))) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1: +C).(csubc g c1 (CHead c k t)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda +(H3: (eq C e2 (CHead x0 k x1))).(\lambda (H4: (eq T t (lift h (r k n) +x1))).(\lambda (H5: (drop h (r k n) c x0)).(let H6 \def (eq_ind C e2 (\lambda +(c0: C).(csubc g e1 c0)) H2 (CHead x0 k x1) H3) in (let H7 \def (eq_ind C e2 +(\lambda (c0: C).(\forall (h0: nat).((drop h0 n (CHead c k t) c0) \to +(\forall (e3: C).((csubc g e3 c0) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 +e3)) (\lambda (c1: C).(csubc g c1 (CHead c k t))))))))) H0 (CHead x0 k x1) +H3) in (let H8 \def (eq_ind T t (\lambda (t0: T).(\forall (h0: nat).((drop h0 +n (CHead c k t0) (CHead x0 k x1)) \to (\forall (e3: C).((csubc g e3 (CHead x0 +k x1)) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 e3)) (\lambda (c1: C).(csubc +g c1 (CHead c k t0))))))))) H7 (lift h (r k n) x1) H4) in (eq_ind_r T (lift h +(r k n) x1) (\lambda (t0: T).(ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) +(\lambda (c1: C).(csubc g c1 (CHead c k t0))))) (let H_x \def +(csubc_gen_head_r g x0 e1 x1 k H6) in (let H9 \def H_x in (or3_ind (ex2 C +(\lambda (c1: C).(eq C e1 (CHead c1 k x1))) (\lambda (c1: C).(csubc g c1 +x0))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k +(Bind Abbr))))) (\lambda (c1: C).(\lambda (v: T).(\lambda (_: A).(eq C e1 +(CHead c1 (Bind Abst) v))))) (\lambda (c1: C).(\lambda (_: T).(\lambda (_: +A).(csubc g c1 x0)))) (\lambda (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g +(asucc g a) c1 v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a +x0 x1))))) (ex4_3 B C T (\lambda (_: B).(\lambda (c1: C).(\lambda (v1: T).(eq +C e1 (CHead c1 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda +(_: T).(eq K k (Bind b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: +T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c1: C).(\lambda (_: +T).(csubc g c1 x0))))) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda +(c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1))))) (\lambda (H10: (ex2 C +(\lambda (c1: C).(eq C e1 (CHead c1 k x1))) (\lambda (c1: C).(csubc g c1 +x0)))).(ex2_ind C (\lambda (c1: C).(eq C e1 (CHead c1 k x1))) (\lambda (c1: +C).(csubc g c1 x0)) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda +(c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1))))) (\lambda (x: +C).(\lambda (H11: (eq C e1 (CHead x k x1))).(\lambda (H12: (csubc g x +x0)).(eq_ind_r C (CHead x k x1) (\lambda (c0: C).(ex2 C (\lambda (c1: +C).(drop h (S n) c1 c0)) (\lambda (c1: C).(csubc g c1 (CHead c k (lift h (r k +n) x1)))))) (let H_x0 \def (H x0 (r k n) h H5 x H12) in (let H13 \def H_x0 in +(ex2_ind C (\lambda (c1: C).(drop h (r k n) c1 x)) (\lambda (c1: C).(csubc g +c1 c)) (ex2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x k x1))) (\lambda +(c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1))))) (\lambda (x2: +C).(\lambda (H14: (drop h (r k n) x2 x)).(\lambda (H15: (csubc g x2 +c)).(ex_intro2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x k x1))) (\lambda +(c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1)))) (CHead x2 k (lift h (r +k n) x1)) (drop_skip k h n x2 x H14 x1) (csubc_head g x2 c H15 k (lift h (r k +n) x1)))))) H13))) e1 H11)))) H10)) (\lambda (H10: (ex5_3 C T A (\lambda (_: +C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c1: +C).(\lambda (v: T).(\lambda (_: A).(eq C e1 (CHead c1 (Bind Abst) v))))) +(\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 x0)))) (\lambda +(c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a x0 x1)))))).(ex5_3_ind C T A +(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) +(\lambda (c1: C).(\lambda (v: T).(\lambda (_: A).(eq C e1 (CHead c1 (Bind +Abst) v))))) (\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 +x0)))) (\lambda (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1 +v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a x0 x1)))) (ex2 +C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead +c k (lift h (r k n) x1))))) (\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: +A).(\lambda (H11: (eq K k (Bind Abbr))).(\lambda (H12: (eq C e1 (CHead x2 +(Bind Abst) x3))).(\lambda (H13: (csubc g x2 x0)).(\lambda (H14: (sc3 g +(asucc g x4) x2 x3)).(\lambda (H15: (sc3 g x4 x0 x1)).(eq_ind_r C (CHead x2 +(Bind Abst) x3) (\lambda (c0: C).(ex2 C (\lambda (c1: C).(drop h (S n) c1 +c0)) (\lambda (c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1)))))) (let +H16 \def (eq_ind K k (\lambda (k0: K).(\forall (h0: nat).((drop h0 n (CHead c +k0 (lift h (r k0 n) x1)) (CHead x0 k0 x1)) \to (\forall (e3: C).((csubc g e3 +(CHead x0 k0 x1)) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 e3)) (\lambda +(c1: C).(csubc g c1 (CHead c k0 (lift h (r k0 n) x1)))))))))) H8 (Bind Abbr) +H11) in (let H17 \def (eq_ind K k (\lambda (k0: K).(drop h (r k0 n) c x0)) H5 +(Bind Abbr) H11) in (eq_ind_r K (Bind Abbr) (\lambda (k0: K).(ex2 C (\lambda +(c1: C).(drop h (S n) c1 (CHead x2 (Bind Abst) x3))) (\lambda (c1: C).(csubc +g c1 (CHead c k0 (lift h (r k0 n) x1)))))) (let H_x0 \def (H x0 (r (Bind +Abbr) n) h H17 x2 H13) in (let H18 \def H_x0 in (ex2_ind C (\lambda (c1: +C).(drop h n c1 x2)) (\lambda (c1: C).(csubc g c1 c)) (ex2 C (\lambda (c1: +C).(drop h (S n) c1 (CHead x2 (Bind Abst) x3))) (\lambda (c1: C).(csubc g c1 +(CHead c (Bind Abbr) (lift h (r (Bind Abbr) n) x1))))) (\lambda (x: +C).(\lambda (H19: (drop h n x x2)).(\lambda (H20: (csubc g x c)).(ex_intro2 C +(\lambda (c1: C).(drop h (S n) c1 (CHead x2 (Bind Abst) x3))) (\lambda (c1: +C).(csubc g c1 (CHead c (Bind Abbr) (lift h (r (Bind Abbr) n) x1)))) (CHead x +(Bind Abst) (lift h n x3)) (drop_skip_bind h n x x2 H19 Abst x3) (csubc_abst +g x c H20 (lift h n x3) x4 (sc3_lift g (asucc g x4) x2 x3 H14 x h n H19) +(lift h (r (Bind Abbr) n) x1) (sc3_lift g x4 x0 x1 H15 c h (r (Bind Abbr) n) +H17)))))) H18))) k H11))) e1 H12))))))))) H10)) (\lambda (H10: (ex4_3 B C T +(\lambda (_: B).(\lambda (c1: C).(\lambda (v1: T).(eq C e1 (CHead c1 (Bind +Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind +b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) +(\lambda (_: B).(\lambda (c1: C).(\lambda (_: T).(csubc g c1 +x0)))))).(ex4_3_ind B C T (\lambda (_: B).(\lambda (c1: C).(\lambda (v1: +T).(eq C e1 (CHead c1 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: +C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: B).(\lambda (_: +C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c1: +C).(\lambda (_: T).(csubc g c1 x0)))) (ex2 C (\lambda (c1: C).(drop h (S n) +c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1))))) +(\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: T).(\lambda (H11: (eq C e1 +(CHead x3 (Bind Void) x4))).(\lambda (H12: (eq K k (Bind x2))).(\lambda (H13: +(not (eq B x2 Void))).(\lambda (H14: (csubc g x3 x0)).(eq_ind_r C (CHead x3 +(Bind Void) x4) (\lambda (c0: C).(ex2 C (\lambda (c1: C).(drop h (S n) c1 +c0)) (\lambda (c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1)))))) (let +H15 \def (eq_ind K k (\lambda (k0: K).(\forall (h0: nat).((drop h0 n (CHead c +k0 (lift h (r k0 n) x1)) (CHead x0 k0 x1)) \to (\forall (e3: C).((csubc g e3 +(CHead x0 k0 x1)) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 e3)) (\lambda +(c1: C).(csubc g c1 (CHead c k0 (lift h (r k0 n) x1)))))))))) H8 (Bind x2) +H12) in (let H16 \def (eq_ind K k (\lambda (k0: K).(drop h (r k0 n) c x0)) H5 +(Bind x2) H12) in (eq_ind_r K (Bind x2) (\lambda (k0: K).(ex2 C (\lambda (c1: +C).(drop h (S n) c1 (CHead x3 (Bind Void) x4))) (\lambda (c1: C).(csubc g c1 +(CHead c k0 (lift h (r k0 n) x1)))))) (let H_x0 \def (H x0 (r (Bind x2) n) h +H16 x3 H14) in (let H17 \def H_x0 in (ex2_ind C (\lambda (c1: C).(drop h n c1 +x3)) (\lambda (c1: C).(csubc g c1 c)) (ex2 C (\lambda (c1: C).(drop h (S n) +c1 (CHead x3 (Bind Void) x4))) (\lambda (c1: C).(csubc g c1 (CHead c (Bind +x2) (lift h (r (Bind x2) n) x1))))) (\lambda (x: C).(\lambda (H18: (drop h n +x x3)).(\lambda (H19: (csubc g x c)).(ex_intro2 C (\lambda (c1: C).(drop h (S +n) c1 (CHead x3 (Bind Void) x4))) (\lambda (c1: C).(csubc g c1 (CHead c (Bind +x2) (lift h (r (Bind x2) n) x1)))) (CHead x (Bind Void) (lift h n x4)) +(drop_skip_bind h n x x3 H18 Void x4) (csubc_void g x c H19 x2 H13 (lift h n +x4) (lift h (r (Bind x2) n) x1)))))) H17))) k H12))) e1 H11)))))))) H10)) +H9))) t H4))))))))) (drop_gen_skip_l c e2 t h n k H1)))))))) d))))))) c2)). +(* COMMENTS +Initial nodes: 3747 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubc/drop1.ma b/matita/matita/contribs/lambdadelta/basic_1/csubc/drop1.ma new file mode 100644 index 000000000..7c539cb4a --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/csubc/drop1.ma @@ -0,0 +1,92 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/csubc/drop.ma". + +theorem drop1_csubc_trans: + \forall (g: G).(\forall (hds: PList).(\forall (c2: C).(\forall (e2: +C).((drop1 hds c2 e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C +(\lambda (c1: C).(drop1 hds c1 e1)) (\lambda (c1: C).(csubc g c2 c1))))))))) +\def + \lambda (g: G).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall +(c2: C).(\forall (e2: C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e2 +e1) \to (ex2 C (\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c2 +c1))))))))) (\lambda (c2: C).(\lambda (e2: C).(\lambda (H: (drop1 PNil c2 +e2)).(\lambda (e1: C).(\lambda (H0: (csubc g e2 e1)).(let H_y \def +(drop1_gen_pnil c2 e2 H) in (let H1 \def (eq_ind_r C e2 (\lambda (c: +C).(csubc g c e1)) H0 c2 H_y) in (ex_intro2 C (\lambda (c1: C).(drop1 PNil c1 +e1)) (\lambda (c1: C).(csubc g c2 c1)) e1 (drop1_nil e1) H1)))))))) (\lambda +(n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c2: +C).(\forall (e2: C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e2 e1) +\to (ex2 C (\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c2 +c1)))))))))).(\lambda (c2: C).(\lambda (e2: C).(\lambda (H0: (drop1 (PCons n +n0 p) c2 e2)).(\lambda (e1: C).(\lambda (H1: (csubc g e2 e1)).(let H_x \def +(drop1_gen_pcons c2 e2 p n n0 H0) in (let H2 \def H_x in (ex2_ind C (\lambda +(c3: C).(drop n n0 c2 c3)) (\lambda (c3: C).(drop1 p c3 e2)) (ex2 C (\lambda +(c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c2 c1))) +(\lambda (x: C).(\lambda (H3: (drop n n0 c2 x)).(\lambda (H4: (drop1 p x +e2)).(let H_x0 \def (H x e2 H4 e1 H1) in (let H5 \def H_x0 in (ex2_ind C +(\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g x c1)) (ex2 C +(\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c2 +c1))) (\lambda (x0: C).(\lambda (H6: (drop1 p x0 e1)).(\lambda (H7: (csubc g +x x0)).(let H_x1 \def (drop_csubc_trans g c2 x n0 n H3 x0 H7) in (let H8 \def +H_x1 in (ex2_ind C (\lambda (c1: C).(drop n n0 c1 x0)) (\lambda (c1: +C).(csubc g c2 c1)) (ex2 C (\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) +(\lambda (c1: C).(csubc g c2 c1))) (\lambda (x1: C).(\lambda (H9: (drop n n0 +x1 x0)).(\lambda (H10: (csubc g c2 x1)).(ex_intro2 C (\lambda (c1: C).(drop1 +(PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c2 c1)) x1 (drop1_cons x1 x0 +n n0 H9 e1 p H6) H10)))) H8)))))) H5)))))) H2)))))))))))) hds)). +(* COMMENTS +Initial nodes: 551 +END *) + +theorem csubc_drop1_conf_rev: + \forall (g: G).(\forall (hds: PList).(\forall (c2: C).(\forall (e2: +C).((drop1 hds c2 e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C +(\lambda (c1: C).(drop1 hds c1 e1)) (\lambda (c1: C).(csubc g c1 c2))))))))) +\def + \lambda (g: G).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall +(c2: C).(\forall (e2: C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e1 +e2) \to (ex2 C (\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c1 +c2))))))))) (\lambda (c2: C).(\lambda (e2: C).(\lambda (H: (drop1 PNil c2 +e2)).(\lambda (e1: C).(\lambda (H0: (csubc g e1 e2)).(let H_y \def +(drop1_gen_pnil c2 e2 H) in (let H1 \def (eq_ind_r C e2 (\lambda (c: +C).(csubc g e1 c)) H0 c2 H_y) in (ex_intro2 C (\lambda (c1: C).(drop1 PNil c1 +e1)) (\lambda (c1: C).(csubc g c1 c2)) e1 (drop1_nil e1) H1)))))))) (\lambda +(n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c2: +C).(\forall (e2: C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e1 e2) +\to (ex2 C (\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c1 +c2)))))))))).(\lambda (c2: C).(\lambda (e2: C).(\lambda (H0: (drop1 (PCons n +n0 p) c2 e2)).(\lambda (e1: C).(\lambda (H1: (csubc g e1 e2)).(let H_x \def +(drop1_gen_pcons c2 e2 p n n0 H0) in (let H2 \def H_x in (ex2_ind C (\lambda +(c3: C).(drop n n0 c2 c3)) (\lambda (c3: C).(drop1 p c3 e2)) (ex2 C (\lambda +(c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c1 c2))) +(\lambda (x: C).(\lambda (H3: (drop n n0 c2 x)).(\lambda (H4: (drop1 p x +e2)).(let H_x0 \def (H x e2 H4 e1 H1) in (let H5 \def H_x0 in (ex2_ind C +(\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c1 x)) (ex2 C +(\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c1 +c2))) (\lambda (x0: C).(\lambda (H6: (drop1 p x0 e1)).(\lambda (H7: (csubc g +x0 x)).(let H_x1 \def (csubc_drop_conf_rev g c2 x n0 n H3 x0 H7) in (let H8 +\def H_x1 in (ex2_ind C (\lambda (c1: C).(drop n n0 c1 x0)) (\lambda (c1: +C).(csubc g c1 c2)) (ex2 C (\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) +(\lambda (c1: C).(csubc g c1 c2))) (\lambda (x1: C).(\lambda (H9: (drop n n0 +x1 x0)).(\lambda (H10: (csubc g x1 c2)).(ex_intro2 C (\lambda (c1: C).(drop1 +(PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c1 c2)) x1 (drop1_cons x1 x0 +n n0 H9 e1 p H6) H10)))) H8)))))) H5)))))) H2)))))))))))) hds)). +(* COMMENTS +Initial nodes: 551 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubc/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/csubc/fwd.ma new file mode 100644 index 000000000..fe04ddd45 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/csubc/fwd.ma @@ -0,0 +1,673 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/csubc/defs.ma". + +theorem csubc_gen_sort_l: + \forall (g: G).(\forall (x: C).(\forall (n: nat).((csubc g (CSort n) x) \to +(eq C x (CSort n))))) +\def + \lambda (g: G).(\lambda (x: C).(\lambda (n: nat).(\lambda (H: (csubc g +(CSort n) x)).(insert_eq C (CSort n) (\lambda (c: C).(csubc g c x)) (\lambda +(c: C).(eq C x c)) (\lambda (y: C).(\lambda (H0: (csubc g y x)).(csubc_ind g +(\lambda (c: C).(\lambda (c0: C).((eq C c (CSort n)) \to (eq C c0 c)))) +(\lambda (n0: nat).(\lambda (H1: (eq C (CSort n0) (CSort n))).(let H2 \def +(f_equal C nat (\lambda (e: C).(match e in C return (\lambda (_: C).nat) with +[(CSort n1) \Rightarrow n1 | (CHead _ _ _) \Rightarrow n0])) (CSort n0) +(CSort n) H1) in (eq_ind_r nat n (\lambda (n1: nat).(eq C (CSort n1) (CSort +n1))) (refl_equal C (CSort n)) n0 H2)))) (\lambda (c1: C).(\lambda (c2: +C).(\lambda (_: (csubc g c1 c2)).(\lambda (_: (((eq C c1 (CSort n)) \to (eq C +c2 c1)))).(\lambda (k: K).(\lambda (v: T).(\lambda (H3: (eq C (CHead c1 k v) +(CSort n))).(let H4 \def (eq_ind C (CHead c1 k v) (\lambda (ee: C).(match ee +in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead +_ _ _) \Rightarrow True])) I (CSort n) H3) in (False_ind (eq C (CHead c2 k v) +(CHead c1 k v)) H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: +(csubc g c1 c2)).(\lambda (_: (((eq C c1 (CSort n)) \to (eq C c2 +c1)))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: +T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u1) (CSort +n))).(let H5 \def (eq_ind C (CHead c1 (Bind Void) u1) (\lambda (ee: C).(match +ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | +(CHead _ _ _) \Rightarrow True])) I (CSort n) H4) in (False_ind (eq C (CHead +c2 (Bind b) u2) (CHead c1 (Bind Void) u1)) H5))))))))))) (\lambda (c1: +C).(\lambda (c2: C).(\lambda (_: (csubc g c1 c2)).(\lambda (_: (((eq C c1 +(CSort n)) \to (eq C c2 c1)))).(\lambda (v: T).(\lambda (a: A).(\lambda (_: +(sc3 g (asucc g a) c1 v)).(\lambda (w: T).(\lambda (_: (sc3 g a c2 +w)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) v) (CSort n))).(let H6 \def +(eq_ind C (CHead c1 (Bind Abst) v) (\lambda (ee: C).(match ee in C return +(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) +\Rightarrow True])) I (CSort n) H5) in (False_ind (eq C (CHead c2 (Bind Abbr) +w) (CHead c1 (Bind Abst) v)) H6)))))))))))) y x H0))) H)))). +(* COMMENTS +Initial nodes: 533 +END *) + +theorem csubc_gen_head_l: + \forall (g: G).(\forall (c1: C).(\forall (x: C).(\forall (v: T).(\forall (k: +K).((csubc g (CHead c1 k v) x) \to (or3 (ex2 C (\lambda (c2: C).(eq C x +(CHead c2 k v))) (\lambda (c2: C).(csubc g c1 c2))) (ex5_3 C T A (\lambda (_: +C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c2: +C).(\lambda (w: T).(\lambda (_: A).(eq C x (CHead c2 (Bind Abbr) w))))) +(\lambda (c2: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c2)))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda +(c2: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T +(\lambda (b: B).(\lambda (c2: C).(\lambda (v2: T).(eq C x (CHead c2 (Bind b) +v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind +Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b +Void))))) (\lambda (_: B).(\lambda (c2: C).(\lambda (_: T).(csubc g c1 +c2))))))))))) +\def + \lambda (g: G).(\lambda (c1: C).(\lambda (x: C).(\lambda (v: T).(\lambda (k: +K).(\lambda (H: (csubc g (CHead c1 k v) x)).(insert_eq C (CHead c1 k v) +(\lambda (c: C).(csubc g c x)) (\lambda (_: C).(or3 (ex2 C (\lambda (c2: +C).(eq C x (CHead c2 k v))) (\lambda (c2: C).(csubc g c1 c2))) (ex5_3 C T A +(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) +(\lambda (c2: C).(\lambda (w: T).(\lambda (_: A).(eq C x (CHead c2 (Bind +Abbr) w))))) (\lambda (c2: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 +c2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 +v)))) (\lambda (c2: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c2 w))))) +(ex4_3 B C T (\lambda (b: B).(\lambda (c2: C).(\lambda (v2: T).(eq C x (CHead +c2 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k +(Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b +Void))))) (\lambda (_: B).(\lambda (c2: C).(\lambda (_: T).(csubc g c1 +c2))))))) (\lambda (y: C).(\lambda (H0: (csubc g y x)).(csubc_ind g (\lambda +(c: C).(\lambda (c0: C).((eq C c (CHead c1 k v)) \to (or3 (ex2 C (\lambda +(c2: C).(eq C c0 (CHead c2 k v))) (\lambda (c2: C).(csubc g c1 c2))) (ex5_3 C +T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) +(\lambda (c2: C).(\lambda (w: T).(\lambda (_: A).(eq C c0 (CHead c2 (Bind +Abbr) w))))) (\lambda (c2: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 +c2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 +v)))) (\lambda (c2: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c2 w))))) +(ex4_3 B C T (\lambda (b: B).(\lambda (c2: C).(\lambda (v2: T).(eq C c0 +(CHead c2 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: +T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: +T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c2: C).(\lambda (_: +T).(csubc g c1 c2))))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) +(CHead c1 k v))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee +in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ +_ _) \Rightarrow False])) I (CHead c1 k v) H1) in (False_ind (or3 (ex2 C +(\lambda (c2: C).(eq C (CSort n) (CHead c2 k v))) (\lambda (c2: C).(csubc g +c1 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k +(Bind Abst))))) (\lambda (c2: C).(\lambda (w: T).(\lambda (_: A).(eq C (CSort +n) (CHead c2 (Bind Abbr) w))))) (\lambda (c2: C).(\lambda (_: T).(\lambda (_: +A).(csubc g c1 c2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g +(asucc g a) c1 v)))) (\lambda (c2: C).(\lambda (w: T).(\lambda (a: A).(sc3 g +a c2 w))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c2: C).(\lambda (v2: +T).(eq C (CSort n) (CHead c2 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: +C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: +C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c2: +C).(\lambda (_: T).(csubc g c1 c2)))))) H2)))) (\lambda (c0: C).(\lambda (c2: +C).(\lambda (H1: (csubc g c0 c2)).(\lambda (H2: (((eq C c0 (CHead c1 k v)) +\to (or3 (ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 k v))) (\lambda (c3: +C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda +(_: A).(eq K k (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: +A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: +T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: C).(\lambda (_: +T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (c3: C).(\lambda (w: +T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda (b: B).(\lambda +(c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_: +B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: +B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: +B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 c3))))))))).(\lambda (k0: +K).(\lambda (v0: T).(\lambda (H3: (eq C (CHead c0 k0 v0) (CHead c1 k +v))).(let H4 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda +(_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) +(CHead c0 k0 v0) (CHead c1 k v) H3) in ((let H5 \def (f_equal C K (\lambda +(e: C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k0 +| (CHead _ k1 _) \Rightarrow k1])) (CHead c0 k0 v0) (CHead c1 k v) H3) in +((let H6 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: +C).T) with [(CSort _) \Rightarrow v0 | (CHead _ _ t) \Rightarrow t])) (CHead +c0 k0 v0) (CHead c1 k v) H3) in (\lambda (H7: (eq K k0 k)).(\lambda (H8: (eq +C c0 c1)).(eq_ind_r T v (\lambda (t: T).(or3 (ex2 C (\lambda (c3: C).(eq C +(CHead c2 k0 t) (CHead c3 k v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C +T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) +(\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C (CHead c2 k0 t) (CHead +c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc +g c1 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g +a) c1 v)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 +w))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C +(CHead c2 k0 t) (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: +C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: +C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: +C).(\lambda (_: T).(csubc g c1 c3))))))) (eq_ind_r K k (\lambda (k1: K).(or3 +(ex2 C (\lambda (c3: C).(eq C (CHead c2 k1 v) (CHead c3 k v))) (\lambda (c3: +C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda +(_: A).(eq K k (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: +A).(eq C (CHead c2 k1 v) (CHead c3 (Bind Abbr) w))))) (\lambda (c3: +C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: +C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (c3: +C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda +(b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C (CHead c2 k1 v) (CHead c3 +(Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k +(Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b +Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 +c3))))))) (let H9 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 k v)) +\to (or3 (ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 k v))) (\lambda (c3: +C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda +(_: A).(eq K k (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: +A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: +T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: C).(\lambda (_: +T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (c3: C).(\lambda (w: +T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda (b: B).(\lambda +(c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_: +B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: +B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: +B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 c3)))))))) H2 c1 H8) in (let +H10 \def (eq_ind C c0 (\lambda (c: C).(csubc g c c2)) H1 c1 H8) in +(or3_intro0 (ex2 C (\lambda (c3: C).(eq C (CHead c2 k v) (CHead c3 k v))) +(\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: +T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3: C).(\lambda (w: +T).(\lambda (_: A).(eq C (CHead c2 k v) (CHead c3 (Bind Abbr) w))))) (\lambda +(c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: +C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (c3: +C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda +(b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C (CHead c2 k v) (CHead c3 (Bind +b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind +Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b +Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 +c3))))) (ex_intro2 C (\lambda (c3: C).(eq C (CHead c2 k v) (CHead c3 k v))) +(\lambda (c3: C).(csubc g c1 c3)) c2 (refl_equal C (CHead c2 k v)) H10)))) k0 +H7) v0 H6)))) H5)) H4))))))))) (\lambda (c0: C).(\lambda (c2: C).(\lambda +(H1: (csubc g c0 c2)).(\lambda (H2: (((eq C c0 (CHead c1 k v)) \to (or3 (ex2 +C (\lambda (c3: C).(eq C c2 (CHead c3 k v))) (\lambda (c3: C).(csubc g c1 +c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k +(Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C c2 +(CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: +A).(csubc g c1 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g +(asucc g a) c1 v)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g +a c3 w))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: +T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: +C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: +C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: +C).(\lambda (_: T).(csubc g c1 c3))))))))).(\lambda (b: B).(\lambda (H3: (not +(eq B b Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead +c0 (Bind Void) u1) (CHead c1 k v))).(let H5 \def (f_equal C C (\lambda (e: +C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | +(CHead c _ _) \Rightarrow c])) (CHead c0 (Bind Void) u1) (CHead c1 k v) H4) +in ((let H6 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda +(_: C).K) with [(CSort _) \Rightarrow (Bind Void) | (CHead _ k0 _) +\Rightarrow k0])) (CHead c0 (Bind Void) u1) (CHead c1 k v) H4) in ((let H7 +\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) +with [(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead c0 +(Bind Void) u1) (CHead c1 k v) H4) in (\lambda (H8: (eq K (Bind Void) +k)).(\lambda (H9: (eq C c0 c1)).(let H10 \def (eq_ind C c0 (\lambda (c: +C).((eq C c (CHead c1 k v)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c2 (CHead +c3 k v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: +C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3: +C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) +(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda +(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T +(\lambda (b0: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind +b0) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind +Void))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b0 +Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 +c3)))))))) H2 c1 H9) in (let H11 \def (eq_ind C c0 (\lambda (c: C).(csubc g c +c2)) H1 c1 H9) in (let H12 \def (eq_ind_r K k (\lambda (k0: K).((eq C c1 +(CHead c1 k0 v)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 k0 v))) +(\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: +T).(\lambda (_: A).(eq K k0 (Bind Abst))))) (\lambda (c3: C).(\lambda (w: +T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: +C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: +C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (c3: +C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda +(b0: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b0) +v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k0 (Bind +Void))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b0 +Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 +c3)))))))) H10 (Bind Void) H8) in (eq_ind K (Bind Void) (\lambda (k0: K).(or3 +(ex2 C (\lambda (c3: C).(eq C (CHead c2 (Bind b) u2) (CHead c3 k0 v))) +(\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: +T).(\lambda (_: A).(eq K k0 (Bind Abst))))) (\lambda (c3: C).(\lambda (w: +T).(\lambda (_: A).(eq C (CHead c2 (Bind b) u2) (CHead c3 (Bind Abbr) w))))) +(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda +(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T +(\lambda (b0: B).(\lambda (c3: C).(\lambda (v2: T).(eq C (CHead c2 (Bind b) +u2) (CHead c3 (Bind b0) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: +T).(eq K k0 (Bind Void))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: +T).(not (eq B b0 Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: +T).(csubc g c1 c3))))))) (or3_intro2 (ex2 C (\lambda (c3: C).(eq C (CHead c2 +(Bind b) u2) (CHead c3 (Bind Void) v))) (\lambda (c3: C).(csubc g c1 c3))) +(ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K (Bind +Void) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C +(CHead c2 (Bind b) u2) (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda +(_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: C).(\lambda (_: +T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (c3: C).(\lambda (w: +T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda (b0: B).(\lambda +(c3: C).(\lambda (v2: T).(eq C (CHead c2 (Bind b) u2) (CHead c3 (Bind b0) +v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K (Bind Void) +(Bind Void))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B +b0 Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 +c3))))) (ex4_3_intro B C T (\lambda (b0: B).(\lambda (c3: C).(\lambda (v2: +T).(eq C (CHead c2 (Bind b) u2) (CHead c3 (Bind b0) v2))))) (\lambda (_: +B).(\lambda (_: C).(\lambda (_: T).(eq K (Bind Void) (Bind Void))))) (\lambda +(b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b0 Void))))) (\lambda (_: +B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 c3)))) b c2 u2 (refl_equal C +(CHead c2 (Bind b) u2)) (refl_equal K (Bind Void)) H3 H11)) k H8))))))) H6)) +H5))))))))))) (\lambda (c0: C).(\lambda (c2: C).(\lambda (H1: (csubc g c0 +c2)).(\lambda (H2: (((eq C c0 (CHead c1 k v)) \to (or3 (ex2 C (\lambda (c3: +C).(eq C c2 (CHead c3 k v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A +(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) +(\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind +Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 +c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 +v)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) +(ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 +(CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: +T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: +T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: +T).(csubc g c1 c3))))))))).(\lambda (v0: T).(\lambda (a: A).(\lambda (H3: +(sc3 g (asucc g a) c0 v0)).(\lambda (w: T).(\lambda (H4: (sc3 g a c2 +w)).(\lambda (H5: (eq C (CHead c0 (Bind Abst) v0) (CHead c1 k v))).(let H6 +\def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) +with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 +(Bind Abst) v0) (CHead c1 k v) H5) in ((let H7 \def (f_equal C K (\lambda (e: +C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow (Bind +Abst) | (CHead _ k0 _) \Rightarrow k0])) (CHead c0 (Bind Abst) v0) (CHead c1 +k v) H5) in ((let H8 \def (f_equal C T (\lambda (e: C).(match e in C return +(\lambda (_: C).T) with [(CSort _) \Rightarrow v0 | (CHead _ _ t) \Rightarrow +t])) (CHead c0 (Bind Abst) v0) (CHead c1 k v) H5) in (\lambda (H9: (eq K +(Bind Abst) k)).(\lambda (H10: (eq C c0 c1)).(let H11 \def (eq_ind T v0 +(\lambda (t: T).(sc3 g (asucc g a) c0 t)) H3 v H8) in (let H12 \def (eq_ind C +c0 (\lambda (c: C).(sc3 g (asucc g a) c v)) H11 c1 H10) in (let H13 \def +(eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 k v)) \to (or3 (ex2 C +(\lambda (c3: C).(eq C c2 (CHead c3 k v))) (\lambda (c3: C).(csubc g c1 c3))) +(ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind +Abst))))) (\lambda (c3: C).(\lambda (w0: T).(\lambda (_: A).(eq C c2 (CHead +c3 (Bind Abbr) w0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: +A).(csubc g c1 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g +(asucc g a0) c1 v)))) (\lambda (c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 +g a0 c3 w0))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: +T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: +C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: +C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: +C).(\lambda (_: T).(csubc g c1 c3)))))))) H2 c1 H10) in (let H14 \def (eq_ind +C c0 (\lambda (c: C).(csubc g c c2)) H1 c1 H10) in (let H15 \def (eq_ind_r K +k (\lambda (k0: K).((eq C c1 (CHead c1 k0 v)) \to (or3 (ex2 C (\lambda (c3: +C).(eq C c2 (CHead c3 k0 v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A +(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k0 (Bind Abst))))) +(\lambda (c3: C).(\lambda (w0: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind +Abbr) w0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 +c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g (asucc g a0) +c1 v)))) (\lambda (c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 g a0 c3 +w0))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C +c2 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: +T).(eq K k0 (Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: +T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: +T).(csubc g c1 c3)))))))) H13 (Bind Abst) H9) in (eq_ind K (Bind Abst) +(\lambda (k0: K).(or3 (ex2 C (\lambda (c3: C).(eq C (CHead c2 (Bind Abbr) w) +(CHead c3 k0 v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda +(_: C).(\lambda (_: T).(\lambda (_: A).(eq K k0 (Bind Abst))))) (\lambda (c3: +C).(\lambda (w0: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) w) (CHead c3 +(Bind Abbr) w0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g +c1 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g (asucc g +a0) c1 v)))) (\lambda (c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 g a0 c3 +w0))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C +(CHead c2 (Bind Abbr) w) (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda +(_: C).(\lambda (_: T).(eq K k0 (Bind Void))))) (\lambda (b: B).(\lambda (_: +C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: +C).(\lambda (_: T).(csubc g c1 c3))))))) (or3_intro1 (ex2 C (\lambda (c3: +C).(eq C (CHead c2 (Bind Abbr) w) (CHead c3 (Bind Abst) v))) (\lambda (c3: +C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda +(_: A).(eq K (Bind Abst) (Bind Abst))))) (\lambda (c3: C).(\lambda (w0: +T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) w) (CHead c3 (Bind Abbr) +w0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g (asucc g a0) c1 v)))) +(\lambda (c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 g a0 c3 w0))))) +(ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C (CHead +c2 (Bind Abbr) w) (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: +C).(\lambda (_: T).(eq K (Bind Abst) (Bind Void))))) (\lambda (b: B).(\lambda +(_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: +C).(\lambda (_: T).(csubc g c1 c3))))) (ex5_3_intro C T A (\lambda (_: +C).(\lambda (_: T).(\lambda (_: A).(eq K (Bind Abst) (Bind Abst))))) (\lambda +(c3: C).(\lambda (w0: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) w) +(CHead c3 (Bind Abbr) w0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: +A).(csubc g c1 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g +(asucc g a0) c1 v)))) (\lambda (c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 +g a0 c3 w0)))) c2 w a (refl_equal K (Bind Abst)) (refl_equal C (CHead c2 +(Bind Abbr) w)) H14 H12 H4)) k H9))))))))) H7)) H6)))))))))))) y x H0))) +H)))))). +(* COMMENTS +Initial nodes: 5205 +END *) + +theorem csubc_gen_sort_r: + \forall (g: G).(\forall (x: C).(\forall (n: nat).((csubc g x (CSort n)) \to +(eq C x (CSort n))))) +\def + \lambda (g: G).(\lambda (x: C).(\lambda (n: nat).(\lambda (H: (csubc g x +(CSort n))).(insert_eq C (CSort n) (\lambda (c: C).(csubc g x c)) (\lambda +(c: C).(eq C x c)) (\lambda (y: C).(\lambda (H0: (csubc g x y)).(csubc_ind g +(\lambda (c: C).(\lambda (c0: C).((eq C c0 (CSort n)) \to (eq C c c0)))) +(\lambda (n0: nat).(\lambda (H1: (eq C (CSort n0) (CSort n))).(let H2 \def +(f_equal C nat (\lambda (e: C).(match e in C return (\lambda (_: C).nat) with +[(CSort n1) \Rightarrow n1 | (CHead _ _ _) \Rightarrow n0])) (CSort n0) +(CSort n) H1) in (eq_ind_r nat n (\lambda (n1: nat).(eq C (CSort n1) (CSort +n1))) (refl_equal C (CSort n)) n0 H2)))) (\lambda (c1: C).(\lambda (c2: +C).(\lambda (_: (csubc g c1 c2)).(\lambda (_: (((eq C c2 (CSort n)) \to (eq C +c1 c2)))).(\lambda (k: K).(\lambda (v: T).(\lambda (H3: (eq C (CHead c2 k v) +(CSort n))).(let H4 \def (eq_ind C (CHead c2 k v) (\lambda (ee: C).(match ee +in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead +_ _ _) \Rightarrow True])) I (CSort n) H3) in (False_ind (eq C (CHead c1 k v) +(CHead c2 k v)) H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: +(csubc g c1 c2)).(\lambda (_: (((eq C c2 (CSort n)) \to (eq C c1 +c2)))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: +T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c2 (Bind b) u2) (CSort +n))).(let H5 \def (eq_ind C (CHead c2 (Bind b) u2) (\lambda (ee: C).(match ee +in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead +_ _ _) \Rightarrow True])) I (CSort n) H4) in (False_ind (eq C (CHead c1 +(Bind Void) u1) (CHead c2 (Bind b) u2)) H5))))))))))) (\lambda (c1: +C).(\lambda (c2: C).(\lambda (_: (csubc g c1 c2)).(\lambda (_: (((eq C c2 +(CSort n)) \to (eq C c1 c2)))).(\lambda (v: T).(\lambda (a: A).(\lambda (_: +(sc3 g (asucc g a) c1 v)).(\lambda (w: T).(\lambda (_: (sc3 g a c2 +w)).(\lambda (H5: (eq C (CHead c2 (Bind Abbr) w) (CSort n))).(let H6 \def +(eq_ind C (CHead c2 (Bind Abbr) w) (\lambda (ee: C).(match ee in C return +(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) +\Rightarrow True])) I (CSort n) H5) in (False_ind (eq C (CHead c1 (Bind Abst) +v) (CHead c2 (Bind Abbr) w)) H6)))))))))))) x y H0))) H)))). +(* COMMENTS +Initial nodes: 533 +END *) + +theorem csubc_gen_head_r: + \forall (g: G).(\forall (c2: C).(\forall (x: C).(\forall (w: T).(\forall (k: +K).((csubc g x (CHead c2 k w)) \to (or3 (ex2 C (\lambda (c1: C).(eq C x +(CHead c1 k w))) (\lambda (c1: C).(csubc g c1 c2))) (ex5_3 C T A (\lambda (_: +C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c1: +C).(\lambda (v: T).(\lambda (_: A).(eq C x (CHead c1 (Bind Abst) v))))) +(\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c2)))) (\lambda +(c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T +(\lambda (_: B).(\lambda (c1: C).(\lambda (v1: T).(eq C x (CHead c1 (Bind +Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind +b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) +(\lambda (_: B).(\lambda (c1: C).(\lambda (_: T).(csubc g c1 c2))))))))))) +\def + \lambda (g: G).(\lambda (c2: C).(\lambda (x: C).(\lambda (w: T).(\lambda (k: +K).(\lambda (H: (csubc g x (CHead c2 k w))).(insert_eq C (CHead c2 k w) +(\lambda (c: C).(csubc g x c)) (\lambda (_: C).(or3 (ex2 C (\lambda (c1: +C).(eq C x (CHead c1 k w))) (\lambda (c1: C).(csubc g c1 c2))) (ex5_3 C T A +(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) +(\lambda (c1: C).(\lambda (v: T).(\lambda (_: A).(eq C x (CHead c1 (Bind +Abst) v))))) (\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 +c2)))) (\lambda (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1 +v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) +(ex4_3 B C T (\lambda (_: B).(\lambda (c1: C).(\lambda (v1: T).(eq C x (CHead +c1 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(eq K +k (Bind b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b +Void))))) (\lambda (_: B).(\lambda (c1: C).(\lambda (_: T).(csubc g c1 +c2))))))) (\lambda (y: C).(\lambda (H0: (csubc g x y)).(csubc_ind g (\lambda +(c: C).(\lambda (c0: C).((eq C c0 (CHead c2 k w)) \to (or3 (ex2 C (\lambda +(c1: C).(eq C c (CHead c1 k w))) (\lambda (c1: C).(csubc g c1 c2))) (ex5_3 C +T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) +(\lambda (c1: C).(\lambda (v: T).(\lambda (_: A).(eq C c (CHead c1 (Bind +Abst) v))))) (\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 +c2)))) (\lambda (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1 +v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) +(ex4_3 B C T (\lambda (_: B).(\lambda (c1: C).(\lambda (v1: T).(eq C c (CHead +c1 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(eq K +k (Bind b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b +Void))))) (\lambda (_: B).(\lambda (c1: C).(\lambda (_: T).(csubc g c1 +c2))))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead c2 k +w))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return +(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) +\Rightarrow False])) I (CHead c2 k w) H1) in (False_ind (or3 (ex2 C (\lambda +(c1: C).(eq C (CSort n) (CHead c1 k w))) (\lambda (c1: C).(csubc g c1 c2))) +(ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind +Abbr))))) (\lambda (c1: C).(\lambda (v: T).(\lambda (_: A).(eq C (CSort n) +(CHead c1 (Bind Abst) v))))) (\lambda (c1: C).(\lambda (_: T).(\lambda (_: +A).(csubc g c1 c2)))) (\lambda (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g +(asucc g a) c1 v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a +c2 w))))) (ex4_3 B C T (\lambda (_: B).(\lambda (c1: C).(\lambda (v1: T).(eq +C (CSort n) (CHead c1 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: +C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: B).(\lambda (_: +C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c1: +C).(\lambda (_: T).(csubc g c1 c2)))))) H2)))) (\lambda (c1: C).(\lambda (c0: +C).(\lambda (H1: (csubc g c1 c0)).(\lambda (H2: (((eq C c0 (CHead c2 k w)) +\to (or3 (ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k w))) (\lambda (c3: +C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda +(_: A).(eq K k (Bind Abbr))))) (\lambda (c3: C).(\lambda (v: T).(\lambda (_: +A).(eq C c1 (CHead c3 (Bind Abst) v))))) (\lambda (c3: C).(\lambda (_: +T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v: +T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) (\lambda (_: C).(\lambda (_: +T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T (\lambda (_: B).(\lambda +(c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind Void) v1))))) (\lambda (b: +B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: +B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: +B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 c2))))))))).(\lambda (k0: +K).(\lambda (v: T).(\lambda (H3: (eq C (CHead c0 k0 v) (CHead c2 k w))).(let +H4 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) +with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k0 +v) (CHead c2 k w) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e +in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k1 +_) \Rightarrow k1])) (CHead c0 k0 v) (CHead c2 k w) H3) in ((let H6 \def +(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with +[(CSort _) \Rightarrow v | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 v) +(CHead c2 k w) H3) in (\lambda (H7: (eq K k0 k)).(\lambda (H8: (eq C c0 +c2)).(eq_ind_r T w (\lambda (t: T).(or3 (ex2 C (\lambda (c3: C).(eq C (CHead +c1 k0 t) (CHead c3 k w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A +(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) +(\lambda (c3: C).(\lambda (v0: T).(\lambda (_: A).(eq C (CHead c1 k0 t) +(CHead c3 (Bind Abst) v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: +A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (a: A).(sc3 +g (asucc g a) c3 v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 +g a c2 w))))) (ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: +T).(eq C (CHead c1 k0 t) (CHead c3 (Bind Void) v1))))) (\lambda (b: +B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: +B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: +B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 c2))))))) (eq_ind_r K k +(\lambda (k1: K).(or3 (ex2 C (\lambda (c3: C).(eq C (CHead c1 k1 w) (CHead c3 +k w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: +C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c3: +C).(\lambda (v0: T).(\lambda (_: A).(eq C (CHead c1 k1 w) (CHead c3 (Bind +Abst) v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 +c2)))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (a: A).(sc3 g (asucc g a) +c3 v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) +(ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead +c1 k1 w) (CHead c3 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: +C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: B).(\lambda (_: +C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: +C).(\lambda (_: T).(csubc g c3 c2))))))) (let H9 \def (eq_ind C c0 (\lambda +(c: C).((eq C c (CHead c2 k w)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c1 +(CHead c3 k w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: +C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c3: +C).(\lambda (v0: T).(\lambda (_: A).(eq C c1 (CHead c3 (Bind Abst) v0))))) +(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda +(c3: C).(\lambda (v0: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v0)))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C +T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind +Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind +b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) +(\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 c2)))))))) H2 c2 +H8) in (let H10 \def (eq_ind C c0 (\lambda (c: C).(csubc g c1 c)) H1 c2 H8) +in (or3_intro0 (ex2 C (\lambda (c3: C).(eq C (CHead c1 k w) (CHead c3 k w))) +(\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: +T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c3: C).(\lambda (v0: +T).(\lambda (_: A).(eq C (CHead c1 k w) (CHead c3 (Bind Abst) v0))))) +(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda +(c3: C).(\lambda (v0: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v0)))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C +T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead c1 k w) +(CHead c3 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: +T).(eq K k (Bind b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not +(eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g +c3 c2))))) (ex_intro2 C (\lambda (c3: C).(eq C (CHead c1 k w) (CHead c3 k +w))) (\lambda (c3: C).(csubc g c3 c2)) c1 (refl_equal C (CHead c1 k w)) +H10)))) k0 H7) v H6)))) H5)) H4))))))))) (\lambda (c1: C).(\lambda (c0: +C).(\lambda (H1: (csubc g c1 c0)).(\lambda (H2: (((eq C c0 (CHead c2 k w)) +\to (or3 (ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k w))) (\lambda (c3: +C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda +(_: A).(eq K k (Bind Abbr))))) (\lambda (c3: C).(\lambda (v: T).(\lambda (_: +A).(eq C c1 (CHead c3 (Bind Abst) v))))) (\lambda (c3: C).(\lambda (_: +T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v: +T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) (\lambda (_: C).(\lambda (_: +T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T (\lambda (_: B).(\lambda +(c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind Void) v1))))) (\lambda (b: +B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: +B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: +B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 c2))))))))).(\lambda (b: +B).(\lambda (H3: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: +T).(\lambda (H4: (eq C (CHead c0 (Bind b) u2) (CHead c2 k w))).(let H5 \def +(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with +[(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 (Bind b) +u2) (CHead c2 k w) H4) in ((let H6 \def (f_equal C K (\lambda (e: C).(match e +in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow (Bind b) | (CHead +_ k0 _) \Rightarrow k0])) (CHead c0 (Bind b) u2) (CHead c2 k w) H4) in ((let +H7 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) +with [(CSort _) \Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c0 +(Bind b) u2) (CHead c2 k w) H4) in (\lambda (H8: (eq K (Bind b) k)).(\lambda +(H9: (eq C c0 c2)).(let H10 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead +c2 k w)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k w))) (\lambda +(c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: +T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c3: C).(\lambda (v: +T).(\lambda (_: A).(eq C c1 (CHead c3 (Bind Abst) v))))) (\lambda (c3: +C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: +C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) (\lambda (_: +C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T (\lambda +(_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind Void) +v1))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind +b0))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b0 +Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 +c2)))))))) H2 c2 H9) in (let H11 \def (eq_ind C c0 (\lambda (c: C).(csubc g +c1 c)) H1 c2 H9) in (let H12 \def (eq_ind_r K k (\lambda (k0: K).((eq C c2 +(CHead c2 k0 w)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k0 w))) +(\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: +T).(\lambda (_: A).(eq K k0 (Bind Abbr))))) (\lambda (c3: C).(\lambda (v: +T).(\lambda (_: A).(eq C c1 (CHead c3 (Bind Abst) v))))) (\lambda (c3: +C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: +C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) (\lambda (_: +C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T (\lambda +(_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind Void) +v1))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(eq K k0 (Bind +b0))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b0 +Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 +c2)))))))) H10 (Bind b) H8) in (eq_ind K (Bind b) (\lambda (k0: K).(or3 (ex2 +C (\lambda (c3: C).(eq C (CHead c1 (Bind Void) u1) (CHead c3 k0 w))) (\lambda +(c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: +T).(\lambda (_: A).(eq K k0 (Bind Abbr))))) (\lambda (c3: C).(\lambda (v: +T).(\lambda (_: A).(eq C (CHead c1 (Bind Void) u1) (CHead c3 (Bind Abst) +v))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) +(\lambda (c3: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C +T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead c1 (Bind +Void) u1) (CHead c3 (Bind Void) v1))))) (\lambda (b0: B).(\lambda (_: +C).(\lambda (_: T).(eq K k0 (Bind b0))))) (\lambda (b0: B).(\lambda (_: +C).(\lambda (_: T).(not (eq B b0 Void))))) (\lambda (_: B).(\lambda (c3: +C).(\lambda (_: T).(csubc g c3 c2))))))) (or3_intro2 (ex2 C (\lambda (c3: +C).(eq C (CHead c1 (Bind Void) u1) (CHead c3 (Bind b) w))) (\lambda (c3: +C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda +(_: A).(eq K (Bind b) (Bind Abbr))))) (\lambda (c3: C).(\lambda (v: +T).(\lambda (_: A).(eq C (CHead c1 (Bind Void) u1) (CHead c3 (Bind Abst) +v))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) +(\lambda (c3: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) +(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C +T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead c1 (Bind +Void) u1) (CHead c3 (Bind Void) v1))))) (\lambda (b0: B).(\lambda (_: +C).(\lambda (_: T).(eq K (Bind b) (Bind b0))))) (\lambda (b0: B).(\lambda (_: +C).(\lambda (_: T).(not (eq B b0 Void))))) (\lambda (_: B).(\lambda (c3: +C).(\lambda (_: T).(csubc g c3 c2))))) (ex4_3_intro B C T (\lambda (_: +B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead c1 (Bind Void) u1) (CHead +c3 (Bind Void) v1))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(eq K +(Bind b) (Bind b0))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not +(eq B b0 Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g +c3 c2)))) b c1 u1 (refl_equal C (CHead c1 (Bind Void) u1)) (refl_equal K +(Bind b)) H3 H11)) k H8))))))) H6)) H5))))))))))) (\lambda (c1: C).(\lambda +(c0: C).(\lambda (H1: (csubc g c1 c0)).(\lambda (H2: (((eq C c0 (CHead c2 k +w)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k w))) (\lambda (c3: +C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda +(_: A).(eq K k (Bind Abbr))))) (\lambda (c3: C).(\lambda (v: T).(\lambda (_: +A).(eq C c1 (CHead c3 (Bind Abst) v))))) (\lambda (c3: C).(\lambda (_: +T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v: +T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) (\lambda (_: C).(\lambda (_: +T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T (\lambda (_: B).(\lambda +(c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind Void) v1))))) (\lambda (b: +B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: +B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: +B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 c2))))))))).(\lambda (v: +T).(\lambda (a: A).(\lambda (H3: (sc3 g (asucc g a) c1 v)).(\lambda (w0: +T).(\lambda (H4: (sc3 g a c0 w0)).(\lambda (H5: (eq C (CHead c0 (Bind Abbr) +w0) (CHead c2 k w))).(let H6 \def (f_equal C C (\lambda (e: C).(match e in C +return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) +\Rightarrow c])) (CHead c0 (Bind Abbr) w0) (CHead c2 k w) H5) in ((let H7 +\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) +with [(CSort _) \Rightarrow (Bind Abbr) | (CHead _ k0 _) \Rightarrow k0])) +(CHead c0 (Bind Abbr) w0) (CHead c2 k w) H5) in ((let H8 \def (f_equal C T +(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) +\Rightarrow w0 | (CHead _ _ t) \Rightarrow t])) (CHead c0 (Bind Abbr) w0) +(CHead c2 k w) H5) in (\lambda (H9: (eq K (Bind Abbr) k)).(\lambda (H10: (eq +C c0 c2)).(let H11 \def (eq_ind T w0 (\lambda (t: T).(sc3 g a c0 t)) H4 w H8) +in (let H12 \def (eq_ind C c0 (\lambda (c: C).(sc3 g a c w)) H11 c2 H10) in +(let H13 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c2 k w)) \to (or3 +(ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k w))) (\lambda (c3: C).(csubc g +c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k +(Bind Abbr))))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (_: A).(eq C c1 +(CHead c3 (Bind Abst) v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: +A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (a0: A).(sc3 +g (asucc g a0) c3 v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: +A).(sc3 g a0 c2 w))))) (ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda +(v1: T).(eq C c1 (CHead c3 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: +C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: B).(\lambda (_: +C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: +C).(\lambda (_: T).(csubc g c3 c2)))))))) H2 c2 H10) in (let H14 \def (eq_ind +C c0 (\lambda (c: C).(csubc g c1 c)) H1 c2 H10) in (let H15 \def (eq_ind_r K +k (\lambda (k0: K).((eq C c2 (CHead c2 k0 w)) \to (or3 (ex2 C (\lambda (c3: +C).(eq C c1 (CHead c3 k0 w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A +(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k0 (Bind Abbr))))) +(\lambda (c3: C).(\lambda (v0: T).(\lambda (_: A).(eq C c1 (CHead c3 (Bind +Abst) v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 +c2)))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (a0: A).(sc3 g (asucc g a0) +c3 v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g a0 c2 +w))))) (ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C +c1 (CHead c3 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: +T).(eq K k0 (Bind b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not +(eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g +c3 c2)))))))) H13 (Bind Abbr) H9) in (eq_ind K (Bind Abbr) (\lambda (k0: +K).(or3 (ex2 C (\lambda (c3: C).(eq C (CHead c1 (Bind Abst) v) (CHead c3 k0 +w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda +(_: T).(\lambda (_: A).(eq K k0 (Bind Abbr))))) (\lambda (c3: C).(\lambda +(v0: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) v) (CHead c3 (Bind Abst) +v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) +(\lambda (c3: C).(\lambda (v0: T).(\lambda (a0: A).(sc3 g (asucc g a0) c3 +v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g a0 c2 w))))) +(ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead +c1 (Bind Abst) v) (CHead c3 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: +C).(\lambda (_: T).(eq K k0 (Bind b))))) (\lambda (b: B).(\lambda (_: +C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: +C).(\lambda (_: T).(csubc g c3 c2))))))) (or3_intro1 (ex2 C (\lambda (c3: +C).(eq C (CHead c1 (Bind Abst) v) (CHead c3 (Bind Abbr) w))) (\lambda (c3: +C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda +(_: A).(eq K (Bind Abbr) (Bind Abbr))))) (\lambda (c3: C).(\lambda (v0: +T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) v) (CHead c3 (Bind Abst) +v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) +(\lambda (c3: C).(\lambda (v0: T).(\lambda (a0: A).(sc3 g (asucc g a0) c3 +v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g a0 c2 w))))) +(ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead +c1 (Bind Abst) v) (CHead c3 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: +C).(\lambda (_: T).(eq K (Bind Abbr) (Bind b))))) (\lambda (b: B).(\lambda +(_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: +C).(\lambda (_: T).(csubc g c3 c2))))) (ex5_3_intro C T A (\lambda (_: +C).(\lambda (_: T).(\lambda (_: A).(eq K (Bind Abbr) (Bind Abbr))))) (\lambda +(c3: C).(\lambda (v0: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) v) +(CHead c3 (Bind Abst) v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: +A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (a0: A).(sc3 +g (asucc g a0) c3 v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: +A).(sc3 g a0 c2 w)))) c1 v a (refl_equal K (Bind Abbr)) (refl_equal C (CHead +c1 (Bind Abst) v)) H14 H3 H12)) k H9))))))))) H7)) H6)))))))))))) x y H0))) +H)))))). +(* COMMENTS +Initial nodes: 5197 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubc/getl.ma b/matita/matita/contribs/lambdadelta/basic_1/csubc/getl.ma new file mode 100644 index 000000000..244c84927 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/csubc/getl.ma @@ -0,0 +1,45 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/csubc/drop.ma". + +include "Basic-1/csubc/clear.ma". + +theorem csubc_getl_conf: + \forall (g: G).(\forall (c1: C).(\forall (e1: C).(\forall (i: nat).((getl i +c1 e1) \to (\forall (c2: C).((csubc g c1 c2) \to (ex2 C (\lambda (e2: +C).(getl i c2 e2)) (\lambda (e2: C).(csubc g e1 e2))))))))) +\def + \lambda (g: G).(\lambda (c1: C).(\lambda (e1: C).(\lambda (i: nat).(\lambda +(H: (getl i c1 e1)).(\lambda (c2: C).(\lambda (H0: (csubc g c1 c2)).(let H1 +\def (getl_gen_all c1 e1 i H) in (ex2_ind C (\lambda (e: C).(drop i O c1 e)) +(\lambda (e: C).(clear e e1)) (ex2 C (\lambda (e2: C).(getl i c2 e2)) +(\lambda (e2: C).(csubc g e1 e2))) (\lambda (x: C).(\lambda (H2: (drop i O c1 +x)).(\lambda (H3: (clear x e1)).(let H_x \def (csubc_drop_conf_O g c1 x i H2 +c2 H0) in (let H4 \def H_x in (ex2_ind C (\lambda (e2: C).(drop i O c2 e2)) +(\lambda (e2: C).(csubc g x e2)) (ex2 C (\lambda (e2: C).(getl i c2 e2)) +(\lambda (e2: C).(csubc g e1 e2))) (\lambda (x0: C).(\lambda (H5: (drop i O +c2 x0)).(\lambda (H6: (csubc g x x0)).(let H_x0 \def (csubc_clear_conf g x e1 +H3 x0 H6) in (let H7 \def H_x0 in (ex2_ind C (\lambda (e2: C).(clear x0 e2)) +(\lambda (e2: C).(csubc g e1 e2)) (ex2 C (\lambda (e2: C).(getl i c2 e2)) +(\lambda (e2: C).(csubc g e1 e2))) (\lambda (x1: C).(\lambda (H8: (clear x0 +x1)).(\lambda (H9: (csubc g e1 x1)).(ex_intro2 C (\lambda (e2: C).(getl i c2 +e2)) (\lambda (e2: C).(csubc g e1 e2)) x1 (getl_intro i c2 x1 x0 H5 H8) +H9)))) H7)))))) H4)))))) H1)))))))). +(* COMMENTS +Initial nodes: 315 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubc/props.ma b/matita/matita/contribs/lambdadelta/basic_1/csubc/props.ma new file mode 100644 index 000000000..d6399181c --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/csubc/props.ma @@ -0,0 +1,30 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/csubc/defs.ma". + +include "Basic-1/sc3/props.ma". + +theorem csubc_refl: + \forall (g: G).(\forall (c: C).(csubc g c c)) +\def + \lambda (g: G).(\lambda (c: C).(C_ind (\lambda (c0: C).(csubc g c0 c0)) +(\lambda (n: nat).(csubc_sort g n)) (\lambda (c0: C).(\lambda (H: (csubc g c0 +c0)).(\lambda (k: K).(\lambda (t: T).(csubc_head g c0 c0 H k t))))) c)). +(* COMMENTS +Initial nodes: 53 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubst0/clear.ma b/matita/matita/contribs/lambdadelta/basic_1/csubst0/clear.ma new file mode 100644 index 000000000..0700d7d48 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/csubst0/clear.ma @@ -0,0 +1,1139 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/csubst0/props.ma". + +include "Basic-1/csubst0/fwd.ma". + +include "Basic-1/clear/fwd.ma". + +theorem csubst0_clear_O: + \forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 O v c1 c2) \to +(\forall (c: C).((clear c1 c) \to (clear c2 c)))))) +\def + \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: +T).((csubst0 O v c c2) \to (\forall (c0: C).((clear c c0) \to (clear c2 +c0))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (H: +(csubst0 O v (CSort n) c2)).(\lambda (c: C).(\lambda (_: (clear (CSort n) +c)).(csubst0_gen_sort c2 v O n H (clear c2 c)))))))) (\lambda (c: C).(\lambda +(H: ((\forall (c2: C).(\forall (v: T).((csubst0 O v c c2) \to (\forall (c0: +C).((clear c c0) \to (clear c2 c0)))))))).(\lambda (k: K).(\lambda (t: +T).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 O v (CHead c k t) +c2)).(\lambda (c0: C).(\lambda (H1: (clear (CHead c k t) c0)).(or3_ind (ex3_2 +T nat (\lambda (_: T).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (u2: +T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: +nat).(subst0 j v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq +nat O (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k +t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (ex4_3 T C nat +(\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j))))) +(\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k +u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t +u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c +c3))))) (clear c2 c0) (\lambda (H2: (ex3_2 T nat (\lambda (_: T).(\lambda (j: +nat).(eq nat O (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead +c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t +u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: nat).(eq nat O (s k +j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda +(u2: T).(\lambda (j: nat).(subst0 j v t u2))) (clear c2 c0) (\lambda (x0: +T).(\lambda (x1: nat).(\lambda (H3: (eq nat O (s k x1))).(\lambda (H4: (eq C +c2 (CHead c k x0))).(\lambda (H5: (subst0 x1 v t x0)).(eq_ind_r C (CHead c k +x0) (\lambda (c3: C).(clear c3 c0)) (K_ind (\lambda (k0: K).((clear (CHead c +k0 t) c0) \to ((eq nat O (s k0 x1)) \to (clear (CHead c k0 x0) c0)))) +(\lambda (b: B).(\lambda (_: (clear (CHead c (Bind b) t) c0)).(\lambda (H7: +(eq nat O (s (Bind b) x1))).(let H8 \def (eq_ind nat O (\lambda (ee: +nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True +| (S _) \Rightarrow False])) I (S x1) H7) in (False_ind (clear (CHead c (Bind +b) x0) c0) H8))))) (\lambda (f: F).(\lambda (H6: (clear (CHead c (Flat f) t) +c0)).(\lambda (H7: (eq nat O (s (Flat f) x1))).(let H8 \def (eq_ind_r nat x1 +(\lambda (n: nat).(subst0 n v t x0)) H5 O H7) in (clear_flat c c0 +(clear_gen_flat f c c0 t H6) f x0))))) k H1 H3) c2 H4)))))) H2)) (\lambda +(H2: (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j)))) +(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: +C).(\lambda (j: nat).(csubst0 j v c c3))))).(ex3_2_ind C nat (\lambda (_: +C).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (c3: C).(\lambda (_: +nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j +v c c3))) (clear c2 c0) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H3: (eq +nat O (s k x1))).(\lambda (H4: (eq C c2 (CHead x0 k t))).(\lambda (H5: +(csubst0 x1 v c x0)).(eq_ind_r C (CHead x0 k t) (\lambda (c3: C).(clear c3 +c0)) (K_ind (\lambda (k0: K).((clear (CHead c k0 t) c0) \to ((eq nat O (s k0 +x1)) \to (clear (CHead x0 k0 t) c0)))) (\lambda (b: B).(\lambda (_: (clear +(CHead c (Bind b) t) c0)).(\lambda (H7: (eq nat O (s (Bind b) x1))).(let H8 +\def (eq_ind nat O (\lambda (ee: nat).(match ee in nat return (\lambda (_: +nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x1) H7) +in (False_ind (clear (CHead x0 (Bind b) t) c0) H8))))) (\lambda (f: +F).(\lambda (H6: (clear (CHead c (Flat f) t) c0)).(\lambda (H7: (eq nat O (s +(Flat f) x1))).(let H8 \def (eq_ind_r nat x1 (\lambda (n: nat).(csubst0 n v c +x0)) H5 O H7) in (clear_flat x0 c0 (H x0 v H8 c0 (clear_gen_flat f c c0 t +H6)) f t))))) k H1 H3) c2 H4)))))) H2)) (\lambda (H2: (ex4_3 T C nat (\lambda +(_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j))))) (\lambda (u2: +T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda +(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: +T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))))).(ex4_3_ind T C +nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j))))) +(\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k +u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t +u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c +c3)))) (clear c2 c0) (\lambda (x0: T).(\lambda (x1: C).(\lambda (x2: +nat).(\lambda (H3: (eq nat O (s k x2))).(\lambda (H4: (eq C c2 (CHead x1 k +x0))).(\lambda (H5: (subst0 x2 v t x0)).(\lambda (H6: (csubst0 x2 v c +x1)).(eq_ind_r C (CHead x1 k x0) (\lambda (c3: C).(clear c3 c0)) (K_ind +(\lambda (k0: K).((clear (CHead c k0 t) c0) \to ((eq nat O (s k0 x2)) \to +(clear (CHead x1 k0 x0) c0)))) (\lambda (b: B).(\lambda (_: (clear (CHead c +(Bind b) t) c0)).(\lambda (H8: (eq nat O (s (Bind b) x2))).(let H9 \def +(eq_ind nat O (\lambda (ee: nat).(match ee in nat return (\lambda (_: +nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x2) H8) +in (False_ind (clear (CHead x1 (Bind b) x0) c0) H9))))) (\lambda (f: +F).(\lambda (H7: (clear (CHead c (Flat f) t) c0)).(\lambda (H8: (eq nat O (s +(Flat f) x2))).(let H9 \def (eq_ind_r nat x2 (\lambda (n: nat).(csubst0 n v c +x1)) H6 O H8) in (let H10 \def (eq_ind_r nat x2 (\lambda (n: nat).(subst0 n v +t x0)) H5 O H8) in (clear_flat x1 c0 (H x1 v H9 c0 (clear_gen_flat f c c0 t +H7)) f x0)))))) k H1 H3) c2 H4)))))))) H2)) (csubst0_gen_head k c c2 t v O +H0))))))))))) c1). +(* COMMENTS +Initial nodes: 1582 +END *) + +theorem csubst0_clear_O_back: + \forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 O v c1 c2) \to +(\forall (c: C).((clear c2 c) \to (clear c1 c)))))) +\def + \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: +T).((csubst0 O v c c2) \to (\forall (c0: C).((clear c2 c0) \to (clear c +c0))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (H: +(csubst0 O v (CSort n) c2)).(\lambda (c: C).(\lambda (_: (clear c2 +c)).(csubst0_gen_sort c2 v O n H (clear (CSort n) c)))))))) (\lambda (c: +C).(\lambda (H: ((\forall (c2: C).(\forall (v: T).((csubst0 O v c c2) \to +(\forall (c0: C).((clear c2 c0) \to (clear c c0)))))))).(\lambda (k: +K).(\lambda (t: T).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 O +v (CHead c k t) c2)).(\lambda (c0: C).(\lambda (H1: (clear c2 c0)).(or3_ind +(ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda +(u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: +T).(\lambda (j: nat).(subst0 j v t u2)))) (ex3_2 C nat (\lambda (_: +C).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (c3: C).(\lambda (_: +nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j +v c c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: +nat).(eq nat O (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: +nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda +(j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: +nat).(csubst0 j v c c3))))) (clear (CHead c k t) c0) (\lambda (H2: (ex3_2 T +nat (\lambda (_: T).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (u2: +T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: +nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: +nat).(eq nat O (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead +c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))) (clear +(CHead c k t) c0) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H3: (eq nat O +(s k x1))).(\lambda (H4: (eq C c2 (CHead c k x0))).(\lambda (H5: (subst0 x1 v +t x0)).(let H6 \def (eq_ind C c2 (\lambda (c3: C).(clear c3 c0)) H1 (CHead c +k x0) H4) in (K_ind (\lambda (k0: K).((eq nat O (s k0 x1)) \to ((clear (CHead +c k0 x0) c0) \to (clear (CHead c k0 t) c0)))) (\lambda (b: B).(\lambda (H7: +(eq nat O (s (Bind b) x1))).(\lambda (_: (clear (CHead c (Bind b) x0) +c0)).(let H9 \def (eq_ind nat O (\lambda (ee: nat).(match ee in nat return +(\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) +I (S x1) H7) in (False_ind (clear (CHead c (Bind b) t) c0) H9))))) (\lambda +(f: F).(\lambda (H7: (eq nat O (s (Flat f) x1))).(\lambda (H8: (clear (CHead +c (Flat f) x0) c0)).(let H9 \def (eq_ind_r nat x1 (\lambda (n: nat).(subst0 n +v t x0)) H5 O H7) in (clear_flat c c0 (clear_gen_flat f c c0 x0 H8) f t))))) +k H3 H6))))))) H2)) (\lambda (H2: (ex3_2 C nat (\lambda (_: C).(\lambda (j: +nat).(eq nat O (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead +c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c +c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat O (s k +j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda +(c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (clear (CHead c k t) c0) +(\lambda (x0: C).(\lambda (x1: nat).(\lambda (H3: (eq nat O (s k +x1))).(\lambda (H4: (eq C c2 (CHead x0 k t))).(\lambda (H5: (csubst0 x1 v c +x0)).(let H6 \def (eq_ind C c2 (\lambda (c3: C).(clear c3 c0)) H1 (CHead x0 k +t) H4) in (K_ind (\lambda (k0: K).((eq nat O (s k0 x1)) \to ((clear (CHead x0 +k0 t) c0) \to (clear (CHead c k0 t) c0)))) (\lambda (b: B).(\lambda (H7: (eq +nat O (s (Bind b) x1))).(\lambda (_: (clear (CHead x0 (Bind b) t) c0)).(let +H9 \def (eq_ind nat O (\lambda (ee: nat).(match ee in nat return (\lambda (_: +nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x1) H7) +in (False_ind (clear (CHead c (Bind b) t) c0) H9))))) (\lambda (f: +F).(\lambda (H7: (eq nat O (s (Flat f) x1))).(\lambda (H8: (clear (CHead x0 +(Flat f) t) c0)).(let H9 \def (eq_ind_r nat x1 (\lambda (n: nat).(csubst0 n v +c x0)) H5 O H7) in (clear_flat c c0 (H x0 v H9 c0 (clear_gen_flat f x0 c0 t +H8)) f t))))) k H3 H6))))))) H2)) (\lambda (H2: (ex4_3 T C nat (\lambda (_: +T).(\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j))))) (\lambda (u2: +T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda +(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: +T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))))).(ex4_3_ind T C +nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j))))) +(\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k +u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t +u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c +c3)))) (clear (CHead c k t) c0) (\lambda (x0: T).(\lambda (x1: C).(\lambda +(x2: nat).(\lambda (H3: (eq nat O (s k x2))).(\lambda (H4: (eq C c2 (CHead x1 +k x0))).(\lambda (H5: (subst0 x2 v t x0)).(\lambda (H6: (csubst0 x2 v c +x1)).(let H7 \def (eq_ind C c2 (\lambda (c3: C).(clear c3 c0)) H1 (CHead x1 k +x0) H4) in (K_ind (\lambda (k0: K).((eq nat O (s k0 x2)) \to ((clear (CHead +x1 k0 x0) c0) \to (clear (CHead c k0 t) c0)))) (\lambda (b: B).(\lambda (H8: +(eq nat O (s (Bind b) x2))).(\lambda (_: (clear (CHead x1 (Bind b) x0) +c0)).(let H10 \def (eq_ind nat O (\lambda (ee: nat).(match ee in nat return +(\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) +I (S x2) H8) in (False_ind (clear (CHead c (Bind b) t) c0) H10))))) (\lambda +(f: F).(\lambda (H8: (eq nat O (s (Flat f) x2))).(\lambda (H9: (clear (CHead +x1 (Flat f) x0) c0)).(let H10 \def (eq_ind_r nat x2 (\lambda (n: +nat).(csubst0 n v c x1)) H6 O H8) in (let H11 \def (eq_ind_r nat x2 (\lambda +(n: nat).(subst0 n v t x0)) H5 O H8) in (clear_flat c c0 (H x1 v H10 c0 +(clear_gen_flat f x1 c0 x0 H9)) f t)))))) k H3 H7))))))))) H2)) +(csubst0_gen_head k c c2 t v O H0))))))))))) c1). +(* COMMENTS +Initial nodes: 1606 +END *) + +theorem csubst0_clear_S: + \forall (c1: C).(\forall (c2: C).(\forall (v: T).(\forall (i: nat).((csubst0 +(S i) v c1 c2) \to (\forall (c: C).((clear c1 c) \to (or4 (clear c2 c) (ex3_4 +B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq +C c (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: +T).(\lambda (u2: T).(clear c2 (CHead e (Bind b) u2)))))) (\lambda (_: +B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) +(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(eq C c (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: +C).(\lambda (e2: C).(\lambda (u: T).(clear c2 (CHead e2 (Bind b) u)))))) +(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i +v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u1: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b) u1))))))) +(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: +C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) +(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 i v e1 e2)))))))))))))) +\def + \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: +T).(\forall (i: nat).((csubst0 (S i) v c c2) \to (\forall (c0: C).((clear c +c0) \to (or4 (clear c2 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: +C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) +(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 +(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: +T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: +B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind +b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: +T).(clear c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T +(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda +(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e2 +(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda +(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda +(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 +e2))))))))))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda +(i: nat).(\lambda (H: (csubst0 (S i) v (CSort n) c2)).(\lambda (c: +C).(\lambda (_: (clear (CSort n) c)).(csubst0_gen_sort c2 v (S i) n H (or4 +(clear c2 c) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: +T).(\lambda (_: T).(eq C c (CHead e (Bind b) u1)))))) (\lambda (b: +B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e (Bind +b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: +T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind b) u)))))) +(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c2 +(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: +B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c +(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) +(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda +(u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda +(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))))))))))) +(\lambda (c: C).(\lambda (H: ((\forall (c2: C).(\forall (v: T).(\forall (i: +nat).((csubst0 (S i) v c c2) \to (\forall (c0: C).((clear c c0) \to (or4 +(clear c2 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: +T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: +B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e (Bind +b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: +T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) +(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c2 +(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: +B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 +(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) +(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda +(u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda +(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 +e2)))))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda +(v: T).(\lambda (i: nat).(\lambda (H0: (csubst0 (S i) v (CHead c k t) +c2)).(\lambda (c0: C).(\lambda (H1: (clear (CHead c k t) c0)).(or3_ind (ex3_2 +T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (S i) (s k j)))) (\lambda +(u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: +T).(\lambda (j: nat).(subst0 j v t u2)))) (ex3_2 C nat (\lambda (_: +C).(\lambda (j: nat).(eq nat (S i) (s k j)))) (\lambda (c3: C).(\lambda (_: +nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j +v c c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: +nat).(eq nat (S i) (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: +nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda +(j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: +nat).(csubst0 j v c c3))))) (or4 (clear c2 c0) (ex3_4 B C T T (\lambda (b: +B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind +b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: +T).(clear c2 (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: +C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T +(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 +(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(clear c2 (CHead e2 (Bind b) u)))))) (\lambda (_: +B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) +(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda +(u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: +B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear +c2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: +C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: +B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 +i v e1 e2)))))))) (\lambda (H2: (ex3_2 T nat (\lambda (_: T).(\lambda (j: +nat).(eq nat (S i) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 +(CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t +u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (S i) (s k +j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda +(u2: T).(\lambda (j: nat).(subst0 j v t u2))) (or4 (clear c2 c0) (ex3_4 B C T +T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 +(CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: +T).(\lambda (u2: T).(clear c2 (CHead e (Bind b) u2)))))) (\lambda (_: +B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) +(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: +C).(\lambda (e2: C).(\lambda (u: T).(clear c2 (CHead e2 (Bind b) u)))))) +(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i +v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) +(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: +C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) +(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 i v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: +nat).(\lambda (H3: (eq nat (S i) (s k x1))).(\lambda (H4: (eq C c2 (CHead c k +x0))).(\lambda (H5: (subst0 x1 v t x0)).(eq_ind_r C (CHead c k x0) (\lambda +(c3: C).(or4 (clear c3 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: +C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) +(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c3 +(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: +T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: +B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind +b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: +T).(clear c3 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T +(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda +(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear c3 (CHead e2 +(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda +(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda +(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 +e2))))))))) (K_ind (\lambda (k0: K).((clear (CHead c k0 t) c0) \to ((eq nat +(S i) (s k0 x1)) \to (or4 (clear (CHead c k0 x0) c0) (ex3_4 B C T T (\lambda +(b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e +(Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda +(u2: T).(clear (CHead c k0 x0) (CHead e (Bind b) u2)))))) (\lambda (_: +B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) +(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: +C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead c k0 x0) (CHead e2 (Bind b) +u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind +b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (u2: T).(clear (CHead c k0 x0) (CHead e2 (Bind b) u2))))))) +(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda +(u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda +(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))))))) +(\lambda (b: B).(\lambda (H6: (clear (CHead c (Bind b) t) c0)).(\lambda (H7: +(eq nat (S i) (s (Bind b) x1))).(let H8 \def (f_equal nat nat (\lambda (e: +nat).(match e in nat return (\lambda (_: nat).nat) with [O \Rightarrow i | (S +n) \Rightarrow n])) (S i) (S x1) H7) in (let H9 \def (eq_ind_r nat x1 +(\lambda (n: nat).(subst0 n v t x0)) H5 i H8) in (eq_ind_r C (CHead c (Bind +b) t) (\lambda (c3: C).(or4 (clear (CHead c (Bind b) x0) c3) (ex3_4 B C T T +(\lambda (b0: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c3 +(CHead e (Bind b0) u1)))))) (\lambda (b0: B).(\lambda (e: C).(\lambda (_: +T).(\lambda (u2: T).(clear (CHead c (Bind b) x0) (CHead e (Bind b0) u2)))))) +(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v +u1 u2)))))) (ex3_4 B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(eq C c3 (CHead e1 (Bind b0) u)))))) (\lambda (b0: +B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead c (Bind b) +x0) (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda +(e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda +(b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq +C c3 (CHead e1 (Bind b0) u1))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda +(e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead c (Bind b) x0) (CHead +e2 (Bind b0) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: +C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: +B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 +i v e1 e2))))))))) (or4_intro1 (clear (CHead c (Bind b) x0) (CHead c (Bind b) +t)) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e: C).(\lambda (u1: T).(\lambda +(_: T).(eq C (CHead c (Bind b) t) (CHead e (Bind b0) u1)))))) (\lambda (b0: +B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead c (Bind b) +x0) (CHead e (Bind b0) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: +T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b0: +B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c (Bind b) +t) (CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(clear (CHead c (Bind b) x0) (CHead e2 (Bind b0) u)))))) +(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i +v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e1 +(Bind b0) u1))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead c (Bind b) x0) (CHead e2 +(Bind b0) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda +(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda +(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 +e2))))))) (ex3_4_intro B C T T (\lambda (b0: B).(\lambda (e: C).(\lambda (u1: +T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e (Bind b0) u1)))))) +(\lambda (b0: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear +(CHead c (Bind b) x0) (CHead e (Bind b0) u2)))))) (\lambda (_: B).(\lambda +(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))) b c t x0 +(refl_equal C (CHead c (Bind b) t)) (clear_bind b c x0) H9)) c0 +(clear_gen_bind b c c0 t H6))))))) (\lambda (f: F).(\lambda (H6: (clear +(CHead c (Flat f) t) c0)).(\lambda (H7: (eq nat (S i) (s (Flat f) x1))).(let +H8 \def (f_equal nat nat (\lambda (e: nat).e) (S i) (s (Flat f) x1) H7) in +(let H9 \def (eq_ind_r nat x1 (\lambda (n: nat).(subst0 n v t x0)) H5 (S i) +H8) in (or4_intro0 (clear (CHead c (Flat f) x0) c0) (ex3_4 B C T T (\lambda +(b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e +(Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda +(u2: T).(clear (CHead c (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_: +B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) +(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: +C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead c (Flat f) x0) (CHead e2 +(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda +(_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda +(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 +(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (u2: T).(clear (CHead c (Flat f) x0) (CHead e2 (Bind b) +u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: +T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) +(clear_flat c c0 (clear_gen_flat f c c0 t H6) f x0))))))) k H1 H3) c2 +H4)))))) H2)) (\lambda (H2: (ex3_2 C nat (\lambda (_: C).(\lambda (j: +nat).(eq nat (S i) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 +(CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c +c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S i) (s k +j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda +(c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (or4 (clear c2 c0) (ex3_4 B C +T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 +(CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: +T).(\lambda (u2: T).(clear c2 (CHead e (Bind b) u2)))))) (\lambda (_: +B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) +(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: +C).(\lambda (e2: C).(\lambda (u: T).(clear c2 (CHead e2 (Bind b) u)))))) +(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i +v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) +(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: +C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) +(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 i v e1 e2)))))))) (\lambda (x0: C).(\lambda (x1: +nat).(\lambda (H3: (eq nat (S i) (s k x1))).(\lambda (H4: (eq C c2 (CHead x0 +k t))).(\lambda (H5: (csubst0 x1 v c x0)).(eq_ind_r C (CHead x0 k t) (\lambda +(c3: C).(or4 (clear c3 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: +C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) +(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c3 +(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: +T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: +B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind +b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: +T).(clear c3 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T +(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda +(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear c3 (CHead e2 +(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda +(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda +(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 +e2))))))))) (K_ind (\lambda (k0: K).((clear (CHead c k0 t) c0) \to ((eq nat +(S i) (s k0 x1)) \to (or4 (clear (CHead x0 k0 t) c0) (ex3_4 B C T T (\lambda +(b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e +(Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda +(u2: T).(clear (CHead x0 k0 t) (CHead e (Bind b) u2)))))) (\lambda (_: +B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) +(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: +C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 k0 t) (CHead e2 (Bind b) +u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind +b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (u2: T).(clear (CHead x0 k0 t) (CHead e2 (Bind b) u2))))))) +(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda +(u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda +(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))))))) +(\lambda (b: B).(\lambda (H6: (clear (CHead c (Bind b) t) c0)).(\lambda (H7: +(eq nat (S i) (s (Bind b) x1))).(let H8 \def (f_equal nat nat (\lambda (e: +nat).(match e in nat return (\lambda (_: nat).nat) with [O \Rightarrow i | (S +n) \Rightarrow n])) (S i) (S x1) H7) in (let H9 \def (eq_ind_r nat x1 +(\lambda (n: nat).(csubst0 n v c x0)) H5 i H8) in (eq_ind_r C (CHead c (Bind +b) t) (\lambda (c3: C).(or4 (clear (CHead x0 (Bind b) t) c3) (ex3_4 B C T T +(\lambda (b0: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c3 +(CHead e (Bind b0) u1)))))) (\lambda (b0: B).(\lambda (e: C).(\lambda (_: +T).(\lambda (u2: T).(clear (CHead x0 (Bind b) t) (CHead e (Bind b0) u2)))))) +(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v +u1 u2)))))) (ex3_4 B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(eq C c3 (CHead e1 (Bind b0) u)))))) (\lambda (b0: +B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Bind b) +t) (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: +B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c3 +(CHead e1 (Bind b0) u1))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Bind b) t) (CHead e2 +(Bind b0) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda +(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda +(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 +e2))))))))) (or4_intro2 (clear (CHead x0 (Bind b) t) (CHead c (Bind b) t)) +(ex3_4 B C T T (\lambda (b0: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: +T).(eq C (CHead c (Bind b) t) (CHead e (Bind b0) u1)))))) (\lambda (b0: +B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Bind b) +t) (CHead e (Bind b0) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: +T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b0: +B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c (Bind b) +t) (CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(clear (CHead x0 (Bind b) t) (CHead e2 (Bind b0) u)))))) +(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i +v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e1 +(Bind b0) u1))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Bind b) t) (CHead e2 +(Bind b0) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda +(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda +(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 +e2))))))) (ex3_4_intro B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(eq C (CHead c (Bind b) t) (CHead e1 (Bind b0) u)))))) +(\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear +(CHead x0 (Bind b) t) (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda +(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2))))) b c x0 t +(refl_equal C (CHead c (Bind b) t)) (clear_bind b x0 t) H9)) c0 +(clear_gen_bind b c c0 t H6))))))) (\lambda (f: F).(\lambda (H6: (clear +(CHead c (Flat f) t) c0)).(\lambda (H7: (eq nat (S i) (s (Flat f) x1))).(let +H8 \def (f_equal nat nat (\lambda (e: nat).e) (S i) (s (Flat f) x1) H7) in +(let H9 \def (eq_ind_r nat x1 (\lambda (n: nat).(csubst0 n v c x0)) H5 (S i) +H8) in (let H10 \def (H x0 v i H9 c0 (clear_gen_flat f c c0 t H6)) in +(or4_ind (clear x0 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: +C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) +(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x0 +(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: +T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: +B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind +b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: +T).(clear x0 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T +(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda +(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear x0 (CHead e2 +(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda +(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda +(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 +e2))))))) (or4 (clear (CHead x0 (Flat f) t) c0) (ex3_4 B C T T (\lambda (b: +B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind +b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: +T).(clear (CHead x0 (Flat f) t) (CHead e (Bind b) u2)))))) (\lambda (_: +B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) +(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: +C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Flat f) t) (CHead e2 +(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda +(_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda +(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 +(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) +u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: +T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 +e2)))))))) (\lambda (H11: (clear x0 c0)).(or4_intro0 (clear (CHead x0 (Flat +f) t) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: +T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: +B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) +t) (CHead e (Bind b) u2)))))) (\lambda (_: 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T (\lambda (b: +B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 +(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 +(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda +(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda +(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 +e2)))))))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: +T).(\lambda (H12: (eq C c0 (CHead x3 (Bind x2) x5))).(\lambda (H13: (clear x0 +(CHead x4 (Bind x2) x5))).(\lambda (H14: (csubst0 i v x3 x4)).(or4_intro2 +(clear (CHead x0 (Flat f) t) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: +C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) +(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear +(CHead x0 (Flat f) t) (CHead e (Bind b) u2)))))) (\lambda (_: 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c0 +(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u)))))) +(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i +v e1 e2))))) x2 x3 x4 x5 H12 (clear_flat x0 (CHead x4 (Bind x2) x5) H13 f t) +H14))))))))) H11)) (\lambda (H11: (ex4_5 B C C T T (\lambda (b: B).(\lambda +(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 +(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (u2: T).(clear x0 (CHead e2 (Bind b) u2))))))) (\lambda (_: +B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 +i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))).(ex4_5_ind B C C T T +(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda +(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: 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e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda +(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 +(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) +u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: +T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 +e2)))))))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: +T).(\lambda (x6: T).(\lambda (H12: (eq C c0 (CHead x3 (Bind x2) +x5))).(\lambda (H13: (clear x0 (CHead x4 (Bind x2) x6))).(\lambda (H14: +(subst0 i v x5 x6)).(\lambda (H15: (csubst0 i v x3 x4)).(or4_intro3 (clear +(CHead x0 (Flat f) t) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: +C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) +(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: 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T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: +T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: +B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear +(CHead x0 (Flat f) t) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda +(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 +u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (_: T).(csubst0 i v e1 e2)))))) x2 x3 x4 x5 x6 H12 (clear_flat x0 +(CHead x4 (Bind x2) x6) H13 f t) H14 H15))))))))))) H11)) H10))))))) k H1 H3) +c2 H4)))))) H2)) (\lambda (H2: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: +C).(\lambda (j: nat).(eq nat (S i) (s k j))))) (\lambda (u2: T).(\lambda (c3: +C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda +(_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: +C).(\lambda (j: nat).(csubst0 j v c c3)))))).(ex4_3_ind T C nat (\lambda (_: +T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S i) (s k j))))) (\lambda (u2: +T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda +(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: +T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (or4 (clear c2 +c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda +(_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: +C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e (Bind b) u2)))))) +(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v +u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: +B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c2 (CHead e2 (Bind +b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind +b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) (\lambda (_: +B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 +i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))) (\lambda (x0: T).(\lambda +(x1: C).(\lambda (x2: nat).(\lambda (H3: (eq nat (S i) (s k x2))).(\lambda +(H4: (eq C c2 (CHead x1 k x0))).(\lambda (H5: (subst0 x2 v t x0)).(\lambda +(H6: (csubst0 x2 v c x1)).(eq_ind_r C (CHead x1 k x0) (\lambda (c3: C).(or4 +(clear c3 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: +T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: +B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c3 (CHead e (Bind +b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: +T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) +(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c3 +(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: +B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 +(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (u2: T).(clear c3 (CHead e2 (Bind b) u2))))))) +(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda +(u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda +(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))))) (K_ind +(\lambda (k0: K).((clear (CHead c k0 t) c0) \to ((eq nat (S i) (s k0 x2)) \to +(or4 (clear (CHead x1 k0 x0) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: +C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) +(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear +(CHead x1 k0 x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: +C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T +(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 +(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(clear (CHead x1 k0 x0) (CHead e2 (Bind b) u)))))) +(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i +v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) +(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(u2: T).(clear (CHead x1 k0 x0) (CHead e2 (Bind b) u2))))))) (\lambda (_: +B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 +i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))))))) (\lambda (b: B).(\lambda +(H7: (clear (CHead c (Bind b) t) c0)).(\lambda (H8: (eq nat (S i) (s (Bind b) +x2))).(let H9 \def (f_equal nat nat (\lambda (e: nat).(match e in nat return +(\lambda (_: nat).nat) with [O \Rightarrow i | (S n) \Rightarrow n])) (S i) +(S x2) H8) in (let H10 \def (eq_ind_r nat x2 (\lambda (n: nat).(csubst0 n v c +x1)) H6 i H9) in (let H11 \def (eq_ind_r nat x2 (\lambda (n: nat).(subst0 n v +t x0)) H5 i H9) in (eq_ind_r C (CHead c (Bind b) t) (\lambda (c3: C).(or4 +(clear (CHead x1 (Bind b) x0) c3) (ex3_4 B C T T (\lambda (b0: B).(\lambda +(e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c3 (CHead e (Bind b0) u1)))))) +(\lambda (b0: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear +(CHead x1 (Bind b) x0) (CHead e (Bind b0) u2)))))) (\lambda (_: B).(\lambda +(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C +T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3 +(CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(clear (CHead x1 (Bind b) x0) (CHead e2 (Bind b0) u)))))) +(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i +v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u1: T).(\lambda (_: T).(eq C c3 (CHead e1 (Bind b0) u1))))))) +(\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(u2: T).(clear (CHead x1 (Bind b) x0) (CHead e2 (Bind b0) u2))))))) (\lambda +(_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: +T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))))) (or4_intro3 +(clear (CHead x1 (Bind b) x0) (CHead c (Bind b) t)) (ex3_4 B C T T (\lambda +(b0: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead c (Bind +b) t) (CHead e (Bind b0) u1)))))) (\lambda (b0: B).(\lambda (e: C).(\lambda +(_: T).(\lambda (u2: T).(clear (CHead x1 (Bind b) x0) (CHead e (Bind b0) +u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: +T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b0: B).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c (Bind b) t) (CHead e1 (Bind +b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: +T).(clear (CHead x1 (Bind b) x0) (CHead e2 (Bind b0) u)))))) (\lambda (_: +B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) +(ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda +(u1: T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e1 (Bind b0) +u1))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (u2: T).(clear (CHead x1 (Bind b) x0) (CHead e2 (Bind b0) +u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: +T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) +(ex4_5_intro B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e1 +(Bind b0) u1))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Bind b) x0) (CHead e2 +(Bind b0) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda +(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda +(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 +e2)))))) b c x1 t x0 (refl_equal C (CHead c (Bind b) t)) (clear_bind b x1 x0) +H11 H10)) c0 (clear_gen_bind b c c0 t H7)))))))) (\lambda (f: F).(\lambda +(H7: (clear (CHead c (Flat f) t) c0)).(\lambda (H8: (eq nat (S i) (s (Flat f) +x2))).(let H9 \def (f_equal nat nat (\lambda (e: nat).e) (S i) (s (Flat f) +x2) H8) in (let H10 \def (eq_ind_r nat x2 (\lambda (n: nat).(csubst0 n v c +x1)) H6 (S i) H9) in (let H11 \def (eq_ind_r nat x2 (\lambda (n: nat).(subst0 +n v t x0)) H5 (S i) H9) in (let H12 \def (H x1 v i H10 c0 (clear_gen_flat f c +c0 t H7)) in (or4_ind (clear x1 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda +(e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) +(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x1 +(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: +T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: +B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind +b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: +T).(clear x1 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T +(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda +(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: +C).(\lambda (e2: C).(\lambda (_: 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(\lambda (b: B).(\lambda +(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 +(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) +u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: +T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 +e2)))))))) (\lambda (H13: (clear x1 c0)).(or4_intro0 (clear (CHead x1 (Flat +f) x0) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: +T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: +B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) +x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: +T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: +B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind +b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: +T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u)))))) (\lambda (_: +B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) +(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda +(u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: +B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear +(CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda +(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 +u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (clear_flat x1 c0 H13 f x0))) +(\lambda (H13: (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: +T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: +B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x1 (CHead e (Bind +b) u2)))))) 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(_: T).(csubst0 i v e1 +e2))))))) (ex3_4_intro B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: +T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: +B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) +x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: +T).(\lambda (u2: T).(subst0 i v u1 u2))))) x3 x4 x5 x6 H14 (clear_flat x1 +(CHead x4 (Bind x3) x6) H15 f x0) H16))))))))) H13)) (\lambda (H13: (ex3_4 B +C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C +c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(clear x1 (CHead e2 (Bind b) u)))))) (\lambda (_: +B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 +e2))))))).(ex3_4_ind B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: +B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x1 (CHead e2 (Bind +b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(csubst0 i v e1 e2))))) (or4 (clear (CHead x1 (Flat f) x0) c0) (ex3_4 B C +T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 +(CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: +T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) +(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v +u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: +B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x1 (Flat f) +x0) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: +B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 +(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 +(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda +(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda +(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 +e2)))))))) (\lambda (x3: B).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: +T).(\lambda (H14: (eq C c0 (CHead x4 (Bind x3) x6))).(\lambda (H15: (clear x1 +(CHead x5 (Bind x3) x6))).(\lambda (H16: (csubst0 i v x4 x5)).(or4_intro2 +(clear (CHead x1 (Flat f) x0) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: +C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) +(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear +(CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda +(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C +T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 +(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u)))))) +(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i +v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) +(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda +(_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: +T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (ex3_4_intro B C +C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 +(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u)))))) +(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i +v e1 e2))))) x3 x4 x5 x6 H14 (clear_flat x1 (CHead x5 (Bind x3) x6) H15 f x0) +H16))))))))) H13)) (\lambda (H13: (ex4_5 B C C T T (\lambda (b: B).(\lambda +(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 +(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (u2: T).(clear x1 (CHead e2 (Bind b) u2))))))) (\lambda (_: +B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 +i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))).(ex4_5_ind B C C T T +(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda +(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear x1 (CHead e2 +(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda +(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda +(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 +e2)))))) (or4 (clear (CHead x1 (Flat f) x0) c0) (ex3_4 B C T T (\lambda (b: +B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind +b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: +T).(clear (CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_: +B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) +(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: +C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2 +(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda +(_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda +(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 +(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) +u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: +T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 +e2)))))))) (\lambda (x3: B).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: +T).(\lambda (x7: T).(\lambda (H14: (eq C c0 (CHead x4 (Bind x3) +x6))).(\lambda (H15: (clear x1 (CHead x5 (Bind x3) x7))).(\lambda (H16: +(subst0 i v x6 x7)).(\lambda (H17: (csubst0 i v x4 x5)).(or4_intro3 (clear +(CHead x1 (Flat f) x0) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: +C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) +(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear +(CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda +(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C +T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 +(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u)))))) +(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i +v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) +(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda +(_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: +T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (ex4_5_intro B C +C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: +T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: +B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear +(CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda +(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 +u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (_: T).(csubst0 i v e1 e2)))))) x3 x4 x5 x6 x7 H14 (clear_flat x1 +(CHead x5 (Bind x3) x7) H15 f x0) H16 H17))))))))))) H13)) H12)))))))) k H1 +H3) c2 H4)))))))) H2)) (csubst0_gen_head k c c2 t v (S i) H0)))))))))))) c1). +(* COMMENTS +Initial nodes: 14968 +END *) + +theorem csubst0_clear_trans: + \forall (c1: C).(\forall (c2: C).(\forall (v: T).(\forall (i: nat).((csubst0 +i v c1 c2) \to (\forall (e2: C).((clear c2 e2) \to (or (clear c1 e2) (ex2 C +(\lambda (e1: C).(csubst0 i v e1 e2)) (\lambda (e1: C).(clear c1 e1)))))))))) +\def + \lambda (c1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda (i: nat).(\lambda +(H: (csubst0 i v c1 c2)).(csubst0_ind (\lambda (n: nat).(\lambda (t: +T).(\lambda (c: C).(\lambda (c0: C).(\forall (e2: C).((clear c0 e2) \to (or +(clear c e2) (ex2 C (\lambda (e1: C).(csubst0 n t e1 e2)) (\lambda (e1: +C).(clear c e1)))))))))) (\lambda (k: K).(\lambda (i0: nat).(\lambda (v0: +T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H0: (subst0 i0 v0 u1 +u2)).(\lambda (c: C).(\lambda (e2: C).(\lambda (H1: (clear (CHead c k u2) +e2)).(K_ind (\lambda (k0: K).((clear (CHead c k0 u2) e2) \to (or (clear +(CHead c k0 u1) e2) (ex2 C (\lambda (e1: C).(csubst0 (s k0 i0) v0 e1 e2)) +(\lambda (e1: C).(clear (CHead c k0 u1) e1)))))) (\lambda (b: B).(\lambda +(H2: (clear (CHead c (Bind b) u2) e2)).(eq_ind_r C (CHead c (Bind b) u2) +(\lambda (c0: C).(or (clear (CHead c (Bind b) u1) c0) (ex2 C (\lambda (e1: +C).(csubst0 (s (Bind b) i0) v0 e1 c0)) (\lambda (e1: C).(clear (CHead c (Bind +b) u1) e1))))) (or_intror (clear (CHead c (Bind b) u1) (CHead c (Bind b) u2)) +(ex2 C (\lambda (e1: C).(csubst0 (S i0) v0 e1 (CHead c (Bind b) u2))) +(\lambda (e1: C).(clear (CHead c (Bind b) u1) e1))) (ex_intro2 C (\lambda +(e1: C).(csubst0 (S i0) v0 e1 (CHead c (Bind b) u2))) (\lambda (e1: C).(clear +(CHead c (Bind b) u1) e1)) (CHead c (Bind b) u1) (csubst0_snd_bind b i0 v0 u1 +u2 H0 c) (clear_bind b c u1))) e2 (clear_gen_bind b c e2 u2 H2)))) (\lambda +(f: F).(\lambda (H2: (clear (CHead c (Flat f) u2) e2)).(or_introl (clear +(CHead c (Flat f) u1) e2) (ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2)) +(\lambda (e1: C).(clear (CHead c (Flat f) u1) e1))) (clear_flat c e2 +(clear_gen_flat f c e2 u2 H2) f u1)))) k H1)))))))))) (\lambda (k: +K).(\lambda (i0: nat).(\lambda (c3: C).(\lambda (c4: C).(\lambda (v0: +T).(\lambda (H0: (csubst0 i0 v0 c3 c4)).(\lambda (H1: ((\forall (e2: +C).((clear c4 e2) \to (or (clear c3 e2) (ex2 C (\lambda (e1: C).(csubst0 i0 +v0 e1 e2)) (\lambda (e1: C).(clear c3 e1)))))))).(\lambda (u: T).(\lambda +(e2: C).(\lambda (H2: (clear (CHead c4 k u) e2)).(K_ind (\lambda (k0: +K).((clear (CHead c4 k0 u) e2) \to (or (clear (CHead c3 k0 u) e2) (ex2 C +(\lambda (e1: C).(csubst0 (s k0 i0) v0 e1 e2)) (\lambda (e1: C).(clear (CHead +c3 k0 u) e1)))))) (\lambda (b: B).(\lambda (H3: (clear (CHead c4 (Bind b) u) +e2)).(eq_ind_r C (CHead c4 (Bind b) u) (\lambda (c: C).(or (clear (CHead c3 +(Bind b) u) c) (ex2 C (\lambda (e1: C).(csubst0 (s (Bind b) i0) v0 e1 c)) +(\lambda (e1: C).(clear (CHead c3 (Bind b) u) e1))))) (or_intror (clear +(CHead c3 (Bind b) u) (CHead c4 (Bind b) u)) (ex2 C (\lambda (e1: C).(csubst0 +(S i0) v0 e1 (CHead c4 (Bind b) u))) (\lambda (e1: C).(clear (CHead c3 (Bind +b) u) e1))) (ex_intro2 C (\lambda (e1: C).(csubst0 (S i0) v0 e1 (CHead c4 +(Bind b) u))) (\lambda (e1: C).(clear (CHead c3 (Bind b) u) e1)) (CHead c3 +(Bind b) u) (csubst0_fst_bind b i0 c3 c4 v0 H0 u) (clear_bind b c3 u))) e2 +(clear_gen_bind b c4 e2 u H3)))) (\lambda (f: F).(\lambda (H3: (clear (CHead +c4 (Flat f) u) e2)).(let H_x \def (H1 e2 (clear_gen_flat f c4 e2 u H3)) in +(let H4 \def H_x in (or_ind (clear c3 e2) (ex2 C (\lambda (e1: C).(csubst0 i0 +v0 e1 e2)) (\lambda (e1: C).(clear c3 e1))) (or (clear (CHead c3 (Flat f) u) +e2) (ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear +(CHead c3 (Flat f) u) e1)))) (\lambda (H5: (clear c3 e2)).(or_introl (clear +(CHead c3 (Flat f) u) e2) (ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2)) +(\lambda (e1: C).(clear (CHead c3 (Flat f) u) e1))) (clear_flat c3 e2 H5 f +u))) (\lambda (H5: (ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda +(e1: C).(clear c3 e1)))).(ex2_ind C (\lambda (e1: C).(csubst0 i0 v0 e1 e2)) +(\lambda (e1: C).(clear c3 e1)) (or (clear (CHead c3 (Flat f) u) e2) (ex2 C +(\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear (CHead c3 +(Flat f) u) e1)))) (\lambda (x: C).(\lambda (H6: (csubst0 i0 v0 x +e2)).(\lambda (H7: (clear c3 x)).(or_intror (clear (CHead c3 (Flat f) u) e2) +(ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear (CHead +c3 (Flat f) u) e1))) (ex_intro2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2)) +(\lambda (e1: C).(clear (CHead c3 (Flat f) u) e1)) x H6 (clear_flat c3 x H7 f +u)))))) H5)) H4))))) k H2))))))))))) (\lambda (k: K).(\lambda (i0: +nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H0: (subst0 +i0 v0 u1 u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (H1: (csubst0 i0 v0 +c3 c4)).(\lambda (H2: ((\forall (e2: C).((clear c4 e2) \to (or (clear c3 e2) +(ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear c3 +e1)))))))).(\lambda (e2: C).(\lambda (H3: (clear (CHead c4 k u2) e2)).(K_ind +(\lambda (k0: K).((clear (CHead c4 k0 u2) e2) \to (or (clear (CHead c3 k0 u1) +e2) (ex2 C (\lambda (e1: C).(csubst0 (s k0 i0) v0 e1 e2)) (\lambda (e1: +C).(clear (CHead c3 k0 u1) e1)))))) (\lambda (b: B).(\lambda (H4: (clear +(CHead c4 (Bind b) u2) e2)).(eq_ind_r C (CHead c4 (Bind b) u2) (\lambda (c: +C).(or (clear (CHead c3 (Bind b) u1) c) (ex2 C (\lambda (e1: C).(csubst0 (s +(Bind b) i0) v0 e1 c)) (\lambda (e1: C).(clear (CHead c3 (Bind b) u1) e1))))) +(or_intror (clear (CHead c3 (Bind b) u1) (CHead c4 (Bind b) u2)) (ex2 C +(\lambda (e1: C).(csubst0 (S i0) v0 e1 (CHead c4 (Bind b) u2))) (\lambda (e1: +C).(clear (CHead c3 (Bind b) u1) e1))) (ex_intro2 C (\lambda (e1: C).(csubst0 +(S i0) v0 e1 (CHead c4 (Bind b) u2))) (\lambda (e1: C).(clear (CHead c3 (Bind +b) u1) e1)) (CHead c3 (Bind b) u1) (csubst0_both_bind b i0 v0 u1 u2 H0 c3 c4 +H1) (clear_bind b c3 u1))) e2 (clear_gen_bind b c4 e2 u2 H4)))) (\lambda (f: +F).(\lambda (H4: (clear (CHead c4 (Flat f) u2) e2)).(let H_x \def (H2 e2 +(clear_gen_flat f c4 e2 u2 H4)) in (let H5 \def H_x in (or_ind (clear c3 e2) +(ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear c3 +e1))) (or (clear (CHead c3 (Flat f) u1) e2) (ex2 C (\lambda (e1: C).(csubst0 +i0 v0 e1 e2)) (\lambda (e1: C).(clear (CHead c3 (Flat f) u1) e1)))) (\lambda +(H6: (clear c3 e2)).(or_introl (clear (CHead c3 (Flat f) u1) e2) (ex2 C +(\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear (CHead c3 +(Flat f) u1) e1))) (clear_flat c3 e2 H6 f u1))) (\lambda (H6: (ex2 C (\lambda +(e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear c3 e1)))).(ex2_ind C +(\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear c3 e1)) (or +(clear (CHead c3 (Flat f) u1) e2) (ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 +e2)) (\lambda (e1: C).(clear (CHead c3 (Flat f) u1) e1)))) (\lambda (x: +C).(\lambda (H7: (csubst0 i0 v0 x e2)).(\lambda (H8: (clear c3 x)).(or_intror +(clear (CHead c3 (Flat f) u1) e2) (ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 +e2)) (\lambda (e1: C).(clear (CHead c3 (Flat f) u1) e1))) (ex_intro2 C +(\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear (CHead c3 +(Flat f) u1) e1)) x H7 (clear_flat c3 x H8 f u1)))))) H6)) H5))))) k +H3))))))))))))) i v c1 c2 H))))). +(* COMMENTS +Initial nodes: 2085 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubst0/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/csubst0/defs.ma new file mode 100644 index 000000000..0068e19f6 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/csubst0/defs.ma @@ -0,0 +1,32 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/subst0/defs.ma". + +include "Basic-1/C/defs.ma". + +inductive csubst0: nat \to (T \to (C \to (C \to Prop))) \def +| csubst0_snd: \forall (k: K).(\forall (i: nat).(\forall (v: T).(\forall (u1: +T).(\forall (u2: T).((subst0 i v u1 u2) \to (\forall (c: C).(csubst0 (s k i) +v (CHead c k u1) (CHead c k u2)))))))) +| csubst0_fst: \forall (k: K).(\forall (i: nat).(\forall (c1: C).(\forall +(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (u: T).(csubst0 (s +k i) v (CHead c1 k u) (CHead c2 k u)))))))) +| csubst0_both: \forall (k: K).(\forall (i: nat).(\forall (v: T).(\forall +(u1: T).(\forall (u2: T).((subst0 i v u1 u2) \to (\forall (c1: C).(\forall +(c2: C).((csubst0 i v c1 c2) \to (csubst0 (s k i) v (CHead c1 k u1) (CHead c2 +k u2)))))))))). + diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubst0/drop.ma b/matita/matita/contribs/lambdadelta/basic_1/csubst0/drop.ma new file mode 100644 index 000000000..b6bc6cae7 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/csubst0/drop.ma @@ -0,0 +1,6294 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/csubst0/fwd.ma". + +include "Basic-1/drop/fwd.ma". + +include "Basic-1/s/props.ma". + +theorem csubst0_drop_gt: + \forall (n: nat).(\forall (i: nat).((lt i n) \to (\forall (c1: C).(\forall +(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n O +c1 e) \to (drop n O c2 e))))))))) +\def + \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (i: nat).((lt i n0) +\to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) +\to (\forall (e: C).((drop n0 O c1 e) \to (drop n0 O c2 e)))))))))) (\lambda +(i: nat).(\lambda (H: (lt i O)).(\lambda (c1: C).(\lambda (c2: C).(\lambda +(v: T).(\lambda (_: (csubst0 i v c1 c2)).(\lambda (e: C).(\lambda (_: (drop O +O c1 e)).(lt_x_O i H (drop O O c2 e)))))))))) (\lambda (n0: nat).(\lambda (H: +((\forall (i: nat).((lt i n0) \to (\forall (c1: C).(\forall (c2: C).(\forall +(v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n0 O c1 e) \to (drop +n0 O c2 e))))))))))).(\lambda (i: nat).(\lambda (H0: (lt i (S n0))).(\lambda +(c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v +c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (drop (S n0) O c2 e))))))) +(\lambda (n1: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (_: (csubst0 i v +(CSort n1) c2)).(\lambda (e: C).(\lambda (H2: (drop (S n0) O (CSort n1) +e)).(and3_ind (eq C e (CSort n1)) (eq nat (S n0) O) (eq nat O O) (drop (S n0) +O c2 e) (\lambda (H3: (eq C e (CSort n1))).(\lambda (H4: (eq nat (S n0) +O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n1) (\lambda (c: C).(drop +(S n0) O c2 c)) (let H6 \def (eq_ind nat (S n0) (\lambda (ee: nat).(match ee +in nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) +\Rightarrow True])) I O H4) in (False_ind (drop (S n0) O c2 (CSort n1)) H6)) +e H3)))) (drop_gen_sort n1 (S n0) O e H2)))))))) (\lambda (c: C).(\lambda +(H1: ((\forall (c2: C).(\forall (v: T).((csubst0 i v c c2) \to (\forall (e: +C).((drop (S n0) O c e) \to (drop (S n0) O c2 e)))))))).(\lambda (k: +K).(\lambda (t: T).(\lambda (c2: C).(\lambda (v: T).(\lambda (H2: (csubst0 i +v (CHead c k t) c2)).(\lambda (e: C).(\lambda (H3: (drop (S n0) O (CHead c k +t) e)).(or3_ind (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k +j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda +(u2: T).(\lambda (j: nat).(subst0 j v t u2)))) (ex3_2 C nat (\lambda (_: +C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_: +nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j +v c c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: +nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: +nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda +(j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: +nat).(csubst0 j v c c3))))) (drop (S n0) O c2 e) (\lambda (H4: (ex3_2 T nat +(\lambda (_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: +T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: +nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: +nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead +c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))) (drop (S +n0) O c2 e) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H5: (eq nat i (s k +x1))).(\lambda (H6: (eq C c2 (CHead c k x0))).(\lambda (_: (subst0 x1 v t +x0)).(eq_ind_r C (CHead c k x0) (\lambda (c0: C).(drop (S n0) O c0 e)) (let +H8 \def (eq_ind nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0: +T).((csubst0 n1 v0 c c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (drop +(S n0) O c3 e0))))))) H1 (s k x1) H5) in (let H9 \def (eq_ind nat i (\lambda +(n1: nat).(lt n1 (S n0))) H0 (s k x1) H5) in (K_ind (\lambda (k0: K).((drop +(r k0 n0) O c e) \to (((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1) +v0 c c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3 +e0))))))) \to ((lt (s k0 x1) (S n0)) \to (drop (S n0) O (CHead c k0 x0) +e))))) (\lambda (b: B).(\lambda (H10: (drop (r (Bind b) n0) O c e)).(\lambda +(_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to +(\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3 +e0)))))))).(\lambda (_: (lt (s (Bind b) x1) (S n0))).(drop_drop (Bind b) n0 c +e H10 x0))))) (\lambda (f: F).(\lambda (H10: (drop (r (Flat f) n0) O c +e)).(\lambda (_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Flat f) x1) +v0 c c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3 +e0)))))))).(\lambda (H12: (lt (s (Flat f) x1) (S n0))).(or_ind (eq nat x1 O) +(ex2 nat (\lambda (m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0))) +(drop (S n0) O (CHead c (Flat f) x0) e) (\lambda (_: (eq nat x1 +O)).(drop_drop (Flat f) n0 c e H10 x0)) (\lambda (H13: (ex2 nat (\lambda (m: +nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda +(m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O +(CHead c (Flat f) x0) e) (\lambda (x: nat).(\lambda (_: (eq nat x1 (S +x))).(\lambda (_: (lt x n0)).(drop_drop (Flat f) n0 c e H10 x0)))) H13)) +(lt_gen_xS x1 n0 H12)))))) k (drop_gen_drop k c e t n0 H3) H8 H9))) c2 +H6)))))) H4)) (\lambda (H4: (ex3_2 C nat (\lambda (_: C).(\lambda (j: +nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead +c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c +c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k +j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda +(c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (drop (S n0) O c2 e) (\lambda +(x0: C).(\lambda (x1: nat).(\lambda (H5: (eq nat i (s k x1))).(\lambda (H6: +(eq C c2 (CHead x0 k t))).(\lambda (H7: (csubst0 x1 v c x0)).(eq_ind_r C +(CHead x0 k t) (\lambda (c0: C).(drop (S n0) O c0 e)) (let H8 \def (eq_ind +nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c +c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3 +e0))))))) H1 (s k x1) H5) in (let H9 \def (eq_ind nat i (\lambda (n1: +nat).(lt n1 (S n0))) H0 (s k x1) H5) in (K_ind (\lambda (k0: K).((drop (r k0 +n0) O c e) \to (((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c +c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3 +e0))))))) \to ((lt (s k0 x1) (S n0)) \to (drop (S n0) O (CHead x0 k0 t) +e))))) (\lambda (b: B).(\lambda (H10: (drop (r (Bind b) n0) O c e)).(\lambda +(_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to +(\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3 +e0)))))))).(\lambda (H12: (lt (s (Bind b) x1) (S n0))).(drop_drop (Bind b) n0 +x0 e (H x1 (lt_S_n x1 n0 H12) c x0 v H7 e H10) t))))) (\lambda (f: +F).(\lambda (H10: (drop (r (Flat f) n0) O c e)).(\lambda (H11: ((\forall (c3: +C).(\forall (v0: T).((csubst0 (s (Flat f) x1) v0 c c3) \to (\forall (e0: +C).((drop (S n0) O c e0) \to (drop (S n0) O c3 e0)))))))).(\lambda (H12: (lt +(s (Flat f) x1) (S n0))).(or_ind (eq nat x1 O) (ex2 nat (\lambda (m: nat).(eq +nat x1 (S m))) (\lambda (m: nat).(lt m n0))) (drop (S n0) O (CHead x0 (Flat +f) t) e) (\lambda (_: (eq nat x1 O)).(drop_drop (Flat f) n0 x0 e (H11 x0 v H7 +e H10) t)) (\lambda (H13: (ex2 nat (\lambda (m: nat).(eq nat x1 (S m))) +(\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda (m: nat).(eq nat x1 (S +m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O (CHead x0 (Flat f) t) e) +(\lambda (x: nat).(\lambda (_: (eq nat x1 (S x))).(\lambda (_: (lt x +n0)).(drop_drop (Flat f) n0 x0 e (H11 x0 v H7 e H10) t)))) H13)) (lt_gen_xS +x1 n0 H12)))))) k (drop_gen_drop k c e t n0 H3) H8 H9))) c2 H6)))))) H4)) +(\lambda (H4: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: +nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: +nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda +(j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: +nat).(csubst0 j v c c3)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_: +C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: +C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda +(_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: +C).(\lambda (j: nat).(csubst0 j v c c3)))) (drop (S n0) O c2 e) (\lambda (x0: +T).(\lambda (x1: C).(\lambda (x2: nat).(\lambda (H5: (eq nat i (s k +x2))).(\lambda (H6: (eq C c2 (CHead x1 k x0))).(\lambda (_: (subst0 x2 v t +x0)).(\lambda (H8: (csubst0 x2 v c x1)).(eq_ind_r C (CHead x1 k x0) (\lambda +(c0: C).(drop (S n0) O c0 e)) (let H9 \def (eq_ind nat i (\lambda (n1: +nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall +(e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3 e0))))))) H1 (s k x2) H5) +in (let H10 \def (eq_ind nat i (\lambda (n1: nat).(lt n1 (S n0))) H0 (s k x2) +H5) in (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to (((\forall (c3: +C).(\forall (v0: T).((csubst0 (s k0 x2) v0 c c3) \to (\forall (e0: C).((drop +(S n0) O c e0) \to (drop (S n0) O c3 e0))))))) \to ((lt (s k0 x2) (S n0)) \to +(drop (S n0) O (CHead x1 k0 x0) e))))) (\lambda (b: B).(\lambda (H11: (drop +(r (Bind b) n0) O c e)).(\lambda (_: ((\forall (c3: C).(\forall (v0: +T).((csubst0 (s (Bind b) x2) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c +e0) \to (drop (S n0) O c3 e0)))))))).(\lambda (H13: (lt (s (Bind b) x2) (S +n0))).(drop_drop (Bind b) n0 x1 e (H x2 (lt_S_n x2 n0 H13) c x1 v H8 e H11) +x0))))) (\lambda (f: F).(\lambda (H11: (drop (r (Flat f) n0) O c e)).(\lambda +(H12: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Flat f) x2) v0 c c3) +\to (\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3 +e0)))))))).(\lambda (H13: (lt (s (Flat f) x2) (S n0))).(or_ind (eq nat x2 O) +(ex2 nat (\lambda (m: nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m n0))) +(drop (S n0) O (CHead x1 (Flat f) x0) e) (\lambda (_: (eq nat x2 +O)).(drop_drop (Flat f) n0 x1 e (H12 x1 v H8 e H11) x0)) (\lambda (H14: (ex2 +nat (\lambda (m: nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m +n0)))).(ex2_ind nat (\lambda (m: nat).(eq nat x2 (S m))) (\lambda (m: +nat).(lt m n0)) (drop (S n0) O (CHead x1 (Flat f) x0) e) (\lambda (x: +nat).(\lambda (_: (eq nat x2 (S x))).(\lambda (_: (lt x n0)).(drop_drop (Flat +f) n0 x1 e (H12 x1 v H8 e H11) x0)))) H14)) (lt_gen_xS x2 n0 H13)))))) k +(drop_gen_drop k c e t n0 H3) H9 H10))) c2 H6)))))))) H4)) (csubst0_gen_head +k c c2 t v i H2))))))))))) c1)))))) n). +(* COMMENTS +Initial nodes: 3092 +END *) + +theorem csubst0_drop_gt_back: + \forall (n: nat).(\forall (i: nat).((lt i n) \to (\forall (c1: C).(\forall +(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n O +c2 e) \to (drop n O c1 e))))))))) +\def + \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (i: nat).((lt i n0) +\to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) +\to (\forall (e: C).((drop n0 O c2 e) \to (drop n0 O c1 e)))))))))) (\lambda +(i: nat).(\lambda (H: (lt i O)).(\lambda (c1: C).(\lambda (c2: C).(\lambda +(v: T).(\lambda (_: (csubst0 i v c1 c2)).(\lambda (e: C).(\lambda (_: (drop O +O c2 e)).(lt_x_O i H (drop O O c1 e)))))))))) (\lambda (n0: nat).(\lambda (H: +((\forall (i: nat).((lt i n0) \to (\forall (c1: C).(\forall (c2: C).(\forall +(v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n0 O c2 e) \to (drop +n0 O c1 e))))))))))).(\lambda (i: nat).(\lambda (H0: (lt i (S n0))).(\lambda +(c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v +c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c e))))))) +(\lambda (n1: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (H1: (csubst0 i +v (CSort n1) c2)).(\lambda (e: C).(\lambda (_: (drop (S n0) O c2 +e)).(csubst0_gen_sort c2 v i n1 H1 (drop (S n0) O (CSort n1) e)))))))) +(\lambda (c: C).(\lambda (H1: ((\forall (c2: C).(\forall (v: T).((csubst0 i v +c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c +e)))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (v: +T).(\lambda (H2: (csubst0 i v (CHead c k t) c2)).(\lambda (e: C).(\lambda +(H3: (drop (S n0) O c2 e)).(or3_ind (ex3_2 T nat (\lambda (_: T).(\lambda (j: +nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead +c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2)))) (ex3_2 C +nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: +C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: +nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: +C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: +C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda +(_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: +C).(\lambda (j: nat).(csubst0 j v c c3))))) (drop (S n0) O (CHead c k t) e) +(\lambda (H4: (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k +j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda +(u2: T).(\lambda (j: nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: +T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: +nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j +v t u2))) (drop (S n0) O (CHead c k t) e) (\lambda (x0: T).(\lambda (x1: +nat).(\lambda (H5: (eq nat i (s k x1))).(\lambda (H6: (eq C c2 (CHead c k +x0))).(\lambda (_: (subst0 x1 v t x0)).(let H8 \def (eq_ind C c2 (\lambda +(c0: C).(drop (S n0) O c0 e)) H3 (CHead c k x0) H6) in (let H9 \def (eq_ind +nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c +c3) \to (\forall (e0: C).((drop (S n0) O c3 e0) \to (drop (S n0) O c +e0))))))) H1 (s k x1) H5) in (let H10 \def (eq_ind nat i (\lambda (n1: +nat).(lt n1 (S n0))) H0 (s k x1) H5) in (K_ind (\lambda (k0: K).(((\forall +(c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c c3) \to (\forall (e0: +C).((drop (S n0) O c3 e0) \to (drop (S n0) O c e0))))))) \to ((lt (s k0 x1) +(S n0)) \to ((drop (r k0 n0) O c e) \to (drop (S n0) O (CHead c k0 t) e))))) +(\lambda (b: B).(\lambda (_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s +(Bind b) x1) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3 e0) \to (drop +(S n0) O c e0)))))))).(\lambda (_: (lt (s (Bind b) x1) (S n0))).(\lambda +(H13: (drop (r (Bind b) n0) O c e)).(drop_drop (Bind b) n0 c e H13 t))))) +(\lambda (f: F).(\lambda (_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s +(Flat f) x1) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3 e0) \to (drop +(S n0) O c e0)))))))).(\lambda (H12: (lt (s (Flat f) x1) (S n0))).(\lambda +(H13: (drop (r (Flat f) n0) O c e)).(or_ind (eq nat x1 O) (ex2 nat (\lambda +(m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0))) (drop (S n0) O +(CHead c (Flat f) t) e) (\lambda (_: (eq nat x1 O)).(drop_drop (Flat f) n0 c +e H13 t)) (\lambda (H14: (ex2 nat (\lambda (m: nat).(eq nat x1 (S m))) +(\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda (m: nat).(eq nat x1 (S +m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O (CHead c (Flat f) t) e) +(\lambda (x: nat).(\lambda (_: (eq nat x1 (S x))).(\lambda (_: (lt x +n0)).(drop_drop (Flat f) n0 c e H13 t)))) H14)) (lt_gen_xS x1 n0 H12)))))) k +H9 H10 (drop_gen_drop k c e x0 n0 H8)))))))))) H4)) (\lambda (H4: (ex3_2 C +nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: +C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: +nat).(csubst0 j v c c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: +nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead +c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (drop (S +n0) O (CHead c k t) e) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H5: (eq +nat i (s k x1))).(\lambda (H6: (eq C c2 (CHead x0 k t))).(\lambda (H7: +(csubst0 x1 v c x0)).(let H8 \def (eq_ind C c2 (\lambda (c0: C).(drop (S n0) +O c0 e)) H3 (CHead x0 k t) H6) in (let H9 \def (eq_ind nat i (\lambda (n1: +nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall +(e0: C).((drop (S n0) O c3 e0) \to (drop (S n0) O c e0))))))) H1 (s k x1) H5) +in (let H10 \def (eq_ind nat i (\lambda (n1: nat).(lt n1 (S n0))) H0 (s k x1) +H5) in (K_ind (\lambda (k0: K).(((\forall (c3: C).(\forall (v0: T).((csubst0 +(s k0 x1) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3 e0) \to (drop (S +n0) O c e0))))))) \to ((lt (s k0 x1) (S n0)) \to ((drop (r k0 n0) O x0 e) \to +(drop (S n0) O (CHead c k0 t) e))))) (\lambda (b: B).(\lambda (_: ((\forall +(c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to (\forall (e0: +C).((drop (S n0) O c3 e0) \to (drop (S n0) O c e0)))))))).(\lambda (H12: (lt +(s (Bind b) x1) (S n0))).(\lambda (H13: (drop (r (Bind b) n0) O x0 +e)).(drop_drop (Bind b) n0 c e (H x1 (lt_S_n x1 n0 H12) c x0 v H7 e H13) +t))))) (\lambda (f: F).(\lambda (H11: ((\forall (c3: C).(\forall (v0: +T).((csubst0 (s (Flat f) x1) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3 +e0) \to (drop (S n0) O c e0)))))))).(\lambda (H12: (lt (s (Flat f) x1) (S +n0))).(\lambda (H13: (drop (r (Flat f) n0) O x0 e)).(or_ind (eq nat x1 O) +(ex2 nat (\lambda (m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0))) +(drop (S n0) O (CHead c (Flat f) t) e) (\lambda (_: (eq nat x1 O)).(drop_drop +(Flat f) n0 c e (H11 x0 v H7 e H13) t)) (\lambda (H14: (ex2 nat (\lambda (m: +nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda +(m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O +(CHead c (Flat f) t) e) (\lambda (x: nat).(\lambda (_: (eq nat x1 (S +x))).(\lambda (_: (lt x n0)).(drop_drop (Flat f) n0 c e (H11 x0 v H7 e H13) +t)))) H14)) (lt_gen_xS x1 n0 H12)))))) k H9 H10 (drop_gen_drop k x0 e t n0 +H8)))))))))) H4)) (\lambda (H4: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: +C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: +C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda +(_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: +C).(\lambda (j: nat).(csubst0 j v c c3)))))).(ex4_3_ind T C nat (\lambda (_: +T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: +T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda +(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: +T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (drop (S n0) O +(CHead c k t) e) (\lambda (x0: T).(\lambda (x1: C).(\lambda (x2: +nat).(\lambda (H5: (eq nat i (s k x2))).(\lambda (H6: (eq C c2 (CHead x1 k +x0))).(\lambda (_: (subst0 x2 v t x0)).(\lambda (H8: (csubst0 x2 v c +x1)).(let H9 \def (eq_ind C c2 (\lambda (c0: C).(drop (S n0) O c0 e)) H3 +(CHead x1 k x0) H6) in (let H10 \def (eq_ind nat i (\lambda (n1: +nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall +(e0: C).((drop (S n0) O c3 e0) \to (drop (S n0) O c e0))))))) H1 (s k x2) H5) +in (let H11 \def (eq_ind nat i (\lambda (n1: nat).(lt n1 (S n0))) H0 (s k x2) +H5) in (K_ind (\lambda (k0: K).(((\forall (c3: C).(\forall (v0: T).((csubst0 +(s k0 x2) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3 e0) \to (drop (S +n0) O c e0))))))) \to ((lt (s k0 x2) (S n0)) \to ((drop (r k0 n0) O x1 e) \to +(drop (S n0) O (CHead c k0 t) e))))) (\lambda (b: B).(\lambda (_: ((\forall +(c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x2) v0 c c3) \to (\forall (e0: +C).((drop (S n0) O c3 e0) \to (drop (S n0) O c e0)))))))).(\lambda (H13: (lt +(s (Bind b) x2) (S n0))).(\lambda (H14: (drop (r (Bind b) n0) O x1 +e)).(drop_drop (Bind b) n0 c e (H x2 (lt_S_n x2 n0 H13) c x1 v H8 e H14) +t))))) (\lambda (f: F).(\lambda (H12: ((\forall (c3: C).(\forall (v0: +T).((csubst0 (s (Flat f) x2) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3 +e0) \to (drop (S n0) O c e0)))))))).(\lambda (H13: (lt (s (Flat f) x2) (S +n0))).(\lambda (H14: (drop (r (Flat f) n0) O x1 e)).(or_ind (eq nat x2 O) +(ex2 nat (\lambda (m: nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m n0))) +(drop (S n0) O (CHead c (Flat f) t) e) (\lambda (_: (eq nat x2 O)).(drop_drop +(Flat f) n0 c e (H12 x1 v H8 e H14) t)) (\lambda (H15: (ex2 nat (\lambda (m: +nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda +(m: nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O +(CHead c (Flat f) t) e) (\lambda (x: nat).(\lambda (_: (eq nat x2 (S +x))).(\lambda (_: (lt x n0)).(drop_drop (Flat f) n0 c e (H12 x1 v H8 e H14) +t)))) H15)) (lt_gen_xS x2 n0 H13)))))) k H10 H11 (drop_gen_drop k x1 e x0 n0 +H9)))))))))))) H4)) (csubst0_gen_head k c c2 t v i H2))))))))))) c1)))))) n). +(* COMMENTS +Initial nodes: 2989 +END *) + +theorem csubst0_drop_lt: + \forall (n: nat).(\forall (i: nat).((lt n i) \to (\forall (c1: C).(\forall +(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n O +c1 e) \to (or4 (drop n O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: +C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: +K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e0 k +w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 (minus i (s k n)) v u w)))))) (ex3_4 K C C T (\lambda (k: +K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k +u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: +T).(drop n O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda +(e2: C).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1 e2)))))) (ex4_5 K C C +T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda +(_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda +(e2: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e2 k w))))))) +(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 (minus i (s k n)) v u w)))))) (\lambda (k: K).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k +n)) v e1 e2)))))))))))))))) +\def + \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (i: nat).((lt n0 i) +\to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) +\to (\forall (e: C).((drop n0 O c1 e) \to (or4 (drop n0 O c2 e) (ex3_4 K C T +T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e +(CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda +(w: T).(drop n0 O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n0)) v u w)))))) +(ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(drop n0 O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda +(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k n0)) v e1 +e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: +K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 +O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n0)) v u w)))))) +(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 (minus i (s k n0)) v e1 e2))))))))))))))))) (\lambda (i: +nat).(\lambda (_: (lt O i)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (v: +T).(\lambda (H0: (csubst0 i v c1 c2)).(\lambda (e: C).(\lambda (H1: (drop O O +c1 e)).(eq_ind C c1 (\lambda (c: C).(or4 (drop O O c2 c) (ex3_4 K C T T +(\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c +(CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda +(w: T).(drop O O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k O)) v u w)))))) +(ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(eq C c (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(drop O O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda +(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k O)) v e1 +e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 k u))))))) (\lambda (k: +K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O +c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k O)) v u w)))))) +(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 (minus i (s k O)) v e1 e2))))))))) (csubst0_ind (\lambda (n0: +nat).(\lambda (t: T).(\lambda (c: C).(\lambda (c0: C).(or4 (drop O O c0 c) +(ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: +T).(eq C c (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: +T).(\lambda (w: T).(drop O O c0 (CHead e0 k w)))))) (\lambda (k: K).(\lambda +(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n0 (s k O)) t u w)))))) +(ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(eq C c (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(drop O O c0 (CHead e2 k u)))))) (\lambda (k: K).(\lambda +(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n0 (s k O)) t e1 +e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 k u))))))) (\lambda (k: +K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O +c0 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n0 (s k O)) t u w)))))) +(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 (minus n0 (s k O)) t e1 e2)))))))))))) (\lambda (k: +K).(\lambda (i0: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: +T).(\lambda (H2: (subst0 i0 v0 u1 u2)).(\lambda (c: C).(let H3 \def (eq_ind_r +nat i0 (\lambda (n0: nat).(subst0 n0 v0 u1 u2)) H2 (minus (s k i0) (s k O)) +(s_arith0 k i0)) in (or4_intro1 (drop O O (CHead c k u2) (CHead c k u1)) +(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: +T).(eq C (CHead c k u1) (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: +C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c k u2) (CHead e0 k0 +w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 (minus (s k i0) (s k0 O)) v0 u w)))))) (ex3_4 K C C T (\lambda +(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c k u1) +(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(drop O O (CHead c k u2) (CHead e2 k0 u)))))) (\lambda +(k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s +k i0) (s k0 O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda +(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c k u1) +(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c k u2) (CHead e2 k0 +w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 O)) v0 u w)))))) (\lambda +(k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: +T).(csubst0 (minus (s k i0) (s k0 O)) v0 e1 e2))))))) (ex3_4_intro K C T T +(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C +(CHead c k u1) (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: +C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c k u2) (CHead e0 k0 +w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 (minus (s k i0) (s k0 O)) v0 u w))))) k c u1 u2 (refl_equal C +(CHead c k u1)) (drop_refl (CHead c k u2)) H3)))))))))) (\lambda (k: +K).(\lambda (i0: nat).(\lambda (c3: C).(\lambda (c4: C).(\lambda (v0: +T).(\lambda (H2: (csubst0 i0 v0 c3 c4)).(\lambda (H3: (or4 (drop O O c4 c3) +(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: +T).(eq C c3 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda +(_: T).(\lambda (w: T).(drop O O c4 (CHead e0 k0 w)))))) (\lambda (k0: +K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k0 +O)) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda +(_: C).(\lambda (u: T).(eq C c3 (CHead e1 k0 u)))))) (\lambda (k0: +K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O c4 (CHead e2 k0 +u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(csubst0 (minus i0 (s k0 O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: +K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 +(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e2 k0 w))))))) +(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 (minus i0 (s k0 O)) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k0 +O)) v0 e1 e2))))))))).(\lambda (u: T).(let H4 \def (eq_ind_r nat i0 (\lambda +(n0: nat).(csubst0 n0 v0 c3 c4)) H2 (minus (s k i0) (s k O)) (s_arith0 k i0)) +in (let H5 \def (eq_ind_r nat i0 (\lambda (n0: nat).(or4 (drop O O c4 c3) +(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda +(_: T).(eq C c3 (CHead e0 k0 u0)))))) (\lambda (k0: K).(\lambda (e0: +C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e0 k0 w)))))) (\lambda +(k0: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus n0 (s +k0 O)) v0 u0 w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u0: T).(eq C c3 (CHead e1 k0 u0)))))) (\lambda +(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop O O c4 (CHead +e2 k0 u0)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda +(_: T).(csubst0 (minus n0 (s k0 O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda +(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq +C c3 (CHead e1 k0 u0))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e2 k0 w))))))) +(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda +(w: T).(subst0 (minus n0 (s k0 O)) v0 u0 w)))))) (\lambda (k0: K).(\lambda +(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n0 +(s k0 O)) v0 e1 e2))))))))) H3 (minus (s k i0) (s k O)) (s_arith0 k i0)) in +(or4_intro2 (drop O O (CHead c4 k u) (CHead c3 k u)) (ex3_4 K C T T (\lambda +(k0: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 k +u) (CHead e0 k0 u0)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: +T).(\lambda (w: T).(drop O O (CHead c4 k u) (CHead e0 k0 w)))))) (\lambda +(k0: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k +i0) (s k0 O)) v0 u0 w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead c3 k u) (CHead e1 k0 +u0)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: +T).(drop O O (CHead c4 k u) (CHead e2 k0 u0)))))) (\lambda (k0: K).(\lambda +(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 O)) +v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda +(_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 k u) (CHead e1 k0 +u0))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (w: T).(drop O O (CHead c4 k u) (CHead e2 k0 w))))))) (\lambda +(k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: +T).(subst0 (minus (s k i0) (s k0 O)) v0 u0 w)))))) (\lambda (k0: K).(\lambda +(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k +i0) (s k0 O)) v0 e1 e2))))))) (ex3_4_intro K C C T (\lambda (k0: K).(\lambda +(e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead c3 k u) (CHead e1 k0 +u0)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: +T).(drop O O (CHead c4 k u) (CHead e2 k0 u0)))))) (\lambda (k0: K).(\lambda +(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 O)) +v0 e1 e2))))) k c3 c4 u (refl_equal C (CHead c3 k u)) (drop_refl (CHead c4 k +u)) H4)))))))))))) (\lambda (k: K).(\lambda (i0: nat).(\lambda (v0: +T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 i0 v0 u1 +u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (H3: (csubst0 i0 v0 c3 +c4)).(\lambda (_: (or4 (drop O O c4 c3) (ex3_4 K C T T (\lambda (k0: +K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e0 k0 +u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: +T).(drop O O c4 (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k0 O)) v0 u w)))))) +(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(eq C c3 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda +(e2: C).(\lambda (u: T).(drop O O c4 (CHead e2 k0 u)))))) (\lambda (k0: +K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (s k0 +O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 k0 +u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (w: T).(drop O O c4 (CHead e2 k0 w))))))) (\lambda (k0: +K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 +(minus i0 (s k0 O)) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda +(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k0 O)) v0 e1 +e2))))))))).(let H5 \def (eq_ind_r nat i0 (\lambda (n0: nat).(subst0 n0 v0 u1 +u2)) H2 (minus (s k i0) (s k O)) (s_arith0 k i0)) in (let H6 \def (eq_ind_r +nat i0 (\lambda (n0: nat).(csubst0 n0 v0 c3 c4)) H3 (minus (s k i0) (s k O)) +(s_arith0 k i0)) in (or4_intro3 (drop O O (CHead c4 k u2) (CHead c3 k u1)) +(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: +T).(eq C (CHead c3 k u1) (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: +C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 k u2) (CHead e0 k0 +w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 (minus (s k i0) (s k0 O)) v0 u w)))))) (ex3_4 K C C T (\lambda +(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c3 k +u1) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(drop O O (CHead c4 k u2) (CHead e2 k0 u)))))) (\lambda +(k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s +k i0) (s k0 O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda +(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 k u1) +(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 k u2) (CHead e2 k0 +w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 O)) v0 u w)))))) (\lambda +(k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: +T).(csubst0 (minus (s k i0) (s k0 O)) v0 e1 e2))))))) (ex4_5_intro K C C T T +(\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda +(_: T).(eq C (CHead c3 k u1) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda +(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 k +u2) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 O)) v0 u +w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 O)) v0 e1 e2)))))) k c3 c4 +u1 u2 (refl_equal C (CHead c3 k u1)) (drop_refl (CHead c4 k u2)) H5 +H6)))))))))))))) i v c1 c2 H0) e (drop_gen_refl c1 e H1)))))))))) (\lambda +(n0: nat).(\lambda (IHn: ((\forall (i: nat).((lt n0 i) \to (\forall (c1: +C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: +C).((drop n0 O c1 e) \to (or4 (drop n0 O c2 e) (ex3_4 K C T T (\lambda (k: +K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k +u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: +T).(drop n0 O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda +(u: T).(\lambda (w: T).(subst0 (minus i (s k n0)) v u w)))))) (ex3_4 K C C T +(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e +(CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda +(u: T).(drop n0 O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k n0)) v e1 e2)))))) +(ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda +(u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda +(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c2 (CHead +e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 (minus i (s k n0)) v u w)))))) (\lambda (k: +K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 +(minus i (s k n0)) v e1 e2)))))))))))))))))).(\lambda (i: nat).(\lambda (H: +(lt (S n0) i)).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: +C).(\forall (v: T).((csubst0 i v c c2) \to (\forall (e: C).((drop (S n0) O c +e) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: +C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: +K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead +e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 (minus i (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: +K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k +u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: +T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1 +e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: +K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S +n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w)))))) +(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 (minus i (s k (S n0))) v e1 e2)))))))))))))) (\lambda (n1: +nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (_: (csubst0 i v (CSort n1) +c2)).(\lambda (e: C).(\lambda (H1: (drop (S n0) O (CSort n1) e)).(and3_ind +(eq C e (CSort n1)) (eq nat (S n0) O) (eq nat O O) (or4 (drop (S n0) O c2 e) +(ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: +T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: +T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: +K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S +n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda +(_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda +(_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) +(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 +(minus i (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: +K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e +(CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) +(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 (minus i (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k (S +n0))) v e1 e2)))))))) (\lambda (H2: (eq C e (CSort n1))).(\lambda (H3: (eq +nat (S n0) O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n1) (\lambda (c: +C).(or4 (drop (S n0) O c2 c) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: +C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 k u)))))) (\lambda (k: +K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead +e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 (minus i (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: +K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 k +u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: +T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1 +e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 k u))))))) (\lambda (k: +K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S +n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w)))))) +(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 (minus i (s k (S n0))) v e1 e2))))))))) (let H5 \def (eq_ind +nat (S n0) (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) +with [O \Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind +(or4 (drop (S n0) O c2 (CSort n1)) (ex3_4 K C T T (\lambda (k: K).(\lambda +(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CSort n1) (CHead e0 k u)))))) +(\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) +O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w)))))) (ex3_4 K C C T +(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CSort +n1) (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: +K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k +(S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CSort n1) (CHead e1 +k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k: +K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 +(minus i (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda +(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1 +e2)))))))) H5)) e H2)))) (drop_gen_sort n1 (S n0) O e H1)))))))) (\lambda (c: +C).(\lambda (H0: ((\forall (c2: C).(\forall (v: T).((csubst0 i v c c2) \to +(\forall (e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C +T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e +(CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda +(w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w)))))) +(ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: +K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k +(S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k +u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k: +K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 +(minus i (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda +(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1 +e2))))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda +(v: T).(\lambda (H1: (csubst0 i v (CHead c k t) c2)).(\lambda (e: C).(\lambda +(H2: (drop (S n0) O (CHead c k t) e)).(or3_ind (ex3_2 T nat (\lambda (_: +T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: +nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j +v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k +j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda +(c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_: +T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: +T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda +(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: +T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))) (or4 (drop (S +n0) O c2 e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: +T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda +(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k0 +w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: +K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 +u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: +T).(drop (S n0) O c2 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k0 (S n0))) v e1 +e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda +(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: +T).(drop (S n0) O c2 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S +n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (_: T).(csubst0 (minus i (s k0 (S n0))) v e1 e2)))))))) +(\lambda (H3: (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k +j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda +(u2: T).(\lambda (j: nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: +T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: +nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j +v t u2))) (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k0: K).(\lambda +(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda +(k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 +(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (ex3_4 K C C T +(\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e +(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k0 u)))))) (\lambda (k0: +K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k0 +(S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 +u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k0 w))))))) (\lambda (k0: +K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 +(minus i (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k0 +(S n0))) v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H4: +(eq nat i (s k x1))).(\lambda (H5: (eq C c2 (CHead c k x0))).(\lambda (_: +(subst0 x1 v t x0)).(eq_ind_r C (CHead c k x0) (\lambda (c0: C).(or4 (drop (S +n0) O c0 e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: +T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda +(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e0 k0 +w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: +K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 +u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: +T).(drop (S n0) O c0 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k0 (S n0))) v e1 +e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda +(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: +T).(drop (S n0) O c0 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S +n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (_: T).(csubst0 (minus i (s k0 (S n0))) v e1 e2))))))))) (let +H7 \def (eq_ind nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0: +T).((csubst0 n1 v0 c c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (or4 +(drop (S n0) O c3 e0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e1: +C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda +(k0: K).(\lambda (e1: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c3 +(CHead e1 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 (minus n1 (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C +T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e0 +(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(drop (S n0) O c3 (CHead e2 k0 u)))))) (\lambda (k0: +K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n1 (s k0 +(S n0))) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 +u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (w: T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0: +K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 +(minus n1 (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 (s k0 +(S n0))) v0 e1 e2)))))))))))))) H0 (s k x1) H4) in (let H8 \def (eq_ind nat i +(\lambda (n1: nat).(lt (S n0) n1)) H (s k x1) H4) in (eq_ind_r nat (s k x1) +(\lambda (n1: nat).(or4 (drop (S n0) O (CHead c k x0) e) (ex3_4 K C T T +(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e +(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: +T).(\lambda (w: T).(drop (S n0) O (CHead c k x0) (CHead e0 k0 w)))))) +(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 +(minus n1 (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda +(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda +(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O +(CHead c k x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n1 (s k0 (S n0))) v e1 +e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda +(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: +T).(drop (S n0) O (CHead c k x0) (CHead e2 k0 w))))))) (\lambda (k0: +K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 +(minus n1 (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 (s k0 +(S n0))) v e1 e2))))))))) (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to +(((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c c3) \to (\forall +(e0: C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T +(\lambda (k1: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 +(CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: +T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k1 w)))))) (\lambda (k1: +K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 x1) +(s k1 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k1 u)))))) (\lambda +(k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 +(CHead e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(csubst0 (minus (s k0 x1) (s k1 (S n0))) v0 e1 e2)))))) +(ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda +(u: T).(\lambda (_: T).(eq C e0 (CHead e1 k1 u))))))) (\lambda (k1: +K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S +n0) O c3 (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 x1) (s k1 (S n0))) v0 +u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (_: T).(csubst0 (minus (s k0 x1) (s k1 (S n0))) v0 e1 +e2)))))))))))))) \to ((lt (S n0) (s k0 x1)) \to (or4 (drop (S n0) O (CHead c +k0 x0) e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u: +T).(\lambda (_: T).(eq C e (CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda +(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c k0 x0) (CHead +e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 (minus (s k0 x1) (s k1 (S n0))) v u w)))))) (ex3_4 K C C T +(\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e +(CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(drop (S n0) O (CHead c k0 x0) (CHead e2 k1 u)))))) +(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 +(minus (s k0 x1) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: +K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e +(CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c k0 x0) (CHead e2 +k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 (minus (s k0 x1) (s k1 (S n0))) v u w)))))) +(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 (minus (s k0 x1) (s k1 (S n0))) v e1 e2)))))))))))) (\lambda +(b: B).(\lambda (H9: (drop (r (Bind b) n0) O c e)).(\lambda (_: ((\forall +(c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to (\forall (e0: +C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T +(\lambda (k0: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 +(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: +T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0: +K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) +x1) (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda +(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 +(CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v0 e1 +e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u))))))) (\lambda +(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: +T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) +x1) (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda +(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 +(S n0))) v0 e1 e2))))))))))))))).(\lambda (_: (lt (S n0) (s (Bind b) +x1))).(or4_intro0 (drop (S n0) O (CHead c (Bind b) x0) e) (ex3_4 K C T T +(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e +(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: +T).(\lambda (w: T).(drop (S n0) O (CHead c (Bind b) x0) (CHead e0 k0 w)))))) +(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 +(minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda +(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 +u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: +T).(drop (S n0) O (CHead c (Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0: +K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind +b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: +K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e +(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c (Bind b) x0) +(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S +n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1 +e2))))))) (drop_drop (Bind b) n0 c e H9 x0)))))) (\lambda (f: F).(\lambda +(H9: (drop (r (Flat f) n0) O c e)).(\lambda (_: ((\forall (c3: C).(\forall +(v0: T).((csubst0 (s (Flat f) x1) v0 c c3) \to (\forall (e0: C).((drop (S n0) +O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T (\lambda (k0: +K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 +u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: T).(\lambda (w: +T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0: K).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S +n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda +(_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda (k0: +K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 (CHead +e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v0 e1 e2)))))) (ex4_5 K C C +T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda +(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c3 +(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S +n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S +n0))) v0 e1 e2))))))))))))))).(\lambda (_: (lt (S n0) (s (Flat f) +x1))).(or4_intro0 (drop (S n0) O (CHead c (Flat f) x0) e) (ex3_4 K C T T +(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e +(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: +T).(\lambda (w: T).(drop (S n0) O (CHead c (Flat f) x0) (CHead e0 k0 w)))))) +(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 +(minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda +(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 +u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: +T).(drop (S n0) O (CHead c (Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: +K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat +f) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: +K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e +(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c (Flat f) x0) +(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S +n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1 +e2))))))) (drop_drop (Flat f) n0 c e H9 x0)))))) k (drop_gen_drop k c e t n0 +H2) H7 H8) i H4))) c2 H5)))))) H3)) (\lambda (H3: (ex3_2 C nat (\lambda (_: +C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_: +nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j +v c c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k +j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda +(c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (or4 (drop (S n0) O c2 e) +(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: +T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda +(_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k0 w)))))) (\lambda (k0: +K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S +n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda +(_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda +(_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k0 +u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(csubst0 (minus i (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda +(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq +C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k0 w))))))) +(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k0 +(S n0))) v e1 e2)))))))) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H4: +(eq nat i (s k x1))).(\lambda (H5: (eq C c2 (CHead x0 k t))).(\lambda (H6: +(csubst0 x1 v c x0)).(eq_ind_r C (CHead x0 k t) (\lambda (c0: C).(or4 (drop +(S n0) O c0 e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: +T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda +(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e0 k0 +w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: +K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 +u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: +T).(drop (S n0) O c0 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k0 (S n0))) v e1 +e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda +(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: +T).(drop (S n0) O c0 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S +n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (_: T).(csubst0 (minus i (s k0 (S n0))) v e1 e2))))))))) (let +H7 \def (eq_ind nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0: +T).((csubst0 n1 v0 c c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (or4 +(drop (S n0) O c3 e0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e1: +C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda +(k0: K).(\lambda (e1: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c3 +(CHead e1 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 (minus n1 (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C +T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e0 +(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(drop (S n0) O c3 (CHead e2 k0 u)))))) (\lambda (k0: +K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n1 (s k0 +(S n0))) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 +u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (w: T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0: +K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 +(minus n1 (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 (s k0 +(S n0))) v0 e1 e2)))))))))))))) H0 (s k x1) H4) in (let H8 \def (eq_ind nat i +(\lambda (n1: nat).(lt (S n0) n1)) H (s k x1) H4) in (eq_ind_r nat (s k x1) +(\lambda (n1: nat).(or4 (drop (S n0) O (CHead x0 k t) e) (ex3_4 K C T T +(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e +(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: +T).(\lambda (w: T).(drop (S n0) O (CHead x0 k t) (CHead e0 k0 w)))))) +(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 +(minus n1 (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda +(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda +(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O +(CHead x0 k t) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n1 (s k0 (S n0))) v e1 +e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda +(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: +T).(drop (S n0) O (CHead x0 k t) (CHead e2 k0 w))))))) (\lambda (k0: +K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 +(minus n1 (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 (s k0 +(S n0))) v e1 e2))))))))) (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to +(((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c c3) \to (\forall +(e0: C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T +(\lambda (k1: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 +(CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: +T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k1 w)))))) (\lambda (k1: +K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 x1) +(s k1 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k1 u)))))) (\lambda +(k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 +(CHead e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(csubst0 (minus (s k0 x1) (s k1 (S n0))) v0 e1 e2)))))) +(ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda +(u: T).(\lambda (_: T).(eq C e0 (CHead e1 k1 u))))))) (\lambda (k1: +K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S +n0) O c3 (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 x1) (s k1 (S n0))) v0 +u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (_: T).(csubst0 (minus (s k0 x1) (s k1 (S n0))) v0 e1 +e2)))))))))))))) \to ((lt (S n0) (s k0 x1)) \to (or4 (drop (S n0) O (CHead x0 +k0 t) e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u: +T).(\lambda (_: T).(eq C e (CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda +(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 k0 t) (CHead +e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 (minus (s k0 x1) (s k1 (S n0))) v u w)))))) (ex3_4 K C C T +(\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e +(CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(drop (S n0) O (CHead x0 k0 t) (CHead e2 k1 u)))))) +(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 +(minus (s k0 x1) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: +K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e +(CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 k0 t) (CHead e2 +k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 (minus (s k0 x1) (s k1 (S n0))) v u w)))))) +(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 (minus (s k0 x1) (s k1 (S n0))) v e1 e2)))))))))))) (\lambda +(b: B).(\lambda (H9: (drop (r (Bind b) n0) O c e)).(\lambda (_: ((\forall +(c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to (\forall (e0: +C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T +(\lambda (k0: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 +(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: +T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0: +K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) +x1) (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda +(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 +(CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v0 e1 +e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u))))))) (\lambda +(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: +T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) +x1) (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda +(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 +(S n0))) v0 e1 e2))))))))))))))).(\lambda (H11: (lt (S n0) (s (Bind b) +x1))).(let H12 \def (IHn x1 (le_S_n (S n0) x1 H11) c x0 v H6 e H9) in +(or4_ind (drop n0 O x0 e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: +C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: +K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x0 (CHead e0 +k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 (minus x1 (s k0 n0)) v u w)))))) (ex3_4 K C C T (\lambda (k0: +K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 +u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: +T).(drop n0 O x0 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus x1 (s k0 n0)) v e1 +e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda +(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: +T).(drop n0 O x0 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x1 (s k0 +n0)) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (_: T).(csubst0 (minus x1 (s k0 n0)) v e1 e2))))))) (or4 +(drop (S n0) O (CHead x0 (Bind b) t) e) (ex3_4 K C T T (\lambda (k0: +K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 +u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: +T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 k0 w)))))) (\lambda (k0: +K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) +x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: +K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 +(Bind b) t) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda +(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1 +e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda +(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: +T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0: +K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 +(minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda +(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s +(Bind b) x1) (s k0 (S n0))) v e1 e2)))))))) (\lambda (H13: (drop n0 O x0 +e)).(or4_intro0 (drop (S n0) O (CHead x0 (Bind b) t) e) (ex3_4 K C T T +(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: 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(k0: K).(\lambda (e0: C).(\lambda (u: +T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda +(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) +(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) +(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda +(e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 +u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C +T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda +(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead +x0 (Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0: 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C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 +(minus x1 (s k0 n0)) v e1 e2)))))))).(ex4_5_ind K C C T T (\lambda (k0: +K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e +(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x0 (CHead e2 k0 w))))))) +(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 (minus x1 (s k0 n0)) v u w)))))) (\lambda (k0: K).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus x1 (s k0 +n0)) v e1 e2)))))) (or4 (drop (S n0) O (CHead x0 (Bind b) t) e) (ex3_4 K C T +T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e +(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: +T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 k0 w)))))) +(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 +(minus (s (Bind b) x1) (s k0 (S n0))) 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H12)))))) (\lambda (f: +F).(\lambda (H9: (drop (r (Flat f) n0) O c e)).(\lambda (H10: ((\forall (c3: +C).(\forall (v0: T).((csubst0 (s (Flat f) x1) v0 c c3) \to (\forall (e0: +C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T +(\lambda (k0: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 +(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: +T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0: +K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) +x1) (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda +(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 +(CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v0 e1 +e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 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(CHead x3 x2 x4)) (drop_drop (Flat f) n0 x0 (CHead +x3 x2 x5) H15 t) H16)) e H14)))))))) H13)) (\lambda (H13: (ex3_4 K C C T +(\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e +(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(drop (S n0) O x0 (CHead e2 k0 u)))))) (\lambda (k0: +K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus x1 (s k0 +(S n0))) v e1 e2))))))).(ex3_4_ind K C C T (\lambda (k0: K).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: +K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O x0 (CHead +e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(csubst0 (minus x1 (s k0 (S n0))) v e1 e2))))) (or4 (drop (S n0) O (CHead +x0 (Flat f) t) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda +(u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda +(e0: C).(\lambda (_: T).(\lambda 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(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S +n0))) v e1 e2))))))) (ex3_4_intro K C C T (\lambda (k0: K).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 x2 x5) (CHead e1 k0 +u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: +T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 u)))))) (\lambda (k0: +K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat +f) x1) (s k0 (S n0))) v e1 e2))))) x2 x3 x4 x5 (refl_equal C (CHead x3 x2 +x5)) (drop_drop (Flat f) n0 x0 (CHead x4 x2 x5) H15 t) H16)) e H14)))))))) +H13)) (\lambda (H13: (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 +u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (w: T).(drop (S n0) O x0 (CHead e2 k0 w))))))) (\lambda (k0: +K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda 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C).(\lambda (j: nat).(eq nat i (s +k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead +c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t +u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c +c3)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: +nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: +nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda +(j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: +nat).(csubst0 j v c c3)))) (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda +(k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 +u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: +T).(drop (S n0) O c2 (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S n0))) v u +w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: +C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k0 u)))))) +(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 +(minus i (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: +K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e +(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k0 w))))))) +(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k0 +(S n0))) v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: C).(\lambda (x2: +nat).(\lambda (H4: (eq nat i (s k x2))).(\lambda (H5: (eq C c2 (CHead x1 k +x0))).(\lambda (_: (subst0 x2 v t x0)).(\lambda (H7: (csubst0 x2 v c +x1)).(eq_ind_r C (CHead x1 k x0) (\lambda (c0: C).(or4 (drop (S n0) O c0 e) +(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: +T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda +(_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e0 k0 w)))))) (\lambda (k0: +K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S +n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda +(_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda +(_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c0 (CHead e2 k0 +u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(csubst0 (minus i (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda +(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq +C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 k0 w))))))) +(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k0 +(S n0))) v e1 e2))))))))) (let H8 \def (eq_ind nat i (\lambda (n1: +nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall +(e0: C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T +(\lambda (k0: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 +(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: +T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0: +K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n1 (s k0 (S +n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda +(_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda (k0: +K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 (CHead +e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(csubst0 (minus n1 (s k0 (S n0))) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda +(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq +C e0 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c3 (CHead e2 k0 w))))))) +(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 (minus n1 (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda +(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 +(s k0 (S n0))) v0 e1 e2)))))))))))))) H0 (s k x2) H4) in (let H9 \def (eq_ind +nat i (\lambda (n1: nat).(lt (S n0) n1)) H (s k x2) H4) in (eq_ind_r nat (s k +x2) (\lambda (n1: nat).(or4 (drop (S n0) O (CHead x1 k x0) e) (ex3_4 K C T T +(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e +(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: +T).(\lambda (w: T).(drop (S n0) O (CHead x1 k x0) (CHead e0 k0 w)))))) +(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 +(minus n1 (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda +(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda +(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O +(CHead x1 k x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n1 (s k0 (S n0))) v e1 +e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda +(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: +T).(drop (S n0) O (CHead x1 k x0) (CHead e2 k0 w))))))) (\lambda (k0: +K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 +(minus n1 (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 (s k0 +(S n0))) v e1 e2))))))))) (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to +(((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x2) v0 c c3) \to (\forall +(e0: C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T +(\lambda (k1: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 +(CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: +T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k1 w)))))) (\lambda (k1: +K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 x2) +(s k1 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k1 u)))))) (\lambda +(k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 +(CHead e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(csubst0 (minus (s k0 x2) (s k1 (S n0))) v0 e1 e2)))))) +(ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda +(u: T).(\lambda (_: T).(eq C e0 (CHead e1 k1 u))))))) (\lambda (k1: +K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S +n0) O c3 (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 x2) (s k1 (S n0))) v0 +u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (_: T).(csubst0 (minus (s k0 x2) (s k1 (S n0))) v0 e1 +e2)))))))))))))) \to ((lt (S n0) (s k0 x2)) \to (or4 (drop (S n0) O (CHead x1 +k0 x0) e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u: +T).(\lambda (_: T).(eq C e (CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda +(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 k0 x0) +(CHead e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 (minus (s k0 x2) (s k1 (S n0))) v u w)))))) (ex3_4 +K C C T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq +C e (CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(drop (S n0) O (CHead x1 k0 x0) (CHead e2 k1 u)))))) +(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 +(minus (s k0 x2) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: +K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e +(CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 k0 x0) (CHead e2 +k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 (minus (s k0 x2) (s k1 (S n0))) v u w)))))) +(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 (minus (s k0 x2) (s k1 (S n0))) v e1 e2)))))))))))) (\lambda +(b: B).(\lambda (H10: (drop (r (Bind b) n0) O c e)).(\lambda (_: ((\forall +(c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x2) v0 c c3) \to (\forall (e0: +C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T +(\lambda (k0: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 +(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: +T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0: +K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) +x2) (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda +(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 +(CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v0 e1 +e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u))))))) (\lambda +(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: +T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) +x2) (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda +(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 +(S n0))) v0 e1 e2))))))))))))))).(\lambda (H12: (lt (S n0) (s (Bind b) +x2))).(let H13 \def (IHn x2 (le_S_n (S n0) x2 H12) c x1 v H7 e H10) in +(or4_ind (drop n0 O x1 e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: +C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: +K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x1 (CHead e0 +k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 (minus x2 (s k0 n0)) v u w)))))) (ex3_4 K C C T (\lambda (k0: +K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 +u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: +T).(drop n0 O x1 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus x2 (s k0 n0)) v e1 +e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda +(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: +T).(drop n0 O x1 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x2 (s k0 +n0)) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (_: T).(csubst0 (minus x2 (s k0 n0)) v e1 e2))))))) (or4 +(drop (S n0) O (CHead x1 (Bind b) x0) e) (ex3_4 K C T T (\lambda (k0: +K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 +u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: +T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) (\lambda (k0: +K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) +x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: +K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 +(Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda +(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 +e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda +(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: +T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0: +K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 +(minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda +(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s +(Bind b) x2) (s k0 (S n0))) v e1 e2)))))))) (\lambda (H14: (drop n0 O x1 +e)).(or4_intro0 (drop (S n0) O (CHead x1 (Bind b) x0) e) (ex3_4 K C T T +(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e +(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: +T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) +(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 +(minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda +(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 +u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: +T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0: +K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind +b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: +K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e +(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) +(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S +n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 +e2))))))) (drop_drop (Bind b) n0 x1 e H14 x0))) (\lambda (H14: (ex3_4 K C T T +(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e +(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: +T).(\lambda (w: T).(drop n0 O x1 (CHead e0 k0 w)))))) (\lambda (k0: +K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x2 (s k0 +n0)) v u w))))))).(ex3_4_ind K C T T (\lambda (k0: K).(\lambda (e0: +C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: +K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x1 (CHead e0 +k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 (minus x2 (s k0 n0)) v u w))))) (or4 (drop (S n0) O (CHead x1 +(Bind b) x0) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: +T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda +(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) +(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) +(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda +(e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 +u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C +T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda +(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead +x1 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) +x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda +(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 +(S n0))) v e1 e2)))))))) (\lambda (x3: K).(\lambda (x4: C).(\lambda (x5: +T).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x4 x3 x5))).(\lambda (H16: +(drop n0 O x1 (CHead x4 x3 x6))).(\lambda (H17: (subst0 (minus x2 (s x3 n0)) +v x5 x6)).(eq_ind_r C (CHead x4 x3 x5) (\lambda (c0: C).(or4 (drop (S n0) O +(CHead x1 (Bind b) x0) c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: +C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 k0 u)))))) (\lambda +(k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O +(CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S +n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda +(_: C).(\lambda (u: T).(eq C c0 (CHead e1 k0 u)))))) (\lambda (k0: +K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 +(Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda +(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 +e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 k0 u))))))) (\lambda +(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: +T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0: +K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 +(minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda +(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s +(Bind b) x2) (s k0 (S n0))) v e1 e2))))))))) (or4_intro1 (drop (S n0) O +(CHead x1 (Bind b) x0) (CHead x4 x3 x5)) (ex3_4 K C T T (\lambda (k0: +K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x5) +(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: +T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) +(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 +(minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda +(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x4 x3 +x5) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 u)))))) +(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 +(minus (s (Bind b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda +(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq +C (CHead x4 x3 x5) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 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K C C T +(\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e +(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(drop n0 O x1 (CHead e2 k0 u)))))) (\lambda (k0: +K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus x2 (s k0 +n0)) v e1 e2))))))).(ex3_4_ind K C C T (\lambda (k0: K).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: +K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O x1 (CHead e2 +k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(csubst0 (minus x2 (s k0 n0)) v e1 e2))))) (or4 (drop (S n0) O (CHead x1 +(Bind b) x0) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: +T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda +(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) +(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) +(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda +(e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 +u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C +T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda +(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead +x1 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) +x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda +(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 +(S n0))) v e1 e2)))))))) (\lambda (x3: K).(\lambda (x4: C).(\lambda (x5: +C).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x4 x3 x6))).(\lambda (H16: +(drop n0 O x1 (CHead x5 x3 x6))).(\lambda (H17: (csubst0 (minus x2 (s x3 n0)) +v x4 x5)).(eq_ind_r C (CHead x4 x3 x6) (\lambda (c0: C).(or4 (drop (S n0) O +(CHead x1 (Bind b) x0) c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: +C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 k0 u)))))) (\lambda +(k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O +(CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S +n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda +(_: C).(\lambda (u: T).(eq C c0 (CHead e1 k0 u)))))) (\lambda (k0: +K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 +(Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda +(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 +e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 k0 u))))))) (\lambda +(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: +T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0: +K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 +(minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda +(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s +(Bind b) x2) (s k0 (S n0))) v e1 e2))))))))) (or4_intro2 (drop (S n0) O +(CHead x1 (Bind b) x0) (CHead x4 x3 x6)) (ex3_4 K C T T (\lambda (k0: +K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6) +(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: +T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) +(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 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+n0))) v e1 e2))))))) (ex3_4_intro K C C T (\lambda (k0: K).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x4 x3 x6) (CHead e1 k0 +u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: +T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0: +K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind +b) x2) (s k0 (S n0))) v e1 e2))))) x3 x4 x5 x6 (refl_equal C (CHead x4 x3 +x6)) (drop_drop (Bind b) n0 x1 (CHead x5 x3 x6) H16 x0) (eq_ind_r nat (S (s +x3 n0)) (\lambda (n1: nat).(csubst0 (minus (s (Bind b) x2) n1) v x4 x5)) H17 +(s x3 (S n0)) (s_S x3 n0)))) e H15)))))))) H14)) (\lambda (H14: (ex4_5 K C C +T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda +(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x1 (CHead +e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 (minus x2 (s k0 n0)) v u w)))))) (\lambda (k0: +K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 +(minus x2 (s k0 n0)) v e1 e2)))))))).(ex4_5_ind K C C T T (\lambda (k0: +K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e +(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x1 (CHead e2 k0 w))))))) +(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 (minus x2 (s k0 n0)) v u w)))))) (\lambda (k0: K).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus x2 (s k0 +n0)) v e1 e2)))))) (or4 (drop (S n0) O (CHead x1 (Bind b) x0) e) (ex3_4 K C T +T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e +(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: +T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) +(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 +(minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda +(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 +u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: +T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0: +K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind +b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: +K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e +(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) +(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S +n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 +e2)))))))) (\lambda (x3: K).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: +T).(\lambda (x7: T).(\lambda (H15: (eq C e (CHead x4 x3 x6))).(\lambda (H16: +(drop n0 O x1 (CHead x5 x3 x7))).(\lambda (H17: (subst0 (minus x2 (s x3 n0)) +v x6 x7)).(\lambda (H18: (csubst0 (minus x2 (s x3 n0)) v x4 x5)).(eq_ind_r C +(CHead x4 x3 x6) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Bind b) x0) +c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda +(_: T).(eq C c0 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: +C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) +(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) +(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(eq C c0 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda +(e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 +u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C +T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(\lambda (_: T).(eq C c0 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda +(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead +x1 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) +x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda +(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 +(S n0))) v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead x1 (Bind b) x0) +(CHead x4 x3 x6)) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda +(u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6) (CHead e0 k0 u)))))) (\lambda +(k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O +(CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S +n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda +(_: C).(\lambda (u: T).(eq C (CHead x4 x3 x6) (CHead e1 k0 u)))))) (\lambda +(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O +(CHead x1 (Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S +n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6) +(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) +(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S +n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: 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+(u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda +(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) +(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) +(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda +(e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 +u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C +T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda +(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead +x1 (Flat f) x0) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) +x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda +(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 +(S n0))) v e1 e2)))))))) (\lambda (x3: K).(\lambda (x4: C).(\lambda (x5: +C).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x4 x3 x6))).(\lambda (H16: +(drop (S n0) O x1 (CHead x5 x3 x6))).(\lambda (H17: (csubst0 (minus x2 (s x3 +(S n0))) v x4 x5)).(eq_ind_r C (CHead x4 x3 x6) (\lambda (c0: C).(or4 (drop +(S n0) O (CHead x1 (Flat f) x0) c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda +(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 k0 u)))))) +(\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S +n0) O (CHead x1 (Flat f) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda +(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 +(S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 k0 u)))))) (\lambda +(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O +(CHead x1 (Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S +n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 k0 +u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 +w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) +(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2))))))))) +(or4_intro2 (drop (S n0) O (CHead x1 (Flat f) x0) (CHead x4 x3 x6)) (ex3_4 K +C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C +(CHead x4 x3 x6) (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: +C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) +(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) +(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(eq C (CHead x4 x3 x6) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: +C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0) +(CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 +e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6) (CHead e1 k0 +u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 +w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) +(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2))))))) +(ex3_4_intro K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(eq C (CHead x4 x3 x6) (CHead e1 k0 u)))))) (\lambda (k0: +K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 +(Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda +(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 +e2))))) x3 x4 x5 x6 (refl_equal C (CHead x4 x3 x6)) (drop_drop (Flat f) n0 x1 +(CHead x5 x3 x6) H16 x0) H17)) e H15)))))))) H14)) (\lambda (H14: (ex4_5 K C +C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda +(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x1 +(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x2 (s k0 (S n0))) v u +w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (_: T).(csubst0 (minus x2 (s k0 (S n0))) v e1 +e2)))))))).(ex4_5_ind K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda +(_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda +(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: +T).(drop (S n0) O x1 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x2 (s k0 (S +n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (_: T).(csubst0 (minus x2 (s k0 (S n0))) v e1 e2)))))) (or4 +(drop (S n0) O (CHead x1 (Flat f) x0) e) (ex3_4 K C T T (\lambda (k0: +K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 +u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: +T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 k0 w)))))) (\lambda (k0: +K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) +x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: +K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 +(Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda +(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 +e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda +(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: +T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 w))))))) (\lambda (k0: +K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 +(minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda +(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s +(Flat f) x2) (s k0 (S n0))) v e1 e2)))))))) (\lambda (x3: K).(\lambda (x4: +C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (x7: T).(\lambda (H15: (eq C e +(CHead x4 x3 x6))).(\lambda (H16: (drop (S n0) O x1 (CHead x5 x3 +x7))).(\lambda (H17: (subst0 (minus x2 (s x3 (S n0))) v x6 x7)).(\lambda +(H18: (csubst0 (minus x2 (s x3 (S n0))) v x4 x5)).(eq_ind_r C (CHead x4 x3 +x6) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Flat f) x0) c0) (ex3_4 K +C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C +c0 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: +T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 k0 w)))))) +(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 +(minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda +(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 +k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: +T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: +K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat +f) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: +K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 +(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) +(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S +n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 +e2))))))))) (or4_intro3 (drop (S n0) O (CHead x1 (Flat f) x0) (CHead x4 x3 +x6)) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: +T).(\lambda (_: T).(eq C (CHead x4 x3 x6) (CHead e0 k0 u)))))) (\lambda (k0: +K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 +(Flat f) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda +(u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u +w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(eq C (CHead x4 x3 x6) (CHead e1 k0 u)))))) (\lambda (k0: +K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 +(Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda +(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 +e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6) (CHead e1 k0 +u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 +w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) +(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2))))))) +(ex4_5_intro K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6) (CHead e1 k0 +u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 +w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) +(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2)))))) x3 x4 x5 +x6 x7 (refl_equal C (CHead x4 x3 x6)) (drop_drop (Flat f) n0 x1 (CHead x5 x3 +x7) H16 x0) H17 H18)) e H15)))))))))) H14)) H13)))))) k (drop_gen_drop k c e +t n0 H2) H8 H9) i H4))) c2 H5)))))))) H3)) (csubst0_gen_head k c c2 t v i +H1))))))))))) c1)))))) n). +(* COMMENTS +Initial nodes: 39886 +END *) + +theorem csubst0_drop_eq: + \forall (n: nat).(\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 +n v c1 c2) \to (\forall (e: C).((drop n O c1 e) \to (or4 (drop n O c2 e) +(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: +T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: +C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e0 (Flat f) w)))))) +(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u +w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: +F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n O c2 (CHead e2 +(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda +(_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda +(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 +(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (w: T).(drop n O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O +v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))) +\def + \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (c1: C).(\forall (c2: +C).(\forall (v: T).((csubst0 n0 v c1 c2) \to (\forall (e: C).((drop n0 O c1 +e) \to (or4 (drop n0 O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: +C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) +(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O +c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: +F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat +f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: +T).(drop n0 O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T +(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: +T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c2 (CHead e2 +(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda +(u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 +e2))))))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda +(H: (csubst0 O v c1 c2)).(\lambda (e: C).(\lambda (H0: (drop O O c1 +e)).(eq_ind C c1 (\lambda (c: C).(or4 (drop O O c2 c) (ex3_4 F C T T (\lambda +(f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 +(Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda +(w: T).(drop O O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T +(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c +(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(drop O O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: +F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) +(ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda +(u: T).(\lambda (_: T).(eq C c (CHead e1 (Flat f) u))))))) (\lambda (f: +F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O +c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: +F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 +O v e1 e2))))))))) (insert_eq nat O (\lambda (n0: nat).(csubst0 n0 v c1 c2)) +(\lambda (n0: nat).(or4 (drop n0 n0 c2 c1) (ex3_4 F C T T (\lambda (f: +F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c1 (CHead e0 (Flat +f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: +T).(drop n0 n0 c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v u w)))))) (ex3_4 F C C T +(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c1 +(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(drop n0 n0 c2 (CHead e2 (Flat f) u)))))) (\lambda (_: +F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 v e1 +e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (_: T).(eq C c1 (CHead e1 (Flat f) u))))))) +(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: +T).(drop n0 n0 c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v u w)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 n0 v e1 e2))))))))) (\lambda (y: nat).(\lambda (H1: (csubst0 +y v c1 c2)).(csubst0_ind (\lambda (n0: nat).(\lambda (t: T).(\lambda (c: +C).(\lambda (c0: C).((eq nat n0 O) \to (or4 (drop n0 n0 c0 c) (ex3_4 F C T T +(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c +(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: +T).(\lambda (w: T).(drop n0 n0 c0 (CHead e0 (Flat f) w)))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 t u w)))))) +(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(eq C c (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: +C).(\lambda (e2: C).(\lambda (u: T).(drop n0 n0 c0 (CHead e2 (Flat f) u)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 +t e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Flat f) u))))))) +(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: +T).(drop n0 n0 c0 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 t u w)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 n0 t e1 e2))))))))))))) (\lambda (k: K).(K_ind (\lambda (k0: +K).(\forall (i: nat).(\forall (v0: T).(\forall (u1: T).(\forall (u2: +T).((subst0 i v0 u1 u2) \to (\forall (c: C).((eq nat (s k0 i) O) \to (or4 +(drop (s k0 i) (s k0 i) (CHead c k0 u2) (CHead c k0 u1)) (ex3_4 F C T T +(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead +c k0 u1) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda +(_: T).(\lambda (w: T).(drop (s k0 i) (s k0 i) (CHead c k0 u2) (CHead e0 +(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 (s k0 i) v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c k0 u1) (CHead e1 (Flat f) +u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: +T).(drop (s k0 i) (s k0 i) (CHead c k0 u2) (CHead e2 (Flat f) u)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (s +k0 i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c k0 u1) +(CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) (s k0 i) (CHead c k0 u2) +(CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 (s k0 i) v0 u w)))))) (\lambda (_: +F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 +(s k0 i) v0 e1 e2)))))))))))))))) (\lambda (b: B).(\lambda (i: nat).(\lambda +(v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0 i v0 u1 +u2)).(\lambda (c: C).(\lambda (H3: (eq nat (S i) O)).(let H4 \def (eq_ind nat +(S i) (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with +[O \Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind (or4 +(drop (S i) (S i) (CHead c (Bind b) u2) (CHead c (Bind b) u1)) (ex3_4 F C T T +(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead +c (Bind b) u1) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: +C).(\lambda (_: T).(\lambda (w: T).(drop (S i) (S i) (CHead c (Bind b) u2) +(CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 (S i) v0 u w)))))) (ex3_4 F C C T (\lambda (f: +F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c (Bind b) +u1) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(drop (S i) (S i) (CHead c (Bind b) u2) (CHead e2 (Flat f) +u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(csubst0 (S i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda +(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c (Bind +b) u1) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda +(e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S i) (S i) (CHead c (Bind b) +u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 (S i) v0 u w)))))) (\lambda (_: +F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 +(S i) v0 e1 e2)))))))) H4)))))))))) (\lambda (f: F).(\lambda (i: +nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 +i v0 u1 u2)).(\lambda (c: C).(\lambda (H3: (eq nat i O)).(let H4 \def (eq_ind +nat i (\lambda (n0: nat).(subst0 n0 v0 u1 u2)) H2 O H3) in (eq_ind_r nat O +(\lambda (n0: nat).(or4 (drop n0 n0 (CHead c (Flat f) u2) (CHead c (Flat f) +u1)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: +T).(\lambda (_: T).(eq C (CHead c (Flat f) u1) (CHead e0 (Flat f0) u)))))) +(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0 +(CHead c (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda +(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v0 u w)))))) (ex3_4 F C C T +(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C +(CHead c (Flat f) u1) (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda +(_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 n0 (CHead c (Flat f) u2) +(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: +F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C +(CHead c (Flat f) u1) (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda +(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0 (CHead c +(Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v0 u w)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 n0 v0 e1 e2))))))))) (or4_intro1 (drop O O (CHead c (Flat f) +u2) (CHead c (Flat f) u1)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: +C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c (Flat f) u1) (CHead e0 +(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda +(w: T).(drop O O (CHead c (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda +(_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) +(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(eq C (CHead c (Flat f) u1) (CHead e1 (Flat f0) u)))))) (\lambda (f0: +F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O (CHead c (Flat +f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda +(e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda +(f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq +C (CHead c (Flat f) u1) (CHead e1 (Flat f0) u))))))) (\lambda (f0: +F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O +(CHead c (Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda +(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 O v0 e1 e2))))))) (ex3_4_intro F C T T (\lambda (f0: +F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c (Flat f) +u1) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda +(_: T).(\lambda (w: T).(drop O O (CHead c (Flat f) u2) (CHead e0 (Flat f0) +w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 O v0 u w))))) f c u1 u2 (refl_equal C (CHead c (Flat f) u1)) +(drop_refl (CHead c (Flat f) u2)) H4)) i H3)))))))))) k)) (\lambda (k: +K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (c3: C).(\forall (c4: +C).(\forall (v0: T).((csubst0 i v0 c3 c4) \to ((((eq nat i O) \to (or4 (drop +i i c4 c3) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: +T).(\lambda (_: T).(eq C c3 (CHead e0 (Flat f) u)))))) (\lambda (f: +F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop i i c4 (CHead e0 +(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 i v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 (Flat f) u)))))) +(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop i i c4 +(CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(csubst0 i v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: +F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 +(CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (w: T).(drop i i c4 (CHead e2 (Flat f) w))))))) +(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 i v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v0 e1 e2)))))))))) \to (\forall +(u: T).((eq nat (s k0 i) O) \to (or4 (drop (s k0 i) (s k0 i) (CHead c4 k0 u) +(CHead c3 k0 u)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda +(u0: T).(\lambda (_: T).(eq C (CHead c3 k0 u) (CHead e0 (Flat f) u0)))))) +(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 +i) (s k0 i) (CHead c4 k0 u) (CHead e0 (Flat f) w)))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (s k0 i) v0 u0 +w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u0: T).(eq C (CHead c3 k0 u) (CHead e1 (Flat f) u0)))))) +(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s k0 +i) (s k0 i) (CHead c4 k0 u) (CHead e2 (Flat f) u0)))))) (\lambda (_: +F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (s k0 i) v0 e1 +e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 k0 u) (CHead e1 (Flat f) +u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (w: T).(drop (s k0 i) (s k0 i) (CHead c4 k0 u) (CHead e2 (Flat f) +w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: +T).(\lambda (w: T).(subst0 (s k0 i) v0 u0 w)))))) (\lambda (_: F).(\lambda +(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (s k0 i) v0 +e1 e2))))))))))))))))) (\lambda (b: B).(\lambda (i: nat).(\lambda (c3: +C).(\lambda (c4: C).(\lambda (v0: T).(\lambda (_: (csubst0 i v0 c3 +c4)).(\lambda (_: (((eq nat i O) \to (or4 (drop i i c4 c3) (ex3_4 F C T T +(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 +(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: +T).(\lambda (w: T).(drop i i c4 (CHead e0 (Flat f) w)))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w)))))) +(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(eq C c3 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: +C).(\lambda (e2: C).(\lambda (u: T).(drop i i c4 (CHead e2 (Flat f) u)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i +v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f) u))))))) +(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: +T).(drop i i c4 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (u: T).(\lambda (H4: (eq nat +(S i) O)).(let H5 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee in nat +return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow +True])) I O H4) in (False_ind (or4 (drop (S i) (S i) (CHead c4 (Bind b) u) +(CHead c3 (Bind b) u)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: +C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 (Bind b) u) (CHead e0 +(Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda +(w: T).(drop (S i) (S i) (CHead c4 (Bind b) u) (CHead e0 (Flat f) w)))))) +(\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (S +i) v0 u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u0: T).(eq C (CHead c3 (Bind b) u) (CHead e1 (Flat f) u0)))))) +(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S i) +(S i) (CHead c4 (Bind b) u) (CHead e2 (Flat f) u0)))))) (\lambda (_: +F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (S i) v0 e1 +e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 (Bind b) u) (CHead e1 +(Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (w: T).(drop (S i) (S i) (CHead c4 (Bind b) u) (CHead e2 +(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda +(u0: T).(\lambda (w: T).(subst0 (S i) v0 u0 w)))))) (\lambda (_: F).(\lambda +(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (S i) v0 e1 +e2)))))))) H5))))))))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (c3: +C).(\lambda (c4: C).(\lambda (v0: T).(\lambda (H2: (csubst0 i v0 c3 +c4)).(\lambda (H3: (((eq nat i O) \to (or4 (drop i i c4 c3) (ex3_4 F C T T +(\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 +(CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: +T).(\lambda (w: T).(drop i i c4 (CHead e0 (Flat f0) w)))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w)))))) +(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(eq C c3 (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: +C).(\lambda (e2: C).(\lambda (u: T).(drop i i c4 (CHead e2 (Flat f0) u)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i +v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda +(_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f0) u))))))) +(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(w: T).(drop i i c4 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (u: T).(\lambda (H4: (eq nat i +O)).(let H5 \def (eq_ind nat i (\lambda (n0: nat).((eq nat n0 O) \to (or4 +(drop n0 n0 c4 c3) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda +(u0: T).(\lambda (_: T).(eq C c3 (CHead e0 (Flat f0) u0)))))) (\lambda (f0: +F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0 c4 (CHead e0 +(Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda +(w: T).(subst0 n0 v0 u0 w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u0: T).(eq C c3 (CHead e1 (Flat f0) u0)))))) +(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop n0 +n0 c4 (CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda +(e2: C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda +(f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq +C c3 (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda +(e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0 c4 (CHead e2 (Flat f0) +w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: +T).(\lambda (w: T).(subst0 n0 v0 u0 w)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 n0 v0 e1 +e2)))))))))) H3 O H4) in (let H6 \def (eq_ind nat i (\lambda (n0: +nat).(csubst0 n0 v0 c3 c4)) H2 O H4) in (eq_ind_r nat O (\lambda (n0: +nat).(or4 (drop n0 n0 (CHead c4 (Flat f) u) (CHead c3 (Flat f) u)) (ex3_4 F C +T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C +(CHead c3 (Flat f) u) (CHead e0 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda +(e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0 (CHead c4 (Flat f) u) +(CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: +T).(\lambda (w: T).(subst0 n0 v0 u0 w)))))) (ex3_4 F C C T (\lambda (f0: +F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead c3 (Flat f) +u) (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda +(e2: C).(\lambda (u0: T).(drop n0 n0 (CHead c4 (Flat f) u) (CHead e2 (Flat +f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 (Flat f) +u) (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda +(e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0 (CHead c4 (Flat f) u) +(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: +C).(\lambda (u0: T).(\lambda (w: T).(subst0 n0 v0 u0 w)))))) (\lambda (_: +F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 +n0 v0 e1 e2))))))))) (or4_intro2 (drop O O (CHead c4 (Flat f) u) (CHead c3 +(Flat f) u)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: +T).(\lambda (_: T).(eq C (CHead c3 (Flat f) u) (CHead e0 (Flat f0) u0)))))) +(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O +(CHead c4 (Flat f) u) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda +(_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v0 u0 w)))))) (ex3_4 F C C +T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C +(CHead c3 (Flat f) u) (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda +(_: C).(\lambda (e2: C).(\lambda (u0: T).(drop O O (CHead c4 (Flat f) u) +(CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: +F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C +(CHead c3 (Flat f) u) (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda +(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 +(Flat f) u) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v0 u0 w)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 O v0 e1 e2))))))) (ex3_4_intro F C C T (\lambda (f0: +F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead c3 (Flat f) +u) (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda +(e2: C).(\lambda (u0: T).(drop O O (CHead c4 (Flat f) u) (CHead e2 (Flat f0) +u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(csubst0 O v0 e1 e2))))) f c3 c4 u (refl_equal C (CHead c3 (Flat f) u)) +(drop_refl (CHead c4 (Flat f) u)) H6)) i H4)))))))))))) k)) (\lambda (k: +K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (v0: T).(\forall (u1: +T).(\forall (u2: T).((subst0 i v0 u1 u2) \to (\forall (c3: C).(\forall (c4: +C).((csubst0 i v0 c3 c4) \to ((((eq nat i O) \to (or4 (drop i i c4 c3) (ex3_4 +F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq +C c3 (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: +T).(\lambda (w: T).(drop i i c4 (CHead e0 (Flat f) w)))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w)))))) +(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(eq C c3 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: +C).(\lambda (e2: C).(\lambda (u: T).(drop i i c4 (CHead e2 (Flat f) u)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i +v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f) u))))))) +(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: +T).(drop i i c4 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 i v0 e1 e2)))))))))) \to ((eq nat (s k0 i) O) \to (or4 (drop +(s k0 i) (s k0 i) (CHead c4 k0 u2) (CHead c3 k0 u1)) (ex3_4 F C T T (\lambda +(f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 k0 +u1) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: +T).(\lambda (w: T).(drop (s k0 i) (s k0 i) (CHead c4 k0 u2) (CHead e0 (Flat +f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 (s k0 i) v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c3 k0 u1) (CHead e1 (Flat f) +u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: +T).(drop (s k0 i) (s k0 i) (CHead c4 k0 u2) (CHead e2 (Flat f) u)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (s +k0 i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 k0 u1) +(CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) (s k0 i) (CHead c4 k0 u2) +(CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 (s k0 i) v0 u w)))))) (\lambda (_: +F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 +(s k0 i) v0 e1 e2))))))))))))))))))) (\lambda (b: B).(\lambda (i: +nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0 +i v0 u1 u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (_: (csubst0 i v0 c3 +c4)).(\lambda (_: (((eq nat i O) \to (or4 (drop i i c4 c3) (ex3_4 F C T T +(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 +(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: +T).(\lambda (w: T).(drop i i c4 (CHead e0 (Flat f) w)))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w)))))) +(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(eq C c3 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: +C).(\lambda (e2: C).(\lambda (u: T).(drop i i c4 (CHead e2 (Flat f) u)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i +v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f) u))))))) +(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: +T).(drop i i c4 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (H5: (eq nat (S i) O)).(let H6 +\def (eq_ind nat (S i) (\lambda (ee: nat).(match ee in nat return (\lambda +(_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H5) +in (False_ind (or4 (drop (S i) (S i) (CHead c4 (Bind b) u2) (CHead c3 (Bind +b) u1)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: +T).(\lambda (_: T).(eq C (CHead c3 (Bind b) u1) (CHead e0 (Flat f) u)))))) +(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S i) +(S i) (CHead c4 (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (S i) v0 u w)))))) +(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(eq C (CHead c3 (Bind b) u1) (CHead e1 (Flat f) u)))))) (\lambda (f: +F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S i) (S i) (CHead +c4 (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (S i) v0 e1 e2)))))) (ex4_5 F C +C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(\lambda (_: T).(eq C (CHead c3 (Bind b) u1) (CHead e1 (Flat f) u))))))) +(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: +T).(drop (S i) (S i) (CHead c4 (Bind b) u2) (CHead e2 (Flat f) w))))))) +(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 (S i) v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (_: T).(csubst0 (S i) v0 e1 e2)))))))) +H6))))))))))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (v0: T).(\lambda +(u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 i v0 u1 u2)).(\lambda (c3: +C).(\lambda (c4: C).(\lambda (H3: (csubst0 i v0 c3 c4)).(\lambda (H4: (((eq +nat i O) \to (or4 (drop i i c4 c3) (ex3_4 F C T T (\lambda (f0: F).(\lambda +(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e0 (Flat f0) u)))))) +(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop i i +c4 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 i v0 u w)))))) (ex3_4 F C C T (\lambda (f0: +F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 (Flat +f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: +T).(drop i i c4 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v0 e1 e2)))))) (ex4_5 F C C T +T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda +(_: T).(eq C c3 (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop i i c4 (CHead e2 +(Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda +(u: T).(\lambda (w: T).(subst0 i v0 u w)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v0 e1 +e2))))))))))).(\lambda (H5: (eq nat i O)).(let H6 \def (eq_ind nat i (\lambda +(n0: nat).((eq nat n0 O) \to (or4 (drop n0 n0 c4 c3) (ex3_4 F C T T (\lambda +(f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e0 +(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda +(w: T).(drop n0 n0 c4 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda +(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v0 u w)))))) (ex3_4 F C C T +(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3 +(CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(drop n0 n0 c4 (CHead e2 (Flat f0) u)))))) (\lambda (_: +F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 v0 e1 +e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f0) u))))))) +(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(w: T).(drop n0 n0 c4 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda +(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v0 u +w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))))))) H4 O H5) in (let H7 \def +(eq_ind nat i (\lambda (n0: nat).(csubst0 n0 v0 c3 c4)) H3 O H5) in (let H8 +\def (eq_ind nat i (\lambda (n0: nat).(subst0 n0 v0 u1 u2)) H2 O H5) in +(eq_ind_r nat O (\lambda (n0: nat).(or4 (drop n0 n0 (CHead c4 (Flat f) u2) +(CHead c3 (Flat f) u1)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: +C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 (Flat f) u1) (CHead e0 +(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda +(w: T).(drop n0 n0 (CHead c4 (Flat f) u2) (CHead e0 (Flat f0) w)))))) +(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v0 +u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(eq C (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u)))))) +(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 n0 +(CHead c4 (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda +(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F +C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(\lambda (_: T).(eq C (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u))))))) +(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(w: T).(drop n0 n0 (CHead c4 (Flat f) u2) (CHead e2 (Flat f0) w))))))) +(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 n0 v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (_: T).(csubst0 n0 v0 e1 e2))))))))) (or4_intro3 +(drop O O (CHead c4 (Flat f) u2) (CHead c3 (Flat f) u1)) (ex3_4 F C T T +(\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C +(CHead c3 (Flat f) u1) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda +(e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 (Flat f) u2) +(CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 O v0 u w)))))) (ex3_4 F C C T (\lambda (f0: +F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c3 (Flat f) +u1) (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda +(e2: C).(\lambda (u: T).(drop O O (CHead c4 (Flat f) u2) (CHead e2 (Flat f0) +u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 (Flat f) +u1) (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda +(e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 (Flat f) u2) +(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (\lambda (_: +F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 +O v0 e1 e2))))))) (ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 (Flat f) +u1) (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda +(e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 (Flat f) u2) +(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (\lambda (_: +F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 +O v0 e1 e2)))))) f c3 c4 u1 u2 (refl_equal C (CHead c3 (Flat f) u1)) +(drop_refl (CHead c4 (Flat f) u2)) H8 H7)) i H5))))))))))))))) k)) y v c1 c2 +H1))) H) e (drop_gen_refl c1 e H0)))))))) (\lambda (n0: nat).(\lambda (IHn: +((\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 n0 v c1 c2) \to +(\forall (e: C).((drop n0 O c1 e) \to (or4 (drop n0 O c2 e) (ex3_4 F C T T +(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e +(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: +T).(\lambda (w: T).(drop n0 O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) +(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: +C).(\lambda (e2: C).(\lambda (u: 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(Flat f) w)))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) +(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: +C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 (Flat f) +u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) +u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O +v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))) (\lambda (n1: +nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (_: (csubst0 (S n0) v (CSort 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T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) +w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 +e2)))))))) (\lambda (H1: (eq C e (CSort n1))).(\lambda (H2: (eq nat (S n0) +O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n1) (\lambda (c: C).(or4 +(drop (S n0) O c2 c) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda +(u: T).(\lambda (_: T).(eq C c (CHead e0 (Flat f) u)))))) (\lambda (f: +F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead +e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda +(w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Flat f) u)))))) +(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) +O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda +(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda +(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C +c (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) +w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 +e2))))))))) (let H4 \def (eq_ind nat (S n0) (\lambda (ee: nat).(match ee in +nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) +\Rightarrow True])) I O H2) in (False_ind (or4 (drop (S n0) O c2 (CSort n1)) +(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: +T).(eq C (CSort n1) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: +C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 (Flat f) +w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(eq C (CSort n1) (CHead e1 (Flat f) +u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: +T).(drop (S n0) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T +(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: +T).(eq C (CSort n1) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead +e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda +(u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 +e2)))))))) H4)) e H1)))) (drop_gen_sort n1 (S n0) O e H0)))))))) (\lambda (c: +C).(\lambda (H: ((\forall (c2: C).(\forall (v: T).((csubst0 (S n0) v c c2) +\to (\forall (e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 +F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq +C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: +T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) +(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: +C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 (Flat f) +u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) +u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O +v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))))))).(\lambda (k: K).(\lambda +(t: T).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 (S n0) v +(CHead c k t) c2)).(\lambda (e: C).(\lambda (H1: (drop (S n0) O (CHead c k t) +e)).(or3_ind (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (S n0) (s +k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) +(\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2)))) (ex3_2 C nat (\lambda +(_: C).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (c3: C).(\lambda +(_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: +nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: +C).(\lambda (j: nat).(eq nat (S n0) (s k j))))) (\lambda (u2: T).(\lambda +(c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k 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+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) +w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 +e2)))))))) (\lambda (H2: (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq +nat (S n0) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k +u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))))).(ex3_2_ind T +nat (\lambda (_: T).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (u2: +T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: +nat).(subst0 j v t u2))) (or4 (drop (S n0) O c2 e) (ex3_4 F C T T (\lambda +(f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 +(Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda +(w: T).(drop (S n0) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda +(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T +(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e +(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: +F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) +(ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda +(u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: +F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S +n0) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda +(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: +F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 +O v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H3: (eq nat +(S n0) (s k x1))).(\lambda (H4: (eq C c2 (CHead c k x0))).(\lambda (H5: +(subst0 x1 v t x0)).(eq_ind_r C (CHead c k x0) (\lambda (c0: C).(or4 (drop (S +n0) O c0 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: +T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: +F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead +e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda +(w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) +(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) +O c0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda +(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda +(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C +e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 (Flat f) +w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 +e2))))))))) (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to ((eq nat (S +n0) (s k0 x1)) \to (or4 (drop (S n0) O (CHead c k0 x0) e) (ex3_4 F C T T +(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e +(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: +T).(\lambda (w: T).(drop (S n0) O (CHead c k0 x0) (CHead e0 (Flat f) w)))))) +(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u +w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: +F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c +k0 x0) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda +(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda +(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C +e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c k0 x0) (CHead e2 +(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda +(u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 +e2))))))))))) (\lambda (b: B).(\lambda (H6: (drop (r (Bind b) n0) O c +e)).(\lambda (H7: (eq nat (S n0) (s (Bind b) x1))).(let H8 \def (f_equal nat +nat (\lambda (e0: nat).(match e0 in nat return (\lambda (_: nat).nat) with [O +\Rightarrow n0 | (S n1) \Rightarrow n1])) (S n0) (S x1) H7) in (let H9 \def +(eq_ind_r nat x1 (\lambda (n1: nat).(subst0 n1 v t x0)) H5 n0 H8) in +(or4_intro0 (drop (S n0) O (CHead c (Bind b) x0) e) (ex3_4 F C T T (\lambda +(f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 +(Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda +(w: T).(drop (S n0) O (CHead c (Bind b) x0) (CHead e0 (Flat f) w)))))) +(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u +w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: +F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c +(Bind b) x0) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T +(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: +T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c +(Bind b) x0) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 O v e1 e2))))))) (drop_drop (Bind b) n0 c e H6 x0))))))) +(\lambda (f: F).(\lambda (H6: (drop (r (Flat f) n0) O c e)).(\lambda (H7: (eq +nat (S n0) (s (Flat f) x1))).(let H8 \def (f_equal nat nat (\lambda (e0: +nat).e0) (S n0) (s (Flat f) x1) H7) in (let H9 \def (eq_ind_r nat x1 (\lambda +(n1: nat).(subst0 n1 v t x0)) H5 (S n0) H8) in (or4_intro0 (drop (S n0) O +(CHead c (Flat f) x0) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: +C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) +(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S +n0) O (CHead c (Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) +(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: +C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) x0) +(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: +F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e +(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c (Flat f) x0) +(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: +F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 +O v e1 e2))))))) (drop_drop (Flat f) n0 c e H6 x0))))))) k (drop_gen_drop k c +e t n0 H1) H3) c2 H4)))))) H2)) (\lambda (H2: (ex3_2 C nat (\lambda (_: +C).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (c3: C).(\lambda (_: +nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j +v c c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) +(s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) +(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (or4 (drop (S n0) O +c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: +T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: +F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead +e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda +(w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) +(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) +O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda +(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda +(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C +e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) +w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 +e2)))))))) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H3: (eq nat (S n0) +(s k x1))).(\lambda (H4: (eq C c2 (CHead x0 k t))).(\lambda (H5: (csubst0 x1 +v c x0)).(eq_ind_r C (CHead x0 k t) (\lambda (c0: C).(or4 (drop (S n0) O c0 +e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda +(_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: +C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e0 (Flat f) +w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) +(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S 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(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u +w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: +F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 +k0 t) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda +(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda +(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C +e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 k0 t) (CHead e2 +(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda +(u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 +e2))))))))))) (\lambda (b: B).(\lambda (H6: (drop (r (Bind b) n0) O c +e)).(\lambda (H7: (eq nat (S n0) (s (Bind b) x1))).(let H8 \def (f_equal nat +nat (\lambda (e0: nat).(match e0 in nat return (\lambda (_: nat).nat) with [O +\Rightarrow n0 | (S n1) \Rightarrow n1])) (S n0) (S x1) H7) in (let H9 \def +(eq_ind_r nat x1 (\lambda (n1: nat).(csubst0 n1 v c x0)) H5 n0 H8) in (let +H10 \def (IHn c x0 v H9 e H6) in (or4_ind (drop n0 O x0 e) (ex3_4 F C T T +(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e +(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: +T).(\lambda (w: T).(drop n0 O x0 (CHead e0 (Flat f) w)))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) +(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: +C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O x0 (CHead e2 (Flat f) u)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O +v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) +(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: +T).(drop n0 O x0 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead x0 (Bind b) t) e) +(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: +T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: +C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) +(CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: +F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat +f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: +T).(drop (S 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F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) +(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) +O (CHead x0 (Bind b) t) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda +(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C +C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: +F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S +n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O +v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Bind b) n0 x0 e H11 +t))) (\lambda (H11: (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda +(u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: +F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x0 (CHead e0 +(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 O v u w))))))).(ex3_4_ind F C T T (\lambda (f: F).(\lambda (e0: +C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) +(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O +x0 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 O v u w))))) (or4 (drop (S n0) O (CHead x0 (Bind +b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: +T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: +F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 +(Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T +(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e +(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) +u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) +u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) +w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 +e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: +T).(\lambda (H12: (eq C e (CHead x3 (Flat x2) x4))).(\lambda (H13: (drop n0 O +x0 (CHead x3 (Flat x2) x5))).(\lambda (H14: (subst0 O v x4 x5)).(eq_ind_r C +(CHead x3 (Flat x2) x4) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x0 (Bind +b) t) c0) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: +T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f) u)))))) (\lambda (f: +F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 +(Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T +(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 +(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) +u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat +f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 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(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: +F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C +(CHead x3 (Flat x2) x4) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda +(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead +x0 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C T T (\lambda (f: +F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) +x4) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: +T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f) +w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 O v u w))))) x2 x3 x4 x5 (refl_equal C (CHead x3 (Flat x2) x4)) +(drop_drop (Bind b) n0 x0 (CHead x3 (Flat x2) x5) H13 t) H14)) e H12)))))))) +H11)) (\lambda (H11: (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda +(_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: +F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O x0 (CHead e2 +(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda +(_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda +(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) +(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O +x0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(csubst0 O v e1 e2))))) (or4 (drop (S n0) O (CHead x0 +(Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: +T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: +F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 +(Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T +(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e +(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) +u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) +u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) +w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 +e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: +T).(\lambda (H12: (eq C e (CHead x3 (Flat x2) x5))).(\lambda (H13: (drop n0 O +x0 (CHead x4 (Flat x2) x5))).(\lambda (H14: (csubst0 O v x3 x4)).(eq_ind_r C +(CHead x3 (Flat x2) x5) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x0 (Bind +b) t) c0) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: +T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f) u)))))) (\lambda (f: +F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 +(Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T +(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 +(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) +u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat +f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) +w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 +e2))))))))) (or4_intro2 (drop (S n0) O (CHead x0 (Bind b) t) (CHead x3 (Flat +x2) x5)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: +T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0 (Flat f) u)))))) +(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) +O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda +(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T +(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead +x3 (Flat x2) x5) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: +C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) +(CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: +F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C +(CHead x3 (Flat x2) x5) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda +(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead +x0 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C C T (\lambda (f: +F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 (Flat x2) +x5) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) +u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(csubst0 O v e1 e2))))) x2 x3 x4 x5 (refl_equal C (CHead x3 (Flat x2) x5)) +(drop_drop (Bind b) n0 x0 (CHead x4 (Flat x2) x5) H13 t) H14)) e H12)))))))) +H11)) (\lambda (H11: (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) +u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (w: T).(drop n0 O x0 (CHead e2 (Flat f) w))))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O +v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda +(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C +e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x0 (CHead e2 (Flat f) w))))))) +(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) +O (CHead x0 (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: +C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) +(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) +O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda +(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T +(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e +(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) +u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) +u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) +w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 +e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: +T).(\lambda (x6: T).(\lambda (H12: (eq C e (CHead x3 (Flat x2) x5))).(\lambda +(H13: (drop n0 O x0 (CHead x4 (Flat x2) x6))).(\lambda (H14: (subst0 O v x5 +x6)).(\lambda (H15: (csubst0 O v x3 x4)).(eq_ind_r C (CHead x3 (Flat x2) x5) +(\lambda (c0: C).(or4 (drop (S n0) O (CHead x0 (Bind b) t) c0) (ex3_4 F C T T +(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 +(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: +T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f) +w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Flat f) u)))))) +(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) +O (CHead x0 (Bind b) t) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda +(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C +C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat f) u))))))) (\lambda (f: +F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S +n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O +v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O +(CHead x0 (Bind b) t) (CHead x3 (Flat x2) x5)) (ex3_4 F C T T (\lambda (f: +F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) +x5) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: +T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f) +w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1 +(Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda +(u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) u)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O +v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1 +(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat +f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) +(ex4_5_intro F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1 +(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat +f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) +x2 x3 x4 x5 x6 (refl_equal C (CHead x3 (Flat x2) x5)) (drop_drop (Bind b) n0 +x0 (CHead x4 (Flat x2) x6) H13 t) H14 H15)) e H12)))))))))) H11)) H10))))))) +(\lambda (f: F).(\lambda (H6: (drop (r (Flat f) n0) O c e)).(\lambda (H7: (eq +nat (S n0) (s (Flat f) x1))).(let H8 \def (f_equal nat nat (\lambda (e0: +nat).e0) (S n0) (s (Flat f) x1) H7) in (let H9 \def (eq_ind_r nat x1 (\lambda +(n1: nat).(csubst0 n1 v c x0)) H5 (S n0) H8) in (let H10 \def (H x0 v H9 e +H6) in (or4_ind (drop (S n0) O x0 e) (ex3_4 F C T T (\lambda (f0: F).(\lambda +(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) +(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S +n0) O x0 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda +(u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: +F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat +f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: +T).(drop (S n0) O x0 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda +(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C +C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: +F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S +n0) O x0 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead x0 (Flat f) t) e) +(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: +T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: +C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) +(CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: +F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat +f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: +T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u)))))) (\lambda +(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 +e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) +(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w))))))) +(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H11: +(drop (S n0) O x0 e)).(or4_intro0 (drop (S n0) O (CHead x0 (Flat f) t) e) +(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: +T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: +C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) +(CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: +F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat +f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: +T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u)))))) (\lambda +(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 +e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) +(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w))))))) +(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Flat +f) n0 x0 e H11 t))) (\lambda (H11: (ex3_4 F C T T (\lambda (f0: F).(\lambda +(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) +(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S +n0) O x0 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda +(u: T).(\lambda (w: T).(subst0 O v u w))))))).(ex3_4_ind F C T T (\lambda +(f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 +(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda +(w: T).(drop (S n0) O x0 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda +(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))) (or4 (drop (S n0) +O (CHead x0 (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: +C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) +(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S +n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) +(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: +C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) +(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: +F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e +(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) +(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: +F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 +O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: T).(\lambda +(x5: T).(\lambda (H12: (eq C e (CHead x3 (Flat x2) x4))).(\lambda (H13: (drop +(S n0) O x0 (CHead x3 (Flat x2) x5))).(\lambda (H14: (subst0 O v x4 +x5)).(eq_ind_r C (CHead x3 (Flat x2) x4) (\lambda (c0: C).(or4 (drop (S n0) O +(CHead x0 (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: +C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f0) u)))))) +(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S +n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) +(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(eq C c0 (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: +C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) +(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: +F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 +(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) +(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: +F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 +O v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead x0 (Flat f) t) (CHead x3 +(Flat x2) x4)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: +T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x4) (CHead e0 (Flat f0) u)))))) +(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S +n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) +(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(eq C (CHead x3 (Flat x2) x4) (CHead e1 (Flat f0) u)))))) (\lambda (f0: +F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 +(Flat f) t) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T +(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda +(_: T).(eq C (CHead x3 (Flat x2) x4) (CHead e1 (Flat f0) u))))))) (\lambda +(f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: +T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda +(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C +T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C +(CHead x3 (Flat x2) x4) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda +(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) +(CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 O v u w))))) x2 x3 x4 x5 (refl_equal C (CHead x3 +(Flat x2) x4)) (drop_drop (Flat f) n0 x0 (CHead x3 (Flat x2) x5) H13 t) H14)) +e H12)))))))) H11)) (\lambda (H11: (ex3_4 F C C T (\lambda (f0: F).(\lambda +(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u)))))) +(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S +n0) O x0 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C +C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e +(CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(drop (S n0) O x0 (CHead e2 (Flat f0) u)))))) (\lambda (_: +F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) +(or4 (drop (S n0) O (CHead x0 (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: +F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat +f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: +T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda +(_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) +(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: +C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) +(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: +F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e +(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) +(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: +F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 +O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda +(x5: T).(\lambda (H12: (eq C e (CHead x3 (Flat x2) x5))).(\lambda (H13: (drop +(S n0) O x0 (CHead x4 (Flat x2) x5))).(\lambda (H14: (csubst0 O v x3 +x4)).(eq_ind_r C (CHead x3 (Flat x2) x5) (\lambda (c0: C).(or4 (drop (S n0) O +(CHead x0 (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: +C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f0) u)))))) +(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S +n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) +(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(eq C c0 (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: +C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) +(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: +F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 +(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) +(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: +F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 +O v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead x0 (Flat f) t) (CHead x3 +(Flat x2) x5)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: +T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0 (Flat f0) u)))))) +(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S +n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) +(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(eq C (CHead x3 (Flat x2) x5) (CHead e1 (Flat f0) u)))))) (\lambda (f0: +F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 +(Flat f) t) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T +(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda +(_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1 (Flat f0) u))))))) (\lambda +(f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: +T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda +(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C +C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C +(CHead x3 (Flat x2) x5) (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda +(_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) +(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(csubst0 O v e1 e2))))) x2 x3 x4 x5 (refl_equal C (CHead +x3 (Flat x2) x5)) (drop_drop (Flat f) n0 x0 (CHead x4 (Flat x2) x5) H13 t) +H14)) e H12)))))))) H11)) (\lambda (H11: (ex4_5 F C C T T (\lambda (f0: +F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e +(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x0 (CHead e2 (Flat f0) +w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 +e2)))))))).(ex4_5_ind F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda +(_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) +(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(w: T).(drop (S n0) O x0 (CHead e2 (Flat f0) w))))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O +v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) O (CHead x0 +(Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: +T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: +F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 +(Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T +(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e +(CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) +u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat +f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) +w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 +e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: +T).(\lambda (x6: T).(\lambda (H12: (eq C e (CHead x3 (Flat x2) x5))).(\lambda +(H13: (drop (S n0) O x0 (CHead x4 (Flat x2) x6))).(\lambda (H14: (subst0 O v +x5 x6)).(\lambda (H15: (csubst0 O v x3 x4)).(eq_ind_r C (CHead x3 (Flat x2) +x5) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x0 (Flat f) t) c0) (ex3_4 F C +T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C +c0 (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: +T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) +w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Flat f0) u)))))) +(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S +n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u)))))) (\lambda (_: +F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) +(ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda +(u: T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat f0) u))))))) (\lambda (f0: +F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S +n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O +v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O +(CHead x0 (Flat f) t) (CHead x3 (Flat x2) x5)) (ex3_4 F C T T (\lambda (f0: +F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) +x5) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda +(_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat +f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1 +(Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda +(u: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O +v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1 +(Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat +f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) +(ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1 +(Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat +f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) +x2 x3 x4 x5 x6 (refl_equal C (CHead x3 (Flat x2) x5)) (drop_drop (Flat f) n0 +x0 (CHead x4 (Flat x2) x6) H13 t) H14 H15)) e H12)))))))))) H11)) H10))))))) +k (drop_gen_drop k c e t n0 H1) H3) c2 H4)))))) H2)) (\lambda (H2: (ex4_3 T C +nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k +j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 +k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t +u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c +c3)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: +nat).(eq nat (S n0) (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda +(_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: +C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: +C).(\lambda (j: nat).(csubst0 j v c c3)))) (or4 (drop (S n0) O c2 e) (ex3_4 F +C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C +e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: +T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) +(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: +C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 (Flat f) +u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) +u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O +v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: +C).(\lambda (x2: nat).(\lambda (H3: (eq nat (S n0) (s k x2))).(\lambda (H4: +(eq C c2 (CHead x1 k x0))).(\lambda (H5: (subst0 x2 v t x0)).(\lambda (H6: +(csubst0 x2 v c x1)).(eq_ind_r C (CHead x1 k x0) (\lambda (c0: C).(or4 (drop +(S n0) O c0 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: +T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: +F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead +e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda +(w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) +(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) +O c0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda +(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda +(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C +e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 (Flat f) +w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 +e2))))))))) (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to ((eq nat (S +n0) (s k0 x2)) \to (or4 (drop (S n0) O (CHead x1 k0 x0) e) (ex3_4 F C T T +(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e +(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: +T).(\lambda (w: T).(drop (S n0) O (CHead x1 k0 x0) (CHead e0 (Flat f) w)))))) +(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u +w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: +F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 +k0 x0) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda +(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda +(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C +e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 k0 x0) (CHead e2 +(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda +(u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 +e2))))))))))) (\lambda (b: B).(\lambda (H7: (drop (r (Bind b) n0) O c +e)).(\lambda (H8: (eq nat (S n0) (s (Bind b) x2))).(let H9 \def (f_equal nat +nat (\lambda (e0: nat).(match e0 in nat return (\lambda (_: nat).nat) with [O +\Rightarrow n0 | (S n1) \Rightarrow n1])) (S n0) (S x2) H8) in (let H10 \def +(eq_ind_r nat x2 (\lambda (n1: nat).(csubst0 n1 v c x1)) H6 n0 H9) in (let +H11 \def (eq_ind_r nat x2 (\lambda (n1: nat).(subst0 n1 v t x0)) H5 n0 H9) in +(let H12 \def (IHn c x1 v H10 e H7) in (or4_ind (drop n0 O x1 e) (ex3_4 F C T +T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e +(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: +T).(\lambda (w: T).(drop n0 O x1 (CHead e0 (Flat f) w)))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) +(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: +C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O x1 (CHead e2 (Flat f) u)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O +v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) +(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: +T).(drop n0 O x1 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead x1 (Bind b) x0) +e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda +(_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: +C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) +(CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: +F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat +f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: +T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u)))))) (\lambda +(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 +e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) +(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: +T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) w))))))) (\lambda +(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H13: +(drop n0 O x1 e)).(or4_intro0 (drop (S n0) O (CHead x1 (Bind b) x0) e) (ex3_4 +F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq +C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: +T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) +w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) +(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) +O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda +(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C +C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: +F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S +n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) w))))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O +v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Bind b) n0 x1 e H13 +x0))) (\lambda (H13: (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda +(u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: +F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x1 (CHead e0 +(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 O v u w))))))).(ex3_4_ind F C T T (\lambda (f: F).(\lambda (e0: +C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) +(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O +x1 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 O v u w))))) (or4 (drop (S n0) O (CHead x1 (Bind +b) x0) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: +T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: +F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 +(Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T +(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e +(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) +u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) +u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) +w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 +e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: T).(\lambda (x6: +T).(\lambda (H14: (eq C e (CHead x4 (Flat x3) x5))).(\lambda (H15: (drop n0 O +x1 (CHead x4 (Flat x3) x6))).(\lambda (H16: (subst0 O v x5 x6)).(eq_ind_r C +(CHead x4 (Flat x3) x5) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Bind +b) x0) c0) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: +T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f) u)))))) (\lambda (f: +F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 +(Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T +(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 +(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) +u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat +f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) +w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 +e2))))))))) (or4_intro1 (drop (S n0) O (CHead x1 (Bind b) x0) (CHead x4 (Flat +x3) x5)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: +T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x5) (CHead e0 (Flat f) u)))))) +(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) +O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda +(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T +(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead +x4 (Flat x3) x5) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: +C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) +(CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: +F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C +(CHead x4 (Flat x3) x5) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda +(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead +x1 (Bind b) x0) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C T T (\lambda (f: +F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) +x5) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: +T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) +w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 O v u w))))) x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x3) x5)) +(drop_drop (Bind b) n0 x1 (CHead x4 (Flat x3) x6) H15 x0) H16)) e H14)))))))) +H13)) (\lambda (H13: (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda +(_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: +F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O x1 (CHead e2 +(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda +(_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda +(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) +(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O +x1 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(csubst0 O v e1 e2))))) (or4 (drop (S n0) O (CHead x1 +(Bind b) x0) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: +T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: +F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 +(Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T +(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e +(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) +u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) +u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) +w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 +e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: +T).(\lambda (H14: (eq C e (CHead x4 (Flat x3) x6))).(\lambda (H15: (drop n0 O +x1 (CHead x5 (Flat x3) x6))).(\lambda (H16: (csubst0 O v x4 x5)).(eq_ind_r C +(CHead x4 (Flat x3) x6) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Bind +b) x0) c0) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: +T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f) u)))))) (\lambda (f: +F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 +(Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T +(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 +(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) +u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat +f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) +w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 +e2))))))))) (or4_intro2 (drop (S n0) O (CHead x1 (Bind b) x0) (CHead x4 (Flat +x3) x6)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: +T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0 (Flat f) u)))))) +(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) +O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda +(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T +(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead +x4 (Flat x3) x6) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: +C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) +(CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: +F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C +(CHead x4 (Flat x3) x6) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda +(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead +x1 (Bind b) x0) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C C T (\lambda (f: +F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x4 (Flat x3) +x6) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) +u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(csubst0 O v e1 e2))))) x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x3) x6)) +(drop_drop (Bind b) n0 x1 (CHead x5 (Flat x3) x6) H15 x0) H16)) e H14)))))))) +H13)) (\lambda (H13: (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) +u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (w: T).(drop n0 O x1 (CHead e2 (Flat f) w))))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O +v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda +(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C +e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x1 (CHead e2 (Flat f) w))))))) +(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) +O (CHead x1 (Bind b) x0) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: +C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) +(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) +O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda +(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T +(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e +(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) +u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) +u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) +w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 +e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: +T).(\lambda (x7: T).(\lambda (H14: (eq C e (CHead x4 (Flat x3) x6))).(\lambda +(H15: (drop n0 O x1 (CHead x5 (Flat x3) x7))).(\lambda (H16: (subst0 O v x6 +x7)).(\lambda (H17: (csubst0 O v x4 x5)).(eq_ind_r C (CHead x4 (Flat x3) x6) +(\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Bind b) x0) c0) (ex3_4 F C T +T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 +(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: +T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) +w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Flat f) u)))))) +(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) +O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda +(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C +C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat f) u))))))) (\lambda (f: +F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S +n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) w))))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O +v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O +(CHead x1 (Bind b) x0) (CHead x4 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f: +F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) +x6) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: +T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) +w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 +(Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda +(u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O +v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 +(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat +f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) +(ex4_5_intro F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 +(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat +f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) +x3 x4 x5 x6 x7 (refl_equal C (CHead x4 (Flat x3) x6)) (drop_drop (Bind b) n0 +x1 (CHead x5 (Flat x3) x7) H15 x0) H16 H17)) e H14)))))))))) H13)) +H12)))))))) (\lambda (f: F).(\lambda (H7: (drop (r (Flat f) n0) O c +e)).(\lambda (H8: (eq nat (S n0) (s (Flat f) x2))).(let H9 \def (f_equal nat +nat (\lambda (e0: nat).e0) (S n0) (s (Flat f) x2) H8) in (let H10 \def +(eq_ind_r nat x2 (\lambda (n1: nat).(csubst0 n1 v c x1)) H6 (S n0) H9) in +(let H11 \def (eq_ind_r nat x2 (\lambda (n1: nat).(subst0 n1 v t x0)) H5 (S +n0) H9) in (let H12 \def (H x1 v H10 e H7) in (or4_ind (drop (S n0) O x1 e) +(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: +T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: +C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x1 (CHead e0 (Flat f0) +w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u)))))) +(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S +n0) O x1 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T +(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda +(_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x1 (CHead +e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: +F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 +O v e1 e2))))))) (or4 (drop (S n0) O (CHead x1 (Flat f) x0) e) (ex3_4 F C T T +(\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e +(CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: +T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) +w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u)))))) +(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S +n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_: +F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) +(ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda +(u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: +F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S +n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w))))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O +v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H13: (drop (S n0) O +x1 e)).(or4_intro0 (drop (S n0) O (CHead x1 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(\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C +(CHead x4 (Flat x3) x5) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda +(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) +(CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 O v u w))))) x3 x4 x5 x6 (refl_equal C (CHead x4 +(Flat x3) x5)) (drop_drop (Flat f) n0 x1 (CHead x4 (Flat x3) x6) H15 x0) +H16)) e H14)))))))) H13)) (\lambda (H13: (ex3_4 F C C T (\lambda (f0: +F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat +f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: +T).(drop (S n0) O x1 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda +(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind +F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq +C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda +(e2: 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(CHead e1 (Flat f0) u))))))) (\lambda (f0: +F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S +n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w))))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O +v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: +C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (H14: (eq C e (CHead x4 (Flat +x3) x6))).(\lambda (H15: (drop (S n0) O x1 (CHead x5 (Flat x3) x6))).(\lambda +(H16: (csubst0 O v x4 x5)).(eq_ind_r C (CHead x4 (Flat x3) x6) (\lambda (c0: +C).(or4 (drop (S n0) O (CHead x1 (Flat f) x0) c0) (ex3_4 F C T T (\lambda +(f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 +(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda +(w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)))))) +(\lambda (_: F).(\lambda (_: 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(\lambda (f0: F).(\lambda (e0: +C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0 +(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda +(w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)))))) +(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u +w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) u)))))) +(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S +n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_: +F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) +(ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda +(u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) +u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) +w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) +(ex3_4_intro F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) u)))))) +(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S +n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_: +F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) +x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x3) x6)) (drop_drop (Flat f) n0 x1 +(CHead x5 (Flat x3) x6) H15 x0) H16)) e H14)))))))) H13)) (\lambda (H13: +(ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda +(u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: +F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S +n0) O x1 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda (f0: +F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e +(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x1 (CHead e2 (Flat f0) +w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) +(or4 (drop (S n0) O (CHead x1 (Flat f) x0) e) (ex3_4 F C T T (\lambda (f0: +F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat +f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: +T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda +(_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) +(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: +C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0) +(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: +F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e +(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) +(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: +F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 +O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda +(x6: T).(\lambda (x7: T).(\lambda (H14: (eq C e (CHead x4 (Flat x3) +x6))).(\lambda (H15: (drop (S n0) O x1 (CHead x5 (Flat x3) x7))).(\lambda +(H16: (subst0 O v x6 x7)).(\lambda (H17: (csubst0 O v x4 x5)).(eq_ind_r C +(CHead x4 (Flat x3) x6) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Flat +f) x0) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: +T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f0) u)))))) (\lambda (f0: +F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 +(Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T +(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 +(CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) +u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat +f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) +w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 +e2))))))))) (or4_intro3 (drop (S n0) O (CHead x1 (Flat f) x0) (CHead x4 (Flat +x3) x6)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: +T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0 (Flat f0) u)))))) +(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S +n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) +(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) u)))))) (\lambda (f0: +F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 +(Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T +(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda +(_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) u))))))) (\lambda +(f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: +T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w))))))) +(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex4_5_intro F C +C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) u))))))) +(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w))))))) +(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) x3 x4 x5 x6 x7 +(refl_equal C (CHead x4 (Flat x3) x6)) (drop_drop (Flat f) n0 x1 (CHead x5 +(Flat x3) x7) H15 x0) H16 H17)) e H14)))))))))) H13)) H12)))))))) k +(drop_gen_drop k c e t n0 H1) H3) c2 H4)))))))) H2)) (csubst0_gen_head k c c2 +t v (S n0) H0))))))))))) c1)))) n). +(* COMMENTS +Initial nodes: 36162 +END *) + +theorem csubst0_drop_eq_back: + \forall (n: nat).(\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 +n v c1 c2) \to (\forall (e: C).((drop n O c2 e) \to (or4 (drop n O c1 e) +(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: +T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: +C).(\lambda (u1: T).(\lambda (_: T).(drop n O c1 (CHead e0 (Flat f) u1)))))) +(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v +u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: +F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n O c1 (CHead e1 +(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda +(_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat +f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: +T).(\lambda (_: T).(drop n O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 +O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))) +\def + \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (c1: C).(\forall (c2: +C).(\forall (v: T).((csubst0 n0 v c1 c2) \to (\forall (e: C).((drop n0 O c2 +e) \to (or4 (drop n0 O c1 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: +C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) +(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O +c1 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: +T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: +F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat +f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(drop n0 O c1 (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T +(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c1 (CHead e1 +(Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda +(u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda +(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 +e2))))))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda +(H: (csubst0 O v c1 c2)).(\lambda (e: C).(\lambda (H0: (drop O O c2 +e)).(eq_ind C c2 (\lambda (c: C).(or4 (drop O O c1 c) (ex3_4 F C T T (\lambda +(f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c (CHead e0 +(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda +(_: T).(drop O O c1 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T +(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c +(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(drop O O c1 (CHead e1 (Flat f) u)))))) (\lambda (_: +F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) +(ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (u2: T).(eq C c (CHead e2 (Flat f) u2))))))) (\lambda (f: +F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop O +O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda +(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: +F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 +O v e1 e2))))))))) (insert_eq nat O (\lambda (n0: nat).(csubst0 n0 v c1 c2)) +(\lambda (n0: nat).(or4 (drop n0 n0 c1 c2) (ex3_4 F C T T (\lambda (f: +F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c2 (CHead e0 (Flat +f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: +T).(drop n0 n0 c1 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (u1: T).(\lambda (u2: T).(subst0 n0 v u1 u2)))))) (ex3_4 F C C T +(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c2 +(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(drop n0 n0 c1 (CHead e1 (Flat f) u)))))) (\lambda (_: +F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 v e1 +e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (u2: T).(eq C c2 (CHead e2 (Flat f) u2))))))) +(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda +(_: T).(drop n0 n0 c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda +(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 n0 v u1 +u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (_: T).(csubst0 n0 v e1 e2))))))))) (\lambda (y: nat).(\lambda +(H1: (csubst0 y v c1 c2)).(csubst0_ind (\lambda (n0: nat).(\lambda (t: +T).(\lambda (c: C).(\lambda (c0: C).((eq nat n0 O) \to (or4 (drop n0 n0 c c0) +(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: +T).(eq C c0 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: +C).(\lambda (u1: T).(\lambda (_: T).(drop n0 n0 c (CHead e0 (Flat f) u1)))))) +(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 n0 +t u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) (\lambda (f: +F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 n0 c (CHead e1 +(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda +(_: T).(csubst0 n0 t e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda +(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e2 +(Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda +(u1: T).(\lambda (_: T).(drop n0 n0 c (CHead e1 (Flat f) u1))))))) (\lambda +(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: +T).(subst0 n0 t u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (_: T).(csubst0 n0 t e1 e2))))))))))))) (\lambda +(k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (v0: T).(\forall +(u1: T).(\forall (u2: T).((subst0 i v0 u1 u2) \to (\forall (c: C).((eq nat (s +k0 i) O) \to (or4 (drop (s k0 i) (s k0 i) (CHead c k0 u1) (CHead c k0 u2)) +(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: +T).(eq C (CHead c k0 u2) (CHead e0 (Flat f) u4)))))) (\lambda (f: F).(\lambda +(e0: C).(\lambda (u3: T).(\lambda (_: T).(drop (s k0 i) (s k0 i) (CHead c k0 +u1) (CHead e0 (Flat f) u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: +T).(\lambda (u4: T).(subst0 (s k0 i) v0 u3 u4)))))) (ex3_4 F C C T (\lambda +(f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c k0 u2) +(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(drop (s k0 i) (s k0 i) (CHead c k0 u1) (CHead e1 (Flat f) +u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(csubst0 (s k0 i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda +(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c k0 +u2) (CHead e2 (Flat f) u4))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda +(_: C).(\lambda (u3: T).(\lambda (_: T).(drop (s k0 i) (s k0 i) (CHead c k0 +u1) (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: +C).(\lambda (u3: T).(\lambda (u4: T).(subst0 (s k0 i) v0 u3 u4)))))) (\lambda +(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: +T).(csubst0 (s k0 i) v0 e1 e2)))))))))))))))) (\lambda (b: B).(\lambda (i: +nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0 +i v0 u1 u2)).(\lambda (c: C).(\lambda (H3: (eq nat (S i) O)).(let H4 \def +(eq_ind nat (S i) (\lambda (ee: nat).(match ee in nat return (\lambda (_: +nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H3) in +(False_ind (or4 (drop (S i) (S i) (CHead c (Bind b) u1) (CHead c (Bind b) +u2)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda +(u4: T).(eq C (CHead c (Bind b) u2) (CHead e0 (Flat f) u4)))))) (\lambda (f: +F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop (S i) (S i) (CHead +c (Bind b) u1) (CHead e0 (Flat f) u3)))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (u3: T).(\lambda (u4: T).(subst0 (S i) v0 u3 u4)))))) (ex3_4 F C +C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C +(CHead c (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda +(e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S i) (S i) (CHead c (Bind b) +u1) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(csubst0 (S i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda +(f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq +C (CHead c (Bind b) u2) (CHead e2 (Flat f) u4))))))) (\lambda (f: F).(\lambda +(e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (_: T).(drop (S i) (S i) +(CHead c (Bind b) u1) (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda +(_: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 (S i) v0 u3 +u4)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (_: T).(csubst0 (S i) v0 e1 e2)))))))) H4)))))))))) (\lambda (f: +F).(\lambda (i: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: +T).(\lambda (H2: (subst0 i v0 u1 u2)).(\lambda (c: C).(\lambda (H3: (eq nat i +O)).(let H4 \def (eq_ind nat i (\lambda (n0: nat).(subst0 n0 v0 u1 u2)) H2 O +H3) in (eq_ind_r nat O (\lambda (n0: nat).(or4 (drop n0 n0 (CHead c (Flat f) +u1) (CHead c (Flat f) u2)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: +C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c (Flat f) u2) (CHead e0 +(Flat f0) u4)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u3: +T).(\lambda (_: T).(drop n0 n0 (CHead c (Flat f) u1) (CHead e0 (Flat f0) +u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: +T).(subst0 n0 v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: +C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c (Flat f) u2) (CHead e2 +(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda +(u: T).(drop n0 n0 (CHead c (Flat f) u1) (CHead e1 (Flat f0) u)))))) (\lambda +(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 v0 e1 +e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c (Flat f) u2) (CHead e2 +(Flat f0) u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u3: T).(\lambda (_: T).(drop n0 n0 (CHead c (Flat f) u1) (CHead +e1 (Flat f0) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: +C).(\lambda (u3: T).(\lambda (u4: T).(subst0 n0 v0 u3 u4)))))) (\lambda (_: +F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 +n0 v0 e1 e2))))))))) (or4_intro1 (drop O O (CHead c (Flat f) u1) (CHead c +(Flat f) u2)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: +T).(\lambda (u4: T).(eq C (CHead c (Flat f) u2) (CHead e0 (Flat f0) u4)))))) +(\lambda (f0: F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop O O +(CHead c (Flat f) u1) (CHead e0 (Flat f0) u3)))))) (\lambda (_: F).(\lambda +(_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (ex3_4 F C +C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C +(CHead c (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda +(e1: C).(\lambda (_: C).(\lambda (u: T).(drop O O (CHead c (Flat f) u1) +(CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: +F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C +(CHead c (Flat f) u2) (CHead e2 (Flat f0) u4))))))) (\lambda (f0: F).(\lambda +(e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c +(Flat f) u1) (CHead e1 (Flat f0) u3))))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 O v0 e1 e2))))))) (ex3_4_intro F C T T (\lambda (f0: +F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c (Flat f) +u2) (CHead e0 (Flat f0) u4)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda +(u3: T).(\lambda (_: T).(drop O O (CHead c (Flat f) u1) (CHead e0 (Flat f0) +u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: +T).(subst0 O v0 u3 u4))))) f c u1 u2 (refl_equal C (CHead c (Flat f) u2)) +(drop_refl (CHead c (Flat f) u1)) H4)) i H3)))))))))) k)) (\lambda (k: +K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (c3: C).(\forall (c4: +C).(\forall (v0: T).((csubst0 i v0 c3 c4) \to ((((eq nat i O) \to (or4 (drop +i i c3 c4) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: +T).(\lambda (u2: T).(eq C c4 (CHead e0 (Flat f) u2)))))) (\lambda (f: +F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop i i c3 (CHead e0 +(Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda +(u2: T).(subst0 i v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: +C).(\lambda (e2: C).(\lambda (u: T).(eq C c4 (CHead e2 (Flat f) u)))))) +(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop i i c3 +(CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(csubst0 i v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: +F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 +(CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u1: T).(\lambda (_: T).(drop i i c3 (CHead e1 (Flat f) u1))))))) +(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda +(u2: T).(subst0 i v0 u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda +(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v0 e1 e2)))))))))) \to +(\forall (u: T).((eq nat (s k0 i) O) \to (or4 (drop (s k0 i) (s k0 i) (CHead +c3 k0 u) (CHead c4 k0 u)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: +C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 k0 u) (CHead e0 (Flat f) +u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: +T).(drop (s k0 i) (s k0 i) (CHead c3 k0 u) (CHead e0 (Flat f) u1)))))) +(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (s +k0 i) v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda +(e2: C).(\lambda (u0: T).(eq C (CHead c4 k0 u) (CHead e2 (Flat f) u0)))))) +(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(drop (s k0 +i) (s k0 i) (CHead c3 k0 u) (CHead e1 (Flat f) u0)))))) (\lambda (_: +F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (s k0 i) v0 e1 +e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 k0 u) (CHead e2 (Flat f) +u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: +T).(\lambda (_: T).(drop (s k0 i) (s k0 i) (CHead c3 k0 u) (CHead e1 (Flat f) +u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: +T).(\lambda (u2: T).(subst0 (s k0 i) v0 u1 u2)))))) (\lambda (_: F).(\lambda +(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (s k0 i) v0 +e1 e2))))))))))))))))) (\lambda (b: B).(\lambda (i: nat).(\lambda (c3: +C).(\lambda (c4: C).(\lambda (v0: T).(\lambda (_: (csubst0 i v0 c3 +c4)).(\lambda (_: (((eq nat i O) \to (or4 (drop i i c3 c4) (ex3_4 F C T T +(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 +(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: +T).(\lambda (_: T).(drop i i c3 (CHead e0 (Flat f) u1)))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v0 u1 u2)))))) +(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: +T).(eq C c4 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(drop i i c3 (CHead e1 (Flat f) u)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i +v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e2 (Flat f) u2))))))) +(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda +(_: T).(drop i i c3 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v0 u1 u2)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (u: T).(\lambda (H4: (eq nat +(S i) O)).(let H5 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee in nat +return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow +True])) I O H4) in (False_ind (or4 (drop (S i) (S i) (CHead c3 (Bind b) u) +(CHead c4 (Bind b) u)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: +C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 (Bind b) u) (CHead e0 +(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda +(_: T).(drop (S i) (S i) (CHead c3 (Bind b) u) (CHead e0 (Flat f) u1)))))) +(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (S +i) v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda +(e2: C).(\lambda (u0: T).(eq C (CHead c4 (Bind b) u) (CHead e2 (Flat f) +u0)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: +T).(drop (S i) (S i) (CHead c3 (Bind b) u) (CHead e1 (Flat f) u0)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (S +i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda +(e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 (Bind b) u) (CHead +e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u1: T).(\lambda (_: T).(drop (S i) (S i) (CHead c3 (Bind b) u) +(CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: +C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (S i) v0 u1 u2)))))) (\lambda +(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: +T).(csubst0 (S i) v0 e1 e2)))))))) H5))))))))))) (\lambda (f: F).(\lambda (i: +nat).(\lambda (c3: C).(\lambda (c4: C).(\lambda (v0: T).(\lambda (H2: +(csubst0 i v0 c3 c4)).(\lambda (H3: (((eq nat i O) \to (or4 (drop i i c3 c4) +(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda +(u2: T).(eq C c4 (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: +C).(\lambda (u1: T).(\lambda (_: T).(drop i i c3 (CHead e0 (Flat f0) u1)))))) +(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i +v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(eq C c4 (CHead e2 (Flat f0) u)))))) (\lambda (f0: +F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop i i c3 (CHead e1 +(Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda +(_: T).(csubst0 i v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda +(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e2 +(Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u1: T).(\lambda (_: T).(drop i i c3 (CHead e1 (Flat f0) +u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: +T).(\lambda (u2: T).(subst0 i v0 u1 u2)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v0 e1 +e2))))))))))).(\lambda (u: T).(\lambda (H4: (eq nat i O)).(let H5 \def +(eq_ind nat i (\lambda (n0: nat).((eq nat n0 O) \to (or4 (drop n0 n0 c3 c4) +(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda +(u2: T).(eq C c4 (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: +C).(\lambda (u1: T).(\lambda (_: T).(drop n0 n0 c3 (CHead e0 (Flat f0) +u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: +T).(subst0 n0 v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: +C).(\lambda (e2: C).(\lambda (u0: T).(eq C c4 (CHead e2 (Flat f0) u0)))))) +(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(drop n0 +n0 c3 (CHead e1 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda +(e2: C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda +(f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq +C c4 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda +(_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 n0 c3 (CHead e1 (Flat f0) +u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: +T).(\lambda (u2: T).(subst0 n0 v0 u1 u2)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 n0 v0 e1 +e2)))))))))) H3 O H4) in (let H6 \def (eq_ind nat i (\lambda (n0: +nat).(csubst0 n0 v0 c3 c4)) H2 O H4) in (eq_ind_r nat O (\lambda (n0: +nat).(or4 (drop n0 n0 (CHead c3 (Flat f) u) (CHead c4 (Flat f) u)) (ex3_4 F C +T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C +(CHead c4 (Flat f) u) (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda +(e0: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 n0 (CHead c3 (Flat f) u) +(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: +T).(\lambda (u2: T).(subst0 n0 v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f0: +F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(eq C (CHead c4 (Flat f) +u) (CHead e2 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda +(_: C).(\lambda (u0: T).(drop n0 n0 (CHead c3 (Flat f) u) (CHead e1 (Flat f0) +u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 (Flat f) +u) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda +(_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 n0 (CHead c3 (Flat f) u) +(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: +C).(\lambda (u1: T).(\lambda (u2: T).(subst0 n0 v0 u1 u2)))))) (\lambda (_: +F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 +n0 v0 e1 e2))))))))) (or4_intro2 (drop O O (CHead c3 (Flat f) u) (CHead c4 +(Flat f) u)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: +T).(\lambda (u2: T).(eq C (CHead c4 (Flat f) u) (CHead e0 (Flat f0) u2)))))) +(\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop O O +(CHead c3 (Flat f) u) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda +(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (ex3_4 F C +C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(eq C +(CHead c4 (Flat f) u) (CHead e2 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda +(e1: C).(\lambda (_: C).(\lambda (u0: T).(drop O O (CHead c3 (Flat f) u) +(CHead e1 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: +F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C +(CHead c4 (Flat f) u) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda +(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop O O (CHead c3 +(Flat f) u) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 O v0 e1 e2))))))) (ex3_4_intro F C C T (\lambda (f0: +F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(eq C (CHead c4 (Flat f) +u) (CHead e2 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda +(_: C).(\lambda (u0: T).(drop O O (CHead c3 (Flat f) u) (CHead e1 (Flat f0) +u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(csubst0 O v0 e1 e2))))) f c3 c4 u (refl_equal C (CHead c4 (Flat f) u)) +(drop_refl (CHead c3 (Flat f) u)) H6)) i H4)))))))))))) k)) (\lambda (k: +K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (v0: T).(\forall (u1: +T).(\forall (u2: T).((subst0 i v0 u1 u2) \to (\forall (c3: C).(\forall (c4: +C).((csubst0 i v0 c3 c4) \to ((((eq nat i O) \to (or4 (drop i i c3 c4) (ex3_4 +F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq +C c4 (CHead e0 (Flat f) u4)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda +(u3: T).(\lambda (_: T).(drop i i c3 (CHead e0 (Flat f) u3)))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 i v0 u3 u4)))))) +(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: +T).(eq C c4 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(drop i i c3 (CHead e1 (Flat f) u)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i +v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (u4: T).(eq C c4 (CHead e2 (Flat f) u4))))))) +(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda +(_: T).(drop i i c3 (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 i v0 u3 u4)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 i v0 e1 e2)))))))))) \to ((eq nat (s k0 i) O) \to (or4 (drop +(s k0 i) (s k0 i) (CHead c3 k0 u1) (CHead c4 k0 u2)) (ex3_4 F C T T (\lambda +(f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 k0 +u2) (CHead e0 (Flat f) u4)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda +(u3: T).(\lambda (_: T).(drop (s k0 i) (s k0 i) (CHead c3 k0 u1) (CHead e0 +(Flat f) u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda +(u4: T).(subst0 (s k0 i) v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f: +F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c4 k0 u2) +(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(drop (s k0 i) (s k0 i) (CHead c3 k0 u1) (CHead e1 (Flat +f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(csubst0 (s k0 i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda +(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 k0 +u2) (CHead e2 (Flat f) u4))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda +(_: C).(\lambda (u3: T).(\lambda (_: T).(drop (s k0 i) (s k0 i) (CHead c3 k0 +u1) (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: +C).(\lambda (u3: T).(\lambda (u4: T).(subst0 (s k0 i) v0 u3 u4)))))) (\lambda +(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: +T).(csubst0 (s k0 i) v0 e1 e2))))))))))))))))))) (\lambda (b: B).(\lambda (i: +nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0 +i v0 u1 u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (_: (csubst0 i v0 c3 +c4)).(\lambda (_: (((eq nat i O) \to (or4 (drop i i c3 c4) (ex3_4 F C T T +(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C c4 +(CHead e0 (Flat f) u4)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u3: +T).(\lambda (_: T).(drop i i c3 (CHead e0 (Flat f) u3)))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 i v0 u3 u4)))))) +(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: +T).(eq C c4 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(drop i i c3 (CHead e1 (Flat f) u)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i +v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (u4: T).(eq C c4 (CHead e2 (Flat f) u4))))))) +(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda +(_: T).(drop i i c3 (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 i v0 u3 u4)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (H5: (eq nat (S i) O)).(let H6 +\def (eq_ind nat (S i) (\lambda (ee: nat).(match ee in nat return (\lambda +(_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H5) +in (False_ind (or4 (drop (S i) (S i) (CHead c3 (Bind b) u1) (CHead c4 (Bind +b) u2)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: +T).(\lambda (u4: T).(eq C (CHead c4 (Bind b) u2) (CHead e0 (Flat f) u4)))))) +(\lambda (f: F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop (S i) +(S i) (CHead c3 (Bind b) u1) (CHead e0 (Flat f) u3)))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 (S i) v0 u3 +u4)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(eq C (CHead c4 (Bind b) u2) (CHead e2 (Flat f) u)))))) +(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S i) +(S i) (CHead c3 (Bind b) u1) (CHead e1 (Flat f) u)))))) (\lambda (_: +F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (S i) v0 e1 +e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 (Bind b) u2) (CHead e2 +(Flat f) u4))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda +(u3: T).(\lambda (_: T).(drop (S i) (S i) (CHead c3 (Bind b) u1) (CHead e1 +(Flat f) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda +(u3: T).(\lambda (u4: T).(subst0 (S i) v0 u3 u4)))))) (\lambda (_: +F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 +(S i) v0 e1 e2)))))))) H6))))))))))))) (\lambda (f: F).(\lambda (i: +nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 +i v0 u1 u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (H3: (csubst0 i v0 c3 +c4)).(\lambda (H4: (((eq nat i O) \to (or4 (drop i i c3 c4) (ex3_4 F C T T +(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C c4 +(CHead e0 (Flat f0) u4)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u3: +T).(\lambda (_: T).(drop i i c3 (CHead e0 (Flat f0) u3)))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 i v0 u3 u4)))))) +(ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: +T).(eq C c4 (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(drop i i c3 (CHead e1 (Flat f0) u)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i +v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda +(e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C c4 (CHead e2 (Flat f0) +u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: +T).(\lambda (_: T).(drop i i c3 (CHead e1 (Flat f0) u3))))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 +i v0 u3 u4)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (H5: (eq nat i +O)).(let H6 \def (eq_ind nat i (\lambda (n0: nat).((eq nat n0 O) \to (or4 +(drop n0 n0 c3 c4) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda +(_: T).(\lambda (u4: T).(eq C c4 (CHead e0 (Flat f0) u4)))))) (\lambda (f0: +F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop n0 n0 c3 (CHead e0 +(Flat f0) u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda +(u4: T).(subst0 n0 v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda +(_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c4 (CHead e2 (Flat f0) u)))))) +(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 n0 +c3 (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: +F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C c4 +(CHead e2 (Flat f0) u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u3: T).(\lambda (_: T).(drop n0 n0 c3 (CHead e1 (Flat f0) +u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u3: +T).(\lambda (u4: T).(subst0 n0 v0 u3 u4)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 n0 v0 e1 +e2)))))))))) H4 O H5) in (let H7 \def (eq_ind nat i (\lambda (n0: +nat).(csubst0 n0 v0 c3 c4)) H3 O H5) in (let H8 \def (eq_ind nat i (\lambda +(n0: nat).(subst0 n0 v0 u1 u2)) H2 O H5) in (eq_ind_r nat O (\lambda (n0: +nat).(or4 (drop n0 n0 (CHead c3 (Flat f) u1) (CHead c4 (Flat f) u2)) (ex3_4 F +C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq +C (CHead c4 (Flat f) u2) (CHead e0 (Flat f0) u4)))))) (\lambda (f0: +F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop n0 n0 (CHead c3 +(Flat f) u1) (CHead e0 (Flat f0) u3)))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (u3: T).(\lambda (u4: T).(subst0 n0 v0 u3 u4)))))) (ex3_4 F C C T +(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C +(CHead c4 (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda +(e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 n0 (CHead c3 (Flat f) u1) +(CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: +F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C +(CHead c4 (Flat f) u2) (CHead e2 (Flat f0) u4))))))) (\lambda (f0: +F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (_: T).(drop n0 +n0 (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u3))))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 +n0 v0 u3 u4)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (_: T).(csubst0 n0 v0 e1 e2))))))))) (or4_intro3 (drop O O +(CHead c3 (Flat f) u1) (CHead c4 (Flat f) u2)) (ex3_4 F C T T (\lambda (f0: +F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 (Flat f) +u2) (CHead e0 (Flat f0) u4)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda +(u3: T).(\lambda (_: T).(drop O O (CHead c3 (Flat f) u1) (CHead e0 (Flat f0) +u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: +T).(subst0 O v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: +C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c4 (Flat f) u2) (CHead e2 +(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda +(u: T).(drop O O (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u)))))) (\lambda +(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 +e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 (Flat f) u2) (CHead e2 +(Flat f0) u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c3 (Flat f) u1) (CHead +e1 (Flat f0) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: +C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (\lambda (_: +F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 +O v0 e1 e2))))))) (ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (_: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 (Flat f) +u2) (CHead e2 (Flat f0) u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda +(_: C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c3 (Flat f) u1) +(CHead e1 (Flat f0) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: +C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (\lambda (_: +F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 +O v0 e1 e2)))))) f c3 c4 u1 u2 (refl_equal C (CHead c4 (Flat f) u2)) +(drop_refl (CHead c3 (Flat f) u1)) H8 H7)) i H5))))))))))))))) k)) y v c1 c2 +H1))) H) e (drop_gen_refl c2 e H0)))))))) (\lambda (n0: nat).(\lambda (IHn: +((\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 n0 v c1 c2) \to +(\forall (e: C).((drop n0 O c2 e) \to (or4 (drop n0 O c1 e) (ex3_4 F C T T +(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e +(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: +T).(\lambda (_: T).(drop n0 O c1 (CHead e0 (Flat f) u1)))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) +(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: +T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c1 (CHead e1 (Flat f) u)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O +v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) +(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda +(_: T).(drop n0 O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda +(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 +u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))))).(\lambda (c1: C).(C_ind +(\lambda (c: C).(\forall (c2: C).(\forall (v: T).((csubst0 (S n0) v c c2) \to +(\forall (e: C).((drop (S n0) O c2 e) \to (or4 (drop (S n0) O c e) (ex3_4 F C +T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e +(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: +T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f) u1)))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) +(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: +T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O c (CHead e1 (Flat f) +u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat +f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: +T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f) u1))))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 +O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))) (\lambda (n1: +nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (H: (csubst0 (S n0) v (CSort +n1) c2)).(\lambda (e: C).(\lambda (_: (drop (S n0) O c2 e)).(csubst0_gen_sort +c2 v (S n0) n1 H (or4 (drop (S n0) O (CSort n1) e) (ex3_4 F C T T (\lambda +(f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 +(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda +(_: T).(drop (S n0) O (CSort n1) (CHead e0 (Flat f) u1)))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) +(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: +T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CSort n1) (CHead e1 (Flat +f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat +f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: +T).(\lambda (_: T).(drop (S n0) O (CSort n1) (CHead e1 (Flat f) u1))))))) +(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda +(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda +(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))))))) +(\lambda (c: C).(\lambda (H: ((\forall (c2: C).(\forall (v: T).((csubst0 (S +n0) v c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (or4 (drop (S n0) O +c e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda +(u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: +C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f) +u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: +T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: +C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) +(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) +O c (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: +F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e +(CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f) +u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: +T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 +e2))))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda +(v: T).(\lambda (H0: (csubst0 (S n0) v (CHead c k t) c2)).(\lambda (e: +C).(\lambda (H1: (drop (S n0) O c2 e)).(or3_ind (ex3_2 T nat (\lambda (_: +T).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (u2: T).(\lambda (_: +nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j +v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k +j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda +(c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_: +T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k j))))) (\lambda (u2: +T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda +(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: +T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))) (or4 (drop (S +n0) O (CHead c k t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: +C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) +(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S +n0) O (CHead c k t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T +(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e +(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O +v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) +(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda +(_: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u1))))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 +O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H2: (ex3_2 T nat +(\lambda (_: T).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (u2: +T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: +nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: +nat).(eq nat (S n0) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 +(CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))) (or4 +(drop (S n0) O (CHead c k t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: +C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) +(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S +n0) O (CHead c k t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T +(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e +(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O +v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) +(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda +(_: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u1))))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 +O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: T).(\lambda +(x1: nat).(\lambda (H3: (eq nat (S n0) (s k x1))).(\lambda (H4: (eq C c2 +(CHead c k x0))).(\lambda (H5: (subst0 x1 v t x0)).(let H6 \def (eq_ind C c2 +(\lambda (c0: C).(drop (S n0) O c0 e)) H1 (CHead c k x0) H4) in (K_ind +(\lambda (k0: K).((eq nat (S n0) (s k0 x1)) \to ((drop (r k0 n0) O c e) \to +(or4 (drop (S n0) O (CHead c k0 t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda +(e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) +(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S +n0) O (CHead c k0 t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T +(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e +(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(drop (S n0) O (CHead c k0 t) (CHead e1 (Flat f) u)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O +v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) +(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda +(_: T).(drop (S n0) O (CHead c k0 t) (CHead e1 (Flat f) u1))))))) (\lambda +(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: +T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))) (\lambda (b: +B).(\lambda (H7: (eq nat (S n0) (s (Bind b) x1))).(\lambda (H8: (drop (r +(Bind b) n0) O c e)).(let H9 \def (f_equal nat nat (\lambda (e0: nat).(match +e0 in nat return (\lambda (_: nat).nat) with [O \Rightarrow n0 | (S n1) +\Rightarrow n1])) (S n0) (S x1) H7) in (let H10 \def (eq_ind_r nat x1 +(\lambda (n1: nat).(subst0 n1 v t x0)) H5 n0 H9) in (or4_intro0 (drop (S n0) +O (CHead c (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: +C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) +(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S +n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) +(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: +T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead +e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: +F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e +(CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) +(CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: +C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: +F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 +O v e1 e2))))))) (drop_drop (Bind b) n0 c e H8 t))))))) (\lambda (f: +F).(\lambda (H7: (eq nat (S n0) (s (Flat f) x1))).(\lambda (H8: (drop (r +(Flat f) n0) O c e)).(let H9 \def (f_equal nat nat (\lambda (e0: nat).e0) (S +n0) x1 H7) in (let H10 \def (eq_ind_r nat x1 (\lambda (n1: nat).(subst0 n1 v +t x0)) H5 (S n0) H9) in (or4_intro0 (drop (S n0) O (CHead c (Flat f) t) e) +(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda +(u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: +C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) +(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: +T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: +F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat +f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda +(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 +e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) +(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda +(_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) +(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda +(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda +(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop +(Flat f) n0 c e H8 t))))))) k H3 (drop_gen_drop k c e x0 n0 H6)))))))) H2)) +(\lambda (H2: (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) +(s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) +(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))).(ex3_2_ind C nat +(\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (c3: +C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: +nat).(csubst0 j v c c3))) (or4 (drop (S n0) O (CHead c k t) e) (ex3_4 F C T T +(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e +(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: +T).(\lambda (_: T).(drop (S n0) O (CHead c k t) (CHead e0 (Flat f) u1)))))) +(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v +u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: +F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c k +t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: +F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e +(CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k t) (CHead e1 +(Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda +(u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda +(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 +e2)))))))) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H3: (eq nat (S n0) +(s k x1))).(\lambda (H4: (eq C c2 (CHead x0 k t))).(\lambda (H5: (csubst0 x1 +v c x0)).(let H6 \def (eq_ind C c2 (\lambda (c0: C).(drop (S n0) O c0 e)) H1 +(CHead x0 k t) H4) in (K_ind (\lambda (k0: K).((eq nat (S n0) (s k0 x1)) \to +((drop (r k0 n0) O x0 e) \to (or4 (drop (S n0) O (CHead c k0 t) e) (ex3_4 F C +T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e +(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: +T).(\lambda (_: T).(drop (S n0) O (CHead c k0 t) (CHead e0 (Flat f) u1)))))) +(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v +u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: +F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c +k0 t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda +(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda +(f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq +C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda +(_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k0 t) (CHead +e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: +C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: +F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 +O v e1 e2))))))))))) (\lambda (b: B).(\lambda (H7: (eq nat (S n0) (s (Bind b) +x1))).(\lambda (H8: (drop (r (Bind b) n0) O x0 e)).(let H9 \def (f_equal nat +nat (\lambda (e0: nat).(match e0 in nat return (\lambda (_: nat).nat) with [O +\Rightarrow n0 | (S n1) \Rightarrow n1])) (S n0) (S x1) H7) in (let H10 \def +(eq_ind_r nat x1 (\lambda (n1: nat).(csubst0 n1 v c x0)) H5 n0 H9) in (let +H11 \def (IHn c x0 v H10 e H8) in (or4_ind (drop n0 O c e) (ex3_4 F C T T +(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e +(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: +T).(\lambda (_: T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) +(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: +T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1 (Flat f) u)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O +v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) +(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda +(_: T).(drop n0 O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e) +(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: +T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: +C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) +(CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: +T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: +F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat +f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda +(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 +e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) +(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda +(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) +(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda +(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda +(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda +(H12: (drop n0 O c e)).(or4_intro0 (drop (S n0) O (CHead c (Bind b) t) e) +(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: +T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: +C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) +(CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: +T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: +F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat +f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda +(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 +e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) +(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda +(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) +(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda +(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda +(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop +(Bind b) n0 c e H12 t))) (\lambda (H12: (ex3_4 F C T T (\lambda (f: +F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat +f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: +T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))).(ex3_4_ind F C +T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e +(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: +T).(\lambda (_: T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))) +(or4 (drop (S n0) O (CHead c (Bind b) t) e) (ex3_4 F C T T (\lambda (f: +F).(\lambda (e0: 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F).(\lambda (e0: C).(\lambda (_: +T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0 (Flat f) u2)))))) +(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S +n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))) +x2 x3 x4 x5 (refl_equal C (CHead x3 (Flat x2) x5)) (drop_drop (Bind b) n0 c +(CHead x3 (Flat x2) x4) H14 t) H15)) e H13)))))))) H12)) (\lambda (H12: +(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: +T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1 (Flat f) u)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O +v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda (_: C).(\lambda +(e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: +F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c 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u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: +C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) +(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) +O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda +(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C +C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f: +F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S +n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 +O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro2 (drop (S n0) O +(CHead c (Bind b) t) (CHead x4 (Flat x2) x5)) (ex3_4 F C T T (\lambda (f: +F).(\lambda (e0: C).(\lambda (_: 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C).(\lambda (_: C).(\lambda (u1: +T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) +(ex3_4_intro F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 (Flat f) u)))))) +(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) +O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda +(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) x2 x3 x4 x5 +(refl_equal C (CHead x4 (Flat x2) x5)) (drop_drop (Bind b) n0 c (CHead x3 +(Flat x2) x5) H14 t) H15)) e H13)))))))) H12)) (\lambda (H12: (ex4_5 F C C T +T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c (CHead e1 +(Flat f) u1))))))) (\lambda (_: F).(\lambda (_: 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C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c +(Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead c (Bind +b) t) (CHead x4 (Flat x2) x6)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: +C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e0 +(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda +(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) +(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v +u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 (Flat f) u)))))) +(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) +O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda +(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C +C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 (Flat f) +u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: +T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) +u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: +T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) +(ex4_5_intro F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 +(Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda +(u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat +f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: +T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) +x2 x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x2) x6)) (drop_drop (Bind b) n0 +c (CHead x3 (Flat x2) x5) H14 t) H15 H16)) e H13)))))))))) H12)) H11))))))) +(\lambda (f: F).(\lambda (H7: (eq nat (S n0) (s (Flat f) x1))).(\lambda (H8: +(drop (r (Flat f) n0) O x0 e)).(let H9 \def (f_equal nat nat (\lambda (e0: +nat).e0) (S n0) x1 H7) in (let H10 \def (eq_ind_r nat x1 (\lambda (n1: +nat).(csubst0 n1 v c x0)) H5 (S n0) H9) in (let H11 \def (H x0 v H10 e H8) in +(or4_ind (drop (S n0) O c e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: +C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) +(\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: 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(ex3_4 F C T T +(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e +(CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: +T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) +u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: +T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: +C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) +(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S +n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: +F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) +(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: +F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S +n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: 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(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: +F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S +n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 +O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Flat f) n0 c e +H12 t))) (\lambda (H12: (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: +C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) +(\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S +n0) O c (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda +(u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))).(ex3_4_ind F C T T (\lambda +(f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 +(Flat f0) u2)))))) 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(Flat f0) u1)))))) (\lambda +(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 +u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0: +F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c +(Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T +(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(u2: T).(eq C c0 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c +(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 O v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead c 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T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) +(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: +C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: +F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 +O v e1 e2))))))) (ex3_4_intro F C T T (\lambda (f0: F).(\lambda (e0: +C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0 +(Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: +T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) +u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: +T).(subst0 O v u1 u2))))) x2 x3 x4 x5 (refl_equal C (CHead x3 (Flat x2) x5)) +(drop_drop (Flat f) n0 c (CHead x3 (Flat x2) x4) H14 t) H15)) e H13)))))))) +H12)) (\lambda (H12: (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda +(e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: +F).(\lambda (e1: C).(\lambda (_: 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(\lambda (f0: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead +e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: +F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e +(CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) +(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: +C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: +F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 +O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda +(x5: T).(\lambda (H13: (eq C e (CHead x4 (Flat x2) x5))).(\lambda (H14: (drop +(S n0) O c (CHead x3 (Flat x2) x5))).(\lambda (H15: (csubst0 O v x3 +x4)).(eq_ind_r C (CHead x4 (Flat x2) x5) (\lambda (c0: C).(or4 (drop (S n0) O +(CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: +C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat f0) u2)))))) +(\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S +n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) +(ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: +T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead +e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: +F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 +(CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) +(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: +C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: +F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 +O v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead c (Flat f) t) (CHead x4 +(Flat x2) x5)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: +T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x5) (CHead e0 (Flat f0) +u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: +T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda +(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 +u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 (Flat f0) u)))))) +(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S +n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: +F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) +(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 (Flat f0) +u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: +T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) +u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: +T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) +(ex3_4_intro F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 (Flat f0) u)))))) +(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S +n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: +F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) +x2 x3 x4 x5 (refl_equal C (CHead x4 (Flat x2) x5)) (drop_drop (Flat f) n0 c +(CHead x3 (Flat x2) x5) H14 t) H15)) e H13)))))))) H12)) (\lambda (H12: +(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: +F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S +n0) O c (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda (f0: +F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e +(CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f0) +u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: +T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) +(or4 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: +F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat +f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: +T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda +(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 +u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: +F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c +(Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T +(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c +(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda +(x4: C).(\lambda (x5: T).(\lambda (x6: T).(\lambda (H13: (eq C e (CHead x4 +(Flat x2) x6))).(\lambda (H14: (drop (S n0) O c (CHead x3 (Flat x2) +x5))).(\lambda (H15: (subst0 O v x5 x6)).(\lambda (H16: (csubst0 O v x3 +x4)).(eq_ind_r C (CHead x4 (Flat x2) x6) (\lambda (c0: C).(or4 (drop (S n0) O +(CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: +C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat f0) u2)))))) +(\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S +n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) +(ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: +T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead +e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: +F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 +(CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) +(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: +C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: +F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 +O v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead c (Flat f) t) (CHead x4 +(Flat x2) x6)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: +T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e0 (Flat f0) +u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: +T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda +(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 +u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 (Flat f0) u)))))) +(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S +n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: +F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) +(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 (Flat f0) +u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: +T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) +u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: +T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) +(ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 +(Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) +(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: +C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: +F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 +O v e1 e2)))))) x2 x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x2) x6)) +(drop_drop (Flat f) n0 c (CHead x3 (Flat x2) x5) H14 t) H15 H16)) e +H13)))))))))) H12)) H11))))))) k H3 (drop_gen_drop k x0 e t n0 H6)))))))) +H2)) (\lambda (H2: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda +(j: nat).(eq nat (S n0) (s k j))))) (\lambda (u2: T).(\lambda (c3: +C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda +(_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: +C).(\lambda (j: nat).(csubst0 j v c c3)))))).(ex4_3_ind T C nat (\lambda (_: +T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k j))))) (\lambda (u2: +T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda +(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: +T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (or4 (drop (S n0) +O (CHead c k t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda +(_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: +F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c +k t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda +(u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: +F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat +f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u)))))) (\lambda (_: +F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) +(ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: +F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S +n0) O (CHead c k t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: C).(\lambda +(x2: nat).(\lambda (H3: (eq nat (S n0) (s k x2))).(\lambda (H4: (eq C c2 +(CHead x1 k x0))).(\lambda (H5: (subst0 x2 v t x0)).(\lambda (H6: (csubst0 x2 +v c x1)).(let H7 \def (eq_ind C c2 (\lambda (c0: C).(drop (S n0) O c0 e)) H1 +(CHead x1 k x0) H4) in (K_ind (\lambda (k0: K).((eq nat (S n0) (s k0 x2)) \to +((drop (r k0 n0) O x1 e) \to (or4 (drop (S n0) O (CHead c k0 t) e) (ex3_4 F C +T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e +(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: +T).(\lambda (_: T).(drop (S n0) O (CHead c k0 t) (CHead e0 (Flat f) u1)))))) +(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v +u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: +F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c +k0 t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda +(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda +(f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq +C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda +(_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k0 t) (CHead +e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: +C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: +F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 +O v e1 e2))))))))))) (\lambda (b: B).(\lambda (H8: (eq nat (S n0) (s (Bind b) +x2))).(\lambda (H9: (drop (r (Bind b) n0) O x1 e)).(let H10 \def (f_equal nat +nat (\lambda (e0: nat).(match e0 in nat return (\lambda (_: nat).nat) with [O +\Rightarrow n0 | (S n1) \Rightarrow n1])) (S n0) (S x2) H8) in (let H11 \def +(eq_ind_r nat x2 (\lambda (n1: nat).(csubst0 n1 v c x1)) H6 n0 H10) in (let +H12 \def (eq_ind_r nat x2 (\lambda (n1: nat).(subst0 n1 v t x0)) H5 n0 H10) +in (let H13 \def (IHn c x1 v H11 e H9) in (or4_ind (drop n0 O c e) (ex3_4 F C +T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e +(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: +T).(\lambda (_: T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) +(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: +T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1 (Flat f) u)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O +v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) +(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda +(_: T).(drop n0 O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e) +(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: +T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: +C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) +(CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: +T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: +F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat +f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda +(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 +e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) +(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda +(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) +(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda +(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda +(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda +(H14: (drop n0 O c e)).(or4_intro0 (drop (S n0) O (CHead c (Bind b) t) e) +(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: +T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: +C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) +(CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: +T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: +F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat +f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda +(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 +e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) +(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda +(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) +(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda +(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda +(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop +(Bind b) n0 c e H14 t))) (\lambda (H14: (ex3_4 F C T T (\lambda (f: +F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat +f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: +T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))).(ex3_4_ind F C +T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e +(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: +T).(\lambda (_: T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))) +(or4 (drop (S n0) O (CHead c (Bind b) t) e) (ex3_4 F C T T (\lambda (f: +F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat +f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: +T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda +(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 +u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: +F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c +(Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T +(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c +(Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda +(x5: T).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x4 (Flat x3) +x6))).(\lambda (H16: (drop n0 O c (CHead x4 (Flat x3) x5))).(\lambda (H17: +(subst0 O v x5 x6)).(eq_ind_r C (CHead x4 (Flat x3) x6) (\lambda (c0: C).(or4 +(drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T (\lambda (f: +F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat +f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: 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(Bind b) n0 c (CHead x4 +(Flat x3) x6) H16 t) H17)) e H15)))))))) H14)) (\lambda (H14: (ex4_5 F C C T +T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c (CHead e1 +(Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda +(u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda +(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 +e2)))))))).(ex4_5_ind F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) +(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda +(_: T).(drop n0 O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 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(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) (\lambda (f: +F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c +(Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T +(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c +(Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead c (Bind +b) t) (CHead x5 (Flat x3) x7)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: +C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e0 +(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda +(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) +(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v +u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2 (Flat f) u)))))) +(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) +O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda +(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C +C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2 (Flat f) +u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: +T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) +u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: +T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) +(ex4_5_intro F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2 +(Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda +(u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat +f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: +T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) +x3 x4 x5 x6 x7 (refl_equal C (CHead x5 (Flat x3) x7)) (drop_drop (Bind b) n0 +c (CHead x4 (Flat x3) x6) H16 t) H17 H18)) e H15)))))))))) H14)) H13)))))))) +(\lambda (f: F).(\lambda (H8: (eq nat (S n0) (s (Flat f) x2))).(\lambda (H9: +(drop (r (Flat f) n0) O x1 e)).(let H10 \def (f_equal nat nat (\lambda (e0: +nat).e0) (S n0) x2 H8) in (let H11 \def (eq_ind_r nat x2 (\lambda (n1: +nat).(csubst0 n1 v c x1)) H6 (S n0) H10) in (let H12 \def (eq_ind_r nat x2 +(\lambda (n1: nat).(subst0 n1 v t x0)) H5 (S n0) H10) in (let H13 \def (H x1 +v H11 e H9) in (or4_ind (drop (S n0) O c e) (ex3_4 F C T T (\lambda (f0: +F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat +f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: +T).(drop (S n0) O c (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T +(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e +(CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(drop (S n0) O c (CHead e1 (Flat f0) u)))))) (\lambda (_: +F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) +(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: +F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S +n0) O c (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead c (Flat f) t) e) +(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda +(u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: +C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) +(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: +T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: +F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat +f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda +(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 +e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) +(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda +(_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) +(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda +(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda +(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda +(H14: (drop (S n0) O c e)).(or4_intro0 (drop (S n0) O (CHead c (Flat f) t) e) +(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda +(u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: +C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) +(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: +T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: +F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat +f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda +(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 +e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) +(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda +(_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) +(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda +(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda +(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop +(Flat f) n0 c e H14 t))) (\lambda (H14: (ex3_4 F C T T (\lambda (f0: +F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat +f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: +T).(drop (S n0) O c (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))).(ex3_4_ind F C +T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C +e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda +(u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f0) u1)))))) +(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v +u1 u2))))) (or4 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T +(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e +(CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: +T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) +u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: +T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: +C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) +(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S +n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: +F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) +(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: +F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S +n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 +O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x3: F).(\lambda +(x4: C).(\lambda (x5: T).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x4 +(Flat x3) x6))).(\lambda (H16: (drop (S n0) O c (CHead x4 (Flat x3) +x5))).(\lambda (H17: (subst0 O v x5 x6)).(eq_ind_r C (CHead x4 (Flat x3) x6) +(\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Flat f) t) c0) (ex3_4 F C T T +(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 +(CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: +T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) +u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: +T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: +C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f0) u)))))) +(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S +n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: +F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) +(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: +F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S +n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 +O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro1 (drop (S n0) O +(CHead c (Flat f) t) (CHead x4 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f0: +F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat +x3) x6) (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: +C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) +(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: +T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: +F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x4 (Flat x3) +x6) (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda +(_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat +f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat +x3) x6) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c +(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C T T (\lambda (f0: +F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat +x3) x6) (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: +C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) +(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: +T).(\lambda (u2: T).(subst0 O v u1 u2))))) x3 x4 x5 x6 (refl_equal C (CHead +x4 (Flat x3) x6)) (drop_drop (Flat f) n0 c (CHead x4 (Flat x3) x5) H16 t) +H17)) e H15)))))))) H14)) (\lambda (H14: (ex3_4 F C C T (\lambda (f0: +F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat +f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(drop (S n0) O c (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C +C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e +(CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(drop (S n0) O c (CHead e1 (Flat f0) u)))))) (\lambda (_: +F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) +(or4 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: +F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat +f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: +T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda +(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 +u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: +F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c +(Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T +(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c +(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda +(x5: C).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x5 (Flat x3) +x6))).(\lambda (H16: (drop (S n0) O c (CHead x4 (Flat x3) x6))).(\lambda +(H17: (csubst0 O v x4 x5)).(eq_ind_r C (CHead x5 (Flat x3) x6) (\lambda (c0: +C).(or4 (drop (S n0) O (CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: +F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat +f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: +T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda +(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 +u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0: +F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c +(Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T +(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(u2: T).(eq C c0 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c +(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 O v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead c (Flat +f) t) (CHead x5 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: +C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x6) (CHead e0 +(Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: +T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) +u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: +T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: +C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2 +(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda +(u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O +v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2 +(Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) +(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: +C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: +F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 +O v e1 e2))))))) (ex3_4_intro F C C T (\lambda (f0: F).(\lambda (_: +C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2 +(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda +(u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O +v e1 e2))))) x3 x4 x5 x6 (refl_equal C (CHead x5 (Flat x3) x6)) (drop_drop +(Flat f) n0 c (CHead x4 (Flat x3) x6) H16 t) H17)) e H15)))))))) H14)) +(\lambda (H14: (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda +(e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) +u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: +T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f0) u1))))))) (\lambda +(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: +T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C +C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: +F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S +n0) O c (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) O (CHead c (Flat f) t) e) +(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda +(u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: +C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) +(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: +T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: +F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat +f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda +(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 +e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) +(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda +(_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) +(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda +(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda +(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda +(x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (x7: +T).(\lambda (H15: (eq C e (CHead x5 (Flat x3) x7))).(\lambda (H16: (drop (S +n0) O c (CHead x4 (Flat x3) x6))).(\lambda (H17: (subst0 O v x6 x7)).(\lambda +(H18: (csubst0 O v x4 x5)).(eq_ind_r C (CHead x5 (Flat x3) x7) (\lambda (c0: +C).(or4 (drop (S n0) O (CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: +F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat +f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: +T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda +(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 +u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0: +F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c +(Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T +(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(u2: T).(eq C c0 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c +(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead c (Flat +f) t) (CHead x5 (Flat x3) x7)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: +C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e0 +(Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: +T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) +u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: +T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: +C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2 +(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda +(u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O +v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2 +(Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) +(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: +C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: +F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 +O v e1 e2))))))) (ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (_: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat +x3) x7) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c +(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: +C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) +(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 O v e1 e2)))))) x3 x4 x5 x6 x7 (refl_equal C (CHead x5 (Flat +x3) x7)) (drop_drop (Flat f) n0 c (CHead x4 (Flat x3) x6) H16 t) H17 H18)) e +H15)))))))))) H14)) H13)))))))) k H3 (drop_gen_drop k x1 e x0 n0 H7)))))))))) +H2)) (csubst0_gen_head k c c2 t v (S n0) H0))))))))))) c1)))) n). +(* COMMENTS +Initial nodes: 34765 +END *) + +theorem csubst0_drop_lt_back: + \forall (n: nat).(\forall (i: nat).((lt n i) \to (\forall (c1: C).(\forall +(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e2: C).((drop n O +c2 e2) \to (or (drop n O c1 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) +v e1 e2)) (\lambda (e1: C).(drop n O c1 e1)))))))))))) +\def + \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (i: nat).((lt n0 i) +\to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) +\to (\forall (e2: C).((drop n0 O c2 e2) \to (or (drop n0 O c1 e2) (ex2 C +(\lambda (e1: C).(csubst0 (minus i n0) v e1 e2)) (\lambda (e1: C).(drop n0 O +c1 e1))))))))))))) (\lambda (i: nat).(\lambda (_: (lt O i)).(\lambda (c1: +C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 i v c1 +c2)).(\lambda (e2: C).(\lambda (H1: (drop O O c2 e2)).(eq_ind C c2 (\lambda +(c: C).(or (drop O O c1 c) (ex2 C (\lambda (e1: C).(csubst0 (minus i O) v e1 +c)) (\lambda (e1: C).(drop O O c1 e1))))) (eq_ind nat i (\lambda (n0: +nat).(or (drop O O c1 c2) (ex2 C (\lambda (e1: C).(csubst0 n0 v e1 c2)) +(\lambda (e1: C).(drop O O c1 e1))))) (or_intror (drop O O c1 c2) (ex2 C +(\lambda (e1: C).(csubst0 i v e1 c2)) (\lambda (e1: C).(drop O O c1 e1))) +(ex_intro2 C (\lambda (e1: C).(csubst0 i v e1 c2)) (\lambda (e1: C).(drop O O +c1 e1)) c1 H0 (drop_refl c1))) (minus i O) (minus_n_O i)) e2 (drop_gen_refl +c2 e2 H1)))))))))) (\lambda (n0: nat).(\lambda (IHn: ((\forall (i: nat).((lt +n0 i) \to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 +c2) \to (\forall (e2: C).((drop n0 O c2 e2) \to (or (drop n0 O c1 e2) (ex2 C +(\lambda (e1: C).(csubst0 (minus i n0) v e1 e2)) (\lambda (e1: C).(drop n0 O +c1 e1)))))))))))))).(\lambda (i: nat).(\lambda (H: (lt (S n0) i)).(\lambda +(c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v +c c2) \to (\forall (e2: C).((drop (S n0) O c2 e2) \to (or (drop (S n0) O c +e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i (S n0)) v e1 e2)) (\lambda (e1: +C).(drop (S n0) O c e1)))))))))) (\lambda (n1: nat).(\lambda (c2: C).(\lambda +(v: T).(\lambda (H0: (csubst0 i v (CSort n1) c2)).(\lambda (e2: C).(\lambda +(_: (drop (S n0) O c2 e2)).(csubst0_gen_sort c2 v i n1 H0 (or (drop (S n0) O +(CSort n1) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i (S n0)) v e1 e2)) +(\lambda (e1: C).(drop (S n0) O (CSort n1) e1))))))))))) (\lambda (c: +C).(\lambda (H0: ((\forall (c2: C).(\forall (v: T).((csubst0 i v c c2) \to +(\forall (e2: C).((drop (S n0) O c2 e2) \to (or (drop (S n0) O c e2) (ex2 C +(\lambda (e1: C).(csubst0 (minus i (S n0)) v e1 e2)) (\lambda (e1: C).(drop +(S n0) O c e1))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: +C).(\lambda (v: T).(\lambda (H1: (csubst0 i v (CHead c k t) c2)).(\lambda +(e2: C).(\lambda (H2: (drop (S n0) O c2 e2)).(or3_ind (ex3_2 T nat (\lambda +(_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: +nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j +v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k +j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda +(c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_: +T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: +T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda +(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: +T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))) (or (drop (S n0) +O (CHead c k t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i (S n0)) v e1 +e2)) (\lambda (e1: C).(drop (S n0) O (CHead c k t) e1)))) (\lambda (H3: +(ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda +(u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: +T).(\lambda (j: nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: +T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: +nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j +v t u2))) (or (drop (S n0) O (CHead c k t) e2) (ex2 C (\lambda (e1: +C).(csubst0 (minus i (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead +c k t) e1)))) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H4: (eq nat i (s +k x1))).(\lambda (H5: (eq C c2 (CHead c k x0))).(\lambda (_: (subst0 x1 v t +x0)).(let H7 \def (eq_ind C c2 (\lambda (c0: C).(drop (S n0) O c0 e2)) H2 +(CHead c k x0) H5) in (let H8 \def (eq_ind nat i (\lambda (n1: nat).(\forall +(c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall (e3: C).((drop (S +n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: C).(csubst0 +(minus n1 (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop (S n0) O c e1)))))))))) +H0 (s k x1) H4) in (let H9 \def (eq_ind nat i (\lambda (n1: nat).(lt (S n0) +n1)) H (s k x1) H4) in (eq_ind_r nat (s k x1) (\lambda (n1: nat).(or (drop (S +n0) O (CHead c k t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus n1 (S n0)) v +e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c k t) e1))))) (K_ind (\lambda +(k0: K).(((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c c3) \to +(\forall (e3: C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C +(\lambda (e1: C).(csubst0 (minus (s k0 x1) (S n0)) v0 e1 e3)) (\lambda (e1: +C).(drop (S n0) O c e1)))))))))) \to ((lt (S n0) (s k0 x1)) \to ((drop (r k0 +n0) O c e2) \to (or (drop (S n0) O (CHead c k0 t) e2) (ex2 C (\lambda (e1: +C).(csubst0 (minus (s k0 x1) (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) +O (CHead c k0 t) e1)))))))) (\lambda (b: B).(\lambda (_: ((\forall (c3: +C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to (\forall (e3: +C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: +C).(csubst0 (minus (s (Bind b) x1) (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop +(S n0) O c e1))))))))))).(\lambda (_: (lt (S n0) (s (Bind b) x1))).(\lambda +(H12: (drop (r (Bind b) n0) O c e2)).(or_introl (drop (S n0) O (CHead c (Bind +b) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x1 n0) v e1 e2)) (\lambda +(e1: C).(drop (S n0) O (CHead c (Bind b) t) e1))) (drop_drop (Bind b) n0 c e2 +H12 t)))))) (\lambda (f: F).(\lambda (_: ((\forall (c3: C).(\forall (v0: +T).((csubst0 (s (Flat f) x1) v0 c c3) \to (\forall (e3: C).((drop (S n0) O c3 +e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: C).(csubst0 (minus (s +(Flat f) x1) (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop (S n0) O c +e1))))))))))).(\lambda (_: (lt (S n0) (s (Flat f) x1))).(\lambda (H12: (drop +(r (Flat f) n0) O c e2)).(or_introl (drop (S n0) O (CHead c (Flat f) t) e2) +(ex2 C (\lambda (e1: C).(csubst0 (minus x1 (S n0)) v e1 e2)) (\lambda (e1: +C).(drop (S n0) O (CHead c (Flat f) t) e1))) (drop_drop (Flat f) n0 c e2 H12 +t)))))) k H8 H9 (drop_gen_drop k c e2 x0 n0 H7)) i H4))))))))) H3)) (\lambda +(H3: (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j)))) +(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: +C).(\lambda (j: nat).(csubst0 j v c c3))))).(ex3_2_ind C nat (\lambda (_: +C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_: +nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j +v c c3))) (or (drop (S n0) O (CHead c k t) e2) (ex2 C (\lambda (e1: +C).(csubst0 (minus i (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead +c k t) e1)))) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H4: (eq nat i (s +k x1))).(\lambda (H5: (eq C c2 (CHead x0 k t))).(\lambda (H6: (csubst0 x1 v c +x0)).(let H7 \def (eq_ind C c2 (\lambda (c0: C).(drop (S n0) O c0 e2)) H2 +(CHead x0 k t) H5) in (let H8 \def (eq_ind nat i (\lambda (n1: nat).(\forall +(c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall (e3: C).((drop (S +n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: C).(csubst0 +(minus n1 (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop (S n0) O c e1)))))))))) +H0 (s k x1) H4) in (let H9 \def (eq_ind nat i (\lambda (n1: nat).(lt (S n0) +n1)) H (s k x1) H4) in (eq_ind_r nat (s k x1) (\lambda (n1: nat).(or (drop (S +n0) O (CHead c k t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus n1 (S n0)) v +e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c k t) e1))))) (K_ind (\lambda +(k0: K).(((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c c3) \to +(\forall (e3: C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C +(\lambda (e1: C).(csubst0 (minus (s k0 x1) (S n0)) v0 e1 e3)) (\lambda (e1: +C).(drop (S n0) O c e1)))))))))) \to ((lt (S n0) (s k0 x1)) \to ((drop (r k0 +n0) O x0 e2) \to (or (drop (S n0) O (CHead c k0 t) e2) (ex2 C (\lambda (e1: +C).(csubst0 (minus (s k0 x1) (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) +O (CHead c k0 t) e1)))))))) (\lambda (b: B).(\lambda (_: ((\forall (c3: +C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to (\forall (e3: +C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: +C).(csubst0 (minus (s (Bind b) x1) (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop +(S n0) O c e1))))))))))).(\lambda (H11: (lt (S n0) (s (Bind b) x1))).(\lambda +(H12: (drop (r (Bind b) n0) O x0 e2)).(let H_x \def (IHn x1 (lt_S_n n0 x1 +H11) c x0 v H6 e2 H12) in (let H13 \def H_x in (or_ind (drop n0 O c e2) (ex2 +C (\lambda (e1: C).(csubst0 (minus x1 n0) v e1 e2)) (\lambda (e1: C).(drop n0 +O c e1))) (or (drop (S n0) O (CHead c (Bind b) t) e2) (ex2 C (\lambda (e1: +C).(csubst0 (minus x1 n0) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c +(Bind b) t) e1)))) (\lambda (H14: (drop n0 O c e2)).(or_introl (drop (S n0) O +(CHead c (Bind b) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x1 n0) v e1 +e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Bind b) t) e1))) (drop_drop +(Bind b) n0 c e2 H14 t))) (\lambda (H14: (ex2 C (\lambda (e1: C).(csubst0 +(minus x1 n0) v e1 e2)) (\lambda (e1: C).(drop n0 O c e1)))).(ex2_ind C +(\lambda (e1: C).(csubst0 (minus x1 n0) v e1 e2)) (\lambda (e1: C).(drop n0 O +c e1)) (or (drop (S n0) O (CHead c (Bind b) t) e2) (ex2 C (\lambda (e1: +C).(csubst0 (minus x1 n0) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c +(Bind b) t) e1)))) (\lambda (x: C).(\lambda (H15: (csubst0 (minus x1 n0) v x +e2)).(\lambda (H16: (drop n0 O c x)).(or_intror (drop (S n0) O (CHead c (Bind +b) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x1 n0) v e1 e2)) (\lambda +(e1: C).(drop (S n0) O (CHead c (Bind b) t) e1))) (ex_intro2 C (\lambda (e1: +C).(csubst0 (minus x1 n0) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c +(Bind b) t) e1)) x H15 (drop_drop (Bind b) n0 c x H16 t)))))) H14)) +H13))))))) (\lambda (f: F).(\lambda (H10: ((\forall (c3: C).(\forall (v0: +T).((csubst0 (s (Flat f) x1) v0 c c3) \to (\forall (e3: C).((drop (S n0) O c3 +e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: C).(csubst0 (minus (s +(Flat f) x1) (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop (S n0) O c +e1))))))))))).(\lambda (_: (lt (S n0) (s (Flat f) x1))).(\lambda (H12: (drop +(r (Flat f) n0) O x0 e2)).(let H_x \def (H10 x0 v H6 e2 H12) in (let H13 \def +H_x in (or_ind (drop (S n0) O c e2) (ex2 C (\lambda (e1: C).(csubst0 (minus +x1 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O c e1))) (or (drop (S n0) +O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x1 (S n0)) +v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Flat f) t) e1)))) +(\lambda (H14: (drop (S n0) O c e2)).(or_introl (drop (S n0) O (CHead c (Flat +f) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x1 (S n0)) v e1 e2)) +(\lambda (e1: C).(drop (S n0) O (CHead c (Flat f) t) e1))) (drop_drop (Flat +f) n0 c e2 H14 t))) (\lambda (H14: (ex2 C (\lambda (e1: C).(csubst0 (minus x1 +(S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O c e1)))).(ex2_ind C +(\lambda (e1: C).(csubst0 (minus x1 (S n0)) v e1 e2)) (\lambda (e1: C).(drop +(S n0) O c e1)) (or (drop (S n0) O (CHead c (Flat f) t) e2) (ex2 C (\lambda +(e1: C).(csubst0 (minus x1 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O +(CHead c (Flat f) t) e1)))) (\lambda (x: C).(\lambda (H15: (csubst0 (minus x1 +(S n0)) v x e2)).(\lambda (H16: (drop (S n0) O c x)).(or_intror (drop (S n0) +O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x1 (S n0)) +v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Flat f) t) e1))) +(ex_intro2 C (\lambda (e1: C).(csubst0 (minus x1 (S n0)) v e1 e2)) (\lambda +(e1: C).(drop (S n0) O (CHead c (Flat f) t) e1)) x H15 (drop_drop (Flat f) n0 +c x H16 t)))))) H14)) H13))))))) k H8 H9 (drop_gen_drop k x0 e2 t n0 H7)) i +H4))))))))) H3)) (\lambda (H3: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: +C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: +C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda +(_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: +C).(\lambda (j: nat).(csubst0 j v c c3)))))).(ex4_3_ind T C nat (\lambda (_: +T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: +T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda +(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: +T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (or (drop (S n0) +O (CHead c k t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i (S n0)) v e1 +e2)) (\lambda (e1: C).(drop (S n0) O (CHead c k t) e1)))) (\lambda (x0: +T).(\lambda (x1: C).(\lambda (x2: nat).(\lambda (H4: (eq nat i (s k +x2))).(\lambda (H5: (eq C c2 (CHead x1 k x0))).(\lambda (_: (subst0 x2 v t +x0)).(\lambda (H7: (csubst0 x2 v c x1)).(let H8 \def (eq_ind C c2 (\lambda +(c0: C).(drop (S n0) O c0 e2)) H2 (CHead x1 k x0) H5) in (let H9 \def (eq_ind +nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c +c3) \to (\forall (e3: C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) +(ex2 C (\lambda (e1: C).(csubst0 (minus n1 (S n0)) v0 e1 e3)) (\lambda (e1: +C).(drop (S n0) O c e1)))))))))) H0 (s k x2) H4) in (let H10 \def (eq_ind nat +i (\lambda (n1: nat).(lt (S n0) n1)) H (s k x2) H4) in (eq_ind_r nat (s k x2) +(\lambda (n1: nat).(or (drop (S n0) O (CHead c k t) e2) (ex2 C (\lambda (e1: +C).(csubst0 (minus n1 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O +(CHead c k t) e1))))) (K_ind (\lambda (k0: K).(((\forall (c3: C).(\forall +(v0: T).((csubst0 (s k0 x2) v0 c c3) \to (\forall (e3: C).((drop (S n0) O c3 +e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: C).(csubst0 (minus (s +k0 x2) (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop (S n0) O c e1)))))))))) \to +((lt (S n0) (s k0 x2)) \to ((drop (r k0 n0) O x1 e2) \to (or (drop (S n0) O +(CHead c k0 t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus (s k0 x2) (S n0)) +v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c k0 t) e1)))))))) (\lambda +(b: B).(\lambda (_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Bind b) +x2) v0 c c3) \to (\forall (e3: C).((drop (S n0) O c3 e3) \to (or (drop (S n0) +O c e3) (ex2 C (\lambda (e1: C).(csubst0 (minus (s (Bind b) x2) (S n0)) v0 e1 +e3)) (\lambda (e1: C).(drop (S n0) O c e1))))))))))).(\lambda (H12: (lt (S +n0) (s (Bind b) x2))).(\lambda (H13: (drop (r (Bind b) n0) O x1 e2)).(let H_x +\def (IHn x2 (lt_S_n n0 x2 H12) c x1 v H7 e2 H13) in (let H14 \def H_x in +(or_ind (drop n0 O c e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x2 n0) v e1 +e2)) (\lambda (e1: C).(drop n0 O c e1))) (or (drop (S n0) O (CHead c (Bind b) +t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x2 n0) v e1 e2)) (\lambda (e1: +C).(drop (S n0) O (CHead c (Bind b) t) e1)))) (\lambda (H15: (drop n0 O c +e2)).(or_introl (drop (S n0) O (CHead c (Bind b) t) e2) (ex2 C (\lambda (e1: +C).(csubst0 (minus x2 n0) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c +(Bind b) t) e1))) (drop_drop (Bind b) n0 c e2 H15 t))) (\lambda (H15: (ex2 C +(\lambda (e1: C).(csubst0 (minus x2 n0) v e1 e2)) (\lambda (e1: C).(drop n0 O +c e1)))).(ex2_ind C (\lambda (e1: C).(csubst0 (minus x2 n0) v e1 e2)) +(\lambda (e1: C).(drop n0 O c e1)) (or (drop (S n0) O (CHead c (Bind b) t) +e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x2 n0) v e1 e2)) (\lambda (e1: +C).(drop (S n0) O (CHead c (Bind b) t) e1)))) (\lambda (x: C).(\lambda (H16: +(csubst0 (minus x2 n0) v x e2)).(\lambda (H17: (drop n0 O c x)).(or_intror +(drop (S n0) O (CHead c (Bind b) t) e2) (ex2 C (\lambda (e1: C).(csubst0 +(minus x2 n0) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Bind b) t) +e1))) (ex_intro2 C (\lambda (e1: C).(csubst0 (minus x2 n0) v e1 e2)) (\lambda +(e1: C).(drop (S n0) O (CHead c (Bind b) t) e1)) x H16 (drop_drop (Bind b) n0 +c x H17 t)))))) H15)) H14))))))) (\lambda (f: F).(\lambda (H11: ((\forall +(c3: C).(\forall (v0: T).((csubst0 (s (Flat f) x2) v0 c c3) \to (\forall (e3: +C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: +C).(csubst0 (minus (s (Flat f) x2) (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop +(S n0) O c e1))))))))))).(\lambda (_: (lt (S n0) (s (Flat f) x2))).(\lambda +(H13: (drop (r (Flat f) n0) O x1 e2)).(let H_x \def (H11 x1 v H7 e2 H13) in +(let H14 \def H_x in (or_ind (drop (S n0) O c e2) (ex2 C (\lambda (e1: +C).(csubst0 (minus x2 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O c +e1))) (or (drop (S n0) O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1: +C).(csubst0 (minus x2 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O +(CHead c (Flat f) t) e1)))) (\lambda (H15: (drop (S n0) O c e2)).(or_introl +(drop (S n0) O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1: C).(csubst0 +(minus x2 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Flat f) +t) e1))) (drop_drop (Flat f) n0 c e2 H15 t))) (\lambda (H15: (ex2 C (\lambda +(e1: C).(csubst0 (minus x2 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O +c e1)))).(ex2_ind C (\lambda (e1: C).(csubst0 (minus x2 (S n0)) v e1 e2)) +(\lambda (e1: C).(drop (S n0) O c e1)) (or (drop (S n0) O (CHead c (Flat f) +t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x2 (S n0)) v e1 e2)) (\lambda +(e1: C).(drop (S n0) O (CHead c (Flat f) t) e1)))) (\lambda (x: C).(\lambda +(H16: (csubst0 (minus x2 (S n0)) v x e2)).(\lambda (H17: (drop (S n0) O c +x)).(or_intror (drop (S n0) O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1: +C).(csubst0 (minus x2 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O +(CHead c (Flat f) t) e1))) (ex_intro2 C (\lambda (e1: C).(csubst0 (minus x2 +(S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Flat f) t) e1)) x +H16 (drop_drop (Flat f) n0 c x H17 t)))))) H15)) H14))))))) k H9 H10 +(drop_gen_drop k x1 e2 x0 n0 H8)) i H4))))))))))) H3)) (csubst0_gen_head k c +c2 t v i H1))))))))))) c1)))))) n). +(* COMMENTS +Initial nodes: 5939 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubst0/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/csubst0/fwd.ma new file mode 100644 index 000000000..9b3de2983 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/csubst0/fwd.ma @@ -0,0 +1,462 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/csubst0/defs.ma". + +theorem csubst0_gen_sort: + \forall (x: C).(\forall (v: T).(\forall (i: nat).(\forall (n: nat).((csubst0 +i v (CSort n) x) \to (\forall (P: Prop).P))))) +\def + \lambda (x: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda +(H: (csubst0 i v (CSort n) x)).(\lambda (P: Prop).(insert_eq C (CSort n) +(\lambda (c: C).(csubst0 i v c x)) (\lambda (_: C).P) (\lambda (y: +C).(\lambda (H0: (csubst0 i v y x)).(csubst0_ind (\lambda (_: nat).(\lambda +(_: T).(\lambda (c: C).(\lambda (_: C).((eq C c (CSort n)) \to P))))) +(\lambda (k: K).(\lambda (i0: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda +(u2: T).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (c: C).(\lambda (H2: (eq +C (CHead c k u1) (CSort n))).(let H3 \def (eq_ind C (CHead c k u1) (\lambda +(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) +\Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H2) in +(False_ind P H3)))))))))) (\lambda (k: K).(\lambda (i0: nat).(\lambda (c1: +C).(\lambda (c2: C).(\lambda (v0: T).(\lambda (_: (csubst0 i0 v0 c1 +c2)).(\lambda (_: (((eq C c1 (CSort n)) \to P))).(\lambda (u: T).(\lambda +(H3: (eq C (CHead c1 k u) (CSort n))).(let H4 \def (eq_ind C (CHead c1 k u) +(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) +\Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H3) in +(False_ind P H4))))))))))) (\lambda (k: K).(\lambda (i0: nat).(\lambda (v0: +T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0 i0 v0 u1 +u2)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csubst0 i0 v0 c1 +c2)).(\lambda (_: (((eq C c1 (CSort n)) \to P))).(\lambda (H4: (eq C (CHead +c1 k u1) (CSort n))).(let H5 \def (eq_ind C (CHead c1 k u1) (\lambda (ee: +C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow +False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H4) in (False_ind P +H5))))))))))))) i v y x H0))) H)))))). +(* COMMENTS +Initial nodes: 355 +END *) + +theorem csubst0_gen_head: + \forall (k: K).(\forall (c1: C).(\forall (x: C).(\forall (u1: T).(\forall +(v: T).(\forall (i: nat).((csubst0 i v (CHead c1 k u1) x) \to (or3 (ex3_2 T +nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: +T).(\lambda (_: nat).(eq C x (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: +nat).(subst0 j v u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq +nat i (s k j)))) (\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2 k +u1)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 c2)))) (ex4_3 T C +nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) +(\lambda (u2: T).(\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2 k +u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 +u2)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 +c2)))))))))))) +\def + \lambda (k: K).(\lambda (c1: C).(\lambda (x: C).(\lambda (u1: T).(\lambda +(v: T).(\lambda (i: nat).(\lambda (H: (csubst0 i v (CHead c1 k u1) +x)).(insert_eq C (CHead c1 k u1) (\lambda (c: C).(csubst0 i v c x)) (\lambda +(_: C).(or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k +j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C x (CHead c1 k u2)))) (\lambda +(u2: T).(\lambda (j: nat).(subst0 j v u1 u2)))) (ex3_2 C nat (\lambda (_: +C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c2: C).(\lambda (_: +nat).(eq C x (CHead c2 k u1)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j +v c1 c2)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: +nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c2: C).(\lambda (_: +nat).(eq C x (CHead c2 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda +(j: nat).(subst0 j v u1 u2)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: +nat).(csubst0 j v c1 c2))))))) (\lambda (y: C).(\lambda (H0: (csubst0 i v y +x)).(csubst0_ind (\lambda (n: nat).(\lambda (t: T).(\lambda (c: C).(\lambda +(c0: C).((eq C c (CHead c1 k u1)) \to (or3 (ex3_2 T nat (\lambda (_: +T).(\lambda (j: nat).(eq nat n (s k j)))) (\lambda (u2: T).(\lambda (_: +nat).(eq C c0 (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j +t u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat n (s k +j)))) (\lambda (c2: C).(\lambda (_: nat).(eq C c0 (CHead c2 k u1)))) (\lambda +(c2: C).(\lambda (j: nat).(csubst0 j t c1 c2)))) (ex4_3 T C nat (\lambda (_: +T).(\lambda (_: C).(\lambda (j: nat).(eq nat n (s k j))))) (\lambda (u2: +T).(\lambda (c2: C).(\lambda (_: nat).(eq C c0 (CHead c2 k u2))))) (\lambda +(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j t u1 u2)))) (\lambda (_: +T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j t c1 c2))))))))))) (\lambda +(k0: K).(\lambda (i0: nat).(\lambda (v0: T).(\lambda (u0: T).(\lambda (u2: +T).(\lambda (H1: (subst0 i0 v0 u0 u2)).(\lambda (c: C).(\lambda (H2: (eq C +(CHead c k0 u0) (CHead c1 k u1))).(let H3 \def (f_equal C C (\lambda (e: +C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c | +(CHead c0 _ _) \Rightarrow c0])) (CHead c k0 u0) (CHead c1 k u1) H2) in ((let +H4 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) +with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead c k0 +u0) (CHead c1 k u1) H2) in ((let H5 \def (f_equal C T (\lambda (e: C).(match +e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ +t) \Rightarrow t])) (CHead c k0 u0) (CHead c1 k u1) H2) in (\lambda (H6: (eq +K k0 k)).(\lambda (H7: (eq C c c1)).(eq_ind_r C c1 (\lambda (c0: C).(or3 +(ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) +(\lambda (u3: T).(\lambda (_: nat).(eq C (CHead c0 k0 u2) (CHead c1 k u3)))) +(\lambda (u3: T).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (ex3_2 C nat +(\lambda (_: C).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) (\lambda (c2: +C).(\lambda (_: nat).(eq C (CHead c0 k0 u2) (CHead c2 k u1)))) (\lambda (c2: +C).(\lambda (j: nat).(csubst0 j v0 c1 c2)))) (ex4_3 T C nat (\lambda (_: +T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k0 i0) (s k j))))) (\lambda +(u3: T).(\lambda (c2: C).(\lambda (_: nat).(eq C (CHead c0 k0 u2) (CHead c2 k +u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 +u3)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v0 c1 +c2))))))) (let H8 \def (eq_ind T u0 (\lambda (t: T).(subst0 i0 v0 t u2)) H1 +u1 H5) in (eq_ind_r K k (\lambda (k1: K).(or3 (ex3_2 T nat (\lambda (_: +T).(\lambda (j: nat).(eq nat (s k1 i0) (s k j)))) (\lambda (u3: T).(\lambda +(_: nat).(eq C (CHead c1 k1 u2) (CHead c1 k u3)))) (\lambda (u3: T).(\lambda +(j: nat).(subst0 j v0 u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: +nat).(eq nat (s k1 i0) (s k j)))) (\lambda (c2: C).(\lambda (_: nat).(eq C +(CHead c1 k1 u2) (CHead c2 k u1)))) (\lambda (c2: C).(\lambda (j: +nat).(csubst0 j v0 c1 c2)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: +C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j))))) (\lambda (u3: T).(\lambda +(c2: C).(\lambda (_: nat).(eq C (CHead c1 k1 u2) (CHead c2 k u3))))) (\lambda +(u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (\lambda (_: +T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v0 c1 c2))))))) (or3_intro0 +(ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) +(\lambda (u3: T).(\lambda (_: nat).(eq C (CHead c1 k u2) (CHead c1 k u3)))) +(\lambda (u3: T).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (ex3_2 C nat +(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (c2: +C).(\lambda (_: nat).(eq C (CHead c1 k u2) (CHead c2 k u1)))) (\lambda (c2: +C).(\lambda (j: nat).(csubst0 j v0 c1 c2)))) (ex4_3 T C nat (\lambda (_: +T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))) (\lambda +(u3: T).(\lambda (c2: C).(\lambda (_: nat).(eq C (CHead c1 k u2) (CHead c2 k +u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 +u3)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v0 c1 +c2))))) (ex3_2_intro T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i0) +(s k j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C (CHead c1 k u2) (CHead c1 +k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j v0 u1 u3))) u2 i0 +(refl_equal nat (s k i0)) (refl_equal C (CHead c1 k u2)) H8)) k0 H6)) c +H7)))) H4)) H3)))))))))) (\lambda (k0: K).(\lambda (i0: nat).(\lambda (c0: +C).(\lambda (c2: C).(\lambda (v0: T).(\lambda (H1: (csubst0 i0 v0 c0 +c2)).(\lambda (H2: (((eq C c0 (CHead c1 k u1)) \to (or3 (ex3_2 T nat (\lambda +(_: T).(\lambda (j: nat).(eq nat i0 (s k j)))) (\lambda (u2: T).(\lambda (_: +nat).(eq C c2 (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j +v0 u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i0 (s k +j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u1)))) (\lambda +(c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: +T).(\lambda (_: C).(\lambda (j: nat).(eq nat i0 (s k j))))) (\lambda (u2: +T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda +(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 u2)))) (\lambda (_: +T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3))))))))).(\lambda +(u: T).(\lambda (H3: (eq C (CHead c0 k0 u) (CHead c1 k u1))).(let H4 \def +(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with +[(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k0 u) +(CHead c1 k u1) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e in +C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) +\Rightarrow k1])) (CHead c0 k0 u) (CHead c1 k u1) H3) in ((let H6 \def +(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with +[(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 u) +(CHead c1 k u1) H3) in (\lambda (H7: (eq K k0 k)).(\lambda (H8: (eq C c0 +c1)).(eq_ind_r T u1 (\lambda (t: T).(or3 (ex3_2 T nat (\lambda (_: +T).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) (\lambda (u2: T).(\lambda +(_: nat).(eq C (CHead c2 k0 t) (CHead c1 k u2)))) (\lambda (u2: T).(\lambda +(j: nat).(subst0 j v0 u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: +nat).(eq nat (s k0 i0) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C +(CHead c2 k0 t) (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: +nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: +C).(\lambda (j: nat).(eq nat (s k0 i0) (s k j))))) (\lambda (u2: T).(\lambda +(c3: C).(\lambda (_: nat).(eq C (CHead c2 k0 t) (CHead c3 k u2))))) (\lambda +(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 u2)))) (\lambda (_: +T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3))))))) (let H9 \def +(eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 k u1)) \to (or3 (ex3_2 T nat +(\lambda (_: T).(\lambda (j: nat).(eq nat i0 (s k j)))) (\lambda (u2: +T).(\lambda (_: nat).(eq C c2 (CHead c1 k u2)))) (\lambda (u2: T).(\lambda +(j: nat).(subst0 j v0 u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: +nat).(eq nat i0 (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead +c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 +T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat i0 (s k +j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 +k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 +u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 +c3)))))))) H2 c1 H8) in (let H10 \def (eq_ind C c0 (\lambda (c: C).(csubst0 +i0 v0 c c2)) H1 c1 H8) in (eq_ind_r K k (\lambda (k1: K).(or3 (ex3_2 T nat +(\lambda (_: T).(\lambda (j: nat).(eq nat (s k1 i0) (s k j)))) (\lambda (u2: +T).(\lambda (_: nat).(eq C (CHead c2 k1 u1) (CHead c1 k u2)))) (\lambda (u2: +T).(\lambda (j: nat).(subst0 j v0 u1 u2)))) (ex3_2 C nat (\lambda (_: +C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j)))) (\lambda (c3: C).(\lambda +(_: nat).(eq C (CHead c2 k1 u1) (CHead c3 k u1)))) (\lambda (c3: C).(\lambda +(j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: +C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j))))) (\lambda (u2: T).(\lambda +(c3: C).(\lambda (_: nat).(eq C (CHead c2 k1 u1) (CHead c3 k u2))))) (\lambda +(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 u2)))) (\lambda (_: +T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3))))))) (or3_intro1 +(ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) +(\lambda (u2: T).(\lambda (_: nat).(eq C (CHead c2 k u1) (CHead c1 k u2)))) +(\lambda (u2: T).(\lambda (j: nat).(subst0 j v0 u1 u2)))) (ex3_2 C nat +(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (c3: +C).(\lambda (_: nat).(eq C (CHead c2 k u1) (CHead c3 k u1)))) (\lambda (c3: +C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: +T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))) (\lambda +(u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k u1) (CHead c3 k +u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 +u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 +c3))))) (ex3_2_intro C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) +(s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k u1) (CHead c3 +k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3))) c2 i0 +(refl_equal nat (s k i0)) (refl_equal C (CHead c2 k u1)) H10)) k0 H7))) u +H6)))) H5)) H4))))))))))) (\lambda (k0: K).(\lambda (i0: nat).(\lambda (v0: +T).(\lambda (u0: T).(\lambda (u2: T).(\lambda (H1: (subst0 i0 v0 u0 +u2)).(\lambda (c0: C).(\lambda (c2: C).(\lambda (H2: (csubst0 i0 v0 c0 +c2)).(\lambda (H3: (((eq C c0 (CHead c1 k u1)) \to (or3 (ex3_2 T nat (\lambda +(_: T).(\lambda (j: nat).(eq nat i0 (s k j)))) (\lambda (u3: T).(\lambda (_: +nat).(eq C c2 (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j +v0 u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i0 (s k +j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u1)))) (\lambda +(c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: +T).(\lambda (_: C).(\lambda (j: nat).(eq nat i0 (s k j))))) (\lambda (u3: +T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u3))))) (\lambda +(u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (\lambda (_: +T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3))))))))).(\lambda +(H4: (eq C (CHead c0 k0 u0) (CHead c1 k u1))).(let H5 \def (f_equal C C +(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) +\Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k0 u0) (CHead c1 k +u1) H4) in ((let H6 \def (f_equal C K (\lambda (e: C).(match e in C return +(\lambda (_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) +\Rightarrow k1])) (CHead c0 k0 u0) (CHead c1 k u1) H4) in ((let H7 \def +(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with +[(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 u0) +(CHead c1 k u1) H4) in (\lambda (H8: (eq K k0 k)).(\lambda (H9: (eq C c0 +c1)).(let H10 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 k u1)) \to +(or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i0 (s k j)))) +(\lambda (u3: T).(\lambda (_: nat).(eq C c2 (CHead c1 k u3)))) (\lambda (u3: +T).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (ex3_2 C nat (\lambda (_: +C).(\lambda (j: nat).(eq nat i0 (s k j)))) (\lambda (c3: C).(\lambda (_: +nat).(eq C c2 (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 +j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: +nat).(eq nat i0 (s k j))))) (\lambda (u3: T).(\lambda (c3: C).(\lambda (_: +nat).(eq C c2 (CHead c3 k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda +(j: nat).(subst0 j v0 u1 u3)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: +nat).(csubst0 j v0 c1 c3)))))))) H3 c1 H9) in (let H11 \def (eq_ind C c0 +(\lambda (c: C).(csubst0 i0 v0 c c2)) H2 c1 H9) in (let H12 \def (eq_ind T u0 +(\lambda (t: T).(subst0 i0 v0 t u2)) H1 u1 H7) in (eq_ind_r K k (\lambda (k1: +K).(or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k1 i0) (s k +j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C (CHead c2 k1 u2) (CHead c1 k +u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (ex3_2 C nat +(\lambda (_: C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j)))) (\lambda (c3: +C).(\lambda (_: nat).(eq C (CHead c2 k1 u2) (CHead c3 k u1)))) (\lambda (c3: +C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: +T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j))))) (\lambda +(u3: T).(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k1 u2) (CHead c3 k +u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 +u3)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 +c3))))))) (or3_intro2 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat +(s k i0) (s k j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C (CHead c2 k u2) +(CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j v0 u1 u3)))) +(ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) +(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k u2) (CHead c3 k u1)))) +(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat +(\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k +j))))) (\lambda (u3: T).(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k +u2) (CHead c3 k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: +nat).(subst0 j v0 u1 u3)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: +nat).(csubst0 j v0 c1 c3))))) (ex4_3_intro T C nat (\lambda (_: T).(\lambda +(_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))) (\lambda (u3: +T).(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k u2) (CHead c3 k +u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 +u3)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 +c3)))) u2 c2 i0 (refl_equal nat (s k i0)) (refl_equal C (CHead c2 k u2)) H12 +H11)) k0 H8))))))) H6)) H5))))))))))))) i v y x H0))) H))))))). +(* COMMENTS +Initial nodes: 4039 +END *) + +theorem csubst0_gen_S_bind_2: + \forall (b: B).(\forall (x: C).(\forall (c2: C).(\forall (v: T).(\forall +(v2: T).(\forall (i: nat).((csubst0 (S i) v x (CHead c2 (Bind b) v2)) \to +(or3 (ex2 T (\lambda (v1: T).(subst0 i v v1 v2)) (\lambda (v1: T).(eq C x +(CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i v c1 c2)) +(\lambda (c1: C).(eq C x (CHead c1 (Bind b) v2)))) (ex3_2 C T (\lambda (_: +C).(\lambda (v1: T).(subst0 i v v1 v2))) (\lambda (c1: C).(\lambda (_: +T).(csubst0 i v c1 c2))) (\lambda (c1: C).(\lambda (v1: T).(eq C x (CHead c1 +(Bind b) v1)))))))))))) +\def + \lambda (b: B).(\lambda (x: C).(\lambda (c2: C).(\lambda (v: T).(\lambda +(v2: T).(\lambda (i: nat).(\lambda (H: (csubst0 (S i) v x (CHead c2 (Bind b) +v2))).(insert_eq C (CHead c2 (Bind b) v2) (\lambda (c: C).(csubst0 (S i) v x +c)) (\lambda (_: C).(or3 (ex2 T (\lambda (v1: T).(subst0 i v v1 v2)) (\lambda +(v1: T).(eq C x (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i +v c1 c2)) (\lambda (c1: C).(eq C x (CHead c1 (Bind b) v2)))) (ex3_2 C T +(\lambda (_: C).(\lambda (v1: T).(subst0 i v v1 v2))) (\lambda (c1: +C).(\lambda (_: T).(csubst0 i v c1 c2))) (\lambda (c1: C).(\lambda (v1: +T).(eq C x (CHead c1 (Bind b) v1))))))) (\lambda (y: C).(\lambda (H0: +(csubst0 (S i) v x y)).(insert_eq nat (S i) (\lambda (n: nat).(csubst0 n v x +y)) (\lambda (_: nat).((eq C y (CHead c2 (Bind b) v2)) \to (or3 (ex2 T +(\lambda (v1: T).(subst0 i v v1 v2)) (\lambda (v1: T).(eq C x (CHead c2 (Bind +b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i v c1 c2)) (\lambda (c1: C).(eq C +x (CHead c1 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: +T).(subst0 i v v1 v2))) (\lambda (c1: C).(\lambda (_: T).(csubst0 i v c1 +c2))) (\lambda (c1: C).(\lambda (v1: T).(eq C x (CHead c1 (Bind b) v1)))))))) +(\lambda (y0: nat).(\lambda (H1: (csubst0 y0 v x y)).(csubst0_ind (\lambda +(n: nat).(\lambda (t: T).(\lambda (c: C).(\lambda (c0: C).((eq nat n (S i)) +\to ((eq C c0 (CHead c2 (Bind b) v2)) \to (or3 (ex2 T (\lambda (v1: +T).(subst0 i t v1 v2)) (\lambda (v1: T).(eq C c (CHead c2 (Bind b) v1)))) +(ex2 C (\lambda (c1: C).(csubst0 i t c1 c2)) (\lambda (c1: C).(eq C c (CHead +c1 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i t v1 +v2))) (\lambda (c1: C).(\lambda (_: T).(csubst0 i t c1 c2))) (\lambda (c1: +C).(\lambda (v1: T).(eq C c (CHead c1 (Bind b) v1)))))))))))) (\lambda (k: +K).(\lambda (i0: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: +T).(\lambda (H2: (subst0 i0 v0 u1 u2)).(\lambda (c: C).(\lambda (H3: (eq nat +(s k i0) (S i))).(\lambda (H4: (eq C (CHead c k u2) (CHead c2 (Bind b) +v2))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda +(_: C).C) with [(CSort _) \Rightarrow c | (CHead c0 _ _) \Rightarrow c0])) +(CHead c k u2) (CHead c2 (Bind b) v2) H4) in ((let H6 \def (f_equal C K +(\lambda (e: C).(match e in C return (\lambda (_: C).K) with [(CSort _) +\Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c k u2) (CHead c2 +(Bind b) v2) H4) in ((let H7 \def (f_equal C T (\lambda (e: C).(match e in C +return (\lambda (_: C).T) with [(CSort _) \Rightarrow u2 | (CHead _ _ t) +\Rightarrow t])) (CHead c k u2) (CHead c2 (Bind b) v2) H4) in (\lambda (H8: +(eq K k (Bind b))).(\lambda (H9: (eq C c c2)).(eq_ind_r C c2 (\lambda (c0: +C).(or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C +(CHead c0 k u1) (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i +v0 c1 c2)) (\lambda (c1: C).(eq C (CHead c0 k u1) (CHead c1 (Bind b) v2)))) +(ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda +(c1: C).(\lambda (_: T).(csubst0 i v0 c1 c2))) (\lambda (c1: C).(\lambda (v1: +T).(eq C (CHead c0 k u1) (CHead c1 (Bind b) v1))))))) (let H10 \def (eq_ind T +u2 (\lambda (t: T).(subst0 i0 v0 u1 t)) H2 v2 H7) in (let H11 \def (eq_ind K +k (\lambda (k0: K).(eq nat (s k0 i0) (S i))) H3 (Bind b) H8) in (eq_ind_r K +(Bind b) (\lambda (k0: K).(or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) +(\lambda (v1: T).(eq C (CHead c2 k0 u1) (CHead c2 (Bind b) v1)))) (ex2 C +(\lambda (c1: C).(csubst0 i v0 c1 c2)) (\lambda (c1: C).(eq C (CHead c2 k0 +u1) (CHead c1 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: +T).(subst0 i v0 v1 v2))) (\lambda (c1: C).(\lambda (_: T).(csubst0 i v0 c1 +c2))) (\lambda (c1: C).(\lambda (v1: T).(eq C (CHead c2 k0 u1) (CHead c1 +(Bind b) v1))))))) (let H12 \def (f_equal nat nat (\lambda (e: nat).(match e +in nat return (\lambda (_: nat).nat) with [O \Rightarrow i0 | (S n) +\Rightarrow n])) (S i0) (S i) H11) in (let H13 \def (eq_ind nat i0 (\lambda +(n: nat).(subst0 n v0 u1 v2)) H10 i H12) in (or3_intro0 (ex2 T (\lambda (v1: +T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C (CHead c2 (Bind b) u1) (CHead +c2 (Bind b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i v0 c1 c2)) (\lambda +(c1: C).(eq C (CHead c2 (Bind b) u1) (CHead c1 (Bind b) v2)))) (ex3_2 C T +(\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda (c1: +C).(\lambda (_: T).(csubst0 i v0 c1 c2))) (\lambda (c1: C).(\lambda (v1: +T).(eq C (CHead c2 (Bind b) u1) (CHead c1 (Bind b) v1))))) (ex_intro2 T +(\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C (CHead c2 (Bind +b) u1) (CHead c2 (Bind b) v1))) u1 H13 (refl_equal C (CHead c2 (Bind b) +u1)))))) k H8))) c H9)))) H6)) H5))))))))))) (\lambda (k: K).(\lambda (i0: +nat).(\lambda (c1: C).(\lambda (c0: C).(\lambda (v0: T).(\lambda (H2: +(csubst0 i0 v0 c1 c0)).(\lambda (H3: (((eq nat i0 (S i)) \to ((eq C c0 (CHead +c2 (Bind b) v2)) \to (or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) +(\lambda (v1: T).(eq C c1 (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3: +C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C c1 (CHead c3 (Bind b) v2)))) +(ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda +(c3: C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1: +T).(eq C c1 (CHead c3 (Bind b) v1)))))))))).(\lambda (u: T).(\lambda (H4: (eq +nat (s k i0) (S i))).(\lambda (H5: (eq C (CHead c0 k u) (CHead c2 (Bind b) +v2))).(let H6 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda +(_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) +(CHead c0 k u) (CHead c2 (Bind b) v2) H5) in ((let H7 \def (f_equal C K +(\lambda (e: C).(match e in C return (\lambda (_: C).K) with [(CSort _) +\Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c0 k u) (CHead c2 +(Bind b) v2) H5) in ((let H8 \def (f_equal C T (\lambda (e: C).(match e in C +return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) +\Rightarrow t])) (CHead c0 k u) (CHead c2 (Bind b) v2) H5) in (\lambda (H9: +(eq K k (Bind b))).(\lambda (H10: (eq C c0 c2)).(eq_ind_r T v2 (\lambda (t: +T).(or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C +(CHead c1 k t) (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3: C).(csubst0 i +v0 c3 c2)) (\lambda (c3: C).(eq C (CHead c1 k t) (CHead c3 (Bind b) v2)))) +(ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda +(c3: C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1: +T).(eq C (CHead c1 k t) (CHead c3 (Bind b) v1))))))) (let H11 \def (eq_ind C +c0 (\lambda (c: C).((eq nat i0 (S i)) \to ((eq C c (CHead c2 (Bind b) v2)) +\to (or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C +c1 (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3: C).(csubst0 i v0 c3 c2)) +(\lambda (c3: C).(eq C c1 (CHead c3 (Bind b) v2)))) (ex3_2 C T (\lambda (_: +C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda (c3: C).(\lambda (_: +T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1: T).(eq C c1 (CHead +c3 (Bind b) v1))))))))) H3 c2 H10) in (let H12 \def (eq_ind C c0 (\lambda (c: +C).(csubst0 i0 v0 c1 c)) H2 c2 H10) in (let H13 \def (eq_ind K k (\lambda +(k0: K).(eq nat (s k0 i0) (S i))) H4 (Bind b) H9) in (eq_ind_r K (Bind b) +(\lambda (k0: K).(or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda +(v1: T).(eq C (CHead c1 k0 v2) (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3: +C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C (CHead c1 k0 v2) (CHead c3 +(Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 +v2))) (\lambda (c3: C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3: +C).(\lambda (v1: T).(eq C (CHead c1 k0 v2) (CHead c3 (Bind b) v1))))))) (let +H14 \def (f_equal nat nat (\lambda (e: nat).(match e in nat return (\lambda +(_: nat).nat) with [O \Rightarrow i0 | (S n) \Rightarrow n])) (S i0) (S i) +H13) in (let H15 \def (eq_ind nat i0 (\lambda (n: nat).((eq nat n (S i)) \to +((eq C c2 (CHead c2 (Bind b) v2)) \to (or3 (ex2 T (\lambda (v1: T).(subst0 i +v0 v1 v2)) (\lambda (v1: T).(eq C c1 (CHead c2 (Bind b) v1)))) (ex2 C +(\lambda (c3: C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C c1 (CHead c3 +(Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 +v2))) (\lambda (c3: C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3: +C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind b) v1))))))))) H11 i H14) in +(let H16 \def (eq_ind nat i0 (\lambda (n: nat).(csubst0 n v0 c1 c2)) H12 i +H14) in (or3_intro1 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda +(v1: T).(eq C (CHead c1 (Bind b) v2) (CHead c2 (Bind b) v1)))) (ex2 C +(\lambda (c3: C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C (CHead c1 (Bind +b) v2) (CHead c3 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: +T).(subst0 i v0 v1 v2))) (\lambda (c3: C).(\lambda (_: T).(csubst0 i v0 c3 +c2))) (\lambda (c3: C).(\lambda (v1: T).(eq C (CHead c1 (Bind b) v2) (CHead +c3 (Bind b) v1))))) (ex_intro2 C (\lambda (c3: C).(csubst0 i v0 c3 c2)) +(\lambda (c3: C).(eq C (CHead c1 (Bind b) v2) (CHead c3 (Bind b) v2))) c1 H16 +(refl_equal C (CHead c1 (Bind b) v2))))))) k H9)))) u H8)))) H7)) +H6)))))))))))) (\lambda (k: K).(\lambda (i0: nat).(\lambda (v0: T).(\lambda +(u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 i0 v0 u1 u2)).(\lambda (c1: +C).(\lambda (c0: C).(\lambda (H3: (csubst0 i0 v0 c1 c0)).(\lambda (H4: (((eq +nat i0 (S i)) \to ((eq C c0 (CHead c2 (Bind b) v2)) \to (or3 (ex2 T (\lambda +(v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C c1 (CHead c2 (Bind b) +v1)))) (ex2 C (\lambda (c3: C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C +c1 (CHead c3 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: +T).(subst0 i v0 v1 v2))) (\lambda (c3: C).(\lambda (_: T).(csubst0 i v0 c3 +c2))) (\lambda (c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind b) +v1)))))))))).(\lambda (H5: (eq nat (s k i0) (S i))).(\lambda (H6: (eq C +(CHead c0 k u2) (CHead c2 (Bind b) v2))).(let H7 \def (f_equal C C (\lambda +(e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 +| (CHead c _ _) \Rightarrow c])) (CHead c0 k u2) (CHead c2 (Bind b) v2) H6) +in ((let H8 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda +(_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) +(CHead c0 k u2) (CHead c2 (Bind b) v2) H6) in ((let H9 \def (f_equal C T +(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) +\Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c0 k u2) (CHead c2 +(Bind b) v2) H6) in (\lambda (H10: (eq K k (Bind b))).(\lambda (H11: (eq C c0 +c2)).(let H12 \def (eq_ind C c0 (\lambda (c: C).((eq nat i0 (S i)) \to ((eq C +c (CHead c2 (Bind b) v2)) \to (or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 +v2)) (\lambda (v1: T).(eq C c1 (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3: +C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C c1 (CHead c3 (Bind b) v2)))) +(ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda +(c3: C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1: +T).(eq C c1 (CHead c3 (Bind b) v1))))))))) H4 c2 H11) in (let H13 \def +(eq_ind C c0 (\lambda (c: C).(csubst0 i0 v0 c1 c)) H3 c2 H11) in (let H14 +\def (eq_ind T u2 (\lambda (t: T).(subst0 i0 v0 u1 t)) H2 v2 H9) in (let H15 +\def (eq_ind K k (\lambda (k0: K).(eq nat (s k0 i0) (S i))) H5 (Bind b) H10) +in (eq_ind_r K (Bind b) (\lambda (k0: K).(or3 (ex2 T (\lambda (v1: T).(subst0 +i v0 v1 v2)) (\lambda (v1: T).(eq C (CHead c1 k0 u1) (CHead c2 (Bind b) +v1)))) (ex2 C (\lambda (c3: C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C +(CHead c1 k0 u1) (CHead c3 (Bind b) v2)))) (ex3_2 C T (\lambda (_: +C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda (c3: C).(\lambda (_: +T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1: T).(eq C (CHead c1 +k0 u1) (CHead c3 (Bind b) v1))))))) (let H16 \def (f_equal nat nat (\lambda +(e: nat).(match e in nat return (\lambda (_: nat).nat) with [O \Rightarrow i0 +| (S n) \Rightarrow n])) (S i0) (S i) H15) in (let H17 \def (eq_ind nat i0 +(\lambda (n: nat).((eq nat n (S i)) \to ((eq C c2 (CHead c2 (Bind b) v2)) \to +(or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C c1 +(CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3: C).(csubst0 i v0 c3 c2)) +(\lambda (c3: C).(eq C c1 (CHead c3 (Bind b) v2)))) (ex3_2 C T (\lambda (_: +C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda (c3: C).(\lambda (_: +T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1: T).(eq C c1 (CHead +c3 (Bind b) v1))))))))) H12 i H16) in (let H18 \def (eq_ind nat i0 (\lambda +(n: nat).(csubst0 n v0 c1 c2)) H13 i H16) in (let H19 \def (eq_ind nat i0 +(\lambda (n: nat).(subst0 n v0 u1 v2)) H14 i H16) in (or3_intro2 (ex2 T +(\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C (CHead c1 (Bind +b) u1) (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3: C).(csubst0 i v0 c3 +c2)) (\lambda (c3: C).(eq C (CHead c1 (Bind b) u1) (CHead c3 (Bind b) v2)))) +(ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda +(c3: C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1: +T).(eq C (CHead c1 (Bind b) u1) (CHead c3 (Bind b) v1))))) (ex3_2_intro C T +(\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda (c3: +C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1: +T).(eq C (CHead c1 (Bind b) u1) (CHead c3 (Bind b) v1)))) c1 u1 H19 H18 +(refl_equal C (CHead c1 (Bind b) u1)))))))) k H10)))))))) H8)) +H7)))))))))))))) y0 v x y H1))) H0))) H))))))). +(* COMMENTS +Initial nodes: 3878 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubst0/getl.ma b/matita/matita/contribs/lambdadelta/basic_1/csubst0/getl.ma new file mode 100644 index 000000000..2701af000 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/csubst0/getl.ma @@ -0,0 +1,1157 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/csubst0/clear.ma". + +include "Basic-1/csubst0/drop.ma". + +include "Basic-1/getl/fwd.ma". + +theorem csubst0_getl_ge: + \forall (i: nat).(\forall (n: nat).((le i n) \to (\forall (c1: C).(\forall +(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((getl n c1 +e) \to (getl n c2 e))))))))) +\def + \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (le i n)).(\lambda (c1: +C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 i v c1 +c2)).(\lambda (e: C).(\lambda (H1: (getl n c1 e)).(let H2 \def (getl_gen_all +c1 e n H1) in (ex2_ind C (\lambda (e0: C).(drop n O c1 e0)) (\lambda (e0: +C).(clear e0 e)) (getl n c2 e) (\lambda (x: C).(\lambda (H3: (drop n O c1 +x)).(\lambda (H4: (clear x e)).(lt_eq_gt_e i n (getl n c2 e) (\lambda (H5: +(lt i n)).(getl_intro n c2 e x (csubst0_drop_gt n i H5 c1 c2 v H0 x H3) H4)) +(\lambda (H5: (eq nat i n)).(let H6 \def (eq_ind_r nat n (\lambda (n0: +nat).(drop n0 O c1 x)) H3 i H5) in (let H7 \def (eq_ind_r nat n (\lambda (n0: +nat).(le i n0)) H i H5) in (eq_ind nat i (\lambda (n0: nat).(getl n0 c2 e)) +(let H8 \def (csubst0_drop_eq i c1 c2 v H0 x H6) in (or4_ind (drop i O c2 x) +(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: +T).(eq C x (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: +C).(\lambda (_: T).(\lambda (w: T).(drop i O c2 (CHead e0 (Flat f) w)))))) +(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u +w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(eq C x (CHead e1 (Flat f) u)))))) (\lambda (f: +F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop i O c2 (CHead e2 +(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda +(_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda +(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C x (CHead e1 +(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (w: T).(drop i O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O +v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (getl i c2 e) (\lambda (H9: +(drop i O c2 x)).(getl_intro i c2 e x H9 H4)) (\lambda (H9: (ex3_4 F C T T +(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C x +(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: +T).(\lambda (w: T).(drop i O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u +w))))))).(ex3_4_ind F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: +T).(\lambda (_: T).(eq C x (CHead e0 (Flat f) u)))))) (\lambda (f: +F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop i O c2 (CHead e0 +(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 O v u w))))) (getl i c2 e) (\lambda (x0: F).(\lambda (x1: +C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H10: (eq C x (CHead x1 (Flat +x0) x2))).(\lambda (H11: (drop i O c2 (CHead x1 (Flat x0) x3))).(\lambda (_: +(subst0 O v x2 x3)).(let H13 \def (eq_ind C x (\lambda (c: C).(clear c e)) H4 +(CHead x1 (Flat x0) x2) H10) in (getl_intro i c2 e (CHead x1 (Flat x0) x3) +H11 (clear_flat x1 e (clear_gen_flat x0 x1 e x2 H13) x0 x3)))))))))) H9)) +(\lambda (H9: (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(eq C x (CHead e1 (Flat f) u)))))) (\lambda (f: +F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop i O c2 (CHead e2 +(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda +(_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda +(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C x (CHead e1 (Flat f) u)))))) +(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop i O c2 +(CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(csubst0 O v e1 e2))))) (getl i c2 e) (\lambda (x0: +F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H10: (eq C x +(CHead x1 (Flat x0) x3))).(\lambda (H11: (drop i O c2 (CHead x2 (Flat x0) +x3))).(\lambda (H12: (csubst0 O v x1 x2)).(let H13 \def (eq_ind C x (\lambda +(c: C).(clear c e)) H4 (CHead x1 (Flat x0) x3) H10) in (getl_intro i c2 e +(CHead x2 (Flat x0) x3) H11 (clear_flat x2 e (csubst0_clear_O x1 x2 v H12 e +(clear_gen_flat x0 x1 e x3 H13)) x0 x3)))))))))) H9)) (\lambda (H9: (ex4_5 F +C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(\lambda (_: T).(eq C x (CHead e1 (Flat f) u))))))) (\lambda (f: +F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop i O +c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: +F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 +O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda (f: F).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C x (CHead e1 (Flat f) +u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (w: T).(drop i O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O +v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (getl i c2 e) (\lambda (x0: +F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: +T).(\lambda (H10: (eq C x (CHead x1 (Flat x0) x3))).(\lambda (H11: (drop i O +c2 (CHead x2 (Flat x0) x4))).(\lambda (_: (subst0 O v x3 x4)).(\lambda (H13: +(csubst0 O v x1 x2)).(let H14 \def (eq_ind C x (\lambda (c: C).(clear c e)) +H4 (CHead x1 (Flat x0) x3) H10) in (getl_intro i c2 e (CHead x2 (Flat x0) x4) +H11 (clear_flat x2 e (csubst0_clear_O x1 x2 v H13 e (clear_gen_flat x0 x1 e +x3 H14)) x0 x4)))))))))))) H9)) H8)) n H5)))) (\lambda (H5: (lt n +i)).(le_lt_false i n H H5 (getl n c2 e))))))) H2)))))))))). +(* COMMENTS +Initial nodes: 1525 +END *) + +theorem csubst0_getl_lt: + \forall (i: nat).(\forall (n: nat).((lt n i) \to (\forall (c1: C).(\forall +(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((getl n c1 +e) \to (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: +C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) +(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 +(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T +(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e +(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: +B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) +v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) +(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: +T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v +u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))))))))))) +\def + \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (lt n i)).(\lambda (c1: +C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 i v c1 +c2)).(\lambda (e: C).(\lambda (H1: (getl n c1 e)).(let H2 \def (getl_gen_all +c1 e n H1) in (ex2_ind C (\lambda (e0: C).(drop n O c1 e0)) (\lambda (e0: +C).(clear e0 e)) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda +(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) +(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 +(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T +(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e +(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: +B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) +v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) +(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: +T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v +u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x: +C).(\lambda (H3: (drop n O c1 x)).(\lambda (H4: (clear x e)).(let H5 \def +(csubst0_drop_lt n i H c1 c2 v H0 x H3) in (or4_ind (drop n O c2 x) (ex3_4 K +C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C +x (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: +T).(\lambda (w: T).(drop n O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda +(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n)) v u w)))))) +(ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(eq C x (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(drop n O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda +(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1 +e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (_: T).(eq C x (CHead e1 k u))))))) (\lambda (k: +K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n O +c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n)) v u w)))))) +(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 (minus i (s k n)) v e1 e2))))))) (or4 (getl n c2 e) (ex3_4 B +C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C +e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: +T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: +B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v +u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: +B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 +(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda +(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: +B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e +(CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) +(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) +v e1 e2)))))))) (\lambda (H6: (drop n O c2 x)).(or4_intro0 (getl n c2 e) +(ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: +T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: +C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) +(\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 +(minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) +(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 +(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T +(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: +T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 +(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda +(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: +B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 +(minus i (S n)) v e1 e2))))))) (getl_intro n c2 e x H6 H4))) (\lambda (H6: +(ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: +T).(eq C x (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: +T).(\lambda (w: T).(drop n O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda +(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n)) v u +w))))))).(ex3_4_ind K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: +T).(\lambda (_: T).(eq C x (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: +C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e0 k w)))))) (\lambda +(k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k +n)) v u w))))) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda +(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) +(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 +(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T +(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e +(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: +B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) +v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) +(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: +T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v +u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x0: +K).(\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq C x +(CHead x1 x0 x2))).(\lambda (H8: (drop n O c2 (CHead x1 x0 x3))).(\lambda +(H9: (subst0 (minus i (s x0 n)) v x2 x3)).(let H10 \def (eq_ind C x (\lambda +(c: C).(clear c e)) H4 (CHead x1 x0 x2) H7) in (K_ind (\lambda (k: K).((drop +n O c2 (CHead x1 k x3)) \to ((subst0 (minus i (s k n)) v x2 x3) \to ((clear +(CHead x1 k x2) e) \to (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: +B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind +b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: +T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 +B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq +C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: +B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) +v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) +(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: +T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v +u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))))))) (\lambda (b: +B).(\lambda (H11: (drop n O c2 (CHead x1 (Bind b) x3))).(\lambda (H12: +(subst0 (minus i (s (Bind b) n)) v x2 x3)).(\lambda (H13: (clear (CHead x1 +(Bind b) x2) e)).(eq_ind_r C (CHead x1 (Bind b) x2) (\lambda (c: C).(or4 +(getl n c2 c) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e0: C).(\lambda (u: +T).(\lambda (_: T).(eq C c (CHead e0 (Bind b0) u)))))) (\lambda (b0: +B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 +(Bind b0) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda +(w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b0: +B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind +b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: +T).(getl n c2 (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) +(ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda +(u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b0) u))))))) (\lambda (b0: +B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n +c2 (CHead e2 (Bind b0) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) +(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro1 (getl n c2 +(CHead x1 (Bind b) x2)) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e0: +C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Bind b) x2) (CHead e0 +(Bind b0) u)))))) (\lambda (b0: B).(\lambda (e0: C).(\lambda (_: T).(\lambda +(w: T).(getl n c2 (CHead e0 (Bind b0) w)))))) (\lambda (_: B).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 +B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq +C (CHead x1 (Bind b) x2) (CHead e1 (Bind b0) u)))))) (\lambda (b0: +B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 +(Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda +(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: +B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C +(CHead x1 (Bind b) x2) (CHead e1 (Bind b0) u))))))) (\lambda (b0: B).(\lambda +(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 +(Bind b0) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda +(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: +B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 +(minus i (S n)) v e1 e2))))))) (ex3_4_intro B C T T (\lambda (b0: B).(\lambda +(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Bind b) x2) (CHead +e0 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (e0: C).(\lambda (_: +T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b0) w)))))) (\lambda (_: +B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v +u w))))) b x1 x2 x3 (refl_equal C (CHead x1 (Bind b) x2)) (getl_intro n c2 +(CHead x1 (Bind b) x3) (CHead x1 (Bind b) x3) H11 (clear_bind b x1 x3)) H12)) +e (clear_gen_bind b x1 e x2 H13)))))) (\lambda (f: F).(\lambda (H11: (drop n +O c2 (CHead x1 (Flat f) x3))).(\lambda (_: (subst0 (minus i (s (Flat f) n)) v +x2 x3)).(\lambda (H13: (clear (CHead x1 (Flat f) x2) e)).(or4_intro0 (getl n +c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: +T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: +B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 +(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: +B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind +b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: +T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) +(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda +(u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: +B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n +c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) +(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (getl_intro n c2 e (CHead x1 +(Flat f) x3) H11 (clear_flat x1 e (clear_gen_flat f x1 e x2 H13) f x3))))))) +x0 H8 H9 H10))))))))) H6)) (\lambda (H6: (ex3_4 K C C T (\lambda (k: +K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C x (CHead e1 k +u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: +T).(drop n O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda +(e2: C).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1 e2))))))).(ex3_4_ind +K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq +C x (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(drop n O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda +(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1 +e2))))) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: +C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) +(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 +(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T +(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e +(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: +B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) +v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) +(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: +T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v +u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x0: +K).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H7: (eq C x +(CHead x1 x0 x3))).(\lambda (H8: (drop n O c2 (CHead x2 x0 x3))).(\lambda +(H9: (csubst0 (minus i (s x0 n)) v x1 x2)).(let H10 \def (eq_ind C x (\lambda +(c: C).(clear c e)) H4 (CHead x1 x0 x3) H7) in (K_ind (\lambda (k: K).((drop +n O c2 (CHead x2 k x3)) \to ((csubst0 (minus i (s k n)) v x1 x2) \to ((clear +(CHead x1 k x3) e) \to (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: +B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind +b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: +T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 +B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq +C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: +B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) +v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) +(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: +T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v +u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))))))) (\lambda (b: +B).(\lambda (H11: (drop n O c2 (CHead x2 (Bind b) x3))).(\lambda (H12: +(csubst0 (minus i (s (Bind b) n)) v x1 x2)).(\lambda (H13: (clear (CHead x1 +(Bind b) x3) e)).(eq_ind_r C (CHead x1 (Bind b) x3) (\lambda (c: C).(or4 +(getl n c2 c) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e0: C).(\lambda (u: +T).(\lambda (_: T).(eq C c (CHead e0 (Bind b0) u)))))) (\lambda (b0: +B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 +(Bind b0) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda +(w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b0: +B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind +b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: +T).(getl n c2 (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) +(ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda +(u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b0) u))))))) (\lambda (b0: +B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n +c2 (CHead e2 (Bind b0) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) +(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro2 (getl n c2 +(CHead x1 (Bind b) x3)) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e0: +C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Bind b) x3) (CHead e0 +(Bind b0) u)))))) (\lambda (b0: B).(\lambda (e0: C).(\lambda (_: T).(\lambda +(w: T).(getl n c2 (CHead e0 (Bind b0) w)))))) (\lambda (_: B).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 +B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq +C (CHead x1 (Bind b) x3) (CHead e1 (Bind b0) u)))))) (\lambda (b0: +B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 +(Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda +(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: +B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C +(CHead x1 (Bind b) x3) (CHead e1 (Bind b0) u))))))) (\lambda (b0: B).(\lambda +(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 +(Bind b0) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda +(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: +B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 +(minus i (S n)) v e1 e2))))))) (ex3_4_intro B C C T (\lambda (b0: B).(\lambda +(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x1 (Bind b) x3) (CHead +e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b0) u)))))) (\lambda (_: +B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) +v e1 e2))))) b x1 x2 x3 (refl_equal C (CHead x1 (Bind b) x3)) (getl_intro n +c2 (CHead x2 (Bind b) x3) (CHead x2 (Bind b) x3) H11 (clear_bind b x2 x3)) +H12)) e (clear_gen_bind b x1 e x3 H13)))))) (\lambda (f: F).(\lambda (H11: +(drop n O c2 (CHead x2 (Flat f) x3))).(\lambda (H12: (csubst0 (minus i (s +(Flat f) n)) v x1 x2)).(\lambda (H13: (clear (CHead x1 (Flat f) x3) e)).(let +H14 \def (eq_ind nat (minus i n) (\lambda (n0: nat).(csubst0 n0 v x1 x2)) H12 +(S (minus i (S n))) (minus_x_Sy i n H)) in (let H15 \def (csubst0_clear_S x1 +x2 v (minus i (S n)) H14 e (clear_gen_flat f x1 e x3 H13)) in (or4_ind (clear +x2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: +T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u1)))))) (\lambda (b: +B).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e0 +(Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda +(u2: T).(subst0 (minus i (S n)) v u1 u2)))))) (ex3_4 B C C T (\lambda (b: +B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind +b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: +T).(clear x2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) +(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda +(u1: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u1))))))) (\lambda (b: +B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: 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(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e +(CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) +(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) +v e1 e2)))))))) (\lambda (H16: (clear x2 e)).(or4_intro0 (getl n c2 e) (ex3_4 +B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq +C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: +T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: +B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v +u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: +B).(\lambda (_: C).(\lambda (e2: 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u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: +T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 +B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq +C c (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: +B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) +v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b) u))))))) +(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: +T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v +u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (_: 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(\lambda (b: +B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 +(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda +(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: +B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e +(CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) +(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) +v e1 e2)))))))) (\lambda (x4: B).(\lambda (x5: C).(\lambda (x6: C).(\lambda +(x7: T).(\lambda (H17: (eq C e (CHead x5 (Bind x4) x7))).(\lambda (H18: +(clear x2 (CHead x6 (Bind x4) x7))).(\lambda (H19: (csubst0 (minus i (S n)) v +x5 x6)).(eq_ind_r C (CHead x5 (Bind x4) x7) (\lambda (c: C).(or4 (getl n c2 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T (\lambda (b: +B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C e +(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e2 (Bind b) u2))))))) +(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda +(u2: T).(subst0 (minus i (S n)) v u1 u2)))))) (\lambda (_: B).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) +v e1 e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C e (CHead e1 (Bind +b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (u2: T).(clear x2 (CHead e2 (Bind b) u2))))))) (\lambda (_: +B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 +(minus i (S n)) v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) +(or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda +(u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: +B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 +(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: +B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind +b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: +T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) +(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda +(u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: +B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n +c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) +(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x4: B).(\lambda +(x5: C).(\lambda (x6: C).(\lambda (x7: T).(\lambda (x8: T).(\lambda (H17: (eq +C e (CHead x5 (Bind x4) x7))).(\lambda (H18: (clear x2 (CHead x6 (Bind x4) +x8))).(\lambda (H19: (subst0 (minus i (S n)) v x7 x8)).(\lambda (H20: +(csubst0 (minus i (S n)) v x5 x6)).(eq_ind_r C (CHead x5 (Bind x4) x7) +(\lambda (c: C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda (b: B).(\lambda +(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Bind b) u)))))) +(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 +(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T +(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c +(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: +B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) +v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b) u))))))) +(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: +T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v +u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro3 +(getl n c2 (CHead x5 (Bind x4) x7)) (ex3_4 B C T T (\lambda (b: B).(\lambda +(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) x7) (CHead +e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda +(w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 +B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq +C (CHead x5 (Bind x4) x7) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda +(_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) +(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 +(minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) +x7) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) +(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) +v e1 e2))))))) (ex4_5_intro B C C T T (\lambda (b: B).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) +x7) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) +(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) +v e1 e2)))))) x4 x5 x6 x7 x8 (refl_equal C (CHead x5 (Bind x4) x7)) +(getl_intro n c2 (CHead x6 (Bind x4) x8) (CHead x2 (Flat f) x3) H11 +(clear_flat x2 (CHead x6 (Bind x4) x8) H18 f x3)) H19 H20)) e H17)))))))))) +H16)) H15))))))) x0 H8 H9 H10))))))))) H6)) (\lambda (H6: (ex4_5 K C C T T +(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: +T).(eq C x (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e2 k w))))))) (\lambda +(k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 (minus i (s k n)) v u w)))))) (\lambda (k: K).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k +n)) v e1 e2)))))))).(ex4_5_ind K C C T T (\lambda (k: K).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C x (CHead e1 k +u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (w: T).(drop n O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda +(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k +n)) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1 e2)))))) (or4 (getl n +c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: +T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: +B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 +(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: +B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind +b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: +T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) +(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda +(u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: +B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n +c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) +(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x0: K).(\lambda +(x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H7: (eq +C x (CHead x1 x0 x3))).(\lambda (H8: (drop n O c2 (CHead x2 x0 x4))).(\lambda +(H9: (subst0 (minus i (s x0 n)) v x3 x4)).(\lambda (H10: (csubst0 (minus i (s +x0 n)) v x1 x2)).(let H11 \def (eq_ind C x (\lambda (c: C).(clear c e)) H4 +(CHead x1 x0 x3) H7) in (K_ind (\lambda (k: K).((drop n O c2 (CHead x2 k x4)) +\to ((subst0 (minus i (s k n)) v x3 x4) \to ((csubst0 (minus i (s k n)) v x1 +x2) \to ((clear (CHead x1 k x3) e) \to (or4 (getl n c2 e) (ex3_4 B C T T +(\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e +(CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: +T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: +B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v +u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: +B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 +(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda +(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: +B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e +(CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) +(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) +v e1 e2))))))))))))) (\lambda (b: B).(\lambda (H12: (drop n O c2 (CHead x2 +(Bind b) x4))).(\lambda (H13: (subst0 (minus i (s (Bind b) n)) v x3 +x4)).(\lambda (H14: (csubst0 (minus i (s (Bind b) n)) v x1 x2)).(\lambda +(H15: (clear (CHead x1 (Bind b) x3) e)).(eq_ind_r C (CHead x1 (Bind b) x3) +(\lambda (c: C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda (b0: B).(\lambda +(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Bind b0) u)))))) +(\lambda (b0: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 +(CHead e0 (Bind b0) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T +(\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c +(CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b0) u)))))) (\lambda (_: +B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) +v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b0) u))))))) +(\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(w: T).(getl n c2 (CHead e2 (Bind b0) w))))))) (\lambda (_: B).(\lambda (_: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v +u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro3 +(getl n c2 (CHead x1 (Bind b) x3)) (ex3_4 B C T T (\lambda (b0: B).(\lambda +(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Bind b) x3) (CHead +e0 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (e0: C).(\lambda (_: +T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b0) w)))))) (\lambda (_: +B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v +u w)))))) (ex3_4 B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(eq C (CHead x1 (Bind b) x3) (CHead e1 (Bind b0) u)))))) +(\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 +(CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T +(\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda +(_: T).(eq C (CHead x1 (Bind b) x3) (CHead e1 (Bind b0) u))))))) (\lambda +(b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: +T).(getl n c2 (CHead e2 (Bind b0) w))))))) (\lambda (_: B).(\lambda (_: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v +u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (ex4_5_intro B C C +T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: +T).(\lambda (_: T).(eq C (CHead x1 (Bind b) x3) (CHead e1 (Bind b0) u))))))) +(\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(w: T).(getl n c2 (CHead e2 (Bind b0) w))))))) (\lambda (_: B).(\lambda (_: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v +u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) b x1 x2 x3 x4 +(refl_equal C (CHead x1 (Bind b) x3)) (getl_intro n c2 (CHead x2 (Bind b) x4) +(CHead x2 (Bind b) x4) H12 (clear_bind b x2 x4)) H13 H14)) e (clear_gen_bind +b x1 e x3 H15))))))) (\lambda (f: F).(\lambda (H12: (drop n O c2 (CHead x2 +(Flat f) x4))).(\lambda (_: (subst0 (minus i (s (Flat f) n)) v x3 +x4)).(\lambda (H14: (csubst0 (minus i (s (Flat f) n)) v x1 x2)).(\lambda +(H15: (clear (CHead x1 (Flat f) x3) e)).(let H16 \def (eq_ind nat (minus i n) +(\lambda (n0: nat).(csubst0 n0 v x1 x2)) H14 (S (minus i (S n))) (minus_x_Sy +i n H)) in (let H17 \def (csubst0_clear_S x1 x2 v (minus i (S n)) H16 e +(clear_gen_flat f x1 e x3 H15)) in (or4_ind (clear x2 e) (ex3_4 B C T T +(\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C e +(CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: +T).(\lambda (u2: T).(clear x2 (CHead e0 (Bind b) u2)))))) (\lambda (_: +B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (minus i (S n)) +v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: +B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x2 (CHead e2 (Bind +b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: +B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C e +(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e2 (Bind b) u2))))))) +(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda +(u2: T).(subst0 (minus i (S n)) v u1 u2)))))) (\lambda (_: B).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) +v e1 e2))))))) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda +(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) +(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 +(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T +(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e +(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: +B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) +v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) +(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: +T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v +u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (H18: +(clear x2 e)).(or4_intro0 (getl n c2 e) (ex3_4 B C T T (\lambda (b: +B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind +b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: +T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 +B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq +C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: +B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) +v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) +(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: +T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v +u w)))))) (\lambda (_: 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(\lambda (b: B).(\lambda (e0: +C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) +(\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 +(minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) +(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 +(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T +(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: +T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 +(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda +(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: +B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 +(minus i (S n)) v e1 e2)))))))) (\lambda (x5: B).(\lambda (x6: C).(\lambda +(x7: T).(\lambda (x8: T).(\lambda (H19: (eq C e (CHead x6 (Bind x5) +x7))).(\lambda (H20: (clear x2 (CHead x6 (Bind x5) x8))).(\lambda (H21: +(subst0 (minus i (S n)) v x7 x8)).(eq_ind_r C (CHead x6 (Bind x5) x7) +(\lambda (c: C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda (b: B).(\lambda +(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Bind b) u)))))) +(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 +(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T +(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c +(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: +B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) +v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b) u))))))) +(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: +T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v +u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro1 +(getl n c2 (CHead x6 (Bind x5) x7)) (ex3_4 B C T T (\lambda (b: B).(\lambda +(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) x7) (CHead +e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda +(w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 +B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq +C (CHead x6 (Bind x5) x7) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda +(_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) +(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 +(minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) +x7) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) +(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) +v e1 e2))))))) (ex3_4_intro B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda +(u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) x7) (CHead e0 (Bind b) +u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: +T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w))))) x5 x6 +x7 x8 (refl_equal C (CHead x6 (Bind x5) x7)) (getl_intro n c2 (CHead x6 (Bind +x5) x8) (CHead x2 (Flat f) x4) H12 (clear_flat x2 (CHead x6 (Bind x5) x8) H20 +f x4)) H21)) e H19)))))))) H18)) (\lambda (H18: (ex3_4 B C C T (\lambda (b: +B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind +b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: +T).(clear x2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 +e2))))))).(ex3_4_ind B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: +B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x2 (CHead e2 (Bind +b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(csubst0 (minus i (S n)) v e1 e2))))) (or4 (getl n c2 e) (ex3_4 B C T T +(\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e +(CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: +T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: +B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v +u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: +B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 +(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda +(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: +B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e +(CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) +(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) +v e1 e2)))))))) (\lambda (x5: B).(\lambda (x6: C).(\lambda (x7: C).(\lambda +(x8: T).(\lambda (H19: (eq C e (CHead x6 (Bind x5) x8))).(\lambda (H20: +(clear x2 (CHead x7 (Bind x5) x8))).(\lambda (H21: (csubst0 (minus i (S n)) v +x6 x7)).(eq_ind_r C (CHead x6 (Bind x5) x8) (\lambda (c: C).(or4 (getl n c2 +c) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda +(_: T).(eq C c (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: +C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) +(\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 +(minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind b) u)))))) +(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 +(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: 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+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x6 (Bind x5) +x8) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: +B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) +v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) x8) (CHead e1 +(Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: +B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 +(minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))) +(ex3_4_intro B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(eq C (CHead x6 (Bind x5) x8) (CHead e1 (Bind b) u)))))) +(\lambda (b: 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T).(\lambda (_: T).(eq C e (CHead e1 (Bind +b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (u2: T).(clear x2 (CHead e2 (Bind b) u2))))))) (\lambda (_: +B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 +(minus i (S n)) v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) +(or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda +(u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: +B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 +(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: +B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind +b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: +T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) +(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda +(u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: +B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n +c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) +(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x5: B).(\lambda +(x6: C).(\lambda (x7: C).(\lambda (x8: T).(\lambda (x9: T).(\lambda (H19: (eq +C e (CHead x6 (Bind x5) x8))).(\lambda (H20: (clear x2 (CHead x7 (Bind x5) +x9))).(\lambda (H21: (subst0 (minus i (S n)) v x8 x9)).(\lambda (H22: +(csubst0 (minus i (S n)) v x6 x7)).(eq_ind_r C (CHead x6 (Bind x5) x8) +(\lambda (c: C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda (b: B).(\lambda +(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Bind b) u)))))) +(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 +(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: +T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T +(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c +(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: +B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) +v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b) u))))))) +(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: +T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v +u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro3 +(getl n c2 (CHead x6 (Bind x5) x8)) (ex3_4 B C T T (\lambda (b: B).(\lambda +(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) x8) (CHead +e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda +(w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 +B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq +C (CHead x6 (Bind x5) x8) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda +(_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) +(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 +(minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) +x8) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) +(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) +v e1 e2))))))) (ex4_5_intro B C C T T (\lambda (b: B).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) +x8) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) +(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) +v e1 e2)))))) x5 x6 x7 x8 x9 (refl_equal C (CHead x6 (Bind x5) x8)) +(getl_intro n c2 (CHead x7 (Bind x5) x9) (CHead x2 (Flat f) x4) H12 +(clear_flat x2 (CHead x7 (Bind x5) x9) H20 f x4)) H21 H22)) e H19)))))))))) +H18)) H17)))))))) x0 H8 H9 H10 H11))))))))))) H6)) H5))))) H2)))))))))). +(* COMMENTS +Initial nodes: 17179 +END *) + +theorem csubst0_getl_ge_back: + \forall (i: nat).(\forall (n: nat).((le i n) \to (\forall (c1: C).(\forall +(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((getl n c2 +e) \to (getl n c1 e))))))))) +\def + \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (le i n)).(\lambda (c1: +C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 i v c1 +c2)).(\lambda (e: C).(\lambda (H1: (getl n c2 e)).(let H2 \def (getl_gen_all +c2 e n H1) in (ex2_ind C (\lambda (e0: C).(drop n O c2 e0)) (\lambda (e0: +C).(clear e0 e)) (getl n c1 e) (\lambda (x: C).(\lambda (H3: (drop n O c2 +x)).(\lambda (H4: (clear x e)).(lt_eq_gt_e i n (getl n c1 e) (\lambda (H5: +(lt i n)).(getl_intro n c1 e x (csubst0_drop_gt_back n i H5 c1 c2 v H0 x H3) +H4)) (\lambda (H5: (eq nat i n)).(let H6 \def (eq_ind_r nat n (\lambda (n0: +nat).(drop n0 O c2 x)) H3 i H5) in (let H7 \def (eq_ind_r nat n (\lambda (n0: +nat).(le i n0)) H i H5) in (eq_ind nat i (\lambda (n0: nat).(getl n0 c1 e)) +(let H8 \def (csubst0_drop_eq_back i c1 c2 v H0 x H6) in (or4_ind (drop i O +c1 x) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: +T).(\lambda (u2: T).(eq C x (CHead e0 (Flat f) u2)))))) (\lambda (f: +F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop i O c1 (CHead e0 +(Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda +(u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: +C).(\lambda (e2: C).(\lambda (u: T).(eq C x (CHead e2 (Flat f) u)))))) +(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop i O c1 +(CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: +F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C x +(CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u1: T).(\lambda (_: T).(drop i O c1 (CHead e1 (Flat f) u1))))))) +(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda +(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda +(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (getl i c1 +e) (\lambda (H9: (drop i O c1 x)).(getl_intro i c1 e x H9 H4)) (\lambda (H9: +(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: +T).(eq C x (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: +C).(\lambda (u1: T).(\lambda (_: T).(drop i O c1 (CHead e0 (Flat f) u1)))))) +(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v +u1 u2))))))).(ex3_4_ind F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: +T).(\lambda (u2: T).(eq C x (CHead e0 (Flat f) u2)))))) (\lambda (f: +F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop i O c1 (CHead e0 +(Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda +(u2: T).(subst0 O v u1 u2))))) (getl i c1 e) (\lambda (x0: F).(\lambda (x1: +C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H10: (eq C x (CHead x1 (Flat +x0) x3))).(\lambda (H11: (drop i O c1 (CHead x1 (Flat x0) x2))).(\lambda (_: +(subst0 O v x2 x3)).(let H13 \def (eq_ind C x (\lambda (c: C).(clear c e)) H4 +(CHead x1 (Flat x0) x3) H10) in (getl_intro i c1 e (CHead x1 (Flat x0) x2) +H11 (clear_flat x1 e (clear_gen_flat x0 x1 e x3 H13) x0 x2)))))))))) H9)) +(\lambda (H9: (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: +C).(\lambda (u: T).(eq C x (CHead e2 (Flat f) u)))))) (\lambda (f: +F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop i O c1 (CHead e1 +(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda +(_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda +(_: C).(\lambda (e2: C).(\lambda (u: T).(eq C x (CHead e2 (Flat f) u)))))) +(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop i O c1 +(CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(csubst0 O v e1 e2))))) (getl i c1 e) (\lambda (x0: +F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H10: (eq C x +(CHead x2 (Flat x0) x3))).(\lambda (H11: (drop i O c1 (CHead x1 (Flat x0) +x3))).(\lambda (H12: (csubst0 O v x1 x2)).(let H13 \def (eq_ind C x (\lambda +(c: C).(clear c e)) H4 (CHead x2 (Flat x0) x3) H10) in (getl_intro i c1 e +(CHead x1 (Flat x0) x3) H11 (clear_flat x1 e (csubst0_clear_O_back x1 x2 v +H12 e (clear_gen_flat x0 x2 e x3 H13)) x0 x3)))))))))) H9)) (\lambda (H9: +(ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (u2: T).(eq C x (CHead e2 (Flat f) u2))))))) (\lambda (f: +F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop i +O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda +(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: +F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 +O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda (f: F).(\lambda (_: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C x (CHead e2 (Flat +f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: +T).(\lambda (_: T).(drop i O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: +F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 +O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (getl i c1 e) (\lambda (x0: +F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: +T).(\lambda (H10: (eq C x (CHead x2 (Flat x0) x4))).(\lambda (H11: (drop i O +c1 (CHead x1 (Flat x0) x3))).(\lambda (_: (subst0 O v x3 x4)).(\lambda (H13: +(csubst0 O v x1 x2)).(let H14 \def (eq_ind C x (\lambda (c: C).(clear c e)) +H4 (CHead x2 (Flat x0) x4) H10) in (getl_intro i c1 e (CHead x1 (Flat x0) x3) +H11 (clear_flat x1 e (csubst0_clear_O_back x1 x2 v H13 e (clear_gen_flat x0 +x2 e x4 H14)) x0 x3)))))))))))) H9)) H8)) n H5)))) (\lambda (H5: (lt n +i)).(le_lt_false i n H H5 (getl n c1 e))))))) H2)))))))))). +(* COMMENTS +Initial nodes: 1525 +END *) + +theorem csubst0_getl_lt_back: + \forall (n: nat).(\forall (i: nat).((lt n i) \to (\forall (c1: C).(\forall +(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e2: C).((getl n c2 +e2) \to (or (getl n c1 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) v e1 +e2)) (\lambda (e1: C).(getl n c1 e1)))))))))))) +\def + \lambda (n: nat).(\lambda (i: nat).(\lambda (H: (lt n i)).(\lambda (c1: +C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 i v c1 +c2)).(\lambda (e2: C).(\lambda (H1: (getl n c2 e2)).(let H2 \def +(getl_gen_all c2 e2 n H1) in (ex2_ind C (\lambda (e: C).(drop n O c2 e)) +(\lambda (e: C).(clear e e2)) (or (getl n c1 e2) (ex2 C (\lambda (e1: +C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(getl n c1 e1)))) (\lambda +(x: C).(\lambda (H3: (drop n O c2 x)).(\lambda (H4: (clear x e2)).(let H_x +\def (csubst0_drop_lt_back n i H c1 c2 v H0 x H3) in (let H5 \def H_x in +(or_ind (drop n O c1 x) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) v e1 x)) +(\lambda (e1: C).(drop n O c1 e1))) (or (getl n c1 e2) (ex2 C (\lambda (e1: +C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(getl n c1 e1)))) (\lambda +(H6: (drop n O c1 x)).(or_introl (getl n c1 e2) (ex2 C (\lambda (e1: +C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(getl n c1 e1))) +(getl_intro n c1 e2 x H6 H4))) (\lambda (H6: (ex2 C (\lambda (e1: C).(csubst0 +(minus i n) v e1 x)) (\lambda (e1: C).(drop n O c1 e1)))).(ex2_ind C (\lambda +(e1: C).(csubst0 (minus i n) v e1 x)) (\lambda (e1: C).(drop n O c1 e1)) (or +(getl n c1 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) v e1 e2)) +(\lambda (e1: C).(getl n c1 e1)))) (\lambda (x0: C).(\lambda (H7: (csubst0 +(minus i n) v x0 x)).(\lambda (H8: (drop n O c1 x0)).(let H_x0 \def +(csubst0_clear_trans x0 x v (minus i n) H7 e2 H4) in (let H9 \def H_x0 in +(or_ind (clear x0 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) v e1 e2)) +(\lambda (e1: C).(clear x0 e1))) (or (getl n c1 e2) (ex2 C (\lambda (e1: +C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(getl n c1 e1)))) (\lambda +(H10: (clear x0 e2)).(or_introl (getl n c1 e2) (ex2 C (\lambda (e1: +C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(getl n c1 e1))) +(getl_intro n c1 e2 x0 H8 H10))) (\lambda (H10: (ex2 C (\lambda (e1: +C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(clear x0 e1)))).(ex2_ind +C (\lambda (e1: C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(clear x0 +e1)) (or (getl n c1 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) v e1 +e2)) (\lambda (e1: C).(getl n c1 e1)))) (\lambda (x1: C).(\lambda (H11: +(csubst0 (minus i n) v x1 e2)).(\lambda (H12: (clear x0 x1)).(or_intror (getl +n c1 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: +C).(getl n c1 e1))) (ex_intro2 C (\lambda (e1: C).(csubst0 (minus i n) v e1 +e2)) (\lambda (e1: C).(getl n c1 e1)) x1 H11 (getl_intro n c1 x1 x0 H8 +H12)))))) H10)) H9)))))) H6)) H5)))))) H2)))))))))). +(* COMMENTS +Initial nodes: 801 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubst0/props.ma b/matita/matita/contribs/lambdadelta/basic_1/csubst0/props.ma new file mode 100644 index 000000000..28b75ec37 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/csubst0/props.ma @@ -0,0 +1,61 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/csubst0/defs.ma". + +theorem csubst0_snd_bind: + \forall (b: B).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall +(u2: T).((subst0 i v u1 u2) \to (\forall (c: C).(csubst0 (S i) v (CHead c +(Bind b) u1) (CHead c (Bind b) u2)))))))) +\def + \lambda (b: B).(\lambda (i: nat).(\lambda (v: T).(\lambda (u1: T).(\lambda +(u2: T).(\lambda (H: (subst0 i v u1 u2)).(\lambda (c: C).(eq_ind nat (s (Bind +b) i) (\lambda (n: nat).(csubst0 n v (CHead c (Bind b) u1) (CHead c (Bind b) +u2))) (csubst0_snd (Bind b) i v u1 u2 H c) (S i) (refl_equal nat (S +i))))))))). +(* COMMENTS +Initial nodes: 91 +END *) + +theorem csubst0_fst_bind: + \forall (b: B).(\forall (i: nat).(\forall (c1: C).(\forall (c2: C).(\forall +(v: T).((csubst0 i v c1 c2) \to (\forall (u: T).(csubst0 (S i) v (CHead c1 +(Bind b) u) (CHead c2 (Bind b) u)))))))) +\def + \lambda (b: B).(\lambda (i: nat).(\lambda (c1: C).(\lambda (c2: C).(\lambda +(v: T).(\lambda (H: (csubst0 i v c1 c2)).(\lambda (u: T).(eq_ind nat (s (Bind +b) i) (\lambda (n: nat).(csubst0 n v (CHead c1 (Bind b) u) (CHead c2 (Bind b) +u))) (csubst0_fst (Bind b) i c1 c2 v H u) (S i) (refl_equal nat (S i))))))))). +(* COMMENTS +Initial nodes: 91 +END *) + +theorem csubst0_both_bind: + \forall (b: B).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall +(u2: T).((subst0 i v u1 u2) \to (\forall (c1: C).(\forall (c2: C).((csubst0 i +v c1 c2) \to (csubst0 (S i) v (CHead c1 (Bind b) u1) (CHead c2 (Bind b) +u2)))))))))) +\def + \lambda (b: B).(\lambda (i: nat).(\lambda (v: T).(\lambda (u1: T).(\lambda +(u2: T).(\lambda (H: (subst0 i v u1 u2)).(\lambda (c1: C).(\lambda (c2: +C).(\lambda (H0: (csubst0 i v c1 c2)).(eq_ind nat (s (Bind b) i) (\lambda (n: +nat).(csubst0 n v (CHead c1 (Bind b) u1) (CHead c2 (Bind b) u2))) +(csubst0_both (Bind b) i v u1 u2 H c1 c2 H0) (S i) (refl_equal nat (S +i))))))))))). +(* COMMENTS +Initial nodes: 107 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubst1/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/csubst1/defs.ma new file mode 100644 index 000000000..d6ba0a942 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/csubst1/defs.ma @@ -0,0 +1,22 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/csubst0/defs.ma". + +inductive csubst1 (i: nat) (v: T) (c1: C): C \to Prop \def +| csubst1_refl: csubst1 i v c1 c1 +| csubst1_sing: \forall (c2: C).((csubst0 i v c1 c2) \to (csubst1 i v c1 c2)). + diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubst1/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/csubst1/fwd.ma new file mode 100644 index 000000000..fe71a56fd --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/csubst1/fwd.ma @@ -0,0 +1,117 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/csubst1/defs.ma". + +include "Basic-1/csubst0/fwd.ma". + +include "Basic-1/subst1/props.ma". + +theorem csubst1_gen_head: + \forall (k: K).(\forall (c1: C).(\forall (x: C).(\forall (u1: T).(\forall +(v: T).(\forall (i: nat).((csubst1 (s k i) v (CHead c1 k u1) x) \to (ex3_2 T +C (\lambda (u2: T).(\lambda (c2: C).(eq C x (CHead c2 k u2)))) (\lambda (u2: +T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c2: +C).(csubst1 i v c1 c2)))))))))) +\def + \lambda (k: K).(\lambda (c1: C).(\lambda (x: C).(\lambda (u1: T).(\lambda +(v: T).(\lambda (i: nat).(\lambda (H: (csubst1 (s k i) v (CHead c1 k u1) +x)).(csubst1_ind (s k i) v (CHead c1 k u1) (\lambda (c: C).(ex3_2 T C +(\lambda (u2: T).(\lambda (c2: C).(eq C c (CHead c2 k u2)))) (\lambda (u2: +T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c2: +C).(csubst1 i v c1 c2))))) (ex3_2_intro T C (\lambda (u2: T).(\lambda (c2: +C).(eq C (CHead c1 k u1) (CHead c2 k u2)))) (\lambda (u2: T).(\lambda (_: +C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c2: C).(csubst1 i v c1 +c2))) u1 c1 (refl_equal C (CHead c1 k u1)) (subst1_refl i v u1) (csubst1_refl +i v c1)) (\lambda (c2: C).(\lambda (H0: (csubst0 (s k i) v (CHead c1 k u1) +c2)).(or3_ind (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i) +(s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c1 k u2)))) +(\lambda (u2: T).(\lambda (j: nat).(subst0 j v u1 u2)))) (ex3_2 C nat +(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i) (s k j)))) (\lambda (c3: +C).(\lambda (_: nat).(eq C c2 (CHead c3 k u1)))) (\lambda (c3: C).(\lambda +(j: nat).(csubst0 j v c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: +C).(\lambda (j: nat).(eq nat (s k i) (s k j))))) (\lambda (u2: T).(\lambda +(c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: +T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u2)))) (\lambda (_: +T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3))))) (ex3_2 T C +(\lambda (u2: T).(\lambda (c3: C).(eq C c2 (CHead c3 k u2)))) (\lambda (u2: +T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3: +C).(csubst1 i v c1 c3)))) (\lambda (H1: (ex3_2 T nat (\lambda (_: T).(\lambda +(j: nat).(eq nat (s k i) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C +c2 (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v u1 +u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i) (s +k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c1 k u2)))) +(\lambda (u2: T).(\lambda (j: nat).(subst0 j v u1 u2))) (ex3_2 T C (\lambda +(u2: T).(\lambda (c3: C).(eq C c2 (CHead c3 k u2)))) (\lambda (u2: +T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3: +C).(csubst1 i v c1 c3)))) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H2: +(eq nat (s k i) (s k x1))).(\lambda (H3: (eq C c2 (CHead c1 k x0))).(\lambda +(H4: (subst0 x1 v u1 x0)).(eq_ind_r C (CHead c1 k x0) (\lambda (c: C).(ex3_2 +T C (\lambda (u2: T).(\lambda (c3: C).(eq C c (CHead c3 k u2)))) (\lambda +(u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3: +C).(csubst1 i v c1 c3))))) (let H5 \def (eq_ind_r nat x1 (\lambda (n: +nat).(subst0 n v u1 x0)) H4 i (s_inj k i x1 H2)) in (ex3_2_intro T C (\lambda +(u2: T).(\lambda (c3: C).(eq C (CHead c1 k x0) (CHead c3 k u2)))) (\lambda +(u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3: +C).(csubst1 i v c1 c3))) x0 c1 (refl_equal C (CHead c1 k x0)) (subst1_single +i v u1 x0 H5) (csubst1_refl i v c1))) c2 H3)))))) H1)) (\lambda (H1: (ex3_2 C +nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i) (s k j)))) (\lambda +(c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u1)))) (\lambda (c3: +C).(\lambda (j: nat).(csubst0 j v c1 c3))))).(ex3_2_ind C nat (\lambda (_: +C).(\lambda (j: nat).(eq nat (s k i) (s k j)))) (\lambda (c3: C).(\lambda (_: +nat).(eq C c2 (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 +j v c1 c3))) (ex3_2 T C (\lambda (u2: T).(\lambda (c3: C).(eq C c2 (CHead c3 +k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: +T).(\lambda (c3: C).(csubst1 i v c1 c3)))) (\lambda (x0: C).(\lambda (x1: +nat).(\lambda (H2: (eq nat (s k i) (s k x1))).(\lambda (H3: (eq C c2 (CHead +x0 k u1))).(\lambda (H4: (csubst0 x1 v c1 x0)).(eq_ind_r C (CHead x0 k u1) +(\lambda (c: C).(ex3_2 T C (\lambda (u2: T).(\lambda (c3: C).(eq C c (CHead +c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda +(_: T).(\lambda (c3: C).(csubst1 i v c1 c3))))) (let H5 \def (eq_ind_r nat x1 +(\lambda (n: nat).(csubst0 n v c1 x0)) H4 i (s_inj k i x1 H2)) in +(ex3_2_intro T C (\lambda (u2: T).(\lambda (c3: C).(eq C (CHead x0 k u1) +(CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) +(\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1 c3))) u1 x0 (refl_equal C +(CHead x0 k u1)) (subst1_refl i v u1) (csubst1_sing i v c1 x0 H5))) c2 +H3)))))) H1)) (\lambda (H1: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: +C).(\lambda (j: nat).(eq nat (s k i) (s k j))))) (\lambda (u2: T).(\lambda +(c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: +T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u2)))) (\lambda (_: +T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))))).(ex4_3_ind T C +nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i) (s k +j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 +k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 +u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 +c3)))) (ex3_2 T C (\lambda (u2: T).(\lambda (c3: C).(eq C c2 (CHead c3 k +u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: +T).(\lambda (c3: C).(csubst1 i v c1 c3)))) (\lambda (x0: T).(\lambda (x1: +C).(\lambda (x2: nat).(\lambda (H2: (eq nat (s k i) (s k x2))).(\lambda (H3: +(eq C c2 (CHead x1 k x0))).(\lambda (H4: (subst0 x2 v u1 x0)).(\lambda (H5: +(csubst0 x2 v c1 x1)).(eq_ind_r C (CHead x1 k x0) (\lambda (c: C).(ex3_2 T C +(\lambda (u2: T).(\lambda (c3: C).(eq C c (CHead c3 k u2)))) (\lambda (u2: +T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3: +C).(csubst1 i v c1 c3))))) (let H6 \def (eq_ind_r nat x2 (\lambda (n: +nat).(csubst0 n v c1 x1)) H5 i (s_inj k i x2 H2)) in (let H7 \def (eq_ind_r +nat x2 (\lambda (n: nat).(subst0 n v u1 x0)) H4 i (s_inj k i x2 H2)) in +(ex3_2_intro T C (\lambda (u2: T).(\lambda (c3: C).(eq C (CHead x1 k x0) +(CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) +(\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1 c3))) x0 x1 (refl_equal C +(CHead x1 k x0)) (subst1_single i v u1 x0 H7) (csubst1_sing i v c1 x1 H6)))) +c2 H3)))))))) H1)) (csubst0_gen_head k c1 c2 u1 v (s k i) H0)))) x H))))))). +(* COMMENTS +Initial nodes: 1817 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubst1/getl.ma b/matita/matita/contribs/lambdadelta/basic_1/csubst1/getl.ma new file mode 100644 index 000000000..e24ba9533 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/csubst1/getl.ma @@ -0,0 +1,283 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/csubst1/props.ma". + +include "Basic-1/csubst0/getl.ma". + +include "Basic-1/subst1/props.ma". + +include "Basic-1/drop/props.ma". + +theorem csubst1_getl_ge: + \forall (i: nat).(\forall (n: nat).((le i n) \to (\forall (c1: C).(\forall +(c2: C).(\forall (v: T).((csubst1 i v c1 c2) \to (\forall (e: C).((getl n c1 +e) \to (getl n c2 e))))))))) +\def + \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (le i n)).(\lambda (c1: +C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst1 i v c1 +c2)).(csubst1_ind i v c1 (\lambda (c: C).(\forall (e: C).((getl n c1 e) \to +(getl n c e)))) (\lambda (e: C).(\lambda (H1: (getl n c1 e)).H1)) (\lambda +(c3: C).(\lambda (H1: (csubst0 i v c1 c3)).(\lambda (e: C).(\lambda (H2: +(getl n c1 e)).(csubst0_getl_ge i n H c1 c3 v H1 e H2))))) c2 H0))))))). +(* COMMENTS +Initial nodes: 111 +END *) + +theorem csubst1_getl_lt: + \forall (i: nat).(\forall (n: nat).((lt n i) \to (\forall (c1: C).(\forall +(c2: C).(\forall (v: T).((csubst1 i v c1 c2) \to (\forall (e1: C).((getl n c1 +e1) \to (ex2 C (\lambda (e2: C).(csubst1 (minus i n) v e1 e2)) (\lambda (e2: +C).(getl n c2 e2))))))))))) +\def + \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (lt n i)).(\lambda (c1: +C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst1 i v c1 +c2)).(csubst1_ind i v c1 (\lambda (c: C).(\forall (e1: C).((getl n c1 e1) \to +(ex2 C (\lambda (e2: C).(csubst1 (minus i n) v e1 e2)) (\lambda (e2: C).(getl +n c e2)))))) (\lambda (e1: C).(\lambda (H1: (getl n c1 e1)).(eq_ind_r nat (S +(minus i (S n))) (\lambda (n0: nat).(ex2 C (\lambda (e2: C).(csubst1 n0 v e1 +e2)) (\lambda (e2: C).(getl n c1 e2)))) (ex_intro2 C (\lambda (e2: +C).(csubst1 (S (minus i (S n))) v e1 e2)) (\lambda (e2: C).(getl n c1 e2)) e1 +(csubst1_refl (S (minus i (S n))) v e1) H1) (minus i n) (minus_x_Sy i n H)))) +(\lambda (c3: C).(\lambda (H1: (csubst0 i v c1 c3)).(\lambda (e1: C).(\lambda +(H2: (getl n c1 e1)).(eq_ind_r nat (S (minus i (S n))) (\lambda (n0: +nat).(ex2 C (\lambda (e2: C).(csubst1 n0 v e1 e2)) (\lambda (e2: C).(getl n +c3 e2)))) (let H3 \def (csubst0_getl_lt i n H c1 c3 v H1 e1 H2) in (or4_ind +(getl n c3 e1) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: +T).(\lambda (_: T).(eq C e1 (CHead e0 (Bind b) u)))))) (\lambda (b: +B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e0 +(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: +B).(\lambda (e2: C).(\lambda (_: C).(\lambda (u: T).(eq C e1 (CHead e2 (Bind +b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e3: C).(\lambda (u: +T).(getl n c3 (CHead e3 (Bind b) u)))))) (\lambda (_: B).(\lambda (e2: +C).(\lambda (e3: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e2 e3)))))) +(ex4_5 B C C T T (\lambda (b: B).(\lambda (e2: C).(\lambda (_: C).(\lambda +(u: T).(\lambda (_: T).(eq C e1 (CHead e2 (Bind b) u))))))) (\lambda (b: +B).(\lambda (_: C).(\lambda (e3: C).(\lambda (_: T).(\lambda (w: T).(getl n +c3 (CHead e3 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) +(\lambda (_: B).(\lambda (e2: C).(\lambda (e3: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 (minus i (S n)) v e2 e3))))))) (ex2 C (\lambda (e2: +C).(csubst1 (S (minus i (S n))) v e1 e2)) (\lambda (e2: C).(getl n c3 e2))) +(\lambda (H4: (getl n c3 e1)).(ex_intro2 C (\lambda (e2: C).(csubst1 (S +(minus i (S n))) v e1 e2)) (\lambda (e2: C).(getl n c3 e2)) e1 (csubst1_refl +(S (minus i (S n))) v e1) H4)) (\lambda (H4: (ex3_4 B C T T (\lambda (b: +B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e1 (CHead e0 (Bind +b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: +T).(getl n c3 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: +C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u +w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: +T).(\lambda (_: T).(eq C e1 (CHead e0 (Bind b) u)))))) (\lambda (b: +B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e0 +(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: +T).(subst0 (minus i (S n)) v u w))))) (ex2 C (\lambda (e2: C).(csubst1 (S +(minus i (S n))) v e1 e2)) (\lambda (e2: C).(getl n c3 e2))) (\lambda (x0: +B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H5: (eq C e1 +(CHead x1 (Bind x0) x2))).(\lambda (H6: (getl n c3 (CHead x1 (Bind x0) +x3))).(\lambda (H7: (subst0 (minus i (S n)) v x2 x3)).(eq_ind_r C (CHead x1 +(Bind x0) x2) (\lambda (c: C).(ex2 C (\lambda (e2: C).(csubst1 (S (minus i (S +n))) v c e2)) (\lambda (e2: C).(getl n c3 e2)))) (ex_intro2 C (\lambda (e2: +C).(csubst1 (S (minus i (S n))) v (CHead x1 (Bind x0) x2) e2)) (\lambda (e2: +C).(getl n c3 e2)) (CHead x1 (Bind x0) x3) (csubst1_sing (S (minus i (S n))) +v (CHead x1 (Bind x0) x2) (CHead x1 (Bind x0) x3) (csubst0_snd_bind x0 (minus +i (S n)) v x2 x3 H7 x1)) H6) e1 H5)))))))) H4)) (\lambda (H4: (ex3_4 B C C T +(\lambda (b: B).(\lambda (e2: C).(\lambda (_: C).(\lambda (u: T).(eq C e1 +(CHead e2 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e3: +C).(\lambda (u: T).(getl n c3 (CHead e3 (Bind b) u)))))) (\lambda (_: +B).(\lambda (e2: C).(\lambda (e3: C).(\lambda (_: T).(csubst0 (minus i (S n)) +v e2 e3))))))).(ex3_4_ind B C C T (\lambda (b: B).(\lambda (e2: C).(\lambda +(_: C).(\lambda (u: T).(eq C e1 (CHead e2 (Bind b) u)))))) (\lambda (b: +B).(\lambda (_: C).(\lambda (e3: C).(\lambda (u: T).(getl n c3 (CHead e3 +(Bind b) u)))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (e3: C).(\lambda +(_: T).(csubst0 (minus i (S n)) v e2 e3))))) (ex2 C (\lambda (e2: C).(csubst1 +(S (minus i (S n))) v e1 e2)) (\lambda (e2: C).(getl n c3 e2))) (\lambda (x0: +B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H5: (eq C e1 +(CHead x1 (Bind x0) x3))).(\lambda (H6: (getl n c3 (CHead x2 (Bind x0) +x3))).(\lambda (H7: (csubst0 (minus i (S n)) v x1 x2)).(eq_ind_r C (CHead x1 +(Bind x0) x3) (\lambda (c: C).(ex2 C (\lambda (e2: C).(csubst1 (S (minus i (S +n))) v c e2)) (\lambda (e2: C).(getl n c3 e2)))) (ex_intro2 C (\lambda (e2: +C).(csubst1 (S (minus i (S n))) v (CHead x1 (Bind x0) x3) e2)) (\lambda (e2: +C).(getl n c3 e2)) (CHead x2 (Bind x0) x3) (csubst1_sing (S (minus i (S n))) +v (CHead x1 (Bind x0) x3) (CHead x2 (Bind x0) x3) (csubst0_fst_bind x0 (minus +i (S n)) x1 x2 v H7 x3)) H6) e1 H5)))))))) H4)) (\lambda (H4: (ex4_5 B C C T +T (\lambda (b: B).(\lambda (e2: C).(\lambda (_: C).(\lambda (u: T).(\lambda +(_: T).(eq C e1 (CHead e2 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: +C).(\lambda (e3: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e3 +(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda +(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: +B).(\lambda (e2: C).(\lambda (e3: C).(\lambda (_: T).(\lambda (_: T).(csubst0 +(minus i (S n)) v e2 e3)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda +(e2: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e1 (CHead e2 +(Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e3: C).(\lambda +(_: T).(\lambda (w: T).(getl n c3 (CHead e3 (Bind b) w))))))) (\lambda (_: +B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 +(minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (e3: +C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e2 e3)))))) +(ex2 C (\lambda (e2: C).(csubst1 (S (minus i (S n))) v e1 e2)) (\lambda (e2: +C).(getl n c3 e2))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: +C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H5: (eq C e1 (CHead x1 (Bind +x0) x3))).(\lambda (H6: (getl n c3 (CHead x2 (Bind x0) x4))).(\lambda (H7: +(subst0 (minus i (S n)) v x3 x4)).(\lambda (H8: (csubst0 (minus i (S n)) v x1 +x2)).(eq_ind_r C (CHead x1 (Bind x0) x3) (\lambda (c: C).(ex2 C (\lambda (e2: +C).(csubst1 (S (minus i (S n))) v c e2)) (\lambda (e2: C).(getl n c3 e2)))) +(ex_intro2 C (\lambda (e2: C).(csubst1 (S (minus i (S n))) v (CHead x1 (Bind +x0) x3) e2)) (\lambda (e2: C).(getl n c3 e2)) (CHead x2 (Bind x0) x4) +(csubst1_sing (S (minus i (S n))) v (CHead x1 (Bind x0) x3) (CHead x2 (Bind +x0) x4) (csubst0_both_bind x0 (minus i (S n)) v x3 x4 H7 x1 x2 H8)) H6) e1 +H5)))))))))) H4)) H3)) (minus i n) (minus_x_Sy i n H)))))) c2 H0))))))). +(* COMMENTS +Initial nodes: 2035 +END *) + +theorem csubst1_getl_ge_back: + \forall (i: nat).(\forall (n: nat).((le i n) \to (\forall (c1: C).(\forall +(c2: C).(\forall (v: T).((csubst1 i v c1 c2) \to (\forall (e: C).((getl n c2 +e) \to (getl n c1 e))))))))) +\def + \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (le i n)).(\lambda (c1: +C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst1 i v c1 +c2)).(csubst1_ind i v c1 (\lambda (c: C).(\forall (e: C).((getl n c e) \to +(getl n c1 e)))) (\lambda (e: C).(\lambda (H1: (getl n c1 e)).H1)) (\lambda +(c3: C).(\lambda (H1: (csubst0 i v c1 c3)).(\lambda (e: C).(\lambda (H2: +(getl n c3 e)).(csubst0_getl_ge_back i n H c1 c3 v H1 e H2))))) c2 H0))))))). +(* COMMENTS +Initial nodes: 111 +END *) + +theorem getl_csubst1: + \forall (d: nat).(\forall (c: C).(\forall (e: C).(\forall (u: T).((getl d c +(CHead e (Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0: C).(\lambda (_: +C).(csubst1 d u c a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) d a0 +a)))))))) +\def + \lambda (d: nat).(nat_ind (\lambda (n: nat).(\forall (c: C).(\forall (e: +C).(\forall (u: T).((getl n c (CHead e (Bind Abbr) u)) \to (ex2_2 C C +(\lambda (a0: C).(\lambda (_: C).(csubst1 n u c a0))) (\lambda (a0: +C).(\lambda (a: C).(drop (S O) n a0 a))))))))) (\lambda (c: C).(C_ind +(\lambda (c0: C).(\forall (e: C).(\forall (u: T).((getl O c0 (CHead e (Bind +Abbr) u)) \to (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O u c0 +a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0 a)))))))) (\lambda +(n: nat).(\lambda (e: C).(\lambda (u: T).(\lambda (H: (getl O (CSort n) +(CHead e (Bind Abbr) u))).(getl_gen_sort n O (CHead e (Bind Abbr) u) H (ex2_2 +C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O u (CSort n) a0))) (\lambda +(a0: C).(\lambda (a: C).(drop (S O) O a0 a))))))))) (\lambda (c0: C).(\lambda +(H: ((\forall (e: C).(\forall (u: T).((getl O c0 (CHead e (Bind Abbr) u)) \to +(ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O u c0 a0))) (\lambda +(a0: C).(\lambda (a: C).(drop (S O) O a0 a))))))))).(\lambda (k: K).(K_ind +(\lambda (k0: K).(\forall (t: T).(\forall (e: C).(\forall (u: T).((getl O +(CHead c0 k0 t) (CHead e (Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0: +C).(\lambda (_: C).(csubst1 O u (CHead c0 k0 t) a0))) (\lambda (a0: +C).(\lambda (a: C).(drop (S O) O a0 a))))))))) (\lambda (b: B).(\lambda (t: +T).(\lambda (e: C).(\lambda (u: T).(\lambda (H0: (getl O (CHead c0 (Bind b) +t) (CHead e (Bind Abbr) u))).(let H1 \def (f_equal C C (\lambda (e0: +C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow e | +(CHead c1 _ _) \Rightarrow c1])) (CHead e (Bind Abbr) u) (CHead c0 (Bind b) +t) (clear_gen_bind b c0 (CHead e (Bind Abbr) u) t (getl_gen_O (CHead c0 (Bind +b) t) (CHead e (Bind Abbr) u) H0))) in ((let H2 \def (f_equal C B (\lambda +(e0: C).(match e0 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow +Abbr | (CHead _ k0 _) \Rightarrow (match k0 in K return (\lambda (_: K).B) +with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead e +(Bind Abbr) u) (CHead c0 (Bind b) t) (clear_gen_bind b c0 (CHead e (Bind +Abbr) u) t (getl_gen_O (CHead c0 (Bind b) t) (CHead e (Bind Abbr) u) H0))) in +((let H3 \def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda +(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) +(CHead e (Bind Abbr) u) (CHead c0 (Bind b) t) (clear_gen_bind b c0 (CHead e +(Bind Abbr) u) t (getl_gen_O (CHead c0 (Bind b) t) (CHead e (Bind Abbr) u) +H0))) in (\lambda (H4: (eq B Abbr b)).(\lambda (_: (eq C e c0)).(eq_ind_r T t +(\lambda (t0: T).(ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O t0 +(CHead c0 (Bind b) t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0 +a))))) (eq_ind B Abbr (\lambda (b0: B).(ex2_2 C C (\lambda (a0: C).(\lambda +(_: C).(csubst1 O t (CHead c0 (Bind b0) t) a0))) (\lambda (a0: C).(\lambda +(a: C).(drop (S O) O a0 a))))) (ex2_2_intro C C (\lambda (a0: C).(\lambda (_: +C).(csubst1 O t (CHead c0 (Bind Abbr) t) a0))) (\lambda (a0: C).(\lambda (a: +C).(drop (S O) O a0 a))) (CHead c0 (Bind Abbr) t) c0 (csubst1_refl O t (CHead +c0 (Bind Abbr) t)) (drop_drop (Bind Abbr) O c0 c0 (drop_refl c0) t)) b H4) u +H3)))) H2)) H1))))))) (\lambda (f: F).(\lambda (t: T).(\lambda (e: +C).(\lambda (u: T).(\lambda (H0: (getl O (CHead c0 (Flat f) t) (CHead e (Bind +Abbr) u))).(let H_x \def (subst1_ex u t O) in (let H1 \def H_x in (ex_ind T +(\lambda (t2: T).(subst1 O u t (lift (S O) O t2))) (ex2_2 C C (\lambda (a0: +C).(\lambda (_: C).(csubst1 O u (CHead c0 (Flat f) t) a0))) (\lambda (a0: +C).(\lambda (a: C).(drop (S O) O a0 a)))) (\lambda (x: T).(\lambda (H2: +(subst1 O u t (lift (S O) O x))).(let H3 \def (H e u (getl_intro O c0 (CHead +e (Bind Abbr) u) c0 (drop_refl c0) (clear_gen_flat f c0 (CHead e (Bind Abbr) +u) t (getl_gen_O (CHead c0 (Flat f) t) (CHead e (Bind Abbr) u) H0)))) in +(ex2_2_ind C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O u c0 a0))) +(\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0 a))) (ex2_2 C C (\lambda +(a0: C).(\lambda (_: C).(csubst1 O u (CHead c0 (Flat f) t) a0))) (\lambda +(a0: C).(\lambda (a: C).(drop (S O) O a0 a)))) (\lambda (x0: C).(\lambda (x1: +C).(\lambda (H4: (csubst1 O u c0 x0)).(\lambda (H5: (drop (S O) O x0 +x1)).(ex2_2_intro C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O u (CHead c0 +(Flat f) t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0 a))) +(CHead x0 (Flat f) (lift (S O) O x)) x1 (csubst1_flat f O u t (lift (S O) O +x) H2 c0 x0 H4) (drop_drop (Flat f) O x0 x1 H5 (lift (S O) O x))))))) H3)))) +H1)))))))) k)))) c)) (\lambda (n: nat).(\lambda (H: ((\forall (c: C).(\forall +(e: C).(\forall (u: T).((getl n c (CHead e (Bind Abbr) u)) \to (ex2_2 C C +(\lambda (a0: C).(\lambda (_: C).(csubst1 n u c a0))) (\lambda (a0: +C).(\lambda (a: C).(drop (S O) n a0 a)))))))))).(\lambda (c: C).(C_ind +(\lambda (c0: C).(\forall (e: C).(\forall (u: T).((getl (S n) c0 (CHead e +(Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S +n) u c0 a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a)))))))) +(\lambda (n0: nat).(\lambda (e: C).(\lambda (u: T).(\lambda (H0: (getl (S n) +(CSort n0) (CHead e (Bind Abbr) u))).(getl_gen_sort n0 (S n) (CHead e (Bind +Abbr) u) H0 (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u +(CSort n0) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 +a))))))))) (\lambda (c0: C).(\lambda (H0: ((\forall (e: C).(\forall (u: +T).((getl (S n) c0 (CHead e (Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0: +C).(\lambda (_: C).(csubst1 (S n) u c0 a0))) (\lambda (a0: C).(\lambda (a: +C).(drop (S O) (S n) a0 a))))))))).(\lambda (k: K).(K_ind (\lambda (k0: +K).(\forall (t: T).(\forall (e: C).(\forall (u: T).((getl (S n) (CHead c0 k0 +t) (CHead e (Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0: C).(\lambda (_: +C).(csubst1 (S n) u (CHead c0 k0 t) a0))) (\lambda (a0: C).(\lambda (a: +C).(drop (S O) (S n) a0 a))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda +(e: C).(\lambda (u: T).(\lambda (H1: (getl (S n) (CHead c0 (Bind b) t) (CHead +e (Bind Abbr) u))).(let H_x \def (subst1_ex u t n) in (let H2 \def H_x in +(ex_ind T (\lambda (t2: T).(subst1 n u t (lift (S O) n t2))) (ex2_2 C C +(\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u (CHead c0 (Bind b) t) a0))) +(\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a)))) (\lambda (x: +T).(\lambda (H3: (subst1 n u t (lift (S O) n x))).(let H4 \def (H c0 e u +(getl_gen_S (Bind b) c0 (CHead e (Bind Abbr) u) t n H1)) in (ex2_2_ind C C +(\lambda (a0: C).(\lambda (_: C).(csubst1 n u c0 a0))) (\lambda (a0: +C).(\lambda (a: C).(drop (S O) n a0 a))) (ex2_2 C C (\lambda (a0: C).(\lambda +(_: C).(csubst1 (S n) u (CHead c0 (Bind b) t) a0))) (\lambda (a0: C).(\lambda +(a: C).(drop (S O) (S n) a0 a)))) (\lambda (x0: C).(\lambda (x1: C).(\lambda +(H5: (csubst1 n u c0 x0)).(\lambda (H6: (drop (S O) n x0 x1)).(ex2_2_intro C +C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u (CHead c0 (Bind b) t) +a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a))) (CHead x0 +(Bind b) (lift (S O) n x)) (CHead x1 (Bind b) x) (csubst1_bind b n u t (lift +(S O) n x) H3 c0 x0 H5) (drop_skip_bind (S O) n x0 x1 H6 b x)))))) H4)))) +H2)))))))) (\lambda (f: F).(\lambda (t: T).(\lambda (e: C).(\lambda (u: +T).(\lambda (H1: (getl (S n) (CHead c0 (Flat f) t) (CHead e (Bind Abbr) +u))).(let H_x \def (subst1_ex u t (S n)) in (let H2 \def H_x in (ex_ind T +(\lambda (t2: T).(subst1 (S n) u t (lift (S O) (S n) t2))) (ex2_2 C C +(\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u (CHead c0 (Flat f) t) a0))) +(\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a)))) (\lambda (x: +T).(\lambda (H3: (subst1 (S n) u t (lift (S O) (S n) x))).(let H4 \def (H0 e +u (getl_gen_S (Flat f) c0 (CHead e (Bind Abbr) u) t n H1)) in (ex2_2_ind C C +(\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u c0 a0))) (\lambda (a0: +C).(\lambda (a: C).(drop (S O) (S n) a0 a))) (ex2_2 C C (\lambda (a0: +C).(\lambda (_: C).(csubst1 (S n) u (CHead c0 (Flat f) t) a0))) (\lambda (a0: +C).(\lambda (a: C).(drop (S O) (S n) a0 a)))) (\lambda (x0: C).(\lambda (x1: +C).(\lambda (H5: (csubst1 (S n) u c0 x0)).(\lambda (H6: (drop (S O) (S n) x0 +x1)).(ex2_2_intro C C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u +(CHead c0 (Flat f) t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S +n) a0 a))) (CHead x0 (Flat f) (lift (S O) (S n) x)) (CHead x1 (Flat f) x) +(csubst1_flat f (S n) u t (lift (S O) (S n) x) H3 c0 x0 H5) (drop_skip_flat +(S O) n x0 x1 H6 f x)))))) H4)))) H2)))))))) k)))) c)))) d). +(* COMMENTS +Initial nodes: 2467 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubst1/props.ma b/matita/matita/contribs/lambdadelta/basic_1/csubst1/props.ma new file mode 100644 index 000000000..518a86564 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/csubst1/props.ma @@ -0,0 +1,75 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/csubst1/defs.ma". + +include "Basic-1/subst1/defs.ma". + +theorem csubst1_head: + \forall (k: K).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall +(u2: T).((subst1 i v u1 u2) \to (\forall (c1: C).(\forall (c2: C).((csubst1 i +v c1 c2) \to (csubst1 (s k i) v (CHead c1 k u1) (CHead c2 k u2)))))))))) +\def + \lambda (k: K).(\lambda (i: nat).(\lambda (v: T).(\lambda (u1: T).(\lambda +(u2: T).(\lambda (H: (subst1 i v u1 u2)).(subst1_ind i v u1 (\lambda (t: +T).(\forall (c1: C).(\forall (c2: C).((csubst1 i v c1 c2) \to (csubst1 (s k +i) v (CHead c1 k u1) (CHead c2 k t)))))) (\lambda (c1: C).(\lambda (c2: +C).(\lambda (H0: (csubst1 i v c1 c2)).(csubst1_ind i v c1 (\lambda (c: +C).(csubst1 (s k i) v (CHead c1 k u1) (CHead c k u1))) (csubst1_refl (s k i) +v (CHead c1 k u1)) (\lambda (c3: C).(\lambda (H1: (csubst0 i v c1 +c3)).(csubst1_sing (s k i) v (CHead c1 k u1) (CHead c3 k u1) (csubst0_fst k i +c1 c3 v H1 u1)))) c2 H0)))) (\lambda (t2: T).(\lambda (H0: (subst0 i v u1 +t2)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csubst1 i v c1 +c2)).(csubst1_ind i v c1 (\lambda (c: C).(csubst1 (s k i) v (CHead c1 k u1) +(CHead c k t2))) (csubst1_sing (s k i) v (CHead c1 k u1) (CHead c1 k t2) +(csubst0_snd k i v u1 t2 H0 c1)) (\lambda (c3: C).(\lambda (H2: (csubst0 i v +c1 c3)).(csubst1_sing (s k i) v (CHead c1 k u1) (CHead c3 k t2) (csubst0_both +k i v u1 t2 H0 c1 c3 H2)))) c2 H1)))))) u2 H)))))). +(* COMMENTS +Initial nodes: 365 +END *) + +theorem csubst1_bind: + \forall (b: B).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall +(u2: T).((subst1 i v u1 u2) \to (\forall (c1: C).(\forall (c2: C).((csubst1 i +v c1 c2) \to (csubst1 (S i) v (CHead c1 (Bind b) u1) (CHead c2 (Bind b) +u2)))))))))) +\def + \lambda (b: B).(\lambda (i: nat).(\lambda (v: T).(\lambda (u1: T).(\lambda +(u2: T).(\lambda (H: (subst1 i v u1 u2)).(\lambda (c1: C).(\lambda (c2: +C).(\lambda (H0: (csubst1 i v c1 c2)).(eq_ind nat (s (Bind b) i) (\lambda (n: +nat).(csubst1 n v (CHead c1 (Bind b) u1) (CHead c2 (Bind b) u2))) +(csubst1_head (Bind b) i v u1 u2 H c1 c2 H0) (S i) (refl_equal nat (S +i))))))))))). +(* COMMENTS +Initial nodes: 107 +END *) + +theorem csubst1_flat: + \forall (f: F).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall +(u2: T).((subst1 i v u1 u2) \to (\forall (c1: C).(\forall (c2: C).((csubst1 i +v c1 c2) \to (csubst1 i v (CHead c1 (Flat f) u1) (CHead c2 (Flat f) +u2)))))))))) +\def + \lambda (f: F).(\lambda (i: nat).(\lambda (v: T).(\lambda (u1: T).(\lambda +(u2: T).(\lambda (H: (subst1 i v u1 u2)).(\lambda (c1: C).(\lambda (c2: +C).(\lambda (H0: (csubst1 i v c1 c2)).(eq_ind nat (s (Flat f) i) (\lambda (n: +nat).(csubst1 n v (CHead c1 (Flat f) u1) (CHead c2 (Flat f) u2))) +(csubst1_head (Flat f) i v u1 u2 H c1 c2 H0) i (refl_equal nat i)))))))))). +(* COMMENTS +Initial nodes: 103 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubt/clear.ma b/matita/matita/contribs/lambdadelta/basic_1/csubt/clear.ma new file mode 100644 index 000000000..c60700147 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/csubt/clear.ma @@ -0,0 +1,74 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/csubt/defs.ma". + +include "Basic-1/clear/fwd.ma". + +theorem csubt_clear_conf: + \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to +(\forall (e1: C).((clear c1 e1) \to (ex2 C (\lambda (e2: C).(csubt g e1 e2)) +(\lambda (e2: C).(clear c2 e2)))))))) +\def + \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubt g c1 +c2)).(csubt_ind g (\lambda (c: C).(\lambda (c0: C).(\forall (e1: C).((clear c +e1) \to (ex2 C (\lambda (e2: C).(csubt g e1 e2)) (\lambda (e2: C).(clear c0 +e2))))))) (\lambda (n: nat).(\lambda (e1: C).(\lambda (H0: (clear (CSort n) +e1)).(clear_gen_sort e1 n H0 (ex2 C (\lambda (e2: C).(csubt g e1 e2)) +(\lambda (e2: C).(clear (CSort n) e2))))))) (\lambda (c3: C).(\lambda (c4: +C).(\lambda (H0: (csubt g c3 c4)).(\lambda (H1: ((\forall (e1: C).((clear c3 +e1) \to (ex2 C (\lambda (e2: C).(csubt g e1 e2)) (\lambda (e2: C).(clear c4 +e2))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (e1: C).(\lambda (H2: +(clear (CHead c3 k u) e1)).(K_ind (\lambda (k0: K).((clear (CHead c3 k0 u) +e1) \to (ex2 C (\lambda (e2: C).(csubt g e1 e2)) (\lambda (e2: C).(clear +(CHead c4 k0 u) e2))))) (\lambda (b: B).(\lambda (H3: (clear (CHead c3 (Bind +b) u) e1)).(eq_ind_r C (CHead c3 (Bind b) u) (\lambda (c: C).(ex2 C (\lambda +(e2: C).(csubt g c e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b) u) e2)))) +(ex_intro2 C (\lambda (e2: C).(csubt g (CHead c3 (Bind b) u) e2)) (\lambda +(e2: C).(clear (CHead c4 (Bind b) u) e2)) (CHead c4 (Bind b) u) (csubt_head g +c3 c4 H0 (Bind b) u) (clear_bind b c4 u)) e1 (clear_gen_bind b c3 e1 u H3)))) +(\lambda (f: F).(\lambda (H3: (clear (CHead c3 (Flat f) u) e1)).(let H4 \def +(H1 e1 (clear_gen_flat f c3 e1 u H3)) in (ex2_ind C (\lambda (e2: C).(csubt g +e1 e2)) (\lambda (e2: C).(clear c4 e2)) (ex2 C (\lambda (e2: C).(csubt g e1 +e2)) (\lambda (e2: C).(clear (CHead c4 (Flat f) u) e2))) (\lambda (x: +C).(\lambda (H5: (csubt g e1 x)).(\lambda (H6: (clear c4 x)).(ex_intro2 C +(\lambda (e2: C).(csubt g e1 e2)) (\lambda (e2: C).(clear (CHead c4 (Flat f) +u) e2)) x H5 (clear_flat c4 x H6 f u))))) H4)))) k H2))))))))) (\lambda (c3: +C).(\lambda (c4: C).(\lambda (H0: (csubt g c3 c4)).(\lambda (_: ((\forall +(e1: C).((clear c3 e1) \to (ex2 C (\lambda (e2: C).(csubt g e1 e2)) (\lambda +(e2: C).(clear c4 e2))))))).(\lambda (b: B).(\lambda (H2: (not (eq B b +Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (e1: C).(\lambda (H3: +(clear (CHead c3 (Bind Void) u1) e1)).(eq_ind_r C (CHead c3 (Bind Void) u1) +(\lambda (c: C).(ex2 C (\lambda (e2: C).(csubt g c e2)) (\lambda (e2: +C).(clear (CHead c4 (Bind b) u2) e2)))) (ex_intro2 C (\lambda (e2: C).(csubt +g (CHead c3 (Bind Void) u1) e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b) +u2) e2)) (CHead c4 (Bind b) u2) (csubt_void g c3 c4 H0 b H2 u1 u2) +(clear_bind b c4 u2)) e1 (clear_gen_bind Void c3 e1 u1 H3)))))))))))) +(\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csubt g c3 c4)).(\lambda (_: +((\forall (e1: C).((clear c3 e1) \to (ex2 C (\lambda (e2: C).(csubt g e1 e2)) +(\lambda (e2: C).(clear c4 e2))))))).(\lambda (u: T).(\lambda (t: T).(\lambda +(H2: (ty3 g c3 u t)).(\lambda (H3: (ty3 g c4 u t)).(\lambda (e1: C).(\lambda +(H4: (clear (CHead c3 (Bind Abst) t) e1)).(eq_ind_r C (CHead c3 (Bind Abst) +t) (\lambda (c: C).(ex2 C (\lambda (e2: C).(csubt g c e2)) (\lambda (e2: +C).(clear (CHead c4 (Bind Abbr) u) e2)))) (ex_intro2 C (\lambda (e2: +C).(csubt g (CHead c3 (Bind Abst) t) e2)) (\lambda (e2: C).(clear (CHead c4 +(Bind Abbr) u) e2)) (CHead c4 (Bind Abbr) u) (csubt_abst g c3 c4 H0 u t H2 +H3) (clear_bind Abbr c4 u)) e1 (clear_gen_bind Abst c3 e1 t H4)))))))))))) c1 +c2 H)))). +(* COMMENTS +Initial nodes: 929 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubt/csuba.ma b/matita/matita/contribs/lambdadelta/basic_1/csubt/csuba.ma new file mode 100644 index 000000000..b2cf1183f --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/csubt/csuba.ma @@ -0,0 +1,42 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/ty3/arity.ma". + +theorem csubt_csuba: + \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to (csuba +g c1 c2)))) +\def + \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubt g c1 +c2)).(csubt_ind g (\lambda (c: C).(\lambda (c0: C).(csuba g c c0))) (\lambda +(n: nat).(csuba_refl g (CSort n))) (\lambda (c3: C).(\lambda (c4: C).(\lambda +(_: (csubt g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (k: K).(\lambda +(u: T).(csuba_head g c3 c4 H1 k u))))))) (\lambda (c3: C).(\lambda (c4: +C).(\lambda (_: (csubt g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (b: +B).(\lambda (H2: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: +T).(csuba_void g c3 c4 H1 b H2 u1 u2))))))))) (\lambda (c3: C).(\lambda (c4: +C).(\lambda (_: (csubt g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (u: +T).(\lambda (t: T).(\lambda (H2: (ty3 g c3 u t)).(\lambda (_: (ty3 g c4 u +t)).(let H_x \def (ty3_arity g c3 u t H2) in (let H4 \def H_x in (ex2_ind A +(\lambda (a1: A).(arity g c3 u a1)) (\lambda (a1: A).(arity g c3 t (asucc g +a1))) (csuba g (CHead c3 (Bind Abst) t) (CHead c4 (Bind Abbr) u)) (\lambda +(x: A).(\lambda (H5: (arity g c3 u x)).(\lambda (H6: (arity g c3 t (asucc g +x))).(csuba_abst g c3 c4 H1 t x H6 u (csuba_arity g c3 u x H5 c4 H1))))) +H4))))))))))) c1 c2 H)))). +(* COMMENTS +Initial nodes: 313 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubt/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/csubt/defs.ma new file mode 100644 index 000000000..a12c9f829 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/csubt/defs.ma @@ -0,0 +1,29 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/ty3/defs.ma". + +inductive csubt (g: G): C \to (C \to Prop) \def +| csubt_sort: \forall (n: nat).(csubt g (CSort n) (CSort n)) +| csubt_head: \forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to (\forall +(k: K).(\forall (u: T).(csubt g (CHead c1 k u) (CHead c2 k u)))))) +| csubt_void: \forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to (\forall +(b: B).((not (eq B b Void)) \to (\forall (u1: T).(\forall (u2: T).(csubt g +(CHead c1 (Bind Void) u1) (CHead c2 (Bind b) u2)))))))) +| csubt_abst: \forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to (\forall +(u: T).(\forall (t: T).((ty3 g c1 u t) \to ((ty3 g c2 u t) \to (csubt g +(CHead c1 (Bind Abst) t) (CHead c2 (Bind Abbr) u)))))))). + diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubt/drop.ma b/matita/matita/contribs/lambdadelta/basic_1/csubt/drop.ma new file mode 100644 index 000000000..adaedcc9a --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/csubt/drop.ma @@ -0,0 +1,590 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/csubt/fwd.ma". + +include "Basic-1/drop/fwd.ma". + +theorem csubt_drop_flat: + \forall (g: G).(\forall (f: F).(\forall (n: nat).(\forall (c1: C).(\forall +(c2: C).((csubt g c1 c2) \to (\forall (d1: C).(\forall (u: T).((drop n O c1 +(CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda +(d2: C).(drop n O c2 (CHead d2 (Flat f) u)))))))))))) +\def + \lambda (g: G).(\lambda (f: F).(\lambda (n: nat).(nat_ind (\lambda (n0: +nat).(\forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to (\forall (d1: +C).(\forall (u: T).((drop n0 O c1 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda +(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 (Flat f) +u))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubt g c1 +c2)).(\lambda (d1: C).(\lambda (u: T).(\lambda (H0: (drop O O c1 (CHead d1 +(Flat f) u))).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c2)) H +(CHead d1 (Flat f) u) (drop_gen_refl c1 (CHead d1 (Flat f) u) H0)) in (let +H_x \def (csubt_gen_flat g d1 c2 u f H1) in (let H2 \def H_x in (ex2_ind C +(\lambda (e2: C).(eq C c2 (CHead e2 (Flat f) u))) (\lambda (e2: C).(csubt g +d1 e2)) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O +c2 (CHead d2 (Flat f) u)))) (\lambda (x: C).(\lambda (H3: (eq C c2 (CHead x +(Flat f) u))).(\lambda (H4: (csubt g d1 x)).(eq_ind_r C (CHead x (Flat f) u) +(\lambda (c: C).(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: +C).(drop O O c (CHead d2 (Flat f) u))))) (ex_intro2 C (\lambda (d2: C).(csubt +g d1 d2)) (\lambda (d2: C).(drop O O (CHead x (Flat f) u) (CHead d2 (Flat f) +u))) x H4 (drop_refl (CHead x (Flat f) u))) c2 H3)))) H2)))))))))) (\lambda +(n0: nat).(\lambda (H: ((\forall (c1: C).(\forall (c2: C).((csubt g c1 c2) +\to (\forall (d1: C).(\forall (u: T).((drop n0 O c1 (CHead d1 (Flat f) u)) +\to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c2 +(CHead d2 (Flat f) u)))))))))))).(\lambda (c1: C).(\lambda (c2: C).(\lambda +(H0: (csubt g c1 c2)).(csubt_ind g (\lambda (c: C).(\lambda (c0: C).(\forall +(d1: C).(\forall (u: T).((drop (S n0) O c (CHead d1 (Flat f) u)) \to (ex2 C +(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O c0 (CHead +d2 (Flat f) u))))))))) (\lambda (n1: nat).(\lambda (d1: C).(\lambda (u: +T).(\lambda (H1: (drop (S n0) O (CSort n1) (CHead d1 (Flat f) u))).(and3_ind +(eq C (CHead d1 (Flat f) u) (CSort n1)) (eq nat (S n0) O) (eq nat O O) (ex2 C +(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) +(CHead d2 (Flat f) u)))) (\lambda (_: (eq C (CHead d1 (Flat f) u) (CSort +n1))).(\lambda (H3: (eq nat (S n0) O)).(\lambda (_: (eq nat O O)).(let H5 +\def (eq_ind nat (S n0) (\lambda (ee: nat).(match ee in nat return (\lambda +(_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H3) +in (False_ind (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop +(S n0) O (CSort n1) (CHead d2 (Flat f) u)))) H5))))) (drop_gen_sort n1 (S n0) +O (CHead d1 (Flat f) u) H1)))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda +(H1: (csubt g c0 c3)).(\lambda (H2: ((\forall (d1: C).(\forall (u: T).((drop +(S n0) O c0 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 +d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Flat f) +u))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (u: T).(\forall +(d1: C).(\forall (u0: T).((drop (S n0) O (CHead c0 k0 u) (CHead d1 (Flat f) +u0)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S +n0) O (CHead c3 k0 u) (CHead d2 (Flat f) u0))))))))) (\lambda (b: B).(\lambda +(u: T).(\lambda (d1: C).(\lambda (u0: T).(\lambda (H3: (drop (S n0) O (CHead +c0 (Bind b) u) (CHead d1 (Flat f) u0))).(ex2_ind C (\lambda (d2: C).(csubt g +d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Flat f) u0))) (ex2 C +(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 +(Bind b) u) (CHead d2 (Flat f) u0)))) (\lambda (x: C).(\lambda (H4: (csubt g +d1 x)).(\lambda (H5: (drop n0 O c3 (CHead x (Flat f) u0))).(ex_intro2 C +(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 +(Bind b) u) (CHead d2 (Flat f) u0))) x H4 (drop_drop (Bind b) n0 c3 (CHead x +(Flat f) u0) H5 u))))) (H c0 c3 H1 d1 u0 (drop_gen_drop (Bind b) c0 (CHead d1 +(Flat f) u0) u n0 H3)))))))) (\lambda (f0: F).(\lambda (u: T).(\lambda (d1: +C).(\lambda (u0: T).(\lambda (H3: (drop (S n0) O (CHead c0 (Flat f0) u) +(CHead d1 (Flat f) u0))).(ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) +(\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Flat f) u0))) (ex2 C (\lambda +(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f0) +u) (CHead d2 (Flat f) u0)))) (\lambda (x: C).(\lambda (H4: (csubt g d1 +x)).(\lambda (H5: (drop (S n0) O c3 (CHead x (Flat f) u0))).(ex_intro2 C +(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 +(Flat f0) u) (CHead d2 (Flat f) u0))) x H4 (drop_drop (Flat f0) n0 c3 (CHead +x (Flat f) u0) H5 u))))) (H2 d1 u0 (drop_gen_drop (Flat f0) c0 (CHead d1 +(Flat f) u0) u n0 H3)))))))) k)))))) (\lambda (c0: C).(\lambda (c3: +C).(\lambda (H1: (csubt g c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (u: +T).((drop (S n0) O c0 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: +C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Flat f) +u))))))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: +T).(\lambda (u2: T).(\lambda (d1: C).(\lambda (u: T).(\lambda (H4: (drop (S +n0) O (CHead c0 (Bind Void) u1) (CHead d1 (Flat f) u))).(ex2_ind C (\lambda +(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Flat f) +u))) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O +(CHead c3 (Bind b) u2) (CHead d2 (Flat f) u)))) (\lambda (x: C).(\lambda (H5: +(csubt g d1 x)).(\lambda (H6: (drop n0 O c3 (CHead x (Flat f) u))).(ex_intro2 +C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 +(Bind b) u2) (CHead d2 (Flat f) u))) x H5 (drop_drop (Bind b) n0 c3 (CHead x +(Flat f) u) H6 u2))))) (H c0 c3 H1 d1 u (drop_gen_drop (Bind Void) c0 (CHead +d1 (Flat f) u) u1 n0 H4)))))))))))))) (\lambda (c0: C).(\lambda (c3: +C).(\lambda (H1: (csubt g c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (u: +T).((drop (S n0) O c0 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: +C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Flat f) +u))))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u +t)).(\lambda (_: (ty3 g c3 u t)).(\lambda (d1: C).(\lambda (u0: T).(\lambda +(H5: (drop (S n0) O (CHead c0 (Bind Abst) t) (CHead d1 (Flat f) +u0))).(ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 +O c3 (CHead d2 (Flat f) u0))) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) +(\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Flat f) +u0)))) (\lambda (x: C).(\lambda (H6: (csubt g d1 x)).(\lambda (H7: (drop n0 O +c3 (CHead x (Flat f) u0))).(ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) +(\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Flat f) +u0))) x H6 (drop_drop (Bind Abbr) n0 c3 (CHead x (Flat f) u0) H7 u))))) (H c0 +c3 H1 d1 u0 (drop_gen_drop (Bind Abst) c0 (CHead d1 (Flat f) u0) t n0 +H5)))))))))))))) c1 c2 H0)))))) n))). +(* COMMENTS +Initial nodes: 2090 +END *) + +theorem csubt_drop_abbr: + \forall (g: G).(\forall (n: nat).(\forall (c1: C).(\forall (c2: C).((csubt g +c1 c2) \to (\forall (d1: C).(\forall (u: T).((drop n O c1 (CHead d1 (Bind +Abbr) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop +n O c2 (CHead d2 (Bind Abbr) u))))))))))) +\def + \lambda (g: G).(\lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (c1: +C).(\forall (c2: C).((csubt g c1 c2) \to (\forall (d1: C).(\forall (u: +T).((drop n0 O c1 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: +C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 (Bind Abbr) +u))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubt g c1 +c2)).(\lambda (d1: C).(\lambda (u: T).(\lambda (H0: (drop O O c1 (CHead d1 +(Bind Abbr) u))).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c2)) H +(CHead d1 (Bind Abbr) u) (drop_gen_refl c1 (CHead d1 (Bind Abbr) u) H0)) in +(let H2 \def (csubt_gen_abbr g d1 c2 u H1) in (ex2_ind C (\lambda (e2: C).(eq +C c2 (CHead e2 (Bind Abbr) u))) (\lambda (e2: C).(csubt g d1 e2)) (ex2 C +(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 +(Bind Abbr) u)))) (\lambda (x: C).(\lambda (H3: (eq C c2 (CHead x (Bind Abbr) +u))).(\lambda (H4: (csubt g d1 x)).(eq_ind_r C (CHead x (Bind Abbr) u) +(\lambda (c: C).(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: +C).(drop O O c (CHead d2 (Bind Abbr) u))))) (ex_intro2 C (\lambda (d2: +C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O (CHead x (Bind Abbr) u) (CHead +d2 (Bind Abbr) u))) x H4 (drop_refl (CHead x (Bind Abbr) u))) c2 H3)))) +H2))))))))) (\lambda (n0: nat).(\lambda (H: ((\forall (c1: C).(\forall (c2: +C).((csubt g c1 c2) \to (\forall (d1: C).(\forall (u: T).((drop n0 O c1 +(CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) +(\lambda (d2: C).(drop n0 O c2 (CHead d2 (Bind Abbr) u)))))))))))).(\lambda +(c1: C).(\lambda (c2: C).(\lambda (H0: (csubt g c1 c2)).(csubt_ind g (\lambda +(c: C).(\lambda (c0: C).(\forall (d1: C).(\forall (u: T).((drop (S n0) O c +(CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) +(\lambda (d2: C).(drop (S n0) O c0 (CHead d2 (Bind Abbr) u))))))))) (\lambda +(n1: nat).(\lambda (d1: C).(\lambda (u: T).(\lambda (H1: (drop (S n0) O +(CSort n1) (CHead d1 (Bind Abbr) u))).(and3_ind (eq C (CHead d1 (Bind Abbr) +u) (CSort n1)) (eq nat (S n0) O) (eq nat O O) (ex2 C (\lambda (d2: C).(csubt +g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) +u)))) (\lambda (_: (eq C (CHead d1 (Bind Abbr) u) (CSort n1))).(\lambda (H3: +(eq nat (S n0) O)).(\lambda (_: (eq nat O O)).(let H5 \def (eq_ind nat (S n0) +(\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O +\Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind (ex2 C +(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) +(CHead d2 (Bind Abbr) u)))) H5))))) (drop_gen_sort n1 (S n0) O (CHead d1 +(Bind Abbr) u) H1)))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H1: +(csubt g c0 c3)).(\lambda (H2: ((\forall (d1: C).(\forall (u: T).((drop (S +n0) O c0 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 +d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abbr) +u))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (u: T).(\forall +(d1: C).(\forall (u0: T).((drop (S n0) O (CHead c0 k0 u) (CHead d1 (Bind +Abbr) u0)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: +C).(drop (S n0) O (CHead c3 k0 u) (CHead d2 (Bind Abbr) u0))))))))) (\lambda +(b: B).(\lambda (u: T).(\lambda (d1: C).(\lambda (u0: T).(\lambda (H3: (drop +(S n0) O (CHead c0 (Bind b) u) (CHead d1 (Bind Abbr) u0))).(ex2_ind C +(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 +(Bind Abbr) u0))) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: +C).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abbr) u0)))) (\lambda +(x: C).(\lambda (H4: (csubt g d1 x)).(\lambda (H5: (drop n0 O c3 (CHead x +(Bind Abbr) u0))).(ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda +(d2: C).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abbr) u0))) x H4 +(drop_drop (Bind b) n0 c3 (CHead x (Bind Abbr) u0) H5 u))))) (H c0 c3 H1 d1 +u0 (drop_gen_drop (Bind b) c0 (CHead d1 (Bind Abbr) u0) u n0 H3)))))))) +(\lambda (f: F).(\lambda (u: T).(\lambda (d1: C).(\lambda (u0: T).(\lambda +(H3: (drop (S n0) O (CHead c0 (Flat f) u) (CHead d1 (Bind Abbr) +u0))).(ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S +n0) O c3 (CHead d2 (Bind Abbr) u0))) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) +(\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind Abbr) +u0)))) (\lambda (x: C).(\lambda (H4: (csubt g d1 x)).(\lambda (H5: (drop (S +n0) O c3 (CHead x (Bind Abbr) u0))).(ex_intro2 C (\lambda (d2: C).(csubt g d1 +d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind +Abbr) u0))) x H4 (drop_drop (Flat f) n0 c3 (CHead x (Bind Abbr) u0) H5 u))))) +(H2 d1 u0 (drop_gen_drop (Flat f) c0 (CHead d1 (Bind Abbr) u0) u n0 +H3)))))))) k)))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H1: (csubt g +c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (u: T).((drop (S n0) O c0 +(CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) +(\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u))))))))).(\lambda +(b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: +T).(\lambda (d1: C).(\lambda (u: T).(\lambda (H4: (drop (S n0) O (CHead c0 +(Bind Void) u1) (CHead d1 (Bind Abbr) u))).(ex2_ind C (\lambda (d2: C).(csubt +g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abbr) u))) (ex2 C +(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 +(Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda (x: C).(\lambda (H5: (csubt +g d1 x)).(\lambda (H6: (drop n0 O c3 (CHead x (Bind Abbr) u))).(ex_intro2 C +(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 +(Bind b) u2) (CHead d2 (Bind Abbr) u))) x H5 (drop_drop (Bind b) n0 c3 (CHead +x (Bind Abbr) u) H6 u2))))) (H c0 c3 H1 d1 u (drop_gen_drop (Bind Void) c0 +(CHead d1 (Bind Abbr) u) u1 n0 H4)))))))))))))) (\lambda (c0: C).(\lambda +(c3: C).(\lambda (H1: (csubt g c0 c3)).(\lambda (_: ((\forall (d1: +C).(\forall (u: T).((drop (S n0) O c0 (CHead d1 (Bind Abbr) u)) \to (ex2 C +(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead +d2 (Bind Abbr) u))))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (_: (ty3 g +c0 u t)).(\lambda (_: (ty3 g c3 u t)).(\lambda (d1: C).(\lambda (u0: +T).(\lambda (H5: (drop (S n0) O (CHead c0 (Bind Abst) t) (CHead d1 (Bind +Abbr) u0))).(ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: +C).(drop n0 O c3 (CHead d2 (Bind Abbr) u0))) (ex2 C (\lambda (d2: C).(csubt g +d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 +(Bind Abbr) u0)))) (\lambda (x: C).(\lambda (H6: (csubt g d1 x)).(\lambda +(H7: (drop n0 O c3 (CHead x (Bind Abbr) u0))).(ex_intro2 C (\lambda (d2: +C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) +(CHead d2 (Bind Abbr) u0))) x H6 (drop_drop (Bind Abbr) n0 c3 (CHead x (Bind +Abbr) u0) H7 u))))) (H c0 c3 H1 d1 u0 (drop_gen_drop (Bind Abst) c0 (CHead d1 +(Bind Abbr) u0) t n0 H5)))))))))))))) c1 c2 H0)))))) n)). +(* COMMENTS +Initial nodes: 2084 +END *) + +theorem csubt_drop_abst: + \forall (g: G).(\forall (n: nat).(\forall (c1: C).(\forall (c2: C).((csubt g +c1 c2) \to (\forall (d1: C).(\forall (t: T).((drop n O c1 (CHead d1 (Bind +Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: +C).(drop n O c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: +C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop n +O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u +t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))))) +\def + \lambda (g: G).(\lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (c1: +C).(\forall (c2: C).((csubt g c1 c2) \to (\forall (d1: C).(\forall (t: +T).((drop n0 O c1 (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: +C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 (Bind Abst) +t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda +(d2: C).(\lambda (u: T).(drop n0 O c2 (CHead d2 (Bind Abbr) u)))) (\lambda +(_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 +g d2 u t)))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubt g +c1 c2)).(\lambda (d1: C).(\lambda (t: T).(\lambda (H0: (drop O O c1 (CHead d1 +(Bind Abst) t))).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c2)) H +(CHead d1 (Bind Abst) t) (drop_gen_refl c1 (CHead d1 (Bind Abst) t) H0)) in +(let H2 \def (csubt_gen_abst g d1 c2 t H1) in (or_ind (ex2 C (\lambda (e2: +C).(eq C c2 (CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csubt g d1 e2))) +(ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) +v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g d1 e2))) (\lambda (_: +C).(\lambda (v2: T).(ty3 g d1 v2 t))) (\lambda (e2: C).(\lambda (v2: T).(ty3 +g e2 v2 t)))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: +C).(drop O O c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: +C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop O +O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u +t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (H3: (ex2 C +(\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csubt +g d1 e2)))).(ex2_ind C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abst) t))) +(\lambda (e2: C).(csubt g d1 e2)) (or (ex2 C (\lambda (d2: C).(csubt g d1 +d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T +(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda +(u: T).(drop O O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: +T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) +(\lambda (x: C).(\lambda (H4: (eq C c2 (CHead x (Bind Abst) t))).(\lambda +(H5: (csubt g d1 x)).(eq_ind_r C (CHead x (Bind Abst) t) (\lambda (c: C).(or +(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O c (CHead +d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 +d2))) (\lambda (d2: C).(\lambda (u: T).(drop O O c (CHead d2 (Bind Abbr) +u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: +C).(\lambda (u: T).(ty3 g d2 u t)))))) (or_introl (ex2 C (\lambda (d2: +C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O (CHead x (Bind Abst) t) (CHead +d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 +d2))) (\lambda (d2: C).(\lambda (u: T).(drop O O (CHead x (Bind Abst) t) +(CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) +(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: +C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O (CHead x (Bind Abst) t) (CHead +d2 (Bind Abst) t))) x H5 (drop_refl (CHead x (Bind Abst) t)))) c2 H4)))) H3)) +(\lambda (H3: (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 +(Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g d1 e2))) +(\lambda (_: C).(\lambda (v2: T).(ty3 g d1 v2 t))) (\lambda (e2: C).(\lambda +(v2: T).(ty3 g e2 v2 t))))).(ex4_2_ind C T (\lambda (e2: C).(\lambda (v2: +T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: +T).(csubt g d1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g d1 v2 t))) +(\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 t))) (or (ex2 C (\lambda (d2: +C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) t)))) +(ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: +C).(\lambda (u: T).(drop O O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: +C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g +d2 u t))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H4: (eq C c2 (CHead +x0 (Bind Abbr) x1))).(\lambda (H5: (csubt g d1 x0)).(\lambda (H6: (ty3 g d1 +x1 t)).(\lambda (H7: (ty3 g x0 x1 t)).(eq_ind_r C (CHead x0 (Bind Abbr) x1) +(\lambda (c: C).(or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: +C).(drop O O c (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: +C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop O +O c (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u +t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))) (or_intror (ex2 C +(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O (CHead x0 (Bind +Abbr) x1) (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda +(_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop O O (CHead x0 +(Bind Abbr) x1) (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: +T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) +(ex4_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda +(d2: C).(\lambda (u: T).(drop O O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind +Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: +C).(\lambda (u: T).(ty3 g d2 u t))) x0 x1 H5 (drop_refl (CHead x0 (Bind Abbr) +x1)) H6 H7)) c2 H4))))))) H3)) H2))))))))) (\lambda (n0: nat).(\lambda (H: +((\forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to (\forall (d1: +C).(\forall (t: T).((drop n0 O c1 (CHead d1 (Bind Abst) t)) \to (or (ex2 C +(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 +(Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 +d2))) (\lambda (d2: C).(\lambda (u: T).(drop n0 O c2 (CHead d2 (Bind Abbr) +u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: +C).(\lambda (u: T).(ty3 g d2 u t))))))))))))).(\lambda (c1: C).(\lambda (c2: +C).(\lambda (H0: (csubt g c1 c2)).(csubt_ind g (\lambda (c: C).(\lambda (c0: +C).(\forall (d1: C).(\forall (t: T).((drop (S n0) O c (CHead d1 (Bind Abst) +t)) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop +(S n0) O c0 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda +(_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda 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(d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (or (ex2 C (\lambda +(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) +u2) (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: +T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O (CHead +c3 (Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: +T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) +(\lambda (H5: (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop +n0 O c3 (CHead d2 (Bind Abst) t))))).(ex2_ind C (\lambda (d2: C).(csubt g d1 +d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abst) t))) (or (ex2 C +(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 +(Bind b) u2) (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda +(_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O +(CHead c3 (Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda +(u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) +(\lambda (x: C).(\lambda (H6: (csubt g d1 x)).(\lambda (H7: (drop n0 O c3 +(CHead x (Bind Abst) t))).(or_introl (ex2 C (\lambda (d2: C).(csubt g d1 d2)) +(\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abst) +t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda +(d2: C).(\lambda (u: T).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind +Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: +C).(\lambda (u: T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csubt g +d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 +(Bind Abst) t))) x H6 (drop_drop (Bind b) n0 c3 (CHead x (Bind Abst) t) H7 +u2)))))) H5)) (\lambda (H5: (ex4_2 C T (\lambda (d2: C).(\lambda (_: +T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop n0 O c3 (CHead d2 +(Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda +(d2: C).(\lambda (u: T).(ty3 g d2 u t))))).(ex4_2_ind C T (\lambda (d2: +C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop +n0 O c3 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 +u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))) (or (ex2 C (\lambda +(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) +u2) (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: +T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O (CHead +c3 (Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: +T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) +(\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (csubt g d1 x0)).(\lambda +(H7: (drop n0 O c3 (CHead x0 (Bind Abbr) x1))).(\lambda (H8: (ty3 g d1 x1 +t)).(\lambda (H9: (ty3 g x0 x1 t)).(or_intror (ex2 C (\lambda (d2: C).(csubt +g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 +(Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 +d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O (CHead c3 (Bind b) u2) +(CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) +(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (ex4_2_intro C T (\lambda +(d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: +T).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda +(_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 +g d2 u t))) x0 x1 H6 (drop_drop (Bind b) n0 c3 (CHead x0 (Bind Abbr) x1) H7 +u2) H8 H9)))))))) H5)) (H c0 c3 H1 d1 t (drop_gen_drop (Bind Void) c0 (CHead +d1 (Bind Abst) t) u1 n0 H4)))))))))))))) (\lambda (c0: C).(\lambda (c3: +C).(\lambda (H1: (csubt g c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (t: +T).((drop (S n0) O c0 (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: +C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abst) +t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda +(d2: C).(\lambda (u: T).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u)))) +(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda +(u: T).(ty3 g d2 u t)))))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (_: +(ty3 g c0 u t)).(\lambda (_: (ty3 g c3 u t)).(\lambda (d1: C).(\lambda (t0: +T).(\lambda (H5: (drop (S n0) O (CHead c0 (Bind Abst) t) (CHead d1 (Bind +Abst) t0))).(or_ind (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: +C).(drop n0 O c3 (CHead d2 (Bind Abst) t0)))) (ex4_2 C T (\lambda (d2: +C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop +n0 O c3 (CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g +d1 u0 t0))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0)))) (or (ex2 C +(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 +(Bind Abbr) u) (CHead d2 (Bind Abst) t0)))) (ex4_2 C T (\lambda (d2: +C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop +(S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (_: +C).(\lambda (u0: T).(ty3 g d1 u0 t0))) (\lambda (d2: C).(\lambda (u0: T).(ty3 +g d2 u0 t0))))) (\lambda (H6: (ex2 C (\lambda (d2: C).(csubt g d1 d2)) +(\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abst) t0))))).(ex2_ind C +(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 +(Bind Abst) t0))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: +C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abst) t0)))) +(ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: +C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind +Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g d1 u0 t0))) (\lambda +(d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0))))) (\lambda (x: C).(\lambda (H7: +(csubt g d1 x)).(\lambda (H8: (drop n0 O c3 (CHead x (Bind Abst) +t0))).(or_introl (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: +C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abst) t0)))) +(ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: +C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind +Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g d1 u0 t0))) (\lambda +(d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0)))) (ex_intro2 C (\lambda (d2: +C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) +(CHead d2 (Bind Abst) t0))) x H7 (drop_drop (Bind Abbr) n0 c3 (CHead x (Bind +Abst) t0) H8 u)))))) H6)) (\lambda (H6: (ex4_2 C T (\lambda (d2: C).(\lambda +(_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop n0 O c3 +(CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g d1 u0 +t0))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0))))).(ex4_2_ind C T +(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda +(u0: T).(drop n0 O c3 (CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda +(u0: T).(ty3 g d1 u0 t0))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 +t0))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S +n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abst) t0)))) (ex4_2 C T +(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda +(u0: T).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) +(\lambda (_: C).(\lambda (u0: T).(ty3 g d1 u0 t0))) (\lambda (d2: C).(\lambda +(u0: T).(ty3 g d2 u0 t0))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H7: +(csubt g d1 x0)).(\lambda (H8: (drop n0 O c3 (CHead x0 (Bind Abbr) +x1))).(\lambda (H9: (ty3 g d1 x1 t0)).(\lambda (H10: (ty3 g x0 x1 +t0)).(or_intror (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: +C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abst) t0)))) +(ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: +C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind +Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g d1 u0 t0))) (\lambda +(d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0)))) (ex4_2_intro C T (\lambda (d2: +C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop +(S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (_: +C).(\lambda (u0: T).(ty3 g d1 u0 t0))) (\lambda (d2: C).(\lambda (u0: T).(ty3 +g d2 u0 t0))) x0 x1 H7 (drop_drop (Bind Abbr) n0 c3 (CHead x0 (Bind Abbr) x1) +H8 u) H9 H10)))))))) H6)) (H c0 c3 H1 d1 t0 (drop_gen_drop (Bind Abst) c0 +(CHead d1 (Bind Abst) t0) t n0 H5)))))))))))))) c1 c2 H0)))))) n)). +(* COMMENTS +Initial nodes: 7940 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubt/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/csubt/fwd.ma new file mode 100644 index 000000000..63a3eca47 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/csubt/fwd.ma @@ -0,0 +1,398 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/csubt/defs.ma". + +theorem csubt_gen_abbr: + \forall (g: G).(\forall (e1: C).(\forall (c2: C).(\forall (v: T).((csubt g +(CHead e1 (Bind Abbr) v) c2) \to (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 +(Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2))))))) +\def + \lambda (g: G).(\lambda (e1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda +(H: (csubt g (CHead e1 (Bind Abbr) v) c2)).(insert_eq C (CHead e1 (Bind Abbr) +v) (\lambda (c: C).(csubt g c c2)) (\lambda (_: C).(ex2 C (\lambda (e2: +C).(eq C c2 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))) +(\lambda (y: C).(\lambda (H0: (csubt g y c2)).(csubt_ind g (\lambda (c: +C).(\lambda (c0: C).((eq C c (CHead e1 (Bind Abbr) v)) \to (ex2 C (\lambda +(e2: C).(eq C c0 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 +e2)))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead e1 (Bind +Abbr) v))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C +return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) +\Rightarrow False])) I (CHead e1 (Bind Abbr) v) H1) in (False_ind (ex2 C +(\lambda (e2: C).(eq C (CSort n) (CHead e2 (Bind Abbr) v))) (\lambda (e2: +C).(csubt g e1 e2))) H2)))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1: +(csubt g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind Abbr) v)) \to (ex2 +C (\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abbr) v))) (\lambda (e2: +C).(csubt g e1 e2)))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C +(CHead c1 k u) (CHead e1 (Bind Abbr) v))).(let H4 \def (f_equal C C (\lambda +(e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 +| (CHead c _ _) \Rightarrow c])) (CHead c1 k u) (CHead e1 (Bind Abbr) v) H3) +in ((let H5 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda +(_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) +(CHead c1 k u) (CHead e1 (Bind Abbr) v) H3) in ((let H6 \def (f_equal C T +(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) +\Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u) (CHead e1 (Bind +Abbr) v) H3) in (\lambda (H7: (eq K k (Bind Abbr))).(\lambda (H8: (eq C c1 +e1)).(eq_ind_r T v (\lambda (t: T).(ex2 C (\lambda (e2: C).(eq C (CHead c3 k +t) (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))) (eq_ind_r K +(Bind Abbr) (\lambda (k0: K).(ex2 C (\lambda (e2: C).(eq C (CHead c3 k0 v) +(CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))) (let H9 \def +(eq_ind C c1 (\lambda (c: C).((eq C c (CHead e1 (Bind Abbr) v)) \to (ex2 C +(\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt +g e1 e2))))) H2 e1 H8) in (let H10 \def (eq_ind C c1 (\lambda (c: C).(csubt g +c c3)) H1 e1 H8) in (ex_intro2 C (\lambda (e2: C).(eq C (CHead c3 (Bind Abbr) +v) (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)) c3 +(refl_equal C (CHead c3 (Bind Abbr) v)) H10))) k H7) u H6)))) H5)) +H4))))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (_: (csubt g c1 +c3)).(\lambda (_: (((eq C c1 (CHead e1 (Bind Abbr) v)) \to (ex2 C (\lambda +(e2: C).(eq C c3 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 +e2)))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: +T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u1) (CHead e1 +(Bind Abbr) v))).(let H5 \def (eq_ind C (CHead c1 (Bind Void) u1) (\lambda +(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) +\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda +(_: K).Prop) with [(Bind b0) \Rightarrow (match b0 in B return (\lambda (_: +B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False | Void +\Rightarrow True]) | (Flat _) \Rightarrow False])])) I (CHead e1 (Bind Abbr) +v) H4) in (False_ind (ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind b) u2) +(CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2))) H5))))))))))) +(\lambda (c1: C).(\lambda (c3: C).(\lambda (_: (csubt g c1 c3)).(\lambda (_: +(((eq C c1 (CHead e1 (Bind Abbr) v)) \to (ex2 C (\lambda (e2: C).(eq C c3 +(CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))))).(\lambda (u: +T).(\lambda (t: T).(\lambda (_: (ty3 g c1 u t)).(\lambda (_: (ty3 g c3 u +t)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) t) (CHead e1 (Bind Abbr) +v))).(let H6 \def (eq_ind C (CHead c1 (Bind Abst) t) (\lambda (ee: C).(match +ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | +(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with +[(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr +\Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat +_) \Rightarrow False])])) I (CHead e1 (Bind Abbr) v) H5) in (False_ind (ex2 C +(\lambda (e2: C).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 (Bind Abbr) v))) +(\lambda (e2: C).(csubt g e1 e2))) H6))))))))))) y c2 H0))) H))))). +(* COMMENTS +Initial nodes: 1111 +END *) + +theorem csubt_gen_abst: + \forall (g: G).(\forall (e1: C).(\forall (c2: C).(\forall (v1: T).((csubt g +(CHead e1 (Bind Abst) v1) c2) \to (or (ex2 C (\lambda (e2: C).(eq C c2 (CHead +e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda +(e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: +C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g +e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))))))))) +\def + \lambda (g: G).(\lambda (e1: C).(\lambda (c2: C).(\lambda (v1: T).(\lambda +(H: (csubt g (CHead e1 (Bind Abst) v1) c2)).(insert_eq C (CHead e1 (Bind +Abst) v1) (\lambda (c: C).(csubt g c c2)) (\lambda (_: C).(or (ex2 C (\lambda +(e2: C).(eq C c2 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 +e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind +Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: +C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 +g e2 v2 v1)))))) (\lambda (y: C).(\lambda (H0: (csubt g y c2)).(csubt_ind g +(\lambda (c: C).(\lambda (c0: C).((eq C c (CHead e1 (Bind Abst) v1)) \to (or +(ex2 C (\lambda (e2: C).(eq C c0 (CHead e2 (Bind Abst) v1))) (\lambda (e2: +C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c0 +(CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 +e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: +C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))))) (\lambda (n: nat).(\lambda (H1: +(eq C (CSort n) (CHead e1 (Bind Abst) v1))).(let H2 \def (eq_ind C (CSort n) +(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) +\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 (Bind Abst) +v1) H1) in (False_ind (or (ex2 C (\lambda (e2: C).(eq C (CSort n) (CHead e2 +(Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: +C).(\lambda (v2: T).(eq C (CSort n) (CHead e2 (Bind Abbr) v2)))) (\lambda +(e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: +T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))))) +H2)))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1: (csubt g c1 +c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind Abst) v1)) \to (or (ex2 C +(\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt +g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 +(Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) +(\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda +(v2: T).(ty3 g e2 v2 v1)))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: +(eq C (CHead c1 k u) (CHead e1 (Bind Abst) v1))).(let H4 \def (f_equal C C +(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) +\Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u) (CHead e1 +(Bind Abst) v1) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e in +C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) +\Rightarrow k0])) (CHead c1 k u) (CHead e1 (Bind Abst) v1) H3) in ((let H6 +\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) +with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u) +(CHead e1 (Bind Abst) v1) H3) in (\lambda (H7: (eq K k (Bind Abst))).(\lambda +(H8: (eq C c1 e1)).(eq_ind_r T v1 (\lambda (t: T).(or (ex2 C (\lambda (e2: +C).(eq C (CHead c3 k t) (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g +e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 k t) +(CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 +e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: +C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))) (eq_ind_r K (Bind Abst) (\lambda +(k0: K).(or (ex2 C (\lambda (e2: C).(eq C (CHead c3 k0 v1) (CHead e2 (Bind +Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: +C).(\lambda (v2: T).(eq C (CHead c3 k0 v1) (CHead e2 (Bind Abbr) v2)))) +(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda +(v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 +v1)))))) (let H9 \def (eq_ind C c1 (\lambda (c: C).((eq C c (CHead e1 (Bind +Abst) v1)) \to (or (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abst) +v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda +(v2: T).(eq C c3 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: +T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) +(\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))))))) H2 e1 H8) in (let +H10 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H8) in (or_introl +(ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind Abst) v1) (CHead e2 (Bind Abst) +v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda +(v2: T).(eq C (CHead c3 (Bind Abst) v1) (CHead e2 (Bind Abbr) v2)))) (\lambda +(e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: +T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))) +(ex_intro2 C (\lambda (e2: C).(eq C (CHead c3 (Bind Abst) v1) (CHead e2 (Bind +Abst) v1))) (\lambda (e2: C).(csubt g e1 e2)) c3 (refl_equal C (CHead c3 +(Bind Abst) v1)) H10)))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: +C).(\lambda (c3: C).(\lambda (_: (csubt g c1 c3)).(\lambda (_: (((eq C c1 +(CHead e1 (Bind Abst) v1)) \to (or (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 +(Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: +C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: +C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g +e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 +v1)))))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: +T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u1) (CHead e1 +(Bind Abst) v1))).(let H5 \def (eq_ind C (CHead c1 (Bind Void) u1) (\lambda +(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) +\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda +(_: K).Prop) with [(Bind b0) \Rightarrow (match b0 in B return (\lambda (_: +B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False | Void +\Rightarrow True]) | (Flat _) \Rightarrow False])])) I (CHead e1 (Bind Abst) +v1) H4) in (False_ind (or (ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind b) +u2) (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T +(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind b) u2) (CHead e2 +(Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) +(\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda +(v2: T).(ty3 g e2 v2 v1))))) H5))))))))))) (\lambda (c1: C).(\lambda (c3: +C).(\lambda (H1: (csubt g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind +Abst) v1)) \to (or (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abst) +v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda +(v2: T).(eq C c3 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: +T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) +(\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))))).(\lambda (u: +T).(\lambda (t: T).(\lambda (H3: (ty3 g c1 u t)).(\lambda (H4: (ty3 g c3 u +t)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) t) (CHead e1 (Bind Abst) +v1))).(let H6 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda +(_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) +(CHead c1 (Bind Abst) t) (CHead e1 (Bind Abst) v1) H5) in ((let H7 \def +(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with +[(CSort _) \Rightarrow t | (CHead _ _ t0) \Rightarrow t0])) (CHead c1 (Bind +Abst) t) (CHead e1 (Bind Abst) v1) H5) in (\lambda (H8: (eq C c1 e1)).(let H9 +\def (eq_ind T t (\lambda (t0: T).(ty3 g c3 u t0)) H4 v1 H7) in (let H10 \def +(eq_ind T t (\lambda (t0: T).(ty3 g c1 u t0)) H3 v1 H7) in (let H11 \def +(eq_ind C c1 (\lambda (c: C).(ty3 g c u v1)) H10 e1 H8) in (let H12 \def +(eq_ind C c1 (\lambda (c: C).((eq C c (CHead e1 (Bind Abst) v1)) \to (or (ex2 +C (\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abst) v1))) (\lambda (e2: +C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c3 +(CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 +e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: +C).(\lambda (v2: T).(ty3 g e2 v2 v1))))))) H2 e1 H8) in (let H13 \def (eq_ind +C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H8) in (or_intror (ex2 C (\lambda +(e2: C).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 (Bind Abst) v1))) (\lambda +(e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C +(CHead c3 (Bind Abbr) u) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: +C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g +e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))) +(ex4_2_intro C T (\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind +Abbr) u) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt +g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: +C).(\lambda (v2: T).(ty3 g e2 v2 v1))) c3 u (refl_equal C (CHead c3 (Bind +Abbr) u)) H13 H11 H9))))))))) H6))))))))))) y c2 H0))) H))))). +(* COMMENTS +Initial nodes: 2362 +END *) + +theorem csubt_gen_flat: + \forall (g: G).(\forall (e1: C).(\forall (c2: C).(\forall (v: T).(\forall +(f: F).((csubt g (CHead e1 (Flat f) v) c2) \to (ex2 C (\lambda (e2: C).(eq C +c2 (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g e1 e2)))))))) +\def + \lambda (g: G).(\lambda (e1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda +(f: F).(\lambda (H: (csubt g (CHead e1 (Flat f) v) c2)).(insert_eq C (CHead +e1 (Flat f) v) (\lambda (c: C).(csubt g c c2)) (\lambda (_: C).(ex2 C +(\lambda (e2: C).(eq C c2 (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g +e1 e2)))) (\lambda (y: C).(\lambda (H0: (csubt g y c2)).(csubt_ind g (\lambda +(c: C).(\lambda (c0: C).((eq C c (CHead e1 (Flat f) v)) \to (ex2 C (\lambda +(e2: C).(eq C c0 (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g e1 +e2)))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead e1 (Flat f) +v))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return +(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) +\Rightarrow False])) I (CHead e1 (Flat f) v) H1) in (False_ind (ex2 C +(\lambda (e2: C).(eq C (CSort n) (CHead e2 (Flat f) v))) (\lambda (e2: +C).(csubt g e1 e2))) H2)))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1: +(csubt g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Flat f) v)) \to (ex2 C +(\lambda (e2: C).(eq C c3 (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g +e1 e2)))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c1 k +u) (CHead e1 (Flat f) v))).(let H4 \def (f_equal C C (\lambda (e: C).(match e +in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) +\Rightarrow c])) (CHead c1 k u) (CHead e1 (Flat f) v) H3) in ((let H5 \def +(f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with +[(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c1 k u) +(CHead e1 (Flat f) v) H3) in ((let H6 \def (f_equal C T (\lambda (e: +C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | +(CHead _ _ t) \Rightarrow t])) (CHead c1 k u) (CHead e1 (Flat f) v) H3) in +(\lambda (H7: (eq K k (Flat f))).(\lambda (H8: (eq C c1 e1)).(eq_ind_r T v +(\lambda (t: T).(ex2 C (\lambda (e2: C).(eq C (CHead c3 k t) (CHead e2 (Flat +f) v))) (\lambda (e2: C).(csubt g e1 e2)))) (eq_ind_r K (Flat f) (\lambda +(k0: K).(ex2 C (\lambda (e2: C).(eq C (CHead c3 k0 v) (CHead e2 (Flat f) v))) +(\lambda (e2: C).(csubt g e1 e2)))) (let H9 \def (eq_ind C c1 (\lambda (c: +C).((eq C c (CHead e1 (Flat f) v)) \to (ex2 C (\lambda (e2: C).(eq C c3 +(CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g e1 e2))))) H2 e1 H8) in +(let H10 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H8) in +(ex_intro2 C (\lambda (e2: C).(eq C (CHead c3 (Flat f) v) (CHead e2 (Flat f) +v))) (\lambda (e2: C).(csubt g e1 e2)) c3 (refl_equal C (CHead c3 (Flat f) +v)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: C).(\lambda (c3: +C).(\lambda (_: (csubt g c1 c3)).(\lambda (_: (((eq C c1 (CHead e1 (Flat f) +v)) \to (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 (Flat f) v))) (\lambda +(e2: C).(csubt g e1 e2)))))).(\lambda (b: B).(\lambda (_: (not (eq B b +Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind +Void) u1) (CHead e1 (Flat f) v))).(let H5 \def (eq_ind C (CHead c1 (Bind +Void) u1) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with +[(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return +(\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow +False])])) I (CHead e1 (Flat f) v) H4) in (False_ind (ex2 C (\lambda (e2: +C).(eq C (CHead c3 (Bind b) u2) (CHead e2 (Flat f) v))) (\lambda (e2: +C).(csubt g e1 e2))) H5))))))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda +(_: (csubt g c1 c3)).(\lambda (_: (((eq C c1 (CHead e1 (Flat f) v)) \to (ex2 +C (\lambda (e2: C).(eq C c3 (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g +e1 e2)))))).(\lambda (u: T).(\lambda (t: T).(\lambda (_: (ty3 g c1 u +t)).(\lambda (_: (ty3 g c3 u t)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) t) +(CHead e1 (Flat f) v))).(let H6 \def (eq_ind C (CHead c1 (Bind Abst) t) +(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) +\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda +(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow +False])])) I (CHead e1 (Flat f) v) H5) in (False_ind (ex2 C (\lambda (e2: +C).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 (Flat f) v))) (\lambda (e2: +C).(csubt g e1 e2))) H6))))))))))) y c2 H0))) H)))))). +(* COMMENTS +Initial nodes: 1103 +END *) + +theorem csubt_gen_bind: + \forall (g: G).(\forall (b1: B).(\forall (e1: C).(\forall (c2: C).(\forall +(v1: T).((csubt g (CHead e1 (Bind b1) v1) c2) \to (ex2_3 B C T (\lambda (b2: +B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) +(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2)))))))))) +\def + \lambda (g: G).(\lambda (b1: B).(\lambda (e1: C).(\lambda (c2: C).(\lambda +(v1: T).(\lambda (H: (csubt g (CHead e1 (Bind b1) v1) c2)).(insert_eq C +(CHead e1 (Bind b1) v1) (\lambda (c: C).(csubt g c c2)) (\lambda (_: +C).(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 +(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: +T).(csubt g e1 e2)))))) (\lambda (y: C).(\lambda (H0: (csubt g y +c2)).(csubt_ind g (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead e1 (Bind +b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: +T).(eq C c0 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: +C).(\lambda (_: T).(csubt g e1 e2)))))))) (\lambda (n: nat).(\lambda (H1: (eq +C (CSort n) (CHead e1 (Bind b1) v1))).(let H2 \def (eq_ind C (CSort n) +(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) +\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 (Bind b1) +v1) H1) in (False_ind (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda +(v2: T).(eq C (CSort n) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda +(e2: C).(\lambda (_: T).(csubt g e1 e2))))) H2)))) (\lambda (c1: C).(\lambda +(c3: C).(\lambda (H1: (csubt g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 +(Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda +(v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: +C).(\lambda (_: T).(csubt g e1 e2)))))))).(\lambda (k: K).(\lambda (u: +T).(\lambda (H3: (eq C (CHead c1 k u) (CHead e1 (Bind b1) v1))).(let H4 \def +(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with +[(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u) +(CHead e1 (Bind b1) v1) H3) in ((let H5 \def (f_equal C K (\lambda (e: +C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | +(CHead _ k0 _) \Rightarrow k0])) (CHead c1 k u) (CHead e1 (Bind b1) v1) H3) +in ((let H6 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda +(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) +(CHead c1 k u) (CHead e1 (Bind b1) v1) H3) in (\lambda (H7: (eq K k (Bind +b1))).(\lambda (H8: (eq C c1 e1)).(eq_ind_r T v1 (\lambda (t: T).(ex2_3 B C T +(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 k t) +(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: +T).(csubt g e1 e2)))))) (eq_ind_r K (Bind b1) (\lambda (k0: K).(ex2_3 B C T +(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 k0 v1) +(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: +T).(csubt g e1 e2)))))) (let H9 \def (eq_ind C c1 (\lambda (c: C).((eq C c +(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: +C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: +B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))))))) H2 e1 H8) in (let +H10 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H8) in +(ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C +(CHead c3 (Bind b1) v1) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda +(e2: C).(\lambda (_: T).(csubt g e1 e2)))) b1 c3 v1 (refl_equal C (CHead c3 +(Bind b1) v1)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: +C).(\lambda (c3: C).(\lambda (H1: (csubt g c1 c3)).(\lambda (H2: (((eq C c1 +(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: +C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: +B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2)))))))).(\lambda (b: +B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: +T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) +v1))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda +(_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) +(CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) v1) H4) in ((let H6 \def +(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with +[(CSort _) \Rightarrow Void | (CHead _ k _) \Rightarrow (match k in K return +(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow +Void])])) (CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) v1) H4) in ((let H7 +\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) +with [(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead c1 +(Bind Void) u1) (CHead e1 (Bind b1) v1) H4) in (\lambda (H8: (eq B Void +b1)).(\lambda (H9: (eq C c1 e1)).(let H10 \def (eq_ind C c1 (\lambda (c: +C).((eq C c (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: +B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) +(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))))))) H2 e1 +H9) in (let H11 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H9) +in (let H12 \def (eq_ind_r B b1 (\lambda (b0: B).((eq C e1 (CHead e1 (Bind +b0) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: +T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: +C).(\lambda (_: T).(csubt g e1 e2))))))) H10 Void H8) in (ex2_3_intro B C T +(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind b) +u2) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: +T).(csubt g e1 e2)))) b c3 u2 (refl_equal C (CHead c3 (Bind b) u2)) +H11))))))) H6)) H5))))))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1: +(csubt g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind b1) v1)) \to (ex2_3 +B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 +(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g +e1 e2)))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H3: (ty3 g c1 u +t)).(\lambda (H4: (ty3 g c3 u t)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) +t) (CHead e1 (Bind b1) v1))).(let H6 \def (f_equal C C (\lambda (e: C).(match +e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ +_) \Rightarrow c])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H5) in +((let H7 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: +C).B) with [(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match k +in K return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) +\Rightarrow Abst])])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H5) in +((let H8 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: +C).T) with [(CSort _) \Rightarrow t | (CHead _ _ t0) \Rightarrow t0])) (CHead +c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H5) in (\lambda (H9: (eq B Abst +b1)).(\lambda (H10: (eq C c1 e1)).(let H11 \def (eq_ind T t (\lambda (t0: +T).(ty3 g c3 u t0)) H4 v1 H8) in (let H12 \def (eq_ind T t (\lambda (t0: +T).(ty3 g c1 u t0)) H3 v1 H8) in (let H13 \def (eq_ind C c1 (\lambda (c: +C).(ty3 g c u v1)) H12 e1 H10) in (let H14 \def (eq_ind C c1 (\lambda (c: +C).((eq C c (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: +B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) +(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))))))) H2 e1 +H10) in (let H15 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H10) +in (let H16 \def (eq_ind_r B b1 (\lambda (b: B).((eq C e1 (CHead e1 (Bind b) +v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq +C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda +(_: T).(csubt g e1 e2))))))) H14 Abst H9) in (ex2_3_intro B C T (\lambda (b2: +B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 +(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g +e1 e2)))) Abbr c3 u (refl_equal C (CHead c3 (Bind Abbr) u)) H15)))))))))) +H7)) H6))))))))))) y c2 H0))) H)))))). +(* COMMENTS +Initial nodes: 1899 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubt/getl.ma b/matita/matita/contribs/lambdadelta/basic_1/csubt/getl.ma new file mode 100644 index 000000000..df14528ed --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/csubt/getl.ma @@ -0,0 +1,426 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/csubt/clear.ma". + +include "Basic-1/csubt/drop.ma". + +include "Basic-1/getl/clear.ma". + +theorem csubt_getl_abbr: + \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).(\forall +(n: nat).((getl n c1 (CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csubt g +c1 c2) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n +c2 (CHead d2 (Bind Abbr) u))))))))))) +\def + \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u: T).(\lambda +(n: nat).(\lambda (H: (getl n c1 (CHead d1 (Bind Abbr) u))).(let H0 \def +(getl_gen_all c1 (CHead d1 (Bind Abbr) u) n H) in (ex2_ind C (\lambda (e: +C).(drop n O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abbr) u))) +(\forall (c2: C).((csubt g c1 c2) \to (ex2 C (\lambda (d2: C).(csubt g d1 +d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u)))))) (\lambda (x: +C).(\lambda (H1: (drop n O c1 x)).(\lambda (H2: (clear x (CHead d1 (Bind +Abbr) u))).(C_ind (\lambda (c: C).((drop n O c1 c) \to ((clear c (CHead d1 +(Bind Abbr) u)) \to (\forall (c2: C).((csubt g c1 c2) \to (ex2 C (\lambda +(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) +u))))))))) (\lambda (n0: nat).(\lambda (_: (drop n O c1 (CSort n0))).(\lambda +(H4: (clear (CSort n0) (CHead d1 (Bind Abbr) u))).(clear_gen_sort (CHead d1 +(Bind Abbr) u) n0 H4 (\forall (c2: C).((csubt g c1 c2) \to (ex2 C (\lambda +(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) +u)))))))))) (\lambda (x0: C).(\lambda (_: (((drop n O c1 x0) \to ((clear x0 +(CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csubt g c1 c2) \to (ex2 C +(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind +Abbr) u)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (H3: (drop n O c1 +(CHead x0 k t))).(\lambda (H4: (clear (CHead x0 k t) (CHead d1 (Bind Abbr) +u))).(K_ind (\lambda (k0: K).((drop n O c1 (CHead x0 k0 t)) \to ((clear +(CHead x0 k0 t) (CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csubt g c1 +c2) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 +(CHead d2 (Bind Abbr) u))))))))) (\lambda (b: B).(\lambda (H5: (drop n O c1 +(CHead x0 (Bind b) t))).(\lambda (H6: (clear (CHead x0 (Bind b) t) (CHead d1 +(Bind Abbr) u))).(let H7 \def (f_equal C C (\lambda (e: C).(match e in C +return (\lambda (_: C).C) with [(CSort _) \Rightarrow d1 | (CHead c _ _) +\Rightarrow c])) (CHead d1 (Bind Abbr) u) (CHead x0 (Bind b) t) +(clear_gen_bind b x0 (CHead d1 (Bind Abbr) u) t H6)) in ((let H8 \def +(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with +[(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow (match k0 in K +return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) +\Rightarrow Abbr])])) (CHead d1 (Bind Abbr) u) (CHead x0 (Bind b) t) +(clear_gen_bind b x0 (CHead d1 (Bind Abbr) u) t H6)) in ((let H9 \def +(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with +[(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d1 (Bind +Abbr) u) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abbr) u) +t H6)) in (\lambda (H10: (eq B Abbr b)).(\lambda (H11: (eq C d1 x0)).(\lambda +(c2: C).(\lambda (H12: (csubt g c1 c2)).(let H13 \def (eq_ind_r T t (\lambda +(t0: T).(drop n O c1 (CHead x0 (Bind b) t0))) H5 u H9) in (let H14 \def +(eq_ind_r B b (\lambda (b0: B).(drop n O c1 (CHead x0 (Bind b0) u))) H13 Abbr +H10) in (let H15 \def (eq_ind_r C x0 (\lambda (c: C).(drop n O c1 (CHead c +(Bind Abbr) u))) H14 d1 H11) in (ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) +(\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abbr) u))) (ex2 C (\lambda (d2: +C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u)))) +(\lambda (x1: C).(\lambda (H16: (csubt g d1 x1)).(\lambda (H17: (drop n O c2 +(CHead x1 (Bind Abbr) u))).(ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) +(\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))) x1 H16 (getl_intro n +c2 (CHead x1 (Bind Abbr) u) (CHead x1 (Bind Abbr) u) H17 (clear_bind Abbr x1 +u)))))) (csubt_drop_abbr g n c1 c2 H12 d1 u H15)))))))))) H8)) H7))))) +(\lambda (f: F).(\lambda (H5: (drop n O c1 (CHead x0 (Flat f) t))).(\lambda +(H6: (clear (CHead x0 (Flat f) t) (CHead d1 (Bind Abbr) u))).(let H7 \def H5 +in (unintro C c1 (\lambda (c: C).((drop n O c (CHead x0 (Flat f) t)) \to +(\forall (c2: C).((csubt g c c2) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) +(\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u)))))))) (nat_ind (\lambda +(n0: nat).(\forall (x1: C).((drop n0 O x1 (CHead x0 (Flat f) t)) \to (\forall +(c2: C).((csubt g x1 c2) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) +(\lambda (d2: C).(getl n0 c2 (CHead d2 (Bind Abbr) u))))))))) (\lambda (x1: +C).(\lambda (H8: (drop O O x1 (CHead x0 (Flat f) t))).(\lambda (c2: +C).(\lambda (H9: (csubt g x1 c2)).(let H10 \def (eq_ind C x1 (\lambda (c: +C).(csubt g c c2)) H9 (CHead x0 (Flat f) t) (drop_gen_refl x1 (CHead x0 (Flat +f) t) H8)) in (let H_y \def (clear_flat x0 (CHead d1 (Bind Abbr) u) +(clear_gen_flat f x0 (CHead d1 (Bind Abbr) u) t H6) f t) in (let H11 \def +(csubt_clear_conf g (CHead x0 (Flat f) t) c2 H10 (CHead d1 (Bind Abbr) u) +H_y) in (ex2_ind C (\lambda (e2: C).(csubt g (CHead d1 (Bind Abbr) u) e2)) +(\lambda (e2: C).(clear c2 e2)) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) +(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (x2: +C).(\lambda (H12: (csubt g (CHead d1 (Bind Abbr) u) x2)).(\lambda (H13: +(clear c2 x2)).(let H14 \def (csubt_gen_abbr g d1 x2 u H12) in (ex2_ind C +(\lambda (e2: C).(eq C x2 (CHead e2 (Bind Abbr) u))) (\lambda (e2: C).(csubt +g d1 e2)) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl O +c2 (CHead d2 (Bind Abbr) u)))) (\lambda (x3: C).(\lambda (H15: (eq C x2 +(CHead x3 (Bind Abbr) u))).(\lambda (H16: (csubt g d1 x3)).(let H17 \def +(eq_ind C x2 (\lambda (c: C).(clear c2 c)) H13 (CHead x3 (Bind Abbr) u) H15) +in (ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl O c2 +(CHead d2 (Bind Abbr) u))) x3 H16 (getl_intro O c2 (CHead x3 (Bind Abbr) u) +c2 (drop_refl c2) H17)))))) H14))))) H11)))))))) (\lambda (n0: nat).(\lambda +(H8: ((\forall (x1: C).((drop n0 O x1 (CHead x0 (Flat f) t)) \to (\forall +(c2: C).((csubt g x1 c2) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) +(\lambda (d2: C).(getl n0 c2 (CHead d2 (Bind Abbr) u)))))))))).(\lambda (x1: +C).(\lambda (H9: (drop (S n0) O x1 (CHead x0 (Flat f) t))).(\lambda (c2: +C).(\lambda (H10: (csubt g x1 c2)).(let H11 \def (drop_clear x1 (CHead x0 +(Flat f) t) n0 H9) in (ex2_3_ind B C T (\lambda (b: B).(\lambda (e: +C).(\lambda (v: T).(clear x1 (CHead e (Bind b) v))))) (\lambda (_: +B).(\lambda (e: C).(\lambda (_: T).(drop n0 O e (CHead x0 (Flat f) t))))) +(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 +(CHead d2 (Bind Abbr) u)))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: +T).(\lambda (H12: (clear x1 (CHead x3 (Bind x2) x4))).(\lambda (H13: (drop n0 +O x3 (CHead x0 (Flat f) t))).(let H14 \def (csubt_clear_conf g x1 c2 H10 +(CHead x3 (Bind x2) x4) H12) in (ex2_ind C (\lambda (e2: C).(csubt g (CHead +x3 (Bind x2) x4) e2)) (\lambda (e2: C).(clear c2 e2)) (ex2 C (\lambda (d2: +C).(csubt g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr) +u)))) (\lambda (x5: C).(\lambda (H15: (csubt g (CHead x3 (Bind x2) x4) +x5)).(\lambda (H16: (clear c2 x5)).(let H17 \def (csubt_gen_bind g x2 x3 x5 +x4 H15) in (ex2_3_ind B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: +T).(eq C x5 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: +C).(\lambda (_: T).(csubt g x3 e2)))) (ex2 C (\lambda (d2: C).(csubt g d1 +d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda +(x6: B).(\lambda (x7: C).(\lambda (x8: T).(\lambda (H18: (eq C x5 (CHead x7 +(Bind x6) x8))).(\lambda (H19: (csubt g x3 x7)).(let H20 \def (eq_ind C x5 +(\lambda (c: C).(clear c2 c)) H16 (CHead x7 (Bind x6) x8) H18) in (let H21 +\def (H8 x3 H13 x7 H19) in (ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) +(\lambda (d2: C).(getl n0 x7 (CHead d2 (Bind Abbr) u))) (ex2 C (\lambda (d2: +C).(csubt g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr) +u)))) (\lambda (x9: C).(\lambda (H22: (csubt g d1 x9)).(\lambda (H23: (getl +n0 x7 (CHead x9 (Bind Abbr) u))).(ex_intro2 C (\lambda (d2: C).(csubt g d1 +d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr) u))) x9 H22 +(getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Abbr) u) n0 H23))))) +H21)))))))) H17))))) H14))))))) H11)))))))) n) H7))))) k H3 H4))))))) x H1 +H2)))) H0))))))). +(* COMMENTS +Initial nodes: 2313 +END *) + +theorem csubt_getl_abst: + \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (t: T).(\forall +(n: nat).((getl n c1 (CHead d1 (Bind Abst) t)) \to (\forall (c2: C).((csubt g +c1 c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: +C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: +C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n +c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u +t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))))) +\def + \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (t: T).(\lambda +(n: nat).(\lambda (H: (getl n c1 (CHead d1 (Bind Abst) t))).(let H0 \def +(getl_gen_all c1 (CHead d1 (Bind Abst) t) n H) in (ex2_ind C (\lambda (e: +C).(drop n O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abst) t))) +(\forall (c2: C).((csubt g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 +d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T +(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda +(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: +T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))))) +(\lambda (x: C).(\lambda (H1: (drop n O c1 x)).(\lambda (H2: (clear x (CHead +d1 (Bind Abst) t))).(C_ind (\lambda (c: C).((drop n O c1 c) \to ((clear c +(CHead d1 (Bind Abst) t)) \to (\forall (c2: C).((csubt g c1 c2) \to (or (ex2 +C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 +(Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 +d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) +(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda +(u: T).(ty3 g d2 u t)))))))))) (\lambda (n0: nat).(\lambda (_: (drop n O c1 +(CSort n0))).(\lambda (H4: (clear (CSort n0) (CHead d1 (Bind Abst) +t))).(clear_gen_sort (CHead d1 (Bind Abst) t) n0 H4 (\forall (c2: C).((csubt +g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: +C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: +C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n +c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u +t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))))))))) (\lambda (x0: +C).(\lambda (_: (((drop n O c1 x0) \to ((clear x0 (CHead d1 (Bind Abst) t)) +\to (\forall (c2: C).((csubt g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csubt +g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T +(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda +(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: +T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u +t))))))))))).(\lambda (k: K).(\lambda (t0: T).(\lambda (H3: (drop n O c1 +(CHead x0 k t0))).(\lambda (H4: (clear (CHead x0 k t0) (CHead d1 (Bind Abst) +t))).(K_ind (\lambda (k0: K).((drop n O c1 (CHead x0 k0 t0)) \to ((clear +(CHead x0 k0 t0) (CHead d1 (Bind Abst) t)) \to (\forall (c2: C).((csubt g c1 +c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n +c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: +T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 +(Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda +(d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))) (\lambda (b: B).(\lambda (H5: +(drop n O c1 (CHead x0 (Bind b) t0))).(\lambda (H6: (clear (CHead x0 (Bind b) +t0) (CHead d1 (Bind Abst) t))).(let H7 \def (f_equal C C (\lambda (e: +C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d1 | +(CHead c _ _) \Rightarrow c])) (CHead d1 (Bind Abst) t) (CHead x0 (Bind b) +t0) (clear_gen_bind b x0 (CHead d1 (Bind Abst) t) t0 H6)) in ((let H8 \def +(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with +[(CSort _) \Rightarrow Abst | (CHead _ k0 _) \Rightarrow (match k0 in K +return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) +\Rightarrow Abst])])) (CHead d1 (Bind Abst) t) (CHead x0 (Bind b) t0) +(clear_gen_bind b x0 (CHead d1 (Bind Abst) t) t0 H6)) in ((let H9 \def +(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with +[(CSort _) \Rightarrow t | (CHead _ _ t1) \Rightarrow t1])) (CHead d1 (Bind +Abst) t) (CHead x0 (Bind b) t0) (clear_gen_bind b x0 (CHead d1 (Bind Abst) t) +t0 H6)) in (\lambda (H10: (eq B Abst b)).(\lambda (H11: (eq C d1 +x0)).(\lambda (c2: C).(\lambda (H12: (csubt g c1 c2)).(let H13 \def (eq_ind_r +T t0 (\lambda (t1: T).(drop n O c1 (CHead x0 (Bind b) t1))) H5 t H9) in (let +H14 \def (eq_ind_r B b (\lambda (b0: B).(drop n O c1 (CHead x0 (Bind b0) t))) +H13 Abst H10) in (let H15 \def (eq_ind_r C x0 (\lambda (c: C).(drop n O c1 +(CHead c (Bind Abst) t))) H14 d1 H11) in (or_ind (ex2 C (\lambda (d2: +C).(csubt g d1 d2)) (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abst) t)))) +(ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: +C).(\lambda (u: T).(drop n O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: +C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g +d2 u t)))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: +C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: +C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n +c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u +t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (H16: (ex2 +C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n O c2 (CHead d2 +(Bind Abst) t))))).(ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: +C).(drop n O c2 (CHead d2 (Bind Abst) t))) (or (ex2 C (\lambda (d2: C).(csubt +g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T +(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda +(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: +T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) +(\lambda (x1: C).(\lambda (H17: (csubt g d1 x1)).(\lambda (H18: (drop n O c2 +(CHead x1 (Bind Abst) t))).(or_introl (ex2 C (\lambda (d2: C).(csubt g d1 +d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T +(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda +(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: +T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) +(ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 +(CHead d2 (Bind Abst) t))) x1 H17 (getl_intro n c2 (CHead x1 (Bind Abst) t) +(CHead x1 (Bind Abst) t) H18 (clear_bind Abst x1 t))))))) H16)) (\lambda +(H16: (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda +(d2: C).(\lambda (u: T).(drop n O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: +C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g +d2 u t))))).(ex4_2_ind C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) +(\lambda (d2: C).(\lambda (u: T).(drop n O c2 (CHead d2 (Bind Abbr) u)))) +(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda +(u: T).(ty3 g d2 u t))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda +(d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: +C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n +c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u +t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x1: +C).(\lambda (x2: T).(\lambda (H17: (csubt g d1 x1)).(\lambda (H18: (drop n O +c2 (CHead x1 (Bind Abbr) x2))).(\lambda (H19: (ty3 g d1 x2 t)).(\lambda (H20: +(ty3 g x1 x2 t)).(or_intror (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda +(d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: +C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n +c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u +t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (ex4_2_intro C T +(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda +(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: +T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))) x1 x2 +H17 (getl_intro n c2 (CHead x1 (Bind Abbr) x2) (CHead x1 (Bind Abbr) x2) H18 +(clear_bind Abbr x1 x2)) H19 H20)))))))) H16)) (csubt_drop_abst g n c1 c2 H12 +d1 t H15)))))))))) H8)) H7))))) (\lambda (f: F).(\lambda (H5: (drop n O c1 +(CHead x0 (Flat f) t0))).(\lambda (H6: (clear (CHead x0 (Flat f) t0) (CHead +d1 (Bind Abst) t))).(let H7 \def H5 in (unintro C c1 (\lambda (c: C).((drop n +O c (CHead x0 (Flat f) t0)) \to (\forall (c2: C).((csubt g c c2) \to (or (ex2 +C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 +(Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 +d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) +(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda +(u: T).(ty3 g d2 u t))))))))) (nat_ind (\lambda (n0: nat).(\forall (x1: +C).((drop n0 O x1 (CHead x0 (Flat f) t0)) \to (\forall (c2: C).((csubt g x1 +c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl +n0 c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: +T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n0 c2 (CHead d2 +(Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda +(d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))) (\lambda (x1: C).(\lambda +(H8: (drop O O x1 (CHead x0 (Flat f) t0))).(\lambda (c2: C).(\lambda (H9: +(csubt g x1 c2)).(let H10 \def (eq_ind C x1 (\lambda (c: C).(csubt g c c2)) +H9 (CHead x0 (Flat f) t0) (drop_gen_refl x1 (CHead x0 (Flat f) t0) H8)) in +(let H_y \def (clear_flat x0 (CHead d1 (Bind Abst) t) (clear_gen_flat f x0 +(CHead d1 (Bind Abst) t) t0 H6) f t0) in (let H11 \def (csubt_clear_conf g +(CHead x0 (Flat f) t0) c2 H10 (CHead d1 (Bind Abst) t) H_y) in (ex2_ind C +(\lambda (e2: C).(csubt g (CHead d1 (Bind Abst) t) e2)) (\lambda (e2: +C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: +C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: +C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl O +c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u +t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x2: +C).(\lambda (H12: (csubt g (CHead d1 (Bind Abst) t) x2)).(\lambda (H13: +(clear c2 x2)).(let H14 \def (csubt_gen_abst g d1 x2 t H12) in (or_ind (ex2 C +(\lambda (e2: C).(eq C x2 (CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csubt +g d1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C x2 (CHead e2 +(Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g d1 e2))) +(\lambda (_: C).(\lambda (v2: T).(ty3 g d1 v2 t))) (\lambda (e2: C).(\lambda +(v2: T).(ty3 g e2 v2 t)))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) +(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda +(d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: +T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: +T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) +(\lambda (H15: (ex2 C (\lambda (e2: C).(eq C x2 (CHead e2 (Bind Abst) t))) +(\lambda (e2: C).(csubt g d1 e2)))).(ex2_ind C (\lambda (e2: C).(eq C x2 +(CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csubt g d1 e2)) (or (ex2 C +(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind +Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) +(\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr) u)))) +(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda +(u: T).(ty3 g d2 u t))))) (\lambda (x3: C).(\lambda (H16: (eq C x2 (CHead x3 +(Bind Abst) t))).(\lambda (H17: (csubt g d1 x3)).(let H18 \def (eq_ind C x2 +(\lambda (c: C).(clear c2 c)) H13 (CHead x3 (Bind Abst) t) H16) in (or_introl +(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead +d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 +d2))) (\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr) u)))) +(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda +(u: T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) +(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t))) x3 H17 (getl_intro O +c2 (CHead x3 (Bind Abst) t) c2 (drop_refl c2) H18))))))) H15)) (\lambda (H15: +(ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C x2 (CHead e2 (Bind Abbr) +v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g d1 e2))) (\lambda (_: +C).(\lambda (v2: T).(ty3 g d1 v2 t))) (\lambda (e2: C).(\lambda (v2: T).(ty3 +g e2 v2 t))))).(ex4_2_ind C T (\lambda (e2: C).(\lambda (v2: T).(eq C x2 +(CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g d1 +e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g d1 v2 t))) (\lambda (e2: +C).(\lambda (v2: T).(ty3 g e2 v2 t))) (or (ex2 C (\lambda (d2: C).(csubt g d1 +d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T +(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda +(u: T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: +T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) +(\lambda (x3: C).(\lambda (x4: T).(\lambda (H16: (eq C x2 (CHead x3 (Bind +Abbr) x4))).(\lambda (H17: (csubt g d1 x3)).(\lambda (H18: (ty3 g d1 x4 +t)).(\lambda (H19: (ty3 g x3 x4 t)).(let H20 \def (eq_ind C x2 (\lambda (c: +C).(clear c2 c)) H13 (CHead x3 (Bind Abbr) x4) H16) in (or_intror (ex2 C +(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind +Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) +(\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr) u)))) +(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda +(u: T).(ty3 g d2 u t)))) (ex4_2_intro C T (\lambda (d2: C).(\lambda (_: +T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 +(Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda +(d2: C).(\lambda (u: T).(ty3 g d2 u t))) x3 x4 H17 (getl_intro O c2 (CHead x3 +(Bind Abbr) x4) c2 (drop_refl c2) H20) H18 H19))))))))) H15)) H14))))) +H11)))))))) (\lambda (n0: nat).(\lambda (H8: ((\forall (x1: C).((drop n0 O x1 +(CHead x0 (Flat f) t0)) \to (\forall (c2: C).((csubt g x1 c2) \to (or (ex2 C +(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n0 c2 (CHead d2 +(Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 +d2))) (\lambda (d2: C).(\lambda (u: T).(getl n0 c2 (CHead d2 (Bind Abbr) +u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: +C).(\lambda (u: T).(ty3 g d2 u t))))))))))).(\lambda (x1: C).(\lambda (H9: +(drop (S n0) O x1 (CHead x0 (Flat f) t0))).(\lambda (c2: C).(\lambda (H10: +(csubt g x1 c2)).(let H11 \def (drop_clear x1 (CHead x0 (Flat f) t0) n0 H9) +in (ex2_3_ind B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear x1 +(CHead e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: +T).(drop n0 O e (CHead x0 (Flat f) t0))))) (or (ex2 C (\lambda (d2: C).(csubt +g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex4_2 +C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: +C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: +C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g +d2 u t))))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: T).(\lambda (H12: +(clear x1 (CHead x3 (Bind x2) x4))).(\lambda (H13: (drop n0 O x3 (CHead x0 +(Flat f) t0))).(let H14 \def (csubt_clear_conf g x1 c2 H10 (CHead x3 (Bind +x2) x4) H12) in (ex2_ind C (\lambda (e2: C).(csubt g (CHead x3 (Bind x2) x4) +e2)) (\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(csubt g d1 +d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T +(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda +(u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda +(u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) +(\lambda (x5: C).(\lambda (H15: (csubt g (CHead x3 (Bind x2) x4) +x5)).(\lambda (H16: (clear c2 x5)).(let H17 \def (csubt_gen_bind g x2 x3 x5 +x4 H15) in (ex2_3_ind B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: +T).(eq C x5 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: +C).(\lambda (_: T).(csubt g x3 e2)))) (or (ex2 C (\lambda (d2: C).(csubt g d1 +d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T +(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda +(u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda +(u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) +(\lambda (x6: B).(\lambda (x7: C).(\lambda (x8: T).(\lambda (H18: (eq C x5 +(CHead x7 (Bind x6) x8))).(\lambda (H19: (csubt g x3 x7)).(let H20 \def +(eq_ind C x5 (\lambda (c: C).(clear c2 c)) H16 (CHead x7 (Bind x6) x8) H18) +in (let H21 \def (H8 x3 H13 x7 H19) in (or_ind (ex2 C (\lambda (d2: C).(csubt +g d1 d2)) (\lambda (d2: C).(getl n0 x7 (CHead d2 (Bind Abst) t)))) (ex4_2 C T +(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda +(u: T).(getl n0 x7 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: +T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (or +(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 +(CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: +T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead +d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) +(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (H22: (ex2 C +(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n0 x7 (CHead d2 +(Bind Abst) t))))).(ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: +C).(getl n0 x7 (CHead d2 (Bind Abst) t))) (or (ex2 C (\lambda (d2: C).(csubt +g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex4_2 +C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: +C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: +C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g +d2 u t))))) (\lambda (x9: C).(\lambda (H23: (csubt g d1 x9)).(\lambda (H24: +(getl n0 x7 (CHead x9 (Bind Abst) t))).(or_introl (ex2 C (\lambda (d2: +C).(csubt g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) +t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda +(d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda +(_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 +g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: +C).(getl (S n0) c2 (CHead d2 (Bind Abst) t))) x9 H23 (getl_clear_bind x6 c2 +x7 x8 H20 (CHead x9 (Bind Abst) t) n0 H24)))))) H22)) (\lambda (H22: (ex4_2 C +T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: +C).(\lambda (u: T).(getl n0 x7 (CHead d2 (Bind Abbr) u)))) (\lambda (_: +C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g +d2 u t))))).(ex4_2_ind C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) +(\lambda (d2: C).(\lambda (u: T).(getl n0 x7 (CHead d2 (Bind Abbr) u)))) +(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda +(u: T).(ty3 g d2 u t))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda +(d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: +C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl +(S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g +d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x9: +C).(\lambda (x10: T).(\lambda (H23: (csubt g d1 x9)).(\lambda (H24: (getl n0 +x7 (CHead x9 (Bind Abbr) x10))).(\lambda (H25: (ty3 g d1 x10 t)).(\lambda +(H26: (ty3 g x9 x10 t)).(or_intror (ex2 C (\lambda (d2: C).(csubt g d1 d2)) +(\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T +(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda +(u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda +(u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) +(ex4_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda +(d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda +(_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 +g d2 u t))) x9 x10 H23 (getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Abbr) +x10) n0 H24) H25 H26)))))))) H22)) H21)))))))) H17))))) H14))))))) +H11)))))))) n) H7))))) k H3 H4))))))) x H1 H2)))) H0))))))). +(* COMMENTS +Initial nodes: 5861 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubt/pc3.ma b/matita/matita/contribs/lambdadelta/basic_1/csubt/pc3.ma new file mode 100644 index 000000000..900ba49e9 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/csubt/pc3.ma @@ -0,0 +1,62 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/csubt/getl.ma". + +include "Basic-1/pc3/left.ma". + +theorem csubt_pr2: + \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pr2 c1 +t1 t2) \to (\forall (c2: C).((csubt g c1 c2) \to (pr2 c2 t1 t2))))))) +\def + \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda +(H: (pr2 c1 t1 t2)).(pr2_ind (\lambda (c: C).(\lambda (t: T).(\lambda (t0: +T).(\forall (c2: C).((csubt g c c2) \to (pr2 c2 t t0)))))) (\lambda (c: +C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(\lambda (c2: +C).(\lambda (_: (csubt g c c2)).(pr2_free c2 t3 t4 H0))))))) (\lambda (c: +C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c +(CHead d (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: +(pr0 t3 t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (c2: +C).(\lambda (H3: (csubt g c c2)).(let H4 \def (csubt_getl_abbr g c d u i H0 +c2 H3) in (ex2_ind C (\lambda (d2: C).(csubt g d d2)) (\lambda (d2: C).(getl +i c2 (CHead d2 (Bind Abbr) u))) (pr2 c2 t3 t) (\lambda (x: C).(\lambda (_: +(csubt g d x)).(\lambda (H6: (getl i c2 (CHead x (Bind Abbr) u))).(pr2_delta +c2 x u i H6 t3 t4 H1 t H2)))) H4)))))))))))))) c1 t1 t2 H))))). +(* COMMENTS +Initial nodes: 245 +END *) + +theorem csubt_pc3: + \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pc3 c1 +t1 t2) \to (\forall (c2: C).((csubt g c1 c2) \to (pc3 c2 t1 t2))))))) +\def + \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda +(H: (pc3 c1 t1 t2)).(pc3_ind_left c1 (\lambda (t: T).(\lambda (t0: +T).(\forall (c2: C).((csubt g c1 c2) \to (pc3 c2 t t0))))) (\lambda (t: +T).(\lambda (c2: C).(\lambda (_: (csubt g c1 c2)).(pc3_refl c2 t)))) (\lambda +(t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c1 t0 t3)).(\lambda (t4: +T).(\lambda (_: (pc3 c1 t3 t4)).(\lambda (H2: ((\forall (c2: C).((csubt g c1 +c2) \to (pc3 c2 t3 t4))))).(\lambda (c2: C).(\lambda (H3: (csubt g c1 +c2)).(pc3_t t3 c2 t0 (pc3_pr2_r c2 t0 t3 (csubt_pr2 g c1 t0 t3 H0 c2 H3)) t4 +(H2 c2 H3)))))))))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c1 +t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3 c1 t0 t4)).(\lambda (H2: ((\forall +(c2: C).((csubt g c1 c2) \to (pc3 c2 t0 t4))))).(\lambda (c2: C).(\lambda +(H3: (csubt g c1 c2)).(pc3_t t0 c2 t3 (pc3_pr2_x c2 t3 t0 (csubt_pr2 g c1 t0 +t3 H0 c2 H3)) t4 (H2 c2 H3)))))))))) t1 t2 H))))). +(* COMMENTS +Initial nodes: 245 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubt/props.ma b/matita/matita/contribs/lambdadelta/basic_1/csubt/props.ma new file mode 100644 index 000000000..2efc87ccb --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/csubt/props.ma @@ -0,0 +1,28 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/csubt/defs.ma". + +theorem csubt_refl: + \forall (g: G).(\forall (c: C).(csubt g c c)) +\def + \lambda (g: G).(\lambda (c: C).(C_ind (\lambda (c0: C).(csubt g c0 c0)) +(\lambda (n: nat).(csubt_sort g n)) (\lambda (c0: C).(\lambda (H: (csubt g c0 +c0)).(\lambda (k: K).(\lambda (t: T).(csubt_head g c0 c0 H k t))))) c)). +(* COMMENTS +Initial nodes: 53 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubt/ty3.ma b/matita/matita/contribs/lambdadelta/basic_1/csubt/ty3.ma new file mode 100644 index 000000000..e6199ac15 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/csubt/ty3.ma @@ -0,0 +1,102 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/csubt/pc3.ma". + +include "Basic-1/csubt/props.ma". + +theorem csubt_ty3: + \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c1 +t1 t2) \to (\forall (c2: C).((csubt g c1 c2) \to (ty3 g c2 t1 t2))))))) +\def + \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda +(H: (ty3 g c1 t1 t2)).(ty3_ind g (\lambda (c: C).(\lambda (t: T).(\lambda +(t0: T).(\forall (c2: C).((csubt g c c2) \to (ty3 g c2 t t0)))))) (\lambda +(c: C).(\lambda (t0: T).(\lambda (t: T).(\lambda (_: (ty3 g c t0 t)).(\lambda +(H1: ((\forall (c2: C).((csubt g c c2) \to (ty3 g c2 t0 t))))).(\lambda (u: +T).(\lambda (t3: T).(\lambda (_: (ty3 g c u t3)).(\lambda (H3: ((\forall (c2: +C).((csubt g c c2) \to (ty3 g c2 u t3))))).(\lambda (H4: (pc3 c t3 +t0)).(\lambda (c2: C).(\lambda (H5: (csubt g c c2)).(ty3_conv g c2 t0 t (H1 +c2 H5) u t3 (H3 c2 H5) (csubt_pc3 g c t3 t0 H4 c2 H5)))))))))))))) (\lambda +(c: C).(\lambda (m: nat).(\lambda (c2: C).(\lambda (_: (csubt g c +c2)).(ty3_sort g c2 m))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d: +C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind Abbr) u))).(\lambda +(t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2: ((\forall (c2: C).((csubt g +d c2) \to (ty3 g c2 u t))))).(\lambda (c2: C).(\lambda (H3: (csubt g c +c2)).(let H4 \def (csubt_getl_abbr g c d u n H0 c2 H3) in (ex2_ind C (\lambda +(d2: C).(csubt g d d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) +u))) (ty3 g c2 (TLRef n) (lift (S n) O t)) (\lambda (x: C).(\lambda (H5: +(csubt g d x)).(\lambda (H6: (getl n c2 (CHead x (Bind Abbr) u))).(ty3_abbr g +n c2 x u H6 t (H2 x H5))))) H4)))))))))))) (\lambda (n: nat).(\lambda (c: +C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind +Abst) u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2: +((\forall (c2: C).((csubt g d c2) \to (ty3 g c2 u t))))).(\lambda (c2: +C).(\lambda (H3: (csubt g c c2)).(let H4 \def (csubt_getl_abst g c d u n H0 +c2 H3) in (or_ind (ex2 C (\lambda (d2: C).(csubt g d d2)) (\lambda (d2: +C).(getl n c2 (CHead d2 (Bind Abst) u)))) (ex4_2 C T (\lambda (d2: +C).(\lambda (_: T).(csubt g d d2))) (\lambda (d2: C).(\lambda (u0: T).(getl n +c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g d u0 +u))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 u)))) (ty3 g c2 (TLRef n) +(lift (S n) O u)) (\lambda (H5: (ex2 C (\lambda (d2: C).(csubt g d d2)) +(\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) u))))).(ex2_ind C (\lambda +(d2: C).(csubt g d d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) +u))) (ty3 g c2 (TLRef n) (lift (S n) O u)) (\lambda (x: C).(\lambda (H6: +(csubt g d x)).(\lambda (H7: (getl n c2 (CHead x (Bind Abst) u))).(ty3_abst g +n c2 x u H7 t (H2 x H6))))) H5)) (\lambda (H5: (ex4_2 C T (\lambda (d2: +C).(\lambda (_: T).(csubt g d d2))) (\lambda (d2: C).(\lambda (u0: T).(getl n +c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g d u0 +u))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 u))))).(ex4_2_ind C T +(\lambda (d2: C).(\lambda (_: T).(csubt g d d2))) (\lambda (d2: C).(\lambda +(u0: T).(getl n c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: +T).(ty3 g d u0 u))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 u))) (ty3 +g c2 (TLRef n) (lift (S n) O u)) (\lambda (x0: C).(\lambda (x1: T).(\lambda +(_: (csubt g d x0)).(\lambda (H7: (getl n c2 (CHead x0 (Bind Abbr) +x1))).(\lambda (_: (ty3 g d x1 u)).(\lambda (H9: (ty3 g x0 x1 u)).(ty3_abbr g +n c2 x0 x1 H7 u H9))))))) H5)) H4)))))))))))) (\lambda (c: C).(\lambda (u: +T).(\lambda (t: T).(\lambda (_: (ty3 g c u t)).(\lambda (H1: ((\forall (c2: +C).((csubt g c c2) \to (ty3 g c2 u t))))).(\lambda (b: B).(\lambda (t0: +T).(\lambda (t3: T).(\lambda (_: (ty3 g (CHead c (Bind b) u) t0 t3)).(\lambda +(H3: ((\forall (c2: C).((csubt g (CHead c (Bind b) u) c2) \to (ty3 g c2 t0 +t3))))).(\lambda (c2: C).(\lambda (H4: (csubt g c c2)).(ty3_bind g c2 u t (H1 +c2 H4) b t0 t3 (H3 (CHead c2 (Bind b) u) (csubt_head g c c2 H4 (Bind b) +u))))))))))))))) (\lambda (c: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: +(ty3 g c w u)).(\lambda (H1: ((\forall (c2: C).((csubt g c c2) \to (ty3 g c2 +w u))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c v (THead (Bind +Abst) u t))).(\lambda (H3: ((\forall (c2: C).((csubt g c c2) \to (ty3 g c2 v +(THead (Bind Abst) u t)))))).(\lambda (c2: C).(\lambda (H4: (csubt g c +c2)).(ty3_appl g c2 w u (H1 c2 H4) v t (H3 c2 H4))))))))))))) (\lambda (c: +C).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (ty3 g c t0 t3)).(\lambda +(H1: ((\forall (c2: C).((csubt g c c2) \to (ty3 g c2 t0 t3))))).(\lambda (t4: +T).(\lambda (_: (ty3 g c t3 t4)).(\lambda (H3: ((\forall (c2: C).((csubt g c +c2) \to (ty3 g c2 t3 t4))))).(\lambda (c2: C).(\lambda (H4: (csubt g c +c2)).(ty3_cast g c2 t0 t3 (H1 c2 H4) t4 (H3 c2 H4)))))))))))) c1 t1 t2 H))))). +(* COMMENTS +Initial nodes: 1325 +END *) + +theorem csubt_ty3_ld: + \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (v: T).((ty3 g c u +v) \to (\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c (Bind Abst) v) t1 +t2) \to (ty3 g (CHead c (Bind Abbr) u) t1 t2)))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (v: T).(\lambda (H: +(ty3 g c u v)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (ty3 g (CHead +c (Bind Abst) v) t1 t2)).(csubt_ty3 g (CHead c (Bind Abst) v) t1 t2 H0 (CHead +c (Bind Abbr) u) (csubt_abst g c c (csubt_refl g c) u v H H))))))))). +(* COMMENTS +Initial nodes: 91 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubv/clear.ma b/matita/matita/contribs/lambdadelta/basic_1/csubv/clear.ma new file mode 100644 index 000000000..5c54f5b9a --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/csubv/clear.ma @@ -0,0 +1,194 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/csubv/defs.ma". + +include "Basic-1/clear/fwd.ma". + +theorem csubv_clear_conf: + \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (b1: +B).(\forall (d1: C).(\forall (v1: T).((clear c1 (CHead d1 (Bind b1) v1)) \to +(ex2_3 B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 +d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda (v2: T).(clear c2 (CHead d2 +(Bind b2) v2)))))))))))) +\def + \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubv c1 c2)).(csubv_ind +(\lambda (c: C).(\lambda (c0: C).(\forall (b1: B).(\forall (d1: C).(\forall +(v1: T).((clear c (CHead d1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (_: +B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2: +B).(\lambda (d2: C).(\lambda (v2: T).(clear c0 (CHead d2 (Bind b2) +v2)))))))))))) (\lambda (n: nat).(\lambda (b1: B).(\lambda (d1: C).(\lambda +(v1: T).(\lambda (H0: (clear (CSort n) (CHead d1 (Bind b1) +v1))).(clear_gen_sort (CHead d1 (Bind b1) v1) n H0 (ex2_3 B C T (\lambda (_: +B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2: +B).(\lambda (d2: C).(\lambda (v2: T).(clear (CSort n) (CHead d2 (Bind b2) +v2)))))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csubv c3 +c4)).(\lambda (_: ((\forall (b1: B).(\forall (d1: C).(\forall (v1: T).((clear +c3 (CHead d1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (_: B).(\lambda (d2: +C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2: B).(\lambda (d2: +C).(\lambda (v2: T).(clear c4 (CHead d2 (Bind b2) v2)))))))))))).(\lambda +(v1: T).(\lambda (v2: T).(\lambda (b1: B).(\lambda (d1: C).(\lambda (v0: +T).(\lambda (H2: (clear (CHead c3 (Bind Void) v1) (CHead d1 (Bind b1) +v0))).(let H3 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda +(_: C).C) with [(CSort _) \Rightarrow d1 | (CHead c _ _) \Rightarrow c])) +(CHead d1 (Bind b1) v0) (CHead c3 (Bind Void) v1) (clear_gen_bind Void c3 +(CHead d1 (Bind b1) v0) v1 H2)) in ((let H4 \def (f_equal C B (\lambda (e: +C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow b1 | +(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind +b) \Rightarrow b | (Flat _) \Rightarrow b1])])) (CHead d1 (Bind b1) v0) +(CHead c3 (Bind Void) v1) (clear_gen_bind Void c3 (CHead d1 (Bind b1) v0) v1 +H2)) in ((let H5 \def (f_equal C T (\lambda (e: C).(match e in C return +(\lambda (_: C).T) with [(CSort _) \Rightarrow v0 | (CHead _ _ t) \Rightarrow +t])) (CHead d1 (Bind b1) v0) (CHead c3 (Bind Void) v1) (clear_gen_bind Void +c3 (CHead d1 (Bind b1) v0) v1 H2)) in (\lambda (_: (eq B b1 Void)).(\lambda +(H7: (eq C d1 c3)).(eq_ind_r C c3 (\lambda (c: C).(ex2_3 B C T (\lambda (_: +B).(\lambda (d2: C).(\lambda (_: T).(csubv c d2)))) (\lambda (b2: B).(\lambda +(d2: C).(\lambda (v3: T).(clear (CHead c4 (Bind Void) v2) (CHead d2 (Bind b2) +v3))))))) (ex2_3_intro B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: +T).(csubv c3 d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda (v3: T).(clear +(CHead c4 (Bind Void) v2) (CHead d2 (Bind b2) v3))))) Void c4 v2 H0 +(clear_bind Void c4 v2)) d1 H7)))) H4)) H3)))))))))))) (\lambda (c3: +C).(\lambda (c4: C).(\lambda (H0: (csubv c3 c4)).(\lambda (_: ((\forall (b1: +B).(\forall (d1: C).(\forall (v1: T).((clear c3 (CHead d1 (Bind b1) v1)) \to +(ex2_3 B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 +d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda (v2: T).(clear c4 (CHead d2 +(Bind b2) v2)))))))))))).(\lambda (b1: B).(\lambda (_: (not (eq B b1 +Void))).(\lambda (b2: B).(\lambda (v1: T).(\lambda (v2: T).(\lambda (b0: +B).(\lambda (d1: C).(\lambda (v0: T).(\lambda (H3: (clear (CHead c3 (Bind b1) +v1) (CHead d1 (Bind b0) v0))).(let H4 \def (f_equal C C (\lambda (e: +C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d1 | +(CHead c _ _) \Rightarrow c])) (CHead d1 (Bind b0) v0) (CHead c3 (Bind b1) +v1) (clear_gen_bind b1 c3 (CHead d1 (Bind b0) v0) v1 H3)) in ((let H5 \def +(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with +[(CSort _) \Rightarrow b0 | (CHead _ k _) \Rightarrow (match k in K return +(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow +b0])])) (CHead d1 (Bind b0) v0) (CHead c3 (Bind b1) v1) (clear_gen_bind b1 c3 +(CHead d1 (Bind b0) v0) v1 H3)) in ((let H6 \def (f_equal C T (\lambda (e: +C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow v0 | +(CHead _ _ t) \Rightarrow t])) (CHead d1 (Bind b0) v0) (CHead c3 (Bind b1) +v1) (clear_gen_bind b1 c3 (CHead d1 (Bind b0) v0) v1 H3)) in (\lambda (_: (eq +B b0 b1)).(\lambda (H8: (eq C d1 c3)).(eq_ind_r C c3 (\lambda (c: C).(ex2_3 B +C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv c d2)))) (\lambda +(b3: B).(\lambda (d2: C).(\lambda (v3: T).(clear (CHead c4 (Bind b2) v2) +(CHead d2 (Bind b3) v3))))))) (ex2_3_intro B C T (\lambda (_: B).(\lambda +(d2: C).(\lambda (_: T).(csubv c3 d2)))) (\lambda (b3: B).(\lambda (d2: +C).(\lambda (v3: T).(clear (CHead c4 (Bind b2) v2) (CHead d2 (Bind b3) +v3))))) b2 c4 v2 H0 (clear_bind b2 c4 v2)) d1 H8)))) H5)) H4))))))))))))))) +(\lambda (c3: C).(\lambda (c4: C).(\lambda (_: (csubv c3 c4)).(\lambda (H1: +((\forall (b1: B).(\forall (d1: C).(\forall (v1: T).((clear c3 (CHead d1 +(Bind b1) v1)) \to (ex2_3 B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: +T).(csubv d1 d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda (v2: T).(clear +c4 (CHead d2 (Bind b2) v2)))))))))))).(\lambda (f1: F).(\lambda (f2: +F).(\lambda (v1: T).(\lambda (v2: T).(\lambda (b1: B).(\lambda (d1: +C).(\lambda (v0: T).(\lambda (H2: (clear (CHead c3 (Flat f1) v1) (CHead d1 +(Bind b1) v0))).(let H_x \def (H1 b1 d1 v0 (clear_gen_flat f1 c3 (CHead d1 +(Bind b1) v0) v1 H2)) in (let H3 \def H_x in (ex2_3_ind B C T (\lambda (_: +B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2: +B).(\lambda (d2: C).(\lambda (v3: T).(clear c4 (CHead d2 (Bind b2) v3))))) +(ex2_3 B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 +d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda (v3: T).(clear (CHead c4 +(Flat f2) v2) (CHead d2 (Bind b2) v3)))))) (\lambda (x0: B).(\lambda (x1: +C).(\lambda (x2: T).(\lambda (H4: (csubv d1 x1)).(\lambda (H5: (clear c4 +(CHead x1 (Bind x0) x2))).(ex2_3_intro B C T (\lambda (_: B).(\lambda (d2: +C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2: B).(\lambda (d2: +C).(\lambda (v3: T).(clear (CHead c4 (Flat f2) v2) (CHead d2 (Bind b2) +v3))))) x0 x1 x2 H4 (clear_flat c4 (CHead x1 (Bind x0) x2) H5 f2 v2))))))) +H3))))))))))))))) c1 c2 H))). +(* COMMENTS +Initial nodes: 1357 +END *) + +theorem csubv_clear_conf_void: + \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (d1: +C).(\forall (v1: T).((clear c1 (CHead d1 (Bind Void) v1)) \to (ex2_2 C T +(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2))) (\lambda (d2: C).(\lambda +(v2: T).(clear c2 (CHead d2 (Bind Void) v2)))))))))) +\def + \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubv c1 c2)).(csubv_ind +(\lambda (c: C).(\lambda (c0: C).(\forall (d1: C).(\forall (v1: T).((clear c +(CHead d1 (Bind Void) v1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (_: +T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(clear c0 (CHead d2 +(Bind Void) v2)))))))))) (\lambda (n: nat).(\lambda (d1: C).(\lambda (v1: +T).(\lambda (H0: (clear (CSort n) (CHead d1 (Bind Void) v1))).(clear_gen_sort +(CHead d1 (Bind Void) v1) n H0 (ex2_2 C T (\lambda (d2: C).(\lambda (_: +T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(clear (CSort n) (CHead +d2 (Bind Void) v2)))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: +(csubv c3 c4)).(\lambda (_: ((\forall (d1: C).(\forall (v1: T).((clear c3 +(CHead d1 (Bind Void) v1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (_: +T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(clear c4 (CHead d2 +(Bind Void) v2)))))))))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (d1: +C).(\lambda (v0: T).(\lambda (H2: (clear (CHead c3 (Bind Void) v1) (CHead d1 +(Bind Void) v0))).(let H3 \def (f_equal C C (\lambda (e: C).(match e in C +return (\lambda (_: C).C) with [(CSort _) \Rightarrow d1 | (CHead c _ _) +\Rightarrow c])) (CHead d1 (Bind Void) v0) (CHead c3 (Bind Void) v1) +(clear_gen_bind Void c3 (CHead d1 (Bind Void) v0) v1 H2)) in ((let H4 \def +(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with +[(CSort _) \Rightarrow v0 | (CHead _ _ t) \Rightarrow t])) (CHead d1 (Bind +Void) v0) (CHead c3 (Bind Void) v1) (clear_gen_bind Void c3 (CHead d1 (Bind +Void) v0) v1 H2)) in (\lambda (H5: (eq C d1 c3)).(eq_ind_r C c3 (\lambda (c: +C).(ex2_2 C T (\lambda (d2: C).(\lambda (_: T).(csubv c d2))) (\lambda (d2: +C).(\lambda (v3: T).(clear (CHead c4 (Bind Void) v2) (CHead d2 (Bind Void) +v3)))))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csubv c3 d2))) +(\lambda (d2: C).(\lambda (v3: T).(clear (CHead c4 (Bind Void) v2) (CHead d2 +(Bind Void) v3)))) c4 v2 H0 (clear_bind Void c4 v2)) d1 H5))) H3))))))))))) +(\lambda (c3: C).(\lambda (c4: C).(\lambda (_: (csubv c3 c4)).(\lambda (_: +((\forall (d1: C).(\forall (v1: T).((clear c3 (CHead d1 (Bind Void) v1)) \to +(ex2_2 C T (\lambda (d2: C).(\lambda (_: T).(csubv d1 d2))) (\lambda (d2: +C).(\lambda (v2: T).(clear c4 (CHead d2 (Bind Void) v2)))))))))).(\lambda +(b1: B).(\lambda (H2: (not (eq B b1 Void))).(\lambda (b2: B).(\lambda (v1: +T).(\lambda (v2: T).(\lambda (d1: C).(\lambda (v0: T).(\lambda (H3: (clear +(CHead c3 (Bind b1) v1) (CHead d1 (Bind Void) v0))).(let H4 \def (f_equal C C +(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) +\Rightarrow d1 | (CHead c _ _) \Rightarrow c])) (CHead d1 (Bind Void) v0) +(CHead c3 (Bind b1) v1) (clear_gen_bind b1 c3 (CHead d1 (Bind Void) v0) v1 +H3)) in ((let H5 \def (f_equal C B (\lambda (e: C).(match e in C return +(\lambda (_: C).B) with [(CSort _) \Rightarrow Void | (CHead _ k _) +\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) +\Rightarrow b | (Flat _) \Rightarrow Void])])) (CHead d1 (Bind Void) v0) +(CHead c3 (Bind b1) v1) (clear_gen_bind b1 c3 (CHead d1 (Bind Void) v0) v1 +H3)) in ((let H6 \def (f_equal C T (\lambda (e: C).(match e in C return +(\lambda (_: C).T) with [(CSort _) \Rightarrow v0 | (CHead _ _ t) \Rightarrow +t])) (CHead d1 (Bind Void) v0) (CHead c3 (Bind b1) v1) (clear_gen_bind b1 c3 +(CHead d1 (Bind Void) v0) v1 H3)) in (\lambda (H7: (eq B Void b1)).(\lambda +(H8: (eq C d1 c3)).(eq_ind_r C c3 (\lambda (c: C).(ex2_2 C T (\lambda (d2: +C).(\lambda (_: T).(csubv c d2))) (\lambda (d2: C).(\lambda (v3: T).(clear +(CHead c4 (Bind b2) v2) (CHead d2 (Bind Void) v3)))))) (let H9 \def (eq_ind_r +B b1 (\lambda (b: B).(not (eq B b Void))) H2 Void H7) in (let H10 \def (match +(H9 (refl_equal B Void)) in False return (\lambda (_: False).(ex2_2 C T +(\lambda (d2: C).(\lambda (_: T).(csubv c3 d2))) (\lambda (d2: C).(\lambda +(v3: T).(clear (CHead c4 (Bind b2) v2) (CHead d2 (Bind Void) v3)))))) with +[]) in H10)) d1 H8)))) H5)) H4)))))))))))))) (\lambda (c3: C).(\lambda (c4: +C).(\lambda (_: (csubv c3 c4)).(\lambda (H1: ((\forall (d1: C).(\forall (v1: +T).((clear c3 (CHead d1 (Bind Void) v1)) \to (ex2_2 C T (\lambda (d2: +C).(\lambda (_: T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(clear +c4 (CHead d2 (Bind Void) v2)))))))))).(\lambda (f1: F).(\lambda (f2: +F).(\lambda (v1: T).(\lambda (v2: T).(\lambda (d1: C).(\lambda (v0: +T).(\lambda (H2: (clear (CHead c3 (Flat f1) v1) (CHead d1 (Bind Void) +v0))).(let H_x \def (H1 d1 v0 (clear_gen_flat f1 c3 (CHead d1 (Bind Void) v0) +v1 H2)) in (let H3 \def H_x in (ex2_2_ind C T (\lambda (d2: C).(\lambda (_: +T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v3: T).(clear c4 (CHead d2 +(Bind Void) v3)))) (ex2_2 C T (\lambda (d2: C).(\lambda (_: T).(csubv d1 +d2))) (\lambda (d2: C).(\lambda (v3: T).(clear (CHead c4 (Flat f2) v2) (CHead +d2 (Bind Void) v3))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H4: (csubv +d1 x0)).(\lambda (H5: (clear c4 (CHead x0 (Bind Void) x1))).(ex2_2_intro C T +(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2))) (\lambda (d2: C).(\lambda +(v3: T).(clear (CHead c4 (Flat f2) v2) (CHead d2 (Bind Void) v3)))) x0 x1 H4 +(clear_flat c4 (CHead x0 (Bind Void) x1) H5 f2 v2)))))) H3)))))))))))))) c1 +c2 H))). +(* COMMENTS +Initial nodes: 1205 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubv/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/csubv/defs.ma new file mode 100644 index 000000000..6ffb63608 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/csubv/defs.ma @@ -0,0 +1,30 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/C/defs.ma". + +inductive csubv: C \to (C \to Prop) \def +| csubv_sort: \forall (n: nat).(csubv (CSort n) (CSort n)) +| csubv_void: \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall +(v1: T).(\forall (v2: T).(csubv (CHead c1 (Bind Void) v1) (CHead c2 (Bind +Void) v2)))))) +| csubv_bind: \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall +(b1: B).((not (eq B b1 Void)) \to (\forall (b2: B).(\forall (v1: T).(\forall +(v2: T).(csubv (CHead c1 (Bind b1) v1) (CHead c2 (Bind b2) v2))))))))) +| csubv_flat: \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall +(f1: F).(\forall (f2: F).(\forall (v1: T).(\forall (v2: T).(csubv (CHead c1 +(Flat f1) v1) (CHead c2 (Flat f2) v2)))))))). + diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubv/drop.ma b/matita/matita/contribs/lambdadelta/basic_1/csubv/drop.ma new file mode 100644 index 000000000..007670dbb --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/csubv/drop.ma @@ -0,0 +1,115 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/csubv/props.ma". + +include "Basic-1/drop/fwd.ma". + +theorem csubv_drop_conf: + \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (e1: +C).(\forall (h: nat).((drop h O c1 e1) \to (ex2 C (\lambda (e2: C).(csubv e1 +e2)) (\lambda (e2: C).(drop h O c2 e2)))))))) +\def + \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubv c1 c2)).(csubv_ind +(\lambda (c: C).(\lambda (c0: C).(\forall (e1: C).(\forall (h: nat).((drop h +O c e1) \to (ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop h O +c0 e2)))))))) (\lambda (n: nat).(\lambda (e1: C).(\lambda (h: nat).(\lambda +(H0: (drop h O (CSort n) e1)).(and3_ind (eq C e1 (CSort n)) (eq nat h O) (eq +nat O O) (ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop h O +(CSort n) e2))) (\lambda (H1: (eq C e1 (CSort n))).(\lambda (H2: (eq nat h +O)).(\lambda (_: (eq nat O O)).(eq_ind_r nat O (\lambda (n0: nat).(ex2 C +(\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop n0 O (CSort n) e2)))) +(eq_ind_r C (CSort n) (\lambda (c: C).(ex2 C (\lambda (e2: C).(csubv c e2)) +(\lambda (e2: C).(drop O O (CSort n) e2)))) (ex_intro2 C (\lambda (e2: +C).(csubv (CSort n) e2)) (\lambda (e2: C).(drop O O (CSort n) e2)) (CSort n) +(csubv_refl (CSort n)) (drop_refl (CSort n))) e1 H1) h H2)))) (drop_gen_sort +n h O e1 H0)))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csubv c3 +c4)).(\lambda (H1: ((\forall (e1: C).(\forall (h: nat).((drop h O c3 e1) \to +(ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop h O c4 +e2)))))))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (e1: C).(\lambda (h: +nat).(\lambda (H2: (drop h O (CHead c3 (Bind Void) v1) e1)).(nat_ind (\lambda +(n: nat).((drop n O (CHead c3 (Bind Void) v1) e1) \to (ex2 C (\lambda (e2: +C).(csubv e1 e2)) (\lambda (e2: C).(drop n O (CHead c4 (Bind Void) v2) +e2))))) (\lambda (H3: (drop O O (CHead c3 (Bind Void) v1) e1)).(eq_ind C +(CHead c3 (Bind Void) v1) (\lambda (c: C).(ex2 C (\lambda (e2: C).(csubv c +e2)) (\lambda (e2: C).(drop O O (CHead c4 (Bind Void) v2) e2)))) (ex_intro2 C +(\lambda (e2: C).(csubv (CHead c3 (Bind Void) v1) e2)) (\lambda (e2: C).(drop +O O (CHead c4 (Bind Void) v2) e2)) (CHead c4 (Bind Void) v2) (csubv_bind_same +c3 c4 H0 Void v1 v2) (drop_refl (CHead c4 (Bind Void) v2))) e1 (drop_gen_refl +(CHead c3 (Bind Void) v1) e1 H3))) (\lambda (h0: nat).(\lambda (_: (((drop h0 +O (CHead c3 (Bind Void) v1) e1) \to (ex2 C (\lambda (e2: C).(csubv e1 e2)) +(\lambda (e2: C).(drop h0 O (CHead c4 (Bind Void) v2) e2)))))).(\lambda (H3: +(drop (S h0) O (CHead c3 (Bind Void) v1) e1)).(let H_x \def (H1 e1 (r (Bind +Void) h0) (drop_gen_drop (Bind Void) c3 e1 v1 h0 H3)) in (let H4 \def H_x in +(ex2_ind C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop h0 O c4 +e2)) (ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop (S h0) O +(CHead c4 (Bind Void) v2) e2))) (\lambda (x: C).(\lambda (H5: (csubv e1 +x)).(\lambda (H6: (drop h0 O c4 x)).(ex_intro2 C (\lambda (e2: C).(csubv e1 +e2)) (\lambda (e2: C).(drop (S h0) O (CHead c4 (Bind Void) v2) e2)) x H5 +(drop_drop (Bind Void) h0 c4 x H6 v2))))) H4)))))) h H2)))))))))) (\lambda +(c3: C).(\lambda (c4: C).(\lambda (H0: (csubv c3 c4)).(\lambda (H1: ((\forall +(e1: C).(\forall (h: nat).((drop h O c3 e1) \to (ex2 C (\lambda (e2: +C).(csubv e1 e2)) (\lambda (e2: C).(drop h O c4 e2)))))))).(\lambda (b1: +B).(\lambda (H2: (not (eq B b1 Void))).(\lambda (b2: B).(\lambda (v1: +T).(\lambda (v2: T).(\lambda (e1: C).(\lambda (h: nat).(\lambda (H3: (drop h +O (CHead c3 (Bind b1) v1) e1)).(nat_ind (\lambda (n: nat).((drop n O (CHead +c3 (Bind b1) v1) e1) \to (ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: +C).(drop n O (CHead c4 (Bind b2) v2) e2))))) (\lambda (H4: (drop O O (CHead +c3 (Bind b1) v1) e1)).(eq_ind C (CHead c3 (Bind b1) v1) (\lambda (c: C).(ex2 +C (\lambda (e2: C).(csubv c e2)) (\lambda (e2: C).(drop O O (CHead c4 (Bind +b2) v2) e2)))) (ex_intro2 C (\lambda (e2: C).(csubv (CHead c3 (Bind b1) v1) +e2)) (\lambda (e2: C).(drop O O (CHead c4 (Bind b2) v2) e2)) (CHead c4 (Bind +b2) v2) (csubv_bind c3 c4 H0 b1 H2 b2 v1 v2) (drop_refl (CHead c4 (Bind b2) +v2))) e1 (drop_gen_refl (CHead c3 (Bind b1) v1) e1 H4))) (\lambda (h0: +nat).(\lambda (_: (((drop h0 O (CHead c3 (Bind b1) v1) e1) \to (ex2 C +(\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop h0 O (CHead c4 (Bind +b2) v2) e2)))))).(\lambda (H4: (drop (S h0) O (CHead c3 (Bind b1) v1) +e1)).(let H_x \def (H1 e1 (r (Bind b1) h0) (drop_gen_drop (Bind b1) c3 e1 v1 +h0 H4)) in (let H5 \def H_x in (ex2_ind C (\lambda (e2: C).(csubv e1 e2)) +(\lambda (e2: C).(drop h0 O c4 e2)) (ex2 C (\lambda (e2: C).(csubv e1 e2)) +(\lambda (e2: C).(drop (S h0) O (CHead c4 (Bind b2) v2) e2))) (\lambda (x: +C).(\lambda (H6: (csubv e1 x)).(\lambda (H7: (drop h0 O c4 x)).(ex_intro2 C +(\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop (S h0) O (CHead c4 +(Bind b2) v2) e2)) x H6 (drop_drop (Bind b2) h0 c4 x H7 v2))))) H5)))))) h +H3))))))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csubv c3 +c4)).(\lambda (H1: ((\forall (e1: C).(\forall (h: nat).((drop h O c3 e1) \to +(ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop h O c4 +e2)))))))).(\lambda (f1: F).(\lambda (f2: F).(\lambda (v1: T).(\lambda (v2: +T).(\lambda (e1: C).(\lambda (h: nat).(\lambda (H2: (drop h O (CHead c3 (Flat +f1) v1) e1)).(nat_ind (\lambda (n: nat).((drop n O (CHead c3 (Flat f1) v1) +e1) \to (ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop n O +(CHead c4 (Flat f2) v2) e2))))) (\lambda (H3: (drop O O (CHead c3 (Flat f1) +v1) e1)).(eq_ind C (CHead c3 (Flat f1) v1) (\lambda (c: C).(ex2 C (\lambda +(e2: C).(csubv c e2)) (\lambda (e2: C).(drop O O (CHead c4 (Flat f2) v2) +e2)))) (ex_intro2 C (\lambda (e2: C).(csubv (CHead c3 (Flat f1) v1) e2)) +(\lambda (e2: C).(drop O O (CHead c4 (Flat f2) v2) e2)) (CHead c4 (Flat f2) +v2) (csubv_flat c3 c4 H0 f1 f2 v1 v2) (drop_refl (CHead c4 (Flat f2) v2))) e1 +(drop_gen_refl (CHead c3 (Flat f1) v1) e1 H3))) (\lambda (h0: nat).(\lambda +(_: (((drop h0 O (CHead c3 (Flat f1) v1) e1) \to (ex2 C (\lambda (e2: +C).(csubv e1 e2)) (\lambda (e2: C).(drop h0 O (CHead c4 (Flat f2) v2) +e2)))))).(\lambda (H3: (drop (S h0) O (CHead c3 (Flat f1) v1) e1)).(let H_x +\def (H1 e1 (r (Flat f1) h0) (drop_gen_drop (Flat f1) c3 e1 v1 h0 H3)) in +(let H4 \def H_x in (ex2_ind C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: +C).(drop (S h0) O c4 e2)) (ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda +(e2: C).(drop (S h0) O (CHead c4 (Flat f2) v2) e2))) (\lambda (x: C).(\lambda +(H5: (csubv e1 x)).(\lambda (H6: (drop (S h0) O c4 x)).(ex_intro2 C (\lambda +(e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop (S h0) O (CHead c4 (Flat f2) +v2) e2)) x H5 (drop_drop (Flat f2) h0 c4 x H6 v2))))) H4)))))) h +H2)))))))))))) c1 c2 H))). +(* COMMENTS +Initial nodes: 1897 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubv/getl.ma b/matita/matita/contribs/lambdadelta/basic_1/csubv/getl.ma new file mode 100644 index 000000000..bbba95084 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/csubv/getl.ma @@ -0,0 +1,90 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/csubv/clear.ma". + +include "Basic-1/csubv/drop.ma". + +include "Basic-1/getl/fwd.ma". + +theorem csubv_getl_conf: + \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (b1: +B).(\forall (d1: C).(\forall (v1: T).(\forall (i: nat).((getl i c1 (CHead d1 +(Bind b1) v1)) \to (ex2_3 B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: +T).(csubv d1 d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda (v2: T).(getl +i c2 (CHead d2 (Bind b2) v2))))))))))))) +\def + \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubv c1 c2)).(\lambda (b1: +B).(\lambda (d1: C).(\lambda (v1: T).(\lambda (i: nat).(\lambda (H0: (getl i +c1 (CHead d1 (Bind b1) v1))).(let H1 \def (getl_gen_all c1 (CHead d1 (Bind +b1) v1) i H0) in (ex2_ind C (\lambda (e: C).(drop i O c1 e)) (\lambda (e: +C).(clear e (CHead d1 (Bind b1) v1))) (ex2_3 B C T (\lambda (_: B).(\lambda +(d2: C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2: B).(\lambda (d2: +C).(\lambda (v2: T).(getl i c2 (CHead d2 (Bind b2) v2)))))) (\lambda (x: +C).(\lambda (H2: (drop i O c1 x)).(\lambda (H3: (clear x (CHead d1 (Bind b1) +v1))).(let H_x \def (csubv_drop_conf c1 c2 H x i H2) in (let H4 \def H_x in +(ex2_ind C (\lambda (e2: C).(csubv x e2)) (\lambda (e2: C).(drop i O c2 e2)) +(ex2_3 B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 +d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda (v2: T).(getl i c2 (CHead +d2 (Bind b2) v2)))))) (\lambda (x0: C).(\lambda (H5: (csubv x x0)).(\lambda +(H6: (drop i O c2 x0)).(let H_x0 \def (csubv_clear_conf x x0 H5 b1 d1 v1 H3) +in (let H7 \def H_x0 in (ex2_3_ind B C T (\lambda (_: B).(\lambda (d2: +C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2: B).(\lambda (d2: +C).(\lambda (v2: T).(clear x0 (CHead d2 (Bind b2) v2))))) (ex2_3 B C T +(\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2)))) (\lambda +(b2: B).(\lambda (d2: C).(\lambda (v2: T).(getl i c2 (CHead d2 (Bind b2) +v2)))))) (\lambda (x1: B).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H8: +(csubv d1 x2)).(\lambda (H9: (clear x0 (CHead x2 (Bind x1) x3))).(ex2_3_intro +B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2)))) +(\lambda (b2: B).(\lambda (d2: C).(\lambda (v2: T).(getl i c2 (CHead d2 (Bind +b2) v2))))) x1 x2 x3 H8 (getl_intro i c2 (CHead x2 (Bind x1) x3) x0 H6 +H9))))))) H7)))))) H4)))))) H1))))))))). +(* COMMENTS +Initial nodes: 455 +END *) + +theorem csubv_getl_conf_void: + \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (d1: +C).(\forall (v1: T).(\forall (i: nat).((getl i c1 (CHead d1 (Bind Void) v1)) +\to (ex2_2 C T (\lambda (d2: C).(\lambda (_: T).(csubv d1 d2))) (\lambda (d2: +C).(\lambda (v2: T).(getl i c2 (CHead d2 (Bind Void) v2))))))))))) +\def + \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubv c1 c2)).(\lambda (d1: +C).(\lambda (v1: T).(\lambda (i: nat).(\lambda (H0: (getl i c1 (CHead d1 +(Bind Void) v1))).(let H1 \def (getl_gen_all c1 (CHead d1 (Bind Void) v1) i +H0) in (ex2_ind C (\lambda (e: C).(drop i O c1 e)) (\lambda (e: C).(clear e +(CHead d1 (Bind Void) v1))) (ex2_2 C T (\lambda (d2: C).(\lambda (_: +T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(getl i c2 (CHead d2 +(Bind Void) v2))))) (\lambda (x: C).(\lambda (H2: (drop i O c1 x)).(\lambda +(H3: (clear x (CHead d1 (Bind Void) v1))).(let H_x \def (csubv_drop_conf c1 +c2 H x i H2) in (let H4 \def H_x in (ex2_ind C (\lambda (e2: C).(csubv x e2)) +(\lambda (e2: C).(drop i O c2 e2)) (ex2_2 C T (\lambda (d2: C).(\lambda (_: +T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(getl i c2 (CHead d2 +(Bind Void) v2))))) (\lambda (x0: C).(\lambda (H5: (csubv x x0)).(\lambda +(H6: (drop i O c2 x0)).(let H_x0 \def (csubv_clear_conf_void x x0 H5 d1 v1 +H3) in (let H7 \def H_x0 in (ex2_2_ind C T (\lambda (d2: C).(\lambda (_: +T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(clear x0 (CHead d2 +(Bind Void) v2)))) (ex2_2 C T (\lambda (d2: C).(\lambda (_: T).(csubv d1 +d2))) (\lambda (d2: C).(\lambda (v2: T).(getl i c2 (CHead d2 (Bind Void) +v2))))) (\lambda (x1: C).(\lambda (x2: T).(\lambda (H8: (csubv d1 +x1)).(\lambda (H9: (clear x0 (CHead x1 (Bind Void) x2))).(ex2_2_intro C T +(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2))) (\lambda (d2: C).(\lambda +(v2: T).(getl i c2 (CHead d2 (Bind Void) v2)))) x1 x2 H8 (getl_intro i c2 +(CHead x1 (Bind Void) x2) x0 H6 H9)))))) H7)))))) H4)))))) H1)))))))). +(* COMMENTS +Initial nodes: 417 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubv/props.ma b/matita/matita/contribs/lambdadelta/basic_1/csubv/props.ma new file mode 100644 index 000000000..716922824 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/csubv/props.ma @@ -0,0 +1,48 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/csubv/defs.ma". + +include "Basic-1/T/props.ma". + +theorem csubv_bind_same: + \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (b: B).(\forall +(v1: T).(\forall (v2: T).(csubv (CHead c1 (Bind b) v1) (CHead c2 (Bind b) +v2))))))) +\def + \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubv c1 c2)).(\lambda (b: +B).(B_ind (\lambda (b0: B).(\forall (v1: T).(\forall (v2: T).(csubv (CHead c1 +(Bind b0) v1) (CHead c2 (Bind b0) v2))))) (\lambda (v1: T).(\lambda (v2: +T).(csubv_bind c1 c2 H Abbr (\lambda (H0: (eq B Abbr Void)).(not_abbr_void +H0)) Abbr v1 v2))) (\lambda (v1: T).(\lambda (v2: T).(csubv_bind c1 c2 H Abst +(sym_not_eq B Void Abst not_void_abst) Abst v1 v2))) (\lambda (v1: +T).(\lambda (v2: T).(csubv_void c1 c2 H v1 v2))) b)))). +(* COMMENTS +Initial nodes: 121 +END *) + +theorem csubv_refl: + \forall (c: C).(csubv c c) +\def + \lambda (c: C).(C_ind (\lambda (c0: C).(csubv c0 c0)) (\lambda (n: +nat).(csubv_sort n)) (\lambda (c0: C).(\lambda (H: (csubv c0 c0)).(\lambda +(k: K).(K_ind (\lambda (k0: K).(\forall (t: T).(csubv (CHead c0 k0 t) (CHead +c0 k0 t)))) (\lambda (b: B).(\lambda (t: T).(csubv_bind_same c0 c0 H b t t))) +(\lambda (f: F).(\lambda (t: T).(csubv_flat c0 c0 H f f t t))) k)))) c). +(* COMMENTS +Initial nodes: 93 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/definitions.ma b/matita/matita/contribs/lambdadelta/basic_1/definitions.ma new file mode 100644 index 000000000..5863b4243 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/definitions.ma @@ -0,0 +1,68 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/tlt/defs.ma". + +include "Basic-1/iso/defs.ma". + +include "Basic-1/clen/defs.ma". + +include "Basic-1/flt/defs.ma". + +include "Basic-1/app/defs.ma". + +include "Basic-1/cnt/defs.ma". + +include "Basic-1/cimp/defs.ma". + +include "Basic-1/csubv/defs.ma". + +include "Basic-1/subst/defs.ma". + +include "Basic-1/subst1/defs.ma". + +include "Basic-1/csubst1/defs.ma". + +include "Basic-1/fsubst0/defs.ma". + +include "Basic-1/next_plus/defs.ma". + +include "Basic-1/sty1/defs.ma". + +include "Basic-1/llt/defs.ma". + +include "Basic-1/aprem/defs.ma". + +include "Basic-1/ex0/defs.ma". + +include "Basic-1/wcpr0/defs.ma". + +include "Basic-1/csuba/defs.ma". + +include "Basic-1/nf2/defs.ma". + +include "Basic-1/ex2/defs.ma". + +include "Basic-1/csubc/defs.ma". + +include "Basic-1/pc1/defs.ma". + +include "Basic-1/ex1/defs.ma". + +include "Basic-1/csubt/defs.ma". + +include "Basic-1/wf3/defs.ma". + diff --git a/matita/matita/contribs/lambdadelta/basic_1/drop/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/drop/defs.ma new file mode 100644 index 000000000..987ae0ceb --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/drop/defs.ma @@ -0,0 +1,31 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/C/defs.ma". + +include "Basic-1/lift/defs.ma". + +include "Basic-1/r/defs.ma". + +inductive drop: nat \to (nat \to (C \to (C \to Prop))) \def +| drop_refl: \forall (c: C).(drop O O c c) +| drop_drop: \forall (k: K).(\forall (h: nat).(\forall (c: C).(\forall (e: +C).((drop (r k h) O c e) \to (\forall (u: T).(drop (S h) O (CHead c k u) +e)))))) +| drop_skip: \forall (k: K).(\forall (h: nat).(\forall (d: nat).(\forall (c: +C).(\forall (e: C).((drop h (r k d) c e) \to (\forall (u: T).(drop h (S d) +(CHead c k (lift h (r k d) u)) (CHead e k u)))))))). + diff --git a/matita/matita/contribs/lambdadelta/basic_1/drop/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/drop/fwd.ma new file mode 100644 index 000000000..48495c148 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/drop/fwd.ma @@ -0,0 +1,384 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/drop/defs.ma". + +theorem drop_gen_sort: + \forall (n: nat).(\forall (h: nat).(\forall (d: nat).(\forall (x: C).((drop +h d (CSort n) x) \to (and3 (eq C x (CSort n)) (eq nat h O) (eq nat d O)))))) +\def + \lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (x: +C).(\lambda (H: (drop h d (CSort n) x)).(insert_eq C (CSort n) (\lambda (c: +C).(drop h d c x)) (\lambda (c: C).(and3 (eq C x c) (eq nat h O) (eq nat d +O))) (\lambda (y: C).(\lambda (H0: (drop h d y x)).(drop_ind (\lambda (n0: +nat).(\lambda (n1: nat).(\lambda (c: C).(\lambda (c0: C).((eq C c (CSort n)) +\to (and3 (eq C c0 c) (eq nat n0 O) (eq nat n1 O))))))) (\lambda (c: +C).(\lambda (H1: (eq C c (CSort n))).(let H2 \def (f_equal C C (\lambda (e: +C).e) c (CSort n) H1) in (eq_ind_r C (CSort n) (\lambda (c0: C).(and3 (eq C +c0 c0) (eq nat O O) (eq nat O O))) (and3_intro (eq C (CSort n) (CSort n)) (eq +nat O O) (eq nat O O) (refl_equal C (CSort n)) (refl_equal nat O) (refl_equal +nat O)) c H2)))) (\lambda (k: K).(\lambda (h0: nat).(\lambda (c: C).(\lambda +(e: C).(\lambda (_: (drop (r k h0) O c e)).(\lambda (_: (((eq C c (CSort n)) +\to (and3 (eq C e c) (eq nat (r k h0) O) (eq nat O O))))).(\lambda (u: +T).(\lambda (H3: (eq C (CHead c k u) (CSort n))).(let H4 \def (eq_ind C +(CHead c k u) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) +with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I +(CSort n) H3) in (False_ind (and3 (eq C e (CHead c k u)) (eq nat (S h0) O) +(eq nat O O)) H4)))))))))) (\lambda (k: K).(\lambda (h0: nat).(\lambda (d0: +nat).(\lambda (c: C).(\lambda (e: C).(\lambda (_: (drop h0 (r k d0) c +e)).(\lambda (_: (((eq C c (CSort n)) \to (and3 (eq C e c) (eq nat h0 O) (eq +nat (r k d0) O))))).(\lambda (u: T).(\lambda (H3: (eq C (CHead c k (lift h0 +(r k d0) u)) (CSort n))).(let H4 \def (eq_ind C (CHead c k (lift h0 (r k d0) +u)) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort +_) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H3) in +(False_ind (and3 (eq C (CHead e k u) (CHead c k (lift h0 (r k d0) u))) (eq +nat h0 O) (eq nat (S d0) O)) H4))))))))))) h d y x H0))) H))))). +(* COMMENTS +Initial nodes: 595 +END *) + +theorem drop_gen_refl: + \forall (x: C).(\forall (e: C).((drop O O x e) \to (eq C x e))) +\def + \lambda (x: C).(\lambda (e: C).(\lambda (H: (drop O O x e)).(insert_eq nat O +(\lambda (n: nat).(drop n O x e)) (\lambda (_: nat).(eq C x e)) (\lambda (y: +nat).(\lambda (H0: (drop y O x e)).(insert_eq nat O (\lambda (n: nat).(drop y +n x e)) (\lambda (n: nat).((eq nat y n) \to (eq C x e))) (\lambda (y0: +nat).(\lambda (H1: (drop y y0 x e)).(drop_ind (\lambda (n: nat).(\lambda (n0: +nat).(\lambda (c: C).(\lambda (c0: C).((eq nat n0 O) \to ((eq nat n n0) \to +(eq C c c0))))))) (\lambda (c: C).(\lambda (_: (eq nat O O)).(\lambda (_: (eq +nat O O)).(refl_equal C c)))) (\lambda (k: K).(\lambda (h: nat).(\lambda (c: +C).(\lambda (e0: C).(\lambda (_: (drop (r k h) O c e0)).(\lambda (_: (((eq +nat O O) \to ((eq nat (r k h) O) \to (eq C c e0))))).(\lambda (u: T).(\lambda +(_: (eq nat O O)).(\lambda (H5: (eq nat (S h) O)).(let H6 \def (eq_ind nat (S +h) (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O +\Rightarrow False | (S _) \Rightarrow True])) I O H5) in (False_ind (eq C +(CHead c k u) e0) H6))))))))))) (\lambda (k: K).(\lambda (h: nat).(\lambda +(d: nat).(\lambda (c: C).(\lambda (e0: C).(\lambda (H2: (drop h (r k d) c +e0)).(\lambda (H3: (((eq nat (r k d) O) \to ((eq nat h (r k d)) \to (eq C c +e0))))).(\lambda (u: T).(\lambda (H4: (eq nat (S d) O)).(\lambda (H5: (eq nat +h (S d))).(let H6 \def (f_equal nat nat (\lambda (e1: nat).e1) h (S d) H5) in +(let H7 \def (eq_ind nat h (\lambda (n: nat).((eq nat (r k d) O) \to ((eq nat +n (r k d)) \to (eq C c e0)))) H3 (S d) H6) in (let H8 \def (eq_ind nat h +(\lambda (n: nat).(drop n (r k d) c e0)) H2 (S d) H6) in (eq_ind_r nat (S d) +(\lambda (n: nat).(eq C (CHead c k (lift n (r k d) u)) (CHead e0 k u))) (let +H9 \def (eq_ind nat (S d) (\lambda (ee: nat).(match ee in nat return (\lambda +(_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H4) +in (False_ind (eq C (CHead c k (lift (S d) (r k d) u)) (CHead e0 k u)) H9)) h +H6)))))))))))))) y y0 x e H1))) H0))) H))). +(* COMMENTS +Initial nodes: 561 +END *) + +theorem drop_gen_drop: + \forall (k: K).(\forall (c: C).(\forall (x: C).(\forall (u: T).(\forall (h: +nat).((drop (S h) O (CHead c k u) x) \to (drop (r k h) O c x)))))) +\def + \lambda (k: K).(\lambda (c: C).(\lambda (x: C).(\lambda (u: T).(\lambda (h: +nat).(\lambda (H: (drop (S h) O (CHead c k u) x)).(insert_eq C (CHead c k u) +(\lambda (c0: C).(drop (S h) O c0 x)) (\lambda (_: C).(drop (r k h) O c x)) +(\lambda (y: C).(\lambda (H0: (drop (S h) O y x)).(insert_eq nat O (\lambda +(n: nat).(drop (S h) n y x)) (\lambda (n: nat).((eq C y (CHead c k u)) \to +(drop (r k h) n c x))) (\lambda (y0: nat).(\lambda (H1: (drop (S h) y0 y +x)).(insert_eq nat (S h) (\lambda (n: nat).(drop n y0 y x)) (\lambda (_: +nat).((eq nat y0 O) \to ((eq C y (CHead c k u)) \to (drop (r k h) y0 c x)))) +(\lambda (y1: nat).(\lambda (H2: (drop y1 y0 y x)).(drop_ind (\lambda (n: +nat).(\lambda (n0: nat).(\lambda (c0: C).(\lambda (c1: C).((eq nat n (S h)) +\to ((eq nat n0 O) \to ((eq C c0 (CHead c k u)) \to (drop (r k h) n0 c +c1)))))))) (\lambda (c0: C).(\lambda (H3: (eq nat O (S h))).(\lambda (_: (eq +nat O O)).(\lambda (H5: (eq C c0 (CHead c k u))).(eq_ind_r C (CHead c k u) +(\lambda (c1: C).(drop (r k h) O c c1)) (let H6 \def (eq_ind nat O (\lambda +(ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow +True | (S _) \Rightarrow False])) I (S h) H3) in (False_ind (drop (r k h) O c +(CHead c k u)) H6)) c0 H5))))) (\lambda (k0: K).(\lambda (h0: nat).(\lambda +(c0: C).(\lambda (e: C).(\lambda (H3: (drop (r k0 h0) O c0 e)).(\lambda (H4: +(((eq nat (r k0 h0) (S h)) \to ((eq nat O O) \to ((eq C c0 (CHead c k u)) \to +(drop (r k h) O c e)))))).(\lambda (u0: T).(\lambda (H5: (eq nat (S h0) (S +h))).(\lambda (_: (eq nat O O)).(\lambda (H7: (eq C (CHead c0 k0 u0) (CHead c +k u))).(let H8 \def (f_equal C C (\lambda (e0: C).(match e0 in C return +(\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c1 _ _) +\Rightarrow c1])) (CHead c0 k0 u0) (CHead c k u) H7) in ((let H9 \def +(f_equal C K (\lambda (e0: C).(match e0 in C return (\lambda (_: C).K) with +[(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead c0 k0 u0) +(CHead c k u) H7) in ((let H10 \def (f_equal C T (\lambda (e0: C).(match e0 +in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) +\Rightarrow t])) (CHead c0 k0 u0) (CHead c k u) H7) in (\lambda (H11: (eq K +k0 k)).(\lambda (H12: (eq C c0 c)).(let H13 \def (eq_ind C c0 (\lambda (c1: +C).((eq nat (r k0 h0) (S h)) \to ((eq nat O O) \to ((eq C c1 (CHead c k u)) +\to (drop (r k h) O c e))))) H4 c H12) in (let H14 \def (eq_ind C c0 (\lambda +(c1: C).(drop (r k0 h0) O c1 e)) H3 c H12) in (let H15 \def (eq_ind K k0 +(\lambda (k1: K).((eq nat (r k1 h0) (S h)) \to ((eq nat O O) \to ((eq C c +(CHead c k u)) \to (drop (r k h) O c e))))) H13 k H11) in (let H16 \def +(eq_ind K k0 (\lambda (k1: K).(drop (r k1 h0) O c e)) H14 k H11) in (let H17 +\def (f_equal nat nat (\lambda (e0: nat).(match e0 in nat return (\lambda (_: +nat).nat) with [O \Rightarrow h0 | (S n) \Rightarrow n])) (S h0) (S h) H5) in +(let H18 \def (eq_ind nat h0 (\lambda (n: nat).((eq nat (r k n) (S h)) \to +((eq nat O O) \to ((eq C c (CHead c k u)) \to (drop (r k h) O c e))))) H15 h +H17) in (let H19 \def (eq_ind nat h0 (\lambda (n: nat).(drop (r k n) O c e)) +H16 h H17) in H19)))))))))) H9)) H8)))))))))))) (\lambda (k0: K).(\lambda +(h0: nat).(\lambda (d: nat).(\lambda (c0: C).(\lambda (e: C).(\lambda (H3: +(drop h0 (r k0 d) c0 e)).(\lambda (H4: (((eq nat h0 (S h)) \to ((eq nat (r k0 +d) O) \to ((eq C c0 (CHead c k u)) \to (drop (r k h) (r k0 d) c +e)))))).(\lambda (u0: T).(\lambda (H5: (eq nat h0 (S h))).(\lambda (H6: (eq +nat (S d) O)).(\lambda (H7: (eq C (CHead c0 k0 (lift h0 (r k0 d) u0)) (CHead +c k u))).(let H8 \def (eq_ind nat h0 (\lambda (n: nat).(eq C (CHead c0 k0 +(lift n (r k0 d) u0)) (CHead c k u))) H7 (S h) H5) in (let H9 \def (eq_ind +nat h0 (\lambda (n: nat).((eq nat n (S h)) \to ((eq nat (r k0 d) O) \to ((eq +C c0 (CHead c k u)) \to (drop (r k h) (r k0 d) c e))))) H4 (S h) H5) in (let +H10 \def (eq_ind nat h0 (\lambda (n: nat).(drop n (r k0 d) c0 e)) H3 (S h) +H5) in (let H11 \def (f_equal C C (\lambda (e0: C).(match e0 in C return +(\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c1 _ _) +\Rightarrow c1])) (CHead c0 k0 (lift (S h) (r k0 d) u0)) (CHead c k u) H8) in +((let H12 \def (f_equal C K (\lambda (e0: C).(match e0 in C return (\lambda +(_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) +(CHead c0 k0 (lift (S h) (r k0 d) u0)) (CHead c k u) H8) in ((let H13 \def +(f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) with +[(CSort _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d0: nat) (t: +T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) +\Rightarrow (TLRef (match (blt i d0) with [true \Rightarrow i | false +\Rightarrow (f i)])) | (THead k1 u1 t0) \Rightarrow (THead k1 (lref_map f d0 +u1) (lref_map f (s k1 d0) t0))]) in lref_map) (\lambda (x0: nat).(plus x0 (S +h))) (r k0 d) u0) | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 (lift (S h) +(r k0 d) u0)) (CHead c k u) H8) in (\lambda (H14: (eq K k0 k)).(\lambda (H15: +(eq C c0 c)).(let H16 \def (eq_ind C c0 (\lambda (c1: C).((eq nat (S h) (S +h)) \to ((eq nat (r k0 d) O) \to ((eq C c1 (CHead c k u)) \to (drop (r k h) +(r k0 d) c e))))) H9 c H15) in (let H17 \def (eq_ind C c0 (\lambda (c1: +C).(drop (S h) (r k0 d) c1 e)) H10 c H15) in (let H18 \def (eq_ind K k0 +(\lambda (k1: K).(eq T (lift (S h) (r k1 d) u0) u)) H13 k H14) in (let H19 +\def (eq_ind K k0 (\lambda (k1: K).((eq nat (S h) (S h)) \to ((eq nat (r k1 +d) O) \to ((eq C c (CHead c k u)) \to (drop (r k h) (r k1 d) c e))))) H16 k +H14) in (let H20 \def (eq_ind K k0 (\lambda (k1: K).(drop (S h) (r k1 d) c +e)) H17 k H14) in (eq_ind_r K k (\lambda (k1: K).(drop (r k h) (S d) c (CHead +e k1 u0))) (let H21 \def (eq_ind_r T u (\lambda (t: T).((eq nat (S h) (S h)) +\to ((eq nat (r k d) O) \to ((eq C c (CHead c k t)) \to (drop (r k h) (r k d) +c e))))) H19 (lift (S h) (r k d) u0) H18) in (let H22 \def (eq_ind nat (S d) +(\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O +\Rightarrow False | (S _) \Rightarrow True])) I O H6) in (False_ind (drop (r +k h) (S d) c (CHead e k u0)) H22))) k0 H14))))))))) H12)) H11)))))))))))))))) +y1 y0 y x H2))) H1))) H0))) H)))))). +(* COMMENTS +Initial nodes: 1856 +END *) + +theorem drop_gen_skip_r: + \forall (c: C).(\forall (x: C).(\forall (u: T).(\forall (h: nat).(\forall +(d: nat).(\forall (k: K).((drop h (S d) x (CHead c k u)) \to (ex2 C (\lambda +(e: C).(eq C x (CHead e k (lift h (r k d) u)))) (\lambda (e: C).(drop h (r k +d) e c))))))))) +\def + \lambda (c: C).(\lambda (x: C).(\lambda (u: T).(\lambda (h: nat).(\lambda +(d: nat).(\lambda (k: K).(\lambda (H: (drop h (S d) x (CHead c k +u))).(insert_eq C (CHead c k u) (\lambda (c0: C).(drop h (S d) x c0)) +(\lambda (_: C).(ex2 C (\lambda (e: C).(eq C x (CHead e k (lift h (r k d) +u)))) (\lambda (e: C).(drop h (r k d) e c)))) (\lambda (y: C).(\lambda (H0: +(drop h (S d) x y)).(insert_eq nat (S d) (\lambda (n: nat).(drop h n x y)) +(\lambda (_: nat).((eq C y (CHead c k u)) \to (ex2 C (\lambda (e: C).(eq C x +(CHead e k (lift h (r k d) u)))) (\lambda (e: C).(drop h (r k d) e c))))) +(\lambda (y0: nat).(\lambda (H1: (drop h y0 x y)).(drop_ind (\lambda (n: +nat).(\lambda (n0: nat).(\lambda (c0: C).(\lambda (c1: C).((eq nat n0 (S d)) +\to ((eq C c1 (CHead c k u)) \to (ex2 C (\lambda (e: C).(eq C c0 (CHead e k +(lift n (r k d) u)))) (\lambda (e: C).(drop n (r k d) e c))))))))) (\lambda +(c0: C).(\lambda (H2: (eq nat O (S d))).(\lambda (H3: (eq C c0 (CHead c k +u))).(eq_ind_r C (CHead c k u) (\lambda (c1: C).(ex2 C (\lambda (e: C).(eq C +c1 (CHead e k (lift O (r k d) u)))) (\lambda (e: C).(drop O (r k d) e c)))) +(let H4 \def (eq_ind nat O (\lambda (ee: nat).(match ee in nat return +(\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) +I (S d) H2) in (False_ind (ex2 C (\lambda (e: C).(eq C (CHead c k u) (CHead e +k (lift O (r k d) u)))) (\lambda (e: C).(drop O (r k d) e c))) H4)) c0 H3)))) +(\lambda (k0: K).(\lambda (h0: nat).(\lambda (c0: C).(\lambda (e: C).(\lambda +(H2: (drop (r k0 h0) O c0 e)).(\lambda (H3: (((eq nat O (S d)) \to ((eq C e +(CHead c k u)) \to (ex2 C (\lambda (e0: C).(eq C c0 (CHead e0 k (lift (r k0 +h0) (r k d) u)))) (\lambda (e0: C).(drop (r k0 h0) (r k d) e0 +c))))))).(\lambda (u0: T).(\lambda (H4: (eq nat O (S d))).(\lambda (H5: (eq C +e (CHead c k u))).(let H6 \def (eq_ind C e (\lambda (c1: C).((eq nat O (S d)) +\to ((eq C c1 (CHead c k u)) \to (ex2 C (\lambda (e0: C).(eq C c0 (CHead e0 k +(lift (r k0 h0) (r k d) u)))) (\lambda (e0: C).(drop (r k0 h0) (r k d) e0 +c)))))) H3 (CHead c k u) H5) in (let H7 \def (eq_ind C e (\lambda (c1: +C).(drop (r k0 h0) O c0 c1)) H2 (CHead c k u) H5) in (let H8 \def (eq_ind nat +O (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O +\Rightarrow True | (S _) \Rightarrow False])) I (S d) H4) in (False_ind (ex2 +C (\lambda (e0: C).(eq C (CHead c0 k0 u0) (CHead e0 k (lift (S h0) (r k d) +u)))) (\lambda (e0: C).(drop (S h0) (r k d) e0 c))) H8))))))))))))) (\lambda +(k0: K).(\lambda (h0: nat).(\lambda (d0: nat).(\lambda (c0: C).(\lambda (e: +C).(\lambda (H2: (drop h0 (r k0 d0) c0 e)).(\lambda (H3: (((eq nat (r k0 d0) +(S d)) \to ((eq C e (CHead c k u)) \to (ex2 C (\lambda (e0: C).(eq C c0 +(CHead e0 k (lift h0 (r k d) u)))) (\lambda (e0: C).(drop h0 (r k d) e0 +c))))))).(\lambda (u0: T).(\lambda (H4: (eq nat (S d0) (S d))).(\lambda (H5: +(eq C (CHead e k0 u0) (CHead c k u))).(let H6 \def (f_equal C C (\lambda (e0: +C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow e | +(CHead c1 _ _) \Rightarrow c1])) (CHead e k0 u0) (CHead c k u) H5) in ((let +H7 \def (f_equal C K (\lambda (e0: C).(match e0 in C return (\lambda (_: +C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) +(CHead e k0 u0) (CHead c k u) H5) in ((let H8 \def (f_equal C T (\lambda (e0: +C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | +(CHead _ _ t) \Rightarrow t])) (CHead e k0 u0) (CHead c k u) H5) in (\lambda +(H9: (eq K k0 k)).(\lambda (H10: (eq C e c)).(eq_ind_r T u (\lambda (t: +T).(ex2 C (\lambda (e0: C).(eq C (CHead c0 k0 (lift h0 (r k0 d0) t)) (CHead +e0 k (lift h0 (r k d) u)))) (\lambda (e0: C).(drop h0 (r k d) e0 c)))) (let +H11 \def (eq_ind C e (\lambda (c1: C).((eq nat (r k0 d0) (S d)) \to ((eq C c1 +(CHead c k u)) \to (ex2 C (\lambda (e0: C).(eq C c0 (CHead e0 k (lift h0 (r k +d) u)))) (\lambda (e0: C).(drop h0 (r k d) e0 c)))))) H3 c H10) in (let H12 +\def (eq_ind C e (\lambda (c1: C).(drop h0 (r k0 d0) c0 c1)) H2 c H10) in +(let H13 \def (eq_ind K k0 (\lambda (k1: K).((eq nat (r k1 d0) (S d)) \to +((eq C c (CHead c k u)) \to (ex2 C (\lambda (e0: C).(eq C c0 (CHead e0 k +(lift h0 (r k d) u)))) (\lambda (e0: C).(drop h0 (r k d) e0 c)))))) H11 k H9) +in (let H14 \def (eq_ind K k0 (\lambda (k1: K).(drop h0 (r k1 d0) c0 c)) H12 +k H9) in (eq_ind_r K k (\lambda (k1: K).(ex2 C (\lambda (e0: C).(eq C (CHead +c0 k1 (lift h0 (r k1 d0) u)) (CHead e0 k (lift h0 (r k d) u)))) (\lambda (e0: +C).(drop h0 (r k d) e0 c)))) (let H15 \def (f_equal nat nat (\lambda (e0: +nat).(match e0 in nat return (\lambda (_: nat).nat) with [O \Rightarrow d0 | +(S n) \Rightarrow n])) (S d0) (S d) H4) in (let H16 \def (eq_ind nat d0 +(\lambda (n: nat).((eq nat (r k n) (S d)) \to ((eq C c (CHead c k u)) \to +(ex2 C (\lambda (e0: C).(eq C c0 (CHead e0 k (lift h0 (r k d) u)))) (\lambda +(e0: C).(drop h0 (r k d) e0 c)))))) H13 d H15) in (let H17 \def (eq_ind nat +d0 (\lambda (n: nat).(drop h0 (r k n) c0 c)) H14 d H15) in (eq_ind_r nat d +(\lambda (n: nat).(ex2 C (\lambda (e0: C).(eq C (CHead c0 k (lift h0 (r k n) +u)) (CHead e0 k (lift h0 (r k d) u)))) (\lambda (e0: C).(drop h0 (r k d) e0 +c)))) (ex_intro2 C (\lambda (e0: C).(eq C (CHead c0 k (lift h0 (r k d) u)) +(CHead e0 k (lift h0 (r k d) u)))) (\lambda (e0: C).(drop h0 (r k d) e0 c)) +c0 (refl_equal C (CHead c0 k (lift h0 (r k d) u))) H17) d0 H15)))) k0 H9))))) +u0 H8)))) H7)) H6)))))))))))) h y0 x y H1))) H0))) H))))))). +(* COMMENTS +Initial nodes: 1758 +END *) + +theorem drop_gen_skip_l: + \forall (c: C).(\forall (x: C).(\forall (u: T).(\forall (h: nat).(\forall +(d: nat).(\forall (k: K).((drop h (S d) (CHead c k u) x) \to (ex3_2 C T +(\lambda (e: C).(\lambda (v: T).(eq C x (CHead e k v)))) (\lambda (_: +C).(\lambda (v: T).(eq T u (lift h (r k d) v)))) (\lambda (e: C).(\lambda (_: +T).(drop h (r k d) c e)))))))))) +\def + \lambda (c: C).(\lambda (x: C).(\lambda (u: T).(\lambda (h: nat).(\lambda +(d: nat).(\lambda (k: K).(\lambda (H: (drop h (S d) (CHead c k u) +x)).(insert_eq C (CHead c k u) (\lambda (c0: C).(drop h (S d) c0 x)) (\lambda +(_: C).(ex3_2 C T (\lambda (e: C).(\lambda (v: T).(eq C x (CHead e k v)))) +(\lambda (_: C).(\lambda (v: T).(eq T u (lift h (r k d) v)))) (\lambda (e: +C).(\lambda (_: T).(drop h (r k d) c e))))) (\lambda (y: C).(\lambda (H0: +(drop h (S d) y x)).(insert_eq nat (S d) (\lambda (n: nat).(drop h n y x)) +(\lambda (_: nat).((eq C y (CHead c k u)) \to (ex3_2 C T (\lambda (e: +C).(\lambda (v: T).(eq C x (CHead e k v)))) (\lambda (_: C).(\lambda (v: +T).(eq T u (lift h (r k d) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r k +d) c e)))))) (\lambda (y0: nat).(\lambda (H1: (drop h y0 y x)).(drop_ind +(\lambda (n: nat).(\lambda (n0: nat).(\lambda (c0: C).(\lambda (c1: C).((eq +nat n0 (S d)) \to ((eq C c0 (CHead c k u)) \to (ex3_2 C T (\lambda (e: +C).(\lambda (v: T).(eq C c1 (CHead e k v)))) (\lambda (_: C).(\lambda (v: +T).(eq T u (lift n (r k d) v)))) (\lambda (e: C).(\lambda (_: T).(drop n (r k +d) c e)))))))))) (\lambda (c0: C).(\lambda (H2: (eq nat O (S d))).(\lambda +(H3: (eq C c0 (CHead c k u))).(eq_ind_r C (CHead c k u) (\lambda (c1: +C).(ex3_2 C T (\lambda (e: C).(\lambda (v: T).(eq C c1 (CHead e k v)))) +(\lambda (_: C).(\lambda (v: T).(eq T u (lift O (r k d) v)))) (\lambda (e: +C).(\lambda (_: T).(drop O (r k d) c e))))) (let H4 \def (eq_ind nat O +(\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O +\Rightarrow True | (S _) \Rightarrow False])) I (S d) H2) in (False_ind +(ex3_2 C T (\lambda (e: C).(\lambda (v: T).(eq C (CHead c k u) (CHead e k +v)))) (\lambda (_: C).(\lambda (v: T).(eq T u (lift O (r k d) v)))) (\lambda +(e: C).(\lambda (_: T).(drop O (r k d) c e)))) H4)) c0 H3)))) (\lambda (k0: +K).(\lambda (h0: nat).(\lambda (c0: C).(\lambda (e: C).(\lambda (H2: (drop (r +k0 h0) O c0 e)).(\lambda (H3: (((eq nat O (S d)) \to ((eq C c0 (CHead c k u)) +\to (ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 k v)))) +(\lambda (_: C).(\lambda (v: T).(eq T u (lift (r k0 h0) (r k d) v)))) +(\lambda (e0: C).(\lambda (_: T).(drop (r k0 h0) (r k d) c +e0)))))))).(\lambda (u0: T).(\lambda (H4: (eq nat O (S d))).(\lambda (H5: (eq +C (CHead c0 k0 u0) (CHead c k u))).(let H6 \def (f_equal C C (\lambda (e0: +C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | +(CHead c1 _ _) \Rightarrow c1])) (CHead c0 k0 u0) (CHead c k u) H5) in ((let +H7 \def (f_equal C K (\lambda (e0: C).(match e0 in C return (\lambda (_: +C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) +(CHead c0 k0 u0) (CHead c k u) H5) in ((let H8 \def (f_equal C T (\lambda +(e0: C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow +u0 | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 u0) (CHead c k u) H5) in +(\lambda (H9: (eq K k0 k)).(\lambda (H10: (eq C c0 c)).(let H11 \def (eq_ind +C c0 (\lambda (c1: C).((eq nat O (S d)) \to ((eq C c1 (CHead c k u)) \to +(ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 k v)))) +(\lambda (_: C).(\lambda (v: T).(eq T u (lift (r k0 h0) (r k d) v)))) +(\lambda (e0: C).(\lambda (_: T).(drop (r k0 h0) (r k d) c e0))))))) H3 c +H10) in (let H12 \def (eq_ind C c0 (\lambda (c1: C).(drop (r k0 h0) O c1 e)) +H2 c H10) in (let H13 \def (eq_ind K k0 (\lambda (k1: K).((eq nat O (S d)) +\to ((eq C c (CHead c k u)) \to (ex3_2 C T (\lambda (e0: C).(\lambda (v: +T).(eq C e (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T u (lift (r +k1 h0) (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop (r k1 h0) (r k d) +c e0))))))) H11 k H9) in (let H14 \def (eq_ind K k0 (\lambda (k1: K).(drop (r +k1 h0) O c e)) H12 k H9) in (let H15 \def (eq_ind nat O (\lambda (ee: +nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True +| (S _) \Rightarrow False])) I (S d) H4) in (False_ind (ex3_2 C T (\lambda +(e0: C).(\lambda (v: T).(eq C e (CHead e0 k v)))) (\lambda (_: C).(\lambda +(v: T).(eq T u (lift (S h0) (r k d) v)))) (\lambda (e0: C).(\lambda (_: +T).(drop (S h0) (r k d) c e0)))) H15))))))))) H7)) H6))))))))))) (\lambda +(k0: K).(\lambda (h0: nat).(\lambda (d0: nat).(\lambda (c0: C).(\lambda (e: +C).(\lambda (H2: (drop h0 (r k0 d0) c0 e)).(\lambda (H3: (((eq nat (r k0 d0) +(S d)) \to ((eq C c0 (CHead c k u)) \to (ex3_2 C T (\lambda (e0: C).(\lambda +(v: T).(eq C e (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T u +(lift h0 (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h0 (r k d) c +e0)))))))).(\lambda (u0: T).(\lambda (H4: (eq nat (S d0) (S d))).(\lambda +(H5: (eq C (CHead c0 k0 (lift h0 (r k0 d0) u0)) (CHead c k u))).(let H6 \def +(f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with +[(CSort _) \Rightarrow c0 | (CHead c1 _ _) \Rightarrow c1])) (CHead c0 k0 +(lift h0 (r k0 d0) u0)) (CHead c k u) H5) in ((let H7 \def (f_equal C K +(\lambda (e0: C).(match e0 in C return (\lambda (_: C).K) with [(CSort _) +\Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead c0 k0 (lift h0 (r k0 +d0) u0)) (CHead c k u) H5) in ((let H8 \def (f_equal C T (\lambda (e0: +C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow ((let +rec lref_map (f: ((nat \to nat))) (d1: nat) (t: T) on t: T \def (match t with +[(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i +d1) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k1 u1 t0) +\Rightarrow (THead k1 (lref_map f d1 u1) (lref_map f (s k1 d1) t0))]) in +lref_map) (\lambda (x0: nat).(plus x0 h0)) (r k0 d0) u0) | (CHead _ _ t) +\Rightarrow t])) (CHead c0 k0 (lift h0 (r k0 d0) u0)) (CHead c k u) H5) in +(\lambda (H9: (eq K k0 k)).(\lambda (H10: (eq C c0 c)).(let H11 \def (eq_ind +C c0 (\lambda (c1: C).((eq nat (r k0 d0) (S d)) \to ((eq C c1 (CHead c k u)) +\to (ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 k v)))) +(\lambda (_: C).(\lambda (v: T).(eq T u (lift h0 (r k d) v)))) (\lambda (e0: +C).(\lambda (_: T).(drop h0 (r k d) c e0))))))) H3 c H10) in (let H12 \def +(eq_ind C c0 (\lambda (c1: C).(drop h0 (r k0 d0) c1 e)) H2 c H10) in (let H13 +\def (eq_ind K k0 (\lambda (k1: K).(eq T (lift h0 (r k1 d0) u0) u)) H8 k H9) +in (let H14 \def (eq_ind K k0 (\lambda (k1: K).((eq nat (r k1 d0) (S d)) \to +((eq C c (CHead c k u)) \to (ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C +e (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T u (lift h0 (r k d) +v)))) (\lambda (e0: C).(\lambda (_: T).(drop h0 (r k d) c e0))))))) H11 k H9) +in (let H15 \def (eq_ind K k0 (\lambda (k1: K).(drop h0 (r k1 d0) c e)) H12 k +H9) in (eq_ind_r K k (\lambda (k1: K).(ex3_2 C T (\lambda (e0: C).(\lambda +(v: T).(eq C (CHead e k1 u0) (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: +T).(eq T u (lift h0 (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h0 +(r k d) c e0))))) (let H16 \def (eq_ind_r T u (\lambda (t: T).((eq nat (r k +d0) (S d)) \to ((eq C c (CHead c k t)) \to (ex3_2 C T (\lambda (e0: +C).(\lambda (v: T).(eq C e (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: +T).(eq T t (lift h0 (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h0 +(r k d) c e0))))))) H14 (lift h0 (r k d0) u0) H13) in (eq_ind T (lift h0 (r k +d0) u0) (\lambda (t: T).(ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C +(CHead e k u0) (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T t +(lift h0 (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h0 (r k d) c +e0))))) (let H17 \def (f_equal nat nat (\lambda (e0: nat).(match e0 in nat +return (\lambda (_: nat).nat) with [O \Rightarrow d0 | (S n) \Rightarrow n])) +(S d0) (S d) H4) in (let H18 \def (eq_ind nat d0 (\lambda (n: nat).((eq nat +(r k n) (S d)) \to ((eq C c (CHead c k (lift h0 (r k n) u0))) \to (ex3_2 C T +(\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 k v)))) (\lambda (_: +C).(\lambda (v: T).(eq T (lift h0 (r k n) u0) (lift h0 (r k d) v)))) (\lambda +(e0: C).(\lambda (_: T).(drop h0 (r k d) c e0))))))) H16 d H17) in (let H19 +\def (eq_ind nat d0 (\lambda (n: nat).(drop h0 (r k n) c e)) H15 d H17) in +(eq_ind_r nat d (\lambda (n: nat).(ex3_2 C T (\lambda (e0: C).(\lambda (v: +T).(eq C (CHead e k u0) (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq +T (lift h0 (r k n) u0) (lift h0 (r k d) v)))) (\lambda (e0: C).(\lambda (_: +T).(drop h0 (r k d) c e0))))) (ex3_2_intro C T (\lambda (e0: C).(\lambda (v: +T).(eq C (CHead e k u0) (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq +T (lift h0 (r k d) u0) (lift h0 (r k d) v)))) (\lambda (e0: C).(\lambda (_: +T).(drop h0 (r k d) c e0))) e u0 (refl_equal C (CHead e k u0)) (refl_equal T +(lift h0 (r k d) u0)) H19) d0 H17)))) u H13)) k0 H9))))))))) H7)) +H6)))))))))))) h y0 y x H1))) H0))) H))))))). +(* COMMENTS +Initial nodes: 2574 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/drop/props.ma b/matita/matita/contribs/lambdadelta/basic_1/drop/props.ma new file mode 100644 index 000000000..ac802b105 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/drop/props.ma @@ -0,0 +1,737 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/drop/fwd.ma". + +include "Basic-1/lift/props.ma". + +include "Basic-1/r/props.ma". + +theorem drop_skip_bind: + \forall (h: nat).(\forall (d: nat).(\forall (c: C).(\forall (e: C).((drop h +d c e) \to (\forall (b: B).(\forall (u: T).(drop h (S d) (CHead c (Bind b) +(lift h d u)) (CHead e (Bind b) u)))))))) +\def + \lambda (h: nat).(\lambda (d: nat).(\lambda (c: C).(\lambda (e: C).(\lambda +(H: (drop h d c e)).(\lambda (b: B).(\lambda (u: T).(eq_ind nat (r (Bind b) +d) (\lambda (n: nat).(drop h (S d) (CHead c (Bind b) (lift h n u)) (CHead e +(Bind b) u))) (drop_skip (Bind b) h d c e H u) d (refl_equal nat d)))))))). +(* COMMENTS +Initial nodes: 95 +END *) + +theorem drop_skip_flat: + \forall (h: nat).(\forall (d: nat).(\forall (c: C).(\forall (e: C).((drop h +(S d) c e) \to (\forall (f: F).(\forall (u: T).(drop h (S d) (CHead c (Flat +f) (lift h (S d) u)) (CHead e (Flat f) u)))))))) +\def + \lambda (h: nat).(\lambda (d: nat).(\lambda (c: C).(\lambda (e: C).(\lambda +(H: (drop h (S d) c e)).(\lambda (f: F).(\lambda (u: T).(eq_ind nat (r (Flat +f) d) (\lambda (n: nat).(drop h (S d) (CHead c (Flat f) (lift h n u)) (CHead +e (Flat f) u))) (drop_skip (Flat f) h d c e H u) (S d) (refl_equal nat (S +d))))))))). +(* COMMENTS +Initial nodes: 101 +END *) + +theorem drop_S: + \forall (b: B).(\forall (c: C).(\forall (e: C).(\forall (u: T).(\forall (h: +nat).((drop h O c (CHead e (Bind b) u)) \to (drop (S h) O c e)))))) +\def + \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e: +C).(\forall (u: T).(\forall (h: nat).((drop h O c0 (CHead e (Bind b) u)) \to +(drop (S h) O c0 e)))))) (\lambda (n: nat).(\lambda (e: C).(\lambda (u: +T).(\lambda (h: nat).(\lambda (H: (drop h O (CSort n) (CHead e (Bind b) +u))).(and3_ind (eq C (CHead e (Bind b) u) (CSort n)) (eq nat h O) (eq nat O +O) (drop (S h) O (CSort n) e) (\lambda (H0: (eq C (CHead e (Bind b) u) (CSort +n))).(\lambda (H1: (eq nat h O)).(\lambda (_: (eq nat O O)).(eq_ind_r nat O +(\lambda (n0: nat).(drop (S n0) O (CSort n) e)) (let H3 \def (eq_ind C (CHead +e (Bind b) u) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) +with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I +(CSort n) H0) in (False_ind (drop (S O) O (CSort n) e) H3)) h H1)))) +(drop_gen_sort n h O (CHead e (Bind b) u) H))))))) (\lambda (c0: C).(\lambda +(H: ((\forall (e: C).(\forall (u: T).(\forall (h: nat).((drop h O c0 (CHead e +(Bind b) u)) \to (drop (S h) O c0 e))))))).(\lambda (k: K).(\lambda (t: +T).(\lambda (e: C).(\lambda (u: T).(\lambda (h: nat).(nat_ind (\lambda (n: +nat).((drop n O (CHead c0 k t) (CHead e (Bind b) u)) \to (drop (S n) O (CHead +c0 k t) e))) (\lambda (H0: (drop O O (CHead c0 k t) (CHead e (Bind b) +u))).(let H1 \def (f_equal C C (\lambda (e0: C).(match e0 in C return +(\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c1 _ _) +\Rightarrow c1])) (CHead c0 k t) (CHead e (Bind b) u) (drop_gen_refl (CHead +c0 k t) (CHead e (Bind b) u) H0)) in ((let H2 \def (f_equal C K (\lambda (e0: +C).(match e0 in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | +(CHead _ k0 _) \Rightarrow k0])) (CHead c0 k t) (CHead e (Bind b) u) +(drop_gen_refl (CHead c0 k t) (CHead e (Bind b) u) H0)) in ((let H3 \def +(f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) with +[(CSort _) \Rightarrow t | (CHead _ _ t0) \Rightarrow t0])) (CHead c0 k t) +(CHead e (Bind b) u) (drop_gen_refl (CHead c0 k t) (CHead e (Bind b) u) H0)) +in (\lambda (H4: (eq K k (Bind b))).(\lambda (H5: (eq C c0 e)).(eq_ind C c0 +(\lambda (c1: C).(drop (S O) O (CHead c0 k t) c1)) (eq_ind_r K (Bind b) +(\lambda (k0: K).(drop (S O) O (CHead c0 k0 t) c0)) (drop_drop (Bind b) O c0 +c0 (drop_refl c0) t) k H4) e H5)))) H2)) H1))) (\lambda (n: nat).(\lambda (_: +(((drop n O (CHead c0 k t) (CHead e (Bind b) u)) \to (drop (S n) O (CHead c0 +k t) e)))).(\lambda (H1: (drop (S n) O (CHead c0 k t) (CHead e (Bind b) +u))).(drop_drop k (S n) c0 e (eq_ind_r nat (S (r k n)) (\lambda (n0: +nat).(drop n0 O c0 e)) (H e u (r k n) (drop_gen_drop k c0 (CHead e (Bind b) +u) t n H1)) (r k (S n)) (r_S k n)) t)))) h)))))))) c)). +(* COMMENTS +Initial nodes: 807 +END *) + +theorem drop_ctail: + \forall (c1: C).(\forall (c2: C).(\forall (d: nat).(\forall (h: nat).((drop +h d c1 c2) \to (\forall (k: K).(\forall (u: T).(drop h d (CTail k u c1) +(CTail k u c2)))))))) +\def + \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (d: +nat).(\forall (h: nat).((drop h d c c2) \to (\forall (k: K).(\forall (u: +T).(drop h d (CTail k u c) (CTail k u c2))))))))) (\lambda (n: nat).(\lambda +(c2: C).(\lambda (d: nat).(\lambda (h: nat).(\lambda (H: (drop h d (CSort n) +c2)).(\lambda (k: K).(\lambda (u: T).(and3_ind (eq C c2 (CSort n)) (eq nat h +O) (eq nat d O) (drop h d (CTail k u (CSort n)) (CTail k u c2)) (\lambda (H0: +(eq C c2 (CSort n))).(\lambda (H1: (eq nat h O)).(\lambda (H2: (eq nat d +O)).(eq_ind_r nat O (\lambda (n0: nat).(drop n0 d (CTail k u (CSort n)) +(CTail k u c2))) (eq_ind_r nat O (\lambda (n0: nat).(drop O n0 (CTail k u +(CSort n)) (CTail k u c2))) (eq_ind_r C (CSort n) (\lambda (c: C).(drop O O +(CTail k u (CSort n)) (CTail k u c))) (drop_refl (CTail k u (CSort n))) c2 +H0) d H2) h H1)))) (drop_gen_sort n h d c2 H))))))))) (\lambda (c2: +C).(\lambda (IHc: ((\forall (c3: C).(\forall (d: nat).(\forall (h: +nat).((drop h d c2 c3) \to (\forall (k: K).(\forall (u: T).(drop h d (CTail k +u c2) (CTail k u c3)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c3: +C).(\lambda (d: nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n +(CHead c2 k t) c3) \to (\forall (k0: K).(\forall (u: T).(drop h n (CTail k0 u +(CHead c2 k t)) (CTail k0 u c3))))))) (\lambda (h: nat).(nat_ind (\lambda (n: +nat).((drop n O (CHead c2 k t) c3) \to (\forall (k0: K).(\forall (u: T).(drop +n O (CTail k0 u (CHead c2 k t)) (CTail k0 u c3)))))) (\lambda (H: (drop O O +(CHead c2 k t) c3)).(\lambda (k0: K).(\lambda (u: T).(eq_ind C (CHead c2 k t) +(\lambda (c: C).(drop O O (CTail k0 u (CHead c2 k t)) (CTail k0 u c))) +(drop_refl (CTail k0 u (CHead c2 k t))) c3 (drop_gen_refl (CHead c2 k t) c3 +H))))) (\lambda (n: nat).(\lambda (_: (((drop n O (CHead c2 k t) c3) \to +(\forall (k0: K).(\forall (u: T).(drop n O (CTail k0 u (CHead c2 k t)) (CTail +k0 u c3))))))).(\lambda (H0: (drop (S n) O (CHead c2 k t) c3)).(\lambda (k0: +K).(\lambda (u: T).(drop_drop k n (CTail k0 u c2) (CTail k0 u c3) (IHc c3 O +(r k n) (drop_gen_drop k c2 c3 t n H0) k0 u) t)))))) h)) (\lambda (n: +nat).(\lambda (H: ((\forall (h: nat).((drop h n (CHead c2 k t) c3) \to +(\forall (k0: K).(\forall (u: T).(drop h n (CTail k0 u (CHead c2 k t)) (CTail +k0 u c3)))))))).(\lambda (h: nat).(\lambda (H0: (drop h (S n) (CHead c2 k t) +c3)).(\lambda (k0: K).(\lambda (u: T).(ex3_2_ind C T (\lambda (e: C).(\lambda +(v: T).(eq C c3 (CHead e k v)))) (\lambda (_: C).(\lambda (v: T).(eq T t +(lift h (r k n) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r k n) c2 e))) +(drop h (S n) (CTail k0 u (CHead c2 k t)) (CTail k0 u c3)) (\lambda (x0: +C).(\lambda (x1: T).(\lambda (H1: (eq C c3 (CHead x0 k x1))).(\lambda (H2: +(eq T t (lift h (r k n) x1))).(\lambda (H3: (drop h (r k n) c2 x0)).(let H4 +\def (eq_ind C c3 (\lambda (c: C).(\forall (h0: nat).((drop h0 n (CHead c2 k +t) c) \to (\forall (k1: K).(\forall (u0: T).(drop h0 n (CTail k1 u0 (CHead c2 +k t)) (CTail k1 u0 c))))))) H (CHead x0 k x1) H1) in (eq_ind_r C (CHead x0 k +x1) (\lambda (c: C).(drop h (S n) (CTail k0 u (CHead c2 k t)) (CTail k0 u +c))) (let H5 \def (eq_ind T t (\lambda (t0: T).(\forall (h0: nat).((drop h0 n +(CHead c2 k t0) (CHead x0 k x1)) \to (\forall (k1: K).(\forall (u0: T).(drop +h0 n (CTail k1 u0 (CHead c2 k t0)) (CTail k1 u0 (CHead x0 k x1)))))))) H4 +(lift h (r k n) x1) H2) in (eq_ind_r T (lift h (r k n) x1) (\lambda (t0: +T).(drop h (S n) (CTail k0 u (CHead c2 k t0)) (CTail k0 u (CHead x0 k x1)))) +(drop_skip k h n (CTail k0 u c2) (CTail k0 u x0) (IHc x0 (r k n) h H3 k0 u) +x1) t H2)) c3 H1))))))) (drop_gen_skip_l c2 c3 t h n k H0)))))))) d))))))) +c1). +(* COMMENTS +Initial nodes: 1211 +END *) + +theorem drop_mono: + \forall (c: C).(\forall (x1: C).(\forall (d: nat).(\forall (h: nat).((drop h +d c x1) \to (\forall (x2: C).((drop h d c x2) \to (eq C x1 x2))))))) +\def + \lambda (c: C).(C_ind (\lambda (c0: C).(\forall (x1: C).(\forall (d: +nat).(\forall (h: nat).((drop h d c0 x1) \to (\forall (x2: C).((drop h d c0 +x2) \to (eq C x1 x2)))))))) (\lambda (n: nat).(\lambda (x1: C).(\lambda (d: +nat).(\lambda (h: nat).(\lambda (H: (drop h d (CSort n) x1)).(\lambda (x2: +C).(\lambda (H0: (drop h d (CSort n) x2)).(and3_ind (eq C x2 (CSort n)) (eq +nat h O) (eq nat d O) (eq C x1 x2) (\lambda (H1: (eq C x2 (CSort +n))).(\lambda (H2: (eq nat h O)).(\lambda (H3: (eq nat d O)).(and3_ind (eq C +x1 (CSort n)) (eq nat h O) (eq nat d O) (eq C x1 x2) (\lambda (H4: (eq C x1 +(CSort n))).(\lambda (H5: (eq nat h O)).(\lambda (H6: (eq nat d O)).(eq_ind_r +C (CSort n) (\lambda (c0: C).(eq C x1 c0)) (let H7 \def (eq_ind nat h +(\lambda (n0: nat).(eq nat n0 O)) H2 O H5) in (let H8 \def (eq_ind nat d +(\lambda (n0: nat).(eq nat n0 O)) H3 O H6) in (eq_ind_r C (CSort n) (\lambda +(c0: C).(eq C c0 (CSort n))) (refl_equal C (CSort n)) x1 H4))) x2 H1)))) +(drop_gen_sort n h d x1 H))))) (drop_gen_sort n h d x2 H0))))))))) (\lambda +(c0: C).(\lambda (H: ((\forall (x1: C).(\forall (d: nat).(\forall (h: +nat).((drop h d c0 x1) \to (\forall (x2: C).((drop h d c0 x2) \to (eq C x1 +x2))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (x1: C).(\lambda (d: +nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n (CHead c0 k t) +x1) \to (\forall (x2: C).((drop h n (CHead c0 k t) x2) \to (eq C x1 x2)))))) +(\lambda (h: nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c0 k t) x1) +\to (\forall (x2: C).((drop n O (CHead c0 k t) x2) \to (eq C x1 x2))))) +(\lambda (H0: (drop O O (CHead c0 k t) x1)).(\lambda (x2: C).(\lambda (H1: +(drop O O (CHead c0 k t) x2)).(eq_ind C (CHead c0 k t) (\lambda (c1: C).(eq C +x1 c1)) (eq_ind C (CHead c0 k t) (\lambda (c1: C).(eq C c1 (CHead c0 k t))) +(refl_equal C (CHead c0 k t)) x1 (drop_gen_refl (CHead c0 k t) x1 H0)) x2 +(drop_gen_refl (CHead c0 k t) x2 H1))))) (\lambda (n: nat).(\lambda (_: +(((drop n O (CHead c0 k t) x1) \to (\forall (x2: C).((drop n O (CHead c0 k t) +x2) \to (eq C x1 x2)))))).(\lambda (H1: (drop (S n) O (CHead c0 k t) +x1)).(\lambda (x2: C).(\lambda (H2: (drop (S n) O (CHead c0 k t) x2)).(H x1 O +(r k n) (drop_gen_drop k c0 x1 t n H1) x2 (drop_gen_drop k c0 x2 t n +H2))))))) h)) (\lambda (n: nat).(\lambda (H0: ((\forall (h: nat).((drop h n +(CHead c0 k t) x1) \to (\forall (x2: C).((drop h n (CHead c0 k t) x2) \to (eq +C x1 x2))))))).(\lambda (h: nat).(\lambda (H1: (drop h (S n) (CHead c0 k t) +x1)).(\lambda (x2: C).(\lambda (H2: (drop h (S n) (CHead c0 k t) +x2)).(ex3_2_ind C T (\lambda (e: C).(\lambda (v: T).(eq C x2 (CHead e k v)))) +(\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r k n) v)))) (\lambda (e: +C).(\lambda (_: T).(drop h (r k n) c0 e))) (eq C x1 x2) (\lambda (x0: +C).(\lambda (x3: T).(\lambda (H3: (eq C x2 (CHead x0 k x3))).(\lambda (H4: +(eq T t (lift h (r k n) x3))).(\lambda (H5: (drop h (r k n) c0 +x0)).(ex3_2_ind C T (\lambda (e: C).(\lambda (v: T).(eq C x1 (CHead e k v)))) +(\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r k n) v)))) (\lambda (e: +C).(\lambda (_: T).(drop h (r k n) c0 e))) (eq C x1 x2) (\lambda (x4: +C).(\lambda (x5: T).(\lambda (H6: (eq C x1 (CHead x4 k x5))).(\lambda (H7: +(eq T t (lift h (r k n) x5))).(\lambda (H8: (drop h (r k n) c0 x4)).(eq_ind_r +C (CHead x0 k x3) (\lambda (c1: C).(eq C x1 c1)) (let H9 \def (eq_ind C x1 +(\lambda (c1: C).(\forall (h0: nat).((drop h0 n (CHead c0 k t) c1) \to +(\forall (x6: C).((drop h0 n (CHead c0 k t) x6) \to (eq C c1 x6)))))) H0 +(CHead x4 k x5) H6) in (eq_ind_r C (CHead x4 k x5) (\lambda (c1: C).(eq C c1 +(CHead x0 k x3))) (let H10 \def (eq_ind T t (\lambda (t0: T).(\forall (h0: +nat).((drop h0 n (CHead c0 k t0) (CHead x4 k x5)) \to (\forall (x6: C).((drop +h0 n (CHead c0 k t0) x6) \to (eq C (CHead x4 k x5) x6)))))) H9 (lift h (r k +n) x5) H7) in (let H11 \def (eq_ind T t (\lambda (t0: T).(eq T t0 (lift h (r +k n) x3))) H4 (lift h (r k n) x5) H7) in (let H12 \def (eq_ind T x5 (\lambda +(t0: T).(\forall (h0: nat).((drop h0 n (CHead c0 k (lift h (r k n) t0)) +(CHead x4 k t0)) \to (\forall (x6: C).((drop h0 n (CHead c0 k (lift h (r k n) +t0)) x6) \to (eq C (CHead x4 k t0) x6)))))) H10 x3 (lift_inj x5 x3 h (r k n) +H11)) in (eq_ind_r T x3 (\lambda (t0: T).(eq C (CHead x4 k t0) (CHead x0 k +x3))) (f_equal3 C K T C CHead x4 x0 k k x3 x3 (sym_eq C x0 x4 (H x0 (r k n) h +H5 x4 H8)) (refl_equal K k) (refl_equal T x3)) x5 (lift_inj x5 x3 h (r k n) +H11))))) x1 H6)) x2 H3)))))) (drop_gen_skip_l c0 x1 t h n k H1))))))) +(drop_gen_skip_l c0 x2 t h n k H2)))))))) d))))))) c). +(* COMMENTS +Initial nodes: 1539 +END *) + +theorem drop_conf_lt: + \forall (k: K).(\forall (i: nat).(\forall (u: T).(\forall (c0: C).(\forall +(c: C).((drop i O c (CHead c0 k u)) \to (\forall (e: C).(\forall (h: +nat).(\forall (d: nat).((drop h (S (plus i d)) c e) \to (ex3_2 T C (\lambda +(v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda +(e0: C).(drop i O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop +h (r k d) c0 e0))))))))))))) +\def + \lambda (k: K).(\lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (u: +T).(\forall (c0: C).(\forall (c: C).((drop n O c (CHead c0 k u)) \to (\forall +(e: C).(\forall (h: nat).(\forall (d: nat).((drop h (S (plus n d)) c e) \to +(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) +(\lambda (v: T).(\lambda (e0: C).(drop n O e (CHead e0 k v)))) (\lambda (_: +T).(\lambda (e0: C).(drop h (r k d) c0 e0))))))))))))) (\lambda (u: +T).(\lambda (c0: C).(\lambda (c: C).(\lambda (H: (drop O O c (CHead c0 k +u))).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H0: (drop +h (S (plus O d)) c e)).(let H1 \def (eq_ind C c (\lambda (c1: C).(drop h (S +(plus O d)) c1 e)) H0 (CHead c0 k u) (drop_gen_refl c (CHead c0 k u) H)) in +(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 k v)))) +(\lambda (_: C).(\lambda (v: T).(eq T u (lift h (r k (plus O d)) v)))) +(\lambda (e0: C).(\lambda (_: T).(drop h (r k (plus O d)) c0 e0))) (ex3_2 T C +(\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: +T).(\lambda (e0: C).(drop O O e (CHead e0 k v)))) (\lambda (_: T).(\lambda +(e0: C).(drop h (r k d) c0 e0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda +(H2: (eq C e (CHead x0 k x1))).(\lambda (H3: (eq T u (lift h (r k (plus O d)) +x1))).(\lambda (H4: (drop h (r k (plus O d)) c0 x0)).(eq_ind_r C (CHead x0 k +x1) (\lambda (c1: C).(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift +h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop O O c1 (CHead e0 k +v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0))))) (eq_ind_r T +(lift h (r k (plus O d)) x1) (\lambda (t: T).(ex3_2 T C (\lambda (v: +T).(\lambda (_: C).(eq T t (lift h (r k d) v)))) (\lambda (v: T).(\lambda +(e0: C).(drop O O (CHead x0 k x1) (CHead e0 k v)))) (\lambda (_: T).(\lambda +(e0: C).(drop h (r k d) c0 e0))))) (ex3_2_intro T C (\lambda (v: T).(\lambda +(_: C).(eq T (lift h (r k (plus O d)) x1) (lift h (r k d) v)))) (\lambda (v: +T).(\lambda (e0: C).(drop O O (CHead x0 k x1) (CHead e0 k v)))) (\lambda (_: +T).(\lambda (e0: C).(drop h (r k d) c0 e0))) x1 x0 (refl_equal T (lift h (r k +d) x1)) (drop_refl (CHead x0 k x1)) H4) u H3) e H2)))))) (drop_gen_skip_l c0 +e u h (plus O d) k H1))))))))))) (\lambda (i0: nat).(\lambda (H: ((\forall +(u: T).(\forall (c0: C).(\forall (c: C).((drop i0 O c (CHead c0 k u)) \to +(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h (S (plus i0 d)) +c e) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) +v)))) (\lambda (v: T).(\lambda (e0: C).(drop i0 O e (CHead e0 k v)))) +(\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0)))))))))))))).(\lambda +(u: T).(\lambda (c0: C).(\lambda (c: C).(C_ind (\lambda (c1: C).((drop (S i0) +O c1 (CHead c0 k u)) \to (\forall (e: C).(\forall (h: nat).(\forall (d: +nat).((drop h (S (plus (S i0) d)) c1 e) \to (ex3_2 T C (\lambda (v: +T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda +(e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: +C).(drop h (r k d) c0 e0)))))))))) (\lambda (n: nat).(\lambda (_: (drop (S +i0) O (CSort n) (CHead c0 k u))).(\lambda (e: C).(\lambda (h: nat).(\lambda +(d: nat).(\lambda (H1: (drop h (S (plus (S i0) d)) (CSort n) e)).(and3_ind +(eq C e (CSort n)) (eq nat h O) (eq nat (S (plus (S i0) d)) O) (ex3_2 T C +(\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: +T).(\lambda (e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda (_: +T).(\lambda (e0: C).(drop h (r k d) c0 e0)))) (\lambda (_: (eq C e (CSort +n))).(\lambda (_: (eq nat h O)).(\lambda (H4: (eq nat (S (plus (S i0) d)) +O)).(let H5 \def (eq_ind nat (S (plus (S i0) d)) (\lambda (ee: nat).(match ee +in nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) +\Rightarrow True])) I O H4) in (False_ind (ex3_2 T C (\lambda (v: T).(\lambda +(_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop +(S i0) O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) +c0 e0)))) H5))))) (drop_gen_sort n h (S (plus (S i0) d)) e H1)))))))) +(\lambda (c1: C).(\lambda (H0: (((drop (S i0) O c1 (CHead c0 k u)) \to +(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h (S (plus (S i0) +d)) c1 e) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k +d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S i0) O e (CHead e0 k v)))) +(\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0))))))))))).(\lambda +(k0: K).(K_ind (\lambda (k1: K).(\forall (t: T).((drop (S i0) O (CHead c1 k1 +t) (CHead c0 k u)) \to (\forall (e: C).(\forall (h: nat).(\forall (d: +nat).((drop h (S (plus (S i0) d)) (CHead c1 k1 t) e) \to (ex3_2 T C (\lambda +(v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda +(e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: +C).(drop h (r k d) c0 e0))))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda +(H1: (drop (S i0) O (CHead c1 (Bind b) t) (CHead c0 k u))).(\lambda (e: +C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2: (drop h (S (plus (S i0) +d)) (CHead c1 (Bind b) t) e)).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: +T).(eq C e (CHead e0 (Bind b) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t +(lift h (r (Bind b) (plus (S i0) d)) v)))) (\lambda (e0: C).(\lambda (_: +T).(drop h (r (Bind b) (plus (S i0) d)) c1 e0))) (ex3_2 T C (\lambda (v: +T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda +(e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: +C).(drop h (r k d) c0 e0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H3: +(eq C e (CHead x0 (Bind b) x1))).(\lambda (_: (eq T t (lift h (r (Bind b) +(plus (S i0) d)) x1))).(\lambda (H5: (drop h (r (Bind b) (plus (S i0) d)) c1 +x0)).(eq_ind_r C (CHead x0 (Bind b) x1) (\lambda (c2: C).(ex3_2 T C (\lambda +(v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda +(e0: C).(drop (S i0) O c2 (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: +C).(drop h (r k d) c0 e0))))) (let H6 \def (H u c0 c1 (drop_gen_drop (Bind b) +c1 (CHead c0 k u) t i0 H1) x0 h d H5) in (ex3_2_ind T C (\lambda (v: +T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda +(e0: C).(drop i0 O x0 (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: +C).(drop h (r k d) c0 e0))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T +u (lift h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S i0) O +(CHead x0 (Bind b) x1) (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: +C).(drop h (r k d) c0 e0)))) (\lambda (x2: T).(\lambda (x3: C).(\lambda (H7: +(eq T u (lift h (r k d) x2))).(\lambda (H8: (drop i0 O x0 (CHead x3 k +x2))).(\lambda (H9: (drop h (r k d) c0 x3)).(ex3_2_intro T C (\lambda (v: +T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda +(e0: C).(drop (S i0) O (CHead x0 (Bind b) x1) (CHead e0 k v)))) (\lambda (_: +T).(\lambda (e0: C).(drop h (r k d) c0 e0))) x2 x3 H7 (drop_drop (Bind b) i0 +x0 (CHead x3 k x2) H8 x1) H9)))))) H6)) e H3)))))) (drop_gen_skip_l c1 e t h +(plus (S i0) d) (Bind b) H2))))))))) (\lambda (f: F).(\lambda (t: T).(\lambda +(H1: (drop (S i0) O (CHead c1 (Flat f) t) (CHead c0 k u))).(\lambda (e: +C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2: (drop h (S (plus (S i0) +d)) (CHead c1 (Flat f) t) e)).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: +T).(eq C e (CHead e0 (Flat f) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t +(lift h (r (Flat f) (plus (S i0) d)) v)))) (\lambda (e0: C).(\lambda (_: +T).(drop h (r (Flat f) (plus (S i0) d)) c1 e0))) (ex3_2 T C (\lambda (v: +T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda +(e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: +C).(drop h (r k d) c0 e0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H3: +(eq C e (CHead x0 (Flat f) x1))).(\lambda (_: (eq T t (lift h (r (Flat f) +(plus (S i0) d)) x1))).(\lambda (H5: (drop h (r (Flat f) (plus (S i0) d)) c1 +x0)).(eq_ind_r C (CHead x0 (Flat f) x1) (\lambda (c2: C).(ex3_2 T C (\lambda +(v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda +(e0: C).(drop (S i0) O c2 (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: +C).(drop h (r k d) c0 e0))))) (ex3_2_ind T C (\lambda (v: T).(\lambda (_: +C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S +i0) O x0 (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) +c0 e0))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) +v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S i0) O (CHead x0 (Flat f) x1) +(CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0)))) +(\lambda (x2: T).(\lambda (x3: C).(\lambda (H6: (eq T u (lift h (r k d) +x2))).(\lambda (H7: (drop (S i0) O x0 (CHead x3 k x2))).(\lambda (H8: (drop h +(r k d) c0 x3)).(ex3_2_intro T C (\lambda (v: T).(\lambda (_: C).(eq T u +(lift h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S i0) O (CHead +x0 (Flat f) x1) (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r +k d) c0 e0))) x2 x3 H6 (drop_drop (Flat f) i0 x0 (CHead x3 k x2) H7 x1) +H8)))))) (H0 (drop_gen_drop (Flat f) c1 (CHead c0 k u) t i0 H1) x0 h d H5)) e +H3)))))) (drop_gen_skip_l c1 e t h (plus (S i0) d) (Flat f) H2))))))))) +k0)))) c)))))) i)). +(* COMMENTS +Initial nodes: 2972 +END *) + +theorem drop_conf_ge: + \forall (i: nat).(\forall (a: C).(\forall (c: C).((drop i O c a) \to +(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to ((le +(plus d h) i) \to (drop (minus i h) O e a))))))))) +\def + \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (a: C).(\forall (c: +C).((drop n O c a) \to (\forall (e: C).(\forall (h: nat).(\forall (d: +nat).((drop h d c e) \to ((le (plus d h) n) \to (drop (minus n h) O e +a)))))))))) (\lambda (a: C).(\lambda (c: C).(\lambda (H: (drop O O c +a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H0: (drop h +d c e)).(\lambda (H1: (le (plus d h) O)).(let H2 \def (eq_ind C c (\lambda +(c0: C).(drop h d c0 e)) H0 a (drop_gen_refl c a H)) in (let H_y \def +(le_n_O_eq (plus d h) H1) in (land_ind (eq nat d O) (eq nat h O) (drop (minus +O h) O e a) (\lambda (H3: (eq nat d O)).(\lambda (H4: (eq nat h O)).(let H5 +\def (eq_ind nat d (\lambda (n: nat).(drop h n a e)) H2 O H3) in (let H6 \def +(eq_ind nat h (\lambda (n: nat).(drop n O a e)) H5 O H4) in (eq_ind_r nat O +(\lambda (n: nat).(drop (minus O n) O e a)) (eq_ind C a (\lambda (c0: +C).(drop (minus O O) O c0 a)) (drop_refl a) e (drop_gen_refl a e H6)) h +H4))))) (plus_O d h (sym_eq nat O (plus d h) H_y))))))))))))) (\lambda (i0: +nat).(\lambda (H: ((\forall (a: C).(\forall (c: C).((drop i0 O c a) \to +(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to ((le +(plus d h) i0) \to (drop (minus i0 h) O e a))))))))))).(\lambda (a: +C).(\lambda (c: C).(C_ind (\lambda (c0: C).((drop (S i0) O c0 a) \to (\forall +(e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c0 e) \to ((le (plus d +h) (S i0)) \to (drop (minus (S i0) h) O e a)))))))) (\lambda (n: +nat).(\lambda (H0: (drop (S i0) O (CSort n) a)).(\lambda (e: C).(\lambda (h: +nat).(\lambda (d: nat).(\lambda (H1: (drop h d (CSort n) e)).(\lambda (H2: +(le (plus d h) (S i0))).(and3_ind (eq C e (CSort n)) (eq nat h O) (eq nat d +O) (drop (minus (S i0) h) O e a) (\lambda (H3: (eq C e (CSort n))).(\lambda +(H4: (eq nat h O)).(\lambda (H5: (eq nat d O)).(and3_ind (eq C a (CSort n)) +(eq nat (S i0) O) (eq nat O O) (drop (minus (S i0) h) O e a) (\lambda (H6: +(eq C a (CSort n))).(\lambda (H7: (eq nat (S i0) O)).(\lambda (_: (eq nat O +O)).(let H9 \def (eq_ind nat d (\lambda (n0: nat).(le (plus n0 h) (S i0))) H2 +O H5) in (let H10 \def (eq_ind nat h (\lambda (n0: nat).(le (plus O n0) (S +i0))) H9 O H4) in (eq_ind_r nat O (\lambda (n0: nat).(drop (minus (S i0) n0) +O e a)) (eq_ind_r C (CSort n) (\lambda (c0: C).(drop (minus (S i0) O) O c0 +a)) (eq_ind_r C (CSort n) (\lambda (c0: C).(drop (minus (S i0) O) O (CSort n) +c0)) (let H11 \def (eq_ind nat (S i0) (\lambda (ee: nat).(match ee in nat +return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow +True])) I O H7) in (False_ind (drop (minus (S i0) O) O (CSort n) (CSort n)) +H11)) a H6) e H3) h H4)))))) (drop_gen_sort n (S i0) O a H0))))) +(drop_gen_sort n h d e H1))))))))) (\lambda (c0: C).(\lambda (H0: (((drop (S +i0) O c0 a) \to (\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h +d c0 e) \to ((le (plus d h) (S i0)) \to (drop (minus (S i0) h) O e +a))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t: T).((drop (S +i0) O (CHead c0 k0 t) a) \to (\forall (e: C).(\forall (h: nat).(\forall (d: +nat).((drop h d (CHead c0 k0 t) e) \to ((le (plus d h) (S i0)) \to (drop +(minus (S i0) h) O e a))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (H1: +(drop (S i0) O (CHead c0 (Bind b) t) a)).(\lambda (e: C).(\lambda (h: +nat).(\lambda (d: nat).(\lambda (H2: (drop h d (CHead c0 (Bind b) t) +e)).(\lambda (H3: (le (plus d h) (S i0))).(nat_ind (\lambda (n: nat).((drop h +n (CHead c0 (Bind b) t) e) \to ((le (plus n h) (S i0)) \to (drop (minus (S +i0) h) O e a)))) (\lambda (H4: (drop h O (CHead c0 (Bind b) t) e)).(\lambda +(H5: (le (plus O h) (S i0))).(nat_ind (\lambda (n: nat).((drop n O (CHead c0 +(Bind b) t) e) \to ((le (plus O n) (S i0)) \to (drop (minus (S i0) n) O e +a)))) (\lambda (H6: (drop O O (CHead c0 (Bind b) t) e)).(\lambda (_: (le +(plus O O) (S i0))).(eq_ind C (CHead c0 (Bind b) t) (\lambda (c1: C).(drop +(minus (S i0) O) O c1 a)) (drop_drop (Bind b) i0 c0 a (drop_gen_drop (Bind b) +c0 a t i0 H1) t) e (drop_gen_refl (CHead c0 (Bind b) t) e H6)))) (\lambda +(h0: nat).(\lambda (_: (((drop h0 O (CHead c0 (Bind b) t) e) \to ((le (plus O +h0) (S i0)) \to (drop (minus (S i0) h0) O e a))))).(\lambda (H6: (drop (S h0) +O (CHead c0 (Bind b) t) e)).(\lambda (H7: (le (plus O (S h0)) (S i0))).(H a +c0 (drop_gen_drop (Bind b) c0 a t i0 H1) e h0 O (drop_gen_drop (Bind b) c0 e +t h0 H6) (le_S_n (plus O h0) i0 H7)))))) h H4 H5))) (\lambda (d0: +nat).(\lambda (_: (((drop h d0 (CHead c0 (Bind b) t) e) \to ((le (plus d0 h) +(S i0)) \to (drop (minus (S i0) h) O e a))))).(\lambda (H4: (drop h (S d0) +(CHead c0 (Bind b) t) e)).(\lambda (H5: (le (plus (S d0) h) (S +i0))).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 (Bind +b) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r (Bind b) d0) +v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r (Bind b) d0) c0 e0))) (drop +(minus (S i0) h) O e a) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C +e (CHead x0 (Bind b) x1))).(\lambda (_: (eq T t (lift h (r (Bind b) d0) +x1))).(\lambda (H8: (drop h (r (Bind b) d0) c0 x0)).(eq_ind_r C (CHead x0 +(Bind b) x1) (\lambda (c1: C).(drop (minus (S i0) h) O c1 a)) (eq_ind nat (S +(minus i0 h)) (\lambda (n: nat).(drop n O (CHead x0 (Bind b) x1) a)) +(drop_drop (Bind b) (minus i0 h) x0 a (H a c0 (drop_gen_drop (Bind b) c0 a t +i0 H1) x0 h d0 H8 (le_S_n (plus d0 h) i0 H5)) x1) (minus (S i0) h) +(minus_Sn_m i0 h (le_trans_plus_r d0 h i0 (le_S_n (plus d0 h) i0 H5)))) e +H6)))))) (drop_gen_skip_l c0 e t h d0 (Bind b) H4)))))) d H2 H3))))))))) +(\lambda (f: F).(\lambda (t: T).(\lambda (H1: (drop (S i0) O (CHead c0 (Flat +f) t) a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2: +(drop h d (CHead c0 (Flat f) t) e)).(\lambda (H3: (le (plus d h) (S +i0))).(nat_ind (\lambda (n: nat).((drop h n (CHead c0 (Flat f) t) e) \to ((le +(plus n h) (S i0)) \to (drop (minus (S i0) h) O e a)))) (\lambda (H4: (drop h +O (CHead c0 (Flat f) t) e)).(\lambda (H5: (le (plus O h) (S i0))).(nat_ind +(\lambda (n: nat).((drop n O (CHead c0 (Flat f) t) e) \to ((le (plus O n) (S +i0)) \to (drop (minus (S i0) n) O e a)))) (\lambda (H6: (drop O O (CHead c0 +(Flat f) t) e)).(\lambda (_: (le (plus O O) (S i0))).(eq_ind C (CHead c0 +(Flat f) t) (\lambda (c1: C).(drop (minus (S i0) O) O c1 a)) (drop_drop (Flat +f) i0 c0 a (drop_gen_drop (Flat f) c0 a t i0 H1) t) e (drop_gen_refl (CHead +c0 (Flat f) t) e H6)))) (\lambda (h0: nat).(\lambda (_: (((drop h0 O (CHead +c0 (Flat f) t) e) \to ((le (plus O h0) (S i0)) \to (drop (minus (S i0) h0) O +e a))))).(\lambda (H6: (drop (S h0) O (CHead c0 (Flat f) t) e)).(\lambda (H7: +(le (plus O (S h0)) (S i0))).(H0 (drop_gen_drop (Flat f) c0 a t i0 H1) e (S +h0) O (drop_gen_drop (Flat f) c0 e t h0 H6) H7))))) h H4 H5))) (\lambda (d0: +nat).(\lambda (_: (((drop h d0 (CHead c0 (Flat f) t) e) \to ((le (plus d0 h) +(S i0)) \to (drop (minus (S i0) h) O e a))))).(\lambda (H4: (drop h (S d0) +(CHead c0 (Flat f) t) e)).(\lambda (H5: (le (plus (S d0) h) (S +i0))).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 (Flat +f) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r (Flat f) d0) +v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r (Flat f) d0) c0 e0))) (drop +(minus (S i0) h) O e a) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C +e (CHead x0 (Flat f) x1))).(\lambda (_: (eq T t (lift h (r (Flat f) d0) +x1))).(\lambda (H8: (drop h (r (Flat f) d0) c0 x0)).(eq_ind_r C (CHead x0 +(Flat f) x1) (\lambda (c1: C).(drop (minus (S i0) h) O c1 a)) (let H9 \def +(eq_ind_r nat (minus (S i0) h) (\lambda (n: nat).(drop n O x0 a)) (H0 +(drop_gen_drop (Flat f) c0 a t i0 H1) x0 h (S d0) H8 H5) (S (minus i0 h)) +(minus_Sn_m i0 h (le_trans_plus_r d0 h i0 (le_S_n (plus d0 h) i0 H5)))) in +(eq_ind nat (S (minus i0 h)) (\lambda (n: nat).(drop n O (CHead x0 (Flat f) +x1) a)) (drop_drop (Flat f) (minus i0 h) x0 a H9 x1) (minus (S i0) h) +(minus_Sn_m i0 h (le_trans_plus_r d0 h i0 (le_S_n (plus d0 h) i0 H5))))) e +H6)))))) (drop_gen_skip_l c0 e t h d0 (Flat f) H4)))))) d H2 H3))))))))) +k)))) c))))) i). +(* COMMENTS +Initial nodes: 2726 +END *) + +theorem drop_conf_rev: + \forall (j: nat).(\forall (e1: C).(\forall (e2: C).((drop j O e1 e2) \to +(\forall (c2: C).(\forall (i: nat).((drop i O c2 e2) \to (ex2 C (\lambda (c1: +C).(drop j O c1 c2)) (\lambda (c1: C).(drop i j c1 e1))))))))) +\def + \lambda (j: nat).(nat_ind (\lambda (n: nat).(\forall (e1: C).(\forall (e2: +C).((drop n O e1 e2) \to (\forall (c2: C).(\forall (i: nat).((drop i O c2 e2) +\to (ex2 C (\lambda (c1: C).(drop n O c1 c2)) (\lambda (c1: C).(drop i n c1 +e1)))))))))) (\lambda (e1: C).(\lambda (e2: C).(\lambda (H: (drop O O e1 +e2)).(\lambda (c2: C).(\lambda (i: nat).(\lambda (H0: (drop i O c2 e2)).(let +H1 \def (eq_ind_r C e2 (\lambda (c: C).(drop i O c2 c)) H0 e1 (drop_gen_refl +e1 e2 H)) in (ex_intro2 C (\lambda (c1: C).(drop O O c1 c2)) (\lambda (c1: +C).(drop i O c1 e1)) c2 (drop_refl c2) H1)))))))) (\lambda (j0: nat).(\lambda +(IHj: ((\forall (e1: C).(\forall (e2: C).((drop j0 O e1 e2) \to (\forall (c2: +C).(\forall (i: nat).((drop i O c2 e2) \to (ex2 C (\lambda (c1: C).(drop j0 O +c1 c2)) (\lambda (c1: C).(drop i j0 c1 e1))))))))))).(\lambda (e1: C).(C_ind +(\lambda (c: C).(\forall (e2: C).((drop (S j0) O c e2) \to (\forall (c2: +C).(\forall (i: nat).((drop i O c2 e2) \to (ex2 C (\lambda (c1: C).(drop (S +j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 c))))))))) (\lambda (n: +nat).(\lambda (e2: C).(\lambda (H: (drop (S j0) O (CSort n) e2)).(\lambda +(c2: C).(\lambda (i: nat).(\lambda (H0: (drop i O c2 e2)).(and3_ind (eq C e2 +(CSort n)) (eq nat (S j0) O) (eq nat O O) (ex2 C (\lambda (c1: C).(drop (S +j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 (CSort n)))) (\lambda (H1: +(eq C e2 (CSort n))).(\lambda (H2: (eq nat (S j0) O)).(\lambda (_: (eq nat O +O)).(let H4 \def (eq_ind C e2 (\lambda (c: C).(drop i O c2 c)) H0 (CSort n) +H1) in (let H5 \def (eq_ind nat (S j0) (\lambda (ee: nat).(match ee in nat +return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow +True])) I O H2) in (False_ind (ex2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) +(\lambda (c1: C).(drop i (S j0) c1 (CSort n)))) H5)))))) (drop_gen_sort n (S +j0) O e2 H)))))))) (\lambda (e2: C).(\lambda (IHe1: ((\forall (e3: C).((drop +(S j0) O e2 e3) \to (\forall (c2: C).(\forall (i: nat).((drop i O c2 e3) \to +(ex2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S +j0) c1 e2)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e3: C).(\lambda +(H: (drop (S j0) O (CHead e2 k t) e3)).(\lambda (c2: C).(\lambda (i: +nat).(\lambda (H0: (drop i O c2 e3)).(K_ind (\lambda (k0: K).((drop (r k0 j0) +O e2 e3) \to (ex2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: +C).(drop i (S j0) c1 (CHead e2 k0 t)))))) (\lambda (b: B).(\lambda (H1: (drop +(r (Bind b) j0) O e2 e3)).(let H_x \def (IHj e2 e3 H1 c2 i H0) in (let H2 +\def H_x in (ex2_ind C (\lambda (c1: C).(drop j0 O c1 c2)) (\lambda (c1: +C).(drop i j0 c1 e2)) (ex2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda +(c1: C).(drop i (S j0) c1 (CHead e2 (Bind b) t)))) (\lambda (x: C).(\lambda +(H3: (drop j0 O x c2)).(\lambda (H4: (drop i j0 x e2)).(ex_intro2 C (\lambda +(c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 (CHead e2 +(Bind b) t))) (CHead x (Bind b) (lift i (r (Bind b) j0) t)) (drop_drop (Bind +b) j0 x c2 H3 (lift i (r (Bind b) j0) t)) (drop_skip (Bind b) i j0 x e2 H4 +t))))) H2))))) (\lambda (f: F).(\lambda (H1: (drop (r (Flat f) j0) O e2 +e3)).(let H_x \def (IHe1 e3 H1 c2 i H0) in (let H2 \def H_x in (ex2_ind C +(\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 +e2)) (ex2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i +(S j0) c1 (CHead e2 (Flat f) t)))) (\lambda (x: C).(\lambda (H3: (drop (S j0) +O x c2)).(\lambda (H4: (drop i (S j0) x e2)).(ex_intro2 C (\lambda (c1: +C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 (CHead e2 (Flat +f) t))) (CHead x (Flat f) (lift i (r (Flat f) j0) t)) (drop_drop (Flat f) j0 +x c2 H3 (lift i (r (Flat f) j0) t)) (drop_skip (Flat f) i j0 x e2 H4 t))))) +H2))))) k (drop_gen_drop k e2 e3 t j0 H))))))))))) e1)))) j). +(* COMMENTS +Initial nodes: 1154 +END *) + +theorem drop_trans_le: + \forall (i: nat).(\forall (d: nat).((le i d) \to (\forall (c1: C).(\forall +(c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((drop i O +c2 e2) \to (ex2 C (\lambda (e1: C).(drop i O c1 e1)) (\lambda (e1: C).(drop h +(minus d i) e1 e2))))))))))) +\def + \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (d: nat).((le n d) \to +(\forall (c1: C).(\forall (c2: C).(\forall (h: nat).((drop h d c1 c2) \to +(\forall (e2: C).((drop n O c2 e2) \to (ex2 C (\lambda (e1: C).(drop n O c1 +e1)) (\lambda (e1: C).(drop h (minus d n) e1 e2)))))))))))) (\lambda (d: +nat).(\lambda (_: (le O d)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (h: +nat).(\lambda (H0: (drop h d c1 c2)).(\lambda (e2: C).(\lambda (H1: (drop O O +c2 e2)).(let H2 \def (eq_ind C c2 (\lambda (c: C).(drop h d c1 c)) H0 e2 +(drop_gen_refl c2 e2 H1)) in (eq_ind nat d (\lambda (n: nat).(ex2 C (\lambda +(e1: C).(drop O O c1 e1)) (\lambda (e1: C).(drop h n e1 e2)))) (ex_intro2 C +(\lambda (e1: C).(drop O O c1 e1)) (\lambda (e1: C).(drop h d e1 e2)) c1 +(drop_refl c1) H2) (minus d O) (minus_n_O d))))))))))) (\lambda (i0: +nat).(\lambda (IHi: ((\forall (d: nat).((le i0 d) \to (\forall (c1: +C).(\forall (c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: +C).((drop i0 O c2 e2) \to (ex2 C (\lambda (e1: C).(drop i0 O c1 e1)) (\lambda +(e1: C).(drop h (minus d i0) e1 e2))))))))))))).(\lambda (d: nat).(nat_ind +(\lambda (n: nat).((le (S i0) n) \to (\forall (c1: C).(\forall (c2: +C).(\forall (h: nat).((drop h n c1 c2) \to (\forall (e2: C).((drop (S i0) O +c2 e2) \to (ex2 C (\lambda (e1: C).(drop (S i0) O c1 e1)) (\lambda (e1: +C).(drop h (minus n (S i0)) e1 e2))))))))))) (\lambda (H: (le (S i0) +O)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (h: nat).(\lambda (_: (drop h +O c1 c2)).(\lambda (e2: C).(\lambda (_: (drop (S i0) O c2 e2)).(ex2_ind nat +(\lambda (n: nat).(eq nat O (S n))) (\lambda (n: nat).(le i0 n)) (ex2 C +(\lambda (e1: C).(drop (S i0) O c1 e1)) (\lambda (e1: C).(drop h (minus O (S +i0)) e1 e2))) (\lambda (x: nat).(\lambda (H2: (eq nat O (S x))).(\lambda (_: +(le i0 x)).(let H4 \def (eq_ind nat O (\lambda (ee: nat).(match ee in nat +return (\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow +False])) I (S x) H2) in (False_ind (ex2 C (\lambda (e1: C).(drop (S i0) O c1 +e1)) (\lambda (e1: C).(drop h (minus O (S i0)) e1 e2))) H4))))) (le_gen_S i0 +O H))))))))) (\lambda (d0: nat).(\lambda (_: (((le (S i0) d0) \to (\forall +(c1: C).(\forall (c2: C).(\forall (h: nat).((drop h d0 c1 c2) \to (\forall +(e2: C).((drop (S i0) O c2 e2) \to (ex2 C (\lambda (e1: C).(drop (S i0) O c1 +e1)) (\lambda (e1: C).(drop h (minus d0 (S i0)) e1 e2)))))))))))).(\lambda +(H: (le (S i0) (S d0))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: +C).(\forall (h: nat).((drop h (S d0) c c2) \to (\forall (e2: C).((drop (S i0) +O c2 e2) \to (ex2 C (\lambda (e1: C).(drop (S i0) O c e1)) (\lambda (e1: +C).(drop h (minus (S d0) (S i0)) e1 e2))))))))) (\lambda (n: nat).(\lambda +(c2: C).(\lambda (h: nat).(\lambda (H0: (drop h (S d0) (CSort n) +c2)).(\lambda (e2: C).(\lambda (H1: (drop (S i0) O c2 e2)).(and3_ind (eq C c2 +(CSort n)) (eq nat h O) (eq nat (S d0) O) (ex2 C (\lambda (e1: C).(drop (S +i0) O (CSort n) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2))) +(\lambda (H2: (eq C c2 (CSort n))).(\lambda (_: (eq nat h O)).(\lambda (_: +(eq nat (S d0) O)).(let H5 \def (eq_ind C c2 (\lambda (c: C).(drop (S i0) O c +e2)) H1 (CSort n) H2) in (and3_ind (eq C e2 (CSort n)) (eq nat (S i0) O) (eq +nat O O) (ex2 C (\lambda (e1: C).(drop (S i0) O (CSort n) e1)) (\lambda (e1: +C).(drop h (minus (S d0) (S i0)) e1 e2))) (\lambda (H6: (eq C e2 (CSort +n))).(\lambda (H7: (eq nat (S i0) O)).(\lambda (_: (eq nat O O)).(eq_ind_r C +(CSort n) (\lambda (c: C).(ex2 C (\lambda (e1: C).(drop (S i0) O (CSort n) +e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 c)))) (let H9 \def +(eq_ind nat (S i0) (\lambda (ee: nat).(match ee in nat return (\lambda (_: +nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H7) in +(False_ind (ex2 C (\lambda (e1: C).(drop (S i0) O (CSort n) e1)) (\lambda +(e1: C).(drop h (minus (S d0) (S i0)) e1 (CSort n)))) H9)) e2 H6)))) +(drop_gen_sort n (S i0) O e2 H5)))))) (drop_gen_sort n h (S d0) c2 H0)))))))) +(\lambda (c2: C).(\lambda (IHc: ((\forall (c3: C).(\forall (h: nat).((drop h +(S d0) c2 c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to (ex2 C (\lambda +(e1: C).(drop (S i0) O c2 e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) +e1 e2)))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t: +T).(\forall (c3: C).(\forall (h: nat).((drop h (S d0) (CHead c2 k0 t) c3) \to +(\forall (e2: C).((drop (S i0) O c3 e2) \to (ex2 C (\lambda (e1: C).(drop (S +i0) O (CHead c2 k0 t) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 +e2)))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (c3: C).(\lambda (h: +nat).(\lambda (H0: (drop h (S d0) (CHead c2 (Bind b) t) c3)).(\lambda (e2: +C).(\lambda (H1: (drop (S i0) O c3 e2)).(ex3_2_ind C T (\lambda (e: +C).(\lambda (v: T).(eq C c3 (CHead e (Bind b) v)))) (\lambda (_: C).(\lambda +(v: T).(eq T t (lift h (r (Bind b) d0) v)))) (\lambda (e: C).(\lambda (_: +T).(drop h (r (Bind b) d0) c2 e))) (ex2 C (\lambda (e1: C).(drop (S i0) O +(CHead c2 (Bind b) t) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 +e2))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H2: (eq C c3 (CHead x0 +(Bind b) x1))).(\lambda (H3: (eq T t (lift h (r (Bind b) d0) x1))).(\lambda +(H4: (drop h (r (Bind b) d0) c2 x0)).(let H5 \def (eq_ind C c3 (\lambda (c: +C).(drop (S i0) O c e2)) H1 (CHead x0 (Bind b) x1) H2) in (eq_ind_r T (lift h +(r (Bind b) d0) x1) (\lambda (t0: T).(ex2 C (\lambda (e1: C).(drop (S i0) O +(CHead c2 (Bind b) t0) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 +e2)))) (ex2_ind C (\lambda (e1: C).(drop i0 O c2 e1)) (\lambda (e1: C).(drop +h (minus d0 i0) e1 e2)) (ex2 C (\lambda (e1: C).(drop (S i0) O (CHead c2 +(Bind b) (lift h (r (Bind b) d0) x1)) e1)) (\lambda (e1: C).(drop h (minus (S +d0) (S i0)) e1 e2))) (\lambda (x: C).(\lambda (H6: (drop i0 O c2 x)).(\lambda +(H7: (drop h (minus d0 i0) x e2)).(ex_intro2 C (\lambda (e1: C).(drop (S i0) +O (CHead c2 (Bind b) (lift h (r (Bind b) d0) x1)) e1)) (\lambda (e1: C).(drop +h (minus (S d0) (S i0)) e1 e2)) x (drop_drop (Bind b) i0 c2 x H6 (lift h (r +(Bind b) d0) x1)) H7)))) (IHi d0 (le_S_n i0 d0 H) c2 x0 h H4 e2 +(drop_gen_drop (Bind b) x0 e2 x1 i0 H5))) t H3))))))) (drop_gen_skip_l c2 c3 +t h d0 (Bind b) H0))))))))) (\lambda (f: F).(\lambda (t: T).(\lambda (c3: +C).(\lambda (h: nat).(\lambda (H0: (drop h (S d0) (CHead c2 (Flat f) t) +c3)).(\lambda (e2: C).(\lambda (H1: (drop (S i0) O c3 e2)).(ex3_2_ind C T +(\lambda (e: C).(\lambda (v: T).(eq C c3 (CHead e (Flat f) v)))) (\lambda (_: +C).(\lambda (v: T).(eq T t (lift h (r (Flat f) d0) v)))) (\lambda (e: +C).(\lambda (_: T).(drop h (r (Flat f) d0) c2 e))) (ex2 C (\lambda (e1: +C).(drop (S i0) O (CHead c2 (Flat f) t) e1)) (\lambda (e1: C).(drop h (minus +(S d0) (S i0)) e1 e2))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H2: (eq C +c3 (CHead x0 (Flat f) x1))).(\lambda (H3: (eq T t (lift h (r (Flat f) d0) +x1))).(\lambda (H4: (drop h (r (Flat f) d0) c2 x0)).(let H5 \def (eq_ind C c3 +(\lambda (c: C).(drop (S i0) O c e2)) H1 (CHead x0 (Flat f) x1) H2) in +(eq_ind_r T (lift h (r (Flat f) d0) x1) (\lambda (t0: T).(ex2 C (\lambda (e1: +C).(drop (S i0) O (CHead c2 (Flat f) t0) e1)) (\lambda (e1: C).(drop h (minus +(S d0) (S i0)) e1 e2)))) (ex2_ind C (\lambda (e1: C).(drop (S i0) O c2 e1)) +(\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2)) (ex2 C (\lambda (e1: +C).(drop (S i0) O (CHead c2 (Flat f) (lift h (r (Flat f) d0) x1)) e1)) +(\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2))) (\lambda (x: +C).(\lambda (H6: (drop (S i0) O c2 x)).(\lambda (H7: (drop h (minus (S d0) (S +i0)) x e2)).(ex_intro2 C (\lambda (e1: C).(drop (S i0) O (CHead c2 (Flat f) +(lift h (r (Flat f) d0) x1)) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S +i0)) e1 e2)) x (drop_drop (Flat f) i0 c2 x H6 (lift h (r (Flat f) d0) x1)) +H7)))) (IHc x0 h H4 e2 (drop_gen_drop (Flat f) x0 e2 x1 i0 H5))) t H3))))))) +(drop_gen_skip_l c2 c3 t h d0 (Flat f) H0))))))))) k)))) c1))))) d)))) i). +(* COMMENTS +Initial nodes: 2453 +END *) + +theorem drop_trans_ge: + \forall (i: nat).(\forall (c1: C).(\forall (c2: C).(\forall (d: +nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((drop i O c2 +e2) \to ((le d i) \to (drop (plus i h) O c1 e2))))))))) +\def + \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (c2: +C).(\forall (d: nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: +C).((drop n O c2 e2) \to ((le d n) \to (drop (plus n h) O c1 e2)))))))))) +(\lambda (c1: C).(\lambda (c2: C).(\lambda (d: nat).(\lambda (h: +nat).(\lambda (H: (drop h d c1 c2)).(\lambda (e2: C).(\lambda (H0: (drop O O +c2 e2)).(\lambda (H1: (le d O)).(eq_ind C c2 (\lambda (c: C).(drop (plus O h) +O c1 c)) (let H_y \def (le_n_O_eq d H1) in (let H2 \def (eq_ind_r nat d +(\lambda (n: nat).(drop h n c1 c2)) H O H_y) in H2)) e2 (drop_gen_refl c2 e2 +H0)))))))))) (\lambda (i0: nat).(\lambda (IHi: ((\forall (c1: C).(\forall +(c2: C).(\forall (d: nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall +(e2: C).((drop i0 O c2 e2) \to ((le d i0) \to (drop (plus i0 h) O c1 +e2))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: +C).(\forall (d: nat).(\forall (h: nat).((drop h d c c2) \to (\forall (e2: +C).((drop (S i0) O c2 e2) \to ((le d (S i0)) \to (drop (plus (S i0) h) O c +e2))))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (d: nat).(\lambda (h: +nat).(\lambda (H: (drop h d (CSort n) c2)).(\lambda (e2: C).(\lambda (H0: +(drop (S i0) O c2 e2)).(\lambda (H1: (le d (S i0))).(and3_ind (eq C c2 (CSort +n)) (eq nat h O) (eq nat d O) (drop (S (plus i0 h)) O (CSort n) e2) (\lambda +(H2: (eq C c2 (CSort n))).(\lambda (H3: (eq nat h O)).(\lambda (H4: (eq nat d +O)).(eq_ind_r nat O (\lambda (n0: nat).(drop (S (plus i0 n0)) O (CSort n) +e2)) (let H5 \def (eq_ind nat d (\lambda (n0: nat).(le n0 (S i0))) H1 O H4) +in (let H6 \def (eq_ind C c2 (\lambda (c: C).(drop (S i0) O c e2)) H0 (CSort +n) H2) in (and3_ind (eq C e2 (CSort n)) (eq nat (S i0) O) (eq nat O O) (drop +(S (plus i0 O)) O (CSort n) e2) (\lambda (H7: (eq C e2 (CSort n))).(\lambda +(H8: (eq nat (S i0) O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n) +(\lambda (c: C).(drop (S (plus i0 O)) O (CSort n) c)) (let H10 \def (eq_ind +nat (S i0) (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) +with [O \Rightarrow False | (S _) \Rightarrow True])) I O H8) in (False_ind +(drop (S (plus i0 O)) O (CSort n) (CSort n)) H10)) e2 H7)))) (drop_gen_sort n +(S i0) O e2 H6)))) h H3)))) (drop_gen_sort n h d c2 H)))))))))) (\lambda (c2: +C).(\lambda (IHc: ((\forall (c3: C).(\forall (d: nat).(\forall (h: +nat).((drop h d c2 c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to ((le d +(S i0)) \to (drop (S (plus i0 h)) O c2 e2)))))))))).(\lambda (k: K).(\lambda +(t: T).(\lambda (c3: C).(\lambda (d: nat).(nat_ind (\lambda (n: nat).(\forall +(h: nat).((drop h n (CHead c2 k t) c3) \to (\forall (e2: C).((drop (S i0) O +c3 e2) \to ((le n (S i0)) \to (drop (S (plus i0 h)) O (CHead c2 k t) +e2))))))) (\lambda (h: nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c2 k +t) c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to ((le O (S i0)) \to +(drop (S (plus i0 n)) O (CHead c2 k t) e2)))))) (\lambda (H: (drop O O (CHead +c2 k t) c3)).(\lambda (e2: C).(\lambda (H0: (drop (S i0) O c3 e2)).(\lambda +(_: (le O (S i0))).(let H2 \def (eq_ind_r C c3 (\lambda (c: C).(drop (S i0) O +c e2)) H0 (CHead c2 k t) (drop_gen_refl (CHead c2 k t) c3 H)) in (eq_ind nat +i0 (\lambda (n: nat).(drop (S n) O (CHead c2 k t) e2)) (drop_drop k i0 c2 e2 +(drop_gen_drop k c2 e2 t i0 H2) t) (plus i0 O) (plus_n_O i0))))))) (\lambda +(n: nat).(\lambda (_: (((drop n O (CHead c2 k t) c3) \to (\forall (e2: +C).((drop (S i0) O c3 e2) \to ((le O (S i0)) \to (drop (S (plus i0 n)) O +(CHead c2 k t) e2))))))).(\lambda (H0: (drop (S n) O (CHead c2 k t) +c3)).(\lambda (e2: C).(\lambda (H1: (drop (S i0) O c3 e2)).(\lambda (H2: (le +O (S i0))).(eq_ind nat (S (plus i0 n)) (\lambda (n0: nat).(drop (S n0) O +(CHead c2 k t) e2)) (drop_drop k (S (plus i0 n)) c2 e2 (eq_ind_r nat (S (r k +(plus i0 n))) (\lambda (n0: nat).(drop n0 O c2 e2)) (eq_ind_r nat (plus i0 (r +k n)) (\lambda (n0: nat).(drop (S n0) O c2 e2)) (IHc c3 O (r k n) +(drop_gen_drop k c2 c3 t n H0) e2 H1 H2) (r k (plus i0 n)) (r_plus_sym k i0 +n)) (r k (S (plus i0 n))) (r_S k (plus i0 n))) t) (plus i0 (S n)) (plus_n_Sm +i0 n)))))))) h)) (\lambda (d0: nat).(\lambda (IHd: ((\forall (h: nat).((drop +h d0 (CHead c2 k t) c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to ((le +d0 (S i0)) \to (drop (S (plus i0 h)) O (CHead c2 k t) e2)))))))).(\lambda (h: +nat).(\lambda (H: (drop h (S d0) (CHead c2 k t) c3)).(\lambda (e2: +C).(\lambda (H0: (drop (S i0) O c3 e2)).(\lambda (H1: (le (S d0) (S +i0))).(ex3_2_ind C T (\lambda (e: C).(\lambda (v: T).(eq C c3 (CHead e k +v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r k d0) v)))) (\lambda +(e: C).(\lambda (_: T).(drop h (r k d0) c2 e))) (drop (S (plus i0 h)) O +(CHead c2 k t) e2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H2: (eq C c3 +(CHead x0 k x1))).(\lambda (H3: (eq T t (lift h (r k d0) x1))).(\lambda (H4: +(drop h (r k d0) c2 x0)).(let H5 \def (eq_ind C c3 (\lambda (c: C).(\forall +(h0: nat).((drop h0 d0 (CHead c2 k t) c) \to (\forall (e3: C).((drop (S i0) O +c e3) \to ((le d0 (S i0)) \to (drop (S (plus i0 h0)) O (CHead c2 k t) +e3))))))) IHd (CHead x0 k x1) H2) in (let H6 \def (eq_ind C c3 (\lambda (c: +C).(drop (S i0) O c e2)) H0 (CHead x0 k x1) H2) in (let H7 \def (eq_ind T t +(\lambda (t0: T).(\forall (h0: nat).((drop h0 d0 (CHead c2 k t0) (CHead x0 k +x1)) \to (\forall (e3: C).((drop (S i0) O (CHead x0 k x1) e3) \to ((le d0 (S +i0)) \to (drop (S (plus i0 h0)) O (CHead c2 k t0) e3))))))) H5 (lift h (r k +d0) x1) H3) in (eq_ind_r T (lift h (r k d0) x1) (\lambda (t0: T).(drop (S +(plus i0 h)) O (CHead c2 k t0) e2)) (drop_drop k (plus i0 h) c2 e2 (K_ind +(\lambda (k0: K).((drop h (r k0 d0) c2 x0) \to ((drop (r k0 i0) O x0 e2) \to +(drop (r k0 (plus i0 h)) O c2 e2)))) (\lambda (b: B).(\lambda (H8: (drop h (r +(Bind b) d0) c2 x0)).(\lambda (H9: (drop (r (Bind b) i0) O x0 e2)).(IHi c2 x0 +(r (Bind b) d0) h H8 e2 H9 (le_S_n (r (Bind b) d0) i0 H1))))) (\lambda (f: +F).(\lambda (H8: (drop h (r (Flat f) d0) c2 x0)).(\lambda (H9: (drop (r (Flat +f) i0) O x0 e2)).(IHc x0 (r (Flat f) d0) h H8 e2 H9 H1)))) k H4 +(drop_gen_drop k x0 e2 x1 i0 H6)) (lift h (r k d0) x1)) t H3))))))))) +(drop_gen_skip_l c2 c3 t h d0 k H))))))))) d))))))) c1)))) i). +(* COMMENTS +Initial nodes: 2020 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/drop1/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/drop1/defs.ma new file mode 100644 index 000000000..c0e14f438 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/drop1/defs.ma @@ -0,0 +1,35 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/drop/defs.ma". + +include "Basic-1/lift1/defs.ma". + +inductive drop1: PList \to (C \to (C \to Prop)) \def +| drop1_nil: \forall (c: C).(drop1 PNil c c) +| drop1_cons: \forall (c1: C).(\forall (c2: C).(\forall (h: nat).(\forall (d: +nat).((drop h d c1 c2) \to (\forall (c3: C).(\forall (hds: PList).((drop1 hds +c2 c3) \to (drop1 (PCons h d hds) c1 c3)))))))). + +definition ptrans: + PList \to (nat \to PList) +\def + let rec ptrans (hds: PList) on hds: (nat \to PList) \def (\lambda (i: +nat).(match hds with [PNil \Rightarrow PNil | (PCons h d hds0) \Rightarrow +(let j \def (trans hds0 i) in (let q \def (ptrans hds0 i) in (match (blt j d) +with [true \Rightarrow (PCons h (minus d (S j)) q) | false \Rightarrow +q])))])) in ptrans. + diff --git a/matita/matita/contribs/lambdadelta/basic_1/drop1/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/drop1/fwd.ma new file mode 100644 index 000000000..6e4d1789e --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/drop1/fwd.ma @@ -0,0 +1,81 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/drop1/defs.ma". + +theorem drop1_gen_pnil: + \forall (c1: C).(\forall (c2: C).((drop1 PNil c1 c2) \to (eq C c1 c2))) +\def + \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (drop1 PNil c1 c2)).(insert_eq +PList PNil (\lambda (p: PList).(drop1 p c1 c2)) (\lambda (_: PList).(eq C c1 +c2)) (\lambda (y: PList).(\lambda (H0: (drop1 y c1 c2)).(drop1_ind (\lambda +(p: PList).(\lambda (c: C).(\lambda (c0: C).((eq PList p PNil) \to (eq C c +c0))))) (\lambda (c: C).(\lambda (_: (eq PList PNil PNil)).(refl_equal C c))) +(\lambda (c3: C).(\lambda (c4: C).(\lambda (h: nat).(\lambda (d: +nat).(\lambda (_: (drop h d c3 c4)).(\lambda (c5: C).(\lambda (hds: +PList).(\lambda (_: (drop1 hds c4 c5)).(\lambda (_: (((eq PList hds PNil) \to +(eq C c4 c5)))).(\lambda (H4: (eq PList (PCons h d hds) PNil)).(let H5 \def +(eq_ind PList (PCons h d hds) (\lambda (ee: PList).(match ee in PList return +(\lambda (_: PList).Prop) with [PNil \Rightarrow False | (PCons _ _ _) +\Rightarrow True])) I PNil H4) in (False_ind (eq C c3 c5) H5)))))))))))) y c1 +c2 H0))) H))). +(* COMMENTS +Initial nodes: 198 +END *) + +theorem drop1_gen_pcons: + \forall (c1: C).(\forall (c3: C).(\forall (hds: PList).(\forall (h: +nat).(\forall (d: nat).((drop1 (PCons h d hds) c1 c3) \to (ex2 C (\lambda +(c2: C).(drop h d c1 c2)) (\lambda (c2: C).(drop1 hds c2 c3)))))))) +\def + \lambda (c1: C).(\lambda (c3: C).(\lambda (hds: PList).(\lambda (h: +nat).(\lambda (d: nat).(\lambda (H: (drop1 (PCons h d hds) c1 c3)).(insert_eq +PList (PCons h d hds) (\lambda (p: PList).(drop1 p c1 c3)) (\lambda (_: +PList).(ex2 C (\lambda (c2: C).(drop h d c1 c2)) (\lambda (c2: C).(drop1 hds +c2 c3)))) (\lambda (y: PList).(\lambda (H0: (drop1 y c1 c3)).(drop1_ind +(\lambda (p: PList).(\lambda (c: C).(\lambda (c0: C).((eq PList p (PCons h d +hds)) \to (ex2 C (\lambda (c2: C).(drop h d c c2)) (\lambda (c2: C).(drop1 +hds c2 c0))))))) (\lambda (c: C).(\lambda (H1: (eq PList PNil (PCons h d +hds))).(let H2 \def (eq_ind PList PNil (\lambda (ee: PList).(match ee in +PList return (\lambda (_: PList).Prop) with [PNil \Rightarrow True | (PCons _ +_ _) \Rightarrow False])) I (PCons h d hds) H1) in (False_ind (ex2 C (\lambda +(c2: C).(drop h d c c2)) (\lambda (c2: C).(drop1 hds c2 c))) H2)))) (\lambda +(c2: C).(\lambda (c4: C).(\lambda (h0: nat).(\lambda (d0: nat).(\lambda (H1: +(drop h0 d0 c2 c4)).(\lambda (c5: C).(\lambda (hds0: PList).(\lambda (H2: +(drop1 hds0 c4 c5)).(\lambda (H3: (((eq PList hds0 (PCons h d hds)) \to (ex2 +C (\lambda (c6: C).(drop h d c4 c6)) (\lambda (c6: C).(drop1 hds c6 +c5)))))).(\lambda (H4: (eq PList (PCons h0 d0 hds0) (PCons h d hds))).(let H5 +\def (f_equal PList nat (\lambda (e: PList).(match e in PList return (\lambda +(_: PList).nat) with [PNil \Rightarrow h0 | (PCons n _ _) \Rightarrow n])) +(PCons h0 d0 hds0) (PCons h d hds) H4) in ((let H6 \def (f_equal PList nat +(\lambda (e: PList).(match e in PList return (\lambda (_: PList).nat) with +[PNil \Rightarrow d0 | (PCons _ n _) \Rightarrow n])) (PCons h0 d0 hds0) +(PCons h d hds) H4) in ((let H7 \def (f_equal PList PList (\lambda (e: +PList).(match e in PList return (\lambda (_: PList).PList) with [PNil +\Rightarrow hds0 | (PCons _ _ p) \Rightarrow p])) (PCons h0 d0 hds0) (PCons h +d hds) H4) in (\lambda (H8: (eq nat d0 d)).(\lambda (H9: (eq nat h0 h)).(let +H10 \def (eq_ind PList hds0 (\lambda (p: PList).((eq PList p (PCons h d hds)) +\to (ex2 C (\lambda (c6: C).(drop h d c4 c6)) (\lambda (c6: C).(drop1 hds c6 +c5))))) H3 hds H7) in (let H11 \def (eq_ind PList hds0 (\lambda (p: +PList).(drop1 p c4 c5)) H2 hds H7) in (let H12 \def (eq_ind nat d0 (\lambda +(n: nat).(drop h0 n c2 c4)) H1 d H8) in (let H13 \def (eq_ind nat h0 (\lambda +(n: nat).(drop n d c2 c4)) H12 h H9) in (ex_intro2 C (\lambda (c6: C).(drop h +d c2 c6)) (\lambda (c6: C).(drop1 hds c6 c5)) c4 H13 H11)))))))) H6)) +H5)))))))))))) y c1 c3 H0))) H)))))). +(* COMMENTS +Initial nodes: 587 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/drop1/getl.ma b/matita/matita/contribs/lambdadelta/basic_1/drop1/getl.ma new file mode 100644 index 000000000..dbee79850 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/drop1/getl.ma @@ -0,0 +1,110 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/drop1/fwd.ma". + +include "Basic-1/getl/drop.ma". + +theorem drop1_getl_trans: + \forall (hds: PList).(\forall (c1: C).(\forall (c2: C).((drop1 hds c2 c1) +\to (\forall (b: B).(\forall (e1: C).(\forall (v: T).(\forall (i: nat).((getl +i c1 (CHead e1 (Bind b) v)) \to (ex2 C (\lambda (e2: C).(drop1 (ptrans hds i) +e2 e1)) (\lambda (e2: C).(getl (trans hds i) c2 (CHead e2 (Bind b) (lift1 +(ptrans hds i) v))))))))))))) +\def + \lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (c1: +C).(\forall (c2: C).((drop1 p c2 c1) \to (\forall (b: B).(\forall (e1: +C).(\forall (v: T).(\forall (i: nat).((getl i c1 (CHead e1 (Bind b) v)) \to +(ex2 C (\lambda (e2: C).(drop1 (ptrans p i) e2 e1)) (\lambda (e2: C).(getl +(trans p i) c2 (CHead e2 (Bind b) (lift1 (ptrans p i) v)))))))))))))) +(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (drop1 PNil c2 c1)).(\lambda +(b: B).(\lambda (e1: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H0: (getl +i c1 (CHead e1 (Bind b) v))).(let H_y \def (drop1_gen_pnil c2 c1 H) in +(eq_ind_r C c1 (\lambda (c: C).(ex2 C (\lambda (e2: C).(drop1 PNil e2 e1)) +(\lambda (e2: C).(getl i c (CHead e2 (Bind b) v))))) (ex_intro2 C (\lambda +(e2: C).(drop1 PNil e2 e1)) (\lambda (e2: C).(getl i c1 (CHead e2 (Bind b) +v))) e1 (drop1_nil e1) H0) c2 H_y)))))))))) (\lambda (h: nat).(\lambda (d: +nat).(\lambda (hds0: PList).(\lambda (H: ((\forall (c1: C).(\forall (c2: +C).((drop1 hds0 c2 c1) \to (\forall (b: B).(\forall (e1: C).(\forall (v: +T).(\forall (i: nat).((getl i c1 (CHead e1 (Bind b) v)) \to (ex2 C (\lambda +(e2: C).(drop1 (ptrans hds0 i) e2 e1)) (\lambda (e2: C).(getl (trans hds0 i) +c2 (CHead e2 (Bind b) (lift1 (ptrans hds0 i) v))))))))))))))).(\lambda (c1: +C).(\lambda (c2: C).(\lambda (H0: (drop1 (PCons h d hds0) c2 c1)).(\lambda +(b: B).(\lambda (e1: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H1: (getl +i c1 (CHead e1 (Bind b) v))).(let H_x \def (drop1_gen_pcons c2 c1 hds0 h d +H0) in (let H2 \def H_x in (ex2_ind C (\lambda (c3: C).(drop h d c2 c3)) +(\lambda (c3: C).(drop1 hds0 c3 c1)) (ex2 C (\lambda (e2: C).(drop1 (match +(blt (trans hds0 i) d) with [true \Rightarrow (PCons h (minus d (S (trans +hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)]) e2 e1)) +(\lambda (e2: C).(getl (match (blt (trans hds0 i) d) with [true \Rightarrow +(trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 +(Bind b) (lift1 (match (blt (trans hds0 i) d) with [true \Rightarrow (PCons h +(minus d (S (trans hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans +hds0 i)]) v))))) (\lambda (x: C).(\lambda (H3: (drop h d c2 x)).(\lambda (H4: +(drop1 hds0 x c1)).(xinduction bool (blt (trans hds0 i) d) (\lambda (b0: +bool).(ex2 C (\lambda (e2: C).(drop1 (match b0 with [true \Rightarrow (PCons +h (minus d (S (trans hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans +hds0 i)]) e2 e1)) (\lambda (e2: C).(getl (match b0 with [true \Rightarrow +(trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 +(Bind b) (lift1 (match b0 with [true \Rightarrow (PCons h (minus d (S (trans +hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)]) v)))))) +(\lambda (x_x: bool).(bool_ind (\lambda (b0: bool).((eq bool (blt (trans hds0 +i) d) b0) \to (ex2 C (\lambda (e2: C).(drop1 (match b0 with [true \Rightarrow +(PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) | false \Rightarrow +(ptrans hds0 i)]) e2 e1)) (\lambda (e2: C).(getl (match b0 with [true +\Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2 +(CHead e2 (Bind b) (lift1 (match b0 with [true \Rightarrow (PCons h (minus d +(S (trans hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)]) +v))))))) (\lambda (H5: (eq bool (blt (trans hds0 i) d) true)).(let H_x0 \def +(H c1 x H4 b e1 v i H1) in (let H6 \def H_x0 in (ex2_ind C (\lambda (e2: +C).(drop1 (ptrans hds0 i) e2 e1)) (\lambda (e2: C).(getl (trans hds0 i) x +(CHead e2 (Bind b) (lift1 (ptrans hds0 i) v)))) (ex2 C (\lambda (e2: +C).(drop1 (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) e2 e1)) +(\lambda (e2: C).(getl (trans hds0 i) c2 (CHead e2 (Bind b) (lift1 (PCons h +(minus d (S (trans hds0 i))) (ptrans hds0 i)) v))))) (\lambda (x0: +C).(\lambda (H7: (drop1 (ptrans hds0 i) x0 e1)).(\lambda (H8: (getl (trans +hds0 i) x (CHead x0 (Bind b) (lift1 (ptrans hds0 i) v)))).(let H_x1 \def +(drop_getl_trans_lt (trans hds0 i) d (blt_lt d (trans hds0 i) H5) c2 x h H3 b +x0 (lift1 (ptrans hds0 i) v) H8) in (let H9 \def H_x1 in (ex2_ind C (\lambda +(e2: C).(getl (trans hds0 i) c2 (CHead e2 (Bind b) (lift h (minus d (S (trans +hds0 i))) (lift1 (ptrans hds0 i) v))))) (\lambda (e2: C).(drop h (minus d (S +(trans hds0 i))) e2 x0)) (ex2 C (\lambda (e2: C).(drop1 (PCons h (minus d (S +(trans hds0 i))) (ptrans hds0 i)) e2 e1)) (\lambda (e2: C).(getl (trans hds0 +i) c2 (CHead e2 (Bind b) (lift1 (PCons h (minus d (S (trans hds0 i))) (ptrans +hds0 i)) v))))) (\lambda (x1: C).(\lambda (H10: (getl (trans hds0 i) c2 +(CHead x1 (Bind b) (lift h (minus d (S (trans hds0 i))) (lift1 (ptrans hds0 +i) v))))).(\lambda (H11: (drop h (minus d (S (trans hds0 i))) x1 +x0)).(ex_intro2 C (\lambda (e2: C).(drop1 (PCons h (minus d (S (trans hds0 +i))) (ptrans hds0 i)) e2 e1)) (\lambda (e2: C).(getl (trans hds0 i) c2 (CHead +e2 (Bind b) (lift1 (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) +v)))) x1 (drop1_cons x1 x0 h (minus d (S (trans hds0 i))) H11 e1 (ptrans hds0 +i) H7) H10)))) H9)))))) H6)))) (\lambda (H5: (eq bool (blt (trans hds0 i) d) +false)).(let H_x0 \def (H c1 x H4 b e1 v i H1) in (let H6 \def H_x0 in +(ex2_ind C (\lambda (e2: C).(drop1 (ptrans hds0 i) e2 e1)) (\lambda (e2: +C).(getl (trans hds0 i) x (CHead e2 (Bind b) (lift1 (ptrans hds0 i) v)))) +(ex2 C (\lambda (e2: C).(drop1 (ptrans hds0 i) e2 e1)) (\lambda (e2: C).(getl +(plus (trans hds0 i) h) c2 (CHead e2 (Bind b) (lift1 (ptrans hds0 i) v))))) +(\lambda (x0: C).(\lambda (H7: (drop1 (ptrans hds0 i) x0 e1)).(\lambda (H8: +(getl (trans hds0 i) x (CHead x0 (Bind b) (lift1 (ptrans hds0 i) v)))).(let +H9 \def (drop_getl_trans_ge (trans hds0 i) c2 x d h H3 (CHead x0 (Bind b) +(lift1 (ptrans hds0 i) v)) H8) in (ex_intro2 C (\lambda (e2: C).(drop1 +(ptrans hds0 i) e2 e1)) (\lambda (e2: C).(getl (plus (trans hds0 i) h) c2 +(CHead e2 (Bind b) (lift1 (ptrans hds0 i) v)))) x0 H7 (H9 (bge_le d (trans +hds0 i) H5))))))) H6)))) x_x)))))) H2))))))))))))))) hds). +(* COMMENTS +Initial nodes: 1674 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/drop1/props.ma b/matita/matita/contribs/lambdadelta/basic_1/drop1/props.ma new file mode 100644 index 000000000..240219b95 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/drop1/props.ma @@ -0,0 +1,97 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/drop1/fwd.ma". + +include "Basic-1/drop/props.ma". + +include "Basic-1/getl/defs.ma". + +theorem drop1_skip_bind: + \forall (b: B).(\forall (e: C).(\forall (hds: PList).(\forall (c: +C).(\forall (u: T).((drop1 hds c e) \to (drop1 (Ss hds) (CHead c (Bind b) +(lift1 hds u)) (CHead e (Bind b) u))))))) +\def + \lambda (b: B).(\lambda (e: C).(\lambda (hds: PList).(PList_ind (\lambda (p: +PList).(\forall (c: C).(\forall (u: T).((drop1 p c e) \to (drop1 (Ss p) +(CHead c (Bind b) (lift1 p u)) (CHead e (Bind b) u)))))) (\lambda (c: +C).(\lambda (u: T).(\lambda (H: (drop1 PNil c e)).(let H_y \def +(drop1_gen_pnil c e H) in (eq_ind_r C e (\lambda (c0: C).(drop1 PNil (CHead +c0 (Bind b) u) (CHead e (Bind b) u))) (drop1_nil (CHead e (Bind b) u)) c +H_y))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda +(H: ((\forall (c: C).(\forall (u: T).((drop1 p c e) \to (drop1 (Ss p) (CHead +c (Bind b) (lift1 p u)) (CHead e (Bind b) u))))))).(\lambda (c: C).(\lambda +(u: T).(\lambda (H0: (drop1 (PCons n n0 p) c e)).(let H_x \def +(drop1_gen_pcons c e p n n0 H0) in (let H1 \def H_x in (ex2_ind C (\lambda +(c2: C).(drop n n0 c c2)) (\lambda (c2: C).(drop1 p c2 e)) (drop1 (PCons n (S +n0) (Ss p)) (CHead c (Bind b) (lift n n0 (lift1 p u))) (CHead e (Bind b) u)) +(\lambda (x: C).(\lambda (H2: (drop n n0 c x)).(\lambda (H3: (drop1 p x +e)).(drop1_cons (CHead c (Bind b) (lift n n0 (lift1 p u))) (CHead x (Bind b) +(lift1 p u)) n (S n0) (drop_skip_bind n n0 c x H2 b (lift1 p u)) (CHead e +(Bind b) u) (Ss p) (H x u H3))))) H1)))))))))) hds))). +(* COMMENTS +Initial nodes: 379 +END *) + +theorem drop1_cons_tail: + \forall (c2: C).(\forall (c3: C).(\forall (h: nat).(\forall (d: nat).((drop +h d c2 c3) \to (\forall (hds: PList).(\forall (c1: C).((drop1 hds c1 c2) \to +(drop1 (PConsTail hds h d) c1 c3)))))))) +\def + \lambda (c2: C).(\lambda (c3: C).(\lambda (h: nat).(\lambda (d: +nat).(\lambda (H: (drop h d c2 c3)).(\lambda (hds: PList).(PList_ind (\lambda +(p: PList).(\forall (c1: C).((drop1 p c1 c2) \to (drop1 (PConsTail p h d) c1 +c3)))) (\lambda (c1: C).(\lambda (H0: (drop1 PNil c1 c2)).(let H_y \def +(drop1_gen_pnil c1 c2 H0) in (eq_ind_r C c2 (\lambda (c: C).(drop1 (PCons h d +PNil) c c3)) (drop1_cons c2 c3 h d H c3 PNil (drop1_nil c3)) c1 H_y)))) +(\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H0: +((\forall (c1: C).((drop1 p c1 c2) \to (drop1 (PConsTail p h d) c1 +c3))))).(\lambda (c1: C).(\lambda (H1: (drop1 (PCons n n0 p) c1 c2)).(let H_x +\def (drop1_gen_pcons c1 c2 p n n0 H1) in (let H2 \def H_x in (ex2_ind C +(\lambda (c4: C).(drop n n0 c1 c4)) (\lambda (c4: C).(drop1 p c4 c2)) (drop1 +(PCons n n0 (PConsTail p h d)) c1 c3) (\lambda (x: C).(\lambda (H3: (drop n +n0 c1 x)).(\lambda (H4: (drop1 p x c2)).(drop1_cons c1 x n n0 H3 c3 +(PConsTail p h d) (H0 x H4))))) H2))))))))) hds)))))). +(* COMMENTS +Initial nodes: 271 +END *) + +theorem drop1_trans: + \forall (is1: PList).(\forall (c1: C).(\forall (c0: C).((drop1 is1 c1 c0) +\to (\forall (is2: PList).(\forall (c2: C).((drop1 is2 c0 c2) \to (drop1 +(papp is1 is2) c1 c2))))))) +\def + \lambda (is1: PList).(PList_ind (\lambda (p: PList).(\forall (c1: +C).(\forall (c0: C).((drop1 p c1 c0) \to (\forall (is2: PList).(\forall (c2: +C).((drop1 is2 c0 c2) \to (drop1 (papp p is2) c1 c2)))))))) (\lambda (c1: +C).(\lambda (c0: C).(\lambda (H: (drop1 PNil c1 c0)).(\lambda (is2: +PList).(\lambda (c2: C).(\lambda (H0: (drop1 is2 c0 c2)).(let H_y \def +(drop1_gen_pnil c1 c0 H) in (let H1 \def (eq_ind_r C c0 (\lambda (c: +C).(drop1 is2 c c2)) H0 c1 H_y) in H1)))))))) (\lambda (n: nat).(\lambda (n0: +nat).(\lambda (p: PList).(\lambda (H: ((\forall (c1: C).(\forall (c0: +C).((drop1 p c1 c0) \to (\forall (is2: PList).(\forall (c2: C).((drop1 is2 c0 +c2) \to (drop1 (papp p is2) c1 c2))))))))).(\lambda (c1: C).(\lambda (c0: +C).(\lambda (H0: (drop1 (PCons n n0 p) c1 c0)).(\lambda (is2: PList).(\lambda +(c2: C).(\lambda (H1: (drop1 is2 c0 c2)).(let H_x \def (drop1_gen_pcons c1 c0 +p n n0 H0) in (let H2 \def H_x in (ex2_ind C (\lambda (c3: C).(drop n n0 c1 +c3)) (\lambda (c3: C).(drop1 p c3 c0)) (drop1 (PCons n n0 (papp p is2)) c1 +c2) (\lambda (x: C).(\lambda (H3: (drop n n0 c1 x)).(\lambda (H4: (drop1 p x +c0)).(drop1_cons c1 x n n0 H3 c2 (papp p is2) (H x c0 H4 is2 c2 H1))))) +H2))))))))))))) is1). +(* COMMENTS +Initial nodes: 287 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/ex0/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/ex0/defs.ma new file mode 100644 index 000000000..6ab98b1af --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/ex0/defs.ma @@ -0,0 +1,32 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/A/defs.ma". + +include "Basic-1/G/defs.ma". + +definition gz: + G +\def + mk_G S lt_n_Sn. + +inductive leqz: A \to (A \to Prop) \def +| leqz_sort: \forall (h1: nat).(\forall (h2: nat).(\forall (n1: nat).(\forall +(n2: nat).((eq nat (plus h1 n2) (plus h2 n1)) \to (leqz (ASort h1 n1) (ASort +h2 n2)))))) +| leqz_head: \forall (a1: A).(\forall (a2: A).((leqz a1 a2) \to (\forall (a3: +A).(\forall (a4: A).((leqz a3 a4) \to (leqz (AHead a1 a3) (AHead a2 a4))))))). + diff --git a/matita/matita/contribs/lambdadelta/basic_1/ex0/props.ma b/matita/matita/contribs/lambdadelta/basic_1/ex0/props.ma new file mode 100644 index 000000000..96dd77da3 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/ex0/props.ma @@ -0,0 +1,207 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/ex0/defs.ma". + +include "Basic-1/leq/defs.ma". + +include "Basic-1/aplus/props.ma". + +theorem aplus_gz_le: + \forall (k: nat).(\forall (h: nat).(\forall (n: nat).((le h k) \to (eq A +(aplus gz (ASort h n) k) (ASort O (plus (minus k h) n)))))) +\def + \lambda (k: nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).(\forall (n0: +nat).((le h n) \to (eq A (aplus gz (ASort h n0) n) (ASort O (plus (minus n h) +n0))))))) (\lambda (h: nat).(\lambda (n: nat).(\lambda (H: (le h O)).(let H_y +\def (le_n_O_eq h H) in (eq_ind nat O (\lambda (n0: nat).(eq A (ASort n0 n) +(ASort O n))) (refl_equal A (ASort O n)) h H_y))))) (\lambda (k0: +nat).(\lambda (IH: ((\forall (h: nat).(\forall (n: nat).((le h k0) \to (eq A +(aplus gz (ASort h n) k0) (ASort O (plus (minus k0 h) n)))))))).(\lambda (h: +nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).((le n (S k0)) \to (eq A +(asucc gz (aplus gz (ASort n n0) k0)) (ASort O (plus (match n with [O +\Rightarrow (S k0) | (S l) \Rightarrow (minus k0 l)]) n0)))))) (\lambda (n: +nat).(\lambda (_: (le O (S k0))).(eq_ind A (aplus gz (asucc gz (ASort O n)) +k0) (\lambda (a: A).(eq A a (ASort O (S (plus k0 n))))) (eq_ind_r A (ASort O +(plus (minus k0 O) (S n))) (\lambda (a: A).(eq A a (ASort O (S (plus k0 +n))))) (eq_ind nat k0 (\lambda (n0: nat).(eq A (ASort O (plus n0 (S n))) +(ASort O (S (plus k0 n))))) (eq_ind nat (S (plus k0 n)) (\lambda (n0: +nat).(eq A (ASort O n0) (ASort O (S (plus k0 n))))) (refl_equal A (ASort O (S +(plus k0 n)))) (plus k0 (S n)) (plus_n_Sm k0 n)) (minus k0 O) (minus_n_O k0)) +(aplus gz (ASort O (S n)) k0) (IH O (S n) (le_O_n k0))) (asucc gz (aplus gz +(ASort O n) k0)) (aplus_asucc gz k0 (ASort O n))))) (\lambda (n: +nat).(\lambda (_: ((\forall (n0: nat).((le n (S k0)) \to (eq A (asucc gz +(aplus gz (ASort n n0) k0)) (ASort O (plus (match n with [O \Rightarrow (S +k0) | (S l) \Rightarrow (minus k0 l)]) n0))))))).(\lambda (n0: nat).(\lambda +(H0: (le (S n) (S k0))).(let H_y \def (le_S_n n k0 H0) in (eq_ind A (aplus gz +(ASort n n0) k0) (\lambda (a: A).(eq A (asucc gz (aplus gz (ASort (S n) n0) +k0)) a)) (eq_ind A (aplus gz (asucc gz (ASort (S n) n0)) k0) (\lambda (a: +A).(eq A a (aplus gz (ASort n n0) k0))) (refl_equal A (aplus gz (ASort n n0) +k0)) (asucc gz (aplus gz (ASort (S n) n0) k0)) (aplus_asucc gz k0 (ASort (S +n) n0))) (ASort O (plus (minus k0 n) n0)) (IH n n0 H_y))))))) h)))) k). +(* COMMENTS +Initial nodes: 683 +END *) + +theorem aplus_gz_ge: + \forall (n: nat).(\forall (k: nat).(\forall (h: nat).((le k h) \to (eq A +(aplus gz (ASort h n) k) (ASort (minus h k) n))))) +\def + \lambda (n: nat).(\lambda (k: nat).(nat_ind (\lambda (n0: nat).(\forall (h: +nat).((le n0 h) \to (eq A (aplus gz (ASort h n) n0) (ASort (minus h n0) +n))))) (\lambda (h: nat).(\lambda (_: (le O h)).(eq_ind nat h (\lambda (n0: +nat).(eq A (ASort h n) (ASort n0 n))) (refl_equal A (ASort h n)) (minus h O) +(minus_n_O h)))) (\lambda (k0: nat).(\lambda (IH: ((\forall (h: nat).((le k0 +h) \to (eq A (aplus gz (ASort h n) k0) (ASort (minus h k0) n)))))).(\lambda +(h: nat).(nat_ind (\lambda (n0: nat).((le (S k0) n0) \to (eq A (asucc gz +(aplus gz (ASort n0 n) k0)) (ASort (minus n0 (S k0)) n)))) (\lambda (H: (le +(S k0) O)).(ex2_ind nat (\lambda (n0: nat).(eq nat O (S n0))) (\lambda (n0: +nat).(le k0 n0)) (eq A (asucc gz (aplus gz (ASort O n) k0)) (ASort O n)) +(\lambda (x: nat).(\lambda (H0: (eq nat O (S x))).(\lambda (_: (le k0 +x)).(let H2 \def (eq_ind nat O (\lambda (ee: nat).(match ee in nat return +(\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) +I (S x) H0) in (False_ind (eq A (asucc gz (aplus gz (ASort O n) k0)) (ASort O +n)) H2))))) (le_gen_S k0 O H))) (\lambda (n0: nat).(\lambda (_: (((le (S k0) +n0) \to (eq A (asucc gz (aplus gz (ASort n0 n) k0)) (ASort (minus n0 (S k0)) +n))))).(\lambda (H0: (le (S k0) (S n0))).(let H_y \def (le_S_n k0 n0 H0) in +(eq_ind A (aplus gz (ASort n0 n) k0) (\lambda (a: A).(eq A (asucc gz (aplus +gz (ASort (S n0) n) k0)) a)) (eq_ind A (aplus gz (asucc gz (ASort (S n0) n)) +k0) (\lambda (a: A).(eq A a (aplus gz (ASort n0 n) k0))) (refl_equal A (aplus +gz (ASort n0 n) k0)) (asucc gz (aplus gz (ASort (S n0) n) k0)) (aplus_asucc +gz k0 (ASort (S n0) n))) (ASort (minus n0 k0) n) (IH n0 H_y)))))) h)))) k)). +(* COMMENTS +Initial nodes: 524 +END *) + +theorem next_plus_gz: + \forall (n: nat).(\forall (h: nat).(eq nat (next_plus gz n h) (plus h n))) +\def + \lambda (n: nat).(\lambda (h: nat).(nat_ind (\lambda (n0: nat).(eq nat +(next_plus gz n n0) (plus n0 n))) (refl_equal nat n) (\lambda (n0: +nat).(\lambda (H: (eq nat (next_plus gz n n0) (plus n0 n))).(f_equal nat nat +S (next_plus gz n n0) (plus n0 n) H))) h)). +(* COMMENTS +Initial nodes: 77 +END *) + +theorem leqz_leq: + \forall (a1: A).(\forall (a2: A).((leq gz a1 a2) \to (leqz a1 a2))) +\def + \lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq gz a1 a2)).(leq_ind gz +(\lambda (a: A).(\lambda (a0: A).(leqz a a0))) (\lambda (h1: nat).(\lambda +(h2: nat).(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda +(H0: (eq A (aplus gz (ASort h1 n1) k) (aplus gz (ASort h2 n2) k))).(lt_le_e k +h1 (leqz (ASort h1 n1) (ASort h2 n2)) (\lambda (H1: (lt k h1)).(lt_le_e k h2 +(leqz (ASort h1 n1) (ASort h2 n2)) (\lambda (H2: (lt k h2)).(let H3 \def +(eq_ind A (aplus gz (ASort h1 n1) k) (\lambda (a: A).(eq A a (aplus gz (ASort +h2 n2) k))) H0 (ASort (minus h1 k) n1) (aplus_gz_ge n1 k h1 (le_S_n k h1 +(le_S (S k) h1 H1)))) in (let H4 \def (eq_ind A (aplus gz (ASort h2 n2) k) +(\lambda (a: A).(eq A (ASort (minus h1 k) n1) a)) H3 (ASort (minus h2 k) n2) +(aplus_gz_ge n2 k h2 (le_S_n k h2 (le_S (S k) h2 H2)))) in (let H5 \def +(f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with +[(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow ((let rec minus (n: nat) +on n: (nat \to nat) \def (\lambda (m: nat).(match n with [O \Rightarrow O | +(S k0) \Rightarrow (match m with [O \Rightarrow (S k0) | (S l) \Rightarrow +(minus k0 l)])])) in minus) h1 k)])) (ASort (minus h1 k) n1) (ASort (minus h2 +k) n2) H4) in ((let H6 \def (f_equal A nat (\lambda (e: A).(match e in A +return (\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) +\Rightarrow n1])) (ASort (minus h1 k) n1) (ASort (minus h2 k) n2) H4) in +(\lambda (H7: (eq nat (minus h1 k) (minus h2 k))).(eq_ind nat n1 (\lambda (n: +nat).(leqz (ASort h1 n1) (ASort h2 n))) (eq_ind nat h1 (\lambda (n: +nat).(leqz (ASort h1 n1) (ASort n n1))) (leqz_sort h1 h1 n1 n1 (refl_equal +nat (plus h1 n1))) h2 (minus_minus k h1 h2 (le_S_n k h1 (le_S (S k) h1 H1)) +(le_S_n k h2 (le_S (S k) h2 H2)) H7)) n2 H6))) H5))))) (\lambda (H2: (le h2 +k)).(let H3 \def (eq_ind A (aplus gz (ASort h1 n1) k) (\lambda (a: A).(eq A a +(aplus gz (ASort h2 n2) k))) H0 (ASort (minus h1 k) n1) (aplus_gz_ge n1 k h1 +(le_S_n k h1 (le_S (S k) h1 H1)))) in (let H4 \def (eq_ind A (aplus gz (ASort +h2 n2) k) (\lambda (a: A).(eq A (ASort (minus h1 k) n1) a)) H3 (ASort O (plus +(minus k h2) n2)) (aplus_gz_le k h2 n2 H2)) in (let H5 \def (eq_ind nat +(minus h1 k) (\lambda (n: nat).(eq A (ASort n n1) (ASort O (plus (minus k h2) +n2)))) H4 (S (minus h1 (S k))) (minus_x_Sy h1 k H1)) in (let H6 \def (eq_ind +A (ASort (S (minus h1 (S k))) n1) (\lambda (ee: A).(match ee in A return +(\lambda (_: A).Prop) with [(ASort n _) \Rightarrow (match n in nat return +(\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True]) +| (AHead _ _) \Rightarrow False])) I (ASort O (plus (minus k h2) n2)) H5) in +(False_ind (leqz (ASort h1 n1) (ASort h2 n2)) H6)))))))) (\lambda (H1: (le h1 +k)).(lt_le_e k h2 (leqz (ASort h1 n1) (ASort h2 n2)) (\lambda (H2: (lt k +h2)).(let H3 \def (eq_ind A (aplus gz (ASort h1 n1) k) (\lambda (a: A).(eq A +a (aplus gz (ASort h2 n2) k))) H0 (ASort O (plus (minus k h1) n1)) +(aplus_gz_le k h1 n1 H1)) in (let H4 \def (eq_ind A (aplus gz (ASort h2 n2) +k) (\lambda (a: A).(eq A (ASort O (plus (minus k h1) n1)) a)) H3 (ASort +(minus h2 k) n2) (aplus_gz_ge n2 k h2 (le_S_n k h2 (le_S (S k) h2 H2)))) in +(let H5 \def (sym_eq A (ASort O (plus (minus k h1) n1)) (ASort (minus h2 k) +n2) H4) in (let H6 \def (eq_ind nat (minus h2 k) (\lambda (n: nat).(eq A +(ASort n n2) (ASort O (plus (minus k h1) n1)))) H5 (S (minus h2 (S k))) +(minus_x_Sy h2 k H2)) in (let H7 \def (eq_ind A (ASort (S (minus h2 (S k))) +n2) (\lambda (ee: A).(match ee in A return (\lambda (_: A).Prop) with [(ASort +n _) \Rightarrow (match n in nat return (\lambda (_: nat).Prop) with [O +\Rightarrow False | (S _) \Rightarrow True]) | (AHead _ _) \Rightarrow +False])) I (ASort O (plus (minus k h1) n1)) H6) in (False_ind (leqz (ASort h1 +n1) (ASort h2 n2)) H7))))))) (\lambda (H2: (le h2 k)).(let H3 \def (eq_ind A +(aplus gz (ASort h1 n1) k) (\lambda (a: A).(eq A a (aplus gz (ASort h2 n2) +k))) H0 (ASort O (plus (minus k h1) n1)) (aplus_gz_le k h1 n1 H1)) in (let H4 +\def (eq_ind A (aplus gz (ASort h2 n2) k) (\lambda (a: A).(eq A (ASort O +(plus (minus k h1) n1)) a)) H3 (ASort O (plus (minus k h2) n2)) (aplus_gz_le +k h2 n2 H2)) in (let H5 \def (f_equal A nat (\lambda (e: A).(match e in A +return (\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) +\Rightarrow ((let rec plus (n: nat) on n: (nat \to nat) \def (\lambda (m: +nat).(match n with [O \Rightarrow m | (S p) \Rightarrow (S (plus p m))])) in +plus) (minus k h1) n1)])) (ASort O (plus (minus k h1) n1)) (ASort O (plus +(minus k h2) n2)) H4) in (let H_y \def (plus_plus k h1 h2 n1 n2 H1 H2 H5) in +(leqz_sort h1 h2 n1 n2 H_y))))))))))))))) (\lambda (a0: A).(\lambda (a3: +A).(\lambda (_: (leq gz a0 a3)).(\lambda (H1: (leqz a0 a3)).(\lambda (a4: +A).(\lambda (a5: A).(\lambda (_: (leq gz a4 a5)).(\lambda (H3: (leqz a4 +a5)).(leqz_head a0 a3 H1 a4 a5 H3))))))))) a1 a2 H))). +(* COMMENTS +Initial nodes: 1375 +END *) + +theorem leq_leqz: + \forall (a1: A).(\forall (a2: A).((leqz a1 a2) \to (leq gz a1 a2))) +\def + \lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leqz a1 a2)).(leqz_ind +(\lambda (a: A).(\lambda (a0: A).(leq gz a a0))) (\lambda (h1: nat).(\lambda +(h2: nat).(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (H0: (eq nat (plus +h1 n2) (plus h2 n1))).(leq_sort gz h1 h2 n1 n2 (plus h1 h2) (eq_ind_r A +(ASort (minus h1 (plus h1 h2)) (next_plus gz n1 (minus (plus h1 h2) h1))) +(\lambda (a: A).(eq A a (aplus gz (ASort h2 n2) (plus h1 h2)))) (eq_ind_r A +(ASort (minus h2 (plus h1 h2)) (next_plus gz n2 (minus (plus h1 h2) h2))) +(\lambda (a: A).(eq A (ASort (minus h1 (plus h1 h2)) (next_plus gz n1 (minus +(plus h1 h2) h1))) a)) (eq_ind_r nat h2 (\lambda (n: nat).(eq A (ASort (minus +h1 (plus h1 h2)) (next_plus gz n1 n)) (ASort (minus h2 (plus h1 h2)) +(next_plus gz n2 (minus (plus h1 h2) h2))))) (eq_ind_r nat h1 (\lambda (n: +nat).(eq A (ASort (minus h1 (plus h1 h2)) (next_plus gz n1 h2)) (ASort (minus +h2 (plus h1 h2)) (next_plus gz n2 n)))) (eq_ind_r nat O (\lambda (n: nat).(eq +A (ASort n (next_plus gz n1 h2)) (ASort (minus h2 (plus h1 h2)) (next_plus gz +n2 h1)))) (eq_ind_r nat O (\lambda (n: nat).(eq A (ASort O (next_plus gz n1 +h2)) (ASort n (next_plus gz n2 h1)))) (eq_ind_r nat (plus h2 n1) (\lambda (n: +nat).(eq A (ASort O n) (ASort O (next_plus gz n2 h1)))) (eq_ind_r nat (plus +h1 n2) (\lambda (n: nat).(eq A (ASort O (plus h2 n1)) (ASort O n))) (f_equal +nat A (ASort O) (plus h2 n1) (plus h1 n2) (sym_eq nat (plus h1 n2) (plus h2 +n1) H0)) (next_plus gz n2 h1) (next_plus_gz n2 h1)) (next_plus gz n1 h2) +(next_plus_gz n1 h2)) (minus h2 (plus h1 h2)) (O_minus h2 (plus h1 h2) +(le_plus_r h1 h2))) (minus h1 (plus h1 h2)) (O_minus h1 (plus h1 h2) +(le_plus_l h1 h2))) (minus (plus h1 h2) h2) (minus_plus_r h1 h2)) (minus +(plus h1 h2) h1) (minus_plus h1 h2)) (aplus gz (ASort h2 n2) (plus h1 h2)) +(aplus_asort_simpl gz (plus h1 h2) h2 n2)) (aplus gz (ASort h1 n1) (plus h1 +h2)) (aplus_asort_simpl gz (plus h1 h2) h1 n1)))))))) (\lambda (a0: +A).(\lambda (a3: A).(\lambda (_: (leqz a0 a3)).(\lambda (H1: (leq gz a0 +a3)).(\lambda (a4: A).(\lambda (a5: A).(\lambda (_: (leqz a4 a5)).(\lambda +(H3: (leq gz a4 a5)).(leq_head gz a0 a3 H1 a4 a5 H3))))))))) a1 a2 H))). +(* COMMENTS +Initial nodes: 717 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/ex1/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/ex1/defs.ma new file mode 100644 index 000000000..dc18f6fc2 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/ex1/defs.ma @@ -0,0 +1,29 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/C/defs.ma". + +definition ex1_c: + C +\def + CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) +(Bind Abst) (TLRef O). + +definition ex1_t: + T +\def + THead (Flat Appl) (TLRef O) (THead (Bind Abst) (TLRef (S (S O))) (TSort O)). + diff --git a/matita/matita/contribs/lambdadelta/basic_1/ex1/props.ma b/matita/matita/contribs/lambdadelta/basic_1/ex1/props.ma new file mode 100644 index 000000000..5c442f5bb --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/ex1/props.ma @@ -0,0 +1,536 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/ex1/defs.ma". + +include "Basic-1/ty3/fwd.ma". + +include "Basic-1/pc3/fwd.ma". + +include "Basic-1/nf2/pr3.ma". + +include "Basic-1/nf2/props.ma". + +include "Basic-1/arity/defs.ma". + +include "Basic-1/leq/props.ma". + +theorem ex1__leq_sort_SS: + \forall (g: G).(\forall (k: nat).(\forall (n: nat).(leq g (ASort k n) (asucc +g (asucc g (ASort (S (S k)) n)))))) +\def + \lambda (g: G).(\lambda (k: nat).(\lambda (n: nat).(leq_refl g (asucc g +(asucc g (ASort (S (S k)) n)))))). +(* COMMENTS +Initial nodes: 27 +END *) + +theorem ex1_arity: + \forall (g: G).(arity g ex1_c ex1_t (ASort O O)) +\def + \lambda (g: G).(arity_appl g (CHead (CHead (CHead (CSort O) (Bind Abst) +(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef O) (ASort (S +(S O)) O) (arity_abst g (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) +(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead (CHead (CSort O) (Bind +Abst) (TSort O)) (Bind Abst) (TSort O)) (TLRef O) O (getl_refl Abst (CHead +(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (TLRef O)) +(ASort (S (S O)) O) (arity_abst g (CHead (CHead (CSort O) (Bind Abst) (TSort +O)) (Bind Abst) (TSort O)) (CHead (CSort O) (Bind Abst) (TSort O)) (TSort O) +O (getl_refl Abst (CHead (CSort O) (Bind Abst) (TSort O)) (TSort O)) (asucc g +(ASort (S (S O)) O)) (arity_repl g (CHead (CSort O) (Bind Abst) (TSort O)) +(TSort O) (ASort O O) (arity_sort g (CHead (CSort O) (Bind Abst) (TSort O)) +O) (asucc g (asucc g (ASort (S (S O)) O))) (ex1__leq_sort_SS g O O)))) (THead +(Bind Abst) (TLRef (S (S O))) (TSort O)) (ASort O O) (arity_head g (CHead +(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind +Abst) (TLRef O)) (TLRef (S (S O))) (ASort (S (S O)) O) (arity_abst g (CHead +(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind +Abst) (TLRef O)) (CSort O) (TSort O) (S (S O)) (getl_clear_bind Abst (CHead +(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind +Abst) (TLRef O)) (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) +(TSort O)) (TLRef O) (clear_bind Abst (CHead (CHead (CSort O) (Bind Abst) +(TSort O)) (Bind Abst) (TSort O)) (TLRef O)) (CHead (CSort O) (Bind Abst) +(TSort O)) (S O) (getl_head (Bind Abst) O (CHead (CSort O) (Bind Abst) (TSort +O)) (CHead (CSort O) (Bind Abst) (TSort O)) (getl_refl Abst (CSort O) (TSort +O)) (TSort O))) (asucc g (ASort (S (S O)) O)) (arity_repl g (CSort O) (TSort +O) (ASort O O) (arity_sort g (CSort O) O) (asucc g (asucc g (ASort (S (S O)) +O))) (ex1__leq_sort_SS g O O))) (TSort O) (ASort O O) (arity_sort g (CHead +(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) +(Bind Abst) (TLRef O)) (Bind Abst) (TLRef (S (S O)))) O))). +(* COMMENTS +Initial nodes: 753 +END *) + +theorem ex1_ty3: + \forall (g: G).(\forall (u: T).((ty3 g ex1_c ex1_t u) \to (\forall (P: +Prop).P))) +\def + \lambda (g: G).(\lambda (u: T).(\lambda (H: (ty3 g (CHead (CHead (CHead +(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef +O)) (THead (Flat Appl) (TLRef O) (THead (Bind Abst) (TLRef (S (S O))) (TSort +O))) u)).(\lambda (P: Prop).(ex3_2_ind T T (\lambda (u0: T).(\lambda (t: +T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) +(TSort O)) (Bind Abst) (TLRef O)) (THead (Flat Appl) (TLRef O) (THead (Bind +Abst) u0 t)) u))) (\lambda (u0: T).(\lambda (t: T).(ty3 g (CHead (CHead +(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) +(TLRef O)) (THead (Bind Abst) (TLRef (S (S O))) (TSort O)) (THead (Bind Abst) +u0 t)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g (CHead (CHead (CHead (CSort +O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) +(TLRef O) u0))) P (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 (CHead +(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind +Abst) (TLRef O)) (THead (Flat Appl) (TLRef O) (THead (Bind Abst) x0 x1)) +u)).(\lambda (H1: (ty3 g (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort +O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (THead (Bind Abst) (TLRef +(S (S O))) (TSort O)) (THead (Bind Abst) x0 x1))).(\lambda (H2: (ty3 g (CHead +(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind +Abst) (TLRef O)) (TLRef O) x0)).(or_ind (ex3_3 C T T (\lambda (_: C).(\lambda +(_: T).(\lambda (t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort +O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S O) O t) x0)))) +(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl O (CHead (CHead (CHead +(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef +O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda +(t: T).(ty3 g e u0 t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: +T).(\lambda (_: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) +(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S O) O u0) x0)))) +(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl O (CHead (CHead (CHead +(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef +O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda +(t: T).(ty3 g e u0 t))))) P (\lambda (H3: (ex3_3 C T T (\lambda (_: +C).(\lambda (_: T).(\lambda (t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind +Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S O) O +t) x0)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl O (CHead +(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind +Abst) (TLRef O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: +T).(\lambda (t: T).(ty3 g e u0 t)))))).(ex3_3_ind C T T (\lambda (_: +C).(\lambda (_: T).(\lambda (t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind +Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S O) O +t) x0)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl O (CHead +(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind +Abst) (TLRef O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: +T).(\lambda (t: T).(ty3 g e u0 t)))) P (\lambda (x2: C).(\lambda (x3: +T).(\lambda (x4: T).(\lambda (_: (pc3 (CHead (CHead (CHead (CSort O) (Bind +Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S O) O +x4) x0)).(\lambda (H5: (getl O (CHead (CHead (CHead (CSort O) (Bind Abst) +(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead x2 (Bind +Abbr) x3))).(\lambda (_: (ty3 g x2 x3 x4)).(ex3_2_ind T T (\lambda (t2: +T).(\lambda (_: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) +(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (THead (Bind Abst) (TLRef (S (S +O))) t2) (THead (Bind Abst) x0 x1)))) (\lambda (_: T).(\lambda (t: T).(ty3 g +(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) +(Bind Abst) (TLRef O)) (TLRef (S (S O))) t))) (\lambda (t2: T).(\lambda (_: +T).(ty3 g (CHead (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind +Abst) (TSort O)) (Bind Abst) (TLRef O)) (Bind Abst) (TLRef (S (S O)))) (TSort +O) t2))) P (\lambda (x5: T).(\lambda (x6: T).(\lambda (_: (pc3 (CHead (CHead +(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) +(TLRef O)) (THead (Bind Abst) (TLRef (S (S O))) x5) (THead (Bind Abst) x0 +x1))).(\lambda (H8: (ty3 g (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort +O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) +x6)).(\lambda (_: (ty3 g (CHead (CHead (CHead (CHead (CSort O) (Bind Abst) +(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (Bind Abst) (TLRef +(S (S O)))) (TSort O) x5)).(or_ind (ex3_3 C T T (\lambda (_: C).(\lambda (_: +T).(\lambda (t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) +(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O t) x6)))) +(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) (CHead +(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind +Abst) (TLRef O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: +T).(\lambda (t: T).(ty3 g e u0 t))))) (ex3_3 C T T (\lambda (_: C).(\lambda +(u0: T).(\lambda (_: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) +(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) +O u0) x6)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) +(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) +(Bind Abst) (TLRef O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda +(u0: T).(\lambda (t: T).(ty3 g e u0 t))))) P (\lambda (H10: (ex3_3 C T T +(\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 (CHead (CHead (CHead +(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef +O)) (lift (S (S (S O))) O t) x6)))) (\lambda (e: C).(\lambda (u0: T).(\lambda +(_: T).(getl (S (S O)) (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) +(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead e (Bind Abbr) u0))))) +(\lambda (e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 +t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 +(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) +(Bind Abst) (TLRef O)) (lift (S (S (S O))) O t) x6)))) (\lambda (e: +C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) (CHead (CHead (CHead +(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef +O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda +(t: T).(ty3 g e u0 t)))) P (\lambda (x7: C).(\lambda (x8: T).(\lambda (x9: +T).(\lambda (_: (pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) +(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O x9) +x6)).(\lambda (H12: (getl (S (S O)) (CHead (CHead (CHead (CSort O) (Bind +Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead x7 +(Bind Abbr) x8))).(\lambda (_: (ty3 g x7 x8 x9)).(let H14 \def (getl_gen_all +(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead +x7 (Bind Abbr) x8) (r (Bind Abst) (S O)) (getl_gen_S (Bind Abst) (CHead +(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead x7 +(Bind Abbr) x8) (TLRef O) (S O) H12)) in (ex2_ind C (\lambda (e: C).(drop (S +O) O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) +e)) (\lambda (e: C).(clear e (CHead x7 (Bind Abbr) x8))) P (\lambda (x: +C).(\lambda (_: (drop (S O) O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) +(Bind Abst) (TSort O)) x)).(\lambda (_: (clear x (CHead x7 (Bind Abbr) +x8))).(let H17 \def (eq_ind C (CHead x2 (Bind Abbr) x3) (\lambda (ee: +C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow +False | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) +with [(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with +[Abbr \Rightarrow True | Abst \Rightarrow False | Void \Rightarrow False]) | +(Flat _) \Rightarrow False])])) I (CHead (CHead (CHead (CSort O) (Bind Abst) +(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (clear_gen_bind Abst +(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead +x2 (Bind Abbr) x3) (TLRef O) (getl_gen_O (CHead (CHead (CHead (CSort O) (Bind +Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead x2 +(Bind Abbr) x3) H5))) in (False_ind P H17))))) H14)))))))) H10)) (\lambda +(H10: (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 +(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) +(Bind Abst) (TLRef O)) (lift (S (S (S O))) O u0) x6)))) (\lambda (e: +C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) (CHead (CHead (CHead +(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef +O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda +(t: T).(ty3 g e u0 t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (u0: +T).(\lambda (_: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) +(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O u0) +x6)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) +(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) +(Bind Abst) (TLRef O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda +(u0: T).(\lambda (t: T).(ty3 g e u0 t)))) P (\lambda (x7: C).(\lambda (x8: +T).(\lambda (x9: T).(\lambda (_: (pc3 (CHead (CHead (CHead (CSort O) (Bind +Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S +O))) O x8) x6)).(\lambda (H12: (getl (S (S O)) (CHead (CHead (CHead (CSort O) +(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead +x7 (Bind Abst) x8))).(\lambda (_: (ty3 g x7 x8 x9)).(let H14 \def +(getl_gen_all (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) +(TSort O)) (CHead x7 (Bind Abst) x8) (r (Bind Abst) (S O)) (getl_gen_S (Bind +Abst) (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) +(CHead x7 (Bind Abst) x8) (TLRef O) (S O) H12)) in (ex2_ind C (\lambda (e: +C).(drop (S O) O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) +(TSort O)) e)) (\lambda (e: C).(clear e (CHead x7 (Bind Abst) x8))) P +(\lambda (x: C).(\lambda (_: (drop (S O) O (CHead (CHead (CSort O) (Bind +Abst) (TSort O)) (Bind Abst) (TSort O)) x)).(\lambda (_: (clear x (CHead x7 +(Bind Abst) x8))).(let H17 \def (eq_ind C (CHead x2 (Bind Abbr) x3) (\lambda +(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) +\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda +(_: K).Prop) with [(Bind b) \Rightarrow (match b in B return (\lambda (_: +B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | Void +\Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead (CHead (CHead +(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef +O)) (clear_gen_bind Abst (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind +Abst) (TSort O)) (CHead x2 (Bind Abbr) x3) (TLRef O) (getl_gen_O (CHead +(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind +Abst) (TLRef O)) (CHead x2 (Bind Abbr) x3) H5))) in (False_ind P H17))))) +H14)))))))) H10)) (ty3_gen_lref g (CHead (CHead (CHead (CSort O) (Bind Abst) +(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) x6 (S (S O)) +H8))))))) (ty3_gen_bind g Abst (CHead (CHead (CHead (CSort O) (Bind Abst) +(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) +(TSort O) (THead (Bind Abst) x0 x1) H1)))))))) H3)) (\lambda (H3: (ex3_3 C T +T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 (CHead (CHead (CHead +(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef +O)) (lift (S O) O u0) x0)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: +T).(getl O (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) +(TSort O)) (Bind Abst) (TLRef O)) (CHead e (Bind Abst) u0))))) (\lambda (e: +C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t)))))).(ex3_3_ind C T T +(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 (CHead (CHead (CHead +(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef +O)) (lift (S O) O u0) x0)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: +T).(getl O (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) +(TSort O)) (Bind Abst) (TLRef O)) (CHead e (Bind Abst) u0))))) (\lambda (e: +C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t)))) P (\lambda (x2: +C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H4: (pc3 (CHead (CHead (CHead +(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef +O)) (lift (S O) O x3) x0)).(\lambda (H5: (getl O (CHead (CHead (CHead (CSort +O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) +(CHead x2 (Bind Abst) x3))).(\lambda (H6: (ty3 g x2 x3 x4)).(ex3_2_ind T T +(\lambda (t2: T).(\lambda (_: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind +Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (THead (Bind +Abst) (TLRef (S (S O))) t2) (THead (Bind Abst) x0 x1)))) (\lambda (_: +T).(\lambda (t: T).(ty3 g (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort +O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) t))) +(\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead (CHead (CHead (CHead (CSort O) +(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (Bind +Abst) (TLRef (S (S O)))) (TSort O) t2))) P (\lambda (x5: T).(\lambda (x6: +T).(\lambda (H7: (pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) +(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (THead (Bind Abst) (TLRef (S (S +O))) x5) (THead (Bind Abst) x0 x1))).(\lambda (H8: (ty3 g (CHead (CHead +(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) +(TLRef O)) (TLRef (S (S O))) x6)).(\lambda (_: (ty3 g (CHead (CHead (CHead +(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) +(TLRef O)) (Bind Abst) (TLRef (S (S O)))) (TSort O) x5)).(or_ind (ex3_3 C T T +(\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 (CHead (CHead (CHead +(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef +O)) (lift (S (S (S O))) O t) x6)))) (\lambda (e: C).(\lambda (u0: T).(\lambda +(_: T).(getl (S (S O)) (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) +(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead e (Bind Abbr) u0))))) +(\lambda (e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t))))) (ex3_3 C +T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 (CHead (CHead +(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) +(TLRef O)) (lift (S (S (S O))) O u0) x6)))) (\lambda (e: C).(\lambda (u0: +T).(\lambda (_: T).(getl (S (S O)) (CHead (CHead (CHead (CSort O) (Bind Abst) +(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead e (Bind Abst) +u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t))))) P +(\lambda (H10: (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: +T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) +(TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O t) x6)))) (\lambda +(e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) (CHead (CHead (CHead +(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef +O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda +(t: T).(ty3 g e u0 t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (_: +T).(\lambda (t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) +(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O t) x6)))) +(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) (CHead +(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind +Abst) (TLRef O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: +T).(\lambda (t: T).(ty3 g e u0 t)))) P (\lambda (x7: C).(\lambda (x8: +T).(\lambda (x9: T).(\lambda (_: (pc3 (CHead (CHead (CHead (CSort O) (Bind +Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S +O))) O x9) x6)).(\lambda (H12: (getl (S (S O)) (CHead (CHead (CHead (CSort O) +(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead +x7 (Bind Abbr) x8))).(\lambda (_: (ty3 g x7 x8 x9)).(let H14 \def +(getl_gen_all (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) +(TSort O)) (CHead x7 (Bind Abbr) x8) (r (Bind Abst) (S O)) (getl_gen_S (Bind +Abst) (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) +(CHead x7 (Bind Abbr) x8) (TLRef O) (S O) H12)) in (ex2_ind C (\lambda (e: +C).(drop (S O) O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) +(TSort O)) e)) (\lambda (e: C).(clear e (CHead x7 (Bind Abbr) x8))) P +(\lambda (x: C).(\lambda (H15: (drop (S O) O (CHead (CHead (CSort O) (Bind +Abst) (TSort O)) (Bind Abst) (TSort O)) x)).(\lambda (H16: (clear x (CHead x7 +(Bind Abbr) x8))).(let H17 \def (f_equal C C (\lambda (e: C).(match e in C +return (\lambda (_: C).C) with [(CSort _) \Rightarrow x2 | (CHead c _ _) +\Rightarrow c])) (CHead x2 (Bind Abst) x3) (CHead (CHead (CHead (CSort O) +(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) +(clear_gen_bind Abst (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind +Abst) (TSort O)) (CHead x2 (Bind Abst) x3) (TLRef O) (getl_gen_O (CHead +(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind +Abst) (TLRef O)) (CHead x2 (Bind Abst) x3) H5))) in ((let H18 \def (f_equal C +T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) +\Rightarrow x3 | (CHead _ _ t) \Rightarrow t])) (CHead x2 (Bind Abst) x3) +(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) +(Bind Abst) (TLRef O)) (clear_gen_bind Abst (CHead (CHead (CSort O) (Bind +Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead x2 (Bind Abst) x3) (TLRef O) +(getl_gen_O (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) +(TSort O)) (Bind Abst) (TLRef O)) (CHead x2 (Bind Abst) x3) H5))) in (\lambda +(H19: (eq C x2 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) +(TSort O)))).(let H20 \def (eq_ind T x3 (\lambda (t: T).(ty3 g x2 t x4)) H6 +(TLRef O) H18) in (let H21 \def (eq_ind T x3 (\lambda (t: T).(pc3 (CHead +(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind +Abst) (TLRef O)) (lift (S O) O t) x0)) H4 (TLRef O) H18) in (let H22 \def +(eq_ind C x2 (\lambda (c: C).(ty3 g c (TLRef O) x4)) H20 (CHead (CHead (CSort +O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) H19) in (let H23 \def +(eq_ind_r C x (\lambda (c: C).(clear c (CHead x7 (Bind Abbr) x8))) H16 (CHead +(CSort O) (Bind Abst) (TSort O)) (drop_gen_refl (CHead (CSort O) (Bind Abst) +(TSort O)) x (drop_gen_drop (Bind Abst) (CHead (CSort O) (Bind Abst) (TSort +O)) x (TSort O) O H15))) in (let H24 \def (eq_ind C (CHead x7 (Bind Abbr) x8) +(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) +\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda +(_: K).Prop) with [(Bind b) \Rightarrow (match b in B return (\lambda (_: +B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | Void +\Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead (CSort O) +(Bind Abst) (TSort O)) (clear_gen_bind Abst (CSort O) (CHead x7 (Bind Abbr) +x8) (TSort O) H23)) in (False_ind P H24)))))))) H17))))) H14)))))))) H10)) +(\lambda (H10: (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: +T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) +(TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O u0) x6)))) (\lambda +(e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) (CHead (CHead (CHead +(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef +O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda +(t: T).(ty3 g e u0 t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (u0: +T).(\lambda (_: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) +(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O u0) +x6)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) +(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) +(Bind Abst) (TLRef O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda +(u0: T).(\lambda (t: T).(ty3 g e u0 t)))) P (\lambda (x7: C).(\lambda (x8: +T).(\lambda (x9: T).(\lambda (H11: (pc3 (CHead (CHead (CHead (CSort O) (Bind +Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S +O))) O x8) x6)).(\lambda (H12: (getl (S (S O)) (CHead (CHead (CHead (CSort O) +(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead +x7 (Bind Abst) x8))).(\lambda (H13: (ty3 g x7 x8 x9)).(let H14 \def +(getl_gen_all (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) +(TSort O)) (CHead x7 (Bind Abst) x8) (r (Bind Abst) (S O)) (getl_gen_S (Bind +Abst) (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) +(CHead x7 (Bind Abst) x8) (TLRef O) (S O) H12)) in (ex2_ind C (\lambda (e: +C).(drop (S O) O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) +(TSort O)) e)) (\lambda (e: C).(clear e (CHead x7 (Bind Abst) x8))) P +(\lambda (x: C).(\lambda (H15: (drop (S O) O (CHead (CHead (CSort O) (Bind +Abst) (TSort O)) (Bind Abst) (TSort O)) x)).(\lambda (H16: (clear x (CHead x7 +(Bind Abst) x8))).(let H17 \def (f_equal C C (\lambda (e: C).(match e in C +return (\lambda (_: C).C) with [(CSort _) \Rightarrow x2 | (CHead c _ _) +\Rightarrow c])) (CHead x2 (Bind Abst) x3) (CHead (CHead (CHead (CSort O) +(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) +(clear_gen_bind Abst (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind +Abst) (TSort O)) (CHead x2 (Bind Abst) x3) (TLRef O) (getl_gen_O (CHead +(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind +Abst) (TLRef O)) (CHead x2 (Bind Abst) x3) H5))) in ((let H18 \def (f_equal C +T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) +\Rightarrow x3 | (CHead _ _ t) \Rightarrow t])) (CHead x2 (Bind Abst) x3) +(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) +(Bind Abst) (TLRef O)) (clear_gen_bind Abst (CHead (CHead (CSort O) (Bind +Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead x2 (Bind Abst) x3) (TLRef O) +(getl_gen_O (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) +(TSort O)) (Bind Abst) (TLRef O)) (CHead x2 (Bind Abst) x3) H5))) in (\lambda +(H19: (eq C x2 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) +(TSort O)))).(let H20 \def (eq_ind T x3 (\lambda (t: T).(ty3 g x2 t x4)) H6 +(TLRef O) H18) in (let H21 \def (eq_ind T x3 (\lambda (t: T).(pc3 (CHead +(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind +Abst) (TLRef O)) (lift (S O) O t) x0)) H4 (TLRef O) H18) in (let H22 \def +(eq_ind C x2 (\lambda (c: C).(ty3 g c (TLRef O) x4)) H20 (CHead (CHead (CSort +O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) H19) in (let H23 \def +(eq_ind_r C x (\lambda (c: C).(clear c (CHead x7 (Bind Abst) x8))) H16 (CHead +(CSort O) (Bind Abst) (TSort O)) (drop_gen_refl (CHead (CSort O) (Bind Abst) +(TSort O)) x (drop_gen_drop (Bind Abst) (CHead (CSort O) (Bind Abst) (TSort +O)) x (TSort O) O H15))) in (let H24 \def (f_equal C C (\lambda (e: C).(match +e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow x7 | (CHead c _ +_) \Rightarrow c])) (CHead x7 (Bind Abst) x8) (CHead (CSort O) (Bind Abst) +(TSort O)) (clear_gen_bind Abst (CSort O) (CHead x7 (Bind Abst) x8) (TSort O) +H23)) in ((let H25 \def (f_equal C T (\lambda (e: C).(match e in C return +(\lambda (_: C).T) with [(CSort _) \Rightarrow x8 | (CHead _ _ t) \Rightarrow +t])) (CHead x7 (Bind Abst) x8) (CHead (CSort O) (Bind Abst) (TSort O)) +(clear_gen_bind Abst (CSort O) (CHead x7 (Bind Abst) x8) (TSort O) H23)) in +(\lambda (H26: (eq C x7 (CSort O))).(let H27 \def (eq_ind T x8 (\lambda (t: +T).(ty3 g x7 t x9)) H13 (TSort O) H25) in (let H28 \def (eq_ind T x8 (\lambda +(t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) +(TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O t) x6)) H11 (TSort O) +H25) in (let H29 \def (eq_ind C x7 (\lambda (c: C).(ty3 g c (TSort O) x9)) +H27 (CSort O) H26) in (or_ind (ex3_3 C T T (\lambda (_: C).(\lambda (_: +T).(\lambda (t: T).(pc3 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind +Abst) (TSort O)) (lift (S O) O t) x4)))) (\lambda (e: C).(\lambda (u0: +T).(\lambda (_: T).(getl O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) +(Bind Abst) (TSort O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda +(u0: T).(\lambda (t: T).(ty3 g e u0 t))))) (ex3_3 C T T (\lambda (_: +C).(\lambda (u0: T).(\lambda (_: T).(pc3 (CHead (CHead (CSort O) (Bind Abst) +(TSort O)) (Bind Abst) (TSort O)) (lift (S O) O u0) x4)))) (\lambda (e: +C).(\lambda (u0: T).(\lambda (_: T).(getl O (CHead (CHead (CSort O) (Bind +Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead e (Bind Abst) u0))))) (\lambda +(e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t))))) P (\lambda (H30: +(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 (CHead +(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (lift (S O) O +t) x4)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl O (CHead +(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead e (Bind +Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 +t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 +(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (lift +(S O) O t) x4)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl O +(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead +e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g +e u0 t)))) P (\lambda (x10: C).(\lambda (x11: T).(\lambda (x12: T).(\lambda +(_: (pc3 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort +O)) (lift (S O) O x12) x4)).(\lambda (H32: (getl O (CHead (CHead (CSort O) +(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead x10 (Bind Abbr) +x11))).(\lambda (_: (ty3 g x10 x11 x12)).(let H34 \def (eq_ind C (CHead x10 +(Bind Abbr) x11) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) +with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K +return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B return +(\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | +Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead (CHead +(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (clear_gen_bind Abst +(CHead (CSort O) (Bind Abst) (TSort O)) (CHead x10 (Bind Abbr) x11) (TSort O) +(getl_gen_O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort +O)) (CHead x10 (Bind Abbr) x11) H32))) in (False_ind P H34)))))))) H30)) +(\lambda (H30: (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: +T).(pc3 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) +(lift (S O) O u0) x4)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: +T).(getl O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort +O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda +(t: T).(ty3 g e u0 t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (u0: +T).(\lambda (_: T).(pc3 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind +Abst) (TSort O)) (lift (S O) O u0) x4)))) (\lambda (e: C).(\lambda (u0: +T).(\lambda (_: T).(getl O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) +(Bind Abst) (TSort O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda +(u0: T).(\lambda (t: T).(ty3 g e u0 t)))) P (\lambda (x10: C).(\lambda (x11: +T).(\lambda (x12: T).(\lambda (H31: (pc3 (CHead (CHead (CSort O) (Bind Abst) +(TSort O)) (Bind Abst) (TSort O)) (lift (S O) O x11) x4)).(\lambda (H32: +(getl O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) +(CHead x10 (Bind Abst) x11))).(\lambda (H33: (ty3 g x10 x11 x12)).(let H34 +\def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) +with [(CSort _) \Rightarrow x10 | (CHead c _ _) \Rightarrow c])) (CHead x10 +(Bind Abst) x11) (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) +(TSort O)) (clear_gen_bind Abst (CHead (CSort O) (Bind Abst) (TSort O)) +(CHead x10 (Bind Abst) x11) (TSort O) (getl_gen_O (CHead (CHead (CSort O) +(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead x10 (Bind Abst) x11) +H32))) in ((let H35 \def (f_equal C T (\lambda (e: C).(match e in C return +(\lambda (_: C).T) with [(CSort _) \Rightarrow x11 | (CHead _ _ t) +\Rightarrow t])) (CHead x10 (Bind Abst) x11) (CHead (CHead (CSort O) (Bind +Abst) (TSort O)) (Bind Abst) (TSort O)) (clear_gen_bind Abst (CHead (CSort O) +(Bind Abst) (TSort O)) (CHead x10 (Bind Abst) x11) (TSort O) (getl_gen_O +(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead +x10 (Bind Abst) x11) H32))) in (\lambda (H36: (eq C x10 (CHead (CSort O) +(Bind Abst) (TSort O)))).(let H37 \def (eq_ind T x11 (\lambda (t: T).(ty3 g +x10 t x12)) H33 (TSort O) H35) in (let H38 \def (eq_ind T x11 (\lambda (t: +T).(pc3 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) +(lift (S O) O t) x4)) H31 (TSort O) H35) in (let H39 \def (eq_ind C x10 +(\lambda (c: C).(ty3 g c (TSort O) x12)) H37 (CHead (CSort O) (Bind Abst) +(TSort O)) H36) in (land_ind (pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) +(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) +x0) (\forall (b: B).(\forall (u0: T).(pc3 (CHead (CHead (CHead (CHead (CSort +O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (Bind +b) u0) x5 x1))) P (\lambda (H40: (pc3 (CHead (CHead (CHead (CSort O) (Bind +Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S +O))) x0)).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pc3 (CHead (CHead +(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind +Abst) (TLRef O)) (Bind b) u0) x5 x1))))).(let H42 \def (eq_ind T (lift (S O) +O (TLRef O)) (\lambda (t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) +(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) +t)) (pc3_t x0 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind +Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) H40 (lift (S O) O +(TLRef O)) (ex2_sym T (pr3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort +O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S O) O (TLRef O))) +(pr3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort +O)) (Bind Abst) (TLRef O)) x0) H21)) (TLRef (plus O (S O))) (lift_lref_ge O +(S O) O (le_n O))) in (let H43 \def H42 in (ex2_ind T (\lambda (t: T).(pr3 +(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) +(Bind Abst) (TLRef O)) (TLRef (S (S O))) t)) (\lambda (t: T).(pr3 (CHead +(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind +Abst) (TLRef O)) (TLRef (S O)) t)) P (\lambda (x13: T).(\lambda (H44: (pr3 +(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) +(Bind Abst) (TLRef O)) (TLRef (S (S O))) x13)).(\lambda (H45: (pr3 (CHead +(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind +Abst) (TLRef O)) (TLRef (S O)) x13)).(let H46 \def (eq_ind_r T x13 (\lambda +(t: T).(eq T (TLRef (S (S O))) t)) (nf2_pr3_unfold (CHead (CHead (CHead +(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef +O)) (TLRef (S (S O))) x13 H44 (nf2_lref_abst (CHead (CHead (CHead (CSort O) +(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CSort +O) (TSort O) (S (S O)) (getl_clear_bind Abst (CHead (CHead (CHead (CSort O) +(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead +(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (TLRef O) +(clear_bind Abst (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) +(TSort O)) (TLRef O)) (CHead (CSort O) (Bind Abst) (TSort O)) (S O) +(getl_head (Bind Abst) O (CHead (CSort O) (Bind Abst) (TSort O)) (CHead +(CSort O) (Bind Abst) (TSort O)) (getl_refl Abst (CSort O) (TSort O)) (TSort +O))))) (TLRef (S O)) (nf2_pr3_unfold (CHead (CHead (CHead (CSort O) (Bind +Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S O)) +x13 H45 (nf2_lref_abst (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) +(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead (CSort O) (Bind Abst) +(TSort O)) (TSort O) (S O) (getl_head (Bind Abst) O (CHead (CHead (CSort O) +(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead (CHead (CSort O) (Bind +Abst) (TSort O)) (Bind Abst) (TSort O)) (getl_refl Abst (CHead (CSort O) +(Bind Abst) (TSort O)) (TSort O)) (TLRef O))))) in (let H47 \def (eq_ind T +(TLRef (S (S O))) (\lambda (ee: T).(match ee in T return (\lambda (_: +T).Prop) with [(TSort _) \Rightarrow False | (TLRef n) \Rightarrow (match n +in nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S n0) +\Rightarrow (match n0 in nat return (\lambda (_: nat).Prop) with [O +\Rightarrow False | (S _) \Rightarrow True])]) | (THead _ _ _) \Rightarrow +False])) I (TLRef (S O)) H46) in (False_ind P H47)))))) H43))))) +(pc3_gen_abst (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind +Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) x0 x5 x1 H7))))))) +H34)))))))) H30)) (ty3_gen_lref g (CHead (CHead (CSort O) (Bind Abst) (TSort +O)) (Bind Abst) (TSort O)) x4 O H22))))))) H24)))))))) H17))))) H14)))))))) +H10)) (ty3_gen_lref g (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) +(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) x6 (S (S O)) H8))))))) +(ty3_gen_bind g Abst (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) +(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) (TSort O) +(THead (Bind Abst) x0 x1) H1)))))))) H3)) (ty3_gen_lref g (CHead (CHead +(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) +(TLRef O)) x0 O H2))))))) (ty3_gen_appl g (CHead (CHead (CHead (CSort O) +(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef +O) (THead (Bind Abst) (TLRef (S (S O))) (TSort O)) u H))))). +(* COMMENTS +Initial nodes: 9973 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/ex2/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/ex2/defs.ma new file mode 100644 index 000000000..35b5df73e --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/ex2/defs.ma @@ -0,0 +1,28 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/C/defs.ma". + +definition ex2_c: + C +\def + CSort O. + +definition ex2_t: + T +\def + THead (Flat Appl) (TSort O) (TSort O). + diff --git a/matita/matita/contribs/lambdadelta/basic_1/ex2/props.ma b/matita/matita/contribs/lambdadelta/basic_1/ex2/props.ma new file mode 100644 index 000000000..4dfa6a582 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/ex2/props.ma @@ -0,0 +1,159 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/ex2/defs.ma". + +include "Basic-1/nf2/defs.ma". + +include "Basic-1/pr2/fwd.ma". + +include "Basic-1/arity/fwd.ma". + +theorem ex2_nf2: + nf2 ex2_c ex2_t +\def + \lambda (t2: T).(\lambda (H: (pr2 (CSort O) (THead (Flat Appl) (TSort O) +(TSort O)) t2)).(let H0 \def (pr2_gen_appl (CSort O) (TSort O) (TSort O) t2 +H) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead +(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 (CSort O) (TSort +O) u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 (CSort O) (TSort O) t3)))) +(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: +T).(eq T (TSort O) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda +(_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 +t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: +T).(pr2 (CSort O) (TSort O) u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda +(_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead (CSort O) +(Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B +b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(eq T (TSort O) (THead (Bind b) y1 +z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: +T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat +Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 (CSort O) (TSort +O) u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CSort O) y1 y2))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 (CHead (CSort O) (Bind b) y2) z1 z2)))))))) (eq T (THead (Flat +Appl) (TSort O) (TSort O)) t2) (\lambda (H1: (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 (CSort O) (TSort O) u2))) (\lambda (_: T).(\lambda +(t3: T).(pr2 (CSort O) (TSort O) t3))))).(ex3_2_ind T T (\lambda (u2: +T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 (CSort O) (TSort O) u2))) (\lambda (_: T).(\lambda +(t3: T).(pr2 (CSort O) (TSort O) t3))) (eq T (THead (Flat Appl) (TSort O) +(TSort O)) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H2: (eq T t2 +(THead (Flat Appl) x0 x1))).(\lambda (H3: (pr2 (CSort O) (TSort O) +x0)).(\lambda (H4: (pr2 (CSort O) (TSort O) x1)).(let H5 \def (eq_ind T x1 +(\lambda (t: T).(eq T t2 (THead (Flat Appl) x0 t))) H2 (TSort O) +(pr2_gen_sort (CSort O) x1 O H4)) in (let H6 \def (eq_ind T x0 (\lambda (t: +T).(eq T t2 (THead (Flat Appl) t (TSort O)))) H5 (TSort O) (pr2_gen_sort +(CSort O) x0 O H3)) in (eq_ind_r T (THead (Flat Appl) (TSort O) (TSort O)) +(\lambda (t: T).(eq T (THead (Flat Appl) (TSort O) (TSort O)) t)) (refl_equal +T (THead (Flat Appl) (TSort O) (TSort O))) t2 H6)))))))) H1)) (\lambda (H1: +(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: +T).(eq T (TSort O) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda +(_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 +t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: +T).(pr2 (CSort O) (TSort O) u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda +(_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead (CSort O) +(Bind b) u) z1 t3))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(eq T (TSort O) (THead (Bind Abst) y1 +z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: +T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 (CSort O) (TSort O) u2))))) (\lambda +(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: +B).(\forall (u: T).(pr2 (CHead (CSort O) (Bind b) u) z1 t3))))))) (eq T +(THead (Flat Appl) (TSort O) (TSort O)) t2) (\lambda (x0: T).(\lambda (x1: +T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H2: (eq T (TSort O) (THead +(Bind Abst) x0 x1))).(\lambda (H3: (eq T t2 (THead (Bind Abbr) x2 +x3))).(\lambda (H4: (pr2 (CSort O) (TSort O) x2)).(\lambda (_: ((\forall (b: +B).(\forall (u: T).(pr2 (CHead (CSort O) (Bind b) u) x1 x3))))).(let H6 \def +(eq_ind T x2 (\lambda (t: T).(eq T t2 (THead (Bind Abbr) t x3))) H3 (TSort O) +(pr2_gen_sort (CSort O) x2 O H4)) in (eq_ind_r T (THead (Bind Abbr) (TSort O) +x3) (\lambda (t: T).(eq T (THead (Flat Appl) (TSort O) (TSort O)) t)) (let H7 +\def (eq_ind T (TSort O) (\lambda (ee: T).(match ee in T return (\lambda (_: +T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | +(THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) x0 x1) H2) in +(False_ind (eq T (THead (Flat Appl) (TSort O) (TSort O)) (THead (Bind Abbr) +(TSort O) x3)) H7)) t2 H6)))))))))) H1)) (\lambda (H1: (ex6_6 B T T T T T +(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T +(TSort O) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T +t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) +(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr2 (CSort O) (TSort O) u2))))))) (\lambda (_: +B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 (CSort O) y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda +(z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead (CSort +O) (Bind b) y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (TSort O) +(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind +b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr2 (CSort O) (TSort O) u2))))))) (\lambda (_: B).(\lambda (y1: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 +(CSort O) y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: +T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead (CSort O) +(Bind b) y2) z1 z2))))))) (eq T (THead (Flat Appl) (TSort O) (TSort O)) t2) +(\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda +(x4: T).(\lambda (x5: T).(\lambda (_: (not (eq B x0 Abst))).(\lambda (H3: (eq +T (TSort O) (THead (Bind x0) x1 x2))).(\lambda (H4: (eq T t2 (THead (Bind x0) +x5 (THead (Flat Appl) (lift (S O) O x4) x3)))).(\lambda (H5: (pr2 (CSort O) +(TSort O) x4)).(\lambda (H6: (pr2 (CSort O) x1 x5)).(\lambda (_: (pr2 (CHead +(CSort O) (Bind x0) x5) x2 x3)).(let H_y \def (pr2_gen_csort x1 x5 O H6) in +(let H8 \def (eq_ind T x4 (\lambda (t: T).(eq T t2 (THead (Bind x0) x5 (THead +(Flat Appl) (lift (S O) O t) x3)))) H4 (TSort O) (pr2_gen_sort (CSort O) x4 O +H5)) in (eq_ind_r T (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O +(TSort O)) x3)) (\lambda (t: T).(eq T (THead (Flat Appl) (TSort O) (TSort O)) +t)) (let H9 \def (eq_ind T (TSort O) (\lambda (ee: T).(match ee in T return +(\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) +\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Bind x0) x1 +x2) H3) in (False_ind (eq T (THead (Flat Appl) (TSort O) (TSort O)) (THead +(Bind x0) x5 (THead (Flat Appl) (lift (S O) O (TSort O)) x3))) H9)) t2 +H8))))))))))))))) H1)) H0))). +(* COMMENTS +Initial nodes: 1939 +END *) + +theorem ex2_arity: + \forall (g: G).(\forall (a: A).((arity g ex2_c ex2_t a) \to (\forall (P: +Prop).P))) +\def + \lambda (g: G).(\lambda (a: A).(\lambda (H: (arity g (CSort O) (THead (Flat +Appl) (TSort O) (TSort O)) a)).(\lambda (P: Prop).(let H0 \def +(arity_gen_appl g (CSort O) (TSort O) (TSort O) a H) in (ex2_ind A (\lambda +(a1: A).(arity g (CSort O) (TSort O) a1)) (\lambda (a1: A).(arity g (CSort O) +(TSort O) (AHead a1 a))) P (\lambda (x: A).(\lambda (_: (arity g (CSort O) +(TSort O) x)).(\lambda (H2: (arity g (CSort O) (TSort O) (AHead x a))).(let +H_x \def (leq_gen_head1 g x a (ASort O O) (arity_gen_sort g (CSort O) O +(AHead x a) H2)) in (let H3 \def H_x in (ex3_2_ind A A (\lambda (a3: +A).(\lambda (_: A).(leq g x a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a +a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort O O) (AHead a3 a4)))) P +(\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g x x0)).(\lambda (_: +(leq g a x1)).(\lambda (H6: (eq A (ASort O O) (AHead x0 x1))).(let H7 \def +(eq_ind A (ASort O O) (\lambda (ee: A).(match ee in A return (\lambda (_: +A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow +False])) I (AHead x0 x1) H6) in (False_ind P H7))))))) H3)))))) H0))))). +(* COMMENTS +Initial nodes: 289 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/flt/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/flt/defs.ma new file mode 100644 index 000000000..3191e1e38 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/flt/defs.ma @@ -0,0 +1,29 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/C/defs.ma". + +definition fweight: + C \to (T \to nat) +\def + \lambda (c: C).(\lambda (t: T).(plus (cweight c) (tweight t))). + +definition flt: + C \to (T \to (C \to (T \to Prop))) +\def + \lambda (c1: C).(\lambda (t1: T).(\lambda (c2: C).(\lambda (t2: T).(lt +(fweight c1 t1) (fweight c2 t2))))). + diff --git a/matita/matita/contribs/lambdadelta/basic_1/flt/props.ma b/matita/matita/contribs/lambdadelta/basic_1/flt/props.ma new file mode 100644 index 000000000..57df41528 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/flt/props.ma @@ -0,0 +1,154 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/flt/defs.ma". + +include "Basic-1/C/props.ma". + +theorem flt_thead_sx: + \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).(flt c u c +(THead k u t))))) +\def + \lambda (_: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t: +T).(le_lt_plus_plus (cweight c) (cweight c) (tweight u) (S (plus (tweight u) +(tweight t))) (le_n (cweight c)) (le_n_S (tweight u) (plus (tweight u) +(tweight t)) (le_plus_l (tweight u) (tweight t))))))). +(* COMMENTS +Initial nodes: 65 +END *) + +theorem flt_thead_dx: + \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).(flt c t c +(THead k u t))))) +\def + \lambda (_: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t: +T).(le_lt_plus_plus (cweight c) (cweight c) (tweight t) (S (plus (tweight u) +(tweight t))) (le_n (cweight c)) (le_n_S (tweight t) (plus (tweight u) +(tweight t)) (le_plus_r (tweight u) (tweight t))))))). +(* COMMENTS +Initial nodes: 65 +END *) + +theorem flt_shift: + \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).(flt (CHead c +k u) t c (THead k u t))))) +\def + \lambda (_: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(eq_ind nat +(S (plus (cweight c) (plus (tweight u) (tweight t)))) (\lambda (n: nat).(lt +(plus (plus (cweight c) (tweight u)) (tweight t)) n)) (eq_ind_r nat (plus +(plus (cweight c) (tweight u)) (tweight t)) (\lambda (n: nat).(lt (plus (plus +(cweight c) (tweight u)) (tweight t)) (S n))) (le_n (S (plus (plus (cweight +c) (tweight u)) (tweight t)))) (plus (cweight c) (plus (tweight u) (tweight +t))) (plus_assoc_l (cweight c) (tweight u) (tweight t))) (plus (cweight c) (S +(plus (tweight u) (tweight t)))) (plus_n_Sm (cweight c) (plus (tweight u) +(tweight t))))))). +(* COMMENTS +Initial nodes: 179 +END *) + +theorem flt_arith0: + \forall (k: K).(\forall (c: C).(\forall (t: T).(\forall (i: nat).(flt c t +(CHead c k t) (TLRef i))))) +\def + \lambda (_: K).(\lambda (c: C).(\lambda (t: T).(\lambda (_: +nat).(lt_x_plus_x_Sy (plus (cweight c) (tweight t)) O)))). +(* COMMENTS +Initial nodes: 21 +END *) + +theorem flt_arith1: + \forall (k1: K).(\forall (c1: C).(\forall (c2: C).(\forall (t1: T).((cle +(CHead c1 k1 t1) c2) \to (\forall (k2: K).(\forall (t2: T).(\forall (i: +nat).(flt c1 t1 (CHead c2 k2 t2) (TLRef i))))))))) +\def + \lambda (_: K).(\lambda (c1: C).(\lambda (c2: C).(\lambda (t1: T).(\lambda +(H: (le (plus (cweight c1) (tweight t1)) (cweight c2))).(\lambda (_: +K).(\lambda (t2: T).(\lambda (_: nat).(le_lt_trans (plus (cweight c1) +(tweight t1)) (cweight c2) (plus (plus (cweight c2) (tweight t2)) (S O)) H +(eq_ind_r nat (plus (S O) (plus (cweight c2) (tweight t2))) (\lambda (n: +nat).(lt (cweight c2) n)) (le_lt_n_Sm (cweight c2) (plus (cweight c2) +(tweight t2)) (le_plus_l (cweight c2) (tweight t2))) (plus (plus (cweight c2) +(tweight t2)) (S O)) (plus_sym (plus (cweight c2) (tweight t2)) (S +O))))))))))). +(* COMMENTS +Initial nodes: 151 +END *) + +theorem flt_arith2: + \forall (c1: C).(\forall (c2: C).(\forall (t1: T).(\forall (i: nat).((flt c1 +t1 c2 (TLRef i)) \to (\forall (k2: K).(\forall (t2: T).(\forall (j: nat).(flt +c1 t1 (CHead c2 k2 t2) (TLRef j))))))))) +\def + \lambda (c1: C).(\lambda (c2: C).(\lambda (t1: T).(\lambda (_: nat).(\lambda +(H: (lt (plus (cweight c1) (tweight t1)) (plus (cweight c2) (S O)))).(\lambda +(_: K).(\lambda (t2: T).(\lambda (_: nat).(lt_le_trans (plus (cweight c1) +(tweight t1)) (plus (cweight c2) (S O)) (plus (plus (cweight c2) (tweight +t2)) (S O)) H (le_plus_plus (cweight c2) (plus (cweight c2) (tweight t2)) (S +O) (S O) (le_plus_l (cweight c2) (tweight t2)) (le_n (S O))))))))))). +(* COMMENTS +Initial nodes: 115 +END *) + +theorem flt_trans: + \forall (c1: C).(\forall (c2: C).(\forall (t1: T).(\forall (t2: T).((flt c1 +t1 c2 t2) \to (\forall (c3: C).(\forall (t3: T).((flt c2 t2 c3 t3) \to (flt +c1 t1 c3 t3)))))))) +\def + \lambda (c1: C).(\lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda +(H: (lt (fweight c1 t1) (fweight c2 t2))).(\lambda (c3: C).(\lambda (t3: +T).(\lambda (H0: (lt (fweight c2 t2) (fweight c3 t3))).(lt_trans (fweight c1 +t1) (fweight c2 t2) (fweight c3 t3) H H0)))))))). +(* COMMENTS +Initial nodes: 63 +END *) + +theorem flt_wf__q_ind: + \forall (P: ((C \to (T \to Prop)))).(((\forall (n: nat).((\lambda (P0: ((C +\to (T \to Prop)))).(\lambda (n0: nat).(\forall (c: C).(\forall (t: T).((eq +nat (fweight c t) n0) \to (P0 c t)))))) P n))) \to (\forall (c: C).(\forall +(t: T).(P c t)))) +\def + let Q \def (\lambda (P: ((C \to (T \to Prop)))).(\lambda (n: nat).(\forall +(c: C).(\forall (t: T).((eq nat (fweight c t) n) \to (P c t)))))) in (\lambda +(P: ((C \to (T \to Prop)))).(\lambda (H: ((\forall (n: nat).(\forall (c: +C).(\forall (t: T).((eq nat (fweight c t) n) \to (P c t))))))).(\lambda (c: +C).(\lambda (t: T).(H (fweight c t) c t (refl_equal nat (fweight c t))))))). +(* COMMENTS +Initial nodes: 85 +END *) + +theorem flt_wf_ind: + \forall (P: ((C \to (T \to Prop)))).(((\forall (c2: C).(\forall (t2: +T).(((\forall (c1: C).(\forall (t1: T).((flt c1 t1 c2 t2) \to (P c1 t1))))) +\to (P c2 t2))))) \to (\forall (c: C).(\forall (t: T).(P c t)))) +\def + let Q \def (\lambda (P: ((C \to (T \to Prop)))).(\lambda (n: nat).(\forall +(c: C).(\forall (t: T).((eq nat (fweight c t) n) \to (P c t)))))) in (\lambda +(P: ((C \to (T \to Prop)))).(\lambda (H: ((\forall (c2: C).(\forall (t2: +T).(((\forall (c1: C).(\forall (t1: T).((flt c1 t1 c2 t2) \to (P c1 t1))))) +\to (P c2 t2)))))).(\lambda (c: C).(\lambda (t: T).(flt_wf__q_ind P (\lambda +(n: nat).(lt_wf_ind n (Q P) (\lambda (n0: nat).(\lambda (H0: ((\forall (m: +nat).((lt m n0) \to (Q P m))))).(\lambda (c0: C).(\lambda (t0: T).(\lambda +(H1: (eq nat (fweight c0 t0) n0)).(let H2 \def (eq_ind_r nat n0 (\lambda (n1: +nat).(\forall (m: nat).((lt m n1) \to (\forall (c1: C).(\forall (t1: T).((eq +nat (fweight c1 t1) m) \to (P c1 t1))))))) H0 (fweight c0 t0) H1) in (H c0 t0 +(\lambda (c1: C).(\lambda (t1: T).(\lambda (H3: (flt c1 t1 c0 t0)).(H2 +(fweight c1 t1) H3 c1 t1 (refl_equal nat (fweight c1 t1))))))))))))))) c +t))))). +(* COMMENTS +Initial nodes: 211 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/fsubst0/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/fsubst0/defs.ma new file mode 100644 index 000000000..f39baebf7 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/fsubst0/defs.ma @@ -0,0 +1,26 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/csubst0/defs.ma". + +inductive fsubst0 (i: nat) (v: T) (c1: C) (t1: T): C \to (T \to Prop) \def +| fsubst0_snd: \forall (t2: T).((subst0 i v t1 t2) \to (fsubst0 i v c1 t1 c1 +t2)) +| fsubst0_fst: \forall (c2: C).((csubst0 i v c1 c2) \to (fsubst0 i v c1 t1 c2 +t1)) +| fsubst0_both: \forall (t2: T).((subst0 i v t1 t2) \to (\forall (c2: +C).((csubst0 i v c1 c2) \to (fsubst0 i v c1 t1 c2 t2)))). + diff --git a/matita/matita/contribs/lambdadelta/basic_1/fsubst0/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/fsubst0/fwd.ma new file mode 100644 index 000000000..3c1212510 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/fsubst0/fwd.ma @@ -0,0 +1,43 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/fsubst0/defs.ma". + +theorem fsubst0_gen_base: + \forall (c1: C).(\forall (c2: C).(\forall (t1: T).(\forall (t2: T).(\forall +(v: T).(\forall (i: nat).((fsubst0 i v c1 t1 c2 t2) \to (or3 (land (eq C c1 +c2) (subst0 i v t1 t2)) (land (eq T t1 t2) (csubst0 i v c1 c2)) (land (subst0 +i v t1 t2) (csubst0 i v c1 c2))))))))) +\def + \lambda (c1: C).(\lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda +(v: T).(\lambda (i: nat).(\lambda (H: (fsubst0 i v c1 t1 c2 t2)).(fsubst0_ind +i v c1 t1 (\lambda (c: C).(\lambda (t: T).(or3 (land (eq C c1 c) (subst0 i v +t1 t)) (land (eq T t1 t) (csubst0 i v c1 c)) (land (subst0 i v t1 t) (csubst0 +i v c1 c))))) (\lambda (t0: T).(\lambda (H0: (subst0 i v t1 t0)).(or3_intro0 +(land (eq C c1 c1) (subst0 i v t1 t0)) (land (eq T t1 t0) (csubst0 i v c1 +c1)) (land (subst0 i v t1 t0) (csubst0 i v c1 c1)) (conj (eq C c1 c1) (subst0 +i v t1 t0) (refl_equal C c1) H0)))) (\lambda (c0: C).(\lambda (H0: (csubst0 i +v c1 c0)).(or3_intro1 (land (eq C c1 c0) (subst0 i v t1 t1)) (land (eq T t1 +t1) (csubst0 i v c1 c0)) (land (subst0 i v t1 t1) (csubst0 i v c1 c0)) (conj +(eq T t1 t1) (csubst0 i v c1 c0) (refl_equal T t1) H0)))) (\lambda (t0: +T).(\lambda (H0: (subst0 i v t1 t0)).(\lambda (c0: C).(\lambda (H1: (csubst0 +i v c1 c0)).(or3_intro2 (land (eq C c1 c0) (subst0 i v t1 t0)) (land (eq T t1 +t0) (csubst0 i v c1 c0)) (land (subst0 i v t1 t0) (csubst0 i v c1 c0)) (conj +(subst0 i v t1 t0) (csubst0 i v c1 c0) H0 H1)))))) c2 t2 H))))))). +(* COMMENTS +Initial nodes: 431 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/getl/clear.ma b/matita/matita/contribs/lambdadelta/basic_1/getl/clear.ma new file mode 100644 index 000000000..de0a85bd3 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/getl/clear.ma @@ -0,0 +1,153 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/getl/props.ma". + +include "Basic-1/clear/drop.ma". + +theorem clear_getl_trans: + \forall (i: nat).(\forall (c2: C).(\forall (c3: C).((getl i c2 c3) \to +(\forall (c1: C).((clear c1 c2) \to (getl i c1 c3)))))) +\def + \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c2: C).(\forall (c3: +C).((getl n c2 c3) \to (\forall (c1: C).((clear c1 c2) \to (getl n c1 +c3))))))) (\lambda (c2: C).(\lambda (c3: C).(\lambda (H: (getl O c2 +c3)).(\lambda (c1: C).(\lambda (H0: (clear c1 c2)).(getl_intro O c1 c3 c1 +(drop_refl c1) (clear_trans c1 c2 H0 c3 (getl_gen_O c2 c3 H)))))))) (\lambda +(n: nat).(\lambda (_: ((\forall (c2: C).(\forall (c3: C).((getl n c2 c3) \to +(\forall (c1: C).((clear c1 c2) \to (getl n c1 c3)))))))).(\lambda (c2: +C).(C_ind (\lambda (c: C).(\forall (c3: C).((getl (S n) c c3) \to (\forall +(c1: C).((clear c1 c) \to (getl (S n) c1 c3)))))) (\lambda (n0: nat).(\lambda +(c3: C).(\lambda (H0: (getl (S n) (CSort n0) c3)).(\lambda (c1: C).(\lambda +(_: (clear c1 (CSort n0))).(getl_gen_sort n0 (S n) c3 H0 (getl (S n) c1 +c3))))))) (\lambda (c: C).(\lambda (_: ((\forall (c3: C).((getl (S n) c c3) +\to (\forall (c1: C).((clear c1 c) \to (getl (S n) c1 c3))))))).(\lambda (k: +K).(\lambda (t: T).(\lambda (c3: C).(\lambda (H1: (getl (S n) (CHead c k t) +c3)).(\lambda (c1: C).(\lambda (H2: (clear c1 (CHead c k t))).(K_ind (\lambda +(k0: K).((getl (S n) (CHead c k0 t) c3) \to ((clear c1 (CHead c k0 t)) \to +(getl (S n) c1 c3)))) (\lambda (b: B).(\lambda (H3: (getl (S n) (CHead c +(Bind b) t) c3)).(\lambda (H4: (clear c1 (CHead c (Bind b) t))).(let H5 \def +(getl_gen_all c c3 (r (Bind b) n) (getl_gen_S (Bind b) c c3 t n H3)) in +(ex2_ind C (\lambda (e: C).(drop n O c e)) (\lambda (e: C).(clear e c3)) +(getl (S n) c1 c3) (\lambda (x: C).(\lambda (H6: (drop n O c x)).(\lambda +(H7: (clear x c3)).(getl_intro (S n) c1 c3 x (drop_clear_O b c1 c t H4 x n +H6) H7)))) H5))))) (\lambda (f: F).(\lambda (_: (getl (S n) (CHead c (Flat f) +t) c3)).(\lambda (H4: (clear c1 (CHead c (Flat f) t))).(clear_gen_flat_r f c1 +c t H4 (getl (S n) c1 c3))))) k H1 H2))))))))) c2)))) i). +(* COMMENTS +Initial nodes: 525 +END *) + +theorem getl_clear_trans: + \forall (i: nat).(\forall (c1: C).(\forall (c2: C).((getl i c1 c2) \to +(\forall (c3: C).((clear c2 c3) \to (getl i c1 c3)))))) +\def + \lambda (i: nat).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (getl i c1 +c2)).(\lambda (c3: C).(\lambda (H0: (clear c2 c3)).(let H1 \def (getl_gen_all +c1 c2 i H) in (ex2_ind C (\lambda (e: C).(drop i O c1 e)) (\lambda (e: +C).(clear e c2)) (getl i c1 c3) (\lambda (x: C).(\lambda (H2: (drop i O c1 +x)).(\lambda (H3: (clear x c2)).(let H4 \def (clear_gen_all x c2 H3) in +(ex_3_ind B C T (\lambda (b: B).(\lambda (e: C).(\lambda (u: T).(eq C c2 +(CHead e (Bind b) u))))) (getl i c1 c3) (\lambda (x0: B).(\lambda (x1: +C).(\lambda (x2: T).(\lambda (H5: (eq C c2 (CHead x1 (Bind x0) x2))).(let H6 +\def (eq_ind C c2 (\lambda (c: C).(clear x c)) H3 (CHead x1 (Bind x0) x2) H5) +in (let H7 \def (eq_ind C c2 (\lambda (c: C).(clear c c3)) H0 (CHead x1 (Bind +x0) x2) H5) in (eq_ind_r C (CHead x1 (Bind x0) x2) (\lambda (c: C).(getl i c1 +c)) (getl_intro i c1 (CHead x1 (Bind x0) x2) x H2 H6) c3 (clear_gen_bind x0 +x1 c3 x2 H7)))))))) H4))))) H1))))))). +(* COMMENTS +Initial nodes: 269 +END *) + +theorem getl_clear_bind: + \forall (b: B).(\forall (c: C).(\forall (e1: C).(\forall (v: T).((clear c +(CHead e1 (Bind b) v)) \to (\forall (e2: C).(\forall (n: nat).((getl n e1 e2) +\to (getl (S n) c e2)))))))) +\def + \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e1: +C).(\forall (v: T).((clear c0 (CHead e1 (Bind b) v)) \to (\forall (e2: +C).(\forall (n: nat).((getl n e1 e2) \to (getl (S n) c0 e2)))))))) (\lambda +(n: nat).(\lambda (e1: C).(\lambda (v: T).(\lambda (H: (clear (CSort n) +(CHead e1 (Bind b) v))).(\lambda (e2: C).(\lambda (n0: nat).(\lambda (_: +(getl n0 e1 e2)).(clear_gen_sort (CHead e1 (Bind b) v) n H (getl (S n0) +(CSort n) e2))))))))) (\lambda (c0: C).(\lambda (H: ((\forall (e1: +C).(\forall (v: T).((clear c0 (CHead e1 (Bind b) v)) \to (\forall (e2: +C).(\forall (n: nat).((getl n e1 e2) \to (getl (S n) c0 e2))))))))).(\lambda +(k: K).(\lambda (t: T).(\lambda (e1: C).(\lambda (v: T).(\lambda (H0: (clear +(CHead c0 k t) (CHead e1 (Bind b) v))).(\lambda (e2: C).(\lambda (n: +nat).(\lambda (H1: (getl n e1 e2)).(K_ind (\lambda (k0: K).((clear (CHead c0 +k0 t) (CHead e1 (Bind b) v)) \to (getl (S n) (CHead c0 k0 t) e2))) (\lambda +(b0: B).(\lambda (H2: (clear (CHead c0 (Bind b0) t) (CHead e1 (Bind b) +v))).(let H3 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda +(_: C).C) with [(CSort _) \Rightarrow e1 | (CHead c1 _ _) \Rightarrow c1])) +(CHead e1 (Bind b) v) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e1 +(Bind b) v) t H2)) in ((let H4 \def (f_equal C B (\lambda (e: C).(match e in +C return (\lambda (_: C).B) with [(CSort _) \Rightarrow b | (CHead _ k0 _) +\Rightarrow (match k0 in K return (\lambda (_: K).B) with [(Bind b1) +\Rightarrow b1 | (Flat _) \Rightarrow b])])) (CHead e1 (Bind b) v) (CHead c0 +(Bind b0) t) (clear_gen_bind b0 c0 (CHead e1 (Bind b) v) t H2)) in ((let H5 +\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) +with [(CSort _) \Rightarrow v | (CHead _ _ t0) \Rightarrow t0])) (CHead e1 +(Bind b) v) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e1 (Bind b) +v) t H2)) in (\lambda (H6: (eq B b b0)).(\lambda (H7: (eq C e1 c0)).(let H8 +\def (eq_ind C e1 (\lambda (c1: C).(getl n c1 e2)) H1 c0 H7) in (eq_ind B b +(\lambda (b1: B).(getl (S n) (CHead c0 (Bind b1) t) e2)) (getl_head (Bind b) +n c0 e2 H8 t) b0 H6))))) H4)) H3)))) (\lambda (f: F).(\lambda (H2: (clear +(CHead c0 (Flat f) t) (CHead e1 (Bind b) v))).(getl_flat c0 e2 (S n) (H e1 v +(clear_gen_flat f c0 (CHead e1 (Bind b) v) t H2) e2 n H1) f t))) k +H0))))))))))) c)). +(* COMMENTS +Initial nodes: 599 +END *) + +theorem getl_clear_conf: + \forall (i: nat).(\forall (c1: C).(\forall (c3: C).((getl i c1 c3) \to +(\forall (c2: C).((clear c1 c2) \to (getl i c2 c3)))))) +\def + \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (c3: +C).((getl n c1 c3) \to (\forall (c2: C).((clear c1 c2) \to (getl n c2 +c3))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H: (getl O c1 +c3)).(\lambda (c2: C).(\lambda (H0: (clear c1 c2)).(eq_ind C c3 (\lambda (c: +C).(getl O c c3)) (let H1 \def (clear_gen_all c1 c3 (getl_gen_O c1 c3 H)) in +(ex_3_ind B C T (\lambda (b: B).(\lambda (e: C).(\lambda (u: T).(eq C c3 +(CHead e (Bind b) u))))) (getl O c3 c3) (\lambda (x0: B).(\lambda (x1: +C).(\lambda (x2: T).(\lambda (H2: (eq C c3 (CHead x1 (Bind x0) x2))).(let H3 +\def (eq_ind C c3 (\lambda (c: C).(clear c1 c)) (getl_gen_O c1 c3 H) (CHead +x1 (Bind x0) x2) H2) in (eq_ind_r C (CHead x1 (Bind x0) x2) (\lambda (c: +C).(getl O c c)) (getl_refl x0 x1 x2) c3 H2)))))) H1)) c2 (clear_mono c1 c3 +(getl_gen_O c1 c3 H) c2 H0))))))) (\lambda (n: nat).(\lambda (_: ((\forall +(c1: C).(\forall (c3: C).((getl n c1 c3) \to (\forall (c2: C).((clear c1 c2) +\to (getl n c2 c3)))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall +(c3: C).((getl (S n) c c3) \to (\forall (c2: C).((clear c c2) \to (getl (S n) +c2 c3)))))) (\lambda (n0: nat).(\lambda (c3: C).(\lambda (H0: (getl (S n) +(CSort n0) c3)).(\lambda (c2: C).(\lambda (_: (clear (CSort n0) +c2)).(getl_gen_sort n0 (S n) c3 H0 (getl (S n) c2 c3))))))) (\lambda (c: +C).(\lambda (H0: ((\forall (c3: C).((getl (S n) c c3) \to (\forall (c2: +C).((clear c c2) \to (getl (S n) c2 c3))))))).(\lambda (k: K).(\lambda (t: +T).(\lambda (c3: C).(\lambda (H1: (getl (S n) (CHead c k t) c3)).(\lambda +(c2: C).(\lambda (H2: (clear (CHead c k t) c2)).(K_ind (\lambda (k0: +K).((getl (S n) (CHead c k0 t) c3) \to ((clear (CHead c k0 t) c2) \to (getl +(S n) c2 c3)))) (\lambda (b: B).(\lambda (H3: (getl (S n) (CHead c (Bind b) +t) c3)).(\lambda (H4: (clear (CHead c (Bind b) t) c2)).(eq_ind_r C (CHead c +(Bind b) t) (\lambda (c0: C).(getl (S n) c0 c3)) (getl_head (Bind b) n c c3 +(getl_gen_S (Bind b) c c3 t n H3) t) c2 (clear_gen_bind b c c2 t H4))))) +(\lambda (f: F).(\lambda (H3: (getl (S n) (CHead c (Flat f) t) c3)).(\lambda +(H4: (clear (CHead c (Flat f) t) c2)).(H0 c3 (getl_gen_S (Flat f) c c3 t n +H3) c2 (clear_gen_flat f c c2 t H4))))) k H1 H2))))))))) c1)))) i). +(* COMMENTS +Initial nodes: 641 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/getl/dec.ma b/matita/matita/contribs/lambdadelta/basic_1/getl/dec.ma new file mode 100644 index 000000000..278362590 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/getl/dec.ma @@ -0,0 +1,100 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/getl/props.ma". + +theorem getl_dec: + \forall (c: C).(\forall (i: nat).(or (ex_3 C B T (\lambda (e: C).(\lambda +(b: B).(\lambda (v: T).(getl i c (CHead e (Bind b) v)))))) (\forall (d: +C).((getl i c d) \to (\forall (P: Prop).P))))) +\def + \lambda (c: C).(C_ind (\lambda (c0: C).(\forall (i: nat).(or (ex_3 C B T +(\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl i c0 (CHead e (Bind b) +v)))))) (\forall (d: C).((getl i c0 d) \to (\forall (P: Prop).P)))))) +(\lambda (n: nat).(\lambda (i: nat).(or_intror (ex_3 C B T (\lambda (e: +C).(\lambda (b: B).(\lambda (v: T).(getl i (CSort n) (CHead e (Bind b) +v)))))) (\forall (d: C).((getl i (CSort n) d) \to (\forall (P: Prop).P))) +(\lambda (d: C).(\lambda (H: (getl i (CSort n) d)).(\lambda (P: +Prop).(getl_gen_sort n i d H P))))))) (\lambda (c0: C).(\lambda (H: ((\forall +(i: nat).(or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: +T).(getl i c0 (CHead e (Bind b) v)))))) (\forall (d: C).((getl i c0 d) \to +(\forall (P: Prop).P))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (i: +nat).(nat_ind (\lambda (n: nat).(or (ex_3 C B T (\lambda (e: C).(\lambda (b: +B).(\lambda (v: T).(getl n (CHead c0 k t) (CHead e (Bind b) v)))))) (\forall +(d: C).((getl n (CHead c0 k t) d) \to (\forall (P: Prop).P))))) (K_ind +(\lambda (k0: K).(or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: +T).(getl O (CHead c0 k0 t) (CHead e (Bind b) v)))))) (\forall (d: C).((getl O +(CHead c0 k0 t) d) \to (\forall (P: Prop).P))))) (\lambda (b: B).(or_introl +(ex_3 C B T (\lambda (e: C).(\lambda (b0: B).(\lambda (v: T).(getl O (CHead +c0 (Bind b) t) (CHead e (Bind b0) v)))))) (\forall (d: C).((getl O (CHead c0 +(Bind b) t) d) \to (\forall (P: Prop).P))) (ex_3_intro C B T (\lambda (e: +C).(\lambda (b0: B).(\lambda (v: T).(getl O (CHead c0 (Bind b) t) (CHead e +(Bind b0) v))))) c0 b t (getl_refl b c0 t)))) (\lambda (f: F).(let H_x \def +(H O) in (let H0 \def H_x in (or_ind (ex_3 C B T (\lambda (e: C).(\lambda (b: +B).(\lambda (v: T).(getl O c0 (CHead e (Bind b) v)))))) (\forall (d: +C).((getl O c0 d) \to (\forall (P: Prop).P))) (or (ex_3 C B T (\lambda (e: +C).(\lambda (b: B).(\lambda (v: T).(getl O (CHead c0 (Flat f) t) (CHead e +(Bind b) v)))))) (\forall (d: C).((getl O (CHead c0 (Flat f) t) d) \to +(\forall (P: Prop).P)))) (\lambda (H1: (ex_3 C B T (\lambda (e: C).(\lambda +(b: B).(\lambda (v: T).(getl O c0 (CHead e (Bind b) v))))))).(ex_3_ind C B T +(\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl O c0 (CHead e (Bind b) +v))))) (or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl +O (CHead c0 (Flat f) t) (CHead e (Bind b) v)))))) (\forall (d: C).((getl O +(CHead c0 (Flat f) t) d) \to (\forall (P: Prop).P)))) (\lambda (x0: +C).(\lambda (x1: B).(\lambda (x2: T).(\lambda (H2: (getl O c0 (CHead x0 (Bind +x1) x2))).(or_introl (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: +T).(getl O (CHead c0 (Flat f) t) (CHead e (Bind b) v)))))) (\forall (d: +C).((getl O (CHead c0 (Flat f) t) d) \to (\forall (P: Prop).P))) (ex_3_intro +C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl O (CHead c0 (Flat +f) t) (CHead e (Bind b) v))))) x0 x1 x2 (getl_flat c0 (CHead x0 (Bind x1) x2) +O H2 f t))))))) H1)) (\lambda (H1: ((\forall (d: C).((getl O c0 d) \to +(\forall (P: Prop).P))))).(or_intror (ex_3 C B T (\lambda (e: C).(\lambda (b: +B).(\lambda (v: T).(getl O (CHead c0 (Flat f) t) (CHead e (Bind b) v)))))) +(\forall (d: C).((getl O (CHead c0 (Flat f) t) d) \to (\forall (P: Prop).P))) +(\lambda (d: C).(\lambda (H2: (getl O (CHead c0 (Flat f) t) d)).(\lambda (P: +Prop).(H1 d (getl_intro O c0 d c0 (drop_refl c0) (clear_gen_flat f c0 d t +(getl_gen_O (CHead c0 (Flat f) t) d H2))) P)))))) H0)))) k) (\lambda (n: +nat).(\lambda (_: (or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda +(v: T).(getl n (CHead c0 k t) (CHead e (Bind b) v)))))) (\forall (d: +C).((getl n (CHead c0 k t) d) \to (\forall (P: Prop).P))))).(let H_x \def (H +(r k n)) in (let H1 \def H_x in (or_ind (ex_3 C B T (\lambda (e: C).(\lambda +(b: B).(\lambda (v: T).(getl (r k n) c0 (CHead e (Bind b) v)))))) (\forall +(d: C).((getl (r k n) c0 d) \to (\forall (P: Prop).P))) (or (ex_3 C B T +(\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl (S n) (CHead c0 k t) +(CHead e (Bind b) v)))))) (\forall (d: C).((getl (S n) (CHead c0 k t) d) \to +(\forall (P: Prop).P)))) (\lambda (H2: (ex_3 C B T (\lambda (e: C).(\lambda +(b: B).(\lambda (v: T).(getl (r k n) c0 (CHead e (Bind b) v))))))).(ex_3_ind +C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl (r k n) c0 (CHead +e (Bind b) v))))) (or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda +(v: T).(getl (S n) (CHead c0 k t) (CHead e (Bind b) v)))))) (\forall (d: +C).((getl (S n) (CHead c0 k t) d) \to (\forall (P: Prop).P)))) (\lambda (x0: +C).(\lambda (x1: B).(\lambda (x2: T).(\lambda (H3: (getl (r k n) c0 (CHead x0 +(Bind x1) x2))).(or_introl (ex_3 C B T (\lambda (e: C).(\lambda (b: +B).(\lambda (v: T).(getl (S n) (CHead c0 k t) (CHead e (Bind b) v)))))) +(\forall (d: C).((getl (S n) (CHead c0 k t) d) \to (\forall (P: Prop).P))) +(ex_3_intro C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl (S n) +(CHead c0 k t) (CHead e (Bind b) v))))) x0 x1 x2 (getl_head k n c0 (CHead x0 +(Bind x1) x2) H3 t))))))) H2)) (\lambda (H2: ((\forall (d: C).((getl (r k n) +c0 d) \to (\forall (P: Prop).P))))).(or_intror (ex_3 C B T (\lambda (e: +C).(\lambda (b: B).(\lambda (v: T).(getl (S n) (CHead c0 k t) (CHead e (Bind +b) v)))))) (\forall (d: C).((getl (S n) (CHead c0 k t) d) \to (\forall (P: +Prop).P))) (\lambda (d: C).(\lambda (H3: (getl (S n) (CHead c0 k t) +d)).(\lambda (P: Prop).(H2 d (getl_gen_S k c0 d t n H3) P)))))) H1))))) +i)))))) c). +(* COMMENTS +Initial nodes: 1563 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/getl/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/getl/defs.ma new file mode 100644 index 000000000..a6efd2eee --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/getl/defs.ma @@ -0,0 +1,24 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/drop/defs.ma". + +include "Basic-1/clear/defs.ma". + +inductive getl (h: nat) (c1: C) (c2: C): Prop \def +| getl_intro: \forall (e: C).((drop h O c1 e) \to ((clear e c2) \to (getl h +c1 c2))). + diff --git a/matita/matita/contribs/lambdadelta/basic_1/getl/drop.ma b/matita/matita/contribs/lambdadelta/basic_1/getl/drop.ma new file mode 100644 index 000000000..e4404e137 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/getl/drop.ma @@ -0,0 +1,514 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/getl/props.ma". + +include "Basic-1/clear/drop.ma". + +theorem getl_drop: + \forall (b: B).(\forall (c: C).(\forall (e: C).(\forall (u: T).(\forall (h: +nat).((getl h c (CHead e (Bind b) u)) \to (drop (S h) O c e)))))) +\def + \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e: +C).(\forall (u: T).(\forall (h: nat).((getl h c0 (CHead e (Bind b) u)) \to +(drop (S h) O c0 e)))))) (\lambda (n: nat).(\lambda (e: C).(\lambda (u: +T).(\lambda (h: nat).(\lambda (H: (getl h (CSort n) (CHead e (Bind b) +u))).(getl_gen_sort n h (CHead e (Bind b) u) H (drop (S h) O (CSort n) +e))))))) (\lambda (c0: C).(\lambda (H: ((\forall (e: C).(\forall (u: +T).(\forall (h: nat).((getl h c0 (CHead e (Bind b) u)) \to (drop (S h) O c0 +e))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e: C).(\lambda (u: +T).(\lambda (h: nat).(nat_ind (\lambda (n: nat).((getl n (CHead c0 k t) +(CHead e (Bind b) u)) \to (drop (S n) O (CHead c0 k t) e))) (\lambda (H0: +(getl O (CHead c0 k t) (CHead e (Bind b) u))).(K_ind (\lambda (k0: K).((clear +(CHead c0 k0 t) (CHead e (Bind b) u)) \to (drop (S O) O (CHead c0 k0 t) e))) +(\lambda (b0: B).(\lambda (H1: (clear (CHead c0 (Bind b0) t) (CHead e (Bind +b) u))).(let H2 \def (f_equal C C (\lambda (e0: C).(match e0 in C return +(\lambda (_: C).C) with [(CSort _) \Rightarrow e | (CHead c1 _ _) \Rightarrow +c1])) (CHead e (Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 +(CHead e (Bind b) u) t H1)) in ((let H3 \def (f_equal C B (\lambda (e0: +C).(match e0 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow b | +(CHead _ k0 _) \Rightarrow (match k0 in K return (\lambda (_: K).B) with +[(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow b])])) (CHead e (Bind b) u) +(CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e (Bind b) u) t H1)) in +((let H4 \def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda +(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) +(CHead e (Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e +(Bind b) u) t H1)) in (\lambda (H5: (eq B b b0)).(\lambda (H6: (eq C e +c0)).(eq_ind_r C c0 (\lambda (c1: C).(drop (S O) O (CHead c0 (Bind b0) t) +c1)) (eq_ind B b (\lambda (b1: B).(drop (S O) O (CHead c0 (Bind b1) t) c0)) +(drop_drop (Bind b) O c0 c0 (drop_refl c0) t) b0 H5) e H6)))) H3)) H2)))) +(\lambda (f: F).(\lambda (H1: (clear (CHead c0 (Flat f) t) (CHead e (Bind b) +u))).(drop_clear_O b (CHead c0 (Flat f) t) e u (clear_flat c0 (CHead e (Bind +b) u) (clear_gen_flat f c0 (CHead e (Bind b) u) t H1) f t) e O (drop_refl +e)))) k (getl_gen_O (CHead c0 k t) (CHead e (Bind b) u) H0))) (\lambda (n: +nat).(\lambda (_: (((getl n (CHead c0 k t) (CHead e (Bind b) u)) \to (drop (S +n) O (CHead c0 k t) e)))).(\lambda (H1: (getl (S n) (CHead c0 k t) (CHead e +(Bind b) u))).(drop_drop k (S n) c0 e (eq_ind_r nat (S (r k n)) (\lambda (n0: +nat).(drop n0 O c0 e)) (H e u (r k n) (getl_gen_S k c0 (CHead e (Bind b) u) t +n H1)) (r k (S n)) (r_S k n)) t)))) h)))))))) c)). +(* COMMENTS +Initial nodes: 827 +END *) + +theorem getl_drop_conf_lt: + \forall (b: B).(\forall (c: C).(\forall (c0: C).(\forall (u: T).(\forall (i: +nat).((getl i c (CHead c0 (Bind b) u)) \to (\forall (e: C).(\forall (h: +nat).(\forall (d: nat).((drop h (S (plus i d)) c e) \to (ex3_2 T C (\lambda +(v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: +C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop +h d c0 e0))))))))))))) +\def + \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (c1: +C).(\forall (u: T).(\forall (i: nat).((getl i c0 (CHead c1 (Bind b) u)) \to +(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h (S (plus i d)) +c0 e) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) +(\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda +(_: T).(\lambda (e0: C).(drop h d c1 e0))))))))))))) (\lambda (n: +nat).(\lambda (c0: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H: (getl i +(CSort n) (CHead c0 (Bind b) u))).(\lambda (e: C).(\lambda (h: nat).(\lambda +(d: nat).(\lambda (_: (drop h (S (plus i d)) (CSort n) e)).(getl_gen_sort n i +(CHead c0 (Bind b) u) H (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u +(lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) +v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c0 e0)))))))))))))) (\lambda +(c0: C).(\lambda (H: ((\forall (c1: C).(\forall (u: T).(\forall (i: +nat).((getl i c0 (CHead c1 (Bind b) u)) \to (\forall (e: C).(\forall (h: +nat).(\forall (d: nat).((drop h (S (plus i d)) c0 e) \to (ex3_2 T C (\lambda +(v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: +C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop +h d c1 e0)))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c1: +C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i (CHead c0 k t) +(CHead c1 (Bind b) u))).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: +nat).(\lambda (H1: (drop h (S (plus i d)) (CHead c0 k t) e)).(let H2 \def +(getl_gen_all (CHead c0 k t) (CHead c1 (Bind b) u) i H0) in (ex2_ind C +(\lambda (e0: C).(drop i O (CHead c0 k t) e0)) (\lambda (e0: C).(clear e0 +(CHead c1 (Bind b) u))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u +(lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) +v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x: +C).(\lambda (H3: (drop i O (CHead c0 k t) x)).(\lambda (H4: (clear x (CHead +c1 (Bind b) u))).(C_ind (\lambda (c2: C).((drop i O (CHead c0 k t) c2) \to +((clear c2 (CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: +C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead +e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) +(\lambda (n: nat).(\lambda (_: (drop i O (CHead c0 k t) (CSort n))).(\lambda +(H6: (clear (CSort n) (CHead c1 (Bind b) u))).(clear_gen_sort (CHead c1 (Bind +b) u) n H6 (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) +(\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda +(_: T).(\lambda (e0: C).(drop h d c1 e0)))))))) (\lambda (x0: C).(\lambda +(IHx: (((drop i O (CHead c0 k t) x0) \to ((clear x0 (CHead c1 (Bind b) u)) +\to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) +(\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda +(_: T).(\lambda (e0: C).(drop h d c1 e0)))))))).(\lambda (k0: K).(\lambda +(t0: T).(\lambda (H5: (drop i O (CHead c0 k t) (CHead x0 k0 t0))).(\lambda +(H6: (clear (CHead x0 k0 t0) (CHead c1 (Bind b) u))).(K_ind (\lambda (k1: +K).((drop i O (CHead c0 k t) (CHead x0 k1 t0)) \to ((clear (CHead x0 k1 t0) +(CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u +(lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) +v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) (\lambda (b0: +B).(\lambda (H7: (drop i O (CHead c0 k t) (CHead x0 (Bind b0) t0))).(\lambda +(H8: (clear (CHead x0 (Bind b0) t0) (CHead c1 (Bind b) u))).(let H9 \def +(f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with +[(CSort _) \Rightarrow c1 | (CHead c2 _ _) \Rightarrow c2])) (CHead c1 (Bind +b) u) (CHead x0 (Bind b0) t0) (clear_gen_bind b0 x0 (CHead c1 (Bind b) u) t0 +H8)) in ((let H10 \def (f_equal C B (\lambda (e0: C).(match e0 in C return +(\lambda (_: C).B) with [(CSort _) \Rightarrow b | (CHead _ k1 _) \Rightarrow +(match k1 in K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | +(Flat _) \Rightarrow b])])) (CHead c1 (Bind b) u) (CHead x0 (Bind b0) t0) +(clear_gen_bind b0 x0 (CHead c1 (Bind b) u) t0 H8)) in ((let H11 \def +(f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) with +[(CSort _) \Rightarrow u | (CHead _ _ t1) \Rightarrow t1])) (CHead c1 (Bind +b) u) (CHead x0 (Bind b0) t0) (clear_gen_bind b0 x0 (CHead c1 (Bind b) u) t0 +H8)) in (\lambda (H12: (eq B b b0)).(\lambda (H13: (eq C c1 x0)).(let H14 +\def (eq_ind_r T t0 (\lambda (t1: T).(drop i O (CHead c0 k t) (CHead x0 (Bind +b0) t1))) H7 u H11) in (let H15 \def (eq_ind_r B b0 (\lambda (b1: B).(drop i +O (CHead c0 k t) (CHead x0 (Bind b1) u))) H14 b H12) in (let H16 \def +(eq_ind_r C x0 (\lambda (c2: C).((drop i O (CHead c0 k t) c2) \to ((clear c2 +(CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u +(lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) +v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) IHx c1 H13) in +(let H17 \def (eq_ind_r C x0 (\lambda (c2: C).(drop i O (CHead c0 k t) (CHead +c2 (Bind b) u))) H15 c1 H13) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: +C).(eq T u (lift h (r (Bind b) d) v)))) (\lambda (v: T).(\lambda (e0: +C).(drop i O e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: +C).(drop h (r (Bind b) d) c1 e0))) (ex3_2 T C (\lambda (v: T).(\lambda (_: +C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead +e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))) +(\lambda (x1: T).(\lambda (x2: C).(\lambda (H18: (eq T u (lift h (r (Bind b) +d) x1))).(\lambda (H19: (drop i O e (CHead x2 (Bind b) x1))).(\lambda (H20: +(drop h (r (Bind b) d) c1 x2)).(let H21 \def (eq_ind T u (\lambda (t1: +T).((drop i O (CHead c0 k t) c1) \to ((clear c1 (CHead c1 (Bind b) t1)) \to +(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T t1 (lift h d v)))) (\lambda +(v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: +T).(\lambda (e0: C).(drop h d c1 e0))))))) H16 (lift h (r (Bind b) d) x1) +H18) in (eq_ind_r T (lift h (r (Bind b) d) x1) (\lambda (t1: T).(ex3_2 T C +(\lambda (v: T).(\lambda (_: C).(eq T t1 (lift h d v)))) (\lambda (v: +T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: +T).(\lambda (e0: C).(drop h d c1 e0))))) (ex3_2_intro T C (\lambda (v: +T).(\lambda (_: C).(eq T (lift h (r (Bind b) d) x1) (lift h d v)))) (\lambda +(v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: +T).(\lambda (e0: C).(drop h d c1 e0))) x1 x2 (refl_equal T (lift h d x1)) +(getl_intro i e (CHead x2 (Bind b) x1) (CHead x2 (Bind b) x1) H19 (clear_bind +b x2 x1)) H20) u H18))))))) (drop_conf_lt (Bind b) i u c1 (CHead c0 k t) H17 +e h d H1))))))))) H10)) H9))))) (\lambda (f: F).(\lambda (H7: (drop i O +(CHead c0 k t) (CHead x0 (Flat f) t0))).(\lambda (H8: (clear (CHead x0 (Flat +f) t0) (CHead c1 (Bind b) u))).(nat_ind (\lambda (n: nat).((drop h (S (plus n +d)) (CHead c0 k t) e) \to ((drop n O (CHead c0 k t) (CHead x0 (Flat f) t0)) +\to ((((drop n O (CHead c0 k t) x0) \to ((clear x0 (CHead c1 (Bind b) u)) \to +(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda +(v: T).(\lambda (e0: C).(getl n e (CHead e0 (Bind b) v)))) (\lambda (_: +T).(\lambda (e0: C).(drop h d c1 e0))))))) \to (ex3_2 T C (\lambda (v: +T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: +C).(getl n e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop +h d c1 e0)))))))) (\lambda (H9: (drop h (S (plus O d)) (CHead c0 k t) +e)).(\lambda (H10: (drop O O (CHead c0 k t) (CHead x0 (Flat f) t0))).(\lambda +(IHx0: (((drop O O (CHead c0 k t) x0) \to ((clear x0 (CHead c1 (Bind b) u)) +\to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) +(\lambda (v: T).(\lambda (e0: C).(getl O e (CHead e0 (Bind b) v)))) (\lambda +(_: T).(\lambda (e0: C).(drop h d c1 e0)))))))).(let H11 \def (f_equal C C +(\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) +\Rightarrow c0 | (CHead c2 _ _) \Rightarrow c2])) (CHead c0 k t) (CHead x0 +(Flat f) t0) (drop_gen_refl (CHead c0 k t) (CHead x0 (Flat f) t0) H10)) in +((let H12 \def (f_equal C K (\lambda (e0: C).(match e0 in C return (\lambda +(_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k1 _) \Rightarrow k1])) +(CHead c0 k t) (CHead x0 (Flat f) t0) (drop_gen_refl (CHead c0 k t) (CHead x0 +(Flat f) t0) H10)) in ((let H13 \def (f_equal C T (\lambda (e0: C).(match e0 +in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow t | (CHead _ _ t1) +\Rightarrow t1])) (CHead c0 k t) (CHead x0 (Flat f) t0) (drop_gen_refl (CHead +c0 k t) (CHead x0 (Flat f) t0) H10)) in (\lambda (H14: (eq K k (Flat +f))).(\lambda (H15: (eq C c0 x0)).(let H16 \def (eq_ind_r C x0 (\lambda (c2: +C).(clear c2 (CHead c1 (Bind b) u))) (clear_gen_flat f x0 (CHead c1 (Bind b) +u) t0 H8) c0 H15) in (let H17 \def (eq_ind_r C x0 (\lambda (c2: C).((drop O O +(CHead c0 k t) c2) \to ((clear c2 (CHead c1 (Bind b) u)) \to (ex3_2 T C +(\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: +T).(\lambda (e0: C).(getl O e (CHead e0 (Bind b) v)))) (\lambda (_: +T).(\lambda (e0: C).(drop h d c1 e0))))))) IHx0 c0 H15) in (let H18 \def +(eq_ind K k (\lambda (k1: K).((drop O O (CHead c0 k1 t) c0) \to ((clear c0 +(CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u +(lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl O e (CHead e0 (Bind b) +v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) H17 (Flat f) +H14) in (let H19 \def (eq_ind K k (\lambda (k1: K).(drop h (S (plus O d)) +(CHead c0 k1 t) e)) H9 (Flat f) H14) in (ex3_2_ind C T (\lambda (e0: +C).(\lambda (v: T).(eq C e (CHead e0 (Flat f) v)))) (\lambda (_: C).(\lambda +(v: T).(eq T t (lift h (r (Flat f) (plus O d)) v)))) (\lambda (e0: +C).(\lambda (_: T).(drop h (r (Flat f) (plus O d)) c0 e0))) (ex3_2 T C +(\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: +T).(\lambda (e0: C).(getl O e (CHead e0 (Bind b) v)))) (\lambda (_: +T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x1: C).(\lambda (x2: +T).(\lambda (H20: (eq C e (CHead x1 (Flat f) x2))).(\lambda (H21: (eq T t +(lift h (r (Flat f) (plus O d)) x2))).(\lambda (H22: (drop h (r (Flat f) +(plus O d)) c0 x1)).(let H23 \def (f_equal T T (\lambda (e0: T).e0) t (lift h +(r (Flat f) (plus O d)) x2) H21) in (let H24 \def (eq_ind C e (\lambda (c2: +C).((drop O O (CHead c0 (Flat f) t) c0) \to ((clear c0 (CHead c1 (Bind b) u)) +\to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) +(\lambda (v: T).(\lambda (e0: C).(getl O c2 (CHead e0 (Bind b) v)))) (\lambda +(_: T).(\lambda (e0: C).(drop h d c1 e0))))))) H18 (CHead x1 (Flat f) x2) +H20) in (eq_ind_r C (CHead x1 (Flat f) x2) (\lambda (c2: C).(ex3_2 T C +(\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: +T).(\lambda (e0: C).(getl O c2 (CHead e0 (Bind b) v)))) (\lambda (_: +T).(\lambda (e0: C).(drop h d c1 e0))))) (let H25 \def (eq_ind T t (\lambda +(t1: T).((drop O O (CHead c0 (Flat f) t1) c0) \to ((clear c0 (CHead c1 (Bind +b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) +(\lambda (v: T).(\lambda (e0: C).(getl O (CHead x1 (Flat f) x2) (CHead e0 +(Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) H24 +(lift h (S d) x2) H23) in (let H26 \def (H c1 u O (getl_intro O c0 (CHead c1 +(Bind b) u) c0 (drop_refl c0) H16) x1 h d H22) in (ex3_2_ind T C (\lambda (v: +T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: +C).(getl O x1 (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop +h d c1 e0))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d +v)))) (\lambda (v: T).(\lambda (e0: C).(getl O (CHead x1 (Flat f) x2) (CHead +e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))) +(\lambda (x3: T).(\lambda (x4: C).(\lambda (H27: (eq T u (lift h d +x3))).(\lambda (H28: (getl O x1 (CHead x4 (Bind b) x3))).(\lambda (H29: (drop +h d c1 x4)).(let H30 \def (eq_ind T u (\lambda (t1: T).((drop O O (CHead c0 +(Flat f) (lift h (S d) x2)) c0) \to ((clear c0 (CHead c1 (Bind b) t1)) \to +(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T t1 (lift h d v)))) (\lambda +(v: T).(\lambda (e0: C).(getl O (CHead x1 (Flat f) x2) (CHead e0 (Bind b) +v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) H25 (lift h d +x3) H27) in (let H31 \def (eq_ind T u (\lambda (t1: T).(clear c0 (CHead c1 +(Bind b) t1))) H16 (lift h d x3) H27) in (eq_ind_r T (lift h d x3) (\lambda +(t1: T).(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T t1 (lift h d v)))) +(\lambda (v: T).(\lambda (e0: C).(getl O (CHead x1 (Flat f) x2) (CHead e0 +(Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))) +(ex3_2_intro T C (\lambda (v: T).(\lambda (_: C).(eq T (lift h d x3) (lift h +d v)))) (\lambda (v: T).(\lambda (e0: C).(getl O (CHead x1 (Flat f) x2) +(CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))) +x3 x4 (refl_equal T (lift h d x3)) (getl_flat x1 (CHead x4 (Bind b) x3) O H28 +f x2) H29) u H27)))))))) H26))) e H20)))))))) (drop_gen_skip_l c0 e t h (plus +O d) (Flat f) H19))))))))) H12)) H11))))) (\lambda (i0: nat).(\lambda (IHi: +(((drop h (S (plus i0 d)) (CHead c0 k t) e) \to ((drop i0 O (CHead c0 k t) +(CHead x0 (Flat f) t0)) \to ((((drop i0 O (CHead c0 k t) x0) \to ((clear x0 +(CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u +(lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i0 e (CHead e0 (Bind +b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) \to (ex3_2 T +C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: +T).(\lambda (e0: C).(getl i0 e (CHead e0 (Bind b) v)))) (\lambda (_: +T).(\lambda (e0: C).(drop h d c1 e0))))))))).(\lambda (H9: (drop h (S (plus +(S i0) d)) (CHead c0 k t) e)).(\lambda (H10: (drop (S i0) O (CHead c0 k t) +(CHead x0 (Flat f) t0))).(\lambda (IHx0: (((drop (S i0) O (CHead c0 k t) x0) +\to ((clear x0 (CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda +(_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl (S i0) +e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 +e0)))))))).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 +k v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r k (plus (S i0) d)) +v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r k (plus (S i0) d)) c0 e0))) +(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda +(v: T).(\lambda (e0: C).(getl (S i0) e (CHead e0 (Bind b) v)))) (\lambda (_: +T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x1: C).(\lambda (x2: +T).(\lambda (H11: (eq C e (CHead x1 k x2))).(\lambda (H12: (eq T t (lift h (r +k (plus (S i0) d)) x2))).(\lambda (H13: (drop h (r k (plus (S i0) d)) c0 +x1)).(let H14 \def (f_equal T T (\lambda (e0: T).e0) t (lift h (r k (plus (S +i0) d)) x2) H12) in (let H15 \def (eq_ind C e (\lambda (c2: C).((drop h (S +(plus i0 d)) (CHead c0 k t) c2) \to ((drop i0 O (CHead c0 k t) (CHead x0 +(Flat f) t0)) \to ((((drop i0 O (CHead c0 k t) x0) \to ((clear x0 (CHead c1 +(Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d +v)))) (\lambda (v: T).(\lambda (e0: C).(getl i0 c2 (CHead e0 (Bind b) v)))) +(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) \to (ex3_2 T C +(\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: +T).(\lambda (e0: C).(getl i0 c2 (CHead e0 (Bind b) v)))) (\lambda (_: +T).(\lambda (e0: C).(drop h d c1 e0)))))))) IHi (CHead x1 k x2) H11) in (let +H16 \def (eq_ind C e (\lambda (c2: C).((drop (S i0) O (CHead c0 k t) x0) \to +((clear x0 (CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: +C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl (S i0) c2 +(CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 +e0))))))) IHx0 (CHead x1 k x2) H11) in (eq_ind_r C (CHead x1 k x2) (\lambda +(c2: C).(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) +(\lambda (v: T).(\lambda (e0: C).(getl (S i0) c2 (CHead e0 (Bind b) v)))) +(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))) (let H17 \def (eq_ind T +t (\lambda (t1: T).((drop (S i0) O (CHead c0 k t1) x0) \to ((clear x0 (CHead +c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift +h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl (S i0) (CHead x1 k x2) +(CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 +e0))))))) H16 (lift h (r k (S (plus i0 d))) x2) H14) in (let H18 \def (eq_ind +T t (\lambda (t1: T).((drop h (S (plus i0 d)) (CHead c0 k t1) (CHead x1 k +x2)) \to ((drop i0 O (CHead c0 k t1) (CHead x0 (Flat f) t0)) \to ((((drop i0 +O (CHead c0 k t1) x0) \to ((clear x0 (CHead c1 (Bind b) u)) \to (ex3_2 T C +(\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: +T).(\lambda (e0: C).(getl i0 (CHead x1 k x2) (CHead e0 (Bind b) v)))) +(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) \to (ex3_2 T C +(\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: +T).(\lambda (e0: C).(getl i0 (CHead x1 k x2) (CHead e0 (Bind b) v)))) +(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))))))) H15 (lift h (r k (S +(plus i0 d))) x2) H14) in (let H19 \def (eq_ind nat (r k (plus (S i0) d)) +(\lambda (n: nat).(drop h n c0 x1)) H13 (plus (r k (S i0)) d) (r_plus k (S +i0) d)) in (let H20 \def (eq_ind nat (r k (S i0)) (\lambda (n: nat).(drop h +(plus n d) c0 x1)) H19 (S (r k i0)) (r_S k i0)) in (let H21 \def (H c1 u (r k +i0) (getl_intro (r k i0) c0 (CHead c1 (Bind b) u) (CHead x0 (Flat f) t0) +(drop_gen_drop k c0 (CHead x0 (Flat f) t0) t i0 H10) (clear_flat x0 (CHead c1 +(Bind b) u) (clear_gen_flat f x0 (CHead c1 (Bind b) u) t0 H8) f t0)) x1 h d +H20) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d +v)))) (\lambda (v: T).(\lambda (e0: C).(getl (r k i0) x1 (CHead e0 (Bind b) +v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))) (ex3_2 T C (\lambda +(v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: +C).(getl (S i0) (CHead x1 k x2) (CHead e0 (Bind b) v)))) (\lambda (_: +T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x3: T).(\lambda (x4: +C).(\lambda (H22: (eq T u (lift h d x3))).(\lambda (H23: (getl (r k i0) x1 +(CHead x4 (Bind b) x3))).(\lambda (H24: (drop h d c1 x4)).(let H25 \def +(eq_ind T u (\lambda (t1: T).((drop (S i0) O (CHead c0 k (lift h (r k (S +(plus i0 d))) x2)) x0) \to ((clear x0 (CHead c1 (Bind b) t1)) \to (ex3_2 T C +(\lambda (v: T).(\lambda (_: C).(eq T t1 (lift h d v)))) (\lambda (v: +T).(\lambda (e0: C).(getl (S i0) (CHead x1 k x2) (CHead e0 (Bind b) v)))) +(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) H17 (lift h d x3) +H22) in (let H26 \def (eq_ind T u (\lambda (t1: T).(clear x0 (CHead c1 (Bind +b) t1))) (clear_gen_flat f x0 (CHead c1 (Bind b) u) t0 H8) (lift h d x3) H22) +in (eq_ind_r T (lift h d x3) (\lambda (t1: T).(ex3_2 T C (\lambda (v: +T).(\lambda (_: C).(eq T t1 (lift h d v)))) (\lambda (v: T).(\lambda (e0: +C).(getl (S i0) (CHead x1 k x2) (CHead e0 (Bind b) v)))) (\lambda (_: +T).(\lambda (e0: C).(drop h d c1 e0))))) (ex3_2_intro T C (\lambda (v: +T).(\lambda (_: C).(eq T (lift h d x3) (lift h d v)))) (\lambda (v: +T).(\lambda (e0: C).(getl (S i0) (CHead x1 k x2) (CHead e0 (Bind b) v)))) +(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))) x3 x4 (refl_equal T (lift +h d x3)) (getl_head k i0 x1 (CHead x4 (Bind b) x3) H23 x2) H24) u H22)))))))) +H21)))))) e H11))))))))) (drop_gen_skip_l c0 e t h (plus (S i0) d) k +H9))))))) i H1 H7 IHx)))) k0 H5 H6))))))) x H3 H4)))) H2)))))))))))))) c)). +(* COMMENTS +Initial nodes: 6137 +END *) + +theorem getl_drop_conf_ge: + \forall (i: nat).(\forall (a: C).(\forall (c: C).((getl i c a) \to (\forall +(e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to ((le (plus d +h) i) \to (getl (minus i h) e a))))))))) +\def + \lambda (i: nat).(\lambda (a: C).(\lambda (c: C).(\lambda (H: (getl i c +a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H0: (drop h +d c e)).(\lambda (H1: (le (plus d h) i)).(let H2 \def (getl_gen_all c a i H) +in (ex2_ind C (\lambda (e0: C).(drop i O c e0)) (\lambda (e0: C).(clear e0 +a)) (getl (minus i h) e a) (\lambda (x: C).(\lambda (H3: (drop i O c +x)).(\lambda (H4: (clear x a)).(getl_intro (minus i h) e a x (drop_conf_ge i +x c H3 e h d H0 H1) H4)))) H2)))))))))). +(* COMMENTS +Initial nodes: 141 +END *) + +theorem getl_conf_ge_drop: + \forall (b: B).(\forall (c1: C).(\forall (e: C).(\forall (u: T).(\forall (i: +nat).((getl i c1 (CHead e (Bind b) u)) \to (\forall (c2: C).((drop (S O) i c1 +c2) \to (drop i O c2 e)))))))) +\def + \lambda (b: B).(\lambda (c1: C).(\lambda (e: C).(\lambda (u: T).(\lambda (i: +nat).(\lambda (H: (getl i c1 (CHead e (Bind b) u))).(\lambda (c2: C).(\lambda +(H0: (drop (S O) i c1 c2)).(let H3 \def (eq_ind nat (minus (S i) (S O)) +(\lambda (n: nat).(drop n O c2 e)) (drop_conf_ge (S i) e c1 (getl_drop b c1 e +u i H) c2 (S O) i H0 (eq_ind_r nat (plus (S O) i) (\lambda (n: nat).(le n (S +i))) (le_n (S i)) (plus i (S O)) (plus_sym i (S O)))) i (minus_Sx_SO i)) in +H3)))))))). +(* COMMENTS +Initial nodes: 151 +END *) + +theorem getl_drop_conf_rev: + \forall (j: nat).(\forall (e1: C).(\forall (e2: C).((drop j O e1 e2) \to +(\forall (b: B).(\forall (c2: C).(\forall (v2: T).(\forall (i: nat).((getl i +c2 (CHead e2 (Bind b) v2)) \to (ex2 C (\lambda (c1: C).(drop j O c1 c2)) +(\lambda (c1: C).(drop (S i) j c1 e1))))))))))) +\def + \lambda (j: nat).(\lambda (e1: C).(\lambda (e2: C).(\lambda (H: (drop j O e1 +e2)).(\lambda (b: B).(\lambda (c2: C).(\lambda (v2: T).(\lambda (i: +nat).(\lambda (H0: (getl i c2 (CHead e2 (Bind b) v2))).(drop_conf_rev j e1 e2 +H c2 (S i) (getl_drop b c2 e2 v2 i H0)))))))))). +(* COMMENTS +Initial nodes: 69 +END *) + +theorem drop_getl_trans_lt: + \forall (i: nat).(\forall (d: nat).((lt i d) \to (\forall (c1: C).(\forall +(c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (b: B).(\forall (e2: +C).(\forall (v: T).((getl i c2 (CHead e2 (Bind b) v)) \to (ex2 C (\lambda +(e1: C).(getl i c1 (CHead e1 (Bind b) (lift h (minus d (S i)) v)))) (\lambda +(e1: C).(drop h (minus d (S i)) e1 e2))))))))))))) +\def + \lambda (i: nat).(\lambda (d: nat).(\lambda (H: (lt i d)).(\lambda (c1: +C).(\lambda (c2: C).(\lambda (h: nat).(\lambda (H0: (drop h d c1 +c2)).(\lambda (b: B).(\lambda (e2: C).(\lambda (v: T).(\lambda (H1: (getl i +c2 (CHead e2 (Bind b) v))).(let H2 \def (getl_gen_all c2 (CHead e2 (Bind b) +v) i H1) in (ex2_ind C (\lambda (e: C).(drop i O c2 e)) (\lambda (e: +C).(clear e (CHead e2 (Bind b) v))) (ex2 C (\lambda (e1: C).(getl i c1 (CHead +e1 (Bind b) (lift h (minus d (S i)) v)))) (\lambda (e1: C).(drop h (minus d +(S i)) e1 e2))) (\lambda (x: C).(\lambda (H3: (drop i O c2 x)).(\lambda (H4: +(clear x (CHead e2 (Bind b) v))).(ex2_ind C (\lambda (e1: C).(drop i O c1 +e1)) (\lambda (e1: C).(drop h (minus d i) e1 x)) (ex2 C (\lambda (e1: +C).(getl i c1 (CHead e1 (Bind b) (lift h (minus d (S i)) v)))) (\lambda (e1: +C).(drop h (minus d (S i)) e1 e2))) (\lambda (x0: C).(\lambda (H5: (drop i O +c1 x0)).(\lambda (H6: (drop h (minus d i) x0 x)).(let H7 \def (eq_ind nat +(minus d i) (\lambda (n: nat).(drop h n x0 x)) H6 (S (minus d (S i))) +(minus_x_Sy d i H)) in (let H8 \def (drop_clear_S x x0 h (minus d (S i)) H7 b +e2 v H4) in (ex2_ind C (\lambda (c3: C).(clear x0 (CHead c3 (Bind b) (lift h +(minus d (S i)) v)))) (\lambda (c3: C).(drop h (minus d (S i)) c3 e2)) (ex2 C +(\lambda (e1: C).(getl i c1 (CHead e1 (Bind b) (lift h (minus d (S i)) v)))) +(\lambda (e1: C).(drop h (minus d (S i)) e1 e2))) (\lambda (x1: C).(\lambda +(H9: (clear x0 (CHead x1 (Bind b) (lift h (minus d (S i)) v)))).(\lambda +(H10: (drop h (minus d (S i)) x1 e2)).(ex_intro2 C (\lambda (e1: C).(getl i +c1 (CHead e1 (Bind b) (lift h (minus d (S i)) v)))) (\lambda (e1: C).(drop h +(minus d (S i)) e1 e2)) x1 (getl_intro i c1 (CHead x1 (Bind b) (lift h (minus +d (S i)) v)) x0 H5 H9) H10)))) H8)))))) (drop_trans_le i d (le_S_n i d (le_S +(S i) d H)) c1 c2 h H0 x H3))))) H2)))))))))))). +(* COMMENTS +Initial nodes: 627 +END *) + +theorem drop_getl_trans_le: + \forall (i: nat).(\forall (d: nat).((le i d) \to (\forall (c1: C).(\forall +(c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((getl i c2 +e2) \to (ex3_2 C C (\lambda (e0: C).(\lambda (_: C).(drop i O c1 e0))) +(\lambda (e0: C).(\lambda (e1: C).(drop h (minus d i) e0 e1))) (\lambda (_: +C).(\lambda (e1: C).(clear e1 e2)))))))))))) +\def + \lambda (i: nat).(\lambda (d: nat).(\lambda (H: (le i d)).(\lambda (c1: +C).(\lambda (c2: C).(\lambda (h: nat).(\lambda (H0: (drop h d c1 +c2)).(\lambda (e2: C).(\lambda (H1: (getl i c2 e2)).(let H2 \def +(getl_gen_all c2 e2 i H1) in (ex2_ind C (\lambda (e: C).(drop i O c2 e)) +(\lambda (e: C).(clear e e2)) (ex3_2 C C (\lambda (e0: C).(\lambda (_: +C).(drop i O c1 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d i) +e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear e1 e2)))) (\lambda (x: +C).(\lambda (H3: (drop i O c2 x)).(\lambda (H4: (clear x e2)).(let H5 \def +(drop_trans_le i d H c1 c2 h H0 x H3) in (ex2_ind C (\lambda (e1: C).(drop i +O c1 e1)) (\lambda (e1: C).(drop h (minus d i) e1 x)) (ex3_2 C C (\lambda +(e0: C).(\lambda (_: C).(drop i O c1 e0))) (\lambda (e0: C).(\lambda (e1: +C).(drop h (minus d i) e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear e1 +e2)))) (\lambda (x0: C).(\lambda (H6: (drop i O c1 x0)).(\lambda (H7: (drop h +(minus d i) x0 x)).(ex3_2_intro C C (\lambda (e0: C).(\lambda (_: C).(drop i +O c1 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d i) e0 e1))) +(\lambda (_: C).(\lambda (e1: C).(clear e1 e2))) x0 x H6 H7 H4)))) H5))))) +H2)))))))))). +(* COMMENTS +Initial nodes: 323 +END *) + +theorem drop_getl_trans_ge: + \forall (i: nat).(\forall (c1: C).(\forall (c2: C).(\forall (d: +nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((getl i c2 e2) +\to ((le d i) \to (getl (plus i h) c1 e2))))))))) +\def + \lambda (i: nat).(\lambda (c1: C).(\lambda (c2: C).(\lambda (d: +nat).(\lambda (h: nat).(\lambda (H: (drop h d c1 c2)).(\lambda (e2: +C).(\lambda (H0: (getl i c2 e2)).(\lambda (H1: (le d i)).(let H2 \def +(getl_gen_all c2 e2 i H0) in (ex2_ind C (\lambda (e: C).(drop i O c2 e)) +(\lambda (e: C).(clear e e2)) (getl (plus i h) c1 e2) (\lambda (x: +C).(\lambda (H3: (drop i O c2 x)).(\lambda (H4: (clear x e2)).(getl_intro +(plus i h) c1 e2 x (drop_trans_ge i c1 c2 d h H x H3 H1) H4)))) H2)))))))))). +(* COMMENTS +Initial nodes: 137 +END *) + +theorem getl_drop_trans: + \forall (c1: C).(\forall (c2: C).(\forall (h: nat).((getl h c1 c2) \to +(\forall (e2: C).(\forall (i: nat).((drop (S i) O c2 e2) \to (drop (S (plus i +h)) O c1 e2))))))) +\def + \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (h: +nat).((getl h c c2) \to (\forall (e2: C).(\forall (i: nat).((drop (S i) O c2 +e2) \to (drop (S (plus i h)) O c e2)))))))) (\lambda (n: nat).(\lambda (c2: +C).(\lambda (h: nat).(\lambda (H: (getl h (CSort n) c2)).(\lambda (e2: +C).(\lambda (i: nat).(\lambda (_: (drop (S i) O c2 e2)).(getl_gen_sort n h c2 +H (drop (S (plus i h)) O (CSort n) e2))))))))) (\lambda (c2: C).(\lambda +(IHc: ((\forall (c3: C).(\forall (h: nat).((getl h c2 c3) \to (\forall (e2: +C).(\forall (i: nat).((drop (S i) O c3 e2) \to (drop (S (plus i h)) O c2 +e2))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t: T).(\forall +(c3: C).(\forall (h: nat).((getl h (CHead c2 k0 t) c3) \to (\forall (e2: +C).(\forall (i: nat).((drop (S i) O c3 e2) \to (drop (S (plus i h)) O (CHead +c2 k0 t) e2))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (c3: +C).(\lambda (h: nat).(nat_ind (\lambda (n: nat).((getl n (CHead c2 (Bind b) +t) c3) \to (\forall (e2: C).(\forall (i: nat).((drop (S i) O c3 e2) \to (drop +(S (plus i n)) O (CHead c2 (Bind b) t) e2)))))) (\lambda (H: (getl O (CHead +c2 (Bind b) t) c3)).(\lambda (e2: C).(\lambda (i: nat).(\lambda (H0: (drop (S +i) O c3 e2)).(let H1 \def (eq_ind C c3 (\lambda (c: C).(drop (S i) O c e2)) +H0 (CHead c2 (Bind b) t) (clear_gen_bind b c2 c3 t (getl_gen_O (CHead c2 +(Bind b) t) c3 H))) in (eq_ind nat i (\lambda (n: nat).(drop (S n) O (CHead +c2 (Bind b) t) e2)) (drop_drop (Bind b) i c2 e2 (drop_gen_drop (Bind b) c2 e2 +t i H1) t) (plus i O) (plus_n_O i))))))) (\lambda (n: nat).(\lambda (_: +(((getl n (CHead c2 (Bind b) t) c3) \to (\forall (e2: C).(\forall (i: +nat).((drop (S i) O c3 e2) \to (drop (S (plus i n)) O (CHead c2 (Bind b) t) +e2))))))).(\lambda (H0: (getl (S n) (CHead c2 (Bind b) t) c3)).(\lambda (e2: +C).(\lambda (i: nat).(\lambda (H1: (drop (S i) O c3 e2)).(eq_ind nat (plus (S +i) n) (\lambda (n0: nat).(drop (S n0) O (CHead c2 (Bind b) t) e2)) (drop_drop +(Bind b) (plus (S i) n) c2 e2 (IHc c3 n (getl_gen_S (Bind b) c2 c3 t n H0) e2 +i H1) t) (plus i (S n)) (plus_Snm_nSm i n)))))))) h))))) (\lambda (f: +F).(\lambda (t: T).(\lambda (c3: C).(\lambda (h: nat).(nat_ind (\lambda (n: +nat).((getl n (CHead c2 (Flat f) t) c3) \to (\forall (e2: C).(\forall (i: +nat).((drop (S i) O c3 e2) \to (drop (S (plus i n)) O (CHead c2 (Flat f) t) +e2)))))) (\lambda (H: (getl O (CHead c2 (Flat f) t) c3)).(\lambda (e2: +C).(\lambda (i: nat).(\lambda (H0: (drop (S i) O c3 e2)).(drop_drop (Flat f) +(plus i O) c2 e2 (IHc c3 O (getl_intro O c2 c3 c2 (drop_refl c2) +(clear_gen_flat f c2 c3 t (getl_gen_O (CHead c2 (Flat f) t) c3 H))) e2 i H0) +t))))) (\lambda (n: nat).(\lambda (_: (((getl n (CHead c2 (Flat f) t) c3) \to +(\forall (e2: C).(\forall (i: nat).((drop (S i) O c3 e2) \to (drop (S (plus i +n)) O (CHead c2 (Flat f) t) e2))))))).(\lambda (H0: (getl (S n) (CHead c2 +(Flat f) t) c3)).(\lambda (e2: C).(\lambda (i: nat).(\lambda (H1: (drop (S i) +O c3 e2)).(drop_drop (Flat f) (plus i (S n)) c2 e2 (IHc c3 (S n) (getl_gen_S +(Flat f) c2 c3 t n H0) e2 i H1) t))))))) h))))) k)))) c1). +(* COMMENTS +Initial nodes: 953 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/getl/flt.ma b/matita/matita/contribs/lambdadelta/basic_1/getl/flt.ma new file mode 100644 index 000000000..3d6c90f73 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/getl/flt.ma @@ -0,0 +1,67 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/getl/fwd.ma". + +include "Basic-1/clear/props.ma". + +include "Basic-1/flt/props.ma". + +theorem getl_flt: + \forall (b: B).(\forall (c: C).(\forall (e: C).(\forall (u: T).(\forall (i: +nat).((getl i c (CHead e (Bind b) u)) \to (flt e u c (TLRef i))))))) +\def + \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e: +C).(\forall (u: T).(\forall (i: nat).((getl i c0 (CHead e (Bind b) u)) \to +(flt e u c0 (TLRef i))))))) (\lambda (n: nat).(\lambda (e: C).(\lambda (u: +T).(\lambda (i: nat).(\lambda (H: (getl i (CSort n) (CHead e (Bind b) +u))).(getl_gen_sort n i (CHead e (Bind b) u) H (flt e u (CSort n) (TLRef +i)))))))) (\lambda (c0: C).(\lambda (H: ((\forall (e: C).(\forall (u: +T).(\forall (i: nat).((getl i c0 (CHead e (Bind b) u)) \to (flt e u c0 (TLRef +i)))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e: C).(\lambda (u: +T).(\lambda (i: nat).(nat_ind (\lambda (n: nat).((getl n (CHead c0 k t) +(CHead e (Bind b) u)) \to (flt e u (CHead c0 k t) (TLRef n)))) (\lambda (H0: +(getl O (CHead c0 k t) (CHead e (Bind b) u))).(K_ind (\lambda (k0: K).((clear +(CHead c0 k0 t) (CHead e (Bind b) u)) \to (flt e u (CHead c0 k0 t) (TLRef +O)))) (\lambda (b0: B).(\lambda (H1: (clear (CHead c0 (Bind b0) t) (CHead e +(Bind b) u))).(let H2 \def (f_equal C C (\lambda (e0: C).(match e0 in C +return (\lambda (_: C).C) with [(CSort _) \Rightarrow e | (CHead c1 _ _) +\Rightarrow c1])) (CHead e (Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind +b0 c0 (CHead e (Bind b) u) t H1)) in ((let H3 \def (f_equal C B (\lambda (e0: +C).(match e0 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow b | +(CHead _ k0 _) \Rightarrow (match k0 in K return (\lambda (_: K).B) with +[(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow b])])) (CHead e (Bind b) u) +(CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e (Bind b) u) t H1)) in +((let H4 \def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda +(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) +(CHead e (Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e +(Bind b) u) t H1)) in (\lambda (H5: (eq B b b0)).(\lambda (H6: (eq C e +c0)).(eq_ind_r T t (\lambda (t0: T).(flt e t0 (CHead c0 (Bind b0) t) (TLRef +O))) (eq_ind_r C c0 (\lambda (c1: C).(flt c1 t (CHead c0 (Bind b0) t) (TLRef +O))) (eq_ind B b (\lambda (b1: B).(flt c0 t (CHead c0 (Bind b1) t) (TLRef +O))) (flt_arith0 (Bind b) c0 t O) b0 H5) e H6) u H4)))) H3)) H2)))) (\lambda +(f: F).(\lambda (H1: (clear (CHead c0 (Flat f) t) (CHead e (Bind b) +u))).(flt_arith1 (Bind b) e c0 u (clear_cle c0 (CHead e (Bind b) u) +(clear_gen_flat f c0 (CHead e (Bind b) u) t H1)) (Flat f) t O))) k +(getl_gen_O (CHead c0 k t) (CHead e (Bind b) u) H0))) (\lambda (n: +nat).(\lambda (_: (((getl n (CHead c0 k t) (CHead e (Bind b) u)) \to (flt e u +(CHead c0 k t) (TLRef n))))).(\lambda (H1: (getl (S n) (CHead c0 k t) (CHead +e (Bind b) u))).(let H_y \def (H e u (r k n) (getl_gen_S k c0 (CHead e (Bind +b) u) t n H1)) in (flt_arith2 e c0 u (r k n) H_y k t (S n)))))) i)))))))) c)). +(* COMMENTS +Initial nodes: 815 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/getl/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/getl/fwd.ma new file mode 100644 index 000000000..b378f61fc --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/getl/fwd.ma @@ -0,0 +1,148 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/getl/defs.ma". + +include "Basic-1/drop/fwd.ma". + +include "Basic-1/clear/fwd.ma". + +theorem getl_gen_all: + \forall (c1: C).(\forall (c2: C).(\forall (i: nat).((getl i c1 c2) \to (ex2 +C (\lambda (e: C).(drop i O c1 e)) (\lambda (e: C).(clear e c2)))))) +\def + \lambda (c1: C).(\lambda (c2: C).(\lambda (i: nat).(\lambda (H: (getl i c1 +c2)).(getl_ind i c1 c2 (ex2 C (\lambda (e: C).(drop i O c1 e)) (\lambda (e: +C).(clear e c2))) (\lambda (e: C).(\lambda (H0: (drop i O c1 e)).(\lambda +(H1: (clear e c2)).(ex_intro2 C (\lambda (e0: C).(drop i O c1 e0)) (\lambda +(e0: C).(clear e0 c2)) e H0 H1)))) H)))). +(* COMMENTS +Initial nodes: 95 +END *) + +theorem getl_gen_sort: + \forall (n: nat).(\forall (h: nat).(\forall (x: C).((getl h (CSort n) x) \to +(\forall (P: Prop).P)))) +\def + \lambda (n: nat).(\lambda (h: nat).(\lambda (x: C).(\lambda (H: (getl h +(CSort n) x)).(\lambda (P: Prop).(let H0 \def (getl_gen_all (CSort n) x h H) +in (ex2_ind C (\lambda (e: C).(drop h O (CSort n) e)) (\lambda (e: C).(clear +e x)) P (\lambda (x0: C).(\lambda (H1: (drop h O (CSort n) x0)).(\lambda (H2: +(clear x0 x)).(and3_ind (eq C x0 (CSort n)) (eq nat h O) (eq nat O O) P +(\lambda (H3: (eq C x0 (CSort n))).(\lambda (_: (eq nat h O)).(\lambda (_: +(eq nat O O)).(let H6 \def (eq_ind C x0 (\lambda (c: C).(clear c x)) H2 +(CSort n) H3) in (clear_gen_sort x n H6 P))))) (drop_gen_sort n h O x0 +H1))))) H0)))))). +(* COMMENTS +Initial nodes: 179 +END *) + +theorem getl_gen_O: + \forall (e: C).(\forall (x: C).((getl O e x) \to (clear e x))) +\def + \lambda (e: C).(\lambda (x: C).(\lambda (H: (getl O e x)).(let H0 \def +(getl_gen_all e x O H) in (ex2_ind C (\lambda (e0: C).(drop O O e e0)) +(\lambda (e0: C).(clear e0 x)) (clear e x) (\lambda (x0: C).(\lambda (H1: +(drop O O e x0)).(\lambda (H2: (clear x0 x)).(let H3 \def (eq_ind_r C x0 +(\lambda (c: C).(clear c x)) H2 e (drop_gen_refl e x0 H1)) in H3)))) H0)))). +(* COMMENTS +Initial nodes: 99 +END *) + +theorem getl_gen_S: + \forall (k: K).(\forall (c: C).(\forall (x: C).(\forall (u: T).(\forall (h: +nat).((getl (S h) (CHead c k u) x) \to (getl (r k h) c x)))))) +\def + \lambda (k: K).(\lambda (c: C).(\lambda (x: C).(\lambda (u: T).(\lambda (h: +nat).(\lambda (H: (getl (S h) (CHead c k u) x)).(let H0 \def (getl_gen_all +(CHead c k u) x (S h) H) in (ex2_ind C (\lambda (e: C).(drop (S h) O (CHead c +k u) e)) (\lambda (e: C).(clear e x)) (getl (r k h) c x) (\lambda (x0: +C).(\lambda (H1: (drop (S h) O (CHead c k u) x0)).(\lambda (H2: (clear x0 +x)).(getl_intro (r k h) c x x0 (drop_gen_drop k c x0 u h H1) H2)))) H0))))))). +(* COMMENTS +Initial nodes: 145 +END *) + +theorem getl_gen_2: + \forall (c1: C).(\forall (c2: C).(\forall (i: nat).((getl i c1 c2) \to (ex_3 +B C T (\lambda (b: B).(\lambda (c: C).(\lambda (v: T).(eq C c2 (CHead c (Bind +b) v))))))))) +\def + \lambda (c1: C).(\lambda (c2: C).(\lambda (i: nat).(\lambda (H: (getl i c1 +c2)).(let H0 \def (getl_gen_all c1 c2 i H) in (ex2_ind C (\lambda (e: +C).(drop i O c1 e)) (\lambda (e: C).(clear e c2)) (ex_3 B C T (\lambda (b: +B).(\lambda (c: C).(\lambda (v: T).(eq C c2 (CHead c (Bind b) v)))))) +(\lambda (x: C).(\lambda (_: (drop i O c1 x)).(\lambda (H2: (clear x +c2)).(let H3 \def (clear_gen_all x c2 H2) in (ex_3_ind B C T (\lambda (b: +B).(\lambda (e: C).(\lambda (u: T).(eq C c2 (CHead e (Bind b) u))))) (ex_3 B +C T (\lambda (b: B).(\lambda (c: C).(\lambda (v: T).(eq C c2 (CHead c (Bind +b) v)))))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H4: +(eq C c2 (CHead x1 (Bind x0) x2))).(let H5 \def (eq_ind C c2 (\lambda (c: +C).(clear x c)) H2 (CHead x1 (Bind x0) x2) H4) in (eq_ind_r C (CHead x1 (Bind +x0) x2) (\lambda (c: C).(ex_3 B C T (\lambda (b: B).(\lambda (c0: C).(\lambda +(v: T).(eq C c (CHead c0 (Bind b) v))))))) (ex_3_intro B C T (\lambda (b: +B).(\lambda (c: C).(\lambda (v: T).(eq C (CHead x1 (Bind x0) x2) (CHead c +(Bind b) v))))) x0 x1 x2 (refl_equal C (CHead x1 (Bind x0) x2))) c2 H4)))))) +H3))))) H0))))). +(* COMMENTS +Initial nodes: 325 +END *) + +theorem getl_gen_flat: + \forall (f: F).(\forall (e: C).(\forall (d: C).(\forall (v: T).(\forall (i: +nat).((getl i (CHead e (Flat f) v) d) \to (getl i e d)))))) +\def + \lambda (f: F).(\lambda (e: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: +nat).(nat_ind (\lambda (n: nat).((getl n (CHead e (Flat f) v) d) \to (getl n +e d))) (\lambda (H: (getl O (CHead e (Flat f) v) d)).(getl_intro O e d e +(drop_refl e) (clear_gen_flat f e d v (getl_gen_O (CHead e (Flat f) v) d +H)))) (\lambda (n: nat).(\lambda (_: (((getl n (CHead e (Flat f) v) d) \to +(getl n e d)))).(\lambda (H0: (getl (S n) (CHead e (Flat f) v) +d)).(getl_gen_S (Flat f) e d v n H0)))) i))))). +(* COMMENTS +Initial nodes: 155 +END *) + +theorem getl_gen_bind: + \forall (b: B).(\forall (e: C).(\forall (d: C).(\forall (v: T).(\forall (i: +nat).((getl i (CHead e (Bind b) v) d) \to (or (land (eq nat i O) (eq C d +(CHead e (Bind b) v))) (ex2 nat (\lambda (j: nat).(eq nat i (S j))) (\lambda +(j: nat).(getl j e d))))))))) +\def + \lambda (b: B).(\lambda (e: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: +nat).(nat_ind (\lambda (n: nat).((getl n (CHead e (Bind b) v) d) \to (or +(land (eq nat n O) (eq C d (CHead e (Bind b) v))) (ex2 nat (\lambda (j: +nat).(eq nat n (S j))) (\lambda (j: nat).(getl j e d)))))) (\lambda (H: (getl +O (CHead e (Bind b) v) d)).(eq_ind_r C (CHead e (Bind b) v) (\lambda (c: +C).(or (land (eq nat O O) (eq C c (CHead e (Bind b) v))) (ex2 nat (\lambda +(j: nat).(eq nat O (S j))) (\lambda (j: nat).(getl j e c))))) (or_introl +(land (eq nat O O) (eq C (CHead e (Bind b) v) (CHead e (Bind b) v))) (ex2 nat +(\lambda (j: nat).(eq nat O (S j))) (\lambda (j: nat).(getl j e (CHead e +(Bind b) v)))) (conj (eq nat O O) (eq C (CHead e (Bind b) v) (CHead e (Bind +b) v)) (refl_equal nat O) (refl_equal C (CHead e (Bind b) v)))) d +(clear_gen_bind b e d v (getl_gen_O (CHead e (Bind b) v) d H)))) (\lambda (n: +nat).(\lambda (_: (((getl n (CHead e (Bind b) v) d) \to (or (land (eq nat n +O) (eq C d (CHead e (Bind b) v))) (ex2 nat (\lambda (j: nat).(eq nat n (S +j))) (\lambda (j: nat).(getl j e d))))))).(\lambda (H0: (getl (S n) (CHead e +(Bind b) v) d)).(or_intror (land (eq nat (S n) O) (eq C d (CHead e (Bind b) +v))) (ex2 nat (\lambda (j: nat).(eq nat (S n) (S j))) (\lambda (j: nat).(getl +j e d))) (ex_intro2 nat (\lambda (j: nat).(eq nat (S n) (S j))) (\lambda (j: +nat).(getl j e d)) n (refl_equal nat (S n)) (getl_gen_S (Bind b) e d v n +H0)))))) i))))). +(* COMMENTS +Initial nodes: 525 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/getl/getl.ma b/matita/matita/contribs/lambdadelta/basic_1/getl/getl.ma new file mode 100644 index 000000000..71327becf --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/getl/getl.ma @@ -0,0 +1,57 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/getl/drop.ma". + +include "Basic-1/getl/clear.ma". + +theorem getl_conf_le: + \forall (i: nat).(\forall (a: C).(\forall (c: C).((getl i c a) \to (\forall +(e: C).(\forall (h: nat).((getl h c e) \to ((le h i) \to (getl (minus i h) e +a)))))))) +\def + \lambda (i: nat).(\lambda (a: C).(\lambda (c: C).(\lambda (H: (getl i c +a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (H0: (getl h c e)).(\lambda +(H1: (le h i)).(let H2 \def (getl_gen_all c e h H0) in (ex2_ind C (\lambda +(e0: C).(drop h O c e0)) (\lambda (e0: C).(clear e0 e)) (getl (minus i h) e +a) (\lambda (x: C).(\lambda (H3: (drop h O c x)).(\lambda (H4: (clear x +e)).(getl_clear_conf (minus i h) x a (getl_drop_conf_ge i a c H x h O H3 H1) +e H4)))) H2))))))))). +(* COMMENTS +Initial nodes: 133 +END *) + +theorem getl_trans: + \forall (i: nat).(\forall (c1: C).(\forall (c2: C).(\forall (h: nat).((getl +h c1 c2) \to (\forall (e2: C).((getl i c2 e2) \to (getl (plus i h) c1 +e2))))))) +\def + \lambda (i: nat).(\lambda (c1: C).(\lambda (c2: C).(\lambda (h: +nat).(\lambda (H: (getl h c1 c2)).(\lambda (e2: C).(\lambda (H0: (getl i c2 +e2)).(let H1 \def (getl_gen_all c2 e2 i H0) in (ex2_ind C (\lambda (e: +C).(drop i O c2 e)) (\lambda (e: C).(clear e e2)) (getl (plus i h) c1 e2) +(\lambda (x: C).(\lambda (H2: (drop i O c2 x)).(\lambda (H3: (clear x +e2)).(nat_ind (\lambda (n: nat).((drop n O c2 x) \to (getl (plus n h) c1 +e2))) (\lambda (H4: (drop O O c2 x)).(let H5 \def (eq_ind_r C x (\lambda (c: +C).(clear c e2)) H3 c2 (drop_gen_refl c2 x H4)) in (getl_clear_trans (plus O +h) c1 c2 H e2 H5))) (\lambda (i0: nat).(\lambda (_: (((drop i0 O c2 x) \to +(getl (plus i0 h) c1 e2)))).(\lambda (H4: (drop (S i0) O c2 x)).(let H_y \def +(getl_drop_trans c1 c2 h H x i0 H4) in (getl_intro (plus (S i0) h) c1 e2 x +H_y H3))))) i H2)))) H1)))))))). +(* COMMENTS +Initial nodes: 247 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/getl/props.ma b/matita/matita/contribs/lambdadelta/basic_1/getl/props.ma new file mode 100644 index 000000000..9399ad4ab --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/getl/props.ma @@ -0,0 +1,104 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/getl/fwd.ma". + +include "Basic-1/drop/props.ma". + +include "Basic-1/clear/props.ma". + +theorem getl_refl: + \forall (b: B).(\forall (c: C).(\forall (u: T).(getl O (CHead c (Bind b) u) +(CHead c (Bind b) u)))) +\def + \lambda (b: B).(\lambda (c: C).(\lambda (u: T).(getl_intro O (CHead c (Bind +b) u) (CHead c (Bind b) u) (CHead c (Bind b) u) (drop_refl (CHead c (Bind b) +u)) (clear_bind b c u)))). +(* COMMENTS +Initial nodes: 59 +END *) + +theorem getl_head: + \forall (k: K).(\forall (h: nat).(\forall (c: C).(\forall (e: C).((getl (r k +h) c e) \to (\forall (u: T).(getl (S h) (CHead c k u) e)))))) +\def + \lambda (k: K).(\lambda (h: nat).(\lambda (c: C).(\lambda (e: C).(\lambda +(H: (getl (r k h) c e)).(\lambda (u: T).(let H0 \def (getl_gen_all c e (r k +h) H) in (ex2_ind C (\lambda (e0: C).(drop (r k h) O c e0)) (\lambda (e0: +C).(clear e0 e)) (getl (S h) (CHead c k u) e) (\lambda (x: C).(\lambda (H1: +(drop (r k h) O c x)).(\lambda (H2: (clear x e)).(getl_intro (S h) (CHead c k +u) e x (drop_drop k h c x H1 u) H2)))) H0))))))). +(* COMMENTS +Initial nodes: 137 +END *) + +theorem getl_flat: + \forall (c: C).(\forall (e: C).(\forall (h: nat).((getl h c e) \to (\forall +(f: F).(\forall (u: T).(getl h (CHead c (Flat f) u) e)))))) +\def + \lambda (c: C).(\lambda (e: C).(\lambda (h: nat).(\lambda (H: (getl h c +e)).(\lambda (f: F).(\lambda (u: T).(let H0 \def (getl_gen_all c e h H) in +(ex2_ind C (\lambda (e0: C).(drop h O c e0)) (\lambda (e0: C).(clear e0 e)) +(getl h (CHead c (Flat f) u) e) (\lambda (x: C).(\lambda (H1: (drop h O c +x)).(\lambda (H2: (clear x e)).(nat_ind (\lambda (n: nat).((drop n O c x) \to +(getl n (CHead c (Flat f) u) e))) (\lambda (H3: (drop O O c x)).(let H4 \def +(eq_ind_r C x (\lambda (c0: C).(clear c0 e)) H2 c (drop_gen_refl c x H3)) in +(getl_intro O (CHead c (Flat f) u) e (CHead c (Flat f) u) (drop_refl (CHead c +(Flat f) u)) (clear_flat c e H4 f u)))) (\lambda (h0: nat).(\lambda (_: +(((drop h0 O c x) \to (getl h0 (CHead c (Flat f) u) e)))).(\lambda (H3: (drop +(S h0) O c x)).(getl_intro (S h0) (CHead c (Flat f) u) e x (drop_drop (Flat +f) h0 c x H3 u) H2)))) h H1)))) H0))))))). +(* COMMENTS +Initial nodes: 285 +END *) + +theorem getl_ctail: + \forall (b: B).(\forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: +nat).((getl i c (CHead d (Bind b) u)) \to (\forall (k: K).(\forall (v: +T).(getl i (CTail k v c) (CHead (CTail k v d) (Bind b) u))))))))) +\def + \lambda (b: B).(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: +nat).(\lambda (H: (getl i c (CHead d (Bind b) u))).(\lambda (k: K).(\lambda +(v: T).(let H0 \def (getl_gen_all c (CHead d (Bind b) u) i H) in (ex2_ind C +(\lambda (e: C).(drop i O c e)) (\lambda (e: C).(clear e (CHead d (Bind b) +u))) (getl i (CTail k v c) (CHead (CTail k v d) (Bind b) u)) (\lambda (x: +C).(\lambda (H1: (drop i O c x)).(\lambda (H2: (clear x (CHead d (Bind b) +u))).(getl_intro i (CTail k v c) (CHead (CTail k v d) (Bind b) u) (CTail k v +x) (drop_ctail c x O i H1 k v) (clear_ctail b x d u H2 k v))))) H0))))))))). +(* COMMENTS +Initial nodes: 203 +END *) + +theorem getl_mono: + \forall (c: C).(\forall (x1: C).(\forall (h: nat).((getl h c x1) \to +(\forall (x2: C).((getl h c x2) \to (eq C x1 x2)))))) +\def + \lambda (c: C).(\lambda (x1: C).(\lambda (h: nat).(\lambda (H: (getl h c +x1)).(\lambda (x2: C).(\lambda (H0: (getl h c x2)).(let H1 \def (getl_gen_all +c x2 h H0) in (ex2_ind C (\lambda (e: C).(drop h O c e)) (\lambda (e: +C).(clear e x2)) (eq C x1 x2) (\lambda (x: C).(\lambda (H2: (drop h O c +x)).(\lambda (H3: (clear x x2)).(let H4 \def (getl_gen_all c x1 h H) in +(ex2_ind C (\lambda (e: C).(drop h O c e)) (\lambda (e: C).(clear e x1)) (eq +C x1 x2) (\lambda (x0: C).(\lambda (H5: (drop h O c x0)).(\lambda (H6: (clear +x0 x1)).(let H7 \def (eq_ind C x (\lambda (c0: C).(drop h O c c0)) H2 x0 +(drop_mono c x O h H2 x0 H5)) in (let H8 \def (eq_ind_r C x0 (\lambda (c0: +C).(drop h O c c0)) H7 x (drop_mono c x O h H2 x0 H5)) in (let H9 \def +(eq_ind_r C x0 (\lambda (c0: C).(clear c0 x1)) H6 x (drop_mono c x O h H2 x0 +H5)) in (clear_mono x x1 H9 x2 H3))))))) H4))))) H1))))))). +(* COMMENTS +Initial nodes: 269 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/iso/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/iso/defs.ma new file mode 100644 index 000000000..202b8a300 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/iso/defs.ma @@ -0,0 +1,24 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/T/defs.ma". + +inductive iso: T \to (T \to Prop) \def +| iso_sort: \forall (n1: nat).(\forall (n2: nat).(iso (TSort n1) (TSort n2))) +| iso_lref: \forall (i1: nat).(\forall (i2: nat).(iso (TLRef i1) (TLRef i2))) +| iso_head: \forall (v1: T).(\forall (v2: T).(\forall (t1: T).(\forall (t2: +T).(\forall (k: K).(iso (THead k v1 t1) (THead k v2 t2)))))). + diff --git a/matita/matita/contribs/lambdadelta/basic_1/iso/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/iso/fwd.ma new file mode 100644 index 000000000..761e982f7 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/iso/fwd.ma @@ -0,0 +1,191 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/iso/defs.ma". + +include "Basic-1/tlist/defs.ma". + +theorem iso_gen_sort: + \forall (u2: T).(\forall (n1: nat).((iso (TSort n1) u2) \to (ex nat (\lambda +(n2: nat).(eq T u2 (TSort n2)))))) +\def + \lambda (u2: T).(\lambda (n1: nat).(\lambda (H: (iso (TSort n1) +u2)).(insert_eq T (TSort n1) (\lambda (t: T).(iso t u2)) (\lambda (_: T).(ex +nat (\lambda (n2: nat).(eq T u2 (TSort n2))))) (\lambda (y: T).(\lambda (H0: +(iso y u2)).(iso_ind (\lambda (t: T).(\lambda (t0: T).((eq T t (TSort n1)) +\to (ex nat (\lambda (n2: nat).(eq T t0 (TSort n2))))))) (\lambda (n0: +nat).(\lambda (n2: nat).(\lambda (H1: (eq T (TSort n0) (TSort n1))).(let H2 +\def (f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) +with [(TSort n) \Rightarrow n | (TLRef _) \Rightarrow n0 | (THead _ _ _) +\Rightarrow n0])) (TSort n0) (TSort n1) H1) in (ex_intro nat (\lambda (n3: +nat).(eq T (TSort n2) (TSort n3))) n2 (refl_equal T (TSort n2))))))) (\lambda +(i1: nat).(\lambda (i2: nat).(\lambda (H1: (eq T (TLRef i1) (TSort n1))).(let +H2 \def (eq_ind T (TLRef i1) (\lambda (ee: T).(match ee in T return (\lambda +(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | +(THead _ _ _) \Rightarrow False])) I (TSort n1) H1) in (False_ind (ex nat +(\lambda (n2: nat).(eq T (TLRef i2) (TSort n2)))) H2))))) (\lambda (v1: +T).(\lambda (v2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k: +K).(\lambda (H1: (eq T (THead k v1 t1) (TSort n1))).(let H2 \def (eq_ind T +(THead k v1 t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) +with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ +_) \Rightarrow True])) I (TSort n1) H1) in (False_ind (ex nat (\lambda (n2: +nat).(eq T (THead k v2 t2) (TSort n2)))) H2)))))))) y u2 H0))) H))). +(* COMMENTS +Initial nodes: 321 +END *) + +theorem iso_gen_lref: + \forall (u2: T).(\forall (n1: nat).((iso (TLRef n1) u2) \to (ex nat (\lambda +(n2: nat).(eq T u2 (TLRef n2)))))) +\def + \lambda (u2: T).(\lambda (n1: nat).(\lambda (H: (iso (TLRef n1) +u2)).(insert_eq T (TLRef n1) (\lambda (t: T).(iso t u2)) (\lambda (_: T).(ex +nat (\lambda (n2: nat).(eq T u2 (TLRef n2))))) (\lambda (y: T).(\lambda (H0: +(iso y u2)).(iso_ind (\lambda (t: T).(\lambda (t0: T).((eq T t (TLRef n1)) +\to (ex nat (\lambda (n2: nat).(eq T t0 (TLRef n2))))))) (\lambda (n0: +nat).(\lambda (n2: nat).(\lambda (H1: (eq T (TSort n0) (TLRef n1))).(let H2 +\def (eq_ind T (TSort n0) (\lambda (ee: T).(match ee in T return (\lambda (_: +T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | +(THead _ _ _) \Rightarrow False])) I (TLRef n1) H1) in (False_ind (ex nat +(\lambda (n3: nat).(eq T (TSort n2) (TLRef n3)))) H2))))) (\lambda (i1: +nat).(\lambda (i2: nat).(\lambda (H1: (eq T (TLRef i1) (TLRef n1))).(let H2 +\def (f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) +with [(TSort _) \Rightarrow i1 | (TLRef n) \Rightarrow n | (THead _ _ _) +\Rightarrow i1])) (TLRef i1) (TLRef n1) H1) in (ex_intro nat (\lambda (n2: +nat).(eq T (TLRef i2) (TLRef n2))) i2 (refl_equal T (TLRef i2))))))) (\lambda +(v1: T).(\lambda (v2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k: +K).(\lambda (H1: (eq T (THead k v1 t1) (TLRef n1))).(let H2 \def (eq_ind T +(THead k v1 t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) +with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ +_) \Rightarrow True])) I (TLRef n1) H1) in (False_ind (ex nat (\lambda (n2: +nat).(eq T (THead k v2 t2) (TLRef n2)))) H2)))))))) y u2 H0))) H))). +(* COMMENTS +Initial nodes: 321 +END *) + +theorem iso_gen_head: + \forall (k: K).(\forall (v1: T).(\forall (t1: T).(\forall (u2: T).((iso +(THead k v1 t1) u2) \to (ex_2 T T (\lambda (v2: T).(\lambda (t2: T).(eq T u2 +(THead k v2 t2))))))))) +\def + \lambda (k: K).(\lambda (v1: T).(\lambda (t1: T).(\lambda (u2: T).(\lambda +(H: (iso (THead k v1 t1) u2)).(insert_eq T (THead k v1 t1) (\lambda (t: +T).(iso t u2)) (\lambda (_: T).(ex_2 T T (\lambda (v2: T).(\lambda (t2: +T).(eq T u2 (THead k v2 t2)))))) (\lambda (y: T).(\lambda (H0: (iso y +u2)).(iso_ind (\lambda (t: T).(\lambda (t0: T).((eq T t (THead k v1 t1)) \to +(ex_2 T T (\lambda (v2: T).(\lambda (t2: T).(eq T t0 (THead k v2 t2)))))))) +(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (H1: (eq T (TSort n1) (THead k +v1 t1))).(let H2 \def (eq_ind T (TSort n1) (\lambda (ee: T).(match ee in T +return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) +\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead k v1 t1) H1) +in (False_ind (ex_2 T T (\lambda (v2: T).(\lambda (t2: T).(eq T (TSort n2) +(THead k v2 t2))))) H2))))) (\lambda (i1: nat).(\lambda (i2: nat).(\lambda +(H1: (eq T (TLRef i1) (THead k v1 t1))).(let H2 \def (eq_ind T (TLRef i1) +(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) +\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow +False])) I (THead k v1 t1) H1) in (False_ind (ex_2 T T (\lambda (v2: +T).(\lambda (t2: T).(eq T (TLRef i2) (THead k v2 t2))))) H2))))) (\lambda +(v0: T).(\lambda (v2: T).(\lambda (t0: T).(\lambda (t2: T).(\lambda (k0: +K).(\lambda (H1: (eq T (THead k0 v0 t0) (THead k v1 t1))).(let H2 \def +(f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with +[(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _ _) +\Rightarrow k1])) (THead k0 v0 t0) (THead k v1 t1) H1) in ((let H3 \def +(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with +[(TSort _) \Rightarrow v0 | (TLRef _) \Rightarrow v0 | (THead _ t _) +\Rightarrow t])) (THead k0 v0 t0) (THead k v1 t1) H1) in ((let H4 \def +(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with +[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) +\Rightarrow t])) (THead k0 v0 t0) (THead k v1 t1) H1) in (\lambda (_: (eq T +v0 v1)).(\lambda (H6: (eq K k0 k)).(eq_ind_r K k (\lambda (k1: K).(ex_2 T T +(\lambda (v3: T).(\lambda (t3: T).(eq T (THead k1 v2 t2) (THead k v3 t3)))))) +(ex_2_intro T T (\lambda (v3: T).(\lambda (t3: T).(eq T (THead k v2 t2) +(THead k v3 t3)))) v2 t2 (refl_equal T (THead k v2 t2))) k0 H6)))) H3)) +H2)))))))) y u2 H0))) H))))). +(* COMMENTS +Initial nodes: 545 +END *) + +theorem iso_flats_lref_bind_false: + \forall (f: F).(\forall (b: B).(\forall (i: nat).(\forall (v: T).(\forall +(t: T).(\forall (vs: TList).((iso (THeads (Flat f) vs (TLRef i)) (THead (Bind +b) v t)) \to (\forall (P: Prop).P))))))) +\def + \lambda (f: F).(\lambda (b: B).(\lambda (i: nat).(\lambda (v: T).(\lambda +(t: T).(\lambda (vs: TList).(TList_ind (\lambda (t0: TList).((iso (THeads +(Flat f) t0 (TLRef i)) (THead (Bind b) v t)) \to (\forall (P: Prop).P))) +(\lambda (H: (iso (TLRef i) (THead (Bind b) v t))).(\lambda (P: Prop).(let +H_x \def (iso_gen_lref (THead (Bind b) v t) i H) in (let H0 \def H_x in +(ex_ind nat (\lambda (n2: nat).(eq T (THead (Bind b) v t) (TLRef n2))) P +(\lambda (x: nat).(\lambda (H1: (eq T (THead (Bind b) v t) (TLRef x))).(let +H2 \def (eq_ind T (THead (Bind b) v t) (\lambda (ee: T).(match ee in T return +(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef x) H1) in +(False_ind P H2)))) H0))))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda +(_: (((iso (THeads (Flat f) t1 (TLRef i)) (THead (Bind b) v t)) \to (\forall +(P: Prop).P)))).(\lambda (H0: (iso (THead (Flat f) t0 (THeads (Flat f) t1 +(TLRef i))) (THead (Bind b) v t))).(\lambda (P: Prop).(let H_x \def +(iso_gen_head (Flat f) t0 (THeads (Flat f) t1 (TLRef i)) (THead (Bind b) v t) +H0) in (let H1 \def H_x in (ex_2_ind T T (\lambda (v2: T).(\lambda (t2: +T).(eq T (THead (Bind b) v t) (THead (Flat f) v2 t2)))) P (\lambda (x0: +T).(\lambda (x1: T).(\lambda (H2: (eq T (THead (Bind b) v t) (THead (Flat f) +x0 x1))).(let H3 \def (eq_ind T (THead (Bind b) v t) (\lambda (ee: T).(match +ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | +(TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return +(\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow +False])])) I (THead (Flat f) x0 x1) H2) in (False_ind P H3))))) H1)))))))) +vs)))))). +(* COMMENTS +Initial nodes: 391 +END *) + +theorem iso_flats_flat_bind_false: + \forall (f1: F).(\forall (f2: F).(\forall (b: B).(\forall (v: T).(\forall +(v2: T).(\forall (t: T).(\forall (t2: T).(\forall (vs: TList).((iso (THeads +(Flat f1) vs (THead (Flat f2) v2 t2)) (THead (Bind b) v t)) \to (\forall (P: +Prop).P))))))))) +\def + \lambda (f1: F).(\lambda (f2: F).(\lambda (b: B).(\lambda (v: T).(\lambda +(v2: T).(\lambda (t: T).(\lambda (t2: T).(\lambda (vs: TList).(TList_ind +(\lambda (t0: TList).((iso (THeads (Flat f1) t0 (THead (Flat f2) v2 t2)) +(THead (Bind b) v t)) \to (\forall (P: Prop).P))) (\lambda (H: (iso (THead +(Flat f2) v2 t2) (THead (Bind b) v t))).(\lambda (P: Prop).(let H_x \def +(iso_gen_head (Flat f2) v2 t2 (THead (Bind b) v t) H) in (let H0 \def H_x in +(ex_2_ind T T (\lambda (v3: T).(\lambda (t3: T).(eq T (THead (Bind b) v t) +(THead (Flat f2) v3 t3)))) P (\lambda (x0: T).(\lambda (x1: T).(\lambda (H1: +(eq T (THead (Bind b) v t) (THead (Flat f2) x0 x1))).(let H2 \def (eq_ind T +(THead (Bind b) v t) (\lambda (ee: T).(match ee in T return (\lambda (_: +T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | +(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with +[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat +f2) x0 x1) H1) in (False_ind P H2))))) H0))))) (\lambda (t0: T).(\lambda (t1: +TList).(\lambda (_: (((iso (THeads (Flat f1) t1 (THead (Flat f2) v2 t2)) +(THead (Bind b) v t)) \to (\forall (P: Prop).P)))).(\lambda (H0: (iso (THead +(Flat f1) t0 (THeads (Flat f1) t1 (THead (Flat f2) v2 t2))) (THead (Bind b) v +t))).(\lambda (P: Prop).(let H_x \def (iso_gen_head (Flat f1) t0 (THeads +(Flat f1) t1 (THead (Flat f2) v2 t2)) (THead (Bind b) v t) H0) in (let H1 +\def H_x in (ex_2_ind T T (\lambda (v3: T).(\lambda (t3: T).(eq T (THead +(Bind b) v t) (THead (Flat f1) v3 t3)))) P (\lambda (x0: T).(\lambda (x1: +T).(\lambda (H2: (eq T (THead (Bind b) v t) (THead (Flat f1) x0 x1))).(let H3 +\def (eq_ind T (THead (Bind b) v t) (\lambda (ee: T).(match ee in T return +(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda +(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow +False])])) I (THead (Flat f1) x0 x1) H2) in (False_ind P H3))))) H1)))))))) +vs)))))))). +(* COMMENTS +Initial nodes: 461 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/iso/props.ma b/matita/matita/contribs/lambdadelta/basic_1/iso/props.ma new file mode 100644 index 000000000..af521d073 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/iso/props.ma @@ -0,0 +1,56 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/iso/fwd.ma". + +theorem iso_refl: + \forall (t: T).(iso t t) +\def + \lambda (t: T).(T_ind (\lambda (t0: T).(iso t0 t0)) (\lambda (n: +nat).(iso_sort n n)) (\lambda (n: nat).(iso_lref n n)) (\lambda (k: +K).(\lambda (t0: T).(\lambda (_: (iso t0 t0)).(\lambda (t1: T).(\lambda (_: +(iso t1 t1)).(iso_head t0 t0 t1 t1 k)))))) t). +(* COMMENTS +Initial nodes: 59 +END *) + +theorem iso_trans: + \forall (t1: T).(\forall (t2: T).((iso t1 t2) \to (\forall (t3: T).((iso t2 +t3) \to (iso t1 t3))))) +\def + \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (iso t1 t2)).(iso_ind (\lambda +(t: T).(\lambda (t0: T).(\forall (t3: T).((iso t0 t3) \to (iso t t3))))) +(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (t3: T).(\lambda (H0: (iso +(TSort n2) t3)).(let H_x \def (iso_gen_sort t3 n2 H0) in (let H1 \def H_x in +(ex_ind nat (\lambda (n3: nat).(eq T t3 (TSort n3))) (iso (TSort n1) t3) +(\lambda (x: nat).(\lambda (H2: (eq T t3 (TSort x))).(eq_ind_r T (TSort x) +(\lambda (t: T).(iso (TSort n1) t)) (iso_sort n1 x) t3 H2))) H1))))))) +(\lambda (i1: nat).(\lambda (i2: nat).(\lambda (t3: T).(\lambda (H0: (iso +(TLRef i2) t3)).(let H_x \def (iso_gen_lref t3 i2 H0) in (let H1 \def H_x in +(ex_ind nat (\lambda (n2: nat).(eq T t3 (TLRef n2))) (iso (TLRef i1) t3) +(\lambda (x: nat).(\lambda (H2: (eq T t3 (TLRef x))).(eq_ind_r T (TLRef x) +(\lambda (t: T).(iso (TLRef i1) t)) (iso_lref i1 x) t3 H2))) H1))))))) +(\lambda (v1: T).(\lambda (v2: T).(\lambda (t3: T).(\lambda (t4: T).(\lambda +(k: K).(\lambda (t5: T).(\lambda (H0: (iso (THead k v2 t4) t5)).(let H_x \def +(iso_gen_head k v2 t4 t5 H0) in (let H1 \def H_x in (ex_2_ind T T (\lambda +(v3: T).(\lambda (t6: T).(eq T t5 (THead k v3 t6)))) (iso (THead k v1 t3) t5) +(\lambda (x0: T).(\lambda (x1: T).(\lambda (H2: (eq T t5 (THead k x0 +x1))).(eq_ind_r T (THead k x0 x1) (\lambda (t: T).(iso (THead k v1 t3) t)) +(iso_head v1 x0 t3 x1 k) t5 H2)))) H1)))))))))) t1 t2 H))). +(* COMMENTS +Initial nodes: 351 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/leq/asucc.ma b/matita/matita/contribs/lambdadelta/basic_1/leq/asucc.ma new file mode 100644 index 000000000..fd9e7c1d3 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/leq/asucc.ma @@ -0,0 +1,479 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/leq/props.ma". + +theorem asucc_repl: + \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (leq g +(asucc g a1) (asucc g a2))))) +\def + \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1 +a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(leq g (asucc g a) (asucc g +a0)))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: nat).(\lambda (n2: +nat).(\lambda (k: nat).(\lambda (H0: (eq A (aplus g (ASort h1 n1) k) (aplus g +(ASort h2 n2) k))).(nat_ind (\lambda (n: nat).((eq A (aplus g (ASort n n1) k) +(aplus g (ASort h2 n2) k)) \to (leq g (match n with [O \Rightarrow (ASort O +(next g n1)) | (S h) \Rightarrow (ASort h n1)]) (match h2 with [O \Rightarrow +(ASort O (next g n2)) | (S h) \Rightarrow (ASort h n2)])))) (\lambda (H1: (eq +A (aplus g (ASort O n1) k) (aplus g (ASort h2 n2) k))).(nat_ind (\lambda (n: +nat).((eq A (aplus g (ASort O n1) k) (aplus g (ASort n n2) k)) \to (leq g +(ASort O (next g n1)) (match n with [O \Rightarrow (ASort O (next g n2)) | (S +h) \Rightarrow (ASort h n2)])))) (\lambda (H2: (eq A (aplus g (ASort O n1) k) +(aplus g (ASort O n2) k))).(leq_sort g O O (next g n1) (next g n2) k (eq_ind +A (aplus g (ASort O n1) (S k)) (\lambda (a: A).(eq A a (aplus g (ASort O +(next g n2)) k))) (eq_ind A (aplus g (ASort O n2) (S k)) (\lambda (a: A).(eq +A (aplus g (ASort O n1) (S k)) a)) (eq_ind_r A (aplus g (ASort O n2) k) +(\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g (ASort O n2) k)))) +(refl_equal A (asucc g (aplus g (ASort O n2) k))) (aplus g (ASort O n1) k) +H2) (aplus g (ASort O (next g n2)) k) (aplus_sort_O_S_simpl g n2 k)) (aplus g +(ASort O (next g n1)) k) (aplus_sort_O_S_simpl g n1 k)))) (\lambda (h3: +nat).(\lambda (_: (((eq A (aplus g (ASort O n1) k) (aplus g (ASort h3 n2) k)) +\to (leq g (ASort O (next g n1)) (match h3 with [O \Rightarrow (ASort O (next +g n2)) | (S h) \Rightarrow (ASort h n2)]))))).(\lambda (H2: (eq A (aplus g +(ASort O n1) k) (aplus g (ASort (S h3) n2) k))).(leq_sort g O h3 (next g n1) +n2 k (eq_ind A (aplus g (ASort O n1) (S k)) (\lambda (a: A).(eq A a (aplus g +(ASort h3 n2) k))) (eq_ind A (aplus g (ASort (S h3) n2) (S k)) (\lambda (a: +A).(eq A (aplus g (ASort O n1) (S k)) a)) (eq_ind_r A (aplus g (ASort (S h3) +n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g (ASort (S h3) n2) +k)))) (refl_equal A (asucc g (aplus g (ASort (S h3) n2) k))) (aplus g (ASort +O n1) k) H2) (aplus g (ASort h3 n2) k) (aplus_sort_S_S_simpl g n2 h3 k)) +(aplus g (ASort O (next g n1)) k) (aplus_sort_O_S_simpl g n1 k)))))) h2 H1)) +(\lambda (h3: nat).(\lambda (IHh1: (((eq A (aplus g (ASort h3 n1) k) (aplus g +(ASort h2 n2) k)) \to (leq g (match h3 with [O \Rightarrow (ASort O (next g +n1)) | (S h) \Rightarrow (ASort h n1)]) (match h2 with [O \Rightarrow (ASort +O (next g n2)) | (S h) \Rightarrow (ASort h n2)]))))).(\lambda (H1: (eq A +(aplus g (ASort (S h3) n1) k) (aplus g (ASort h2 n2) k))).(nat_ind (\lambda +(n: nat).((eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort n n2) k)) \to +((((eq A (aplus g (ASort h3 n1) k) (aplus g (ASort n n2) k)) \to (leq g +(match h3 with [O \Rightarrow (ASort O (next g n1)) | (S h) \Rightarrow +(ASort h n1)]) (match n with [O \Rightarrow (ASort O (next g n2)) | (S h) +\Rightarrow (ASort h n2)])))) \to (leq g (ASort h3 n1) (match n with [O +\Rightarrow (ASort O (next g n2)) | (S h) \Rightarrow (ASort h n2)]))))) +(\lambda (H2: (eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort O n2) +k))).(\lambda (_: (((eq A (aplus g (ASort h3 n1) k) (aplus g (ASort O n2) k)) +\to (leq g (match h3 with [O \Rightarrow (ASort O (next g n1)) | (S h) +\Rightarrow (ASort h n1)]) (ASort O (next g n2)))))).(leq_sort g h3 O n1 +(next g n2) k (eq_ind A (aplus g (ASort O n2) (S k)) (\lambda (a: A).(eq A +(aplus g (ASort h3 n1) k) a)) (eq_ind A (aplus g (ASort (S h3) n1) (S k)) +(\lambda (a: A).(eq A a (aplus g (ASort O n2) (S k)))) (eq_ind_r A (aplus g +(ASort O n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g (ASort O +n2) k)))) (refl_equal A (asucc g (aplus g (ASort O n2) k))) (aplus g (ASort +(S h3) n1) k) H2) (aplus g (ASort h3 n1) k) (aplus_sort_S_S_simpl g n1 h3 k)) +(aplus g (ASort O (next g n2)) k) (aplus_sort_O_S_simpl g n2 k))))) (\lambda +(h4: nat).(\lambda (_: (((eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort +h4 n2) k)) \to ((((eq A (aplus g (ASort h3 n1) k) (aplus g (ASort h4 n2) k)) +\to (leq g (match h3 with [O \Rightarrow (ASort O (next g n1)) | (S h) +\Rightarrow (ASort h n1)]) (match h4 with [O \Rightarrow (ASort O (next g +n2)) | (S h) \Rightarrow (ASort h n2)])))) \to (leq g (ASort h3 n1) (match h4 +with [O \Rightarrow (ASort O (next g n2)) | (S h) \Rightarrow (ASort h +n2)])))))).(\lambda (H2: (eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort +(S h4) n2) k))).(\lambda (_: (((eq A (aplus g (ASort h3 n1) k) (aplus g +(ASort (S h4) n2) k)) \to (leq g (match h3 with [O \Rightarrow (ASort O (next +g n1)) | (S h) \Rightarrow (ASort h n1)]) (ASort h4 n2))))).(leq_sort g h3 h4 +n1 n2 k (eq_ind A (aplus g (ASort (S h3) n1) (S k)) (\lambda (a: A).(eq A a +(aplus g (ASort h4 n2) k))) (eq_ind A (aplus g (ASort (S h4) n2) (S k)) +(\lambda (a: A).(eq A (aplus g (ASort (S h3) n1) (S k)) a)) (eq_ind_r A +(aplus g (ASort (S h4) n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g +(aplus g (ASort (S h4) n2) k)))) (refl_equal A (asucc g (aplus g (ASort (S +h4) n2) k))) (aplus g (ASort (S h3) n1) k) H2) (aplus g (ASort h4 n2) k) +(aplus_sort_S_S_simpl g n2 h4 k)) (aplus g (ASort h3 n1) k) +(aplus_sort_S_S_simpl g n1 h3 k))))))) h2 H1 IHh1)))) h1 H0))))))) (\lambda +(a3: A).(\lambda (a4: A).(\lambda (H0: (leq g a3 a4)).(\lambda (_: (leq g +(asucc g a3) (asucc g a4))).(\lambda (a5: A).(\lambda (a6: A).(\lambda (_: +(leq g a5 a6)).(\lambda (H3: (leq g (asucc g a5) (asucc g a6))).(leq_head g +a3 a4 H0 (asucc g a5) (asucc g a6) H3))))))))) a1 a2 H)))). +(* COMMENTS +Initial nodes: 1907 +END *) + +theorem asucc_inj: + \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (asucc g a1) (asucc +g a2)) \to (leq g a1 a2)))) +\def + \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2: +A).((leq g (asucc g a) (asucc g a2)) \to (leq g a a2)))) (\lambda (n: +nat).(\lambda (n0: nat).(\lambda (a2: A).(A_ind (\lambda (a: A).((leq g +(asucc g (ASort n n0)) (asucc g a)) \to (leq g (ASort n n0) a))) (\lambda +(n1: nat).(\lambda (n2: nat).(\lambda (H: (leq g (asucc g (ASort n n0)) +(asucc g (ASort n1 n2)))).(nat_ind (\lambda (n3: nat).((leq g (asucc g (ASort +n3 n0)) (asucc g (ASort n1 n2))) \to (leq g (ASort n3 n0) (ASort n1 n2)))) +(\lambda (H0: (leq g (asucc g (ASort O n0)) (asucc g (ASort n1 +n2)))).(nat_ind (\lambda (n3: nat).((leq g (asucc g (ASort O n0)) (asucc g +(ASort n3 n2))) \to (leq g (ASort O n0) (ASort n3 n2)))) (\lambda (H1: (leq g +(asucc g (ASort O n0)) (asucc g (ASort O n2)))).(let H_x \def (leq_gen_sort1 +g O (next g n0) (ASort O (next g n2)) H1) in (let H2 \def H_x in (ex2_3_ind +nat nat nat (\lambda (n3: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A +(aplus g (ASort O (next g n0)) k) (aplus g (ASort h2 n3) k))))) (\lambda (n3: +nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A (ASort O (next g n2)) (ASort +h2 n3))))) (leq g (ASort O n0) (ASort O n2)) (\lambda (x0: nat).(\lambda (x1: +nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort O (next g n0)) +x2) (aplus g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort O (next g n2)) +(ASort x1 x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e in A +return (\lambda (_: A).nat) with [(ASort n3 _) \Rightarrow n3 | (AHead _ _) +\Rightarrow O])) (ASort O (next g n2)) (ASort x1 x0) H4) in ((let H6 \def +(f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with +[(ASort _ n3) \Rightarrow n3 | (AHead _ _) \Rightarrow ((match g with [(mk_G +next _) \Rightarrow next]) n2)])) (ASort O (next g n2)) (ASort x1 x0) H4) in +(\lambda (H7: (eq nat O x1)).(let H8 \def (eq_ind_r nat x1 (\lambda (n3: +nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort n3 x0) x2))) H3 +O H7) in (let H9 \def (eq_ind_r nat x0 (\lambda (n3: nat).(eq A (aplus g +(ASort O (next g n0)) x2) (aplus g (ASort O n3) x2))) H8 (next g n2) H6) in +(let H10 \def (eq_ind_r A (aplus g (ASort O (next g n0)) x2) (\lambda (a: +A).(eq A a (aplus g (ASort O (next g n2)) x2))) H9 (aplus g (ASort O n0) (S +x2)) (aplus_sort_O_S_simpl g n0 x2)) in (let H11 \def (eq_ind_r A (aplus g +(ASort O (next g n2)) x2) (\lambda (a: A).(eq A (aplus g (ASort O n0) (S x2)) +a)) H10 (aplus g (ASort O n2) (S x2)) (aplus_sort_O_S_simpl g n2 x2)) in +(leq_sort g O O n0 n2 (S x2) H11))))))) H5))))))) H2)))) (\lambda (n3: +nat).(\lambda (_: (((leq g (asucc g (ASort O n0)) (asucc g (ASort n3 n2))) +\to (leq g (ASort O n0) (ASort n3 n2))))).(\lambda (H1: (leq g (asucc g +(ASort O n0)) (asucc g (ASort (S n3) n2)))).(let H_x \def (leq_gen_sort1 g O +(next g n0) (ASort n3 n2) H1) in (let H2 \def H_x in (ex2_3_ind nat nat nat +(\lambda (n4: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort +O (next g n0)) k) (aplus g (ASort h2 n4) k))))) (\lambda (n4: nat).(\lambda +(h2: nat).(\lambda (_: nat).(eq A (ASort n3 n2) (ASort h2 n4))))) (leq g +(ASort O n0) (ASort (S n3) n2)) (\lambda (x0: nat).(\lambda (x1: +nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort O (next g n0)) +x2) (aplus g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort n3 n2) (ASort x1 +x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e in A return +(\lambda (_: A).nat) with [(ASort n4 _) \Rightarrow n4 | (AHead _ _) +\Rightarrow n3])) (ASort n3 n2) (ASort x1 x0) H4) in ((let H6 \def (f_equal A +nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort _ +n4) \Rightarrow n4 | (AHead _ _) \Rightarrow n2])) (ASort n3 n2) (ASort x1 +x0) H4) in (\lambda (H7: (eq nat n3 x1)).(let H8 \def (eq_ind_r nat x1 +(\lambda (n4: nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort +n4 x0) x2))) H3 n3 H7) in (let H9 \def (eq_ind_r nat x0 (\lambda (n4: +nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort n3 n4) x2))) H8 +n2 H6) in (let H10 \def (eq_ind_r A (aplus g (ASort O (next g n0)) x2) +(\lambda (a: A).(eq A a (aplus g (ASort n3 n2) x2))) H9 (aplus g (ASort O n0) +(S x2)) (aplus_sort_O_S_simpl g n0 x2)) in (let H11 \def (eq_ind_r A (aplus g +(ASort n3 n2) x2) (\lambda (a: A).(eq A (aplus g (ASort O n0) (S x2)) a)) H10 +(aplus g (ASort (S n3) n2) (S x2)) (aplus_sort_S_S_simpl g n2 n3 x2)) in +(leq_sort g O (S n3) n0 n2 (S x2) H11))))))) H5))))))) H2)))))) n1 H0)) +(\lambda (n3: nat).(\lambda (IHn: (((leq g (asucc g (ASort n3 n0)) (asucc g +(ASort n1 n2))) \to (leq g (ASort n3 n0) (ASort n1 n2))))).(\lambda (H0: (leq +g (asucc g (ASort (S n3) n0)) (asucc g (ASort n1 n2)))).(nat_ind (\lambda +(n4: nat).((leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort n4 n2))) \to +((((leq g (asucc g (ASort n3 n0)) (asucc g (ASort n4 n2))) \to (leq g (ASort +n3 n0) (ASort n4 n2)))) \to (leq g (ASort (S n3) n0) (ASort n4 n2))))) +(\lambda (H1: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort O +n2)))).(\lambda (_: (((leq g (asucc g (ASort n3 n0)) (asucc g (ASort O n2))) +\to (leq g (ASort n3 n0) (ASort O n2))))).(let H_x \def (leq_gen_sort1 g n3 +n0 (ASort O (next g n2)) H1) in (let H2 \def H_x in (ex2_3_ind nat nat nat +(\lambda (n4: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort +n3 n0) k) (aplus g (ASort h2 n4) k))))) (\lambda (n4: nat).(\lambda (h2: +nat).(\lambda (_: nat).(eq A (ASort O (next g n2)) (ASort h2 n4))))) (leq g +(ASort (S n3) n0) (ASort O n2)) (\lambda (x0: nat).(\lambda (x1: +nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort n3 n0) x2) (aplus +g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort O (next g n2)) (ASort x1 +x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e in A return +(\lambda (_: A).nat) with [(ASort n4 _) \Rightarrow n4 | (AHead _ _) +\Rightarrow O])) (ASort O (next g n2)) (ASort x1 x0) H4) in ((let H6 \def +(f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with +[(ASort _ n4) \Rightarrow n4 | (AHead _ _) \Rightarrow ((match g with [(mk_G +next _) \Rightarrow next]) n2)])) (ASort O (next g n2)) (ASort x1 x0) H4) in +(\lambda (H7: (eq nat O x1)).(let H8 \def (eq_ind_r nat x1 (\lambda (n4: +nat).(eq A (aplus g (ASort n3 n0) x2) (aplus g (ASort n4 x0) x2))) H3 O H7) +in (let H9 \def (eq_ind_r nat x0 (\lambda (n4: nat).(eq A (aplus g (ASort n3 +n0) x2) (aplus g (ASort O n4) x2))) H8 (next g n2) H6) in (let H10 \def +(eq_ind_r A (aplus g (ASort n3 n0) x2) (\lambda (a: A).(eq A a (aplus g +(ASort O (next g n2)) x2))) H9 (aplus g (ASort (S n3) n0) (S x2)) +(aplus_sort_S_S_simpl g n0 n3 x2)) in (let H11 \def (eq_ind_r A (aplus g +(ASort O (next g n2)) x2) (\lambda (a: A).(eq A (aplus g (ASort (S n3) n0) (S +x2)) a)) H10 (aplus g (ASort O n2) (S x2)) (aplus_sort_O_S_simpl g n2 x2)) in +(leq_sort g (S n3) O n0 n2 (S x2) H11))))))) H5))))))) H2))))) (\lambda (n4: +nat).(\lambda (_: (((leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort n4 +n2))) \to ((((leq g (asucc g (ASort n3 n0)) (asucc g (ASort n4 n2))) \to (leq +g (ASort n3 n0) (ASort n4 n2)))) \to (leq g (ASort (S n3) n0) (ASort n4 +n2)))))).(\lambda (H1: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort (S +n4) n2)))).(\lambda (_: (((leq g (asucc g (ASort n3 n0)) (asucc g (ASort (S +n4) n2))) \to (leq g (ASort n3 n0) (ASort (S n4) n2))))).(let H_x \def +(leq_gen_sort1 g n3 n0 (ASort n4 n2) H1) in (let H2 \def H_x in (ex2_3_ind +nat nat nat (\lambda (n5: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A +(aplus g (ASort n3 n0) k) (aplus g (ASort h2 n5) k))))) (\lambda (n5: +nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A (ASort n4 n2) (ASort h2 +n5))))) (leq g (ASort (S n3) n0) (ASort (S n4) n2)) (\lambda (x0: +nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g +(ASort n3 n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort n4 +n2) (ASort x1 x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e in A +return (\lambda (_: A).nat) with [(ASort n5 _) \Rightarrow n5 | (AHead _ _) +\Rightarrow n4])) (ASort n4 n2) (ASort x1 x0) H4) in ((let H6 \def (f_equal A +nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort _ +n5) \Rightarrow n5 | (AHead _ _) \Rightarrow n2])) (ASort n4 n2) (ASort x1 +x0) H4) in (\lambda (H7: (eq nat n4 x1)).(let H8 \def (eq_ind_r nat x1 +(\lambda (n5: nat).(eq A (aplus g (ASort n3 n0) x2) (aplus g (ASort n5 x0) +x2))) H3 n4 H7) in (let H9 \def (eq_ind_r nat x0 (\lambda (n5: nat).(eq A +(aplus g (ASort n3 n0) x2) (aplus g (ASort n4 n5) x2))) H8 n2 H6) in (let H10 +\def (eq_ind_r A (aplus g (ASort n3 n0) x2) (\lambda (a: A).(eq A a (aplus g +(ASort n4 n2) x2))) H9 (aplus g (ASort (S n3) n0) (S x2)) +(aplus_sort_S_S_simpl g n0 n3 x2)) in (let H11 \def (eq_ind_r A (aplus g +(ASort n4 n2) x2) (\lambda (a: A).(eq A (aplus g (ASort (S n3) n0) (S x2)) +a)) H10 (aplus g (ASort (S n4) n2) (S x2)) (aplus_sort_S_S_simpl g n2 n4 x2)) +in (leq_sort g (S n3) (S n4) n0 n2 (S x2) H11))))))) H5))))))) H2))))))) n1 +H0 IHn)))) n H)))) (\lambda (a: A).(\lambda (H: (((leq g (asucc g (ASort n +n0)) (asucc g a)) \to (leq g (ASort n n0) a)))).(\lambda (a0: A).(\lambda +(H0: (((leq g (asucc g (ASort n n0)) (asucc g a0)) \to (leq g (ASort n n0) +a0)))).(\lambda (H1: (leq g (asucc g (ASort n n0)) (asucc g (AHead a +a0)))).(nat_ind (\lambda (n1: nat).((((leq g (asucc g (ASort n1 n0)) (asucc g +a)) \to (leq g (ASort n1 n0) a))) \to ((((leq g (asucc g (ASort n1 n0)) +(asucc g a0)) \to (leq g (ASort n1 n0) a0))) \to ((leq g (asucc g (ASort n1 +n0)) (asucc g (AHead a a0))) \to (leq g (ASort n1 n0) (AHead a a0)))))) +(\lambda (_: (((leq g (asucc g (ASort O n0)) (asucc g a)) \to (leq g (ASort O +n0) a)))).(\lambda (_: (((leq g (asucc g (ASort O n0)) (asucc g a0)) \to (leq +g (ASort O n0) a0)))).(\lambda (H4: (leq g (asucc g (ASort O n0)) (asucc g +(AHead a a0)))).(let H_x \def (leq_gen_sort1 g O (next g n0) (AHead a (asucc +g a0)) H4) in (let H5 \def H_x in (ex2_3_ind nat nat nat (\lambda (n2: +nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort O (next g +n0)) k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: +nat).(\lambda (_: nat).(eq A (AHead a (asucc g a0)) (ASort h2 n2))))) (leq g +(ASort O n0) (AHead a a0)) (\lambda (x0: nat).(\lambda (x1: nat).(\lambda +(x2: nat).(\lambda (_: (eq A (aplus g (ASort O (next g n0)) x2) (aplus g +(ASort x1 x0) x2))).(\lambda (H7: (eq A (AHead a (asucc g a0)) (ASort x1 +x0))).(let H8 \def (eq_ind A (AHead a (asucc g a0)) (\lambda (ee: A).(match +ee in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | +(AHead _ _) \Rightarrow True])) I (ASort x1 x0) H7) in (False_ind (leq g +(ASort O n0) (AHead a a0)) H8))))))) H5)))))) (\lambda (n1: nat).(\lambda (_: +(((((leq g (asucc g (ASort n1 n0)) (asucc g a)) \to (leq g (ASort n1 n0) a))) +\to ((((leq g (asucc g (ASort n1 n0)) (asucc g a0)) \to (leq g (ASort n1 n0) +a0))) \to ((leq g (asucc g (ASort n1 n0)) (asucc g (AHead a a0))) \to (leq g +(ASort n1 n0) (AHead a a0))))))).(\lambda (_: (((leq g (asucc g (ASort (S n1) +n0)) (asucc g a)) \to (leq g (ASort (S n1) n0) a)))).(\lambda (_: (((leq g +(asucc g (ASort (S n1) n0)) (asucc g a0)) \to (leq g (ASort (S n1) n0) +a0)))).(\lambda (H4: (leq g (asucc g (ASort (S n1) n0)) (asucc g (AHead a +a0)))).(let H_x \def (leq_gen_sort1 g n1 n0 (AHead a (asucc g a0)) H4) in +(let H5 \def H_x in (ex2_3_ind nat nat nat (\lambda (n2: nat).(\lambda (h2: +nat).(\lambda (k: nat).(eq A (aplus g (ASort n1 n0) k) (aplus g (ASort h2 n2) +k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A (AHead a +(asucc g a0)) (ASort h2 n2))))) (leq g (ASort (S n1) n0) (AHead a a0)) +(\lambda (x0: nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (_: (eq A +(aplus g (ASort n1 n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H7: (eq A +(AHead a (asucc g a0)) (ASort x1 x0))).(let H8 \def (eq_ind A (AHead a (asucc +g a0)) (\lambda (ee: A).(match ee in A return (\lambda (_: A).Prop) with +[(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort x1 +x0) H7) in (False_ind (leq g (ASort (S n1) n0) (AHead a a0)) H8))))))) +H5)))))))) n H H0 H1)))))) a2)))) (\lambda (a: A).(\lambda (_: ((\forall (a2: +A).((leq g (asucc g a) (asucc g a2)) \to (leq g a a2))))).(\lambda (a0: +A).(\lambda (H0: ((\forall (a2: A).((leq g (asucc g a0) (asucc g a2)) \to +(leq g a0 a2))))).(\lambda (a2: A).(A_ind (\lambda (a3: A).((leq g (asucc g +(AHead a a0)) (asucc g a3)) \to (leq g (AHead a a0) a3))) (\lambda (n: +nat).(\lambda (n0: nat).(\lambda (H1: (leq g (asucc g (AHead a a0)) (asucc g +(ASort n n0)))).(nat_ind (\lambda (n1: nat).((leq g (asucc g (AHead a a0)) +(asucc g (ASort n1 n0))) \to (leq g (AHead a a0) (ASort n1 n0)))) (\lambda +(H2: (leq g (asucc g (AHead a a0)) (asucc g (ASort O n0)))).(let H_x \def +(leq_gen_head1 g a (asucc g a0) (ASort O (next g n0)) H2) in (let H3 \def H_x +in (ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq g a a3))) (\lambda +(_: A).(\lambda (a4: A).(leq g (asucc g a0) a4))) (\lambda (a3: A).(\lambda +(a4: A).(eq A (ASort O (next g n0)) (AHead a3 a4)))) (leq g (AHead a a0) +(ASort O n0)) (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g a +x0)).(\lambda (_: (leq g (asucc g a0) x1)).(\lambda (H6: (eq A (ASort O (next +g n0)) (AHead x0 x1))).(let H7 \def (eq_ind A (ASort O (next g n0)) (\lambda +(ee: A).(match ee in A return (\lambda (_: A).Prop) with [(ASort _ _) +\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H6) in +(False_ind (leq g (AHead a a0) (ASort O n0)) H7))))))) H3)))) (\lambda (n1: +nat).(\lambda (_: (((leq g (asucc g (AHead a a0)) (asucc g (ASort n1 n0))) +\to (leq g (AHead a a0) (ASort n1 n0))))).(\lambda (H2: (leq g (asucc g +(AHead a a0)) (asucc g (ASort (S n1) n0)))).(let H_x \def (leq_gen_head1 g a +(asucc g a0) (ASort n1 n0) H2) in (let H3 \def H_x in (ex3_2_ind A A (\lambda +(a3: A).(\lambda (_: A).(leq g a a3))) (\lambda (_: A).(\lambda (a4: A).(leq +g (asucc g a0) a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort n1 n0) +(AHead a3 a4)))) (leq g (AHead a a0) (ASort (S n1) n0)) (\lambda (x0: +A).(\lambda (x1: A).(\lambda (_: (leq g a x0)).(\lambda (_: (leq g (asucc g +a0) x1)).(\lambda (H6: (eq A (ASort n1 n0) (AHead x0 x1))).(let H7 \def +(eq_ind A (ASort n1 n0) (\lambda (ee: A).(match ee in A return (\lambda (_: +A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow +False])) I (AHead x0 x1) H6) in (False_ind (leq g (AHead a a0) (ASort (S n1) +n0)) H7))))))) H3)))))) n H1)))) (\lambda (a3: A).(\lambda (_: (((leq g +(asucc g (AHead a a0)) (asucc g a3)) \to (leq g (AHead a a0) a3)))).(\lambda +(a4: A).(\lambda (_: (((leq g (asucc g (AHead a a0)) (asucc g a4)) \to (leq g +(AHead a a0) a4)))).(\lambda (H3: (leq g (asucc g (AHead a a0)) (asucc g +(AHead a3 a4)))).(let H_x \def (leq_gen_head1 g a (asucc g a0) (AHead a3 +(asucc g a4)) H3) in (let H4 \def H_x in (ex3_2_ind A A (\lambda (a5: +A).(\lambda (_: A).(leq g a a5))) (\lambda (_: A).(\lambda (a6: A).(leq g +(asucc g a0) a6))) (\lambda (a5: A).(\lambda (a6: A).(eq A (AHead a3 (asucc g +a4)) (AHead a5 a6)))) (leq g (AHead a a0) (AHead a3 a4)) (\lambda (x0: +A).(\lambda (x1: A).(\lambda (H5: (leq g a x0)).(\lambda (H6: (leq g (asucc g +a0) x1)).(\lambda (H7: (eq A (AHead a3 (asucc g a4)) (AHead x0 x1))).(let H8 +\def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) +with [(ASort _ _) \Rightarrow a3 | (AHead a5 _) \Rightarrow a5])) (AHead a3 +(asucc g a4)) (AHead x0 x1) H7) in ((let H9 \def (f_equal A A (\lambda (e: +A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow +((let rec asucc (g0: G) (l: A) on l: A \def (match l with [(ASort n0 n) +\Rightarrow (match n0 with [O \Rightarrow (ASort O (next g0 n)) | (S h) +\Rightarrow (ASort h n)]) | (AHead a5 a6) \Rightarrow (AHead a5 (asucc g0 +a6))]) in asucc) g a4) | (AHead _ a5) \Rightarrow a5])) (AHead a3 (asucc g +a4)) (AHead x0 x1) H7) in (\lambda (H10: (eq A a3 x0)).(let H11 \def +(eq_ind_r A x1 (\lambda (a5: A).(leq g (asucc g a0) a5)) H6 (asucc g a4) H9) +in (let H12 \def (eq_ind_r A x0 (\lambda (a5: A).(leq g a a5)) H5 a3 H10) in +(leq_head g a a3 H12 a0 a4 (H0 a4 H11)))))) H8))))))) H4)))))))) a2)))))) +a1)). +(* COMMENTS +Initial nodes: 4697 +END *) + +theorem leq_asucc: + \forall (g: G).(\forall (a: A).(ex A (\lambda (a0: A).(leq g a (asucc g +a0))))) +\def + \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(ex A (\lambda (a1: +A).(leq g a0 (asucc g a1))))) (\lambda (n: nat).(\lambda (n0: nat).(ex_intro +A (\lambda (a0: A).(leq g (ASort n n0) (asucc g a0))) (ASort (S n) n0) +(leq_refl g (ASort n n0))))) (\lambda (a0: A).(\lambda (_: (ex A (\lambda +(a1: A).(leq g a0 (asucc g a1))))).(\lambda (a1: A).(\lambda (H0: (ex A +(\lambda (a2: A).(leq g a1 (asucc g a2))))).(let H1 \def H0 in (ex_ind A +(\lambda (a2: A).(leq g a1 (asucc g a2))) (ex A (\lambda (a2: A).(leq g +(AHead a0 a1) (asucc g a2)))) (\lambda (x: A).(\lambda (H2: (leq g a1 (asucc +g x))).(ex_intro A (\lambda (a2: A).(leq g (AHead a0 a1) (asucc g a2))) +(AHead a0 x) (leq_head g a0 a0 (leq_refl g a0) a1 (asucc g x) H2)))) H1)))))) +a)). +(* COMMENTS +Initial nodes: 221 +END *) + +theorem leq_ahead_asucc_false: + \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (AHead a1 a2) +(asucc g a1)) \to (\forall (P: Prop).P)))) +\def + \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2: +A).((leq g (AHead a a2) (asucc g a)) \to (\forall (P: Prop).P)))) (\lambda +(n: nat).(\lambda (n0: nat).(\lambda (a2: A).(\lambda (H: (leq g (AHead +(ASort n n0) a2) (match n with [O \Rightarrow (ASort O (next g n0)) | (S h) +\Rightarrow (ASort h n0)]))).(\lambda (P: Prop).(nat_ind (\lambda (n1: +nat).((leq g (AHead (ASort n1 n0) a2) (match n1 with [O \Rightarrow (ASort O +(next g n0)) | (S h) \Rightarrow (ASort h n0)])) \to P)) (\lambda (H0: (leq g +(AHead (ASort O n0) a2) (ASort O (next g n0)))).(let H_x \def (leq_gen_head1 +g (ASort O n0) a2 (ASort O (next g n0)) H0) in (let H1 \def H_x in (ex3_2_ind +A A (\lambda (a3: A).(\lambda (_: A).(leq g (ASort O n0) a3))) (\lambda (_: +A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A +(ASort O (next g n0)) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: +A).(\lambda (_: (leq g (ASort O n0) x0)).(\lambda (_: (leq g a2 x1)).(\lambda +(H4: (eq A (ASort O (next g n0)) (AHead x0 x1))).(let H5 \def (eq_ind A +(ASort O (next g n0)) (\lambda (ee: A).(match ee in A return (\lambda (_: +A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow +False])) I (AHead x0 x1) H4) in (False_ind P H5))))))) H1)))) (\lambda (n1: +nat).(\lambda (_: (((leq g (AHead (ASort n1 n0) a2) (match n1 with [O +\Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)])) \to +P))).(\lambda (H0: (leq g (AHead (ASort (S n1) n0) a2) (ASort n1 n0))).(let +H_x \def (leq_gen_head1 g (ASort (S n1) n0) a2 (ASort n1 n0) H0) in (let H1 +\def H_x in (ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq g (ASort (S +n1) n0) a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: +A).(\lambda (a4: A).(eq A (ASort n1 n0) (AHead a3 a4)))) P (\lambda (x0: +A).(\lambda (x1: A).(\lambda (_: (leq g (ASort (S n1) n0) x0)).(\lambda (_: +(leq g a2 x1)).(\lambda (H4: (eq A (ASort n1 n0) (AHead x0 x1))).(let H5 \def +(eq_ind A (ASort n1 n0) (\lambda (ee: A).(match ee in A return (\lambda (_: +A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow +False])) I (AHead x0 x1) H4) in (False_ind P H5))))))) H1)))))) n H)))))) +(\lambda (a: A).(\lambda (_: ((\forall (a2: A).((leq g (AHead a a2) (asucc g +a)) \to (\forall (P: Prop).P))))).(\lambda (a0: A).(\lambda (_: ((\forall +(a2: A).((leq g (AHead a0 a2) (asucc g a0)) \to (\forall (P: +Prop).P))))).(\lambda (a2: A).(\lambda (H1: (leq g (AHead (AHead a a0) a2) +(AHead a (asucc g a0)))).(\lambda (P: Prop).(let H_x \def (leq_gen_head1 g +(AHead a a0) a2 (AHead a (asucc g a0)) H1) in (let H2 \def H_x in (ex3_2_ind +A A (\lambda (a3: A).(\lambda (_: A).(leq g (AHead a a0) a3))) (\lambda (_: +A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A +(AHead a (asucc g a0)) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: +A).(\lambda (H3: (leq g (AHead a a0) x0)).(\lambda (H4: (leq g a2 +x1)).(\lambda (H5: (eq A (AHead a (asucc g a0)) (AHead x0 x1))).(let H6 \def +(f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with +[(ASort _ _) \Rightarrow a | (AHead a3 _) \Rightarrow a3])) (AHead a (asucc g +a0)) (AHead x0 x1) H5) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e +in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow ((let rec asucc +(g0: G) (l: A) on l: A \def (match l with [(ASort n0 n) \Rightarrow (match n0 +with [O \Rightarrow (ASort O (next g0 n)) | (S h) \Rightarrow (ASort h n)]) | +(AHead a3 a4) \Rightarrow (AHead a3 (asucc g0 a4))]) in asucc) g a0) | (AHead +_ a3) \Rightarrow a3])) (AHead a (asucc g a0)) (AHead x0 x1) H5) in (\lambda +(H8: (eq A a x0)).(let H9 \def (eq_ind_r A x1 (\lambda (a3: A).(leq g a2 a3)) +H4 (asucc g a0) H7) in (let H10 \def (eq_ind_r A x0 (\lambda (a3: A).(leq g +(AHead a a0) a3)) H3 a H8) in (leq_ahead_false_1 g a a0 H10 P))))) H6))))))) +H2)))))))))) a1)). +(* COMMENTS +Initial nodes: 927 +END *) + +theorem leq_asucc_false: + \forall (g: G).(\forall (a: A).((leq g (asucc g a) a) \to (\forall (P: +Prop).P))) +\def + \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).((leq g (asucc g a0) +a0) \to (\forall (P: Prop).P))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda +(H: (leq g (match n with [O \Rightarrow (ASort O (next g n0)) | (S h) +\Rightarrow (ASort h n0)]) (ASort n n0))).(\lambda (P: Prop).(nat_ind +(\lambda (n1: nat).((leq g (match n1 with [O \Rightarrow (ASort O (next g +n0)) | (S h) \Rightarrow (ASort h n0)]) (ASort n1 n0)) \to P)) (\lambda (H0: +(leq g (ASort O (next g n0)) (ASort O n0))).(let H_x \def (leq_gen_sort1 g O +(next g n0) (ASort O n0) H0) in (let H1 \def H_x in (ex2_3_ind nat nat nat +(\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort +O (next g n0)) k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda +(h2: nat).(\lambda (_: nat).(eq A (ASort O n0) (ASort h2 n2))))) P (\lambda +(x0: nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (H2: (eq A (aplus g +(ASort O (next g n0)) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H3: (eq A +(ASort O n0) (ASort x1 x0))).(let H4 \def (f_equal A nat (\lambda (e: +A).(match e in A return (\lambda (_: A).nat) with [(ASort n1 _) \Rightarrow +n1 | (AHead _ _) \Rightarrow O])) (ASort O n0) (ASort x1 x0) H3) in ((let H5 +\def (f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) +with [(ASort _ n1) \Rightarrow n1 | (AHead _ _) \Rightarrow n0])) (ASort O +n0) (ASort x1 x0) H3) in (\lambda (H6: (eq nat O x1)).(let H7 \def (eq_ind_r +nat x1 (\lambda (n1: nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g +(ASort n1 x0) x2))) H2 O H6) in (let H8 \def (eq_ind_r nat x0 (\lambda (n1: +nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort O n1) x2))) H7 +n0 H5) in (let H9 \def (eq_ind_r A (aplus g (ASort O (next g n0)) x2) +(\lambda (a0: A).(eq A a0 (aplus g (ASort O n0) x2))) H8 (aplus g (ASort O +n0) (S x2)) (aplus_sort_O_S_simpl g n0 x2)) in (let H_y \def (aplus_inj g (S +x2) x2 (ASort O n0) H9) in (le_Sx_x x2 (eq_ind_r nat x2 (\lambda (n1: +nat).(le n1 x2)) (le_n x2) (S x2) H_y) P))))))) H4))))))) H1)))) (\lambda +(n1: nat).(\lambda (_: (((leq g (match n1 with [O \Rightarrow (ASort O (next +g n0)) | (S h) \Rightarrow (ASort h n0)]) (ASort n1 n0)) \to P))).(\lambda +(H0: (leq g (ASort n1 n0) (ASort (S n1) n0))).(let H_x \def (leq_gen_sort1 g +n1 n0 (ASort (S n1) n0) H0) in (let H1 \def H_x in (ex2_3_ind nat nat nat +(\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort +n1 n0) k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: +nat).(\lambda (_: nat).(eq A (ASort (S n1) n0) (ASort h2 n2))))) P (\lambda +(x0: nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (H2: (eq A (aplus g +(ASort n1 n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H3: (eq A (ASort (S +n1) n0) (ASort x1 x0))).(let H4 \def (f_equal A nat (\lambda (e: A).(match e +in A return (\lambda (_: A).nat) with [(ASort n2 _) \Rightarrow n2 | (AHead _ +_) \Rightarrow (S n1)])) (ASort (S n1) n0) (ASort x1 x0) H3) in ((let H5 \def +(f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with +[(ASort _ n2) \Rightarrow n2 | (AHead _ _) \Rightarrow n0])) (ASort (S n1) +n0) (ASort x1 x0) H3) in (\lambda (H6: (eq nat (S n1) x1)).(let H7 \def +(eq_ind_r nat x1 (\lambda (n2: nat).(eq A (aplus g (ASort n1 n0) x2) (aplus g +(ASort n2 x0) x2))) H2 (S n1) H6) in (let H8 \def (eq_ind_r nat x0 (\lambda +(n2: nat).(eq A (aplus g (ASort n1 n0) x2) (aplus g (ASort (S n1) n2) x2))) +H7 n0 H5) in (let H9 \def (eq_ind_r A (aplus g (ASort n1 n0) x2) (\lambda +(a0: A).(eq A a0 (aplus g (ASort (S n1) n0) x2))) H8 (aplus g (ASort (S n1) +n0) (S x2)) (aplus_sort_S_S_simpl g n0 n1 x2)) in (let H_y \def (aplus_inj g +(S x2) x2 (ASort (S n1) n0) H9) in (le_Sx_x x2 (eq_ind_r nat x2 (\lambda (n2: +nat).(le n2 x2)) (le_n x2) (S x2) H_y) P))))))) H4))))))) H1)))))) n H))))) +(\lambda (a0: A).(\lambda (_: (((leq g (asucc g a0) a0) \to (\forall (P: +Prop).P)))).(\lambda (a1: A).(\lambda (H0: (((leq g (asucc g a1) a1) \to +(\forall (P: Prop).P)))).(\lambda (H1: (leq g (AHead a0 (asucc g a1)) (AHead +a0 a1))).(\lambda (P: Prop).(let H_x \def (leq_gen_head1 g a0 (asucc g a1) +(AHead a0 a1) H1) in (let H2 \def H_x in (ex3_2_ind A A (\lambda (a3: +A).(\lambda (_: A).(leq g a0 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g +(asucc g a1) a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (AHead a0 a1) +(AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (H3: (leq g a0 +x0)).(\lambda (H4: (leq g (asucc g a1) x1)).(\lambda (H5: (eq A (AHead a0 a1) +(AHead x0 x1))).(let H6 \def (f_equal A A (\lambda (e: A).(match e in A +return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead a2 _) +\Rightarrow a2])) (AHead a0 a1) (AHead x0 x1) H5) in ((let H7 \def (f_equal A +A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) +\Rightarrow a1 | (AHead _ a2) \Rightarrow a2])) (AHead a0 a1) (AHead x0 x1) +H5) in (\lambda (H8: (eq A a0 x0)).(let H9 \def (eq_ind_r A x1 (\lambda (a2: +A).(leq g (asucc g a1) a2)) H4 a1 H7) in (let H10 \def (eq_ind_r A x0 +(\lambda (a2: A).(leq g a0 a2)) H3 a0 H8) in (H0 H9 P))))) H6))))))) +H2))))))))) a)). +(* COMMENTS +Initial nodes: 1327 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/leq/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/leq/defs.ma new file mode 100644 index 000000000..5a5308280 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/leq/defs.ma @@ -0,0 +1,26 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/aplus/defs.ma". + +inductive leq (g: G): A \to (A \to Prop) \def +| leq_sort: \forall (h1: nat).(\forall (h2: nat).(\forall (n1: nat).(\forall +(n2: nat).(\forall (k: nat).((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort +h2 n2) k)) \to (leq g (ASort h1 n1) (ASort h2 n2))))))) +| leq_head: \forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall (a3: +A).(\forall (a4: A).((leq g a3 a4) \to (leq g (AHead a1 a3) (AHead a2 +a4))))))). + diff --git a/matita/matita/contribs/lambdadelta/basic_1/leq/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/leq/fwd.ma new file mode 100644 index 000000000..e259fa19e --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/leq/fwd.ma @@ -0,0 +1,244 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/leq/defs.ma". + +theorem leq_gen_sort1: + \forall (g: G).(\forall (h1: nat).(\forall (n1: nat).(\forall (a2: A).((leq +g (ASort h1 n1) a2) \to (ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: +nat).(\lambda (k: nat).(eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) +k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a2 +(ASort h2 n2)))))))))) +\def + \lambda (g: G).(\lambda (h1: nat).(\lambda (n1: nat).(\lambda (a2: +A).(\lambda (H: (leq g (ASort h1 n1) a2)).(insert_eq A (ASort h1 n1) (\lambda +(a: A).(leq g a a2)) (\lambda (a: A).(ex2_3 nat nat nat (\lambda (n2: +nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g a k) (aplus g (ASort +h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A +a2 (ASort h2 n2))))))) (\lambda (y: A).(\lambda (H0: (leq g y a2)).(leq_ind g +(\lambda (a: A).(\lambda (a0: A).((eq A a (ASort h1 n1)) \to (ex2_3 nat nat +nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g a +k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: +nat).(\lambda (_: nat).(eq A a0 (ASort h2 n2))))))))) (\lambda (h0: +nat).(\lambda (h2: nat).(\lambda (n0: nat).(\lambda (n2: nat).(\lambda (k: +nat).(\lambda (H1: (eq A (aplus g (ASort h0 n0) k) (aplus g (ASort h2 n2) +k))).(\lambda (H2: (eq A (ASort h0 n0) (ASort h1 n1))).(let H3 \def (f_equal +A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort +n _) \Rightarrow n | (AHead _ _) \Rightarrow h0])) (ASort h0 n0) (ASort h1 +n1) H2) in ((let H4 \def (f_equal A nat (\lambda (e: A).(match e in A return +(\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) +\Rightarrow n0])) (ASort h0 n0) (ASort h1 n1) H2) in (\lambda (H5: (eq nat h0 +h1)).(let H6 \def (eq_ind nat n0 (\lambda (n: nat).(eq A (aplus g (ASort h0 +n) k) (aplus g (ASort h2 n2) k))) H1 n1 H4) in (eq_ind_r nat n1 (\lambda (n: +nat).(ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (k0: +nat).(eq A (aplus g (ASort h0 n) k0) (aplus g (ASort h3 n3) k0))))) (\lambda +(n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A (ASort h2 n2) (ASort h3 +n3))))))) (let H7 \def (eq_ind nat h0 (\lambda (n: nat).(eq A (aplus g (ASort +n n1) k) (aplus g (ASort h2 n2) k))) H6 h1 H5) in (eq_ind_r nat h1 (\lambda +(n: nat).(ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda +(k0: nat).(eq A (aplus g (ASort n n1) k0) (aplus g (ASort h3 n3) k0))))) +(\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A (ASort h2 n2) +(ASort h3 n3))))))) (ex2_3_intro nat nat nat (\lambda (n3: nat).(\lambda (h3: +nat).(\lambda (k0: nat).(eq A (aplus g (ASort h1 n1) k0) (aplus g (ASort h3 +n3) k0))))) (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A +(ASort h2 n2) (ASort h3 n3))))) n2 h2 k H7 (refl_equal A (ASort h2 n2))) h0 +H5)) n0 H4)))) H3))))))))) (\lambda (a1: A).(\lambda (a3: A).(\lambda (_: +(leq g a1 a3)).(\lambda (_: (((eq A a1 (ASort h1 n1)) \to (ex2_3 nat nat nat +(\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g a1 k) +(aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda +(_: nat).(eq A a3 (ASort h2 n2))))))))).(\lambda (a4: A).(\lambda (a5: +A).(\lambda (_: (leq g a4 a5)).(\lambda (_: (((eq A a4 (ASort h1 n1)) \to +(ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: +nat).(eq A (aplus g a4 k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: +nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a5 (ASort h2 +n2))))))))).(\lambda (H5: (eq A (AHead a1 a4) (ASort h1 n1))).(let H6 \def +(eq_ind A (AHead a1 a4) (\lambda (ee: A).(match ee in A return (\lambda (_: +A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow +True])) I (ASort h1 n1) H5) in (False_ind (ex2_3 nat nat nat (\lambda (n2: +nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (AHead a1 a4) k) +(aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda +(_: nat).(eq A (AHead a3 a5) (ASort h2 n2)))))) H6))))))))))) y a2 H0))) +H))))). +(* COMMENTS +Initial nodes: 913 +END *) + +theorem leq_gen_head1: + \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a: A).((leq g +(AHead a1 a2) a) \to (ex3_2 A A (\lambda (a3: A).(\lambda (_: A).(leq g a1 +a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: +A).(\lambda (a4: A).(eq A a (AHead a3 a4))))))))) +\def + \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a: A).(\lambda +(H: (leq g (AHead a1 a2) a)).(insert_eq A (AHead a1 a2) (\lambda (a0: A).(leq +g a0 a)) (\lambda (_: A).(ex3_2 A A (\lambda (a3: A).(\lambda (_: A).(leq g +a1 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: +A).(\lambda (a4: A).(eq A a (AHead a3 a4)))))) (\lambda (y: A).(\lambda (H0: +(leq g y a)).(leq_ind g (\lambda (a0: A).(\lambda (a3: A).((eq A a0 (AHead a1 +a2)) \to (ex3_2 A A (\lambda (a4: A).(\lambda (_: A).(leq g a1 a4))) (\lambda +(_: A).(\lambda (a5: A).(leq g a2 a5))) (\lambda (a4: A).(\lambda (a5: A).(eq +A a3 (AHead a4 a5)))))))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: +nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (_: (eq A (aplus g (ASort +h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (H2: (eq A (ASort h1 n1) +(AHead a1 a2))).(let H3 \def (eq_ind A (ASort h1 n1) (\lambda (ee: A).(match +ee in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | +(AHead _ _) \Rightarrow False])) I (AHead a1 a2) H2) in (False_ind (ex3_2 A A +(\lambda (a3: A).(\lambda (_: A).(leq g a1 a3))) (\lambda (_: A).(\lambda +(a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort h2 n2) +(AHead a3 a4))))) H3))))))))) (\lambda (a0: A).(\lambda (a3: A).(\lambda (H1: +(leq g a0 a3)).(\lambda (H2: (((eq A a0 (AHead a1 a2)) \to (ex3_2 A A +(\lambda (a4: A).(\lambda (_: A).(leq g a1 a4))) (\lambda (_: A).(\lambda +(a5: A).(leq g a2 a5))) (\lambda (a4: A).(\lambda (a5: A).(eq A a3 (AHead a4 +a5)))))))).(\lambda (a4: A).(\lambda (a5: A).(\lambda (H3: (leq g a4 +a5)).(\lambda (H4: (((eq A a4 (AHead a1 a2)) \to (ex3_2 A A (\lambda (a6: +A).(\lambda (_: A).(leq g a1 a6))) (\lambda (_: A).(\lambda (a7: A).(leq g a2 +a7))) (\lambda (a6: A).(\lambda (a7: A).(eq A a5 (AHead a6 +a7)))))))).(\lambda (H5: (eq A (AHead a0 a4) (AHead a1 a2))).(let H6 \def +(f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with +[(ASort _ _) \Rightarrow a0 | (AHead a6 _) \Rightarrow a6])) (AHead a0 a4) +(AHead a1 a2) H5) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e in A +return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a4 | (AHead _ a6) +\Rightarrow a6])) (AHead a0 a4) (AHead a1 a2) H5) in (\lambda (H8: (eq A a0 +a1)).(let H9 \def (eq_ind A a4 (\lambda (a6: A).((eq A a6 (AHead a1 a2)) \to +(ex3_2 A A (\lambda (a7: A).(\lambda (_: A).(leq g a1 a7))) (\lambda (_: +A).(\lambda (a8: A).(leq g a2 a8))) (\lambda (a7: A).(\lambda (a8: A).(eq A +a5 (AHead a7 a8))))))) H4 a2 H7) in (let H10 \def (eq_ind A a4 (\lambda (a6: +A).(leq g a6 a5)) H3 a2 H7) in (let H11 \def (eq_ind A a0 (\lambda (a6: +A).((eq A a6 (AHead a1 a2)) \to (ex3_2 A A (\lambda (a7: A).(\lambda (_: +A).(leq g a1 a7))) (\lambda (_: A).(\lambda (a8: A).(leq g a2 a8))) (\lambda +(a7: A).(\lambda (a8: A).(eq A a3 (AHead a7 a8))))))) H2 a1 H8) in (let H12 +\def (eq_ind A a0 (\lambda (a6: A).(leq g a6 a3)) H1 a1 H8) in (ex3_2_intro A +A (\lambda (a6: A).(\lambda (_: A).(leq g a1 a6))) (\lambda (_: A).(\lambda +(a7: A).(leq g a2 a7))) (\lambda (a6: A).(\lambda (a7: A).(eq A (AHead a3 a5) +(AHead a6 a7)))) a3 a5 H12 H10 (refl_equal A (AHead a3 a5))))))))) +H6))))))))))) y a H0))) H))))). +(* COMMENTS +Initial nodes: 797 +END *) + +theorem leq_gen_sort2: + \forall (g: G).(\forall (h1: nat).(\forall (n1: nat).(\forall (a2: A).((leq +g a2 (ASort h1 n1)) \to (ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: +nat).(\lambda (k: nat).(eq A (aplus g (ASort h2 n2) k) (aplus g (ASort h1 n1) +k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a2 +(ASort h2 n2)))))))))) +\def + \lambda (g: G).(\lambda (h1: nat).(\lambda (n1: nat).(\lambda (a2: +A).(\lambda (H: (leq g a2 (ASort h1 n1))).(insert_eq A (ASort h1 n1) (\lambda +(a: A).(leq g a2 a)) (\lambda (a: A).(ex2_3 nat nat nat (\lambda (n2: +nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort h2 n2) k) +(aplus g a k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq +A a2 (ASort h2 n2))))))) (\lambda (y: A).(\lambda (H0: (leq g a2 y)).(leq_ind +g (\lambda (a: A).(\lambda (a0: A).((eq A a0 (ASort h1 n1)) \to (ex2_3 nat +nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus +g (ASort h2 n2) k) (aplus g a0 k))))) (\lambda (n2: nat).(\lambda (h2: +nat).(\lambda (_: nat).(eq A a (ASort h2 n2))))))))) (\lambda (h0: +nat).(\lambda (h2: nat).(\lambda (n0: nat).(\lambda (n2: nat).(\lambda (k: +nat).(\lambda (H1: (eq A (aplus g (ASort h0 n0) k) (aplus g (ASort h2 n2) +k))).(\lambda (H2: (eq A (ASort h2 n2) (ASort h1 n1))).(let H3 \def (f_equal +A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort +n _) \Rightarrow n | (AHead _ _) \Rightarrow h2])) (ASort h2 n2) (ASort h1 +n1) H2) in ((let H4 \def (f_equal A nat (\lambda (e: A).(match e in A return +(\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) +\Rightarrow n2])) (ASort h2 n2) (ASort h1 n1) H2) in (\lambda (H5: (eq nat h2 +h1)).(let H6 \def (eq_ind nat n2 (\lambda (n: nat).(eq A (aplus g (ASort h0 +n0) k) (aplus g (ASort h2 n) k))) H1 n1 H4) in (eq_ind_r nat n1 (\lambda (n: +nat).(ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (k0: +nat).(eq A (aplus g (ASort h3 n3) k0) (aplus g (ASort h2 n) k0))))) (\lambda +(n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A (ASort h0 n0) (ASort h3 +n3))))))) (let H7 \def (eq_ind nat h2 (\lambda (n: nat).(eq A (aplus g (ASort +h0 n0) k) (aplus g (ASort n n1) k))) H6 h1 H5) in (eq_ind_r nat h1 (\lambda +(n: nat).(ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda +(k0: nat).(eq A (aplus g (ASort h3 n3) k0) (aplus g (ASort n n1) k0))))) +(\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A (ASort h0 n0) +(ASort h3 n3))))))) (ex2_3_intro nat nat nat (\lambda (n3: nat).(\lambda (h3: +nat).(\lambda (k0: nat).(eq A (aplus g (ASort h3 n3) k0) (aplus g (ASort h1 +n1) k0))))) (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A +(ASort h0 n0) (ASort h3 n3))))) n0 h0 k H7 (refl_equal A (ASort h0 n0))) h2 +H5)) n2 H4)))) H3))))))))) (\lambda (a1: A).(\lambda (a3: A).(\lambda (_: +(leq g a1 a3)).(\lambda (_: (((eq A a3 (ASort h1 n1)) \to (ex2_3 nat nat nat +(\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort +h2 n2) k) (aplus g a3 k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda +(_: nat).(eq A a1 (ASort h2 n2))))))))).(\lambda (a4: A).(\lambda (a5: +A).(\lambda (_: (leq g a4 a5)).(\lambda (_: (((eq A a5 (ASort h1 n1)) \to +(ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: +nat).(eq A (aplus g (ASort h2 n2) k) (aplus g a5 k))))) (\lambda (n2: +nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a4 (ASort h2 +n2))))))))).(\lambda (H5: (eq A (AHead a3 a5) (ASort h1 n1))).(let H6 \def +(eq_ind A (AHead a3 a5) (\lambda (ee: A).(match ee in A return (\lambda (_: +A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow +True])) I (ASort h1 n1) H5) in (False_ind (ex2_3 nat nat nat (\lambda (n2: +nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort h2 n2) k) +(aplus g (AHead a3 a5) k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda +(_: nat).(eq A (AHead a1 a4) (ASort h2 n2)))))) H6))))))))))) a2 y H0))) +H))))). +(* COMMENTS +Initial nodes: 913 +END *) + +theorem leq_gen_head2: + \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a: A).((leq g a +(AHead a1 a2)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (_: A).(leq g a3 +a1))) (\lambda (_: A).(\lambda (a4: A).(leq g a4 a2))) (\lambda (a3: +A).(\lambda (a4: A).(eq A a (AHead a3 a4))))))))) +\def + \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a: A).(\lambda +(H: (leq g a (AHead a1 a2))).(insert_eq A (AHead a1 a2) (\lambda (a0: A).(leq +g a a0)) (\lambda (_: A).(ex3_2 A A (\lambda (a3: A).(\lambda (_: A).(leq g +a3 a1))) (\lambda (_: A).(\lambda (a4: A).(leq g a4 a2))) (\lambda (a3: +A).(\lambda (a4: A).(eq A a (AHead a3 a4)))))) (\lambda (y: A).(\lambda (H0: +(leq g a y)).(leq_ind g (\lambda (a0: A).(\lambda (a3: A).((eq A a3 (AHead a1 +a2)) \to (ex3_2 A A (\lambda (a4: A).(\lambda (_: A).(leq g a4 a1))) (\lambda +(_: A).(\lambda (a5: A).(leq g a5 a2))) (\lambda (a4: A).(\lambda (a5: A).(eq +A a0 (AHead a4 a5)))))))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: +nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (_: (eq A (aplus g (ASort +h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (H2: (eq A (ASort h2 n2) +(AHead a1 a2))).(let H3 \def (eq_ind A (ASort h2 n2) (\lambda (ee: A).(match +ee in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | +(AHead _ _) \Rightarrow False])) I (AHead a1 a2) H2) in (False_ind (ex3_2 A A +(\lambda (a3: A).(\lambda (_: A).(leq g a3 a1))) (\lambda (_: A).(\lambda +(a4: A).(leq g a4 a2))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort h1 n1) +(AHead a3 a4))))) H3))))))))) (\lambda (a0: A).(\lambda (a3: A).(\lambda (H1: +(leq g a0 a3)).(\lambda (H2: (((eq A a3 (AHead a1 a2)) \to (ex3_2 A A +(\lambda (a4: A).(\lambda (_: A).(leq g a4 a1))) (\lambda (_: A).(\lambda +(a5: A).(leq g a5 a2))) (\lambda (a4: A).(\lambda (a5: A).(eq A a0 (AHead a4 +a5)))))))).(\lambda (a4: A).(\lambda (a5: A).(\lambda (H3: (leq g a4 +a5)).(\lambda (H4: (((eq A a5 (AHead a1 a2)) \to (ex3_2 A A (\lambda (a6: +A).(\lambda (_: A).(leq g a6 a1))) (\lambda (_: A).(\lambda (a7: A).(leq g a7 +a2))) (\lambda (a6: A).(\lambda (a7: A).(eq A a4 (AHead a6 +a7)))))))).(\lambda (H5: (eq A (AHead a3 a5) (AHead a1 a2))).(let H6 \def +(f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with +[(ASort _ _) \Rightarrow a3 | (AHead a6 _) \Rightarrow a6])) (AHead a3 a5) +(AHead a1 a2) H5) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e in A +return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a5 | (AHead _ a6) +\Rightarrow a6])) (AHead a3 a5) (AHead a1 a2) H5) in (\lambda (H8: (eq A a3 +a1)).(let H9 \def (eq_ind A a5 (\lambda (a6: A).((eq A a6 (AHead a1 a2)) \to +(ex3_2 A A (\lambda (a7: A).(\lambda (_: A).(leq g a7 a1))) (\lambda (_: +A).(\lambda (a8: A).(leq g a8 a2))) (\lambda (a7: A).(\lambda (a8: A).(eq A +a4 (AHead a7 a8))))))) H4 a2 H7) in (let H10 \def (eq_ind A a5 (\lambda (a6: +A).(leq g a4 a6)) H3 a2 H7) in (let H11 \def (eq_ind A a3 (\lambda (a6: +A).((eq A a6 (AHead a1 a2)) \to (ex3_2 A A (\lambda (a7: A).(\lambda (_: +A).(leq g a7 a1))) (\lambda (_: A).(\lambda (a8: A).(leq g a8 a2))) (\lambda +(a7: A).(\lambda (a8: A).(eq A a0 (AHead a7 a8))))))) H2 a1 H8) in (let H12 +\def (eq_ind A a3 (\lambda (a6: A).(leq g a0 a6)) H1 a1 H8) in (ex3_2_intro A +A (\lambda (a6: A).(\lambda (_: A).(leq g a6 a1))) (\lambda (_: A).(\lambda +(a7: A).(leq g a7 a2))) (\lambda (a6: A).(\lambda (a7: A).(eq A (AHead a0 a4) +(AHead a6 a7)))) a0 a4 H12 H10 (refl_equal A (AHead a0 a4))))))))) +H6))))))))))) a y H0))) H))))). +(* COMMENTS +Initial nodes: 797 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/leq/props.ma b/matita/matita/contribs/lambdadelta/basic_1/leq/props.ma new file mode 100644 index 000000000..b83fc503e --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/leq/props.ma @@ -0,0 +1,233 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/leq/fwd.ma". + +include "Basic-1/aplus/props.ma". + +theorem ahead_inj_snd: + \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a3: A).(\forall +(a4: A).((leq g (AHead a1 a2) (AHead a3 a4)) \to (leq g a2 a4)))))) +\def + \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a3: A).(\lambda +(a4: A).(\lambda (H: (leq g (AHead a1 a2) (AHead a3 a4))).(let H_x \def +(leq_gen_head1 g a1 a2 (AHead a3 a4) H) in (let H0 \def H_x in (ex3_2_ind A A +(\lambda (a5: A).(\lambda (_: A).(leq g a1 a5))) (\lambda (_: A).(\lambda +(a6: A).(leq g a2 a6))) (\lambda (a5: A).(\lambda (a6: A).(eq A (AHead a3 a4) +(AHead a5 a6)))) (leq g a2 a4) (\lambda (x0: A).(\lambda (x1: A).(\lambda +(H1: (leq g a1 x0)).(\lambda (H2: (leq g a2 x1)).(\lambda (H3: (eq A (AHead +a3 a4) (AHead x0 x1))).(let H4 \def (f_equal A A (\lambda (e: A).(match e in +A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a3 | (AHead a _) +\Rightarrow a])) (AHead a3 a4) (AHead x0 x1) H3) in ((let H5 \def (f_equal A +A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) +\Rightarrow a4 | (AHead _ a) \Rightarrow a])) (AHead a3 a4) (AHead x0 x1) H3) +in (\lambda (H6: (eq A a3 x0)).(let H7 \def (eq_ind_r A x1 (\lambda (a: +A).(leq g a2 a)) H2 a4 H5) in (let H8 \def (eq_ind_r A x0 (\lambda (a: +A).(leq g a1 a)) H1 a3 H6) in H7)))) H4))))))) H0)))))))). +(* COMMENTS +Initial nodes: 259 +END *) + +theorem leq_refl: + \forall (g: G).(\forall (a: A).(leq g a a)) +\def + \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(leq g a0 a0)) +(\lambda (n: nat).(\lambda (n0: nat).(leq_sort g n n n0 n0 O (refl_equal A +(aplus g (ASort n n0) O))))) (\lambda (a0: A).(\lambda (H: (leq g a0 +a0)).(\lambda (a1: A).(\lambda (H0: (leq g a1 a1)).(leq_head g a0 a0 H a1 a1 +H0))))) a)). +(* COMMENTS +Initial nodes: 87 +END *) + +theorem leq_eq: + \forall (g: G).(\forall (a1: A).(\forall (a2: A).((eq A a1 a2) \to (leq g a1 +a2)))) +\def + \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (eq A a1 +a2)).(eq_ind A a1 (\lambda (a: A).(leq g a1 a)) (leq_refl g a1) a2 H)))). +(* COMMENTS +Initial nodes: 39 +END *) + +theorem leq_sym: + \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (leq g +a2 a1)))) +\def + \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1 +a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(leq g a0 a))) (\lambda (h1: +nat).(\lambda (h2: nat).(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (k: +nat).(\lambda (H0: (eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) +k))).(leq_sort g h2 h1 n2 n1 k (sym_eq A (aplus g (ASort h1 n1) k) (aplus g +(ASort h2 n2) k) H0)))))))) (\lambda (a3: A).(\lambda (a4: A).(\lambda (_: +(leq g a3 a4)).(\lambda (H1: (leq g a4 a3)).(\lambda (a5: A).(\lambda (a6: +A).(\lambda (_: (leq g a5 a6)).(\lambda (H3: (leq g a6 a5)).(leq_head g a4 a3 +H1 a6 a5 H3))))))))) a1 a2 H)))). +(* COMMENTS +Initial nodes: 173 +END *) + +theorem leq_trans: + \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall +(a3: A).((leq g a2 a3) \to (leq g a1 a3)))))) +\def + \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1 +a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(\forall (a3: A).((leq g a0 +a3) \to (leq g a a3))))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: +nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (H0: (eq A (aplus g (ASort +h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (a3: A).(\lambda (H1: (leq g +(ASort h2 n2) a3)).(let H_x \def (leq_gen_sort1 g h2 n2 a3 H1) in (let H2 +\def H_x in (ex2_3_ind nat nat nat (\lambda (n3: nat).(\lambda (h3: +nat).(\lambda (k0: nat).(eq A (aplus g (ASort h2 n2) k0) (aplus g (ASort h3 +n3) k0))))) (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A a3 +(ASort h3 n3))))) (leq g (ASort h1 n1) a3) (\lambda (x0: nat).(\lambda (x1: +nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort h2 n2) x2) (aplus +g (ASort x1 x0) x2))).(\lambda (H4: (eq A a3 (ASort x1 x0))).(let H5 \def +(f_equal A A (\lambda (e: A).e) a3 (ASort x1 x0) H4) in (eq_ind_r A (ASort x1 +x0) (\lambda (a: A).(leq g (ASort h1 n1) a)) (lt_le_e k x2 (leq g (ASort h1 +n1) (ASort x1 x0)) (\lambda (H6: (lt k x2)).(let H_y \def (aplus_reg_r g +(ASort h1 n1) (ASort h2 n2) k k H0 (minus x2 k)) in (let H7 \def (eq_ind_r +nat (plus (minus x2 k) k) (\lambda (n: nat).(eq A (aplus g (ASort h1 n1) n) +(aplus g (ASort h2 n2) n))) H_y x2 (le_plus_minus_sym k x2 (le_trans k (S k) +x2 (le_S k k (le_n k)) H6))) in (leq_sort g h1 x1 n1 x0 x2 (trans_eq A (aplus +g (ASort h1 n1) x2) (aplus g (ASort h2 n2) x2) (aplus g (ASort x1 x0) x2) H7 +H3))))) (\lambda (H6: (le x2 k)).(let H_y \def (aplus_reg_r g (ASort h2 n2) +(ASort x1 x0) x2 x2 H3 (minus k x2)) in (let H7 \def (eq_ind_r nat (plus +(minus k x2) x2) (\lambda (n: nat).(eq A (aplus g (ASort h2 n2) n) (aplus g +(ASort x1 x0) n))) H_y k (le_plus_minus_sym x2 k H6)) in (leq_sort g h1 x1 n1 +x0 k (trans_eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k) (aplus g +(ASort x1 x0) k) H0 H7)))))) a3 H5))))))) H2))))))))))) (\lambda (a3: +A).(\lambda (a4: A).(\lambda (_: (leq g a3 a4)).(\lambda (H1: ((\forall (a5: +A).((leq g a4 a5) \to (leq g a3 a5))))).(\lambda (a5: A).(\lambda (a6: +A).(\lambda (_: (leq g a5 a6)).(\lambda (H3: ((\forall (a7: A).((leq g a6 a7) +\to (leq g a5 a7))))).(\lambda (a0: A).(\lambda (H4: (leq g (AHead a4 a6) +a0)).(let H_x \def (leq_gen_head1 g a4 a6 a0 H4) in (let H5 \def H_x in +(ex3_2_ind A A (\lambda (a7: A).(\lambda (_: A).(leq g a4 a7))) (\lambda (_: +A).(\lambda (a8: A).(leq g a6 a8))) (\lambda (a7: A).(\lambda (a8: A).(eq A +a0 (AHead a7 a8)))) (leq g (AHead a3 a5) a0) (\lambda (x0: A).(\lambda (x1: +A).(\lambda (H6: (leq g a4 x0)).(\lambda (H7: (leq g a6 x1)).(\lambda (H8: +(eq A a0 (AHead x0 x1))).(let H9 \def (f_equal A A (\lambda (e: A).e) a0 +(AHead x0 x1) H8) in (eq_ind_r A (AHead x0 x1) (\lambda (a: A).(leq g (AHead +a3 a5) a)) (leq_head g a3 x0 (H1 x0 H6) a5 x1 (H3 x1 H7)) a0 H9))))))) +H5))))))))))))) a1 a2 H)))). +(* COMMENTS +Initial nodes: 869 +END *) + +theorem leq_ahead_false_1: + \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (AHead a1 a2) a1) +\to (\forall (P: Prop).P)))) +\def + \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2: +A).((leq g (AHead a a2) a) \to (\forall (P: Prop).P)))) (\lambda (n: +nat).(\lambda (n0: nat).(\lambda (a2: A).(\lambda (H: (leq g (AHead (ASort n +n0) a2) (ASort n n0))).(\lambda (P: Prop).(nat_ind (\lambda (n1: nat).((leq g +(AHead (ASort n1 n0) a2) (ASort n1 n0)) \to P)) (\lambda (H0: (leq g (AHead +(ASort O n0) a2) (ASort O n0))).(let H_x \def (leq_gen_head1 g (ASort O n0) +a2 (ASort O n0) H0) in (let H1 \def H_x in (ex3_2_ind A A (\lambda (a3: +A).(\lambda (_: A).(leq g (ASort O n0) a3))) (\lambda (_: A).(\lambda (a4: +A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort O n0) +(AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g +(ASort O n0) x0)).(\lambda (_: (leq g a2 x1)).(\lambda (H4: (eq A (ASort O +n0) (AHead x0 x1))).(let H5 \def (eq_ind A (ASort O n0) (\lambda (ee: +A).(match ee in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow +True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H4) in (False_ind P +H5))))))) H1)))) (\lambda (n1: nat).(\lambda (_: (((leq g (AHead (ASort n1 +n0) a2) (ASort n1 n0)) \to P))).(\lambda (H0: (leq g (AHead (ASort (S n1) n0) +a2) (ASort (S n1) n0))).(let H_x \def (leq_gen_head1 g (ASort (S n1) n0) a2 +(ASort (S n1) n0) H0) in (let H1 \def H_x in (ex3_2_ind A A (\lambda (a3: +A).(\lambda (_: A).(leq g (ASort (S n1) n0) a3))) (\lambda (_: A).(\lambda +(a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort (S n1) +n0) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g +(ASort (S n1) n0) x0)).(\lambda (_: (leq g a2 x1)).(\lambda (H4: (eq A (ASort +(S n1) n0) (AHead x0 x1))).(let H5 \def (eq_ind A (ASort (S n1) n0) (\lambda +(ee: A).(match ee in A return (\lambda (_: A).Prop) with [(ASort _ _) +\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H4) in +(False_ind P H5))))))) H1)))))) n H)))))) (\lambda (a: A).(\lambda (H: +((\forall (a2: A).((leq g (AHead a a2) a) \to (\forall (P: +Prop).P))))).(\lambda (a0: A).(\lambda (_: ((\forall (a2: A).((leq g (AHead +a0 a2) a0) \to (\forall (P: Prop).P))))).(\lambda (a2: A).(\lambda (H1: (leq +g (AHead (AHead a a0) a2) (AHead a a0))).(\lambda (P: Prop).(let H_x \def +(leq_gen_head1 g (AHead a a0) a2 (AHead a a0) H1) in (let H2 \def H_x in +(ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq g (AHead a a0) a3))) +(\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda +(a4: A).(eq A (AHead a a0) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: +A).(\lambda (H3: (leq g (AHead a a0) x0)).(\lambda (H4: (leq g a2 +x1)).(\lambda (H5: (eq A (AHead a a0) (AHead x0 x1))).(let H6 \def (f_equal A +A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) +\Rightarrow a | (AHead a3 _) \Rightarrow a3])) (AHead a a0) (AHead x0 x1) H5) +in ((let H7 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda +(_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead _ a3) \Rightarrow a3])) +(AHead a a0) (AHead x0 x1) H5) in (\lambda (H8: (eq A a x0)).(let H9 \def +(eq_ind_r A x1 (\lambda (a3: A).(leq g a2 a3)) H4 a0 H7) in (let H10 \def +(eq_ind_r A x0 (\lambda (a3: A).(leq g (AHead a a0) a3)) H3 a H8) in (H a0 +H10 P))))) H6))))))) H2)))))))))) a1)). +(* COMMENTS +Initial nodes: 797 +END *) + +theorem leq_ahead_false_2: + \forall (g: G).(\forall (a2: A).(\forall (a1: A).((leq g (AHead a1 a2) a2) +\to (\forall (P: Prop).P)))) +\def + \lambda (g: G).(\lambda (a2: A).(A_ind (\lambda (a: A).(\forall (a1: +A).((leq g (AHead a1 a) a) \to (\forall (P: Prop).P)))) (\lambda (n: +nat).(\lambda (n0: nat).(\lambda (a1: A).(\lambda (H: (leq g (AHead a1 (ASort +n n0)) (ASort n n0))).(\lambda (P: Prop).(nat_ind (\lambda (n1: nat).((leq g +(AHead a1 (ASort n1 n0)) (ASort n1 n0)) \to P)) (\lambda (H0: (leq g (AHead +a1 (ASort O n0)) (ASort O n0))).(let H_x \def (leq_gen_head1 g a1 (ASort O +n0) (ASort O n0) H0) in (let H1 \def H_x in (ex3_2_ind A A (\lambda (a3: +A).(\lambda (_: A).(leq g a1 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g +(ASort O n0) a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort O n0) +(AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g a1 +x0)).(\lambda (_: (leq g (ASort O n0) x1)).(\lambda (H4: (eq A (ASort O n0) +(AHead x0 x1))).(let H5 \def (eq_ind A (ASort O n0) (\lambda (ee: A).(match +ee in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | +(AHead _ _) \Rightarrow False])) I (AHead x0 x1) H4) in (False_ind P +H5))))))) H1)))) (\lambda (n1: nat).(\lambda (_: (((leq g (AHead a1 (ASort n1 +n0)) (ASort n1 n0)) \to P))).(\lambda (H0: (leq g (AHead a1 (ASort (S n1) +n0)) (ASort (S n1) n0))).(let H_x \def (leq_gen_head1 g a1 (ASort (S n1) n0) +(ASort (S n1) n0) H0) in (let H1 \def H_x in (ex3_2_ind A A (\lambda (a3: +A).(\lambda (_: A).(leq g a1 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g +(ASort (S n1) n0) a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort (S n1) +n0) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g +a1 x0)).(\lambda (_: (leq g (ASort (S n1) n0) x1)).(\lambda (H4: (eq A (ASort +(S n1) n0) (AHead x0 x1))).(let H5 \def (eq_ind A (ASort (S n1) n0) (\lambda +(ee: A).(match ee in A return (\lambda (_: A).Prop) with [(ASort _ _) +\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H4) in +(False_ind P H5))))))) H1)))))) n H)))))) (\lambda (a: A).(\lambda (_: +((\forall (a1: A).((leq g (AHead a1 a) a) \to (\forall (P: +Prop).P))))).(\lambda (a0: A).(\lambda (H0: ((\forall (a1: A).((leq g (AHead +a1 a0) a0) \to (\forall (P: Prop).P))))).(\lambda (a1: A).(\lambda (H1: (leq +g (AHead a1 (AHead a a0)) (AHead a a0))).(\lambda (P: Prop).(let H_x \def +(leq_gen_head1 g a1 (AHead a a0) (AHead a a0) H1) in (let H2 \def H_x in +(ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq g a1 a3))) (\lambda (_: +A).(\lambda (a4: A).(leq g (AHead a a0) a4))) (\lambda (a3: A).(\lambda (a4: +A).(eq A (AHead a a0) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: +A).(\lambda (H3: (leq g a1 x0)).(\lambda (H4: (leq g (AHead a a0) +x1)).(\lambda (H5: (eq A (AHead a a0) (AHead x0 x1))).(let H6 \def (f_equal A +A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) +\Rightarrow a | (AHead a3 _) \Rightarrow a3])) (AHead a a0) (AHead x0 x1) H5) +in ((let H7 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda +(_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead _ a3) \Rightarrow a3])) +(AHead a a0) (AHead x0 x1) H5) in (\lambda (H8: (eq A a x0)).(let H9 \def +(eq_ind_r A x1 (\lambda (a3: A).(leq g (AHead a a0) a3)) H4 a0 H7) in (let +H10 \def (eq_ind_r A x0 (\lambda (a3: A).(leq g a1 a3)) H3 a H8) in (H0 a H9 +P))))) H6))))))) H2)))))))))) a2)). +(* COMMENTS +Initial nodes: 797 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/lift/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/lift/defs.ma new file mode 100644 index 000000000..046506672 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/lift/defs.ma @@ -0,0 +1,42 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/tlist/defs.ma". + +include "Basic-1/s/defs.ma". + +definition lref_map: + ((nat \to nat)) \to (nat \to (T \to T)) +\def + let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t +with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match +(blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u +t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in +lref_map. + +definition lift: + nat \to (nat \to (T \to T)) +\def + \lambda (h: nat).(\lambda (i: nat).(\lambda (t: T).(lref_map (\lambda (x: +nat).(plus x h)) i t))). + +definition lifts: + nat \to (nat \to (TList \to TList)) +\def + let rec lifts (h: nat) (d: nat) (ts: TList) on ts: TList \def (match ts with +[TNil \Rightarrow TNil | (TCons t ts0) \Rightarrow (TCons (lift h d t) (lifts +h d ts0))]) in lifts. + diff --git a/matita/matita/contribs/lambdadelta/basic_1/lift/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/lift/fwd.ma new file mode 100644 index 000000000..324fed2fb --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/lift/fwd.ma @@ -0,0 +1,434 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/lift/defs.ma". + +theorem lift_sort: + \forall (n: nat).(\forall (h: nat).(\forall (d: nat).(eq T (lift h d (TSort +n)) (TSort n)))) +\def + \lambda (n: nat).(\lambda (_: nat).(\lambda (_: nat).(refl_equal T (TSort +n)))). +(* COMMENTS +Initial nodes: 13 +END *) + +theorem lift_lref_lt: + \forall (n: nat).(\forall (h: nat).(\forall (d: nat).((lt n d) \to (eq T +(lift h d (TLRef n)) (TLRef n))))) +\def + \lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (lt n +d)).(eq_ind bool true (\lambda (b: bool).(eq T (TLRef (match b with [true +\Rightarrow n | false \Rightarrow (plus n h)])) (TLRef n))) (refl_equal T +(TLRef n)) (blt n d) (sym_eq bool (blt n d) true (lt_blt d n H)))))). +(* COMMENTS +Initial nodes: 72 +END *) + +theorem lift_lref_ge: + \forall (n: nat).(\forall (h: nat).(\forall (d: nat).((le d n) \to (eq T +(lift h d (TLRef n)) (TLRef (plus n h)))))) +\def + \lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (le d +n)).(eq_ind bool false (\lambda (b: bool).(eq T (TLRef (match b with [true +\Rightarrow n | false \Rightarrow (plus n h)])) (TLRef (plus n h)))) +(refl_equal T (TLRef (plus n h))) (blt n d) (sym_eq bool (blt n d) false +(le_bge d n H)))))). +(* COMMENTS +Initial nodes: 80 +END *) + +theorem lift_head: + \forall (k: K).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall +(d: nat).(eq T (lift h d (THead k u t)) (THead k (lift h d u) (lift h (s k d) +t))))))) +\def + \lambda (k: K).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda +(d: nat).(refl_equal T (THead k (lift h d u) (lift h (s k d) t))))))). +(* COMMENTS +Initial nodes: 37 +END *) + +theorem lift_bind: + \forall (b: B).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall +(d: nat).(eq T (lift h d (THead (Bind b) u t)) (THead (Bind b) (lift h d u) +(lift h (S d) t))))))) +\def + \lambda (b: B).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda +(d: nat).(refl_equal T (THead (Bind b) (lift h d u) (lift h (S d) t))))))). +(* COMMENTS +Initial nodes: 37 +END *) + +theorem lift_flat: + \forall (f: F).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall +(d: nat).(eq T (lift h d (THead (Flat f) u t)) (THead (Flat f) (lift h d u) +(lift h d t))))))) +\def + \lambda (f: F).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda +(d: nat).(refl_equal T (THead (Flat f) (lift h d u) (lift h d t))))))). +(* COMMENTS +Initial nodes: 35 +END *) + +theorem lift_gen_sort: + \forall (h: nat).(\forall (d: nat).(\forall (n: nat).(\forall (t: T).((eq T +(TSort n) (lift h d t)) \to (eq T t (TSort n)))))) +\def + \lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (t: T).(T_ind +(\lambda (t0: T).((eq T (TSort n) (lift h d t0)) \to (eq T t0 (TSort n)))) +(\lambda (n0: nat).(\lambda (H: (eq T (TSort n) (lift h d (TSort +n0)))).(sym_eq T (TSort n) (TSort n0) H))) (\lambda (n0: nat).(\lambda (H: +(eq T (TSort n) (lift h d (TLRef n0)))).(lt_le_e n0 d (eq T (TLRef n0) (TSort +n)) (\lambda (_: (lt n0 d)).(let H1 \def (eq_ind T (lift h d (TLRef n0)) +(\lambda (t0: T).(eq T (TSort n) t0)) H (TLRef n0) (lift_lref_lt n0 h d (let +H1 \def (eq_ind T (TSort n) (\lambda (ee: T).(match ee in T return (\lambda +(_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | +(THead _ _ _) \Rightarrow False])) I (lift h d (TLRef n0)) H) in (False_ind +(lt n0 d) H1)))) in (let H2 \def (eq_ind T (TSort n) (\lambda (ee: T).(match +ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | +(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef n0) +H1) in (False_ind (eq T (TLRef n0) (TSort n)) H2)))) (\lambda (_: (le d +n0)).(let H1 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t0: T).(eq T +(TSort n) t0)) H (TLRef (plus n0 h)) (lift_lref_ge n0 h d (let H1 \def +(eq_ind T (TSort n) (\lambda (ee: T).(match ee in T return (\lambda (_: +T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | +(THead _ _ _) \Rightarrow False])) I (lift h d (TLRef n0)) H) in (False_ind +(le d n0) H1)))) in (let H2 \def (eq_ind T (TSort n) (\lambda (ee: T).(match +ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | +(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef +(plus n0 h)) H1) in (False_ind (eq T (TLRef n0) (TSort n)) H2))))))) (\lambda +(k: K).(\lambda (t0: T).(\lambda (_: (((eq T (TSort n) (lift h d t0)) \to (eq +T t0 (TSort n))))).(\lambda (t1: T).(\lambda (_: (((eq T (TSort n) (lift h d +t1)) \to (eq T t1 (TSort n))))).(\lambda (H1: (eq T (TSort n) (lift h d +(THead k t0 t1)))).(let H2 \def (eq_ind T (lift h d (THead k t0 t1)) (\lambda +(t2: T).(eq T (TSort n) t2)) H1 (THead k (lift h d t0) (lift h (s k d) t1)) +(lift_head k t0 t1 h d)) in (let H3 \def (eq_ind T (TSort n) (\lambda (ee: +T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow +True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I +(THead k (lift h d t0) (lift h (s k d) t1)) H2) in (False_ind (eq T (THead k +t0 t1) (TSort n)) H3))))))))) t)))). +(* COMMENTS +Initial nodes: 613 +END *) + +theorem lift_gen_lref: + \forall (t: T).(\forall (d: nat).(\forall (h: nat).(\forall (i: nat).((eq T +(TLRef i) (lift h d t)) \to (or (land (lt i d) (eq T t (TLRef i))) (land (le +(plus d h) i) (eq T t (TLRef (minus i h))))))))) +\def + \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d: nat).(\forall (h: +nat).(\forall (i: nat).((eq T (TLRef i) (lift h d t0)) \to (or (land (lt i d) +(eq T t0 (TLRef i))) (land (le (plus d h) i) (eq T t0 (TLRef (minus i +h)))))))))) (\lambda (n: nat).(\lambda (d: nat).(\lambda (h: nat).(\lambda +(i: nat).(\lambda (H: (eq T (TLRef i) (lift h d (TSort n)))).(let H0 \def +(eq_ind T (lift h d (TSort n)) (\lambda (t0: T).(eq T (TLRef i) t0)) H (TSort +n) (lift_sort n h d)) in (let H1 \def (eq_ind T (TLRef i) (\lambda (ee: +T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow +False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I +(TSort n) H0) in (False_ind (or (land (lt i d) (eq T (TSort n) (TLRef i))) +(land (le (plus d h) i) (eq T (TSort n) (TLRef (minus i h))))) H1)))))))) +(\lambda (n: nat).(\lambda (d: nat).(\lambda (h: nat).(\lambda (i: +nat).(\lambda (H: (eq T (TLRef i) (lift h d (TLRef n)))).(lt_le_e n d (or +(land (lt i d) (eq T (TLRef n) (TLRef i))) (land (le (plus d h) i) (eq T +(TLRef n) (TLRef (minus i h))))) (\lambda (H0: (lt n d)).(let H1 \def (eq_ind +T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (TLRef i) t0)) H (TLRef n) +(lift_lref_lt n h d H0)) in (let H2 \def (f_equal T nat (\lambda (e: +T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow i | +(TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow i])) (TLRef i) (TLRef +n) H1) in (eq_ind_r nat n (\lambda (n0: nat).(or (land (lt n0 d) (eq T (TLRef +n) (TLRef n0))) (land (le (plus d h) n0) (eq T (TLRef n) (TLRef (minus n0 +h)))))) (or_introl (land (lt n d) (eq T (TLRef n) (TLRef n))) (land (le (plus +d h) n) (eq T (TLRef n) (TLRef (minus n h)))) (conj (lt n d) (eq T (TLRef n) +(TLRef n)) H0 (refl_equal T (TLRef n)))) i H2)))) (\lambda (H0: (le d +n)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (TLRef +i) t0)) H (TLRef (plus n h)) (lift_lref_ge n h d H0)) in (let H2 \def +(f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with +[(TSort _) \Rightarrow i | (TLRef n0) \Rightarrow n0 | (THead _ _ _) +\Rightarrow i])) (TLRef i) (TLRef (plus n h)) H1) in (eq_ind_r nat (plus n h) +(\lambda (n0: nat).(or (land (lt n0 d) (eq T (TLRef n) (TLRef n0))) (land (le +(plus d h) n0) (eq T (TLRef n) (TLRef (minus n0 h)))))) (eq_ind_r nat n +(\lambda (n0: nat).(or (land (lt (plus n h) d) (eq T (TLRef n) (TLRef (plus n +h)))) (land (le (plus d h) (plus n h)) (eq T (TLRef n) (TLRef n0))))) +(or_intror (land (lt (plus n h) d) (eq T (TLRef n) (TLRef (plus n h)))) (land +(le (plus d h) (plus n h)) (eq T (TLRef n) (TLRef n))) (conj (le (plus d h) +(plus n h)) (eq T (TLRef n) (TLRef n)) (le_plus_plus d n h h H0 (le_n h)) +(refl_equal T (TLRef n)))) (minus (plus n h) h) (minus_plus_r n h)) i +H2)))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (_: ((\forall (d: +nat).(\forall (h: nat).(\forall (i: nat).((eq T (TLRef i) (lift h d t0)) \to +(or (land (lt i d) (eq T t0 (TLRef i))) (land (le (plus d h) i) (eq T t0 +(TLRef (minus i h))))))))))).(\lambda (t1: T).(\lambda (_: ((\forall (d: +nat).(\forall (h: nat).(\forall (i: nat).((eq T (TLRef i) (lift h d t1)) \to +(or (land (lt i d) (eq T t1 (TLRef i))) (land (le (plus d h) i) (eq T t1 +(TLRef (minus i h))))))))))).(\lambda (d: nat).(\lambda (h: nat).(\lambda (i: +nat).(\lambda (H1: (eq T (TLRef i) (lift h d (THead k t0 t1)))).(let H2 \def +(eq_ind T (lift h d (THead k t0 t1)) (\lambda (t2: T).(eq T (TLRef i) t2)) H1 +(THead k (lift h d t0) (lift h (s k d) t1)) (lift_head k t0 t1 h d)) in (let +H3 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda +(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | +(THead _ _ _) \Rightarrow False])) I (THead k (lift h d t0) (lift h (s k d) +t1)) H2) in (False_ind (or (land (lt i d) (eq T (THead k t0 t1) (TLRef i))) +(land (le (plus d h) i) (eq T (THead k t0 t1) (TLRef (minus i h))))) +H3)))))))))))) t). +(* COMMENTS +Initial nodes: 1221 +END *) + +theorem lift_gen_lref_lt: + \forall (h: nat).(\forall (d: nat).(\forall (n: nat).((lt n d) \to (\forall +(t: T).((eq T (TLRef n) (lift h d t)) \to (eq T t (TLRef n))))))) +\def + \lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (H: (lt n +d)).(\lambda (t: T).(\lambda (H0: (eq T (TLRef n) (lift h d t))).(let H_x +\def (lift_gen_lref t d h n H0) in (let H1 \def H_x in (or_ind (land (lt n d) +(eq T t (TLRef n))) (land (le (plus d h) n) (eq T t (TLRef (minus n h)))) (eq +T t (TLRef n)) (\lambda (H2: (land (lt n d) (eq T t (TLRef n)))).(land_ind +(lt n d) (eq T t (TLRef n)) (eq T t (TLRef n)) (\lambda (_: (lt n +d)).(\lambda (H4: (eq T t (TLRef n))).(eq_ind_r T (TLRef n) (\lambda (t0: +T).(eq T t0 (TLRef n))) (refl_equal T (TLRef n)) t H4))) H2)) (\lambda (H2: +(land (le (plus d h) n) (eq T t (TLRef (minus n h))))).(land_ind (le (plus d +h) n) (eq T t (TLRef (minus n h))) (eq T t (TLRef n)) (\lambda (H3: (le (plus +d h) n)).(\lambda (H4: (eq T t (TLRef (minus n h)))).(eq_ind_r T (TLRef +(minus n h)) (\lambda (t0: T).(eq T t0 (TLRef n))) (le_false (plus d h) n (eq +T (TLRef (minus n h)) (TLRef n)) H3 (lt_le_S n (plus d h) (le_plus_trans (S +n) d h H))) t H4))) H2)) H1)))))))). +(* COMMENTS +Initial nodes: 363 +END *) + +theorem lift_gen_lref_false: + \forall (h: nat).(\forall (d: nat).(\forall (n: nat).((le d n) \to ((lt n +(plus d h)) \to (\forall (t: T).((eq T (TLRef n) (lift h d t)) \to (\forall +(P: Prop).P))))))) +\def + \lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (H: (le d +n)).(\lambda (H0: (lt n (plus d h))).(\lambda (t: T).(\lambda (H1: (eq T +(TLRef n) (lift h d t))).(\lambda (P: Prop).(let H_x \def (lift_gen_lref t d +h n H1) in (let H2 \def H_x in (or_ind (land (lt n d) (eq T t (TLRef n))) +(land (le (plus d h) n) (eq T t (TLRef (minus n h)))) P (\lambda (H3: (land +(lt n d) (eq T t (TLRef n)))).(land_ind (lt n d) (eq T t (TLRef n)) P +(\lambda (H4: (lt n d)).(\lambda (_: (eq T t (TLRef n))).(le_false d n P H +H4))) H3)) (\lambda (H3: (land (le (plus d h) n) (eq T t (TLRef (minus n +h))))).(land_ind (le (plus d h) n) (eq T t (TLRef (minus n h))) P (\lambda +(H4: (le (plus d h) n)).(\lambda (_: (eq T t (TLRef (minus n h)))).(le_false +(plus d h) n P H4 H0))) H3)) H2)))))))))). +(* COMMENTS +Initial nodes: 269 +END *) + +theorem lift_gen_lref_ge: + \forall (h: nat).(\forall (d: nat).(\forall (n: nat).((le d n) \to (\forall +(t: T).((eq T (TLRef (plus n h)) (lift h d t)) \to (eq T t (TLRef n))))))) +\def + \lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (H: (le d +n)).(\lambda (t: T).(\lambda (H0: (eq T (TLRef (plus n h)) (lift h d +t))).(let H_x \def (lift_gen_lref t d h (plus n h) H0) in (let H1 \def H_x in +(or_ind (land (lt (plus n h) d) (eq T t (TLRef (plus n h)))) (land (le (plus +d h) (plus n h)) (eq T t (TLRef (minus (plus n h) h)))) (eq T t (TLRef n)) +(\lambda (H2: (land (lt (plus n h) d) (eq T t (TLRef (plus n h))))).(land_ind +(lt (plus n h) d) (eq T t (TLRef (plus n h))) (eq T t (TLRef n)) (\lambda +(H3: (lt (plus n h) d)).(\lambda (H4: (eq T t (TLRef (plus n h)))).(eq_ind_r +T (TLRef (plus n h)) (\lambda (t0: T).(eq T t0 (TLRef n))) (le_false d n (eq +T (TLRef (plus n h)) (TLRef n)) H (lt_le_S n d (simpl_lt_plus_r h n d +(lt_le_trans (plus n h) d (plus d h) H3 (le_plus_l d h))))) t H4))) H2)) +(\lambda (H2: (land (le (plus d h) (plus n h)) (eq T t (TLRef (minus (plus n +h) h))))).(land_ind (le (plus d h) (plus n h)) (eq T t (TLRef (minus (plus n +h) h))) (eq T t (TLRef n)) (\lambda (_: (le (plus d h) (plus n h))).(\lambda +(H4: (eq T t (TLRef (minus (plus n h) h)))).(eq_ind_r T (TLRef (minus (plus n +h) h)) (\lambda (t0: T).(eq T t0 (TLRef n))) (f_equal nat T TLRef (minus +(plus n h) h) n (minus_plus_r n h)) t H4))) H2)) H1)))))))). +(* COMMENTS +Initial nodes: 473 +END *) + +theorem lift_gen_head: + \forall (k: K).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h: +nat).(\forall (d: nat).((eq T (THead k u t) (lift h d x)) \to (ex3_2 T T +(\lambda (y: T).(\lambda (z: T).(eq T x (THead k y z)))) (\lambda (y: +T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: +T).(eq T t (lift h (s k d) z))))))))))) +\def + \lambda (k: K).(\lambda (u: T).(\lambda (t: T).(\lambda (x: T).(T_ind +(\lambda (t0: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k u t) +(lift h d t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead +k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda +(_: T).(\lambda (z: T).(eq T t (lift h (s k d) z))))))))) (\lambda (n: +nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (eq T (THead k u t) +(lift h d (TSort n)))).(let H0 \def (eq_ind T (lift h d (TSort n)) (\lambda +(t0: T).(eq T (THead k u t) t0)) H (TSort n) (lift_sort n h d)) in (let H1 +\def (eq_ind T (THead k u t) (\lambda (ee: T).(match ee in T return (\lambda +(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False +| (THead _ _ _) \Rightarrow True])) I (TSort n) H0) in (False_ind (ex3_2 T T +(\lambda (y: T).(\lambda (z: T).(eq T (TSort n) (THead k y z)))) (\lambda (y: +T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: +T).(eq T t (lift h (s k d) z))))) H1))))))) (\lambda (n: nat).(\lambda (h: +nat).(\lambda (d: nat).(\lambda (H: (eq T (THead k u t) (lift h d (TLRef +n)))).(lt_le_e n d (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) +(THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) +(\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k d) z))))) (\lambda (H0: +(lt n d)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T +(THead k u t) t0)) H (TLRef n) (lift_lref_lt n h d H0)) in (let H2 \def +(eq_ind T (THead k u t) (\lambda (ee: T).(match ee in T return (\lambda (_: +T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | +(THead _ _ _) \Rightarrow True])) I (TLRef n) H1) in (False_ind (ex3_2 T T +(\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead k y z)))) (\lambda (y: +T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: +T).(eq T t (lift h (s k d) z))))) H2)))) (\lambda (H0: (le d n)).(let H1 \def +(eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (THead k u t) t0)) H +(TLRef (plus n h)) (lift_lref_ge n h d H0)) in (let H2 \def (eq_ind T (THead +k u t) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with +[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) +\Rightarrow True])) I (TLRef (plus n h)) H1) in (False_ind (ex3_2 T T +(\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead k y z)))) (\lambda (y: +T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: +T).(eq T t (lift h (s k d) z))))) H2))))))))) (\lambda (k0: K).(\lambda (t0: +T).(\lambda (H: ((\forall (h: nat).(\forall (d: nat).((eq T (THead k u t) +(lift h d t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead +k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda +(_: T).(\lambda (z: T).(eq T t (lift h (s k d) z)))))))))).(\lambda (t1: +T).(\lambda (H0: ((\forall (h: nat).(\forall (d: nat).((eq T (THead k u t) +(lift h d t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t1 (THead +k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda +(_: T).(\lambda (z: T).(eq T t (lift h (s k d) z)))))))))).(\lambda (h: +nat).(\lambda (d: nat).(\lambda (H1: (eq T (THead k u t) (lift h d (THead k0 +t0 t1)))).(let H2 \def (eq_ind T (lift h d (THead k0 t0 t1)) (\lambda (t2: +T).(eq T (THead k u t) t2)) H1 (THead k0 (lift h d t0) (lift h (s k0 d) t1)) +(lift_head k0 t0 t1 h d)) in (let H3 \def (f_equal T K (\lambda (e: T).(match +e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) +\Rightarrow k | (THead k1 _ _) \Rightarrow k1])) (THead k u t) (THead k0 +(lift h d t0) (lift h (s k0 d) t1)) H2) in ((let H4 \def (f_equal T T +(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t2 _) \Rightarrow t2])) +(THead k u t) (THead k0 (lift h d t0) (lift h (s k0 d) t1)) H2) in ((let H5 +\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) +with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t2) +\Rightarrow t2])) (THead k u t) (THead k0 (lift h d t0) (lift h (s k0 d) t1)) +H2) in (\lambda (H6: (eq T u (lift h d t0))).(\lambda (H7: (eq K k k0)).(let +H8 \def (eq_ind_r K k0 (\lambda (k1: K).(eq T t (lift h (s k1 d) t1))) H5 k +H7) in (eq_ind K k (\lambda (k1: K).(ex3_2 T T (\lambda (y: T).(\lambda (z: +T).(eq T (THead k1 t0 t1) (THead k y z)))) (\lambda (y: T).(\lambda (_: +T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s +k d) z)))))) (let H9 \def (eq_ind T t (\lambda (t2: T).(\forall (h0: +nat).(\forall (d0: nat).((eq T (THead k u t2) (lift h0 d0 t1)) \to (ex3_2 T T +(\lambda (y: T).(\lambda (z: T).(eq T t1 (THead k y z)))) (\lambda (y: +T).(\lambda (_: T).(eq T u (lift h0 d0 y)))) (\lambda (_: T).(\lambda (z: +T).(eq T t2 (lift h0 (s k d0) z))))))))) H0 (lift h (s k d) t1) H8) in (let +H10 \def (eq_ind T t (\lambda (t2: T).(\forall (h0: nat).(\forall (d0: +nat).((eq T (THead k u t2) (lift h0 d0 t0)) \to (ex3_2 T T (\lambda (y: +T).(\lambda (z: T).(eq T t0 (THead k y z)))) (\lambda (y: T).(\lambda (_: +T).(eq T u (lift h0 d0 y)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift +h0 (s k d0) z))))))))) H (lift h (s k d) t1) H8) in (eq_ind_r T (lift h (s k +d) t1) (\lambda (t2: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T +(THead k t0 t1) (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u +(lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h (s k d) +z)))))) (let H11 \def (eq_ind T u (\lambda (t2: T).(\forall (h0: +nat).(\forall (d0: nat).((eq T (THead k t2 (lift h (s k d) t1)) (lift h0 d0 +t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead k y z)))) +(\lambda (y: T).(\lambda (_: T).(eq T t2 (lift h0 d0 y)))) (\lambda (_: +T).(\lambda (z: T).(eq T (lift h (s k d) t1) (lift h0 (s k d0) z))))))))) H10 +(lift h d t0) H6) in (let H12 \def (eq_ind T u (\lambda (t2: T).(\forall (h0: +nat).(\forall (d0: nat).((eq T (THead k t2 (lift h (s k d) t1)) (lift h0 d0 +t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t1 (THead k y z)))) +(\lambda (y: T).(\lambda (_: T).(eq T t2 (lift h0 d0 y)))) (\lambda (_: +T).(\lambda (z: T).(eq T (lift h (s k d) t1) (lift h0 (s k d0) z))))))))) H9 +(lift h d t0) H6) in (eq_ind_r T (lift h d t0) (\lambda (t2: T).(ex3_2 T T +(\lambda (y: T).(\lambda (z: T).(eq T (THead k t0 t1) (THead k y z)))) +(\lambda (y: T).(\lambda (_: T).(eq T t2 (lift h d y)))) (\lambda (_: +T).(\lambda (z: T).(eq T (lift h (s k d) t1) (lift h (s k d) z)))))) +(ex3_2_intro T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k t0 t1) (THead +k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t0) (lift h d y)))) +(\lambda (_: T).(\lambda (z: T).(eq T (lift h (s k d) t1) (lift h (s k d) +z)))) t0 t1 (refl_equal T (THead k t0 t1)) (refl_equal T (lift h d t0)) +(refl_equal T (lift h (s k d) t1))) u H6))) t H8))) k0 H7))))) H4)) +H3))))))))))) x)))). +(* COMMENTS +Initial nodes: 2083 +END *) + +theorem lift_gen_bind: + \forall (b: B).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h: +nat).(\forall (d: nat).((eq T (THead (Bind b) u t) (lift h d x)) \to (ex3_2 T +T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Bind b) y z)))) (\lambda +(y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: +T).(eq T t (lift h (S d) z))))))))))) +\def + \lambda (b: B).(\lambda (u: T).(\lambda (t: T).(\lambda (x: T).(\lambda (h: +nat).(\lambda (d: nat).(\lambda (H: (eq T (THead (Bind b) u t) (lift h d +x))).(let H_x \def (lift_gen_head (Bind b) u t x h d H) in (let H0 \def H_x +in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Bind b) y +z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: +T).(\lambda (z: T).(eq T t (lift h (S d) z)))) (ex3_2 T T (\lambda (y: +T).(\lambda (z: T).(eq T x (THead (Bind b) y z)))) (\lambda (y: T).(\lambda +(_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift +h (S d) z))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H1: (eq T x (THead +(Bind b) x0 x1))).(\lambda (H2: (eq T u (lift h d x0))).(\lambda (H3: (eq T t +(lift h (S d) x1))).(eq_ind_r T (THead (Bind b) x0 x1) (\lambda (t0: +T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead (Bind b) y +z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: +T).(\lambda (z: T).(eq T t (lift h (S d) z)))))) (eq_ind_r T (lift h (S d) +x1) (\lambda (t0: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead +(Bind b) x0 x1) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T +u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t0 (lift h (S d) +z)))))) (eq_ind_r T (lift h d x0) (\lambda (t0: T).(ex3_2 T T (\lambda (y: +T).(\lambda (z: T).(eq T (THead (Bind b) x0 x1) (THead (Bind b) y z)))) +(\lambda (y: T).(\lambda (_: T).(eq T t0 (lift h d y)))) (\lambda (_: +T).(\lambda (z: T).(eq T (lift h (S d) x1) (lift h (S d) z)))))) (ex3_2_intro +T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Bind b) x0 x1) (THead (Bind +b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d x0) (lift h d +y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h (S d) x1) (lift h (S d) +z)))) x0 x1 (refl_equal T (THead (Bind b) x0 x1)) (refl_equal T (lift h d +x0)) (refl_equal T (lift h (S d) x1))) u H2) t H3) x H1)))))) H0))))))))). +(* COMMENTS +Initial nodes: 637 +END *) + +theorem lift_gen_flat: + \forall (f: F).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h: +nat).(\forall (d: nat).((eq T (THead (Flat f) u t) (lift h d x)) \to (ex3_2 T +T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Flat f) y z)))) (\lambda +(y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: +T).(eq T t (lift h d z))))))))))) +\def + \lambda (f: F).(\lambda (u: T).(\lambda (t: T).(\lambda (x: T).(\lambda (h: +nat).(\lambda (d: nat).(\lambda (H: (eq T (THead (Flat f) u t) (lift h d +x))).(let H_x \def (lift_gen_head (Flat f) u t x h d H) in (let H0 \def H_x +in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Flat f) y +z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: +T).(\lambda (z: T).(eq T t (lift h d z)))) (ex3_2 T T (\lambda (y: +T).(\lambda (z: T).(eq T x (THead (Flat f) y z)))) (\lambda (y: T).(\lambda +(_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift +h d z))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H1: (eq T x (THead +(Flat f) x0 x1))).(\lambda (H2: (eq T u (lift h d x0))).(\lambda (H3: (eq T t +(lift h d x1))).(eq_ind_r T (THead (Flat f) x0 x1) (\lambda (t0: T).(ex3_2 T +T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead (Flat f) y z)))) (\lambda +(y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: +T).(eq T t (lift h d z)))))) (eq_ind_r T (lift h d x1) (\lambda (t0: +T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Flat f) x0 x1) +(THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d +y)))) (\lambda (_: T).(\lambda (z: T).(eq T t0 (lift h d z)))))) (eq_ind_r T +(lift h d x0) (\lambda (t0: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq +T (THead (Flat f) x0 x1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: +T).(eq T t0 (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h d +x1) (lift h d z)))))) (ex3_2_intro T T (\lambda (y: T).(\lambda (z: T).(eq T +(THead (Flat f) x0 x1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: +T).(eq T (lift h d x0) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T +(lift h d x1) (lift h d z)))) x0 x1 (refl_equal T (THead (Flat f) x0 x1)) +(refl_equal T (lift h d x0)) (refl_equal T (lift h d x1))) u H2) t H3) x +H1)))))) H0))))))))). +(* COMMENTS +Initial nodes: 615 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/lift/props.ma b/matita/matita/contribs/lambdadelta/basic_1/lift/props.ma new file mode 100644 index 000000000..f0ed22451 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/lift/props.ma @@ -0,0 +1,592 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/lift/fwd.ma". + +include "Basic-1/s/props.ma". + +theorem thead_x_lift_y_y: + \forall (k: K).(\forall (t: T).(\forall (v: T).(\forall (h: nat).(\forall +(d: nat).((eq T (THead k v (lift h d t)) t) \to (\forall (P: Prop).P)))))) +\def + \lambda (k: K).(\lambda (t: T).(T_ind (\lambda (t0: T).(\forall (v: +T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k v (lift h d t0)) t0) +\to (\forall (P: Prop).P)))))) (\lambda (n: nat).(\lambda (v: T).(\lambda (h: +nat).(\lambda (d: nat).(\lambda (H: (eq T (THead k v (lift h d (TSort n))) +(TSort n))).(\lambda (P: Prop).(let H0 \def (eq_ind T (THead k v (lift h d +(TSort n))) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with +[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) +\Rightarrow True])) I (TSort n) H) in (False_ind P H0)))))))) (\lambda (n: +nat).(\lambda (v: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (eq T +(THead k v (lift h d (TLRef n))) (TLRef n))).(\lambda (P: Prop).(let H0 \def +(eq_ind T (THead k v (lift h d (TLRef n))) (\lambda (ee: T).(match ee in T +return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H) in +(False_ind P H0)))))))) (\lambda (k0: K).(\lambda (t0: T).(\lambda (_: +((\forall (v: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k v (lift +h d t0)) t0) \to (\forall (P: Prop).P))))))).(\lambda (t1: T).(\lambda (H0: +((\forall (v: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k v (lift +h d t1)) t1) \to (\forall (P: Prop).P))))))).(\lambda (v: T).(\lambda (h: +nat).(\lambda (d: nat).(\lambda (H1: (eq T (THead k v (lift h d (THead k0 t0 +t1))) (THead k0 t0 t1))).(\lambda (P: Prop).(let H2 \def (f_equal T K +(\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) +\Rightarrow k | (TLRef _) \Rightarrow k | (THead k1 _ _) \Rightarrow k1])) +(THead k v (lift h d (THead k0 t0 t1))) (THead k0 t0 t1) H1) in ((let H3 \def +(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with +[(TSort _) \Rightarrow v | (TLRef _) \Rightarrow v | (THead _ t2 _) +\Rightarrow t2])) (THead k v (lift h d (THead k0 t0 t1))) (THead k0 t0 t1) +H1) in ((let H4 \def (f_equal T T (\lambda (e: T).(match e in T return +(\lambda (_: T).T) with [(TSort _) \Rightarrow (THead k0 ((let rec lref_map +(f: ((nat \to nat))) (d0: nat) (t2: T) on t2: T \def (match t2 with [(TSort +n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) +with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k1 u t3) +\Rightarrow (THead k1 (lref_map f d0 u) (lref_map f (s k1 d0) t3))]) in +lref_map) (\lambda (x: nat).(plus x h)) d t0) ((let rec lref_map (f: ((nat +\to nat))) (d0: nat) (t2: T) on t2: T \def (match t2 with [(TSort n) +\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) with +[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k1 u t3) +\Rightarrow (THead k1 (lref_map f d0 u) (lref_map f (s k1 d0) t3))]) in +lref_map) (\lambda (x: nat).(plus x h)) (s k0 d) t1)) | (TLRef _) \Rightarrow +(THead k0 ((let rec lref_map (f: ((nat \to nat))) (d0: nat) (t2: T) on t2: T +\def (match t2 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow +(TLRef (match (blt i d0) with [true \Rightarrow i | false \Rightarrow (f +i)])) | (THead k1 u t3) \Rightarrow (THead k1 (lref_map f d0 u) (lref_map f +(s k1 d0) t3))]) in lref_map) (\lambda (x: nat).(plus x h)) d t0) ((let rec +lref_map (f: ((nat \to nat))) (d0: nat) (t2: T) on t2: T \def (match t2 with +[(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i +d0) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k1 u t3) +\Rightarrow (THead k1 (lref_map f d0 u) (lref_map f (s k1 d0) t3))]) in +lref_map) (\lambda (x: nat).(plus x h)) (s k0 d) t1)) | (THead _ _ t2) +\Rightarrow t2])) (THead k v (lift h d (THead k0 t0 t1))) (THead k0 t0 t1) +H1) in (\lambda (_: (eq T v t0)).(\lambda (H6: (eq K k k0)).(let H7 \def +(eq_ind K k (\lambda (k1: K).(\forall (v0: T).(\forall (h0: nat).(\forall +(d0: nat).((eq T (THead k1 v0 (lift h0 d0 t1)) t1) \to (\forall (P0: +Prop).P0)))))) H0 k0 H6) in (let H8 \def (eq_ind T (lift h d (THead k0 t0 +t1)) (\lambda (t2: T).(eq T t2 t1)) H4 (THead k0 (lift h d t0) (lift h (s k0 +d) t1)) (lift_head k0 t0 t1 h d)) in (H7 (lift h d t0) h (s k0 d) H8 P)))))) +H3)) H2)))))))))))) t)). +(* COMMENTS +Initial nodes: 887 +END *) + +theorem lift_r: + \forall (t: T).(\forall (d: nat).(eq T (lift O d t) t)) +\def + \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d: nat).(eq T (lift O d t0) +t0))) (\lambda (n: nat).(\lambda (_: nat).(refl_equal T (TSort n)))) (\lambda +(n: nat).(\lambda (d: nat).(lt_le_e n d (eq T (lift O d (TLRef n)) (TLRef n)) +(\lambda (H: (lt n d)).(eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 (TLRef +n))) (refl_equal T (TLRef n)) (lift O d (TLRef n)) (lift_lref_lt n O d H))) +(\lambda (H: (le d n)).(eq_ind_r T (TLRef (plus n O)) (\lambda (t0: T).(eq T +t0 (TLRef n))) (f_equal nat T TLRef (plus n O) n (sym_eq nat n (plus n O) +(plus_n_O n))) (lift O d (TLRef n)) (lift_lref_ge n O d H)))))) (\lambda (k: +K).(\lambda (t0: T).(\lambda (H: ((\forall (d: nat).(eq T (lift O d t0) +t0)))).(\lambda (t1: T).(\lambda (H0: ((\forall (d: nat).(eq T (lift O d t1) +t1)))).(\lambda (d: nat).(eq_ind_r T (THead k (lift O d t0) (lift O (s k d) +t1)) (\lambda (t2: T).(eq T t2 (THead k t0 t1))) (f_equal3 K T T T THead k k +(lift O d t0) t0 (lift O (s k d) t1) t1 (refl_equal K k) (H d) (H0 (s k d))) +(lift O d (THead k t0 t1)) (lift_head k t0 t1 O d)))))))) t). +(* COMMENTS +Initial nodes: 367 +END *) + +theorem lift_lref_gt: + \forall (d: nat).(\forall (n: nat).((lt d n) \to (eq T (lift (S O) d (TLRef +(pred n))) (TLRef n)))) +\def + \lambda (d: nat).(\lambda (n: nat).(\lambda (H: (lt d n)).(eq_ind_r T (TLRef +(plus (pred n) (S O))) (\lambda (t: T).(eq T t (TLRef n))) (eq_ind nat (plus +(S O) (pred n)) (\lambda (n0: nat).(eq T (TLRef n0) (TLRef n))) (eq_ind nat n +(\lambda (n0: nat).(eq T (TLRef n0) (TLRef n))) (refl_equal T (TLRef n)) (S +(pred n)) (S_pred n d H)) (plus (pred n) (S O)) (plus_sym (S O) (pred n))) +(lift (S O) d (TLRef (pred n))) (lift_lref_ge (pred n) (S O) d (le_S_n d +(pred n) (eq_ind nat n (\lambda (n0: nat).(le (S d) n0)) H (S (pred n)) +(S_pred n d H))))))). +(* COMMENTS +Initial nodes: 193 +END *) + +theorem lifts_tapp: + \forall (h: nat).(\forall (d: nat).(\forall (v: T).(\forall (vs: TList).(eq +TList (lifts h d (TApp vs v)) (TApp (lifts h d vs) (lift h d v)))))) +\def + \lambda (h: nat).(\lambda (d: nat).(\lambda (v: T).(\lambda (vs: +TList).(TList_ind (\lambda (t: TList).(eq TList (lifts h d (TApp t v)) (TApp +(lifts h d t) (lift h d v)))) (refl_equal TList (TCons (lift h d v) TNil)) +(\lambda (t: T).(\lambda (t0: TList).(\lambda (H: (eq TList (lifts h d (TApp +t0 v)) (TApp (lifts h d t0) (lift h d v)))).(eq_ind_r TList (TApp (lifts h d +t0) (lift h d v)) (\lambda (t1: TList).(eq TList (TCons (lift h d t) t1) +(TCons (lift h d t) (TApp (lifts h d t0) (lift h d v))))) (refl_equal TList +(TCons (lift h d t) (TApp (lifts h d t0) (lift h d v)))) (lifts h d (TApp t0 +v)) H)))) vs)))). +(* COMMENTS +Initial nodes: 215 +END *) + +theorem lift_inj: + \forall (x: T).(\forall (t: T).(\forall (h: nat).(\forall (d: nat).((eq T +(lift h d x) (lift h d t)) \to (eq T x t))))) +\def + \lambda (x: T).(T_ind (\lambda (t: T).(\forall (t0: T).(\forall (h: +nat).(\forall (d: nat).((eq T (lift h d t) (lift h d t0)) \to (eq T t +t0)))))) (\lambda (n: nat).(\lambda (t: T).(\lambda (h: nat).(\lambda (d: +nat).(\lambda (H: (eq T (lift h d (TSort n)) (lift h d t))).(let H0 \def +(eq_ind T (lift h d (TSort n)) (\lambda (t0: T).(eq T t0 (lift h d t))) H +(TSort n) (lift_sort n h d)) in (sym_eq T t (TSort n) (lift_gen_sort h d n t +H0)))))))) (\lambda (n: nat).(\lambda (t: T).(\lambda (h: nat).(\lambda (d: +nat).(\lambda (H: (eq T (lift h d (TLRef n)) (lift h d t))).(lt_le_e n d (eq +T (TLRef n) t) (\lambda (H0: (lt n d)).(let H1 \def (eq_ind T (lift h d +(TLRef n)) (\lambda (t0: T).(eq T t0 (lift h d t))) H (TLRef n) (lift_lref_lt +n h d H0)) in (sym_eq T t (TLRef n) (lift_gen_lref_lt h d n (lt_le_trans n d +d H0 (le_n d)) t H1)))) (\lambda (H0: (le d n)).(let H1 \def (eq_ind T (lift +h d (TLRef n)) (\lambda (t0: T).(eq T t0 (lift h d t))) H (TLRef (plus n h)) +(lift_lref_ge n h d H0)) in (sym_eq T t (TLRef n) (lift_gen_lref_ge h d n H0 +t H1)))))))))) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t: +T).(((\forall (t0: T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t) +(lift h d t0)) \to (eq T t t0)))))) \to (\forall (t0: T).(((\forall (t1: +T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t0) (lift h d t1)) +\to (eq T t0 t1)))))) \to (\forall (t1: T).(\forall (h: nat).(\forall (d: +nat).((eq T (lift h d (THead k0 t t0)) (lift h d t1)) \to (eq T (THead k0 t +t0) t1)))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (H: ((\forall (t0: +T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t) (lift h d t0)) \to +(eq T t t0))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (t1: T).(\forall +(h: nat).(\forall (d: nat).((eq T (lift h d t0) (lift h d t1)) \to (eq T t0 +t1))))))).(\lambda (t1: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: +(eq T (lift h d (THead (Bind b) t t0)) (lift h d t1))).(let H2 \def (eq_ind T +(lift h d (THead (Bind b) t t0)) (\lambda (t2: T).(eq T t2 (lift h d t1))) H1 +(THead (Bind b) (lift h d t) (lift h (S d) t0)) (lift_bind b t t0 h d)) in +(ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T t1 (THead (Bind b) y +z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t) (lift h d y)))) +(\lambda (_: T).(\lambda (z: T).(eq T (lift h (S d) t0) (lift h (S d) z)))) +(eq T (THead (Bind b) t t0) t1) (\lambda (x0: T).(\lambda (x1: T).(\lambda +(H3: (eq T t1 (THead (Bind b) x0 x1))).(\lambda (H4: (eq T (lift h d t) (lift +h d x0))).(\lambda (H5: (eq T (lift h (S d) t0) (lift h (S d) x1))).(eq_ind_r +T (THead (Bind b) x0 x1) (\lambda (t2: T).(eq T (THead (Bind b) t t0) t2)) +(f_equal3 K T T T THead (Bind b) (Bind b) t x0 t0 x1 (refl_equal K (Bind b)) +(H x0 h d H4) (H0 x1 h (S d) H5)) t1 H3)))))) (lift_gen_bind b (lift h d t) +(lift h (S d) t0) t1 h d H2)))))))))))) (\lambda (f: F).(\lambda (t: +T).(\lambda (H: ((\forall (t0: T).(\forall (h: nat).(\forall (d: nat).((eq T +(lift h d t) (lift h d t0)) \to (eq T t t0))))))).(\lambda (t0: T).(\lambda +(H0: ((\forall (t1: T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d +t0) (lift h d t1)) \to (eq T t0 t1))))))).(\lambda (t1: T).(\lambda (h: +nat).(\lambda (d: nat).(\lambda (H1: (eq T (lift h d (THead (Flat f) t t0)) +(lift h d t1))).(let H2 \def (eq_ind T (lift h d (THead (Flat f) t t0)) +(\lambda (t2: T).(eq T t2 (lift h d t1))) H1 (THead (Flat f) (lift h d t) +(lift h d t0)) (lift_flat f t t0 h d)) in (ex3_2_ind T T (\lambda (y: +T).(\lambda (z: T).(eq T t1 (THead (Flat f) y z)))) (\lambda (y: T).(\lambda +(_: T).(eq T (lift h d t) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq +T (lift h d t0) (lift h d z)))) (eq T (THead (Flat f) t t0) t1) (\lambda (x0: +T).(\lambda (x1: T).(\lambda (H3: (eq T t1 (THead (Flat f) x0 x1))).(\lambda +(H4: (eq T (lift h d t) (lift h d x0))).(\lambda (H5: (eq T (lift h d t0) +(lift h d x1))).(eq_ind_r T (THead (Flat f) x0 x1) (\lambda (t2: T).(eq T +(THead (Flat f) t t0) t2)) (f_equal3 K T T T THead (Flat f) (Flat f) t x0 t0 +x1 (refl_equal K (Flat f)) (H x0 h d H4) (H0 x1 h d H5)) t1 H3)))))) +(lift_gen_flat f (lift h d t) (lift h d t0) t1 h d H2)))))))))))) k)) x). +(* COMMENTS +Initial nodes: 1391 +END *) + +theorem lift_gen_lift: + \forall (t1: T).(\forall (x: T).(\forall (h1: nat).(\forall (h2: +nat).(\forall (d1: nat).(\forall (d2: nat).((le d1 d2) \to ((eq T (lift h1 d1 +t1) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1 +t2))) (\lambda (t2: T).(eq T t1 (lift h2 d2 t2))))))))))) +\def + \lambda (t1: T).(T_ind (\lambda (t: T).(\forall (x: T).(\forall (h1: +nat).(\forall (h2: nat).(\forall (d1: nat).(\forall (d2: nat).((le d1 d2) \to +((eq T (lift h1 d1 t) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2: +T).(eq T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T t (lift h2 d2 +t2)))))))))))) (\lambda (n: nat).(\lambda (x: T).(\lambda (h1: nat).(\lambda +(h2: nat).(\lambda (d1: nat).(\lambda (d2: nat).(\lambda (_: (le d1 +d2)).(\lambda (H0: (eq T (lift h1 d1 (TSort n)) (lift h2 (plus d2 h1) +x))).(let H1 \def (eq_ind T (lift h1 d1 (TSort n)) (\lambda (t: T).(eq T t +(lift h2 (plus d2 h1) x))) H0 (TSort n) (lift_sort n h1 d1)) in (eq_ind_r T +(TSort n) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h1 d1 t2))) +(\lambda (t2: T).(eq T (TSort n) (lift h2 d2 t2))))) (ex_intro2 T (\lambda +(t2: T).(eq T (TSort n) (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TSort n) +(lift h2 d2 t2))) (TSort n) (eq_ind_r T (TSort n) (\lambda (t: T).(eq T +(TSort n) t)) (refl_equal T (TSort n)) (lift h1 d1 (TSort n)) (lift_sort n h1 +d1)) (eq_ind_r T (TSort n) (\lambda (t: T).(eq T (TSort n) t)) (refl_equal T +(TSort n)) (lift h2 d2 (TSort n)) (lift_sort n h2 d2))) x (lift_gen_sort h2 +(plus d2 h1) n x H1))))))))))) (\lambda (n: nat).(\lambda (x: T).(\lambda +(h1: nat).(\lambda (h2: nat).(\lambda (d1: nat).(\lambda (d2: nat).(\lambda +(H: (le d1 d2)).(\lambda (H0: (eq T (lift h1 d1 (TLRef n)) (lift h2 (plus d2 +h1) x))).(lt_le_e n d1 (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1 t2))) +(\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2)))) (\lambda (H1: (lt n +d1)).(let H2 \def (eq_ind T (lift h1 d1 (TLRef n)) (\lambda (t: T).(eq T t +(lift h2 (plus d2 h1) x))) H0 (TLRef n) (lift_lref_lt n h1 d1 H1)) in +(eq_ind_r T (TLRef n) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift +h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2))))) (ex_intro2 T +(\lambda (t2: T).(eq T (TLRef n) (lift h1 d1 t2))) (\lambda (t2: T).(eq T +(TLRef n) (lift h2 d2 t2))) (TLRef n) (eq_ind_r T (TLRef n) (\lambda (t: +T).(eq T (TLRef n) t)) (refl_equal T (TLRef n)) (lift h1 d1 (TLRef n)) +(lift_lref_lt n h1 d1 H1)) (eq_ind_r T (TLRef n) (\lambda (t: T).(eq T (TLRef +n) t)) (refl_equal T (TLRef n)) (lift h2 d2 (TLRef n)) (lift_lref_lt n h2 d2 +(lt_le_trans n d1 d2 H1 H)))) x (lift_gen_lref_lt h2 (plus d2 h1) n +(lt_le_trans n d1 (plus d2 h1) H1 (le_plus_trans d1 d2 h1 H)) x H2)))) +(\lambda (H1: (le d1 n)).(let H2 \def (eq_ind T (lift h1 d1 (TLRef n)) +(\lambda (t: T).(eq T t (lift h2 (plus d2 h1) x))) H0 (TLRef (plus n h1)) +(lift_lref_ge n h1 d1 H1)) in (lt_le_e n d2 (ex2 T (\lambda (t2: T).(eq T x +(lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2)))) +(\lambda (H3: (lt n d2)).(eq_ind_r T (TLRef (plus n h1)) (\lambda (t: T).(ex2 +T (\lambda (t2: T).(eq T t (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) +(lift h2 d2 t2))))) (ex_intro2 T (\lambda (t2: T).(eq T (TLRef (plus n h1)) +(lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2))) (TLRef +n) (eq_ind_r T (TLRef (plus n h1)) (\lambda (t: T).(eq T (TLRef (plus n h1)) +t)) (refl_equal T (TLRef (plus n h1))) (lift h1 d1 (TLRef n)) (lift_lref_ge n +h1 d1 H1)) (eq_ind_r T (TLRef n) (\lambda (t: T).(eq T (TLRef n) t)) +(refl_equal T (TLRef n)) (lift h2 d2 (TLRef n)) (lift_lref_lt n h2 d2 H3))) x +(lift_gen_lref_lt h2 (plus d2 h1) (plus n h1) (lt_reg_r n d2 h1 H3) x H2))) +(\lambda (H3: (le d2 n)).(lt_le_e n (plus d2 h2) (ex2 T (\lambda (t2: T).(eq +T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2)))) +(\lambda (H4: (lt n (plus d2 h2))).(lift_gen_lref_false h2 (plus d2 h1) (plus +n h1) (le_plus_plus d2 n h1 h1 H3 (le_n h1)) (eq_ind_r nat (plus (plus d2 h2) +h1) (\lambda (n0: nat).(lt (plus n h1) n0)) (lt_reg_r n (plus d2 h2) h1 H4) +(plus (plus d2 h1) h2) (plus_permute_2_in_3 d2 h1 h2)) x H2 (ex2 T (\lambda +(t2: T).(eq T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 +d2 t2)))))) (\lambda (H4: (le (plus d2 h2) n)).(let H5 \def (eq_ind nat (plus +n h1) (\lambda (n0: nat).(eq T (TLRef n0) (lift h2 (plus d2 h1) x))) H2 (plus +(minus (plus n h1) h2) h2) (le_plus_minus_sym h2 (plus n h1) (le_plus_trans +h2 n h1 (le_trans h2 (plus d2 h2) n (le_plus_r d2 h2) H4)))) in (eq_ind_r T +(TLRef (minus (plus n h1) h2)) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T +t (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2))))) +(ex_intro2 T (\lambda (t2: T).(eq T (TLRef (minus (plus n h1) h2)) (lift h1 +d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2))) (TLRef (minus n +h2)) (eq_ind_r nat (plus (minus n h2) h1) (\lambda (n0: nat).(eq T (TLRef n0) +(lift h1 d1 (TLRef (minus n h2))))) (eq_ind_r T (TLRef (plus (minus n h2) +h1)) (\lambda (t: T).(eq T (TLRef (plus (minus n h2) h1)) t)) (refl_equal T +(TLRef (plus (minus n h2) h1))) (lift h1 d1 (TLRef (minus n h2))) +(lift_lref_ge (minus n h2) h1 d1 (le_trans d1 d2 (minus n h2) H (le_minus d2 +n h2 H4)))) (minus (plus n h1) h2) (le_minus_plus h2 n (le_trans h2 (plus d2 +h2) n (le_plus_r d2 h2) H4) h1)) (eq_ind_r nat (plus (minus n h2) h2) +(\lambda (n0: nat).(eq T (TLRef n0) (lift h2 d2 (TLRef (minus n0 h2))))) +(eq_ind_r T (TLRef (plus (minus (plus (minus n h2) h2) h2) h2)) (\lambda (t: +T).(eq T (TLRef (plus (minus n h2) h2)) t)) (f_equal nat T TLRef (plus (minus +n h2) h2) (plus (minus (plus (minus n h2) h2) h2) h2) (f_equal2 nat nat nat +plus (minus n h2) (minus (plus (minus n h2) h2) h2) h2 h2 (sym_eq nat (minus +(plus (minus n h2) h2) h2) (minus n h2) (minus_plus_r (minus n h2) h2)) +(refl_equal nat h2))) (lift h2 d2 (TLRef (minus (plus (minus n h2) h2) h2))) +(lift_lref_ge (minus (plus (minus n h2) h2) h2) h2 d2 (le_minus d2 (plus +(minus n h2) h2) h2 (le_plus_plus d2 (minus n h2) h2 h2 (le_minus d2 n h2 H4) +(le_n h2))))) n (le_plus_minus_sym h2 n (le_trans h2 (plus d2 h2) n +(le_plus_r d2 h2) H4)))) x (lift_gen_lref_ge h2 (plus d2 h1) (minus (plus n +h1) h2) (arith0 h2 d2 n H4 h1) x H5)))))))))))))))))) (\lambda (k: +K).(\lambda (t: T).(\lambda (H: ((\forall (x: T).(\forall (h1: nat).(\forall +(h2: nat).(\forall (d1: nat).(\forall (d2: nat).((le d1 d2) \to ((eq T (lift +h1 d1 t) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2: T).(eq T x (lift +h1 d1 t2))) (\lambda (t2: T).(eq T t (lift h2 d2 t2))))))))))))).(\lambda +(t0: T).(\lambda (H0: ((\forall (x: T).(\forall (h1: nat).(\forall (h2: +nat).(\forall (d1: nat).(\forall (d2: nat).((le d1 d2) \to ((eq T (lift h1 d1 +t0) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1 +t2))) (\lambda (t2: T).(eq T t0 (lift h2 d2 t2))))))))))))).(\lambda (x: +T).(\lambda (h1: nat).(\lambda (h2: nat).(\lambda (d1: nat).(\lambda (d2: +nat).(\lambda (H1: (le d1 d2)).(\lambda (H2: (eq T (lift h1 d1 (THead k t +t0)) (lift h2 (plus d2 h1) x))).(K_ind (\lambda (k0: K).((eq T (lift h1 d1 +(THead k0 t t0)) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2: T).(eq T +x (lift h1 d1 t2))) (\lambda (t2: T).(eq T (THead k0 t t0) (lift h2 d2 +t2)))))) (\lambda (b: B).(\lambda (H3: (eq T (lift h1 d1 (THead (Bind b) t +t0)) (lift h2 (plus d2 h1) x))).(let H4 \def (eq_ind T (lift h1 d1 (THead +(Bind b) t t0)) (\lambda (t2: T).(eq T t2 (lift h2 (plus d2 h1) x))) H3 +(THead (Bind b) (lift h1 d1 t) (lift h1 (S d1) t0)) (lift_bind b t t0 h1 d1)) +in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Bind b) y +z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h1 d1 t) (lift h2 (plus d2 +h1) y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h1 (S d1) t0) (lift h2 +(S (plus d2 h1)) z)))) (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1 t2))) +(\lambda (t2: T).(eq T (THead (Bind b) t t0) (lift h2 d2 t2)))) (\lambda (x0: +T).(\lambda (x1: T).(\lambda (H5: (eq T x (THead (Bind b) x0 x1))).(\lambda +(H6: (eq T (lift h1 d1 t) (lift h2 (plus d2 h1) x0))).(\lambda (H7: (eq T +(lift h1 (S d1) t0) (lift h2 (S (plus d2 h1)) x1))).(eq_ind_r T (THead (Bind +b) x0 x1) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h1 d1 t3))) +(\lambda (t3: T).(eq T (THead (Bind b) t t0) (lift h2 d2 t3))))) (ex2_ind T +(\lambda (t2: T).(eq T x0 (lift h1 d1 t2))) (\lambda (t2: T).(eq T t (lift h2 +d2 t2))) (ex2 T (\lambda (t2: T).(eq T (THead (Bind b) x0 x1) (lift h1 d1 +t2))) (\lambda (t2: T).(eq T (THead (Bind b) t t0) (lift h2 d2 t2)))) +(\lambda (x2: T).(\lambda (H8: (eq T x0 (lift h1 d1 x2))).(\lambda (H9: (eq T +t (lift h2 d2 x2))).(eq_ind_r T (lift h1 d1 x2) (\lambda (t2: T).(ex2 T +(\lambda (t3: T).(eq T (THead (Bind b) t2 x1) (lift h1 d1 t3))) (\lambda (t3: +T).(eq T (THead (Bind b) t t0) (lift h2 d2 t3))))) (eq_ind_r T (lift h2 d2 +x2) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead (Bind b) (lift h1 +d1 x2) x1) (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead (Bind b) t2 t0) +(lift h2 d2 t3))))) (let H10 \def (refl_equal nat (plus (S d2) h1)) in (let +H11 \def (eq_ind nat (S (plus d2 h1)) (\lambda (n: nat).(eq T (lift h1 (S d1) +t0) (lift h2 n x1))) H7 (plus (S d2) h1) H10) in (ex2_ind T (\lambda (t2: +T).(eq T x1 (lift h1 (S d1) t2))) (\lambda (t2: T).(eq T t0 (lift h2 (S d2) +t2))) (ex2 T (\lambda (t2: T).(eq T (THead (Bind b) (lift h1 d1 x2) x1) (lift +h1 d1 t2))) (\lambda (t2: T).(eq T (THead (Bind b) (lift h2 d2 x2) t0) (lift +h2 d2 t2)))) (\lambda (x3: T).(\lambda (H12: (eq T x1 (lift h1 (S d1) +x3))).(\lambda (H13: (eq T t0 (lift h2 (S d2) x3))).(eq_ind_r T (lift h1 (S +d1) x3) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead (Bind b) (lift +h1 d1 x2) t2) (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead (Bind b) (lift +h2 d2 x2) t0) (lift h2 d2 t3))))) (eq_ind_r T (lift h2 (S d2) x3) (\lambda +(t2: T).(ex2 T (\lambda (t3: T).(eq T (THead (Bind b) (lift h1 d1 x2) (lift +h1 (S d1) x3)) (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead (Bind b) (lift +h2 d2 x2) t2) (lift h2 d2 t3))))) (ex_intro2 T (\lambda (t2: T).(eq T (THead +(Bind b) (lift h1 d1 x2) (lift h1 (S d1) x3)) (lift h1 d1 t2))) (\lambda (t2: +T).(eq T (THead (Bind b) (lift h2 d2 x2) (lift h2 (S d2) x3)) (lift h2 d2 +t2))) (THead (Bind b) x2 x3) (eq_ind_r T (THead (Bind b) (lift h1 d1 x2) +(lift h1 (S d1) x3)) (\lambda (t2: T).(eq T (THead (Bind b) (lift h1 d1 x2) +(lift h1 (S d1) x3)) t2)) (refl_equal T (THead (Bind b) (lift h1 d1 x2) (lift +h1 (S d1) x3))) (lift h1 d1 (THead (Bind b) x2 x3)) (lift_bind b x2 x3 h1 +d1)) (eq_ind_r T (THead (Bind b) (lift h2 d2 x2) (lift h2 (S d2) x3)) +(\lambda (t2: T).(eq T (THead (Bind b) (lift h2 d2 x2) (lift h2 (S d2) x3)) +t2)) (refl_equal T (THead (Bind b) (lift h2 d2 x2) (lift h2 (S d2) x3))) +(lift h2 d2 (THead (Bind b) x2 x3)) (lift_bind b x2 x3 h2 d2))) t0 H13) x1 +H12)))) (H0 x1 h1 h2 (S d1) (S d2) (le_n_S d1 d2 H1) H11)))) t H9) x0 H8)))) +(H x0 h1 h2 d1 d2 H1 H6)) x H5)))))) (lift_gen_bind b (lift h1 d1 t) (lift h1 +(S d1) t0) x h2 (plus d2 h1) H4))))) (\lambda (f: F).(\lambda (H3: (eq T +(lift h1 d1 (THead (Flat f) t t0)) (lift h2 (plus d2 h1) x))).(let H4 \def +(eq_ind T (lift h1 d1 (THead (Flat f) t t0)) (\lambda (t2: T).(eq T t2 (lift +h2 (plus d2 h1) x))) H3 (THead (Flat f) (lift h1 d1 t) (lift h1 d1 t0)) +(lift_flat f t t0 h1 d1)) in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: +T).(eq T x (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T +(lift h1 d1 t) (lift h2 (plus d2 h1) y)))) (\lambda (_: T).(\lambda (z: +T).(eq T (lift h1 d1 t0) (lift h2 (plus d2 h1) z)))) (ex2 T (\lambda (t2: +T).(eq T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T (THead (Flat f) t t0) +(lift h2 d2 t2)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T x +(THead (Flat f) x0 x1))).(\lambda (H6: (eq T (lift h1 d1 t) (lift h2 (plus d2 +h1) x0))).(\lambda (H7: (eq T (lift h1 d1 t0) (lift h2 (plus d2 h1) +x1))).(eq_ind_r T (THead (Flat f) x0 x1) (\lambda (t2: T).(ex2 T (\lambda +(t3: T).(eq T t2 (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead (Flat f) t +t0) (lift h2 d2 t3))))) (ex2_ind T (\lambda (t2: T).(eq T x0 (lift h1 d1 +t2))) (\lambda (t2: T).(eq T t (lift h2 d2 t2))) (ex2 T (\lambda (t2: T).(eq +T (THead (Flat f) x0 x1) (lift h1 d1 t2))) (\lambda (t2: T).(eq T (THead +(Flat f) t t0) (lift h2 d2 t2)))) (\lambda (x2: T).(\lambda (H8: (eq T x0 +(lift h1 d1 x2))).(\lambda (H9: (eq T t (lift h2 d2 x2))).(eq_ind_r T (lift +h1 d1 x2) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead (Flat f) t2 +x1) (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead (Flat f) t t0) (lift h2 +d2 t3))))) (eq_ind_r T (lift h2 d2 x2) (\lambda (t2: T).(ex2 T (\lambda (t3: +T).(eq T (THead (Flat f) (lift h1 d1 x2) x1) (lift h1 d1 t3))) (\lambda (t3: +T).(eq T (THead (Flat f) t2 t0) (lift h2 d2 t3))))) (ex2_ind T (\lambda (t2: +T).(eq T x1 (lift h1 d1 t2))) (\lambda (t2: T).(eq T t0 (lift h2 d2 t2))) +(ex2 T (\lambda (t2: T).(eq T (THead (Flat f) (lift h1 d1 x2) x1) (lift h1 d1 +t2))) (\lambda (t2: T).(eq T (THead (Flat f) (lift h2 d2 x2) t0) (lift h2 d2 +t2)))) (\lambda (x3: T).(\lambda (H10: (eq T x1 (lift h1 d1 x3))).(\lambda +(H11: (eq T t0 (lift h2 d2 x3))).(eq_ind_r T (lift h1 d1 x3) (\lambda (t2: +T).(ex2 T (\lambda (t3: T).(eq T (THead (Flat f) (lift h1 d1 x2) t2) (lift h1 +d1 t3))) (\lambda (t3: T).(eq T (THead (Flat f) (lift h2 d2 x2) t0) (lift h2 +d2 t3))))) (eq_ind_r T (lift h2 d2 x3) (\lambda (t2: T).(ex2 T (\lambda (t3: +T).(eq T (THead (Flat f) (lift h1 d1 x2) (lift h1 d1 x3)) (lift h1 d1 t3))) +(\lambda (t3: T).(eq T (THead (Flat f) (lift h2 d2 x2) t2) (lift h2 d2 +t3))))) (ex_intro2 T (\lambda (t2: T).(eq T (THead (Flat f) (lift h1 d1 x2) +(lift h1 d1 x3)) (lift h1 d1 t2))) (\lambda (t2: T).(eq T (THead (Flat f) +(lift h2 d2 x2) (lift h2 d2 x3)) (lift h2 d2 t2))) (THead (Flat f) x2 x3) +(eq_ind_r T (THead (Flat f) (lift h1 d1 x2) (lift h1 d1 x3)) (\lambda (t2: +T).(eq T (THead (Flat f) (lift h1 d1 x2) (lift h1 d1 x3)) t2)) (refl_equal T +(THead (Flat f) (lift h1 d1 x2) (lift h1 d1 x3))) (lift h1 d1 (THead (Flat f) +x2 x3)) (lift_flat f x2 x3 h1 d1)) (eq_ind_r T (THead (Flat f) (lift h2 d2 +x2) (lift h2 d2 x3)) (\lambda (t2: T).(eq T (THead (Flat f) (lift h2 d2 x2) +(lift h2 d2 x3)) t2)) (refl_equal T (THead (Flat f) (lift h2 d2 x2) (lift h2 +d2 x3))) (lift h2 d2 (THead (Flat f) x2 x3)) (lift_flat f x2 x3 h2 d2))) t0 +H11) x1 H10)))) (H0 x1 h1 h2 d1 d2 H1 H7)) t H9) x0 H8)))) (H x0 h1 h2 d1 d2 +H1 H6)) x H5)))))) (lift_gen_flat f (lift h1 d1 t) (lift h1 d1 t0) x h2 (plus +d2 h1) H4))))) k H2))))))))))))) t1). +(* COMMENTS +Initial nodes: 5037 +END *) + +theorem lifts_inj: + \forall (xs: TList).(\forall (ts: TList).(\forall (h: nat).(\forall (d: +nat).((eq TList (lifts h d xs) (lifts h d ts)) \to (eq TList xs ts))))) +\def + \lambda (xs: TList).(TList_ind (\lambda (t: TList).(\forall (ts: +TList).(\forall (h: nat).(\forall (d: nat).((eq TList (lifts h d t) (lifts h +d ts)) \to (eq TList t ts)))))) (\lambda (ts: TList).(TList_ind (\lambda (t: +TList).(\forall (h: nat).(\forall (d: nat).((eq TList (lifts h d TNil) (lifts +h d t)) \to (eq TList TNil t))))) (\lambda (_: nat).(\lambda (_: +nat).(\lambda (_: (eq TList TNil TNil)).(refl_equal TList TNil)))) (\lambda +(t: T).(\lambda (t0: TList).(\lambda (_: ((\forall (h: nat).(\forall (d: +nat).((eq TList TNil (lifts h d t0)) \to (eq TList TNil t0)))))).(\lambda (h: +nat).(\lambda (d: nat).(\lambda (H0: (eq TList TNil (TCons (lift h d t) +(lifts h d t0)))).(let H1 \def (eq_ind TList TNil (\lambda (ee: TList).(match +ee in TList return (\lambda (_: TList).Prop) with [TNil \Rightarrow True | +(TCons _ _) \Rightarrow False])) I (TCons (lift h d t) (lifts h d t0)) H0) in +(False_ind (eq TList TNil (TCons t t0)) H1)))))))) ts)) (\lambda (t: +T).(\lambda (t0: TList).(\lambda (H: ((\forall (ts: TList).(\forall (h: +nat).(\forall (d: nat).((eq TList (lifts h d t0) (lifts h d ts)) \to (eq +TList t0 ts))))))).(\lambda (ts: TList).(TList_ind (\lambda (t1: +TList).(\forall (h: nat).(\forall (d: nat).((eq TList (lifts h d (TCons t +t0)) (lifts h d t1)) \to (eq TList (TCons t t0) t1))))) (\lambda (h: +nat).(\lambda (d: nat).(\lambda (H0: (eq TList (TCons (lift h d t) (lifts h d +t0)) TNil)).(let H1 \def (eq_ind TList (TCons (lift h d t) (lifts h d t0)) +(\lambda (ee: TList).(match ee in TList return (\lambda (_: TList).Prop) with +[TNil \Rightarrow False | (TCons _ _) \Rightarrow True])) I TNil H0) in +(False_ind (eq TList (TCons t t0) TNil) H1))))) (\lambda (t1: T).(\lambda +(t2: TList).(\lambda (_: ((\forall (h: nat).(\forall (d: nat).((eq TList +(TCons (lift h d t) (lifts h d t0)) (lifts h d t2)) \to (eq TList (TCons t +t0) t2)))))).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: (eq TList +(TCons (lift h d t) (lifts h d t0)) (TCons (lift h d t1) (lifts h d +t2)))).(let H2 \def (f_equal TList T (\lambda (e: TList).(match e in TList +return (\lambda (_: TList).T) with [TNil \Rightarrow ((let rec lref_map (f: +((nat \to nat))) (d0: nat) (t3: T) on t3: T \def (match t3 with [(TSort n) +\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) with +[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t4) \Rightarrow +(THead k (lref_map f d0 u) (lref_map f (s k d0) t4))]) in lref_map) (\lambda +(x: nat).(plus x h)) d t) | (TCons t3 _) \Rightarrow t3])) (TCons (lift h d +t) (lifts h d t0)) (TCons (lift h d t1) (lifts h d t2)) H1) in ((let H3 \def +(f_equal TList TList (\lambda (e: TList).(match e in TList return (\lambda +(_: TList).TList) with [TNil \Rightarrow ((let rec lifts (h0: nat) (d0: nat) +(ts0: TList) on ts0: TList \def (match ts0 with [TNil \Rightarrow TNil | +(TCons t3 ts1) \Rightarrow (TCons (lift h0 d0 t3) (lifts h0 d0 ts1))]) in +lifts) h d t0) | (TCons _ t3) \Rightarrow t3])) (TCons (lift h d t) (lifts h +d t0)) (TCons (lift h d t1) (lifts h d t2)) H1) in (\lambda (H4: (eq T (lift +h d t) (lift h d t1))).(eq_ind T t (\lambda (t3: T).(eq TList (TCons t t0) +(TCons t3 t2))) (f_equal2 T TList TList TCons t t t0 t2 (refl_equal T t) (H +t2 h d H3)) t1 (lift_inj t t1 h d H4)))) H2)))))))) ts))))) xs). +(* COMMENTS +Initial nodes: 772 +END *) + +theorem lift_free: + \forall (t: T).(\forall (h: nat).(\forall (k: nat).(\forall (d: +nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to (eq T (lift k e +(lift h d t)) (lift (plus k h) d t)))))))) +\def + \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (h: nat).(\forall (k: +nat).(\forall (d: nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to +(eq T (lift k e (lift h d t0)) (lift (plus k h) d t0))))))))) (\lambda (n: +nat).(\lambda (h: nat).(\lambda (k: nat).(\lambda (d: nat).(\lambda (e: +nat).(\lambda (_: (le e (plus d h))).(\lambda (_: (le d e)).(eq_ind_r T +(TSort n) (\lambda (t0: T).(eq T (lift k e t0) (lift (plus k h) d (TSort +n)))) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq T t0 (lift (plus k h) d +(TSort n)))) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq T (TSort n) t0)) +(refl_equal T (TSort n)) (lift (plus k h) d (TSort n)) (lift_sort n (plus k +h) d)) (lift k e (TSort n)) (lift_sort n k e)) (lift h d (TSort n)) +(lift_sort n h d))))))))) (\lambda (n: nat).(\lambda (h: nat).(\lambda (k: +nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (H: (le e (plus d +h))).(\lambda (H0: (le d e)).(lt_le_e n d (eq T (lift k e (lift h d (TLRef +n))) (lift (plus k h) d (TLRef n))) (\lambda (H1: (lt n d)).(eq_ind_r T +(TLRef n) (\lambda (t0: T).(eq T (lift k e t0) (lift (plus k h) d (TLRef +n)))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 (lift (plus k h) d +(TLRef n)))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (TLRef n) t0)) +(refl_equal T (TLRef n)) (lift (plus k h) d (TLRef n)) (lift_lref_lt n (plus +k h) d H1)) (lift k e (TLRef n)) (lift_lref_lt n k e (lt_le_trans n d e H1 +H0))) (lift h d (TLRef n)) (lift_lref_lt n h d H1))) (\lambda (H1: (le d +n)).(eq_ind_r T (TLRef (plus n h)) (\lambda (t0: T).(eq T (lift k e t0) (lift +(plus k h) d (TLRef n)))) (eq_ind_r T (TLRef (plus (plus n h) k)) (\lambda +(t0: T).(eq T t0 (lift (plus k h) d (TLRef n)))) (eq_ind_r T (TLRef (plus n +(plus k h))) (\lambda (t0: T).(eq T (TLRef (plus (plus n h) k)) t0)) (f_equal +nat T TLRef (plus (plus n h) k) (plus n (plus k h)) +(plus_permute_2_in_3_assoc n h k)) (lift (plus k h) d (TLRef n)) +(lift_lref_ge n (plus k h) d H1)) (lift k e (TLRef (plus n h))) (lift_lref_ge +(plus n h) k e (le_trans e (plus d h) (plus n h) H (le_plus_plus d n h h H1 +(le_n h))))) (lift h d (TLRef n)) (lift_lref_ge n h d H1))))))))))) (\lambda +(k: K).(\lambda (t0: T).(\lambda (H: ((\forall (h: nat).(\forall (k0: +nat).(\forall (d: nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to +(eq T (lift k0 e (lift h d t0)) (lift (plus k0 h) d t0)))))))))).(\lambda +(t1: T).(\lambda (H0: ((\forall (h: nat).(\forall (k0: nat).(\forall (d: +nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to (eq T (lift k0 e +(lift h d t1)) (lift (plus k0 h) d t1)))))))))).(\lambda (h: nat).(\lambda +(k0: nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (H1: (le e (plus d +h))).(\lambda (H2: (le d e)).(eq_ind_r T (THead k (lift h d t0) (lift h (s k +d) t1)) (\lambda (t2: T).(eq T (lift k0 e t2) (lift (plus k0 h) d (THead k t0 +t1)))) (eq_ind_r T (THead k (lift k0 e (lift h d t0)) (lift k0 (s k e) (lift +h (s k d) t1))) (\lambda (t2: T).(eq T t2 (lift (plus k0 h) d (THead k t0 +t1)))) (eq_ind_r T (THead k (lift (plus k0 h) d t0) (lift (plus k0 h) (s k d) +t1)) (\lambda (t2: T).(eq T (THead k (lift k0 e (lift h d t0)) (lift k0 (s k +e) (lift h (s k d) t1))) t2)) (f_equal3 K T T T THead k k (lift k0 e (lift h +d t0)) (lift (plus k0 h) d t0) (lift k0 (s k e) (lift h (s k d) t1)) (lift +(plus k0 h) (s k d) t1) (refl_equal K k) (H h k0 d e H1 H2) (H0 h k0 (s k d) +(s k e) (eq_ind nat (s k (plus d h)) (\lambda (n: nat).(le (s k e) n)) (s_le +k e (plus d h) H1) (plus (s k d) h) (s_plus k d h)) (s_le k d e H2))) (lift +(plus k0 h) d (THead k t0 t1)) (lift_head k t0 t1 (plus k0 h) d)) (lift k0 e +(THead k (lift h d t0) (lift h (s k d) t1))) (lift_head k (lift h d t0) (lift +h (s k d) t1) k0 e)) (lift h d (THead k t0 t1)) (lift_head k t0 t1 h +d))))))))))))) t). +(* COMMENTS +Initial nodes: 1407 +END *) + +theorem lift_d: + \forall (t: T).(\forall (h: nat).(\forall (k: nat).(\forall (d: +nat).(\forall (e: nat).((le e d) \to (eq T (lift h (plus k d) (lift k e t)) +(lift k e (lift h d t)))))))) +\def + \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (h: nat).(\forall (k: +nat).(\forall (d: nat).(\forall (e: nat).((le e d) \to (eq T (lift h (plus k +d) (lift k e t0)) (lift k e (lift h d t0))))))))) (\lambda (n: nat).(\lambda +(h: nat).(\lambda (k: nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (_: +(le e d)).(eq_ind_r T (TSort n) (\lambda (t0: T).(eq T (lift h (plus k d) t0) +(lift k e (lift h d (TSort n))))) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq +T t0 (lift k e (lift h d (TSort n))))) (eq_ind_r T (TSort n) (\lambda (t0: +T).(eq T (TSort n) (lift k e t0))) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq +T (TSort n) t0)) (refl_equal T (TSort n)) (lift k e (TSort n)) (lift_sort n k +e)) (lift h d (TSort n)) (lift_sort n h d)) (lift h (plus k d) (TSort n)) +(lift_sort n h (plus k d))) (lift k e (TSort n)) (lift_sort n k e)))))))) +(\lambda (n: nat).(\lambda (h: nat).(\lambda (k: nat).(\lambda (d: +nat).(\lambda (e: nat).(\lambda (H: (le e d)).(lt_le_e n e (eq T (lift h +(plus k d) (lift k e (TLRef n))) (lift k e (lift h d (TLRef n)))) (\lambda +(H0: (lt n e)).(let H1 \def (lt_le_trans n e d H0 H) in (eq_ind_r T (TLRef n) +(\lambda (t0: T).(eq T (lift h (plus k d) t0) (lift k e (lift h d (TLRef +n))))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 (lift k e (lift h d +(TLRef n))))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (TLRef n) (lift k +e t0))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (TLRef n) t0)) +(refl_equal T (TLRef n)) (lift k e (TLRef n)) (lift_lref_lt n k e H0)) (lift +h d (TLRef n)) (lift_lref_lt n h d H1)) (lift h (plus k d) (TLRef n)) +(lift_lref_lt n h (plus k d) (lt_le_trans n d (plus k d) H1 (le_plus_r k +d)))) (lift k e (TLRef n)) (lift_lref_lt n k e H0)))) (\lambda (H0: (le e +n)).(eq_ind_r T (TLRef (plus n k)) (\lambda (t0: T).(eq T (lift h (plus k d) +t0) (lift k e (lift h d (TLRef n))))) (eq_ind_r nat (plus d k) (\lambda (n0: +nat).(eq T (lift h n0 (TLRef (plus n k))) (lift k e (lift h d (TLRef n))))) +(lt_le_e n d (eq T (lift h (plus d k) (TLRef (plus n k))) (lift k e (lift h d +(TLRef n)))) (\lambda (H1: (lt n d)).(eq_ind_r T (TLRef (plus n k)) (\lambda +(t0: T).(eq T t0 (lift k e (lift h d (TLRef n))))) (eq_ind_r T (TLRef n) +(\lambda (t0: T).(eq T (TLRef (plus n k)) (lift k e t0))) (eq_ind_r T (TLRef +(plus n k)) (\lambda (t0: T).(eq T (TLRef (plus n k)) t0)) (refl_equal T +(TLRef (plus n k))) (lift k e (TLRef n)) (lift_lref_ge n k e H0)) (lift h d +(TLRef n)) (lift_lref_lt n h d H1)) (lift h (plus d k) (TLRef (plus n k))) +(lift_lref_lt (plus n k) h (plus d k) (lt_reg_r n d k H1)))) (\lambda (H1: +(le d n)).(eq_ind_r T (TLRef (plus (plus n k) h)) (\lambda (t0: T).(eq T t0 +(lift k e (lift h d (TLRef n))))) (eq_ind_r T (TLRef (plus n h)) (\lambda +(t0: T).(eq T (TLRef (plus (plus n k) h)) (lift k e t0))) (eq_ind_r T (TLRef +(plus (plus n h) k)) (\lambda (t0: T).(eq T (TLRef (plus (plus n k) h)) t0)) +(f_equal nat T TLRef (plus (plus n k) h) (plus (plus n h) k) (sym_eq nat +(plus (plus n h) k) (plus (plus n k) h) (plus_permute_2_in_3 n h k))) (lift k +e (TLRef (plus n h))) (lift_lref_ge (plus n h) k e (le_plus_trans e n h H0))) +(lift h d (TLRef n)) (lift_lref_ge n h d H1)) (lift h (plus d k) (TLRef (plus +n k))) (lift_lref_ge (plus n k) h (plus d k) (le_plus_plus d n k k H1 (le_n +k)))))) (plus k d) (plus_sym k d)) (lift k e (TLRef n)) (lift_lref_ge n k e +H0)))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (H: ((\forall (h: +nat).(\forall (k0: nat).(\forall (d: nat).(\forall (e: nat).((le e d) \to (eq +T (lift h (plus k0 d) (lift k0 e t0)) (lift k0 e (lift h d +t0)))))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (h: nat).(\forall (k0: +nat).(\forall (d: nat).(\forall (e: nat).((le e d) \to (eq T (lift h (plus k0 +d) (lift k0 e t1)) (lift k0 e (lift h d t1)))))))))).(\lambda (h: +nat).(\lambda (k0: nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (H1: (le +e d)).(eq_ind_r T (THead k (lift k0 e t0) (lift k0 (s k e) t1)) (\lambda (t2: +T).(eq T (lift h (plus k0 d) t2) (lift k0 e (lift h d (THead k t0 t1))))) +(eq_ind_r T (THead k (lift h (plus k0 d) (lift k0 e t0)) (lift h (s k (plus +k0 d)) (lift k0 (s k e) t1))) (\lambda (t2: T).(eq T t2 (lift k0 e (lift h d +(THead k t0 t1))))) (eq_ind_r T (THead k (lift h d t0) (lift h (s k d) t1)) +(\lambda (t2: T).(eq T (THead k (lift h (plus k0 d) (lift k0 e t0)) (lift h +(s k (plus k0 d)) (lift k0 (s k e) t1))) (lift k0 e t2))) (eq_ind_r T (THead +k (lift k0 e (lift h d t0)) (lift k0 (s k e) (lift h (s k d) t1))) (\lambda +(t2: T).(eq T (THead k (lift h (plus k0 d) (lift k0 e t0)) (lift h (s k (plus +k0 d)) (lift k0 (s k e) t1))) t2)) (eq_ind_r nat (plus k0 (s k d)) (\lambda +(n: nat).(eq T (THead k (lift h (plus k0 d) (lift k0 e t0)) (lift h n (lift +k0 (s k e) t1))) (THead k (lift k0 e (lift h d t0)) (lift k0 (s k e) (lift h +(s k d) t1))))) (f_equal3 K T T T THead k k (lift h (plus k0 d) (lift k0 e +t0)) (lift k0 e (lift h d t0)) (lift h (plus k0 (s k d)) (lift k0 (s k e) +t1)) (lift k0 (s k e) (lift h (s k d) t1)) (refl_equal K k) (H h k0 d e H1) +(H0 h k0 (s k d) (s k e) (s_le k e d H1))) (s k (plus k0 d)) (s_plus_sym k k0 +d)) (lift k0 e (THead k (lift h d t0) (lift h (s k d) t1))) (lift_head k +(lift h d t0) (lift h (s k d) t1) k0 e)) (lift h d (THead k t0 t1)) +(lift_head k t0 t1 h d)) (lift h (plus k0 d) (THead k (lift k0 e t0) (lift k0 +(s k e) t1))) (lift_head k (lift k0 e t0) (lift k0 (s k e) t1) h (plus k0 +d))) (lift k0 e (THead k t0 t1)) (lift_head k t0 t1 k0 e)))))))))))) t). +(* COMMENTS +Initial nodes: 2143 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/lift/tlt.ma b/matita/matita/contribs/lambdadelta/basic_1/lift/tlt.ma new file mode 100644 index 000000000..1d8edc7df --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/lift/tlt.ma @@ -0,0 +1,299 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/lift/fwd.ma". + +include "Basic-1/tlt/props.ma". + +theorem lift_weight_map: + \forall (t: T).(\forall (h: nat).(\forall (d: nat).(\forall (f: ((nat \to +nat))).(((\forall (m: nat).((le d m) \to (eq nat (f m) O)))) \to (eq nat +(weight_map f (lift h d t)) (weight_map f t)))))) +\def + \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (h: nat).(\forall (d: +nat).(\forall (f: ((nat \to nat))).(((\forall (m: nat).((le d m) \to (eq nat +(f m) O)))) \to (eq nat (weight_map f (lift h d t0)) (weight_map f t0))))))) +(\lambda (n: nat).(\lambda (_: nat).(\lambda (d: nat).(\lambda (f: ((nat \to +nat))).(\lambda (_: ((\forall (m: nat).((le d m) \to (eq nat (f m) +O))))).(refl_equal nat (weight_map f (TSort n)))))))) (\lambda (n: +nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (f: ((nat \to +nat))).(\lambda (H: ((\forall (m: nat).((le d m) \to (eq nat (f m) +O))))).(lt_le_e n d (eq nat (weight_map f (lift h d (TLRef n))) (weight_map f +(TLRef n))) (\lambda (H0: (lt n d)).(eq_ind_r T (TLRef n) (\lambda (t0: +T).(eq nat (weight_map f t0) (weight_map f (TLRef n)))) (refl_equal nat +(weight_map f (TLRef n))) (lift h d (TLRef n)) (lift_lref_lt n h d H0))) +(\lambda (H0: (le d n)).(eq_ind_r T (TLRef (plus n h)) (\lambda (t0: T).(eq +nat (weight_map f t0) (weight_map f (TLRef n)))) (eq_ind_r nat O (\lambda +(n0: nat).(eq nat (f (plus n h)) n0)) (H (plus n h) (le_plus_trans d n h H0)) +(f n) (H n H0)) (lift h d (TLRef n)) (lift_lref_ge n h d H0))))))))) (\lambda +(k: K).(\lambda (t0: T).(\lambda (H: ((\forall (h: nat).(\forall (d: +nat).(\forall (f: ((nat \to nat))).(((\forall (m: nat).((le d m) \to (eq nat +(f m) O)))) \to (eq nat (weight_map f (lift h d t0)) (weight_map f +t0)))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (h: nat).(\forall (d: +nat).(\forall (f: ((nat \to nat))).(((\forall (m: nat).((le d m) \to (eq nat +(f m) O)))) \to (eq nat (weight_map f (lift h d t1)) (weight_map f +t1)))))))).(\lambda (h: nat).(\lambda (d: nat).(\lambda (f: ((nat \to +nat))).(\lambda (H1: ((\forall (m: nat).((le d m) \to (eq nat (f m) +O))))).(K_ind (\lambda (k0: K).(eq nat (weight_map f (lift h d (THead k0 t0 +t1))) (weight_map f (THead k0 t0 t1)))) (\lambda (b: B).(eq_ind_r T (THead +(Bind b) (lift h d t0) (lift h (s (Bind b) d) t1)) (\lambda (t2: T).(eq nat +(weight_map f t2) (weight_map f (THead (Bind b) t0 t1)))) (B_ind (\lambda +(b0: B).(eq nat (match b0 with [Abbr \Rightarrow (S (plus (weight_map f (lift +h d t0)) (weight_map (wadd f (S (weight_map f (lift h d t0)))) (lift h (S d) +t1)))) | Abst \Rightarrow (S (plus (weight_map f (lift h d t0)) (weight_map +(wadd f O) (lift h (S d) t1)))) | Void \Rightarrow (S (plus (weight_map f +(lift h d t0)) (weight_map (wadd f O) (lift h (S d) t1))))]) (match b0 with +[Abbr \Rightarrow (S (plus (weight_map f t0) (weight_map (wadd f (S +(weight_map f t0))) t1))) | Abst \Rightarrow (S (plus (weight_map f t0) +(weight_map (wadd f O) t1))) | Void \Rightarrow (S (plus (weight_map f t0) +(weight_map (wadd f O) t1)))]))) (eq_ind_r nat (weight_map f t0) (\lambda (n: +nat).(eq nat (S (plus n (weight_map (wadd f (S n)) (lift h (S d) t1)))) (S +(plus (weight_map f t0) (weight_map (wadd f (S (weight_map f t0))) t1))))) +(eq_ind_r nat (weight_map (wadd f (S (weight_map f t0))) t1) (\lambda (n: +nat).(eq nat (S (plus (weight_map f t0) n)) (S (plus (weight_map f t0) +(weight_map (wadd f (S (weight_map f t0))) t1))))) (refl_equal nat (S (plus +(weight_map f t0) (weight_map (wadd f (S (weight_map f t0))) t1)))) +(weight_map (wadd f (S (weight_map f t0))) (lift h (S d) t1)) (H0 h (S d) +(wadd f (S (weight_map f t0))) (\lambda (m: nat).(\lambda (H2: (le (S d) +m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S n))) (\lambda (n: nat).(le d +n)) (eq nat (wadd f (S (weight_map f t0)) m) O) (\lambda (x: nat).(\lambda +(H3: (eq nat m (S x))).(\lambda (H4: (le d x)).(eq_ind_r nat (S x) (\lambda +(n: nat).(eq nat (wadd f (S (weight_map f t0)) n) O)) (H1 x H4) m H3)))) +(le_gen_S d m H2)))))) (weight_map f (lift h d t0)) (H h d f H1)) (eq_ind_r +nat (weight_map (wadd f O) t1) (\lambda (n: nat).(eq nat (S (plus (weight_map +f (lift h d t0)) n)) (S (plus (weight_map f t0) (weight_map (wadd f O) +t1))))) (f_equal nat nat S (plus (weight_map f (lift h d t0)) (weight_map +(wadd f O) t1)) (plus (weight_map f t0) (weight_map (wadd f O) t1)) (f_equal2 +nat nat nat plus (weight_map f (lift h d t0)) (weight_map f t0) (weight_map +(wadd f O) t1) (weight_map (wadd f O) t1) (H h d f H1) (refl_equal nat +(weight_map (wadd f O) t1)))) (weight_map (wadd f O) (lift h (S d) t1)) (H0 h +(S d) (wadd f O) (\lambda (m: nat).(\lambda (H2: (le (S d) m)).(ex2_ind nat +(\lambda (n: nat).(eq nat m (S n))) (\lambda (n: nat).(le d n)) (eq nat (wadd +f O m) O) (\lambda (x: nat).(\lambda (H3: (eq nat m (S x))).(\lambda (H4: (le +d x)).(eq_ind_r nat (S x) (\lambda (n: nat).(eq nat (wadd f O n) O)) (H1 x +H4) m H3)))) (le_gen_S d m H2)))))) (eq_ind_r nat (weight_map (wadd f O) t1) +(\lambda (n: nat).(eq nat (S (plus (weight_map f (lift h d t0)) n)) (S (plus +(weight_map f t0) (weight_map (wadd f O) t1))))) (f_equal nat nat S (plus +(weight_map f (lift h d t0)) (weight_map (wadd f O) t1)) (plus (weight_map f +t0) (weight_map (wadd f O) t1)) (f_equal2 nat nat nat plus (weight_map f +(lift h d t0)) (weight_map f t0) (weight_map (wadd f O) t1) (weight_map (wadd +f O) t1) (H h d f H1) (refl_equal nat (weight_map (wadd f O) t1)))) +(weight_map (wadd f O) (lift h (S d) t1)) (H0 h (S d) (wadd f O) (\lambda (m: +nat).(\lambda (H2: (le (S d) m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S +n))) (\lambda (n: nat).(le d n)) (eq nat (wadd f O m) O) (\lambda (x: +nat).(\lambda (H3: (eq nat m (S x))).(\lambda (H4: (le d x)).(eq_ind_r nat (S +x) (\lambda (n: nat).(eq nat (wadd f O n) O)) (H1 x H4) m H3)))) (le_gen_S d +m H2)))))) b) (lift h d (THead (Bind b) t0 t1)) (lift_head (Bind b) t0 t1 h +d))) (\lambda (f0: F).(eq_ind_r T (THead (Flat f0) (lift h d t0) (lift h (s +(Flat f0) d) t1)) (\lambda (t2: T).(eq nat (weight_map f t2) (weight_map f +(THead (Flat f0) t0 t1)))) (f_equal nat nat S (plus (weight_map f (lift h d +t0)) (weight_map f (lift h d t1))) (plus (weight_map f t0) (weight_map f t1)) +(f_equal2 nat nat nat plus (weight_map f (lift h d t0)) (weight_map f t0) +(weight_map f (lift h d t1)) (weight_map f t1) (H h d f H1) (H0 h d f H1))) +(lift h d (THead (Flat f0) t0 t1)) (lift_head (Flat f0) t0 t1 h d))) +k)))))))))) t). +(* COMMENTS +Initial nodes: 1969 +END *) + +theorem lift_weight: + \forall (t: T).(\forall (h: nat).(\forall (d: nat).(eq nat (weight (lift h d +t)) (weight t)))) +\def + \lambda (t: T).(\lambda (h: nat).(\lambda (d: nat).(lift_weight_map t h d +(\lambda (_: nat).O) (\lambda (m: nat).(\lambda (_: (le d m)).(refl_equal nat +O)))))). +(* COMMENTS +Initial nodes: 31 +END *) + +theorem lift_weight_add: + \forall (w: nat).(\forall (t: T).(\forall (h: nat).(\forall (d: +nat).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall +(m: nat).((lt m d) \to (eq nat (g m) (f m))))) \to ((eq nat (g d) w) \to +(((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f m))))) \to (eq nat +(weight_map f (lift h d t)) (weight_map g (lift (S h) d t))))))))))) +\def + \lambda (w: nat).(\lambda (t: T).(T_ind (\lambda (t0: T).(\forall (h: +nat).(\forall (d: nat).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to +nat))).(((\forall (m: nat).((lt m d) \to (eq nat (g m) (f m))))) \to ((eq nat +(g d) w) \to (((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f m))))) +\to (eq nat (weight_map f (lift h d t0)) (weight_map g (lift (S h) d +t0))))))))))) (\lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda +(f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (_: ((\forall (m: +nat).((lt m d) \to (eq nat (g m) (f m)))))).(\lambda (_: (eq nat (g d) +w)).(\lambda (_: ((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f +m)))))).(refl_equal nat (weight_map g (lift (S h) d (TSort n)))))))))))) +(\lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (f: ((nat \to +nat))).(\lambda (g: ((nat \to nat))).(\lambda (H: ((\forall (m: nat).((lt m +d) \to (eq nat (g m) (f m)))))).(\lambda (_: (eq nat (g d) w)).(\lambda (H1: +((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f m)))))).(lt_le_e n d +(eq nat (weight_map f (lift h d (TLRef n))) (weight_map g (lift (S h) d +(TLRef n)))) (\lambda (H2: (lt n d)).(eq_ind_r T (TLRef n) (\lambda (t0: +T).(eq nat (weight_map f t0) (weight_map g (lift (S h) d (TLRef n))))) +(eq_ind_r T (TLRef n) (\lambda (t0: T).(eq nat (weight_map f (TLRef n)) +(weight_map g t0))) (sym_eq nat (g n) (f n) (H n H2)) (lift (S h) d (TLRef +n)) (lift_lref_lt n (S h) d H2)) (lift h d (TLRef n)) (lift_lref_lt n h d +H2))) (\lambda (H2: (le d n)).(eq_ind_r T (TLRef (plus n h)) (\lambda (t0: +T).(eq nat (weight_map f t0) (weight_map g (lift (S h) d (TLRef n))))) +(eq_ind_r T (TLRef (plus n (S h))) (\lambda (t0: T).(eq nat (weight_map f +(TLRef (plus n h))) (weight_map g t0))) (eq_ind nat (S (plus n h)) (\lambda +(n0: nat).(eq nat (f (plus n h)) (g n0))) (sym_eq nat (g (S (plus n h))) (f +(plus n h)) (H1 (plus n h) (le_plus_trans d n h H2))) (plus n (S h)) +(plus_n_Sm n h)) (lift (S h) d (TLRef n)) (lift_lref_ge n (S h) d H2)) (lift +h d (TLRef n)) (lift_lref_ge n h d H2)))))))))))) (\lambda (k: K).(\lambda +(t0: T).(\lambda (H: ((\forall (h: nat).(\forall (d: nat).(\forall (f: ((nat +\to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).((lt m d) \to +(eq nat (g m) (f m))))) \to ((eq nat (g d) w) \to (((\forall (m: nat).((le d +m) \to (eq nat (g (S m)) (f m))))) \to (eq nat (weight_map f (lift h d t0)) +(weight_map g (lift (S h) d t0)))))))))))).(\lambda (t1: T).(\lambda (H0: +((\forall (h: nat).(\forall (d: nat).(\forall (f: ((nat \to nat))).(\forall +(g: ((nat \to nat))).(((\forall (m: nat).((lt m d) \to (eq nat (g m) (f +m))))) \to ((eq nat (g d) w) \to (((\forall (m: nat).((le d m) \to (eq nat (g +(S m)) (f m))))) \to (eq nat (weight_map f (lift h d t1)) (weight_map g (lift +(S h) d t1)))))))))))).(\lambda (h: nat).(\lambda (d: nat).(\lambda (f: ((nat +\to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H1: ((\forall (m: +nat).((lt m d) \to (eq nat (g m) (f m)))))).(\lambda (H2: (eq nat (g d) +w)).(\lambda (H3: ((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f +m)))))).(K_ind (\lambda (k0: K).(eq nat (weight_map f (lift h d (THead k0 t0 +t1))) (weight_map g (lift (S h) d (THead k0 t0 t1))))) (\lambda (b: +B).(eq_ind_r T (THead (Bind b) (lift h d t0) (lift h (s (Bind b) d) t1)) +(\lambda (t2: T).(eq nat (weight_map f t2) (weight_map g (lift (S h) d (THead +(Bind b) t0 t1))))) (eq_ind_r T (THead (Bind b) (lift (S h) d t0) (lift (S h) +(s (Bind b) d) t1)) (\lambda (t2: T).(eq nat (weight_map f (THead (Bind b) +(lift h d t0) (lift h (s (Bind b) d) t1))) (weight_map g t2))) (B_ind +(\lambda (b0: B).(eq nat (match b0 with [Abbr \Rightarrow (S (plus +(weight_map f (lift h d t0)) (weight_map (wadd f (S (weight_map f (lift h d +t0)))) (lift h (S d) t1)))) | Abst \Rightarrow (S (plus (weight_map f (lift h +d t0)) (weight_map (wadd f O) (lift h (S d) t1)))) | Void \Rightarrow (S +(plus (weight_map f (lift h d t0)) (weight_map (wadd f O) (lift h (S d) +t1))))]) (match b0 with [Abbr \Rightarrow (S (plus (weight_map g (lift (S h) +d t0)) (weight_map (wadd g (S (weight_map g (lift (S h) d t0)))) (lift (S h) +(S d) t1)))) | Abst \Rightarrow (S (plus (weight_map g (lift (S h) d t0)) +(weight_map (wadd g O) (lift (S h) (S d) t1)))) | Void \Rightarrow (S (plus +(weight_map g (lift (S h) d t0)) (weight_map (wadd g O) (lift (S h) (S d) +t1))))]))) (f_equal nat nat S (plus (weight_map f (lift h d t0)) (weight_map +(wadd f (S (weight_map f (lift h d t0)))) (lift h (S d) t1))) (plus +(weight_map g (lift (S h) d t0)) (weight_map (wadd g (S (weight_map g (lift +(S h) d t0)))) (lift (S h) (S d) t1))) (f_equal2 nat nat nat plus (weight_map +f (lift h d t0)) (weight_map g (lift (S h) d t0)) (weight_map (wadd f (S +(weight_map f (lift h d t0)))) (lift h (S d) t1)) (weight_map (wadd g (S +(weight_map g (lift (S h) d t0)))) (lift (S h) (S d) t1)) (H h d f g H1 H2 +H3) (H0 h (S d) (wadd f (S (weight_map f (lift h d t0)))) (wadd g (S +(weight_map g (lift (S h) d t0)))) (\lambda (m: nat).(\lambda (H4: (lt m (S +d))).(or_ind (eq nat m O) (ex2 nat (\lambda (m0: nat).(eq nat m (S m0))) +(\lambda (m0: nat).(lt m0 d))) (eq nat (wadd g (S (weight_map g (lift (S h) d +t0))) m) (wadd f (S (weight_map f (lift h d t0))) m)) (\lambda (H5: (eq nat m +O)).(eq_ind_r nat O (\lambda (n: nat).(eq nat (wadd g (S (weight_map g (lift +(S h) d t0))) n) (wadd f (S (weight_map f (lift h d t0))) n))) (f_equal nat +nat S (weight_map g (lift (S h) d t0)) (weight_map f (lift h d t0)) (sym_eq +nat (weight_map f (lift h d t0)) (weight_map g (lift (S h) d t0)) (H h d f g +H1 H2 H3))) m H5)) (\lambda (H5: (ex2 nat (\lambda (m0: nat).(eq nat m (S +m0))) (\lambda (m0: nat).(lt m0 d)))).(ex2_ind nat (\lambda (m0: nat).(eq nat +m (S m0))) (\lambda (m0: nat).(lt m0 d)) (eq nat (wadd g (S (weight_map g +(lift (S h) d t0))) m) (wadd f (S (weight_map f (lift h d t0))) m)) (\lambda +(x: nat).(\lambda (H6: (eq nat m (S x))).(\lambda (H7: (lt x d)).(eq_ind_r +nat (S x) (\lambda (n: nat).(eq nat (wadd g (S (weight_map g (lift (S h) d +t0))) n) (wadd f (S (weight_map f (lift h d t0))) n))) (H1 x H7) m H6)))) +H5)) (lt_gen_xS m d H4)))) H2 (\lambda (m: nat).(\lambda (H4: (le (S d) +m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S n))) (\lambda (n: nat).(le d +n)) (eq nat (g m) (wadd f (S (weight_map f (lift h d t0))) m)) (\lambda (x: +nat).(\lambda (H5: (eq nat m (S x))).(\lambda (H6: (le d x)).(eq_ind_r nat (S +x) (\lambda (n: nat).(eq nat (g n) (wadd f (S (weight_map f (lift h d t0))) +n))) (H3 x H6) m H5)))) (le_gen_S d m H4))))))) (f_equal nat nat S (plus +(weight_map f (lift h d t0)) (weight_map (wadd f O) (lift h (S d) t1))) (plus +(weight_map g (lift (S h) d t0)) (weight_map (wadd g O) (lift (S h) (S d) +t1))) (f_equal2 nat nat nat plus (weight_map f (lift h d t0)) (weight_map g +(lift (S h) d t0)) (weight_map (wadd f O) (lift h (S d) t1)) (weight_map +(wadd g O) (lift (S h) (S d) t1)) (H h d f g H1 H2 H3) (H0 h (S d) (wadd f O) +(wadd g O) (\lambda (m: nat).(\lambda (H4: (lt m (S d))).(or_ind (eq nat m O) +(ex2 nat (\lambda (m0: nat).(eq nat m (S m0))) (\lambda (m0: nat).(lt m0 d))) +(eq nat (wadd g O m) (wadd f O m)) (\lambda (H5: (eq nat m O)).(eq_ind_r nat +O (\lambda (n: nat).(eq nat (wadd g O n) (wadd f O n))) (refl_equal nat O) m +H5)) (\lambda (H5: (ex2 nat (\lambda (m0: nat).(eq nat m (S m0))) (\lambda +(m0: nat).(lt m0 d)))).(ex2_ind nat (\lambda (m0: nat).(eq nat m (S m0))) +(\lambda (m0: nat).(lt m0 d)) (eq nat (wadd g O m) (wadd f O m)) (\lambda (x: +nat).(\lambda (H6: (eq nat m (S x))).(\lambda (H7: (lt x d)).(eq_ind_r nat (S +x) (\lambda (n: nat).(eq nat (wadd g O n) (wadd f O n))) (H1 x H7) m H6)))) +H5)) (lt_gen_xS m d H4)))) H2 (\lambda (m: nat).(\lambda (H4: (le (S d) +m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S n))) (\lambda (n: nat).(le d +n)) (eq nat (g m) (wadd f O m)) (\lambda (x: nat).(\lambda (H5: (eq nat m (S +x))).(\lambda (H6: (le d x)).(eq_ind_r nat (S x) (\lambda (n: nat).(eq nat (g +n) (wadd f O n))) (H3 x H6) m H5)))) (le_gen_S d m H4))))))) (f_equal nat nat +S (plus (weight_map f (lift h d t0)) (weight_map (wadd f O) (lift h (S d) +t1))) (plus (weight_map g (lift (S h) d t0)) (weight_map (wadd g O) (lift (S +h) (S d) t1))) (f_equal2 nat nat nat plus (weight_map f (lift h d t0)) +(weight_map g (lift (S h) d t0)) (weight_map (wadd f O) (lift h (S d) t1)) +(weight_map (wadd g O) (lift (S h) (S d) t1)) (H h d f g H1 H2 H3) (H0 h (S +d) (wadd f O) (wadd g O) (\lambda (m: nat).(\lambda (H4: (lt m (S +d))).(or_ind (eq nat m O) (ex2 nat (\lambda (m0: nat).(eq nat m (S m0))) +(\lambda (m0: nat).(lt m0 d))) (eq nat (wadd g O m) (wadd f O m)) (\lambda +(H5: (eq nat m O)).(eq_ind_r nat O (\lambda (n: nat).(eq nat (wadd g O n) +(wadd f O n))) (refl_equal nat O) m H5)) (\lambda (H5: (ex2 nat (\lambda (m0: +nat).(eq nat m (S m0))) (\lambda (m0: nat).(lt m0 d)))).(ex2_ind nat (\lambda +(m0: nat).(eq nat m (S m0))) (\lambda (m0: nat).(lt m0 d)) (eq nat (wadd g O +m) (wadd f O m)) (\lambda (x: nat).(\lambda (H6: (eq nat m (S x))).(\lambda +(H7: (lt x d)).(eq_ind_r nat (S x) (\lambda (n: nat).(eq nat (wadd g O n) +(wadd f O n))) (H1 x H7) m H6)))) H5)) (lt_gen_xS m d H4)))) H2 (\lambda (m: +nat).(\lambda (H4: (le (S d) m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S +n))) (\lambda (n: nat).(le d n)) (eq nat (g m) (wadd f O m)) (\lambda (x: +nat).(\lambda (H5: (eq nat m (S x))).(\lambda (H6: (le d x)).(eq_ind_r nat (S +x) (\lambda (n: nat).(eq nat (g n) (wadd f O n))) (H3 x H6) m H5)))) +(le_gen_S d m H4))))))) b) (lift (S h) d (THead (Bind b) t0 t1)) (lift_head +(Bind b) t0 t1 (S h) d)) (lift h d (THead (Bind b) t0 t1)) (lift_head (Bind +b) t0 t1 h d))) (\lambda (f0: F).(eq_ind_r T (THead (Flat f0) (lift h d t0) +(lift h (s (Flat f0) d) t1)) (\lambda (t2: T).(eq nat (weight_map f t2) +(weight_map g (lift (S h) d (THead (Flat f0) t0 t1))))) (eq_ind_r T (THead +(Flat f0) (lift (S h) d t0) (lift (S h) (s (Flat f0) d) t1)) (\lambda (t2: +T).(eq nat (weight_map f (THead (Flat f0) (lift h d t0) (lift h (s (Flat f0) +d) t1))) (weight_map g t2))) (f_equal nat nat S (plus (weight_map f (lift h d +t0)) (weight_map f (lift h d t1))) (plus (weight_map g (lift (S h) d t0)) +(weight_map g (lift (S h) d t1))) (f_equal2 nat nat nat plus (weight_map f +(lift h d t0)) (weight_map g (lift (S h) d t0)) (weight_map f (lift h d t1)) +(weight_map g (lift (S h) d t1)) (H h d f g H1 H2 H3) (H0 h d f g H1 H2 H3))) +(lift (S h) d (THead (Flat f0) t0 t1)) (lift_head (Flat f0) t0 t1 (S h) d)) +(lift h d (THead (Flat f0) t0 t1)) (lift_head (Flat f0) t0 t1 h d))) +k))))))))))))) t)). +(* COMMENTS +Initial nodes: 3697 +END *) + +theorem lift_weight_add_O: + \forall (w: nat).(\forall (t: T).(\forall (h: nat).(\forall (f: ((nat \to +nat))).(eq nat (weight_map f (lift h O t)) (weight_map (wadd f w) (lift (S h) +O t)))))) +\def + \lambda (w: nat).(\lambda (t: T).(\lambda (h: nat).(\lambda (f: ((nat \to +nat))).(lift_weight_add (plus (wadd f w O) O) t h O f (wadd f w) (\lambda (m: +nat).(\lambda (H: (lt m O)).(lt_x_O m H (eq nat (wadd f w m) (f m))))) +(plus_n_O (wadd f w O)) (\lambda (m: nat).(\lambda (_: (le O m)).(refl_equal +nat (f m)))))))). +(* COMMENTS +Initial nodes: 93 +END *) + +theorem lift_tlt_dx: + \forall (k: K).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall +(d: nat).(tlt t (THead k u (lift h d t))))))) +\def + \lambda (k: K).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda +(d: nat).(eq_ind nat (weight (lift h d t)) (\lambda (n: nat).(lt n (weight +(THead k u (lift h d t))))) (tlt_head_dx k u (lift h d t)) (weight t) +(lift_weight t h d)))))). +(* COMMENTS +Initial nodes: 71 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/lift1/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/lift1/defs.ma new file mode 100644 index 000000000..1f473cbce --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/lift1/defs.ma @@ -0,0 +1,40 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/lift/defs.ma". + +definition trans: + PList \to (nat \to nat) +\def + let rec trans (hds: PList) on hds: (nat \to nat) \def (\lambda (i: +nat).(match hds with [PNil \Rightarrow i | (PCons h d hds0) \Rightarrow (let +j \def (trans hds0 i) in (match (blt j d) with [true \Rightarrow j | false +\Rightarrow (plus j h)]))])) in trans. + +definition lift1: + PList \to (T \to T) +\def + let rec lift1 (hds: PList) on hds: (T \to T) \def (\lambda (t: T).(match hds +with [PNil \Rightarrow t | (PCons h d hds0) \Rightarrow (lift h d (lift1 hds0 +t))])) in lift1. + +definition lifts1: + PList \to (TList \to TList) +\def + let rec lifts1 (hds: PList) (ts: TList) on ts: TList \def (match ts with +[TNil \Rightarrow TNil | (TCons t ts0) \Rightarrow (TCons (lift1 hds t) +(lifts1 hds ts0))]) in lifts1. + diff --git a/matita/matita/contribs/lambdadelta/basic_1/lift1/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/lift1/fwd.ma new file mode 100644 index 000000000..e9ae2d11c --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/lift1/fwd.ma @@ -0,0 +1,164 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/lift1/defs.ma". + +include "Basic-1/lift/fwd.ma". + +theorem lift1_sort: + \forall (n: nat).(\forall (is: PList).(eq T (lift1 is (TSort n)) (TSort n))) +\def + \lambda (n: nat).(\lambda (is: PList).(PList_ind (\lambda (p: PList).(eq T +(lift1 p (TSort n)) (TSort n))) (refl_equal T (TSort n)) (\lambda (n0: +nat).(\lambda (n1: nat).(\lambda (p: PList).(\lambda (H: (eq T (lift1 p +(TSort n)) (TSort n))).(eq_ind_r T (TSort n) (\lambda (t: T).(eq T (lift n0 +n1 t) (TSort n))) (refl_equal T (TSort n)) (lift1 p (TSort n)) H))))) is)). +(* COMMENTS +Initial nodes: 99 +END *) + +theorem lift1_lref: + \forall (hds: PList).(\forall (i: nat).(eq T (lift1 hds (TLRef i)) (TLRef +(trans hds i)))) +\def + \lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (i: nat).(eq T +(lift1 p (TLRef i)) (TLRef (trans p i))))) (\lambda (i: nat).(refl_equal T +(TLRef i))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda +(H: ((\forall (i: nat).(eq T (lift1 p (TLRef i)) (TLRef (trans p +i)))))).(\lambda (i: nat).(eq_ind_r T (TLRef (trans p i)) (\lambda (t: T).(eq +T (lift n n0 t) (TLRef (match (blt (trans p i) n0) with [true \Rightarrow +(trans p i) | false \Rightarrow (plus (trans p i) n)])))) (refl_equal T +(TLRef (match (blt (trans p i) n0) with [true \Rightarrow (trans p i) | false +\Rightarrow (plus (trans p i) n)]))) (lift1 p (TLRef i)) (H i))))))) hds). +(* COMMENTS +Initial nodes: 165 +END *) + +theorem lift1_bind: + \forall (b: B).(\forall (hds: PList).(\forall (u: T).(\forall (t: T).(eq T +(lift1 hds (THead (Bind b) u t)) (THead (Bind b) (lift1 hds u) (lift1 (Ss +hds) t)))))) +\def + \lambda (b: B).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall +(u: T).(\forall (t: T).(eq T (lift1 p (THead (Bind b) u t)) (THead (Bind b) +(lift1 p u) (lift1 (Ss p) t)))))) (\lambda (u: T).(\lambda (t: T).(refl_equal +T (THead (Bind b) u t)))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: +PList).(\lambda (H: ((\forall (u: T).(\forall (t: T).(eq T (lift1 p (THead +(Bind b) u t)) (THead (Bind b) (lift1 p u) (lift1 (Ss p) t))))))).(\lambda +(u: T).(\lambda (t: T).(eq_ind_r T (THead (Bind b) (lift1 p u) (lift1 (Ss p) +t)) (\lambda (t0: T).(eq T (lift n n0 t0) (THead (Bind b) (lift n n0 (lift1 p +u)) (lift n (S n0) (lift1 (Ss p) t))))) (eq_ind_r T (THead (Bind b) (lift n +n0 (lift1 p u)) (lift n (S n0) (lift1 (Ss p) t))) (\lambda (t0: T).(eq T t0 +(THead (Bind b) (lift n n0 (lift1 p u)) (lift n (S n0) (lift1 (Ss p) t))))) +(refl_equal T (THead (Bind b) (lift n n0 (lift1 p u)) (lift n (S n0) (lift1 +(Ss p) t)))) (lift n n0 (THead (Bind b) (lift1 p u) (lift1 (Ss p) t))) +(lift_bind b (lift1 p u) (lift1 (Ss p) t) n n0)) (lift1 p (THead (Bind b) u +t)) (H u t)))))))) hds)). +(* COMMENTS +Initial nodes: 379 +END *) + +theorem lift1_flat: + \forall (f: F).(\forall (hds: PList).(\forall (u: T).(\forall (t: T).(eq T +(lift1 hds (THead (Flat f) u t)) (THead (Flat f) (lift1 hds u) (lift1 hds +t)))))) +\def + \lambda (f: F).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall +(u: T).(\forall (t: T).(eq T (lift1 p (THead (Flat f) u t)) (THead (Flat f) +(lift1 p u) (lift1 p t)))))) (\lambda (u: T).(\lambda (t: T).(refl_equal T +(THead (Flat f) u t)))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: +PList).(\lambda (H: ((\forall (u: T).(\forall (t: T).(eq T (lift1 p (THead +(Flat f) u t)) (THead (Flat f) (lift1 p u) (lift1 p t))))))).(\lambda (u: +T).(\lambda (t: T).(eq_ind_r T (THead (Flat f) (lift1 p u) (lift1 p t)) +(\lambda (t0: T).(eq T (lift n n0 t0) (THead (Flat f) (lift n n0 (lift1 p u)) +(lift n n0 (lift1 p t))))) (eq_ind_r T (THead (Flat f) (lift n n0 (lift1 p +u)) (lift n n0 (lift1 p t))) (\lambda (t0: T).(eq T t0 (THead (Flat f) (lift +n n0 (lift1 p u)) (lift n n0 (lift1 p t))))) (refl_equal T (THead (Flat f) +(lift n n0 (lift1 p u)) (lift n n0 (lift1 p t)))) (lift n n0 (THead (Flat f) +(lift1 p u) (lift1 p t))) (lift_flat f (lift1 p u) (lift1 p t) n n0)) (lift1 +p (THead (Flat f) u t)) (H u t)))))))) hds)). +(* COMMENTS +Initial nodes: 353 +END *) + +theorem lift1_cons_tail: + \forall (t: T).(\forall (h: nat).(\forall (d: nat).(\forall (hds: PList).(eq +T (lift1 (PConsTail hds h d) t) (lift1 hds (lift h d t)))))) +\def + \lambda (t: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (hds: +PList).(PList_ind (\lambda (p: PList).(eq T (lift1 (PConsTail p h d) t) +(lift1 p (lift h d t)))) (refl_equal T (lift h d t)) (\lambda (n: +nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: (eq T (lift1 +(PConsTail p h d) t) (lift1 p (lift h d t)))).(eq_ind_r T (lift1 p (lift h d +t)) (\lambda (t0: T).(eq T (lift n n0 t0) (lift n n0 (lift1 p (lift h d +t))))) (refl_equal T (lift n n0 (lift1 p (lift h d t)))) (lift1 (PConsTail p +h d) t) H))))) hds)))). +(* COMMENTS +Initial nodes: 171 +END *) + +theorem lifts1_flat: + \forall (f: F).(\forall (hds: PList).(\forall (t: T).(\forall (ts: +TList).(eq T (lift1 hds (THeads (Flat f) ts t)) (THeads (Flat f) (lifts1 hds +ts) (lift1 hds t)))))) +\def + \lambda (f: F).(\lambda (hds: PList).(\lambda (t: T).(\lambda (ts: +TList).(TList_ind (\lambda (t0: TList).(eq T (lift1 hds (THeads (Flat f) t0 +t)) (THeads (Flat f) (lifts1 hds t0) (lift1 hds t)))) (refl_equal T (lift1 +hds t)) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H: (eq T (lift1 hds +(THeads (Flat f) t1 t)) (THeads (Flat f) (lifts1 hds t1) (lift1 hds +t)))).(eq_ind_r T (THead (Flat f) (lift1 hds t0) (lift1 hds (THeads (Flat f) +t1 t))) (\lambda (t2: T).(eq T t2 (THead (Flat f) (lift1 hds t0) (THeads +(Flat f) (lifts1 hds t1) (lift1 hds t))))) (eq_ind_r T (THeads (Flat f) +(lifts1 hds t1) (lift1 hds t)) (\lambda (t2: T).(eq T (THead (Flat f) (lift1 +hds t0) t2) (THead (Flat f) (lift1 hds t0) (THeads (Flat f) (lifts1 hds t1) +(lift1 hds t))))) (refl_equal T (THead (Flat f) (lift1 hds t0) (THeads (Flat +f) (lifts1 hds t1) (lift1 hds t)))) (lift1 hds (THeads (Flat f) t1 t)) H) +(lift1 hds (THead (Flat f) t0 (THeads (Flat f) t1 t))) (lift1_flat f hds t0 +(THeads (Flat f) t1 t)))))) ts)))). +(* COMMENTS +Initial nodes: 329 +END *) + +theorem lifts1_nil: + \forall (ts: TList).(eq TList (lifts1 PNil ts) ts) +\def + \lambda (ts: TList).(TList_ind (\lambda (t: TList).(eq TList (lifts1 PNil t) +t)) (refl_equal TList TNil) (\lambda (t: T).(\lambda (t0: TList).(\lambda (H: +(eq TList (lifts1 PNil t0) t0)).(eq_ind_r TList t0 (\lambda (t1: TList).(eq +TList (TCons t t1) (TCons t t0))) (refl_equal TList (TCons t t0)) (lifts1 +PNil t0) H)))) ts). +(* COMMENTS +Initial nodes: 83 +END *) + +theorem lifts1_cons: + \forall (h: nat).(\forall (d: nat).(\forall (hds: PList).(\forall (ts: +TList).(eq TList (lifts1 (PCons h d hds) ts) (lifts h d (lifts1 hds ts)))))) +\def + \lambda (h: nat).(\lambda (d: nat).(\lambda (hds: PList).(\lambda (ts: +TList).(TList_ind (\lambda (t: TList).(eq TList (lifts1 (PCons h d hds) t) +(lifts h d (lifts1 hds t)))) (refl_equal TList TNil) (\lambda (t: T).(\lambda +(t0: TList).(\lambda (H: (eq TList (lifts1 (PCons h d hds) t0) (lifts h d +(lifts1 hds t0)))).(eq_ind_r TList (lifts h d (lifts1 hds t0)) (\lambda (t1: +TList).(eq TList (TCons (lift h d (lift1 hds t)) t1) (TCons (lift h d (lift1 +hds t)) (lifts h d (lifts1 hds t0))))) (refl_equal TList (TCons (lift h d +(lift1 hds t)) (lifts h d (lifts1 hds t0)))) (lifts1 (PCons h d hds) t0) +H)))) ts)))). +(* COMMENTS +Initial nodes: 187 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/lift1/props.ma b/matita/matita/contribs/lambdadelta/basic_1/lift1/props.ma new file mode 100644 index 000000000..ebda0267b --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/lift1/props.ma @@ -0,0 +1,139 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/lift/props.ma". + +include "Basic-1/drop1/defs.ma". + +theorem lift1_lift1: + \forall (is1: PList).(\forall (is2: PList).(\forall (t: T).(eq T (lift1 is1 +(lift1 is2 t)) (lift1 (papp is1 is2) t)))) +\def + \lambda (is1: PList).(PList_ind (\lambda (p: PList).(\forall (is2: +PList).(\forall (t: T).(eq T (lift1 p (lift1 is2 t)) (lift1 (papp p is2) +t))))) (\lambda (is2: PList).(\lambda (t: T).(refl_equal T (lift1 is2 t)))) +(\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: +((\forall (is2: PList).(\forall (t: T).(eq T (lift1 p (lift1 is2 t)) (lift1 +(papp p is2) t)))))).(\lambda (is2: PList).(\lambda (t: T).(f_equal3 nat nat +T T lift n n n0 n0 (lift1 p (lift1 is2 t)) (lift1 (papp p is2) t) (refl_equal +nat n) (refl_equal nat n0) (H is2 t)))))))) is1). +(* COMMENTS +Initial nodes: 145 +END *) + +theorem lift1_xhg: + \forall (hds: PList).(\forall (t: T).(eq T (lift1 (Ss hds) (lift (S O) O t)) +(lift (S O) O (lift1 hds t)))) +\def + \lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (t: T).(eq T +(lift1 (Ss p) (lift (S O) O t)) (lift (S O) O (lift1 p t))))) (\lambda (t: +T).(refl_equal T (lift (S O) O t))) (\lambda (h: nat).(\lambda (d: +nat).(\lambda (p: PList).(\lambda (H: ((\forall (t: T).(eq T (lift1 (Ss p) +(lift (S O) O t)) (lift (S O) O (lift1 p t)))))).(\lambda (t: T).(eq_ind_r T +(lift (S O) O (lift1 p t)) (\lambda (t0: T).(eq T (lift h (S d) t0) (lift (S +O) O (lift h d (lift1 p t))))) (eq_ind nat (plus (S O) d) (\lambda (n: +nat).(eq T (lift h n (lift (S O) O (lift1 p t))) (lift (S O) O (lift h d +(lift1 p t))))) (eq_ind_r T (lift (S O) O (lift h d (lift1 p t))) (\lambda +(t0: T).(eq T t0 (lift (S O) O (lift h d (lift1 p t))))) (refl_equal T (lift +(S O) O (lift h d (lift1 p t)))) (lift h (plus (S O) d) (lift (S O) O (lift1 +p t))) (lift_d (lift1 p t) h (S O) d O (le_O_n d))) (S d) (refl_equal nat (S +d))) (lift1 (Ss p) (lift (S O) O t)) (H t))))))) hds). +(* COMMENTS +Initial nodes: 371 +END *) + +theorem lifts1_xhg: + \forall (hds: PList).(\forall (ts: TList).(eq TList (lifts1 (Ss hds) (lifts +(S O) O ts)) (lifts (S O) O (lifts1 hds ts)))) +\def + \lambda (hds: PList).(\lambda (ts: TList).(TList_ind (\lambda (t: TList).(eq +TList (lifts1 (Ss hds) (lifts (S O) O t)) (lifts (S O) O (lifts1 hds t)))) +(refl_equal TList TNil) (\lambda (t: T).(\lambda (t0: TList).(\lambda (H: (eq +TList (lifts1 (Ss hds) (lifts (S O) O t0)) (lifts (S O) O (lifts1 hds +t0)))).(eq_ind_r T (lift (S O) O (lift1 hds t)) (\lambda (t1: T).(eq TList +(TCons t1 (lifts1 (Ss hds) (lifts (S O) O t0))) (TCons (lift (S O) O (lift1 +hds t)) (lifts (S O) O (lifts1 hds t0))))) (eq_ind_r TList (lifts (S O) O +(lifts1 hds t0)) (\lambda (t1: TList).(eq TList (TCons (lift (S O) O (lift1 +hds t)) t1) (TCons (lift (S O) O (lift1 hds t)) (lifts (S O) O (lifts1 hds +t0))))) (refl_equal TList (TCons (lift (S O) O (lift1 hds t)) (lifts (S O) O +(lifts1 hds t0)))) (lifts1 (Ss hds) (lifts (S O) O t0)) H) (lift1 (Ss hds) +(lift (S O) O t)) (lift1_xhg hds t))))) ts)). +(* COMMENTS +Initial nodes: 307 +END *) + +theorem lift1_free: + \forall (hds: PList).(\forall (i: nat).(\forall (t: T).(eq T (lift1 hds +(lift (S i) O t)) (lift (S (trans hds i)) O (lift1 (ptrans hds i) t))))) +\def + \lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (i: +nat).(\forall (t: T).(eq T (lift1 p (lift (S i) O t)) (lift (S (trans p i)) O +(lift1 (ptrans p i) t)))))) (\lambda (i: nat).(\lambda (t: T).(refl_equal T +(lift (S i) O t)))) (\lambda (h: nat).(\lambda (d: nat).(\lambda (hds0: +PList).(\lambda (H: ((\forall (i: nat).(\forall (t: T).(eq T (lift1 hds0 +(lift (S i) O t)) (lift (S (trans hds0 i)) O (lift1 (ptrans hds0 i) +t))))))).(\lambda (i: nat).(\lambda (t: T).(eq_ind_r T (lift (S (trans hds0 +i)) O (lift1 (ptrans hds0 i) t)) (\lambda (t0: T).(eq T (lift h d t0) (lift +(S (match (blt (trans hds0 i) d) with [true \Rightarrow (trans hds0 i) | +false \Rightarrow (plus (trans hds0 i) h)])) O (lift1 (match (blt (trans hds0 +i) d) with [true \Rightarrow (PCons h (minus d (S (trans hds0 i))) (ptrans +hds0 i)) | false \Rightarrow (ptrans hds0 i)]) t)))) (xinduction bool (blt +(trans hds0 i) d) (\lambda (b: bool).(eq T (lift h d (lift (S (trans hds0 i)) +O (lift1 (ptrans hds0 i) t))) (lift (S (match b with [true \Rightarrow (trans +hds0 i) | false \Rightarrow (plus (trans hds0 i) h)])) O (lift1 (match b with +[true \Rightarrow (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) | +false \Rightarrow (ptrans hds0 i)]) t)))) (\lambda (x_x: bool).(bool_ind +(\lambda (b: bool).((eq bool (blt (trans hds0 i) d) b) \to (eq T (lift h d +(lift (S (trans hds0 i)) O (lift1 (ptrans hds0 i) t))) (lift (S (match b with +[true \Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i) +h)])) O (lift1 (match b with [true \Rightarrow (PCons h (minus d (S (trans +hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)]) t))))) +(\lambda (H0: (eq bool (blt (trans hds0 i) d) true)).(eq_ind_r nat (plus (S +(trans hds0 i)) (minus d (S (trans hds0 i)))) (\lambda (n: nat).(eq T (lift h +n (lift (S (trans hds0 i)) O (lift1 (ptrans hds0 i) t))) (lift (S (trans hds0 +i)) O (lift1 (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) t)))) +(eq_ind_r T (lift (S (trans hds0 i)) O (lift h (minus d (S (trans hds0 i))) +(lift1 (ptrans hds0 i) t))) (\lambda (t0: T).(eq T t0 (lift (S (trans hds0 +i)) O (lift1 (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) t)))) +(refl_equal T (lift (S (trans hds0 i)) O (lift1 (PCons h (minus d (S (trans +hds0 i))) (ptrans hds0 i)) t))) (lift h (plus (S (trans hds0 i)) (minus d (S +(trans hds0 i)))) (lift (S (trans hds0 i)) O (lift1 (ptrans hds0 i) t))) +(lift_d (lift1 (ptrans hds0 i) t) h (S (trans hds0 i)) (minus d (S (trans +hds0 i))) O (le_O_n (minus d (S (trans hds0 i)))))) d (le_plus_minus (S +(trans hds0 i)) d (bge_le (S (trans hds0 i)) d (le_bge (S (trans hds0 i)) d +(lt_le_S (trans hds0 i) d (blt_lt d (trans hds0 i) H0))))))) (\lambda (H0: +(eq bool (blt (trans hds0 i) d) false)).(eq_ind_r T (lift (plus h (S (trans +hds0 i))) O (lift1 (ptrans hds0 i) t)) (\lambda (t0: T).(eq T t0 (lift (S +(plus (trans hds0 i) h)) O (lift1 (ptrans hds0 i) t)))) (eq_ind nat (S (plus +h (trans hds0 i))) (\lambda (n: nat).(eq T (lift n O (lift1 (ptrans hds0 i) +t)) (lift (S (plus (trans hds0 i) h)) O (lift1 (ptrans hds0 i) t)))) +(eq_ind_r nat (plus (trans hds0 i) h) (\lambda (n: nat).(eq T (lift (S n) O +(lift1 (ptrans hds0 i) t)) (lift (S (plus (trans hds0 i) h)) O (lift1 (ptrans +hds0 i) t)))) (refl_equal T (lift (S (plus (trans hds0 i) h)) O (lift1 +(ptrans hds0 i) t))) (plus h (trans hds0 i)) (plus_sym h (trans hds0 i))) +(plus h (S (trans hds0 i))) (plus_n_Sm h (trans hds0 i))) (lift h d (lift (S +(trans hds0 i)) O (lift1 (ptrans hds0 i) t))) (lift_free (lift1 (ptrans hds0 +i) t) (S (trans hds0 i)) h O d (eq_ind nat (S (plus O (trans hds0 i))) +(\lambda (n: nat).(le d n)) (eq_ind_r nat (plus (trans hds0 i) O) (\lambda +(n: nat).(le d (S n))) (le_S d (plus (trans hds0 i) O) (le_plus_trans d +(trans hds0 i) O (bge_le d (trans hds0 i) H0))) (plus O (trans hds0 i)) +(plus_sym O (trans hds0 i))) (plus O (S (trans hds0 i))) (plus_n_Sm O (trans +hds0 i))) (le_O_n d)))) x_x))) (lift1 hds0 (lift (S i) O t)) (H i t)))))))) +hds). +(* COMMENTS +Initial nodes: 1339 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/llt/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/llt/defs.ma new file mode 100644 index 000000000..96d869935 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/llt/defs.ma @@ -0,0 +1,30 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/A/defs.ma". + +definition lweight: + A \to nat +\def + let rec lweight (a: A) on a: nat \def (match a with [(ASort _ _) \Rightarrow +O | (AHead a1 a2) \Rightarrow (S (plus (lweight a1) (lweight a2)))]) in +lweight. + +definition llt: + A \to (A \to Prop) +\def + \lambda (a1: A).(\lambda (a2: A).(lt (lweight a1) (lweight a2))). + diff --git a/matita/matita/contribs/lambdadelta/basic_1/llt/props.ma b/matita/matita/contribs/lambdadelta/basic_1/llt/props.ma new file mode 100644 index 000000000..ef1f15a96 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/llt/props.ma @@ -0,0 +1,114 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/llt/defs.ma". + +include "Basic-1/leq/defs.ma". + +theorem lweight_repl: + \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (eq nat +(lweight a1) (lweight a2))))) +\def + \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1 +a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(eq nat (lweight a) (lweight +a0)))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: nat).(\lambda (n2: +nat).(\lambda (k: nat).(\lambda (_: (eq A (aplus g (ASort h1 n1) k) (aplus g +(ASort h2 n2) k))).(refl_equal nat O))))))) (\lambda (a0: A).(\lambda (a3: +A).(\lambda (_: (leq g a0 a3)).(\lambda (H1: (eq nat (lweight a0) (lweight +a3))).(\lambda (a4: A).(\lambda (a5: A).(\lambda (_: (leq g a4 a5)).(\lambda +(H3: (eq nat (lweight a4) (lweight a5))).(f_equal nat nat S (plus (lweight +a0) (lweight a4)) (plus (lweight a3) (lweight a5)) (f_equal2 nat nat nat plus +(lweight a0) (lweight a3) (lweight a4) (lweight a5) H1 H3)))))))))) a1 a2 +H)))). +(* COMMENTS +Initial nodes: 189 +END *) + +theorem llt_repl: + \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall +(a3: A).((llt a1 a3) \to (llt a2 a3)))))) +\def + \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1 +a2)).(\lambda (a3: A).(\lambda (H0: (lt (lweight a1) (lweight a3))).(let H1 +\def (eq_ind nat (lweight a1) (\lambda (n: nat).(lt n (lweight a3))) H0 +(lweight a2) (lweight_repl g a1 a2 H)) in H1)))))). +(* COMMENTS +Initial nodes: 61 +END *) + +theorem llt_trans: + \forall (a1: A).(\forall (a2: A).(\forall (a3: A).((llt a1 a2) \to ((llt a2 +a3) \to (llt a1 a3))))) +\def + \lambda (a1: A).(\lambda (a2: A).(\lambda (a3: A).(\lambda (H: (lt (lweight +a1) (lweight a2))).(\lambda (H0: (lt (lweight a2) (lweight a3))).(lt_trans +(lweight a1) (lweight a2) (lweight a3) H H0))))). +(* COMMENTS +Initial nodes: 43 +END *) + +theorem llt_head_sx: + \forall (a1: A).(\forall (a2: A).(llt a1 (AHead a1 a2))) +\def + \lambda (a1: A).(\lambda (a2: A).(le_n_S (lweight a1) (plus (lweight a1) +(lweight a2)) (le_plus_l (lweight a1) (lweight a2)))). +(* COMMENTS +Initial nodes: 29 +END *) + +theorem llt_head_dx: + \forall (a1: A).(\forall (a2: A).(llt a2 (AHead a1 a2))) +\def + \lambda (a1: A).(\lambda (a2: A).(le_n_S (lweight a2) (plus (lweight a1) +(lweight a2)) (le_plus_r (lweight a1) (lweight a2)))). +(* COMMENTS +Initial nodes: 29 +END *) + +theorem llt_wf__q_ind: + \forall (P: ((A \to Prop))).(((\forall (n: nat).((\lambda (P0: ((A \to +Prop))).(\lambda (n0: nat).(\forall (a: A).((eq nat (lweight a) n0) \to (P0 +a))))) P n))) \to (\forall (a: A).(P a))) +\def + let Q \def (\lambda (P: ((A \to Prop))).(\lambda (n: nat).(\forall (a: +A).((eq nat (lweight a) n) \to (P a))))) in (\lambda (P: ((A \to +Prop))).(\lambda (H: ((\forall (n: nat).(\forall (a: A).((eq nat (lweight a) +n) \to (P a)))))).(\lambda (a: A).(H (lweight a) a (refl_equal nat (lweight +a)))))). +(* COMMENTS +Initial nodes: 61 +END *) + +theorem llt_wf_ind: + \forall (P: ((A \to Prop))).(((\forall (a2: A).(((\forall (a1: A).((llt a1 +a2) \to (P a1)))) \to (P a2)))) \to (\forall (a: A).(P a))) +\def + let Q \def (\lambda (P: ((A \to Prop))).(\lambda (n: nat).(\forall (a: +A).((eq nat (lweight a) n) \to (P a))))) in (\lambda (P: ((A \to +Prop))).(\lambda (H: ((\forall (a2: A).(((\forall (a1: A).((lt (lweight a1) +(lweight a2)) \to (P a1)))) \to (P a2))))).(\lambda (a: A).(llt_wf__q_ind +(\lambda (a0: A).(P a0)) (\lambda (n: nat).(lt_wf_ind n (Q (\lambda (a0: +A).(P a0))) (\lambda (n0: nat).(\lambda (H0: ((\forall (m: nat).((lt m n0) +\to (Q (\lambda (a0: A).(P a0)) m))))).(\lambda (a0: A).(\lambda (H1: (eq nat +(lweight a0) n0)).(let H2 \def (eq_ind_r nat n0 (\lambda (n1: nat).(\forall +(m: nat).((lt m n1) \to (\forall (a1: A).((eq nat (lweight a1) m) \to (P +a1)))))) H0 (lweight a0) H1) in (H a0 (\lambda (a1: A).(\lambda (H3: (lt +(lweight a1) (lweight a0))).(H2 (lweight a1) H3 a1 (refl_equal nat (lweight +a1))))))))))))) a)))). +(* COMMENTS +Initial nodes: 179 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/next_plus/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/next_plus/defs.ma new file mode 100644 index 000000000..1ff7ecdff --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/next_plus/defs.ma @@ -0,0 +1,24 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/G/defs.ma". + +definition next_plus: + G \to (nat \to (nat \to nat)) +\def + let rec next_plus (g: G) (n: nat) (i: nat) on i: nat \def (match i with [O +\Rightarrow n | (S i0) \Rightarrow (next g (next_plus g n i0))]) in next_plus. + diff --git a/matita/matita/contribs/lambdadelta/basic_1/next_plus/props.ma b/matita/matita/contribs/lambdadelta/basic_1/next_plus/props.ma new file mode 100644 index 000000000..258c3a711 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/next_plus/props.ma @@ -0,0 +1,68 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/next_plus/defs.ma". + +theorem next_plus_assoc: + \forall (g: G).(\forall (n: nat).(\forall (h1: nat).(\forall (h2: nat).(eq +nat (next_plus g (next_plus g n h1) h2) (next_plus g n (plus h1 h2)))))) +\def + \lambda (g: G).(\lambda (n: nat).(\lambda (h1: nat).(nat_ind (\lambda (n0: +nat).(\forall (h2: nat).(eq nat (next_plus g (next_plus g n n0) h2) +(next_plus g n (plus n0 h2))))) (\lambda (h2: nat).(refl_equal nat (next_plus +g n h2))) (\lambda (n0: nat).(\lambda (_: ((\forall (h2: nat).(eq nat +(next_plus g (next_plus g n n0) h2) (next_plus g n (plus n0 h2)))))).(\lambda +(h2: nat).(nat_ind (\lambda (n1: nat).(eq nat (next_plus g (next g (next_plus +g n n0)) n1) (next g (next_plus g n (plus n0 n1))))) (eq_ind nat n0 (\lambda +(n1: nat).(eq nat (next g (next_plus g n n0)) (next g (next_plus g n n1)))) +(refl_equal nat (next g (next_plus g n n0))) (plus n0 O) (plus_n_O n0)) +(\lambda (n1: nat).(\lambda (H0: (eq nat (next_plus g (next g (next_plus g n +n0)) n1) (next g (next_plus g n (plus n0 n1))))).(eq_ind nat (S (plus n0 n1)) +(\lambda (n2: nat).(eq nat (next g (next_plus g (next g (next_plus g n n0)) +n1)) (next g (next_plus g n n2)))) (f_equal nat nat (next g) (next_plus g +(next g (next_plus g n n0)) n1) (next g (next_plus g n (plus n0 n1))) H0) +(plus n0 (S n1)) (plus_n_Sm n0 n1)))) h2)))) h1))). +(* COMMENTS +Initial nodes: 351 +END *) + +theorem next_plus_next: + \forall (g: G).(\forall (n: nat).(\forall (h: nat).(eq nat (next_plus g +(next g n) h) (next g (next_plus g n h))))) +\def + \lambda (g: G).(\lambda (n: nat).(\lambda (h: nat).(eq_ind_r nat (next_plus +g n (plus (S O) h)) (\lambda (n0: nat).(eq nat n0 (next g (next_plus g n +h)))) (refl_equal nat (next g (next_plus g n h))) (next_plus g (next_plus g n +(S O)) h) (next_plus_assoc g n (S O) h)))). +(* COMMENTS +Initial nodes: 87 +END *) + +theorem next_plus_lt: + \forall (g: G).(\forall (h: nat).(\forall (n: nat).(lt n (next_plus g (next +g n) h)))) +\def + \lambda (g: G).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall (n0: +nat).(lt n0 (next_plus g (next g n0) n)))) (\lambda (n: nat).(next_lt g n)) +(\lambda (n: nat).(\lambda (H: ((\forall (n0: nat).(lt n0 (next_plus g (next +g n0) n))))).(\lambda (n0: nat).(eq_ind nat (next_plus g (next g (next g n0)) +n) (\lambda (n1: nat).(lt n0 n1)) (lt_trans n0 (next g n0) (next_plus g (next +g (next g n0)) n) (next_lt g n0) (H (next g n0))) (next g (next_plus g (next +g n0) n)) (next_plus_next g (next g n0) n))))) h)). +(* COMMENTS +Initial nodes: 153 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/nf2/arity.ma b/matita/matita/contribs/lambdadelta/basic_1/nf2/arity.ma new file mode 100644 index 000000000..98770d9e9 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/nf2/arity.ma @@ -0,0 +1,496 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/nf2/fwd.ma". + +include "Basic-1/arity/subst0.ma". + +theorem arity_nf2_inv_all: + \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t +a) \to ((nf2 c t) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t +(THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c w))) +(\lambda (w: T).(\lambda (u: T).(nf2 (CHead c (Bind Abst) w) u)))) (ex nat +(\lambda (n: nat).(eq T t (TSort n)))) (ex3_2 TList nat (\lambda (ws: +TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i))))) +(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: +TList).(\lambda (i: nat).(nf2 c (TLRef i))))))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: +(arity g c t a)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (_: +A).((nf2 c0 t0) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t0 +(THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) +(\lambda (w: T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat +(\lambda (n: nat).(eq T t0 (TSort n)))) (ex3_2 TList nat (\lambda (ws: +TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i))))) +(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: +TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))))))) (\lambda (c0: C).(\lambda +(n: nat).(\lambda (_: (nf2 c0 (TSort n))).(or3_intro1 (ex3_2 T T (\lambda (w: +T).(\lambda (u: T).(eq T (TSort n) (THead (Bind Abst) w u)))) (\lambda (w: +T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead +c0 (Bind Abst) w) u)))) (ex nat (\lambda (n0: nat).(eq T (TSort n) (TSort +n0)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (TSort +n) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: +nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef +i))))) (ex_intro nat (\lambda (n0: nat).(eq T (TSort n) (TSort n0))) n +(refl_equal T (TSort n))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: +T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr) +u))).(\lambda (a0: A).(\lambda (_: (arity g d u a0)).(\lambda (_: (((nf2 d u) +\to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u (THead (Bind +Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 d w))) (\lambda (w: +T).(\lambda (u0: T).(nf2 (CHead d (Bind Abst) w) u0)))) (ex nat (\lambda (n: +nat).(eq T u (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i0: +nat).(eq T u (THeads (Flat Appl) ws (TLRef i0))))) (\lambda (ws: +TList).(\lambda (_: nat).(nfs2 d ws))) (\lambda (_: TList).(\lambda (i0: +nat).(nf2 d (TLRef i0))))))))).(\lambda (H3: (nf2 c0 (TLRef +i))).(nf2_gen_lref c0 d u i H0 H3 (or3 (ex3_2 T T (\lambda (w: T).(\lambda +(u0: T).(eq T (TLRef i) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda +(_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind +Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (TLRef i) (TSort n)))) (ex3_2 +TList nat (\lambda (ws: TList).(\lambda (i0: nat).(eq T (TLRef i) (THeads +(Flat Appl) ws (TLRef i0))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 +ws))) (\lambda (_: TList).(\lambda (i0: nat).(nf2 c0 (TLRef +i0)))))))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda +(i: nat).(\lambda (_: (getl i c0 (CHead d (Bind Abst) u))).(\lambda (a0: +A).(\lambda (_: (arity g d u (asucc g a0))).(\lambda (_: (((nf2 d u) \to (or3 +(ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u (THead (Bind Abst) w +u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 d w))) (\lambda (w: T).(\lambda +(u0: T).(nf2 (CHead d (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T u +(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i0: nat).(eq T u +(THeads (Flat Appl) ws (TLRef i0))))) (\lambda (ws: TList).(\lambda (_: +nat).(nfs2 d ws))) (\lambda (_: TList).(\lambda (i0: nat).(nf2 d (TLRef +i0))))))))).(\lambda (H3: (nf2 c0 (TLRef i))).(or3_intro2 (ex3_2 T T (\lambda +(w: T).(\lambda (u0: T).(eq T (TLRef i) (THead (Bind Abst) w u0)))) (\lambda +(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 +(CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (TLRef i) +(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i0: nat).(eq T +(TLRef i) (THeads (Flat Appl) ws (TLRef i0))))) (\lambda (ws: TList).(\lambda +(_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i0: nat).(nf2 c0 (TLRef +i0))))) (ex3_2_intro TList nat (\lambda (ws: TList).(\lambda (i0: nat).(eq T +(TLRef i) (THeads (Flat Appl) ws (TLRef i0))))) (\lambda (ws: TList).(\lambda +(_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i0: nat).(nf2 c0 (TLRef +i0)))) TNil i (refl_equal T (TLRef i)) I H3))))))))))) (\lambda (b: +B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c0: C).(\lambda (u: +T).(\lambda (a1: A).(\lambda (_: (arity g c0 u a1)).(\lambda (_: (((nf2 c0 u) +\to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u (THead (Bind +Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: +T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: +nat).(eq T u (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: +nat).(eq T u (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: +TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: +nat).(nf2 c0 (TLRef i))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda +(H3: (arity g (CHead c0 (Bind b) u) t0 a2)).(\lambda (_: (((nf2 (CHead c0 +(Bind b) u) t0) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T t0 +(THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 (CHead c0 +(Bind b) u) w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead (CHead c0 (Bind +b) u) (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort n)))) +(ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads +(Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 +(CHead c0 (Bind b) u) ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 (CHead +c0 (Bind b) u) (TLRef i))))))))).(\lambda (H5: (nf2 c0 (THead (Bind b) u +t0))).(B_ind (\lambda (b0: B).((not (eq B b0 Abst)) \to ((arity g (CHead c0 +(Bind b0) u) t0 a2) \to ((nf2 c0 (THead (Bind b0) u t0)) \to (or3 (ex3_2 T T +(\lambda (w: T).(\lambda (u0: T).(eq T (THead (Bind b0) u t0) (THead (Bind +Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: +T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: +nat).(eq T (THead (Bind b0) u t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws: +TList).(\lambda (i: nat).(eq T (THead (Bind b0) u t0) (THeads (Flat Appl) ws +(TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda +(_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))))))) (\lambda (_: (not (eq +B Abbr Abst))).(\lambda (_: (arity g (CHead c0 (Bind Abbr) u) t0 +a2)).(\lambda (H8: (nf2 c0 (THead (Bind Abbr) u t0))).(nf2_gen_abbr c0 u t0 +H8 (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Bind Abbr) +u t0) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 +w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) +(ex nat (\lambda (n: nat).(eq T (THead (Bind Abbr) u t0) (TSort n)))) (ex3_2 +TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Bind Abbr) u +t0) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: +nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef +i)))))))))) (\lambda (H6: (not (eq B Abst Abst))).(\lambda (_: (arity g +(CHead c0 (Bind Abst) u) t0 a2)).(\lambda (_: (nf2 c0 (THead (Bind Abst) u +t0))).(let H9 \def (match (H6 (refl_equal B Abst)) in False return (\lambda +(_: False).(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead +(Bind Abst) u t0) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: +T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) +w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind Abst) u t0) (TSort +n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead +(Bind Abst) u t0) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: +TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: +nat).(nf2 c0 (TLRef i))))))) with []) in H9)))) (\lambda (_: (not (eq B Void +Abst))).(\lambda (H7: (arity g (CHead c0 (Bind Void) u) t0 a2)).(\lambda (H8: +(nf2 c0 (THead (Bind Void) u t0))).(let H9 \def (arity_gen_cvoid g (CHead c0 +(Bind Void) u) t0 a2 H7 c0 u O (getl_refl Void c0 u)) in (ex_ind T (\lambda +(v: T).(eq T t0 (lift (S O) O v))) (or3 (ex3_2 T T (\lambda (w: T).(\lambda +(u0: T).(eq T (THead (Bind Void) u t0) (THead (Bind Abst) w u0)))) (\lambda +(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 +(CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind +Void) u t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: +nat).(eq T (THead (Bind Void) u t0) (THeads (Flat Appl) ws (TLRef i))))) +(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: +TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))) (\lambda (x: T).(\lambda +(H10: (eq T t0 (lift (S O) O x))).(let H11 \def (eq_ind T t0 (\lambda (t1: +T).(nf2 c0 (THead (Bind Void) u t1))) H8 (lift (S O) O x) H10) in (eq_ind_r T +(lift (S O) O x) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (w: T).(\lambda +(u0: T).(eq T (THead (Bind Void) u t1) (THead (Bind Abst) w u0)))) (\lambda +(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 +(CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind +Void) u t1) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: +nat).(eq T (THead (Bind Void) u t1) (THeads (Flat Appl) ws (TLRef i))))) +(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: +TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))) (nf2_gen_void c0 u x H11 +(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Bind Void) u +(lift (S O) O x)) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: +T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) +w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind Void) u (lift (S O) O +x)) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq +T (THead (Bind Void) u (lift (S O) O x)) (THeads (Flat Appl) ws (TLRef i))))) +(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: +TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))) t0 H10)))) H9))))) b H0 H3 +H5))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda +(_: (arity g c0 u (asucc g a1))).(\lambda (_: (((nf2 c0 u) \to (or3 (ex3_2 T +T (\lambda (w: T).(\lambda (u0: T).(eq T u (THead (Bind Abst) w u0)))) +(\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: +T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T u +(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T u +(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: +nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef +i))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c0 +(Bind Abst) u) t0 a2)).(\lambda (_: (((nf2 (CHead c0 (Bind Abst) u) t0) \to +(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) +w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 (CHead c0 (Bind Abst) u) w))) +(\lambda (w: T).(\lambda (u0: T).(nf2 (CHead (CHead c0 (Bind Abst) u) (Bind +Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort n)))) (ex3_2 TList +nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws +(TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 (CHead c0 (Bind +Abst) u) ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 (CHead c0 (Bind +Abst) u) (TLRef i))))))))).(\lambda (H4: (nf2 c0 (THead (Bind Abst) u +t0))).(let H5 \def (nf2_gen_abst c0 u t0 H4) in (land_ind (nf2 c0 u) (nf2 +(CHead c0 (Bind Abst) u) t0) (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: +T).(eq T (THead (Bind Abst) u t0) (THead (Bind Abst) w u0)))) (\lambda (w: +T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead +c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind Abst) u +t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq +T (THead (Bind Abst) u t0) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: +TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: +nat).(nf2 c0 (TLRef i)))))) (\lambda (H6: (nf2 c0 u)).(\lambda (H7: (nf2 +(CHead c0 (Bind Abst) u) t0)).(or3_intro0 (ex3_2 T T (\lambda (w: T).(\lambda +(u0: T).(eq T (THead (Bind Abst) u t0) (THead (Bind Abst) w u0)))) (\lambda +(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 +(CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind +Abst) u t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: +nat).(eq T (THead (Bind Abst) u t0) (THeads (Flat Appl) ws (TLRef i))))) +(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: +TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))) (ex3_2_intro T T (\lambda (w: +T).(\lambda (u0: T).(eq T (THead (Bind Abst) u t0) (THead (Bind Abst) w +u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda +(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0))) u t0 (refl_equal T (THead (Bind +Abst) u t0)) H6 H7)))) H5)))))))))))) (\lambda (c0: C).(\lambda (u: +T).(\lambda (a1: A).(\lambda (_: (arity g c0 u a1)).(\lambda (_: (((nf2 c0 u) +\to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u (THead (Bind +Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: +T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: +nat).(eq T u (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: +nat).(eq T u (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: +TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: +nat).(nf2 c0 (TLRef i))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda +(H2: (arity g c0 t0 (AHead a1 a2))).(\lambda (H3: (((nf2 c0 t0) \to (or3 +(ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w +u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda +(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T +t0 (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T +t0 (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: +nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef +i))))))))).(\lambda (H4: (nf2 c0 (THead (Flat Appl) u t0))).(let H5 \def +(nf2_gen_flat Appl c0 u t0 H4) in (land_ind (nf2 c0 u) (nf2 c0 t0) (or3 +(ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t0) +(THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) +(\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat +(\lambda (n: nat).(eq T (THead (Flat Appl) u t0) (TSort n)))) (ex3_2 TList +nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u t0) +(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: +nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef +i)))))) (\lambda (H6: (nf2 c0 u)).(\lambda (H7: (nf2 c0 t0)).(let H_x \def +(H3 H7) in (let H8 \def H_x in (or3_ind (ex3_2 T T (\lambda (w: T).(\lambda +(u0: T).(eq T t0 (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: +T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) +w) u0)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort n)))) (ex3_2 TList nat +(\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef +i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: +TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))) (or3 (ex3_2 T T (\lambda (w: +T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t0) (THead (Bind Abst) w +u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda +(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T +(THead (Flat Appl) u t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws: +TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u t0) (THeads (Flat Appl) +ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) +(\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))) (\lambda (H9: +(ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w +u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda +(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0))))).(ex3_2_ind T T (\lambda (w: +T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w u0)))) (\lambda (w: +T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead +c0 (Bind Abst) w) u0))) (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq +T (THead (Flat Appl) u t0) (THead (Bind Abst) w u0)))) (\lambda (w: +T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead +c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u +t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq +T (THead (Flat Appl) u t0) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: +TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: +nat).(nf2 c0 (TLRef i)))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H10: +(eq T t0 (THead (Bind Abst) x0 x1))).(\lambda (_: (nf2 c0 x0)).(\lambda (_: +(nf2 (CHead c0 (Bind Abst) x0) x1)).(let H13 \def (eq_ind T t0 (\lambda (t1: +T).(nf2 c0 (THead (Flat Appl) u t1))) H4 (THead (Bind Abst) x0 x1) H10) in +(let H14 \def (eq_ind T t0 (\lambda (t1: T).(arity g c0 t1 (AHead a1 a2))) H2 +(THead (Bind Abst) x0 x1) H10) in (eq_ind_r T (THead (Bind Abst) x0 x1) +(\lambda (t1: T).(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T +(THead (Flat Appl) u t1) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda +(_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind +Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u t1) +(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T +(THead (Flat Appl) u t1) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: +TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: +nat).(nf2 c0 (TLRef i))))))) (nf2_gen_beta c0 u x0 x1 H13 (or3 (ex3_2 T T +(\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl) u (THead (Bind +Abst) x0 x1)) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: +T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) +w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u (THead (Bind +Abst) x0 x1)) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: +nat).(eq T (THead (Flat Appl) u (THead (Bind Abst) x0 x1)) (THeads (Flat +Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) +(\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))) t0 H10)))))))) +H9)) (\lambda (H9: (ex nat (\lambda (n: nat).(eq T t0 (TSort n))))).(ex_ind +nat (\lambda (n: nat).(eq T t0 (TSort n))) (or3 (ex3_2 T T (\lambda (w: +T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t0) (THead (Bind Abst) w +u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda +(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T +(THead (Flat Appl) u t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws: +TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u t0) (THeads (Flat Appl) +ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) +(\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))) (\lambda (x: +nat).(\lambda (H10: (eq T t0 (TSort x))).(let H11 \def (eq_ind T t0 (\lambda +(t1: T).(nf2 c0 (THead (Flat Appl) u t1))) H4 (TSort x) H10) in (let H12 \def +(eq_ind T t0 (\lambda (t1: T).(arity g c0 t1 (AHead a1 a2))) H2 (TSort x) +H10) in (eq_ind_r T (TSort x) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (w: +T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t1) (THead (Bind Abst) w +u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda +(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T +(THead (Flat Appl) u t1) (TSort n)))) (ex3_2 TList nat (\lambda (ws: +TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u t1) (THeads (Flat Appl) +ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) +(\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))) (let H_x0 \def +(leq_gen_head1 g a1 a2 (ASort O x) (arity_gen_sort g c0 x (AHead a1 a2) H12)) +in (let H13 \def H_x0 in (ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq +g a1 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: +A).(\lambda (a4: A).(eq A (ASort O x) (AHead a3 a4)))) (or3 (ex3_2 T T +(\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl) u (TSort x)) (THead +(Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda +(w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda +(n: nat).(eq T (THead (Flat Appl) u (TSort x)) (TSort n)))) (ex3_2 TList nat +(\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u (TSort x)) +(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: +nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef +i)))))) (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g a1 +x0)).(\lambda (_: (leq g a2 x1)).(\lambda (H16: (eq A (ASort O x) (AHead x0 +x1))).(let H17 \def (eq_ind A (ASort O x) (\lambda (ee: A).(match ee in A +return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) +\Rightarrow False])) I (AHead x0 x1) H16) in (False_ind (or3 (ex3_2 T T +(\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl) u (TSort x)) (THead +(Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda +(w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda +(n: nat).(eq T (THead (Flat Appl) u (TSort x)) (TSort n)))) (ex3_2 TList nat +(\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u (TSort x)) +(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: +nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef +i)))))) H17))))))) H13))) t0 H10))))) H9)) (\lambda (H9: (ex3_2 TList nat +(\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef +i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: +TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))).(ex3_2_ind TList nat (\lambda +(ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i))))) +(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: +TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))) (or3 (ex3_2 T T (\lambda (w: +T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t0) (THead (Bind Abst) w +u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda +(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T +(THead (Flat Appl) u t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws: +TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u t0) (THeads (Flat Appl) +ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) +(\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))) (\lambda (x0: +TList).(\lambda (x1: nat).(\lambda (H10: (eq T t0 (THeads (Flat Appl) x0 +(TLRef x1)))).(\lambda (H11: (nfs2 c0 x0)).(\lambda (H12: (nf2 c0 (TLRef +x1))).(let H13 \def (eq_ind T t0 (\lambda (t1: T).(nf2 c0 (THead (Flat Appl) +u t1))) H4 (THeads (Flat Appl) x0 (TLRef x1)) H10) in (let H14 \def (eq_ind T +t0 (\lambda (t1: T).(arity g c0 t1 (AHead a1 a2))) H2 (THeads (Flat Appl) x0 +(TLRef x1)) H10) in (eq_ind_r T (THeads (Flat Appl) x0 (TLRef x1)) (\lambda +(t1: T).(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat +Appl) u t1) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 +c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) +(ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u t1) (TSort n)))) (ex3_2 +TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u +t1) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: +nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef +i))))))) (or3_intro2 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead +(Flat Appl) u (THeads (Flat Appl) x0 (TLRef x1))) (THead (Bind Abst) w u0)))) +(\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: +T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T +(THead (Flat Appl) u (THeads (Flat Appl) x0 (TLRef x1))) (TSort n)))) (ex3_2 +TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u +(THeads (Flat Appl) x0 (TLRef x1))) (THeads (Flat Appl) ws (TLRef i))))) +(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: +TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))) (ex3_2_intro TList nat +(\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u (THeads +(Flat Appl) x0 (TLRef x1))) (THeads (Flat Appl) ws (TLRef i))))) (\lambda +(ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: +nat).(nf2 c0 (TLRef i)))) (TCons u x0) x1 (refl_equal T (THead (Flat Appl) u +(THeads (Flat Appl) x0 (TLRef x1)))) (conj (nf2 c0 u) (nfs2 c0 x0) H6 H11) +H12)) t0 H10)))))))) H9)) H8))))) H5)))))))))))) (\lambda (c0: C).(\lambda +(u: T).(\lambda (a0: A).(\lambda (_: (arity g c0 u (asucc g a0))).(\lambda +(_: (((nf2 c0 u) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u +(THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) +(\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat +(\lambda (n: nat).(eq T u (TSort n)))) (ex3_2 TList nat (\lambda (ws: +TList).(\lambda (i: nat).(eq T u (THeads (Flat Appl) ws (TLRef i))))) +(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: +TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))))).(\lambda (t0: T).(\lambda +(_: (arity g c0 t0 a0)).(\lambda (_: (((nf2 c0 t0) \to (or3 (ex3_2 T T +(\lambda (w: T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w u0)))) +(\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: +T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T t0 +(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 +(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: +nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef +i))))))))).(\lambda (H4: (nf2 c0 (THead (Flat Cast) u t0))).(nf2_gen_cast c0 +u t0 H4 (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat +Cast) u t0) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 +c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) +(ex nat (\lambda (n: nat).(eq T (THead (Flat Cast) u t0) (TSort n)))) (ex3_2 +TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Cast) u +t0) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: +nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef +i)))))))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda +(_: (arity g c0 t0 a1)).(\lambda (H1: (((nf2 c0 t0) \to (or3 (ex3_2 T T +(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda +(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 +(CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort +n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 +(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: +nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef +i))))))))).(\lambda (a2: A).(\lambda (_: (leq g a1 a2)).(\lambda (H3: (nf2 c0 +t0)).(let H_x \def (H1 H3) in (let H4 \def H_x in (or3_ind (ex3_2 T T +(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda +(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 +(CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort +n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 +(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: +nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef +i))))) (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind +Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: +T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: +nat).(eq T t0 (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: +nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: +TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: +nat).(nf2 c0 (TLRef i)))))) (\lambda (H5: (ex3_2 T T (\lambda (w: T).(\lambda +(u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: +T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) +u))))).(ex3_2_ind T T (\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind +Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: +T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u))) (or3 (ex3_2 T T +(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda +(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 +(CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort +n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 +(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: +nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef +i)))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H6: (eq T t0 (THead (Bind +Abst) x0 x1))).(\lambda (H7: (nf2 c0 x0)).(\lambda (H8: (nf2 (CHead c0 (Bind +Abst) x0) x1)).(eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t1: T).(or3 +(ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t1 (THead (Bind Abst) w +u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda +(u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t1 +(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t1 +(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: +nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef +i))))))) (or3_intro0 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T (THead +(Bind Abst) x0 x1) (THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: +T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) +u)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind Abst) x0 x1) (TSort n)))) +(ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Bind +Abst) x0 x1) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: +TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: +nat).(nf2 c0 (TLRef i))))) (ex3_2_intro T T (\lambda (w: T).(\lambda (u: +T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) w u)))) (\lambda (w: +T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead +c0 (Bind Abst) w) u))) x0 x1 (refl_equal T (THead (Bind Abst) x0 x1)) H7 H8)) +t0 H6)))))) H5)) (\lambda (H5: (ex nat (\lambda (n: nat).(eq T t0 (TSort +n))))).(ex_ind nat (\lambda (n: nat).(eq T t0 (TSort n))) (or3 (ex3_2 T T +(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda +(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 +(CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort +n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 +(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: +nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef +i)))))) (\lambda (x: nat).(\lambda (H6: (eq T t0 (TSort x))).(eq_ind_r T +(TSort x) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u: +T).(eq T t1 (THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 +c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) +(ex nat (\lambda (n: nat).(eq T t1 (TSort n)))) (ex3_2 TList nat (\lambda +(ws: TList).(\lambda (i: nat).(eq T t1 (THeads (Flat Appl) ws (TLRef i))))) +(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: +TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))) (or3_intro1 (ex3_2 T T +(\lambda (w: T).(\lambda (u: T).(eq T (TSort x) (THead (Bind Abst) w u)))) +(\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: +T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T (TSort +x) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T +(TSort x) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda +(_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef +i))))) (ex_intro nat (\lambda (n: nat).(eq T (TSort x) (TSort n))) x +(refl_equal T (TSort x)))) t0 H6))) H5)) (\lambda (H5: (ex3_2 TList nat +(\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef +i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: +TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))).(ex3_2_ind TList nat (\lambda +(ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i))))) +(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: +TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))) (or3 (ex3_2 T T (\lambda (w: +T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda (w: +T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead +c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort n)))) +(ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads +(Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 +ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))) (\lambda +(x0: TList).(\lambda (x1: nat).(\lambda (H6: (eq T t0 (THeads (Flat Appl) x0 +(TLRef x1)))).(\lambda (H7: (nfs2 c0 x0)).(\lambda (H8: (nf2 c0 (TLRef +x1))).(eq_ind_r T (THeads (Flat Appl) x0 (TLRef x1)) (\lambda (t1: T).(or3 +(ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t1 (THead (Bind Abst) w +u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda +(u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t1 +(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t1 +(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: +nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef +i))))))) (or3_intro2 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T (THeads +(Flat Appl) x0 (TLRef x1)) (THead (Bind Abst) w u)))) (\lambda (w: +T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead +c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T (THeads (Flat Appl) +x0 (TLRef x1)) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda +(i: nat).(eq T (THeads (Flat Appl) x0 (TLRef x1)) (THeads (Flat Appl) ws +(TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda +(_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))) (ex3_2_intro TList nat +(\lambda (ws: TList).(\lambda (i: nat).(eq T (THeads (Flat Appl) x0 (TLRef +x1)) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: +nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef +i)))) x0 x1 (refl_equal T (THeads (Flat Appl) x0 (TLRef x1))) H7 H8)) t0 +H6)))))) H5)) H4))))))))))) c t a H))))). +(* COMMENTS +Initial nodes: 9193 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/nf2/dec.ma b/matita/matita/contribs/lambdadelta/basic_1/nf2/dec.ma new file mode 100644 index 000000000..33b652baf --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/nf2/dec.ma @@ -0,0 +1,200 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/nf2/defs.ma". + +include "Basic-1/pr2/clen.ma". + +include "Basic-1/pr2/fwd.ma". + +include "Basic-1/pr0/dec.ma". + +include "Basic-1/C/props.ma". + +theorem nf2_dec: + \forall (c: C).(\forall (t1: T).(or (nf2 c t1) (ex2 T (\lambda (t2: T).((eq +T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c t1 t2))))) +\def + \lambda (c: C).(c_tail_ind (\lambda (c0: C).(\forall (t1: T).(or (\forall +(t2: T).((pr2 c0 t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 +t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c0 t1 t2)))))) (\lambda +(n: nat).(\lambda (t1: T).(let H_x \def (nf0_dec t1) in (let H \def H_x in +(or_ind (\forall (t2: T).((pr0 t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: +T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t1 t2))) +(or (\forall (t2: T).((pr2 (CSort n) t1 t2) \to (eq T t1 t2))) (ex2 T +(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: +T).(pr2 (CSort n) t1 t2)))) (\lambda (H0: ((\forall (t2: T).((pr0 t1 t2) \to +(eq T t1 t2))))).(or_introl (\forall (t2: T).((pr2 (CSort n) t1 t2) \to (eq T +t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) +(\lambda (t2: T).(pr2 (CSort n) t1 t2))) (\lambda (t2: T).(\lambda (H1: (pr2 +(CSort n) t1 t2)).(let H_y \def (pr2_gen_csort t1 t2 n H1) in (H0 t2 +H_y)))))) (\lambda (H0: (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall +(P: Prop).P))) (\lambda (t2: T).(pr0 t1 t2)))).(ex2_ind T (\lambda (t2: +T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t1 t2)) +(or (\forall (t2: T).((pr2 (CSort n) t1 t2) \to (eq T t1 t2))) (ex2 T +(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: +T).(pr2 (CSort n) t1 t2)))) (\lambda (x: T).(\lambda (H1: (((eq T t1 x) \to +(\forall (P: Prop).P)))).(\lambda (H2: (pr0 t1 x)).(or_intror (\forall (t2: +T).((pr2 (CSort n) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T +t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CSort n) t1 t2))) +(ex_intro2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) +(\lambda (t2: T).(pr2 (CSort n) t1 t2)) x H1 (pr2_free (CSort n) t1 x +H2)))))) H0)) H))))) (\lambda (c0: C).(\lambda (H: ((\forall (t1: T).(or +(\forall (t2: T).((pr2 c0 t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: +T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c0 t1 +t2))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (t1: T).(let H_x \def (H +t1) in (let H0 \def H_x in (or_ind (\forall (t2: T).((pr2 c0 t1 t2) \to (eq T +t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) +(\lambda (t2: T).(pr2 c0 t1 t2))) (or (\forall (t2: T).((pr2 (CTail k t c0) +t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall +(P: Prop).P))) (\lambda (t2: T).(pr2 (CTail k t c0) t1 t2)))) (\lambda (H1: +((\forall (t2: T).((pr2 c0 t1 t2) \to (eq T t1 t2))))).(K_ind (\lambda (k0: +K).(or (\forall (t2: T).((pr2 (CTail k0 t c0) t1 t2) \to (eq T t1 t2))) (ex2 +T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: +T).(pr2 (CTail k0 t c0) t1 t2))))) (\lambda (b: B).(B_ind (\lambda (b0: +B).(or (\forall (t2: T).((pr2 (CTail (Bind b0) t c0) t1 t2) \to (eq T t1 +t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) +(\lambda (t2: T).(pr2 (CTail (Bind b0) t c0) t1 t2))))) (let H_x0 \def +(dnf_dec t t1 (clen c0)) in (let H2 \def H_x0 in (ex_ind T (\lambda (v: +T).(or (subst0 (clen c0) t t1 (lift (S O) (clen c0) v)) (eq T t1 (lift (S O) +(clen c0) v)))) (or (\forall (t2: T).((pr2 (CTail (Bind Abbr) t c0) t1 t2) +\to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: +Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) t1 t2)))) (\lambda +(x: T).(\lambda (H3: (or (subst0 (clen c0) t t1 (lift (S O) (clen c0) x)) (eq +T t1 (lift (S O) (clen c0) x)))).(or_ind (subst0 (clen c0) t t1 (lift (S O) +(clen c0) x)) (eq T t1 (lift (S O) (clen c0) x)) (or (\forall (t2: T).((pr2 +(CTail (Bind Abbr) t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: +T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail +(Bind Abbr) t c0) t1 t2)))) (\lambda (H4: (subst0 (clen c0) t t1 (lift (S O) +(clen c0) x))).(let H_x1 \def (getl_ctail_clen Abbr t c0) in (let H5 \def +H_x1 in (ex_ind nat (\lambda (n: nat).(getl (clen c0) (CTail (Bind Abbr) t +c0) (CHead (CSort n) (Bind Abbr) t))) (or (\forall (t2: T).((pr2 (CTail (Bind +Abbr) t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) +\to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) t1 +t2)))) (\lambda (x0: nat).(\lambda (H6: (getl (clen c0) (CTail (Bind Abbr) t +c0) (CHead (CSort x0) (Bind Abbr) t))).(or_intror (\forall (t2: T).((pr2 +(CTail (Bind Abbr) t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: +T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail +(Bind Abbr) t c0) t1 t2))) (ex_intro2 T (\lambda (t2: T).((eq T t1 t2) \to +(\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) t1 +t2)) (lift (S O) (clen c0) x) (\lambda (H7: (eq T t1 (lift (S O) (clen c0) +x))).(\lambda (P: Prop).(let H8 \def (eq_ind T t1 (\lambda (t0: T).(subst0 +(clen c0) t t0 (lift (S O) (clen c0) x))) H4 (lift (S O) (clen c0) x) H7) in +(subst0_gen_lift_false x t (lift (S O) (clen c0) x) (S O) (clen c0) (clen c0) +(le_n (clen c0)) (eq_ind_r nat (plus (S O) (clen c0)) (\lambda (n: nat).(lt +(clen c0) n)) (le_n (plus (S O) (clen c0))) (plus (clen c0) (S O)) (plus_sym +(clen c0) (S O))) H8 P)))) (pr2_delta (CTail (Bind Abbr) t c0) (CSort x0) t +(clen c0) H6 t1 t1 (pr0_refl t1) (lift (S O) (clen c0) x) H4))))) H5)))) +(\lambda (H4: (eq T t1 (lift (S O) (clen c0) x))).(let H5 \def (eq_ind T t1 +(\lambda (t0: T).(\forall (t2: T).((pr2 c0 t0 t2) \to (eq T t0 t2)))) H1 +(lift (S O) (clen c0) x) H4) in (eq_ind_r T (lift (S O) (clen c0) x) (\lambda +(t0: T).(or (\forall (t2: T).((pr2 (CTail (Bind Abbr) t c0) t0 t2) \to (eq T +t0 t2))) (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: Prop).P))) +(\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) t0 t2))))) (or_introl (\forall +(t2: T).((pr2 (CTail (Bind Abbr) t c0) (lift (S O) (clen c0) x) t2) \to (eq T +(lift (S O) (clen c0) x) t2))) (ex2 T (\lambda (t2: T).((eq T (lift (S O) +(clen c0) x) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail +(Bind Abbr) t c0) (lift (S O) (clen c0) x) t2))) (\lambda (t2: T).(\lambda +(H6: (pr2 (CTail (Bind Abbr) t c0) (lift (S O) (clen c0) x) t2)).(let H_x1 +\def (pr2_gen_ctail (Bind Abbr) c0 t (lift (S O) (clen c0) x) t2 H6) in (let +H7 \def H_x1 in (or_ind (pr2 c0 (lift (S O) (clen c0) x) t2) (ex3 T (\lambda +(_: T).(eq K (Bind Abbr) (Bind Abbr))) (\lambda (t0: T).(pr0 (lift (S O) +(clen c0) x) t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2))) (eq T (lift +(S O) (clen c0) x) t2) (\lambda (H8: (pr2 c0 (lift (S O) (clen c0) x) +t2)).(H5 t2 H8)) (\lambda (H8: (ex3 T (\lambda (_: T).(eq K (Bind Abbr) (Bind +Abbr))) (\lambda (t0: T).(pr0 (lift (S O) (clen c0) x) t0)) (\lambda (t0: +T).(subst0 (clen c0) t t0 t2)))).(ex3_ind T (\lambda (_: T).(eq K (Bind Abbr) +(Bind Abbr))) (\lambda (t0: T).(pr0 (lift (S O) (clen c0) x) t0)) (\lambda +(t0: T).(subst0 (clen c0) t t0 t2)) (eq T (lift (S O) (clen c0) x) t2) +(\lambda (x0: T).(\lambda (_: (eq K (Bind Abbr) (Bind Abbr))).(\lambda (H10: +(pr0 (lift (S O) (clen c0) x) x0)).(\lambda (H11: (subst0 (clen c0) t x0 +t2)).(ex2_ind T (\lambda (t3: T).(eq T x0 (lift (S O) (clen c0) t3))) +(\lambda (t3: T).(pr0 x t3)) (eq T (lift (S O) (clen c0) x) t2) (\lambda (x1: +T).(\lambda (H12: (eq T x0 (lift (S O) (clen c0) x1))).(\lambda (_: (pr0 x +x1)).(let H14 \def (eq_ind T x0 (\lambda (t0: T).(subst0 (clen c0) t t0 t2)) +H11 (lift (S O) (clen c0) x1) H12) in (subst0_gen_lift_false x1 t t2 (S O) +(clen c0) (clen c0) (le_n (clen c0)) (eq_ind_r nat (plus (S O) (clen c0)) +(\lambda (n: nat).(lt (clen c0) n)) (le_n (plus (S O) (clen c0))) (plus (clen +c0) (S O)) (plus_sym (clen c0) (S O))) H14 (eq T (lift (S O) (clen c0) x) +t2)))))) (pr0_gen_lift x x0 (S O) (clen c0) H10)))))) H8)) H7)))))) t1 H4))) +H3))) H2))) (or_introl (\forall (t2: T).((pr2 (CTail (Bind Abst) t c0) t1 t2) +\to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: +Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abst) t c0) t1 t2))) (\lambda +(t2: T).(\lambda (H2: (pr2 (CTail (Bind Abst) t c0) t1 t2)).(let H_x0 \def +(pr2_gen_ctail (Bind Abst) c0 t t1 t2 H2) in (let H3 \def H_x0 in (or_ind +(pr2 c0 t1 t2) (ex3 T (\lambda (_: T).(eq K (Bind Abst) (Bind Abbr))) +(\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2))) +(eq T t1 t2) (\lambda (H4: (pr2 c0 t1 t2)).(H1 t2 H4)) (\lambda (H4: (ex3 T +(\lambda (_: T).(eq K (Bind Abst) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) +(\lambda (t0: T).(subst0 (clen c0) t t0 t2)))).(ex3_ind T (\lambda (_: T).(eq +K (Bind Abst) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: +T).(subst0 (clen c0) t t0 t2)) (eq T t1 t2) (\lambda (x0: T).(\lambda (H5: +(eq K (Bind Abst) (Bind Abbr))).(\lambda (_: (pr0 t1 x0)).(\lambda (_: +(subst0 (clen c0) t x0 t2)).(let H8 \def (eq_ind K (Bind Abst) (\lambda (ee: +K).(match ee in K return (\lambda (_: K).Prop) with [(Bind b0) \Rightarrow +(match b0 in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | +Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) \Rightarrow +False])) I (Bind Abbr) H5) in (False_ind (eq T t1 t2) H8)))))) H4)) H3)))))) +(or_introl (\forall (t2: T).((pr2 (CTail (Bind Void) t c0) t1 t2) \to (eq T +t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) +(\lambda (t2: T).(pr2 (CTail (Bind Void) t c0) t1 t2))) (\lambda (t2: +T).(\lambda (H2: (pr2 (CTail (Bind Void) t c0) t1 t2)).(let H_x0 \def +(pr2_gen_ctail (Bind Void) c0 t t1 t2 H2) in (let H3 \def H_x0 in (or_ind +(pr2 c0 t1 t2) (ex3 T (\lambda (_: T).(eq K (Bind Void) (Bind Abbr))) +(\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2))) +(eq T t1 t2) (\lambda (H4: (pr2 c0 t1 t2)).(H1 t2 H4)) (\lambda (H4: (ex3 T +(\lambda (_: T).(eq K (Bind Void) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) +(\lambda (t0: T).(subst0 (clen c0) t t0 t2)))).(ex3_ind T (\lambda (_: T).(eq +K (Bind Void) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: +T).(subst0 (clen c0) t t0 t2)) (eq T t1 t2) (\lambda (x0: T).(\lambda (H5: +(eq K (Bind Void) (Bind Abbr))).(\lambda (_: (pr0 t1 x0)).(\lambda (_: +(subst0 (clen c0) t x0 t2)).(let H8 \def (eq_ind K (Bind Void) (\lambda (ee: +K).(match ee in K return (\lambda (_: K).Prop) with [(Bind b0) \Rightarrow +(match b0 in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | +Abst \Rightarrow False | Void \Rightarrow True]) | (Flat _) \Rightarrow +False])) I (Bind Abbr) H5) in (False_ind (eq T t1 t2) H8)))))) H4)) H3)))))) +b)) (\lambda (f: F).(or_introl (\forall (t2: T).((pr2 (CTail (Flat f) t c0) +t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall +(P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Flat f) t c0) t1 t2))) (\lambda +(t2: T).(\lambda (H2: (pr2 (CTail (Flat f) t c0) t1 t2)).(let H_x0 \def +(pr2_gen_ctail (Flat f) c0 t t1 t2 H2) in (let H3 \def H_x0 in (or_ind (pr2 +c0 t1 t2) (ex3 T (\lambda (_: T).(eq K (Flat f) (Bind Abbr))) (\lambda (t0: +T).(pr0 t1 t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2))) (eq T t1 t2) +(\lambda (H4: (pr2 c0 t1 t2)).(H1 t2 H4)) (\lambda (H4: (ex3 T (\lambda (_: +T).(eq K (Flat f) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: +T).(subst0 (clen c0) t t0 t2)))).(ex3_ind T (\lambda (_: T).(eq K (Flat f) +(Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(subst0 (clen +c0) t t0 t2)) (eq T t1 t2) (\lambda (x0: T).(\lambda (H5: (eq K (Flat f) +(Bind Abbr))).(\lambda (_: (pr0 t1 x0)).(\lambda (_: (subst0 (clen c0) t x0 +t2)).(let H8 \def (eq_ind K (Flat f) (\lambda (ee: K).(match ee in K return +(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow +True])) I (Bind Abbr) H5) in (False_ind (eq T t1 t2) H8)))))) H4)) H3))))))) +k)) (\lambda (H1: (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: +Prop).P))) (\lambda (t2: T).(pr2 c0 t1 t2)))).(ex2_ind T (\lambda (t2: +T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c0 t1 t2)) +(or (\forall (t2: T).((pr2 (CTail k t c0) t1 t2) \to (eq T t1 t2))) (ex2 T +(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: +T).(pr2 (CTail k t c0) t1 t2)))) (\lambda (x: T).(\lambda (H2: (((eq T t1 x) +\to (\forall (P: Prop).P)))).(\lambda (H3: (pr2 c0 t1 x)).(or_intror (\forall +(t2: T).((pr2 (CTail k t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: +T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail k t +c0) t1 t2))) (ex_intro2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: +Prop).P))) (\lambda (t2: T).(pr2 (CTail k t c0) t1 t2)) x H2 (pr2_ctail c0 t1 +x H3 k t)))))) H1)) H0)))))))) c). +(* COMMENTS +Initial nodes: 3653 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/nf2/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/nf2/defs.ma new file mode 100644 index 000000000..98e931c0b --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/nf2/defs.ma @@ -0,0 +1,31 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/pr2/defs.ma". + +definition nf2: + C \to (T \to Prop) +\def + \lambda (c: C).(\lambda (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (eq T t1 +t2)))). + +definition nfs2: + C \to (TList \to Prop) +\def + let rec nfs2 (c: C) (ts: TList) on ts: Prop \def (match ts with [TNil +\Rightarrow True | (TCons t ts0) \Rightarrow (land (nf2 c t) (nfs2 c ts0))]) +in nfs2. + diff --git a/matita/matita/contribs/lambdadelta/basic_1/nf2/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/nf2/fwd.ma new file mode 100644 index 000000000..9138ff2fa --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/nf2/fwd.ma @@ -0,0 +1,220 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/nf2/defs.ma". + +include "Basic-1/pr2/clen.ma". + +include "Basic-1/subst0/dec.ma". + +include "Basic-1/T/props.ma". + +theorem nf2_gen_lref: + \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c +(CHead d (Bind Abbr) u)) \to ((nf2 c (TLRef i)) \to (\forall (P: Prop).P)))))) +\def + \lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda +(H: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H0: ((\forall (t2: T).((pr2 +c (TLRef i) t2) \to (eq T (TLRef i) t2))))).(\lambda (P: +Prop).(lift_gen_lref_false (S i) O i (le_O_n i) (le_n (plus O (S i))) u (H0 +(lift (S i) O u) (pr2_delta c d u i H (TLRef i) (TLRef i) (pr0_refl (TLRef +i)) (lift (S i) O u) (subst0_lref u i))) P))))))). +(* COMMENTS +Initial nodes: 129 +END *) + +theorem nf2_gen_abst: + \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Bind Abst) u +t)) \to (land (nf2 c u) (nf2 (CHead c (Bind Abst) u) t))))) +\def + \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: ((\forall (t2: +T).((pr2 c (THead (Bind Abst) u t) t2) \to (eq T (THead (Bind Abst) u t) +t2))))).(conj (\forall (t2: T).((pr2 c u t2) \to (eq T u t2))) (\forall (t2: +T).((pr2 (CHead c (Bind Abst) u) t t2) \to (eq T t t2))) (\lambda (t2: +T).(\lambda (H0: (pr2 c u t2)).(let H1 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | +(TLRef _) \Rightarrow u | (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abst) +u t) (THead (Bind Abst) t2 t) (H (THead (Bind Abst) t2 t) (pr2_head_1 c u t2 +H0 (Bind Abst) t))) in (let H2 \def (eq_ind_r T t2 (\lambda (t0: T).(pr2 c u +t0)) H0 u H1) in (eq_ind T u (\lambda (t0: T).(eq T u t0)) (refl_equal T u) +t2 H1))))) (\lambda (t2: T).(\lambda (H0: (pr2 (CHead c (Bind Abst) u) t +t2)).(let H1 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda +(_: T).T) with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ +_ t0) \Rightarrow t0])) (THead (Bind Abst) u t) (THead (Bind Abst) u t2) (H +(THead (Bind Abst) u t2) (let H_y \def (pr2_gen_cbind Abst c u t t2 H0) in +H_y))) in (let H2 \def (eq_ind_r T t2 (\lambda (t0: T).(pr2 (CHead c (Bind +Abst) u) t t0)) H0 t H1) in (eq_ind T t (\lambda (t0: T).(eq T t t0)) +(refl_equal T t) t2 H1))))))))). +(* COMMENTS +Initial nodes: 353 +END *) + +theorem nf2_gen_cast: + \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Flat Cast) u +t)) \to (\forall (P: Prop).P)))) +\def + \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (nf2 c (THead +(Flat Cast) u t))).(\lambda (P: Prop).(thead_x_y_y (Flat Cast) u t (H t +(pr2_free c (THead (Flat Cast) u t) t (pr0_tau t t (pr0_refl t) u))) P))))). +(* COMMENTS +Initial nodes: 65 +END *) + +theorem nf2_gen_beta: + \forall (c: C).(\forall (u: T).(\forall (v: T).(\forall (t: T).((nf2 c +(THead (Flat Appl) u (THead (Bind Abst) v t))) \to (\forall (P: Prop).P))))) +\def + \lambda (c: C).(\lambda (u: T).(\lambda (v: T).(\lambda (t: T).(\lambda (H: +((\forall (t2: T).((pr2 c (THead (Flat Appl) u (THead (Bind Abst) v t)) t2) +\to (eq T (THead (Flat Appl) u (THead (Bind Abst) v t)) t2))))).(\lambda (P: +Prop).(let H0 \def (eq_ind T (THead (Flat Appl) u (THead (Bind Abst) v t)) +(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) +\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow +(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | +(Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u t) (H (THead (Bind +Abbr) u t) (pr2_free c (THead (Flat Appl) u (THead (Bind Abst) v t)) (THead +(Bind Abbr) u t) (pr0_beta v u u (pr0_refl u) t t (pr0_refl t))))) in +(False_ind P H0))))))). +(* COMMENTS +Initial nodes: 183 +END *) + +theorem nf2_gen_flat: + \forall (f: F).(\forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c +(THead (Flat f) u t)) \to (land (nf2 c u) (nf2 c t)))))) +\def + \lambda (f: F).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: +((\forall (t2: T).((pr2 c (THead (Flat f) u t) t2) \to (eq T (THead (Flat f) +u t) t2))))).(conj (\forall (t2: T).((pr2 c u t2) \to (eq T u t2))) (\forall +(t2: T).((pr2 c t t2) \to (eq T t t2))) (\lambda (t2: T).(\lambda (H0: (pr2 c +u t2)).(let H1 \def (f_equal T T (\lambda (e: T).(match e in T return +(\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | +(THead _ t0 _) \Rightarrow t0])) (THead (Flat f) u t) (THead (Flat f) t2 t) +(H (THead (Flat f) t2 t) (pr2_head_1 c u t2 H0 (Flat f) t))) in H1))) +(\lambda (t2: T).(\lambda (H0: (pr2 c t t2)).(let H1 \def (f_equal T T +(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t0) \Rightarrow t0])) +(THead (Flat f) u t) (THead (Flat f) u t2) (H (THead (Flat f) u t2) +(pr2_head_2 c u t t2 (Flat f) (pr2_cflat c t t2 H0 f u)))) in H1)))))))). +(* COMMENTS +Initial nodes: 251 +END *) + +theorem nf2_gen__nf2_gen_aux: + \forall (b: B).(\forall (x: T).(\forall (u: T).(\forall (d: nat).((eq T +(THead (Bind b) u (lift (S O) d x)) x) \to (\forall (P: Prop).P))))) +\def + \lambda (b: B).(\lambda (x: T).(T_ind (\lambda (t: T).(\forall (u: +T).(\forall (d: nat).((eq T (THead (Bind b) u (lift (S O) d t)) t) \to +(\forall (P: Prop).P))))) (\lambda (n: nat).(\lambda (u: T).(\lambda (d: +nat).(\lambda (H: (eq T (THead (Bind b) u (lift (S O) d (TSort n))) (TSort +n))).(\lambda (P: Prop).(let H0 \def (eq_ind T (THead (Bind b) u (lift (S O) +d (TSort n))) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) +with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ +_) \Rightarrow True])) I (TSort n) H) in (False_ind P H0))))))) (\lambda (n: +nat).(\lambda (u: T).(\lambda (d: nat).(\lambda (H: (eq T (THead (Bind b) u +(lift (S O) d (TLRef n))) (TLRef n))).(\lambda (P: Prop).(let H0 \def (eq_ind +T (THead (Bind b) u (lift (S O) d (TLRef n))) (\lambda (ee: T).(match ee in T +return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H) in +(False_ind P H0))))))) (\lambda (k: K).(\lambda (t: T).(\lambda (_: ((\forall +(u: T).(\forall (d: nat).((eq T (THead (Bind b) u (lift (S O) d t)) t) \to +(\forall (P: Prop).P)))))).(\lambda (t0: T).(\lambda (H0: ((\forall (u: +T).(\forall (d: nat).((eq T (THead (Bind b) u (lift (S O) d t0)) t0) \to +(\forall (P: Prop).P)))))).(\lambda (u: T).(\lambda (d: nat).(\lambda (H1: +(eq T (THead (Bind b) u (lift (S O) d (THead k t t0))) (THead k t +t0))).(\lambda (P: Prop).(let H2 \def (f_equal T K (\lambda (e: T).(match e +in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow (Bind b) | (TLRef +_) \Rightarrow (Bind b) | (THead k0 _ _) \Rightarrow k0])) (THead (Bind b) u +(lift (S O) d (THead k t t0))) (THead k t t0) H1) in ((let H3 \def (f_equal T +T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t1 _) \Rightarrow t1])) +(THead (Bind b) u (lift (S O) d (THead k t t0))) (THead k t t0) H1) in ((let +H4 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) +with [(TSort _) \Rightarrow (THead k ((let rec lref_map (f: ((nat \to nat))) +(d0: nat) (t1: T) on t1: T \def (match t1 with [(TSort n) \Rightarrow (TSort +n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) with [true \Rightarrow i +| false \Rightarrow (f i)])) | (THead k0 u0 t2) \Rightarrow (THead k0 +(lref_map f d0 u0) (lref_map f (s k0 d0) t2))]) in lref_map) (\lambda (x0: +nat).(plus x0 (S O))) d t) ((let rec lref_map (f: ((nat \to nat))) (d0: nat) +(t1: T) on t1: T \def (match t1 with [(TSort n) \Rightarrow (TSort n) | +(TLRef i) \Rightarrow (TLRef (match (blt i d0) with [true \Rightarrow i | +false \Rightarrow (f i)])) | (THead k0 u0 t2) \Rightarrow (THead k0 (lref_map +f d0 u0) (lref_map f (s k0 d0) t2))]) in lref_map) (\lambda (x0: nat).(plus +x0 (S O))) (s k d) t0)) | (TLRef _) \Rightarrow (THead k ((let rec lref_map +(f: ((nat \to nat))) (d0: nat) (t1: T) on t1: T \def (match t1 with [(TSort +n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) +with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k0 u0 t2) +\Rightarrow (THead k0 (lref_map f d0 u0) (lref_map f (s k0 d0) t2))]) in +lref_map) (\lambda (x0: nat).(plus x0 (S O))) d t) ((let rec lref_map (f: +((nat \to nat))) (d0: nat) (t1: T) on t1: T \def (match t1 with [(TSort n) +\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) with +[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k0 u0 t2) +\Rightarrow (THead k0 (lref_map f d0 u0) (lref_map f (s k0 d0) t2))]) in +lref_map) (\lambda (x0: nat).(plus x0 (S O))) (s k d) t0)) | (THead _ _ t1) +\Rightarrow t1])) (THead (Bind b) u (lift (S O) d (THead k t t0))) (THead k t +t0) H1) in (\lambda (_: (eq T u t)).(\lambda (H6: (eq K (Bind b) k)).(let H7 +\def (eq_ind_r K k (\lambda (k0: K).(eq T (lift (S O) d (THead k0 t t0)) t0)) +H4 (Bind b) H6) in (let H8 \def (eq_ind T (lift (S O) d (THead (Bind b) t +t0)) (\lambda (t1: T).(eq T t1 t0)) H7 (THead (Bind b) (lift (S O) d t) (lift +(S O) (S d) t0)) (lift_bind b t t0 (S O) d)) in (H0 (lift (S O) d t) (S d) H8 +P)))))) H3)) H2))))))))))) x)). +(* COMMENTS +Initial nodes: 935 +END *) + +theorem nf2_gen_abbr: + \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Bind Abbr) u +t)) \to (\forall (P: Prop).P)))) +\def + \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: ((\forall (t2: +T).((pr2 c (THead (Bind Abbr) u t) t2) \to (eq T (THead (Bind Abbr) u t) +t2))))).(\lambda (P: Prop).(let H_x \def (dnf_dec u t O) in (let H0 \def H_x +in (ex_ind T (\lambda (v: T).(or (subst0 O u t (lift (S O) O v)) (eq T t +(lift (S O) O v)))) P (\lambda (x: T).(\lambda (H1: (or (subst0 O u t (lift +(S O) O x)) (eq T t (lift (S O) O x)))).(or_ind (subst0 O u t (lift (S O) O +x)) (eq T t (lift (S O) O x)) P (\lambda (H2: (subst0 O u t (lift (S O) O +x))).(let H3 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda +(_: T).T) with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ +_ t0) \Rightarrow t0])) (THead (Bind Abbr) u t) (THead (Bind Abbr) u (lift (S +O) O x)) (H (THead (Bind Abbr) u (lift (S O) O x)) (pr2_free c (THead (Bind +Abbr) u t) (THead (Bind Abbr) u (lift (S O) O x)) (pr0_delta u u (pr0_refl u) +t t (pr0_refl t) (lift (S O) O x) H2)))) in (let H4 \def (eq_ind T t (\lambda +(t0: T).(subst0 O u t0 (lift (S O) O x))) H2 (lift (S O) O x) H3) in +(subst0_refl u (lift (S O) O x) O H4 P)))) (\lambda (H2: (eq T t (lift (S O) +O x))).(let H3 \def (eq_ind T t (\lambda (t0: T).(\forall (t2: T).((pr2 c +(THead (Bind Abbr) u t0) t2) \to (eq T (THead (Bind Abbr) u t0) t2)))) H +(lift (S O) O x) H2) in (nf2_gen__nf2_gen_aux Abbr x u O (H3 x (pr2_free c +(THead (Bind Abbr) u (lift (S O) O x)) x (pr0_zeta Abbr not_abbr_abst x x +(pr0_refl x) u))) P))) H1))) H0))))))). +(* COMMENTS +Initial nodes: 511 +END *) + +theorem nf2_gen_void: + \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Bind Void) u +(lift (S O) O t))) \to (\forall (P: Prop).P)))) +\def + \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: ((\forall (t2: +T).((pr2 c (THead (Bind Void) u (lift (S O) O t)) t2) \to (eq T (THead (Bind +Void) u (lift (S O) O t)) t2))))).(\lambda (P: Prop).(nf2_gen__nf2_gen_aux +Void t u O (H t (pr2_free c (THead (Bind Void) u (lift (S O) O t)) t +(pr0_zeta Void (sym_not_eq B Abst Void not_abst_void) t t (pr0_refl t) u))) +P))))). +(* COMMENTS +Initial nodes: 121 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/nf2/iso.ma b/matita/matita/contribs/lambdadelta/basic_1/nf2/iso.ma new file mode 100644 index 000000000..6a2ce00f8 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/nf2/iso.ma @@ -0,0 +1,130 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/nf2/pr3.ma". + +include "Basic-1/pr3/fwd.ma". + +include "Basic-1/iso/props.ma". + +theorem nf2_iso_appls_lref: + \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (vs: +TList).(\forall (u: T).((pr3 c (THeads (Flat Appl) vs (TLRef i)) u) \to (iso +(THeads (Flat Appl) vs (TLRef i)) u)))))) +\def + \lambda (c: C).(\lambda (i: nat).(\lambda (H: (nf2 c (TLRef i))).(\lambda +(vs: TList).(TList_ind (\lambda (t: TList).(\forall (u: T).((pr3 c (THeads +(Flat Appl) t (TLRef i)) u) \to (iso (THeads (Flat Appl) t (TLRef i)) u)))) +(\lambda (u: T).(\lambda (H0: (pr3 c (TLRef i) u)).(let H_y \def +(nf2_pr3_unfold c (TLRef i) u H0 H) in (let H1 \def (eq_ind_r T u (\lambda +(t: T).(pr3 c (TLRef i) t)) H0 (TLRef i) H_y) in (eq_ind T (TLRef i) (\lambda +(t: T).(iso (TLRef i) t)) (iso_refl (TLRef i)) u H_y))))) (\lambda (t: +T).(\lambda (t0: TList).(\lambda (H0: ((\forall (u: T).((pr3 c (THeads (Flat +Appl) t0 (TLRef i)) u) \to (iso (THeads (Flat Appl) t0 (TLRef i)) +u))))).(\lambda (u: T).(\lambda (H1: (pr3 c (THead (Flat Appl) t (THeads +(Flat Appl) t0 (TLRef i))) u)).(let H2 \def (pr3_gen_appl c t (THeads (Flat +Appl) t0 (TLRef i)) u H1) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda +(t2: T).(eq T u (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: +T).(pr3 c t u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) +t0 (TLRef i)) t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda +(u2: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) u))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))) (\lambda +(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat +Appl) t0 (TLRef i)) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u0: +T).(pr3 (CHead c (Bind b) u0) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat +Appl) t0 (TLRef i)) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda +(_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: +T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) +u))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))))) (\lambda (_: +B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: +T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) +y2) z1 z2)))))))) (iso (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef +i))) u) (\lambda (H3: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T u +(THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) +(\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) +t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T u (THead (Flat +Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_: +T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) t2))) (iso +(THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) u) (\lambda (x0: +T).(\lambda (x1: T).(\lambda (H4: (eq T u (THead (Flat Appl) x0 +x1))).(\lambda (_: (pr3 c t x0)).(\lambda (_: (pr3 c (THeads (Flat Appl) t0 +(TLRef i)) x1)).(eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda (t1: T).(iso +(THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) t1)) (iso_head t x0 +(THeads (Flat Appl) t0 (TLRef i)) x1 (Flat Appl)) u H4)))))) H3)) (\lambda +(H3: (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(t2: T).(pr3 c (THead (Bind Abbr) u2 t2) u))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))) (\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat +Appl) t0 (TLRef i)) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u0: +T).(pr3 (CHead c (Bind b) u0) z1 t2))))))))).(ex4_4_ind T T T T (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind +Abbr) u2 t2) u))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr3 c t u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind +Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda +(t2: T).(\forall (b: B).(\forall (u0: T).(pr3 (CHead c (Bind b) u0) z1 +t2))))))) (iso (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) u) +(\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda +(_: (pr3 c (THead (Bind Abbr) x2 x3) u)).(\lambda (_: (pr3 c t x2)).(\lambda +(H6: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind Abst) x0 +x1))).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pr3 (CHead c (Bind b) +u0) x1 x3))))).(let H_y \def (H0 (THead (Bind Abst) x0 x1) H6) in +(iso_flats_lref_bind_false Appl Abst i x0 x1 t0 H_y (iso (THead (Flat Appl) t +(THeads (Flat Appl) t0 (TLRef i))) u))))))))))) H3)) (\lambda (H3: (ex6_6 B T +T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: +B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind b) y1 z1)))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda +(u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift +(S O) O u2) z2)) u))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))))) +(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 +(CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat +Appl) t0 (TLRef i)) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda +(_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: +T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) +u))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))))) (\lambda (_: +B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: +T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) +y2) z1 z2))))))) (iso (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) +u) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: +T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (not (eq B x0 +Abst))).(\lambda (H5: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind +x0) x1 x2))).(\lambda (_: (pr3 c (THead (Bind x0) x5 (THead (Flat Appl) (lift +(S O) O x4) x3)) u)).(\lambda (_: (pr3 c t x4)).(\lambda (_: (pr3 c x1 +x5)).(\lambda (_: (pr3 (CHead c (Bind x0) x5) x2 x3)).(let H_y \def (H0 +(THead (Bind x0) x1 x2) H5) in (iso_flats_lref_bind_false Appl x0 i x1 x2 t0 +H_y (iso (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) +u))))))))))))))) H3)) H2))))))) vs)))). +(* COMMENTS +Initial nodes: 1817 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/nf2/lift1.ma b/matita/matita/contribs/lambdadelta/basic_1/nf2/lift1.ma new file mode 100644 index 000000000..d50790336 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/nf2/lift1.ma @@ -0,0 +1,41 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/nf2/props.ma". + +include "Basic-1/drop1/fwd.ma". + +theorem nf2_lift1: + \forall (e: C).(\forall (hds: PList).(\forall (c: C).(\forall (t: T).((drop1 +hds c e) \to ((nf2 e t) \to (nf2 c (lift1 hds t))))))) +\def + \lambda (e: C).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall +(c: C).(\forall (t: T).((drop1 p c e) \to ((nf2 e t) \to (nf2 c (lift1 p +t))))))) (\lambda (c: C).(\lambda (t: T).(\lambda (H: (drop1 PNil c +e)).(\lambda (H0: (nf2 e t)).(let H_y \def (drop1_gen_pnil c e H) in +(eq_ind_r C e (\lambda (c0: C).(nf2 c0 t)) H0 c H_y)))))) (\lambda (n: +nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c: +C).(\forall (t: T).((drop1 p c e) \to ((nf2 e t) \to (nf2 c (lift1 p +t)))))))).(\lambda (c: C).(\lambda (t: T).(\lambda (H0: (drop1 (PCons n n0 p) +c e)).(\lambda (H1: (nf2 e t)).(let H_x \def (drop1_gen_pcons c e p n n0 H0) +in (let H2 \def H_x in (ex2_ind C (\lambda (c2: C).(drop n n0 c c2)) (\lambda +(c2: C).(drop1 p c2 e)) (nf2 c (lift n n0 (lift1 p t))) (\lambda (x: +C).(\lambda (H3: (drop n n0 c x)).(\lambda (H4: (drop1 p x e)).(nf2_lift x +(lift1 p t) (H x t H4 H1) c n n0 H3)))) H2))))))))))) hds)). +(* COMMENTS +Initial nodes: 249 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/nf2/pr3.ma b/matita/matita/contribs/lambdadelta/basic_1/nf2/pr3.ma new file mode 100644 index 000000000..3db24223e --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/nf2/pr3.ma @@ -0,0 +1,56 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/nf2/defs.ma". + +include "Basic-1/pr3/pr3.ma". + +theorem nf2_pr3_unfold: + \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to ((nf2 c +t1) \to (eq T t1 t2))))) +\def + \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 +t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).((nf2 c t) \to (eq T t +t0)))) (\lambda (t: T).(\lambda (H0: (nf2 c t)).(H0 t (pr2_free c t t +(pr0_refl t))))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c t3 +t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2: (((nf2 c t0) +\to (eq T t0 t4)))).(\lambda (H3: (nf2 c t3)).(let H4 \def H3 in (let H5 \def +(eq_ind T t3 (\lambda (t: T).(nf2 c t)) H3 t0 (H4 t0 H0)) in (let H6 \def +(eq_ind T t3 (\lambda (t: T).(pr2 c t t0)) H0 t0 (H4 t0 H0)) in (eq_ind_r T +t0 (\lambda (t: T).(eq T t t4)) (H2 H5) t3 (H4 t0 H0)))))))))))) t1 t2 H)))). +(* COMMENTS +Initial nodes: 187 +END *) + +theorem nf2_pr3_confluence: + \forall (c: C).(\forall (t1: T).((nf2 c t1) \to (\forall (t2: T).((nf2 c t2) +\to (\forall (t: T).((pr3 c t t1) \to ((pr3 c t t2) \to (eq T t1 t2)))))))) +\def + \lambda (c: C).(\lambda (t1: T).(\lambda (H: (nf2 c t1)).(\lambda (t2: +T).(\lambda (H0: (nf2 c t2)).(\lambda (t: T).(\lambda (H1: (pr3 c t +t1)).(\lambda (H2: (pr3 c t t2)).(ex2_ind T (\lambda (t0: T).(pr3 c t2 t0)) +(\lambda (t0: T).(pr3 c t1 t0)) (eq T t1 t2) (\lambda (x: T).(\lambda (H3: +(pr3 c t2 x)).(\lambda (H4: (pr3 c t1 x)).(let H_y \def (nf2_pr3_unfold c t1 +x H4 H) in (let H5 \def (eq_ind_r T x (\lambda (t0: T).(pr3 c t1 t0)) H4 t1 +H_y) in (let H6 \def (eq_ind_r T x (\lambda (t0: T).(pr3 c t2 t0)) H3 t1 H_y) +in (let H_y0 \def (nf2_pr3_unfold c t2 t1 H6 H0) in (let H7 \def (eq_ind T t2 +(\lambda (t0: T).(pr3 c t0 t1)) H6 t1 H_y0) in (eq_ind_r T t1 (\lambda (t0: +T).(eq T t1 t0)) (refl_equal T t1) t2 H_y0))))))))) (pr3_confluence c t t2 H2 +t1 H1))))))))). +(* COMMENTS +Initial nodes: 215 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/nf2/props.ma b/matita/matita/contribs/lambdadelta/basic_1/nf2/props.ma new file mode 100644 index 000000000..2f0f092f2 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/nf2/props.ma @@ -0,0 +1,341 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/nf2/defs.ma". + +include "Basic-1/pr2/fwd.ma". + +theorem nf2_sort: + \forall (c: C).(\forall (n: nat).(nf2 c (TSort n))) +\def + \lambda (c: C).(\lambda (n: nat).(\lambda (t2: T).(\lambda (H: (pr2 c (TSort +n) t2)).(eq_ind_r T (TSort n) (\lambda (t: T).(eq T (TSort n) t)) (refl_equal +T (TSort n)) t2 (pr2_gen_sort c t2 n H))))). +(* COMMENTS +Initial nodes: 55 +END *) + +theorem nf2_csort_lref: + \forall (n: nat).(\forall (i: nat).(nf2 (CSort n) (TLRef i))) +\def + \lambda (n: nat).(\lambda (i: nat).(\lambda (t2: T).(\lambda (H: (pr2 (CSort +n) (TLRef i) t2)).(let H0 \def (pr2_gen_lref (CSort n) t2 i H) in (or_ind (eq +T t2 (TLRef i)) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i (CSort n) +(CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T t2 (lift (S +i) O u))))) (eq T (TLRef i) t2) (\lambda (H1: (eq T t2 (TLRef i))).(eq_ind_r +T (TLRef i) (\lambda (t: T).(eq T (TLRef i) t)) (refl_equal T (TLRef i)) t2 +H1)) (\lambda (H1: (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i (CSort +n) (CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T t2 (lift +(S i) O u)))))).(ex2_2_ind C T (\lambda (d: C).(\lambda (u: T).(getl i (CSort +n) (CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T t2 (lift +(S i) O u)))) (eq T (TLRef i) t2) (\lambda (x0: C).(\lambda (x1: T).(\lambda +(H2: (getl i (CSort n) (CHead x0 (Bind Abbr) x1))).(\lambda (H3: (eq T t2 +(lift (S i) O x1))).(eq_ind_r T (lift (S i) O x1) (\lambda (t: T).(eq T +(TLRef i) t)) (getl_gen_sort n i (CHead x0 (Bind Abbr) x1) H2 (eq T (TLRef i) +(lift (S i) O x1))) t2 H3))))) H1)) H0))))). +(* COMMENTS +Initial nodes: 355 +END *) + +theorem nf2_abst: + \forall (c: C).(\forall (u: T).((nf2 c u) \to (\forall (b: B).(\forall (v: +T).(\forall (t: T).((nf2 (CHead c (Bind b) v) t) \to (nf2 c (THead (Bind +Abst) u t)))))))) +\def + \lambda (c: C).(\lambda (u: T).(\lambda (H: ((\forall (t2: T).((pr2 c u t2) +\to (eq T u t2))))).(\lambda (b: B).(\lambda (v: T).(\lambda (t: T).(\lambda +(H0: ((\forall (t2: T).((pr2 (CHead c (Bind b) v) t t2) \to (eq T t +t2))))).(\lambda (t2: T).(\lambda (H1: (pr2 c (THead (Bind Abst) u t) +t2)).(let H2 \def (pr2_gen_abst c u t t2 H1) in (ex3_2_ind T T (\lambda (u2: +T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c u u2))) (\lambda (_: T).(\lambda (t3: T).(\forall +(b0: B).(\forall (u0: T).(pr2 (CHead c (Bind b0) u0) t t3))))) (eq T (THead +(Bind Abst) u t) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H3: (eq T t2 +(THead (Bind Abst) x0 x1))).(\lambda (H4: (pr2 c u x0)).(\lambda (H5: +((\forall (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind b0) u0) t +x1))))).(eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t0: T).(eq T (THead +(Bind Abst) u t) t0)) (f_equal3 K T T T THead (Bind Abst) (Bind Abst) u x0 t +x1 (refl_equal K (Bind Abst)) (H x0 H4) (H0 x1 (H5 b v))) t2 H3)))))) +H2)))))))))). +(* COMMENTS +Initial nodes: 299 +END *) + +theorem nf2_abst_shift: + \forall (c: C).(\forall (u: T).((nf2 c u) \to (\forall (t: T).((nf2 (CHead c +(Bind Abst) u) t) \to (nf2 c (THead (Bind Abst) u t)))))) +\def + \lambda (c: C).(\lambda (u: T).(\lambda (H: ((\forall (t2: T).((pr2 c u t2) +\to (eq T u t2))))).(\lambda (t: T).(\lambda (H0: ((\forall (t2: T).((pr2 +(CHead c (Bind Abst) u) t t2) \to (eq T t t2))))).(\lambda (t2: T).(\lambda +(H1: (pr2 c (THead (Bind Abst) u t) t2)).(let H2 \def (pr2_gen_abst c u t t2 +H1) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind +Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u u2))) (\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) +u0) t t3))))) (eq T (THead (Bind Abst) u t) t2) (\lambda (x0: T).(\lambda +(x1: T).(\lambda (H3: (eq T t2 (THead (Bind Abst) x0 x1))).(\lambda (H4: (pr2 +c u x0)).(\lambda (H5: ((\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind +b) u0) t x1))))).(eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t0: T).(eq T +(THead (Bind Abst) u t) t0)) (f_equal3 K T T T THead (Bind Abst) (Bind Abst) +u x0 t x1 (refl_equal K (Bind Abst)) (H x0 H4) (H0 x1 (H5 Abst u))) t2 +H3)))))) H2)))))))). +(* COMMENTS +Initial nodes: 295 +END *) + +theorem nfs2_tapp: + \forall (c: C).(\forall (t: T).(\forall (ts: TList).((nfs2 c (TApp ts t)) +\to (land (nfs2 c ts) (nf2 c t))))) +\def + \lambda (c: C).(\lambda (t: T).(\lambda (ts: TList).(TList_ind (\lambda (t0: +TList).((nfs2 c (TApp t0 t)) \to (land (nfs2 c t0) (nf2 c t)))) (\lambda (H: +(land (nf2 c t) True)).(let H0 \def H in (land_ind (nf2 c t) True (land True +(nf2 c t)) (\lambda (H1: (nf2 c t)).(\lambda (_: True).(conj True (nf2 c t) I +H1))) H0))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H: (((nfs2 c +(TApp t1 t)) \to (land (nfs2 c t1) (nf2 c t))))).(\lambda (H0: (land (nf2 c +t0) (nfs2 c (TApp t1 t)))).(let H1 \def H0 in (land_ind (nf2 c t0) (nfs2 c +(TApp t1 t)) (land (land (nf2 c t0) (nfs2 c t1)) (nf2 c t)) (\lambda (H2: +(nf2 c t0)).(\lambda (H3: (nfs2 c (TApp t1 t))).(let H_x \def (H H3) in (let +H4 \def H_x in (land_ind (nfs2 c t1) (nf2 c t) (land (land (nf2 c t0) (nfs2 c +t1)) (nf2 c t)) (\lambda (H5: (nfs2 c t1)).(\lambda (H6: (nf2 c t)).(conj +(land (nf2 c t0) (nfs2 c t1)) (nf2 c t) (conj (nf2 c t0) (nfs2 c t1) H2 H5) +H6))) H4))))) H1)))))) ts))). +(* COMMENTS +Initial nodes: 295 +END *) + +theorem nf2_appls_lref: + \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (vs: +TList).((nfs2 c vs) \to (nf2 c (THeads (Flat Appl) vs (TLRef i))))))) +\def + \lambda (c: C).(\lambda (i: nat).(\lambda (H: (nf2 c (TLRef i))).(\lambda +(vs: TList).(TList_ind (\lambda (t: TList).((nfs2 c t) \to (nf2 c (THeads +(Flat Appl) t (TLRef i))))) (\lambda (_: True).H) (\lambda (t: T).(\lambda +(t0: TList).(\lambda (H0: (((nfs2 c t0) \to (nf2 c (THeads (Flat Appl) t0 +(TLRef i)))))).(\lambda (H1: (land (nf2 c t) (nfs2 c t0))).(let H2 \def H1 in +(land_ind (nf2 c t) (nfs2 c t0) (nf2 c (THead (Flat Appl) t (THeads (Flat +Appl) t0 (TLRef i)))) (\lambda (H3: (nf2 c t)).(\lambda (H4: (nfs2 c +t0)).(let H_y \def (H0 H4) in (\lambda (t2: T).(\lambda (H5: (pr2 c (THead +(Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) t2)).(let H6 \def +(pr2_gen_appl c t (THeads (Flat Appl) t0 (TLRef i)) t2 H5) in (or3_ind (ex3_2 +T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c t u2))) (\lambda (_: T).(\lambda (t3: +T).(pr2 c (THeads (Flat Appl) t0 (TLRef i)) t3)))) (ex4_4 T T T T (\lambda +(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THeads (Flat +Appl) t0 (TLRef i)) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda +(_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 +t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: +T).(pr2 c t u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda +(t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 +t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) +(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(eq T (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind +b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead +(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t +u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (eq T (THead (Flat Appl) t +(THeads (Flat Appl) t0 (TLRef i))) t2) (\lambda (H7: (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c t u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c +(THeads (Flat Appl) t0 (TLRef i)) t3))))).(ex3_2_ind T T (\lambda (u2: +T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c t u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c +(THeads (Flat Appl) t0 (TLRef i)) t3))) (eq T (THead (Flat Appl) t (THeads +(Flat Appl) t0 (TLRef i))) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda +(H8: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H9: (pr2 c t +x0)).(\lambda (H10: (pr2 c (THeads (Flat Appl) t0 (TLRef i)) x1)).(eq_ind_r T +(THead (Flat Appl) x0 x1) (\lambda (t1: T).(eq T (THead (Flat Appl) t (THeads +(Flat Appl) t0 (TLRef i))) t1)) (let H11 \def (eq_ind_r T x1 (\lambda (t1: +T).(pr2 c (THeads (Flat Appl) t0 (TLRef i)) t1)) H10 (THeads (Flat Appl) t0 +(TLRef i)) (H_y x1 H10)) in (eq_ind T (THeads (Flat Appl) t0 (TLRef i)) +(\lambda (t1: T).(eq T (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef +i))) (THead (Flat Appl) x0 t1))) (let H12 \def (eq_ind_r T x0 (\lambda (t1: +T).(pr2 c t t1)) H9 t (H3 x0 H9)) in (eq_ind T t (\lambda (t1: T).(eq T +(THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) (THead (Flat Appl) t1 +(THeads (Flat Appl) t0 (TLRef i))))) (refl_equal T (THead (Flat Appl) t +(THeads (Flat Appl) t0 (TLRef i)))) x0 (H3 x0 H9))) x1 (H_y x1 H10))) t2 +H8)))))) H7)) (\lambda (H7: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(eq T (THeads (Flat Appl) t0 (TLRef i)) +(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t u2))))) (\lambda +(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: +B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3))))))))).(ex4_4_ind T T T +T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T +(THeads (Flat Appl) t0 (TLRef i)) (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind +Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr2 c t u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) +z1 t3))))))) (eq T (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) +t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: +T).(\lambda (H8: (eq T (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind Abst) +x0 x1))).(\lambda (H9: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (_: (pr2 +c t x2)).(\lambda (_: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) +u) x1 x3))))).(eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda (t1: T).(eq T +(THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) t1)) (TList_ind +(\lambda (t1: TList).((nf2 c (THeads (Flat Appl) t1 (TLRef i))) \to ((eq T +(THeads (Flat Appl) t1 (TLRef i)) (THead (Bind Abst) x0 x1)) \to (eq T (THead +(Flat Appl) t (THeads (Flat Appl) t1 (TLRef i))) (THead (Bind Abbr) x2 +x3))))) (\lambda (_: (nf2 c (THeads (Flat Appl) TNil (TLRef i)))).(\lambda +(H13: (eq T (THeads (Flat Appl) TNil (TLRef i)) (THead (Bind Abst) x0 +x1))).(let H14 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T +return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) x0 +x1) H13) in (False_ind (eq T (THead (Flat Appl) t (THeads (Flat Appl) TNil +(TLRef i))) (THead (Bind Abbr) x2 x3)) H14)))) (\lambda (t1: T).(\lambda (t3: +TList).(\lambda (_: (((nf2 c (THeads (Flat Appl) t3 (TLRef i))) \to ((eq T +(THeads (Flat Appl) t3 (TLRef i)) (THead (Bind Abst) x0 x1)) \to (eq T (THead +(Flat Appl) t (THeads (Flat Appl) t3 (TLRef i))) (THead (Bind Abbr) x2 +x3)))))).(\lambda (_: (nf2 c (THeads (Flat Appl) (TCons t1 t3) (TLRef +i)))).(\lambda (H13: (eq T (THeads (Flat Appl) (TCons t1 t3) (TLRef i)) +(THead (Bind Abst) x0 x1))).(let H14 \def (eq_ind T (THead (Flat Appl) t1 +(THeads (Flat Appl) t3 (TLRef i))) (\lambda (ee: T).(match ee in T return +(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda +(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow +True])])) I (THead (Bind Abst) x0 x1) H13) in (False_ind (eq T (THead (Flat +Appl) t (THeads (Flat Appl) (TCons t1 t3) (TLRef i))) (THead (Bind Abbr) x2 +x3)) H14))))))) t0 H_y H8) t2 H9))))))))) H7)) (\lambda (H7: (ex6_6 B T T T T +T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda +(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq +T (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind b) y1 z1)))))))) (\lambda +(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: +T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S +O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t u2))))))) (\lambda (_: +B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: +T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) +y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B +b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(eq T (THeads (Flat Appl) t0 (TLRef i)) +(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind +b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr2 c t u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda +(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) (eq T (THead +(Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) t2) (\lambda (x0: +B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: +T).(\lambda (x5: T).(\lambda (_: (not (eq B x0 Abst))).(\lambda (H9: (eq T +(THeads (Flat Appl) t0 (TLRef i)) (THead (Bind x0) x1 x2))).(\lambda (H10: +(eq T t2 (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) +x3)))).(\lambda (_: (pr2 c t x4)).(\lambda (_: (pr2 c x1 x5)).(\lambda (_: +(pr2 (CHead c (Bind x0) x5) x2 x3)).(eq_ind_r T (THead (Bind x0) x5 (THead +(Flat Appl) (lift (S O) O x4) x3)) (\lambda (t1: T).(eq T (THead (Flat Appl) +t (THeads (Flat Appl) t0 (TLRef i))) t1)) (TList_ind (\lambda (t1: +TList).((nf2 c (THeads (Flat Appl) t1 (TLRef i))) \to ((eq T (THeads (Flat +Appl) t1 (TLRef i)) (THead (Bind x0) x1 x2)) \to (eq T (THead (Flat Appl) t +(THeads (Flat Appl) t1 (TLRef i))) (THead (Bind x0) x5 (THead (Flat Appl) +(lift (S O) O x4) x3)))))) (\lambda (_: (nf2 c (THeads (Flat Appl) TNil +(TLRef i)))).(\lambda (H15: (eq T (THeads (Flat Appl) TNil (TLRef i)) (THead +(Bind x0) x1 x2))).(let H16 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match +ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | +(TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead +(Bind x0) x1 x2) H15) in (False_ind (eq T (THead (Flat Appl) t (THeads (Flat +Appl) TNil (TLRef i))) (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O +x4) x3))) H16)))) (\lambda (t1: T).(\lambda (t3: TList).(\lambda (_: (((nf2 c +(THeads (Flat Appl) t3 (TLRef i))) \to ((eq T (THeads (Flat Appl) t3 (TLRef +i)) (THead (Bind x0) x1 x2)) \to (eq T (THead (Flat Appl) t (THeads (Flat +Appl) t3 (TLRef i))) (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) +x3))))))).(\lambda (_: (nf2 c (THeads (Flat Appl) (TCons t1 t3) (TLRef +i)))).(\lambda (H15: (eq T (THeads (Flat Appl) (TCons t1 t3) (TLRef i)) +(THead (Bind x0) x1 x2))).(let H16 \def (eq_ind T (THead (Flat Appl) t1 +(THeads (Flat Appl) t3 (TLRef i))) (\lambda (ee: T).(match ee in T return +(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda +(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow +True])])) I (THead (Bind x0) x1 x2) H15) in (False_ind (eq T (THead (Flat +Appl) t (THeads (Flat Appl) (TCons t1 t3) (TLRef i))) (THead (Bind x0) x5 +(THead (Flat Appl) (lift (S O) O x4) x3))) H16))))))) t0 H_y H9) t2 +H10))))))))))))) H7)) H6))))))) H2)))))) vs)))). +(* COMMENTS +Initial nodes: 2915 +END *) + +theorem nf2_appl_lref: + \forall (c: C).(\forall (u: T).((nf2 c u) \to (\forall (i: nat).((nf2 c +(TLRef i)) \to (nf2 c (THead (Flat Appl) u (TLRef i))))))) +\def + \lambda (c: C).(\lambda (u: T).(\lambda (H: (nf2 c u)).(\lambda (i: +nat).(\lambda (H0: (nf2 c (TLRef i))).(let H_y \def (nf2_appls_lref c i H0 +(TCons u TNil)) in (H_y (conj (nf2 c u) True H I))))))). +(* COMMENTS +Initial nodes: 49 +END *) + +theorem nf2_lref_abst: + \forall (c: C).(\forall (e: C).(\forall (u: T).(\forall (i: nat).((getl i c +(CHead e (Bind Abst) u)) \to (nf2 c (TLRef i)))))) +\def + \lambda (c: C).(\lambda (e: C).(\lambda (u: T).(\lambda (i: nat).(\lambda +(H: (getl i c (CHead e (Bind Abst) u))).(\lambda (t2: T).(\lambda (H0: (pr2 c +(TLRef i) t2)).(let H1 \def (pr2_gen_lref c t2 i H0) in (or_ind (eq T t2 +(TLRef i)) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c (CHead d +(Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T t2 (lift (S i) O +u0))))) (eq T (TLRef i) t2) (\lambda (H2: (eq T t2 (TLRef i))).(eq_ind_r T +(TLRef i) (\lambda (t: T).(eq T (TLRef i) t)) (refl_equal T (TLRef i)) t2 +H2)) (\lambda (H2: (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c +(CHead d (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T t2 (lift +(S i) O u0)))))).(ex2_2_ind C T (\lambda (d: C).(\lambda (u0: T).(getl i c +(CHead d (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T t2 (lift +(S i) O u0)))) (eq T (TLRef i) t2) (\lambda (x0: C).(\lambda (x1: T).(\lambda +(H3: (getl i c (CHead x0 (Bind Abbr) x1))).(\lambda (H4: (eq T t2 (lift (S i) +O x1))).(eq_ind_r T (lift (S i) O x1) (\lambda (t: T).(eq T (TLRef i) t)) +(let H5 \def (eq_ind C (CHead e (Bind Abst) u) (\lambda (c0: C).(getl i c +c0)) H (CHead x0 (Bind Abbr) x1) (getl_mono c (CHead e (Bind Abst) u) i H +(CHead x0 (Bind Abbr) x1) H3)) in (let H6 \def (eq_ind C (CHead e (Bind Abst) +u) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort +_) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return +(\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B return +(\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow True | +Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead x0 (Bind +Abbr) x1) (getl_mono c (CHead e (Bind Abst) u) i H (CHead x0 (Bind Abbr) x1) +H3)) in (False_ind (eq T (TLRef i) (lift (S i) O x1)) H6))) t2 H4))))) H2)) +H1)))))))). +(* COMMENTS +Initial nodes: 494 +END *) + +theorem nf2_lift: + \forall (d: C).(\forall (t: T).((nf2 d t) \to (\forall (c: C).(\forall (h: +nat).(\forall (i: nat).((drop h i c d) \to (nf2 c (lift h i t)))))))) +\def + \lambda (d: C).(\lambda (t: T).(\lambda (H: ((\forall (t2: T).((pr2 d t t2) +\to (eq T t t2))))).(\lambda (c: C).(\lambda (h: nat).(\lambda (i: +nat).(\lambda (H0: (drop h i c d)).(\lambda (t2: T).(\lambda (H1: (pr2 c +(lift h i t) t2)).(let H2 \def (pr2_gen_lift c t t2 h i H1 d H0) in (ex2_ind +T (\lambda (t3: T).(eq T t2 (lift h i t3))) (\lambda (t3: T).(pr2 d t t3)) +(eq T (lift h i t) t2) (\lambda (x: T).(\lambda (H3: (eq T t2 (lift h i +x))).(\lambda (H4: (pr2 d t x)).(eq_ind_r T (lift h i x) (\lambda (t0: T).(eq +T (lift h i t) t0)) (let H_y \def (H x H4) in (let H5 \def (eq_ind_r T x +(\lambda (t0: T).(pr2 d t t0)) H4 t H_y) in (eq_ind T t (\lambda (t0: T).(eq +T (lift h i t) (lift h i t0))) (refl_equal T (lift h i t)) x H_y))) t2 H3)))) +H2)))))))))). +(* COMMENTS +Initial nodes: 245 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/pc1/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/pc1/defs.ma new file mode 100644 index 000000000..80726a188 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/pc1/defs.ma @@ -0,0 +1,24 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/pr1/defs.ma". + +definition pc1: + T \to (T \to Prop) +\def + \lambda (t1: T).(\lambda (t2: T).(ex2 T (\lambda (t: T).(pr1 t1 t)) (\lambda +(t: T).(pr1 t2 t)))). + diff --git a/matita/matita/contribs/lambdadelta/basic_1/pc1/props.ma b/matita/matita/contribs/lambdadelta/basic_1/pc1/props.ma new file mode 100644 index 000000000..0da1a5aca --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/pc1/props.ma @@ -0,0 +1,146 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/pc1/defs.ma". + +include "Basic-1/pr1/pr1.ma". + +theorem pc1_pr0_r: + \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (pc1 t1 t2))) +\def + \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(ex_intro2 T +(\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)) t2 (pr1_pr0 t1 t2 H) +(pr1_refl t2)))). +(* COMMENTS +Initial nodes: 43 +END *) + +theorem pc1_pr0_x: + \forall (t1: T).(\forall (t2: T).((pr0 t2 t1) \to (pc1 t1 t2))) +\def + \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t2 t1)).(ex_intro2 T +(\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)) t1 (pr1_refl t1) +(pr1_pr0 t2 t1 H)))). +(* COMMENTS +Initial nodes: 43 +END *) + +theorem pc1_refl: + \forall (t: T).(pc1 t t) +\def + \lambda (t: T).(ex_intro2 T (\lambda (t0: T).(pr1 t t0)) (\lambda (t0: +T).(pr1 t t0)) t (pr1_refl t) (pr1_refl t)). +(* COMMENTS +Initial nodes: 31 +END *) + +theorem pc1_pr0_u: + \forall (t2: T).(\forall (t1: T).((pr0 t1 t2) \to (\forall (t3: T).((pc1 t2 +t3) \to (pc1 t1 t3))))) +\def + \lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pr0 t1 t2)).(\lambda (t3: +T).(\lambda (H0: (pc1 t2 t3)).(let H1 \def H0 in (ex2_ind T (\lambda (t: +T).(pr1 t2 t)) (\lambda (t: T).(pr1 t3 t)) (pc1 t1 t3) (\lambda (x: +T).(\lambda (H2: (pr1 t2 x)).(\lambda (H3: (pr1 t3 x)).(ex_intro2 T (\lambda +(t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t3 t)) x (pr1_sing t2 t1 H x H2) +H3)))) H1)))))). +(* COMMENTS +Initial nodes: 97 +END *) + +theorem pc1_s: + \forall (t2: T).(\forall (t1: T).((pc1 t1 t2) \to (pc1 t2 t1))) +\def + \lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pc1 t1 t2)).(let H0 \def H in +(ex2_ind T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)) (pc1 t2 +t1) (\lambda (x: T).(\lambda (H1: (pr1 t1 x)).(\lambda (H2: (pr1 t2 +x)).(ex_intro2 T (\lambda (t: T).(pr1 t2 t)) (\lambda (t: T).(pr1 t1 t)) x H2 +H1)))) H0)))). +(* COMMENTS +Initial nodes: 79 +END *) + +theorem pc1_head_1: + \forall (u1: T).(\forall (u2: T).((pc1 u1 u2) \to (\forall (t: T).(\forall +(k: K).(pc1 (THead k u1 t) (THead k u2 t)))))) +\def + \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pc1 u1 u2)).(\lambda (t: +T).(\lambda (k: K).(let H0 \def H in (ex2_ind T (\lambda (t0: T).(pr1 u1 t0)) +(\lambda (t0: T).(pr1 u2 t0)) (pc1 (THead k u1 t) (THead k u2 t)) (\lambda +(x: T).(\lambda (H1: (pr1 u1 x)).(\lambda (H2: (pr1 u2 x)).(ex_intro2 T +(\lambda (t0: T).(pr1 (THead k u1 t) t0)) (\lambda (t0: T).(pr1 (THead k u2 +t) t0)) (THead k x t) (pr1_head_1 u1 x H1 t k) (pr1_head_1 u2 x H2 t k))))) +H0)))))). +(* COMMENTS +Initial nodes: 133 +END *) + +theorem pc1_head_2: + \forall (t1: T).(\forall (t2: T).((pc1 t1 t2) \to (\forall (u: T).(\forall +(k: K).(pc1 (THead k u t1) (THead k u t2)))))) +\def + \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc1 t1 t2)).(\lambda (u: +T).(\lambda (k: K).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr1 t1 t)) +(\lambda (t: T).(pr1 t2 t)) (pc1 (THead k u t1) (THead k u t2)) (\lambda (x: +T).(\lambda (H1: (pr1 t1 x)).(\lambda (H2: (pr1 t2 x)).(ex_intro2 T (\lambda +(t: T).(pr1 (THead k u t1) t)) (\lambda (t: T).(pr1 (THead k u t2) t)) (THead +k u x) (pr1_head_2 t1 x H1 u k) (pr1_head_2 t2 x H2 u k))))) H0)))))). +(* COMMENTS +Initial nodes: 133 +END *) + +theorem pc1_t: + \forall (t2: T).(\forall (t1: T).((pc1 t1 t2) \to (\forall (t3: T).((pc1 t2 +t3) \to (pc1 t1 t3))))) +\def + \lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pc1 t1 t2)).(\lambda (t3: +T).(\lambda (H0: (pc1 t2 t3)).(let H1 \def H0 in (ex2_ind T (\lambda (t: +T).(pr1 t2 t)) (\lambda (t: T).(pr1 t3 t)) (pc1 t1 t3) (\lambda (x: +T).(\lambda (H2: (pr1 t2 x)).(\lambda (H3: (pr1 t3 x)).(let H4 \def H in +(ex2_ind T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)) (pc1 t1 +t3) (\lambda (x0: T).(\lambda (H5: (pr1 t1 x0)).(\lambda (H6: (pr1 t2 +x0)).(ex2_ind T (\lambda (t: T).(pr1 x0 t)) (\lambda (t: T).(pr1 x t)) (pc1 +t1 t3) (\lambda (x1: T).(\lambda (H7: (pr1 x0 x1)).(\lambda (H8: (pr1 x +x1)).(ex_intro2 T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t3 t)) x1 +(pr1_t x0 t1 H5 x1 H7) (pr1_t x t3 H3 x1 H8))))) (pr1_confluence t2 x0 H6 x +H2))))) H4))))) H1)))))). +(* COMMENTS +Initial nodes: 203 +END *) + +theorem pc1_pr0_u2: + \forall (t0: T).(\forall (t1: T).((pr0 t0 t1) \to (\forall (t2: T).((pc1 t0 +t2) \to (pc1 t1 t2))))) +\def + \lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr0 t0 t1)).(\lambda (t2: +T).(\lambda (H0: (pc1 t0 t2)).(pc1_t t0 t1 (pc1_pr0_x t1 t0 H) t2 H0))))). +(* COMMENTS +Initial nodes: 35 +END *) + +theorem pc1_head: + \forall (u1: T).(\forall (u2: T).((pc1 u1 u2) \to (\forall (t1: T).(\forall +(t2: T).((pc1 t1 t2) \to (\forall (k: K).(pc1 (THead k u1 t1) (THead k u2 +t2)))))))) +\def + \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pc1 u1 u2)).(\lambda (t1: +T).(\lambda (t2: T).(\lambda (H0: (pc1 t1 t2)).(\lambda (k: K).(pc1_t (THead +k u2 t1) (THead k u1 t1) (pc1_head_1 u1 u2 H t1 k) (THead k u2 t2) +(pc1_head_2 t1 t2 H0 u2 k)))))))). +(* COMMENTS +Initial nodes: 71 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/pc3/dec.ma b/matita/matita/contribs/lambdadelta/basic_1/pc3/dec.ma new file mode 100644 index 000000000..aa7a4d89e --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/pc3/dec.ma @@ -0,0 +1,152 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/ty3/arity_props.ma". + +include "Basic-1/nf2/fwd.ma". + +theorem pc3_dec: + \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (t1: T).((ty3 g c +u1 t1) \to (\forall (u2: T).(\forall (t2: T).((ty3 g c u2 t2) \to (or (pc3 c +u1 u2) ((pc3 c u1 u2) \to False))))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda +(H: (ty3 g c u1 t1)).(\lambda (u2: T).(\lambda (t2: T).(\lambda (H0: (ty3 g c +u2 t2)).(let H_y \def (ty3_sn3 g c u1 t1 H) in (let H_y0 \def (ty3_sn3 g c u2 +t2 H0) in (let H_x \def (nf2_sn3 c u1 H_y) in (let H1 \def H_x in (ex2_ind T +(\lambda (u: T).(pr3 c u1 u)) (\lambda (u: T).(nf2 c u)) (or (pc3 c u1 u2) +((pc3 c u1 u2) \to False)) (\lambda (x: T).(\lambda (H2: (pr3 c u1 +x)).(\lambda (H3: (nf2 c x)).(let H_x0 \def (nf2_sn3 c u2 H_y0) in (let H4 +\def H_x0 in (ex2_ind T (\lambda (u: T).(pr3 c u2 u)) (\lambda (u: T).(nf2 c +u)) (or (pc3 c u1 u2) ((pc3 c u1 u2) \to False)) (\lambda (x0: T).(\lambda +(H5: (pr3 c u2 x0)).(\lambda (H6: (nf2 c x0)).(let H_x1 \def (term_dec x x0) +in (let H7 \def H_x1 in (or_ind (eq T x x0) ((eq T x x0) \to (\forall (P: +Prop).P)) (or (pc3 c u1 u2) ((pc3 c u1 u2) \to False)) (\lambda (H8: (eq T x +x0)).(let H9 \def (eq_ind_r T x0 (\lambda (t: T).(nf2 c t)) H6 x H8) in (let +H10 \def (eq_ind_r T x0 (\lambda (t: T).(pr3 c u2 t)) H5 x H8) in (or_introl +(pc3 c u1 u2) ((pc3 c u1 u2) \to False) (pc3_pr3_t c u1 x H2 u2 H10))))) +(\lambda (H8: (((eq T x x0) \to (\forall (P: Prop).P)))).(or_intror (pc3 c u1 +u2) ((pc3 c u1 u2) \to False) (\lambda (H9: (pc3 c u1 u2)).(let H10 \def H9 +in (ex2_ind T (\lambda (t: T).(pr3 c u1 t)) (\lambda (t: T).(pr3 c u2 t)) +False (\lambda (x1: T).(\lambda (H11: (pr3 c u1 x1)).(\lambda (H12: (pr3 c u2 +x1)).(let H_x2 \def (pr3_confluence c u2 x0 H5 x1 H12) in (let H13 \def H_x2 +in (ex2_ind T (\lambda (t: T).(pr3 c x0 t)) (\lambda (t: T).(pr3 c x1 t)) +False (\lambda (x2: T).(\lambda (H14: (pr3 c x0 x2)).(\lambda (H15: (pr3 c x1 +x2)).(let H_y1 \def (nf2_pr3_unfold c x0 x2 H14 H6) in (let H16 \def +(eq_ind_r T x2 (\lambda (t: T).(pr3 c x1 t)) H15 x0 H_y1) in (let H17 \def +(nf2_pr3_confluence c x H3 x0 H6 u1 H2) in (H8 (H17 (pr3_t x1 u1 c H11 x0 +H16)) False))))))) H13)))))) H10))))) H7)))))) H4)))))) H1)))))))))))). +(* COMMENTS +Initial nodes: 551 +END *) + +theorem pc3_abst_dec: + \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (t1: T).((ty3 g c +u1 t1) \to (\forall (u2: T).(\forall (t2: T).((ty3 g c u2 t2) \to (or (ex4_2 +T T (\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst) u2 u)))) +(\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u) t1))) +(\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_: T).(\lambda +(v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind Abst) u2 u)) +\to False)))))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda +(H: (ty3 g c u1 t1)).(\lambda (u2: T).(\lambda (t2: T).(\lambda (H0: (ty3 g c +u2 t2)).(let H1 \def (ty3_sn3 g c u1 t1 H) in (let H2 \def (ty3_sn3 g c u2 t2 +H0) in (let H_x \def (nf2_sn3 c u1 H1) in (let H3 \def H_x in (ex2_ind T +(\lambda (u: T).(pr3 c u1 u)) (\lambda (u: T).(nf2 c u)) (or (ex4_2 T T +(\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst) u2 u)))) +(\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u) t1))) +(\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_: T).(\lambda +(v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind Abst) u2 u)) +\to False))) (\lambda (x: T).(\lambda (H4: (pr3 c u1 x)).(\lambda (H5: (nf2 c +x)).(let H_x0 \def (nf2_sn3 c u2 H2) in (let H6 \def H_x0 in (ex2_ind T +(\lambda (u: T).(pr3 c u2 u)) (\lambda (u: T).(nf2 c u)) (or (ex4_2 T T +(\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst) u2 u)))) +(\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u) t1))) +(\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_: T).(\lambda +(v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind Abst) u2 u)) +\to False))) (\lambda (x0: T).(\lambda (H7: (pr3 c u2 x0)).(\lambda (H8: (nf2 +c x0)).(let H_x1 \def (abst_dec x x0) in (let H9 \def H_x1 in (or_ind (ex T +(\lambda (t: T).(eq T x (THead (Bind Abst) x0 t)))) (\forall (t: T).((eq T x +(THead (Bind Abst) x0 t)) \to (\forall (P: Prop).P))) (or (ex4_2 T T (\lambda +(u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst) u2 u)))) (\lambda (u: +T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u) t1))) (\lambda (_: +T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_: T).(\lambda (v2: T).(nf2 c +v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind Abst) u2 u)) \to False))) +(\lambda (H10: (ex T (\lambda (t: T).(eq T x (THead (Bind Abst) x0 +t))))).(ex_ind T (\lambda (t: T).(eq T x (THead (Bind Abst) x0 t))) (or +(ex4_2 T T (\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst) u2 +u)))) (\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u) +t1))) (\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_: +T).(\lambda (v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind +Abst) u2 u)) \to False))) (\lambda (x1: T).(\lambda (H11: (eq T x (THead +(Bind Abst) x0 x1))).(let H12 \def (eq_ind T x (\lambda (t: T).(nf2 c t)) H5 +(THead (Bind Abst) x0 x1) H11) in (let H13 \def (eq_ind T x (\lambda (t: +T).(pr3 c u1 t)) H4 (THead (Bind Abst) x0 x1) H11) in (let H_y \def +(ty3_sred_pr3 c u1 (THead (Bind Abst) x0 x1) H13 g t1 H) in (or_introl (ex4_2 +T T (\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst) u2 u)))) +(\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u) t1))) +(\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_: T).(\lambda +(v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind Abst) u2 u)) +\to False)) (ex4_2_intro T T (\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead +(Bind Abst) u2 u)))) (\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind +Abst) v2 u) t1))) (\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda +(_: T).(\lambda (v2: T).(nf2 c v2))) x1 x0 (pc3_pr3_t c u1 (THead (Bind Abst) +x0 x1) H13 (THead (Bind Abst) u2 x1) (pr3_head_12 c u2 x0 H7 (Bind Abst) x1 +x1 (pr3_refl (CHead c (Bind Abst) x0) x1))) H_y H7 H8))))))) H10)) (\lambda +(H10: ((\forall (t: T).((eq T x (THead (Bind Abst) x0 t)) \to (\forall (P: +Prop).P))))).(or_intror (ex4_2 T T (\lambda (u: T).(\lambda (_: T).(pc3 c u1 +(THead (Bind Abst) u2 u)))) (\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead +(Bind Abst) v2 u) t1))) (\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) +(\lambda (_: T).(\lambda (v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1 +(THead (Bind Abst) u2 u)) \to False)) (\lambda (u: T).(\lambda (H11: (pc3 c +u1 (THead (Bind Abst) u2 u))).(let H12 \def H11 in (ex2_ind T (\lambda (t: +T).(pr3 c u1 t)) (\lambda (t: T).(pr3 c (THead (Bind Abst) u2 u) t)) False +(\lambda (x1: T).(\lambda (H13: (pr3 c u1 x1)).(\lambda (H14: (pr3 c (THead +(Bind Abst) u2 u) x1)).(ex2_ind T (\lambda (t: T).(pr3 c x1 t)) (\lambda (t: +T).(pr3 c x t)) False (\lambda (x2: T).(\lambda (H15: (pr3 c x1 x2)).(\lambda +(H16: (pr3 c x x2)).(let H_y \def (nf2_pr3_unfold c x x2 H16 H5) in (let H17 +\def (eq_ind_r T x2 (\lambda (t: T).(pr3 c x1 t)) H15 x H_y) in (let H18 \def +(pr3_gen_abst c u2 u x1 H14) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: +T).(eq T x1 (THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: +T).(pr3 c u2 u3))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall +(u0: T).(pr3 (CHead c (Bind b) u0) u t3))))) False (\lambda (x3: T).(\lambda +(x4: T).(\lambda (H19: (eq T x1 (THead (Bind Abst) x3 x4))).(\lambda (H20: +(pr3 c u2 x3)).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pr3 (CHead c +(Bind b) u0) u x4))))).(let H22 \def (eq_ind T x1 (\lambda (t: T).(pr3 c t +x)) H17 (THead (Bind Abst) x3 x4) H19) in (let H23 \def (pr3_gen_abst c x3 x4 +x H22) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead +(Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c x3 u3))) +(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr3 (CHead +c (Bind b) u0) x4 t3))))) False (\lambda (x5: T).(\lambda (x6: T).(\lambda +(H24: (eq T x (THead (Bind Abst) x5 x6))).(\lambda (H25: (pr3 c x3 +x5)).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pr3 (CHead c (Bind b) +u0) x4 x6))))).(let H27 \def (eq_ind T x (\lambda (t: T).(\forall (t0: +T).((eq T t (THead (Bind Abst) x0 t0)) \to (\forall (P: Prop).P)))) H10 +(THead (Bind Abst) x5 x6) H24) in (let H28 \def (eq_ind T x (\lambda (t: +T).(nf2 c t)) H5 (THead (Bind Abst) x5 x6) H24) in (let H29 \def +(nf2_gen_abst c x5 x6 H28) in (land_ind (nf2 c x5) (nf2 (CHead c (Bind Abst) +x5) x6) False (\lambda (H30: (nf2 c x5)).(\lambda (_: (nf2 (CHead c (Bind +Abst) x5) x6)).(let H32 \def (nf2_pr3_confluence c x0 H8 x5 H30 u2 H7) in +(H27 x6 (sym_eq T (THead (Bind Abst) x0 x6) (THead (Bind Abst) x5 x6) +(f_equal3 K T T T THead (Bind Abst) (Bind Abst) x0 x5 x6 x6 (refl_equal K +(Bind Abst)) (H32 (pr3_t x3 u2 c H20 x5 H25)) (refl_equal T x6))) False)))) +H29))))))))) H23)))))))) H18))))))) (pr3_confluence c u1 x1 H13 x H4))))) +H12)))))) H9)))))) H6)))))) H3)))))))))))). +(* COMMENTS +Initial nodes: 1759 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/pc3/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/pc3/defs.ma new file mode 100644 index 000000000..e7ea2b24f --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/pc3/defs.ma @@ -0,0 +1,31 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/pr3/defs.ma". + +definition pc3: + C \to (T \to (T \to Prop)) +\def + \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(ex2 T (\lambda (t: T).(pr3 +c t1 t)) (\lambda (t: T).(pr3 c t2 t))))). + +inductive pc3_left (c: C): T \to (T \to Prop) \def +| pc3_left_r: \forall (t: T).(pc3_left c t t) +| pc3_left_ur: \forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall +(t3: T).((pc3_left c t2 t3) \to (pc3_left c t1 t3))))) +| pc3_left_ux: \forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall +(t3: T).((pc3_left c t1 t3) \to (pc3_left c t2 t3))))). + diff --git a/matita/matita/contribs/lambdadelta/basic_1/pc3/fsubst0.ma b/matita/matita/contribs/lambdadelta/basic_1/pc3/fsubst0.ma new file mode 100644 index 000000000..c563ca397 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/pc3/fsubst0.ma @@ -0,0 +1,726 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/pc3/left.ma". + +include "Basic-1/fsubst0/defs.ma". + +include "Basic-1/csubst0/getl.ma". + +theorem pc3_pr2_fsubst0: + \forall (c1: C).(\forall (t1: T).(\forall (t: T).((pr2 c1 t1 t) \to (\forall +(i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t2: T).((fsubst0 i u c1 +t1 c2 t2) \to (\forall (e: C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3 +c2 t2 t))))))))))) +\def + \lambda (c1: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H: (pr2 c1 t1 +t)).(pr2_ind (\lambda (c: C).(\lambda (t0: T).(\lambda (t2: T).(\forall (i: +nat).(\forall (u: T).(\forall (c2: C).(\forall (t3: T).((fsubst0 i u c t0 c2 +t3) \to (\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) \to (pc3 c2 t3 +t2))))))))))) (\lambda (c: C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H0: +(pr0 t2 t3)).(\lambda (i: nat).(\lambda (u: T).(\lambda (c2: C).(\lambda (t0: +T).(\lambda (H1: (fsubst0 i u c t2 c2 t0)).(fsubst0_ind i u c t2 (\lambda +(c0: C).(\lambda (t4: T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) +\to (pc3 c0 t4 t3))))) (\lambda (t4: T).(\lambda (H2: (subst0 i u t2 +t4)).(\lambda (e: C).(\lambda (H3: (getl i c (CHead e (Bind Abbr) +u))).(or_ind (pr0 t4 t3) (ex2 T (\lambda (w2: T).(pr0 t4 w2)) (\lambda (w2: +T).(subst0 i u t3 w2))) (pc3 c t4 t3) (\lambda (H4: (pr0 t4 t3)).(pc3_pr2_r c +t4 t3 (pr2_free c t4 t3 H4))) (\lambda (H4: (ex2 T (\lambda (w2: T).(pr0 t4 +w2)) (\lambda (w2: T).(subst0 i u t3 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 +t4 w2)) (\lambda (w2: T).(subst0 i u t3 w2)) (pc3 c t4 t3) (\lambda (x: +T).(\lambda (H5: (pr0 t4 x)).(\lambda (H6: (subst0 i u t3 x)).(pc3_pr2_u c x +t4 (pr2_free c t4 x H5) t3 (pc3_pr2_x c x t3 (pr2_delta c e u i H3 t3 t3 +(pr0_refl t3) x H6)))))) H4)) (pr0_subst0 t2 t3 H0 u t4 i H2 u (pr0_refl +u))))))) (\lambda (c0: C).(\lambda (_: (csubst0 i u c c0)).(\lambda (e: +C).(\lambda (_: (getl i c (CHead e (Bind Abbr) u))).(pc3_pr2_r c0 t2 t3 +(pr2_free c0 t2 t3 H0)))))) (\lambda (t4: T).(\lambda (H2: (subst0 i u t2 +t4)).(\lambda (c0: C).(\lambda (H3: (csubst0 i u c c0)).(\lambda (e: +C).(\lambda (H4: (getl i c (CHead e (Bind Abbr) u))).(or_ind (pr0 t4 t3) (ex2 +T (\lambda (w2: T).(pr0 t4 w2)) (\lambda (w2: T).(subst0 i u t3 w2))) (pc3 c0 +t4 t3) (\lambda (H5: (pr0 t4 t3)).(pc3_pr2_r c0 t4 t3 (pr2_free c0 t4 t3 +H5))) (\lambda (H5: (ex2 T (\lambda (w2: T).(pr0 t4 w2)) (\lambda (w2: +T).(subst0 i u t3 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t4 w2)) (\lambda +(w2: T).(subst0 i u t3 w2)) (pc3 c0 t4 t3) (\lambda (x: T).(\lambda (H6: (pr0 +t4 x)).(\lambda (H7: (subst0 i u t3 x)).(pc3_pr2_u c0 x t4 (pr2_free c0 t4 x +H6) t3 (pc3_pr2_x c0 x t3 (pr2_delta c0 e u i (csubst0_getl_ge i i (le_n i) c +c0 u H3 (CHead e (Bind Abbr) u) H4) t3 t3 (pr0_refl t3) x H7)))))) H5)) +(pr0_subst0 t2 t3 H0 u t4 i H2 u (pr0_refl u))))))))) c2 t0 H1)))))))))) +(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda +(H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (t2: T).(\lambda (t3: +T).(\lambda (H1: (pr0 t2 t3)).(\lambda (t0: T).(\lambda (H2: (subst0 i u t3 +t0)).(\lambda (i0: nat).(\lambda (u0: T).(\lambda (c2: C).(\lambda (t4: +T).(\lambda (H3: (fsubst0 i0 u0 c t2 c2 t4)).(fsubst0_ind i0 u0 c t2 (\lambda +(c0: C).(\lambda (t5: T).(\forall (e: C).((getl i0 c (CHead e (Bind Abbr) +u0)) \to (pc3 c0 t5 t0))))) (\lambda (t5: T).(\lambda (H4: (subst0 i0 u0 t2 +t5)).(\lambda (e: C).(\lambda (H5: (getl i0 c (CHead e (Bind Abbr) +u0))).(pc3_t t2 c t5 (pc3_s c t5 t2 (pc3_pr2_r c t2 t5 (pr2_delta c e u0 i0 +H5 t2 t2 (pr0_refl t2) t5 H4))) t0 (pc3_pr2_r c t2 t0 (pr2_delta c d u i H0 +t2 t3 H1 t0 H2))))))) (\lambda (c0: C).(\lambda (H4: (csubst0 i0 u0 c +c0)).(\lambda (e: C).(\lambda (H5: (getl i0 c (CHead e (Bind Abbr) +u0))).(lt_le_e i i0 (pc3 c0 t2 t0) (\lambda (H6: (lt i i0)).(let H7 \def +(csubst0_getl_lt i0 i H6 c c0 u0 H4 (CHead d (Bind Abbr) u) H0) in (or4_ind +(getl i c0 (CHead d (Bind Abbr) u)) (ex3_4 B C T T (\lambda (b: B).(\lambda +(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead +e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: +T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: +B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) +u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) +(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c0 +(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) (ex4_5 B C C T T +(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda +(_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda +(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl +i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: +C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w)))))) +(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))) (pc3 c0 t2 t0) (\lambda (H8: +(getl i c0 (CHead d (Bind Abbr) u))).(pc3_pr2_r c0 t2 t0 (pr2_delta c0 d u i +H8 t2 t3 H1 t0 H2))) (\lambda (H8: (ex3_4 B C T T (\lambda (b: B).(\lambda +(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead +e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: +T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: +B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) +u0 u1 w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda +(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b) +u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: +T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: +C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w))))) +(pc3 c0 t2 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda +(x3: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) +x2))).(\lambda (H10: (getl i c0 (CHead x1 (Bind x0) x3))).(\lambda (H11: +(subst0 (minus i0 (S i)) u0 x2 x3)).(let H12 \def (f_equal C C (\lambda (e0: +C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | +(CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) +x2) H9) in ((let H13 \def (f_equal C B (\lambda (e0: C).(match e0 in C return +(\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) +\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) +\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead +x1 (Bind x0) x2) H9) in ((let H14 \def (f_equal C T (\lambda (e0: C).(match +e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ +t5) \Rightarrow t5])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H9) in +(\lambda (H15: (eq B Abbr x0)).(\lambda (H16: (eq C d x1)).(let H17 \def +(eq_ind_r T x2 (\lambda (t5: T).(subst0 (minus i0 (S i)) u0 t5 x3)) H11 u +H14) in (let H18 \def (eq_ind_r C x1 (\lambda (c3: C).(getl i c0 (CHead c3 +(Bind x0) x3))) H10 d H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b: +B).(getl i c0 (CHead d (Bind b) x3))) H18 Abbr H15) in (ex2_ind T (\lambda +(t5: T).(subst0 i x3 t3 t5)) (\lambda (t5: T).(subst0 (S (plus (minus i0 (S +i)) i)) u0 t0 t5)) (pc3 c0 t2 t0) (\lambda (x: T).(\lambda (H20: (subst0 i x3 +t3 x)).(\lambda (H21: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let +H22 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n: +nat).(subst0 n u0 t0 x)) H21 i0 (lt_plus_minus_r i i0 H6)) in (pc3_pr2_u c0 x +t2 (pr2_delta c0 d x3 i H19 t2 t3 H1 x H20) t0 (pc3_pr2_x c0 x t0 (pr2_delta +c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H4 (CHead e (Bind Abbr) +u0) H5) t0 t0 (pr0_refl t0) x H22))))))) (subst0_subst0_back t3 t0 u i H2 x3 +u0 (minus i0 (S i)) H17)))))))) H13)) H12))))))))) H8)) (\lambda (H8: (ex3_4 +B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq +C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: +B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 +(Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda +(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))).(ex3_4_ind B C C T (\lambda +(b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind +Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda +(e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 (Bind b) u1)))))) (\lambda (_: +B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S +i)) u0 e1 e2))))) (pc3 c0 t2 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda +(x2: C).(\lambda (x3: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead +x1 (Bind x0) x3))).(\lambda (H10: (getl i c0 (CHead x2 (Bind x0) +x3))).(\lambda (H11: (csubst0 (minus i0 (S i)) u0 x1 x2)).(let H12 \def +(f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with +[(CSort _) \Rightarrow d | (CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind +Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H13 \def (f_equal C B (\lambda +(e0: C).(match e0 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow +Abbr | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with +[(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) +u) (CHead x1 (Bind x0) x3) H9) in ((let H14 \def (f_equal C T (\lambda (e0: +C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | +(CHead _ _ t5) \Rightarrow t5])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) +x3) H9) in (\lambda (H15: (eq B Abbr x0)).(\lambda (H16: (eq C d x1)).(let +H17 \def (eq_ind_r T x3 (\lambda (t5: T).(getl i c0 (CHead x2 (Bind x0) t5))) +H10 u H14) in (let H18 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus +i0 (S i)) u0 c3 x2)) H11 d H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b: +B).(getl i c0 (CHead x2 (Bind b) u))) H17 Abbr H15) in (pc3_pr2_r c0 t2 t0 +(pr2_delta c0 x2 u i H19 t2 t3 H1 t0 H2)))))))) H13)) H12))))))))) H8)) +(\lambda (H8: (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 +(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: +B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 +(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 +e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 +(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: +B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 +(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) +(pc3 c0 t2 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda +(x3: T).(\lambda (x4: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead +x1 (Bind x0) x3))).(\lambda (H10: (getl i c0 (CHead x2 (Bind x0) +x4))).(\lambda (H11: (subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H12: +(csubst0 (minus i0 (S i)) u0 x1 x2)).(let H13 \def (f_equal C C (\lambda (e0: +C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | +(CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) +x3) H9) in ((let H14 \def (f_equal C B (\lambda (e0: C).(match e0 in C return +(\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) +\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) +\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead +x1 (Bind x0) x3) H9) in ((let H15 \def (f_equal C T (\lambda (e0: C).(match +e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ +t5) \Rightarrow t5])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in +(\lambda (H16: (eq B Abbr x0)).(\lambda (H17: (eq C d x1)).(let H18 \def +(eq_ind_r T x3 (\lambda (t5: T).(subst0 (minus i0 (S i)) u0 t5 x4)) H11 u +H15) in (let H19 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus i0 (S +i)) u0 c3 x2)) H12 d H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b: +B).(getl i c0 (CHead x2 (Bind b) x4))) H10 Abbr H16) in (ex2_ind T (\lambda +(t5: T).(subst0 i x4 t3 t5)) (\lambda (t5: T).(subst0 (S (plus (minus i0 (S +i)) i)) u0 t0 t5)) (pc3 c0 t2 t0) (\lambda (x: T).(\lambda (H21: (subst0 i x4 +t3 x)).(\lambda (H22: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let +H23 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n: +nat).(subst0 n u0 t0 x)) H22 i0 (lt_plus_minus_r i i0 H6)) in (pc3_pr2_u c0 x +t2 (pr2_delta c0 x2 x4 i H20 t2 t3 H1 x H21) t0 (pc3_pr2_x c0 x t0 (pr2_delta +c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H4 (CHead e (Bind Abbr) +u0) H5) t0 t0 (pr0_refl t0) x H23))))))) (subst0_subst0_back t3 t0 u i H2 x4 +u0 (minus i0 (S i)) H18)))))))) H14)) H13))))))))))) H8)) H7))) (\lambda (H6: +(le i0 i)).(pc3_pr2_r c0 t2 t0 (pr2_delta c0 d u i (csubst0_getl_ge i0 i H6 c +c0 u0 H4 (CHead d (Bind Abbr) u) H0) t2 t3 H1 t0 H2)))))))) (\lambda (t5: +T).(\lambda (H4: (subst0 i0 u0 t2 t5)).(\lambda (c0: C).(\lambda (H5: +(csubst0 i0 u0 c c0)).(\lambda (e: C).(\lambda (H6: (getl i0 c (CHead e (Bind +Abbr) u0))).(lt_le_e i i0 (pc3 c0 t5 t0) (\lambda (H7: (lt i i0)).(let H8 +\def (csubst0_getl_lt i0 i H7 c c0 u0 H5 (CHead d (Bind Abbr) u) H0) in +(or4_ind (getl i c0 (CHead d (Bind Abbr) u)) (ex3_4 B C T T (\lambda (b: +B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind +Abbr) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: +C).(\lambda (_: T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) +(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 +(minus i0 (S i)) u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 +(Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda +(u1: T).(getl i c0 (CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) +(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda +(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) +u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: +B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 +(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))) +(pc3 c0 t5 t0) (\lambda (H9: (getl i c0 (CHead d (Bind Abbr) u))).(pc3_pr2_u2 +c0 t2 t5 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 +(CHead e (Bind Abbr) u0) H6) t2 t2 (pr0_refl t2) t5 H4) t0 (pc3_pr2_r c0 t2 +t0 (pr2_delta c0 d u i H9 t2 t3 H1 t0 H2)))) (\lambda (H9: (ex3_4 B C T T +(\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C +(CHead d (Bind Abbr) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda +(e0: C).(\lambda (_: T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) +(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 +(minus i0 (S i)) u0 u1 w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda +(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead +e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: +T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: +B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) +u0 u1 w))))) (pc3 c0 t5 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: +T).(\lambda (x3: T).(\lambda (H10: (eq C (CHead d (Bind Abbr) u) (CHead x1 +(Bind x0) x2))).(\lambda (H11: (getl i c0 (CHead x1 (Bind x0) x3))).(\lambda +(H12: (subst0 (minus i0 (S i)) u0 x2 x3)).(let H13 \def (f_equal C C (\lambda +(e0: C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow +d | (CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind +x0) x2) H10) in ((let H14 \def (f_equal C B (\lambda (e0: C).(match e0 in C +return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) +\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) +\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead +x1 (Bind x0) x2) H10) in ((let H15 \def (f_equal C T (\lambda (e0: C).(match +e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ +t6) \Rightarrow t6])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H10) in +(\lambda (H16: (eq B Abbr x0)).(\lambda (H17: (eq C d x1)).(let H18 \def +(eq_ind_r T x2 (\lambda (t6: T).(subst0 (minus i0 (S i)) u0 t6 x3)) H12 u +H15) in (let H19 \def (eq_ind_r C x1 (\lambda (c3: C).(getl i c0 (CHead c3 +(Bind x0) x3))) H11 d H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b: +B).(getl i c0 (CHead d (Bind b) x3))) H19 Abbr H16) in (ex2_ind T (\lambda +(t6: T).(subst0 i x3 t3 t6)) (\lambda (t6: T).(subst0 (S (plus (minus i0 (S +i)) i)) u0 t0 t6)) (pc3 c0 t5 t0) (\lambda (x: T).(\lambda (H21: (subst0 i x3 +t3 x)).(\lambda (H22: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let +H23 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n: +nat).(subst0 n u0 t0 x)) H22 i0 (lt_plus_minus_r i i0 H7)) in (pc3_pr2_u2 c0 +t2 t5 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 +(CHead e (Bind Abbr) u0) H6) t2 t2 (pr0_refl t2) t5 H4) t0 (pc3_pr2_u c0 x t2 +(pr2_delta c0 d x3 i H20 t2 t3 H1 x H21) t0 (pc3_pr2_x c0 x t0 (pr2_delta c0 +e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e (Bind Abbr) u0) +H6) t0 t0 (pr0_refl t0) x H23)))))))) (subst0_subst0_back t3 t0 u i H2 x3 u0 +(minus i0 (S i)) H18)))))))) H14)) H13))))))))) H9)) (\lambda (H9: (ex3_4 B C +C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C +(CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda +(_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 (Bind b) +u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))).(ex3_4_ind B C C T (\lambda (b: +B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind +Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda +(e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 (Bind b) u1)))))) (\lambda (_: +B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S +i)) u0 e1 e2))))) (pc3 c0 t5 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda +(x2: C).(\lambda (x3: T).(\lambda (H10: (eq C (CHead d (Bind Abbr) u) (CHead +x1 (Bind x0) x3))).(\lambda (H11: (getl i c0 (CHead x2 (Bind x0) +x3))).(\lambda (H12: (csubst0 (minus i0 (S i)) u0 x1 x2)).(let H13 \def +(f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with +[(CSort _) \Rightarrow d | (CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind +Abbr) u) (CHead x1 (Bind x0) x3) H10) in ((let H14 \def (f_equal C B (\lambda +(e0: C).(match e0 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow +Abbr | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with +[(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) +u) (CHead x1 (Bind x0) x3) H10) in ((let H15 \def (f_equal C T (\lambda (e0: +C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | +(CHead _ _ t6) \Rightarrow t6])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) +x3) H10) in (\lambda (H16: (eq B Abbr x0)).(\lambda (H17: (eq C d x1)).(let +H18 \def (eq_ind_r T x3 (\lambda (t6: T).(getl i c0 (CHead x2 (Bind x0) t6))) +H11 u H15) in (let H19 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus +i0 (S i)) u0 c3 x2)) H12 d H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b: +B).(getl i c0 (CHead x2 (Bind b) u))) H18 Abbr H16) in (pc3_pr2_u2 c0 t2 t5 +(pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e +(Bind Abbr) u0) H6) t2 t2 (pr0_refl t2) t5 H4) t0 (pc3_pr2_r c0 t2 t0 +(pr2_delta c0 x2 u i H20 t2 t3 H1 t0 H2))))))))) H14)) H13))))))))) H9)) +(\lambda (H9: (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 +(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: +B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 +(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 +e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 +(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: +B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 +(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) +(pc3 c0 t5 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda +(x3: T).(\lambda (x4: T).(\lambda (H10: (eq C (CHead d (Bind Abbr) u) (CHead +x1 (Bind x0) x3))).(\lambda (H11: (getl i c0 (CHead x2 (Bind x0) +x4))).(\lambda (H12: (subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H13: +(csubst0 (minus i0 (S i)) u0 x1 x2)).(let H14 \def (f_equal C C (\lambda (e0: +C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | +(CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) +x3) H10) in ((let H15 \def (f_equal C B (\lambda (e0: C).(match e0 in C +return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) +\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) +\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead +x1 (Bind x0) x3) H10) in ((let H16 \def (f_equal C T (\lambda (e0: C).(match +e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ +t6) \Rightarrow t6])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H10) in +(\lambda (H17: (eq B Abbr x0)).(\lambda (H18: (eq C d x1)).(let H19 \def +(eq_ind_r T x3 (\lambda (t6: T).(subst0 (minus i0 (S i)) u0 t6 x4)) H12 u +H16) in (let H20 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus i0 (S +i)) u0 c3 x2)) H13 d H18) in (let H21 \def (eq_ind_r B x0 (\lambda (b: +B).(getl i c0 (CHead x2 (Bind b) x4))) H11 Abbr H17) in (ex2_ind T (\lambda +(t6: T).(subst0 i x4 t3 t6)) (\lambda (t6: T).(subst0 (S (plus (minus i0 (S +i)) i)) u0 t0 t6)) (pc3 c0 t5 t0) (\lambda (x: T).(\lambda (H22: (subst0 i x4 +t3 x)).(\lambda (H23: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let +H24 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n: +nat).(subst0 n u0 t0 x)) H23 i0 (lt_plus_minus_r i i0 H7)) in (pc3_pr2_u2 c0 +t2 t5 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 +(CHead e (Bind Abbr) u0) H6) t2 t2 (pr0_refl t2) t5 H4) t0 (pc3_pr2_u c0 x t2 +(pr2_delta c0 x2 x4 i H21 t2 t3 H1 x H22) t0 (pc3_pr2_x c0 x t0 (pr2_delta c0 +e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e (Bind Abbr) u0) +H6) t0 t0 (pr0_refl t0) x H24)))))))) (subst0_subst0_back t3 t0 u i H2 x4 u0 +(minus i0 (S i)) H19)))))))) H15)) H14))))))))))) H9)) H8))) (\lambda (H7: +(le i0 i)).(pc3_pr2_u2 c0 t2 t5 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 +(le_n i0) c c0 u0 H5 (CHead e (Bind Abbr) u0) H6) t2 t2 (pr0_refl t2) t5 H4) +t0 (pc3_pr2_r c0 t2 t0 (pr2_delta c0 d u i (csubst0_getl_ge i0 i H7 c c0 u0 +H5 (CHead d (Bind Abbr) u) H0) t2 t3 H1 t0 H2))))))))))) c2 t4 +H3)))))))))))))))) c1 t1 t H)))). +(* COMMENTS +Initial nodes: 6455 +END *) + +theorem pc3_pr2_fsubst0_back: + \forall (c1: C).(\forall (t: T).(\forall (t1: T).((pr2 c1 t t1) \to (\forall +(i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t2: T).((fsubst0 i u c1 +t1 c2 t2) \to (\forall (e: C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3 +c2 t t2))))))))))) +\def + \lambda (c1: C).(\lambda (t: T).(\lambda (t1: T).(\lambda (H: (pr2 c1 t +t1)).(pr2_ind (\lambda (c: C).(\lambda (t0: T).(\lambda (t2: T).(\forall (i: +nat).(\forall (u: T).(\forall (c2: C).(\forall (t3: T).((fsubst0 i u c t2 c2 +t3) \to (\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) \to (pc3 c2 t0 +t3))))))))))) (\lambda (c: C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H0: +(pr0 t2 t3)).(\lambda (i: nat).(\lambda (u: T).(\lambda (c2: C).(\lambda (t0: +T).(\lambda (H1: (fsubst0 i u c t3 c2 t0)).(fsubst0_ind i u c t3 (\lambda +(c0: C).(\lambda (t4: T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) +\to (pc3 c0 t2 t4))))) (\lambda (t4: T).(\lambda (H2: (subst0 i u t3 +t4)).(\lambda (e: C).(\lambda (H3: (getl i c (CHead e (Bind Abbr) +u))).(pc3_pr2_u c t3 t2 (pr2_free c t2 t3 H0) t4 (pc3_pr2_r c t3 t4 +(pr2_delta c e u i H3 t3 t3 (pr0_refl t3) t4 H2))))))) (\lambda (c0: +C).(\lambda (_: (csubst0 i u c c0)).(\lambda (e: C).(\lambda (_: (getl i c +(CHead e (Bind Abbr) u))).(pc3_pr2_r c0 t2 t3 (pr2_free c0 t2 t3 H0)))))) +(\lambda (t4: T).(\lambda (H2: (subst0 i u t3 t4)).(\lambda (c0: C).(\lambda +(H3: (csubst0 i u c c0)).(\lambda (e: C).(\lambda (H4: (getl i c (CHead e +(Bind Abbr) u))).(pc3_pr2_u c0 t3 t2 (pr2_free c0 t2 t3 H0) t4 (pc3_pr2_r c0 +t3 t4 (pr2_delta c0 e u i (csubst0_getl_ge i i (le_n i) c c0 u H3 (CHead e +(Bind Abbr) u) H4) t3 t3 (pr0_refl t3) t4 H2))))))))) c2 t0 H1)))))))))) +(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda +(H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (t2: T).(\lambda (t3: +T).(\lambda (H1: (pr0 t2 t3)).(\lambda (t0: T).(\lambda (H2: (subst0 i u t3 +t0)).(\lambda (i0: nat).(\lambda (u0: T).(\lambda (c2: C).(\lambda (t4: +T).(\lambda (H3: (fsubst0 i0 u0 c t0 c2 t4)).(fsubst0_ind i0 u0 c t0 (\lambda +(c0: C).(\lambda (t5: T).(\forall (e: C).((getl i0 c (CHead e (Bind Abbr) +u0)) \to (pc3 c0 t2 t5))))) (\lambda (t5: T).(\lambda (H4: (subst0 i0 u0 t0 +t5)).(\lambda (e: C).(\lambda (H5: (getl i0 c (CHead e (Bind Abbr) +u0))).(pc3_t t3 c t2 (pc3_pr3_r c t2 t3 (pr3_pr2 c t2 t3 (pr2_free c t2 t3 +H1))) t5 (pc3_pr3_r c t3 t5 (pr3_sing c t0 t3 (pr2_delta c d u i H0 t3 t3 +(pr0_refl t3) t0 H2) t5 (pr3_pr2 c t0 t5 (pr2_delta c e u0 i0 H5 t0 t0 +(pr0_refl t0) t5 H4))))))))) (\lambda (c0: C).(\lambda (H4: (csubst0 i0 u0 c +c0)).(\lambda (e: C).(\lambda (H5: (getl i0 c (CHead e (Bind Abbr) +u0))).(lt_le_e i i0 (pc3 c0 t2 t0) (\lambda (H6: (lt i i0)).(let H7 \def +(csubst0_getl_lt i0 i H6 c c0 u0 H4 (CHead d (Bind Abbr) u) H0) in (or4_ind +(getl i c0 (CHead d (Bind Abbr) u)) (ex3_4 B C T T (\lambda (b: B).(\lambda +(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead +e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: +T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: +B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) +u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) +(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c0 +(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) (ex4_5 B C C T T +(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda +(_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda +(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl +i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: +C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w)))))) +(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))) (pc3 c0 t2 t0) (\lambda (H8: +(getl i c0 (CHead d (Bind Abbr) u))).(pc3_pr2_r c0 t2 t0 (pr2_delta c0 d u i +H8 t2 t3 H1 t0 H2))) (\lambda (H8: (ex3_4 B C T T (\lambda (b: B).(\lambda +(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead +e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: +T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: +B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) +u0 u1 w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda +(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b) +u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: +T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: +C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w))))) +(pc3 c0 t2 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda +(x3: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) +x2))).(\lambda (H10: (getl i c0 (CHead x1 (Bind x0) x3))).(\lambda (H11: +(subst0 (minus i0 (S i)) u0 x2 x3)).(let H12 \def (f_equal C C (\lambda (e0: +C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | +(CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) +x2) H9) in ((let H13 \def (f_equal C B (\lambda (e0: C).(match e0 in C return +(\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) +\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) +\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead +x1 (Bind x0) x2) H9) in ((let H14 \def (f_equal C T (\lambda (e0: C).(match +e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ +t5) \Rightarrow t5])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H9) in +(\lambda (H15: (eq B Abbr x0)).(\lambda (H16: (eq C d x1)).(let H17 \def +(eq_ind_r T x2 (\lambda (t5: T).(subst0 (minus i0 (S i)) u0 t5 x3)) H11 u +H14) in (let H18 \def (eq_ind_r C x1 (\lambda (c3: C).(getl i c0 (CHead c3 +(Bind x0) x3))) H10 d H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b: +B).(getl i c0 (CHead d (Bind b) x3))) H18 Abbr H15) in (ex2_ind T (\lambda +(t5: T).(subst0 i x3 t3 t5)) (\lambda (t5: T).(subst0 (S (plus (minus i0 (S +i)) i)) u0 t0 t5)) (pc3 c0 t2 t0) (\lambda (x: T).(\lambda (H20: (subst0 i x3 +t3 x)).(\lambda (H21: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let +H22 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n: +nat).(subst0 n u0 t0 x)) H21 i0 (lt_plus_minus_r i i0 H6)) in (pc3_pr2_u c0 x +t2 (pr2_delta c0 d x3 i H19 t2 t3 H1 x H20) t0 (pc3_pr2_x c0 x t0 (pr2_delta +c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H4 (CHead e (Bind Abbr) +u0) H5) t0 t0 (pr0_refl t0) x H22))))))) (subst0_subst0_back t3 t0 u i H2 x3 +u0 (minus i0 (S i)) H17)))))))) H13)) H12))))))))) H8)) (\lambda (H8: (ex3_4 +B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq +C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: +B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 +(Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda +(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))).(ex3_4_ind B C C T (\lambda +(b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind +Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda +(e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 (Bind b) u1)))))) (\lambda (_: +B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S +i)) u0 e1 e2))))) (pc3 c0 t2 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda +(x2: C).(\lambda (x3: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead +x1 (Bind x0) x3))).(\lambda (H10: (getl i c0 (CHead x2 (Bind x0) +x3))).(\lambda (H11: (csubst0 (minus i0 (S i)) u0 x1 x2)).(let H12 \def +(f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with +[(CSort _) \Rightarrow d | (CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind +Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H13 \def (f_equal C B (\lambda +(e0: C).(match e0 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow +Abbr | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with +[(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) +u) (CHead x1 (Bind x0) x3) H9) in ((let H14 \def (f_equal C T (\lambda (e0: +C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | +(CHead _ _ t5) \Rightarrow t5])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) +x3) H9) in (\lambda (H15: (eq B Abbr x0)).(\lambda (H16: (eq C d x1)).(let +H17 \def (eq_ind_r T x3 (\lambda (t5: T).(getl i c0 (CHead x2 (Bind x0) t5))) +H10 u H14) in (let H18 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus +i0 (S i)) u0 c3 x2)) H11 d H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b: +B).(getl i c0 (CHead x2 (Bind b) u))) H17 Abbr H15) in (pc3_pr2_r c0 t2 t0 +(pr2_delta c0 x2 u i H19 t2 t3 H1 t0 H2)))))))) H13)) H12))))))))) H8)) +(\lambda (H8: (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 +(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: +B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 +(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 +e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 +(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: +B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 +(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) +(pc3 c0 t2 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda +(x3: T).(\lambda (x4: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead +x1 (Bind x0) x3))).(\lambda (H10: (getl i c0 (CHead x2 (Bind x0) +x4))).(\lambda (H11: (subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H12: +(csubst0 (minus i0 (S i)) u0 x1 x2)).(let H13 \def (f_equal C C (\lambda (e0: +C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | +(CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) +x3) H9) in ((let H14 \def (f_equal C B (\lambda (e0: C).(match e0 in C return +(\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) +\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) +\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead +x1 (Bind x0) x3) H9) in ((let H15 \def (f_equal C T (\lambda (e0: C).(match +e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ +t5) \Rightarrow t5])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in +(\lambda (H16: (eq B Abbr x0)).(\lambda (H17: (eq C d x1)).(let H18 \def +(eq_ind_r T x3 (\lambda (t5: T).(subst0 (minus i0 (S i)) u0 t5 x4)) H11 u +H15) in (let H19 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus i0 (S +i)) u0 c3 x2)) H12 d H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b: +B).(getl i c0 (CHead x2 (Bind b) x4))) H10 Abbr H16) in (ex2_ind T (\lambda +(t5: T).(subst0 i x4 t3 t5)) (\lambda (t5: T).(subst0 (S (plus (minus i0 (S +i)) i)) u0 t0 t5)) (pc3 c0 t2 t0) (\lambda (x: T).(\lambda (H21: (subst0 i x4 +t3 x)).(\lambda (H22: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let +H23 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n: +nat).(subst0 n u0 t0 x)) H22 i0 (lt_plus_minus_r i i0 H6)) in (pc3_pr2_u c0 x +t2 (pr2_delta c0 x2 x4 i H20 t2 t3 H1 x H21) t0 (pc3_pr2_x c0 x t0 (pr2_delta +c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H4 (CHead e (Bind Abbr) +u0) H5) t0 t0 (pr0_refl t0) x H23))))))) (subst0_subst0_back t3 t0 u i H2 x4 +u0 (minus i0 (S i)) H18)))))))) H14)) H13))))))))))) H8)) H7))) (\lambda (H6: +(le i0 i)).(pc3_pr2_r c0 t2 t0 (pr2_delta c0 d u i (csubst0_getl_ge i0 i H6 c +c0 u0 H4 (CHead d (Bind Abbr) u) H0) t2 t3 H1 t0 H2)))))))) (\lambda (t5: +T).(\lambda (H4: (subst0 i0 u0 t0 t5)).(\lambda (c0: C).(\lambda (H5: +(csubst0 i0 u0 c c0)).(\lambda (e: C).(\lambda (H6: (getl i0 c (CHead e (Bind +Abbr) u0))).(lt_le_e i i0 (pc3 c0 t2 t5) (\lambda (H7: (lt i i0)).(let H8 +\def (csubst0_getl_lt i0 i H7 c c0 u0 H5 (CHead d (Bind Abbr) u) H0) in +(or4_ind (getl i c0 (CHead d (Bind Abbr) u)) (ex3_4 B C T T (\lambda (b: +B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind +Abbr) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: +C).(\lambda (_: T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) +(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 +(minus i0 (S i)) u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: +C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 +(Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda +(u1: T).(getl i c0 (CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: +C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) +(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda +(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) +u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: +B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 +(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))) +(pc3 c0 t2 t5) (\lambda (H9: (getl i c0 (CHead d (Bind Abbr) u))).(pc3_pr2_u +c0 t3 t2 (pr2_free c0 t2 t3 H1) t5 (pc3_pr3_r c0 t3 t5 (pr3_sing c0 t0 t3 +(pr2_delta c0 d u i H9 t3 t3 (pr0_refl t3) t0 H2) t5 (pr3_pr2 c0 t0 t5 +(pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e +(Bind Abbr) u0) H6) t0 t0 (pr0_refl t0) t5 H4)))))) (\lambda (H9: (ex3_4 B C +T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C +(CHead d (Bind Abbr) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda +(e0: C).(\lambda (_: T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) +(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 +(minus i0 (S i)) u0 u1 w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda +(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead +e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: +T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: +B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) +u0 u1 w))))) (pc3 c0 t2 t5) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: +T).(\lambda (x3: T).(\lambda (H10: (eq C (CHead d (Bind Abbr) u) (CHead x1 +(Bind x0) x2))).(\lambda (H11: (getl i c0 (CHead x1 (Bind x0) x3))).(\lambda +(H12: (subst0 (minus i0 (S i)) u0 x2 x3)).(let H13 \def (f_equal C C (\lambda +(e0: C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow +d | (CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind +x0) x2) H10) in ((let H14 \def (f_equal C B (\lambda (e0: C).(match e0 in C +return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) +\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) +\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead +x1 (Bind x0) x2) H10) in ((let H15 \def (f_equal C T (\lambda (e0: C).(match +e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ +t6) \Rightarrow t6])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H10) in +(\lambda (H16: (eq B Abbr x0)).(\lambda (H17: (eq C d x1)).(let H18 \def +(eq_ind_r T x2 (\lambda (t6: T).(subst0 (minus i0 (S i)) u0 t6 x3)) H12 u +H15) in (let H19 \def (eq_ind_r C x1 (\lambda (c3: C).(getl i c0 (CHead c3 +(Bind x0) x3))) H11 d H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b: +B).(getl i c0 (CHead d (Bind b) x3))) H19 Abbr H16) in (ex2_ind T (\lambda +(t6: T).(subst0 i x3 t3 t6)) (\lambda (t6: T).(subst0 (S (plus (minus i0 (S +i)) i)) u0 t0 t6)) (pc3 c0 t2 t5) (\lambda (x: T).(\lambda (H21: (subst0 i x3 +t3 x)).(\lambda (H22: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let +H23 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n: +nat).(subst0 n u0 t0 x)) H22 i0 (lt_plus_minus_r i i0 H7)) in (pc3_pr2_u c0 x +t2 (pr2_delta c0 d x3 i H20 t2 t3 H1 x H21) t5 (pc3_pr2_u2 c0 t0 x (pr2_delta +c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e (Bind Abbr) +u0) H6) t0 t0 (pr0_refl t0) x H23) t5 (pc3_pr2_r c0 t0 t5 (pr2_delta c0 e u0 +i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e (Bind Abbr) u0) H6) +t0 t0 (pr0_refl t0) t5 H4)))))))) (subst0_subst0_back t3 t0 u i H2 x3 u0 +(minus i0 (S i)) H18)))))))) H14)) H13))))))))) H9)) (\lambda (H9: (ex3_4 B C +C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C +(CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda +(_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 (Bind b) +u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))).(ex3_4_ind B C C T (\lambda (b: +B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind +Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda +(e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 (Bind b) u1)))))) (\lambda (_: +B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S +i)) u0 e1 e2))))) (pc3 c0 t2 t5) (\lambda (x0: B).(\lambda (x1: C).(\lambda +(x2: C).(\lambda (x3: T).(\lambda (H10: (eq C (CHead d (Bind Abbr) u) (CHead +x1 (Bind x0) x3))).(\lambda (H11: (getl i c0 (CHead x2 (Bind x0) +x3))).(\lambda (H12: (csubst0 (minus i0 (S i)) u0 x1 x2)).(let H13 \def +(f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with +[(CSort _) \Rightarrow d | (CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind +Abbr) u) (CHead x1 (Bind x0) x3) H10) in ((let H14 \def (f_equal C B (\lambda +(e0: C).(match e0 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow +Abbr | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with +[(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) +u) (CHead x1 (Bind x0) x3) H10) in ((let H15 \def (f_equal C T (\lambda (e0: +C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | +(CHead _ _ t6) \Rightarrow t6])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) +x3) H10) in (\lambda (H16: (eq B Abbr x0)).(\lambda (H17: (eq C d x1)).(let +H18 \def (eq_ind_r T x3 (\lambda (t6: T).(getl i c0 (CHead x2 (Bind x0) t6))) +H11 u H15) in (let H19 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus +i0 (S i)) u0 c3 x2)) H12 d H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b: +B).(getl i c0 (CHead x2 (Bind b) u))) H18 Abbr H16) in (pc3_pr2_u c0 t0 t2 +(pr2_delta c0 x2 u i H20 t2 t3 H1 t0 H2) t5 (pc3_pr2_r c0 t0 t5 (pr2_delta c0 +e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e (Bind Abbr) u0) +H6) t0 t0 (pr0_refl t0) t5 H4))))))))) H14)) H13))))))))) H9)) (\lambda (H9: +(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda +(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) +u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: +T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: +B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 +(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 +e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 +(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda +(_: T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: +B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 +(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) +(pc3 c0 t2 t5) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda +(x3: T).(\lambda (x4: T).(\lambda (H10: (eq C (CHead d (Bind Abbr) u) (CHead +x1 (Bind x0) x3))).(\lambda (H11: (getl i c0 (CHead x2 (Bind x0) +x4))).(\lambda (H12: (subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H13: +(csubst0 (minus i0 (S i)) u0 x1 x2)).(let H14 \def (f_equal C C (\lambda (e0: +C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | +(CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) +x3) H10) in ((let H15 \def (f_equal C B (\lambda (e0: C).(match e0 in C +return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) +\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) +\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead +x1 (Bind x0) x3) H10) in ((let H16 \def (f_equal C T (\lambda (e0: C).(match +e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ +t6) \Rightarrow t6])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H10) in +(\lambda (H17: (eq B Abbr x0)).(\lambda (H18: (eq C d x1)).(let H19 \def +(eq_ind_r T x3 (\lambda (t6: T).(subst0 (minus i0 (S i)) u0 t6 x4)) H12 u +H16) in (let H20 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus i0 (S +i)) u0 c3 x2)) H13 d H18) in (let H21 \def (eq_ind_r B x0 (\lambda (b: +B).(getl i c0 (CHead x2 (Bind b) x4))) H11 Abbr H17) in (ex2_ind T (\lambda +(t6: T).(subst0 i x4 t3 t6)) (\lambda (t6: T).(subst0 (S (plus (minus i0 (S +i)) i)) u0 t0 t6)) (pc3 c0 t2 t5) (\lambda (x: T).(\lambda (H22: (subst0 i x4 +t3 x)).(\lambda (H23: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let +H24 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n: +nat).(subst0 n u0 t0 x)) H23 i0 (lt_plus_minus_r i i0 H7)) in (pc3_pr2_u c0 x +t2 (pr2_delta c0 x2 x4 i H21 t2 t3 H1 x H22) t5 (pc3_pr2_u2 c0 t0 x +(pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e +(Bind Abbr) u0) H6) t0 t0 (pr0_refl t0) x H24) t5 (pc3_pr2_r c0 t0 t5 +(pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e +(Bind Abbr) u0) H6) t0 t0 (pr0_refl t0) t5 H4)))))))) (subst0_subst0_back t3 +t0 u i H2 x4 u0 (minus i0 (S i)) H19)))))))) H15)) H14))))))))))) H9)) H8))) +(\lambda (H7: (le i0 i)).(pc3_pr2_u c0 t0 t2 (pr2_delta c0 d u i +(csubst0_getl_ge i0 i H7 c c0 u0 H5 (CHead d (Bind Abbr) u) H0) t2 t3 H1 t0 +H2) t5 (pc3_pr2_r c0 t0 t5 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n +i0) c c0 u0 H5 (CHead e (Bind Abbr) u0) H6) t0 t0 (pr0_refl t0) t5 +H4))))))))))) c2 t4 H3)))))))))))))))) c1 t t1 H)))). +(* COMMENTS +Initial nodes: 6191 +END *) + +theorem pc3_fsubst0: + \forall (c1: C).(\forall (t1: T).(\forall (t: T).((pc3 c1 t1 t) \to (\forall +(i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t2: T).((fsubst0 i u c1 +t1 c2 t2) \to (\forall (e: C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3 +c2 t2 t))))))))))) +\def + \lambda (c1: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H: (pc3 c1 t1 +t)).(pc3_ind_left c1 (\lambda (t0: T).(\lambda (t2: T).(\forall (i: +nat).(\forall (u: T).(\forall (c2: C).(\forall (t3: T).((fsubst0 i u c1 t0 c2 +t3) \to (\forall (e: C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3 c2 t3 +t2)))))))))) (\lambda (t0: T).(\lambda (i: nat).(\lambda (u: T).(\lambda (c2: +C).(\lambda (t2: T).(\lambda (H0: (fsubst0 i u c1 t0 c2 t2)).(fsubst0_ind i u +c1 t0 (\lambda (c: C).(\lambda (t3: T).(\forall (e: C).((getl i c1 (CHead e +(Bind Abbr) u)) \to (pc3 c t3 t0))))) (\lambda (t3: T).(\lambda (H1: (subst0 +i u t0 t3)).(\lambda (e: C).(\lambda (H2: (getl i c1 (CHead e (Bind Abbr) +u))).(pc3_pr2_x c1 t3 t0 (pr2_delta c1 e u i H2 t0 t0 (pr0_refl t0) t3 +H1)))))) (\lambda (c0: C).(\lambda (_: (csubst0 i u c1 c0)).(\lambda (e: +C).(\lambda (_: (getl i c1 (CHead e (Bind Abbr) u))).(pc3_refl c0 t0))))) +(\lambda (t3: T).(\lambda (H1: (subst0 i u t0 t3)).(\lambda (c0: C).(\lambda +(H2: (csubst0 i u c1 c0)).(\lambda (e: C).(\lambda (H3: (getl i c1 (CHead e +(Bind Abbr) u))).(pc3_pr2_x c0 t3 t0 (pr2_delta c0 e u i (csubst0_getl_ge i i +(le_n i) c1 c0 u H2 (CHead e (Bind Abbr) u) H3) t0 t0 (pr0_refl t0) t3 +H1)))))))) c2 t2 H0))))))) (\lambda (t0: T).(\lambda (t2: T).(\lambda (H0: +(pr2 c1 t0 t2)).(\lambda (t3: T).(\lambda (H1: (pc3 c1 t2 t3)).(\lambda (H2: +((\forall (i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t4: +T).((fsubst0 i u c1 t2 c2 t4) \to (\forall (e: C).((getl i c1 (CHead e (Bind +Abbr) u)) \to (pc3 c2 t4 t3)))))))))).(\lambda (i: nat).(\lambda (u: +T).(\lambda (c2: C).(\lambda (t4: T).(\lambda (H3: (fsubst0 i u c1 t0 c2 +t4)).(fsubst0_ind i u c1 t0 (\lambda (c: C).(\lambda (t5: T).(\forall (e: +C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3 c t5 t3))))) (\lambda (t5: +T).(\lambda (H4: (subst0 i u t0 t5)).(\lambda (e: C).(\lambda (H5: (getl i c1 +(CHead e (Bind Abbr) u))).(pc3_t t2 c1 t5 (pc3_pr2_fsubst0 c1 t0 t2 H0 i u c1 +t5 (fsubst0_snd i u c1 t0 t5 H4) e H5) t3 H1))))) (\lambda (c0: C).(\lambda +(H4: (csubst0 i u c1 c0)).(\lambda (e: C).(\lambda (H5: (getl i c1 (CHead e +(Bind Abbr) u))).(pc3_t t2 c0 t0 (pc3_pr2_fsubst0 c1 t0 t2 H0 i u c0 t0 +(fsubst0_fst i u c1 t0 c0 H4) e H5) t3 (H2 i u c0 t2 (fsubst0_fst i u c1 t2 +c0 H4) e H5)))))) (\lambda (t5: T).(\lambda (H4: (subst0 i u t0 t5)).(\lambda +(c0: C).(\lambda (H5: (csubst0 i u c1 c0)).(\lambda (e: C).(\lambda (H6: +(getl i c1 (CHead e (Bind Abbr) u))).(pc3_t t2 c0 t5 (pc3_pr2_fsubst0 c1 t0 +t2 H0 i u c0 t5 (fsubst0_both i u c1 t0 t5 H4 c0 H5) e H6) t3 (H2 i u c0 t2 +(fsubst0_fst i u c1 t2 c0 H5) e H6)))))))) c2 t4 H3)))))))))))) (\lambda (t0: +T).(\lambda (t2: T).(\lambda (H0: (pr2 c1 t0 t2)).(\lambda (t3: T).(\lambda +(H1: (pc3 c1 t0 t3)).(\lambda (H2: ((\forall (i: nat).(\forall (u: +T).(\forall (c2: C).(\forall (t4: T).((fsubst0 i u c1 t0 c2 t4) \to (\forall +(e: C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3 c2 t4 +t3)))))))))).(\lambda (i: nat).(\lambda (u: T).(\lambda (c2: C).(\lambda (t4: +T).(\lambda (H3: (fsubst0 i u c1 t2 c2 t4)).(fsubst0_ind i u c1 t2 (\lambda +(c: C).(\lambda (t5: T).(\forall (e: C).((getl i c1 (CHead e (Bind Abbr) u)) +\to (pc3 c t5 t3))))) (\lambda (t5: T).(\lambda (H4: (subst0 i u t2 +t5)).(\lambda (e: C).(\lambda (H5: (getl i c1 (CHead e (Bind Abbr) +u))).(pc3_t t0 c1 t5 (pc3_s c1 t5 t0 (pc3_pr2_fsubst0_back c1 t0 t2 H0 i u c1 +t5 (fsubst0_snd i u c1 t2 t5 H4) e H5)) t3 H1))))) (\lambda (c0: C).(\lambda +(H4: (csubst0 i u c1 c0)).(\lambda (e: C).(\lambda (H5: (getl i c1 (CHead e +(Bind Abbr) u))).(pc3_t t0 c0 t2 (pc3_s c0 t2 t0 (pc3_pr2_fsubst0_back c1 t0 +t2 H0 i u c0 t2 (fsubst0_fst i u c1 t2 c0 H4) e H5)) t3 (H2 i u c0 t0 +(fsubst0_fst i u c1 t0 c0 H4) e H5)))))) (\lambda (t5: T).(\lambda (H4: +(subst0 i u t2 t5)).(\lambda (c0: C).(\lambda (H5: (csubst0 i u c1 +c0)).(\lambda (e: C).(\lambda (H6: (getl i c1 (CHead e (Bind Abbr) +u))).(pc3_t t0 c0 t5 (pc3_s c0 t5 t0 (pc3_pr2_fsubst0_back c1 t0 t2 H0 i u c0 +t5 (fsubst0_both i u c1 t2 t5 H4 c0 H5) e H6)) t3 (H2 i u c0 t0 (fsubst0_fst +i u c1 t0 c0 H5) e H6)))))))) c2 t4 H3)))))))))))) t1 t H)))). +(* COMMENTS +Initial nodes: 1249 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/pc3/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/pc3/fwd.ma new file mode 100644 index 000000000..c89e4563c --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/pc3/fwd.ma @@ -0,0 +1,333 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/pc3/props.ma". + +include "Basic-1/pr3/fwd.ma". + +theorem pc3_gen_sort: + \forall (c: C).(\forall (m: nat).(\forall (n: nat).((pc3 c (TSort m) (TSort +n)) \to (eq nat m n)))) +\def + \lambda (c: C).(\lambda (m: nat).(\lambda (n: nat).(\lambda (H: (pc3 c +(TSort m) (TSort n))).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr3 c +(TSort m) t)) (\lambda (t: T).(pr3 c (TSort n) t)) (eq nat m n) (\lambda (x: +T).(\lambda (H1: (pr3 c (TSort m) x)).(\lambda (H2: (pr3 c (TSort n) x)).(let +H3 \def (eq_ind T x (\lambda (t: T).(eq T t (TSort n))) (pr3_gen_sort c x n +H2) (TSort m) (pr3_gen_sort c x m H1)) in (let H4 \def (f_equal T nat +(\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort n0) +\Rightarrow n0 | (TLRef _) \Rightarrow m | (THead _ _ _) \Rightarrow m])) +(TSort m) (TSort n) H3) in H4))))) H0))))). +(* COMMENTS +Initial nodes: 153 +END *) + +theorem pc3_gen_abst: + \forall (c: C).(\forall (u1: T).(\forall (u2: T).(\forall (t1: T).(\forall +(t2: T).((pc3 c (THead (Bind Abst) u1 t1) (THead (Bind Abst) u2 t2)) \to +(land (pc3 c u1 u2) (\forall (b: B).(\forall (u: T).(pc3 (CHead c (Bind b) u) +t1 t2))))))))) +\def + \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (t1: T).(\lambda +(t2: T).(\lambda (H: (pc3 c (THead (Bind Abst) u1 t1) (THead (Bind Abst) u2 +t2))).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr3 c (THead (Bind Abst) +u1 t1) t)) (\lambda (t: T).(pr3 c (THead (Bind Abst) u2 t2) t)) (land (pc3 c +u1 u2) (\forall (b: B).(\forall (u: T).(pc3 (CHead c (Bind b) u) t1 t2)))) +(\lambda (x: T).(\lambda (H1: (pr3 c (THead (Bind Abst) u1 t1) x)).(\lambda +(H2: (pr3 c (THead (Bind Abst) u2 t2) x)).(let H3 \def (pr3_gen_abst c u2 t2 +x H2) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead +(Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c u2 u3))) +(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead +c (Bind b) u) t2 t3))))) (land (pc3 c u1 u2) (\forall (b: B).(\forall (u: +T).(pc3 (CHead c (Bind b) u) t1 t2)))) (\lambda (x0: T).(\lambda (x1: +T).(\lambda (H4: (eq T x (THead (Bind Abst) x0 x1))).(\lambda (H5: (pr3 c u2 +x0)).(\lambda (H6: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) +t2 x1))))).(let H7 \def (pr3_gen_abst c u1 t1 x H1) in (ex3_2_ind T T +(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u3 t3)))) +(\lambda (u3: T).(\lambda (_: T).(pr3 c u1 u3))) (\lambda (_: T).(\lambda +(t3: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t1 t3))))) +(land (pc3 c u1 u2) (\forall (b: B).(\forall (u: T).(pc3 (CHead c (Bind b) u) +t1 t2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H8: (eq T x (THead +(Bind Abst) x2 x3))).(\lambda (H9: (pr3 c u1 x2)).(\lambda (H10: ((\forall +(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t1 x3))))).(let H11 \def +(eq_ind T x (\lambda (t: T).(eq T t (THead (Bind Abst) x0 x1))) H4 (THead +(Bind Abst) x2 x3) H8) in (let H12 \def (f_equal T T (\lambda (e: T).(match e +in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x2 | (TLRef _) +\Rightarrow x2 | (THead _ t _) \Rightarrow t])) (THead (Bind Abst) x2 x3) +(THead (Bind Abst) x0 x1) H11) in ((let H13 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x3 | +(TLRef _) \Rightarrow x3 | (THead _ _ t) \Rightarrow t])) (THead (Bind Abst) +x2 x3) (THead (Bind Abst) x0 x1) H11) in (\lambda (H14: (eq T x2 x0)).(let +H15 \def (eq_ind T x3 (\lambda (t: T).(\forall (b: B).(\forall (u: T).(pr3 +(CHead c (Bind b) u) t1 t)))) H10 x1 H13) in (let H16 \def (eq_ind T x2 +(\lambda (t: T).(pr3 c u1 t)) H9 x0 H14) in (conj (pc3 c u1 u2) (\forall (b: +B).(\forall (u: T).(pc3 (CHead c (Bind b) u) t1 t2))) (pc3_pr3_t c u1 x0 H16 +u2 H5) (\lambda (b: B).(\lambda (u: T).(pc3_pr3_t (CHead c (Bind b) u) t1 x1 +(H15 b u) t2 (H6 b u))))))))) H12)))))))) H7))))))) H3))))) H0))))))). +(* COMMENTS +Initial nodes: 715 +END *) + +theorem pc3_gen_abst_shift: + \forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).((pc3 c +(THead (Bind Abst) u t1) (THead (Bind Abst) u t2)) \to (pc3 (CHead c (Bind +Abst) u) t1 t2))))) +\def + \lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda +(H: (pc3 c (THead (Bind Abst) u t1) (THead (Bind Abst) u t2))).(let H_x \def +(pc3_gen_abst c u u t1 t2 H) in (let H0 \def H_x in (land_ind (pc3 c u u) +(\forall (b: B).(\forall (u0: T).(pc3 (CHead c (Bind b) u0) t1 t2))) (pc3 +(CHead c (Bind Abst) u) t1 t2) (\lambda (_: (pc3 c u u)).(\lambda (H2: +((\forall (b: B).(\forall (u0: T).(pc3 (CHead c (Bind b) u0) t1 t2))))).(H2 +Abst u))) H0))))))). +(* COMMENTS +Initial nodes: 129 +END *) + +theorem pc3_gen_lift: + \forall (c: C).(\forall (t1: T).(\forall (t2: T).(\forall (h: nat).(\forall +(d: nat).((pc3 c (lift h d t1) (lift h d t2)) \to (\forall (e: C).((drop h d +c e) \to (pc3 e t1 t2)))))))) +\def + \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (h: nat).(\lambda +(d: nat).(\lambda (H: (pc3 c (lift h d t1) (lift h d t2))).(\lambda (e: +C).(\lambda (H0: (drop h d c e)).(let H1 \def H in (ex2_ind T (\lambda (t: +T).(pr3 c (lift h d t1) t)) (\lambda (t: T).(pr3 c (lift h d t2) t)) (pc3 e +t1 t2) (\lambda (x: T).(\lambda (H2: (pr3 c (lift h d t1) x)).(\lambda (H3: +(pr3 c (lift h d t2) x)).(let H4 \def (pr3_gen_lift c t2 x h d H3 e H0) in +(ex2_ind T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr3 e +t2 t3)) (pc3 e t1 t2) (\lambda (x0: T).(\lambda (H5: (eq T x (lift h d +x0))).(\lambda (H6: (pr3 e t2 x0)).(let H7 \def (pr3_gen_lift c t1 x h d H2 e +H0) in (ex2_ind T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: +T).(pr3 e t1 t3)) (pc3 e t1 t2) (\lambda (x1: T).(\lambda (H8: (eq T x (lift +h d x1))).(\lambda (H9: (pr3 e t1 x1)).(let H10 \def (eq_ind T x (\lambda (t: +T).(eq T t (lift h d x0))) H5 (lift h d x1) H8) in (let H11 \def (eq_ind T x1 +(\lambda (t: T).(pr3 e t1 t)) H9 x0 (lift_inj x1 x0 h d H10)) in (pc3_pr3_t e +t1 x0 H11 t2 H6)))))) H7))))) H4))))) H1))))))))). +(* COMMENTS +Initial nodes: 363 +END *) + +theorem pc3_gen_not_abst: + \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (t1: +T).(\forall (t2: T).(\forall (u1: T).(\forall (u2: T).((pc3 c (THead (Bind b) +u1 t1) (THead (Bind Abst) u2 t2)) \to (pc3 (CHead c (Bind b) u1) t1 (lift (S +O) O (THead (Bind Abst) u2 t2)))))))))) +\def + \lambda (b: B).(B_ind (\lambda (b0: B).((not (eq B b0 Abst)) \to (\forall +(c: C).(\forall (t1: T).(\forall (t2: T).(\forall (u1: T).(\forall (u2: +T).((pc3 c (THead (Bind b0) u1 t1) (THead (Bind Abst) u2 t2)) \to (pc3 (CHead +c (Bind b0) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2))))))))))) (\lambda +(_: (not (eq B Abbr Abst))).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: +T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H0: (pc3 c (THead (Bind Abbr) +u1 t1) (THead (Bind Abst) u2 t2))).(let H1 \def H0 in (ex2_ind T (\lambda (t: +T).(pr3 c (THead (Bind Abbr) u1 t1) t)) (\lambda (t: T).(pr3 c (THead (Bind +Abst) u2 t2) t)) (pc3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O (THead (Bind +Abst) u2 t2))) (\lambda (x: T).(\lambda (H2: (pr3 c (THead (Bind Abbr) u1 t1) +x)).(\lambda (H3: (pr3 c (THead (Bind Abst) u2 t2) x)).(let H4 \def +(pr3_gen_abbr c u1 t1 x H2) in (or_ind (ex3_2 T T (\lambda (u3: T).(\lambda +(t3: T).(eq T x (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: +T).(pr3 c u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr3 (CHead c (Bind Abbr) +u1) t1 t3)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O x)) (pc3 (CHead +c (Bind Abbr) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2))) (\lambda (H5: +(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3 +t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c u1 u3))) (\lambda (_: +T).(\lambda (t3: T).(pr3 (CHead c (Bind Abbr) u1) t1 t3))))).(ex3_2_ind T T +(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3 t3)))) +(\lambda (u3: T).(\lambda (_: T).(pr3 c u1 u3))) (\lambda (_: T).(\lambda +(t3: T).(pr3 (CHead c (Bind Abbr) u1) t1 t3))) (pc3 (CHead c (Bind Abbr) u1) +t1 (lift (S O) O (THead (Bind Abst) u2 t2))) (\lambda (x0: T).(\lambda (x1: +T).(\lambda (H6: (eq T x (THead (Bind Abbr) x0 x1))).(\lambda (_: (pr3 c u1 +x0)).(\lambda (_: (pr3 (CHead c (Bind Abbr) u1) t1 x1)).(let H9 \def +(pr3_gen_abst c u2 t2 x H3) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: +T).(eq T x (THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 +c u2 u3))) (\lambda (_: T).(\lambda (t3: T).(\forall (b0: B).(\forall (u: +T).(pr3 (CHead c (Bind b0) u) t2 t3))))) (pc3 (CHead c (Bind Abbr) u1) t1 +(lift (S O) O (THead (Bind Abst) u2 t2))) (\lambda (x2: T).(\lambda (x3: +T).(\lambda (H10: (eq T x (THead (Bind Abst) x2 x3))).(\lambda (_: (pr3 c u2 +x2)).(\lambda (_: ((\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) +u) t2 x3))))).(let H13 \def (eq_ind T x (\lambda (t: T).(eq T t (THead (Bind +Abbr) x0 x1))) H6 (THead (Bind Abst) x2 x3) H10) in (let H14 \def (eq_ind T +(THead (Bind Abst) x2 x3) (\lambda (ee: T).(match ee in T return (\lambda (_: +T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | +(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with +[(Bind b0) \Rightarrow (match b0 in B return (\lambda (_: B).Prop) with [Abbr +\Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat +_) \Rightarrow False])])) I (THead (Bind Abbr) x0 x1) H13) in (False_ind (pc3 +(CHead c (Bind Abbr) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2))) +H14)))))))) H9))))))) H5)) (\lambda (H5: (pr3 (CHead c (Bind Abbr) u1) t1 +(lift (S O) O x))).(let H6 \def (pr3_gen_abst c u2 t2 x H3) in (ex3_2_ind T T +(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u3 t3)))) +(\lambda (u3: T).(\lambda (_: T).(pr3 c u2 u3))) (\lambda (_: T).(\lambda +(t3: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t2 +t3))))) (pc3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O (THead (Bind Abst) u2 +t2))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: (eq T x (THead (Bind +Abst) x0 x1))).(\lambda (H8: (pr3 c u2 x0)).(\lambda (H9: ((\forall (b0: +B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t2 x1))))).(let H10 \def +(eq_ind T x (\lambda (t: T).(pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O +t))) H5 (THead (Bind Abst) x0 x1) H7) in (pc3_pr3_t (CHead c (Bind Abbr) u1) +t1 (lift (S O) O (THead (Bind Abst) x0 x1)) H10 (lift (S O) O (THead (Bind +Abst) u2 t2)) (pr3_lift (CHead c (Bind Abbr) u1) c (S O) O (drop_drop (Bind +Abbr) O c c (drop_refl c) u1) (THead (Bind Abst) u2 t2) (THead (Bind Abst) x0 +x1) (pr3_head_12 c u2 x0 H8 (Bind Abst) t2 x1 (H9 Abst x0)))))))))) H6))) +H4))))) H1))))))))) (\lambda (H: (not (eq B Abst Abst))).(\lambda (c: +C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (u1: T).(\lambda (u2: +T).(\lambda (_: (pc3 c (THead (Bind Abst) u1 t1) (THead (Bind Abst) u2 +t2))).(let H1 \def (match (H (refl_equal B Abst)) in False return (\lambda +(_: False).(pc3 (CHead c (Bind Abst) u1) t1 (lift (S O) O (THead (Bind Abst) +u2 t2)))) with []) in H1)))))))) (\lambda (_: (not (eq B Void +Abst))).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (u1: +T).(\lambda (u2: T).(\lambda (H0: (pc3 c (THead (Bind Void) u1 t1) (THead +(Bind Abst) u2 t2))).(let H1 \def H0 in (ex2_ind T (\lambda (t: T).(pr3 c +(THead (Bind Void) u1 t1) t)) (\lambda (t: T).(pr3 c (THead (Bind Abst) u2 +t2) t)) (pc3 (CHead c (Bind Void) u1) t1 (lift (S O) O (THead (Bind Abst) u2 +t2))) (\lambda (x: T).(\lambda (H2: (pr3 c (THead (Bind Void) u1 t1) +x)).(\lambda (H3: (pr3 c (THead (Bind Abst) u2 t2) x)).(let H4 \def +(pr3_gen_void c u1 t1 x H2) in (or_ind (ex3_2 T T (\lambda (u3: T).(\lambda +(t3: T).(eq T x (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: +T).(pr3 c u1 u3))) (\lambda (_: T).(\lambda (t3: T).(\forall (b0: B).(\forall +(u: T).(pr3 (CHead c (Bind b0) u) t1 t3)))))) (pr3 (CHead c (Bind Void) u1) +t1 (lift (S O) O x)) (pc3 (CHead c (Bind Void) u1) t1 (lift (S O) O (THead +(Bind Abst) u2 t2))) (\lambda (H5: (ex3_2 T T (\lambda (u3: T).(\lambda (t3: +T).(eq T x (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 +c u1 u3))) (\lambda (_: T).(\lambda (t3: T).(\forall (b0: B).(\forall (u: +T).(pr3 (CHead c (Bind b0) u) t1 t3))))))).(ex3_2_ind T T (\lambda (u3: +T).(\lambda (t3: T).(eq T x (THead (Bind Void) u3 t3)))) (\lambda (u3: +T).(\lambda (_: T).(pr3 c u1 u3))) (\lambda (_: T).(\lambda (t3: T).(\forall +(b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t1 t3))))) (pc3 (CHead c +(Bind Void) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2))) (\lambda (x0: +T).(\lambda (x1: T).(\lambda (H6: (eq T x (THead (Bind Void) x0 +x1))).(\lambda (_: (pr3 c u1 x0)).(\lambda (_: ((\forall (b0: B).(\forall (u: +T).(pr3 (CHead c (Bind b0) u) t1 x1))))).(let H9 \def (pr3_gen_abst c u2 t2 x +H3) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind +Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c u2 u3))) (\lambda (_: +T).(\lambda (t3: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) +u) t2 t3))))) (pc3 (CHead c (Bind Void) u1) t1 (lift (S O) O (THead (Bind +Abst) u2 t2))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H10: (eq T x +(THead (Bind Abst) x2 x3))).(\lambda (_: (pr3 c u2 x2)).(\lambda (_: +((\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t2 x3))))).(let +H13 \def (eq_ind T x (\lambda (t: T).(eq T t (THead (Bind Void) x0 x1))) H6 +(THead (Bind Abst) x2 x3) H10) in (let H14 \def (eq_ind T (THead (Bind Abst) +x2 x3) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with +[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) +\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b0) +\Rightarrow (match b0 in B return (\lambda (_: B).Prop) with [Abbr +\Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat +_) \Rightarrow False])])) I (THead (Bind Void) x0 x1) H13) in (False_ind (pc3 +(CHead c (Bind Void) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2))) +H14)))))))) H9))))))) H5)) (\lambda (H5: (pr3 (CHead c (Bind Void) u1) t1 +(lift (S O) O x))).(let H6 \def (pr3_gen_abst c u2 t2 x H3) in (ex3_2_ind T T +(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u3 t3)))) +(\lambda (u3: T).(\lambda (_: T).(pr3 c u2 u3))) (\lambda (_: T).(\lambda +(t3: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t2 +t3))))) (pc3 (CHead c (Bind Void) u1) t1 (lift (S O) O (THead (Bind Abst) u2 +t2))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: (eq T x (THead (Bind +Abst) x0 x1))).(\lambda (H8: (pr3 c u2 x0)).(\lambda (H9: ((\forall (b0: +B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t2 x1))))).(let H10 \def +(eq_ind T x (\lambda (t: T).(pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O +t))) H5 (THead (Bind Abst) x0 x1) H7) in (pc3_pr3_t (CHead c (Bind Void) u1) +t1 (lift (S O) O (THead (Bind Abst) x0 x1)) H10 (lift (S O) O (THead (Bind +Abst) u2 t2)) (pr3_lift (CHead c (Bind Void) u1) c (S O) O (drop_drop (Bind +Void) O c c (drop_refl c) u1) (THead (Bind Abst) u2 t2) (THead (Bind Abst) x0 +x1) (pr3_head_12 c u2 x0 H8 (Bind Abst) t2 x1 (H9 Abst x0)))))))))) H6))) +H4))))) H1))))))))) b). +(* COMMENTS +Initial nodes: 2427 +END *) + +theorem pc3_gen_lift_abst: + \forall (c: C).(\forall (t: T).(\forall (t2: T).(\forall (u2: T).(\forall +(h: nat).(\forall (d: nat).((pc3 c (lift h d t) (THead (Bind Abst) u2 t2)) +\to (\forall (e: C).((drop h d c e) \to (ex3_2 T T (\lambda (u1: T).(\lambda +(t1: T).(pr3 e t (THead (Bind Abst) u1 t1)))) (\lambda (u1: T).(\lambda (_: +T).(pr3 c u2 (lift h d u1)))) (\lambda (_: T).(\lambda (t1: T).(\forall (b: +B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t2 (lift h (S d) +t1))))))))))))))) +\def + \lambda (c: C).(\lambda (t: T).(\lambda (t2: T).(\lambda (u2: T).(\lambda +(h: nat).(\lambda (d: nat).(\lambda (H: (pc3 c (lift h d t) (THead (Bind +Abst) u2 t2))).(\lambda (e: C).(\lambda (H0: (drop h d c e)).(let H1 \def H +in (ex2_ind T (\lambda (t0: T).(pr3 c (lift h d t) t0)) (\lambda (t0: T).(pr3 +c (THead (Bind Abst) u2 t2) t0)) (ex3_2 T T (\lambda (u1: T).(\lambda (t1: +T).(pr3 e t (THead (Bind Abst) u1 t1)))) (\lambda (u1: T).(\lambda (_: +T).(pr3 c u2 (lift h d u1)))) (\lambda (_: T).(\lambda (t1: T).(\forall (b: +B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t2 (lift h (S d) t1))))))) +(\lambda (x: T).(\lambda (H2: (pr3 c (lift h d t) x)).(\lambda (H3: (pr3 c +(THead (Bind Abst) u2 t2) x)).(let H4 \def (pr3_gen_lift c t x h d H2 e H0) +in (ex2_ind T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr3 +e t t3)) (ex3_2 T T (\lambda (u1: T).(\lambda (t1: T).(pr3 e t (THead (Bind +Abst) u1 t1)))) (\lambda (u1: T).(\lambda (_: T).(pr3 c u2 (lift h d u1)))) +(\lambda (_: T).(\lambda (t1: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead +c (Bind b) u) t2 (lift h (S d) t1))))))) (\lambda (x0: T).(\lambda (H5: (eq T +x (lift h d x0))).(\lambda (H6: (pr3 e t x0)).(let H7 \def (pr3_gen_abst c u2 +t2 x H3) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead +(Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c u2 u3))) +(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead +c (Bind b) u) t2 t3))))) (ex3_2 T T (\lambda (u1: T).(\lambda (t1: T).(pr3 e +t (THead (Bind Abst) u1 t1)))) (\lambda (u1: T).(\lambda (_: T).(pr3 c u2 +(lift h d u1)))) (\lambda (_: T).(\lambda (t1: T).(\forall (b: B).(\forall +(u: T).(pr3 (CHead c (Bind b) u) t2 (lift h (S d) t1))))))) (\lambda (x1: +T).(\lambda (x2: T).(\lambda (H8: (eq T x (THead (Bind Abst) x1 +x2))).(\lambda (H9: (pr3 c u2 x1)).(\lambda (H10: ((\forall (b: B).(\forall +(u: T).(pr3 (CHead c (Bind b) u) t2 x2))))).(let H11 \def (eq_ind T x +(\lambda (t0: T).(eq T t0 (lift h d x0))) H5 (THead (Bind Abst) x1 x2) H8) in +(ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x0 (THead (Bind Abst) y +z)))) (\lambda (y: T).(\lambda (_: T).(eq T x1 (lift h d y)))) (\lambda (_: +T).(\lambda (z: T).(eq T x2 (lift h (S d) z)))) (ex3_2 T T (\lambda (u1: +T).(\lambda (t1: T).(pr3 e t (THead (Bind Abst) u1 t1)))) (\lambda (u1: +T).(\lambda (_: T).(pr3 c u2 (lift h d u1)))) (\lambda (_: T).(\lambda (t1: +T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t2 (lift h (S d) +t1))))))) (\lambda (x3: T).(\lambda (x4: T).(\lambda (H12: (eq T x0 (THead +(Bind Abst) x3 x4))).(\lambda (H13: (eq T x1 (lift h d x3))).(\lambda (H14: +(eq T x2 (lift h (S d) x4))).(let H15 \def (eq_ind T x2 (\lambda (t0: +T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t2 t0)))) H10 +(lift h (S d) x4) H14) in (let H16 \def (eq_ind T x1 (\lambda (t0: T).(pr3 c +u2 t0)) H9 (lift h d x3) H13) in (let H17 \def (eq_ind T x0 (\lambda (t0: +T).(pr3 e t t0)) H6 (THead (Bind Abst) x3 x4) H12) in (ex3_2_intro T T +(\lambda (u1: T).(\lambda (t1: T).(pr3 e t (THead (Bind Abst) u1 t1)))) +(\lambda (u1: T).(\lambda (_: T).(pr3 c u2 (lift h d u1)))) (\lambda (_: +T).(\lambda (t1: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) +t2 (lift h (S d) t1)))))) x3 x4 H17 H16 H15))))))))) (lift_gen_bind Abst x1 +x2 x0 h d H11)))))))) H7))))) H4))))) H1)))))))))). +(* COMMENTS +Initial nodes: 973 +END *) + +theorem pc3_gen_sort_abst: + \forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (n: nat).((pc3 c +(TSort n) (THead (Bind Abst) u t)) \to (\forall (P: Prop).P))))) +\def + \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (n: nat).(\lambda +(H: (pc3 c (TSort n) (THead (Bind Abst) u t))).(\lambda (P: Prop).(let H0 +\def H in (ex2_ind T (\lambda (t0: T).(pr3 c (TSort n) t0)) (\lambda (t0: +T).(pr3 c (THead (Bind Abst) u t) t0)) P (\lambda (x: T).(\lambda (H1: (pr3 c +(TSort n) x)).(\lambda (H2: (pr3 c (THead (Bind Abst) u t) x)).(let H3 \def +(pr3_gen_abst c u t x H2) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: +T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 +c u u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u0: +T).(pr3 (CHead c (Bind b) u0) t t2))))) P (\lambda (x0: T).(\lambda (x1: +T).(\lambda (H4: (eq T x (THead (Bind Abst) x0 x1))).(\lambda (_: (pr3 c u +x0)).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pr3 (CHead c (Bind b) +u0) t x1))))).(let H7 \def (eq_ind T x (\lambda (t0: T).(eq T t0 (TSort n))) +(pr3_gen_sort c x n H1) (THead (Bind Abst) x0 x1) H4) in (let H8 \def (eq_ind +T (THead (Bind Abst) x0 x1) (\lambda (ee: T).(match ee in T return (\lambda +(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False +| (THead _ _ _) \Rightarrow True])) I (TSort n) H7) in (False_ind P +H8)))))))) H3))))) H0))))))). +(* COMMENTS +Initial nodes: 303 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/pc3/left.ma b/matita/matita/contribs/lambdadelta/basic_1/pc3/left.ma new file mode 100644 index 000000000..12d22d10c --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/pc3/left.ma @@ -0,0 +1,125 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/pc3/props.ma". + +theorem pc3_ind_left__pc3_left_pr3: + \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to +(pc3_left c t1 t2)))) +\def + \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 +t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(pc3_left c t t0))) (\lambda +(t: T).(pc3_left_r c t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 +c t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2: +(pc3_left c t0 t4)).(pc3_left_ur c t3 t0 H0 t4 H2))))))) t1 t2 H)))). +(* COMMENTS +Initial nodes: 87 +END *) + +theorem pc3_ind_left__pc3_left_trans: + \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3_left c t1 t2) \to +(\forall (t3: T).((pc3_left c t2 t3) \to (pc3_left c t1 t3)))))) +\def + \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3_left c t1 +t2)).(pc3_left_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (t3: +T).((pc3_left c t0 t3) \to (pc3_left c t t3))))) (\lambda (t: T).(\lambda +(t3: T).(\lambda (H0: (pc3_left c t t3)).H0))) (\lambda (t0: T).(\lambda (t3: +T).(\lambda (H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t3 +t4)).(\lambda (H2: ((\forall (t5: T).((pc3_left c t4 t5) \to (pc3_left c t3 +t5))))).(\lambda (t5: T).(\lambda (H3: (pc3_left c t4 t5)).(pc3_left_ur c t0 +t3 H0 t5 (H2 t5 H3)))))))))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: +(pr2 c t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t0 t4)).(\lambda +(H2: ((\forall (t5: T).((pc3_left c t4 t5) \to (pc3_left c t0 +t5))))).(\lambda (t5: T).(\lambda (H3: (pc3_left c t4 t5)).(pc3_left_ux c t0 +t3 H0 t5 (H2 t5 H3)))))))))) t1 t2 H)))). +(* COMMENTS +Initial nodes: 195 +END *) + +theorem pc3_ind_left__pc3_left_sym: + \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3_left c t1 t2) \to +(pc3_left c t2 t1)))) +\def + \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3_left c t1 +t2)).(pc3_left_ind c (\lambda (t: T).(\lambda (t0: T).(pc3_left c t0 t))) +(\lambda (t: T).(pc3_left_r c t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda +(H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t3 +t4)).(\lambda (H2: (pc3_left c t4 t3)).(pc3_ind_left__pc3_left_trans c t4 t3 +H2 t0 (pc3_left_ux c t0 t3 H0 t0 (pc3_left_r c t0))))))))) (\lambda (t0: +T).(\lambda (t3: T).(\lambda (H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda +(_: (pc3_left c t0 t4)).(\lambda (H2: (pc3_left c t4 +t0)).(pc3_ind_left__pc3_left_trans c t4 t0 H2 t3 (pc3_left_ur c t0 t3 H0 t3 +(pc3_left_r c t3))))))))) t1 t2 H)))). +(* COMMENTS +Initial nodes: 163 +END *) + +theorem pc3_ind_left__pc3_left_pc3: + \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to +(pc3_left c t1 t2)))) +\def + \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c t1 +t2)).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: +T).(pr3 c t2 t)) (pc3_left c t1 t2) (\lambda (x: T).(\lambda (H1: (pr3 c t1 +x)).(\lambda (H2: (pr3 c t2 x)).(pc3_ind_left__pc3_left_trans c t1 x +(pc3_ind_left__pc3_left_pr3 c t1 x H1) t2 (pc3_ind_left__pc3_left_sym c t2 x +(pc3_ind_left__pc3_left_pr3 c t2 x H2)))))) H0))))). +(* COMMENTS +Initial nodes: 105 +END *) + +theorem pc3_ind_left__pc3_pc3_left: + \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3_left c t1 t2) \to +(pc3 c t1 t2)))) +\def + \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3_left c t1 +t2)).(pc3_left_ind c (\lambda (t: T).(\lambda (t0: T).(pc3 c t t0))) (\lambda +(t: T).(pc3_refl c t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c +t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t3 t4)).(\lambda (H2: (pc3 +c t3 t4)).(pc3_t t3 c t0 (pc3_pr2_r c t0 t3 H0) t4 H2))))))) (\lambda (t0: +T).(\lambda (t3: T).(\lambda (H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda +(_: (pc3_left c t0 t4)).(\lambda (H2: (pc3 c t0 t4)).(pc3_t t0 c t3 +(pc3_pr2_x c t3 t0 H0) t4 H2))))))) t1 t2 H)))). +(* COMMENTS +Initial nodes: 147 +END *) + +theorem pc3_ind_left: + \forall (c: C).(\forall (P: ((T \to (T \to Prop)))).(((\forall (t: T).(P t +t))) \to (((\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall (t3: +T).((pc3 c t2 t3) \to ((P t2 t3) \to (P t1 t3)))))))) \to (((\forall (t1: +T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall (t3: T).((pc3 c t1 t3) \to +((P t1 t3) \to (P t2 t3)))))))) \to (\forall (t: T).(\forall (t0: T).((pc3 c +t t0) \to (P t t0)))))))) +\def + \lambda (c: C).(\lambda (P: ((T \to (T \to Prop)))).(\lambda (H: ((\forall +(t: T).(P t t)))).(\lambda (H0: ((\forall (t1: T).(\forall (t2: T).((pr2 c t1 +t2) \to (\forall (t3: T).((pc3 c t2 t3) \to ((P t2 t3) \to (P t1 +t3))))))))).(\lambda (H1: ((\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) +\to (\forall (t3: T).((pc3 c t1 t3) \to ((P t1 t3) \to (P t2 +t3))))))))).(\lambda (t: T).(\lambda (t0: T).(\lambda (H2: (pc3 c t +t0)).(pc3_left_ind c (\lambda (t1: T).(\lambda (t2: T).(P t1 t2))) H (\lambda +(t1: T).(\lambda (t2: T).(\lambda (H3: (pr2 c t1 t2)).(\lambda (t3: +T).(\lambda (H4: (pc3_left c t2 t3)).(\lambda (H5: (P t2 t3)).(H0 t1 t2 H3 t3 +(pc3_ind_left__pc3_pc3_left c t2 t3 H4) H5))))))) (\lambda (t1: T).(\lambda +(t2: T).(\lambda (H3: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda (H4: (pc3_left +c t1 t3)).(\lambda (H5: (P t1 t3)).(H1 t1 t2 H3 t3 +(pc3_ind_left__pc3_pc3_left c t1 t3 H4) H5))))))) t t0 +(pc3_ind_left__pc3_left_pc3 c t t0 H2))))))))). +(* COMMENTS +Initial nodes: 225 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/pc3/nf2.ma b/matita/matita/contribs/lambdadelta/basic_1/pc3/nf2.ma new file mode 100644 index 000000000..ed930b85e --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/pc3/nf2.ma @@ -0,0 +1,52 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/pc3/defs.ma". + +include "Basic-1/nf2/pr3.ma". + +theorem pc3_nf2: + \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to ((nf2 c +t1) \to ((nf2 c t2) \to (eq T t1 t2)))))) +\def + \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c t1 +t2)).(\lambda (H0: (nf2 c t1)).(\lambda (H1: (nf2 c t2)).(let H2 \def H in +(ex2_ind T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) (eq T +t1 t2) (\lambda (x: T).(\lambda (H3: (pr3 c t1 x)).(\lambda (H4: (pr3 c t2 +x)).(let H_y \def (nf2_pr3_unfold c t1 x H3 H0) in (let H5 \def (eq_ind_r T x +(\lambda (t: T).(pr3 c t2 t)) H4 t1 H_y) in (let H6 \def (eq_ind_r T x +(\lambda (t: T).(pr3 c t1 t)) H3 t1 H_y) in (let H_y0 \def (nf2_pr3_unfold c +t2 t1 H5 H1) in (let H7 \def (eq_ind T t2 (\lambda (t: T).(pr3 c t t1)) H5 t1 +H_y0) in (eq_ind_r T t1 (\lambda (t: T).(eq T t1 t)) (refl_equal T t1) t2 +H_y0))))))))) H2))))))). +(* COMMENTS +Initial nodes: 195 +END *) + +theorem pc3_nf2_unfold: + \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to ((nf2 c +t2) \to (pr3 c t1 t2))))) +\def + \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c t1 +t2)).(\lambda (H0: (nf2 c t2)).(let H1 \def H in (ex2_ind T (\lambda (t: +T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) (pr3 c t1 t2) (\lambda (x: +T).(\lambda (H2: (pr3 c t1 x)).(\lambda (H3: (pr3 c t2 x)).(let H_y \def +(nf2_pr3_unfold c t2 x H3 H0) in (let H4 \def (eq_ind_r T x (\lambda (t: +T).(pr3 c t1 t)) H2 t2 H_y) in H4))))) H1)))))). +(* COMMENTS +Initial nodes: 109 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/pc3/pc1.ma b/matita/matita/contribs/lambdadelta/basic_1/pc3/pc1.ma new file mode 100644 index 000000000..dd89f951d --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/pc3/pc1.ma @@ -0,0 +1,36 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/pc3/defs.ma". + +include "Basic-1/pc1/defs.ma". + +include "Basic-1/pr3/pr1.ma". + +theorem pc3_pc1: + \forall (t1: T).(\forall (t2: T).((pc1 t1 t2) \to (\forall (c: C).(pc3 c t1 +t2)))) +\def + \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc1 t1 t2)).(\lambda (c: +C).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: +T).(pr1 t2 t)) (pc3 c t1 t2) (\lambda (x: T).(\lambda (H1: (pr1 t1 +x)).(\lambda (H2: (pr1 t2 x)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) +(\lambda (t: T).(pr3 c t2 t)) x (pr3_pr1 t1 x H1 c) (pr3_pr1 t2 x H2 c))))) +H0))))). +(* COMMENTS +Initial nodes: 103 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/pc3/props.ma b/matita/matita/contribs/lambdadelta/basic_1/pc3/props.ma new file mode 100644 index 000000000..b0a9a2f50 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/pc3/props.ma @@ -0,0 +1,483 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/pc3/defs.ma". + +include "Basic-1/pr3/pr3.ma". + +theorem clear_pc3_trans: + \forall (c2: C).(\forall (t1: T).(\forall (t2: T).((pc3 c2 t1 t2) \to +(\forall (c1: C).((clear c1 c2) \to (pc3 c1 t1 t2)))))) +\def + \lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c2 t1 +t2)).(\lambda (c1: C).(\lambda (H0: (clear c1 c2)).(let H1 \def H in (ex2_ind +T (\lambda (t: T).(pr3 c2 t1 t)) (\lambda (t: T).(pr3 c2 t2 t)) (pc3 c1 t1 +t2) (\lambda (x: T).(\lambda (H2: (pr3 c2 t1 x)).(\lambda (H3: (pr3 c2 t2 +x)).(ex_intro2 T (\lambda (t: T).(pr3 c1 t1 t)) (\lambda (t: T).(pr3 c1 t2 +t)) x (clear_pr3_trans c2 t1 x H2 c1 H0) (clear_pr3_trans c2 t2 x H3 c1 +H0))))) H1))))))). +(* COMMENTS +Initial nodes: 129 +END *) + +theorem pc3_pr2_r: + \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (pc3 c +t1 t2)))) +\def + \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 +t2)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) +t2 (pr3_pr2 c t1 t2 H) (pr3_refl c t2))))). +(* COMMENTS +Initial nodes: 55 +END *) + +theorem pc3_pr2_x: + \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t2 t1) \to (pc3 c +t1 t2)))) +\def + \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t2 +t1)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) +t1 (pr3_refl c t1) (pr3_pr2 c t2 t1 H))))). +(* COMMENTS +Initial nodes: 55 +END *) + +theorem pc3_pr3_r: + \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (pc3 c +t1 t2)))) +\def + \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 +t2)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) +t2 H (pr3_refl c t2))))). +(* COMMENTS +Initial nodes: 47 +END *) + +theorem pc3_pr3_x: + \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t2 t1) \to (pc3 c +t1 t2)))) +\def + \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t2 +t1)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) +t1 (pr3_refl c t1) H)))). +(* COMMENTS +Initial nodes: 47 +END *) + +theorem pc3_pr3_t: + \forall (c: C).(\forall (t1: T).(\forall (t0: T).((pr3 c t1 t0) \to (\forall +(t2: T).((pr3 c t2 t0) \to (pc3 c t1 t2)))))) +\def + \lambda (c: C).(\lambda (t1: T).(\lambda (t0: T).(\lambda (H: (pr3 c t1 +t0)).(\lambda (t2: T).(\lambda (H0: (pr3 c t2 t0)).(ex_intro2 T (\lambda (t: +T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) t0 H H0)))))). +(* COMMENTS +Initial nodes: 53 +END *) + +theorem pc3_refl: + \forall (c: C).(\forall (t: T).(pc3 c t t)) +\def + \lambda (c: C).(\lambda (t: T).(ex_intro2 T (\lambda (t0: T).(pr3 c t t0)) +(\lambda (t0: T).(pr3 c t t0)) t (pr3_refl c t) (pr3_refl c t))). +(* COMMENTS +Initial nodes: 41 +END *) + +theorem pc3_s: + \forall (c: C).(\forall (t2: T).(\forall (t1: T).((pc3 c t1 t2) \to (pc3 c +t2 t1)))) +\def + \lambda (c: C).(\lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pc3 c t1 +t2)).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: +T).(pr3 c t2 t)) (pc3 c t2 t1) (\lambda (x: T).(\lambda (H1: (pr3 c t1 +x)).(\lambda (H2: (pr3 c t2 x)).(ex_intro2 T (\lambda (t: T).(pr3 c t2 t)) +(\lambda (t: T).(pr3 c t1 t)) x H2 H1)))) H0))))). +(* COMMENTS +Initial nodes: 97 +END *) + +theorem pc3_thin_dx: + \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to (\forall +(u: T).(\forall (f: F).(pc3 c (THead (Flat f) u t1) (THead (Flat f) u +t2))))))) +\def + \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c t1 +t2)).(\lambda (u: T).(\lambda (f: F).(let H0 \def H in (ex2_ind T (\lambda +(t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) (pc3 c (THead (Flat f) u +t1) (THead (Flat f) u t2)) (\lambda (x: T).(\lambda (H1: (pr3 c t1 +x)).(\lambda (H2: (pr3 c t2 x)).(ex_intro2 T (\lambda (t: T).(pr3 c (THead +(Flat f) u t1) t)) (\lambda (t: T).(pr3 c (THead (Flat f) u t2) t)) (THead +(Flat f) u x) (pr3_thin_dx c t1 x H1 u f) (pr3_thin_dx c t2 x H2 u f))))) +H0))))))). +(* COMMENTS +Initial nodes: 165 +END *) + +theorem pc3_head_1: + \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pc3 c u1 u2) \to (\forall +(k: K).(\forall (t: T).(pc3 c (THead k u1 t) (THead k u2 t))))))) +\def + \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pc3 c u1 +u2)).(\lambda (k: K).(\lambda (t: T).(let H0 \def H in (ex2_ind T (\lambda +(t0: T).(pr3 c u1 t0)) (\lambda (t0: T).(pr3 c u2 t0)) (pc3 c (THead k u1 t) +(THead k u2 t)) (\lambda (x: T).(\lambda (H1: (pr3 c u1 x)).(\lambda (H2: +(pr3 c u2 x)).(ex_intro2 T (\lambda (t0: T).(pr3 c (THead k u1 t) t0)) +(\lambda (t0: T).(pr3 c (THead k u2 t) t0)) (THead k x t) (pr3_head_12 c u1 x +H1 k t t (pr3_refl (CHead c k x) t)) (pr3_head_12 c u2 x H2 k t t (pr3_refl +(CHead c k x) t)))))) H0))))))). +(* COMMENTS +Initial nodes: 183 +END *) + +theorem pc3_head_2: + \forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).(\forall +(k: K).((pc3 (CHead c k u) t1 t2) \to (pc3 c (THead k u t1) (THead k u +t2))))))) +\def + \lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda +(k: K).(\lambda (H: (pc3 (CHead c k u) t1 t2)).(let H0 \def H in (ex2_ind T +(\lambda (t: T).(pr3 (CHead c k u) t1 t)) (\lambda (t: T).(pr3 (CHead c k u) +t2 t)) (pc3 c (THead k u t1) (THead k u t2)) (\lambda (x: T).(\lambda (H1: +(pr3 (CHead c k u) t1 x)).(\lambda (H2: (pr3 (CHead c k u) t2 x)).(ex_intro2 +T (\lambda (t: T).(pr3 c (THead k u t1) t)) (\lambda (t: T).(pr3 c (THead k u +t2) t)) (THead k u x) (pr3_head_12 c u u (pr3_refl c u) k t1 x H1) +(pr3_head_12 c u u (pr3_refl c u) k t2 x H2))))) H0))))))). +(* COMMENTS +Initial nodes: 201 +END *) + +theorem pc3_pr2_u: + \forall (c: C).(\forall (t2: T).(\forall (t1: T).((pr2 c t1 t2) \to (\forall +(t3: T).((pc3 c t2 t3) \to (pc3 c t1 t3)))))) +\def + \lambda (c: C).(\lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pr2 c t1 +t2)).(\lambda (t3: T).(\lambda (H0: (pc3 c t2 t3)).(let H1 \def H0 in +(ex2_ind T (\lambda (t: T).(pr3 c t2 t)) (\lambda (t: T).(pr3 c t3 t)) (pc3 c +t1 t3) (\lambda (x: T).(\lambda (H2: (pr3 c t2 x)).(\lambda (H3: (pr3 c t3 +x)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t3 t)) +x (pr3_sing c t2 t1 H x H2) H3)))) H1))))))). +(* COMMENTS +Initial nodes: 119 +END *) + +theorem pc3_t: + \forall (t2: T).(\forall (c: C).(\forall (t1: T).((pc3 c t1 t2) \to (\forall +(t3: T).((pc3 c t2 t3) \to (pc3 c t1 t3)))))) +\def + \lambda (t2: T).(\lambda (c: C).(\lambda (t1: T).(\lambda (H: (pc3 c t1 +t2)).(\lambda (t3: T).(\lambda (H0: (pc3 c t2 t3)).(let H1 \def H0 in +(ex2_ind T (\lambda (t: T).(pr3 c t2 t)) (\lambda (t: T).(pr3 c t3 t)) (pc3 c +t1 t3) (\lambda (x: T).(\lambda (H2: (pr3 c t2 x)).(\lambda (H3: (pr3 c t3 +x)).(let H4 \def H in (ex2_ind T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: +T).(pr3 c t2 t)) (pc3 c t1 t3) (\lambda (x0: T).(\lambda (H5: (pr3 c t1 +x0)).(\lambda (H6: (pr3 c t2 x0)).(ex2_ind T (\lambda (t: T).(pr3 c x0 t)) +(\lambda (t: T).(pr3 c x t)) (pc3 c t1 t3) (\lambda (x1: T).(\lambda (H7: +(pr3 c x0 x1)).(\lambda (H8: (pr3 c x x1)).(pc3_pr3_t c t1 x1 (pr3_t x0 t1 c +H5 x1 H7) t3 (pr3_t x t3 c H3 x1 H8))))) (pr3_confluence c t2 x0 H6 x H2))))) +H4))))) H1))))))). +(* COMMENTS +Initial nodes: 233 +END *) + +theorem pc3_pr2_u2: + \forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr2 c t0 t1) \to (\forall +(t2: T).((pc3 c t0 t2) \to (pc3 c t1 t2)))))) +\def + \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr2 c t0 +t1)).(\lambda (t2: T).(\lambda (H0: (pc3 c t0 t2)).(pc3_t t0 c t1 (pc3_pr2_x +c t1 t0 H) t2 H0)))))). +(* COMMENTS +Initial nodes: 45 +END *) + +theorem pc3_pr3_conf: + \forall (c: C).(\forall (t: T).(\forall (t1: T).((pc3 c t t1) \to (\forall +(t2: T).((pr3 c t t2) \to (pc3 c t2 t1)))))) +\def + \lambda (c: C).(\lambda (t: T).(\lambda (t1: T).(\lambda (H: (pc3 c t +t1)).(\lambda (t2: T).(\lambda (H0: (pr3 c t t2)).(pc3_t t c t2 (pc3_pr3_x c +t2 t H0) t1 H)))))). +(* COMMENTS +Initial nodes: 45 +END *) + +theorem pc3_head_12: + \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pc3 c u1 u2) \to (\forall +(k: K).(\forall (t1: T).(\forall (t2: T).((pc3 (CHead c k u2) t1 t2) \to (pc3 +c (THead k u1 t1) (THead k u2 t2))))))))) +\def + \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pc3 c u1 +u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pc3 +(CHead c k u2) t1 t2)).(pc3_t (THead k u2 t1) c (THead k u1 t1) (pc3_head_1 c +u1 u2 H k t1) (THead k u2 t2) (pc3_head_2 c u2 t1 t2 k H0))))))))). +(* COMMENTS +Initial nodes: 89 +END *) + +theorem pc3_head_21: + \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pc3 c u1 u2) \to (\forall +(k: K).(\forall (t1: T).(\forall (t2: T).((pc3 (CHead c k u1) t1 t2) \to (pc3 +c (THead k u1 t1) (THead k u2 t2))))))))) +\def + \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pc3 c u1 +u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pc3 +(CHead c k u1) t1 t2)).(pc3_t (THead k u1 t2) c (THead k u1 t1) (pc3_head_2 c +u1 t1 t2 k H0) (THead k u2 t2) (pc3_head_1 c u1 u2 H k t2))))))))). +(* COMMENTS +Initial nodes: 89 +END *) + +theorem pc3_pr0_pr2_t: + \forall (u1: T).(\forall (u2: T).((pr0 u2 u1) \to (\forall (c: C).(\forall +(t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pc3 +(CHead c k u1) t1 t2)))))))) +\def + \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr0 u2 u1)).(\lambda (c: +C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k: K).(\lambda (H0: (pr2 +(CHead c k u2) t1 t2)).(insert_eq C (CHead c k u2) (\lambda (c0: C).(pr2 c0 +t1 t2)) (\lambda (_: C).(pc3 (CHead c k u1) t1 t2)) (\lambda (y: C).(\lambda +(H1: (pr2 y t1 t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: +T).((eq C c0 (CHead c k u2)) \to (pc3 (CHead c k u1) t t0))))) (\lambda (c0: +C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (H3: +(eq C c0 (CHead c k u2))).(let H4 \def (f_equal C C (\lambda (e: C).e) c0 +(CHead c k u2) H3) in (pc3_pr2_r (CHead c k u1) t3 t4 (pr2_free (CHead c k +u1) t3 t4 H2)))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: +T).(\lambda (i: nat).(\lambda (H2: (getl i c0 (CHead d (Bind Abbr) +u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda +(t: T).(\lambda (H4: (subst0 i u t4 t)).(\lambda (H5: (eq C c0 (CHead c k +u2))).(let H6 \def (f_equal C C (\lambda (e: C).e) c0 (CHead c k u2) H5) in +(let H7 \def (eq_ind C c0 (\lambda (c1: C).(getl i c1 (CHead d (Bind Abbr) +u))) H2 (CHead c k u2) H6) in (nat_ind (\lambda (n: nat).((getl n (CHead c k +u2) (CHead d (Bind Abbr) u)) \to ((subst0 n u t4 t) \to (pc3 (CHead c k u1) +t3 t)))) (\lambda (H8: (getl O (CHead c k u2) (CHead d (Bind Abbr) +u))).(\lambda (H9: (subst0 O u t4 t)).(K_ind (\lambda (k0: K).((clear (CHead +c k0 u2) (CHead d (Bind Abbr) u)) \to (pc3 (CHead c k0 u1) t3 t))) (\lambda +(b: B).(\lambda (H10: (clear (CHead c (Bind b) u2) (CHead d (Bind Abbr) +u))).(let H11 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda +(_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) +(CHead d (Bind Abbr) u) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d +(Bind Abbr) u) u2 H10)) in ((let H12 \def (f_equal C B (\lambda (e: C).(match +e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ +k0 _) \Rightarrow (match k0 in K return (\lambda (_: K).B) with [(Bind b0) +\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) +(CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 H10)) in +((let H13 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: +C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead +d (Bind Abbr) u) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind +Abbr) u) u2 H10)) in (\lambda (H14: (eq B Abbr b)).(\lambda (_: (eq C d +c)).(let H16 \def (eq_ind T u (\lambda (t0: T).(subst0 O t0 t4 t)) H9 u2 H13) +in (eq_ind B Abbr (\lambda (b0: B).(pc3 (CHead c (Bind b0) u1) t3 t)) +(ex2_ind T (\lambda (t0: T).(subst0 O u1 t4 t0)) (\lambda (t0: T).(pr0 t t0)) +(pc3 (CHead c (Bind Abbr) u1) t3 t) (\lambda (x: T).(\lambda (H17: (subst0 O +u1 t4 x)).(\lambda (H18: (pr0 t x)).(pc3_pr3_t (CHead c (Bind Abbr) u1) t3 x +(pr3_pr2 (CHead c (Bind Abbr) u1) t3 x (pr2_delta (CHead c (Bind Abbr) u1) c +u1 O (getl_refl Abbr c u1) t3 t4 H3 x H17)) t (pr3_pr2 (CHead c (Bind Abbr) +u1) t x (pr2_free (CHead c (Bind Abbr) u1) t x H18)))))) (pr0_subst0_fwd u2 +t4 t O H16 u1 H)) b H14))))) H12)) H11)))) (\lambda (f: F).(\lambda (H10: +(clear (CHead c (Flat f) u2) (CHead d (Bind Abbr) u))).(clear_pc3_trans +(CHead d (Bind Abbr) u) t3 t (pc3_pr2_r (CHead d (Bind Abbr) u) t3 t +(pr2_delta (CHead d (Bind Abbr) u) d u O (getl_refl Abbr d u) t3 t4 H3 t H9)) +(CHead c (Flat f) u1) (clear_flat c (CHead d (Bind Abbr) u) (clear_gen_flat f +c (CHead d (Bind Abbr) u) u2 H10) f u1)))) k (getl_gen_O (CHead c k u2) +(CHead d (Bind Abbr) u) H8)))) (\lambda (i0: nat).(\lambda (IHi: (((getl i0 +(CHead c k u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t4 t) \to (pc3 +(CHead c k u1) t3 t))))).(\lambda (H8: (getl (S i0) (CHead c k u2) (CHead d +(Bind Abbr) u))).(\lambda (H9: (subst0 (S i0) u t4 t)).(K_ind (\lambda (k0: +K).((((getl i0 (CHead c k0 u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t4 +t) \to (pc3 (CHead c k0 u1) t3 t)))) \to ((getl (r k0 i0) c (CHead d (Bind +Abbr) u)) \to (pc3 (CHead c k0 u1) t3 t)))) (\lambda (b: B).(\lambda (_: +(((getl i0 (CHead c (Bind b) u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u +t4 t) \to (pc3 (CHead c (Bind b) u1) t3 t))))).(\lambda (H10: (getl (r (Bind +b) i0) c (CHead d (Bind Abbr) u))).(pc3_pr2_r (CHead c (Bind b) u1) t3 t +(pr2_delta (CHead c (Bind b) u1) d u (S i0) (getl_head (Bind b) i0 c (CHead d +(Bind Abbr) u) H10 u1) t3 t4 H3 t H9))))) (\lambda (f: F).(\lambda (_: +(((getl i0 (CHead c (Flat f) u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u +t4 t) \to (pc3 (CHead c (Flat f) u1) t3 t))))).(\lambda (H10: (getl (r (Flat +f) i0) c (CHead d (Bind Abbr) u))).(pc3_pr2_r (CHead c (Flat f) u1) t3 t +(pr2_cflat c t3 t (pr2_delta c d u (r (Flat f) i0) H10 t3 t4 H3 t H9) f +u1))))) k IHi (getl_gen_S k c (CHead d (Bind Abbr) u) u2 i0 H8)))))) i H7 +H4)))))))))))))) y t1 t2 H1))) H0)))))))). +(* COMMENTS +Initial nodes: 1533 +END *) + +theorem pc3_pr2_pr2_t: + \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr2 c u2 u1) \to (\forall +(t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pc3 +(CHead c k u1) t1 t2)))))))) +\def + \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr2 c u2 +u1)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (t1: +T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c0 k t) t1 t2) \to (pc3 +(CHead c0 k t0) t1 t2)))))))) (\lambda (c0: C).(\lambda (t1: T).(\lambda (t2: +T).(\lambda (H0: (pr0 t1 t2)).(\lambda (t0: T).(\lambda (t3: T).(\lambda (k: +K).(\lambda (H1: (pr2 (CHead c0 k t1) t0 t3)).(pc3_pr0_pr2_t t2 t1 H0 c0 t0 +t3 k H1))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda +(i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t1: +T).(\lambda (t2: T).(\lambda (H1: (pr0 t1 t2)).(\lambda (t: T).(\lambda (H2: +(subst0 i u t2 t)).(\lambda (t0: T).(\lambda (t3: T).(\lambda (k: K).(\lambda +(H3: (pr2 (CHead c0 k t1) t0 t3)).(insert_eq C (CHead c0 k t1) (\lambda (c1: +C).(pr2 c1 t0 t3)) (\lambda (_: C).(pc3 (CHead c0 k t) t0 t3)) (\lambda (y: +C).(\lambda (H4: (pr2 y t0 t3)).(pr2_ind (\lambda (c1: C).(\lambda (t4: +T).(\lambda (t5: T).((eq C c1 (CHead c0 k t1)) \to (pc3 (CHead c0 k t) t4 +t5))))) (\lambda (c1: C).(\lambda (t4: T).(\lambda (t5: T).(\lambda (H5: (pr0 +t4 t5)).(\lambda (_: (eq C c1 (CHead c0 k t1))).(pc3_pr2_r (CHead c0 k t) t4 +t5 (pr2_free (CHead c0 k t) t4 t5 H5))))))) (\lambda (c1: C).(\lambda (d0: +C).(\lambda (u0: T).(\lambda (i0: nat).(\lambda (H5: (getl i0 c1 (CHead d0 +(Bind Abbr) u0))).(\lambda (t4: T).(\lambda (t5: T).(\lambda (H6: (pr0 t4 +t5)).(\lambda (t6: T).(\lambda (H7: (subst0 i0 u0 t5 t6)).(\lambda (H8: (eq C +c1 (CHead c0 k t1))).(let H9 \def (eq_ind C c1 (\lambda (c2: C).(getl i0 c2 +(CHead d0 (Bind Abbr) u0))) H5 (CHead c0 k t1) H8) in (nat_ind (\lambda (n: +nat).((getl n (CHead c0 k t1) (CHead d0 (Bind Abbr) u0)) \to ((subst0 n u0 t5 +t6) \to (pc3 (CHead c0 k t) t4 t6)))) (\lambda (H10: (getl O (CHead c0 k t1) +(CHead d0 (Bind Abbr) u0))).(\lambda (H11: (subst0 O u0 t5 t6)).(K_ind +(\lambda (k0: K).((clear (CHead c0 k0 t1) (CHead d0 (Bind Abbr) u0)) \to (pc3 +(CHead c0 k0 t) t4 t6))) (\lambda (b: B).(\lambda (H12: (clear (CHead c0 +(Bind b) t1) (CHead d0 (Bind Abbr) u0))).(let H13 \def (f_equal C C (\lambda +(e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d0 +| (CHead c2 _ _) \Rightarrow c2])) (CHead d0 (Bind Abbr) u0) (CHead c0 (Bind +b) t1) (clear_gen_bind b c0 (CHead d0 (Bind Abbr) u0) t1 H12)) in ((let H14 +\def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) +with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow (match k0 in K +return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) +\Rightarrow Abbr])])) (CHead d0 (Bind Abbr) u0) (CHead c0 (Bind b) t1) +(clear_gen_bind b c0 (CHead d0 (Bind Abbr) u0) t1 H12)) in ((let H15 \def +(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with +[(CSort _) \Rightarrow u0 | (CHead _ _ t7) \Rightarrow t7])) (CHead d0 (Bind +Abbr) u0) (CHead c0 (Bind b) t1) (clear_gen_bind b c0 (CHead d0 (Bind Abbr) +u0) t1 H12)) in (\lambda (H16: (eq B Abbr b)).(\lambda (_: (eq C d0 c0)).(let +H18 \def (eq_ind T u0 (\lambda (t7: T).(subst0 O t7 t5 t6)) H11 t1 H15) in +(eq_ind B Abbr (\lambda (b0: B).(pc3 (CHead c0 (Bind b0) t) t4 t6)) (ex2_ind +T (\lambda (t7: T).(subst0 O t2 t5 t7)) (\lambda (t7: T).(pr0 t6 t7)) (pc3 +(CHead c0 (Bind Abbr) t) t4 t6) (\lambda (x: T).(\lambda (H19: (subst0 O t2 +t5 x)).(\lambda (H20: (pr0 t6 x)).(ex2_ind T (\lambda (t7: T).(subst0 O t t5 +t7)) (\lambda (t7: T).(subst0 (S (plus i O)) u x t7)) (pc3 (CHead c0 (Bind +Abbr) t) t4 t6) (\lambda (x0: T).(\lambda (H21: (subst0 O t t5 x0)).(\lambda +(H22: (subst0 (S (plus i O)) u x x0)).(let H23 \def (f_equal nat nat S (plus +i O) i (sym_eq nat i (plus i O) (plus_n_O i))) in (let H24 \def (eq_ind nat +(S (plus i O)) (\lambda (n: nat).(subst0 n u x x0)) H22 (S i) H23) in +(pc3_pr2_u (CHead c0 (Bind Abbr) t) x0 t4 (pr2_delta (CHead c0 (Bind Abbr) t) +c0 t O (getl_refl Abbr c0 t) t4 t5 H6 x0 H21) t6 (pc3_pr2_x (CHead c0 (Bind +Abbr) t) x0 t6 (pr2_delta (CHead c0 (Bind Abbr) t) d u (S i) (getl_head (Bind +Abbr) i c0 (CHead d (Bind Abbr) u) H0 t) t6 x H20 x0 H24)))))))) +(subst0_subst0_back t5 x t2 O H19 t u i H2))))) (pr0_subst0_fwd t1 t5 t6 O +H18 t2 H1)) b H16))))) H14)) H13)))) (\lambda (f: F).(\lambda (H12: (clear +(CHead c0 (Flat f) t1) (CHead d0 (Bind Abbr) u0))).(clear_pc3_trans (CHead d0 +(Bind Abbr) u0) t4 t6 (pc3_pr2_r (CHead d0 (Bind Abbr) u0) t4 t6 (pr2_delta +(CHead d0 (Bind Abbr) u0) d0 u0 O (getl_refl Abbr d0 u0) t4 t5 H6 t6 H11)) +(CHead c0 (Flat f) t) (clear_flat c0 (CHead d0 (Bind Abbr) u0) +(clear_gen_flat f c0 (CHead d0 (Bind Abbr) u0) t1 H12) f t)))) k (getl_gen_O +(CHead c0 k t1) (CHead d0 (Bind Abbr) u0) H10)))) (\lambda (i1: nat).(\lambda +(_: (((getl i1 (CHead c0 k t1) (CHead d0 (Bind Abbr) u0)) \to ((subst0 i1 u0 +t5 t6) \to (pc3 (CHead c0 k t) t4 t6))))).(\lambda (H10: (getl (S i1) (CHead +c0 k t1) (CHead d0 (Bind Abbr) u0))).(\lambda (H11: (subst0 (S i1) u0 t5 +t6)).(K_ind (\lambda (k0: K).((getl (r k0 i1) c0 (CHead d0 (Bind Abbr) u0)) +\to (pc3 (CHead c0 k0 t) t4 t6))) (\lambda (b: B).(\lambda (H12: (getl (r +(Bind b) i1) c0 (CHead d0 (Bind Abbr) u0))).(pc3_pr2_r (CHead c0 (Bind b) t) +t4 t6 (pr2_delta (CHead c0 (Bind b) t) d0 u0 (S i1) (getl_head (Bind b) i1 c0 +(CHead d0 (Bind Abbr) u0) H12 t) t4 t5 H6 t6 H11)))) (\lambda (f: F).(\lambda +(H12: (getl (r (Flat f) i1) c0 (CHead d0 (Bind Abbr) u0))).(pc3_pr2_r (CHead +c0 (Flat f) t) t4 t6 (pr2_cflat c0 t4 t6 (pr2_delta c0 d0 u0 (r (Flat f) i1) +H12 t4 t5 H6 t6 H11) f t)))) k (getl_gen_S k c0 (CHead d0 (Bind Abbr) u0) t1 +i1 H10)))))) i0 H9 H7))))))))))))) y t0 t3 H4))) H3))))))))))))))) c u2 u1 +H)))). +(* COMMENTS +Initial nodes: 1671 +END *) + +theorem pc3_pr2_pr3_t: + \forall (c: C).(\forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall +(k: K).((pr3 (CHead c k u2) t1 t2) \to (\forall (u1: T).((pr2 c u2 u1) \to +(pc3 (CHead c k u1) t1 t2)))))))) +\def + \lambda (c: C).(\lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda +(k: K).(\lambda (H: (pr3 (CHead c k u2) t1 t2)).(pr3_ind (CHead c k u2) +(\lambda (t: T).(\lambda (t0: T).(\forall (u1: T).((pr2 c u2 u1) \to (pc3 +(CHead c k u1) t t0))))) (\lambda (t: T).(\lambda (u1: T).(\lambda (_: (pr2 c +u2 u1)).(pc3_refl (CHead c k u1) t)))) (\lambda (t0: T).(\lambda (t3: +T).(\lambda (H0: (pr2 (CHead c k u2) t3 t0)).(\lambda (t4: T).(\lambda (_: +(pr3 (CHead c k u2) t0 t4)).(\lambda (H2: ((\forall (u1: T).((pr2 c u2 u1) +\to (pc3 (CHead c k u1) t0 t4))))).(\lambda (u1: T).(\lambda (H3: (pr2 c u2 +u1)).(pc3_t t0 (CHead c k u1) t3 (pc3_pr2_pr2_t c u1 u2 H3 t3 t0 k H0) t4 (H2 +u1 H3)))))))))) t1 t2 H)))))). +(* COMMENTS +Initial nodes: 199 +END *) + +theorem pc3_pr3_pc3_t: + \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u2 u1) \to (\forall +(t1: T).(\forall (t2: T).(\forall (k: K).((pc3 (CHead c k u2) t1 t2) \to (pc3 +(CHead c k u1) t1 t2)))))))) +\def + \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u2 +u1)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (t1: T).(\forall +(t2: T).(\forall (k: K).((pc3 (CHead c k t) t1 t2) \to (pc3 (CHead c k t0) t1 +t2))))))) (\lambda (t: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k: +K).(\lambda (H0: (pc3 (CHead c k t) t1 t2)).H0))))) (\lambda (t2: T).(\lambda +(t1: T).(\lambda (H0: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda (_: (pr3 c t2 +t3)).(\lambda (H2: ((\forall (t4: T).(\forall (t5: T).(\forall (k: K).((pc3 +(CHead c k t2) t4 t5) \to (pc3 (CHead c k t3) t4 t5))))))).(\lambda (t0: +T).(\lambda (t4: T).(\lambda (k: K).(\lambda (H3: (pc3 (CHead c k t1) t0 +t4)).(H2 t0 t4 k (let H4 \def H3 in (ex2_ind T (\lambda (t: T).(pr3 (CHead c +k t1) t0 t)) (\lambda (t: T).(pr3 (CHead c k t1) t4 t)) (pc3 (CHead c k t2) +t0 t4) (\lambda (x: T).(\lambda (H5: (pr3 (CHead c k t1) t0 x)).(\lambda (H6: +(pr3 (CHead c k t1) t4 x)).(pc3_t x (CHead c k t2) t0 (pc3_pr2_pr3_t c t1 t0 +x k H5 t2 H0) t4 (pc3_s (CHead c k t2) x t4 (pc3_pr2_pr3_t c t1 t4 x k H6 t2 +H0)))))) H4))))))))))))) u2 u1 H)))). +(* COMMENTS +Initial nodes: 319 +END *) + +theorem pc3_lift: + \forall (c: C).(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h +d c e) \to (\forall (t1: T).(\forall (t2: T).((pc3 e t1 t2) \to (pc3 c (lift +h d t1) (lift h d t2))))))))) +\def + \lambda (c: C).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda +(H: (drop h d c e)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pc3 e t1 +t2)).(let H1 \def H0 in (ex2_ind T (\lambda (t: T).(pr3 e t1 t)) (\lambda (t: +T).(pr3 e t2 t)) (pc3 c (lift h d t1) (lift h d t2)) (\lambda (x: T).(\lambda +(H2: (pr3 e t1 x)).(\lambda (H3: (pr3 e t2 x)).(pc3_pr3_t c (lift h d t1) +(lift h d x) (pr3_lift c e h d H t1 x H2) (lift h d t2) (pr3_lift c e h d H +t2 x H3))))) H1))))))))). +(* COMMENTS +Initial nodes: 159 +END *) + +theorem pc3_eta: + \forall (c: C).(\forall (t: T).(\forall (w: T).(\forall (u: T).((pc3 c t +(THead (Bind Abst) w u)) \to (\forall (v: T).((pc3 c v w) \to (pc3 c (THead +(Bind Abst) v (THead (Flat Appl) (TLRef O) (lift (S O) O t))) t))))))) +\def + \lambda (c: C).(\lambda (t: T).(\lambda (w: T).(\lambda (u: T).(\lambda (H: +(pc3 c t (THead (Bind Abst) w u))).(\lambda (v: T).(\lambda (H0: (pc3 c v +w)).(pc3_t (THead (Bind Abst) w (THead (Flat Appl) (TLRef O) (lift (S O) O +(THead (Bind Abst) w u)))) c (THead (Bind Abst) v (THead (Flat Appl) (TLRef +O) (lift (S O) O t))) (pc3_head_21 c v w H0 (Bind Abst) (THead (Flat Appl) +(TLRef O) (lift (S O) O t)) (THead (Flat Appl) (TLRef O) (lift (S O) O (THead +(Bind Abst) w u))) (pc3_thin_dx (CHead c (Bind Abst) v) (lift (S O) O t) +(lift (S O) O (THead (Bind Abst) w u)) (pc3_lift (CHead c (Bind Abst) v) c (S +O) O (drop_drop (Bind Abst) O c c (drop_refl c) v) t (THead (Bind Abst) w u) +H) (TLRef O) Appl)) t (pc3_t (THead (Bind Abst) w u) c (THead (Bind Abst) w +(THead (Flat Appl) (TLRef O) (lift (S O) O (THead (Bind Abst) w u)))) +(pc3_pr3_r c (THead (Bind Abst) w (THead (Flat Appl) (TLRef O) (lift (S O) O +(THead (Bind Abst) w u)))) (THead (Bind Abst) w u) (pr3_eta c w u w (pr3_refl +c w))) t (pc3_s c (THead (Bind Abst) w u) t H))))))))). +(* COMMENTS +Initial nodes: 399 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/pc3/subst1.ma b/matita/matita/contribs/lambdadelta/basic_1/pc3/subst1.ma new file mode 100644 index 000000000..3af245b3a --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/pc3/subst1.ma @@ -0,0 +1,48 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/pc3/props.ma". + +include "Basic-1/pr3/subst1.ma". + +theorem pc3_gen_cabbr: + \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to (\forall +(e: C).(\forall (u: T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) +\to (\forall (a0: C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d +a0 a) \to (\forall (x1: T).((subst1 d u t1 (lift (S O) d x1)) \to (\forall +(x2: T).((subst1 d u t2 (lift (S O) d x2)) \to (pc3 a x1 x2)))))))))))))))) +\def + \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c t1 +t2)).(\lambda (e: C).(\lambda (u: T).(\lambda (d: nat).(\lambda (H0: (getl d +c (CHead e (Bind Abbr) u))).(\lambda (a0: C).(\lambda (H1: (csubst1 d u c +a0)).(\lambda (a: C).(\lambda (H2: (drop (S O) d a0 a)).(\lambda (x1: +T).(\lambda (H3: (subst1 d u t1 (lift (S O) d x1))).(\lambda (x2: T).(\lambda +(H4: (subst1 d u t2 (lift (S O) d x2))).(let H5 \def H in (ex2_ind T (\lambda +(t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) (pc3 a x1 x2) (\lambda (x: +T).(\lambda (H6: (pr3 c t1 x)).(\lambda (H7: (pr3 c t2 x)).(ex2_ind T +(\lambda (x3: T).(subst1 d u x (lift (S O) d x3))) (\lambda (x3: T).(pr3 a x2 +x3)) (pc3 a x1 x2) (\lambda (x0: T).(\lambda (H8: (subst1 d u x (lift (S O) d +x0))).(\lambda (H9: (pr3 a x2 x0)).(ex2_ind T (\lambda (x3: T).(subst1 d u x +(lift (S O) d x3))) (\lambda (x3: T).(pr3 a x1 x3)) (pc3 a x1 x2) (\lambda +(x3: T).(\lambda (H10: (subst1 d u x (lift (S O) d x3))).(\lambda (H11: (pr3 +a x1 x3)).(let H12 \def (eq_ind T x3 (\lambda (t: T).(pr3 a x1 t)) H11 x0 +(subst1_confluence_lift x x3 u d H10 x0 H8)) in (pc3_pr3_t a x1 x0 H12 x2 +H9))))) (pr3_gen_cabbr c t1 x H6 e u d H0 a0 H1 a H2 x1 H3))))) +(pr3_gen_cabbr c t2 x H7 e u d H0 a0 H1 a H2 x2 H4))))) H5))))))))))))))))). +(* COMMENTS +Initial nodes: 405 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/pc3/wcpr0.ma b/matita/matita/contribs/lambdadelta/basic_1/pc3/wcpr0.ma new file mode 100644 index 000000000..2b51edf62 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/pc3/wcpr0.ma @@ -0,0 +1,96 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/pc3/props.ma". + +include "Basic-1/wcpr0/getl.ma". + +theorem pc3_wcpr0__pc3_wcpr0_t_aux: + \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (k: K).(\forall +(u: T).(\forall (t1: T).(\forall (t2: T).((pr3 (CHead c1 k u) t1 t2) \to (pc3 +(CHead c2 k u) t1 t2)))))))) +\def + \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(\lambda (k: +K).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 +(CHead c1 k u) t1 t2)).(pr3_ind (CHead c1 k u) (\lambda (t: T).(\lambda (t0: +T).(pc3 (CHead c2 k u) t t0))) (\lambda (t: T).(pc3_refl (CHead c2 k u) t)) +(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr2 (CHead c1 k u) t4 +t3)).(\lambda (t5: T).(\lambda (_: (pr3 (CHead c1 k u) t3 t5)).(\lambda (H3: +(pc3 (CHead c2 k u) t3 t5)).(pc3_t t3 (CHead c2 k u) t4 (insert_eq C (CHead +c1 k u) (\lambda (c: C).(pr2 c t4 t3)) (\lambda (_: C).(pc3 (CHead c2 k u) t4 +t3)) (\lambda (y: C).(\lambda (H4: (pr2 y t4 t3)).(pr2_ind (\lambda (c: +C).(\lambda (t: T).(\lambda (t0: T).((eq C c (CHead c1 k u)) \to (pc3 (CHead +c2 k u) t t0))))) (\lambda (c: C).(\lambda (t6: T).(\lambda (t0: T).(\lambda +(H5: (pr0 t6 t0)).(\lambda (_: (eq C c (CHead c1 k u))).(pc3_pr2_r (CHead c2 +k u) t6 t0 (pr2_free (CHead c2 k u) t6 t0 H5))))))) (\lambda (c: C).(\lambda +(d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H5: (getl i c (CHead d +(Bind Abbr) u0))).(\lambda (t6: T).(\lambda (t0: T).(\lambda (H6: (pr0 t6 +t0)).(\lambda (t: T).(\lambda (H7: (subst0 i u0 t0 t)).(\lambda (H8: (eq C c +(CHead c1 k u))).(let H9 \def (eq_ind C c (\lambda (c0: C).(getl i c0 (CHead +d (Bind Abbr) u0))) H5 (CHead c1 k u) H8) in (ex3_2_ind C T (\lambda (e2: +C).(\lambda (u2: T).(getl i (CHead c2 k u) (CHead e2 (Bind Abbr) u2)))) +(\lambda (e2: C).(\lambda (_: T).(wcpr0 d e2))) (\lambda (_: C).(\lambda (u2: +T).(pr0 u0 u2))) (pc3 (CHead c2 k u) t6 t) (\lambda (x0: C).(\lambda (x1: +T).(\lambda (H10: (getl i (CHead c2 k u) (CHead x0 (Bind Abbr) x1))).(\lambda +(_: (wcpr0 d x0)).(\lambda (H12: (pr0 u0 x1)).(ex2_ind T (\lambda (t7: +T).(subst0 i x1 t0 t7)) (\lambda (t7: T).(pr0 t t7)) (pc3 (CHead c2 k u) t6 +t) (\lambda (x: T).(\lambda (H13: (subst0 i x1 t0 x)).(\lambda (H14: (pr0 t +x)).(pc3_pr2_u (CHead c2 k u) x t6 (pr2_delta (CHead c2 k u) x0 x1 i H10 t6 +t0 H6 x H13) t (pc3_pr2_x (CHead c2 k u) x t (pr2_free (CHead c2 k u) t x +H14)))))) (pr0_subst0_fwd u0 t0 t i H7 x1 H12))))))) (wcpr0_getl (CHead c1 k +u) (CHead c2 k u) (wcpr0_comp c1 c2 H u u (pr0_refl u) k) i d u0 (Bind Abbr) +H9)))))))))))))) y t4 t3 H4))) H1) t5 H3))))))) t1 t2 H0)))))))). +(* COMMENTS +Initial nodes: 689 +END *) + +theorem pc3_wcpr0_t: + \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (t1: +T).(\forall (t2: T).((pr3 c1 t1 t2) \to (pc3 c2 t1 t2)))))) +\def + \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(wcpr0_ind +(\lambda (c: C).(\lambda (c0: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 +t2) \to (pc3 c0 t1 t2)))))) (\lambda (c: C).(\lambda (t1: T).(\lambda (t2: +T).(\lambda (H0: (pr3 c t1 t2)).(pc3_pr3_r c t1 t2 H0))))) (\lambda (c0: +C).(\lambda (c3: C).(\lambda (H0: (wcpr0 c0 c3)).(\lambda (_: ((\forall (t1: +T).(\forall (t2: T).((pr3 c0 t1 t2) \to (pc3 c3 t1 t2)))))).(\lambda (u1: +T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 u2)).(\lambda (k: K).(\lambda (t1: +T).(\lambda (t2: T).(\lambda (H3: (pr3 (CHead c0 k u1) t1 t2)).(let H4 \def +(pc3_pr2_pr3_t c0 u1 t1 t2 k H3 u2 (pr2_free c0 u1 u2 H2)) in (ex2_ind T +(\lambda (t: T).(pr3 (CHead c0 k u2) t1 t)) (\lambda (t: T).(pr3 (CHead c0 k +u2) t2 t)) (pc3 (CHead c3 k u2) t1 t2) (\lambda (x: T).(\lambda (H5: (pr3 +(CHead c0 k u2) t1 x)).(\lambda (H6: (pr3 (CHead c0 k u2) t2 x)).(pc3_t x +(CHead c3 k u2) t1 (pc3_wcpr0__pc3_wcpr0_t_aux c0 c3 H0 k u2 t1 x H5) t2 +(pc3_s (CHead c3 k u2) x t2 (pc3_wcpr0__pc3_wcpr0_t_aux c0 c3 H0 k u2 t2 x +H6)))))) H4))))))))))))) c1 c2 H))). +(* COMMENTS +Initial nodes: 299 +END *) + +theorem pc3_wcpr0: + \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (t1: +T).(\forall (t2: T).((pc3 c1 t1 t2) \to (pc3 c2 t1 t2)))))) +\def + \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(\lambda (t1: +T).(\lambda (t2: T).(\lambda (H0: (pc3 c1 t1 t2)).(let H1 \def H0 in (ex2_ind +T (\lambda (t: T).(pr3 c1 t1 t)) (\lambda (t: T).(pr3 c1 t2 t)) (pc3 c2 t1 +t2) (\lambda (x: T).(\lambda (H2: (pr3 c1 t1 x)).(\lambda (H3: (pr3 c1 t2 +x)).(pc3_t x c2 t1 (pc3_wcpr0_t c1 c2 H t1 x H2) t2 (pc3_s c2 x t2 +(pc3_wcpr0_t c1 c2 H t2 x H3)))))) H1))))))). +(* COMMENTS +Initial nodes: 121 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr0/dec.ma b/matita/matita/contribs/lambdadelta/basic_1/pr0/dec.ma new file mode 100644 index 000000000..c28504bee --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/pr0/dec.ma @@ -0,0 +1,529 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/pr0/fwd.ma". + +include "Basic-1/subst0/dec.ma". + +include "Basic-1/T/dec.ma". + +include "Basic-1/T/props.ma". + +theorem nf0_dec: + \forall (t1: T).(or (\forall (t2: T).((pr0 t1 t2) \to (eq T t1 t2))) (ex2 T +(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: +T).(pr0 t1 t2)))) +\def + \lambda (t1: T).(T_ind (\lambda (t: T).(or (\forall (t2: T).((pr0 t t2) \to +(eq T t t2))) (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P: +Prop).P))) (\lambda (t2: T).(pr0 t t2))))) (\lambda (n: nat).(or_introl +(\forall (t2: T).((pr0 (TSort n) t2) \to (eq T (TSort n) t2))) (ex2 T +(\lambda (t2: T).((eq T (TSort n) t2) \to (\forall (P: Prop).P))) (\lambda +(t2: T).(pr0 (TSort n) t2))) (\lambda (t2: T).(\lambda (H: (pr0 (TSort n) +t2)).(eq_ind_r T (TSort n) (\lambda (t: T).(eq T (TSort n) t)) (refl_equal T +(TSort n)) t2 (pr0_gen_sort t2 n H)))))) (\lambda (n: nat).(or_introl +(\forall (t2: T).((pr0 (TLRef n) t2) \to (eq T (TLRef n) t2))) (ex2 T +(\lambda (t2: T).((eq T (TLRef n) t2) \to (\forall (P: Prop).P))) (\lambda +(t2: T).(pr0 (TLRef n) t2))) (\lambda (t2: T).(\lambda (H: (pr0 (TLRef n) +t2)).(eq_ind_r T (TLRef n) (\lambda (t: T).(eq T (TLRef n) t)) (refl_equal T +(TLRef n)) t2 (pr0_gen_lref t2 n H)))))) (\lambda (k: K).(\lambda (t: +T).(\lambda (H: (or (\forall (t2: T).((pr0 t t2) \to (eq T t t2))) (ex2 T +(\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: +T).(pr0 t t2))))).(\lambda (t0: T).(\lambda (H0: (or (\forall (t2: T).((pr0 +t0 t2) \to (eq T t0 t2))) (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall +(P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2))))).(K_ind (\lambda (k0: K).(or +(\forall (t2: T).((pr0 (THead k0 t t0) t2) \to (eq T (THead k0 t t0) t2))) +(ex2 T (\lambda (t2: T).((eq T (THead k0 t t0) t2) \to (\forall (P: +Prop).P))) (\lambda (t2: T).(pr0 (THead k0 t t0) t2))))) (\lambda (b: +B).(B_ind (\lambda (b0: B).(or (\forall (t2: T).((pr0 (THead (Bind b0) t t0) +t2) \to (eq T (THead (Bind b0) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T +(THead (Bind b0) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 +(THead (Bind b0) t t0) t2))))) (or_intror (\forall (t2: T).((pr0 (THead (Bind +Abbr) t t0) t2) \to (eq T (THead (Bind Abbr) t t0) t2))) (ex2 T (\lambda (t2: +T).((eq T (THead (Bind Abbr) t t0) t2) \to (\forall (P: Prop).P))) (\lambda +(t2: T).(pr0 (THead (Bind Abbr) t t0) t2))) (let H_x \def (dnf_dec t t0 O) in +(let H1 \def H_x in (ex_ind T (\lambda (v: T).(or (subst0 O t t0 (lift (S O) +O v)) (eq T t0 (lift (S O) O v)))) (ex2 T (\lambda (t2: T).((eq T (THead +(Bind Abbr) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 +(THead (Bind Abbr) t t0) t2))) (\lambda (x: T).(\lambda (H2: (or (subst0 O t +t0 (lift (S O) O x)) (eq T t0 (lift (S O) O x)))).(or_ind (subst0 O t t0 +(lift (S O) O x)) (eq T t0 (lift (S O) O x)) (ex2 T (\lambda (t2: T).((eq T +(THead (Bind Abbr) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: +T).(pr0 (THead (Bind Abbr) t t0) t2))) (\lambda (H3: (subst0 O t t0 (lift (S +O) O x))).(ex_intro2 T (\lambda (t2: T).((eq T (THead (Bind Abbr) t t0) t2) +\to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abbr) t t0) +t2)) (THead (Bind Abbr) t (lift (S O) O x)) (\lambda (H4: (eq T (THead (Bind +Abbr) t t0) (THead (Bind Abbr) t (lift (S O) O x)))).(\lambda (P: Prop).(let +H5 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) +with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t2) +\Rightarrow t2])) (THead (Bind Abbr) t t0) (THead (Bind Abbr) t (lift (S O) O +x)) H4) in (let H6 \def (eq_ind T t0 (\lambda (t2: T).(subst0 O t t2 (lift (S +O) O x))) H3 (lift (S O) O x) H5) in (subst0_refl t (lift (S O) O x) O H6 +P))))) (pr0_delta t t (pr0_refl t) t0 t0 (pr0_refl t0) (lift (S O) O x) H3))) +(\lambda (H3: (eq T t0 (lift (S O) O x))).(eq_ind_r T (lift (S O) O x) +(\lambda (t2: T).(ex2 T (\lambda (t3: T).((eq T (THead (Bind Abbr) t t2) t3) +\to (\forall (P: Prop).P))) (\lambda (t3: T).(pr0 (THead (Bind Abbr) t t2) +t3)))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Bind Abbr) t (lift (S O) +O x)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind +Abbr) t (lift (S O) O x)) t2)) x (\lambda (H4: (eq T (THead (Bind Abbr) t +(lift (S O) O x)) x)).(\lambda (P: Prop).(thead_x_lift_y_y (Bind Abbr) x t (S +O) O H4 P))) (pr0_zeta Abbr not_abbr_abst x x (pr0_refl x) t)) t0 H3)) H2))) +H1)))) (let H1 \def H in (or_ind (\forall (t2: T).((pr0 t t2) \to (eq T t +t2))) (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) +(\lambda (t2: T).(pr0 t t2))) (or (\forall (t2: T).((pr0 (THead (Bind Abst) t +t0) t2) \to (eq T (THead (Bind Abst) t t0) t2))) (ex2 T (\lambda (t2: T).((eq +T (THead (Bind Abst) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: +T).(pr0 (THead (Bind Abst) t t0) t2)))) (\lambda (H2: ((\forall (t2: T).((pr0 +t t2) \to (eq T t t2))))).(let H3 \def H0 in (or_ind (\forall (t2: T).((pr0 +t0 t2) \to (eq T t0 t2))) (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall +(P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2))) (or (\forall (t2: T).((pr0 +(THead (Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0) t2))) (ex2 T +(\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P: +Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2)))) (\lambda +(H4: ((\forall (t2: T).((pr0 t0 t2) \to (eq T t0 t2))))).(or_introl (\forall +(t2: T).((pr0 (THead (Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0) +t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to +(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2))) +(\lambda (t2: T).(\lambda (H5: (pr0 (THead (Bind Abst) t t0) t2)).(ex3_2_ind +T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda (t3: +T).(pr0 t0 t3))) (eq T (THead (Bind Abst) t t0) t2) (\lambda (x0: T).(\lambda +(x1: T).(\lambda (H6: (eq T t2 (THead (Bind Abst) x0 x1))).(\lambda (H7: (pr0 +t x0)).(\lambda (H8: (pr0 t0 x1)).(let H_y \def (H4 x1 H8) in (let H_y0 \def +(H2 x0 H7) in (let H9 \def (eq_ind_r T x1 (\lambda (t3: T).(pr0 t0 t3)) H8 t0 +H_y) in (let H10 \def (eq_ind_r T x1 (\lambda (t3: T).(eq T t2 (THead (Bind +Abst) x0 t3))) H6 t0 H_y) in (let H11 \def (eq_ind_r T x0 (\lambda (t3: +T).(pr0 t t3)) H7 t H_y0) in (let H12 \def (eq_ind_r T x0 (\lambda (t3: +T).(eq T t2 (THead (Bind Abst) t3 t0))) H10 t H_y0) in (eq_ind_r T (THead +(Bind Abst) t t0) (\lambda (t3: T).(eq T (THead (Bind Abst) t t0) t3)) +(refl_equal T (THead (Bind Abst) t t0)) t2 H12)))))))))))) (pr0_gen_abst t t0 +t2 H5)))))) (\lambda (H4: (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall +(P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2)))).(ex2_ind T (\lambda (t2: +T).((eq T t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2)) +(or (\forall (t2: T).((pr0 (THead (Bind Abst) t t0) t2) \to (eq T (THead +(Bind Abst) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abst) t +t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) +t t0) t2)))) (\lambda (x: T).(\lambda (H5: (((eq T t0 x) \to (\forall (P: +Prop).P)))).(\lambda (H6: (pr0 t0 x)).(or_intror (\forall (t2: T).((pr0 +(THead (Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0) t2))) (ex2 T +(\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P: +Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2))) (ex_intro2 T +(\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P: +Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2)) (THead (Bind +Abst) t x) (\lambda (H7: (eq T (THead (Bind Abst) t t0) (THead (Bind Abst) t +x))).(\lambda (P: Prop).(let H8 \def (f_equal T T (\lambda (e: T).(match e in +T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) +\Rightarrow t0 | (THead _ _ t2) \Rightarrow t2])) (THead (Bind Abst) t t0) +(THead (Bind Abst) t x) H7) in (let H9 \def (eq_ind_r T x (\lambda (t2: +T).(pr0 t0 t2)) H6 t0 H8) in (let H10 \def (eq_ind_r T x (\lambda (t2: +T).((eq T t0 t2) \to (\forall (P0: Prop).P0))) H5 t0 H8) in (H10 (refl_equal +T t0) P)))))) (pr0_comp t t (pr0_refl t) t0 x H6 (Bind Abst))))))) H4)) H3))) +(\lambda (H2: (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P: +Prop).P))) (\lambda (t2: T).(pr0 t t2)))).(ex2_ind T (\lambda (t2: T).((eq T +t t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2)) (or (\forall +(t2: T).((pr0 (THead (Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0) +t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to +(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2)))) +(\lambda (x: T).(\lambda (H3: (((eq T t x) \to (\forall (P: +Prop).P)))).(\lambda (H4: (pr0 t x)).(or_intror (\forall (t2: T).((pr0 (THead +(Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0) t2))) (ex2 T +(\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P: +Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2))) (ex_intro2 T +(\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P: +Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2)) (THead (Bind +Abst) x t0) (\lambda (H5: (eq T (THead (Bind Abst) t t0) (THead (Bind Abst) x +t0))).(\lambda (P: Prop).(let H6 \def (f_equal T T (\lambda (e: T).(match e +in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t | (TLRef _) +\Rightarrow t | (THead _ t2 _) \Rightarrow t2])) (THead (Bind Abst) t t0) +(THead (Bind Abst) x t0) H5) in (let H7 \def (eq_ind_r T x (\lambda (t2: +T).(pr0 t t2)) H4 t H6) in (let H8 \def (eq_ind_r T x (\lambda (t2: T).((eq T +t t2) \to (\forall (P0: Prop).P0))) H3 t H6) in (H8 (refl_equal T t) P)))))) +(pr0_comp t x H4 t0 t0 (pr0_refl t0) (Bind Abst))))))) H2)) H1)) (let H_x +\def (dnf_dec t t0 O) in (let H1 \def H_x in (ex_ind T (\lambda (v: T).(or +(subst0 O t t0 (lift (S O) O v)) (eq T t0 (lift (S O) O v)))) (or (\forall +(t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) +t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to +(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)))) +(\lambda (x: T).(\lambda (H2: (or (subst0 O t t0 (lift (S O) O x)) (eq T t0 +(lift (S O) O x)))).(or_ind (subst0 O t t0 (lift (S O) O x)) (eq T t0 (lift +(S O) O x)) (or (\forall (t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T +(THead (Bind Void) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind +Void) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead +(Bind Void) t t0) t2)))) (\lambda (H3: (subst0 O t t0 (lift (S O) O x))).(let +H4 \def H in (or_ind (\forall (t2: T).((pr0 t t2) \to (eq T t t2))) (ex2 T +(\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: +T).(pr0 t t2))) (or (\forall (t2: T).((pr0 (THead (Bind Void) t t0) t2) \to +(eq T (THead (Bind Void) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead +(Bind Void) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 +(THead (Bind Void) t t0) t2)))) (\lambda (H5: ((\forall (t2: T).((pr0 t t2) +\to (eq T t t2))))).(let H6 \def H0 in (or_ind (\forall (t2: T).((pr0 t0 t2) +\to (eq T t0 t2))) (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: +Prop).P))) (\lambda (t2: T).(pr0 t0 t2))) (or (\forall (t2: T).((pr0 (THead +(Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) t2))) (ex2 T +(\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P: +Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)))) (\lambda +(H7: ((\forall (t2: T).((pr0 t0 t2) \to (eq T t0 t2))))).(or_introl (\forall +(t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) +t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to +(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2))) +(\lambda (t2: T).(\lambda (H8: (pr0 (THead (Bind Void) t t0) t2)).(or_ind +(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 +t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda +(t3: T).(pr0 t0 t3)))) (pr0 t0 (lift (S O) O t2)) (eq T (THead (Bind Void) t +t0) t2) (\lambda (H9: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 +(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) +(\lambda (_: T).(\lambda (t3: T).(pr0 t0 t3))))).(ex3_2_ind T T (\lambda (u2: +T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t0 +t3))) (eq T (THead (Bind Void) t t0) t2) (\lambda (x0: T).(\lambda (x1: +T).(\lambda (H10: (eq T t2 (THead (Bind Void) x0 x1))).(\lambda (H11: (pr0 t +x0)).(\lambda (H12: (pr0 t0 x1)).(let H_y \def (H7 x1 H12) in (let H_y0 \def +(H5 x0 H11) in (let H13 \def (eq_ind_r T x1 (\lambda (t3: T).(pr0 t0 t3)) H12 +t0 H_y) in (let H14 \def (eq_ind_r T x1 (\lambda (t3: T).(eq T t2 (THead +(Bind Void) x0 t3))) H10 t0 H_y) in (let H15 \def (eq_ind_r T x0 (\lambda +(t3: T).(pr0 t t3)) H11 t H_y0) in (let H16 \def (eq_ind_r T x0 (\lambda (t3: +T).(eq T t2 (THead (Bind Void) t3 t0))) H14 t H_y0) in (eq_ind_r T (THead +(Bind Void) t t0) (\lambda (t3: T).(eq T (THead (Bind Void) t t0) t3)) +(refl_equal T (THead (Bind Void) t t0)) t2 H16)))))))))))) H9)) (\lambda (H9: +(pr0 t0 (lift (S O) O t2))).(let H_y \def (H7 (lift (S O) O t2) H9) in (let +H10 \def (eq_ind T t0 (\lambda (t3: T).(subst0 O t t3 (lift (S O) O x))) H3 +(lift (S O) O t2) H_y) in (eq_ind_r T (lift (S O) O t2) (\lambda (t3: T).(eq +T (THead (Bind Void) t t3) t2)) (subst0_gen_lift_false t2 t (lift (S O) O x) +(S O) O O (le_n O) (eq_ind_r nat (plus (S O) O) (\lambda (n: nat).(lt O n)) +(le_n (plus (S O) O)) (plus O (S O)) (plus_sym O (S O))) H10 (eq T (THead +(Bind Void) t (lift (S O) O t2)) t2)) t0 H_y)))) (pr0_gen_void t t0 t2 +H8)))))) (\lambda (H7: (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: +Prop).P))) (\lambda (t2: T).(pr0 t0 t2)))).(ex2_ind T (\lambda (t2: T).((eq T +t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2)) (or (\forall +(t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) +t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to +(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)))) +(\lambda (x0: T).(\lambda (H8: (((eq T t0 x0) \to (\forall (P: +Prop).P)))).(\lambda (H9: (pr0 t0 x0)).(or_intror (\forall (t2: T).((pr0 +(THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) t2))) (ex2 T +(\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P: +Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2))) (ex_intro2 T +(\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P: +Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)) (THead (Bind +Void) t x0) (\lambda (H10: (eq T (THead (Bind Void) t t0) (THead (Bind Void) +t x0))).(\lambda (P: Prop).(let H11 \def (f_equal T T (\lambda (e: T).(match +e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) +\Rightarrow t0 | (THead _ _ t2) \Rightarrow t2])) (THead (Bind Void) t t0) +(THead (Bind Void) t x0) H10) in (let H12 \def (eq_ind_r T x0 (\lambda (t2: +T).(pr0 t0 t2)) H9 t0 H11) in (let H13 \def (eq_ind_r T x0 (\lambda (t2: +T).((eq T t0 t2) \to (\forall (P0: Prop).P0))) H8 t0 H11) in (H13 (refl_equal +T t0) P)))))) (pr0_comp t t (pr0_refl t) t0 x0 H9 (Bind Void))))))) H7)) +H6))) (\lambda (H5: (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P: +Prop).P))) (\lambda (t2: T).(pr0 t t2)))).(ex2_ind T (\lambda (t2: T).((eq T +t t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2)) (or (\forall +(t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) +t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to +(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)))) +(\lambda (x0: T).(\lambda (H6: (((eq T t x0) \to (\forall (P: +Prop).P)))).(\lambda (H7: (pr0 t x0)).(or_intror (\forall (t2: T).((pr0 +(THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) t2))) (ex2 T +(\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P: +Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2))) (ex_intro2 T +(\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P: +Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)) (THead (Bind +Void) x0 t0) (\lambda (H8: (eq T (THead (Bind Void) t t0) (THead (Bind Void) +x0 t0))).(\lambda (P: Prop).(let H9 \def (f_equal T T (\lambda (e: T).(match +e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t | (TLRef _) +\Rightarrow t | (THead _ t2 _) \Rightarrow t2])) (THead (Bind Void) t t0) +(THead (Bind Void) x0 t0) H8) in (let H10 \def (eq_ind_r T x0 (\lambda (t2: +T).(pr0 t t2)) H7 t H9) in (let H11 \def (eq_ind_r T x0 (\lambda (t2: T).((eq +T t t2) \to (\forall (P0: Prop).P0))) H6 t H9) in (H11 (refl_equal T t) +P)))))) (pr0_comp t x0 H7 t0 t0 (pr0_refl t0) (Bind Void))))))) H5)) H4))) +(\lambda (H3: (eq T t0 (lift (S O) O x))).(let H4 \def (eq_ind T t0 (\lambda +(t2: T).(or (\forall (t3: T).((pr0 t2 t3) \to (eq T t2 t3))) (ex2 T (\lambda +(t3: T).((eq T t2 t3) \to (\forall (P: Prop).P))) (\lambda (t3: T).(pr0 t2 +t3))))) H0 (lift (S O) O x) H3) in (eq_ind_r T (lift (S O) O x) (\lambda (t2: +T).(or (\forall (t3: T).((pr0 (THead (Bind Void) t t2) t3) \to (eq T (THead +(Bind Void) t t2) t3))) (ex2 T (\lambda (t3: T).((eq T (THead (Bind Void) t +t2) t3) \to (\forall (P: Prop).P))) (\lambda (t3: T).(pr0 (THead (Bind Void) +t t2) t3))))) (or_intror (\forall (t2: T).((pr0 (THead (Bind Void) t (lift (S +O) O x)) t2) \to (eq T (THead (Bind Void) t (lift (S O) O x)) t2))) (ex2 T +(\lambda (t2: T).((eq T (THead (Bind Void) t (lift (S O) O x)) t2) \to +(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t (lift (S +O) O x)) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Bind Void) t +(lift (S O) O x)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 +(THead (Bind Void) t (lift (S O) O x)) t2)) x (\lambda (H5: (eq T (THead +(Bind Void) t (lift (S O) O x)) x)).(\lambda (P: Prop).(thead_x_lift_y_y +(Bind Void) x t (S O) O H5 P))) (pr0_zeta Void (sym_not_eq B Abst Void +not_abst_void) x x (pr0_refl x) t))) t0 H3))) H2))) H1))) b)) (\lambda (f: +F).(F_ind (\lambda (f0: F).(or (\forall (t2: T).((pr0 (THead (Flat f0) t t0) +t2) \to (eq T (THead (Flat f0) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T +(THead (Flat f0) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 +(THead (Flat f0) t t0) t2))))) (let H_x \def (binder_dec t0) in (let H1 \def +H_x in (or_ind (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: +T).(eq T t0 (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: +T).(\forall (u: T).((eq T t0 (THead (Bind b) w u)) \to (\forall (P: +Prop).P))))) (or (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq +T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat +Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead +(Flat Appl) t t0) t2)))) (\lambda (H2: (ex_3 B T T (\lambda (b: B).(\lambda +(w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w u))))))).(ex_3_ind B T T +(\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w +u))))) (or (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T +(THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat +Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead +(Flat Appl) t t0) t2)))) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: +T).(\lambda (H3: (eq T t0 (THead (Bind x0) x1 x2))).(let H4 \def (eq_ind T t0 +(\lambda (t2: T).(or (\forall (t3: T).((pr0 t2 t3) \to (eq T t2 t3))) (ex2 T +(\lambda (t3: T).((eq T t2 t3) \to (\forall (P: Prop).P))) (\lambda (t3: +T).(pr0 t2 t3))))) H0 (THead (Bind x0) x1 x2) H3) in (eq_ind_r T (THead (Bind +x0) x1 x2) (\lambda (t2: T).(or (\forall (t3: T).((pr0 (THead (Flat Appl) t +t2) t3) \to (eq T (THead (Flat Appl) t t2) t3))) (ex2 T (\lambda (t3: T).((eq +T (THead (Flat Appl) t t2) t3) \to (\forall (P: Prop).P))) (\lambda (t3: +T).(pr0 (THead (Flat Appl) t t2) t3))))) (B_ind (\lambda (b: B).((or (\forall +(t2: T).((pr0 (THead (Bind b) x1 x2) t2) \to (eq T (THead (Bind b) x1 x2) +t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind b) x1 x2) t2) \to (\forall +(P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind b) x1 x2) t2)))) \to (or +(\forall (t2: T).((pr0 (THead (Flat Appl) t (THead (Bind b) x1 x2)) t2) \to +(eq T (THead (Flat Appl) t (THead (Bind b) x1 x2)) t2))) (ex2 T (\lambda (t2: +T).((eq T (THead (Flat Appl) t (THead (Bind b) x1 x2)) t2) \to (\forall (P: +Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead (Bind b) x1 x2)) +t2)))))) (\lambda (_: (or (\forall (t2: T).((pr0 (THead (Bind Abbr) x1 x2) +t2) \to (eq T (THead (Bind Abbr) x1 x2) t2))) (ex2 T (\lambda (t2: T).((eq T +(THead (Bind Abbr) x1 x2) t2) \to (\forall (P: Prop).P))) (\lambda (t2: +T).(pr0 (THead (Bind Abbr) x1 x2) t2))))).(or_intror (\forall (t2: T).((pr0 +(THead (Flat Appl) t (THead (Bind Abbr) x1 x2)) t2) \to (eq T (THead (Flat +Appl) t (THead (Bind Abbr) x1 x2)) t2))) (ex2 T (\lambda (t2: T).((eq T +(THead (Flat Appl) t (THead (Bind Abbr) x1 x2)) t2) \to (\forall (P: +Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead (Bind Abbr) x1 +x2)) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t (THead +(Bind Abbr) x1 x2)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 +(THead (Flat Appl) t (THead (Bind Abbr) x1 x2)) t2)) (THead (Bind Abbr) x1 +(THead (Flat Appl) (lift (S O) O t) x2)) (\lambda (H6: (eq T (THead (Flat +Appl) t (THead (Bind Abbr) x1 x2)) (THead (Bind Abbr) x1 (THead (Flat Appl) +(lift (S O) O t) x2)))).(\lambda (P: Prop).(let H7 \def (eq_ind T (THead +(Flat Appl) t (THead (Bind Abbr) x1 x2)) (\lambda (ee: T).(match ee in T +return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead _ _ t2) \Rightarrow (match t2 in T return (\lambda +(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False +| (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda (_: K).Prop) with +[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])])) I (THead (Bind +Abbr) x1 (THead (Flat Appl) (lift (S O) O t) x2)) H6) in (False_ind P H7)))) +(pr0_upsilon Abbr not_abbr_abst t t (pr0_refl t) x1 x1 (pr0_refl x1) x2 x2 +(pr0_refl x2))))) (\lambda (_: (or (\forall (t2: T).((pr0 (THead (Bind Abst) +x1 x2) t2) \to (eq T (THead (Bind Abst) x1 x2) t2))) (ex2 T (\lambda (t2: +T).((eq T (THead (Bind Abst) x1 x2) t2) \to (\forall (P: Prop).P))) (\lambda +(t2: T).(pr0 (THead (Bind Abst) x1 x2) t2))))).(or_intror (\forall (t2: +T).((pr0 (THead (Flat Appl) t (THead (Bind Abst) x1 x2)) t2) \to (eq T (THead +(Flat Appl) t (THead (Bind Abst) x1 x2)) t2))) (ex2 T (\lambda (t2: T).((eq T +(THead (Flat Appl) t (THead (Bind Abst) x1 x2)) t2) \to (\forall (P: +Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead (Bind Abst) x1 +x2)) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t (THead +(Bind Abst) x1 x2)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 +(THead (Flat Appl) t (THead (Bind Abst) x1 x2)) t2)) (THead (Bind Abbr) t x2) +(\lambda (H6: (eq T (THead (Flat Appl) t (THead (Bind Abst) x1 x2)) (THead +(Bind Abbr) t x2))).(\lambda (P: Prop).(let H7 \def (eq_ind T (THead (Flat +Appl) t (THead (Bind Abst) x1 x2)) (\lambda (ee: T).(match ee in T return +(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda +(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow +True])])) I (THead (Bind Abbr) t x2) H6) in (False_ind P H7)))) (pr0_beta x1 +t t (pr0_refl t) x2 x2 (pr0_refl x2))))) (\lambda (_: (or (\forall (t2: +T).((pr0 (THead (Bind Void) x1 x2) t2) \to (eq T (THead (Bind Void) x1 x2) +t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) x1 x2) t2) \to +(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) x1 x2) +t2))))).(or_intror (\forall (t2: T).((pr0 (THead (Flat Appl) t (THead (Bind +Void) x1 x2)) t2) \to (eq T (THead (Flat Appl) t (THead (Bind Void) x1 x2)) +t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t (THead (Bind Void) +x1 x2)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat +Appl) t (THead (Bind Void) x1 x2)) t2))) (ex_intro2 T (\lambda (t2: T).((eq T +(THead (Flat Appl) t (THead (Bind Void) x1 x2)) t2) \to (\forall (P: +Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead (Bind Void) x1 +x2)) t2)) (THead (Bind Void) x1 (THead (Flat Appl) (lift (S O) O t) x2)) +(\lambda (H6: (eq T (THead (Flat Appl) t (THead (Bind Void) x1 x2)) (THead +(Bind Void) x1 (THead (Flat Appl) (lift (S O) O t) x2)))).(\lambda (P: +Prop).(let H7 \def (eq_ind T (THead (Flat Appl) t (THead (Bind Void) x1 x2)) +(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) +\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ t2) \Rightarrow +(match t2 in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False +| (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K +return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) +\Rightarrow False])])])) I (THead (Bind Void) x1 (THead (Flat Appl) (lift (S +O) O t) x2)) H6) in (False_ind P H7)))) (pr0_upsilon Void (sym_not_eq B Abst +Void not_abst_void) t t (pr0_refl t) x1 x1 (pr0_refl x1) x2 x2 (pr0_refl +x2))))) x0 H4) t0 H3)))))) H2)) (\lambda (H2: ((\forall (b: B).(\forall (w: +T).(\forall (u: T).((eq T t0 (THead (Bind b) w u)) \to (\forall (P: +Prop).P))))))).(let H3 \def H in (or_ind (\forall (t2: T).((pr0 t t2) \to (eq +T t t2))) (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) +(\lambda (t2: T).(pr0 t t2))) (or (\forall (t2: T).((pr0 (THead (Flat Appl) t +t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq +T (THead (Flat Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: +T).(pr0 (THead (Flat Appl) t t0) t2)))) (\lambda (H4: ((\forall (t2: T).((pr0 +t t2) \to (eq T t t2))))).(let H5 \def H0 in (or_ind (\forall (t2: T).((pr0 +t0 t2) \to (eq T t0 t2))) (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall +(P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2))) (or (\forall (t2: T).((pr0 +(THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T +(\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P: +Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)))) (\lambda +(H6: ((\forall (t2: T).((pr0 t0 t2) \to (eq T t0 t2))))).(or_introl (\forall +(t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) +t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to +(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2))) +(\lambda (t2: T).(\lambda (H7: (pr0 (THead (Flat Appl) t t0) t2)).(or3_ind +(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 +t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda +(t3: T).(pr0 t0 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(eq T t0 (THead (Bind Abst) y1 z1)))))) +(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 +(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr0 t u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda +(_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t0 (THead (Bind +b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead +(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 t +u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(t3: T).(pr0 z1 t3)))))))) (eq T (THead (Flat Appl) t t0) t2) (\lambda (H8: +(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 +t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda +(t3: T).(pr0 t0 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq +T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t +u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t0 t3))) (eq T (THead (Flat Appl) +t t0) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H9: (eq T t2 (THead +(Flat Appl) x0 x1))).(\lambda (H10: (pr0 t x0)).(\lambda (H11: (pr0 t0 +x1)).(let H_y \def (H6 x1 H11) in (let H_y0 \def (H4 x0 H10) in (let H12 \def +(eq_ind_r T x1 (\lambda (t3: T).(pr0 t0 t3)) H11 t0 H_y) in (let H13 \def +(eq_ind_r T x1 (\lambda (t3: T).(eq T t2 (THead (Flat Appl) x0 t3))) H9 t0 +H_y) in (let H14 \def (eq_ind_r T x0 (\lambda (t3: T).(pr0 t t3)) H10 t H_y0) +in (let H15 \def (eq_ind_r T x0 (\lambda (t3: T).(eq T t2 (THead (Flat Appl) +t3 t0))) H13 t H_y0) in (eq_ind_r T (THead (Flat Appl) t t0) (\lambda (t3: +T).(eq T (THead (Flat Appl) t t0) t3)) (refl_equal T (THead (Flat Appl) t +t0)) t2 H15)))))))))))) H8)) (\lambda (H8: (ex4_4 T T T T (\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t0 (THead (Bind +Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr0 t u2))))) (\lambda (_: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))).(ex4_4_ind T T T T +(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t0 +(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 t u2))))) (\lambda +(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))) (eq +T (THead (Flat Appl) t t0) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda +(x2: T).(\lambda (x3: T).(\lambda (H9: (eq T t0 (THead (Bind Abst) x0 +x1))).(\lambda (H10: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (_: (pr0 t +x2)).(\lambda (_: (pr0 x1 x3)).(eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda +(t3: T).(eq T (THead (Flat Appl) t t0) t3)) (let H13 \def (eq_ind T t0 +(\lambda (t3: T).(\forall (t4: T).((pr0 t3 t4) \to (eq T t3 t4)))) H6 (THead +(Bind Abst) x0 x1) H9) in (let H14 \def (eq_ind T t0 (\lambda (t3: +T).(\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t3 (THead (Bind b) +w u)) \to (\forall (P: Prop).P)))))) H2 (THead (Bind Abst) x0 x1) H9) in +(eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t3: T).(eq T (THead (Flat +Appl) t t3) (THead (Bind Abbr) x2 x3))) (H14 Abst x0 x1 (H13 (THead (Bind +Abst) x0 x1) (pr0_refl (THead (Bind Abst) x0 x1))) (eq T (THead (Flat Appl) t +(THead (Bind Abst) x0 x1)) (THead (Bind Abbr) x2 x3))) t0 H9))) t2 +H10))))))))) H8)) (\lambda (H8: (ex6_6 B T T T T T (\lambda (b: B).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not +(eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t0 (THead (Bind b) +y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead (Flat +Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 t u2))))))) +(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 +t3))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b +Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(eq T t0 (THead (Bind b) y1 z1)))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead (Flat Appl) (lift +(S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 t u2))))))) (\lambda +(_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: +T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))) +(eq T (THead (Flat Appl) t t0) t2) (\lambda (x0: B).(\lambda (x1: T).(\lambda +(x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (not +(eq B x0 Abst))).(\lambda (H10: (eq T t0 (THead (Bind x0) x1 x2))).(\lambda +(H11: (eq T t2 (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) +x5)))).(\lambda (_: (pr0 t x3)).(\lambda (_: (pr0 x1 x4)).(\lambda (_: (pr0 +x2 x5)).(eq_ind_r T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) +x5)) (\lambda (t3: T).(eq T (THead (Flat Appl) t t0) t3)) (let H15 \def +(eq_ind T t0 (\lambda (t3: T).(\forall (t4: T).((pr0 t3 t4) \to (eq T t3 +t4)))) H6 (THead (Bind x0) x1 x2) H10) in (let H16 \def (eq_ind T t0 (\lambda +(t3: T).(\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t3 (THead +(Bind b) w u)) \to (\forall (P: Prop).P)))))) H2 (THead (Bind x0) x1 x2) H10) +in (eq_ind_r T (THead (Bind x0) x1 x2) (\lambda (t3: T).(eq T (THead (Flat +Appl) t t3) (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)))) +(H16 x0 x1 x2 (H15 (THead (Bind x0) x1 x2) (pr0_refl (THead (Bind x0) x1 +x2))) (eq T (THead (Flat Appl) t (THead (Bind x0) x1 x2)) (THead (Bind x0) x4 +(THead (Flat Appl) (lift (S O) O x3) x5)))) t0 H10))) t2 H11))))))))))))) +H8)) (pr0_gen_appl t t0 t2 H7)))))) (\lambda (H6: (ex2 T (\lambda (t2: +T).((eq T t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t0 +t2)))).(ex2_ind T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: Prop).P))) +(\lambda (t2: T).(pr0 t0 t2)) (or (\forall (t2: T).((pr0 (THead (Flat Appl) t +t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq +T (THead (Flat Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: +T).(pr0 (THead (Flat Appl) t t0) t2)))) (\lambda (x: T).(\lambda (H7: (((eq T +t0 x) \to (\forall (P: Prop).P)))).(\lambda (H8: (pr0 t0 x)).(or_intror +(\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat +Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) +\to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) +t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to +(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)) +(THead (Flat Appl) t x) (\lambda (H9: (eq T (THead (Flat Appl) t t0) (THead +(Flat Appl) t x))).(\lambda (P: Prop).(let H10 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | +(TLRef _) \Rightarrow t0 | (THead _ _ t2) \Rightarrow t2])) (THead (Flat +Appl) t t0) (THead (Flat Appl) t x) H9) in (let H11 \def (eq_ind_r T x +(\lambda (t2: T).(pr0 t0 t2)) H8 t0 H10) in (let H12 \def (eq_ind_r T x +(\lambda (t2: T).((eq T t0 t2) \to (\forall (P0: Prop).P0))) H7 t0 H10) in +(H12 (refl_equal T t0) P)))))) (pr0_comp t t (pr0_refl t) t0 x H8 (Flat +Appl))))))) H6)) H5))) (\lambda (H4: (ex2 T (\lambda (t2: T).((eq T t t2) \to +(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2)))).(ex2_ind T (\lambda +(t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2)) +(or (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead +(Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t +t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) +t t0) t2)))) (\lambda (x: T).(\lambda (H5: (((eq T t x) \to (\forall (P: +Prop).P)))).(\lambda (H6: (pr0 t x)).(or_intror (\forall (t2: T).((pr0 (THead +(Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T +(\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P: +Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2))) (ex_intro2 T +(\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P: +Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)) (THead (Flat +Appl) x t0) (\lambda (H7: (eq T (THead (Flat Appl) t t0) (THead (Flat Appl) x +t0))).(\lambda (P: Prop).(let H8 \def (f_equal T T (\lambda (e: T).(match e +in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t | (TLRef _) +\Rightarrow t | (THead _ t2 _) \Rightarrow t2])) (THead (Flat Appl) t t0) +(THead (Flat Appl) x t0) H7) in (let H9 \def (eq_ind_r T x (\lambda (t2: +T).(pr0 t t2)) H6 t H8) in (let H10 \def (eq_ind_r T x (\lambda (t2: T).((eq +T t t2) \to (\forall (P0: Prop).P0))) H5 t H8) in (H10 (refl_equal T t) +P)))))) (pr0_comp t x H6 t0 t0 (pr0_refl t0) (Flat Appl))))))) H4)) H3))) +H1))) (or_intror (\forall (t2: T).((pr0 (THead (Flat Cast) t t0) t2) \to (eq +T (THead (Flat Cast) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat +Cast) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead +(Flat Cast) t t0) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Flat +Cast) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead +(Flat Cast) t t0) t2)) t0 (\lambda (H1: (eq T (THead (Flat Cast) t t0) +t0)).(\lambda (P: Prop).(thead_x_y_y (Flat Cast) t t0 H1 P))) (pr0_tau t0 t0 +(pr0_refl t0) t))) f)) k)))))) t1). +(* COMMENTS +Initial nodes: 10459 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr0/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/pr0/defs.ma new file mode 100644 index 000000000..0568e070c --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/pr0/defs.ma @@ -0,0 +1,40 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/subst0/defs.ma". + +inductive pr0: T \to (T \to Prop) \def +| pr0_refl: \forall (t: T).(pr0 t t) +| pr0_comp: \forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (t1: +T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (k: K).(pr0 (THead k u1 t1) +(THead k u2 t2)))))))) +| pr0_beta: \forall (u: T).(\forall (v1: T).(\forall (v2: T).((pr0 v1 v2) \to +(\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (pr0 (THead (Flat Appl) v1 +(THead (Bind Abst) u t1)) (THead (Bind Abbr) v2 t2)))))))) +| pr0_upsilon: \forall (b: B).((not (eq B b Abst)) \to (\forall (v1: +T).(\forall (v2: T).((pr0 v1 v2) \to (\forall (u1: T).(\forall (u2: T).((pr0 +u1 u2) \to (\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (pr0 (THead +(Flat Appl) v1 (THead (Bind b) u1 t1)) (THead (Bind b) u2 (THead (Flat Appl) +(lift (S O) O v2) t2))))))))))))) +| pr0_delta: \forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (t1: +T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (w: T).((subst0 O u2 t2 w) \to +(pr0 (THead (Bind Abbr) u1 t1) (THead (Bind Abbr) u2 w))))))))) +| pr0_zeta: \forall (b: B).((not (eq B b Abst)) \to (\forall (t1: T).(\forall +(t2: T).((pr0 t1 t2) \to (\forall (u: T).(pr0 (THead (Bind b) u (lift (S O) O +t1)) t2)))))) +| pr0_tau: \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (u: +T).(pr0 (THead (Flat Cast) u t1) t2)))). + diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr0/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/pr0/fwd.ma new file mode 100644 index 000000000..46caceab4 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/pr0/fwd.ma @@ -0,0 +1,2018 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/pr0/props.ma". + +theorem pr0_gen_sort: + \forall (x: T).(\forall (n: nat).((pr0 (TSort n) x) \to (eq T x (TSort n)))) +\def + \lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr0 (TSort n) x)).(insert_eq +T (TSort n) (\lambda (t: T).(pr0 t x)) (\lambda (t: T).(eq T x t)) (\lambda +(y: T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: T).(\lambda (t0: +T).((eq T t (TSort n)) \to (eq T t0 t)))) (\lambda (t: T).(\lambda (H1: (eq T +t (TSort n))).(let H2 \def (f_equal T T (\lambda (e: T).e) t (TSort n) H1) in +(eq_ind_r T (TSort n) (\lambda (t0: T).(eq T t0 t0)) (refl_equal T (TSort n)) +t H2)))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda +(_: (((eq T u1 (TSort n)) \to (eq T u2 u1)))).(\lambda (t1: T).(\lambda (t2: +T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2 +t1)))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u1 t1) (TSort n))).(let +H6 \def (eq_ind T (THead k u1 t1) (\lambda (ee: T).(match ee in T return +(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H5) in +(False_ind (eq T (THead k u2 t2) (THead k u1 t1)) H6)))))))))))) (\lambda (u: +T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: +(((eq T v1 (TSort n)) \to (eq T v2 v1)))).(\lambda (t1: T).(\lambda (t2: +T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2 +t1)))).(\lambda (H5: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t1)) +(TSort n))).(let H6 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u +t1)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with +[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) +\Rightarrow True])) I (TSort n) H5) in (False_ind (eq T (THead (Bind Abbr) v2 +t2) (THead (Flat Appl) v1 (THead (Bind Abst) u t1))) H6)))))))))))) (\lambda +(b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: +T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (TSort n)) \to (eq T v2 +v1)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda +(_: (((eq T u1 (TSort n)) \to (eq T u2 u1)))).(\lambda (t1: T).(\lambda (t2: +T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2 +t1)))).(\lambda (H8: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t1)) +(TSort n))).(let H9 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 +t1)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with +[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) +\Rightarrow True])) I (TSort n) H8) in (False_ind (eq T (THead (Bind b) u2 +(THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Flat Appl) v1 (THead (Bind +b) u1 t1))) H9))))))))))))))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda +(_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 (TSort n)) \to (eq T u2 +u1)))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda +(_: (((eq T t1 (TSort n)) \to (eq T t2 t1)))).(\lambda (w: T).(\lambda (_: +(subst0 O u2 t2 w)).(\lambda (H6: (eq T (THead (Bind Abbr) u1 t1) (TSort +n))).(let H7 \def (eq_ind T (THead (Bind Abbr) u1 t1) (\lambda (ee: T).(match +ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | +(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) +H6) in (False_ind (eq T (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u1 t1)) +H7))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda +(t1: T).(\lambda (t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 +(TSort n)) \to (eq T t2 t1)))).(\lambda (u: T).(\lambda (H4: (eq T (THead +(Bind b) u (lift (S O) O t1)) (TSort n))).(let H5 \def (eq_ind T (THead (Bind +b) u (lift (S O) O t1)) (\lambda (ee: T).(match ee in T return (\lambda (_: +T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | +(THead _ _ _) \Rightarrow True])) I (TSort n) H4) in (False_ind (eq T t2 +(THead (Bind b) u (lift (S O) O t1))) H5)))))))))) (\lambda (t1: T).(\lambda +(t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq +T t2 t1)))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u t1) +(TSort n))).(let H4 \def (eq_ind T (THead (Flat Cast) u t1) (\lambda (ee: +T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow +False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I +(TSort n) H3) in (False_ind (eq T t2 (THead (Flat Cast) u t1)) H4)))))))) y x +H0))) H))). +(* COMMENTS +Initial nodes: 1045 +END *) + +theorem pr0_gen_lref: + \forall (x: T).(\forall (n: nat).((pr0 (TLRef n) x) \to (eq T x (TLRef n)))) +\def + \lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr0 (TLRef n) x)).(insert_eq +T (TLRef n) (\lambda (t: T).(pr0 t x)) (\lambda (t: T).(eq T x t)) (\lambda +(y: T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: T).(\lambda (t0: +T).((eq T t (TLRef n)) \to (eq T t0 t)))) (\lambda (t: T).(\lambda (H1: (eq T +t (TLRef n))).(let H2 \def (f_equal T T (\lambda (e: T).e) t (TLRef n) H1) in +(eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 t0)) (refl_equal T (TLRef n)) +t H2)))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda +(_: (((eq T u1 (TLRef n)) \to (eq T u2 u1)))).(\lambda (t1: T).(\lambda (t2: +T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (eq T t2 +t1)))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u1 t1) (TLRef n))).(let +H6 \def (eq_ind T (THead k u1 t1) (\lambda (ee: T).(match ee in T return +(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H5) in +(False_ind (eq T (THead k u2 t2) (THead k u1 t1)) H6)))))))))))) (\lambda (u: +T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: +(((eq T v1 (TLRef n)) \to (eq T v2 v1)))).(\lambda (t1: T).(\lambda (t2: +T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (eq T t2 +t1)))).(\lambda (H5: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t1)) +(TLRef n))).(let H6 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u +t1)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with +[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) +\Rightarrow True])) I (TLRef n) H5) in (False_ind (eq T (THead (Bind Abbr) v2 +t2) (THead (Flat Appl) v1 (THead (Bind Abst) u t1))) H6)))))))))))) (\lambda +(b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: +T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (TLRef n)) \to (eq T v2 +v1)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda +(_: (((eq T u1 (TLRef n)) \to (eq T u2 u1)))).(\lambda (t1: T).(\lambda (t2: +T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (eq T t2 +t1)))).(\lambda (H8: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t1)) +(TLRef n))).(let H9 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 +t1)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with +[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) +\Rightarrow True])) I (TLRef n) H8) in (False_ind (eq T (THead (Bind b) u2 +(THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Flat Appl) v1 (THead (Bind +b) u1 t1))) H9))))))))))))))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda +(_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 (TLRef n)) \to (eq T u2 +u1)))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda +(_: (((eq T t1 (TLRef n)) \to (eq T t2 t1)))).(\lambda (w: T).(\lambda (_: +(subst0 O u2 t2 w)).(\lambda (H6: (eq T (THead (Bind Abbr) u1 t1) (TLRef +n))).(let H7 \def (eq_ind T (THead (Bind Abbr) u1 t1) (\lambda (ee: T).(match +ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | +(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) +H6) in (False_ind (eq T (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u1 t1)) +H7))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda +(t1: T).(\lambda (t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 +(TLRef n)) \to (eq T t2 t1)))).(\lambda (u: T).(\lambda (H4: (eq T (THead +(Bind b) u (lift (S O) O t1)) (TLRef n))).(let H5 \def (eq_ind T (THead (Bind +b) u (lift (S O) O t1)) (\lambda (ee: T).(match ee in T return (\lambda (_: +T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | +(THead _ _ _) \Rightarrow True])) I (TLRef n) H4) in (False_ind (eq T t2 +(THead (Bind b) u (lift (S O) O t1))) H5)))))))))) (\lambda (t1: T).(\lambda +(t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (eq +T t2 t1)))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u t1) +(TLRef n))).(let H4 \def (eq_ind T (THead (Flat Cast) u t1) (\lambda (ee: +T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow +False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I +(TLRef n) H3) in (False_ind (eq T t2 (THead (Flat Cast) u t1)) H4)))))))) y x +H0))) H))). +(* COMMENTS +Initial nodes: 1045 +END *) + +theorem pr0_gen_abst: + \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Bind Abst) u1 +t1) x) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind +Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: +T).(\lambda (t2: T).(pr0 t1 t2))))))) +\def + \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead +(Bind Abst) u1 t1) x)).(insert_eq T (THead (Bind Abst) u1 t1) (\lambda (t: +T).(pr0 t x)) (\lambda (_: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t2: +T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 +u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))))) (\lambda (y: +T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: T).(\lambda (t0: T).((eq T +t (THead (Bind Abst) u1 t1)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: +T).(eq T t0 (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: +T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))))))) (\lambda +(t: T).(\lambda (H1: (eq T t (THead (Bind Abst) u1 t1))).(let H2 \def +(f_equal T T (\lambda (e: T).e) t (THead (Bind Abst) u1 t1) H1) in (eq_ind_r +T (THead (Bind Abst) u1 t1) (\lambda (t0: T).(ex3_2 T T (\lambda (u2: +T).(\lambda (t2: T).(eq T t0 (THead (Bind Abst) u2 t2)))) (\lambda (u2: +T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 +t2))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Bind +Abst) u1 t1) (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: +T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))) u1 t1 +(refl_equal T (THead (Bind Abst) u1 t1)) (pr0_refl u1) (pr0_refl t1)) t +H2)))) (\lambda (u0: T).(\lambda (u2: T).(\lambda (H1: (pr0 u0 u2)).(\lambda +(H2: (((eq T u0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T (\lambda (u3: +T).(\lambda (t2: T).(eq T u2 (THead (Bind Abst) u3 t2)))) (\lambda (u3: +T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 +t2))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H3: (pr0 t0 +t2)).(\lambda (H4: (((eq T t0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T +(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u3 t3)))) +(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: +T).(pr0 t1 t3))))))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u0 t0) +(THead (Bind Abst) u1 t1))).(let H6 \def (f_equal T K (\lambda (e: T).(match +e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) +\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u0 t0) (THead (Bind +Abst) u1 t1) H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T +return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) +\Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead k u0 t0) (THead (Bind +Abst) u1 t1) H5) in ((let H8 \def (f_equal T T (\lambda (e: T).(match e in T +return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) +\Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k u0 t0) (THead (Bind +Abst) u1 t1) H5) in (\lambda (H9: (eq T u0 u1)).(\lambda (H10: (eq K k (Bind +Abst))).(eq_ind_r K (Bind Abst) (\lambda (k0: K).(ex3_2 T T (\lambda (u3: +T).(\lambda (t3: T).(eq T (THead k0 u2 t2) (THead (Bind Abst) u3 t3)))) +(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: +T).(pr0 t1 t3))))) (let H11 \def (eq_ind T t0 (\lambda (t: T).((eq T t (THead +(Bind Abst) u1 t1)) \to (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 +(THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) +(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))))) H4 t1 H8) in (let H12 \def +(eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H3 t1 H8) in (let H13 \def (eq_ind T +u0 (\lambda (t: T).((eq T t (THead (Bind Abst) u1 t1)) \to (ex3_2 T T +(\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead (Bind Abst) u3 t3)))) +(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: +T).(pr0 t1 t3)))))) H2 u1 H9) in (let H14 \def (eq_ind T u0 (\lambda (t: +T).(pr0 t u2)) H1 u1 H9) in (ex3_2_intro T T (\lambda (u3: T).(\lambda (t3: +T).(eq T (THead (Bind Abst) u2 t2) (THead (Bind Abst) u3 t3)))) (\lambda (u3: +T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 +t3))) u2 t2 (refl_equal T (THead (Bind Abst) u2 t2)) H14 H12))))) k H10)))) +H7)) H6)))))))))))) (\lambda (u: T).(\lambda (v1: T).(\lambda (v2: +T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind Abst) u1 +t1)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind +Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: +T).(\lambda (t2: T).(pr0 t1 t2))))))).(\lambda (t0: T).(\lambda (t2: +T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Abst) u1 +t1)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind +Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(pr0 t1 t3))))))).(\lambda (H5: (eq T (THead (Flat Appl) +v1 (THead (Bind Abst) u t0)) (THead (Bind Abst) u1 t1))).(let H6 \def (eq_ind +T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (ee: T).(match ee +in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef +_) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return +(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow +True])])) I (THead (Bind Abst) u1 t1) H5) in (False_ind (ex3_2 T T (\lambda +(u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) (THead (Bind Abst) +u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(pr0 t1 t3)))) H6)))))))))))) (\lambda (b: B).(\lambda +(_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 +v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T +(\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind Abst) u2 t2)))) +(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: +T).(pr0 t1 t2))))))).(\lambda (u0: T).(\lambda (u2: T).(\lambda (_: (pr0 u0 +u2)).(\lambda (_: (((eq T u0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T +(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Bind Abst) u3 t2)))) +(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: +T).(pr0 t1 t2))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 +t2)).(\lambda (_: (((eq T t0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T +(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u3 t3)))) +(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: +T).(pr0 t1 t3))))))).(\lambda (H8: (eq T (THead (Flat Appl) v1 (THead (Bind +b) u0 t0)) (THead (Bind Abst) u1 t1))).(let H9 \def (eq_ind T (THead (Flat +Appl) v1 (THead (Bind b) u0 t0)) (\lambda (ee: T).(match ee in T return +(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda +(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow +True])])) I (THead (Bind Abst) u1 t1) H8) in (False_ind (ex3_2 T T (\lambda +(u3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S +O) O v2) t2)) (THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: +T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) +H9))))))))))))))))) (\lambda (u0: T).(\lambda (u2: T).(\lambda (_: (pr0 u0 +u2)).(\lambda (_: (((eq T u0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T +(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Bind Abst) u3 t2)))) +(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: +T).(pr0 t1 t2))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 +t2)).(\lambda (_: (((eq T t0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T +(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u3 t3)))) +(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: +T).(pr0 t1 t3))))))).(\lambda (w: T).(\lambda (_: (subst0 O u2 t2 +w)).(\lambda (H6: (eq T (THead (Bind Abbr) u0 t0) (THead (Bind Abst) u1 +t1))).(let H7 \def (eq_ind T (THead (Bind Abbr) u0 t0) (\lambda (ee: +T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow +False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K +return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B return +(\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | +Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (THead (Bind +Abst) u1 t1) H6) in (False_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t3: +T).(eq T (THead (Bind Abbr) u2 w) (THead (Bind Abst) u3 t3)))) (\lambda (u3: +T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 +t3)))) H7))))))))))))) (\lambda (b: B).(\lambda (H1: (not (eq B b +Abst))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda +(H3: (((eq T t0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 +t3))))))).(\lambda (u: T).(\lambda (H4: (eq T (THead (Bind b) u (lift (S O) O +t0)) (THead (Bind Abst) u1 t1))).(let H5 \def (f_equal T B (\lambda (e: +T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | +(TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k in K return +(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow +b])])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abst) u1 t1) H4) in +((let H6 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: +T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) +\Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abst) u1 +t1) H4) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T return +(\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat +\to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow +(TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true +\Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t3) \Rightarrow +(THead k (lref_map f d u0) (lref_map f (s k d) t3))]) in lref_map) (\lambda +(x0: nat).(plus x0 (S O))) O t0) | (TLRef _) \Rightarrow ((let rec lref_map +(f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) +\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with +[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t3) +\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t3))]) in +lref_map) (\lambda (x0: nat).(plus x0 (S O))) O t0) | (THead _ _ t) +\Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abst) u1 +t1) H4) in (\lambda (_: (eq T u u1)).(\lambda (H9: (eq B b Abst)).(let H10 +\def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H1 Abst H9) in (let +H11 \def (eq_ind_r T t1 (\lambda (t: T).((eq T t0 (THead (Bind Abst) u1 t)) +\to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) +u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(pr0 t t3)))))) H3 (lift (S O) O t0) H7) in (eq_ind T +(lift (S O) O t0) (\lambda (t: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t3: +T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: +T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t t3))))) (let H12 +\def (match (H10 (refl_equal B Abst)) in False return (\lambda (_: +False).(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind +Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(pr0 (lift (S O) O t0) t3))))) with []) in H12) t1 +H7)))))) H6)) H5)))))))))) (\lambda (t0: T).(\lambda (t2: T).(\lambda (_: +(pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T +T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: +T).(pr0 t1 t3))))))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u +t0) (THead (Bind Abst) u1 t1))).(let H4 \def (eq_ind T (THead (Flat Cast) u +t0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort +_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) +\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) +\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u1 +t1) H3) in (False_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 +(THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) +(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) H4)))))))) y x H0))) H)))). +(* COMMENTS +Initial nodes: 2838 +END *) + +theorem pr0_gen_appl: + \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Flat Appl) u1 +t1) x) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead +(Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda +(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind +Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T +(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 +(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T x (THead (Bind b) +v2 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: B).(\lambda +(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 +u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(t2: T).(pr0 z1 t2)))))))))))) +\def + \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead +(Flat Appl) u1 t1) x)).(insert_eq T (THead (Flat Appl) u1 t1) (\lambda (t: +T).(pr0 t x)) (\lambda (_: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: +T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 +u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T +(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 +(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda +(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) +(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: +B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T x +(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda +(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: +T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 +v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))))))) (\lambda (y: +T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: T).(\lambda (t0: T).((eq T +t (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda +(t2: T).(eq T t0 (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: +T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T +T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 +(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda +(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) +(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: +B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T +t0 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) +(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: +T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 +v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))))))))) (\lambda (t: +T).(\lambda (H1: (eq T t (THead (Flat Appl) u1 t1))).(let H2 \def (f_equal T +T (\lambda (e: T).e) t (THead (Flat Appl) u1 t1) H1) in (eq_ind_r T (THead +(Flat Appl) u1 t1) (\lambda (t0: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda +(t2: T).(eq T t0 (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: +T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T +T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 +(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda +(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) +(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: +B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T +t0 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) +(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: +T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 +v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))))))) (or3_intro0 (ex3_2 T +T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Appl) u1 t1) (THead +(Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda +(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind +Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(t2: T).(eq T (THead (Flat Appl) u1 t1) (THead (Bind Abbr) u2 t2)))))) +(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 +u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: +T).(pr0 z1 t2)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B +b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(v2: T).(\lambda (t2: T).(eq T (THead (Flat Appl) u1 t1) (THead (Bind b) v2 +(THead (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 +u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(t2: T).(pr0 z1 t2)))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: +T).(eq T (THead (Flat Appl) u1 t1) (THead (Flat Appl) u2 t2)))) (\lambda (u2: +T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 +t2))) u1 t1 (refl_equal T (THead (Flat Appl) u1 t1)) (pr0_refl u1) (pr0_refl +t1))) t H2)))) (\lambda (u0: T).(\lambda (u2: T).(\lambda (H1: (pr0 u0 +u2)).(\lambda (H2: (((eq T u0 (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T +(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) +(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: +T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda +(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Bind +Abbr) u3 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda +(_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind +b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(u3: T).(\lambda (v2: T).(\lambda (t2: T).(eq T u2 (THead (Bind b) v2 (THead +(Flat Appl) (lift (S O) O u3) t2))))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 +u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(t2: T).(pr0 z1 t2)))))))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda +(H3: (pr0 t0 t2)).(\lambda (H4: (((eq T t0 (THead (Flat Appl) u1 t1)) \to +(or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) +u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: +T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 +z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: +T).(eq T t2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T +(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 +(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind +b) v2 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda +(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) +(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (t3: T).(pr0 z1 t3)))))))))))).(\lambda (k: K).(\lambda (H5: (eq +T (THead k u0 t0) (THead (Flat Appl) u1 t1))).(let H6 \def (f_equal T K +(\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) +\Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) +(THead k u0 t0) (THead (Flat Appl) u1 t1) H5) in ((let H7 \def (f_equal T T +(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) +(THead k u0 t0) (THead (Flat Appl) u1 t1) H5) in ((let H8 \def (f_equal T T +(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) +(THead k u0 t0) (THead (Flat Appl) u1 t1) H5) in (\lambda (H9: (eq T u0 +u1)).(\lambda (H10: (eq K k (Flat Appl))).(eq_ind_r K (Flat Appl) (\lambda +(k0: K).(or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead k0 u2 +t2) (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 +u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda +(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead +(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: +T).(\lambda (t3: T).(eq T (THead k0 u2 t2) (THead (Bind Abbr) u3 t3)))))) +(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 +u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: +T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B +b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda +(v2: T).(\lambda (t3: T).(eq T (THead k0 u2 t2) (THead (Bind b) v2 (THead +(Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 +u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(t3: T).(pr0 z1 t3)))))))))) (let H11 \def (eq_ind T t0 (\lambda (t: T).((eq +T t (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda +(t3: T).(eq T t2 (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: +T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T +T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 +(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: +T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda +(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) +(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: +B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T +t2 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) +(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: +T).(\lambda (_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 +v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))))))) H4 t1 H8) in (let +H12 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H3 t1 H8) in (let H13 \def +(eq_ind T u0 (\lambda (t: T).((eq T t (THead (Flat Appl) u1 t1)) \to (or3 +(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead (Flat Appl) u3 +t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: +T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 +z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: +T).(eq T u2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T +(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 +(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T u2 (THead (Bind +b) v2 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda +(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) +(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (t3: T).(pr0 z1 t3))))))))))) H2 u1 H9) in (let H14 \def (eq_ind +T u0 (\lambda (t: T).(pr0 t u2)) H1 u1 H9) in (or3_intro0 (ex3_2 T T (\lambda +(u3: T).(\lambda (t3: T).(eq T (THead (Flat Appl) u2 t2) (THead (Flat Appl) +u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: +T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 +z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: +T).(eq T (THead (Flat Appl) u2 t2) (THead (Bind Abbr) u3 t3)))))) (\lambda +(_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 +t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) +(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (v2: T).(\lambda +(t3: T).(eq T (THead (Flat Appl) u2 t2) (THead (Bind b) v2 (THead (Flat Appl) +(lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3))))))) +(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 +t3)))))))) (ex3_2_intro T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead +(Flat Appl) u2 t2) (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: +T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) u2 t2 +(refl_equal T (THead (Flat Appl) u2 t2)) H14 H12)))))) k H10)))) H7)) +H6)))))))))))) (\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda +(H1: (pr0 v1 v2)).(\lambda (H2: (((eq T v1 (THead (Flat Appl) u1 t1)) \to +(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Flat Appl) +u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: +T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 +z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: +T).(eq T v2 (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T +(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 +(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (v3: T).(\lambda (t2: T).(eq T v2 (THead (Bind +b) v3 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda +(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) +(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (t2: T).(pr0 z1 t2)))))))))))).(\lambda (t0: T).(\lambda (t2: +T).(\lambda (H3: (pr0 t0 t2)).(\lambda (H4: (((eq T t0 (THead (Flat Appl) u1 +t1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead +(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda +(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind +Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T +(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 +(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T t2 (THead (Bind +b) v3 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda +(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) +(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (t3: T).(pr0 z1 t3)))))))))))).(\lambda (H5: (eq T (THead (Flat +Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat Appl) u1 t1))).(let H6 \def +(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with +[(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t _) +\Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat +Appl) u1 t1) H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T +return (\lambda (_: T).T) with [(TSort _) \Rightarrow (THead (Bind Abst) u +t0) | (TLRef _) \Rightarrow (THead (Bind Abst) u t0) | (THead _ _ t) +\Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat +Appl) u1 t1) H5) in (\lambda (H8: (eq T v1 u1)).(let H9 \def (eq_ind T v1 +(\lambda (t: T).((eq T t (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T v2 (THead (Flat Appl) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: +T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda +(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T v2 (THead (Bind +Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind +b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T v2 (THead (Bind b) v3 (THead +(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 +u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(t3: T).(pr0 z1 t3))))))))))) H2 u1 H8) in (let H10 \def (eq_ind T v1 +(\lambda (t: T).(pr0 t v2)) H1 u1 H8) in (let H11 \def (eq_ind_r T t1 +(\lambda (t: T).((eq T t0 (THead (Flat Appl) u1 t)) \to (or3 (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: +T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda +(_: T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind +Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind +b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v3 (THead +(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 +u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(t3: T).(pr0 z1 t3))))))))))) H4 (THead (Bind Abst) u t0) H7) in (let H12 +\def (eq_ind_r T t1 (\lambda (t: T).((eq T u1 (THead (Flat Appl) u1 t)) \to +(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T v2 (THead (Flat Appl) +u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 +z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: +T).(eq T v2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T +(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t +(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T v2 (THead (Bind +b) v3 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda +(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) +(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (t3: T).(pr0 z1 t3))))))))))) H9 (THead (Bind Abst) u t0) H7) in +(eq_ind T (THead (Bind Abst) u t0) (\lambda (t: T).(or3 (ex3_2 T T (\lambda +(u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) (THead (Flat Appl) +u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 +z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: +T).(eq T (THead (Bind Abbr) v2 t2) (THead (Bind Abbr) u2 t3)))))) (\lambda +(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 +t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) +(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(eq T t (THead (Bind b) y1 z1)))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v3: T).(\lambda +(t3: T).(eq T (THead (Bind Abbr) v2 t2) (THead (Bind b) v3 (THead (Flat Appl) +(lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) +(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 +t3)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T +(THead (Bind Abbr) v2 t2) (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 (THead +(Bind Abst) u t0) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) u t0) (THead +(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) (THead (Bind Abbr) u2 +t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: +T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda +(t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B +b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) u t0) (THead +(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T (THead (Bind +Abbr) v2 t2) (THead (Bind b) v3 (THead (Flat Appl) (lift (S O) O u2) +t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda +(y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 +y1 v3))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))))) (ex4_4_intro T T T T +(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T +(THead (Bind Abst) u t0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) +v2 t2) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))) u t0 v2 t2 +(refl_equal T (THead (Bind Abst) u t0)) (refl_equal T (THead (Bind Abbr) v2 +t2)) H10 H3)) t1 H7))))))) H6)))))))))))) (\lambda (b: B).(\lambda (H1: (not +(eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H2: (pr0 v1 +v2)).(\lambda (H3: (((eq T v1 (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T +(\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Flat Appl) u2 t2)))) +(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: +T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda +(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind +Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T (\lambda (b0: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind +b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(u2: T).(\lambda (v3: T).(\lambda (t2: T).(eq T v2 (THead (Bind b0) v3 (THead +(Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 +u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(t2: T).(pr0 z1 t2)))))))))))).(\lambda (u0: T).(\lambda (u2: T).(\lambda +(H4: (pr0 u0 u2)).(\lambda (H5: (((eq T u0 (THead (Flat Appl) u1 t1)) \to +(or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) +u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: +T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 +z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: +T).(eq T u2 (THead (Bind Abbr) u3 t2)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T +(\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 +(THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t2: T).(eq T u2 (THead (Bind +b0) v3 (THead (Flat Appl) (lift (S O) O u3) t2))))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda +(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) +(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (t2: T).(pr0 z1 t2)))))))))))).(\lambda (t0: T).(\lambda (t2: +T).(\lambda (H6: (pr0 t0 t2)).(\lambda (H7: (((eq T t0 (THead (Flat Appl) u1 +t1)) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead +(Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda +(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind +Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda +(t3: T).(eq T t2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T +(\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 +(THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T t2 (THead (Bind +b0) v3 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda +(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) +(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (t3: T).(pr0 z1 t3)))))))))))).(\lambda (H8: (eq T (THead (Flat +Appl) v1 (THead (Bind b) u0 t0)) (THead (Flat Appl) u1 t1))).(let H9 \def +(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with +[(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t _) +\Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead (Flat +Appl) u1 t1) H8) in ((let H10 \def (f_equal T T (\lambda (e: T).(match e in T +return (\lambda (_: T).T) with [(TSort _) \Rightarrow (THead (Bind b) u0 t0) +| (TLRef _) \Rightarrow (THead (Bind b) u0 t0) | (THead _ _ t) \Rightarrow +t])) (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead (Flat Appl) u1 t1) +H8) in (\lambda (H11: (eq T v1 u1)).(let H12 \def (eq_ind T v1 (\lambda (t: +T).((eq T t (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T (\lambda (u3: +T).(\lambda (t3: T).(eq T v2 (THead (Flat Appl) u3 t3)))) (\lambda (u3: +T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 +t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T v2 (THead (Bind +Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda +(_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b0: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind +b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T v2 (THead (Bind b0) v3 (THead +(Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 +u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(t3: T).(pr0 z1 t3))))))))))) H3 u1 H11) in (let H13 \def (eq_ind T v1 +(\lambda (t: T).(pr0 t v2)) H2 u1 H11) in (let H14 \def (eq_ind_r T t1 +(\lambda (t: T).((eq T t0 (THead (Flat Appl) u1 t)) \to (or3 (ex3_2 T T +(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u3 t3)))) +(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: +T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda +(_: T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind +Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda +(_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b0: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind +b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T t2 (THead (Bind b0) v3 (THead +(Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 +u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(t3: T).(pr0 z1 t3))))))))))) H7 (THead (Bind b) u0 t0) H10) in (let H15 \def +(eq_ind_r T t1 (\lambda (t: T).((eq T u0 (THead (Flat Appl) u1 t)) \to (or3 +(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead (Flat Appl) u3 +t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: +T).(\lambda (t3: T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 +z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: +T).(eq T u2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T +(\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t +(THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T u2 (THead (Bind +b0) v3 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda +(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) +(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (t3: T).(pr0 z1 t3))))))))))) H5 (THead (Bind b) u0 t0) H10) in +(let H16 \def (eq_ind_r T t1 (\lambda (t: T).((eq T u1 (THead (Flat Appl) u1 +t)) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T v2 (THead +(Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda +(_: T).(\lambda (t3: T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind +Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda +(t3: T).(eq T v2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T +(\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t +(THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T v2 (THead (Bind +b0) v3 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda +(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) +(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (t3: T).(pr0 z1 t3))))))))))) H12 (THead (Bind b) u0 t0) H10) in +(eq_ind T (THead (Bind b) u0 t0) (\lambda (t: T).(or3 (ex3_2 T T (\lambda +(u3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S +O) O v2) t2)) (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: +T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t t3)))) (ex4_4 T T T +T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t +(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: +T).(\lambda (t3: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O +v2) t2)) (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T +(\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t +(THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T (THead (Bind b) +u2 (THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Bind b0) v3 (THead (Flat +Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3))))))) +(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 +t3)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T +(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Flat +Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: +T).(\lambda (t3: T).(pr0 (THead (Bind b) u0 t0) t3)))) (ex4_4 T T T T +(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T +(THead (Bind b) u0 t0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 +(THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Bind Abbr) u3 t3)))))) +(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 +u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: +T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B +b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) u0 t0) (THead +(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T (THead (Bind b) +u2 (THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Bind b0) v3 (THead (Flat +Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3))))))) +(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 +t3)))))))) (ex6_6_intro B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 +Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) u0 t0) (THead (Bind +b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 (THead +(Flat Appl) (lift (S O) O v2) t2)) (THead (Bind b0) v3 (THead (Flat Appl) +(lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3))))))) +(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 +t3))))))) b u0 t0 v2 u2 t2 H1 (refl_equal T (THead (Bind b) u0 t0)) +(refl_equal T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2))) +H13 H4 H6)) t1 H10)))))))) H9))))))))))))))))) (\lambda (u0: T).(\lambda (u2: +T).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (((eq T u0 (THead (Flat Appl) u1 +t1)) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead +(Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda +(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind +Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda +(t2: T).(eq T u2 (THead (Bind Abbr) u3 t2)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T +(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 +(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t2: T).(eq T u2 (THead (Bind +b) v2 (THead (Flat Appl) (lift (S O) O u3) t2))))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda +(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) +(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (t2: T).(pr0 z1 t2)))))))))))).(\lambda (t0: T).(\lambda (t2: +T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u1 +t1)) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead +(Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda +(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind +Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda +(t3: T).(eq T t2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T +(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 +(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind +b) v2 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda +(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) +(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (t3: T).(pr0 z1 t3)))))))))))).(\lambda (w: T).(\lambda (_: +(subst0 O u2 t2 w)).(\lambda (H6: (eq T (THead (Bind Abbr) u0 t0) (THead +(Flat Appl) u1 t1))).(let H7 \def (eq_ind T (THead (Bind Abbr) u0 t0) +(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) +\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow +(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | +(Flat _) \Rightarrow False])])) I (THead (Flat Appl) u1 t1) H6) in (False_ind +(or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 +w) (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 +u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda +(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead +(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: +T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u3 +t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: +T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda +(t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B +b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda +(v2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 w) (THead (Bind b) v2 +(THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 +u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(t3: T).(pr0 z1 t3))))))))) H7))))))))))))) (\lambda (b: B).(\lambda (_: (not +(eq B b Abst))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 +t2)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: +T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda +(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind +Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b0: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind +b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b0) v2 (THead +(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 +u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(t3: T).(pr0 z1 t3)))))))))))).(\lambda (u: T).(\lambda (H4: (eq T (THead +(Bind b) u (lift (S O) O t0)) (THead (Flat Appl) u1 t1))).(let H5 \def +(eq_ind T (THead (Bind b) u (lift (S O) O t0)) (\lambda (ee: T).(match ee in +T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda +(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow +False])])) I (THead (Flat Appl) u1 t1) H4) in (False_ind (or3 (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: +T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda +(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind +Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b0: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind +b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b0) v2 (THead +(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 +u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(t3: T).(pr0 z1 t3))))))))) H5)))))))))) (\lambda (t0: T).(\lambda (t2: +T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u1 +t1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead +(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda +(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind +Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T +(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 +(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind +b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda +(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) +(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (t3: T).(pr0 z1 t3)))))))))))).(\lambda (u: T).(\lambda (H3: (eq +T (THead (Flat Cast) u t0) (THead (Flat Appl) u1 t1))).(let H4 \def (eq_ind T +(THead (Flat Cast) u t0) (\lambda (ee: T).(match ee in T return (\lambda (_: +T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | +(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with +[(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f in F return +(\lambda (_: F).Prop) with [Appl \Rightarrow False | Cast \Rightarrow +True])])])) I (THead (Flat Appl) u1 t1) H3) in (False_ind (or3 (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: +T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda +(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind +Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind +b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead +(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 +u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(t3: T).(pr0 z1 t3))))))))) H4)))))))) y x H0))) H)))). +(* COMMENTS +Initial nodes: 12299 +END *) + +theorem pr0_gen_cast: + \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Flat Cast) u1 +t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead +(Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda +(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 x))))) +\def + \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead +(Flat Cast) u1 t1) x)).(insert_eq T (THead (Flat Cast) u1 t1) (\lambda (t: +T).(pr0 t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: +T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 +u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 x))) +(\lambda (y: T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: T).(\lambda +(t0: T).((eq T t (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: +T).(\lambda (t2: T).(eq T t0 (THead (Flat Cast) u2 t2)))) (\lambda (u2: +T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 +t2)))) (pr0 t1 t0))))) (\lambda (t: T).(\lambda (H1: (eq T t (THead (Flat +Cast) u1 t1))).(let H2 \def (f_equal T T (\lambda (e: T).e) t (THead (Flat +Cast) u1 t1) H1) in (eq_ind_r T (THead (Flat Cast) u1 t1) (\lambda (t0: +T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Flat +Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: +T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 t0))) (or_introl (ex3_2 T T +(\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Cast) u1 t1) (THead +(Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda +(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (THead (Flat Cast) u1 t1)) +(ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Cast) +u1 t1) (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 +u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))) u1 t1 (refl_equal T +(THead (Flat Cast) u1 t1)) (pr0_refl u1) (pr0_refl t1))) t H2)))) (\lambda +(u0: T).(\lambda (u2: T).(\lambda (H1: (pr0 u0 u2)).(\lambda (H2: (((eq T u0 +(THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: +T).(eq T u2 (THead (Flat Cast) u3 t2)))) (\lambda (u3: T).(\lambda (_: +T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 +u2))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H3: (pr0 t0 t2)).(\lambda +(H4: (((eq T t0 (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: +T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u3 t3)))) (\lambda (u3: +T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 +t3)))) (pr0 t1 t2))))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u0 t0) +(THead (Flat Cast) u1 t1))).(let H6 \def (f_equal T K (\lambda (e: T).(match +e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) +\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u0 t0) (THead (Flat +Cast) u1 t1) H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T +return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) +\Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead k u0 t0) (THead (Flat +Cast) u1 t1) H5) in ((let H8 \def (f_equal T T (\lambda (e: T).(match e in T +return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) +\Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k u0 t0) (THead (Flat +Cast) u1 t1) H5) in (\lambda (H9: (eq T u0 u1)).(\lambda (H10: (eq K k (Flat +Cast))).(eq_ind_r K (Flat Cast) (\lambda (k0: K).(or (ex3_2 T T (\lambda (u3: +T).(\lambda (t3: T).(eq T (THead k0 u2 t2) (THead (Flat Cast) u3 t3)))) +(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: +T).(pr0 t1 t3)))) (pr0 t1 (THead k0 u2 t2)))) (let H11 \def (eq_ind T t0 +(\lambda (t: T).((eq T t (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T +(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u3 t3)))) +(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: +T).(pr0 t1 t3)))) (pr0 t1 t2)))) H4 t1 H8) in (let H12 \def (eq_ind T t0 +(\lambda (t: T).(pr0 t t2)) H3 t1 H8) in (let H13 \def (eq_ind T u0 (\lambda +(t: T).((eq T t (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: +T).(\lambda (t3: T).(eq T u2 (THead (Flat Cast) u3 t3)))) (\lambda (u3: +T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 +t3)))) (pr0 t1 u2)))) H2 u1 H9) in (let H14 \def (eq_ind T u0 (\lambda (t: +T).(pr0 t u2)) H1 u1 H9) in (or_introl (ex3_2 T T (\lambda (u3: T).(\lambda +(t3: T).(eq T (THead (Flat Cast) u2 t2) (THead (Flat Cast) u3 t3)))) (\lambda +(u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 +t1 t3)))) (pr0 t1 (THead (Flat Cast) u2 t2)) (ex3_2_intro T T (\lambda (u3: +T).(\lambda (t3: T).(eq T (THead (Flat Cast) u2 t2) (THead (Flat Cast) u3 +t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: +T).(\lambda (t3: T).(pr0 t1 t3))) u2 t2 (refl_equal T (THead (Flat Cast) u2 +t2)) H14 H12)))))) k H10)))) H7)) H6)))))))))))) (\lambda (u: T).(\lambda +(v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 +(THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: +T).(eq T v2 (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: +T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 +v2))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda +(_: (((eq T t0 (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 +t3)))) (pr0 t1 t2))))).(\lambda (H5: (eq T (THead (Flat Appl) v1 (THead (Bind +Abst) u t0)) (THead (Flat Cast) u1 t1))).(let H6 \def (eq_ind T (THead (Flat +Appl) v1 (THead (Bind Abst) u t0)) (\lambda (ee: T).(match ee in T return +(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda +(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f +in F return (\lambda (_: F).Prop) with [Appl \Rightarrow True | Cast +\Rightarrow False])])])) I (THead (Flat Cast) u1 t1) H5) in (False_ind (or +(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) +(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) +(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (THead (Bind Abbr) v2 +t2))) H6)))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b +Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda +(_: (((eq T v1 (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: +T).(\lambda (t2: T).(eq T v2 (THead (Flat Cast) u2 t2)))) (\lambda (u2: +T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 +t2)))) (pr0 t1 v2))))).(\lambda (u0: T).(\lambda (u2: T).(\lambda (_: (pr0 u0 +u2)).(\lambda (_: (((eq T u0 (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T +(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Cast) u3 t2)))) +(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: +T).(pr0 t1 t2)))) (pr0 t1 u2))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda +(_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Flat Cast) u1 t1)) \to (or +(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u3 +t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: +T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2))))).(\lambda (H8: (eq T (THead +(Flat Appl) v1 (THead (Bind b) u0 t0)) (THead (Flat Cast) u1 t1))).(let H9 +\def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (\lambda (ee: +T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow +False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K +return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) +\Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl \Rightarrow +True | Cast \Rightarrow False])])])) I (THead (Flat Cast) u1 t1) H8) in +(False_ind (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Flat Cast) u3 +t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: +T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (THead (Bind b) u2 (THead (Flat +Appl) (lift (S O) O v2) t2)))) H9))))))))))))))))) (\lambda (u0: T).(\lambda +(u2: T).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (((eq T u0 (THead (Flat Cast) +u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead +(Flat Cast) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda +(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 u2))))).(\lambda (t0: +T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead +(Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq +T t2 (THead (Flat Cast) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 +u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 +t2))))).(\lambda (w: T).(\lambda (_: (subst0 O u2 t2 w)).(\lambda (H6: (eq T +(THead (Bind Abbr) u0 t0) (THead (Flat Cast) u1 t1))).(let H7 \def (eq_ind T +(THead (Bind Abbr) u0 t0) (\lambda (ee: T).(match ee in T return (\lambda (_: +T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | +(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with +[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat +Cast) u1 t1) H6) in (False_ind (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: +T).(eq T (THead (Bind Abbr) u2 w) (THead (Flat Cast) u3 t3)))) (\lambda (u3: +T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 +t3)))) (pr0 t1 (THead (Bind Abbr) u2 w))) H7))))))))))))) (\lambda (b: +B).(\lambda (_: (not (eq B b Abst))).(\lambda (t0: T).(\lambda (t2: +T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Flat Cast) u1 +t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead +(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda +(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2))))).(\lambda (u: +T).(\lambda (H4: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead (Flat +Cast) u1 t1))).(let H5 \def (eq_ind T (THead (Bind b) u (lift (S O) O t0)) +(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) +\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow +(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | +(Flat _) \Rightarrow False])])) I (THead (Flat Cast) u1 t1) H4) in (False_ind +(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) +u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2)) H5)))))))))) (\lambda (t0: +T).(\lambda (t2: T).(\lambda (H1: (pr0 t0 t2)).(\lambda (H2: (((eq T t0 +(THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: +T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: +T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 +t2))))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u t0) (THead +(Flat Cast) u1 t1))).(let H4 \def (f_equal T T (\lambda (e: T).(match e in T +return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) +\Rightarrow u | (THead _ t _) \Rightarrow t])) (THead (Flat Cast) u t0) +(THead (Flat Cast) u1 t1) H3) in ((let H5 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | +(TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Flat Cast) +u t0) (THead (Flat Cast) u1 t1) H3) in (\lambda (_: (eq T u u1)).(let H7 \def +(eq_ind T t0 (\lambda (t: T).((eq T t (THead (Flat Cast) u1 t1)) \to (or +(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 +t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2)))) H2 t1 H5) in (let H8 \def +(eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H1 t1 H5) in (or_intror (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: +T).(pr0 t1 t3)))) (pr0 t1 t2) H8))))) H4)))))))) y x H0))) H)))). +(* COMMENTS +Initial nodes: 2911 +END *) + +theorem pr0_gen_abbr: + \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Bind Abbr) u1 +t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead +(Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda +(u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y: T).(pr0 t1 y)) +(\lambda (y: T).(subst0 O u2 y t2))))))) (pr0 t1 (lift (S O) O x)))))) +\def + \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead +(Bind Abbr) u1 t1) x)).(insert_eq T (THead (Bind Abbr) u1 t1) (\lambda (t: +T).(pr0 t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: +T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 +u1 u2))) (\lambda (u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda +(y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y t2))))))) (pr0 t1 (lift (S +O) O x)))) (\lambda (y: T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: +T).(\lambda (t0: T).((eq T t (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T +(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))) +(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: +T).(or (pr0 t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: +T).(subst0 O u2 y0 t2))))))) (pr0 t1 (lift (S O) O t0)))))) (\lambda (t: +T).(\lambda (H1: (eq T t (THead (Bind Abbr) u1 t1))).(let H2 \def (f_equal T +T (\lambda (e: T).e) t (THead (Bind Abbr) u1 t1) H1) in (eq_ind_r T (THead +(Bind Abbr) u1 t1) (\lambda (t0: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda +(t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: +T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T +(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 t2))))))) (pr0 +t1 (lift (S O) O t0)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2: +T).(eq T (THead (Bind Abbr) u1 t1) (THead (Bind Abbr) u2 t2)))) (\lambda (u2: +T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: T).(or (pr0 +t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 +t2))))))) (pr0 t1 (lift (S O) O (THead (Bind Abbr) u1 t1))) (ex3_2_intro T T +(\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Bind Abbr) u1 t1) (THead +(Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda +(u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) +(\lambda (y0: T).(subst0 O u2 y0 t2)))))) u1 t1 (refl_equal T (THead (Bind +Abbr) u1 t1)) (pr0_refl u1) (or_introl (pr0 t1 t1) (ex2 T (\lambda (y0: +T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u1 y0 t1))) (pr0_refl t1)))) t +H2)))) (\lambda (u0: T).(\lambda (u2: T).(\lambda (H1: (pr0 u0 u2)).(\lambda +(H2: (((eq T u0 (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: +T).(\lambda (t2: T).(eq T u2 (THead (Bind Abbr) u3 t2)))) (\lambda (u3: +T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t2: T).(or (pr0 +t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 +t2))))))) (pr0 t1 (lift (S O) O u2)))))).(\lambda (t0: T).(\lambda (t2: +T).(\lambda (H3: (pr0 t0 t2)).(\lambda (H4: (((eq T t0 (THead (Bind Abbr) u1 +t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead +(Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda +(u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) +(\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O +t2)))))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u0 t0) (THead (Bind +Abbr) u1 t1))).(let H6 \def (f_equal T K (\lambda (e: T).(match e in T return +(\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | +(THead k0 _ _) \Rightarrow k0])) (THead k u0 t0) (THead (Bind Abbr) u1 t1) +H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T return +(\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 +| (THead _ t _) \Rightarrow t])) (THead k u0 t0) (THead (Bind Abbr) u1 t1) +H5) in ((let H8 \def (f_equal T T (\lambda (e: T).(match e in T return +(\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 +| (THead _ _ t) \Rightarrow t])) (THead k u0 t0) (THead (Bind Abbr) u1 t1) +H5) in (\lambda (H9: (eq T u0 u1)).(\lambda (H10: (eq K k (Bind +Abbr))).(eq_ind_r K (Bind Abbr) (\lambda (k0: K).(or (ex3_2 T T (\lambda (u3: +T).(\lambda (t3: T).(eq T (THead k0 u2 t2) (THead (Bind Abbr) u3 t3)))) +(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: +T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: +T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O (THead k0 u2 t2))))) (let +H11 \def (eq_ind T t0 (\lambda (t: T).((eq T t (THead (Bind Abbr) u1 t1)) \to +(or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) +u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: +T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) +(\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O t2))))) H4 +t1 H8) in (let H12 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H3 t1 H8) in +(let H13 \def (eq_ind T u0 (\lambda (t: T).((eq T t (THead (Bind Abbr) u1 +t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead +(Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda +(u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) +(\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O u2))))) H2 +u1 H9) in (let H14 \def (eq_ind T u0 (\lambda (t: T).(pr0 t u2)) H1 u1 H9) in +(or_introl (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind +Abbr) u2 t2) (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: +T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T +(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 +t1 (lift (S O) O (THead (Bind Abbr) u2 t2))) (ex3_2_intro T T (\lambda (u3: +T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 t2) (THead (Bind Abbr) u3 +t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: +T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) +(\lambda (y0: T).(subst0 O u3 y0 t3)))))) u2 t2 (refl_equal T (THead (Bind +Abbr) u2 t2)) H14 (or_introl (pr0 t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) +(\lambda (y0: T).(subst0 O u2 y0 t2))) H12))))))) k H10)))) H7)) +H6)))))))))))) (\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: +(pr0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind Abbr) u1 t1)) \to (or +(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind Abbr) u2 +t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: +T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) +(\lambda (y0: T).(subst0 O u2 y0 t2))))))) (pr0 t1 (lift (S O) O +v2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda +(_: (((eq T t0 (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0 +t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 +t3))))))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (H5: (eq T (THead (Flat +Appl) v1 (THead (Bind Abst) u t0)) (THead (Bind Abbr) u1 t1))).(let H6 \def +(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (ee: +T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow +False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K +return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) +\Rightarrow True])])) I (THead (Bind Abbr) u1 t1) H5) in (False_ind (or +(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) +(THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) +(\lambda (u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: +T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S +O) O (THead (Bind Abbr) v2 t2)))) H6)))))))))))) (\lambda (b: B).(\lambda (_: +(not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 +v2)).(\lambda (_: (((eq T v1 (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T +(\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind Abbr) u2 t2)))) +(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: +T).(or (pr0 t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: +T).(subst0 O u2 y0 t2))))))) (pr0 t1 (lift (S O) O v2)))))).(\lambda (u0: +T).(\lambda (u2: T).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (((eq T u0 (THead +(Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq +T u2 (THead (Bind Abbr) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 +u3))) (\lambda (u3: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y0: +T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 t2))))))) (pr0 t1 (lift (S +O) O u2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 +t2)).(\lambda (_: (((eq T t0 (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T +(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u3 t3)))) +(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: +T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: +T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (H8: (eq +T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead (Bind Abbr) u1 +t1))).(let H9 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) +(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) +\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow +(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | +(Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u1 t1) H8) in (False_ind +(or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 +(THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Bind Abbr) u3 t3)))) +(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: +T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: +T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O (THead (Bind b) u2 (THead +(Flat Appl) (lift (S O) O v2) t2))))) H9))))))))))))))))) (\lambda (u0: +T).(\lambda (u2: T).(\lambda (H1: (pr0 u0 u2)).(\lambda (H2: (((eq T u0 +(THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: +T).(eq T u2 (THead (Bind Abbr) u3 t2)))) (\lambda (u3: T).(\lambda (_: +T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T +(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 t2))))))) (pr0 +t1 (lift (S O) O u2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H3: +(pr0 t0 t2)).(\lambda (H4: (((eq T t0 (THead (Bind Abbr) u1 t1)) \to (or +(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u3 +t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: +T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) +(\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O +t2)))))).(\lambda (w: T).(\lambda (H5: (subst0 O u2 t2 w)).(\lambda (H6: (eq +T (THead (Bind Abbr) u0 t0) (THead (Bind Abbr) u1 t1))).(let H7 \def (f_equal +T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) +(THead (Bind Abbr) u0 t0) (THead (Bind Abbr) u1 t1) H6) in ((let H8 \def +(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with +[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) +\Rightarrow t])) (THead (Bind Abbr) u0 t0) (THead (Bind Abbr) u1 t1) H6) in +(\lambda (H9: (eq T u0 u1)).(let H10 \def (eq_ind T t0 (\lambda (t: T).((eq T +t (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda +(t3: T).(eq T t2 (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: +T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T +(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 +t1 (lift (S O) O t2))))) H4 t1 H8) in (let H11 \def (eq_ind T t0 (\lambda (t: +T).(pr0 t t2)) H3 t1 H8) in (let H12 \def (eq_ind T u0 (\lambda (t: T).((eq T +t (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda +(t3: T).(eq T u2 (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: +T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T +(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 +t1 (lift (S O) O u2))))) H2 u1 H9) in (let H13 \def (eq_ind T u0 (\lambda (t: +T).(pr0 t u2)) H1 u1 H9) in (or_introl (ex3_2 T T (\lambda (u3: T).(\lambda +(t3: T).(eq T (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u3 t3)))) (\lambda +(u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or +(pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O +u3 y0 t3))))))) (pr0 t1 (lift (S O) O (THead (Bind Abbr) u2 w))) (ex3_2_intro +T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 w) (THead +(Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda +(u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) +(\lambda (y0: T).(subst0 O u3 y0 t3)))))) u2 w (refl_equal T (THead (Bind +Abbr) u2 w)) H13 (or_intror (pr0 t1 w) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) +(\lambda (y0: T).(subst0 O u2 y0 w))) (ex_intro2 T (\lambda (y0: T).(pr0 t1 +y0)) (\lambda (y0: T).(subst0 O u2 y0 w)) t2 H11 H5)))))))))) H7))))))))))))) +(\lambda (b: B).(\lambda (H1: (not (eq B b Abst))).(\lambda (t0: T).(\lambda +(t2: T).(\lambda (H2: (pr0 t0 t2)).(\lambda (H3: (((eq T t0 (THead (Bind +Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 +(THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) +(\lambda (u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: +T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S +O) O t2)))))).(\lambda (u: T).(\lambda (H4: (eq T (THead (Bind b) u (lift (S +O) O t0)) (THead (Bind Abbr) u1 t1))).(let H5 \def (f_equal T B (\lambda (e: +T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | +(TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k in K return +(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow +b])])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abbr) u1 t1) H4) in +((let H6 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: +T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) +\Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abbr) u1 +t1) H4) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T return +(\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat +\to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow +(TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true +\Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t3) \Rightarrow +(THead k (lref_map f d u0) (lref_map f (s k d) t3))]) in lref_map) (\lambda +(x0: nat).(plus x0 (S O))) O t0) | (TLRef _) \Rightarrow ((let rec lref_map +(f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) +\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with +[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t3) +\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t3))]) in +lref_map) (\lambda (x0: nat).(plus x0 (S O))) O t0) | (THead _ _ t) +\Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abbr) u1 +t1) H4) in (\lambda (_: (eq T u u1)).(\lambda (H9: (eq B b Abbr)).(let H10 +\def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H1 Abbr H9) in (let +H11 \def (eq_ind_r T t1 (\lambda (t: T).((eq T t0 (THead (Bind Abbr) u1 t)) +\to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind +Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: +T).(\lambda (t3: T).(or (pr0 t t3) (ex2 T (\lambda (y0: T).(pr0 t y0)) +(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t (lift (S O) O t2))))) H3 +(lift (S O) O t0) H7) in (eq_ind T (lift (S O) O t0) (\lambda (t: T).(or +(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 +t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: +T).(\lambda (t3: T).(or (pr0 t t3) (ex2 T (\lambda (y0: T).(pr0 t y0)) +(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t (lift (S O) O t2)))) +(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind +Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: +T).(\lambda (t3: T).(or (pr0 (lift (S O) O t0) t3) (ex2 T (\lambda (y0: +T).(pr0 (lift (S O) O t0) y0)) (\lambda (y0: T).(subst0 O u2 y0 t3))))))) +(pr0 (lift (S O) O t0) (lift (S O) O t2)) (pr0_lift t0 t2 H2 (S O) O)) t1 +H7)))))) H6)) H5)))))))))) (\lambda (t0: T).(\lambda (t2: T).(\lambda (_: +(pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Abbr) u1 t1)) \to (or +(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 +t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: +T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) +(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S O) O +t2)))))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u t0) (THead +(Bind Abbr) u1 t1))).(let H4 \def (eq_ind T (THead (Flat Cast) u t0) (\lambda +(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) +\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow +(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | +(Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u1 t1) H3) in (False_ind +(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) +u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: +T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) +(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S O) O t2))) +H4)))))))) y x H0))) H)))). +(* COMMENTS +Initial nodes: 4711 +END *) + +theorem pr0_gen_void: + \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Bind Void) u1 +t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead +(Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda +(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O x)))))) +\def + \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead +(Bind Void) u1 t1) x)).(insert_eq T (THead (Bind Void) u1 t1) (\lambda (t: +T).(pr0 t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: +T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 +u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) +O x)))) (\lambda (y: T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: +T).(\lambda (t0: T).((eq T t (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T +(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Void) u2 t2)))) +(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: +T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O t0)))))) (\lambda (t: T).(\lambda +(H1: (eq T t (THead (Bind Void) u1 t1))).(let H2 \def (f_equal T T (\lambda +(e: T).e) t (THead (Bind Void) u1 t1) H1) in (eq_ind_r T (THead (Bind Void) +u1 t1) (\lambda (t0: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq +T t0 (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 +u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O +t0)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead +(Bind Void) u1 t1) (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: +T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 +(lift (S O) O (THead (Bind Void) u1 t1))) (ex3_2_intro T T (\lambda (u2: +T).(\lambda (t2: T).(eq T (THead (Bind Void) u1 t1) (THead (Bind Void) u2 +t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: +T).(\lambda (t2: T).(pr0 t1 t2))) u1 t1 (refl_equal T (THead (Bind Void) u1 +t1)) (pr0_refl u1) (pr0_refl t1))) t H2)))) (\lambda (u0: T).(\lambda (u2: +T).(\lambda (H1: (pr0 u0 u2)).(\lambda (H2: (((eq T u0 (THead (Bind Void) u1 +t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead +(Bind Void) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda +(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O +u2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H3: (pr0 t0 +t2)).(\lambda (H4: (((eq T t0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T +(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u3 t3)))) +(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: +T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (k: K).(\lambda +(H5: (eq T (THead k u0 t0) (THead (Bind Void) u1 t1))).(let H6 \def (f_equal +T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) +\Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) +(THead k u0 t0) (THead (Bind Void) u1 t1) H5) in ((let H7 \def (f_equal T T +(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) +(THead k u0 t0) (THead (Bind Void) u1 t1) H5) in ((let H8 \def (f_equal T T +(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) +(THead k u0 t0) (THead (Bind Void) u1 t1) H5) in (\lambda (H9: (eq T u0 +u1)).(\lambda (H10: (eq K k (Bind Void))).(eq_ind_r K (Bind Void) (\lambda +(k0: K).(or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead k0 u2 +t2) (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 +u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O +(THead k0 u2 t2))))) (let H11 \def (eq_ind T t0 (\lambda (t: T).((eq T t +(THead (Bind Void) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: +T).(eq T t2 (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: +T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 +(lift (S O) O t2))))) H4 t1 H8) in (let H12 \def (eq_ind T t0 (\lambda (t: +T).(pr0 t t2)) H3 t1 H8) in (let H13 \def (eq_ind T u0 (\lambda (t: T).((eq T +t (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda +(t3: T).(eq T u2 (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: +T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 +(lift (S O) O u2))))) H2 u1 H9) in (let H14 \def (eq_ind T u0 (\lambda (t: +T).(pr0 t u2)) H1 u1 H9) in (or_introl (ex3_2 T T (\lambda (u3: T).(\lambda +(t3: T).(eq T (THead (Bind Void) u2 t2) (THead (Bind Void) u3 t3)))) (\lambda +(u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 +t1 t3)))) (pr0 t1 (lift (S O) O (THead (Bind Void) u2 t2))) (ex3_2_intro T T +(\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Void) u2 t2) (THead +(Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda +(_: T).(\lambda (t3: T).(pr0 t1 t3))) u2 t2 (refl_equal T (THead (Bind Void) +u2 t2)) H14 H12)))))) k H10)))) H7)) H6)))))))))))) (\lambda (u: T).(\lambda +(v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 +(THead (Bind Void) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: +T).(eq T v2 (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: +T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 +(lift (S O) O v2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 +t2)).(\lambda (_: (((eq T t0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: +T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (H5: (eq T (THead +(Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Bind Void) u1 t1))).(let H6 +\def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (ee: +T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow +False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K +return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) +\Rightarrow True])])) I (THead (Bind Void) u1 t1) H5) in (False_ind (or +(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) +(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) +(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O (THead +(Bind Abbr) v2 t2)))) H6)))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B +b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 +v2)).(\lambda (_: (((eq T v1 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T +(\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind Void) u2 t2)))) +(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: +T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O v2)))))).(\lambda (u0: T).(\lambda +(u2: T).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (((eq T u0 (THead (Bind Void) +u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead +(Bind Void) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda +(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O +u2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda +(_: (((eq T t0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: +T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u3 t3)))) (\lambda (u3: +T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 +t3)))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (H8: (eq T (THead (Flat Appl) +v1 (THead (Bind b) u0 t0)) (THead (Bind Void) u1 t1))).(let H9 \def (eq_ind T +(THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (\lambda (ee: T).(match ee in T +return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda +(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow +True])])) I (THead (Bind Void) u1 t1) H8) in (False_ind (or (ex3_2 T T +(\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) +(lift (S O) O v2) t2)) (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda +(_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 +(lift (S O) O (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2))))) +H9))))))))))))))))) (\lambda (u0: T).(\lambda (u2: T).(\lambda (_: (pr0 u0 +u2)).(\lambda (_: (((eq T u0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T +(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Bind Void) u3 t2)))) +(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: +T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O u2)))))).(\lambda (t0: T).(\lambda +(t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Void) +u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead +(Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda +(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O +t2)))))).(\lambda (w: T).(\lambda (_: (subst0 O u2 t2 w)).(\lambda (H6: (eq T +(THead (Bind Abbr) u0 t0) (THead (Bind Void) u1 t1))).(let H7 \def (eq_ind T +(THead (Bind Abbr) u0 t0) (\lambda (ee: T).(match ee in T return (\lambda (_: +T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | +(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with +[(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr +\Rightarrow True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat +_) \Rightarrow False])])) I (THead (Bind Void) u1 t1) H6) in (False_ind (or +(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 w) +(THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) +(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O (THead +(Bind Abbr) u2 w)))) H7))))))))))))) (\lambda (b: B).(\lambda (H1: (not (eq B +b Abst))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H2: (pr0 t0 +t2)).(\lambda (H3: (((eq T t0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: +T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (u: T).(\lambda +(H4: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Void) u1 +t1))).(let H5 \def (f_equal T B (\lambda (e: T).(match e in T return (\lambda +(_: T).B) with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead k +_ _) \Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b0) +\Rightarrow b0 | (Flat _) \Rightarrow b])])) (THead (Bind b) u (lift (S O) O +t0)) (THead (Bind Void) u1 t1) H4) in ((let H6 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | +(TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t])) (THead (Bind b) u +(lift (S O) O t0)) (THead (Bind Void) u1 t1) H4) in ((let H7 \def (f_equal T +T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T +\def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow +(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) +| (THead k u0 t3) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) +t3))]) in lref_map) (\lambda (x0: nat).(plus x0 (S O))) O t0) | (TLRef _) +\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T +\def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow +(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) +| (THead k u0 t3) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) +t3))]) in lref_map) (\lambda (x0: nat).(plus x0 (S O))) O t0) | (THead _ _ t) +\Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Void) u1 +t1) H4) in (\lambda (_: (eq T u u1)).(\lambda (H9: (eq B b Void)).(let H10 +\def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H1 Void H9) in (let +H11 \def (eq_ind_r T t1 (\lambda (t: T).((eq T t0 (THead (Bind Void) u1 t)) +\to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind +Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(pr0 t t3)))) (pr0 t (lift (S O) O t2))))) H3 (lift (S O) +O t0) H7) in (eq_ind T (lift (S O) O t0) (\lambda (t: T).(or (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: +T).(pr0 t t3)))) (pr0 t (lift (S O) O t2)))) (or_intror (ex3_2 T T (\lambda +(u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 (lift +(S O) O t0) t3)))) (pr0 (lift (S O) O t0) (lift (S O) O t2)) (pr0_lift t0 t2 +H2 (S O) O)) t1 H7)))))) H6)) H5)))))))))) (\lambda (t0: T).(\lambda (t2: +T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Void) u1 +t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead +(Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda +(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O +t2)))))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u t0) (THead +(Bind Void) u1 t1))).(let H4 \def (eq_ind T (THead (Flat Cast) u t0) (\lambda +(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) +\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow +(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | +(Flat _) \Rightarrow True])])) I (THead (Bind Void) u1 t1) H3) in (False_ind +(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) +u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2))) H4)))))))) y x +H0))) H)))). +(* COMMENTS +Initial nodes: 3436 +END *) + +theorem pr0_gen_lift: + \forall (t1: T).(\forall (x: T).(\forall (h: nat).(\forall (d: nat).((pr0 +(lift h d t1) x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda +(t2: T).(pr0 t1 t2))))))) +\def + \lambda (t1: T).(\lambda (x: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda +(H: (pr0 (lift h d t1) x)).(insert_eq T (lift h d t1) (\lambda (t: T).(pr0 t +x)) (\lambda (_: T).(ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda +(t2: T).(pr0 t1 t2)))) (\lambda (y: T).(\lambda (H0: (pr0 y x)).(unintro nat +d (\lambda (n: nat).((eq T y (lift h n t1)) \to (ex2 T (\lambda (t2: T).(eq T +x (lift h n t2))) (\lambda (t2: T).(pr0 t1 t2))))) (unintro T t1 (\lambda (t: +T).(\forall (x0: nat).((eq T y (lift h x0 t)) \to (ex2 T (\lambda (t2: T).(eq +T x (lift h x0 t2))) (\lambda (t2: T).(pr0 t t2)))))) (pr0_ind (\lambda (t: +T).(\lambda (t0: T).(\forall (x0: T).(\forall (x1: nat).((eq T t (lift h x1 +x0)) \to (ex2 T (\lambda (t2: T).(eq T t0 (lift h x1 t2))) (\lambda (t2: +T).(pr0 x0 t2)))))))) (\lambda (t: T).(\lambda (x0: T).(\lambda (x1: +nat).(\lambda (H1: (eq T t (lift h x1 x0))).(ex_intro2 T (\lambda (t2: T).(eq +T t (lift h x1 t2))) (\lambda (t2: T).(pr0 x0 t2)) x0 H1 (pr0_refl x0)))))) +(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda (H2: +((\forall (x0: T).(\forall (x1: nat).((eq T u1 (lift h x1 x0)) \to (ex2 T +(\lambda (t2: T).(eq T u2 (lift h x1 t2))) (\lambda (t2: T).(pr0 x0 +t2)))))))).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2 +t3)).(\lambda (H4: ((\forall (x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 +x0)) \to (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: +T).(pr0 x0 t4)))))))).(\lambda (k: K).(\lambda (x0: T).(\lambda (x1: +nat).(\lambda (H5: (eq T (THead k u1 t2) (lift h x1 x0))).(K_ind (\lambda +(k0: K).((eq T (THead k0 u1 t2) (lift h x1 x0)) \to (ex2 T (\lambda (t4: +T).(eq T (THead k0 u2 t3) (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4))))) +(\lambda (b: B).(\lambda (H6: (eq T (THead (Bind b) u1 t2) (lift h x1 +x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Bind +b) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u1 (lift h x1 y0)))) +(\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h (S x1) z)))) (ex2 T (\lambda +(t4: T).(eq T (THead (Bind b) u2 t3) (lift h x1 t4))) (\lambda (t4: T).(pr0 +x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq T x0 (THead +(Bind b) x2 x3))).(\lambda (H8: (eq T u1 (lift h x1 x2))).(\lambda (H9: (eq T +t2 (lift h (S x1) x3))).(eq_ind_r T (THead (Bind b) x2 x3) (\lambda (t: +T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 t3) (lift h x1 t4))) +(\lambda (t4: T).(pr0 t t4)))) (ex2_ind T (\lambda (t4: T).(eq T t3 (lift h +(S x1) t4))) (\lambda (t4: T).(pr0 x3 t4)) (ex2 T (\lambda (t4: T).(eq T +(THead (Bind b) u2 t3) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind b) +x2 x3) t4))) (\lambda (x4: T).(\lambda (H_x: (eq T t3 (lift h (S x1) +x4))).(\lambda (H10: (pr0 x3 x4)).(eq_ind_r T (lift h (S x1) x4) (\lambda (t: +T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 t) (lift h x1 t4))) +(\lambda (t4: T).(pr0 (THead (Bind b) x2 x3) t4)))) (ex2_ind T (\lambda (t4: +T).(eq T u2 (lift h x1 t4))) (\lambda (t4: T).(pr0 x2 t4)) (ex2 T (\lambda +(t4: T).(eq T (THead (Bind b) u2 (lift h (S x1) x4)) (lift h x1 t4))) +(\lambda (t4: T).(pr0 (THead (Bind b) x2 x3) t4))) (\lambda (x5: T).(\lambda +(H_x0: (eq T u2 (lift h x1 x5))).(\lambda (H11: (pr0 x2 x5)).(eq_ind_r T +(lift h x1 x5) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) +t (lift h (S x1) x4)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind b) +x2 x3) t4)))) (ex_intro2 T (\lambda (t4: T).(eq T (THead (Bind b) (lift h x1 +x5) (lift h (S x1) x4)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind +b) x2 x3) t4)) (THead (Bind b) x5 x4) (sym_eq T (lift h x1 (THead (Bind b) x5 +x4)) (THead (Bind b) (lift h x1 x5) (lift h (S x1) x4)) (lift_bind b x5 x4 h +x1)) (pr0_comp x2 x5 H11 x3 x4 H10 (Bind b))) u2 H_x0)))) (H2 x2 x1 H8)) t3 +H_x)))) (H4 x3 (S x1) H9)) x0 H7)))))) (lift_gen_bind b u1 t2 x0 h x1 H6)))) +(\lambda (f: F).(\lambda (H6: (eq T (THead (Flat f) u1 t2) (lift h x1 +x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Flat +f) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u1 (lift h x1 y0)))) +(\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h x1 z)))) (ex2 T (\lambda +(t4: T).(eq T (THead (Flat f) u2 t3) (lift h x1 t4))) (\lambda (t4: T).(pr0 +x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq T x0 (THead +(Flat f) x2 x3))).(\lambda (H8: (eq T u1 (lift h x1 x2))).(\lambda (H9: (eq T +t2 (lift h x1 x3))).(eq_ind_r T (THead (Flat f) x2 x3) (\lambda (t: T).(ex2 T +(\lambda (t4: T).(eq T (THead (Flat f) u2 t3) (lift h x1 t4))) (\lambda (t4: +T).(pr0 t t4)))) (ex2_ind T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) +(\lambda (t4: T).(pr0 x3 t4)) (ex2 T (\lambda (t4: T).(eq T (THead (Flat f) +u2 t3) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat f) x2 x3) t4))) +(\lambda (x4: T).(\lambda (H_x: (eq T t3 (lift h x1 x4))).(\lambda (H10: (pr0 +x3 x4)).(eq_ind_r T (lift h x1 x4) (\lambda (t: T).(ex2 T (\lambda (t4: +T).(eq T (THead (Flat f) u2 t) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead +(Flat f) x2 x3) t4)))) (ex2_ind T (\lambda (t4: T).(eq T u2 (lift h x1 t4))) +(\lambda (t4: T).(pr0 x2 t4)) (ex2 T (\lambda (t4: T).(eq T (THead (Flat f) +u2 (lift h x1 x4)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat f) x2 +x3) t4))) (\lambda (x5: T).(\lambda (H_x0: (eq T u2 (lift h x1 x5))).(\lambda +(H11: (pr0 x2 x5)).(eq_ind_r T (lift h x1 x5) (\lambda (t: T).(ex2 T (\lambda +(t4: T).(eq T (THead (Flat f) t (lift h x1 x4)) (lift h x1 t4))) (\lambda +(t4: T).(pr0 (THead (Flat f) x2 x3) t4)))) (ex_intro2 T (\lambda (t4: T).(eq +T (THead (Flat f) (lift h x1 x5) (lift h x1 x4)) (lift h x1 t4))) (\lambda +(t4: T).(pr0 (THead (Flat f) x2 x3) t4)) (THead (Flat f) x5 x4) (sym_eq T +(lift h x1 (THead (Flat f) x5 x4)) (THead (Flat f) (lift h x1 x5) (lift h x1 +x4)) (lift_flat f x5 x4 h x1)) (pr0_comp x2 x5 H11 x3 x4 H10 (Flat f))) u2 +H_x0)))) (H2 x2 x1 H8)) t3 H_x)))) (H4 x3 x1 H9)) x0 H7)))))) (lift_gen_flat +f u1 t2 x0 h x1 H6)))) k H5))))))))))))) (\lambda (u: T).(\lambda (v1: +T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (H2: ((\forall (x0: +T).(\forall (x1: nat).((eq T v1 (lift h x1 x0)) \to (ex2 T (\lambda (t2: +T).(eq T v2 (lift h x1 t2))) (\lambda (t2: T).(pr0 x0 t2)))))))).(\lambda +(t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2 t3)).(\lambda (H4: ((\forall +(x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to (ex2 T (\lambda (t4: +T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4)))))))).(\lambda +(x0: T).(\lambda (x1: nat).(\lambda (H5: (eq T (THead (Flat Appl) v1 (THead +(Bind Abst) u t2)) (lift h x1 x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda +(z: T).(eq T x0 (THead (Flat Appl) y0 z)))) (\lambda (y0: T).(\lambda (_: +T).(eq T v1 (lift h x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T (THead +(Bind Abst) u t2) (lift h x1 z)))) (ex2 T (\lambda (t4: T).(eq T (THead (Bind +Abbr) v2 t3) (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4))) (\lambda (x2: +T).(\lambda (x3: T).(\lambda (H6: (eq T x0 (THead (Flat Appl) x2 +x3))).(\lambda (H7: (eq T v1 (lift h x1 x2))).(\lambda (H8: (eq T (THead +(Bind Abst) u t2) (lift h x1 x3))).(eq_ind_r T (THead (Flat Appl) x2 x3) +(\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) v2 t3) (lift +h x1 t4))) (\lambda (t4: T).(pr0 t t4)))) (ex3_2_ind T T (\lambda (y0: +T).(\lambda (z: T).(eq T x3 (THead (Bind Abst) y0 z)))) (\lambda (y0: +T).(\lambda (_: T).(eq T u (lift h x1 y0)))) (\lambda (_: T).(\lambda (z: +T).(eq T t2 (lift h (S x1) z)))) (ex2 T (\lambda (t4: T).(eq T (THead (Bind +Abbr) v2 t3) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 x3) +t4))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H9: (eq T x3 (THead (Bind +Abst) x4 x5))).(\lambda (_: (eq T u (lift h x1 x4))).(\lambda (H11: (eq T t2 +(lift h (S x1) x5))).(eq_ind_r T (THead (Bind Abst) x4 x5) (\lambda (t: +T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) v2 t3) (lift h x1 t4))) +(\lambda (t4: T).(pr0 (THead (Flat Appl) x2 t) t4)))) (ex2_ind T (\lambda +(t4: T).(eq T t3 (lift h (S x1) t4))) (\lambda (t4: T).(pr0 x5 t4)) (ex2 T +(\lambda (t4: T).(eq T (THead (Bind Abbr) v2 t3) (lift h x1 t4))) (\lambda +(t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind Abst) x4 x5)) t4))) (\lambda +(x6: T).(\lambda (H_x: (eq T t3 (lift h (S x1) x6))).(\lambda (H12: (pr0 x5 +x6)).(eq_ind_r T (lift h (S x1) x6) (\lambda (t: T).(ex2 T (\lambda (t4: +T).(eq T (THead (Bind Abbr) v2 t) (lift h x1 t4))) (\lambda (t4: T).(pr0 +(THead (Flat Appl) x2 (THead (Bind Abst) x4 x5)) t4)))) (ex2_ind T (\lambda +(t4: T).(eq T v2 (lift h x1 t4))) (\lambda (t4: T).(pr0 x2 t4)) (ex2 T +(\lambda (t4: T).(eq T (THead (Bind Abbr) v2 (lift h (S x1) x6)) (lift h x1 +t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind Abst) x4 x5)) +t4))) (\lambda (x7: T).(\lambda (H_x0: (eq T v2 (lift h x1 x7))).(\lambda +(H13: (pr0 x2 x7)).(eq_ind_r T (lift h x1 x7) (\lambda (t: T).(ex2 T (\lambda +(t4: T).(eq T (THead (Bind Abbr) t (lift h (S x1) x6)) (lift h x1 t4))) +(\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind Abst) x4 x5)) t4)))) +(ex_intro2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) (lift h x1 x7) (lift h +(S x1) x6)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 +(THead (Bind Abst) x4 x5)) t4)) (THead (Bind Abbr) x7 x6) (sym_eq T (lift h +x1 (THead (Bind Abbr) x7 x6)) (THead (Bind Abbr) (lift h x1 x7) (lift h (S +x1) x6)) (lift_bind Abbr x7 x6 h x1)) (pr0_beta x4 x2 x7 H13 x5 x6 H12)) v2 +H_x0)))) (H2 x2 x1 H7)) t3 H_x)))) (H4 x5 (S x1) H11)) x3 H9)))))) +(lift_gen_bind Abst u t2 x3 h x1 H8)) x0 H6)))))) (lift_gen_flat Appl v1 +(THead (Bind Abst) u t2) x0 h x1 H5)))))))))))))) (\lambda (b: B).(\lambda +(H1: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 +v1 v2)).(\lambda (H3: ((\forall (x0: T).(\forall (x1: nat).((eq T v1 (lift h +x1 x0)) \to (ex2 T (\lambda (t2: T).(eq T v2 (lift h x1 t2))) (\lambda (t2: +T).(pr0 x0 t2)))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 +u2)).(\lambda (H5: ((\forall (x0: T).(\forall (x1: nat).((eq T u1 (lift h x1 +x0)) \to (ex2 T (\lambda (t2: T).(eq T u2 (lift h x1 t2))) (\lambda (t2: +T).(pr0 x0 t2)))))))).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2 +t3)).(\lambda (H7: ((\forall (x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 +x0)) \to (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: +T).(pr0 x0 t4)))))))).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H8: (eq T +(THead (Flat Appl) v1 (THead (Bind b) u1 t2)) (lift h x1 x0))).(ex3_2_ind T T +(\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Flat Appl) y0 z)))) +(\lambda (y0: T).(\lambda (_: T).(eq T v1 (lift h x1 y0)))) (\lambda (_: +T).(\lambda (z: T).(eq T (THead (Bind b) u1 t2) (lift h x1 z)))) (ex2 T +(\lambda (t4: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O +v2) t3)) (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4))) (\lambda (x2: +T).(\lambda (x3: T).(\lambda (H9: (eq T x0 (THead (Flat Appl) x2 +x3))).(\lambda (H10: (eq T v1 (lift h x1 x2))).(\lambda (H11: (eq T (THead +(Bind b) u1 t2) (lift h x1 x3))).(eq_ind_r T (THead (Flat Appl) x2 x3) +(\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 (THead (Flat +Appl) (lift (S O) O v2) t3)) (lift h x1 t4))) (\lambda (t4: T).(pr0 t t4)))) +(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x3 (THead (Bind b) y0 +z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u1 (lift h x1 y0)))) (\lambda +(_: T).(\lambda (z: T).(eq T t2 (lift h (S x1) z)))) (ex2 T (\lambda (t4: +T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) (lift h +x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 x3) t4))) (\lambda (x4: +T).(\lambda (x5: T).(\lambda (H12: (eq T x3 (THead (Bind b) x4 x5))).(\lambda +(H13: (eq T u1 (lift h x1 x4))).(\lambda (H14: (eq T t2 (lift h (S x1) +x5))).(eq_ind_r T (THead (Bind b) x4 x5) (\lambda (t: T).(ex2 T (\lambda (t4: +T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) (lift h +x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 t) t4)))) (ex2_ind T +(\lambda (t4: T).(eq T t3 (lift h (S x1) t4))) (\lambda (t4: T).(pr0 x5 t4)) +(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S +O) O v2) t3)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 +(THead (Bind b) x4 x5)) t4))) (\lambda (x6: T).(\lambda (H_x: (eq T t3 (lift +h (S x1) x6))).(\lambda (H15: (pr0 x5 x6)).(eq_ind_r T (lift h (S x1) x6) +(\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 (THead (Flat +Appl) (lift (S O) O v2) t)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead +(Flat Appl) x2 (THead (Bind b) x4 x5)) t4)))) (ex2_ind T (\lambda (t4: T).(eq +T u2 (lift h x1 t4))) (\lambda (t4: T).(pr0 x4 t4)) (ex2 T (\lambda (t4: +T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (lift h (S +x1) x6))) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead +(Bind b) x4 x5)) t4))) (\lambda (x7: T).(\lambda (H_x0: (eq T u2 (lift h x1 +x7))).(\lambda (H16: (pr0 x4 x7)).(eq_ind_r T (lift h x1 x7) (\lambda (t: +T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) t (THead (Flat Appl) (lift +(S O) O v2) (lift h (S x1) x6))) (lift h x1 t4))) (\lambda (t4: T).(pr0 +(THead (Flat Appl) x2 (THead (Bind b) x4 x5)) t4)))) (ex2_ind T (\lambda (t4: +T).(eq T v2 (lift h x1 t4))) (\lambda (t4: T).(pr0 x2 t4)) (ex2 T (\lambda +(t4: T).(eq T (THead (Bind b) (lift h x1 x7) (THead (Flat Appl) (lift (S O) O +v2) (lift h (S x1) x6))) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat +Appl) x2 (THead (Bind b) x4 x5)) t4))) (\lambda (x8: T).(\lambda (H_x1: (eq T +v2 (lift h x1 x8))).(\lambda (H17: (pr0 x2 x8)).(eq_ind_r T (lift h x1 x8) +(\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) (lift h x1 x7) +(THead (Flat Appl) (lift (S O) O t) (lift h (S x1) x6))) (lift h x1 t4))) +(\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind b) x4 x5)) t4)))) +(eq_ind T (lift h (plus (S O) x1) (lift (S O) O x8)) (\lambda (t: T).(ex2 T +(\lambda (t4: T).(eq T (THead (Bind b) (lift h x1 x7) (THead (Flat Appl) t +(lift h (S x1) x6))) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat +Appl) x2 (THead (Bind b) x4 x5)) t4)))) (eq_ind T (lift h (S x1) (THead (Flat +Appl) (lift (S O) O x8) x6)) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T +(THead (Bind b) (lift h x1 x7) t) (lift h x1 t4))) (\lambda (t4: T).(pr0 +(THead (Flat Appl) x2 (THead (Bind b) x4 x5)) t4)))) (ex_intro2 T (\lambda +(t4: T).(eq T (THead (Bind b) (lift h x1 x7) (lift h (S x1) (THead (Flat +Appl) (lift (S O) O x8) x6))) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead +(Flat Appl) x2 (THead (Bind b) x4 x5)) t4)) (THead (Bind b) x7 (THead (Flat +Appl) (lift (S O) O x8) x6)) (sym_eq T (lift h x1 (THead (Bind b) x7 (THead +(Flat Appl) (lift (S O) O x8) x6))) (THead (Bind b) (lift h x1 x7) (lift h (S +x1) (THead (Flat Appl) (lift (S O) O x8) x6))) (lift_bind b x7 (THead (Flat +Appl) (lift (S O) O x8) x6) h x1)) (pr0_upsilon b H1 x2 x8 H17 x4 x7 H16 x5 +x6 H15)) (THead (Flat Appl) (lift h (S x1) (lift (S O) O x8)) (lift h (S x1) +x6)) (lift_flat Appl (lift (S O) O x8) x6 h (S x1))) (lift (S O) O (lift h x1 +x8)) (lift_d x8 h (S O) x1 O (le_O_n x1))) v2 H_x1)))) (H3 x2 x1 H10)) u2 +H_x0)))) (H5 x4 x1 H13)) t3 H_x)))) (H7 x5 (S x1) H14)) x3 H12)))))) +(lift_gen_bind b u1 t2 x3 h x1 H11)) x0 H9)))))) (lift_gen_flat Appl v1 +(THead (Bind b) u1 t2) x0 h x1 H8))))))))))))))))))) (\lambda (u1: +T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda (H2: ((\forall (x0: +T).(\forall (x1: nat).((eq T u1 (lift h x1 x0)) \to (ex2 T (\lambda (t2: +T).(eq T u2 (lift h x1 t2))) (\lambda (t2: T).(pr0 x0 t2)))))))).(\lambda +(t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2 t3)).(\lambda (H4: ((\forall +(x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to (ex2 T (\lambda (t4: +T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4)))))))).(\lambda (w: +T).(\lambda (H5: (subst0 O u2 t3 w)).(\lambda (x0: T).(\lambda (x1: +nat).(\lambda (H6: (eq T (THead (Bind Abbr) u1 t2) (lift h x1 +x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Bind +Abbr) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u1 (lift h x1 y0)))) +(\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h (S x1) z)))) (ex2 T (\lambda +(t4: T).(eq T (THead (Bind Abbr) u2 w) (lift h x1 t4))) (\lambda (t4: T).(pr0 +x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq T x0 (THead +(Bind Abbr) x2 x3))).(\lambda (H8: (eq T u1 (lift h x1 x2))).(\lambda (H9: +(eq T t2 (lift h (S x1) x3))).(eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda +(t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) u2 w) (lift h x1 +t4))) (\lambda (t4: T).(pr0 t t4)))) (ex2_ind T (\lambda (t4: T).(eq T t3 +(lift h (S x1) t4))) (\lambda (t4: T).(pr0 x3 t4)) (ex2 T (\lambda (t4: +T).(eq T (THead (Bind Abbr) u2 w) (lift h x1 t4))) (\lambda (t4: T).(pr0 +(THead (Bind Abbr) x2 x3) t4))) (\lambda (x4: T).(\lambda (H_x: (eq T t3 +(lift h (S x1) x4))).(\lambda (H10: (pr0 x3 x4)).(let H11 \def (eq_ind T t3 +(\lambda (t: T).(subst0 O u2 t w)) H5 (lift h (S x1) x4) H_x) in (ex2_ind T +(\lambda (t4: T).(eq T u2 (lift h x1 t4))) (\lambda (t4: T).(pr0 x2 t4)) (ex2 +T (\lambda (t4: T).(eq T (THead (Bind Abbr) u2 w) (lift h x1 t4))) (\lambda +(t4: T).(pr0 (THead (Bind Abbr) x2 x3) t4))) (\lambda (x5: T).(\lambda (H_x0: +(eq T u2 (lift h x1 x5))).(\lambda (H12: (pr0 x2 x5)).(eq_ind_r T (lift h x1 +x5) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) t w) +(lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind Abbr) x2 x3) t4)))) (let +H13 \def (eq_ind T u2 (\lambda (t: T).(subst0 O t (lift h (S x1) x4) w)) H11 +(lift h x1 x5) H_x0) in (let H14 \def (refl_equal nat (S (plus O x1))) in +(let H15 \def (eq_ind nat (S x1) (\lambda (n: nat).(subst0 O (lift h x1 x5) +(lift h n x4) w)) H13 (S (plus O x1)) H14) in (ex2_ind T (\lambda (t4: T).(eq +T w (lift h (S (plus O x1)) t4))) (\lambda (t4: T).(subst0 O x5 x4 t4)) (ex2 +T (\lambda (t4: T).(eq T (THead (Bind Abbr) (lift h x1 x5) w) (lift h x1 +t4))) (\lambda (t4: T).(pr0 (THead (Bind Abbr) x2 x3) t4))) (\lambda (x6: +T).(\lambda (H16: (eq T w (lift h (S (plus O x1)) x6))).(\lambda (H17: +(subst0 O x5 x4 x6)).(eq_ind_r T (lift h (S (plus O x1)) x6) (\lambda (t: +T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) (lift h x1 x5) t) (lift h +x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind Abbr) x2 x3) t4)))) (ex_intro2 T +(\lambda (t4: T).(eq T (THead (Bind Abbr) (lift h x1 x5) (lift h (S (plus O +x1)) x6)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind Abbr) x2 x3) +t4)) (THead (Bind Abbr) x5 x6) (sym_eq T (lift h x1 (THead (Bind Abbr) x5 +x6)) (THead (Bind Abbr) (lift h x1 x5) (lift h (S (plus O x1)) x6)) +(lift_bind Abbr x5 x6 h (plus O x1))) (pr0_delta x2 x5 H12 x3 x4 H10 x6 H17)) +w H16)))) (subst0_gen_lift_lt x5 x4 w O h x1 H15))))) u2 H_x0)))) (H2 x2 x1 +H8)))))) (H4 x3 (S x1) H9)) x0 H7)))))) (lift_gen_bind Abbr u1 t2 x0 h x1 +H6))))))))))))))) (\lambda (b: B).(\lambda (H1: (not (eq B b Abst))).(\lambda +(t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2 t3)).(\lambda (H3: ((\forall +(x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to (ex2 T (\lambda (t4: +T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4)))))))).(\lambda (u: +T).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H4: (eq T (THead (Bind b) u +(lift (S O) O t2)) (lift h x1 x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda +(z: T).(eq T x0 (THead (Bind b) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq +T u (lift h x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift (S O) O t2) +(lift h (S x1) z)))) (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) +(\lambda (t4: T).(pr0 x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda +(H5: (eq T x0 (THead (Bind b) x2 x3))).(\lambda (_: (eq T u (lift h x1 +x2))).(\lambda (H7: (eq T (lift (S O) O t2) (lift h (S x1) x3))).(eq_ind_r T +(THead (Bind b) x2 x3) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T t3 (lift +h x1 t4))) (\lambda (t4: T).(pr0 t t4)))) (let H8 \def (eq_ind_r nat (plus (S +O) x1) (\lambda (n: nat).(eq nat (S x1) n)) (refl_equal nat (plus (S O) x1)) +(plus x1 (S O)) (plus_sym x1 (S O))) in (let H9 \def (eq_ind nat (S x1) +(\lambda (n: nat).(eq T (lift (S O) O t2) (lift h n x3))) H7 (plus x1 (S O)) +H8) in (ex2_ind T (\lambda (t4: T).(eq T x3 (lift (S O) O t4))) (\lambda (t4: +T).(eq T t2 (lift h x1 t4))) (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 +t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 x3) t4))) (\lambda (x4: +T).(\lambda (H10: (eq T x3 (lift (S O) O x4))).(\lambda (H11: (eq T t2 (lift +h x1 x4))).(eq_ind_r T (lift (S O) O x4) (\lambda (t: T).(ex2 T (\lambda (t4: +T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 t) +t4)))) (ex2_ind T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: +T).(pr0 x4 t4)) (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda +(t4: T).(pr0 (THead (Bind b) x2 (lift (S O) O x4)) t4))) (\lambda (x5: +T).(\lambda (H_x: (eq T t3 (lift h x1 x5))).(\lambda (H12: (pr0 x4 +x5)).(eq_ind_r T (lift h x1 x5) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T +t (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 (lift (S O) O +x4)) t4)))) (ex_intro2 T (\lambda (t4: T).(eq T (lift h x1 x5) (lift h x1 +t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 (lift (S O) O x4)) t4)) x5 +(refl_equal T (lift h x1 x5)) (pr0_zeta b H1 x4 x5 H12 x2)) t3 H_x)))) (H3 x4 +x1 H11)) x3 H10)))) (lift_gen_lift t2 x3 (S O) h O x1 (le_O_n x1) H9)))) x0 +H5)))))) (lift_gen_bind b u (lift (S O) O t2) x0 h x1 H4)))))))))))) (\lambda +(t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2 t3)).(\lambda (H2: ((\forall +(x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to (ex2 T (\lambda (t4: +T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4)))))))).(\lambda (u: +T).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H3: (eq T (THead (Flat Cast) +u t2) (lift h x1 x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T +x0 (THead (Flat Cast) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u (lift +h x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h x1 z)))) (ex2 T +(\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4))) +(\lambda (x2: T).(\lambda (x3: T).(\lambda (H4: (eq T x0 (THead (Flat Cast) +x2 x3))).(\lambda (_: (eq T u (lift h x1 x2))).(\lambda (H6: (eq T t2 (lift h +x1 x3))).(eq_ind_r T (THead (Flat Cast) x2 x3) (\lambda (t: T).(ex2 T +(\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 t t4)))) +(ex2_ind T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 +x3 t4)) (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: +T).(pr0 (THead (Flat Cast) x2 x3) t4))) (\lambda (x4: T).(\lambda (H_x: (eq T +t3 (lift h x1 x4))).(\lambda (H7: (pr0 x3 x4)).(eq_ind_r T (lift h x1 x4) +(\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T t (lift h x1 t4))) (\lambda +(t4: T).(pr0 (THead (Flat Cast) x2 x3) t4)))) (ex_intro2 T (\lambda (t4: +T).(eq T (lift h x1 x4) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat +Cast) x2 x3) t4)) x4 (refl_equal T (lift h x1 x4)) (pr0_tau x3 x4 H7 x2)) t3 +H_x)))) (H2 x3 x1 H6)) x0 H4)))))) (lift_gen_flat Cast u t2 x0 h x1 +H3)))))))))) y x H0))))) H))))). +(* COMMENTS +Initial nodes: 7569 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr0/pr0.ma b/matita/matita/contribs/lambdadelta/basic_1/pr0/pr0.ma new file mode 100644 index 000000000..9a3b397fe --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/pr0/pr0.ma @@ -0,0 +1,2507 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/pr0/fwd.ma". + +include "Basic-1/lift/tlt.ma". + +theorem pr0_confluence__pr0_cong_upsilon_refl: + \forall (b: B).((not (eq B b Abst)) \to (\forall (u0: T).(\forall (u3: +T).((pr0 u0 u3) \to (\forall (t4: T).(\forall (t5: T).((pr0 t4 t5) \to +(\forall (u2: T).(\forall (v2: T).(\forall (x: T).((pr0 u2 x) \to ((pr0 v2 x) +\to (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u0 t4)) +t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O +v2) t5)) t))))))))))))))) +\def + \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (u0: T).(\lambda +(u3: T).(\lambda (H0: (pr0 u0 u3)).(\lambda (t4: T).(\lambda (t5: T).(\lambda +(H1: (pr0 t4 t5)).(\lambda (u2: T).(\lambda (v2: T).(\lambda (x: T).(\lambda +(H2: (pr0 u2 x)).(\lambda (H3: (pr0 v2 x)).(ex_intro2 T (\lambda (t: T).(pr0 +(THead (Flat Appl) u2 (THead (Bind b) u0 t4)) t)) (\lambda (t: T).(pr0 (THead +(Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t)) (THead (Bind b) u3 +(THead (Flat Appl) (lift (S O) O x) t5)) (pr0_upsilon b H u2 x H2 u0 u3 H0 t4 +t5 H1) (pr0_comp u3 u3 (pr0_refl u3) (THead (Flat Appl) (lift (S O) O v2) t5) +(THead (Flat Appl) (lift (S O) O x) t5) (pr0_comp (lift (S O) O v2) (lift (S +O) O x) (pr0_lift v2 x H3 (S O) O) t5 t5 (pr0_refl t5) (Flat Appl)) (Bind +b))))))))))))))). +(* COMMENTS +Initial nodes: 257 +END *) + +theorem pr0_confluence__pr0_cong_upsilon_cong: + \forall (b: B).((not (eq B b Abst)) \to (\forall (u2: T).(\forall (v2: +T).(\forall (x: T).((pr0 u2 x) \to ((pr0 v2 x) \to (\forall (t2: T).(\forall +(t5: T).(\forall (x0: T).((pr0 t2 x0) \to ((pr0 t5 x0) \to (\forall (u5: +T).(\forall (u3: T).(\forall (x1: T).((pr0 u5 x1) \to ((pr0 u3 x1) \to (ex2 T +(\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 t2)) t)) +(\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) +t5)) t))))))))))))))))))) +\def + \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (u2: T).(\lambda +(v2: T).(\lambda (x: T).(\lambda (H0: (pr0 u2 x)).(\lambda (H1: (pr0 v2 +x)).(\lambda (t2: T).(\lambda (t5: T).(\lambda (x0: T).(\lambda (H2: (pr0 t2 +x0)).(\lambda (H3: (pr0 t5 x0)).(\lambda (u5: T).(\lambda (u3: T).(\lambda +(x1: T).(\lambda (H4: (pr0 u5 x1)).(\lambda (H5: (pr0 u3 x1)).(ex_intro2 T +(\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 t2)) t)) +(\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) +t5)) t)) (THead (Bind b) x1 (THead (Flat Appl) (lift (S O) O x) x0)) +(pr0_upsilon b H u2 x H0 u5 x1 H4 t2 x0 H2) (pr0_comp u3 x1 H5 (THead (Flat +Appl) (lift (S O) O v2) t5) (THead (Flat Appl) (lift (S O) O x) x0) (pr0_comp +(lift (S O) O v2) (lift (S O) O x) (pr0_lift v2 x H1 (S O) O) t5 x0 H3 (Flat +Appl)) (Bind b))))))))))))))))))). +(* COMMENTS +Initial nodes: 269 +END *) + +theorem pr0_confluence__pr0_cong_upsilon_delta: + (not (eq B Abbr Abst)) \to (\forall (u5: T).(\forall (t2: T).(\forall (w: +T).((subst0 O u5 t2 w) \to (\forall (u2: T).(\forall (v2: T).(\forall (x: +T).((pr0 u2 x) \to ((pr0 v2 x) \to (\forall (t5: T).(\forall (x0: T).((pr0 t2 +x0) \to ((pr0 t5 x0) \to (\forall (u3: T).(\forall (x1: T).((pr0 u5 x1) \to +((pr0 u3 x1) \to (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead +(Bind Abbr) u5 w)) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 (THead +(Flat Appl) (lift (S O) O v2) t5)) t)))))))))))))))))))) +\def + \lambda (H: (not (eq B Abbr Abst))).(\lambda (u5: T).(\lambda (t2: +T).(\lambda (w: T).(\lambda (H0: (subst0 O u5 t2 w)).(\lambda (u2: +T).(\lambda (v2: T).(\lambda (x: T).(\lambda (H1: (pr0 u2 x)).(\lambda (H2: +(pr0 v2 x)).(\lambda (t5: T).(\lambda (x0: T).(\lambda (H3: (pr0 t2 +x0)).(\lambda (H4: (pr0 t5 x0)).(\lambda (u3: T).(\lambda (x1: T).(\lambda +(H5: (pr0 u5 x1)).(\lambda (H6: (pr0 u3 x1)).(or_ind (pr0 w x0) (ex2 T +(\lambda (w2: T).(pr0 w w2)) (\lambda (w2: T).(subst0 O x1 x0 w2))) (ex2 T +(\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abbr) u5 w)) t)) +(\lambda (t: T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O +v2) t5)) t))) (\lambda (H7: (pr0 w x0)).(ex_intro2 T (\lambda (t: T).(pr0 +(THead (Flat Appl) u2 (THead (Bind Abbr) u5 w)) t)) (\lambda (t: T).(pr0 +(THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t)) (THead +(Bind Abbr) x1 (THead (Flat Appl) (lift (S O) O x) x0)) (pr0_upsilon Abbr H +u2 x H1 u5 x1 H5 w x0 H7) (pr0_comp u3 x1 H6 (THead (Flat Appl) (lift (S O) O +v2) t5) (THead (Flat Appl) (lift (S O) O x) x0) (pr0_comp (lift (S O) O v2) +(lift (S O) O x) (pr0_lift v2 x H2 (S O) O) t5 x0 H4 (Flat Appl)) (Bind +Abbr)))) (\lambda (H7: (ex2 T (\lambda (w2: T).(pr0 w w2)) (\lambda (w2: +T).(subst0 O x1 x0 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 w w2)) (\lambda +(w2: T).(subst0 O x1 x0 w2)) (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) +u2 (THead (Bind Abbr) u5 w)) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 +(THead (Flat Appl) (lift (S O) O v2) t5)) t))) (\lambda (x2: T).(\lambda (H8: +(pr0 w x2)).(\lambda (H9: (subst0 O x1 x0 x2)).(ex_intro2 T (\lambda (t: +T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abbr) u5 w)) t)) (\lambda (t: +T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t)) +(THead (Bind Abbr) x1 (THead (Flat Appl) (lift (S O) O x) x2)) (pr0_upsilon +Abbr H u2 x H1 u5 x1 H5 w x2 H8) (pr0_delta u3 x1 H6 (THead (Flat Appl) (lift +(S O) O v2) t5) (THead (Flat Appl) (lift (S O) O x) x0) (pr0_comp (lift (S O) +O v2) (lift (S O) O x) (pr0_lift v2 x H2 (S O) O) t5 x0 H4 (Flat Appl)) +(THead (Flat Appl) (lift (S O) O x) x2) (subst0_snd (Flat Appl) x1 x2 x0 O H9 +(lift (S O) O x))))))) H7)) (pr0_subst0 t2 x0 H3 u5 w O H0 x1 +H5))))))))))))))))))). +(* COMMENTS +Initial nodes: 769 +END *) + +theorem pr0_confluence__pr0_cong_upsilon_zeta: + \forall (b: B).((not (eq B b Abst)) \to (\forall (u0: T).(\forall (u3: +T).((pr0 u0 u3) \to (\forall (u2: T).(\forall (v2: T).(\forall (x0: T).((pr0 +u2 x0) \to ((pr0 v2 x0) \to (\forall (x: T).(\forall (t3: T).(\forall (x1: +T).((pr0 x x1) \to ((pr0 t3 x1) \to (ex2 T (\lambda (t: T).(pr0 (THead (Flat +Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) +(lift (S O) O v2) (lift (S O) O x))) t))))))))))))))))) +\def + \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (u0: T).(\lambda +(u3: T).(\lambda (_: (pr0 u0 u3)).(\lambda (u2: T).(\lambda (v2: T).(\lambda +(x0: T).(\lambda (H1: (pr0 u2 x0)).(\lambda (H2: (pr0 v2 x0)).(\lambda (x: +T).(\lambda (t3: T).(\lambda (x1: T).(\lambda (H3: (pr0 x x1)).(\lambda (H4: +(pr0 t3 x1)).(eq_ind T (lift (S O) O (THead (Flat Appl) v2 x)) (\lambda (t: +T).(ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3) t0)) (\lambda (t0: +T).(pr0 (THead (Bind b) u3 t) t0)))) (ex_intro2 T (\lambda (t: T).(pr0 (THead +(Flat Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (lift (S O) O +(THead (Flat Appl) v2 x))) t)) (THead (Flat Appl) x0 x1) (pr0_comp u2 x0 H1 +t3 x1 H4 (Flat Appl)) (pr0_zeta b H (THead (Flat Appl) v2 x) (THead (Flat +Appl) x0 x1) (pr0_comp v2 x0 H2 x x1 H3 (Flat Appl)) u3)) (THead (Flat Appl) +(lift (S O) O v2) (lift (S O) O x)) (lift_flat Appl v2 x (S O) +O)))))))))))))))). +(* COMMENTS +Initial nodes: 283 +END *) + +theorem pr0_confluence__pr0_cong_delta: + \forall (u3: T).(\forall (t5: T).(\forall (w: T).((subst0 O u3 t5 w) \to +(\forall (u2: T).(\forall (x: T).((pr0 u2 x) \to ((pr0 u3 x) \to (\forall +(t3: T).(\forall (x0: T).((pr0 t3 x0) \to ((pr0 t5 x0) \to (ex2 T (\lambda +(t: T).(pr0 (THead (Bind Abbr) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind +Abbr) u3 w) t)))))))))))))) +\def + \lambda (u3: T).(\lambda (t5: T).(\lambda (w: T).(\lambda (H: (subst0 O u3 +t5 w)).(\lambda (u2: T).(\lambda (x: T).(\lambda (H0: (pr0 u2 x)).(\lambda +(H1: (pr0 u3 x)).(\lambda (t3: T).(\lambda (x0: T).(\lambda (H2: (pr0 t3 +x0)).(\lambda (H3: (pr0 t5 x0)).(or_ind (pr0 w x0) (ex2 T (\lambda (w2: +T).(pr0 w w2)) (\lambda (w2: T).(subst0 O x x0 w2))) (ex2 T (\lambda (t: +T).(pr0 (THead (Bind Abbr) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) +u3 w) t))) (\lambda (H4: (pr0 w x0)).(ex_intro2 T (\lambda (t: T).(pr0 (THead +(Bind Abbr) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w) t)) +(THead (Bind Abbr) x x0) (pr0_comp u2 x H0 t3 x0 H2 (Bind Abbr)) (pr0_comp u3 +x H1 w x0 H4 (Bind Abbr)))) (\lambda (H4: (ex2 T (\lambda (w2: T).(pr0 w w2)) +(\lambda (w2: T).(subst0 O x x0 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 w +w2)) (\lambda (w2: T).(subst0 O x x0 w2)) (ex2 T (\lambda (t: T).(pr0 (THead +(Bind Abbr) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w) t))) +(\lambda (x1: T).(\lambda (H5: (pr0 w x1)).(\lambda (H6: (subst0 O x x0 +x1)).(ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 t3) t)) (\lambda +(t: T).(pr0 (THead (Bind Abbr) u3 w) t)) (THead (Bind Abbr) x x1) (pr0_delta +u2 x H0 t3 x0 H2 x1 H6) (pr0_comp u3 x H1 w x1 H5 (Bind Abbr)))))) H4)) +(pr0_subst0 t5 x0 H3 u3 w O H x H1))))))))))))). +(* COMMENTS +Initial nodes: 409 +END *) + +theorem pr0_confluence__pr0_upsilon_upsilon: + \forall (b: B).((not (eq B b Abst)) \to (\forall (v1: T).(\forall (v2: +T).(\forall (x0: T).((pr0 v1 x0) \to ((pr0 v2 x0) \to (\forall (u1: +T).(\forall (u2: T).(\forall (x1: T).((pr0 u1 x1) \to ((pr0 u2 x1) \to +(\forall (t1: T).(\forall (t2: T).(\forall (x2: T).((pr0 t1 x2) \to ((pr0 t2 +x2) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u1 (THead (Flat Appl) +(lift (S O) O v1) t1)) t)) (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead +(Flat Appl) (lift (S O) O v2) t2)) t))))))))))))))))))) +\def + \lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda +(v2: T).(\lambda (x0: T).(\lambda (H0: (pr0 v1 x0)).(\lambda (H1: (pr0 v2 +x0)).(\lambda (u1: T).(\lambda (u2: T).(\lambda (x1: T).(\lambda (H2: (pr0 u1 +x1)).(\lambda (H3: (pr0 u2 x1)).(\lambda (t1: T).(\lambda (t2: T).(\lambda +(x2: T).(\lambda (H4: (pr0 t1 x2)).(\lambda (H5: (pr0 t2 x2)).(ex_intro2 T +(\lambda (t: T).(pr0 (THead (Bind b) u1 (THead (Flat Appl) (lift (S O) O v1) +t1)) t)) (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S +O) O v2) t2)) t)) (THead (Bind b) x1 (THead (Flat Appl) (lift (S O) O x0) +x2)) (pr0_comp u1 x1 H2 (THead (Flat Appl) (lift (S O) O v1) t1) (THead (Flat +Appl) (lift (S O) O x0) x2) (pr0_comp (lift (S O) O v1) (lift (S O) O x0) +(pr0_lift v1 x0 H0 (S O) O) t1 x2 H4 (Flat Appl)) (Bind b)) (pr0_comp u2 x1 +H3 (THead (Flat Appl) (lift (S O) O v2) t2) (THead (Flat Appl) (lift (S O) O +x0) x2) (pr0_comp (lift (S O) O v2) (lift (S O) O x0) (pr0_lift v2 x0 H1 (S +O) O) t2 x2 H5 (Flat Appl)) (Bind b))))))))))))))))))). +(* COMMENTS +Initial nodes: 347 +END *) + +theorem pr0_confluence__pr0_delta_delta: + \forall (u2: T).(\forall (t3: T).(\forall (w: T).((subst0 O u2 t3 w) \to +(\forall (u3: T).(\forall (t5: T).(\forall (w0: T).((subst0 O u3 t5 w0) \to +(\forall (x: T).((pr0 u2 x) \to ((pr0 u3 x) \to (\forall (x0: T).((pr0 t3 x0) +\to ((pr0 t5 x0) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) +(\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t)))))))))))))))) +\def + \lambda (u2: T).(\lambda (t3: T).(\lambda (w: T).(\lambda (H: (subst0 O u2 +t3 w)).(\lambda (u3: T).(\lambda (t5: T).(\lambda (w0: T).(\lambda (H0: +(subst0 O u3 t5 w0)).(\lambda (x: T).(\lambda (H1: (pr0 u2 x)).(\lambda (H2: +(pr0 u3 x)).(\lambda (x0: T).(\lambda (H3: (pr0 t3 x0)).(\lambda (H4: (pr0 t5 +x0)).(or_ind (pr0 w0 x0) (ex2 T (\lambda (w2: T).(pr0 w0 w2)) (\lambda (w2: +T).(subst0 O x x0 w2))) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) +t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (H5: (pr0 w0 +x0)).(or_ind (pr0 w x0) (ex2 T (\lambda (w2: T).(pr0 w w2)) (\lambda (w2: +T).(subst0 O x x0 w2))) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) +t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (H6: (pr0 w +x0)).(ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda +(t: T).(pr0 (THead (Bind Abbr) u3 w0) t)) (THead (Bind Abbr) x x0) (pr0_comp +u2 x H1 w x0 H6 (Bind Abbr)) (pr0_comp u3 x H2 w0 x0 H5 (Bind Abbr)))) +(\lambda (H6: (ex2 T (\lambda (w2: T).(pr0 w w2)) (\lambda (w2: T).(subst0 O +x x0 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 w w2)) (\lambda (w2: T).(subst0 +O x x0 w2)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda +(t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (x1: T).(\lambda (H7: +(pr0 w x1)).(\lambda (H8: (subst0 O x x0 x1)).(ex_intro2 T (\lambda (t: +T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) +u3 w0) t)) (THead (Bind Abbr) x x1) (pr0_comp u2 x H1 w x1 H7 (Bind Abbr)) +(pr0_delta u3 x H2 w0 x0 H5 x1 H8))))) H6)) (pr0_subst0 t3 x0 H3 u2 w O H x +H1))) (\lambda (H5: (ex2 T (\lambda (w2: T).(pr0 w0 w2)) (\lambda (w2: +T).(subst0 O x x0 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 w0 w2)) (\lambda +(w2: T).(subst0 O x x0 w2)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 +w) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (x1: +T).(\lambda (H6: (pr0 w0 x1)).(\lambda (H7: (subst0 O x x0 x1)).(or_ind (pr0 +w x0) (ex2 T (\lambda (w2: T).(pr0 w w2)) (\lambda (w2: T).(subst0 O x x0 +w2))) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: +T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (H8: (pr0 w x0)).(ex_intro2 T +(\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead +(Bind Abbr) u3 w0) t)) (THead (Bind Abbr) x x1) (pr0_delta u2 x H1 w x0 H8 x1 +H7) (pr0_comp u3 x H2 w0 x1 H6 (Bind Abbr)))) (\lambda (H8: (ex2 T (\lambda +(w2: T).(pr0 w w2)) (\lambda (w2: T).(subst0 O x x0 w2)))).(ex2_ind T +(\lambda (w2: T).(pr0 w w2)) (\lambda (w2: T).(subst0 O x x0 w2)) (ex2 T +(\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead +(Bind Abbr) u3 w0) t))) (\lambda (x2: T).(\lambda (H9: (pr0 w x2)).(\lambda +(H10: (subst0 O x x0 x2)).(or4_ind (eq T x2 x1) (ex2 T (\lambda (t: +T).(subst0 O x x2 t)) (\lambda (t: T).(subst0 O x x1 t))) (subst0 O x x2 x1) +(subst0 O x x1 x2) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) +(\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (H11: (eq T x2 +x1)).(let H12 \def (eq_ind T x2 (\lambda (t: T).(pr0 w t)) H9 x1 H11) in +(ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: +T).(pr0 (THead (Bind Abbr) u3 w0) t)) (THead (Bind Abbr) x x1) (pr0_comp u2 x +H1 w x1 H12 (Bind Abbr)) (pr0_comp u3 x H2 w0 x1 H6 (Bind Abbr))))) (\lambda +(H11: (ex2 T (\lambda (t: T).(subst0 O x x2 t)) (\lambda (t: T).(subst0 O x +x1 t)))).(ex2_ind T (\lambda (t: T).(subst0 O x x2 t)) (\lambda (t: +T).(subst0 O x x1 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) +t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (x3: +T).(\lambda (H12: (subst0 O x x2 x3)).(\lambda (H13: (subst0 O x x1 +x3)).(ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda +(t: T).(pr0 (THead (Bind Abbr) u3 w0) t)) (THead (Bind Abbr) x x3) (pr0_delta +u2 x H1 w x2 H9 x3 H12) (pr0_delta u3 x H2 w0 x1 H6 x3 H13))))) H11)) +(\lambda (H11: (subst0 O x x2 x1)).(ex_intro2 T (\lambda (t: T).(pr0 (THead +(Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t)) +(THead (Bind Abbr) x x1) (pr0_delta u2 x H1 w x2 H9 x1 H11) (pr0_comp u3 x H2 +w0 x1 H6 (Bind Abbr)))) (\lambda (H11: (subst0 O x x1 x2)).(ex_intro2 T +(\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead +(Bind Abbr) u3 w0) t)) (THead (Bind Abbr) x x2) (pr0_comp u2 x H1 w x2 H9 +(Bind Abbr)) (pr0_delta u3 x H2 w0 x1 H6 x2 H11))) (subst0_confluence_eq x0 +x2 x O H10 x1 H7))))) H8)) (pr0_subst0 t3 x0 H3 u2 w O H x H1))))) H5)) +(pr0_subst0 t5 x0 H4 u3 w0 O H0 x H2))))))))))))))). +(* COMMENTS +Initial nodes: 1501 +END *) + +theorem pr0_confluence__pr0_delta_tau: + \forall (u2: T).(\forall (t3: T).(\forall (w: T).((subst0 O u2 t3 w) \to +(\forall (t4: T).((pr0 (lift (S O) O t4) t3) \to (\forall (t2: T).(ex2 T +(\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 t2 +t))))))))) +\def + \lambda (u2: T).(\lambda (t3: T).(\lambda (w: T).(\lambda (H: (subst0 O u2 +t3 w)).(\lambda (t4: T).(\lambda (H0: (pr0 (lift (S O) O t4) t3)).(\lambda +(t2: T).(ex2_ind T (\lambda (t5: T).(eq T t3 (lift (S O) O t5))) (\lambda +(t5: T).(pr0 t4 t5)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) +(\lambda (t: T).(pr0 t2 t))) (\lambda (x: T).(\lambda (H1: (eq T t3 (lift (S +O) O x))).(\lambda (_: (pr0 t4 x)).(let H3 \def (eq_ind T t3 (\lambda (t: +T).(subst0 O u2 t w)) H (lift (S O) O x) H1) in (subst0_gen_lift_false x u2 w +(S O) O O (le_n O) (eq_ind_r nat (plus (S O) O) (\lambda (n: nat).(lt O n)) +(le_n (plus (S O) O)) (plus O (S O)) (plus_sym O (S O))) H3 (ex2 T (\lambda +(t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 t2 t)))))))) +(pr0_gen_lift t4 t3 (S O) O H0)))))))). +(* COMMENTS +Initial nodes: 257 +END *) + +theorem pr0_confluence: + \forall (t0: T).(\forall (t1: T).((pr0 t0 t1) \to (\forall (t2: T).((pr0 t0 +t2) \to (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t))))))) +\def + \lambda (t0: T).(tlt_wf_ind (\lambda (t: T).(\forall (t1: T).((pr0 t t1) \to +(\forall (t2: T).((pr0 t t2) \to (ex2 T (\lambda (t3: T).(pr0 t1 t3)) +(\lambda (t3: T).(pr0 t2 t3)))))))) (\lambda (t: T).(\lambda (H: ((\forall +(v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 +v t2) \to (ex2 T (\lambda (t3: T).(pr0 t1 t3)) (\lambda (t3: T).(pr0 t2 +t3))))))))))).(\lambda (t1: T).(\lambda (H0: (pr0 t t1)).(\lambda (t2: +T).(\lambda (H1: (pr0 t t2)).(let H2 \def (match H0 in pr0 return (\lambda +(t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).((eq T t3 t) \to ((eq T t4 +t1) \to (ex2 T (\lambda (t5: T).(pr0 t1 t5)) (\lambda (t5: T).(pr0 t2 +t5)))))))) with [(pr0_refl t3) \Rightarrow (\lambda (H2: (eq T t3 +t)).(\lambda (H3: (eq T t3 t1)).(eq_ind T t (\lambda (t4: T).((eq T t4 t1) +\to (ex2 T (\lambda (t5: T).(pr0 t1 t5)) (\lambda (t5: T).(pr0 t2 t5))))) +(\lambda (H4: (eq T t t1)).(eq_ind T t1 (\lambda (_: T).(ex2 T (\lambda (t5: +T).(pr0 t1 t5)) (\lambda (t5: T).(pr0 t2 t5)))) (let H5 \def (match H1 in pr0 +return (\lambda (t4: T).(\lambda (t5: T).(\lambda (_: (pr0 t4 t5)).((eq T t4 +t) \to ((eq T t5 t2) \to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: +T).(pr0 t2 t6)))))))) with [(pr0_refl t4) \Rightarrow (\lambda (H5: (eq T t4 +t)).(\lambda (H6: (eq T t4 t2)).(eq_ind T t (\lambda (t5: T).((eq T t5 t2) +\to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6))))) +(\lambda (H7: (eq T t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t6: +T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6)))) (let H8 \def (eq_ind T t +(\lambda (t5: T).(eq T t4 t5)) H5 t2 H7) in (let H9 \def (eq_ind T t (\lambda +(t5: T).(eq T t5 t1)) H4 t2 H7) in (let H10 \def (eq_ind T t (\lambda (t5: +T).(eq T t3 t5)) H2 t2 H7) in (let H11 \def (eq_ind T t (\lambda (t5: +T).(\forall (v: T).((tlt v t5) \to (\forall (t6: T).((pr0 v t6) \to (\forall +(t7: T).((pr0 v t7) \to (ex2 T (\lambda (t8: T).(pr0 t6 t8)) (\lambda (t8: +T).(pr0 t7 t8)))))))))) H t2 H7) in (let H12 \def (eq_ind T t2 (\lambda (t5: +T).(\forall (v: T).((tlt v t5) \to (\forall (t6: T).((pr0 v t6) \to (\forall +(t7: T).((pr0 v t7) \to (ex2 T (\lambda (t8: T).(pr0 t6 t8)) (\lambda (t8: +T).(pr0 t7 t8)))))))))) H11 t1 H9) in (eq_ind_r T t1 (\lambda (t5: T).(ex2 T +(\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t5 t6)))) (let H13 \def +(eq_ind T t2 (\lambda (t5: T).(eq T t3 t5)) H10 t1 H9) in (ex_intro2 T +(\lambda (t5: T).(pr0 t1 t5)) (\lambda (t5: T).(pr0 t1 t5)) t1 (pr0_refl t1) +(pr0_refl t1))) t2 H9)))))) t (sym_eq T t t2 H7))) t4 (sym_eq T t4 t H5) +H6))) | (pr0_comp u1 u2 H5 t4 t5 H6 k) \Rightarrow (\lambda (H7: (eq T (THead +k u1 t4) t)).(\lambda (H8: (eq T (THead k u2 t5) t2)).(eq_ind T (THead k u1 +t4) (\lambda (_: T).((eq T (THead k u2 t5) t2) \to ((pr0 u1 u2) \to ((pr0 t4 +t5) \to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 +t7))))))) (\lambda (H9: (eq T (THead k u2 t5) t2)).(eq_ind T (THead k u2 t5) +(\lambda (t6: T).((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t7: +T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t6 t7)))))) (\lambda (H10: (pr0 u1 +u2)).(\lambda (H11: (pr0 t4 t5)).(let H12 \def (eq_ind_r T t (\lambda (t6: +T).(eq T t6 t1)) H4 (THead k u1 t4) H7) in (eq_ind T (THead k u1 t4) (\lambda +(t6: T).(ex2 T (\lambda (t7: T).(pr0 t6 t7)) (\lambda (t7: T).(pr0 (THead k +u2 t5) t7)))) (let H13 \def (eq_ind_r T t (\lambda (t6: T).(eq T t3 t6)) H2 +(THead k u1 t4) H7) in (let H14 \def (eq_ind_r T t (\lambda (t6: T).(\forall +(v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall (t8: +T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 +t8 t9)))))))))) H (THead k u1 t4) H7) in (ex_intro2 T (\lambda (t6: T).(pr0 +(THead k u1 t4) t6)) (\lambda (t6: T).(pr0 (THead k u2 t5) t6)) (THead k u2 +t5) (pr0_comp u1 u2 H10 t4 t5 H11 k) (pr0_refl (THead k u2 t5))))) t1 H12)))) +t2 H9)) t H7 H8 H5 H6))) | (pr0_beta u v1 v2 H5 t4 t5 H6) \Rightarrow +(\lambda (H7: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t4)) +t)).(\lambda (H8: (eq T (THead (Bind Abbr) v2 t5) t2)).(eq_ind T (THead (Flat +Appl) v1 (THead (Bind Abst) u t4)) (\lambda (_: T).((eq T (THead (Bind Abbr) +v2 t5) t2) \to ((pr0 v1 v2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t7: T).(pr0 +t1 t7)) (\lambda (t7: T).(pr0 t2 t7))))))) (\lambda (H9: (eq T (THead (Bind +Abbr) v2 t5) t2)).(eq_ind T (THead (Bind Abbr) v2 t5) (\lambda (t6: T).((pr0 +v1 v2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda +(t7: T).(pr0 t6 t7)))))) (\lambda (H10: (pr0 v1 v2)).(\lambda (H11: (pr0 t4 +t5)).(let H12 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 t1)) H4 (THead +(Flat Appl) v1 (THead (Bind Abst) u t4)) H7) in (eq_ind T (THead (Flat Appl) +v1 (THead (Bind Abst) u t4)) (\lambda (t6: T).(ex2 T (\lambda (t7: T).(pr0 t6 +t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t5) t7)))) (let H13 \def +(eq_ind_r T t (\lambda (t6: T).(eq T t3 t6)) H2 (THead (Flat Appl) v1 (THead +(Bind Abst) u t4)) H7) in (let H14 \def (eq_ind_r T t (\lambda (t6: +T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall +(t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: +T).(pr0 t8 t9)))))))))) H (THead (Flat Appl) v1 (THead (Bind Abst) u t4)) H7) +in (ex_intro2 T (\lambda (t6: T).(pr0 (THead (Flat Appl) v1 (THead (Bind +Abst) u t4)) t6)) (\lambda (t6: T).(pr0 (THead (Bind Abbr) v2 t5) t6)) (THead +(Bind Abbr) v2 t5) (pr0_beta u v1 v2 H10 t4 t5 H11) (pr0_refl (THead (Bind +Abbr) v2 t5))))) t1 H12)))) t2 H9)) t H7 H8 H5 H6))) | (pr0_upsilon b H5 v1 +v2 H6 u1 u2 H7 t4 t5 H8) \Rightarrow (\lambda (H9: (eq T (THead (Flat Appl) +v1 (THead (Bind b) u1 t4)) t)).(\lambda (H10: (eq T (THead (Bind b) u2 (THead +(Flat Appl) (lift (S O) O v2) t5)) t2)).(eq_ind T (THead (Flat Appl) v1 +(THead (Bind b) u1 t4)) (\lambda (_: T).((eq T (THead (Bind b) u2 (THead +(Flat Appl) (lift (S O) O v2) t5)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1 +v2) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t7: T).(pr0 t1 +t7)) (\lambda (t7: T).(pr0 t2 t7))))))))) (\lambda (H11: (eq T (THead (Bind +b) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) t2)).(eq_ind T (THead (Bind +b) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) (\lambda (t6: T).((not (eq B +b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T +(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t6 t7)))))))) (\lambda +(H12: (not (eq B b Abst))).(\lambda (H13: (pr0 v1 v2)).(\lambda (H14: (pr0 u1 +u2)).(\lambda (H15: (pr0 t4 t5)).(let H16 \def (eq_ind_r T t (\lambda (t6: +T).(eq T t6 t1)) H4 (THead (Flat Appl) v1 (THead (Bind b) u1 t4)) H9) in +(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t4)) (\lambda (t6: T).(ex2 +T (\lambda (t7: T).(pr0 t6 t7)) (\lambda (t7: T).(pr0 (THead (Bind b) u2 +(THead (Flat Appl) (lift (S O) O v2) t5)) t7)))) (let H17 \def (eq_ind_r T t +(\lambda (t6: T).(eq T t3 t6)) H2 (THead (Flat Appl) v1 (THead (Bind b) u1 +t4)) H9) in (let H18 \def (eq_ind_r T t (\lambda (t6: T).(\forall (v: +T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall (t8: T).((pr0 v +t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8 +t9)))))))))) H (THead (Flat Appl) v1 (THead (Bind b) u1 t4)) H9) in +(pr0_confluence__pr0_cong_upsilon_refl b H12 u1 u2 H14 t4 t5 H15 v1 v2 v2 H13 +(pr0_refl v2)))) t1 H16)))))) t2 H11)) t H9 H10 H5 H6 H7 H8))) | (pr0_delta +u1 u2 H5 t4 t5 H6 w H7) \Rightarrow (\lambda (H8: (eq T (THead (Bind Abbr) u1 +t4) t)).(\lambda (H9: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead +(Bind Abbr) u1 t4) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t2) \to +((pr0 u1 u2) \to ((pr0 t4 t5) \to ((subst0 O u2 t5 w) \to (ex2 T (\lambda +(t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)))))))) (\lambda (H10: (eq T +(THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda +(t6: T).((pr0 u1 u2) \to ((pr0 t4 t5) \to ((subst0 O u2 t5 w) \to (ex2 T +(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t6 t7))))))) (\lambda +(H11: (pr0 u1 u2)).(\lambda (H12: (pr0 t4 t5)).(\lambda (H13: (subst0 O u2 t5 +w)).(let H14 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 t1)) H4 (THead +(Bind Abbr) u1 t4) H8) in (eq_ind T (THead (Bind Abbr) u1 t4) (\lambda (t6: +T).(ex2 T (\lambda (t7: T).(pr0 t6 t7)) (\lambda (t7: T).(pr0 (THead (Bind +Abbr) u2 w) t7)))) (let H15 \def (eq_ind_r T t (\lambda (t6: T).(eq T t3 t6)) +H2 (THead (Bind Abbr) u1 t4) H8) in (let H16 \def (eq_ind_r T t (\lambda (t6: +T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall +(t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: +T).(pr0 t8 t9)))))))))) H (THead (Bind Abbr) u1 t4) H8) in (ex_intro2 T +(\lambda (t6: T).(pr0 (THead (Bind Abbr) u1 t4) t6)) (\lambda (t6: T).(pr0 +(THead (Bind Abbr) u2 w) t6)) (THead (Bind Abbr) u2 w) (pr0_delta u1 u2 H11 +t4 t5 H12 w H13) (pr0_refl (THead (Bind Abbr) u2 w))))) t1 H14))))) t2 H10)) +t H8 H9 H5 H6 H7))) | (pr0_zeta b H5 t4 t5 H6 u) \Rightarrow (\lambda (H7: +(eq T (THead (Bind b) u (lift (S O) O t4)) t)).(\lambda (H8: (eq T t5 +t2)).(eq_ind T (THead (Bind b) u (lift (S O) O t4)) (\lambda (_: T).((eq T t5 +t2) \to ((not (eq B b Abst)) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t7: +T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))))))) (\lambda (H9: (eq T t5 +t2)).(eq_ind T t2 (\lambda (t6: T).((not (eq B b Abst)) \to ((pr0 t4 t6) \to +(ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)))))) +(\lambda (H10: (not (eq B b Abst))).(\lambda (H11: (pr0 t4 t2)).(let H12 \def +(eq_ind_r T t (\lambda (t6: T).(eq T t6 t1)) H4 (THead (Bind b) u (lift (S O) +O t4)) H7) in (eq_ind T (THead (Bind b) u (lift (S O) O t4)) (\lambda (t6: +T).(ex2 T (\lambda (t7: T).(pr0 t6 t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let +H13 \def (eq_ind_r T t (\lambda (t6: T).(eq T t3 t6)) H2 (THead (Bind b) u +(lift (S O) O t4)) H7) in (let H14 \def (eq_ind_r T t (\lambda (t6: +T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall +(t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: +T).(pr0 t8 t9)))))))))) H (THead (Bind b) u (lift (S O) O t4)) H7) in +(ex_intro2 T (\lambda (t6: T).(pr0 (THead (Bind b) u (lift (S O) O t4)) t6)) +(\lambda (t6: T).(pr0 t2 t6)) t2 (pr0_zeta b H10 t4 t2 H11 u) (pr0_refl +t2)))) t1 H12)))) t5 (sym_eq T t5 t2 H9))) t H7 H8 H5 H6))) | (pr0_tau t4 t5 +H5 u) \Rightarrow (\lambda (H6: (eq T (THead (Flat Cast) u t4) t)).(\lambda +(H7: (eq T t5 t2)).(eq_ind T (THead (Flat Cast) u t4) (\lambda (_: T).((eq T +t5 t2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda +(t7: T).(pr0 t2 t7)))))) (\lambda (H8: (eq T t5 t2)).(eq_ind T t2 (\lambda +(t6: T).((pr0 t4 t6) \to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: +T).(pr0 t2 t7))))) (\lambda (H9: (pr0 t4 t2)).(let H10 \def (eq_ind_r T t +(\lambda (t6: T).(eq T t6 t1)) H4 (THead (Flat Cast) u t4) H6) in (eq_ind T +(THead (Flat Cast) u t4) (\lambda (t6: T).(ex2 T (\lambda (t7: T).(pr0 t6 +t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let H11 \def (eq_ind_r T t (\lambda +(t6: T).(eq T t3 t6)) H2 (THead (Flat Cast) u t4) H6) in (let H12 \def +(eq_ind_r T t (\lambda (t6: T).(\forall (v: T).((tlt v t6) \to (\forall (t7: +T).((pr0 v t7) \to (\forall (t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: +T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead (Flat Cast) u +t4) H6) in (ex_intro2 T (\lambda (t6: T).(pr0 (THead (Flat Cast) u t4) t6)) +(\lambda (t6: T).(pr0 t2 t6)) t2 (pr0_tau t4 t2 H9 u) (pr0_refl t2)))) t1 +H10))) t5 (sym_eq T t5 t2 H8))) t H6 H7 H5)))]) in (H5 (refl_equal T t) +(refl_equal T t2))) t (sym_eq T t t1 H4))) t3 (sym_eq T t3 t H2) H3))) | +(pr0_comp u1 u2 H2 t3 t4 H3 k) \Rightarrow (\lambda (H4: (eq T (THead k u1 +t3) t)).(\lambda (H5: (eq T (THead k u2 t4) t1)).(eq_ind T (THead k u1 t3) +(\lambda (_: T).((eq T (THead k u2 t4) t1) \to ((pr0 u1 u2) \to ((pr0 t3 t4) +\to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6))))))) +(\lambda (H6: (eq T (THead k u2 t4) t1)).(eq_ind T (THead k u2 t4) (\lambda +(t5: T).((pr0 u1 u2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t5 +t6)) (\lambda (t6: T).(pr0 t2 t6)))))) (\lambda (H7: (pr0 u1 u2)).(\lambda +(H8: (pr0 t3 t4)).(let H9 \def (match H1 in pr0 return (\lambda (t5: +T).(\lambda (t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 t) \to ((eq T t6 t2) +\to (ex2 T (\lambda (t7: T).(pr0 (THead k u2 t4) t7)) (\lambda (t7: T).(pr0 +t2 t7)))))))) with [(pr0_refl t5) \Rightarrow (\lambda (H9: (eq T t5 +t)).(\lambda (H10: (eq T t5 t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2) +\to (ex2 T (\lambda (t7: T).(pr0 (THead k u2 t4) t7)) (\lambda (t7: T).(pr0 +t2 t7))))) (\lambda (H11: (eq T t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T +(\lambda (t7: T).(pr0 (THead k u2 t4) t7)) (\lambda (t7: T).(pr0 t2 t7)))) +(let H12 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 t2)) H11 (THead k u1 +t3) H4) in (eq_ind T (THead k u1 t3) (\lambda (t6: T).(ex2 T (\lambda (t7: +T).(pr0 (THead k u2 t4) t7)) (\lambda (t7: T).(pr0 t6 t7)))) (let H13 \def +(eq_ind_r T t (\lambda (t6: T).(eq T t5 t6)) H9 (THead k u1 t3) H4) in (let +H14 \def (eq_ind_r T t (\lambda (t6: T).(\forall (v: T).((tlt v t6) \to +(\forall (t7: T).((pr0 v t7) \to (\forall (t8: T).((pr0 v t8) \to (ex2 T +(\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead +k u1 t3) H4) in (ex_intro2 T (\lambda (t6: T).(pr0 (THead k u2 t4) t6)) +(\lambda (t6: T).(pr0 (THead k u1 t3) t6)) (THead k u2 t4) (pr0_refl (THead k +u2 t4)) (pr0_comp u1 u2 H7 t3 t4 H8 k)))) t2 H12)) t (sym_eq T t t2 H11))) t5 +(sym_eq T t5 t H9) H10))) | (pr0_comp u0 u3 H9 t5 t6 H10 k0) \Rightarrow +(\lambda (H11: (eq T (THead k0 u0 t5) t)).(\lambda (H12: (eq T (THead k0 u3 +t6) t2)).(eq_ind T (THead k0 u0 t5) (\lambda (_: T).((eq T (THead k0 u3 t6) +t2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead +k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H13: (eq T (THead +k0 u3 t6) t2)).(eq_ind T (THead k0 u3 t6) (\lambda (t7: T).((pr0 u0 u3) \to +((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda +(t8: T).(pr0 t7 t8)))))) (\lambda (H14: (pr0 u0 u3)).(\lambda (H15: (pr0 t5 +t6)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead k u1 t3) t7)) +H4 (THead k0 u0 t5) H11) in (let H17 \def (f_equal T K (\lambda (e: T).(match +e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) +\Rightarrow k | (THead k1 _ _) \Rightarrow k1])) (THead k u1 t3) (THead k0 u0 +t5) H16) in ((let H18 \def (f_equal T T (\lambda (e: T).(match e in T return +(\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 +| (THead _ t7 _) \Rightarrow t7])) (THead k u1 t3) (THead k0 u0 t5) H16) in +((let H19 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: +T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ +t7) \Rightarrow t7])) (THead k u1 t3) (THead k0 u0 t5) H16) in (\lambda (H20: +(eq T u1 u0)).(\lambda (H21: (eq K k k0)).(let H22 \def (eq_ind_r T t +(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) +\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) +(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead k0 u0 t5) H11) in (eq_ind_r +K k0 (\lambda (k1: K).(ex2 T (\lambda (t7: T).(pr0 (THead k1 u2 t4) t7)) +(\lambda (t7: T).(pr0 (THead k0 u3 t6) t7)))) (let H23 \def (eq_ind T u1 +(\lambda (t7: T).(pr0 t7 u2)) H7 u0 H20) in (let H24 \def (eq_ind T t3 +(\lambda (t7: T).(pr0 t7 t4)) H8 t5 H19) in (ex2_ind T (\lambda (t7: T).(pr0 +t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead k0 +u2 t4) t7)) (\lambda (t7: T).(pr0 (THead k0 u3 t6) t7))) (\lambda (x: +T).(\lambda (H25: (pr0 t4 x)).(\lambda (H26: (pr0 t6 x)).(ex2_ind T (\lambda +(t7: T).(pr0 u2 t7)) (\lambda (t7: T).(pr0 u3 t7)) (ex2 T (\lambda (t7: +T).(pr0 (THead k0 u2 t4) t7)) (\lambda (t7: T).(pr0 (THead k0 u3 t6) t7))) +(\lambda (x0: T).(\lambda (H27: (pr0 u2 x0)).(\lambda (H28: (pr0 u3 +x0)).(ex_intro2 T (\lambda (t7: T).(pr0 (THead k0 u2 t4) t7)) (\lambda (t7: +T).(pr0 (THead k0 u3 t6) t7)) (THead k0 x0 x) (pr0_comp u2 x0 H27 t4 x H25 +k0) (pr0_comp u3 x0 H28 t6 x H26 k0))))) (H22 u0 (tlt_head_sx k0 u0 t5) u2 +H23 u3 H14))))) (H22 t5 (tlt_head_dx k0 u0 t5) t4 H24 t6 H15)))) k H21))))) +H18)) H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_beta u v1 v2 H9 t5 t6 H10) +\Rightarrow (\lambda (H11: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u +t5)) t)).(\lambda (H12: (eq T (THead (Bind Abbr) v2 t6) t2)).(eq_ind T (THead +(Flat Appl) v1 (THead (Bind Abst) u t5)) (\lambda (_: T).((eq T (THead (Bind +Abbr) v2 t6) t2) \to ((pr0 v1 v2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: +T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda +(H13: (eq T (THead (Bind Abbr) v2 t6) t2)).(eq_ind T (THead (Bind Abbr) v2 +t6) (\lambda (t7: T).((pr0 v1 v2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: +T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (H14: +(pr0 v1 v2)).(\lambda (H15: (pr0 t5 t6)).(let H16 \def (eq_ind_r T t (\lambda +(t7: T).(eq T (THead k u1 t3) t7)) H4 (THead (Flat Appl) v1 (THead (Bind +Abst) u t5)) H11) in (let H17 \def (f_equal T K (\lambda (e: T).(match e in T +return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) +\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) (THead (Flat +Appl) v1 (THead (Bind Abst) u t5)) H16) in ((let H18 \def (f_equal T T +(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) +(THead k u1 t3) (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) H16) in ((let +H19 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) +with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7) +\Rightarrow t7])) (THead k u1 t3) (THead (Flat Appl) v1 (THead (Bind Abst) u +t5)) H16) in (\lambda (H20: (eq T u1 v1)).(\lambda (H21: (eq K k (Flat +Appl))).(let H22 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v +t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to +(ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 +t10)))))))))) H (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) H11) in +(eq_ind_r K (Flat Appl) (\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 (THead +k0 u2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t6) t7)))) (let +H23 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H7 v1 H20) in (let H24 +\def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H8 (THead (Bind Abst) u t5) +H19) in (let H25 \def (match H24 in pr0 return (\lambda (t7: T).(\lambda (t8: +T).(\lambda (_: (pr0 t7 t8)).((eq T t7 (THead (Bind Abst) u t5)) \to ((eq T +t8 t4) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) +(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9)))))))) with [(pr0_refl +t7) \Rightarrow (\lambda (H25: (eq T t7 (THead (Bind Abst) u t5))).(\lambda +(H26: (eq T t7 t4)).(eq_ind T (THead (Bind Abst) u t5) (\lambda (t8: T).((eq +T t8 t4) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) +(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9))))) (\lambda (H27: (eq T +(THead (Bind Abst) u t5) t4)).(eq_ind T (THead (Bind Abst) u t5) (\lambda +(t8: T).(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t8) t9)) (\lambda +(t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9)))) (ex2_ind T (\lambda (t8: +T).(pr0 u2 t8)) (\lambda (t8: T).(pr0 v2 t8)) (ex2 T (\lambda (t8: T).(pr0 +(THead (Flat Appl) u2 (THead (Bind Abst) u t5)) t8)) (\lambda (t8: T).(pr0 +(THead (Bind Abbr) v2 t6) t8))) (\lambda (x: T).(\lambda (H28: (pr0 u2 +x)).(\lambda (H29: (pr0 v2 x)).(ex_intro2 T (\lambda (t8: T).(pr0 (THead +(Flat Appl) u2 (THead (Bind Abst) u t5)) t8)) (\lambda (t8: T).(pr0 (THead +(Bind Abbr) v2 t6) t8)) (THead (Bind Abbr) x t6) (pr0_beta u u2 x H28 t5 t6 +H15) (pr0_comp v2 x H29 t6 t6 (pr0_refl t6) (Bind Abbr)))))) (H22 v1 +(tlt_head_sx (Flat Appl) v1 (THead (Bind Abst) u t5)) u2 H23 v2 H14)) t4 +H27)) t7 (sym_eq T t7 (THead (Bind Abst) u t5) H25) H26))) | (pr0_comp u0 u3 +H25 t7 t8 H26 k0) \Rightarrow (\lambda (H27: (eq T (THead k0 u0 t7) (THead +(Bind Abst) u t5))).(\lambda (H28: (eq T (THead k0 u3 t8) t4)).((let H29 \def +(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with +[(TSort _) \Rightarrow t7 | (TLRef _) \Rightarrow t7 | (THead _ _ t9) +\Rightarrow t9])) (THead k0 u0 t7) (THead (Bind Abst) u t5) H27) in ((let H30 +\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) +with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t9 _) +\Rightarrow t9])) (THead k0 u0 t7) (THead (Bind Abst) u t5) H27) in ((let H31 +\def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) +with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _ _) +\Rightarrow k1])) (THead k0 u0 t7) (THead (Bind Abst) u t5) H27) in (eq_ind K +(Bind Abst) (\lambda (k1: K).((eq T u0 u) \to ((eq T t7 t5) \to ((eq T (THead +k1 u3 t8) t4) \to ((pr0 u0 u3) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: +T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind +Abbr) v2 t6) t9))))))))) (\lambda (H32: (eq T u0 u)).(eq_ind T u (\lambda +(t9: T).((eq T t7 t5) \to ((eq T (THead (Bind Abst) u3 t8) t4) \to ((pr0 t9 +u3) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 +t4) t10)) (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t6) t10)))))))) +(\lambda (H33: (eq T t7 t5)).(eq_ind T t5 (\lambda (t9: T).((eq T (THead +(Bind Abst) u3 t8) t4) \to ((pr0 u u3) \to ((pr0 t9 t8) \to (ex2 T (\lambda +(t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda (t10: T).(pr0 (THead +(Bind Abbr) v2 t6) t10))))))) (\lambda (H34: (eq T (THead (Bind Abst) u3 t8) +t4)).(eq_ind T (THead (Bind Abst) u3 t8) (\lambda (t9: T).((pr0 u u3) \to +((pr0 t5 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t9) +t10)) (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t6) t10)))))) (\lambda (_: +(pr0 u u3)).(\lambda (H36: (pr0 t5 t8)).(ex2_ind T (\lambda (t9: T).(pr0 t8 +t9)) (\lambda (t9: T).(pr0 t6 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat +Appl) u2 (THead (Bind Abst) u3 t8)) t9)) (\lambda (t9: T).(pr0 (THead (Bind +Abbr) v2 t6) t9))) (\lambda (x: T).(\lambda (H37: (pr0 t8 x)).(\lambda (H38: +(pr0 t6 x)).(ex2_ind T (\lambda (t9: T).(pr0 u2 t9)) (\lambda (t9: T).(pr0 v2 +t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u3 +t8)) t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9))) (\lambda (x0: +T).(\lambda (H39: (pr0 u2 x0)).(\lambda (H40: (pr0 v2 x0)).(ex_intro2 T +(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u3 t8)) t9)) +(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9)) (THead (Bind Abbr) x0 x) +(pr0_beta u3 u2 x0 H39 t8 x H37) (pr0_comp v2 x0 H40 t6 x H38 (Bind +Abbr)))))) (H22 v1 (tlt_head_sx (Flat Appl) v1 (THead (Bind Abst) u t5)) u2 +H23 v2 H14))))) (H22 t5 (tlt_trans (THead (Bind Abst) u t5) t5 (THead (Flat +Appl) v1 (THead (Bind Abst) u t5)) (tlt_head_dx (Bind Abst) u t5) +(tlt_head_dx (Flat Appl) v1 (THead (Bind Abst) u t5))) t8 H36 t6 H15)))) t4 +H34)) t7 (sym_eq T t7 t5 H33))) u0 (sym_eq T u0 u H32))) k0 (sym_eq K k0 +(Bind Abst) H31))) H30)) H29)) H28 H25 H26))) | (pr0_beta u0 v0 v3 H25 t7 t8 +H26) \Rightarrow (\lambda (H27: (eq T (THead (Flat Appl) v0 (THead (Bind +Abst) u0 t7)) (THead (Bind Abst) u t5))).(\lambda (H28: (eq T (THead (Bind +Abbr) v3 t8) t4)).((let H29 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind +Abst) u0 t7)) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with +[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _) +\Rightarrow (match k0 in K return (\lambda (_: K).Prop) with [(Bind _) +\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u t5) +H27) in (False_ind ((eq T (THead (Bind Abbr) v3 t8) t4) \to ((pr0 v0 v3) \to +((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) +(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9)))))) H29)) H28 H25 H26))) +| (pr0_upsilon b H25 v0 v3 H26 u0 u3 H27 t7 t8 H28) \Rightarrow (\lambda +(H29: (eq T (THead (Flat Appl) v0 (THead (Bind b) u0 t7)) (THead (Bind Abst) +u t5))).(\lambda (H30: (eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S +O) O v3) t8)) t4)).((let H31 \def (eq_ind T (THead (Flat Appl) v0 (THead +(Bind b) u0 t7)) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) +with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ +_) \Rightarrow (match k0 in K return (\lambda (_: K).Prop) with [(Bind _) +\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u t5) +H29) in (False_ind ((eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O +v3) t8)) t4) \to ((not (eq B b Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3) \to +((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) +(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9)))))))) H31)) H30 H25 H26 +H27 H28))) | (pr0_delta u0 u3 H25 t7 t8 H26 w H27) \Rightarrow (\lambda (H28: +(eq T (THead (Bind Abbr) u0 t7) (THead (Bind Abst) u t5))).(\lambda (H29: (eq +T (THead (Bind Abbr) u3 w) t4)).((let H30 \def (eq_ind T (THead (Bind Abbr) +u0 t7) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with +[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _) +\Rightarrow (match k0 in K return (\lambda (_: K).Prop) with [(Bind b) +\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow +True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _) +\Rightarrow False])])) I (THead (Bind Abst) u t5) H28) in (False_ind ((eq T +(THead (Bind Abbr) u3 w) t4) \to ((pr0 u0 u3) \to ((pr0 t7 t8) \to ((subst0 O +u3 t8 w) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) +(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9))))))) H30)) H29 H25 H26 +H27))) | (pr0_zeta b H25 t7 t8 H26 u0) \Rightarrow (\lambda (H27: (eq T +(THead (Bind b) u0 (lift (S O) O t7)) (THead (Bind Abst) u t5))).(\lambda +(H28: (eq T t8 t4)).((let H29 \def (f_equal T T (\lambda (e: T).(match e in T +return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: +((nat \to nat))) (d: nat) (t9: T) on t9: T \def (match t9 with [(TSort n) +\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with +[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k0 u3 t10) +\Rightarrow (THead k0 (lref_map f d u3) (lref_map f (s k0 d) t10))]) in +lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (TLRef _) \Rightarrow +((let rec lref_map (f: ((nat \to nat))) (d: nat) (t9: T) on t9: T \def (match +t9 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef +(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | +(THead k0 u3 t10) \Rightarrow (THead k0 (lref_map f d u3) (lref_map f (s k0 +d) t10))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (THead _ _ +t9) \Rightarrow t9])) (THead (Bind b) u0 (lift (S O) O t7)) (THead (Bind +Abst) u t5) H27) in ((let H30 \def (f_equal T T (\lambda (e: T).(match e in T +return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) +\Rightarrow u0 | (THead _ t9 _) \Rightarrow t9])) (THead (Bind b) u0 (lift (S +O) O t7)) (THead (Bind Abst) u t5) H27) in ((let H31 \def (f_equal T B +(\lambda (e: T).(match e in T return (\lambda (_: T).B) with [(TSort _) +\Rightarrow b | (TLRef _) \Rightarrow b | (THead k0 _ _) \Rightarrow (match +k0 in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) +\Rightarrow b])])) (THead (Bind b) u0 (lift (S O) O t7)) (THead (Bind Abst) u +t5) H27) in (eq_ind B Abst (\lambda (b0: B).((eq T u0 u) \to ((eq T (lift (S +O) O t7) t5) \to ((eq T t8 t4) \to ((not (eq B b0 Abst)) \to ((pr0 t7 t8) \to +(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: +T).(pr0 (THead (Bind Abbr) v2 t6) t9))))))))) (\lambda (H32: (eq T u0 +u)).(eq_ind T u (\lambda (_: T).((eq T (lift (S O) O t7) t5) \to ((eq T t8 +t4) \to ((not (eq B Abst Abst)) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: +T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda (t10: T).(pr0 (THead (Bind +Abbr) v2 t6) t10)))))))) (\lambda (H33: (eq T (lift (S O) O t7) t5)).(eq_ind +T (lift (S O) O t7) (\lambda (_: T).((eq T t8 t4) \to ((not (eq B Abst Abst)) +\to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) +t10)) (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t6) t10))))))) (\lambda +(H34: (eq T t8 t4)).(eq_ind T t4 (\lambda (t9: T).((not (eq B Abst Abst)) \to +((pr0 t7 t9) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) +t10)) (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t6) t10)))))) (\lambda +(H35: (not (eq B Abst Abst))).(\lambda (_: (pr0 t7 t4)).(let H37 \def (match +(H35 (refl_equal B Abst)) in False return (\lambda (_: False).(ex2 T (\lambda +(t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead +(Bind Abbr) v2 t6) t9)))) with []) in H37))) t8 (sym_eq T t8 t4 H34))) t5 +H33)) u0 (sym_eq T u0 u H32))) b (sym_eq B b Abst H31))) H30)) H29)) H28 H25 +H26))) | (pr0_tau t7 t8 H25 u0) \Rightarrow (\lambda (H26: (eq T (THead (Flat +Cast) u0 t7) (THead (Bind Abst) u t5))).(\lambda (H27: (eq T t8 t4)).((let +H28 \def (eq_ind T (THead (Flat Cast) u0 t7) (\lambda (e: T).(match e in T +return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda +(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow +True])])) I (THead (Bind Abst) u t5) H26) in (False_ind ((eq T t8 t4) \to +((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) +(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9))))) H28)) H27 H25)))]) in +(H25 (refl_equal T (THead (Bind Abst) u t5)) (refl_equal T t4))))) k H21))))) +H18)) H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_upsilon b H9 v1 v2 H10 u0 +u3 H11 t5 t6 H12) \Rightarrow (\lambda (H13: (eq T (THead (Flat Appl) v1 +(THead (Bind b) u0 t5)) t)).(\lambda (H14: (eq T (THead (Bind b) u3 (THead +(Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Flat Appl) v1 +(THead (Bind b) u0 t5)) (\lambda (_: T).((eq T (THead (Bind b) u3 (THead +(Flat Appl) (lift (S O) O v2) t6)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1 +v2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead +k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H15: (eq T +(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T +(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) (\lambda (t7: +T).((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) +\to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 +t7 t8)))))))) (\lambda (H16: (not (eq B b Abst))).(\lambda (H17: (pr0 v1 +v2)).(\lambda (H18: (pr0 u0 u3)).(\lambda (H19: (pr0 t5 t6)).(let H20 \def +(eq_ind_r T t (\lambda (t7: T).(eq T (THead k u1 t3) t7)) H4 (THead (Flat +Appl) v1 (THead (Bind b) u0 t5)) H13) in (let H21 \def (f_equal T K (\lambda +(e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k +| (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) +(THead (Flat Appl) v1 (THead (Bind b) u0 t5)) H20) in ((let H22 \def (f_equal +T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) +(THead k u1 t3) (THead (Flat Appl) v1 (THead (Bind b) u0 t5)) H20) in ((let +H23 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) +with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7) +\Rightarrow t7])) (THead k u1 t3) (THead (Flat Appl) v1 (THead (Bind b) u0 +t5)) H20) in (\lambda (H24: (eq T u1 v1)).(\lambda (H25: (eq K k (Flat +Appl))).(let H26 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v +t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to +(ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 +t10)))))))))) H (THead (Flat Appl) v1 (THead (Bind b) u0 t5)) H13) in +(eq_ind_r K (Flat Appl) (\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 (THead +k0 u2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) +(lift (S O) O v2) t6)) t7)))) (let H27 \def (eq_ind T u1 (\lambda (t7: +T).(pr0 t7 u2)) H7 v1 H24) in (let H28 \def (eq_ind T t3 (\lambda (t7: +T).(pr0 t7 t4)) H8 (THead (Bind b) u0 t5) H23) in (let H29 \def (match H28 in +pr0 return (\lambda (t7: T).(\lambda (t8: T).(\lambda (_: (pr0 t7 t8)).((eq T +t7 (THead (Bind b) u0 t5)) \to ((eq T t8 t4) \to (ex2 T (\lambda (t9: T).(pr0 +(THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 +(THead (Flat Appl) (lift (S O) O v2) t6)) t9)))))))) with [(pr0_refl t7) +\Rightarrow (\lambda (H29: (eq T t7 (THead (Bind b) u0 t5))).(\lambda (H30: +(eq T t7 t4)).(eq_ind T (THead (Bind b) u0 t5) (\lambda (t8: T).((eq T t8 t4) +\to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: +T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9))))) +(\lambda (H31: (eq T (THead (Bind b) u0 t5) t4)).(eq_ind T (THead (Bind b) u0 +t5) (\lambda (t8: T).(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t8) +t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) +O v2) t6)) t9)))) (ex2_ind T (\lambda (t8: T).(pr0 u2 t8)) (\lambda (t8: +T).(pr0 v2 t8)) (ex2 T (\lambda (t8: T).(pr0 (THead (Flat Appl) u2 (THead +(Bind b) u0 t5)) t8)) (\lambda (t8: T).(pr0 (THead (Bind b) u3 (THead (Flat +Appl) (lift (S O) O v2) t6)) t8))) (\lambda (x: T).(\lambda (H32: (pr0 u2 +x)).(\lambda (H33: (pr0 v2 x)).(pr0_confluence__pr0_cong_upsilon_refl b H16 +u0 u3 H18 t5 t6 H19 u2 v2 x H32 H33)))) (H26 v1 (tlt_head_sx (Flat Appl) v1 +(THead (Bind b) u0 t5)) u2 H27 v2 H17)) t4 H31)) t7 (sym_eq T t7 (THead (Bind +b) u0 t5) H29) H30))) | (pr0_comp u4 u5 H29 t7 t8 H30 k0) \Rightarrow +(\lambda (H31: (eq T (THead k0 u4 t7) (THead (Bind b) u0 t5))).(\lambda (H32: +(eq T (THead k0 u5 t8) t4)).((let H33 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t7 | +(TLRef _) \Rightarrow t7 | (THead _ _ t9) \Rightarrow t9])) (THead k0 u4 t7) +(THead (Bind b) u0 t5) H31) in ((let H34 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u4 | +(TLRef _) \Rightarrow u4 | (THead _ t9 _) \Rightarrow t9])) (THead k0 u4 t7) +(THead (Bind b) u0 t5) H31) in ((let H35 \def (f_equal T K (\lambda (e: +T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k0 | +(TLRef _) \Rightarrow k0 | (THead k1 _ _) \Rightarrow k1])) (THead k0 u4 t7) +(THead (Bind b) u0 t5) H31) in (eq_ind K (Bind b) (\lambda (k1: K).((eq T u4 +u0) \to ((eq T t7 t5) \to ((eq T (THead k1 u5 t8) t4) \to ((pr0 u4 u5) \to +((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) +(\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) +t6)) t9))))))))) (\lambda (H36: (eq T u4 u0)).(eq_ind T u0 (\lambda (t9: +T).((eq T t7 t5) \to ((eq T (THead (Bind b) u5 t8) t4) \to ((pr0 t9 u5) \to +((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) +t10)) (\lambda (t10: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S +O) O v2) t6)) t10)))))))) (\lambda (H37: (eq T t7 t5)).(eq_ind T t5 (\lambda +(t9: T).((eq T (THead (Bind b) u5 t8) t4) \to ((pr0 u0 u5) \to ((pr0 t9 t8) +\to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda +(t10: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) +t10))))))) (\lambda (H38: (eq T (THead (Bind b) u5 t8) t4)).(eq_ind T (THead +(Bind b) u5 t8) (\lambda (t9: T).((pr0 u0 u5) \to ((pr0 t5 t8) \to (ex2 T +(\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t9) t10)) (\lambda (t10: T).(pr0 +(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t10)))))) +(\lambda (H39: (pr0 u0 u5)).(\lambda (H40: (pr0 t5 t8)).(ex2_ind T (\lambda +(t9: T).(pr0 t8 t9)) (\lambda (t9: T).(pr0 t6 t9)) (ex2 T (\lambda (t9: +T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 t8)) t9)) (\lambda (t9: +T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9))) +(\lambda (x: T).(\lambda (H41: (pr0 t8 x)).(\lambda (H42: (pr0 t6 +x)).(ex2_ind T (\lambda (t9: T).(pr0 u5 t9)) (\lambda (t9: T).(pr0 u3 t9)) +(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 t8)) +t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) +O v2) t6)) t9))) (\lambda (x0: T).(\lambda (H43: (pr0 u5 x0)).(\lambda (H44: +(pr0 u3 x0)).(ex2_ind T (\lambda (t9: T).(pr0 u2 t9)) (\lambda (t9: T).(pr0 +v2 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 +t8)) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift +(S O) O v2) t6)) t9))) (\lambda (x1: T).(\lambda (H45: (pr0 u2 x1)).(\lambda +(H46: (pr0 v2 x1)).(pr0_confluence__pr0_cong_upsilon_cong b H16 u2 v2 x1 H45 +H46 t8 t6 x H41 H42 u5 u3 x0 H43 H44)))) (H26 v1 (tlt_head_sx (Flat Appl) v1 +(THead (Bind b) u0 t5)) u2 H27 v2 H17))))) (H26 u0 (tlt_trans (THead (Bind b) +u0 t5) u0 (THead (Flat Appl) v1 (THead (Bind b) u0 t5)) (tlt_head_sx (Bind b) +u0 t5) (tlt_head_dx (Flat Appl) v1 (THead (Bind b) u0 t5))) u5 H39 u3 +H18))))) (H26 t5 (tlt_trans (THead (Bind b) u0 t5) t5 (THead (Flat Appl) v1 +(THead (Bind b) u0 t5)) (tlt_head_dx (Bind b) u0 t5) (tlt_head_dx (Flat Appl) +v1 (THead (Bind b) u0 t5))) t8 H40 t6 H19)))) t4 H38)) t7 (sym_eq T t7 t5 +H37))) u4 (sym_eq T u4 u0 H36))) k0 (sym_eq K k0 (Bind b) H35))) H34)) H33)) +H32 H29 H30))) | (pr0_beta u v0 v3 H29 t7 t8 H30) \Rightarrow (\lambda (H31: +(eq T (THead (Flat Appl) v0 (THead (Bind Abst) u t7)) (THead (Bind b) u0 +t5))).(\lambda (H32: (eq T (THead (Bind Abbr) v3 t8) t4)).((let H33 \def +(eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u t7)) (\lambda (e: +T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow +False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in +K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) +\Rightarrow True])])) I (THead (Bind b) u0 t5) H31) in (False_ind ((eq T +(THead (Bind Abbr) v3 t8) t4) \to ((pr0 v0 v3) \to ((pr0 t7 t8) \to (ex2 T +(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 +(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9)))))) H33)) +H32 H29 H30))) | (pr0_upsilon b0 H29 v0 v3 H30 u4 u5 H31 t7 t8 H32) +\Rightarrow (\lambda (H33: (eq T (THead (Flat Appl) v0 (THead (Bind b0) u4 +t7)) (THead (Bind b) u0 t5))).(\lambda (H34: (eq T (THead (Bind b0) u5 (THead +(Flat Appl) (lift (S O) O v3) t8)) t4)).((let H35 \def (eq_ind T (THead (Flat +Appl) v0 (THead (Bind b0) u4 t7)) (\lambda (e: T).(match e in T return +(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda +(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow +True])])) I (THead (Bind b) u0 t5) H33) in (False_ind ((eq T (THead (Bind b0) +u5 (THead (Flat Appl) (lift (S O) O v3) t8)) t4) \to ((not (eq B b0 Abst)) +\to ((pr0 v0 v3) \to ((pr0 u4 u5) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: +T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) +u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9)))))))) H35)) H34 H29 H30 H31 +H32))) | (pr0_delta u4 u5 H29 t7 t8 H30 w H31) \Rightarrow (\lambda (H32: (eq +T (THead (Bind Abbr) u4 t7) (THead (Bind b) u0 t5))).(\lambda (H33: (eq T +(THead (Bind Abbr) u5 w) t4)).((let H34 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t7 | +(TLRef _) \Rightarrow t7 | (THead _ _ t9) \Rightarrow t9])) (THead (Bind +Abbr) u4 t7) (THead (Bind b) u0 t5) H32) in ((let H35 \def (f_equal T T +(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow u4 | (TLRef _) \Rightarrow u4 | (THead _ t9 _) \Rightarrow t9])) +(THead (Bind Abbr) u4 t7) (THead (Bind b) u0 t5) H32) in ((let H36 \def +(f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with +[(TSort _) \Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | (THead k0 _ _) +\Rightarrow (match k0 in K return (\lambda (_: K).B) with [(Bind b0) +\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (THead (Bind Abbr) u4 t7) +(THead (Bind b) u0 t5) H32) in (eq_ind B Abbr (\lambda (b0: B).((eq T u4 u0) +\to ((eq T t7 t5) \to ((eq T (THead (Bind Abbr) u5 w) t4) \to ((pr0 u4 u5) +\to ((pr0 t7 t8) \to ((subst0 O u5 t8 w) \to (ex2 T (\lambda (t9: T).(pr0 +(THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind b0) u3 +(THead (Flat Appl) (lift (S O) O v2) t6)) t9)))))))))) (\lambda (H37: (eq T +u4 u0)).(eq_ind T u0 (\lambda (t9: T).((eq T t7 t5) \to ((eq T (THead (Bind +Abbr) u5 w) t4) \to ((pr0 t9 u5) \to ((pr0 t7 t8) \to ((subst0 O u5 t8 w) \to +(ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda (t10: +T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) +t10))))))))) (\lambda (H38: (eq T t7 t5)).(eq_ind T t5 (\lambda (t9: T).((eq +T (THead (Bind Abbr) u5 w) t4) \to ((pr0 u0 u5) \to ((pr0 t9 t8) \to ((subst0 +O u5 t8 w) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) +(\lambda (t10: T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O +v2) t6)) t10)))))))) (\lambda (H39: (eq T (THead (Bind Abbr) u5 w) +t4)).(eq_ind T (THead (Bind Abbr) u5 w) (\lambda (t9: T).((pr0 u0 u5) \to +((pr0 t5 t8) \to ((subst0 O u5 t8 w) \to (ex2 T (\lambda (t10: T).(pr0 (THead +(Flat Appl) u2 t9) t10)) (\lambda (t10: T).(pr0 (THead (Bind Abbr) u3 (THead +(Flat Appl) (lift (S O) O v2) t6)) t10))))))) (\lambda (H40: (pr0 u0 +u5)).(\lambda (H41: (pr0 t5 t8)).(\lambda (H42: (subst0 O u5 t8 w)).(let H43 +\def (eq_ind_r B b (\lambda (b0: B).(\forall (v: T).((tlt v (THead (Flat +Appl) v1 (THead (Bind b0) u0 t5))) \to (\forall (t9: T).((pr0 v t9) \to +(\forall (t10: T).((pr0 v t10) \to (ex2 T (\lambda (t11: T).(pr0 t9 t11)) +(\lambda (t11: T).(pr0 t10 t11)))))))))) H26 Abbr H36) in (let H44 \def +(eq_ind_r B b (\lambda (b0: B).(eq T t3 (THead (Bind b0) u0 t5))) H23 Abbr +H36) in (let H45 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 Abst))) +H16 Abbr H36) in (ex2_ind T (\lambda (t9: T).(pr0 t8 t9)) (\lambda (t9: +T).(pr0 t6 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead +(Bind Abbr) u5 w)) t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) u3 (THead +(Flat Appl) (lift (S O) O v2) t6)) t9))) (\lambda (x: T).(\lambda (H46: (pr0 +t8 x)).(\lambda (H47: (pr0 t6 x)).(ex2_ind T (\lambda (t9: T).(pr0 u5 t9)) +(\lambda (t9: T).(pr0 u3 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) +u2 (THead (Bind Abbr) u5 w)) t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) u3 +(THead (Flat Appl) (lift (S O) O v2) t6)) t9))) (\lambda (x0: T).(\lambda +(H48: (pr0 u5 x0)).(\lambda (H49: (pr0 u3 x0)).(ex2_ind T (\lambda (t9: +T).(pr0 u2 t9)) (\lambda (t9: T).(pr0 v2 t9)) (ex2 T (\lambda (t9: T).(pr0 +(THead (Flat Appl) u2 (THead (Bind Abbr) u5 w)) t9)) (\lambda (t9: T).(pr0 +(THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9))) +(\lambda (x1: T).(\lambda (H50: (pr0 u2 x1)).(\lambda (H51: (pr0 v2 +x1)).(pr0_confluence__pr0_cong_upsilon_delta H45 u5 t8 w H42 u2 v2 x1 H50 H51 +t6 x H46 H47 u3 x0 H48 H49)))) (H43 v1 (tlt_head_sx (Flat Appl) v1 (THead +(Bind Abbr) u0 t5)) u2 H27 v2 H17))))) (H43 u0 (tlt_trans (THead (Bind Abbr) +u0 t5) u0 (THead (Flat Appl) v1 (THead (Bind Abbr) u0 t5)) (tlt_head_sx (Bind +Abbr) u0 t5) (tlt_head_dx (Flat Appl) v1 (THead (Bind Abbr) u0 t5))) u5 H40 +u3 H18))))) (H43 t5 (tlt_trans (THead (Bind Abbr) u0 t5) t5 (THead (Flat +Appl) v1 (THead (Bind Abbr) u0 t5)) (tlt_head_dx (Bind Abbr) u0 t5) +(tlt_head_dx (Flat Appl) v1 (THead (Bind Abbr) u0 t5))) t8 H41 t6 H19)))))))) +t4 H39)) t7 (sym_eq T t7 t5 H38))) u4 (sym_eq T u4 u0 H37))) b H36)) H35)) +H34)) H33 H29 H30 H31))) | (pr0_zeta b0 H29 t7 t8 H30 u) \Rightarrow (\lambda +(H31: (eq T (THead (Bind b0) u (lift (S O) O t7)) (THead (Bind b) u0 +t5))).(\lambda (H32: (eq T t8 t4)).((let H33 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let +rec lref_map (f: ((nat \to nat))) (d: nat) (t9: T) on t9: T \def (match t9 +with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match +(blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k0 +u4 t10) \Rightarrow (THead k0 (lref_map f d u4) (lref_map f (s k0 d) t10))]) +in lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (TLRef _) \Rightarrow +((let rec lref_map (f: ((nat \to nat))) (d: nat) (t9: T) on t9: T \def (match +t9 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef +(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | +(THead k0 u4 t10) \Rightarrow (THead k0 (lref_map f d u4) (lref_map f (s k0 +d) t10))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (THead _ _ +t9) \Rightarrow t9])) (THead (Bind b0) u (lift (S O) O t7)) (THead (Bind b) +u0 t5) H31) in ((let H34 \def (f_equal T T (\lambda (e: T).(match e in T +return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) +\Rightarrow u | (THead _ t9 _) \Rightarrow t9])) (THead (Bind b0) u (lift (S +O) O t7)) (THead (Bind b) u0 t5) H31) in ((let H35 \def (f_equal T B (\lambda +(e: T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b0 +| (TLRef _) \Rightarrow b0 | (THead k0 _ _) \Rightarrow (match k0 in K return +(\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow +b0])])) (THead (Bind b0) u (lift (S O) O t7)) (THead (Bind b) u0 t5) H31) in +(eq_ind B b (\lambda (b1: B).((eq T u u0) \to ((eq T (lift (S O) O t7) t5) +\to ((eq T t8 t4) \to ((not (eq B b1 Abst)) \to ((pr0 t7 t8) \to (ex2 T +(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 +(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9))))))))) +(\lambda (H36: (eq T u u0)).(eq_ind T u0 (\lambda (_: T).((eq T (lift (S O) O +t7) t5) \to ((eq T t8 t4) \to ((not (eq B b Abst)) \to ((pr0 t7 t8) \to (ex2 +T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda (t10: +T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) +t10)))))))) (\lambda (H37: (eq T (lift (S O) O t7) t5)).(eq_ind T (lift (S O) +O t7) (\lambda (_: T).((eq T t8 t4) \to ((not (eq B b Abst)) \to ((pr0 t7 t8) +\to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda +(t10: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) +t10))))))) (\lambda (H38: (eq T t8 t4)).(eq_ind T t4 (\lambda (t9: T).((not +(eq B b Abst)) \to ((pr0 t7 t9) \to (ex2 T (\lambda (t10: T).(pr0 (THead +(Flat Appl) u2 t4) t10)) (\lambda (t10: T).(pr0 (THead (Bind b) u3 (THead +(Flat Appl) (lift (S O) O v2) t6)) t10)))))) (\lambda (H39: (not (eq B b +Abst))).(\lambda (H40: (pr0 t7 t4)).(let H41 \def (eq_ind_r T t5 (\lambda +(t9: T).(\forall (v: T).((tlt v (THead (Flat Appl) v1 (THead (Bind b) u0 +t9))) \to (\forall (t10: T).((pr0 v t10) \to (\forall (t11: T).((pr0 v t11) +\to (ex2 T (\lambda (t12: T).(pr0 t10 t12)) (\lambda (t12: T).(pr0 t11 +t12)))))))))) H26 (lift (S O) O t7) H37) in (let H42 \def (eq_ind_r T t5 +(\lambda (t9: T).(eq T t3 (THead (Bind b) u0 t9))) H23 (lift (S O) O t7) H37) +in (let H43 \def (eq_ind_r T t5 (\lambda (t9: T).(pr0 t9 t6)) H19 (lift (S O) +O t7) H37) in (ex2_ind T (\lambda (t9: T).(eq T t6 (lift (S O) O t9))) +(\lambda (t9: T).(pr0 t7 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) +u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift +(S O) O v2) t6)) t9))) (\lambda (x: T).(\lambda (H44: (eq T t6 (lift (S O) O +x))).(\lambda (H45: (pr0 t7 x)).(eq_ind_r T (lift (S O) O x) (\lambda (t9: +T).(ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda +(t10: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t9)) +t10)))) (ex2_ind T (\lambda (t9: T).(pr0 x t9)) (\lambda (t9: T).(pr0 t4 t9)) +(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: +T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) (lift (S O) O +x))) t9))) (\lambda (x0: T).(\lambda (H46: (pr0 x x0)).(\lambda (H47: (pr0 t4 +x0)).(ex2_ind T (\lambda (t9: T).(pr0 u2 t9)) (\lambda (t9: T).(pr0 v2 t9)) +(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: +T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) (lift (S O) O +x))) t9))) (\lambda (x1: T).(\lambda (H48: (pr0 u2 x1)).(\lambda (H49: (pr0 +v2 x1)).(pr0_confluence__pr0_cong_upsilon_zeta b H39 u0 u3 H18 u2 v2 x1 H48 +H49 x t4 x0 H46 H47)))) (H41 v1 (tlt_head_sx (Flat Appl) v1 (THead (Bind b) +u0 (lift (S O) O t7))) u2 H27 v2 H17))))) (H41 t7 (tlt_trans (THead (Bind b) +u0 (lift (S O) O t7)) t7 (THead (Flat Appl) v1 (THead (Bind b) u0 (lift (S O) +O t7))) (lift_tlt_dx (Bind b) u0 t7 (S O) O) (tlt_head_dx (Flat Appl) v1 +(THead (Bind b) u0 (lift (S O) O t7)))) x H45 t4 H40)) t6 H44)))) +(pr0_gen_lift t7 t6 (S O) O H43))))))) t8 (sym_eq T t8 t4 H38))) t5 H37)) u +(sym_eq T u u0 H36))) b0 (sym_eq B b0 b H35))) H34)) H33)) H32 H29 H30))) | +(pr0_tau t7 t8 H29 u) \Rightarrow (\lambda (H30: (eq T (THead (Flat Cast) u +t7) (THead (Bind b) u0 t5))).(\lambda (H31: (eq T t8 t4)).((let H32 \def +(eq_ind T (THead (Flat Cast) u t7) (\lambda (e: T).(match e in T return +(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda +(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow +True])])) I (THead (Bind b) u0 t5) H30) in (False_ind ((eq T t8 t4) \to ((pr0 +t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) +(\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) +t6)) t9))))) H32)) H31 H29)))]) in (H29 (refl_equal T (THead (Bind b) u0 t5)) +(refl_equal T t4))))) k H25))))) H22)) H21))))))) t2 H15)) t H13 H14 H9 H10 +H11 H12))) | (pr0_delta u0 u3 H9 t5 t6 H10 w H11) \Rightarrow (\lambda (H12: +(eq T (THead (Bind Abbr) u0 t5) t)).(\lambda (H13: (eq T (THead (Bind Abbr) +u3 w) t2)).(eq_ind T (THead (Bind Abbr) u0 t5) (\lambda (_: T).((eq T (THead +(Bind Abbr) u3 w) t2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to ((subst0 O u3 t6 +w) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: +T).(pr0 t2 t8)))))))) (\lambda (H14: (eq T (THead (Bind Abbr) u3 w) +t2)).(eq_ind T (THead (Bind Abbr) u3 w) (\lambda (t7: T).((pr0 u0 u3) \to +((pr0 t5 t6) \to ((subst0 O u3 t6 w) \to (ex2 T (\lambda (t8: T).(pr0 (THead +k u2 t4) t8)) (\lambda (t8: T).(pr0 t7 t8))))))) (\lambda (H15: (pr0 u0 +u3)).(\lambda (H16: (pr0 t5 t6)).(\lambda (H17: (subst0 O u3 t6 w)).(let H18 +\def (eq_ind_r T t (\lambda (t7: T).(eq T (THead k u1 t3) t7)) H4 (THead +(Bind Abbr) u0 t5) H12) in (let H19 \def (f_equal T K (\lambda (e: T).(match +e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) +\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) (THead (Bind +Abbr) u0 t5) H18) in ((let H20 \def (f_equal T T (\lambda (e: T).(match e in +T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) +\Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t3) (THead +(Bind Abbr) u0 t5) H18) in ((let H21 \def (f_equal T T (\lambda (e: T).(match +e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) +\Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t3) (THead +(Bind Abbr) u0 t5) H18) in (\lambda (H22: (eq T u1 u0)).(\lambda (H23: (eq K +k (Bind Abbr))).(let H24 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: +T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v +t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 +t10)))))))))) H (THead (Bind Abbr) u0 t5) H12) in (eq_ind_r K (Bind Abbr) +(\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 (THead k0 u2 t4) t7)) (\lambda +(t7: T).(pr0 (THead (Bind Abbr) u3 w) t7)))) (let H25 \def (eq_ind T u1 +(\lambda (t7: T).(pr0 t7 u2)) H7 u0 H22) in (let H26 \def (eq_ind T t3 +(\lambda (t7: T).(pr0 t7 t4)) H8 t5 H21) in (ex2_ind T (\lambda (t7: T).(pr0 +t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead +(Bind Abbr) u2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u3 w) t7))) +(\lambda (x: T).(\lambda (H27: (pr0 t4 x)).(\lambda (H28: (pr0 t6 +x)).(ex2_ind T (\lambda (t7: T).(pr0 u2 t7)) (\lambda (t7: T).(pr0 u3 t7)) +(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 t4) t7)) (\lambda (t7: +T).(pr0 (THead (Bind Abbr) u3 w) t7))) (\lambda (x0: T).(\lambda (H29: (pr0 +u2 x0)).(\lambda (H30: (pr0 u3 x0)).(pr0_confluence__pr0_cong_delta u3 t6 w +H17 u2 x0 H29 H30 t4 x H27 H28)))) (H24 u0 (tlt_head_sx (Bind Abbr) u0 t5) u2 +H25 u3 H15))))) (H24 t5 (tlt_head_dx (Bind Abbr) u0 t5) t4 H26 t6 H16)))) k +H23))))) H20)) H19)))))) t2 H14)) t H12 H13 H9 H10 H11))) | (pr0_zeta b H9 t5 +t6 H10 u) \Rightarrow (\lambda (H11: (eq T (THead (Bind b) u (lift (S O) O +t5)) t)).(\lambda (H12: (eq T t6 t2)).(eq_ind T (THead (Bind b) u (lift (S O) +O t5)) (\lambda (_: T).((eq T t6 t2) \to ((not (eq B b Abst)) \to ((pr0 t5 +t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: +T).(pr0 t2 t8))))))) (\lambda (H13: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: +T).((not (eq B b Abst)) \to ((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 +(THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H14: (not +(eq B b Abst))).(\lambda (H15: (pr0 t5 t2)).(let H16 \def (eq_ind_r T t +(\lambda (t7: T).(eq T (THead k u1 t3) t7)) H4 (THead (Bind b) u (lift (S O) +O t5)) H11) in (let H17 \def (f_equal T K (\lambda (e: T).(match e in T +return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) +\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) (THead (Bind +b) u (lift (S O) O t5)) H16) in ((let H18 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | +(TLRef _) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t3) +(THead (Bind b) u (lift (S O) O t5)) H16) in ((let H19 \def (f_equal T T +(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) +(THead k u1 t3) (THead (Bind b) u (lift (S O) O t5)) H16) in (\lambda (H20: +(eq T u1 u)).(\lambda (H21: (eq K k (Bind b))).(let H22 \def (eq_ind_r T t +(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) +\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) +(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Bind b) u (lift (S O) O +t5)) H11) in (eq_ind_r K (Bind b) (\lambda (k0: K).(ex2 T (\lambda (t7: +T).(pr0 (THead k0 u2 t4) t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let H23 \def +(eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H7 u H20) in (let H24 \def (eq_ind +T t3 (\lambda (t7: T).(pr0 t7 t4)) H8 (lift (S O) O t5) H19) in (ex2_ind T +(\lambda (t7: T).(eq T t4 (lift (S O) O t7))) (\lambda (t7: T).(pr0 t5 t7)) +(ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 t4) t7)) (\lambda (t7: +T).(pr0 t2 t7))) (\lambda (x: T).(\lambda (H25: (eq T t4 (lift (S O) O +x))).(\lambda (H26: (pr0 t5 x)).(eq_ind_r T (lift (S O) O x) (\lambda (t7: +T).(ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 t7) t8)) (\lambda (t8: +T).(pr0 t2 t8)))) (ex2_ind T (\lambda (t7: T).(pr0 x t7)) (\lambda (t7: +T).(pr0 t2 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (lift (S O) O +x)) t7)) (\lambda (t7: T).(pr0 t2 t7))) (\lambda (x0: T).(\lambda (H27: (pr0 +x x0)).(\lambda (H28: (pr0 t2 x0)).(ex_intro2 T (\lambda (t7: T).(pr0 (THead +(Bind b) u2 (lift (S O) O x)) t7)) (\lambda (t7: T).(pr0 t2 t7)) x0 (pr0_zeta +b H14 x x0 H27 u2) H28)))) (H22 t5 (lift_tlt_dx (Bind b) u t5 (S O) O) x H26 +t2 H15)) t4 H25)))) (pr0_gen_lift t5 t4 (S O) O H24)))) k H21))))) H18)) +H17))))) t6 (sym_eq T t6 t2 H13))) t H11 H12 H9 H10))) | (pr0_tau t5 t6 H9 u) +\Rightarrow (\lambda (H10: (eq T (THead (Flat Cast) u t5) t)).(\lambda (H11: +(eq T t6 t2)).(eq_ind T (THead (Flat Cast) u t5) (\lambda (_: T).((eq T t6 +t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) +(\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H12: (eq T t6 t2)).(eq_ind T t2 +(\lambda (t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 +t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))) (\lambda (H13: (pr0 t5 t2)).(let +H14 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead k u1 t3) t7)) H4 (THead +(Flat Cast) u t5) H10) in (let H15 \def (f_equal T K (\lambda (e: T).(match e +in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) +\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) (THead (Flat +Cast) u t5) H14) in ((let H16 \def (f_equal T T (\lambda (e: T).(match e in T +return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) +\Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t3) (THead +(Flat Cast) u t5) H14) in ((let H17 \def (f_equal T T (\lambda (e: T).(match +e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) +\Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t3) (THead +(Flat Cast) u t5) H14) in (\lambda (H18: (eq T u1 u)).(\lambda (H19: (eq K k +(Flat Cast))).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: +T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v +t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 +t10)))))))))) H (THead (Flat Cast) u t5) H10) in (eq_ind_r K (Flat Cast) +(\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 (THead k0 u2 t4) t7)) (\lambda +(t7: T).(pr0 t2 t7)))) (let H21 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 +u2)) H7 u H18) in (let H22 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H8 +t5 H17) in (ex2_ind T (\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t2 +t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Flat Cast) u2 t4) t7)) (\lambda +(t7: T).(pr0 t2 t7))) (\lambda (x: T).(\lambda (H23: (pr0 t4 x)).(\lambda +(H24: (pr0 t2 x)).(ex_intro2 T (\lambda (t7: T).(pr0 (THead (Flat Cast) u2 +t4) t7)) (\lambda (t7: T).(pr0 t2 t7)) x (pr0_tau t4 x H23 u2) H24)))) (H20 +t5 (tlt_head_dx (Flat Cast) u t5) t4 H22 t2 H13)))) k H19))))) H16)) H15)))) +t6 (sym_eq T t6 t2 H12))) t H10 H11 H9)))]) in (H9 (refl_equal T t) +(refl_equal T t2))))) t1 H6)) t H4 H5 H2 H3))) | (pr0_beta u v1 v2 H2 t3 t4 +H3) \Rightarrow (\lambda (H4: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) +u t3)) t)).(\lambda (H5: (eq T (THead (Bind Abbr) v2 t4) t1)).(eq_ind T +(THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (\lambda (_: T).((eq T (THead +(Bind Abbr) v2 t4) t1) \to ((pr0 v1 v2) \to ((pr0 t3 t4) \to (ex2 T (\lambda +(t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6))))))) (\lambda (H6: (eq T +(THead (Bind Abbr) v2 t4) t1)).(eq_ind T (THead (Bind Abbr) v2 t4) (\lambda +(t5: T).((pr0 v1 v2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t5 +t6)) (\lambda (t6: T).(pr0 t2 t6)))))) (\lambda (H7: (pr0 v1 v2)).(\lambda +(H8: (pr0 t3 t4)).(let H9 \def (match H1 in pr0 return (\lambda (t5: +T).(\lambda (t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 t) \to ((eq T t6 t2) +\to (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: +T).(pr0 t2 t7)))))))) with [(pr0_refl t5) \Rightarrow (\lambda (H9: (eq T t5 +t)).(\lambda (H10: (eq T t5 t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2) +\to (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: +T).(pr0 t2 t7))))) (\lambda (H11: (eq T t t2)).(eq_ind T t2 (\lambda (_: +T).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: +T).(pr0 t2 t7)))) (let H12 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 t2)) +H11 (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) H4) in (eq_ind T (THead +(Flat Appl) v1 (THead (Bind Abst) u t3)) (\lambda (t6: T).(ex2 T (\lambda +(t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 t6 t7)))) +(let H13 \def (eq_ind_r T t (\lambda (t6: T).(eq T t5 t6)) H9 (THead (Flat +Appl) v1 (THead (Bind Abst) u t3)) H4) in (let H14 \def (eq_ind_r T t +(\lambda (t6: T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) +\to (\forall (t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) +(\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead (Flat Appl) v1 (THead (Bind +Abst) u t3)) H4) in (ex_intro2 T (\lambda (t6: T).(pr0 (THead (Bind Abbr) v2 +t4) t6)) (\lambda (t6: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u +t3)) t6)) (THead (Bind Abbr) v2 t4) (pr0_refl (THead (Bind Abbr) v2 t4)) +(pr0_beta u v1 v2 H7 t3 t4 H8)))) t2 H12)) t (sym_eq T t t2 H11))) t5 (sym_eq +T t5 t H9) H10))) | (pr0_comp u1 u2 H9 t5 t6 H10 k) \Rightarrow (\lambda +(H11: (eq T (THead k u1 t5) t)).(\lambda (H12: (eq T (THead k u2 t6) +t2)).(eq_ind T (THead k u1 t5) (\lambda (_: T).((eq T (THead k u2 t6) t2) \to +((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind +Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H13: (eq T +(THead k u2 t6) t2)).(eq_ind T (THead k u2 t6) (\lambda (t7: T).((pr0 u1 u2) +\to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) +t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (H14: (pr0 u1 u2)).(\lambda +(H15: (pr0 t5 t6)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead +(Flat Appl) v1 (THead (Bind Abst) u t3)) t7)) H4 (THead k u1 t5) H11) in (let +H17 \def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) +with [(TSort _) \Rightarrow (Flat Appl) | (TLRef _) \Rightarrow (Flat Appl) | +(THead k0 _ _) \Rightarrow k0])) (THead (Flat Appl) v1 (THead (Bind Abst) u +t3)) (THead k u1 t5) H16) in ((let H18 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | +(TLRef _) \Rightarrow v1 | (THead _ t7 _) \Rightarrow t7])) (THead (Flat +Appl) v1 (THead (Bind Abst) u t3)) (THead k u1 t5) H16) in ((let H19 \def +(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with +[(TSort _) \Rightarrow (THead (Bind Abst) u t3) | (TLRef _) \Rightarrow +(THead (Bind Abst) u t3) | (THead _ _ t7) \Rightarrow t7])) (THead (Flat +Appl) v1 (THead (Bind Abst) u t3)) (THead k u1 t5) H16) in (\lambda (H20: (eq +T v1 u1)).(\lambda (H21: (eq K (Flat Appl) k)).(eq_ind K (Flat Appl) (\lambda +(k0: K).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda +(t7: T).(pr0 (THead k0 u2 t6) t7)))) (let H22 \def (eq_ind_r K k (\lambda +(k0: K).(eq T (THead k0 u1 t5) t)) H11 (Flat Appl) H21) in (let H23 \def +(eq_ind_r T t5 (\lambda (t7: T).(pr0 t7 t6)) H15 (THead (Bind Abst) u t3) +H19) in (let H24 \def (match H23 in pr0 return (\lambda (t7: T).(\lambda (t8: +T).(\lambda (_: (pr0 t7 t8)).((eq T t7 (THead (Bind Abst) u t3)) \to ((eq T +t8 t6) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) +(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t6) t9)))))))) with [(pr0_refl +t7) \Rightarrow (\lambda (H24: (eq T t7 (THead (Bind Abst) u t3))).(\lambda +(H25: (eq T t7 t6)).(eq_ind T (THead (Bind Abst) u t3) (\lambda (t8: T).((eq +T t8 t6) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) +(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t6) t9))))) (\lambda (H26: (eq T +(THead (Bind Abst) u t3) t6)).(eq_ind T (THead (Bind Abst) u t3) (\lambda +(t8: T).(ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda +(t9: T).(pr0 (THead (Flat Appl) u2 t8) t9)))) (let H27 \def (eq_ind_r T t5 +(\lambda (t8: T).(eq T (THead (Flat Appl) u1 t8) t)) H22 (THead (Bind Abst) u +t3) H19) in (let H28 \def (eq_ind_r T t (\lambda (t8: T).(\forall (v: +T).((tlt v t8) \to (\forall (t9: T).((pr0 v t9) \to (\forall (t10: T).((pr0 v +t10) \to (ex2 T (\lambda (t11: T).(pr0 t9 t11)) (\lambda (t11: T).(pr0 t10 +t11)))))))))) H (THead (Flat Appl) u1 (THead (Bind Abst) u t3)) H27) in (let +H29 \def (eq_ind T v1 (\lambda (t8: T).(pr0 t8 v2)) H7 u1 H20) in (ex2_ind T +(\lambda (t8: T).(pr0 v2 t8)) (\lambda (t8: T).(pr0 u2 t8)) (ex2 T (\lambda +(t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 (THead +(Flat Appl) u2 (THead (Bind Abst) u t3)) t8))) (\lambda (x: T).(\lambda (H30: +(pr0 v2 x)).(\lambda (H31: (pr0 u2 x)).(ex_intro2 T (\lambda (t8: T).(pr0 +(THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 (THead (Flat Appl) u2 +(THead (Bind Abst) u t3)) t8)) (THead (Bind Abbr) x t4) (pr0_comp v2 x H30 t4 +t4 (pr0_refl t4) (Bind Abbr)) (pr0_beta u u2 x H31 t3 t4 H8))))) (H28 u1 +(tlt_head_sx (Flat Appl) u1 (THead (Bind Abst) u t3)) v2 H29 u2 H14))))) t6 +H26)) t7 (sym_eq T t7 (THead (Bind Abst) u t3) H24) H25))) | (pr0_comp u0 u3 +H24 t7 t8 H25 k0) \Rightarrow (\lambda (H26: (eq T (THead k0 u0 t7) (THead +(Bind Abst) u t3))).(\lambda (H27: (eq T (THead k0 u3 t8) t6)).((let H28 \def +(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with +[(TSort _) \Rightarrow t7 | (TLRef _) \Rightarrow t7 | (THead _ _ t9) +\Rightarrow t9])) (THead k0 u0 t7) (THead (Bind Abst) u t3) H26) in ((let H29 +\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) +with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t9 _) +\Rightarrow t9])) (THead k0 u0 t7) (THead (Bind Abst) u t3) H26) in ((let H30 +\def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) +with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _ _) +\Rightarrow k1])) (THead k0 u0 t7) (THead (Bind Abst) u t3) H26) in (eq_ind K +(Bind Abst) (\lambda (k1: K).((eq T u0 u) \to ((eq T t7 t3) \to ((eq T (THead +k1 u3 t8) t6) \to ((pr0 u0 u3) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: +T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Flat +Appl) u2 t6) t9))))))))) (\lambda (H31: (eq T u0 u)).(eq_ind T u (\lambda +(t9: T).((eq T t7 t3) \to ((eq T (THead (Bind Abst) u3 t8) t6) \to ((pr0 t9 +u3) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 +t4) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t6) t10)))))))) +(\lambda (H32: (eq T t7 t3)).(eq_ind T t3 (\lambda (t9: T).((eq T (THead +(Bind Abst) u3 t8) t6) \to ((pr0 u u3) \to ((pr0 t9 t8) \to (ex2 T (\lambda +(t10: T).(pr0 (THead (Bind Abbr) v2 t4) t10)) (\lambda (t10: T).(pr0 (THead +(Flat Appl) u2 t6) t10))))))) (\lambda (H33: (eq T (THead (Bind Abst) u3 t8) +t6)).(eq_ind T (THead (Bind Abst) u3 t8) (\lambda (t9: T).((pr0 u u3) \to +((pr0 t3 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t4) +t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t9) t10)))))) (\lambda (_: +(pr0 u u3)).(\lambda (H35: (pr0 t3 t8)).(let H36 \def (eq_ind_r T t5 (\lambda +(t9: T).(eq T (THead (Flat Appl) u1 t9) t)) H22 (THead (Bind Abst) u t3) H19) +in (let H37 \def (eq_ind_r T t (\lambda (t9: T).(\forall (v: T).((tlt v t9) +\to (\forall (t10: T).((pr0 v t10) \to (\forall (t11: T).((pr0 v t11) \to +(ex2 T (\lambda (t12: T).(pr0 t10 t12)) (\lambda (t12: T).(pr0 t11 +t12)))))))))) H (THead (Flat Appl) u1 (THead (Bind Abst) u t3)) H36) in (let +H38 \def (eq_ind T v1 (\lambda (t9: T).(pr0 t9 v2)) H7 u1 H20) in (ex2_ind T +(\lambda (t9: T).(pr0 v2 t9)) (\lambda (t9: T).(pr0 u2 t9)) (ex2 T (\lambda +(t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead +(Flat Appl) u2 (THead (Bind Abst) u3 t8)) t9))) (\lambda (x: T).(\lambda +(H39: (pr0 v2 x)).(\lambda (H40: (pr0 u2 x)).(ex2_ind T (\lambda (t9: T).(pr0 +t8 t9)) (\lambda (t9: T).(pr0 t4 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead +(Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead +(Bind Abst) u3 t8)) t9))) (\lambda (x0: T).(\lambda (H41: (pr0 t8 +x0)).(\lambda (H42: (pr0 t4 x0)).(ex_intro2 T (\lambda (t9: T).(pr0 (THead +(Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead +(Bind Abst) u3 t8)) t9)) (THead (Bind Abbr) x x0) (pr0_comp v2 x H39 t4 x0 +H42 (Bind Abbr)) (pr0_beta u3 u2 x H40 t8 x0 H41))))) (H37 t3 (tlt_trans +(THead (Bind Abst) u t3) t3 (THead (Flat Appl) u1 (THead (Bind Abst) u t3)) +(tlt_head_dx (Bind Abst) u t3) (tlt_head_dx (Flat Appl) u1 (THead (Bind Abst) +u t3))) t8 H35 t4 H8))))) (H37 u1 (tlt_head_sx (Flat Appl) u1 (THead (Bind +Abst) u t3)) v2 H38 u2 H14))))))) t6 H33)) t7 (sym_eq T t7 t3 H32))) u0 +(sym_eq T u0 u H31))) k0 (sym_eq K k0 (Bind Abst) H30))) H29)) H28)) H27 H24 +H25))) | (pr0_beta u0 v0 v3 H24 t7 t8 H25) \Rightarrow (\lambda (H26: (eq T +(THead (Flat Appl) v0 (THead (Bind Abst) u0 t7)) (THead (Bind Abst) u +t3))).(\lambda (H27: (eq T (THead (Bind Abbr) v3 t8) t6)).((let H28 \def +(eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u0 t7)) (\lambda (e: +T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow +False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in +K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) +\Rightarrow True])])) I (THead (Bind Abst) u t3) H26) in (False_ind ((eq T +(THead (Bind Abbr) v3 t8) t6) \to ((pr0 v0 v3) \to ((pr0 t7 t8) \to (ex2 T +(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 +(THead (Flat Appl) u2 t6) t9)))))) H28)) H27 H24 H25))) | (pr0_upsilon b H24 +v0 v3 H25 u0 u3 H26 t7 t8 H27) \Rightarrow (\lambda (H28: (eq T (THead (Flat +Appl) v0 (THead (Bind b) u0 t7)) (THead (Bind Abst) u t3))).(\lambda (H29: +(eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v3) t8)) t6)).((let +H30 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind b) u0 t7)) (\lambda (e: +T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow +False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in +K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) +\Rightarrow True])])) I (THead (Bind Abst) u t3) H28) in (False_ind ((eq T +(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v3) t8)) t6) \to ((not +(eq B b Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3) \to ((pr0 t7 t8) \to (ex2 T +(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 +(THead (Flat Appl) u2 t6) t9)))))))) H30)) H29 H24 H25 H26 H27))) | +(pr0_delta u0 u3 H24 t7 t8 H25 w H26) \Rightarrow (\lambda (H27: (eq T (THead +(Bind Abbr) u0 t7) (THead (Bind Abst) u t3))).(\lambda (H28: (eq T (THead +(Bind Abbr) u3 w) t6)).((let H29 \def (eq_ind T (THead (Bind Abbr) u0 t7) +(\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) +\Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow +(match k0 in K return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match +b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst +\Rightarrow False | Void \Rightarrow False]) | (Flat _) \Rightarrow +False])])) I (THead (Bind Abst) u t3) H27) in (False_ind ((eq T (THead (Bind +Abbr) u3 w) t6) \to ((pr0 u0 u3) \to ((pr0 t7 t8) \to ((subst0 O u3 t8 w) \to +(ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: +T).(pr0 (THead (Flat Appl) u2 t6) t9))))))) H29)) H28 H24 H25 H26))) | +(pr0_zeta b H24 t7 t8 H25 u0) \Rightarrow (\lambda (H26: (eq T (THead (Bind +b) u0 (lift (S O) O t7)) (THead (Bind Abst) u t3))).(\lambda (H27: (eq T t8 +t6)).((let H28 \def (f_equal T T (\lambda (e: T).(match e in T return +(\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat +\to nat))) (d: nat) (t9: T) on t9: T \def (match t9 with [(TSort n) +\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with +[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k0 u3 t10) +\Rightarrow (THead k0 (lref_map f d u3) (lref_map f (s k0 d) t10))]) in +lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (TLRef _) \Rightarrow +((let rec lref_map (f: ((nat \to nat))) (d: nat) (t9: T) on t9: T \def (match +t9 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef +(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | +(THead k0 u3 t10) \Rightarrow (THead k0 (lref_map f d u3) (lref_map f (s k0 +d) t10))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (THead _ _ +t9) \Rightarrow t9])) (THead (Bind b) u0 (lift (S O) O t7)) (THead (Bind +Abst) u t3) H26) in ((let H29 \def (f_equal T T (\lambda (e: T).(match e in T +return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) +\Rightarrow u0 | (THead _ t9 _) \Rightarrow t9])) (THead (Bind b) u0 (lift (S +O) O t7)) (THead (Bind Abst) u t3) H26) in ((let H30 \def (f_equal T B +(\lambda (e: T).(match e in T return (\lambda (_: T).B) with [(TSort _) +\Rightarrow b | (TLRef _) \Rightarrow b | (THead k0 _ _) \Rightarrow (match +k0 in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) +\Rightarrow b])])) (THead (Bind b) u0 (lift (S O) O t7)) (THead (Bind Abst) u +t3) H26) in (eq_ind B Abst (\lambda (b0: B).((eq T u0 u) \to ((eq T (lift (S +O) O t7) t3) \to ((eq T t8 t6) \to ((not (eq B b0 Abst)) \to ((pr0 t7 t8) \to +(ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: +T).(pr0 (THead (Flat Appl) u2 t6) t9))))))))) (\lambda (H31: (eq T u0 +u)).(eq_ind T u (\lambda (_: T).((eq T (lift (S O) O t7) t3) \to ((eq T t8 +t6) \to ((not (eq B Abst Abst)) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: +T).(pr0 (THead (Bind Abbr) v2 t4) t10)) (\lambda (t10: T).(pr0 (THead (Flat +Appl) u2 t6) t10)))))))) (\lambda (H32: (eq T (lift (S O) O t7) t3)).(eq_ind +T (lift (S O) O t7) (\lambda (_: T).((eq T t8 t6) \to ((not (eq B Abst Abst)) +\to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t4) +t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t6) t10))))))) (\lambda +(H33: (eq T t8 t6)).(eq_ind T t6 (\lambda (t9: T).((not (eq B Abst Abst)) \to +((pr0 t7 t9) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t4) +t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t6) t10)))))) (\lambda +(H34: (not (eq B Abst Abst))).(\lambda (_: (pr0 t7 t6)).(let H36 \def (match +(H34 (refl_equal B Abst)) in False return (\lambda (_: False).(ex2 T (\lambda +(t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead +(Flat Appl) u2 t6) t9)))) with []) in H36))) t8 (sym_eq T t8 t6 H33))) t3 +H32)) u0 (sym_eq T u0 u H31))) b (sym_eq B b Abst H30))) H29)) H28)) H27 H24 +H25))) | (pr0_tau t7 t8 H24 u0) \Rightarrow (\lambda (H25: (eq T (THead (Flat +Cast) u0 t7) (THead (Bind Abst) u t3))).(\lambda (H26: (eq T t8 t6)).((let +H27 \def (eq_ind T (THead (Flat Cast) u0 t7) (\lambda (e: T).(match e in T +return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda +(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow +True])])) I (THead (Bind Abst) u t3) H25) in (False_ind ((eq T t8 t6) \to +((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) +(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t6) t9))))) H27)) H26 H24)))]) in +(H24 (refl_equal T (THead (Bind Abst) u t3)) (refl_equal T t6))))) k H21)))) +H18)) H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_beta u0 v0 v3 H9 t5 t6 +H10) \Rightarrow (\lambda (H11: (eq T (THead (Flat Appl) v0 (THead (Bind +Abst) u0 t5)) t)).(\lambda (H12: (eq T (THead (Bind Abbr) v3 t6) t2)).(eq_ind +T (THead (Flat Appl) v0 (THead (Bind Abst) u0 t5)) (\lambda (_: T).((eq T +(THead (Bind Abbr) v3 t6) t2) \to ((pr0 v0 v3) \to ((pr0 t5 t6) \to (ex2 T +(\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 +t8))))))) (\lambda (H13: (eq T (THead (Bind Abbr) v3 t6) t2)).(eq_ind T +(THead (Bind Abbr) v3 t6) (\lambda (t7: T).((pr0 v0 v3) \to ((pr0 t5 t6) \to +(ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: +T).(pr0 t7 t8)))))) (\lambda (H14: (pr0 v0 v3)).(\lambda (H15: (pr0 t5 +t6)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1 +(THead (Bind Abst) u t3)) t7)) H4 (THead (Flat Appl) v0 (THead (Bind Abst) u0 +t5)) H11) in (let H17 \def (f_equal T T (\lambda (e: T).(match e in T return +(\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 +| (THead _ t7 _) \Rightarrow t7])) (THead (Flat Appl) v1 (THead (Bind Abst) u +t3)) (THead (Flat Appl) v0 (THead (Bind Abst) u0 t5)) H16) in ((let H18 \def +(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with +[(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ _ t7) +\Rightarrow (match t7 in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t8 _) \Rightarrow t8])])) +(THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (THead (Flat Appl) v0 (THead +(Bind Abst) u0 t5)) H16) in ((let H19 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | +(TLRef _) \Rightarrow t3 | (THead _ _ t7) \Rightarrow (match t7 in T return +(\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 +| (THead _ _ t8) \Rightarrow t8])])) (THead (Flat Appl) v1 (THead (Bind Abst) +u t3)) (THead (Flat Appl) v0 (THead (Bind Abst) u0 t5)) H16) in (\lambda (_: +(eq T u u0)).(\lambda (H21: (eq T v1 v0)).(let H22 \def (eq_ind_r T t +(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) +\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) +(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Appl) v0 (THead (Bind +Abst) u0 t5)) H11) in (let H23 \def (eq_ind T v1 (\lambda (t7: T).(pr0 t7 +v2)) H7 v0 H21) in (let H24 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) +H8 t5 H19) in (ex2_ind T (\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 +t6 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda +(t7: T).(pr0 (THead (Bind Abbr) v3 t6) t7))) (\lambda (x: T).(\lambda (H25: +(pr0 t4 x)).(\lambda (H26: (pr0 t6 x)).(ex2_ind T (\lambda (t7: T).(pr0 v2 +t7)) (\lambda (t7: T).(pr0 v3 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind +Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v3 t6) t7))) +(\lambda (x0: T).(\lambda (H27: (pr0 v2 x0)).(\lambda (H28: (pr0 v3 +x0)).(ex_intro2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) +(\lambda (t7: T).(pr0 (THead (Bind Abbr) v3 t6) t7)) (THead (Bind Abbr) x0 x) +(pr0_comp v2 x0 H27 t4 x H25 (Bind Abbr)) (pr0_comp v3 x0 H28 t6 x H26 (Bind +Abbr)))))) (H22 v0 (tlt_head_sx (Flat Appl) v0 (THead (Bind Abst) u0 t5)) v2 +H23 v3 H14))))) (H22 t5 (tlt_trans (THead (Bind Abst) u0 t5) t5 (THead (Flat +Appl) v0 (THead (Bind Abst) u0 t5)) (tlt_head_dx (Bind Abst) u0 t5) +(tlt_head_dx (Flat Appl) v0 (THead (Bind Abst) u0 t5))) t4 H24 t6 H15)))))))) +H18)) H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_upsilon b H9 v0 v3 H10 u1 +u2 H11 t5 t6 H12) \Rightarrow (\lambda (H13: (eq T (THead (Flat Appl) v0 +(THead (Bind b) u1 t5)) t)).(\lambda (H14: (eq T (THead (Bind b) u2 (THead +(Flat Appl) (lift (S O) O v3) t6)) t2)).(eq_ind T (THead (Flat Appl) v0 +(THead (Bind b) u1 t5)) (\lambda (_: T).((eq T (THead (Bind b) u2 (THead +(Flat Appl) (lift (S O) O v3) t6)) t2) \to ((not (eq B b Abst)) \to ((pr0 v0 +v3) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead +(Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H15: +(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v3) t6)) +t2)).(eq_ind T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v3) t6)) +(\lambda (t7: T).((not (eq B b Abst)) \to ((pr0 v0 v3) \to ((pr0 u1 u2) \to +((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) +(\lambda (t8: T).(pr0 t7 t8)))))))) (\lambda (H16: (not (eq B b +Abst))).(\lambda (_: (pr0 v0 v3)).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (pr0 +t5 t6)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) +v1 (THead (Bind Abst) u t3)) t7)) H4 (THead (Flat Appl) v0 (THead (Bind b) u1 +t5)) H13) in (let H21 \def (f_equal T T (\lambda (e: T).(match e in T return +(\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 +| (THead _ t7 _) \Rightarrow t7])) (THead (Flat Appl) v1 (THead (Bind Abst) u +t3)) (THead (Flat Appl) v0 (THead (Bind b) u1 t5)) H20) in ((let H22 \def +(f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with +[(TSort _) \Rightarrow Abst | (TLRef _) \Rightarrow Abst | (THead _ _ t7) +\Rightarrow (match t7 in T return (\lambda (_: T).B) with [(TSort _) +\Rightarrow Abst | (TLRef _) \Rightarrow Abst | (THead k _ _) \Rightarrow +(match k in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | +(Flat _) \Rightarrow Abst])])])) (THead (Flat Appl) v1 (THead (Bind Abst) u +t3)) (THead (Flat Appl) v0 (THead (Bind b) u1 t5)) H20) in ((let H23 \def +(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with +[(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ _ t7) +\Rightarrow (match t7 in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t8 _) \Rightarrow t8])])) +(THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (THead (Flat Appl) v0 (THead +(Bind b) u1 t5)) H20) in ((let H24 \def (f_equal T T (\lambda (e: T).(match e +in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) +\Rightarrow t3 | (THead _ _ t7) \Rightarrow (match t7 in T return (\lambda +(_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead +_ _ t8) \Rightarrow t8])])) (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) +(THead (Flat Appl) v0 (THead (Bind b) u1 t5)) H20) in (\lambda (_: (eq T u +u1)).(\lambda (H26: (eq B Abst b)).(\lambda (_: (eq T v1 v0)).(eq_ind B Abst +(\lambda (b0: B).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) +(\lambda (t7: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O +v3) t6)) t7)))) (let H28 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 +Abst))) H16 Abst H26) in (let H29 \def (match (H28 (refl_equal B Abst)) in +False return (\lambda (_: False).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind +Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abst) u2 (THead (Flat +Appl) (lift (S O) O v3) t6)) t7)))) with []) in H29)) b H26))))) H23)) H22)) +H21))))))) t2 H15)) t H13 H14 H9 H10 H11 H12))) | (pr0_delta u1 u2 H9 t5 t6 +H10 w H11) \Rightarrow (\lambda (H12: (eq T (THead (Bind Abbr) u1 t5) +t)).(\lambda (H13: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind +Abbr) u1 t5) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t2) \to ((pr0 u1 +u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T (\lambda (t8: T).(pr0 +(THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8)))))))) (\lambda +(H14: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u2 w) +(\lambda (t7: T).((pr0 u1 u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to +(ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: +T).(pr0 t7 t8))))))) (\lambda (_: (pr0 u1 u2)).(\lambda (_: (pr0 t5 +t6)).(\lambda (_: (subst0 O u2 t6 w)).(let H18 \def (eq_ind_r T t (\lambda +(t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) t7)) H4 (THead +(Bind Abbr) u1 t5) H12) in (let H19 \def (eq_ind T (THead (Flat Appl) v1 +(THead (Bind Abst) u t3)) (\lambda (ee: T).(match ee in T return (\lambda (_: +T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | +(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with +[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind +Abbr) u1 t5) H18) in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind +Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7))) +H19)))))) t2 H14)) t H12 H13 H9 H10 H11))) | (pr0_zeta b H9 t5 t6 H10 u0) +\Rightarrow (\lambda (H11: (eq T (THead (Bind b) u0 (lift (S O) O t5)) +t)).(\lambda (H12: (eq T t6 t2)).(eq_ind T (THead (Bind b) u0 (lift (S O) O +t5)) (\lambda (_: T).((eq T t6 t2) \to ((not (eq B b Abst)) \to ((pr0 t5 t6) +\to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: +T).(pr0 t2 t8))))))) (\lambda (H13: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: +T).((not (eq B b Abst)) \to ((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 +(THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (_: +(not (eq B b Abst))).(\lambda (_: (pr0 t5 t2)).(let H16 \def (eq_ind_r T t +(\lambda (t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) t7)) +H4 (THead (Bind b) u0 (lift (S O) O t5)) H11) in (let H17 \def (eq_ind T +(THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (\lambda (ee: T).(match ee in +T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda +(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow +True])])) I (THead (Bind b) u0 (lift (S O) O t5)) H16) in (False_ind (ex2 T +(\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 t2 +t7))) H17))))) t6 (sym_eq T t6 t2 H13))) t H11 H12 H9 H10))) | (pr0_tau t5 t6 +H9 u0) \Rightarrow (\lambda (H10: (eq T (THead (Flat Cast) u0 t5) +t)).(\lambda (H11: (eq T t6 t2)).(eq_ind T (THead (Flat Cast) u0 t5) (\lambda +(_: T).((eq T t6 t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead +(Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H12: (eq +T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8: +T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))) +(\lambda (_: (pr0 t5 t2)).(let H14 \def (eq_ind_r T t (\lambda (t7: T).(eq T +(THead (Flat Appl) v1 (THead (Bind Abst) u t3)) t7)) H4 (THead (Flat Cast) u0 +t5) H10) in (let H15 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) +u t3)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with +[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) +\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) +\Rightarrow False | (Flat f) \Rightarrow (match f in F return (\lambda (_: +F).Prop) with [Appl \Rightarrow True | Cast \Rightarrow False])])])) I (THead +(Flat Cast) u0 t5) H14) in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead +(Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 t2 t7))) H15)))) t6 (sym_eq T +t6 t2 H12))) t H10 H11 H9)))]) in (H9 (refl_equal T t) (refl_equal T t2))))) +t1 H6)) t H4 H5 H2 H3))) | (pr0_upsilon b H2 v1 v2 H3 u1 u2 H4 t3 t4 H5) +\Rightarrow (\lambda (H6: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) +t)).(\lambda (H7: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O +v2) t4)) t1)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) +(\lambda (_: T).((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O +v2) t4)) t1) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to +((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 +t2 t6))))))))) (\lambda (H8: (eq T (THead (Bind b) u2 (THead (Flat Appl) +(lift (S O) O v2) t4)) t1)).(eq_ind T (THead (Bind b) u2 (THead (Flat Appl) +(lift (S O) O v2) t4)) (\lambda (t5: T).((not (eq B b Abst)) \to ((pr0 v1 v2) +\to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t5 t6)) +(\lambda (t6: T).(pr0 t2 t6)))))))) (\lambda (H9: (not (eq B b +Abst))).(\lambda (H10: (pr0 v1 v2)).(\lambda (H11: (pr0 u1 u2)).(\lambda +(H12: (pr0 t3 t4)).(let H13 \def (match H1 in pr0 return (\lambda (t5: +T).(\lambda (t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 t) \to ((eq T t6 t2) +\to (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift +(S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 t2 t7)))))))) with [(pr0_refl t5) +\Rightarrow (\lambda (H13: (eq T t5 t)).(\lambda (H14: (eq T t5 t2)).(eq_ind +T t (\lambda (t6: T).((eq T t6 t2) \to (ex2 T (\lambda (t7: T).(pr0 (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: +T).(pr0 t2 t7))))) (\lambda (H15: (eq T t t2)).(eq_ind T t2 (\lambda (_: +T).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift +(S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let H16 \def (eq_ind_r +T t (\lambda (t6: T).(eq T t6 t2)) H15 (THead (Flat Appl) v1 (THead (Bind b) +u1 t3)) H6) in (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) +(\lambda (t6: T).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat +Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 t6 t7)))) (let H17 +\def (eq_ind_r T t (\lambda (t6: T).(eq T t5 t6)) H13 (THead (Flat Appl) v1 +(THead (Bind b) u1 t3)) H6) in (let H18 \def (eq_ind_r T t (\lambda (t6: +T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall +(t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: +T).(pr0 t8 t9)))))))))) H (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) H6) +in (ex2_sym T (pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 t3))) (pr0 (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) +(pr0_confluence__pr0_cong_upsilon_refl b H9 u1 u2 H11 t3 t4 H12 v1 v2 v2 H10 +(pr0_refl v2))))) t2 H16)) t (sym_eq T t t2 H15))) t5 (sym_eq T t5 t H13) +H14))) | (pr0_comp u0 u3 H13 t5 t6 H14 k) \Rightarrow (\lambda (H15: (eq T +(THead k u0 t5) t)).(\lambda (H16: (eq T (THead k u3 t6) t2)).(eq_ind T +(THead k u0 t5) (\lambda (_: T).((eq T (THead k u3 t6) t2) \to ((pr0 u0 u3) +\to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead +(Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) +(\lambda (H17: (eq T (THead k u3 t6) t2)).(eq_ind T (THead k u3 t6) (\lambda +(t7: T).((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: +T).(pr0 t7 t8)))))) (\lambda (H18: (pr0 u0 u3)).(\lambda (H19: (pr0 t5 +t6)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1 +(THead (Bind b) u1 t3)) t7)) H6 (THead k u0 t5) H15) in (let H21 \def +(f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with +[(TSort _) \Rightarrow (Flat Appl) | (TLRef _) \Rightarrow (Flat Appl) | +(THead k0 _ _) \Rightarrow k0])) (THead (Flat Appl) v1 (THead (Bind b) u1 +t3)) (THead k u0 t5) H20) in ((let H22 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | +(TLRef _) \Rightarrow v1 | (THead _ t7 _) \Rightarrow t7])) (THead (Flat +Appl) v1 (THead (Bind b) u1 t3)) (THead k u0 t5) H20) in ((let H23 \def +(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with +[(TSort _) \Rightarrow (THead (Bind b) u1 t3) | (TLRef _) \Rightarrow (THead +(Bind b) u1 t3) | (THead _ _ t7) \Rightarrow t7])) (THead (Flat Appl) v1 +(THead (Bind b) u1 t3)) (THead k u0 t5) H20) in (\lambda (H24: (eq T v1 +u0)).(\lambda (H25: (eq K (Flat Appl) k)).(eq_ind K (Flat Appl) (\lambda (k0: +K).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift +(S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 (THead k0 u3 t6) t7)))) (let H26 +\def (eq_ind_r K k (\lambda (k0: K).(eq T (THead k0 u0 t5) t)) H15 (Flat +Appl) H25) in (let H27 \def (eq_ind_r T t5 (\lambda (t7: T).(pr0 t7 t6)) H19 +(THead (Bind b) u1 t3) H23) in (let H28 \def (match H27 in pr0 return +(\lambda (t7: T).(\lambda (t8: T).(\lambda (_: (pr0 t7 t8)).((eq T t7 (THead +(Bind b) u1 t3)) \to ((eq T t8 t6) \to (ex2 T (\lambda (t9: T).(pr0 (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: +T).(pr0 (THead (Flat Appl) u3 t6) t9)))))))) with [(pr0_refl t7) \Rightarrow +(\lambda (H28: (eq T t7 (THead (Bind b) u1 t3))).(\lambda (H29: (eq T t7 +t6)).(eq_ind T (THead (Bind b) u1 t3) (\lambda (t8: T).((eq T t8 t6) \to (ex2 +T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O +v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9))))) +(\lambda (H30: (eq T (THead (Bind b) u1 t3) t6)).(eq_ind T (THead (Bind b) u1 +t3) (\lambda (t8: T).(ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead +(Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat +Appl) u3 t8) t9)))) (let H31 \def (eq_ind_r T t5 (\lambda (t8: T).(eq T +(THead (Flat Appl) u0 t8) t)) H26 (THead (Bind b) u1 t3) H23) in (let H32 +\def (eq_ind_r T t (\lambda (t8: T).(\forall (v: T).((tlt v t8) \to (\forall +(t9: T).((pr0 v t9) \to (\forall (t10: T).((pr0 v t10) \to (ex2 T (\lambda +(t11: T).(pr0 t9 t11)) (\lambda (t11: T).(pr0 t10 t11)))))))))) H (THead +(Flat Appl) u0 (THead (Bind b) u1 t3)) H31) in (let H33 \def (eq_ind T v1 +(\lambda (t8: T).(pr0 t8 v2)) H10 u0 H24) in (ex2_ind T (\lambda (t8: T).(pr0 +v2 t8)) (\lambda (t8: T).(pr0 u3 t8)) (ex2 T (\lambda (t8: T).(pr0 (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: +T).(pr0 (THead (Flat Appl) u3 (THead (Bind b) u1 t3)) t8))) (\lambda (x: +T).(\lambda (H34: (pr0 v2 x)).(\lambda (H35: (pr0 u3 x)).(ex2_sym T (pr0 +(THead (Flat Appl) u3 (THead (Bind b) u1 t3))) (pr0 (THead (Bind b) u2 (THead +(Flat Appl) (lift (S O) O v2) t4))) (pr0_confluence__pr0_cong_upsilon_refl b +H9 u1 u2 H11 t3 t4 H12 u3 v2 x H35 H34))))) (H32 u0 (tlt_head_sx (Flat Appl) +u0 (THead (Bind b) u1 t3)) v2 H33 u3 H18))))) t6 H30)) t7 (sym_eq T t7 (THead +(Bind b) u1 t3) H28) H29))) | (pr0_comp u4 u5 H28 t7 t8 H29 k0) \Rightarrow +(\lambda (H30: (eq T (THead k0 u4 t7) (THead (Bind b) u1 t3))).(\lambda (H31: +(eq T (THead k0 u5 t8) t6)).((let H32 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t7 | +(TLRef _) \Rightarrow t7 | (THead _ _ t9) \Rightarrow t9])) (THead k0 u4 t7) +(THead (Bind b) u1 t3) H30) in ((let H33 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u4 | +(TLRef _) \Rightarrow u4 | (THead _ t9 _) \Rightarrow t9])) (THead k0 u4 t7) +(THead (Bind b) u1 t3) H30) in ((let H34 \def (f_equal T K (\lambda (e: +T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k0 | +(TLRef _) \Rightarrow k0 | (THead k1 _ _) \Rightarrow k1])) (THead k0 u4 t7) +(THead (Bind b) u1 t3) H30) in (eq_ind K (Bind b) (\lambda (k1: K).((eq T u4 +u1) \to ((eq T t7 t3) \to ((eq T (THead k1 u5 t8) t6) \to ((pr0 u4 u5) \to +((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat +Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 +t6) t9))))))))) (\lambda (H35: (eq T u4 u1)).(eq_ind T u1 (\lambda (t9: +T).((eq T t7 t3) \to ((eq T (THead (Bind b) u5 t8) t6) \to ((pr0 t9 u5) \to +((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind b) u2 (THead +(Flat Appl) (lift (S O) O v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat +Appl) u3 t6) t10)))))))) (\lambda (H36: (eq T t7 t3)).(eq_ind T t3 (\lambda +(t9: T).((eq T (THead (Bind b) u5 t8) t6) \to ((pr0 u1 u5) \to ((pr0 t9 t8) +\to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift +(S O) O v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6) +t10))))))) (\lambda (H37: (eq T (THead (Bind b) u5 t8) t6)).(eq_ind T (THead +(Bind b) u5 t8) (\lambda (t9: T).((pr0 u1 u5) \to ((pr0 t3 t8) \to (ex2 T +(\lambda (t10: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O +v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t9) t10)))))) +(\lambda (H38: (pr0 u1 u5)).(\lambda (H39: (pr0 t3 t8)).(let H40 \def +(eq_ind_r T t5 (\lambda (t9: T).(eq T (THead (Flat Appl) u0 t9) t)) H26 +(THead (Bind b) u1 t3) H23) in (let H41 \def (eq_ind_r T t (\lambda (t9: +T).(\forall (v: T).((tlt v t9) \to (\forall (t10: T).((pr0 v t10) \to +(\forall (t11: T).((pr0 v t11) \to (ex2 T (\lambda (t12: T).(pr0 t10 t12)) +(\lambda (t12: T).(pr0 t11 t12)))))))))) H (THead (Flat Appl) u0 (THead (Bind +b) u1 t3)) H40) in (let H42 \def (eq_ind T v1 (\lambda (t9: T).(pr0 t9 v2)) +H10 u0 H24) in (ex2_ind T (\lambda (t9: T).(pr0 v2 t9)) (\lambda (t9: T).(pr0 +u3 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) +(lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 +(THead (Bind b) u5 t8)) t9))) (\lambda (x: T).(\lambda (H43: (pr0 v2 +x)).(\lambda (H44: (pr0 u3 x)).(ex2_ind T (\lambda (t9: T).(pr0 t8 t9)) +(\lambda (t9: T).(pr0 t4 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 +(THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead +(Flat Appl) u3 (THead (Bind b) u5 t8)) t9))) (\lambda (x0: T).(\lambda (H45: +(pr0 t8 x0)).(\lambda (H46: (pr0 t4 x0)).(ex2_ind T (\lambda (t9: T).(pr0 u5 +t9)) (\lambda (t9: T).(pr0 u2 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Bind +b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 +(THead (Flat Appl) u3 (THead (Bind b) u5 t8)) t9))) (\lambda (x1: T).(\lambda +(H47: (pr0 u5 x1)).(\lambda (H48: (pr0 u2 x1)).(ex2_sym T (pr0 (THead (Flat +Appl) u3 (THead (Bind b) u5 t8))) (pr0 (THead (Bind b) u2 (THead (Flat Appl) +(lift (S O) O v2) t4))) (pr0_confluence__pr0_cong_upsilon_cong b H9 u3 v2 x +H44 H43 t8 t4 x0 H45 H46 u5 u2 x1 H47 H48))))) (H41 u1 (tlt_trans (THead +(Bind b) u1 t3) u1 (THead (Flat Appl) u0 (THead (Bind b) u1 t3)) (tlt_head_sx +(Bind b) u1 t3) (tlt_head_dx (Flat Appl) u0 (THead (Bind b) u1 t3))) u5 H38 +u2 H11))))) (H41 t3 (tlt_trans (THead (Bind b) u1 t3) t3 (THead (Flat Appl) +u0 (THead (Bind b) u1 t3)) (tlt_head_dx (Bind b) u1 t3) (tlt_head_dx (Flat +Appl) u0 (THead (Bind b) u1 t3))) t8 H39 t4 H12))))) (H41 u0 (tlt_head_sx +(Flat Appl) u0 (THead (Bind b) u1 t3)) v2 H42 u3 H18))))))) t6 H37)) t7 +(sym_eq T t7 t3 H36))) u4 (sym_eq T u4 u1 H35))) k0 (sym_eq K k0 (Bind b) +H34))) H33)) H32)) H31 H28 H29))) | (pr0_beta u v0 v3 H28 t7 t8 H29) +\Rightarrow (\lambda (H30: (eq T (THead (Flat Appl) v0 (THead (Bind Abst) u +t7)) (THead (Bind b) u1 t3))).(\lambda (H31: (eq T (THead (Bind Abbr) v3 t8) +t6)).((let H32 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u t7)) +(\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) +\Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow +(match k0 in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False +| (Flat _) \Rightarrow True])])) I (THead (Bind b) u1 t3) H30) in (False_ind +((eq T (THead (Bind Abbr) v3 t8) t6) \to ((pr0 v0 v3) \to ((pr0 t7 t8) \to +(ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S +O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9)))))) +H32)) H31 H28 H29))) | (pr0_upsilon b0 H28 v0 v3 H29 u4 u5 H30 t7 t8 H31) +\Rightarrow (\lambda (H32: (eq T (THead (Flat Appl) v0 (THead (Bind b0) u4 +t7)) (THead (Bind b) u1 t3))).(\lambda (H33: (eq T (THead (Bind b0) u5 (THead +(Flat Appl) (lift (S O) O v3) t8)) t6)).((let H34 \def (eq_ind T (THead (Flat +Appl) v0 (THead (Bind b0) u4 t7)) (\lambda (e: T).(match e in T return +(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda +(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow +True])])) I (THead (Bind b) u1 t3) H32) in (False_ind ((eq T (THead (Bind b0) +u5 (THead (Flat Appl) (lift (S O) O v3) t8)) t6) \to ((not (eq B b0 Abst)) +\to ((pr0 v0 v3) \to ((pr0 u4 u5) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: +T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) +(\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9)))))))) H34)) H33 H28 H29 +H30 H31))) | (pr0_delta u4 u5 H28 t7 t8 H29 w H30) \Rightarrow (\lambda (H31: +(eq T (THead (Bind Abbr) u4 t7) (THead (Bind b) u1 t3))).(\lambda (H32: (eq T +(THead (Bind Abbr) u5 w) t6)).((let H33 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t7 | +(TLRef _) \Rightarrow t7 | (THead _ _ t9) \Rightarrow t9])) (THead (Bind +Abbr) u4 t7) (THead (Bind b) u1 t3) H31) in ((let H34 \def (f_equal T T +(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow u4 | (TLRef _) \Rightarrow u4 | (THead _ t9 _) \Rightarrow t9])) +(THead (Bind Abbr) u4 t7) (THead (Bind b) u1 t3) H31) in ((let H35 \def +(f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with +[(TSort _) \Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | (THead k0 _ _) +\Rightarrow (match k0 in K return (\lambda (_: K).B) with [(Bind b0) +\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (THead (Bind Abbr) u4 t7) +(THead (Bind b) u1 t3) H31) in (eq_ind B Abbr (\lambda (b0: B).((eq T u4 u1) +\to ((eq T t7 t3) \to ((eq T (THead (Bind Abbr) u5 w) t6) \to ((pr0 u4 u5) +\to ((pr0 t7 t8) \to ((subst0 O u5 t8 w) \to (ex2 T (\lambda (t9: T).(pr0 +(THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda +(t9: T).(pr0 (THead (Flat Appl) u3 t6) t9)))))))))) (\lambda (H36: (eq T u4 +u1)).(eq_ind T u1 (\lambda (t9: T).((eq T t7 t3) \to ((eq T (THead (Bind +Abbr) u5 w) t6) \to ((pr0 t9 u5) \to ((pr0 t7 t8) \to ((subst0 O u5 t8 w) \to +(ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift +(S O) O v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6) +t10))))))))) (\lambda (H37: (eq T t7 t3)).(eq_ind T t3 (\lambda (t9: T).((eq +T (THead (Bind Abbr) u5 w) t6) \to ((pr0 u1 u5) \to ((pr0 t9 t8) \to ((subst0 +O u5 t8 w) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) u2 (THead +(Flat Appl) (lift (S O) O v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat +Appl) u3 t6) t10)))))))) (\lambda (H38: (eq T (THead (Bind Abbr) u5 w) +t6)).(eq_ind T (THead (Bind Abbr) u5 w) (\lambda (t9: T).((pr0 u1 u5) \to +((pr0 t3 t8) \to ((subst0 O u5 t8 w) \to (ex2 T (\lambda (t10: T).(pr0 (THead +(Bind Abbr) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t10)) (\lambda (t10: +T).(pr0 (THead (Flat Appl) u3 t9) t10))))))) (\lambda (H39: (pr0 u1 +u5)).(\lambda (H40: (pr0 t3 t8)).(\lambda (H41: (subst0 O u5 t8 w)).(let H42 +\def (eq_ind_r B b (\lambda (b0: B).(eq T (THead (Bind b0) u1 t3) t5)) H23 +Abbr H35) in (let H43 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 +Abst))) H9 Abbr H35) in (let H44 \def (eq_ind_r B b (\lambda (b0: B).(eq T +(THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t1)) H8 Abbr +H35) in (let H45 \def (eq_ind_r T t5 (\lambda (t9: T).(eq T (THead (Flat +Appl) u0 t9) t)) H26 (THead (Bind Abbr) u1 t3) H42) in (let H46 \def +(eq_ind_r T t (\lambda (t9: T).(\forall (v: T).((tlt v t9) \to (\forall (t10: +T).((pr0 v t10) \to (\forall (t11: T).((pr0 v t11) \to (ex2 T (\lambda (t12: +T).(pr0 t10 t12)) (\lambda (t12: T).(pr0 t11 t12)))))))))) H (THead (Flat +Appl) u0 (THead (Bind Abbr) u1 t3)) H45) in (let H47 \def (eq_ind T v1 +(\lambda (t9: T).(pr0 t9 v2)) H10 u0 H24) in (ex2_ind T (\lambda (t9: T).(pr0 +v2 t9)) (\lambda (t9: T).(pr0 u3 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead +(Bind Abbr) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: +T).(pr0 (THead (Flat Appl) u3 (THead (Bind Abbr) u5 w)) t9))) (\lambda (x: +T).(\lambda (H48: (pr0 v2 x)).(\lambda (H49: (pr0 u3 x)).(ex2_ind T (\lambda +(t9: T).(pr0 t8 t9)) (\lambda (t9: T).(pr0 t4 t9)) (ex2 T (\lambda (t9: +T).(pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) +(\lambda (t9: T).(pr0 (THead (Flat Appl) u3 (THead (Bind Abbr) u5 w)) t9))) +(\lambda (x0: T).(\lambda (H50: (pr0 t8 x0)).(\lambda (H51: (pr0 t4 +x0)).(ex2_ind T (\lambda (t9: T).(pr0 u5 t9)) (\lambda (t9: T).(pr0 u2 t9)) +(ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift +(S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 (THead +(Bind Abbr) u5 w)) t9))) (\lambda (x1: T).(\lambda (H52: (pr0 u5 +x1)).(\lambda (H53: (pr0 u2 x1)).(ex2_sym T (pr0 (THead (Flat Appl) u3 (THead +(Bind Abbr) u5 w))) (pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift (S O) +O v2) t4))) (pr0_confluence__pr0_cong_upsilon_delta H43 u5 t8 w H41 u3 v2 x +H49 H48 t4 x0 H50 H51 u2 x1 H52 H53))))) (H46 u1 (tlt_trans (THead (Bind +Abbr) u1 t3) u1 (THead (Flat Appl) u0 (THead (Bind Abbr) u1 t3)) (tlt_head_sx +(Bind Abbr) u1 t3) (tlt_head_dx (Flat Appl) u0 (THead (Bind Abbr) u1 t3))) u5 +H39 u2 H11))))) (H46 t3 (tlt_trans (THead (Bind Abbr) u1 t3) t3 (THead (Flat +Appl) u0 (THead (Bind Abbr) u1 t3)) (tlt_head_dx (Bind Abbr) u1 t3) +(tlt_head_dx (Flat Appl) u0 (THead (Bind Abbr) u1 t3))) t8 H40 t4 H12))))) +(H46 u0 (tlt_head_sx (Flat Appl) u0 (THead (Bind Abbr) u1 t3)) v2 H47 u3 +H18))))))))))) t6 H38)) t7 (sym_eq T t7 t3 H37))) u4 (sym_eq T u4 u1 H36))) b +H35)) H34)) H33)) H32 H28 H29 H30))) | (pr0_zeta b0 H28 t7 t8 H29 u) +\Rightarrow (\lambda (H30: (eq T (THead (Bind b0) u (lift (S O) O t7)) (THead +(Bind b) u1 t3))).(\lambda (H31: (eq T t8 t6)).((let H32 \def (f_equal T T +(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t9: T) on t9: T +\def (match t9 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow +(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) +| (THead k0 u4 t10) \Rightarrow (THead k0 (lref_map f d u4) (lref_map f (s k0 +d) t10))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (TLRef _) +\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t9: T) on t9: T +\def (match t9 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow +(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) +| (THead k0 u4 t10) \Rightarrow (THead k0 (lref_map f d u4) (lref_map f (s k0 +d) t10))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (THead _ _ +t9) \Rightarrow t9])) (THead (Bind b0) u (lift (S O) O t7)) (THead (Bind b) +u1 t3) H30) in ((let H33 \def (f_equal T T (\lambda (e: T).(match e in T +return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) +\Rightarrow u | (THead _ t9 _) \Rightarrow t9])) (THead (Bind b0) u (lift (S +O) O t7)) (THead (Bind b) u1 t3) H30) in ((let H34 \def (f_equal T B (\lambda +(e: T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b0 +| (TLRef _) \Rightarrow b0 | (THead k0 _ _) \Rightarrow (match k0 in K return +(\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow +b0])])) (THead (Bind b0) u (lift (S O) O t7)) (THead (Bind b) u1 t3) H30) in +(eq_ind B b (\lambda (b1: B).((eq T u u1) \to ((eq T (lift (S O) O t7) t3) +\to ((eq T t8 t6) \to ((not (eq B b1 Abst)) \to ((pr0 t7 t8) \to (ex2 T +(\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) +t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9))))))))) +(\lambda (H35: (eq T u u1)).(eq_ind T u1 (\lambda (_: T).((eq T (lift (S O) O +t7) t3) \to ((eq T t8 t6) \to ((not (eq B b Abst)) \to ((pr0 t7 t8) \to (ex2 +T (\lambda (t10: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O +v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6) t10)))))))) +(\lambda (H36: (eq T (lift (S O) O t7) t3)).(eq_ind T (lift (S O) O t7) +(\lambda (_: T).((eq T t8 t6) \to ((not (eq B b Abst)) \to ((pr0 t7 t8) \to +(ex2 T (\lambda (t10: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S +O) O v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6) +t10))))))) (\lambda (H37: (eq T t8 t6)).(eq_ind T t6 (\lambda (t9: T).((not +(eq B b Abst)) \to ((pr0 t7 t9) \to (ex2 T (\lambda (t10: T).(pr0 (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t10)) (\lambda (t10: +T).(pr0 (THead (Flat Appl) u3 t6) t10)))))) (\lambda (H38: (not (eq B b +Abst))).(\lambda (H39: (pr0 t7 t6)).(let H40 \def (eq_ind_r T t3 (\lambda +(t9: T).(eq T (THead (Bind b) u1 t9) t5)) H23 (lift (S O) O t7) H36) in (let +H41 \def (eq_ind_r T t5 (\lambda (t9: T).(eq T (THead (Flat Appl) u0 t9) t)) +H26 (THead (Bind b) u1 (lift (S O) O t7)) H40) in (let H42 \def (eq_ind_r T t +(\lambda (t9: T).(\forall (v: T).((tlt v t9) \to (\forall (t10: T).((pr0 v +t10) \to (\forall (t11: T).((pr0 v t11) \to (ex2 T (\lambda (t12: T).(pr0 t10 +t12)) (\lambda (t12: T).(pr0 t11 t12)))))))))) H (THead (Flat Appl) u0 (THead +(Bind b) u1 (lift (S O) O t7))) H41) in (let H43 \def (eq_ind_r T t3 (\lambda +(t9: T).(pr0 t9 t4)) H12 (lift (S O) O t7) H36) in (ex2_ind T (\lambda (t9: +T).(eq T t4 (lift (S O) O t9))) (\lambda (t9: T).(pr0 t7 t9)) (ex2 T (\lambda +(t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) +t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9))) (\lambda (x: +T).(\lambda (H44: (eq T t4 (lift (S O) O x))).(\lambda (H45: (pr0 t7 +x)).(eq_ind_r T (lift (S O) O x) (\lambda (t9: T).(ex2 T (\lambda (t10: +T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t9)) t10)) +(\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6) t10)))) (let H46 \def +(eq_ind T v1 (\lambda (t9: T).(pr0 t9 v2)) H10 u0 H24) in (ex2_ind T (\lambda +(t9: T).(pr0 v2 t9)) (\lambda (t9: T).(pr0 u3 t9)) (ex2 T (\lambda (t9: +T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (lift (S O) O +x))) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9))) (\lambda (x0: +T).(\lambda (H47: (pr0 v2 x0)).(\lambda (H48: (pr0 u3 x0)).(ex2_ind T +(\lambda (t9: T).(pr0 x t9)) (\lambda (t9: T).(pr0 t6 t9)) (ex2 T (\lambda +(t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (lift (S +O) O x))) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9))) (\lambda +(x1: T).(\lambda (H49: (pr0 x x1)).(\lambda (H50: (pr0 t6 x1)).(ex2_sym T +(pr0 (THead (Flat Appl) u3 t6)) (pr0 (THead (Bind b) u2 (THead (Flat Appl) +(lift (S O) O v2) (lift (S O) O x)))) (pr0_confluence__pr0_cong_upsilon_zeta +b H38 u1 u2 H11 u3 v2 x0 H48 H47 x t6 x1 H49 H50))))) (H42 t7 (tlt_trans +(THead (Bind b) u1 (lift (S O) O t7)) t7 (THead (Flat Appl) u0 (THead (Bind +b) u1 (lift (S O) O t7))) (lift_tlt_dx (Bind b) u1 t7 (S O) O) (tlt_head_dx +(Flat Appl) u0 (THead (Bind b) u1 (lift (S O) O t7)))) x H45 t6 H39))))) (H42 +u0 (tlt_head_sx (Flat Appl) u0 (THead (Bind b) u1 (lift (S O) O t7))) v2 H46 +u3 H18))) t4 H44)))) (pr0_gen_lift t7 t4 (S O) O H43)))))))) t8 (sym_eq T t8 +t6 H37))) t3 H36)) u (sym_eq T u u1 H35))) b0 (sym_eq B b0 b H34))) H33)) +H32)) H31 H28 H29))) | (pr0_tau t7 t8 H28 u) \Rightarrow (\lambda (H29: (eq T +(THead (Flat Cast) u t7) (THead (Bind b) u1 t3))).(\lambda (H30: (eq T t8 +t6)).((let H31 \def (eq_ind T (THead (Flat Cast) u t7) (\lambda (e: T).(match +e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | +(TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K +return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) +\Rightarrow True])])) I (THead (Bind b) u1 t3) H29) in (False_ind ((eq T t8 +t6) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 +(THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead +(Flat Appl) u3 t6) t9))))) H31)) H30 H28)))]) in (H28 (refl_equal T (THead +(Bind b) u1 t3)) (refl_equal T t6))))) k H25)))) H22)) H21))))) t2 H17)) t +H15 H16 H13 H14))) | (pr0_beta u v0 v3 H13 t5 t6 H14) \Rightarrow (\lambda +(H15: (eq T (THead (Flat Appl) v0 (THead (Bind Abst) u t5)) t)).(\lambda +(H16: (eq T (THead (Bind Abbr) v3 t6) t2)).(eq_ind T (THead (Flat Appl) v0 +(THead (Bind Abst) u t5)) (\lambda (_: T).((eq T (THead (Bind Abbr) v3 t6) +t2) \to ((pr0 v0 v3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: +T).(pr0 t2 t8))))))) (\lambda (H17: (eq T (THead (Bind Abbr) v3 t6) +t2)).(eq_ind T (THead (Bind Abbr) v3 t6) (\lambda (t7: T).((pr0 v0 v3) \to +((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat +Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda +(_: (pr0 v0 v3)).(\lambda (_: (pr0 t5 t6)).(let H20 \def (eq_ind_r T t +(\lambda (t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) t7)) H6 +(THead (Flat Appl) v0 (THead (Bind Abst) u t5)) H15) in (let H21 \def +(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with +[(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t7 _) +\Rightarrow t7])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat +Appl) v0 (THead (Bind Abst) u t5)) H20) in ((let H22 \def (f_equal T B +(\lambda (e: T).(match e in T return (\lambda (_: T).B) with [(TSort _) +\Rightarrow b | (TLRef _) \Rightarrow b | (THead _ _ t7) \Rightarrow (match +t7 in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | (TLRef _) +\Rightarrow b | (THead k _ _) \Rightarrow (match k in K return (\lambda (_: +K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow b])])])) (THead +(Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) v0 (THead (Bind +Abst) u t5)) H20) in ((let H23 \def (f_equal T T (\lambda (e: T).(match e in +T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) +\Rightarrow u1 | (THead _ _ t7) \Rightarrow (match t7 in T return (\lambda +(_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead +_ t8 _) \Rightarrow t8])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) +(THead (Flat Appl) v0 (THead (Bind Abst) u t5)) H20) in ((let H24 \def +(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with +[(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7) +\Rightarrow (match t7 in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t8) \Rightarrow +t8])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) v0 +(THead (Bind Abst) u t5)) H20) in (\lambda (_: (eq T u1 u)).(\lambda (H26: +(eq B b Abst)).(\lambda (H27: (eq T v1 v0)).(let H28 \def (eq_ind_r T t +(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) +\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) +(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Appl) v0 (THead (Bind +Abst) u t5)) H15) in (let H29 \def (eq_ind T v1 (\lambda (t7: T).(pr0 t7 v2)) +H10 v0 H27) in (eq_ind_r B Abst (\lambda (b0: B).(ex2 T (\lambda (t7: T).(pr0 +(THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda +(t7: T).(pr0 (THead (Bind Abbr) v3 t6) t7)))) (let H30 \def (eq_ind B b +(\lambda (b0: B).(not (eq B b0 Abst))) H9 Abst H26) in (let H31 \def (match +(H30 (refl_equal B Abst)) in False return (\lambda (_: False).(ex2 T (\lambda +(t7: T).(pr0 (THead (Bind Abst) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) +t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v3 t6) t7)))) with []) in H31)) +b H26))))))) H23)) H22)) H21))))) t2 H17)) t H15 H16 H13 H14))) | +(pr0_upsilon b0 H13 v0 v3 H14 u0 u3 H15 t5 t6 H16) \Rightarrow (\lambda (H17: +(eq T (THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) t)).(\lambda (H18: (eq T +(THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) t2)).(eq_ind T +(THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) (\lambda (_: T).((eq T (THead +(Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) t2) \to ((not (eq B b0 +Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda +(t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) +t8)) (\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H19: (eq T (THead (Bind +b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) t2)).(eq_ind T (THead (Bind +b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) (\lambda (t7: T).((not (eq B +b0 Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T +(\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) +t4)) t8)) (\lambda (t8: T).(pr0 t7 t8)))))))) (\lambda (_: (not (eq B b0 +Abst))).(\lambda (H21: (pr0 v0 v3)).(\lambda (H22: (pr0 u0 u3)).(\lambda +(H23: (pr0 t5 t6)).(let H24 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead +(Flat Appl) v1 (THead (Bind b) u1 t3)) t7)) H6 (THead (Flat Appl) v0 (THead +(Bind b0) u0 t5)) H17) in (let H25 \def (f_equal T T (\lambda (e: T).(match e +in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) +\Rightarrow v1 | (THead _ t7 _) \Rightarrow t7])) (THead (Flat Appl) v1 +(THead (Bind b) u1 t3)) (THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) H24) +in ((let H26 \def (f_equal T B (\lambda (e: T).(match e in T return (\lambda +(_: T).B) with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead _ +_ t7) \Rightarrow (match t7 in T return (\lambda (_: T).B) with [(TSort _) +\Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k +in K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) +\Rightarrow b])])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead +(Flat Appl) v0 (THead (Bind b0) u0 t5)) H24) in ((let H27 \def (f_equal T T +(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ _ t7) \Rightarrow (match +t7 in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) +\Rightarrow u1 | (THead _ t8 _) \Rightarrow t8])])) (THead (Flat Appl) v1 +(THead (Bind b) u1 t3)) (THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) H24) +in ((let H28 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda +(_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead +_ _ t7) \Rightarrow (match t7 in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t8) \Rightarrow +t8])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) v0 +(THead (Bind b0) u0 t5)) H24) in (\lambda (H29: (eq T u1 u0)).(\lambda (H30: +(eq B b b0)).(\lambda (H31: (eq T v1 v0)).(let H32 \def (eq_ind_r T t +(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) +\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) +(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Appl) v0 (THead (Bind +b0) u0 t5)) H17) in (let H33 \def (eq_ind T v1 (\lambda (t7: T).(pr0 t7 v2)) +H10 v0 H31) in (eq_ind_r B b0 (\lambda (b1: B).(ex2 T (\lambda (t7: T).(pr0 +(THead (Bind b1) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda +(t7: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) +t7)))) (let H34 \def (eq_ind B b (\lambda (b1: B).(not (eq B b1 Abst))) H9 b0 +H30) in (let H35 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H11 u0 H29) +in (let H36 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H12 t5 H28) in +(ex2_ind T (\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T +(\lambda (t7: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O +v2) t4)) t7)) (\lambda (t7: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) +(lift (S O) O v3) t6)) t7))) (\lambda (x: T).(\lambda (H37: (pr0 t4 +x)).(\lambda (H38: (pr0 t6 x)).(ex2_ind T (\lambda (t7: T).(pr0 u2 t7)) +(\lambda (t7: T).(pr0 u3 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b0) +u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 +(THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) t7))) (\lambda +(x0: T).(\lambda (H39: (pr0 u2 x0)).(\lambda (H40: (pr0 u3 x0)).(ex2_ind T +(\lambda (t7: T).(pr0 v2 t7)) (\lambda (t7: T).(pr0 v3 t7)) (ex2 T (\lambda +(t7: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) +t7)) (\lambda (t7: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) +O v3) t6)) t7))) (\lambda (x1: T).(\lambda (H41: (pr0 v2 x1)).(\lambda (H42: +(pr0 v3 x1)).(pr0_confluence__pr0_upsilon_upsilon b0 H34 v2 v3 x1 H41 H42 u2 +u3 x0 H39 H40 t4 t6 x H37 H38)))) (H32 v0 (tlt_head_sx (Flat Appl) v0 (THead +(Bind b0) u0 t5)) v2 H33 v3 H21))))) (H32 u0 (tlt_trans (THead (Bind b0) u0 +t5) u0 (THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) (tlt_head_sx (Bind b0) +u0 t5) (tlt_head_dx (Flat Appl) v0 (THead (Bind b0) u0 t5))) u2 H35 u3 +H22))))) (H32 t5 (tlt_trans (THead (Bind b0) u0 t5) t5 (THead (Flat Appl) v0 +(THead (Bind b0) u0 t5)) (tlt_head_dx (Bind b0) u0 t5) (tlt_head_dx (Flat +Appl) v0 (THead (Bind b0) u0 t5))) t4 H36 t6 H23))))) b H30))))))) H27)) +H26)) H25))))))) t2 H19)) t H17 H18 H13 H14 H15 H16))) | (pr0_delta u0 u3 H13 +t5 t6 H14 w H15) \Rightarrow (\lambda (H16: (eq T (THead (Bind Abbr) u0 t5) +t)).(\lambda (H17: (eq T (THead (Bind Abbr) u3 w) t2)).(eq_ind T (THead (Bind +Abbr) u0 t5) (\lambda (_: T).((eq T (THead (Bind Abbr) u3 w) t2) \to ((pr0 u0 +u3) \to ((pr0 t5 t6) \to ((subst0 O u3 t6 w) \to (ex2 T (\lambda (t8: T).(pr0 +(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda +(t8: T).(pr0 t2 t8)))))))) (\lambda (H18: (eq T (THead (Bind Abbr) u3 w) +t2)).(eq_ind T (THead (Bind Abbr) u3 w) (\lambda (t7: T).((pr0 u0 u3) \to +((pr0 t5 t6) \to ((subst0 O u3 t6 w) \to (ex2 T (\lambda (t8: T).(pr0 (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: +T).(pr0 t7 t8))))))) (\lambda (_: (pr0 u0 u3)).(\lambda (_: (pr0 t5 +t6)).(\lambda (_: (subst0 O u3 t6 w)).(let H22 \def (eq_ind_r T t (\lambda +(t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) t7)) H6 (THead +(Bind Abbr) u0 t5) H16) in (let H23 \def (eq_ind T (THead (Flat Appl) v1 +(THead (Bind b) u1 t3)) (\lambda (ee: T).(match ee in T return (\lambda (_: +T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | +(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with +[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind +Abbr) u0 t5) H22) in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) +u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 +(THead (Bind Abbr) u3 w) t7))) H23)))))) t2 H18)) t H16 H17 H13 H14 H15))) | +(pr0_zeta b0 H13 t5 t6 H14 u) \Rightarrow (\lambda (H15: (eq T (THead (Bind +b0) u (lift (S O) O t5)) t)).(\lambda (H16: (eq T t6 t2)).(eq_ind T (THead +(Bind b0) u (lift (S O) O t5)) (\lambda (_: T).((eq T t6 t2) \to ((not (eq B +b0 Abst)) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) +u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t2 +t8))))))) (\lambda (H17: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((not +(eq B b0 Abst)) \to ((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: +T).(pr0 t2 t8)))))) (\lambda (_: (not (eq B b0 Abst))).(\lambda (_: (pr0 t5 +t2)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1 +(THead (Bind b) u1 t3)) t7)) H6 (THead (Bind b0) u (lift (S O) O t5)) H15) in +(let H21 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) +(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) +\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow +(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | +(Flat _) \Rightarrow True])])) I (THead (Bind b0) u (lift (S O) O t5)) H20) +in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat +Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 t2 t7))) H21))))) t6 +(sym_eq T t6 t2 H17))) t H15 H16 H13 H14))) | (pr0_tau t5 t6 H13 u) +\Rightarrow (\lambda (H14: (eq T (THead (Flat Cast) u t5) t)).(\lambda (H15: +(eq T t6 t2)).(eq_ind T (THead (Flat Cast) u t5) (\lambda (_: T).((eq T t6 +t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 +(THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t2 +t8)))))) (\lambda (H16: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5 +t7) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) +(lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t2 t8))))) (\lambda (_: +(pr0 t5 t2)).(let H18 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat +Appl) v1 (THead (Bind b) u1 t3)) t7)) H6 (THead (Flat Cast) u t5) H14) in +(let H19 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) +(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) +\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow +(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | +(Flat f) \Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl +\Rightarrow True | Cast \Rightarrow False])])])) I (THead (Flat Cast) u t5) +H18) in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead +(Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 t2 t7))) +H19)))) t6 (sym_eq T t6 t2 H16))) t H14 H15 H13)))]) in (H13 (refl_equal T t) +(refl_equal T t2))))))) t1 H8)) t H6 H7 H2 H3 H4 H5))) | (pr0_delta u1 u2 H2 +t3 t4 H3 w H4) \Rightarrow (\lambda (H5: (eq T (THead (Bind Abbr) u1 t3) +t)).(\lambda (H6: (eq T (THead (Bind Abbr) u2 w) t1)).(eq_ind T (THead (Bind +Abbr) u1 t3) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t1) \to ((pr0 u1 +u2) \to ((pr0 t3 t4) \to ((subst0 O u2 t4 w) \to (ex2 T (\lambda (t6: T).(pr0 +t1 t6)) (\lambda (t6: T).(pr0 t2 t6)))))))) (\lambda (H7: (eq T (THead (Bind +Abbr) u2 w) t1)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t5: T).((pr0 u1 +u2) \to ((pr0 t3 t4) \to ((subst0 O u2 t4 w) \to (ex2 T (\lambda (t6: T).(pr0 +t5 t6)) (\lambda (t6: T).(pr0 t2 t6))))))) (\lambda (H8: (pr0 u1 +u2)).(\lambda (H9: (pr0 t3 t4)).(\lambda (H10: (subst0 O u2 t4 w)).(let H11 +\def (match H1 in pr0 return (\lambda (t5: T).(\lambda (t6: T).(\lambda (_: +(pr0 t5 t6)).((eq T t5 t) \to ((eq T t6 t2) \to (ex2 T (\lambda (t7: T).(pr0 +(THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7)))))))) with +[(pr0_refl t5) \Rightarrow (\lambda (H11: (eq T t5 t)).(\lambda (H12: (eq T +t5 t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2) \to (ex2 T (\lambda (t7: +T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7))))) +(\lambda (H13: (eq T t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t7: +T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let +H14 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 t2)) H13 (THead (Bind Abbr) +u1 t3) H5) in (eq_ind T (THead (Bind Abbr) u1 t3) (\lambda (t6: T).(ex2 T +(\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t6 +t7)))) (let H15 \def (eq_ind_r T t (\lambda (t6: T).(eq T t5 t6)) H11 (THead +(Bind Abbr) u1 t3) H5) in (let H16 \def (eq_ind_r T t (\lambda (t6: +T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall +(t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: +T).(pr0 t8 t9)))))))))) H (THead (Bind Abbr) u1 t3) H5) in (ex_intro2 T +(\lambda (t6: T).(pr0 (THead (Bind Abbr) u2 w) t6)) (\lambda (t6: T).(pr0 +(THead (Bind Abbr) u1 t3) t6)) (THead (Bind Abbr) u2 w) (pr0_refl (THead +(Bind Abbr) u2 w)) (pr0_delta u1 u2 H8 t3 t4 H9 w H10)))) t2 H14)) t (sym_eq +T t t2 H13))) t5 (sym_eq T t5 t H11) H12))) | (pr0_comp u0 u3 H11 t5 t6 H12 +k) \Rightarrow (\lambda (H13: (eq T (THead k u0 t5) t)).(\lambda (H14: (eq T +(THead k u3 t6) t2)).(eq_ind T (THead k u0 t5) (\lambda (_: T).((eq T (THead +k u3 t6) t2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: +T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) +(\lambda (H15: (eq T (THead k u3 t6) t2)).(eq_ind T (THead k u3 t6) (\lambda +(t7: T).((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead +(Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (H16: (pr0 +u0 u3)).(\lambda (H17: (pr0 t5 t6)).(let H18 \def (eq_ind_r T t (\lambda (t7: +T).(eq T (THead (Bind Abbr) u1 t3) t7)) H5 (THead k u0 t5) H13) in (let H19 +\def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) +with [(TSort _) \Rightarrow (Bind Abbr) | (TLRef _) \Rightarrow (Bind Abbr) | +(THead k0 _ _) \Rightarrow k0])) (THead (Bind Abbr) u1 t3) (THead k u0 t5) +H18) in ((let H20 \def (f_equal T T (\lambda (e: T).(match e in T return +(\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 +| (THead _ t7 _) \Rightarrow t7])) (THead (Bind Abbr) u1 t3) (THead k u0 t5) +H18) in ((let H21 \def (f_equal T T (\lambda (e: T).(match e in T return +(\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 +| (THead _ _ t7) \Rightarrow t7])) (THead (Bind Abbr) u1 t3) (THead k u0 t5) +H18) in (\lambda (H22: (eq T u1 u0)).(\lambda (H23: (eq K (Bind Abbr) +k)).(eq_ind K (Bind Abbr) (\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 +(THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 (THead k0 u3 t6) t7)))) +(let H24 \def (eq_ind_r K k (\lambda (k0: K).(eq T (THead k0 u0 t5) t)) H13 +(Bind Abbr) H23) in (let H25 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: +T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v +t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 +t10)))))))))) H (THead (Bind Abbr) u0 t5) H24) in (let H26 \def (eq_ind T u1 +(\lambda (t7: T).(pr0 t7 u2)) H8 u0 H22) in (let H27 \def (eq_ind T t3 +(\lambda (t7: T).(pr0 t7 t4)) H9 t5 H21) in (ex2_ind T (\lambda (t7: T).(pr0 +t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead +(Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u3 t6) t7))) +(\lambda (x: T).(\lambda (H28: (pr0 t4 x)).(\lambda (H29: (pr0 t6 +x)).(ex2_ind T (\lambda (t7: T).(pr0 u2 t7)) (\lambda (t7: T).(pr0 u3 t7)) +(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: +T).(pr0 (THead (Bind Abbr) u3 t6) t7))) (\lambda (x0: T).(\lambda (H30: (pr0 +u2 x0)).(\lambda (H31: (pr0 u3 x0)).(ex2_sym T (pr0 (THead (Bind Abbr) u3 +t6)) (pr0 (THead (Bind Abbr) u2 w)) (pr0_confluence__pr0_cong_delta u2 t4 w +H10 u3 x0 H31 H30 t6 x H29 H28))))) (H25 u0 (tlt_head_sx (Bind Abbr) u0 t5) +u2 H26 u3 H16))))) (H25 t5 (tlt_head_dx (Bind Abbr) u0 t5) t4 H27 t6 +H17)))))) k H23)))) H20)) H19))))) t2 H15)) t H13 H14 H11 H12))) | (pr0_beta +u v1 v2 H11 t5 t6 H12) \Rightarrow (\lambda (H13: (eq T (THead (Flat Appl) v1 +(THead (Bind Abst) u t5)) t)).(\lambda (H14: (eq T (THead (Bind Abbr) v2 t6) +t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) (\lambda (_: +T).((eq T (THead (Bind Abbr) v2 t6) t2) \to ((pr0 v1 v2) \to ((pr0 t5 t6) \to +(ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: +T).(pr0 t2 t8))))))) (\lambda (H15: (eq T (THead (Bind Abbr) v2 t6) +t2)).(eq_ind T (THead (Bind Abbr) v2 t6) (\lambda (t7: T).((pr0 v1 v2) \to +((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) +(\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 +t5 t6)).(let H18 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind Abbr) +u1 t3) t7)) H5 (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) H13) in (let +H19 \def (eq_ind T (THead (Bind Abbr) u1 t3) (\lambda (ee: T).(match ee in T +return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda +(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow +False])])) I (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) H18) in +(False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) +(\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t6) t7))) H19))))) t2 H15)) t H13 +H14 H11 H12))) | (pr0_upsilon b H11 v1 v2 H12 u0 u3 H13 t5 t6 H14) +\Rightarrow (\lambda (H15: (eq T (THead (Flat Appl) v1 (THead (Bind b) u0 +t5)) t)).(\lambda (H16: (eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S +O) O v2) t6)) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u0 t5)) +(\lambda (_: T).((eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O +v2) t6)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u3) \to +((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) +(\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H17: (eq T (THead (Bind b) u3 +(THead (Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Bind b) u3 +(THead (Flat Appl) (lift (S O) O v2) t6)) (\lambda (t7: T).((not (eq B b +Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda +(t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t7 +t8)))))))) (\lambda (_: (not (eq B b Abst))).(\lambda (_: (pr0 v1 +v2)).(\lambda (_: (pr0 u0 u3)).(\lambda (_: (pr0 t5 t6)).(let H22 \def +(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind Abbr) u1 t3) t7)) H5 (THead +(Flat Appl) v1 (THead (Bind b) u0 t5)) H15) in (let H23 \def (eq_ind T (THead +(Bind Abbr) u1 t3) (\lambda (ee: T).(match ee in T return (\lambda (_: +T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | +(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with +[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat +Appl) v1 (THead (Bind b) u0 t5)) H22) in (False_ind (ex2 T (\lambda (t7: +T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 (THead (Bind b) +u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t7))) H23))))))) t2 H17)) t H15 +H16 H11 H12 H13 H14))) | (pr0_delta u0 u3 H11 t5 t6 H12 w0 H13) \Rightarrow +(\lambda (H14: (eq T (THead (Bind Abbr) u0 t5) t)).(\lambda (H15: (eq T +(THead (Bind Abbr) u3 w0) t2)).(eq_ind T (THead (Bind Abbr) u0 t5) (\lambda +(_: T).((eq T (THead (Bind Abbr) u3 w0) t2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) +\to ((subst0 O u3 t6 w0) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) +u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8)))))))) (\lambda (H16: (eq T (THead +(Bind Abbr) u3 w0) t2)).(eq_ind T (THead (Bind Abbr) u3 w0) (\lambda (t7: +T).((pr0 u0 u3) \to ((pr0 t5 t6) \to ((subst0 O u3 t6 w0) \to (ex2 T (\lambda +(t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t7 t8))))))) +(\lambda (H17: (pr0 u0 u3)).(\lambda (H18: (pr0 t5 t6)).(\lambda (H19: +(subst0 O u3 t6 w0)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T +(THead (Bind Abbr) u1 t3) t7)) H5 (THead (Bind Abbr) u0 t5) H14) in (let H21 +\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) +with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t7 _) +\Rightarrow t7])) (THead (Bind Abbr) u1 t3) (THead (Bind Abbr) u0 t5) H20) in +((let H22 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: +T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ +t7) \Rightarrow t7])) (THead (Bind Abbr) u1 t3) (THead (Bind Abbr) u0 t5) +H20) in (\lambda (H23: (eq T u1 u0)).(let H24 \def (eq_ind_r T t (\lambda +(t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to +(\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) +(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Bind Abbr) u0 t5) H14) in +(let H25 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H8 u0 H23) in (let +H26 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H9 t5 H22) in (ex2_ind T +(\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda +(t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 (THead (Bind +Abbr) u3 w0) t7))) (\lambda (x: T).(\lambda (H27: (pr0 t4 x)).(\lambda (H28: +(pr0 t6 x)).(ex2_ind T (\lambda (t7: T).(pr0 u2 t7)) (\lambda (t7: T).(pr0 u3 +t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: +T).(pr0 (THead (Bind Abbr) u3 w0) t7))) (\lambda (x0: T).(\lambda (H29: (pr0 +u2 x0)).(\lambda (H30: (pr0 u3 x0)).(pr0_confluence__pr0_delta_delta u2 t4 w +H10 u3 t6 w0 H19 x0 H29 H30 x H27 H28)))) (H24 u0 (tlt_head_sx (Bind Abbr) u0 +t5) u2 H25 u3 H17))))) (H24 t5 (tlt_head_dx (Bind Abbr) u0 t5) t4 H26 t6 +H18))))))) H21)))))) t2 H16)) t H14 H15 H11 H12 H13))) | (pr0_zeta b H11 t5 +t6 H12 u) \Rightarrow (\lambda (H13: (eq T (THead (Bind b) u (lift (S O) O +t5)) t)).(\lambda (H14: (eq T t6 t2)).(eq_ind T (THead (Bind b) u (lift (S O) +O t5)) (\lambda (_: T).((eq T t6 t2) \to ((not (eq B b Abst)) \to ((pr0 t5 +t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda +(t8: T).(pr0 t2 t8))))))) (\lambda (H15: (eq T t6 t2)).(eq_ind T t2 (\lambda +(t7: T).((not (eq B b Abst)) \to ((pr0 t5 t7) \to (ex2 T (\lambda (t8: +T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) +(\lambda (H16: (not (eq B b Abst))).(\lambda (H17: (pr0 t5 t2)).(let H18 \def +(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind Abbr) u1 t3) t7)) H5 (THead +(Bind b) u (lift (S O) O t5)) H13) in (let H19 \def (f_equal T B (\lambda (e: +T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow Abbr | +(TLRef _) \Rightarrow Abbr | (THead k _ _) \Rightarrow (match k in K return +(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow +Abbr])])) (THead (Bind Abbr) u1 t3) (THead (Bind b) u (lift (S O) O t5)) H18) +in ((let H20 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda +(_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead +_ t7 _) \Rightarrow t7])) (THead (Bind Abbr) u1 t3) (THead (Bind b) u (lift +(S O) O t5)) H18) in ((let H21 \def (f_equal T T (\lambda (e: T).(match e in +T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) +\Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead (Bind Abbr) u1 t3) +(THead (Bind b) u (lift (S O) O t5)) H18) in (\lambda (H22: (eq T u1 +u)).(\lambda (H23: (eq B Abbr b)).(let H24 \def (eq_ind_r B b (\lambda (b0: +B).(not (eq B b0 Abst))) H16 Abbr H23) in (let H25 \def (eq_ind_r B b +(\lambda (b0: B).(eq T (THead (Bind b0) u (lift (S O) O t5)) t)) H13 Abbr +H23) in (let H26 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v +t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to +(ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 +t10)))))))))) H (THead (Bind Abbr) u (lift (S O) O t5)) H25) in (let H27 \def +(eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H8 u H22) in (let H28 \def (eq_ind +T t3 (\lambda (t7: T).(pr0 t7 t4)) H9 (lift (S O) O t5) H21) in (ex2_ind T +(\lambda (t7: T).(eq T t4 (lift (S O) O t7))) (\lambda (t7: T).(pr0 t5 t7)) +(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: +T).(pr0 t2 t7))) (\lambda (x: T).(\lambda (H29: (eq T t4 (lift (S O) O +x))).(\lambda (H30: (pr0 t5 x)).(let H31 \def (eq_ind T t4 (\lambda (t7: +T).(subst0 O u2 t7 w)) H10 (lift (S O) O x) H29) in (ex2_ind T (\lambda (t7: +T).(pr0 x t7)) (\lambda (t7: T).(pr0 t2 t7)) (ex2 T (\lambda (t7: T).(pr0 +(THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7))) (\lambda (x0: +T).(\lambda (_: (pr0 x x0)).(\lambda (_: (pr0 t2 +x0)).(pr0_confluence__pr0_delta_tau u2 (lift (S O) O x) w H31 x (pr0_refl +(lift (S O) O x)) t2)))) (H26 t5 (lift_tlt_dx (Bind Abbr) u t5 (S O) O) x H30 +t2 H17)))))) (pr0_gen_lift t5 t4 (S O) O H28)))))))))) H20)) H19))))) t6 +(sym_eq T t6 t2 H15))) t H13 H14 H11 H12))) | (pr0_tau t5 t6 H11 u) +\Rightarrow (\lambda (H12: (eq T (THead (Flat Cast) u t5) t)).(\lambda (H13: +(eq T t6 t2)).(eq_ind T (THead (Flat Cast) u t5) (\lambda (_: T).((eq T t6 +t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 +w) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H14: (eq T t6 +t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8: +T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8))))) +(\lambda (_: (pr0 t5 t2)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T +(THead (Bind Abbr) u1 t3) t7)) H5 (THead (Flat Cast) u t5) H12) in (let H17 +\def (eq_ind T (THead (Bind Abbr) u1 t3) (\lambda (ee: T).(match ee in T +return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda +(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow +False])])) I (THead (Flat Cast) u t5) H16) in (False_ind (ex2 T (\lambda (t7: +T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7))) H17)))) +t6 (sym_eq T t6 t2 H14))) t H12 H13 H11)))]) in (H11 (refl_equal T t) +(refl_equal T t2)))))) t1 H7)) t H5 H6 H2 H3 H4))) | (pr0_zeta b H2 t3 t4 H3 +u) \Rightarrow (\lambda (H4: (eq T (THead (Bind b) u (lift (S O) O t3)) +t)).(\lambda (H5: (eq T t4 t1)).(eq_ind T (THead (Bind b) u (lift (S O) O +t3)) (\lambda (_: T).((eq T t4 t1) \to ((not (eq B b Abst)) \to ((pr0 t3 t4) +\to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6))))))) +(\lambda (H6: (eq T t4 t1)).(eq_ind T t1 (\lambda (t5: T).((not (eq B b +Abst)) \to ((pr0 t3 t5) \to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda +(t6: T).(pr0 t2 t6)))))) (\lambda (H7: (not (eq B b Abst))).(\lambda (H8: +(pr0 t3 t1)).(let H9 \def (match H1 in pr0 return (\lambda (t5: T).(\lambda +(t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 t) \to ((eq T t6 t2) \to (ex2 T +(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)))))))) with +[(pr0_refl t5) \Rightarrow (\lambda (H9: (eq T t5 t)).(\lambda (H10: (eq T t5 +t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2) \to (ex2 T (\lambda (t7: +T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))))) (\lambda (H11: (eq T t +t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t7: T).(pr0 t1 t7)) +(\lambda (t7: T).(pr0 t2 t7)))) (let H12 \def (eq_ind_r T t (\lambda (t6: +T).(eq T t6 t2)) H11 (THead (Bind b) u (lift (S O) O t3)) H4) in (eq_ind T +(THead (Bind b) u (lift (S O) O t3)) (\lambda (t6: T).(ex2 T (\lambda (t7: +T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t6 t7)))) (let H13 \def (eq_ind_r T t +(\lambda (t6: T).(eq T t5 t6)) H9 (THead (Bind b) u (lift (S O) O t3)) H4) in +(let H14 \def (eq_ind_r T t (\lambda (t6: T).(\forall (v: T).((tlt v t6) \to +(\forall (t7: T).((pr0 v t7) \to (\forall (t8: T).((pr0 v t8) \to (ex2 T +(\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead +(Bind b) u (lift (S O) O t3)) H4) in (ex_intro2 T (\lambda (t6: T).(pr0 t1 +t6)) (\lambda (t6: T).(pr0 (THead (Bind b) u (lift (S O) O t3)) t6)) t1 +(pr0_refl t1) (pr0_zeta b H7 t3 t1 H8 u)))) t2 H12)) t (sym_eq T t t2 H11))) +t5 (sym_eq T t5 t H9) H10))) | (pr0_comp u1 u2 H9 t5 t6 H10 k) \Rightarrow +(\lambda (H11: (eq T (THead k u1 t5) t)).(\lambda (H12: (eq T (THead k u2 t6) +t2)).(eq_ind T (THead k u1 t5) (\lambda (_: T).((eq T (THead k u2 t6) t2) \to +((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) +(\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H13: (eq T (THead k u2 t6) +t2)).(eq_ind T (THead k u2 t6) (\lambda (t7: T).((pr0 u1 u2) \to ((pr0 t5 t6) +\to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 t8)))))) +(\lambda (_: (pr0 u1 u2)).(\lambda (H15: (pr0 t5 t6)).(let H16 \def (eq_ind_r +T t (\lambda (t7: T).(eq T (THead (Bind b) u (lift (S O) O t3)) t7)) H4 +(THead k u1 t5) H11) in (let H17 \def (f_equal T K (\lambda (e: T).(match e +in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow (Bind b) | (TLRef +_) \Rightarrow (Bind b) | (THead k0 _ _) \Rightarrow k0])) (THead (Bind b) u +(lift (S O) O t3)) (THead k u1 t5) H16) in ((let H18 \def (f_equal T T +(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t7 _) \Rightarrow t7])) +(THead (Bind b) u (lift (S O) O t3)) (THead k u1 t5) H16) in ((let H19 \def +(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with +[(TSort _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t7: +T) on t7: T \def (match t7 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) +\Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false +\Rightarrow (f i)])) | (THead k0 u0 t8) \Rightarrow (THead k0 (lref_map f d +u0) (lref_map f (s k0 d) t8))]) in lref_map) (\lambda (x: nat).(plus x (S +O))) O t3) | (TLRef _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) +(d: nat) (t7: T) on t7: T \def (match t7 with [(TSort n) \Rightarrow (TSort +n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | +false \Rightarrow (f i)])) | (THead k0 u0 t8) \Rightarrow (THead k0 (lref_map +f d u0) (lref_map f (s k0 d) t8))]) in lref_map) (\lambda (x: nat).(plus x (S +O))) O t3) | (THead _ _ t7) \Rightarrow t7])) (THead (Bind b) u (lift (S O) O +t3)) (THead k u1 t5) H16) in (\lambda (_: (eq T u u1)).(\lambda (H21: (eq K +(Bind b) k)).(eq_ind K (Bind b) (\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 +t1 t7)) (\lambda (t7: T).(pr0 (THead k0 u2 t6) t7)))) (let H22 \def (eq_ind_r +K k (\lambda (k0: K).(eq T (THead k0 u1 t5) t)) H11 (Bind b) H21) in (let H23 +\def (eq_ind_r T t5 (\lambda (t7: T).(pr0 t7 t6)) H15 (lift (S O) O t3) H19) +in (ex2_ind T (\lambda (t7: T).(eq T t6 (lift (S O) O t7))) (\lambda (t7: +T).(pr0 t3 t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 +(THead (Bind b) u2 t6) t7))) (\lambda (x: T).(\lambda (H24: (eq T t6 (lift (S +O) O x))).(\lambda (H25: (pr0 t3 x)).(let H26 \def (eq_ind_r T t5 (\lambda +(t7: T).(eq T (THead (Bind b) u1 t7) t)) H22 (lift (S O) O t3) H19) in (let +H27 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to +(\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T +(\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H +(THead (Bind b) u1 (lift (S O) O t3)) H26) in (eq_ind_r T (lift (S O) O x) +(\lambda (t7: T).(ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 +(THead (Bind b) u2 t7) t8)))) (ex2_ind T (\lambda (t7: T).(pr0 x t7)) +(\lambda (t7: T).(pr0 t1 t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda +(t7: T).(pr0 (THead (Bind b) u2 (lift (S O) O x)) t7))) (\lambda (x0: +T).(\lambda (H28: (pr0 x x0)).(\lambda (H29: (pr0 t1 x0)).(ex_intro2 T +(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Bind b) u2 (lift +(S O) O x)) t7)) x0 H29 (pr0_zeta b H7 x x0 H28 u2))))) (H27 t3 (lift_tlt_dx +(Bind b) u1 t3 (S O) O) x H25 t1 H8)) t6 H24)))))) (pr0_gen_lift t3 t6 (S O) +O H23)))) k H21)))) H18)) H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_beta +u0 v1 v2 H9 t5 t6 H10) \Rightarrow (\lambda (H11: (eq T (THead (Flat Appl) v1 +(THead (Bind Abst) u0 t5)) t)).(\lambda (H12: (eq T (THead (Bind Abbr) v2 t6) +t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t5)) (\lambda (_: +T).((eq T (THead (Bind Abbr) v2 t6) t2) \to ((pr0 v1 v2) \to ((pr0 t5 t6) \to +(ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))))) +(\lambda (H13: (eq T (THead (Bind Abbr) v2 t6) t2)).(eq_ind T (THead (Bind +Abbr) v2 t6) (\lambda (t7: T).((pr0 v1 v2) \to ((pr0 t5 t6) \to (ex2 T +(\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (_: +(pr0 v1 v2)).(\lambda (_: (pr0 t5 t6)).(let H16 \def (eq_ind_r T t (\lambda +(t7: T).(eq T (THead (Bind b) u (lift (S O) O t3)) t7)) H4 (THead (Flat Appl) +v1 (THead (Bind Abst) u0 t5)) H11) in (let H17 \def (eq_ind T (THead (Bind b) +u (lift (S O) O t3)) (\lambda (ee: T).(match ee in T return (\lambda (_: +T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | +(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with +[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat +Appl) v1 (THead (Bind Abst) u0 t5)) H16) in (False_ind (ex2 T (\lambda (t7: +T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t6) t7))) +H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_upsilon b0 H9 v1 v2 H10 u1 u2 +H11 t5 t6 H12) \Rightarrow (\lambda (H13: (eq T (THead (Flat Appl) v1 (THead +(Bind b0) u1 t5)) t)).(\lambda (H14: (eq T (THead (Bind b0) u2 (THead (Flat +Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Flat Appl) v1 (THead +(Bind b0) u1 t5)) (\lambda (_: T).((eq T (THead (Bind b0) u2 (THead (Flat +Appl) (lift (S O) O v2) t6)) t2) \to ((not (eq B b0 Abst)) \to ((pr0 v1 v2) +\to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) +(\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H15: (eq T (THead (Bind b0) u2 +(THead (Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Bind b0) u2 +(THead (Flat Appl) (lift (S O) O v2) t6)) (\lambda (t7: T).((not (eq B b0 +Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda +(t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 t8)))))))) (\lambda (_: (not +(eq B b0 Abst))).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 u1 u2)).(\lambda +(_: (pr0 t5 t6)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead +(Bind b) u (lift (S O) O t3)) t7)) H4 (THead (Flat Appl) v1 (THead (Bind b0) +u1 t5)) H13) in (let H21 \def (eq_ind T (THead (Bind b) u (lift (S O) O t3)) +(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) +\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow +(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | +(Flat _) \Rightarrow False])])) I (THead (Flat Appl) v1 (THead (Bind b0) u1 +t5)) H20) in (False_ind (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: +T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t6)) t7))) +H21))))))) t2 H15)) t H13 H14 H9 H10 H11 H12))) | (pr0_delta u1 u2 H9 t5 t6 +H10 w H11) \Rightarrow (\lambda (H12: (eq T (THead (Bind Abbr) u1 t5) +t)).(\lambda (H13: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind +Abbr) u1 t5) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t2) \to ((pr0 u1 +u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T (\lambda (t8: T).(pr0 +t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))))) (\lambda (H14: (eq T (THead (Bind +Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t7: T).((pr0 u1 +u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T (\lambda (t8: T).(pr0 +t1 t8)) (\lambda (t8: T).(pr0 t7 t8))))))) (\lambda (_: (pr0 u1 u2)).(\lambda +(H16: (pr0 t5 t6)).(\lambda (H17: (subst0 O u2 t6 w)).(let H18 \def (eq_ind_r +T t (\lambda (t7: T).(eq T (THead (Bind b) u (lift (S O) O t3)) t7)) H4 +(THead (Bind Abbr) u1 t5) H12) in (let H19 \def (f_equal T B (\lambda (e: +T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | +(TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k in K return +(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow +b])])) (THead (Bind b) u (lift (S O) O t3)) (THead (Bind Abbr) u1 t5) H18) in +((let H20 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: +T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t7 +_) \Rightarrow t7])) (THead (Bind b) u (lift (S O) O t3)) (THead (Bind Abbr) +u1 t5) H18) in ((let H21 \def (f_equal T T (\lambda (e: T).(match e in T +return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: +((nat \to nat))) (d: nat) (t7: T) on t7: T \def (match t7 with [(TSort n) +\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with +[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t8) +\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t8))]) in +lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (TLRef _) \Rightarrow +((let rec lref_map (f: ((nat \to nat))) (d: nat) (t7: T) on t7: T \def (match +t7 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef +(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | +(THead k u0 t8) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) +t8))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (THead _ _ t7) +\Rightarrow t7])) (THead (Bind b) u (lift (S O) O t3)) (THead (Bind Abbr) u1 +t5) H18) in (\lambda (_: (eq T u u1)).(\lambda (H23: (eq B b Abbr)).(let H24 +\def (eq_ind_r T t5 (\lambda (t7: T).(pr0 t7 t6)) H16 (lift (S O) O t3) H21) +in (ex2_ind T (\lambda (t7: T).(eq T t6 (lift (S O) O t7))) (\lambda (t7: +T).(pr0 t3 t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 +(THead (Bind Abbr) u2 w) t7))) (\lambda (x: T).(\lambda (H25: (eq T t6 (lift +(S O) O x))).(\lambda (H26: (pr0 t3 x)).(let H27 \def (eq_ind_r T t5 (\lambda +(t7: T).(eq T (THead (Bind Abbr) u1 t7) t)) H12 (lift (S O) O t3) H21) in +(let H28 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to +(\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T +(\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H +(THead (Bind Abbr) u1 (lift (S O) O t3)) H27) in (let H29 \def (eq_ind T t6 +(\lambda (t7: T).(subst0 O u2 t7 w)) H17 (lift (S O) O x) H25) in (let H30 +\def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H7 Abbr H23) in +(ex2_ind T (\lambda (t7: T).(pr0 x t7)) (\lambda (t7: T).(pr0 t1 t7)) (ex2 T +(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) +t7))) (\lambda (x0: T).(\lambda (_: (pr0 x x0)).(\lambda (_: (pr0 t1 +x0)).(ex2_sym T (pr0 (THead (Bind Abbr) u2 w)) (pr0 t1) +(pr0_confluence__pr0_delta_tau u2 (lift (S O) O x) w H29 x (pr0_refl (lift (S +O) O x)) t1))))) (H28 t3 (lift_tlt_dx (Bind Abbr) u1 t3 (S O) O) x H26 t1 +H8))))))))) (pr0_gen_lift t3 t6 (S O) O H24)))))) H20)) H19)))))) t2 H14)) t +H12 H13 H9 H10 H11))) | (pr0_zeta b0 H9 t5 t6 H10 u0) \Rightarrow (\lambda +(H11: (eq T (THead (Bind b0) u0 (lift (S O) O t5)) t)).(\lambda (H12: (eq T +t6 t2)).(eq_ind T (THead (Bind b0) u0 (lift (S O) O t5)) (\lambda (_: T).((eq +T t6 t2) \to ((not (eq B b0 Abst)) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: +T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H13: (eq T t6 +t2)).(eq_ind T t2 (\lambda (t7: T).((not (eq B b0 Abst)) \to ((pr0 t5 t7) \to +(ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))) +(\lambda (_: (not (eq B b0 Abst))).(\lambda (H15: (pr0 t5 t2)).(let H16 \def +(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind b) u (lift (S O) O t3)) +t7)) H4 (THead (Bind b0) u0 (lift (S O) O t5)) H11) in (let H17 \def (f_equal +T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with [(TSort _) +\Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k +in K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) +\Rightarrow b])])) (THead (Bind b) u (lift (S O) O t3)) (THead (Bind b0) u0 +(lift (S O) O t5)) H16) in ((let H18 \def (f_equal T T (\lambda (e: T).(match +e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) +\Rightarrow u | (THead _ t7 _) \Rightarrow t7])) (THead (Bind b) u (lift (S +O) O t3)) (THead (Bind b0) u0 (lift (S O) O t5)) H16) in ((let H19 \def +(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with +[(TSort _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t7: +T) on t7: T \def (match t7 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) +\Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false +\Rightarrow (f i)])) | (THead k u1 t8) \Rightarrow (THead k (lref_map f d u1) +(lref_map f (s k d) t8))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O +t3) | (TLRef _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) +(t7: T) on t7: T \def (match t7 with [(TSort n) \Rightarrow (TSort n) | +(TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | +false \Rightarrow (f i)])) | (THead k u1 t8) \Rightarrow (THead k (lref_map f +d u1) (lref_map f (s k d) t8))]) in lref_map) (\lambda (x: nat).(plus x (S +O))) O t3) | (THead _ _ t7) \Rightarrow t7])) (THead (Bind b) u (lift (S O) O +t3)) (THead (Bind b0) u0 (lift (S O) O t5)) H16) in (\lambda (_: (eq T u +u0)).(\lambda (H21: (eq B b b0)).(let H22 \def (eq_ind_r T t (\lambda (t7: +T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall +(t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: +T).(pr0 t9 t10)))))))))) H (THead (Bind b0) u0 (lift (S O) O t5)) H11) in +(let H23 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t1)) H8 t5 (lift_inj t3 +t5 (S O) O H19)) in (let H24 \def (eq_ind B b (\lambda (b1: B).(not (eq B b1 +Abst))) H7 b0 H21) in (ex2_ind T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: +T).(pr0 t2 t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 +t7))) (\lambda (x: T).(\lambda (H25: (pr0 t1 x)).(\lambda (H26: (pr0 t2 +x)).(ex_intro2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)) +x H25 H26)))) (H22 t5 (lift_tlt_dx (Bind b0) u0 t5 (S O) O) t1 H23 t2 +H15)))))))) H18)) H17))))) t6 (sym_eq T t6 t2 H13))) t H11 H12 H9 H10))) | +(pr0_tau t5 t6 H9 u0) \Rightarrow (\lambda (H10: (eq T (THead (Flat Cast) u0 +t5) t)).(\lambda (H11: (eq T t6 t2)).(eq_ind T (THead (Flat Cast) u0 t5) +(\lambda (_: T).((eq T t6 t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: +T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H12: (eq T t6 +t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8: +T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))) (\lambda (_: (pr0 t5 +t2)).(let H14 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind b) u +(lift (S O) O t3)) t7)) H4 (THead (Flat Cast) u0 t5) H10) in (let H15 \def +(eq_ind T (THead (Bind b) u (lift (S O) O t3)) (\lambda (ee: T).(match ee in +T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda +(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow +False])])) I (THead (Flat Cast) u0 t5) H14) in (False_ind (ex2 T (\lambda +(t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))) H15)))) t6 (sym_eq T t6 +t2 H12))) t H10 H11 H9)))]) in (H9 (refl_equal T t) (refl_equal T t2))))) t4 +(sym_eq T t4 t1 H6))) t H4 H5 H2 H3))) | (pr0_tau t3 t4 H2 u) \Rightarrow +(\lambda (H3: (eq T (THead (Flat Cast) u t3) t)).(\lambda (H4: (eq T t4 +t1)).(eq_ind T (THead (Flat Cast) u t3) (\lambda (_: T).((eq T t4 t1) \to +((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 +t2 t6)))))) (\lambda (H5: (eq T t4 t1)).(eq_ind T t1 (\lambda (t5: T).((pr0 +t3 t5) \to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 +t6))))) (\lambda (H6: (pr0 t3 t1)).(let H7 \def (match H1 in pr0 return +(\lambda (t5: T).(\lambda (t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 t) \to +((eq T t6 t2) \to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 +t2 t7)))))))) with [(pr0_refl t5) \Rightarrow (\lambda (H7: (eq T t5 +t)).(\lambda (H8: (eq T t5 t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2) +\to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))))) +(\lambda (H9: (eq T t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t7: +T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let H10 \def (eq_ind_r T t +(\lambda (t6: T).(eq T t6 t2)) H9 (THead (Flat Cast) u t3) H3) in (eq_ind T +(THead (Flat Cast) u t3) (\lambda (t6: T).(ex2 T (\lambda (t7: T).(pr0 t1 +t7)) (\lambda (t7: T).(pr0 t6 t7)))) (let H11 \def (eq_ind_r T t (\lambda +(t6: T).(eq T t5 t6)) H7 (THead (Flat Cast) u t3) H3) in (let H12 \def +(eq_ind_r T t (\lambda (t6: T).(\forall (v: T).((tlt v t6) \to (\forall (t7: +T).((pr0 v t7) \to (\forall (t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: +T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead (Flat Cast) u +t3) H3) in (ex_intro2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 +(THead (Flat Cast) u t3) t6)) t1 (pr0_refl t1) (pr0_tau t3 t1 H6 u)))) t2 +H10)) t (sym_eq T t t2 H9))) t5 (sym_eq T t5 t H7) H8))) | (pr0_comp u1 u2 H7 +t5 t6 H8 k) \Rightarrow (\lambda (H9: (eq T (THead k u1 t5) t)).(\lambda +(H10: (eq T (THead k u2 t6) t2)).(eq_ind T (THead k u1 t5) (\lambda (_: +T).((eq T (THead k u2 t6) t2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T +(\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda +(H11: (eq T (THead k u2 t6) t2)).(eq_ind T (THead k u2 t6) (\lambda (t7: +T).((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) +(\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (_: (pr0 u1 u2)).(\lambda (H13: +(pr0 t5 t6)).(let H14 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat +Cast) u t3) t7)) H3 (THead k u1 t5) H9) in (let H15 \def (f_equal T K +(\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) +\Rightarrow (Flat Cast) | (TLRef _) \Rightarrow (Flat Cast) | (THead k0 _ _) +\Rightarrow k0])) (THead (Flat Cast) u t3) (THead k u1 t5) H14) in ((let H16 +\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) +with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t7 _) +\Rightarrow t7])) (THead (Flat Cast) u t3) (THead k u1 t5) H14) in ((let H17 +\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) +with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7) +\Rightarrow t7])) (THead (Flat Cast) u t3) (THead k u1 t5) H14) in (\lambda +(_: (eq T u u1)).(\lambda (H19: (eq K (Flat Cast) k)).(eq_ind K (Flat Cast) +(\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 +(THead k0 u2 t6) t7)))) (let H20 \def (eq_ind_r K k (\lambda (k0: K).(eq T +(THead k0 u1 t5) t)) H9 (Flat Cast) H19) in (let H21 \def (eq_ind_r T t +(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) +\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) +(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Cast) u1 t5) H20) in +(let H22 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t1)) H6 t5 H17) in +(ex2_ind T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T +(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Flat Cast) u2 t6) +t7))) (\lambda (x: T).(\lambda (H23: (pr0 t1 x)).(\lambda (H24: (pr0 t6 +x)).(ex_intro2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead +(Flat Cast) u2 t6) t7)) x H23 (pr0_tau t6 x H24 u2))))) (H21 t5 (tlt_head_dx +(Flat Cast) u1 t5) t1 H22 t6 H13))))) k H19)))) H16)) H15))))) t2 H11)) t H9 +H10 H7 H8))) | (pr0_beta u0 v1 v2 H7 t5 t6 H8) \Rightarrow (\lambda (H9: (eq +T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t5)) t)).(\lambda (H10: (eq T +(THead (Bind Abbr) v2 t6) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind +Abst) u0 t5)) (\lambda (_: T).((eq T (THead (Bind Abbr) v2 t6) t2) \to ((pr0 +v1 v2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda +(t8: T).(pr0 t2 t8))))))) (\lambda (H11: (eq T (THead (Bind Abbr) v2 t6) +t2)).(eq_ind T (THead (Bind Abbr) v2 t6) (\lambda (t7: T).((pr0 v1 v2) \to +((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 +t7 t8)))))) (\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 t5 t6)).(let H14 \def +(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Cast) u t3) t7)) H3 (THead +(Flat Appl) v1 (THead (Bind Abst) u0 t5)) H9) in (let H15 \def (eq_ind T +(THead (Flat Cast) u t3) (\lambda (ee: T).(match ee in T return (\lambda (_: +T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | +(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with +[(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f in F return +(\lambda (_: F).Prop) with [Appl \Rightarrow False | Cast \Rightarrow +True])])])) I (THead (Flat Appl) v1 (THead (Bind Abst) u0 t5)) H14) in +(False_ind (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead +(Bind Abbr) v2 t6) t7))) H15))))) t2 H11)) t H9 H10 H7 H8))) | (pr0_upsilon b +H7 v1 v2 H8 u1 u2 H9 t5 t6 H10) \Rightarrow (\lambda (H11: (eq T (THead (Flat +Appl) v1 (THead (Bind b) u1 t5)) t)).(\lambda (H12: (eq T (THead (Bind b) u2 +(THead (Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Flat Appl) +v1 (THead (Bind b) u1 t5)) (\lambda (_: T).((eq T (THead (Bind b) u2 (THead +(Flat Appl) (lift (S O) O v2) t6)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1 +v2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 +t8)) (\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H13: (eq T (THead (Bind +b) u2 (THead (Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Bind +b) u2 (THead (Flat Appl) (lift (S O) O v2) t6)) (\lambda (t7: T).((not (eq B +b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T +(\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 t8)))))))) (\lambda +(_: (not (eq B b Abst))).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 u1 +u2)).(\lambda (_: (pr0 t5 t6)).(let H18 \def (eq_ind_r T t (\lambda (t7: +T).(eq T (THead (Flat Cast) u t3) t7)) H3 (THead (Flat Appl) v1 (THead (Bind +b) u1 t5)) H11) in (let H19 \def (eq_ind T (THead (Flat Cast) u t3) (\lambda +(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) +\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow +(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | +(Flat f) \Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl +\Rightarrow False | Cast \Rightarrow True])])])) I (THead (Flat Appl) v1 +(THead (Bind b) u1 t5)) H18) in (False_ind (ex2 T (\lambda (t7: T).(pr0 t1 +t7)) (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) +O v2) t6)) t7))) H19))))))) t2 H13)) t H11 H12 H7 H8 H9 H10))) | (pr0_delta +u1 u2 H7 t5 t6 H8 w H9) \Rightarrow (\lambda (H10: (eq T (THead (Bind Abbr) +u1 t5) t)).(\lambda (H11: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T +(THead (Bind Abbr) u1 t5) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t2) +\to ((pr0 u1 u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T (\lambda +(t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))))) (\lambda (H12: (eq T +(THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda +(t7: T).((pr0 u1 u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T +(\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 t8))))))) (\lambda (_: +(pr0 u1 u2)).(\lambda (_: (pr0 t5 t6)).(\lambda (_: (subst0 O u2 t6 w)).(let +H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Cast) u t3) t7)) +H3 (THead (Bind Abbr) u1 t5) H10) in (let H17 \def (eq_ind T (THead (Flat +Cast) u t3) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with +[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) +\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) +\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u1 +t5) H16) in (False_ind (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: +T).(pr0 (THead (Bind Abbr) u2 w) t7))) H17)))))) t2 H12)) t H10 H11 H7 H8 +H9))) | (pr0_zeta b H7 t5 t6 H8 u0) \Rightarrow (\lambda (H9: (eq T (THead +(Bind b) u0 (lift (S O) O t5)) t)).(\lambda (H10: (eq T t6 t2)).(eq_ind T +(THead (Bind b) u0 (lift (S O) O t5)) (\lambda (_: T).((eq T t6 t2) \to ((not +(eq B b Abst)) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) +(\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H11: (eq T t6 t2)).(eq_ind T t2 +(\lambda (t7: T).((not (eq B b Abst)) \to ((pr0 t5 t7) \to (ex2 T (\lambda +(t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (_: (not (eq +B b Abst))).(\lambda (_: (pr0 t5 t2)).(let H14 \def (eq_ind_r T t (\lambda +(t7: T).(eq T (THead (Flat Cast) u t3) t7)) H3 (THead (Bind b) u0 (lift (S O) +O t5)) H9) in (let H15 \def (eq_ind T (THead (Flat Cast) u t3) (\lambda (ee: +T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow +False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K +return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) +\Rightarrow True])])) I (THead (Bind b) u0 (lift (S O) O t5)) H14) in +(False_ind (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 +t7))) H15))))) t6 (sym_eq T t6 t2 H11))) t H9 H10 H7 H8))) | (pr0_tau t5 t6 +H7 u0) \Rightarrow (\lambda (H8: (eq T (THead (Flat Cast) u0 t5) t)).(\lambda +(H9: (eq T t6 t2)).(eq_ind T (THead (Flat Cast) u0 t5) (\lambda (_: T).((eq T +t6 t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda +(t8: T).(pr0 t2 t8)))))) (\lambda (H10: (eq T t6 t2)).(eq_ind T t2 (\lambda +(t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: +T).(pr0 t2 t8))))) (\lambda (H11: (pr0 t5 t2)).(let H12 \def (eq_ind_r T t +(\lambda (t7: T).(eq T (THead (Flat Cast) u t3) t7)) H3 (THead (Flat Cast) u0 +t5) H8) in (let H13 \def (f_equal T T (\lambda (e: T).(match e in T return +(\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | +(THead _ t7 _) \Rightarrow t7])) (THead (Flat Cast) u t3) (THead (Flat Cast) +u0 t5) H12) in ((let H14 \def (f_equal T T (\lambda (e: T).(match e in T +return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) +\Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead (Flat Cast) u t3) +(THead (Flat Cast) u0 t5) H12) in (\lambda (_: (eq T u u0)).(let H16 \def +(eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: +T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: +T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Cast) +u0 t5) H8) in (let H17 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t1)) H6 t5 +H14) in (ex2_ind T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 +t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))) +(\lambda (x: T).(\lambda (H18: (pr0 t1 x)).(\lambda (H19: (pr0 t2 +x)).(ex_intro2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)) +x H18 H19)))) (H16 t5 (tlt_head_dx (Flat Cast) u0 t5) t1 H17 t2 H11)))))) +H13)))) t6 (sym_eq T t6 t2 H10))) t H8 H9 H7)))]) in (H7 (refl_equal T t) +(refl_equal T t2)))) t4 (sym_eq T t4 t1 H5))) t H3 H4 H2)))]) in (H2 +(refl_equal T t) (refl_equal T t1))))))))) t0). +(* COMMENTS +Initial nodes: 46103 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr0/props.ma b/matita/matita/contribs/lambdadelta/basic_1/pr0/props.ma new file mode 100644 index 000000000..d7f69d691 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/pr0/props.ma @@ -0,0 +1,1758 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/pr0/defs.ma". + +include "Basic-1/subst0/subst0.ma". + +theorem pr0_lift: + \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (h: nat).(\forall +(d: nat).(pr0 (lift h d t1) (lift h d t2)))))) +\def + \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(pr0_ind (\lambda +(t: T).(\lambda (t0: T).(\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t) +(lift h d t0)))))) (\lambda (t: T).(\lambda (h: nat).(\lambda (d: +nat).(pr0_refl (lift h d t))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda +(_: (pr0 u1 u2)).(\lambda (H1: ((\forall (h: nat).(\forall (d: nat).(pr0 +(lift h d u1) (lift h d u2)))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda +(_: (pr0 t3 t4)).(\lambda (H3: ((\forall (h: nat).(\forall (d: nat).(pr0 +(lift h d t3) (lift h d t4)))))).(\lambda (k: K).(\lambda (h: nat).(\lambda +(d: nat).(eq_ind_r T (THead k (lift h d u1) (lift h (s k d) t3)) (\lambda (t: +T).(pr0 t (lift h d (THead k u2 t4)))) (eq_ind_r T (THead k (lift h d u2) +(lift h (s k d) t4)) (\lambda (t: T).(pr0 (THead k (lift h d u1) (lift h (s k +d) t3)) t)) (pr0_comp (lift h d u1) (lift h d u2) (H1 h d) (lift h (s k d) +t3) (lift h (s k d) t4) (H3 h (s k d)) k) (lift h d (THead k u2 t4)) +(lift_head k u2 t4 h d)) (lift h d (THead k u1 t3)) (lift_head k u1 t3 h +d))))))))))))) (\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: +(pr0 v1 v2)).(\lambda (H1: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h +d v1) (lift h d v2)))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 +t3 t4)).(\lambda (H3: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t3) +(lift h d t4)))))).(\lambda (h: nat).(\lambda (d: nat).(eq_ind_r T (THead +(Flat Appl) (lift h d v1) (lift h (s (Flat Appl) d) (THead (Bind Abst) u +t3))) (\lambda (t: T).(pr0 t (lift h d (THead (Bind Abbr) v2 t4)))) (eq_ind_r +T (THead (Bind Abst) (lift h (s (Flat Appl) d) u) (lift h (s (Bind Abst) (s +(Flat Appl) d)) t3)) (\lambda (t: T).(pr0 (THead (Flat Appl) (lift h d v1) t) +(lift h d (THead (Bind Abbr) v2 t4)))) (eq_ind_r T (THead (Bind Abbr) (lift h +d v2) (lift h (s (Bind Abbr) d) t4)) (\lambda (t: T).(pr0 (THead (Flat Appl) +(lift h d v1) (THead (Bind Abst) (lift h (s (Flat Appl) d) u) (lift h (s +(Bind Abst) (s (Flat Appl) d)) t3))) t)) (pr0_beta (lift h (s (Flat Appl) d) +u) (lift h d v1) (lift h d v2) (H1 h d) (lift h (s (Bind Abst) (s (Flat Appl) +d)) t3) (lift h (s (Bind Abbr) d) t4) (H3 h (s (Bind Abbr) d))) (lift h d +(THead (Bind Abbr) v2 t4)) (lift_head (Bind Abbr) v2 t4 h d)) (lift h (s +(Flat Appl) d) (THead (Bind Abst) u t3)) (lift_head (Bind Abst) u t3 h (s +(Flat Appl) d))) (lift h d (THead (Flat Appl) v1 (THead (Bind Abst) u t3))) +(lift_head (Flat Appl) v1 (THead (Bind Abst) u t3) h d))))))))))))) (\lambda +(b: B).(\lambda (H0: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: +T).(\lambda (_: (pr0 v1 v2)).(\lambda (H2: ((\forall (h: nat).(\forall (d: +nat).(pr0 (lift h d v1) (lift h d v2)))))).(\lambda (u1: T).(\lambda (u2: +T).(\lambda (_: (pr0 u1 u2)).(\lambda (H4: ((\forall (h: nat).(\forall (d: +nat).(pr0 (lift h d u1) (lift h d u2)))))).(\lambda (t3: T).(\lambda (t4: +T).(\lambda (_: (pr0 t3 t4)).(\lambda (H6: ((\forall (h: nat).(\forall (d: +nat).(pr0 (lift h d t3) (lift h d t4)))))).(\lambda (h: nat).(\lambda (d: +nat).(eq_ind_r T (THead (Flat Appl) (lift h d v1) (lift h (s (Flat Appl) d) +(THead (Bind b) u1 t3))) (\lambda (t: T).(pr0 t (lift h d (THead (Bind b) u2 +(THead (Flat Appl) (lift (S O) O v2) t4))))) (eq_ind_r T (THead (Bind b) +(lift h (s (Flat Appl) d) u1) (lift h (s (Bind b) (s (Flat Appl) d)) t3)) +(\lambda (t: T).(pr0 (THead (Flat Appl) (lift h d v1) t) (lift h d (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))))) (eq_ind_r T (THead +(Bind b) (lift h d u2) (lift h (s (Bind b) d) (THead (Flat Appl) (lift (S O) +O v2) t4))) (\lambda (t: T).(pr0 (THead (Flat Appl) (lift h d v1) (THead +(Bind b) (lift h (s (Flat Appl) d) u1) (lift h (s (Bind b) (s (Flat Appl) d)) +t3))) t)) (eq_ind_r T (THead (Flat Appl) (lift h (s (Bind b) d) (lift (S O) O +v2)) (lift h (s (Flat Appl) (s (Bind b) d)) t4)) (\lambda (t: T).(pr0 (THead +(Flat Appl) (lift h d v1) (THead (Bind b) (lift h (s (Flat Appl) d) u1) (lift +h (s (Bind b) (s (Flat Appl) d)) t3))) (THead (Bind b) (lift h d u2) t))) +(eq_ind nat (plus (S O) d) (\lambda (n: nat).(pr0 (THead (Flat Appl) (lift h +d v1) (THead (Bind b) (lift h d u1) (lift h n t3))) (THead (Bind b) (lift h d +u2) (THead (Flat Appl) (lift h n (lift (S O) O v2)) (lift h n t4))))) +(eq_ind_r T (lift (S O) O (lift h d v2)) (\lambda (t: T).(pr0 (THead (Flat +Appl) (lift h d v1) (THead (Bind b) (lift h d u1) (lift h (plus (S O) d) +t3))) (THead (Bind b) (lift h d u2) (THead (Flat Appl) t (lift h (plus (S O) +d) t4))))) (pr0_upsilon b H0 (lift h d v1) (lift h d v2) (H2 h d) (lift h d +u1) (lift h d u2) (H4 h d) (lift h (plus (S O) d) t3) (lift h (plus (S O) d) +t4) (H6 h (plus (S O) d))) (lift h (plus (S O) d) (lift (S O) O v2)) (lift_d +v2 h (S O) d O (le_O_n d))) (S d) (refl_equal nat (S d))) (lift h (s (Bind b) +d) (THead (Flat Appl) (lift (S O) O v2) t4)) (lift_head (Flat Appl) (lift (S +O) O v2) t4 h (s (Bind b) d))) (lift h d (THead (Bind b) u2 (THead (Flat +Appl) (lift (S O) O v2) t4))) (lift_head (Bind b) u2 (THead (Flat Appl) (lift +(S O) O v2) t4) h d)) (lift h (s (Flat Appl) d) (THead (Bind b) u1 t3)) +(lift_head (Bind b) u1 t3 h (s (Flat Appl) d))) (lift h d (THead (Flat Appl) +v1 (THead (Bind b) u1 t3))) (lift_head (Flat Appl) v1 (THead (Bind b) u1 t3) +h d)))))))))))))))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 +u2)).(\lambda (H1: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d u1) +(lift h d u2)))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 +t4)).(\lambda (H3: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t3) +(lift h d t4)))))).(\lambda (w: T).(\lambda (H4: (subst0 O u2 t4 w)).(\lambda +(h: nat).(\lambda (d: nat).(eq_ind_r T (THead (Bind Abbr) (lift h d u1) (lift +h (s (Bind Abbr) d) t3)) (\lambda (t: T).(pr0 t (lift h d (THead (Bind Abbr) +u2 w)))) (eq_ind_r T (THead (Bind Abbr) (lift h d u2) (lift h (s (Bind Abbr) +d) w)) (\lambda (t: T).(pr0 (THead (Bind Abbr) (lift h d u1) (lift h (s (Bind +Abbr) d) t3)) t)) (pr0_delta (lift h d u1) (lift h d u2) (H1 h d) (lift h (S +d) t3) (lift h (S d) t4) (H3 h (S d)) (lift h (S d) w) (let d' \def (S d) in +(eq_ind nat (minus (S d) (S O)) (\lambda (n: nat).(subst0 O (lift h n u2) +(lift h d' t4) (lift h d' w))) (subst0_lift_lt t4 w u2 O H4 (S d) (le_n_S O d +(le_O_n d)) h) d (eq_ind nat d (\lambda (n: nat).(eq nat n d)) (refl_equal +nat d) (minus d O) (minus_n_O d))))) (lift h d (THead (Bind Abbr) u2 w)) +(lift_head (Bind Abbr) u2 w h d)) (lift h d (THead (Bind Abbr) u1 t3)) +(lift_head (Bind Abbr) u1 t3 h d)))))))))))))) (\lambda (b: B).(\lambda (H0: +(not (eq B b Abst))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 +t4)).(\lambda (H2: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t3) +(lift h d t4)))))).(\lambda (u: T).(\lambda (h: nat).(\lambda (d: +nat).(eq_ind_r T (THead (Bind b) (lift h d u) (lift h (s (Bind b) d) (lift (S +O) O t3))) (\lambda (t: T).(pr0 t (lift h d t4))) (eq_ind nat (plus (S O) d) +(\lambda (n: nat).(pr0 (THead (Bind b) (lift h d u) (lift h n (lift (S O) O +t3))) (lift h d t4))) (eq_ind_r T (lift (S O) O (lift h d t3)) (\lambda (t: +T).(pr0 (THead (Bind b) (lift h d u) t) (lift h d t4))) (pr0_zeta b H0 (lift +h d t3) (lift h d t4) (H2 h d) (lift h d u)) (lift h (plus (S O) d) (lift (S +O) O t3)) (lift_d t3 h (S O) d O (le_O_n d))) (S d) (refl_equal nat (S d))) +(lift h d (THead (Bind b) u (lift (S O) O t3))) (lift_head (Bind b) u (lift +(S O) O t3) h d))))))))))) (\lambda (t3: T).(\lambda (t4: T).(\lambda (_: +(pr0 t3 t4)).(\lambda (H1: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h +d t3) (lift h d t4)))))).(\lambda (u: T).(\lambda (h: nat).(\lambda (d: +nat).(eq_ind_r T (THead (Flat Cast) (lift h d u) (lift h (s (Flat Cast) d) +t3)) (\lambda (t: T).(pr0 t (lift h d t4))) (pr0_tau (lift h (s (Flat Cast) +d) t3) (lift h d t4) (H1 h d) (lift h d u)) (lift h d (THead (Flat Cast) u +t3)) (lift_head (Flat Cast) u t3 h d))))))))) t1 t2 H))). +(* COMMENTS +Initial nodes: 2845 +END *) + +theorem pr0_subst0_back: + \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst0 +i u2 t1 t2) \to (\forall (u1: T).((pr0 u1 u2) \to (ex2 T (\lambda (t: +T).(subst0 i u1 t1 t)) (\lambda (t: T).(pr0 t t2))))))))) +\def + \lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (i: nat).(\lambda +(H: (subst0 i u2 t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t: +T).(\lambda (t0: T).(\lambda (t3: T).(\forall (u1: T).((pr0 u1 t) \to (ex2 T +(\lambda (t4: T).(subst0 n u1 t0 t4)) (\lambda (t4: T).(pr0 t4 t3))))))))) +(\lambda (v: T).(\lambda (i0: nat).(\lambda (u1: T).(\lambda (H0: (pr0 u1 +v)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1 (TLRef i0) t)) (\lambda (t: +T).(pr0 t (lift (S i0) O v))) (lift (S i0) O u1) (subst0_lref u1 i0) +(pr0_lift u1 v H0 (S i0) O)))))) (\lambda (v: T).(\lambda (u3: T).(\lambda +(u1: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v u1 u3)).(\lambda (H1: +((\forall (u4: T).((pr0 u4 v) \to (ex2 T (\lambda (t: T).(subst0 i0 u4 u1 t)) +(\lambda (t: T).(pr0 t u3))))))).(\lambda (t: T).(\lambda (k: K).(\lambda +(u0: T).(\lambda (H2: (pr0 u0 v)).(ex2_ind T (\lambda (t0: T).(subst0 i0 u0 +u1 t0)) (\lambda (t0: T).(pr0 t0 u3)) (ex2 T (\lambda (t0: T).(subst0 i0 u0 +(THead k u1 t) t0)) (\lambda (t0: T).(pr0 t0 (THead k u3 t)))) (\lambda (x: +T).(\lambda (H3: (subst0 i0 u0 u1 x)).(\lambda (H4: (pr0 x u3)).(ex_intro2 T +(\lambda (t0: T).(subst0 i0 u0 (THead k u1 t) t0)) (\lambda (t0: T).(pr0 t0 +(THead k u3 t))) (THead k x t) (subst0_fst u0 x u1 i0 H3 t k) (pr0_comp x u3 +H4 t t (pr0_refl t) k))))) (H1 u0 H2)))))))))))) (\lambda (k: K).(\lambda (v: +T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (i0: nat).(\lambda (_: (subst0 +(s k i0) v t4 t3)).(\lambda (H1: ((\forall (u1: T).((pr0 u1 v) \to (ex2 T +(\lambda (t: T).(subst0 (s k i0) u1 t4 t)) (\lambda (t: T).(pr0 t +t3))))))).(\lambda (u: T).(\lambda (u1: T).(\lambda (H2: (pr0 u1 v)).(ex2_ind +T (\lambda (t: T).(subst0 (s k i0) u1 t4 t)) (\lambda (t: T).(pr0 t t3)) (ex2 +T (\lambda (t: T).(subst0 i0 u1 (THead k u t4) t)) (\lambda (t: T).(pr0 t +(THead k u t3)))) (\lambda (x: T).(\lambda (H3: (subst0 (s k i0) u1 t4 +x)).(\lambda (H4: (pr0 x t3)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1 +(THead k u t4) t)) (\lambda (t: T).(pr0 t (THead k u t3))) (THead k u x) +(subst0_snd k u1 x t4 i0 H3 u) (pr0_comp u u (pr0_refl u) x t3 H4 k))))) (H1 +u1 H2)))))))))))) (\lambda (v: T).(\lambda (u1: T).(\lambda (u3: T).(\lambda +(i0: nat).(\lambda (_: (subst0 i0 v u1 u3)).(\lambda (H1: ((\forall (u4: +T).((pr0 u4 v) \to (ex2 T (\lambda (t: T).(subst0 i0 u4 u1 t)) (\lambda (t: +T).(pr0 t u3))))))).(\lambda (k: K).(\lambda (t3: T).(\lambda (t4: +T).(\lambda (_: (subst0 (s k i0) v t3 t4)).(\lambda (H3: ((\forall (u4: +T).((pr0 u4 v) \to (ex2 T (\lambda (t: T).(subst0 (s k i0) u4 t3 t)) (\lambda +(t: T).(pr0 t t4))))))).(\lambda (u0: T).(\lambda (H4: (pr0 u0 v)).(ex2_ind T +(\lambda (t: T).(subst0 (s k i0) u0 t3 t)) (\lambda (t: T).(pr0 t t4)) (ex2 T +(\lambda (t: T).(subst0 i0 u0 (THead k u1 t3) t)) (\lambda (t: T).(pr0 t +(THead k u3 t4)))) (\lambda (x: T).(\lambda (H5: (subst0 (s k i0) u0 t3 +x)).(\lambda (H6: (pr0 x t4)).(ex2_ind T (\lambda (t: T).(subst0 i0 u0 u1 t)) +(\lambda (t: T).(pr0 t u3)) (ex2 T (\lambda (t: T).(subst0 i0 u0 (THead k u1 +t3) t)) (\lambda (t: T).(pr0 t (THead k u3 t4)))) (\lambda (x0: T).(\lambda +(H7: (subst0 i0 u0 u1 x0)).(\lambda (H8: (pr0 x0 u3)).(ex_intro2 T (\lambda +(t: T).(subst0 i0 u0 (THead k u1 t3) t)) (\lambda (t: T).(pr0 t (THead k u3 +t4))) (THead k x0 x) (subst0_both u0 u1 x0 i0 H7 k t3 x H5) (pr0_comp x0 u3 +H8 x t4 H6 k))))) (H1 u0 H4))))) (H3 u0 H4))))))))))))))) i u2 t1 t2 H))))). +(* COMMENTS +Initial nodes: 979 +END *) + +theorem pr0_subst0_fwd: + \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst0 +i u2 t1 t2) \to (\forall (u1: T).((pr0 u2 u1) \to (ex2 T (\lambda (t: +T).(subst0 i u1 t1 t)) (\lambda (t: T).(pr0 t2 t))))))))) +\def + \lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (i: nat).(\lambda +(H: (subst0 i u2 t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t: +T).(\lambda (t0: T).(\lambda (t3: T).(\forall (u1: T).((pr0 t u1) \to (ex2 T +(\lambda (t4: T).(subst0 n u1 t0 t4)) (\lambda (t4: T).(pr0 t3 t4))))))))) +(\lambda (v: T).(\lambda (i0: nat).(\lambda (u1: T).(\lambda (H0: (pr0 v +u1)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1 (TLRef i0) t)) (\lambda (t: +T).(pr0 (lift (S i0) O v) t)) (lift (S i0) O u1) (subst0_lref u1 i0) +(pr0_lift v u1 H0 (S i0) O)))))) (\lambda (v: T).(\lambda (u3: T).(\lambda +(u1: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v u1 u3)).(\lambda (H1: +((\forall (u4: T).((pr0 v u4) \to (ex2 T (\lambda (t: T).(subst0 i0 u4 u1 t)) +(\lambda (t: T).(pr0 u3 t))))))).(\lambda (t: T).(\lambda (k: K).(\lambda +(u0: T).(\lambda (H2: (pr0 v u0)).(ex2_ind T (\lambda (t0: T).(subst0 i0 u0 +u1 t0)) (\lambda (t0: T).(pr0 u3 t0)) (ex2 T (\lambda (t0: T).(subst0 i0 u0 +(THead k u1 t) t0)) (\lambda (t0: T).(pr0 (THead k u3 t) t0))) (\lambda (x: +T).(\lambda (H3: (subst0 i0 u0 u1 x)).(\lambda (H4: (pr0 u3 x)).(ex_intro2 T +(\lambda (t0: T).(subst0 i0 u0 (THead k u1 t) t0)) (\lambda (t0: T).(pr0 +(THead k u3 t) t0)) (THead k x t) (subst0_fst u0 x u1 i0 H3 t k) (pr0_comp u3 +x H4 t t (pr0_refl t) k))))) (H1 u0 H2)))))))))))) (\lambda (k: K).(\lambda +(v: T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (i0: nat).(\lambda (_: +(subst0 (s k i0) v t4 t3)).(\lambda (H1: ((\forall (u1: T).((pr0 v u1) \to +(ex2 T (\lambda (t: T).(subst0 (s k i0) u1 t4 t)) (\lambda (t: T).(pr0 t3 +t))))))).(\lambda (u: T).(\lambda (u1: T).(\lambda (H2: (pr0 v u1)).(ex2_ind +T (\lambda (t: T).(subst0 (s k i0) u1 t4 t)) (\lambda (t: T).(pr0 t3 t)) (ex2 +T (\lambda (t: T).(subst0 i0 u1 (THead k u t4) t)) (\lambda (t: T).(pr0 +(THead k u t3) t))) (\lambda (x: T).(\lambda (H3: (subst0 (s k i0) u1 t4 +x)).(\lambda (H4: (pr0 t3 x)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1 +(THead k u t4) t)) (\lambda (t: T).(pr0 (THead k u t3) t)) (THead k u x) +(subst0_snd k u1 x t4 i0 H3 u) (pr0_comp u u (pr0_refl u) t3 x H4 k))))) (H1 +u1 H2)))))))))))) (\lambda (v: T).(\lambda (u1: T).(\lambda (u3: T).(\lambda +(i0: nat).(\lambda (_: (subst0 i0 v u1 u3)).(\lambda (H1: ((\forall (u4: +T).((pr0 v u4) \to (ex2 T (\lambda (t: T).(subst0 i0 u4 u1 t)) (\lambda (t: +T).(pr0 u3 t))))))).(\lambda (k: K).(\lambda (t3: T).(\lambda (t4: +T).(\lambda (_: (subst0 (s k i0) v t3 t4)).(\lambda (H3: ((\forall (u4: +T).((pr0 v u4) \to (ex2 T (\lambda (t: T).(subst0 (s k i0) u4 t3 t)) (\lambda +(t: T).(pr0 t4 t))))))).(\lambda (u0: T).(\lambda (H4: (pr0 v u0)).(ex2_ind T +(\lambda (t: T).(subst0 (s k i0) u0 t3 t)) (\lambda (t: T).(pr0 t4 t)) (ex2 T +(\lambda (t: T).(subst0 i0 u0 (THead k u1 t3) t)) (\lambda (t: T).(pr0 (THead +k u3 t4) t))) (\lambda (x: T).(\lambda (H5: (subst0 (s k i0) u0 t3 +x)).(\lambda (H6: (pr0 t4 x)).(ex2_ind T (\lambda (t: T).(subst0 i0 u0 u1 t)) +(\lambda (t: T).(pr0 u3 t)) (ex2 T (\lambda (t: T).(subst0 i0 u0 (THead k u1 +t3) t)) (\lambda (t: T).(pr0 (THead k u3 t4) t))) (\lambda (x0: T).(\lambda +(H7: (subst0 i0 u0 u1 x0)).(\lambda (H8: (pr0 u3 x0)).(ex_intro2 T (\lambda +(t: T).(subst0 i0 u0 (THead k u1 t3) t)) (\lambda (t: T).(pr0 (THead k u3 t4) +t)) (THead k x0 x) (subst0_both u0 u1 x0 i0 H7 k t3 x H5) (pr0_comp u3 x0 H8 +t4 x H6 k))))) (H1 u0 H4))))) (H3 u0 H4))))))))))))))) i u2 t1 t2 H))))). +(* COMMENTS +Initial nodes: 979 +END *) + +theorem pr0_subst0: + \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (v1: T).(\forall +(w1: T).(\forall (i: nat).((subst0 i v1 t1 w1) \to (\forall (v2: T).((pr0 v1 +v2) \to (or (pr0 w1 t2) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: +T).(subst0 i v2 t2 w2)))))))))))) +\def + \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(pr0_ind (\lambda +(t: T).(\lambda (t0: T).(\forall (v1: T).(\forall (w1: T).(\forall (i: +nat).((subst0 i v1 t w1) \to (\forall (v2: T).((pr0 v1 v2) \to (or (pr0 w1 +t0) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t0 +w2)))))))))))) (\lambda (t: T).(\lambda (v1: T).(\lambda (w1: T).(\lambda (i: +nat).(\lambda (H0: (subst0 i v1 t w1)).(\lambda (v2: T).(\lambda (H1: (pr0 v1 +v2)).(or_intror (pr0 w1 t) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: +T).(subst0 i v2 t w2))) (ex2_sym T (subst0 i v2 t) (pr0 w1) (pr0_subst0_fwd +v1 t w1 i H0 v2 H1)))))))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (H0: +(pr0 u1 u2)).(\lambda (H1: ((\forall (v1: T).(\forall (w1: T).(\forall (i: +nat).((subst0 i v1 u1 w1) \to (\forall (v2: T).((pr0 v1 v2) \to (or (pr0 w1 +u2) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 u2 +w2)))))))))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 +t4)).(\lambda (H3: ((\forall (v1: T).(\forall (w1: T).(\forall (i: +nat).((subst0 i v1 t3 w1) \to (\forall (v2: T).((pr0 v1 v2) \to (or (pr0 w1 +t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 +w2)))))))))))).(\lambda (k: K).(\lambda (v1: T).(\lambda (w1: T).(\lambda (i: +nat).(\lambda (H4: (subst0 i v1 (THead k u1 t3) w1)).(\lambda (v2: +T).(\lambda (H5: (pr0 v1 v2)).(or3_ind (ex2 T (\lambda (u3: T).(eq T w1 +(THead k u3 t3))) (\lambda (u3: T).(subst0 i v1 u1 u3))) (ex2 T (\lambda (t5: +T).(eq T w1 (THead k u1 t5))) (\lambda (t5: T).(subst0 (s k i) v1 t3 t5))) +(ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T w1 (THead k u3 t5)))) +(\lambda (u3: T).(\lambda (_: T).(subst0 i v1 u1 u3))) (\lambda (_: +T).(\lambda (t5: T).(subst0 (s k i) v1 t3 t5)))) (or (pr0 w1 (THead k u2 t4)) +(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 (THead k +u2 t4) w2)))) (\lambda (H6: (ex2 T (\lambda (u3: T).(eq T w1 (THead k u3 +t3))) (\lambda (u3: T).(subst0 i v1 u1 u3)))).(ex2_ind T (\lambda (u3: T).(eq +T w1 (THead k u3 t3))) (\lambda (u3: T).(subst0 i v1 u1 u3)) (or (pr0 w1 +(THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: +T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x: T).(\lambda (H7: (eq T w1 +(THead k x t3))).(\lambda (H8: (subst0 i v1 u1 x)).(eq_ind_r T (THead k x t3) +(\lambda (t: T).(or (pr0 t (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 t +w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))))) (or_ind (pr0 x u2) +(ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 u2 w2))) +(or (pr0 (THead k x t3) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead +k x t3) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda +(H9: (pr0 x u2)).(or_introl (pr0 (THead k x t3) (THead k u2 t4)) (ex2 T +(\lambda (w2: T).(pr0 (THead k x t3) w2)) (\lambda (w2: T).(subst0 i v2 +(THead k u2 t4) w2))) (pr0_comp x u2 H9 t3 t4 H2 k))) (\lambda (H9: (ex2 T +(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 u2 w2)))).(ex2_ind +T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 u2 w2)) (or (pr0 +(THead k x t3) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x t3) +w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x0: +T).(\lambda (H10: (pr0 x x0)).(\lambda (H11: (subst0 i v2 u2 x0)).(or_intror +(pr0 (THead k x t3) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x +t3) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))) (ex_intro2 T +(\lambda (w2: T).(pr0 (THead k x t3) w2)) (\lambda (w2: T).(subst0 i v2 +(THead k u2 t4) w2)) (THead k x0 t4) (pr0_comp x x0 H10 t3 t4 H2 k) +(subst0_fst v2 x0 u2 i H11 t4 k)))))) H9)) (H1 v1 x i H8 v2 H5)) w1 H7)))) +H6)) (\lambda (H6: (ex2 T (\lambda (t5: T).(eq T w1 (THead k u1 t5))) +(\lambda (t5: T).(subst0 (s k i) v1 t3 t5)))).(ex2_ind T (\lambda (t5: T).(eq +T w1 (THead k u1 t5))) (\lambda (t5: T).(subst0 (s k i) v1 t3 t5)) (or (pr0 +w1 (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: +T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x: T).(\lambda (H7: (eq T w1 +(THead k u1 x))).(\lambda (H8: (subst0 (s k i) v1 t3 x)).(eq_ind_r T (THead k +u1 x) (\lambda (t: T).(or (pr0 t (THead k u2 t4)) (ex2 T (\lambda (w2: +T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))))) (or_ind +(pr0 x t4) (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s k +i) v2 t4 w2))) (or (pr0 (THead k u1 x) (THead k u2 t4)) (ex2 T (\lambda (w2: +T).(pr0 (THead k u1 x) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) +w2)))) (\lambda (H9: (pr0 x t4)).(or_introl (pr0 (THead k u1 x) (THead k u2 +t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k u1 x) w2)) (\lambda (w2: +T).(subst0 i v2 (THead k u2 t4) w2))) (pr0_comp u1 u2 H0 x t4 H9 k))) +(\lambda (H9: (ex2 T (\lambda (w2: T).(pr0 x w2)) 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w2)) (THead k x t4) (pr0_comp x0 x H12 x1 t4 H10 k) (subst0_fst v2 x u2 i +H13 t4 k)))))) H11)) (H1 v1 x0 i H8 v2 H5))) (\lambda (H10: (ex2 T (\lambda +(w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s k i) v2 t4 w2)))).(ex2_ind T +(\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s k i) v2 t4 w2)) (or +(pr0 (THead k x0 x1) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k +x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x: +T).(\lambda (H11: (pr0 x1 x)).(\lambda (H12: (subst0 (s k i) v2 t4 +x)).(or_ind (pr0 x0 u2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: +T).(subst0 i v2 u2 w2))) (or (pr0 (THead k x0 x1) (THead k u2 t4)) (ex2 T +(\lambda (w2: T).(pr0 (THead k x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 +(THead k u2 t4) w2)))) (\lambda (H13: (pr0 x0 u2)).(or_intror (pr0 (THead k +x0 x1) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x0 x1) w2)) +(\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))) (ex_intro2 T (\lambda +(w2: T).(pr0 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(subst0_gen_head k v1 u1 t3 w1 i H4))))))))))))))))) +(\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H0: (pr0 v1 +v2)).(\lambda (H1: ((\forall (v3: T).(\forall (w1: T).(\forall (i: +nat).((subst0 i v3 v1 w1) \to (\forall (v4: T).((pr0 v3 v4) \to (or (pr0 w1 +v2) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v4 v2 +w2)))))))))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 +t4)).(\lambda (H3: ((\forall (v3: T).(\forall (w1: T).(\forall (i: +nat).((subst0 i v3 t3 w1) \to (\forall (v4: T).((pr0 v3 v4) \to (or (pr0 w1 +t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v4 t4 +w2)))))))))))).(\lambda (v0: T).(\lambda (w1: T).(\lambda (i: nat).(\lambda +(H4: (subst0 i v0 (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) +w1)).(\lambda (v3: T).(\lambda (H5: (pr0 v0 v3)).(or3_ind (ex2 T (\lambda +(u2: T).(eq T w1 (THead (Flat Appl) u2 (THead (Bind Abst) u t3)))) (\lambda +(u2: T).(subst0 i v0 v1 u2))) (ex2 T (\lambda (t5: T).(eq T w1 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T).(subst0 i v3 v2 w2)) (or (pr0 (THead (Flat Appl) x (THead +(Bind Abst) u t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 +(THead (Flat Appl) x (THead (Bind Abst) u t3)) w2)) (\lambda (w2: T).(subst0 +i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (x0: T).(\lambda (H10: (pr0 x +x0)).(\lambda (H11: (subst0 i v3 v2 x0)).(or_intror (pr0 (THead (Flat Appl) x +(THead (Bind Abst) u t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: +T).(pr0 (THead (Flat Appl) x (THead (Bind Abst) u t3)) w2)) (\lambda (w2: +T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: +T).(pr0 (THead (Flat Appl) x (THead (Bind Abst) u t3)) w2)) (\lambda (w2: +T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)) (THead (Bind Abbr) x0 t4) +(pr0_beta u x x0 H10 t3 t4 H2) (subst0_fst v3 x0 v2 i H11 t4 (Bind +Abbr))))))) H9)) (H1 v0 x i H8 v3 H5)) w1 H7)))) H6)) (\lambda (H6: (ex2 T +(\lambda (t5: T).(eq T w1 (THead (Flat Appl) v1 t5))) (\lambda (t5: +T).(subst0 (s (Flat Appl) i) v0 (THead 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(\lambda (w2: T).(pr0 +w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) +(\lambda (H9: (ex2 T (\lambda (u2: T).(eq T x (THead (Bind Abst) u2 t3))) +(\lambda (u2: T).(subst0 (s (Flat Appl) i) v0 u u2)))).(ex2_ind T (\lambda +(u2: T).(eq T x (THead (Bind Abst) u2 t3))) (\lambda (u2: T).(subst0 (s (Flat +Appl) i) v0 u u2)) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda +(w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) +w2)))) (\lambda (x0: T).(\lambda (H10: (eq T x (THead (Bind Abst) x0 +t3))).(\lambda (_: (subst0 (s (Flat Appl) i) v0 u x0)).(let H12 \def (eq_ind +T x (\lambda (t: T).(eq T w1 (THead (Flat Appl) v1 t))) H7 (THead (Bind Abst) +x0 t3) H10) in (eq_ind_r T (THead (Flat Appl) v1 (THead (Bind Abst) x0 t3)) +(\lambda (t: T).(or (pr0 t (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: +T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) +w2))))) (or_introl (pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 t3)) +(THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 +(THead (Bind Abst) x0 t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind +Abbr) v2 t4) w2))) (pr0_beta x0 v1 v2 H0 t3 t4 H2)) w1 H12))))) H9)) (\lambda +(H9: (ex2 T (\lambda (t5: T).(eq T x (THead (Bind Abst) u t5))) (\lambda (t5: +T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5)))).(ex2_ind T (\lambda +(t5: T).(eq T x (THead (Bind Abst) u t5))) (\lambda (t5: T).(subst0 (s (Bind +Abst) (s (Flat Appl) i)) v0 t3 t5)) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) +(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead +(Bind Abbr) v2 t4) w2)))) (\lambda (x0: T).(\lambda (H10: (eq T x (THead +(Bind Abst) u x0))).(\lambda (H11: (subst0 (s (Bind Abst) (s (Flat Appl) i)) +v0 t3 x0)).(let H12 \def (eq_ind T x (\lambda (t: T).(eq T w1 (THead (Flat +Appl) v1 t))) H7 (THead (Bind Abst) u x0) H10) in (eq_ind_r T (THead (Flat +Appl) v1 (THead (Bind Abst) u x0)) (\lambda (t: T).(or (pr0 t (THead (Bind +Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i +v3 (THead (Bind Abbr) v2 t4) w2))))) (or_ind (pr0 x0 t4) (ex2 T (\lambda (w2: +T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 +t4 w2))) (or (pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u x0)) (THead +(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead +(Bind Abst) u x0)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 +t4) w2)))) (\lambda (H13: (pr0 x0 t4)).(or_introl (pr0 (THead (Flat Appl) v1 +(THead (Bind Abst) u x0)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: +T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u x0)) w2)) (\lambda (w2: +T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (pr0_beta u v1 v2 H0 x0 t4 +H13))) (\lambda (H13: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: +T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda +(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) +i)) v3 t4 w2)) (or (pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u x0)) +(THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 +(THead (Bind Abst) u x0)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind +Abbr) v2 t4) w2)))) (\lambda (x1: T).(\lambda (H14: (pr0 x0 x1)).(\lambda +(H15: (subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 x1)).(or_intror (pr0 +(THead (Flat Appl) v1 (THead (Bind Abst) u x0)) (THead (Bind Abbr) v2 t4)) +(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u x0)) +w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (ex_intro2 +T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u x0)) w2)) +(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)) (THead (Bind +Abbr) v2 x1) (pr0_beta u v1 v2 H0 x0 x1 H14) (subst0_snd (Bind Abbr) v3 x1 t4 +i H15 v2)))))) H13)) (H3 v0 x0 (s (Bind Abst) (s (Flat Appl) i)) H11 v3 H5)) +w1 H12))))) H9)) (\lambda (H9: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: +T).(eq T x (THead (Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: +T).(subst0 (s (Flat Appl) i) v0 u u2))) (\lambda (_: T).(\lambda (t5: +T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5))))).(ex3_2_ind T T +(\lambda (u2: T).(\lambda (t5: T).(eq T x (THead (Bind Abst) u2 t5)))) +(\lambda (u2: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u u2))) +(\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 +t3 t5))) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 +w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) +(\lambda (x0: T).(\lambda (x1: T).(\lambda (H10: (eq T x (THead (Bind Abst) +x0 x1))).(\lambda (_: (subst0 (s (Flat Appl) i) v0 u x0)).(\lambda (H12: +(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 x1)).(let H13 \def (eq_ind T +x (\lambda (t: T).(eq T w1 (THead (Flat Appl) v1 t))) H7 (THead (Bind Abst) +x0 x1) H10) in (eq_ind_r T (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) +(\lambda (t: T).(or (pr0 t (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: +T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) +w2))))) (or_ind (pr0 x1 t4) (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda +(w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2))) (or (pr0 (THead +(Flat Appl) v1 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) v2 t4)) (ex2 T +(\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) w2)) +(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H14: +(pr0 x1 t4)).(or_introl (pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) +(THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 +(THead (Bind Abst) x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind +Abbr) v2 t4) w2))) (pr0_beta x0 v1 v2 H0 x1 t4 H14))) (\lambda (H14: (ex2 T +(\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s +(Flat Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x1 w2)) +(\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2)) (or +(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) v2 +t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 +x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) +(\lambda (x2: T).(\lambda (H15: (pr0 x1 x2)).(\lambda (H16: (subst0 (s (Bind +Abst) (s (Flat Appl) i)) v3 t4 x2)).(or_intror (pr0 (THead (Flat Appl) v1 +(THead (Bind Abst) x0 x1)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: +T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) w2)) (\lambda (w2: +T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: +T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) w2)) (\lambda (w2: +T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)) (THead (Bind Abbr) v2 x2) +(pr0_beta x0 v1 v2 H0 x1 x2 H15) (subst0_snd (Bind Abbr) v3 x2 t4 i H16 +v2)))))) H14)) (H3 v0 x1 (s (Bind Abst) (s (Flat Appl) i)) H12 v3 H5)) w1 +H13))))))) H9)) (subst0_gen_head (Bind Abst) v0 u t3 x (s (Flat Appl) i) +H8))))) H6)) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T +w1 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 +v1 u2))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Flat Appl) i) v0 (THead +(Bind Abst) u t3) t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: +T).(eq T w1 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: +T).(subst0 i v0 v1 u2))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Flat +Appl) i) v0 (THead (Bind Abst) u t3) t5))) (or (pr0 w1 (THead (Bind Abbr) v2 +t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 +(THead (Bind Abbr) v2 t4) w2)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda +(H7: (eq T w1 (THead (Flat Appl) x0 x1))).(\lambda (H8: (subst0 i v0 v1 +x0)).(\lambda (H9: (subst0 (s (Flat Appl) i) v0 (THead (Bind Abst) u t3) +x1)).(or3_ind (ex2 T (\lambda (u2: T).(eq T x1 (THead (Bind Abst) u2 t3))) +(\lambda (u2: T).(subst0 (s (Flat Appl) i) v0 u u2))) (ex2 T (\lambda (t5: +T).(eq T x1 (THead (Bind Abst) u t5))) (\lambda (t5: T).(subst0 (s (Bind +Abst) (s (Flat Appl) i)) v0 t3 t5))) (ex3_2 T T (\lambda (u2: T).(\lambda +(t5: T).(eq T x1 (THead (Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: +T).(subst0 (s (Flat Appl) i) v0 u u2))) (\lambda (_: T).(\lambda (t5: +T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5)))) (or (pr0 w1 (THead +(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: +T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H10: (ex2 T +(\lambda (u2: T).(eq T x1 (THead (Bind Abst) u2 t3))) (\lambda (u2: +T).(subst0 (s (Flat Appl) i) v0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T x1 +(THead (Bind Abst) u2 t3))) (\lambda (u2: T).(subst0 (s (Flat Appl) i) v0 u +u2)) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 +w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda +(x: T).(\lambda (H11: (eq T x1 (THead (Bind Abst) x t3))).(\lambda (_: +(subst0 (s (Flat Appl) i) v0 u x)).(let H13 \def (eq_ind T x1 (\lambda (t: +T).(eq T w1 (THead (Flat Appl) x0 t))) H7 (THead (Bind Abst) x t3) H11) in +(eq_ind_r T (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) (\lambda (t: +T).(or (pr0 t (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 t w2)) +(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))))) (or_ind (pr0 +x0 v2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 +w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) (THead (Bind +Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind +Abst) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) +w2)))) (\lambda (H14: (pr0 x0 v2)).(or_introl (pr0 (THead (Flat Appl) x0 +(THead (Bind Abst) x t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: +T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) w2)) (\lambda (w2: +T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (pr0_beta x x0 v2 H14 t3 t4 +H2))) (\lambda (H14: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: +T).(subst0 i v3 v2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda +(w2: T).(subst0 i v3 v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind +Abst) x t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead +(Flat Appl) x0 (THead (Bind Abst) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 +(THead (Bind Abbr) v2 t4) w2)))) (\lambda (x2: T).(\lambda (H15: (pr0 x0 +x2)).(\lambda (H16: (subst0 i v3 v2 x2)).(or_intror (pr0 (THead (Flat Appl) +x0 (THead (Bind Abst) x t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: +T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) w2)) (\lambda (w2: +T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: +T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) w2)) (\lambda (w2: +T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)) (THead (Bind Abbr) x2 t4) +(pr0_beta x x0 x2 H15 t3 t4 H2) (subst0_fst v3 x2 v2 i H16 t4 (Bind +Abbr))))))) H14)) (H1 v0 x0 i H8 v3 H5)) w1 H13))))) H10)) (\lambda (H10: +(ex2 T (\lambda (t5: T).(eq T x1 (THead (Bind Abst) u t5))) (\lambda (t5: +T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5)))).(ex2_ind T (\lambda +(t5: T).(eq T x1 (THead (Bind Abst) u t5))) (\lambda (t5: T).(subst0 (s (Bind +Abst) (s (Flat Appl) i)) v0 t3 t5)) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) +(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead +(Bind Abbr) v2 t4) w2)))) (\lambda (x: T).(\lambda (H11: (eq T x1 (THead +(Bind Abst) u x))).(\lambda (H12: (subst0 (s (Bind Abst) (s (Flat Appl) i)) +v0 t3 x)).(let H13 \def (eq_ind T x1 (\lambda (t: T).(eq T w1 (THead (Flat +Appl) x0 t))) H7 (THead (Bind Abst) u x) H11) in (eq_ind_r T (THead (Flat +Appl) x0 (THead (Bind Abst) u x)) (\lambda (t: T).(or (pr0 t (THead (Bind +Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i +v3 (THead (Bind Abbr) v2 t4) w2))))) (or_ind (pr0 x t4) (ex2 T (\lambda (w2: +T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 +t4 w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind +Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind +Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) +w2)))) (\lambda (H14: (pr0 x t4)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2: +T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat +Appl) x0 (THead (Bind Abst) u x)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda +(w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) w2)) (\lambda +(w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H15: (pr0 x0 +v2)).(or_introl (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead +(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead +(Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 +t4) w2))) (pr0_beta u x0 v2 H15 x t4 H14))) (\lambda (H15: (ex2 T (\lambda +(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)))).(ex2_ind T +(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)) (or (pr0 +(THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind Abbr) v2 t4)) +(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) +w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda +(x2: T).(\lambda (H16: (pr0 x0 x2)).(\lambda (H17: (subst0 i v3 v2 +x2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead +(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead +(Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 +t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead +(Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 +t4) w2)) (THead (Bind Abbr) x2 t4) (pr0_beta u x0 x2 H16 x t4 H14) +(subst0_fst v3 x2 v2 i H17 t4 (Bind Abbr))))))) H15)) (H1 v0 x0 i H8 v3 H5))) +(\lambda (H14: (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 +(s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda (w2: +T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 +t4 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind +Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind +Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) +w2)))) (\lambda (x2: T).(\lambda (H15: (pr0 x x2)).(\lambda (H16: (subst0 (s +(Bind Abst) (s (Flat Appl) i)) v3 t4 x2)).(or_ind (pr0 x0 v2) (ex2 T (\lambda +(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead +(Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind Abbr) v2 t4)) (ex2 T +(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) w2)) +(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H17: +(pr0 x0 v2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) +(THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 +(THead (Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind +Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 +(THead (Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind +Abbr) v2 t4) w2)) (THead (Bind Abbr) v2 x2) (pr0_beta u x0 v2 H17 x x2 H15) +(subst0_snd (Bind Abbr) v3 x2 t4 i H16 v2)))) (\lambda (H17: (ex2 T (\lambda +(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)))).(ex2_ind T +(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)) (or (pr0 +(THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind Abbr) v2 t4)) +(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) +w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda +(x3: T).(\lambda (H18: (pr0 x0 x3)).(\lambda (H19: (subst0 i v3 v2 +x3)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead +(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead +(Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 +t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead +(Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 +t4) w2)) (THead (Bind Abbr) x3 x2) (pr0_beta u x0 x3 H18 x x2 H15) +(subst0_both v3 v2 x3 i H19 (Bind Abbr) t4 x2 H16)))))) H17)) (H1 v0 x0 i H8 +v3 H5))))) H14)) (H3 v0 x (s (Bind Abst) (s (Flat Appl) i)) H12 v3 H5)) w1 +H13))))) H10)) (\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: +T).(eq T x1 (THead (Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: +T).(subst0 (s (Flat Appl) i) v0 u u2))) (\lambda (_: T).(\lambda (t5: +T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5))))).(ex3_2_ind T T +(\lambda (u2: T).(\lambda (t5: T).(eq T x1 (THead (Bind Abst) u2 t5)))) +(\lambda (u2: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u u2))) +(\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 +t3 t5))) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 +w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) +(\lambda (x2: T).(\lambda (x3: T).(\lambda (H11: (eq T x1 (THead (Bind Abst) +x2 x3))).(\lambda (_: (subst0 (s (Flat Appl) i) v0 u x2)).(\lambda (H13: +(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 x3)).(let H14 \def (eq_ind T +x1 (\lambda (t: T).(eq T w1 (THead (Flat Appl) x0 t))) H7 (THead (Bind Abst) +x2 x3) H11) in (eq_ind_r T (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) +(\lambda (t: T).(or (pr0 t (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: +T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) +w2))))) (or_ind (pr0 x3 t4) (ex2 T (\lambda (w2: T).(pr0 x3 w2)) (\lambda +(w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2))) (or (pr0 (THead +(Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead (Bind Abbr) v2 t4)) (ex2 T +(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) w2)) +(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H15: +(pr0 x3 t4)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) +(\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat Appl) x0 (THead +(Bind Abst) x2 x3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 +(THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) w2)) (\lambda (w2: +T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H16: (pr0 x0 +v2)).(or_introl (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead +(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead +(Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 +t4) w2))) (pr0_beta x2 x0 v2 H16 x3 t4 H15))) (\lambda (H16: (ex2 T (\lambda +(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)))).(ex2_ind T +(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)) (or (pr0 +(THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead (Bind Abbr) v2 t4)) +(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) +w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda +(x: T).(\lambda (H17: (pr0 x0 x)).(\lambda (H18: (subst0 i v3 v2 +x)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead +(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead +(Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 +t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead +(Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 +t4) w2)) (THead (Bind Abbr) x t4) (pr0_beta x2 x0 x H17 x3 t4 H15) +(subst0_fst v3 x v2 i H18 t4 (Bind Abbr))))))) H16)) (H1 v0 x0 i H8 v3 H5))) +(\lambda (H15: (ex2 T (\lambda (w2: T).(pr0 x3 w2)) (\lambda (w2: T).(subst0 +(s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda (w2: +T).(pr0 x3 w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 +t4 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead +(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead +(Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 +t4) w2)))) (\lambda (x: T).(\lambda (H16: (pr0 x3 x)).(\lambda (H17: (subst0 +(s (Bind Abst) (s (Flat Appl) i)) v3 t4 x)).(or_ind (pr0 x0 v2) (ex2 T +(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 +(THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead (Bind Abbr) v2 t4)) +(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) +w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda +(H18: (pr0 x0 v2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) +x2 x3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat +Appl) x0 (THead (Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead +(Bind Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat +Appl) x0 (THead (Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead +(Bind Abbr) v2 t4) w2)) (THead (Bind Abbr) v2 x) (pr0_beta x2 x0 v2 H18 x3 x +H16) (subst0_snd (Bind Abbr) v3 x t4 i H17 v2)))) (\lambda (H18: (ex2 T +(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 +w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 +v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead +(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead +(Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 +t4) w2)))) (\lambda (x4: T).(\lambda (H19: (pr0 x0 x4)).(\lambda (H20: +(subst0 i v3 v2 x4)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) +x2 x3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat +Appl) x0 (THead (Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead +(Bind Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat +Appl) x0 (THead (Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead +(Bind Abbr) v2 t4) w2)) (THead (Bind Abbr) x4 x) (pr0_beta x2 x0 x4 H19 x3 x +H16) (subst0_both v3 v2 x4 i H20 (Bind Abbr) t4 x H17)))))) H18)) (H1 v0 x0 i +H8 v3 H5))))) H15)) (H3 v0 x3 (s (Bind Abst) (s (Flat Appl) i)) H13 v3 H5)) +w1 H14))))))) H10)) (subst0_gen_head (Bind Abst) v0 u t3 x1 (s (Flat Appl) i) +H9))))))) H6)) (subst0_gen_head (Flat Appl) v0 v1 (THead (Bind Abst) u t3) w1 +i H4))))))))))))))))) (\lambda (b: B).(\lambda (H0: (not (eq B b +Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H1: (pr0 v1 v2)).(\lambda +(H2: ((\forall (v3: T).(\forall (w1: T).(\forall (i: nat).((subst0 i v3 v1 +w1) \to (\forall (v4: T).((pr0 v3 v4) \to (or (pr0 w1 v2) (ex2 T (\lambda +(w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v4 v2 w2)))))))))))).(\lambda +(u1: T).(\lambda (u2: T).(\lambda (H3: (pr0 u1 u2)).(\lambda (H4: ((\forall +(v3: T).(\forall (w1: T).(\forall (i: nat).((subst0 i v3 u1 w1) \to (\forall +(v4: T).((pr0 v3 v4) \to (or (pr0 w1 u2) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) +(\lambda (w2: T).(subst0 i v4 u2 w2)))))))))))).(\lambda (t3: T).(\lambda +(t4: T).(\lambda (H5: (pr0 t3 t4)).(\lambda (H6: ((\forall (v3: T).(\forall +(w1: T).(\forall (i: nat).((subst0 i v3 t3 w1) \to (\forall (v4: T).((pr0 v3 +v4) \to (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: +T).(subst0 i v4 t4 w2)))))))))))).(\lambda (v0: T).(\lambda (w1: T).(\lambda +(i: nat).(\lambda (H7: (subst0 i v0 (THead (Flat Appl) v1 (THead (Bind b) u1 +t3)) w1)).(\lambda (v3: T).(\lambda (H8: (pr0 v0 v3)).(or3_ind (ex2 T +(\lambda (u3: T).(eq T w1 (THead (Flat Appl) u3 (THead (Bind b) u1 t3)))) +(\lambda (u3: T).(subst0 i v0 v1 u3))) (ex2 T (\lambda (t5: T).(eq T w1 +(THead (Flat Appl) v1 t5))) (\lambda (t5: T).(subst0 (s (Flat Appl) i) v0 +(THead (Bind b) u1 t3) t5))) (ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq +T w1 (THead (Flat Appl) u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i +v0 v1 u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s 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t4))) +(ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2))))) (or_ind (pr0 x +v2) (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v3 v2 +w2))) (or (pr0 (THead (Flat Appl) x (THead (Bind b) u1 t3)) (THead (Bind b) +u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 +(THead (Flat Appl) x (THead (Bind b) u1 t3)) w2)) (\lambda (w2: T).(subst0 i +v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) +(\lambda (H12: (pr0 x v2)).(or_introl (pr0 (THead (Flat Appl) x (THead (Bind +b) u1 t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 +T (\lambda (w2: T).(pr0 (THead (Flat Appl) x (THead (Bind b) u1 t3)) w2)) +(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S +O) O v2) t4)) w2))) (pr0_upsilon b H0 x v2 H12 u1 u2 H3 t3 t4 H5))) (\lambda +(H12: (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v3 v2 +w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v3 +v2 w2)) (or (pr0 (THead (Flat Appl) x (THead (Bind b) u1 t3)) (THead (Bind b) +u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 +(THead (Flat Appl) x (THead (Bind b) u1 t3)) w2)) (\lambda (w2: T).(subst0 i +v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) +(\lambda (x0: T).(\lambda (H13: (pr0 x x0)).(\lambda (H14: (subst0 i v3 v2 +x0)).(or_intror (pr0 (THead (Flat Appl) x (THead (Bind b) u1 t3)) (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: +T).(pr0 (THead (Flat Appl) x (THead (Bind b) u1 t3)) w2)) (\lambda (w2: +T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) +w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x (THead (Bind b) +u1 t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat +Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 (THead (Flat Appl) (lift +(S O) O x0) t4)) (pr0_upsilon b H0 x x0 H13 u1 u2 H3 t3 t4 H5) (subst0_snd +(Bind b) v3 (THead (Flat Appl) (lift (S O) O x0) t4) (THead (Flat Appl) (lift +(S O) O v2) t4) i (subst0_fst v3 (lift (S O) O x0) (lift (S O) O v2) (s (Bind +b) i) (subst0_lift_ge_s v2 x0 v3 i H14 O (le_O_n i) b) t4 (Flat Appl)) +u2)))))) H12)) (H2 v0 x i H11 v3 H8)) w1 H10)))) H9)) (\lambda (H9: (ex2 T +(\lambda (t5: T).(eq T w1 (THead (Flat Appl) v1 t5))) (\lambda (t5: +T).(subst0 (s (Flat Appl) i) v0 (THead (Bind b) u1 t3) t5)))).(ex2_ind T +(\lambda (t5: T).(eq T w1 (THead (Flat Appl) v1 t5))) (\lambda (t5: +T).(subst0 (s (Flat Appl) i) v0 (THead (Bind b) u1 t3) t5)) (or (pr0 w1 +(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda +(w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead +(Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x: T).(\lambda (H10: (eq +T w1 (THead (Flat Appl) v1 x))).(\lambda (H11: (subst0 (s (Flat Appl) i) v0 +(THead (Bind b) u1 t3) x)).(or3_ind (ex2 T (\lambda (u3: T).(eq T x (THead +(Bind b) u3 t3))) (\lambda (u3: T).(subst0 (s (Flat Appl) i) v0 u1 u3))) (ex2 +T (\lambda (t5: T).(eq T x (THead (Bind b) u1 t5))) (\lambda (t5: T).(subst0 +(s (Bind b) (s (Flat Appl) i)) v0 t3 t5))) (ex3_2 T T (\lambda (u3: +T).(\lambda (t5: T).(eq T x (THead (Bind b) u3 t5)))) (\lambda (u3: +T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u1 u3))) (\lambda (_: +T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 t5)))) (or +(pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T +(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) +u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H12: (ex2 T +(\lambda (u3: T).(eq T x (THead (Bind b) u3 t3))) (\lambda (u3: T).(subst0 (s +(Flat Appl) i) v0 u1 u3)))).(ex2_ind T (\lambda (u3: T).(eq T x (THead (Bind +b) u3 t3))) (\lambda (u3: T).(subst0 (s (Flat Appl) i) v0 u1 u3)) (or (pr0 w1 +(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda +(w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead +(Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x0: T).(\lambda (H13: (eq +T x (THead (Bind b) x0 t3))).(\lambda (H14: (subst0 (s (Flat Appl) i) v0 u1 +x0)).(let H15 \def (eq_ind T x (\lambda (t: T).(eq T w1 (THead (Flat Appl) v1 +t))) H10 (THead (Bind b) x0 t3) H13) in (eq_ind_r T (THead (Flat Appl) v1 +(THead (Bind b) x0 t3)) (\lambda (t: T).(or (pr0 t (THead (Bind b) u2 (THead +(Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 t w2)) +(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S +O) O v2) t4)) w2))))) (or_ind (pr0 x0 u2) (ex2 T (\lambda (w2: T).(pr0 x0 +w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 u2 w2))) (or (pr0 (THead +(Flat Appl) v1 (THead (Bind b) x0 t3)) (THead (Bind b) u2 (THead (Flat Appl) +(lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 +(THead (Bind b) x0 t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 +(THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H16: (pr0 x0 +u2)).(or_introl (pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 t3)) (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: +T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 t3)) w2)) (\lambda (w2: +T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) +w2))) (pr0_upsilon b H0 v1 v2 H1 x0 u2 H16 t3 t4 H5))) (\lambda (H16: (ex2 T +(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 +u2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 +(s (Flat Appl) i) v3 u2 w2)) (or (pr0 (THead (Flat Appl) v1 (THead (Bind b) +x0 t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T +(\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 t3)) w2)) +(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S +O) O v2) t4)) w2)))) (\lambda (x1: T).(\lambda (H17: (pr0 x0 x1)).(\lambda +(H18: (subst0 (s (Flat Appl) i) v3 u2 x1)).(or_intror (pr0 (THead (Flat Appl) +v1 (THead (Bind b) x0 t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) +O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind +b) x0 t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat +Appl) (lift (S O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead +(Flat Appl) v1 (THead (Bind b) x0 t3)) w2)) (\lambda (w2: T).(subst0 i v3 +(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)) (THead +(Bind b) x1 (THead (Flat Appl) (lift (S O) O v2) t4)) (pr0_upsilon b H0 v1 v2 +H1 x0 x1 H17 t3 t4 H5) (subst0_fst v3 x1 u2 i H18 (THead (Flat Appl) (lift (S +O) O v2) t4) (Bind b))))))) H16)) (H4 v0 x0 (s (Flat Appl) i) H14 v3 H8)) w1 +H15))))) H12)) (\lambda (H12: (ex2 T (\lambda (t5: T).(eq T x (THead (Bind b) +u1 t5))) (\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 +t5)))).(ex2_ind T (\lambda (t5: T).(eq T x (THead (Bind b) u1 t5))) (\lambda +(t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 t5)) (or (pr0 w1 (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: +T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat +Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x0: T).(\lambda (H13: (eq T x +(THead (Bind b) u1 x0))).(\lambda (H14: (subst0 (s (Bind b) (s (Flat Appl) +i)) v0 t3 x0)).(let H15 \def (eq_ind T x (\lambda (t: T).(eq T w1 (THead +(Flat Appl) v1 t))) H10 (THead (Bind b) u1 x0) H13) in (eq_ind_r T (THead +(Flat Appl) v1 (THead (Bind b) u1 x0)) (\lambda (t: T).(or (pr0 t (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: +T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat +Appl) (lift (S O) O v2) t4)) w2))))) (or_ind (pr0 x0 t4) (ex2 T (\lambda (w2: +T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 +w2))) (or (pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 x0)) (THead (Bind b) +u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 +(THead (Flat Appl) v1 (THead (Bind b) u1 x0)) w2)) (\lambda (w2: T).(subst0 i +v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) +(\lambda (H16: (pr0 x0 t4)).(or_introl (pr0 (THead (Flat Appl) v1 (THead +(Bind b) u1 x0)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) +t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 +x0)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) +(lift (S O) O v2) t4)) w2))) (pr0_upsilon b H0 v1 v2 H1 u1 u2 H3 x0 t4 H16))) +(\lambda (H16: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 +(s (Bind b) (s (Flat Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 +x0 w2)) (\lambda (w2: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 w2)) +(or (pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 x0)) (THead (Bind b) u2 +(THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 +(THead (Flat Appl) v1 (THead (Bind b) u1 x0)) w2)) (\lambda (w2: T).(subst0 i +v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) +(\lambda (x1: T).(\lambda (H17: (pr0 x0 x1)).(\lambda (H18: (subst0 (s (Bind +b) (s (Flat Appl) i)) v3 t4 x1)).(or_intror (pr0 (THead (Flat Appl) v1 (THead +(Bind b) u1 x0)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) +t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 +x0)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) +(lift (S O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat +Appl) v1 (THead (Bind b) u1 x0)) w2)) (\lambda (w2: T).(subst0 i v3 (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 +(THead (Flat Appl) (lift (S O) O v2) x1)) (pr0_upsilon b H0 v1 v2 H1 u1 u2 H3 +x0 x1 H17) (subst0_snd (Bind b) v3 (THead (Flat Appl) (lift (S O) O v2) x1) +(THead (Flat Appl) (lift (S O) O v2) t4) i (subst0_snd (Flat Appl) v3 x1 t4 +(s (Bind b) i) H18 (lift (S O) O v2)) u2)))))) H16)) (H6 v0 x0 (s (Bind b) (s +(Flat Appl) i)) H14 v3 H8)) w1 H15))))) H12)) (\lambda (H12: (ex3_2 T T +(\lambda (u3: T).(\lambda (t5: T).(eq T x (THead (Bind b) u3 t5)))) (\lambda +(u3: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u1 u3))) (\lambda (_: +T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 +t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t5: T).(eq T x (THead (Bind +b) u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u1 +u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) +v0 t3 t5))) (or (pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O +v2) t4))) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 +(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda +(x0: T).(\lambda (x1: T).(\lambda (H13: (eq T x (THead (Bind b) x0 +x1))).(\lambda (H14: (subst0 (s (Flat Appl) i) v0 u1 x0)).(\lambda (H15: +(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 x1)).(let H16 \def (eq_ind T x +(\lambda (t: T).(eq T w1 (THead (Flat Appl) v1 t))) H10 (THead (Bind b) x0 +x1) H13) in (eq_ind_r T (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) +(\lambda (t: T).(or (pr0 t (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) +O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 +(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2))))) (or_ind +(pr0 x1 t4) (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s +(Bind b) (s (Flat Appl) i)) v3 t4 w2))) (or (pr0 (THead (Flat Appl) v1 (THead +(Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) +t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 +x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) +(lift (S O) O v2) t4)) w2)))) (\lambda (H17: (pr0 x1 t4)).(or_ind (pr0 x0 u2) +(ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) +i) v3 u2 w2))) (or (pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: +T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) w2)) (\lambda (w2: +T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) +w2)))) (\lambda (H18: (pr0 x0 u2)).(or_introl (pr0 (THead (Flat Appl) v1 +(THead (Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O +v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) +x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat +Appl) (lift (S O) O v2) t4)) w2))) (pr0_upsilon b H0 v1 v2 H1 x0 u2 H18 x1 t4 +H17))) (\lambda (H18: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: +T).(subst0 (s (Flat Appl) i) v3 u2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 +w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 u2 w2)) (or (pr0 (THead +(Flat Appl) v1 (THead (Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat Appl) +(lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 +(THead (Bind b) x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 +(THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x2: T).(\lambda +(H19: (pr0 x0 x2)).(\lambda (H20: (subst0 (s (Flat Appl) i) v3 u2 +x2)).(or_intror (pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: +T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) w2)) (\lambda (w2: +T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) +w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind +b) x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat +Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) x2 (THead (Flat Appl) (lift +(S O) O v2) t4)) (pr0_upsilon b H0 v1 v2 H1 x0 x2 H19 x1 t4 H17) (subst0_fst +v3 x2 u2 i H20 (THead (Flat Appl) (lift (S O) O v2) t4) (Bind b))))))) H18)) +(H4 v0 x0 (s (Flat Appl) i) H14 v3 H8))) (\lambda (H17: (ex2 T (\lambda (w2: +T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 +w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s +(Bind b) (s (Flat Appl) i)) v3 t4 w2)) (or (pr0 (THead (Flat Appl) v1 (THead +(Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) +t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 +x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) +(lift (S O) O v2) t4)) w2)))) (\lambda (x2: T).(\lambda (H18: (pr0 x1 +x2)).(\lambda (H19: (subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 x2)).(or_ind +(pr0 x0 u2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s +(Flat Appl) i) v3 u2 w2))) (or (pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 +x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T +(\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) w2)) +(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S +O) O v2) t4)) w2)))) (\lambda (H20: (pr0 x0 u2)).(or_intror (pr0 (THead (Flat +Appl) v1 (THead (Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift +(S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead +(Bind b) x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead +(Flat Appl) (lift (S O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 +(THead (Flat Appl) v1 (THead (Bind b) x0 x1)) w2)) (\lambda (w2: T).(subst0 i +v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)) (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) x2)) (pr0_upsilon b H0 v1 v2 +H1 x0 u2 H20 x1 x2 H18) (subst0_snd (Bind b) v3 (THead (Flat Appl) (lift (S +O) O v2) x2) (THead (Flat Appl) (lift (S O) O v2) t4) i (subst0_snd (Flat +Appl) v3 x2 t4 (s (Bind b) i) H19 (lift (S O) O v2)) u2)))) (\lambda (H20: +(ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) +i) v3 u2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: +T).(subst0 (s (Flat Appl) i) v3 u2 w2)) (or (pr0 (THead (Flat Appl) v1 (THead +(Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) +t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 +x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) +(lift (S O) O v2) t4)) w2)))) (\lambda (x3: T).(\lambda (H21: (pr0 x0 +x3)).(\lambda (H22: (subst0 (s (Flat Appl) i) v3 u2 x3)).(or_intror (pr0 +(THead (Flat Appl) v1 (THead (Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat +Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) +v1 (THead (Bind b) x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) +u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: +T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) w2)) (\lambda (w2: +T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) +w2)) (THead (Bind b) x3 (THead (Flat Appl) (lift (S O) O v2) x2)) +(pr0_upsilon b H0 v1 v2 H1 x0 x3 H21 x1 x2 H18) (subst0_both v3 u2 x3 i H22 +(Bind b) (THead (Flat Appl) (lift (S O) O v2) t4) (THead (Flat Appl) (lift (S +O) O v2) x2) (subst0_snd (Flat Appl) v3 x2 t4 (s (Bind b) i) H19 (lift (S O) +O v2)))))))) H20)) (H4 v0 x0 (s (Flat Appl) i) H14 v3 H8))))) H17)) (H6 v0 x1 +(s (Bind b) (s (Flat Appl) i)) H15 v3 H8)) w1 H16))))))) H12)) +(subst0_gen_head (Bind b) v0 u1 t3 x (s (Flat Appl) i) H11))))) H9)) (\lambda +(H9: (ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T w1 (THead (Flat Appl) +u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v0 v1 u3))) (\lambda (_: +T).(\lambda (t5: T).(subst0 (s (Flat Appl) i) v0 (THead (Bind b) u1 t3) +t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t5: T).(eq T w1 (THead +(Flat Appl) u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v0 v1 u3))) +(\lambda (_: T).(\lambda (t5: T).(subst0 (s (Flat Appl) i) v0 (THead (Bind b) +u1 t3) t5))) (or (pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O +v2) t4))) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 +(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda +(x0: T).(\lambda (x1: T).(\lambda (H10: (eq T w1 (THead (Flat Appl) x0 +x1))).(\lambda (H11: (subst0 i v0 v1 x0)).(\lambda (H12: (subst0 (s (Flat +Appl) i) v0 (THead (Bind b) u1 t3) x1)).(or3_ind (ex2 T (\lambda (u3: T).(eq +T x1 (THead (Bind b) u3 t3))) (\lambda (u3: T).(subst0 (s (Flat Appl) i) v0 +u1 u3))) (ex2 T (\lambda (t5: T).(eq T x1 (THead (Bind b) u1 t5))) (\lambda +(t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 t5))) (ex3_2 T T +(\lambda (u3: T).(\lambda (t5: T).(eq T x1 (THead (Bind b) u3 t5)))) (\lambda +(u3: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u1 u3))) (\lambda (_: +T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 t5)))) (or +(pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T +(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) +u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H13: (ex2 T +(\lambda (u3: T).(eq T x1 (THead (Bind b) u3 t3))) (\lambda (u3: T).(subst0 +(s (Flat Appl) i) v0 u1 u3)))).(ex2_ind T (\lambda (u3: T).(eq T x1 (THead +(Bind b) u3 t3))) (\lambda (u3: T).(subst0 (s (Flat Appl) i) v0 u1 u3)) (or +(pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T +(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) +u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x: T).(\lambda +(H14: (eq T x1 (THead (Bind b) x t3))).(\lambda (H15: (subst0 (s (Flat Appl) +i) v0 u1 x)).(let H16 \def (eq_ind T x1 (\lambda (t: T).(eq T w1 (THead (Flat +Appl) x0 t))) H10 (THead (Bind b) x t3) H14) in (eq_ind_r T (THead (Flat +Appl) x0 (THead (Bind b) x t3)) (\lambda (t: T).(or (pr0 t (THead (Bind b) u2 +(THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 t +w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) +(lift (S O) O v2) t4)) w2))))) (or_ind (pr0 x u2) (ex2 T (\lambda (w2: +T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 u2 w2))) (or +(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) (THead (Bind b) u2 (THead +(Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat +Appl) x0 (THead (Bind b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H17: +(pr0 x u2)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda +(w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) +x t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T +(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) +(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S +O) O v2) t4)) w2)))) (\lambda (H18: (pr0 x0 v2)).(or_introl (pr0 (THead (Flat +Appl) x0 (THead (Bind b) x t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift +(S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead +(Bind b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead +(Flat Appl) (lift (S O) O v2) t4)) w2))) (pr0_upsilon b H0 x0 v2 H18 x u2 H17 +t3 t4 H5))) (\lambda (H18: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: +T).(subst0 i v3 v2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda +(w2: T).(subst0 i v3 v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x +t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T +(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) +(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S +O) O v2) t4)) w2)))) (\lambda (x2: T).(\lambda (H19: (pr0 x0 x2)).(\lambda +(H20: (subst0 i v3 v2 x2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind +b) x t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 +T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) +(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S +O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 +(THead (Bind b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 +(THead (Flat Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 (THead +(Flat Appl) (lift (S O) O x2) t4)) (pr0_upsilon b H0 x0 x2 H19 x u2 H17 t3 t4 +H5) (subst0_snd (Bind b) v3 (THead (Flat Appl) (lift (S O) O x2) t4) (THead +(Flat Appl) (lift (S O) O v2) t4) i (subst0_fst v3 (lift (S O) O x2) (lift (S +O) O v2) (s (Bind b) i) (subst0_lift_ge_s v2 x2 v3 i H20 O (le_O_n i) b) t4 +(Flat Appl)) u2)))))) H18)) (H2 v0 x0 i H11 v3 H8))) (\lambda (H17: (ex2 T +(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 u2 +w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s +(Flat Appl) i) v3 u2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x +t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T +(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) +(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S +O) O v2) t4)) w2)))) (\lambda (x2: T).(\lambda (H18: (pr0 x x2)).(\lambda +(H19: (subst0 (s (Flat Appl) i) v3 u2 x2)).(or_ind (pr0 x0 v2) (ex2 T +(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 +(THead (Flat Appl) x0 (THead (Bind b) x t3)) (THead (Bind b) u2 (THead (Flat +Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) +x0 (THead (Bind b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) +u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H20: (pr0 x0 +v2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: +T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) (\lambda (w2: +T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) +w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind +b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat +Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) x2 (THead (Flat Appl) (lift +(S O) O v2) t4)) (pr0_upsilon b H0 x0 v2 H20 x x2 H18 t3 t4 H5) (subst0_fst +v3 x2 u2 i H19 (THead (Flat Appl) (lift (S O) O v2) t4) (Bind b))))) (\lambda +(H20: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 +w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 +v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) (THead (Bind b) +u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 +(THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) (\lambda (w2: T).(subst0 i +v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) +(\lambda (x3: T).(\lambda (H21: (pr0 x0 x3)).(\lambda (H22: (subst0 i v3 v2 +x3)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: +T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) (\lambda (w2: +T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) +w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind +b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat +Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) x2 (THead (Flat Appl) (lift +(S O) O x3) t4)) (pr0_upsilon b H0 x0 x3 H21 x x2 H18 t3 t4 H5) (subst0_both +v3 u2 x2 i H19 (Bind b) (THead (Flat Appl) (lift (S O) O v2) t4) (THead (Flat +Appl) (lift (S O) O x3) t4) (subst0_fst v3 (lift (S O) O x3) (lift (S O) O +v2) (s (Bind b) i) (subst0_lift_ge_s v2 x3 v3 i H22 O (le_O_n i) b) t4 (Flat +Appl)))))))) H20)) (H2 v0 x0 i H11 v3 H8))))) H17)) (H4 v0 x (s (Flat Appl) +i) H15 v3 H8)) w1 H16))))) H13)) (\lambda (H13: (ex2 T (\lambda (t5: T).(eq T +x1 (THead (Bind b) u1 t5))) (\lambda (t5: T).(subst0 (s (Bind b) (s (Flat +Appl) i)) v0 t3 t5)))).(ex2_ind T (\lambda (t5: T).(eq T x1 (THead (Bind b) +u1 t5))) (\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 t5)) +(or (pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) +(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x: +T).(\lambda (H14: (eq T x1 (THead (Bind b) u1 x))).(\lambda (H15: (subst0 (s +(Bind b) (s (Flat Appl) i)) v0 t3 x)).(let H16 \def (eq_ind T x1 (\lambda (t: +T).(eq T w1 (THead (Flat Appl) x0 t))) H10 (THead (Bind b) u1 x) H14) in +(eq_ind_r T (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (\lambda (t: T).(or +(pr0 t (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T +(\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 +(THead (Flat Appl) (lift (S O) O v2) t4)) w2))))) (or_ind (pr0 x t4) (ex2 T +(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind b) (s (Flat +Appl) i)) v3 t4 w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) +(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda +(w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: +T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) +w2)))) (\lambda (H17: (pr0 x t4)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2: +T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat +Appl) x0 (THead (Bind b) u1 x)) (THead (Bind b) u2 (THead (Flat Appl) (lift +(S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead +(Bind b) u1 x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead +(Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H18: (pr0 x0 +v2)).(or_introl (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: +T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: +T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) +w2))) (pr0_upsilon b H0 x0 v2 H18 u1 u2 H3 x t4 H17))) (\lambda (H18: (ex2 T +(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 +w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 +v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead (Bind b) +u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 +(THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: T).(subst0 i +v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) +(\lambda (x2: T).(\lambda (H19: (pr0 x0 x2)).(\lambda (H20: (subst0 i v3 v2 +x2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: +T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: +T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) +w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind +b) u1 x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat +Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 (THead (Flat Appl) (lift +(S O) O x2) t4)) (pr0_upsilon b H0 x0 x2 H19 u1 u2 H3 x t4 H17) (subst0_snd +(Bind b) v3 (THead (Flat Appl) (lift (S O) O x2) t4) (THead (Flat Appl) (lift +(S O) O v2) t4) i (subst0_fst v3 (lift (S O) O x2) (lift (S O) O v2) (s (Bind +b) i) (subst0_lift_ge_s v2 x2 v3 i H20 O (le_O_n i) b) t4 (Flat Appl)) +u2)))))) H18)) (H2 v0 x0 i H11 v3 H8))) (\lambda (H17: (ex2 T (\lambda (w2: +T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 +w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s +(Bind b) (s (Flat Appl) i)) v3 t4 w2)) (or (pr0 (THead (Flat Appl) x0 (THead +(Bind b) u1 x)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) +(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) +w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) +(lift (S O) O v2) t4)) w2)))) (\lambda (x2: T).(\lambda (H18: (pr0 x +x2)).(\lambda (H19: (subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 x2)).(or_ind +(pr0 x0 v2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i +v3 v2 w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: +T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: +T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) +w2)))) (\lambda (H20: (pr0 x0 v2)).(or_intror (pr0 (THead (Flat Appl) x0 +(THead (Bind b) u1 x)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O +v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) +u1 x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat +Appl) (lift (S O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead +(Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: T).(subst0 i v3 +(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)) (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) x2)) (pr0_upsilon b H0 x0 v2 +H20 u1 u2 H3 x x2 H18) (subst0_snd (Bind b) v3 (THead (Flat Appl) (lift (S O) +O v2) x2) (THead (Flat Appl) (lift (S O) O v2) t4) i (subst0_snd (Flat Appl) +v3 x2 t4 (s (Bind b) i) H19 (lift (S O) O v2)) u2)))) (\lambda (H20: (ex2 T +(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 +w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 +v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead (Bind b) +u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 +(THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: T).(subst0 i +v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) +(\lambda (x3: T).(\lambda (H21: (pr0 x0 x3)).(\lambda (H22: (subst0 i v3 v2 +x3)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: +T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: +T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) +w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind +b) u1 x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat +Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 (THead (Flat Appl) (lift +(S O) O x3) x2)) (pr0_upsilon b H0 x0 x3 H21 u1 u2 H3 x x2 H18) (subst0_snd +(Bind b) v3 (THead (Flat Appl) (lift (S O) O x3) x2) (THead (Flat Appl) (lift +(S O) O v2) t4) i (subst0_both v3 (lift (S O) O v2) (lift (S O) O x3) (s +(Bind b) i) (subst0_lift_ge_s v2 x3 v3 i H22 O (le_O_n i) b) (Flat Appl) t4 +x2 H19) u2)))))) H20)) (H2 v0 x0 i H11 v3 H8))))) H17)) (H6 v0 x (s (Bind b) +(s (Flat Appl) i)) H15 v3 H8)) w1 H16))))) H13)) (\lambda (H13: (ex3_2 T T +(\lambda (u3: T).(\lambda (t5: T).(eq T x1 (THead (Bind b) u3 t5)))) (\lambda +(u3: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u1 u3))) (\lambda (_: +T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 +t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t5: T).(eq T x1 (THead +(Bind b) u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) +v0 u1 u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat +Appl) i)) v0 t3 t5))) (or (pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift +(S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: +T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) +w2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H14: (eq T x1 (THead (Bind +b) x2 x3))).(\lambda (H15: (subst0 (s (Flat Appl) i) v0 u1 x2)).(\lambda +(H16: (subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 x3)).(let H17 \def (eq_ind +T x1 (\lambda (t: T).(eq T w1 (THead (Flat Appl) x0 t))) H10 (THead (Bind b) +x2 x3) H14) in (eq_ind_r T (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) +(\lambda (t: T).(or (pr0 t (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) +O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 +(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2))))) (or_ind +(pr0 x3 t4) (ex2 T (\lambda (w2: T).(pr0 x3 w2)) (\lambda (w2: T).(subst0 (s +(Bind b) (s (Flat Appl) i)) v3 t4 w2))) (or (pr0 (THead (Flat Appl) x0 (THead +(Bind b) x2 x3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) +t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 +x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) +(lift (S O) O v2) t4)) w2)))) (\lambda (H18: (pr0 x3 t4)).(or_ind (pr0 x2 u2) +(ex2 T (\lambda (w2: T).(pr0 x2 w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) +i) v3 u2 w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead +(Bind b) u2 (THead (Flat Appl) 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i H12))))) H10)) (H1 v1 x i H9 v2 H6)) +w1 H8)))) H7)) (\lambda (H7: (ex2 T (\lambda (t5: T).(eq T w1 (THead (Bind +Abbr) u1 t5))) (\lambda (t5: T).(subst0 (s (Bind Abbr) i) v1 t3 +t5)))).(ex2_ind T (\lambda (t5: T).(eq T w1 (THead (Bind Abbr) u1 t5))) +(\lambda (t5: T).(subst0 (s (Bind Abbr) i) v1 t3 t5)) (or (pr0 w1 (THead +(Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: +T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x: T).(\lambda (H8: +(eq T w1 (THead (Bind Abbr) u1 x))).(\lambda (H9: (subst0 (s (Bind Abbr) i) +v1 t3 x)).(eq_ind_r T (THead (Bind Abbr) u1 x) (\lambda (t: T).(or (pr0 t +(THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: +T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))))) (or_ind (pr0 x t4) (ex2 T +(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind Abbr) i) v2 t4 +w2))) (or (pr0 (THead (Bind Abbr) u1 x) (THead (Bind Abbr) u2 w)) (ex2 T +(\lambda (w2: T).(pr0 (THead (Bind Abbr) u1 x) w2)) (\lambda 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x)).(\lambda (H14: (subst0 i v2 u2 x)).(ex2_ind T +(\lambda (t: T).(subst0 O x t4 t)) (\lambda (t: T).(subst0 (S (plus i O)) v2 +w t)) (or (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T +(\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 +i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x2: T).(\lambda (H15: (subst0 +O x t4 x2)).(\lambda (H16: (subst0 (S (plus i O)) v2 w x2)).(let H17 \def +(f_equal nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O i))) in +(let H18 \def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n v2 w +x2)) H16 (S i) H17) in (or_intror (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind +Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) +(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))) (ex_intro2 T +(\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 +i v2 (THead (Bind Abbr) u2 w) w2)) (THead (Bind Abbr) x x2) (pr0_delta x0 x +H13 x1 t4 H11 x2 H15) (subst0_both v2 u2 x i H14 (Bind Abbr) w x2 H18)))))))) +(subst0_subst0_back t4 w u2 O H4 x v2 i H14))))) H12)) (H1 v1 x0 i H9 v2 +H6))) (\lambda (H11: (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: +T).(subst0 (s (Bind Abbr) i) v2 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x1 +w2)) (\lambda (w2: T).(subst0 (s (Bind Abbr) i) v2 t4 w2)) (or (pr0 (THead +(Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 +(THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind +Abbr) u2 w) w2)))) (\lambda (x: T).(\lambda (H12: (pr0 x1 x)).(\lambda (H13: +(subst0 (s (Bind Abbr) i) v2 t4 x)).(or_ind (pr0 x0 u2) (ex2 T (\lambda (w2: +T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 u2 w2))) (or (pr0 (THead (Bind +Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead +(Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 +w) w2)))) (\lambda (H14: (pr0 x0 u2)).(ex2_ind T (\lambda (t: T).(subst0 O u2 +x t)) (\lambda (t: 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(\lambda (w2: T).(subst0 i v2 +u2 w2)) (or (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T +(\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 +i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x2: T).(\lambda (H15: (pr0 x0 +x2)).(\lambda (H16: (subst0 i v2 u2 x2)).(ex2_ind T (\lambda (t: T).(subst0 O +x2 t4 t)) (\lambda (t: T).(subst0 (S (plus i O)) v2 w t)) (or (pr0 (THead +(Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 +(THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind +Abbr) u2 w) w2)))) (\lambda (x3: T).(\lambda (H17: (subst0 O x2 t4 +x3)).(\lambda (H18: (subst0 (S (plus i O)) v2 w x3)).(let H19 \def (f_equal +nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O i))) in (let H20 +\def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n v2 w x3)) H18 (S +i) H19) in (ex2_ind T (\lambda (t: T).(subst0 (s (Bind Abbr) i) v2 x3 t)) +(\lambda (t: T).(subst0 O x2 x t)) (or (pr0 (THead (Bind Abbr) x0 x1) (THead +(Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) +w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda +(x4: T).(\lambda (H21: (subst0 (s (Bind Abbr) i) v2 x3 x4)).(\lambda (H22: +(subst0 O x2 x x4)).(or_intror (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind +Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) +(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))) (ex_intro2 T +(\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 +i v2 (THead (Bind Abbr) u2 w) w2)) (THead (Bind Abbr) x2 x4) (pr0_delta x0 x2 +H15 x1 x H12 x4 H22) (subst0_both v2 u2 x2 i H16 (Bind Abbr) w x4 +(subst0_trans x3 w v2 (s (Bind Abbr) i) H20 x4 H21))))))) +(subst0_confluence_neq t4 x3 x2 O H17 x v2 (s (Bind Abbr) i) H13 (O_S +i)))))))) (subst0_subst0_back t4 w u2 O H4 x2 v2 i H16))))) H14)) (H1 v1 x0 i +H9 v2 H6))))) H11)) (H3 v1 x1 (s (Bind Abbr) i) H10 v2 H6)) w1 H8)))))) H7)) +(subst0_gen_head (Bind Abbr) v1 u1 t3 w1 i H5)))))))))))))))))) (\lambda (b: +B).(\lambda (H0: (not (eq B b Abst))).(\lambda (t3: T).(\lambda (t4: +T).(\lambda (H1: (pr0 t3 t4)).(\lambda (H2: ((\forall (v1: T).(\forall (w1: +T).(\forall (i: nat).((subst0 i v1 t3 w1) \to (\forall (v2: T).((pr0 v1 v2) +\to (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: +T).(subst0 i v2 t4 w2)))))))))))).(\lambda (u: T).(\lambda (v1: T).(\lambda +(w1: T).(\lambda (i: nat).(\lambda (H3: (subst0 i v1 (THead (Bind b) u (lift +(S O) O t3)) w1)).(\lambda (v2: T).(\lambda (H4: (pr0 v1 v2)).(or3_ind (ex2 T +(\lambda (u2: T).(eq T w1 (THead (Bind b) u2 (lift (S O) O t3)))) (\lambda +(u2: T).(subst0 i v1 u u2))) (ex2 T (\lambda (t5: T).(eq T w1 (THead (Bind b) +u t5))) (\lambda (t5: T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5))) +(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T w1 (THead (Bind b) u2 +t5)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: +T).(\lambda (t5: T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5)))) (or +(pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i +v2 t4 w2)))) (\lambda (H5: (ex2 T (\lambda (u2: T).(eq T w1 (THead (Bind b) +u2 (lift (S O) O t3)))) (\lambda (u2: T).(subst0 i v1 u u2)))).(ex2_ind T +(\lambda (u2: T).(eq T w1 (THead (Bind b) u2 (lift (S O) O t3)))) (\lambda +(u2: T).(subst0 i v1 u u2)) (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 +w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H6: +(eq T w1 (THead (Bind b) x (lift (S O) O t3)))).(\lambda (_: (subst0 i v1 u +x)).(eq_ind_r T (THead (Bind b) x (lift (S O) O t3)) (\lambda (t: T).(or (pr0 +t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4 +w2))))) (or_introl (pr0 (THead (Bind b) x (lift (S O) O t3)) t4) (ex2 T +(\lambda (w2: T).(pr0 (THead (Bind b) x (lift (S O) O t3)) w2)) (\lambda (w2: +T).(subst0 i v2 t4 w2))) (pr0_zeta b H0 t3 t4 H1 x)) w1 H6)))) H5)) (\lambda +(H5: (ex2 T (\lambda (t5: T).(eq T w1 (THead (Bind b) u t5))) (\lambda (t5: +T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5)))).(ex2_ind T (\lambda +(t5: T).(eq T w1 (THead (Bind b) u t5))) (\lambda (t5: T).(subst0 (s (Bind b) +i) v1 (lift (S O) O t3) t5)) (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 +w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H6: +(eq T w1 (THead (Bind b) u x))).(\lambda (H7: (subst0 (s (Bind b) i) v1 (lift +(S O) O t3) x)).(ex2_ind T (\lambda (t5: T).(eq T x (lift (S O) O t5))) +(\lambda (t5: T).(subst0 (minus (s (Bind b) i) (S O)) v1 t3 t5)) (or (pr0 w1 +t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 +w2)))) (\lambda (x0: T).(\lambda (H8: (eq T x (lift (S O) O x0))).(\lambda +(H9: (subst0 (minus (s (Bind b) i) (S O)) v1 t3 x0)).(let H10 \def (eq_ind T +x (\lambda (t: T).(eq T w1 (THead (Bind b) u t))) H6 (lift (S O) O x0) H8) in +(eq_ind_r T (THead (Bind b) u (lift (S O) O x0)) (\lambda (t: T).(or (pr0 t +t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4 +w2))))) (let H11 \def (eq_ind_r nat (minus i O) (\lambda (n: nat).(subst0 n +v1 t3 x0)) H9 i (minus_n_O i)) in (or_ind (pr0 x0 t4) (ex2 T (\lambda (w2: +T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (or (pr0 (THead (Bind +b) u (lift (S O) O x0)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) u +(lift (S O) O x0)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda +(H12: (pr0 x0 t4)).(or_introl (pr0 (THead (Bind b) u (lift (S O) O x0)) t4) +(ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) u (lift (S O) O x0)) w2)) +(\lambda (w2: T).(subst0 i v2 t4 w2))) (pr0_zeta b H0 x0 t4 H12 u))) (\lambda +(H12: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 t4 +w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 +t4 w2)) (or (pr0 (THead (Bind b) u (lift (S O) O x0)) t4) (ex2 T (\lambda +(w2: T).(pr0 (THead (Bind b) u (lift (S O) O x0)) w2)) (\lambda (w2: +T).(subst0 i v2 t4 w2)))) (\lambda (x1: T).(\lambda (H13: (pr0 x0 +x1)).(\lambda (H14: (subst0 i v2 t4 x1)).(or_intror (pr0 (THead (Bind b) u +(lift (S O) O x0)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) u (lift +(S O) O x0)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (ex_intro2 T +(\lambda (w2: T).(pr0 (THead (Bind b) u (lift (S O) O x0)) w2)) (\lambda (w2: +T).(subst0 i v2 t4 w2)) x1 (pr0_zeta b H0 x0 x1 H13 u) H14))))) H12)) (H2 v1 +x0 i H11 v2 H4))) w1 H10))))) (subst0_gen_lift_ge v1 t3 x (s (Bind b) i) (S +O) O H7 (le_n_S O i (le_O_n i))))))) H5)) (\lambda (H5: (ex3_2 T T (\lambda +(u2: T).(\lambda (t5: T).(eq T w1 (THead (Bind b) u2 t5)))) (\lambda (u2: +T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: +T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5))))).(ex3_2_ind T T +(\lambda (u2: T).(\lambda (t5: T).(eq T w1 (THead (Bind b) u2 t5)))) (\lambda +(u2: T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: +T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5))) (or (pr0 w1 t4) (ex2 T +(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) +(\lambda (x0: T).(\lambda (x1: T).(\lambda (H6: (eq T w1 (THead (Bind b) x0 +x1))).(\lambda (_: (subst0 i v1 u x0)).(\lambda (H8: (subst0 (s (Bind b) i) +v1 (lift (S O) O t3) x1)).(ex2_ind T (\lambda (t5: T).(eq T x1 (lift (S O) O +t5))) (\lambda (t5: T).(subst0 (minus (s (Bind b) i) (S O)) v1 t3 t5)) (or +(pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i +v2 t4 w2)))) (\lambda (x: T).(\lambda (H9: (eq T x1 (lift (S O) O +x))).(\lambda (H10: (subst0 (minus (s (Bind b) i) (S O)) v1 t3 x)).(let H11 +\def (eq_ind T x1 (\lambda (t: T).(eq T w1 (THead (Bind b) x0 t))) H6 (lift +(S O) O x) H9) in (eq_ind_r T (THead (Bind b) x0 (lift (S O) O x)) (\lambda +(t: T).(or (pr0 t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: +T).(subst0 i v2 t4 w2))))) (let H12 \def (eq_ind_r nat (minus i O) (\lambda +(n: nat).(subst0 n v1 t3 x)) H10 i (minus_n_O i)) in (or_ind (pr0 x t4) (ex2 +T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (or +(pr0 (THead (Bind b) x0 (lift (S O) O x)) t4) (ex2 T (\lambda (w2: T).(pr0 +(THead (Bind b) x0 (lift (S O) O x)) w2)) (\lambda (w2: T).(subst0 i v2 t4 +w2)))) (\lambda (H13: (pr0 x t4)).(or_introl (pr0 (THead (Bind b) x0 (lift (S +O) O x)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) x0 (lift (S O) O +x)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (pr0_zeta b H0 x t4 H13 x0))) +(\lambda (H13: (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i +v2 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 +i v2 t4 w2)) (or (pr0 (THead (Bind b) x0 (lift (S O) O x)) t4) (ex2 T +(\lambda (w2: T).(pr0 (THead (Bind b) x0 (lift (S O) O x)) w2)) (\lambda (w2: +T).(subst0 i v2 t4 w2)))) (\lambda (x2: T).(\lambda (H14: (pr0 x +x2)).(\lambda (H15: (subst0 i v2 t4 x2)).(or_intror (pr0 (THead (Bind b) x0 +(lift (S O) O x)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) x0 (lift +(S O) O x)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (ex_intro2 T (\lambda +(w2: T).(pr0 (THead (Bind b) x0 (lift (S O) O x)) w2)) (\lambda (w2: +T).(subst0 i v2 t4 w2)) x2 (pr0_zeta b H0 x x2 H14 x0) H15))))) H13)) (H2 v1 +x i H12 v2 H4))) w1 H11))))) (subst0_gen_lift_ge v1 t3 x1 (s (Bind b) i) (S +O) O H8 (le_n_S O i (le_O_n i))))))))) H5)) (subst0_gen_head (Bind b) v1 u +(lift (S O) O t3) w1 i H3))))))))))))))) (\lambda (t3: T).(\lambda (t4: +T).(\lambda (H0: (pr0 t3 t4)).(\lambda (H1: ((\forall (v1: T).(\forall (w1: +T).(\forall (i: nat).((subst0 i v1 t3 w1) \to (\forall (v2: T).((pr0 v1 v2) +\to (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: +T).(subst0 i v2 t4 w2)))))))))))).(\lambda (u: T).(\lambda (v1: T).(\lambda +(w1: T).(\lambda (i: nat).(\lambda (H2: (subst0 i v1 (THead (Flat Cast) u t3) +w1)).(\lambda (v2: T).(\lambda (H3: (pr0 v1 v2)).(or3_ind (ex2 T (\lambda +(u2: T).(eq T w1 (THead (Flat Cast) u2 t3))) (\lambda (u2: T).(subst0 i v1 u +u2))) (ex2 T (\lambda (t5: T).(eq T w1 (THead (Flat Cast) u t5))) (\lambda +(t5: T).(subst0 (s (Flat Cast) i) v1 t3 t5))) (ex3_2 T T (\lambda (u2: +T).(\lambda (t5: T).(eq T w1 (THead (Flat Cast) u2 t5)))) (\lambda (u2: +T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: +T).(subst0 (s (Flat Cast) i) v1 t3 t5)))) (or (pr0 w1 t4) (ex2 T (\lambda +(w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (H4: +(ex2 T (\lambda (u2: T).(eq T w1 (THead (Flat Cast) u2 t3))) (\lambda (u2: +T).(subst0 i v1 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T w1 (THead (Flat +Cast) u2 t3))) (\lambda (u2: T).(subst0 i v1 u u2)) (or (pr0 w1 t4) (ex2 T +(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) +(\lambda (x: T).(\lambda (H5: (eq T w1 (THead (Flat Cast) x t3))).(\lambda +(_: (subst0 i v1 u x)).(eq_ind_r T (THead (Flat Cast) x t3) (\lambda (t: +T).(or (pr0 t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: +T).(subst0 i v2 t4 w2))))) (or_introl (pr0 (THead (Flat Cast) x t3) t4) (ex2 +T (\lambda (w2: T).(pr0 (THead (Flat Cast) x t3) w2)) (\lambda (w2: +T).(subst0 i v2 t4 w2))) (pr0_tau t3 t4 H0 x)) w1 H5)))) H4)) (\lambda (H4: +(ex2 T (\lambda (t5: T).(eq T w1 (THead (Flat Cast) u t5))) (\lambda (t5: +T).(subst0 (s (Flat Cast) i) v1 t3 t5)))).(ex2_ind T (\lambda (t5: T).(eq T +w1 (THead (Flat Cast) u t5))) (\lambda (t5: T).(subst0 (s (Flat Cast) i) v1 +t3 t5)) (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: +T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H5: (eq T w1 (THead (Flat +Cast) u x))).(\lambda (H6: (subst0 (s (Flat Cast) i) v1 t3 x)).(eq_ind_r T +(THead (Flat Cast) u x) (\lambda (t: T).(or (pr0 t t4) (ex2 T (\lambda (w2: +T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))))) (or_ind (pr0 x t4) +(ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Flat Cast) +i) v2 t4 w2))) (or (pr0 (THead (Flat Cast) u x) t4) (ex2 T (\lambda (w2: +T).(pr0 (THead (Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) +(\lambda (H7: (pr0 x t4)).(or_introl (pr0 (THead (Flat Cast) u x) t4) (ex2 T +(\lambda (w2: T).(pr0 (THead (Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i +v2 t4 w2))) (pr0_tau x t4 H7 u))) (\lambda (H7: (ex2 T (\lambda (w2: T).(pr0 +x w2)) (\lambda (w2: T).(subst0 (s (Flat Cast) i) v2 t4 w2)))).(ex2_ind T +(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Flat Cast) i) v2 t4 +w2)) (or (pr0 (THead (Flat Cast) u x) t4) (ex2 T (\lambda (w2: T).(pr0 (THead +(Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x0: +T).(\lambda (H8: (pr0 x x0)).(\lambda (H9: (subst0 (s (Flat Cast) i) v2 t4 +x0)).(or_intror (pr0 (THead (Flat Cast) u x) t4) (ex2 T (\lambda (w2: T).(pr0 +(THead (Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) +(ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) u x) w2)) (\lambda (w2: +T).(subst0 i v2 t4 w2)) x0 (pr0_tau x x0 H8 u) H9))))) H7)) (H1 v1 x (s (Flat +Cast) i) H6 v2 H3)) w1 H5)))) H4)) (\lambda (H4: (ex3_2 T T (\lambda (u2: +T).(\lambda (t5: T).(eq T w1 (THead (Flat Cast) u2 t5)))) (\lambda (u2: +T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: +T).(subst0 (s (Flat Cast) i) v1 t3 t5))))).(ex3_2_ind T T (\lambda (u2: +T).(\lambda (t5: T).(eq T w1 (THead (Flat Cast) u2 t5)))) (\lambda (u2: +T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: +T).(subst0 (s (Flat Cast) i) v1 t3 t5))) (or (pr0 w1 t4) (ex2 T (\lambda (w2: +T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x0: +T).(\lambda (x1: T).(\lambda (H5: (eq T w1 (THead (Flat Cast) x0 +x1))).(\lambda (_: (subst0 i v1 u x0)).(\lambda (H7: (subst0 (s (Flat Cast) +i) v1 t3 x1)).(eq_ind_r T (THead (Flat Cast) x0 x1) (\lambda (t: T).(or (pr0 +t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4 +w2))))) (or_ind (pr0 x1 t4) (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda +(w2: T).(subst0 (s (Flat Cast) i) v2 t4 w2))) (or (pr0 (THead (Flat Cast) x0 +x1) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) x0 x1) w2)) (\lambda +(w2: T).(subst0 i v2 t4 w2)))) (\lambda (H8: (pr0 x1 t4)).(or_introl (pr0 +(THead (Flat Cast) x0 x1) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) +x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (pr0_tau x1 t4 H8 x0))) +(\lambda (H8: (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 +(s (Flat Cast) i) v2 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x1 w2)) +(\lambda (w2: T).(subst0 (s (Flat Cast) i) v2 t4 w2)) (or (pr0 (THead (Flat +Cast) x0 x1) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) x0 x1) w2)) +(\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H9: (pr0 x1 +x)).(\lambda (H10: (subst0 (s (Flat Cast) i) v2 t4 x)).(or_intror (pr0 (THead +(Flat Cast) x0 x1) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) x0 x1) +w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (ex_intro2 T (\lambda (w2: +T).(pr0 (THead (Flat Cast) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)) +x (pr0_tau x1 x H9 x0) H10))))) H8)) (H1 v1 x1 (s (Flat Cast) i) H7 v2 H3)) +w1 H5)))))) H4)) (subst0_gen_head (Flat Cast) v1 u t3 w1 i H2))))))))))))) t1 +t2 H))). +(* COMMENTS +Initial nodes: 38857 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr0/subst1.ma b/matita/matita/contribs/lambdadelta/basic_1/pr0/subst1.ma new file mode 100644 index 000000000..877f87f01 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/pr0/subst1.ma @@ -0,0 +1,105 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/pr0/props.ma". + +include "Basic-1/subst1/defs.ma". + +theorem pr0_delta1: + \forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (t1: T).(\forall +(t2: T).((pr0 t1 t2) \to (\forall (w: T).((subst1 O u2 t2 w) \to (pr0 (THead +(Bind Abbr) u1 t1) (THead (Bind Abbr) u2 w))))))))) +\def + \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr0 u1 u2)).(\lambda (t1: +T).(\lambda (t2: T).(\lambda (H0: (pr0 t1 t2)).(\lambda (w: T).(\lambda (H1: +(subst1 O u2 t2 w)).(subst1_ind O u2 t2 (\lambda (t: T).(pr0 (THead (Bind +Abbr) u1 t1) (THead (Bind Abbr) u2 t))) (pr0_comp u1 u2 H t1 t2 H0 (Bind +Abbr)) (\lambda (t0: T).(\lambda (H2: (subst0 O u2 t2 t0)).(pr0_delta u1 u2 H +t1 t2 H0 t0 H2))) w H1)))))))). +(* COMMENTS +Initial nodes: 115 +END *) + +theorem pr0_subst1_back: + \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst1 +i u2 t1 t2) \to (\forall (u1: T).((pr0 u1 u2) \to (ex2 T (\lambda (t: +T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t t2))))))))) +\def + \lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (i: nat).(\lambda +(H: (subst1 i u2 t1 t2)).(subst1_ind i u2 t1 (\lambda (t: T).(\forall (u1: +T).((pr0 u1 u2) \to (ex2 T (\lambda (t0: T).(subst1 i u1 t1 t0)) (\lambda +(t0: T).(pr0 t0 t)))))) (\lambda (u1: T).(\lambda (_: (pr0 u1 u2)).(ex_intro2 +T (\lambda (t: T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t t1)) t1 +(subst1_refl i u1 t1) (pr0_refl t1)))) (\lambda (t0: T).(\lambda (H0: (subst0 +i u2 t1 t0)).(\lambda (u1: T).(\lambda (H1: (pr0 u1 u2)).(ex2_ind T (\lambda +(t: T).(subst0 i u1 t1 t)) (\lambda (t: T).(pr0 t t0)) (ex2 T (\lambda (t: +T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t t0))) (\lambda (x: T).(\lambda +(H2: (subst0 i u1 t1 x)).(\lambda (H3: (pr0 x t0)).(ex_intro2 T (\lambda (t: +T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t t0)) x (subst1_single i u1 t1 x +H2) H3)))) (pr0_subst0_back u2 t1 t0 i H0 u1 H1)))))) t2 H))))). +(* COMMENTS +Initial nodes: 251 +END *) + +theorem pr0_subst1_fwd: + \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst1 +i u2 t1 t2) \to (\forall (u1: T).((pr0 u2 u1) \to (ex2 T (\lambda (t: +T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t2 t))))))))) +\def + \lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (i: nat).(\lambda +(H: (subst1 i u2 t1 t2)).(subst1_ind i u2 t1 (\lambda (t: T).(\forall (u1: +T).((pr0 u2 u1) \to (ex2 T (\lambda (t0: T).(subst1 i u1 t1 t0)) (\lambda +(t0: T).(pr0 t t0)))))) (\lambda (u1: T).(\lambda (_: (pr0 u2 u1)).(ex_intro2 +T (\lambda (t: T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t1 t)) t1 +(subst1_refl i u1 t1) (pr0_refl t1)))) (\lambda (t0: T).(\lambda (H0: (subst0 +i u2 t1 t0)).(\lambda (u1: T).(\lambda (H1: (pr0 u2 u1)).(ex2_ind T (\lambda +(t: T).(subst0 i u1 t1 t)) (\lambda (t: T).(pr0 t0 t)) (ex2 T (\lambda (t: +T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t0 t))) (\lambda (x: T).(\lambda +(H2: (subst0 i u1 t1 x)).(\lambda (H3: (pr0 t0 x)).(ex_intro2 T (\lambda (t: +T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t0 t)) x (subst1_single i u1 t1 x +H2) H3)))) (pr0_subst0_fwd u2 t1 t0 i H0 u1 H1)))))) t2 H))))). +(* COMMENTS +Initial nodes: 251 +END *) + +theorem pr0_subst1: + \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (v1: T).(\forall +(w1: T).(\forall (i: nat).((subst1 i v1 t1 w1) \to (\forall (v2: T).((pr0 v1 +v2) \to (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst1 i v2 t2 +w2))))))))))) +\def + \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(\lambda (v1: +T).(\lambda (w1: T).(\lambda (i: nat).(\lambda (H0: (subst1 i v1 t1 +w1)).(subst1_ind i v1 t1 (\lambda (t: T).(\forall (v2: T).((pr0 v1 v2) \to +(ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst1 i v2 t2 w2)))))) +(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(ex_intro2 T (\lambda (w2: T).(pr0 +t1 w2)) (\lambda (w2: T).(subst1 i v2 t2 w2)) t2 H (subst1_refl i v2 t2)))) +(\lambda (t0: T).(\lambda (H1: (subst0 i v1 t1 t0)).(\lambda (v2: T).(\lambda +(H2: (pr0 v1 v2)).(or_ind (pr0 t0 t2) (ex2 T (\lambda (w2: T).(pr0 t0 w2)) +(\lambda (w2: T).(subst0 i v2 t2 w2))) (ex2 T (\lambda (w2: T).(pr0 t0 w2)) +(\lambda (w2: T).(subst1 i v2 t2 w2))) (\lambda (H3: (pr0 t0 t2)).(ex_intro2 +T (\lambda (w2: T).(pr0 t0 w2)) (\lambda (w2: T).(subst1 i v2 t2 w2)) t2 H3 +(subst1_refl i v2 t2))) (\lambda (H3: (ex2 T (\lambda (w2: T).(pr0 t0 w2)) +(\lambda (w2: T).(subst0 i v2 t2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t0 +w2)) (\lambda (w2: T).(subst0 i v2 t2 w2)) (ex2 T (\lambda (w2: T).(pr0 t0 +w2)) (\lambda (w2: T).(subst1 i v2 t2 w2))) (\lambda (x: T).(\lambda (H4: +(pr0 t0 x)).(\lambda (H5: (subst0 i v2 t2 x)).(ex_intro2 T (\lambda (w2: +T).(pr0 t0 w2)) (\lambda (w2: T).(subst1 i v2 t2 w2)) x H4 (subst1_single i +v2 t2 x H5))))) H3)) (pr0_subst0 t1 t2 H v1 t0 i H1 v2 H2)))))) w1 H0))))))). +(* COMMENTS +Initial nodes: 385 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr1/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/pr1/defs.ma new file mode 100644 index 000000000..84db71acb --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/pr1/defs.ma @@ -0,0 +1,23 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/pr0/defs.ma". + +inductive pr1: T \to (T \to Prop) \def +| pr1_refl: \forall (t: T).(pr1 t t) +| pr1_sing: \forall (t2: T).(\forall (t1: T).((pr0 t1 t2) \to (\forall (t3: +T).((pr1 t2 t3) \to (pr1 t1 t3))))). + diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr1/pr1.ma b/matita/matita/contribs/lambdadelta/basic_1/pr1/pr1.ma new file mode 100644 index 000000000..ec469da50 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/pr1/pr1.ma @@ -0,0 +1,70 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/pr1/props.ma". + +include "Basic-1/pr0/pr0.ma". + +theorem pr1_strip: + \forall (t0: T).(\forall (t1: T).((pr1 t0 t1) \to (\forall (t2: T).((pr0 t0 +t2) \to (ex2 T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t))))))) +\def + \lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr1 t0 t1)).(pr1_ind (\lambda +(t: T).(\lambda (t2: T).(\forall (t3: T).((pr0 t t3) \to (ex2 T (\lambda (t4: +T).(pr1 t2 t4)) (\lambda (t4: T).(pr1 t3 t4))))))) (\lambda (t: T).(\lambda +(t2: T).(\lambda (H0: (pr0 t t2)).(ex_intro2 T (\lambda (t3: T).(pr1 t t3)) +(\lambda (t3: T).(pr1 t2 t3)) t2 (pr1_pr0 t t2 H0) (pr1_refl t2))))) (\lambda +(t2: T).(\lambda (t3: T).(\lambda (H0: (pr0 t3 t2)).(\lambda (t4: T).(\lambda +(_: (pr1 t2 t4)).(\lambda (H2: ((\forall (t5: T).((pr0 t2 t5) \to (ex2 T +(\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t))))))).(\lambda (t5: +T).(\lambda (H3: (pr0 t3 t5)).(ex2_ind T (\lambda (t: T).(pr0 t5 t)) (\lambda +(t: T).(pr0 t2 t)) (ex2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 +t))) (\lambda (x: T).(\lambda (H4: (pr0 t5 x)).(\lambda (H5: (pr0 t2 x)).(let +H6 \def (H2 x H5) in (ex2_ind T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: +T).(pr1 x t)) (ex2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t))) +(\lambda (x0: T).(\lambda (H7: (pr1 t4 x0)).(\lambda (H8: (pr1 x +x0)).(ex_intro2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t)) x0 +H7 (pr1_t x t5 (pr1_pr0 t5 x H4) x0 H8))))) H6))))) (pr0_confluence t3 t5 H3 +t2 H0)))))))))) t0 t1 H))). +(* COMMENTS +Initial nodes: 317 +END *) + +theorem pr1_confluence: + \forall (t0: T).(\forall (t1: T).((pr1 t0 t1) \to (\forall (t2: T).((pr1 t0 +t2) \to (ex2 T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t))))))) +\def + \lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr1 t0 t1)).(pr1_ind (\lambda +(t: T).(\lambda (t2: T).(\forall (t3: T).((pr1 t t3) \to (ex2 T (\lambda (t4: +T).(pr1 t2 t4)) (\lambda (t4: T).(pr1 t3 t4))))))) (\lambda (t: T).(\lambda +(t2: T).(\lambda (H0: (pr1 t t2)).(ex_intro2 T (\lambda (t3: T).(pr1 t t3)) +(\lambda (t3: T).(pr1 t2 t3)) t2 H0 (pr1_refl t2))))) (\lambda (t2: +T).(\lambda (t3: T).(\lambda (H0: (pr0 t3 t2)).(\lambda (t4: T).(\lambda (_: +(pr1 t2 t4)).(\lambda (H2: ((\forall (t5: T).((pr1 t2 t5) \to (ex2 T (\lambda +(t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t))))))).(\lambda (t5: T).(\lambda +(H3: (pr1 t3 t5)).(let H_x \def (pr1_strip t3 t5 H3 t2 H0) in (let H4 \def +H_x in (ex2_ind T (\lambda (t: T).(pr1 t5 t)) (\lambda (t: T).(pr1 t2 t)) +(ex2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t))) (\lambda (x: +T).(\lambda (H5: (pr1 t5 x)).(\lambda (H6: (pr1 t2 x)).(let H_x0 \def (H2 x +H6) in (let H7 \def H_x0 in (ex2_ind T (\lambda (t: T).(pr1 t4 t)) (\lambda +(t: T).(pr1 x t)) (ex2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 +t))) (\lambda (x0: T).(\lambda (H8: (pr1 t4 x0)).(\lambda (H9: (pr1 x +x0)).(ex_intro2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t)) x0 +H8 (pr1_t x t5 H5 x0 H9))))) H7)))))) H4))))))))))) t0 t1 H))). +(* COMMENTS +Initial nodes: 311 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr1/props.ma b/matita/matita/contribs/lambdadelta/basic_1/pr1/props.ma new file mode 100644 index 000000000..0615bed96 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/pr1/props.ma @@ -0,0 +1,126 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/pr1/defs.ma". + +include "Basic-1/pr0/subst1.ma". + +include "Basic-1/subst1/props.ma". + +include "Basic-1/T/props.ma". + +theorem pr1_pr0: + \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (pr1 t1 t2))) +\def + \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(pr1_sing t2 t1 H +t2 (pr1_refl t2)))). +(* COMMENTS +Initial nodes: 23 +END *) + +theorem pr1_t: + \forall (t2: T).(\forall (t1: T).((pr1 t1 t2) \to (\forall (t3: T).((pr1 t2 +t3) \to (pr1 t1 t3))))) +\def + \lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pr1 t1 t2)).(pr1_ind (\lambda +(t: T).(\lambda (t0: T).(\forall (t3: T).((pr1 t0 t3) \to (pr1 t t3))))) +(\lambda (t: T).(\lambda (t3: T).(\lambda (H0: (pr1 t t3)).H0))) (\lambda +(t0: T).(\lambda (t3: T).(\lambda (H0: (pr0 t3 t0)).(\lambda (t4: T).(\lambda +(_: (pr1 t0 t4)).(\lambda (H2: ((\forall (t5: T).((pr1 t4 t5) \to (pr1 t0 +t5))))).(\lambda (t5: T).(\lambda (H3: (pr1 t4 t5)).(pr1_sing t0 t3 H0 t5 (H2 +t5 H3)))))))))) t1 t2 H))). +(* COMMENTS +Initial nodes: 103 +END *) + +theorem pr1_head_1: + \forall (u1: T).(\forall (u2: T).((pr1 u1 u2) \to (\forall (t: T).(\forall +(k: K).(pr1 (THead k u1 t) (THead k u2 t)))))) +\def + \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr1 u1 u2)).(\lambda (t: +T).(\lambda (k: K).(pr1_ind (\lambda (t0: T).(\lambda (t1: T).(pr1 (THead k +t0 t) (THead k t1 t)))) (\lambda (t0: T).(pr1_refl (THead k t0 t))) (\lambda +(t2: T).(\lambda (t1: T).(\lambda (H0: (pr0 t1 t2)).(\lambda (t3: T).(\lambda +(_: (pr1 t2 t3)).(\lambda (H2: (pr1 (THead k t2 t) (THead k t3 t))).(pr1_sing +(THead k t2 t) (THead k t1 t) (pr0_comp t1 t2 H0 t t (pr0_refl t) k) (THead k +t3 t) H2))))))) u1 u2 H))))). +(* COMMENTS +Initial nodes: 137 +END *) + +theorem pr1_head_2: + \forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall (u: T).(\forall +(k: K).(pr1 (THead k u t1) (THead k u t2)))))) +\def + \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr1 t1 t2)).(\lambda (u: +T).(\lambda (k: K).(pr1_ind (\lambda (t: T).(\lambda (t0: T).(pr1 (THead k u +t) (THead k u t0)))) (\lambda (t: T).(pr1_refl (THead k u t))) (\lambda (t0: +T).(\lambda (t3: T).(\lambda (H0: (pr0 t3 t0)).(\lambda (t4: T).(\lambda (_: +(pr1 t0 t4)).(\lambda (H2: (pr1 (THead k u t0) (THead k u t4))).(pr1_sing +(THead k u t0) (THead k u t3) (pr0_comp u u (pr0_refl u) t3 t0 H0 k) (THead k +u t4) H2))))))) t1 t2 H))))). +(* COMMENTS +Initial nodes: 137 +END *) + +theorem pr1_comp: + \forall (v: T).(\forall (w: T).((pr1 v w) \to (\forall (t: T).(\forall (u: +T).((pr1 t u) \to (\forall (k: K).(pr1 (THead k v t) (THead k w u)))))))) +\def + \lambda (v: T).(\lambda (w: T).(\lambda (H: (pr1 v w)).(pr1_ind (\lambda (t: +T).(\lambda (t0: T).(\forall (t1: T).(\forall (u: T).((pr1 t1 u) \to (\forall +(k: K).(pr1 (THead k t t1) (THead k t0 u)))))))) (\lambda (t: T).(\lambda +(t0: T).(\lambda (u: T).(\lambda (H0: (pr1 t0 u)).(\lambda (k: K).(pr1_head_2 +t0 u H0 t k)))))) (\lambda (t2: T).(\lambda (t1: T).(\lambda (H0: (pr0 t1 +t2)).(\lambda (t3: T).(\lambda (H1: (pr1 t2 t3)).(\lambda (_: ((\forall (t: +T).(\forall (u: T).((pr1 t u) \to (\forall (k: K).(pr1 (THead k t2 t) (THead +k t3 u)))))))).(\lambda (t: T).(\lambda (u: T).(\lambda (H3: (pr1 t +u)).(\lambda (k: K).(pr1_ind (\lambda (t0: T).(\lambda (t4: T).(pr1 (THead k +t1 t0) (THead k t3 t4)))) (\lambda (t0: T).(pr1_head_1 t1 t3 (pr1_sing t2 t1 +H0 t3 H1) t0 k)) (\lambda (t0: T).(\lambda (t4: T).(\lambda (H4: (pr0 t4 +t0)).(\lambda (t5: T).(\lambda (_: (pr1 t0 t5)).(\lambda (H6: (pr1 (THead k +t1 t0) (THead k t3 t5))).(pr1_sing (THead k t1 t0) (THead k t1 t4) (pr0_comp +t1 t1 (pr0_refl t1) t4 t0 H4 k) (THead k t3 t5) H6))))))) t u H3))))))))))) v +w H))). +(* COMMENTS +Initial nodes: 273 +END *) + +theorem pr1_eta: + \forall (w: T).(\forall (u: T).(let t \def (THead (Bind Abst) w u) in +(\forall (v: T).((pr1 v w) \to (pr1 (THead (Bind Abst) v (THead (Flat Appl) +(TLRef O) (lift (S O) O t))) t))))) +\def + \lambda (w: T).(\lambda (u: T).(let t \def (THead (Bind Abst) w u) in +(\lambda (v: T).(\lambda (H: (pr1 v w)).(eq_ind_r T (THead (Bind Abst) (lift +(S O) O w) (lift (S O) (S O) u)) (\lambda (t0: T).(pr1 (THead (Bind Abst) v +(THead (Flat Appl) (TLRef O) t0)) (THead (Bind Abst) w u))) (pr1_comp v w H +(THead (Flat Appl) (TLRef O) (THead (Bind Abst) (lift (S O) O w) (lift (S O) +(S O) u))) u (pr1_sing (THead (Bind Abbr) (TLRef O) (lift (S O) (S O) u)) +(THead (Flat Appl) (TLRef O) (THead (Bind Abst) (lift (S O) O w) (lift (S O) +(S O) u))) (pr0_beta (lift (S O) O w) (TLRef O) (TLRef O) (pr0_refl (TLRef +O)) (lift (S O) (S O) u) (lift (S O) (S O) u) (pr0_refl (lift (S O) (S O) +u))) u (pr1_sing (THead (Bind Abbr) (TLRef O) (lift (S O) O u)) (THead (Bind +Abbr) (TLRef O) (lift (S O) (S O) u)) (pr0_delta1 (TLRef O) (TLRef O) +(pr0_refl (TLRef O)) (lift (S O) (S O) u) (lift (S O) (S O) u) (pr0_refl +(lift (S O) (S O) u)) (lift (S O) O u) (subst1_lift_S u O O (le_n O))) u +(pr1_pr0 (THead (Bind Abbr) (TLRef O) (lift (S O) O u)) u (pr0_zeta Abbr +not_abbr_abst u u (pr0_refl u) (TLRef O))))) (Bind Abst)) (lift (S O) O +(THead (Bind Abst) w u)) (lift_bind Abst w u (S O) O)))))). +(* COMMENTS +Initial nodes: 463 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr2/clen.ma b/matita/matita/contribs/lambdadelta/basic_1/pr2/clen.ma new file mode 100644 index 000000000..df9ea1fd9 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/pr2/clen.ma @@ -0,0 +1,161 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/pr2/props.ma". + +include "Basic-1/clen/getl.ma". + +theorem pr2_gen_ctail: + \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall +(t2: T).((pr2 (CTail k u c) t1 t2) \to (or (pr2 c t1 t2) (ex3 T (\lambda (_: +T).(eq K k (Bind Abbr))) (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(subst0 +(clen c) u t t2))))))))) +\def + \lambda (k: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda +(t2: T).(\lambda (H: (pr2 (CTail k u c) t1 t2)).(insert_eq C (CTail k u c) +(\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(or (pr2 c t1 t2) (ex3 T +(\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t: T).(pr0 t1 t)) (\lambda +(t: T).(subst0 (clen c) u t t2))))) (\lambda (y: C).(\lambda (H0: (pr2 y t1 +t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 +(CTail k u c)) \to (or (pr2 c t t0) (ex3 T (\lambda (_: T).(eq K k (Bind +Abbr))) (\lambda (t3: T).(pr0 t t3)) (\lambda (t3: T).(subst0 (clen c) u t3 +t0)))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: +(pr0 t3 t4)).(\lambda (_: (eq C c0 (CTail k u c))).(or_introl (pr2 c t3 t4) +(ex3 T (\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t: T).(pr0 t3 t)) +(\lambda (t: T).(subst0 (clen c) u t t4))) (pr2_free c t3 t4 H1))))))) +(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda +(H1: (getl i c0 (CHead d (Bind Abbr) u0))).(\lambda (t3: T).(\lambda (t4: +T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: (subst0 i u0 t4 +t)).(\lambda (H4: (eq C c0 (CTail k u c))).(let H5 \def (eq_ind C c0 (\lambda +(c1: C).(getl i c1 (CHead d (Bind Abbr) u0))) H1 (CTail k u c) H4) in (let +H_x \def (getl_gen_tail k Abbr u u0 d c i H5) in (let H6 \def H_x in (or_ind +(ex2 C (\lambda (e: C).(eq C d (CTail k u e))) (\lambda (e: C).(getl i c +(CHead e (Bind Abbr) u0)))) (ex4 nat (\lambda (_: nat).(eq nat i (clen c))) +(\lambda (_: nat).(eq K k (Bind Abbr))) (\lambda (_: nat).(eq T u u0)) +(\lambda (n: nat).(eq C d (CSort n)))) (or (pr2 c t3 t) (ex3 T (\lambda (_: +T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0: +T).(subst0 (clen c) u t0 t)))) (\lambda (H7: (ex2 C (\lambda (e: C).(eq C d +(CTail k u e))) (\lambda (e: C).(getl i c (CHead e (Bind Abbr) +u0))))).(ex2_ind C (\lambda (e: C).(eq C d (CTail k u e))) (\lambda (e: +C).(getl i c (CHead e (Bind Abbr) u0))) (or (pr2 c t3 t) (ex3 T (\lambda (_: +T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0: +T).(subst0 (clen c) u t0 t)))) (\lambda (x: C).(\lambda (_: (eq C d (CTail k +u x))).(\lambda (H9: (getl i c (CHead x (Bind Abbr) u0))).(or_introl (pr2 c +t3 t) (ex3 T (\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3 +t0)) (\lambda (t0: T).(subst0 (clen c) u t0 t))) (pr2_delta c x u0 i H9 t3 t4 +H2 t H3))))) H7)) (\lambda (H7: (ex4 nat (\lambda (_: nat).(eq nat i (clen +c))) (\lambda (_: nat).(eq K k (Bind Abbr))) (\lambda (_: nat).(eq T u u0)) +(\lambda (n: nat).(eq C d (CSort n))))).(ex4_ind nat (\lambda (_: nat).(eq +nat i (clen c))) (\lambda (_: nat).(eq K k (Bind Abbr))) (\lambda (_: +nat).(eq T u u0)) (\lambda (n: nat).(eq C d (CSort n))) (or (pr2 c t3 t) (ex3 +T (\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3 t0)) +(\lambda (t0: T).(subst0 (clen c) u t0 t)))) (\lambda (x0: nat).(\lambda (H8: +(eq nat i (clen c))).(\lambda (H9: (eq K k (Bind Abbr))).(\lambda (H10: (eq T +u u0)).(\lambda (_: (eq C d (CSort x0))).(let H12 \def (eq_ind nat i (\lambda +(n: nat).(subst0 n u0 t4 t)) H3 (clen c) H8) in (let H13 \def (eq_ind_r T u0 +(\lambda (t0: T).(subst0 (clen c) t0 t4 t)) H12 u H10) in (eq_ind_r K (Bind +Abbr) (\lambda (k0: K).(or (pr2 c t3 t) (ex3 T (\lambda (_: T).(eq K k0 (Bind +Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0 +t))))) (or_intror (pr2 c t3 t) (ex3 T (\lambda (_: T).(eq K (Bind Abbr) (Bind +Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0 +t))) (ex3_intro T (\lambda (_: T).(eq K (Bind Abbr) (Bind Abbr))) (\lambda +(t0: T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0 t)) t4 +(refl_equal K (Bind Abbr)) H2 H13)) k H9)))))))) H7)) H6))))))))))))))) y t1 +t2 H0))) H)))))). +(* COMMENTS +Initial nodes: 1161 +END *) + +theorem pr2_gen_cbind: + \forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall +(t2: T).((pr2 (CHead c (Bind b) v) t1 t2) \to (pr2 c (THead (Bind b) v t1) +(THead (Bind b) v t2))))))) +\def + \lambda (b: B).(\lambda (c: C).(\lambda (v: T).(\lambda (t1: T).(\lambda +(t2: T).(\lambda (H: (pr2 (CHead c (Bind b) v) t1 t2)).(insert_eq C (CHead c +(Bind b) v) (\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 c (THead +(Bind b) v t1) (THead (Bind b) v t2))) (\lambda (y: C).(\lambda (H0: (pr2 y +t1 t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 +(CHead c (Bind b) v)) \to (pr2 c (THead (Bind b) v t) (THead (Bind b) v +t0)))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: +(pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c (Bind b) v))).(pr2_free c (THead +(Bind b) v t3) (THead (Bind b) v t4) (pr0_comp v v (pr0_refl v) t3 t4 H1 +(Bind b)))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: +nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t3: +T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: +(subst0 i u t4 t)).(\lambda (H4: (eq C c0 (CHead c (Bind b) v))).(let H5 \def +(eq_ind C c0 (\lambda (c1: C).(getl i c1 (CHead d (Bind Abbr) u))) H1 (CHead +c (Bind b) v) H4) in (let H_x \def (getl_gen_bind b c (CHead d (Bind Abbr) u) +v i H5) in (let H6 \def H_x in (or_ind (land (eq nat i O) (eq C (CHead d +(Bind Abbr) u) (CHead c (Bind b) v))) (ex2 nat (\lambda (j: nat).(eq nat i (S +j))) (\lambda (j: nat).(getl j c (CHead d (Bind Abbr) u)))) (pr2 c (THead +(Bind b) v t3) (THead (Bind b) v t)) (\lambda (H7: (land (eq nat i O) (eq C +(CHead d (Bind Abbr) u) (CHead c (Bind b) v)))).(land_ind (eq nat i O) (eq C +(CHead d (Bind Abbr) u) (CHead c (Bind b) v)) (pr2 c (THead (Bind b) v t3) +(THead (Bind b) v t)) (\lambda (H8: (eq nat i O)).(\lambda (H9: (eq C (CHead +d (Bind Abbr) u) (CHead c (Bind b) v))).(let H10 \def (f_equal C C (\lambda +(e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d +| (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u) (CHead c (Bind b) +v) H9) in ((let H11 \def (f_equal C B (\lambda (e: C).(match e in C return +(\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) +\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b0) +\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) +(CHead c (Bind b) v) H9) in ((let H12 \def (f_equal C T (\lambda (e: +C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | +(CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind Abbr) u) (CHead c (Bind b) v) +H9) in (\lambda (H13: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H15 \def +(eq_ind nat i (\lambda (n: nat).(subst0 n u t4 t)) H3 O H8) in (let H16 \def +(eq_ind T u (\lambda (t0: T).(subst0 O t0 t4 t)) H15 v H12) in (eq_ind B Abbr +(\lambda (b0: B).(pr2 c (THead (Bind b0) v t3) (THead (Bind b0) v t))) +(pr2_free c (THead (Bind Abbr) v t3) (THead (Bind Abbr) v t) (pr0_delta v v +(pr0_refl v) t3 t4 H2 t H16)) b H13)))))) H11)) H10)))) H7)) (\lambda (H7: +(ex2 nat (\lambda (j: nat).(eq nat i (S j))) (\lambda (j: nat).(getl j c +(CHead d (Bind Abbr) u))))).(ex2_ind nat (\lambda (j: nat).(eq nat i (S j))) +(\lambda (j: nat).(getl j c (CHead d (Bind Abbr) u))) (pr2 c (THead (Bind b) +v t3) (THead (Bind b) v t)) (\lambda (x: nat).(\lambda (H8: (eq nat i (S +x))).(\lambda (H9: (getl x c (CHead d (Bind Abbr) u))).(let H10 \def (f_equal +nat nat (\lambda (e: nat).e) i (S x) H8) in (let H11 \def (eq_ind nat i +(\lambda (n: nat).(subst0 n u t4 t)) H3 (S x) H10) in (pr2_head_2 c v t3 t +(Bind b) (pr2_delta (CHead c (Bind b) v) d u (S x) (getl_clear_bind b (CHead +c (Bind b) v) c v (clear_bind b c v) (CHead d (Bind Abbr) u) x H9) t3 t4 H2 t +H11))))))) H7)) H6))))))))))))))) y t1 t2 H0))) H)))))). +(* COMMENTS +Initial nodes: 1085 +END *) + +theorem pr2_gen_cflat: + \forall (f: F).(\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall +(t2: T).((pr2 (CHead c (Flat f) v) t1 t2) \to (pr2 c t1 t2)))))) +\def + \lambda (f: F).(\lambda (c: C).(\lambda (v: T).(\lambda (t1: T).(\lambda +(t2: T).(\lambda (H: (pr2 (CHead c (Flat f) v) t1 t2)).(insert_eq C (CHead c +(Flat f) v) (\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 c t1 t2)) +(\lambda (y: C).(\lambda (H0: (pr2 y t1 t2)).(pr2_ind (\lambda (c0: +C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CHead c (Flat f) v)) \to (pr2 +c t t0))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: +(pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c (Flat f) v))).(pr2_free c t3 t4 +H1)))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: +nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t3: +T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: +(subst0 i u t4 t)).(\lambda (H4: (eq C c0 (CHead c (Flat f) v))).(let H5 \def +(eq_ind C c0 (\lambda (c1: C).(getl i c1 (CHead d (Bind Abbr) u))) H1 (CHead +c (Flat f) v) H4) in (let H_y \def (getl_gen_flat f c (CHead d (Bind Abbr) u) +v i H5) in (pr2_delta c d u i H_y t3 t4 H2 t H3)))))))))))))) y t1 t2 H0))) +H)))))). +(* COMMENTS +Initial nodes: 293 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr2/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/pr2/defs.ma new file mode 100644 index 000000000..e21568232 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/pr2/defs.ma @@ -0,0 +1,28 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/pr0/defs.ma". + +include "Basic-1/getl/defs.ma". + +inductive pr2: C \to (T \to (T \to Prop)) \def +| pr2_free: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to +(pr2 c t1 t2)))) +| pr2_delta: \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: +nat).((getl i c (CHead d (Bind Abbr) u)) \to (\forall (t1: T).(\forall (t2: +T).((pr0 t1 t2) \to (\forall (t: T).((subst0 i u t2 t) \to (pr2 c t1 +t)))))))))). + diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr2/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/pr2/fwd.ma new file mode 100644 index 000000000..1a18f7dbc --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/pr2/fwd.ma @@ -0,0 +1,3343 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/pr2/defs.ma". + +include "Basic-1/pr0/fwd.ma". + +include "Basic-1/getl/drop.ma". + +include "Basic-1/getl/clear.ma". + +theorem pr2_gen_sort: + \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr2 c (TSort n) x) \to +(eq T x (TSort n))))) +\def + \lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr2 c (TSort +n) x)).(insert_eq T (TSort n) (\lambda (t: T).(pr2 c t x)) (\lambda (t: +T).(eq T x t)) (\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda +(_: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (TSort n)) \to (eq T t0 +t))))) (\lambda (_: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H1: (pr0 +t1 t2)).(\lambda (H2: (eq T t1 (TSort n))).(let H3 \def (eq_ind T t1 (\lambda +(t: T).(pr0 t t2)) H1 (TSort n) H2) in (eq_ind_r T (TSort n) (\lambda (t: +T).(eq T t2 t)) (eq_ind_r T (TSort n) (\lambda (t: T).(eq T t (TSort n))) +(refl_equal T (TSort n)) t2 (pr0_gen_sort t2 n H3)) t1 H2))))))) (\lambda +(c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (_: (getl +i c0 (CHead d (Bind Abbr) u))).(\lambda (t1: T).(\lambda (t2: T).(\lambda +(H2: (pr0 t1 t2)).(\lambda (t: T).(\lambda (H3: (subst0 i u t2 t)).(\lambda +(H4: (eq T t1 (TSort n))).(let H5 \def (eq_ind T t1 (\lambda (t0: T).(pr0 t0 +t2)) H2 (TSort n) H4) in (eq_ind_r T (TSort n) (\lambda (t0: T).(eq T t t0)) +(let H6 \def (eq_ind T t2 (\lambda (t0: T).(subst0 i u t0 t)) H3 (TSort n) +(pr0_gen_sort t2 n H5)) in (subst0_gen_sort u t i n H6 (eq T t (TSort n)))) +t1 H4))))))))))))) c y x H0))) H)))). +(* COMMENTS +Initial nodes: 347 +END *) + +theorem pr2_gen_lref: + \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr2 c (TLRef n) x) \to +(or (eq T x (TLRef n)) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c +(CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T x (lift (S +n) O u))))))))) +\def + \lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr2 c (TLRef +n) x)).(insert_eq T (TLRef n) (\lambda (t: T).(pr2 c t x)) (\lambda (t: +T).(or (eq T x t) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c (CHead +d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T x (lift (S n) O +u))))))) (\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0: +C).(\lambda (t: T).(\lambda (t0: T).((eq T t (TLRef n)) \to (or (eq T t0 t) +(ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c0 (CHead d (Bind Abbr) +u)))) (\lambda (_: C).(\lambda (u: T).(eq T t0 (lift (S n) O u)))))))))) +(\lambda (c0: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H1: (pr0 t1 +t2)).(\lambda (H2: (eq T t1 (TLRef n))).(let H3 \def (eq_ind T t1 (\lambda +(t: T).(pr0 t t2)) H1 (TLRef n) H2) in (eq_ind_r T (TLRef n) (\lambda (t: +T).(or (eq T t2 t) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c0 +(CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T t2 (lift (S +n) O u))))))) (eq_ind_r T (TLRef n) (\lambda (t: T).(or (eq T t (TLRef n)) +(ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c0 (CHead d (Bind Abbr) +u)))) (\lambda (_: C).(\lambda (u: T).(eq T t (lift (S n) O u))))))) +(or_introl (eq T (TLRef n) (TLRef n)) (ex2_2 C T (\lambda (d: C).(\lambda (u: +T).(getl n c0 (CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq +T (TLRef n) (lift (S n) O u))))) (refl_equal T (TLRef n))) t2 (pr0_gen_lref +t2 n H3)) t1 H2))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: +T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr) +u))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H2: (pr0 t1 t2)).(\lambda +(t: T).(\lambda (H3: (subst0 i u t2 t)).(\lambda (H4: (eq T t1 (TLRef +n))).(let H5 \def (eq_ind T t1 (\lambda (t0: T).(pr0 t0 t2)) H2 (TLRef n) H4) +in (eq_ind_r T (TLRef n) (\lambda (t0: T).(or (eq T t t0) (ex2_2 C T (\lambda +(d0: C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind Abbr) u0)))) (\lambda (_: +C).(\lambda (u0: T).(eq T t (lift (S n) O u0))))))) (let H6 \def (eq_ind T t2 +(\lambda (t0: T).(subst0 i u t0 t)) H3 (TLRef n) (pr0_gen_lref t2 n H5)) in +(land_ind (eq nat n i) (eq T t (lift (S n) O u)) (or (eq T t (TLRef n)) +(ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind Abbr) +u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T t (lift (S n) O u0)))))) +(\lambda (H7: (eq nat n i)).(\lambda (H8: (eq T t (lift (S n) O +u))).(eq_ind_r T (lift (S n) O u) (\lambda (t0: T).(or (eq T t0 (TLRef n)) +(ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind Abbr) +u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T t0 (lift (S n) O u0))))))) (let +H9 \def (eq_ind_r nat i (\lambda (n0: nat).(getl n0 c0 (CHead d (Bind Abbr) +u))) H1 n H7) in (or_intror (eq T (lift (S n) O u) (TLRef n)) (ex2_2 C T +(\lambda (d0: C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind Abbr) u0)))) +(\lambda (_: C).(\lambda (u0: T).(eq T (lift (S n) O u) (lift (S n) O u0))))) +(ex2_2_intro C T (\lambda (d0: C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind +Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T (lift (S n) O u) (lift (S +n) O u0)))) d u H9 (refl_equal T (lift (S n) O u))))) t H8))) +(subst0_gen_lref u t i n H6))) t1 H4))))))))))))) c y x H0))) H)))). +(* COMMENTS +Initial nodes: 1003 +END *) + +theorem pr2_gen_abst: + \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c +(THead (Bind Abst) u1 t1) x) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: +T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 +c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: +T).(pr2 (CHead c (Bind b) u) t1 t2)))))))))) +\def + \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda +(H: (pr2 c (THead (Bind Abst) u1 t1) x)).(insert_eq T (THead (Bind Abst) u1 +t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(ex3_2 T T (\lambda (u2: +T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall +(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2))))))) (\lambda (y: +T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0: C).(\lambda (t: +T).(\lambda (t0: T).((eq T t (THead (Bind Abst) u1 t1)) \to (ex3_2 T T +(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abst) u2 t2)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda +(t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 +t2)))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H1: +(pr0 t0 t2)).(\lambda (H2: (eq T t0 (THead (Bind Abst) u1 t1))).(let H3 \def +(eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Bind Abst) u1 t1) H2) in +(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) +u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(pr0 t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: +T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: +T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall +(u: T).(pr2 (CHead c0 (Bind b) u) t1 t3)))))) (\lambda (x0: T).(\lambda (x1: +T).(\lambda (H4: (eq T t2 (THead (Bind Abst) x0 x1))).(\lambda (H5: (pr0 u1 +x0)).(\lambda (H6: (pr0 t1 x1)).(eq_ind_r T (THead (Bind Abst) x0 x1) +(\lambda (t: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead +(Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) +(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead +c0 (Bind b) u) t1 t3))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: +T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall +(b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 t3))))) x0 x1 +(refl_equal T (THead (Bind Abst) x0 x1)) (pr2_free c0 u1 x0 H5) (\lambda (b: +B).(\lambda (u: T).(pr2_free (CHead c0 (Bind b) u) t1 x1 H6)))) t2 H4)))))) +(pr0_gen_abst u1 t1 t2 H3)))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda +(u: T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr) +u))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H2: (pr0 t0 t2)).(\lambda +(t: T).(\lambda (H3: (subst0 i u t2 t)).(\lambda (H4: (eq T t0 (THead (Bind +Abst) u1 t1))).(let H5 \def (eq_ind T t0 (\lambda (t3: T).(pr0 t3 t2)) H2 +(THead (Bind Abst) u1 t1) H4) in (ex3_2_ind T T (\lambda (u2: T).(\lambda +(t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: +T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda +(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 +t3)))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H6: (eq T t2 (THead +(Bind Abst) x0 x1))).(\lambda (H7: (pr0 u1 x0)).(\lambda (H8: (pr0 t1 +x1)).(let H9 \def (eq_ind T t2 (\lambda (t3: T).(subst0 i u t3 t)) H3 (THead +(Bind Abst) x0 x1) H6) in (or3_ind (ex2 T (\lambda (u2: T).(eq T t (THead +(Bind Abst) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2))) (ex2 T (\lambda +(t3: T).(eq T t (THead (Bind Abst) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind +Abst) i) u x1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t +(THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 +u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u x1 t3)))) +(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst) u2 +t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) +u0) t1 t3)))))) (\lambda (H10: (ex2 T (\lambda (u2: T).(eq T t (THead (Bind +Abst) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2)))).(ex2_ind T (\lambda +(u2: T).(eq T t (THead (Bind Abst) u2 x1))) (\lambda (u2: T).(subst0 i u x0 +u2)) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst) +u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) +u0) t1 t3)))))) (\lambda (x2: T).(\lambda (H11: (eq T t (THead (Bind Abst) x2 +x1))).(\lambda (H12: (subst0 i u x0 x2)).(eq_ind_r T (THead (Bind Abst) x2 +x1) (\lambda (t3: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 +(THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 +u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 +(CHead c0 (Bind b) u0) t1 t4))))))) (ex3_2_intro T T (\lambda (u2: +T).(\lambda (t3: T).(eq T (THead (Bind Abst) x2 x1) (THead (Bind Abst) u2 +t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) +u0) t1 t3))))) x2 x1 (refl_equal T (THead (Bind Abst) x2 x1)) (pr2_delta c0 d +u i H1 u1 x0 H7 x2 H12) (\lambda (b: B).(\lambda (u0: T).(pr2_free (CHead c0 +(Bind b) u0) t1 x1 H8)))) t H11)))) H10)) (\lambda (H10: (ex2 T (\lambda (t3: +T).(eq T t (THead (Bind Abst) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind +Abst) i) u x1 t3)))).(ex2_ind T (\lambda (t3: T).(eq T t (THead (Bind Abst) +x0 t3))) (\lambda (t3: T).(subst0 (s (Bind Abst) i) u x1 t3)) (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda +(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 +t3)))))) (\lambda (x2: T).(\lambda (H11: (eq T t (THead (Bind Abst) x0 +x2))).(\lambda (H12: (subst0 (s (Bind Abst) i) u x1 x2)).(eq_ind_r T (THead +(Bind Abst) x0 x2) (\lambda (t3: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t4: +T).(eq T t3 (THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: +T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall +(u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t4))))))) (ex3_2_intro T T (\lambda +(u2: T).(\lambda (t3: T).(eq T (THead (Bind Abst) x0 x2) (THead (Bind Abst) +u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) +u0) t1 t3))))) x0 x2 (refl_equal T (THead (Bind Abst) x0 x2)) (pr2_free c0 u1 +x0 H7) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u +(S i) (getl_head (Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 x1 H8 x2 +H12)))) t H11)))) H10)) (\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda +(t3: T).(eq T t (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: +T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind +Abst) i) u x1 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T +t (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u +x0 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u x1 +t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst) +u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) +u0) t1 t3)))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H11: (eq T t +(THead (Bind Abst) x2 x3))).(\lambda (H12: (subst0 i u x0 x2)).(\lambda (H13: +(subst0 (s (Bind Abst) i) u x1 x3)).(eq_ind_r T (THead (Bind Abst) x2 x3) +(\lambda (t3: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead +(Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) +(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead +c0 (Bind b) u0) t1 t4))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: +T).(eq T (THead (Bind Abst) x2 x3) (THead (Bind Abst) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall +(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))))) x2 x3 +(refl_equal T (THead (Bind Abst) x2 x3)) (pr2_delta c0 d u i H1 u1 x0 H7 x2 +H12) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u +(S i) (getl_head (Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 x1 H8 x3 +H13)))) t H11)))))) H10)) (subst0_gen_head (Bind Abst) u x0 x1 t i H9)))))))) +(pr0_gen_abst u1 t1 t2 H5)))))))))))))) c y x H0))) H))))). +(* COMMENTS +Initial nodes: 2383 +END *) + +theorem pr2_gen_cast: + \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c +(THead (Flat Cast) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda +(t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: +T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c t1 t2)))) (pr2 c +t1 x)))))) +\def + \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda +(H: (pr2 c (THead (Flat Cast) u1 t1) x)).(insert_eq T (THead (Flat Cast) u1 +t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: +T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c t1 +t2)))) (pr2 c t1 x))) (\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind +(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Flat Cast) +u1 t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead +(Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) +(\lambda (_: T).(\lambda (t2: T).(pr2 c0 t1 t2)))) (pr2 c0 t1 t0)))))) +(\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H1: (pr0 t0 +t2)).(\lambda (H2: (eq T t0 (THead (Flat Cast) u1 t1))).(let H3 \def (eq_ind +T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Flat Cast) u1 t1) H2) in (or_ind +(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 +t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2) (or (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 +t1 t3)))) (pr2 c0 t1 t2)) (\lambda (H4: (ex3_2 T T (\lambda (u2: T).(\lambda +(t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: +T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind +T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: +T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 +(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 +u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t2)) +(\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T t2 (THead (Flat Cast) +x0 x1))).(\lambda (H6: (pr0 u1 x0)).(\lambda (H7: (pr0 t1 x1)).(eq_ind_r T +(THead (Flat Cast) x0 x1) (\lambda (t: T).(or (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 +t1 t3)))) (pr2 c0 t1 t))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda +(t3: T).(eq T (THead (Flat Cast) x0 x1) (THead (Flat Cast) u2 t3)))) (\lambda +(u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: +T).(pr2 c0 t1 t3)))) (pr2 c0 t1 (THead (Flat Cast) x0 x1)) (ex3_2_intro T T +(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Cast) x0 x1) (THead +(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) +(\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3))) x0 x1 (refl_equal T (THead +(Flat Cast) x0 x1)) (pr2_free c0 u1 x0 H6) (pr2_free c0 t1 x1 H7))) t2 +H5)))))) H4)) (\lambda (H4: (pr0 t1 t2)).(or_intror (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 +t1 t3)))) (pr2 c0 t1 t2) (pr2_free c0 t1 t2 H4))) (pr0_gen_cast u1 t1 t2 +H3)))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: +nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t0: +T).(\lambda (t2: T).(\lambda (H2: (pr0 t0 t2)).(\lambda (t: T).(\lambda (H3: +(subst0 i u t2 t)).(\lambda (H4: (eq T t0 (THead (Flat Cast) u1 t1))).(let H5 +\def (eq_ind T t0 (\lambda (t3: T).(pr0 t3 t2)) H2 (THead (Flat Cast) u1 t1) +H4) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead +(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda +(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2) (or (ex3_2 T T (\lambda +(u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 +t1 t3)))) (pr2 c0 t1 t)) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda +(t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: +T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind +T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: +T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t +(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 +u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t)) +(\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: (eq T t2 (THead (Flat Cast) +x0 x1))).(\lambda (H8: (pr0 u1 x0)).(\lambda (H9: (pr0 t1 x1)).(let H10 \def +(eq_ind T t2 (\lambda (t3: T).(subst0 i u t3 t)) H3 (THead (Flat Cast) x0 x1) +H7) in (or3_ind (ex2 T (\lambda (u2: T).(eq T t (THead (Flat Cast) u2 x1))) +(\lambda (u2: T).(subst0 i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T t (THead +(Flat Cast) x0 t3))) (\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3))) +(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 +t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: +T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3)))) (or (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda +(t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t)) (\lambda (H11: (ex2 T (\lambda (u2: +T).(eq T t (THead (Flat Cast) u2 x1))) (\lambda (u2: T).(subst0 i u x0 +u2)))).(ex2_ind T (\lambda (u2: T).(eq T t (THead (Flat Cast) u2 x1))) +(\lambda (u2: T).(subst0 i u x0 u2)) (or (ex3_2 T T (\lambda (u2: T).(\lambda +(t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: +T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 +c0 t1 t)) (\lambda (x2: T).(\lambda (H12: (eq T t (THead (Flat Cast) x2 +x1))).(\lambda (H13: (subst0 i u x0 x2)).(or_introl (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 +t1 t3)))) (pr2 c0 t1 t) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: +T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 +c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3))) x2 x1 H12 +(pr2_delta c0 d u i H1 u1 x0 H8 x2 H13) (pr2_free c0 t1 x1 H9)))))) H11)) +(\lambda (H11: (ex2 T (\lambda (t3: T).(eq T t (THead (Flat Cast) x0 t3))) +(\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3)))).(ex2_ind T (\lambda +(t3: T).(eq T t (THead (Flat Cast) x0 t3))) (\lambda (t3: T).(subst0 (s (Flat +Cast) i) u x1 t3)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t +(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 +u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t)) +(\lambda (x2: T).(\lambda (H12: (eq T t (THead (Flat Cast) x0 x2))).(\lambda +(H13: (subst0 (s (Flat Cast) i) u x1 x2)).(or_introl (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 +t1 t3)))) (pr2 c0 t1 t) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: +T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 +c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3))) x0 x2 H12 +(pr2_free c0 u1 x0 H8) (pr2_delta c0 d u i H1 t1 x1 H9 x2 H13)))))) H11)) +(\lambda (H11: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead +(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) +(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 +t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat +Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) +(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3))) (or +(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 +t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t)) (\lambda (x2: +T).(\lambda (x3: T).(\lambda (H12: (eq T t (THead (Flat Cast) x2 +x3))).(\lambda (H13: (subst0 i u x0 x2)).(\lambda (H14: (subst0 (s (Flat +Cast) i) u x1 x3)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: +T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 +c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t) +(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast) +u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(pr2 c0 t1 t3))) x2 x3 H12 (pr2_delta c0 d u i H1 u1 x0 +H8 x2 H13) (pr2_delta c0 d u i H1 t1 x1 H9 x3 H14)))))))) H11)) +(subst0_gen_head (Flat Cast) u x0 x1 t i H10)))))))) H6)) (\lambda (H6: (pr0 +t1 t2)).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t +(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 +u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t) +(pr2_delta c0 d u i H1 t1 t2 H6 t H3))) (pr0_gen_cast u1 t1 t2 +H5)))))))))))))) c y x H0))) H))))). +(* COMMENTS +Initial nodes: 2659 +END *) + +theorem pr2_gen_csort: + \forall (t1: T).(\forall (t2: T).(\forall (n: nat).((pr2 (CSort n) t1 t2) +\to (pr0 t1 t2)))) +\def + \lambda (t1: T).(\lambda (t2: T).(\lambda (n: nat).(\lambda (H: (pr2 (CSort +n) t1 t2)).(insert_eq C (CSort n) (\lambda (c: C).(pr2 c t1 t2)) (\lambda (_: +C).(pr0 t1 t2)) (\lambda (y: C).(\lambda (H0: (pr2 y t1 t2)).(pr2_ind +(\lambda (c: C).(\lambda (t: T).(\lambda (t0: T).((eq C c (CSort n)) \to (pr0 +t t0))))) (\lambda (c: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: +(pr0 t3 t4)).(\lambda (_: (eq C c (CSort n))).H1))))) (\lambda (c: +C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H1: (getl i c +(CHead d (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 +t3 t4)).(\lambda (t: T).(\lambda (_: (subst0 i u t4 t)).(\lambda (H4: (eq C c +(CSort n))).(let H5 \def (eq_ind C c (\lambda (c0: C).(getl i c0 (CHead d +(Bind Abbr) u))) H1 (CSort n) H4) in (getl_gen_sort n i (CHead d (Bind Abbr) +u) H5 (pr0 t3 t)))))))))))))) y t1 t2 H0))) H)))). +(* COMMENTS +Initial nodes: 221 +END *) + +theorem pr2_gen_appl: + \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c +(THead (Flat Appl) u1 t1) x) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda +(t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: +T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c t1 t2)))) (ex4_4 T +T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T +t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda +(u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall +(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T +T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda +(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq +T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead +(Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda +(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))))))) +\def + \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda +(H: (pr2 c (THead (Flat Appl) u1 t1) x)).(insert_eq T (THead (Flat Appl) u1 +t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(or3 (ex3_2 T T (\lambda +(u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c t1 +t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind +Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) +z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) +(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda +(y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) +z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: +B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: +T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) +y2) z1 z2)))))))))) (\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind +(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Flat Appl) +u1 t1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead +(Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) +(\lambda (_: T).(\lambda (t2: T).(pr2 c0 t1 t2)))) (ex4_4 T T T T (\lambda +(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead +(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall +(b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) z1 t2)))))))) (ex6_6 B T T +T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: +B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T +t0 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) +(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 +y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: +T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 +z2))))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: T).(\lambda +(H1: (pr0 t0 t2)).(\lambda (H2: (eq T t0 (THead (Flat Appl) u1 t1))).(let H3 +\def (eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Flat Appl) u1 t1) +H2) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead +(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda +(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind +Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T +(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 +(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind +b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda +(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) +(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (t3: T).(pr0 z1 t3)))))))) (or3 (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 +t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind +Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) +u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b +Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda +(u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift +(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) +(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 +(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (H4: (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 +t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead +(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda +(_: T).(\lambda (t3: T).(pr0 t1 t3))) (or3 (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 +t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind +Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) +u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b +Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda +(u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift +(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) +(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 +(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x0: T).(\lambda (x1: +T).(\lambda (H5: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H6: (pr0 u1 +x0)).(\lambda (H7: (pr0 t1 x1)).(eq_ind_r T (THead (Flat Appl) x0 x1) +(\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t +(THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 +u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T +(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 +(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall +(b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T +T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: +B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t +(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda +(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 +y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: +T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 +z2)))))))))) (or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T +(THead (Flat Appl) x0 x1) (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 +t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) +x0 x1) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall +(u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda +(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 +(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Flat +Appl) x0 x1) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) +z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: +B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: +T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) +y2) z1 z2)))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T +(THead (Flat Appl) x0 x1) (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 +t1 t3))) x0 x1 (refl_equal T (THead (Flat Appl) x0 x1)) (pr2_free c0 u1 x0 +H6) (pr2_free c0 t1 x1 H7))) t2 H5)))))) H4)) (\lambda (H4: (ex4_4 T T T T +(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 +(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda +(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 +t3))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind +Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(pr0 z1 t3))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda +(t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: +T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 +T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq +T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda +(u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall +(b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T +T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: +B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T +t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) +(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 +y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: +T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 +z2))))))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: +T).(\lambda (H5: (eq T t1 (THead (Bind Abst) x0 x1))).(\lambda (H6: (eq T t2 +(THead (Bind Abbr) x2 x3))).(\lambda (H7: (pr0 u1 x2)).(\lambda (H8: (pr0 x1 +x3)).(eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda (t: T).(or3 (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda +(t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) +(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t +(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 +(CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind +b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead +(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 +u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (eq_ind_r T (THead (Bind +Abst) x0 x1) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: +T).(eq T (THead (Bind Abbr) x2 x3) (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 +t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) +x2 x3) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall +(u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda +(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t +(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind +Abbr) x2 x3) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) +z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: +B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: +T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) +y2) z1 z2)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: +T).(eq T (THead (Bind Abbr) x2 x3) (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 +(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) +(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) u2 +t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: +T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) +u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b +Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead +(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind +Abbr) x2 x3) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) +z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: +B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: +T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) +y2) z1 z2)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead +(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) u2 +t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: +T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) +u) z1 t3))))))) x0 x1 x2 x3 (refl_equal T (THead (Bind Abst) x0 x1)) +(refl_equal T (THead (Bind Abbr) x2 x3)) (pr2_free c0 u1 x2 H7) (\lambda (b: +B).(\lambda (u: T).(pr2_free (CHead c0 (Bind b) u) x1 x3 H8))))) t1 H5) t2 +H6))))))))) H4)) (\lambda (H4: (ex6_6 B T T T T T (\lambda (b: B).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not +(eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) +y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead (Flat +Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) +(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 +t3))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b +Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead (Flat Appl) (lift +(S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) +(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 +t3))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead +(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) +(\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda +(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead +(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall +(b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T +T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: +B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T +t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) +(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 +y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: +T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 +z2))))))))) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: +T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H5: (not (eq B x0 +Abst))).(\lambda (H6: (eq T t1 (THead (Bind x0) x1 x2))).(\lambda (H7: (eq T +t2 (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)))).(\lambda +(H8: (pr0 u1 x3)).(\lambda (H9: (pr0 x1 x4)).(\lambda (H10: (pr0 x2 +x5)).(eq_ind_r T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) +x5)) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T +t (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 +u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T +(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 +(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall +(b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T +T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: +B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t +(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda +(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 +y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: +T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 +z2)))))))))) (eq_ind_r T (THead (Bind x0) x1 x2) (\lambda (t: T).(or3 (ex3_2 +T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat +Appl) (lift (S O) O x3) x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 +t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) +x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind Abbr) u2 t3)))))) +(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 +u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: +T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) +(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: +B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(eq T t (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T +(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind b) +y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 +c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat +Appl) (lift (S O) O x3) x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 +(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind +Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) +(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 +(CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind +x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T +(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind b) +y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 +c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T +(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T +(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda +(y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) +(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda +(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 +y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: +T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))) +x0 x1 x2 x5 x3 x4 H5 (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T +(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5))) (pr2_free c0 +u1 x3 H8) (pr2_free c0 x1 x4 H9) (pr2_free (CHead c0 (Bind x0) x4) x2 x5 +H10))) t1 H6) t2 H7))))))))))))) H4)) (pr0_gen_appl u1 t1 t2 H3)))))))) +(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda +(H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t0: T).(\lambda (t2: +T).(\lambda (H2: (pr0 t0 t2)).(\lambda (t: T).(\lambda (H3: (subst0 i u t2 +t)).(\lambda (H4: (eq T t0 (THead (Flat Appl) u1 t1))).(let H5 \def (eq_ind T +t0 (\lambda (t3: T).(pr0 t3 t2)) H2 (THead (Flat Appl) u1 t1) H4) in (or3_ind +(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 +t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 +z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: +T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T +(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 +(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind +b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda +(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) +(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (t3: T).(pr0 z1 t3)))))))) (or3 (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 +t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind +Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) +u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b +Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda +(u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift +(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) +(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 +(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (H6: (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 +t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead +(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda +(_: T).(\lambda (t3: T).(pr0 t1 t3))) (or3 (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 +t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind +Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) +u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b +Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda +(u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift +(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) +(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 +(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x0: T).(\lambda (x1: +T).(\lambda (H7: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H8: (pr0 u1 +x0)).(\lambda (H9: (pr0 t1 x1)).(let H10 \def (eq_ind T t2 (\lambda (t3: +T).(subst0 i u t3 t)) H3 (THead (Flat Appl) x0 x1) H7) in (or3_ind (ex2 T +(\lambda (u2: T).(eq T t (THead (Flat Appl) u2 x1))) (\lambda (u2: T).(subst0 +i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T t (THead (Flat Appl) x0 t3))) +(\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 t3))) (ex3_2 T T (\lambda +(u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3: +T).(subst0 (s (Flat Appl) i) u x1 t3)))) (or3 (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 +t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind +Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) +u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b +Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda +(u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift +(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) +(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 +(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (H11: (ex2 T (\lambda (u2: +T).(eq T t (THead (Flat Appl) u2 x1))) (\lambda (u2: T).(subst0 i u x0 +u2)))).(ex2_ind T (\lambda (u2: T).(eq T t (THead (Flat Appl) u2 x1))) +(\lambda (u2: T).(subst0 i u x0 u2)) (or3 (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 +t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind +Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) +u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b +Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda +(u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift +(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) +(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 +(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x2: T).(\lambda (H12: (eq T t +(THead (Flat Appl) x2 x1))).(\lambda (H13: (subst0 i u x0 x2)).(eq_ind_r T +(THead (Flat Appl) x2 x1) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: +T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 +t1 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind +Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) +u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b +Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda +(u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift +(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) +(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 +(CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro0 (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x1) (THead (Flat Appl) u2 +t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind +Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(t3: T).(eq T (THead (Flat Appl) x2 x1) (THead (Bind Abbr) u2 t3)))))) +(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 +u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: +T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) +(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: +B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T +(THead (Flat Appl) x2 x1) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O +u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: +B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: +T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) +y2) z1 z2)))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T +(THead (Flat Appl) x2 x1) (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 +t1 t3))) x2 x1 (refl_equal T (THead (Flat Appl) x2 x1)) (pr2_delta c0 d u i +H1 u1 x0 H8 x2 H13) (pr2_free c0 t1 x1 H9))) t H12)))) H11)) (\lambda (H11: +(ex2 T (\lambda (t3: T).(eq T t (THead (Flat Appl) x0 t3))) (\lambda (t3: +T).(subst0 (s (Flat Appl) i) u x1 t3)))).(ex2_ind T (\lambda (t3: T).(eq T t +(THead (Flat Appl) x0 t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 +t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat +Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda +(_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind +Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall +(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda +(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 +(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) +y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 +c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x2: T).(\lambda +(H12: (eq T t (THead (Flat Appl) x0 x2))).(\lambda (H13: (subst0 (s (Flat +Appl) i) u x1 x2)).(eq_ind_r T (THead (Flat Appl) x0 x2) (\lambda (t3: +T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat +Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda +(_: T).(\lambda (t4: T).(pr2 c0 t1 t4)))) (ex4_4 T T T T (\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind +Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall +(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda +(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 +(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind +b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda +(_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) +(or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat +Appl) x0 x2) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: +T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 +T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq +T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda +(u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x0 x2) (THead (Bind Abbr) +u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: +T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) +u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b +Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda +(u2: T).(\lambda (y2: T).(eq T (THead (Flat Appl) x0 x2) (THead (Bind b) y2 +(THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 +u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) (ex3_2_intro T T (\lambda +(u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x0 x2) (THead (Flat Appl) +u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(pr2 c0 t1 t3))) x0 x2 (refl_equal T (THead (Flat Appl) +x0 x2)) (pr2_free c0 u1 x0 H8) (pr2_delta c0 d u i H1 t1 x1 H9 x2 H13))) t +H12)))) H11)) (\lambda (H11: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq +T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u +x0 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 +t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat +Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) +(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 t3))) (or3 +(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 +t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind +Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall +(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda +(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 +(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) +y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 +c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x2: T).(\lambda +(x3: T).(\lambda (H12: (eq T t (THead (Flat Appl) x2 x3))).(\lambda (H13: +(subst0 i u x0 x2)).(\lambda (H14: (subst0 (s (Flat Appl) i) u x1 +x3)).(eq_ind_r T (THead (Flat Appl) x2 x3) (\lambda (t3: T).(or3 (ex3_2 T T +(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda +(t4: T).(pr2 c0 t1 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) +(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 +(THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 +(CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind +b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead +(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 +u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro0 (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x3) (THead +(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) +(\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda +(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead +(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x3) (THead (Bind Abbr) u2 +t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: +T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) +u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b +Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda +(u2: T).(\lambda (y2: T).(eq T (THead (Flat Appl) x2 x3) (THead (Bind b) y2 +(THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 +u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) (ex3_2_intro T T (\lambda +(u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x3) (THead (Flat Appl) +u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(pr2 c0 t1 t3))) x2 x3 (refl_equal T (THead (Flat Appl) +x2 x3)) (pr2_delta c0 d u i H1 u1 x0 H8 x2 H13) (pr2_delta c0 d u i H1 t1 x1 +H9 x3 H14))) t H12)))))) H11)) (subst0_gen_head (Flat Appl) u x0 x1 t i +H10)))))))) H6)) (\lambda (H6: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) +(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 +(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda +(_: T).(\lambda (t3: T).(pr0 z1 t3))))))).(ex4_4_ind T T T T (\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind +Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))) (or3 (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda +(t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) +(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t +(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 +(CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind +b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead +(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 +u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x0: T).(\lambda +(x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq T t1 (THead (Bind +Abst) x0 x1))).(\lambda (H8: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda +(H9: (pr0 u1 x2)).(\lambda (H10: (pr0 x1 x3)).(let H11 \def (eq_ind T t2 +(\lambda (t3: T).(subst0 i u t3 t)) H3 (THead (Bind Abbr) x2 x3) H8) in +(eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t3: T).(or3 (ex3_2 T T +(\lambda (u2: T).(\lambda (t4: T).(eq T t (THead (Flat Appl) u2 t4)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda +(t4: T).(pr2 c0 t3 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(eq T t3 (THead (Bind Abst) y1 z1)))))) +(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t +(THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 +(CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t3 (THead (Bind +b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead +(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 +u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_ind (ex2 T (\lambda +(u2: T).(eq T t (THead (Bind Abbr) u2 x3))) (\lambda (u2: T).(subst0 i u x2 +u2))) (ex2 T (\lambda (t3: T).(eq T t (THead (Bind Abbr) x2 t3))) (\lambda +(t3: T).(subst0 (s (Bind Abbr) i) u x3 t3))) (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(subst0 i u x2 u2))) (\lambda (_: T).(\lambda (t3: +T).(subst0 (s (Bind Abbr) i) u x3 t3)))) (or3 (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 +(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) +(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall +(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T +T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: +B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda +(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: +T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) +O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: +B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: +T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) +y2) z1 z2))))))))) (\lambda (H12: (ex2 T (\lambda (u2: T).(eq T t (THead +(Bind Abbr) u2 x3))) (\lambda (u2: T).(subst0 i u x2 u2)))).(ex2_ind T +(\lambda (u2: T).(eq T t (THead (Bind Abbr) u2 x3))) (\lambda (u2: T).(subst0 +i u x2 u2)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead +(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) +(\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind Abst) x0 x1) t3)))) +(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: +T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda +(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind +Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) +u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b +Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead +(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) +y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 +c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x4: T).(\lambda +(H13: (eq T t (THead (Bind Abbr) x4 x3))).(\lambda (H14: (subst0 i u x2 +x4)).(eq_ind_r T (THead (Bind Abbr) x4 x3) (\lambda (t3: T).(or3 (ex3_2 T T +(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda +(t4: T).(pr2 c0 (THead (Bind Abst) x0 x1) t4)))) (ex4_4 T T T T (\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) +x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) +(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 +u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: +T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) +(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: +B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda +(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: +T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S +O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: +B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: +T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) +y2) z1 z2)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: +T).(eq T (THead (Bind Abbr) x4 x3) (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 +(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) +(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x3) (THead (Bind Abbr) u2 +t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: +T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) +u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b +Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead +(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind +Abbr) x4 x3) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) +z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: +B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: +T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) +y2) z1 z2)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead +(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x3) (THead (Bind Abbr) u2 +t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: +T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) +u0) z1 t3))))))) x0 x1 x4 x3 (refl_equal T (THead (Bind Abst) x0 x1)) +(refl_equal T (THead (Bind Abbr) x4 x3)) (pr2_delta c0 d u i H1 u1 x2 H9 x4 +H14) (\lambda (b: B).(\lambda (u0: T).(pr2_free (CHead c0 (Bind b) u0) x1 x3 +H10))))) t H13)))) H12)) (\lambda (H12: (ex2 T (\lambda (t3: T).(eq T t +(THead (Bind Abbr) x2 t3))) (\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x3 +t3)))).(ex2_ind T (\lambda (t3: T).(eq T t (THead (Bind Abbr) x2 t3))) +(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x3 t3)) (or3 (ex3_2 T T (\lambda +(u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 +(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) +(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall +(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T +T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: +B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda +(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: +T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) +O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: +B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: +T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) +y2) z1 z2))))))))) (\lambda (x4: T).(\lambda (H13: (eq T t (THead (Bind Abbr) +x2 x4))).(\lambda (H14: (subst0 (s (Bind Abbr) i) u x3 x4)).(eq_ind_r T +(THead (Bind Abbr) x2 x4) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: +T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 +(THead (Bind Abst) x0 x1) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) +(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall +(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T +T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: +B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda +(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: +T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S +O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: +B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: +T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) +y2) z1 z2)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: +T).(eq T (THead (Bind Abbr) x2 x4) (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 +(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) +(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x4) (THead (Bind Abbr) u2 +t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: +T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) +u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b +Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead +(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind +Abbr) x2 x4) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) +z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: +B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: +T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) +y2) z1 z2)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead +(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x4) (THead (Bind Abbr) u2 +t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: +T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) +u0) z1 t3))))))) x0 x1 x2 x4 (refl_equal T (THead (Bind Abst) x0 x1)) +(refl_equal T (THead (Bind Abbr) x2 x4)) (pr2_free c0 u1 x2 H9) (\lambda (b: +B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) +(getl_clear_bind b (CHead c0 (Bind b) u0) c0 u0 (clear_bind b c0 u0) (CHead d +(Bind Abbr) u) i H1) x1 x3 H10 x4 H14))))) t H13)))) H12)) (\lambda (H12: +(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 +t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x2 u2))) (\lambda (_: +T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x3 t3))))).(ex3_2_ind T T +(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(subst0 i u x2 u2))) (\lambda (_: +T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x3 t3))) (or3 (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda +(t3: T).(pr2 c0 (THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) +x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) +(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 +u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: +T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) +(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: +B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda +(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: +T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) +O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: +B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: +T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) +y2) z1 z2))))))))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H13: (eq T t +(THead (Bind Abbr) x4 x5))).(\lambda (H14: (subst0 i u x2 x4)).(\lambda (H15: +(subst0 (s (Bind Abbr) i) u x3 x5)).(eq_ind_r T (THead (Bind Abbr) x4 x5) +(\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 +(THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 +u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind Abst) x0 x1) +t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1 +z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: +T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall +(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda +(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T +(THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda +(y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) +z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: +B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: +T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) +y2) z1 z2)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: +T).(eq T (THead (Bind Abbr) x4 x5) (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 +(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) +(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2 +t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: +T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) +u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b +Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead +(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind +Abbr) x4 x5) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) +z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: +B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: +T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) +y2) z1 z2)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead +(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2 +t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: +T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) +u0) z1 t3))))))) x0 x1 x4 x5 (refl_equal T (THead (Bind Abst) x0 x1)) +(refl_equal T (THead (Bind Abbr) x4 x5)) (pr2_delta c0 d u i H1 u1 x2 H9 x4 +H14) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u +(S i) (getl_clear_bind b (CHead c0 (Bind b) u0) c0 u0 (clear_bind b c0 u0) +(CHead d (Bind Abbr) u) i H1) x1 x3 H10 x5 H15))))) t H13)))))) H12)) +(subst0_gen_head (Bind Abbr) u x2 x3 t i H11)) t1 H7)))))))))) H6)) (\lambda +(H6: (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) +(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda +(t3: T).(eq T t2 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) +t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda +(y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 +y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))))).(ex6_6_ind B T T T T +T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda +(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq +T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead +(Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda +(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) +(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (t3: T).(pr0 z1 t3))))))) (or3 (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 +t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind +Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) +u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b +Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda +(u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift +(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) +(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 +(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x0: B).(\lambda (x1: +T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: +T).(\lambda (H7: (not (eq B x0 Abst))).(\lambda (H8: (eq T t1 (THead (Bind +x0) x1 x2))).(\lambda (H9: (eq T t2 (THead (Bind x0) x4 (THead (Flat Appl) +(lift (S O) O x3) x5)))).(\lambda (H10: (pr0 u1 x3)).(\lambda (H11: (pr0 x1 +x4)).(\lambda (H12: (pr0 x2 x5)).(let H13 \def (eq_ind T t2 (\lambda (t3: +T).(subst0 i u t3 t)) H3 (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O +x3) x5)) H9) in (eq_ind_r T (THead (Bind x0) x1 x2) (\lambda (t3: T).(or3 +(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t (THead (Flat Appl) u2 +t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: +T).(\lambda (t4: T).(pr2 c0 t3 t4)))) (ex4_4 T T T T (\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t3 (THead (Bind +Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(t4: T).(eq T t (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall +(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda +(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t3 +(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) +y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 +c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_ind (ex2 T (\lambda +(u2: T).(eq T t (THead (Bind x0) u2 (THead (Flat Appl) (lift (S O) O x3) +x5)))) (\lambda (u2: T).(subst0 i u x4 u2))) (ex2 T (\lambda (t3: T).(eq T t +(THead (Bind x0) x4 t3))) (\lambda (t3: T).(subst0 (s (Bind x0) i) u (THead +(Flat Appl) (lift (S O) O x3) x5) t3))) (ex3_2 T T (\lambda (u2: T).(\lambda +(t3: T).(eq T t (THead (Bind x0) u2 t3)))) (\lambda (u2: T).(\lambda (_: +T).(subst0 i u x4 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind x0) +i) u (THead (Flat Appl) (lift (S O) O x3) x5) t3)))) (or3 (ex3_2 T T (\lambda +(u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 +(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind +Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall +(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda +(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T +(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda +(y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) +z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: +B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: +T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) +y2) z1 z2))))))))) (\lambda (H14: (ex2 T (\lambda (u2: T).(eq T t (THead +(Bind x0) u2 (THead (Flat Appl) (lift (S O) O x3) x5)))) (\lambda (u2: +T).(subst0 i u x4 u2)))).(ex2_ind T (\lambda (u2: T).(eq T t (THead (Bind x0) +u2 (THead (Flat Appl) (lift (S O) O x3) x5)))) (\lambda (u2: T).(subst0 i u +x4 u2)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead +(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) +(\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) +(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: +T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind +Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) +u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b +Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind +b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead +(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 +u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x6: T).(\lambda +(H15: (eq T t (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) +x5)))).(\lambda (H16: (subst0 i u x4 x6)).(eq_ind_r T (THead (Bind x0) x6 +(THead (Flat Appl) (lift (S O) O x3) x5)) (\lambda (t3: T).(or3 (ex3_2 T T +(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda +(t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 +x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda +(u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall +(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T +T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: +B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda +(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: +T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S +O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: +B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: +T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) +y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: +T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead +(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) +(\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) +(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: +T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) +x6 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind Abbr) u2 t3)))))) +(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 +u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: +T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) +(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: +B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda +(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: +T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) +O x3) x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) +z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: +B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: +T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) +y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B +b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind +b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead +(Flat Appl) (lift (S O) O x3) x5)) (THead (Bind b) y2 (THead (Flat Appl) +(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) +(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 +(CHead c0 (Bind b) y2) z1 z2))))))) x0 x1 x2 x5 x3 x6 H7 (refl_equal T (THead +(Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x6 (THead (Flat Appl) (lift +(S O) O x3) x5))) (pr2_free c0 u1 x3 H10) (pr2_delta c0 d u i H1 x1 x4 H11 x6 +H16) (pr2_free (CHead c0 (Bind x0) x6) x2 x5 H12))) t H15)))) H14)) (\lambda +(H14: (ex2 T (\lambda (t3: T).(eq T t (THead (Bind x0) x4 t3))) (\lambda (t3: +T).(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) O x3) x5) +t3)))).(ex2_ind T (\lambda (t3: T).(eq T t (THead (Bind x0) x4 t3))) (\lambda +(t3: T).(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) O x3) x5) +t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat +Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda +(_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T +(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T +(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind +Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) +u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b +Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind +b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead +(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 +u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x6: T).(\lambda +(H15: (eq T t (THead (Bind x0) x4 x6))).(\lambda (H16: (subst0 (s (Bind x0) +i) u (THead (Flat Appl) (lift (S O) O x3) x5) x6)).(eq_ind_r T (THead (Bind +x0) x4 x6) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: +T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: +T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) +x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 +z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 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y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: +T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) +y2) z1 z2)))))))))) (or3_ind (ex2 T (\lambda (u2: T).(eq T x6 (THead (Flat +Appl) u2 x5))) (\lambda (u2: T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) +u2))) (ex2 T (\lambda (t3: T).(eq T x6 (THead (Flat Appl) (lift (S O) O x3) +t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3))) +(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x6 (THead (Flat Appl) u2 +t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0) i) u (lift (S O) +O x3) u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind +x0) i)) u x5 t3)))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T +(THead (Bind x0) x4 x6) (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 +(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: 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B).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b +Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind +b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead +(Flat Appl) x7 x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) +z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: +B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: +T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) +y2) z1 z2))))))))) (\lambda (x8: T).(\lambda (H20: (eq T x7 (lift (S O) O +x8))).(\lambda (H21: (subst0 (minus (s (Bind x0) i) (S O)) u x3 x8)).(let H22 +\def (eq_ind nat (minus (s (Bind x0) i) (S O)) (\lambda (n: nat).(subst0 n u +x3 x8)) H21 i (s_arith1 x0 i)) in (eq_ind_r T (lift (S O) O x8) (\lambda (t3: +T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) +x4 (THead (Flat Appl) t3 x5)) (THead (Flat Appl) u2 t4)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 +(THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind +Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(t4: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) t3 x5)) (THead (Bind +Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) +u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b +Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind +b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead +(Flat Appl) t3 x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) +z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: +B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: +T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) +y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: +T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x8) x5)) (THead 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t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) +(s (Bind x0) i)) u x5 t3)))).(ex2_ind T (\lambda (t3: T).(eq T x6 (THead +(Flat Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) +(s (Bind x0) i)) u x5 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: +T).(eq T (THead (Bind x0) x4 x6) (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 +(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind +Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(t3: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind Abbr) u2 t3)))))) (\lambda +(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall +(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T +T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: +B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda +(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: +T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind b) y2 (THead +(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 +u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x7: T).(\lambda +(H18: (eq T x6 (THead (Flat Appl) (lift (S O) O x3) x7))).(\lambda (H19: +(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 x7)).(eq_ind_r T (THead (Flat +Appl) (lift (S O) O x3) x7) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: +T).(\lambda (t4: T).(eq T (THead (Bind x0) x4 t3) (THead (Flat Appl) u2 +t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: +T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T +(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T +(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) +x4 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall +(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda +(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T +(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda +(y2: T).(eq T (THead (Bind x0) x4 t3) (THead (Bind b) y2 (THead (Flat Appl) +(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) +(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 +(CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) +O x3) x7)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 +c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) +t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 +z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: +T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x7)) (THead +(Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 +(CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind +x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T +(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x7)) (THead (Bind b) +y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 +c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T +(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T +(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda +(y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x7)) +(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda +(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 +y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: +T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))) +x0 x1 x2 x7 x3 x4 H7 (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T +(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x7))) (pr2_free c0 +u1 x3 H10) (pr2_free c0 x1 x4 H11) (pr2_delta (CHead c0 (Bind x0) x4) d u (S +i) (getl_clear_bind x0 (CHead c0 (Bind x0) x4) c0 x4 (clear_bind x0 c0 x4) +(CHead d (Bind Abbr) u) i H1) x2 x5 H12 x7 H19))) x6 H18)))) H17)) (\lambda +(H17: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x6 (THead (Flat +Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0) i) u +(lift (S O) O x3) u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat +Appl) (s (Bind x0) i)) u x5 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda +(t3: T).(eq T x6 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: +T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) u2))) (\lambda (_: T).(\lambda +(t3: T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3))) (or3 (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 x6) (THead (Flat +Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda +(_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T +(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T +(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) +x4 x6) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall +(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda +(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T +(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda +(y2: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind b) y2 (THead (Flat Appl) +(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) +(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 +(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x7: T).(\lambda (x8: +T).(\lambda (H18: (eq T x6 (THead (Flat Appl) x7 x8))).(\lambda (H19: (subst0 +(s (Bind x0) i) u (lift (S O) O x3) x7)).(\lambda (H20: (subst0 (s (Flat +Appl) (s (Bind x0) i)) u x5 x8)).(eq_ind_r T (THead (Flat Appl) x7 x8) +(\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T +(THead (Bind x0) x4 t3) (THead (Flat Appl) u2 t4)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 +(THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind +Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(t4: T).(eq T (THead (Bind x0) x4 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda +(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall +(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T +T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: +B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda +(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: +T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 t3) (THead (Bind b) y2 (THead +(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 +u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (ex2_ind T (\lambda (t3: +T).(eq T x7 (lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus (s (Bind x0) +i) (S O)) u x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T +(THead (Bind x0) x4 (THead (Flat Appl) x7 x8)) (THead (Flat Appl) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda +(t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 +x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda +(u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) x7 x8)) +(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 +(CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind +x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T +(THead (Bind x0) x4 (THead (Flat Appl) x7 x8)) (THead (Bind b) y2 (THead +(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 +u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x9: T).(\lambda +(H21: (eq T x7 (lift (S O) O x9))).(\lambda (H22: (subst0 (minus (s (Bind x0) +i) (S O)) u x3 x9)).(let H23 \def (eq_ind nat (minus (s (Bind x0) i) (S O)) +(\lambda (n: nat).(subst0 n u x3 x9)) H22 i (s_arith1 x0 i)) in (eq_ind_r T +(lift (S O) O x9) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda +(t4: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) t3 x8)) (THead (Flat +Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda +(_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T +(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T +(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) +x4 (THead (Flat Appl) t3 x8)) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall +(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T +T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: +B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda +(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: +T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) t3 x8)) +(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda +(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 +y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: +T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 +z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T +(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x9) x8)) (THead (Flat +Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda +(_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T +(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T +(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) +x4 (THead (Flat Appl) (lift (S O) O x9) x8)) (THead (Bind Abbr) u2 t3)))))) +(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 +u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: +T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) +(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: +B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda +(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: +T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) +O x9) x8)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) +z2))))))))) (\lambda (_: B).(\lambda (_: 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T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: +B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda +(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: +T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S +O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: +B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: +T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) +y2) z1 z2)))))))))) (or3_ind (ex2 T (\lambda (u2: T).(eq T x7 (THead (Flat +Appl) u2 x5))) (\lambda (u2: T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) +u2))) (ex2 T (\lambda (t3: T).(eq T x7 (THead (Flat Appl) (lift (S O) O x3) +t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3))) +(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x7 (THead (Flat Appl) u2 +t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0) i) u (lift (S O) +O x3) u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind +x0) i)) u x5 t3)))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T +(THead (Bind x0) x6 x7) (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 +(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind +Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind Abbr) u2 t3)))))) (\lambda +(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall +(b: B).(\forall (u0: 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c0 +(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind +Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x8 x5)) (THead (Bind +Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) +u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b +Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind +b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead 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+(THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind +Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(t4: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t3 x5)) (THead (Bind +Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) +u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b +Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind +b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead +(Flat 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Appl) (lift (S O) O x9) x5)) (THead (Bind Abbr) u2 t3)))))) +(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 +u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: +T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) +(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: +B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda +(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: +T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) +O x9) x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) +z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: +B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: +T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) +y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B +b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind +b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead +(Flat Appl) (lift (S O) O x9) x5)) (THead (Bind b) y2 (THead (Flat Appl) +(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) +(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 +(CHead c0 (Bind b) y2) z1 z2))))))) x0 x1 x2 x5 x9 x6 H7 (refl_equal T (THead +(Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x6 (THead (Flat Appl) (lift +(S O) O x9) x5))) (pr2_delta c0 d u i H1 u1 x3 H10 x9 H23) (pr2_delta c0 d u +i H1 x1 x4 H11 x6 H16) (pr2_free (CHead c0 (Bind x0) x6) x2 x5 H12))) x8 +H21))))) (subst0_gen_lift_ge u x3 x8 (s (Bind x0) i) (S O) O H20 (le_n_S O i +(le_O_n i)))) x7 H19)))) H18)) (\lambda (H18: (ex2 T (\lambda (t3: T).(eq T +x7 (THead (Flat Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s +(Flat Appl) (s (Bind x0) i)) u x5 t3)))).(ex2_ind T (\lambda (t3: T).(eq T x7 +(THead (Flat Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat +Appl) (s (Bind x0) i)) u x5 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda +(t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Flat Appl) u2 t3)))) (\lambda +(u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: +T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 +x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda +(u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind Abbr) u2 +t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: +T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) +u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b +Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind +b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 x7) +(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda +(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 +y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: +T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 +z2))))))))) (\lambda (x8: T).(\lambda (H19: (eq T x7 (THead (Flat Appl) (lift +(S O) O x3) x8))).(\lambda (H20: (subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 +x8)).(eq_ind_r T (THead (Flat Appl) (lift (S O) O x3) x8) (\lambda (t3: +T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) +x6 t3) (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 +u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) +t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 +z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: +T).(eq T (THead (Bind x0) x6 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall +(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T +T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: +B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda +(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: +T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 t3) (THead (Bind b) y2 (THead +(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 +u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat +Appl) (lift (S O) O x3) x8)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 +(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind +Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8)) +(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 +(CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind +x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T +(THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8)) (THead (Bind b) +y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 +c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T +(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T +(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda +(y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8)) +(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda +(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 +y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: +T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))) +x0 x1 x2 x8 x3 x6 H7 (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T +(THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8))) (pr2_free c0 +u1 x3 H10) (pr2_delta c0 d u i H1 x1 x4 H11 x6 H16) (pr2_delta (CHead c0 +(Bind x0) x6) d u (S i) (getl_clear_bind x0 (CHead c0 (Bind x0) x6) c0 x6 +(clear_bind x0 c0 x6) (CHead d (Bind Abbr) u) i H1) x2 x5 H12 x8 H20))) x7 +H19)))) H18)) (\lambda (H18: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq +T x7 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s +(Bind x0) i) u (lift (S O) O x3) u2))) (\lambda (_: T).(\lambda (t3: +T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3))))).(ex3_2_ind T T +(\lambda (u2: T).(\lambda (t3: T).(eq T x7 (THead (Flat Appl) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) +u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind x0) +i)) u x5 t3))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead +(Bind x0) x6 x7) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: +T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) +x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 +z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: +T).(eq T (THead (Bind x0) x6 x7) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall +(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T +T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: +B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda +(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: +T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind b) y2 (THead +(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 +u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x8: T).(\lambda +(x9: T).(\lambda (H19: (eq T x7 (THead (Flat Appl) x8 x9))).(\lambda (H20: +(subst0 (s (Bind x0) i) u (lift (S O) O x3) x8)).(\lambda (H21: (subst0 (s +(Flat Appl) (s (Bind x0) i)) u x5 x9)).(eq_ind_r T (THead (Flat Appl) x8 x9) +(\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T +(THead (Bind x0) x6 t3) (THead (Flat Appl) u2 t4)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 +(THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind +Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(t4: T).(eq T (THead (Bind x0) x6 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda +(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall +(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T +T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: +B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda +(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: +T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 t3) (THead (Bind b) y2 (THead +(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 +u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (ex2_ind T (\lambda (t3: +T).(eq T x8 (lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus (s (Bind x0) +i) (S O)) u x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T +(THead (Bind x0) x6 (THead (Flat Appl) x8 x9)) (THead (Flat Appl) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda +(t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 +x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda +(u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x8 x9)) +(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 +(CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind +x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T +(THead (Bind x0) x6 (THead (Flat Appl) x8 x9)) (THead (Bind b) y2 (THead +(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 +u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x10: T).(\lambda +(H22: (eq T x8 (lift (S O) O x10))).(\lambda (H23: (subst0 (minus (s (Bind +x0) i) (S O)) u x3 x10)).(let H24 \def (eq_ind nat (minus (s (Bind x0) i) (S +O)) (\lambda (n: nat).(subst0 n u x3 x10)) H23 i (s_arith1 x0 i)) in +(eq_ind_r T (lift (S O) O x10) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: +T).(\lambda (t4: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t3 x9)) +(THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 +u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) +(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: +T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) +x6 (THead (Flat Appl) t3 x9)) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall +(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T +T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: +B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda +(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: +T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t3 x9)) +(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda +(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 +y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: +T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 +z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T +(THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x10) x9)) (THead (Flat +Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda +(_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T +(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T +(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) +x6 (THead (Flat Appl) (lift (S O) O x10) x9)) (THead (Bind Abbr) u2 t3)))))) +(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 +u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: +T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) +(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: +B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda +(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: +T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) +O x10) x9)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) +z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: +B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: +T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) +y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B +b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind +b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead +(Flat Appl) (lift (S O) O x10) x9)) (THead (Bind b) y2 (THead (Flat Appl) +(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) +(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 +(CHead c0 (Bind b) y2) z1 z2))))))) x0 x1 x2 x9 x10 x6 H7 (refl_equal T +(THead (Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x6 (THead (Flat Appl) +(lift (S O) O x10) x9))) (pr2_delta c0 d u i H1 u1 x3 H10 x10 H24) (pr2_delta +c0 d u i H1 x1 x4 H11 x6 H16) (pr2_delta (CHead c0 (Bind x0) x6) d u (S i) +(getl_clear_bind x0 (CHead c0 (Bind x0) x6) c0 x6 (clear_bind x0 c0 x6) +(CHead d (Bind Abbr) u) i H1) x2 x5 H12 x9 H21))) x8 H22))))) +(subst0_gen_lift_ge u x3 x8 (s (Bind x0) i) (S O) O H20 (le_n_S O i (le_O_n +i)))) x7 H19)))))) H18)) (subst0_gen_head (Flat Appl) u (lift (S O) O x3) x5 +x7 (s (Bind x0) i) H17)) t H15)))))) H14)) (subst0_gen_head (Bind x0) u x4 +(THead (Flat Appl) (lift (S O) O x3) x5) t i H13)) t1 H8)))))))))))))) H6)) +(pr0_gen_appl u1 t1 t2 H5)))))))))))))) c y x H0))) H))))). +(* COMMENTS +Initial nodes: 38859 +END *) + +theorem pr2_gen_abbr: + \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c +(THead (Bind Abbr) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda +(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: +T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(or3 (\forall (b: +B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2))) (ex2 T (\lambda (u: +T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t2))) (ex3_2 T +T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) +(\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: +T).(pr2 (CHead c (Bind Abbr) u1) z t2)))))))) (\forall (b: B).(\forall (u: +T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))))))) +\def + \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda +(H: (pr2 c (THead (Bind Abbr) u1 t1) x)).(insert_eq T (THead (Bind Abbr) u1 +t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: +T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(or3 +(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2))) (ex2 T +(\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 +t2))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) +u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: +T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t2)))))))) (\forall (b: +B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))) +(\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0: C).(\lambda +(t: T).(\lambda (t0: T).((eq T t (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T +T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda +(t2: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 +t2))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind +Abbr) u) t1 t2))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 +(Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) +(\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t2)))))))) +(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O +t0))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H1: +(pr0 t0 t2)).(\lambda (H2: (eq T t0 (THead (Bind Abbr) u1 t1))).(let H3 \def +(eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Bind Abbr) u1 t1) H2) in +(or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind +Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: +T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) +(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S O) O t2)) (or +(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 +t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind +b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead +c0 (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 +(CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 +z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z +t3)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 +(lift (S O) O t2))))) (\lambda (H4: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: +T).(eq T t2 (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: +T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T +(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 +t3)))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead +(Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda +(u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) +(\lambda (y0: T).(subst0 O u2 y0 t3)))))) (or (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 +(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 t3))) (ex2 T +(\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind Abbr) u) t1 +t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) +u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: +T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: +B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O t2))))) +(\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T t2 (THead (Bind Abbr) +x0 x1))).(\lambda (H6: (pr0 u1 x0)).(\lambda (H_x: (or (pr0 t1 x1) (ex2 T +(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O x0 y0 +x1))))).(or_ind (pr0 t1 x1) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda +(y0: T).(subst0 O x0 y0 x1))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: +T).(eq T t2 (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: +T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: +B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 t3))) (ex2 T (\lambda (u: +T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind Abbr) u) t1 t3))) (ex3_2 +T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) +(\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: +T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u: +T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O t2))))) (\lambda (H7: (pr0 t1 +x1)).(eq_ind_r T (THead (Bind Abbr) x0 x1) (\lambda (t: T).(or (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda +(t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 +t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind +Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 +(Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) +(\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) +(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O +t)))))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead +(Bind Abbr) x0 x1) (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: +T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: +B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 t3))) (ex2 T (\lambda (u: +T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind Abbr) u) t1 t3))) (ex3_2 +T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) +(\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: +T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u: +T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O (THead (Bind Abbr) x0 x1))))) +(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) +x0 x1) (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 +u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u: +T).(pr2 (CHead c0 (Bind b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) +(\lambda (u: T).(pr2 (CHead c0 (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda +(y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: +T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead +c0 (Bind Abbr) u1) z t3))))))) x0 x1 (refl_equal T (THead (Bind Abbr) x0 x1)) +(pr2_free c0 u1 x0 H6) (or3_intro0 (\forall (b: B).(\forall (u: T).(pr2 +(CHead c0 (Bind b) u) t1 x1))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda +(u: T).(pr2 (CHead c0 (Bind Abbr) u) t1 x1))) (ex3_2 T T (\lambda (y0: +T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: +T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead +c0 (Bind Abbr) u1) z x1)))) (\lambda (b: B).(\lambda (u: T).(pr2_free (CHead +c0 (Bind b) u) t1 x1 H7)))))) t2 H5)) (\lambda (H_x0: (ex2 T (\lambda (y0: +T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O x0 y0 x1)))).(ex2_ind T (\lambda +(y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O x0 y0 x1)) (or (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda +(t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 +t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind +Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 +(Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) +(\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) +(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O +t2))))) (\lambda (x2: T).(\lambda (H7: (pr0 t1 x2)).(\lambda (H8: (subst0 O +x0 x2 x1)).(eq_ind_r T (THead (Bind Abbr) x0 x1) (\lambda (t: T).(or (ex3_2 T +T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) +(\lambda (u2: T).(\lambda 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(\lambda (_: +T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3))))))) x0 x3 H13 +(pr2_free c0 u1 x0 H8) (or3_intro2 (\forall (b: B).(\forall (u0: T).(pr2 +(CHead c0 (Bind b) u0) t1 x3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda +(u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 x3))) (ex3_2 T T (\lambda (y0: +T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: +T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead +c0 (Bind Abbr) u1) z x3)))) (ex3_2_intro T T (\lambda (y0: T).(\lambda (_: +T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: +T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) +u1) z x3))) x4 x1 (pr2_delta (CHead c0 (Bind Abbr) u1) c0 u1 O (getl_refl +Abbr c0 u1) t1 x2 H9 x4 H15) H16 (pr2_delta (CHead c0 (Bind Abbr) u1) d u (S +i) (getl_head (Bind Abbr) i c0 (CHead d (Bind Abbr) u) H1 u1) x1 x1 (pr0_refl +x1) x3 H14)))))))) (pr0_subst0_back x0 x2 x1 O H10 u1 H8))))) H12)) (\lambda 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B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 +(lift (S O) O t))))) (\lambda (x3: T).(\lambda (x4: T).(\lambda (H13: (eq T t +(THead (Bind Abbr) x3 x4))).(\lambda (H14: (subst0 i u x0 x3)).(\lambda (H15: +(subst0 (s (Bind Abbr) i) u x1 x4)).(ex2_ind T (\lambda (t3: T).(subst0 O u1 +x2 t3)) (\lambda (t3: T).(pr0 t3 x1)) (or (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 +(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T +(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) +t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind +Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: +T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: +B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))) +(\lambda (x5: T).(\lambda (H16: (subst0 O u1 x2 x5)).(\lambda (H17: (pr0 x5 +x1)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead +(Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) +(\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 +(CHead c0 (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda +(u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y0: +T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: +T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead +c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead +c0 (Bind b) u0) t1 (lift (S O) O t)))) (ex3_2_intro T T (\lambda (u2: +T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 +(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T +(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) +t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind +Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: +T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3))))))) x3 x4 H13 +(pr2_delta c0 d u i H1 u1 x0 H8 x3 H14) (or3_intro2 (\forall (b: B).(\forall +(u0: T).(pr2 (CHead c0 (Bind b) u0) t1 x4))) (ex2 T (\lambda (u0: T).(pr0 u1 +u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 x4))) (ex3_2 T T +(\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) +(\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: +T).(pr2 (CHead c0 (Bind Abbr) u1) z x4)))) (ex3_2_intro T T (\lambda (y0: +T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: +T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead +c0 (Bind Abbr) u1) z x4))) x5 x1 (pr2_delta (CHead c0 (Bind Abbr) u1) c0 u1 O +(getl_refl Abbr c0 u1) t1 x2 H9 x5 H16) H17 (pr2_delta (CHead c0 (Bind Abbr) +u1) d u (S i) (getl_head (Bind Abbr) i c0 (CHead d (Bind Abbr) u) H1 u1) x1 +x1 (pr0_refl x1) x4 H15)))))))) (pr0_subst0_back x0 x2 x1 O H10 u1 H8))))))) +H12)) (subst0_gen_head (Bind Abbr) u x0 x1 t i H11)))))) H_x0)) H_x)))))) +H6)) (\lambda (H6: (pr0 t1 (lift (S O) O t2))).(or_intror (ex3_2 T T (\lambda +(u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 +(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T +(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) +t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind +Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: +T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: +B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))) +(\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) +(getl_head (Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 (lift (S O) O t2) +H6 (lift (S O) O t) (subst0_lift_ge_S t2 t u i H3 O (le_O_n i))))))) +(pr0_gen_abbr u1 t1 t2 H5)))))))))))))) c y x H0))) H))))). +(* COMMENTS +Initial nodes: 11671 +END *) + +theorem pr2_gen_void: + \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c +(THead (Bind Void) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda +(t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: +T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall +(u: T).(pr2 (CHead c (Bind b) u) t1 t2)))))) (\forall (b: B).(\forall (u: +T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))))))) +\def + \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda +(H: (pr2 c (THead (Bind Void) u1 t1) x)).(insert_eq T (THead (Bind Void) u1 +t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: +T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall +(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2)))))) (\forall (b: +B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))) +(\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0: C).(\lambda +(t: T).(\lambda (t0: T).((eq T t (THead (Bind Void) u1 t1)) \to (or (ex3_2 T +T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Void) u2 t2)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda +(t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 +t2)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 (lift +(S O) O t0))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: +T).(\lambda (H1: (pr0 t0 t2)).(\lambda (H2: (eq T t0 (THead (Bind Void) u1 +t1))).(let H3 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Bind +Void) u1 t1) H2) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq +T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 +u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O +t2)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind +Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda +(_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind +b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) +t1 (lift (S O) O t2))))) (\lambda (H4: (ex3_2 T T (\lambda (u2: T).(\lambda +(t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: +T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind +T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: +T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 +(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 +u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 +(CHead c0 (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead +c0 (Bind b) u) t1 (lift (S O) O t2))))) (\lambda (x0: T).(\lambda (x1: +T).(\lambda (H5: (eq T t2 (THead (Bind Void) x0 x1))).(\lambda (H6: (pr0 u1 +x0)).(\lambda (H7: (pr0 t1 x1)).(eq_ind_r T (THead (Bind Void) x0 x1) +(\lambda (t: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t +(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 +u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 +(CHead c0 (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead +c0 (Bind b) u) t1 (lift (S O) O t)))))) (or_introl (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T (THead (Bind Void) x0 x1) (THead (Bind Void) u2 +t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) +u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 +(lift (S O) O (THead (Bind Void) x0 x1))))) (ex3_2_intro T T (\lambda (u2: +T).(\lambda (t3: T).(eq T (THead (Bind Void) x0 x1) (THead (Bind Void) u2 +t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) +u) t1 t3))))) x0 x1 (refl_equal T (THead (Bind Void) x0 x1)) (pr2_free c0 u1 +x0 H6) (\lambda (b: B).(\lambda (u: T).(pr2_free (CHead c0 (Bind b) u) t1 x1 +H7))))) t2 H5)))))) H4)) (\lambda (H4: (pr0 t1 (lift (S O) O t2))).(or_intror +(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 +t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) +u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 +(lift (S O) O t2)))) (\lambda (b: B).(\lambda (u: T).(pr2_free (CHead c0 +(Bind b) u) t1 (lift (S O) O t2) H4))))) (pr0_gen_void u1 t1 t2 H3)))))))) +(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda +(H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t0: T).(\lambda (t2: +T).(\lambda (H2: (pr0 t0 t2)).(\lambda (t: T).(\lambda (H3: (subst0 i u t2 +t)).(\lambda (H4: (eq T t0 (THead (Bind Void) u1 t1))).(let H5 \def (eq_ind T +t0 (\lambda (t3: T).(pr0 t3 t2)) H2 (THead (Bind Void) u1 t1) H4) in (or_ind +(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 +t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2)) (or (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda +(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 +t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 +(lift (S O) O t))))) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: +T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: +T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind +T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: +T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t +(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 +u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 +(CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 +(CHead c0 (Bind b) u0) t1 (lift (S O) O t))))) (\lambda (x0: T).(\lambda (x1: +T).(\lambda (H7: (eq T t2 (THead (Bind Void) x0 x1))).(\lambda (H8: (pr0 u1 +x0)).(\lambda (H9: (pr0 t1 x1)).(let H10 \def (eq_ind T t2 (\lambda (t3: +T).(subst0 i u t3 t)) H3 (THead (Bind Void) x0 x1) H7) in (or3_ind (ex2 T +(\lambda (u2: T).(eq T t (THead (Bind Void) u2 x1))) (\lambda (u2: T).(subst0 +i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T t (THead (Bind Void) x0 t3))) +(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3))) (ex3_2 T T (\lambda +(u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3: +T).(subst0 (s (Bind Void) i) u x1 t3)))) (or (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall +(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: +B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))) +(\lambda (H11: (ex2 T (\lambda (u2: T).(eq T t (THead (Bind Void) u2 x1))) +(\lambda (u2: T).(subst0 i u x0 u2)))).(ex2_ind T (\lambda (u2: T).(eq T t +(THead (Bind Void) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2)) (or (ex3_2 T +T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda +(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 +t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 +(lift (S O) O t))))) (\lambda (x2: T).(\lambda (H12: (eq T t (THead (Bind +Void) x2 x1))).(\lambda (H13: (subst0 i u x0 x2)).(or_introl (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda +(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 +t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 +(lift (S O) O t)))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T +t (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 +u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 +(CHead c0 (Bind b) u0) t1 t3))))) x2 x1 H12 (pr2_delta c0 d u i H1 u1 x0 H8 +x2 H13) (\lambda (b: B).(\lambda (u0: T).(pr2_free (CHead c0 (Bind b) u0) t1 +x1 H9)))))))) H11)) (\lambda (H11: (ex2 T (\lambda (t3: T).(eq T t (THead +(Bind Void) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 +t3)))).(ex2_ind T (\lambda (t3: T).(eq T t (THead (Bind Void) x0 t3))) +(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3)) (or (ex3_2 T T (\lambda +(u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall +(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: +B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))) +(\lambda (x2: T).(\lambda (H12: (eq T t (THead (Bind Void) x0 x2))).(\lambda +(H13: (subst0 (s (Bind Void) i) u x1 x2)).(or_introl (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall +(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: +B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))) +(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) +u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) +u0) t1 t3))))) x0 x2 H12 (pr2_free c0 u1 x0 H8) (\lambda (b: B).(\lambda (u0: +T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) (getl_head (Bind b) i c0 +(CHead d (Bind Abbr) u) H1 u0) t1 x1 H9 x2 H13)))))))) H11)) (\lambda (H11: +(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 +t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: +T).(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3))))).(ex3_2_ind T T +(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: +T).(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3))) (or (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda +(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 +t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 +(lift (S O) O t))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (eq T t +(THead (Bind Void) x2 x3))).(\lambda (H13: (subst0 i u x0 x2)).(\lambda (H14: +(subst0 (s (Bind Void) i) u x1 x3)).(or_introl (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall +(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: +B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))) +(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) +u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) +u0) t1 t3))))) x2 x3 H12 (pr2_delta c0 d u i H1 u1 x0 H8 x2 H13) (\lambda (b: +B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) (getl_head +(Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 x1 H9 x3 H14)))))))))) H11)) +(subst0_gen_head (Bind Void) u x0 x1 t i H10)))))))) H6)) (\lambda (H6: (pr0 +t1 (lift (S O) O t2))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: +T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 +c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: +T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0: +T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))) (\lambda (b: +B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) (getl_head +(Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 (lift (S O) O t2) H6 (lift (S +O) O t) (subst0_lift_ge_S t2 t u i H3 O (le_O_n i))))))) (pr0_gen_void u1 t1 +t2 H5)))))))))))))) c y x H0))) H))))). +(* COMMENTS +Initial nodes: 3467 +END *) + +theorem pr2_gen_lift: + \forall (c: C).(\forall (t1: T).(\forall (x: T).(\forall (h: nat).(\forall +(d: nat).((pr2 c (lift h d t1) x) \to (\forall (e: C).((drop h d c e) \to +(ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr2 e t1 +t2)))))))))) +\def + \lambda (c: C).(\lambda (t1: T).(\lambda (x: T).(\lambda (h: nat).(\lambda +(d: nat).(\lambda (H: (pr2 c (lift h d t1) x)).(insert_eq T (lift h d t1) +(\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(\forall (e: C).((drop h d c e) +\to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr2 e +t1 t2)))))) (\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0: +C).(\lambda (t: T).(\lambda (t0: T).((eq T t (lift h d t1)) \to (\forall (e: +C).((drop h d c0 e) \to (ex2 T (\lambda (t2: T).(eq T t0 (lift h d t2))) +(\lambda (t2: T).(pr2 e t1 t2))))))))) (\lambda (c0: C).(\lambda (t0: +T).(\lambda (t2: T).(\lambda (H1: (pr0 t0 t2)).(\lambda (H2: (eq T t0 (lift h +d t1))).(\lambda (e: C).(\lambda (_: (drop h d c0 e)).(let H4 \def (eq_ind T +t0 (\lambda (t: T).(pr0 t t2)) H1 (lift h d t1) H2) in (ex2_ind T (\lambda +(t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(pr0 t1 t3)) (ex2 T +(\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))) +(\lambda (x0: T).(\lambda (H5: (eq T t2 (lift h d x0))).(\lambda (H6: (pr0 t1 +x0)).(eq_ind_r T (lift h d x0) (\lambda (t: T).(ex2 T (\lambda (t3: T).(eq T +t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)))) (ex_intro2 T (\lambda +(t3: T).(eq T (lift h d x0) (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)) +x0 (refl_equal T (lift h d x0)) (pr2_free e t1 x0 H6)) t2 H5)))) +(pr0_gen_lift t1 t2 h d H4)))))))))) (\lambda (c0: C).(\lambda (d0: +C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d0 (Bind +Abbr) u))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H2: (pr0 t0 +t2)).(\lambda (t: T).(\lambda (H3: (subst0 i u t2 t)).(\lambda (H4: (eq T t0 +(lift h d t1))).(\lambda (e: C).(\lambda (H5: (drop h d c0 e)).(let H6 \def +(eq_ind T t0 (\lambda (t3: T).(pr0 t3 t2)) H2 (lift h d t1) H4) in (ex2_ind T +(\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(pr0 t1 t3)) (ex2 +T (\lambda (t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))) +(\lambda (x0: T).(\lambda (H7: (eq T t2 (lift h d x0))).(\lambda (H8: (pr0 t1 +x0)).(let H9 \def (eq_ind T t2 (\lambda (t3: T).(subst0 i u t3 t)) H3 (lift h +d x0) H7) in (lt_le_e i d (ex2 T (\lambda (t3: T).(eq T t (lift h d t3))) +(\lambda (t3: T).(pr2 e t1 t3))) (\lambda (H10: (lt i d)).(let H11 \def +(eq_ind nat d (\lambda (n: nat).(subst0 i u (lift h n x0) t)) H9 (S (plus i +(minus d (S i)))) (lt_plus_minus i d H10)) in (let H12 \def (eq_ind nat d +(\lambda (n: nat).(drop h n c0 e)) H5 (S (plus i (minus d (S i)))) +(lt_plus_minus i d H10)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: +C).(eq T u (lift h (minus d (S i)) v)))) (\lambda (v: T).(\lambda (e0: +C).(getl i e (CHead e0 (Bind Abbr) v)))) (\lambda (_: T).(\lambda (e0: +C).(drop h (minus d (S i)) d0 e0))) (ex2 T (\lambda (t3: T).(eq T t (lift h d +t3))) (\lambda (t3: T).(pr2 e t1 t3))) (\lambda (x1: T).(\lambda (x2: +C).(\lambda (H13: (eq T u (lift h (minus d (S i)) x1))).(\lambda (H14: (getl +i e (CHead x2 (Bind Abbr) x1))).(\lambda (_: (drop h (minus d (S i)) d0 +x2)).(let H16 \def (eq_ind T u (\lambda (t3: T).(subst0 i t3 (lift h (S (plus +i (minus d (S i)))) x0) t)) H11 (lift h (minus d (S i)) x1) H13) in (ex2_ind +T (\lambda (t3: T).(eq T t (lift h (S (plus i (minus d (S i)))) t3))) +(\lambda (t3: T).(subst0 i x1 x0 t3)) (ex2 T (\lambda (t3: T).(eq T t (lift h +d t3))) (\lambda (t3: T).(pr2 e t1 t3))) (\lambda (x3: T).(\lambda (H17: (eq +T t (lift h (S (plus i (minus d (S i)))) x3))).(\lambda (H18: (subst0 i x1 x0 +x3)).(let H19 \def (eq_ind_r nat (S (plus i (minus d (S i)))) (\lambda (n: +nat).(eq T t (lift h n x3))) H17 d (lt_plus_minus i d H10)) in (ex_intro2 T +(\lambda (t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)) x3 +H19 (pr2_delta e x2 x1 i H14 t1 x0 H8 x3 H18)))))) (subst0_gen_lift_lt x1 x0 +t i h (minus d (S i)) H16)))))))) (getl_drop_conf_lt Abbr c0 d0 u i H1 e h +(minus d (S i)) H12))))) (\lambda (H10: (le d i)).(lt_le_e i (plus d h) (ex2 +T (\lambda (t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))) +(\lambda (H11: (lt i (plus d h))).(subst0_gen_lift_false x0 u t h d i H10 H11 +H9 (ex2 T (\lambda (t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 +t3))))) (\lambda (H11: (le (plus d h) i)).(ex2_ind T (\lambda (t3: T).(eq T t +(lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u x0 t3)) (ex2 T +(\lambda (t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))) +(\lambda (x1: T).(\lambda (H12: (eq T t (lift h d x1))).(\lambda (H13: +(subst0 (minus i h) u x0 x1)).(ex_intro2 T (\lambda (t3: T).(eq T t (lift h d +t3))) (\lambda (t3: T).(pr2 e t1 t3)) x1 H12 (pr2_delta e d0 u (minus i h) +(getl_drop_conf_ge i (CHead d0 (Bind Abbr) u) c0 H1 e h d H5 H11) t1 x0 H8 x1 +H13))))) (subst0_gen_lift_ge u x0 t i h d H9 H11)))))))))) (pr0_gen_lift t1 +t2 h d H6)))))))))))))))) c y x H0))) H)))))). +(* COMMENTS +Initial nodes: 1579 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr2/pr2.ma b/matita/matita/contribs/lambdadelta/basic_1/pr2/pr2.ma new file mode 100644 index 000000000..f8df4e9e2 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/pr2/pr2.ma @@ -0,0 +1,258 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/pr2/defs.ma". + +include "Basic-1/pr0/pr0.ma". + +include "Basic-1/getl/props.ma". + +theorem pr2_confluence__pr2_free_free: + \forall (c: C).(\forall (t0: T).(\forall (t1: T).(\forall (t2: T).((pr0 t0 +t1) \to ((pr0 t0 t2) \to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: +T).(pr2 c t2 t)))))))) +\def + \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda +(H: (pr0 t0 t1)).(\lambda (H0: (pr0 t0 t2)).(ex2_ind T (\lambda (t: T).(pr0 +t2 t)) (\lambda (t: T).(pr0 t1 t)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) +(\lambda (t: T).(pr2 c t2 t))) (\lambda (x: T).(\lambda (H1: (pr0 t2 +x)).(\lambda (H2: (pr0 t1 x)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) +(\lambda (t: T).(pr2 c t2 t)) x (pr2_free c t1 x H2) (pr2_free c t2 x H1))))) +(pr0_confluence t0 t2 H0 t1 H))))))). +(* COMMENTS +Initial nodes: 135 +END *) + +theorem pr2_confluence__pr2_free_delta: + \forall (c: C).(\forall (d: C).(\forall (t0: T).(\forall (t1: T).(\forall +(t2: T).(\forall (t4: T).(\forall (u: T).(\forall (i: nat).((pr0 t0 t1) \to +((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t4) \to ((subst0 i u t4 t2) +\to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 +t)))))))))))))) +\def + \lambda (c: C).(\lambda (d: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda +(t2: T).(\lambda (t4: T).(\lambda (u: T).(\lambda (i: nat).(\lambda (H: (pr0 +t0 t1)).(\lambda (H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H1: (pr0 +t0 t4)).(\lambda (H2: (subst0 i u t4 t2)).(ex2_ind T (\lambda (t: T).(pr0 t4 +t)) (\lambda (t: T).(pr0 t1 t)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda +(t: T).(pr2 c t2 t))) (\lambda (x: T).(\lambda (H3: (pr0 t4 x)).(\lambda (H4: +(pr0 t1 x)).(or_ind (pr0 t2 x) (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda +(w2: T).(subst0 i u x w2))) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: +T).(pr2 c t2 t))) (\lambda (H5: (pr0 t2 x)).(ex_intro2 T (\lambda (t: T).(pr2 +c t1 t)) (\lambda (t: T).(pr2 c t2 t)) x (pr2_free c t1 x H4) (pr2_free c t2 +x H5))) (\lambda (H5: (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: +T).(subst0 i u x w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t2 w2)) (\lambda +(w2: T).(subst0 i u x w2)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: +T).(pr2 c t2 t))) (\lambda (x0: T).(\lambda (H6: (pr0 t2 x0)).(\lambda (H7: +(subst0 i u x x0)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: +T).(pr2 c t2 t)) x0 (pr2_delta c d u i H0 t1 x H4 x0 H7) (pr2_free c t2 x0 +H6))))) H5)) (pr0_subst0 t4 x H3 u t2 i H2 u (pr0_refl u)))))) +(pr0_confluence t0 t4 H1 t1 H))))))))))))). +(* COMMENTS +Initial nodes: 403 +END *) + +theorem pr2_confluence__pr2_delta_delta: + \forall (c: C).(\forall (d: C).(\forall (d0: C).(\forall (t0: T).(\forall +(t1: T).(\forall (t2: T).(\forall (t3: T).(\forall (t4: T).(\forall (u: +T).(\forall (u0: T).(\forall (i: nat).(\forall (i0: nat).((getl i c (CHead d +(Bind Abbr) u)) \to ((pr0 t0 t3) \to ((subst0 i u t3 t1) \to ((getl i0 c +(CHead d0 (Bind Abbr) u0)) \to ((pr0 t0 t4) \to ((subst0 i0 u0 t4 t2) \to +(ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 +t)))))))))))))))))))) +\def + \lambda (c: C).(\lambda (d: C).(\lambda (d0: C).(\lambda (t0: T).(\lambda +(t1: T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (u: +T).(\lambda (u0: T).(\lambda (i: nat).(\lambda (i0: nat).(\lambda (H: (getl i +c (CHead d (Bind Abbr) u))).(\lambda (H0: (pr0 t0 t3)).(\lambda (H1: (subst0 +i u t3 t1)).(\lambda (H2: (getl i0 c (CHead d0 (Bind Abbr) u0))).(\lambda +(H3: (pr0 t0 t4)).(\lambda (H4: (subst0 i0 u0 t4 t2)).(ex2_ind T (\lambda (t: +T).(pr0 t4 t)) (\lambda (t: T).(pr0 t3 t)) (ex2 T (\lambda (t: T).(pr2 c t1 +t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (x: T).(\lambda (H5: (pr0 t4 +x)).(\lambda (H6: (pr0 t3 x)).(or_ind (pr0 t1 x) (ex2 T (\lambda (w2: T).(pr0 +t1 w2)) (\lambda (w2: T).(subst0 i u x w2))) (ex2 T (\lambda (t: T).(pr2 c t1 +t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (H7: (pr0 t1 x)).(or_ind (pr0 t2 +x) (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: T).(subst0 i0 u0 x +w2))) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) +(\lambda (H8: (pr0 t2 x)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda +(t: T).(pr2 c t2 t)) x (pr2_free c t1 x H7) (pr2_free c t2 x H8))) (\lambda +(H8: (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: T).(subst0 i0 u0 x +w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: T).(subst0 i0 +u0 x w2)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) +(\lambda (x0: T).(\lambda (H9: (pr0 t2 x0)).(\lambda (H10: (subst0 i0 u0 x +x0)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t)) +x0 (pr2_delta c d0 u0 i0 H2 t1 x H7 x0 H10) (pr2_free c t2 x0 H9))))) H8)) +(pr0_subst0 t4 x H5 u0 t2 i0 H4 u0 (pr0_refl u0)))) (\lambda (H7: (ex2 T +(\lambda (w2: T).(pr0 t1 w2)) (\lambda (w2: T).(subst0 i u x w2)))).(ex2_ind +T (\lambda (w2: T).(pr0 t1 w2)) (\lambda (w2: T).(subst0 i u x w2)) (ex2 T +(\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (x0: +T).(\lambda (H8: (pr0 t1 x0)).(\lambda (H9: (subst0 i u x x0)).(or_ind (pr0 +t2 x) (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: T).(subst0 i0 u0 x +w2))) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) +(\lambda (H10: (pr0 t2 x)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) +(\lambda (t: T).(pr2 c t2 t)) x0 (pr2_free c t1 x0 H8) (pr2_delta c d u i H +t2 x H10 x0 H9))) (\lambda (H10: (ex2 T (\lambda (w2: T).(pr0 t2 w2)) +(\lambda (w2: T).(subst0 i0 u0 x w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t2 +w2)) (\lambda (w2: T).(subst0 i0 u0 x w2)) (ex2 T (\lambda (t: T).(pr2 c t1 +t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (x1: T).(\lambda (H11: (pr0 t2 +x1)).(\lambda (H12: (subst0 i0 u0 x x1)).(neq_eq_e i i0 (ex2 T (\lambda (t: +T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (H13: (not (eq nat i +i0))).(ex2_ind T (\lambda (t: T).(subst0 i u x1 t)) (\lambda (t: T).(subst0 +i0 u0 x0 t)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 +t))) (\lambda (x2: T).(\lambda (H14: (subst0 i u x1 x2)).(\lambda (H15: +(subst0 i0 u0 x0 x2)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: +T).(pr2 c t2 t)) x2 (pr2_delta c d0 u0 i0 H2 t1 x0 H8 x2 H15) (pr2_delta c d +u i H t2 x1 H11 x2 H14))))) (subst0_confluence_neq x x1 u0 i0 H12 x0 u i H9 +(sym_not_eq nat i i0 H13)))) (\lambda (H13: (eq nat i i0)).(let H14 \def +(eq_ind_r nat i0 (\lambda (n: nat).(subst0 n u0 x x1)) H12 i H13) in (let H15 +\def (eq_ind_r nat i0 (\lambda (n: nat).(getl n c (CHead d0 (Bind Abbr) u0))) +H2 i H13) in (let H16 \def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c0: +C).(getl i c c0)) H (CHead d0 (Bind Abbr) u0) (getl_mono c (CHead d (Bind +Abbr) u) i H (CHead d0 (Bind Abbr) u0) H15)) in (let H17 \def (f_equal C C +(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) +\Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abbr) u) +(CHead d0 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) i H (CHead d0 +(Bind Abbr) u0) H15)) in ((let H18 \def (f_equal C T (\lambda (e: C).(match e +in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) +\Rightarrow t])) (CHead d (Bind Abbr) u) (CHead d0 (Bind Abbr) u0) (getl_mono +c (CHead d (Bind Abbr) u) i H (CHead d0 (Bind Abbr) u0) H15)) in (\lambda +(H19: (eq C d d0)).(let H20 \def (eq_ind_r T u0 (\lambda (t: T).(subst0 i t x +x1)) H14 u H18) in (let H21 \def (eq_ind_r T u0 (\lambda (t: T).(getl i c +(CHead d0 (Bind Abbr) t))) H16 u H18) in (let H22 \def (eq_ind_r C d0 +(\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr) u))) H21 d H19) in (or4_ind +(eq T x1 x0) (ex2 T (\lambda (t: T).(subst0 i u x1 t)) (\lambda (t: +T).(subst0 i u x0 t))) (subst0 i u x1 x0) (subst0 i u x0 x1) (ex2 T (\lambda +(t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (H23: (eq T x1 +x0)).(let H24 \def (eq_ind T x1 (\lambda (t: T).(pr0 t2 t)) H11 x0 H23) in +(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t)) x0 +(pr2_free c t1 x0 H8) (pr2_free c t2 x0 H24)))) (\lambda (H23: (ex2 T +(\lambda (t: T).(subst0 i u x1 t)) (\lambda (t: T).(subst0 i u x0 +t)))).(ex2_ind T (\lambda (t: T).(subst0 i u x1 t)) (\lambda (t: T).(subst0 i +u x0 t)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) +(\lambda (x2: T).(\lambda (H24: (subst0 i u x1 x2)).(\lambda (H25: (subst0 i +u x0 x2)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c +t2 t)) x2 (pr2_delta c d u i H22 t1 x0 H8 x2 H25) (pr2_delta c d u i H22 t2 +x1 H11 x2 H24))))) H23)) (\lambda (H23: (subst0 i u x1 x0)).(ex_intro2 T +(\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t)) x0 (pr2_free c t1 +x0 H8) (pr2_delta c d u i H22 t2 x1 H11 x0 H23))) (\lambda (H23: (subst0 i u +x0 x1)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 +t)) x1 (pr2_delta c d u i H22 t1 x0 H8 x1 H23) (pr2_free c t2 x1 H11))) +(subst0_confluence_eq x x1 u i H20 x0 H9))))))) H17)))))))))) H10)) +(pr0_subst0 t4 x H5 u0 t2 i0 H4 u0 (pr0_refl u0)))))) H7)) (pr0_subst0 t3 x +H6 u t1 i H1 u (pr0_refl u)))))) (pr0_confluence t0 t4 H3 t3 +H0))))))))))))))))))). +(* COMMENTS +Initial nodes: 1901 +END *) + +theorem pr2_confluence: + \forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr2 c t0 t1) \to (\forall +(t2: T).((pr2 c t0 t2) \to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: +T).(pr2 c t2 t)))))))) +\def + \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr2 c t0 +t1)).(\lambda (t2: T).(\lambda (H0: (pr2 c t0 t2)).(let H1 \def (match H in +pr2 return (\lambda (c0: C).(\lambda (t: T).(\lambda (t3: T).(\lambda (_: +(pr2 c0 t t3)).((eq C c0 c) \to ((eq T t t0) \to ((eq T t3 t1) \to (ex2 T +(\lambda (t4: T).(pr2 c t1 t4)) (\lambda (t4: T).(pr2 c t2 t4)))))))))) with +[(pr2_free c0 t3 t4 H1) \Rightarrow (\lambda (H2: (eq C c0 c)).(\lambda (H3: +(eq T t3 t0)).(\lambda (H4: (eq T t4 t1)).(eq_ind C c (\lambda (_: C).((eq T +t3 t0) \to ((eq T t4 t1) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t: T).(pr2 c +t1 t)) (\lambda (t: T).(pr2 c t2 t))))))) (\lambda (H5: (eq T t3 t0)).(eq_ind +T t0 (\lambda (t: T).((eq T t4 t1) \to ((pr0 t t4) \to (ex2 T (\lambda (t5: +T).(pr2 c t1 t5)) (\lambda (t5: T).(pr2 c t2 t5)))))) (\lambda (H6: (eq T t4 +t1)).(eq_ind T t1 (\lambda (t: T).((pr0 t0 t) \to (ex2 T (\lambda (t5: +T).(pr2 c t1 t5)) (\lambda (t5: T).(pr2 c t2 t5))))) (\lambda (H7: (pr0 t0 +t1)).(let H8 \def (match H0 in pr2 return (\lambda (c1: C).(\lambda (t: +T).(\lambda (t5: T).(\lambda (_: (pr2 c1 t t5)).((eq C c1 c) \to ((eq T t t0) +\to ((eq T t5 t2) \to (ex2 T (\lambda (t6: T).(pr2 c t1 t6)) (\lambda (t6: +T).(pr2 c t2 t6)))))))))) with [(pr2_free c1 t5 t6 H8) \Rightarrow (\lambda +(H9: (eq C c1 c)).(\lambda (H10: (eq T t5 t0)).(\lambda (H11: (eq T t6 +t2)).(eq_ind C c (\lambda (_: C).((eq T t5 t0) \to ((eq T t6 t2) \to ((pr0 t5 +t6) \to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 +t))))))) (\lambda (H12: (eq T t5 t0)).(eq_ind T t0 (\lambda (t: T).((eq T t6 +t2) \to ((pr0 t t6) \to (ex2 T (\lambda (t7: T).(pr2 c t1 t7)) (\lambda (t7: +T).(pr2 c t2 t7)))))) (\lambda (H13: (eq T t6 t2)).(eq_ind T t2 (\lambda (t: +T).((pr0 t0 t) \to (ex2 T (\lambda (t7: T).(pr2 c t1 t7)) (\lambda (t7: +T).(pr2 c t2 t7))))) (\lambda (H14: (pr0 t0 +t2)).(pr2_confluence__pr2_free_free c t0 t1 t2 H7 H14)) t6 (sym_eq T t6 t2 +H13))) t5 (sym_eq T t5 t0 H12))) c1 (sym_eq C c1 c H9) H10 H11 H8)))) | +(pr2_delta c1 d u i H8 t5 t6 H9 t H10) \Rightarrow (\lambda (H11: (eq C c1 +c)).(\lambda (H12: (eq T t5 t0)).(\lambda (H13: (eq T t t2)).(eq_ind C c +(\lambda (c2: C).((eq T t5 t0) \to ((eq T t t2) \to ((getl i c2 (CHead d +(Bind Abbr) u)) \to ((pr0 t5 t6) \to ((subst0 i u t6 t) \to (ex2 T (\lambda +(t7: T).(pr2 c t1 t7)) (\lambda (t7: T).(pr2 c t2 t7))))))))) (\lambda (H14: +(eq T t5 t0)).(eq_ind T t0 (\lambda (t7: T).((eq T t t2) \to ((getl i c +(CHead d (Bind Abbr) u)) \to ((pr0 t7 t6) \to ((subst0 i u t6 t) \to (ex2 T +(\lambda (t8: T).(pr2 c t1 t8)) (\lambda (t8: T).(pr2 c t2 t8)))))))) +(\lambda (H15: (eq T t t2)).(eq_ind T t2 (\lambda (t7: T).((getl i c (CHead d +(Bind Abbr) u)) \to ((pr0 t0 t6) \to ((subst0 i u t6 t7) \to (ex2 T (\lambda +(t8: T).(pr2 c t1 t8)) (\lambda (t8: T).(pr2 c t2 t8))))))) (\lambda (H16: +(getl i c (CHead d (Bind Abbr) u))).(\lambda (H17: (pr0 t0 t6)).(\lambda +(H18: (subst0 i u t6 t2)).(pr2_confluence__pr2_free_delta c d t0 t1 t2 t6 u i +H7 H16 H17 H18)))) t (sym_eq T t t2 H15))) t5 (sym_eq T t5 t0 H14))) c1 +(sym_eq C c1 c H11) H12 H13 H8 H9 H10))))]) in (H8 (refl_equal C c) +(refl_equal T t0) (refl_equal T t2)))) t4 (sym_eq T t4 t1 H6))) t3 (sym_eq T +t3 t0 H5))) c0 (sym_eq C c0 c H2) H3 H4 H1)))) | (pr2_delta c0 d u i H1 t3 t4 +H2 t H3) \Rightarrow (\lambda (H4: (eq C c0 c)).(\lambda (H5: (eq T t3 +t0)).(\lambda (H6: (eq T t t1)).(eq_ind C c (\lambda (c1: C).((eq T t3 t0) +\to ((eq T t t1) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t3 t4) +\to ((subst0 i u t4 t) \to (ex2 T (\lambda (t5: T).(pr2 c t1 t5)) (\lambda +(t5: T).(pr2 c t2 t5))))))))) (\lambda (H7: (eq T t3 t0)).(eq_ind T t0 +(\lambda (t5: T).((eq T t t1) \to ((getl i c (CHead d (Bind Abbr) u)) \to +((pr0 t5 t4) \to ((subst0 i u t4 t) \to (ex2 T (\lambda (t6: T).(pr2 c t1 +t6)) (\lambda (t6: T).(pr2 c t2 t6)))))))) (\lambda (H8: (eq T t t1)).(eq_ind +T t1 (\lambda (t5: T).((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t4) +\to ((subst0 i u t4 t5) \to (ex2 T (\lambda (t6: T).(pr2 c t1 t6)) (\lambda +(t6: T).(pr2 c t2 t6))))))) (\lambda (H9: (getl i c (CHead d (Bind Abbr) +u))).(\lambda (H10: (pr0 t0 t4)).(\lambda (H11: (subst0 i u t4 t1)).(let H12 +\def (match H0 in pr2 return (\lambda (c1: C).(\lambda (t5: T).(\lambda (t6: +T).(\lambda (_: (pr2 c1 t5 t6)).((eq C c1 c) \to ((eq T t5 t0) \to ((eq T t6 +t2) \to (ex2 T (\lambda (t7: T).(pr2 c t1 t7)) (\lambda (t7: T).(pr2 c t2 +t7)))))))))) with [(pr2_free c1 t5 t6 H12) \Rightarrow (\lambda (H13: (eq C +c1 c)).(\lambda (H14: (eq T t5 t0)).(\lambda (H15: (eq T t6 t2)).(eq_ind C c +(\lambda (_: C).((eq T t5 t0) \to ((eq T t6 t2) \to ((pr0 t5 t6) \to (ex2 T +(\lambda (t7: T).(pr2 c t1 t7)) (\lambda (t7: T).(pr2 c t2 t7))))))) (\lambda +(H16: (eq T t5 t0)).(eq_ind T t0 (\lambda (t7: T).((eq T t6 t2) \to ((pr0 t7 +t6) \to (ex2 T (\lambda (t8: T).(pr2 c t1 t8)) (\lambda (t8: T).(pr2 c t2 +t8)))))) (\lambda (H17: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t0 +t7) \to (ex2 T (\lambda (t8: T).(pr2 c t1 t8)) (\lambda (t8: T).(pr2 c t2 +t8))))) (\lambda (H18: (pr0 t0 t2)).(ex2_sym T (pr2 c t2) (pr2 c t1) +(pr2_confluence__pr2_free_delta c d t0 t2 t1 t4 u i H18 H9 H10 H11))) t6 +(sym_eq T t6 t2 H17))) t5 (sym_eq T t5 t0 H16))) c1 (sym_eq C c1 c H13) H14 +H15 H12)))) | (pr2_delta c1 d0 u0 i0 H12 t5 t6 H13 t7 H14) \Rightarrow +(\lambda (H15: (eq C c1 c)).(\lambda (H16: (eq T t5 t0)).(\lambda (H17: (eq T +t7 t2)).(eq_ind C c (\lambda (c2: C).((eq T t5 t0) \to ((eq T t7 t2) \to +((getl i0 c2 (CHead d0 (Bind Abbr) u0)) \to ((pr0 t5 t6) \to ((subst0 i0 u0 +t6 t7) \to (ex2 T (\lambda (t8: T).(pr2 c t1 t8)) (\lambda (t8: T).(pr2 c t2 +t8))))))))) (\lambda (H18: (eq T t5 t0)).(eq_ind T t0 (\lambda (t8: T).((eq T +t7 t2) \to ((getl i0 c (CHead d0 (Bind Abbr) u0)) \to ((pr0 t8 t6) \to +((subst0 i0 u0 t6 t7) \to (ex2 T (\lambda (t9: T).(pr2 c t1 t9)) (\lambda +(t9: T).(pr2 c t2 t9)))))))) (\lambda (H19: (eq T t7 t2)).(eq_ind T t2 +(\lambda (t8: T).((getl i0 c (CHead d0 (Bind Abbr) u0)) \to ((pr0 t0 t6) \to +((subst0 i0 u0 t6 t8) \to (ex2 T (\lambda (t9: T).(pr2 c t1 t9)) (\lambda +(t9: T).(pr2 c t2 t9))))))) (\lambda (H20: (getl i0 c (CHead d0 (Bind Abbr) +u0))).(\lambda (H21: (pr0 t0 t6)).(\lambda (H22: (subst0 i0 u0 t6 +t2)).(pr2_confluence__pr2_delta_delta c d d0 t0 t1 t2 t4 t6 u u0 i i0 H9 H10 +H11 H20 H21 H22)))) t7 (sym_eq T t7 t2 H19))) t5 (sym_eq T t5 t0 H18))) c1 +(sym_eq C c1 c H15) H16 H17 H12 H13 H14))))]) in (H12 (refl_equal C c) +(refl_equal T t0) (refl_equal T t2)))))) t (sym_eq T t t1 H8))) t3 (sym_eq T +t3 t0 H7))) c0 (sym_eq C c0 c H4) H5 H6 H1 H2 H3))))]) in (H1 (refl_equal C +c) (refl_equal T t0) (refl_equal T t1)))))))). +(* COMMENTS +Initial nodes: 2087 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr2/props.ma b/matita/matita/contribs/lambdadelta/basic_1/pr2/props.ma new file mode 100644 index 000000000..2faeb6ebd --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/pr2/props.ma @@ -0,0 +1,307 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/pr2/defs.ma". + +include "Basic-1/pr0/props.ma". + +include "Basic-1/getl/drop.ma". + +include "Basic-1/getl/clear.ma". + +theorem pr2_thin_dx: + \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall +(u: T).(\forall (f: F).(pr2 c (THead (Flat f) u t1) (THead (Flat f) u +t2))))))) +\def + \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 +t2)).(\lambda (u: T).(\lambda (f: F).(pr2_ind (\lambda (c0: C).(\lambda (t: +T).(\lambda (t0: T).(pr2 c0 (THead (Flat f) u t) (THead (Flat f) u t0))))) +(\lambda (c0: C).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr0 t0 +t3)).(pr2_free c0 (THead (Flat f) u t0) (THead (Flat f) u t3) (pr0_comp u u +(pr0_refl u) t0 t3 H0 (Flat f))))))) (\lambda (c0: C).(\lambda (d: +C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind +Abbr) u0))).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H1: (pr0 t0 +t3)).(\lambda (t: T).(\lambda (H2: (subst0 i u0 t3 t)).(pr2_delta c0 d u0 i +H0 (THead (Flat f) u t0) (THead (Flat f) u t3) (pr0_comp u u (pr0_refl u) t0 +t3 H1 (Flat f)) (THead (Flat f) u t) (subst0_snd (Flat f) u0 t t3 i H2 +u)))))))))))) c t1 t2 H)))))). +(* COMMENTS +Initial nodes: 239 +END *) + +theorem pr2_head_1: + \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr2 c u1 u2) \to (\forall +(k: K).(\forall (t: T).(pr2 c (THead k u1 t) (THead k u2 t))))))) +\def + \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr2 c u1 +u2)).(\lambda (k: K).(\lambda (t: T).(pr2_ind (\lambda (c0: C).(\lambda (t0: +T).(\lambda (t1: T).(pr2 c0 (THead k t0 t) (THead k t1 t))))) (\lambda (c0: +C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr0 t1 t2)).(pr2_free c0 +(THead k t1 t) (THead k t2 t) (pr0_comp t1 t2 H0 t t (pr0_refl t) k)))))) +(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda +(H0: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t1: T).(\lambda (t2: +T).(\lambda (H1: (pr0 t1 t2)).(\lambda (t0: T).(\lambda (H2: (subst0 i u t2 +t0)).(pr2_delta c0 d u i H0 (THead k t1 t) (THead k t2 t) (pr0_comp t1 t2 H1 +t t (pr0_refl t) k) (THead k t0 t) (subst0_fst u t0 t2 i H2 t k)))))))))))) c +u1 u2 H)))))). +(* COMMENTS +Initial nodes: 219 +END *) + +theorem pr2_head_2: + \forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).(\forall +(k: K).((pr2 (CHead c k u) t1 t2) \to (pr2 c (THead k u t1) (THead k u +t2))))))) +\def + \lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda +(k: K).(\lambda (H: (pr2 (CHead c k u) t1 t2)).(insert_eq C (CHead c k u) +(\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 c (THead k u t1) (THead +k u t2))) (\lambda (y: C).(\lambda (H0: (pr2 y t1 t2)).(pr2_ind (\lambda (c0: +C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CHead c k u)) \to (pr2 c +(THead k u t) (THead k u t0)))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda +(t4: T).(\lambda (H1: (pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c k +u))).(pr2_free c (THead k u t3) (THead k u t4) (pr0_comp u u (pr0_refl u) t3 +t4 H1 k))))))) (K_ind (\lambda (k0: K).(\forall (c0: C).(\forall (d: +C).(\forall (u0: T).(\forall (i: nat).((getl i c0 (CHead d (Bind Abbr) u0)) +\to (\forall (t3: T).(\forall (t4: T).((pr0 t3 t4) \to (\forall (t: +T).((subst0 i u0 t4 t) \to ((eq C c0 (CHead c k0 u)) \to (pr2 c (THead k0 u +t3) (THead k0 u t)))))))))))))) (\lambda (b: B).(\lambda (c0: C).(\lambda (d: +C).(\lambda (u0: T).(\lambda (i: nat).(nat_ind (\lambda (n: nat).((getl n c0 +(CHead d (Bind Abbr) u0)) \to (\forall (t3: T).(\forall (t4: T).((pr0 t3 t4) +\to (\forall (t: T).((subst0 n u0 t4 t) \to ((eq C c0 (CHead c (Bind b) u)) +\to (pr2 c (THead (Bind b) u t3) (THead (Bind b) u t)))))))))) (\lambda (H1: +(getl O c0 (CHead d (Bind Abbr) u0))).(\lambda (t3: T).(\lambda (t4: +T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: (subst0 O u0 t4 +t)).(\lambda (H4: (eq C c0 (CHead c (Bind b) u))).(let H5 \def (eq_ind C c0 +(\lambda (c1: C).(getl O c1 (CHead d (Bind Abbr) u0))) H1 (CHead c (Bind b) +u) H4) in (let H6 \def (f_equal C C (\lambda (e: C).(match e in C return +(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow +c1])) (CHead d (Bind Abbr) u0) (CHead c (Bind b) u) (clear_gen_bind b c +(CHead d (Bind Abbr) u0) u (getl_gen_O (CHead c (Bind b) u) (CHead d (Bind +Abbr) u0) H5))) in ((let H7 \def (f_equal C B (\lambda (e: C).(match e in C +return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _) +\Rightarrow (match k0 in K return (\lambda (_: K).B) with [(Bind b0) +\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u0) +(CHead c (Bind b) u) (clear_gen_bind b c (CHead d (Bind Abbr) u0) u +(getl_gen_O (CHead c (Bind b) u) (CHead d (Bind Abbr) u0) H5))) in ((let H8 +\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) +with [(CSort _) \Rightarrow u0 | (CHead _ _ t0) \Rightarrow t0])) (CHead d +(Bind Abbr) u0) (CHead c (Bind b) u) (clear_gen_bind b c (CHead d (Bind Abbr) +u0) u (getl_gen_O (CHead c (Bind b) u) (CHead d (Bind Abbr) u0) H5))) in +(\lambda (H9: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H11 \def (eq_ind T +u0 (\lambda (t0: T).(subst0 O t0 t4 t)) H3 u H8) in (eq_ind B Abbr (\lambda +(b0: B).(pr2 c (THead (Bind b0) u t3) (THead (Bind b0) u t))) (pr2_free c +(THead (Bind Abbr) u t3) (THead (Bind Abbr) u t) (pr0_delta u u (pr0_refl u) +t3 t4 H2 t H11)) b H9))))) H7)) H6)))))))))) (\lambda (n: nat).(\lambda (H1: +(((getl n c0 (CHead d (Bind Abbr) u0)) \to (\forall (t3: T).(\forall (t4: +T).((pr0 t3 t4) \to (\forall (t: T).((subst0 n u0 t4 t) \to ((eq C c0 (CHead +c (Bind b) u)) \to (pr2 c (THead (Bind b) u t3) (THead (Bind b) u +t))))))))))).(\lambda (H2: (getl (S n) c0 (CHead d (Bind Abbr) u0))).(\lambda +(t3: T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda (t: T).(\lambda +(H4: (subst0 (S n) u0 t4 t)).(\lambda (H5: (eq C c0 (CHead c (Bind b) +u))).(let H6 \def (eq_ind C c0 (\lambda (c1: C).(getl (S n) c1 (CHead d (Bind +Abbr) u0))) H2 (CHead c (Bind b) u) H5) in (let H7 \def (eq_ind C c0 (\lambda +(c1: C).((getl n c1 (CHead d (Bind Abbr) u0)) \to (\forall (t5: T).(\forall +(t6: T).((pr0 t5 t6) \to (\forall (t0: T).((subst0 n u0 t6 t0) \to ((eq C c1 +(CHead c (Bind b) u)) \to (pr2 c (THead (Bind b) u t5) (THead (Bind b) u +t0)))))))))) H1 (CHead c (Bind b) u) H5) in (pr2_delta c d u0 (r (Bind b) n) +(getl_gen_S (Bind b) c (CHead d (Bind Abbr) u0) u n H6) (THead (Bind b) u t3) +(THead (Bind b) u t4) (pr0_comp u u (pr0_refl u) t3 t4 H3 (Bind b)) (THead +(Bind b) u t) (subst0_snd (Bind b) u0 t t4 (r (Bind b) n) H4 u))))))))))))) +i)))))) (\lambda (f: F).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: +T).(\lambda (i: nat).(nat_ind (\lambda (n: nat).((getl n c0 (CHead d (Bind +Abbr) u0)) \to (\forall (t3: T).(\forall (t4: T).((pr0 t3 t4) \to (\forall +(t: T).((subst0 n u0 t4 t) \to ((eq C c0 (CHead c (Flat f) u)) \to (pr2 c +(THead (Flat f) u t3) (THead (Flat f) u t)))))))))) (\lambda (H1: (getl O c0 +(CHead d (Bind Abbr) u0))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: +(pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: (subst0 O u0 t4 t)).(\lambda (H4: +(eq C c0 (CHead c (Flat f) u))).(let H5 \def (eq_ind C c0 (\lambda (c1: +C).(getl O c1 (CHead d (Bind Abbr) u0))) H1 (CHead c (Flat f) u) H4) in +(pr2_delta c d u0 O (getl_intro O c (CHead d (Bind Abbr) u0) c (drop_refl c) +(clear_gen_flat f c (CHead d (Bind Abbr) u0) u (getl_gen_O (CHead c (Flat f) +u) (CHead d (Bind Abbr) u0) H5))) (THead (Flat f) u t3) (THead (Flat f) u t4) +(pr0_comp u u (pr0_refl u) t3 t4 H2 (Flat f)) (THead (Flat f) u t) +(subst0_snd (Flat f) u0 t t4 O H3 u)))))))))) (\lambda (n: nat).(\lambda (H1: +(((getl n c0 (CHead d (Bind Abbr) u0)) \to (\forall (t3: T).(\forall (t4: +T).((pr0 t3 t4) \to (\forall (t: T).((subst0 n u0 t4 t) \to ((eq C c0 (CHead +c (Flat f) u)) \to (pr2 c (THead (Flat f) u t3) (THead (Flat f) u +t))))))))))).(\lambda (H2: (getl (S n) c0 (CHead d (Bind Abbr) u0))).(\lambda +(t3: T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda (t: T).(\lambda +(H4: (subst0 (S n) u0 t4 t)).(\lambda (H5: (eq C c0 (CHead c (Flat f) +u))).(let H6 \def (eq_ind C c0 (\lambda (c1: C).(getl (S n) c1 (CHead d (Bind +Abbr) u0))) H2 (CHead c (Flat f) u) H5) in (let H7 \def (eq_ind C c0 (\lambda +(c1: C).((getl n c1 (CHead d (Bind Abbr) u0)) \to (\forall (t5: T).(\forall +(t6: T).((pr0 t5 t6) \to (\forall (t0: T).((subst0 n u0 t6 t0) \to ((eq C c1 +(CHead c (Flat f) u)) \to (pr2 c (THead (Flat f) u t5) (THead (Flat f) u +t0)))))))))) H1 (CHead c (Flat f) u) H5) in (pr2_delta c d u0 (r (Flat f) n) +(getl_gen_S (Flat f) c (CHead d (Bind Abbr) u0) u n H6) (THead (Flat f) u t3) +(THead (Flat f) u t4) (pr0_comp u u (pr0_refl u) t3 t4 H3 (Flat f)) (THead +(Flat f) u t) (subst0_snd (Flat f) u0 t t4 (r (Flat f) n) H4 u))))))))))))) +i)))))) k) y t1 t2 H0))) H)))))). +(* COMMENTS +Initial nodes: 1947 +END *) + +theorem clear_pr2_trans: + \forall (c2: C).(\forall (t1: T).(\forall (t2: T).((pr2 c2 t1 t2) \to +(\forall (c1: C).((clear c1 c2) \to (pr2 c1 t1 t2)))))) +\def + \lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c2 t1 +t2)).(pr2_ind (\lambda (c: C).(\lambda (t: T).(\lambda (t0: T).(\forall (c1: +C).((clear c1 c) \to (pr2 c1 t t0)))))) (\lambda (c: C).(\lambda (t3: +T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(\lambda (c1: C).(\lambda (_: +(clear c1 c)).(pr2_free c1 t3 t4 H0))))))) (\lambda (c: C).(\lambda (d: +C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d (Bind +Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 +t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (c1: +C).(\lambda (H3: (clear c1 c)).(pr2_delta c1 d u i (clear_getl_trans i c +(CHead d (Bind Abbr) u) H0 c1 H3) t3 t4 H1 t H2))))))))))))) c2 t1 t2 H)))). +(* COMMENTS +Initial nodes: 171 +END *) + +theorem pr2_cflat: + \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall +(f: F).(\forall (v: T).(pr2 (CHead c (Flat f) v) t1 t2)))))) +\def + \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 +t2)).(\lambda (f: F).(\lambda (v: T).(pr2_ind (\lambda (c0: C).(\lambda (t: +T).(\lambda (t0: T).(pr2 (CHead c0 (Flat f) v) t t0)))) (\lambda (c0: +C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(pr2_free +(CHead c0 (Flat f) v) t3 t4 H0))))) (\lambda (c0: C).(\lambda (d: C).(\lambda +(u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr) +u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 t4)).(\lambda +(t: T).(\lambda (H2: (subst0 i u t4 t)).(pr2_delta (CHead c0 (Flat f) v) d u +i (getl_flat c0 (CHead d (Bind Abbr) u) i H0 f v) t3 t4 H1 t H2))))))))))) c +t1 t2 H)))))). +(* COMMENTS +Initial nodes: 175 +END *) + +theorem pr2_ctail: + \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall +(k: K).(\forall (u: T).(pr2 (CTail k u c) t1 t2)))))) +\def + \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 +t2)).(\lambda (k: K).(\lambda (u: T).(pr2_ind (\lambda (c0: C).(\lambda (t: +T).(\lambda (t0: T).(pr2 (CTail k u c0) t t0)))) (\lambda (c0: C).(\lambda +(t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(pr2_free (CTail k u c0) +t3 t4 H0))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: +nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr) u0))).(\lambda (t3: +T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H2: +(subst0 i u0 t4 t)).(pr2_delta (CTail k u c0) (CTail k u d) u0 i (getl_ctail +Abbr c0 d u0 i H0 k u) t3 t4 H1 t H2))))))))))) c t1 t2 H)))))). +(* COMMENTS +Initial nodes: 171 +END *) + +theorem pr2_change: + \forall (b: B).((not (eq B b Abbr)) \to (\forall (c: C).(\forall (v1: +T).(\forall (t1: T).(\forall (t2: T).((pr2 (CHead c (Bind b) v1) t1 t2) \to +(\forall (v2: T).(pr2 (CHead c (Bind b) v2) t1 t2)))))))) +\def + \lambda (b: B).(\lambda (H: (not (eq B b Abbr))).(\lambda (c: C).(\lambda +(v1: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr2 (CHead c (Bind +b) v1) t1 t2)).(\lambda (v2: T).(insert_eq C (CHead c (Bind b) v1) (\lambda +(c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 (CHead c (Bind b) v2) t1 t2)) +(\lambda (y: C).(\lambda (H1: (pr2 y t1 t2)).(pr2_ind (\lambda (c0: +C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CHead c (Bind b) v1)) \to (pr2 +(CHead c (Bind b) v2) t t0))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda +(t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c (Bind b) +v1))).(pr2_free (CHead c (Bind b) v2) t3 t4 H2)))))) (\lambda (c0: +C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H2: (getl i c0 +(CHead d (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H3: +(pr0 t3 t4)).(\lambda (t: T).(\lambda (H4: (subst0 i u t4 t)).(\lambda (H5: +(eq C c0 (CHead c (Bind b) v1))).(let H6 \def (eq_ind C c0 (\lambda (c1: +C).(getl i c1 (CHead d (Bind Abbr) u))) H2 (CHead c (Bind b) v1) H5) in +(nat_ind (\lambda (n: nat).((getl n (CHead c (Bind b) v1) (CHead d (Bind +Abbr) u)) \to ((subst0 n u t4 t) \to (pr2 (CHead c (Bind b) v2) t3 t)))) +(\lambda (H7: (getl O (CHead c (Bind b) v1) (CHead d (Bind Abbr) +u))).(\lambda (H8: (subst0 O u t4 t)).(let H9 \def (f_equal C C (\lambda (e: +C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | +(CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u) (CHead c (Bind b) +v1) (clear_gen_bind b c (CHead d (Bind Abbr) u) v1 (getl_gen_O (CHead c (Bind +b) v1) (CHead d (Bind Abbr) u) H7))) in ((let H10 \def (f_equal C B (\lambda +(e: C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow +Abbr | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with +[(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind +Abbr) u) (CHead c (Bind b) v1) (clear_gen_bind b c (CHead d (Bind Abbr) u) v1 +(getl_gen_O (CHead c (Bind b) v1) (CHead d (Bind Abbr) u) H7))) in ((let H11 +\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) +with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d +(Bind Abbr) u) (CHead c (Bind b) v1) (clear_gen_bind b c (CHead d (Bind Abbr) +u) v1 (getl_gen_O (CHead c (Bind b) v1) (CHead d (Bind Abbr) u) H7))) in +(\lambda (H12: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H14 \def (eq_ind +T u (\lambda (t0: T).(subst0 O t0 t4 t)) H8 v1 H11) in (let H15 \def +(eq_ind_r B b (\lambda (b0: B).(not (eq B b0 Abbr))) H Abbr H12) in (eq_ind B +Abbr (\lambda (b0: B).(pr2 (CHead c (Bind b0) v2) t3 t)) (let H16 \def (match +(H15 (refl_equal B Abbr)) in False return (\lambda (_: False).(pr2 (CHead c +(Bind Abbr) v2) t3 t)) with []) in H16) b H12)))))) H10)) H9)))) (\lambda +(i0: nat).(\lambda (_: (((getl i0 (CHead c (Bind b) v1) (CHead d (Bind Abbr) +u)) \to ((subst0 i0 u t4 t) \to (pr2 (CHead c (Bind b) v2) t3 t))))).(\lambda +(H7: (getl (S i0) (CHead c (Bind b) v1) (CHead d (Bind Abbr) u))).(\lambda +(H8: (subst0 (S i0) u t4 t)).(pr2_delta (CHead c (Bind b) v2) d u (S i0) +(getl_head (Bind b) i0 c (CHead d (Bind Abbr) u) (getl_gen_S (Bind b) c +(CHead d (Bind Abbr) u) v1 i0 H7) v2) t3 t4 H3 t H8))))) i H6 H4))))))))))))) +y t1 t2 H1))) H0)))))))). +(* COMMENTS +Initial nodes: 913 +END *) + +theorem pr2_lift: + \forall (c: C).(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h +d c e) \to (\forall (t1: T).(\forall (t2: T).((pr2 e t1 t2) \to (pr2 c (lift +h d t1) (lift h d t2))))))))) +\def + \lambda (c: C).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda +(H: (drop h d c e)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr2 e t1 +t2)).(insert_eq C e (\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 c +(lift h d t1) (lift h d t2))) (\lambda (y: C).(\lambda (H1: (pr2 y t1 +t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 e) +\to (pr2 c (lift h d t) (lift h d t0)))))) (\lambda (c0: C).(\lambda (t3: +T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (_: (eq C c0 +e)).(pr2_free c (lift h d t3) (lift h d t4) (pr0_lift t3 t4 H2 h d))))))) +(\lambda (c0: C).(\lambda (d0: C).(\lambda (u: T).(\lambda (i: nat).(\lambda +(H2: (getl i c0 (CHead d0 (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: +T).(\lambda (H3: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H4: (subst0 i u t4 +t)).(\lambda (H5: (eq C c0 e)).(let H6 \def (eq_ind C c0 (\lambda (c1: +C).(getl i c1 (CHead d0 (Bind Abbr) u))) H2 e H5) in (lt_le_e i d (pr2 c +(lift h d t3) (lift h d t)) (\lambda (H7: (lt i d)).(let H8 \def +(drop_getl_trans_le i d (le_S_n i d (le_S (S i) d H7)) c e h H (CHead d0 +(Bind Abbr) u) H6) in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop i +O c e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d i) e0 e1))) +(\lambda (_: C).(\lambda (e1: C).(clear e1 (CHead d0 (Bind Abbr) u)))) (pr2 c +(lift h d t3) (lift h d t)) (\lambda (x0: C).(\lambda (x1: C).(\lambda (H9: +(drop i O c x0)).(\lambda (H10: (drop h (minus d i) x0 x1)).(\lambda (H11: +(clear x1 (CHead d0 (Bind Abbr) u))).(let H12 \def (eq_ind nat (minus d i) +(\lambda (n: nat).(drop h n x0 x1)) H10 (S (minus d (S i))) (minus_x_Sy d i +H7)) in (let H13 \def (drop_clear_S x1 x0 h (minus d (S i)) H12 Abbr d0 u +H11) in (ex2_ind C (\lambda (c1: C).(clear x0 (CHead c1 (Bind Abbr) (lift h +(minus d (S i)) u)))) (\lambda (c1: C).(drop h (minus d (S i)) c1 d0)) (pr2 c +(lift h d t3) (lift h d t)) (\lambda (x: C).(\lambda (H14: (clear x0 (CHead x +(Bind Abbr) (lift h (minus d (S i)) u)))).(\lambda (_: (drop h (minus d (S +i)) x d0)).(pr2_delta c x (lift h (minus d (S i)) u) i (getl_intro i c (CHead +x (Bind Abbr) (lift h (minus d (S i)) u)) x0 H9 H14) (lift h d t3) (lift h d +t4) (pr0_lift t3 t4 H3 h d) (lift h d t) (subst0_lift_lt t4 t u i H4 d H7 +h))))) H13)))))))) H8))) (\lambda (H7: (le d i)).(pr2_delta c d0 u (plus i h) +(drop_getl_trans_ge i c e d h H (CHead d0 (Bind Abbr) u) H6 H7) (lift h d t3) +(lift h d t4) (pr0_lift t3 t4 H3 h d) (lift h d t) (subst0_lift_ge t4 t u i h +H4 d H7)))))))))))))))) y t1 t2 H1))) H0)))))))). +(* COMMENTS +Initial nodes: 849 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr2/subst1.ma b/matita/matita/contribs/lambdadelta/basic_1/pr2/subst1.ma new file mode 100644 index 000000000..47704dd52 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/pr2/subst1.ma @@ -0,0 +1,281 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/pr2/defs.ma". + +include "Basic-1/pr0/subst1.ma". + +include "Basic-1/pr0/fwd.ma". + +include "Basic-1/csubst1/getl.ma". + +include "Basic-1/csubst1/fwd.ma". + +include "Basic-1/subst1/subst1.ma". + +include "Basic-1/getl/drop.ma". + +theorem pr2_delta1: + \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c +(CHead d (Bind Abbr) u)) \to (\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) +\to (\forall (t: T).((subst1 i u t2 t) \to (pr2 c t1 t)))))))))) +\def + \lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda +(H: (getl i c (CHead d (Bind Abbr) u))).(\lambda (t1: T).(\lambda (t2: +T).(\lambda (H0: (pr0 t1 t2)).(\lambda (t: T).(\lambda (H1: (subst1 i u t2 +t)).(subst1_ind i u t2 (\lambda (t0: T).(pr2 c t1 t0)) (pr2_free c t1 t2 H0) +(\lambda (t0: T).(\lambda (H2: (subst0 i u t2 t0)).(pr2_delta c d u i H t1 t2 +H0 t0 H2))) t H1)))))))))). +(* COMMENTS +Initial nodes: 111 +END *) + +theorem pr2_subst1: + \forall (c: C).(\forall (e: C).(\forall (v: T).(\forall (i: nat).((getl i c +(CHead e (Bind Abbr) v)) \to (\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) +\to (\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T (\lambda (w2: T).(pr2 c +w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2)))))))))))) +\def + \lambda (c: C).(\lambda (e: C).(\lambda (v: T).(\lambda (i: nat).(\lambda +(H: (getl i c (CHead e (Bind Abbr) v))).(\lambda (t1: T).(\lambda (t2: +T).(\lambda (H0: (pr2 c t1 t2)).(insert_eq C c (\lambda (c0: C).(pr2 c0 t1 +t2)) (\lambda (c0: C).(\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T +(\lambda (w2: T).(pr2 c0 w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2)))))) +(\lambda (y: C).(\lambda (H1: (pr2 y t1 t2)).(pr2_ind (\lambda (c0: +C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 c) \to (\forall (w1: +T).((subst1 i v t w1) \to (ex2 T (\lambda (w2: T).(pr2 c0 w1 w2)) (\lambda +(w2: T).(subst1 i v t0 w2))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda +(t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (H3: (eq C c0 c)).(\lambda (w1: +T).(\lambda (H4: (subst1 i v t3 w1)).(eq_ind_r C c (\lambda (c1: C).(ex2 T +(\lambda (w2: T).(pr2 c1 w1 w2)) (\lambda (w2: T).(subst1 i v t4 w2)))) +(ex2_ind T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst1 i v t4 w2)) +(ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t4 w2))) +(\lambda (x: T).(\lambda (H5: (pr0 w1 x)).(\lambda (H6: (subst1 i v t4 +x)).(ex_intro2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v +t4 w2)) x (pr2_free c w1 x H5) H6)))) (pr0_subst1 t3 t4 H2 v w1 i H4 v +(pr0_refl v))) c0 H3)))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: +T).(\lambda (i0: nat).(\lambda (H2: (getl i0 c0 (CHead d (Bind Abbr) +u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda +(t: T).(\lambda (H4: (subst0 i0 u t4 t)).(\lambda (H5: (eq C c0 c)).(\lambda +(w1: T).(\lambda (H6: (subst1 i v t3 w1)).(let H7 \def (eq_ind C c0 (\lambda +(c1: C).(getl i0 c1 (CHead d (Bind Abbr) u))) H2 c H5) in (eq_ind_r C c +(\lambda (c1: C).(ex2 T (\lambda (w2: T).(pr2 c1 w1 w2)) (\lambda (w2: +T).(subst1 i v t w2)))) (ex2_ind T (\lambda (w2: T).(pr0 w1 w2)) (\lambda +(w2: T).(subst1 i v t4 w2)) (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda +(w2: T).(subst1 i v t w2))) (\lambda (x: T).(\lambda (H8: (pr0 w1 +x)).(\lambda (H9: (subst1 i v t4 x)).(neq_eq_e i i0 (ex2 T (\lambda (w2: +T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t w2))) (\lambda (H10: (not +(eq nat i i0))).(ex2_ind T (\lambda (t0: T).(subst1 i v t t0)) (\lambda (t0: +T).(subst1 i0 u x t0)) (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: +T).(subst1 i v t w2))) (\lambda (x0: T).(\lambda (H11: (subst1 i v t +x0)).(\lambda (H12: (subst1 i0 u x x0)).(ex_intro2 T (\lambda (w2: T).(pr2 c +w1 w2)) (\lambda (w2: T).(subst1 i v t w2)) x0 (pr2_delta1 c d u i0 H7 w1 x +H8 x0 H12) H11)))) (subst1_confluence_neq t4 t u i0 (subst1_single i0 u t4 t +H4) x v i H9 (sym_not_eq nat i i0 H10)))) (\lambda (H10: (eq nat i i0)).(let +H11 \def (eq_ind_r nat i0 (\lambda (n: nat).(subst0 n u t4 t)) H4 i H10) in +(let H12 \def (eq_ind_r nat i0 (\lambda (n: nat).(getl n c (CHead d (Bind +Abbr) u))) H7 i H10) in (let H13 \def (eq_ind C (CHead e (Bind Abbr) v) +(\lambda (c1: C).(getl i c c1)) H (CHead d (Bind Abbr) u) (getl_mono c (CHead +e (Bind Abbr) v) i H (CHead d (Bind Abbr) u) H12)) in (let H14 \def (f_equal +C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) +\Rightarrow e | (CHead c1 _ _) \Rightarrow c1])) (CHead e (Bind Abbr) v) +(CHead d (Bind Abbr) u) (getl_mono c (CHead e (Bind Abbr) v) i H (CHead d +(Bind Abbr) u) H12)) in ((let H15 \def (f_equal C T (\lambda (e0: C).(match +e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow v | (CHead _ _ +t0) \Rightarrow t0])) (CHead e (Bind Abbr) v) (CHead d (Bind Abbr) u) +(getl_mono c (CHead e (Bind Abbr) v) i H (CHead d (Bind Abbr) u) H12)) in +(\lambda (H16: (eq C e d)).(let H17 \def (eq_ind_r T u (\lambda (t0: T).(getl +i c (CHead d (Bind Abbr) t0))) H13 v H15) in (let H18 \def (eq_ind_r T u +(\lambda (t0: T).(subst0 i t0 t4 t)) H11 v H15) in (let H19 \def (eq_ind_r C +d (\lambda (c1: C).(getl i c (CHead c1 (Bind Abbr) v))) H17 e H16) in +(ex2_ind T (\lambda (t0: T).(subst1 i v t t0)) (\lambda (t0: T).(subst1 i v x +t0)) (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t +w2))) (\lambda (x0: T).(\lambda (H20: (subst1 i v t x0)).(\lambda (H21: +(subst1 i v x x0)).(ex_intro2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: +T).(subst1 i v t w2)) x0 (pr2_delta1 c e v i H19 w1 x H8 x0 H21) H20)))) +(subst1_confluence_eq t4 t v i (subst1_single i v t4 t H18) x H9))))))) +H14)))))))))) (pr0_subst1 t3 t4 H3 v w1 i H6 v (pr0_refl v))) c0 +H5))))))))))))))) y t1 t2 H1))) H0)))))))). +(* COMMENTS +Initial nodes: 1311 +END *) + +theorem pr2_gen_cabbr: + \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall +(e: C).(\forall (u: T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) +\to (\forall (a0: C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d +a0 a) \to (\forall (x1: T).((subst1 d u t1 (lift (S O) d x1)) \to (ex2 T +(\lambda (x2: T).(subst1 d u t2 (lift (S O) d x2))) (\lambda (x2: T).(pr2 a +x1 x2)))))))))))))))) +\def + \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 +t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (e: +C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u)) \to +(\forall (a0: C).((csubst1 d u c0 a0) \to (\forall (a: C).((drop (S O) d a0 +a) \to (\forall (x1: T).((subst1 d u t (lift (S O) d x1)) \to (ex2 T (\lambda +(x2: T).(subst1 d u t0 (lift (S O) d x2))) (\lambda (x2: T).(pr2 a x1 +x2)))))))))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: +T).(\lambda (H0: (pr0 t3 t4)).(\lambda (e: C).(\lambda (u: T).(\lambda (d: +nat).(\lambda (_: (getl d c0 (CHead e (Bind Abbr) u))).(\lambda (a0: +C).(\lambda (_: (csubst1 d u c0 a0)).(\lambda (a: C).(\lambda (_: (drop (S O) +d a0 a)).(\lambda (x1: T).(\lambda (H4: (subst1 d u t3 (lift (S O) d +x1))).(ex2_ind T (\lambda (w2: T).(pr0 (lift (S O) d x1) w2)) (\lambda (w2: +T).(subst1 d u t4 w2)) (ex2 T (\lambda (x2: T).(subst1 d u t4 (lift (S O) d +x2))) (\lambda (x2: T).(pr2 a x1 x2))) (\lambda (x: T).(\lambda (H5: (pr0 +(lift (S O) d x1) x)).(\lambda (H6: (subst1 d u t4 x)).(ex2_ind T (\lambda +(t5: T).(eq T x (lift (S O) d t5))) (\lambda (t5: T).(pr0 x1 t5)) (ex2 T +(\lambda (x2: T).(subst1 d u t4 (lift (S O) d x2))) (\lambda (x2: T).(pr2 a +x1 x2))) (\lambda (x0: T).(\lambda (H7: (eq T x (lift (S O) d x0))).(\lambda +(H8: (pr0 x1 x0)).(let H9 \def (eq_ind T x (\lambda (t: T).(subst1 d u t4 t)) +H6 (lift (S O) d x0) H7) in (ex_intro2 T (\lambda (x2: T).(subst1 d u t4 +(lift (S O) d x2))) (\lambda (x2: T).(pr2 a x1 x2)) x0 H9 (pr2_free a x1 x0 +H8)))))) (pr0_gen_lift x1 x (S O) d H5))))) (pr0_subst1 t3 t4 H0 u (lift (S +O) d x1) d H4 u (pr0_refl u))))))))))))))))) (\lambda (c0: C).(\lambda (d: +C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind +Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 +t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (e: +C).(\lambda (u0: T).(\lambda (d0: nat).(\lambda (H3: (getl d0 c0 (CHead e +(Bind Abbr) u0))).(\lambda (a0: C).(\lambda (H4: (csubst1 d0 u0 c0 +a0)).(\lambda (a: C).(\lambda (H5: (drop (S O) d0 a0 a)).(\lambda (x1: +T).(\lambda (H6: (subst1 d0 u0 t3 (lift (S O) d0 x1))).(ex2_ind T (\lambda +(w2: T).(pr0 (lift (S O) d0 x1) w2)) (\lambda (w2: T).(subst1 d0 u0 t4 w2)) +(ex2 T (\lambda (x2: T).(subst1 d0 u0 t (lift (S O) d0 x2))) (\lambda (x2: +T).(pr2 a x1 x2))) (\lambda (x: T).(\lambda (H7: (pr0 (lift (S O) d0 x1) +x)).(\lambda (H8: (subst1 d0 u0 t4 x)).(ex2_ind T (\lambda (t5: T).(eq T x +(lift (S O) d0 t5))) (\lambda (t5: T).(pr0 x1 t5)) (ex2 T (\lambda (x2: +T).(subst1 d0 u0 t (lift (S O) d0 x2))) (\lambda (x2: T).(pr2 a x1 x2))) +(\lambda (x0: T).(\lambda (H9: (eq T x (lift (S O) d0 x0))).(\lambda (H10: +(pr0 x1 x0)).(let H11 \def (eq_ind T x (\lambda (t0: T).(subst1 d0 u0 t4 t0)) +H8 (lift (S O) d0 x0) H9) in (lt_eq_gt_e i d0 (ex2 T (\lambda (x2: T).(subst1 +d0 u0 t (lift (S O) d0 x2))) (\lambda (x2: T).(pr2 a x1 x2))) (\lambda (H12: +(lt i d0)).(ex2_ind T (\lambda (t0: T).(subst1 d0 u0 t t0)) (\lambda (t0: +T).(subst1 i u (lift (S O) d0 x0) t0)) (ex2 T (\lambda (x2: T).(subst1 d0 u0 +t (lift (S O) d0 x2))) (\lambda (x2: T).(pr2 a x1 x2))) (\lambda (x2: +T).(\lambda (H13: (subst1 d0 u0 t x2)).(\lambda (H14: (subst1 i u (lift (S O) +d0 x0) x2)).(ex2_ind C (\lambda (e2: C).(csubst1 (minus d0 i) u0 (CHead d +(Bind Abbr) u) e2)) (\lambda (e2: C).(getl i a0 e2)) (ex2 T (\lambda (x3: +T).(subst1 d0 u0 t (lift (S O) d0 x3))) (\lambda (x3: T).(pr2 a x1 x3))) +(\lambda (x3: C).(\lambda (H15: (csubst1 (minus d0 i) u0 (CHead d (Bind Abbr) +u) x3)).(\lambda (H16: (getl i a0 x3)).(let H17 \def (eq_ind nat (minus d0 i) +(\lambda (n: nat).(csubst1 n u0 (CHead d (Bind Abbr) u) x3)) H15 (S (minus d0 +(S i))) (minus_x_Sy d0 i H12)) in (let H18 \def (csubst1_gen_head (Bind Abbr) +d x3 u u0 (minus d0 (S i)) H17) in (ex3_2_ind T C (\lambda (u2: T).(\lambda +(c2: C).(eq C x3 (CHead c2 (Bind Abbr) u2)))) (\lambda (u2: T).(\lambda (_: +C).(subst1 (minus d0 (S i)) u0 u u2))) (\lambda (_: T).(\lambda (c2: +C).(csubst1 (minus d0 (S i)) u0 d c2))) (ex2 T (\lambda (x4: T).(subst1 d0 u0 +t (lift (S O) d0 x4))) (\lambda (x4: T).(pr2 a x1 x4))) (\lambda (x4: +T).(\lambda (x5: C).(\lambda (H19: (eq C x3 (CHead x5 (Bind Abbr) +x4))).(\lambda (H20: (subst1 (minus d0 (S i)) u0 u x4)).(\lambda (_: (csubst1 +(minus d0 (S i)) u0 d x5)).(let H22 \def (eq_ind C x3 (\lambda (c1: C).(getl +i a0 c1)) H16 (CHead x5 (Bind Abbr) x4) H19) in (let H23 \def (eq_ind nat d0 +(\lambda (n: nat).(drop (S O) n a0 a)) H5 (S (plus i (minus d0 (S i)))) +(lt_plus_minus i d0 H12)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: +C).(eq T x4 (lift (S O) (minus d0 (S i)) v)))) (\lambda (v: T).(\lambda (e0: +C).(getl i a (CHead e0 (Bind Abbr) v)))) (\lambda (_: T).(\lambda (e0: +C).(drop (S O) (minus d0 (S i)) x5 e0))) (ex2 T (\lambda (x6: T).(subst1 d0 +u0 t (lift (S O) d0 x6))) (\lambda (x6: T).(pr2 a x1 x6))) (\lambda (x6: +T).(\lambda (x7: C).(\lambda (H24: (eq T x4 (lift (S O) (minus d0 (S i)) +x6))).(\lambda (H25: (getl i a (CHead x7 (Bind Abbr) x6))).(\lambda (_: (drop +(S O) (minus d0 (S i)) x5 x7)).(let H27 \def (eq_ind T x4 (\lambda (t0: +T).(subst1 (minus d0 (S i)) u0 u t0)) H20 (lift (S O) (minus d0 (S i)) x6) +H24) in (ex2_ind T (\lambda (t0: T).(subst1 i (lift (S O) (minus d0 (S i)) +x6) (lift (S O) d0 x0) t0)) (\lambda (t0: T).(subst1 (S (plus (minus d0 (S +i)) i)) u0 x2 t0)) (ex2 T (\lambda (x8: T).(subst1 d0 u0 t (lift (S O) d0 +x8))) (\lambda (x8: T).(pr2 a x1 x8))) (\lambda (x8: T).(\lambda (H28: +(subst1 i (lift (S O) (minus d0 (S i)) x6) (lift (S O) d0 x0) x8)).(\lambda +(H29: (subst1 (S (plus (minus d0 (S i)) i)) u0 x2 x8)).(let H30 \def (eq_ind +nat d0 (\lambda (n: nat).(subst1 i (lift (S O) (minus d0 (S i)) x6) (lift (S +O) n x0) x8)) H28 (S (plus i (minus d0 (S i)))) (lt_plus_minus i d0 H12)) in +(ex2_ind T (\lambda (t5: T).(eq T x8 (lift (S O) (S (plus i (minus d0 (S +i)))) t5))) (\lambda (t5: T).(subst1 i x6 x0 t5)) (ex2 T (\lambda (x9: +T).(subst1 d0 u0 t (lift (S O) d0 x9))) (\lambda (x9: T).(pr2 a x1 x9))) +(\lambda (x9: T).(\lambda (H31: (eq T x8 (lift (S O) (S (plus i (minus d0 (S +i)))) x9))).(\lambda (H32: (subst1 i x6 x0 x9)).(let H33 \def (eq_ind T x8 +(\lambda (t0: T).(subst1 (S (plus (minus d0 (S i)) i)) u0 x2 t0)) H29 (lift +(S O) (S (plus i (minus d0 (S i)))) x9) H31) in (let H34 \def (eq_ind_r nat +(S (plus i (minus d0 (S i)))) (\lambda (n: nat).(subst1 (S (plus (minus d0 (S +i)) i)) u0 x2 (lift (S O) n x9))) H33 d0 (lt_plus_minus i d0 H12)) in (let +H35 \def (eq_ind_r nat (S (plus (minus d0 (S i)) i)) (\lambda (n: +nat).(subst1 n u0 x2 (lift (S O) d0 x9))) H34 d0 (lt_plus_minus_r i d0 H12)) +in (ex_intro2 T (\lambda (x10: T).(subst1 d0 u0 t (lift (S O) d0 x10))) +(\lambda (x10: T).(pr2 a x1 x10)) x9 (subst1_trans x2 t u0 d0 H13 (lift (S O) +d0 x9) H35) (pr2_delta1 a x7 x6 i H25 x1 x0 H10 x9 H32)))))))) +(subst1_gen_lift_lt x6 x0 x8 i (S O) (minus d0 (S i)) H30)))))) +(subst1_subst1_back (lift (S O) d0 x0) x2 u i H14 (lift (S O) (minus d0 (S +i)) x6) u0 (minus d0 (S i)) H27)))))))) (getl_drop_conf_lt Abbr a0 x5 x4 i +H22 a (S O) (minus d0 (S i)) H23))))))))) H18)))))) (csubst1_getl_lt d0 i H12 +c0 a0 u0 H4 (CHead d (Bind Abbr) u) H0))))) (subst1_confluence_neq t4 t u i +(subst1_single i u t4 t H2) (lift (S O) d0 x0) u0 d0 H11 (lt_neq i d0 H12)))) +(\lambda (H12: (eq nat i d0)).(let H13 \def (eq_ind_r nat d0 (\lambda (n: +nat).(subst1 n u0 t4 (lift (S O) n x0))) H11 i H12) in (let H14 \def +(eq_ind_r nat d0 (\lambda (n: nat).(drop (S O) n a0 a)) H5 i H12) in (let H15 +\def (eq_ind_r nat d0 (\lambda (n: nat).(csubst1 n u0 c0 a0)) H4 i H12) in +(let H16 \def (eq_ind_r nat d0 (\lambda (n: nat).(getl n c0 (CHead e (Bind +Abbr) u0))) H3 i H12) in (eq_ind nat i (\lambda (n: nat).(ex2 T (\lambda (x2: +T).(subst1 n u0 t (lift (S O) n x2))) (\lambda (x2: T).(pr2 a x1 x2)))) (let +H17 \def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c1: C).(getl i c0 c1)) +H0 (CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead +e (Bind Abbr) u0) H16)) in (let H18 \def (f_equal C C (\lambda (e0: C).(match +e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ +_) \Rightarrow c1])) (CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0) +(getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead e (Bind Abbr) u0) H16)) in +((let H19 \def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda +(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) +(CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d (Bind +Abbr) u) i H0 (CHead e (Bind Abbr) u0) H16)) in (\lambda (H20: (eq C d +e)).(let H21 \def (eq_ind_r T u0 (\lambda (t0: T).(getl i c0 (CHead e (Bind +Abbr) t0))) H17 u H19) in (let H22 \def (eq_ind_r T u0 (\lambda (t0: +T).(subst1 i t0 t4 (lift (S O) i x0))) H13 u H19) in (let H23 \def (eq_ind_r +T u0 (\lambda (t0: T).(csubst1 i t0 c0 a0)) H15 u H19) in (eq_ind T u +(\lambda (t0: T).(ex2 T (\lambda (x2: T).(subst1 i t0 t (lift (S O) i x2))) +(\lambda (x2: T).(pr2 a x1 x2)))) (let H24 \def (eq_ind_r C e (\lambda (c1: +C).(getl i c0 (CHead c1 (Bind Abbr) u))) H21 d H20) in (ex2_ind T (\lambda +(t0: T).(subst1 i u t t0)) (\lambda (t0: T).(subst1 i u (lift (S O) i x0) +t0)) (ex2 T (\lambda (x2: T).(subst1 i u t (lift (S O) i x2))) (\lambda (x2: +T).(pr2 a x1 x2))) (\lambda (x2: T).(\lambda (H25: (subst1 i u t +x2)).(\lambda (H26: (subst1 i u (lift (S O) i x0) x2)).(let H27 \def (eq_ind +T x2 (\lambda (t0: T).(subst1 i u t t0)) H25 (lift (S O) i x0) +(subst1_gen_lift_eq x0 u x2 (S O) i i (le_n i) (eq_ind_r nat (plus (S O) i) +(\lambda (n: nat).(lt i n)) (le_n (plus (S O) i)) (plus i (S O)) (plus_sym i +(S O))) H26)) in (ex_intro2 T (\lambda (x3: T).(subst1 i u t (lift (S O) i +x3))) (\lambda (x3: T).(pr2 a x1 x3)) x0 H27 (pr2_free a x1 x0 H10)))))) +(subst1_confluence_eq t4 t u i (subst1_single i u t4 t H2) (lift (S O) i x0) +H22))) u0 H19)))))) H18))) d0 H12)))))) (\lambda (H12: (lt d0 i)).(ex2_ind T +(\lambda (t0: T).(subst1 d0 u0 t t0)) (\lambda (t0: T).(subst1 i u (lift (S +O) d0 x0) t0)) (ex2 T (\lambda (x2: T).(subst1 d0 u0 t (lift (S O) d0 x2))) +(\lambda (x2: T).(pr2 a x1 x2))) (\lambda (x2: T).(\lambda (H13: (subst1 d0 +u0 t x2)).(\lambda (H14: (subst1 i u (lift (S O) d0 x0) x2)).(ex2_ind T +(\lambda (t5: T).(eq T x2 (lift (S O) d0 t5))) (\lambda (t5: T).(subst1 +(minus i (S O)) u x0 t5)) (ex2 T (\lambda (x3: T).(subst1 d0 u0 t (lift (S O) +d0 x3))) (\lambda (x3: T).(pr2 a x1 x3))) (\lambda (x3: T).(\lambda (H15: (eq +T x2 (lift (S O) d0 x3))).(\lambda (H16: (subst1 (minus i (S O)) u x0 +x3)).(let H17 \def (eq_ind T x2 (\lambda (t0: T).(subst1 d0 u0 t t0)) H13 +(lift (S O) d0 x3) H15) in (ex_intro2 T (\lambda (x4: T).(subst1 d0 u0 t +(lift (S O) d0 x4))) (\lambda (x4: T).(pr2 a x1 x4)) x3 H17 (pr2_delta1 a d u +(minus i (S O)) (getl_drop_conf_ge i (CHead d (Bind Abbr) u) a0 +(csubst1_getl_ge d0 i (le_S_n d0 i (le_S (S d0) i H12)) c0 a0 u0 H4 (CHead d +(Bind Abbr) u) H0) a (S O) d0 H5 (eq_ind_r nat (plus (S O) d0) (\lambda (n: +nat).(le n i)) H12 (plus d0 (S O)) (plus_sym d0 (S O)))) x1 x0 H10 x3 +H16)))))) (subst1_gen_lift_ge u x0 x2 i (S O) d0 H14 (eq_ind_r nat (plus (S +O) d0) (\lambda (n: nat).(le n i)) H12 (plus d0 (S O)) (plus_sym d0 (S +O)))))))) (subst1_confluence_neq t4 t u i (subst1_single i u t4 t H2) (lift +(S O) d0 x0) u0 d0 H11 (sym_not_eq nat d0 i (lt_neq d0 i H12)))))))))) +(pr0_gen_lift x1 x (S O) d0 H7))))) (pr0_subst1 t3 t4 H1 u0 (lift (S O) d0 +x1) d0 H6 u0 (pr0_refl u0))))))))))))))))))))))) c t1 t2 H)))). +(* COMMENTS +Initial nodes: 3757 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr3/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/pr3/defs.ma new file mode 100644 index 000000000..9619ee1d7 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/pr3/defs.ma @@ -0,0 +1,23 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/pr2/defs.ma". + +inductive pr3 (c: C): T \to (T \to Prop) \def +| pr3_refl: \forall (t: T).(pr3 c t t) +| pr3_sing: \forall (t2: T).(\forall (t1: T).((pr2 c t1 t2) \to (\forall (t3: +T).((pr3 c t2 t3) \to (pr3 c t1 t3))))). + diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr3/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/pr3/fwd.ma new file mode 100644 index 000000000..726c420f0 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/pr3/fwd.ma @@ -0,0 +1,1604 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/pr3/props.ma". + +include "Basic-1/pr2/fwd.ma". + +theorem pr3_gen_sort: + \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr3 c (TSort n) x) \to +(eq T x (TSort n))))) +\def + \lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr3 c (TSort +n) x)).(insert_eq T (TSort n) (\lambda (t: T).(pr3 c t x)) (\lambda (t: +T).(eq T x t)) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(pr3_ind c (\lambda +(t: T).(\lambda (t0: T).((eq T t (TSort n)) \to (eq T t0 t)))) (\lambda (t: +T).(\lambda (_: (eq T t (TSort n))).(refl_equal T t))) (\lambda (t2: +T).(\lambda (t1: T).(\lambda (H1: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda +(_: (pr3 c t2 t3)).(\lambda (H3: (((eq T t2 (TSort n)) \to (eq T t3 +t2)))).(\lambda (H4: (eq T t1 (TSort n))).(let H5 \def (eq_ind T t1 (\lambda +(t: T).(pr2 c t t2)) H1 (TSort n) H4) in (eq_ind_r T (TSort n) (\lambda (t: +T).(eq T t3 t)) (let H6 \def (eq_ind T t2 (\lambda (t: T).((eq T t (TSort n)) +\to (eq T t3 t))) H3 (TSort n) (pr2_gen_sort c t2 n H5)) in (H6 (refl_equal T +(TSort n)))) t1 H4))))))))) y x H0))) H)))). +(* COMMENTS +Initial nodes: 253 +END *) + +theorem pr3_gen_abst: + \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c +(THead (Bind Abst) u1 t1) x) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: +T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 +c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: +T).(pr3 (CHead c (Bind b) u) t1 t2)))))))))) +\def + \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda +(H: (pr3 c (THead (Bind Abst) u1 t1) x)).(insert_eq T (THead (Bind Abst) u1 +t1) (\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(ex3_2 T T (\lambda (u2: +T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall +(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t1 t2))))))) (\lambda (y: +T).(\lambda (H0: (pr3 c y x)).(unintro T t1 (\lambda (t: T).((eq T y (THead +(Bind Abst) u1 t)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x +(THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) +(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead +c (Bind b) u) t t2)))))))) (unintro T u1 (\lambda (t: T).(\forall (x0: +T).((eq T y (THead (Bind Abst) t x0)) \to (ex3_2 T T (\lambda (u2: +T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_: T).(\lambda (t2: T).(\forall +(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x0 t2))))))))) (pr3_ind c +(\lambda (t: T).(\lambda (t0: T).(\forall (x0: T).(\forall (x1: T).((eq T t +(THead (Bind Abst) x0 x1)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: +T).(eq T t0 (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: +T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall +(u: T).(pr3 (CHead c (Bind b) u) x1 t2))))))))))) (\lambda (t: T).(\lambda +(x0: T).(\lambda (x1: T).(\lambda (H1: (eq T t (THead (Bind Abst) x0 +x1))).(ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T t (THead (Bind +Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: +T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) +x1 t2))))) x0 x1 H1 (pr3_refl c x0) (\lambda (b: B).(\lambda (u: T).(pr3_refl +(CHead c (Bind b) u) x1)))))))) (\lambda (t2: T).(\lambda (t3: T).(\lambda +(H1: (pr2 c t3 t2)).(\lambda (t4: T).(\lambda (_: (pr3 c t2 t4)).(\lambda +(H3: ((\forall (x0: T).(\forall (x1: T).((eq T t2 (THead (Bind Abst) x0 x1)) +\to (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst) +u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: +T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) +x1 t5))))))))))).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t3 +(THead (Bind Abst) x0 x1))).(let H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c +t t2)) H1 (THead (Bind Abst) x0 x1) H4) in (let H6 \def (pr2_gen_abst c x0 x1 +t2 H5) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead +(Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) +(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead +c (Bind b) u) x1 t5))))) (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T +t4 (THead (Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 +u2))) (\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 +(CHead c (Bind b) u) x1 t5)))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda +(H7: (eq T t2 (THead (Bind Abst) x2 x3))).(\lambda (H8: (pr2 c x0 +x2)).(\lambda (H9: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) +x1 x3))))).(let H10 \def (eq_ind T t2 (\lambda (t: T).(\forall (x4: +T).(\forall (x5: T).((eq T t (THead (Bind Abst) x4 x5)) \to (ex3_2 T T +(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst) u2 t5)))) +(\lambda (u2: T).(\lambda (_: T).(pr3 c x4 u2))) (\lambda (_: T).(\lambda +(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x5 +t5)))))))))) H3 (THead (Bind Abst) x2 x3) H7) in (let H11 \def (H10 x2 x3 +(refl_equal T (THead (Bind Abst) x2 x3))) in (ex3_2_ind T T (\lambda (u2: +T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst) u2 t5)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall +(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 t5))))) (ex3_2 T T +(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst) u2 t5)))) +(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda +(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) +(\lambda (x4: T).(\lambda (x5: T).(\lambda (H12: (eq T t4 (THead (Bind Abst) +x4 x5))).(\lambda (H13: (pr3 c x2 x4)).(\lambda (H14: ((\forall (b: +B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 x5))))).(ex3_2_intro T T +(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst) u2 t5)))) +(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda +(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5))))) +x4 x5 H12 (pr3_sing c x2 x0 H8 x4 H13) (\lambda (b: B).(\lambda (u: +T).(pr3_sing (CHead c (Bind b) u) x3 x1 (H9 b u) x5 (H14 b u)))))))))) +H11)))))))) H6)))))))))))) y x H0))))) H))))). +(* COMMENTS +Initial nodes: 1261 +END *) + +theorem pr3_gen_cast: + \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c +(THead (Flat Cast) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda +(t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: +T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t1 t2)))) (pr3 c +t1 x)))))) +\def + \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda +(H: (pr3 c (THead (Flat Cast) u1 t1) x)).(insert_eq T (THead (Flat Cast) u1 +t1) (\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: +T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t1 +t2)))) (pr3 c t1 x))) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(unintro T +t1 (\lambda (t: T).((eq T y (THead (Flat Cast) u1 t)) \to (or (ex3_2 T T +(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) +(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda +(t2: T).(pr3 c t t2)))) (pr3 c t x)))) (unintro T u1 (\lambda (t: T).(\forall +(x0: T).((eq T y (THead (Flat Cast) t x0)) \to (or (ex3_2 T T (\lambda (u2: +T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c x0 +t2)))) (pr3 c x0 x))))) (pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall +(x0: T).(\forall (x1: T).((eq T t (THead (Flat Cast) x0 x1)) \to (or (ex3_2 T +T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Flat Cast) u2 t2)))) +(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda +(t2: T).(pr3 c x1 t2)))) (pr3 c x1 t0))))))) (\lambda (t: T).(\lambda (x0: +T).(\lambda (x1: T).(\lambda (H1: (eq T t (THead (Flat Cast) x0 +x1))).(eq_ind_r T (THead (Flat Cast) x0 x1) (\lambda (t0: T).(or (ex3_2 T T +(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Flat Cast) u2 t2)))) +(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda +(t2: T).(pr3 c x1 t2)))) (pr3 c x1 t0))) (or_introl (ex3_2 T T (\lambda (u2: +T).(\lambda (t2: T).(eq T (THead (Flat Cast) x0 x1) (THead (Flat Cast) u2 +t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: +T).(\lambda (t2: T).(pr3 c x1 t2)))) (pr3 c x1 (THead (Flat Cast) x0 x1)) +(ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Cast) +x0 x1) (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c +x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c x1 t2))) x0 x1 (refl_equal T +(THead (Flat Cast) x0 x1)) (pr3_refl c x0) (pr3_refl c x1))) t H1))))) +(\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c t3 t2)).(\lambda (t4: +T).(\lambda (H2: (pr3 c t2 t4)).(\lambda (H3: ((\forall (x0: T).(\forall (x1: +T).((eq T t2 (THead (Flat Cast) x0 x1)) \to (or (ex3_2 T T (\lambda (u2: +T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 t5)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 +t5)))) (pr3 c x1 t4))))))).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: +(eq T t3 (THead (Flat Cast) x0 x1))).(let H5 \def (eq_ind T t3 (\lambda (t: +T).(pr2 c t t2)) H1 (THead (Flat Cast) x0 x1) H4) in (let H6 \def +(pr2_gen_cast c x0 x1 t2 H5) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda +(t5: T).(eq T t2 (THead (Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: +T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr2 c x1 t5)))) (pr2 c +x1 t2) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat +Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: +T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4)) (\lambda (H7: (ex3_2 T T +(\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Flat Cast) u2 t5)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda +(t5: T).(pr2 c x1 t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: +T).(eq T t2 (THead (Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: +T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr2 c x1 t5))) (or +(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 +t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: +T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4)) (\lambda (x2: T).(\lambda +(x3: T).(\lambda (H8: (eq T t2 (THead (Flat Cast) x2 x3))).(\lambda (H9: (pr2 +c x0 x2)).(\lambda (H10: (pr2 c x1 x3)).(let H11 \def (eq_ind T t2 (\lambda +(t: T).(\forall (x4: T).(\forall (x5: T).((eq T t (THead (Flat Cast) x4 x5)) +\to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat +Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x4 u2))) (\lambda (_: +T).(\lambda (t5: T).(pr3 c x5 t5)))) (pr3 c x5 t4)))))) H3 (THead (Flat Cast) +x2 x3) H8) in (let H12 \def (H11 x2 x3 (refl_equal T (THead (Flat Cast) x2 +x3))) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead +(Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) +(\lambda (_: T).(\lambda (t5: T).(pr3 c x3 t5)))) (pr3 c x3 t4) (or (ex3_2 T +T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 t5)))) +(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda +(t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4)) (\lambda (H13: (ex3_2 T T (\lambda +(u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 t5)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x3 +t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead +(Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) +(\lambda (_: T).(\lambda (t5: T).(pr3 c x3 t5))) (or (ex3_2 T T (\lambda (u2: +T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 t5)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 +t5)))) (pr3 c x1 t4)) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H14: (eq T +t4 (THead (Flat Cast) x4 x5))).(\lambda (H15: (pr3 c x2 x4)).(\lambda (H16: +(pr3 c x3 x5)).(eq_ind_r T (THead (Flat Cast) x4 x5) (\lambda (t: T).(or +(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t (THead (Flat Cast) u2 +t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: +T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t))) (or_introl (ex3_2 T T +(\lambda (u2: T).(\lambda (t5: T).(eq T (THead (Flat Cast) x4 x5) (THead +(Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) +(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 (THead (Flat +Cast) x4 x5)) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t5: T).(eq T (THead +(Flat Cast) x4 x5) (THead (Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: +T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5))) x4 x5 +(refl_equal T (THead (Flat Cast) x4 x5)) (pr3_sing c x2 x0 H9 x4 H15) +(pr3_sing c x3 x1 H10 x5 H16))) t4 H14)))))) H13)) (\lambda (H13: (pr3 c x3 +t4)).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead +(Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) +(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4) (pr3_sing c +x3 x1 H10 t4 H13))) H12)))))))) H7)) (\lambda (H7: (pr2 c x1 t2)).(or_intror +(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 +t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: +T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4) (pr3_sing c t2 x1 H7 t4 +H2))) H6)))))))))))) y x H0))))) H))))). +(* COMMENTS +Initial nodes: 2001 +END *) + +theorem pr3_gen_lift: + \forall (c: C).(\forall (t1: T).(\forall (x: T).(\forall (h: nat).(\forall +(d: nat).((pr3 c (lift h d t1) x) \to (\forall (e: C).((drop h d c e) \to +(ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr3 e t1 +t2)))))))))) +\def + \lambda (c: C).(\lambda (t1: T).(\lambda (x: T).(\lambda (h: nat).(\lambda +(d: nat).(\lambda (H: (pr3 c (lift h d t1) x)).(insert_eq T (lift h d t1) +(\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(\forall (e: C).((drop h d c e) +\to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr3 e +t1 t2)))))) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(unintro T t1 (\lambda +(t: T).((eq T y (lift h d t)) \to (\forall (e: C).((drop h d c e) \to (ex2 T +(\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr3 e t t2))))))) +(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (x0: T).((eq T t (lift h +d x0)) \to (\forall (e: C).((drop h d c e) \to (ex2 T (\lambda (t2: T).(eq T +t0 (lift h d t2))) (\lambda (t2: T).(pr3 e x0 t2))))))))) (\lambda (t: +T).(\lambda (x0: T).(\lambda (H1: (eq T t (lift h d x0))).(\lambda (e: +C).(\lambda (_: (drop h d c e)).(ex_intro2 T (\lambda (t2: T).(eq T t (lift h +d t2))) (\lambda (t2: T).(pr3 e x0 t2)) x0 H1 (pr3_refl e x0))))))) (\lambda +(t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c t3 t2)).(\lambda (t4: +T).(\lambda (_: (pr3 c t2 t4)).(\lambda (H3: ((\forall (x0: T).((eq T t2 +(lift h d x0)) \to (\forall (e: C).((drop h d c e) \to (ex2 T (\lambda (t5: +T).(eq T t4 (lift h d t5))) (\lambda (t5: T).(pr3 e x0 t5))))))))).(\lambda +(x0: T).(\lambda (H4: (eq T t3 (lift h d x0))).(\lambda (e: C).(\lambda (H5: +(drop h d c e)).(let H6 \def (eq_ind T t3 (\lambda (t: T).(pr2 c t t2)) H1 +(lift h d x0) H4) in (let H7 \def (pr2_gen_lift c x0 t2 h d H6 e H5) in +(ex2_ind T (\lambda (t5: T).(eq T t2 (lift h d t5))) (\lambda (t5: T).(pr2 e +x0 t5)) (ex2 T (\lambda (t5: T).(eq T t4 (lift h d t5))) (\lambda (t5: +T).(pr3 e x0 t5))) (\lambda (x1: T).(\lambda (H8: (eq T t2 (lift h d +x1))).(\lambda (H9: (pr2 e x0 x1)).(ex2_ind T (\lambda (t5: T).(eq T t4 (lift +h d t5))) (\lambda (t5: T).(pr3 e x1 t5)) (ex2 T (\lambda (t5: T).(eq T t4 +(lift h d t5))) (\lambda (t5: T).(pr3 e x0 t5))) (\lambda (x2: T).(\lambda +(H10: (eq T t4 (lift h d x2))).(\lambda (H11: (pr3 e x1 x2)).(ex_intro2 T +(\lambda (t5: T).(eq T t4 (lift h d t5))) (\lambda (t5: T).(pr3 e x0 t5)) x2 +H10 (pr3_sing e x1 x0 H9 x2 H11))))) (H3 x1 H8 e H5))))) H7))))))))))))) y x +H0)))) H)))))). +(* COMMENTS +Initial nodes: 689 +END *) + +theorem pr3_gen_lref: + \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr3 c (TLRef n) x) \to +(or (eq T x (TLRef n)) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda +(_: T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: +T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda +(v: T).(eq T x (lift (S n) O v)))))))))) +\def + \lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr3 c (TLRef +n) x)).(insert_eq T (TLRef n) (\lambda (t: T).(pr3 c t x)) (\lambda (t: +T).(or (eq T x t) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: +T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: +T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda +(v: T).(eq T x (lift (S n) O v)))))))) (\lambda (y: T).(\lambda (H0: (pr3 c y +x)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).((eq T t (TLRef n)) \to (or +(eq T t0 t) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: +T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: +T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda +(v: T).(eq T t0 (lift (S n) O v)))))))))) (\lambda (t: T).(\lambda (_: (eq T +t (TLRef n))).(or_introl (eq T t t) (ex3_3 C T T (\lambda (d: C).(\lambda (u: +T).(\lambda (_: T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: +C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda +(_: T).(\lambda (v: T).(eq T t (lift (S n) O v)))))) (refl_equal T t)))) +(\lambda (t2: T).(\lambda (t1: T).(\lambda (H1: (pr2 c t1 t2)).(\lambda (t3: +T).(\lambda (H2: (pr3 c t2 t3)).(\lambda (H3: (((eq T t2 (TLRef n)) \to (or +(eq T t3 t2) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: +T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: +T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda +(v: T).(eq T t3 (lift (S n) O v)))))))))).(\lambda (H4: (eq T t1 (TLRef +n))).(let H5 \def (eq_ind T t1 (\lambda (t: T).(pr2 c t t2)) H1 (TLRef n) H4) +in (eq_ind_r T (TLRef n) (\lambda (t: T).(or (eq T t3 t) (ex3_3 C T T +(\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead d (Bind +Abbr) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) +(\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T t3 (lift (S n) O +v)))))))) (let H6 \def (pr2_gen_lref c t2 n H5) in (or_ind (eq T t2 (TLRef +n)) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) +u)))) (\lambda (_: C).(\lambda (u: T).(eq T t2 (lift (S n) O u))))) (or (eq T +t3 (TLRef n)) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: +T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: +T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda +(v: T).(eq T t3 (lift (S n) O v))))))) (\lambda (H7: (eq T t2 (TLRef +n))).(let H8 \def (eq_ind T t2 (\lambda (t: T).((eq T t (TLRef n)) \to (or +(eq T t3 t) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: +T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: +T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda +(v: T).(eq T t3 (lift (S n) O v))))))))) H3 (TLRef n) H7) in (let H9 \def +(eq_ind T t2 (\lambda (t: T).(pr3 c t t3)) H2 (TLRef n) H7) in (H8 +(refl_equal T (TLRef n)))))) (\lambda (H7: (ex2_2 C T (\lambda (d: +C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda (_: +C).(\lambda (u: T).(eq T t2 (lift (S n) O u)))))).(ex2_2_ind C T (\lambda (d: +C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda (_: +C).(\lambda (u: T).(eq T t2 (lift (S n) O u)))) (or (eq T t3 (TLRef n)) +(ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead +d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (v: T).(pr3 d u +v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T t3 (lift (S n) O +v))))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H8: (getl n c (CHead x0 +(Bind Abbr) x1))).(\lambda (H9: (eq T t2 (lift (S n) O x1))).(let H10 \def +(eq_ind T t2 (\lambda (t: T).((eq T t (TLRef n)) \to (or (eq T t3 t) (ex3_3 C +T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead d (Bind +Abbr) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) +(\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T t3 (lift (S n) O +v))))))))) H3 (lift (S n) O x1) H9) in (let H11 \def (eq_ind T t2 (\lambda +(t: T).(pr3 c t t3)) H2 (lift (S n) O x1) H9) in (let H12 \def (pr3_gen_lift +c x1 t3 (S n) O H11 x0 (getl_drop Abbr c x0 x1 n H8)) in (ex2_ind T (\lambda +(t4: T).(eq T t3 (lift (S n) O t4))) (\lambda (t4: T).(pr3 x0 x1 t4)) (or (eq +T t3 (TLRef n)) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: +T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: +T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda +(v: T).(eq T t3 (lift (S n) O v))))))) (\lambda (x2: T).(\lambda (H13: (eq T +t3 (lift (S n) O x2))).(\lambda (H14: (pr3 x0 x1 x2)).(or_intror (eq T t3 +(TLRef n)) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(getl +n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (v: +T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T t3 +(lift (S n) O v)))))) (ex3_3_intro C T T (\lambda (d: C).(\lambda (u: +T).(\lambda (_: T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: +C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda +(_: T).(\lambda (v: T).(eq T t3 (lift (S n) O v))))) x0 x1 x2 H8 H14 H13))))) +H12)))))))) H7)) H6)) t1 H4))))))))) y x H0))) H)))). +(* COMMENTS +Initial nodes: 1515 +END *) + +theorem pr3_gen_void: + \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c +(THead (Bind Void) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda +(t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: +T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall +(u: T).(pr3 (CHead c (Bind b) u) t1 t2)))))) (pr3 (CHead c (Bind Void) u1) t1 +(lift (S O) O x))))))) +\def + \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda +(H: (pr3 c (THead (Bind Void) u1 t1) x)).(insert_eq T (THead (Bind Void) u1 +t1) (\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: +T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall +(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t1 t2)))))) (pr3 (CHead c +(Bind Void) u1) t1 (lift (S O) O x)))) (\lambda (y: T).(\lambda (H0: (pr3 c y +x)).(unintro T t1 (\lambda (t: T).((eq T y (THead (Bind Void) u1 t)) \to (or +(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 +t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: +T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) +t t2)))))) (pr3 (CHead c (Bind Void) u1) t (lift (S O) O x))))) (unintro T u1 +(\lambda (t: T).(\forall (x0: T).((eq T y (THead (Bind Void) t x0)) \to (or +(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 +t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_: +T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) +x0 t2)))))) (pr3 (CHead c (Bind Void) t) x0 (lift (S O) O x)))))) (pr3_ind c +(\lambda (t: T).(\lambda (t0: T).(\forall (x0: T).(\forall (x1: T).((eq T t +(THead (Bind Void) x0 x1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: +T).(eq T t0 (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: +T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall +(u: T).(pr3 (CHead c (Bind b) u) x1 t2)))))) (pr3 (CHead c (Bind Void) x0) x1 +(lift (S O) O t0)))))))) (\lambda (t: T).(\lambda (x0: T).(\lambda (x1: +T).(\lambda (H1: (eq T t (THead (Bind Void) x0 x1))).(eq_ind_r T (THead (Bind +Void) x0 x1) (\lambda (t0: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: +T).(eq T t0 (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: +T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall +(u: T).(pr3 (CHead c (Bind b) u) x1 t2)))))) (pr3 (CHead c (Bind Void) x0) x1 +(lift (S O) O t0)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2: +T).(eq T (THead (Bind Void) x0 x1) (THead (Bind Void) u2 t2)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall +(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t2)))))) (pr3 (CHead c +(Bind Void) x0) x1 (lift (S O) O (THead (Bind Void) x0 x1))) (ex3_2_intro T T +(\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Bind Void) x0 x1) (THead +(Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) +(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead +c (Bind b) u) x1 t2))))) x0 x1 (refl_equal T (THead (Bind Void) x0 x1)) +(pr3_refl c x0) (\lambda (b: B).(\lambda (u: T).(pr3_refl (CHead c (Bind b) +u) x1))))) t H1))))) (\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c +t3 t2)).(\lambda (t4: T).(\lambda (H2: (pr3 c t2 t4)).(\lambda (H3: ((\forall +(x0: T).(\forall (x1: T).((eq T t2 (THead (Bind Void) x0 x1)) \to (or (ex3_2 +T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) +(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda +(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) +(pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4)))))))).(\lambda (x0: +T).(\lambda (x1: T).(\lambda (H4: (eq T t3 (THead (Bind Void) x0 x1))).(let +H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c t t2)) H1 (THead (Bind Void) x0 +x1) H4) in (let H6 \def (pr2_gen_void c x0 x1 t2 H5) in (or_ind (ex3_2 T T +(\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind Void) u2 t5)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda +(t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 t5)))))) +(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 (lift (S O) O +t2)))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind +Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: +T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) +x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4))) (\lambda +(H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind Void) +u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: +T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) +x1 t5))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead +(Bind Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) +(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead +c (Bind b) u) x1 t5))))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq +T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 +u2))) (\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 +(CHead c (Bind b) u) x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) +O t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H8: (eq T t2 (THead (Bind +Void) x2 x3))).(\lambda (H9: (pr2 c x0 x2)).(\lambda (H10: ((\forall (b: +B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 x3))))).(let H11 \def (eq_ind +T t2 (\lambda (t: T).(\forall (x4: T).(\forall (x5: T).((eq T t (THead (Bind +Void) x4 x5)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 +(THead (Bind Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x4 u2))) +(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead +c (Bind b) u) x5 t5)))))) (pr3 (CHead c (Bind Void) x4) x5 (lift (S O) O +t4))))))) H3 (THead (Bind Void) x2 x3) H8) in (let H12 \def (H11 x2 x3 +(refl_equal T (THead (Bind Void) x2 x3))) in (or_ind (ex3_2 T T (\lambda (u2: +T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall +(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 t5)))))) (pr3 (CHead c +(Bind Void) x2) x3 (lift (S O) O t4)) (or (ex3_2 T T (\lambda (u2: +T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall +(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) (pr3 (CHead c +(Bind Void) x0) x1 (lift (S O) O t4))) (\lambda (H13: (ex3_2 T T (\lambda +(u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall +(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 t5))))))).(ex3_2_ind T T +(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) +(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda +(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 t5))))) +(or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) +u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: +T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) +x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4))) (\lambda +(x4: T).(\lambda (x5: T).(\lambda (H14: (eq T t4 (THead (Bind Void) x4 +x5))).(\lambda (H15: (pr3 c x2 x4)).(\lambda (H16: ((\forall (b: B).(\forall +(u: T).(pr3 (CHead c (Bind b) u) x3 x5))))).(or_introl (ex3_2 T T (\lambda +(u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall +(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) (pr3 (CHead c +(Bind Void) x0) x1 (lift (S O) O t4)) (ex3_2_intro T T (\lambda (u2: +T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall +(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5))))) x4 x5 H14 +(pr3_sing c x2 x0 H9 x4 H15) (\lambda (b: B).(\lambda (u: T).(pr3_sing (CHead +c (Bind b) u) x3 x1 (H10 b u) x5 (H16 b u))))))))))) H13)) (\lambda (H13: +(pr3 (CHead c (Bind Void) x2) x3 (lift (S O) O t4))).(or_intror (ex3_2 T T +(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) +(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda +(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) +(pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4)) (pr3_sing (CHead c (Bind +Void) x0) x3 x1 (H10 Void x0) (lift (S O) O t4) (pr3_pr2_pr3_t c x2 x3 (lift +(S O) O t4) (Bind Void) H13 x0 H9)))) H12)))))))) H7)) (\lambda (H7: +((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 (lift (S O) O +t2)))))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 +(THead (Bind Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) +(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead +c (Bind b) u) x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4)) +(pr3_sing (CHead c (Bind Void) x0) (lift (S O) O t2) x1 (H7 Void x0) (lift (S +O) O t4) (pr3_lift (CHead c (Bind Void) x0) c (S O) O (drop_drop (Bind Void) +O c c (drop_refl c) x0) t2 t4 H2)))) H6)))))))))))) y x H0))))) H))))). +(* COMMENTS +Initial nodes: 2645 +END *) + +theorem pr3_gen_abbr: + \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c +(THead (Bind Abbr) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda +(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: +T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) +u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O x))))))) +\def + \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda +(H: (pr3 c (THead (Bind Abbr) u1 t1) x)).(insert_eq T (THead (Bind Abbr) u1 +t1) (\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: +T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 +(CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S +O) O x)))) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(unintro T t1 (\lambda +(t: T).((eq T y (THead (Bind Abbr) u1 t)) \to (or (ex3_2 T T (\lambda (u2: +T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 +(CHead c (Bind Abbr) u1) t t2)))) (pr3 (CHead c (Bind Abbr) u1) t (lift (S O) +O x))))) (unintro T u1 (\lambda (t: T).(\forall (x0: T).((eq T y (THead (Bind +Abbr) t x0)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x +(THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) +(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) t) x0 t2)))) (pr3 +(CHead c (Bind Abbr) t) x0 (lift (S O) O x)))))) (pr3_ind c (\lambda (t: +T).(\lambda (t0: T).(\forall (x0: T).(\forall (x1: T).((eq T t (THead (Bind +Abbr) x0 x1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 +(THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) +(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) x0) x1 t2)))) (pr3 +(CHead c (Bind Abbr) x0) x1 (lift (S O) O t0)))))))) (\lambda (t: T).(\lambda +(x0: T).(\lambda (x1: T).(\lambda (H1: (eq T t (THead (Bind Abbr) x0 +x1))).(eq_ind_r T (THead (Bind Abbr) x0 x1) (\lambda (t0: T).(or (ex3_2 T T +(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))) +(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda +(t2: T).(pr3 (CHead c (Bind Abbr) x0) x1 t2)))) (pr3 (CHead c (Bind Abbr) x0) +x1 (lift (S O) O t0)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2: +T).(eq T (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 t2)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 +(CHead c (Bind Abbr) x0) x1 t2)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S +O) O (THead (Bind Abbr) x0 x1))) (ex3_2_intro T T (\lambda (u2: T).(\lambda +(t2: T).(eq T (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 t2)))) (\lambda +(u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 +(CHead c (Bind Abbr) x0) x1 t2))) x0 x1 (refl_equal T (THead (Bind Abbr) x0 +x1)) (pr3_refl c x0) (pr3_refl (CHead c (Bind Abbr) x0) x1))) t H1))))) +(\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c t3 t2)).(\lambda (t4: +T).(\lambda (H2: (pr3 c t2 t4)).(\lambda (H3: ((\forall (x0: T).(\forall (x1: +T).((eq T t2 (THead (Bind Abbr) x0 x1)) \to (or (ex3_2 T T (\lambda (u2: +T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 +(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S +O) O t4)))))))).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t3 +(THead (Bind Abbr) x0 x1))).(let H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c +t t2)) H1 (THead (Bind Abbr) x0 x1) H4) in (let H6 \def (pr2_gen_abbr c x0 x1 +t2 H5) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 +(THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) +(\lambda (_: T).(\lambda (t5: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 +(CHead c (Bind b) u) x1 t5))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: +T).(pr2 (CHead c (Bind Abbr) u) x1 t5))) (ex3_2 T T (\lambda (y0: T).(\lambda +(_: T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: +T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) +z t5)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 +(lift (S O) O t2)))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T +t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 +u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 +t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (H7: +(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind Abbr) u2 +t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: +T).(\lambda (t5: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind +b) u) x1 t5))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead +c (Bind Abbr) u) x1 t5))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 +(CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 +z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) z +t5))))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead +(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) +(\lambda (_: T).(\lambda (t5: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 +(CHead c (Bind b) u) x1 t5))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: +T).(pr2 (CHead c (Bind Abbr) u) x1 t5))) (ex3_2 T T (\lambda (y0: T).(\lambda +(_: T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: +T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) +z t5))))))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead +(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) +(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 +(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (x2: T).(\lambda +(x3: T).(\lambda (H8: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (H9: (pr2 +c x0 x2)).(\lambda (H10: (or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c +(Bind b) u) x1 x3))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 +(CHead c (Bind Abbr) u) x1 x3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: +T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: +T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) +z x3)))))).(or3_ind (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) +x1 x3))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead c +(Bind Abbr) u) x1 x3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 +(CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 +z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) z x3)))) +(or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) +u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: +T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c +(Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (H11: ((\forall (b: +B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 x3))))).(let H12 \def (eq_ind +T t2 (\lambda (t: T).(\forall (x4: T).(\forall (x5: T).((eq T t (THead (Bind +Abbr) x4 x5)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 +(THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x4 u2))) +(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x4) x5 t5)))) (pr3 +(CHead c (Bind Abbr) x4) x5 (lift (S O) O t4))))))) H3 (THead (Bind Abbr) x2 +x3) H8) in (let H13 \def (H12 x2 x3 (refl_equal T (THead (Bind Abbr) x2 x3))) +in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind +Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: +T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 t5)))) (pr3 (CHead c +(Bind Abbr) x2) x3 (lift (S O) O t4)) (or (ex3_2 T T (\lambda (u2: +T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 +(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S +O) O t4))) (\lambda (H14: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T +t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 +u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 +t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead +(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) +(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 t5))) (or +(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 +t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: +T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c +(Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (x4: T).(\lambda (x5: +T).(\lambda (H15: (eq T t4 (THead (Bind Abbr) x4 x5))).(\lambda (H16: (pr3 c +x2 x4)).(\lambda (H17: (pr3 (CHead c (Bind Abbr) x2) x3 x5)).(eq_ind_r T +(THead (Bind Abbr) x4 x5) (\lambda (t: T).(or (ex3_2 T T (\lambda (u2: +T).(\lambda (t5: T).(eq T t (THead (Bind Abbr) u2 t5)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 +(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S +O) O t)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T +(THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2 t5)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 +(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S +O) O (THead (Bind Abbr) x4 x5))) (ex3_2_intro T T (\lambda (u2: T).(\lambda +(t5: T).(eq T (THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2 t5)))) (\lambda +(u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 +(CHead c (Bind Abbr) x0) x1 t5))) x4 x5 (refl_equal T (THead (Bind Abbr) x4 +x5)) (pr3_sing c x2 x0 H9 x4 H16) (pr3_sing (CHead c (Bind Abbr) x0) x3 x1 +(H11 Abbr x0) x5 (pr3_pr2_pr3_t c x2 x3 x5 (Bind Abbr) H17 x0 H9)))) t4 +H15)))))) H14)) (\lambda (H14: (pr3 (CHead c (Bind Abbr) x2) x3 (lift (S O) O +t4))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead +(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) +(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 +(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4)) (pr3_sing (CHead c (Bind Abbr) +x0) x3 x1 (H11 Abbr x0) (lift (S O) O t4) (pr3_pr2_pr3_t c x2 x3 (lift (S O) +O t4) (Bind Abbr) H14 x0 H9)))) H13)))) (\lambda (H11: (ex2 T (\lambda (u: +T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) x1 +x3)))).(ex2_ind T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead c +(Bind Abbr) u) x1 x3)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T +t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 +u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 +t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (x4: +T).(\lambda (H12: (pr0 x0 x4)).(\lambda (H13: (pr2 (CHead c (Bind Abbr) x4) +x1 x3)).(let H14 \def (eq_ind T t2 (\lambda (t: T).(\forall (x5: T).(\forall +(x6: T).((eq T t (THead (Bind Abbr) x5 x6)) \to (or (ex3_2 T T (\lambda (u2: +T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c x5 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 +(CHead c (Bind Abbr) x5) x6 t5)))) (pr3 (CHead c (Bind Abbr) x5) x6 (lift (S +O) O t4))))))) H3 (THead (Bind Abbr) x2 x3) H8) in (let H15 \def (H14 x2 x3 +(refl_equal T (THead (Bind Abbr) x2 x3))) in (or_ind (ex3_2 T T (\lambda (u2: +T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 +(CHead c (Bind Abbr) x2) x3 t5)))) (pr3 (CHead c (Bind Abbr) x2) x3 (lift (S +O) O t4)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead +(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) +(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 +(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (H16: (ex3_2 T T +(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) +(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda +(t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 t5))))).(ex3_2_ind T T (\lambda (u2: +T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 +(CHead c (Bind Abbr) x2) x3 t5))) (or (ex3_2 T T (\lambda (u2: T).(\lambda +(t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: +T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) +x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda +(x5: T).(\lambda (x6: T).(\lambda (H17: (eq T t4 (THead (Bind Abbr) x5 +x6))).(\lambda (H18: (pr3 c x2 x5)).(\lambda (H19: (pr3 (CHead c (Bind Abbr) +x2) x3 x6)).(eq_ind_r T (THead (Bind Abbr) x5 x6) (\lambda (t: T).(or (ex3_2 +T T (\lambda (u2: T).(\lambda (t5: T).(eq T t (THead (Bind Abbr) u2 t5)))) +(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda +(t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) +x1 (lift (S O) O t)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t5: +T).(eq T (THead (Bind Abbr) x5 x6) (THead (Bind Abbr) u2 t5)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 +(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S +O) O (THead (Bind Abbr) x5 x6))) (ex3_2_intro T T (\lambda (u2: T).(\lambda +(t5: T).(eq T (THead (Bind Abbr) x5 x6) (THead (Bind Abbr) u2 t5)))) (\lambda +(u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 +(CHead c (Bind Abbr) x0) x1 t5))) x5 x6 (refl_equal T (THead (Bind Abbr) x5 +x6)) (pr3_sing c x2 x0 H9 x5 H18) (pr3_t x3 x1 (CHead c (Bind Abbr) x0) +(pr3_pr0_pr2_t x0 x4 H12 c x1 x3 (Bind Abbr) H13) x6 (pr3_pr2_pr3_t c x2 x3 +x6 (Bind Abbr) H19 x0 H9)))) t4 H17)))))) H16)) (\lambda (H16: (pr3 (CHead c +(Bind Abbr) x2) x3 (lift (S O) O t4))).(or_intror (ex3_2 T T (\lambda (u2: +T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 +(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S +O) O t4)) (pr3_t x3 x1 (CHead c (Bind Abbr) x0) (pr3_pr0_pr2_t x0 x4 H12 c x1 +x3 (Bind Abbr) H13) (lift (S O) O t4) (pr3_pr2_pr3_t c x2 x3 (lift (S O) O +t4) (Bind Abbr) H16 x0 H9)))) H15)))))) H11)) (\lambda (H11: (ex3_2 T T +(\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) +(\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: +T).(pr2 (CHead c (Bind Abbr) x0) z x3))))).(ex3_2_ind T T (\lambda (y0: +T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: +T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c +(Bind Abbr) x0) z x3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq +T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 +u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 +t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (x4: +T).(\lambda (x5: T).(\lambda (H12: (pr2 (CHead c (Bind Abbr) x0) x1 +x4)).(\lambda (H13: (pr0 x4 x5)).(\lambda (H14: (pr2 (CHead c (Bind Abbr) x0) +x5 x3)).(let H15 \def (eq_ind T t2 (\lambda (t: T).(\forall (x6: T).(\forall +(x7: T).((eq T t (THead (Bind Abbr) x6 x7)) \to (or (ex3_2 T T (\lambda (u2: +T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c x6 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 +(CHead c (Bind Abbr) x6) x7 t5)))) (pr3 (CHead c (Bind Abbr) x6) x7 (lift (S +O) O t4))))))) H3 (THead (Bind Abbr) x2 x3) H8) in (let H16 \def (H15 x2 x3 +(refl_equal T (THead (Bind Abbr) x2 x3))) in (or_ind (ex3_2 T T (\lambda (u2: +T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 +(CHead c (Bind Abbr) x2) x3 t5)))) (pr3 (CHead c (Bind Abbr) x2) x3 (lift (S +O) O t4)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead +(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) +(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 +(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (H17: (ex3_2 T T +(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) +(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda +(t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 t5))))).(ex3_2_ind T T (\lambda (u2: +T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 +(CHead c (Bind Abbr) x2) x3 t5))) (or (ex3_2 T T (\lambda (u2: T).(\lambda +(t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: +T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) +x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda +(x6: T).(\lambda (x7: T).(\lambda (H18: (eq T t4 (THead (Bind Abbr) x6 +x7))).(\lambda (H19: (pr3 c x2 x6)).(\lambda (H20: (pr3 (CHead c (Bind Abbr) +x2) x3 x7)).(eq_ind_r T (THead (Bind Abbr) x6 x7) (\lambda (t: T).(or (ex3_2 +T T (\lambda (u2: T).(\lambda (t5: T).(eq T t (THead (Bind Abbr) u2 t5)))) +(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda +(t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) +x1 (lift (S O) O t)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t5: +T).(eq T (THead (Bind Abbr) x6 x7) (THead (Bind Abbr) u2 t5)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 +(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S +O) O (THead (Bind Abbr) x6 x7))) (ex3_2_intro T T (\lambda (u2: T).(\lambda +(t5: T).(eq T (THead (Bind Abbr) x6 x7) (THead (Bind Abbr) u2 t5)))) (\lambda +(u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 +(CHead c (Bind Abbr) x0) x1 t5))) x6 x7 (refl_equal T (THead (Bind Abbr) x6 +x7)) (pr3_sing c x2 x0 H9 x6 H19) (pr3_sing (CHead c (Bind Abbr) x0) x4 x1 +H12 x7 (pr3_sing (CHead c (Bind Abbr) x0) x5 x4 (pr2_free (CHead c (Bind +Abbr) x0) x4 x5 H13) x7 (pr3_sing (CHead c (Bind Abbr) x0) x3 x5 H14 x7 +(pr3_pr2_pr3_t c x2 x3 x7 (Bind Abbr) H20 x0 H9)))))) t4 H18)))))) H17)) +(\lambda (H17: (pr3 (CHead c (Bind Abbr) x2) x3 (lift (S O) O +t4))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead +(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) +(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 +(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4)) (pr3_sing (CHead c (Bind Abbr) +x0) x4 x1 H12 (lift (S O) O t4) (pr3_sing (CHead c (Bind Abbr) x0) x5 x4 +(pr2_free (CHead c (Bind Abbr) x0) x4 x5 H13) (lift (S O) O t4) (pr3_sing +(CHead c (Bind Abbr) x0) x3 x5 H14 (lift (S O) O t4) (pr3_pr2_pr3_t c x2 x3 +(lift (S O) O t4) (Bind Abbr) H17 x0 H9)))))) H16)))))))) H11)) H10)))))) +H7)) (\lambda (H7: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) +x1 (lift (S O) O t2)))))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda +(t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: +T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) +x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4)) (pr3_sing +(CHead c (Bind Abbr) x0) (lift (S O) O t2) x1 (H7 Abbr x0) (lift (S O) O t4) +(pr3_lift (CHead c (Bind Abbr) x0) c (S O) O (drop_drop (Bind Abbr) O c c +(drop_refl c) x0) t2 t4 H2)))) H6)))))))))))) y x H0))))) H))))). +(* COMMENTS +Initial nodes: 5983 +END *) + +theorem pr3_gen_appl: + \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c +(THead (Flat Appl) u1 t1) x) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda +(t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: +T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t1 t2)))) (ex4_4 T +T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 +c (THead (Bind Abbr) u2 t2) x))))) (\lambda (_: T).(\lambda (_: T).(\lambda +(u2: T).(\lambda (_: T).(pr3 c u1 u2))))) (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(pr3 c t1 (THead (Bind Abst) y1 z1)))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall +(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T +T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda +(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 +c t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind +b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) x))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr3 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda +(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))))) +\def + \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda +(H: (pr3 c (THead (Flat Appl) u1 t1) x)).(insert_eq T (THead (Flat Appl) u1 +t1) (\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(or3 (ex3_2 T T (\lambda +(u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t1 +t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) x))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))))) +(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c t1 +(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) +z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) +(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(pr3 c t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda +(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) +x))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))))))) (\lambda (_: +B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: +T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) +y2) z1 z2)))))))))) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(unintro T t1 +(\lambda (t: T).((eq T y (THead (Flat Appl) u1 t)) \to (or3 (ex3_2 T T +(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) +(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda +(t2: T).(pr3 c t t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) x))))) +(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 +u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: +T).(pr3 c t (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 +(CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c t (THead (Bind +b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead +(Flat Appl) (lift (S O) O u2) z2)) x))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 +u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda +(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))) (unintro T u1 (\lambda +(t: T).(\forall (x0: T).((eq T y (THead (Flat Appl) t x0)) \to (or3 (ex3_2 T +T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) +(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_: T).(\lambda (t2: +T).(pr3 c x0 t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda +(u2: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) x))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))) (\lambda +(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x0 (THead +(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) +z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) +(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(pr3 c x0 (THead (Bind b) y1 z1)))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda +(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) +x))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))))) (\lambda (_: +B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: +T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) +y2) z1 z2)))))))))))) (pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall +(x0: T).(\forall (x1: T).((eq T t (THead (Flat Appl) x0 x1)) \to (or3 (ex3_2 +T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Flat Appl) u2 t2)))) +(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda +(t2: T).(pr3 c x1 t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) t0))))) +(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 +u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: +T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 +(CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead +(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) +y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t0))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda +(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))))))) +(\lambda (t: T).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H1: (eq T t +(THead (Flat Appl) x0 x1))).(eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda +(t0: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead +(Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) +(\lambda (_: T).(\lambda (t2: T).(pr3 c x1 t2)))) (ex4_4 T T T T (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind +Abbr) u2 t2) t0))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall +(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T +T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda +(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 +c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind +b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t0))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda +(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))) +(or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat +Appl) x0 x1) (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: +T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c x1 t2)))) (ex4_4 T +T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 +c (THead (Bind Abbr) u2 t2) (THead (Flat Appl) x0 x1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))) +(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 +(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) +z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) +(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda +(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) +(THead (Flat Appl) x0 x1)))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))))) +(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 +(CHead c (Bind b) y2) z1 z2)))))))) (ex3_2_intro T T (\lambda (u2: +T).(\lambda (t2: T).(eq T (THead (Flat Appl) x0 x1) (THead (Flat Appl) u2 +t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: +T).(\lambda (t2: T).(pr3 c x1 t2))) x0 x1 (refl_equal T (THead (Flat Appl) x0 +x1)) (pr3_refl c x0) (pr3_refl c x1))) t H1))))) (\lambda (t2: T).(\lambda +(t3: T).(\lambda (H1: (pr2 c t3 t2)).(\lambda (t4: T).(\lambda (H2: (pr3 c t2 +t4)).(\lambda (H3: ((\forall (x0: T).(\forall (x1: T).((eq T t2 (THead (Flat +Appl) x0 x1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 +(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) +(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind +Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall +(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T +T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda +(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 +c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind +b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda +(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 +z2)))))))))))))).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t3 +(THead (Flat Appl) x0 x1))).(let H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c +t t2)) H1 (THead (Flat Appl) x0 x1) H4) in (let H6 \def (pr2_gen_appl c x0 x1 +t2 H5) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 +(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) +(\lambda (_: T).(\lambda (t5: T).(pr2 c x1 t5)))) (ex4_4 T T T T (\lambda +(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x1 (THead +(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t5: T).(eq T t2 (THead (Bind Abbr) u2 t5)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall +(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T +T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda +(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq +T x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead +(Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr2 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda +(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (or3 (ex3_2 +T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) +(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda +(t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) +(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 +u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: +T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 +(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead +(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) +y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda +(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda +(H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Flat Appl) +u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: +T).(\lambda (t5: T).(pr2 c x1 t5))))).(ex3_2_ind T T (\lambda (u2: +T).(\lambda (t5: T).(eq T t2 (THead (Flat Appl) u2 t5)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr2 c x1 +t5))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat +Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: +T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda +(_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) +t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 +c x0 u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: +T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 +(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead +(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) +y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda +(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda +(x2: T).(\lambda (x3: T).(\lambda (H8: (eq T t2 (THead (Flat Appl) x2 +x3))).(\lambda (H9: (pr2 c x0 x2)).(\lambda (H10: (pr2 c x1 x3)).(let H11 +\def (eq_ind T t2 (\lambda (t: T).(\forall (x4: T).(\forall (x5: T).((eq T t +(THead (Flat Appl) x4 x5)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: +T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: +T).(pr3 c x4 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x5 t5)))) (ex4_4 T +T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 +c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda +(u2: T).(\lambda (_: T).(pr3 c x4 u2))))) (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(pr3 c x5 (THead (Bind Abst) y1 z1)))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall +(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T +T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda +(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 +c x5 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind +b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr3 c x4 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda +(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))))) H3 +(THead (Flat Appl) x2 x3) H8) in (let H12 \def (eq_ind T t2 (\lambda (t: +T).(pr3 c t t4)) H2 (THead (Flat Appl) x2 x3) H8) in (let H13 \def (H11 x2 x3 +(refl_equal T (THead (Flat Appl) x2 x3))) in (or3_ind (ex3_2 T T (\lambda +(u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x3 +t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))))) +(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x3 +(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) +z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) +(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(pr3 c x3 (THead (Bind b) y1 z1)))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda +(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) +t4))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))))))) (\lambda (_: +B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: +T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) +y2) z1 z2)))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 +(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) +(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind +Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall +(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T +T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda +(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 +c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind +b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda +(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda +(H14: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat +Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: +T).(\lambda (t5: T).(pr3 c x3 t5))))).(ex3_2_ind T T (\lambda (u2: +T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x3 +t5))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat +Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: +T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda +(_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) +t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 +c x0 u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: +T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 +(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead +(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) +y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda +(_: 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T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) +(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda +(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) +t))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))))) (\lambda (_: +B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: +T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) +y2) z1 z2)))))))))) (or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: +T).(eq T (THead (Flat Appl) x4 x5) (THead (Flat Appl) u2 t5)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 +t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) (THead (Flat Appl) x4 +x5)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: +T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall +(u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda +(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 +(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) +y2 (THead (Flat Appl) (lift (S O) O u2) z2)) (THead (Flat Appl) x4 x5)))))))) +(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 +y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: +T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) +(ex3_2_intro T T (\lambda (u2: T).(\lambda (t5: T).(eq T (THead (Flat Appl) +x4 x5) (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c +x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5))) x4 x5 (refl_equal T +(THead (Flat Appl) x4 x5)) (pr3_sing c x2 x0 H9 x4 H16) (pr3_sing c x3 x1 H10 +x5 H17))) t4 H15)))))) H14)) (\lambda (H14: (ex4_4 T T T T (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind +Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr3 c x2 u2))))) (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(pr3 c x3 (THead (Bind Abst) y1 z1)))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall +(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5))))))))).(ex4_4_ind T +T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 +c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda +(u2: T).(\lambda (_: T).(pr3 c x2 u2))))) (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(pr3 c x3 (THead (Bind Abst) y1 z1)))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall +(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5))))))) (or3 (ex3_2 T T +(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) +(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda +(t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) +(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 +u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: +T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 +(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead +(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) +y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda +(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda +(x4: T).(\lambda (x5: T).(\lambda (x6: T).(\lambda (x7: T).(\lambda (H15: +(pr3 c (THead (Bind Abbr) x6 x7) t4)).(\lambda (H16: (pr3 c x2 x6)).(\lambda +(H17: (pr3 c x3 (THead (Bind Abst) x4 x5))).(\lambda (H18: ((\forall (b: +B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x5 x7))))).(or3_intro1 (ex3_2 T +T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) +(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda +(t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) +(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 +u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: +T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 +(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead +(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) +y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda +(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (ex4_4_intro +T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: +T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind +Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda +(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 +t5))))))) x4 x5 x6 x7 H15 (pr3_sing c x2 x0 H9 x6 H16) (pr3_sing c x3 x1 H10 +(THead (Bind Abst) x4 x5) H17) H18)))))))))) H14)) (\lambda (H14: (ex6_6 B T +T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: +B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(pr3 c x3 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c +(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) +(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr3 c x2 u2))))))) (\lambda (_: B).(\lambda (y1: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 +y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: +T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 +z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b +Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(pr3 c x3 (THead (Bind b) y1 z1)))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda +(u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift +(S O) O u2) z2)) t4))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))))))) +(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 +(CHead c (Bind b) y2) z1 z2))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda +(t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: +T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T +T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 +c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda +(u2: T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall +(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T +T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda +(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 +c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind +b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda +(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda +(x4: B).(\lambda (x5: T).(\lambda (x6: T).(\lambda (x7: T).(\lambda (x8: +T).(\lambda (x9: T).(\lambda (H15: (not (eq B x4 Abst))).(\lambda (H16: (pr3 +c x3 (THead (Bind x4) x5 x6))).(\lambda (H17: (pr3 c (THead (Bind x4) x9 +(THead (Flat Appl) (lift (S O) O x8) x7)) t4)).(\lambda (H18: (pr3 c x2 +x8)).(\lambda (H19: (pr3 c x5 x9)).(\lambda (H20: (pr3 (CHead c (Bind x4) x9) +x6 x7)).(or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 +(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) +(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind +Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall +(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T +T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda +(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 +c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind +b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda +(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (ex6_6_intro +B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: +B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(pr3 c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c +(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) +(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 +y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: +T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))) +x4 x5 x6 x7 x8 x9 H15 (pr3_sing c x3 x1 H10 (THead (Bind x4) x5 x6) H16) H17 +(pr3_sing c x2 x0 H9 x8 H18) H19 H20)))))))))))))) H14)) H13))))))))) H7)) +(\lambda (H7: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(eq T x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind +Abbr) u2 t5)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr2 c x0 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) +z1 t5))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda +(_: T).(\lambda (_: T).(eq T x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind +Abbr) u2 t5)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr2 c x0 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) +z1 t5))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 +(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) +(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind +Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall +(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T +T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda +(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 +c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind +b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda +(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda +(x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H8: (eq +T x1 (THead (Bind Abst) x2 x3))).(\lambda (H9: (eq T t2 (THead (Bind Abbr) x4 +x5))).(\lambda (H10: (pr2 c x0 x4)).(\lambda (H11: ((\forall (b: B).(\forall +(u: T).(pr2 (CHead c (Bind b) u) x3 x5))))).(eq_ind_r T (THead (Bind Abst) x2 +x3) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T +t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 +u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c t t5)))) (ex4_4 T T T T +(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c +(THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda +(u2: T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(pr3 c t (THead (Bind Abst) y1 z1)))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall +(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T +T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda +(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 +c t (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind +b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda +(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))) (let H12 +\def (eq_ind T t2 (\lambda (t: T).(\forall (x6: T).(\forall (x7: T).((eq T t +(THead (Flat Appl) x6 x7)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: +T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: +T).(pr3 c x6 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x7 t5)))) (ex4_4 T +T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 +c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda +(u2: T).(\lambda (_: T).(pr3 c x6 u2))))) (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(pr3 c x7 (THead (Bind Abst) y1 z1)))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall +(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T +T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda +(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 +c x7 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind +b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr3 c x6 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda +(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))))) H3 +(THead (Bind Abbr) x4 x5) H9) in (let H13 \def (eq_ind T t2 (\lambda (t: +T).(pr3 c t t4)) H2 (THead (Bind Abbr) x4 x5) H9) in (or3_intro1 (ex3_2 T T +(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) +(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda +(t5: T).(pr3 c (THead (Bind Abst) x2 x3) t5)))) (ex4_4 T T T T (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind +Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) x2 x3) (THead +(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) +z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) +(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) x2 x3) (THead (Bind b) y1 +z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: +T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat +Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 +u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda +(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (ex4_4_intro T T T T +(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c +(THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda +(u2: T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) x2 x3) (THead +(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) +z1 t5))))))) x2 x3 x4 x5 H13 (pr3_pr2 c x0 x4 H10) (pr3_refl c (THead (Bind +Abst) x2 x3)) (\lambda (b: B).(\lambda (u: T).(pr3_pr2 (CHead c (Bind b) u) +x3 x5 (H11 b u)))))))) x1 H8))))))))) H7)) (\lambda (H7: (ex6_6 B T T T T T +(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T x1 +(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind +b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr2 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda +(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind +B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: +B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(eq T x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T +t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) +(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr2 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 +y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: +T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) +(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) +u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: +T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda +(_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) +t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 +c x0 u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: +T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 +(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead +(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) +y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda +(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda +(x2: B).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (x6: +T).(\lambda (x7: T).(\lambda (H8: (not (eq B x2 Abst))).(\lambda (H9: (eq T +x1 (THead (Bind x2) x3 x4))).(\lambda (H10: (eq T t2 (THead (Bind x2) x7 +(THead (Flat Appl) (lift (S O) O x6) x5)))).(\lambda (H11: (pr2 c x0 +x6)).(\lambda (H12: (pr2 c x3 x7)).(\lambda (H13: (pr2 (CHead c (Bind x2) x7) +x4 x5)).(eq_ind_r T (THead (Bind x2) x3 x4) (\lambda (t: T).(or3 (ex3_2 T T +(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) +(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda +(t5: T).(pr3 c t t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) +(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 +u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: +T).(pr3 c t (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 +(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c t (THead (Bind +b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead +(Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 +u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda +(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))) (let H14 \def (eq_ind T t2 +(\lambda (t: T).(\forall (x8: T).(\forall (x9: T).((eq T t (THead (Flat Appl) +x8 x9)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead +(Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x8 u2))) +(\lambda (_: T).(\lambda (t5: T).(pr3 c x9 t5)))) (ex4_4 T T T T (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind +Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr3 c x8 u2))))) (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(pr3 c x9 (THead (Bind Abst) y1 z1)))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall +(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T +T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda +(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 +c x9 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind +b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr3 c x8 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda +(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))))) H3 +(THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) O x6) x5)) H10) in (let +H15 \def (eq_ind T t2 (\lambda (t: T).(pr3 c t t4)) H2 (THead (Bind x2) x7 +(THead (Flat Appl) (lift (S O) O x6) x5)) H10) in (or3_intro2 (ex3_2 T T +(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) +(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda +(t5: T).(pr3 c (THead (Bind x2) x3 x4) t5)))) (ex4_4 T T T T (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind +Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind x2) x3 x4) (THead +(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) +z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) +(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(pr3 c (THead (Bind x2) x3 x4) (THead (Bind b) y1 +z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: +T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat +Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 +u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda +(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T +(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c +(THead (Bind x2) x3 x4) (THead (Bind b) y1 z1)))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda +(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) +t4))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))))) (\lambda (_: +B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: +T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) +y2) z1 z2))))))) x2 x3 x4 x5 x6 x7 H8 (pr3_refl c (THead (Bind x2) x3 x4)) +H15 (pr3_pr2 c x0 x6 H11) (pr3_pr2 c x3 x7 H12) (pr3_pr2 (CHead c (Bind x2) +x7) x4 x5 H13))))) x1 H9))))))))))))) H7)) H6)))))))))))) y x H0))))) H))))). +(* COMMENTS +Initial nodes: 12691 +END *) + +theorem pr3_gen_bind: + \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (u1: +T).(\forall (t1: T).(\forall (x: T).((pr3 c (THead (Bind b) u1 t1) x) \to (or +(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind b) u2 +t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: +T).(\lambda (t2: T).(pr3 (CHead c (Bind b) u1) t1 t2)))) (pr3 (CHead c (Bind +b) u1) t1 (lift (S O) O x))))))))) +\def + \lambda (b: B).(B_ind (\lambda (b0: B).((not (eq B b0 Abst)) \to (\forall +(c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c (THead (Bind +b0) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x +(THead (Bind b0) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) +(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind b0) u1) t1 t2)))) (pr3 +(CHead c (Bind b0) u1) t1 (lift (S O) O x)))))))))) (\lambda (_: (not (eq B +Abbr Abst))).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: +T).(\lambda (H0: (pr3 c (THead (Bind Abbr) u1 t1) x)).(let H1 \def +(pr3_gen_abbr c u1 t1 x H0) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda +(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: +T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) +u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O x)) (or (ex3_2 T +T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) +(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda +(t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) +t1 (lift (S O) O x))) (\lambda (H2: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: +T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 +c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 +t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind +Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: +T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2))) (or (ex3_2 T T +(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) +(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda +(t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) +t1 (lift (S O) O x))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H3: (eq T x +(THead (Bind Abbr) x0 x1))).(\lambda (H4: (pr3 c u1 x0)).(\lambda (H5: (pr3 +(CHead c (Bind Abbr) u1) t1 x1)).(or_introl (ex3_2 T T (\lambda (u2: +T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 +(CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S +O) O x)) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead +(Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) +(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2))) x0 x1 +H3 H4 H5))))))) H2)) (\lambda (H2: (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S +O) O x))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x +(THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) +(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2)))) (pr3 +(CHead c (Bind Abbr) u1) t1 (lift (S O) O x)) H2)) H1)))))))) (\lambda (H: +(not (eq B Abst Abst))).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: +T).(\lambda (x: T).(\lambda (_: (pr3 c (THead (Bind Abst) u1 t1) x)).(let H1 +\def (match (H (refl_equal B Abst)) in False return (\lambda (_: False).(or +(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 +t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: +T).(\lambda (t2: T).(pr3 (CHead c (Bind Abst) u1) t1 t2)))) (pr3 (CHead c +(Bind Abst) u1) t1 (lift (S O) O x)))) with []) in H1))))))) (\lambda (_: +(not (eq B Void Abst))).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: +T).(\lambda (x: T).(\lambda (H0: (pr3 c (THead (Bind Void) u1 t1) x)).(let H1 +\def (pr3_gen_void c u1 t1 x H0) in (or_ind (ex3_2 T T (\lambda (u2: +T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall +(b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t1 t2)))))) (pr3 (CHead c +(Bind Void) u1) t1 (lift (S O) O x)) (or (ex3_2 T T (\lambda (u2: T).(\lambda +(t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: +T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) +u1) t1 t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O x))) (\lambda +(H2: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) +u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: +T).(\lambda (t2: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) +u) t1 t2))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T x +(THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) +(\lambda (_: T).(\lambda (t2: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead +c (Bind b0) u) t1 t2))))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: +T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 +c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) u1) t1 +t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O x))) (\lambda (x0: +T).(\lambda (x1: T).(\lambda (H3: (eq T x (THead (Bind Void) x0 +x1))).(\lambda (H4: (pr3 c u1 x0)).(\lambda (H5: ((\forall (b0: B).(\forall +(u: T).(pr3 (CHead c (Bind b0) u) t1 x1))))).(or_introl (ex3_2 T T (\lambda +(u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 +(CHead c (Bind Void) u1) t1 t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S +O) O x)) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead +(Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) +(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) u1) t1 t2))) x0 x1 +H3 H4 (H5 Void u1)))))))) H2)) (\lambda (H2: (pr3 (CHead c (Bind Void) u1) t1 +(lift (S O) O x))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t2: +T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 +c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) u1) t1 +t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O x)) H2)) H1)))))))) b). +(* COMMENTS +Initial nodes: 1721 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr3/iso.ma b/matita/matita/contribs/lambdadelta/basic_1/pr3/iso.ma new file mode 100644 index 000000000..c7b2fb7aa --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/pr3/iso.ma @@ -0,0 +1,1155 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/pr3/fwd.ma". + +include "Basic-1/iso/props.ma". + +include "Basic-1/tlist/props.ma". + +theorem pr3_iso_appls_abbr: + \forall (c: C).(\forall (d: C).(\forall (w: T).(\forall (i: nat).((getl i c +(CHead d (Bind Abbr) w)) \to (\forall (vs: TList).(let u1 \def (THeads (Flat +Appl) vs (TLRef i)) in (\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to +(\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) vs (lift (S i) O w)) +u2)))))))))) +\def + \lambda (c: C).(\lambda (d: C).(\lambda (w: T).(\lambda (i: nat).(\lambda +(H: (getl i c (CHead d (Bind Abbr) w))).(\lambda (vs: TList).(TList_ind +(\lambda (t: TList).(let u1 \def (THeads (Flat Appl) t (TLRef i)) in (\forall +(u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to +(pr3 c (THeads (Flat Appl) t (lift (S i) O w)) u2)))))) (\lambda (u2: +T).(\lambda (H0: (pr3 c (TLRef i) u2)).(\lambda (H1: (((iso (TLRef i) u2) \to +(\forall (P: Prop).P)))).(let H2 \def (pr3_gen_lref c u2 i H0) in (or_ind (eq +T u2 (TLRef i)) (ex3_3 C T T (\lambda (d0: C).(\lambda (u: T).(\lambda (_: +T).(getl i c (CHead d0 (Bind Abbr) u))))) (\lambda (d0: C).(\lambda (u: +T).(\lambda (v: T).(pr3 d0 u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda +(v: T).(eq T u2 (lift (S i) O v)))))) (pr3 c (lift (S i) O w) u2) (\lambda +(H3: (eq T u2 (TLRef i))).(let H4 \def (eq_ind T u2 (\lambda (t: T).((iso +(TLRef i) t) \to (\forall (P: Prop).P))) H1 (TLRef i) H3) in (eq_ind_r T +(TLRef i) (\lambda (t: T).(pr3 c (lift (S i) O w) t)) (H4 (iso_refl (TLRef +i)) (pr3 c (lift (S i) O w) (TLRef i))) u2 H3))) (\lambda (H3: (ex3_3 C T T +(\lambda (d0: C).(\lambda (u: T).(\lambda (_: T).(getl i c (CHead d0 (Bind +Abbr) u))))) (\lambda (d0: C).(\lambda (u: T).(\lambda (v: T).(pr3 d0 u v)))) +(\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T u2 (lift (S i) O +v))))))).(ex3_3_ind C T T (\lambda (d0: C).(\lambda (u: T).(\lambda (_: +T).(getl i c (CHead d0 (Bind Abbr) u))))) (\lambda (d0: C).(\lambda (u: +T).(\lambda (v: T).(pr3 d0 u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda +(v: T).(eq T u2 (lift (S i) O v))))) (pr3 c (lift (S i) O w) u2) (\lambda +(x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H4: (getl i c (CHead x0 +(Bind Abbr) x1))).(\lambda (H5: (pr3 x0 x1 x2)).(\lambda (H6: (eq T u2 (lift +(S i) O x2))).(let H7 \def (eq_ind T u2 (\lambda (t: T).((iso (TLRef i) t) +\to (\forall (P: Prop).P))) H1 (lift (S i) O x2) H6) in (eq_ind_r T (lift (S +i) O x2) (\lambda (t: T).(pr3 c (lift (S i) O w) t)) (let H8 \def (eq_ind C +(CHead d (Bind Abbr) w) (\lambda (c0: C).(getl i c c0)) H (CHead x0 (Bind +Abbr) x1) (getl_mono c (CHead d (Bind Abbr) w) i H (CHead x0 (Bind Abbr) x1) +H4)) in (let H9 \def (f_equal C C (\lambda (e: C).(match e in C return +(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow +c0])) (CHead d (Bind Abbr) w) (CHead x0 (Bind Abbr) x1) (getl_mono c (CHead d +(Bind Abbr) w) i H (CHead x0 (Bind Abbr) x1) H4)) in ((let H10 \def (f_equal +C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) +\Rightarrow w | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abbr) w) (CHead +x0 (Bind Abbr) x1) (getl_mono c (CHead d (Bind Abbr) w) i H (CHead x0 (Bind +Abbr) x1) H4)) in (\lambda (H11: (eq C d x0)).(let H12 \def (eq_ind_r T x1 +(\lambda (t: T).(getl i c (CHead x0 (Bind Abbr) t))) H8 w H10) in (let H13 +\def (eq_ind_r T x1 (\lambda (t: T).(pr3 x0 t x2)) H5 w H10) in (let H14 \def +(eq_ind_r C x0 (\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr) w))) H12 d +H11) in (let H15 \def (eq_ind_r C x0 (\lambda (c0: C).(pr3 c0 w x2)) H13 d +H11) in (pr3_lift c d (S i) O (getl_drop Abbr c d w i H14) w x2 H15))))))) +H9))) u2 H6)))))))) H3)) H2))))) (\lambda (t: T).(\lambda (t0: +TList).(\lambda (H0: ((\forall (u2: T).((pr3 c (THeads (Flat Appl) t0 (TLRef +i)) u2) \to ((((iso (THeads (Flat Appl) t0 (TLRef i)) u2) \to (\forall (P: +Prop).P))) \to (pr3 c (THeads (Flat Appl) t0 (lift (S i) O w)) +u2)))))).(\lambda (u2: T).(\lambda (H1: (pr3 c (THead (Flat Appl) t (THeads +(Flat Appl) t0 (TLRef i))) u2)).(\lambda (H2: (((iso (THead (Flat Appl) t +(THeads (Flat Appl) t0 (TLRef i))) u2) \to (\forall (P: Prop).P)))).(let H3 +\def (pr3_gen_appl c t (THeads (Flat Appl) t0 (TLRef i)) u2 H1) in (or3_ind +(ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 +t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t u3))) (\lambda (_: +T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) t2)))) (ex4_4 T +T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 +c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda +(u3: T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) +(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) +z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) +(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind +b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead +(Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t +u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda +(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (pr3 c (THead (Flat Appl) t +(THeads (Flat Appl) t0 (lift (S i) O w))) u2) (\lambda (H4: (ex3_2 T T +(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) +(\lambda (u3: T).(\lambda (_: T).(pr3 c t u3))) (\lambda (_: T).(\lambda (t2: +T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) t2))))).(ex3_2_ind T T (\lambda +(u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: +T).(\lambda (_: T).(pr3 c t u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c +(THeads (Flat Appl) t0 (TLRef i)) t2))) (pr3 c (THead (Flat Appl) t (THeads +(Flat Appl) t0 (lift (S i) O w))) u2) (\lambda (x0: T).(\lambda (x1: +T).(\lambda (H5: (eq T u2 (THead (Flat Appl) x0 x1))).(\lambda (_: (pr3 c t +x0)).(\lambda (_: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) x1)).(let H8 \def +(eq_ind T u2 (\lambda (t1: T).((iso (THead (Flat Appl) t (THeads (Flat Appl) +t0 (TLRef i))) t1) \to (\forall (P: Prop).P))) H2 (THead (Flat Appl) x0 x1) +H5) in (eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda (t1: T).(pr3 c (THead +(Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O w))) t1)) (H8 (iso_head t +x0 (THeads (Flat Appl) t0 (TLRef i)) x1 (Flat Appl)) (pr3 c (THead (Flat +Appl) t (THeads (Flat Appl) t0 (lift (S i) O w))) (THead (Flat Appl) x0 x1))) +u2 H5))))))) H4)) (\lambda (H4: (ex4_4 T T T T (\lambda (_: T).(\lambda (_: +T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) +(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t +u3))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: +T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind Abst) y1 z1)))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall +(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2))))))))).(ex4_4_ind T +T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 +c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda +(u3: T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) +(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) +z1 t2))))))) (pr3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O +w))) u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: +T).(\lambda (H5: (pr3 c (THead (Bind Abbr) x2 x3) u2)).(\lambda (H6: (pr3 c t +x2)).(\lambda (H7: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind +Abst) x0 x1))).(\lambda (H8: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c +(Bind b) u) x1 x3))))).(pr3_t (THead (Bind Abbr) t x1) (THead (Flat Appl) t +(THeads (Flat Appl) t0 (lift (S i) O w))) c (pr3_t (THead (Flat Appl) t +(THead (Bind Abst) x0 x1)) (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift +(S i) O w))) c (pr3_thin_dx c (THeads (Flat Appl) t0 (lift (S i) O w)) (THead +(Bind Abst) x0 x1) (H0 (THead (Bind Abst) x0 x1) H7 (\lambda (H9: (iso +(THeads (Flat Appl) t0 (TLRef i)) (THead (Bind Abst) x0 x1))).(\lambda (P: +Prop).(iso_flats_lref_bind_false Appl Abst i x0 x1 t0 H9 P)))) t Appl) (THead +(Bind Abbr) t x1) (pr3_pr2 c (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) +(THead (Bind Abbr) t x1) (pr2_free c (THead (Flat Appl) t (THead (Bind Abst) +x0 x1)) (THead (Bind Abbr) t x1) (pr0_beta x0 t t (pr0_refl t) x1 x1 +(pr0_refl x1))))) u2 (pr3_t (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) t +x1) c (pr3_head_12 c t x2 H6 (Bind Abbr) x1 x3 (H8 Abbr x2)) u2 H5)))))))))) +H4)) (\lambda (H4: (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B +b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) +(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) +y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda +(_: T).(pr3 c t u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda +(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind +B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: +B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind b) y1 z1)))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda +(u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift +(S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t u3))))))) +(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 +(CHead c (Bind b) y2) z1 z2))))))) (pr3 c (THead (Flat Appl) t (THeads (Flat +Appl) t0 (lift (S i) O w))) u2) (\lambda (x0: B).(\lambda (x1: T).(\lambda +(x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H5: (not +(eq B x0 Abst))).(\lambda (H6: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) +(THead (Bind x0) x1 x2))).(\lambda (H7: (pr3 c (THead (Bind x0) x5 (THead +(Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda (H8: (pr3 c t x4)).(\lambda +(H9: (pr3 c x1 x5)).(\lambda (H10: (pr3 (CHead c (Bind x0) x5) x2 x3)).(pr3_t +(THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) (THead (Flat +Appl) t (THeads (Flat Appl) t0 (lift (S i) O w))) c (pr3_t (THead (Bind x0) +x1 (THead (Flat Appl) (lift (S O) O t) x2)) (THead (Flat Appl) t (THeads +(Flat Appl) t0 (lift (S i) O w))) c (pr3_t (THead (Flat Appl) t (THead (Bind +x0) x1 x2)) (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O w))) c +(pr3_thin_dx c (THeads (Flat Appl) t0 (lift (S i) O w)) (THead (Bind x0) x1 +x2) (H0 (THead (Bind x0) x1 x2) H6 (\lambda (H11: (iso (THeads (Flat Appl) t0 +(TLRef i)) (THead (Bind x0) x1 x2))).(\lambda (P: +Prop).(iso_flats_lref_bind_false Appl x0 i x1 x2 t0 H11 P)))) t Appl) (THead +(Bind x0) x1 (THead (Flat Appl) (lift (S O) O t) x2)) (pr3_pr2 c (THead (Flat +Appl) t (THead (Bind x0) x1 x2)) (THead (Bind x0) x1 (THead (Flat Appl) (lift +(S O) O t) x2)) (pr2_free c (THead (Flat Appl) t (THead (Bind x0) x1 x2)) +(THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O t) x2)) (pr0_upsilon x0 +H5 t t (pr0_refl t) x1 x1 (pr0_refl x1) x2 x2 (pr0_refl x2))))) (THead (Bind +x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) (pr3_head_12 c x1 x1 +(pr3_refl c x1) (Bind x0) (THead (Flat Appl) (lift (S O) O t) x2) (THead +(Flat Appl) (lift (S O) O x4) x2) (pr3_head_12 (CHead c (Bind x0) x1) (lift +(S O) O t) (lift (S O) O x4) (pr3_lift (CHead c (Bind x0) x1) c (S O) O +(drop_drop (Bind x0) O c c (drop_refl c) x1) t x4 H8) (Flat Appl) x2 x2 +(pr3_refl (CHead (CHead c (Bind x0) x1) (Flat Appl) (lift (S O) O x4)) x2)))) +u2 (pr3_t (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) +(THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) c (pr3_head_12 +c x1 x5 H9 (Bind x0) (THead (Flat Appl) (lift (S O) O x4) x2) (THead (Flat +Appl) (lift (S O) O x4) x3) (pr3_thin_dx (CHead c (Bind x0) x5) x2 x3 H10 +(lift (S O) O x4) Appl)) u2 H7)))))))))))))) H4)) H3)))))))) vs)))))). +(* COMMENTS +Initial nodes: 3759 +END *) + +theorem pr3_iso_appls_cast: + \forall (c: C).(\forall (v: T).(\forall (t: T).(\forall (vs: TList).(let u1 +\def (THeads (Flat Appl) vs (THead (Flat Cast) v t)) in (\forall (u2: +T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to (pr3 c +(THeads (Flat Appl) vs t) u2)))))))) +\def + \lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (vs: +TList).(TList_ind (\lambda (t0: TList).(let u1 \def (THeads (Flat Appl) t0 +(THead (Flat Cast) v t)) in (\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 +u2) \to (\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) t0 t) u2)))))) +(\lambda (u2: T).(\lambda (H: (pr3 c (THead (Flat Cast) v t) u2)).(\lambda +(H0: (((iso (THead (Flat Cast) v t) u2) \to (\forall (P: Prop).P)))).(let H1 +\def (pr3_gen_cast c v t u2 H) in (or_ind (ex3_2 T T (\lambda (u3: +T).(\lambda (t2: T).(eq T u2 (THead (Flat Cast) u3 t2)))) (\lambda (u3: +T).(\lambda (_: T).(pr3 c v u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t +t2)))) (pr3 c t u2) (pr3 c t u2) (\lambda (H2: (ex3_2 T T (\lambda (u3: +T).(\lambda (t2: T).(eq T u2 (THead (Flat Cast) u3 t2)))) (\lambda (u3: +T).(\lambda (_: T).(pr3 c v u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t +t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead +(Flat Cast) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c v u3))) +(\lambda (_: T).(\lambda (t2: T).(pr3 c t t2))) (pr3 c t u2) (\lambda (x0: +T).(\lambda (x1: T).(\lambda (H3: (eq T u2 (THead (Flat Cast) x0 +x1))).(\lambda (_: (pr3 c v x0)).(\lambda (_: (pr3 c t x1)).(let H6 \def +(eq_ind T u2 (\lambda (t0: T).((iso (THead (Flat Cast) v t) t0) \to (\forall +(P: Prop).P))) H0 (THead (Flat Cast) x0 x1) H3) in (eq_ind_r T (THead (Flat +Cast) x0 x1) (\lambda (t0: T).(pr3 c t t0)) (H6 (iso_head v x0 t x1 (Flat +Cast)) (pr3 c t (THead (Flat Cast) x0 x1))) u2 H3))))))) H2)) (\lambda (H2: +(pr3 c t u2)).H2) H1))))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H: +((\forall (u2: T).((pr3 c (THeads (Flat Appl) t1 (THead (Flat Cast) v t)) u2) +\to ((((iso (THeads (Flat Appl) t1 (THead (Flat Cast) v t)) u2) \to (\forall +(P: Prop).P))) \to (pr3 c (THeads (Flat Appl) t1 t) u2)))))).(\lambda (u2: +T).(\lambda (H0: (pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead +(Flat Cast) v t))) u2)).(\lambda (H1: (((iso (THead (Flat Appl) t0 (THeads +(Flat Appl) t1 (THead (Flat Cast) v t))) u2) \to (\forall (P: +Prop).P)))).(let H2 \def (pr3_gen_appl c t0 (THeads (Flat Appl) t1 (THead +(Flat Cast) v t)) u2 H0) in (or3_ind (ex3_2 T T (\lambda (u3: T).(\lambda +(t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: +T).(pr3 c t0 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat +Appl) t1 (THead (Flat Cast) v t)) t2)))) (ex4_4 T T T T (\lambda (_: +T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind +Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: +T).(\lambda (_: T).(pr3 c t0 u3))))) (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat +Cast) v t)) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 +(CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat +Appl) t1 (THead (Flat Cast) v t)) (THead (Bind b) y1 z1)))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u3: T).(\lambda +(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u3) z2)) +u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3))))))) (\lambda (_: +B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: +T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) +y2) z1 z2)))))))) (pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 t)) u2) +(\lambda (H3: (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead +(Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3))) +(\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat +Cast) v t)) t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 +(THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3))) +(\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat +Cast) v t)) t2))) (pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 t)) u2) +(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T u2 (THead (Flat Appl) +x0 x1))).(\lambda (_: (pr3 c t0 x0)).(\lambda (_: (pr3 c (THeads (Flat Appl) +t1 (THead (Flat Cast) v t)) x1)).(let H7 \def (eq_ind T u2 (\lambda (t2: +T).((iso (THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Flat Cast) v +t))) t2) \to (\forall (P: Prop).P))) H1 (THead (Flat Appl) x0 x1) H4) in +(eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda (t2: T).(pr3 c (THead (Flat +Appl) t0 (THeads (Flat Appl) t1 t)) t2)) (H7 (iso_head t0 x0 (THeads (Flat +Appl) t1 (THead (Flat Cast) v t)) x1 (Flat Appl)) (pr3 c (THead (Flat Appl) +t0 (THeads (Flat Appl) t1 t)) (THead (Flat Appl) x0 x1))) u2 H4))))))) H3)) +(\lambda (H3: (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: +T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3))))) +(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c +(THeads (Flat Appl) t1 (THead (Flat Cast) v t)) (THead (Bind Abst) y1 +z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: +T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 +t2))))))))).(ex4_4_ind T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: +T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3))))) +(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c +(THeads (Flat Appl) t1 (THead (Flat Cast) v t)) (THead (Bind Abst) y1 +z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: +T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2))))))) +(pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 t)) u2) (\lambda (x0: +T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H4: (pr3 c +(THead (Bind Abbr) x2 x3) u2)).(\lambda (H5: (pr3 c t0 x2)).(\lambda (H6: +(pr3 c (THeads (Flat Appl) t1 (THead (Flat Cast) v t)) (THead (Bind Abst) x0 +x1))).(\lambda (H7: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) +u) x1 x3))))).(pr3_t (THead (Bind Abbr) t0 x1) (THead (Flat Appl) t0 (THeads +(Flat Appl) t1 t)) c (pr3_t (THead (Flat Appl) t0 (THead (Bind Abst) x0 x1)) +(THead (Flat Appl) t0 (THeads (Flat Appl) t1 t)) c (pr3_thin_dx c (THeads +(Flat Appl) t1 t) (THead (Bind Abst) x0 x1) (H (THead (Bind Abst) x0 x1) H6 +(\lambda (H8: (iso (THeads (Flat Appl) t1 (THead (Flat Cast) v t)) (THead +(Bind Abst) x0 x1))).(\lambda (P: Prop).(iso_flats_flat_bind_false Appl Cast +Abst x0 v x1 t t1 H8 P)))) t0 Appl) (THead (Bind Abbr) t0 x1) (pr3_pr2 c +(THead (Flat Appl) t0 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) t0 x1) +(pr2_free c (THead (Flat Appl) t0 (THead (Bind Abst) x0 x1)) (THead (Bind +Abbr) t0 x1) (pr0_beta x0 t0 t0 (pr0_refl t0) x1 x1 (pr0_refl x1))))) u2 +(pr3_t (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) t0 x1) c (pr3_head_12 c +t0 x2 H5 (Bind Abbr) x1 x3 (H7 Abbr x2)) u2 H4)))))))))) H3)) (\lambda (H3: +(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: +B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat Cast) v t)) (THead (Bind b) +y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: +T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat +Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t0 +u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda +(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind B T T T T T +(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c +(THeads (Flat Appl) t1 (THead (Flat Cast) v t)) (THead (Bind b) y1 z1)))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda +(u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift +(S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3))))))) +(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 +(CHead c (Bind b) y2) z1 z2))))))) (pr3 c (THead (Flat Appl) t0 (THeads (Flat +Appl) t1 t)) u2) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda +(x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H4: (not (eq B x0 +Abst))).(\lambda (H5: (pr3 c (THeads (Flat Appl) t1 (THead (Flat Cast) v t)) +(THead (Bind x0) x1 x2))).(\lambda (H6: (pr3 c (THead (Bind x0) x5 (THead +(Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda (H7: (pr3 c t0 x4)).(\lambda +(H8: (pr3 c x1 x5)).(\lambda (H9: (pr3 (CHead c (Bind x0) x5) x2 x3)).(pr3_t +(THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) (THead (Flat +Appl) t0 (THeads (Flat Appl) t1 t)) c (pr3_t (THead (Bind x0) x1 (THead (Flat +Appl) (lift (S O) O t0) x2)) (THead (Flat Appl) t0 (THeads (Flat Appl) t1 t)) +c (pr3_t (THead (Flat Appl) t0 (THead (Bind x0) x1 x2)) (THead (Flat Appl) t0 +(THeads (Flat Appl) t1 t)) c (pr3_thin_dx c (THeads (Flat Appl) t1 t) (THead +(Bind x0) x1 x2) (H (THead (Bind x0) x1 x2) H5 (\lambda (H10: (iso (THeads +(Flat Appl) t1 (THead (Flat Cast) v t)) (THead (Bind x0) x1 x2))).(\lambda +(P: Prop).(iso_flats_flat_bind_false Appl Cast x0 x1 v x2 t t1 H10 P)))) t0 +Appl) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O t0) x2)) (pr3_pr2 +c (THead (Flat Appl) t0 (THead (Bind x0) x1 x2)) (THead (Bind x0) x1 (THead +(Flat Appl) (lift (S O) O t0) x2)) (pr2_free c (THead (Flat Appl) t0 (THead +(Bind x0) x1 x2)) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O t0) +x2)) (pr0_upsilon x0 H4 t0 t0 (pr0_refl t0) x1 x1 (pr0_refl x1) x2 x2 +(pr0_refl x2))))) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) +x2)) (pr3_head_12 c x1 x1 (pr3_refl c x1) (Bind x0) (THead (Flat Appl) (lift +(S O) O t0) x2) (THead (Flat Appl) (lift (S O) O x4) x2) (pr3_head_12 (CHead +c (Bind x0) x1) (lift (S O) O t0) (lift (S O) O x4) (pr3_lift (CHead c (Bind +x0) x1) c (S O) O (drop_drop (Bind x0) O c c (drop_refl c) x1) t0 x4 H7) +(Flat Appl) x2 x2 (pr3_refl (CHead (CHead c (Bind x0) x1) (Flat Appl) (lift +(S O) O x4)) x2)))) u2 (pr3_t (THead (Bind x0) x5 (THead (Flat Appl) (lift (S +O) O x4) x3)) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) c +(pr3_head_12 c x1 x5 H8 (Bind x0) (THead (Flat Appl) (lift (S O) O x4) x2) +(THead (Flat Appl) (lift (S O) O x4) x3) (pr3_thin_dx (CHead c (Bind x0) x5) +x2 x3 H9 (lift (S O) O x4) Appl)) u2 H6)))))))))))))) H3)) H2)))))))) vs)))). +(* COMMENTS +Initial nodes: 3297 +END *) + +theorem pr3_iso_appl_bind: + \forall (b: B).((not (eq B b Abst)) \to (\forall (v1: T).(\forall (v2: +T).(\forall (t: T).(let u1 \def (THead (Flat Appl) v1 (THead (Bind b) v2 t)) +in (\forall (c: C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to +(\forall (P: Prop).P))) \to (pr3 c (THead (Bind b) v2 (THead (Flat Appl) +(lift (S O) O v1) t)) u2)))))))))) +\def + \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (v1: T).(\lambda +(v2: T).(\lambda (t: T).(\lambda (c: C).(\lambda (u2: T).(\lambda (H0: (pr3 c +(THead (Flat Appl) v1 (THead (Bind b) v2 t)) u2)).(\lambda (H1: (((iso (THead +(Flat Appl) v1 (THead (Bind b) v2 t)) u2) \to (\forall (P: Prop).P)))).(let +H2 \def (pr3_gen_appl c v1 (THead (Bind b) v2 t) u2 H0) in (or3_ind (ex3_2 T +T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) +(\lambda (u3: T).(\lambda (_: T).(pr3 c v1 u3))) (\lambda (_: T).(\lambda +(t2: T).(pr3 c (THead (Bind b) v2 t) t2)))) (ex4_4 T T T T (\lambda (_: +T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind +Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: +T).(\lambda (_: T).(pr3 c v1 u3))))) (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind b) v2 t) (THead (Bind +Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda +(t2: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) z1 +t2)))))))) (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) +(\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(pr3 c (THead (Bind b) v2 t) (THead (Bind b0) y1 +z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: +T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b0) y2 (THead (Flat +Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v1 +u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b0: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda +(y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2)))))))) (pr3 c (THead (Bind b) v2 +(THead (Flat Appl) (lift (S O) O v1) t)) u2) (\lambda (H3: (ex3_2 T T +(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) +(\lambda (u3: T).(\lambda (_: T).(pr3 c v1 u3))) (\lambda (_: T).(\lambda +(t2: T).(pr3 c (THead (Bind b) v2 t) t2))))).(ex3_2_ind T T (\lambda (u3: +T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: +T).(\lambda (_: T).(pr3 c v1 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c +(THead (Bind b) v2 t) t2))) (pr3 c (THead (Bind b) v2 (THead (Flat Appl) +(lift (S O) O v1) t)) u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq +T u2 (THead (Flat Appl) x0 x1))).(\lambda (_: (pr3 c v1 x0)).(\lambda (_: +(pr3 c (THead (Bind b) v2 t) x1)).(let H7 \def (eq_ind T u2 (\lambda (t0: +T).((iso (THead (Flat Appl) v1 (THead (Bind b) v2 t)) t0) \to (\forall (P: +Prop).P))) H1 (THead (Flat Appl) x0 x1) H4) in (eq_ind_r T (THead (Flat Appl) +x0 x1) (\lambda (t0: T).(pr3 c (THead (Bind b) v2 (THead (Flat Appl) (lift (S +O) O v1) t)) t0)) (H7 (iso_head v1 x0 (THead (Bind b) v2 t) x1 (Flat Appl)) +(pr3 c (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) t)) (THead +(Flat Appl) x0 x1))) u2 H4))))))) H3)) (\lambda (H3: (ex4_4 T T T T (\lambda +(_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind +Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: +T).(\lambda (_: T).(pr3 c v1 u3))))) (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind b) v2 t) (THead (Bind +Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda +(t2: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) z1 +t2))))))))).(ex4_4_ind T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: +T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v1 u3))))) +(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c +(THead (Bind b) v2 t) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b0: B).(\forall (u: +T).(pr3 (CHead c (Bind b0) u) z1 t2))))))) (pr3 c (THead (Bind b) v2 (THead +(Flat Appl) (lift (S O) O v1) t)) u2) (\lambda (x0: T).(\lambda (x1: +T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H4: (pr3 c (THead (Bind Abbr) +x2 x3) u2)).(\lambda (H5: (pr3 c v1 x2)).(\lambda (H6: (pr3 c (THead (Bind b) +v2 t) (THead (Bind Abst) x0 x1))).(\lambda (H7: ((\forall (b0: B).(\forall +(u: T).(pr3 (CHead c (Bind b0) u) x1 x3))))).(pr3_t (THead (Bind Abbr) x2 x3) +(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) t)) c (let H_x \def +(pr3_gen_bind b H c v2 t (THead (Bind Abst) x0 x1) H6) in (let H8 \def H_x in +(or_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind Abst) +x0 x1) (THead (Bind b) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c v2 +u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind b) v2) t t2)))) +(pr3 (CHead c (Bind b) v2) t (lift (S O) O (THead (Bind Abst) x0 x1))) (pr3 c +(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) t)) (THead (Bind +Abbr) x2 x3)) (\lambda (H9: (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq +T (THead (Bind Abst) x0 x1) (THead (Bind b) u3 t2)))) (\lambda (u3: +T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 +(CHead c (Bind b) v2) t t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: +T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) u3 t2)))) (\lambda (u3: +T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 +(CHead c (Bind b) v2) t t2))) (pr3 c (THead (Bind b) v2 (THead (Flat Appl) +(lift (S O) O v1) t)) (THead (Bind Abbr) x2 x3)) (\lambda (x4: T).(\lambda +(x5: T).(\lambda (H10: (eq T (THead (Bind Abst) x0 x1) (THead (Bind b) x4 +x5))).(\lambda (H11: (pr3 c v2 x4)).(\lambda (H12: (pr3 (CHead c (Bind b) v2) +t x5)).(let H13 \def (f_equal T B (\lambda (e: T).(match e in T return +(\lambda (_: T).B) with [(TSort _) \Rightarrow Abst | (TLRef _) \Rightarrow +Abst | (THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).B) with +[(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abst])])) (THead (Bind Abst) +x0 x1) (THead (Bind b) x4 x5) H10) in ((let H14 \def (f_equal T T (\lambda +(e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 +| (TLRef _) \Rightarrow x0 | (THead _ t0 _) \Rightarrow t0])) (THead (Bind +Abst) x0 x1) (THead (Bind b) x4 x5) H10) in ((let H15 \def (f_equal T T +(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow x1 | (TLRef _) \Rightarrow x1 | (THead _ _ t0) \Rightarrow t0])) +(THead (Bind Abst) x0 x1) (THead (Bind b) x4 x5) H10) in (\lambda (H16: (eq T +x0 x4)).(\lambda (H17: (eq B Abst b)).(let H18 \def (eq_ind_r T x5 (\lambda +(t0: T).(pr3 (CHead c (Bind b) v2) t t0)) H12 x1 H15) in (let H19 \def +(eq_ind_r T x4 (\lambda (t0: T).(pr3 c v2 t0)) H11 x0 H16) in (let H20 \def +(eq_ind_r B b (\lambda (b0: B).(pr3 (CHead c (Bind b0) v2) t x1)) H18 Abst +H17) in (let H21 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 Abst))) H +Abst H17) in (eq_ind B Abst (\lambda (b0: B).(pr3 c (THead (Bind b0) v2 +(THead (Flat Appl) (lift (S O) O v1) t)) (THead (Bind Abbr) x2 x3))) (let H22 +\def (match (H21 (refl_equal B Abst)) in False return (\lambda (_: +False).(pr3 c (THead (Bind Abst) v2 (THead (Flat Appl) (lift (S O) O v1) t)) +(THead (Bind Abbr) x2 x3))) with []) in H22) b H17)))))))) H14)) H13))))))) +H9)) (\lambda (H9: (pr3 (CHead c (Bind b) v2) t (lift (S O) O (THead (Bind +Abst) x0 x1)))).(pr3_t (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O +x2) (lift (S O) O (THead (Bind Abst) x0 x1)))) (THead (Bind b) v2 (THead +(Flat Appl) (lift (S O) O v1) t)) c (pr3_head_2 c v2 (THead (Flat Appl) (lift +(S O) O v1) t) (THead (Flat Appl) (lift (S O) O x2) (lift (S O) O (THead +(Bind Abst) x0 x1))) (Bind b) (pr3_flat (CHead c (Bind b) v2) (lift (S O) O +v1) (lift (S O) O x2) (pr3_lift (CHead c (Bind b) v2) c (S O) O (drop_drop +(Bind b) O c c (drop_refl c) v2) v1 x2 H5) t (lift (S O) O (THead (Bind Abst) +x0 x1)) H9 Appl)) (THead (Bind Abbr) x2 x3) (eq_ind T (lift (S O) O (THead +(Flat Appl) x2 (THead (Bind Abst) x0 x1))) (\lambda (t0: T).(pr3 c (THead +(Bind b) v2 t0) (THead (Bind Abbr) x2 x3))) (pr3_sing c (THead (Bind Abbr) x2 +x1) (THead (Bind b) v2 (lift (S O) O (THead (Flat Appl) x2 (THead (Bind Abst) +x0 x1)))) (pr2_free c (THead (Bind b) v2 (lift (S O) O (THead (Flat Appl) x2 +(THead (Bind Abst) x0 x1)))) (THead (Bind Abbr) x2 x1) (pr0_zeta b H (THead +(Flat Appl) x2 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) x2 x1) (pr0_beta +x0 x2 x2 (pr0_refl x2) x1 x1 (pr0_refl x1)) v2)) (THead (Bind Abbr) x2 x3) +(pr3_head_12 c x2 x2 (pr3_refl c x2) (Bind Abbr) x1 x3 (H7 Abbr x2))) (THead +(Flat Appl) (lift (S O) O x2) (lift (S O) O (THead (Bind Abst) x0 x1))) +(lift_flat Appl x2 (THead (Bind Abst) x0 x1) (S O) O)))) H8))) u2 H4))))))))) +H3)) (\lambda (H3: (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B +b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind b) v2 t) (THead +(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b0) +y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda +(_: T).(pr3 c v1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) +(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda +(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2))))))))).(ex6_6_ind +B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: +B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(pr3 c (THead (Bind b) v2 t) (THead (Bind b0) y1 z1)))))))) (\lambda +(b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u3: +T).(\lambda (y2: T).(pr3 c (THead (Bind b0) y2 (THead (Flat Appl) (lift (S O) +O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v1 u3))))))) (\lambda (_: +B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(y2: T).(pr3 c y1 y2))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (z1: +T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) +y2) z1 z2))))))) (pr3 c (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O +v1) t)) u2) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: +T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H4: (not (eq B x0 +Abst))).(\lambda (H5: (pr3 c (THead (Bind b) v2 t) (THead (Bind x0) x1 +x2))).(\lambda (H6: (pr3 c (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) +O x4) x3)) u2)).(\lambda (H7: (pr3 c v1 x4)).(\lambda (H8: (pr3 c x1 +x5)).(\lambda (H9: (pr3 (CHead c (Bind x0) x5) x2 x3)).(pr3_t (THead (Bind +x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) (THead (Bind b) v2 (THead +(Flat Appl) (lift (S O) O v1) t)) c (let H_x \def (pr3_gen_bind b H c v2 t +(THead (Bind x0) x1 x2) H5) in (let H10 \def H_x in (or_ind (ex3_2 T T +(\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind +b) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: +T).(\lambda (t2: T).(pr3 (CHead c (Bind b) v2) t t2)))) (pr3 (CHead c (Bind +b) v2) t (lift (S O) O (THead (Bind x0) x1 x2))) (pr3 c (THead (Bind b) v2 +(THead (Flat Appl) (lift (S O) O v1) t)) (THead (Bind x0) x5 (THead (Flat +Appl) (lift (S O) O x4) x3))) (\lambda (H11: (ex3_2 T T (\lambda (u3: +T).(\lambda (t2: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) u3 t2)))) +(\lambda (u3: T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: T).(\lambda +(t2: T).(pr3 (CHead c (Bind b) v2) t t2))))).(ex3_2_ind T T (\lambda (u3: +T).(\lambda (t2: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) u3 t2)))) +(\lambda (u3: T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: T).(\lambda +(t2: T).(pr3 (CHead c (Bind b) v2) t t2))) (pr3 c (THead (Bind b) v2 (THead +(Flat Appl) (lift (S O) O v1) t)) (THead (Bind x0) x5 (THead (Flat Appl) +(lift (S O) O x4) x3))) (\lambda (x6: T).(\lambda (x7: T).(\lambda (H12: (eq +T (THead (Bind x0) x1 x2) (THead (Bind b) x6 x7))).(\lambda (H13: (pr3 c v2 +x6)).(\lambda (H14: (pr3 (CHead c (Bind b) v2) t x7)).(let H15 \def (f_equal +T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with [(TSort _) +\Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead k _ _) \Rightarrow (match +k in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) +\Rightarrow x0])])) (THead (Bind x0) x1 x2) (THead (Bind b) x6 x7) H12) in +((let H16 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: +T).T) with [(TSort _) \Rightarrow x1 | (TLRef _) \Rightarrow x1 | (THead _ t0 +_) \Rightarrow t0])) (THead (Bind x0) x1 x2) (THead (Bind b) x6 x7) H12) in +((let H17 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: +T).T) with [(TSort _) \Rightarrow x2 | (TLRef _) \Rightarrow x2 | (THead _ _ +t0) \Rightarrow t0])) (THead (Bind x0) x1 x2) (THead (Bind b) x6 x7) H12) in +(\lambda (H18: (eq T x1 x6)).(\lambda (H19: (eq B x0 b)).(let H20 \def +(eq_ind_r T x7 (\lambda (t0: T).(pr3 (CHead c (Bind b) v2) t t0)) H14 x2 H17) +in (let H21 \def (eq_ind_r T x6 (\lambda (t0: T).(pr3 c v2 t0)) H13 x1 H18) +in (let H22 \def (eq_ind B x0 (\lambda (b0: B).(pr3 (CHead c (Bind b0) x5) x2 +x3)) H9 b H19) in (let H23 \def (eq_ind B x0 (\lambda (b0: B).(not (eq B b0 +Abst))) H4 b H19) in (eq_ind_r B b (\lambda (b0: B).(pr3 c (THead (Bind b) v2 +(THead (Flat Appl) (lift (S O) O v1) t)) (THead (Bind b0) x5 (THead (Flat +Appl) (lift (S O) O x4) x3)))) (pr3_head_21 c v2 x5 (pr3_t x1 v2 c H21 x5 H8) +(Bind b) (THead (Flat Appl) (lift (S O) O v1) t) (THead (Flat Appl) (lift (S +O) O x4) x3) (pr3_flat (CHead c (Bind b) v2) (lift (S O) O v1) (lift (S O) O +x4) (pr3_lift (CHead c (Bind b) v2) c (S O) O (drop_drop (Bind b) O c c +(drop_refl c) v2) v1 x4 H7) t x3 (pr3_t x2 t (CHead c (Bind b) v2) H20 x3 +(pr3_pr3_pr3_t c v2 x1 H21 x2 x3 (Bind b) (pr3_pr3_pr3_t c x1 x5 H8 x2 x3 +(Bind b) H22))) Appl)) x0 H19)))))))) H16)) H15))))))) H11)) (\lambda (H11: +(pr3 (CHead c (Bind b) v2) t (lift (S O) O (THead (Bind x0) x1 x2)))).(pr3_t +(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O x4) (lift (S O) O (THead +(Bind x0) x1 x2)))) (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) +t)) c (pr3_head_2 c v2 (THead (Flat Appl) (lift (S O) O v1) t) (THead (Flat +Appl) (lift (S O) O x4) (lift (S O) O (THead (Bind x0) x1 x2))) (Bind b) +(pr3_flat (CHead c (Bind b) v2) (lift (S O) O v1) (lift (S O) O x4) (pr3_lift +(CHead c (Bind b) v2) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) v2) +v1 x4 H7) t (lift (S O) O (THead (Bind x0) x1 x2)) H11 Appl)) (THead (Bind +x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) (eq_ind T (lift (S O) O +(THead (Flat Appl) x4 (THead (Bind x0) x1 x2))) (\lambda (t0: T).(pr3 c +(THead (Bind b) v2 t0) (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O +x4) x3)))) (pr3_sing c (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O +x4) x2)) (THead (Bind b) v2 (lift (S O) O (THead (Flat Appl) x4 (THead (Bind +x0) x1 x2)))) (pr2_free c (THead (Bind b) v2 (lift (S O) O (THead (Flat Appl) +x4 (THead (Bind x0) x1 x2)))) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S +O) O x4) x2)) (pr0_zeta b H (THead (Flat Appl) x4 (THead (Bind x0) x1 x2)) +(THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) (pr0_upsilon x0 +H4 x4 x4 (pr0_refl x4) x1 x1 (pr0_refl x1) x2 x2 (pr0_refl x2)) v2)) (THead +(Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) (pr3_head_12 c x1 x5 +H8 (Bind x0) (THead (Flat Appl) (lift (S O) O x4) x2) (THead (Flat Appl) +(lift (S O) O x4) x3) (pr3_thin_dx (CHead c (Bind x0) x5) x2 x3 H9 (lift (S +O) O x4) Appl))) (THead (Flat Appl) (lift (S O) O x4) (lift (S O) O (THead +(Bind x0) x1 x2))) (lift_flat Appl x4 (THead (Bind x0) x1 x2) (S O) O)))) +H10))) u2 H6))))))))))))) H3)) H2)))))))))). +(* COMMENTS +Initial nodes: 4805 +END *) + +theorem pr3_iso_appls_appl_bind: + \forall (b: B).((not (eq B b Abst)) \to (\forall (v: T).(\forall (u: +T).(\forall (t: T).(\forall (vs: TList).(let u1 \def (THeads (Flat Appl) vs +(THead (Flat Appl) v (THead (Bind b) u t))) in (\forall (c: C).(\forall (u2: +T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to (pr3 c +(THeads (Flat Appl) vs (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) +t))) u2))))))))))) +\def + \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (v: T).(\lambda +(u: T).(\lambda (t: T).(\lambda (vs: TList).(TList_ind (\lambda (t0: +TList).(let u1 \def (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind +b) u t))) in (\forall (c: C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 +u2) \to (\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) t0 (THead +(Bind b) u (THead (Flat Appl) (lift (S O) O v) t))) u2))))))) (\lambda (c: +C).(\lambda (u2: T).(\lambda (H0: (pr3 c (THead (Flat Appl) v (THead (Bind b) +u t)) u2)).(\lambda (H1: (((iso (THead (Flat Appl) v (THead (Bind b) u t)) +u2) \to (\forall (P: Prop).P)))).(pr3_iso_appl_bind b H v u t c u2 H0 H1))))) +(\lambda (t0: T).(\lambda (t1: TList).(\lambda (H0: ((\forall (c: C).(\forall +(u2: T).((pr3 c (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u +t))) u2) \to ((((iso (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind +b) u t))) u2) \to (\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) t1 +(THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t))) u2))))))).(\lambda +(c: C).(\lambda (u2: T).(\lambda (H1: (pr3 c (THead (Flat Appl) t0 (THeads +(Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t)))) u2)).(\lambda +(H2: (((iso (THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Flat Appl) v +(THead (Bind b) u t)))) u2) \to (\forall (P: Prop).P)))).(let H3 \def +(pr3_gen_appl c t0 (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind +b) u t))) u2 H1) in (or3_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq +T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t0 +u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t1 (THead +(Flat Appl) v (THead (Bind b) u t))) t2)))) (ex4_4 T T T T (\lambda (_: +T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind +Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: +T).(\lambda (_: T).(pr3 c t0 u3))))) (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat +Appl) v (THead (Bind b) u t))) (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b0: +B).(\forall (u0: T).(pr3 (CHead c (Bind b0) u0) z1 t2)))))))) (ex6_6 B T T T +T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda +(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 +c (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t))) (THead +(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b0) +y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda +(_: T).(pr3 c t0 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) +(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda +(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2)))))))) (pr3 c +(THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat +Appl) (lift (S O) O v) t)))) u2) (\lambda (H4: (ex3_2 T T (\lambda (u3: +T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: +T).(\lambda (_: T).(pr3 c t0 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c +(THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t))) +t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead +(Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3))) +(\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat +Appl) v (THead (Bind b) u t))) t2))) (pr3 c (THead (Flat Appl) t0 (THeads +(Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) +u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T u2 (THead (Flat +Appl) x0 x1))).(\lambda (_: (pr3 c t0 x0)).(\lambda (_: (pr3 c (THeads (Flat +Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t))) x1)).(let H8 \def +(eq_ind T u2 (\lambda (t2: T).((iso (THead (Flat Appl) t0 (THeads (Flat Appl) +t1 (THead (Flat Appl) v (THead (Bind b) u t)))) t2) \to (\forall (P: +Prop).P))) H2 (THead (Flat Appl) x0 x1) H5) in (eq_ind_r T (THead (Flat Appl) +x0 x1) (\lambda (t2: T).(pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 +(THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) t2)) (H8 +(iso_head t0 x0 (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u +t))) x1 (Flat Appl)) (pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 +(THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) (THead (Flat +Appl) x0 x1))) u2 H5))))))) H4)) (\lambda (H4: (ex4_4 T T T T (\lambda (_: +T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind +Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: +T).(\lambda (_: T).(pr3 c t0 u3))))) (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat +Appl) v (THead (Bind b) u t))) (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b0: +B).(\forall (u0: T).(pr3 (CHead c (Bind b0) u0) z1 t2))))))))).(ex4_4_ind T T +T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c +(THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda +(u3: T).(\lambda (_: T).(pr3 c t0 u3))))) (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat +Appl) v (THead (Bind b) u t))) (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b0: +B).(\forall (u0: T).(pr3 (CHead c (Bind b0) u0) z1 t2))))))) (pr3 c (THead +(Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) +(lift (S O) O v) t)))) u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: +T).(\lambda (x3: T).(\lambda (H5: (pr3 c (THead (Bind Abbr) x2 x3) +u2)).(\lambda (H6: (pr3 c t0 x2)).(\lambda (H7: (pr3 c (THeads (Flat Appl) t1 +(THead (Flat Appl) v (THead (Bind b) u t))) (THead (Bind Abst) x0 +x1))).(\lambda (H8: ((\forall (b0: B).(\forall (u0: T).(pr3 (CHead c (Bind +b0) u0) x1 x3))))).(pr3_t (THead (Bind Abbr) t0 x1) (THead (Flat Appl) t0 +(THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) +t)))) c (pr3_t (THead (Flat Appl) t0 (THead (Bind Abst) x0 x1)) (THead (Flat +Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) (lift (S +O) O v) t)))) c (pr3_thin_dx c (THeads (Flat Appl) t1 (THead (Bind b) u +(THead (Flat Appl) (lift (S O) O v) t))) (THead (Bind Abst) x0 x1) (H0 c +(THead (Bind Abst) x0 x1) H7 (\lambda (H9: (iso (THeads (Flat Appl) t1 (THead +(Flat Appl) v (THead (Bind b) u t))) (THead (Bind Abst) x0 x1))).(\lambda (P: +Prop).(iso_flats_flat_bind_false Appl Appl Abst x0 v x1 (THead (Bind b) u t) +t1 H9 P)))) t0 Appl) (THead (Bind Abbr) t0 x1) (pr3_pr2 c (THead (Flat Appl) +t0 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) t0 x1) (pr2_free c (THead +(Flat Appl) t0 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) t0 x1) (pr0_beta +x0 t0 t0 (pr0_refl t0) x1 x1 (pr0_refl x1))))) u2 (pr3_t (THead (Bind Abbr) +x2 x3) (THead (Bind Abbr) t0 x1) c (pr3_head_12 c t0 x2 H6 (Bind Abbr) x1 x3 +(H8 Abbr x2)) u2 H5)))))))))) H4)) (\lambda (H4: (ex6_6 B T T T T T (\lambda +(b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c +(THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t))) (THead +(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b0) +y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda +(_: T).(pr3 c t0 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) +(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda +(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2))))))))).(ex6_6_ind +B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: +B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u +t))) (THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind +b0) y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda +(_: T).(pr3 c t0 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) +(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda +(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2))))))) (pr3 c +(THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat +Appl) (lift (S O) O v) t)))) u2) (\lambda (x0: B).(\lambda (x1: T).(\lambda +(x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H5: (not +(eq B x0 Abst))).(\lambda (H6: (pr3 c (THeads (Flat Appl) t1 (THead (Flat +Appl) v (THead (Bind b) u t))) (THead (Bind x0) x1 x2))).(\lambda (H7: (pr3 c +(THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda +(H8: (pr3 c t0 x4)).(\lambda (H9: (pr3 c x1 x5)).(\lambda (H10: (pr3 (CHead c +(Bind x0) x5) x2 x3)).(pr3_t (THead (Bind x0) x1 (THead (Flat Appl) (lift (S +O) O x4) x2)) (THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u +(THead (Flat Appl) (lift (S O) O v) t)))) c (pr3_t (THead (Bind x0) x1 (THead +(Flat Appl) (lift (S O) O t0) x2)) (THead (Flat Appl) t0 (THeads (Flat Appl) +t1 (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) c (pr3_t +(THead (Flat Appl) t0 (THead (Bind x0) x1 x2)) (THead (Flat Appl) t0 (THeads +(Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) c +(pr3_thin_dx c (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) +(lift (S O) O v) t))) (THead (Bind x0) x1 x2) (H0 c (THead (Bind x0) x1 x2) +H6 (\lambda (H11: (iso (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead +(Bind b) u t))) (THead (Bind x0) x1 x2))).(\lambda (P: +Prop).(iso_flats_flat_bind_false Appl Appl x0 x1 v x2 (THead (Bind b) u t) t1 +H11 P)))) t0 Appl) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O t0) +x2)) (pr3_pr2 c (THead (Flat Appl) t0 (THead (Bind x0) x1 x2)) (THead (Bind +x0) x1 (THead (Flat Appl) (lift (S O) O t0) x2)) (pr2_free c (THead (Flat +Appl) t0 (THead (Bind x0) x1 x2)) (THead (Bind x0) x1 (THead (Flat Appl) +(lift (S O) O t0) x2)) (pr0_upsilon x0 H5 t0 t0 (pr0_refl t0) x1 x1 (pr0_refl +x1) x2 x2 (pr0_refl x2))))) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S +O) O x4) x2)) (pr3_head_12 c x1 x1 (pr3_refl c x1) (Bind x0) (THead (Flat +Appl) (lift (S O) O t0) x2) (THead (Flat Appl) (lift (S O) O x4) x2) +(pr3_head_12 (CHead c (Bind x0) x1) (lift (S O) O t0) (lift (S O) O x4) +(pr3_lift (CHead c (Bind x0) x1) c (S O) O (drop_drop (Bind x0) O c c +(drop_refl c) x1) t0 x4 H8) (Flat Appl) x2 x2 (pr3_refl (CHead (CHead c (Bind +x0) x1) (Flat Appl) (lift (S O) O x4)) x2)))) u2 (pr3_t (THead (Bind x0) x5 +(THead (Flat Appl) (lift (S O) O x4) x3)) (THead (Bind x0) x1 (THead (Flat +Appl) (lift (S O) O x4) x2)) c (pr3_head_12 c x1 x5 H9 (Bind x0) (THead (Flat +Appl) (lift (S O) O x4) x2) (THead (Flat Appl) (lift (S O) O x4) x3) +(pr3_thin_dx (CHead c (Bind x0) x5) x2 x3 H10 (lift (S O) O x4) Appl)) u2 +H7)))))))))))))) H4)) H3))))))))) vs)))))). +(* COMMENTS +Initial nodes: 3571 +END *) + +theorem pr3_iso_appls_bind: + \forall (b: B).((not (eq B b Abst)) \to (\forall (vs: TList).(\forall (u: +T).(\forall (t: T).(let u1 \def (THeads (Flat Appl) vs (THead (Bind b) u t)) +in (\forall (c: C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to +(\forall (P: Prop).P))) \to (pr3 c (THead (Bind b) u (THeads (Flat Appl) +(lifts (S O) O vs) t)) u2)))))))))) +\def + \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (vs: +TList).(tlist_ind_rev (\lambda (t: TList).(\forall (u: T).(\forall (t0: +T).(let u1 \def (THeads (Flat Appl) t (THead (Bind b) u t0)) in (\forall (c: +C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: +Prop).P))) \to (pr3 c (THead (Bind b) u (THeads (Flat Appl) (lifts (S O) O t) +t0)) u2))))))))) (\lambda (u: T).(\lambda (t: T).(\lambda (c: C).(\lambda +(u2: T).(\lambda (H0: (pr3 c (THead (Bind b) u t) u2)).(\lambda (_: (((iso +(THead (Bind b) u t) u2) \to (\forall (P: Prop).P)))).H0)))))) (\lambda (ts: +TList).(\lambda (t: T).(\lambda (H0: ((\forall (u: T).(\forall (t0: +T).(\forall (c: C).(\forall (u2: T).((pr3 c (THeads (Flat Appl) ts (THead +(Bind b) u t0)) u2) \to ((((iso (THeads (Flat Appl) ts (THead (Bind b) u t0)) +u2) \to (\forall (P: Prop).P))) \to (pr3 c (THead (Bind b) u (THeads (Flat +Appl) (lifts (S O) O ts) t0)) u2))))))))).(\lambda (u: T).(\lambda (t0: +T).(\lambda (c: C).(\lambda (u2: T).(\lambda (H1: (pr3 c (THeads (Flat Appl) +(TApp ts t) (THead (Bind b) u t0)) u2)).(\lambda (H2: (((iso (THeads (Flat +Appl) (TApp ts t) (THead (Bind b) u t0)) u2) \to (\forall (P: +Prop).P)))).(eq_ind_r TList (TApp (lifts (S O) O ts) (lift (S O) O t)) +(\lambda (t1: TList).(pr3 c (THead (Bind b) u (THeads (Flat Appl) t1 t0)) +u2)) (eq_ind_r T (THeads (Flat Appl) (lifts (S O) O ts) (THead (Flat Appl) +(lift (S O) O t) t0)) (\lambda (t1: T).(pr3 c (THead (Bind b) u t1) u2)) (let +H3 \def (eq_ind T (THeads (Flat Appl) (TApp ts t) (THead (Bind b) u t0)) +(\lambda (t1: T).(pr3 c t1 u2)) H1 (THeads (Flat Appl) ts (THead (Flat Appl) +t (THead (Bind b) u t0))) (theads_tapp (Flat Appl) t (THead (Bind b) u t0) +ts)) in (let H4 \def (eq_ind T (THeads (Flat Appl) (TApp ts t) (THead (Bind +b) u t0)) (\lambda (t1: T).((iso t1 u2) \to (\forall (P: Prop).P))) H2 +(THeads (Flat Appl) ts (THead (Flat Appl) t (THead (Bind b) u t0))) +(theads_tapp (Flat Appl) t (THead (Bind b) u t0) ts)) in (TList_ind (\lambda +(t1: TList).(((\forall (u0: T).(\forall (t2: T).(\forall (c0: C).(\forall +(u3: T).((pr3 c0 (THeads (Flat Appl) t1 (THead (Bind b) u0 t2)) u3) \to +((((iso (THeads (Flat Appl) t1 (THead (Bind b) u0 t2)) u3) \to (\forall (P: +Prop).P))) \to (pr3 c0 (THead (Bind b) u0 (THeads (Flat Appl) (lifts (S O) O +t1) t2)) u3)))))))) \to ((pr3 c (THeads (Flat Appl) t1 (THead (Flat Appl) t +(THead (Bind b) u t0))) u2) \to ((((iso (THeads (Flat Appl) t1 (THead (Flat +Appl) t (THead (Bind b) u t0))) u2) \to (\forall (P: Prop).P))) \to (pr3 c +(THead (Bind b) u (THeads (Flat Appl) (lifts (S O) O t1) (THead (Flat Appl) +(lift (S O) O t) t0))) u2))))) (\lambda (_: ((\forall (u0: T).(\forall (t1: +T).(\forall (c0: C).(\forall (u3: T).((pr3 c0 (THeads (Flat Appl) TNil (THead +(Bind b) u0 t1)) u3) \to ((((iso (THeads (Flat Appl) TNil (THead (Bind b) u0 +t1)) u3) \to (\forall (P: Prop).P))) \to (pr3 c0 (THead (Bind b) u0 (THeads +(Flat Appl) (lifts (S O) O TNil) t1)) u3))))))))).(\lambda (H6: (pr3 c +(THeads (Flat Appl) TNil (THead (Flat Appl) t (THead (Bind b) u t0))) +u2)).(\lambda (H7: (((iso (THeads (Flat Appl) TNil (THead (Flat Appl) t +(THead (Bind b) u t0))) u2) \to (\forall (P: Prop).P)))).(pr3_iso_appl_bind b +H t u t0 c u2 H6 H7)))) (\lambda (t1: T).(\lambda (ts0: TList).(\lambda (_: +((((\forall (u0: T).(\forall (t2: T).(\forall (c0: C).(\forall (u3: T).((pr3 +c0 (THeads (Flat Appl) ts0 (THead (Bind b) u0 t2)) u3) \to ((((iso (THeads +(Flat Appl) ts0 (THead (Bind b) u0 t2)) u3) \to (\forall (P: Prop).P))) \to +(pr3 c0 (THead (Bind b) u0 (THeads (Flat Appl) (lifts (S O) O ts0) t2)) +u3)))))))) \to ((pr3 c (THeads (Flat Appl) ts0 (THead (Flat Appl) t (THead +(Bind b) u t0))) u2) \to ((((iso (THeads (Flat Appl) ts0 (THead (Flat Appl) t +(THead (Bind b) u t0))) u2) \to (\forall (P: Prop).P))) \to (pr3 c (THead +(Bind b) u (THeads (Flat Appl) (lifts (S O) O ts0) (THead (Flat Appl) (lift +(S O) O t) t0))) u2)))))).(\lambda (H5: ((\forall (u0: T).(\forall (t2: +T).(\forall (c0: C).(\forall (u3: T).((pr3 c0 (THeads (Flat Appl) (TCons t1 +ts0) (THead (Bind b) u0 t2)) u3) \to ((((iso (THeads (Flat Appl) (TCons t1 +ts0) (THead (Bind b) u0 t2)) u3) \to (\forall (P: Prop).P))) \to (pr3 c0 +(THead (Bind b) u0 (THeads (Flat Appl) (lifts (S O) O (TCons t1 ts0)) t2)) +u3))))))))).(\lambda (H6: (pr3 c (THeads (Flat Appl) (TCons t1 ts0) (THead +(Flat Appl) t (THead (Bind b) u t0))) u2)).(\lambda (H7: (((iso (THeads (Flat +Appl) (TCons t1 ts0) (THead (Flat Appl) t (THead (Bind b) u t0))) u2) \to +(\forall (P: Prop).P)))).(H5 u (THead (Flat Appl) (lift (S O) O t) t0) c u2 +(pr3_iso_appls_appl_bind b H t u t0 (TCons t1 ts0) c u2 H6 H7) (\lambda (H8: +(iso (THeads (Flat Appl) (TCons t1 ts0) (THead (Bind b) u (THead (Flat Appl) +(lift (S O) O t) t0))) u2)).(\lambda (P: Prop).(H7 (iso_trans (THeads (Flat +Appl) (TCons t1 ts0) (THead (Flat Appl) t (THead (Bind b) u t0))) (THeads +(Flat Appl) (TCons t1 ts0) (THead (Bind b) u (THead (Flat Appl) (lift (S O) O +t) t0))) (iso_head t1 t1 (THeads (Flat Appl) ts0 (THead (Flat Appl) t (THead +(Bind b) u t0))) (THeads (Flat Appl) ts0 (THead (Bind b) u (THead (Flat Appl) +(lift (S O) O t) t0))) (Flat Appl)) u2 H8) P)))))))))) ts H0 H3 H4))) (THeads +(Flat Appl) (TApp (lifts (S O) O ts) (lift (S O) O t)) t0) (theads_tapp (Flat +Appl) (lift (S O) O t) t0 (lifts (S O) O ts))) (lifts (S O) O (TApp ts t)) +(lifts_tapp (S O) O t ts))))))))))) vs))). +(* COMMENTS +Initial nodes: 1681 +END *) + +theorem pr3_iso_beta: + \forall (v: T).(\forall (w: T).(\forall (t: T).(let u1 \def (THead (Flat +Appl) v (THead (Bind Abst) w t)) in (\forall (c: C).(\forall (u2: T).((pr3 c +u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to (pr3 c (THead (Bind +Abbr) v t) u2)))))))) +\def + \lambda (v: T).(\lambda (w: T).(\lambda (t: T).(\lambda (c: C).(\lambda (u2: +T).(\lambda (H: (pr3 c (THead (Flat Appl) v (THead (Bind Abst) w t)) +u2)).(\lambda (H0: (((iso (THead (Flat Appl) v (THead (Bind Abst) w t)) u2) +\to (\forall (P: Prop).P)))).(let H1 \def (pr3_gen_appl c v (THead (Bind +Abst) w t) u2 H) in (or3_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq +T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c v +u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THead (Bind Abst) w t) t2)))) +(ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: +T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v u3))))) (\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) +w t) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda +(_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind +b) u) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B +b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) w t) (THead +(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) +y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda +(_: T).(pr3 c v u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda +(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (pr3 c +(THead (Bind Abbr) v t) u2) (\lambda (H2: (ex3_2 T T (\lambda (u3: +T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: +T).(\lambda (_: T).(pr3 c v u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c +(THead (Bind Abst) w t) t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: +T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: +T).(pr3 c v u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THead (Bind Abst) +w t) t2))) (pr3 c (THead (Bind Abbr) v t) u2) (\lambda (x0: T).(\lambda (x1: +T).(\lambda (H3: (eq T u2 (THead (Flat Appl) x0 x1))).(\lambda (_: (pr3 c v +x0)).(\lambda (_: (pr3 c (THead (Bind Abst) w t) x1)).(let H6 \def (eq_ind T +u2 (\lambda (t0: T).((iso (THead (Flat Appl) v (THead (Bind Abst) w t)) t0) +\to (\forall (P: Prop).P))) H0 (THead (Flat Appl) x0 x1) H3) in (eq_ind_r T +(THead (Flat Appl) x0 x1) (\lambda (t0: T).(pr3 c (THead (Bind Abbr) v t) +t0)) (H6 (iso_head v x0 (THead (Bind Abst) w t) x1 (Flat Appl)) (pr3 c (THead +(Bind Abbr) v t) (THead (Flat Appl) x0 x1))) u2 H3))))))) H2)) (\lambda (H2: +(ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: +T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v u3))))) (\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) +w t) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda +(_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind +b) u) z1 t2))))))))).(ex4_4_ind T T T T (\lambda (_: T).(\lambda (_: +T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) +(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v +u3))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: +T).(pr3 c (THead (Bind Abst) w t) (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall +(u: T).(pr3 (CHead c (Bind b) u) z1 t2))))))) (pr3 c (THead (Bind Abbr) v t) +u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: +T).(\lambda (H3: (pr3 c (THead (Bind Abbr) x2 x3) u2)).(\lambda (H4: (pr3 c v +x2)).(\lambda (H5: (pr3 c (THead (Bind Abst) w t) (THead (Bind Abst) x0 +x1))).(\lambda (H6: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) +u) x1 x3))))).(let H7 \def (pr3_gen_abst c w t (THead (Bind Abst) x0 x1) H5) +in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind Abst) +x0 x1) (THead (Bind Abst) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c w +u3))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 +(CHead c (Bind b) u) t t2))))) (pr3 c (THead (Bind Abbr) v t) u2) (\lambda +(x4: T).(\lambda (x5: T).(\lambda (H8: (eq T (THead (Bind Abst) x0 x1) (THead +(Bind Abst) x4 x5))).(\lambda (H9: (pr3 c w x4)).(\lambda (H10: ((\forall (b: +B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t x5))))).(let H11 \def (f_equal +T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ t0 _) \Rightarrow t0])) +(THead (Bind Abst) x0 x1) (THead (Bind Abst) x4 x5) H8) in ((let H12 \def +(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with +[(TSort _) \Rightarrow x1 | (TLRef _) \Rightarrow x1 | (THead _ _ t0) +\Rightarrow t0])) (THead (Bind Abst) x0 x1) (THead (Bind Abst) x4 x5) H8) in +(\lambda (H13: (eq T x0 x4)).(let H14 \def (eq_ind_r T x5 (\lambda (t0: +T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t t0)))) H10 x1 +H12) in (let H15 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 c w t0)) H9 x0 +H13) in (pr3_t (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) v t) c +(pr3_head_12 c v x2 H4 (Bind Abbr) t x3 (pr3_t x1 t (CHead c (Bind Abbr) x2) +(H14 Abbr x2) x3 (H6 Abbr x2))) u2 H3))))) H11))))))) H7)))))))))) H2)) +(\lambda (H2: (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) +(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) w t) (THead (Bind b) y1 +z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: +T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat +Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v +u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda +(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind B T T T T T +(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c +(THead (Bind Abst) w t) (THead (Bind b) y1 z1)))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u3: T).(\lambda +(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u3) z2)) +u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v u3))))))) (\lambda (_: +B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: +T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) +y2) z1 z2))))))) (pr3 c (THead (Bind Abbr) v t) u2) (\lambda (x0: B).(\lambda +(x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: +T).(\lambda (H3: (not (eq B x0 Abst))).(\lambda (H4: (pr3 c (THead (Bind +Abst) w t) (THead (Bind x0) x1 x2))).(\lambda (H5: (pr3 c (THead (Bind x0) x5 +(THead (Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda (_: (pr3 c v +x4)).(\lambda (_: (pr3 c x1 x5)).(\lambda (H8: (pr3 (CHead c (Bind x0) x5) x2 +x3)).(let H9 \def (pr3_gen_abst c w t (THead (Bind x0) x1 x2) H4) in +(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind x0) x1 +x2) (THead (Bind Abst) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c w +u3))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 +(CHead c (Bind b) u) t t2))))) (pr3 c (THead (Bind Abbr) v t) u2) (\lambda +(x6: T).(\lambda (x7: T).(\lambda (H10: (eq T (THead (Bind x0) x1 x2) (THead +(Bind Abst) x6 x7))).(\lambda (H11: (pr3 c w x6)).(\lambda (H12: ((\forall +(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t x7))))).(let H13 \def +(f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with +[(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead k _ _) +\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) +\Rightarrow b | (Flat _) \Rightarrow x0])])) (THead (Bind x0) x1 x2) (THead +(Bind Abst) x6 x7) H10) in ((let H14 \def (f_equal T T (\lambda (e: T).(match +e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x1 | (TLRef _) +\Rightarrow x1 | (THead _ t0 _) \Rightarrow t0])) (THead (Bind x0) x1 x2) +(THead (Bind Abst) x6 x7) H10) in ((let H15 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x2 | +(TLRef _) \Rightarrow x2 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind x0) +x1 x2) (THead (Bind Abst) x6 x7) H10) in (\lambda (H16: (eq T x1 +x6)).(\lambda (H17: (eq B x0 Abst)).(let H18 \def (eq_ind_r T x7 (\lambda +(t0: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t t0)))) +H12 x2 H15) in (let H19 \def (eq_ind_r T x6 (\lambda (t0: T).(pr3 c w t0)) +H11 x1 H16) in (let H20 \def (eq_ind B x0 (\lambda (b: B).(pr3 (CHead c (Bind +b) x5) x2 x3)) H8 Abst H17) in (let H21 \def (eq_ind B x0 (\lambda (b: +B).(pr3 c (THead (Bind b) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) u2)) +H5 Abst H17) in (let H22 \def (eq_ind B x0 (\lambda (b: B).(not (eq B b +Abst))) H3 Abst H17) in (let H23 \def (match (H22 (refl_equal B Abst)) in +False return (\lambda (_: False).(pr3 c (THead (Bind Abbr) v t) u2)) with []) +in H23))))))))) H14)) H13))))))) H9)))))))))))))) H2)) H1)))))))). +(* COMMENTS +Initial nodes: 2459 +END *) + +theorem pr3_iso_appls_beta: + \forall (us: TList).(\forall (v: T).(\forall (w: T).(\forall (t: T).(let u1 +\def (THeads (Flat Appl) us (THead (Flat Appl) v (THead (Bind Abst) w t))) in +(\forall (c: C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to +(\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) us (THead (Bind Abbr) +v t)) u2))))))))) +\def + \lambda (us: TList).(TList_ind (\lambda (t: TList).(\forall (v: T).(\forall +(w: T).(\forall (t0: T).(let u1 \def (THeads (Flat Appl) t (THead (Flat Appl) +v (THead (Bind Abst) w t0))) in (\forall (c: C).(\forall (u2: T).((pr3 c u1 +u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to (pr3 c (THeads (Flat +Appl) t (THead (Bind Abbr) v t0)) u2)))))))))) (\lambda (v: T).(\lambda (w: +T).(\lambda (t: T).(\lambda (c: C).(\lambda (u2: T).(\lambda (H: (pr3 c +(THead (Flat Appl) v (THead (Bind Abst) w t)) u2)).(\lambda (H0: (((iso +(THead (Flat Appl) v (THead (Bind Abst) w t)) u2) \to (\forall (P: +Prop).P)))).(pr3_iso_beta v w t c u2 H H0)))))))) (\lambda (t: T).(\lambda +(t0: TList).(\lambda (H: ((\forall (v: T).(\forall (w: T).(\forall (t1: +T).(\forall (c: C).(\forall (u2: T).((pr3 c (THeads (Flat Appl) t0 (THead +(Flat Appl) v (THead (Bind Abst) w t1))) u2) \to ((((iso (THeads (Flat Appl) +t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) u2) \to (\forall (P: +Prop).P))) \to (pr3 c (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1)) +u2)))))))))).(\lambda (v: T).(\lambda (w: T).(\lambda (t1: T).(\lambda (c: +C).(\lambda (u2: T).(\lambda (H0: (pr3 c (THead (Flat Appl) t (THeads (Flat +Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1)))) u2)).(\lambda (H1: +(((iso (THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Flat Appl) v +(THead (Bind Abst) w t1)))) u2) \to (\forall (P: Prop).P)))).(let H2 \def +(pr3_gen_appl c t (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind +Abst) w t1))) u2 H0) in (or3_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t2: +T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: +T).(pr3 c t u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) +t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) t2)))) (ex4_4 T T T T +(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c +(THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda +(u3: T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat +Appl) v (THead (Bind Abst) w t1))) (THead (Bind Abst) y1 z1)))))) (\lambda +(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: +B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T T T +(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c +(THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) (THead +(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) +y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda +(_: T).(pr3 c t u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda +(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (pr3 c +(THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) u2) +(\lambda (H3: (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead +(Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t u3))) +(\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat +Appl) v (THead (Bind Abst) w t1))) t2))))).(ex3_2_ind T T (\lambda (u3: +T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: +T).(\lambda (_: T).(pr3 c t u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c +(THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) t2))) +(pr3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) +u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T u2 (THead (Flat +Appl) x0 x1))).(\lambda (_: (pr3 c t x0)).(\lambda (_: (pr3 c (THeads (Flat +Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) x1)).(let H7 \def +(eq_ind T u2 (\lambda (t2: T).((iso (THead (Flat Appl) t (THeads (Flat Appl) +t0 (THead (Flat Appl) v (THead (Bind Abst) w t1)))) t2) \to (\forall (P: +Prop).P))) H1 (THead (Flat Appl) x0 x1) H4) in (eq_ind_r T (THead (Flat Appl) +x0 x1) (\lambda (t2: T).(pr3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 +(THead (Bind Abbr) v t1))) t2)) (H7 (iso_head t x0 (THeads (Flat Appl) t0 +(THead (Flat Appl) v (THead (Bind Abst) w t1))) x1 (Flat Appl)) (pr3 c (THead +(Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) (THead (Flat +Appl) x0 x1))) u2 H4))))))) H3)) (\lambda (H3: (ex4_4 T T T T (\lambda (_: +T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind +Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: +T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat +Appl) v (THead (Bind Abst) w t1))) (THead (Bind Abst) y1 z1)))))) (\lambda +(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: +B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2))))))))).(ex4_4_ind T T T +T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c +(THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda +(u3: T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat +Appl) v (THead (Bind Abst) w t1))) (THead (Bind Abst) y1 z1)))))) (\lambda +(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: +B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2))))))) (pr3 c (THead (Flat +Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) u2) (\lambda (x0: +T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H4: (pr3 c +(THead (Bind Abbr) x2 x3) u2)).(\lambda (H5: (pr3 c t x2)).(\lambda (H6: (pr3 +c (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) +(THead (Bind Abst) x0 x1))).(\lambda (H7: ((\forall (b: B).(\forall (u: +T).(pr3 (CHead c (Bind b) u) x1 x3))))).(pr3_t (THead (Bind Abbr) t x1) +(THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) c +(pr3_t (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) (THead (Flat Appl) t +(THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) c (pr3_thin_dx c (THeads +(Flat Appl) t0 (THead (Bind Abbr) v t1)) (THead (Bind Abst) x0 x1) (H v w t1 +c (THead (Bind Abst) x0 x1) H6 (\lambda (H8: (iso (THeads (Flat Appl) t0 +(THead (Flat Appl) v (THead (Bind Abst) w t1))) (THead (Bind Abst) x0 +x1))).(\lambda (P: Prop).(iso_flats_flat_bind_false Appl Appl Abst x0 v x1 +(THead (Bind Abst) w t1) t0 H8 P)))) t Appl) (THead (Bind Abbr) t x1) +(pr3_pr2 c (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) +t x1) (pr2_free c (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) (THead +(Bind Abbr) t x1) (pr0_beta x0 t t (pr0_refl t) x1 x1 (pr0_refl x1))))) u2 +(pr3_t (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) t x1) c (pr3_head_12 c t +x2 H5 (Bind Abbr) x1 x3 (H7 Abbr x2)) u2 H4)))))))))) H3)) (\lambda (H3: +(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: +B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) +w t1))) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c +(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) +(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: +T).(\lambda (_: T).(pr3 c t u3))))))) (\lambda (_: B).(\lambda (y1: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 +y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: +T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 +z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b +Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat +Appl) v (THead (Bind Abst) w t1))) (THead (Bind b) y1 z1)))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u3: T).(\lambda +(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u3) z2)) +u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t u3))))))) (\lambda (_: +B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: +T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) +y2) z1 z2))))))) (pr3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (THead +(Bind Abbr) v t1))) u2) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: +T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H4: (not (eq +B x0 Abst))).(\lambda (H5: (pr3 c (THeads (Flat Appl) t0 (THead (Flat Appl) v +(THead (Bind Abst) w t1))) (THead (Bind x0) x1 x2))).(\lambda (H6: (pr3 c +(THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda +(H7: (pr3 c t x4)).(\lambda (H8: (pr3 c x1 x5)).(\lambda (H9: (pr3 (CHead c +(Bind x0) x5) x2 x3)).(pr3_t (THead (Bind x0) x1 (THead (Flat Appl) (lift (S +O) O x4) x2)) (THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) +v t1))) c (pr3_t (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O t) x2)) +(THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) c +(pr3_t (THead (Flat Appl) t (THead (Bind x0) x1 x2)) (THead (Flat Appl) t +(THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) c (pr3_thin_dx c (THeads +(Flat Appl) t0 (THead (Bind Abbr) v t1)) (THead (Bind x0) x1 x2) (H v w t1 c +(THead (Bind x0) x1 x2) H5 (\lambda (H10: (iso (THeads (Flat Appl) t0 (THead +(Flat Appl) v (THead (Bind Abst) w t1))) (THead (Bind x0) x1 x2))).(\lambda +(P: Prop).(iso_flats_flat_bind_false Appl Appl x0 x1 v x2 (THead (Bind Abst) +w t1) t0 H10 P)))) t Appl) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) +O t) x2)) (pr3_pr2 c (THead (Flat Appl) t (THead (Bind x0) x1 x2)) (THead +(Bind x0) x1 (THead (Flat Appl) (lift (S O) O t) x2)) (pr2_free c (THead +(Flat Appl) t (THead (Bind x0) x1 x2)) (THead (Bind x0) x1 (THead (Flat Appl) +(lift (S O) O t) x2)) (pr0_upsilon x0 H4 t t (pr0_refl t) x1 x1 (pr0_refl x1) +x2 x2 (pr0_refl x2))))) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O +x4) x2)) (pr3_head_12 c x1 x1 (pr3_refl c x1) (Bind x0) (THead (Flat Appl) +(lift (S O) O t) x2) (THead (Flat Appl) (lift (S O) O x4) x2) (pr3_head_12 +(CHead c (Bind x0) x1) (lift (S O) O t) (lift (S O) O x4) (pr3_lift (CHead c +(Bind x0) x1) c (S O) O (drop_drop (Bind x0) O c c (drop_refl c) x1) t x4 H7) +(Flat Appl) x2 x2 (pr3_refl (CHead (CHead c (Bind x0) x1) (Flat Appl) (lift +(S O) O x4)) x2)))) u2 (pr3_t (THead (Bind x0) x5 (THead (Flat Appl) (lift (S +O) O x4) x3)) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) c +(pr3_head_12 c x1 x5 H8 (Bind x0) (THead (Flat Appl) (lift (S O) O x4) x2) +(THead (Flat Appl) (lift (S O) O x4) x3) (pr3_thin_dx (CHead c (Bind x0) x5) +x2 x3 H9 (lift (S O) O x4) Appl)) u2 H6)))))))))))))) H3)) H2)))))))))))) us). +(* COMMENTS +Initial nodes: 3345 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr3/pr1.ma b/matita/matita/contribs/lambdadelta/basic_1/pr3/pr1.ma new file mode 100644 index 000000000..e428daf37 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/pr3/pr1.ma @@ -0,0 +1,34 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/pr3/defs.ma". + +include "Basic-1/pr1/defs.ma". + +theorem pr3_pr1: + \forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall (c: C).(pr3 c t1 +t2)))) +\def + \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr1 t1 t2)).(pr1_ind (\lambda +(t: T).(\lambda (t0: T).(\forall (c: C).(pr3 c t t0)))) (\lambda (t: +T).(\lambda (c: C).(pr3_refl c t))) (\lambda (t0: T).(\lambda (t3: +T).(\lambda (H0: (pr0 t3 t0)).(\lambda (t4: T).(\lambda (_: (pr1 t0 +t4)).(\lambda (H2: ((\forall (c: C).(pr3 c t0 t4)))).(\lambda (c: +C).(pr3_sing c t0 t3 (pr2_free c t3 t0 H0) t4 (H2 c))))))))) t1 t2 H))). +(* COMMENTS +Initial nodes: 95 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr3/pr3.ma b/matita/matita/contribs/lambdadelta/basic_1/pr3/pr3.ma new file mode 100644 index 000000000..935850b1c --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/pr3/pr3.ma @@ -0,0 +1,74 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/pr3/props.ma". + +include "Basic-1/pr2/pr2.ma". + +theorem pr3_strip: + \forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr3 c t0 t1) \to (\forall +(t2: T).((pr2 c t0 t2) \to (ex2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: +T).(pr3 c t2 t)))))))) +\def + \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr3 c t0 +t1)).(pr3_ind c (\lambda (t: T).(\lambda (t2: T).(\forall (t3: T).((pr2 c t +t3) \to (ex2 T (\lambda (t4: T).(pr3 c t2 t4)) (\lambda (t4: T).(pr3 c t3 +t4))))))) (\lambda (t: T).(\lambda (t2: T).(\lambda (H0: (pr2 c t +t2)).(ex_intro2 T (\lambda (t3: T).(pr3 c t t3)) (\lambda (t3: T).(pr3 c t2 +t3)) t2 (pr3_pr2 c t t2 H0) (pr3_refl c t2))))) (\lambda (t2: T).(\lambda +(t3: T).(\lambda (H0: (pr2 c t3 t2)).(\lambda (t4: T).(\lambda (_: (pr3 c t2 +t4)).(\lambda (H2: ((\forall (t5: T).((pr2 c t2 t5) \to (ex2 T (\lambda (t: +T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t))))))).(\lambda (t5: T).(\lambda +(H3: (pr2 c t3 t5)).(ex2_ind T (\lambda (t: T).(pr2 c t5 t)) (\lambda (t: +T).(pr2 c t2 t)) (ex2 T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c +t5 t))) (\lambda (x: T).(\lambda (H4: (pr2 c t5 x)).(\lambda (H5: (pr2 c t2 +x)).(ex2_ind T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c x t)) +(ex2 T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t))) (\lambda +(x0: T).(\lambda (H6: (pr3 c t4 x0)).(\lambda (H7: (pr3 c x x0)).(ex_intro2 T +(\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t)) x0 H6 (pr3_sing c +x t5 H4 x0 H7))))) (H2 x H5))))) (pr2_confluence c t3 t5 H3 t2 H0)))))))))) +t0 t1 H)))). +(* COMMENTS +Initial nodes: 375 +END *) + +theorem pr3_confluence: + \forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr3 c t0 t1) \to (\forall +(t2: T).((pr3 c t0 t2) \to (ex2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: +T).(pr3 c t2 t)))))))) +\def + \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr3 c t0 +t1)).(pr3_ind c (\lambda (t: T).(\lambda (t2: T).(\forall (t3: T).((pr3 c t +t3) \to (ex2 T (\lambda (t4: T).(pr3 c t2 t4)) (\lambda (t4: T).(pr3 c t3 +t4))))))) (\lambda (t: T).(\lambda (t2: T).(\lambda (H0: (pr3 c t +t2)).(ex_intro2 T (\lambda (t3: T).(pr3 c t t3)) (\lambda (t3: T).(pr3 c t2 +t3)) t2 H0 (pr3_refl c t2))))) (\lambda (t2: T).(\lambda (t3: T).(\lambda +(H0: (pr2 c t3 t2)).(\lambda (t4: T).(\lambda (_: (pr3 c t2 t4)).(\lambda +(H2: ((\forall (t5: T).((pr3 c t2 t5) \to (ex2 T (\lambda (t: T).(pr3 c t4 +t)) (\lambda (t: T).(pr3 c t5 t))))))).(\lambda (t5: T).(\lambda (H3: (pr3 c +t3 t5)).(ex2_ind T (\lambda (t: T).(pr3 c t5 t)) (\lambda (t: T).(pr3 c t2 +t)) (ex2 T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t))) +(\lambda (x: T).(\lambda (H4: (pr3 c t5 x)).(\lambda (H5: (pr3 c t2 +x)).(ex2_ind T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c x t)) +(ex2 T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t))) (\lambda +(x0: T).(\lambda (H6: (pr3 c t4 x0)).(\lambda (H7: (pr3 c x x0)).(ex_intro2 T +(\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t)) x0 H6 (pr3_t x t5 +c H4 x0 H7))))) (H2 x H5))))) (pr3_strip c t3 t5 H3 t2 H0)))))))))) t0 t1 +H)))). +(* COMMENTS +Initial nodes: 367 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr3/props.ma b/matita/matita/contribs/lambdadelta/basic_1/pr3/props.ma new file mode 100644 index 000000000..777f5d85d --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/pr3/props.ma @@ -0,0 +1,407 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/pr3/pr1.ma". + +include "Basic-1/pr2/props.ma". + +include "Basic-1/pr1/props.ma". + +theorem clear_pr3_trans: + \forall (c2: C).(\forall (t1: T).(\forall (t2: T).((pr3 c2 t1 t2) \to +(\forall (c1: C).((clear c1 c2) \to (pr3 c1 t1 t2)))))) +\def + \lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c2 t1 +t2)).(\lambda (c1: C).(\lambda (H0: (clear c1 c2)).(pr3_ind c2 (\lambda (t: +T).(\lambda (t0: T).(pr3 c1 t t0))) (\lambda (t: T).(pr3_refl c1 t)) (\lambda +(t3: T).(\lambda (t4: T).(\lambda (H1: (pr2 c2 t4 t3)).(\lambda (t5: +T).(\lambda (_: (pr3 c2 t3 t5)).(\lambda (H3: (pr3 c1 t3 t5)).(pr3_sing c1 t3 +t4 (clear_pr2_trans c2 t4 t3 H1 c1 H0) t5 H3))))))) t1 t2 H)))))). +(* COMMENTS +Initial nodes: 107 +END *) + +theorem pr3_pr2: + \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (pr3 c +t1 t2)))) +\def + \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 +t2)).(pr3_sing c t2 t1 H t2 (pr3_refl c t2))))). +(* COMMENTS +Initial nodes: 31 +END *) + +theorem pr3_t: + \forall (t2: T).(\forall (t1: T).(\forall (c: C).((pr3 c t1 t2) \to (\forall +(t3: T).((pr3 c t2 t3) \to (pr3 c t1 t3)))))) +\def + \lambda (t2: T).(\lambda (t1: T).(\lambda (c: C).(\lambda (H: (pr3 c t1 +t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (t3: T).((pr3 c t0 +t3) \to (pr3 c t t3))))) (\lambda (t: T).(\lambda (t3: T).(\lambda (H0: (pr3 +c t t3)).H0))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c t3 +t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2: ((\forall +(t5: T).((pr3 c t4 t5) \to (pr3 c t0 t5))))).(\lambda (t5: T).(\lambda (H3: +(pr3 c t4 t5)).(pr3_sing c t0 t3 H0 t5 (H2 t5 H3)))))))))) t1 t2 H)))). +(* COMMENTS +Initial nodes: 127 +END *) + +theorem pr3_thin_dx: + \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall +(u: T).(\forall (f: F).(pr3 c (THead (Flat f) u t1) (THead (Flat f) u +t2))))))) +\def + \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 +t2)).(\lambda (u: T).(\lambda (f: F).(pr3_ind c (\lambda (t: T).(\lambda (t0: +T).(pr3 c (THead (Flat f) u t) (THead (Flat f) u t0)))) (\lambda (t: +T).(pr3_refl c (THead (Flat f) u t))) (\lambda (t0: T).(\lambda (t3: +T).(\lambda (H0: (pr2 c t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0 +t4)).(\lambda (H2: (pr3 c (THead (Flat f) u t0) (THead (Flat f) u +t4))).(pr3_sing c (THead (Flat f) u t0) (THead (Flat f) u t3) (pr2_thin_dx c +t3 t0 H0 u f) (THead (Flat f) u t4) H2))))))) t1 t2 H)))))). +(* COMMENTS +Initial nodes: 167 +END *) + +theorem pr3_head_1: + \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall +(k: K).(\forall (t: T).(pr3 c (THead k u1 t) (THead k u2 t))))))) +\def + \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1 +u2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (k: K).(\forall +(t1: T).(pr3 c (THead k t t1) (THead k t0 t1)))))) (\lambda (t: T).(\lambda +(k: K).(\lambda (t0: T).(pr3_refl c (THead k t t0))))) (\lambda (t2: +T).(\lambda (t1: T).(\lambda (H0: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda +(_: (pr3 c t2 t3)).(\lambda (H2: ((\forall (k: K).(\forall (t: T).(pr3 c +(THead k t2 t) (THead k t3 t)))))).(\lambda (k: K).(\lambda (t: T).(pr3_sing +c (THead k t2 t) (THead k t1 t) (pr2_head_1 c t1 t2 H0 k t) (THead k t3 t) +(H2 k t)))))))))) u1 u2 H)))). +(* COMMENTS +Initial nodes: 167 +END *) + +theorem pr3_head_2: + \forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).(\forall +(k: K).((pr3 (CHead c k u) t1 t2) \to (pr3 c (THead k u t1) (THead k u +t2))))))) +\def + \lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda +(k: K).(\lambda (H: (pr3 (CHead c k u) t1 t2)).(pr3_ind (CHead c k u) +(\lambda (t: T).(\lambda (t0: T).(pr3 c (THead k u t) (THead k u t0)))) +(\lambda (t: T).(pr3_refl c (THead k u t))) (\lambda (t0: T).(\lambda (t3: +T).(\lambda (H0: (pr2 (CHead c k u) t3 t0)).(\lambda (t4: T).(\lambda (_: +(pr3 (CHead c k u) t0 t4)).(\lambda (H2: (pr3 c (THead k u t0) (THead k u +t4))).(pr3_sing c (THead k u t0) (THead k u t3) (pr2_head_2 c u t3 t0 k H0) +(THead k u t4) H2))))))) t1 t2 H)))))). +(* COMMENTS +Initial nodes: 175 +END *) + +theorem pr3_head_21: + \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall +(k: K).(\forall (t1: T).(\forall (t2: T).((pr3 (CHead c k u1) t1 t2) \to (pr3 +c (THead k u1 t1) (THead k u2 t2))))))))) +\def + \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1 +u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 +(CHead c k u1) t1 t2)).(pr3_t (THead k u1 t2) (THead k u1 t1) c (pr3_head_2 c +u1 t1 t2 k H0) (THead k u2 t2) (pr3_head_1 c u1 u2 H k t2))))))))). +(* COMMENTS +Initial nodes: 89 +END *) + +theorem pr3_head_12: + \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall +(k: K).(\forall (t1: T).(\forall (t2: T).((pr3 (CHead c k u2) t1 t2) \to (pr3 +c (THead k u1 t1) (THead k u2 t2))))))))) +\def + \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1 +u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 +(CHead c k u2) t1 t2)).(pr3_t (THead k u2 t1) (THead k u1 t1) c (pr3_head_1 c +u1 u2 H k t1) (THead k u2 t2) (pr3_head_2 c u2 t1 t2 k H0))))))))). +(* COMMENTS +Initial nodes: 89 +END *) + +theorem pr3_cflat: + \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall +(f: F).(\forall (v: T).(pr3 (CHead c (Flat f) v) t1 t2)))))) +\def + \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 +t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (f: F).(\forall (v: +T).(pr3 (CHead c (Flat f) v) t t0))))) (\lambda (t: T).(\lambda (f: +F).(\lambda (v: T).(pr3_refl (CHead c (Flat f) v) t)))) (\lambda (t3: +T).(\lambda (t4: T).(\lambda (H0: (pr2 c t4 t3)).(\lambda (t5: T).(\lambda +(_: (pr3 c t3 t5)).(\lambda (H2: ((\forall (f: F).(\forall (v: T).(pr3 (CHead +c (Flat f) v) t3 t5))))).(\lambda (f: F).(\lambda (v: T).(pr3_sing (CHead c +(Flat f) v) t3 t4 (pr2_cflat c t4 t3 H0 f v) t5 (H2 f v)))))))))) t1 t2 H)))). +(* COMMENTS +Initial nodes: 151 +END *) + +theorem pr3_flat: + \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall +(t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall (f: F).(pr3 c (THead +(Flat f) u1 t1) (THead (Flat f) u2 t2))))))))) +\def + \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1 +u2)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 c t1 t2)).(\lambda +(f: F).(pr3_head_12 c u1 u2 H (Flat f) t1 t2 (pr3_cflat c t1 t2 H0 f +u2))))))))). +(* COMMENTS +Initial nodes: 59 +END *) + +theorem pr3_pr0_pr2_t: + \forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (c: C).(\forall +(t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pr3 +(CHead c k u1) t1 t2)))))))) +\def + \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr0 u1 u2)).(\lambda (c: +C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k: K).(\lambda (H0: (pr2 +(CHead c k u2) t1 t2)).(insert_eq C (CHead c k u2) (\lambda (c0: C).(pr2 c0 +t1 t2)) (\lambda (_: C).(pr3 (CHead c k u1) t1 t2)) (\lambda (y: C).(\lambda +(H1: (pr2 y t1 t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: +T).((eq C c0 (CHead c k u2)) \to (pr3 (CHead c k u1) t t0))))) (\lambda (c0: +C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (_: +(eq C c0 (CHead c k u2))).(pr3_pr2 (CHead c k u1) t3 t4 (pr2_free (CHead c k +u1) t3 t4 H2))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda +(i: nat).(\lambda (H2: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t3: +T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H4: +(subst0 i u t4 t)).(\lambda (H5: (eq C c0 (CHead c k u2))).(let H6 \def +(eq_ind C c0 (\lambda (c1: C).(getl i c1 (CHead d (Bind Abbr) u))) H2 (CHead +c k u2) H5) in (nat_ind (\lambda (n: nat).((getl n (CHead c k u2) (CHead d +(Bind Abbr) u)) \to ((subst0 n u t4 t) \to (pr3 (CHead c k u1) t3 t)))) +(\lambda (H7: (getl O (CHead c k u2) (CHead d (Bind Abbr) u))).(\lambda (H8: +(subst0 O u t4 t)).(K_ind (\lambda (k0: K).((getl O (CHead c k0 u2) (CHead d +(Bind Abbr) u)) \to (pr3 (CHead c k0 u1) t3 t))) (\lambda (b: B).(\lambda +(H9: (getl O (CHead c (Bind b) u2) (CHead d (Bind Abbr) u))).(let H10 \def +(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with +[(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind +Abbr) u) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 +(getl_gen_O (CHead c (Bind b) u2) (CHead d (Bind Abbr) u) H9))) in ((let H11 +\def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) +with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow (match k0 in K +return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) +\Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead c (Bind b) u2) +(clear_gen_bind b c (CHead d (Bind Abbr) u) u2 (getl_gen_O (CHead c (Bind b) +u2) (CHead d (Bind Abbr) u) H9))) in ((let H12 \def (f_equal C T (\lambda (e: +C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | +(CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind Abbr) u) (CHead c (Bind b) +u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 (getl_gen_O (CHead c (Bind +b) u2) (CHead d (Bind Abbr) u) H9))) in (\lambda (H13: (eq B Abbr +b)).(\lambda (_: (eq C d c)).(let H15 \def (eq_ind T u (\lambda (t0: +T).(subst0 O t0 t4 t)) H8 u2 H12) in (eq_ind B Abbr (\lambda (b0: B).(pr3 +(CHead c (Bind b0) u1) t3 t)) (ex2_ind T (\lambda (t0: T).(subst0 O u1 t4 +t0)) (\lambda (t0: T).(pr0 t0 t)) (pr3 (CHead c (Bind Abbr) u1) t3 t) +(\lambda (x: T).(\lambda (H16: (subst0 O u1 t4 x)).(\lambda (H17: (pr0 x +t)).(pr3_sing (CHead c (Bind Abbr) u1) x t3 (pr2_delta (CHead c (Bind Abbr) +u1) c u1 O (getl_refl Abbr c u1) t3 t4 H3 x H16) t (pr3_pr2 (CHead c (Bind +Abbr) u1) x t (pr2_free (CHead c (Bind Abbr) u1) x t H17)))))) +(pr0_subst0_back u2 t4 t O H15 u1 H)) b H13))))) H11)) H10)))) (\lambda (f: +F).(\lambda (H9: (getl O (CHead c (Flat f) u2) (CHead d (Bind Abbr) +u))).(pr3_pr2 (CHead c (Flat f) u1) t3 t (pr2_cflat c t3 t (pr2_delta c d u O +(getl_intro O c (CHead d (Bind Abbr) u) c (drop_refl c) (clear_gen_flat f c +(CHead d (Bind Abbr) u) u2 (getl_gen_O (CHead c (Flat f) u2) (CHead d (Bind +Abbr) u) H9))) t3 t4 H3 t H8) f u1)))) k H7))) (\lambda (i0: nat).(\lambda +(IHi: (((getl i0 (CHead c k u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t4 +t) \to (pr3 (CHead c k u1) t3 t))))).(\lambda (H7: (getl (S i0) (CHead c k +u2) (CHead d (Bind Abbr) u))).(\lambda (H8: (subst0 (S i0) u t4 t)).(K_ind +(\lambda (k0: K).((getl (S i0) (CHead c k0 u2) (CHead d (Bind Abbr) u)) \to +((((getl i0 (CHead c k0 u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t4 t) +\to (pr3 (CHead c k0 u1) t3 t)))) \to (pr3 (CHead c k0 u1) t3 t)))) (\lambda +(b: B).(\lambda (H9: (getl (S i0) (CHead c (Bind b) u2) (CHead d (Bind Abbr) +u))).(\lambda (_: (((getl i0 (CHead c (Bind b) u2) (CHead d (Bind Abbr) u)) +\to ((subst0 i0 u t4 t) \to (pr3 (CHead c (Bind b) u1) t3 t))))).(pr3_pr2 +(CHead c (Bind b) u1) t3 t (pr2_delta (CHead c (Bind b) u1) d u (S i0) +(getl_head (Bind b) i0 c (CHead d (Bind Abbr) u) (getl_gen_S (Bind b) c +(CHead d (Bind Abbr) u) u2 i0 H9) u1) t3 t4 H3 t H8))))) (\lambda (f: +F).(\lambda (H9: (getl (S i0) (CHead c (Flat f) u2) (CHead d (Bind Abbr) +u))).(\lambda (_: (((getl i0 (CHead c (Flat f) u2) (CHead d (Bind Abbr) u)) +\to ((subst0 i0 u t4 t) \to (pr3 (CHead c (Flat f) u1) t3 t))))).(pr3_pr2 +(CHead c (Flat f) u1) t3 t (pr2_cflat c t3 t (pr2_delta c d u (r (Flat f) i0) +(getl_gen_S (Flat f) c (CHead d (Bind Abbr) u) u2 i0 H9) t3 t4 H3 t H8) f +u1))))) k H7 IHi))))) i H6 H4))))))))))))) y t1 t2 H1))) H0)))))))). +(* COMMENTS +Initial nodes: 1557 +END *) + +theorem pr3_pr2_pr2_t: + \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr2 c u1 u2) \to (\forall +(t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pr3 +(CHead c k u1) t1 t2)))))))) +\def + \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr2 c u1 +u2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (t1: +T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c0 k t0) t1 t2) \to (pr3 +(CHead c0 k t) t1 t2)))))))) (\lambda (c0: C).(\lambda (t1: T).(\lambda (t2: +T).(\lambda (H0: (pr0 t1 t2)).(\lambda (t0: T).(\lambda (t3: T).(\lambda (k: +K).(\lambda (H1: (pr2 (CHead c0 k t2) t0 t3)).(pr3_pr0_pr2_t t1 t2 H0 c0 t0 +t3 k H1))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda +(i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t1: +T).(\lambda (t2: T).(\lambda (H1: (pr0 t1 t2)).(\lambda (t: T).(\lambda (H2: +(subst0 i u t2 t)).(\lambda (t0: T).(\lambda (t3: T).(\lambda (k: K).(\lambda +(H3: (pr2 (CHead c0 k t) t0 t3)).(insert_eq C (CHead c0 k t) (\lambda (c1: +C).(pr2 c1 t0 t3)) (\lambda (_: C).(pr3 (CHead c0 k t1) t0 t3)) (\lambda (y: +C).(\lambda (H4: (pr2 y t0 t3)).(pr2_ind (\lambda (c1: C).(\lambda (t4: +T).(\lambda (t5: T).((eq C c1 (CHead c0 k t)) \to (pr3 (CHead c0 k t1) t4 +t5))))) (\lambda (c1: C).(\lambda (t4: T).(\lambda (t5: T).(\lambda (H5: (pr0 +t4 t5)).(\lambda (_: (eq C c1 (CHead c0 k t))).(pr3_pr2 (CHead c0 k t1) t4 t5 +(pr2_free (CHead c0 k t1) t4 t5 H5))))))) (\lambda (c1: C).(\lambda (d0: +C).(\lambda (u0: T).(\lambda (i0: nat).(\lambda (H5: (getl i0 c1 (CHead d0 +(Bind Abbr) u0))).(\lambda (t4: T).(\lambda (t5: T).(\lambda (H6: (pr0 t4 +t5)).(\lambda (t6: T).(\lambda (H7: (subst0 i0 u0 t5 t6)).(\lambda (H8: (eq C +c1 (CHead c0 k t))).(let H9 \def (eq_ind C c1 (\lambda (c2: C).(getl i0 c2 +(CHead d0 (Bind Abbr) u0))) H5 (CHead c0 k t) H8) in (nat_ind (\lambda (n: +nat).((getl n (CHead c0 k t) (CHead d0 (Bind Abbr) u0)) \to ((subst0 n u0 t5 +t6) \to (pr3 (CHead c0 k t1) t4 t6)))) (\lambda (H10: (getl O (CHead c0 k t) +(CHead d0 (Bind Abbr) u0))).(\lambda (H11: (subst0 O u0 t5 t6)).(K_ind +(\lambda (k0: K).((clear (CHead c0 k0 t) (CHead d0 (Bind Abbr) u0)) \to (pr3 +(CHead c0 k0 t1) t4 t6))) (\lambda (b: B).(\lambda (H12: (clear (CHead c0 +(Bind b) t) (CHead d0 (Bind Abbr) u0))).(let H13 \def (f_equal C C (\lambda +(e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d0 +| (CHead c2 _ _) \Rightarrow c2])) (CHead d0 (Bind Abbr) u0) (CHead c0 (Bind +b) t) (clear_gen_bind b c0 (CHead d0 (Bind Abbr) u0) t H12)) in ((let H14 +\def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) +with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow (match k0 in K +return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) +\Rightarrow Abbr])])) (CHead d0 (Bind Abbr) u0) (CHead c0 (Bind b) t) +(clear_gen_bind b c0 (CHead d0 (Bind Abbr) u0) t H12)) in ((let H15 \def +(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with +[(CSort _) \Rightarrow u0 | (CHead _ _ t7) \Rightarrow t7])) (CHead d0 (Bind +Abbr) u0) (CHead c0 (Bind b) t) (clear_gen_bind b c0 (CHead d0 (Bind Abbr) +u0) t H12)) in (\lambda (H16: (eq B Abbr b)).(\lambda (_: (eq C d0 c0)).(let +H18 \def (eq_ind T u0 (\lambda (t7: T).(subst0 O t7 t5 t6)) H11 t H15) in +(eq_ind B Abbr (\lambda (b0: B).(pr3 (CHead c0 (Bind b0) t1) t4 t6)) (ex2_ind +T (\lambda (t7: T).(subst0 O t2 t5 t7)) (\lambda (t7: T).(subst0 (S (plus i +O)) u t7 t6)) (pr3 (CHead c0 (Bind Abbr) t1) t4 t6) (\lambda (x: T).(\lambda +(H19: (subst0 O t2 t5 x)).(\lambda (H20: (subst0 (S (plus i O)) u x t6)).(let +H21 \def (f_equal nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O +i))) in (let H22 \def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n +u x t6)) H20 (S i) H21) in (ex2_ind T (\lambda (t7: T).(subst0 O t1 t5 t7)) +(\lambda (t7: T).(pr0 t7 x)) (pr3 (CHead c0 (Bind Abbr) t1) t4 t6) (\lambda +(x0: T).(\lambda (H23: (subst0 O t1 t5 x0)).(\lambda (H24: (pr0 x0 +x)).(pr3_sing (CHead c0 (Bind Abbr) t1) x0 t4 (pr2_delta (CHead c0 (Bind +Abbr) t1) c0 t1 O (getl_refl Abbr c0 t1) t4 t5 H6 x0 H23) t6 (pr3_pr2 (CHead +c0 (Bind Abbr) t1) x0 t6 (pr2_delta (CHead c0 (Bind Abbr) t1) d u (S i) +(getl_clear_bind Abbr (CHead c0 (Bind Abbr) t1) c0 t1 (clear_bind Abbr c0 t1) +(CHead d (Bind Abbr) u) i H0) x0 x H24 t6 H22)))))) (pr0_subst0_back t2 t5 x +O H19 t1 H1))))))) (subst0_subst0 t5 t6 t O H18 t2 u i H2)) b H16))))) H14)) +H13)))) (\lambda (f: F).(\lambda (H12: (clear (CHead c0 (Flat f) t) (CHead d0 +(Bind Abbr) u0))).(pr3_pr2 (CHead c0 (Flat f) t1) t4 t6 (pr2_cflat c0 t4 t6 +(pr2_delta c0 d0 u0 O (getl_intro O c0 (CHead d0 (Bind Abbr) u0) c0 +(drop_refl c0) (clear_gen_flat f c0 (CHead d0 (Bind Abbr) u0) t H12)) t4 t5 +H6 t6 H11) f t1)))) k (getl_gen_O (CHead c0 k t) (CHead d0 (Bind Abbr) u0) +H10)))) (\lambda (i1: nat).(\lambda (_: (((getl i1 (CHead c0 k t) (CHead d0 +(Bind Abbr) u0)) \to ((subst0 i1 u0 t5 t6) \to (pr3 (CHead c0 k t1) t4 +t6))))).(\lambda (H10: (getl (S i1) (CHead c0 k t) (CHead d0 (Bind Abbr) +u0))).(\lambda (H11: (subst0 (S i1) u0 t5 t6)).(K_ind (\lambda (k0: K).((getl +(S i1) (CHead c0 k0 t) (CHead d0 (Bind Abbr) u0)) \to (pr3 (CHead c0 k0 t1) +t4 t6))) (\lambda (b: B).(\lambda (H12: (getl (S i1) (CHead c0 (Bind b) t) +(CHead d0 (Bind Abbr) u0))).(pr3_pr2 (CHead c0 (Bind b) t1) t4 t6 (pr2_delta +(CHead c0 (Bind b) t1) d0 u0 (S i1) (getl_head (Bind b) i1 c0 (CHead d0 (Bind +Abbr) u0) (getl_gen_S (Bind b) c0 (CHead d0 (Bind Abbr) u0) t i1 H12) t1) t4 +t5 H6 t6 H11)))) (\lambda (f: F).(\lambda (H12: (getl (S i1) (CHead c0 (Flat +f) t) (CHead d0 (Bind Abbr) u0))).(pr3_pr2 (CHead c0 (Flat f) t1) t4 t6 +(pr2_cflat c0 t4 t6 (pr2_delta c0 d0 u0 (r (Flat f) i1) (getl_gen_S (Flat f) +c0 (CHead d0 (Bind Abbr) u0) t i1 H12) t4 t5 H6 t6 H11) f t1)))) k H10))))) +i0 H9 H7))))))))))))) y t0 t3 H4))) H3))))))))))))))) c u1 u2 H)))). +(* COMMENTS +Initial nodes: 1697 +END *) + +theorem pr3_pr2_pr3_t: + \forall (c: C).(\forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall +(k: K).((pr3 (CHead c k u2) t1 t2) \to (\forall (u1: T).((pr2 c u1 u2) \to +(pr3 (CHead c k u1) t1 t2)))))))) +\def + \lambda (c: C).(\lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda +(k: K).(\lambda (H: (pr3 (CHead c k u2) t1 t2)).(pr3_ind (CHead c k u2) +(\lambda (t: T).(\lambda (t0: T).(\forall (u1: T).((pr2 c u1 u2) \to (pr3 +(CHead c k u1) t t0))))) (\lambda (t: T).(\lambda (u1: T).(\lambda (_: (pr2 c +u1 u2)).(pr3_refl (CHead c k u1) t)))) (\lambda (t0: T).(\lambda (t3: +T).(\lambda (H0: (pr2 (CHead c k u2) t3 t0)).(\lambda (t4: T).(\lambda (_: +(pr3 (CHead c k u2) t0 t4)).(\lambda (H2: ((\forall (u1: T).((pr2 c u1 u2) +\to (pr3 (CHead c k u1) t0 t4))))).(\lambda (u1: T).(\lambda (H3: (pr2 c u1 +u2)).(pr3_t t0 t3 (CHead c k u1) (pr3_pr2_pr2_t c u1 u2 H3 t3 t0 k H0) t4 (H2 +u1 H3)))))))))) t1 t2 H)))))). +(* COMMENTS +Initial nodes: 199 +END *) + +theorem pr3_pr3_pr3_t: + \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall +(t1: T).(\forall (t2: T).(\forall (k: K).((pr3 (CHead c k u2) t1 t2) \to (pr3 +(CHead c k u1) t1 t2)))))))) +\def + \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1 +u2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (t1: T).(\forall +(t2: T).(\forall (k: K).((pr3 (CHead c k t0) t1 t2) \to (pr3 (CHead c k t) t1 +t2))))))) (\lambda (t: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k: +K).(\lambda (H0: (pr3 (CHead c k t) t1 t2)).H0))))) (\lambda (t2: T).(\lambda +(t1: T).(\lambda (H0: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda (_: (pr3 c t2 +t3)).(\lambda (H2: ((\forall (t4: T).(\forall (t5: T).(\forall (k: K).((pr3 +(CHead c k t3) t4 t5) \to (pr3 (CHead c k t2) t4 t5))))))).(\lambda (t0: +T).(\lambda (t4: T).(\lambda (k: K).(\lambda (H3: (pr3 (CHead c k t3) t0 +t4)).(pr3_pr2_pr3_t c t2 t0 t4 k (H2 t0 t4 k H3) t1 H0))))))))))) u1 u2 H)))). +(* COMMENTS +Initial nodes: 187 +END *) + +theorem pr3_lift: + \forall (c: C).(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h +d c e) \to (\forall (t1: T).(\forall (t2: T).((pr3 e t1 t2) \to (pr3 c (lift +h d t1) (lift h d t2))))))))) +\def + \lambda (c: C).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda +(H: (drop h d c e)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 e t1 +t2)).(pr3_ind e (\lambda (t: T).(\lambda (t0: T).(pr3 c (lift h d t) (lift h +d t0)))) (\lambda (t: T).(pr3_refl c (lift h d t))) (\lambda (t0: T).(\lambda +(t3: T).(\lambda (H1: (pr2 e t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 e t0 +t4)).(\lambda (H3: (pr3 c (lift h d t0) (lift h d t4))).(pr3_sing c (lift h d +t0) (lift h d t3) (pr2_lift c e h d H t3 t0 H1) (lift h d t4) H3))))))) t1 t2 +H0)))))))). +(* COMMENTS +Initial nodes: 167 +END *) + +theorem pr3_eta: + \forall (c: C).(\forall (w: T).(\forall (u: T).(let t \def (THead (Bind +Abst) w u) in (\forall (v: T).((pr3 c v w) \to (pr3 c (THead (Bind Abst) v +(THead (Flat Appl) (TLRef O) (lift (S O) O t))) t)))))) +\def + \lambda (c: C).(\lambda (w: T).(\lambda (u: T).(let t \def (THead (Bind +Abst) w u) in (\lambda (v: T).(\lambda (H: (pr3 c v w)).(eq_ind_r T (THead +(Bind Abst) (lift (S O) O w) (lift (S O) (S O) u)) (\lambda (t0: T).(pr3 c +(THead (Bind Abst) v (THead (Flat Appl) (TLRef O) t0)) (THead (Bind Abst) w +u))) (pr3_head_12 c v w H (Bind Abst) (THead (Flat Appl) (TLRef O) (THead +(Bind Abst) (lift (S O) O w) (lift (S O) (S O) u))) u (pr3_pr1 (THead (Flat +Appl) (TLRef O) (THead (Bind Abst) (lift (S O) O w) (lift (S O) (S O) u))) u +(pr1_sing (THead (Bind Abbr) (TLRef O) (lift (S O) (S O) u)) (THead (Flat +Appl) (TLRef O) (THead (Bind Abst) (lift (S O) O w) (lift (S O) (S O) u))) +(pr0_beta (lift (S O) O w) (TLRef O) (TLRef O) (pr0_refl (TLRef O)) (lift (S +O) (S O) u) (lift (S O) (S O) u) (pr0_refl (lift (S O) (S O) u))) u (pr1_sing +(THead (Bind Abbr) (TLRef O) (lift (S O) O u)) (THead (Bind Abbr) (TLRef O) +(lift (S O) (S O) u)) (pr0_delta1 (TLRef O) (TLRef O) (pr0_refl (TLRef O)) +(lift (S O) (S O) u) (lift (S O) (S O) u) (pr0_refl (lift (S O) (S O) u)) +(lift (S O) O u) (subst1_lift_S u O O (le_n O))) u (pr1_pr0 (THead (Bind +Abbr) (TLRef O) (lift (S O) O u)) u (pr0_zeta Abbr not_abbr_abst u u +(pr0_refl u) (TLRef O))))) (CHead c (Bind Abst) w))) (lift (S O) O (THead +(Bind Abst) w u)) (lift_bind Abst w u (S O) O))))))). +(* COMMENTS +Initial nodes: 523 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr3/subst1.ma b/matita/matita/contribs/lambdadelta/basic_1/pr3/subst1.ma new file mode 100644 index 000000000..3bcbd4965 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/pr3/subst1.ma @@ -0,0 +1,95 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/pr3/defs.ma". + +include "Basic-1/pr2/subst1.ma". + +theorem pr3_subst1: + \forall (c: C).(\forall (e: C).(\forall (v: T).(\forall (i: nat).((getl i c +(CHead e (Bind Abbr) v)) \to (\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) +\to (\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T (\lambda (w2: T).(pr3 c +w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2)))))))))))) +\def + \lambda (c: C).(\lambda (e: C).(\lambda (v: T).(\lambda (i: nat).(\lambda +(H: (getl i c (CHead e (Bind Abbr) v))).(\lambda (t1: T).(\lambda (t2: +T).(\lambda (H0: (pr3 c t1 t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: +T).(\forall (w1: T).((subst1 i v t w1) \to (ex2 T (\lambda (w2: T).(pr3 c w1 +w2)) (\lambda (w2: T).(subst1 i v t0 w2))))))) (\lambda (t: T).(\lambda (w1: +T).(\lambda (H1: (subst1 i v t w1)).(ex_intro2 T (\lambda (w2: T).(pr3 c w1 +w2)) (\lambda (w2: T).(subst1 i v t w2)) w1 (pr3_refl c w1) H1)))) (\lambda +(t3: T).(\lambda (t4: T).(\lambda (H1: (pr2 c t4 t3)).(\lambda (t5: +T).(\lambda (_: (pr3 c t3 t5)).(\lambda (H3: ((\forall (w1: T).((subst1 i v +t3 w1) \to (ex2 T (\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1 i +v t5 w2))))))).(\lambda (w1: T).(\lambda (H4: (subst1 i v t4 w1)).(ex2_ind T +(\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t3 w2)) (ex2 T +(\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1 i v t5 w2))) +(\lambda (x: T).(\lambda (H5: (pr2 c w1 x)).(\lambda (H6: (subst1 i v t3 +x)).(ex2_ind T (\lambda (w2: T).(pr3 c x w2)) (\lambda (w2: T).(subst1 i v t5 +w2)) (ex2 T (\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1 i v t5 +w2))) (\lambda (x0: T).(\lambda (H7: (pr3 c x x0)).(\lambda (H8: (subst1 i v +t5 x0)).(ex_intro2 T (\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1 +i v t5 w2)) x0 (pr3_sing c x w1 H5 x0 H7) H8)))) (H3 x H6))))) (pr2_subst1 c +e v i H t4 t3 H1 w1 H4)))))))))) t1 t2 H0)))))))). +(* COMMENTS +Initial nodes: 425 +END *) + +theorem pr3_gen_cabbr: + \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall +(e: C).(\forall (u: T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) +\to (\forall (a0: C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d +a0 a) \to (\forall (x1: T).((subst1 d u t1 (lift (S O) d x1)) \to (ex2 T +(\lambda (x2: T).(subst1 d u t2 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a +x1 x2)))))))))))))))) +\def + \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 +t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (e: C).(\forall (u: +T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) \to (\forall (a0: +C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (\forall +(x1: T).((subst1 d u t (lift (S O) d x1)) \to (ex2 T (\lambda (x2: T).(subst1 +d u t0 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x1 x2))))))))))))))) +(\lambda (t: T).(\lambda (e: C).(\lambda (u: T).(\lambda (d: nat).(\lambda +(_: (getl d c (CHead e (Bind Abbr) u))).(\lambda (a0: C).(\lambda (_: +(csubst1 d u c a0)).(\lambda (a: C).(\lambda (_: (drop (S O) d a0 +a)).(\lambda (x1: T).(\lambda (H3: (subst1 d u t (lift (S O) d +x1))).(ex_intro2 T (\lambda (x2: T).(subst1 d u t (lift (S O) d x2))) +(\lambda (x2: T).(pr3 a x1 x2)) x1 H3 (pr3_refl a x1))))))))))))) (\lambda +(t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c t3 t0)).(\lambda (t4: +T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2: ((\forall (e: C).(\forall (u: +T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) \to (\forall (a0: +C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (\forall +(x1: T).((subst1 d u t0 (lift (S O) d x1)) \to (ex2 T (\lambda (x2: +T).(subst1 d u t4 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x1 +x2))))))))))))))).(\lambda (e: C).(\lambda (u: T).(\lambda (d: nat).(\lambda +(H3: (getl d c (CHead e (Bind Abbr) u))).(\lambda (a0: C).(\lambda (H4: +(csubst1 d u c a0)).(\lambda (a: C).(\lambda (H5: (drop (S O) d a0 +a)).(\lambda (x1: T).(\lambda (H6: (subst1 d u t3 (lift (S O) d +x1))).(ex2_ind T (\lambda (x2: T).(subst1 d u t0 (lift (S O) d x2))) (\lambda +(x2: T).(pr2 a x1 x2)) (ex2 T (\lambda (x2: T).(subst1 d u t4 (lift (S O) d +x2))) (\lambda (x2: T).(pr3 a x1 x2))) (\lambda (x: T).(\lambda (H7: (subst1 +d u t0 (lift (S O) d x))).(\lambda (H8: (pr2 a x1 x)).(ex2_ind T (\lambda +(x2: T).(subst1 d u t4 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x x2)) +(ex2 T (\lambda (x2: T).(subst1 d u t4 (lift (S O) d x2))) (\lambda (x2: +T).(pr3 a x1 x2))) (\lambda (x0: T).(\lambda (H9: (subst1 d u t4 (lift (S O) +d x0))).(\lambda (H10: (pr3 a x x0)).(ex_intro2 T (\lambda (x2: T).(subst1 d +u t4 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x1 x2)) x0 H9 (pr3_sing a x +x1 H8 x0 H10))))) (H2 e u d H3 a0 H4 a H5 x H7))))) (pr2_gen_cabbr c t3 t0 H0 +e u d H3 a0 H4 a H5 x1 H6)))))))))))))))))) t1 t2 H)))). +(* COMMENTS +Initial nodes: 731 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr3/wcpr0.ma b/matita/matita/contribs/lambdadelta/basic_1/pr3/wcpr0.ma new file mode 100644 index 000000000..052f4e5b2 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/pr3/wcpr0.ma @@ -0,0 +1,66 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/pr3/props.ma". + +include "Basic-1/wcpr0/getl.ma". + +theorem pr3_wcpr0_t: + \forall (c1: C).(\forall (c2: C).((wcpr0 c2 c1) \to (\forall (t1: +T).(\forall (t2: T).((pr3 c1 t1 t2) \to (pr3 c2 t1 t2)))))) +\def + \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c2 c1)).(wcpr0_ind +(\lambda (c: C).(\lambda (c0: C).(\forall (t1: T).(\forall (t2: T).((pr3 c0 +t1 t2) \to (pr3 c t1 t2)))))) (\lambda (c: C).(\lambda (t1: T).(\lambda (t2: +T).(\lambda (H0: (pr3 c t1 t2)).H0)))) (\lambda (c0: C).(\lambda (c3: +C).(\lambda (H0: (wcpr0 c0 c3)).(\lambda (_: ((\forall (t1: T).(\forall (t2: +T).((pr3 c3 t1 t2) \to (pr3 c0 t1 t2)))))).(\lambda (u1: T).(\lambda (u2: +T).(\lambda (H2: (pr0 u1 u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: +T).(\lambda (H3: (pr3 (CHead c3 k u2) t1 t2)).(pr3_ind (CHead c3 k u1) +(\lambda (t: T).(\lambda (t0: T).(pr3 (CHead c0 k u1) t t0))) (\lambda (t: +T).(pr3_refl (CHead c0 k u1) t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda +(H4: (pr2 (CHead c3 k u1) t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 (CHead +c3 k u1) t0 t4)).(\lambda (H6: (pr3 (CHead c0 k u1) t0 t4)).(pr3_t t0 t3 +(CHead c0 k u1) (insert_eq C (CHead c3 k u1) (\lambda (c: C).(pr2 c t3 t0)) +(\lambda (_: C).(pr3 (CHead c0 k u1) t3 t0)) (\lambda (y: C).(\lambda (H7: +(pr2 y t3 t0)).(pr2_ind (\lambda (c: C).(\lambda (t: T).(\lambda (t5: T).((eq +C c (CHead c3 k u1)) \to (pr3 (CHead c0 k u1) t t5))))) (\lambda (c: +C).(\lambda (t5: T).(\lambda (t6: T).(\lambda (H8: (pr0 t5 t6)).(\lambda (_: +(eq C c (CHead c3 k u1))).(pr3_pr2 (CHead c0 k u1) t5 t6 (pr2_free (CHead c0 +k u1) t5 t6 H8))))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: +T).(\lambda (i: nat).(\lambda (H8: (getl i c (CHead d (Bind Abbr) +u))).(\lambda (t5: T).(\lambda (t6: T).(\lambda (H9: (pr0 t5 t6)).(\lambda +(t: T).(\lambda (H10: (subst0 i u t6 t)).(\lambda (H11: (eq C c (CHead c3 k +u1))).(let H12 \def (eq_ind C c (\lambda (c4: C).(getl i c4 (CHead d (Bind +Abbr) u))) H8 (CHead c3 k u1) H11) in (ex3_2_ind C T (\lambda (e2: +C).(\lambda (u3: T).(getl i (CHead c0 k u1) (CHead e2 (Bind Abbr) u3)))) +(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 d))) (\lambda (_: C).(\lambda (u3: +T).(pr0 u3 u))) (pr3 (CHead c0 k u1) t5 t) (\lambda (x0: C).(\lambda (x1: +T).(\lambda (H13: (getl i (CHead c0 k u1) (CHead x0 (Bind Abbr) +x1))).(\lambda (_: (wcpr0 x0 d)).(\lambda (H15: (pr0 x1 u)).(ex2_ind T +(\lambda (t7: T).(subst0 i x1 t6 t7)) (\lambda (t7: T).(pr0 t7 t)) (pr3 +(CHead c0 k u1) t5 t) (\lambda (x: T).(\lambda (H16: (subst0 i x1 t6 +x)).(\lambda (H17: (pr0 x t)).(pr3_sing (CHead c0 k u1) x t5 (pr2_delta +(CHead c0 k u1) x0 x1 i H13 t5 t6 H9 x H16) t (pr3_pr2 (CHead c0 k u1) x t +(pr2_free (CHead c0 k u1) x t H17)))))) (pr0_subst0_back u t6 t i H10 x1 +H15))))))) (wcpr0_getl_back (CHead c3 k u1) (CHead c0 k u1) (wcpr0_comp c0 c3 +H0 u1 u1 (pr0_refl u1) k) i d u (Bind Abbr) H12)))))))))))))) y t3 t0 H7))) +H4) t4 H6))))))) t1 t2 (pr3_pr2_pr3_t c3 u2 t1 t2 k H3 u1 (pr2_free c3 u1 u2 +H2)))))))))))))) c2 c1 H))). +(* COMMENTS +Initial nodes: 799 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/preamble.ma b/matita/matita/contribs/lambdadelta/basic_1/preamble.ma new file mode 100644 index 000000000..c9d6a172a --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/preamble.ma @@ -0,0 +1,15 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "Ground-1/theory.ma". diff --git a/matita/matita/contribs/lambdadelta/basic_1/r/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/r/defs.ma new file mode 100644 index 000000000..cde40fdf8 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/r/defs.ma @@ -0,0 +1,24 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/T/defs.ma". + +definition r: + K \to (nat \to nat) +\def + \lambda (k: K).(\lambda (i: nat).(match k with [(Bind _) \Rightarrow i | +(Flat _) \Rightarrow (S i)])). + diff --git a/matita/matita/contribs/lambdadelta/basic_1/r/props.ma b/matita/matita/contribs/lambdadelta/basic_1/r/props.ma new file mode 100644 index 000000000..0815aaf5e --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/r/props.ma @@ -0,0 +1,117 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/r/defs.ma". + +include "Basic-1/s/defs.ma". + +theorem r_S: + \forall (k: K).(\forall (i: nat).(eq nat (r k (S i)) (S (r k i)))) +\def + \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(eq nat (r k0 (S +i)) (S (r k0 i))))) (\lambda (b: B).(\lambda (i: nat).(refl_equal nat (S (r +(Bind b) i))))) (\lambda (f: F).(\lambda (i: nat).(refl_equal nat (S (r (Flat +f) i))))) k). +(* COMMENTS +Initial nodes: 65 +END *) + +theorem r_plus: + \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (r k (plus i j)) +(plus (r k i) j)))) +\def + \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: +nat).(eq nat (r k0 (plus i j)) (plus (r k0 i) j))))) (\lambda (b: B).(\lambda +(i: nat).(\lambda (j: nat).(refl_equal nat (plus (r (Bind b) i) j))))) +(\lambda (f: F).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (plus (r +(Flat f) i) j))))) k). +(* COMMENTS +Initial nodes: 79 +END *) + +theorem r_plus_sym: + \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (r k (plus i j)) +(plus i (r k j))))) +\def + \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: +nat).(eq nat (r k0 (plus i j)) (plus i (r k0 j)))))) (\lambda (_: B).(\lambda +(i: nat).(\lambda (j: nat).(refl_equal nat (plus i j))))) (\lambda (_: +F).(\lambda (i: nat).(\lambda (j: nat).(plus_n_Sm i j)))) k). +(* COMMENTS +Initial nodes: 63 +END *) + +theorem r_minus: + \forall (i: nat).(\forall (n: nat).((lt n i) \to (\forall (k: K).(eq nat +(minus (r k i) (S n)) (r k (minus i (S n))))))) +\def + \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (lt n i)).(\lambda (k: +K).(K_ind (\lambda (k0: K).(eq nat (minus (r k0 i) (S n)) (r k0 (minus i (S +n))))) (\lambda (_: B).(refl_equal nat (minus i (S n)))) (\lambda (_: +F).(minus_x_Sy i n H)) k)))). +(* COMMENTS +Initial nodes: 69 +END *) + +theorem r_dis: + \forall (k: K).(\forall (P: Prop).(((((\forall (i: nat).(eq nat (r k i) i))) +\to P)) \to (((((\forall (i: nat).(eq nat (r k i) (S i)))) \to P)) \to P))) +\def + \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (P: Prop).(((((\forall (i: +nat).(eq nat (r k0 i) i))) \to P)) \to (((((\forall (i: nat).(eq nat (r k0 i) +(S i)))) \to P)) \to P)))) (\lambda (b: B).(\lambda (P: Prop).(\lambda (H: +((((\forall (i: nat).(eq nat (r (Bind b) i) i))) \to P))).(\lambda (_: +((((\forall (i: nat).(eq nat (r (Bind b) i) (S i)))) \to P))).(H (\lambda (i: +nat).(refl_equal nat i))))))) (\lambda (f: F).(\lambda (P: Prop).(\lambda (_: +((((\forall (i: nat).(eq nat (r (Flat f) i) i))) \to P))).(\lambda (H0: +((((\forall (i: nat).(eq nat (r (Flat f) i) (S i)))) \to P))).(H0 (\lambda +(i: nat).(refl_equal nat (S i)))))))) k). +(* COMMENTS +Initial nodes: 151 +END *) + +theorem s_r: + \forall (k: K).(\forall (i: nat).(eq nat (s k (r k i)) (S i))) +\def + \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(eq nat (s k0 (r k0 +i)) (S i)))) (\lambda (_: B).(\lambda (i: nat).(refl_equal nat (S i)))) +(\lambda (_: F).(\lambda (i: nat).(refl_equal nat (S i)))) k). +(* COMMENTS +Initial nodes: 51 +END *) + +theorem r_arith0: + \forall (k: K).(\forall (i: nat).(eq nat (minus (r k (S i)) (S O)) (r k i))) +\def + \lambda (k: K).(\lambda (i: nat).(eq_ind_r nat (S (r k i)) (\lambda (n: +nat).(eq nat (minus n (S O)) (r k i))) (eq_ind_r nat (r k i) (\lambda (n: +nat).(eq nat n (r k i))) (refl_equal nat (r k i)) (minus (S (r k i)) (S O)) +(minus_Sx_SO (r k i))) (r k (S i)) (r_S k i))). +(* COMMENTS +Initial nodes: 105 +END *) + +theorem r_arith1: + \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (minus (r k (S +i)) (S j)) (minus (r k i) j)))) +\def + \lambda (k: K).(\lambda (i: nat).(\lambda (j: nat).(eq_ind_r nat (S (r k i)) +(\lambda (n: nat).(eq nat (minus n (S j)) (minus (r k i) j))) (refl_equal nat +(minus (r k i) j)) (r k (S i)) (r_S k i)))). +(* COMMENTS +Initial nodes: 69 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/s/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/s/defs.ma new file mode 100644 index 000000000..0d1fb7914 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/s/defs.ma @@ -0,0 +1,24 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/T/defs.ma". + +definition s: + K \to (nat \to nat) +\def + \lambda (k: K).(\lambda (i: nat).(match k with [(Bind _) \Rightarrow (S i) | +(Flat _) \Rightarrow i])). + diff --git a/matita/matita/contribs/lambdadelta/basic_1/s/props.ma b/matita/matita/contribs/lambdadelta/basic_1/s/props.ma new file mode 100644 index 000000000..3cb4fbd74 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/s/props.ma @@ -0,0 +1,151 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/s/defs.ma". + +theorem s_S: + \forall (k: K).(\forall (i: nat).(eq nat (s k (S i)) (S (s k i)))) +\def + \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(eq nat (s k0 (S +i)) (S (s k0 i))))) (\lambda (b: B).(\lambda (i: nat).(refl_equal nat (S (s +(Bind b) i))))) (\lambda (f: F).(\lambda (i: nat).(refl_equal nat (S (s (Flat +f) i))))) k). +(* COMMENTS +Initial nodes: 65 +END *) + +theorem s_plus: + \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (s k (plus i j)) +(plus (s k i) j)))) +\def + \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: +nat).(eq nat (s k0 (plus i j)) (plus (s k0 i) j))))) (\lambda (b: B).(\lambda +(i: nat).(\lambda (j: nat).(refl_equal nat (plus (s (Bind b) i) j))))) +(\lambda (f: F).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (plus (s +(Flat f) i) j))))) k). +(* COMMENTS +Initial nodes: 79 +END *) + +theorem s_plus_sym: + \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (s k (plus i j)) +(plus i (s k j))))) +\def + \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: +nat).(eq nat (s k0 (plus i j)) (plus i (s k0 j)))))) (\lambda (_: B).(\lambda +(i: nat).(\lambda (j: nat).(eq_ind_r nat (plus i (S j)) (\lambda (n: nat).(eq +nat n (plus i (S j)))) (refl_equal nat (plus i (S j))) (S (plus i j)) +(plus_n_Sm i j))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (j: +nat).(refl_equal nat (plus i (s (Flat f) j)))))) k). +(* COMMENTS +Initial nodes: 117 +END *) + +theorem s_minus: + \forall (k: K).(\forall (i: nat).(\forall (j: nat).((le j i) \to (eq nat (s +k (minus i j)) (minus (s k i) j))))) +\def + \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: +nat).((le j i) \to (eq nat (s k0 (minus i j)) (minus (s k0 i) j)))))) +(\lambda (_: B).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (le j +i)).(eq_ind_r nat (minus (S i) j) (\lambda (n: nat).(eq nat n (minus (S i) +j))) (refl_equal nat (minus (S i) j)) (S (minus i j)) (minus_Sn_m i j H)))))) +(\lambda (f: F).(\lambda (i: nat).(\lambda (j: nat).(\lambda (_: (le j +i)).(refl_equal nat (minus (s (Flat f) i) j)))))) k). +(* COMMENTS +Initial nodes: 137 +END *) + +theorem minus_s_s: + \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (minus (s k i) (s +k j)) (minus i j)))) +\def + \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: +nat).(eq nat (minus (s k0 i) (s k0 j)) (minus i j))))) (\lambda (_: +B).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (minus i j))))) +(\lambda (_: F).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (minus i +j))))) k). +(* COMMENTS +Initial nodes: 67 +END *) + +theorem s_le: + \forall (k: K).(\forall (i: nat).(\forall (j: nat).((le i j) \to (le (s k i) +(s k j))))) +\def + \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: +nat).((le i j) \to (le (s k0 i) (s k0 j)))))) (\lambda (_: B).(\lambda (i: +nat).(\lambda (j: nat).(\lambda (H: (le i j)).(le_n_S i j H))))) (\lambda (_: +F).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (le i j)).H)))) k). +(* COMMENTS +Initial nodes: 65 +END *) + +theorem s_lt: + \forall (k: K).(\forall (i: nat).(\forall (j: nat).((lt i j) \to (lt (s k i) +(s k j))))) +\def + \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: +nat).((lt i j) \to (lt (s k0 i) (s k0 j)))))) (\lambda (_: B).(\lambda (i: +nat).(\lambda (j: nat).(\lambda (H: (lt i j)).(le_n_S (S i) j H))))) (\lambda +(_: F).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (lt i j)).H)))) k). +(* COMMENTS +Initial nodes: 67 +END *) + +theorem s_inj: + \forall (k: K).(\forall (i: nat).(\forall (j: nat).((eq nat (s k i) (s k j)) +\to (eq nat i j)))) +\def + \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: +nat).((eq nat (s k0 i) (s k0 j)) \to (eq nat i j))))) (\lambda (b: +B).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (eq nat (s (Bind b) i) (s +(Bind b) j))).(eq_add_S i j H))))) (\lambda (f: F).(\lambda (i: nat).(\lambda +(j: nat).(\lambda (H: (eq nat (s (Flat f) i) (s (Flat f) j))).H)))) k). +(* COMMENTS +Initial nodes: 97 +END *) + +theorem s_inc: + \forall (k: K).(\forall (i: nat).(le i (s k i))) +\def + \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(le i (s k0 i)))) +(\lambda (b: B).(\lambda (i: nat).(le_S_n i (s (Bind b) i) (le_S (S i) (s +(Bind b) i) (le_n (s (Bind b) i)))))) (\lambda (f: F).(\lambda (i: nat).(le_n +(s (Flat f) i)))) k). +(* COMMENTS +Initial nodes: 73 +END *) + +theorem s_arith0: + \forall (k: K).(\forall (i: nat).(eq nat (minus (s k i) (s k O)) i)) +\def + \lambda (k: K).(\lambda (i: nat).(eq_ind_r nat (minus i O) (\lambda (n: +nat).(eq nat n i)) (eq_ind nat i (\lambda (n: nat).(eq nat n i)) (refl_equal +nat i) (minus i O) (minus_n_O i)) (minus (s k i) (s k O)) (minus_s_s k i O))). +(* COMMENTS +Initial nodes: 77 +END *) + +theorem s_arith1: + \forall (b: B).(\forall (i: nat).(eq nat (minus (s (Bind b) i) (S O)) i)) +\def + \lambda (_: B).(\lambda (i: nat).(eq_ind nat i (\lambda (n: nat).(eq nat n +i)) (refl_equal nat i) (minus i O) (minus_n_O i))). +(* COMMENTS +Initial nodes: 35 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/sc3/arity.ma b/matita/matita/contribs/lambdadelta/basic_1/sc3/arity.ma new file mode 100644 index 000000000..651321580 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/sc3/arity.ma @@ -0,0 +1,322 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/csubc/arity.ma". + +include "Basic-1/csubc/getl.ma". + +include "Basic-1/csubc/drop1.ma". + +include "Basic-1/csubc/props.ma". + +theorem sc3_arity_csubc: + \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1 +t a) \to (\forall (d1: C).(\forall (is: PList).((drop1 is d1 c1) \to (\forall +(c2: C).((csubc g d1 c2) \to (sc3 g a c2 (lift1 is t))))))))))) +\def + \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: +(arity g c1 t a)).(arity_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda (a0: +A).(\forall (d1: C).(\forall (is: PList).((drop1 is d1 c) \to (\forall (c2: +C).((csubc g d1 c2) \to (sc3 g a0 c2 (lift1 is t0)))))))))) (\lambda (c: +C).(\lambda (n: nat).(\lambda (d1: C).(\lambda (is: PList).(\lambda (_: +(drop1 is d1 c)).(\lambda (c2: C).(\lambda (_: (csubc g d1 c2)).(eq_ind_r T +(TSort n) (\lambda (t0: T).(land (arity g c2 t0 (ASort O n)) (sn3 c2 t0))) +(conj (arity g c2 (TSort n) (ASort O n)) (sn3 c2 (TSort n)) (arity_sort g c2 +n) (sn3_nf2 c2 (TSort n) (nf2_sort c2 n))) (lift1 is (TSort n)) (lift1_sort n +is))))))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: +nat).(\lambda (H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (a0: +A).(\lambda (_: (arity g d u a0)).(\lambda (H2: ((\forall (d1: C).(\forall +(is: PList).((drop1 is d1 d) \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g +a0 c2 (lift1 is u))))))))).(\lambda (d1: C).(\lambda (is: PList).(\lambda +(H3: (drop1 is d1 c)).(\lambda (c2: C).(\lambda (H4: (csubc g d1 c2)).(let +H_x \def (drop1_getl_trans is c d1 H3 Abbr d u i H0) in (let H5 \def H_x in +(ex2_ind C (\lambda (e2: C).(drop1 (ptrans is i) e2 d)) (\lambda (e2: +C).(getl (trans is i) d1 (CHead e2 (Bind Abbr) (lift1 (ptrans is i) u)))) +(sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x: C).(\lambda (_: (drop1 +(ptrans is i) x d)).(\lambda (H7: (getl (trans is i) d1 (CHead x (Bind Abbr) +(lift1 (ptrans is i) u)))).(let H_x0 \def (csubc_getl_conf g d1 (CHead x +(Bind Abbr) (lift1 (ptrans is i) u)) (trans is i) H7 c2 H4) in (let H8 \def +H_x0 in (ex2_ind C (\lambda (e2: C).(getl (trans is i) c2 e2)) (\lambda (e2: +C).(csubc g (CHead x (Bind Abbr) (lift1 (ptrans is i) u)) e2)) (sc3 g a0 c2 +(lift1 is (TLRef i))) (\lambda (x0: C).(\lambda (H9: (getl (trans is i) c2 +x0)).(\lambda (H10: (csubc g (CHead x (Bind Abbr) (lift1 (ptrans is i) u)) +x0)).(let H_x1 \def (csubc_gen_head_l g x x0 (lift1 (ptrans is i) u) (Bind +Abbr) H10) in (let H11 \def H_x1 in (or3_ind (ex2 C (\lambda (c3: C).(eq C x0 +(CHead c3 (Bind Abbr) (lift1 (ptrans is i) u)))) (\lambda (c3: C).(csubc g x +c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K +(Bind Abbr) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: +A).(eq C x0 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: +T).(\lambda (_: A).(csubc g x c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans is i) u))))) (\lambda (c3: +C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3 w))))) (ex4_3 B C T (\lambda +(b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C x0 (CHead c3 (Bind b) v2))))) +(\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K (Bind Abbr) (Bind +Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b +Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g x c3))))) +(sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (H12: (ex2 C (\lambda (c3: C).(eq +C x0 (CHead c3 (Bind Abbr) (lift1 (ptrans is i) u)))) (\lambda (c3: C).(csubc +g x c3)))).(ex2_ind C (\lambda (c3: C).(eq C x0 (CHead c3 (Bind Abbr) (lift1 +(ptrans is i) u)))) (\lambda (c3: C).(csubc g x c3)) (sc3 g a0 c2 (lift1 is +(TLRef i))) (\lambda (x1: C).(\lambda (H13: (eq C x0 (CHead x1 (Bind Abbr) +(lift1 (ptrans is i) u)))).(\lambda (_: (csubc g x x1)).(let H15 \def (eq_ind +C x0 (\lambda (c0: C).(getl (trans is i) c2 c0)) H9 (CHead x1 (Bind Abbr) +(lift1 (ptrans is i) u)) H13) in (let H_y \def (sc3_abbr g a0 TNil) in +(eq_ind_r T (TLRef (trans is i)) (\lambda (t0: T).(sc3 g a0 c2 t0)) (H_y +(trans is i) x1 (lift1 (ptrans is i) u) c2 (eq_ind T (lift1 is (lift (S i) O +u)) (\lambda (t0: T).(sc3 g a0 c2 t0)) (eq_ind T (lift1 (PConsTail is (S i) +O) u) (\lambda (t0: T).(sc3 g a0 c2 t0)) (H2 d1 (PConsTail is (S i) O) +(drop1_cons_tail c d (S i) O (getl_drop Abbr c d u i H0) is d1 H3) c2 H4) +(lift1 is (lift (S i) O u)) (lift1_cons_tail u (S i) O is)) (lift (S (trans +is i)) O (lift1 (ptrans is i) u)) (lift1_free is i u)) H15) (lift1 is (TLRef +i)) (lift1_lref is i))))))) H12)) (\lambda (H12: (ex5_3 C T A (\lambda (_: +C).(\lambda (_: T).(\lambda (_: A).(eq K (Bind Abbr) (Bind Abst))))) (\lambda +(c3: C).(\lambda (w: T).(\lambda (_: A).(eq C x0 (CHead c3 (Bind Abbr) w))))) +(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g x c3)))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans +is i) u))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3 +w)))))).(ex5_3_ind C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq +K (Bind Abbr) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: +A).(eq C x0 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: +T).(\lambda (_: A).(csubc g x c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans is i) u))))) (\lambda (c3: +C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3 w)))) (sc3 g a0 c2 (lift1 is +(TLRef i))) (\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: A).(\lambda (H13: +(eq K (Bind Abbr) (Bind Abst))).(\lambda (H14: (eq C x0 (CHead x1 (Bind Abbr) +x2))).(\lambda (_: (csubc g x x1)).(\lambda (_: (sc3 g (asucc g x3) x (lift1 +(ptrans is i) u))).(\lambda (_: (sc3 g x3 x1 x2)).(let H18 \def (eq_ind C x0 +(\lambda (c0: C).(getl (trans is i) c2 c0)) H9 (CHead x1 (Bind Abbr) x2) H14) +in (let H19 \def (eq_ind K (Bind Abbr) (\lambda (ee: K).(match ee in K return +(\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B return +(\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | +Void \Rightarrow False]) | (Flat _) \Rightarrow False])) I (Bind Abst) H13) +in (False_ind (sc3 g a0 c2 (lift1 is (TLRef i))) H19))))))))))) H12)) +(\lambda (H12: (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: +T).(eq C x0 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: +C).(\lambda (_: T).(eq K (Bind Abbr) (Bind Void))))) (\lambda (b: B).(\lambda +(_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: +C).(\lambda (_: T).(csubc g x c3)))))).(ex4_3_ind B C T (\lambda (b: +B).(\lambda (c3: C).(\lambda (v2: T).(eq C x0 (CHead c3 (Bind b) v2))))) +(\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K (Bind Abbr) (Bind +Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b +Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g x c3)))) +(sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x1: B).(\lambda (x2: C).(\lambda +(x3: T).(\lambda (H13: (eq C x0 (CHead x2 (Bind x1) x3))).(\lambda (H14: (eq +K (Bind Abbr) (Bind Void))).(\lambda (_: (not (eq B x1 Void))).(\lambda (_: +(csubc g x x2)).(let H17 \def (eq_ind C x0 (\lambda (c0: C).(getl (trans is +i) c2 c0)) H9 (CHead x2 (Bind x1) x3) H13) in (let H18 \def (eq_ind K (Bind +Abbr) (\lambda (ee: K).(match ee in K return (\lambda (_: K).Prop) with +[(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr +\Rightarrow True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat +_) \Rightarrow False])) I (Bind Void) H14) in (False_ind (sc3 g a0 c2 (lift1 +is (TLRef i))) H18)))))))))) H12)) H11)))))) H8)))))) H5)))))))))))))))) +(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda +(H0: (getl i c (CHead d (Bind Abst) u))).(\lambda (a0: A).(\lambda (H1: +(arity g d u (asucc g a0))).(\lambda (_: ((\forall (d1: C).(\forall (is: +PList).((drop1 is d1 d) \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g +(asucc g a0) c2 (lift1 is u))))))))).(\lambda (d1: C).(\lambda (is: +PList).(\lambda (H3: (drop1 is d1 c)).(\lambda (c2: C).(\lambda (H4: (csubc g +d1 c2)).(let H5 \def H0 in (let H_x \def (drop1_getl_trans is c d1 H3 Abst d +u i H5) in (let H6 \def H_x in (ex2_ind C (\lambda (e2: C).(drop1 (ptrans is +i) e2 d)) (\lambda (e2: C).(getl (trans is i) d1 (CHead e2 (Bind Abst) (lift1 +(ptrans is i) u)))) (sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x: +C).(\lambda (H7: (drop1 (ptrans is i) x d)).(\lambda (H8: (getl (trans is i) +d1 (CHead x (Bind Abst) (lift1 (ptrans is i) u)))).(let H_x0 \def +(csubc_getl_conf g d1 (CHead x (Bind Abst) (lift1 (ptrans is i) u)) (trans is +i) H8 c2 H4) in (let H9 \def H_x0 in (ex2_ind C (\lambda (e2: C).(getl (trans +is i) c2 e2)) (\lambda (e2: C).(csubc g (CHead x (Bind Abst) (lift1 (ptrans +is i) u)) e2)) (sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x0: C).(\lambda +(H10: (getl (trans is i) c2 x0)).(\lambda (H11: (csubc g (CHead x (Bind Abst) +(lift1 (ptrans is i) u)) x0)).(let H_x1 \def (csubc_gen_head_l g x x0 (lift1 +(ptrans is i) u) (Bind Abst) H11) in (let H12 \def H_x1 in (or3_ind (ex2 C +(\lambda (c3: C).(eq C x0 (CHead c3 (Bind Abst) (lift1 (ptrans is i) u)))) +(\lambda (c3: C).(csubc g x c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: +T).(\lambda (_: A).(eq K (Bind Abst) (Bind Abst))))) (\lambda (c3: +C).(\lambda (w: T).(\lambda (_: A).(eq C x0 (CHead c3 (Bind Abbr) w))))) +(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g x c3)))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans +is i) u))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3 +w))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C +x0 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: +T).(eq K (Bind Abst) (Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda +(_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: +T).(csubc g x c3))))) (sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (H13: (ex2 +C (\lambda (c3: C).(eq C x0 (CHead c3 (Bind Abst) (lift1 (ptrans is i) u)))) +(\lambda (c3: C).(csubc g x c3)))).(ex2_ind C (\lambda (c3: C).(eq C x0 +(CHead c3 (Bind Abst) (lift1 (ptrans is i) u)))) (\lambda (c3: C).(csubc g x +c3)) (sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x1: C).(\lambda (H14: (eq C +x0 (CHead x1 (Bind Abst) (lift1 (ptrans is i) u)))).(\lambda (_: (csubc g x +x1)).(let H16 \def (eq_ind C x0 (\lambda (c0: C).(getl (trans is i) c2 c0)) +H10 (CHead x1 (Bind Abst) (lift1 (ptrans is i) u)) H14) in (let H_y \def +(sc3_abst g a0 TNil) in (eq_ind_r T (TLRef (trans is i)) (\lambda (t0: +T).(sc3 g a0 c2 t0)) (H_y c2 (trans is i) (csubc_arity_conf g d1 c2 H4 (TLRef +(trans is i)) a0 (eq_ind T (lift1 is (TLRef i)) (\lambda (t0: T).(arity g d1 +t0 a0)) (arity_lift1 g a0 c is d1 (TLRef i) H3 (arity_abst g c d u i H0 a0 +H1)) (TLRef (trans is i)) (lift1_lref is i))) (nf2_lref_abst c2 x1 (lift1 +(ptrans is i) u) (trans is i) H16) I) (lift1 is (TLRef i)) (lift1_lref is +i))))))) H13)) (\lambda (H13: (ex5_3 C T A (\lambda (_: C).(\lambda (_: +T).(\lambda (_: A).(eq K (Bind Abst) (Bind Abst))))) (\lambda (c3: +C).(\lambda (w: T).(\lambda (_: A).(eq C x0 (CHead c3 (Bind Abbr) w))))) +(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g x c3)))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans +is i) u))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3 +w)))))).(ex5_3_ind C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq +K (Bind Abst) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: +A).(eq C x0 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: +T).(\lambda (_: A).(csubc g x c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans is i) u))))) (\lambda (c3: +C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3 w)))) (sc3 g a0 c2 (lift1 is +(TLRef i))) (\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: A).(\lambda (_: +(eq K (Bind Abst) (Bind Abst))).(\lambda (H15: (eq C x0 (CHead x1 (Bind Abbr) +x2))).(\lambda (_: (csubc g x x1)).(\lambda (H17: (sc3 g (asucc g x3) x +(lift1 (ptrans is i) u))).(\lambda (H18: (sc3 g x3 x1 x2)).(let H19 \def +(eq_ind C x0 (\lambda (c0: C).(getl (trans is i) c2 c0)) H10 (CHead x1 (Bind +Abbr) x2) H15) in (let H_y \def (sc3_abbr g a0 TNil) in (eq_ind_r T (TLRef +(trans is i)) (\lambda (t0: T).(sc3 g a0 c2 t0)) (H_y (trans is i) x1 x2 c2 +(let H_y0 \def (arity_lift1 g (asucc g a0) d (ptrans is i) x u H7 H1) in (let +H_y1 \def (sc3_arity_gen g x (lift1 (ptrans is i) u) (asucc g x3) H17) in +(sc3_repl g x3 c2 (lift (S (trans is i)) O x2) (sc3_lift g x3 x1 x2 H18 c2 (S +(trans is i)) O (getl_drop Abbr c2 x1 x2 (trans is i) H19)) a0 (asucc_inj g +x3 a0 (arity_mono g x (lift1 (ptrans is i) u) (asucc g x3) H_y1 (asucc g a0) +H_y0))))) H19) (lift1 is (TLRef i)) (lift1_lref is i)))))))))))) H13)) +(\lambda (H13: (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: +T).(eq C x0 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: +C).(\lambda (_: T).(eq K (Bind Abst) (Bind Void))))) (\lambda (b: B).(\lambda +(_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: +C).(\lambda (_: T).(csubc g x c3)))))).(ex4_3_ind B C T (\lambda (b: +B).(\lambda (c3: C).(\lambda (v2: T).(eq C x0 (CHead c3 (Bind b) v2))))) +(\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K (Bind Abst) (Bind +Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b +Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g x c3)))) +(sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x1: B).(\lambda (x2: C).(\lambda +(x3: T).(\lambda (H14: (eq C x0 (CHead x2 (Bind x1) x3))).(\lambda (H15: (eq +K (Bind Abst) (Bind Void))).(\lambda (_: (not (eq B x1 Void))).(\lambda (_: +(csubc g x x2)).(let H18 \def (eq_ind C x0 (\lambda (c0: C).(getl (trans is +i) c2 c0)) H10 (CHead x2 (Bind x1) x3) H14) in (let H19 \def (eq_ind K (Bind +Abst) (\lambda (ee: K).(match ee in K return (\lambda (_: K).Prop) with +[(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr +\Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat +_) \Rightarrow False])) I (Bind Void) H15) in (False_ind (sc3 g a0 c2 (lift1 +is (TLRef i))) H19)))))))))) H13)) H12)))))) H9)))))) H6))))))))))))))))) +(\lambda (b: B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c: C).(\lambda +(u: T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H2: +((\forall (d1: C).(\forall (is: PList).((drop1 is d1 c) \to (\forall (c2: +C).((csubc g d1 c2) \to (sc3 g a1 c2 (lift1 is u))))))))).(\lambda (t0: +T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c (Bind b) u) t0 +a2)).(\lambda (H4: ((\forall (d1: C).(\forall (is: PList).((drop1 is d1 +(CHead c (Bind b) u)) \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g a2 c2 +(lift1 is t0))))))))).(\lambda (d1: C).(\lambda (is: PList).(\lambda (H5: +(drop1 is d1 c)).(\lambda (c2: C).(\lambda (H6: (csubc g d1 c2)).(let H_y +\def (sc3_bind g b H0 a1 a2 TNil) in (eq_ind_r T (THead (Bind b) (lift1 is u) +(lift1 (Ss is) t0)) (\lambda (t1: T).(sc3 g a2 c2 t1)) (H_y c2 (lift1 is u) +(lift1 (Ss is) t0) (H4 (CHead d1 (Bind b) (lift1 is u)) (Ss is) +(drop1_skip_bind b c is d1 u H5) (CHead c2 (Bind b) (lift1 is u)) (csubc_head +g d1 c2 H6 (Bind b) (lift1 is u))) (H2 d1 is H5 c2 H6)) (lift1 is (THead +(Bind b) u t0)) (lift1_bind b is u t0))))))))))))))))))) (\lambda (c: +C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H0: (arity g c u (asucc g +a1))).(\lambda (H1: ((\forall (d1: C).(\forall (is: PList).((drop1 is d1 c) +\to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g (asucc g a1) c2 (lift1 is +u))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (H2: (arity g (CHead c +(Bind Abst) u) t0 a2)).(\lambda (H3: ((\forall (d1: C).(\forall (is: +PList).((drop1 is d1 (CHead c (Bind Abst) u)) \to (\forall (c2: C).((csubc g +d1 c2) \to (sc3 g a2 c2 (lift1 is t0))))))))).(\lambda (d1: C).(\lambda (is: +PList).(\lambda (H4: (drop1 is d1 c)).(\lambda (c2: C).(\lambda (H5: (csubc g +d1 c2)).(eq_ind_r T (THead (Bind Abst) (lift1 is u) (lift1 (Ss is) t0)) +(\lambda (t1: T).(land (arity g c2 t1 (AHead a1 a2)) (\forall (d: C).(\forall +(w: T).((sc3 g a1 d w) \to (\forall (is0: PList).((drop1 is0 d c2) \to (sc3 g +a2 d (THead (Flat Appl) w (lift1 is0 t1)))))))))) (conj (arity g c2 (THead +(Bind Abst) (lift1 is u) (lift1 (Ss is) t0)) (AHead a1 a2)) (\forall (d: +C).(\forall (w: T).((sc3 g a1 d w) \to (\forall (is0: PList).((drop1 is0 d +c2) \to (sc3 g a2 d (THead (Flat Appl) w (lift1 is0 (THead (Bind Abst) (lift1 +is u) (lift1 (Ss is) t0)))))))))) (csubc_arity_conf g d1 c2 H5 (THead (Bind +Abst) (lift1 is u) (lift1 (Ss is) t0)) (AHead a1 a2) (arity_head g d1 (lift1 +is u) a1 (arity_lift1 g (asucc g a1) c is d1 u H4 H0) (lift1 (Ss is) t0) a2 +(arity_lift1 g a2 (CHead c (Bind Abst) u) (Ss is) (CHead d1 (Bind Abst) +(lift1 is u)) t0 (drop1_skip_bind Abst c is d1 u H4) H2))) (\lambda (d: +C).(\lambda (w: T).(\lambda (H6: (sc3 g a1 d w)).(\lambda (is0: +PList).(\lambda (H7: (drop1 is0 d c2)).(eq_ind_r T (THead (Bind Abst) (lift1 +is0 (lift1 is u)) (lift1 (Ss is0) (lift1 (Ss is) t0))) (\lambda (t1: T).(sc3 +g a2 d (THead (Flat Appl) w t1))) (let H8 \def (sc3_appl g a1 a2 TNil) in (H8 +d w (lift1 (Ss is0) (lift1 (Ss is) t0)) (let H_y \def (sc3_bind g Abbr +(\lambda (H9: (eq B Abbr Abst)).(not_abbr_abst H9)) a1 a2 TNil) in (H_y d w +(lift1 (Ss is0) (lift1 (Ss is) t0)) (let H_x \def (csubc_drop1_conf_rev g is0 +d c2 H7 d1 H5) in (let H9 \def H_x in (ex2_ind C (\lambda (c3: C).(drop1 is0 +c3 d1)) (\lambda (c3: C).(csubc g c3 d)) (sc3 g a2 (CHead d (Bind Abbr) w) +(lift1 (Ss is0) (lift1 (Ss is) t0))) (\lambda (x: C).(\lambda (H10: (drop1 +is0 x d1)).(\lambda (H11: (csubc g x d)).(eq_ind_r T (lift1 (papp (Ss is0) +(Ss is)) t0) (\lambda (t1: T).(sc3 g a2 (CHead d (Bind Abbr) w) t1)) +(eq_ind_r PList (Ss (papp is0 is)) (\lambda (p: PList).(sc3 g a2 (CHead d +(Bind Abbr) w) (lift1 p t0))) (H3 (CHead x (Bind Abst) (lift1 (papp is0 is) +u)) (Ss (papp is0 is)) (drop1_skip_bind Abst c (papp is0 is) x u (drop1_trans +is0 x d1 H10 is c H4)) (CHead d (Bind Abbr) w) (csubc_abst g x d H11 (lift1 +(papp is0 is) u) a1 (H1 x (papp is0 is) (drop1_trans is0 x d1 H10 is c H4) x +(csubc_refl g x)) w H6)) (papp (Ss is0) (Ss is)) (papp_ss is0 is)) (lift1 (Ss +is0) (lift1 (Ss is) t0)) (lift1_lift1 (Ss is0) (Ss is) t0))))) H9))) H6)) H6 +(lift1 is0 (lift1 is u)) (sc3_lift1 g c2 (asucc g a1) is0 d (lift1 is u) (H1 +d1 is H4 c2 H5) H7))) (lift1 is0 (THead (Bind Abst) (lift1 is u) (lift1 (Ss +is) t0))) (lift1_bind Abst is0 (lift1 is u) (lift1 (Ss is) t0))))))))) (lift1 +is (THead (Bind Abst) u t0)) (lift1_bind Abst is u t0)))))))))))))))) +(\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c u +a1)).(\lambda (H1: ((\forall (d1: C).(\forall (is: PList).((drop1 is d1 c) +\to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g a1 c2 (lift1 is +u))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g c t0 +(AHead a1 a2))).(\lambda (H3: ((\forall (d1: C).(\forall (is: PList).((drop1 +is d1 c) \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g (AHead a1 a2) c2 +(lift1 is t0))))))))).(\lambda (d1: C).(\lambda (is: PList).(\lambda (H4: +(drop1 is d1 c)).(\lambda (c2: C).(\lambda (H5: (csubc g d1 c2)).(let H_y +\def (H1 d1 is H4 c2 H5) in (let H_y0 \def (H3 d1 is H4 c2 H5) in (let H6 +\def H_y0 in (land_ind (arity g c2 (lift1 is t0) (AHead a1 a2)) (\forall (d: +C).(\forall (w: T).((sc3 g a1 d w) \to (\forall (is0: PList).((drop1 is0 d +c2) \to (sc3 g a2 d (THead (Flat Appl) w (lift1 is0 (lift1 is t0))))))))) +(sc3 g a2 c2 (lift1 is (THead (Flat Appl) u t0))) (\lambda (_: (arity g c2 +(lift1 is t0) (AHead a1 a2))).(\lambda (H8: ((\forall (d: C).(\forall (w: +T).((sc3 g a1 d w) \to (\forall (is0: PList).((drop1 is0 d c2) \to (sc3 g a2 +d (THead (Flat Appl) w (lift1 is0 (lift1 is t0))))))))))).(let H_y1 \def (H8 +c2 (lift1 is u) H_y PNil) in (eq_ind_r T (THead (Flat Appl) (lift1 is u) +(lift1 is t0)) (\lambda (t1: T).(sc3 g a2 c2 t1)) (H_y1 (drop1_nil c2)) +(lift1 is (THead (Flat Appl) u t0)) (lift1_flat Appl is u t0))))) +H6)))))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a0: +A).(\lambda (_: (arity g c u (asucc g a0))).(\lambda (H1: ((\forall (d1: +C).(\forall (is: PList).((drop1 is d1 c) \to (\forall (c2: C).((csubc g d1 +c2) \to (sc3 g (asucc g a0) c2 (lift1 is u))))))))).(\lambda (t0: T).(\lambda +(_: (arity g c t0 a0)).(\lambda (H3: ((\forall (d1: C).(\forall (is: +PList).((drop1 is d1 c) \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g a0 +c2 (lift1 is t0))))))))).(\lambda (d1: C).(\lambda (is: PList).(\lambda (H4: +(drop1 is d1 c)).(\lambda (c2: C).(\lambda (H5: (csubc g d1 c2)).(let H_y +\def (sc3_cast g a0 TNil) in (eq_ind_r T (THead (Flat Cast) (lift1 is u) +(lift1 is t0)) (\lambda (t1: T).(sc3 g a0 c2 t1)) (H_y c2 (lift1 is u) (H1 d1 +is H4 c2 H5) (lift1 is t0) (H3 d1 is H4 c2 H5)) (lift1 is (THead (Flat Cast) +u t0)) (lift1_flat Cast is u t0)))))))))))))))) (\lambda (c: C).(\lambda (t0: +T).(\lambda (a1: A).(\lambda (_: (arity g c t0 a1)).(\lambda (H1: ((\forall +(d1: C).(\forall (is: PList).((drop1 is d1 c) \to (\forall (c2: C).((csubc g +d1 c2) \to (sc3 g a1 c2 (lift1 is t0))))))))).(\lambda (a2: A).(\lambda (H2: +(leq g a1 a2)).(\lambda (d1: C).(\lambda (is: PList).(\lambda (H3: (drop1 is +d1 c)).(\lambda (c2: C).(\lambda (H4: (csubc g d1 c2)).(sc3_repl g a1 c2 +(lift1 is t0) (H1 d1 is H3 c2 H4) a2 H2))))))))))))) c1 t a H))))). +(* COMMENTS +Initial nodes: 5940 +END *) + +theorem sc3_arity: + \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t +a) \to (sc3 g a c t))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: +(arity g c t a)).(let H_y \def (sc3_arity_csubc g c t a H c PNil) in (H_y +(drop1_nil c) c (csubc_refl g c))))))). +(* COMMENTS +Initial nodes: 47 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/sc3/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/sc3/defs.ma new file mode 100644 index 000000000..32fdfda84 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/sc3/defs.ma @@ -0,0 +1,32 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/sn3/defs.ma". + +include "Basic-1/arity/defs.ma". + +include "Basic-1/drop1/defs.ma". + +definition sc3: + G \to (A \to (C \to (T \to Prop))) +\def + let rec sc3 (g: G) (a: A) on a: (C \to (T \to Prop)) \def (\lambda (c: +C).(\lambda (t: T).(match a with [(ASort h n) \Rightarrow (land (arity g c t +(ASort h n)) (sn3 c t)) | (AHead a1 a2) \Rightarrow (land (arity g c t (AHead +a1 a2)) (\forall (d: C).(\forall (w: T).((sc3 g a1 d w) \to (\forall (is: +PList).((drop1 is d c) \to (sc3 g a2 d (THead (Flat Appl) w (lift1 is +t)))))))))]))) in sc3. + diff --git a/matita/matita/contribs/lambdadelta/basic_1/sc3/props.ma b/matita/matita/contribs/lambdadelta/basic_1/sc3/props.ma new file mode 100644 index 000000000..e1d909251 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/sc3/props.ma @@ -0,0 +1,728 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/sc3/defs.ma". + +include "Basic-1/sn3/lift1.ma". + +include "Basic-1/nf2/lift1.ma". + +include "Basic-1/csuba/arity.ma". + +include "Basic-1/arity/lift1.ma". + +include "Basic-1/arity/aprem.ma". + +include "Basic-1/llt/props.ma". + +include "Basic-1/drop1/getl.ma". + +include "Basic-1/drop1/props.ma". + +include "Basic-1/lift1/props.ma". + +theorem sc3_arity_gen: + \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((sc3 g a c +t) \to (arity g c t a))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(A_ind +(\lambda (a0: A).((sc3 g a0 c t) \to (arity g c t a0))) (\lambda (n: +nat).(\lambda (n0: nat).(\lambda (H: (land (arity g c t (ASort n n0)) (sn3 c +t))).(let H0 \def H in (land_ind (arity g c t (ASort n n0)) (sn3 c t) (arity +g c t (ASort n n0)) (\lambda (H1: (arity g c t (ASort n n0))).(\lambda (_: +(sn3 c t)).H1)) H0))))) (\lambda (a0: A).(\lambda (_: (((sc3 g a0 c t) \to +(arity g c t a0)))).(\lambda (a1: A).(\lambda (_: (((sc3 g a1 c t) \to (arity +g c t a1)))).(\lambda (H1: (land (arity g c t (AHead a0 a1)) (\forall (d: +C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) +\to (sc3 g a1 d (THead (Flat Appl) w (lift1 is t)))))))))).(let H2 \def H1 in +(land_ind (arity g c t (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g +a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat +Appl) w (lift1 is t)))))))) (arity g c t (AHead a0 a1)) (\lambda (H3: (arity +g c t (AHead a0 a1))).(\lambda (_: ((\forall (d: C).(\forall (w: T).((sc3 g +a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat +Appl) w (lift1 is t)))))))))).H3)) H2))))))) a)))). +(* COMMENTS +Initial nodes: 369 +END *) + +theorem sc3_repl: + \forall (g: G).(\forall (a1: A).(\forall (c: C).(\forall (t: T).((sc3 g a1 c +t) \to (\forall (a2: A).((leq g a1 a2) \to (sc3 g a2 c t))))))) +\def + \lambda (g: G).(\lambda (a1: A).(llt_wf_ind (\lambda (a: A).(\forall (c: +C).(\forall (t: T).((sc3 g a c t) \to (\forall (a2: A).((leq g a a2) \to (sc3 +g a2 c t))))))) (\lambda (a2: A).(A_ind (\lambda (a: A).(((\forall (a3: +A).((llt a3 a) \to (\forall (c: C).(\forall (t: T).((sc3 g a3 c t) \to +(\forall (a4: A).((leq g a3 a4) \to (sc3 g a4 c t))))))))) \to (\forall (c: +C).(\forall (t: T).((sc3 g a c t) \to (\forall (a3: A).((leq g a a3) \to (sc3 +g a3 c t)))))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (_: ((\forall +(a3: A).((llt a3 (ASort n n0)) \to (\forall (c: C).(\forall (t: T).((sc3 g a3 +c t) \to (\forall (a4: A).((leq g a3 a4) \to (sc3 g a4 c t)))))))))).(\lambda +(c: C).(\lambda (t: T).(\lambda (H0: (land (arity g c t (ASort n n0)) (sn3 c +t))).(\lambda (a3: A).(\lambda (H1: (leq g (ASort n n0) a3)).(let H2 \def H0 +in (land_ind (arity g c t (ASort n n0)) (sn3 c t) (sc3 g a3 c t) (\lambda +(H3: (arity g c t (ASort n n0))).(\lambda (H4: (sn3 c t)).(let H_y \def +(arity_repl g c t (ASort n n0) H3 a3 H1) in (let H_x \def (leq_gen_sort1 g n +n0 a3 H1) in (let H5 \def H_x in (ex2_3_ind nat nat nat (\lambda (n2: +nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort n n0) k) +(aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda +(_: nat).(eq A a3 (ASort h2 n2))))) (sc3 g a3 c t) (\lambda (x0: +nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (_: (eq A (aplus g (ASort +n n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H7: (eq A a3 (ASort x1 +x0))).(let H8 \def (f_equal A A (\lambda (e: A).e) a3 (ASort x1 x0) H7) in +(let H9 \def (eq_ind A a3 (\lambda (a: A).(arity g c t a)) H_y (ASort x1 x0) +H8) in (eq_ind_r A (ASort x1 x0) (\lambda (a: A).(sc3 g a c t)) (conj (arity +g c t (ASort x1 x0)) (sn3 c t) H9 H4) a3 H8)))))))) H5)))))) H2)))))))))) +(\lambda (a: A).(\lambda (_: ((((\forall (a3: A).((llt a3 a) \to (\forall (c: +C).(\forall (t: T).((sc3 g a3 c t) \to (\forall (a4: A).((leq g a3 a4) \to +(sc3 g a4 c t))))))))) \to (\forall (c: C).(\forall (t: T).((sc3 g a c t) \to +(\forall (a3: A).((leq g a a3) \to (sc3 g a3 c t))))))))).(\lambda (a0: +A).(\lambda (H0: ((((\forall (a3: A).((llt a3 a0) \to (\forall (c: +C).(\forall (t: T).((sc3 g a3 c t) \to (\forall (a4: A).((leq g a3 a4) \to +(sc3 g a4 c t))))))))) \to (\forall (c: C).(\forall (t: T).((sc3 g a0 c t) +\to (\forall (a3: A).((leq g a0 a3) \to (sc3 g a3 c t))))))))).(\lambda (H1: +((\forall (a3: A).((llt a3 (AHead a a0)) \to (\forall (c: C).(\forall (t: +T).((sc3 g a3 c t) \to (\forall (a4: A).((leq g a3 a4) \to (sc3 g a4 c +t)))))))))).(\lambda (c: C).(\lambda (t: T).(\lambda (H2: (land (arity g c t +(AHead a a0)) (\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall +(is: PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w (lift1 is +t)))))))))).(\lambda (a3: A).(\lambda (H3: (leq g (AHead a a0) a3)).(let H4 +\def H2 in (land_ind (arity g c t (AHead a a0)) (\forall (d: C).(\forall (w: +T).((sc3 g a d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a0 d +(THead (Flat Appl) w (lift1 is t)))))))) (sc3 g a3 c t) (\lambda (H5: (arity +g c t (AHead a a0))).(\lambda (H6: ((\forall (d: C).(\forall (w: T).((sc3 g a +d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat +Appl) w (lift1 is t)))))))))).(let H_x \def (leq_gen_head1 g a a0 a3 H3) in +(let H7 \def H_x in (ex3_2_ind A A (\lambda (a4: A).(\lambda (_: A).(leq g a +a4))) (\lambda (_: A).(\lambda (a5: A).(leq g a0 a5))) (\lambda (a4: +A).(\lambda (a5: A).(eq A a3 (AHead a4 a5)))) (sc3 g a3 c t) (\lambda (x0: +A).(\lambda (x1: A).(\lambda (H8: (leq g a x0)).(\lambda (H9: (leq g a0 +x1)).(\lambda (H10: (eq A a3 (AHead x0 x1))).(let H11 \def (f_equal A A +(\lambda (e: A).e) a3 (AHead x0 x1) H10) in (eq_ind_r A (AHead x0 x1) +(\lambda (a4: A).(sc3 g a4 c t)) (conj (arity g c t (AHead x0 x1)) (\forall +(d: C).(\forall (w: T).((sc3 g x0 d w) \to (\forall (is: PList).((drop1 is d +c) \to (sc3 g x1 d (THead (Flat Appl) w (lift1 is t)))))))) (arity_repl g c t +(AHead a a0) H5 (AHead x0 x1) (leq_head g a x0 H8 a0 x1 H9)) (\lambda (d: +C).(\lambda (w: T).(\lambda (H12: (sc3 g x0 d w)).(\lambda (is: +PList).(\lambda (H13: (drop1 is d c)).(H0 (\lambda (a4: A).(\lambda (H14: +(llt a4 a0)).(\lambda (c0: C).(\lambda (t0: T).(\lambda (H15: (sc3 g a4 c0 +t0)).(\lambda (a5: A).(\lambda (H16: (leq g a4 a5)).(H1 a4 (llt_trans a4 a0 +(AHead a a0) H14 (llt_head_dx a a0)) c0 t0 H15 a5 H16)))))))) d (THead (Flat +Appl) w (lift1 is t)) (H6 d w (H1 x0 (llt_repl g a x0 H8 (AHead a a0) +(llt_head_sx a a0)) d w H12 a (leq_sym g a x0 H8)) is H13) x1 H9))))))) a3 +H11))))))) H7))))) H4)))))))))))) a2)) a1)). +(* COMMENTS +Initial nodes: 1359 +END *) + +theorem sc3_lift: + \forall (g: G).(\forall (a: A).(\forall (e: C).(\forall (t: T).((sc3 g a e +t) \to (\forall (c: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) +\to (sc3 g a c (lift h d t)))))))))) +\def + \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(\forall (e: +C).(\forall (t: T).((sc3 g a0 e t) \to (\forall (c: C).(\forall (h: +nat).(\forall (d: nat).((drop h d c e) \to (sc3 g a0 c (lift h d t)))))))))) +(\lambda (n: nat).(\lambda (n0: nat).(\lambda (e: C).(\lambda (t: T).(\lambda +(H: (land (arity g e t (ASort n n0)) (sn3 e t))).(\lambda (c: C).(\lambda (h: +nat).(\lambda (d: nat).(\lambda (H0: (drop h d c e)).(let H1 \def H in +(land_ind (arity g e t (ASort n n0)) (sn3 e t) (land (arity g c (lift h d t) +(ASort n n0)) (sn3 c (lift h d t))) (\lambda (H2: (arity g e t (ASort n +n0))).(\lambda (H3: (sn3 e t)).(conj (arity g c (lift h d t) (ASort n n0)) +(sn3 c (lift h d t)) (arity_lift g e t (ASort n n0) H2 c h d H0) (sn3_lift e +t H3 c h d H0)))) H1))))))))))) (\lambda (a0: A).(\lambda (_: ((\forall (e: +C).(\forall (t: T).((sc3 g a0 e t) \to (\forall (c: C).(\forall (h: +nat).(\forall (d: nat).((drop h d c e) \to (sc3 g a0 c (lift h d +t))))))))))).(\lambda (a1: A).(\lambda (_: ((\forall (e: C).(\forall (t: +T).((sc3 g a1 e t) \to (\forall (c: C).(\forall (h: nat).(\forall (d: +nat).((drop h d c e) \to (sc3 g a1 c (lift h d t))))))))))).(\lambda (e: +C).(\lambda (t: T).(\lambda (H1: (land (arity g e t (AHead a0 a1)) (\forall +(d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d +e) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is t)))))))))).(\lambda (c: +C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2: (drop h d c e)).(let H3 +\def H1 in (land_ind (arity g e t (AHead a0 a1)) (\forall (d0: C).(\forall +(w: T).((sc3 g a0 d0 w) \to (\forall (is: PList).((drop1 is d0 e) \to (sc3 g +a1 d0 (THead (Flat Appl) w (lift1 is t)))))))) (land (arity g c (lift h d t) +(AHead a0 a1)) (\forall (d0: C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall +(is: PList).((drop1 is d0 c) \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is +(lift h d t)))))))))) (\lambda (H4: (arity g e t (AHead a0 a1))).(\lambda +(H5: ((\forall (d0: C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: +PList).((drop1 is d0 e) \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is +t)))))))))).(conj (arity g c (lift h d t) (AHead a0 a1)) (\forall (d0: +C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: PList).((drop1 is d0 c) +\to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is (lift h d t))))))))) +(arity_lift g e t (AHead a0 a1) H4 c h d H2) (\lambda (d0: C).(\lambda (w: +T).(\lambda (H6: (sc3 g a0 d0 w)).(\lambda (is: PList).(\lambda (H7: (drop1 +is d0 c)).(let H_y \def (H5 d0 w H6 (PConsTail is h d)) in (eq_ind T (lift1 +(PConsTail is h d) t) (\lambda (t0: T).(sc3 g a1 d0 (THead (Flat Appl) w +t0))) (H_y (drop1_cons_tail c e h d H2 is d0 H7)) (lift1 is (lift h d t)) +(lift1_cons_tail t h d is))))))))))) H3))))))))))))) a)). +(* COMMENTS +Initial nodes: 849 +END *) + +theorem sc3_lift1: + \forall (g: G).(\forall (e: C).(\forall (a: A).(\forall (hds: +PList).(\forall (c: C).(\forall (t: T).((sc3 g a e t) \to ((drop1 hds c e) +\to (sc3 g a c (lift1 hds t))))))))) +\def + \lambda (g: G).(\lambda (e: C).(\lambda (a: A).(\lambda (hds: +PList).(PList_ind (\lambda (p: PList).(\forall (c: C).(\forall (t: T).((sc3 g +a e t) \to ((drop1 p c e) \to (sc3 g a c (lift1 p t))))))) (\lambda (c: +C).(\lambda (t: T).(\lambda (H: (sc3 g a e t)).(\lambda (H0: (drop1 PNil c +e)).(let H_y \def (drop1_gen_pnil c e H0) in (eq_ind_r C e (\lambda (c0: +C).(sc3 g a c0 t)) H c H_y)))))) (\lambda (n: nat).(\lambda (n0: +nat).(\lambda (p: PList).(\lambda (H: ((\forall (c: C).(\forall (t: T).((sc3 +g a e t) \to ((drop1 p c e) \to (sc3 g a c (lift1 p t)))))))).(\lambda (c: +C).(\lambda (t: T).(\lambda (H0: (sc3 g a e t)).(\lambda (H1: (drop1 (PCons n +n0 p) c e)).(let H_x \def (drop1_gen_pcons c e p n n0 H1) in (let H2 \def H_x +in (ex2_ind C (\lambda (c2: C).(drop n n0 c c2)) (\lambda (c2: C).(drop1 p c2 +e)) (sc3 g a c (lift n n0 (lift1 p t))) (\lambda (x: C).(\lambda (H3: (drop n +n0 c x)).(\lambda (H4: (drop1 p x e)).(sc3_lift g a x (lift1 p t) (H x t H0 +H4) c n n0 H3)))) H2))))))))))) hds)))). +(* COMMENTS +Initial nodes: 289 +END *) + +theorem sc3_abbr: + \forall (g: G).(\forall (a: A).(\forall (vs: TList).(\forall (i: +nat).(\forall (d: C).(\forall (v: T).(\forall (c: C).((sc3 g a c (THeads +(Flat Appl) vs (lift (S i) O v))) \to ((getl i c (CHead d (Bind Abbr) v)) \to +(sc3 g a c (THeads (Flat Appl) vs (TLRef i))))))))))) +\def + \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(\forall (vs: +TList).(\forall (i: nat).(\forall (d: C).(\forall (v: T).(\forall (c: +C).((sc3 g a0 c (THeads (Flat Appl) vs (lift (S i) O v))) \to ((getl i c +(CHead d (Bind Abbr) v)) \to (sc3 g a0 c (THeads (Flat Appl) vs (TLRef +i))))))))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (vs: +TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(\lambda (c: +C).(\lambda (H: (land (arity g c (THeads (Flat Appl) vs (lift (S i) O v)) +(ASort n n0)) (sn3 c (THeads (Flat Appl) vs (lift (S i) O v))))).(\lambda +(H0: (getl i c (CHead d (Bind Abbr) v))).(let H1 \def H in (land_ind (arity g +c (THeads (Flat Appl) vs (lift (S i) O v)) (ASort n n0)) (sn3 c (THeads (Flat +Appl) vs (lift (S i) O v))) (land (arity g c (THeads (Flat Appl) vs (TLRef +i)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (TLRef i)))) (\lambda (H2: +(arity g c (THeads (Flat Appl) vs (lift (S i) O v)) (ASort n n0))).(\lambda +(H3: (sn3 c (THeads (Flat Appl) vs (lift (S i) O v)))).(conj (arity g c +(THeads (Flat Appl) vs (TLRef i)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs +(TLRef i))) (arity_appls_abbr g c d v i H0 vs (ASort n n0) H2) +(sn3_appls_abbr c d v i H0 vs H3)))) H1))))))))))) (\lambda (a0: A).(\lambda +(_: ((\forall (vs: TList).(\forall (i: nat).(\forall (d: C).(\forall (v: +T).(\forall (c: C).((sc3 g a0 c (THeads (Flat Appl) vs (lift (S i) O v))) \to +((getl i c (CHead d (Bind Abbr) v)) \to (sc3 g a0 c (THeads (Flat Appl) vs +(TLRef i)))))))))))).(\lambda (a1: A).(\lambda (H0: ((\forall (vs: +TList).(\forall (i: nat).(\forall (d: C).(\forall (v: T).(\forall (c: +C).((sc3 g a1 c (THeads (Flat Appl) vs (lift (S i) O v))) \to ((getl i c +(CHead d (Bind Abbr) v)) \to (sc3 g a1 c (THeads (Flat Appl) vs (TLRef +i)))))))))))).(\lambda (vs: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda +(v: T).(\lambda (c: C).(\lambda (H1: (land (arity g c (THeads (Flat Appl) vs +(lift (S i) O v)) (AHead a0 a1)) (\forall (d0: C).(\forall (w: T).((sc3 g a0 +d0 w) \to (\forall (is: PList).((drop1 is d0 c) \to (sc3 g a1 d0 (THead (Flat +Appl) w (lift1 is (THeads (Flat Appl) vs (lift (S i) O v)))))))))))).(\lambda +(H2: (getl i c (CHead d (Bind Abbr) v))).(let H3 \def H1 in (land_ind (arity +g c (THeads (Flat Appl) vs (lift (S i) O v)) (AHead a0 a1)) (\forall (d0: +C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: PList).((drop1 is d0 c) +\to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs (lift +(S i) O v)))))))))) (land (arity g c (THeads (Flat Appl) vs (TLRef i)) (AHead +a0 a1)) (\forall (d0: C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: +PList).((drop1 is d0 c) \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is +(THeads (Flat Appl) vs (TLRef i))))))))))) (\lambda (H4: (arity g c (THeads +(Flat Appl) vs (lift (S i) O v)) (AHead a0 a1))).(\lambda (H5: ((\forall (d0: +C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: PList).((drop1 is d0 c) +\to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs (lift +(S i) O v)))))))))))).(conj (arity g c (THeads (Flat Appl) vs (TLRef i)) +(AHead a0 a1)) (\forall (d0: C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall +(is: PList).((drop1 is d0 c) \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is +(THeads (Flat Appl) vs (TLRef i)))))))))) (arity_appls_abbr g c d v i H2 vs +(AHead a0 a1) H4) (\lambda (d0: C).(\lambda (w: T).(\lambda (H6: (sc3 g a0 d0 +w)).(\lambda (is: PList).(\lambda (H7: (drop1 is d0 c)).(let H_x \def +(drop1_getl_trans is c d0 H7 Abbr d v i H2) in (let H8 \def H_x in (ex2_ind C +(\lambda (e2: C).(drop1 (ptrans is i) e2 d)) (\lambda (e2: C).(getl (trans is +i) d0 (CHead e2 (Bind Abbr) (lift1 (ptrans is i) v)))) (sc3 g a1 d0 (THead +(Flat Appl) w (lift1 is (THeads (Flat Appl) vs (TLRef i))))) (\lambda (x: +C).(\lambda (_: (drop1 (ptrans is i) x d)).(\lambda (H10: (getl (trans is i) +d0 (CHead x (Bind Abbr) (lift1 (ptrans is i) v)))).(let H_y \def (H0 (TCons w +(lifts1 is vs))) in (eq_ind_r T (THeads (Flat Appl) (lifts1 is vs) (lift1 is +(TLRef i))) (\lambda (t: T).(sc3 g a1 d0 (THead (Flat Appl) w t))) (eq_ind_r +T (TLRef (trans is i)) (\lambda (t: T).(sc3 g a1 d0 (THead (Flat Appl) w +(THeads (Flat Appl) (lifts1 is vs) t)))) (H_y (trans is i) x (lift1 (ptrans +is i) v) d0 (eq_ind T (lift1 is (lift (S i) O v)) (\lambda (t: T).(sc3 g a1 +d0 (THead (Flat Appl) w (THeads (Flat Appl) (lifts1 is vs) t)))) (eq_ind T +(lift1 is (THeads (Flat Appl) vs (lift (S i) O v))) (\lambda (t: T).(sc3 g a1 +d0 (THead (Flat Appl) w t))) (H5 d0 w H6 is H7) (THeads (Flat Appl) (lifts1 +is vs) (lift1 is (lift (S i) O v))) (lifts1_flat Appl is (lift (S i) O v) +vs)) (lift (S (trans is i)) O (lift1 (ptrans is i) v)) (lift1_free is i v)) +H10) (lift1 is (TLRef i)) (lift1_lref is i)) (lift1 is (THeads (Flat Appl) vs +(TLRef i))) (lifts1_flat Appl is (TLRef i) vs)))))) H8))))))))))) +H3))))))))))))) a)). +(* COMMENTS +Initial nodes: 1563 +END *) + +theorem sc3_cast: + \forall (g: G).(\forall (a: A).(\forall (vs: TList).(\forall (c: C).(\forall +(u: T).((sc3 g (asucc g a) c (THeads (Flat Appl) vs u)) \to (\forall (t: +T).((sc3 g a c (THeads (Flat Appl) vs t)) \to (sc3 g a c (THeads (Flat Appl) +vs (THead (Flat Cast) u t)))))))))) +\def + \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(\forall (vs: +TList).(\forall (c: C).(\forall (u: T).((sc3 g (asucc g a0) c (THeads (Flat +Appl) vs u)) \to (\forall (t: T).((sc3 g a0 c (THeads (Flat Appl) vs t)) \to +(sc3 g a0 c (THeads (Flat Appl) vs (THead (Flat Cast) u t)))))))))) (\lambda +(n: nat).(\lambda (n0: nat).(\lambda (vs: TList).(\lambda (c: C).(\lambda (u: +T).(\lambda (H: (sc3 g (match n with [O \Rightarrow (ASort O (next g n0)) | +(S h) \Rightarrow (ASort h n0)]) c (THeads (Flat Appl) vs u))).(\lambda (t: +T).(\lambda (H0: (land (arity g c (THeads (Flat Appl) vs t) (ASort n n0)) +(sn3 c (THeads (Flat Appl) vs t)))).(nat_ind (\lambda (n1: nat).((sc3 g +(match n1 with [O \Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow +(ASort h n0)]) c (THeads (Flat Appl) vs u)) \to ((land (arity g c (THeads +(Flat Appl) vs t) (ASort n1 n0)) (sn3 c (THeads (Flat Appl) vs t))) \to (land +(arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort n1 n0)) +(sn3 c (THeads (Flat Appl) vs (THead (Flat Cast) u t))))))) (\lambda (H1: +(sc3 g (ASort O (next g n0)) c (THeads (Flat Appl) vs u))).(\lambda (H2: +(land (arity g c (THeads (Flat Appl) vs t) (ASort O n0)) (sn3 c (THeads (Flat +Appl) vs t)))).(let H3 \def H1 in (land_ind (arity g c (THeads (Flat Appl) vs +u) (ASort O (next g n0))) (sn3 c (THeads (Flat Appl) vs u)) (land (arity g c +(THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort O n0)) (sn3 c (THeads +(Flat Appl) vs (THead (Flat Cast) u t)))) (\lambda (H4: (arity g c (THeads +(Flat Appl) vs u) (ASort O (next g n0)))).(\lambda (H5: (sn3 c (THeads (Flat +Appl) vs u))).(let H6 \def H2 in (land_ind (arity g c (THeads (Flat Appl) vs +t) (ASort O n0)) (sn3 c (THeads (Flat Appl) vs t)) (land (arity g c (THeads +(Flat Appl) vs (THead (Flat Cast) u t)) (ASort O n0)) (sn3 c (THeads (Flat +Appl) vs (THead (Flat Cast) u t)))) (\lambda (H7: (arity g c (THeads (Flat +Appl) vs t) (ASort O n0))).(\lambda (H8: (sn3 c (THeads (Flat Appl) vs +t))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort +O n0)) (sn3 c (THeads (Flat Appl) vs (THead (Flat Cast) u t))) +(arity_appls_cast g c u t vs (ASort O n0) H4 H7) (sn3_appls_cast c vs u H5 t +H8)))) H6)))) H3)))) (\lambda (n1: nat).(\lambda (_: (((sc3 g (match n1 with +[O \Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)]) c +(THeads (Flat Appl) vs u)) \to ((land (arity g c (THeads (Flat Appl) vs t) +(ASort n1 n0)) (sn3 c (THeads (Flat Appl) vs t))) \to (land (arity g c +(THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort n1 n0)) (sn3 c (THeads +(Flat Appl) vs (THead (Flat Cast) u t)))))))).(\lambda (H1: (sc3 g (ASort n1 +n0) c (THeads (Flat Appl) vs u))).(\lambda (H2: (land (arity g c (THeads +(Flat Appl) vs t) (ASort (S n1) n0)) (sn3 c (THeads (Flat Appl) vs t)))).(let +H3 \def H1 in (land_ind (arity g c (THeads (Flat Appl) vs u) (ASort n1 n0)) +(sn3 c (THeads (Flat Appl) vs u)) (land (arity g c (THeads (Flat Appl) vs +(THead (Flat Cast) u t)) (ASort (S n1) n0)) (sn3 c (THeads (Flat Appl) vs +(THead (Flat Cast) u t)))) (\lambda (H4: (arity g c (THeads (Flat Appl) vs u) +(ASort n1 n0))).(\lambda (H5: (sn3 c (THeads (Flat Appl) vs u))).(let H6 \def +H2 in (land_ind (arity g c (THeads (Flat Appl) vs t) (ASort (S n1) n0)) (sn3 +c (THeads (Flat Appl) vs t)) (land (arity g c (THeads (Flat Appl) vs (THead +(Flat Cast) u t)) (ASort (S n1) n0)) (sn3 c (THeads (Flat Appl) vs (THead +(Flat Cast) u t)))) (\lambda (H7: (arity g c (THeads (Flat Appl) vs t) (ASort +(S n1) n0))).(\lambda (H8: (sn3 c (THeads (Flat Appl) vs t))).(conj (arity g +c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort (S n1) n0)) (sn3 c +(THeads (Flat Appl) vs (THead (Flat Cast) u t))) (arity_appls_cast g c u t vs +(ASort (S n1) n0) H4 H7) (sn3_appls_cast c vs u H5 t H8)))) H6)))) H3)))))) n +H H0))))))))) (\lambda (a0: A).(\lambda (_: ((\forall (vs: TList).(\forall +(c: C).(\forall (u: T).((sc3 g (asucc g a0) c (THeads (Flat Appl) vs u)) \to +(\forall (t: T).((sc3 g a0 c (THeads (Flat Appl) vs t)) \to (sc3 g a0 c +(THeads (Flat Appl) vs (THead (Flat Cast) u t))))))))))).(\lambda (a1: +A).(\lambda (H0: ((\forall (vs: TList).(\forall (c: C).(\forall (u: T).((sc3 +g (asucc g a1) c (THeads (Flat Appl) vs u)) \to (\forall (t: T).((sc3 g a1 c +(THeads (Flat Appl) vs t)) \to (sc3 g a1 c (THeads (Flat Appl) vs (THead +(Flat Cast) u t))))))))))).(\lambda (vs: TList).(\lambda (c: C).(\lambda (u: +T).(\lambda (H1: (land (arity g c (THeads (Flat Appl) vs u) (AHead a0 (asucc +g a1))) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: +PList).((drop1 is d c) \to (sc3 g (asucc g a1) d (THead (Flat Appl) w (lift1 +is (THeads (Flat Appl) vs u))))))))))).(\lambda (t: T).(\lambda (H2: (land +(arity g c (THeads (Flat Appl) vs t) (AHead a0 a1)) (\forall (d: C).(\forall +(w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 +d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs t))))))))))).(let H3 +\def H1 in (land_ind (arity g c (THeads (Flat Appl) vs u) (AHead a0 (asucc g +a1))) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: +PList).((drop1 is d c) \to (sc3 g (asucc g a1) d (THead (Flat Appl) w (lift1 +is (THeads (Flat Appl) vs u))))))))) (land (arity g c (THeads (Flat Appl) vs +(THead (Flat Cast) u t)) (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 +g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead +(Flat Appl) w (lift1 is (THeads (Flat Appl) vs (THead (Flat Cast) u +t))))))))))) (\lambda (H4: (arity g c (THeads (Flat Appl) vs u) (AHead a0 +(asucc g a1)))).(\lambda (H5: ((\forall (d: C).(\forall (w: T).((sc3 g a0 d +w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g (asucc g a1) d (THead +(Flat Appl) w (lift1 is (THeads (Flat Appl) vs u))))))))))).(let H6 \def H2 +in (land_ind (arity g c (THeads (Flat Appl) vs t) (AHead a0 a1)) (\forall (d: +C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) +\to (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs +t))))))))) (land (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) +(AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall +(is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is +(THeads (Flat Appl) vs (THead (Flat Cast) u t))))))))))) (\lambda (H7: (arity +g c (THeads (Flat Appl) vs t) (AHead a0 a1))).(\lambda (H8: ((\forall (d: +C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) +\to (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs +t))))))))))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) +(AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall +(is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is +(THeads (Flat Appl) vs (THead (Flat Cast) u t)))))))))) (arity_appls_cast g c +u t vs (AHead a0 a1) H4 H7) (\lambda (d: C).(\lambda (w: T).(\lambda (H9: +(sc3 g a0 d w)).(\lambda (is: PList).(\lambda (H10: (drop1 is d c)).(let H_y +\def (H0 (TCons w (lifts1 is vs))) in (eq_ind_r T (THeads (Flat Appl) (lifts1 +is vs) (lift1 is (THead (Flat Cast) u t))) (\lambda (t0: T).(sc3 g a1 d +(THead (Flat Appl) w t0))) (eq_ind_r T (THead (Flat Cast) (lift1 is u) (lift1 +is t)) (\lambda (t0: T).(sc3 g a1 d (THead (Flat Appl) w (THeads (Flat Appl) +(lifts1 is vs) t0)))) (H_y d (lift1 is u) (eq_ind T (lift1 is (THeads (Flat +Appl) vs u)) (\lambda (t0: T).(sc3 g (asucc g a1) d (THead (Flat Appl) w +t0))) (H5 d w H9 is H10) (THeads (Flat Appl) (lifts1 is vs) (lift1 is u)) +(lifts1_flat Appl is u vs)) (lift1 is t) (eq_ind T (lift1 is (THeads (Flat +Appl) vs t)) (\lambda (t0: T).(sc3 g a1 d (THead (Flat Appl) w t0))) (H8 d w +H9 is H10) (THeads (Flat Appl) (lifts1 is vs) (lift1 is t)) (lifts1_flat Appl +is t vs))) (lift1 is (THead (Flat Cast) u t)) (lift1_flat Cast is u t)) +(lift1 is (THeads (Flat Appl) vs (THead (Flat Cast) u t))) (lifts1_flat Appl +is (THead (Flat Cast) u t) vs))))))))))) H6)))) H3)))))))))))) a)). +(* COMMENTS +Initial nodes: 2625 +END *) + +theorem sc3_props__sc3_sn3_abst: + \forall (g: G).(\forall (a: A).(land (\forall (c: C).(\forall (t: T).((sc3 g +a c t) \to (sn3 c t)))) (\forall (vs: TList).(\forall (i: nat).(let t \def +(THeads (Flat Appl) vs (TLRef i)) in (\forall (c: C).((arity g c t a) \to +((nf2 c (TLRef i)) \to ((sns3 c vs) \to (sc3 g a c t)))))))))) +\def + \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(land (\forall (c: +C).(\forall (t: T).((sc3 g a0 c t) \to (sn3 c t)))) (\forall (vs: +TList).(\forall (i: nat).(let t \def (THeads (Flat Appl) vs (TLRef i)) in +(\forall (c: C).((arity g c t a0) \to ((nf2 c (TLRef i)) \to ((sns3 c vs) \to +(sc3 g a0 c t)))))))))) (\lambda (n: nat).(\lambda (n0: nat).(conj (\forall +(c: C).(\forall (t: T).((land (arity g c t (ASort n n0)) (sn3 c t)) \to (sn3 +c t)))) (\forall (vs: TList).(\forall (i: nat).(\forall (c: C).((arity g c +(THeads (Flat Appl) vs (TLRef i)) (ASort n n0)) \to ((nf2 c (TLRef i)) \to +((sns3 c vs) \to (land (arity g c (THeads (Flat Appl) vs (TLRef i)) (ASort n +n0)) (sn3 c (THeads (Flat Appl) vs (TLRef i)))))))))) (\lambda (c: +C).(\lambda (t: T).(\lambda (H: (land (arity g c t (ASort n n0)) (sn3 c +t))).(let H0 \def H in (land_ind (arity g c t (ASort n n0)) (sn3 c t) (sn3 c +t) (\lambda (_: (arity g c t (ASort n n0))).(\lambda (H2: (sn3 c t)).H2)) +H0))))) (\lambda (vs: TList).(\lambda (i: nat).(\lambda (c: C).(\lambda (H: +(arity g c (THeads (Flat Appl) vs (TLRef i)) (ASort n n0))).(\lambda (H0: +(nf2 c (TLRef i))).(\lambda (H1: (sns3 c vs)).(conj (arity g c (THeads (Flat +Appl) vs (TLRef i)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (TLRef i))) H +(sn3_appls_lref c i H0 vs H1))))))))))) (\lambda (a0: A).(\lambda (H: (land +(\forall (c: C).(\forall (t: T).((sc3 g a0 c t) \to (sn3 c t)))) (\forall +(vs: TList).(\forall (i: nat).(\forall (c: C).((arity g c (THeads (Flat Appl) +vs (TLRef i)) a0) \to ((nf2 c (TLRef i)) \to ((sns3 c vs) \to (sc3 g a0 c +(THeads (Flat Appl) vs (TLRef i))))))))))).(\lambda (a1: A).(\lambda (H0: +(land (\forall (c: C).(\forall (t: T).((sc3 g a1 c t) \to (sn3 c t)))) +(\forall (vs: TList).(\forall (i: nat).(\forall (c: C).((arity g c (THeads +(Flat Appl) vs (TLRef i)) a1) \to ((nf2 c (TLRef i)) \to ((sns3 c vs) \to +(sc3 g a1 c (THeads (Flat Appl) vs (TLRef i))))))))))).(conj (\forall (c: +C).(\forall (t: T).((land (arity g c t (AHead a0 a1)) (\forall (d: +C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) +\to (sc3 g a1 d (THead (Flat Appl) w (lift1 is t))))))))) \to (sn3 c t)))) +(\forall (vs: TList).(\forall (i: nat).(\forall (c: C).((arity g c (THeads +(Flat Appl) vs (TLRef i)) (AHead a0 a1)) \to ((nf2 c (TLRef i)) \to ((sns3 c +vs) \to (land (arity g c (THeads (Flat Appl) vs (TLRef i)) (AHead a0 a1)) +(\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: +PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads +(Flat Appl) vs (TLRef i))))))))))))))))) (\lambda (c: C).(\lambda (t: +T).(\lambda (H1: (land (arity g c t (AHead a0 a1)) (\forall (d: C).(\forall +(w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 +d (THead (Flat Appl) w (lift1 is t)))))))))).(let H2 \def H in (land_ind +(\forall (c0: C).(\forall (t0: T).((sc3 g a0 c0 t0) \to (sn3 c0 t0)))) +(\forall (vs: TList).(\forall (i: nat).(\forall (c0: C).((arity g c0 (THeads +(Flat Appl) vs (TLRef i)) a0) \to ((nf2 c0 (TLRef i)) \to ((sns3 c0 vs) \to +(sc3 g a0 c0 (THeads (Flat Appl) vs (TLRef i))))))))) (sn3 c t) (\lambda (_: +((\forall (c0: C).(\forall (t0: T).((sc3 g a0 c0 t0) \to (sn3 c0 +t0)))))).(\lambda (H4: ((\forall (vs: TList).(\forall (i: nat).(\forall (c0: +C).((arity g c0 (THeads (Flat Appl) vs (TLRef i)) a0) \to ((nf2 c0 (TLRef i)) +\to ((sns3 c0 vs) \to (sc3 g a0 c0 (THeads (Flat Appl) vs (TLRef +i))))))))))).(let H5 \def H0 in (land_ind (\forall (c0: C).(\forall (t0: +T).((sc3 g a1 c0 t0) \to (sn3 c0 t0)))) (\forall (vs: TList).(\forall (i: +nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) vs (TLRef i)) a1) \to +((nf2 c0 (TLRef i)) \to ((sns3 c0 vs) \to (sc3 g a1 c0 (THeads (Flat Appl) vs +(TLRef i))))))))) (sn3 c t) (\lambda (H6: ((\forall (c0: C).(\forall (t0: +T).((sc3 g a1 c0 t0) \to (sn3 c0 t0)))))).(\lambda (_: ((\forall (vs: +TList).(\forall (i: nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) vs +(TLRef i)) a1) \to ((nf2 c0 (TLRef i)) \to ((sns3 c0 vs) \to (sc3 g a1 c0 +(THeads (Flat Appl) vs (TLRef i))))))))))).(let H8 \def H1 in (land_ind +(arity g c t (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) +\to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w +(lift1 is t)))))))) (sn3 c t) (\lambda (H9: (arity g c t (AHead a0 +a1))).(\lambda (H10: ((\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to +(\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w +(lift1 is t)))))))))).(let H_y \def (arity_aprem g c t (AHead a0 a1) H9 O a0) +in (let H11 \def (H_y (aprem_zero a0 a1)) in (ex2_3_ind C T nat (\lambda (d: +C).(\lambda (_: T).(\lambda (j: nat).(drop j O d c)))) (\lambda (d: +C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g a0))))) (sn3 c t) +(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H12: (drop x2 +O x0 c)).(\lambda (H13: (arity g x0 x1 (asucc g a0))).(let H_y0 \def (H10 +(CHead x0 (Bind Abst) x1) (TLRef O) (H4 TNil O (CHead x0 (Bind Abst) x1) +(arity_abst g (CHead x0 (Bind Abst) x1) x0 x1 O (getl_refl Abst x0 x1) a0 +H13) (nf2_lref_abst (CHead x0 (Bind Abst) x1) x0 x1 O (getl_refl Abst x0 x1)) +I) (PCons (S x2) O PNil)) in (let H_y1 \def (H6 (CHead x0 (Bind Abst) x1) +(THead (Flat Appl) (TLRef O) (lift (S x2) O t)) (H_y0 (drop1_cons (CHead x0 +(Bind Abst) x1) c (S x2) O (drop_drop (Bind Abst) x2 x0 c H12 x1) c PNil +(drop1_nil c)))) in (let H_x \def (sn3_gen_flat Appl (CHead x0 (Bind Abst) +x1) (TLRef O) (lift (S x2) O t) H_y1) in (let H14 \def H_x in (land_ind (sn3 +(CHead x0 (Bind Abst) x1) (TLRef O)) (sn3 (CHead x0 (Bind Abst) x1) (lift (S +x2) O t)) (sn3 c t) (\lambda (_: (sn3 (CHead x0 (Bind Abst) x1) (TLRef +O))).(\lambda (H16: (sn3 (CHead x0 (Bind Abst) x1) (lift (S x2) O +t))).(sn3_gen_lift (CHead x0 (Bind Abst) x1) t (S x2) O H16 c (drop_drop +(Bind Abst) x2 x0 c H12 x1)))) H14)))))))))) H11))))) H8)))) H5)))) H2))))) +(\lambda (vs: TList).(\lambda (i: nat).(\lambda (c: C).(\lambda (H1: (arity g +c (THeads (Flat Appl) vs (TLRef i)) (AHead a0 a1))).(\lambda (H2: (nf2 c +(TLRef i))).(\lambda (H3: (sns3 c vs)).(conj (arity g c (THeads (Flat Appl) +vs (TLRef i)) (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) +\to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w +(lift1 is (THeads (Flat Appl) vs (TLRef i)))))))))) H1 (\lambda (d: +C).(\lambda (w: T).(\lambda (H4: (sc3 g a0 d w)).(\lambda (is: +PList).(\lambda (H5: (drop1 is d c)).(let H6 \def H in (land_ind (\forall +(c0: C).(\forall (t: T).((sc3 g a0 c0 t) \to (sn3 c0 t)))) (\forall (vs0: +TList).(\forall (i0: nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) +vs0 (TLRef i0)) a0) \to ((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a0 +c0 (THeads (Flat Appl) vs0 (TLRef i0))))))))) (sc3 g a1 d (THead (Flat Appl) +w (lift1 is (THeads (Flat Appl) vs (TLRef i))))) (\lambda (H7: ((\forall (c0: +C).(\forall (t: T).((sc3 g a0 c0 t) \to (sn3 c0 t)))))).(\lambda (_: +((\forall (vs0: TList).(\forall (i0: nat).(\forall (c0: C).((arity g c0 +(THeads (Flat Appl) vs0 (TLRef i0)) a0) \to ((nf2 c0 (TLRef i0)) \to ((sns3 +c0 vs0) \to (sc3 g a0 c0 (THeads (Flat Appl) vs0 (TLRef i0))))))))))).(let H9 +\def H0 in (land_ind (\forall (c0: C).(\forall (t: T).((sc3 g a1 c0 t) \to +(sn3 c0 t)))) (\forall (vs0: TList).(\forall (i0: nat).(\forall (c0: +C).((arity g c0 (THeads (Flat Appl) vs0 (TLRef i0)) a1) \to ((nf2 c0 (TLRef +i0)) \to ((sns3 c0 vs0) \to (sc3 g a1 c0 (THeads (Flat Appl) vs0 (TLRef +i0))))))))) (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs +(TLRef i))))) (\lambda (_: ((\forall (c0: C).(\forall (t: T).((sc3 g a1 c0 t) +\to (sn3 c0 t)))))).(\lambda (H11: ((\forall (vs0: TList).(\forall (i0: +nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) vs0 (TLRef i0)) a1) +\to ((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a1 c0 (THeads (Flat +Appl) vs0 (TLRef i0))))))))))).(let H_y \def (H11 (TCons w (lifts1 is vs))) +in (eq_ind_r T (THeads (Flat Appl) (lifts1 is vs) (lift1 is (TLRef i))) +(\lambda (t: T).(sc3 g a1 d (THead (Flat Appl) w t))) (eq_ind_r T (TLRef +(trans is i)) (\lambda (t: T).(sc3 g a1 d (THead (Flat Appl) w (THeads (Flat +Appl) (lifts1 is vs) t)))) (H_y (trans is i) d (eq_ind T (lift1 is (TLRef i)) +(\lambda (t: T).(arity g d (THead (Flat Appl) w (THeads (Flat Appl) (lifts1 +is vs) t)) a1)) (eq_ind T (lift1 is (THeads (Flat Appl) vs (TLRef i))) +(\lambda (t: T).(arity g d (THead (Flat Appl) w t) a1)) (arity_appl g d w a0 +(sc3_arity_gen g d w a0 H4) (lift1 is (THeads (Flat Appl) vs (TLRef i))) a1 +(arity_lift1 g (AHead a0 a1) c is d (THeads (Flat Appl) vs (TLRef i)) H5 H1)) +(THeads (Flat Appl) (lifts1 is vs) (lift1 is (TLRef i))) (lifts1_flat Appl is +(TLRef i) vs)) (TLRef (trans is i)) (lift1_lref is i)) (eq_ind T (lift1 is +(TLRef i)) (\lambda (t: T).(nf2 d t)) (nf2_lift1 c is d (TLRef i) H5 H2) +(TLRef (trans is i)) (lift1_lref is i)) (conj (sn3 d w) (sns3 d (lifts1 is +vs)) (H7 d w H4) (sns3_lifts1 c is d H5 vs H3))) (lift1 is (TLRef i)) +(lift1_lref is i)) (lift1 is (THeads (Flat Appl) vs (TLRef i))) (lifts1_flat +Appl is (TLRef i) vs))))) H9)))) H6))))))))))))))))))) a)). +(* COMMENTS +Initial nodes: 2737 +END *) + +theorem sc3_sn3: + \forall (g: G).(\forall (a: A).(\forall (c: C).(\forall (t: T).((sc3 g a c +t) \to (sn3 c t))))) +\def + \lambda (g: G).(\lambda (a: A).(\lambda (c: C).(\lambda (t: T).(\lambda (H: +(sc3 g a c t)).(let H_x \def (sc3_props__sc3_sn3_abst g a) in (let H0 \def +H_x in (land_ind (\forall (c0: C).(\forall (t0: T).((sc3 g a c0 t0) \to (sn3 +c0 t0)))) (\forall (vs: TList).(\forall (i: nat).(\forall (c0: C).((arity g +c0 (THeads (Flat Appl) vs (TLRef i)) a) \to ((nf2 c0 (TLRef i)) \to ((sns3 c0 +vs) \to (sc3 g a c0 (THeads (Flat Appl) vs (TLRef i))))))))) (sn3 c t) +(\lambda (H1: ((\forall (c0: C).(\forall (t0: T).((sc3 g a c0 t0) \to (sn3 c0 +t0)))))).(\lambda (_: ((\forall (vs: TList).(\forall (i: nat).(\forall (c0: +C).((arity g c0 (THeads (Flat Appl) vs (TLRef i)) a) \to ((nf2 c0 (TLRef i)) +\to ((sns3 c0 vs) \to (sc3 g a c0 (THeads (Flat Appl) vs (TLRef +i))))))))))).(H1 c t H))) H0))))))). +(* COMMENTS +Initial nodes: 203 +END *) + +theorem sc3_abst: + \forall (g: G).(\forall (a: A).(\forall (vs: TList).(\forall (c: C).(\forall +(i: nat).((arity g c (THeads (Flat Appl) vs (TLRef i)) a) \to ((nf2 c (TLRef +i)) \to ((sns3 c vs) \to (sc3 g a c (THeads (Flat Appl) vs (TLRef i)))))))))) +\def + \lambda (g: G).(\lambda (a: A).(\lambda (vs: TList).(\lambda (c: C).(\lambda +(i: nat).(\lambda (H: (arity g c (THeads (Flat Appl) vs (TLRef i)) +a)).(\lambda (H0: (nf2 c (TLRef i))).(\lambda (H1: (sns3 c vs)).(let H_x \def +(sc3_props__sc3_sn3_abst g a) in (let H2 \def H_x in (land_ind (\forall (c0: +C).(\forall (t: T).((sc3 g a c0 t) \to (sn3 c0 t)))) (\forall (vs0: +TList).(\forall (i0: nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) +vs0 (TLRef i0)) a) \to ((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a +c0 (THeads (Flat Appl) vs0 (TLRef i0))))))))) (sc3 g a c (THeads (Flat Appl) +vs (TLRef i))) (\lambda (_: ((\forall (c0: C).(\forall (t: T).((sc3 g a c0 t) +\to (sn3 c0 t)))))).(\lambda (H4: ((\forall (vs0: TList).(\forall (i0: +nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) vs0 (TLRef i0)) a) \to +((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a c0 (THeads (Flat Appl) +vs0 (TLRef i0))))))))))).(H4 vs i c H H0 H1))) H2)))))))))). +(* COMMENTS +Initial nodes: 249 +END *) + +theorem sc3_bind: + \forall (g: G).(\forall (b: B).((not (eq B b Abst)) \to (\forall (a1: +A).(\forall (a2: A).(\forall (vs: TList).(\forall (c: C).(\forall (v: +T).(\forall (t: T).((sc3 g a2 (CHead c (Bind b) v) (THeads (Flat Appl) (lifts +(S O) O vs) t)) \to ((sc3 g a1 c v) \to (sc3 g a2 c (THeads (Flat Appl) vs +(THead (Bind b) v t))))))))))))) +\def + \lambda (g: G).(\lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda +(a1: A).(\lambda (a2: A).(A_ind (\lambda (a: A).(\forall (vs: TList).(\forall +(c: C).(\forall (v: T).(\forall (t: T).((sc3 g a (CHead c (Bind b) v) (THeads +(Flat Appl) (lifts (S O) O vs) t)) \to ((sc3 g a1 c v) \to (sc3 g a c (THeads +(Flat Appl) vs (THead (Bind b) v t)))))))))) (\lambda (n: nat).(\lambda (n0: +nat).(\lambda (vs: TList).(\lambda (c: C).(\lambda (v: T).(\lambda (t: +T).(\lambda (H0: (land (arity g (CHead c (Bind b) v) (THeads (Flat Appl) +(lifts (S O) O vs) t) (ASort n n0)) (sn3 (CHead c (Bind b) v) (THeads (Flat +Appl) (lifts (S O) O vs) t)))).(\lambda (H1: (sc3 g a1 c v)).(let H2 \def H0 +in (land_ind (arity g (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O +vs) t) (ASort n n0)) (sn3 (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S +O) O vs) t)) (land (arity g c (THeads (Flat Appl) vs (THead (Bind b) v t)) +(ASort n n0)) (sn3 c (THeads (Flat Appl) vs (THead (Bind b) v t)))) (\lambda +(H3: (arity g (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O vs) t) +(ASort n n0))).(\lambda (H4: (sn3 (CHead c (Bind b) v) (THeads (Flat Appl) +(lifts (S O) O vs) t))).(conj (arity g c (THeads (Flat Appl) vs (THead (Bind +b) v t)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (THead (Bind b) v t))) +(arity_appls_bind g b H c v a1 (sc3_arity_gen g c v a1 H1) t vs (ASort n n0) +H3) (sn3_appls_bind b H c v (sc3_sn3 g a1 c v H1) vs t H4)))) H2)))))))))) +(\lambda (a: A).(\lambda (_: ((\forall (vs: TList).(\forall (c: C).(\forall +(v: T).(\forall (t: T).((sc3 g a (CHead c (Bind b) v) (THeads (Flat Appl) +(lifts (S O) O vs) t)) \to ((sc3 g a1 c v) \to (sc3 g a c (THeads (Flat Appl) +vs (THead (Bind b) v t))))))))))).(\lambda (a0: A).(\lambda (H1: ((\forall +(vs: TList).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sc3 g a0 (CHead +c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O vs) t)) \to ((sc3 g a1 c v) +\to (sc3 g a0 c (THeads (Flat Appl) vs (THead (Bind b) v +t))))))))))).(\lambda (vs: TList).(\lambda (c: C).(\lambda (v: T).(\lambda +(t: T).(\lambda (H2: (land (arity g (CHead c (Bind b) v) (THeads (Flat Appl) +(lifts (S O) O vs) t) (AHead a a0)) (\forall (d: C).(\forall (w: T).((sc3 g a +d w) \to (\forall (is: PList).((drop1 is d (CHead c (Bind b) v)) \to (sc3 g +a0 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) (lifts (S O) O vs) +t))))))))))).(\lambda (H3: (sc3 g a1 c v)).(let H4 \def H2 in (land_ind +(arity g (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O vs) t) +(AHead a a0)) (\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall +(is: PList).((drop1 is d (CHead c (Bind b) v)) \to (sc3 g a0 d (THead (Flat +Appl) w (lift1 is (THeads (Flat Appl) (lifts (S O) O vs) t))))))))) (land +(arity g c (THeads (Flat Appl) vs (THead (Bind b) v t)) (AHead a a0)) +(\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall (is: +PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w (lift1 is (THeads +(Flat Appl) vs (THead (Bind b) v t))))))))))) (\lambda (H5: (arity g (CHead c +(Bind b) v) (THeads (Flat Appl) (lifts (S O) O vs) t) (AHead a a0))).(\lambda +(H6: ((\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall (is: +PList).((drop1 is d (CHead c (Bind b) v)) \to (sc3 g a0 d (THead (Flat Appl) +w (lift1 is (THeads (Flat Appl) (lifts (S O) O vs) t))))))))))).(conj (arity +g c (THeads (Flat Appl) vs (THead (Bind b) v t)) (AHead a a0)) (\forall (d: +C).(\forall (w: T).((sc3 g a d w) \to (\forall (is: PList).((drop1 is d c) +\to (sc3 g a0 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs (THead +(Bind b) v t)))))))))) (arity_appls_bind g b H c v a1 (sc3_arity_gen g c v a1 +H3) t vs (AHead a a0) H5) (\lambda (d: C).(\lambda (w: T).(\lambda (H7: (sc3 +g a d w)).(\lambda (is: PList).(\lambda (H8: (drop1 is d c)).(let H_y \def +(H1 (TCons w (lifts1 is vs))) in (eq_ind_r T (THeads (Flat Appl) (lifts1 is +vs) (lift1 is (THead (Bind b) v t))) (\lambda (t0: T).(sc3 g a0 d (THead +(Flat Appl) w t0))) (eq_ind_r T (THead (Bind b) (lift1 is v) (lift1 (Ss is) +t)) (\lambda (t0: T).(sc3 g a0 d (THead (Flat Appl) w (THeads (Flat Appl) +(lifts1 is vs) t0)))) (H_y d (lift1 is v) (lift1 (Ss is) t) (eq_ind TList +(lifts1 (Ss is) (lifts (S O) O vs)) (\lambda (t0: TList).(sc3 g a0 (CHead d +(Bind b) (lift1 is v)) (THead (Flat Appl) (lift (S O) O w) (THeads (Flat +Appl) t0 (lift1 (Ss is) t))))) (eq_ind T (lift1 (Ss is) (THeads (Flat Appl) +(lifts (S O) O vs) t)) (\lambda (t0: T).(sc3 g a0 (CHead d (Bind b) (lift1 is +v)) (THead (Flat Appl) (lift (S O) O w) t0))) (H6 (CHead d (Bind b) (lift1 is +v)) (lift (S O) O w) (sc3_lift g a d w H7 (CHead d (Bind b) (lift1 is v)) (S +O) O (drop_drop (Bind b) O d d (drop_refl d) (lift1 is v))) (Ss is) +(drop1_skip_bind b c is d v H8)) (THeads (Flat Appl) (lifts1 (Ss is) (lifts +(S O) O vs)) (lift1 (Ss is) t)) (lifts1_flat Appl (Ss is) t (lifts (S O) O +vs))) (lifts (S O) O (lifts1 is vs)) (lifts1_xhg is vs)) (sc3_lift1 g c a1 is +d v H3 H8)) (lift1 is (THead (Bind b) v t)) (lift1_bind b is v t)) (lift1 is +(THeads (Flat Appl) vs (THead (Bind b) v t))) (lifts1_flat Appl is (THead +(Bind b) v t) vs))))))))))) H4)))))))))))) a2))))). +(* COMMENTS +Initial nodes: 1797 +END *) + +theorem sc3_appl: + \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (vs: +TList).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sc3 g a2 c (THeads +(Flat Appl) vs (THead (Bind Abbr) v t))) \to ((sc3 g a1 c v) \to (\forall (w: +T).((sc3 g (asucc g a1) c w) \to (sc3 g a2 c (THeads (Flat Appl) vs (THead +(Flat Appl) v (THead (Bind Abst) w t)))))))))))))) +\def + \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(A_ind (\lambda (a: +A).(\forall (vs: TList).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sc3 +g a c (THeads (Flat Appl) vs (THead (Bind Abbr) v t))) \to ((sc3 g a1 c v) +\to (\forall (w: T).((sc3 g (asucc g a1) c w) \to (sc3 g a c (THeads (Flat +Appl) vs (THead (Flat Appl) v (THead (Bind Abst) w t))))))))))))) (\lambda +(n: nat).(\lambda (n0: nat).(\lambda (vs: TList).(\lambda (c: C).(\lambda (v: +T).(\lambda (t: T).(\lambda (H: (land (arity g c (THeads (Flat Appl) vs +(THead (Bind Abbr) v t)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (THead +(Bind Abbr) v t))))).(\lambda (H0: (sc3 g a1 c v)).(\lambda (w: T).(\lambda +(H1: (sc3 g (asucc g a1) c w)).(let H2 \def H in (land_ind (arity g c (THeads +(Flat Appl) vs (THead (Bind Abbr) v t)) (ASort n n0)) (sn3 c (THeads (Flat +Appl) vs (THead (Bind Abbr) v t))) (land (arity g c (THeads (Flat Appl) vs +(THead (Flat Appl) v (THead (Bind Abst) w t))) (ASort n n0)) (sn3 c (THeads +(Flat Appl) vs (THead (Flat Appl) v (THead (Bind Abst) w t))))) (\lambda (H3: +(arity g c (THeads (Flat Appl) vs (THead (Bind Abbr) v t)) (ASort n +n0))).(\lambda (H4: (sn3 c (THeads (Flat Appl) vs (THead (Bind Abbr) v +t)))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat Appl) v (THead +(Bind Abst) w t))) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (THead (Flat +Appl) v (THead (Bind Abst) w t)))) (arity_appls_appl g c v a1 (sc3_arity_gen +g c v a1 H0) w (sc3_arity_gen g c w (asucc g a1) H1) t vs (ASort n n0) H3) +(sn3_appls_beta c v t vs H4 w (sc3_sn3 g (asucc g a1) c w H1))))) +H2)))))))))))) (\lambda (a: A).(\lambda (_: ((\forall (vs: TList).(\forall +(c: C).(\forall (v: T).(\forall (t: T).((sc3 g a c (THeads (Flat Appl) vs +(THead (Bind Abbr) v t))) \to ((sc3 g a1 c v) \to (\forall (w: T).((sc3 g +(asucc g a1) c w) \to (sc3 g a c (THeads (Flat Appl) vs (THead (Flat Appl) v +(THead (Bind Abst) w t)))))))))))))).(\lambda (a0: A).(\lambda (H0: ((\forall +(vs: TList).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sc3 g a0 c +(THeads (Flat Appl) vs (THead (Bind Abbr) v t))) \to ((sc3 g a1 c v) \to +(\forall (w: T).((sc3 g (asucc g a1) c w) \to (sc3 g a0 c (THeads (Flat Appl) +vs (THead (Flat Appl) v (THead (Bind Abst) w t)))))))))))))).(\lambda (vs: +TList).(\lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (H1: (land +(arity g c (THeads (Flat Appl) vs (THead (Bind Abbr) v t)) (AHead a a0)) +(\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall (is: +PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w (lift1 is (THeads +(Flat Appl) vs (THead (Bind Abbr) v t)))))))))))).(\lambda (H2: (sc3 g a1 c +v)).(\lambda (w: T).(\lambda (H3: (sc3 g (asucc g a1) c w)).(let H4 \def H1 +in (land_ind (arity g c (THeads (Flat Appl) vs (THead (Bind Abbr) v t)) +(AHead a a0)) (\forall (d: C).(\forall (w0: T).((sc3 g a d w0) \to (\forall +(is: PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w0 (lift1 is +(THeads (Flat Appl) vs (THead (Bind Abbr) v t)))))))))) (land (arity g c +(THeads (Flat Appl) vs (THead (Flat Appl) v (THead (Bind Abst) w t))) (AHead +a a0)) (\forall (d: C).(\forall (w0: T).((sc3 g a d w0) \to (\forall (is: +PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w0 (lift1 is +(THeads (Flat Appl) vs (THead (Flat Appl) v (THead (Bind Abst) w +t)))))))))))) (\lambda (H5: (arity g c (THeads (Flat Appl) vs (THead (Bind +Abbr) v t)) (AHead a a0))).(\lambda (H6: ((\forall (d: C).(\forall (w0: +T).((sc3 g a d w0) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a0 d +(THead (Flat Appl) w0 (lift1 is (THeads (Flat Appl) vs (THead (Bind Abbr) v +t)))))))))))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat Appl) v +(THead (Bind Abst) w t))) (AHead a a0)) (\forall (d: C).(\forall (w0: +T).((sc3 g a d w0) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a0 d +(THead (Flat Appl) w0 (lift1 is (THeads (Flat Appl) vs (THead (Flat Appl) v +(THead (Bind Abst) w t))))))))))) (arity_appls_appl g c v a1 (sc3_arity_gen g +c v a1 H2) w (sc3_arity_gen g c w (asucc g a1) H3) t vs (AHead a a0) H5) +(\lambda (d: C).(\lambda (w0: T).(\lambda (H7: (sc3 g a d w0)).(\lambda (is: +PList).(\lambda (H8: (drop1 is d c)).(eq_ind_r T (THeads (Flat Appl) (lifts1 +is vs) (lift1 is (THead (Flat Appl) v (THead (Bind Abst) w t)))) (\lambda +(t0: T).(sc3 g a0 d (THead (Flat Appl) w0 t0))) (eq_ind_r T (THead (Flat +Appl) (lift1 is v) (lift1 is (THead (Bind Abst) w t))) (\lambda (t0: T).(sc3 +g a0 d (THead (Flat Appl) w0 (THeads (Flat Appl) (lifts1 is vs) t0)))) +(eq_ind_r T (THead (Bind Abst) (lift1 is w) (lift1 (Ss is) t)) (\lambda (t0: +T).(sc3 g a0 d (THead (Flat Appl) w0 (THeads (Flat Appl) (lifts1 is vs) +(THead (Flat Appl) (lift1 is v) t0))))) (let H_y \def (H0 (TCons w0 (lifts1 +is vs))) in (H_y d (lift1 is v) (lift1 (Ss is) t) (eq_ind T (lift1 is (THead +(Bind Abbr) v t)) (\lambda (t0: T).(sc3 g a0 d (THead (Flat Appl) w0 (THeads +(Flat Appl) (lifts1 is vs) t0)))) (eq_ind T (lift1 is (THeads (Flat Appl) vs +(THead (Bind Abbr) v t))) (\lambda (t0: T).(sc3 g a0 d (THead (Flat Appl) w0 +t0))) (H6 d w0 H7 is H8) (THeads (Flat Appl) (lifts1 is vs) (lift1 is (THead +(Bind Abbr) v t))) (lifts1_flat Appl is (THead (Bind Abbr) v t) vs)) (THead +(Bind Abbr) (lift1 is v) (lift1 (Ss is) t)) (lift1_bind Abbr is v t)) +(sc3_lift1 g c a1 is d v H2 H8) (lift1 is w) (sc3_lift1 g c (asucc g a1) is d +w H3 H8))) (lift1 is (THead (Bind Abst) w t)) (lift1_bind Abst is w t)) +(lift1 is (THead (Flat Appl) v (THead (Bind Abst) w t))) (lift1_flat Appl is +v (THead (Bind Abst) w t))) (lift1 is (THeads (Flat Appl) vs (THead (Flat +Appl) v (THead (Bind Abst) w t)))) (lifts1_flat Appl is (THead (Flat Appl) v +(THead (Bind Abst) w t)) vs)))))))))) H4)))))))))))))) a2))). +(* COMMENTS +Initial nodes: 1901 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/sn3/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/sn3/defs.ma new file mode 100644 index 000000000..686e6c673 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/sn3/defs.ma @@ -0,0 +1,29 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/pr3/defs.ma". + +inductive sn3 (c: C): T \to Prop \def +| sn3_sing: \forall (t1: T).(((\forall (t2: T).((((eq T t1 t2) \to (\forall +(P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2))))) \to (sn3 c t1)). + +definition sns3: + C \to (TList \to Prop) +\def + let rec sns3 (c: C) (ts: TList) on ts: Prop \def (match ts with [TNil +\Rightarrow True | (TCons t ts0) \Rightarrow (land (sn3 c t) (sns3 c ts0))]) +in sns3. + diff --git a/matita/matita/contribs/lambdadelta/basic_1/sn3/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/sn3/fwd.ma new file mode 100644 index 000000000..68276fe9f --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/sn3/fwd.ma @@ -0,0 +1,197 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/sn3/defs.ma". + +include "Basic-1/pr3/props.ma". + +theorem sn3_gen_bind: + \forall (b: B).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 c +(THead (Bind b) u t)) \to (land (sn3 c u) (sn3 (CHead c (Bind b) u) t)))))) +\def + \lambda (b: B).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: +(sn3 c (THead (Bind b) u t))).(insert_eq T (THead (Bind b) u t) (\lambda (t0: +T).(sn3 c t0)) (\lambda (_: T).(land (sn3 c u) (sn3 (CHead c (Bind b) u) t))) +(\lambda (y: T).(\lambda (H0: (sn3 c y)).(unintro T t (\lambda (t0: T).((eq T +y (THead (Bind b) u t0)) \to (land (sn3 c u) (sn3 (CHead c (Bind b) u) t0)))) +(unintro T u (\lambda (t0: T).(\forall (x: T).((eq T y (THead (Bind b) t0 x)) +\to (land (sn3 c t0) (sn3 (CHead c (Bind b) t0) x))))) (sn3_ind c (\lambda +(t0: T).(\forall (x: T).(\forall (x0: T).((eq T t0 (THead (Bind b) x x0)) \to +(land (sn3 c x) (sn3 (CHead c (Bind b) x) x0)))))) (\lambda (t1: T).(\lambda +(H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 +c t1 t2) \to (sn3 c t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2) +\to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (\forall (x: T).(\forall +(x0: T).((eq T t2 (THead (Bind b) x x0)) \to (land (sn3 c x) (sn3 (CHead c +(Bind b) x) x0)))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: (eq T +t1 (THead (Bind b) x x0))).(let H4 \def (eq_ind T t1 (\lambda (t0: +T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c +t0 t2) \to (\forall (x1: T).(\forall (x2: T).((eq T t2 (THead (Bind b) x1 +x2)) \to (land (sn3 c x1) (sn3 (CHead c (Bind b) x1) x2))))))))) H2 (THead +(Bind b) x x0) H3) in (let H5 \def (eq_ind T t1 (\lambda (t0: T).(\forall +(t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to +(sn3 c t2))))) H1 (THead (Bind b) x x0) H3) in (conj (sn3 c x) (sn3 (CHead c +(Bind b) x) x0) (sn3_sing c x (\lambda (t2: T).(\lambda (H6: (((eq T x t2) +\to (\forall (P: Prop).P)))).(\lambda (H7: (pr3 c x t2)).(let H8 \def (H4 +(THead (Bind b) t2 x0) (\lambda (H8: (eq T (THead (Bind b) x x0) (THead (Bind +b) t2 x0))).(\lambda (P: Prop).(let H9 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x | +(TLRef _) \Rightarrow x | (THead _ t0 _) \Rightarrow t0])) (THead (Bind b) x +x0) (THead (Bind b) t2 x0) H8) in (let H10 \def (eq_ind_r T t2 (\lambda (t0: +T).(pr3 c x t0)) H7 x H9) in (let H11 \def (eq_ind_r T t2 (\lambda (t0: +T).((eq T x t0) \to (\forall (P0: Prop).P0))) H6 x H9) in (H11 (refl_equal T +x) P)))))) (pr3_head_12 c x t2 H7 (Bind b) x0 x0 (pr3_refl (CHead c (Bind b) +t2) x0)) t2 x0 (refl_equal T (THead (Bind b) t2 x0))) in (land_ind (sn3 c t2) +(sn3 (CHead c (Bind b) t2) x0) (sn3 c t2) (\lambda (H9: (sn3 c t2)).(\lambda +(_: (sn3 (CHead c (Bind b) t2) x0)).H9)) H8)))))) (sn3_sing (CHead c (Bind b) +x) x0 (\lambda (t2: T).(\lambda (H6: (((eq T x0 t2) \to (\forall (P: +Prop).P)))).(\lambda (H7: (pr3 (CHead c (Bind b) x) x0 t2)).(let H8 \def (H4 +(THead (Bind b) x t2) (\lambda (H8: (eq T (THead (Bind b) x x0) (THead (Bind +b) x t2))).(\lambda (P: Prop).(let H9 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | +(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind b) x +x0) (THead (Bind b) x t2) H8) in (let H10 \def (eq_ind_r T t2 (\lambda (t0: +T).(pr3 (CHead c (Bind b) x) x0 t0)) H7 x0 H9) in (let H11 \def (eq_ind_r T +t2 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H6 x0 H9) in +(H11 (refl_equal T x0) P)))))) (pr3_head_12 c x x (pr3_refl c x) (Bind b) x0 +t2 H7) x t2 (refl_equal T (THead (Bind b) x t2))) in (land_ind (sn3 c x) (sn3 +(CHead c (Bind b) x) t2) (sn3 (CHead c (Bind b) x) t2) (\lambda (_: (sn3 c +x)).(\lambda (H10: (sn3 (CHead c (Bind b) x) t2)).H10)) H8))))))))))))))) y +H0))))) H))))). +(* COMMENTS +Initial nodes: 1055 +END *) + +theorem sn3_gen_flat: + \forall (f: F).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 c +(THead (Flat f) u t)) \to (land (sn3 c u) (sn3 c t)))))) +\def + \lambda (f: F).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: +(sn3 c (THead (Flat f) u t))).(insert_eq T (THead (Flat f) u t) (\lambda (t0: +T).(sn3 c t0)) (\lambda (_: T).(land (sn3 c u) (sn3 c t))) (\lambda (y: +T).(\lambda (H0: (sn3 c y)).(unintro T t (\lambda (t0: T).((eq T y (THead +(Flat f) u t0)) \to (land (sn3 c u) (sn3 c t0)))) (unintro T u (\lambda (t0: +T).(\forall (x: T).((eq T y (THead (Flat f) t0 x)) \to (land (sn3 c t0) (sn3 +c x))))) (sn3_ind c (\lambda (t0: T).(\forall (x: T).(\forall (x0: T).((eq T +t0 (THead (Flat f) x x0)) \to (land (sn3 c x) (sn3 c x0)))))) (\lambda (t1: +T).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: +Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H2: ((\forall +(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to +(\forall (x: T).(\forall (x0: T).((eq T t2 (THead (Flat f) x x0)) \to (land +(sn3 c x) (sn3 c x0)))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: +(eq T t1 (THead (Flat f) x x0))).(let H4 \def (eq_ind T t1 (\lambda (t0: +T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c +t0 t2) \to (\forall (x1: T).(\forall (x2: T).((eq T t2 (THead (Flat f) x1 +x2)) \to (land (sn3 c x1) (sn3 c x2))))))))) H2 (THead (Flat f) x x0) H3) in +(let H5 \def (eq_ind T t1 (\lambda (t0: T).(\forall (t2: T).((((eq T t0 t2) +\to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (sn3 c t2))))) H1 (THead +(Flat f) x x0) H3) in (conj (sn3 c x) (sn3 c x0) (sn3_sing c x (\lambda (t2: +T).(\lambda (H6: (((eq T x t2) \to (\forall (P: Prop).P)))).(\lambda (H7: +(pr3 c x t2)).(let H8 \def (H4 (THead (Flat f) t2 x0) (\lambda (H8: (eq T +(THead (Flat f) x x0) (THead (Flat f) t2 x0))).(\lambda (P: Prop).(let H9 +\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) +with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t0 _) +\Rightarrow t0])) (THead (Flat f) x x0) (THead (Flat f) t2 x0) H8) in (let +H10 \def (eq_ind_r T t2 (\lambda (t0: T).(pr3 c x t0)) H7 x H9) in (let H11 +\def (eq_ind_r T t2 (\lambda (t0: T).((eq T x t0) \to (\forall (P0: +Prop).P0))) H6 x H9) in (H11 (refl_equal T x) P)))))) (pr3_head_12 c x t2 H7 +(Flat f) x0 x0 (pr3_refl (CHead c (Flat f) t2) x0)) t2 x0 (refl_equal T +(THead (Flat f) t2 x0))) in (land_ind (sn3 c t2) (sn3 c x0) (sn3 c t2) +(\lambda (H9: (sn3 c t2)).(\lambda (_: (sn3 c x0)).H9)) H8)))))) (sn3_sing c +x0 (\lambda (t2: T).(\lambda (H6: (((eq T x0 t2) \to (\forall (P: +Prop).P)))).(\lambda (H7: (pr3 c x0 t2)).(let H8 \def (H4 (THead (Flat f) x +t2) (\lambda (H8: (eq T (THead (Flat f) x x0) (THead (Flat f) x +t2))).(\lambda (P: Prop).(let H9 \def (f_equal T T (\lambda (e: T).(match e +in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) +\Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat f) x x0) +(THead (Flat f) x t2) H8) in (let H10 \def (eq_ind_r T t2 (\lambda (t0: +T).(pr3 c x0 t0)) H7 x0 H9) in (let H11 \def (eq_ind_r T t2 (\lambda (t0: +T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H6 x0 H9) in (H11 (refl_equal +T x0) P)))))) (pr3_thin_dx c x0 t2 H7 x f) x t2 (refl_equal T (THead (Flat f) +x t2))) in (land_ind (sn3 c x) (sn3 c t2) (sn3 c t2) (\lambda (_: (sn3 c +x)).(\lambda (H10: (sn3 c t2)).H10)) H8))))))))))))))) y H0))))) H))))). +(* COMMENTS +Initial nodes: 925 +END *) + +theorem sn3_gen_head: + \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 c +(THead k u t)) \to (sn3 c u))))) +\def + \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (c: C).(\forall (u: +T).(\forall (t: T).((sn3 c (THead k0 u t)) \to (sn3 c u)))))) (\lambda (b: +B).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (sn3 c (THead +(Bind b) u t))).(let H_x \def (sn3_gen_bind b c u t H) in (let H0 \def H_x in +(land_ind (sn3 c u) (sn3 (CHead c (Bind b) u) t) (sn3 c u) (\lambda (H1: (sn3 +c u)).(\lambda (_: (sn3 (CHead c (Bind b) u) t)).H1)) H0)))))))) (\lambda (f: +F).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (sn3 c (THead +(Flat f) u t))).(let H_x \def (sn3_gen_flat f c u t H) in (let H0 \def H_x in +(land_ind (sn3 c u) (sn3 c t) (sn3 c u) (\lambda (H1: (sn3 c u)).(\lambda (_: +(sn3 c t)).H1)) H0)))))))) k). +(* COMMENTS +Initial nodes: 191 +END *) + +theorem sn3_gen_cflat: + \forall (f: F).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 (CHead +c (Flat f) u) t) \to (sn3 c t))))) +\def + \lambda (f: F).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: +(sn3 (CHead c (Flat f) u) t)).(sn3_ind (CHead c (Flat f) u) (\lambda (t0: +T).(sn3 c t0)) (\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 +t2) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Flat f) u) t1 t2) \to +(sn3 (CHead c (Flat f) u) t2)))))).(\lambda (H1: ((\forall (t2: T).((((eq T +t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Flat f) u) t1 t2) \to +(sn3 c t2)))))).(sn3_sing c t1 (\lambda (t2: T).(\lambda (H2: (((eq T t1 t2) +\to (\forall (P: Prop).P)))).(\lambda (H3: (pr3 c t1 t2)).(H1 t2 H2 +(pr3_cflat c t1 t2 H3 f u))))))))) t H))))). +(* COMMENTS +Initial nodes: 175 +END *) + +theorem sn3_gen_lift: + \forall (c1: C).(\forall (t: T).(\forall (h: nat).(\forall (d: nat).((sn3 c1 +(lift h d t)) \to (\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 t))))))) +\def + \lambda (c1: C).(\lambda (t: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda +(H: (sn3 c1 (lift h d t))).(insert_eq T (lift h d t) (\lambda (t0: T).(sn3 c1 +t0)) (\lambda (_: T).(\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 t)))) +(\lambda (y: T).(\lambda (H0: (sn3 c1 y)).(unintro T t (\lambda (t0: T).((eq +T y (lift h d t0)) \to (\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 t0))))) +(sn3_ind c1 (\lambda (t0: T).(\forall (x: T).((eq T t0 (lift h d x)) \to +(\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 x)))))) (\lambda (t1: +T).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: +Prop).P))) \to ((pr3 c1 t1 t2) \to (sn3 c1 t2)))))).(\lambda (H2: ((\forall +(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c1 t1 t2) \to +(\forall (x: T).((eq T t2 (lift h d x)) \to (\forall (c2: C).((drop h d c1 +c2) \to (sn3 c2 x)))))))))).(\lambda (x: T).(\lambda (H3: (eq T t1 (lift h d +x))).(\lambda (c2: C).(\lambda (H4: (drop h d c1 c2)).(let H5 \def (eq_ind T +t1 (\lambda (t0: T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P: +Prop).P))) \to ((pr3 c1 t0 t2) \to (\forall (x0: T).((eq T t2 (lift h d x0)) +\to (\forall (c3: C).((drop h d c1 c3) \to (sn3 c3 x0))))))))) H2 (lift h d +x) H3) in (let H6 \def (eq_ind T t1 (\lambda (t0: T).(\forall (t2: T).((((eq +T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c1 t0 t2) \to (sn3 c1 t2))))) +H1 (lift h d x) H3) in (sn3_sing c2 x (\lambda (t2: T).(\lambda (H7: (((eq T +x t2) \to (\forall (P: Prop).P)))).(\lambda (H8: (pr3 c2 x t2)).(H5 (lift h d +t2) (\lambda (H9: (eq T (lift h d x) (lift h d t2))).(\lambda (P: Prop).(let +H10 \def (eq_ind_r T t2 (\lambda (t0: T).(pr3 c2 x t0)) H8 x (lift_inj x t2 h +d H9)) in (let H11 \def (eq_ind_r T t2 (\lambda (t0: T).((eq T x t0) \to +(\forall (P0: Prop).P0))) H7 x (lift_inj x t2 h d H9)) in (H11 (refl_equal T +x) P))))) (pr3_lift c1 c2 h d H4 x t2 H8) t2 (refl_equal T (lift h d t2)) c2 +H4)))))))))))))) y H0)))) H))))). +(* COMMENTS +Initial nodes: 565 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/sn3/lift1.ma b/matita/matita/contribs/lambdadelta/basic_1/sn3/lift1.ma new file mode 100644 index 000000000..1b64c22bb --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/sn3/lift1.ma @@ -0,0 +1,46 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/sn3/props.ma". + +include "Basic-1/drop1/fwd.ma". + +include "Basic-1/lift1/fwd.ma". + +theorem sns3_lifts1: + \forall (e: C).(\forall (hds: PList).(\forall (c: C).((drop1 hds c e) \to +(\forall (ts: TList).((sns3 e ts) \to (sns3 c (lifts1 hds ts))))))) +\def + \lambda (e: C).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall +(c: C).((drop1 p c e) \to (\forall (ts: TList).((sns3 e ts) \to (sns3 c +(lifts1 p ts))))))) (\lambda (c: C).(\lambda (H: (drop1 PNil c e)).(\lambda +(ts: TList).(\lambda (H0: (sns3 e ts)).(let H_y \def (drop1_gen_pnil c e H) +in (eq_ind_r C e (\lambda (c0: C).(sns3 c0 (lifts1 PNil ts))) (eq_ind_r TList +ts (\lambda (t: TList).(sns3 e t)) H0 (lifts1 PNil ts) (lifts1_nil ts)) c +H_y)))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda +(H: ((\forall (c: C).((drop1 p c e) \to (\forall (ts: TList).((sns3 e ts) \to +(sns3 c (lifts1 p ts)))))))).(\lambda (c: C).(\lambda (H0: (drop1 (PCons n n0 +p) c e)).(\lambda (ts: TList).(\lambda (H1: (sns3 e ts)).(let H_x \def +(drop1_gen_pcons c e p n n0 H0) in (let H2 \def H_x in (ex2_ind C (\lambda +(c2: C).(drop n n0 c c2)) (\lambda (c2: C).(drop1 p c2 e)) (sns3 c (lifts1 +(PCons n n0 p) ts)) (\lambda (x: C).(\lambda (H3: (drop n n0 c x)).(\lambda +(H4: (drop1 p x e)).(eq_ind_r TList (lifts n n0 (lifts1 p ts)) (\lambda (t: +TList).(sns3 c t)) (sns3_lifts c x n n0 H3 (lifts1 p ts) (H x H4 ts H1)) +(lifts1 (PCons n n0 p) ts) (lifts1_cons n n0 p ts))))) H2))))))))))) hds)). +(* COMMENTS +Initial nodes: 323 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/sn3/nf2.ma b/matita/matita/contribs/lambdadelta/basic_1/sn3/nf2.ma new file mode 100644 index 000000000..824428084 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/sn3/nf2.ma @@ -0,0 +1,66 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/sn3/defs.ma". + +include "Basic-1/nf2/dec.ma". + +include "Basic-1/nf2/pr3.ma". + +theorem sn3_nf2: + \forall (c: C).(\forall (t: T).((nf2 c t) \to (sn3 c t))) +\def + \lambda (c: C).(\lambda (t: T).(\lambda (H: (nf2 c t)).(sn3_sing c t +(\lambda (t2: T).(\lambda (H0: (((eq T t t2) \to (\forall (P: +Prop).P)))).(\lambda (H1: (pr3 c t t2)).(let H_y \def (nf2_pr3_unfold c t t2 +H1 H) in (let H2 \def (eq_ind_r T t2 (\lambda (t0: T).(pr3 c t t0)) H1 t H_y) +in (let H3 \def (eq_ind_r T t2 (\lambda (t0: T).((eq T t t0) \to (\forall (P: +Prop).P))) H0 t H_y) in (eq_ind T t (\lambda (t0: T).(sn3 c t0)) (H3 +(refl_equal T t) (sn3 c t)) t2 H_y)))))))))). +(* COMMENTS +Initial nodes: 129 +END *) + +theorem nf2_sn3: + \forall (c: C).(\forall (t: T).((sn3 c t) \to (ex2 T (\lambda (u: T).(pr3 c +t u)) (\lambda (u: T).(nf2 c u))))) +\def + \lambda (c: C).(\lambda (t: T).(\lambda (H: (sn3 c t)).(sn3_ind c (\lambda +(t0: T).(ex2 T (\lambda (u: T).(pr3 c t0 u)) (\lambda (u: T).(nf2 c u)))) +(\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) \to (\forall +(P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H1: ((\forall +(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to +(ex2 T (\lambda (u: T).(pr3 c t2 u)) (\lambda (u: T).(nf2 c u)))))))).(let +H_x \def (nf2_dec c t1) in (let H2 \def H_x in (or_ind (nf2 c t1) (ex2 T +(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: +T).(pr2 c t1 t2))) (ex2 T (\lambda (u: T).(pr3 c t1 u)) (\lambda (u: T).(nf2 +c u))) (\lambda (H3: (nf2 c t1)).(ex_intro2 T (\lambda (u: T).(pr3 c t1 u)) +(\lambda (u: T).(nf2 c u)) t1 (pr3_refl c t1) H3)) (\lambda (H3: (ex2 T +(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: +T).(pr2 c t1 t2)))).(ex2_ind T (\lambda (t2: T).((eq T t1 t2) \to (\forall +(P: Prop).P))) (\lambda (t2: T).(pr2 c t1 t2)) (ex2 T (\lambda (u: T).(pr3 c +t1 u)) (\lambda (u: T).(nf2 c u))) (\lambda (x: T).(\lambda (H4: (((eq T t1 +x) \to (\forall (P: Prop).P)))).(\lambda (H5: (pr2 c t1 x)).(let H_y \def (H1 +x H4) in (let H6 \def (H_y (pr3_pr2 c t1 x H5)) in (ex2_ind T (\lambda (u: +T).(pr3 c x u)) (\lambda (u: T).(nf2 c u)) (ex2 T (\lambda (u: T).(pr3 c t1 +u)) (\lambda (u: T).(nf2 c u))) (\lambda (x0: T).(\lambda (H7: (pr3 c x +x0)).(\lambda (H8: (nf2 c x0)).(ex_intro2 T (\lambda (u: T).(pr3 c t1 u)) +(\lambda (u: T).(nf2 c u)) x0 (pr3_sing c x t1 H5 x0 H7) H8)))) H6)))))) H3)) +H2)))))) t H))). +(* COMMENTS +Initial nodes: 443 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/sn3/props.ma b/matita/matita/contribs/lambdadelta/basic_1/sn3/props.ma new file mode 100644 index 000000000..ea72c8869 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/sn3/props.ma @@ -0,0 +1,2575 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/sn3/nf2.ma". + +include "Basic-1/sn3/fwd.ma". + +include "Basic-1/nf2/iso.ma". + +include "Basic-1/pr3/iso.ma". + +theorem sn3_pr3_trans: + \forall (c: C).(\forall (t1: T).((sn3 c t1) \to (\forall (t2: T).((pr3 c t1 +t2) \to (sn3 c t2))))) +\def + \lambda (c: C).(\lambda (t1: T).(\lambda (H: (sn3 c t1)).(sn3_ind c (\lambda +(t: T).(\forall (t2: T).((pr3 c t t2) \to (sn3 c t2)))) (\lambda (t2: +T).(\lambda (H0: ((\forall (t3: T).((((eq T t2 t3) \to (\forall (P: +Prop).P))) \to ((pr3 c t2 t3) \to (sn3 c t3)))))).(\lambda (H1: ((\forall +(t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to +(\forall (t4: T).((pr3 c t3 t4) \to (sn3 c t4)))))))).(\lambda (t3: +T).(\lambda (H2: (pr3 c t2 t3)).(sn3_sing c t3 (\lambda (t0: T).(\lambda (H3: +(((eq T t3 t0) \to (\forall (P: Prop).P)))).(\lambda (H4: (pr3 c t3 t0)).(let +H_x \def (term_dec t2 t3) in (let H5 \def H_x in (or_ind (eq T t2 t3) ((eq T +t2 t3) \to (\forall (P: Prop).P)) (sn3 c t0) (\lambda (H6: (eq T t2 t3)).(let +H7 \def (eq_ind_r T t3 (\lambda (t: T).(pr3 c t t0)) H4 t2 H6) in (let H8 +\def (eq_ind_r T t3 (\lambda (t: T).((eq T t t0) \to (\forall (P: Prop).P))) +H3 t2 H6) in (let H9 \def (eq_ind_r T t3 (\lambda (t: T).(pr3 c t2 t)) H2 t2 +H6) in (H0 t0 H8 H7))))) (\lambda (H6: (((eq T t2 t3) \to (\forall (P: +Prop).P)))).(H1 t3 H6 H2 t0 H4)) H5)))))))))))) t1 H))). +(* COMMENTS +Initial nodes: 289 +END *) + +theorem sn3_pr2_intro: + \forall (c: C).(\forall (t1: T).(((\forall (t2: T).((((eq T t1 t2) \to +(\forall (P: Prop).P))) \to ((pr2 c t1 t2) \to (sn3 c t2))))) \to (sn3 c t1))) +\def + \lambda (c: C).(\lambda (t1: T).(\lambda (H: ((\forall (t2: T).((((eq T t1 +t2) \to (\forall (P: Prop).P))) \to ((pr2 c t1 t2) \to (sn3 c +t2)))))).(sn3_sing c t1 (\lambda (t2: T).(\lambda (H0: (((eq T t1 t2) \to +(\forall (P: Prop).P)))).(\lambda (H1: (pr3 c t1 t2)).(let H2 \def H0 in +((let H3 \def H in (pr3_ind c (\lambda (t: T).(\lambda (t0: T).(((\forall +(t3: T).((((eq T t t3) \to (\forall (P: Prop).P))) \to ((pr2 c t t3) \to (sn3 +c t3))))) \to ((((eq T t t0) \to (\forall (P: Prop).P))) \to (sn3 c t0))))) +(\lambda (t: T).(\lambda (H4: ((\forall (t3: T).((((eq T t t3) \to (\forall +(P: Prop).P))) \to ((pr2 c t t3) \to (sn3 c t3)))))).(\lambda (H5: (((eq T t +t) \to (\forall (P: Prop).P)))).(H4 t H5 (pr2_free c t t (pr0_refl t)))))) +(\lambda (t3: T).(\lambda (t4: T).(\lambda (H4: (pr2 c t4 t3)).(\lambda (t5: +T).(\lambda (H5: (pr3 c t3 t5)).(\lambda (H6: ((((\forall (t6: T).((((eq T t3 +t6) \to (\forall (P: Prop).P))) \to ((pr2 c t3 t6) \to (sn3 c t6))))) \to +((((eq T t3 t5) \to (\forall (P: Prop).P))) \to (sn3 c t5))))).(\lambda (H7: +((\forall (t6: T).((((eq T t4 t6) \to (\forall (P: Prop).P))) \to ((pr2 c t4 +t6) \to (sn3 c t6)))))).(\lambda (H8: (((eq T t4 t5) \to (\forall (P: +Prop).P)))).(let H_x \def (term_dec t4 t3) in (let H9 \def H_x in (or_ind (eq +T t4 t3) ((eq T t4 t3) \to (\forall (P: Prop).P)) (sn3 c t5) (\lambda (H10: +(eq T t4 t3)).(let H11 \def (eq_ind T t4 (\lambda (t: T).((eq T t t5) \to +(\forall (P: Prop).P))) H8 t3 H10) in (let H12 \def (eq_ind T t4 (\lambda (t: +T).(\forall (t6: T).((((eq T t t6) \to (\forall (P: Prop).P))) \to ((pr2 c t +t6) \to (sn3 c t6))))) H7 t3 H10) in (let H13 \def (eq_ind T t4 (\lambda (t: +T).(pr2 c t t3)) H4 t3 H10) in (H6 H12 H11))))) (\lambda (H10: (((eq T t4 t3) +\to (\forall (P: Prop).P)))).(sn3_pr3_trans c t3 (H7 t3 H10 H4) t5 H5)) +H9))))))))))) t1 t2 H1 H3)) H2)))))))). +(* COMMENTS +Initial nodes: 467 +END *) + +theorem sn3_cast: + \forall (c: C).(\forall (u: T).((sn3 c u) \to (\forall (t: T).((sn3 c t) \to +(sn3 c (THead (Flat Cast) u t)))))) +\def + \lambda (c: C).(\lambda (u: T).(\lambda (H: (sn3 c u)).(sn3_ind c (\lambda +(t: T).(\forall (t0: T).((sn3 c t0) \to (sn3 c (THead (Flat Cast) t t0))))) +(\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) \to (\forall +(P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H1: ((\forall +(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to +(\forall (t: T).((sn3 c t) \to (sn3 c (THead (Flat Cast) t2 +t))))))))).(\lambda (t: T).(\lambda (H2: (sn3 c t)).(sn3_ind c (\lambda (t0: +T).(sn3 c (THead (Flat Cast) t1 t0))) (\lambda (t0: T).(\lambda (H3: +((\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t0 +t2) \to (sn3 c t2)))))).(\lambda (H4: ((\forall (t2: T).((((eq T t0 t2) \to +(\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (sn3 c (THead (Flat Cast) t1 +t2))))))).(sn3_pr2_intro c (THead (Flat Cast) t1 t0) (\lambda (t2: +T).(\lambda (H5: (((eq T (THead (Flat Cast) t1 t0) t2) \to (\forall (P: +Prop).P)))).(\lambda (H6: (pr2 c (THead (Flat Cast) t1 t0) t2)).(let H7 \def +(pr2_gen_cast c t1 t0 t2 H6) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda +(t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: +T).(pr2 c t1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t0 t3)))) (pr2 c +t0 t2) (sn3 c t2) (\lambda (H8: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: +T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: +T).(pr2 c t1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t0 +t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead +(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2))) +(\lambda (_: T).(\lambda (t3: T).(pr2 c t0 t3))) (sn3 c t2) (\lambda (x0: +T).(\lambda (x1: T).(\lambda (H9: (eq T t2 (THead (Flat Cast) x0 +x1))).(\lambda (H10: (pr2 c t1 x0)).(\lambda (H11: (pr2 c t0 x1)).(let H12 +\def (eq_ind T t2 (\lambda (t3: T).((eq T (THead (Flat Cast) t1 t0) t3) \to +(\forall (P: Prop).P))) H5 (THead (Flat Cast) x0 x1) H9) in (eq_ind_r T +(THead (Flat Cast) x0 x1) (\lambda (t3: T).(sn3 c t3)) (let H_x \def +(term_dec x0 t1) in (let H13 \def H_x in (or_ind (eq T x0 t1) ((eq T x0 t1) +\to (\forall (P: Prop).P)) (sn3 c (THead (Flat Cast) x0 x1)) (\lambda (H14: +(eq T x0 t1)).(let H15 \def (eq_ind T x0 (\lambda (t3: T).((eq T (THead (Flat +Cast) t1 t0) (THead (Flat Cast) t3 x1)) \to (\forall (P: Prop).P))) H12 t1 +H14) in (let H16 \def (eq_ind T x0 (\lambda (t3: T).(pr2 c t1 t3)) H10 t1 +H14) in (eq_ind_r T t1 (\lambda (t3: T).(sn3 c (THead (Flat Cast) t3 x1))) +(let H_x0 \def (term_dec t0 x1) in (let H17 \def H_x0 in (or_ind (eq T t0 x1) +((eq T t0 x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Cast) t1 x1)) +(\lambda (H18: (eq T t0 x1)).(let H19 \def (eq_ind_r T x1 (\lambda (t3: +T).((eq T (THead (Flat Cast) t1 t0) (THead (Flat Cast) t1 t3)) \to (\forall +(P: Prop).P))) H15 t0 H18) in (let H20 \def (eq_ind_r T x1 (\lambda (t3: +T).(pr2 c t0 t3)) H11 t0 H18) in (eq_ind T t0 (\lambda (t3: T).(sn3 c (THead +(Flat Cast) t1 t3))) (H19 (refl_equal T (THead (Flat Cast) t1 t0)) (sn3 c +(THead (Flat Cast) t1 t0))) x1 H18)))) (\lambda (H18: (((eq T t0 x1) \to +(\forall (P: Prop).P)))).(H4 x1 H18 (pr3_pr2 c t0 x1 H11))) H17))) x0 H14)))) +(\lambda (H14: (((eq T x0 t1) \to (\forall (P: Prop).P)))).(H1 x0 (\lambda +(H15: (eq T t1 x0)).(\lambda (P: Prop).(let H16 \def (eq_ind_r T x0 (\lambda +(t3: T).((eq T t3 t1) \to (\forall (P0: Prop).P0))) H14 t1 H15) in (let H17 +\def (eq_ind_r T x0 (\lambda (t3: T).((eq T (THead (Flat Cast) t1 t0) (THead +(Flat Cast) t3 x1)) \to (\forall (P0: Prop).P0))) H12 t1 H15) in (let H18 +\def (eq_ind_r T x0 (\lambda (t3: T).(pr2 c t1 t3)) H10 t1 H15) in (H16 +(refl_equal T t1) P)))))) (pr3_pr2 c t1 x0 H10) x1 (let H_x0 \def (term_dec +t0 x1) in (let H15 \def H_x0 in (or_ind (eq T t0 x1) ((eq T t0 x1) \to +(\forall (P: Prop).P)) (sn3 c x1) (\lambda (H16: (eq T t0 x1)).(let H17 \def +(eq_ind_r T x1 (\lambda (t3: T).((eq T (THead (Flat Cast) t1 t0) (THead (Flat +Cast) x0 t3)) \to (\forall (P: Prop).P))) H12 t0 H16) in (let H18 \def +(eq_ind_r T x1 (\lambda (t3: T).(pr2 c t0 t3)) H11 t0 H16) in (eq_ind T t0 +(\lambda (t3: T).(sn3 c t3)) (sn3_sing c t0 H3) x1 H16)))) (\lambda (H16: +(((eq T t0 x1) \to (\forall (P: Prop).P)))).(H3 x1 H16 (pr3_pr2 c t0 x1 +H11))) H15))))) H13))) t2 H9))))))) H8)) (\lambda (H8: (pr2 c t0 +t2)).(sn3_pr3_trans c t0 (sn3_sing c t0 H3) t2 (pr3_pr2 c t0 t2 H8))) +H7))))))))) t H2)))))) u H))). +(* COMMENTS +Initial nodes: 1239 +END *) + +theorem sn3_cflat: + \forall (c: C).(\forall (t: T).((sn3 c t) \to (\forall (f: F).(\forall (u: +T).(sn3 (CHead c (Flat f) u) t))))) +\def + \lambda (c: C).(\lambda (t: T).(\lambda (H: (sn3 c t)).(\lambda (f: +F).(\lambda (u: T).(sn3_ind c (\lambda (t0: T).(sn3 (CHead c (Flat f) u) t0)) +(\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) \to (\forall +(P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H1: ((\forall +(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to +(sn3 (CHead c (Flat f) u) t2)))))).(sn3_pr2_intro (CHead c (Flat f) u) t1 +(\lambda (t2: T).(\lambda (H2: (((eq T t1 t2) \to (\forall (P: +Prop).P)))).(\lambda (H3: (pr2 (CHead c (Flat f) u) t1 t2)).(H1 t2 H2 +(pr3_pr2 c t1 t2 (pr2_gen_cflat f c u t1 t2 H3)))))))))) t H))))). +(* COMMENTS +Initial nodes: 175 +END *) + +theorem sn3_shift: + \forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sn3 c +(THead (Bind b) v t)) \to (sn3 (CHead c (Bind b) v) t))))) +\def + \lambda (b: B).(\lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (H: +(sn3 c (THead (Bind b) v t))).(let H_x \def (sn3_gen_bind b c v t H) in (let +H0 \def H_x in (land_ind (sn3 c v) (sn3 (CHead c (Bind b) v) t) (sn3 (CHead c +(Bind b) v) t) (\lambda (_: (sn3 c v)).(\lambda (H2: (sn3 (CHead c (Bind b) +v) t)).H2)) H0))))))). +(* COMMENTS +Initial nodes: 95 +END *) + +theorem sn3_change: + \forall (b: B).((not (eq B b Abbr)) \to (\forall (c: C).(\forall (v1: +T).(\forall (t: T).((sn3 (CHead c (Bind b) v1) t) \to (\forall (v2: T).(sn3 +(CHead c (Bind b) v2) t))))))) +\def + \lambda (b: B).(\lambda (H: (not (eq B b Abbr))).(\lambda (c: C).(\lambda +(v1: T).(\lambda (t: T).(\lambda (H0: (sn3 (CHead c (Bind b) v1) t)).(\lambda +(v2: T).(sn3_ind (CHead c (Bind b) v1) (\lambda (t0: T).(sn3 (CHead c (Bind +b) v2) t0)) (\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) +\to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) v1) t1 t2) \to (sn3 +(CHead c (Bind b) v1) t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 +t2) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) v1) t1 t2) \to +(sn3 (CHead c (Bind b) v2) t2)))))).(sn3_pr2_intro (CHead c (Bind b) v2) t1 +(\lambda (t2: T).(\lambda (H3: (((eq T t1 t2) \to (\forall (P: +Prop).P)))).(\lambda (H4: (pr2 (CHead c (Bind b) v2) t1 t2)).(H2 t2 H3 +(pr3_pr2 (CHead c (Bind b) v1) t1 t2 (pr2_change b H c v2 t1 t2 H4 +v1)))))))))) t H0))))))). +(* COMMENTS +Initial nodes: 239 +END *) + +theorem sn3_gen_def: + \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c +(CHead d (Bind Abbr) v)) \to ((sn3 c (TLRef i)) \to (sn3 d v)))))) +\def + \lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda +(H: (getl i c (CHead d (Bind Abbr) v))).(\lambda (H0: (sn3 c (TLRef +i))).(sn3_gen_lift c v (S i) O (sn3_pr3_trans c (TLRef i) H0 (lift (S i) O v) +(pr3_pr2 c (TLRef i) (lift (S i) O v) (pr2_delta c d v i H (TLRef i) (TLRef +i) (pr0_refl (TLRef i)) (lift (S i) O v) (subst0_lref v i)))) d (getl_drop +Abbr c d v i H))))))). +(* COMMENTS +Initial nodes: 139 +END *) + +theorem sn3_cdelta: + \forall (v: T).(\forall (t: T).(\forall (i: nat).(((\forall (w: T).(ex T +(\lambda (u: T).(subst0 i w t u))))) \to (\forall (c: C).(\forall (d: +C).((getl i c (CHead d (Bind Abbr) v)) \to ((sn3 c t) \to (sn3 d v)))))))) +\def + \lambda (v: T).(\lambda (t: T).(\lambda (i: nat).(\lambda (H: ((\forall (w: +T).(ex T (\lambda (u: T).(subst0 i w t u)))))).(let H_x \def (H v) in (let H0 +\def H_x in (ex_ind T (\lambda (u: T).(subst0 i v t u)) (\forall (c: +C).(\forall (d: C).((getl i c (CHead d (Bind Abbr) v)) \to ((sn3 c t) \to +(sn3 d v))))) (\lambda (x: T).(\lambda (H1: (subst0 i v t x)).(subst0_ind +(\lambda (n: nat).(\lambda (t0: T).(\lambda (t1: T).(\lambda (_: T).(\forall +(c: C).(\forall (d: C).((getl n c (CHead d (Bind Abbr) t0)) \to ((sn3 c t1) +\to (sn3 d t0))))))))) (\lambda (v0: T).(\lambda (i0: nat).(\lambda (c: +C).(\lambda (d: C).(\lambda (H2: (getl i0 c (CHead d (Bind Abbr) +v0))).(\lambda (H3: (sn3 c (TLRef i0))).(sn3_gen_def c d v0 i0 H2 H3))))))) +(\lambda (v0: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i0: +nat).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (H3: ((\forall (c: +C).(\forall (d: C).((getl i0 c (CHead d (Bind Abbr) v0)) \to ((sn3 c u1) \to +(sn3 d v0))))))).(\lambda (t0: T).(\lambda (k: K).(\lambda (c: C).(\lambda +(d: C).(\lambda (H4: (getl i0 c (CHead d (Bind Abbr) v0))).(\lambda (H5: (sn3 +c (THead k u1 t0))).(let H_y \def (sn3_gen_head k c u1 t0 H5) in (H3 c d H4 +H_y)))))))))))))) (\lambda (k: K).(\lambda (v0: T).(\lambda (t2: T).(\lambda +(t1: T).(\lambda (i0: nat).(\lambda (H2: (subst0 (s k i0) v0 t1 t2)).(\lambda +(H3: ((\forall (c: C).(\forall (d: C).((getl (s k i0) c (CHead d (Bind Abbr) +v0)) \to ((sn3 c t1) \to (sn3 d v0))))))).(\lambda (u: T).(\lambda (c: +C).(\lambda (d: C).(\lambda (H4: (getl i0 c (CHead d (Bind Abbr) +v0))).(\lambda (H5: (sn3 c (THead k u t1))).(K_ind (\lambda (k0: K).((subst0 +(s k0 i0) v0 t1 t2) \to (((\forall (c0: C).(\forall (d0: C).((getl (s k0 i0) +c0 (CHead d0 (Bind Abbr) v0)) \to ((sn3 c0 t1) \to (sn3 d0 v0)))))) \to ((sn3 +c (THead k0 u t1)) \to (sn3 d v0))))) (\lambda (b: B).(\lambda (_: (subst0 (s +(Bind b) i0) v0 t1 t2)).(\lambda (H7: ((\forall (c0: C).(\forall (d0: +C).((getl (s (Bind b) i0) c0 (CHead d0 (Bind Abbr) v0)) \to ((sn3 c0 t1) \to +(sn3 d0 v0))))))).(\lambda (H8: (sn3 c (THead (Bind b) u t1))).(let H_x0 \def +(sn3_gen_bind b c u t1 H8) in (let H9 \def H_x0 in (land_ind (sn3 c u) (sn3 +(CHead c (Bind b) u) t1) (sn3 d v0) (\lambda (_: (sn3 c u)).(\lambda (H11: +(sn3 (CHead c (Bind b) u) t1)).(H7 (CHead c (Bind b) u) d (getl_clear_bind b +(CHead c (Bind b) u) c u (clear_bind b c u) (CHead d (Bind Abbr) v0) i0 H4) +H11))) H9))))))) (\lambda (f: F).(\lambda (_: (subst0 (s (Flat f) i0) v0 t1 +t2)).(\lambda (H7: ((\forall (c0: C).(\forall (d0: C).((getl (s (Flat f) i0) +c0 (CHead d0 (Bind Abbr) v0)) \to ((sn3 c0 t1) \to (sn3 d0 v0))))))).(\lambda +(H8: (sn3 c (THead (Flat f) u t1))).(let H_x0 \def (sn3_gen_flat f c u t1 H8) +in (let H9 \def H_x0 in (land_ind (sn3 c u) (sn3 c t1) (sn3 d v0) (\lambda +(_: (sn3 c u)).(\lambda (H11: (sn3 c t1)).(H7 c d H4 H11))) H9))))))) k H2 H3 +H5))))))))))))) (\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda +(i0: nat).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (H3: ((\forall (c: +C).(\forall (d: C).((getl i0 c (CHead d (Bind Abbr) v0)) \to ((sn3 c u1) \to +(sn3 d v0))))))).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda +(_: (subst0 (s k i0) v0 t1 t2)).(\lambda (_: ((\forall (c: C).(\forall (d: +C).((getl (s k i0) c (CHead d (Bind Abbr) v0)) \to ((sn3 c t1) \to (sn3 d +v0))))))).(\lambda (c: C).(\lambda (d: C).(\lambda (H6: (getl i0 c (CHead d +(Bind Abbr) v0))).(\lambda (H7: (sn3 c (THead k u1 t1))).(let H_y \def +(sn3_gen_head k c u1 t1 H7) in (H3 c d H6 H_y))))))))))))))))) i v t x H1))) +H0)))))). +(* COMMENTS +Initial nodes: 949 +END *) + +theorem sn3_cpr3_trans: + \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall +(k: K).(\forall (t: T).((sn3 (CHead c k u1) t) \to (sn3 (CHead c k u2) +t))))))) +\def + \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1 +u2)).(\lambda (k: K).(\lambda (t: T).(\lambda (H0: (sn3 (CHead c k u1) +t)).(sn3_ind (CHead c k u1) (\lambda (t0: T).(sn3 (CHead c k u2) t0)) +(\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) \to (\forall +(P: Prop).P))) \to ((pr3 (CHead c k u1) t1 t2) \to (sn3 (CHead c k u1) +t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: +Prop).P))) \to ((pr3 (CHead c k u1) t1 t2) \to (sn3 (CHead c k u2) +t2)))))).(sn3_sing (CHead c k u2) t1 (\lambda (t2: T).(\lambda (H3: (((eq T +t1 t2) \to (\forall (P: Prop).P)))).(\lambda (H4: (pr3 (CHead c k u2) t1 +t2)).(H2 t2 H3 (pr3_pr3_pr3_t c u1 u2 H t1 t2 k H4))))))))) t H0))))))). +(* COMMENTS +Initial nodes: 203 +END *) + +theorem sn3_bind: + \forall (b: B).(\forall (c: C).(\forall (u: T).((sn3 c u) \to (\forall (t: +T).((sn3 (CHead c (Bind b) u) t) \to (sn3 c (THead (Bind b) u t))))))) +\def + \lambda (b: B).(\lambda (c: C).(\lambda (u: T).(\lambda (H: (sn3 c +u)).(sn3_ind c (\lambda (t: T).(\forall (t0: T).((sn3 (CHead c (Bind b) t) +t0) \to (sn3 c (THead (Bind b) t t0))))) (\lambda (t1: T).(\lambda (_: +((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 +t2) \to (sn3 c t2)))))).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to +(\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (\forall (t: T).((sn3 (CHead c +(Bind b) t2) t) \to (sn3 c (THead (Bind b) t2 t))))))))).(\lambda (t: +T).(\lambda (H2: (sn3 (CHead c (Bind b) t1) t)).(sn3_ind (CHead c (Bind b) +t1) (\lambda (t0: T).(sn3 c (THead (Bind b) t1 t0))) (\lambda (t2: +T).(\lambda (H3: ((\forall (t3: T).((((eq T t2 t3) \to (\forall (P: +Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to (sn3 (CHead c (Bind b) +t1) t3)))))).(\lambda (H4: ((\forall (t3: T).((((eq T t2 t3) \to (\forall (P: +Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to (sn3 c (THead (Bind b) +t1 t3))))))).(sn3_sing c (THead (Bind b) t1 t2) (\lambda (t3: T).(\lambda +(H5: (((eq T (THead (Bind b) t1 t2) t3) \to (\forall (P: Prop).P)))).(\lambda +(H6: (pr3 c (THead (Bind b) t1 t2) t3)).(let H_x \def (bind_dec_not b Abst) +in (let H7 \def H_x in (or_ind (eq B b Abst) (not (eq B b Abst)) (sn3 c t3) +(\lambda (H8: (eq B b Abst)).(let H9 \def (eq_ind B b (\lambda (b0: B).(pr3 c +(THead (Bind b0) t1 t2) t3)) H6 Abst H8) in (let H10 \def (eq_ind B b +(\lambda (b0: B).((eq T (THead (Bind b0) t1 t2) t3) \to (\forall (P: +Prop).P))) H5 Abst H8) in (let H11 \def (eq_ind B b (\lambda (b0: B).(\forall +(t4: T).((((eq T t2 t4) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind +b0) t1) t2 t4) \to (sn3 c (THead (Bind b0) t1 t4)))))) H4 Abst H8) in (let +H12 \def (eq_ind B b (\lambda (b0: B).(\forall (t4: T).((((eq T t2 t4) \to +(\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b0) t1) t2 t4) \to (sn3 +(CHead c (Bind b0) t1) t4))))) H3 Abst H8) in (let H13 \def (eq_ind B b +(\lambda (b0: B).(\forall (t4: T).((((eq T t1 t4) \to (\forall (P: Prop).P))) +\to ((pr3 c t1 t4) \to (\forall (t0: T).((sn3 (CHead c (Bind b0) t4) t0) \to +(sn3 c (THead (Bind b0) t4 t0)))))))) H1 Abst H8) in (let H14 \def +(pr3_gen_abst c t1 t2 t3 H9) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: +T).(eq T t3 (THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: +T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b0: B).(\forall +(u0: T).(pr3 (CHead c (Bind b0) u0) t2 t4))))) (sn3 c t3) (\lambda (x0: +T).(\lambda (x1: T).(\lambda (H15: (eq T t3 (THead (Bind Abst) x0 +x1))).(\lambda (H16: (pr3 c t1 x0)).(\lambda (H17: ((\forall (b0: B).(\forall +(u0: T).(pr3 (CHead c (Bind b0) u0) t2 x1))))).(let H18 \def (eq_ind T t3 +(\lambda (t0: T).((eq T (THead (Bind Abst) t1 t2) t0) \to (\forall (P: +Prop).P))) H10 (THead (Bind Abst) x0 x1) H15) in (eq_ind_r T (THead (Bind +Abst) x0 x1) (\lambda (t0: T).(sn3 c t0)) (let H_x0 \def (term_dec t1 x0) in +(let H19 \def H_x0 in (or_ind (eq T t1 x0) ((eq T t1 x0) \to (\forall (P: +Prop).P)) (sn3 c (THead (Bind Abst) x0 x1)) (\lambda (H20: (eq T t1 x0)).(let +H21 \def (eq_ind_r T x0 (\lambda (t0: T).((eq T (THead (Bind Abst) t1 t2) +(THead (Bind Abst) t0 x1)) \to (\forall (P: Prop).P))) H18 t1 H20) in (let +H22 \def (eq_ind_r T x0 (\lambda (t0: T).(pr3 c t1 t0)) H16 t1 H20) in +(eq_ind T t1 (\lambda (t0: T).(sn3 c (THead (Bind Abst) t0 x1))) (let H_x1 +\def (term_dec t2 x1) in (let H23 \def H_x1 in (or_ind (eq T t2 x1) ((eq T t2 +x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Bind Abst) t1 x1)) (\lambda +(H24: (eq T t2 x1)).(let H25 \def (eq_ind_r T x1 (\lambda (t0: T).((eq T +(THead (Bind Abst) t1 t2) (THead (Bind Abst) t1 t0)) \to (\forall (P: +Prop).P))) H21 t2 H24) in (let H26 \def (eq_ind_r T x1 (\lambda (t0: +T).(\forall (b0: B).(\forall (u0: T).(pr3 (CHead c (Bind b0) u0) t2 t0)))) +H17 t2 H24) in (eq_ind T t2 (\lambda (t0: T).(sn3 c (THead (Bind Abst) t1 +t0))) (H25 (refl_equal T (THead (Bind Abst) t1 t2)) (sn3 c (THead (Bind Abst) +t1 t2))) x1 H24)))) (\lambda (H24: (((eq T t2 x1) \to (\forall (P: +Prop).P)))).(H11 x1 H24 (H17 Abst t1))) H23))) x0 H20)))) (\lambda (H20: +(((eq T t1 x0) \to (\forall (P: Prop).P)))).(let H_x1 \def (term_dec t2 x1) +in (let H21 \def H_x1 in (or_ind (eq T t2 x1) ((eq T t2 x1) \to (\forall (P: +Prop).P)) (sn3 c (THead (Bind Abst) x0 x1)) (\lambda (H22: (eq T t2 x1)).(let +H23 \def (eq_ind_r T x1 (\lambda (t0: T).(\forall (b0: B).(\forall (u0: +T).(pr3 (CHead c (Bind b0) u0) t2 t0)))) H17 t2 H22) in (eq_ind T t2 (\lambda +(t0: T).(sn3 c (THead (Bind Abst) x0 t0))) (H13 x0 H20 H16 t2 (sn3_cpr3_trans +c t1 x0 H16 (Bind Abst) t2 (sn3_sing (CHead c (Bind Abst) t1) t2 H12))) x1 +H22))) (\lambda (H22: (((eq T t2 x1) \to (\forall (P: Prop).P)))).(H13 x0 H20 +H16 x1 (sn3_cpr3_trans c t1 x0 H16 (Bind Abst) x1 (H12 x1 H22 (H17 Abst +t1))))) H21)))) H19))) t3 H15))))))) H14)))))))) (\lambda (H8: (not (eq B b +Abst))).(let H_x0 \def (pr3_gen_bind b H8 c t1 t2 t3 H6) in (let H9 \def H_x0 +in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind +b) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_: +T).(\lambda (t4: T).(pr3 (CHead c (Bind b) t1) t2 t4)))) (pr3 (CHead c (Bind +b) t1) t2 (lift (S O) O t3)) (sn3 c t3) (\lambda (H10: (ex3_2 T T (\lambda +(u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr3 +(CHead c (Bind b) t1) t2 t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda +(t4: T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda (u2: T).(\lambda (_: +T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr3 (CHead c (Bind b) +t1) t2 t4))) (sn3 c t3) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq +T t3 (THead (Bind b) x0 x1))).(\lambda (H12: (pr3 c t1 x0)).(\lambda (H13: +(pr3 (CHead c (Bind b) t1) t2 x1)).(let H14 \def (eq_ind T t3 (\lambda (t0: +T).((eq T (THead (Bind b) t1 t2) t0) \to (\forall (P: Prop).P))) H5 (THead +(Bind b) x0 x1) H11) in (eq_ind_r T (THead (Bind b) x0 x1) (\lambda (t0: +T).(sn3 c t0)) (let H_x1 \def (term_dec t1 x0) in (let H15 \def H_x1 in +(or_ind (eq T t1 x0) ((eq T t1 x0) \to (\forall (P: Prop).P)) (sn3 c (THead +(Bind b) x0 x1)) (\lambda (H16: (eq T t1 x0)).(let H17 \def (eq_ind_r T x0 +(\lambda (t0: T).((eq T (THead (Bind b) t1 t2) (THead (Bind b) t0 x1)) \to +(\forall (P: Prop).P))) H14 t1 H16) in (let H18 \def (eq_ind_r T x0 (\lambda +(t0: T).(pr3 c t1 t0)) H12 t1 H16) in (eq_ind T t1 (\lambda (t0: T).(sn3 c +(THead (Bind b) t0 x1))) (let H_x2 \def (term_dec t2 x1) in (let H19 \def +H_x2 in (or_ind (eq T t2 x1) ((eq T t2 x1) \to (\forall (P: Prop).P)) (sn3 c +(THead (Bind b) t1 x1)) (\lambda (H20: (eq T t2 x1)).(let H21 \def (eq_ind_r +T x1 (\lambda (t0: T).((eq T (THead (Bind b) t1 t2) (THead (Bind b) t1 t0)) +\to (\forall (P: Prop).P))) H17 t2 H20) in (let H22 \def (eq_ind_r T x1 +(\lambda (t0: T).(pr3 (CHead c (Bind b) t1) t2 t0)) H13 t2 H20) in (eq_ind T +t2 (\lambda (t0: T).(sn3 c (THead (Bind b) t1 t0))) (H21 (refl_equal T (THead +(Bind b) t1 t2)) (sn3 c (THead (Bind b) t1 t2))) x1 H20)))) (\lambda (H20: +(((eq T t2 x1) \to (\forall (P: Prop).P)))).(H4 x1 H20 H13)) H19))) x0 +H16)))) (\lambda (H16: (((eq T t1 x0) \to (\forall (P: Prop).P)))).(let H_x2 +\def (term_dec t2 x1) in (let H17 \def H_x2 in (or_ind (eq T t2 x1) ((eq T t2 +x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Bind b) x0 x1)) (\lambda (H18: +(eq T t2 x1)).(let H19 \def (eq_ind_r T x1 (\lambda (t0: T).(pr3 (CHead c +(Bind b) t1) t2 t0)) H13 t2 H18) in (eq_ind T t2 (\lambda (t0: T).(sn3 c +(THead (Bind b) x0 t0))) (H1 x0 H16 H12 t2 (sn3_cpr3_trans c t1 x0 H12 (Bind +b) t2 (sn3_sing (CHead c (Bind b) t1) t2 H3))) x1 H18))) (\lambda (H18: (((eq +T t2 x1) \to (\forall (P: Prop).P)))).(H1 x0 H16 H12 x1 (sn3_cpr3_trans c t1 +x0 H12 (Bind b) x1 (H3 x1 H18 H13)))) H17)))) H15))) t3 H11))))))) H10)) +(\lambda (H10: (pr3 (CHead c (Bind b) t1) t2 (lift (S O) O +t3))).(sn3_gen_lift (CHead c (Bind b) t1) t3 (S O) O (sn3_pr3_trans (CHead c +(Bind b) t1) t2 (sn3_sing (CHead c (Bind b) t1) t2 H3) (lift (S O) O t3) H10) +c (drop_drop (Bind b) O c c (drop_refl c) t1))) H9)))) H7)))))))))) t +H2)))))) u H)))). +(* COMMENTS +Initial nodes: 2401 +END *) + +theorem sn3_beta: + \forall (c: C).(\forall (v: T).(\forall (t: T).((sn3 c (THead (Bind Abbr) v +t)) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) v (THead +(Bind Abst) w t)))))))) +\def + \lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (H: (sn3 c (THead +(Bind Abbr) v t))).(insert_eq T (THead (Bind Abbr) v t) (\lambda (t0: T).(sn3 +c t0)) (\lambda (_: T).(\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat +Appl) v (THead (Bind Abst) w t)))))) (\lambda (y: T).(\lambda (H0: (sn3 c +y)).(unintro T t (\lambda (t0: T).((eq T y (THead (Bind Abbr) v t0)) \to +(\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) v (THead (Bind Abst) +w t0))))))) (unintro T v (\lambda (t0: T).(\forall (x: T).((eq T y (THead +(Bind Abbr) t0 x)) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat +Appl) t0 (THead (Bind Abst) w x)))))))) (sn3_ind c (\lambda (t0: T).(\forall +(x: T).(\forall (x0: T).((eq T t0 (THead (Bind Abbr) x x0)) \to (\forall (w: +T).((sn3 c w) \to (sn3 c (THead (Flat Appl) x (THead (Bind Abst) w +x0))))))))) (\lambda (t1: T).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) +\to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda +(H2: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 +c t1 t2) \to (\forall (x: T).(\forall (x0: T).((eq T t2 (THead (Bind Abbr) x +x0)) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) x (THead +(Bind Abst) w x0))))))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: +(eq T t1 (THead (Bind Abbr) x x0))).(\lambda (w: T).(\lambda (H4: (sn3 c +w)).(let H5 \def (eq_ind T t1 (\lambda (t0: T).(\forall (t2: T).((((eq T t0 +t2) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (\forall (x1: +T).(\forall (x2: T).((eq T t2 (THead (Bind Abbr) x1 x2)) \to (\forall (w0: +T).((sn3 c w0) \to (sn3 c (THead (Flat Appl) x1 (THead (Bind Abst) w0 +x2)))))))))))) H2 (THead (Bind Abbr) x x0) H3) in (let H6 \def (eq_ind T t1 +(\lambda (t0: T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) +\to ((pr3 c t0 t2) \to (sn3 c t2))))) H1 (THead (Bind Abbr) x x0) H3) in +(sn3_ind c (\lambda (t0: T).(sn3 c (THead (Flat Appl) x (THead (Bind Abst) t0 +x0)))) (\lambda (t2: T).(\lambda (H7: ((\forall (t3: T).((((eq T t2 t3) \to +(\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to (sn3 c t3)))))).(\lambda (H8: +((\forall (t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t2 +t3) \to (sn3 c (THead (Flat Appl) x (THead (Bind Abst) t3 +x0)))))))).(sn3_pr2_intro c (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) +(\lambda (t3: T).(\lambda (H9: (((eq T (THead (Flat Appl) x (THead (Bind +Abst) t2 x0)) t3) \to (\forall (P: Prop).P)))).(\lambda (H10: (pr2 c (THead +(Flat Appl) x (THead (Bind Abst) t2 x0)) t3)).(let H11 \def (pr2_gen_appl c x +(THead (Bind Abst) t2 x0) t3 H10) in (or3_ind (ex3_2 T T (\lambda (u2: +T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c +(THead (Bind Abst) t2 x0) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) t2 x0) +(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda +(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: +B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4)))))))) (ex6_6 B T T T T T +(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T +(THead (Bind Abst) t2 x0) (THead (Bind b) y1 z1)))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda +(y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) +z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: +B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: +T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) +y2) z1 z2)))))))) (sn3 c t3) (\lambda (H12: (ex3_2 T T (\lambda (u2: +T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c +(THead (Bind Abst) t2 x0) t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda +(t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: +T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind Abst) +t2 x0) t4))) (sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (H13: (eq +T t3 (THead (Flat Appl) x1 x2))).(\lambda (H14: (pr2 c x x1)).(\lambda (H15: +(pr2 c (THead (Bind Abst) t2 x0) x2)).(let H16 \def (eq_ind T t3 (\lambda +(t0: T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) t0) \to +(\forall (P: Prop).P))) H9 (THead (Flat Appl) x1 x2) H13) in (eq_ind_r T +(THead (Flat Appl) x1 x2) (\lambda (t0: T).(sn3 c t0)) (let H17 \def +(pr2_gen_abst c t2 x0 x2 H15) in (ex3_2_ind T T (\lambda (u2: T).(\lambda +(t4: T).(eq T x2 (THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: +T).(pr2 c t2 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall +(u: T).(pr2 (CHead c (Bind b) u) x0 t4))))) (sn3 c (THead (Flat Appl) x1 x2)) +(\lambda (x3: T).(\lambda (x4: T).(\lambda (H18: (eq T x2 (THead (Bind Abst) +x3 x4))).(\lambda (H19: (pr2 c t2 x3)).(\lambda (H20: ((\forall (b: +B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x0 x4))))).(let H21 \def (eq_ind +T x2 (\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) +(THead (Flat Appl) x1 t0)) \to (\forall (P: Prop).P))) H16 (THead (Bind Abst) +x3 x4) H18) in (eq_ind_r T (THead (Bind Abst) x3 x4) (\lambda (t0: T).(sn3 c +(THead (Flat Appl) x1 t0))) (let H_x \def (term_dec t2 x3) in (let H22 \def +H_x in (or_ind (eq T t2 x3) ((eq T t2 x3) \to (\forall (P: Prop).P)) (sn3 c +(THead (Flat Appl) x1 (THead (Bind Abst) x3 x4))) (\lambda (H23: (eq T t2 +x3)).(let H24 \def (eq_ind_r T x3 (\lambda (t0: T).((eq T (THead (Flat Appl) +x (THead (Bind Abst) t2 x0)) (THead (Flat Appl) x1 (THead (Bind Abst) t0 +x4))) \to (\forall (P: Prop).P))) H21 t2 H23) in (let H25 \def (eq_ind_r T x3 +(\lambda (t0: T).(pr2 c t2 t0)) H19 t2 H23) in (eq_ind T t2 (\lambda (t0: +T).(sn3 c (THead (Flat Appl) x1 (THead (Bind Abst) t0 x4)))) (let H_x0 \def +(term_dec x x1) in (let H26 \def H_x0 in (or_ind (eq T x x1) ((eq T x x1) \to +(\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Bind Abst) t2 +x4))) (\lambda (H27: (eq T x x1)).(let H28 \def (eq_ind_r T x1 (\lambda (t0: +T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) (THead (Flat Appl) +t0 (THead (Bind Abst) t2 x4))) \to (\forall (P: Prop).P))) H24 x H27) in (let +H29 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14 x H27) in (eq_ind +T x (\lambda (t0: T).(sn3 c (THead (Flat Appl) t0 (THead (Bind Abst) t2 +x4)))) (let H_x1 \def (term_dec x0 x4) in (let H30 \def H_x1 in (or_ind (eq T +x0 x4) ((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x +(THead (Bind Abst) t2 x4))) (\lambda (H31: (eq T x0 x4)).(let H32 \def +(eq_ind_r T x4 (\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind +Abst) t2 x0)) (THead (Flat Appl) x (THead (Bind Abst) t2 t0))) \to (\forall +(P: Prop).P))) H28 x0 H31) in (let H33 \def (eq_ind_r T x4 (\lambda (t0: +T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 +H31) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Flat Appl) x (THead +(Bind Abst) t2 t0)))) (H32 (refl_equal T (THead (Flat Appl) x (THead (Bind +Abst) t2 x0))) (sn3 c (THead (Flat Appl) x (THead (Bind Abst) t2 x0)))) x4 +H31)))) (\lambda (H31: (((eq T x0 x4) \to (\forall (P: Prop).P)))).(H5 (THead +(Bind Abbr) x x4) (\lambda (H32: (eq T (THead (Bind Abbr) x x0) (THead (Bind +Abbr) x x4))).(\lambda (P: Prop).(let H33 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | +(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind +Abbr) x x0) (THead (Bind Abbr) x x4) H32) in (let H34 \def (eq_ind_r T x4 +(\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H31 x0 H33) in +(let H35 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: +T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H33) in (H34 (refl_equal T x0) +P)))))) (pr3_pr2 c (THead (Bind Abbr) x x0) (THead (Bind Abbr) x x4) +(pr2_head_2 c x x0 x4 (Bind Abbr) (H20 Abbr x))) x x4 (refl_equal T (THead +(Bind Abbr) x x4)) t2 (sn3_sing c t2 H7))) H30))) x1 H27)))) (\lambda (H27: +(((eq T x x1) \to (\forall (P: Prop).P)))).(H5 (THead (Bind Abbr) x1 x4) +(\lambda (H28: (eq T (THead (Bind Abbr) x x0) (THead (Bind Abbr) x1 +x4))).(\lambda (P: Prop).(let H29 \def (f_equal T T (\lambda (e: T).(match e +in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) +\Rightarrow x | (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abbr) x x0) +(THead (Bind Abbr) x1 x4) H28) in ((let H30 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | +(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind +Abbr) x x0) (THead (Bind Abbr) x1 x4) H28) in (\lambda (H31: (eq T x +x1)).(let H32 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall +(u: T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H30) in (let H33 \def +(eq_ind_r T x1 (\lambda (t0: T).((eq T x t0) \to (\forall (P0: Prop).P0))) +H27 x H31) in (let H34 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14 +x H31) in (H33 (refl_equal T x) P)))))) H29)))) (pr3_head_12 c x x1 (pr3_pr2 +c x x1 H14) (Bind Abbr) x0 x4 (pr3_pr2 (CHead c (Bind Abbr) x1) x0 x4 (H20 +Abbr x1))) x1 x4 (refl_equal T (THead (Bind Abbr) x1 x4)) t2 (sn3_sing c t2 +H7))) H26))) x3 H23)))) (\lambda (H23: (((eq T t2 x3) \to (\forall (P: +Prop).P)))).(let H_x0 \def (term_dec x x1) in (let H24 \def H_x0 in (or_ind +(eq T x x1) ((eq T x x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) +x1 (THead (Bind Abst) x3 x4))) (\lambda (H25: (eq T x x1)).(let H26 \def +(eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14 x H25) in (eq_ind T x +(\lambda (t0: T).(sn3 c (THead (Flat Appl) t0 (THead (Bind Abst) x3 x4)))) +(let H_x1 \def (term_dec x0 x4) in (let H27 \def H_x1 in (or_ind (eq T x0 x4) +((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x (THead +(Bind Abst) x3 x4))) (\lambda (H28: (eq T x0 x4)).(let H29 \def (eq_ind_r T +x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) +x0 t0)))) H20 x0 H28) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Flat +Appl) x (THead (Bind Abst) x3 t0)))) (H8 x3 H23 (pr3_pr2 c t2 x3 H19)) x4 +H28))) (\lambda (H28: (((eq T x0 x4) \to (\forall (P: Prop).P)))).(H5 (THead +(Bind Abbr) x x4) (\lambda (H29: (eq T (THead (Bind Abbr) x x0) (THead (Bind +Abbr) x x4))).(\lambda (P: Prop).(let H30 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | +(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind +Abbr) x x0) (THead (Bind Abbr) x x4) H29) in (let H31 \def (eq_ind_r T x4 +(\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H28 x0 H30) in +(let H32 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: +T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H30) in (H31 (refl_equal T x0) +P)))))) (pr3_pr2 c (THead (Bind Abbr) x x0) (THead (Bind Abbr) x x4) +(pr2_head_2 c x x0 x4 (Bind Abbr) (H20 Abbr x))) x x4 (refl_equal T (THead +(Bind Abbr) x x4)) x3 (H7 x3 H23 (pr3_pr2 c t2 x3 H19)))) H27))) x1 H25))) +(\lambda (H25: (((eq T x x1) \to (\forall (P: Prop).P)))).(H5 (THead (Bind +Abbr) x1 x4) (\lambda (H26: (eq T (THead (Bind Abbr) x x0) (THead (Bind Abbr) +x1 x4))).(\lambda (P: Prop).(let H27 \def (f_equal T T (\lambda (e: T).(match +e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) +\Rightarrow x | (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abbr) x x0) +(THead (Bind Abbr) x1 x4) H26) in ((let H28 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | +(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind +Abbr) x x0) (THead (Bind Abbr) x1 x4) H26) in (\lambda (H29: (eq T x +x1)).(let H30 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall +(u: T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H28) in (let H31 \def +(eq_ind_r T x1 (\lambda (t0: T).((eq T x t0) \to (\forall (P0: Prop).P0))) +H25 x H29) in (let H32 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14 +x H29) in (H31 (refl_equal T x) P)))))) H27)))) (pr3_head_12 c x x1 (pr3_pr2 +c x x1 H14) (Bind Abbr) x0 x4 (pr3_pr2 (CHead c (Bind Abbr) x1) x0 x4 (H20 +Abbr x1))) x1 x4 (refl_equal T (THead (Bind Abbr) x1 x4)) x3 (H7 x3 H23 +(pr3_pr2 c t2 x3 H19)))) H24)))) H22))) x2 H18))))))) H17)) t3 H13))))))) +H12)) (\lambda (H12: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) t2 x0) (THead +(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda +(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: +B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4))))))))).(ex4_4_ind T T T +T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T +(THead (Bind Abst) t2 x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind +Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t4: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) +z1 t4))))))) (sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: +T).(\lambda (x4: T).(\lambda (H13: (eq T (THead (Bind Abst) t2 x0) (THead +(Bind Abst) x1 x2))).(\lambda (H14: (eq T t3 (THead (Bind Abbr) x3 +x4))).(\lambda (H15: (pr2 c x x3)).(\lambda (H16: ((\forall (b: B).(\forall +(u: T).(pr2 (CHead c (Bind b) u) x2 x4))))).(let H17 \def (eq_ind T t3 +(\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) t0) +\to (\forall (P: Prop).P))) H9 (THead (Bind Abbr) x3 x4) H14) in (eq_ind_r T +(THead (Bind Abbr) x3 x4) (\lambda (t0: T).(sn3 c t0)) (let H18 \def (f_equal +T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow t2 | (TLRef _) \Rightarrow t2 | (THead _ t0 _) \Rightarrow t0])) +(THead (Bind Abst) t2 x0) (THead (Bind Abst) x1 x2) H13) in ((let H19 \def +(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with +[(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ t0) +\Rightarrow t0])) (THead (Bind Abst) t2 x0) (THead (Bind Abst) x1 x2) H13) in +(\lambda (_: (eq T t2 x1)).(let H21 \def (eq_ind_r T x2 (\lambda (t0: +T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t0 x4)))) H16 x0 +H19) in (let H_x \def (term_dec x x3) in (let H22 \def H_x in (or_ind (eq T x +x3) ((eq T x x3) \to (\forall (P: Prop).P)) (sn3 c (THead (Bind Abbr) x3 x4)) +(\lambda (H23: (eq T x x3)).(let H24 \def (eq_ind_r T x3 (\lambda (t0: +T).(pr2 c x t0)) H15 x H23) in (eq_ind T x (\lambda (t0: T).(sn3 c (THead +(Bind Abbr) t0 x4))) (let H_x0 \def (term_dec x0 x4) in (let H25 \def H_x0 in +(or_ind (eq T x0 x4) ((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead +(Bind Abbr) x x4)) (\lambda (H26: (eq T x0 x4)).(let H27 \def (eq_ind_r T x4 +(\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x0 +t0)))) H21 x0 H26) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Bind Abbr) +x t0))) (sn3_sing c (THead (Bind Abbr) x x0) H6) x4 H26))) (\lambda (H26: +(((eq T x0 x4) \to (\forall (P: Prop).P)))).(H6 (THead (Bind Abbr) x x4) +(\lambda (H27: (eq T (THead (Bind Abbr) x x0) (THead (Bind Abbr) x +x4))).(\lambda (P: Prop).(let H28 \def (f_equal T T (\lambda (e: T).(match e +in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) +\Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind Abbr) x x0) +(THead (Bind Abbr) x x4) H27) in (let H29 \def (eq_ind_r T x4 (\lambda (t0: +T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H26 x0 H28) in (let H30 \def +(eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c +(Bind b) u) x0 t0)))) H21 x0 H28) in (H29 (refl_equal T x0) P)))))) (pr3_pr2 +c (THead (Bind Abbr) x x0) (THead (Bind Abbr) x x4) (pr2_head_2 c x x0 x4 +(Bind Abbr) (H21 Abbr x))))) H25))) x3 H23))) (\lambda (H23: (((eq T x x3) +\to (\forall (P: Prop).P)))).(H6 (THead (Bind Abbr) x3 x4) (\lambda (H24: (eq +T (THead (Bind Abbr) x x0) (THead (Bind Abbr) x3 x4))).(\lambda (P: +Prop).(let H25 \def (f_equal T T (\lambda (e: T).(match e in T return +(\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | +(THead _ t0 _) \Rightarrow t0])) (THead (Bind Abbr) x x0) (THead (Bind Abbr) +x3 x4) H24) in ((let H26 \def (f_equal T T (\lambda (e: T).(match e in T +return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) +\Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind Abbr) x x0) +(THead (Bind Abbr) x3 x4) H24) in (\lambda (H27: (eq T x x3)).(let H28 \def +(eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c +(Bind b) u) x0 t0)))) H21 x0 H26) in (let H29 \def (eq_ind_r T x3 (\lambda +(t0: T).((eq T x t0) \to (\forall (P0: Prop).P0))) H23 x H27) in (let H30 +\def (eq_ind_r T x3 (\lambda (t0: T).(pr2 c x t0)) H15 x H27) in (H29 +(refl_equal T x) P)))))) H25)))) (pr3_head_12 c x x3 (pr3_pr2 c x x3 H15) +(Bind Abbr) x0 x4 (pr3_pr2 (CHead c (Bind Abbr) x3) x0 x4 (H21 Abbr x3))))) +H22)))))) H18)) t3 H14)))))))))) H12)) (\lambda (H12: (ex6_6 B T T T T T +(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T +(THead (Bind Abst) t2 x0) (THead (Bind b) y1 z1)))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda +(y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) +z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: +B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: +T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) +y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B +b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) t2 x0) (THead +(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind +b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda +(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) (sn3 c t3) +(\lambda (x1: B).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda +(x5: T).(\lambda (x6: T).(\lambda (H13: (not (eq B x1 Abst))).(\lambda (H14: +(eq T (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3))).(\lambda (H15: (eq +T t3 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))).(\lambda +(_: (pr2 c x x5)).(\lambda (H17: (pr2 c x2 x6)).(\lambda (H18: (pr2 (CHead c +(Bind x1) x6) x3 x4)).(let H19 \def (eq_ind T t3 (\lambda (t0: T).((eq T +(THead (Flat Appl) x (THead (Bind Abst) t2 x0)) t0) \to (\forall (P: +Prop).P))) H9 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)) +H15) in (eq_ind_r T (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) +x4)) (\lambda (t0: T).(sn3 c t0)) (let H20 \def (f_equal T B (\lambda (e: +T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow Abst | +(TLRef _) \Rightarrow Abst | (THead k _ _) \Rightarrow (match k in K return +(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow +Abst])])) (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3) H14) in ((let H21 +\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) +with [(TSort _) \Rightarrow t2 | (TLRef _) \Rightarrow t2 | (THead _ t0 _) +\Rightarrow t0])) (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3) H14) in +((let H22 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: +T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ +t0) \Rightarrow t0])) (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3) H14) +in (\lambda (H23: (eq T t2 x2)).(\lambda (H24: (eq B Abst x1)).(let H25 \def +(eq_ind_r T x3 (\lambda (t0: T).(pr2 (CHead c (Bind x1) x6) t0 x4)) H18 x0 +H22) in (let H26 \def (eq_ind_r T x2 (\lambda (t0: T).(pr2 c t0 x6)) H17 t2 +H23) in (let H27 \def (eq_ind_r B x1 (\lambda (b: B).(pr2 (CHead c (Bind b) +x6) x0 x4)) H25 Abst H24) in (let H28 \def (eq_ind_r B x1 (\lambda (b: +B).(not (eq B b Abst))) H13 Abst H24) in (eq_ind B Abst (\lambda (b: B).(sn3 +c (THead (Bind b) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))) (let H29 +\def (match (H28 (refl_equal B Abst)) in False return (\lambda (_: +False).(sn3 c (THead (Bind Abst) x6 (THead (Flat Appl) (lift (S O) O x5) +x4)))) with []) in H29) x1 H24)))))))) H21)) H20)) t3 H15)))))))))))))) H12)) +H11))))))))) w H4))))))))))) y H0))))) H)))). +(* COMMENTS +Initial nodes: 5699 +END *) + +theorem sn3_appl_lref: + \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (v: +T).((sn3 c v) \to (sn3 c (THead (Flat Appl) v (TLRef i))))))) +\def + \lambda (c: C).(\lambda (i: nat).(\lambda (H: (nf2 c (TLRef i))).(\lambda +(v: T).(\lambda (H0: (sn3 c v)).(sn3_ind c (\lambda (t: T).(sn3 c (THead +(Flat Appl) t (TLRef i)))) (\lambda (t1: T).(\lambda (_: ((\forall (t2: +T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c +t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: +Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c (THead (Flat Appl) t2 (TLRef +i)))))))).(sn3_pr2_intro c (THead (Flat Appl) t1 (TLRef i)) (\lambda (t2: +T).(\lambda (H3: (((eq T (THead (Flat Appl) t1 (TLRef i)) t2) \to (\forall +(P: Prop).P)))).(\lambda (H4: (pr2 c (THead (Flat Appl) t1 (TLRef i)) +t2)).(let H5 \def (pr2_gen_appl c t1 (TLRef i) t2 H4) in (or3_ind (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2))) (\lambda (_: T).(\lambda +(t3: T).(pr2 c (TLRef i) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i) (THead (Bind Abst) y1 +z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: +T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2))))) (\lambda (_: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall +(u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda +(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T +(TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T +t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) +(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr2 c t1 u2))))))) (\lambda (_: B).(\lambda (y1: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 +y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: +T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) +(sn3 c t2) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T +t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c t1 +u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (TLRef i) t3))))).(ex3_2_ind T +T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2))) (\lambda (_: T).(\lambda +(t3: T).(pr2 c (TLRef i) t3))) (sn3 c t2) (\lambda (x0: T).(\lambda (x1: +T).(\lambda (H7: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H8: (pr2 c t1 +x0)).(\lambda (H9: (pr2 c (TLRef i) x1)).(let H10 \def (eq_ind T t2 (\lambda +(t: T).((eq T (THead (Flat Appl) t1 (TLRef i)) t) \to (\forall (P: Prop).P))) +H3 (THead (Flat Appl) x0 x1) H7) in (eq_ind_r T (THead (Flat Appl) x0 x1) +(\lambda (t: T).(sn3 c t)) (let H11 \def (eq_ind_r T x1 (\lambda (t: T).((eq +T (THead (Flat Appl) t1 (TLRef i)) (THead (Flat Appl) x0 t)) \to (\forall (P: +Prop).P))) H10 (TLRef i) (H x1 H9)) in (let H12 \def (eq_ind_r T x1 (\lambda +(t: T).(pr2 c (TLRef i) t)) H9 (TLRef i) (H x1 H9)) in (eq_ind T (TLRef i) +(\lambda (t: T).(sn3 c (THead (Flat Appl) x0 t))) (let H_x \def (term_dec t1 +x0) in (let H13 \def H_x in (or_ind (eq T t1 x0) ((eq T t1 x0) \to (\forall +(P: Prop).P)) (sn3 c (THead (Flat Appl) x0 (TLRef i))) (\lambda (H14: (eq T +t1 x0)).(let H15 \def (eq_ind_r T x0 (\lambda (t: T).((eq T (THead (Flat +Appl) t1 (TLRef i)) (THead (Flat Appl) t (TLRef i))) \to (\forall (P: +Prop).P))) H11 t1 H14) in (let H16 \def (eq_ind_r T x0 (\lambda (t: T).(pr2 c +t1 t)) H8 t1 H14) in (eq_ind T t1 (\lambda (t: T).(sn3 c (THead (Flat Appl) t +(TLRef i)))) (H15 (refl_equal T (THead (Flat Appl) t1 (TLRef i))) (sn3 c +(THead (Flat Appl) t1 (TLRef i)))) x0 H14)))) (\lambda (H14: (((eq T t1 x0) +\to (\forall (P: Prop).P)))).(H2 x0 H14 (pr3_pr2 c t1 x0 H8))) H13))) x1 (H +x1 H9)))) t2 H7))))))) H6)) (\lambda (H6: (ex4_4 T T T T (\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i) (THead +(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall +(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3))))))))).(ex4_4_ind T +T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T +(TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) +(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 +u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: +T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3))))))) +(sn3 c t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: +T).(\lambda (H7: (eq T (TLRef i) (THead (Bind Abst) x0 x1))).(\lambda (H8: +(eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (_: (pr2 c t1 x2)).(\lambda (_: +((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 x3))))).(let +H11 \def (eq_ind T t2 (\lambda (t: T).((eq T (THead (Flat Appl) t1 (TLRef i)) +t) \to (\forall (P: Prop).P))) H3 (THead (Bind Abbr) x2 x3) H8) in (eq_ind_r +T (THead (Bind Abbr) x2 x3) (\lambda (t: T).(sn3 c t)) (let H12 \def (eq_ind +T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with +[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) +\Rightarrow False])) I (THead (Bind Abst) x0 x1) H7) in (False_ind (sn3 c +(THead (Bind Abbr) x2 x3)) H12)) t2 H8)))))))))) H6)) (\lambda (H6: (ex6_6 B +T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: +B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(eq T (TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda +(_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq +T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) +(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr2 c t1 u2))))))) (\lambda (_: B).(\lambda (y1: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 +y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: +T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 +z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b +Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i) (THead (Bind b) y1 +z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: +T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat +Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2))))))) +(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 +(CHead c (Bind b) y2) z1 z2))))))) (sn3 c t2) (\lambda (x0: B).(\lambda (x1: +T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: +T).(\lambda (_: (not (eq B x0 Abst))).(\lambda (H8: (eq T (TLRef i) (THead +(Bind x0) x1 x2))).(\lambda (H9: (eq T t2 (THead (Bind x0) x5 (THead (Flat +Appl) (lift (S O) O x4) x3)))).(\lambda (_: (pr2 c t1 x4)).(\lambda (_: (pr2 +c x1 x5)).(\lambda (_: (pr2 (CHead c (Bind x0) x5) x2 x3)).(let H13 \def +(eq_ind T t2 (\lambda (t: T).((eq T (THead (Flat Appl) t1 (TLRef i)) t) \to +(\forall (P: Prop).P))) H3 (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) +O x4) x3)) H9) in (eq_ind_r T (THead (Bind x0) x5 (THead (Flat Appl) (lift (S +O) O x4) x3)) (\lambda (t: T).(sn3 c t)) (let H14 \def (eq_ind T (TLRef i) +(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) +\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow +False])) I (THead (Bind x0) x1 x2) H8) in (False_ind (sn3 c (THead (Bind x0) +x5 (THead (Flat Appl) (lift (S O) O x4) x3))) H14)) t2 H9)))))))))))))) H6)) +H5))))))))) v H0))))). +(* COMMENTS +Initial nodes: 2125 +END *) + +theorem sn3_appl_abbr: + \forall (c: C).(\forall (d: C).(\forall (w: T).(\forall (i: nat).((getl i c +(CHead d (Bind Abbr) w)) \to (\forall (v: T).((sn3 c (THead (Flat Appl) v +(lift (S i) O w))) \to (sn3 c (THead (Flat Appl) v (TLRef i))))))))) +\def + \lambda (c: C).(\lambda (d: C).(\lambda (w: T).(\lambda (i: nat).(\lambda +(H: (getl i c (CHead d (Bind Abbr) w))).(\lambda (v: T).(\lambda (H0: (sn3 c +(THead (Flat Appl) v (lift (S i) O w)))).(insert_eq T (THead (Flat Appl) v +(lift (S i) O w)) (\lambda (t: T).(sn3 c t)) (\lambda (_: T).(sn3 c (THead +(Flat Appl) v (TLRef i)))) (\lambda (y: T).(\lambda (H1: (sn3 c y)).(unintro +T v (\lambda (t: T).((eq T y (THead (Flat Appl) t (lift (S i) O w))) \to (sn3 +c (THead (Flat Appl) t (TLRef i))))) (sn3_ind c (\lambda (t: T).(\forall (x: +T).((eq T t (THead (Flat Appl) x (lift (S i) O w))) \to (sn3 c (THead (Flat +Appl) x (TLRef i)))))) (\lambda (t1: T).(\lambda (H2: ((\forall (t2: +T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c +t2)))))).(\lambda (H3: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: +Prop).P))) \to ((pr3 c t1 t2) \to (\forall (x: T).((eq T t2 (THead (Flat +Appl) x (lift (S i) O w))) \to (sn3 c (THead (Flat Appl) x (TLRef +i)))))))))).(\lambda (x: T).(\lambda (H4: (eq T t1 (THead (Flat Appl) x (lift +(S i) O w)))).(let H5 \def (eq_ind T t1 (\lambda (t: T).(\forall (t2: +T).((((eq T t t2) \to (\forall (P: Prop).P))) \to ((pr3 c t t2) \to (\forall +(x0: T).((eq T t2 (THead (Flat Appl) x0 (lift (S i) O w))) \to (sn3 c (THead +(Flat Appl) x0 (TLRef i))))))))) H3 (THead (Flat Appl) x (lift (S i) O w)) +H4) in (let H6 \def (eq_ind T t1 (\lambda (t: T).(\forall (t2: T).((((eq T t +t2) \to (\forall (P: Prop).P))) \to ((pr3 c t t2) \to (sn3 c t2))))) H2 +(THead (Flat Appl) x (lift (S i) O w)) H4) in (sn3_pr2_intro c (THead (Flat +Appl) x (TLRef i)) (\lambda (t2: T).(\lambda (H7: (((eq T (THead (Flat Appl) +x (TLRef i)) t2) \to (\forall (P: Prop).P)))).(\lambda (H8: (pr2 c (THead +(Flat Appl) x (TLRef i)) t2)).(let H9 \def (pr2_gen_appl c x (TLRef i) t2 H8) +in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead +(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) +(\lambda (_: T).(\lambda (t3: T).(pr2 c (TLRef i) t3)))) (ex4_4 T T T T +(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T +(TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) +(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x +u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: +T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) +(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: +B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(eq T (TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda +(_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq +T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) +(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 +y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: +T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) +(sn3 c t2) (\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T +t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x +u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (TLRef i) t3))))).(ex3_2_ind T +T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t3: +T).(pr2 c (TLRef i) t3))) (sn3 c t2) (\lambda (x0: T).(\lambda (x1: +T).(\lambda (H11: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H12: (pr2 c +x x0)).(\lambda (H13: (pr2 c (TLRef i) x1)).(let H14 \def (eq_ind T t2 +(\lambda (t: T).((eq T (THead (Flat Appl) x (TLRef i)) t) \to (\forall (P: +Prop).P))) H7 (THead (Flat Appl) x0 x1) H11) in (eq_ind_r T (THead (Flat +Appl) x0 x1) (\lambda (t: T).(sn3 c t)) (let H15 \def (pr2_gen_lref c x1 i +H13) in (or_ind (eq T x1 (TLRef i)) (ex2_2 C T (\lambda (d0: C).(\lambda (u: +T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq +T x1 (lift (S i) O u))))) (sn3 c (THead (Flat Appl) x0 x1)) (\lambda (H16: +(eq T x1 (TLRef i))).(let H17 \def (eq_ind T x1 (\lambda (t: T).((eq T (THead +(Flat Appl) x (TLRef i)) (THead (Flat Appl) x0 t)) \to (\forall (P: +Prop).P))) H14 (TLRef i) H16) in (eq_ind_r T (TLRef i) (\lambda (t: T).(sn3 c +(THead (Flat Appl) x0 t))) (let H_x \def (term_dec x x0) in (let H18 \def H_x +in (or_ind (eq T x x0) ((eq T x x0) \to (\forall (P: Prop).P)) (sn3 c (THead +(Flat Appl) x0 (TLRef i))) (\lambda (H19: (eq T x x0)).(let H20 \def +(eq_ind_r T x0 (\lambda (t: T).((eq T (THead (Flat Appl) x (TLRef i)) (THead +(Flat Appl) t (TLRef i))) \to (\forall (P: Prop).P))) H17 x H19) in (let H21 +\def (eq_ind_r T x0 (\lambda (t: T).(pr2 c x t)) H12 x H19) in (eq_ind T x +(\lambda (t: T).(sn3 c (THead (Flat Appl) t (TLRef i)))) (H20 (refl_equal T +(THead (Flat Appl) x (TLRef i))) (sn3 c (THead (Flat Appl) x (TLRef i)))) x0 +H19)))) (\lambda (H19: (((eq T x x0) \to (\forall (P: Prop).P)))).(H5 (THead +(Flat Appl) x0 (lift (S i) O w)) (\lambda (H20: (eq T (THead (Flat Appl) x +(lift (S i) O w)) (THead (Flat Appl) x0 (lift (S i) O w)))).(\lambda (P: +Prop).(let H21 \def (f_equal T T (\lambda (e: T).(match e in T return +(\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | +(THead _ t _) \Rightarrow t])) (THead (Flat Appl) x (lift (S i) O w)) (THead +(Flat Appl) x0 (lift (S i) O w)) H20) in (let H22 \def (eq_ind_r T x0 +(\lambda (t: T).((eq T x t) \to (\forall (P0: Prop).P0))) H19 x H21) in (let +H23 \def (eq_ind_r T x0 (\lambda (t: T).(pr2 c x t)) H12 x H21) in (H22 +(refl_equal T x) P)))))) (pr3_pr2 c (THead (Flat Appl) x (lift (S i) O w)) +(THead (Flat Appl) x0 (lift (S i) O w)) (pr2_head_1 c x x0 H12 (Flat Appl) +(lift (S i) O w))) x0 (refl_equal T (THead (Flat Appl) x0 (lift (S i) O +w))))) H18))) x1 H16))) (\lambda (H16: (ex2_2 C T (\lambda (d0: C).(\lambda +(u: T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: +T).(eq T x1 (lift (S i) O u)))))).(ex2_2_ind C T (\lambda (d0: C).(\lambda +(u: T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: +T).(eq T x1 (lift (S i) O u)))) (sn3 c (THead (Flat Appl) x0 x1)) (\lambda +(x2: C).(\lambda (x3: T).(\lambda (H17: (getl i c (CHead x2 (Bind Abbr) +x3))).(\lambda (H18: (eq T x1 (lift (S i) O x3))).(let H19 \def (eq_ind T x1 +(\lambda (t: T).((eq T (THead (Flat Appl) x (TLRef i)) (THead (Flat Appl) x0 +t)) \to (\forall (P: Prop).P))) H14 (lift (S i) O x3) H18) in (eq_ind_r T +(lift (S i) O x3) (\lambda (t: T).(sn3 c (THead (Flat Appl) x0 t))) (let H20 +\def (eq_ind C (CHead d (Bind Abbr) w) (\lambda (c0: C).(getl i c c0)) H +(CHead x2 (Bind Abbr) x3) (getl_mono c (CHead d (Bind Abbr) w) i H (CHead x2 +(Bind Abbr) x3) H17)) in (let H21 \def (f_equal C C (\lambda (e: C).(match e +in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) +\Rightarrow c0])) (CHead d (Bind Abbr) w) (CHead x2 (Bind Abbr) x3) +(getl_mono c (CHead d (Bind Abbr) w) i H (CHead x2 (Bind Abbr) x3) H17)) in +((let H22 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: +C).T) with [(CSort _) \Rightarrow w | (CHead _ _ t) \Rightarrow t])) (CHead d +(Bind Abbr) w) (CHead x2 (Bind Abbr) x3) (getl_mono c (CHead d (Bind Abbr) w) +i H (CHead x2 (Bind Abbr) x3) H17)) in (\lambda (H23: (eq C d x2)).(let H24 +\def (eq_ind_r T x3 (\lambda (t: T).(getl i c (CHead x2 (Bind Abbr) t))) H20 +w H22) in (eq_ind T w (\lambda (t: T).(sn3 c (THead (Flat Appl) x0 (lift (S +i) O t)))) (let H25 \def (eq_ind_r C x2 (\lambda (c0: C).(getl i c (CHead c0 +(Bind Abbr) w))) H24 d H23) in (let H_x \def (term_dec x x0) in (let H26 \def +H_x in (or_ind (eq T x x0) ((eq T x x0) \to (\forall (P: Prop).P)) (sn3 c +(THead (Flat Appl) x0 (lift (S i) O w))) (\lambda (H27: (eq T x x0)).(let H28 +\def (eq_ind_r T x0 (\lambda (t: T).(pr2 c x t)) H12 x H27) in (eq_ind T x +(\lambda (t: T).(sn3 c (THead (Flat Appl) t (lift (S i) O w)))) (sn3_sing c +(THead (Flat Appl) x (lift (S i) O w)) H6) x0 H27))) (\lambda (H27: (((eq T x +x0) \to (\forall (P: Prop).P)))).(H6 (THead (Flat Appl) x0 (lift (S i) O w)) +(\lambda (H28: (eq T (THead (Flat Appl) x (lift (S i) O w)) (THead (Flat +Appl) x0 (lift (S i) O w)))).(\lambda (P: Prop).(let H29 \def (f_equal T T +(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t _) \Rightarrow t])) +(THead (Flat Appl) x (lift (S i) O w)) (THead (Flat Appl) x0 (lift (S i) O +w)) H28) in (let H30 \def (eq_ind_r T x0 (\lambda (t: T).((eq T x t) \to +(\forall (P0: Prop).P0))) H27 x H29) in (let H31 \def (eq_ind_r T x0 (\lambda +(t: T).(pr2 c x t)) H12 x H29) in (H30 (refl_equal T x) P)))))) (pr3_pr2 c +(THead (Flat Appl) x (lift (S i) O w)) (THead (Flat Appl) x0 (lift (S i) O +w)) (pr2_head_1 c x x0 H12 (Flat Appl) (lift (S i) O w))))) H26)))) x3 +H22)))) H21))) x1 H18)))))) H16)) H15)) t2 H11))))))) H10)) (\lambda (H10: +(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: +T).(eq T (TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda +(_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 +t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: +T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda +(t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 +t3))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(eq T (TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda +(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead +(Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda +(_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind +b) u) z1 t3))))))) (sn3 c t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: +T).(\lambda (x3: T).(\lambda (H11: (eq T (TLRef i) (THead (Bind Abst) x0 +x1))).(\lambda (H12: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (_: (pr2 c +x x2)).(\lambda (_: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) +u) x1 x3))))).(let H15 \def (eq_ind T t2 (\lambda (t: T).((eq T (THead (Flat +Appl) x (TLRef i)) t) \to (\forall (P: Prop).P))) H7 (THead (Bind Abbr) x2 +x3) H12) in (eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda (t: T).(sn3 c t)) +(let H16 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return +(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) x0 +x1) H11) in (False_ind (sn3 c (THead (Bind Abbr) x2 x3)) H16)) t2 +H12)))))))))) H10)) (\lambda (H10: (ex6_6 B T T T T T (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i) +(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind +b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda +(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind +B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: +B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(eq T (TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda +(_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq +T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) +(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 +y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: +T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) +(sn3 c t2) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: +T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (not (eq B x0 +Abst))).(\lambda (H12: (eq T (TLRef i) (THead (Bind x0) x1 x2))).(\lambda +(H13: (eq T t2 (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) +x3)))).(\lambda (_: (pr2 c x x4)).(\lambda (_: (pr2 c x1 x5)).(\lambda (_: +(pr2 (CHead c (Bind x0) x5) x2 x3)).(let H17 \def (eq_ind T t2 (\lambda (t: +T).((eq T (THead (Flat Appl) x (TLRef i)) t) \to (\forall (P: Prop).P))) H7 +(THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) H13) in +(eq_ind_r T (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) +(\lambda (t: T).(sn3 c t)) (let H18 \def (eq_ind T (TLRef i) (\lambda (ee: +T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow +False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I +(THead (Bind x0) x1 x2) H12) in (False_ind (sn3 c (THead (Bind x0) x5 (THead +(Flat Appl) (lift (S O) O x4) x3))) H18)) t2 H13)))))))))))))) H10)) +H9))))))))))))) y H1)))) H0))))))). +(* COMMENTS +Initial nodes: 3727 +END *) + +theorem sn3_appl_cast: + \forall (c: C).(\forall (v: T).(\forall (u: T).((sn3 c (THead (Flat Appl) v +u)) \to (\forall (t: T).((sn3 c (THead (Flat Appl) v t)) \to (sn3 c (THead +(Flat Appl) v (THead (Flat Cast) u t)))))))) +\def + \lambda (c: C).(\lambda (v: T).(\lambda (u: T).(\lambda (H: (sn3 c (THead +(Flat Appl) v u))).(insert_eq T (THead (Flat Appl) v u) (\lambda (t: T).(sn3 +c t)) (\lambda (_: T).(\forall (t0: T).((sn3 c (THead (Flat Appl) v t0)) \to +(sn3 c (THead (Flat Appl) v (THead (Flat Cast) u t0)))))) (\lambda (y: +T).(\lambda (H0: (sn3 c y)).(unintro T u (\lambda (t: T).((eq T y (THead +(Flat Appl) v t)) \to (\forall (t0: T).((sn3 c (THead (Flat Appl) v t0)) \to +(sn3 c (THead (Flat Appl) v (THead (Flat Cast) t t0))))))) (unintro T v +(\lambda (t: T).(\forall (x: T).((eq T y (THead (Flat Appl) t x)) \to +(\forall (t0: T).((sn3 c (THead (Flat Appl) t t0)) \to (sn3 c (THead (Flat +Appl) t (THead (Flat Cast) x t0)))))))) (sn3_ind c (\lambda (t: T).(\forall +(x: T).(\forall (x0: T).((eq T t (THead (Flat Appl) x x0)) \to (\forall (t0: +T).((sn3 c (THead (Flat Appl) x t0)) \to (sn3 c (THead (Flat Appl) x (THead +(Flat Cast) x0 t0))))))))) (\lambda (t1: T).(\lambda (H1: ((\forall (t2: +T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c +t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: +Prop).P))) \to ((pr3 c t1 t2) \to (\forall (x: T).(\forall (x0: T).((eq T t2 +(THead (Flat Appl) x x0)) \to (\forall (t: T).((sn3 c (THead (Flat Appl) x +t)) \to (sn3 c (THead (Flat Appl) x (THead (Flat Cast) x0 +t))))))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: (eq T t1 (THead +(Flat Appl) x x0))).(\lambda (t: T).(\lambda (H4: (sn3 c (THead (Flat Appl) x +t))).(insert_eq T (THead (Flat Appl) x t) (\lambda (t0: T).(sn3 c t0)) +(\lambda (_: T).(sn3 c (THead (Flat Appl) x (THead (Flat Cast) x0 t)))) +(\lambda (y0: T).(\lambda (H5: (sn3 c y0)).(unintro T t (\lambda (t0: T).((eq +T y0 (THead (Flat Appl) x t0)) \to (sn3 c (THead (Flat Appl) x (THead (Flat +Cast) x0 t0))))) (sn3_ind c (\lambda (t0: T).(\forall (x1: T).((eq T t0 +(THead (Flat Appl) x x1)) \to (sn3 c (THead (Flat Appl) x (THead (Flat Cast) +x0 x1)))))) (\lambda (t0: T).(\lambda (H6: ((\forall (t2: T).((((eq T t0 t2) +\to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (sn3 c t2)))))).(\lambda +(H7: ((\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 +c t0 t2) \to (\forall (x1: T).((eq T t2 (THead (Flat Appl) x x1)) \to (sn3 c +(THead (Flat Appl) x (THead (Flat Cast) x0 x1)))))))))).(\lambda (x1: +T).(\lambda (H8: (eq T t0 (THead (Flat Appl) x x1))).(let H9 \def (eq_ind T +t0 (\lambda (t2: T).(\forall (t3: T).((((eq T t2 t3) \to (\forall (P: +Prop).P))) \to ((pr3 c t2 t3) \to (\forall (x2: T).((eq T t3 (THead (Flat +Appl) x x2)) \to (sn3 c (THead (Flat Appl) x (THead (Flat Cast) x0 +x2))))))))) H7 (THead (Flat Appl) x x1) H8) in (let H10 \def (eq_ind T t0 +(\lambda (t2: T).(\forall (t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) +\to ((pr3 c t2 t3) \to (sn3 c t3))))) H6 (THead (Flat Appl) x x1) H8) in (let +H11 \def (eq_ind T t1 (\lambda (t2: T).(\forall (t3: T).((((eq T t2 t3) \to +(\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to (\forall (x2: T).(\forall (x3: +T).((eq T t3 (THead (Flat Appl) x2 x3)) \to (\forall (t4: T).((sn3 c (THead +(Flat Appl) x2 t4)) \to (sn3 c (THead (Flat Appl) x2 (THead (Flat Cast) x3 +t4)))))))))))) H2 (THead (Flat Appl) x x0) H3) in (let H12 \def (eq_ind T t1 +(\lambda (t2: T).(\forall (t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) +\to ((pr3 c t2 t3) \to (sn3 c t3))))) H1 (THead (Flat Appl) x x0) H3) in +(sn3_pr2_intro c (THead (Flat Appl) x (THead (Flat Cast) x0 x1)) (\lambda +(t2: T).(\lambda (H13: (((eq T (THead (Flat Appl) x (THead (Flat Cast) x0 +x1)) t2) \to (\forall (P: Prop).P)))).(\lambda (H14: (pr2 c (THead (Flat +Appl) x (THead (Flat Cast) x0 x1)) t2)).(let H15 \def (pr2_gen_appl c x +(THead (Flat Cast) x0 x1) t2 H14) in (or3_ind (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c +(THead (Flat Cast) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Cast) x0 x1) +(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda +(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: +B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T +T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda +(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq +T (THead (Flat Cast) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda +(y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) +z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: +B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: +T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) +y2) z1 z2)))))))) (sn3 c t2) (\lambda (H16: (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c +(THead (Flat Cast) x0 x1) t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda +(t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: +T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Flat Cast) +x0 x1) t3))) (sn3 c t2) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H17: (eq +T t2 (THead (Flat Appl) x2 x3))).(\lambda (H18: (pr2 c x x2)).(\lambda (H19: +(pr2 c (THead (Flat Cast) x0 x1) x3)).(let H20 \def (eq_ind T t2 (\lambda +(t3: T).((eq T (THead (Flat Appl) x (THead (Flat Cast) x0 x1)) t3) \to +(\forall (P: Prop).P))) H13 (THead (Flat Appl) x2 x3) H17) in (eq_ind_r T +(THead (Flat Appl) x2 x3) (\lambda (t3: T).(sn3 c t3)) (let H21 \def +(pr2_gen_cast c x0 x1 x3 H19) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda +(t3: T).(eq T x3 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: +T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c x1 t3)))) (pr2 c +x1 x3) (sn3 c (THead (Flat Appl) x2 x3)) (\lambda (H22: (ex3_2 T T (\lambda +(u2: T).(\lambda (t3: T).(eq T x3 (THead (Flat Cast) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c x1 +t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x3 (THead +(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) +(\lambda (_: T).(\lambda (t3: T).(pr2 c x1 t3))) (sn3 c (THead (Flat Appl) x2 +x3)) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H23: (eq T x3 (THead (Flat +Cast) x4 x5))).(\lambda (H24: (pr2 c x0 x4)).(\lambda (H25: (pr2 c x1 +x5)).(let H26 \def (eq_ind T x3 (\lambda (t3: T).((eq T (THead (Flat Appl) x +(THead (Flat Cast) x0 x1)) (THead (Flat Appl) x2 t3)) \to (\forall (P: +Prop).P))) H20 (THead (Flat Cast) x4 x5) H23) in (eq_ind_r T (THead (Flat +Cast) x4 x5) (\lambda (t3: T).(sn3 c (THead (Flat Appl) x2 t3))) (let H_x +\def (term_dec (THead (Flat Appl) x x0) (THead (Flat Appl) x2 x4)) in (let +H27 \def H_x in (or_ind (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x2 +x4)) ((eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x2 x4)) \to (\forall +(P: Prop).P)) (sn3 c (THead (Flat Appl) x2 (THead (Flat Cast) x4 x5))) +(\lambda (H28: (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x2 +x4))).(let H29 \def (f_equal T T (\lambda (e: T).(match e in T return +(\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | +(THead _ t3 _) \Rightarrow t3])) (THead (Flat Appl) x x0) (THead (Flat Appl) +x2 x4) H28) in ((let H30 \def (f_equal T T (\lambda (e: T).(match e in T +return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) +\Rightarrow x0 | (THead _ _ t3) \Rightarrow t3])) (THead (Flat Appl) x x0) +(THead (Flat Appl) x2 x4) H28) in (\lambda (H31: (eq T x x2)).(let H32 \def +(eq_ind_r T x4 (\lambda (t3: T).((eq T (THead (Flat Appl) x (THead (Flat +Cast) x0 x1)) (THead (Flat Appl) x2 (THead (Flat Cast) t3 x5))) \to (\forall +(P: Prop).P))) H26 x0 H30) in (let H33 \def (eq_ind_r T x4 (\lambda (t3: +T).(pr2 c x0 t3)) H24 x0 H30) in (eq_ind T x0 (\lambda (t3: T).(sn3 c (THead +(Flat Appl) x2 (THead (Flat Cast) t3 x5)))) (let H34 \def (eq_ind_r T x2 +(\lambda (t3: T).((eq T (THead (Flat Appl) x (THead (Flat Cast) x0 x1)) +(THead (Flat Appl) t3 (THead (Flat Cast) x0 x5))) \to (\forall (P: Prop).P))) +H32 x H31) in (let H35 \def (eq_ind_r T x2 (\lambda (t3: T).(pr2 c x t3)) H18 +x H31) in (eq_ind T x (\lambda (t3: T).(sn3 c (THead (Flat Appl) t3 (THead +(Flat Cast) x0 x5)))) (let H_x0 \def (term_dec (THead (Flat Appl) x x1) +(THead (Flat Appl) x x5)) in (let H36 \def H_x0 in (or_ind (eq T (THead (Flat +Appl) x x1) (THead (Flat Appl) x x5)) ((eq T (THead (Flat Appl) x x1) (THead +(Flat Appl) x x5)) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x +(THead (Flat Cast) x0 x5))) (\lambda (H37: (eq T (THead (Flat Appl) x x1) +(THead (Flat Appl) x x5))).(let H38 \def (f_equal T T (\lambda (e: T).(match +e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x1 | (TLRef _) +\Rightarrow x1 | (THead _ _ t3) \Rightarrow t3])) (THead (Flat Appl) x x1) +(THead (Flat Appl) x x5) H37) in (let H39 \def (eq_ind_r T x5 (\lambda (t3: +T).((eq T (THead (Flat Appl) x (THead (Flat Cast) x0 x1)) (THead (Flat Appl) +x (THead (Flat Cast) x0 t3))) \to (\forall (P: Prop).P))) H34 x1 H38) in (let +H40 \def (eq_ind_r T x5 (\lambda (t3: T).(pr2 c x1 t3)) H25 x1 H38) in +(eq_ind T x1 (\lambda (t3: T).(sn3 c (THead (Flat Appl) x (THead (Flat Cast) +x0 t3)))) (H39 (refl_equal T (THead (Flat Appl) x (THead (Flat Cast) x0 x1))) +(sn3 c (THead (Flat Appl) x (THead (Flat Cast) x0 x1)))) x5 H38))))) (\lambda +(H37: (((eq T (THead (Flat Appl) x x1) (THead (Flat Appl) x x5)) \to (\forall +(P: Prop).P)))).(H9 (THead (Flat Appl) x x5) H37 (pr3_pr2 c (THead (Flat +Appl) x x1) (THead (Flat Appl) x x5) (pr2_thin_dx c x1 x5 H25 x Appl)) x5 +(refl_equal T (THead (Flat Appl) x x5)))) H36))) x2 H31))) x4 H30))))) H29))) +(\lambda (H28: (((eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x2 x4)) +\to (\forall (P: Prop).P)))).(let H_x0 \def (term_dec (THead (Flat Appl) x +x1) (THead (Flat Appl) x2 x5)) in (let H29 \def H_x0 in (or_ind (eq T (THead +(Flat Appl) x x1) (THead (Flat Appl) x2 x5)) ((eq T (THead (Flat Appl) x x1) +(THead (Flat Appl) x2 x5)) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat +Appl) x2 (THead (Flat Cast) x4 x5))) (\lambda (H30: (eq T (THead (Flat Appl) +x x1) (THead (Flat Appl) x2 x5))).(let H31 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x | +(TLRef _) \Rightarrow x | (THead _ t3 _) \Rightarrow t3])) (THead (Flat Appl) +x x1) (THead (Flat Appl) x2 x5) H30) in ((let H32 \def (f_equal T T (\lambda +(e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x1 +| (TLRef _) \Rightarrow x1 | (THead _ _ t3) \Rightarrow t3])) (THead (Flat +Appl) x x1) (THead (Flat Appl) x2 x5) H30) in (\lambda (H33: (eq T x +x2)).(let H34 \def (eq_ind_r T x5 (\lambda (t3: T).(pr2 c x1 t3)) H25 x1 H32) +in (eq_ind T x1 (\lambda (t3: T).(sn3 c (THead (Flat Appl) x2 (THead (Flat +Cast) x4 t3)))) (let H35 \def (eq_ind_r T x2 (\lambda (t3: T).((eq T (THead +(Flat Appl) x x0) (THead (Flat Appl) t3 x4)) \to (\forall (P: Prop).P))) H28 +x H33) in (let H36 \def (eq_ind_r T x2 (\lambda (t3: T).(pr2 c x t3)) H18 x +H33) in (eq_ind T x (\lambda (t3: T).(sn3 c (THead (Flat Appl) t3 (THead +(Flat Cast) x4 x1)))) (H11 (THead (Flat Appl) x x4) H35 (pr3_pr2 c (THead +(Flat Appl) x x0) (THead (Flat Appl) x x4) (pr2_thin_dx c x0 x4 H24 x Appl)) +x x4 (refl_equal T (THead (Flat Appl) x x4)) x1 (sn3_sing c (THead (Flat +Appl) x x1) H10)) x2 H33))) x5 H32)))) H31))) (\lambda (H30: (((eq T (THead +(Flat Appl) x x1) (THead (Flat Appl) x2 x5)) \to (\forall (P: +Prop).P)))).(H11 (THead (Flat Appl) x2 x4) H28 (pr3_flat c x x2 (pr3_pr2 c x +x2 H18) x0 x4 (pr3_pr2 c x0 x4 H24) Appl) x2 x4 (refl_equal T (THead (Flat +Appl) x2 x4)) x5 (H10 (THead (Flat Appl) x2 x5) H30 (pr3_flat c x x2 (pr3_pr2 +c x x2 H18) x1 x5 (pr3_pr2 c x1 x5 H25) Appl)))) H29)))) H27))) x3 H23))))))) +H22)) (\lambda (H22: (pr2 c x1 x3)).(let H_x \def (term_dec (THead (Flat +Appl) x x1) (THead (Flat Appl) x2 x3)) in (let H23 \def H_x in (or_ind (eq T +(THead (Flat Appl) x x1) (THead (Flat Appl) x2 x3)) ((eq T (THead (Flat Appl) +x x1) (THead (Flat Appl) x2 x3)) \to (\forall (P: Prop).P)) (sn3 c (THead +(Flat Appl) x2 x3)) (\lambda (H24: (eq T (THead (Flat Appl) x x1) (THead +(Flat Appl) x2 x3))).(let H25 \def (f_equal T T (\lambda (e: T).(match e in T +return (\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) +\Rightarrow x | (THead _ t3 _) \Rightarrow t3])) (THead (Flat Appl) x x1) +(THead (Flat Appl) x2 x3) H24) in ((let H26 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x1 | +(TLRef _) \Rightarrow x1 | (THead _ _ t3) \Rightarrow t3])) (THead (Flat +Appl) x x1) (THead (Flat Appl) x2 x3) H24) in (\lambda (H27: (eq T x +x2)).(let H28 \def (eq_ind_r T x3 (\lambda (t3: T).(pr2 c x1 t3)) H22 x1 H26) +in (let H29 \def (eq_ind_r T x3 (\lambda (t3: T).((eq T (THead (Flat Appl) x +(THead (Flat Cast) x0 x1)) (THead (Flat Appl) x2 t3)) \to (\forall (P: +Prop).P))) H20 x1 H26) in (eq_ind T x1 (\lambda (t3: T).(sn3 c (THead (Flat +Appl) x2 t3))) (let H30 \def (eq_ind_r T x2 (\lambda (t3: T).((eq T (THead +(Flat Appl) x (THead (Flat Cast) x0 x1)) (THead (Flat Appl) t3 x1)) \to +(\forall (P: Prop).P))) H29 x H27) in (let H31 \def (eq_ind_r T x2 (\lambda +(t3: T).(pr2 c x t3)) H18 x H27) in (eq_ind T x (\lambda (t3: T).(sn3 c +(THead (Flat Appl) t3 x1))) (sn3_sing c (THead (Flat Appl) x x1) H10) x2 +H27))) x3 H26))))) H25))) (\lambda (H24: (((eq T (THead (Flat Appl) x x1) +(THead (Flat Appl) x2 x3)) \to (\forall (P: Prop).P)))).(H10 (THead (Flat +Appl) x2 x3) H24 (pr3_flat c x x2 (pr3_pr2 c x x2 H18) x1 x3 (pr3_pr2 c x1 x3 +H22) Appl))) H23)))) H21)) t2 H17))))))) H16)) (\lambda (H16: (ex4_4 T T T T +(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T +(THead (Flat Cast) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind +Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) +u0) z1 t3))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Cast) x0 x1) (THead +(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda +(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: +B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3))))))) (sn3 c t2) +(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda +(H17: (eq T (THead (Flat Cast) x0 x1) (THead (Bind Abst) x2 x3))).(\lambda +(H18: (eq T t2 (THead (Bind Abbr) x4 x5))).(\lambda (_: (pr2 c x +x4)).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) +u0) x3 x5))))).(let H21 \def (eq_ind T t2 (\lambda (t3: T).((eq T (THead +(Flat Appl) x (THead (Flat Cast) x0 x1)) t3) \to (\forall (P: Prop).P))) H13 +(THead (Bind Abbr) x4 x5) H18) in (eq_ind_r T (THead (Bind Abbr) x4 x5) +(\lambda (t3: T).(sn3 c t3)) (let H22 \def (eq_ind T (THead (Flat Cast) x0 +x1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort +_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) +\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) +\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) x2 +x3) H17) in (False_ind (sn3 c (THead (Bind Abbr) x4 x5)) H22)) t2 +H18)))))))))) H16)) (\lambda (H16: (ex6_6 B T T T T T (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat +Cast) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T +t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) +(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 +y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: +T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 +z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b +Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Cast) x0 x1) (THead +(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind +b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda +(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) (sn3 c t2) +(\lambda (x2: B).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda +(x6: T).(\lambda (x7: T).(\lambda (_: (not (eq B x2 Abst))).(\lambda (H18: +(eq T (THead (Flat Cast) x0 x1) (THead (Bind x2) x3 x4))).(\lambda (H19: (eq +T t2 (THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) O x6) x5)))).(\lambda +(_: (pr2 c x x6)).(\lambda (_: (pr2 c x3 x7)).(\lambda (_: (pr2 (CHead c +(Bind x2) x7) x4 x5)).(let H23 \def (eq_ind T t2 (\lambda (t3: T).((eq T +(THead (Flat Appl) x (THead (Flat Cast) x0 x1)) t3) \to (\forall (P: +Prop).P))) H13 (THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) O x6) x5)) +H19) in (eq_ind_r T (THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) O x6) +x5)) (\lambda (t3: T).(sn3 c t3)) (let H24 \def (eq_ind T (THead (Flat Cast) +x0 x1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with +[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) +\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) +\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind x2) x3 x4) +H18) in (False_ind (sn3 c (THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) +O x6) x5))) H24)) t2 H19)))))))))))))) H16)) H15))))))))))))))) y0 H5)))) +H4))))))))) y H0))))) H)))). +(* COMMENTS +Initial nodes: 5149 +END *) + +theorem sn3_appl_bind: + \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (u: +T).((sn3 c u) \to (\forall (t: T).(\forall (v: T).((sn3 (CHead c (Bind b) u) +(THead (Flat Appl) (lift (S O) O v) t)) \to (sn3 c (THead (Flat Appl) v +(THead (Bind b) u t)))))))))) +\def + \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (c: C).(\lambda +(u: T).(\lambda (H0: (sn3 c u)).(sn3_ind c (\lambda (t: T).(\forall (t0: +T).(\forall (v: T).((sn3 (CHead c (Bind b) t) (THead (Flat Appl) (lift (S O) +O v) t0)) \to (sn3 c (THead (Flat Appl) v (THead (Bind b) t t0))))))) +(\lambda (t1: T).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall +(P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H2: ((\forall +(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to +(\forall (t: T).(\forall (v: T).((sn3 (CHead c (Bind b) t2) (THead (Flat +Appl) (lift (S O) O v) t)) \to (sn3 c (THead (Flat Appl) v (THead (Bind b) t2 +t))))))))))).(\lambda (t: T).(\lambda (v: T).(\lambda (H3: (sn3 (CHead c +(Bind b) t1) (THead (Flat Appl) (lift (S O) O v) t))).(insert_eq T (THead +(Flat Appl) (lift (S O) O v) t) (\lambda (t0: T).(sn3 (CHead c (Bind b) t1) +t0)) (\lambda (_: T).(sn3 c (THead (Flat Appl) v (THead (Bind b) t1 t)))) +(\lambda (y: T).(\lambda (H4: (sn3 (CHead c (Bind b) t1) y)).(unintro T t +(\lambda (t0: T).((eq T y (THead (Flat Appl) (lift (S O) O v) t0)) \to (sn3 c +(THead (Flat Appl) v (THead (Bind b) t1 t0))))) (unintro T v (\lambda (t0: +T).(\forall (x: T).((eq T y (THead (Flat Appl) (lift (S O) O t0) x)) \to (sn3 +c (THead (Flat Appl) t0 (THead (Bind b) t1 x)))))) (sn3_ind (CHead c (Bind b) +t1) (\lambda (t0: T).(\forall (x: T).(\forall (x0: T).((eq T t0 (THead (Flat +Appl) (lift (S O) O x) x0)) \to (sn3 c (THead (Flat Appl) x (THead (Bind b) +t1 x0))))))) (\lambda (t2: T).(\lambda (H5: ((\forall (t3: T).((((eq T t2 t3) +\to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to (sn3 +(CHead c (Bind b) t1) t3)))))).(\lambda (H6: ((\forall (t3: T).((((eq T t2 +t3) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to +(\forall (x: T).(\forall (x0: T).((eq T t3 (THead (Flat Appl) (lift (S O) O +x) x0)) \to (sn3 c (THead (Flat Appl) x (THead (Bind b) t1 +x0))))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H7: (eq T t2 (THead +(Flat Appl) (lift (S O) O x) x0))).(let H8 \def (eq_ind T t2 (\lambda (t0: +T).(\forall (t3: T).((((eq T t0 t3) \to (\forall (P: Prop).P))) \to ((pr3 +(CHead c (Bind b) t1) t0 t3) \to (\forall (x1: T).(\forall (x2: T).((eq T t3 +(THead (Flat Appl) (lift (S O) O x1) x2)) \to (sn3 c (THead (Flat Appl) x1 +(THead (Bind b) t1 x2)))))))))) H6 (THead (Flat Appl) (lift (S O) O x) x0) +H7) in (let H9 \def (eq_ind T t2 (\lambda (t0: T).(\forall (t3: T).((((eq T +t0 t3) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t0 t3) \to +(sn3 (CHead c (Bind b) t1) t3))))) H5 (THead (Flat Appl) (lift (S O) O x) x0) +H7) in (sn3_pr2_intro c (THead (Flat Appl) x (THead (Bind b) t1 x0)) (\lambda +(t3: T).(\lambda (H10: (((eq T (THead (Flat Appl) x (THead (Bind b) t1 x0)) +t3) \to (\forall (P: Prop).P)))).(\lambda (H11: (pr2 c (THead (Flat Appl) x +(THead (Bind b) t1 x0)) t3)).(let H12 \def (pr2_gen_appl c x (THead (Bind b) +t1 x0) t3 H11) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T +t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x +u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind b) t1 x0) t4)))) +(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: +T).(eq T (THead (Bind b) t1 x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind +Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t4: T).(\forall (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind b0) +u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B +b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) t1 x0) (THead +(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind +b0) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) +(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda +(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b0) y2) z1 z2)))))))) (sn3 c t3) +(\lambda (H13: (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead +(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) +(\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind b) t1 x0) +t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead +(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) +(\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind b) t1 x0) t4))) (sn3 c +t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (H14: (eq T t3 (THead (Flat +Appl) x1 x2))).(\lambda (H15: (pr2 c x x1)).(\lambda (H16: (pr2 c (THead +(Bind b) t1 x0) x2)).(let H17 \def (eq_ind T t3 (\lambda (t0: T).((eq T +(THead (Flat Appl) x (THead (Bind b) t1 x0)) t0) \to (\forall (P: Prop).P))) +H10 (THead (Flat Appl) x1 x2) H14) in (eq_ind_r T (THead (Flat Appl) x1 x2) +(\lambda (t0: T).(sn3 c t0)) (let H_x \def (pr3_gen_bind b H c t1 x0 x2) in +(let H18 \def (H_x (pr3_pr2 c (THead (Bind b) t1 x0) x2 H16)) in (or_ind +(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead (Bind b) u2 +t4)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_: +T).(\lambda (t4: T).(pr3 (CHead c (Bind b) t1) x0 t4)))) (pr3 (CHead c (Bind +b) t1) x0 (lift (S O) O x2)) (sn3 c (THead (Flat Appl) x1 x2)) (\lambda (H19: +(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead (Bind b) u2 +t4)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_: +T).(\lambda (t4: T).(pr3 (CHead c (Bind b) t1) x0 t4))))).(ex3_2_ind T T +(\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead (Bind b) u2 t4)))) (\lambda +(u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr3 +(CHead c (Bind b) t1) x0 t4))) (sn3 c (THead (Flat Appl) x1 x2)) (\lambda +(x3: T).(\lambda (x4: T).(\lambda (H20: (eq T x2 (THead (Bind b) x3 +x4))).(\lambda (H21: (pr3 c t1 x3)).(\lambda (H22: (pr3 (CHead c (Bind b) t1) +x0 x4)).(let H23 \def (eq_ind T x2 (\lambda (t0: T).((eq T (THead (Flat Appl) +x (THead (Bind b) t1 x0)) (THead (Flat Appl) x1 t0)) \to (\forall (P: +Prop).P))) H17 (THead (Bind b) x3 x4) H20) in (eq_ind_r T (THead (Bind b) x3 +x4) (\lambda (t0: T).(sn3 c (THead (Flat Appl) x1 t0))) (let H_x0 \def +(term_dec t1 x3) in (let H24 \def H_x0 in (or_ind (eq T t1 x3) ((eq T t1 x3) +\to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Bind b) x3 +x4))) (\lambda (H25: (eq T t1 x3)).(let H26 \def (eq_ind_r T x3 (\lambda (t0: +T).((eq T (THead (Flat Appl) x (THead (Bind b) t1 x0)) (THead (Flat Appl) x1 +(THead (Bind b) t0 x4))) \to (\forall (P: Prop).P))) H23 t1 H25) in (let H27 +\def (eq_ind_r T x3 (\lambda (t0: T).(pr3 c t1 t0)) H21 t1 H25) in (eq_ind T +t1 (\lambda (t0: T).(sn3 c (THead (Flat Appl) x1 (THead (Bind b) t0 x4)))) +(let H_x1 \def (term_dec x0 x4) in (let H28 \def H_x1 in (or_ind (eq T x0 x4) +((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead +(Bind b) t1 x4))) (\lambda (H29: (eq T x0 x4)).(let H30 \def (eq_ind_r T x4 +(\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind b) t1 x0)) (THead +(Flat Appl) x1 (THead (Bind b) t1 t0))) \to (\forall (P: Prop).P))) H26 x0 +H29) in (let H31 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 (CHead c (Bind b) +t1) x0 t0)) H22 x0 H29) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Flat +Appl) x1 (THead (Bind b) t1 t0)))) (let H_x2 \def (term_dec x x1) in (let H32 +\def H_x2 in (or_ind (eq T x x1) ((eq T x x1) \to (\forall (P: Prop).P)) (sn3 +c (THead (Flat Appl) x1 (THead (Bind b) t1 x0))) (\lambda (H33: (eq T x +x1)).(let H34 \def (eq_ind_r T x1 (\lambda (t0: T).((eq T (THead (Flat Appl) +x (THead (Bind b) t1 x0)) (THead (Flat Appl) t0 (THead (Bind b) t1 x0))) \to +(\forall (P: Prop).P))) H30 x H33) in (let H35 \def (eq_ind_r T x1 (\lambda +(t0: T).(pr2 c x t0)) H15 x H33) in (eq_ind T x (\lambda (t0: T).(sn3 c +(THead (Flat Appl) t0 (THead (Bind b) t1 x0)))) (H34 (refl_equal T (THead +(Flat Appl) x (THead (Bind b) t1 x0))) (sn3 c (THead (Flat Appl) x (THead +(Bind b) t1 x0)))) x1 H33)))) (\lambda (H33: (((eq T x x1) \to (\forall (P: +Prop).P)))).(H8 (THead (Flat Appl) (lift (S O) O x1) x0) (\lambda (H34: (eq T +(THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) +x0))).(\lambda (P: Prop).(let H35 \def (f_equal T T (\lambda (e: T).(match e +in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map +(f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n) +\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with +[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4) +\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in +lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow +((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match +t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef +(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | +(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) +t4))]) in lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _) +\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) +(lift (S O) O x1) x0) H34) in (let H36 \def (eq_ind_r T x1 (\lambda (t0: +T).((eq T x t0) \to (\forall (P0: Prop).P0))) H33 x (lift_inj x x1 (S O) O +H35)) in (let H37 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x +(lift_inj x x1 (S O) O H35)) in (H36 (refl_equal T x) P)))))) (pr3_flat +(CHead c (Bind b) t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c +(Bind b) t1) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1 +(pr3_pr2 c x x1 H15)) x0 x0 (pr3_refl (CHead c (Bind b) t1) x0) Appl) x1 x0 +(refl_equal T (THead (Flat Appl) (lift (S O) O x1) x0)))) H32))) x4 H29)))) +(\lambda (H29: (((eq T x0 x4) \to (\forall (P: Prop).P)))).(H8 (THead (Flat +Appl) (lift (S O) O x1) x4) (\lambda (H30: (eq T (THead (Flat Appl) (lift (S +O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) x4))).(\lambda (P: +Prop).(let H31 \def (f_equal T T (\lambda (e: T).(match e in T return +(\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat +\to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n) +\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with +[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4) +\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in +lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow +((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match +t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef +(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | +(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) +t4))]) in lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _) +\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) +(lift (S O) O x1) x4) H30) in ((let H32 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | +(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat +Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) x4) H30) in +(\lambda (H33: (eq T (lift (S O) O x) (lift (S O) O x1))).(let H34 \def +(eq_ind_r T x4 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) +H29 x0 H32) in (let H35 \def (eq_ind_r T x4 (\lambda (t0: T).((eq T (THead +(Flat Appl) x (THead (Bind b) t1 x0)) (THead (Flat Appl) x1 (THead (Bind b) +t1 t0))) \to (\forall (P0: Prop).P0))) H26 x0 H32) in (let H36 \def (eq_ind_r +T x4 (\lambda (t0: T).(pr3 (CHead c (Bind b) t1) x0 t0)) H22 x0 H32) in (let +H37 \def (eq_ind_r T x1 (\lambda (t0: T).((eq T (THead (Flat Appl) x (THead +(Bind b) t1 x0)) (THead (Flat Appl) t0 (THead (Bind b) t1 x0))) \to (\forall +(P0: Prop).P0))) H35 x (lift_inj x x1 (S O) O H33)) in (let H38 \def +(eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x (lift_inj x x1 (S O) O +H33)) in (H34 (refl_equal T x0) P)))))))) H31)))) (pr3_flat (CHead c (Bind b) +t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c (Bind b) t1) c (S +O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1 (pr3_pr2 c x x1 H15)) +x0 x4 H22 Appl) x1 x4 (refl_equal T (THead (Flat Appl) (lift (S O) O x1) +x4)))) H28))) x3 H25)))) (\lambda (H25: (((eq T t1 x3) \to (\forall (P: +Prop).P)))).(H2 x3 H25 H21 x4 x1 (sn3_cpr3_trans c t1 x3 H21 (Bind b) (THead +(Flat Appl) (lift (S O) O x1) x4) (let H_x1 \def (term_dec x0 x4) in (let H26 +\def H_x1 in (or_ind (eq T x0 x4) ((eq T x0 x4) \to (\forall (P: Prop).P)) +(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x1) x4)) (\lambda +(H27: (eq T x0 x4)).(let H28 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 (CHead +c (Bind b) t1) x0 t0)) H22 x0 H27) in (eq_ind T x0 (\lambda (t0: T).(sn3 +(CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x1) t0))) (let H_x2 +\def (term_dec x x1) in (let H29 \def H_x2 in (or_ind (eq T x x1) ((eq T x +x1) \to (\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat Appl) +(lift (S O) O x1) x0)) (\lambda (H30: (eq T x x1)).(let H31 \def (eq_ind_r T +x1 (\lambda (t0: T).(pr2 c x t0)) H15 x H30) in (eq_ind T x (\lambda (t0: +T).(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O t0) x0))) +(sn3_sing (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x) x0) H9) +x1 H30))) (\lambda (H30: (((eq T x x1) \to (\forall (P: Prop).P)))).(H9 +(THead (Flat Appl) (lift (S O) O x1) x0) (\lambda (H31: (eq T (THead (Flat +Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) +x0))).(\lambda (P: Prop).(let H32 \def (f_equal T T (\lambda (e: T).(match e +in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map +(f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n) +\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with +[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4) +\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in +lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow +((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match +t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef +(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | +(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) +t4))]) in lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _) +\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) +(lift (S O) O x1) x0) H31) in (let H33 \def (eq_ind_r T x1 (\lambda (t0: +T).((eq T x t0) \to (\forall (P0: Prop).P0))) H30 x (lift_inj x x1 (S O) O +H32)) in (let H34 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x +(lift_inj x x1 (S O) O H32)) in (H33 (refl_equal T x) P)))))) (pr3_flat +(CHead c (Bind b) t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c +(Bind b) t1) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1 +(pr3_pr2 c x x1 H15)) x0 x0 (pr3_refl (CHead c (Bind b) t1) x0) Appl))) +H29))) x4 H27))) (\lambda (H27: (((eq T x0 x4) \to (\forall (P: +Prop).P)))).(H9 (THead (Flat Appl) (lift (S O) O x1) x4) (\lambda (H28: (eq T +(THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) +x4))).(\lambda (P: Prop).(let H29 \def (f_equal T T (\lambda (e: T).(match e +in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map +(f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n) +\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with +[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4) +\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in +lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow +((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match +t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef +(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | +(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) +t4))]) in lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _) +\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) +(lift (S O) O x1) x4) H28) in ((let H30 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | +(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat +Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) x4) H28) in +(\lambda (H31: (eq T (lift (S O) O x) (lift (S O) O x1))).(let H32 \def +(eq_ind_r T x4 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) +H27 x0 H30) in (let H33 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 (CHead c +(Bind b) t1) x0 t0)) H22 x0 H30) in (let H34 \def (eq_ind_r T x1 (\lambda +(t0: T).(pr2 c x t0)) H15 x (lift_inj x x1 (S O) O H31)) in (H32 (refl_equal +T x0) P)))))) H29)))) (pr3_flat (CHead c (Bind b) t1) (lift (S O) O x) (lift +(S O) O x1) (pr3_lift (CHead c (Bind b) t1) c (S O) O (drop_drop (Bind b) O c +c (drop_refl c) t1) x x1 (pr3_pr2 c x x1 H15)) x0 x4 H22 Appl))) H26)))))) +H24))) x2 H20))))))) H19)) (\lambda (H19: (pr3 (CHead c (Bind b) t1) x0 (lift +(S O) O x2))).(sn3_gen_lift (CHead c (Bind b) t1) (THead (Flat Appl) x1 x2) +(S O) O (eq_ind_r T (THead (Flat Appl) (lift (S O) O x1) (lift (S O) (s (Flat +Appl) O) x2)) (\lambda (t0: T).(sn3 (CHead c (Bind b) t1) t0)) (sn3_pr3_trans +(CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x1) x0) (let H_x0 \def +(term_dec x x1) in (let H20 \def H_x0 in (or_ind (eq T x x1) ((eq T x x1) \to +(\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S +O) O x1) x0)) (\lambda (H21: (eq T x x1)).(let H22 \def (eq_ind_r T x1 +(\lambda (t0: T).(pr2 c x t0)) H15 x H21) in (eq_ind T x (\lambda (t0: +T).(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O t0) x0))) +(sn3_sing (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x) x0) H9) +x1 H21))) (\lambda (H21: (((eq T x x1) \to (\forall (P: Prop).P)))).(H9 +(THead (Flat Appl) (lift (S O) O x1) x0) (\lambda (H22: (eq T (THead (Flat +Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) +x0))).(\lambda (P: Prop).(let H23 \def (f_equal T T (\lambda (e: T).(match e +in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map +(f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n) +\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with +[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4) +\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in +lref_map) (\lambda (x3: nat).(plus x3 (S O))) O x) | (TLRef _) \Rightarrow +((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match +t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef +(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | +(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) +t4))]) in lref_map) (\lambda (x3: nat).(plus x3 (S O))) O x) | (THead _ t0 _) +\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) +(lift (S O) O x1) x0) H22) in (let H24 \def (eq_ind_r T x1 (\lambda (t0: +T).((eq T x t0) \to (\forall (P0: Prop).P0))) H21 x (lift_inj x x1 (S O) O +H23)) in (let H25 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x +(lift_inj x x1 (S O) O H23)) in (H24 (refl_equal T x) P)))))) (pr3_flat +(CHead c (Bind b) t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c +(Bind b) t1) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1 +(pr3_pr2 c x x1 H15)) x0 x0 (pr3_refl (CHead c (Bind b) t1) x0) Appl))) +H20))) (THead (Flat Appl) (lift (S O) O x1) (lift (S O) O x2)) (pr3_thin_dx +(CHead c (Bind b) t1) x0 (lift (S O) O x2) H19 (lift (S O) O x1) Appl)) (lift +(S O) O (THead (Flat Appl) x1 x2)) (lift_head (Flat Appl) x1 x2 (S O) O)) c +(drop_drop (Bind b) O c c (drop_refl c) t1))) H18))) t3 H14))))))) H13)) +(\lambda (H13: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(eq T (THead (Bind b) t1 x0) (THead (Bind Abst) y1 +z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: +T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b0: B).(\forall (u0: +T).(pr2 (CHead c (Bind b0) u0) z1 t4))))))))).(ex4_4_ind T T T T (\lambda +(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind +b) t1 x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) +(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x +u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: +T).(\forall (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind b0) u0) z1 t4))))))) +(sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: +T).(\lambda (H14: (eq T (THead (Bind b) t1 x0) (THead (Bind Abst) x1 +x2))).(\lambda (H15: (eq T t3 (THead (Bind Abbr) x3 x4))).(\lambda (_: (pr2 c +x x3)).(\lambda (H17: ((\forall (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind +b0) u0) x2 x4))))).(let H18 \def (eq_ind T t3 (\lambda (t0: T).((eq T (THead +(Flat Appl) x (THead (Bind b) t1 x0)) t0) \to (\forall (P: Prop).P))) H10 +(THead (Bind Abbr) x3 x4) H15) in (eq_ind_r T (THead (Bind Abbr) x3 x4) +(\lambda (t0: T).(sn3 c t0)) (let H19 \def (f_equal T B (\lambda (e: +T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | +(TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k in K return +(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow +b])])) (THead (Bind b) t1 x0) (THead (Bind Abst) x1 x2) H14) in ((let H20 +\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) +with [(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ t0 _) +\Rightarrow t0])) (THead (Bind b) t1 x0) (THead (Bind Abst) x1 x2) H14) in +((let H21 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: +T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ +t0) \Rightarrow t0])) (THead (Bind b) t1 x0) (THead (Bind Abst) x1 x2) H14) +in (\lambda (_: (eq T t1 x1)).(\lambda (H23: (eq B b Abst)).(let H24 \def +(eq_ind_r T x2 (\lambda (t0: T).(\forall (b0: B).(\forall (u0: T).(pr2 (CHead +c (Bind b0) u0) t0 x4)))) H17 x0 H21) in (let H25 \def (eq_ind B b (\lambda +(b0: B).((eq T (THead (Flat Appl) x (THead (Bind b0) t1 x0)) (THead (Bind +Abbr) x3 x4)) \to (\forall (P: Prop).P))) H18 Abst H23) in (let H26 \def +(eq_ind B b (\lambda (b0: B).(\forall (t4: T).((((eq T (THead (Flat Appl) +(lift (S O) O x) x0) t4) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind +b0) t1) (THead (Flat Appl) (lift (S O) O x) x0) t4) \to (sn3 (CHead c (Bind +b0) t1) t4))))) H9 Abst H23) in (let H27 \def (eq_ind B b (\lambda (b0: +B).(\forall (t4: T).((((eq T (THead (Flat Appl) (lift (S O) O x) x0) t4) \to +(\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b0) t1) (THead (Flat Appl) +(lift (S O) O x) x0) t4) \to (\forall (x5: T).(\forall (x6: T).((eq T t4 +(THead (Flat Appl) (lift (S O) O x5) x6)) \to (sn3 c (THead (Flat Appl) x5 +(THead (Bind b0) t1 x6)))))))))) H8 Abst H23) in (let H28 \def (eq_ind B b +(\lambda (b0: B).(\forall (t4: T).((((eq T t1 t4) \to (\forall (P: Prop).P))) +\to ((pr3 c t1 t4) \to (\forall (t0: T).(\forall (v0: T).((sn3 (CHead c (Bind +b0) t4) (THead (Flat Appl) (lift (S O) O v0) t0)) \to (sn3 c (THead (Flat +Appl) v0 (THead (Bind b0) t4 t0)))))))))) H2 Abst H23) in (let H29 \def +(eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H Abst H23) in (let H30 +\def (match (H29 (refl_equal B Abst)) in False return (\lambda (_: +False).(sn3 c (THead (Bind Abbr) x3 x4))) with []) in H30)))))))))) H20)) +H19)) t3 H15)))))))))) H13)) (\lambda (H13: (ex6_6 B T T T T T (\lambda (b0: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) +t1 x0) (THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T +t3 (THead (Bind b0) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) +(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 +y2))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: +T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b0) y2) z1 +z2))))))))).(ex6_6_ind B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 +Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) t1 x0) (THead (Bind +b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind b0) y2 (THead +(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x +u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b0: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 (CHead c (Bind b0) y2) z1 z2))))))) (sn3 c t3) (\lambda (x1: +B).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: +T).(\lambda (x6: T).(\lambda (_: (not (eq B x1 Abst))).(\lambda (H15: (eq T +(THead (Bind b) t1 x0) (THead (Bind x1) x2 x3))).(\lambda (H16: (eq T t3 +(THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))).(\lambda +(H17: (pr2 c x x5)).(\lambda (H18: (pr2 c x2 x6)).(\lambda (H19: (pr2 (CHead +c (Bind x1) x6) x3 x4)).(let H20 \def (eq_ind T t3 (\lambda (t0: T).((eq T +(THead (Flat Appl) x (THead (Bind b) t1 x0)) t0) \to (\forall (P: Prop).P))) +H10 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)) H16) in +(eq_ind_r T (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)) +(\lambda (t0: T).(sn3 c t0)) (let H21 \def (f_equal T B (\lambda (e: +T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | +(TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k in K return +(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow +b])])) (THead (Bind b) t1 x0) (THead (Bind x1) x2 x3) H15) in ((let H22 \def +(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with +[(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ t0 _) +\Rightarrow t0])) (THead (Bind b) t1 x0) (THead (Bind x1) x2 x3) H15) in +((let H23 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: +T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ +t0) \Rightarrow t0])) (THead (Bind b) t1 x0) (THead (Bind x1) x2 x3) H15) in +(\lambda (H24: (eq T t1 x2)).(\lambda (H25: (eq B b x1)).(let H26 \def +(eq_ind_r T x3 (\lambda (t0: T).(pr2 (CHead c (Bind x1) x6) t0 x4)) H19 x0 +H23) in (let H27 \def (eq_ind_r T x2 (\lambda (t0: T).(pr2 c t0 x6)) H18 t1 +H24) in (let H28 \def (eq_ind_r B x1 (\lambda (b0: B).(pr2 (CHead c (Bind b0) +x6) x0 x4)) H26 b H25) in (eq_ind B b (\lambda (b0: B).(sn3 c (THead (Bind +b0) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))) (sn3_pr3_trans c (THead +(Bind b) t1 (THead (Flat Appl) (lift (S O) O x5) x4)) (sn3_bind b c t1 +(sn3_sing c t1 H1) (THead (Flat Appl) (lift (S O) O x5) x4) (let H_x \def +(term_dec x x5) in (let H29 \def H_x in (or_ind (eq T x x5) ((eq T x x5) \to +(\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S +O) O x5) x4)) (\lambda (H30: (eq T x x5)).(let H31 \def (eq_ind_r T x5 +(\lambda (t0: T).(pr2 c x t0)) H17 x H30) in (eq_ind T x (\lambda (t0: +T).(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O t0) x4))) (let +H_x0 \def (term_dec x0 x4) in (let H32 \def H_x0 in (or_ind (eq T x0 x4) ((eq +T x0 x4) \to (\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat +Appl) (lift (S O) O x) x4)) (\lambda (H33: (eq T x0 x4)).(let H34 \def +(eq_ind_r T x4 (\lambda (t0: T).(pr2 (CHead c (Bind b) x6) x0 t0)) H28 x0 +H33) in (eq_ind T x0 (\lambda (t0: T).(sn3 (CHead c (Bind b) t1) (THead (Flat +Appl) (lift (S O) O x) t0))) (sn3_sing (CHead c (Bind b) t1) (THead (Flat +Appl) (lift (S O) O x) x0) H9) x4 H33))) (\lambda (H33: (((eq T x0 x4) \to +(\forall (P: Prop).P)))).(H9 (THead (Flat Appl) (lift (S O) O x) x4) (\lambda +(H34: (eq T (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift +(S O) O x) x4))).(\lambda (P: Prop).(let H35 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | +(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat +Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x) x4) H34) in +(let H36 \def (eq_ind_r T x4 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: +Prop).P0))) H33 x0 H35) in (let H37 \def (eq_ind_r T x4 (\lambda (t0: T).(pr2 +(CHead c (Bind b) x6) x0 t0)) H28 x0 H35) in (H36 (refl_equal T x0) P)))))) +(pr3_pr3_pr3_t c t1 x6 (pr3_pr2 c t1 x6 H27) (THead (Flat Appl) (lift (S O) O +x) x0) (THead (Flat Appl) (lift (S O) O x) x4) (Bind b) (pr3_pr2 (CHead c +(Bind b) x6) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift +(S O) O x) x4) (pr2_thin_dx (CHead c (Bind b) x6) x0 x4 H28 (lift (S O) O x) +Appl))))) H32))) x5 H30))) (\lambda (H30: (((eq T x x5) \to (\forall (P: +Prop).P)))).(H9 (THead (Flat Appl) (lift (S O) O x5) x4) (\lambda (H31: (eq T +(THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x5) +x4))).(\lambda (P: Prop).(let H32 \def (f_equal T T (\lambda (e: T).(match e +in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map +(f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n) +\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with +[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4) +\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in +lref_map) (\lambda (x7: nat).(plus x7 (S O))) O x) | (TLRef _) \Rightarrow +((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match +t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef +(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | +(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) +t4))]) in lref_map) (\lambda (x7: nat).(plus x7 (S O))) O x) | (THead _ t0 _) +\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) +(lift (S O) O x5) x4) H31) in ((let H33 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | +(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat +Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x5) x4) H31) in +(\lambda (H34: (eq T (lift (S O) O x) (lift (S O) O x5))).(let H35 \def +(eq_ind_r T x5 (\lambda (t0: T).((eq T x t0) \to (\forall (P0: Prop).P0))) +H30 x (lift_inj x x5 (S O) O H34)) in (let H36 \def (eq_ind_r T x5 (\lambda +(t0: T).(pr2 c x t0)) H17 x (lift_inj x x5 (S O) O H34)) in (let H37 \def +(eq_ind_r T x4 (\lambda (t0: T).(pr2 (CHead c (Bind b) x6) x0 t0)) H28 x0 +H33) in (H35 (refl_equal T x) P)))))) H32)))) (pr3_pr3_pr3_t c t1 x6 (pr3_pr2 +c t1 x6 H27) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift +(S O) O x5) x4) (Bind b) (pr3_flat (CHead c (Bind b) x6) (lift (S O) O x) +(lift (S O) O x5) (pr3_lift (CHead c (Bind b) x6) c (S O) O (drop_drop (Bind +b) O c c (drop_refl c) x6) x x5 (pr3_pr2 c x x5 H17)) x0 x4 (pr3_pr2 (CHead c +(Bind b) x6) x0 x4 H28) Appl)))) H29)))) (THead (Bind b) x6 (THead (Flat +Appl) (lift (S O) O x5) x4)) (pr3_pr2 c (THead (Bind b) t1 (THead (Flat Appl) +(lift (S O) O x5) x4)) (THead (Bind b) x6 (THead (Flat Appl) (lift (S O) O +x5) x4)) (pr2_head_1 c t1 x6 H27 (Bind b) (THead (Flat Appl) (lift (S O) O +x5) x4)))) x1 H25))))))) H22)) H21)) t3 H16)))))))))))))) H13)) +H12)))))))))))))) y H4))))) H3))))))) u H0))))). +(* COMMENTS +Initial nodes: 9191 +END *) + +theorem sn3_appl_appl: + \forall (v1: T).(\forall (t1: T).(let u1 \def (THead (Flat Appl) v1 t1) in +(\forall (c: C).((sn3 c u1) \to (\forall (v2: T).((sn3 c v2) \to (((\forall +(u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to +(sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c (THead (Flat Appl) v2 +u1))))))))) +\def + \lambda (v1: T).(\lambda (t1: T).(let u1 \def (THead (Flat Appl) v1 t1) in +(\lambda (c: C).(\lambda (H: (sn3 c (THead (Flat Appl) v1 t1))).(insert_eq T +(THead (Flat Appl) v1 t1) (\lambda (t: T).(sn3 c t)) (\lambda (t: T).(\forall +(v2: T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c t u2) \to ((((iso t u2) +\to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to +(sn3 c (THead (Flat Appl) v2 t)))))) (\lambda (y: T).(\lambda (H0: (sn3 c +y)).(unintro T t1 (\lambda (t: T).((eq T y (THead (Flat Appl) v1 t)) \to +(\forall (v2: T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c y u2) \to ((((iso +y u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) +\to (sn3 c (THead (Flat Appl) v2 y))))))) (unintro T v1 (\lambda (t: +T).(\forall (x: T).((eq T y (THead (Flat Appl) t x)) \to (\forall (v2: +T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c y u2) \to ((((iso y u2) \to +(\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c +(THead (Flat Appl) v2 y)))))))) (sn3_ind c (\lambda (t: T).(\forall (x: +T).(\forall (x0: T).((eq T t (THead (Flat Appl) x x0)) \to (\forall (v2: +T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c t u2) \to ((((iso t u2) \to +(\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c +(THead (Flat Appl) v2 t))))))))) (\lambda (t2: T).(\lambda (H1: ((\forall +(t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to +(sn3 c t3)))))).(\lambda (H2: ((\forall (t3: T).((((eq T t2 t3) \to (\forall +(P: Prop).P))) \to ((pr3 c t2 t3) \to (\forall (x: T).(\forall (x0: T).((eq T +t3 (THead (Flat Appl) x x0)) \to (\forall (v2: T).((sn3 c v2) \to (((\forall +(u2: T).((pr3 c t3 u2) \to ((((iso t3 u2) \to (\forall (P: Prop).P))) \to +(sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c (THead (Flat Appl) v2 +t3))))))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: (eq T t2 +(THead (Flat Appl) x x0))).(\lambda (v2: T).(\lambda (H4: (sn3 c +v2)).(sn3_ind c (\lambda (t: T).(((\forall (u2: T).((pr3 c t2 u2) \to ((((iso +t2 u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) t u2)))))) +\to (sn3 c (THead (Flat Appl) t t2)))) (\lambda (t0: T).(\lambda (H5: +((\forall (t3: T).((((eq T t0 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t0 +t3) \to (sn3 c t3)))))).(\lambda (H6: ((\forall (t3: T).((((eq T t0 t3) \to +(\forall (P: Prop).P))) \to ((pr3 c t0 t3) \to (((\forall (u2: T).((pr3 c t2 +u2) \to ((((iso t2 u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat +Appl) t3 u2)))))) \to (sn3 c (THead (Flat Appl) t3 t2)))))))).(\lambda (H7: +((\forall (u2: T).((pr3 c t2 u2) \to ((((iso t2 u2) \to (\forall (P: +Prop).P))) \to (sn3 c (THead (Flat Appl) t0 u2))))))).(let H8 \def (eq_ind T +t2 (\lambda (t: T).(\forall (u2: T).((pr3 c t u2) \to ((((iso t u2) \to +(\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) t0 u2)))))) H7 (THead +(Flat Appl) x x0) H3) in (let H9 \def (eq_ind T t2 (\lambda (t: T).(\forall +(t3: T).((((eq T t0 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t3) \to +(((\forall (u2: T).((pr3 c t u2) \to ((((iso t u2) \to (\forall (P: +Prop).P))) \to (sn3 c (THead (Flat Appl) t3 u2)))))) \to (sn3 c (THead (Flat +Appl) t3 t))))))) H6 (THead (Flat Appl) x x0) H3) in (let H10 \def (eq_ind T +t2 (\lambda (t: T).(\forall (t3: T).((((eq T t t3) \to (\forall (P: +Prop).P))) \to ((pr3 c t t3) \to (\forall (x1: T).(\forall (x2: T).((eq T t3 +(THead (Flat Appl) x1 x2)) \to (\forall (v3: T).((sn3 c v3) \to (((\forall +(u2: T).((pr3 c t3 u2) \to ((((iso t3 u2) \to (\forall (P: Prop).P))) \to +(sn3 c (THead (Flat Appl) v3 u2)))))) \to (sn3 c (THead (Flat Appl) v3 +t3)))))))))))) H2 (THead (Flat Appl) x x0) H3) in (let H11 \def (eq_ind T t2 +(\lambda (t: T).(\forall (t3: T).((((eq T t t3) \to (\forall (P: Prop).P))) +\to ((pr3 c t t3) \to (sn3 c t3))))) H1 (THead (Flat Appl) x x0) H3) in +(eq_ind_r T (THead (Flat Appl) x x0) (\lambda (t: T).(sn3 c (THead (Flat +Appl) t0 t))) (sn3_pr2_intro c (THead (Flat Appl) t0 (THead (Flat Appl) x +x0)) (\lambda (t3: T).(\lambda (H12: (((eq T (THead (Flat Appl) t0 (THead +(Flat Appl) x x0)) t3) \to (\forall (P: Prop).P)))).(\lambda (H13: (pr2 c +(THead (Flat Appl) t0 (THead (Flat Appl) x x0)) t3)).(let H14 \def +(pr2_gen_appl c t0 (THead (Flat Appl) x x0) t3 H13) in (or3_ind (ex3_2 T T +(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))) (\lambda (_: T).(\lambda +(t4: T).(pr2 c (THead (Flat Appl) x x0) t4)))) (ex4_4 T T T T (\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Appl) +x x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda +(u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall +(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4)))))))) (ex6_6 B T T T +T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda +(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq +T (THead (Flat Appl) x x0) (THead (Bind b) y1 z1)))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda +(y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) +z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))))))) (\lambda (_: +B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: +T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) +y2) z1 z2)))))))) (sn3 c t3) (\lambda (H15: (ex3_2 T T (\lambda (u2: +T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c t0 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c +(THead (Flat Appl) x x0) t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda +(t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: +T).(pr2 c t0 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Flat Appl) +x x0) t4))) (sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (H16: (eq T +t3 (THead (Flat Appl) x1 x2))).(\lambda (H17: (pr2 c t0 x1)).(\lambda (H18: +(pr2 c (THead (Flat Appl) x x0) x2)).(let H19 \def (eq_ind T t3 (\lambda (t: +T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) t) \to (\forall (P: +Prop).P))) H12 (THead (Flat Appl) x1 x2) H16) in (eq_ind_r T (THead (Flat +Appl) x1 x2) (\lambda (t: T).(sn3 c t)) (let H20 \def (pr2_gen_appl c x x0 x2 +H18) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead +(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) +(\lambda (_: T).(\lambda (t4: T).(pr2 c x0 t4)))) (ex4_4 T T T T (\lambda +(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x0 (THead +(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t4: T).(eq T x2 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda +(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: +B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4)))))))) (ex6_6 B T T T T T +(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T x0 +(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x2 (THead (Bind +b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda +(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (sn3 c +(THead (Flat Appl) x1 x2)) (\lambda (H21: (ex3_2 T T (\lambda (u2: +T).(\lambda (t4: T).(eq T x2 (THead (Flat Appl) u2 t4)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c x0 +t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead +(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) +(\lambda (_: T).(\lambda (t4: T).(pr2 c x0 t4))) (sn3 c (THead (Flat Appl) x1 +x2)) (\lambda (x3: T).(\lambda (x4: T).(\lambda (H22: (eq T x2 (THead (Flat +Appl) x3 x4))).(\lambda (H23: (pr2 c x x3)).(\lambda (H24: (pr2 c x0 +x4)).(let H25 \def (eq_ind T x2 (\lambda (t: T).((eq T (THead (Flat Appl) t0 +(THead (Flat Appl) x x0)) (THead (Flat Appl) x1 t)) \to (\forall (P: +Prop).P))) H19 (THead (Flat Appl) x3 x4) H22) in (eq_ind_r T (THead (Flat +Appl) x3 x4) (\lambda (t: T).(sn3 c (THead (Flat Appl) x1 t))) (let H_x \def +(term_dec (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4)) in (let H26 +\def H_x in (or_ind (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4)) +((eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4)) \to (\forall (P: +Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Flat Appl) x3 x4))) (\lambda +(H27: (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4))).(let H28 +\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) +with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t _) +\Rightarrow t])) (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4) H27) in +((let H29 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: +T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ +t) \Rightarrow t])) (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4) H27) +in (\lambda (H30: (eq T x x3)).(let H31 \def (eq_ind_r T x4 (\lambda (t: +T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) (THead (Flat Appl) +x1 (THead (Flat Appl) x3 t))) \to (\forall (P: Prop).P))) H25 x0 H29) in (let +H32 \def (eq_ind_r T x4 (\lambda (t: T).(pr2 c x0 t)) H24 x0 H29) in (eq_ind +T x0 (\lambda (t: T).(sn3 c (THead (Flat Appl) x1 (THead (Flat Appl) x3 t)))) +(let H33 \def (eq_ind_r T x3 (\lambda (t: T).((eq T (THead (Flat Appl) t0 +(THead (Flat Appl) x x0)) (THead (Flat Appl) x1 (THead (Flat Appl) t x0))) +\to (\forall (P: Prop).P))) H31 x H30) in (let H34 \def (eq_ind_r T x3 +(\lambda (t: T).(pr2 c x t)) H23 x H30) in (eq_ind T x (\lambda (t: T).(sn3 c +(THead (Flat Appl) x1 (THead (Flat Appl) t x0)))) (let H_x0 \def (term_dec t0 +x1) in (let H35 \def H_x0 in (or_ind (eq T t0 x1) ((eq T t0 x1) \to (\forall +(P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Flat Appl) x x0))) +(\lambda (H36: (eq T t0 x1)).(let H37 \def (eq_ind_r T x1 (\lambda (t: +T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) (THead (Flat Appl) +t (THead (Flat Appl) x x0))) \to (\forall (P: Prop).P))) H33 t0 H36) in (let +H38 \def (eq_ind_r T x1 (\lambda (t: T).(pr2 c t0 t)) H17 t0 H36) in (eq_ind +T t0 (\lambda (t: T).(sn3 c (THead (Flat Appl) t (THead (Flat Appl) x x0)))) +(H37 (refl_equal T (THead (Flat Appl) t0 (THead (Flat Appl) x x0))) (sn3 c +(THead (Flat Appl) t0 (THead (Flat Appl) x x0)))) x1 H36)))) (\lambda (H36: +(((eq T t0 x1) \to (\forall (P: Prop).P)))).(H9 x1 H36 (pr3_pr2 c t0 x1 H17) +(\lambda (u2: T).(\lambda (H37: (pr3 c (THead (Flat Appl) x x0) u2)).(\lambda +(H38: (((iso (THead (Flat Appl) x x0) u2) \to (\forall (P: +Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) t0 u2) (H8 u2 H37 H38) (THead +(Flat Appl) x1 u2) (pr3_pr2 c (THead (Flat Appl) t0 u2) (THead (Flat Appl) x1 +u2) (pr2_head_1 c t0 x1 H17 (Flat Appl) u2)))))))) H35))) x3 H30))) x4 +H29))))) H28))) (\lambda (H27: (((eq T (THead (Flat Appl) x x0) (THead (Flat +Appl) x3 x4)) \to (\forall (P: Prop).P)))).(H10 (THead (Flat Appl) x3 x4) H27 +(pr3_flat c x x3 (pr3_pr2 c x x3 H23) x0 x4 (pr3_pr2 c x0 x4 H24) Appl) x3 x4 +(refl_equal T (THead (Flat Appl) x3 x4)) x1 (sn3_pr3_trans c t0 (sn3_sing c +t0 H5) x1 (pr3_pr2 c t0 x1 H17)) (\lambda (u2: T).(\lambda (H28: (pr3 c +(THead (Flat Appl) x3 x4) u2)).(\lambda (H29: (((iso (THead (Flat Appl) x3 +x4) u2) \to (\forall (P: Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) t0 +u2) (H8 u2 (pr3_sing c (THead (Flat Appl) x x4) (THead (Flat Appl) x x0) +(pr2_thin_dx c x0 x4 H24 x Appl) u2 (pr3_sing c (THead (Flat Appl) x3 x4) +(THead (Flat Appl) x x4) (pr2_head_1 c x x3 H23 (Flat Appl) x4) u2 H28)) +(\lambda (H30: (iso (THead (Flat Appl) x x0) u2)).(\lambda (P: Prop).(H29 +(iso_trans (THead (Flat Appl) x3 x4) (THead (Flat Appl) x x0) (iso_head x3 x +x4 x0 (Flat Appl)) u2 H30) P)))) (THead (Flat Appl) x1 u2) (pr3_pr2 c (THead +(Flat Appl) t0 u2) (THead (Flat Appl) x1 u2) (pr2_head_1 c t0 x1 H17 (Flat +Appl) u2)))))))) H26))) x2 H22))))))) H21)) (\lambda (H21: (ex4_4 T T T T +(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x0 +(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t4: T).(eq T x2 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda +(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: +B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4))))))))).(ex4_4_ind T T T +T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x0 +(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t4: T).(eq T x2 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda +(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: +B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4))))))) (sn3 c (THead (Flat +Appl) x1 x2)) (\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda +(x6: T).(\lambda (H22: (eq T x0 (THead (Bind Abst) x3 x4))).(\lambda (H23: +(eq T x2 (THead (Bind Abbr) x5 x6))).(\lambda (H24: (pr2 c x x5)).(\lambda +(H25: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x4 +x6))))).(let H26 \def (eq_ind T x2 (\lambda (t: T).((eq T (THead (Flat Appl) +t0 (THead (Flat Appl) x x0)) (THead (Flat Appl) x1 t)) \to (\forall (P: +Prop).P))) H19 (THead (Bind Abbr) x5 x6) H23) in (eq_ind_r T (THead (Bind +Abbr) x5 x6) (\lambda (t: T).(sn3 c (THead (Flat Appl) x1 t))) (let H27 \def +(eq_ind T x0 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) +x t)) (THead (Flat Appl) x1 (THead (Bind Abbr) x5 x6))) \to (\forall (P: +Prop).P))) H26 (THead (Bind Abst) x3 x4) H22) in (let H28 \def (eq_ind T x0 +(\lambda (t: T).(\forall (t4: T).((((eq T (THead (Flat Appl) x t) t4) \to +(\forall (P: Prop).P))) \to ((pr3 c (THead (Flat Appl) x t) t4) \to (sn3 c +t4))))) H11 (THead (Bind Abst) x3 x4) H22) in (let H29 \def (eq_ind T x0 +(\lambda (t: T).(\forall (t4: T).((((eq T (THead (Flat Appl) x t) t4) \to +(\forall (P: Prop).P))) \to ((pr3 c (THead (Flat Appl) x t) t4) \to (\forall +(x7: T).(\forall (x8: T).((eq T t4 (THead (Flat Appl) x7 x8)) \to (\forall +(v3: T).((sn3 c v3) \to (((\forall (u2: T).((pr3 c t4 u2) \to ((((iso t4 u2) +\to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v3 u2)))))) \to +(sn3 c (THead (Flat Appl) v3 t4)))))))))))) H10 (THead (Bind Abst) x3 x4) +H22) in (let H30 \def (eq_ind T x0 (\lambda (t: T).(\forall (u2: T).((pr3 c +(THead (Flat Appl) x t) u2) \to ((((iso (THead (Flat Appl) x t) u2) \to +(\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) t0 u2)))))) H8 (THead +(Bind Abst) x3 x4) H22) in (let H31 \def (eq_ind T x0 (\lambda (t: +T).(\forall (t4: T).((((eq T t0 t4) \to (\forall (P: Prop).P))) \to ((pr3 c +t0 t4) \to (((\forall (u2: T).((pr3 c (THead (Flat Appl) x t) u2) \to ((((iso +(THead (Flat Appl) x t) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead +(Flat Appl) t4 u2)))))) \to (sn3 c (THead (Flat Appl) t4 (THead (Flat Appl) x +t)))))))) H9 (THead (Bind Abst) x3 x4) H22) in (sn3_pr3_trans c (THead (Flat +Appl) t0 (THead (Bind Abbr) x5 x6)) (H30 (THead (Bind Abbr) x5 x6) (pr3_sing +c (THead (Bind Abbr) x x4) (THead (Flat Appl) x (THead (Bind Abst) x3 x4)) +(pr2_free c (THead (Flat Appl) x (THead (Bind Abst) x3 x4)) (THead (Bind +Abbr) x x4) (pr0_beta x3 x x (pr0_refl x) x4 x4 (pr0_refl x4))) (THead (Bind +Abbr) x5 x6) (pr3_head_12 c x x5 (pr3_pr2 c x x5 H24) (Bind Abbr) x4 x6 +(pr3_pr2 (CHead c (Bind Abbr) x5) x4 x6 (H25 Abbr x5)))) (\lambda (H32: (iso +(THead (Flat Appl) x (THead (Bind Abst) x3 x4)) (THead (Bind Abbr) x5 +x6))).(\lambda (P: Prop).(let H33 \def (match H32 in iso return (\lambda (t: +T).(\lambda (t4: T).(\lambda (_: (iso t t4)).((eq T t (THead (Flat Appl) x +(THead (Bind Abst) x3 x4))) \to ((eq T t4 (THead (Bind Abbr) x5 x6)) \to +P))))) with [(iso_sort n1 n2) \Rightarrow (\lambda (H33: (eq T (TSort n1) +(THead (Flat Appl) x (THead (Bind Abst) x3 x4)))).(\lambda (H34: (eq T (TSort +n2) (THead (Bind Abbr) x5 x6))).((let H35 \def (eq_ind T (TSort n1) (\lambda +(e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow +True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I +(THead (Flat Appl) x (THead (Bind Abst) x3 x4)) H33) in (False_ind ((eq T +(TSort n2) (THead (Bind Abbr) x5 x6)) \to P) H35)) H34))) | (iso_lref i1 i2) +\Rightarrow (\lambda (H33: (eq T (TLRef i1) (THead (Flat Appl) x (THead (Bind +Abst) x3 x4)))).(\lambda (H34: (eq T (TLRef i2) (THead (Bind Abbr) x5 +x6))).((let H35 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e in T +return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) x +(THead (Bind Abst) x3 x4)) H33) in (False_ind ((eq T (TLRef i2) (THead (Bind +Abbr) x5 x6)) \to P) H35)) H34))) | (iso_head v4 v5 t4 t5 k) \Rightarrow +(\lambda (H33: (eq T (THead k v4 t4) (THead (Flat Appl) x (THead (Bind Abst) +x3 x4)))).(\lambda (H34: (eq T (THead k v5 t5) (THead (Bind Abbr) x5 +x6))).((let H35 \def (f_equal T T (\lambda (e: T).(match e in T return +(\lambda (_: T).T) with [(TSort _) \Rightarrow t4 | (TLRef _) \Rightarrow t4 +| (THead _ _ t) \Rightarrow t])) (THead k v4 t4) (THead (Flat Appl) x (THead +(Bind Abst) x3 x4)) H33) in ((let H36 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v4 | +(TLRef _) \Rightarrow v4 | (THead _ t _) \Rightarrow t])) (THead k v4 t4) +(THead (Flat Appl) x (THead (Bind Abst) x3 x4)) H33) in ((let H37 \def +(f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with +[(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) +\Rightarrow k0])) (THead k v4 t4) (THead (Flat Appl) x (THead (Bind Abst) x3 +x4)) H33) in (eq_ind K (Flat Appl) (\lambda (k0: K).((eq T v4 x) \to ((eq T +t4 (THead (Bind Abst) x3 x4)) \to ((eq T (THead k0 v5 t5) (THead (Bind Abbr) +x5 x6)) \to P)))) (\lambda (H38: (eq T v4 x)).(eq_ind T x (\lambda (_: +T).((eq T t4 (THead (Bind Abst) x3 x4)) \to ((eq T (THead (Flat Appl) v5 t5) +(THead (Bind Abbr) x5 x6)) \to P))) (\lambda (H39: (eq T t4 (THead (Bind +Abst) x3 x4))).(eq_ind T (THead (Bind Abst) x3 x4) (\lambda (_: T).((eq T +(THead (Flat Appl) v5 t5) (THead (Bind Abbr) x5 x6)) \to P)) (\lambda (H40: +(eq T (THead (Flat Appl) v5 t5) (THead (Bind Abbr) x5 x6))).(let H41 \def +(eq_ind T (THead (Flat Appl) v5 t5) (\lambda (e: T).(match e in T return +(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda +(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow +True])])) I (THead (Bind Abbr) x5 x6) H40) in (False_ind P H41))) t4 (sym_eq +T t4 (THead (Bind Abst) x3 x4) H39))) v4 (sym_eq T v4 x H38))) k (sym_eq K k +(Flat Appl) H37))) H36)) H35)) H34)))]) in (H33 (refl_equal T (THead (Flat +Appl) x (THead (Bind Abst) x3 x4))) (refl_equal T (THead (Bind Abbr) x5 +x6))))))) (THead (Flat Appl) x1 (THead (Bind Abbr) x5 x6)) (pr3_pr2 c (THead +(Flat Appl) t0 (THead (Bind Abbr) x5 x6)) (THead (Flat Appl) x1 (THead (Bind +Abbr) x5 x6)) (pr2_head_1 c t0 x1 H17 (Flat Appl) (THead (Bind Abbr) x5 +x6))))))))) x2 H23)))))))))) H21)) (\lambda (H21: (ex6_6 B T T T T T (\lambda +(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T x0 +(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x2 (THead (Bind +b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda +(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind +B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: +B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(eq T x0 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T +x2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) +(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 +y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: +T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) +(sn3 c (THead (Flat Appl) x1 x2)) (\lambda (x3: B).(\lambda (x4: T).(\lambda +(x5: T).(\lambda (x6: T).(\lambda (x7: T).(\lambda (x8: T).(\lambda (H22: +(not (eq B x3 Abst))).(\lambda (H23: (eq T x0 (THead (Bind x3) x4 +x5))).(\lambda (H24: (eq T x2 (THead (Bind x3) x8 (THead (Flat Appl) (lift (S +O) O x7) x6)))).(\lambda (H25: (pr2 c x x7)).(\lambda (H26: (pr2 c x4 +x8)).(\lambda (H27: (pr2 (CHead c (Bind x3) x8) x5 x6)).(let H28 \def (eq_ind +T x2 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) +(THead (Flat Appl) x1 t)) \to (\forall (P: Prop).P))) H19 (THead (Bind x3) x8 +(THead (Flat Appl) (lift (S O) O x7) x6)) H24) in (eq_ind_r T (THead (Bind +x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)) (\lambda (t: T).(sn3 c +(THead (Flat Appl) x1 t))) (let H29 \def (eq_ind T x0 (\lambda (t: T).((eq T +(THead (Flat Appl) t0 (THead (Flat Appl) x t)) (THead (Flat Appl) x1 (THead +(Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))) \to (\forall (P: +Prop).P))) H28 (THead (Bind x3) x4 x5) H23) in (let H30 \def (eq_ind T x0 +(\lambda (t: T).(\forall (t4: T).((((eq T (THead (Flat Appl) x t) t4) \to +(\forall (P: Prop).P))) \to ((pr3 c (THead (Flat Appl) x t) t4) \to (sn3 c +t4))))) H11 (THead (Bind x3) x4 x5) H23) in (let H31 \def (eq_ind T x0 +(\lambda (t: T).(\forall (t4: T).((((eq T (THead (Flat Appl) x t) t4) \to +(\forall (P: Prop).P))) \to ((pr3 c (THead (Flat Appl) x t) t4) \to (\forall +(x9: T).(\forall (x10: T).((eq T t4 (THead (Flat Appl) x9 x10)) \to (\forall +(v3: T).((sn3 c v3) \to (((\forall (u2: T).((pr3 c t4 u2) \to ((((iso t4 u2) +\to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v3 u2)))))) \to +(sn3 c (THead (Flat Appl) v3 t4)))))))))))) H10 (THead (Bind x3) x4 x5) H23) +in (let H32 \def (eq_ind T x0 (\lambda (t: T).(\forall (u2: T).((pr3 c (THead +(Flat Appl) x t) u2) \to ((((iso (THead (Flat Appl) x t) u2) \to (\forall (P: +Prop).P))) \to (sn3 c (THead (Flat Appl) t0 u2)))))) H8 (THead (Bind x3) x4 +x5) H23) in (let H33 \def (eq_ind T x0 (\lambda (t: T).(\forall (t4: +T).((((eq T t0 t4) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t4) \to +(((\forall (u2: T).((pr3 c (THead (Flat Appl) x t) u2) \to ((((iso (THead +(Flat Appl) x t) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat +Appl) t4 u2)))))) \to (sn3 c (THead (Flat Appl) t4 (THead (Flat Appl) x +t)))))))) H9 (THead (Bind x3) x4 x5) H23) in (sn3_pr3_trans c (THead (Flat +Appl) t0 (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) (H32 +(THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)) (pr3_sing c +(THead (Bind x3) x4 (THead (Flat Appl) (lift (S O) O x) x5)) (THead (Flat +Appl) x (THead (Bind x3) x4 x5)) (pr2_free c (THead (Flat Appl) x (THead +(Bind x3) x4 x5)) (THead (Bind x3) x4 (THead (Flat Appl) (lift (S O) O x) +x5)) (pr0_upsilon x3 H22 x x (pr0_refl x) x4 x4 (pr0_refl x4) x5 x5 (pr0_refl +x5))) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)) +(pr3_head_12 c x4 x8 (pr3_pr2 c x4 x8 H26) (Bind x3) (THead (Flat Appl) (lift +(S O) O x) x5) (THead (Flat Appl) (lift (S O) O x7) x6) (pr3_head_12 (CHead c +(Bind x3) x8) (lift (S O) O x) (lift (S O) O x7) (pr3_lift (CHead c (Bind x3) +x8) c (S O) O (drop_drop (Bind x3) O c c (drop_refl c) x8) x x7 (pr3_pr2 c x +x7 H25)) (Flat Appl) x5 x6 (pr3_pr2 (CHead (CHead c (Bind x3) x8) (Flat Appl) +(lift (S O) O x7)) x5 x6 (pr2_cflat (CHead c (Bind x3) x8) x5 x6 H27 Appl +(lift (S O) O x7)))))) (\lambda (H34: (iso (THead (Flat Appl) x (THead (Bind +x3) x4 x5)) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) +x6)))).(\lambda (P: Prop).(let H35 \def (match H34 in iso return (\lambda (t: +T).(\lambda (t4: T).(\lambda (_: (iso t t4)).((eq T t (THead (Flat Appl) x +(THead (Bind x3) x4 x5))) \to ((eq T t4 (THead (Bind x3) x8 (THead (Flat +Appl) (lift (S O) O x7) x6))) \to P))))) with [(iso_sort n1 n2) \Rightarrow +(\lambda (H35: (eq T (TSort n1) (THead (Flat Appl) x (THead (Bind x3) x4 +x5)))).(\lambda (H36: (eq T (TSort n2) (THead (Bind x3) x8 (THead (Flat Appl) +(lift (S O) O x7) x6)))).((let H37 \def (eq_ind T (TSort n1) (\lambda (e: +T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow +True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I +(THead (Flat Appl) x (THead (Bind x3) x4 x5)) H35) in (False_ind ((eq T +(TSort n2) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to +P) H37)) H36))) | (iso_lref i1 i2) \Rightarrow (\lambda (H35: (eq T (TLRef +i1) (THead (Flat Appl) x (THead (Bind x3) x4 x5)))).(\lambda (H36: (eq T +(TLRef i2) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) +x6)))).((let H37 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e in T +return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) x +(THead (Bind x3) x4 x5)) H35) in (False_ind ((eq T (TLRef i2) (THead (Bind +x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to P) H37)) H36))) | +(iso_head v4 v5 t4 t5 k) \Rightarrow (\lambda (H35: (eq T (THead k v4 t4) +(THead (Flat Appl) x (THead (Bind x3) x4 x5)))).(\lambda (H36: (eq T (THead k +v5 t5) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))).((let +H37 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) +with [(TSort _) \Rightarrow t4 | (TLRef _) \Rightarrow t4 | (THead _ _ t) +\Rightarrow t])) (THead k v4 t4) (THead (Flat Appl) x (THead (Bind x3) x4 +x5)) H35) in ((let H38 \def (f_equal T T (\lambda (e: T).(match e in T return +(\lambda (_: T).T) with [(TSort _) \Rightarrow v4 | (TLRef _) \Rightarrow v4 +| (THead _ t _) \Rightarrow t])) (THead k v4 t4) (THead (Flat Appl) x (THead +(Bind x3) x4 x5)) H35) in ((let H39 \def (f_equal T K (\lambda (e: T).(match +e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) +\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k v4 t4) (THead (Flat +Appl) x (THead (Bind x3) x4 x5)) H35) in (eq_ind K (Flat Appl) (\lambda (k0: +K).((eq T v4 x) \to ((eq T t4 (THead (Bind x3) x4 x5)) \to ((eq T (THead k0 +v5 t5) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to +P)))) (\lambda (H40: (eq T v4 x)).(eq_ind T x (\lambda (_: T).((eq T t4 +(THead (Bind x3) x4 x5)) \to ((eq T (THead (Flat Appl) v5 t5) (THead (Bind +x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to P))) (\lambda (H41: (eq +T t4 (THead (Bind x3) x4 x5))).(eq_ind T (THead (Bind x3) x4 x5) (\lambda (_: +T).((eq T (THead (Flat Appl) v5 t5) (THead (Bind x3) x8 (THead (Flat Appl) +(lift (S O) O x7) x6))) \to P)) (\lambda (H42: (eq T (THead (Flat Appl) v5 +t5) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))).(let H43 +\def (eq_ind T (THead (Flat Appl) v5 t5) (\lambda (e: T).(match e in T return +(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda +(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow +True])])) I (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)) +H42) in (False_ind P H43))) t4 (sym_eq T t4 (THead (Bind x3) x4 x5) H41))) v4 +(sym_eq T v4 x H40))) k (sym_eq K k (Flat Appl) H39))) H38)) H37)) H36)))]) +in (H35 (refl_equal T (THead (Flat Appl) x (THead (Bind x3) x4 x5))) +(refl_equal T (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) +x6)))))))) (THead (Flat Appl) x1 (THead (Bind x3) x8 (THead (Flat Appl) (lift +(S O) O x7) x6))) (pr3_pr2 c (THead (Flat Appl) t0 (THead (Bind x3) x8 (THead +(Flat Appl) (lift (S O) O x7) x6))) (THead (Flat Appl) x1 (THead (Bind x3) x8 +(THead (Flat Appl) (lift (S O) O x7) x6))) (pr2_head_1 c t0 x1 H17 (Flat +Appl) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))))))))) +x2 H24)))))))))))))) H21)) H20)) t3 H16))))))) H15)) (\lambda (H15: (ex4_4 T +T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T +(THead (Flat Appl) x x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind +Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr2 c t0 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t4: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) +z1 t4))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda +(_: T).(\lambda (_: T).(eq T (THead (Flat Appl) x x0) (THead (Bind Abst) y1 +z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: +T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))))) (\lambda (_: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall +(u: T).(pr2 (CHead c (Bind b) u) z1 t4))))))) (sn3 c t3) (\lambda (x1: +T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H16: (eq T +(THead (Flat Appl) x x0) (THead (Bind Abst) x1 x2))).(\lambda (H17: (eq T t3 +(THead (Bind Abbr) x3 x4))).(\lambda (_: (pr2 c t0 x3)).(\lambda (_: +((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x2 x4))))).(let +H20 \def (eq_ind T t3 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead +(Flat Appl) x x0)) t) \to (\forall (P: Prop).P))) H12 (THead (Bind Abbr) x3 +x4) H17) in (eq_ind_r T (THead (Bind Abbr) x3 x4) (\lambda (t: T).(sn3 c t)) +(let H21 \def (eq_ind T (THead (Flat Appl) x x0) (\lambda (ee: T).(match ee +in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef +_) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return +(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow +True])])) I (THead (Bind Abst) x1 x2) H16) in (False_ind (sn3 c (THead (Bind +Abbr) x3 x4)) H21)) t3 H17)))))))))) H15)) (\lambda (H15: (ex6_6 B T T T T T +(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T +(THead (Flat Appl) x x0) (THead (Bind b) y1 z1)))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda +(y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) +z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))))))) (\lambda (_: +B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: +T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) +y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B +b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Appl) x x0) (THead +(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind +b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr2 c t0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda +(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) (sn3 c t3) +(\lambda (x1: B).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda +(x5: T).(\lambda (x6: T).(\lambda (_: (not (eq B x1 Abst))).(\lambda (H17: +(eq T (THead (Flat Appl) x x0) (THead (Bind x1) x2 x3))).(\lambda (H18: (eq T +t3 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))).(\lambda +(_: (pr2 c t0 x5)).(\lambda (_: (pr2 c x2 x6)).(\lambda (_: (pr2 (CHead c +(Bind x1) x6) x3 x4)).(let H22 \def (eq_ind T t3 (\lambda (t: T).((eq T +(THead (Flat Appl) t0 (THead (Flat Appl) x x0)) t) \to (\forall (P: +Prop).P))) H12 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)) +H18) in (eq_ind_r T (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) +x4)) (\lambda (t: T).(sn3 c t)) (let H23 \def (eq_ind T (THead (Flat Appl) x +x0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort +_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) +\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) +\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind x1) x2 x3) +H17) in (False_ind (sn3 c (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) +O x5) x4))) H23)) t3 H18)))))))))))))) H15)) H14)))))) t2 H3))))))))) v2 +H4))))))))) y H0))))) H))))). +(* COMMENTS +Initial nodes: 9317 +END *) + +theorem sn3_appl_beta: + \forall (c: C).(\forall (u: T).(\forall (v: T).(\forall (t: T).((sn3 c +(THead (Flat Appl) u (THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w) +\to (sn3 c (THead (Flat Appl) u (THead (Flat Appl) v (THead (Bind Abst) w +t)))))))))) +\def + \lambda (c: C).(\lambda (u: T).(\lambda (v: T).(\lambda (t: T).(\lambda (H: +(sn3 c (THead (Flat Appl) u (THead (Bind Abbr) v t)))).(\lambda (w: +T).(\lambda (H0: (sn3 c w)).(let H_x \def (sn3_gen_flat Appl c u (THead (Bind +Abbr) v t) H) in (let H1 \def H_x in (land_ind (sn3 c u) (sn3 c (THead (Bind +Abbr) v t)) (sn3 c (THead (Flat Appl) u (THead (Flat Appl) v (THead (Bind +Abst) w t)))) (\lambda (H2: (sn3 c u)).(\lambda (H3: (sn3 c (THead (Bind +Abbr) v t))).(sn3_appl_appl v (THead (Bind Abst) w t) c (sn3_beta c v t H3 w +H0) u H2 (\lambda (u2: T).(\lambda (H4: (pr3 c (THead (Flat Appl) v (THead +(Bind Abst) w t)) u2)).(\lambda (H5: (((iso (THead (Flat Appl) v (THead (Bind +Abst) w t)) u2) \to (\forall (P: Prop).P)))).(sn3_pr3_trans c (THead (Flat +Appl) u (THead (Bind Abbr) v t)) H (THead (Flat Appl) u u2) (pr3_thin_dx c +(THead (Bind Abbr) v t) u2 (pr3_iso_beta v w t c u2 H4 H5) u Appl)))))))) +H1))))))))). +(* COMMENTS +Initial nodes: 289 +END *) + +theorem sn3_appl_appls: + \forall (v1: T).(\forall (t1: T).(\forall (vs: TList).(let u1 \def (THeads +(Flat Appl) (TCons v1 vs) t1) in (\forall (c: C).((sn3 c u1) \to (\forall +(v2: T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) +\to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to +(sn3 c (THead (Flat Appl) v2 u1)))))))))) +\def + \lambda (v1: T).(\lambda (t1: T).(\lambda (vs: TList).(let u1 \def (THeads +(Flat Appl) (TCons v1 vs) t1) in (\lambda (c: C).(\lambda (H: (sn3 c (THead +(Flat Appl) v1 (THeads (Flat Appl) vs t1)))).(\lambda (v2: T).(\lambda (H0: +(sn3 c v2)).(\lambda (H1: ((\forall (u2: T).((pr3 c (THead (Flat Appl) v1 +(THeads (Flat Appl) vs t1)) u2) \to ((((iso (THead (Flat Appl) v1 (THeads +(Flat Appl) vs t1)) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat +Appl) v2 u2))))))).(sn3_appl_appl v1 (THeads (Flat Appl) vs t1) c H v2 H0 +H1))))))))). +(* COMMENTS +Initial nodes: 141 +END *) + +theorem sn3_appls_lref: + \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (us: +TList).((sns3 c us) \to (sn3 c (THeads (Flat Appl) us (TLRef i))))))) +\def + \lambda (c: C).(\lambda (i: nat).(\lambda (H: (nf2 c (TLRef i))).(\lambda +(us: TList).(TList_ind (\lambda (t: TList).((sns3 c t) \to (sn3 c (THeads +(Flat Appl) t (TLRef i))))) (\lambda (_: True).(sn3_nf2 c (TLRef i) H)) +(\lambda (t: T).(\lambda (t0: TList).(TList_ind (\lambda (t1: TList).((((sns3 +c t1) \to (sn3 c (THeads (Flat Appl) t1 (TLRef i))))) \to ((land (sn3 c t) +(sns3 c t1)) \to (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t1 (TLRef +i))))))) (\lambda (_: (((sns3 c TNil) \to (sn3 c (THeads (Flat Appl) TNil +(TLRef i)))))).(\lambda (H1: (land (sn3 c t) (sns3 c TNil))).(let H2 \def H1 +in (land_ind (sn3 c t) True (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) +TNil (TLRef i)))) (\lambda (H3: (sn3 c t)).(\lambda (_: True).(sn3_appl_lref +c i H t H3))) H2)))) (\lambda (t1: T).(\lambda (t2: TList).(\lambda (_: +(((((sns3 c t2) \to (sn3 c (THeads (Flat Appl) t2 (TLRef i))))) \to ((land +(sn3 c t) (sns3 c t2)) \to (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t2 +(TLRef i)))))))).(\lambda (H1: (((sns3 c (TCons t1 t2)) \to (sn3 c (THeads +(Flat Appl) (TCons t1 t2) (TLRef i)))))).(\lambda (H2: (land (sn3 c t) (sns3 +c (TCons t1 t2)))).(let H3 \def H2 in (land_ind (sn3 c t) (land (sn3 c t1) +(sns3 c t2)) (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) (TCons t1 t2) +(TLRef i)))) (\lambda (H4: (sn3 c t)).(\lambda (H5: (land (sn3 c t1) (sns3 c +t2))).(land_ind (sn3 c t1) (sns3 c t2) (sn3 c (THead (Flat Appl) t (THeads +(Flat Appl) (TCons t1 t2) (TLRef i)))) (\lambda (H6: (sn3 c t1)).(\lambda +(H7: (sns3 c t2)).(sn3_appl_appls t1 (TLRef i) t2 c (H1 (conj (sn3 c t1) +(sns3 c t2) H6 H7)) t H4 (\lambda (u2: T).(\lambda (H8: (pr3 c (THeads (Flat +Appl) (TCons t1 t2) (TLRef i)) u2)).(\lambda (H9: (((iso (THeads (Flat Appl) +(TCons t1 t2) (TLRef i)) u2) \to (\forall (P: Prop).P)))).(H9 +(nf2_iso_appls_lref c i H (TCons t1 t2) u2 H8) (sn3 c (THead (Flat Appl) t +u2))))))))) H5))) H3))))))) t0))) us)))). +(* COMMENTS +Initial nodes: 577 +END *) + +theorem sn3_appls_cast: + \forall (c: C).(\forall (vs: TList).(\forall (u: T).((sn3 c (THeads (Flat +Appl) vs u)) \to (\forall (t: T).((sn3 c (THeads (Flat Appl) vs t)) \to (sn3 +c (THeads (Flat Appl) vs (THead (Flat Cast) u t)))))))) +\def + \lambda (c: C).(\lambda (vs: TList).(TList_ind (\lambda (t: TList).(\forall +(u: T).((sn3 c (THeads (Flat Appl) t u)) \to (\forall (t0: T).((sn3 c (THeads +(Flat Appl) t t0)) \to (sn3 c (THeads (Flat Appl) t (THead (Flat Cast) u +t0)))))))) (\lambda (u: T).(\lambda (H: (sn3 c u)).(\lambda (t: T).(\lambda +(H0: (sn3 c t)).(sn3_cast c u H t H0))))) (\lambda (t: T).(\lambda (t0: +TList).(TList_ind (\lambda (t1: TList).(((\forall (u: T).((sn3 c (THeads +(Flat Appl) t1 u)) \to (\forall (t2: T).((sn3 c (THeads (Flat Appl) t1 t2)) +\to (sn3 c (THeads (Flat Appl) t1 (THead (Flat Cast) u t2)))))))) \to +(\forall (u: T).((sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t1 u))) \to +(\forall (t2: T).((sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t1 t2))) +\to (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t1 (THead (Flat Cast) u +t2)))))))))) (\lambda (_: ((\forall (u: T).((sn3 c (THeads (Flat Appl) TNil +u)) \to (\forall (t1: T).((sn3 c (THeads (Flat Appl) TNil t1)) \to (sn3 c +(THeads (Flat Appl) TNil (THead (Flat Cast) u t1))))))))).(\lambda (u: +T).(\lambda (H0: (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) TNil +u)))).(\lambda (t1: T).(\lambda (H1: (sn3 c (THead (Flat Appl) t (THeads +(Flat Appl) TNil t1)))).(sn3_appl_cast c t u H0 t1 H1)))))) (\lambda (t1: +T).(\lambda (t2: TList).(\lambda (_: ((((\forall (u: T).((sn3 c (THeads (Flat +Appl) t2 u)) \to (\forall (t3: T).((sn3 c (THeads (Flat Appl) t2 t3)) \to +(sn3 c (THeads (Flat Appl) t2 (THead (Flat Cast) u t3)))))))) \to (\forall +(u: T).((sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t2 u))) \to (\forall +(t3: T).((sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t2 t3))) \to (sn3 c +(THead (Flat Appl) t (THeads (Flat Appl) t2 (THead (Flat Cast) u +t3))))))))))).(\lambda (H0: ((\forall (u: T).((sn3 c (THeads (Flat Appl) +(TCons t1 t2) u)) \to (\forall (t3: T).((sn3 c (THeads (Flat Appl) (TCons t1 +t2) t3)) \to (sn3 c (THeads (Flat Appl) (TCons t1 t2) (THead (Flat Cast) u +t3))))))))).(\lambda (u: T).(\lambda (H1: (sn3 c (THead (Flat Appl) t (THeads +(Flat Appl) (TCons t1 t2) u)))).(\lambda (t3: T).(\lambda (H2: (sn3 c (THead +(Flat Appl) t (THeads (Flat Appl) (TCons t1 t2) t3)))).(let H_x \def +(sn3_gen_flat Appl c t (THeads (Flat Appl) (TCons t1 t2) t3) H2) in (let H3 +\def H_x in (land_ind (sn3 c t) (sn3 c (THead (Flat Appl) t1 (THeads (Flat +Appl) t2 t3))) (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) (TCons t1 t2) +(THead (Flat Cast) u t3)))) (\lambda (_: (sn3 c t)).(\lambda (H5: (sn3 c +(THead (Flat Appl) t1 (THeads (Flat Appl) t2 t3)))).(let H6 \def H5 in (let +H_x0 \def (sn3_gen_flat Appl c t (THeads (Flat Appl) (TCons t1 t2) u) H1) in +(let H7 \def H_x0 in (land_ind (sn3 c t) (sn3 c (THead (Flat Appl) t1 (THeads +(Flat Appl) t2 u))) (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) (TCons t1 +t2) (THead (Flat Cast) u t3)))) (\lambda (H8: (sn3 c t)).(\lambda (H9: (sn3 c +(THead (Flat Appl) t1 (THeads (Flat Appl) t2 u)))).(let H10 \def H9 in +(sn3_appl_appls t1 (THead (Flat Cast) u t3) t2 c (H0 u H10 t3 H6) t H8 +(\lambda (u2: T).(\lambda (H11: (pr3 c (THeads (Flat Appl) (TCons t1 t2) +(THead (Flat Cast) u t3)) u2)).(\lambda (H12: (((iso (THeads (Flat Appl) +(TCons t1 t2) (THead (Flat Cast) u t3)) u2) \to (\forall (P: +Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) t (THeads (Flat Appl) (TCons +t1 t2) t3)) H2 (THead (Flat Appl) t u2) (pr3_thin_dx c (THeads (Flat Appl) +(TCons t1 t2) t3) u2 (pr3_iso_appls_cast c u t3 (TCons t1 t2) u2 H11 H12) t +Appl))))))))) H7)))))) H3))))))))))) t0))) vs)). +(* COMMENTS +Initial nodes: 1025 +END *) + +theorem sn3_appls_bind: + \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (u: +T).((sn3 c u) \to (\forall (vs: TList).(\forall (t: T).((sn3 (CHead c (Bind +b) u) (THeads (Flat Appl) (lifts (S O) O vs) t)) \to (sn3 c (THeads (Flat +Appl) vs (THead (Bind b) u t)))))))))) +\def + \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (c: C).(\lambda +(u: T).(\lambda (H0: (sn3 c u)).(\lambda (vs: TList).(TList_ind (\lambda (t: +TList).(\forall (t0: T).((sn3 (CHead c (Bind b) u) (THeads (Flat Appl) (lifts +(S O) O t) t0)) \to (sn3 c (THeads (Flat Appl) t (THead (Bind b) u t0)))))) +(\lambda (t: T).(\lambda (H1: (sn3 (CHead c (Bind b) u) t)).(sn3_bind b c u +H0 t H1))) (\lambda (v: T).(\lambda (vs0: TList).(TList_ind (\lambda (t: +TList).(((\forall (t0: T).((sn3 (CHead c (Bind b) u) (THeads (Flat Appl) +(lifts (S O) O t) t0)) \to (sn3 c (THeads (Flat Appl) t (THead (Bind b) u +t0)))))) \to (\forall (t0: T).((sn3 (CHead c (Bind b) u) (THead (Flat Appl) +(lift (S O) O v) (THeads (Flat Appl) (lifts (S O) O t) t0))) \to (sn3 c +(THead (Flat Appl) v (THeads (Flat Appl) t (THead (Bind b) u t0)))))))) +(\lambda (_: ((\forall (t: T).((sn3 (CHead c (Bind b) u) (THeads (Flat Appl) +(lifts (S O) O TNil) t)) \to (sn3 c (THeads (Flat Appl) TNil (THead (Bind b) +u t))))))).(\lambda (t: T).(\lambda (H2: (sn3 (CHead c (Bind b) u) (THead +(Flat Appl) (lift (S O) O v) (THeads (Flat Appl) (lifts (S O) O TNil) +t)))).(sn3_appl_bind b H c u H0 t v H2)))) (\lambda (t: T).(\lambda (t0: +TList).(\lambda (_: ((((\forall (t1: T).((sn3 (CHead c (Bind b) u) (THeads +(Flat Appl) (lifts (S O) O t0) t1)) \to (sn3 c (THeads (Flat Appl) t0 (THead +(Bind b) u t1)))))) \to (\forall (t1: T).((sn3 (CHead c (Bind b) u) (THead +(Flat Appl) (lift (S O) O v) (THeads (Flat Appl) (lifts (S O) O t0) t1))) \to +(sn3 c (THead (Flat Appl) v (THeads (Flat Appl) t0 (THead (Bind b) u +t1))))))))).(\lambda (H2: ((\forall (t1: T).((sn3 (CHead c (Bind b) u) +(THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1)) \to (sn3 c (THeads +(Flat Appl) (TCons t t0) (THead (Bind b) u t1))))))).(\lambda (t1: +T).(\lambda (H3: (sn3 (CHead c (Bind b) u) (THead (Flat Appl) (lift (S O) O +v) (THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1)))).(let H_x \def +(sn3_gen_flat Appl (CHead c (Bind b) u) (lift (S O) O v) (THeads (Flat Appl) +(lifts (S O) O (TCons t t0)) t1) H3) in (let H4 \def H_x in (land_ind (sn3 +(CHead c (Bind b) u) (lift (S O) O v)) (sn3 (CHead c (Bind b) u) (THead (Flat +Appl) (lift (S O) O t) (THeads (Flat Appl) (lifts (S O) O t0) t1))) (sn3 c +(THead (Flat Appl) v (THeads (Flat Appl) (TCons t t0) (THead (Bind b) u +t1)))) (\lambda (H5: (sn3 (CHead c (Bind b) u) (lift (S O) O v))).(\lambda +(H6: (sn3 (CHead c (Bind b) u) (THead (Flat Appl) (lift (S O) O t) (THeads +(Flat Appl) (lifts (S O) O t0) t1)))).(let H_y \def (sn3_gen_lift (CHead c +(Bind b) u) v (S O) O H5 c) in (sn3_appl_appls t (THead (Bind b) u t1) t0 c +(H2 t1 H6) v (H_y (drop_drop (Bind b) O c c (drop_refl c) u)) (\lambda (u2: +T).(\lambda (H7: (pr3 c (THeads (Flat Appl) (TCons t t0) (THead (Bind b) u +t1)) u2)).(\lambda (H8: (((iso (THeads (Flat Appl) (TCons t t0) (THead (Bind +b) u t1)) u2) \to (\forall (P: Prop).P)))).(let H9 \def (pr3_iso_appls_bind b +H (TCons t t0) u t1 c u2 H7 H8) in (sn3_pr3_trans c (THead (Flat Appl) v +(THead (Bind b) u (THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1))) +(sn3_appl_bind b H c u H0 (THeads (Flat Appl) (lifts (S O) O (TCons t t0)) +t1) v H3) (THead (Flat Appl) v u2) (pr3_flat c v v (pr3_refl c v) (THead +(Bind b) u (THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1)) u2 H9 +Appl)))))))))) H4))))))))) vs0))) vs)))))). +(* COMMENTS +Initial nodes: 1143 +END *) + +theorem sn3_appls_beta: + \forall (c: C).(\forall (v: T).(\forall (t: T).(\forall (us: TList).((sn3 c +(THeads (Flat Appl) us (THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c +w) \to (sn3 c (THeads (Flat Appl) us (THead (Flat Appl) v (THead (Bind Abst) +w t)))))))))) +\def + \lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (us: +TList).(TList_ind (\lambda (t0: TList).((sn3 c (THeads (Flat Appl) t0 (THead +(Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THeads (Flat +Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t)))))))) (\lambda (H: +(sn3 c (THead (Bind Abbr) v t))).(\lambda (w: T).(\lambda (H0: (sn3 c +w)).(sn3_beta c v t H w H0)))) (\lambda (u: T).(\lambda (us0: +TList).(TList_ind (\lambda (t0: TList).((((sn3 c (THeads (Flat Appl) t0 +(THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THeads +(Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t)))))))) \to ((sn3 +c (THead (Flat Appl) u (THeads (Flat Appl) t0 (THead (Bind Abbr) v t)))) \to +(\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) u (THeads (Flat +Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t)))))))))) (\lambda (_: +(((sn3 c (THeads (Flat Appl) TNil (THead (Bind Abbr) v t))) \to (\forall (w: +T).((sn3 c w) \to (sn3 c (THeads (Flat Appl) TNil (THead (Flat Appl) v (THead +(Bind Abst) w t))))))))).(\lambda (H0: (sn3 c (THead (Flat Appl) u (THeads +(Flat Appl) TNil (THead (Bind Abbr) v t))))).(\lambda (w: T).(\lambda (H1: +(sn3 c w)).(sn3_appl_beta c u v t H0 w H1))))) (\lambda (t0: T).(\lambda (t1: +TList).(\lambda (_: (((((sn3 c (THeads (Flat Appl) t1 (THead (Bind Abbr) v +t))) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THeads (Flat Appl) t1 (THead +(Flat Appl) v (THead (Bind Abst) w t)))))))) \to ((sn3 c (THead (Flat Appl) u +(THeads (Flat Appl) t1 (THead (Bind Abbr) v t)))) \to (\forall (w: T).((sn3 c +w) \to (sn3 c (THead (Flat Appl) u (THeads (Flat Appl) t1 (THead (Flat Appl) +v (THead (Bind Abst) w t))))))))))).(\lambda (H0: (((sn3 c (THeads (Flat +Appl) (TCons t0 t1) (THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w) +\to (sn3 c (THeads (Flat Appl) (TCons t0 t1) (THead (Flat Appl) v (THead +(Bind Abst) w t))))))))).(\lambda (H1: (sn3 c (THead (Flat Appl) u (THeads +(Flat Appl) (TCons t0 t1) (THead (Bind Abbr) v t))))).(\lambda (w: +T).(\lambda (H2: (sn3 c w)).(let H_x \def (sn3_gen_flat Appl c u (THeads +(Flat Appl) (TCons t0 t1) (THead (Bind Abbr) v t)) H1) in (let H3 \def H_x in +(land_ind (sn3 c u) (sn3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 +(THead (Bind Abbr) v t)))) (sn3 c (THead (Flat Appl) u (THeads (Flat Appl) +(TCons t0 t1) (THead (Flat Appl) v (THead (Bind Abst) w t))))) (\lambda (H4: +(sn3 c u)).(\lambda (H5: (sn3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 +(THead (Bind Abbr) v t))))).(sn3_appl_appls t0 (THead (Flat Appl) v (THead +(Bind Abst) w t)) t1 c (H0 H5 w H2) u H4 (\lambda (u2: T).(\lambda (H6: (pr3 +c (THeads (Flat Appl) (TCons t0 t1) (THead (Flat Appl) v (THead (Bind Abst) w +t))) u2)).(\lambda (H7: (((iso (THeads (Flat Appl) (TCons t0 t1) (THead (Flat +Appl) v (THead (Bind Abst) w t))) u2) \to (\forall (P: Prop).P)))).(let H8 +\def (pr3_iso_appls_beta (TCons t0 t1) v w t c u2 H6 H7) in (sn3_pr3_trans c +(THead (Flat Appl) u (THeads (Flat Appl) (TCons t0 t1) (THead (Bind Abbr) v +t))) H1 (THead (Flat Appl) u u2) (pr3_thin_dx c (THeads (Flat Appl) (TCons t0 +t1) (THead (Bind Abbr) v t)) u2 H8 u Appl))))))))) H3)))))))))) us0))) us)))). +(* COMMENTS +Initial nodes: 987 +END *) + +theorem sn3_lift: + \forall (d: C).(\forall (t: T).((sn3 d t) \to (\forall (c: C).(\forall (h: +nat).(\forall (i: nat).((drop h i c d) \to (sn3 c (lift h i t)))))))) +\def + \lambda (d: C).(\lambda (t: T).(\lambda (H: (sn3 d t)).(sn3_ind d (\lambda +(t0: T).(\forall (c: C).(\forall (h: nat).(\forall (i: nat).((drop h i c d) +\to (sn3 c (lift h i t0))))))) (\lambda (t1: T).(\lambda (_: ((\forall (t2: +T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 d t1 t2) \to (sn3 d +t2)))))).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: +Prop).P))) \to ((pr3 d t1 t2) \to (\forall (c: C).(\forall (h: nat).(\forall +(i: nat).((drop h i c d) \to (sn3 c (lift h i t2))))))))))).(\lambda (c: +C).(\lambda (h: nat).(\lambda (i: nat).(\lambda (H2: (drop h i c +d)).(sn3_pr2_intro c (lift h i t1) (\lambda (t2: T).(\lambda (H3: (((eq T +(lift h i t1) t2) \to (\forall (P: Prop).P)))).(\lambda (H4: (pr2 c (lift h i +t1) t2)).(let H5 \def (pr2_gen_lift c t1 t2 h i H4 d H2) in (ex2_ind T +(\lambda (t3: T).(eq T t2 (lift h i t3))) (\lambda (t3: T).(pr2 d t1 t3)) +(sn3 c t2) (\lambda (x: T).(\lambda (H6: (eq T t2 (lift h i x))).(\lambda +(H7: (pr2 d t1 x)).(let H8 \def (eq_ind T t2 (\lambda (t0: T).((eq T (lift h +i t1) t0) \to (\forall (P: Prop).P))) H3 (lift h i x) H6) in (eq_ind_r T +(lift h i x) (\lambda (t0: T).(sn3 c t0)) (H1 x (\lambda (H9: (eq T t1 +x)).(\lambda (P: Prop).(let H10 \def (eq_ind_r T x (\lambda (t0: T).((eq T +(lift h i t1) (lift h i t0)) \to (\forall (P0: Prop).P0))) H8 t1 H9) in (let +H11 \def (eq_ind_r T x (\lambda (t0: T).(pr2 d t1 t0)) H7 t1 H9) in (H10 +(refl_equal T (lift h i t1)) P))))) (pr3_pr2 d t1 x H7) c h i H2) t2 H6))))) +H5))))))))))))) t H))). +(* COMMENTS +Initial nodes: 439 +END *) + +theorem sn3_abbr: + \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c +(CHead d (Bind Abbr) v)) \to ((sn3 d v) \to (sn3 c (TLRef i))))))) +\def + \lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda +(H: (getl i c (CHead d (Bind Abbr) v))).(\lambda (H0: (sn3 d +v)).(sn3_pr2_intro c (TLRef i) (\lambda (t2: T).(\lambda (H1: (((eq T (TLRef +i) t2) \to (\forall (P: Prop).P)))).(\lambda (H2: (pr2 c (TLRef i) t2)).(let +H3 \def (pr2_gen_lref c t2 i H2) in (or_ind (eq T t2 (TLRef i)) (ex2_2 C T +(\lambda (d0: C).(\lambda (u: T).(getl i c (CHead d0 (Bind Abbr) u)))) +(\lambda (_: C).(\lambda (u: T).(eq T t2 (lift (S i) O u))))) (sn3 c t2) +(\lambda (H4: (eq T t2 (TLRef i))).(let H5 \def (eq_ind T t2 (\lambda (t: +T).((eq T (TLRef i) t) \to (\forall (P: Prop).P))) H1 (TLRef i) H4) in +(eq_ind_r T (TLRef i) (\lambda (t: T).(sn3 c t)) (H5 (refl_equal T (TLRef i)) +(sn3 c (TLRef i))) t2 H4))) (\lambda (H4: (ex2_2 C T (\lambda (d0: +C).(\lambda (u: T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_: +C).(\lambda (u: T).(eq T t2 (lift (S i) O u)))))).(ex2_2_ind C T (\lambda +(d0: C).(\lambda (u: T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_: +C).(\lambda (u: T).(eq T t2 (lift (S i) O u)))) (sn3 c t2) (\lambda (x0: +C).(\lambda (x1: T).(\lambda (H5: (getl i c (CHead x0 (Bind Abbr) +x1))).(\lambda (H6: (eq T t2 (lift (S i) O x1))).(let H7 \def (eq_ind T t2 +(\lambda (t: T).((eq T (TLRef i) t) \to (\forall (P: Prop).P))) H1 (lift (S +i) O x1) H6) in (eq_ind_r T (lift (S i) O x1) (\lambda (t: T).(sn3 c t)) (let +H8 \def (eq_ind C (CHead d (Bind Abbr) v) (\lambda (c0: C).(getl i c c0)) H +(CHead x0 (Bind Abbr) x1) (getl_mono c (CHead d (Bind Abbr) v) i H (CHead x0 +(Bind Abbr) x1) H5)) in (let H9 \def (f_equal C C (\lambda (e: C).(match e in +C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) +\Rightarrow c0])) (CHead d (Bind Abbr) v) (CHead x0 (Bind Abbr) x1) +(getl_mono c (CHead d (Bind Abbr) v) i H (CHead x0 (Bind Abbr) x1) H5)) in +((let H10 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: +C).T) with [(CSort _) \Rightarrow v | (CHead _ _ t) \Rightarrow t])) (CHead d +(Bind Abbr) v) (CHead x0 (Bind Abbr) x1) (getl_mono c (CHead d (Bind Abbr) v) +i H (CHead x0 (Bind Abbr) x1) H5)) in (\lambda (H11: (eq C d x0)).(let H12 +\def (eq_ind_r T x1 (\lambda (t: T).(getl i c (CHead x0 (Bind Abbr) t))) H8 v +H10) in (eq_ind T v (\lambda (t: T).(sn3 c (lift (S i) O t))) (let H13 \def +(eq_ind_r C x0 (\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr) v))) H12 d +H11) in (sn3_lift d v H0 c (S i) O (getl_drop Abbr c d v i H13))) x1 H10)))) +H9))) t2 H6)))))) H4)) H3))))))))))). +(* COMMENTS +Initial nodes: 743 +END *) + +theorem sn3_appls_abbr: + \forall (c: C).(\forall (d: C).(\forall (w: T).(\forall (i: nat).((getl i c +(CHead d (Bind Abbr) w)) \to (\forall (vs: TList).((sn3 c (THeads (Flat Appl) +vs (lift (S i) O w))) \to (sn3 c (THeads (Flat Appl) vs (TLRef i))))))))) +\def + \lambda (c: C).(\lambda (d: C).(\lambda (w: T).(\lambda (i: nat).(\lambda +(H: (getl i c (CHead d (Bind Abbr) w))).(\lambda (vs: TList).(TList_ind +(\lambda (t: TList).((sn3 c (THeads (Flat Appl) t (lift (S i) O w))) \to (sn3 +c (THeads (Flat Appl) t (TLRef i))))) (\lambda (H0: (sn3 c (lift (S i) O +w))).(let H_y \def (sn3_gen_lift c w (S i) O H0 d (getl_drop Abbr c d w i H)) +in (sn3_abbr c d w i H H_y))) (\lambda (v: T).(\lambda (vs0: +TList).(TList_ind (\lambda (t: TList).((((sn3 c (THeads (Flat Appl) t (lift +(S i) O w))) \to (sn3 c (THeads (Flat Appl) t (TLRef i))))) \to ((sn3 c +(THead (Flat Appl) v (THeads (Flat Appl) t (lift (S i) O w)))) \to (sn3 c +(THead (Flat Appl) v (THeads (Flat Appl) t (TLRef i))))))) (\lambda (_: +(((sn3 c (THeads (Flat Appl) TNil (lift (S i) O w))) \to (sn3 c (THeads (Flat +Appl) TNil (TLRef i)))))).(\lambda (H1: (sn3 c (THead (Flat Appl) v (THeads +(Flat Appl) TNil (lift (S i) O w))))).(sn3_appl_abbr c d w i H v H1))) +(\lambda (t: T).(\lambda (t0: TList).(\lambda (_: (((((sn3 c (THeads (Flat +Appl) t0 (lift (S i) O w))) \to (sn3 c (THeads (Flat Appl) t0 (TLRef i))))) +\to ((sn3 c (THead (Flat Appl) v (THeads (Flat Appl) t0 (lift (S i) O w)))) +\to (sn3 c (THead (Flat Appl) v (THeads (Flat Appl) t0 (TLRef +i)))))))).(\lambda (H1: (((sn3 c (THeads (Flat Appl) (TCons t t0) (lift (S i) +O w))) \to (sn3 c (THeads (Flat Appl) (TCons t t0) (TLRef i)))))).(\lambda +(H2: (sn3 c (THead (Flat Appl) v (THeads (Flat Appl) (TCons t t0) (lift (S i) +O w))))).(let H_x \def (sn3_gen_flat Appl c v (THeads (Flat Appl) (TCons t +t0) (lift (S i) O w)) H2) in (let H3 \def H_x in (land_ind (sn3 c v) (sn3 c +(THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O w)))) (sn3 c (THead +(Flat Appl) v (THeads (Flat Appl) (TCons t t0) (TLRef i)))) (\lambda (H4: +(sn3 c v)).(\lambda (H5: (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 +(lift (S i) O w))))).(sn3_appl_appls t (TLRef i) t0 c (H1 H5) v H4 (\lambda +(u2: T).(\lambda (H6: (pr3 c (THeads (Flat Appl) (TCons t t0) (TLRef i)) +u2)).(\lambda (H7: (((iso (THeads (Flat Appl) (TCons t t0) (TLRef i)) u2) \to +(\forall (P: Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) v (THeads (Flat +Appl) (TCons t t0) (lift (S i) O w))) H2 (THead (Flat Appl) v u2) +(pr3_thin_dx c (THeads (Flat Appl) (TCons t t0) (lift (S i) O w)) u2 +(pr3_iso_appls_abbr c d w i H (TCons t t0) u2 H6 H7) v Appl)))))))) +H3)))))))) vs0))) vs)))))). +(* COMMENTS +Initial nodes: 797 +END *) + +theorem sns3_lifts: + \forall (c: C).(\forall (d: C).(\forall (h: nat).(\forall (i: nat).((drop h +i c d) \to (\forall (ts: TList).((sns3 d ts) \to (sns3 c (lifts h i ts)))))))) +\def + \lambda (c: C).(\lambda (d: C).(\lambda (h: nat).(\lambda (i: nat).(\lambda +(H: (drop h i c d)).(\lambda (ts: TList).(TList_ind (\lambda (t: +TList).((sns3 d t) \to (sns3 c (lifts h i t)))) (\lambda (H0: True).H0) +(\lambda (t: T).(\lambda (t0: TList).(\lambda (H0: (((sns3 d t0) \to (sns3 c +(lifts h i t0))))).(\lambda (H1: (land (sn3 d t) (sns3 d t0))).(let H2 \def +H1 in (land_ind (sn3 d t) (sns3 d t0) (land (sn3 c (lift h i t)) (sns3 c +(lifts h i t0))) (\lambda (H3: (sn3 d t)).(\lambda (H4: (sns3 d t0)).(conj +(sn3 c (lift h i t)) (sns3 c (lifts h i t0)) (sn3_lift d t H3 c h i H) (H0 +H4)))) H2)))))) ts)))))). +(* COMMENTS +Initial nodes: 185 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/spare.ma b/matita/matita/contribs/lambdadelta/basic_1/spare.ma new file mode 100644 index 000000000..95d2c24ee --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/spare.ma @@ -0,0 +1,38 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/theory.ma". + +axiom pc3_gen_appls_sort_abst: + \forall (c: C).(\forall (vs: TList).(\forall (w: T).(\forall (u: T).(\forall +(n: nat).((pc3 c (THeads (Flat Appl) vs (TSort n)) (THead (Bind Abst) w u)) +\to False))))) +. + +axiom pc3_gen_appls_lref_abst: + \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c +(CHead d (Bind Abst) v)) \to (\forall (vs: TList).(\forall (w: T).(\forall +(u: T).((pc3 c (THeads (Flat Appl) vs (TLRef i)) (THead (Bind Abst) w u)) \to +False)))))))) +. + +axiom pc3_gen_appls_lref_sort: + \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c +(CHead d (Bind Abst) v)) \to (\forall (vs: TList).(\forall (ws: +TList).(\forall (n: nat).((pc3 c (THeads (Flat Appl) vs (TLRef i)) (THeads +(Flat Appl) ws (TSort n))) \to False)))))))) +. + diff --git a/matita/matita/contribs/lambdadelta/basic_1/sty0/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/sty0/defs.ma new file mode 100644 index 000000000..6468d9dbb --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/sty0/defs.ma @@ -0,0 +1,39 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/G/defs.ma". + +include "Basic-1/getl/defs.ma". + +inductive sty0 (g: G): C \to (T \to (T \to Prop)) \def +| sty0_sort: \forall (c: C).(\forall (n: nat).(sty0 g c (TSort n) (TSort +(next g n)))) +| sty0_abbr: \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: +nat).((getl i c (CHead d (Bind Abbr) v)) \to (\forall (w: T).((sty0 g d v w) +\to (sty0 g c (TLRef i) (lift (S i) O w)))))))) +| sty0_abst: \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: +nat).((getl i c (CHead d (Bind Abst) v)) \to (\forall (w: T).((sty0 g d v w) +\to (sty0 g c (TLRef i) (lift (S i) O v)))))))) +| sty0_bind: \forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t1: +T).(\forall (t2: T).((sty0 g (CHead c (Bind b) v) t1 t2) \to (sty0 g c (THead +(Bind b) v t1) (THead (Bind b) v t2))))))) +| sty0_appl: \forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall (t2: +T).((sty0 g c t1 t2) \to (sty0 g c (THead (Flat Appl) v t1) (THead (Flat +Appl) v t2)))))) +| sty0_cast: \forall (c: C).(\forall (v1: T).(\forall (v2: T).((sty0 g c v1 +v2) \to (\forall (t1: T).(\forall (t2: T).((sty0 g c t1 t2) \to (sty0 g c +(THead (Flat Cast) v1 t1) (THead (Flat Cast) v2 t2)))))))). + diff --git a/matita/matita/contribs/lambdadelta/basic_1/sty0/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/sty0/fwd.ma new file mode 100644 index 000000000..134ec3c10 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/sty0/fwd.ma @@ -0,0 +1,562 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/sty0/defs.ma". + +theorem sty0_gen_sort: + \forall (g: G).(\forall (c: C).(\forall (x: T).(\forall (n: nat).((sty0 g c +(TSort n) x) \to (eq T x (TSort (next g n))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda +(H: (sty0 g c (TSort n) x)).(insert_eq T (TSort n) (\lambda (t: T).(sty0 g c +t x)) (\lambda (_: T).(eq T x (TSort (next g n)))) (\lambda (y: T).(\lambda +(H0: (sty0 g c y x)).(sty0_ind g (\lambda (_: C).(\lambda (t: T).(\lambda +(t0: T).((eq T t (TSort n)) \to (eq T t0 (TSort (next g n))))))) (\lambda (_: +C).(\lambda (n0: nat).(\lambda (H1: (eq T (TSort n0) (TSort n))).(let H2 \def +(f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with +[(TSort n1) \Rightarrow n1 | (TLRef _) \Rightarrow n0 | (THead _ _ _) +\Rightarrow n0])) (TSort n0) (TSort n) H1) in (eq_ind_r nat n (\lambda (n1: +nat).(eq T (TSort (next g n1)) (TSort (next g n)))) (refl_equal T (TSort +(next g n))) n0 H2))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (v: +T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind Abbr) +v))).(\lambda (w: T).(\lambda (_: (sty0 g d v w)).(\lambda (_: (((eq T v +(TSort n)) \to (eq T w (TSort (next g n)))))).(\lambda (H4: (eq T (TLRef i) +(TSort n))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T +return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (TSort n) H4) in +(False_ind (eq T (lift (S i) O w) (TSort (next g n))) H5))))))))))) (\lambda +(c0: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (_: (getl +i c0 (CHead d (Bind Abst) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v +w)).(\lambda (_: (((eq T v (TSort n)) \to (eq T w (TSort (next g +n)))))).(\lambda (H4: (eq T (TLRef i) (TSort n))).(let H5 \def (eq_ind T +(TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with +[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) +\Rightarrow False])) I (TSort n) H4) in (False_ind (eq T (lift (S i) O v) +(TSort (next g n))) H5))))))))))) (\lambda (b: B).(\lambda (c0: C).(\lambda +(v: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (sty0 g (CHead c0 (Bind +b) v) t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2 (TSort (next g +n)))))).(\lambda (H3: (eq T (THead (Bind b) v t1) (TSort n))).(let H4 \def +(eq_ind T (THead (Bind b) v t1) (\lambda (ee: T).(match ee in T return +(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H3) in +(False_ind (eq T (THead (Bind b) v t2) (TSort (next g n))) H4)))))))))) +(\lambda (c0: C).(\lambda (v: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda +(_: (sty0 g c0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2 (TSort +(next g n)))))).(\lambda (H3: (eq T (THead (Flat Appl) v t1) (TSort n))).(let +H4 \def (eq_ind T (THead (Flat Appl) v t1) (\lambda (ee: T).(match ee in T +return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H3) in +(False_ind (eq T (THead (Flat Appl) v t2) (TSort (next g n))) H4))))))))) +(\lambda (c0: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (sty0 g c0 v1 +v2)).(\lambda (_: (((eq T v1 (TSort n)) \to (eq T v2 (TSort (next g +n)))))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (sty0 g c0 t1 +t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2 (TSort (next g +n)))))).(\lambda (H5: (eq T (THead (Flat Cast) v1 t1) (TSort n))).(let H6 +\def (eq_ind T (THead (Flat Cast) v1 t1) (\lambda (ee: T).(match ee in T +return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H5) in +(False_ind (eq T (THead (Flat Cast) v2 t2) (TSort (next g n))) H6)))))))))))) +c y x H0))) H))))). +(* COMMENTS +Initial nodes: 869 +END *) + +theorem sty0_gen_lref: + \forall (g: G).(\forall (c: C).(\forall (x: T).(\forall (n: nat).((sty0 g c +(TLRef n) x) \to (or (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda +(_: T).(getl n c (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: +T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda +(t: T).(eq T x (lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda +(u: T).(\lambda (_: T).(getl n c (CHead e (Bind Abst) u))))) (\lambda (e: +C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda +(u: T).(\lambda (_: T).(eq T x (lift (S n) O u))))))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda +(H: (sty0 g c (TLRef n) x)).(insert_eq T (TLRef n) (\lambda (t: T).(sty0 g c +t x)) (\lambda (_: T).(or (ex3_3 C T T (\lambda (e: C).(\lambda (u: +T).(\lambda (_: T).(getl n c (CHead e (Bind Abbr) u))))) (\lambda (e: +C).(\lambda (u: T).(\lambda (t0: T).(sty0 g e u t0)))) (\lambda (_: +C).(\lambda (_: T).(\lambda (t0: T).(eq T x (lift (S n) O t0)))))) (ex3_3 C T +T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead e (Bind +Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t0: T).(sty0 g e u +t0)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T x (lift (S n) O +u)))))))) (\lambda (y: T).(\lambda (H0: (sty0 g c y x)).(sty0_ind g (\lambda +(c0: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (TLRef n)) \to (or (ex3_3 C +T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind +Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t1: T).(sty0 g e u +t1)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t1: T).(eq T t0 (lift (S n) +O t1)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl +n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda +(t1: T).(sty0 g e u t1)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: +T).(eq T t0 (lift (S n) O u))))))))))) (\lambda (c0: C).(\lambda (n0: +nat).(\lambda (H1: (eq T (TSort n0) (TLRef n))).(let H2 \def (eq_ind T (TSort +n0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort +_) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow +False])) I (TLRef n) H1) in (False_ind (or (ex3_3 C T T (\lambda (e: +C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u))))) +(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda +(_: C).(\lambda (_: T).(\lambda (t: T).(eq T (TSort (next g n0)) (lift (S n) +O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl +n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: +T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T +(TSort (next g n0)) (lift (S n) O u))))))) H2))))) (\lambda (c0: C).(\lambda +(d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d +(Bind Abbr) v))).(\lambda (w: T).(\lambda (H2: (sty0 g d v w)).(\lambda (_: +(((eq T v (TLRef n)) \to (or (ex3_3 C T T (\lambda (e: C).(\lambda (u: +T).(\lambda (_: T).(getl n d (CHead e (Bind Abbr) u))))) (\lambda (e: +C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda +(_: T).(\lambda (t: T).(eq T w (lift (S n) O t)))))) (ex3_3 C T T (\lambda +(e: C).(\lambda (u: T).(\lambda (_: T).(getl n d (CHead e (Bind Abst) u))))) +(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda +(_: C).(\lambda (u: T).(\lambda (_: T).(eq T w (lift (S n) O +u)))))))))).(\lambda (H4: (eq T (TLRef i) (TLRef n))).(let H5 \def (f_equal T +nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort _) +\Rightarrow i | (TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow i])) +(TLRef i) (TLRef n) H4) in (let H6 \def (eq_ind nat i (\lambda (n0: +nat).(getl n0 c0 (CHead d (Bind Abbr) v))) H1 n H5) in (eq_ind_r nat n +(\lambda (n0: nat).(or (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda +(_: T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: +T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda +(t: T).(eq T (lift (S n0) O w) (lift (S n) O t)))))) (ex3_3 C T T (\lambda +(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) +(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda +(_: C).(\lambda (u: T).(\lambda (_: T).(eq T (lift (S n0) O w) (lift (S n) O +u)))))))) (or_introl (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda +(_: T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: +T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda +(t: T).(eq T (lift (S n) O w) (lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: +C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) +(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda +(_: C).(\lambda (u: T).(\lambda (_: T).(eq T (lift (S n) O w) (lift (S n) O +u)))))) (ex3_3_intro C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: +T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: +T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda +(t: T).(eq T (lift (S n) O w) (lift (S n) O t))))) d v w H6 H2 (refl_equal T +(lift (S n) O w)))) i H5)))))))))))) (\lambda (c0: C).(\lambda (d: +C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d (Bind +Abst) v))).(\lambda (w: T).(\lambda (H2: (sty0 g d v w)).(\lambda (_: (((eq T +v (TLRef n)) \to (or (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda +(_: T).(getl n d (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: +T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda +(t: T).(eq T w (lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda +(u: T).(\lambda (_: T).(getl n d (CHead e (Bind Abst) u))))) (\lambda (e: +C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda +(u: T).(\lambda (_: T).(eq T w (lift (S n) O u)))))))))).(\lambda (H4: (eq T +(TLRef i) (TLRef n))).(let H5 \def (f_equal T nat (\lambda (e: T).(match e in +T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow i | (TLRef n0) +\Rightarrow n0 | (THead _ _ _) \Rightarrow i])) (TLRef i) (TLRef n) H4) in +(let H6 \def (eq_ind nat i (\lambda (n0: nat).(getl n0 c0 (CHead d (Bind +Abst) v))) H1 n H5) in (eq_ind_r nat n (\lambda (n0: nat).(or (ex3_3 C T T +(\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind +Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u +t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T (lift (S n0) O v) +(lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda +(_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: +T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda +(_: T).(eq T (lift (S n0) O v) (lift (S n) O u)))))))) (or_intror (ex3_3 C T +T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind +Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u +t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T (lift (S n) O v) +(lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda +(_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: +T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda +(_: T).(eq T (lift (S n) O v) (lift (S n) O u)))))) (ex3_3_intro C T T +(\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind +Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u +t)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T (lift (S n) O v) +(lift (S n) O u))))) d v w H6 H2 (refl_equal T (lift (S n) O v)))) i +H5)))))))))))) (\lambda (b: B).(\lambda (c0: C).(\lambda (v: T).(\lambda (t1: +T).(\lambda (t2: T).(\lambda (_: (sty0 g (CHead c0 (Bind b) v) t1 +t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (or (ex3_3 C T T (\lambda (e: +C).(\lambda (u: T).(\lambda (_: T).(getl n (CHead c0 (Bind b) v) (CHead e +(Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e +u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T t2 (lift (S n) +O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl +n (CHead c0 (Bind b) v) (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda +(u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: +T).(\lambda (_: T).(eq T t2 (lift (S n) O u)))))))))).(\lambda (H3: (eq T +(THead (Bind b) v t1) (TLRef n))).(let H4 \def (eq_ind T (THead (Bind b) v +t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort +_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) +\Rightarrow True])) I (TLRef n) H3) in (False_ind (or (ex3_3 C T T (\lambda +(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u))))) +(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda +(_: C).(\lambda (_: T).(\lambda (t: T).(eq T (THead (Bind b) v t2) (lift (S +n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: +T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: +T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda +(_: T).(eq T (THead (Bind b) v t2) (lift (S n) O u))))))) H4)))))))))) +(\lambda (c0: C).(\lambda (v: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda +(_: (sty0 g c0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (or (ex3_3 C T +T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind +Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u +t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T t2 (lift (S n) O +t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n +c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: +T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T t2 +(lift (S n) O u)))))))))).(\lambda (H3: (eq T (THead (Flat Appl) v t1) (TLRef +n))).(let H4 \def (eq_ind T (THead (Flat Appl) v t1) (\lambda (ee: T).(match +ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | +(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) +H3) in (False_ind (or (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda +(_: T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: +T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda +(t: T).(eq T (THead (Flat Appl) v t2) (lift (S n) O t)))))) (ex3_3 C T T +(\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind +Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u +t)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T (THead (Flat +Appl) v t2) (lift (S n) O u))))))) H4))))))))) (\lambda (c0: C).(\lambda (v1: +T).(\lambda (v2: T).(\lambda (_: (sty0 g c0 v1 v2)).(\lambda (_: (((eq T v1 +(TLRef n)) \to (or (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: +T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: +T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda +(t: T).(eq T v2 (lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda +(u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: +C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda +(u: T).(\lambda (_: T).(eq T v2 (lift (S n) O u)))))))))).(\lambda (t1: +T).(\lambda (t2: T).(\lambda (_: (sty0 g c0 t1 t2)).(\lambda (_: (((eq T t1 +(TLRef n)) \to (or (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: +T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: +T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda +(t: T).(eq T t2 (lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda +(u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: +C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda +(u: T).(\lambda (_: T).(eq T t2 (lift (S n) O u)))))))))).(\lambda (H5: (eq T +(THead (Flat Cast) v1 t1) (TLRef n))).(let H6 \def (eq_ind T (THead (Flat +Cast) v1 t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) +with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ +_) \Rightarrow True])) I (TLRef n) H5) in (False_ind (or (ex3_3 C T T +(\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind +Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u +t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T (THead (Flat +Cast) v2 t2) (lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: +T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: +C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda +(u: T).(\lambda (_: T).(eq T (THead (Flat Cast) v2 t2) (lift (S n) O u))))))) +H6)))))))))))) c y x H0))) H))))). +(* COMMENTS +Initial nodes: 3231 +END *) + +theorem sty0_gen_bind: + \forall (g: G).(\forall (b: B).(\forall (c: C).(\forall (u: T).(\forall (t1: +T).(\forall (x: T).((sty0 g c (THead (Bind b) u t1) x) \to (ex2 T (\lambda +(t2: T).(sty0 g (CHead c (Bind b) u) t1 t2)) (\lambda (t2: T).(eq T x (THead +(Bind b) u t2)))))))))) +\def + \lambda (g: G).(\lambda (b: B).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: +T).(\lambda (x: T).(\lambda (H: (sty0 g c (THead (Bind b) u t1) +x)).(insert_eq T (THead (Bind b) u t1) (\lambda (t: T).(sty0 g c t x)) +(\lambda (_: T).(ex2 T (\lambda (t2: T).(sty0 g (CHead c (Bind b) u) t1 t2)) +(\lambda (t2: T).(eq T x (THead (Bind b) u t2))))) (\lambda (y: T).(\lambda +(H0: (sty0 g c y x)).(sty0_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda +(t0: T).((eq T t (THead (Bind b) u t1)) \to (ex2 T (\lambda (t2: T).(sty0 g +(CHead c0 (Bind b) u) t1 t2)) (\lambda (t2: T).(eq T t0 (THead (Bind b) u +t2)))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (H1: (eq T (TSort n) +(THead (Bind b) u t1))).(let H2 \def (eq_ind T (TSort n) (\lambda (ee: +T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow +True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I +(THead (Bind b) u t1) H1) in (False_ind (ex2 T (\lambda (t2: T).(sty0 g +(CHead c0 (Bind b) u) t1 t2)) (\lambda (t2: T).(eq T (TSort (next g n)) +(THead (Bind b) u t2)))) H2))))) (\lambda (c0: C).(\lambda (d: C).(\lambda +(v: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind Abbr) +v))).(\lambda (w: T).(\lambda (_: (sty0 g d v w)).(\lambda (_: (((eq T v +(THead (Bind b) u t1)) \to (ex2 T (\lambda (t2: T).(sty0 g (CHead d (Bind b) +u) t1 t2)) (\lambda (t2: T).(eq T w (THead (Bind b) u t2))))))).(\lambda (H4: +(eq T (TLRef i) (THead (Bind b) u t1))).(let H5 \def (eq_ind T (TLRef i) +(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) +\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow +False])) I (THead (Bind b) u t1) H4) in (False_ind (ex2 T (\lambda (t2: +T).(sty0 g (CHead c0 (Bind b) u) t1 t2)) (\lambda (t2: T).(eq T (lift (S i) O +w) (THead (Bind b) u t2)))) H5))))))))))) (\lambda (c0: C).(\lambda (d: +C).(\lambda (v: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind +Abst) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v w)).(\lambda (_: (((eq T +v (THead (Bind b) u t1)) \to (ex2 T (\lambda (t2: T).(sty0 g (CHead d (Bind +b) u) t1 t2)) (\lambda (t2: T).(eq T w (THead (Bind b) u t2))))))).(\lambda +(H4: (eq T (TLRef i) (THead (Bind b) u t1))).(let H5 \def (eq_ind T (TLRef i) +(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) +\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow +False])) I (THead (Bind b) u t1) H4) in (False_ind (ex2 T (\lambda (t2: +T).(sty0 g (CHead c0 (Bind b) u) t1 t2)) (\lambda (t2: T).(eq T (lift (S i) O +v) (THead (Bind b) u t2)))) H5))))))))))) (\lambda (b0: B).(\lambda (c0: +C).(\lambda (v: T).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H1: (sty0 g +(CHead c0 (Bind b0) v) t0 t2)).(\lambda (H2: (((eq T t0 (THead (Bind b) u +t1)) \to (ex2 T (\lambda (t3: T).(sty0 g (CHead (CHead c0 (Bind b0) v) (Bind +b) u) t1 t3)) (\lambda (t3: T).(eq T t2 (THead (Bind b) u t3))))))).(\lambda +(H3: (eq T (THead (Bind b0) v t0) (THead (Bind b) u t1))).(let H4 \def +(f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with +[(TSort _) \Rightarrow b0 | (TLRef _) \Rightarrow b0 | (THead k _ _) +\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b1) +\Rightarrow b1 | (Flat _) \Rightarrow b0])])) (THead (Bind b0) v t0) (THead +(Bind b) u t1) H3) in ((let H5 \def (f_equal T T (\lambda (e: T).(match e in +T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v | (TLRef _) +\Rightarrow v | (THead _ t _) \Rightarrow t])) (THead (Bind b0) v t0) (THead +(Bind b) u t1) H3) in ((let H6 \def (f_equal T T (\lambda (e: T).(match e in +T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) +\Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Bind b0) v t0) (THead +(Bind b) u t1) H3) in (\lambda (H7: (eq T v u)).(\lambda (H8: (eq B b0 +b)).(let H9 \def (eq_ind T t0 (\lambda (t: T).((eq T t (THead (Bind b) u t1)) +\to (ex2 T (\lambda (t3: T).(sty0 g (CHead (CHead c0 (Bind b0) v) (Bind b) u) +t1 t3)) (\lambda (t3: T).(eq T t2 (THead (Bind b) u t3)))))) H2 t1 H6) in +(let H10 \def (eq_ind T t0 (\lambda (t: T).(sty0 g (CHead c0 (Bind b0) v) t +t2)) H1 t1 H6) in (let H11 \def (eq_ind T v (\lambda (t: T).((eq T t1 (THead +(Bind b) u t1)) \to (ex2 T (\lambda (t3: T).(sty0 g (CHead (CHead c0 (Bind +b0) t) (Bind b) u) t1 t3)) (\lambda (t3: T).(eq T t2 (THead (Bind b) u +t3)))))) H9 u H7) in (let H12 \def (eq_ind T v (\lambda (t: T).(sty0 g (CHead +c0 (Bind b0) t) t1 t2)) H10 u H7) in (eq_ind_r T u (\lambda (t: T).(ex2 T +(\lambda (t3: T).(sty0 g (CHead c0 (Bind b) u) t1 t3)) (\lambda (t3: T).(eq T +(THead (Bind b0) t t2) (THead (Bind b) u t3))))) (let H13 \def (eq_ind B b0 +(\lambda (b1: B).((eq T t1 (THead (Bind b) u t1)) \to (ex2 T (\lambda (t3: +T).(sty0 g (CHead (CHead c0 (Bind b1) u) (Bind b) u) t1 t3)) (\lambda (t3: +T).(eq T t2 (THead (Bind b) u t3)))))) H11 b H8) in (let H14 \def (eq_ind B +b0 (\lambda (b1: B).(sty0 g (CHead c0 (Bind b1) u) t1 t2)) H12 b H8) in +(eq_ind_r B b (\lambda (b1: B).(ex2 T (\lambda (t3: T).(sty0 g (CHead c0 +(Bind b) u) t1 t3)) (\lambda (t3: T).(eq T (THead (Bind b1) u t2) (THead +(Bind b) u t3))))) (ex_intro2 T (\lambda (t3: T).(sty0 g (CHead c0 (Bind b) +u) t1 t3)) (\lambda (t3: T).(eq T (THead (Bind b) u t2) (THead (Bind b) u +t3))) t2 H14 (refl_equal T (THead (Bind b) u t2))) b0 H8))) v H7)))))))) H5)) +H4)))))))))) (\lambda (c0: C).(\lambda (v: T).(\lambda (t0: T).(\lambda (t2: +T).(\lambda (_: (sty0 g c0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind b) u +t1)) \to (ex2 T (\lambda (t3: T).(sty0 g (CHead c0 (Bind b) u) t1 t3)) +(\lambda (t3: T).(eq T t2 (THead (Bind b) u t3))))))).(\lambda (H3: (eq T +(THead (Flat Appl) v t0) (THead (Bind b) u t1))).(let H4 \def (eq_ind T +(THead (Flat Appl) v t0) (\lambda (ee: T).(match ee in T return (\lambda (_: +T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | +(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with +[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind +b) u t1) H3) in (False_ind (ex2 T (\lambda (t3: T).(sty0 g (CHead c0 (Bind b) +u) t1 t3)) (\lambda (t3: T).(eq T (THead (Flat Appl) v t2) (THead (Bind b) u +t3)))) H4))))))))) (\lambda (c0: C).(\lambda (v1: T).(\lambda (v2: +T).(\lambda (_: (sty0 g c0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind b) u +t1)) \to (ex2 T (\lambda (t2: T).(sty0 g (CHead c0 (Bind b) u) t1 t2)) +(\lambda (t2: T).(eq T v2 (THead (Bind b) u t2))))))).(\lambda (t0: +T).(\lambda (t2: T).(\lambda (_: (sty0 g c0 t0 t2)).(\lambda (_: (((eq T t0 +(THead (Bind b) u t1)) \to (ex2 T (\lambda (t3: T).(sty0 g (CHead c0 (Bind b) +u) t1 t3)) (\lambda (t3: T).(eq T t2 (THead (Bind b) u t3))))))).(\lambda +(H5: (eq T (THead (Flat Cast) v1 t0) (THead (Bind b) u t1))).(let H6 \def +(eq_ind T (THead (Flat Cast) v1 t0) (\lambda (ee: T).(match ee in T return +(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda +(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow +True])])) I (THead (Bind b) u t1) H5) in (False_ind (ex2 T (\lambda (t3: +T).(sty0 g (CHead c0 (Bind b) u) t1 t3)) (\lambda (t3: T).(eq T (THead (Flat +Cast) v2 t2) (THead (Bind b) u t3)))) H6)))))))))))) c y x H0))) H))))))). +(* COMMENTS +Initial nodes: 1975 +END *) + +theorem sty0_gen_appl: + \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (x: +T).((sty0 g c (THead (Flat Appl) u t1) x) \to (ex2 T (\lambda (t2: T).(sty0 g +c t1 t2)) (\lambda (t2: T).(eq T x (THead (Flat Appl) u t2))))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (x: +T).(\lambda (H: (sty0 g c (THead (Flat Appl) u t1) x)).(insert_eq T (THead +(Flat Appl) u t1) (\lambda (t: T).(sty0 g c t x)) (\lambda (_: T).(ex2 T +(\lambda (t2: T).(sty0 g c t1 t2)) (\lambda (t2: T).(eq T x (THead (Flat +Appl) u t2))))) (\lambda (y: T).(\lambda (H0: (sty0 g c y x)).(sty0_ind g +(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Flat Appl) +u t1)) \to (ex2 T (\lambda (t2: T).(sty0 g c0 t1 t2)) (\lambda (t2: T).(eq T +t0 (THead (Flat Appl) u t2)))))))) (\lambda (c0: C).(\lambda (n: +nat).(\lambda (H1: (eq T (TSort n) (THead (Flat Appl) u t1))).(let H2 \def +(eq_ind T (TSort n) (\lambda (ee: T).(match ee in T return (\lambda (_: +T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | +(THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) u t1) H1) in +(False_ind (ex2 T (\lambda (t2: T).(sty0 g c0 t1 t2)) (\lambda (t2: T).(eq T +(TSort (next g n)) (THead (Flat Appl) u t2)))) H2))))) (\lambda (c0: +C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (_: (getl i c0 +(CHead d (Bind Abbr) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v +w)).(\lambda (_: (((eq T v (THead (Flat Appl) u t1)) \to (ex2 T (\lambda (t2: +T).(sty0 g d t1 t2)) (\lambda (t2: T).(eq T w (THead (Flat Appl) u +t2))))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat Appl) u t1))).(let H5 +\def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_: +T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | +(THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) u t1) H4) in +(False_ind (ex2 T (\lambda (t2: T).(sty0 g c0 t1 t2)) (\lambda (t2: T).(eq T +(lift (S i) O w) (THead (Flat Appl) u t2)))) H5))))))))))) (\lambda (c0: +C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (_: (getl i c0 +(CHead d (Bind Abst) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v +w)).(\lambda (_: (((eq T v (THead (Flat Appl) u t1)) \to (ex2 T (\lambda (t2: +T).(sty0 g d t1 t2)) (\lambda (t2: T).(eq T w (THead (Flat Appl) u +t2))))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat Appl) u t1))).(let H5 +\def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_: +T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | +(THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) u t1) H4) in +(False_ind (ex2 T (\lambda (t2: T).(sty0 g c0 t1 t2)) (\lambda (t2: T).(eq T +(lift (S i) O v) (THead (Flat Appl) u t2)))) H5))))))))))) (\lambda (b: +B).(\lambda (c0: C).(\lambda (v: T).(\lambda (t0: T).(\lambda (t2: +T).(\lambda (_: (sty0 g (CHead c0 (Bind b) v) t0 t2)).(\lambda (_: (((eq T t0 +(THead (Flat Appl) u t1)) \to (ex2 T (\lambda (t3: T).(sty0 g (CHead c0 (Bind +b) v) t1 t3)) (\lambda (t3: T).(eq T t2 (THead (Flat Appl) u +t3))))))).(\lambda (H3: (eq T (THead (Bind b) v t0) (THead (Flat Appl) u +t1))).(let H4 \def (eq_ind T (THead (Bind b) v t0) (\lambda (ee: T).(match ee +in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef +_) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return +(\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow +False])])) I (THead (Flat Appl) u t1) H3) in (False_ind (ex2 T (\lambda (t3: +T).(sty0 g c0 t1 t3)) (\lambda (t3: T).(eq T (THead (Bind b) v t2) (THead +(Flat Appl) u t3)))) H4)))))))))) (\lambda (c0: C).(\lambda (v: T).(\lambda +(t0: T).(\lambda (t2: T).(\lambda (H1: (sty0 g c0 t0 t2)).(\lambda (H2: (((eq +T t0 (THead (Flat Appl) u t1)) \to (ex2 T (\lambda (t3: T).(sty0 g c0 t1 t3)) +(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u t3))))))).(\lambda (H3: (eq T +(THead (Flat Appl) v t0) (THead (Flat Appl) u t1))).(let H4 \def (f_equal T T +(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow v | (TLRef _) \Rightarrow v | (THead _ t _) \Rightarrow t])) +(THead (Flat Appl) v t0) (THead (Flat Appl) u t1) H3) in ((let H5 \def +(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with +[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) +\Rightarrow t])) (THead (Flat Appl) v t0) (THead (Flat Appl) u t1) H3) in +(\lambda (H6: (eq T v u)).(let H7 \def (eq_ind T t0 (\lambda (t: T).((eq T t +(THead (Flat Appl) u t1)) \to (ex2 T (\lambda (t3: T).(sty0 g c0 t1 t3)) +(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u t3)))))) H2 t1 H5) in (let H8 +\def (eq_ind T t0 (\lambda (t: T).(sty0 g c0 t t2)) H1 t1 H5) in (eq_ind_r T +u (\lambda (t: T).(ex2 T (\lambda (t3: T).(sty0 g c0 t1 t3)) (\lambda (t3: +T).(eq T (THead (Flat Appl) t t2) (THead (Flat Appl) u t3))))) (ex_intro2 T +(\lambda (t3: T).(sty0 g c0 t1 t3)) (\lambda (t3: T).(eq T (THead (Flat Appl) +u t2) (THead (Flat Appl) u t3))) t2 H8 (refl_equal T (THead (Flat Appl) u +t2))) v H6))))) H4))))))))) (\lambda (c0: C).(\lambda (v1: T).(\lambda (v2: +T).(\lambda (_: (sty0 g c0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Flat Appl) +u t1)) \to (ex2 T (\lambda (t2: T).(sty0 g c0 t1 t2)) (\lambda (t2: T).(eq T +v2 (THead (Flat Appl) u t2))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda +(_: (sty0 g c0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u t1)) \to +(ex2 T (\lambda (t3: T).(sty0 g c0 t1 t3)) (\lambda (t3: T).(eq T t2 (THead +(Flat Appl) u t3))))))).(\lambda (H5: (eq T (THead (Flat Cast) v1 t0) (THead +(Flat Appl) u t1))).(let H6 \def (eq_ind T (THead (Flat Cast) v1 t0) (\lambda +(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) +\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow +(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | +(Flat f) \Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl +\Rightarrow False | Cast \Rightarrow True])])])) I (THead (Flat Appl) u t1) +H5) in (False_ind (ex2 T (\lambda (t3: T).(sty0 g c0 t1 t3)) (\lambda (t3: +T).(eq T (THead (Flat Cast) v2 t2) (THead (Flat Appl) u t3)))) H6)))))))))))) +c y x H0))) H)))))). +(* COMMENTS +Initial nodes: 1489 +END *) + +theorem sty0_gen_cast: + \forall (g: G).(\forall (c: C).(\forall (v1: T).(\forall (t1: T).(\forall +(x: T).((sty0 g c (THead (Flat Cast) v1 t1) x) \to (ex3_2 T T (\lambda (v2: +T).(\lambda (_: T).(sty0 g c v1 v2))) (\lambda (_: T).(\lambda (t2: T).(sty0 +g c t1 t2))) (\lambda (v2: T).(\lambda (t2: T).(eq T x (THead (Flat Cast) v2 +t2)))))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (v1: T).(\lambda (t1: T).(\lambda +(x: T).(\lambda (H: (sty0 g c (THead (Flat Cast) v1 t1) x)).(insert_eq T +(THead (Flat Cast) v1 t1) (\lambda (t: T).(sty0 g c t x)) (\lambda (_: +T).(ex3_2 T T (\lambda (v2: T).(\lambda (_: T).(sty0 g c v1 v2))) (\lambda +(_: T).(\lambda (t2: T).(sty0 g c t1 t2))) (\lambda (v2: T).(\lambda (t2: +T).(eq T x (THead (Flat Cast) v2 t2)))))) (\lambda (y: T).(\lambda (H0: (sty0 +g c y x)).(sty0_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq +T t (THead (Flat Cast) v1 t1)) \to (ex3_2 T T (\lambda (v2: T).(\lambda (_: +T).(sty0 g c0 v1 v2))) (\lambda (_: T).(\lambda (t2: T).(sty0 g c0 t1 t2))) +(\lambda (v2: T).(\lambda (t2: T).(eq T t0 (THead (Flat Cast) v2 t2))))))))) +(\lambda (c0: C).(\lambda (n: nat).(\lambda (H1: (eq T (TSort n) (THead (Flat +Cast) v1 t1))).(let H2 \def (eq_ind T (TSort n) (\lambda (ee: T).(match ee in +T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) +\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) +v1 t1) H1) in (False_ind (ex3_2 T T (\lambda (v2: T).(\lambda (_: T).(sty0 g +c0 v1 v2))) (\lambda (_: T).(\lambda (t2: T).(sty0 g c0 t1 t2))) (\lambda +(v2: T).(\lambda (t2: T).(eq T (TSort (next g n)) (THead (Flat Cast) v2 +t2))))) H2))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: +nat).(\lambda (_: (getl i c0 (CHead d (Bind Abbr) v))).(\lambda (w: +T).(\lambda (_: (sty0 g d v w)).(\lambda (_: (((eq T v (THead (Flat Cast) v1 +t1)) \to (ex3_2 T T (\lambda (v2: T).(\lambda (_: T).(sty0 g d v1 v2))) +(\lambda (_: T).(\lambda (t2: T).(sty0 g d t1 t2))) (\lambda (v2: T).(\lambda +(t2: T).(eq T w (THead (Flat Cast) v2 t2)))))))).(\lambda (H4: (eq T (TLRef +i) (THead (Flat Cast) v1 t1))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: +T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow +False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I +(THead (Flat Cast) v1 t1) H4) in (False_ind (ex3_2 T T (\lambda (v2: +T).(\lambda (_: T).(sty0 g c0 v1 v2))) (\lambda (_: T).(\lambda (t2: T).(sty0 +g c0 t1 t2))) (\lambda (v2: T).(\lambda (t2: T).(eq T (lift (S i) O w) (THead +(Flat Cast) v2 t2))))) H5))))))))))) (\lambda (c0: C).(\lambda (d: +C).(\lambda (v: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind +Abst) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v w)).(\lambda (_: (((eq T +v (THead (Flat Cast) v1 t1)) \to (ex3_2 T T (\lambda (v2: T).(\lambda (_: +T).(sty0 g d v1 v2))) (\lambda (_: T).(\lambda (t2: T).(sty0 g d t1 t2))) +(\lambda (v2: T).(\lambda (t2: T).(eq T w (THead (Flat Cast) v2 +t2)))))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat Cast) v1 t1))).(let H5 +\def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_: +T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | +(THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) v1 t1) H4) in +(False_ind (ex3_2 T T (\lambda (v2: T).(\lambda (_: T).(sty0 g c0 v1 v2))) +(\lambda (_: T).(\lambda (t2: T).(sty0 g c0 t1 t2))) (\lambda (v2: +T).(\lambda (t2: T).(eq T (lift (S i) O v) (THead (Flat Cast) v2 t2))))) +H5))))))))))) (\lambda (b: B).(\lambda (c0: C).(\lambda (v: T).(\lambda (t0: +T).(\lambda (t2: T).(\lambda (_: (sty0 g (CHead c0 (Bind b) v) t0 +t2)).(\lambda (_: (((eq T t0 (THead (Flat Cast) v1 t1)) \to (ex3_2 T T +(\lambda (v2: T).(\lambda (_: T).(sty0 g (CHead c0 (Bind b) v) v1 v2))) +(\lambda (_: T).(\lambda (t3: T).(sty0 g (CHead c0 (Bind b) v) t1 t3))) +(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) v2 +t3)))))))).(\lambda (H3: (eq T (THead (Bind b) v t0) (THead (Flat Cast) v1 +t1))).(let H4 \def (eq_ind T (THead (Bind b) v t0) (\lambda (ee: T).(match ee +in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef +_) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return +(\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow +False])])) I (THead (Flat Cast) v1 t1) H3) in (False_ind (ex3_2 T T (\lambda +(v2: T).(\lambda (_: T).(sty0 g c0 v1 v2))) (\lambda (_: T).(\lambda (t3: +T).(sty0 g c0 t1 t3))) (\lambda (v2: T).(\lambda (t3: T).(eq T (THead (Bind +b) v t2) (THead (Flat Cast) v2 t3))))) H4)))))))))) (\lambda (c0: C).(\lambda +(v: T).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (sty0 g c0 t0 +t2)).(\lambda (_: (((eq T t0 (THead (Flat Cast) v1 t1)) \to (ex3_2 T T +(\lambda (v2: T).(\lambda (_: T).(sty0 g c0 v1 v2))) (\lambda (_: T).(\lambda +(t3: T).(sty0 g c0 t1 t3))) (\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead +(Flat Cast) v2 t3)))))))).(\lambda (H3: (eq T (THead (Flat Appl) v t0) (THead +(Flat Cast) v1 t1))).(let H4 \def (eq_ind T (THead (Flat Appl) v t0) (\lambda +(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) +\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow +(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | +(Flat f) \Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl +\Rightarrow True | Cast \Rightarrow False])])])) I (THead (Flat Cast) v1 t1) +H3) in (False_ind (ex3_2 T T (\lambda (v2: T).(\lambda (_: T).(sty0 g c0 v1 +v2))) (\lambda (_: T).(\lambda (t3: T).(sty0 g c0 t1 t3))) (\lambda (v2: +T).(\lambda (t3: T).(eq T (THead (Flat Appl) v t2) (THead (Flat Cast) v2 +t3))))) H4))))))))) (\lambda (c0: C).(\lambda (v0: T).(\lambda (v2: +T).(\lambda (H1: (sty0 g c0 v0 v2)).(\lambda (H2: (((eq T v0 (THead (Flat +Cast) v1 t1)) \to (ex3_2 T T (\lambda (v3: T).(\lambda (_: T).(sty0 g c0 v1 +v3))) (\lambda (_: T).(\lambda (t2: T).(sty0 g c0 t1 t2))) (\lambda (v3: +T).(\lambda (t2: T).(eq T v2 (THead (Flat Cast) v3 t2)))))))).(\lambda (t0: +T).(\lambda (t2: T).(\lambda (H3: (sty0 g c0 t0 t2)).(\lambda (H4: (((eq T t0 +(THead (Flat Cast) v1 t1)) \to (ex3_2 T T (\lambda (v3: T).(\lambda (_: +T).(sty0 g c0 v1 v3))) (\lambda (_: T).(\lambda (t3: T).(sty0 g c0 t1 t3))) +(\lambda (v3: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) v3 +t3)))))))).(\lambda (H5: (eq T (THead (Flat Cast) v0 t0) (THead (Flat Cast) +v1 t1))).(let H6 \def (f_equal T T (\lambda (e: T).(match e in T return +(\lambda (_: T).T) with [(TSort _) \Rightarrow v0 | (TLRef _) \Rightarrow v0 +| (THead _ t _) \Rightarrow t])) (THead (Flat Cast) v0 t0) (THead (Flat Cast) +v1 t1) H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T return +(\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 +| (THead _ _ t) \Rightarrow t])) (THead (Flat Cast) v0 t0) (THead (Flat Cast) +v1 t1) H5) in (\lambda (H8: (eq T v0 v1)).(let H9 \def (eq_ind T t0 (\lambda +(t: T).((eq T t (THead (Flat Cast) v1 t1)) \to (ex3_2 T T (\lambda (v3: +T).(\lambda (_: T).(sty0 g c0 v1 v3))) (\lambda (_: T).(\lambda (t3: T).(sty0 +g c0 t1 t3))) (\lambda (v3: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) +v3 t3))))))) H4 t1 H7) in (let H10 \def (eq_ind T t0 (\lambda (t: T).(sty0 g +c0 t t2)) H3 t1 H7) in (let H11 \def (eq_ind T v0 (\lambda (t: T).((eq T t +(THead (Flat Cast) v1 t1)) \to (ex3_2 T T (\lambda (v3: T).(\lambda (_: +T).(sty0 g c0 v1 v3))) (\lambda (_: T).(\lambda (t3: T).(sty0 g c0 t1 t3))) +(\lambda (v3: T).(\lambda (t3: T).(eq T v2 (THead (Flat Cast) v3 t3))))))) H2 +v1 H8) in (let H12 \def (eq_ind T v0 (\lambda (t: T).(sty0 g c0 t v2)) H1 v1 +H8) in (ex3_2_intro T T (\lambda (v3: T).(\lambda (_: T).(sty0 g c0 v1 v3))) +(\lambda (_: T).(\lambda (t3: T).(sty0 g c0 t1 t3))) (\lambda (v3: +T).(\lambda (t3: T).(eq T (THead (Flat Cast) v2 t2) (THead (Flat Cast) v3 +t3)))) v2 t2 H12 H10 (refl_equal T (THead (Flat Cast) v2 t2))))))))) +H6)))))))))))) c y x H0))) H)))))). +(* COMMENTS +Initial nodes: 1855 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/sty0/props.ma b/matita/matita/contribs/lambdadelta/basic_1/sty0/props.ma new file mode 100644 index 000000000..00c9f6818 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/sty0/props.ma @@ -0,0 +1,217 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/sty0/defs.ma". + +include "Basic-1/getl/drop.ma". + +theorem sty0_lift: + \forall (g: G).(\forall (e: C).(\forall (t1: T).(\forall (t2: T).((sty0 g e +t1 t2) \to (\forall (c: C).(\forall (h: nat).(\forall (d: nat).((drop h d c +e) \to (sty0 g c (lift h d t1) (lift h d t2)))))))))) +\def + \lambda (g: G).(\lambda (e: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda +(H: (sty0 g e t1 t2)).(sty0_ind g (\lambda (c: C).(\lambda (t: T).(\lambda +(t0: T).(\forall (c0: C).(\forall (h: nat).(\forall (d: nat).((drop h d c0 c) +\to (sty0 g c0 (lift h d t) (lift h d t0))))))))) (\lambda (c: C).(\lambda +(n: nat).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (_: +(drop h d c0 c)).(eq_ind_r T (TSort n) (\lambda (t: T).(sty0 g c0 t (lift h d +(TSort (next g n))))) (eq_ind_r T (TSort (next g n)) (\lambda (t: T).(sty0 g +c0 (TSort n) t)) (sty0_sort g c0 n) (lift h d (TSort (next g n))) (lift_sort +(next g n) h d)) (lift h d (TSort n)) (lift_sort n h d)))))))) (\lambda (c: +C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H0: (getl i c +(CHead d (Bind Abbr) v))).(\lambda (w: T).(\lambda (H1: (sty0 g d v +w)).(\lambda (H2: ((\forall (c0: C).(\forall (h: nat).(\forall (d0: +nat).((drop h d0 c0 d) \to (sty0 g c0 (lift h d0 v) (lift h d0 +w)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d0: nat).(\lambda (H3: +(drop h d0 c0 c)).(lt_le_e i d0 (sty0 g c0 (lift h d0 (TLRef i)) (lift h d0 +(lift (S i) O w))) (\lambda (H4: (lt i d0)).(let H5 \def (drop_getl_trans_le +i d0 (le_S_n i d0 (le_S (S i) d0 H4)) c0 c h H3 (CHead d (Bind Abbr) v) H0) +in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop i O c0 e0))) +(\lambda (e0: C).(\lambda (e1: C).(drop h (minus d0 i) e0 e1))) (\lambda (_: +C).(\lambda (e1: C).(clear e1 (CHead d (Bind Abbr) v)))) (sty0 g c0 (lift h +d0 (TLRef i)) (lift h d0 (lift (S i) O w))) (\lambda (x0: C).(\lambda (x1: +C).(\lambda (H6: (drop i O c0 x0)).(\lambda (H7: (drop h (minus d0 i) x0 +x1)).(\lambda (H8: (clear x1 (CHead d (Bind Abbr) v))).(let H9 \def (eq_ind +nat (minus d0 i) (\lambda (n: nat).(drop h n x0 x1)) H7 (S (minus d0 (S i))) +(minus_x_Sy d0 i H4)) in (let H10 \def (drop_clear_S x1 x0 h (minus d0 (S i)) +H9 Abbr d v H8) in (ex2_ind C (\lambda (c1: C).(clear x0 (CHead c1 (Bind +Abbr) (lift h (minus d0 (S i)) v)))) (\lambda (c1: C).(drop h (minus d0 (S +i)) c1 d)) (sty0 g c0 (lift h d0 (TLRef i)) (lift h d0 (lift (S i) O w))) +(\lambda (x: C).(\lambda (H11: (clear x0 (CHead x (Bind Abbr) (lift h (minus +d0 (S i)) v)))).(\lambda (H12: (drop h (minus d0 (S i)) x d)).(eq_ind_r T +(TLRef i) (\lambda (t: T).(sty0 g c0 t (lift h d0 (lift (S i) O w)))) (eq_ind +nat (plus (S i) (minus d0 (S i))) (\lambda (n: nat).(sty0 g c0 (TLRef i) +(lift h n (lift (S i) O w)))) (eq_ind_r T (lift (S i) O (lift h (minus d0 (S +i)) w)) (\lambda (t: T).(sty0 g c0 (TLRef i) t)) (eq_ind nat d0 (\lambda (_: +nat).(sty0 g c0 (TLRef i) (lift (S i) O (lift h (minus d0 (S i)) w)))) +(sty0_abbr g c0 x (lift h (minus d0 (S i)) v) i (getl_intro i c0 (CHead x +(Bind Abbr) (lift h (minus d0 (S i)) v)) x0 H6 H11) (lift h (minus d0 (S i)) +w) (H2 x h (minus d0 (S i)) H12)) (plus (S i) (minus d0 (S i))) +(le_plus_minus (S i) d0 H4)) (lift h (plus (S i) (minus d0 (S i))) (lift (S +i) O w)) (lift_d w h (S i) (minus d0 (S i)) O (le_O_n (minus d0 (S i))))) d0 +(le_plus_minus_r (S i) d0 H4)) (lift h d0 (TLRef i)) (lift_lref_lt i h d0 +H4))))) H10)))))))) H5))) (\lambda (H4: (le d0 i)).(eq_ind_r T (TLRef (plus i +h)) (\lambda (t: T).(sty0 g c0 t (lift h d0 (lift (S i) O w)))) (eq_ind nat +(S i) (\lambda (_: nat).(sty0 g c0 (TLRef (plus i h)) (lift h d0 (lift (S i) +O w)))) (eq_ind_r T (lift (plus h (S i)) O w) (\lambda (t: T).(sty0 g c0 +(TLRef (plus i h)) t)) (eq_ind_r nat (plus (S i) h) (\lambda (n: nat).(sty0 g +c0 (TLRef (plus i h)) (lift n O w))) (sty0_abbr g c0 d v (plus i h) +(drop_getl_trans_ge i c0 c d0 h H3 (CHead d (Bind Abbr) v) H0 H4) w H1) (plus +h (S i)) (plus_sym h (S i))) (lift h d0 (lift (S i) O w)) (lift_free w (S i) +h O d0 (le_S d0 i H4) (le_O_n d0))) (plus i (S O)) (eq_ind_r nat (plus (S O) +i) (\lambda (n: nat).(eq nat (S i) n)) (refl_equal nat (plus (S O) i)) (plus +i (S O)) (plus_sym i (S O)))) (lift h d0 (TLRef i)) (lift_lref_ge i h d0 +H4)))))))))))))))) (\lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda +(i: nat).(\lambda (H0: (getl i c (CHead d (Bind Abst) v))).(\lambda (w: +T).(\lambda (H1: (sty0 g d v w)).(\lambda (H2: ((\forall (c0: C).(\forall (h: +nat).(\forall (d0: nat).((drop h d0 c0 d) \to (sty0 g c0 (lift h d0 v) (lift +h d0 w)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d0: nat).(\lambda +(H3: (drop h d0 c0 c)).(lt_le_e i d0 (sty0 g c0 (lift h d0 (TLRef i)) (lift h +d0 (lift (S i) O v))) (\lambda (H4: (lt i d0)).(let H5 \def +(drop_getl_trans_le i d0 (le_S_n i d0 (le_S (S i) d0 H4)) c0 c h H3 (CHead d +(Bind Abst) v) H0) in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop i +O c0 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d0 i) e0 e1))) +(\lambda (_: C).(\lambda (e1: C).(clear e1 (CHead d (Bind Abst) v)))) (sty0 g +c0 (lift h d0 (TLRef i)) (lift h d0 (lift (S i) O v))) (\lambda (x0: +C).(\lambda (x1: C).(\lambda (H6: (drop i O c0 x0)).(\lambda (H7: (drop h +(minus d0 i) x0 x1)).(\lambda (H8: (clear x1 (CHead d (Bind Abst) v))).(let +H9 \def (eq_ind nat (minus d0 i) (\lambda (n: nat).(drop h n x0 x1)) H7 (S +(minus d0 (S i))) (minus_x_Sy d0 i H4)) in (let H10 \def (drop_clear_S x1 x0 +h (minus d0 (S i)) H9 Abst d v H8) in (ex2_ind C (\lambda (c1: C).(clear x0 +(CHead c1 (Bind Abst) (lift h (minus d0 (S i)) v)))) (\lambda (c1: C).(drop h +(minus d0 (S i)) c1 d)) (sty0 g c0 (lift h d0 (TLRef i)) (lift h d0 (lift (S +i) O v))) (\lambda (x: C).(\lambda (H11: (clear x0 (CHead x (Bind Abst) (lift +h (minus d0 (S i)) v)))).(\lambda (H12: (drop h (minus d0 (S i)) x +d)).(eq_ind_r T (TLRef i) (\lambda (t: T).(sty0 g c0 t (lift h d0 (lift (S i) +O v)))) (eq_ind nat (plus (S i) (minus d0 (S i))) (\lambda (n: nat).(sty0 g +c0 (TLRef i) (lift h n (lift (S i) O v)))) (eq_ind_r T (lift (S i) O (lift h +(minus d0 (S i)) v)) (\lambda (t: T).(sty0 g c0 (TLRef i) t)) (eq_ind nat d0 +(\lambda (_: nat).(sty0 g c0 (TLRef i) (lift (S i) O (lift h (minus d0 (S i)) +v)))) (sty0_abst g c0 x (lift h (minus d0 (S i)) v) i (getl_intro i c0 (CHead +x (Bind Abst) (lift h (minus d0 (S i)) v)) x0 H6 H11) (lift h (minus d0 (S +i)) w) (H2 x h (minus d0 (S i)) H12)) (plus (S i) (minus d0 (S i))) +(le_plus_minus (S i) d0 H4)) (lift h (plus (S i) (minus d0 (S i))) (lift (S +i) O v)) (lift_d v h (S i) (minus d0 (S i)) O (le_O_n (minus d0 (S i))))) d0 +(le_plus_minus_r (S i) d0 H4)) (lift h d0 (TLRef i)) (lift_lref_lt i h d0 +H4))))) H10)))))))) H5))) (\lambda (H4: (le d0 i)).(eq_ind_r T (TLRef (plus i +h)) (\lambda (t: T).(sty0 g c0 t (lift h d0 (lift (S i) O v)))) (eq_ind nat +(S i) (\lambda (_: nat).(sty0 g c0 (TLRef (plus i h)) (lift h d0 (lift (S i) +O v)))) (eq_ind_r T (lift (plus h (S i)) O v) (\lambda (t: T).(sty0 g c0 +(TLRef (plus i h)) t)) (eq_ind_r nat (plus (S i) h) (\lambda (n: nat).(sty0 g +c0 (TLRef (plus i h)) (lift n O v))) (sty0_abst g c0 d v (plus i h) +(drop_getl_trans_ge i c0 c d0 h H3 (CHead d (Bind Abst) v) H0 H4) w H1) (plus +h (S i)) (plus_sym h (S i))) (lift h d0 (lift (S i) O v)) (lift_free v (S i) +h O d0 (le_S d0 i H4) (le_O_n d0))) (plus i (S O)) (eq_ind_r nat (plus (S O) +i) (\lambda (n: nat).(eq nat (S i) n)) (refl_equal nat (plus (S O) i)) (plus +i (S O)) (plus_sym i (S O)))) (lift h d0 (TLRef i)) (lift_lref_ge i h d0 +H4)))))))))))))))) (\lambda (b: B).(\lambda (c: C).(\lambda (v: T).(\lambda +(t3: T).(\lambda (t4: T).(\lambda (_: (sty0 g (CHead c (Bind b) v) t3 +t4)).(\lambda (H1: ((\forall (c0: C).(\forall (h: nat).(\forall (d: +nat).((drop h d c0 (CHead c (Bind b) v)) \to (sty0 g c0 (lift h d t3) (lift h +d t4)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda +(H2: (drop h d c0 c)).(eq_ind_r T (THead (Bind b) (lift h d v) (lift h (s +(Bind b) d) t3)) (\lambda (t: T).(sty0 g c0 t (lift h d (THead (Bind b) v +t4)))) (eq_ind_r T (THead (Bind b) (lift h d v) (lift h (s (Bind b) d) t4)) +(\lambda (t: T).(sty0 g c0 (THead (Bind b) (lift h d v) (lift h (s (Bind b) +d) t3)) t)) (sty0_bind g b c0 (lift h d v) (lift h (S d) t3) (lift h (S d) +t4) (H1 (CHead c0 (Bind b) (lift h d v)) h (S d) (drop_skip_bind h d c0 c H2 +b v))) (lift h d (THead (Bind b) v t4)) (lift_head (Bind b) v t4 h d)) (lift +h d (THead (Bind b) v t3)) (lift_head (Bind b) v t3 h d))))))))))))) (\lambda +(c: C).(\lambda (v: T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (sty0 g +c t3 t4)).(\lambda (H1: ((\forall (c0: C).(\forall (h: nat).(\forall (d: +nat).((drop h d c0 c) \to (sty0 g c0 (lift h d t3) (lift h d +t4)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2: +(drop h d c0 c)).(eq_ind_r T (THead (Flat Appl) (lift h d v) (lift h (s (Flat +Appl) d) t3)) (\lambda (t: T).(sty0 g c0 t (lift h d (THead (Flat Appl) v +t4)))) (eq_ind_r T (THead (Flat Appl) (lift h d v) (lift h (s (Flat Appl) d) +t4)) (\lambda (t: T).(sty0 g c0 (THead (Flat Appl) (lift h d v) (lift h (s +(Flat Appl) d) t3)) t)) (sty0_appl g c0 (lift h d v) (lift h (s (Flat Appl) +d) t3) (lift h (s (Flat Appl) d) t4) (H1 c0 h (s (Flat Appl) d) H2)) (lift h +d (THead (Flat Appl) v t4)) (lift_head (Flat Appl) v t4 h d)) (lift h d +(THead (Flat Appl) v t3)) (lift_head (Flat Appl) v t3 h d)))))))))))) +(\lambda (c: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (sty0 g c v1 +v2)).(\lambda (H1: ((\forall (c0: C).(\forall (h: nat).(\forall (d: +nat).((drop h d c0 c) \to (sty0 g c0 (lift h d v1) (lift h d +v2)))))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (sty0 g c t3 +t4)).(\lambda (H3: ((\forall (c0: C).(\forall (h: nat).(\forall (d: +nat).((drop h d c0 c) \to (sty0 g c0 (lift h d t3) (lift h d +t4)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H4: +(drop h d c0 c)).(eq_ind_r T (THead (Flat Cast) (lift h d v1) (lift h (s +(Flat Cast) d) t3)) (\lambda (t: T).(sty0 g c0 t (lift h d (THead (Flat Cast) +v2 t4)))) (eq_ind_r T (THead (Flat Cast) (lift h d v2) (lift h (s (Flat Cast) +d) t4)) (\lambda (t: T).(sty0 g c0 (THead (Flat Cast) (lift h d v1) (lift h +(s (Flat Cast) d) t3)) t)) (sty0_cast g c0 (lift h d v1) (lift h d v2) (H1 c0 +h d H4) (lift h (s (Flat Cast) d) t3) (lift h (s (Flat Cast) d) t4) (H3 c0 h +(s (Flat Cast) d) H4)) (lift h d (THead (Flat Cast) v2 t4)) (lift_head (Flat +Cast) v2 t4 h d)) (lift h d (THead (Flat Cast) v1 t3)) (lift_head (Flat Cast) +v1 t3 h d))))))))))))))) e t1 t2 H))))). +(* COMMENTS +Initial nodes: 3677 +END *) + +theorem sty0_correct: + \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((sty0 g c +t1 t) \to (ex T (\lambda (t2: T).(sty0 g c t t2))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H: +(sty0 g c t1 t)).(sty0_ind g (\lambda (c0: C).(\lambda (_: T).(\lambda (t2: +T).(ex T (\lambda (t3: T).(sty0 g c0 t2 t3)))))) (\lambda (c0: C).(\lambda +(n: nat).(ex_intro T (\lambda (t2: T).(sty0 g c0 (TSort (next g n)) t2)) +(TSort (next g (next g n))) (sty0_sort g c0 (next g n))))) (\lambda (c0: +C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 +(CHead d (Bind Abbr) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v +w)).(\lambda (H2: (ex T (\lambda (t2: T).(sty0 g d w t2)))).(let H3 \def H2 +in (ex_ind T (\lambda (t2: T).(sty0 g d w t2)) (ex T (\lambda (t2: T).(sty0 g +c0 (lift (S i) O w) t2))) (\lambda (x: T).(\lambda (H4: (sty0 g d w +x)).(ex_intro T (\lambda (t2: T).(sty0 g c0 (lift (S i) O w) t2)) (lift (S i) +O x) (sty0_lift g d w x H4 c0 (S i) O (getl_drop Abbr c0 d v i H0))))) +H3)))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: +nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abst) v))).(\lambda (w: +T).(\lambda (H1: (sty0 g d v w)).(\lambda (H2: (ex T (\lambda (t2: T).(sty0 g +d w t2)))).(let H3 \def H2 in (ex_ind T (\lambda (t2: T).(sty0 g d w t2)) (ex +T (\lambda (t2: T).(sty0 g c0 (lift (S i) O v) t2))) (\lambda (x: T).(\lambda +(_: (sty0 g d w x)).(ex_intro T (\lambda (t2: T).(sty0 g c0 (lift (S i) O v) +t2)) (lift (S i) O w) (sty0_lift g d v w H1 c0 (S i) O (getl_drop Abst c0 d v +i H0))))) H3)))))))))) (\lambda (b: B).(\lambda (c0: C).(\lambda (v: +T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (sty0 g (CHead c0 (Bind b) +v) t2 t3)).(\lambda (H1: (ex T (\lambda (t4: T).(sty0 g (CHead c0 (Bind b) v) +t3 t4)))).(let H2 \def H1 in (ex_ind T (\lambda (t4: T).(sty0 g (CHead c0 +(Bind b) v) t3 t4)) (ex T (\lambda (t4: T).(sty0 g c0 (THead (Bind b) v t3) +t4))) (\lambda (x: T).(\lambda (H3: (sty0 g (CHead c0 (Bind b) v) t3 +x)).(ex_intro T (\lambda (t4: T).(sty0 g c0 (THead (Bind b) v t3) t4)) (THead +(Bind b) v x) (sty0_bind g b c0 v t3 x H3)))) H2))))))))) (\lambda (c0: +C).(\lambda (v: T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (sty0 g c0 +t2 t3)).(\lambda (H1: (ex T (\lambda (t4: T).(sty0 g c0 t3 t4)))).(let H2 +\def H1 in (ex_ind T (\lambda (t4: T).(sty0 g c0 t3 t4)) (ex T (\lambda (t4: +T).(sty0 g c0 (THead (Flat Appl) v t3) t4))) (\lambda (x: T).(\lambda (H3: +(sty0 g c0 t3 x)).(ex_intro T (\lambda (t4: T).(sty0 g c0 (THead (Flat Appl) +v t3) t4)) (THead (Flat Appl) v x) (sty0_appl g c0 v t3 x H3)))) H2)))))))) +(\lambda (c0: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (sty0 g c0 v1 +v2)).(\lambda (H1: (ex T (\lambda (t2: T).(sty0 g c0 v2 t2)))).(\lambda (t2: +T).(\lambda (t3: T).(\lambda (_: (sty0 g c0 t2 t3)).(\lambda (H3: (ex T +(\lambda (t4: T).(sty0 g c0 t3 t4)))).(let H4 \def H1 in (ex_ind T (\lambda +(t4: T).(sty0 g c0 v2 t4)) (ex T (\lambda (t4: T).(sty0 g c0 (THead (Flat +Cast) v2 t3) t4))) (\lambda (x: T).(\lambda (H5: (sty0 g c0 v2 x)).(let H6 +\def H3 in (ex_ind T (\lambda (t4: T).(sty0 g c0 t3 t4)) (ex T (\lambda (t4: +T).(sty0 g c0 (THead (Flat Cast) v2 t3) t4))) (\lambda (x0: T).(\lambda (H7: +(sty0 g c0 t3 x0)).(ex_intro T (\lambda (t4: T).(sty0 g c0 (THead (Flat Cast) +v2 t3) t4)) (THead (Flat Cast) x x0) (sty0_cast g c0 v2 x H5 t3 x0 H7)))) +H6)))) H4))))))))))) c t1 t H))))). +(* COMMENTS +Initial nodes: 991 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/sty1/cnt.ma b/matita/matita/contribs/lambdadelta/basic_1/sty1/cnt.ma new file mode 100644 index 000000000..0f39bdfd9 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/sty1/cnt.ma @@ -0,0 +1,89 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/sty1/props.ma". + +include "Basic-1/cnt/props.ma". + +theorem sty1_cnt: + \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((sty0 g c +t1 t) \to (ex2 T (\lambda (t2: T).(sty1 g c t1 t2)) (\lambda (t2: T).(cnt +t2))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H: +(sty0 g c t1 t)).(sty0_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (_: +T).(ex2 T (\lambda (t3: T).(sty1 g c0 t0 t3)) (\lambda (t3: T).(cnt t3)))))) +(\lambda (c0: C).(\lambda (n: nat).(ex_intro2 T (\lambda (t2: T).(sty1 g c0 +(TSort n) t2)) (\lambda (t2: T).(cnt t2)) (TSort (next g n)) (sty1_sty0 g c0 +(TSort n) (TSort (next g n)) (sty0_sort g c0 n)) (cnt_sort (next g n))))) +(\lambda (c0: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda +(H0: (getl i c0 (CHead d (Bind Abbr) v))).(\lambda (w: T).(\lambda (_: (sty0 +g d v w)).(\lambda (H2: (ex2 T (\lambda (t2: T).(sty1 g d v t2)) (\lambda +(t2: T).(cnt t2)))).(let H3 \def H2 in (ex2_ind T (\lambda (t2: T).(sty1 g d +v t2)) (\lambda (t2: T).(cnt t2)) (ex2 T (\lambda (t2: T).(sty1 g c0 (TLRef +i) t2)) (\lambda (t2: T).(cnt t2))) (\lambda (x: T).(\lambda (H4: (sty1 g d v +x)).(\lambda (H5: (cnt x)).(ex_intro2 T (\lambda (t2: T).(sty1 g c0 (TLRef i) +t2)) (\lambda (t2: T).(cnt t2)) (lift (S i) O x) (sty1_abbr g c0 d v i H0 x +H4) (cnt_lift x H5 (S i) O))))) H3)))))))))) (\lambda (c0: C).(\lambda (d: +C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind +Abst) v))).(\lambda (w: T).(\lambda (H1: (sty0 g d v w)).(\lambda (H2: (ex2 T +(\lambda (t2: T).(sty1 g d v t2)) (\lambda (t2: T).(cnt t2)))).(let H3 \def +H2 in (ex2_ind T (\lambda (t2: T).(sty1 g d v t2)) (\lambda (t2: T).(cnt t2)) +(ex2 T (\lambda (t2: T).(sty1 g c0 (TLRef i) t2)) (\lambda (t2: T).(cnt t2))) +(\lambda (x: T).(\lambda (H4: (sty1 g d v x)).(\lambda (H5: (cnt +x)).(ex_intro2 T (\lambda (t2: T).(sty1 g c0 (TLRef i) t2)) (\lambda (t2: +T).(cnt t2)) (lift (S i) O x) (sty1_trans g c0 (TLRef i) (lift (S i) O v) +(sty1_sty0 g c0 (TLRef i) (lift (S i) O v) (sty0_abst g c0 d v i H0 w H1)) +(lift (S i) O x) (sty1_lift g d v x H4 c0 (S i) O (getl_drop Abst c0 d v i +H0))) (cnt_lift x H5 (S i) O))))) H3)))))))))) (\lambda (b: B).(\lambda (c0: +C).(\lambda (v: T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (sty0 g +(CHead c0 (Bind b) v) t2 t3)).(\lambda (H1: (ex2 T (\lambda (t4: T).(sty1 g +(CHead c0 (Bind b) v) t2 t4)) (\lambda (t4: T).(cnt t4)))).(let H2 \def H1 in +(ex2_ind T (\lambda (t4: T).(sty1 g (CHead c0 (Bind b) v) t2 t4)) (\lambda +(t4: T).(cnt t4)) (ex2 T (\lambda (t4: T).(sty1 g c0 (THead (Bind b) v t2) +t4)) (\lambda (t4: T).(cnt t4))) (\lambda (x: T).(\lambda (H3: (sty1 g (CHead +c0 (Bind b) v) t2 x)).(\lambda (H4: (cnt x)).(ex_intro2 T (\lambda (t4: +T).(sty1 g c0 (THead (Bind b) v t2) t4)) (\lambda (t4: T).(cnt t4)) (THead +(Bind b) v x) (sty1_bind g b c0 v t2 x H3) (cnt_head x H4 (Bind b) v))))) +H2))))))))) (\lambda (c0: C).(\lambda (v: T).(\lambda (t2: T).(\lambda (t3: +T).(\lambda (_: (sty0 g c0 t2 t3)).(\lambda (H1: (ex2 T (\lambda (t4: +T).(sty1 g c0 t2 t4)) (\lambda (t4: T).(cnt t4)))).(let H2 \def H1 in +(ex2_ind T (\lambda (t4: T).(sty1 g c0 t2 t4)) (\lambda (t4: T).(cnt t4)) +(ex2 T (\lambda (t4: T).(sty1 g c0 (THead (Flat Appl) v t2) t4)) (\lambda +(t4: T).(cnt t4))) (\lambda (x: T).(\lambda (H3: (sty1 g c0 t2 x)).(\lambda +(H4: (cnt x)).(ex_intro2 T (\lambda (t4: T).(sty1 g c0 (THead (Flat Appl) v +t2) t4)) (\lambda (t4: T).(cnt t4)) (THead (Flat Appl) v x) (sty1_appl g c0 v +t2 x H3) (cnt_head x H4 (Flat Appl) v))))) H2)))))))) (\lambda (c0: +C).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H0: (sty0 g c0 v1 +v2)).(\lambda (_: (ex2 T (\lambda (t2: T).(sty1 g c0 v1 t2)) (\lambda (t2: +T).(cnt t2)))).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (sty0 g c0 t2 +t3)).(\lambda (H3: (ex2 T (\lambda (t4: T).(sty1 g c0 t2 t4)) (\lambda (t4: +T).(cnt t4)))).(let H4 \def H3 in (ex2_ind T (\lambda (t4: T).(sty1 g c0 t2 +t4)) (\lambda (t4: T).(cnt t4)) (ex2 T (\lambda (t4: T).(sty1 g c0 (THead +(Flat Cast) v1 t2) t4)) (\lambda (t4: T).(cnt t4))) (\lambda (x: T).(\lambda +(H5: (sty1 g c0 t2 x)).(\lambda (H6: (cnt x)).(let H_x \def (sty1_cast2 g c0 +t2 x H5 v1 v2 H0) in (let H7 \def H_x in (ex2_ind T (\lambda (v3: T).(sty1 g +c0 v1 v3)) (\lambda (v3: T).(sty1 g c0 (THead (Flat Cast) v1 t2) (THead (Flat +Cast) v3 x))) (ex2 T (\lambda (t4: T).(sty1 g c0 (THead (Flat Cast) v1 t2) +t4)) (\lambda (t4: T).(cnt t4))) (\lambda (x0: T).(\lambda (_: (sty1 g c0 v1 +x0)).(\lambda (H9: (sty1 g c0 (THead (Flat Cast) v1 t2) (THead (Flat Cast) x0 +x))).(ex_intro2 T (\lambda (t4: T).(sty1 g c0 (THead (Flat Cast) v1 t2) t4)) +(\lambda (t4: T).(cnt t4)) (THead (Flat Cast) x0 x) H9 (cnt_head x H6 (Flat +Cast) x0))))) H7)))))) H4))))))))))) c t1 t H))))). +(* COMMENTS +Initial nodes: 1313 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/sty1/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/sty1/defs.ma new file mode 100644 index 000000000..8fd219524 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/sty1/defs.ma @@ -0,0 +1,23 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/sty0/defs.ma". + +inductive sty1 (g: G) (c: C) (t1: T): T \to Prop \def +| sty1_sty0: \forall (t2: T).((sty0 g c t1 t2) \to (sty1 g c t1 t2)) +| sty1_sing: \forall (t: T).((sty1 g c t1 t) \to (\forall (t2: T).((sty0 g c +t t2) \to (sty1 g c t1 t2)))). + diff --git a/matita/matita/contribs/lambdadelta/basic_1/sty1/props.ma b/matita/matita/contribs/lambdadelta/basic_1/sty1/props.ma new file mode 100644 index 000000000..53061952e --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/sty1/props.ma @@ -0,0 +1,163 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/sty1/defs.ma". + +include "Basic-1/sty0/props.ma". + +theorem sty1_trans: + \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((sty1 g c +t1 t) \to (\forall (t2: T).((sty1 g c t t2) \to (sty1 g c t1 t2))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H: +(sty1 g c t1 t)).(\lambda (t2: T).(\lambda (H0: (sty1 g c t t2)).(sty1_ind g +c t (\lambda (t0: T).(sty1 g c t1 t0)) (\lambda (t3: T).(\lambda (H1: (sty0 g +c t t3)).(sty1_sing g c t1 t H t3 H1))) (\lambda (t0: T).(\lambda (_: (sty1 g +c t t0)).(\lambda (H2: (sty1 g c t1 t0)).(\lambda (t3: T).(\lambda (H3: (sty0 +g c t0 t3)).(sty1_sing g c t1 t0 H2 t3 H3)))))) t2 H0))))))). +(* COMMENTS +Initial nodes: 131 +END *) + +theorem sty1_bind: + \forall (g: G).(\forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t1: +T).(\forall (t2: T).((sty1 g (CHead c (Bind b) v) t1 t2) \to (sty1 g c (THead +(Bind b) v t1) (THead (Bind b) v t2)))))))) +\def + \lambda (g: G).(\lambda (b: B).(\lambda (c: C).(\lambda (v: T).(\lambda (t1: +T).(\lambda (t2: T).(\lambda (H: (sty1 g (CHead c (Bind b) v) t1 +t2)).(sty1_ind g (CHead c (Bind b) v) t1 (\lambda (t: T).(sty1 g c (THead +(Bind b) v t1) (THead (Bind b) v t))) (\lambda (t3: T).(\lambda (H0: (sty0 g +(CHead c (Bind b) v) t1 t3)).(sty1_sty0 g c (THead (Bind b) v t1) (THead +(Bind b) v t3) (sty0_bind g b c v t1 t3 H0)))) (\lambda (t: T).(\lambda (_: +(sty1 g (CHead c (Bind b) v) t1 t)).(\lambda (H1: (sty1 g c (THead (Bind b) v +t1) (THead (Bind b) v t))).(\lambda (t3: T).(\lambda (H2: (sty0 g (CHead c +(Bind b) v) t t3)).(sty1_sing g c (THead (Bind b) v t1) (THead (Bind b) v t) +H1 (THead (Bind b) v t3) (sty0_bind g b c v t t3 H2))))))) t2 H))))))). +(* COMMENTS +Initial nodes: 259 +END *) + +theorem sty1_appl: + \forall (g: G).(\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall +(t2: T).((sty1 g c t1 t2) \to (sty1 g c (THead (Flat Appl) v t1) (THead (Flat +Appl) v t2))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (v: T).(\lambda (t1: T).(\lambda +(t2: T).(\lambda (H: (sty1 g c t1 t2)).(sty1_ind g c t1 (\lambda (t: T).(sty1 +g c (THead (Flat Appl) v t1) (THead (Flat Appl) v t))) (\lambda (t3: +T).(\lambda (H0: (sty0 g c t1 t3)).(sty1_sty0 g c (THead (Flat Appl) v t1) +(THead (Flat Appl) v t3) (sty0_appl g c v t1 t3 H0)))) (\lambda (t: +T).(\lambda (_: (sty1 g c t1 t)).(\lambda (H1: (sty1 g c (THead (Flat Appl) v +t1) (THead (Flat Appl) v t))).(\lambda (t3: T).(\lambda (H2: (sty0 g c t +t3)).(sty1_sing g c (THead (Flat Appl) v t1) (THead (Flat Appl) v t) H1 +(THead (Flat Appl) v t3) (sty0_appl g c v t t3 H2))))))) t2 H)))))). +(* COMMENTS +Initial nodes: 213 +END *) + +theorem sty1_lift: + \forall (g: G).(\forall (e: C).(\forall (t1: T).(\forall (t2: T).((sty1 g e +t1 t2) \to (\forall (c: C).(\forall (h: nat).(\forall (d: nat).((drop h d c +e) \to (sty1 g c (lift h d t1) (lift h d t2)))))))))) +\def + \lambda (g: G).(\lambda (e: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda +(H: (sty1 g e t1 t2)).(sty1_ind g e t1 (\lambda (t: T).(\forall (c: +C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to (sty1 g c (lift h +d t1) (lift h d t))))))) (\lambda (t3: T).(\lambda (H0: (sty0 g e t1 +t3)).(\lambda (c: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: (drop +h d c e)).(sty1_sty0 g c (lift h d t1) (lift h d t3) (sty0_lift g e t1 t3 H0 +c h d H1)))))))) (\lambda (t: T).(\lambda (_: (sty1 g e t1 t)).(\lambda (H1: +((\forall (c: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to +(sty1 g c (lift h d t1) (lift h d t)))))))).(\lambda (t3: T).(\lambda (H2: +(sty0 g e t t3)).(\lambda (c: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda +(H3: (drop h d c e)).(sty1_sing g c (lift h d t1) (lift h d t) (H1 c h d H3) +(lift h d t3) (sty0_lift g e t t3 H2 c h d H3))))))))))) t2 H))))). +(* COMMENTS +Initial nodes: 277 +END *) + +theorem sty1_correct: + \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((sty1 g c +t1 t) \to (ex T (\lambda (t2: T).(sty0 g c t t2))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H: +(sty1 g c t1 t)).(sty1_ind g c t1 (\lambda (t0: T).(ex T (\lambda (t2: +T).(sty0 g c t0 t2)))) (\lambda (t2: T).(\lambda (H0: (sty0 g c t1 +t2)).(sty0_correct g c t1 t2 H0))) (\lambda (t0: T).(\lambda (_: (sty1 g c t1 +t0)).(\lambda (_: (ex T (\lambda (t2: T).(sty0 g c t0 t2)))).(\lambda (t2: +T).(\lambda (H2: (sty0 g c t0 t2)).(sty0_correct g c t0 t2 H2)))))) t H))))). +(* COMMENTS +Initial nodes: 123 +END *) + +theorem sty1_abbr: + \forall (g: G).(\forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: +nat).((getl i c (CHead d (Bind Abbr) v)) \to (\forall (w: T).((sty1 g d v w) +\to (sty1 g c (TLRef i) (lift (S i) O w))))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: +nat).(\lambda (H: (getl i c (CHead d (Bind Abbr) v))).(\lambda (w: +T).(\lambda (H0: (sty1 g d v w)).(sty1_ind g d v (\lambda (t: T).(sty1 g c +(TLRef i) (lift (S i) O t))) (\lambda (t2: T).(\lambda (H1: (sty0 g d v +t2)).(sty1_sty0 g c (TLRef i) (lift (S i) O t2) (sty0_abbr g c d v i H t2 +H1)))) (\lambda (t: T).(\lambda (_: (sty1 g d v t)).(\lambda (H2: (sty1 g c +(TLRef i) (lift (S i) O t))).(\lambda (t2: T).(\lambda (H3: (sty0 g d t +t2)).(sty1_sing g c (TLRef i) (lift (S i) O t) H2 (lift (S i) O t2) +(sty0_lift g d t t2 H3 c (S i) O (getl_drop Abbr c d v i H)))))))) w +H0)))))))). +(* COMMENTS +Initial nodes: 231 +END *) + +theorem sty1_cast2: + \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((sty1 g c +t1 t2) \to (\forall (v1: T).(\forall (v2: T).((sty0 g c v1 v2) \to (ex2 T +(\lambda (v3: T).(sty1 g c v1 v3)) (\lambda (v3: T).(sty1 g c (THead (Flat +Cast) v1 t1) (THead (Flat Cast) v3 t2))))))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda +(H: (sty1 g c t1 t2)).(sty1_ind g c t1 (\lambda (t: T).(\forall (v1: +T).(\forall (v2: T).((sty0 g c v1 v2) \to (ex2 T (\lambda (v3: T).(sty1 g c +v1 v3)) (\lambda (v3: T).(sty1 g c (THead (Flat Cast) v1 t1) (THead (Flat +Cast) v3 t)))))))) (\lambda (t3: T).(\lambda (H0: (sty0 g c t1 t3)).(\lambda +(v1: T).(\lambda (v2: T).(\lambda (H1: (sty0 g c v1 v2)).(ex_intro2 T +(\lambda (v3: T).(sty1 g c v1 v3)) (\lambda (v3: T).(sty1 g c (THead (Flat +Cast) v1 t1) (THead (Flat Cast) v3 t3))) v2 (sty1_sty0 g c v1 v2 H1) +(sty1_sty0 g c (THead (Flat Cast) v1 t1) (THead (Flat Cast) v2 t3) (sty0_cast +g c v1 v2 H1 t1 t3 H0)))))))) (\lambda (t: T).(\lambda (_: (sty1 g c t1 +t)).(\lambda (H1: ((\forall (v1: T).(\forall (v2: T).((sty0 g c v1 v2) \to +(ex2 T (\lambda (v3: T).(sty1 g c v1 v3)) (\lambda (v3: T).(sty1 g c (THead +(Flat Cast) v1 t1) (THead (Flat Cast) v3 t))))))))).(\lambda (t3: T).(\lambda +(H2: (sty0 g c t t3)).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H3: (sty0 g +c v1 v2)).(let H_x \def (H1 v1 v2 H3) in (let H4 \def H_x in (ex2_ind T +(\lambda (v3: T).(sty1 g c v1 v3)) (\lambda (v3: T).(sty1 g c (THead (Flat +Cast) v1 t1) (THead (Flat Cast) v3 t))) (ex2 T (\lambda (v3: T).(sty1 g c v1 +v3)) (\lambda (v3: T).(sty1 g c (THead (Flat Cast) v1 t1) (THead (Flat Cast) +v3 t3)))) (\lambda (x: T).(\lambda (H5: (sty1 g c v1 x)).(\lambda (H6: (sty1 +g c (THead (Flat Cast) v1 t1) (THead (Flat Cast) x t))).(let H_x0 \def +(sty1_correct g c v1 x H5) in (let H7 \def H_x0 in (ex_ind T (\lambda (t4: +T).(sty0 g c x t4)) (ex2 T (\lambda (v3: T).(sty1 g c v1 v3)) (\lambda (v3: +T).(sty1 g c (THead (Flat Cast) v1 t1) (THead (Flat Cast) v3 t3)))) (\lambda +(x0: T).(\lambda (H8: (sty0 g c x x0)).(ex_intro2 T (\lambda (v3: T).(sty1 g +c v1 v3)) (\lambda (v3: T).(sty1 g c (THead (Flat Cast) v1 t1) (THead (Flat +Cast) v3 t3))) x0 (sty1_sing g c v1 x H5 x0 H8) (sty1_sing g c (THead (Flat +Cast) v1 t1) (THead (Flat Cast) x t) H6 (THead (Flat Cast) x0 t3) (sty0_cast +g c x x0 H8 t t3 H2))))) H7)))))) H4))))))))))) t2 H))))). +(* COMMENTS +Initial nodes: 657 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/subst/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/subst/defs.ma new file mode 100644 index 000000000..e0ed86b52 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/subst/defs.ma @@ -0,0 +1,27 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/lift/defs.ma". + +definition subst: + nat \to (T \to (T \to T)) +\def + let rec subst (d: nat) (v: T) (t: T) on t: T \def (match t with [(TSort n) +\Rightarrow (TSort n) | (TLRef i) \Rightarrow (match (blt i d) with [true +\Rightarrow (TLRef i) | false \Rightarrow (match (blt d i) with [true +\Rightarrow (TLRef (pred i)) | false \Rightarrow (lift d O v)])]) | (THead k +u t0) \Rightarrow (THead k (subst d v u) (subst (s k d) v t0))]) in subst. + diff --git a/matita/matita/contribs/lambdadelta/basic_1/subst/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/subst/fwd.ma new file mode 100644 index 000000000..a0678e5fb --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/subst/fwd.ma @@ -0,0 +1,79 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/subst/defs.ma". + +theorem subst_sort: + \forall (v: T).(\forall (d: nat).(\forall (k: nat).(eq T (subst d v (TSort +k)) (TSort k)))) +\def + \lambda (_: T).(\lambda (_: nat).(\lambda (k: nat).(refl_equal T (TSort +k)))). +(* COMMENTS +Initial nodes: 13 +END *) + +theorem subst_lref_lt: + \forall (v: T).(\forall (d: nat).(\forall (i: nat).((lt i d) \to (eq T +(subst d v (TLRef i)) (TLRef i))))) +\def + \lambda (v: T).(\lambda (d: nat).(\lambda (i: nat).(\lambda (H: (lt i +d)).(eq_ind_r bool true (\lambda (b: bool).(eq T (match b with [true +\Rightarrow (TLRef i) | false \Rightarrow (match (blt d i) with [true +\Rightarrow (TLRef (pred i)) | false \Rightarrow (lift d O v)])]) (TLRef i))) +(refl_equal T (TLRef i)) (blt i d) (lt_blt d i H))))). +(* COMMENTS +Initial nodes: 73 +END *) + +theorem subst_lref_eq: + \forall (v: T).(\forall (i: nat).(eq T (subst i v (TLRef i)) (lift i O v))) +\def + \lambda (v: T).(\lambda (i: nat).(eq_ind_r bool false (\lambda (b: bool).(eq +T (match b with [true \Rightarrow (TLRef i) | false \Rightarrow (match b with +[true \Rightarrow (TLRef (pred i)) | false \Rightarrow (lift i O v)])]) (lift +i O v))) (refl_equal T (lift i O v)) (blt i i) (le_bge i i (le_n i)))). +(* COMMENTS +Initial nodes: 71 +END *) + +theorem subst_lref_gt: + \forall (v: T).(\forall (d: nat).(\forall (i: nat).((lt d i) \to (eq T +(subst d v (TLRef i)) (TLRef (pred i)))))) +\def + \lambda (v: T).(\lambda (d: nat).(\lambda (i: nat).(\lambda (H: (lt d +i)).(eq_ind_r bool false (\lambda (b: bool).(eq T (match b with [true +\Rightarrow (TLRef i) | false \Rightarrow (match (blt d i) with [true +\Rightarrow (TLRef (pred i)) | false \Rightarrow (lift d O v)])]) (TLRef +(pred i)))) (eq_ind_r bool true (\lambda (b: bool).(eq T (match b with [true +\Rightarrow (TLRef (pred i)) | false \Rightarrow (lift d O v)]) (TLRef (pred +i)))) (refl_equal T (TLRef (pred i))) (blt d i) (lt_blt i d H)) (blt i d) +(le_bge d i (lt_le_weak d i H)))))). +(* COMMENTS +Initial nodes: 130 +END *) + +theorem subst_head: + \forall (k: K).(\forall (w: T).(\forall (u: T).(\forall (t: T).(\forall (d: +nat).(eq T (subst d w (THead k u t)) (THead k (subst d w u) (subst (s k d) w +t))))))) +\def + \lambda (k: K).(\lambda (w: T).(\lambda (u: T).(\lambda (t: T).(\lambda (d: +nat).(refl_equal T (THead k (subst d w u) (subst (s k d) w t))))))). +(* COMMENTS +Initial nodes: 37 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/subst/props.ma b/matita/matita/contribs/lambdadelta/basic_1/subst/props.ma new file mode 100644 index 000000000..3bad044e3 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/subst/props.ma @@ -0,0 +1,116 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/subst/fwd.ma". + +include "Basic-1/subst0/defs.ma". + +include "Basic-1/lift/props.ma". + +theorem subst_lift_SO: + \forall (v: T).(\forall (t: T).(\forall (d: nat).(eq T (subst d v (lift (S +O) d t)) t))) +\def + \lambda (v: T).(\lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d: nat).(eq +T (subst d v (lift (S O) d t0)) t0))) (\lambda (n: nat).(\lambda (d: +nat).(eq_ind_r T (TSort n) (\lambda (t0: T).(eq T (subst d v t0) (TSort n))) +(eq_ind_r T (TSort n) (\lambda (t0: T).(eq T t0 (TSort n))) (refl_equal T +(TSort n)) (subst d v (TSort n)) (subst_sort v d n)) (lift (S O) d (TSort n)) +(lift_sort n (S O) d)))) (\lambda (n: nat).(\lambda (d: nat).(lt_le_e n d (eq +T (subst d v (lift (S O) d (TLRef n))) (TLRef n)) (\lambda (H: (lt n +d)).(eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (subst d v t0) (TLRef n))) +(eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 (TLRef n))) (refl_equal T +(TLRef n)) (subst d v (TLRef n)) (subst_lref_lt v d n H)) (lift (S O) d +(TLRef n)) (lift_lref_lt n (S O) d H))) (\lambda (H: (le d n)).(eq_ind_r T +(TLRef (plus n (S O))) (\lambda (t0: T).(eq T (subst d v t0) (TLRef n))) +(eq_ind nat (S (plus n O)) (\lambda (n0: nat).(eq T (subst d v (TLRef n0)) +(TLRef n))) (eq_ind_r T (TLRef (pred (S (plus n O)))) (\lambda (t0: T).(eq T +t0 (TLRef n))) (eq_ind nat (plus n O) (\lambda (n0: nat).(eq T (TLRef n0) +(TLRef n))) (f_equal nat T TLRef (plus n O) n (sym_eq nat n (plus n O) +(plus_n_O n))) (pred (S (plus n O))) (pred_Sn (plus n O))) (subst d v (TLRef +(S (plus n O)))) (subst_lref_gt v d (S (plus n O)) (le_n_S d (plus n O) +(le_plus_trans d n O H)))) (plus n (S O)) (plus_n_Sm n O)) (lift (S O) d +(TLRef n)) (lift_lref_ge n (S O) d H)))))) (\lambda (k: K).(\lambda (t0: +T).(\lambda (H: ((\forall (d: nat).(eq T (subst d v (lift (S O) d t0)) +t0)))).(\lambda (t1: T).(\lambda (H0: ((\forall (d: nat).(eq T (subst d v +(lift (S O) d t1)) t1)))).(\lambda (d: nat).(eq_ind_r T (THead k (lift (S O) +d t0) (lift (S O) (s k d) t1)) (\lambda (t2: T).(eq T (subst d v t2) (THead k +t0 t1))) (eq_ind_r T (THead k (subst d v (lift (S O) d t0)) (subst (s k d) v +(lift (S O) (s k d) t1))) (\lambda (t2: T).(eq T t2 (THead k t0 t1))) +(f_equal3 K T T T THead k k (subst d v (lift (S O) d t0)) t0 (subst (s k d) v +(lift (S O) (s k d) t1)) t1 (refl_equal K k) (H d) (H0 (s k d))) (subst d v +(THead k (lift (S O) d t0) (lift (S O) (s k d) t1))) (subst_head k v (lift (S +O) d t0) (lift (S O) (s k d) t1) d)) (lift (S O) d (THead k t0 t1)) +(lift_head k t0 t1 (S O) d)))))))) t)). +(* COMMENTS +Initial nodes: 879 +END *) + +theorem subst_subst0: + \forall (v: T).(\forall (t1: T).(\forall (t2: T).(\forall (d: nat).((subst0 +d v t1 t2) \to (eq T (subst d v t1) (subst d v t2)))))) +\def + \lambda (v: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (d: nat).(\lambda +(H: (subst0 d v t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t: +T).(\lambda (t0: T).(\lambda (t3: T).(eq T (subst n t t0) (subst n t t3)))))) +(\lambda (v0: T).(\lambda (i: nat).(eq_ind_r T (lift i O v0) (\lambda (t: +T).(eq T t (subst i v0 (lift (S i) O v0)))) (eq_ind nat (plus (S O) i) +(\lambda (n: nat).(eq T (lift i O v0) (subst i v0 (lift n O v0)))) (eq_ind T +(lift (S O) i (lift i O v0)) (\lambda (t: T).(eq T (lift i O v0) (subst i v0 +t))) (eq_ind_r T (lift i O v0) (\lambda (t: T).(eq T (lift i O v0) t)) +(refl_equal T (lift i O v0)) (subst i v0 (lift (S O) i (lift i O v0))) +(subst_lift_SO v0 (lift i O v0) i)) (lift (plus (S O) i) O v0) (lift_free v0 +i (S O) O i (le_n (plus O i)) (le_O_n i))) (S i) (refl_equal nat (S i))) +(subst i v0 (TLRef i)) (subst_lref_eq v0 i)))) (\lambda (v0: T).(\lambda (u2: +T).(\lambda (u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v0 u1 +u2)).(\lambda (H1: (eq T (subst i v0 u1) (subst i v0 u2))).(\lambda (t: +T).(\lambda (k: K).(eq_ind_r T (THead k (subst i v0 u1) (subst (s k i) v0 t)) +(\lambda (t0: T).(eq T t0 (subst i v0 (THead k u2 t)))) (eq_ind_r T (THead k +(subst i v0 u2) (subst (s k i) v0 t)) (\lambda (t0: T).(eq T (THead k (subst +i v0 u1) (subst (s k i) v0 t)) t0)) (eq_ind_r T (subst i v0 u2) (\lambda (t0: +T).(eq T (THead k t0 (subst (s k i) v0 t)) (THead k (subst i v0 u2) (subst (s +k i) v0 t)))) (refl_equal T (THead k (subst i v0 u2) (subst (s k i) v0 t))) +(subst i v0 u1) H1) (subst i v0 (THead k u2 t)) (subst_head k v0 u2 t i)) +(subst i v0 (THead k u1 t)) (subst_head k v0 u1 t i)))))))))) (\lambda (k: +K).(\lambda (v0: T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (i: +nat).(\lambda (_: (subst0 (s k i) v0 t4 t3)).(\lambda (H1: (eq T (subst (s k +i) v0 t4) (subst (s k i) v0 t3))).(\lambda (u: T).(eq_ind_r T (THead k (subst +i v0 u) (subst (s k i) v0 t4)) (\lambda (t: T).(eq T t (subst i v0 (THead k u +t3)))) (eq_ind_r T (THead k (subst i v0 u) (subst (s k i) v0 t3)) (\lambda +(t: T).(eq T (THead k (subst i v0 u) (subst (s k i) v0 t4)) t)) (eq_ind_r T +(subst (s k i) v0 t3) (\lambda (t: T).(eq T (THead k (subst i v0 u) t) (THead +k (subst i v0 u) (subst (s k i) v0 t3)))) (refl_equal T (THead k (subst i v0 +u) (subst (s k i) v0 t3))) (subst (s k i) v0 t4) H1) (subst i v0 (THead k u +t3)) (subst_head k v0 u t3 i)) (subst i v0 (THead k u t4)) (subst_head k v0 u +t4 i)))))))))) (\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda +(i: nat).(\lambda (_: (subst0 i v0 u1 u2)).(\lambda (H1: (eq T (subst i v0 +u1) (subst i v0 u2))).(\lambda (k: K).(\lambda (t3: T).(\lambda (t4: +T).(\lambda (_: (subst0 (s k i) v0 t3 t4)).(\lambda (H3: (eq T (subst (s k i) +v0 t3) (subst (s k i) v0 t4))).(eq_ind_r T (THead k (subst i v0 u1) (subst (s +k i) v0 t3)) (\lambda (t: T).(eq T t (subst i v0 (THead k u2 t4)))) (eq_ind_r +T (THead k (subst i v0 u2) (subst (s k i) v0 t4)) (\lambda (t: T).(eq T +(THead k (subst i v0 u1) (subst (s k i) v0 t3)) t)) (eq_ind_r T (subst i v0 +u2) (\lambda (t: T).(eq T (THead k t (subst (s k i) v0 t3)) (THead k (subst i +v0 u2) (subst (s k i) v0 t4)))) (eq_ind_r T (subst (s k i) v0 t4) (\lambda +(t: T).(eq T (THead k (subst i v0 u2) t) (THead k (subst i v0 u2) (subst (s k +i) v0 t4)))) (refl_equal T (THead k (subst i v0 u2) (subst (s k i) v0 t4))) +(subst (s k i) v0 t3) H3) (subst i v0 u1) H1) (subst i v0 (THead k u2 t4)) +(subst_head k v0 u2 t4 i)) (subst i v0 (THead k u1 t3)) (subst_head k v0 u1 +t3 i))))))))))))) d v t1 t2 H))))). +(* COMMENTS +Initial nodes: 1363 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/subst0/dec.ma b/matita/matita/contribs/lambdadelta/basic_1/subst0/dec.ma new file mode 100644 index 000000000..0234ff06c --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/subst0/dec.ma @@ -0,0 +1,182 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/subst0/defs.ma". + +include "Basic-1/lift/props.ma". + +theorem dnf_dec2: + \forall (t: T).(\forall (d: nat).(or (\forall (w: T).(ex T (\lambda (v: +T).(subst0 d w t (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T t (lift (S +O) d v)))))) +\def + \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d: nat).(or (\forall (w: +T).(ex T (\lambda (v: T).(subst0 d w t0 (lift (S O) d v))))) (ex T (\lambda +(v: T).(eq T t0 (lift (S O) d v))))))) (\lambda (n: nat).(\lambda (d: +nat).(or_intror (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (TSort n) +(lift (S O) d v))))) (ex T (\lambda (v: T).(eq T (TSort n) (lift (S O) d +v)))) (ex_intro T (\lambda (v: T).(eq T (TSort n) (lift (S O) d v))) (TSort +n) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq T (TSort n) t0)) (refl_equal T +(TSort n)) (lift (S O) d (TSort n)) (lift_sort n (S O) d)))))) (\lambda (n: +nat).(\lambda (d: nat).(lt_eq_gt_e n d (or (\forall (w: T).(ex T (\lambda (v: +T).(subst0 d w (TLRef n) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T +(TLRef n) (lift (S O) d v))))) (\lambda (H: (lt n d)).(or_intror (\forall (w: +T).(ex T (\lambda (v: T).(subst0 d w (TLRef n) (lift (S O) d v))))) (ex T +(\lambda (v: T).(eq T (TLRef n) (lift (S O) d v)))) (ex_intro T (\lambda (v: +T).(eq T (TLRef n) (lift (S O) d v))) (TLRef n) (eq_ind_r T (TLRef n) +(\lambda (t0: T).(eq T (TLRef n) t0)) (refl_equal T (TLRef n)) (lift (S O) d +(TLRef n)) (lift_lref_lt n (S O) d H))))) (\lambda (H: (eq nat n d)).(eq_ind +nat n (\lambda (n0: nat).(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 n0 +w (TLRef n) (lift (S O) n0 v))))) (ex T (\lambda (v: T).(eq T (TLRef n) (lift +(S O) n0 v)))))) (or_introl (\forall (w: T).(ex T (\lambda (v: T).(subst0 n w +(TLRef n) (lift (S O) n v))))) (ex T (\lambda (v: T).(eq T (TLRef n) (lift (S +O) n v)))) (\lambda (w: T).(ex_intro T (\lambda (v: T).(subst0 n w (TLRef n) +(lift (S O) n v))) (lift n O w) (eq_ind_r T (lift (plus (S O) n) O w) +(\lambda (t0: T).(subst0 n w (TLRef n) t0)) (subst0_lref w n) (lift (S O) n +(lift n O w)) (lift_free w n (S O) O n (le_n (plus O n)) (le_O_n n)))))) d +H)) (\lambda (H: (lt d n)).(or_intror (\forall (w: T).(ex T (\lambda (v: +T).(subst0 d w (TLRef n) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T +(TLRef n) (lift (S O) d v)))) (ex_intro T (\lambda (v: T).(eq T (TLRef n) +(lift (S O) d v))) (TLRef (pred n)) (eq_ind_r T (TLRef n) (\lambda (t0: +T).(eq T (TLRef n) t0)) (refl_equal T (TLRef n)) (lift (S O) d (TLRef (pred +n))) (lift_lref_gt d n H)))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda +(H: ((\forall (d: nat).(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w +t0 (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T t0 (lift (S O) d +v)))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (d: nat).(or (\forall (w: +T).(ex T (\lambda (v: T).(subst0 d w t1 (lift (S O) d v))))) (ex T (\lambda +(v: T).(eq T t1 (lift (S O) d v)))))))).(\lambda (d: nat).(let H_x \def (H d) +in (let H1 \def H_x in (or_ind (\forall (w: T).(ex T (\lambda (v: T).(subst0 +d w t0 (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T t0 (lift (S O) d +v)))) (or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) +(lift (S O) d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 t1) (lift (S O) +d v))))) (\lambda (H2: ((\forall (w: T).(ex T (\lambda (v: T).(subst0 d w t0 +(lift (S O) d v))))))).(let H_x0 \def (H0 (s k d)) in (let H3 \def H_x0 in +(or_ind (\forall (w: T).(ex T (\lambda (v: T).(subst0 (s k d) w t1 (lift (S +O) (s k d) v))))) (ex T (\lambda (v: T).(eq T t1 (lift (S O) (s k d) v)))) +(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) (lift +(S O) d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 t1) (lift (S O) d +v))))) (\lambda (H4: ((\forall (w: T).(ex T (\lambda (v: T).(subst0 (s k d) w +t1 (lift (S O) (s k d) v))))))).(or_introl (\forall (w: T).(ex T (\lambda (v: +T).(subst0 d w (THead k t0 t1) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq +T (THead k t0 t1) (lift (S O) d v)))) (\lambda (w: T).(let H_x1 \def (H4 w) +in (let H5 \def H_x1 in (ex_ind T (\lambda (v: T).(subst0 (s k d) w t1 (lift +(S O) (s k d) v))) (ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) (lift (S +O) d v)))) (\lambda (x: T).(\lambda (H6: (subst0 (s k d) w t1 (lift (S O) (s +k d) x))).(let H_x2 \def (H2 w) in (let H7 \def H_x2 in (ex_ind T (\lambda +(v: T).(subst0 d w t0 (lift (S O) d v))) (ex T (\lambda (v: T).(subst0 d w +(THead k t0 t1) (lift (S O) d v)))) (\lambda (x0: T).(\lambda (H8: (subst0 d +w t0 (lift (S O) d x0))).(ex_intro T (\lambda (v: T).(subst0 d w (THead k t0 +t1) (lift (S O) d v))) (THead k x0 x) (eq_ind_r T (THead k (lift (S O) d x0) +(lift (S O) (s k d) x)) (\lambda (t2: T).(subst0 d w (THead k t0 t1) t2)) +(subst0_both w t0 (lift (S O) d x0) d H8 k t1 (lift (S O) (s k d) x) H6) +(lift (S O) d (THead k x0 x)) (lift_head k x0 x (S O) d))))) H7))))) H5)))))) +(\lambda (H4: (ex T (\lambda (v: T).(eq T t1 (lift (S O) (s k d) +v))))).(ex_ind T (\lambda (v: T).(eq T t1 (lift (S O) (s k d) v))) (or +(\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) (lift (S O) +d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 t1) (lift (S O) d v))))) +(\lambda (x: T).(\lambda (H5: (eq T t1 (lift (S O) (s k d) x))).(eq_ind_r T +(lift (S O) (s k d) x) (\lambda (t2: T).(or (\forall (w: T).(ex T (\lambda +(v: T).(subst0 d w (THead k t0 t2) (lift (S O) d v))))) (ex T (\lambda (v: +T).(eq T (THead k t0 t2) (lift (S O) d v)))))) (or_introl (\forall (w: T).(ex +T (\lambda (v: T).(subst0 d w (THead k t0 (lift (S O) (s k d) x)) (lift (S O) +d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 (lift (S O) (s k d) x)) +(lift (S O) d v)))) (\lambda (w: T).(let H_x1 \def (H2 w) in (let H6 \def +H_x1 in (ex_ind T (\lambda (v: T).(subst0 d w t0 (lift (S O) d v))) (ex T +(\lambda (v: T).(subst0 d w (THead k t0 (lift (S O) (s k d) x)) (lift (S O) d +v)))) (\lambda (x0: T).(\lambda (H7: (subst0 d w t0 (lift (S O) d +x0))).(ex_intro T (\lambda (v: T).(subst0 d w (THead k t0 (lift (S O) (s k d) +x)) (lift (S O) d v))) (THead k x0 x) (eq_ind_r T (THead k (lift (S O) d x0) +(lift (S O) (s k d) x)) (\lambda (t2: T).(subst0 d w (THead k t0 (lift (S O) +(s k d) x)) t2)) (subst0_fst w (lift (S O) d x0) t0 d H7 (lift (S O) (s k d) +x) k) (lift (S O) d (THead k x0 x)) (lift_head k x0 x (S O) d))))) H6))))) t1 +H5))) H4)) H3)))) (\lambda (H2: (ex T (\lambda (v: T).(eq T t0 (lift (S O) d +v))))).(ex_ind T (\lambda (v: T).(eq T t0 (lift (S O) d v))) (or (\forall (w: +T).(ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) (lift (S O) d v))))) (ex +T (\lambda (v: T).(eq T (THead k t0 t1) (lift (S O) d v))))) (\lambda (x: +T).(\lambda (H3: (eq T t0 (lift (S O) d x))).(let H_x0 \def (H0 (s k d)) in +(let H4 \def H_x0 in (or_ind (\forall (w: T).(ex T (\lambda (v: T).(subst0 (s +k d) w t1 (lift (S O) (s k d) v))))) (ex T (\lambda (v: T).(eq T t1 (lift (S +O) (s k d) v)))) (or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead +k t0 t1) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 t1) +(lift (S O) d v))))) (\lambda (H5: ((\forall (w: T).(ex T (\lambda (v: +T).(subst0 (s k d) w t1 (lift (S O) (s k d) v))))))).(eq_ind_r T (lift (S O) +d x) (\lambda (t2: T).(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w +(THead k t2 t1) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T (THead k t2 +t1) (lift (S O) d v)))))) (or_introl (\forall (w: T).(ex T (\lambda (v: +T).(subst0 d w (THead k (lift (S O) d x) t1) (lift (S O) d v))))) (ex T +(\lambda (v: T).(eq T (THead k (lift (S O) d x) t1) (lift (S O) d v)))) +(\lambda (w: T).(let H_x1 \def (H5 w) in (let H6 \def H_x1 in (ex_ind T +(\lambda (v: T).(subst0 (s k d) w t1 (lift (S O) (s k d) v))) (ex T (\lambda +(v: T).(subst0 d w (THead k (lift (S O) d x) t1) (lift (S O) d v)))) (\lambda +(x0: T).(\lambda (H7: (subst0 (s k d) w t1 (lift (S O) (s k d) +x0))).(ex_intro T (\lambda (v: T).(subst0 d w (THead k (lift (S O) d x) t1) +(lift (S O) d v))) (THead k x x0) (eq_ind_r T (THead k (lift (S O) d x) (lift +(S O) (s k d) x0)) (\lambda (t2: T).(subst0 d w (THead k (lift (S O) d x) t1) +t2)) (subst0_snd k w (lift (S O) (s k d) x0) t1 d H7 (lift (S O) d x)) (lift +(S O) d (THead k x x0)) (lift_head k x x0 (S O) d))))) H6))))) t0 H3)) +(\lambda (H5: (ex T (\lambda (v: T).(eq T t1 (lift (S O) (s k d) +v))))).(ex_ind T (\lambda (v: T).(eq T t1 (lift (S O) (s k d) v))) (or +(\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) (lift (S O) +d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 t1) (lift (S O) d v))))) +(\lambda (x0: T).(\lambda (H6: (eq T t1 (lift (S O) (s k d) x0))).(eq_ind_r T +(lift (S O) (s k d) x0) (\lambda (t2: T).(or (\forall (w: T).(ex T (\lambda +(v: T).(subst0 d w (THead k t0 t2) (lift (S O) d v))))) (ex T (\lambda (v: +T).(eq T (THead k t0 t2) (lift (S O) d v)))))) (eq_ind_r T (lift (S O) d x) +(\lambda (t2: T).(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead +k t2 (lift (S O) (s k d) x0)) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq +T (THead k t2 (lift (S O) (s k d) x0)) (lift (S O) d v)))))) (or_intror +(\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead k (lift (S O) d x) +(lift (S O) (s k d) x0)) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T +(THead k (lift (S O) d x) (lift (S O) (s k d) x0)) (lift (S O) d v)))) +(ex_intro T (\lambda (v: T).(eq T (THead k (lift (S O) d x) (lift (S O) (s k +d) x0)) (lift (S O) d v))) (THead k x x0) (eq_ind_r T (THead k (lift (S O) d +x) (lift (S O) (s k d) x0)) (\lambda (t2: T).(eq T (THead k (lift (S O) d x) +(lift (S O) (s k d) x0)) t2)) (refl_equal T (THead k (lift (S O) d x) (lift +(S O) (s k d) x0))) (lift (S O) d (THead k x x0)) (lift_head k x x0 (S O) +d)))) t0 H3) t1 H6))) H5)) H4))))) H2)) H1))))))))) t). +(* COMMENTS +Initial nodes: 3549 +END *) + +theorem dnf_dec: + \forall (w: T).(\forall (t: T).(\forall (d: nat).(ex T (\lambda (v: T).(or +(subst0 d w t (lift (S O) d v)) (eq T t (lift (S O) d v))))))) +\def + \lambda (w: T).(\lambda (t: T).(\lambda (d: nat).(let H_x \def (dnf_dec2 t +d) in (let H \def H_x in (or_ind (\forall (w0: T).(ex T (\lambda (v: +T).(subst0 d w0 t (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T t (lift (S +O) d v)))) (ex T (\lambda (v: T).(or (subst0 d w t (lift (S O) d v)) (eq T t +(lift (S O) d v))))) (\lambda (H0: ((\forall (w0: T).(ex T (\lambda (v: +T).(subst0 d w0 t (lift (S O) d v))))))).(let H_x0 \def (H0 w) in (let H1 +\def H_x0 in (ex_ind T (\lambda (v: T).(subst0 d w t (lift (S O) d v))) (ex T +(\lambda (v: T).(or (subst0 d w t (lift (S O) d v)) (eq T t (lift (S O) d +v))))) (\lambda (x: T).(\lambda (H2: (subst0 d w t (lift (S O) d +x))).(ex_intro T (\lambda (v: T).(or (subst0 d w t (lift (S O) d v)) (eq T t +(lift (S O) d v)))) x (or_introl (subst0 d w t (lift (S O) d x)) (eq T t +(lift (S O) d x)) H2)))) H1)))) (\lambda (H0: (ex T (\lambda (v: T).(eq T t +(lift (S O) d v))))).(ex_ind T (\lambda (v: T).(eq T t (lift (S O) d v))) (ex +T (\lambda (v: T).(or (subst0 d w t (lift (S O) d v)) (eq T t (lift (S O) d +v))))) (\lambda (x: T).(\lambda (H1: (eq T t (lift (S O) d x))).(eq_ind_r T +(lift (S O) d x) (\lambda (t0: T).(ex T (\lambda (v: T).(or (subst0 d w t0 +(lift (S O) d v)) (eq T t0 (lift (S O) d v)))))) (ex_intro T (\lambda (v: +T).(or (subst0 d w (lift (S O) d x) (lift (S O) d v)) (eq T (lift (S O) d x) +(lift (S O) d v)))) x (or_intror (subst0 d w (lift (S O) d x) (lift (S O) d +x)) (eq T (lift (S O) d x) (lift (S O) d x)) (refl_equal T (lift (S O) d +x)))) t H1))) H0)) H))))). +(* COMMENTS +Initial nodes: 603 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/subst0/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/subst0/defs.ma new file mode 100644 index 000000000..a493a7ac2 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/subst0/defs.ma @@ -0,0 +1,32 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/lift/defs.ma". + +inductive subst0: nat \to (T \to (T \to (T \to Prop))) \def +| subst0_lref: \forall (v: T).(\forall (i: nat).(subst0 i v (TLRef i) (lift +(S i) O v))) +| subst0_fst: \forall (v: T).(\forall (u2: T).(\forall (u1: T).(\forall (i: +nat).((subst0 i v u1 u2) \to (\forall (t: T).(\forall (k: K).(subst0 i v +(THead k u1 t) (THead k u2 t)))))))) +| subst0_snd: \forall (k: K).(\forall (v: T).(\forall (t2: T).(\forall (t1: +T).(\forall (i: nat).((subst0 (s k i) v t1 t2) \to (\forall (u: T).(subst0 i +v (THead k u t1) (THead k u t2)))))))) +| subst0_both: \forall (v: T).(\forall (u1: T).(\forall (u2: T).(\forall (i: +nat).((subst0 i v u1 u2) \to (\forall (k: K).(\forall (t1: T).(\forall (t2: +T).((subst0 (s k i) v t1 t2) \to (subst0 i v (THead k u1 t1) (THead k u2 +t2)))))))))). + diff --git a/matita/matita/contribs/lambdadelta/basic_1/subst0/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/subst0/fwd.ma new file mode 100644 index 000000000..165555fe2 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/subst0/fwd.ma @@ -0,0 +1,728 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/subst0/defs.ma". + +include "Basic-1/lift/props.ma". + +theorem subst0_gen_sort: + \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst0 +i v (TSort n) x) \to (\forall (P: Prop).P))))) +\def + \lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda +(H: (subst0 i v (TSort n) x)).(\lambda (P: Prop).(insert_eq T (TSort n) +(\lambda (t: T).(subst0 i v t x)) (\lambda (_: T).P) (\lambda (y: T).(\lambda +(H0: (subst0 i v y x)).(subst0_ind (\lambda (_: nat).(\lambda (_: T).(\lambda +(t0: T).(\lambda (_: T).((eq T t0 (TSort n)) \to P))))) (\lambda (_: +T).(\lambda (i0: nat).(\lambda (H1: (eq T (TLRef i0) (TSort n))).(let H2 \def +(eq_ind T (TLRef i0) (\lambda (ee: T).(match ee in T return (\lambda (_: +T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | +(THead _ _ _) \Rightarrow False])) I (TSort n) H1) in (False_ind P H2))))) +(\lambda (v0: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i0: +nat).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (_: (((eq T u1 (TSort n)) +\to P))).(\lambda (t: T).(\lambda (k: K).(\lambda (H3: (eq T (THead k u1 t) +(TSort n))).(let H4 \def (eq_ind T (THead k u1 t) (\lambda (ee: T).(match ee +in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef +_) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H3) in +(False_ind P H4))))))))))) (\lambda (k: K).(\lambda (v0: T).(\lambda (t2: +T).(\lambda (t1: T).(\lambda (i0: nat).(\lambda (_: (subst0 (s k i0) v0 t1 +t2)).(\lambda (_: (((eq T t1 (TSort n)) \to P))).(\lambda (u: T).(\lambda +(H3: (eq T (THead k u t1) (TSort n))).(let H4 \def (eq_ind T (THead k u t1) +(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) +\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow +True])) I (TSort n) H3) in (False_ind P H4))))))))))) (\lambda (v0: +T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i0: nat).(\lambda (_: (subst0 +i0 v0 u1 u2)).(\lambda (_: (((eq T u1 (TSort n)) \to P))).(\lambda (k: +K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (subst0 (s k i0) v0 t1 +t2)).(\lambda (_: (((eq T t1 (TSort n)) \to P))).(\lambda (H5: (eq T (THead k +u1 t1) (TSort n))).(let H6 \def (eq_ind T (THead k u1 t1) (\lambda (ee: +T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow +False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I +(TSort n) H5) in (False_ind P H6)))))))))))))) i v y x H0))) H)))))). +(* COMMENTS +Initial nodes: 445 +END *) + +theorem subst0_gen_lref: + \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst0 +i v (TLRef n) x) \to (land (eq nat n i) (eq T x (lift (S n) O v))))))) +\def + \lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda +(H: (subst0 i v (TLRef n) x)).(insert_eq T (TLRef n) (\lambda (t: T).(subst0 +i v t x)) (\lambda (_: T).(land (eq nat n i) (eq T x (lift (S n) O v)))) +(\lambda (y: T).(\lambda (H0: (subst0 i v y x)).(subst0_ind (\lambda (n0: +nat).(\lambda (t: T).(\lambda (t0: T).(\lambda (t1: T).((eq T t0 (TLRef n)) +\to (land (eq nat n n0) (eq T t1 (lift (S n) O t)))))))) (\lambda (v0: +T).(\lambda (i0: nat).(\lambda (H1: (eq T (TLRef i0) (TLRef n))).(let H2 \def +(f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with +[(TSort _) \Rightarrow i0 | (TLRef n0) \Rightarrow n0 | (THead _ _ _) +\Rightarrow i0])) (TLRef i0) (TLRef n) H1) in (eq_ind_r nat n (\lambda (n0: +nat).(land (eq nat n n0) (eq T (lift (S n0) O v0) (lift (S n) O v0)))) (conj +(eq nat n n) (eq T (lift (S n) O v0) (lift (S n) O v0)) (refl_equal nat n) +(refl_equal T (lift (S n) O v0))) i0 H2))))) (\lambda (v0: T).(\lambda (u2: +T).(\lambda (u1: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v0 u1 +u2)).(\lambda (_: (((eq T u1 (TLRef n)) \to (land (eq nat n i0) (eq T u2 +(lift (S n) O v0)))))).(\lambda (t: T).(\lambda (k: K).(\lambda (H3: (eq T +(THead k u1 t) (TLRef n))).(let H4 \def (eq_ind T (THead k u1 t) (\lambda +(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) +\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow +True])) I (TLRef n) H3) in (False_ind (land (eq nat n i0) (eq T (THead k u2 +t) (lift (S n) O v0))) H4))))))))))) (\lambda (k: K).(\lambda (v0: +T).(\lambda (t2: T).(\lambda (t1: T).(\lambda (i0: nat).(\lambda (_: (subst0 +(s k i0) v0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (land (eq nat n (s +k i0)) (eq T t2 (lift (S n) O v0)))))).(\lambda (u: T).(\lambda (H3: (eq T +(THead k u t1) (TLRef n))).(let H4 \def (eq_ind T (THead k u t1) (\lambda +(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) +\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow +True])) I (TLRef n) H3) in (False_ind (land (eq nat n i0) (eq T (THead k u +t2) (lift (S n) O v0))) H4))))))))))) (\lambda (v0: T).(\lambda (u1: +T).(\lambda (u2: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v0 u1 +u2)).(\lambda (_: (((eq T u1 (TLRef n)) \to (land (eq nat n i0) (eq T u2 +(lift (S n) O v0)))))).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: +T).(\lambda (_: (subst0 (s k i0) v0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef +n)) \to (land (eq nat n (s k i0)) (eq T t2 (lift (S n) O v0)))))).(\lambda +(H5: (eq T (THead k u1 t1) (TLRef n))).(let H6 \def (eq_ind T (THead k u1 t1) +(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) +\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow +True])) I (TLRef n) H5) in (False_ind (land (eq nat n i0) (eq T (THead k u2 +t2) (lift (S n) O v0))) H6)))))))))))))) i v y x H0))) H))))). +(* COMMENTS +Initial nodes: 779 +END *) + +theorem subst0_gen_head: + \forall (k: K).(\forall (v: T).(\forall (u1: T).(\forall (t1: T).(\forall +(x: T).(\forall (i: nat).((subst0 i v (THead k u1 t1) x) \to (or3 (ex2 T +(\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 +u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k u1 t2))) (\lambda (t2: +T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: +T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1 +u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1 t2))))))))))) +\def + \lambda (k: K).(\lambda (v: T).(\lambda (u1: T).(\lambda (t1: T).(\lambda +(x: T).(\lambda (i: nat).(\lambda (H: (subst0 i v (THead k u1 t1) +x)).(insert_eq T (THead k u1 t1) (\lambda (t: T).(subst0 i v t x)) (\lambda +(_: T).(or3 (ex2 T (\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2: +T).(subst0 i v u1 u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k u1 t2))) +(\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u2: +T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: +T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1 +t2)))))) (\lambda (y: T).(\lambda (H0: (subst0 i v y x)).(subst0_ind (\lambda +(n: nat).(\lambda (t: T).(\lambda (t0: T).(\lambda (t2: T).((eq T t0 (THead k +u1 t1)) \to (or3 (ex2 T (\lambda (u2: T).(eq T t2 (THead k u2 t1))) (\lambda +(u2: T).(subst0 n t u1 u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead k u1 +t3))) (\lambda (t3: T).(subst0 (s k n) t t1 t3))) (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: +T).(subst0 n t u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k n) t t1 +t3)))))))))) (\lambda (v0: T).(\lambda (i0: nat).(\lambda (H1: (eq T (TLRef +i0) (THead k u1 t1))).(let H2 \def (eq_ind T (TLRef i0) (\lambda (ee: +T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow +False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I +(THead k u1 t1) H1) in (False_ind (or3 (ex2 T (\lambda (u2: T).(eq T (lift (S +i0) O v0) (THead k u2 t1))) (\lambda (u2: T).(subst0 i0 v0 u1 u2))) (ex2 T +(\lambda (t2: T).(eq T (lift (S i0) O v0) (THead k u1 t2))) (\lambda (t2: +T).(subst0 (s k i0) v0 t1 t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: +T).(eq T (lift (S i0) O v0) (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: +T).(subst0 i0 v0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i0) +v0 t1 t2))))) H2))))) (\lambda (v0: T).(\lambda (u2: T).(\lambda (u0: +T).(\lambda (i0: nat).(\lambda (H1: (subst0 i0 v0 u0 u2)).(\lambda (H2: (((eq +T u0 (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T u2 (THead k u3 +t1))) (\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t2: T).(eq T +u2 (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))) (ex3_2 T T +(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead k u3 t2)))) (\lambda (u3: +T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t2: +T).(subst0 (s k i0) v0 t1 t2)))))))).(\lambda (t: T).(\lambda (k0: +K).(\lambda (H3: (eq T (THead k0 u0 t) (THead k u1 t1))).(let H4 \def +(f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with +[(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _ _) +\Rightarrow k1])) (THead k0 u0 t) (THead k u1 t1) H3) in ((let H5 \def +(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with +[(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t0 _) +\Rightarrow t0])) (THead k0 u0 t) (THead k u1 t1) H3) in ((let H6 \def +(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with +[(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t0) +\Rightarrow t0])) (THead k0 u0 t) (THead k u1 t1) H3) in (\lambda (H7: (eq T +u0 u1)).(\lambda (H8: (eq K k0 k)).(eq_ind_r K k (\lambda (k1: K).(or3 (ex2 T +(\lambda (u3: T).(eq T (THead k1 u2 t) (THead k u3 t1))) (\lambda (u3: +T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t2: T).(eq T (THead k1 u2 t) +(THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))) (ex3_2 T T +(\lambda (u3: T).(\lambda (t2: T).(eq T (THead k1 u2 t) (THead k u3 t2)))) +(\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: +T).(\lambda (t2: T).(subst0 (s k i0) v0 t1 t2)))))) (eq_ind_r T t1 (\lambda +(t0: T).(or3 (ex2 T (\lambda (u3: T).(eq T (THead k u2 t0) (THead k u3 t1))) +(\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t2: T).(eq T (THead +k u2 t0) (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))) +(ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead k u2 t0) (THead k +u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda +(_: T).(\lambda (t2: T).(subst0 (s k i0) v0 t1 t2)))))) (let H9 \def (eq_ind +T u0 (\lambda (t0: T).((eq T t0 (THead k u1 t1)) \to (or3 (ex2 T (\lambda +(u3: T).(eq T u2 (THead k u3 t1))) (\lambda (u3: T).(subst0 i0 v0 u1 u3))) +(ex2 T (\lambda (t2: T).(eq T u2 (THead k u1 t2))) (\lambda (t2: T).(subst0 +(s k i0) v0 t1 t2))) (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 +(THead k u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) +(\lambda (_: T).(\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))))))) H2 u1 H7) +in (let H10 \def (eq_ind T u0 (\lambda (t0: T).(subst0 i0 v0 t0 u2)) H1 u1 +H7) in (or3_intro0 (ex2 T (\lambda (u3: T).(eq T (THead k u2 t1) (THead k u3 +t1))) (\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t2: T).(eq T +(THead k u2 t1) (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1 +t2))) (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead k u2 t1) +(THead k u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) +(\lambda (_: T).(\lambda (t2: T).(subst0 (s k i0) v0 t1 t2)))) (ex_intro2 T +(\lambda (u3: T).(eq T (THead k u2 t1) (THead k u3 t1))) (\lambda (u3: +T).(subst0 i0 v0 u1 u3)) u2 (refl_equal T (THead k u2 t1)) H10)))) t H6) k0 +H8)))) H5)) H4))))))))))) (\lambda (k0: K).(\lambda (v0: T).(\lambda (t2: +T).(\lambda (t0: T).(\lambda (i0: nat).(\lambda (H1: (subst0 (s k0 i0) v0 t0 +t2)).(\lambda (H2: (((eq T t0 (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u2: +T).(eq T t2 (THead k u2 t1))) (\lambda (u2: T).(subst0 (s k0 i0) v0 u1 u2))) +(ex2 T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 +(s k (s k0 i0)) v0 t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq +T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s k0 i0) v0 +u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k0 i0)) v0 t1 +t3)))))))).(\lambda (u: T).(\lambda (H3: (eq T (THead k0 u t0) (THead k u1 +t1))).(let H4 \def (f_equal T K (\lambda (e: T).(match e in T return (\lambda +(_: T).K) with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead +k1 _ _) \Rightarrow k1])) (THead k0 u t0) (THead k u1 t1) H3) in ((let H5 +\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) +with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) +\Rightarrow t])) (THead k0 u t0) (THead k u1 t1) H3) in ((let H6 \def +(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with +[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) +\Rightarrow t])) (THead k0 u t0) (THead k u1 t1) H3) in (\lambda (H7: (eq T u +u1)).(\lambda (H8: (eq K k0 k)).(eq_ind_r T u1 (\lambda (t: T).(or3 (ex2 T +(\lambda (u2: T).(eq T (THead k0 t t2) (THead k u2 t1))) (\lambda (u2: +T).(subst0 i0 v0 u1 u2))) (ex2 T (\lambda (t3: T).(eq T (THead k0 t t2) +(THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T (THead k0 t t2) (THead k u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(subst0 (s k i0) v0 t1 t3)))))) (let H9 \def (eq_ind T t0 +(\lambda (t: T).((eq T t (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u2: +T).(eq T t2 (THead k u2 t1))) (\lambda (u2: T).(subst0 (s k0 i0) v0 u1 u2))) +(ex2 T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 +(s k (s k0 i0)) v0 t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq +T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s k0 i0) v0 +u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k0 i0)) v0 t1 +t3))))))) H2 t1 H6) in (let H10 \def (eq_ind T t0 (\lambda (t: T).(subst0 (s +k0 i0) v0 t t2)) H1 t1 H6) in (let H11 \def (eq_ind K k0 (\lambda (k1: +K).((eq T t1 (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u2: T).(eq T t2 +(THead k u2 t1))) (\lambda (u2: T).(subst0 (s k1 i0) v0 u1 u2))) (ex2 T +(\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k (s +k1 i0)) v0 t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 +(THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s k1 i0) v0 u1 +u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k1 i0)) v0 t1 +t3))))))) H9 k H8) in (let H12 \def (eq_ind K k0 (\lambda (k1: K).(subst0 (s +k1 i0) v0 t1 t2)) H10 k H8) in (eq_ind_r K k (\lambda (k1: K).(or3 (ex2 T +(\lambda (u2: T).(eq T (THead k1 u1 t2) (THead k u2 t1))) (\lambda (u2: +T).(subst0 i0 v0 u1 u2))) (ex2 T (\lambda (t3: T).(eq T (THead k1 u1 t2) +(THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T (THead k1 u1 t2) (THead k u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(subst0 (s k i0) v0 t1 t3)))))) (or3_intro1 (ex2 T +(\lambda (u2: T).(eq T (THead k u1 t2) (THead k u2 t1))) (\lambda (u2: +T).(subst0 i0 v0 u1 u2))) (ex2 T (\lambda (t3: T).(eq T (THead k u1 t2) +(THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T (THead k u1 t2) (THead k u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(subst0 (s k i0) v0 t1 t3)))) (ex_intro2 T (\lambda (t3: +T).(eq T (THead k u1 t2) (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) +v0 t1 t3)) t2 (refl_equal T (THead k u1 t2)) H12)) k0 H8))))) u H7)))) H5)) +H4))))))))))) (\lambda (v0: T).(\lambda (u0: T).(\lambda (u2: T).(\lambda +(i0: nat).(\lambda (H1: (subst0 i0 v0 u0 u2)).(\lambda (H2: (((eq T u0 (THead +k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T u2 (THead k u3 t1))) +(\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t2: T).(eq T u2 +(THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))) (ex3_2 T T +(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead k u3 t2)))) (\lambda (u3: +T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t2: +T).(subst0 (s k i0) v0 t1 t2)))))))).(\lambda (k0: K).(\lambda (t0: +T).(\lambda (t2: T).(\lambda (H3: (subst0 (s k0 i0) v0 t0 t2)).(\lambda (H4: +(((eq T t0 (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T t2 (THead +k u3 t1))) (\lambda (u3: T).(subst0 (s k0 i0) v0 u1 u3))) (ex2 T (\lambda +(t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k (s k0 i0)) +v0 t1 t3))) (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead k u3 +t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 (s k0 i0) v0 u1 u3))) +(\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k0 i0)) v0 t1 +t3)))))))).(\lambda (H5: (eq T (THead k0 u0 t0) (THead k u1 t1))).(let H6 +\def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) +with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _ _) +\Rightarrow k1])) (THead k0 u0 t0) (THead k u1 t1) H5) in ((let H7 \def +(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with +[(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) +\Rightarrow t])) (THead k0 u0 t0) (THead k u1 t1) H5) in ((let H8 \def +(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with +[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) +\Rightarrow t])) (THead k0 u0 t0) (THead k u1 t1) H5) in (\lambda (H9: (eq T +u0 u1)).(\lambda (H10: (eq K k0 k)).(let H11 \def (eq_ind T t0 (\lambda (t: +T).((eq T t (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T t2 (THead +k u3 t1))) (\lambda (u3: T).(subst0 (s k0 i0) v0 u1 u3))) (ex2 T (\lambda +(t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k (s k0 i0)) +v0 t1 t3))) (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead k u3 +t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 (s k0 i0) v0 u1 u3))) +(\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k0 i0)) v0 t1 t3))))))) H4 +t1 H8) in (let H12 \def (eq_ind T t0 (\lambda (t: T).(subst0 (s k0 i0) v0 t +t2)) H3 t1 H8) in (let H13 \def (eq_ind K k0 (\lambda (k1: K).((eq T t1 +(THead k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T t2 (THead k u3 t1))) +(\lambda (u3: T).(subst0 (s k1 i0) v0 u1 u3))) (ex2 T (\lambda (t3: T).(eq T +t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k (s k1 i0)) v0 t1 t3))) +(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead k u3 t3)))) +(\lambda (u3: T).(\lambda (_: T).(subst0 (s k1 i0) v0 u1 u3))) (\lambda (_: +T).(\lambda (t3: T).(subst0 (s k (s k1 i0)) v0 t1 t3))))))) H11 k H10) in +(let H14 \def (eq_ind K k0 (\lambda (k1: K).(subst0 (s k1 i0) v0 t1 t2)) H12 +k H10) in (eq_ind_r K k (\lambda (k1: K).(or3 (ex2 T (\lambda (u3: T).(eq T +(THead k1 u2 t2) (THead k u3 t1))) (\lambda (u3: T).(subst0 i0 v0 u1 u3))) +(ex2 T (\lambda (t3: T).(eq T (THead k1 u2 t2) (THead k u1 t3))) (\lambda +(t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T (\lambda (u3: T).(\lambda +(t3: T).(eq T (THead k1 u2 t2) (THead k u3 t3)))) (\lambda (u3: T).(\lambda +(_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k +i0) v0 t1 t3)))))) (let H15 \def (eq_ind T u0 (\lambda (t: T).((eq T t (THead +k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T u2 (THead k u3 t1))) +(\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t3: T).(eq T u2 +(THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T +(\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead k u3 t3)))) (\lambda (u3: +T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t3: +T).(subst0 (s k i0) v0 t1 t3))))))) H2 u1 H9) in (let H16 \def (eq_ind T u0 +(\lambda (t: T).(subst0 i0 v0 t u2)) H1 u1 H9) in (or3_intro2 (ex2 T (\lambda +(u3: T).(eq T (THead k u2 t2) (THead k u3 t1))) (\lambda (u3: T).(subst0 i0 +v0 u1 u3))) (ex2 T (\lambda (t3: T).(eq T (THead k u2 t2) (THead k u1 t3))) +(\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T (\lambda (u3: +T).(\lambda (t3: T).(eq T (THead k u2 t2) (THead k u3 t3)))) (\lambda (u3: +T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t3: +T).(subst0 (s k i0) v0 t1 t3)))) (ex3_2_intro T T (\lambda (u3: T).(\lambda +(t3: T).(eq T (THead k u2 t2) (THead k u3 t3)))) (\lambda (u3: T).(\lambda +(_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k +i0) v0 t1 t3))) u2 t2 (refl_equal T (THead k u2 t2)) H16 H14)))) k0 +H10)))))))) H7)) H6)))))))))))))) i v y x H0))) H))))))). +(* COMMENTS +Initial nodes: 4255 +END *) + +theorem subst0_gen_lift_lt: + \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall +(h: nat).(\forall (d: nat).((subst0 i (lift h d u) (lift h (S (plus i d)) t1) +x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda +(t2: T).(subst0 i u t1 t2))))))))) +\def + \lambda (u: T).(\lambda (t1: T).(T_ind (\lambda (t: T).(\forall (x: +T).(\forall (i: nat).(\forall (h: nat).(\forall (d: nat).((subst0 i (lift h d +u) (lift h (S (plus i d)) t) x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h +(S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u t t2))))))))) (\lambda (n: +nat).(\lambda (x: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda (d: +nat).(\lambda (H: (subst0 i (lift h d u) (lift h (S (plus i d)) (TSort n)) +x)).(let H0 \def (eq_ind T (lift h (S (plus i d)) (TSort n)) (\lambda (t: +T).(subst0 i (lift h d u) t x)) H (TSort n) (lift_sort n h (S (plus i d)))) +in (subst0_gen_sort (lift h d u) x i n H0 (ex2 T (\lambda (t2: T).(eq T x +(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TSort n) +t2))))))))))) (\lambda (n: nat).(\lambda (x: T).(\lambda (i: nat).(\lambda +(h: nat).(\lambda (d: nat).(\lambda (H: (subst0 i (lift h d u) (lift h (S +(plus i d)) (TLRef n)) x)).(lt_le_e n (S (plus i d)) (ex2 T (\lambda (t2: +T).(eq T x (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef +n) t2))) (\lambda (H0: (lt n (S (plus i d)))).(let H1 \def (eq_ind T (lift h +(S (plus i d)) (TLRef n)) (\lambda (t: T).(subst0 i (lift h d u) t x)) H +(TLRef n) (lift_lref_lt n h (S (plus i d)) H0)) in (land_ind (eq nat n i) (eq +T x (lift (S n) O (lift h d u))) (ex2 T (\lambda (t2: T).(eq T x (lift h (S +(plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef n) t2))) (\lambda (H2: +(eq nat n i)).(\lambda (H3: (eq T x (lift (S n) O (lift h d u)))).(eq_ind_r T +(lift (S n) O (lift h d u)) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t +(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef n) t2)))) +(eq_ind_r nat i (\lambda (n0: nat).(ex2 T (\lambda (t2: T).(eq T (lift (S n0) +O (lift h d u)) (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u +(TLRef n0) t2)))) (eq_ind T (lift h (plus (S i) d) (lift (S i) O u)) (\lambda +(t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h (S (plus i d)) t2))) (\lambda +(t2: T).(subst0 i u (TLRef i) t2)))) (ex_intro2 T (\lambda (t2: T).(eq T +(lift h (S (plus i d)) (lift (S i) O u)) (lift h (S (plus i d)) t2))) +(\lambda (t2: T).(subst0 i u (TLRef i) t2)) (lift (S i) O u) (refl_equal T +(lift h (S (plus i d)) (lift (S i) O u))) (subst0_lref u i)) (lift (S i) O +(lift h d u)) (lift_d u h (S i) d O (le_O_n d))) n H2) x H3))) +(subst0_gen_lref (lift h d u) x i n H1)))) (\lambda (H0: (le (S (plus i d)) +n)).(let H1 \def (eq_ind T (lift h (S (plus i d)) (TLRef n)) (\lambda (t: +T).(subst0 i (lift h d u) t x)) H (TLRef (plus n h)) (lift_lref_ge n h (S +(plus i d)) H0)) in (land_ind (eq nat (plus n h) i) (eq T x (lift (S (plus n +h)) O (lift h d u))) (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) +t2))) (\lambda (t2: T).(subst0 i u (TLRef n) t2))) (\lambda (H2: (eq nat +(plus n h) i)).(\lambda (_: (eq T x (lift (S (plus n h)) O (lift h d +u)))).(let H4 \def (eq_ind_r nat i (\lambda (n0: nat).(le (S (plus n0 d)) n)) +H0 (plus n h) H2) in (le_false n (plus (plus n h) d) (ex2 T (\lambda (t2: +T).(eq T x (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef +n) t2))) (le_plus_trans n (plus n h) d (le_plus_l n h)) H4)))) +(subst0_gen_lref (lift h d u) x i (plus n h) H1))))))))))) (\lambda (k: +K).(\lambda (t: T).(\lambda (H: ((\forall (x: T).(\forall (i: nat).(\forall +(h: nat).(\forall (d: nat).((subst0 i (lift h d u) (lift h (S (plus i d)) t) +x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda +(t2: T).(subst0 i u t t2)))))))))).(\lambda (t0: T).(\lambda (H0: ((\forall +(x: T).(\forall (i: nat).(\forall (h: nat).(\forall (d: nat).((subst0 i (lift +h d u) (lift h (S (plus i d)) t0) x) \to (ex2 T (\lambda (t2: T).(eq T x +(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u t0 +t2)))))))))).(\lambda (x: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda (d: +nat).(\lambda (H1: (subst0 i (lift h d u) (lift h (S (plus i d)) (THead k t +t0)) x)).(let H2 \def (eq_ind T (lift h (S (plus i d)) (THead k t t0)) +(\lambda (t2: T).(subst0 i (lift h d u) t2 x)) H1 (THead k (lift h (S (plus i +d)) t) (lift h (s k (S (plus i d))) t0)) (lift_head k t t0 h (S (plus i d)))) +in (or3_ind (ex2 T (\lambda (u2: T).(eq T x (THead k u2 (lift h (s k (S (plus +i d))) t0)))) (\lambda (u2: T).(subst0 i (lift h d u) (lift h (S (plus i d)) +t) u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h (S (plus i d)) t) +t2))) (\lambda (t2: T).(subst0 (s k i) (lift h d u) (lift h (s k (S (plus i +d))) t0) t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k +u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i (lift h d u) (lift h (S +(plus i d)) t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) (lift h +d u) (lift h (s k (S (plus i d))) t0) t2)))) (ex2 T (\lambda (t2: T).(eq T x +(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) +t2))) (\lambda (H3: (ex2 T (\lambda (u2: T).(eq T x (THead k u2 (lift h (s k +(S (plus i d))) t0)))) (\lambda (u2: T).(subst0 i (lift h d u) (lift h (S +(plus i d)) t) u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead k u2 (lift h +(s k (S (plus i d))) t0)))) (\lambda (u2: T).(subst0 i (lift h d u) (lift h +(S (plus i d)) t) u2)) (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) +t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2))) (\lambda (x0: +T).(\lambda (H4: (eq T x (THead k x0 (lift h (s k (S (plus i d))) +t0)))).(\lambda (H5: (subst0 i (lift h d u) (lift h (S (plus i d)) t) +x0)).(eq_ind_r T (THead k x0 (lift h (s k (S (plus i d))) t0)) (\lambda (t2: +T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda +(t3: T).(subst0 i u (THead k t t0) t3)))) (ex2_ind T (\lambda (t2: T).(eq T +x0 (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u t t2)) (ex2 T +(\lambda (t2: T).(eq T (THead k x0 (lift h (s k (S (plus i d))) t0)) (lift h +(S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2))) +(\lambda (x1: T).(\lambda (H6: (eq T x0 (lift h (S (plus i d)) x1))).(\lambda +(H7: (subst0 i u t x1)).(eq_ind_r T (lift h (S (plus i d)) x1) (\lambda (t2: +T).(ex2 T (\lambda (t3: T).(eq T (THead k t2 (lift h (s k (S (plus i d))) +t0)) (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u (THead k t t0) +t3)))) (eq_ind T (lift h (S (plus i d)) (THead k x1 t0)) (\lambda (t2: +T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda +(t3: T).(subst0 i u (THead k t t0) t3)))) (ex_intro2 T (\lambda (t2: T).(eq T +(lift h (S (plus i d)) (THead k x1 t0)) (lift h (S (plus i d)) t2))) (\lambda +(t2: T).(subst0 i u (THead k t t0) t2)) (THead k x1 t0) (refl_equal T (lift h +(S (plus i d)) (THead k x1 t0))) (subst0_fst u x1 t i H7 t0 k)) (THead k +(lift h (S (plus i d)) x1) (lift h (s k (S (plus i d))) t0)) (lift_head k x1 +t0 h (S (plus i d)))) x0 H6)))) (H x0 i h d H5)) x H4)))) H3)) (\lambda (H3: +(ex2 T (\lambda (t2: T).(eq T x (THead k (lift h (S (plus i d)) t) t2))) +(\lambda (t2: T).(subst0 (s k i) (lift h d u) (lift h (s k (S (plus i d))) +t0) t2)))).(ex2_ind T (\lambda (t2: T).(eq T x (THead k (lift h (S (plus i +d)) t) t2))) (\lambda (t2: T).(subst0 (s k i) (lift h d u) (lift h (s k (S +(plus i d))) t0) t2)) (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) +t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2))) (\lambda (x0: +T).(\lambda (H4: (eq T x (THead k (lift h (S (plus i d)) t) x0))).(\lambda +(H5: (subst0 (s k i) (lift h d u) (lift h (s k (S (plus i d))) t0) +x0)).(eq_ind_r T (THead k (lift h (S (plus i d)) t) x0) (\lambda (t2: T).(ex2 +T (\lambda (t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda (t3: +T).(subst0 i u (THead k t t0) t3)))) (let H6 \def (eq_ind nat (s k (S (plus i +d))) (\lambda (n: nat).(subst0 (s k i) (lift h d u) (lift h n t0) x0)) H5 (S +(s k (plus i d))) (s_S k (plus i d))) in (let H7 \def (eq_ind nat (s k (plus +i d)) (\lambda (n: nat).(subst0 (s k i) (lift h d u) (lift h (S n) t0) x0)) +H6 (plus (s k i) d) (s_plus k i d)) in (ex2_ind T (\lambda (t2: T).(eq T x0 +(lift h (S (plus (s k i) d)) t2))) (\lambda (t2: T).(subst0 (s k i) u t0 t2)) +(ex2 T (\lambda (t2: T).(eq T (THead k (lift h (S (plus i d)) t) x0) (lift h +(S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2))) +(\lambda (x1: T).(\lambda (H8: (eq T x0 (lift h (S (plus (s k i) d)) +x1))).(\lambda (H9: (subst0 (s k i) u t0 x1)).(eq_ind_r T (lift h (S (plus (s +k i) d)) x1) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k (lift h +(S (plus i d)) t) t2) (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i +u (THead k t t0) t3)))) (eq_ind nat (s k (plus i d)) (\lambda (n: nat).(ex2 T +(\lambda (t2: T).(eq T (THead k (lift h (S (plus i d)) t) (lift h (S n) x1)) +(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) +t2)))) (eq_ind nat (s k (S (plus i d))) (\lambda (n: nat).(ex2 T (\lambda +(t2: T).(eq T (THead k (lift h (S (plus i d)) t) (lift h n x1)) (lift h (S +(plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2)))) (eq_ind +T (lift h (S (plus i d)) (THead k t x1)) (\lambda (t2: T).(ex2 T (\lambda +(t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u +(THead k t t0) t3)))) (ex_intro2 T (\lambda (t2: T).(eq T (lift h (S (plus i +d)) (THead k t x1)) (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u +(THead k t t0) t2)) (THead k t x1) (refl_equal T (lift h (S (plus i d)) +(THead k t x1))) (subst0_snd k u x1 t0 i H9 t)) (THead k (lift h (S (plus i +d)) t) (lift h (s k (S (plus i d))) x1)) (lift_head k t x1 h (S (plus i d)))) +(S (s k (plus i d))) (s_S k (plus i d))) (plus (s k i) d) (s_plus k i d)) x0 +H8)))) (H0 x0 (s k i) h d H7)))) x H4)))) H3)) (\lambda (H3: (ex3_2 T T +(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: +T).(\lambda (_: T).(subst0 i (lift h d u) (lift h (S (plus i d)) t) u2))) +(\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) (lift h d u) (lift h (s k (S +(plus i d))) t0) t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq +T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i (lift h d +u) (lift h (S (plus i d)) t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 +(s k i) (lift h d u) (lift h (s k (S (plus i d))) t0) t2))) (ex2 T (\lambda +(t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u +(THead k t t0) t2))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T x +(THead k x0 x1))).(\lambda (H5: (subst0 i (lift h d u) (lift h (S (plus i d)) +t) x0)).(\lambda (H6: (subst0 (s k i) (lift h d u) (lift h (s k (S (plus i +d))) t0) x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t2: T).(ex2 T (\lambda +(t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u +(THead k t t0) t3)))) (let H7 \def (eq_ind nat (s k (S (plus i d))) (\lambda +(n: nat).(subst0 (s k i) (lift h d u) (lift h n t0) x1)) H6 (S (s k (plus i +d))) (s_S k (plus i d))) in (let H8 \def (eq_ind nat (s k (plus i d)) +(\lambda (n: nat).(subst0 (s k i) (lift h d u) (lift h (S n) t0) x1)) H7 +(plus (s k i) d) (s_plus k i d)) in (ex2_ind T (\lambda (t2: T).(eq T x1 +(lift h (S (plus (s k i) d)) t2))) (\lambda (t2: T).(subst0 (s k i) u t0 t2)) +(ex2 T (\lambda (t2: T).(eq T (THead k x0 x1) (lift h (S (plus i d)) t2))) +(\lambda (t2: T).(subst0 i u (THead k t t0) t2))) (\lambda (x2: T).(\lambda +(H9: (eq T x1 (lift h (S (plus (s k i) d)) x2))).(\lambda (H10: (subst0 (s k +i) u t0 x2)).(ex2_ind T (\lambda (t2: T).(eq T x0 (lift h (S (plus i d)) +t2))) (\lambda (t2: T).(subst0 i u t t2)) (ex2 T (\lambda (t2: T).(eq T +(THead k x0 x1) (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u +(THead k t t0) t2))) (\lambda (x3: T).(\lambda (H11: (eq T x0 (lift h (S +(plus i d)) x3))).(\lambda (H12: (subst0 i u t x3)).(eq_ind_r T (lift h (S +(plus i d)) x3) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k t2 +x1) (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u (THead k t t0) +t3)))) (eq_ind_r T (lift h (S (plus (s k i) d)) x2) (\lambda (t2: T).(ex2 T +(\lambda (t3: T).(eq T (THead k (lift h (S (plus i d)) x3) t2) (lift h (S +(plus i d)) t3))) (\lambda (t3: T).(subst0 i u (THead k t t0) t3)))) (eq_ind +nat (s k (plus i d)) (\lambda (n: nat).(ex2 T (\lambda (t2: T).(eq T (THead k +(lift h (S (plus i d)) x3) (lift h (S n) x2)) (lift h (S (plus i d)) t2))) +(\lambda (t2: T).(subst0 i u (THead k t t0) t2)))) (eq_ind nat (s k (S (plus +i d))) (\lambda (n: nat).(ex2 T (\lambda (t2: T).(eq T (THead k (lift h (S +(plus i d)) x3) (lift h n x2)) (lift h (S (plus i d)) t2))) (\lambda (t2: +T).(subst0 i u (THead k t t0) t2)))) (eq_ind T (lift h (S (plus i d)) (THead +k x3 x2)) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h (S (plus +i d)) t3))) (\lambda (t3: T).(subst0 i u (THead k t t0) t3)))) (ex_intro2 T +(\lambda (t2: T).(eq T (lift h (S (plus i d)) (THead k x3 x2)) (lift h (S +(plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2)) (THead k +x3 x2) (refl_equal T (lift h (S (plus i d)) (THead k x3 x2))) (subst0_both u +t x3 i H12 k t0 x2 H10)) (THead k (lift h (S (plus i d)) x3) (lift h (s k (S +(plus i d))) x2)) (lift_head k x3 x2 h (S (plus i d)))) (S (s k (plus i d))) +(s_S k (plus i d))) (plus (s k i) d) (s_plus k i d)) x1 H9) x0 H11)))) (H x0 +i h d H5))))) (H0 x1 (s k i) h d H8)))) x H4)))))) H3)) (subst0_gen_head k +(lift h d u) (lift h (S (plus i d)) t) (lift h (s k (S (plus i d))) t0) x i +H2))))))))))))) t1)). +(* COMMENTS +Initial nodes: 5157 +END *) + +theorem subst0_gen_lift_false: + \forall (t: T).(\forall (u: T).(\forall (x: T).(\forall (h: nat).(\forall +(d: nat).(\forall (i: nat).((le d i) \to ((lt i (plus d h)) \to ((subst0 i u +(lift h d t) x) \to (\forall (P: Prop).P))))))))) +\def + \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (u: T).(\forall (x: +T).(\forall (h: nat).(\forall (d: nat).(\forall (i: nat).((le d i) \to ((lt i +(plus d h)) \to ((subst0 i u (lift h d t0) x) \to (\forall (P: +Prop).P)))))))))) (\lambda (n: nat).(\lambda (u: T).(\lambda (x: T).(\lambda +(h: nat).(\lambda (d: nat).(\lambda (i: nat).(\lambda (_: (le d i)).(\lambda +(_: (lt i (plus d h))).(\lambda (H1: (subst0 i u (lift h d (TSort n)) +x)).(\lambda (P: Prop).(let H2 \def (eq_ind T (lift h d (TSort n)) (\lambda +(t0: T).(subst0 i u t0 x)) H1 (TSort n) (lift_sort n h d)) in +(subst0_gen_sort u x i n H2 P)))))))))))) (\lambda (n: nat).(\lambda (u: +T).(\lambda (x: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (i: +nat).(\lambda (H: (le d i)).(\lambda (H0: (lt i (plus d h))).(\lambda (H1: +(subst0 i u (lift h d (TLRef n)) x)).(\lambda (P: Prop).(lt_le_e n d P +(\lambda (H2: (lt n d)).(let H3 \def (eq_ind T (lift h d (TLRef n)) (\lambda +(t0: T).(subst0 i u t0 x)) H1 (TLRef n) (lift_lref_lt n h d H2)) in (land_ind +(eq nat n i) (eq T x (lift (S n) O u)) P (\lambda (H4: (eq nat n i)).(\lambda +(_: (eq T x (lift (S n) O u))).(let H6 \def (eq_ind nat n (\lambda (n0: +nat).(lt n0 d)) H2 i H4) in (le_false d i P H H6)))) (subst0_gen_lref u x i n +H3)))) (\lambda (H2: (le d n)).(let H3 \def (eq_ind T (lift h d (TLRef n)) +(\lambda (t0: T).(subst0 i u t0 x)) H1 (TLRef (plus n h)) (lift_lref_ge n h d +H2)) in (land_ind (eq nat (plus n h) i) (eq T x (lift (S (plus n h)) O u)) P +(\lambda (H4: (eq nat (plus n h) i)).(\lambda (_: (eq T x (lift (S (plus n +h)) O u))).(let H6 \def (eq_ind_r nat i (\lambda (n0: nat).(lt n0 (plus d +h))) H0 (plus n h) H4) in (le_false d n P H2 (lt_le_S n d (simpl_lt_plus_r h +n d H6)))))) (subst0_gen_lref u x i (plus n h) H3))))))))))))))) (\lambda (k: +K).(\lambda (t0: T).(\lambda (H: ((\forall (u: T).(\forall (x: T).(\forall +(h: nat).(\forall (d: nat).(\forall (i: nat).((le d i) \to ((lt i (plus d h)) +\to ((subst0 i u (lift h d t0) x) \to (\forall (P: +Prop).P))))))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (u: T).(\forall +(x: T).(\forall (h: nat).(\forall (d: nat).(\forall (i: nat).((le d i) \to +((lt i (plus d h)) \to ((subst0 i u (lift h d t1) x) \to (\forall (P: +Prop).P))))))))))).(\lambda (u: T).(\lambda (x: T).(\lambda (h: nat).(\lambda +(d: nat).(\lambda (i: nat).(\lambda (H1: (le d i)).(\lambda (H2: (lt i (plus +d h))).(\lambda (H3: (subst0 i u (lift h d (THead k t0 t1)) x)).(\lambda (P: +Prop).(let H4 \def (eq_ind T (lift h d (THead k t0 t1)) (\lambda (t2: +T).(subst0 i u t2 x)) H3 (THead k (lift h d t0) (lift h (s k d) t1)) +(lift_head k t0 t1 h d)) in (or3_ind (ex2 T (\lambda (u2: T).(eq T x (THead k +u2 (lift h (s k d) t1)))) (\lambda (u2: T).(subst0 i u (lift h d t0) u2))) +(ex2 T (\lambda (t2: T).(eq T x (THead k (lift h d t0) t2))) (\lambda (t2: +T).(subst0 (s k i) u (lift h (s k d) t1) t2))) (ex3_2 T T (\lambda (u2: +T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: +T).(subst0 i u (lift h d t0) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 +(s k i) u (lift h (s k d) t1) t2)))) P (\lambda (H5: (ex2 T (\lambda (u2: +T).(eq T x (THead k u2 (lift h (s k d) t1)))) (\lambda (u2: T).(subst0 i u +(lift h d t0) u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead k u2 (lift h +(s k d) t1)))) (\lambda (u2: T).(subst0 i u (lift h d t0) u2)) P (\lambda +(x0: T).(\lambda (_: (eq T x (THead k x0 (lift h (s k d) t1)))).(\lambda (H7: +(subst0 i u (lift h d t0) x0)).(H u x0 h d i H1 H2 H7 P)))) H5)) (\lambda +(H5: (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h d t0) t2))) (\lambda +(t2: T).(subst0 (s k i) u (lift h (s k d) t1) t2)))).(ex2_ind T (\lambda (t2: +T).(eq T x (THead k (lift h d t0) t2))) (\lambda (t2: T).(subst0 (s k i) u +(lift h (s k d) t1) t2)) P (\lambda (x0: T).(\lambda (_: (eq T x (THead k +(lift h d t0) x0))).(\lambda (H7: (subst0 (s k i) u (lift h (s k d) t1) +x0)).(H0 u x0 h (s k d) (s k i) (s_le k d i H1) (eq_ind nat (s k (plus d h)) +(\lambda (n: nat).(lt (s k i) n)) (lt_le_S (s k i) (s k (plus d h)) (s_lt k i +(plus d h) H2)) (plus (s k d) h) (s_plus k d h)) H7 P)))) H5)) (\lambda (H5: +(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) +(\lambda (u2: T).(\lambda (_: T).(subst0 i u (lift h d t0) u2))) (\lambda (_: +T).(\lambda (t2: T).(subst0 (s k i) u (lift h (s k d) t1) t2))))).(ex3_2_ind +T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda +(u2: T).(\lambda (_: T).(subst0 i u (lift h d t0) u2))) (\lambda (_: +T).(\lambda (t2: T).(subst0 (s k i) u (lift h (s k d) t1) t2))) P (\lambda +(x0: T).(\lambda (x1: T).(\lambda (_: (eq T x (THead k x0 x1))).(\lambda (H7: +(subst0 i u (lift h d t0) x0)).(\lambda (_: (subst0 (s k i) u (lift h (s k d) +t1) x1)).(H u x0 h d i H1 H2 H7 P)))))) H5)) (subst0_gen_head k u (lift h d +t0) (lift h (s k d) t1) x i H4))))))))))))))))) t). +(* COMMENTS +Initial nodes: 1621 +END *) + +theorem subst0_gen_lift_ge: + \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall +(h: nat).(\forall (d: nat).((subst0 i u (lift h d t1) x) \to ((le (plus d h) +i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: +T).(subst0 (minus i h) u t1 t2)))))))))) +\def + \lambda (u: T).(\lambda (t1: T).(T_ind (\lambda (t: T).(\forall (x: +T).(\forall (i: nat).(\forall (h: nat).(\forall (d: nat).((subst0 i u (lift h +d t) x) \to ((le (plus d h) i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d +t2))) (\lambda (t2: T).(subst0 (minus i h) u t t2)))))))))) (\lambda (n: +nat).(\lambda (x: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda (d: +nat).(\lambda (H: (subst0 i u (lift h d (TSort n)) x)).(\lambda (_: (le (plus +d h) i)).(let H1 \def (eq_ind T (lift h d (TSort n)) (\lambda (t: T).(subst0 +i u t x)) H (TSort n) (lift_sort n h d)) in (subst0_gen_sort u x i n H1 (ex2 +T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i +h) u (TSort n) t2)))))))))))) (\lambda (n: nat).(\lambda (x: T).(\lambda (i: +nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (subst0 i u (lift h d +(TLRef n)) x)).(\lambda (H0: (le (plus d h) i)).(lt_le_e n d (ex2 T (\lambda +(t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (TLRef +n) t2))) (\lambda (H1: (lt n d)).(let H2 \def (eq_ind T (lift h d (TLRef n)) +(\lambda (t: T).(subst0 i u t x)) H (TLRef n) (lift_lref_lt n h d H1)) in +(land_ind (eq nat n i) (eq T x (lift (S n) O u)) (ex2 T (\lambda (t2: T).(eq +T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (TLRef n) t2))) +(\lambda (H3: (eq nat n i)).(\lambda (_: (eq T x (lift (S n) O u))).(let H5 +\def (eq_ind nat n (\lambda (n0: nat).(lt n0 d)) H1 i H3) in (le_false (plus +d h) i (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: +T).(subst0 (minus i h) u (TLRef n) t2))) H0 (le_plus_trans (S i) d h H5))))) +(subst0_gen_lref u x i n H2)))) (\lambda (H1: (le d n)).(let H2 \def (eq_ind +T (lift h d (TLRef n)) (\lambda (t: T).(subst0 i u t x)) H (TLRef (plus n h)) +(lift_lref_ge n h d H1)) in (land_ind (eq nat (plus n h) i) (eq T x (lift (S +(plus n h)) O u)) (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda +(t2: T).(subst0 (minus i h) u (TLRef n) t2))) (\lambda (H3: (eq nat (plus n +h) i)).(\lambda (H4: (eq T x (lift (S (plus n h)) O u))).(eq_ind nat (plus n +h) (\lambda (n0: nat).(ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) +(\lambda (t2: T).(subst0 (minus n0 h) u (TLRef n) t2)))) (eq_ind_r T (lift (S +(plus n h)) O u) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h d +t2))) (\lambda (t2: T).(subst0 (minus (plus n h) h) u (TLRef n) t2)))) +(eq_ind_r nat n (\lambda (n0: nat).(ex2 T (\lambda (t2: T).(eq T (lift (S +(plus n h)) O u) (lift h d t2))) (\lambda (t2: T).(subst0 n0 u (TLRef n) +t2)))) (ex_intro2 T (\lambda (t2: T).(eq T (lift (S (plus n h)) O u) (lift h +d t2))) (\lambda (t2: T).(subst0 n u (TLRef n) t2)) (lift (S n) O u) +(eq_ind_r T (lift (plus h (S n)) O u) (\lambda (t: T).(eq T (lift (S (plus n +h)) O u) t)) (eq_ind_r nat (plus h n) (\lambda (n0: nat).(eq T (lift (S n0) O +u) (lift (plus h (S n)) O u))) (eq_ind_r nat (plus h (S n)) (\lambda (n0: +nat).(eq T (lift n0 O u) (lift (plus h (S n)) O u))) (refl_equal T (lift +(plus h (S n)) O u)) (S (plus h n)) (plus_n_Sm h n)) (plus n h) (plus_sym n +h)) (lift h d (lift (S n) O u)) (lift_free u (S n) h O d (le_trans_plus_r O d +(plus O (S n)) (le_plus_plus O O d (S n) (le_n O) (le_S d n H1))) (le_O_n +d))) (subst0_lref u n)) (minus (plus n h) h) (minus_plus_r n h)) x H4) i +H3))) (subst0_gen_lref u x i (plus n h) H2)))))))))))) (\lambda (k: +K).(\lambda (t: T).(\lambda (H: ((\forall (x: T).(\forall (i: nat).(\forall +(h: nat).(\forall (d: nat).((subst0 i u (lift h d t) x) \to ((le (plus d h) +i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: +T).(subst0 (minus i h) u t t2))))))))))).(\lambda (t0: T).(\lambda (H0: +((\forall (x: T).(\forall (i: nat).(\forall (h: nat).(\forall (d: +nat).((subst0 i u (lift h d t0) x) \to ((le (plus d h) i) \to (ex2 T (\lambda +(t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u t0 +t2))))))))))).(\lambda (x: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda +(d: nat).(\lambda (H1: (subst0 i u (lift h d (THead k t t0)) x)).(\lambda +(H2: (le (plus d h) i)).(let H3 \def (eq_ind T (lift h d (THead k t t0)) +(\lambda (t2: T).(subst0 i u t2 x)) H1 (THead k (lift h d t) (lift h (s k d) +t0)) (lift_head k t t0 h d)) in (or3_ind (ex2 T (\lambda (u2: T).(eq T x +(THead k u2 (lift h (s k d) t0)))) (\lambda (u2: T).(subst0 i u (lift h d t) +u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h d t) t2))) (\lambda +(t2: T).(subst0 (s k i) u (lift h (s k d) t0) t2))) (ex3_2 T T (\lambda (u2: +T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: +T).(subst0 i u (lift h d t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s +k i) u (lift h (s k d) t0) t2)))) (ex2 T (\lambda (t2: T).(eq T x (lift h d +t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda +(H4: (ex2 T (\lambda (u2: T).(eq T x (THead k u2 (lift h (s k d) t0)))) +(\lambda (u2: T).(subst0 i u (lift h d t) u2)))).(ex2_ind T (\lambda (u2: +T).(eq T x (THead k u2 (lift h (s k d) t0)))) (\lambda (u2: T).(subst0 i u +(lift h d t) u2)) (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda +(t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (x0: T).(\lambda +(H5: (eq T x (THead k x0 (lift h (s k d) t0)))).(\lambda (H6: (subst0 i u +(lift h d t) x0)).(eq_ind_r T (THead k x0 (lift h (s k d) t0)) (\lambda (t2: +T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst0 +(minus i h) u (THead k t t0) t3)))) (ex2_ind T (\lambda (t2: T).(eq T x0 +(lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u t t2)) (ex2 T (\lambda +(t2: T).(eq T (THead k x0 (lift h (s k d) t0)) (lift h d t2))) (\lambda (t2: +T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (x1: T).(\lambda (H7: +(eq T x0 (lift h d x1))).(\lambda (H8: (subst0 (minus i h) u t x1)).(eq_ind_r +T (lift h d x1) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k t2 +(lift h (s k d) t0)) (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u +(THead k t t0) t3)))) (eq_ind T (lift h d (THead k x1 t0)) (\lambda (t2: +T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst0 +(minus i h) u (THead k t t0) t3)))) (ex_intro2 T (\lambda (t2: T).(eq T (lift +h d (THead k x1 t0)) (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u +(THead k t t0) t2)) (THead k x1 t0) (refl_equal T (lift h d (THead k x1 t0))) +(subst0_fst u x1 t (minus i h) H8 t0 k)) (THead k (lift h d x1) (lift h (s k +d) t0)) (lift_head k x1 t0 h d)) x0 H7)))) (H x0 i h d H6 H2)) x H5)))) H4)) +(\lambda (H4: (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h d t) t2))) +(\lambda (t2: T).(subst0 (s k i) u (lift h (s k d) t0) t2)))).(ex2_ind T +(\lambda (t2: T).(eq T x (THead k (lift h d t) t2))) (\lambda (t2: T).(subst0 +(s k i) u (lift h (s k d) t0) t2)) (ex2 T (\lambda (t2: T).(eq T x (lift h d +t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda +(x0: T).(\lambda (H5: (eq T x (THead k (lift h d t) x0))).(\lambda (H6: +(subst0 (s k i) u (lift h (s k d) t0) x0)).(eq_ind_r T (THead k (lift h d t) +x0) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3))) +(\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) (ex2_ind T +(\lambda (t2: T).(eq T x0 (lift h (s k d) t2))) (\lambda (t2: T).(subst0 +(minus (s k i) h) u t0 t2)) (ex2 T (\lambda (t2: T).(eq T (THead k (lift h d +t) x0) (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead k t t0) +t2))) (\lambda (x1: T).(\lambda (H7: (eq T x0 (lift h (s k d) x1))).(\lambda +(H8: (subst0 (minus (s k i) h) u t0 x1)).(eq_ind_r T (lift h (s k d) x1) +(\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k (lift h d t) t2) +(lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) +(eq_ind T (lift h d (THead k t x1)) (\lambda (t2: T).(ex2 T (\lambda (t3: +T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t +t0) t3)))) (let H9 \def (eq_ind_r nat (minus (s k i) h) (\lambda (n: +nat).(subst0 n u t0 x1)) H8 (s k (minus i h)) (s_minus k i h (le_trans_plus_r +d h i H2))) in (ex_intro2 T (\lambda (t2: T).(eq T (lift h d (THead k t x1)) +(lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead k t t0) t2)) +(THead k t x1) (refl_equal T (lift h d (THead k t x1))) (subst0_snd k u x1 t0 +(minus i h) H9 t))) (THead k (lift h d t) (lift h (s k d) x1)) (lift_head k t +x1 h d)) x0 H7)))) (H0 x0 (s k i) h (s k d) H6 (eq_ind nat (s k (plus d h)) +(\lambda (n: nat).(le n (s k i))) (s_le k (plus d h) i H2) (plus (s k d) h) +(s_plus k d h)))) x H5)))) H4)) (\lambda (H4: (ex3_2 T T (\lambda (u2: +T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: +T).(subst0 i u (lift h d t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s +k i) u (lift h (s k d) t0) t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda +(t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i +u (lift h d t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) u (lift +h (s k d) t0) t2))) (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda +(t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (x0: T).(\lambda +(x1: T).(\lambda (H5: (eq T x (THead k x0 x1))).(\lambda (H6: (subst0 i u +(lift h d t) x0)).(\lambda (H7: (subst0 (s k i) u (lift h (s k d) t0) +x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq +T t2 (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) +t3)))) (ex2_ind T (\lambda (t2: T).(eq T x1 (lift h (s k d) t2))) (\lambda +(t2: T).(subst0 (minus (s k i) h) u t0 t2)) (ex2 T (\lambda (t2: T).(eq T +(THead k x0 x1) (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead +k t t0) t2))) (\lambda (x2: T).(\lambda (H8: (eq T x1 (lift h (s k d) +x2))).(\lambda (H9: (subst0 (minus (s k i) h) u t0 x2)).(ex2_ind T (\lambda +(t2: T).(eq T x0 (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u t +t2)) (ex2 T (\lambda (t2: T).(eq T (THead k x0 x1) (lift h d t2))) (\lambda +(t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (x3: T).(\lambda +(H10: (eq T x0 (lift h d x3))).(\lambda (H11: (subst0 (minus i h) u t +x3)).(eq_ind_r T (lift h d x3) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T +(THead k t2 x1) (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead +k t t0) t3)))) (eq_ind_r T (lift h (s k d) x2) (\lambda (t2: T).(ex2 T +(\lambda (t3: T).(eq T (THead k (lift h d x3) t2) (lift h d t3))) (\lambda +(t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) (eq_ind T (lift h d +(THead k x3 x2)) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d +t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) (let H12 +\def (eq_ind_r nat (minus (s k i) h) (\lambda (n: nat).(subst0 n u t0 x2)) H9 +(s k (minus i h)) (s_minus k i h (le_trans_plus_r d h i H2))) in (ex_intro2 T +(\lambda (t2: T).(eq T (lift h d (THead k x3 x2)) (lift h d t2))) (\lambda +(t2: T).(subst0 (minus i h) u (THead k t t0) t2)) (THead k x3 x2) (refl_equal +T (lift h d (THead k x3 x2))) (subst0_both u t x3 (minus i h) H11 k t0 x2 +H12))) (THead k (lift h d x3) (lift h (s k d) x2)) (lift_head k x3 x2 h d)) +x1 H8) x0 H10)))) (H x0 i h d H6 H2))))) (H0 x1 (s k i) h (s k d) H7 (eq_ind +nat (s k (plus d h)) (\lambda (n: nat).(le n (s k i))) (s_le k (plus d h) i +H2) (plus (s k d) h) (s_plus k d h)))) x H5)))))) H4)) (subst0_gen_head k u +(lift h d t) (lift h (s k d) t0) x i H3)))))))))))))) t1)). +(* COMMENTS +Initial nodes: 4191 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/subst0/props.ma b/matita/matita/contribs/lambdadelta/basic_1/subst0/props.ma new file mode 100644 index 000000000..5da05fa2a --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/subst0/props.ma @@ -0,0 +1,241 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/subst0/fwd.ma". + +theorem subst0_refl: + \forall (u: T).(\forall (t: T).(\forall (d: nat).((subst0 d u t t) \to +(\forall (P: Prop).P)))) +\def + \lambda (u: T).(\lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d: +nat).((subst0 d u t0 t0) \to (\forall (P: Prop).P)))) (\lambda (n: +nat).(\lambda (d: nat).(\lambda (H: (subst0 d u (TSort n) (TSort +n))).(\lambda (P: Prop).(subst0_gen_sort u (TSort n) d n H P))))) (\lambda +(n: nat).(\lambda (d: nat).(\lambda (H: (subst0 d u (TLRef n) (TLRef +n))).(\lambda (P: Prop).(land_ind (eq nat n d) (eq T (TLRef n) (lift (S n) O +u)) P (\lambda (_: (eq nat n d)).(\lambda (H1: (eq T (TLRef n) (lift (S n) O +u))).(lift_gen_lref_false (S n) O n (le_O_n n) (le_n (plus O (S n))) u H1 +P))) (subst0_gen_lref u (TLRef n) d n H)))))) (\lambda (k: K).(\lambda (t0: +T).(\lambda (H: ((\forall (d: nat).((subst0 d u t0 t0) \to (\forall (P: +Prop).P))))).(\lambda (t1: T).(\lambda (H0: ((\forall (d: nat).((subst0 d u +t1 t1) \to (\forall (P: Prop).P))))).(\lambda (d: nat).(\lambda (H1: (subst0 +d u (THead k t0 t1) (THead k t0 t1))).(\lambda (P: Prop).(or3_ind (ex2 T +(\lambda (u2: T).(eq T (THead k t0 t1) (THead k u2 t1))) (\lambda (u2: +T).(subst0 d u t0 u2))) (ex2 T (\lambda (t2: T).(eq T (THead k t0 t1) (THead +k t0 t2))) (\lambda (t2: T).(subst0 (s k d) u t1 t2))) (ex3_2 T T (\lambda +(u2: T).(\lambda (t2: T).(eq T (THead k t0 t1) (THead k u2 t2)))) (\lambda +(u2: T).(\lambda (_: T).(subst0 d u t0 u2))) (\lambda (_: T).(\lambda (t2: +T).(subst0 (s k d) u t1 t2)))) P (\lambda (H2: (ex2 T (\lambda (u2: T).(eq T +(THead k t0 t1) (THead k u2 t1))) (\lambda (u2: T).(subst0 d u t0 +u2)))).(ex2_ind T (\lambda (u2: T).(eq T (THead k t0 t1) (THead k u2 t1))) +(\lambda (u2: T).(subst0 d u t0 u2)) P (\lambda (x: T).(\lambda (H3: (eq T +(THead k t0 t1) (THead k x t1))).(\lambda (H4: (subst0 d u t0 x)).(let H5 +\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) +with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ t2 _) +\Rightarrow t2])) (THead k t0 t1) (THead k x t1) H3) in (let H6 \def +(eq_ind_r T x (\lambda (t2: T).(subst0 d u t0 t2)) H4 t0 H5) in (H d H6 +P)))))) H2)) (\lambda (H2: (ex2 T (\lambda (t2: T).(eq T (THead k t0 t1) +(THead k t0 t2))) (\lambda (t2: T).(subst0 (s k d) u t1 t2)))).(ex2_ind T +(\lambda (t2: T).(eq T (THead k t0 t1) (THead k t0 t2))) (\lambda (t2: +T).(subst0 (s k d) u t1 t2)) P (\lambda (x: T).(\lambda (H3: (eq T (THead k +t0 t1) (THead k t0 x))).(\lambda (H4: (subst0 (s k d) u t1 x)).(let H5 \def +(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with +[(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ _ t2) +\Rightarrow t2])) (THead k t0 t1) (THead k t0 x) H3) in (let H6 \def +(eq_ind_r T x (\lambda (t2: T).(subst0 (s k d) u t1 t2)) H4 t1 H5) in (H0 (s +k d) H6 P)))))) H2)) (\lambda (H2: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: +T).(eq T (THead k t0 t1) (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: +T).(subst0 d u t0 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k d) u t1 +t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead k t0 +t1) (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 d u t0 u2))) +(\lambda (_: T).(\lambda (t2: T).(subst0 (s k d) u t1 t2))) P (\lambda (x0: +T).(\lambda (x1: T).(\lambda (H3: (eq T (THead k t0 t1) (THead k x0 +x1))).(\lambda (H4: (subst0 d u t0 x0)).(\lambda (H5: (subst0 (s k d) u t1 +x1)).(let H6 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda +(_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead +_ t2 _) \Rightarrow t2])) (THead k t0 t1) (THead k x0 x1) H3) in ((let H7 +\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) +with [(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ _ t2) +\Rightarrow t2])) (THead k t0 t1) (THead k x0 x1) H3) in (\lambda (H8: (eq T +t0 x0)).(let H9 \def (eq_ind_r T x1 (\lambda (t2: T).(subst0 (s k d) u t1 +t2)) H5 t1 H7) in (let H10 \def (eq_ind_r T x0 (\lambda (t2: T).(subst0 d u +t0 t2)) H4 t0 H8) in (H d H10 P))))) H6))))))) H2)) (subst0_gen_head k u t0 +t1 (THead k t0 t1) d H1)))))))))) t)). +(* COMMENTS +Initial nodes: 1119 +END *) + +theorem subst0_lift_lt: + \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).((subst0 +i u t1 t2) \to (\forall (d: nat).((lt i d) \to (\forall (h: nat).(subst0 i +(lift h (minus d (S i)) u) (lift h d t1) (lift h d t2))))))))) +\def + \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda +(H: (subst0 i u t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t: +T).(\lambda (t0: T).(\lambda (t3: T).(\forall (d: nat).((lt n d) \to (\forall +(h: nat).(subst0 n (lift h (minus d (S n)) t) (lift h d t0) (lift h d +t3))))))))) (\lambda (v: T).(\lambda (i0: nat).(\lambda (d: nat).(\lambda +(H0: (lt i0 d)).(\lambda (h: nat).(eq_ind_r T (TLRef i0) (\lambda (t: +T).(subst0 i0 (lift h (minus d (S i0)) v) t (lift h d (lift (S i0) O v)))) +(let w \def (minus d (S i0)) in (eq_ind nat (plus (S i0) (minus d (S i0))) +(\lambda (n: nat).(subst0 i0 (lift h w v) (TLRef i0) (lift h n (lift (S i0) O +v)))) (eq_ind_r T (lift (S i0) O (lift h (minus d (S i0)) v)) (\lambda (t: +T).(subst0 i0 (lift h w v) (TLRef i0) t)) (subst0_lref (lift h (minus d (S +i0)) v) i0) (lift h (plus (S i0) (minus d (S i0))) (lift (S i0) O v)) (lift_d +v h (S i0) (minus d (S i0)) O (le_O_n (minus d (S i0))))) d (le_plus_minus_r +(S i0) d H0))) (lift h d (TLRef i0)) (lift_lref_lt i0 h d H0))))))) (\lambda +(v: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i0: nat).(\lambda (_: +(subst0 i0 v u1 u2)).(\lambda (H1: ((\forall (d: nat).((lt i0 d) \to (\forall +(h: nat).(subst0 i0 (lift h (minus d (S i0)) v) (lift h d u1) (lift h d +u2))))))).(\lambda (t: T).(\lambda (k: K).(\lambda (d: nat).(\lambda (H2: (lt +i0 d)).(\lambda (h: nat).(eq_ind_r T (THead k (lift h d u1) (lift h (s k d) +t)) (\lambda (t0: T).(subst0 i0 (lift h (minus d (S i0)) v) t0 (lift h d +(THead k u2 t)))) (eq_ind_r T (THead k (lift h d u2) (lift h (s k d) t)) +(\lambda (t0: T).(subst0 i0 (lift h (minus d (S i0)) v) (THead k (lift h d +u1) (lift h (s k d) t)) t0)) (subst0_fst (lift h (minus d (S i0)) v) (lift h +d u2) (lift h d u1) i0 (H1 d H2 h) (lift h (s k d) t) k) (lift h d (THead k +u2 t)) (lift_head k u2 t h d)) (lift h d (THead k u1 t)) (lift_head k u1 t h +d))))))))))))) (\lambda (k: K).(\lambda (v: T).(\lambda (t0: T).(\lambda (t3: +T).(\lambda (i0: nat).(\lambda (_: (subst0 (s k i0) v t3 t0)).(\lambda (H1: +((\forall (d: nat).((lt (s k i0) d) \to (\forall (h: nat).(subst0 (s k i0) +(lift h (minus d (S (s k i0))) v) (lift h d t3) (lift h d t0))))))).(\lambda +(u0: T).(\lambda (d: nat).(\lambda (H2: (lt i0 d)).(\lambda (h: nat).(let H3 +\def (eq_ind_r nat (S (s k i0)) (\lambda (n: nat).(\forall (d0: nat).((lt (s +k i0) d0) \to (\forall (h0: nat).(subst0 (s k i0) (lift h0 (minus d0 n) v) +(lift h0 d0 t3) (lift h0 d0 t0)))))) H1 (s k (S i0)) (s_S k i0)) in (eq_ind_r +T (THead k (lift h d u0) (lift h (s k d) t3)) (\lambda (t: T).(subst0 i0 +(lift h (minus d (S i0)) v) t (lift h d (THead k u0 t0)))) (eq_ind_r T (THead +k (lift h d u0) (lift h (s k d) t0)) (\lambda (t: T).(subst0 i0 (lift h +(minus d (S i0)) v) (THead k (lift h d u0) (lift h (s k d) t3)) t)) (eq_ind +nat (minus (s k d) (s k (S i0))) (\lambda (n: nat).(subst0 i0 (lift h n v) +(THead k (lift h d u0) (lift h (s k d) t3)) (THead k (lift h d u0) (lift h (s +k d) t0)))) (subst0_snd k (lift h (minus (s k d) (s k (S i0))) v) (lift h (s +k d) t0) (lift h (s k d) t3) i0 (H3 (s k d) (s_lt k i0 d H2) h) (lift h d +u0)) (minus d (S i0)) (minus_s_s k d (S i0))) (lift h d (THead k u0 t0)) +(lift_head k u0 t0 h d)) (lift h d (THead k u0 t3)) (lift_head k u0 t3 h +d)))))))))))))) (\lambda (v: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda +(i0: nat).(\lambda (_: (subst0 i0 v u1 u2)).(\lambda (H1: ((\forall (d: +nat).((lt i0 d) \to (\forall (h: nat).(subst0 i0 (lift h (minus d (S i0)) v) +(lift h d u1) (lift h d u2))))))).(\lambda (k: K).(\lambda (t0: T).(\lambda +(t3: T).(\lambda (_: (subst0 (s k i0) v t0 t3)).(\lambda (H3: ((\forall (d: +nat).((lt (s k i0) d) \to (\forall (h: nat).(subst0 (s k i0) (lift h (minus d +(S (s k i0))) v) (lift h d t0) (lift h d t3))))))).(\lambda (d: nat).(\lambda +(H4: (lt i0 d)).(\lambda (h: nat).(let H5 \def (eq_ind_r nat (S (s k i0)) +(\lambda (n: nat).(\forall (d0: nat).((lt (s k i0) d0) \to (\forall (h0: +nat).(subst0 (s k i0) (lift h0 (minus d0 n) v) (lift h0 d0 t0) (lift h0 d0 +t3)))))) H3 (s k (S i0)) (s_S k i0)) in (eq_ind_r T (THead k (lift h d u1) +(lift h (s k d) t0)) (\lambda (t: T).(subst0 i0 (lift h (minus d (S i0)) v) t +(lift h d (THead k u2 t3)))) (eq_ind_r T (THead k (lift h d u2) (lift h (s k +d) t3)) (\lambda (t: T).(subst0 i0 (lift h (minus d (S i0)) v) (THead k (lift +h d u1) (lift h (s k d) t0)) t)) (subst0_both (lift h (minus d (S i0)) v) +(lift h d u1) (lift h d u2) i0 (H1 d H4 h) k (lift h (s k d) t0) (lift h (s k +d) t3) (eq_ind nat (minus (s k d) (s k (S i0))) (\lambda (n: nat).(subst0 (s +k i0) (lift h n v) (lift h (s k d) t0) (lift h (s k d) t3))) (H5 (s k d) +(s_lt k i0 d H4) h) (minus d (S i0)) (minus_s_s k d (S i0)))) (lift h d +(THead k u2 t3)) (lift_head k u2 t3 h d)) (lift h d (THead k u1 t0)) +(lift_head k u1 t0 h d))))))))))))))))) i u t1 t2 H))))). +(* COMMENTS +Initial nodes: 1805 +END *) + +theorem subst0_lift_ge: + \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).(\forall +(h: nat).((subst0 i u t1 t2) \to (\forall (d: nat).((le d i) \to (subst0 +(plus i h) u (lift h d t1) (lift h d t2))))))))) +\def + \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda +(h: nat).(\lambda (H: (subst0 i u t1 t2)).(subst0_ind (\lambda (n: +nat).(\lambda (t: T).(\lambda (t0: T).(\lambda (t3: T).(\forall (d: nat).((le +d n) \to (subst0 (plus n h) t (lift h d t0) (lift h d t3)))))))) (\lambda (v: +T).(\lambda (i0: nat).(\lambda (d: nat).(\lambda (H0: (le d i0)).(eq_ind_r T +(TLRef (plus i0 h)) (\lambda (t: T).(subst0 (plus i0 h) v t (lift h d (lift +(S i0) O v)))) (eq_ind_r T (lift (plus h (S i0)) O v) (\lambda (t: T).(subst0 +(plus i0 h) v (TLRef (plus i0 h)) t)) (eq_ind nat (S (plus h i0)) (\lambda +(n: nat).(subst0 (plus i0 h) v (TLRef (plus i0 h)) (lift n O v))) (eq_ind_r +nat (plus h i0) (\lambda (n: nat).(subst0 n v (TLRef n) (lift (S (plus h i0)) +O v))) (subst0_lref v (plus h i0)) (plus i0 h) (plus_sym i0 h)) (plus h (S +i0)) (plus_n_Sm h i0)) (lift h d (lift (S i0) O v)) (lift_free v (S i0) h O d +(le_S d i0 H0) (le_O_n d))) (lift h d (TLRef i0)) (lift_lref_ge i0 h d +H0)))))) (\lambda (v: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i0: +nat).(\lambda (_: (subst0 i0 v u1 u2)).(\lambda (H1: ((\forall (d: nat).((le +d i0) \to (subst0 (plus i0 h) v (lift h d u1) (lift h d u2)))))).(\lambda (t: +T).(\lambda (k: K).(\lambda (d: nat).(\lambda (H2: (le d i0)).(eq_ind_r T +(THead k (lift h d u1) (lift h (s k d) t)) (\lambda (t0: T).(subst0 (plus i0 +h) v t0 (lift h d (THead k u2 t)))) (eq_ind_r T (THead k (lift h d u2) (lift +h (s k d) t)) (\lambda (t0: T).(subst0 (plus i0 h) v (THead k (lift h d u1) +(lift h (s k d) t)) t0)) (subst0_fst v (lift h d u2) (lift h d u1) (plus i0 +h) (H1 d H2) (lift h (s k d) t) k) (lift h d (THead k u2 t)) (lift_head k u2 +t h d)) (lift h d (THead k u1 t)) (lift_head k u1 t h d)))))))))))) (\lambda +(k: K).(\lambda (v: T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (i0: +nat).(\lambda (_: (subst0 (s k i0) v t3 t0)).(\lambda (H1: ((\forall (d: +nat).((le d (s k i0)) \to (subst0 (plus (s k i0) h) v (lift h d t3) (lift h d +t0)))))).(\lambda (u0: T).(\lambda (d: nat).(\lambda (H2: (le d i0)).(let H3 +\def (eq_ind_r nat (plus (s k i0) h) (\lambda (n: nat).(\forall (d0: +nat).((le d0 (s k i0)) \to (subst0 n v (lift h d0 t3) (lift h d0 t0))))) H1 +(s k (plus i0 h)) (s_plus k i0 h)) in (eq_ind_r T (THead k (lift h d u0) +(lift h (s k d) t3)) (\lambda (t: T).(subst0 (plus i0 h) v t (lift h d (THead +k u0 t0)))) (eq_ind_r T (THead k (lift h d u0) (lift h (s k d) t0)) (\lambda +(t: T).(subst0 (plus i0 h) v (THead k (lift h d u0) (lift h (s k d) t3)) t)) +(subst0_snd k v (lift h (s k d) t0) (lift h (s k d) t3) (plus i0 h) (H3 (s k +d) (s_le k d i0 H2)) (lift h d u0)) (lift h d (THead k u0 t0)) (lift_head k +u0 t0 h d)) (lift h d (THead k u0 t3)) (lift_head k u0 t3 h d))))))))))))) +(\lambda (v: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i0: nat).(\lambda +(_: (subst0 i0 v u1 u2)).(\lambda (H1: ((\forall (d: nat).((le d i0) \to +(subst0 (plus i0 h) v (lift h d u1) (lift h d u2)))))).(\lambda (k: +K).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (subst0 (s k i0) v t0 +t3)).(\lambda (H3: ((\forall (d: nat).((le d (s k i0)) \to (subst0 (plus (s k +i0) h) v (lift h d t0) (lift h d t3)))))).(\lambda (d: nat).(\lambda (H4: (le +d i0)).(let H5 \def (eq_ind_r nat (plus (s k i0) h) (\lambda (n: +nat).(\forall (d0: nat).((le d0 (s k i0)) \to (subst0 n v (lift h d0 t0) +(lift h d0 t3))))) H3 (s k (plus i0 h)) (s_plus k i0 h)) in (eq_ind_r T +(THead k (lift h d u1) (lift h (s k d) t0)) (\lambda (t: T).(subst0 (plus i0 +h) v t (lift h d (THead k u2 t3)))) (eq_ind_r T (THead k (lift h d u2) (lift +h (s k d) t3)) (\lambda (t: T).(subst0 (plus i0 h) v (THead k (lift h d u1) +(lift h (s k d) t0)) t)) (subst0_both v (lift h d u1) (lift h d u2) (plus i0 +h) (H1 d H4) k (lift h (s k d) t0) (lift h (s k d) t3) (H5 (s k d) (s_le k d +i0 H4))) (lift h d (THead k u2 t3)) (lift_head k u2 t3 h d)) (lift h d (THead +k u1 t0)) (lift_head k u1 t0 h d)))))))))))))))) i u t1 t2 H)))))). +(* COMMENTS +Initial nodes: 1449 +END *) + +theorem subst0_lift_ge_S: + \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).((subst0 +i u t1 t2) \to (\forall (d: nat).((le d i) \to (subst0 (S i) u (lift (S O) d +t1) (lift (S O) d t2)))))))) +\def + \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda +(H: (subst0 i u t1 t2)).(\lambda (d: nat).(\lambda (H0: (le d i)).(eq_ind nat +(plus i (S O)) (\lambda (n: nat).(subst0 n u (lift (S O) d t1) (lift (S O) d +t2))) (subst0_lift_ge t1 t2 u i (S O) H d H0) (S i) (eq_ind_r nat (plus (S O) +i) (\lambda (n: nat).(eq nat n (S i))) (refl_equal nat (S i)) (plus i (S O)) +(plus_sym i (S O)))))))))). +(* COMMENTS +Initial nodes: 137 +END *) + +theorem subst0_lift_ge_s: + \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).((subst0 +i u t1 t2) \to (\forall (d: nat).((le d i) \to (\forall (b: B).(subst0 (s +(Bind b) i) u (lift (S O) d t1) (lift (S O) d t2))))))))) +\def + \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda +(H: (subst0 i u t1 t2)).(\lambda (d: nat).(\lambda (H0: (le d i)).(\lambda +(_: B).(subst0_lift_ge_S t1 t2 u i H d H0)))))))). +(* COMMENTS +Initial nodes: 43 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/subst0/subst0.ma b/matita/matita/contribs/lambdadelta/basic_1/subst0/subst0.ma new file mode 100644 index 000000000..66c167d2b --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/subst0/subst0.ma @@ -0,0 +1,1407 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/subst0/props.ma". + +theorem subst0_subst0: + \forall (t1: T).(\forall (t2: T).(\forall (u2: T).(\forall (j: nat).((subst0 +j u2 t1 t2) \to (\forall (u1: T).(\forall (u: T).(\forall (i: nat).((subst0 i +u u1 u2) \to (ex2 T (\lambda (t: T).(subst0 j u1 t1 t)) (\lambda (t: +T).(subst0 (S (plus i j)) u t t2))))))))))) +\def + \lambda (t1: T).(\lambda (t2: T).(\lambda (u2: T).(\lambda (j: nat).(\lambda +(H: (subst0 j u2 t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t: +T).(\lambda (t0: T).(\lambda (t3: T).(\forall (u1: T).(\forall (u: +T).(\forall (i: nat).((subst0 i u u1 t) \to (ex2 T (\lambda (t4: T).(subst0 n +u1 t0 t4)) (\lambda (t4: T).(subst0 (S (plus i n)) u t4 t3))))))))))) +(\lambda (v: T).(\lambda (i: nat).(\lambda (u1: T).(\lambda (u: T).(\lambda +(i0: nat).(\lambda (H0: (subst0 i0 u u1 v)).(eq_ind nat (plus i0 (S i)) +(\lambda (n: nat).(ex2 T (\lambda (t: T).(subst0 i u1 (TLRef i) t)) (\lambda +(t: T).(subst0 n u t (lift (S i) O v))))) (ex_intro2 T (\lambda (t: +T).(subst0 i u1 (TLRef i) t)) (\lambda (t: T).(subst0 (plus i0 (S i)) u t +(lift (S i) O v))) (lift (S i) O u1) (subst0_lref u1 i) (subst0_lift_ge u1 v +u i0 (S i) H0 O (le_O_n i0))) (S (plus i0 i)) (sym_eq nat (S (plus i0 i)) +(plus i0 (S i)) (plus_n_Sm i0 i))))))))) (\lambda (v: T).(\lambda (u0: +T).(\lambda (u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1 +u0)).(\lambda (H1: ((\forall (u3: T).(\forall (u: T).(\forall (i0: +nat).((subst0 i0 u u3 v) \to (ex2 T (\lambda (t: T).(subst0 i u3 u1 t)) +(\lambda (t: T).(subst0 (S (plus i0 i)) u t u0))))))))).(\lambda (t: +T).(\lambda (k: K).(\lambda (u3: T).(\lambda (u: T).(\lambda (i0: +nat).(\lambda (H2: (subst0 i0 u u3 v)).(ex2_ind T (\lambda (t0: T).(subst0 i +u3 u1 t0)) (\lambda (t0: T).(subst0 (S (plus i0 i)) u t0 u0)) (ex2 T (\lambda +(t0: T).(subst0 i u3 (THead k u1 t) t0)) (\lambda (t0: T).(subst0 (S (plus i0 +i)) u t0 (THead k u0 t)))) (\lambda (x: T).(\lambda (H3: (subst0 i u3 u1 +x)).(\lambda (H4: (subst0 (S (plus i0 i)) u x u0)).(ex_intro2 T (\lambda (t0: +T).(subst0 i u3 (THead k u1 t) t0)) (\lambda (t0: T).(subst0 (S (plus i0 i)) +u t0 (THead k u0 t))) (THead k x t) (subst0_fst u3 x u1 i H3 t k) (subst0_fst +u u0 x (S (plus i0 i)) H4 t k))))) (H1 u3 u i0 H2)))))))))))))) (\lambda (k: +K).(\lambda (v: T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (i: +nat).(\lambda (_: (subst0 (s k i) v t3 t0)).(\lambda (H1: ((\forall (u1: +T).(\forall (u: T).(\forall (i0: nat).((subst0 i0 u u1 v) \to (ex2 T (\lambda +(t: T).(subst0 (s k i) u1 t3 t)) (\lambda (t: T).(subst0 (S (plus i0 (s k +i))) u t t0))))))))).(\lambda (u: T).(\lambda (u1: T).(\lambda (u0: +T).(\lambda (i0: nat).(\lambda (H2: (subst0 i0 u0 u1 v)).(ex2_ind T (\lambda +(t: T).(subst0 (s k i) u1 t3 t)) (\lambda (t: T).(subst0 (S (plus i0 (s k +i))) u0 t t0)) (ex2 T (\lambda (t: T).(subst0 i u1 (THead k u t3) t)) +(\lambda (t: T).(subst0 (S (plus i0 i)) u0 t (THead k u t0)))) (\lambda (x: +T).(\lambda (H3: (subst0 (s k i) u1 t3 x)).(\lambda (H4: (subst0 (S (plus i0 +(s k i))) u0 x t0)).(let H5 \def (eq_ind_r nat (plus i0 (s k i)) (\lambda (n: +nat).(subst0 (S n) u0 x t0)) H4 (s k (plus i0 i)) (s_plus_sym k i0 i)) in +(let H6 \def (eq_ind_r nat (S (s k (plus i0 i))) (\lambda (n: nat).(subst0 n +u0 x t0)) H5 (s k (S (plus i0 i))) (s_S k (plus i0 i))) in (ex_intro2 T +(\lambda (t: T).(subst0 i u1 (THead k u t3) t)) (\lambda (t: T).(subst0 (S +(plus i0 i)) u0 t (THead k u t0))) (THead k u x) (subst0_snd k u1 x t3 i H3 +u) (subst0_snd k u0 t0 x (S (plus i0 i)) H6 u))))))) (H1 u1 u0 i0 +H2)))))))))))))) (\lambda (v: T).(\lambda (u1: T).(\lambda (u0: T).(\lambda +(i: nat).(\lambda (_: (subst0 i v u1 u0)).(\lambda (H1: ((\forall (u3: +T).(\forall (u: T).(\forall (i0: nat).((subst0 i0 u u3 v) \to (ex2 T (\lambda +(t: T).(subst0 i u3 u1 t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u t +u0))))))))).(\lambda (k: K).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: +(subst0 (s k i) v t0 t3)).(\lambda (H3: ((\forall (u3: T).(\forall (u: +T).(\forall (i0: nat).((subst0 i0 u u3 v) \to (ex2 T (\lambda (t: T).(subst0 +(s k i) u3 t0 t)) (\lambda (t: T).(subst0 (S (plus i0 (s k i))) u t +t3))))))))).(\lambda (u3: T).(\lambda (u: T).(\lambda (i0: nat).(\lambda (H4: +(subst0 i0 u u3 v)).(ex2_ind T (\lambda (t: T).(subst0 (s k i) u3 t0 t)) +(\lambda (t: T).(subst0 (S (plus i0 (s k i))) u t t3)) (ex2 T (\lambda (t: +T).(subst0 i u3 (THead k u1 t0) t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u +t (THead k u0 t3)))) (\lambda (x: T).(\lambda (H5: (subst0 (s k i) u3 t0 +x)).(\lambda (H6: (subst0 (S (plus i0 (s k i))) u x t3)).(ex2_ind T (\lambda +(t: T).(subst0 i u3 u1 t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u t u0)) +(ex2 T (\lambda (t: T).(subst0 i u3 (THead k u1 t0) t)) (\lambda (t: +T).(subst0 (S (plus i0 i)) u t (THead k u0 t3)))) (\lambda (x0: T).(\lambda +(H7: (subst0 i u3 u1 x0)).(\lambda (H8: (subst0 (S (plus i0 i)) u x0 +u0)).(let H9 \def (eq_ind_r nat (plus i0 (s k i)) (\lambda (n: nat).(subst0 +(S n) u x t3)) H6 (s k (plus i0 i)) (s_plus_sym k i0 i)) in (let H10 \def +(eq_ind_r nat (S (s k (plus i0 i))) (\lambda (n: nat).(subst0 n u x t3)) H9 +(s k (S (plus i0 i))) (s_S k (plus i0 i))) in (ex_intro2 T (\lambda (t: +T).(subst0 i u3 (THead k u1 t0) t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u +t (THead k u0 t3))) (THead k x0 x) (subst0_both u3 u1 x0 i H7 k t0 x H5) +(subst0_both u x0 u0 (S (plus i0 i)) H8 k x t3 H10))))))) (H1 u3 u i0 H4))))) +(H3 u3 u i0 H4))))))))))))))))) j u2 t1 t2 H))))). +(* COMMENTS +Initial nodes: 1613 +END *) + +theorem subst0_subst0_back: + \forall (t1: T).(\forall (t2: T).(\forall (u2: T).(\forall (j: nat).((subst0 +j u2 t1 t2) \to (\forall (u1: T).(\forall (u: T).(\forall (i: nat).((subst0 i +u u2 u1) \to (ex2 T (\lambda (t: T).(subst0 j u1 t1 t)) (\lambda (t: +T).(subst0 (S (plus i j)) u t2 t))))))))))) +\def + \lambda (t1: T).(\lambda (t2: T).(\lambda (u2: T).(\lambda (j: nat).(\lambda +(H: (subst0 j u2 t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t: +T).(\lambda (t0: T).(\lambda (t3: T).(\forall (u1: T).(\forall (u: +T).(\forall (i: nat).((subst0 i u t u1) \to (ex2 T (\lambda (t4: T).(subst0 n +u1 t0 t4)) (\lambda (t4: T).(subst0 (S (plus i n)) u t3 t4))))))))))) +(\lambda (v: T).(\lambda (i: nat).(\lambda (u1: T).(\lambda (u: T).(\lambda +(i0: nat).(\lambda (H0: (subst0 i0 u v u1)).(eq_ind nat (plus i0 (S i)) +(\lambda (n: nat).(ex2 T (\lambda (t: T).(subst0 i u1 (TLRef i) t)) (\lambda +(t: T).(subst0 n u (lift (S i) O v) t)))) (ex_intro2 T (\lambda (t: +T).(subst0 i u1 (TLRef i) t)) (\lambda (t: T).(subst0 (plus i0 (S i)) u (lift +(S i) O v) t)) (lift (S i) O u1) (subst0_lref u1 i) (subst0_lift_ge v u1 u i0 +(S i) H0 O (le_O_n i0))) (S (plus i0 i)) (sym_eq nat (S (plus i0 i)) (plus i0 +(S i)) (plus_n_Sm i0 i))))))))) (\lambda (v: T).(\lambda (u0: T).(\lambda +(u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1 u0)).(\lambda (H1: +((\forall (u3: T).(\forall (u: T).(\forall (i0: nat).((subst0 i0 u v u3) \to +(ex2 T (\lambda (t: T).(subst0 i u3 u1 t)) (\lambda (t: T).(subst0 (S (plus +i0 i)) u u0 t))))))))).(\lambda (t: T).(\lambda (k: K).(\lambda (u3: +T).(\lambda (u: T).(\lambda (i0: nat).(\lambda (H2: (subst0 i0 u v +u3)).(ex2_ind T (\lambda (t0: T).(subst0 i u3 u1 t0)) (\lambda (t0: +T).(subst0 (S (plus i0 i)) u u0 t0)) (ex2 T (\lambda (t0: T).(subst0 i u3 +(THead k u1 t) t0)) (\lambda (t0: T).(subst0 (S (plus i0 i)) u (THead k u0 t) +t0))) (\lambda (x: T).(\lambda (H3: (subst0 i u3 u1 x)).(\lambda (H4: (subst0 +(S (plus i0 i)) u u0 x)).(ex_intro2 T (\lambda (t0: T).(subst0 i u3 (THead k +u1 t) t0)) (\lambda (t0: T).(subst0 (S (plus i0 i)) u (THead k u0 t) t0)) +(THead k x t) (subst0_fst u3 x u1 i H3 t k) (subst0_fst u x u0 (S (plus i0 +i)) H4 t k))))) (H1 u3 u i0 H2)))))))))))))) (\lambda (k: K).(\lambda (v: +T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (i: nat).(\lambda (_: (subst0 +(s k i) v t3 t0)).(\lambda (H1: ((\forall (u1: T).(\forall (u: T).(\forall +(i0: nat).((subst0 i0 u v u1) \to (ex2 T (\lambda (t: T).(subst0 (s k i) u1 +t3 t)) (\lambda (t: T).(subst0 (S (plus i0 (s k i))) u t0 t))))))))).(\lambda +(u: T).(\lambda (u1: T).(\lambda (u0: T).(\lambda (i0: nat).(\lambda (H2: +(subst0 i0 u0 v u1)).(ex2_ind T (\lambda (t: T).(subst0 (s k i) u1 t3 t)) +(\lambda (t: T).(subst0 (S (plus i0 (s k i))) u0 t0 t)) (ex2 T (\lambda (t: +T).(subst0 i u1 (THead k u t3) t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u0 +(THead k u t0) t))) (\lambda (x: T).(\lambda (H3: (subst0 (s k i) u1 t3 +x)).(\lambda (H4: (subst0 (S (plus i0 (s k i))) u0 t0 x)).(let H5 \def +(eq_ind_r nat (plus i0 (s k i)) (\lambda (n: nat).(subst0 (S n) u0 t0 x)) H4 +(s k (plus i0 i)) (s_plus_sym k i0 i)) in (let H6 \def (eq_ind_r nat (S (s k +(plus i0 i))) (\lambda (n: nat).(subst0 n u0 t0 x)) H5 (s k (S (plus i0 i))) +(s_S k (plus i0 i))) in (ex_intro2 T (\lambda (t: T).(subst0 i u1 (THead k u +t3) t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u0 (THead k u t0) t)) (THead +k u x) (subst0_snd k u1 x t3 i H3 u) (subst0_snd k u0 x t0 (S (plus i0 i)) H6 +u))))))) (H1 u1 u0 i0 H2)))))))))))))) (\lambda (v: T).(\lambda (u1: +T).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1 +u0)).(\lambda (H1: ((\forall (u3: T).(\forall (u: T).(\forall (i0: +nat).((subst0 i0 u v u3) \to (ex2 T (\lambda (t: T).(subst0 i u3 u1 t)) +(\lambda (t: T).(subst0 (S (plus i0 i)) u u0 t))))))))).(\lambda (k: +K).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (subst0 (s k i) v t0 +t3)).(\lambda (H3: ((\forall (u3: T).(\forall (u: T).(\forall (i0: +nat).((subst0 i0 u v u3) \to (ex2 T (\lambda (t: T).(subst0 (s k i) u3 t0 t)) +(\lambda (t: T).(subst0 (S (plus i0 (s k i))) u t3 t))))))))).(\lambda (u3: +T).(\lambda (u: T).(\lambda (i0: nat).(\lambda (H4: (subst0 i0 u v +u3)).(ex2_ind T (\lambda (t: T).(subst0 (s k i) u3 t0 t)) (\lambda (t: +T).(subst0 (S (plus i0 (s k i))) u t3 t)) (ex2 T (\lambda (t: T).(subst0 i u3 +(THead k u1 t0) t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u (THead k u0 t3) +t))) (\lambda (x: T).(\lambda (H5: (subst0 (s k i) u3 t0 x)).(\lambda (H6: +(subst0 (S (plus i0 (s k i))) u t3 x)).(ex2_ind T (\lambda (t: T).(subst0 i +u3 u1 t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u u0 t)) (ex2 T (\lambda +(t: T).(subst0 i u3 (THead k u1 t0) t)) (\lambda (t: T).(subst0 (S (plus i0 +i)) u (THead k u0 t3) t))) (\lambda (x0: T).(\lambda (H7: (subst0 i u3 u1 +x0)).(\lambda (H8: (subst0 (S (plus i0 i)) u u0 x0)).(let H9 \def (eq_ind_r +nat (plus i0 (s k i)) (\lambda (n: nat).(subst0 (S n) u t3 x)) H6 (s k (plus +i0 i)) (s_plus_sym k i0 i)) in (let H10 \def (eq_ind_r nat (S (s k (plus i0 +i))) (\lambda (n: nat).(subst0 n u t3 x)) H9 (s k (S (plus i0 i))) (s_S k +(plus i0 i))) in (ex_intro2 T (\lambda (t: T).(subst0 i u3 (THead k u1 t0) +t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u (THead k u0 t3) t)) (THead k x0 +x) (subst0_both u3 u1 x0 i H7 k t0 x H5) (subst0_both u u0 x0 (S (plus i0 i)) +H8 k t3 x H10))))))) (H1 u3 u i0 H4))))) (H3 u3 u i0 H4))))))))))))))))) j u2 +t1 t2 H))))). +(* COMMENTS +Initial nodes: 1613 +END *) + +theorem subst0_trans: + \forall (t2: T).(\forall (t1: T).(\forall (v: T).(\forall (i: nat).((subst0 +i v t1 t2) \to (\forall (t3: T).((subst0 i v t2 t3) \to (subst0 i v t1 +t3))))))) +\def + \lambda (t2: T).(\lambda (t1: T).(\lambda (v: T).(\lambda (i: nat).(\lambda +(H: (subst0 i v t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t: +T).(\lambda (t0: T).(\lambda (t3: T).(\forall (t4: T).((subst0 n t t3 t4) \to +(subst0 n t t0 t4))))))) (\lambda (v0: T).(\lambda (i0: nat).(\lambda (t3: +T).(\lambda (H0: (subst0 i0 v0 (lift (S i0) O v0) t3)).(subst0_gen_lift_false +v0 v0 t3 (S i0) O i0 (le_O_n i0) (le_n (plus O (S i0))) H0 (subst0 i0 v0 +(TLRef i0) t3)))))) (\lambda (v0: T).(\lambda (u2: T).(\lambda (u1: +T).(\lambda (i0: nat).(\lambda (H0: (subst0 i0 v0 u1 u2)).(\lambda (H1: +((\forall (t3: T).((subst0 i0 v0 u2 t3) \to (subst0 i0 v0 u1 t3))))).(\lambda +(t: T).(\lambda (k: K).(\lambda (t3: T).(\lambda (H2: (subst0 i0 v0 (THead k +u2 t) t3)).(or3_ind (ex2 T (\lambda (u3: T).(eq T t3 (THead k u3 t))) +(\lambda (u3: T).(subst0 i0 v0 u2 u3))) (ex2 T (\lambda (t4: T).(eq T t3 +(THead k u2 t4))) (\lambda (t4: T).(subst0 (s k i0) v0 t t4))) (ex3_2 T T +(\lambda (u3: T).(\lambda (t4: T).(eq T t3 (THead k u3 t4)))) (\lambda (u3: +T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) (\lambda (_: T).(\lambda (t4: +T).(subst0 (s k i0) v0 t t4)))) (subst0 i0 v0 (THead k u1 t) t3) (\lambda +(H3: (ex2 T (\lambda (u3: T).(eq T t3 (THead k u3 t))) (\lambda (u3: +T).(subst0 i0 v0 u2 u3)))).(ex2_ind T (\lambda (u3: T).(eq T t3 (THead k u3 +t))) (\lambda (u3: T).(subst0 i0 v0 u2 u3)) (subst0 i0 v0 (THead k u1 t) t3) +(\lambda (x: T).(\lambda (H4: (eq T t3 (THead k x t))).(\lambda (H5: (subst0 +i0 v0 u2 x)).(eq_ind_r T (THead k x t) (\lambda (t0: T).(subst0 i0 v0 (THead +k u1 t) t0)) (subst0_fst v0 x u1 i0 (H1 x H5) t k) t3 H4)))) H3)) (\lambda +(H3: (ex2 T (\lambda (t4: T).(eq T t3 (THead k u2 t4))) (\lambda (t4: +T).(subst0 (s k i0) v0 t t4)))).(ex2_ind T (\lambda (t4: T).(eq T t3 (THead k +u2 t4))) (\lambda (t4: T).(subst0 (s k i0) v0 t t4)) (subst0 i0 v0 (THead k +u1 t) t3) (\lambda (x: T).(\lambda (H4: (eq T t3 (THead k u2 x))).(\lambda +(H5: (subst0 (s k i0) v0 t x)).(eq_ind_r T (THead k u2 x) (\lambda (t0: +T).(subst0 i0 v0 (THead k u1 t) t0)) (subst0_both v0 u1 u2 i0 H0 k t x H5) t3 +H4)))) H3)) (\lambda (H3: (ex3_2 T T (\lambda (u3: T).(\lambda (t4: T).(eq T +t3 (THead k u3 t4)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) +(\lambda (_: T).(\lambda (t4: T).(subst0 (s k i0) v0 t t4))))).(ex3_2_ind T T +(\lambda (u3: T).(\lambda (t4: T).(eq T t3 (THead k u3 t4)))) (\lambda (u3: +T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) (\lambda (_: T).(\lambda (t4: +T).(subst0 (s k i0) v0 t t4))) (subst0 i0 v0 (THead k u1 t) t3) (\lambda (x0: +T).(\lambda (x1: T).(\lambda (H4: (eq T t3 (THead k x0 x1))).(\lambda (H5: +(subst0 i0 v0 u2 x0)).(\lambda (H6: (subst0 (s k i0) v0 t x1)).(eq_ind_r T +(THead k x0 x1) (\lambda (t0: T).(subst0 i0 v0 (THead k u1 t) t0)) +(subst0_both v0 u1 x0 i0 (H1 x0 H5) k t x1 H6) t3 H4)))))) H3)) +(subst0_gen_head k v0 u2 t t3 i0 H2)))))))))))) (\lambda (k: K).(\lambda (v0: +T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (i0: nat).(\lambda (H0: (subst0 +(s k i0) v0 t3 t0)).(\lambda (H1: ((\forall (t4: T).((subst0 (s k i0) v0 t0 +t4) \to (subst0 (s k i0) v0 t3 t4))))).(\lambda (u: T).(\lambda (t4: +T).(\lambda (H2: (subst0 i0 v0 (THead k u t0) t4)).(or3_ind (ex2 T (\lambda +(u2: T).(eq T t4 (THead k u2 t0))) (\lambda (u2: T).(subst0 i0 v0 u u2))) +(ex2 T (\lambda (t5: T).(eq T t4 (THead k u t5))) (\lambda (t5: T).(subst0 (s +k i0) v0 t0 t5))) (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 +(THead k u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u u2))) +(\lambda (_: T).(\lambda (t5: T).(subst0 (s k i0) v0 t0 t5)))) (subst0 i0 v0 +(THead k u t3) t4) (\lambda (H3: (ex2 T (\lambda (u2: T).(eq T t4 (THead k u2 +t0))) (\lambda (u2: T).(subst0 i0 v0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq +T t4 (THead k u2 t0))) (\lambda (u2: T).(subst0 i0 v0 u u2)) (subst0 i0 v0 +(THead k u t3) t4) (\lambda (x: T).(\lambda (H4: (eq T t4 (THead k x +t0))).(\lambda (H5: (subst0 i0 v0 u x)).(eq_ind_r T (THead k x t0) (\lambda +(t: T).(subst0 i0 v0 (THead k u t3) t)) (subst0_both v0 u x i0 H5 k t3 t0 H0) +t4 H4)))) H3)) (\lambda (H3: (ex2 T (\lambda (t5: T).(eq T t4 (THead k u +t5))) (\lambda (t5: T).(subst0 (s k i0) v0 t0 t5)))).(ex2_ind T (\lambda (t5: +T).(eq T t4 (THead k u t5))) (\lambda (t5: T).(subst0 (s k i0) v0 t0 t5)) +(subst0 i0 v0 (THead k u t3) t4) (\lambda (x: T).(\lambda (H4: (eq T t4 +(THead k u x))).(\lambda (H5: (subst0 (s k i0) v0 t0 x)).(eq_ind_r T (THead k +u x) (\lambda (t: T).(subst0 i0 v0 (THead k u t3) t)) (subst0_snd k v0 x t3 +i0 (H1 x H5) u) t4 H4)))) H3)) (\lambda (H3: (ex3_2 T T (\lambda (u2: +T).(\lambda (t5: T).(eq T t4 (THead k u2 t5)))) (\lambda (u2: T).(\lambda (_: +T).(subst0 i0 v0 u u2))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s k i0) v0 +t0 t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead k +u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u u2))) (\lambda (_: +T).(\lambda (t5: T).(subst0 (s k i0) v0 t0 t5))) (subst0 i0 v0 (THead k u t3) +t4) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t4 (THead k x0 +x1))).(\lambda (H5: (subst0 i0 v0 u x0)).(\lambda (H6: (subst0 (s k i0) v0 t0 +x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t: T).(subst0 i0 v0 (THead k u t3) +t)) (subst0_both v0 u x0 i0 H5 k t3 x1 (H1 x1 H6)) t4 H4)))))) H3)) +(subst0_gen_head k v0 u t0 t4 i0 H2)))))))))))) (\lambda (v0: T).(\lambda +(u1: T).(\lambda (u2: T).(\lambda (i0: nat).(\lambda (H0: (subst0 i0 v0 u1 +u2)).(\lambda (H1: ((\forall (t3: T).((subst0 i0 v0 u2 t3) \to (subst0 i0 v0 +u1 t3))))).(\lambda (k: K).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H2: +(subst0 (s k i0) v0 t0 t3)).(\lambda (H3: ((\forall (t4: T).((subst0 (s k i0) +v0 t3 t4) \to (subst0 (s k i0) v0 t0 t4))))).(\lambda (t4: T).(\lambda (H4: +(subst0 i0 v0 (THead k u2 t3) t4)).(or3_ind (ex2 T (\lambda (u3: T).(eq T t4 +(THead k u3 t3))) (\lambda (u3: T).(subst0 i0 v0 u2 u3))) (ex2 T (\lambda +(t5: T).(eq T t4 (THead k u2 t5))) (\lambda (t5: T).(subst0 (s k i0) v0 t3 +t5))) (ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T t4 (THead k u3 +t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) (\lambda (_: +T).(\lambda (t5: T).(subst0 (s k i0) v0 t3 t5)))) (subst0 i0 v0 (THead k u1 +t0) t4) (\lambda (H5: (ex2 T (\lambda (u3: T).(eq T t4 (THead k u3 t3))) +(\lambda (u3: T).(subst0 i0 v0 u2 u3)))).(ex2_ind T (\lambda (u3: T).(eq T t4 +(THead k u3 t3))) (\lambda (u3: T).(subst0 i0 v0 u2 u3)) (subst0 i0 v0 (THead +k u1 t0) t4) (\lambda (x: T).(\lambda (H6: (eq T t4 (THead k x t3))).(\lambda +(H7: (subst0 i0 v0 u2 x)).(eq_ind_r T (THead k x t3) (\lambda (t: T).(subst0 +i0 v0 (THead k u1 t0) t)) (subst0_both v0 u1 x i0 (H1 x H7) k t0 t3 H2) t4 +H6)))) H5)) (\lambda (H5: (ex2 T (\lambda (t5: T).(eq T t4 (THead k u2 t5))) +(\lambda (t5: T).(subst0 (s k i0) v0 t3 t5)))).(ex2_ind T (\lambda (t5: +T).(eq T t4 (THead k u2 t5))) (\lambda (t5: T).(subst0 (s k i0) v0 t3 t5)) +(subst0 i0 v0 (THead k u1 t0) t4) (\lambda (x: T).(\lambda (H6: (eq T t4 +(THead k u2 x))).(\lambda (H7: (subst0 (s k i0) v0 t3 x)).(eq_ind_r T (THead +k u2 x) (\lambda (t: T).(subst0 i0 v0 (THead k u1 t0) t)) (subst0_both v0 u1 +u2 i0 H0 k t0 x (H3 x H7)) t4 H6)))) H5)) (\lambda (H5: (ex3_2 T T (\lambda +(u3: T).(\lambda (t5: T).(eq T t4 (THead k u3 t5)))) (\lambda (u3: +T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) (\lambda (_: T).(\lambda (t5: +T).(subst0 (s k i0) v0 t3 t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda +(t5: T).(eq T t4 (THead k u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 +i0 v0 u2 u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s k i0) v0 t3 t5))) +(subst0 i0 v0 (THead k u1 t0) t4) (\lambda (x0: T).(\lambda (x1: T).(\lambda +(H6: (eq T t4 (THead k x0 x1))).(\lambda (H7: (subst0 i0 v0 u2 x0)).(\lambda +(H8: (subst0 (s k i0) v0 t3 x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t: +T).(subst0 i0 v0 (THead k u1 t0) t)) (subst0_both v0 u1 x0 i0 (H1 x0 H7) k t0 +x1 (H3 x1 H8)) t4 H6)))))) H5)) (subst0_gen_head k v0 u2 t3 t4 i0 +H4))))))))))))))) i v t1 t2 H))))). +(* COMMENTS +Initial nodes: 2555 +END *) + +theorem subst0_confluence_neq: + \forall (t0: T).(\forall (t1: T).(\forall (u1: T).(\forall (i1: +nat).((subst0 i1 u1 t0 t1) \to (\forall (t2: T).(\forall (u2: T).(\forall +(i2: nat).((subst0 i2 u2 t0 t2) \to ((not (eq nat i1 i2)) \to (ex2 T (\lambda +(t: T).(subst0 i2 u2 t1 t)) (\lambda (t: T).(subst0 i1 u1 t2 t)))))))))))) +\def + \lambda (t0: T).(\lambda (t1: T).(\lambda (u1: T).(\lambda (i1: +nat).(\lambda (H: (subst0 i1 u1 t0 t1)).(subst0_ind (\lambda (n: +nat).(\lambda (t: T).(\lambda (t2: T).(\lambda (t3: T).(\forall (t4: +T).(\forall (u2: T).(\forall (i2: nat).((subst0 i2 u2 t2 t4) \to ((not (eq +nat n i2)) \to (ex2 T (\lambda (t5: T).(subst0 i2 u2 t3 t5)) (\lambda (t5: +T).(subst0 n t t4 t5)))))))))))) (\lambda (v: T).(\lambda (i: nat).(\lambda +(t2: T).(\lambda (u2: T).(\lambda (i2: nat).(\lambda (H0: (subst0 i2 u2 +(TLRef i) t2)).(\lambda (H1: (not (eq nat i i2))).(land_ind (eq nat i i2) (eq +T t2 (lift (S i) O u2)) (ex2 T (\lambda (t: T).(subst0 i2 u2 (lift (S i) O v) +t)) (\lambda (t: T).(subst0 i v t2 t))) (\lambda (H2: (eq nat i i2)).(\lambda +(H3: (eq T t2 (lift (S i) O u2))).(let H4 \def (eq_ind nat i (\lambda (n: +nat).(not (eq nat n i2))) H1 i2 H2) in (eq_ind_r T (lift (S i) O u2) (\lambda +(t: T).(ex2 T (\lambda (t3: T).(subst0 i2 u2 (lift (S i) O v) t3)) (\lambda +(t3: T).(subst0 i v t t3)))) (let H5 \def (match (H4 (refl_equal nat i2)) in +False return (\lambda (_: False).(ex2 T (\lambda (t: T).(subst0 i2 u2 (lift +(S i) O v) t)) (\lambda (t: T).(subst0 i v (lift (S i) O u2) t)))) with []) +in H5) t2 H3)))) (subst0_gen_lref u2 t2 i2 i H0))))))))) (\lambda (v: +T).(\lambda (u2: T).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H0: (subst0 +i v u0 u2)).(\lambda (H1: ((\forall (t2: T).(\forall (u3: T).(\forall (i2: +nat).((subst0 i2 u3 u0 t2) \to ((not (eq nat i i2)) \to (ex2 T (\lambda (t: +T).(subst0 i2 u3 u2 t)) (\lambda (t: T).(subst0 i v t2 t)))))))))).(\lambda +(t: T).(\lambda (k: K).(\lambda (t2: T).(\lambda (u3: T).(\lambda (i2: +nat).(\lambda (H2: (subst0 i2 u3 (THead k u0 t) t2)).(\lambda (H3: (not (eq +nat i i2))).(or3_ind (ex2 T (\lambda (u4: T).(eq T t2 (THead k u4 t))) +(\lambda (u4: T).(subst0 i2 u3 u0 u4))) (ex2 T (\lambda (t3: T).(eq T t2 +(THead k u0 t3))) (\lambda (t3: T).(subst0 (s k i2) u3 t t3))) (ex3_2 T T +(\lambda (u4: T).(\lambda (t3: T).(eq T t2 (THead k u4 t3)))) (\lambda (u4: +T).(\lambda (_: T).(subst0 i2 u3 u0 u4))) (\lambda (_: T).(\lambda (t3: +T).(subst0 (s k i2) u3 t t3)))) (ex2 T (\lambda (t3: T).(subst0 i2 u3 (THead +k u2 t) t3)) (\lambda (t3: T).(subst0 i v t2 t3))) (\lambda (H4: (ex2 T +(\lambda (u4: T).(eq T t2 (THead k u4 t))) (\lambda (u4: T).(subst0 i2 u3 u0 +u4)))).(ex2_ind T (\lambda (u4: T).(eq T t2 (THead k u4 t))) (\lambda (u4: +T).(subst0 i2 u3 u0 u4)) (ex2 T (\lambda (t3: T).(subst0 i2 u3 (THead k u2 t) +t3)) (\lambda (t3: T).(subst0 i v t2 t3))) (\lambda (x: T).(\lambda (H5: (eq +T t2 (THead k x t))).(\lambda (H6: (subst0 i2 u3 u0 x)).(eq_ind_r T (THead k +x t) (\lambda (t3: T).(ex2 T (\lambda (t4: T).(subst0 i2 u3 (THead k u2 t) +t4)) (\lambda (t4: T).(subst0 i v t3 t4)))) (ex2_ind T (\lambda (t3: +T).(subst0 i2 u3 u2 t3)) (\lambda (t3: T).(subst0 i v x t3)) (ex2 T (\lambda +(t3: T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i v (THead +k x t) t3))) (\lambda (x0: T).(\lambda (H7: (subst0 i2 u3 u2 x0)).(\lambda +(H8: (subst0 i v x x0)).(ex_intro2 T (\lambda (t3: T).(subst0 i2 u3 (THead k +u2 t) t3)) (\lambda (t3: T).(subst0 i v (THead k x t) t3)) (THead k x0 t) +(subst0_fst u3 x0 u2 i2 H7 t k) (subst0_fst v x0 x i H8 t k))))) (H1 x u3 i2 +H6 H3)) t2 H5)))) H4)) (\lambda (H4: (ex2 T (\lambda (t3: T).(eq T t2 (THead +k u0 t3))) (\lambda (t3: T).(subst0 (s k i2) u3 t t3)))).(ex2_ind T (\lambda +(t3: T).(eq T t2 (THead k u0 t3))) (\lambda (t3: T).(subst0 (s k i2) u3 t +t3)) (ex2 T (\lambda (t3: T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda (t3: +T).(subst0 i v t2 t3))) (\lambda (x: T).(\lambda (H5: (eq T t2 (THead k u0 +x))).(\lambda (H6: (subst0 (s k i2) u3 t x)).(eq_ind_r T (THead k u0 x) +(\lambda (t3: T).(ex2 T (\lambda (t4: T).(subst0 i2 u3 (THead k u2 t) t4)) +(\lambda (t4: T).(subst0 i v t3 t4)))) (ex_intro2 T (\lambda (t3: T).(subst0 +i2 u3 (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i v (THead k u0 x) t3)) +(THead k u2 x) (subst0_snd k u3 x t i2 H6 u2) (subst0_fst v u2 u0 i H0 x k)) +t2 H5)))) H4)) (\lambda (H4: (ex3_2 T T (\lambda (u4: T).(\lambda (t3: T).(eq +T t2 (THead k u4 t3)))) (\lambda (u4: T).(\lambda (_: T).(subst0 i2 u3 u0 +u4))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i2) u3 t +t3))))).(ex3_2_ind T T (\lambda (u4: T).(\lambda (t3: T).(eq T t2 (THead k u4 +t3)))) (\lambda (u4: T).(\lambda (_: T).(subst0 i2 u3 u0 u4))) (\lambda (_: +T).(\lambda (t3: T).(subst0 (s k i2) u3 t t3))) (ex2 T (\lambda (t3: +T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i v t2 t3))) +(\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T t2 (THead k x0 +x1))).(\lambda (H6: (subst0 i2 u3 u0 x0)).(\lambda (H7: (subst0 (s k i2) u3 t +x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t3: T).(ex2 T (\lambda (t4: +T).(subst0 i2 u3 (THead k u2 t) t4)) (\lambda (t4: T).(subst0 i v t3 t4)))) +(ex2_ind T (\lambda (t3: T).(subst0 i2 u3 u2 t3)) (\lambda (t3: T).(subst0 i +v x0 t3)) (ex2 T (\lambda (t3: T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda +(t3: T).(subst0 i v (THead k x0 x1) t3))) (\lambda (x: T).(\lambda (H8: +(subst0 i2 u3 u2 x)).(\lambda (H9: (subst0 i v x0 x)).(ex_intro2 T (\lambda +(t3: T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i v (THead +k x0 x1) t3)) (THead k x x1) (subst0_both u3 u2 x i2 H8 k t x1 H7) +(subst0_fst v x x0 i H9 x1 k))))) (H1 x0 u3 i2 H6 H3)) t2 H5)))))) H4)) +(subst0_gen_head k u3 u0 t t2 i2 H2))))))))))))))) (\lambda (k: K).(\lambda +(v: T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (i: nat).(\lambda (H0: +(subst0 (s k i) v t3 t2)).(\lambda (H1: ((\forall (t4: T).(\forall (u2: +T).(\forall (i2: nat).((subst0 i2 u2 t3 t4) \to ((not (eq nat (s k i) i2)) +\to (ex2 T (\lambda (t: T).(subst0 i2 u2 t2 t)) (\lambda (t: T).(subst0 (s k +i) v t4 t)))))))))).(\lambda (u: T).(\lambda (t4: T).(\lambda (u2: +T).(\lambda (i2: nat).(\lambda (H2: (subst0 i2 u2 (THead k u t3) +t4)).(\lambda (H3: (not (eq nat i i2))).(or3_ind (ex2 T (\lambda (u3: T).(eq +T t4 (THead k u3 t3))) (\lambda (u3: T).(subst0 i2 u2 u u3))) (ex2 T (\lambda +(t5: T).(eq T t4 (THead k u t5))) (\lambda (t5: T).(subst0 (s k i2) u2 t3 +t5))) (ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T t4 (THead k u3 +t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i2 u2 u u3))) (\lambda (_: +T).(\lambda (t5: T).(subst0 (s k i2) u2 t3 t5)))) (ex2 T (\lambda (t: +T).(subst0 i2 u2 (THead k u t2) t)) (\lambda (t: T).(subst0 i v t4 t))) +(\lambda (H4: (ex2 T (\lambda (u3: T).(eq T t4 (THead k u3 t3))) (\lambda +(u3: T).(subst0 i2 u2 u u3)))).(ex2_ind T (\lambda (u3: T).(eq T t4 (THead k +u3 t3))) (\lambda (u3: T).(subst0 i2 u2 u u3)) (ex2 T (\lambda (t: T).(subst0 +i2 u2 (THead k u t2) t)) (\lambda (t: T).(subst0 i v t4 t))) (\lambda (x: +T).(\lambda (H5: (eq T t4 (THead k x t3))).(\lambda (H6: (subst0 i2 u2 u +x)).(eq_ind_r T (THead k x t3) (\lambda (t: T).(ex2 T (\lambda (t5: +T).(subst0 i2 u2 (THead k u t2) t5)) (\lambda (t5: T).(subst0 i v t t5)))) +(ex_intro2 T (\lambda (t: T).(subst0 i2 u2 (THead k u t2) t)) (\lambda (t: +T).(subst0 i v (THead k x t3) t)) (THead k x t2) (subst0_fst u2 x u i2 H6 t2 +k) (subst0_snd k v t2 t3 i H0 x)) t4 H5)))) H4)) (\lambda (H4: (ex2 T +(\lambda (t5: T).(eq T t4 (THead k u t5))) (\lambda (t5: T).(subst0 (s k i2) +u2 t3 t5)))).(ex2_ind T (\lambda (t5: T).(eq T t4 (THead k u t5))) (\lambda +(t5: T).(subst0 (s k i2) u2 t3 t5)) (ex2 T (\lambda (t: T).(subst0 i2 u2 +(THead k u t2) t)) (\lambda (t: T).(subst0 i v t4 t))) (\lambda (x: +T).(\lambda (H5: (eq T t4 (THead k u x))).(\lambda (H6: (subst0 (s k i2) u2 +t3 x)).(eq_ind_r T (THead k u x) (\lambda (t: T).(ex2 T (\lambda (t5: +T).(subst0 i2 u2 (THead k u t2) t5)) (\lambda (t5: T).(subst0 i v t t5)))) +(ex2_ind T (\lambda (t: T).(subst0 (s k i2) u2 t2 t)) (\lambda (t: T).(subst0 +(s k i) v x t)) (ex2 T (\lambda (t: T).(subst0 i2 u2 (THead k u t2) t)) +(\lambda (t: T).(subst0 i v (THead k u x) t))) (\lambda (x0: T).(\lambda (H7: +(subst0 (s k i2) u2 t2 x0)).(\lambda (H8: (subst0 (s k i) v x x0)).(ex_intro2 +T (\lambda (t: T).(subst0 i2 u2 (THead k u t2) t)) (\lambda (t: T).(subst0 i +v (THead k u x) t)) (THead k u x0) (subst0_snd k u2 x0 t2 i2 H7 u) +(subst0_snd k v x0 x i H8 u))))) (H1 x u2 (s k i2) H6 (ex2_ind T (\lambda (t: +T).(subst0 (s k i2) u2 t2 t)) (\lambda (t: T).(subst0 (s k i) v x t)) ((eq +nat (s k i) (s k i2)) \to False) (\lambda (x0: T).(\lambda (_: (subst0 (s k +i2) u2 t2 x0)).(\lambda (_: (subst0 (s k i) v x x0)).(\lambda (H9: (eq nat (s +k i) (s k i2))).(H3 (s_inj k i i2 H9)))))) (H1 x u2 (s k i2) H6 (\lambda (H7: +(eq nat (s k i) (s k i2))).(H3 (s_inj k i i2 H7))))))) t4 H5)))) H4)) +(\lambda (H4: (ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T t4 (THead k +u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i2 u2 u u3))) (\lambda (_: +T).(\lambda (t5: T).(subst0 (s k i2) u2 t3 t5))))).(ex3_2_ind T T (\lambda +(u3: T).(\lambda (t5: T).(eq T t4 (THead k u3 t5)))) (\lambda (u3: +T).(\lambda (_: T).(subst0 i2 u2 u u3))) (\lambda (_: T).(\lambda (t5: +T).(subst0 (s k i2) u2 t3 t5))) (ex2 T (\lambda (t: T).(subst0 i2 u2 (THead k +u t2) t)) (\lambda (t: T).(subst0 i v t4 t))) (\lambda (x0: T).(\lambda (x1: +T).(\lambda (H5: (eq T t4 (THead k x0 x1))).(\lambda (H6: (subst0 i2 u2 u +x0)).(\lambda (H7: (subst0 (s k i2) u2 t3 x1)).(eq_ind_r T (THead k x0 x1) +(\lambda (t: T).(ex2 T (\lambda (t5: T).(subst0 i2 u2 (THead k u t2) t5)) +(\lambda (t5: T).(subst0 i v t t5)))) (ex2_ind T (\lambda (t: T).(subst0 (s k +i2) u2 t2 t)) (\lambda (t: T).(subst0 (s k i) v x1 t)) (ex2 T (\lambda (t: +T).(subst0 i2 u2 (THead k u t2) t)) (\lambda (t: T).(subst0 i v (THead k x0 +x1) t))) (\lambda (x: T).(\lambda (H8: (subst0 (s k i2) u2 t2 x)).(\lambda +(H9: (subst0 (s k i) v x1 x)).(ex_intro2 T (\lambda (t: T).(subst0 i2 u2 +(THead k u t2) t)) (\lambda (t: T).(subst0 i v (THead k x0 x1) t)) (THead k +x0 x) (subst0_both u2 u x0 i2 H6 k t2 x H8) (subst0_snd k v x x1 i H9 x0))))) +(H1 x1 u2 (s k i2) H7 (ex2_ind T (\lambda (t: T).(subst0 (s k i2) u2 t2 t)) +(\lambda (t: T).(subst0 (s k i) v x1 t)) ((eq nat (s k i) (s k i2)) \to +False) (\lambda (x: T).(\lambda (_: (subst0 (s k i2) u2 t2 x)).(\lambda (_: +(subst0 (s k i) v x1 x)).(\lambda (H10: (eq nat (s k i) (s k i2))).(H3 (s_inj +k i i2 H10)))))) (H1 x1 u2 (s k i2) H7 (\lambda (H8: (eq nat (s k i) (s k +i2))).(H3 (s_inj k i i2 H8))))))) t4 H5)))))) H4)) (subst0_gen_head k u2 u t3 +t4 i2 H2))))))))))))))) (\lambda (v: T).(\lambda (u0: T).(\lambda (u2: +T).(\lambda (i: nat).(\lambda (H0: (subst0 i v u0 u2)).(\lambda (H1: +((\forall (t2: T).(\forall (u3: T).(\forall (i2: nat).((subst0 i2 u3 u0 t2) +\to ((not (eq nat i i2)) \to (ex2 T (\lambda (t: T).(subst0 i2 u3 u2 t)) +(\lambda (t: T).(subst0 i v t2 t)))))))))).(\lambda (k: K).(\lambda (t2: +T).(\lambda (t3: T).(\lambda (H2: (subst0 (s k i) v t2 t3)).(\lambda (H3: +((\forall (t4: T).(\forall (u3: T).(\forall (i2: nat).((subst0 i2 u3 t2 t4) +\to ((not (eq nat (s k i) i2)) \to (ex2 T (\lambda (t: T).(subst0 i2 u3 t3 +t)) (\lambda (t: T).(subst0 (s k i) v t4 t)))))))))).(\lambda (t4: +T).(\lambda (u3: T).(\lambda (i2: nat).(\lambda (H4: (subst0 i2 u3 (THead k +u0 t2) t4)).(\lambda (H5: (not (eq nat i i2))).(or3_ind (ex2 T (\lambda (u4: +T).(eq T t4 (THead k u4 t2))) (\lambda (u4: T).(subst0 i2 u3 u0 u4))) (ex2 T +(\lambda (t5: T).(eq T t4 (THead k u0 t5))) (\lambda (t5: T).(subst0 (s k i2) +u3 t2 t5))) (ex3_2 T T (\lambda (u4: T).(\lambda (t5: T).(eq T t4 (THead k u4 +t5)))) (\lambda (u4: T).(\lambda (_: T).(subst0 i2 u3 u0 u4))) (\lambda (_: +T).(\lambda (t5: T).(subst0 (s k i2) u3 t2 t5)))) (ex2 T (\lambda (t: +T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v t4 t))) +(\lambda (H6: (ex2 T (\lambda (u4: T).(eq T t4 (THead k u4 t2))) (\lambda +(u4: T).(subst0 i2 u3 u0 u4)))).(ex2_ind T (\lambda (u4: T).(eq T t4 (THead k +u4 t2))) (\lambda (u4: T).(subst0 i2 u3 u0 u4)) (ex2 T (\lambda (t: +T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v t4 t))) +(\lambda (x: T).(\lambda (H7: (eq T t4 (THead k x t2))).(\lambda (H8: (subst0 +i2 u3 u0 x)).(eq_ind_r T (THead k x t2) (\lambda (t: T).(ex2 T (\lambda (t5: +T).(subst0 i2 u3 (THead k u2 t3) t5)) (\lambda (t5: T).(subst0 i v t t5)))) +(ex2_ind T (\lambda (t: T).(subst0 i2 u3 u2 t)) (\lambda (t: T).(subst0 i v x +t)) (ex2 T (\lambda (t: T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: +T).(subst0 i v (THead k x t2) t))) (\lambda (x0: T).(\lambda (H9: (subst0 i2 +u3 u2 x0)).(\lambda (H10: (subst0 i v x x0)).(ex_intro2 T (\lambda (t: +T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v (THead k x +t2) t)) (THead k x0 t3) (subst0_fst u3 x0 u2 i2 H9 t3 k) (subst0_both v x x0 +i H10 k t2 t3 H2))))) (H1 x u3 i2 H8 H5)) t4 H7)))) H6)) (\lambda (H6: (ex2 T +(\lambda (t5: T).(eq T t4 (THead k u0 t5))) (\lambda (t5: T).(subst0 (s k i2) +u3 t2 t5)))).(ex2_ind T (\lambda (t5: T).(eq T t4 (THead k u0 t5))) (\lambda +(t5: T).(subst0 (s k i2) u3 t2 t5)) (ex2 T (\lambda (t: T).(subst0 i2 u3 +(THead k u2 t3) t)) (\lambda (t: T).(subst0 i v t4 t))) (\lambda (x: +T).(\lambda (H7: (eq T t4 (THead k u0 x))).(\lambda (H8: (subst0 (s k i2) u3 +t2 x)).(eq_ind_r T (THead k u0 x) (\lambda (t: T).(ex2 T (\lambda (t5: +T).(subst0 i2 u3 (THead k u2 t3) t5)) (\lambda (t5: T).(subst0 i v t t5)))) +(ex2_ind T (\lambda (t: T).(subst0 (s k i2) u3 t3 t)) (\lambda (t: T).(subst0 +(s k i) v x t)) (ex2 T (\lambda (t: T).(subst0 i2 u3 (THead k u2 t3) t)) +(\lambda (t: T).(subst0 i v (THead k u0 x) t))) (\lambda (x0: T).(\lambda +(H9: (subst0 (s k i2) u3 t3 x0)).(\lambda (H10: (subst0 (s k i) v x +x0)).(ex_intro2 T (\lambda (t: T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda +(t: T).(subst0 i v (THead k u0 x) t)) (THead k u2 x0) (subst0_snd k u3 x0 t3 +i2 H9 u2) (subst0_both v u0 u2 i H0 k x x0 H10))))) (H3 x u3 (s k i2) H8 +(ex2_ind T (\lambda (t: T).(subst0 (s k i2) u3 t3 t)) (\lambda (t: T).(subst0 +(s k i) v x t)) ((eq nat (s k i) (s k i2)) \to False) (\lambda (x0: +T).(\lambda (_: (subst0 (s k i2) u3 t3 x0)).(\lambda (_: (subst0 (s k i) v x +x0)).(\lambda (H11: (eq nat (s k i) (s k i2))).(H5 (s_inj k i i2 H11)))))) +(H3 x u3 (s k i2) H8 (\lambda (H9: (eq nat (s k i) (s k i2))).(H5 (s_inj k i +i2 H9))))))) t4 H7)))) H6)) (\lambda (H6: (ex3_2 T T (\lambda (u4: +T).(\lambda (t5: T).(eq T t4 (THead k u4 t5)))) (\lambda (u4: T).(\lambda (_: +T).(subst0 i2 u3 u0 u4))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s k i2) +u3 t2 t5))))).(ex3_2_ind T T (\lambda (u4: T).(\lambda (t5: T).(eq T t4 +(THead k u4 t5)))) (\lambda (u4: T).(\lambda (_: T).(subst0 i2 u3 u0 u4))) +(\lambda (_: T).(\lambda (t5: T).(subst0 (s k i2) u3 t2 t5))) (ex2 T (\lambda +(t: T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v t4 t))) +(\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: (eq T t4 (THead k x0 +x1))).(\lambda (H8: (subst0 i2 u3 u0 x0)).(\lambda (H9: (subst0 (s k i2) u3 +t2 x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t: T).(ex2 T (\lambda (t5: +T).(subst0 i2 u3 (THead k u2 t3) t5)) (\lambda (t5: T).(subst0 i v t t5)))) +(ex2_ind T (\lambda (t: T).(subst0 i2 u3 u2 t)) (\lambda (t: T).(subst0 i v +x0 t)) (ex2 T (\lambda (t: T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: +T).(subst0 i v (THead k x0 x1) t))) (\lambda (x: T).(\lambda (H10: (subst0 i2 +u3 u2 x)).(\lambda (H11: (subst0 i v x0 x)).(ex2_ind T (\lambda (t: +T).(subst0 (s k i2) u3 t3 t)) (\lambda (t: T).(subst0 (s k i) v x1 t)) (ex2 T +(\lambda (t: T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v +(THead k x0 x1) t))) (\lambda (x2: T).(\lambda (H12: (subst0 (s k i2) u3 t3 +x2)).(\lambda (H13: (subst0 (s k i) v x1 x2)).(ex_intro2 T (\lambda (t: +T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v (THead k x0 +x1) t)) (THead k x x2) (subst0_both u3 u2 x i2 H10 k t3 x2 H12) (subst0_both +v x0 x i H11 k x1 x2 H13))))) (H3 x1 u3 (s k i2) H9 (ex2_ind T (\lambda (t: +T).(subst0 (s k i2) u3 t3 t)) (\lambda (t: T).(subst0 (s k i) v x1 t)) ((eq +nat (s k i) (s k i2)) \to False) (\lambda (x2: T).(\lambda (_: (subst0 (s k +i2) u3 t3 x2)).(\lambda (_: (subst0 (s k i) v x1 x2)).(\lambda (H14: (eq nat +(s k i) (s k i2))).(H5 (s_inj k i i2 H14)))))) (H3 x1 u3 (s k i2) H9 (\lambda +(H12: (eq nat (s k i) (s k i2))).(H5 (s_inj k i i2 H12)))))))))) (H1 x0 u3 i2 +H8 H5)) t4 H7)))))) H6)) (subst0_gen_head k u3 u0 t2 t4 i2 +H4)))))))))))))))))) i1 u1 t0 t1 H))))). +(* COMMENTS +Initial nodes: 5375 +END *) + +theorem subst0_confluence_eq: + \forall (t0: T).(\forall (t1: T).(\forall (u: T).(\forall (i: nat).((subst0 +i u t0 t1) \to (\forall (t2: T).((subst0 i u t0 t2) \to (or4 (eq T t1 t2) +(ex2 T (\lambda (t: T).(subst0 i u t1 t)) (\lambda (t: T).(subst0 i u t2 t))) +(subst0 i u t1 t2) (subst0 i u t2 t1)))))))) +\def + \lambda (t0: T).(\lambda (t1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda +(H: (subst0 i u t0 t1)).(subst0_ind (\lambda (n: nat).(\lambda (t: +T).(\lambda (t2: T).(\lambda (t3: T).(\forall (t4: T).((subst0 n t t2 t4) \to +(or4 (eq T t3 t4) (ex2 T (\lambda (t5: T).(subst0 n t t3 t5)) (\lambda (t5: +T).(subst0 n t t4 t5))) (subst0 n t t3 t4) (subst0 n t t4 t3)))))))) (\lambda +(v: T).(\lambda (i0: nat).(\lambda (t2: T).(\lambda (H0: (subst0 i0 v (TLRef +i0) t2)).(land_ind (eq nat i0 i0) (eq T t2 (lift (S i0) O v)) (or4 (eq T +(lift (S i0) O v) t2) (ex2 T (\lambda (t: T).(subst0 i0 v (lift (S i0) O v) +t)) (\lambda (t: T).(subst0 i0 v t2 t))) (subst0 i0 v (lift (S i0) O v) t2) +(subst0 i0 v t2 (lift (S i0) O v))) (\lambda (_: (eq nat i0 i0)).(\lambda +(H2: (eq T t2 (lift (S i0) O v))).(or4_intro0 (eq T (lift (S i0) O v) t2) +(ex2 T (\lambda (t: T).(subst0 i0 v (lift (S i0) O v) t)) (\lambda (t: +T).(subst0 i0 v t2 t))) (subst0 i0 v (lift (S i0) O v) t2) (subst0 i0 v t2 +(lift (S i0) O v)) (sym_eq T t2 (lift (S i0) O v) H2)))) (subst0_gen_lref v +t2 i0 i0 H0)))))) (\lambda (v: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda +(i0: nat).(\lambda (H0: (subst0 i0 v u1 u2)).(\lambda (H1: ((\forall (t2: +T).((subst0 i0 v u1 t2) \to (or4 (eq T u2 t2) (ex2 T (\lambda (t: T).(subst0 +i0 v u2 t)) (\lambda (t: T).(subst0 i0 v t2 t))) (subst0 i0 v u2 t2) (subst0 +i0 v t2 u2)))))).(\lambda (t: T).(\lambda (k: K).(\lambda (t2: T).(\lambda +(H2: (subst0 i0 v (THead k u1 t) t2)).(or3_ind (ex2 T (\lambda (u3: T).(eq T +t2 (THead k u3 t))) (\lambda (u3: T).(subst0 i0 v u1 u3))) (ex2 T (\lambda +(t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v t +t3))) (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead k u3 +t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v u1 u3))) (\lambda (_: +T).(\lambda (t3: T).(subst0 (s k i0) v t t3)))) (or4 (eq T (THead k u2 t) t2) +(ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: +T).(subst0 i0 v t2 t3))) (subst0 i0 v (THead k u2 t) t2) (subst0 i0 v t2 +(THead k u2 t))) (\lambda (H3: (ex2 T (\lambda (u3: T).(eq T t2 (THead k u3 +t))) (\lambda (u3: T).(subst0 i0 v u1 u3)))).(ex2_ind T (\lambda (u3: T).(eq +T t2 (THead k u3 t))) (\lambda (u3: T).(subst0 i0 v u1 u3)) (or4 (eq T (THead +k u2 t) t2) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda +(t3: T).(subst0 i0 v t2 t3))) (subst0 i0 v (THead k u2 t) t2) (subst0 i0 v t2 +(THead k u2 t))) (\lambda (x: T).(\lambda (H4: (eq T t2 (THead k x +t))).(\lambda (H5: (subst0 i0 v u1 x)).(eq_ind_r T (THead k x t) (\lambda +(t3: T).(or4 (eq T (THead k u2 t) t3) (ex2 T (\lambda (t4: T).(subst0 i0 v +(THead k u2 t) t4)) (\lambda (t4: T).(subst0 i0 v t3 t4))) (subst0 i0 v +(THead k u2 t) t3) (subst0 i0 v t3 (THead k u2 t)))) (or4_ind (eq T u2 x) +(ex2 T (\lambda (t3: T).(subst0 i0 v u2 t3)) (\lambda (t3: T).(subst0 i0 v x +t3))) (subst0 i0 v u2 x) (subst0 i0 v x u2) (or4 (eq T (THead k u2 t) (THead +k x t)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda +(t3: T).(subst0 i0 v (THead k x t) t3))) (subst0 i0 v (THead k u2 t) (THead k +x t)) (subst0 i0 v (THead k x t) (THead k u2 t))) (\lambda (H6: (eq T u2 +x)).(eq_ind_r T x (\lambda (t3: T).(or4 (eq T (THead k t3 t) (THead k x t)) +(ex2 T (\lambda (t4: T).(subst0 i0 v (THead k t3 t) t4)) (\lambda (t4: +T).(subst0 i0 v (THead k x t) t4))) (subst0 i0 v (THead k t3 t) (THead k x +t)) (subst0 i0 v (THead k x t) (THead k t3 t)))) (or4_intro0 (eq T (THead k x +t) (THead k x t)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k x t) t3)) +(\lambda (t3: T).(subst0 i0 v (THead k x t) t3))) (subst0 i0 v (THead k x t) +(THead k x t)) (subst0 i0 v (THead k x t) (THead k x t)) (refl_equal T (THead +k x t))) u2 H6)) (\lambda (H6: (ex2 T (\lambda (t3: T).(subst0 i0 v u2 t3)) +(\lambda (t3: T).(subst0 i0 v x t3)))).(ex2_ind T (\lambda (t3: T).(subst0 i0 +v u2 t3)) (\lambda (t3: T).(subst0 i0 v x t3)) (or4 (eq T (THead k u2 t) +(THead k x t)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) +(\lambda (t3: T).(subst0 i0 v (THead k x t) t3))) (subst0 i0 v (THead k u2 t) +(THead k x t)) (subst0 i0 v (THead k x t) (THead k u2 t))) (\lambda (x0: +T).(\lambda (H7: (subst0 i0 v u2 x0)).(\lambda (H8: (subst0 i0 v x +x0)).(or4_intro1 (eq T (THead k u2 t) (THead k x t)) (ex2 T (\lambda (t3: +T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k x +t) t3))) (subst0 i0 v (THead k u2 t) (THead k x t)) (subst0 i0 v (THead k x +t) (THead k u2 t)) (ex_intro2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) +t3)) (\lambda (t3: T).(subst0 i0 v (THead k x t) t3)) (THead k x0 t) +(subst0_fst v x0 u2 i0 H7 t k) (subst0_fst v x0 x i0 H8 t k)))))) H6)) +(\lambda (H6: (subst0 i0 v u2 x)).(or4_intro2 (eq T (THead k u2 t) (THead k x +t)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: +T).(subst0 i0 v (THead k x t) t3))) (subst0 i0 v (THead k u2 t) (THead k x +t)) (subst0 i0 v (THead k x t) (THead k u2 t)) (subst0_fst v x u2 i0 H6 t +k))) (\lambda (H6: (subst0 i0 v x u2)).(or4_intro3 (eq T (THead k u2 t) +(THead k x t)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) +(\lambda (t3: T).(subst0 i0 v (THead k x t) t3))) (subst0 i0 v (THead k u2 t) +(THead k x t)) (subst0 i0 v (THead k x t) (THead k u2 t)) (subst0_fst v u2 x +i0 H6 t k))) (H1 x H5)) t2 H4)))) H3)) (\lambda (H3: (ex2 T (\lambda (t3: +T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v t +t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: +T).(subst0 (s k i0) v t t3)) (or4 (eq T (THead k u2 t) t2) (ex2 T (\lambda +(t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0 v t2 +t3))) (subst0 i0 v (THead k u2 t) t2) (subst0 i0 v t2 (THead k u2 t))) +(\lambda (x: T).(\lambda (H4: (eq T t2 (THead k u1 x))).(\lambda (H5: (subst0 +(s k i0) v t x)).(eq_ind_r T (THead k u1 x) (\lambda (t3: T).(or4 (eq T +(THead k u2 t) t3) (ex2 T (\lambda (t4: T).(subst0 i0 v (THead k u2 t) t4)) +(\lambda (t4: T).(subst0 i0 v t3 t4))) (subst0 i0 v (THead k u2 t) t3) +(subst0 i0 v t3 (THead k u2 t)))) (or4_ind (eq T u2 u2) (ex2 T (\lambda (t3: +T).(subst0 i0 v u2 t3)) (\lambda (t3: T).(subst0 i0 v u2 t3))) (subst0 i0 v +u2 u2) (subst0 i0 v u2 u2) (or4 (eq T (THead k u2 t) (THead k u1 x)) (ex2 T +(\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0 +v (THead k u1 x) t3))) (subst0 i0 v (THead k u2 t) (THead k u1 x)) (subst0 i0 +v (THead k u1 x) (THead k u2 t))) (\lambda (_: (eq T u2 u2)).(or4_intro1 (eq +T (THead k u2 t) (THead k u1 x)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead +k u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k u1 x) t3))) (subst0 i0 v +(THead k u2 t) (THead k u1 x)) (subst0 i0 v (THead k u1 x) (THead k u2 t)) +(ex_intro2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: +T).(subst0 i0 v (THead k u1 x) t3)) (THead k u2 x) (subst0_snd k v x t i0 H5 +u2) (subst0_fst v u2 u1 i0 H0 x k)))) (\lambda (H6: (ex2 T (\lambda (t3: +T).(subst0 i0 v u2 t3)) (\lambda (t3: T).(subst0 i0 v u2 t3)))).(ex2_ind T +(\lambda (t3: T).(subst0 i0 v u2 t3)) (\lambda (t3: T).(subst0 i0 v u2 t3)) +(or4 (eq T (THead k u2 t) (THead k u1 x)) (ex2 T (\lambda (t3: T).(subst0 i0 +v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k u1 x) t3))) +(subst0 i0 v (THead k u2 t) (THead k u1 x)) (subst0 i0 v (THead k u1 x) +(THead k u2 t))) (\lambda (x0: T).(\lambda (_: (subst0 i0 v u2 x0)).(\lambda +(_: (subst0 i0 v u2 x0)).(or4_intro1 (eq T (THead k u2 t) (THead k u1 x)) +(ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: +T).(subst0 i0 v (THead k u1 x) t3))) (subst0 i0 v (THead k u2 t) (THead k u1 +x)) (subst0 i0 v (THead k u1 x) (THead k u2 t)) (ex_intro2 T (\lambda (t3: +T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k u1 +x) t3)) (THead k u2 x) (subst0_snd k v x t i0 H5 u2) (subst0_fst v u2 u1 i0 +H0 x k)))))) H6)) (\lambda (_: (subst0 i0 v u2 u2)).(or4_intro1 (eq T (THead +k u2 t) (THead k u1 x)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) +t3)) (\lambda (t3: T).(subst0 i0 v (THead k u1 x) t3))) (subst0 i0 v (THead k +u2 t) (THead k u1 x)) (subst0 i0 v (THead k u1 x) (THead k u2 t)) (ex_intro2 +T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 +i0 v (THead k u1 x) t3)) (THead k u2 x) (subst0_snd k v x t i0 H5 u2) +(subst0_fst v u2 u1 i0 H0 x k)))) (\lambda (_: (subst0 i0 v u2 +u2)).(or4_intro1 (eq T (THead k u2 t) (THead k u1 x)) (ex2 T (\lambda (t3: +T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k u1 +x) t3))) (subst0 i0 v (THead k u2 t) (THead k u1 x)) (subst0 i0 v (THead k u1 +x) (THead k u2 t)) (ex_intro2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) +t3)) (\lambda (t3: T).(subst0 i0 v (THead k u1 x) t3)) (THead k u2 x) +(subst0_snd k v x t i0 H5 u2) (subst0_fst v u2 u1 i0 H0 x k)))) (H1 u2 H0)) +t2 H4)))) H3)) (\lambda (H3: (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq +T t2 (THead k u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v u1 +u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i0) v t +t3))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead k u3 +t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v u1 u3))) (\lambda (_: +T).(\lambda (t3: T).(subst0 (s k i0) v t t3))) (or4 (eq T (THead k u2 t) t2) +(ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: +T).(subst0 i0 v t2 t3))) (subst0 i0 v (THead k u2 t) t2) (subst0 i0 v t2 +(THead k u2 t))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t2 +(THead k x0 x1))).(\lambda (H5: (subst0 i0 v u1 x0)).(\lambda (H6: (subst0 (s +k i0) v t x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t3: T).(or4 (eq T (THead +k u2 t) t3) (ex2 T (\lambda (t4: T).(subst0 i0 v (THead k u2 t) t4)) (\lambda +(t4: T).(subst0 i0 v t3 t4))) (subst0 i0 v (THead k u2 t) t3) (subst0 i0 v t3 +(THead k u2 t)))) (or4_ind (eq T u2 x0) (ex2 T (\lambda (t3: T).(subst0 i0 v +u2 t3)) (\lambda (t3: T).(subst0 i0 v x0 t3))) (subst0 i0 v u2 x0) (subst0 i0 +v x0 u2) (or4 (eq T (THead k u2 t) (THead k x0 x1)) (ex2 T (\lambda (t3: +T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k x0 +x1) t3))) (subst0 i0 v (THead k u2 t) (THead k x0 x1)) (subst0 i0 v (THead k +x0 x1) (THead k u2 t))) (\lambda (H7: (eq T u2 x0)).(eq_ind_r T x0 (\lambda +(t3: T).(or4 (eq T (THead k t3 t) (THead k x0 x1)) (ex2 T (\lambda (t4: +T).(subst0 i0 v (THead k t3 t) t4)) (\lambda (t4: T).(subst0 i0 v (THead k x0 +x1) t4))) (subst0 i0 v (THead k t3 t) (THead k x0 x1)) (subst0 i0 v (THead k +x0 x1) (THead k t3 t)))) (or4_intro2 (eq T (THead k x0 t) (THead k x0 x1)) +(ex2 T (\lambda (t3: T).(subst0 i0 v (THead k x0 t) t3)) (\lambda (t3: +T).(subst0 i0 v (THead k x0 x1) t3))) (subst0 i0 v (THead k x0 t) (THead k x0 +x1)) (subst0 i0 v (THead k x0 x1) (THead k x0 t)) (subst0_snd k v x1 t i0 H6 +x0)) u2 H7)) (\lambda (H7: (ex2 T (\lambda (t3: T).(subst0 i0 v u2 t3)) +(\lambda (t3: T).(subst0 i0 v x0 t3)))).(ex2_ind T (\lambda (t3: T).(subst0 +i0 v u2 t3)) (\lambda (t3: T).(subst0 i0 v x0 t3)) (or4 (eq T (THead k u2 t) +(THead k x0 x1)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) +(\lambda (t3: T).(subst0 i0 v (THead k x0 x1) t3))) (subst0 i0 v (THead k u2 +t) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t))) (\lambda +(x: T).(\lambda (H8: (subst0 i0 v u2 x)).(\lambda (H9: (subst0 i0 v x0 +x)).(or4_intro1 (eq T (THead k u2 t) (THead k x0 x1)) (ex2 T (\lambda (t3: +T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k x0 +x1) t3))) (subst0 i0 v (THead k u2 t) (THead k x0 x1)) (subst0 i0 v (THead k +x0 x1) (THead k u2 t)) (ex_intro2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 +t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k x0 x1) t3)) (THead k x x1) +(subst0_both v u2 x i0 H8 k t x1 H6) (subst0_fst v x x0 i0 H9 x1 k)))))) H7)) +(\lambda (H7: (subst0 i0 v u2 x0)).(or4_intro2 (eq T (THead k u2 t) (THead k +x0 x1)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda +(t3: T).(subst0 i0 v (THead k x0 x1) t3))) (subst0 i0 v (THead k u2 t) (THead +k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t)) (subst0_both v u2 x0 +i0 H7 k t x1 H6))) (\lambda (H7: (subst0 i0 v x0 u2)).(or4_intro1 (eq T +(THead k u2 t) (THead k x0 x1)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k +u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k x0 x1) t3))) (subst0 i0 v +(THead k u2 t) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t)) +(ex_intro2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: +T).(subst0 i0 v (THead k x0 x1) t3)) (THead k u2 x1) (subst0_snd k v x1 t i0 +H6 u2) (subst0_fst v u2 x0 i0 H7 x1 k)))) (H1 x0 H5)) t2 H4)))))) H3)) +(subst0_gen_head k v u1 t t2 i0 H2)))))))))))) (\lambda (k: K).(\lambda (v: +T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (i0: nat).(\lambda (H0: (subst0 +(s k i0) v t3 t2)).(\lambda (H1: ((\forall (t4: T).((subst0 (s k i0) v t3 t4) +\to (or4 (eq T t2 t4) (ex2 T (\lambda (t: T).(subst0 (s k i0) v t2 t)) +(\lambda (t: T).(subst0 (s k i0) v t4 t))) (subst0 (s k i0) v t2 t4) (subst0 +(s k i0) v t4 t2)))))).(\lambda (u0: T).(\lambda (t4: T).(\lambda (H2: +(subst0 i0 v (THead k u0 t3) t4)).(or3_ind (ex2 T (\lambda (u2: T).(eq T t4 +(THead k u2 t3))) (\lambda (u2: T).(subst0 i0 v u0 u2))) (ex2 T (\lambda (t5: +T).(eq T t4 (THead k u0 t5))) (\lambda (t5: T).(subst0 (s k i0) v t3 t5))) +(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead k u2 t5)))) +(\lambda (u2: T).(\lambda (_: T).(subst0 i0 v u0 u2))) (\lambda (_: +T).(\lambda (t5: T).(subst0 (s k i0) v t3 t5)))) (or4 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i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k u1 +x) t))) (subst0 i0 v (THead k u2 t3) (THead k u1 x)) (subst0 i0 v (THead k u1 +x) (THead k u2 t3)) (subst0_both v u1 u2 i0 H0 k x t3 H8))) (H1 u2 H0))) (H3 +x H7)) t4 H6)))) H5)) (\lambda (H5: (ex3_2 T T (\lambda (u3: T).(\lambda (t5: +T).(eq T t4 (THead k u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v +u1 u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s k i0) v t2 +t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t5: T).(eq T t4 (THead k u3 +t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v u1 u3))) (\lambda (_: +T).(\lambda (t5: T).(subst0 (s k i0) v t2 t5))) (or4 (eq T (THead k u2 t3) +t4) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: +T).(subst0 i0 v t4 t))) (subst0 i0 v (THead k u2 t3) t4) (subst0 i0 v t4 +(THead k u2 t3))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H6: (eq T t4 +(THead k x0 x1))).(\lambda (H7: (subst0 i0 v u1 x0)).(\lambda (H8: (subst0 (s +k i0) v t2 x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t: T).(or4 (eq T (THead +k u2 t3) t) (ex2 T (\lambda (t5: T).(subst0 i0 v (THead k u2 t3) t5)) +(\lambda (t5: T).(subst0 i0 v t t5))) (subst0 i0 v (THead k u2 t3) t) (subst0 +i0 v t (THead k u2 t3)))) (or4_ind (eq T t3 x1) (ex2 T (\lambda (t: +T).(subst0 (s k i0) v t3 t)) (\lambda (t: T).(subst0 (s k i0) v x1 t))) +(subst0 (s k i0) v t3 x1) (subst0 (s k i0) v x1 t3) (or4 (eq T (THead k u2 +t3) (THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) +(\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 +t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t3))) (\lambda +(H9: (eq T t3 x1)).(eq_ind_r T x1 (\lambda (t: T).(or4 (eq T (THead k u2 t) +(THead k x0 x1)) (ex2 T (\lambda (t5: T).(subst0 i0 v (THead k u2 t) t5)) +(\lambda (t5: T).(subst0 i0 v (THead k x0 x1) t5))) (subst0 i0 v (THead k u2 +t) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t)))) (or4_ind +(eq T u2 x0) (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: +T).(subst0 i0 v x0 t))) (subst0 i0 v u2 x0) (subst0 i0 v x0 u2) (or4 (eq T +(THead k u2 x1) (THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k +u2 x1) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v +(THead k u2 x1) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 +x1))) (\lambda (H10: (eq T u2 x0)).(eq_ind_r T x0 (\lambda (t: T).(or4 (eq T +(THead k t x1) (THead k x0 x1)) (ex2 T (\lambda (t5: T).(subst0 i0 v (THead k +t x1) t5)) (\lambda (t5: T).(subst0 i0 v (THead k x0 x1) t5))) (subst0 i0 v +(THead k t x1) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k t +x1)))) (or4_intro0 (eq T (THead k x0 x1) (THead k x0 x1)) (ex2 T (\lambda (t: +T).(subst0 i0 v (THead k x0 x1) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 +x1) t))) (subst0 i0 v (THead k x0 x1) (THead k x0 x1)) (subst0 i0 v (THead k +x0 x1) (THead k x0 x1)) (refl_equal T (THead k x0 x1))) u2 H10)) (\lambda +(H10: (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0 i0 v +x0 t)))).(ex2_ind T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: +T).(subst0 i0 v x0 t)) (or4 (eq T (THead k u2 x1) (THead k x0 x1)) (ex2 T +(\lambda (t: T).(subst0 i0 v (THead k u2 x1) t)) (\lambda (t: T).(subst0 i0 v +(THead k x0 x1) t))) (subst0 i0 v (THead k u2 x1) (THead k x0 x1)) (subst0 i0 +v (THead k x0 x1) (THead k u2 x1))) (\lambda (x: T).(\lambda (H11: (subst0 i0 +v u2 x)).(\lambda (H12: (subst0 i0 v x0 x)).(or4_intro1 (eq T (THead k u2 x1) +(THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 x1) t)) +(\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 +x1) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 x1)) (ex_intro2 +T (\lambda (t: T).(subst0 i0 v (THead k u2 x1) t)) (\lambda (t: T).(subst0 i0 +v (THead k x0 x1) t)) (THead k x x1) (subst0_fst v x u2 i0 H11 x1 k) +(subst0_fst v x x0 i0 H12 x1 k)))))) H10)) (\lambda (H10: (subst0 i0 v u2 +x0)).(or4_intro2 (eq T (THead k u2 x1) (THead k x0 x1)) (ex2 T (\lambda (t: +T).(subst0 i0 v (THead k u2 x1) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 +x1) t))) (subst0 i0 v (THead k u2 x1) (THead k x0 x1)) (subst0 i0 v (THead k +x0 x1) (THead k u2 x1)) (subst0_fst v x0 u2 i0 H10 x1 k))) (\lambda (H10: +(subst0 i0 v x0 u2)).(or4_intro3 (eq T (THead k u2 x1) (THead k x0 x1)) (ex2 +T (\lambda (t: T).(subst0 i0 v (THead k u2 x1) t)) (\lambda (t: T).(subst0 i0 +v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 x1) (THead k x0 x1)) (subst0 +i0 v (THead k x0 x1) (THead k u2 x1)) (subst0_fst v u2 x0 i0 H10 x1 k))) (H1 +x0 H7)) t3 H9)) (\lambda (H9: (ex2 T (\lambda (t: T).(subst0 (s k i0) v t3 +t)) (\lambda (t: T).(subst0 (s k i0) v x1 t)))).(ex2_ind T (\lambda (t: +T).(subst0 (s k i0) v t3 t)) (\lambda (t: T).(subst0 (s k i0) v x1 t)) (or4 +(eq T (THead k u2 t3) (THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v +(THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 +i0 v (THead k u2 t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k +u2 t3))) (\lambda (x: T).(\lambda (H10: (subst0 (s k i0) v t3 x)).(\lambda +(H11: (subst0 (s k i0) v x1 x)).(or4_ind (eq T u2 x0) (ex2 T (\lambda (t: +T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0 i0 v x0 t))) (subst0 i0 v u2 +x0) (subst0 i0 v x0 u2) (or4 (eq T (THead k u2 t3) (THead k x0 x1)) (ex2 T +(\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v +(THead k x0 x1) t))) (subst0 i0 v (THead k u2 t3) (THead k x0 x1)) (subst0 i0 +v (THead k x0 x1) (THead k u2 t3))) (\lambda (H12: (eq T u2 x0)).(eq_ind_r T +x0 (\lambda (t: T).(or4 (eq T (THead k t t3) (THead k x0 x1)) (ex2 T (\lambda +(t5: T).(subst0 i0 v (THead k t t3) t5)) (\lambda (t5: T).(subst0 i0 v (THead +k x0 x1) t5))) (subst0 i0 v (THead k t t3) (THead k x0 x1)) (subst0 i0 v +(THead k x0 x1) (THead k t t3)))) (or4_intro1 (eq T (THead k x0 t3) (THead k +x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k x0 t3) t)) (\lambda (t: +T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k x0 t3) (THead k x0 +x1)) (subst0 i0 v (THead k x0 x1) (THead k x0 t3)) (ex_intro2 T (\lambda (t: +T).(subst0 i0 v (THead k x0 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 +x1) t)) (THead k x0 x) (subst0_snd k v x t3 i0 H10 x0) (subst0_snd k v x x1 +i0 H11 x0))) u2 H12)) (\lambda (H12: (ex2 T (\lambda (t: T).(subst0 i0 v u2 +t)) (\lambda (t: T).(subst0 i0 v x0 t)))).(ex2_ind T (\lambda (t: T).(subst0 +i0 v u2 t)) (\lambda (t: T).(subst0 i0 v x0 t)) (or4 (eq T (THead k u2 t3) +(THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) +(\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 +t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t3))) (\lambda +(x2: T).(\lambda (H13: (subst0 i0 v u2 x2)).(\lambda (H14: (subst0 i0 v x0 +x2)).(or4_intro1 (eq T (THead k u2 t3) (THead k x0 x1)) (ex2 T (\lambda (t: +T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 +x1) t))) (subst0 i0 v (THead k u2 t3) (THead k x0 x1)) (subst0 i0 v (THead k +x0 x1) (THead k u2 t3)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k u2 +t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t)) (THead k x2 x) +(subst0_both v u2 x2 i0 H13 k t3 x H10) (subst0_both v x0 x2 i0 H14 k x1 x +H11)))))) H12)) (\lambda (H12: (subst0 i0 v u2 x0)).(or4_intro1 (eq T (THead +k u2 t3) (THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) +t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k +u2 t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t3)) +(ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: +T).(subst0 i0 v (THead k x0 x1) t)) (THead k x0 x) (subst0_both v u2 x0 i0 +H12 k t3 x H10) (subst0_snd k v x x1 i0 H11 x0)))) (\lambda (H12: (subst0 i0 +v x0 u2)).(or4_intro1 (eq T (THead k u2 t3) (THead k x0 x1)) (ex2 T (\lambda +(t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k +x0 x1) t))) (subst0 i0 v (THead k u2 t3) (THead k x0 x1)) (subst0 i0 v (THead +k x0 x1) (THead k u2 t3)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k +u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t)) (THead k u2 x) +(subst0_snd k v x t3 i0 H10 u2) (subst0_both v x0 u2 i0 H12 k x1 x H11)))) +(H1 x0 H7))))) H9)) (\lambda (H9: (subst0 (s k i0) v t3 x1)).(or4_ind (eq T +u2 x0) (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0 i0 +v x0 t))) (subst0 i0 v u2 x0) (subst0 i0 v x0 u2) (or4 (eq T (THead k u2 t3) +(THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) +(\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 +t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t3))) (\lambda +(H10: (eq T u2 x0)).(eq_ind_r T x0 (\lambda (t: T).(or4 (eq T (THead k t t3) +(THead k x0 x1)) (ex2 T (\lambda (t5: T).(subst0 i0 v (THead k t t3) t5)) +(\lambda (t5: T).(subst0 i0 v (THead k x0 x1) t5))) (subst0 i0 v (THead k t +t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k t t3)))) +(or4_intro2 (eq T (THead k x0 t3) (THead k x0 x1)) (ex2 T (\lambda (t: +T).(subst0 i0 v (THead k x0 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 +x1) t))) (subst0 i0 v (THead k x0 t3) (THead k x0 x1)) (subst0 i0 v (THead k +x0 x1) (THead k x0 t3)) (subst0_snd k v x1 t3 i0 H9 x0)) u2 H10)) (\lambda +(H10: (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0 i0 v +x0 t)))).(ex2_ind T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: +T).(subst0 i0 v x0 t)) (or4 (eq T (THead k u2 t3) (THead k x0 x1)) (ex2 T +(\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v +(THead k x0 x1) t))) (subst0 i0 v (THead k u2 t3) (THead k x0 x1)) (subst0 i0 +v (THead k x0 x1) (THead k u2 t3))) (\lambda (x: T).(\lambda (H11: (subst0 i0 +v u2 x)).(\lambda (H12: (subst0 i0 v x0 x)).(or4_intro1 (eq T (THead k u2 t3) +(THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) +(\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 +t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t3)) (ex_intro2 +T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 +v (THead k x0 x1) t)) (THead k x x1) (subst0_both v u2 x i0 H11 k t3 x1 H9) +(subst0_fst v x x0 i0 H12 x1 k)))))) H10)) (\lambda (H10: (subst0 i0 v u2 +x0)).(or4_intro2 (eq T (THead k u2 t3) (THead k x0 x1)) (ex2 T (\lambda (t: +T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 +x1) t))) (subst0 i0 v (THead k u2 t3) (THead k x0 x1)) (subst0 i0 v (THead k +x0 x1) (THead k u2 t3)) (subst0_both v u2 x0 i0 H10 k t3 x1 H9))) (\lambda +(H10: (subst0 i0 v x0 u2)).(or4_intro1 (eq T (THead k u2 t3) (THead k x0 x1)) +(ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: +T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 t3) (THead k x0 +x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t3)) (ex_intro2 T (\lambda (t: +T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 +x1) t)) (THead k u2 x1) (subst0_snd k v x1 t3 i0 H9 u2) (subst0_fst v u2 x0 +i0 H10 x1 k)))) (H1 x0 H7))) (\lambda (H9: (subst0 (s k i0) v x1 +t3)).(or4_ind (eq T u2 x0) (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) +(\lambda (t: T).(subst0 i0 v x0 t))) (subst0 i0 v u2 x0) (subst0 i0 v x0 u2) +(or4 (eq T (THead k u2 t3) (THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 +v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) +(subst0 i0 v (THead k u2 t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) +(THead k u2 t3))) (\lambda (H10: (eq T u2 x0)).(eq_ind_r T x0 (\lambda (t: +T).(or4 (eq T (THead k t t3) (THead k x0 x1)) (ex2 T (\lambda (t5: T).(subst0 +i0 v (THead k t t3) t5)) (\lambda (t5: T).(subst0 i0 v (THead k x0 x1) t5))) +(subst0 i0 v (THead k t t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) +(THead k t t3)))) (or4_intro3 (eq T (THead k x0 t3) (THead k x0 x1)) (ex2 T +(\lambda (t: T).(subst0 i0 v (THead k x0 t3) t)) (\lambda (t: T).(subst0 i0 v +(THead k x0 x1) t))) (subst0 i0 v (THead k x0 t3) (THead k x0 x1)) (subst0 i0 +v (THead k x0 x1) (THead k x0 t3)) (subst0_snd k v t3 x1 i0 H9 x0)) u2 H10)) +(\lambda (H10: (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: +T).(subst0 i0 v x0 t)))).(ex2_ind T (\lambda (t: T).(subst0 i0 v u2 t)) +(\lambda (t: T).(subst0 i0 v x0 t)) (or4 (eq T (THead k u2 t3) (THead k x0 +x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: +T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 t3) (THead k x0 +x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t3))) (\lambda (x: T).(\lambda +(H11: (subst0 i0 v u2 x)).(\lambda (H12: (subst0 i0 v x0 x)).(or4_intro1 (eq +T (THead k u2 t3) (THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead +k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v +(THead k u2 t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 +t3)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda +(t: T).(subst0 i0 v (THead k x0 x1) t)) (THead k x t3) (subst0_fst v x u2 i0 +H11 t3 k) (subst0_both v x0 x i0 H12 k x1 t3 H9)))))) H10)) (\lambda (H10: +(subst0 i0 v u2 x0)).(or4_intro1 (eq T (THead k u2 t3) (THead k x0 x1)) (ex2 +T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 +v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 t3) (THead k x0 x1)) (subst0 +i0 v (THead k x0 x1) (THead k u2 t3)) (ex_intro2 T (\lambda (t: T).(subst0 i0 +v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t)) (THead +k x0 t3) (subst0_fst v x0 u2 i0 H10 t3 k) (subst0_snd k v t3 x1 i0 H9 x0)))) +(\lambda (H10: (subst0 i0 v x0 u2)).(or4_intro3 (eq T (THead k u2 t3) (THead +k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda +(t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 t3) (THead +k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t3)) (subst0_both v x0 u2 +i0 H10 k x1 t3 H9))) (H1 x0 H7))) (H3 x1 H8)) t4 H6)))))) H5)) +(subst0_gen_head k v u1 t2 t4 i0 H4))))))))))))))) i u t0 t1 H))))). +(* COMMENTS +Initial nodes: 25595 +END *) + +theorem subst0_confluence_lift: + \forall (t0: T).(\forall (t1: T).(\forall (u: T).(\forall (i: nat).((subst0 +i u t0 (lift (S O) i t1)) \to (\forall (t2: T).((subst0 i u t0 (lift (S O) i +t2)) \to (eq T t1 t2))))))) +\def + \lambda (t0: T).(\lambda (t1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda +(H: (subst0 i u t0 (lift (S O) i t1))).(\lambda (t2: T).(\lambda (H0: (subst0 +i u t0 (lift (S O) i t2))).(or4_ind (eq T (lift (S O) i t2) (lift (S O) i +t1)) (ex2 T (\lambda (t: T).(subst0 i u (lift (S O) i t2) t)) (\lambda (t: +T).(subst0 i u (lift (S O) i t1) t))) (subst0 i u (lift (S O) i t2) (lift (S +O) i t1)) (subst0 i u (lift (S O) i t1) (lift (S O) i t2)) (eq T t1 t2) +(\lambda (H1: (eq T (lift (S O) i t2) (lift (S O) i t1))).(let H2 \def +(sym_eq T (lift (S O) i t2) (lift (S O) i t1) H1) in (lift_inj t1 t2 (S O) i +H2))) (\lambda (H1: (ex2 T (\lambda (t: T).(subst0 i u (lift (S O) i t2) t)) +(\lambda (t: T).(subst0 i u (lift (S O) i t1) t)))).(ex2_ind T (\lambda (t: +T).(subst0 i u (lift (S O) i t2) t)) (\lambda (t: T).(subst0 i u (lift (S O) +i t1) t)) (eq T t1 t2) (\lambda (x: T).(\lambda (_: (subst0 i u (lift (S O) i +t2) x)).(\lambda (H3: (subst0 i u (lift (S O) i t1) +x)).(subst0_gen_lift_false t1 u x (S O) i i (le_n i) (eq_ind_r nat (plus (S +O) i) (\lambda (n: nat).(lt i n)) (le_n (plus (S O) i)) (plus i (S O)) +(plus_sym i (S O))) H3 (eq T t1 t2))))) H1)) (\lambda (H1: (subst0 i u (lift +(S O) i t2) (lift (S O) i t1))).(subst0_gen_lift_false t2 u (lift (S O) i t1) +(S O) i i (le_n i) (eq_ind_r nat (plus (S O) i) (\lambda (n: nat).(lt i n)) +(le_n (plus (S O) i)) (plus i (S O)) (plus_sym i (S O))) H1 (eq T t1 t2))) +(\lambda (H1: (subst0 i u (lift (S O) i t1) (lift (S O) i +t2))).(subst0_gen_lift_false t1 u (lift (S O) i t2) (S O) i i (le_n i) +(eq_ind_r nat (plus (S O) i) (\lambda (n: nat).(lt i n)) (le_n (plus (S O) +i)) (plus i (S O)) (plus_sym i (S O))) H1 (eq T t1 t2))) +(subst0_confluence_eq t0 (lift (S O) i t2) u i H0 (lift (S O) i t1) H)))))))). +(* COMMENTS +Initial nodes: 703 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/subst0/tlt.ma b/matita/matita/contribs/lambdadelta/basic_1/subst0/tlt.ma new file mode 100644 index 000000000..c8e8420bb --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/subst0/tlt.ma @@ -0,0 +1,468 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/subst0/defs.ma". + +include "Basic-1/lift/props.ma". + +include "Basic-1/lift/tlt.ma". + +theorem subst0_weight_le: + \forall (u: T).(\forall (t: T).(\forall (z: T).(\forall (d: nat).((subst0 d +u t z) \to (\forall (f: ((nat \to nat))).(\forall (g: ((nat \to +nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S +d) O u)) (g d)) \to (le (weight_map f z) (weight_map g t)))))))))) +\def + \lambda (u: T).(\lambda (t: T).(\lambda (z: T).(\lambda (d: nat).(\lambda +(H: (subst0 d u t z)).(subst0_ind (\lambda (n: nat).(\lambda (t0: T).(\lambda +(t1: T).(\lambda (t2: T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to +nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S +n) O t0)) (g n)) \to (le (weight_map f t2) (weight_map g t1)))))))))) +(\lambda (v: T).(\lambda (i: nat).(\lambda (f: ((nat \to nat))).(\lambda (g: +((nat \to nat))).(\lambda (_: ((\forall (m: nat).(le (f m) (g m))))).(\lambda +(H1: (lt (weight_map f (lift (S i) O v)) (g i))).(le_S_n (weight_map f (lift +(S i) O v)) (weight_map g (TLRef i)) (le_S (S (weight_map f (lift (S i) O +v))) (weight_map g (TLRef i)) H1)))))))) (\lambda (v: T).(\lambda (u2: +T).(\lambda (u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1 +u2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to +nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S +i) O v)) (g i)) \to (le (weight_map f u2) (weight_map g u1)))))))).(\lambda +(t0: T).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (f: ((nat \to +nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) +\to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (le (weight_map f (THead +k0 u2 t0)) (weight_map g (THead k0 u1 t0)))))))) (\lambda (b: B).(B_ind +(\lambda (b0: B).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to +nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S +i) O v)) (g i)) \to (le (weight_map f (THead (Bind b0) u2 t0)) (weight_map g +(THead (Bind b0) u1 t0)))))))) (\lambda (f: ((nat \to nat))).(\lambda (g: +((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g +m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g i))).(le_n_S +(plus (weight_map f u2) (weight_map (wadd f (S (weight_map f u2))) t0)) (plus +(weight_map g u1) (weight_map (wadd g (S (weight_map g u1))) t0)) +(le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f (S +(weight_map f u2))) t0) (weight_map (wadd g (S (weight_map g u1))) t0) (H1 f +g H2 H3) (weight_le t0 (wadd f (S (weight_map f u2))) (wadd g (S (weight_map +g u1))) (\lambda (n: nat).(wadd_le f g H2 (S (weight_map f u2)) (S +(weight_map g u1)) (le_n_S (weight_map f u2) (weight_map g u1) (H1 f g H2 +H3)) n))))))))) (\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to +nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3: (lt +(weight_map f (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map f u2) +(weight_map (wadd f O) t0)) (plus (weight_map g u1) (weight_map (wadd g O) +t0)) (le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f O) +t0) (weight_map (wadd g O) t0) (H1 f g H2 H3) (weight_le t0 (wadd f O) (wadd +g O) (\lambda (n: nat).(wadd_le f g H2 O O (le_n O) n))))))))) (\lambda (f: +((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: +nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g +i))).(le_n_S (plus (weight_map f u2) (weight_map (wadd f O) t0)) (plus +(weight_map g u1) (weight_map (wadd g O) t0)) (le_plus_plus (weight_map f u2) +(weight_map g u1) (weight_map (wadd f O) t0) (weight_map (wadd g O) t0) (H1 f +g H2 H3) (weight_le t0 (wadd f O) (wadd g O) (\lambda (n: nat).(wadd_le f g +H2 O O (le_n O) n))))))))) b)) (\lambda (_: F).(\lambda (f0: ((nat \to +nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f0 +m) (g m))))).(\lambda (H3: (lt (weight_map f0 (lift (S i) O v)) (g +i))).(le_n_S (plus (weight_map f0 u2) (weight_map f0 t0)) (plus (weight_map g +u1) (weight_map g t0)) (le_plus_plus (weight_map f0 u2) (weight_map g u1) +(weight_map f0 t0) (weight_map g t0) (H1 f0 g H2 H3) (weight_le t0 f0 g +H2)))))))) k))))))))) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (v: +T).(\forall (t2: T).(\forall (t1: T).(\forall (i: nat).((subst0 (s k0 i) v t1 +t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to +nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S +(s k0 i)) O v)) (g (s k0 i))) \to (le (weight_map f t2) (weight_map g +t1))))))) \to (\forall (u0: T).(\forall (f: ((nat \to nat))).(\forall (g: +((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map +f (lift (S i) O v)) (g i)) \to (le (weight_map f (THead k0 u0 t2)) +(weight_map g (THead k0 u0 t1))))))))))))))) (\lambda (b: B).(B_ind (\lambda +(b0: B).(\forall (v: T).(\forall (t2: T).(\forall (t1: T).(\forall (i: +nat).((subst0 (s (Bind b0) i) v t1 t2) \to (((\forall (f: ((nat \to +nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) +\to ((lt (weight_map f (lift (S (s (Bind b0) i)) O v)) (g (s (Bind b0) i))) +\to (le (weight_map f t2) (weight_map g t1))))))) \to (\forall (u0: +T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: +nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to +(le (weight_map f (THead (Bind b0) u0 t2)) (weight_map g (THead (Bind b0) u0 +t1))))))))))))))) (\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda +(i: nat).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H1: ((\forall (f: +((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) +(g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (le +(weight_map f t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f: +((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: +nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g +i))).(le_n_S (plus (weight_map f u0) (weight_map (wadd f (S (weight_map f +u0))) t2)) (plus (weight_map g u0) (weight_map (wadd g (S (weight_map g u0))) +t1)) (le_plus_plus (weight_map f u0) (weight_map g u0) (weight_map (wadd f (S +(weight_map f u0))) t2) (weight_map (wadd g (S (weight_map g u0))) t1) +(weight_le u0 f g H2) (H1 (wadd f (S (weight_map f u0))) (wadd g (S +(weight_map g u0))) (\lambda (m: nat).(wadd_le f g H2 (S (weight_map f u0)) +(S (weight_map g u0)) (le_n_S (weight_map f u0) (weight_map g u0) (weight_le +u0 f g H2)) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda (n: +nat).(lt n (g i))) H3 (weight_map (wadd f (S (weight_map f u0))) (lift (S (S +i)) O v)) (lift_weight_add_O (S (weight_map f u0)) v (S i) f)))))))))))))))) +(\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda (i: nat).(\lambda +(_: (subst0 (S i) v t1 t2)).(\lambda (H1: ((\forall (f: ((nat \to +nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) +\to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (le (weight_map f +t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f: ((nat \to +nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f +m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g +i))).(le_n_S (plus (weight_map f u0) (weight_map (wadd f O) t2)) (plus +(weight_map g u0) (weight_map (wadd g O) t1)) (le_plus_plus (weight_map f u0) +(weight_map g u0) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1) +(weight_le u0 f g H2) (H1 (wadd f O) (wadd g O) (\lambda (m: nat).(wadd_le f +g H2 O O (le_n O) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda +(n: nat).(lt n (g i))) H3 (weight_map (wadd f O) (lift (S (S i)) O v)) +(lift_weight_add_O O v (S i) f)))))))))))))))) (\lambda (v: T).(\lambda (t2: +T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (_: (subst0 (S i) v t1 +t2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to +nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S +(S i)) O v)) (g (S i))) \to (le (weight_map f t2) (weight_map g +t1)))))))).(\lambda (u0: T).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat +\to nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3: +(lt (weight_map f (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map f u0) +(weight_map (wadd f O) t2)) (plus (weight_map g u0) (weight_map (wadd g O) +t1)) (le_plus_plus (weight_map f u0) (weight_map g u0) (weight_map (wadd f O) +t2) (weight_map (wadd g O) t1) (weight_le u0 f g H2) (H1 (wadd f O) (wadd g +O) (\lambda (m: nat).(wadd_le f g H2 O O (le_n O) m)) (eq_ind nat (weight_map +f (lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H3 (weight_map (wadd f O) +(lift (S (S i)) O v)) (lift_weight_add_O O v (S i) f)))))))))))))))) b)) +(\lambda (_: F).(\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda +(i: nat).(\lambda (_: (subst0 i v t1 t2)).(\lambda (H1: ((\forall (f0: ((nat +\to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f0 m) (g +m)))) \to ((lt (weight_map f0 (lift (S i) O v)) (g i)) \to (le (weight_map f0 +t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f0: ((nat \to +nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f0 +m) (g m))))).(\lambda (H3: (lt (weight_map f0 (lift (S i) O v)) (g +i))).(le_n_S (plus (weight_map f0 u0) (weight_map f0 t2)) (plus (weight_map g +u0) (weight_map g t1)) (le_plus_plus (weight_map f0 u0) (weight_map g u0) +(weight_map f0 t2) (weight_map g t1) (weight_le u0 f0 g H2) (H1 f0 g H2 +H3))))))))))))))) k)) (\lambda (v: T).(\lambda (u1: T).(\lambda (u2: +T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1 u2)).(\lambda (H1: ((\forall +(f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f +m) (g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (le +(weight_map f u2) (weight_map g u1)))))))).(\lambda (k: K).(K_ind (\lambda +(k0: K).(\forall (t1: T).(\forall (t2: T).((subst0 (s k0 i) v t1 t2) \to +(((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: +nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S (s k0 i)) O v)) (g (s +k0 i))) \to (le (weight_map f t2) (weight_map g t1))))))) \to (\forall (f: +((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) +(g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (le (weight_map +f (THead k0 u2 t2)) (weight_map g (THead k0 u1 t1)))))))))))) (\lambda (b: +B).(B_ind (\lambda (b0: B).(\forall (t1: T).(\forall (t2: T).((subst0 (s +(Bind b0) i) v t1 t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat +\to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f +(lift (S (s (Bind b0) i)) O v)) (g (s (Bind b0) i))) \to (le (weight_map f +t2) (weight_map g t1))))))) \to (\forall (f: ((nat \to nat))).(\forall (g: +((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map +f (lift (S i) O v)) (g i)) \to (le (weight_map f (THead (Bind b0) u2 t2)) +(weight_map g (THead (Bind b0) u1 t1)))))))))))) (\lambda (t1: T).(\lambda +(t2: T).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H3: ((\forall (f: +((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) +(g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (le +(weight_map f t2) (weight_map g t1)))))))).(\lambda (f: ((nat \to +nat))).(\lambda (g: ((nat \to nat))).(\lambda (H4: ((\forall (m: nat).(le (f +m) (g m))))).(\lambda (H5: (lt (weight_map f (lift (S i) O v)) (g +i))).(le_n_S (plus (weight_map f u2) (weight_map (wadd f (S (weight_map f +u2))) t2)) (plus (weight_map g u1) (weight_map (wadd g (S (weight_map g u1))) +t1)) (le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f (S +(weight_map f u2))) t2) (weight_map (wadd g (S (weight_map g u1))) t1) (H1 f +g H4 H5) (H3 (wadd f (S (weight_map f u2))) (wadd g (S (weight_map g u1))) +(\lambda (m: nat).(wadd_le f g H4 (S (weight_map f u2)) (S (weight_map g u1)) +(le_n_S (weight_map f u2) (weight_map g u1) (H1 f g H4 H5)) m)) (eq_ind nat +(weight_map f (lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H5 +(weight_map (wadd f (S (weight_map f u2))) (lift (S (S i)) O v)) +(lift_weight_add_O (S (weight_map f u2)) v (S i) f))))))))))))) (\lambda (t1: +T).(\lambda (t2: T).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H3: +((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: +nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S +i))) \to (le (weight_map f t2) (weight_map g t1)))))))).(\lambda (f: ((nat +\to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H4: ((\forall (m: nat).(le +(f m) (g m))))).(\lambda (H5: (lt (weight_map f (lift (S i) O v)) (g +i))).(le_n_S (plus (weight_map f u2) (weight_map (wadd f O) t2)) (plus +(weight_map g u1) (weight_map (wadd g O) t1)) (le_plus_plus (weight_map f u2) +(weight_map g u1) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1) (H1 f +g H4 H5) (H3 (wadd f O) (wadd g O) (\lambda (m: nat).(wadd_le f g H4 O O +(le_n O) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda (n: +nat).(lt n (g i))) H5 (weight_map (wadd f O) (lift (S (S i)) O v)) +(lift_weight_add_O O v (S i) f))))))))))))) (\lambda (t1: T).(\lambda (t2: +T).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H3: ((\forall (f: ((nat \to +nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) +\to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (le (weight_map f +t2) (weight_map g t1)))))))).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat +\to nat))).(\lambda (H4: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H5: +(lt (weight_map f (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map f u2) +(weight_map (wadd f O) t2)) (plus (weight_map g u1) (weight_map (wadd g O) +t1)) (le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f O) +t2) (weight_map (wadd g O) t1) (H1 f g H4 H5) (H3 (wadd f O) (wadd g O) +(\lambda (m: nat).(wadd_le f g H4 O O (le_n O) m)) (eq_ind nat (weight_map f +(lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H5 (weight_map (wadd f O) +(lift (S (S i)) O v)) (lift_weight_add_O O v (S i) f))))))))))))) b)) +(\lambda (_: F).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (subst0 i v t1 +t2)).(\lambda (H3: ((\forall (f0: ((nat \to nat))).(\forall (g: ((nat \to +nat))).(((\forall (m: nat).(le (f0 m) (g m)))) \to ((lt (weight_map f0 (lift +(S i) O v)) (g i)) \to (le (weight_map f0 t2) (weight_map g +t1)))))))).(\lambda (f0: ((nat \to nat))).(\lambda (g: ((nat \to +nat))).(\lambda (H4: ((\forall (m: nat).(le (f0 m) (g m))))).(\lambda (H5: +(lt (weight_map f0 (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map f0 u2) +(weight_map f0 t2)) (plus (weight_map g u1) (weight_map g t1)) (le_plus_plus +(weight_map f0 u2) (weight_map g u1) (weight_map f0 t2) (weight_map g t1) (H1 +f0 g H4 H5) (H3 f0 g H4 H5)))))))))))) k)))))))) d u t z H))))). +(* COMMENTS +Initial nodes: 4101 +END *) + +theorem subst0_weight_lt: + \forall (u: T).(\forall (t: T).(\forall (z: T).(\forall (d: nat).((subst0 d +u t z) \to (\forall (f: ((nat \to nat))).(\forall (g: ((nat \to +nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S +d) O u)) (g d)) \to (lt (weight_map f z) (weight_map g t)))))))))) +\def + \lambda (u: T).(\lambda (t: T).(\lambda (z: T).(\lambda (d: nat).(\lambda +(H: (subst0 d u t z)).(subst0_ind (\lambda (n: nat).(\lambda (t0: T).(\lambda +(t1: T).(\lambda (t2: T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to +nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S +n) O t0)) (g n)) \to (lt (weight_map f t2) (weight_map g t1)))))))))) +(\lambda (v: T).(\lambda (i: nat).(\lambda (f: ((nat \to nat))).(\lambda (g: +((nat \to nat))).(\lambda (_: ((\forall (m: nat).(le (f m) (g m))))).(\lambda +(H1: (lt (weight_map f (lift (S i) O v)) (g i))).H1)))))) (\lambda (v: +T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i +v u1 u2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to +nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S +i) O v)) (g i)) \to (lt (weight_map f u2) (weight_map g u1)))))))).(\lambda +(t0: T).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (f: ((nat \to +nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) +\to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (lt (weight_map f (THead +k0 u2 t0)) (weight_map g (THead k0 u1 t0)))))))) (\lambda (b: B).(B_ind +(\lambda (b0: B).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to +nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S +i) O v)) (g i)) \to (lt (weight_map f (THead (Bind b0) u2 t0)) (weight_map g +(THead (Bind b0) u1 t0)))))))) (\lambda (f: ((nat \to nat))).(\lambda (g: +((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g +m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g i))).(lt_n_S +(plus (weight_map f u2) (weight_map (wadd f (S (weight_map f u2))) t0)) (plus +(weight_map g u1) (weight_map (wadd g (S (weight_map g u1))) t0)) +(lt_le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f (S +(weight_map f u2))) t0) (weight_map (wadd g (S (weight_map g u1))) t0) (H1 f +g H2 H3) (weight_le t0 (wadd f (S (weight_map f u2))) (wadd g (S (weight_map +g u1))) (\lambda (n: nat).(wadd_lt f g H2 (S (weight_map f u2)) (S +(weight_map g u1)) (lt_n_S (weight_map f u2) (weight_map g u1) (H1 f g H2 +H3)) n))))))))) (\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to +nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3: (lt +(weight_map f (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map f u2) +(weight_map (wadd f O) t0)) (plus (weight_map g u1) (weight_map (wadd g O) +t0)) (lt_le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f +O) t0) (weight_map (wadd g O) t0) (H1 f g H2 H3) (weight_le t0 (wadd f O) +(wadd g O) (\lambda (n: nat).(le_S_n (wadd f O n) (wadd g O n) (le_n_S (wadd +f O n) (wadd g O n) (wadd_le f g H2 O O (le_n O) n))))))))))) (\lambda (f: +((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: +nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g +i))).(lt_n_S (plus (weight_map f u2) (weight_map (wadd f O) t0)) (plus +(weight_map g u1) (weight_map (wadd g O) t0)) (lt_le_plus_plus (weight_map f +u2) (weight_map g u1) (weight_map (wadd f O) t0) (weight_map (wadd g O) t0) +(H1 f g H2 H3) (weight_le t0 (wadd f O) (wadd g O) (\lambda (n: nat).(le_S_n +(wadd f O n) (wadd g O n) (le_n_S (wadd f O n) (wadd g O n) (wadd_le f g H2 O +O (le_n O) n))))))))))) b)) (\lambda (_: F).(\lambda (f0: ((nat \to +nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f0 +m) (g m))))).(\lambda (H3: (lt (weight_map f0 (lift (S i) O v)) (g +i))).(lt_n_S (plus (weight_map f0 u2) (weight_map f0 t0)) (plus (weight_map g +u1) (weight_map g t0)) (lt_le_plus_plus (weight_map f0 u2) (weight_map g u1) +(weight_map f0 t0) (weight_map g t0) (H1 f0 g H2 H3) (weight_le t0 f0 g +H2)))))))) k))))))))) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (v: +T).(\forall (t2: T).(\forall (t1: T).(\forall (i: nat).((subst0 (s k0 i) v t1 +t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to +nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S +(s k0 i)) O v)) (g (s k0 i))) \to (lt (weight_map f t2) (weight_map g +t1))))))) \to (\forall (u0: T).(\forall (f: ((nat \to nat))).(\forall (g: +((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map +f (lift (S i) O v)) (g i)) \to (lt (weight_map f (THead k0 u0 t2)) +(weight_map g (THead k0 u0 t1))))))))))))))) (\lambda (b: B).(B_ind (\lambda +(b0: B).(\forall (v: T).(\forall (t2: T).(\forall (t1: T).(\forall (i: +nat).((subst0 (s (Bind b0) i) v t1 t2) \to (((\forall (f: ((nat \to +nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) +\to ((lt (weight_map f (lift (S (s (Bind b0) i)) O v)) (g (s (Bind b0) i))) +\to (lt (weight_map f t2) (weight_map g t1))))))) \to (\forall (u0: +T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: +nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to +(lt (weight_map f (THead (Bind b0) u0 t2)) (weight_map g (THead (Bind b0) u0 +t1))))))))))))))) (\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda +(i: nat).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H1: ((\forall (f: +((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) +(g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (lt +(weight_map f t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f: +((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: +nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g +i))).(lt_n_S (plus (weight_map f u0) (weight_map (wadd f (S (weight_map f +u0))) t2)) (plus (weight_map g u0) (weight_map (wadd g (S (weight_map g u0))) +t1)) (le_lt_plus_plus (weight_map f u0) (weight_map g u0) (weight_map (wadd f +(S (weight_map f u0))) t2) (weight_map (wadd g (S (weight_map g u0))) t1) +(weight_le u0 f g H2) (H1 (wadd f (S (weight_map f u0))) (wadd g (S +(weight_map g u0))) (\lambda (m: nat).(wadd_le f g H2 (S (weight_map f u0)) +(S (weight_map g u0)) (le_n_S (weight_map f u0) (weight_map g u0) (weight_le +u0 f g H2)) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda (n: +nat).(lt n (g i))) H3 (weight_map (wadd f (S (weight_map f u0))) (lift (S (S +i)) O v)) (lift_weight_add_O (S (weight_map f u0)) v (S i) f)))))))))))))))) +(\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda (i: nat).(\lambda +(_: (subst0 (S i) v t1 t2)).(\lambda (H1: ((\forall (f: ((nat \to +nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) +\to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (lt (weight_map f +t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f: ((nat \to +nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f +m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g +i))).(lt_n_S (plus (weight_map f u0) (weight_map (wadd f O) t2)) (plus +(weight_map g u0) (weight_map (wadd g O) t1)) (le_lt_plus_plus (weight_map f +u0) (weight_map g u0) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1) +(weight_le u0 f g H2) (H1 (wadd f O) (wadd g O) (\lambda (m: nat).(wadd_le f +g H2 O O (le_n O) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda +(n: nat).(lt n (g i))) H3 (weight_map (wadd f O) (lift (S (S i)) O v)) +(lift_weight_add_O O v (S i) f)))))))))))))))) (\lambda (v: T).(\lambda (t2: +T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (_: (subst0 (S i) v t1 +t2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to +nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S +(S i)) O v)) (g (S i))) \to (lt (weight_map f t2) (weight_map g +t1)))))))).(\lambda (u0: T).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat +\to nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3: +(lt (weight_map f (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map f u0) +(weight_map (wadd f O) t2)) (plus (weight_map g u0) (weight_map (wadd g O) +t1)) (le_lt_plus_plus (weight_map f u0) (weight_map g u0) (weight_map (wadd f +O) t2) (weight_map (wadd g O) t1) (weight_le u0 f g H2) (H1 (wadd f O) (wadd +g O) (\lambda (m: nat).(wadd_le f g H2 O O (le_n O) m)) (eq_ind nat +(weight_map f (lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H3 +(weight_map (wadd f O) (lift (S (S i)) O v)) (lift_weight_add_O O v (S i) +f)))))))))))))))) b)) (\lambda (_: F).(\lambda (v: T).(\lambda (t2: +T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v t1 +t2)).(\lambda (H1: ((\forall (f0: ((nat \to nat))).(\forall (g: ((nat \to +nat))).(((\forall (m: nat).(le (f0 m) (g m)))) \to ((lt (weight_map f0 (lift +(S i) O v)) (g i)) \to (lt (weight_map f0 t2) (weight_map g +t1)))))))).(\lambda (u0: T).(\lambda (f0: ((nat \to nat))).(\lambda (g: ((nat +\to nat))).(\lambda (H2: ((\forall (m: nat).(le (f0 m) (g m))))).(\lambda +(H3: (lt (weight_map f0 (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map +f0 u0) (weight_map f0 t2)) (plus (weight_map g u0) (weight_map g t1)) +(le_lt_plus_plus (weight_map f0 u0) (weight_map g u0) (weight_map f0 t2) +(weight_map g t1) (weight_le u0 f0 g H2) (H1 f0 g H2 H3))))))))))))))) k)) +(\lambda (v: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i: nat).(\lambda +(_: (subst0 i v u1 u2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall +(g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt +(weight_map f (lift (S i) O v)) (g i)) \to (lt (weight_map f u2) (weight_map +g u1)))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t1: +T).(\forall (t2: T).((subst0 (s k0 i) v t1 t2) \to (((\forall (f: ((nat \to +nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) +\to ((lt (weight_map f (lift (S (s k0 i)) O v)) (g (s k0 i))) \to (lt +(weight_map f t2) (weight_map g t1))))))) \to (\forall (f: ((nat \to +nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) +\to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (lt (weight_map f (THead +k0 u2 t2)) (weight_map g (THead k0 u1 t1)))))))))))) (\lambda (b: B).(B_ind +(\lambda (b0: B).(\forall (t1: T).(\forall (t2: T).((subst0 (s (Bind b0) i) v +t1 t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to +nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S +(s (Bind b0) i)) O v)) (g (s (Bind b0) i))) \to (lt (weight_map f t2) +(weight_map g t1))))))) \to (\forall (f: ((nat \to nat))).(\forall (g: ((nat +\to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f +(lift (S i) O v)) (g i)) \to (lt (weight_map f (THead (Bind b0) u2 t2)) +(weight_map g (THead (Bind b0) u1 t1)))))))))))) (\lambda (t1: T).(\lambda +(t2: T).(\lambda (H2: (subst0 (S i) v t1 t2)).(\lambda (_: ((\forall (f: +((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) +(g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (lt +(weight_map f t2) (weight_map g t1)))))))).(\lambda (f: ((nat \to +nat))).(\lambda (g: ((nat \to nat))).(\lambda (H4: ((\forall (m: nat).(le (f +m) (g m))))).(\lambda (H5: (lt (weight_map f (lift (S i) O v)) (g +i))).(lt_n_S (plus (weight_map f u2) (weight_map (wadd f (S (weight_map f +u2))) t2)) (plus (weight_map g u1) (weight_map (wadd g (S (weight_map g u1))) +t1)) (lt_le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f +(S (weight_map f u2))) t2) (weight_map (wadd g (S (weight_map g u1))) t1) (H1 +f g H4 H5) (subst0_weight_le v t1 t2 (S i) H2 (wadd f (S (weight_map f u2))) +(wadd g (S (weight_map g u1))) (\lambda (m: nat).(wadd_lt f g H4 (S +(weight_map f u2)) (S (weight_map g u1)) (lt_n_S (weight_map f u2) +(weight_map g u1) (H1 f g H4 H5)) m)) (eq_ind nat (weight_map f (lift (S i) O +v)) (\lambda (n: nat).(lt n (g i))) H5 (weight_map (wadd f (S (weight_map f +u2))) (lift (S (S i)) O v)) (lift_weight_add_O (S (weight_map f u2)) v (S i) +f))))))))))))) (\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (subst0 (S i) v +t1 t2)).(\lambda (H3: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to +nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S +(S i)) O v)) (g (S i))) \to (lt (weight_map f t2) (weight_map g +t1)))))))).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to +nat))).(\lambda (H4: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H5: (lt +(weight_map f (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map f u2) +(weight_map (wadd f O) t2)) (plus (weight_map g u1) (weight_map (wadd g O) +t1)) (lt_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f O) +t2) (weight_map (wadd g O) t1) (H1 f g H4 H5) (H3 (wadd f O) (wadd g O) +(\lambda (m: nat).(le_S_n (wadd f O m) (wadd g O m) (le_n_S (wadd f O m) +(wadd g O m) (wadd_le f g H4 O O (le_n O) m)))) (eq_ind nat (weight_map f +(lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H5 (weight_map (wadd f O) +(lift (S (S i)) O v)) (lift_weight_add_O O v (S i) f))))))))))))) (\lambda +(t1: T).(\lambda (t2: T).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H3: +((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: +nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S +i))) \to (lt (weight_map f t2) (weight_map g t1)))))))).(\lambda (f: ((nat +\to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H4: ((\forall (m: nat).(le +(f m) (g m))))).(\lambda (H5: (lt (weight_map f (lift (S i) O v)) (g +i))).(lt_n_S (plus (weight_map f u2) (weight_map (wadd f O) t2)) (plus +(weight_map g u1) (weight_map (wadd g O) t1)) (lt_plus_plus (weight_map f u2) +(weight_map g u1) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1) (H1 f +g H4 H5) (H3 (wadd f O) (wadd g O) (\lambda (m: nat).(le_S_n (wadd f O m) +(wadd g O m) (le_n_S (wadd f O m) (wadd g O m) (wadd_le f g H4 O O (le_n O) +m)))) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda (n: nat).(lt n (g +i))) H5 (weight_map (wadd f O) (lift (S (S i)) O v)) (lift_weight_add_O O v +(S i) f))))))))))))) b)) (\lambda (_: F).(\lambda (t1: T).(\lambda (t2: +T).(\lambda (_: (subst0 i v t1 t2)).(\lambda (H3: ((\forall (f0: ((nat \to +nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f0 m) (g m)))) +\to ((lt (weight_map f0 (lift (S i) O v)) (g i)) \to (lt (weight_map f0 t2) +(weight_map g t1)))))))).(\lambda (f0: ((nat \to nat))).(\lambda (g: ((nat +\to nat))).(\lambda (H4: ((\forall (m: nat).(le (f0 m) (g m))))).(\lambda +(H5: (lt (weight_map f0 (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map +f0 u2) (weight_map f0 t2)) (plus (weight_map g u1) (weight_map g t1)) +(lt_plus_plus (weight_map f0 u2) (weight_map g u1) (weight_map f0 t2) +(weight_map g t1) (H1 f0 g H4 H5) (H3 f0 g H4 H5)))))))))))) k)))))))) d u t +z H))))). +(* COMMENTS +Initial nodes: 4207 +END *) + +theorem subst0_tlt_head: + \forall (u: T).(\forall (t: T).(\forall (z: T).((subst0 O u t z) \to (tlt +(THead (Bind Abbr) u z) (THead (Bind Abbr) u t))))) +\def + \lambda (u: T).(\lambda (t: T).(\lambda (z: T).(\lambda (H: (subst0 O u t +z)).(lt_n_S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd +(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) z)) (plus +(weight_map (\lambda (_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) (S +(weight_map (\lambda (_: nat).O) u))) t)) (le_lt_plus_plus (weight_map +(\lambda (_: nat).O) u) (weight_map (\lambda (_: nat).O) u) (weight_map (wadd +(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) z) (weight_map +(wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t) (le_n +(weight_map (\lambda (_: nat).O) u)) (subst0_weight_lt u t z O H (wadd +(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) (wadd (\lambda +(_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) (\lambda (m: nat).(le_n +(wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u)) m))) +(eq_ind nat (weight_map (\lambda (_: nat).O) (lift O O u)) (\lambda (n: +nat).(lt n (S (weight_map (\lambda (_: nat).O) u)))) (eq_ind_r T u (\lambda +(t0: T).(lt (weight_map (\lambda (_: nat).O) t0) (S (weight_map (\lambda (_: +nat).O) u)))) (le_n (S (weight_map (\lambda (_: nat).O) u))) (lift O O u) +(lift_r u O)) (weight_map (wadd (\lambda (_: nat).O) (S (weight_map (\lambda +(_: nat).O) u))) (lift (S O) O u)) (lift_weight_add_O (S (weight_map (\lambda +(_: nat).O) u)) u O (\lambda (_: nat).O))))))))). +(* COMMENTS +Initial nodes: 347 +END *) + +theorem subst0_tlt: + \forall (u: T).(\forall (t: T).(\forall (z: T).((subst0 O u t z) \to (tlt z +(THead (Bind Abbr) u t))))) +\def + \lambda (u: T).(\lambda (t: T).(\lambda (z: T).(\lambda (H: (subst0 O u t +z)).(tlt_trans (THead (Bind Abbr) u z) z (THead (Bind Abbr) u t) (tlt_head_dx +(Bind Abbr) u z) (subst0_tlt_head u t z H))))). +(* COMMENTS +Initial nodes: 59 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/subst1/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/subst1/defs.ma new file mode 100644 index 000000000..6a51bcfb5 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/subst1/defs.ma @@ -0,0 +1,22 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/subst0/defs.ma". + +inductive subst1 (i: nat) (v: T) (t1: T): T \to Prop \def +| subst1_refl: subst1 i v t1 t1 +| subst1_single: \forall (t2: T).((subst0 i v t1 t2) \to (subst1 i v t1 t2)). + diff --git a/matita/matita/contribs/lambdadelta/basic_1/subst1/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/subst1/fwd.ma new file mode 100644 index 000000000..a2bc1edd6 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/subst1/fwd.ma @@ -0,0 +1,182 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/subst1/defs.ma". + +include "Basic-1/subst0/props.ma". + +theorem subst1_gen_sort: + \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst1 +i v (TSort n) x) \to (eq T x (TSort n)))))) +\def + \lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda +(H: (subst1 i v (TSort n) x)).(subst1_ind i v (TSort n) (\lambda (t: T).(eq T +t (TSort n))) (refl_equal T (TSort n)) (\lambda (t2: T).(\lambda (H0: (subst0 +i v (TSort n) t2)).(subst0_gen_sort v t2 i n H0 (eq T t2 (TSort n))))) x +H))))). +(* COMMENTS +Initial nodes: 89 +END *) + +theorem subst1_gen_lref: + \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst1 +i v (TLRef n) x) \to (or (eq T x (TLRef n)) (land (eq nat n i) (eq T x (lift +(S n) O v)))))))) +\def + \lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda +(H: (subst1 i v (TLRef n) x)).(subst1_ind i v (TLRef n) (\lambda (t: T).(or +(eq T t (TLRef n)) (land (eq nat n i) (eq T t (lift (S n) O v))))) (or_introl +(eq T (TLRef n) (TLRef n)) (land (eq nat n i) (eq T (TLRef n) (lift (S n) O +v))) (refl_equal T (TLRef n))) (\lambda (t2: T).(\lambda (H0: (subst0 i v +(TLRef n) t2)).(land_ind (eq nat n i) (eq T t2 (lift (S n) O v)) (or (eq T t2 +(TLRef n)) (land (eq nat n i) (eq T t2 (lift (S n) O v)))) (\lambda (H1: (eq +nat n i)).(\lambda (H2: (eq T t2 (lift (S n) O v))).(or_intror (eq T t2 +(TLRef n)) (land (eq nat n i) (eq T t2 (lift (S n) O v))) (conj (eq nat n i) +(eq T t2 (lift (S n) O v)) H1 H2)))) (subst0_gen_lref v t2 i n H0)))) x +H))))). +(* COMMENTS +Initial nodes: 305 +END *) + +theorem subst1_gen_head: + \forall (k: K).(\forall (v: T).(\forall (u1: T).(\forall (t1: T).(\forall +(x: T).(\forall (i: nat).((subst1 i v (THead k u1 t1) x) \to (ex3_2 T T +(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: +T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (t2: +T).(subst1 (s k i) v t1 t2)))))))))) +\def + \lambda (k: K).(\lambda (v: T).(\lambda (u1: T).(\lambda (t1: T).(\lambda +(x: T).(\lambda (i: nat).(\lambda (H: (subst1 i v (THead k u1 t1) +x)).(subst1_ind i v (THead k u1 t1) (\lambda (t: T).(ex3_2 T T (\lambda (u2: +T).(\lambda (t2: T).(eq T t (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: +T).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (t2: T).(subst1 (s k i) v t1 +t2))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead k u1 +t1) (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2))) +(\lambda (_: T).(\lambda (t2: T).(subst1 (s k i) v t1 t2))) u1 t1 (refl_equal +T (THead k u1 t1)) (subst1_refl i v u1) (subst1_refl (s k i) v t1)) (\lambda +(t2: T).(\lambda (H0: (subst0 i v (THead k u1 t1) t2)).(or3_ind (ex2 T +(\lambda (u2: T).(eq T t2 (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 +u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: +T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: +T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v +u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3)))) (ex3_2 +T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) (\lambda +(u2: T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (t3: +T).(subst1 (s k i) v t1 t3)))) (\lambda (H1: (ex2 T (\lambda (u2: T).(eq T t2 +(THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2)))).(ex2_ind T (\lambda +(u2: T).(eq T t2 (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2)) +(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(subst1 (s k i) v t1 t3)))) (\lambda (x0: T).(\lambda +(H2: (eq T t2 (THead k x0 t1))).(\lambda (H3: (subst0 i v u1 +x0)).(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 +t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(subst1 (s k i) v t1 t3))) x0 t1 H2 (subst1_single i v u1 +x0 H3) (subst1_refl (s k i) v t1))))) H1)) (\lambda (H1: (ex2 T (\lambda (t3: +T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 +t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: +T).(subst0 (s k i) v t1 t3)) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq +T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2))) +(\lambda (_: T).(\lambda (t3: T).(subst1 (s k i) v t1 t3)))) (\lambda (x0: +T).(\lambda (H2: (eq T t2 (THead k u1 x0))).(\lambda (H3: (subst0 (s k i) v +t1 x0)).(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k +u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(subst1 (s k i) v t1 t3))) u1 x0 H2 (subst1_refl i v u1) +(subst1_single (s k i) v t1 x0 H3))))) H1)) (\lambda (H1: (ex3_2 T T (\lambda +(u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t3: +T).(subst0 (s k i) v t1 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: +T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v +u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T +T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (t3: +T).(subst1 (s k i) v t1 t3)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda +(H2: (eq T t2 (THead k x0 x1))).(\lambda (H3: (subst0 i v u1 x0)).(\lambda +(H4: (subst0 (s k i) v t1 x1)).(ex3_2_intro T T (\lambda (u2: T).(\lambda +(t3: T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1 +i v u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst1 (s k i) v t1 t3))) x0 +x1 H2 (subst1_single i v u1 x0 H3) (subst1_single (s k i) v t1 x1 H4))))))) +H1)) (subst0_gen_head k v u1 t1 t2 i H0)))) x H))))))). +(* COMMENTS +Initial nodes: 1199 +END *) + +theorem subst1_gen_lift_lt: + \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall +(h: nat).(\forall (d: nat).((subst1 i (lift h d u) (lift h (S (plus i d)) t1) +x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda +(t2: T).(subst1 i u t1 t2))))))))) +\def + \lambda (u: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (i: nat).(\lambda +(h: nat).(\lambda (d: nat).(\lambda (H: (subst1 i (lift h d u) (lift h (S +(plus i d)) t1) x)).(subst1_ind i (lift h d u) (lift h (S (plus i d)) t1) +(\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h (S (plus i d)) t2))) +(\lambda (t2: T).(subst1 i u t1 t2)))) (ex_intro2 T (\lambda (t2: T).(eq T +(lift h (S (plus i d)) t1) (lift h (S (plus i d)) t2))) (\lambda (t2: +T).(subst1 i u t1 t2)) t1 (refl_equal T (lift h (S (plus i d)) t1)) +(subst1_refl i u t1)) (\lambda (t2: T).(\lambda (H0: (subst0 i (lift h d u) +(lift h (S (plus i d)) t1) t2)).(ex2_ind T (\lambda (t3: T).(eq T t2 (lift h +(S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u t1 t3)) (ex2 T (\lambda +(t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst1 i u t1 +t3))) (\lambda (x0: T).(\lambda (H1: (eq T t2 (lift h (S (plus i d)) +x0))).(\lambda (H2: (subst0 i u t1 x0)).(ex_intro2 T (\lambda (t3: T).(eq T +t2 (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst1 i u t1 t3)) x0 H1 +(subst1_single i u t1 x0 H2))))) (subst0_gen_lift_lt u t1 t2 i h d H0)))) x +H))))))). +(* COMMENTS +Initial nodes: 395 +END *) + +theorem subst1_gen_lift_eq: + \forall (t: T).(\forall (u: T).(\forall (x: T).(\forall (h: nat).(\forall +(d: nat).(\forall (i: nat).((le d i) \to ((lt i (plus d h)) \to ((subst1 i u +(lift h d t) x) \to (eq T x (lift h d t)))))))))) +\def + \lambda (t: T).(\lambda (u: T).(\lambda (x: T).(\lambda (h: nat).(\lambda +(d: nat).(\lambda (i: nat).(\lambda (H: (le d i)).(\lambda (H0: (lt i (plus d +h))).(\lambda (H1: (subst1 i u (lift h d t) x)).(subst1_ind i u (lift h d t) +(\lambda (t0: T).(eq T t0 (lift h d t))) (refl_equal T (lift h d t)) (\lambda +(t2: T).(\lambda (H2: (subst0 i u (lift h d t) t2)).(subst0_gen_lift_false t +u t2 h d i H H0 H2 (eq T t2 (lift h d t))))) x H1))))))))). +(* COMMENTS +Initial nodes: 141 +END *) + +theorem subst1_gen_lift_ge: + \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall +(h: nat).(\forall (d: nat).((subst1 i u (lift h d t1) x) \to ((le (plus d h) +i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: +T).(subst1 (minus i h) u t1 t2)))))))))) +\def + \lambda (u: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (i: nat).(\lambda +(h: nat).(\lambda (d: nat).(\lambda (H: (subst1 i u (lift h d t1) +x)).(\lambda (H0: (le (plus d h) i)).(subst1_ind i u (lift h d t1) (\lambda +(t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h d t2))) (\lambda (t2: +T).(subst1 (minus i h) u t1 t2)))) (ex_intro2 T (\lambda (t2: T).(eq T (lift +h d t1) (lift h d t2))) (\lambda (t2: T).(subst1 (minus i h) u t1 t2)) t1 +(refl_equal T (lift h d t1)) (subst1_refl (minus i h) u t1)) (\lambda (t2: +T).(\lambda (H1: (subst0 i u (lift h d t1) t2)).(ex2_ind T (\lambda (t3: +T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u t1 t3)) +(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst1 +(minus i h) u t1 t3))) (\lambda (x0: T).(\lambda (H2: (eq T t2 (lift h d +x0))).(\lambda (H3: (subst0 (minus i h) u t1 x0)).(ex_intro2 T (\lambda (t3: +T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst1 (minus i h) u t1 t3)) x0 +H2 (subst1_single (minus i h) u t1 x0 H3))))) (subst0_gen_lift_ge u t1 t2 i h +d H1 H0)))) x H)))))))). +(* COMMENTS +Initial nodes: 355 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/subst1/props.ma b/matita/matita/contribs/lambdadelta/basic_1/subst1/props.ma new file mode 100644 index 000000000..ac8dea954 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/subst1/props.ma @@ -0,0 +1,179 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/subst1/defs.ma". + +include "Basic-1/subst0/props.ma". + +theorem subst1_head: + \forall (v: T).(\forall (u1: T).(\forall (u2: T).(\forall (i: nat).((subst1 +i v u1 u2) \to (\forall (k: K).(\forall (t1: T).(\forall (t2: T).((subst1 (s +k i) v t1 t2) \to (subst1 i v (THead k u1 t1) (THead k u2 t2)))))))))) +\def + \lambda (v: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i: nat).(\lambda +(H: (subst1 i v u1 u2)).(subst1_ind i v u1 (\lambda (t: T).(\forall (k: +K).(\forall (t1: T).(\forall (t2: T).((subst1 (s k i) v t1 t2) \to (subst1 i +v (THead k u1 t1) (THead k t t2))))))) (\lambda (k: K).(\lambda (t1: +T).(\lambda (t2: T).(\lambda (H0: (subst1 (s k i) v t1 t2)).(subst1_ind (s k +i) v t1 (\lambda (t: T).(subst1 i v (THead k u1 t1) (THead k u1 t))) +(subst1_refl i v (THead k u1 t1)) (\lambda (t3: T).(\lambda (H1: (subst0 (s k +i) v t1 t3)).(subst1_single i v (THead k u1 t1) (THead k u1 t3) (subst0_snd k +v t3 t1 i H1 u1)))) t2 H0))))) (\lambda (t2: T).(\lambda (H0: (subst0 i v u1 +t2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t0: T).(\lambda (H1: (subst1 +(s k i) v t1 t0)).(subst1_ind (s k i) v t1 (\lambda (t: T).(subst1 i v (THead +k u1 t1) (THead k t2 t))) (subst1_single i v (THead k u1 t1) (THead k t2 t1) +(subst0_fst v t2 u1 i H0 t1 k)) (\lambda (t3: T).(\lambda (H2: (subst0 (s k +i) v t1 t3)).(subst1_single i v (THead k u1 t1) (THead k t2 t3) (subst0_both +v u1 t2 i H0 k t1 t3 H2)))) t0 H1))))))) u2 H))))). +(* COMMENTS +Initial nodes: 369 +END *) + +theorem subst1_lift_lt: + \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).((subst1 +i u t1 t2) \to (\forall (d: nat).((lt i d) \to (\forall (h: nat).(subst1 i +(lift h (minus d (S i)) u) (lift h d t1) (lift h d t2))))))))) +\def + \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda +(H: (subst1 i u t1 t2)).(subst1_ind i u t1 (\lambda (t: T).(\forall (d: +nat).((lt i d) \to (\forall (h: nat).(subst1 i (lift h (minus d (S i)) u) +(lift h d t1) (lift h d t)))))) (\lambda (d: nat).(\lambda (_: (lt i +d)).(\lambda (h: nat).(subst1_refl i (lift h (minus d (S i)) u) (lift h d +t1))))) (\lambda (t3: T).(\lambda (H0: (subst0 i u t1 t3)).(\lambda (d: +nat).(\lambda (H1: (lt i d)).(\lambda (h: nat).(subst1_single i (lift h +(minus d (S i)) u) (lift h d t1) (lift h d t3) (subst0_lift_lt t1 t3 u i H0 d +H1 h))))))) t2 H))))). +(* COMMENTS +Initial nodes: 185 +END *) + +theorem subst1_lift_ge: + \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).(\forall +(h: nat).((subst1 i u t1 t2) \to (\forall (d: nat).((le d i) \to (subst1 +(plus i h) u (lift h d t1) (lift h d t2))))))))) +\def + \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda +(h: nat).(\lambda (H: (subst1 i u t1 t2)).(subst1_ind i u t1 (\lambda (t: +T).(\forall (d: nat).((le d i) \to (subst1 (plus i h) u (lift h d t1) (lift h +d t))))) (\lambda (d: nat).(\lambda (_: (le d i)).(subst1_refl (plus i h) u +(lift h d t1)))) (\lambda (t3: T).(\lambda (H0: (subst0 i u t1 t3)).(\lambda +(d: nat).(\lambda (H1: (le d i)).(subst1_single (plus i h) u (lift h d t1) +(lift h d t3) (subst0_lift_ge t1 t3 u i h H0 d H1)))))) t2 H)))))). +(* COMMENTS +Initial nodes: 157 +END *) + +theorem subst1_ex: + \forall (u: T).(\forall (t1: T).(\forall (d: nat).(ex T (\lambda (t2: +T).(subst1 d u t1 (lift (S O) d t2)))))) +\def + \lambda (u: T).(\lambda (t1: T).(T_ind (\lambda (t: T).(\forall (d: nat).(ex +T (\lambda (t2: T).(subst1 d u t (lift (S O) d t2)))))) (\lambda (n: +nat).(\lambda (d: nat).(ex_intro T (\lambda (t2: T).(subst1 d u (TSort n) +(lift (S O) d t2))) (TSort n) (eq_ind_r T (TSort n) (\lambda (t: T).(subst1 d +u (TSort n) t)) (subst1_refl d u (TSort n)) (lift (S O) d (TSort n)) +(lift_sort n (S O) d))))) (\lambda (n: nat).(\lambda (d: nat).(lt_eq_gt_e n d +(ex T (\lambda (t2: T).(subst1 d u (TLRef n) (lift (S O) d t2)))) (\lambda +(H: (lt n d)).(ex_intro T (\lambda (t2: T).(subst1 d u (TLRef n) (lift (S O) +d t2))) (TLRef n) (eq_ind_r T (TLRef n) (\lambda (t: T).(subst1 d u (TLRef n) +t)) (subst1_refl d u (TLRef n)) (lift (S O) d (TLRef n)) (lift_lref_lt n (S +O) d H)))) (\lambda (H: (eq nat n d)).(eq_ind nat n (\lambda (n0: nat).(ex T +(\lambda (t2: T).(subst1 n0 u (TLRef n) (lift (S O) n0 t2))))) (ex_intro T +(\lambda (t2: T).(subst1 n u (TLRef n) (lift (S O) n t2))) (lift n O u) +(eq_ind_r T (lift (plus (S O) n) O u) (\lambda (t: T).(subst1 n u (TLRef n) +t)) (subst1_single n u (TLRef n) (lift (S n) O u) (subst0_lref u n)) (lift (S +O) n (lift n O u)) (lift_free u n (S O) O n (le_n (plus O n)) (le_O_n n)))) d +H)) (\lambda (H: (lt d n)).(ex_intro T (\lambda (t2: T).(subst1 d u (TLRef n) +(lift (S O) d t2))) (TLRef (pred n)) (eq_ind_r T (TLRef n) (\lambda (t: +T).(subst1 d u (TLRef n) t)) (subst1_refl d u (TLRef n)) (lift (S O) d (TLRef +(pred n))) (lift_lref_gt d n H))))))) (\lambda (k: K).(\lambda (t: +T).(\lambda (H: ((\forall (d: nat).(ex T (\lambda (t2: T).(subst1 d u t (lift +(S O) d t2))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (d: nat).(ex T +(\lambda (t2: T).(subst1 d u t0 (lift (S O) d t2))))))).(\lambda (d: +nat).(let H_x \def (H d) in (let H1 \def H_x in (ex_ind T (\lambda (t2: +T).(subst1 d u t (lift (S O) d t2))) (ex T (\lambda (t2: T).(subst1 d u +(THead k t t0) (lift (S O) d t2)))) (\lambda (x: T).(\lambda (H2: (subst1 d u +t (lift (S O) d x))).(let H_x0 \def (H0 (s k d)) in (let H3 \def H_x0 in +(ex_ind T (\lambda (t2: T).(subst1 (s k d) u t0 (lift (S O) (s k d) t2))) (ex +T (\lambda (t2: T).(subst1 d u (THead k t t0) (lift (S O) d t2)))) (\lambda +(x0: T).(\lambda (H4: (subst1 (s k d) u t0 (lift (S O) (s k d) +x0))).(ex_intro T (\lambda (t2: T).(subst1 d u (THead k t t0) (lift (S O) d +t2))) (THead k x x0) (eq_ind_r T (THead k (lift (S O) d x) (lift (S O) (s k +d) x0)) (\lambda (t2: T).(subst1 d u (THead k t t0) t2)) (subst1_head u t +(lift (S O) d x) d H2 k t0 (lift (S O) (s k d) x0) H4) (lift (S O) d (THead k +x x0)) (lift_head k x x0 (S O) d))))) H3))))) H1))))))))) t1)). +(* COMMENTS +Initial nodes: 925 +END *) + +theorem subst1_lift_S: + \forall (u: T).(\forall (i: nat).(\forall (h: nat).((le h i) \to (subst1 i +(TLRef h) (lift (S h) (S i) u) (lift (S h) i u))))) +\def + \lambda (u: T).(T_ind (\lambda (t: T).(\forall (i: nat).(\forall (h: +nat).((le h i) \to (subst1 i (TLRef h) (lift (S h) (S i) t) (lift (S h) i +t)))))) (\lambda (n: nat).(\lambda (i: nat).(\lambda (h: nat).(\lambda (_: +(le h i)).(eq_ind_r T (TSort n) (\lambda (t: T).(subst1 i (TLRef h) t (lift +(S h) i (TSort n)))) (eq_ind_r T (TSort n) (\lambda (t: T).(subst1 i (TLRef +h) (TSort n) t)) (subst1_refl i (TLRef h) (TSort n)) (lift (S h) i (TSort n)) +(lift_sort n (S h) i)) (lift (S h) (S i) (TSort n)) (lift_sort n (S h) (S +i))))))) (\lambda (n: nat).(\lambda (i: nat).(\lambda (h: nat).(\lambda (H: +(le h i)).(lt_eq_gt_e n i (subst1 i (TLRef h) (lift (S h) (S i) (TLRef n)) +(lift (S h) i (TLRef n))) (\lambda (H0: (lt n i)).(eq_ind_r T (TLRef n) +(\lambda (t: T).(subst1 i (TLRef h) t (lift (S h) i (TLRef n)))) (eq_ind_r T +(TLRef n) (\lambda (t: T).(subst1 i (TLRef h) (TLRef n) t)) (subst1_refl i +(TLRef h) (TLRef n)) (lift (S h) i (TLRef n)) (lift_lref_lt n (S h) i H0)) +(lift (S h) (S i) (TLRef n)) (lift_lref_lt n (S h) (S i) (le_S (S n) i H0)))) +(\lambda (H0: (eq nat n i)).(let H1 \def (eq_ind_r nat i (\lambda (n0: +nat).(le h n0)) H n H0) in (eq_ind nat n (\lambda (n0: nat).(subst1 n0 (TLRef +h) (lift (S h) (S n0) (TLRef n)) (lift (S h) n0 (TLRef n)))) (eq_ind_r T +(TLRef n) (\lambda (t: T).(subst1 n (TLRef h) t (lift (S h) n (TLRef n)))) +(eq_ind_r T (TLRef (plus n (S h))) (\lambda (t: T).(subst1 n (TLRef h) (TLRef +n) t)) (eq_ind nat (S (plus n h)) (\lambda (n0: nat).(subst1 n (TLRef h) +(TLRef n) (TLRef n0))) (eq_ind_r nat (plus h n) (\lambda (n0: nat).(subst1 n +(TLRef h) (TLRef n) (TLRef (S n0)))) (eq_ind nat (plus h (S n)) (\lambda (n0: +nat).(subst1 n (TLRef h) (TLRef n) (TLRef n0))) (eq_ind T (lift (S n) O +(TLRef h)) (\lambda (t: T).(subst1 n (TLRef h) (TLRef n) t)) (subst1_single n +(TLRef h) (TLRef n) (lift (S n) O (TLRef h)) (subst0_lref (TLRef h) n)) +(TLRef (plus h (S n))) (lift_lref_ge h (S n) O (le_O_n h))) (S (plus h n)) +(sym_eq nat (S (plus h n)) (plus h (S n)) (plus_n_Sm h n))) (plus n h) +(plus_sym n h)) (plus n (S h)) (plus_n_Sm n h)) (lift (S h) n (TLRef n)) +(lift_lref_ge n (S h) n (le_n n))) (lift (S h) (S n) (TLRef n)) (lift_lref_lt +n (S h) (S n) (le_n (S n)))) i H0))) (\lambda (H0: (lt i n)).(eq_ind_r T +(TLRef (plus n (S h))) (\lambda (t: T).(subst1 i (TLRef h) t (lift (S h) i +(TLRef n)))) (eq_ind_r T (TLRef (plus n (S h))) (\lambda (t: T).(subst1 i +(TLRef h) (TLRef (plus n (S h))) t)) (subst1_refl i (TLRef h) (TLRef (plus n +(S h)))) (lift (S h) i (TLRef n)) (lift_lref_ge n (S h) i (le_S_n i n (le_S +(S i) n H0)))) (lift (S h) (S i) (TLRef n)) (lift_lref_ge n (S h) (S i) +H0)))))))) (\lambda (k: K).(\lambda (t: T).(\lambda (H: ((\forall (i: +nat).(\forall (h: nat).((le h i) \to (subst1 i (TLRef h) (lift (S h) (S i) t) +(lift (S h) i t))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (i: +nat).(\forall (h: nat).((le h i) \to (subst1 i (TLRef h) (lift (S h) (S i) +t0) (lift (S h) i t0))))))).(\lambda (i: nat).(\lambda (h: nat).(\lambda (H1: +(le h i)).(eq_ind_r T (THead k (lift (S h) (S i) t) (lift (S h) (s k (S i)) +t0)) (\lambda (t1: T).(subst1 i (TLRef h) t1 (lift (S h) i (THead k t t0)))) +(eq_ind_r T (THead k (lift (S h) i t) (lift (S h) (s k i) t0)) (\lambda (t1: +T).(subst1 i (TLRef h) (THead k (lift (S h) (S i) t) (lift (S h) (s k (S i)) +t0)) t1)) (subst1_head (TLRef h) (lift (S h) (S i) t) (lift (S h) i t) i (H i +h H1) k (lift (S h) (s k (S i)) t0) (lift (S h) (s k i) t0) (eq_ind_r nat (S +(s k i)) (\lambda (n: nat).(subst1 (s k i) (TLRef h) (lift (S h) n t0) (lift +(S h) (s k i) t0))) (H0 (s k i) h (le_trans h i (s k i) H1 (s_inc k i))) (s k +(S i)) (s_S k i))) (lift (S h) i (THead k t t0)) (lift_head k t t0 (S h) i)) +(lift (S h) (S i) (THead k t t0)) (lift_head k t t0 (S h) (S i))))))))))) u). +(* COMMENTS +Initial nodes: 1421 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/subst1/subst1.ma b/matita/matita/contribs/lambdadelta/basic_1/subst1/subst1.ma new file mode 100644 index 000000000..e6dae0dc9 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/subst1/subst1.ma @@ -0,0 +1,214 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/subst1/fwd.ma". + +include "Basic-1/subst0/subst0.ma". + +theorem subst1_subst1: + \forall (t1: T).(\forall (t2: T).(\forall (u2: T).(\forall (j: nat).((subst1 +j u2 t1 t2) \to (\forall (u1: T).(\forall (u: T).(\forall (i: nat).((subst1 i +u u1 u2) \to (ex2 T (\lambda (t: T).(subst1 j u1 t1 t)) (\lambda (t: +T).(subst1 (S (plus i j)) u t t2))))))))))) +\def + \lambda (t1: T).(\lambda (t2: T).(\lambda (u2: T).(\lambda (j: nat).(\lambda +(H: (subst1 j u2 t1 t2)).(subst1_ind j u2 t1 (\lambda (t: T).(\forall (u1: +T).(\forall (u: T).(\forall (i: nat).((subst1 i u u1 u2) \to (ex2 T (\lambda +(t0: T).(subst1 j u1 t1 t0)) (\lambda (t0: T).(subst1 (S (plus i j)) u t0 +t)))))))) (\lambda (u1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda (_: +(subst1 i u u1 u2)).(ex_intro2 T (\lambda (t: T).(subst1 j u1 t1 t)) (\lambda +(t: T).(subst1 (S (plus i j)) u t t1)) t1 (subst1_refl j u1 t1) (subst1_refl +(S (plus i j)) u t1)))))) (\lambda (t3: T).(\lambda (H0: (subst0 j u2 t1 +t3)).(\lambda (u1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda (H1: (subst1 +i u u1 u2)).(insert_eq T u2 (\lambda (t: T).(subst1 i u u1 t)) (\lambda (_: +T).(ex2 T (\lambda (t0: T).(subst1 j u1 t1 t0)) (\lambda (t0: T).(subst1 (S +(plus i j)) u t0 t3)))) (\lambda (y: T).(\lambda (H2: (subst1 i u u1 +y)).(subst1_ind i u u1 (\lambda (t: T).((eq T t u2) \to (ex2 T (\lambda (t0: +T).(subst1 j u1 t1 t0)) (\lambda (t0: T).(subst1 (S (plus i j)) u t0 t3))))) +(\lambda (H3: (eq T u1 u2)).(eq_ind_r T u2 (\lambda (t: T).(ex2 T (\lambda +(t0: T).(subst1 j t t1 t0)) (\lambda (t0: T).(subst1 (S (plus i j)) u t0 +t3)))) (ex_intro2 T (\lambda (t: T).(subst1 j u2 t1 t)) (\lambda (t: +T).(subst1 (S (plus i j)) u t t3)) t3 (subst1_single j u2 t1 t3 H0) +(subst1_refl (S (plus i j)) u t3)) u1 H3)) (\lambda (t0: T).(\lambda (H3: +(subst0 i u u1 t0)).(\lambda (H4: (eq T t0 u2)).(let H5 \def (eq_ind T t0 +(\lambda (t: T).(subst0 i u u1 t)) H3 u2 H4) in (ex2_ind T (\lambda (t: +T).(subst0 j u1 t1 t)) (\lambda (t: T).(subst0 (S (plus i j)) u t t3)) (ex2 T +(\lambda (t: T).(subst1 j u1 t1 t)) (\lambda (t: T).(subst1 (S (plus i j)) u +t t3))) (\lambda (x: T).(\lambda (H6: (subst0 j u1 t1 x)).(\lambda (H7: +(subst0 (S (plus i j)) u x t3)).(ex_intro2 T (\lambda (t: T).(subst1 j u1 t1 +t)) (\lambda (t: T).(subst1 (S (plus i j)) u t t3)) x (subst1_single j u1 t1 +x H6) (subst1_single (S (plus i j)) u x t3 H7))))) (subst0_subst0 t1 t3 u2 j +H0 u1 u i H5)))))) y H2))) H1))))))) t2 H))))). +(* COMMENTS +Initial nodes: 649 +END *) + +theorem subst1_subst1_back: + \forall (t1: T).(\forall (t2: T).(\forall (u2: T).(\forall (j: nat).((subst1 +j u2 t1 t2) \to (\forall (u1: T).(\forall (u: T).(\forall (i: nat).((subst1 i +u u2 u1) \to (ex2 T (\lambda (t: T).(subst1 j u1 t1 t)) (\lambda (t: +T).(subst1 (S (plus i j)) u t2 t))))))))))) +\def + \lambda (t1: T).(\lambda (t2: T).(\lambda (u2: T).(\lambda (j: nat).(\lambda +(H: (subst1 j u2 t1 t2)).(subst1_ind j u2 t1 (\lambda (t: T).(\forall (u1: +T).(\forall (u: T).(\forall (i: nat).((subst1 i u u2 u1) \to (ex2 T (\lambda +(t0: T).(subst1 j u1 t1 t0)) (\lambda (t0: T).(subst1 (S (plus i j)) u t +t0)))))))) (\lambda (u1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda (_: +(subst1 i u u2 u1)).(ex_intro2 T (\lambda (t: T).(subst1 j u1 t1 t)) (\lambda +(t: T).(subst1 (S (plus i j)) u t1 t)) t1 (subst1_refl j u1 t1) (subst1_refl +(S (plus i j)) u t1)))))) (\lambda (t3: T).(\lambda (H0: (subst0 j u2 t1 +t3)).(\lambda (u1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda (H1: (subst1 +i u u2 u1)).(subst1_ind i u u2 (\lambda (t: T).(ex2 T (\lambda (t0: +T).(subst1 j t t1 t0)) (\lambda (t0: T).(subst1 (S (plus i j)) u t3 t0)))) +(ex_intro2 T (\lambda (t: T).(subst1 j u2 t1 t)) (\lambda (t: T).(subst1 (S +(plus i j)) u t3 t)) t3 (subst1_single j u2 t1 t3 H0) (subst1_refl (S (plus i +j)) u t3)) (\lambda (t0: T).(\lambda (H2: (subst0 i u u2 t0)).(ex2_ind T +(\lambda (t: T).(subst0 j t0 t1 t)) (\lambda (t: T).(subst0 (S (plus i j)) u +t3 t)) (ex2 T (\lambda (t: T).(subst1 j t0 t1 t)) (\lambda (t: T).(subst1 (S +(plus i j)) u t3 t))) (\lambda (x: T).(\lambda (H3: (subst0 j t0 t1 +x)).(\lambda (H4: (subst0 (S (plus i j)) u t3 x)).(ex_intro2 T (\lambda (t: +T).(subst1 j t0 t1 t)) (\lambda (t: T).(subst1 (S (plus i j)) u t3 t)) x +(subst1_single j t0 t1 x H3) (subst1_single (S (plus i j)) u t3 x H4))))) +(subst0_subst0_back t1 t3 u2 j H0 t0 u i H2)))) u1 H1))))))) t2 H))))). +(* COMMENTS +Initial nodes: 487 +END *) + +theorem subst1_trans: + \forall (t2: T).(\forall (t1: T).(\forall (v: T).(\forall (i: nat).((subst1 +i v t1 t2) \to (\forall (t3: T).((subst1 i v t2 t3) \to (subst1 i v t1 +t3))))))) +\def + \lambda (t2: T).(\lambda (t1: T).(\lambda (v: T).(\lambda (i: nat).(\lambda +(H: (subst1 i v t1 t2)).(subst1_ind i v t1 (\lambda (t: T).(\forall (t3: +T).((subst1 i v t t3) \to (subst1 i v t1 t3)))) (\lambda (t3: T).(\lambda +(H0: (subst1 i v t1 t3)).H0)) (\lambda (t3: T).(\lambda (H0: (subst0 i v t1 +t3)).(\lambda (t4: T).(\lambda (H1: (subst1 i v t3 t4)).(subst1_ind i v t3 +(\lambda (t: T).(subst1 i v t1 t)) (subst1_single i v t1 t3 H0) (\lambda (t0: +T).(\lambda (H2: (subst0 i v t3 t0)).(subst1_single i v t1 t0 (subst0_trans +t3 t1 v i H0 t0 H2)))) t4 H1))))) t2 H))))). +(* COMMENTS +Initial nodes: 165 +END *) + +theorem subst1_confluence_neq: + \forall (t0: T).(\forall (t1: T).(\forall (u1: T).(\forall (i1: +nat).((subst1 i1 u1 t0 t1) \to (\forall (t2: T).(\forall (u2: T).(\forall +(i2: nat).((subst1 i2 u2 t0 t2) \to ((not (eq nat i1 i2)) \to (ex2 T (\lambda +(t: T).(subst1 i2 u2 t1 t)) (\lambda (t: T).(subst1 i1 u1 t2 t)))))))))))) +\def + \lambda (t0: T).(\lambda (t1: T).(\lambda (u1: T).(\lambda (i1: +nat).(\lambda (H: (subst1 i1 u1 t0 t1)).(subst1_ind i1 u1 t0 (\lambda (t: +T).(\forall (t2: T).(\forall (u2: T).(\forall (i2: nat).((subst1 i2 u2 t0 t2) +\to ((not (eq nat i1 i2)) \to (ex2 T (\lambda (t3: T).(subst1 i2 u2 t t3)) +(\lambda (t3: T).(subst1 i1 u1 t2 t3))))))))) (\lambda (t2: T).(\lambda (u2: +T).(\lambda (i2: nat).(\lambda (H0: (subst1 i2 u2 t0 t2)).(\lambda (_: (not +(eq nat i1 i2))).(ex_intro2 T (\lambda (t: T).(subst1 i2 u2 t0 t)) (\lambda +(t: T).(subst1 i1 u1 t2 t)) t2 H0 (subst1_refl i1 u1 t2))))))) (\lambda (t2: +T).(\lambda (H0: (subst0 i1 u1 t0 t2)).(\lambda (t3: T).(\lambda (u2: +T).(\lambda (i2: nat).(\lambda (H1: (subst1 i2 u2 t0 t3)).(\lambda (H2: (not +(eq nat i1 i2))).(subst1_ind i2 u2 t0 (\lambda (t: T).(ex2 T (\lambda (t4: +T).(subst1 i2 u2 t2 t4)) (\lambda (t4: T).(subst1 i1 u1 t t4)))) (ex_intro2 T +(\lambda (t: T).(subst1 i2 u2 t2 t)) (\lambda (t: T).(subst1 i1 u1 t0 t)) t2 +(subst1_refl i2 u2 t2) (subst1_single i1 u1 t0 t2 H0)) (\lambda (t4: +T).(\lambda (H3: (subst0 i2 u2 t0 t4)).(ex2_ind T (\lambda (t: T).(subst0 i1 +u1 t4 t)) (\lambda (t: T).(subst0 i2 u2 t2 t)) (ex2 T (\lambda (t: T).(subst1 +i2 u2 t2 t)) (\lambda (t: T).(subst1 i1 u1 t4 t))) (\lambda (x: T).(\lambda +(H4: (subst0 i1 u1 t4 x)).(\lambda (H5: (subst0 i2 u2 t2 x)).(ex_intro2 T +(\lambda (t: T).(subst1 i2 u2 t2 t)) (\lambda (t: T).(subst1 i1 u1 t4 t)) x +(subst1_single i2 u2 t2 x H5) (subst1_single i1 u1 t4 x H4))))) +(subst0_confluence_neq t0 t4 u2 i2 H3 t2 u1 i1 H0 (sym_not_eq nat i1 i2 +H2))))) t3 H1)))))))) t1 H))))). +(* COMMENTS +Initial nodes: 455 +END *) + +theorem subst1_confluence_eq: + \forall (t0: T).(\forall (t1: T).(\forall (u: T).(\forall (i: nat).((subst1 +i u t0 t1) \to (\forall (t2: T).((subst1 i u t0 t2) \to (ex2 T (\lambda (t: +T).(subst1 i u t1 t)) (\lambda (t: T).(subst1 i u t2 t))))))))) +\def + \lambda (t0: T).(\lambda (t1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda +(H: (subst1 i u t0 t1)).(subst1_ind i u t0 (\lambda (t: T).(\forall (t2: +T).((subst1 i u t0 t2) \to (ex2 T (\lambda (t3: T).(subst1 i u t t3)) +(\lambda (t3: T).(subst1 i u t2 t3)))))) (\lambda (t2: T).(\lambda (H0: +(subst1 i u t0 t2)).(ex_intro2 T (\lambda (t: T).(subst1 i u t0 t)) (\lambda +(t: T).(subst1 i u t2 t)) t2 H0 (subst1_refl i u t2)))) (\lambda (t2: +T).(\lambda (H0: (subst0 i u t0 t2)).(\lambda (t3: T).(\lambda (H1: (subst1 i +u t0 t3)).(subst1_ind i u t0 (\lambda (t: T).(ex2 T (\lambda (t4: T).(subst1 +i u t2 t4)) (\lambda (t4: T).(subst1 i u t t4)))) (ex_intro2 T (\lambda (t: +T).(subst1 i u t2 t)) (\lambda (t: T).(subst1 i u t0 t)) t2 (subst1_refl i u +t2) (subst1_single i u t0 t2 H0)) (\lambda (t4: T).(\lambda (H2: (subst0 i u +t0 t4)).(or4_ind (eq T t4 t2) (ex2 T (\lambda (t: T).(subst0 i u t4 t)) +(\lambda (t: T).(subst0 i u t2 t))) (subst0 i u t4 t2) (subst0 i u t2 t4) +(ex2 T (\lambda (t: T).(subst1 i u t2 t)) (\lambda (t: T).(subst1 i u t4 t))) +(\lambda (H3: (eq T t4 t2)).(eq_ind_r T t2 (\lambda (t: T).(ex2 T (\lambda +(t5: T).(subst1 i u t2 t5)) (\lambda (t5: T).(subst1 i u t t5)))) (ex_intro2 +T (\lambda (t: T).(subst1 i u t2 t)) (\lambda (t: T).(subst1 i u t2 t)) t2 +(subst1_refl i u t2) (subst1_refl i u t2)) t4 H3)) (\lambda (H3: (ex2 T +(\lambda (t: T).(subst0 i u t4 t)) (\lambda (t: T).(subst0 i u t2 +t)))).(ex2_ind T (\lambda (t: T).(subst0 i u t4 t)) (\lambda (t: T).(subst0 i +u t2 t)) (ex2 T (\lambda (t: T).(subst1 i u t2 t)) (\lambda (t: T).(subst1 i +u t4 t))) (\lambda (x: T).(\lambda (H4: (subst0 i u t4 x)).(\lambda (H5: +(subst0 i u t2 x)).(ex_intro2 T (\lambda (t: T).(subst1 i u t2 t)) (\lambda +(t: T).(subst1 i u t4 t)) x (subst1_single i u t2 x H5) (subst1_single i u t4 +x H4))))) H3)) (\lambda (H3: (subst0 i u t4 t2)).(ex_intro2 T (\lambda (t: +T).(subst1 i u t2 t)) (\lambda (t: T).(subst1 i u t4 t)) t2 (subst1_refl i u +t2) (subst1_single i u t4 t2 H3))) (\lambda (H3: (subst0 i u t2 +t4)).(ex_intro2 T (\lambda (t: T).(subst1 i u t2 t)) (\lambda (t: T).(subst1 +i u t4 t)) t4 (subst1_single i u t2 t4 H3) (subst1_refl i u t4))) +(subst0_confluence_eq t0 t4 u i H2 t2 H0)))) t3 H1))))) t1 H))))). +(* COMMENTS +Initial nodes: 729 +END *) + +theorem subst1_confluence_lift: + \forall (t0: T).(\forall (t1: T).(\forall (u: T).(\forall (i: nat).((subst1 +i u t0 (lift (S O) i t1)) \to (\forall (t2: T).((subst1 i u t0 (lift (S O) i +t2)) \to (eq T t1 t2))))))) +\def + \lambda (t0: T).(\lambda (t1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda +(H: (subst1 i u t0 (lift (S O) i t1))).(insert_eq T (lift (S O) i t1) +(\lambda (t: T).(subst1 i u t0 t)) (\lambda (_: T).(\forall (t2: T).((subst1 +i u t0 (lift (S O) i t2)) \to (eq T t1 t2)))) (\lambda (y: T).(\lambda (H0: +(subst1 i u t0 y)).(subst1_ind i u t0 (\lambda (t: T).((eq T t (lift (S O) i +t1)) \to (\forall (t2: T).((subst1 i u t0 (lift (S O) i t2)) \to (eq T t1 +t2))))) (\lambda (H1: (eq T t0 (lift (S O) i t1))).(\lambda (t2: T).(\lambda +(H2: (subst1 i u t0 (lift (S O) i t2))).(let H3 \def (eq_ind T t0 (\lambda +(t: T).(subst1 i u t (lift (S O) i t2))) H2 (lift (S O) i t1) H1) in (let H4 +\def (sym_eq T (lift (S O) i t2) (lift (S O) i t1) (subst1_gen_lift_eq t1 u +(lift (S O) i t2) (S O) i i (le_n i) (eq_ind_r nat (plus (S O) i) (\lambda +(n: nat).(lt i n)) (le_n (plus (S O) i)) (plus i (S O)) (plus_sym i (S O))) +H3)) in (lift_inj t1 t2 (S O) i H4)))))) (\lambda (t2: T).(\lambda (H1: +(subst0 i u t0 t2)).(\lambda (H2: (eq T t2 (lift (S O) i t1))).(\lambda (t3: +T).(\lambda (H3: (subst1 i u t0 (lift (S O) i t3))).(let H4 \def (eq_ind T t2 +(\lambda (t: T).(subst0 i u t0 t)) H1 (lift (S O) i t1) H2) in (insert_eq T +(lift (S O) i t3) (\lambda (t: T).(subst1 i u t0 t)) (\lambda (_: T).(eq T t1 +t3)) (\lambda (y0: T).(\lambda (H5: (subst1 i u t0 y0)).(subst1_ind i u t0 +(\lambda (t: T).((eq T t (lift (S O) i t3)) \to (eq T t1 t3))) (\lambda (H6: +(eq T t0 (lift (S O) i t3))).(let H7 \def (eq_ind T t0 (\lambda (t: +T).(subst0 i u t (lift (S O) i t1))) H4 (lift (S O) i t3) H6) in +(subst0_gen_lift_false t3 u (lift (S O) i t1) (S O) i i (le_n i) (eq_ind_r +nat (plus (S O) i) (\lambda (n: nat).(lt i n)) (le_n (plus (S O) i)) (plus i +(S O)) (plus_sym i (S O))) H7 (eq T t1 t3)))) (\lambda (t4: T).(\lambda (H6: +(subst0 i u t0 t4)).(\lambda (H7: (eq T t4 (lift (S O) i t3))).(let H8 \def +(eq_ind T t4 (\lambda (t: T).(subst0 i u t0 t)) H6 (lift (S O) i t3) H7) in +(sym_eq T t3 t1 (subst0_confluence_lift t0 t3 u i H8 t1 H4)))))) y0 H5))) +H3))))))) y H0))) H))))). +(* COMMENTS +Initial nodes: 735 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/theory.ma b/matita/matita/contribs/lambdadelta/basic_1/theory.ma new file mode 100644 index 000000000..f405be588 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/theory.ma @@ -0,0 +1,42 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/subst0/tlt.ma". + +include "Basic-1/subst/props.ma". + +include "Basic-1/sty1/cnt.ma". + +include "Basic-1/ex0/props.ma". + +include "Basic-1/wcpr0/fwd.ma". + +include "Basic-1/pr3/wcpr0.ma". + +include "Basic-1/ex2/props.ma". + +include "Basic-1/ex1/props.ma". + +include "Basic-1/ty3/sty0.ma". + +include "Basic-1/csubt/csuba.ma". + +include "Basic-1/ty3/fwd_nf2.ma". + +include "Basic-1/ty3/nf2.ma". + +include "Basic-1/wf3/props.ma". + diff --git a/matita/matita/contribs/lambdadelta/basic_1/tlist/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/tlist/defs.ma new file mode 100644 index 000000000..9445013d4 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/tlist/defs.ma @@ -0,0 +1,47 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/T/defs.ma". + +inductive TList: Set \def +| TNil: TList +| TCons: T \to (TList \to TList). + +definition THeads: + K \to (TList \to (T \to T)) +\def + let rec THeads (k: K) (us: TList) on us: (T \to T) \def (\lambda (t: +T).(match us with [TNil \Rightarrow t | (TCons u ul) \Rightarrow (THead k u +(THeads k ul t))])) in THeads. + +definition TApp: + TList \to (T \to TList) +\def + let rec TApp (ts: TList) on ts: (T \to TList) \def (\lambda (v: T).(match ts +with [TNil \Rightarrow (TCons v TNil) | (TCons t ts0) \Rightarrow (TCons t +(TApp ts0 v))])) in TApp. + +definition tslen: + TList \to nat +\def + let rec tslen (ts: TList) on ts: nat \def (match ts with [TNil \Rightarrow O +| (TCons _ ts0) \Rightarrow (S (tslen ts0))]) in tslen. + +definition tslt: + TList \to (TList \to Prop) +\def + \lambda (ts1: TList).(\lambda (ts2: TList).(lt (tslen ts1) (tslen ts2))). + diff --git a/matita/matita/contribs/lambdadelta/basic_1/tlist/props.ma b/matita/matita/contribs/lambdadelta/basic_1/tlist/props.ma new file mode 100644 index 000000000..92b0d05ff --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/tlist/props.ma @@ -0,0 +1,131 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/tlist/defs.ma". + +theorem tslt_wf__q_ind: + \forall (P: ((TList \to Prop))).(((\forall (n: nat).((\lambda (P0: ((TList +\to Prop))).(\lambda (n0: nat).(\forall (ts: TList).((eq nat (tslen ts) n0) +\to (P0 ts))))) P n))) \to (\forall (ts: TList).(P ts))) +\def + let Q \def (\lambda (P: ((TList \to Prop))).(\lambda (n: nat).(\forall (ts: +TList).((eq nat (tslen ts) n) \to (P ts))))) in (\lambda (P: ((TList \to +Prop))).(\lambda (H: ((\forall (n: nat).(\forall (ts: TList).((eq nat (tslen +ts) n) \to (P ts)))))).(\lambda (ts: TList).(H (tslen ts) ts (refl_equal nat +(tslen ts)))))). +(* COMMENTS +Initial nodes: 61 +END *) + +theorem tslt_wf_ind: + \forall (P: ((TList \to Prop))).(((\forall (ts2: TList).(((\forall (ts1: +TList).((tslt ts1 ts2) \to (P ts1)))) \to (P ts2)))) \to (\forall (ts: +TList).(P ts))) +\def + let Q \def (\lambda (P: ((TList \to Prop))).(\lambda (n: nat).(\forall (ts: +TList).((eq nat (tslen ts) n) \to (P ts))))) in (\lambda (P: ((TList \to +Prop))).(\lambda (H: ((\forall (ts2: TList).(((\forall (ts1: TList).((lt +(tslen ts1) (tslen ts2)) \to (P ts1)))) \to (P ts2))))).(\lambda (ts: +TList).(tslt_wf__q_ind (\lambda (t: TList).(P t)) (\lambda (n: +nat).(lt_wf_ind n (Q (\lambda (t: TList).(P t))) (\lambda (n0: nat).(\lambda +(H0: ((\forall (m: nat).((lt m n0) \to (Q (\lambda (t: TList).(P t)) +m))))).(\lambda (ts0: TList).(\lambda (H1: (eq nat (tslen ts0) n0)).(let H2 +\def (eq_ind_r nat n0 (\lambda (n1: nat).(\forall (m: nat).((lt m n1) \to +(\forall (ts1: TList).((eq nat (tslen ts1) m) \to (P ts1)))))) H0 (tslen ts0) +H1) in (H ts0 (\lambda (ts1: TList).(\lambda (H3: (lt (tslen ts1) (tslen +ts0))).(H2 (tslen ts1) H3 ts1 (refl_equal nat (tslen ts1))))))))))))) ts)))). +(* COMMENTS +Initial nodes: 179 +END *) + +theorem theads_tapp: + \forall (k: K).(\forall (v: T).(\forall (t: T).(\forall (vs: TList).(eq T +(THeads k (TApp vs v) t) (THeads k vs (THead k v t)))))) +\def + \lambda (k: K).(\lambda (v: T).(\lambda (t: T).(\lambda (vs: +TList).(TList_ind (\lambda (t0: TList).(eq T (THeads k (TApp t0 v) t) (THeads +k t0 (THead k v t)))) (refl_equal T (THead k v t)) (\lambda (t0: T).(\lambda +(t1: TList).(\lambda (H: (eq T (THeads k (TApp t1 v) t) (THeads k t1 (THead k +v t)))).(eq_ind T (THeads k (TApp t1 v) t) (\lambda (t2: T).(eq T (THead k t0 +(THeads k (TApp t1 v) t)) (THead k t0 t2))) (refl_equal T (THead k t0 (THeads +k (TApp t1 v) t))) (THeads k t1 (THead k v t)) H)))) vs)))). +(* COMMENTS +Initial nodes: 175 +END *) + +theorem tcons_tapp_ex: + \forall (ts1: TList).(\forall (t1: T).(ex2_2 TList T (\lambda (ts2: +TList).(\lambda (t2: T).(eq TList (TCons t1 ts1) (TApp ts2 t2)))) (\lambda +(ts2: TList).(\lambda (_: T).(eq nat (tslen ts1) (tslen ts2)))))) +\def + \lambda (ts1: TList).(TList_ind (\lambda (t: TList).(\forall (t1: T).(ex2_2 +TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons t1 t) (TApp +ts2 t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (tslen t) (tslen +ts2))))))) (\lambda (t1: T).(ex2_2_intro TList T (\lambda (ts2: +TList).(\lambda (t2: T).(eq TList (TCons t1 TNil) (TApp ts2 t2)))) (\lambda +(ts2: TList).(\lambda (_: T).(eq nat O (tslen ts2)))) TNil t1 (refl_equal +TList (TApp TNil t1)) (refl_equal nat (tslen TNil)))) (\lambda (t: +T).(\lambda (t0: TList).(\lambda (H: ((\forall (t1: T).(ex2_2 TList T +(\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons t1 t0) (TApp ts2 +t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (tslen t0) (tslen +ts2)))))))).(\lambda (t1: T).(let H_x \def (H t) in (let H0 \def H_x in +(ex2_2_ind TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons t +t0) (TApp ts2 t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (tslen t0) +(tslen ts2)))) (ex2_2 TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq +TList (TCons t1 (TCons t t0)) (TApp ts2 t2)))) (\lambda (ts2: TList).(\lambda +(_: T).(eq nat (S (tslen t0)) (tslen ts2))))) (\lambda (x0: TList).(\lambda +(x1: T).(\lambda (H1: (eq TList (TCons t t0) (TApp x0 x1))).(\lambda (H2: (eq +nat (tslen t0) (tslen x0))).(eq_ind_r TList (TApp x0 x1) (\lambda (t2: +TList).(ex2_2 TList T (\lambda (ts2: TList).(\lambda (t3: T).(eq TList (TCons +t1 t2) (TApp ts2 t3)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (S +(tslen t0)) (tslen ts2)))))) (eq_ind_r nat (tslen x0) (\lambda (n: +nat).(ex2_2 TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons +t1 (TApp x0 x1)) (TApp ts2 t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq +nat (S n) (tslen ts2)))))) (ex2_2_intro TList T (\lambda (ts2: +TList).(\lambda (t2: T).(eq TList (TCons t1 (TApp x0 x1)) (TApp ts2 t2)))) +(\lambda (ts2: TList).(\lambda (_: T).(eq nat (S (tslen x0)) (tslen ts2)))) +(TCons t1 x0) x1 (refl_equal TList (TApp (TCons t1 x0) x1)) (refl_equal nat +(tslen (TCons t1 x0)))) (tslen t0) H2) (TCons t t0) H1))))) H0))))))) ts1). +(* COMMENTS +Initial nodes: 503 +END *) + +theorem tlist_ind_rev: + \forall (P: ((TList \to Prop))).((P TNil) \to (((\forall (ts: +TList).(\forall (t: T).((P ts) \to (P (TApp ts t)))))) \to (\forall (ts: +TList).(P ts)))) +\def + \lambda (P: ((TList \to Prop))).(\lambda (H: (P TNil)).(\lambda (H0: +((\forall (ts: TList).(\forall (t: T).((P ts) \to (P (TApp ts +t))))))).(\lambda (ts: TList).(tslt_wf_ind (\lambda (t: TList).(P t)) +(\lambda (ts2: TList).(TList_ind (\lambda (t: TList).(((\forall (ts1: +TList).((tslt ts1 t) \to (P ts1)))) \to (P t))) (\lambda (_: ((\forall (ts1: +TList).((tslt ts1 TNil) \to (P ts1))))).H) (\lambda (t: T).(\lambda (t0: +TList).(\lambda (_: ((((\forall (ts1: TList).((tslt ts1 t0) \to (P ts1)))) +\to (P t0)))).(\lambda (H2: ((\forall (ts1: TList).((tslt ts1 (TCons t t0)) +\to (P ts1))))).(let H_x \def (tcons_tapp_ex t0 t) in (let H3 \def H_x in +(ex2_2_ind TList T (\lambda (ts3: TList).(\lambda (t2: T).(eq TList (TCons t +t0) (TApp ts3 t2)))) (\lambda (ts3: TList).(\lambda (_: T).(eq nat (tslen t0) +(tslen ts3)))) (P (TCons t t0)) (\lambda (x0: TList).(\lambda (x1: +T).(\lambda (H4: (eq TList (TCons t t0) (TApp x0 x1))).(\lambda (H5: (eq nat +(tslen t0) (tslen x0))).(eq_ind_r TList (TApp x0 x1) (\lambda (t1: TList).(P +t1)) (H0 x0 x1 (H2 x0 (eq_ind nat (tslen t0) (\lambda (n: nat).(lt n (tslen +(TCons t t0)))) (le_n (tslen (TCons t t0))) (tslen x0) H5))) (TCons t t0) +H4))))) H3))))))) ts2)) ts)))). +(* COMMENTS +Initial nodes: 273 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/tlt/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/tlt/defs.ma new file mode 100644 index 000000000..546bdb78d --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/tlt/defs.ma @@ -0,0 +1,46 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/T/defs.ma". + +definition wadd: + ((nat \to nat)) \to (nat \to (nat \to nat)) +\def + \lambda (f: ((nat \to nat))).(\lambda (w: nat).(\lambda (n: nat).(match n +with [O \Rightarrow w | (S m) \Rightarrow (f m)]))). + +definition weight_map: + ((nat \to nat)) \to (T \to nat) +\def + let rec weight_map (f: ((nat \to nat))) (t: T) on t: nat \def (match t with +[(TSort _) \Rightarrow O | (TLRef n) \Rightarrow (f n) | (THead k u t0) +\Rightarrow (match k with [(Bind b) \Rightarrow (match b with [Abbr +\Rightarrow (S (plus (weight_map f u) (weight_map (wadd f (S (weight_map f +u))) t0))) | Abst \Rightarrow (S (plus (weight_map f u) (weight_map (wadd f +O) t0))) | Void \Rightarrow (S (plus (weight_map f u) (weight_map (wadd f O) +t0)))]) | (Flat _) \Rightarrow (S (plus (weight_map f u) (weight_map f +t0)))])]) in weight_map. + +definition weight: + T \to nat +\def + weight_map (\lambda (_: nat).O). + +definition tlt: + T \to (T \to Prop) +\def + \lambda (t1: T).(\lambda (t2: T).(lt (weight t1) (weight t2))). + diff --git a/matita/matita/contribs/lambdadelta/basic_1/tlt/props.ma b/matita/matita/contribs/lambdadelta/basic_1/tlt/props.ma new file mode 100644 index 000000000..0b4f16d41 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/tlt/props.ma @@ -0,0 +1,300 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/tlt/defs.ma". + +theorem wadd_le: + \forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (n: +nat).(le (f n) (g n)))) \to (\forall (v: nat).(\forall (w: nat).((le v w) \to +(\forall (n: nat).(le (wadd f v n) (wadd g w n)))))))) +\def + \lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H: +((\forall (n: nat).(le (f n) (g n))))).(\lambda (v: nat).(\lambda (w: +nat).(\lambda (H0: (le v w)).(\lambda (n: nat).(nat_ind (\lambda (n0: +nat).(le (wadd f v n0) (wadd g w n0))) H0 (\lambda (n0: nat).(\lambda (_: (le +(wadd f v n0) (wadd g w n0))).(H n0))) n))))))). +(* COMMENTS +Initial nodes: 81 +END *) + +theorem wadd_lt: + \forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (n: +nat).(le (f n) (g n)))) \to (\forall (v: nat).(\forall (w: nat).((lt v w) \to +(\forall (n: nat).(le (wadd f v n) (wadd g w n)))))))) +\def + \lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H: +((\forall (n: nat).(le (f n) (g n))))).(\lambda (v: nat).(\lambda (w: +nat).(\lambda (H0: (lt v w)).(\lambda (n: nat).(nat_ind (\lambda (n0: +nat).(le (wadd f v n0) (wadd g w n0))) (le_S_n v w (le_S (S v) w H0)) +(\lambda (n0: nat).(\lambda (_: (le (wadd f v n0) (wadd g w n0))).(H n0))) +n))))))). +(* COMMENTS +Initial nodes: 95 +END *) + +theorem wadd_O: + \forall (n: nat).(eq nat (wadd (\lambda (_: nat).O) O n) O) +\def + \lambda (n: nat).(nat_ind (\lambda (n0: nat).(eq nat (wadd (\lambda (_: +nat).O) O n0) O)) (refl_equal nat O) (\lambda (n0: nat).(\lambda (_: (eq nat +(wadd (\lambda (_: nat).O) O n0) O)).(refl_equal nat O))) n). +(* COMMENTS +Initial nodes: 53 +END *) + +theorem weight_le: + \forall (t: T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to +nat))).(((\forall (n: nat).(le (f n) (g n)))) \to (le (weight_map f t) +(weight_map g t))))) +\def + \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (f: ((nat \to +nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g n)))) +\to (le (weight_map f t0) (weight_map g t0)))))) (\lambda (n: nat).(\lambda +(f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (_: ((\forall +(n0: nat).(le (f n0) (g n0))))).(le_n (weight_map g (TSort n))))))) (\lambda +(n: nat).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda +(H: ((\forall (n0: nat).(le (f n0) (g n0))))).(H n))))) (\lambda (k: +K).(K_ind (\lambda (k0: K).(\forall (t0: T).(((\forall (f: ((nat \to +nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g n)))) +\to (le (weight_map f t0) (weight_map g t0)))))) \to (\forall (t1: +T).(((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall +(n: nat).(le (f n) (g n)))) \to (le (weight_map f t1) (weight_map g t1)))))) +\to (\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall +(n: nat).(le (f n) (g n)))) \to (le (weight_map f (THead k0 t0 t1)) +(weight_map g (THead k0 t0 t1))))))))))) (\lambda (b: B).(B_ind (\lambda (b0: +B).(\forall (t0: T).(((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to +nat))).(((\forall (n: nat).(le (f n) (g n)))) \to (le (weight_map f t0) +(weight_map g t0)))))) \to (\forall (t1: T).(((\forall (f: ((nat \to +nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g n)))) +\to (le (weight_map f t1) (weight_map g t1)))))) \to (\forall (f: ((nat \to +nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g n)))) +\to (le (match b0 with [Abbr \Rightarrow (S (plus (weight_map f t0) +(weight_map (wadd f (S (weight_map f t0))) t1))) | Abst \Rightarrow (S (plus +(weight_map f t0) (weight_map (wadd f O) t1))) | Void \Rightarrow (S (plus +(weight_map f t0) (weight_map (wadd f O) t1)))]) (match b0 with [Abbr +\Rightarrow (S (plus (weight_map g t0) (weight_map (wadd g (S (weight_map g +t0))) t1))) | Abst \Rightarrow (S (plus (weight_map g t0) (weight_map (wadd g +O) t1))) | Void \Rightarrow (S (plus (weight_map g t0) (weight_map (wadd g O) +t1)))])))))))))) (\lambda (t0: T).(\lambda (H: ((\forall (f: ((nat \to +nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g n)))) +\to (le (weight_map f t0) (weight_map g t0))))))).(\lambda (t1: T).(\lambda +(H0: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall +(n: nat).(le (f n) (g n)))) \to (le (weight_map f t1) (weight_map g +t1))))))).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to +nat))).(\lambda (H1: ((\forall (n: nat).(le (f n) (g n))))).(le_n_S (plus +(weight_map f t0) (weight_map (wadd f (S (weight_map f t0))) t1)) (plus +(weight_map g t0) (weight_map (wadd g (S (weight_map g t0))) t1)) +(le_plus_plus (weight_map f t0) (weight_map g t0) (weight_map (wadd f (S +(weight_map f t0))) t1) (weight_map (wadd g (S (weight_map g t0))) t1) (H f g +H1) (H0 (wadd f (S (weight_map f t0))) (wadd g (S (weight_map g t0))) +(\lambda (n: nat).(wadd_le f g H1 (S (weight_map f t0)) (S (weight_map g t0)) +(le_n_S (weight_map f t0) (weight_map g t0) (H f g H1)) n)))))))))))) +(\lambda (t0: T).(\lambda (H: ((\forall (f: ((nat \to nat))).(\forall (g: +((nat \to nat))).(((\forall (n: nat).(le (f n) (g n)))) \to (le (weight_map f +t0) (weight_map g t0))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (f: +((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) +(g n)))) \to (le (weight_map f t1) (weight_map g t1))))))).(\lambda (f: ((nat +\to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H1: ((\forall (n: nat).(le +(f n) (g n))))).(le_n_S (plus (weight_map f t0) (weight_map (wadd f O) t1)) +(plus (weight_map g t0) (weight_map (wadd g O) t1)) (le_plus_plus (weight_map +f t0) (weight_map g t0) (weight_map (wadd f O) t1) (weight_map (wadd g O) t1) +(H f g H1) (H0 (wadd f O) (wadd g O) (\lambda (n: nat).(wadd_le f g H1 O O +(le_n O) n)))))))))))) (\lambda (t0: T).(\lambda (H: ((\forall (f: ((nat \to +nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g n)))) +\to (le (weight_map f t0) (weight_map g t0))))))).(\lambda (t1: T).(\lambda +(H0: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall +(n: nat).(le (f n) (g n)))) \to (le (weight_map f t1) (weight_map g +t1))))))).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to +nat))).(\lambda (H1: ((\forall (n: nat).(le (f n) (g n))))).(le_n_S (plus +(weight_map f t0) (weight_map (wadd f O) t1)) (plus (weight_map g t0) +(weight_map (wadd g O) t1)) (le_plus_plus (weight_map f t0) (weight_map g t0) +(weight_map (wadd f O) t1) (weight_map (wadd g O) t1) (H f g H1) (H0 (wadd f +O) (wadd g O) (\lambda (n: nat).(wadd_le f g H1 O O (le_n O) n)))))))))))) +b)) (\lambda (_: F).(\lambda (t0: T).(\lambda (H: ((\forall (f0: ((nat \to +nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f0 n) (g n)))) +\to (le (weight_map f0 t0) (weight_map g t0))))))).(\lambda (t1: T).(\lambda +(H0: ((\forall (f0: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall +(n: nat).(le (f0 n) (g n)))) \to (le (weight_map f0 t1) (weight_map g +t1))))))).(\lambda (f0: ((nat \to nat))).(\lambda (g: ((nat \to +nat))).(\lambda (H1: ((\forall (n: nat).(le (f0 n) (g n))))).(le_n_S (plus +(weight_map f0 t0) (weight_map f0 t1)) (plus (weight_map g t0) (weight_map g +t1)) (le_plus_plus (weight_map f0 t0) (weight_map g t0) (weight_map f0 t1) +(weight_map g t1) (H f0 g H1) (H0 f0 g H1))))))))))) k)) t). +(* COMMENTS +Initial nodes: 1309 +END *) + +theorem weight_eq: + \forall (t: T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to +nat))).(((\forall (n: nat).(eq nat (f n) (g n)))) \to (eq nat (weight_map f +t) (weight_map g t))))) +\def + \lambda (t: T).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to +nat))).(\lambda (H: ((\forall (n: nat).(eq nat (f n) (g n))))).(le_antisym +(weight_map f t) (weight_map g t) (weight_le t f g (\lambda (n: +nat).(eq_ind_r nat (g n) (\lambda (n0: nat).(le n0 (g n))) (le_n (g n)) (f n) +(H n)))) (weight_le t g f (\lambda (n: nat).(eq_ind_r nat (g n) (\lambda (n0: +nat).(le (g n) n0)) (le_n (g n)) (f n) (H n)))))))). +(* COMMENTS +Initial nodes: 121 +END *) + +theorem weight_add_O: + \forall (t: T).(eq nat (weight_map (wadd (\lambda (_: nat).O) O) t) +(weight_map (\lambda (_: nat).O) t)) +\def + \lambda (t: T).(weight_eq t (wadd (\lambda (_: nat).O) O) (\lambda (_: +nat).O) (\lambda (n: nat).(wadd_O n))). +(* COMMENTS +Initial nodes: 23 +END *) + +theorem weight_add_S: + \forall (t: T).(\forall (m: nat).(le (weight_map (wadd (\lambda (_: nat).O) +O) t) (weight_map (wadd (\lambda (_: nat).O) (S m)) t))) +\def + \lambda (t: T).(\lambda (m: nat).(weight_le t (wadd (\lambda (_: nat).O) O) +(wadd (\lambda (_: nat).O) (S m)) (\lambda (n: nat).(wadd_le (\lambda (_: +nat).O) (\lambda (_: nat).O) (\lambda (_: nat).(le_n O)) O (S m) (le_S O m +(le_O_n m)) n)))). +(* COMMENTS +Initial nodes: 61 +END *) + +theorem tlt_trans: + \forall (v: T).(\forall (u: T).(\forall (t: T).((tlt u v) \to ((tlt v t) \to +(tlt u t))))) +\def + \lambda (v: T).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (lt (weight u) +(weight v))).(\lambda (H0: (lt (weight v) (weight t))).(lt_trans (weight u) +(weight v) (weight t) H H0))))). +(* COMMENTS +Initial nodes: 43 +END *) + +theorem tlt_head_sx: + \forall (k: K).(\forall (u: T).(\forall (t: T).(tlt u (THead k u t)))) +\def + \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (u: T).(\forall (t: T).(lt +(weight_map (\lambda (_: nat).O) u) (weight_map (\lambda (_: nat).O) (THead +k0 u t)))))) (\lambda (b: B).(B_ind (\lambda (b0: B).(\forall (u: T).(\forall +(t: T).(lt (weight_map (\lambda (_: nat).O) u) (match b0 with [Abbr +\Rightarrow (S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd +(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t))) | Abst +\Rightarrow (S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd +(\lambda (_: nat).O) O) t))) | Void \Rightarrow (S (plus (weight_map (\lambda +(_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) O) t)))]))))) (\lambda +(u: T).(\lambda (t: T).(le_n_S (weight_map (\lambda (_: nat).O) u) (plus +(weight_map (\lambda (_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) (S +(weight_map (\lambda (_: nat).O) u))) t)) (le_plus_l (weight_map (\lambda (_: +nat).O) u) (weight_map (wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: +nat).O) u))) t))))) (\lambda (u: T).(\lambda (t: T).(le_n_S (weight_map +(\lambda (_: nat).O) u) (plus (weight_map (\lambda (_: nat).O) u) (weight_map +(wadd (\lambda (_: nat).O) O) t)) (le_plus_l (weight_map (\lambda (_: nat).O) +u) (weight_map (wadd (\lambda (_: nat).O) O) t))))) (\lambda (u: T).(\lambda +(t: T).(le_n_S (weight_map (\lambda (_: nat).O) u) (plus (weight_map (\lambda +(_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) O) t)) (le_plus_l +(weight_map (\lambda (_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) O) +t))))) b)) (\lambda (_: F).(\lambda (u: T).(\lambda (t: T).(le_n_S +(weight_map (\lambda (_: nat).O) u) (plus (weight_map (\lambda (_: nat).O) u) +(weight_map (\lambda (_: nat).O) t)) (le_plus_l (weight_map (\lambda (_: +nat).O) u) (weight_map (\lambda (_: nat).O) t)))))) k). +(* COMMENTS +Initial nodes: 379 +END *) + +theorem tlt_head_dx: + \forall (k: K).(\forall (u: T).(\forall (t: T).(tlt t (THead k u t)))) +\def + \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (u: T).(\forall (t: T).(lt +(weight_map (\lambda (_: nat).O) t) (weight_map (\lambda (_: nat).O) (THead +k0 u t)))))) (\lambda (b: B).(B_ind (\lambda (b0: B).(\forall (u: T).(\forall +(t: T).(lt (weight_map (\lambda (_: nat).O) t) (match b0 with [Abbr +\Rightarrow (S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd +(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t))) | Abst +\Rightarrow (S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd +(\lambda (_: nat).O) O) t))) | Void \Rightarrow (S (plus (weight_map (\lambda +(_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) O) t)))]))))) (\lambda +(u: T).(\lambda (t: T).(lt_le_trans (weight_map (\lambda (_: nat).O) t) (S +(weight_map (\lambda (_: nat).O) t)) (S (plus (weight_map (\lambda (_: +nat).O) u) (weight_map (wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: +nat).O) u))) t))) (lt_n_Sn (weight_map (\lambda (_: nat).O) t)) (le_n_S +(weight_map (\lambda (_: nat).O) t) (plus (weight_map (\lambda (_: nat).O) u) +(weight_map (wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) +u))) t)) (le_trans (weight_map (\lambda (_: nat).O) t) (weight_map (wadd +(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t) (plus +(weight_map (\lambda (_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) (S +(weight_map (\lambda (_: nat).O) u))) t)) (eq_ind nat (weight_map (wadd +(\lambda (_: nat).O) O) t) (\lambda (n: nat).(le n (weight_map (wadd (\lambda +(_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t))) (weight_add_S t +(weight_map (\lambda (_: nat).O) u)) (weight_map (\lambda (_: nat).O) t) +(weight_add_O t)) (le_plus_r (weight_map (\lambda (_: nat).O) u) (weight_map +(wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t))))))) +(\lambda (u: T).(\lambda (t: T).(eq_ind_r nat (weight_map (\lambda (_: +nat).O) t) (\lambda (n: nat).(lt (weight_map (\lambda (_: nat).O) t) (S (plus +(weight_map (\lambda (_: nat).O) u) n)))) (le_n_S (weight_map (\lambda (_: +nat).O) t) (plus (weight_map (\lambda (_: nat).O) u) (weight_map (\lambda (_: +nat).O) t)) (le_plus_r (weight_map (\lambda (_: nat).O) u) (weight_map +(\lambda (_: nat).O) t))) (weight_map (wadd (\lambda (_: nat).O) O) t) +(weight_add_O t)))) (\lambda (u: T).(\lambda (t: T).(eq_ind_r nat (weight_map +(\lambda (_: nat).O) t) (\lambda (n: nat).(lt (weight_map (\lambda (_: +nat).O) t) (S (plus (weight_map (\lambda (_: nat).O) u) n)))) (le_n_S +(weight_map (\lambda (_: nat).O) t) (plus (weight_map (\lambda (_: nat).O) u) +(weight_map (\lambda (_: nat).O) t)) (le_plus_r (weight_map (\lambda (_: +nat).O) u) (weight_map (\lambda (_: nat).O) t))) (weight_map (wadd (\lambda +(_: nat).O) O) t) (weight_add_O t)))) b)) (\lambda (_: F).(\lambda (u: +T).(\lambda (t: T).(le_n_S (weight_map (\lambda (_: nat).O) t) (plus +(weight_map (\lambda (_: nat).O) u) (weight_map (\lambda (_: nat).O) t)) +(le_plus_r (weight_map (\lambda (_: nat).O) u) (weight_map (\lambda (_: +nat).O) t)))))) k). +(* COMMENTS +Initial nodes: 659 +END *) + +theorem tlt_wf__q_ind: + \forall (P: ((T \to Prop))).(((\forall (n: nat).((\lambda (P0: ((T \to +Prop))).(\lambda (n0: nat).(\forall (t: T).((eq nat (weight t) n0) \to (P0 +t))))) P n))) \to (\forall (t: T).(P t))) +\def + let Q \def (\lambda (P: ((T \to Prop))).(\lambda (n: nat).(\forall (t: +T).((eq nat (weight t) n) \to (P t))))) in (\lambda (P: ((T \to +Prop))).(\lambda (H: ((\forall (n: nat).(\forall (t: T).((eq nat (weight t) +n) \to (P t)))))).(\lambda (t: T).(H (weight t) t (refl_equal nat (weight +t)))))). +(* COMMENTS +Initial nodes: 61 +END *) + +theorem tlt_wf_ind: + \forall (P: ((T \to Prop))).(((\forall (t: T).(((\forall (v: T).((tlt v t) +\to (P v)))) \to (P t)))) \to (\forall (t: T).(P t))) +\def + let Q \def (\lambda (P: ((T \to Prop))).(\lambda (n: nat).(\forall (t: +T).((eq nat (weight t) n) \to (P t))))) in (\lambda (P: ((T \to +Prop))).(\lambda (H: ((\forall (t: T).(((\forall (v: T).((lt (weight v) +(weight t)) \to (P v)))) \to (P t))))).(\lambda (t: T).(tlt_wf__q_ind +(\lambda (t0: T).(P t0)) (\lambda (n: nat).(lt_wf_ind n (Q (\lambda (t0: +T).(P t0))) (\lambda (n0: nat).(\lambda (H0: ((\forall (m: nat).((lt m n0) +\to (Q (\lambda (t0: T).(P t0)) m))))).(\lambda (t0: T).(\lambda (H1: (eq nat +(weight t0) n0)).(let H2 \def (eq_ind_r nat n0 (\lambda (n1: nat).(\forall +(m: nat).((lt m n1) \to (\forall (t1: T).((eq nat (weight t1) m) \to (P +t1)))))) H0 (weight t0) H1) in (H t0 (\lambda (v: T).(\lambda (H3: (lt +(weight v) (weight t0))).(H2 (weight v) H3 v (refl_equal nat (weight +v))))))))))))) t)))). +(* COMMENTS +Initial nodes: 179 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/ty3/arity.ma b/matita/matita/contribs/lambdadelta/basic_1/ty3/arity.ma new file mode 100644 index 000000000..4400e1540 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/ty3/arity.ma @@ -0,0 +1,186 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/ty3/pr3_props.ma". + +include "Basic-1/arity/pr3.ma". + +include "Basic-1/asucc/fwd.ma". + +theorem ty3_arity: + \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c +t1 t2) \to (ex2 A (\lambda (a1: A).(arity g c t1 a1)) (\lambda (a1: A).(arity +g c t2 (asucc g a1)))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda +(H: (ty3 g c t1 t2)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda +(t0: T).(ex2 A (\lambda (a1: A).(arity g c0 t a1)) (\lambda (a1: A).(arity g +c0 t0 (asucc g a1))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t: +T).(\lambda (_: (ty3 g c0 t3 t)).(\lambda (H1: (ex2 A (\lambda (a1: A).(arity +g c0 t3 a1)) (\lambda (a1: A).(arity g c0 t (asucc g a1))))).(\lambda (u: +T).(\lambda (t4: T).(\lambda (_: (ty3 g c0 u t4)).(\lambda (H3: (ex2 A +(\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t4 (asucc g +a1))))).(\lambda (H4: (pc3 c0 t4 t3)).(let H5 \def H1 in (ex2_ind A (\lambda +(a1: A).(arity g c0 t3 a1)) (\lambda (a1: A).(arity g c0 t (asucc g a1))) +(ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t3 +(asucc g a1)))) (\lambda (x: A).(\lambda (H6: (arity g c0 t3 x)).(\lambda (_: +(arity g c0 t (asucc g x))).(let H8 \def H3 in (ex2_ind A (\lambda (a1: +A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t4 (asucc g a1))) (ex2 A +(\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t3 (asucc g +a1)))) (\lambda (x0: A).(\lambda (H9: (arity g c0 u x0)).(\lambda (H10: +(arity g c0 t4 (asucc g x0))).(let H11 \def H4 in (ex2_ind T (\lambda (t0: +T).(pr3 c0 t4 t0)) (\lambda (t0: T).(pr3 c0 t3 t0)) (ex2 A (\lambda (a1: +A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t3 (asucc g a1)))) +(\lambda (x1: T).(\lambda (H12: (pr3 c0 t4 x1)).(\lambda (H13: (pr3 c0 t3 +x1)).(ex_intro2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity +g c0 t3 (asucc g a1))) x0 H9 (arity_repl g c0 t3 x H6 (asucc g x0) (leq_sym g +(asucc g x0) x (arity_mono g c0 x1 (asucc g x0) (arity_sred_pr3 c0 t4 x1 H12 +g (asucc g x0) H10) x (arity_sred_pr3 c0 t3 x1 H13 g x H6)))))))) H11))))) +H8))))) H5)))))))))))) (\lambda (c0: C).(\lambda (m: nat).(ex_intro2 A +(\lambda (a1: A).(arity g c0 (TSort m) a1)) (\lambda (a1: A).(arity g c0 +(TSort (next g m)) (asucc g a1))) (ASort O m) (arity_sort g c0 m) (arity_sort +g c0 (next g m))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: +C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abbr) +u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2: (ex2 A +(\lambda (a1: A).(arity g d u a1)) (\lambda (a1: A).(arity g d t (asucc g +a1))))).(let H3 \def H2 in (ex2_ind A (\lambda (a1: A).(arity g d u a1)) +(\lambda (a1: A).(arity g d t (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g +c0 (TLRef n) a1)) (\lambda (a1: A).(arity g c0 (lift (S n) O t) (asucc g +a1)))) (\lambda (x: A).(\lambda (H4: (arity g d u x)).(\lambda (H5: (arity g +d t (asucc g x))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (TLRef n) a1)) +(\lambda (a1: A).(arity g c0 (lift (S n) O t) (asucc g a1))) x (arity_abbr g +c0 d u n H0 x H4) (arity_lift g d t (asucc g x) H5 c0 (S n) O (getl_drop Abbr +c0 d u n H0)))))) H3)))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda +(d: C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abst) +u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2: (ex2 A +(\lambda (a1: A).(arity g d u a1)) (\lambda (a1: A).(arity g d t (asucc g +a1))))).(let H3 \def H2 in (ex2_ind A (\lambda (a1: A).(arity g d u a1)) +(\lambda (a1: A).(arity g d t (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g +c0 (TLRef n) a1)) (\lambda (a1: A).(arity g c0 (lift (S n) O u) (asucc g +a1)))) (\lambda (x: A).(\lambda (H4: (arity g d u x)).(\lambda (_: (arity g d +t (asucc g x))).(let H_x \def (leq_asucc g x) in (let H6 \def H_x in (ex_ind +A (\lambda (a0: A).(leq g x (asucc g a0))) (ex2 A (\lambda (a1: A).(arity g +c0 (TLRef n) a1)) (\lambda (a1: A).(arity g c0 (lift (S n) O u) (asucc g +a1)))) (\lambda (x0: A).(\lambda (H7: (leq g x (asucc g x0))).(ex_intro2 A +(\lambda (a1: A).(arity g c0 (TLRef n) a1)) (\lambda (a1: A).(arity g c0 +(lift (S n) O u) (asucc g a1))) x0 (arity_abst g c0 d u n H0 x0 (arity_repl g +d u x H4 (asucc g x0) H7)) (arity_repl g c0 (lift (S n) O u) x (arity_lift g +d u x H4 c0 (S n) O (getl_drop Abst c0 d u n H0)) (asucc g x0) H7)))) +H6)))))) H3)))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (t: +T).(\lambda (_: (ty3 g c0 u t)).(\lambda (H1: (ex2 A (\lambda (a1: A).(arity +g c0 u a1)) (\lambda (a1: A).(arity g c0 t (asucc g a1))))).(\lambda (b: +B).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) +u) t3 t4)).(\lambda (H3: (ex2 A (\lambda (a1: A).(arity g (CHead c0 (Bind b) +u) t3 a1)) (\lambda (a1: A).(arity g (CHead c0 (Bind b) u) t4 (asucc g +a1))))).(let H4 \def H1 in (ex2_ind A (\lambda (a1: A).(arity g c0 u a1)) +(\lambda (a1: A).(arity g c0 t (asucc g a1))) (ex2 A (\lambda (a1: A).(arity +g c0 (THead (Bind b) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind b) +u t4) (asucc g a1)))) (\lambda (x: A).(\lambda (H5: (arity g c0 u +x)).(\lambda (_: (arity g c0 t (asucc g x))).(let H7 \def H3 in (ex2_ind A +(\lambda (a1: A).(arity g (CHead c0 (Bind b) u) t3 a1)) (\lambda (a1: +A).(arity g (CHead c0 (Bind b) u) t4 (asucc g a1))) (ex2 A (\lambda (a1: +A).(arity g c0 (THead (Bind b) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead +(Bind b) u t4) (asucc g a1)))) (\lambda (x0: A).(\lambda (H8: (arity g (CHead +c0 (Bind b) u) t3 x0)).(\lambda (H9: (arity g (CHead c0 (Bind b) u) t4 (asucc +g x0))).(let H_x \def (leq_asucc g x) in (let H10 \def H_x in (ex_ind A +(\lambda (a0: A).(leq g x (asucc g a0))) (ex2 A (\lambda (a1: A).(arity g c0 +(THead (Bind b) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind b) u t4) +(asucc g a1)))) (\lambda (x1: A).(\lambda (H11: (leq g x (asucc g +x1))).(B_ind (\lambda (b0: B).((arity g (CHead c0 (Bind b0) u) t3 x0) \to +((arity g (CHead c0 (Bind b0) u) t4 (asucc g x0)) \to (ex2 A (\lambda (a1: +A).(arity g c0 (THead (Bind b0) u t3) a1)) (\lambda (a1: A).(arity g c0 +(THead (Bind b0) u t4) (asucc g a1))))))) (\lambda (H12: (arity g (CHead c0 +(Bind Abbr) u) t3 x0)).(\lambda (H13: (arity g (CHead c0 (Bind Abbr) u) t4 +(asucc g x0))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Bind Abbr) u +t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind Abbr) u t4) (asucc g a1))) +x0 (arity_bind g Abbr not_abbr_abst c0 u x H5 t3 x0 H12) (arity_bind g Abbr +not_abbr_abst c0 u x H5 t4 (asucc g x0) H13)))) (\lambda (H12: (arity g +(CHead c0 (Bind Abst) u) t3 x0)).(\lambda (H13: (arity g (CHead c0 (Bind +Abst) u) t4 (asucc g x0))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (THead +(Bind Abst) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind Abst) u t4) +(asucc g a1))) (AHead x1 x0) (arity_head g c0 u x1 (arity_repl g c0 u x H5 +(asucc g x1) H11) t3 x0 H12) (arity_repl g c0 (THead (Bind Abst) u t4) (AHead +x1 (asucc g x0)) (arity_head g c0 u x1 (arity_repl g c0 u x H5 (asucc g x1) +H11) t4 (asucc g x0) H13) (asucc g (AHead x1 x0)) (leq_refl g (asucc g (AHead +x1 x0))))))) (\lambda (H12: (arity g (CHead c0 (Bind Void) u) t3 +x0)).(\lambda (H13: (arity g (CHead c0 (Bind Void) u) t4 (asucc g +x0))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Bind Void) u t3) a1)) +(\lambda (a1: A).(arity g c0 (THead (Bind Void) u t4) (asucc g a1))) x0 +(arity_bind g Void (sym_not_eq B Abst Void not_abst_void) c0 u x H5 t3 x0 +H12) (arity_bind g Void (sym_not_eq B Abst Void not_abst_void) c0 u x H5 t4 +(asucc g x0) H13)))) b H8 H9))) H10)))))) H7))))) H4)))))))))))) (\lambda +(c0: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda +(H1: (ex2 A (\lambda (a1: A).(arity g c0 w a1)) (\lambda (a1: A).(arity g c0 +u (asucc g a1))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v +(THead (Bind Abst) u t))).(\lambda (H3: (ex2 A (\lambda (a1: A).(arity g c0 v +a1)) (\lambda (a1: A).(arity g c0 (THead (Bind Abst) u t) (asucc g +a1))))).(let H4 \def H1 in (ex2_ind A (\lambda (a1: A).(arity g c0 w a1)) +(\lambda (a1: A).(arity g c0 u (asucc g a1))) (ex2 A (\lambda (a1: A).(arity +g c0 (THead (Flat Appl) w v) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat +Appl) w (THead (Bind Abst) u t)) (asucc g a1)))) (\lambda (x: A).(\lambda +(H5: (arity g c0 w x)).(\lambda (H6: (arity g c0 u (asucc g x))).(let H7 \def +H3 in (ex2_ind A (\lambda (a1: A).(arity g c0 v a1)) (\lambda (a1: A).(arity +g c0 (THead (Bind Abst) u t) (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g +c0 (THead (Flat Appl) w v) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat +Appl) w (THead (Bind Abst) u t)) (asucc g a1)))) (\lambda (x0: A).(\lambda +(H8: (arity g c0 v x0)).(\lambda (H9: (arity g c0 (THead (Bind Abst) u t) +(asucc g x0))).(let H10 \def (arity_gen_abst g c0 u t (asucc g x0) H9) in +(ex3_2_ind A A (\lambda (a1: A).(\lambda (a2: A).(eq A (asucc g x0) (AHead a1 +a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g c0 u (asucc g a1)))) +(\lambda (_: A).(\lambda (a2: A).(arity g (CHead c0 (Bind Abst) u) t a2))) +(ex2 A (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w v) a1)) (\lambda +(a1: A).(arity g c0 (THead (Flat Appl) w (THead (Bind Abst) u t)) (asucc g +a1)))) (\lambda (x1: A).(\lambda (x2: A).(\lambda (H11: (eq A (asucc g x0) +(AHead x1 x2))).(\lambda (H12: (arity g c0 u (asucc g x1))).(\lambda (H13: +(arity g (CHead c0 (Bind Abst) u) t x2)).(let H14 \def (sym_eq A (asucc g x0) +(AHead x1 x2) H11) in (let H15 \def (asucc_gen_head g x1 x2 x0 H14) in +(ex2_ind A (\lambda (a0: A).(eq A x0 (AHead x1 a0))) (\lambda (a0: A).(eq A +x2 (asucc g a0))) (ex2 A (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w v) +a1)) (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w (THead (Bind Abst) u +t)) (asucc g a1)))) (\lambda (x3: A).(\lambda (H16: (eq A x0 (AHead x1 +x3))).(\lambda (H17: (eq A x2 (asucc g x3))).(let H18 \def (eq_ind A x2 +(\lambda (a: A).(arity g (CHead c0 (Bind Abst) u) t a)) H13 (asucc g x3) H17) +in (let H19 \def (eq_ind A x0 (\lambda (a: A).(arity g c0 v a)) H8 (AHead x1 +x3) H16) in (ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w v) +a1)) (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w (THead (Bind Abst) u +t)) (asucc g a1))) x3 (arity_appl g c0 w x1 (arity_repl g c0 w x H5 x1 +(leq_sym g x1 x (asucc_inj g x1 x (arity_mono g c0 u (asucc g x1) H12 (asucc +g x) H6)))) v x3 H19) (arity_appl g c0 w x H5 (THead (Bind Abst) u t) (asucc +g x3) (arity_head g c0 u x H6 t (asucc g x3) H18)))))))) H15)))))))) H10))))) +H7))))) H4))))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: +T).(\lambda (_: (ty3 g c0 t3 t4)).(\lambda (H1: (ex2 A (\lambda (a1: +A).(arity g c0 t3 a1)) (\lambda (a1: A).(arity g c0 t4 (asucc g +a1))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t4 t0)).(\lambda (H3: (ex2 A +(\lambda (a1: A).(arity g c0 t4 a1)) (\lambda (a1: A).(arity g c0 t0 (asucc g +a1))))).(let H4 \def H1 in (ex2_ind A (\lambda (a1: A).(arity g c0 t3 a1)) +(\lambda (a1: A).(arity g c0 t4 (asucc g a1))) (ex2 A (\lambda (a1: A).(arity +g c0 (THead (Flat Cast) t4 t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat +Cast) t0 t4) (asucc g a1)))) (\lambda (x: A).(\lambda (H5: (arity g c0 t3 +x)).(\lambda (H6: (arity g c0 t4 (asucc g x))).(let H7 \def H3 in (ex2_ind A +(\lambda (a1: A).(arity g c0 t4 a1)) (\lambda (a1: A).(arity g c0 t0 (asucc g +a1))) (ex2 A (\lambda (a1: A).(arity g c0 (THead (Flat Cast) t4 t3) a1)) +(\lambda (a1: A).(arity g c0 (THead (Flat Cast) t0 t4) (asucc g a1)))) +(\lambda (x0: A).(\lambda (H8: (arity g c0 t4 x0)).(\lambda (H9: (arity g c0 +t0 (asucc g x0))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Flat +Cast) t4 t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat Cast) t0 t4) +(asucc g a1))) x (arity_cast g c0 t4 x H6 t3 H5) (arity_cast g c0 t0 (asucc g +x) (arity_repl g c0 t0 (asucc g x0) H9 (asucc g (asucc g x)) (asucc_repl g x0 +(asucc g x) (arity_mono g c0 t4 x0 H8 (asucc g x) H6))) t4 H6))))) H7))))) +H4)))))))))) c t1 t2 H))))). +(* COMMENTS +Initial nodes: 3761 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/ty3/arity_props.ma b/matita/matita/contribs/lambdadelta/basic_1/ty3/arity_props.ma new file mode 100644 index 000000000..49d6c0572 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/ty3/arity_props.ma @@ -0,0 +1,117 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/ty3/arity.ma". + +include "Basic-1/sc3/arity.ma". + +theorem ty3_predicative: + \forall (g: G).(\forall (c: C).(\forall (v: T).(\forall (t: T).(\forall (u: +T).((ty3 g c (THead (Bind Abst) v t) u) \to ((pc3 c u v) \to (\forall (P: +Prop).P))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (u: +T).(\lambda (H: (ty3 g c (THead (Bind Abst) v t) u)).(\lambda (H0: (pc3 c u +v)).(\lambda (P: Prop).(let H1 \def H in (ex3_2_ind T T (\lambda (t2: +T).(\lambda (_: T).(pc3 c (THead (Bind Abst) v t2) u))) (\lambda (_: +T).(\lambda (t0: T).(ty3 g c v t0))) (\lambda (t2: T).(\lambda (_: T).(ty3 g +(CHead c (Bind Abst) v) t t2))) P (\lambda (x0: T).(\lambda (x1: T).(\lambda +(_: (pc3 c (THead (Bind Abst) v x0) u)).(\lambda (H3: (ty3 g c v +x1)).(\lambda (_: (ty3 g (CHead c (Bind Abst) v) t x0)).(let H_y \def +(ty3_conv g c v x1 H3 (THead (Bind Abst) v t) u H H0) in (let H_x \def +(ty3_arity g c (THead (Bind Abst) v t) v H_y) in (let H5 \def H_x in (ex2_ind +A (\lambda (a1: A).(arity g c (THead (Bind Abst) v t) a1)) (\lambda (a1: +A).(arity g c v (asucc g a1))) P (\lambda (x: A).(\lambda (H6: (arity g c +(THead (Bind Abst) v t) x)).(\lambda (H7: (arity g c v (asucc g x))).(let H8 +\def (arity_gen_abst g c v t x H6) in (ex3_2_ind A A (\lambda (a1: +A).(\lambda (a2: A).(eq A x (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: +A).(arity g c v (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g +(CHead c (Bind Abst) v) t a2))) P (\lambda (x2: A).(\lambda (x3: A).(\lambda +(H9: (eq A x (AHead x2 x3))).(\lambda (H10: (arity g c v (asucc g +x2))).(\lambda (_: (arity g (CHead c (Bind Abst) v) t x3)).(let H12 \def +(eq_ind A x (\lambda (a: A).(arity g c v (asucc g a))) H7 (AHead x2 x3) H9) +in (leq_ahead_asucc_false g x2 (asucc g x3) (arity_mono g c v (asucc g (AHead +x2 x3)) H12 (asucc g x2) H10) P))))))) H8))))) H5))))))))) (ty3_gen_bind g +Abst c v t u H1)))))))))). +(* COMMENTS +Initial nodes: 497 +END *) + +theorem ty3_repellent: + \forall (g: G).(\forall (c: C).(\forall (w: T).(\forall (t: T).(\forall (u1: +T).((ty3 g c (THead (Bind Abst) w t) u1) \to (\forall (u2: T).((ty3 g (CHead +c (Bind Abst) w) t (lift (S O) O u2)) \to ((pc3 c u1 u2) \to (\forall (P: +Prop).P))))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (w: T).(\lambda (t: T).(\lambda (u1: +T).(\lambda (H: (ty3 g c (THead (Bind Abst) w t) u1)).(\lambda (u2: +T).(\lambda (H0: (ty3 g (CHead c (Bind Abst) w) t (lift (S O) O +u2))).(\lambda (H1: (pc3 c u1 u2)).(\lambda (P: Prop).(ex_ind T (\lambda (t0: +T).(ty3 g (CHead c (Bind Abst) w) (lift (S O) O u2) t0)) P (\lambda (x: +T).(\lambda (H2: (ty3 g (CHead c (Bind Abst) w) (lift (S O) O u2) x)).(let H3 +\def (ty3_gen_lift g (CHead c (Bind Abst) w) u2 x (S O) O H2 c (drop_drop +(Bind Abst) O c c (drop_refl c) w)) in (ex2_ind T (\lambda (t2: T).(pc3 +(CHead c (Bind Abst) w) (lift (S O) O t2) x)) (\lambda (t2: T).(ty3 g c u2 +t2)) P (\lambda (x0: T).(\lambda (_: (pc3 (CHead c (Bind Abst) w) (lift (S O) +O x0) x)).(\lambda (H5: (ty3 g c u2 x0)).(let H_y \def (ty3_conv g c u2 x0 H5 +(THead (Bind Abst) w t) u1 H H1) in (let H_x \def (ty3_arity g (CHead c (Bind +Abst) w) t (lift (S O) O u2) H0) in (let H6 \def H_x in (ex2_ind A (\lambda +(a1: A).(arity g (CHead c (Bind Abst) w) t a1)) (\lambda (a1: A).(arity g +(CHead c (Bind Abst) w) (lift (S O) O u2) (asucc g a1))) P (\lambda (x1: +A).(\lambda (H7: (arity g (CHead c (Bind Abst) w) t x1)).(\lambda (H8: (arity +g (CHead c (Bind Abst) w) (lift (S O) O u2) (asucc g x1))).(let H_x0 \def +(ty3_arity g c (THead (Bind Abst) w t) u2 H_y) in (let H9 \def H_x0 in +(ex2_ind A (\lambda (a1: A).(arity g c (THead (Bind Abst) w t) a1)) (\lambda +(a1: A).(arity g c u2 (asucc g a1))) P (\lambda (x2: A).(\lambda (H10: (arity +g c (THead (Bind Abst) w t) x2)).(\lambda (H11: (arity g c u2 (asucc g +x2))).(arity_repellent g c w t x1 H7 x2 H10 (asucc_inj g x1 x2 (arity_mono g +c u2 (asucc g x1) (arity_gen_lift g (CHead c (Bind Abst) w) u2 (asucc g x1) +(S O) O H8 c (drop_drop (Bind Abst) O c c (drop_refl c) w)) (asucc g x2) +H11)) P)))) H9)))))) H6))))))) H3)))) (ty3_correct g (CHead c (Bind Abst) w) +t (lift (S O) O u2) H0))))))))))). +(* COMMENTS +Initial nodes: 651 +END *) + +theorem ty3_acyclic: + \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t +u) \to ((pc3 c u t) \to (\forall (P: Prop).P)))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H: +(ty3 g c t u)).(\lambda (H0: (pc3 c u t)).(\lambda (P: Prop).(let H_y \def +(ty3_conv g c t u H t u H H0) in (let H_x \def (ty3_arity g c t t H_y) in +(let H1 \def H_x in (ex2_ind A (\lambda (a1: A).(arity g c t a1)) (\lambda +(a1: A).(arity g c t (asucc g a1))) P (\lambda (x: A).(\lambda (H2: (arity g +c t x)).(\lambda (H3: (arity g c t (asucc g x))).(leq_asucc_false g x +(arity_mono g c t (asucc g x) H3 x H2) P)))) H1)))))))))). +(* COMMENTS +Initial nodes: 151 +END *) + +theorem ty3_sn3: + \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t +u) \to (sn3 c t))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H: +(ty3 g c t u)).(let H_x \def (ty3_arity g c t u H) in (let H0 \def H_x in +(ex2_ind A (\lambda (a1: A).(arity g c t a1)) (\lambda (a1: A).(arity g c u +(asucc g a1))) (sn3 c t) (\lambda (x: A).(\lambda (H1: (arity g c t +x)).(\lambda (_: (arity g c u (asucc g x))).(sc3_sn3 g x c t (sc3_arity g c t +x H1))))) H0))))))). +(* COMMENTS +Initial nodes: 119 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/ty3/dec.ma b/matita/matita/contribs/lambdadelta/basic_1/ty3/dec.ma new file mode 100644 index 000000000..c176d635e --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/ty3/dec.ma @@ -0,0 +1,438 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/pc3/dec.ma". + +include "Basic-1/getl/flt.ma". + +include "Basic-1/getl/dec.ma". + +theorem ty3_inference: + \forall (g: G).(\forall (c: C).(\forall (t1: T).(or (ex T (\lambda (t2: +T).(ty3 g c t1 t2))) (\forall (t2: T).((ty3 g c t1 t2) \to False))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(flt_wf_ind (\lambda (c0: +C).(\lambda (t: T).(or (ex T (\lambda (t2: T).(ty3 g c0 t t2))) (\forall (t2: +T).((ty3 g c0 t t2) \to False))))) (\lambda (c2: C).(\lambda (t2: T).(T_ind +(\lambda (t: T).(((\forall (c1: C).(\forall (t3: T).((flt c1 t3 c2 t) \to (or +(ex T (\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4) +\to False))))))) \to (or (ex T (\lambda (t3: T).(ty3 g c2 t t3))) (\forall +(t3: T).((ty3 g c2 t t3) \to False))))) (\lambda (n: nat).(\lambda (_: +((\forall (c1: C).(\forall (t3: T).((flt c1 t3 c2 (TSort n)) \to (or (ex T +(\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to +False)))))))).(or_introl (ex T (\lambda (t3: T).(ty3 g c2 (TSort n) t3))) +(\forall (t3: T).((ty3 g c2 (TSort n) t3) \to False)) (ex_intro T (\lambda +(t3: T).(ty3 g c2 (TSort n) t3)) (TSort (next g n)) (ty3_sort g c2 n))))) +(\lambda (n: nat).(\lambda (H: ((\forall (c1: C).(\forall (t3: T).((flt c1 t3 +c2 (TLRef n)) \to (or (ex T (\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: +T).((ty3 g c1 t3 t4) \to False)))))))).(let H_x \def (getl_dec c2 n) in (let +H0 \def H_x in (or_ind (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda +(v: T).(getl n c2 (CHead e (Bind b) v)))))) (\forall (d: C).((getl n c2 d) +\to (\forall (P: Prop).P))) (or (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) +t3))) (\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to False))) (\lambda (H1: +(ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl n c2 (CHead +e (Bind b) v))))))).(ex_3_ind C B T (\lambda (e: C).(\lambda (b: B).(\lambda +(v: T).(getl n c2 (CHead e (Bind b) v))))) (or (ex T (\lambda (t3: T).(ty3 g +c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to False))) +(\lambda (x0: C).(\lambda (x1: B).(\lambda (x2: T).(\lambda (H2: (getl n c2 +(CHead x0 (Bind x1) x2))).(let H3 \def (H x0 x2 (getl_flt x1 c2 x0 x2 n H2)) +in (or_ind (ex T (\lambda (t3: T).(ty3 g x0 x2 t3))) (\forall (t3: T).((ty3 g +x0 x2 t3) \to False)) (or (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) +(\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to False))) (\lambda (H4: (ex T +(\lambda (t3: T).(ty3 g x0 x2 t3)))).(ex_ind T (\lambda (t3: T).(ty3 g x0 x2 +t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: +T).((ty3 g c2 (TLRef n) t3) \to False))) (\lambda (x: T).(\lambda (H5: (ty3 g +x0 x2 x)).(B_ind (\lambda (b: B).((getl n c2 (CHead x0 (Bind b) x2)) \to (or +(ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 +(TLRef n) t3) \to False))))) (\lambda (H6: (getl n c2 (CHead x0 (Bind Abbr) +x2))).(or_introl (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall +(t3: T).((ty3 g c2 (TLRef n) t3) \to False)) (ex_intro T (\lambda (t3: +T).(ty3 g c2 (TLRef n) t3)) (lift (S n) O x) (ty3_abbr g n c2 x0 x2 H6 x +H5)))) (\lambda (H6: (getl n c2 (CHead x0 (Bind Abst) x2))).(or_introl (ex T +(\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef +n) t3) \to False)) (ex_intro T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3)) +(lift (S n) O x2) (ty3_abst g n c2 x0 x2 H6 x H5)))) (\lambda (H6: (getl n c2 +(CHead x0 (Bind Void) x2))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 +(TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to False)) +(\lambda (t3: T).(\lambda (H7: (ty3 g c2 (TLRef n) t3)).(or_ind (ex3_3 C T T +(\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) +t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e +(Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u +t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 +(lift (S n) O u) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl +n c2 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: +T).(ty3 g e u t))))) False (\lambda (H8: (ex3_3 C T T (\lambda (_: +C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda +(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u))))) +(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind +C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O +t) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e +(Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u +t)))) False (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: +(pc3 c2 (lift (S n) O x5) t3)).(\lambda (H10: (getl n c2 (CHead x3 (Bind +Abbr) x4))).(\lambda (_: (ty3 g x3 x4 x5)).(let H12 \def (eq_ind C (CHead x0 +(Bind Void) x2) (\lambda (c0: C).(getl n c2 c0)) H6 (CHead x3 (Bind Abbr) x4) +(getl_mono c2 (CHead x0 (Bind Void) x2) n H6 (CHead x3 (Bind Abbr) x4) H10)) +in (let H13 \def (eq_ind C (CHead x0 (Bind Void) x2) (\lambda (ee: C).(match +ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | +(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with +[(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr +\Rightarrow False | Abst \Rightarrow False | Void \Rightarrow True]) | (Flat +_) \Rightarrow False])])) I (CHead x3 (Bind Abbr) x4) (getl_mono c2 (CHead x0 +(Bind Void) x2) n H6 (CHead x3 (Bind Abbr) x4) H10)) in (False_ind False +H13))))))))) H8)) (\lambda (H8: (ex3_3 C T T (\lambda (_: C).(\lambda (u: +T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e: C).(\lambda +(u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u))))) (\lambda (e: +C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T +(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) +t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e +(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u +t)))) False (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: +(pc3 c2 (lift (S n) O x4) t3)).(\lambda (H10: (getl n c2 (CHead x3 (Bind +Abst) x4))).(\lambda (_: (ty3 g x3 x4 x5)).(let H12 \def (eq_ind C (CHead x0 +(Bind Void) x2) (\lambda (c0: C).(getl n c2 c0)) H6 (CHead x3 (Bind Abst) x4) +(getl_mono c2 (CHead x0 (Bind Void) x2) n H6 (CHead x3 (Bind Abst) x4) H10)) +in (let H13 \def (eq_ind C (CHead x0 (Bind Void) x2) (\lambda (ee: C).(match +ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | +(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with +[(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr +\Rightarrow False | Abst \Rightarrow False | Void \Rightarrow True]) | (Flat +_) \Rightarrow False])])) I (CHead x3 (Bind Abst) x4) (getl_mono c2 (CHead x0 +(Bind Void) x2) n H6 (CHead x3 (Bind Abst) x4) H10)) in (False_ind False +H13))))))))) H8)) (ty3_gen_lref g c2 t3 n H7)))))) x1 H2))) H4)) (\lambda +(H4: ((\forall (t3: T).((ty3 g x0 x2 t3) \to False)))).(or_intror (ex T +(\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef +n) t3) \to False)) (\lambda (t3: T).(\lambda (H5: (ty3 g c2 (TLRef n) +t3)).(or_ind (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: +T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda +(_: T).(getl n c2 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: +T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda +(u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e: +C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u))))) +(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))) False +(\lambda (H6: (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: +T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda +(_: T).(getl n c2 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: +T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T (\lambda (_: +C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda +(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u))))) +(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))) False +(\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (pc3 c2 (lift +(S n) O x5) t3)).(\lambda (H8: (getl n c2 (CHead x3 (Bind Abbr) +x4))).(\lambda (H9: (ty3 g x3 x4 x5)).(let H10 \def (eq_ind C (CHead x0 (Bind +x1) x2) (\lambda (c0: C).(getl n c2 c0)) H2 (CHead x3 (Bind Abbr) x4) +(getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind Abbr) x4) H8)) in +(let H11 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: +C).C) with [(CSort _) \Rightarrow x0 | (CHead c0 _ _) \Rightarrow c0])) +(CHead x0 (Bind x1) x2) (CHead x3 (Bind Abbr) x4) (getl_mono c2 (CHead x0 +(Bind x1) x2) n H2 (CHead x3 (Bind Abbr) x4) H8)) in ((let H12 \def (f_equal +C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with [(CSort _) +\Rightarrow x1 | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: +K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow x1])])) (CHead x0 +(Bind x1) x2) (CHead x3 (Bind Abbr) x4) (getl_mono c2 (CHead x0 (Bind x1) x2) +n H2 (CHead x3 (Bind Abbr) x4) H8)) in ((let H13 \def (f_equal C T (\lambda +(e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow x2 +| (CHead _ _ t) \Rightarrow t])) (CHead x0 (Bind x1) x2) (CHead x3 (Bind +Abbr) x4) (getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind Abbr) +x4) H8)) in (\lambda (_: (eq B x1 Abbr)).(\lambda (H15: (eq C x0 x3)).(let +H16 \def (eq_ind_r T x4 (\lambda (t: T).(getl n c2 (CHead x3 (Bind Abbr) t))) +H10 x2 H13) in (let H17 \def (eq_ind_r T x4 (\lambda (t: T).(ty3 g x3 t x5)) +H9 x2 H13) in (let H18 \def (eq_ind_r C x3 (\lambda (c0: C).(getl n c2 (CHead +c0 (Bind Abbr) x2))) H16 x0 H15) in (let H19 \def (eq_ind_r C x3 (\lambda +(c0: C).(ty3 g c0 x2 x5)) H17 x0 H15) in (H4 x5 H19)))))))) H12)) +H11))))))))) H6)) (\lambda (H6: (ex3_3 C T T (\lambda (_: C).(\lambda (u: +T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e: C).(\lambda +(u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u))))) (\lambda (e: +C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T +(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) +t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e +(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u +t)))) False (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H7: +(pc3 c2 (lift (S n) O x4) t3)).(\lambda (H8: (getl n c2 (CHead x3 (Bind Abst) +x4))).(\lambda (H9: (ty3 g x3 x4 x5)).(let H10 \def (eq_ind C (CHead x0 (Bind +x1) x2) (\lambda (c0: C).(getl n c2 c0)) H2 (CHead x3 (Bind Abst) x4) +(getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind Abst) x4) H8)) in +(let H11 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: +C).C) with [(CSort _) \Rightarrow x0 | (CHead c0 _ _) \Rightarrow c0])) +(CHead x0 (Bind x1) x2) (CHead x3 (Bind Abst) x4) (getl_mono c2 (CHead x0 +(Bind x1) x2) n H2 (CHead x3 (Bind Abst) x4) H8)) in ((let H12 \def (f_equal +C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with [(CSort _) +\Rightarrow x1 | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: +K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow x1])])) (CHead x0 +(Bind x1) x2) (CHead x3 (Bind Abst) x4) (getl_mono c2 (CHead x0 (Bind x1) x2) +n H2 (CHead x3 (Bind Abst) x4) H8)) in ((let H13 \def (f_equal C T (\lambda +(e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow x2 +| (CHead _ _ t) \Rightarrow t])) (CHead x0 (Bind x1) x2) (CHead x3 (Bind +Abst) x4) (getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind Abst) +x4) H8)) in (\lambda (_: (eq B x1 Abst)).(\lambda (H15: (eq C x0 x3)).(let +H16 \def (eq_ind_r T x4 (\lambda (t: T).(getl n c2 (CHead x3 (Bind Abst) t))) +H10 x2 H13) in (let H17 \def (eq_ind_r T x4 (\lambda (t: T).(ty3 g x3 t x5)) +H9 x2 H13) in (let H18 \def (eq_ind_r T x4 (\lambda (t: T).(pc3 c2 (lift (S +n) O t) t3)) H7 x2 H13) in (let H19 \def (eq_ind_r C x3 (\lambda (c0: +C).(getl n c2 (CHead c0 (Bind Abst) x2))) H16 x0 H15) in (let H20 \def +(eq_ind_r C x3 (\lambda (c0: C).(ty3 g c0 x2 x5)) H17 x0 H15) in (H4 x5 +H20))))))))) H12)) H11))))))))) H6)) (ty3_gen_lref g c2 t3 n H5)))))) +H3)))))) H1)) (\lambda (H1: ((\forall (d: C).((getl n c2 d) \to (\forall (P: +Prop).P))))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) +(\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to False)) (\lambda (t3: +T).(\lambda (H2: (ty3 g c2 (TLRef n) t3)).(or_ind (ex3_3 C T T (\lambda (_: +C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda +(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u))))) +(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T +T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) +t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e +(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u +t))))) False (\lambda (H3: (ex3_3 C T T (\lambda (_: C).(\lambda (_: +T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda (e: C).(\lambda +(u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u))))) (\lambda (e: +C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T +(\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) +t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e +(Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u +t)))) False (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: +(pc3 c2 (lift (S n) O x2) t3)).(\lambda (H5: (getl n c2 (CHead x0 (Bind Abbr) +x1))).(\lambda (_: (ty3 g x0 x1 x2)).(H1 (CHead x0 (Bind Abbr) x1) H5 +False))))))) H3)) (\lambda (H3: (ex3_3 C T T (\lambda (_: C).(\lambda (u: +T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e: C).(\lambda +(u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u))))) (\lambda (e: +C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T +(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) +t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e +(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u +t)))) False (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: +(pc3 c2 (lift (S n) O x1) t3)).(\lambda (H5: (getl n c2 (CHead x0 (Bind Abst) +x1))).(\lambda (_: (ty3 g x0 x1 x2)).(H1 (CHead x0 (Bind Abst) x1) H5 +False))))))) H3)) (ty3_gen_lref g c2 t3 n H2)))))) H0))))) (\lambda (k: +K).(\lambda (t: T).(\lambda (_: ((((\forall (c1: C).(\forall (t3: T).((flt c1 +t3 c2 t) \to (or (ex T (\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: +T).((ty3 g c1 t3 t4) \to False))))))) \to (or (ex T (\lambda (t3: T).(ty3 g +c2 t t3))) (\forall (t3: T).((ty3 g c2 t t3) \to False)))))).(\lambda (t0: +T).(\lambda (_: ((((\forall (c1: C).(\forall (t3: T).((flt c1 t3 c2 t0) \to +(or (ex T (\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3 g c1 t3 +t4) \to False))))))) \to (or (ex T (\lambda (t3: T).(ty3 g c2 t0 t3))) +(\forall (t3: T).((ty3 g c2 t0 t3) \to False)))))).(\lambda (H1: ((\forall +(c1: C).(\forall (t3: T).((flt c1 t3 c2 (THead k t t0)) \to (or (ex T +(\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to +False)))))))).(K_ind (\lambda (k0: K).(((\forall (c1: C).(\forall (t3: +T).((flt c1 t3 c2 (THead k0 t t0)) \to (or (ex T (\lambda (t4: T).(ty3 g c1 +t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to False))))))) \to (or (ex T +(\lambda (t3: T).(ty3 g c2 (THead k0 t t0) t3))) (\forall (t3: T).((ty3 g c2 +(THead k0 t t0) t3) \to False))))) (\lambda (b: B).(\lambda (H2: ((\forall +(c1: C).(\forall (t3: T).((flt c1 t3 c2 (THead (Bind b) t t0)) \to (or (ex T +(\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to +False)))))))).(let H3 \def (H2 c2 t (flt_thead_sx (Bind b) c2 t t0)) in +(or_ind (ex T (\lambda (t3: T).(ty3 g c2 t t3))) (\forall (t3: T).((ty3 g c2 +t t3) \to False)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Bind b) t t0) +t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0) t3) \to False))) +(\lambda (H4: (ex T (\lambda (t3: T).(ty3 g c2 t t3)))).(ex_ind T (\lambda +(t3: T).(ty3 g c2 t t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Bind b) +t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0) t3) \to +False))) (\lambda (x: T).(\lambda (H5: (ty3 g c2 t x)).(let H6 \def (H2 +(CHead c2 (Bind b) t) t0 (flt_shift (Bind b) c2 t t0)) in (or_ind (ex T +(\lambda (t3: T).(ty3 g (CHead c2 (Bind b) t) t0 t3))) (\forall (t3: T).((ty3 +g (CHead c2 (Bind b) t) t0 t3) \to False)) (or (ex T (\lambda (t3: T).(ty3 g +c2 (THead (Bind b) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t +t0) t3) \to False))) (\lambda (H7: (ex T (\lambda (t3: T).(ty3 g (CHead c2 +(Bind b) t) t0 t3)))).(ex_ind T (\lambda (t3: T).(ty3 g (CHead c2 (Bind b) t) +t0 t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Bind b) t t0) t3))) +(\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0) t3) \to False))) (\lambda +(x0: T).(\lambda (H8: (ty3 g (CHead c2 (Bind b) t) t0 x0)).(or_introl (ex T +(\lambda (t3: T).(ty3 g c2 (THead (Bind b) t t0) t3))) (\forall (t3: T).((ty3 +g c2 (THead (Bind b) t t0) t3) \to False)) (ex_intro T (\lambda (t3: T).(ty3 +g c2 (THead (Bind b) t t0) t3)) (THead (Bind b) t x0) (ty3_bind g c2 t x H5 b +t0 x0 H8))))) H7)) (\lambda (H7: ((\forall (t3: T).((ty3 g (CHead c2 (Bind b) +t) t0 t3) \to False)))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead +(Bind b) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0) t3) +\to False)) (\lambda (t3: T).(\lambda (H8: (ty3 g c2 (THead (Bind b) t t0) +t3)).(ex3_2_ind T T (\lambda (t4: T).(\lambda (_: T).(pc3 c2 (THead (Bind b) +t t4) t3))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c2 t t5))) (\lambda (t4: +T).(\lambda (_: T).(ty3 g (CHead c2 (Bind b) t) t0 t4))) False (\lambda (x0: +T).(\lambda (x1: T).(\lambda (_: (pc3 c2 (THead (Bind b) t x0) t3)).(\lambda +(_: (ty3 g c2 t x1)).(\lambda (H11: (ty3 g (CHead c2 (Bind b) t) t0 x0)).(H7 +x0 H11)))))) (ty3_gen_bind g b c2 t t0 t3 H8)))))) H6)))) H4)) (\lambda (H4: +((\forall (t3: T).((ty3 g c2 t t3) \to False)))).(or_intror (ex T (\lambda +(t3: T).(ty3 g c2 (THead (Bind b) t t0) t3))) (\forall (t3: T).((ty3 g c2 +(THead (Bind b) t t0) t3) \to False)) (\lambda (t3: T).(\lambda (H5: (ty3 g +c2 (THead (Bind b) t t0) t3)).(ex3_2_ind T T (\lambda (t4: T).(\lambda (_: +T).(pc3 c2 (THead (Bind b) t t4) t3))) (\lambda (_: T).(\lambda (t5: T).(ty3 +g c2 t t5))) (\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead c2 (Bind b) t) t0 +t4))) False (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 c2 (THead +(Bind b) t x0) t3)).(\lambda (H7: (ty3 g c2 t x1)).(\lambda (_: (ty3 g (CHead +c2 (Bind b) t) t0 x0)).(H4 x1 H7)))))) (ty3_gen_bind g b c2 t t0 t3 H5)))))) +H3)))) (\lambda (f: F).(\lambda (H2: ((\forall (c1: C).(\forall (t3: T).((flt +c1 t3 c2 (THead (Flat f) t t0)) \to (or (ex T (\lambda (t4: T).(ty3 g c1 t3 +t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to False)))))))).(F_ind (\lambda +(f0: F).(((\forall (c1: C).(\forall (t3: T).((flt c1 t3 c2 (THead (Flat f0) t +t0)) \to (or (ex T (\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3 +g c1 t3 t4) \to False))))))) \to (or (ex T (\lambda (t3: T).(ty3 g c2 (THead +(Flat f0) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat f0) t t0) t3) +\to False))))) (\lambda (H3: ((\forall (c1: C).(\forall (t3: T).((flt c1 t3 +c2 (THead (Flat Appl) t t0)) \to (or (ex T (\lambda (t4: T).(ty3 g c1 t3 +t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to False)))))))).(let H4 \def (H3 +c2 t (flt_thead_sx (Flat Appl) c2 t t0)) in (or_ind (ex T (\lambda (t3: +T).(ty3 g c2 t t3))) (\forall (t3: T).((ty3 g c2 t t3) \to False)) (or (ex T +(\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: +T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False))) (\lambda (H5: (ex T +(\lambda (t3: T).(ty3 g c2 t t3)))).(ex_ind T (\lambda (t3: T).(ty3 g c2 t +t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) +(\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False))) +(\lambda (x: T).(\lambda (H6: (ty3 g c2 t x)).(let H7 \def (H3 c2 t0 +(flt_thead_dx (Flat Appl) c2 t t0)) in (or_ind (ex T (\lambda (t3: T).(ty3 g +c2 t0 t3))) (\forall (t3: T).((ty3 g c2 t0 t3) \to False)) (or (ex T (\lambda +(t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 +(THead (Flat Appl) t t0) t3) \to False))) (\lambda (H8: (ex T (\lambda (t3: +T).(ty3 g c2 t0 t3)))).(ex_ind T (\lambda (t3: T).(ty3 g c2 t0 t3)) (or (ex T +(\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: +T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False))) (\lambda (x0: +T).(\lambda (H9: (ty3 g c2 t0 x0)).(ex_ind T (\lambda (t3: T).(ty3 g c2 x0 +t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) +(\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False))) +(\lambda (x1: T).(\lambda (H10: (ty3 g c2 x0 x1)).(ex_ind T (\lambda (t3: +T).(ty3 g c2 x t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t +t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to +False))) (\lambda (x2: T).(\lambda (H11: (ty3 g c2 x x2)).(let H12 \def +(ty3_sn3 g c2 x x2 H11) in (let H_x \def (nf2_sn3 c2 x H12) in (let H13 \def +H_x in (ex2_ind T (\lambda (u: T).(pr3 c2 x u)) (\lambda (u: T).(nf2 c2 u)) +(or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall +(t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False))) (\lambda (x3: +T).(\lambda (H14: (pr3 c2 x x3)).(\lambda (H15: (nf2 c2 x3)).(let H16 \def +(ty3_sred_pr3 c2 x x3 H14 g x2 H11) in (let H_x0 \def (pc3_abst_dec g c2 x0 +x1 H10 x3 x2 H16) in (let H17 \def H_x0 in (or_ind (ex4_2 T T (\lambda (u: +T).(\lambda (_: T).(pc3 c2 x0 (THead (Bind Abst) x3 u)))) (\lambda (u: +T).(\lambda (v2: T).(ty3 g c2 (THead (Bind Abst) v2 u) x1))) (\lambda (_: +T).(\lambda (v2: T).(pr3 c2 x3 v2))) (\lambda (_: T).(\lambda (v2: T).(nf2 c2 +v2)))) (\forall (u: T).((pc3 c2 x0 (THead (Bind Abst) x3 u)) \to False)) (or +(ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: +T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False))) (\lambda (H18: (ex4_2 +T T (\lambda (u: T).(\lambda (_: T).(pc3 c2 x0 (THead (Bind Abst) x3 u)))) +(\lambda (u: T).(\lambda (v2: T).(ty3 g c2 (THead (Bind Abst) v2 u) x1))) +(\lambda (_: T).(\lambda (v2: T).(pr3 c2 x3 v2))) (\lambda (_: T).(\lambda +(v2: T).(nf2 c2 v2))))).(ex4_2_ind T T (\lambda (u: T).(\lambda (_: T).(pc3 +c2 x0 (THead (Bind Abst) x3 u)))) (\lambda (u: T).(\lambda (v2: T).(ty3 g c2 +(THead (Bind Abst) v2 u) x1))) (\lambda (_: T).(\lambda (v2: T).(pr3 c2 x3 +v2))) (\lambda (_: T).(\lambda (v2: T).(nf2 c2 v2))) (or (ex T (\lambda (t3: +T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 +(THead (Flat Appl) t t0) t3) \to False))) (\lambda (x4: T).(\lambda (x5: +T).(\lambda (H19: (pc3 c2 x0 (THead (Bind Abst) x3 x4))).(\lambda (H20: (ty3 +g c2 (THead (Bind Abst) x5 x4) x1)).(\lambda (H21: (pr3 c2 x3 x5)).(\lambda +(_: (nf2 c2 x5)).(let H_y \def (nf2_pr3_unfold c2 x3 x5 H21 H15) in (let H23 +\def (eq_ind_r T x5 (\lambda (t3: T).(pr3 c2 x3 t3)) H21 x3 H_y) in (let H24 +\def (eq_ind_r T x5 (\lambda (t3: T).(ty3 g c2 (THead (Bind Abst) t3 x4) x1)) +H20 x3 H_y) in (or_introl (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) +t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to +False)) (ex_intro T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3)) +(THead (Flat Appl) t (THead (Bind Abst) x3 x4)) (ty3_appl g c2 t x3 (ty3_tred +g c2 t x H6 x3 H14) t0 x4 (ty3_conv g c2 (THead (Bind Abst) x3 x4) x1 H24 t0 +x0 H9 H19))))))))))))) H18)) (\lambda (H18: ((\forall (u: T).((pc3 c2 x0 +(THead (Bind Abst) x3 u)) \to False)))).(or_intror (ex T (\lambda (t3: +T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 +(THead (Flat Appl) t t0) t3) \to False)) (\lambda (t3: T).(\lambda (H19: (ty3 +g c2 (THead (Flat Appl) t t0) t3)).(ex3_2_ind T T (\lambda (u: T).(\lambda +(t4: T).(pc3 c2 (THead (Flat Appl) t (THead (Bind Abst) u t4)) t3))) (\lambda +(u: T).(\lambda (t4: T).(ty3 g c2 t0 (THead (Bind Abst) u t4)))) (\lambda (u: +T).(\lambda (_: T).(ty3 g c2 t u))) False (\lambda (x4: T).(\lambda (x5: +T).(\lambda (_: (pc3 c2 (THead (Flat Appl) t (THead (Bind Abst) x4 x5)) +t3)).(\lambda (H21: (ty3 g c2 t0 (THead (Bind Abst) x4 x5))).(\lambda (H22: +(ty3 g c2 t x4)).(let H_y \def (ty3_unique g c2 t x4 H22 x H6) in (let H_y0 +\def (ty3_unique g c2 t0 (THead (Bind Abst) x4 x5) H21 x0 H9) in (H18 x5 +(pc3_t (THead (Bind Abst) x4 x5) c2 x0 (pc3_s c2 x0 (THead (Bind Abst) x4 x5) +H_y0) (THead (Bind Abst) x3 x5) (pc3_head_1 c2 x4 x3 (pc3_t x c2 x4 H_y x3 +(pc3_pr3_r c2 x x3 H14)) (Bind Abst) x5)))))))))) (ty3_gen_appl g c2 t t0 t3 +H19)))))) H17))))))) H13)))))) (ty3_correct g c2 t x H6)))) (ty3_correct g c2 +t0 x0 H9)))) H8)) (\lambda (H8: ((\forall (t3: T).((ty3 g c2 t0 t3) \to +False)))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t +t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to +False)) (\lambda (t3: T).(\lambda (H9: (ty3 g c2 (THead (Flat Appl) t t0) +t3)).(ex3_2_ind T T (\lambda (u: T).(\lambda (t4: T).(pc3 c2 (THead (Flat +Appl) t (THead (Bind Abst) u t4)) t3))) (\lambda (u: T).(\lambda (t4: T).(ty3 +g c2 t0 (THead (Bind Abst) u t4)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c2 +t u))) False (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 c2 (THead +(Flat Appl) t (THead (Bind Abst) x0 x1)) t3)).(\lambda (H11: (ty3 g c2 t0 +(THead (Bind Abst) x0 x1))).(\lambda (_: (ty3 g c2 t x0)).(H8 (THead (Bind +Abst) x0 x1) H11)))))) (ty3_gen_appl g c2 t t0 t3 H9)))))) H7)))) H5)) +(\lambda (H5: ((\forall (t3: T).((ty3 g c2 t t3) \to False)))).(or_intror (ex +T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: +T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False)) (\lambda (t3: +T).(\lambda (H6: (ty3 g c2 (THead (Flat Appl) t t0) t3)).(ex3_2_ind T T +(\lambda (u: T).(\lambda (t4: T).(pc3 c2 (THead (Flat Appl) t (THead (Bind +Abst) u t4)) t3))) (\lambda (u: T).(\lambda (t4: T).(ty3 g c2 t0 (THead (Bind +Abst) u t4)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c2 t u))) False +(\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 c2 (THead (Flat Appl) t +(THead (Bind Abst) x0 x1)) t3)).(\lambda (_: (ty3 g c2 t0 (THead (Bind Abst) +x0 x1))).(\lambda (H9: (ty3 g c2 t x0)).(H5 x0 H9)))))) (ty3_gen_appl g c2 t +t0 t3 H6)))))) H4))) (\lambda (H3: ((\forall (c1: C).(\forall (t3: T).((flt +c1 t3 c2 (THead (Flat Cast) t t0)) \to (or (ex T (\lambda (t4: T).(ty3 g c1 +t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to False)))))))).(let H4 \def +(H3 c2 t (flt_thead_sx (Flat Cast) c2 t t0)) in (or_ind (ex T (\lambda (t3: +T).(ty3 g c2 t t3))) (\forall (t3: T).((ty3 g c2 t t3) \to False)) (or (ex T +(\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3: +T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False))) (\lambda (H5: (ex T +(\lambda (t3: T).(ty3 g c2 t t3)))).(ex_ind T (\lambda (t3: T).(ty3 g c2 t +t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) +(\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False))) +(\lambda (x: T).(\lambda (H6: (ty3 g c2 t x)).(let H7 \def (H3 c2 t0 +(flt_thead_dx (Flat Cast) c2 t t0)) in (or_ind (ex T (\lambda (t3: T).(ty3 g +c2 t0 t3))) (\forall (t3: T).((ty3 g c2 t0 t3) \to False)) (or (ex T (\lambda +(t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3: T).((ty3 g c2 +(THead (Flat Cast) t t0) t3) \to False))) (\lambda (H8: (ex T (\lambda (t3: +T).(ty3 g c2 t0 t3)))).(ex_ind T (\lambda (t3: T).(ty3 g c2 t0 t3)) (or (ex T +(\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3: +T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False))) (\lambda (x0: +T).(\lambda (H9: (ty3 g c2 t0 x0)).(ex_ind T (\lambda (t3: T).(ty3 g c2 x0 +t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) +(\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False))) +(\lambda (x1: T).(\lambda (H10: (ty3 g c2 x0 x1)).(let H_x \def (pc3_dec g c2 +x0 x1 H10 t x H6) in (let H11 \def H_x in (or_ind (pc3 c2 x0 t) ((pc3 c2 x0 +t) \to False) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) +t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False))) +(\lambda (H12: (pc3 c2 x0 t)).(or_introl (ex T (\lambda (t3: T).(ty3 g c2 +(THead (Flat Cast) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) +t t0) t3) \to False)) (ex_intro T (\lambda (t3: T).(ty3 g c2 (THead (Flat +Cast) t t0) t3)) (THead (Flat Cast) x t) (ty3_cast g c2 t0 t (ty3_conv g c2 t +x H6 t0 x0 H9 H12) x H6)))) (\lambda (H12: (((pc3 c2 x0 t) \to +False))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) +t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False)) +(\lambda (t3: T).(\lambda (H13: (ty3 g c2 (THead (Flat Cast) t t0) +t3)).(ex3_ind T (\lambda (t4: T).(pc3 c2 (THead (Flat Cast) t4 t) t3)) +(\lambda (_: T).(ty3 g c2 t0 t)) (\lambda (t4: T).(ty3 g c2 t t4)) False +(\lambda (x2: T).(\lambda (_: (pc3 c2 (THead (Flat Cast) x2 t) t3)).(\lambda +(H15: (ty3 g c2 t0 t)).(\lambda (H16: (ty3 g c2 t x2)).(let H_y \def +(ty3_unique g c2 t x2 H16 x H6) in (let H_y0 \def (ty3_unique g c2 t0 t H15 +x0 H9) in (H12 (ex2_sym T (pr3 c2 t) (pr3 c2 x0) H_y0)))))))) (ty3_gen_cast g +c2 t0 t t3 H13)))))) H11))))) (ty3_correct g c2 t0 x0 H9)))) H8)) (\lambda +(H8: ((\forall (t3: T).((ty3 g c2 t0 t3) \to False)))).(or_intror (ex T +(\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3: +T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False)) (\lambda (t3: +T).(\lambda (H9: (ty3 g c2 (THead (Flat Cast) t t0) t3)).(ex3_ind T (\lambda +(t4: T).(pc3 c2 (THead (Flat Cast) t4 t) t3)) (\lambda (_: T).(ty3 g c2 t0 +t)) (\lambda (t4: T).(ty3 g c2 t t4)) False (\lambda (x0: T).(\lambda (_: +(pc3 c2 (THead (Flat Cast) x0 t) t3)).(\lambda (H11: (ty3 g c2 t0 +t)).(\lambda (_: (ty3 g c2 t x0)).(H8 t H11))))) (ty3_gen_cast g c2 t0 t t3 +H9)))))) H7)))) H5)) (\lambda (H5: ((\forall (t3: T).((ty3 g c2 t t3) \to +False)))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t +t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to +False)) (\lambda (t3: T).(\lambda (H6: (ty3 g c2 (THead (Flat Cast) t t0) +t3)).(ex3_ind T (\lambda (t4: T).(pc3 c2 (THead (Flat Cast) t4 t) t3)) +(\lambda (_: T).(ty3 g c2 t0 t)) (\lambda (t4: T).(ty3 g c2 t t4)) False +(\lambda (x0: T).(\lambda (_: (pc3 c2 (THead (Flat Cast) x0 t) t3)).(\lambda +(_: (ty3 g c2 t0 t)).(\lambda (H9: (ty3 g c2 t x0)).(ex_ind T (\lambda (t4: +T).(ty3 g c2 x0 t4)) False (\lambda (x: T).(\lambda (_: (ty3 g c2 x0 x)).(H5 +x0 H9))) (ty3_correct g c2 t x0 H9)))))) (ty3_gen_cast g c2 t0 t t3 H6)))))) +H4))) f H2))) k H1))))))) t2))) c t1))). +(* COMMENTS +Initial nodes: 9001 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/ty3/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/ty3/defs.ma new file mode 100644 index 000000000..d1f500cdf --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/ty3/defs.ma @@ -0,0 +1,49 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/G/defs.ma". + +include "Basic-1/pc3/defs.ma". + +inductive ty3 (g: G): C \to (T \to (T \to Prop)) \def +| ty3_conv: \forall (c: C).(\forall (t2: T).(\forall (t: T).((ty3 g c t2 t) +\to (\forall (u: T).(\forall (t1: T).((ty3 g c u t1) \to ((pc3 c t1 t2) \to +(ty3 g c u t2)))))))) +| ty3_sort: \forall (c: C).(\forall (m: nat).(ty3 g c (TSort m) (TSort (next +g m)))) +| ty3_abbr: \forall (n: nat).(\forall (c: C).(\forall (d: C).(\forall (u: +T).((getl n c (CHead d (Bind Abbr) u)) \to (\forall (t: T).((ty3 g d u t) \to +(ty3 g c (TLRef n) (lift (S n) O t)))))))) +| ty3_abst: \forall (n: nat).(\forall (c: C).(\forall (d: C).(\forall (u: +T).((getl n c (CHead d (Bind Abst) u)) \to (\forall (t: T).((ty3 g d u t) \to +(ty3 g c (TLRef n) (lift (S n) O u)))))))) +| ty3_bind: \forall (c: C).(\forall (u: T).(\forall (t: T).((ty3 g c u t) \to +(\forall (b: B).(\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c (Bind b) +u) t1 t2) \to (ty3 g c (THead (Bind b) u t1) (THead (Bind b) u t2))))))))) +| ty3_appl: \forall (c: C).(\forall (w: T).(\forall (u: T).((ty3 g c w u) \to +(\forall (v: T).(\forall (t: T).((ty3 g c v (THead (Bind Abst) u t)) \to (ty3 +g c (THead (Flat Appl) w v) (THead (Flat Appl) w (THead (Bind Abst) u +t))))))))) +| ty3_cast: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c t1 t2) +\to (\forall (t0: T).((ty3 g c t2 t0) \to (ty3 g c (THead (Flat Cast) t2 t1) +(THead (Flat Cast) t0 t2))))))). + +inductive tys3 (g: G) (c: C): TList \to (T \to Prop) \def +| tys3_nil: \forall (u: T).(\forall (u0: T).((ty3 g c u u0) \to (tys3 g c +TNil u))) +| tys3_cons: \forall (t: T).(\forall (u: T).((ty3 g c t u) \to (\forall (ts: +TList).((tys3 g c ts u) \to (tys3 g c (TCons t ts) u))))). + diff --git a/matita/matita/contribs/lambdadelta/basic_1/ty3/fsubst0.ma b/matita/matita/contribs/lambdadelta/basic_1/ty3/fsubst0.ma new file mode 100644 index 000000000..f92366a26 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/ty3/fsubst0.ma @@ -0,0 +1,995 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/ty3/props.ma". + +include "Basic-1/pc3/fsubst0.ma". + +include "Basic-1/getl/getl.ma". + +theorem ty3_fsubst0: + \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t: T).((ty3 g c1 +t1 t) \to (\forall (i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t2: +T).((fsubst0 i u c1 t1 c2 t2) \to (\forall (e: C).((getl i c1 (CHead e (Bind +Abbr) u)) \to (ty3 g c2 t2 t)))))))))))) +\def + \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t: T).(\lambda +(H: (ty3 g c1 t1 t)).(ty3_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda +(t2: T).(\forall (i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t3: +T).((fsubst0 i u c t0 c2 t3) \to (\forall (e: C).((getl i c (CHead e (Bind +Abbr) u)) \to (ty3 g c2 t3 t2))))))))))) (\lambda (c: C).(\lambda (t2: +T).(\lambda (t0: T).(\lambda (H0: (ty3 g c t2 t0)).(\lambda (H1: ((\forall +(i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t3: T).((fsubst0 i u c t2 +c2 t3) \to (\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) \to (ty3 g c2 +t3 t0)))))))))).(\lambda (u: T).(\lambda (t3: T).(\lambda (_: (ty3 g c u +t3)).(\lambda (H3: ((\forall (i: nat).(\forall (u0: T).(\forall (c2: +C).(\forall (t4: T).((fsubst0 i u0 c u c2 t4) \to (\forall (e: C).((getl i c +(CHead e (Bind Abbr) u0)) \to (ty3 g c2 t4 t3)))))))))).(\lambda (H4: (pc3 c +t3 t2)).(\lambda (i: nat).(\lambda (u0: T).(\lambda (c2: C).(\lambda (t4: +T).(\lambda (H5: (fsubst0 i u0 c u c2 t4)).(fsubst0_ind i u0 c u (\lambda +(c0: C).(\lambda (t5: T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u0)) +\to (ty3 g c0 t5 t2))))) (\lambda (t5: T).(\lambda (H6: (subst0 i u0 u +t5)).(\lambda (e: C).(\lambda (H7: (getl i c (CHead e (Bind Abbr) +u0))).(ty3_conv g c t2 t0 H0 t5 t3 (H3 i u0 c t5 (fsubst0_snd i u0 c u t5 H6) +e H7) H4))))) (\lambda (c3: C).(\lambda (H6: (csubst0 i u0 c c3)).(\lambda +(e: C).(\lambda (H7: (getl i c (CHead e (Bind Abbr) u0))).(ty3_conv g c3 t2 +t0 (H1 i u0 c3 t2 (fsubst0_fst i u0 c t2 c3 H6) e H7) u t3 (H3 i u0 c3 u +(fsubst0_fst i u0 c u c3 H6) e H7) (pc3_fsubst0 c t3 t2 H4 i u0 c3 t3 +(fsubst0_fst i u0 c t3 c3 H6) e H7)))))) (\lambda (t5: T).(\lambda (H6: +(subst0 i u0 u t5)).(\lambda (c3: C).(\lambda (H7: (csubst0 i u0 c +c3)).(\lambda (e: C).(\lambda (H8: (getl i c (CHead e (Bind Abbr) +u0))).(ty3_conv g c3 t2 t0 (H1 i u0 c3 t2 (fsubst0_fst i u0 c t2 c3 H7) e H8) +t5 t3 (H3 i u0 c3 t5 (fsubst0_both i u0 c u t5 H6 c3 H7) e H8) (pc3_fsubst0 c +t3 t2 H4 i u0 c3 t3 (fsubst0_fst i u0 c t3 c3 H7) e H8)))))))) c2 t4 +H5)))))))))))))))) (\lambda (c: C).(\lambda (m: nat).(\lambda (i: +nat).(\lambda (u: T).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H0: (fsubst0 +i u c (TSort m) c2 t2)).(fsubst0_ind i u c (TSort m) (\lambda (c0: +C).(\lambda (t0: T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) \to +(ty3 g c0 t0 (TSort (next g m))))))) (\lambda (t3: T).(\lambda (H1: (subst0 i +u (TSort m) t3)).(\lambda (e: C).(\lambda (_: (getl i c (CHead e (Bind Abbr) +u))).(subst0_gen_sort u t3 i m H1 (ty3 g c t3 (TSort (next g m)))))))) +(\lambda (c3: C).(\lambda (_: (csubst0 i u c c3)).(\lambda (e: C).(\lambda +(_: (getl i c (CHead e (Bind Abbr) u))).(ty3_sort g c3 m))))) (\lambda (t3: +T).(\lambda (H1: (subst0 i u (TSort m) t3)).(\lambda (c3: C).(\lambda (_: +(csubst0 i u c c3)).(\lambda (e: C).(\lambda (_: (getl i c (CHead e (Bind +Abbr) u))).(subst0_gen_sort u t3 i m H1 (ty3 g c3 t3 (TSort (next g +m)))))))))) c2 t2 H0)))))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d: +C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind Abbr) u))).(\lambda +(t0: T).(\lambda (H1: (ty3 g d u t0)).(\lambda (H2: ((\forall (i: +nat).(\forall (u0: T).(\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 d u c2 +t2) \to (\forall (e: C).((getl i d (CHead e (Bind Abbr) u0)) \to (ty3 g c2 t2 +t0)))))))))).(\lambda (i: nat).(\lambda (u0: T).(\lambda (c2: C).(\lambda +(t2: T).(\lambda (H3: (fsubst0 i u0 c (TLRef n) c2 t2)).(fsubst0_ind i u0 c +(TLRef n) (\lambda (c0: C).(\lambda (t3: T).(\forall (e: C).((getl i c (CHead +e (Bind Abbr) u0)) \to (ty3 g c0 t3 (lift (S n) O t0)))))) (\lambda (t3: +T).(\lambda (H4: (subst0 i u0 (TLRef n) t3)).(\lambda (e: C).(\lambda (H5: +(getl i c (CHead e (Bind Abbr) u0))).(land_ind (eq nat n i) (eq T t3 (lift (S +n) O u0)) (ty3 g c t3 (lift (S n) O t0)) (\lambda (H6: (eq nat n i)).(\lambda +(H7: (eq T t3 (lift (S n) O u0))).(eq_ind_r T (lift (S n) O u0) (\lambda (t4: +T).(ty3 g c t4 (lift (S n) O t0))) (let H8 \def (eq_ind_r nat i (\lambda (n0: +nat).(getl n0 c (CHead e (Bind Abbr) u0))) H5 n H6) in (let H9 \def (eq_ind C +(CHead d (Bind Abbr) u) (\lambda (c0: C).(getl n c c0)) H0 (CHead e (Bind +Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) n H0 (CHead e (Bind Abbr) u0) +H8)) in (let H10 \def (f_equal C C (\lambda (e0: C).(match e0 in C return +(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow +c0])) (CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono c (CHead d +(Bind Abbr) u) n H0 (CHead e (Bind Abbr) u0) H8)) in ((let H11 \def (f_equal +C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) +\Rightarrow u | (CHead _ _ t4) \Rightarrow t4])) (CHead d (Bind Abbr) u) +(CHead e (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) n H0 (CHead e +(Bind Abbr) u0) H8)) in (\lambda (H12: (eq C d e)).(let H13 \def (eq_ind_r C +e (\lambda (c0: C).(getl n c (CHead c0 (Bind Abbr) u0))) H9 d H12) in (let +H14 \def (eq_ind_r T u0 (\lambda (t4: T).(getl n c (CHead d (Bind Abbr) t4))) +H13 u H11) in (eq_ind T u (\lambda (t4: T).(ty3 g c (lift (S n) O t4) (lift +(S n) O t0))) (ty3_lift g d u t0 H1 c O (S n) (getl_drop Abbr c d u n H14)) +u0 H11))))) H10)))) t3 H7))) (subst0_gen_lref u0 t3 i n H4)))))) (\lambda +(c3: C).(\lambda (H4: (csubst0 i u0 c c3)).(\lambda (e: C).(\lambda (H5: +(getl i c (CHead e (Bind Abbr) u0))).(lt_le_e n i (ty3 g c3 (TLRef n) (lift +(S n) O t0)) (\lambda (H6: (lt n i)).(let H7 \def (csubst0_getl_lt i n H6 c +c3 u0 H4 (CHead d (Bind Abbr) u) H0) in (or4_ind (getl n c3 (CHead d (Bind +Abbr) u)) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: +T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b) u1)))))) +(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 +(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: +T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w)))))) (ex3_4 B C C T +(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C +(CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda +(_: C).(\lambda (e2: C).(\lambda (u1: T).(getl n c3 (CHead e2 (Bind b) +u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: +T).(csubst0 (minus i (S n)) u0 e1 e2)))))) (ex4_5 B C C T T (\lambda (b: +B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C +(CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda +(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e2 +(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda +(u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w)))))) (\lambda (_: +B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 +(minus i (S n)) u0 e1 e2))))))) (ty3 g c3 (TLRef n) (lift (S n) O t0)) +(\lambda (H8: (getl n c3 (CHead d (Bind Abbr) u))).(ty3_abbr g n c3 d u H8 t0 +H1)) (\lambda (H8: (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda +(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b) +u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: +T).(getl n c3 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: +C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 +w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: +T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b) u1)))))) +(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 +(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: +T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w))))) (ty3 g c3 (TLRef n) +(lift (S n) O t0)) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: +T).(\lambda (x3: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead x1 +(Bind x0) x2))).(\lambda (H10: (getl n c3 (CHead x1 (Bind x0) x3))).(\lambda +(H11: (subst0 (minus i (S n)) u0 x2 x3)).(let H12 \def (f_equal C C (\lambda +(e0: C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow +d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abbr) u) (CHead x1 (Bind +x0) x2) H9) in ((let H13 \def (f_equal C B (\lambda (e0: C).(match e0 in C +return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) +\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) +\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead +x1 (Bind x0) x2) H9) in ((let H14 \def (f_equal C T (\lambda (e0: C).(match +e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ +t3) \Rightarrow t3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H9) in +(\lambda (H15: (eq B Abbr x0)).(\lambda (H16: (eq C d x1)).(let H17 \def +(eq_ind_r T x2 (\lambda (t3: T).(subst0 (minus i (S n)) u0 t3 x3)) H11 u H14) +in (let H18 \def (eq_ind_r C x1 (\lambda (c0: C).(getl n c3 (CHead c0 (Bind +x0) x3))) H10 d H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b: B).(getl n +c3 (CHead d (Bind b) x3))) H18 Abbr H15) in (let H20 \def (eq_ind nat (minus +i n) (\lambda (n0: nat).(getl n0 (CHead d (Bind Abbr) x3) (CHead e (Bind +Abbr) u0))) (getl_conf_le i (CHead e (Bind Abbr) u0) c3 (csubst0_getl_ge i i +(le_n i) c c3 u0 H4 (CHead e (Bind Abbr) u0) H5) (CHead d (Bind Abbr) x3) n +H19 (le_S_n n i (le_S (S n) i H6))) (S (minus i (S n))) (minus_x_Sy i n H6)) +in (ty3_abbr g n c3 d x3 H19 t0 (H2 (minus i (S n)) u0 d x3 (fsubst0_snd +(minus i (S n)) u0 d u x3 H17) e (getl_gen_S (Bind Abbr) d (CHead e (Bind +Abbr) u0) x3 (minus i (S n)) H20)))))))))) H13)) H12))))))))) H8)) (\lambda +(H8: (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda +(u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda +(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl n c3 (CHead e2 +(Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda +(_: T).(csubst0 (minus i (S n)) u0 e1 e2))))))).(ex3_4_ind B C C T (\lambda +(b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind +Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda +(e2: C).(\lambda (u1: T).(getl n c3 (CHead e2 (Bind b) u1)))))) (\lambda (_: +B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) +u0 e1 e2))))) (ty3 g c3 (TLRef n) (lift (S n) O t0)) (\lambda (x0: +B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H9: (eq C +(CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3))).(\lambda (H10: (getl n c3 +(CHead x2 (Bind x0) x3))).(\lambda (H11: (csubst0 (minus i (S n)) u0 x1 +x2)).(let H12 \def (f_equal C C (\lambda (e0: C).(match e0 in C return +(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow +c0])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H13 \def +(f_equal C B (\lambda (e0: C).(match e0 in C return (\lambda (_: C).B) with +[(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k in K return +(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow +Abbr])])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H14 +\def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) +with [(CSort _) \Rightarrow u | (CHead _ _ t3) \Rightarrow t3])) (CHead d +(Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in (\lambda (H15: (eq B Abbr +x0)).(\lambda (H16: (eq C d x1)).(let H17 \def (eq_ind_r T x3 (\lambda (t3: +T).(getl n c3 (CHead x2 (Bind x0) t3))) H10 u H14) in (let H18 \def (eq_ind_r +C x1 (\lambda (c0: C).(csubst0 (minus i (S n)) u0 c0 x2)) H11 d H16) in (let +H19 \def (eq_ind_r B x0 (\lambda (b: B).(getl n c3 (CHead x2 (Bind b) u))) +H17 Abbr H15) in (let H20 \def (eq_ind nat (minus i n) (\lambda (n0: +nat).(getl n0 (CHead x2 (Bind Abbr) u) (CHead e (Bind Abbr) u0))) +(getl_conf_le i (CHead e (Bind Abbr) u0) c3 (csubst0_getl_ge i i (le_n i) c +c3 u0 H4 (CHead e (Bind Abbr) u0) H5) (CHead x2 (Bind Abbr) u) n H19 (le_S_n +n i (le_S (S n) i H6))) (S (minus i (S n))) (minus_x_Sy i n H6)) in (ty3_abbr +g n c3 x2 u H19 t0 (H2 (minus i (S n)) u0 x2 u (fsubst0_fst (minus i (S n)) +u0 d u x2 H18) e (csubst0_getl_ge_back (minus i (S n)) (minus i (S n)) (le_n +(minus i (S n))) d x2 u0 H18 (CHead e (Bind Abbr) u0) (getl_gen_S (Bind Abbr) +x2 (CHead e (Bind Abbr) u0) u (minus i (S n)) H20))))))))))) H13)) +H12))))))))) H8)) (\lambda (H8: (ex4_5 B C C T T (\lambda (b: B).(\lambda +(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind +Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: +C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e2 +(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda +(u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w)))))) (\lambda (_: +B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 +(minus i (S n)) u0 e1 e2)))))))).(ex4_5_ind B C C T T (\lambda (b: +B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C +(CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda +(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e2 +(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda +(u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w)))))) (\lambda (_: +B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 +(minus i (S n)) u0 e1 e2)))))) (ty3 g c3 (TLRef n) (lift (S n) O t0)) +(\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda +(x4: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) +x3))).(\lambda (H10: (getl n c3 (CHead x2 (Bind x0) x4))).(\lambda (H11: +(subst0 (minus i (S n)) u0 x3 x4)).(\lambda (H12: (csubst0 (minus i (S n)) u0 +x1 x2)).(let H13 \def (f_equal C C (\lambda (e0: C).(match e0 in C return +(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow +c0])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H14 \def +(f_equal C B (\lambda (e0: C).(match e0 in C return (\lambda (_: C).B) with +[(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k in K return +(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow +Abbr])])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H15 +\def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) +with [(CSort _) \Rightarrow u | (CHead _ _ t3) \Rightarrow t3])) (CHead d +(Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in (\lambda (H16: (eq B Abbr +x0)).(\lambda (H17: (eq C d x1)).(let H18 \def (eq_ind_r T x3 (\lambda (t3: +T).(subst0 (minus i (S n)) u0 t3 x4)) H11 u H15) in (let H19 \def (eq_ind_r C +x1 (\lambda (c0: C).(csubst0 (minus i (S n)) u0 c0 x2)) H12 d H17) in (let +H20 \def (eq_ind_r B x0 (\lambda (b: B).(getl n c3 (CHead x2 (Bind b) x4))) +H10 Abbr H16) in (let H21 \def (eq_ind nat (minus i n) (\lambda (n0: +nat).(getl n0 (CHead x2 (Bind Abbr) x4) (CHead e (Bind Abbr) u0))) +(getl_conf_le i (CHead e (Bind Abbr) u0) c3 (csubst0_getl_ge i i (le_n i) c +c3 u0 H4 (CHead e (Bind Abbr) u0) H5) (CHead x2 (Bind Abbr) x4) n H20 (le_S_n +n i (le_S (S n) i H6))) (S (minus i (S n))) (minus_x_Sy i n H6)) in (ty3_abbr +g n c3 x2 x4 H20 t0 (H2 (minus i (S n)) u0 x2 x4 (fsubst0_both (minus i (S +n)) u0 d u x4 H18 x2 H19) e (csubst0_getl_ge_back (minus i (S n)) (minus i (S +n)) (le_n (minus i (S n))) d x2 u0 H19 (CHead e (Bind Abbr) u0) (getl_gen_S +(Bind Abbr) x2 (CHead e (Bind Abbr) u0) x4 (minus i (S n)) H21))))))))))) +H14)) H13))))))))))) H8)) H7))) (\lambda (H6: (le i n)).(ty3_abbr g n c3 d u +(csubst0_getl_ge i n H6 c c3 u0 H4 (CHead d (Bind Abbr) u) H0) t0 H1))))))) +(\lambda (t3: T).(\lambda (H4: (subst0 i u0 (TLRef n) t3)).(\lambda (c3: +C).(\lambda (H5: (csubst0 i u0 c c3)).(\lambda (e: C).(\lambda (H6: (getl i c +(CHead e (Bind Abbr) u0))).(land_ind (eq nat n i) (eq T t3 (lift (S n) O u0)) +(ty3 g c3 t3 (lift (S n) O t0)) (\lambda (H7: (eq nat n i)).(\lambda (H8: (eq +T t3 (lift (S n) O u0))).(eq_ind_r T (lift (S n) O u0) (\lambda (t4: T).(ty3 +g c3 t4 (lift (S n) O t0))) (let H9 \def (eq_ind_r nat i (\lambda (n0: +nat).(getl n0 c (CHead e (Bind Abbr) u0))) H6 n H7) in (let H10 \def +(eq_ind_r nat i (\lambda (n0: nat).(csubst0 n0 u0 c c3)) H5 n H7) in (let H11 +\def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c0: C).(getl n c c0)) H0 +(CHead e (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) n H0 (CHead e +(Bind Abbr) u0) H9)) in (let H12 \def (f_equal C C (\lambda (e0: C).(match e0 +in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) +\Rightarrow c0])) (CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono +c (CHead d (Bind Abbr) u) n H0 (CHead e (Bind Abbr) u0) H9)) in ((let H13 +\def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) +with [(CSort _) \Rightarrow u | (CHead _ _ t4) \Rightarrow t4])) (CHead d +(Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) +n H0 (CHead e (Bind Abbr) u0) H9)) in (\lambda (H14: (eq C d e)).(let H15 +\def (eq_ind_r C e (\lambda (c0: C).(getl n c (CHead c0 (Bind Abbr) u0))) H11 +d H14) in (let H16 \def (eq_ind_r T u0 (\lambda (t4: T).(getl n c (CHead d +(Bind Abbr) t4))) H15 u H13) in (let H17 \def (eq_ind_r T u0 (\lambda (t4: +T).(csubst0 n t4 c c3)) H10 u H13) in (eq_ind T u (\lambda (t4: T).(ty3 g c3 +(lift (S n) O t4) (lift (S n) O t0))) (ty3_lift g d u t0 H1 c3 O (S n) +(getl_drop Abbr c3 d u n (csubst0_getl_ge n n (le_n n) c c3 u H17 (CHead d +(Bind Abbr) u) H16))) u0 H13)))))) H12))))) t3 H8))) (subst0_gen_lref u0 t3 i +n H4)))))))) c2 t2 H3)))))))))))))) (\lambda (n: nat).(\lambda (c: +C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind +Abst) u))).(\lambda (t0: T).(\lambda (H1: (ty3 g d u t0)).(\lambda (H2: +((\forall (i: nat).(\forall (u0: T).(\forall (c2: C).(\forall (t2: +T).((fsubst0 i u0 d u c2 t2) \to (\forall (e: C).((getl i d (CHead e (Bind +Abbr) u0)) \to (ty3 g c2 t2 t0)))))))))).(\lambda (i: nat).(\lambda (u0: +T).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H3: (fsubst0 i u0 c (TLRef n) +c2 t2)).(fsubst0_ind i u0 c (TLRef n) (\lambda (c0: C).(\lambda (t3: +T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u0)) \to (ty3 g c0 t3 +(lift (S n) O u)))))) (\lambda (t3: T).(\lambda (H4: (subst0 i u0 (TLRef n) +t3)).(\lambda (e: C).(\lambda (H5: (getl i c (CHead e (Bind Abbr) +u0))).(land_ind (eq nat n i) (eq T t3 (lift (S n) O u0)) (ty3 g c t3 (lift (S +n) O u)) (\lambda (H6: (eq nat n i)).(\lambda (H7: (eq T t3 (lift (S n) O +u0))).(eq_ind_r T (lift (S n) O u0) (\lambda (t4: T).(ty3 g c t4 (lift (S n) +O u))) (let H8 \def (eq_ind_r nat i (\lambda (n0: nat).(getl n0 c (CHead e +(Bind Abbr) u0))) H5 n H6) in (let H9 \def (eq_ind C (CHead d (Bind Abst) u) +(\lambda (c0: C).(getl n c c0)) H0 (CHead e (Bind Abbr) u0) (getl_mono c +(CHead d (Bind Abst) u) n H0 (CHead e (Bind Abbr) u0) H8)) in (let H10 \def +(eq_ind C (CHead d (Bind Abst) u) (\lambda (ee: C).(match ee in C return +(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) +\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b) +\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow +False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) +\Rightarrow False])])) I (CHead e (Bind Abbr) u0) (getl_mono c (CHead d (Bind +Abst) u) n H0 (CHead e (Bind Abbr) u0) H8)) in (False_ind (ty3 g c (lift (S +n) O u0) (lift (S n) O u)) H10)))) t3 H7))) (subst0_gen_lref u0 t3 i n +H4)))))) (\lambda (c3: C).(\lambda (H4: (csubst0 i u0 c c3)).(\lambda (e: +C).(\lambda (H5: (getl i c (CHead e (Bind Abbr) u0))).(lt_le_e n i (ty3 g c3 +(TLRef n) (lift (S n) O u)) (\lambda (H6: (lt n i)).(let H7 \def +(csubst0_getl_lt i n H6 c c3 u0 H4 (CHead d (Bind Abst) u) H0) in (or4_ind +(getl n c3 (CHead d (Bind Abst) u)) (ex3_4 B C T T (\lambda (b: B).(\lambda +(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead +e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: +T).(\lambda (w: T).(getl n c3 (CHead e0 (Bind b) w)))))) (\lambda (_: +B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i (S n)) +u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: +C).(\lambda (u1: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1)))))) +(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl n c3 +(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: +C).(\lambda (_: T).(csubst0 (minus i (S n)) u0 e1 e2)))))) (ex4_5 B C C T T +(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda +(_: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1))))))) (\lambda +(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl +n c3 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: +C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w)))))) +(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 (minus i (S n)) u0 e1 e2))))))) (ty3 g c3 (TLRef n) (lift (S +n) O u)) (\lambda (H8: (getl n c3 (CHead d (Bind Abst) u))).(ty3_abst g n c3 +d u H8 t0 H1)) (\lambda (H8: (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: +C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead e0 +(Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda +(w: T).(getl n c3 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: +C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 +w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: +T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead e0 (Bind b) u1)))))) +(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 +(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: +T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w))))) (ty3 g c3 (TLRef n) +(lift (S n) O u)) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda +(x3: T).(\lambda (H9: (eq C (CHead d (Bind Abst) u) (CHead x1 (Bind x0) +x2))).(\lambda (H10: (getl n c3 (CHead x1 (Bind x0) x3))).(\lambda (H11: +(subst0 (minus i (S n)) u0 x2 x3)).(let H12 \def (f_equal C C (\lambda (e0: +C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | +(CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) +x2) H9) in ((let H13 \def (f_equal C B (\lambda (e0: C).(match e0 in C return +(\lambda (_: C).B) with [(CSort _) \Rightarrow Abst | (CHead _ k _) +\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) +\Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead d (Bind Abst) u) (CHead +x1 (Bind x0) x2) H9) in ((let H14 \def (f_equal C T (\lambda (e0: C).(match +e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ +t3) \Rightarrow t3])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x2) H9) in +(\lambda (H15: (eq B Abst x0)).(\lambda (H16: (eq C d x1)).(let H17 \def +(eq_ind_r T x2 (\lambda (t3: T).(subst0 (minus i (S n)) u0 t3 x3)) H11 u H14) +in (let H18 \def (eq_ind_r C x1 (\lambda (c0: C).(getl n c3 (CHead c0 (Bind +x0) x3))) H10 d H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b: B).(getl n +c3 (CHead d (Bind b) x3))) H18 Abst H15) in (let H20 \def (eq_ind nat (minus +i n) (\lambda (n0: nat).(getl n0 (CHead d (Bind Abst) x3) (CHead e (Bind +Abbr) u0))) (getl_conf_le i (CHead e (Bind Abbr) u0) c3 (csubst0_getl_ge i i +(le_n i) c c3 u0 H4 (CHead e (Bind Abbr) u0) H5) (CHead d (Bind Abst) x3) n +H19 (le_S_n n i (le_S (S n) i H6))) (S (minus i (S n))) (minus_x_Sy i n H6)) +in (ty3_conv g c3 (lift (S n) O u) (lift (S n) O t0) (ty3_lift g d u t0 H1 c3 +O (S n) (getl_drop Abst c3 d x3 n H19)) (TLRef n) (lift (S n) O x3) (ty3_abst +g n c3 d x3 H19 t0 (H2 (minus i (S n)) u0 d x3 (fsubst0_snd (minus i (S n)) +u0 d u x3 H17) e (getl_gen_S (Bind Abst) d (CHead e (Bind Abbr) u0) x3 (minus +i (S n)) H20))) (pc3_lift c3 d (S n) O (getl_drop Abst c3 d x3 n H19) x3 u +(pc3_pr2_x d x3 u (pr2_delta d e u0 (r (Bind Abst) (minus i (S n))) +(getl_gen_S (Bind Abst) d (CHead e (Bind Abbr) u0) x3 (minus i (S n)) H20) u +u (pr0_refl u) x3 H17))))))))))) H13)) H12))))))))) H8)) (\lambda (H8: (ex3_4 +B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq +C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: +B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl n c3 (CHead e2 +(Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda +(_: T).(csubst0 (minus i (S n)) u0 e1 e2))))))).(ex3_4_ind B C C T (\lambda +(b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind +Abst) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda +(e2: C).(\lambda (u1: T).(getl n c3 (CHead e2 (Bind b) u1)))))) (\lambda (_: +B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) +u0 e1 e2))))) (ty3 g c3 (TLRef n) (lift (S n) O u)) (\lambda (x0: B).(\lambda +(x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H9: (eq C (CHead d (Bind +Abst) u) (CHead x1 (Bind x0) x3))).(\lambda (H10: (getl n c3 (CHead x2 (Bind +x0) x3))).(\lambda (H11: (csubst0 (minus i (S n)) u0 x1 x2)).(let H12 \def +(f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with +[(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind +Abst) u) (CHead x1 (Bind x0) x3) H9) in ((let H13 \def (f_equal C B (\lambda +(e0: C).(match e0 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow +Abst | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with +[(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead d (Bind Abst) +u) (CHead x1 (Bind x0) x3) H9) in ((let H14 \def (f_equal C T (\lambda (e0: +C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | +(CHead _ _ t3) \Rightarrow t3])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) +x3) H9) in (\lambda (H15: (eq B Abst x0)).(\lambda (H16: (eq C d x1)).(let +H17 \def (eq_ind_r T x3 (\lambda (t3: T).(getl n c3 (CHead x2 (Bind x0) t3))) +H10 u H14) in (let H18 \def (eq_ind_r C x1 (\lambda (c0: C).(csubst0 (minus i +(S n)) u0 c0 x2)) H11 d H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b: +B).(getl n c3 (CHead x2 (Bind b) u))) H17 Abst H15) in (let H20 \def (eq_ind +nat (minus i n) (\lambda (n0: nat).(getl n0 (CHead x2 (Bind Abst) u) (CHead e +(Bind Abbr) u0))) (getl_conf_le i (CHead e (Bind Abbr) u0) c3 +(csubst0_getl_ge i i (le_n i) c c3 u0 H4 (CHead e (Bind Abbr) u0) H5) (CHead +x2 (Bind Abst) u) n H19 (le_S_n n i (le_S (S n) i H6))) (S (minus i (S n))) +(minus_x_Sy i n H6)) in (ty3_abst g n c3 x2 u H19 t0 (H2 (minus i (S n)) u0 +x2 u (fsubst0_fst (minus i (S n)) u0 d u x2 H18) e (csubst0_getl_ge_back +(minus i (S n)) (minus i (S n)) (le_n (minus i (S n))) d x2 u0 H18 (CHead e +(Bind Abbr) u0) (getl_gen_S (Bind Abst) x2 (CHead e (Bind Abbr) u0) u (minus +i (S n)) H20))))))))))) H13)) H12))))))))) H8)) (\lambda (H8: (ex4_5 B C C T +T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda +(_: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1))))))) (\lambda +(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl +n c3 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: +C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w)))))) +(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 (minus i (S n)) u0 e1 e2)))))))).(ex4_5_ind B C C T T +(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda +(_: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1))))))) (\lambda +(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl +n c3 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: +C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w)))))) +(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda +(_: T).(csubst0 (minus i (S n)) u0 e1 e2)))))) (ty3 g c3 (TLRef n) (lift (S +n) O u)) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: +T).(\lambda (x4: T).(\lambda (H9: (eq C (CHead d (Bind Abst) u) (CHead x1 +(Bind x0) x3))).(\lambda (H10: (getl n c3 (CHead x2 (Bind x0) x4))).(\lambda +(H11: (subst0 (minus i (S n)) u0 x3 x4)).(\lambda (H12: (csubst0 (minus i (S +n)) u0 x1 x2)).(let H13 \def (f_equal C C (\lambda (e0: C).(match e0 in C +return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) +\Rightarrow c0])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3) H9) in +((let H14 \def (f_equal C B (\lambda (e0: C).(match e0 in C return (\lambda +(_: C).B) with [(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match +k in K return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) +\Rightarrow Abst])])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3) H9) in +((let H15 \def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda +(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t3) \Rightarrow t3])) +(CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3) H9) in (\lambda (H16: (eq B +Abst x0)).(\lambda (H17: (eq C d x1)).(let H18 \def (eq_ind_r T x3 (\lambda +(t3: T).(subst0 (minus i (S n)) u0 t3 x4)) H11 u H15) in (let H19 \def +(eq_ind_r C x1 (\lambda (c0: C).(csubst0 (minus i (S n)) u0 c0 x2)) H12 d +H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b: B).(getl n c3 (CHead x2 +(Bind b) x4))) H10 Abst H16) in (let H21 \def (eq_ind nat (minus i n) +(\lambda (n0: nat).(getl n0 (CHead x2 (Bind Abst) x4) (CHead e (Bind Abbr) +u0))) (getl_conf_le i (CHead e (Bind Abbr) u0) c3 (csubst0_getl_ge i i (le_n +i) c c3 u0 H4 (CHead e (Bind Abbr) u0) H5) (CHead x2 (Bind Abst) x4) n H20 +(le_S_n n i (le_S (S n) i H6))) (S (minus i (S n))) (minus_x_Sy i n H6)) in +(ty3_conv g c3 (lift (S n) O u) (lift (S n) O t0) (ty3_lift g x2 u t0 (H2 +(minus i (S n)) u0 x2 u (fsubst0_fst (minus i (S n)) u0 d u x2 H19) e +(csubst0_getl_ge_back (minus i (S n)) (minus i (S n)) (le_n (minus i (S n))) +d x2 u0 H19 (CHead e (Bind Abbr) u0) (getl_gen_S (Bind Abst) x2 (CHead e +(Bind Abbr) u0) x4 (minus i (S n)) H21))) c3 O (S n) (getl_drop Abst c3 x2 x4 +n H20)) (TLRef n) (lift (S n) O x4) (ty3_abst g n c3 x2 x4 H20 t0 (H2 (minus +i (S n)) u0 x2 x4 (fsubst0_both (minus i (S n)) u0 d u x4 H18 x2 H19) e +(csubst0_getl_ge_back (minus i (S n)) (minus i (S n)) (le_n (minus i (S n))) +d x2 u0 H19 (CHead e (Bind Abbr) u0) (getl_gen_S (Bind Abst) x2 (CHead e +(Bind Abbr) u0) x4 (minus i (S n)) H21)))) (pc3_lift c3 x2 (S n) O (getl_drop +Abst c3 x2 x4 n H20) x4 u (pc3_fsubst0 d u u (pc3_refl d u) (minus i (S n)) +u0 x2 x4 (fsubst0_both (minus i (S n)) u0 d u x4 H18 x2 H19) e +(csubst0_getl_ge_back (minus i (S n)) (minus i (S n)) (le_n (minus i (S n))) +d x2 u0 H19 (CHead e (Bind Abbr) u0) (getl_gen_S (Bind Abst) x2 (CHead e +(Bind Abbr) u0) x4 (minus i (S n)) H21)))))))))))) H14)) H13))))))))))) H8)) +H7))) (\lambda (H6: (le i n)).(ty3_abst g n c3 d u (csubst0_getl_ge i n H6 c +c3 u0 H4 (CHead d (Bind Abst) u) H0) t0 H1))))))) (\lambda (t3: T).(\lambda +(H4: (subst0 i u0 (TLRef n) t3)).(\lambda (c3: C).(\lambda (H5: (csubst0 i u0 +c c3)).(\lambda (e: C).(\lambda (H6: (getl i c (CHead e (Bind Abbr) +u0))).(land_ind (eq nat n i) (eq T t3 (lift (S n) O u0)) (ty3 g c3 t3 (lift +(S n) O u)) (\lambda (H7: (eq nat n i)).(\lambda (H8: (eq T t3 (lift (S n) O +u0))).(eq_ind_r T (lift (S n) O u0) (\lambda (t4: T).(ty3 g c3 t4 (lift (S n) +O u))) (let H9 \def (eq_ind_r nat i (\lambda (n0: nat).(getl n0 c (CHead e +(Bind Abbr) u0))) H6 n H7) in (let H10 \def (eq_ind_r nat i (\lambda (n0: +nat).(csubst0 n0 u0 c c3)) H5 n H7) in (let H11 \def (eq_ind C (CHead d (Bind +Abst) u) (\lambda (c0: C).(getl n c c0)) H0 (CHead e (Bind Abbr) u0) +(getl_mono c (CHead d (Bind Abst) u) n H0 (CHead e (Bind Abbr) u0) H9)) in +(let H12 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (ee: C).(match ee in +C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k +_) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b) +\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow +False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) +\Rightarrow False])])) I (CHead e (Bind Abbr) u0) (getl_mono c (CHead d (Bind +Abst) u) n H0 (CHead e (Bind Abbr) u0) H9)) in (False_ind (ty3 g c3 (lift (S +n) O u0) (lift (S n) O u)) H12))))) t3 H8))) (subst0_gen_lref u0 t3 i n +H4)))))))) c2 t2 H3)))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda +(t0: T).(\lambda (H0: (ty3 g c u t0)).(\lambda (H1: ((\forall (i: +nat).(\forall (u0: T).(\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 c u c2 +t2) \to (\forall (e: C).((getl i c (CHead e (Bind Abbr) u0)) \to (ty3 g c2 t2 +t0)))))))))).(\lambda (b: B).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H2: +(ty3 g (CHead c (Bind b) u) t2 t3)).(\lambda (H3: ((\forall (i: nat).(\forall +(u0: T).(\forall (c2: C).(\forall (t4: T).((fsubst0 i u0 (CHead c (Bind b) u) +t2 c2 t4) \to (\forall (e: C).((getl i (CHead c (Bind b) u) (CHead e (Bind +Abbr) u0)) \to (ty3 g c2 t4 t3)))))))))).(\lambda (i: nat).(\lambda (u0: +T).(\lambda (c2: C).(\lambda (t4: T).(\lambda (H4: (fsubst0 i u0 c (THead +(Bind b) u t2) c2 t4)).(fsubst0_ind i u0 c (THead (Bind b) u t2) (\lambda +(c0: C).(\lambda (t5: T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u0)) +\to (ty3 g c0 t5 (THead (Bind b) u t3)))))) (\lambda (t5: T).(\lambda (H5: +(subst0 i u0 (THead (Bind b) u t2) t5)).(\lambda (e: C).(\lambda (H6: (getl i +c (CHead e (Bind Abbr) u0))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t5 (THead +(Bind b) u2 t2))) (\lambda (u2: T).(subst0 i u0 u u2))) (ex2 T (\lambda (t6: +T).(eq T t5 (THead (Bind b) u t6))) (\lambda (t6: T).(subst0 (s (Bind b) i) +u0 t2 t6))) (ex3_2 T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead +(Bind b) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) +(\lambda (_: T).(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 t6)))) (ty3 g c +t5 (THead (Bind b) u t3)) (\lambda (H7: (ex2 T (\lambda (u2: T).(eq T t5 +(THead (Bind b) u2 t2))) (\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T +(\lambda (u2: T).(eq T t5 (THead (Bind b) u2 t2))) (\lambda (u2: T).(subst0 i +u0 u u2)) (ty3 g c t5 (THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H8: +(eq T t5 (THead (Bind b) x t2))).(\lambda (H9: (subst0 i u0 u x)).(eq_ind_r T +(THead (Bind b) x t2) (\lambda (t6: T).(ty3 g c t6 (THead (Bind b) u t3))) +(ex_ind T (\lambda (t6: T).(ty3 g (CHead c (Bind b) u) t3 t6)) (ty3 g c +(THead (Bind b) x t2) (THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (H10: +(ty3 g (CHead c (Bind b) u) t3 x0)).(ex_ind T (\lambda (t6: T).(ty3 g (CHead +c (Bind b) x) t3 t6)) (ty3 g c (THead (Bind b) x t2) (THead (Bind b) u t3)) +(\lambda (x1: T).(\lambda (_: (ty3 g (CHead c (Bind b) x) t3 x1)).(ty3_conv g +c (THead (Bind b) u t3) (THead (Bind b) u x0) (ty3_bind g c u t0 H0 b t3 x0 +H10) (THead (Bind b) x t2) (THead (Bind b) x t3) (ty3_bind g c x t0 (H1 i u0 +c x (fsubst0_snd i u0 c u x H9) e H6) b t2 t3 (H3 (S i) u0 (CHead c (Bind b) +x) t2 (fsubst0_fst (S i) u0 (CHead c (Bind b) u) t2 (CHead c (Bind b) x) +(csubst0_snd_bind b i u0 u x H9 c)) e (getl_head (Bind b) i c (CHead e (Bind +Abbr) u0) H6 u))) (pc3_fsubst0 c (THead (Bind b) u t3) (THead (Bind b) u t3) +(pc3_refl c (THead (Bind b) u t3)) i u0 c (THead (Bind b) x t3) (fsubst0_snd +i u0 c (THead (Bind b) u t3) (THead (Bind b) x t3) (subst0_fst u0 x u i H9 t3 +(Bind b))) e H6)))) (ty3_correct g (CHead c (Bind b) x) t2 t3 (H3 (S i) u0 +(CHead c (Bind b) x) t2 (fsubst0_fst (S i) u0 (CHead c (Bind b) u) t2 (CHead +c (Bind b) x) (csubst0_snd_bind b i u0 u x H9 c)) e (getl_head (Bind b) i c +(CHead e (Bind Abbr) u0) H6 u)))))) (ty3_correct g (CHead c (Bind b) u) t2 t3 +H2)) t5 H8)))) H7)) (\lambda (H7: (ex2 T (\lambda (t6: T).(eq T t5 (THead +(Bind b) u t6))) (\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 +t6)))).(ex2_ind T (\lambda (t6: T).(eq T t5 (THead (Bind b) u t6))) (\lambda +(t6: T).(subst0 (s (Bind b) i) u0 t2 t6)) (ty3 g c t5 (THead (Bind b) u t3)) +(\lambda (x: T).(\lambda (H8: (eq T t5 (THead (Bind b) u x))).(\lambda (H9: +(subst0 (s (Bind b) i) u0 t2 x)).(eq_ind_r T (THead (Bind b) u x) (\lambda +(t6: T).(ty3 g c t6 (THead (Bind b) u t3))) (ex_ind T (\lambda (t6: T).(ty3 g +(CHead c (Bind b) u) t3 t6)) (ty3 g c (THead (Bind b) u x) (THead (Bind b) u +t3)) (\lambda (x0: T).(\lambda (_: (ty3 g (CHead c (Bind b) u) t3 +x0)).(ty3_bind g c u t0 H0 b x t3 (H3 (S i) u0 (CHead c (Bind b) u) x +(fsubst0_snd (S i) u0 (CHead c (Bind b) u) t2 x H9) e (getl_head (Bind b) i c +(CHead e (Bind Abbr) u0) H6 u))))) (ty3_correct g (CHead c (Bind b) u) x t3 +(H3 (S i) u0 (CHead c (Bind b) u) x (fsubst0_snd (S i) u0 (CHead c (Bind b) +u) t2 x H9) e (getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H6 u)))) t5 +H8)))) H7)) (\lambda (H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t6: T).(eq T +t5 (THead (Bind b) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u +u2))) (\lambda (_: T).(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 +t6))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead +(Bind b) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) +(\lambda (_: T).(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 t6))) (ty3 g c +t5 (THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H8: (eq +T t5 (THead (Bind b) x0 x1))).(\lambda (H9: (subst0 i u0 u x0)).(\lambda +(H10: (subst0 (s (Bind b) i) u0 t2 x1)).(eq_ind_r T (THead (Bind b) x0 x1) +(\lambda (t6: T).(ty3 g c t6 (THead (Bind b) u t3))) (ex_ind T (\lambda (t6: +T).(ty3 g (CHead c (Bind b) u) t3 t6)) (ty3 g c (THead (Bind b) x0 x1) (THead +(Bind b) u t3)) (\lambda (x: T).(\lambda (H11: (ty3 g (CHead c (Bind b) u) t3 +x)).(ex_ind T (\lambda (t6: T).(ty3 g (CHead c (Bind b) x0) t3 t6)) (ty3 g c +(THead (Bind b) x0 x1) (THead (Bind b) u t3)) (\lambda (x2: T).(\lambda (_: +(ty3 g (CHead c (Bind b) x0) t3 x2)).(ty3_conv g c (THead (Bind b) u t3) +(THead (Bind b) u x) (ty3_bind g c u t0 H0 b t3 x H11) (THead (Bind b) x0 x1) +(THead (Bind b) x0 t3) (ty3_bind g c x0 t0 (H1 i u0 c x0 (fsubst0_snd i u0 c +u x0 H9) e H6) b x1 t3 (H3 (S i) u0 (CHead c (Bind b) x0) x1 (fsubst0_both (S +i) u0 (CHead c (Bind b) u) t2 x1 H10 (CHead c (Bind b) x0) (csubst0_snd_bind +b i u0 u x0 H9 c)) e (getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H6 u))) +(pc3_fsubst0 c (THead (Bind b) u t3) (THead (Bind b) u t3) (pc3_refl c (THead +(Bind b) u t3)) i u0 c (THead (Bind b) x0 t3) (fsubst0_snd i u0 c (THead +(Bind b) u t3) (THead (Bind b) x0 t3) (subst0_fst u0 x0 u i H9 t3 (Bind b))) +e H6)))) (ty3_correct g (CHead c (Bind b) x0) x1 t3 (H3 (S i) u0 (CHead c +(Bind b) x0) x1 (fsubst0_both (S i) u0 (CHead c (Bind b) u) t2 x1 H10 (CHead +c (Bind b) x0) (csubst0_snd_bind b i u0 u x0 H9 c)) e (getl_head (Bind b) i c +(CHead e (Bind Abbr) u0) H6 u)))))) (ty3_correct g (CHead c (Bind b) u) t2 t3 +H2)) t5 H8)))))) H7)) (subst0_gen_head (Bind b) u0 u t2 t5 i H5)))))) +(\lambda (c3: C).(\lambda (H5: (csubst0 i u0 c c3)).(\lambda (e: C).(\lambda +(H6: (getl i c (CHead e (Bind Abbr) u0))).(ex_ind T (\lambda (t5: T).(ty3 g +(CHead c3 (Bind b) u) t3 t5)) (ty3 g c3 (THead (Bind b) u t2) (THead (Bind b) +u t3)) (\lambda (x: T).(\lambda (_: (ty3 g (CHead c3 (Bind b) u) t3 +x)).(ty3_bind g c3 u t0 (H1 i u0 c3 u (fsubst0_fst i u0 c u c3 H5) e H6) b t2 +t3 (H3 (S i) u0 (CHead c3 (Bind b) u) t2 (fsubst0_fst (S i) u0 (CHead c (Bind +b) u) t2 (CHead c3 (Bind b) u) (csubst0_fst_bind b i c c3 u0 H5 u)) e +(getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H6 u))))) (ty3_correct g +(CHead c3 (Bind b) u) t2 t3 (H3 (S i) u0 (CHead c3 (Bind b) u) t2 +(fsubst0_fst (S i) u0 (CHead c (Bind b) u) t2 (CHead c3 (Bind b) u) +(csubst0_fst_bind b i c c3 u0 H5 u)) e (getl_head (Bind b) i c (CHead e (Bind +Abbr) u0) H6 u)))))))) (\lambda (t5: T).(\lambda (H5: (subst0 i u0 (THead +(Bind b) u t2) t5)).(\lambda (c3: C).(\lambda (H6: (csubst0 i u0 c +c3)).(\lambda (e: C).(\lambda (H7: (getl i c (CHead e (Bind Abbr) +u0))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t5 (THead (Bind b) u2 t2))) +(\lambda (u2: T).(subst0 i u0 u u2))) (ex2 T (\lambda (t6: T).(eq T t5 (THead +(Bind b) u t6))) (\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 t6))) (ex3_2 T +T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead (Bind b) u2 t6)))) +(\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: +T).(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 t6)))) (ty3 g c3 t5 (THead +(Bind b) u t3)) (\lambda (H8: (ex2 T (\lambda (u2: T).(eq T t5 (THead (Bind +b) u2 t2))) (\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T (\lambda (u2: +T).(eq T t5 (THead (Bind b) u2 t2))) (\lambda (u2: T).(subst0 i u0 u u2)) +(ty3 g c3 t5 (THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H9: (eq T t5 +(THead (Bind b) x t2))).(\lambda (H10: (subst0 i u0 u x)).(eq_ind_r T (THead +(Bind b) x t2) (\lambda (t6: T).(ty3 g c3 t6 (THead (Bind b) u t3))) (ex_ind +T (\lambda (t6: T).(ty3 g (CHead c3 (Bind b) u) t3 t6)) (ty3 g c3 (THead +(Bind b) x t2) (THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (H11: (ty3 g +(CHead c3 (Bind b) u) t3 x0)).(ex_ind T (\lambda (t6: T).(ty3 g (CHead c3 +(Bind b) u) x0 t6)) (ty3 g c3 (THead (Bind b) x t2) (THead (Bind b) u t3)) +(\lambda (x1: T).(\lambda (_: (ty3 g (CHead c3 (Bind b) u) x0 x1)).(ex_ind T +(\lambda (t6: T).(ty3 g (CHead c3 (Bind b) x) t3 t6)) (ty3 g c3 (THead (Bind +b) x t2) (THead (Bind b) u t3)) (\lambda (x2: T).(\lambda (_: (ty3 g (CHead +c3 (Bind b) x) t3 x2)).(ty3_conv g c3 (THead (Bind b) u t3) (THead (Bind b) u +x0) (ty3_bind g c3 u t0 (H1 i u0 c3 u (fsubst0_fst i u0 c u c3 H6) e H7) b t3 +x0 H11) (THead (Bind b) x t2) (THead (Bind b) x t3) (ty3_bind g c3 x t0 (H1 i +u0 c3 x (fsubst0_both i u0 c u x H10 c3 H6) e H7) b t2 t3 (H3 (S i) u0 (CHead +c3 (Bind b) x) t2 (fsubst0_fst (S i) u0 (CHead c (Bind b) u) t2 (CHead c3 +(Bind b) x) (csubst0_both_bind b i u0 u x H10 c c3 H6)) e (getl_head (Bind b) +i c (CHead e (Bind Abbr) u0) H7 u))) (pc3_fsubst0 c (THead (Bind b) u t3) +(THead (Bind b) u t3) (pc3_refl c (THead (Bind b) u t3)) i u0 c3 (THead (Bind +b) x t3) (fsubst0_both i u0 c (THead (Bind b) u t3) (THead (Bind b) x t3) +(subst0_fst u0 x u i H10 t3 (Bind b)) c3 H6) e H7)))) (ty3_correct g (CHead +c3 (Bind b) x) t2 t3 (H3 (S i) u0 (CHead c3 (Bind b) x) t2 (fsubst0_fst (S i) +u0 (CHead c (Bind b) u) t2 (CHead c3 (Bind b) x) (csubst0_both_bind b i u0 u +x H10 c c3 H6)) e (getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H7 u)))))) +(ty3_correct g (CHead c3 (Bind b) u) t3 x0 H11)))) (ty3_correct g (CHead c3 +(Bind b) u) t2 t3 (H3 (S i) u0 (CHead c3 (Bind b) u) t2 (fsubst0_fst (S i) u0 +(CHead c (Bind b) u) t2 (CHead c3 (Bind b) u) (csubst0_fst_bind b i c c3 u0 +H6 u)) e (getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H7 u)))) t5 H9)))) +H8)) (\lambda (H8: (ex2 T (\lambda (t6: T).(eq T t5 (THead (Bind b) u t6))) +(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 t6)))).(ex2_ind T (\lambda (t6: +T).(eq T t5 (THead (Bind b) u t6))) (\lambda (t6: T).(subst0 (s (Bind b) i) +u0 t2 t6)) (ty3 g c3 t5 (THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H9: +(eq T t5 (THead (Bind b) u x))).(\lambda (H10: (subst0 (s (Bind b) i) u0 t2 +x)).(eq_ind_r T (THead (Bind b) u x) (\lambda (t6: T).(ty3 g c3 t6 (THead +(Bind b) u t3))) (ex_ind T (\lambda (t6: T).(ty3 g (CHead c3 (Bind b) u) t3 +t6)) (ty3 g c3 (THead (Bind b) u x) (THead (Bind b) u t3)) (\lambda (x0: +T).(\lambda (_: (ty3 g (CHead c3 (Bind b) u) t3 x0)).(ty3_bind g c3 u t0 (H1 +i u0 c3 u (fsubst0_fst i u0 c u c3 H6) e H7) b x t3 (H3 (S i) u0 (CHead c3 +(Bind b) u) x (fsubst0_both (S i) u0 (CHead c (Bind b) u) t2 x H10 (CHead c3 +(Bind b) u) (csubst0_fst_bind b i c c3 u0 H6 u)) e (getl_head (Bind b) i c +(CHead e (Bind Abbr) u0) H7 u))))) (ty3_correct g (CHead c3 (Bind b) u) x t3 +(H3 (S i) u0 (CHead c3 (Bind b) u) x (fsubst0_both (S i) u0 (CHead c (Bind b) +u) t2 x H10 (CHead c3 (Bind b) u) (csubst0_fst_bind b i c c3 u0 H6 u)) e +(getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H7 u)))) t5 H9)))) H8)) +(\lambda (H8: (ex3_2 T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead +(Bind b) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) +(\lambda (_: T).(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 +t6))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead +(Bind b) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) +(\lambda (_: T).(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 t6))) (ty3 g c3 +t5 (THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H9: (eq +T t5 (THead (Bind b) x0 x1))).(\lambda (H10: (subst0 i u0 u x0)).(\lambda +(H11: (subst0 (s (Bind b) i) u0 t2 x1)).(eq_ind_r T (THead (Bind b) x0 x1) +(\lambda (t6: T).(ty3 g c3 t6 (THead (Bind b) u t3))) (ex_ind T (\lambda (t6: +T).(ty3 g (CHead c3 (Bind b) u) t3 t6)) (ty3 g c3 (THead (Bind b) x0 x1) +(THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H12: (ty3 g (CHead c3 (Bind +b) u) t3 x)).(ex_ind T (\lambda (t6: T).(ty3 g (CHead c3 (Bind b) u) x t6)) +(ty3 g c3 (THead (Bind b) x0 x1) (THead (Bind b) u t3)) (\lambda (x2: +T).(\lambda (_: (ty3 g (CHead c3 (Bind b) u) x x2)).(ex_ind T (\lambda (t6: +T).(ty3 g (CHead c3 (Bind b) x0) t3 t6)) (ty3 g c3 (THead (Bind b) x0 x1) +(THead (Bind b) u t3)) (\lambda (x3: T).(\lambda (_: (ty3 g (CHead c3 (Bind +b) x0) t3 x3)).(ty3_conv g c3 (THead (Bind b) u t3) (THead (Bind b) u x) +(ty3_bind g c3 u t0 (H1 i u0 c3 u (fsubst0_fst i u0 c u c3 H6) e H7) b t3 x +H12) (THead (Bind b) x0 x1) (THead (Bind b) x0 t3) (ty3_bind g c3 x0 t0 (H1 i +u0 c3 x0 (fsubst0_both i u0 c u x0 H10 c3 H6) e H7) b x1 t3 (H3 (S i) u0 +(CHead c3 (Bind b) x0) x1 (fsubst0_both (S i) u0 (CHead c (Bind b) u) t2 x1 +H11 (CHead c3 (Bind b) x0) (csubst0_both_bind b i u0 u x0 H10 c c3 H6)) e +(getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H7 u))) (pc3_fsubst0 c +(THead (Bind b) u t3) (THead (Bind b) u t3) (pc3_refl c (THead (Bind b) u +t3)) i u0 c3 (THead (Bind b) x0 t3) (fsubst0_both i u0 c (THead (Bind b) u +t3) (THead (Bind b) x0 t3) (subst0_fst u0 x0 u i H10 t3 (Bind b)) c3 H6) e +H7)))) (ty3_correct g (CHead c3 (Bind b) x0) x1 t3 (H3 (S i) u0 (CHead c3 +(Bind b) x0) x1 (fsubst0_both (S i) u0 (CHead c (Bind b) u) t2 x1 H11 (CHead +c3 (Bind b) x0) (csubst0_both_bind b i u0 u x0 H10 c c3 H6)) e (getl_head +(Bind b) i c (CHead e (Bind Abbr) u0) H7 u)))))) (ty3_correct g (CHead c3 +(Bind b) u) t3 x H12)))) (ty3_correct g (CHead c3 (Bind b) u) t2 t3 (H3 (S i) +u0 (CHead c3 (Bind b) u) t2 (fsubst0_fst (S i) u0 (CHead c (Bind b) u) t2 +(CHead c3 (Bind b) u) (csubst0_fst_bind b i c c3 u0 H6 u)) e (getl_head (Bind +b) i c (CHead e (Bind Abbr) u0) H7 u)))) t5 H9)))))) H8)) (subst0_gen_head +(Bind b) u0 u t2 t5 i H5)))))))) c2 t4 H4)))))))))))))))) (\lambda (c: +C).(\lambda (w: T).(\lambda (u: T).(\lambda (H0: (ty3 g c w u)).(\lambda (H1: +((\forall (i: nat).(\forall (u0: T).(\forall (c2: C).(\forall (t2: +T).((fsubst0 i u0 c w c2 t2) \to (\forall (e: C).((getl i c (CHead e (Bind +Abbr) u0)) \to (ty3 g c2 t2 u)))))))))).(\lambda (v: T).(\lambda (t0: +T).(\lambda (H2: (ty3 g c v (THead (Bind Abst) u t0))).(\lambda (H3: +((\forall (i: nat).(\forall (u0: T).(\forall (c2: C).(\forall (t2: +T).((fsubst0 i u0 c v c2 t2) \to (\forall (e: C).((getl i c (CHead e (Bind +Abbr) u0)) \to (ty3 g c2 t2 (THead (Bind Abst) u t0))))))))))).(\lambda (i: +nat).(\lambda (u0: T).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H4: +(fsubst0 i u0 c (THead (Flat Appl) w v) c2 t2)).(fsubst0_ind i u0 c (THead +(Flat Appl) w v) (\lambda (c0: C).(\lambda (t3: T).(\forall (e: C).((getl i c +(CHead e (Bind Abbr) u0)) \to (ty3 g c0 t3 (THead (Flat Appl) w (THead (Bind +Abst) u t0))))))) (\lambda (t3: T).(\lambda (H5: (subst0 i u0 (THead (Flat +Appl) w v) t3)).(\lambda (e: C).(\lambda (H6: (getl i c (CHead e (Bind Abbr) +u0))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t3 (THead (Flat Appl) u2 v))) +(\lambda (u2: T).(subst0 i u0 w u2))) (ex2 T (\lambda (t4: T).(eq T t3 (THead +(Flat Appl) w t4))) (\lambda (t4: T).(subst0 (s (Flat Appl) i) u0 v t4))) +(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 +t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 w u2))) (\lambda (_: +T).(\lambda (t4: T).(subst0 (s (Flat Appl) i) u0 v t4)))) (ty3 g c t3 (THead +(Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (H7: (ex2 T (\lambda (u2: +T).(eq T t3 (THead (Flat Appl) u2 v))) (\lambda (u2: T).(subst0 i u0 w +u2)))).(ex2_ind T (\lambda (u2: T).(eq T t3 (THead (Flat Appl) u2 v))) +(\lambda (u2: T).(subst0 i u0 w u2)) (ty3 g c t3 (THead (Flat Appl) w (THead +(Bind Abst) u t0))) (\lambda (x: T).(\lambda (H8: (eq T t3 (THead (Flat Appl) +x v))).(\lambda (H9: (subst0 i u0 w x)).(eq_ind_r T (THead (Flat Appl) x v) +(\lambda (t4: T).(ty3 g c t4 (THead (Flat Appl) w (THead (Bind Abst) u t0)))) +(ex_ind T (\lambda (t4: T).(ty3 g c (THead (Bind Abst) u t0) t4)) (ty3 g c +(THead (Flat Appl) x v) (THead (Flat Appl) w (THead (Bind Abst) u t0))) +(\lambda (x0: T).(\lambda (H10: (ty3 g c (THead (Bind Abst) u t0) +x0)).(ex3_2_ind T T (\lambda (t4: T).(\lambda (_: T).(pc3 c (THead (Bind +Abst) u t4) x0))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c u t5))) (\lambda +(t4: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t0 t4))) (ty3 g c +(THead (Flat Appl) x v) (THead (Flat Appl) w (THead (Bind Abst) u t0))) +(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 c (THead (Bind Abst) u +x1) x0)).(\lambda (_: (ty3 g c u x2)).(\lambda (H13: (ty3 g (CHead c (Bind +Abst) u) t0 x1)).(ex_ind T (\lambda (t4: T).(ty3 g c u t4)) (ty3 g c (THead +(Flat Appl) x v) (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda +(x3: T).(\lambda (H14: (ty3 g c u x3)).(ty3_conv g c (THead (Flat Appl) w +(THead (Bind Abst) u t0)) (THead (Flat Appl) w (THead (Bind Abst) u x1)) +(ty3_appl g c w u H0 (THead (Bind Abst) u t0) x1 (ty3_bind g c u x3 H14 Abst +t0 x1 H13)) (THead (Flat Appl) x v) (THead (Flat Appl) x (THead (Bind Abst) u +t0)) (ty3_appl g c x u (H1 i u0 c x (fsubst0_snd i u0 c w x H9) e H6) v t0 +H2) (pc3_fsubst0 c (THead (Flat Appl) w (THead (Bind Abst) u t0)) (THead +(Flat Appl) w (THead (Bind Abst) u t0)) (pc3_refl c (THead (Flat Appl) w +(THead (Bind Abst) u t0))) i u0 c (THead (Flat Appl) x (THead (Bind Abst) u +t0)) (fsubst0_snd i u0 c (THead (Flat Appl) w (THead (Bind Abst) u t0)) +(THead (Flat Appl) x (THead (Bind Abst) u t0)) (subst0_fst u0 x w i H9 (THead +(Bind Abst) u t0) (Flat Appl))) e H6)))) (ty3_correct g c x u (H1 i u0 c x +(fsubst0_snd i u0 c w x H9) e H6)))))))) (ty3_gen_bind g Abst c u t0 x0 +H10)))) (ty3_correct g c v (THead (Bind Abst) u t0) H2)) t3 H8)))) H7)) +(\lambda (H7: (ex2 T (\lambda (t4: T).(eq T t3 (THead (Flat Appl) w t4))) +(\lambda (t4: T).(subst0 (s (Flat Appl) i) u0 v t4)))).(ex2_ind T (\lambda +(t4: T).(eq T t3 (THead (Flat Appl) w t4))) (\lambda (t4: T).(subst0 (s (Flat +Appl) i) u0 v t4)) (ty3 g c t3 (THead (Flat Appl) w (THead (Bind Abst) u +t0))) (\lambda (x: T).(\lambda (H8: (eq T t3 (THead (Flat Appl) w +x))).(\lambda (H9: (subst0 (s (Flat Appl) i) u0 v x)).(eq_ind_r T (THead +(Flat Appl) w x) (\lambda (t4: T).(ty3 g c t4 (THead (Flat Appl) w (THead +(Bind Abst) u t0)))) (ty3_appl g c w u H0 x t0 (H3 (s (Flat Appl) i) u0 c x +(fsubst0_snd (s (Flat Appl) i) u0 c v x H9) e H6)) t3 H8)))) H7)) (\lambda +(H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) +u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 w u2))) (\lambda (_: +T).(\lambda (t4: T).(subst0 (s (Flat Appl) i) u0 v t4))))).(ex3_2_ind T T +(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) +(\lambda (u2: T).(\lambda (_: T).(subst0 i u0 w u2))) (\lambda (_: +T).(\lambda (t4: T).(subst0 (s (Flat Appl) i) u0 v t4))) (ty3 g c t3 (THead +(Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x0: T).(\lambda (x1: +T).(\lambda (H8: (eq T t3 (THead (Flat Appl) x0 x1))).(\lambda (H9: (subst0 i +u0 w x0)).(\lambda (H10: (subst0 (s (Flat Appl) i) u0 v x1)).(eq_ind_r T +(THead (Flat Appl) x0 x1) (\lambda (t4: T).(ty3 g c t4 (THead (Flat Appl) w +(THead (Bind Abst) u t0)))) (ex_ind T (\lambda (t4: T).(ty3 g c (THead (Bind +Abst) u t0) t4)) (ty3 g c (THead (Flat Appl) x0 x1) (THead (Flat Appl) w +(THead (Bind Abst) u t0))) (\lambda (x: T).(\lambda (H11: (ty3 g c (THead +(Bind Abst) u t0) x)).(ex3_2_ind T T (\lambda (t4: T).(\lambda (_: T).(pc3 c +(THead (Bind Abst) u t4) x))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c u +t5))) (\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t0 +t4))) (ty3 g c (THead (Flat Appl) x0 x1) (THead (Flat Appl) w (THead (Bind +Abst) u t0))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (_: (pc3 c (THead +(Bind Abst) u x2) x)).(\lambda (_: (ty3 g c u x3)).(\lambda (H14: (ty3 g +(CHead c (Bind Abst) u) t0 x2)).(ex_ind T (\lambda (t4: T).(ty3 g c u t4)) +(ty3 g c (THead (Flat Appl) x0 x1) (THead (Flat Appl) w (THead (Bind Abst) u +t0))) (\lambda (x4: T).(\lambda (H15: (ty3 g c u x4)).(ty3_conv g c (THead +(Flat Appl) w (THead (Bind Abst) u t0)) (THead (Flat Appl) w (THead (Bind +Abst) u x2)) (ty3_appl g c w u H0 (THead (Bind Abst) u t0) x2 (ty3_bind g c u +x4 H15 Abst t0 x2 H14)) (THead (Flat Appl) x0 x1) (THead (Flat Appl) x0 +(THead (Bind Abst) u t0)) (ty3_appl g c x0 u (H1 i u0 c x0 (fsubst0_snd i u0 +c w x0 H9) e H6) x1 t0 (H3 (s (Flat Appl) i) u0 c x1 (fsubst0_snd (s (Flat +Appl) i) u0 c v x1 H10) e H6)) (pc3_fsubst0 c (THead (Flat Appl) w (THead +(Bind Abst) u t0)) (THead (Flat Appl) w (THead (Bind Abst) u t0)) (pc3_refl c +(THead (Flat Appl) w (THead (Bind Abst) u t0))) i u0 c (THead (Flat Appl) x0 +(THead (Bind Abst) u t0)) (fsubst0_snd i u0 c (THead (Flat Appl) w (THead +(Bind Abst) u t0)) (THead (Flat Appl) x0 (THead (Bind Abst) u t0)) +(subst0_fst u0 x0 w i H9 (THead (Bind Abst) u t0) (Flat Appl))) e H6)))) +(ty3_correct g c w u H0))))))) (ty3_gen_bind g Abst c u t0 x H11)))) +(ty3_correct g c v (THead (Bind Abst) u t0) H2)) t3 H8)))))) H7)) +(subst0_gen_head (Flat Appl) u0 w v t3 i H5)))))) (\lambda (c3: C).(\lambda +(H5: (csubst0 i u0 c c3)).(\lambda (e: C).(\lambda (H6: (getl i c (CHead e +(Bind Abbr) u0))).(ty3_appl g c3 w u (H1 i u0 c3 w (fsubst0_fst i u0 c w c3 +H5) e H6) v t0 (H3 i u0 c3 v (fsubst0_fst i u0 c v c3 H5) e H6)))))) (\lambda +(t3: T).(\lambda (H5: (subst0 i u0 (THead (Flat Appl) w v) t3)).(\lambda (c3: +C).(\lambda (H6: (csubst0 i u0 c c3)).(\lambda (e: C).(\lambda (H7: (getl i c +(CHead e (Bind Abbr) u0))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t3 (THead +(Flat Appl) u2 v))) (\lambda (u2: T).(subst0 i u0 w u2))) (ex2 T (\lambda +(t4: T).(eq T t3 (THead (Flat Appl) w t4))) (\lambda (t4: T).(subst0 (s (Flat +Appl) i) u0 v t4))) (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 +(THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 w +u2))) (\lambda (_: T).(\lambda (t4: T).(subst0 (s (Flat Appl) i) u0 v t4)))) +(ty3 g c3 t3 (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (H8: +(ex2 T (\lambda (u2: T).(eq T t3 (THead (Flat Appl) u2 v))) (\lambda (u2: +T).(subst0 i u0 w u2)))).(ex2_ind T (\lambda (u2: T).(eq T t3 (THead (Flat +Appl) u2 v))) (\lambda (u2: T).(subst0 i u0 w u2)) (ty3 g c3 t3 (THead (Flat +Appl) w (THead (Bind Abst) u t0))) (\lambda (x: T).(\lambda (H9: (eq T t3 +(THead (Flat Appl) x v))).(\lambda (H10: (subst0 i u0 w x)).(eq_ind_r T +(THead (Flat Appl) x v) (\lambda (t4: T).(ty3 g c3 t4 (THead (Flat Appl) w +(THead (Bind Abst) u t0)))) (ex_ind T (\lambda (t4: T).(ty3 g c3 (THead (Bind +Abst) u t0) t4)) (ty3 g c3 (THead (Flat Appl) x v) (THead (Flat Appl) w +(THead (Bind Abst) u t0))) (\lambda (x0: T).(\lambda (H11: (ty3 g c3 (THead +(Bind Abst) u t0) x0)).(ex3_2_ind T T (\lambda (t4: T).(\lambda (_: T).(pc3 +c3 (THead (Bind Abst) u t4) x0))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c3 +u t5))) (\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead c3 (Bind Abst) u) t0 +t4))) (ty3 g c3 (THead (Flat Appl) x v) (THead (Flat Appl) w (THead (Bind +Abst) u t0))) (\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 c3 (THead +(Bind Abst) u x1) x0)).(\lambda (H13: (ty3 g c3 u x2)).(\lambda (H14: (ty3 g +(CHead c3 (Bind Abst) u) t0 x1)).(ty3_conv g c3 (THead (Flat Appl) w (THead +(Bind Abst) u t0)) (THead (Flat Appl) w (THead (Bind Abst) u x1)) (ty3_appl g +c3 w u (H1 i u0 c3 w (fsubst0_fst i u0 c w c3 H6) e H7) (THead (Bind Abst) u +t0) x1 (ty3_bind g c3 u x2 H13 Abst t0 x1 H14)) (THead (Flat Appl) x v) +(THead (Flat Appl) x (THead (Bind Abst) u t0)) (ty3_appl g c3 x u (H1 i u0 c3 +x (fsubst0_both i u0 c w x H10 c3 H6) e H7) v t0 (H3 i u0 c3 v (fsubst0_fst i +u0 c v c3 H6) e H7)) (pc3_fsubst0 c (THead (Flat Appl) w (THead (Bind Abst) u +t0)) (THead (Flat Appl) w (THead (Bind Abst) u t0)) (pc3_refl c (THead (Flat +Appl) w (THead (Bind Abst) u t0))) i u0 c3 (THead (Flat Appl) x (THead (Bind +Abst) u t0)) (fsubst0_both i u0 c (THead (Flat Appl) w (THead (Bind Abst) u +t0)) (THead (Flat Appl) x (THead (Bind Abst) u t0)) (subst0_fst u0 x w i H10 +(THead (Bind Abst) u t0) (Flat Appl)) c3 H6) e H7))))))) (ty3_gen_bind g Abst +c3 u t0 x0 H11)))) (ty3_correct g c3 v (THead (Bind Abst) u t0) (H3 i u0 c3 v +(fsubst0_fst i u0 c v c3 H6) e H7))) t3 H9)))) H8)) (\lambda (H8: (ex2 T +(\lambda (t4: T).(eq T t3 (THead (Flat Appl) w t4))) (\lambda (t4: T).(subst0 +(s (Flat Appl) i) u0 v t4)))).(ex2_ind T (\lambda (t4: T).(eq T t3 (THead +(Flat Appl) w t4))) (\lambda (t4: T).(subst0 (s (Flat Appl) i) u0 v t4)) (ty3 +g c3 t3 (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x: +T).(\lambda (H9: (eq T t3 (THead (Flat Appl) w x))).(\lambda (H10: (subst0 (s +(Flat Appl) i) u0 v x)).(eq_ind_r T (THead (Flat Appl) w x) (\lambda (t4: +T).(ty3 g c3 t4 (THead (Flat Appl) w (THead (Bind Abst) u t0)))) (ty3_appl g +c3 w u (H1 i u0 c3 w (fsubst0_fst i u0 c w c3 H6) e H7) x t0 (H3 i u0 c3 x +(fsubst0_both i u0 c v x H10 c3 H6) e H7)) t3 H9)))) H8)) (\lambda (H8: +(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 +t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 w u2))) (\lambda (_: +T).(\lambda (t4: T).(subst0 (s (Flat Appl) i) u0 v t4))))).(ex3_2_ind T T +(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) +(\lambda (u2: T).(\lambda (_: T).(subst0 i u0 w u2))) (\lambda (_: +T).(\lambda (t4: T).(subst0 (s (Flat Appl) i) u0 v t4))) (ty3 g c3 t3 (THead +(Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x0: T).(\lambda (x1: +T).(\lambda (H9: (eq T t3 (THead (Flat Appl) x0 x1))).(\lambda (H10: (subst0 +i u0 w x0)).(\lambda (H11: (subst0 (s (Flat Appl) i) u0 v x1)).(eq_ind_r T +(THead (Flat Appl) x0 x1) (\lambda (t4: T).(ty3 g c3 t4 (THead (Flat Appl) w +(THead (Bind Abst) u t0)))) (ex_ind T (\lambda (t4: T).(ty3 g c3 (THead (Bind +Abst) u t0) t4)) (ty3 g c3 (THead (Flat Appl) x0 x1) (THead (Flat Appl) w +(THead (Bind Abst) u t0))) (\lambda (x: T).(\lambda (H12: (ty3 g c3 (THead +(Bind Abst) u t0) x)).(ex3_2_ind T T (\lambda (t4: T).(\lambda (_: T).(pc3 c3 +(THead (Bind Abst) u t4) x))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c3 u +t5))) (\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead c3 (Bind Abst) u) t0 +t4))) (ty3 g c3 (THead (Flat Appl) x0 x1) (THead (Flat Appl) w (THead (Bind +Abst) u t0))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (_: (pc3 c3 (THead +(Bind Abst) u x2) x)).(\lambda (_: (ty3 g c3 u x3)).(\lambda (H15: (ty3 g +(CHead c3 (Bind Abst) u) t0 x2)).(ex_ind T (\lambda (t4: T).(ty3 g c3 u t4)) +(ty3 g c3 (THead (Flat Appl) x0 x1) (THead (Flat Appl) w (THead (Bind Abst) u +t0))) (\lambda (x4: T).(\lambda (H16: (ty3 g c3 u x4)).(ty3_conv g c3 (THead +(Flat Appl) w (THead (Bind Abst) u t0)) (THead (Flat Appl) w (THead (Bind +Abst) u x2)) (ty3_appl g c3 w u (H1 i u0 c3 w (fsubst0_fst i u0 c w c3 H6) e +H7) (THead (Bind Abst) u t0) x2 (ty3_bind g c3 u x4 H16 Abst t0 x2 H15)) +(THead (Flat Appl) x0 x1) (THead (Flat Appl) x0 (THead (Bind Abst) u t0)) +(ty3_appl g c3 x0 u (H1 i u0 c3 x0 (fsubst0_both i u0 c w x0 H10 c3 H6) e H7) +x1 t0 (H3 i u0 c3 x1 (fsubst0_both i u0 c v x1 H11 c3 H6) e H7)) (pc3_fsubst0 +c (THead (Flat Appl) w (THead (Bind Abst) u t0)) (THead (Flat Appl) w (THead +(Bind Abst) u t0)) (pc3_refl c (THead (Flat Appl) w (THead (Bind Abst) u +t0))) i u0 c3 (THead (Flat Appl) x0 (THead (Bind Abst) u t0)) (fsubst0_both i +u0 c (THead (Flat Appl) w (THead (Bind Abst) u t0)) (THead (Flat Appl) x0 +(THead (Bind Abst) u t0)) (subst0_fst u0 x0 w i H10 (THead (Bind Abst) u t0) +(Flat Appl)) c3 H6) e H7)))) (ty3_correct g c3 w u (H1 i u0 c3 w (fsubst0_fst +i u0 c w c3 H6) e H7)))))))) (ty3_gen_bind g Abst c3 u t0 x H12)))) +(ty3_correct g c3 v (THead (Bind Abst) u t0) (H3 i u0 c3 v (fsubst0_fst i u0 +c v c3 H6) e H7))) t3 H9)))))) H8)) (subst0_gen_head (Flat Appl) u0 w v t3 i +H5)))))))) c2 t2 H4))))))))))))))) (\lambda (c: C).(\lambda (t2: T).(\lambda +(t3: T).(\lambda (H0: (ty3 g c t2 t3)).(\lambda (H1: ((\forall (i: +nat).(\forall (u: T).(\forall (c2: C).(\forall (t4: T).((fsubst0 i u c t2 c2 +t4) \to (\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) \to (ty3 g c2 t4 +t3)))))))))).(\lambda (t0: T).(\lambda (H2: (ty3 g c t3 t0)).(\lambda (H3: +((\forall (i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t4: +T).((fsubst0 i u c t3 c2 t4) \to (\forall (e: C).((getl i c (CHead e (Bind +Abbr) u)) \to (ty3 g c2 t4 t0)))))))))).(\lambda (i: nat).(\lambda (u: +T).(\lambda (c2: C).(\lambda (t4: T).(\lambda (H4: (fsubst0 i u c (THead +(Flat Cast) t3 t2) c2 t4)).(fsubst0_ind i u c (THead (Flat Cast) t3 t2) +(\lambda (c0: C).(\lambda (t5: T).(\forall (e: C).((getl i c (CHead e (Bind +Abbr) u)) \to (ty3 g c0 t5 (THead (Flat Cast) t0 t3)))))) (\lambda (t5: +T).(\lambda (H5: (subst0 i u (THead (Flat Cast) t3 t2) t5)).(\lambda (e: +C).(\lambda (H6: (getl i c (CHead e (Bind Abbr) u))).(or3_ind (ex2 T (\lambda +(u2: T).(eq T t5 (THead (Flat Cast) u2 t2))) (\lambda (u2: T).(subst0 i u t3 +u2))) (ex2 T (\lambda (t6: T).(eq T t5 (THead (Flat Cast) t3 t6))) (\lambda +(t6: T).(subst0 (s (Flat Cast) i) u t2 t6))) (ex3_2 T T (\lambda (u2: +T).(\lambda (t6: T).(eq T t5 (THead (Flat Cast) u2 t6)))) (\lambda (u2: +T).(\lambda (_: T).(subst0 i u t3 u2))) (\lambda (_: T).(\lambda (t6: +T).(subst0 (s (Flat Cast) i) u t2 t6)))) (ty3 g c t5 (THead (Flat Cast) t0 +t3)) (\lambda (H7: (ex2 T (\lambda (u2: T).(eq T t5 (THead (Flat Cast) u2 +t2))) (\lambda (u2: T).(subst0 i u t3 u2)))).(ex2_ind T (\lambda (u2: T).(eq +T t5 (THead (Flat Cast) u2 t2))) (\lambda (u2: T).(subst0 i u t3 u2)) (ty3 g +c t5 (THead (Flat Cast) t0 t3)) (\lambda (x: T).(\lambda (H8: (eq T t5 (THead +(Flat Cast) x t2))).(\lambda (H9: (subst0 i u t3 x)).(eq_ind_r T (THead (Flat +Cast) x t2) (\lambda (t6: T).(ty3 g c t6 (THead (Flat Cast) t0 t3))) (ex_ind +T (\lambda (t6: T).(ty3 g c t0 t6)) (ty3 g c (THead (Flat Cast) x t2) (THead +(Flat Cast) t0 t3)) (\lambda (x0: T).(\lambda (H10: (ty3 g c t0 +x0)).(ty3_conv g c (THead (Flat Cast) t0 t3) (THead (Flat Cast) x0 t0) +(ty3_cast g c t3 t0 H2 x0 H10) (THead (Flat Cast) x t2) (THead (Flat Cast) t0 +x) (ty3_cast g c t2 x (ty3_conv g c x t0 (H3 i u c x (fsubst0_snd i u c t3 x +H9) e H6) t2 t3 H0 (pc3_s c t3 x (pc3_fsubst0 c t3 t3 (pc3_refl c t3) i u c x +(fsubst0_snd i u c t3 x H9) e H6))) t0 (H3 i u c x (fsubst0_snd i u c t3 x +H9) e H6)) (pc3_fsubst0 c (THead (Flat Cast) t0 t3) (THead (Flat Cast) t0 t3) +(pc3_refl c (THead (Flat Cast) t0 t3)) i u c (THead (Flat Cast) t0 x) +(fsubst0_snd i u c (THead (Flat Cast) t0 t3) (THead (Flat Cast) t0 x) +(subst0_snd (Flat Cast) u x t3 i H9 t0)) e H6)))) (ty3_correct g c x t0 (H3 i +u c x (fsubst0_snd i u c t3 x H9) e H6))) t5 H8)))) H7)) (\lambda (H7: (ex2 T +(\lambda (t6: T).(eq T t5 (THead (Flat Cast) t3 t6))) (\lambda (t6: +T).(subst0 (s (Flat Cast) i) u t2 t6)))).(ex2_ind T (\lambda (t6: T).(eq T t5 +(THead (Flat Cast) t3 t6))) (\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 +t6)) (ty3 g c t5 (THead (Flat Cast) t0 t3)) (\lambda (x: T).(\lambda (H8: (eq +T t5 (THead (Flat Cast) t3 x))).(\lambda (H9: (subst0 (s (Flat Cast) i) u t2 +x)).(eq_ind_r T (THead (Flat Cast) t3 x) (\lambda (t6: T).(ty3 g c t6 (THead +(Flat Cast) t0 t3))) (ty3_cast g c x t3 (H1 (s (Flat Cast) i) u c x +(fsubst0_snd (s (Flat Cast) i) u c t2 x H9) e H6) t0 H2) t5 H8)))) H7)) +(\lambda (H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead +(Flat Cast) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u t3 u2))) +(\lambda (_: T).(\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 +t6))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead +(Flat Cast) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u t3 u2))) +(\lambda (_: T).(\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 t6))) (ty3 g +c t5 (THead (Flat Cast) t0 t3)) (\lambda (x0: T).(\lambda (x1: T).(\lambda +(H8: (eq T t5 (THead (Flat Cast) x0 x1))).(\lambda (H9: (subst0 i u t3 +x0)).(\lambda (H10: (subst0 (s (Flat Cast) i) u t2 x1)).(eq_ind_r T (THead +(Flat Cast) x0 x1) (\lambda (t6: T).(ty3 g c t6 (THead (Flat Cast) t0 t3))) +(ex_ind T (\lambda (t6: T).(ty3 g c t0 t6)) (ty3 g c (THead (Flat Cast) x0 +x1) (THead (Flat Cast) t0 t3)) (\lambda (x: T).(\lambda (H11: (ty3 g c t0 +x)).(ty3_conv g c (THead (Flat Cast) t0 t3) (THead (Flat Cast) x t0) +(ty3_cast g c t3 t0 H2 x H11) (THead (Flat Cast) x0 x1) (THead (Flat Cast) t0 +x0) (ty3_cast g c x1 x0 (ty3_conv g c x0 t0 (H3 i u c x0 (fsubst0_snd i u c +t3 x0 H9) e H6) x1 t3 (H1 (s (Flat Cast) i) u c x1 (fsubst0_snd (s (Flat +Cast) i) u c t2 x1 H10) e H6) (pc3_s c t3 x0 (pc3_fsubst0 c t3 t3 (pc3_refl c +t3) i u c x0 (fsubst0_snd i u c t3 x0 H9) e H6))) t0 (H3 i u c x0 +(fsubst0_snd i u c t3 x0 H9) e H6)) (pc3_fsubst0 c (THead (Flat Cast) t0 t3) +(THead (Flat Cast) t0 t3) (pc3_refl c (THead (Flat Cast) t0 t3)) i u c (THead +(Flat Cast) t0 x0) (fsubst0_snd i u c (THead (Flat Cast) t0 t3) (THead (Flat +Cast) t0 x0) (subst0_snd (Flat Cast) u x0 t3 i H9 t0)) e H6)))) (ty3_correct +g c x0 t0 (H3 i u c x0 (fsubst0_snd i u c t3 x0 H9) e H6))) t5 H8)))))) H7)) +(subst0_gen_head (Flat Cast) u t3 t2 t5 i H5)))))) (\lambda (c3: C).(\lambda +(H5: (csubst0 i u c c3)).(\lambda (e: C).(\lambda (H6: (getl i c (CHead e +(Bind Abbr) u))).(ty3_cast g c3 t2 t3 (H1 i u c3 t2 (fsubst0_fst i u c t2 c3 +H5) e H6) t0 (H3 i u c3 t3 (fsubst0_fst i u c t3 c3 H5) e H6)))))) (\lambda +(t5: T).(\lambda (H5: (subst0 i u (THead (Flat Cast) t3 t2) t5)).(\lambda +(c3: C).(\lambda (H6: (csubst0 i u c c3)).(\lambda (e: C).(\lambda (H7: (getl +i c (CHead e (Bind Abbr) u))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t5 +(THead (Flat Cast) u2 t2))) (\lambda (u2: T).(subst0 i u t3 u2))) (ex2 T +(\lambda (t6: T).(eq T t5 (THead (Flat Cast) t3 t6))) (\lambda (t6: +T).(subst0 (s (Flat Cast) i) u t2 t6))) (ex3_2 T T (\lambda (u2: T).(\lambda +(t6: T).(eq T t5 (THead (Flat Cast) u2 t6)))) (\lambda (u2: T).(\lambda (_: +T).(subst0 i u t3 u2))) (\lambda (_: T).(\lambda (t6: T).(subst0 (s (Flat +Cast) i) u t2 t6)))) (ty3 g c3 t5 (THead (Flat Cast) t0 t3)) (\lambda (H8: +(ex2 T (\lambda (u2: T).(eq T t5 (THead (Flat Cast) u2 t2))) (\lambda (u2: +T).(subst0 i u t3 u2)))).(ex2_ind T (\lambda (u2: T).(eq T t5 (THead (Flat +Cast) u2 t2))) (\lambda (u2: T).(subst0 i u t3 u2)) (ty3 g c3 t5 (THead (Flat +Cast) t0 t3)) (\lambda (x: T).(\lambda (H9: (eq T t5 (THead (Flat Cast) x +t2))).(\lambda (H10: (subst0 i u t3 x)).(eq_ind_r T (THead (Flat Cast) x t2) +(\lambda (t6: T).(ty3 g c3 t6 (THead (Flat Cast) t0 t3))) (ex_ind T (\lambda +(t6: T).(ty3 g c3 t0 t6)) (ty3 g c3 (THead (Flat Cast) x t2) (THead (Flat +Cast) t0 t3)) (\lambda (x0: T).(\lambda (H11: (ty3 g c3 t0 x0)).(ty3_conv g +c3 (THead (Flat Cast) t0 t3) (THead (Flat Cast) x0 t0) (ty3_cast g c3 t3 t0 +(H3 i u c3 t3 (fsubst0_fst i u c t3 c3 H6) e H7) x0 H11) (THead (Flat Cast) x +t2) (THead (Flat Cast) t0 x) (ty3_cast g c3 t2 x (ty3_conv g c3 x t0 (H3 i u +c3 x (fsubst0_both i u c t3 x H10 c3 H6) e H7) t2 t3 (H1 i u c3 t2 +(fsubst0_fst i u c t2 c3 H6) e H7) (pc3_s c3 t3 x (pc3_fsubst0 c t3 t3 +(pc3_refl c t3) i u c3 x (fsubst0_both i u c t3 x H10 c3 H6) e H7))) t0 (H3 i +u c3 x (fsubst0_both i u c t3 x H10 c3 H6) e H7)) (pc3_fsubst0 c (THead (Flat +Cast) t0 t3) (THead (Flat Cast) t0 t3) (pc3_refl c (THead (Flat Cast) t0 t3)) +i u c3 (THead (Flat Cast) t0 x) (fsubst0_both i u c (THead (Flat Cast) t0 t3) +(THead (Flat Cast) t0 x) (subst0_snd (Flat Cast) u x t3 i H10 t0) c3 H6) e +H7)))) (ty3_correct g c3 t3 t0 (H3 i u c3 t3 (fsubst0_fst i u c t3 c3 H6) e +H7))) t5 H9)))) H8)) (\lambda (H8: (ex2 T (\lambda (t6: T).(eq T t5 (THead +(Flat Cast) t3 t6))) (\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 +t6)))).(ex2_ind T (\lambda (t6: T).(eq T t5 (THead (Flat Cast) t3 t6))) +(\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 t6)) (ty3 g c3 t5 (THead +(Flat Cast) t0 t3)) (\lambda (x: T).(\lambda (H9: (eq T t5 (THead (Flat Cast) +t3 x))).(\lambda (H10: (subst0 (s (Flat Cast) i) u t2 x)).(eq_ind_r T (THead +(Flat Cast) t3 x) (\lambda (t6: T).(ty3 g c3 t6 (THead (Flat Cast) t0 t3))) +(ty3_cast g c3 x t3 (H1 i u c3 x (fsubst0_both i u c t2 x H10 c3 H6) e H7) t0 +(H3 i u c3 t3 (fsubst0_fst i u c t3 c3 H6) e H7)) t5 H9)))) H8)) (\lambda +(H8: (ex3_2 T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead (Flat Cast) +u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u t3 u2))) (\lambda (_: +T).(\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 t6))))).(ex3_2_ind T T +(\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead (Flat Cast) u2 t6)))) +(\lambda (u2: T).(\lambda (_: T).(subst0 i u t3 u2))) (\lambda (_: +T).(\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 t6))) (ty3 g c3 t5 (THead +(Flat Cast) t0 t3)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H9: (eq T t5 +(THead (Flat Cast) x0 x1))).(\lambda (H10: (subst0 i u t3 x0)).(\lambda (H11: +(subst0 (s (Flat Cast) i) u t2 x1)).(eq_ind_r T (THead (Flat Cast) x0 x1) +(\lambda (t6: T).(ty3 g c3 t6 (THead (Flat Cast) t0 t3))) (ex_ind T (\lambda +(t6: T).(ty3 g c3 t0 t6)) (ty3 g c3 (THead (Flat Cast) x0 x1) (THead (Flat +Cast) t0 t3)) (\lambda (x: T).(\lambda (H12: (ty3 g c3 t0 x)).(ty3_conv g c3 +(THead (Flat Cast) t0 t3) (THead (Flat Cast) x t0) (ty3_cast g c3 t3 t0 (H3 i +u c3 t3 (fsubst0_fst i u c t3 c3 H6) e H7) x H12) (THead (Flat Cast) x0 x1) +(THead (Flat Cast) t0 x0) (ty3_cast g c3 x1 x0 (ty3_conv g c3 x0 t0 (H3 i u +c3 x0 (fsubst0_both i u c t3 x0 H10 c3 H6) e H7) x1 t3 (H1 i u c3 x1 +(fsubst0_both i u c t2 x1 H11 c3 H6) e H7) (pc3_s c3 t3 x0 (pc3_fsubst0 c t3 +t3 (pc3_refl c t3) i u c3 x0 (fsubst0_both i u c t3 x0 H10 c3 H6) e H7))) t0 +(H3 i u c3 x0 (fsubst0_both i u c t3 x0 H10 c3 H6) e H7)) (pc3_fsubst0 c +(THead (Flat Cast) t0 t3) (THead (Flat Cast) t0 t3) (pc3_refl c (THead (Flat +Cast) t0 t3)) i u c3 (THead (Flat Cast) t0 x0) (fsubst0_both i u c (THead +(Flat Cast) t0 t3) (THead (Flat Cast) t0 x0) (subst0_snd (Flat Cast) u x0 t3 +i H10 t0) c3 H6) e H7)))) (ty3_correct g c3 t3 t0 (H3 i u c3 t3 (fsubst0_fst +i u c t3 c3 H6) e H7))) t5 H9)))))) H8)) (subst0_gen_head (Flat Cast) u t3 t2 +t5 i H5)))))))) c2 t4 H4)))))))))))))) c1 t1 t H))))). +(* COMMENTS +Initial nodes: 23439 +END *) + +theorem ty3_csubst0: + \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c1 +t1 t2) \to (\forall (e: C).(\forall (u: T).(\forall (i: nat).((getl i c1 +(CHead e (Bind Abbr) u)) \to (\forall (c2: C).((csubst0 i u c1 c2) \to (ty3 g +c2 t1 t2))))))))))) +\def + \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda +(H: (ty3 g c1 t1 t2)).(\lambda (e: C).(\lambda (u: T).(\lambda (i: +nat).(\lambda (H0: (getl i c1 (CHead e (Bind Abbr) u))).(\lambda (c2: +C).(\lambda (H1: (csubst0 i u c1 c2)).(ty3_fsubst0 g c1 t1 t2 H i u c2 t1 +(fsubst0_fst i u c1 t1 c2 H1) e H0))))))))))). +(* COMMENTS +Initial nodes: 89 +END *) + +theorem ty3_subst0: + \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((ty3 g c t1 +t) \to (\forall (e: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead e +(Bind Abbr) u)) \to (\forall (t2: T).((subst0 i u t1 t2) \to (ty3 g c t2 +t))))))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H: +(ty3 g c t1 t)).(\lambda (e: C).(\lambda (u: T).(\lambda (i: nat).(\lambda +(H0: (getl i c (CHead e (Bind Abbr) u))).(\lambda (t2: T).(\lambda (H1: +(subst0 i u t1 t2)).(ty3_fsubst0 g c t1 t H i u c t2 (fsubst0_snd i u c t1 t2 +H1) e H0))))))))))). +(* COMMENTS +Initial nodes: 89 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/ty3/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/ty3/fwd.ma new file mode 100644 index 000000000..bf6634e45 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/ty3/fwd.ma @@ -0,0 +1,922 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/ty3/defs.ma". + +include "Basic-1/pc3/props.ma". + +theorem ty3_gen_sort: + \forall (g: G).(\forall (c: C).(\forall (x: T).(\forall (n: nat).((ty3 g c +(TSort n) x) \to (pc3 c (TSort (next g n)) x))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda +(H: (ty3 g c (TSort n) x)).(insert_eq T (TSort n) (\lambda (t: T).(ty3 g c t +x)) (\lambda (_: T).(pc3 c (TSort (next g n)) x)) (\lambda (y: T).(\lambda +(H0: (ty3 g c y x)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: +T).((eq T t (TSort n)) \to (pc3 c0 (TSort (next g n)) t0))))) (\lambda (c0: +C).(\lambda (t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda +(_: (((eq T t2 (TSort n)) \to (pc3 c0 (TSort (next g n)) t)))).(\lambda (u: +T).(\lambda (t1: T).(\lambda (H3: (ty3 g c0 u t1)).(\lambda (H4: (((eq T u +(TSort n)) \to (pc3 c0 (TSort (next g n)) t1)))).(\lambda (H5: (pc3 c0 t1 +t2)).(\lambda (H6: (eq T u (TSort n))).(let H7 \def (f_equal T T (\lambda (e: +T).e) u (TSort n) H6) in (let H8 \def (eq_ind T u (\lambda (t0: T).((eq T t0 +(TSort n)) \to (pc3 c0 (TSort (next g n)) t1))) H4 (TSort n) H7) in (let H9 +\def (eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 t1)) H3 (TSort n) H7) in +(pc3_t t1 c0 (TSort (next g n)) (H8 (refl_equal T (TSort n))) t2 +H5))))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq T +(TSort m) (TSort n))).(let H2 \def (f_equal T nat (\lambda (e: T).(match e in +T return (\lambda (_: T).nat) with [(TSort n0) \Rightarrow n0 | (TLRef _) +\Rightarrow m | (THead _ _ _) \Rightarrow m])) (TSort m) (TSort n) H1) in +(eq_ind_r nat n (\lambda (n0: nat).(pc3 c0 (TSort (next g n)) (TSort (next g +n0)))) (pc3_refl c0 (TSort (next g n))) m H2))))) (\lambda (n0: nat).(\lambda +(c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (_: (getl n0 c0 (CHead d +(Bind Abbr) u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (_: +(((eq T u (TSort n)) \to (pc3 d (TSort (next g n)) t)))).(\lambda (H4: (eq T +(TLRef n0) (TSort n))).(let H5 \def (eq_ind T (TLRef n0) (\lambda (ee: +T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow +False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I +(TSort n) H4) in (False_ind (pc3 c0 (TSort (next g n)) (lift (S n0) O t)) +H5))))))))))) (\lambda (n0: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda +(u: T).(\lambda (_: (getl n0 c0 (CHead d (Bind Abst) u))).(\lambda (t: +T).(\lambda (_: (ty3 g d u t)).(\lambda (_: (((eq T u (TSort n)) \to (pc3 d +(TSort (next g n)) t)))).(\lambda (H4: (eq T (TLRef n0) (TSort n))).(let H5 +\def (eq_ind T (TLRef n0) (\lambda (ee: T).(match ee in T return (\lambda (_: +T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | +(THead _ _ _) \Rightarrow False])) I (TSort n) H4) in (False_ind (pc3 c0 +(TSort (next g n)) (lift (S n0) O u)) H5))))))))))) (\lambda (c0: C).(\lambda +(u: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((eq T u +(TSort n)) \to (pc3 c0 (TSort (next g n)) t)))).(\lambda (b: B).(\lambda (t1: +T).(\lambda (t2: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t1 +t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (pc3 (CHead c0 (Bind b) u) (TSort +(next g n)) t2)))).(\lambda (H5: (eq T (THead (Bind b) u t1) (TSort n))).(let +H6 \def (eq_ind T (THead (Bind b) u t1) (\lambda (ee: T).(match ee in T +return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H5) in +(False_ind (pc3 c0 (TSort (next g n)) (THead (Bind b) u t2)) H6))))))))))))) +(\lambda (c0: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w +u)).(\lambda (_: (((eq T w (TSort n)) \to (pc3 c0 (TSort (next g n)) +u)))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead (Bind +Abst) u t))).(\lambda (_: (((eq T v (TSort n)) \to (pc3 c0 (TSort (next g n)) +(THead (Bind Abst) u t))))).(\lambda (H5: (eq T (THead (Flat Appl) w v) +(TSort n))).(let H6 \def (eq_ind T (THead (Flat Appl) w v) (\lambda (ee: +T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow +False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I +(TSort n) H5) in (False_ind (pc3 c0 (TSort (next g n)) (THead (Flat Appl) w +(THead (Bind Abst) u t))) H6)))))))))))) (\lambda (c0: C).(\lambda (t1: +T).(\lambda (t2: T).(\lambda (_: (ty3 g c0 t1 t2)).(\lambda (_: (((eq T t1 +(TSort n)) \to (pc3 c0 (TSort (next g n)) t2)))).(\lambda (t0: T).(\lambda +(_: (ty3 g c0 t2 t0)).(\lambda (_: (((eq T t2 (TSort n)) \to (pc3 c0 (TSort +(next g n)) t0)))).(\lambda (H5: (eq T (THead (Flat Cast) t2 t1) (TSort +n))).(let H6 \def (eq_ind T (THead (Flat Cast) t2 t1) (\lambda (ee: T).(match +ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | +(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) +H5) in (False_ind (pc3 c0 (TSort (next g n)) (THead (Flat Cast) t0 t2)) +H6))))))))))) c y x H0))) H))))). +(* COMMENTS +Initial nodes: 1179 +END *) + +theorem ty3_gen_lref: + \forall (g: G).(\forall (c: C).(\forall (x: T).(\forall (n: nat).((ty3 g c +(TLRef n) x) \to (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda +(t: T).(pc3 c (lift (S n) O t) x)))) (\lambda (e: C).(\lambda (u: T).(\lambda +(_: T).(getl n c (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: +T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda +(u: T).(\lambda (_: T).(pc3 c (lift (S n) O u) x)))) (\lambda (e: C).(\lambda +(u: T).(\lambda (_: T).(getl n c (CHead e (Bind Abst) u))))) (\lambda (e: +C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda +(H: (ty3 g c (TLRef n) x)).(insert_eq T (TLRef n) (\lambda (t: T).(ty3 g c t +x)) (\lambda (_: T).(or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda +(t0: T).(pc3 c (lift (S n) O t0) x)))) (\lambda (e: C).(\lambda (u: +T).(\lambda (_: T).(getl n c (CHead e (Bind Abbr) u))))) (\lambda (e: +C).(\lambda (u: T).(\lambda (t0: T).(ty3 g e u t0))))) (ex3_3 C T T (\lambda +(_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c (lift (S n) O u) x)))) (\lambda +(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead e (Bind Abst) u))))) +(\lambda (e: C).(\lambda (u: T).(\lambda (t0: T).(ty3 g e u t0))))))) +(\lambda (y: T).(\lambda (H0: (ty3 g c y x)).(ty3_ind g (\lambda (c0: +C).(\lambda (t: T).(\lambda (t0: T).((eq T t (TLRef n)) \to (or (ex3_3 C T T +(\lambda (_: C).(\lambda (_: T).(\lambda (t1: T).(pc3 c0 (lift (S n) O t1) +t0)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e +(Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t1: T).(ty3 g e +u t1))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 +c0 (lift (S n) O u) t0)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: +T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: +T).(\lambda (t1: T).(ty3 g e u t1)))))))))) (\lambda (c0: C).(\lambda (t2: +T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda (_: (((eq T t2 +(TLRef n)) \to (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: +T).(pc3 c0 (lift (S n) O t0) t)))) (\lambda (e: C).(\lambda (u: T).(\lambda +(_: T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: +T).(\lambda (t0: T).(ty3 g e u t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda +(u: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u) t)))) (\lambda (e: +C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) +(\lambda (e: C).(\lambda (u: T).(\lambda (t0: T).(ty3 g e u +t0))))))))).(\lambda (u: T).(\lambda (t1: T).(\lambda (H3: (ty3 g c0 u +t1)).(\lambda (H4: (((eq T u (TLRef n)) \to (or (ex3_3 C T T (\lambda (_: +C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t1)))) (\lambda +(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) +u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) +(ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift +(S n) O u0) t1)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n +c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda +(t0: T).(ty3 g e u0 t0))))))))).(\lambda (H5: (pc3 c0 t1 t2)).(\lambda (H6: +(eq T u (TLRef n))).(let H7 \def (f_equal T T (\lambda (e: T).e) u (TLRef n) +H6) in (let H8 \def (eq_ind T u (\lambda (t0: T).((eq T t0 (TLRef n)) \to (or +(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t3: T).(pc3 c0 (lift +(S n) O t3) t1)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n +c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda +(t3: T).(ty3 g e u0 t3))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: +T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) t1)))) (\lambda (e: C).(\lambda +(u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: +C).(\lambda (u0: T).(\lambda (t3: T).(ty3 g e u0 t3)))))))) H4 (TLRef n) H7) +in (let H9 \def (eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 t1)) H3 (TLRef n) +H7) in (let H10 \def (H8 (refl_equal T (TLRef n))) in (or_ind (ex3_3 C T T +(\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) +t1)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e +(Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g +e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: +T).(pc3 c0 (lift (S n) O u0) t1)))) (\lambda (e: C).(\lambda (u0: T).(\lambda +(_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: +T).(\lambda (t0: T).(ty3 g e u0 t0))))) (or (ex3_3 C T T (\lambda (_: +C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda +(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) +u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) +(ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift +(S n) O u0) t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n +c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda +(t0: T).(ty3 g e u0 t0)))))) (\lambda (H11: (ex3_3 C T T (\lambda (_: +C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t1)))) (\lambda +(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) +u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 +t0)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: +T).(pc3 c0 (lift (S n) O t0) t1)))) (\lambda (e: C).(\lambda (u0: T).(\lambda +(_: T).(getl n c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: +T).(\lambda (t0: T).(ty3 g e u0 t0)))) (or (ex3_3 C T T (\lambda (_: +C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda +(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) +u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) +(ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift +(S n) O u0) t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n +c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda +(t0: T).(ty3 g e u0 t0)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: +T).(\lambda (H12: (pc3 c0 (lift (S n) O x2) t1)).(\lambda (H13: (getl n c0 +(CHead x0 (Bind Abbr) x1))).(\lambda (H14: (ty3 g x0 x1 x2)).(or_introl +(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift +(S n) O t0) t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n +c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda +(t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: +T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) t2)))) (\lambda (e: C).(\lambda +(u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: +C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3_intro C T T +(\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) +t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e +(Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g +e u0 t0)))) x0 x1 x2 (pc3_t t1 c0 (lift (S n) O x2) H12 t2 H5) H13 +H14)))))))) H11)) (\lambda (H11: (ex3_3 C T T (\lambda (_: C).(\lambda (u0: +T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) t1)))) (\lambda (e: C).(\lambda +(u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: +C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))))).(ex3_3_ind C T T +(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) +t1)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e +(Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g +e u0 t0)))) (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: +T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda +(_: T).(getl n c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: +T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda +(u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) t2)))) (\lambda (e: +C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) +(\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))))) +(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H12: (pc3 c0 +(lift (S n) O x1) t1)).(\lambda (H13: (getl n c0 (CHead x0 (Bind Abst) +x1))).(\lambda (H14: (ty3 g x0 x1 x2)).(or_intror (ex3_3 C T T (\lambda (_: +C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda +(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) +u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) +(ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift +(S n) O u0) t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n +c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda +(t0: T).(ty3 g e u0 t0))))) (ex3_3_intro C T T (\lambda (_: C).(\lambda (u0: +T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) t2)))) (\lambda (e: C).(\lambda +(u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: +C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))) x0 x1 x2 (pc3_t t1 c0 +(lift (S n) O x1) H12 t2 H5) H13 H14)))))))) H11)) H10)))))))))))))))) +(\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq T (TSort m) (TLRef +n))).(let H2 \def (eq_ind T (TSort m) (\lambda (ee: T).(match ee in T return +(\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) +\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef n) H1) in +(False_ind (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: +T).(pc3 c0 (lift (S n) O t) (TSort (next g m)))))) (\lambda (e: C).(\lambda +(u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: +C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T T (\lambda +(_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u) (TSort (next +g m)))))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e +(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u +t)))))) H2))))) (\lambda (n0: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda +(u: T).(\lambda (H1: (getl n0 c0 (CHead d (Bind Abbr) u))).(\lambda (t: +T).(\lambda (H2: (ty3 g d u t)).(\lambda (_: (((eq T u (TLRef n)) \to (or +(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 d (lift (S +n) O t0) t)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n d +(CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: +T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda +(_: T).(pc3 d (lift (S n) O u0) t)))) (\lambda (e: C).(\lambda (u0: +T).(\lambda (_: T).(getl n d (CHead e (Bind Abst) u0))))) (\lambda (e: +C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))))))).(\lambda (H4: +(eq T (TLRef n0) (TLRef n))).(let H5 \def (f_equal T nat (\lambda (e: +T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow n0 | +(TLRef n1) \Rightarrow n1 | (THead _ _ _) \Rightarrow n0])) (TLRef n0) (TLRef +n) H4) in (let H6 \def (eq_ind nat n0 (\lambda (n1: nat).(getl n1 c0 (CHead d +(Bind Abbr) u))) H1 n H5) in (eq_ind_r nat n (\lambda (n1: nat).(or (ex3_3 C +T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O +t0) (lift (S n1) O t))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: +T).(getl n c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: +T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda +(u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) (lift (S n1) O t))))) +(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind +Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 +t0))))))) (or_introl (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda +(t0: T).(pc3 c0 (lift (S n) O t0) (lift (S n) O t))))) (\lambda (e: +C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u0))))) +(\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3 +C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O +u0) (lift (S n) O t))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: +T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: +T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3_intro C T T (\lambda (_: +C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) (lift (S n) O +t))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e +(Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g +e u0 t0)))) d u t (pc3_refl c0 (lift (S n) O t)) H6 H2)) n0 H5)))))))))))) +(\lambda (n0: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda +(H1: (getl n0 c0 (CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (H2: (ty3 +g d u t)).(\lambda (_: (((eq T u (TLRef n)) \to (or (ex3_3 C T T (\lambda (_: +C).(\lambda (_: T).(\lambda (t0: T).(pc3 d (lift (S n) O t0) t)))) (\lambda +(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n d (CHead e (Bind Abbr) +u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) +(ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 d (lift (S +n) O u0) t)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n d +(CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: +T).(ty3 g e u0 t0))))))))).(\lambda (H4: (eq T (TLRef n0) (TLRef n))).(let H5 +\def (f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) +with [(TSort _) \Rightarrow n0 | (TLRef n1) \Rightarrow n1 | (THead _ _ _) +\Rightarrow n0])) (TLRef n0) (TLRef n) H4) in (let H6 \def (eq_ind nat n0 +(\lambda (n1: nat).(getl n1 c0 (CHead d (Bind Abst) u))) H1 n H5) in +(eq_ind_r nat n (\lambda (n1: nat).(or (ex3_3 C T T (\lambda (_: C).(\lambda +(_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) (lift (S n1) O u))))) +(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind +Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 +t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 +(lift (S n) O u0) (lift (S n1) O u))))) (\lambda (e: C).(\lambda (u0: +T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: +C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))))) (or_intror (ex3_3 +C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O +t0) (lift (S n) O u))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: +T).(getl n c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: +T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda +(u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) (lift (S n) O u))))) +(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind +Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 +t0))))) (ex3_3_intro C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: +T).(pc3 c0 (lift (S n) O u0) (lift (S n) O u))))) (\lambda (e: C).(\lambda +(u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: +C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))) d u t (pc3_refl c0 +(lift (S n) O u)) H6 H2)) n0 H5)))))))))))) (\lambda (c0: C).(\lambda (u: +T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((eq T u (TLRef +n)) \to (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: +T).(pc3 c0 (lift (S n) O t0) t)))) (\lambda (e: C).(\lambda (u0: T).(\lambda +(_: T).(getl n c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: +T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda +(u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) t)))) (\lambda (e: +C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) +(\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 +t0))))))))).(\lambda (b: B).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: +(ty3 g (CHead c0 (Bind b) u) t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to +(or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 (CHead +c0 (Bind b) u) (lift (S n) O t0) t2)))) (\lambda (e: C).(\lambda (u0: +T).(\lambda (_: T).(getl n (CHead c0 (Bind b) u) (CHead e (Bind Abbr) u0))))) +(\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3 +C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 (CHead c0 (Bind +b) u) (lift (S n) O u0) t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: +T).(getl n (CHead c0 (Bind b) u) (CHead e (Bind Abst) u0))))) (\lambda (e: +C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))))))).(\lambda (H5: +(eq T (THead (Bind b) u t1) (TLRef n))).(let H6 \def (eq_ind T (THead (Bind +b) u t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with +[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) +\Rightarrow True])) I (TLRef n) H5) in (False_ind (or (ex3_3 C T T (\lambda +(_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) (THead +(Bind b) u t2))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n +c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda +(t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: +T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) (THead (Bind b) u t2))))) +(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind +Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 +t0)))))) H6))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda (u: +T).(\lambda (_: (ty3 g c0 w u)).(\lambda (_: (((eq T w (TLRef n)) \to (or +(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c0 (lift (S +n) O t) u)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 +(CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t: +T).(ty3 g e u0 t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda +(_: T).(pc3 c0 (lift (S n) O u0) u)))) (\lambda (e: C).(\lambda (u0: +T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: +C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t))))))))).(\lambda (v: +T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead (Bind Abst) u +t))).(\lambda (_: (((eq T v (TLRef n)) \to (or (ex3_3 C T T (\lambda (_: +C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) (THead (Bind +Abst) u t))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 +(CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: +T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda +(_: T).(pc3 c0 (lift (S n) O u0) (THead (Bind Abst) u t))))) (\lambda (e: +C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) +(\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 +t0))))))))).(\lambda (H5: (eq T (THead (Flat Appl) w v) (TLRef n))).(let H6 +\def (eq_ind T (THead (Flat Appl) w v) (\lambda (ee: T).(match ee in T return +(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H5) in +(False_ind (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: +T).(pc3 c0 (lift (S n) O t0) (THead (Flat Appl) w (THead (Bind Abst) u +t)))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e +(Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g +e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: +T).(pc3 c0 (lift (S n) O u0) (THead (Flat Appl) w (THead (Bind Abst) u +t)))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e +(Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g +e u0 t0)))))) H6)))))))))))) (\lambda (c0: C).(\lambda (t1: T).(\lambda (t2: +T).(\lambda (_: (ty3 g c0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (or +(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c0 (lift (S +n) O t) t2)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 +(CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: +T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda +(_: T).(pc3 c0 (lift (S n) O u) t2)))) (\lambda (e: C).(\lambda (u: +T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: +C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))))))).(\lambda (t0: +T).(\lambda (_: (ty3 g c0 t2 t0)).(\lambda (_: (((eq T t2 (TLRef n)) \to (or +(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c0 (lift (S +n) O t) t0)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 +(CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: +T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda +(_: T).(pc3 c0 (lift (S n) O u) t0)))) (\lambda (e: C).(\lambda (u: +T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: +C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))))))).(\lambda (H5: (eq T +(THead (Flat Cast) t2 t1) (TLRef n))).(let H6 \def (eq_ind T (THead (Flat +Cast) t2 t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) +with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ +_) \Rightarrow True])) I (TLRef n) H5) in (False_ind (or (ex3_3 C T T +(\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c0 (lift (S n) O t) +(THead (Flat Cast) t0 t2))))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: +T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: +T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda +(u: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u) (THead (Flat Cast) t0 t2))))) +(\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind +Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u +t)))))) H6))))))))))) c y x H0))) H))))). +(* COMMENTS +Initial nodes: 5569 +END *) + +theorem ty3_gen_bind: + \forall (g: G).(\forall (b: B).(\forall (c: C).(\forall (u: T).(\forall (t1: +T).(\forall (x: T).((ty3 g c (THead (Bind b) u t1) x) \to (ex3_2 T T (\lambda +(t2: T).(\lambda (_: T).(pc3 c (THead (Bind b) u t2) x))) (\lambda (_: +T).(\lambda (t: T).(ty3 g c u t))) (\lambda (t2: T).(\lambda (_: T).(ty3 g +(CHead c (Bind b) u) t1 t2)))))))))) +\def + \lambda (g: G).(\lambda (b: B).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: +T).(\lambda (x: T).(\lambda (H: (ty3 g c (THead (Bind b) u t1) x)).(insert_eq +T (THead (Bind b) u t1) (\lambda (t: T).(ty3 g c t x)) (\lambda (_: T).(ex3_2 +T T (\lambda (t2: T).(\lambda (_: T).(pc3 c (THead (Bind b) u t2) x))) +(\lambda (_: T).(\lambda (t0: T).(ty3 g c u t0))) (\lambda (t2: T).(\lambda +(_: T).(ty3 g (CHead c (Bind b) u) t1 t2))))) (\lambda (y: T).(\lambda (H0: +(ty3 g c y x)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: +T).((eq T t (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda (t2: T).(\lambda +(_: T).(pc3 c0 (THead (Bind b) u t2) t0))) (\lambda (_: T).(\lambda (t3: +T).(ty3 g c0 u t3))) (\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind +b) u) t1 t2)))))))) (\lambda (c0: C).(\lambda (t2: T).(\lambda (t: +T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda (_: (((eq T t2 (THead (Bind b) u +t1)) \to (ex3_2 T T (\lambda (t3: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) +u t3) t))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u t0))) (\lambda (t3: +T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3))))))).(\lambda (u0: +T).(\lambda (t0: T).(\lambda (H3: (ty3 g c0 u0 t0)).(\lambda (H4: (((eq T u0 +(THead (Bind b) u t1)) \to (ex3_2 T T (\lambda (t3: T).(\lambda (_: T).(pc3 +c0 (THead (Bind b) u t3) t0))) (\lambda (_: T).(\lambda (t4: T).(ty3 g c0 u +t4))) (\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 +t3))))))).(\lambda (H5: (pc3 c0 t0 t2)).(\lambda (H6: (eq T u0 (THead (Bind +b) u t1))).(let H7 \def (f_equal T T (\lambda (e: T).e) u0 (THead (Bind b) u +t1) H6) in (let H8 \def (eq_ind T u0 (\lambda (t3: T).((eq T t3 (THead (Bind +b) u t1)) \to (ex3_2 T T (\lambda (t4: T).(\lambda (_: T).(pc3 c0 (THead +(Bind b) u t4) t0))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c0 u t5))) +(\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t4)))))) H4 +(THead (Bind b) u t1) H7) in (let H9 \def (eq_ind T u0 (\lambda (t3: T).(ty3 +g c0 t3 t0)) H3 (THead (Bind b) u t1) H7) in (let H10 \def (H8 (refl_equal T +(THead (Bind b) u t1))) in (ex3_2_ind T T (\lambda (t3: T).(\lambda (_: +T).(pc3 c0 (THead (Bind b) u t3) t0))) (\lambda (_: T).(\lambda (t4: T).(ty3 +g c0 u t4))) (\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 +t3))) (ex3_2 T T (\lambda (t3: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u +t3) t2))) (\lambda (_: T).(\lambda (t4: T).(ty3 g c0 u t4))) (\lambda (t3: +T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3)))) (\lambda (x0: +T).(\lambda (x1: T).(\lambda (H11: (pc3 c0 (THead (Bind b) u x0) +t0)).(\lambda (H12: (ty3 g c0 u x1)).(\lambda (H13: (ty3 g (CHead c0 (Bind b) +u) t1 x0)).(ex3_2_intro T T (\lambda (t3: T).(\lambda (_: T).(pc3 c0 (THead +(Bind b) u t3) t2))) (\lambda (_: T).(\lambda (t4: T).(ty3 g c0 u t4))) +(\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3))) x0 x1 +(pc3_t t0 c0 (THead (Bind b) u x0) H11 t2 H5) H12 H13)))))) +H10)))))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq T +(TSort m) (THead (Bind b) u t1))).(let H2 \def (eq_ind T (TSort m) (\lambda +(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) +\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow +False])) I (THead (Bind b) u t1) H1) in (False_ind (ex3_2 T T (\lambda (t2: +T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t2) (TSort (next g m))))) +(\lambda (_: T).(\lambda (t: T).(ty3 g c0 u t))) (\lambda (t2: T).(\lambda +(_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2)))) H2))))) (\lambda (n: +nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (_: (getl n +c0 (CHead d (Bind Abbr) u0))).(\lambda (t: T).(\lambda (_: (ty3 g d u0 +t)).(\lambda (_: (((eq T u0 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda +(t2: T).(\lambda (_: T).(pc3 d (THead (Bind b) u t2) t))) (\lambda (_: +T).(\lambda (t0: T).(ty3 g d u t0))) (\lambda (t2: T).(\lambda (_: T).(ty3 g +(CHead d (Bind b) u) t1 t2))))))).(\lambda (H4: (eq T (TLRef n) (THead (Bind +b) u t1))).(let H5 \def (eq_ind T (TLRef n) (\lambda (ee: T).(match ee in T +return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t1) +H4) in (False_ind (ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3 c0 (THead +(Bind b) u t2) (lift (S n) O t)))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c0 +u t0))) (\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 +t2)))) H5))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: +C).(\lambda (u0: T).(\lambda (_: (getl n c0 (CHead d (Bind Abst) +u0))).(\lambda (t: T).(\lambda (_: (ty3 g d u0 t)).(\lambda (_: (((eq T u0 +(THead (Bind b) u t1)) \to (ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3 d +(THead (Bind b) u t2) t))) (\lambda (_: T).(\lambda (t0: T).(ty3 g d u t0))) +(\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead d (Bind b) u) t1 +t2))))))).(\lambda (H4: (eq T (TLRef n) (THead (Bind b) u t1))).(let H5 \def +(eq_ind T (TLRef n) (\lambda (ee: T).(match ee in T return (\lambda (_: +T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | +(THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t1) H4) in (False_ind +(ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t2) +(lift (S n) O u0)))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u t0))) +(\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2)))) +H5))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (t: T).(\lambda (H1: +(ty3 g c0 u0 t)).(\lambda (H2: (((eq T u0 (THead (Bind b) u t1)) \to (ex3_2 T +T (\lambda (t2: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t2) t))) +(\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u t0))) (\lambda (t2: T).(\lambda +(_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2))))))).(\lambda (b0: B).(\lambda +(t0: T).(\lambda (t2: T).(\lambda (H3: (ty3 g (CHead c0 (Bind b0) u0) t0 +t2)).(\lambda (H4: (((eq T t0 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda +(t3: T).(\lambda (_: T).(pc3 (CHead c0 (Bind b0) u0) (THead (Bind b) u t3) +t2))) (\lambda (_: T).(\lambda (t4: T).(ty3 g (CHead c0 (Bind b0) u0) u t4))) +(\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead (CHead c0 (Bind b0) u0) (Bind +b) u) t1 t3))))))).(\lambda (H5: (eq T (THead (Bind b0) u0 t0) (THead (Bind +b) u t1))).(let H6 \def (f_equal T B (\lambda (e: T).(match e in T return +(\lambda (_: T).B) with [(TSort _) \Rightarrow b0 | (TLRef _) \Rightarrow b0 +| (THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).B) with +[(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow b0])])) (THead (Bind b0) u0 +t0) (THead (Bind b) u t1) H5) in ((let H7 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | +(TLRef _) \Rightarrow u0 | (THead _ t3 _) \Rightarrow t3])) (THead (Bind b0) +u0 t0) (THead (Bind b) u t1) H5) in ((let H8 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | +(TLRef _) \Rightarrow t0 | (THead _ _ t3) \Rightarrow t3])) (THead (Bind b0) +u0 t0) (THead (Bind b) u t1) H5) in (\lambda (H9: (eq T u0 u)).(\lambda (H10: +(eq B b0 b)).(let H11 \def (eq_ind T t0 (\lambda (t3: T).((eq T t3 (THead +(Bind b) u t1)) \to (ex3_2 T T (\lambda (t4: T).(\lambda (_: T).(pc3 (CHead +c0 (Bind b0) u0) (THead (Bind b) u t4) t2))) (\lambda (_: T).(\lambda (t5: +T).(ty3 g (CHead c0 (Bind b0) u0) u t5))) (\lambda (t4: T).(\lambda (_: +T).(ty3 g (CHead (CHead c0 (Bind b0) u0) (Bind b) u) t1 t4)))))) H4 t1 H8) in +(let H12 \def (eq_ind T t0 (\lambda (t3: T).(ty3 g (CHead c0 (Bind b0) u0) t3 +t2)) H3 t1 H8) in (let H13 \def (eq_ind B b0 (\lambda (b1: B).((eq T t1 +(THead (Bind b) u t1)) \to (ex3_2 T T (\lambda (t3: T).(\lambda (_: T).(pc3 +(CHead c0 (Bind b1) u0) (THead (Bind b) u t3) t2))) (\lambda (_: T).(\lambda +(t4: T).(ty3 g (CHead c0 (Bind b1) u0) u t4))) (\lambda (t3: T).(\lambda (_: +T).(ty3 g (CHead (CHead c0 (Bind b1) u0) (Bind b) u) t1 t3)))))) H11 b H10) +in (let H14 \def (eq_ind B b0 (\lambda (b1: B).(ty3 g (CHead c0 (Bind b1) u0) +t1 t2)) H12 b H10) in (eq_ind_r B b (\lambda (b1: B).(ex3_2 T T (\lambda (t3: +T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t3) (THead (Bind b1) u0 t2)))) +(\lambda (_: T).(\lambda (t4: T).(ty3 g c0 u t4))) (\lambda (t3: T).(\lambda +(_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3))))) (let H15 \def (eq_ind T u0 +(\lambda (t3: T).((eq T t1 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda +(t4: T).(\lambda (_: T).(pc3 (CHead c0 (Bind b) t3) (THead (Bind b) u t4) +t2))) (\lambda (_: T).(\lambda (t5: T).(ty3 g (CHead c0 (Bind b) t3) u t5))) +(\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead (CHead c0 (Bind b) t3) (Bind +b) u) t1 t4)))))) H13 u H9) in (let H16 \def (eq_ind T u0 (\lambda (t3: +T).(ty3 g (CHead c0 (Bind b) t3) t1 t2)) H14 u H9) in (let H17 \def (eq_ind T +u0 (\lambda (t3: T).((eq T t3 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda +(t4: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t4) t))) (\lambda (_: +T).(\lambda (t5: T).(ty3 g c0 u t5))) (\lambda (t4: T).(\lambda (_: T).(ty3 g +(CHead c0 (Bind b) u) t1 t4)))))) H2 u H9) in (let H18 \def (eq_ind T u0 +(\lambda (t3: T).(ty3 g c0 t3 t)) H1 u H9) in (eq_ind_r T u (\lambda (t3: +T).(ex3_2 T T (\lambda (t4: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t4) +(THead (Bind b) t3 t2)))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c0 u t5))) +(\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t4))))) +(ex3_2_intro T T (\lambda (t3: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u +t3) (THead (Bind b) u t2)))) (\lambda (_: T).(\lambda (t4: T).(ty3 g c0 u +t4))) (\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3))) +t2 t (pc3_refl c0 (THead (Bind b) u t2)) H18 H16) u0 H9))))) b0 H10)))))))) +H7)) H6))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda (u0: +T).(\lambda (_: (ty3 g c0 w u0)).(\lambda (_: (((eq T w (THead (Bind b) u +t1)) \to (ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) +u t2) u0))) (\lambda (_: T).(\lambda (t: T).(ty3 g c0 u t))) (\lambda (t2: +T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2))))))).(\lambda (v: +T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead (Bind Abst) u0 +t))).(\lambda (_: (((eq T v (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda +(t2: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t2) (THead (Bind Abst) u0 +t)))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u t0))) (\lambda (t2: +T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2))))))).(\lambda (H5: +(eq T (THead (Flat Appl) w v) (THead (Bind b) u t1))).(let H6 \def (eq_ind T +(THead (Flat Appl) w v) (\lambda (ee: T).(match ee in T return (\lambda (_: +T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | +(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with +[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind +b) u t1) H5) in (False_ind (ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3 +c0 (THead (Bind b) u t2) (THead (Flat Appl) w (THead (Bind Abst) u0 t))))) +(\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u t0))) (\lambda (t2: T).(\lambda +(_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2)))) H6)))))))))))) (\lambda (c0: +C).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (ty3 g c0 t0 t2)).(\lambda +(_: (((eq T t0 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda (t3: +T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t3) t2))) (\lambda (_: +T).(\lambda (t: T).(ty3 g c0 u t))) (\lambda (t3: T).(\lambda (_: T).(ty3 g +(CHead c0 (Bind b) u) t1 t3))))))).(\lambda (t3: T).(\lambda (_: (ty3 g c0 t2 +t3)).(\lambda (_: (((eq T t2 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda +(t4: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t4) t3))) (\lambda (_: +T).(\lambda (t: T).(ty3 g c0 u t))) (\lambda (t4: T).(\lambda (_: T).(ty3 g +(CHead c0 (Bind b) u) t1 t4))))))).(\lambda (H5: (eq T (THead (Flat Cast) t2 +t0) (THead (Bind b) u t1))).(let H6 \def (eq_ind T (THead (Flat Cast) t2 t0) +(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) +\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow +(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | +(Flat _) \Rightarrow True])])) I (THead (Bind b) u t1) H5) in (False_ind +(ex3_2 T T (\lambda (t4: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t4) +(THead (Flat Cast) t3 t2)))) (\lambda (_: T).(\lambda (t: T).(ty3 g c0 u t))) +(\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t4)))) +H6))))))))))) c y x H0))) H))))))). +(* COMMENTS +Initial nodes: 3389 +END *) + +theorem ty3_gen_appl: + \forall (g: G).(\forall (c: C).(\forall (w: T).(\forall (v: T).(\forall (x: +T).((ty3 g c (THead (Flat Appl) w v) x) \to (ex3_2 T T (\lambda (u: +T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x))) +(\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) +(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (w: T).(\lambda (v: T).(\lambda (x: +T).(\lambda (H: (ty3 g c (THead (Flat Appl) w v) x)).(insert_eq T (THead +(Flat Appl) w v) (\lambda (t: T).(ty3 g c t x)) (\lambda (_: T).(ex3_2 T T +(\lambda (u: T).(\lambda (t0: T).(pc3 c (THead (Flat Appl) w (THead (Bind +Abst) u t0)) x))) (\lambda (u: T).(\lambda (t0: T).(ty3 g c v (THead (Bind +Abst) u t0)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c w u))))) (\lambda (y: +T).(\lambda (H0: (ty3 g c y x)).(ty3_ind g (\lambda (c0: C).(\lambda (t: +T).(\lambda (t0: T).((eq T t (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda +(u: T).(\lambda (t1: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u +t1)) t0))) (\lambda (u: T).(\lambda (t1: T).(ty3 g c0 v (THead (Bind Abst) u +t1)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c0 w u)))))))) (\lambda (c0: +C).(\lambda (t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda +(_: (((eq T t2 (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda (u: +T).(\lambda (t0: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u t0)) +t))) (\lambda (u: T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u t0)))) +(\lambda (u: T).(\lambda (_: T).(ty3 g c0 w u))))))).(\lambda (u: T).(\lambda +(t1: T).(\lambda (H3: (ty3 g c0 u t1)).(\lambda (H4: (((eq T u (THead (Flat +Appl) w v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0 (THead +(Flat Appl) w (THead (Bind Abst) u0 t0)) t1))) (\lambda (u0: T).(\lambda (t0: +T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_: +T).(ty3 g c0 w u0))))))).(\lambda (H5: (pc3 c0 t1 t2)).(\lambda (H6: (eq T u +(THead (Flat Appl) w v))).(let H7 \def (f_equal T T (\lambda (e: T).e) u +(THead (Flat Appl) w v) H6) in (let H8 \def (eq_ind T u (\lambda (t0: T).((eq +T t0 (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t3: +T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t3)) t1))) (\lambda +(u0: T).(\lambda (t3: T).(ty3 g c0 v (THead (Bind Abst) u0 t3)))) (\lambda +(u0: T).(\lambda (_: T).(ty3 g c0 w u0)))))) H4 (THead (Flat Appl) w v) H7) +in (let H9 \def (eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 t1)) H3 (THead +(Flat Appl) w v) H7) in (let H10 \def (H8 (refl_equal T (THead (Flat Appl) w +v))) in (ex3_2_ind T T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0 (THead (Flat +Appl) w (THead (Bind Abst) u0 t0)) t1))) (\lambda (u0: T).(\lambda (t0: +T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_: +T).(ty3 g c0 w u0))) (ex3_2 T T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0 +(THead (Flat Appl) w (THead (Bind Abst) u0 t0)) t2))) (\lambda (u0: +T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: +T).(\lambda (_: T).(ty3 g c0 w u0)))) (\lambda (x0: T).(\lambda (x1: +T).(\lambda (H11: (pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) x0 x1)) +t1)).(\lambda (H12: (ty3 g c0 v (THead (Bind Abst) x0 x1))).(\lambda (H13: +(ty3 g c0 w x0)).(ex3_2_intro T T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0 +(THead (Flat Appl) w (THead (Bind Abst) u0 t0)) t2))) (\lambda (u0: +T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: +T).(\lambda (_: T).(ty3 g c0 w u0))) x0 x1 (pc3_t t1 c0 (THead (Flat Appl) w +(THead (Bind Abst) x0 x1)) H11 t2 H5) H12 H13)))))) H10)))))))))))))))) +(\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq T (TSort m) (THead (Flat +Appl) w v))).(let H2 \def (eq_ind T (TSort m) (\lambda (ee: T).(match ee in T +return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) +\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) w +v) H1) in (False_ind (ex3_2 T T (\lambda (u: T).(\lambda (t: T).(pc3 c0 +(THead (Flat Appl) w (THead (Bind Abst) u t)) (TSort (next g m))))) (\lambda +(u: T).(\lambda (t: T).(ty3 g c0 v (THead (Bind Abst) u t)))) (\lambda (u: +T).(\lambda (_: T).(ty3 g c0 w u)))) H2))))) (\lambda (n: nat).(\lambda (c0: +C).(\lambda (d: C).(\lambda (u: T).(\lambda (_: (getl n c0 (CHead d (Bind +Abbr) u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (_: (((eq T u +(THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t0: +T).(pc3 d (THead (Flat Appl) w (THead (Bind Abst) u0 t0)) t))) (\lambda (u0: +T).(\lambda (t0: T).(ty3 g d v (THead (Bind Abst) u0 t0)))) (\lambda (u0: +T).(\lambda (_: T).(ty3 g d w u0))))))).(\lambda (H4: (eq T (TLRef n) (THead +(Flat Appl) w v))).(let H5 \def (eq_ind T (TLRef n) (\lambda (ee: T).(match +ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | +(TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead +(Flat Appl) w v) H4) in (False_ind (ex3_2 T T (\lambda (u0: T).(\lambda (t0: +T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t0)) (lift (S n) O +t)))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u0 +t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w u0)))) H5))))))))))) +(\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda +(_: (getl n c0 (CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (_: (ty3 g +d u t)).(\lambda (_: (((eq T u (THead (Flat Appl) w v)) \to (ex3_2 T T +(\lambda (u0: T).(\lambda (t0: T).(pc3 d (THead (Flat Appl) w (THead (Bind +Abst) u0 t0)) t))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g d v (THead (Bind +Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g d w +u0))))))).(\lambda (H4: (eq T (TLRef n) (THead (Flat Appl) w v))).(let H5 +\def (eq_ind T (TLRef n) (\lambda (ee: T).(match ee in T return (\lambda (_: +T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | +(THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) w v) H4) in +(False_ind (ex3_2 T T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0 (THead (Flat +Appl) w (THead (Bind Abst) u0 t0)) (lift (S n) O u)))) (\lambda (u0: +T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: +T).(\lambda (_: T).(ty3 g c0 w u0)))) H5))))))))))) (\lambda (c0: C).(\lambda +(u: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((eq T u +(THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t0: +T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t0)) t))) (\lambda (u0: +T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: +T).(\lambda (_: T).(ty3 g c0 w u0))))))).(\lambda (b: B).(\lambda (t1: +T).(\lambda (t2: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t1 +t2)).(\lambda (_: (((eq T t1 (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda +(u0: T).(\lambda (t0: T).(pc3 (CHead c0 (Bind b) u) (THead (Flat Appl) w +(THead (Bind Abst) u0 t0)) t2))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g +(CHead c0 (Bind b) u) v (THead (Bind Abst) u0 t0)))) (\lambda (u0: +T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) w u0))))))).(\lambda (H5: (eq +T (THead (Bind b) u t1) (THead (Flat Appl) w v))).(let H6 \def (eq_ind T +(THead (Bind b) u t1) (\lambda (ee: T).(match ee in T return (\lambda (_: +T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | +(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with +[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat +Appl) w v) H5) in (False_ind (ex3_2 T T (\lambda (u0: T).(\lambda (t0: +T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t0)) (THead (Bind b) u +t2)))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u0 +t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w u0)))) H6))))))))))))) +(\lambda (c0: C).(\lambda (w0: T).(\lambda (u: T).(\lambda (H1: (ty3 g c0 w0 +u)).(\lambda (H2: (((eq T w0 (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda +(u0: T).(\lambda (t: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 +t)) u))) (\lambda (u0: T).(\lambda (t: T).(ty3 g c0 v (THead (Bind Abst) u0 +t)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w u0))))))).(\lambda (v0: +T).(\lambda (t: T).(\lambda (H3: (ty3 g c0 v0 (THead (Bind Abst) u +t))).(\lambda (H4: (((eq T v0 (THead (Flat Appl) w v)) \to (ex3_2 T T +(\lambda (u0: T).(\lambda (t0: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind +Abst) u0 t0)) (THead (Bind Abst) u t)))) (\lambda (u0: T).(\lambda (t0: +T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_: +T).(ty3 g c0 w u0))))))).(\lambda (H5: (eq T (THead (Flat Appl) w0 v0) (THead +(Flat Appl) w v))).(let H6 \def (f_equal T T (\lambda (e: T).(match e in T +return (\lambda (_: T).T) with [(TSort _) \Rightarrow w0 | (TLRef _) +\Rightarrow w0 | (THead _ t0 _) \Rightarrow t0])) (THead (Flat Appl) w0 v0) +(THead (Flat Appl) w v) H5) in ((let H7 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v0 | +(TLRef _) \Rightarrow v0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat +Appl) w0 v0) (THead (Flat Appl) w v) H5) in (\lambda (H8: (eq T w0 w)).(let +H9 \def (eq_ind T v0 (\lambda (t0: T).((eq T t0 (THead (Flat Appl) w v)) \to +(ex3_2 T T (\lambda (u0: T).(\lambda (t1: T).(pc3 c0 (THead (Flat Appl) w +(THead (Bind Abst) u0 t1)) (THead (Bind Abst) u t)))) (\lambda (u0: +T).(\lambda (t1: T).(ty3 g c0 v (THead (Bind Abst) u0 t1)))) (\lambda (u0: +T).(\lambda (_: T).(ty3 g c0 w u0)))))) H4 v H7) in (let H10 \def (eq_ind T +v0 (\lambda (t0: T).(ty3 g c0 t0 (THead (Bind Abst) u t))) H3 v H7) in (let +H11 \def (eq_ind T w0 (\lambda (t0: T).((eq T t0 (THead (Flat Appl) w v)) \to +(ex3_2 T T (\lambda (u0: T).(\lambda (t1: T).(pc3 c0 (THead (Flat Appl) w +(THead (Bind Abst) u0 t1)) u))) (\lambda (u0: T).(\lambda (t1: T).(ty3 g c0 v +(THead (Bind Abst) u0 t1)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w +u0)))))) H2 w H8) in (let H12 \def (eq_ind T w0 (\lambda (t0: T).(ty3 g c0 t0 +u)) H1 w H8) in (eq_ind_r T w (\lambda (t0: T).(ex3_2 T T (\lambda (u0: +T).(\lambda (t1: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t1)) +(THead (Flat Appl) t0 (THead (Bind Abst) u t))))) (\lambda (u0: T).(\lambda +(t1: T).(ty3 g c0 v (THead (Bind Abst) u0 t1)))) (\lambda (u0: T).(\lambda +(_: T).(ty3 g c0 w u0))))) (ex3_2_intro T T (\lambda (u0: T).(\lambda (t0: +T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t0)) (THead (Flat Appl) +w (THead (Bind Abst) u t))))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g c0 v +(THead (Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w +u0))) u t (pc3_refl c0 (THead (Flat Appl) w (THead (Bind Abst) u t))) H10 +H12) w0 H8))))))) H6)))))))))))) (\lambda (c0: C).(\lambda (t1: T).(\lambda +(t2: T).(\lambda (_: (ty3 g c0 t1 t2)).(\lambda (_: (((eq T t1 (THead (Flat +Appl) w v)) \to (ex3_2 T T (\lambda (u: T).(\lambda (t: T).(pc3 c0 (THead +(Flat Appl) w (THead (Bind Abst) u t)) t2))) (\lambda (u: T).(\lambda (t: +T).(ty3 g c0 v (THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (_: +T).(ty3 g c0 w u))))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t2 +t0)).(\lambda (_: (((eq T t2 (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda +(u: T).(\lambda (t: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u t)) +t0))) (\lambda (u: T).(\lambda (t: T).(ty3 g c0 v (THead (Bind Abst) u t)))) +(\lambda (u: T).(\lambda (_: T).(ty3 g c0 w u))))))).(\lambda (H5: (eq T +(THead (Flat Cast) t2 t1) (THead (Flat Appl) w v))).(let H6 \def (eq_ind T +(THead (Flat Cast) t2 t1) (\lambda (ee: T).(match ee in T return (\lambda (_: +T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | +(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with +[(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f in F return +(\lambda (_: F).Prop) with [Appl \Rightarrow False | Cast \Rightarrow +True])])])) I (THead (Flat Appl) w v) H5) in (False_ind (ex3_2 T T (\lambda +(u: T).(\lambda (t: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u t)) +(THead (Flat Cast) t0 t2)))) (\lambda (u: T).(\lambda (t: T).(ty3 g c0 v +(THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c0 w u)))) +H6))))))))))) c y x H0))) H)))))). +(* COMMENTS +Initial nodes: 3171 +END *) + +theorem ty3_gen_cast: + \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).(\forall +(x: T).((ty3 g c (THead (Flat Cast) t2 t1) x) \to (ex3 T (\lambda (t0: +T).(pc3 c (THead (Flat Cast) t0 t2) x)) (\lambda (_: T).(ty3 g c t1 t2)) +(\lambda (t0: T).(ty3 g c t2 t0)))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda +(x: T).(\lambda (H: (ty3 g c (THead (Flat Cast) t2 t1) x)).(insert_eq T +(THead (Flat Cast) t2 t1) (\lambda (t: T).(ty3 g c t x)) (\lambda (_: T).(ex3 +T (\lambda (t0: T).(pc3 c (THead (Flat Cast) t0 t2) x)) (\lambda (_: T).(ty3 +g c t1 t2)) (\lambda (t0: T).(ty3 g c t2 t0)))) (\lambda (y: T).(\lambda (H0: +(ty3 g c y x)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: +T).((eq T t (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t3: T).(pc3 c0 +(THead (Flat Cast) t3 t2) t0)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda +(t3: T).(ty3 g c0 t2 t3))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t: +T).(\lambda (_: (ty3 g c0 t0 t)).(\lambda (_: (((eq T t0 (THead (Flat Cast) +t2 t1)) \to (ex3 T (\lambda (t3: T).(pc3 c0 (THead (Flat Cast) t3 t2) t)) +(\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t3: T).(ty3 g c0 t2 +t3)))))).(\lambda (u: T).(\lambda (t3: T).(\lambda (H3: (ty3 g c0 u +t3)).(\lambda (H4: (((eq T u (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda +(t4: T).(pc3 c0 (THead (Flat Cast) t4 t2) t3)) (\lambda (_: T).(ty3 g c0 t1 +t2)) (\lambda (t4: T).(ty3 g c0 t2 t4)))))).(\lambda (H5: (pc3 c0 t3 +t0)).(\lambda (H6: (eq T u (THead (Flat Cast) t2 t1))).(let H7 \def (f_equal +T T (\lambda (e: T).e) u (THead (Flat Cast) t2 t1) H6) in (let H8 \def +(eq_ind T u (\lambda (t4: T).((eq T t4 (THead (Flat Cast) t2 t1)) \to (ex3 T +(\lambda (t5: T).(pc3 c0 (THead (Flat Cast) t5 t2) t3)) (\lambda (_: T).(ty3 +g c0 t1 t2)) (\lambda (t5: T).(ty3 g c0 t2 t5))))) H4 (THead (Flat Cast) t2 +t1) H7) in (let H9 \def (eq_ind T u (\lambda (t4: T).(ty3 g c0 t4 t3)) H3 +(THead (Flat Cast) t2 t1) H7) in (let H10 \def (H8 (refl_equal T (THead (Flat +Cast) t2 t1))) in (ex3_ind T (\lambda (t4: T).(pc3 c0 (THead (Flat Cast) t4 +t2) t3)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t4: T).(ty3 g c0 t2 t4)) +(ex3 T (\lambda (t4: T).(pc3 c0 (THead (Flat Cast) t4 t2) t0)) (\lambda (_: +T).(ty3 g c0 t1 t2)) (\lambda (t4: T).(ty3 g c0 t2 t4))) (\lambda (x0: +T).(\lambda (H11: (pc3 c0 (THead (Flat Cast) x0 t2) t3)).(\lambda (H12: (ty3 +g c0 t1 t2)).(\lambda (H13: (ty3 g c0 t2 x0)).(ex3_intro T (\lambda (t4: +T).(pc3 c0 (THead (Flat Cast) t4 t2) t0)) (\lambda (_: T).(ty3 g c0 t1 t2)) +(\lambda (t4: T).(ty3 g c0 t2 t4)) x0 (pc3_t t3 c0 (THead (Flat Cast) x0 t2) +H11 t0 H5) H12 H13))))) H10)))))))))))))))) (\lambda (c0: C).(\lambda (m: +nat).(\lambda (H1: (eq T (TSort m) (THead (Flat Cast) t2 t1))).(let H2 \def +(eq_ind T (TSort m) (\lambda (ee: T).(match ee in T return (\lambda (_: +T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | +(THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) t2 t1) H1) in +(False_ind (ex3 T (\lambda (t0: T).(pc3 c0 (THead (Flat Cast) t0 t2) (TSort +(next g m)))) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0 t2 +t0))) H2))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: +T).(\lambda (_: (getl n c0 (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda +(_: (ty3 g d u t)).(\lambda (_: (((eq T u (THead (Flat Cast) t2 t1)) \to (ex3 +T (\lambda (t0: T).(pc3 d (THead (Flat Cast) t0 t2) t)) (\lambda (_: T).(ty3 +g d t1 t2)) (\lambda (t0: T).(ty3 g d t2 t0)))))).(\lambda (H4: (eq T (TLRef +n) (THead (Flat Cast) t2 t1))).(let H5 \def (eq_ind T (TLRef n) (\lambda (ee: +T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow +False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I +(THead (Flat Cast) t2 t1) H4) in (False_ind (ex3 T (\lambda (t0: T).(pc3 c0 +(THead (Flat Cast) t0 t2) (lift (S n) O t))) (\lambda (_: T).(ty3 g c0 t1 +t2)) (\lambda (t0: T).(ty3 g c0 t2 t0))) H5))))))))))) (\lambda (n: +nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (_: (getl n c0 +(CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (_: (ty3 g d u +t)).(\lambda (_: (((eq T u (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda +(t0: T).(pc3 d (THead (Flat Cast) t0 t2) t)) (\lambda (_: T).(ty3 g d t1 t2)) +(\lambda (t0: T).(ty3 g d t2 t0)))))).(\lambda (H4: (eq T (TLRef n) (THead +(Flat Cast) t2 t1))).(let H5 \def (eq_ind T (TLRef n) (\lambda (ee: T).(match +ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | +(TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead +(Flat Cast) t2 t1) H4) in (False_ind (ex3 T (\lambda (t0: T).(pc3 c0 (THead +(Flat Cast) t0 t2) (lift (S n) O u))) (\lambda (_: T).(ty3 g c0 t1 t2)) +(\lambda (t0: T).(ty3 g c0 t2 t0))) H5))))))))))) (\lambda (c0: C).(\lambda +(u: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((eq T u +(THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t0: T).(pc3 c0 (THead (Flat +Cast) t0 t2) t)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0 +t2 t0)))))).(\lambda (b: B).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: +(ty3 g (CHead c0 (Bind b) u) t0 t3)).(\lambda (_: (((eq T t0 (THead (Flat +Cast) t2 t1)) \to (ex3 T (\lambda (t4: T).(pc3 (CHead c0 (Bind b) u) (THead +(Flat Cast) t4 t2) t3)) (\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2)) +(\lambda (t4: T).(ty3 g (CHead c0 (Bind b) u) t2 t4)))))).(\lambda (H5: (eq T +(THead (Bind b) u t0) (THead (Flat Cast) t2 t1))).(let H6 \def (eq_ind T +(THead (Bind b) u t0) (\lambda (ee: T).(match ee in T return (\lambda (_: +T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | +(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with +[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat +Cast) t2 t1) H5) in (False_ind (ex3 T (\lambda (t4: T).(pc3 c0 (THead (Flat +Cast) t4 t2) (THead (Bind b) u t3))) (\lambda (_: T).(ty3 g c0 t1 t2)) +(\lambda (t4: T).(ty3 g c0 t2 t4))) H6))))))))))))) (\lambda (c0: C).(\lambda +(w: T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda (_: (((eq T w +(THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t0: T).(pc3 c0 (THead (Flat +Cast) t0 t2) u)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0 +t2 t0)))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead +(Bind Abst) u t))).(\lambda (_: (((eq T v (THead (Flat Cast) t2 t1)) \to (ex3 +T (\lambda (t0: T).(pc3 c0 (THead (Flat Cast) t0 t2) (THead (Bind Abst) u +t))) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0 t2 +t0)))))).(\lambda (H5: (eq T (THead (Flat Appl) w v) (THead (Flat Cast) t2 +t1))).(let H6 \def (eq_ind T (THead (Flat Appl) w v) (\lambda (ee: T).(match +ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | +(TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return +(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow +(match f in F return (\lambda (_: F).Prop) with [Appl \Rightarrow True | Cast +\Rightarrow False])])])) I (THead (Flat Cast) t2 t1) H5) in (False_ind (ex3 T +(\lambda (t0: T).(pc3 c0 (THead (Flat Cast) t0 t2) (THead (Flat Appl) w +(THead (Bind Abst) u t)))) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0: +T).(ty3 g c0 t2 t0))) H6)))))))))))) (\lambda (c0: C).(\lambda (t0: +T).(\lambda (t3: T).(\lambda (H1: (ty3 g c0 t0 t3)).(\lambda (H2: (((eq T t0 +(THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t4: T).(pc3 c0 (THead (Flat +Cast) t4 t2) t3)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t4: T).(ty3 g +c0 t2 t4)))))).(\lambda (t4: T).(\lambda (H3: (ty3 g c0 t3 t4)).(\lambda (H4: +(((eq T t3 (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t5: T).(pc3 c0 +(THead (Flat Cast) t5 t2) t4)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda +(t5: T).(ty3 g c0 t2 t5)))))).(\lambda (H5: (eq T (THead (Flat Cast) t3 t0) +(THead (Flat Cast) t2 t1))).(let H6 \def (f_equal T T (\lambda (e: T).(match +e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) +\Rightarrow t3 | (THead _ t _) \Rightarrow t])) (THead (Flat Cast) t3 t0) +(THead (Flat Cast) t2 t1) H5) in ((let H7 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | +(TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Flat Cast) +t3 t0) (THead (Flat Cast) t2 t1) H5) in (\lambda (H8: (eq T t3 t2)).(let H9 +\def (eq_ind T t3 (\lambda (t: T).((eq T t (THead (Flat Cast) t2 t1)) \to +(ex3 T (\lambda (t5: T).(pc3 c0 (THead (Flat Cast) t5 t2) t4)) (\lambda (_: +T).(ty3 g c0 t1 t2)) (\lambda (t5: T).(ty3 g c0 t2 t5))))) H4 t2 H8) in (let +H10 \def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 t t4)) H3 t2 H8) in (let H11 +\def (eq_ind T t3 (\lambda (t: T).((eq T t0 (THead (Flat Cast) t2 t1)) \to +(ex3 T (\lambda (t5: T).(pc3 c0 (THead (Flat Cast) t5 t2) t)) (\lambda (_: +T).(ty3 g c0 t1 t2)) (\lambda (t5: T).(ty3 g c0 t2 t5))))) H2 t2 H8) in (let +H12 \def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 t0 t)) H1 t2 H8) in (eq_ind_r +T t2 (\lambda (t: T).(ex3 T (\lambda (t5: T).(pc3 c0 (THead (Flat Cast) t5 +t2) (THead (Flat Cast) t4 t))) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda +(t5: T).(ty3 g c0 t2 t5)))) (let H13 \def (eq_ind T t0 (\lambda (t: T).((eq T +t (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t5: T).(pc3 c0 (THead (Flat +Cast) t5 t2) t2)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t5: T).(ty3 g +c0 t2 t5))))) H11 t1 H7) in (let H14 \def (eq_ind T t0 (\lambda (t: T).(ty3 g +c0 t t2)) H12 t1 H7) in (ex3_intro T (\lambda (t5: T).(pc3 c0 (THead (Flat +Cast) t5 t2) (THead (Flat Cast) t4 t2))) (\lambda (_: T).(ty3 g c0 t1 t2)) +(\lambda (t5: T).(ty3 g c0 t2 t5)) t4 (pc3_refl c0 (THead (Flat Cast) t4 t2)) +H14 H10))) t3 H8))))))) H6))))))))))) c y x H0))) H)))))). +(* COMMENTS +Initial nodes: 2609 +END *) + +theorem tys3_gen_nil: + \forall (g: G).(\forall (c: C).(\forall (u: T).((tys3 g c TNil u) \to (ex T +(\lambda (u0: T).(ty3 g c u u0)))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (H: (tys3 g c TNil +u)).(insert_eq TList TNil (\lambda (t: TList).(tys3 g c t u)) (\lambda (_: +TList).(ex T (\lambda (u0: T).(ty3 g c u u0)))) (\lambda (y: TList).(\lambda +(H0: (tys3 g c y u)).(tys3_ind g c (\lambda (t: TList).(\lambda (t0: T).((eq +TList t TNil) \to (ex T (\lambda (u0: T).(ty3 g c t0 u0)))))) (\lambda (u0: +T).(\lambda (u1: T).(\lambda (H1: (ty3 g c u0 u1)).(\lambda (_: (eq TList +TNil TNil)).(ex_intro T (\lambda (u2: T).(ty3 g c u0 u2)) u1 H1))))) (\lambda +(t: T).(\lambda (u0: T).(\lambda (_: (ty3 g c t u0)).(\lambda (ts: +TList).(\lambda (_: (tys3 g c ts u0)).(\lambda (_: (((eq TList ts TNil) \to +(ex T (\lambda (u1: T).(ty3 g c u0 u1)))))).(\lambda (H4: (eq TList (TCons t +ts) TNil)).(let H5 \def (eq_ind TList (TCons t ts) (\lambda (ee: +TList).(match ee in TList return (\lambda (_: TList).Prop) with [TNil +\Rightarrow False | (TCons _ _) \Rightarrow True])) I TNil H4) in (False_ind +(ex T (\lambda (u1: T).(ty3 g c u0 u1))) H5))))))))) y u H0))) H)))). +(* COMMENTS +Initial nodes: 255 +END *) + +theorem tys3_gen_cons: + \forall (g: G).(\forall (c: C).(\forall (ts: TList).(\forall (t: T).(\forall +(u: T).((tys3 g c (TCons t ts) u) \to (land (ty3 g c t u) (tys3 g c ts +u))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (ts: TList).(\lambda (t: T).(\lambda +(u: T).(\lambda (H: (tys3 g c (TCons t ts) u)).(insert_eq TList (TCons t ts) +(\lambda (t0: TList).(tys3 g c t0 u)) (\lambda (_: TList).(land (ty3 g c t u) +(tys3 g c ts u))) (\lambda (y: TList).(\lambda (H0: (tys3 g c y u)).(tys3_ind +g c (\lambda (t0: TList).(\lambda (t1: T).((eq TList t0 (TCons t ts)) \to +(land (ty3 g c t t1) (tys3 g c ts t1))))) (\lambda (u0: T).(\lambda (u1: +T).(\lambda (_: (ty3 g c u0 u1)).(\lambda (H2: (eq TList TNil (TCons t +ts))).(let H3 \def (eq_ind TList TNil (\lambda (ee: TList).(match ee in TList +return (\lambda (_: TList).Prop) with [TNil \Rightarrow True | (TCons _ _) +\Rightarrow False])) I (TCons t ts) H2) in (False_ind (land (ty3 g c t u0) +(tys3 g c ts u0)) H3)))))) (\lambda (t0: T).(\lambda (u0: T).(\lambda (H1: +(ty3 g c t0 u0)).(\lambda (ts0: TList).(\lambda (H2: (tys3 g c ts0 +u0)).(\lambda (H3: (((eq TList ts0 (TCons t ts)) \to (land (ty3 g c t u0) +(tys3 g c ts u0))))).(\lambda (H4: (eq TList (TCons t0 ts0) (TCons t +ts))).(let H5 \def (f_equal TList T (\lambda (e: TList).(match e in TList +return (\lambda (_: TList).T) with [TNil \Rightarrow t0 | (TCons t1 _) +\Rightarrow t1])) (TCons t0 ts0) (TCons t ts) H4) in ((let H6 \def (f_equal +TList TList (\lambda (e: TList).(match e in TList return (\lambda (_: +TList).TList) with [TNil \Rightarrow ts0 | (TCons _ t1) \Rightarrow t1])) +(TCons t0 ts0) (TCons t ts) H4) in (\lambda (H7: (eq T t0 t)).(let H8 \def +(eq_ind TList ts0 (\lambda (t1: TList).((eq TList t1 (TCons t ts)) \to (land +(ty3 g c t u0) (tys3 g c ts u0)))) H3 ts H6) in (let H9 \def (eq_ind TList +ts0 (\lambda (t1: TList).(tys3 g c t1 u0)) H2 ts H6) in (let H10 \def (eq_ind +T t0 (\lambda (t1: T).(ty3 g c t1 u0)) H1 t H7) in (conj (ty3 g c t u0) (tys3 +g c ts u0) H10 H9)))))) H5))))))))) y u H0))) H)))))). +(* COMMENTS +Initial nodes: 479 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/ty3/fwd_nf2.ma b/matita/matita/contribs/lambdadelta/basic_1/ty3/fwd_nf2.ma new file mode 100644 index 000000000..ca4e40c7b --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/ty3/fwd_nf2.ma @@ -0,0 +1,301 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/ty3/arity_props.ma". + +include "Basic-1/pc3/nf2.ma". + +include "Basic-1/nf2/fwd.ma". + +theorem ty3_gen_appl_nf2: + \forall (g: G).(\forall (c: C).(\forall (w: T).(\forall (v: T).(\forall (x: +T).((ty3 g c (THead (Flat Appl) w v) x) \to (ex4_2 T T (\lambda (u: +T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x))) +(\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) +(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t: +T).(nf2 c (THead (Bind Abst) u t)))))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (w: T).(\lambda (v: T).(\lambda (x: +T).(\lambda (H: (ty3 g c (THead (Flat Appl) w v) x)).(ex3_2_ind T T (\lambda +(u: T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) +x))) (\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) +(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (ex4_2 T T (\lambda (u: +T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x))) +(\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) +(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t: +T).(nf2 c (THead (Bind Abst) u t))))) (\lambda (x0: T).(\lambda (x1: +T).(\lambda (H0: (pc3 c (THead (Flat Appl) w (THead (Bind Abst) x0 x1)) +x)).(\lambda (H1: (ty3 g c v (THead (Bind Abst) x0 x1))).(\lambda (H2: (ty3 g +c w x0)).(let H_x \def (ty3_correct g c v (THead (Bind Abst) x0 x1) H1) in +(let H3 \def H_x in (ex_ind T (\lambda (t: T).(ty3 g c (THead (Bind Abst) x0 +x1) t)) (ex4_2 T T (\lambda (u: T).(\lambda (t: T).(pc3 c (THead (Flat Appl) +w (THead (Bind Abst) u t)) x))) (\lambda (u: T).(\lambda (t: T).(ty3 g c v +(THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) +(\lambda (u: T).(\lambda (t: T).(nf2 c (THead (Bind Abst) u t))))) (\lambda +(x2: T).(\lambda (H4: (ty3 g c (THead (Bind Abst) x0 x1) x2)).(let H_x0 \def +(ty3_correct g c w x0 H2) in (let H5 \def H_x0 in (ex_ind T (\lambda (t: +T).(ty3 g c x0 t)) (ex4_2 T T (\lambda (u: T).(\lambda (t: T).(pc3 c (THead +(Flat Appl) w (THead (Bind Abst) u t)) x))) (\lambda (u: T).(\lambda (t: +T).(ty3 g c v (THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3 +g c w u))) (\lambda (u: T).(\lambda (t: T).(nf2 c (THead (Bind Abst) u t))))) +(\lambda (x3: T).(\lambda (H6: (ty3 g c x0 x3)).(let H7 \def (ty3_sn3 g c +(THead (Bind Abst) x0 x1) x2 H4) in (let H_x1 \def (nf2_sn3 c (THead (Bind +Abst) x0 x1) H7) in (let H8 \def H_x1 in (ex2_ind T (\lambda (u: T).(pr3 c +(THead (Bind Abst) x0 x1) u)) (\lambda (u: T).(nf2 c u)) (ex4_2 T T (\lambda +(u: T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) +x))) (\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) +(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t: +T).(nf2 c (THead (Bind Abst) u t))))) (\lambda (x4: T).(\lambda (H9: (pr3 c +(THead (Bind Abst) x0 x1) x4)).(\lambda (H10: (nf2 c x4)).(let H11 \def +(pr3_gen_abst c x0 x1 x4 H9) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: +T).(eq T x4 (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: +T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall +(u: T).(pr3 (CHead c (Bind b) u) x1 t2))))) (ex4_2 T T (\lambda (u: +T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x))) +(\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) +(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t: +T).(nf2 c (THead (Bind Abst) u t))))) (\lambda (x5: T).(\lambda (x6: +T).(\lambda (H12: (eq T x4 (THead (Bind Abst) x5 x6))).(\lambda (H13: (pr3 c +x0 x5)).(\lambda (H14: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind +b) u) x1 x6))))).(let H15 \def (eq_ind T x4 (\lambda (t: T).(nf2 c t)) H10 +(THead (Bind Abst) x5 x6) H12) in (let H16 \def (pr3_head_12 c x0 x5 H13 +(Bind Abst) x1 x6 (H14 Abst x5)) in (ex4_2_intro T T (\lambda (u: T).(\lambda +(t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x))) (\lambda (u: +T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) (\lambda (u: +T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t: T).(nf2 c +(THead (Bind Abst) u t)))) x5 x6 (pc3_pr3_conf c (THead (Flat Appl) w (THead +(Bind Abst) x0 x1)) x H0 (THead (Flat Appl) w (THead (Bind Abst) x5 x6)) +(pr3_thin_dx c (THead (Bind Abst) x0 x1) (THead (Bind Abst) x5 x6) H16 w +Appl)) (ty3_conv g c (THead (Bind Abst) x5 x6) x2 (ty3_sred_pr3 c (THead +(Bind Abst) x0 x1) (THead (Bind Abst) x5 x6) H16 g x2 H4) v (THead (Bind +Abst) x0 x1) H1 (pc3_pr3_r c (THead (Bind Abst) x0 x1) (THead (Bind Abst) x5 +x6) H16)) (ty3_conv g c x5 x3 (ty3_sred_pr3 c x0 x5 H13 g x3 H6) w x0 H2 +(pc3_pr3_r c x0 x5 H13)) H15)))))))) H11))))) H8)))))) H5))))) H3)))))))) +(ty3_gen_appl g c w v x H))))))). +(* COMMENTS +Initial nodes: 1289 +END *) + +theorem ty3_inv_lref_nf2_pc3: + \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (i: nat).((ty3 g c +(TLRef i) u1) \to ((nf2 c (TLRef i)) \to (\forall (u2: T).((nf2 c u2) \to +((pc3 c u1 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift (S i) O u)))))))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (u1: T).(\lambda (i: nat).(\lambda +(H: (ty3 g c (TLRef i) u1)).(insert_eq T (TLRef i) (\lambda (t: T).(ty3 g c t +u1)) (\lambda (t: T).((nf2 c t) \to (\forall (u2: T).((nf2 c u2) \to ((pc3 c +u1 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift (S i) O u))))))))) (\lambda +(y: T).(\lambda (H0: (ty3 g c y u1)).(ty3_ind g (\lambda (c0: C).(\lambda (t: +T).(\lambda (t0: T).((eq T t (TLRef i)) \to ((nf2 c0 t) \to (\forall (u2: +T).((nf2 c0 u2) \to ((pc3 c0 t0 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift +(S i) O u)))))))))))) (\lambda (c0: C).(\lambda (t2: T).(\lambda (t: +T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda (_: (((eq T t2 (TLRef i)) \to ((nf2 +c0 t2) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 t u2) \to (ex T +(\lambda (u: T).(eq T u2 (lift (S i) O u))))))))))).(\lambda (u: T).(\lambda +(t1: T).(\lambda (H3: (ty3 g c0 u t1)).(\lambda (H4: (((eq T u (TLRef i)) \to +((nf2 c0 u) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 t1 u2) \to (ex T +(\lambda (u0: T).(eq T u2 (lift (S i) O u0))))))))))).(\lambda (H5: (pc3 c0 +t1 t2)).(\lambda (H6: (eq T u (TLRef i))).(\lambda (H7: (nf2 c0 u)).(\lambda +(u2: T).(\lambda (H8: (nf2 c0 u2)).(\lambda (H9: (pc3 c0 t2 u2)).(let H10 +\def (eq_ind T u (\lambda (t0: T).(nf2 c0 t0)) H7 (TLRef i) H6) in (let H11 +\def (eq_ind T u (\lambda (t0: T).((eq T t0 (TLRef i)) \to ((nf2 c0 t0) \to +(\forall (u3: T).((nf2 c0 u3) \to ((pc3 c0 t1 u3) \to (ex T (\lambda (u0: +T).(eq T u3 (lift (S i) O u0)))))))))) H4 (TLRef i) H6) in (let H12 \def +(eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 t1)) H3 (TLRef i) H6) in (let H_y +\def (H11 (refl_equal T (TLRef i)) H10 u2 H8) in (H_y (pc3_t t2 c0 t1 H5 u2 +H9))))))))))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq +T (TSort m) (TLRef i))).(\lambda (_: (nf2 c0 (TSort m))).(\lambda (u2: +T).(\lambda (_: (nf2 c0 u2)).(\lambda (_: (pc3 c0 (TSort (next g m)) +u2)).(let H5 \def (eq_ind T (TSort m) (\lambda (ee: T).(match ee in T return +(\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) +\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef i) H1) in +(False_ind (ex T (\lambda (u: T).(eq T u2 (lift (S i) O u)))) H5))))))))) +(\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda +(H1: (getl n c0 (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda (_: (ty3 g +d u t)).(\lambda (_: (((eq T u (TLRef i)) \to ((nf2 d u) \to (\forall (u2: +T).((nf2 d u2) \to ((pc3 d t u2) \to (ex T (\lambda (u0: T).(eq T u2 (lift (S +i) O u0))))))))))).(\lambda (H4: (eq T (TLRef n) (TLRef i))).(\lambda (H5: +(nf2 c0 (TLRef n))).(\lambda (u2: T).(\lambda (_: (nf2 c0 u2)).(\lambda (H7: +(pc3 c0 (lift (S n) O t) u2)).(let H8 \def (f_equal T nat (\lambda (e: +T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow n | +(TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef +i) H4) in (let H9 \def (eq_ind nat n (\lambda (n0: nat).(pc3 c0 (lift (S n0) +O t) u2)) H7 i H8) in (let H10 \def (eq_ind nat n (\lambda (n0: nat).(nf2 c0 +(TLRef n0))) H5 i H8) in (let H11 \def (eq_ind nat n (\lambda (n0: nat).(getl +n0 c0 (CHead d (Bind Abbr) u))) H1 i H8) in (nf2_gen_lref c0 d u i H11 H10 +(ex T (\lambda (u0: T).(eq T u2 (lift (S i) O u0)))))))))))))))))))))) +(\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda +(H1: (getl n c0 (CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (_: (ty3 g +d u t)).(\lambda (_: (((eq T u (TLRef i)) \to ((nf2 d u) \to (\forall (u2: +T).((nf2 d u2) \to ((pc3 d t u2) \to (ex T (\lambda (u0: T).(eq T u2 (lift (S +i) O u0))))))))))).(\lambda (H4: (eq T (TLRef n) (TLRef i))).(\lambda (H5: +(nf2 c0 (TLRef n))).(\lambda (u2: T).(\lambda (H6: (nf2 c0 u2)).(\lambda (H7: +(pc3 c0 (lift (S n) O u) u2)).(let H8 \def (f_equal T nat (\lambda (e: +T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow n | +(TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef +i) H4) in (let H9 \def (eq_ind nat n (\lambda (n0: nat).(pc3 c0 (lift (S n0) +O u) u2)) H7 i H8) in (let H10 \def (eq_ind nat n (\lambda (n0: nat).(nf2 c0 +(TLRef n0))) H5 i H8) in (let H11 \def (eq_ind nat n (\lambda (n0: nat).(getl +n0 c0 (CHead d (Bind Abst) u))) H1 i H8) in (let H_y \def (pc3_nf2_unfold c0 +(lift (S i) O u) u2 H9 H6) in (let H12 \def (pr3_gen_lift c0 u u2 (S i) O H_y +d (getl_drop Abst c0 d u i H11)) in (ex2_ind T (\lambda (t2: T).(eq T u2 +(lift (S i) O t2))) (\lambda (t2: T).(pr3 d u t2)) (ex T (\lambda (u0: T).(eq +T u2 (lift (S i) O u0)))) (\lambda (x: T).(\lambda (H13: (eq T u2 (lift (S i) +O x))).(\lambda (_: (pr3 d u x)).(eq_ind_r T (lift (S i) O x) (\lambda (t0: +T).(ex T (\lambda (u0: T).(eq T t0 (lift (S i) O u0))))) (ex_intro T (\lambda +(u0: T).(eq T (lift (S i) O x) (lift (S i) O u0))) x (refl_equal T (lift (S +i) O x))) u2 H13)))) H12)))))))))))))))))))) (\lambda (c0: C).(\lambda (u: +T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((eq T u (TLRef +i)) \to ((nf2 c0 u) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 t u2) \to +(ex T (\lambda (u0: T).(eq T u2 (lift (S i) O u0))))))))))).(\lambda (b: +B).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) +u) t1 t2)).(\lambda (_: (((eq T t1 (TLRef i)) \to ((nf2 (CHead c0 (Bind b) u) +t1) \to (\forall (u2: T).((nf2 (CHead c0 (Bind b) u) u2) \to ((pc3 (CHead c0 +(Bind b) u) t2 u2) \to (ex T (\lambda (u0: T).(eq T u2 (lift (S i) O +u0))))))))))).(\lambda (H5: (eq T (THead (Bind b) u t1) (TLRef i))).(\lambda +(_: (nf2 c0 (THead (Bind b) u t1))).(\lambda (u2: T).(\lambda (_: (nf2 c0 +u2)).(\lambda (_: (pc3 c0 (THead (Bind b) u t2) u2)).(let H9 \def (eq_ind T +(THead (Bind b) u t1) (\lambda (ee: T).(match ee in T return (\lambda (_: +T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | +(THead _ _ _) \Rightarrow True])) I (TLRef i) H5) in (False_ind (ex T +(\lambda (u0: T).(eq T u2 (lift (S i) O u0)))) H9))))))))))))))))) (\lambda +(c0: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda +(_: (((eq T w (TLRef i)) \to ((nf2 c0 w) \to (\forall (u2: T).((nf2 c0 u2) +\to ((pc3 c0 u u2) \to (ex T (\lambda (u0: T).(eq T u2 (lift (S i) O +u0))))))))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead +(Bind Abst) u t))).(\lambda (_: (((eq T v (TLRef i)) \to ((nf2 c0 v) \to +(\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 (THead (Bind Abst) u t) u2) \to +(ex T (\lambda (u0: T).(eq T u2 (lift (S i) O u0))))))))))).(\lambda (H5: (eq +T (THead (Flat Appl) w v) (TLRef i))).(\lambda (_: (nf2 c0 (THead (Flat Appl) +w v))).(\lambda (u2: T).(\lambda (_: (nf2 c0 u2)).(\lambda (_: (pc3 c0 (THead +(Flat Appl) w (THead (Bind Abst) u t)) u2)).(let H9 \def (eq_ind T (THead +(Flat Appl) w v) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) +with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ +_) \Rightarrow True])) I (TLRef i) H5) in (False_ind (ex T (\lambda (u0: +T).(eq T u2 (lift (S i) O u0)))) H9)))))))))))))))) (\lambda (c0: C).(\lambda +(t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g c0 t1 t2)).(\lambda (_: (((eq T +t1 (TLRef i)) \to ((nf2 c0 t1) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 +t2 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift (S i) O u))))))))))).(\lambda +(t0: T).(\lambda (_: (ty3 g c0 t2 t0)).(\lambda (_: (((eq T t2 (TLRef i)) \to +((nf2 c0 t2) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 t0 u2) \to (ex T +(\lambda (u: T).(eq T u2 (lift (S i) O u))))))))))).(\lambda (H5: (eq T +(THead (Flat Cast) t2 t1) (TLRef i))).(\lambda (_: (nf2 c0 (THead (Flat Cast) +t2 t1))).(\lambda (u2: T).(\lambda (_: (nf2 c0 u2)).(\lambda (_: (pc3 c0 +(THead (Flat Cast) t0 t2) u2)).(let H9 \def (eq_ind T (THead (Flat Cast) t2 +t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort +_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) +\Rightarrow True])) I (TLRef i) H5) in (False_ind (ex T (\lambda (u: T).(eq T +u2 (lift (S i) O u)))) H9))))))))))))))) c y u1 H0))) H))))). +(* COMMENTS +Initial nodes: 2175 +END *) + +theorem ty3_inv_lref_nf2: + \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (i: nat).((ty3 g c +(TLRef i) u) \to ((nf2 c (TLRef i)) \to ((nf2 c u) \to (ex T (\lambda (u0: +T).(eq T u (lift (S i) O u0)))))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (i: nat).(\lambda +(H: (ty3 g c (TLRef i) u)).(\lambda (H0: (nf2 c (TLRef i))).(\lambda (H1: +(nf2 c u)).(ty3_inv_lref_nf2_pc3 g c u i H H0 u H1 (pc3_refl c u)))))))). +(* COMMENTS +Initial nodes: 57 +END *) + +theorem ty3_inv_appls_lref_nf2: + \forall (g: G).(\forall (c: C).(\forall (vs: TList).(\forall (u1: +T).(\forall (i: nat).((ty3 g c (THeads (Flat Appl) vs (TLRef i)) u1) \to +((nf2 c (TLRef i)) \to ((nf2 c u1) \to (ex2 T (\lambda (u: T).(nf2 c (lift (S +i) O u))) (\lambda (u: T).(pc3 c (THeads (Flat Appl) vs (lift (S i) O u)) +u1)))))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (vs: TList).(TList_ind (\lambda (t: +TList).(\forall (u1: T).(\forall (i: nat).((ty3 g c (THeads (Flat Appl) t +(TLRef i)) u1) \to ((nf2 c (TLRef i)) \to ((nf2 c u1) \to (ex2 T (\lambda (u: +T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3 c (THeads (Flat Appl) t +(lift (S i) O u)) u1))))))))) (\lambda (u1: T).(\lambda (i: nat).(\lambda (H: +(ty3 g c (TLRef i) u1)).(\lambda (H0: (nf2 c (TLRef i))).(\lambda (H1: (nf2 c +u1)).(let H_x \def (ty3_inv_lref_nf2 g c u1 i H H0 H1) in (let H2 \def H_x in +(ex_ind T (\lambda (u0: T).(eq T u1 (lift (S i) O u0))) (ex2 T (\lambda (u: +T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3 c (lift (S i) O u) u1))) +(\lambda (x: T).(\lambda (H3: (eq T u1 (lift (S i) O x))).(let H4 \def +(eq_ind T u1 (\lambda (t: T).(nf2 c t)) H1 (lift (S i) O x) H3) in (eq_ind_r +T (lift (S i) O x) (\lambda (t: T).(ex2 T (\lambda (u: T).(nf2 c (lift (S i) +O u))) (\lambda (u: T).(pc3 c (lift (S i) O u) t)))) (ex_intro2 T (\lambda +(u: T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3 c (lift (S i) O u) +(lift (S i) O x))) x H4 (pc3_refl c (lift (S i) O x))) u1 H3)))) H2)))))))) +(\lambda (t: T).(\lambda (t0: TList).(\lambda (H: ((\forall (u1: T).(\forall +(i: nat).((ty3 g c (THeads (Flat Appl) t0 (TLRef i)) u1) \to ((nf2 c (TLRef +i)) \to ((nf2 c u1) \to (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O u))) +(\lambda (u: T).(pc3 c (THeads (Flat Appl) t0 (lift (S i) O u)) +u1)))))))))).(\lambda (u1: T).(\lambda (i: nat).(\lambda (H0: (ty3 g c (THead +(Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) u1)).(\lambda (H1: (nf2 c +(TLRef i))).(\lambda (_: (nf2 c u1)).(let H_x \def (ty3_gen_appl_nf2 g c t +(THeads (Flat Appl) t0 (TLRef i)) u1 H0) in (let H3 \def H_x in (ex4_2_ind T +T (\lambda (u: T).(\lambda (t1: T).(pc3 c (THead (Flat Appl) t (THead (Bind +Abst) u t1)) u1))) (\lambda (u: T).(\lambda (t1: T).(ty3 g c (THeads (Flat +Appl) t0 (TLRef i)) (THead (Bind Abst) u t1)))) (\lambda (u: T).(\lambda (_: +T).(ty3 g c t u))) (\lambda (u: T).(\lambda (t1: T).(nf2 c (THead (Bind Abst) +u t1)))) (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O u))) (\lambda (u: +T).(pc3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O u))) +u1))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (pc3 c (THead (Flat +Appl) t (THead (Bind Abst) x0 x1)) u1)).(\lambda (H5: (ty3 g c (THeads (Flat +Appl) t0 (TLRef i)) (THead (Bind Abst) x0 x1))).(\lambda (_: (ty3 g c t +x0)).(\lambda (H7: (nf2 c (THead (Bind Abst) x0 x1))).(let H8 \def +(nf2_gen_abst c x0 x1 H7) in (land_ind (nf2 c x0) (nf2 (CHead c (Bind Abst) +x0) x1) (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3 +c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O u))) u1))) +(\lambda (H9: (nf2 c x0)).(\lambda (H10: (nf2 (CHead c (Bind Abst) x0) +x1)).(let H_y \def (H (THead (Bind Abst) x0 x1) i H5 H1) in (let H11 \def +(H_y (nf2_abst_shift c x0 H9 x1 H10)) in (ex2_ind T (\lambda (u: T).(nf2 c +(lift (S i) O u))) (\lambda (u: T).(pc3 c (THeads (Flat Appl) t0 (lift (S i) +O u)) (THead (Bind Abst) x0 x1))) (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O +u))) (\lambda (u: T).(pc3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift +(S i) O u))) u1))) (\lambda (x: T).(\lambda (H12: (nf2 c (lift (S i) O +x))).(\lambda (H13: (pc3 c (THeads (Flat Appl) t0 (lift (S i) O x)) (THead +(Bind Abst) x0 x1))).(ex_intro2 T (\lambda (u: T).(nf2 c (lift (S i) O u))) +(\lambda (u: T).(pc3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S +i) O u))) u1)) x H12 (pc3_t (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) c +(THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O x))) (pc3_thin_dx c +(THeads (Flat Appl) t0 (lift (S i) O x)) (THead (Bind Abst) x0 x1) H13 t +Appl) u1 H4))))) H11))))) H8)))))))) H3))))))))))) vs))). +(* COMMENTS +Initial nodes: 1213 +END *) + +theorem ty3_inv_lref_lref_nf2: + \forall (g: G).(\forall (c: C).(\forall (i: nat).(\forall (j: nat).((ty3 g c +(TLRef i) (TLRef j)) \to ((nf2 c (TLRef i)) \to ((nf2 c (TLRef j)) \to (lt i +j))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (i: nat).(\lambda (j: nat).(\lambda +(H: (ty3 g c (TLRef i) (TLRef j))).(\lambda (H0: (nf2 c (TLRef i))).(\lambda +(H1: (nf2 c (TLRef j))).(let H_x \def (ty3_inv_lref_nf2 g c (TLRef j) i H H0 +H1) in (let H2 \def H_x in (ex_ind T (\lambda (u0: T).(eq T (TLRef j) (lift +(S i) O u0))) (lt i j) (\lambda (x: T).(\lambda (H3: (eq T (TLRef j) (lift (S +i) O x))).(let H_x0 \def (lift_gen_lref x O (S i) j H3) in (let H4 \def H_x0 +in (or_ind (land (lt j O) (eq T x (TLRef j))) (land (le (S i) j) (eq T x +(TLRef (minus j (S i))))) (lt i j) (\lambda (H5: (land (lt j O) (eq T x +(TLRef j)))).(land_ind (lt j O) (eq T x (TLRef j)) (lt i j) (\lambda (H6: (lt +j O)).(\lambda (_: (eq T x (TLRef j))).(lt_x_O j H6 (lt i j)))) H5)) (\lambda +(H5: (land (le (S i) j) (eq T x (TLRef (minus j (S i)))))).(land_ind (le (S +i) j) (eq T x (TLRef (minus j (S i)))) (lt i j) (\lambda (H6: (le (S i) +j)).(\lambda (_: (eq T x (TLRef (minus j (S i))))).H6)) H5)) H4))))) +H2))))))))). +(* COMMENTS +Initial nodes: 337 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/ty3/nf2.ma b/matita/matita/contribs/lambdadelta/basic_1/ty3/nf2.ma new file mode 100644 index 000000000..47b675663 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/ty3/nf2.ma @@ -0,0 +1,472 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/ty3/arity.ma". + +include "Basic-1/pc3/nf2.ma". + +include "Basic-1/nf2/arity.ma". + +definition ty3_nf2_inv_abst_premise: + C \to (T \to (T \to Prop)) +\def + \lambda (c: C).(\lambda (w: T).(\lambda (u: T).(\forall (d: C).(\forall (wi: +T).(\forall (i: nat).((getl i c (CHead d (Bind Abst) wi)) \to (\forall (vs: +TList).((pc3 c (THeads (Flat Appl) vs (lift (S i) O wi)) (THead (Bind Abst) w +u)) \to False)))))))). + +theorem ty3_nf2_inv_abst_premise_csort: + \forall (w: T).(\forall (u: T).(\forall (m: nat).(ty3_nf2_inv_abst_premise +(CSort m) w u))) +\def + \lambda (w: T).(\lambda (u: T).(\lambda (m: nat).(\lambda (d: C).(\lambda +(wi: T).(\lambda (i: nat).(\lambda (H: (getl i (CSort m) (CHead d (Bind Abst) +wi))).(\lambda (vs: TList).(\lambda (_: (pc3 (CSort m) (THeads (Flat Appl) vs +(lift (S i) O wi)) (THead (Bind Abst) w u))).(getl_gen_sort m i (CHead d +(Bind Abst) wi) H False))))))))). +(* COMMENTS +Initial nodes: 85 +END *) + +theorem ty3_nf2_inv_all: + \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t +u) \to ((nf2 c t) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T +t (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c w))) +(\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c (Bind Abst) w) u0)))) (ex nat +(\lambda (n: nat).(eq T t (TSort n)))) (ex3_2 TList nat (\lambda (ws: +TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i))))) +(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: +TList).(\lambda (i: nat).(nf2 c (TLRef i))))))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H: +(ty3 g c t u)).(\lambda (H0: (nf2 c t)).(let H_x \def (ty3_arity g c t u H) +in (let H1 \def H_x in (ex2_ind A (\lambda (a1: A).(arity g c t a1)) (\lambda +(a1: A).(arity g c u (asucc g a1))) (or3 (ex3_2 T T (\lambda (w: T).(\lambda +(u0: T).(eq T t (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: +T).(nf2 c w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c (Bind Abst) w) +u0)))) (ex nat (\lambda (n: nat).(eq T t (TSort n)))) (ex3_2 TList nat +(\lambda (ws: TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef +i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: +TList).(\lambda (i: nat).(nf2 c (TLRef i)))))) (\lambda (x: A).(\lambda (H2: +(arity g c t x)).(\lambda (_: (arity g c u (asucc g x))).(arity_nf2_inv_all g +c t x H2 H0)))) H1)))))))). +(* COMMENTS +Initial nodes: 233 +END *) + +theorem ty3_nf2_inv_sort: + \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (m: nat).((ty3 g c t +(TSort m)) \to ((nf2 c t) \to (or (ex2 nat (\lambda (n: nat).(eq T t (TSort +n))) (\lambda (n: nat).(eq nat m (next g n)))) (ex3_2 TList nat (\lambda (ws: +TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i))))) +(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: +TList).(\lambda (i: nat).(nf2 c (TLRef i))))))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (m: nat).(\lambda +(H: (ty3 g c t (TSort m))).(\lambda (H0: (nf2 c t)).(let H_x \def +(ty3_nf2_inv_all g c t (TSort m) H H0) in (let H1 \def H_x in (or3_ind (ex3_2 +T T (\lambda (w: T).(\lambda (u: T).(eq T t (THead (Bind Abst) w u)))) +(\lambda (w: T).(\lambda (_: T).(nf2 c w))) (\lambda (w: T).(\lambda (u: +T).(nf2 (CHead c (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t +(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t +(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: +nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef i))))) +(or (ex2 nat (\lambda (n: nat).(eq T t (TSort n))) (\lambda (n: nat).(eq nat +m (next g n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T +t (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: +nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef +i)))))) (\lambda (H2: (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t +(THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c w))) +(\lambda (w: T).(\lambda (u: T).(nf2 (CHead c (Bind Abst) w) +u))))).(ex3_2_ind T T (\lambda (w: T).(\lambda (u: T).(eq T t (THead (Bind +Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c w))) (\lambda (w: +T).(\lambda (u: T).(nf2 (CHead c (Bind Abst) w) u))) (or (ex2 nat (\lambda +(n: nat).(eq T t (TSort n))) (\lambda (n: nat).(eq nat m (next g n)))) (ex3_2 +TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) +ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) +(\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef i)))))) (\lambda (x0: +T).(\lambda (x1: T).(\lambda (H3: (eq T t (THead (Bind Abst) x0 +x1))).(\lambda (_: (nf2 c x0)).(\lambda (_: (nf2 (CHead c (Bind Abst) x0) +x1)).(let H6 \def (eq_ind T t (\lambda (t0: T).(ty3 g c t0 (TSort m))) H +(THead (Bind Abst) x0 x1) H3) in (eq_ind_r T (THead (Bind Abst) x0 x1) +(\lambda (t0: T).(or (ex2 nat (\lambda (n: nat).(eq T t0 (TSort n))) (\lambda +(n: nat).(eq nat m (next g n)))) (ex3_2 TList nat (\lambda (ws: +TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i))))) +(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: +TList).(\lambda (i: nat).(nf2 c (TLRef i))))))) (ex3_2_ind T T (\lambda (t2: +T).(\lambda (_: T).(pc3 c (THead (Bind Abst) x0 t2) (TSort m)))) (\lambda (_: +T).(\lambda (t0: T).(ty3 g c x0 t0))) (\lambda (t2: T).(\lambda (_: T).(ty3 g +(CHead c (Bind Abst) x0) x1 t2))) (or (ex2 nat (\lambda (n: nat).(eq T (THead +(Bind Abst) x0 x1) (TSort n))) (\lambda (n: nat).(eq nat m (next g n)))) +(ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Bind +Abst) x0 x1) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: +TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: +nat).(nf2 c (TLRef i)))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: +(pc3 c (THead (Bind Abst) x0 x2) (TSort m))).(\lambda (_: (ty3 g c x0 +x3)).(\lambda (_: (ty3 g (CHead c (Bind Abst) x0) x1 x2)).(pc3_gen_sort_abst +c x0 x2 m (pc3_s c (TSort m) (THead (Bind Abst) x0 x2) H7) (or (ex2 nat +(\lambda (n: nat).(eq T (THead (Bind Abst) x0 x1) (TSort n))) (\lambda (n: +nat).(eq nat m (next g n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda +(i: nat).(eq T (THead (Bind Abst) x0 x1) (THeads (Flat Appl) ws (TLRef i))))) +(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: +TList).(\lambda (i: nat).(nf2 c (TLRef i)))))))))))) (ty3_gen_bind g Abst c +x0 x1 (TSort m) H6)) t H3))))))) H2)) (\lambda (H2: (ex nat (\lambda (n: +nat).(eq T t (TSort n))))).(ex_ind nat (\lambda (n: nat).(eq T t (TSort n))) +(or (ex2 nat (\lambda (n: nat).(eq T t (TSort n))) (\lambda (n: nat).(eq nat +m (next g n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T +t (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: +nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef +i)))))) (\lambda (x: nat).(\lambda (H3: (eq T t (TSort x))).(let H4 \def +(eq_ind T t (\lambda (t0: T).(ty3 g c t0 (TSort m))) H (TSort x) H3) in +(eq_ind_r T (TSort x) (\lambda (t0: T).(or (ex2 nat (\lambda (n: nat).(eq T +t0 (TSort n))) (\lambda (n: nat).(eq nat m (next g n)))) (ex3_2 TList nat +(\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef +i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: +TList).(\lambda (i: nat).(nf2 c (TLRef i))))))) (eq_ind nat (next g x) +(\lambda (n: nat).(or (ex2 nat (\lambda (n0: nat).(eq T (TSort x) (TSort +n0))) (\lambda (n0: nat).(eq nat n (next g n0)))) (ex3_2 TList nat (\lambda +(ws: TList).(\lambda (i: nat).(eq T (TSort x) (THeads (Flat Appl) ws (TLRef +i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: +TList).(\lambda (i: nat).(nf2 c (TLRef i))))))) (or_introl (ex2 nat (\lambda +(n: nat).(eq T (TSort x) (TSort n))) (\lambda (n: nat).(eq nat (next g x) +(next g n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T +(TSort x) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda +(_: nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef +i))))) (ex_intro2 nat (\lambda (n: nat).(eq T (TSort x) (TSort n))) (\lambda +(n: nat).(eq nat (next g x) (next g n))) x (refl_equal T (TSort x)) +(refl_equal nat (next g x)))) m (pc3_gen_sort c (next g x) m (ty3_gen_sort g +c (TSort m) x H4))) t H3)))) H2)) (\lambda (H2: (ex3_2 TList nat (\lambda +(ws: TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i))))) +(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: +TList).(\lambda (i: nat).(nf2 c (TLRef i)))))).(ex3_2_ind TList nat (\lambda +(ws: TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i))))) +(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: +TList).(\lambda (i: nat).(nf2 c (TLRef i)))) (or (ex2 nat (\lambda (n: +nat).(eq T t (TSort n))) (\lambda (n: nat).(eq nat m (next g n)))) (ex3_2 +TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) +ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) +(\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef i)))))) (\lambda (x0: +TList).(\lambda (x1: nat).(\lambda (H3: (eq T t (THeads (Flat Appl) x0 (TLRef +x1)))).(\lambda (H4: (nfs2 c x0)).(\lambda (H5: (nf2 c (TLRef x1))).(let H6 +\def (eq_ind T t (\lambda (t0: T).(ty3 g c t0 (TSort m))) H (THeads (Flat +Appl) x0 (TLRef x1)) H3) in (eq_ind_r T (THeads (Flat Appl) x0 (TLRef x1)) +(\lambda (t0: T).(or (ex2 nat (\lambda (n: nat).(eq T t0 (TSort n))) (\lambda +(n: nat).(eq nat m (next g n)))) (ex3_2 TList nat (\lambda (ws: +TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i))))) +(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: +TList).(\lambda (i: nat).(nf2 c (TLRef i))))))) (or_intror (ex2 nat (\lambda +(n: nat).(eq T (THeads (Flat Appl) x0 (TLRef x1)) (TSort n))) (\lambda (n: +nat).(eq nat m (next g n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda +(i: nat).(eq T (THeads (Flat Appl) x0 (TLRef x1)) (THeads (Flat Appl) ws +(TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda +(_: TList).(\lambda (i: nat).(nf2 c (TLRef i))))) (ex3_2_intro TList nat +(\lambda (ws: TList).(\lambda (i: nat).(eq T (THeads (Flat Appl) x0 (TLRef +x1)) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: +nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef i)))) +x0 x1 (refl_equal T (THeads (Flat Appl) x0 (TLRef x1))) H4 H5)) t H3))))))) +H2)) H1)))))))). +(* COMMENTS +Initial nodes: 2045 +END *) + +theorem ty3_nf2_gen__ty3_nf2_inv_abst_aux: + \forall (c: C).(\forall (w1: T).(\forall (u1: T).((ty3_nf2_inv_abst_premise +c w1 u1) \to (\forall (t: T).(\forall (w2: T).(\forall (u2: T).((pc3 c (THead +(Flat Appl) t (THead (Bind Abst) w2 u2)) (THead (Bind Abst) w1 u1)) \to +(ty3_nf2_inv_abst_premise c w2 u2)))))))) +\def + \lambda (c: C).(\lambda (w1: T).(\lambda (u1: T).(\lambda (H: ((\forall (d: +C).(\forall (wi: T).(\forall (i: nat).((getl i c (CHead d (Bind Abst) wi)) +\to (\forall (vs: TList).((pc3 c (THeads (Flat Appl) vs (lift (S i) O wi)) +(THead (Bind Abst) w1 u1)) \to False)))))))).(\lambda (t: T).(\lambda (w2: +T).(\lambda (u2: T).(\lambda (H0: (pc3 c (THead (Flat Appl) t (THead (Bind +Abst) w2 u2)) (THead (Bind Abst) w1 u1))).(\lambda (d: C).(\lambda (wi: +T).(\lambda (i: nat).(\lambda (H1: (getl i c (CHead d (Bind Abst) +wi))).(\lambda (vs: TList).(\lambda (H2: (pc3 c (THeads (Flat Appl) vs (lift +(S i) O wi)) (THead (Bind Abst) w2 u2))).(H d wi i H1 (TCons t vs) (pc3_t +(THead (Flat Appl) t (THead (Bind Abst) w2 u2)) c (THead (Flat Appl) t +(THeads (Flat Appl) vs (lift (S i) O wi))) (pc3_thin_dx c (THeads (Flat Appl) +vs (lift (S i) O wi)) (THead (Bind Abst) w2 u2) H2 t Appl) (THead (Bind Abst) +w1 u1) H0))))))))))))))). +(* COMMENTS +Initial nodes: 271 +END *) + +theorem ty3_nf2_inv_abst: + \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (w: T).(\forall (u: +T).((ty3 g c t (THead (Bind Abst) w u)) \to ((nf2 c t) \to ((nf2 c w) \to +((ty3_nf2_inv_abst_premise c w u) \to (ex4_2 T T (\lambda (v: T).(\lambda (_: +T).(eq T t (THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g +c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v +u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) +v)))))))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (w: T).(\lambda (u: +T).(\lambda (H: (ty3 g c t (THead (Bind Abst) w u))).(\lambda (H0: (nf2 c +t)).(\lambda (H1: (nf2 c w)).(\lambda (H2: (ty3_nf2_inv_abst_premise c w +u)).(let H_x \def (ty3_nf2_inv_all g c t (THead (Bind Abst) w u) H H0) in +(let H3 \def H_x in (or3_ind (ex3_2 T T (\lambda (w0: T).(\lambda (u0: T).(eq +T t (THead (Bind Abst) w0 u0)))) (\lambda (w0: T).(\lambda (_: T).(nf2 c +w0))) (\lambda (w0: T).(\lambda (u0: T).(nf2 (CHead c (Bind Abst) w0) u0)))) +(ex nat (\lambda (n: nat).(eq T t (TSort n)))) (ex3_2 TList nat (\lambda (ws: +TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i))))) +(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: +TList).(\lambda (i: nat).(nf2 c (TLRef i))))) (ex4_2 T T (\lambda (v: +T).(\lambda (_: T).(eq T t (THead (Bind Abst) w v)))) (\lambda (_: +T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g +(CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c +(Bind Abst) w) v)))) (\lambda (H4: (ex3_2 T T (\lambda (w0: T).(\lambda (u0: +T).(eq T t (THead (Bind Abst) w0 u0)))) (\lambda (w0: T).(\lambda (_: T).(nf2 +c w0))) (\lambda (w0: T).(\lambda (u0: T).(nf2 (CHead c (Bind Abst) w0) +u0))))).(ex3_2_ind T T (\lambda (w0: T).(\lambda (u0: T).(eq T t (THead (Bind +Abst) w0 u0)))) (\lambda (w0: T).(\lambda (_: T).(nf2 c w0))) (\lambda (w0: +T).(\lambda (u0: T).(nf2 (CHead c (Bind Abst) w0) u0))) (ex4_2 T T (\lambda +(v: T).(\lambda (_: T).(eq T t (THead (Bind Abst) w v)))) (\lambda (_: +T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g +(CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c +(Bind Abst) w) v)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T t +(THead (Bind Abst) x0 x1))).(\lambda (H6: (nf2 c x0)).(\lambda (H7: (nf2 +(CHead c (Bind Abst) x0) x1)).(let H8 \def (eq_ind T t (\lambda (t0: T).(ty3 +g c t0 (THead (Bind Abst) w u))) H (THead (Bind Abst) x0 x1) H5) in (eq_ind_r +T (THead (Bind Abst) x0 x1) (\lambda (t0: T).(ex4_2 T T (\lambda (v: +T).(\lambda (_: T).(eq T t0 (THead (Bind Abst) w v)))) (\lambda (_: +T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g +(CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c +(Bind Abst) w) v))))) (ex_ind T (\lambda (t0: T).(ty3 g c (THead (Bind Abst) +w u) t0)) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (THead (Bind Abst) +x0 x1) (THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w +w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u))) +(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v)))) (\lambda +(x: T).(\lambda (H9: (ty3 g c (THead (Bind Abst) w u) x)).(ex3_2_ind T T +(\lambda (t2: T).(\lambda (_: T).(pc3 c (THead (Bind Abst) w t2) x))) +(\lambda (_: T).(\lambda (t0: T).(ty3 g c w t0))) (\lambda (t2: T).(\lambda +(_: T).(ty3 g (CHead c (Bind Abst) w) u t2))) (ex4_2 T T (\lambda (v: +T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) w v)))) +(\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda +(_: T).(ty3 g (CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: +T).(nf2 (CHead c (Bind Abst) w) v)))) (\lambda (x2: T).(\lambda (x3: +T).(\lambda (_: (pc3 c (THead (Bind Abst) w x2) x)).(\lambda (H11: (ty3 g c w +x3)).(\lambda (H12: (ty3 g (CHead c (Bind Abst) w) u x2)).(ex3_2_ind T T +(\lambda (t2: T).(\lambda (_: T).(pc3 c (THead (Bind Abst) x0 t2) (THead +(Bind Abst) w u)))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c x0 t0))) +(\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x0) x1 t2))) +(ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) +(THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) +(\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u))) +(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v)))) (\lambda +(x4: T).(\lambda (x5: T).(\lambda (H13: (pc3 c (THead (Bind Abst) x0 x4) +(THead (Bind Abst) w u))).(\lambda (_: (ty3 g c x0 x5)).(\lambda (H15: (ty3 g +(CHead c (Bind Abst) x0) x1 x4)).(land_ind (pc3 c x0 w) (\forall (b: +B).(\forall (u0: T).(pc3 (CHead c (Bind b) u0) x4 u))) (ex4_2 T T (\lambda +(v: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) w +v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: +T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u))) (\lambda (v: +T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v)))) (\lambda (H16: (pc3 c +x0 w)).(\lambda (H17: ((\forall (b: B).(\forall (u0: T).(pc3 (CHead c (Bind +b) u0) x4 u))))).(let H_y \def (pc3_nf2 c x0 w H16 H6 H1) in (let H18 \def +(eq_ind T x0 (\lambda (t0: T).(ty3 g (CHead c (Bind Abst) t0) x1 x4)) H15 w +H_y) in (let H19 \def (eq_ind T x0 (\lambda (t0: T).(nf2 (CHead c (Bind Abst) +t0) x1)) H7 w H_y) in (eq_ind_r T w (\lambda (t0: T).(ex4_2 T T (\lambda (v: +T).(\lambda (_: T).(eq T (THead (Bind Abst) t0 x1) (THead (Bind Abst) w v)))) +(\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda +(_: T).(ty3 g (CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: +T).(nf2 (CHead c (Bind Abst) w) v))))) (ex4_2_intro T T (\lambda (v: +T).(\lambda (_: T).(eq T (THead (Bind Abst) w x1) (THead (Bind Abst) w v)))) +(\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda +(_: T).(ty3 g (CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: +T).(nf2 (CHead c (Bind Abst) w) v))) x1 x3 (refl_equal T (THead (Bind Abst) w +x1)) H11 (ty3_conv g (CHead c (Bind Abst) w) u x2 H12 x1 x4 H18 (H17 Abst w)) +H19) x0 H_y)))))) (pc3_gen_abst c x0 w x4 u H13))))))) (ty3_gen_bind g Abst c +x0 x1 (THead (Bind Abst) w u) H8))))))) (ty3_gen_bind g Abst c w u x H9)))) +(ty3_correct g c (THead (Bind Abst) x0 x1) (THead (Bind Abst) w u) H8)) t +H5))))))) H4)) (\lambda (H4: (ex nat (\lambda (n: nat).(eq T t (TSort +n))))).(ex_ind nat (\lambda (n: nat).(eq T t (TSort n))) (ex4_2 T T (\lambda +(v: T).(\lambda (_: T).(eq T t (THead (Bind Abst) w v)))) (\lambda (_: +T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g +(CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c +(Bind Abst) w) v)))) (\lambda (x: nat).(\lambda (H5: (eq T t (TSort x))).(let +H6 \def (eq_ind T t (\lambda (t0: T).(ty3 g c t0 (THead (Bind Abst) w u))) H +(TSort x) H5) in (eq_ind_r T (TSort x) (\lambda (t0: T).(ex4_2 T T (\lambda +(v: T).(\lambda (_: T).(eq T t0 (THead (Bind Abst) w v)))) (\lambda (_: +T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g +(CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c +(Bind Abst) w) v))))) (pc3_gen_sort_abst c w u (next g x) (ty3_gen_sort g c +(THead (Bind Abst) w u) x H6) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq +T (TSort x) (THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 +g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v +u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v))))) t +H5)))) H4)) (\lambda (H4: (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: +nat).(eq T t (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: +TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: +nat).(nf2 c (TLRef i)))))).(ex3_2_ind TList nat (\lambda (ws: TList).(\lambda +(i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: +TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: +nat).(nf2 c (TLRef i)))) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T t +(THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) +(\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u))) +(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v)))) (\lambda +(x0: TList).(\lambda (x1: nat).(\lambda (H5: (eq T t (THeads (Flat Appl) x0 +(TLRef x1)))).(\lambda (_: (nfs2 c x0)).(\lambda (H7: (nf2 c (TLRef +x1))).(let H8 \def (eq_ind T t (\lambda (t0: T).(ty3 g c t0 (THead (Bind +Abst) w u))) H (THeads (Flat Appl) x0 (TLRef x1)) H5) in (eq_ind_r T (THeads +(Flat Appl) x0 (TLRef x1)) (\lambda (t0: T).(ex4_2 T T (\lambda (v: +T).(\lambda (_: T).(eq T t0 (THead (Bind Abst) w v)))) (\lambda (_: +T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g +(CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c +(Bind Abst) w) v))))) (let H9 \def H2 in ((let H10 \def H8 in (unintro T u +(\lambda (t0: T).((ty3 g c (THeads (Flat Appl) x0 (TLRef x1)) (THead (Bind +Abst) w t0)) \to ((ty3_nf2_inv_abst_premise c w t0) \to (ex4_2 T T (\lambda +(v: T).(\lambda (_: T).(eq T (THeads (Flat Appl) x0 (TLRef x1)) (THead (Bind +Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: +T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v t0))) (\lambda (v: +T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v))))))) (unintro T w +(\lambda (t0: T).(\forall (x: T).((ty3 g c (THeads (Flat Appl) x0 (TLRef x1)) +(THead (Bind Abst) t0 x)) \to ((ty3_nf2_inv_abst_premise c t0 x) \to (ex4_2 T +T (\lambda (v: T).(\lambda (_: T).(eq T (THeads (Flat Appl) x0 (TLRef x1)) +(THead (Bind Abst) t0 v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c t0 +w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) t0) v x))) +(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) t0) v)))))))) +(TList_ind (\lambda (t0: TList).(\forall (x: T).(\forall (x2: T).((ty3 g c +(THeads (Flat Appl) t0 (TLRef x1)) (THead (Bind Abst) x x2)) \to +((ty3_nf2_inv_abst_premise c x x2) \to (ex4_2 T T (\lambda (v: T).(\lambda +(_: T).(eq T (THeads (Flat Appl) t0 (TLRef x1)) (THead (Bind Abst) x v)))) +(\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda +(_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_: +T).(nf2 (CHead c (Bind Abst) x) v))))))))) (\lambda (x: T).(\lambda (x2: +T).(\lambda (H11: (ty3 g c (TLRef x1) (THead (Bind Abst) x x2))).(\lambda +(H12: (ty3_nf2_inv_abst_premise c x x2)).(or_ind (ex3_3 C T T (\lambda (_: +C).(\lambda (_: T).(\lambda (t0: T).(pc3 c (lift (S x1) O t0) (THead (Bind +Abst) x x2))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl x1 c +(CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: +T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda +(_: T).(pc3 c (lift (S x1) O u0) (THead (Bind Abst) x x2))))) (\lambda (e: +C).(\lambda (u0: T).(\lambda (_: T).(getl x1 c (CHead e (Bind Abst) u0))))) +(\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex4_2 +T T (\lambda (v: T).(\lambda (_: T).(eq T (TLRef x1) (THead (Bind Abst) x +v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: +T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v: +T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) v)))) (\lambda (H13: (ex3_3 C +T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c (lift (S x1) O +t0) (THead (Bind Abst) x x2))))) (\lambda (e: C).(\lambda (u0: T).(\lambda +(_: T).(getl x1 c (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: +T).(\lambda (t0: T).(ty3 g e u0 t0)))))).(ex3_3_ind C T T (\lambda (_: +C).(\lambda (_: T).(\lambda (t0: T).(pc3 c (lift (S x1) O t0) (THead (Bind +Abst) x x2))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl x1 c +(CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: +T).(ty3 g e u0 t0)))) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (TLRef +x1) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c x +w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) +(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) v)))) (\lambda +(x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (pc3 c (lift (S x1) O +x5) (THead (Bind Abst) x x2))).(\lambda (H15: (getl x1 c (CHead x3 (Bind +Abbr) x4))).(\lambda (_: (ty3 g x3 x4 x5)).(nf2_gen_lref c x3 x4 x1 H15 H7 +(ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (TLRef x1) (THead (Bind +Abst) x v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: +T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v: +T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) v))))))))))) H13)) (\lambda +(H13: (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c +(lift (S x1) O u0) (THead (Bind Abst) x x2))))) (\lambda (e: C).(\lambda (u0: +T).(\lambda (_: T).(getl x1 c (CHead e (Bind Abst) u0))))) (\lambda (e: +C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))))).(ex3_3_ind C T T +(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c (lift (S x1) O u0) +(THead (Bind Abst) x x2))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: +T).(getl x1 c (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: +T).(\lambda (t0: T).(ty3 g e u0 t0)))) (ex4_2 T T (\lambda (v: T).(\lambda +(_: T).(eq T (TLRef x1) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda +(w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c +(Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind +Abst) x) v)))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda +(H14: (pc3 c (lift (S x1) O x4) (THead (Bind Abst) x x2))).(\lambda (H15: +(getl x1 c (CHead x3 (Bind Abst) x4))).(\lambda (_: (ty3 g x3 x4 x5)).(let +H_x0 \def (H12 x3 x4 x1 H15 TNil H14) in (let H17 \def H_x0 in (False_ind +(ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (TLRef x1) (THead (Bind +Abst) x v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: +T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v: +T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) v)))) H17))))))))) H13)) +(ty3_gen_lref g c (THead (Bind Abst) x x2) x1 H11)))))) (\lambda (t0: +T).(\lambda (t1: TList).(\lambda (H11: ((\forall (x: T).(\forall (x2: +T).((ty3 g c (THeads (Flat Appl) t1 (TLRef x1)) (THead (Bind Abst) x x2)) \to +((ty3_nf2_inv_abst_premise c x x2) \to (ex4_2 T T (\lambda (v: T).(\lambda +(_: T).(eq T (THeads (Flat Appl) t1 (TLRef x1)) (THead (Bind Abst) x v)))) +(\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda +(_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_: +T).(nf2 (CHead c (Bind Abst) x) v)))))))))).(\lambda (x: T).(\lambda (x2: +T).(\lambda (H12: (ty3 g c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 +(TLRef x1))) (THead (Bind Abst) x x2))).(\lambda (H13: +(ty3_nf2_inv_abst_premise c x x2)).(ex3_2_ind T T (\lambda (u0: T).(\lambda +(t2: T).(pc3 c (THead (Flat Appl) t0 (THead (Bind Abst) u0 t2)) (THead (Bind +Abst) x x2)))) (\lambda (u0: T).(\lambda (t2: T).(ty3 g c (THeads (Flat Appl) +t1 (TLRef x1)) (THead (Bind Abst) u0 t2)))) (\lambda (u0: T).(\lambda (_: +T).(ty3 g c t0 u0))) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (THead +(Flat Appl) t0 (THeads (Flat Appl) t1 (TLRef x1))) (THead (Bind Abst) x v)))) +(\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda +(_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_: +T).(nf2 (CHead c (Bind Abst) x) v)))) (\lambda (x3: T).(\lambda (x4: +T).(\lambda (H14: (pc3 c (THead (Flat Appl) t0 (THead (Bind Abst) x3 x4)) +(THead (Bind Abst) x x2))).(\lambda (H15: (ty3 g c (THeads (Flat Appl) t1 +(TLRef x1)) (THead (Bind Abst) x3 x4))).(\lambda (_: (ty3 g c t0 x3)).(let +H_y \def (ty3_nf2_gen__ty3_nf2_inv_abst_aux c x x2 H13 t0 x3 x4 H14) in (let +H_x0 \def (H11 x3 x4 H15 H_y) in (let H17 \def H_x0 in (ex4_2_ind T T +(\lambda (v: T).(\lambda (_: T).(eq T (THeads (Flat Appl) t1 (TLRef x1)) +(THead (Bind Abst) x3 v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c x3 +w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x3) v x4))) +(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x3) v))) (ex4_2 T T +(\lambda (v: T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (THeads (Flat +Appl) t1 (TLRef x1))) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda +(w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c +(Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind +Abst) x) v)))) (\lambda (x5: T).(\lambda (x6: T).(\lambda (H18: (eq T (THeads +(Flat Appl) t1 (TLRef x1)) (THead (Bind Abst) x3 x5))).(\lambda (_: (ty3 g c +x3 x6)).(\lambda (_: (ty3 g (CHead c (Bind Abst) x3) x5 x4)).(\lambda (_: +(nf2 (CHead c (Bind Abst) x3) x5)).(TList_ind (\lambda (t2: TList).((eq T +(THeads (Flat Appl) t2 (TLRef x1)) (THead (Bind Abst) x3 x5)) \to (ex4_2 T T +(\lambda (v: T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (THeads (Flat +Appl) t2 (TLRef x1))) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda +(w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c +(Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind +Abst) x) v)))))) (\lambda (H22: (eq T (THeads (Flat Appl) TNil (TLRef x1)) +(THead (Bind Abst) x3 x5))).(let H23 \def (eq_ind T (TLRef x1) (\lambda (ee: +T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow +False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I +(THead (Bind Abst) x3 x5) H22) in (False_ind (ex4_2 T T (\lambda (v: +T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (THeads (Flat Appl) TNil +(TLRef x1))) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 +g c x w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v +x2))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) v)))) +H23))) (\lambda (t2: T).(\lambda (t3: TList).(\lambda (_: (((eq T (THeads +(Flat Appl) t3 (TLRef x1)) (THead (Bind Abst) x3 x5)) \to (ex4_2 T T (\lambda +(v: T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (THeads (Flat Appl) t3 +(TLRef x1))) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 +g c x w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v +x2))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) +v))))))).(\lambda (H22: (eq T (THeads (Flat Appl) (TCons t2 t3) (TLRef x1)) +(THead (Bind Abst) x3 x5))).(let H23 \def (eq_ind T (THead (Flat Appl) t2 +(THeads (Flat Appl) t3 (TLRef x1))) (\lambda (ee: T).(match ee in T return +(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda +(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow +True])])) I (THead (Bind Abst) x3 x5) H22) in (False_ind (ex4_2 T T (\lambda +(v: T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (THeads (Flat Appl) (TCons +t2 t3) (TLRef x1))) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda (w0: +T).(ty3 g c x w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind +Abst) x) v x2))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) +v)))) H23)))))) t1 H18))))))) H17))))))))) (ty3_gen_appl g c t0 (THeads (Flat +Appl) t1 (TLRef x1)) (THead (Bind Abst) x x2) H12))))))))) x0)) H10)) H9)) t +H5))))))) H4)) H3))))))))))). +(* COMMENTS +Initial nodes: 5333 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/ty3/pr3.ma b/matita/matita/contribs/lambdadelta/basic_1/ty3/pr3.ma new file mode 100644 index 000000000..8d184aa58 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/ty3/pr3.ma @@ -0,0 +1,728 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/csubt/ty3.ma". + +include "Basic-1/ty3/subst1.ma". + +include "Basic-1/ty3/fsubst0.ma". + +include "Basic-1/pc3/pc1.ma". + +include "Basic-1/pc3/wcpr0.ma". + +include "Basic-1/pc1/props.ma". + +theorem ty3_sred_wcpr0_pr0: + \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t: T).((ty3 g c1 +t1 t) \to (\forall (c2: C).((wcpr0 c1 c2) \to (\forall (t2: T).((pr0 t1 t2) +\to (ty3 g c2 t2 t))))))))) +\def + \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t: T).(\lambda +(H: (ty3 g c1 t1 t)).(ty3_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda +(t2: T).(\forall (c2: C).((wcpr0 c c2) \to (\forall (t3: T).((pr0 t0 t3) \to +(ty3 g c2 t3 t2)))))))) (\lambda (c: C).(\lambda (t2: T).(\lambda (t0: +T).(\lambda (_: (ty3 g c t2 t0)).(\lambda (H1: ((\forall (c2: C).((wcpr0 c +c2) \to (\forall (t3: T).((pr0 t2 t3) \to (ty3 g c2 t3 t0))))))).(\lambda (u: +T).(\lambda (t3: T).(\lambda (_: (ty3 g c u t3)).(\lambda (H3: ((\forall (c2: +C).((wcpr0 c c2) \to (\forall (t4: T).((pr0 u t4) \to (ty3 g c2 t4 +t3))))))).(\lambda (H4: (pc3 c t3 t2)).(\lambda (c2: C).(\lambda (H5: (wcpr0 +c c2)).(\lambda (t4: T).(\lambda (H6: (pr0 u t4)).(ty3_conv g c2 t2 t0 (H1 c2 +H5 t2 (pr0_refl t2)) t4 t3 (H3 c2 H5 t4 H6) (pc3_wcpr0 c c2 H5 t3 t2 +H4)))))))))))))))) (\lambda (c: C).(\lambda (m: nat).(\lambda (c2: +C).(\lambda (_: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H1: (pr0 (TSort m) +t2)).(eq_ind_r T (TSort m) (\lambda (t0: T).(ty3 g c2 t0 (TSort (next g m)))) +(ty3_sort g c2 m) t2 (pr0_gen_sort t2 m H1)))))))) (\lambda (n: nat).(\lambda +(c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind +Abbr) u))).(\lambda (t0: T).(\lambda (_: (ty3 g d u t0)).(\lambda (H2: +((\forall (c2: C).((wcpr0 d c2) \to (\forall (t2: T).((pr0 u t2) \to (ty3 g +c2 t2 t0))))))).(\lambda (c2: C).(\lambda (H3: (wcpr0 c c2)).(\lambda (t2: +T).(\lambda (H4: (pr0 (TLRef n) t2)).(eq_ind_r T (TLRef n) (\lambda (t3: +T).(ty3 g c2 t3 (lift (S n) O t0))) (ex3_2_ind C T (\lambda (e2: C).(\lambda +(u2: T).(getl n c2 (CHead e2 (Bind Abbr) u2)))) (\lambda (e2: C).(\lambda (_: +T).(wcpr0 d e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u u2))) (ty3 g c2 +(TLRef n) (lift (S n) O t0)) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: +(getl n c2 (CHead x0 (Bind Abbr) x1))).(\lambda (H6: (wcpr0 d x0)).(\lambda +(H7: (pr0 u x1)).(ty3_abbr g n c2 x0 x1 H5 t0 (H2 x0 H6 x1 H7))))))) +(wcpr0_getl c c2 H3 n d u (Bind Abbr) H0)) t2 (pr0_gen_lref t2 n +H4)))))))))))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d: C).(\lambda +(u: T).(\lambda (H0: (getl n c (CHead d (Bind Abst) u))).(\lambda (t0: +T).(\lambda (_: (ty3 g d u t0)).(\lambda (H2: ((\forall (c2: C).((wcpr0 d c2) +\to (\forall (t2: T).((pr0 u t2) \to (ty3 g c2 t2 t0))))))).(\lambda (c2: +C).(\lambda (H3: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H4: (pr0 (TLRef n) +t2)).(eq_ind_r T (TLRef n) (\lambda (t3: T).(ty3 g c2 t3 (lift (S n) O u))) +(ex3_2_ind C T (\lambda (e2: C).(\lambda (u2: T).(getl n c2 (CHead e2 (Bind +Abst) u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 d e2))) (\lambda (_: +C).(\lambda (u2: T).(pr0 u u2))) (ty3 g c2 (TLRef n) (lift (S n) O u)) +(\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: (getl n c2 (CHead x0 (Bind +Abst) x1))).(\lambda (H6: (wcpr0 d x0)).(\lambda (H7: (pr0 u x1)).(ty3_conv g +c2 (lift (S n) O u) (lift (S n) O t0) (ty3_lift g x0 u t0 (H2 x0 H6 u +(pr0_refl u)) c2 O (S n) (getl_drop Abst c2 x0 x1 n H5)) (TLRef n) (lift (S +n) O x1) (ty3_abst g n c2 x0 x1 H5 t0 (H2 x0 H6 x1 H7)) (pc3_lift c2 x0 (S n) +O (getl_drop Abst c2 x0 x1 n H5) x1 u (pc3_pr2_x x0 x1 u (pr2_free x0 u x1 +H7))))))))) (wcpr0_getl c c2 H3 n d u (Bind Abst) H0)) t2 (pr0_gen_lref t2 n +H4)))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (t0: T).(\lambda +(_: (ty3 g c u t0)).(\lambda (H1: ((\forall (c2: C).((wcpr0 c c2) \to +(\forall (t2: T).((pr0 u t2) \to (ty3 g c2 t2 t0))))))).(\lambda (b: +B).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H2: (ty3 g (CHead c (Bind b) +u) t2 t3)).(\lambda (H3: ((\forall (c2: C).((wcpr0 (CHead c (Bind b) u) c2) +\to (\forall (t4: T).((pr0 t2 t4) \to (ty3 g c2 t4 t3))))))).(\lambda (c2: +C).(\lambda (H4: (wcpr0 c c2)).(\lambda (t4: T).(\lambda (H5: (pr0 (THead +(Bind b) u t2) t4)).(let H6 \def (match H5 in pr0 return (\lambda (t5: +T).(\lambda (t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 (THead (Bind b) u +t2)) \to ((eq T t6 t4) \to (ty3 g c2 t4 (THead (Bind b) u t3))))))) with +[(pr0_refl t5) \Rightarrow (\lambda (H6: (eq T t5 (THead (Bind b) u +t2))).(\lambda (H7: (eq T t5 t4)).(eq_ind T (THead (Bind b) u t2) (\lambda +(t6: T).((eq T t6 t4) \to (ty3 g c2 t4 (THead (Bind b) u t3)))) (\lambda (H8: +(eq T (THead (Bind b) u t2) t4)).(eq_ind T (THead (Bind b) u t2) (\lambda +(t6: T).(ty3 g c2 t6 (THead (Bind b) u t3))) (ty3_bind g c2 u t0 (H1 c2 H4 u +(pr0_refl u)) b t2 t3 (H3 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H4 u u +(pr0_refl u) (Bind b)) t2 (pr0_refl t2))) t4 H8)) t5 (sym_eq T t5 (THead +(Bind b) u t2) H6) H7))) | (pr0_comp u1 u2 H6 t5 t6 H7 k) \Rightarrow +(\lambda (H8: (eq T (THead k u1 t5) (THead (Bind b) u t2))).(\lambda (H9: (eq +T (THead k u2 t6) t4)).((let H10 \def (f_equal T T (\lambda (e: T).(match e +in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t5 | (TLRef _) +\Rightarrow t5 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t5) (THead +(Bind b) u t2) H8) in ((let H11 \def (f_equal T T (\lambda (e: T).(match e in +T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) +\Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t5) (THead +(Bind b) u t2) H8) in ((let H12 \def (f_equal T K (\lambda (e: T).(match e in +T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) +\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t5) (THead (Bind +b) u t2) H8) in (eq_ind K (Bind b) (\lambda (k0: K).((eq T u1 u) \to ((eq T +t5 t2) \to ((eq T (THead k0 u2 t6) t4) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to +(ty3 g c2 t4 (THead (Bind b) u t3)))))))) (\lambda (H13: (eq T u1 u)).(eq_ind +T u (\lambda (t7: T).((eq T t5 t2) \to ((eq T (THead (Bind b) u2 t6) t4) \to +((pr0 t7 u2) \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3))))))) +(\lambda (H14: (eq T t5 t2)).(eq_ind T t2 (\lambda (t7: T).((eq T (THead +(Bind b) u2 t6) t4) \to ((pr0 u u2) \to ((pr0 t7 t6) \to (ty3 g c2 t4 (THead +(Bind b) u t3)))))) (\lambda (H15: (eq T (THead (Bind b) u2 t6) t4)).(eq_ind +T (THead (Bind b) u2 t6) (\lambda (t7: T).((pr0 u u2) \to ((pr0 t2 t6) \to +(ty3 g c2 t7 (THead (Bind b) u t3))))) (\lambda (H16: (pr0 u u2)).(\lambda +(H17: (pr0 t2 t6)).(ex_ind T (\lambda (t7: T).(ty3 g (CHead c2 (Bind b) u) t3 +t7)) (ty3 g c2 (THead (Bind b) u2 t6) (THead (Bind b) u t3)) (\lambda (x: +T).(\lambda (H18: (ty3 g (CHead c2 (Bind b) u) t3 x)).(ex_ind T (\lambda (t7: +T).(ty3 g (CHead c2 (Bind b) u2) t3 t7)) (ty3 g c2 (THead (Bind b) u2 t6) +(THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (_: (ty3 g (CHead c2 (Bind +b) u2) t3 x0)).(ty3_conv g c2 (THead (Bind b) u t3) (THead (Bind b) u x) +(ty3_bind g c2 u t0 (H1 c2 H4 u (pr0_refl u)) b t3 x H18) (THead (Bind b) u2 +t6) (THead (Bind b) u2 t3) (ty3_bind g c2 u2 t0 (H1 c2 H4 u2 H16) b t6 t3 (H3 +(CHead c2 (Bind b) u2) (wcpr0_comp c c2 H4 u u2 H16 (Bind b)) t6 H17)) +(pc3_pr2_x c2 (THead (Bind b) u2 t3) (THead (Bind b) u t3) (pr2_head_1 c2 u +u2 (pr2_free c2 u u2 H16) (Bind b) t3))))) (ty3_correct g (CHead c2 (Bind b) +u2) t6 t3 (H3 (CHead c2 (Bind b) u2) (wcpr0_comp c c2 H4 u u2 H16 (Bind b)) +t6 H17))))) (ty3_correct g (CHead c2 (Bind b) u) t2 t3 (H3 (CHead c2 (Bind b) +u) (wcpr0_comp c c2 H4 u u (pr0_refl u) (Bind b)) t2 (pr0_refl t2)))))) t4 +H15)) t5 (sym_eq T t5 t2 H14))) u1 (sym_eq T u1 u H13))) k (sym_eq K k (Bind +b) H12))) H11)) H10)) H9 H6 H7))) | (pr0_beta u0 v1 v2 H6 t5 t6 H7) +\Rightarrow (\lambda (H8: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u0 +t5)) (THead (Bind b) u t2))).(\lambda (H9: (eq T (THead (Bind Abbr) v2 t6) +t4)).((let H10 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u0 +t5)) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort +_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) +\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) +\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u t2) +H8) in (False_ind ((eq T (THead (Bind Abbr) v2 t6) t4) \to ((pr0 v1 v2) \to +((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3))))) H10)) H9 H6 H7))) | +(pr0_upsilon b0 H6 v1 v2 H7 u1 u2 H8 t5 t6 H9) \Rightarrow (\lambda (H10: (eq +T (THead (Flat Appl) v1 (THead (Bind b0) u1 t5)) (THead (Bind b) u +t2))).(\lambda (H11: (eq T (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) +O v2) t6)) t4)).((let H12 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind +b0) u1 t5)) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with +[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) +\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) +\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u t2) +H10) in (False_ind ((eq T (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) +O v2) t6)) t4) \to ((not (eq B b0 Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) +\to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3))))))) H12)) H11 H6 H7 +H8 H9))) | (pr0_delta u1 u2 H6 t5 t6 H7 w H8) \Rightarrow (\lambda (H9: (eq T +(THead (Bind Abbr) u1 t5) (THead (Bind b) u t2))).(\lambda (H10: (eq T (THead +(Bind Abbr) u2 w) t4)).((let H11 \def (f_equal T T (\lambda (e: T).(match e +in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t5 | (TLRef _) +\Rightarrow t5 | (THead _ _ t7) \Rightarrow t7])) (THead (Bind Abbr) u1 t5) +(THead (Bind b) u t2) H9) in ((let H12 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | +(TLRef _) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead (Bind +Abbr) u1 t5) (THead (Bind b) u t2) H9) in ((let H13 \def (f_equal T B +(\lambda (e: T).(match e in T return (\lambda (_: T).B) with [(TSort _) +\Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | (THead k _ _) \Rightarrow +(match k in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | +(Flat _) \Rightarrow Abbr])])) (THead (Bind Abbr) u1 t5) (THead (Bind b) u +t2) H9) in (eq_ind B Abbr (\lambda (b0: B).((eq T u1 u) \to ((eq T t5 t2) \to +((eq T (THead (Bind Abbr) u2 w) t4) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to +((subst0 O u2 t6 w) \to (ty3 g c2 t4 (THead (Bind b0) u t3))))))))) (\lambda +(H14: (eq T u1 u)).(eq_ind T u (\lambda (t7: T).((eq T t5 t2) \to ((eq T +(THead (Bind Abbr) u2 w) t4) \to ((pr0 t7 u2) \to ((pr0 t5 t6) \to ((subst0 O +u2 t6 w) \to (ty3 g c2 t4 (THead (Bind Abbr) u t3)))))))) (\lambda (H15: (eq +T t5 t2)).(eq_ind T t2 (\lambda (t7: T).((eq T (THead (Bind Abbr) u2 w) t4) +\to ((pr0 u u2) \to ((pr0 t7 t6) \to ((subst0 O u2 t6 w) \to (ty3 g c2 t4 +(THead (Bind Abbr) u t3))))))) (\lambda (H16: (eq T (THead (Bind Abbr) u2 w) +t4)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t7: T).((pr0 u u2) \to +((pr0 t2 t6) \to ((subst0 O u2 t6 w) \to (ty3 g c2 t7 (THead (Bind Abbr) u +t3)))))) (\lambda (H17: (pr0 u u2)).(\lambda (H18: (pr0 t2 t6)).(\lambda +(H19: (subst0 O u2 t6 w)).(let H20 \def (eq_ind_r B b (\lambda (b0: +B).(\forall (c3: C).((wcpr0 (CHead c (Bind b0) u) c3) \to (\forall (t7: +T).((pr0 t2 t7) \to (ty3 g c3 t7 t3)))))) H3 Abbr H13) in (let H21 \def +(eq_ind_r B b (\lambda (b0: B).(ty3 g (CHead c (Bind b0) u) t2 t3)) H2 Abbr +H13) in (ex_ind T (\lambda (t7: T).(ty3 g (CHead c2 (Bind Abbr) u) t3 t7)) +(ty3 g c2 (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u t3)) (\lambda (x: +T).(\lambda (H22: (ty3 g (CHead c2 (Bind Abbr) u) t3 x)).(ex_ind T (\lambda +(t7: T).(ty3 g (CHead c2 (Bind Abbr) u2) t3 t7)) (ty3 g c2 (THead (Bind Abbr) +u2 w) (THead (Bind Abbr) u t3)) (\lambda (x0: T).(\lambda (_: (ty3 g (CHead +c2 (Bind Abbr) u2) t3 x0)).(ty3_conv g c2 (THead (Bind Abbr) u t3) (THead +(Bind Abbr) u x) (ty3_bind g c2 u t0 (H1 c2 H4 u (pr0_refl u)) Abbr t3 x H22) +(THead (Bind Abbr) u2 w) (THead (Bind Abbr) u2 t3) (ty3_bind g c2 u2 t0 (H1 +c2 H4 u2 H17) Abbr w t3 (ty3_subst0 g (CHead c2 (Bind Abbr) u2) t6 t3 (H20 +(CHead c2 (Bind Abbr) u2) (wcpr0_comp c c2 H4 u u2 H17 (Bind Abbr)) t6 H18) +c2 u2 O (getl_refl Abbr c2 u2) w H19)) (pc3_pr2_x c2 (THead (Bind Abbr) u2 +t3) (THead (Bind Abbr) u t3) (pr2_head_1 c2 u u2 (pr2_free c2 u u2 H17) (Bind +Abbr) t3))))) (ty3_correct g (CHead c2 (Bind Abbr) u2) t6 t3 (H20 (CHead c2 +(Bind Abbr) u2) (wcpr0_comp c c2 H4 u u2 H17 (Bind Abbr)) t6 H18))))) +(ty3_correct g (CHead c2 (Bind Abbr) u) t2 t3 (H20 (CHead c2 (Bind Abbr) u) +(wcpr0_comp c c2 H4 u u (pr0_refl u) (Bind Abbr)) t2 (pr0_refl t2))))))))) t4 +H16)) t5 (sym_eq T t5 t2 H15))) u1 (sym_eq T u1 u H14))) b H13)) H12)) H11)) +H10 H6 H7 H8))) | (pr0_zeta b0 H6 t5 t6 H7 u0) \Rightarrow (\lambda (H8: (eq +T (THead (Bind b0) u0 (lift (S O) O t5)) (THead (Bind b) u t2))).(\lambda +(H9: (eq T t6 t4)).((let H10 \def (f_equal T T (\lambda (e: T).(match e in T +return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: +((nat \to nat))) (d: nat) (t7: T) on t7: T \def (match t7 with [(TSort n) +\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with +[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u1 t8) +\Rightarrow (THead k (lref_map f d u1) (lref_map f (s k d) t8))]) in +lref_map) (\lambda (x: nat).(plus x (S O))) O t5) | (TLRef _) \Rightarrow +((let rec lref_map (f: ((nat \to nat))) (d: nat) (t7: T) on t7: T \def (match +t7 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef +(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | +(THead k u1 t8) \Rightarrow (THead k (lref_map f d u1) (lref_map f (s k d) +t8))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t5) | (THead _ _ t7) +\Rightarrow t7])) (THead (Bind b0) u0 (lift (S O) O t5)) (THead (Bind b) u +t2) H8) in ((let H11 \def (f_equal T T (\lambda (e: T).(match e in T return +(\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 +| (THead _ t7 _) \Rightarrow t7])) (THead (Bind b0) u0 (lift (S O) O t5)) +(THead (Bind b) u t2) H8) in ((let H12 \def (f_equal T B (\lambda (e: +T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b0 | +(TLRef _) \Rightarrow b0 | (THead k _ _) \Rightarrow (match k in K return +(\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow +b0])])) (THead (Bind b0) u0 (lift (S O) O t5)) (THead (Bind b) u t2) H8) in +(eq_ind B b (\lambda (b1: B).((eq T u0 u) \to ((eq T (lift (S O) O t5) t2) +\to ((eq T t6 t4) \to ((not (eq B b1 Abst)) \to ((pr0 t5 t6) \to (ty3 g c2 t4 +(THead (Bind b) u t3)))))))) (\lambda (H13: (eq T u0 u)).(eq_ind T u (\lambda +(_: T).((eq T (lift (S O) O t5) t2) \to ((eq T t6 t4) \to ((not (eq B b +Abst)) \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3))))))) (\lambda +(H14: (eq T (lift (S O) O t5) t2)).(eq_ind T (lift (S O) O t5) (\lambda (_: +T).((eq T t6 t4) \to ((not (eq B b Abst)) \to ((pr0 t5 t6) \to (ty3 g c2 t4 +(THead (Bind b) u t3)))))) (\lambda (H15: (eq T t6 t4)).(eq_ind T t4 (\lambda +(t7: T).((not (eq B b Abst)) \to ((pr0 t5 t7) \to (ty3 g c2 t4 (THead (Bind +b) u t3))))) (\lambda (H16: (not (eq B b Abst))).(\lambda (H17: (pr0 t5 +t4)).(let H18 \def (eq_ind_r T t2 (\lambda (t7: T).(\forall (c3: C).((wcpr0 +(CHead c (Bind b) u) c3) \to (\forall (t8: T).((pr0 t7 t8) \to (ty3 g c3 t8 +t3)))))) H3 (lift (S O) O t5) H14) in (let H19 \def (eq_ind_r T t2 (\lambda +(t7: T).(ty3 g (CHead c (Bind b) u) t7 t3)) H2 (lift (S O) O t5) H14) in +(ex_ind T (\lambda (t7: T).(ty3 g (CHead c2 (Bind b) u) t3 t7)) (ty3 g c2 t4 +(THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H20: (ty3 g (CHead c2 (Bind +b) u) t3 x)).(B_ind (\lambda (b1: B).((not (eq B b1 Abst)) \to ((ty3 g (CHead +c2 (Bind b1) u) t3 x) \to ((ty3 g (CHead c2 (Bind b1) u) (lift (S O) O t4) +t3) \to (ty3 g c2 t4 (THead (Bind b1) u t3)))))) (\lambda (H21: (not (eq B +Abbr Abst))).(\lambda (H22: (ty3 g (CHead c2 (Bind Abbr) u) t3 x)).(\lambda +(H23: (ty3 g (CHead c2 (Bind Abbr) u) (lift (S O) O t4) t3)).(let H24 \def +(ty3_gen_cabbr g (CHead c2 (Bind Abbr) u) (lift (S O) O t4) t3 H23 c2 u O +(getl_refl Abbr c2 u) (CHead c2 (Bind Abbr) u) (csubst1_refl O u (CHead c2 +(Bind Abbr) u)) c2 (drop_drop (Bind Abbr) O c2 c2 (drop_refl c2) u)) in +(ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(subst1 O u (lift (S O) O t4) +(lift (S O) O y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 O u t3 (lift (S +O) O y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g c2 y1 y2))) (ty3 g c2 t4 +(THead (Bind Abbr) u t3)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H25: +(subst1 O u (lift (S O) O t4) (lift (S O) O x0))).(\lambda (H26: (subst1 O u +t3 (lift (S O) O x1))).(\lambda (H27: (ty3 g c2 x0 x1)).(let H28 \def (eq_ind +T x0 (\lambda (t7: T).(ty3 g c2 t7 x1)) H27 t4 (lift_inj x0 t4 (S O) O +(subst1_gen_lift_eq t4 u (lift (S O) O x0) (S O) O O (le_n O) (eq_ind_r nat +(plus (S O) O) (\lambda (n: nat).(lt O n)) (le_n (plus (S O) O)) (plus O (S +O)) (plus_sym O (S O))) H25))) in (ty3_conv g c2 (THead (Bind Abbr) u t3) +(THead (Bind Abbr) u x) (ty3_bind g c2 u t0 (H1 c2 H4 u (pr0_refl u)) Abbr t3 +x H22) t4 x1 H28 (pc3_pr3_x c2 x1 (THead (Bind Abbr) u t3) (pr3_t (THead +(Bind Abbr) u (lift (S O) O x1)) (THead (Bind Abbr) u t3) c2 (pr3_pr2 c2 +(THead (Bind Abbr) u t3) (THead (Bind Abbr) u (lift (S O) O x1)) (pr2_free c2 +(THead (Bind Abbr) u t3) (THead (Bind Abbr) u (lift (S O) O x1)) (pr0_delta1 +u u (pr0_refl u) t3 t3 (pr0_refl t3) (lift (S O) O x1) H26))) x1 (pr3_pr2 c2 +(THead (Bind Abbr) u (lift (S O) O x1)) x1 (pr2_free c2 (THead (Bind Abbr) u +(lift (S O) O x1)) x1 (pr0_zeta Abbr H21 x1 x1 (pr0_refl x1) u)))))))))))) +H24))))) (\lambda (H21: (not (eq B Abst Abst))).(\lambda (_: (ty3 g (CHead c2 +(Bind Abst) u) t3 x)).(\lambda (_: (ty3 g (CHead c2 (Bind Abst) u) (lift (S +O) O t4) t3)).(let H24 \def (match (H21 (refl_equal B Abst)) in False return +(\lambda (_: False).(ty3 g c2 t4 (THead (Bind Abst) u t3))) with []) in +H24)))) (\lambda (H21: (not (eq B Void Abst))).(\lambda (H22: (ty3 g (CHead +c2 (Bind Void) u) t3 x)).(\lambda (H23: (ty3 g (CHead c2 (Bind Void) u) (lift +(S O) O t4) t3)).(let H24 \def (ty3_gen_cvoid g (CHead c2 (Bind Void) u) +(lift (S O) O t4) t3 H23 c2 u O (getl_refl Void c2 u) c2 (drop_drop (Bind +Void) O c2 c2 (drop_refl c2) u)) in (ex3_2_ind T T (\lambda (y1: T).(\lambda +(_: T).(eq T (lift (S O) O t4) (lift (S O) O y1)))) (\lambda (_: T).(\lambda +(y2: T).(eq T t3 (lift (S O) O y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 +g c2 y1 y2))) (ty3 g c2 t4 (THead (Bind Void) u t3)) (\lambda (x0: +T).(\lambda (x1: T).(\lambda (H25: (eq T (lift (S O) O t4) (lift (S O) O +x0))).(\lambda (H26: (eq T t3 (lift (S O) O x1))).(\lambda (H27: (ty3 g c2 x0 +x1)).(let H28 \def (eq_ind T t3 (\lambda (t7: T).(ty3 g (CHead c2 (Bind Void) +u) t7 x)) H22 (lift (S O) O x1) H26) in (eq_ind_r T (lift (S O) O x1) +(\lambda (t7: T).(ty3 g c2 t4 (THead (Bind Void) u t7))) (let H29 \def +(eq_ind_r T x0 (\lambda (t7: T).(ty3 g c2 t7 x1)) H27 t4 (lift_inj t4 x0 (S +O) O H25)) in (ty3_conv g c2 (THead (Bind Void) u (lift (S O) O x1)) (THead +(Bind Void) u x) (ty3_bind g c2 u t0 (H1 c2 H4 u (pr0_refl u)) Void (lift (S +O) O x1) x H28) t4 x1 H29 (pc3_s c2 x1 (THead (Bind Void) u (lift (S O) O +x1)) (pc3_pr2_r c2 (THead (Bind Void) u (lift (S O) O x1)) x1 (pr2_free c2 +(THead (Bind Void) u (lift (S O) O x1)) x1 (pr0_zeta Void H21 x1 x1 (pr0_refl +x1) u)))))) t3 H26))))))) H24))))) b H16 H20 (H18 (CHead c2 (Bind b) u) +(wcpr0_comp c c2 H4 u u (pr0_refl u) (Bind b)) (lift (S O) O t4) (pr0_lift t5 +t4 H17 (S O) O))))) (ty3_correct g (CHead c2 (Bind b) u) (lift (S O) O t4) t3 +(H18 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H4 u u (pr0_refl u) (Bind b)) +(lift (S O) O t4) (pr0_lift t5 t4 H17 (S O) O)))))))) t6 (sym_eq T t6 t4 +H15))) t2 H14)) u0 (sym_eq T u0 u H13))) b0 (sym_eq B b0 b H12))) H11)) H10)) +H9 H6 H7))) | (pr0_tau t5 t6 H6 u0) \Rightarrow (\lambda (H7: (eq T (THead +(Flat Cast) u0 t5) (THead (Bind b) u t2))).(\lambda (H8: (eq T t6 t4)).((let +H9 \def (eq_ind T (THead (Flat Cast) u0 t5) (\lambda (e: T).(match e in T +return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda +(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow +True])])) I (THead (Bind b) u t2) H7) in (False_ind ((eq T t6 t4) \to ((pr0 +t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3)))) H9)) H8 H6)))]) in (H6 +(refl_equal T (THead (Bind b) u t2)) (refl_equal T t4))))))))))))))))) +(\lambda (c: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c w +u)).(\lambda (H1: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 +w t2) \to (ty3 g c2 t2 u))))))).(\lambda (v: T).(\lambda (t0: T).(\lambda +(H2: (ty3 g c v (THead (Bind Abst) u t0))).(\lambda (H3: ((\forall (c2: +C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 v t2) \to (ty3 g c2 t2 (THead +(Bind Abst) u t0)))))))).(\lambda (c2: C).(\lambda (H4: (wcpr0 c +c2)).(\lambda (t2: T).(\lambda (H5: (pr0 (THead (Flat Appl) w v) t2)).(let H6 +\def (match H5 in pr0 return (\lambda (t3: T).(\lambda (t4: T).(\lambda (_: +(pr0 t3 t4)).((eq T t3 (THead (Flat Appl) w v)) \to ((eq T t4 t2) \to (ty3 g +c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u t0)))))))) with [(pr0_refl +t3) \Rightarrow (\lambda (H6: (eq T t3 (THead (Flat Appl) w v))).(\lambda +(H7: (eq T t3 t2)).(eq_ind T (THead (Flat Appl) w v) (\lambda (t4: T).((eq T +t4 t2) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u t0))))) +(\lambda (H8: (eq T (THead (Flat Appl) w v) t2)).(eq_ind T (THead (Flat Appl) +w v) (\lambda (t4: T).(ty3 g c2 t4 (THead (Flat Appl) w (THead (Bind Abst) u +t0)))) (ty3_appl g c2 w u (H1 c2 H4 w (pr0_refl w)) v t0 (H3 c2 H4 v +(pr0_refl v))) t2 H8)) t3 (sym_eq T t3 (THead (Flat Appl) w v) H6) H7))) | +(pr0_comp u1 u2 H6 t3 t4 H7 k) \Rightarrow (\lambda (H8: (eq T (THead k u1 +t3) (THead (Flat Appl) w v))).(\lambda (H9: (eq T (THead k u2 t4) t2)).((let +H10 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) +with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t5) +\Rightarrow t5])) (THead k u1 t3) (THead (Flat Appl) w v) H8) in ((let H11 +\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) +with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t5 _) +\Rightarrow t5])) (THead k u1 t3) (THead (Flat Appl) w v) H8) in ((let H12 +\def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) +with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) +\Rightarrow k0])) (THead k u1 t3) (THead (Flat Appl) w v) H8) in (eq_ind K +(Flat Appl) (\lambda (k0: K).((eq T u1 w) \to ((eq T t3 v) \to ((eq T (THead +k0 u2 t4) t2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat +Appl) w (THead (Bind Abst) u t0))))))))) (\lambda (H13: (eq T u1 w)).(eq_ind +T w (\lambda (t5: T).((eq T t3 v) \to ((eq T (THead (Flat Appl) u2 t4) t2) +\to ((pr0 t5 u2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w +(THead (Bind Abst) u t0)))))))) (\lambda (H14: (eq T t3 v)).(eq_ind T v +(\lambda (t5: T).((eq T (THead (Flat Appl) u2 t4) t2) \to ((pr0 w u2) \to +((pr0 t5 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u +t0))))))) (\lambda (H15: (eq T (THead (Flat Appl) u2 t4) t2)).(eq_ind T +(THead (Flat Appl) u2 t4) (\lambda (t5: T).((pr0 w u2) \to ((pr0 v t4) \to +(ty3 g c2 t5 (THead (Flat Appl) w (THead (Bind Abst) u t0)))))) (\lambda +(H16: (pr0 w u2)).(\lambda (H17: (pr0 v t4)).(ex_ind T (\lambda (t5: T).(ty3 +g c2 (THead (Bind Abst) u t0) t5)) (ty3 g c2 (THead (Flat Appl) u2 t4) (THead +(Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x: T).(\lambda (H18: (ty3 +g c2 (THead (Bind Abst) u t0) x)).(ex3_2_ind T T (\lambda (t5: T).(\lambda +(_: T).(pc3 c2 (THead (Bind Abst) u t5) x))) (\lambda (_: T).(\lambda (t6: +T).(ty3 g c2 u t6))) (\lambda (t5: T).(\lambda (_: T).(ty3 g (CHead c2 (Bind +Abst) u) t0 t5))) (ty3 g c2 (THead (Flat Appl) u2 t4) (THead (Flat Appl) w +(THead (Bind Abst) u t0))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: +(pc3 c2 (THead (Bind Abst) u x0) x)).(\lambda (H20: (ty3 g c2 u x1)).(\lambda +(H21: (ty3 g (CHead c2 (Bind Abst) u) t0 x0)).(ty3_conv g c2 (THead (Flat +Appl) w (THead (Bind Abst) u t0)) (THead (Flat Appl) w (THead (Bind Abst) u +x0)) (ty3_appl g c2 w u (H1 c2 H4 w (pr0_refl w)) (THead (Bind Abst) u t0) x0 +(ty3_bind g c2 u x1 H20 Abst t0 x0 H21)) (THead (Flat Appl) u2 t4) (THead +(Flat Appl) u2 (THead (Bind Abst) u t0)) (ty3_appl g c2 u2 u (H1 c2 H4 u2 +H16) t4 t0 (H3 c2 H4 t4 H17)) (pc3_pr2_x c2 (THead (Flat Appl) u2 (THead +(Bind Abst) u t0)) (THead (Flat Appl) w (THead (Bind Abst) u t0)) (pr2_head_1 +c2 w u2 (pr2_free c2 w u2 H16) (Flat Appl) (THead (Bind Abst) u t0))))))))) +(ty3_gen_bind g Abst c2 u t0 x H18)))) (ty3_correct g c2 v (THead (Bind Abst) +u t0) (H3 c2 H4 v (pr0_refl v)))))) t2 H15)) t3 (sym_eq T t3 v H14))) u1 +(sym_eq T u1 w H13))) k (sym_eq K k (Flat Appl) H12))) H11)) H10)) H9 H6 +H7))) | (pr0_beta u0 v1 v2 H6 t3 t4 H7) \Rightarrow (\lambda (H8: (eq T +(THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) (THead (Flat Appl) w +v))).(\lambda (H9: (eq T (THead (Bind Abbr) v2 t4) t2)).((let H10 \def +(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with +[(TSort _) \Rightarrow (THead (Bind Abst) u0 t3) | (TLRef _) \Rightarrow +(THead (Bind Abst) u0 t3) | (THead _ _ t5) \Rightarrow t5])) (THead (Flat +Appl) v1 (THead (Bind Abst) u0 t3)) (THead (Flat Appl) w v) H8) in ((let H11 +\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) +with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t5 _) +\Rightarrow t5])) (THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) (THead +(Flat Appl) w v) H8) in (eq_ind T w (\lambda (t5: T).((eq T (THead (Bind +Abst) u0 t3) v) \to ((eq T (THead (Bind Abbr) v2 t4) t2) \to ((pr0 t5 v2) \to +((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u +t0)))))))) (\lambda (H12: (eq T (THead (Bind Abst) u0 t3) v)).(eq_ind T +(THead (Bind Abst) u0 t3) (\lambda (_: T).((eq T (THead (Bind Abbr) v2 t4) +t2) \to ((pr0 w v2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w +(THead (Bind Abst) u t0))))))) (\lambda (H13: (eq T (THead (Bind Abbr) v2 t4) +t2)).(eq_ind T (THead (Bind Abbr) v2 t4) (\lambda (t5: T).((pr0 w v2) \to +((pr0 t3 t4) \to (ty3 g c2 t5 (THead (Flat Appl) w (THead (Bind Abst) u +t0)))))) (\lambda (H14: (pr0 w v2)).(\lambda (H15: (pr0 t3 t4)).(let H16 \def +(eq_ind_r T v (\lambda (t5: T).(\forall (c3: C).((wcpr0 c c3) \to (\forall +(t6: T).((pr0 t5 t6) \to (ty3 g c3 t6 (THead (Bind Abst) u t0))))))) H3 +(THead (Bind Abst) u0 t3) H12) in (let H17 \def (eq_ind_r T v (\lambda (t5: +T).(ty3 g c t5 (THead (Bind Abst) u t0))) H2 (THead (Bind Abst) u0 t3) H12) +in (ex_ind T (\lambda (t5: T).(ty3 g c2 (THead (Bind Abst) u t0) t5)) (ty3 g +c2 (THead (Bind Abbr) v2 t4) (THead (Flat Appl) w (THead (Bind Abst) u t0))) +(\lambda (x: T).(\lambda (H18: (ty3 g c2 (THead (Bind Abst) u t0) +x)).(ex3_2_ind T T (\lambda (t5: T).(\lambda (_: T).(pc3 c2 (THead (Bind +Abst) u t5) x))) (\lambda (_: T).(\lambda (t6: T).(ty3 g c2 u t6))) (\lambda +(t5: T).(\lambda (_: T).(ty3 g (CHead c2 (Bind Abst) u) t0 t5))) (ty3 g c2 +(THead (Bind Abbr) v2 t4) (THead (Flat Appl) w (THead (Bind Abst) u t0))) +(\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 c2 (THead (Bind Abst) u +x0) x)).(\lambda (H20: (ty3 g c2 u x1)).(\lambda (H21: (ty3 g (CHead c2 (Bind +Abst) u) t0 x0)).(ex3_2_ind T T (\lambda (t5: T).(\lambda (_: T).(pc3 c2 +(THead (Bind Abst) u0 t5) (THead (Bind Abst) u t0)))) (\lambda (_: +T).(\lambda (t6: T).(ty3 g c2 u0 t6))) (\lambda (t5: T).(\lambda (_: T).(ty3 +g (CHead c2 (Bind Abst) u0) t4 t5))) (ty3 g c2 (THead (Bind Abbr) v2 t4) +(THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x2: T).(\lambda +(x3: T).(\lambda (H22: (pc3 c2 (THead (Bind Abst) u0 x2) (THead (Bind Abst) u +t0))).(\lambda (H23: (ty3 g c2 u0 x3)).(\lambda (H24: (ty3 g (CHead c2 (Bind +Abst) u0) t4 x2)).(land_ind (pc3 c2 u0 u) (\forall (b: B).(\forall (u1: +T).(pc3 (CHead c2 (Bind b) u1) x2 t0))) (ty3 g c2 (THead (Bind Abbr) v2 t4) +(THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (H25: (pc3 c2 u0 +u)).(\lambda (H26: ((\forall (b: B).(\forall (u1: T).(pc3 (CHead c2 (Bind b) +u1) x2 t0))))).(ty3_conv g c2 (THead (Flat Appl) w (THead (Bind Abst) u t0)) +(THead (Flat Appl) w (THead (Bind Abst) u x0)) (ty3_appl g c2 w u (H1 c2 H4 w +(pr0_refl w)) (THead (Bind Abst) u t0) x0 (ty3_bind g c2 u x1 H20 Abst t0 x0 +H21)) (THead (Bind Abbr) v2 t4) (THead (Bind Abbr) v2 x2) (ty3_bind g c2 v2 u +(H1 c2 H4 v2 H14) Abbr t4 x2 (csubt_ty3_ld g c2 v2 u0 (ty3_conv g c2 u0 x3 +H23 v2 u (H1 c2 H4 v2 H14) (pc3_s c2 u u0 H25)) t4 x2 H24)) (pc3_t (THead +(Bind Abbr) v2 t0) c2 (THead (Bind Abbr) v2 x2) (pc3_head_2 c2 v2 x2 t0 (Bind +Abbr) (H26 Abbr v2)) (THead (Flat Appl) w (THead (Bind Abst) u t0)) +(pc3_pr2_x c2 (THead (Bind Abbr) v2 t0) (THead (Flat Appl) w (THead (Bind +Abst) u t0)) (pr2_free c2 (THead (Flat Appl) w (THead (Bind Abst) u t0)) +(THead (Bind Abbr) v2 t0) (pr0_beta u w v2 H14 t0 t0 (pr0_refl t0)))))))) +(pc3_gen_abst c2 u0 u x2 t0 H22))))))) (ty3_gen_bind g Abst c2 u0 t4 (THead +(Bind Abst) u t0) (H16 c2 H4 (THead (Bind Abst) u0 t4) (pr0_comp u0 u0 +(pr0_refl u0) t3 t4 H15 (Bind Abst)))))))))) (ty3_gen_bind g Abst c2 u t0 x +H18)))) (ty3_correct g c2 (THead (Bind Abst) u0 t3) (THead (Bind Abst) u t0) +(H16 c2 H4 (THead (Bind Abst) u0 t3) (pr0_refl (THead (Bind Abst) u0 +t3))))))))) t2 H13)) v H12)) v1 (sym_eq T v1 w H11))) H10)) H9 H6 H7))) | +(pr0_upsilon b H6 v1 v2 H7 u1 u2 H8 t3 t4 H9) \Rightarrow (\lambda (H10: (eq +T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) w +v))).(\lambda (H11: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O +v2) t4)) t2)).((let H12 \def (f_equal T T (\lambda (e: T).(match e in T +return (\lambda (_: T).T) with [(TSort _) \Rightarrow (THead (Bind b) u1 t3) +| (TLRef _) \Rightarrow (THead (Bind b) u1 t3) | (THead _ _ t5) \Rightarrow +t5])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) w v) +H10) in ((let H13 \def (f_equal T T (\lambda (e: T).(match e in T return +(\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 +| (THead _ t5 _) \Rightarrow t5])) (THead (Flat Appl) v1 (THead (Bind b) u1 +t3)) (THead (Flat Appl) w v) H10) in (eq_ind T w (\lambda (t5: T).((eq T +(THead (Bind b) u1 t3) v) \to ((eq T (THead (Bind b) u2 (THead (Flat Appl) +(lift (S O) O v2) t4)) t2) \to ((not (eq B b Abst)) \to ((pr0 t5 v2) \to +((pr0 u1 u2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead +(Bind Abst) u t0)))))))))) (\lambda (H14: (eq T (THead (Bind b) u1 t3) +v)).(eq_ind T (THead (Bind b) u1 t3) (\lambda (_: T).((eq T (THead (Bind b) +u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t2) \to ((not (eq B b Abst)) \to +((pr0 w v2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat +Appl) w (THead (Bind Abst) u t0))))))))) (\lambda (H15: (eq T (THead (Bind b) +u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t2)).(eq_ind T (THead (Bind b) +u2 (THead (Flat Appl) (lift (S O) O v2) t4)) (\lambda (t5: T).((not (eq B b +Abst)) \to ((pr0 w v2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ty3 g c2 t5 +(THead (Flat Appl) w (THead (Bind Abst) u t0)))))))) (\lambda (H16: (not (eq +B b Abst))).(\lambda (H17: (pr0 w v2)).(\lambda (H18: (pr0 u1 u2)).(\lambda +(H19: (pr0 t3 t4)).(let H20 \def (eq_ind_r T v (\lambda (t5: T).(\forall (c3: +C).((wcpr0 c c3) \to (\forall (t6: T).((pr0 t5 t6) \to (ty3 g c3 t6 (THead +(Bind Abst) u t0))))))) H3 (THead (Bind b) u1 t3) H14) in (let H21 \def +(eq_ind_r T v (\lambda (t5: T).(ty3 g c t5 (THead (Bind Abst) u t0))) H2 +(THead (Bind b) u1 t3) H14) in (ex_ind T (\lambda (t5: T).(ty3 g c2 (THead +(Bind Abst) u t0) t5)) (ty3 g c2 (THead (Bind b) u2 (THead (Flat Appl) (lift +(S O) O v2) t4)) (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x: +T).(\lambda (H22: (ty3 g c2 (THead (Bind Abst) u t0) x)).(let H23 \def H22 in +(ex3_2_ind T T (\lambda (t5: T).(\lambda (_: T).(pc3 c2 (THead (Bind Abst) u +t5) x))) (\lambda (_: T).(\lambda (t6: T).(ty3 g c2 u t6))) (\lambda (t5: +T).(\lambda (_: T).(ty3 g (CHead c2 (Bind Abst) u) t0 t5))) (ty3 g c2 (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) (THead (Flat Appl) w +(THead (Bind Abst) u t0))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: +(pc3 c2 (THead (Bind Abst) u x0) x)).(\lambda (H25: (ty3 g c2 u x1)).(\lambda +(H26: (ty3 g (CHead c2 (Bind Abst) u) t0 x0)).(ex3_2_ind T T (\lambda (t5: +T).(\lambda (_: T).(pc3 c2 (THead (Bind b) u2 t5) (THead (Bind Abst) u t0)))) +(\lambda (_: T).(\lambda (t6: T).(ty3 g c2 u2 t6))) (\lambda (t5: T).(\lambda +(_: T).(ty3 g (CHead c2 (Bind b) u2) t4 t5))) (ty3 g c2 (THead (Bind b) u2 +(THead (Flat Appl) (lift (S O) O v2) t4)) (THead (Flat Appl) w (THead (Bind +Abst) u t0))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H27: (pc3 c2 (THead +(Bind b) u2 x2) (THead (Bind Abst) u t0))).(\lambda (H28: (ty3 g c2 u2 +x3)).(\lambda (H29: (ty3 g (CHead c2 (Bind b) u2) t4 x2)).(let H30 \def +(eq_ind T (lift (S O) O (THead (Bind Abst) u t0)) (\lambda (t5: T).(pc3 +(CHead c2 (Bind b) u2) x2 t5)) (pc3_gen_not_abst b H16 c2 x2 t0 u2 u H27) +(THead (Bind Abst) (lift (S O) O u) (lift (S O) (S O) t0)) (lift_bind Abst u +t0 (S O) O)) in (let H31 \def (eq_ind T (lift (S O) O (THead (Bind Abst) u +t0)) (\lambda (t5: T).(ty3 g (CHead c2 (Bind b) u2) t5 (lift (S O) O x))) +(ty3_lift g c2 (THead (Bind Abst) u t0) x H22 (CHead c2 (Bind b) u2) O (S O) +(drop_drop (Bind b) O c2 c2 (drop_refl c2) u2)) (THead (Bind Abst) (lift (S +O) O u) (lift (S O) (S O) t0)) (lift_bind Abst u t0 (S O) O)) in (ex3_2_ind T +T (\lambda (t5: T).(\lambda (_: T).(pc3 (CHead c2 (Bind b) u2) (THead (Bind +Abst) (lift (S O) O u) t5) (lift (S O) O x)))) (\lambda (_: T).(\lambda (t6: +T).(ty3 g (CHead c2 (Bind b) u2) (lift (S O) O u) t6))) (\lambda (t5: +T).(\lambda (_: T).(ty3 g (CHead (CHead c2 (Bind b) u2) (Bind Abst) (lift (S +O) O u)) (lift (S O) (S O) t0) t5))) (ty3 g c2 (THead (Bind b) u2 (THead +(Flat Appl) (lift (S O) O v2) t4)) (THead (Flat Appl) w (THead (Bind Abst) u +t0))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (pc3 (CHead c2 (Bind b) +u2) (THead (Bind Abst) (lift (S O) O u) x4) (lift (S O) O x))).(\lambda (H33: +(ty3 g (CHead c2 (Bind b) u2) (lift (S O) O u) x5)).(\lambda (H34: (ty3 g +(CHead (CHead c2 (Bind b) u2) (Bind Abst) (lift (S O) O u)) (lift (S O) (S O) +t0) x4)).(ty3_conv g c2 (THead (Flat Appl) w (THead (Bind Abst) u t0)) (THead +(Flat Appl) w (THead (Bind Abst) u x0)) (ty3_appl g c2 w u (H1 c2 H4 w +(pr0_refl w)) (THead (Bind Abst) u t0) x0 (ty3_bind g c2 u x1 H25 Abst t0 x0 +H26)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (THead (Bind Abst) (lift (S +O) O u) (lift (S O) (S O) t0)))) (ty3_bind g c2 u2 x3 H28 b (THead (Flat +Appl) (lift (S O) O v2) t4) (THead (Flat Appl) (lift (S O) O v2) (THead (Bind +Abst) (lift (S O) O u) (lift (S O) (S O) t0))) (ty3_appl g (CHead c2 (Bind b) +u2) (lift (S O) O v2) (lift (S O) O u) (ty3_lift g c2 v2 u (H1 c2 H4 v2 H17) +(CHead c2 (Bind b) u2) O (S O) (drop_drop (Bind b) O c2 c2 (drop_refl c2) +u2)) t4 (lift (S O) (S O) t0) (ty3_conv g (CHead c2 (Bind b) u2) (THead (Bind +Abst) (lift (S O) O u) (lift (S O) (S O) t0)) (THead (Bind Abst) (lift (S O) +O u) x4) (ty3_bind g (CHead c2 (Bind b) u2) (lift (S O) O u) x5 H33 Abst +(lift (S O) (S O) t0) x4 H34) t4 x2 H29 H30))) (eq_ind T (lift (S O) O (THead +(Bind Abst) u t0)) (\lambda (t5: T).(pc3 c2 (THead (Bind b) u2 (THead (Flat +Appl) (lift (S O) O v2) t5)) (THead (Flat Appl) w (THead (Bind Abst) u t0)))) +(pc3_pc1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (lift (S O) +O (THead (Bind Abst) u t0)))) (THead (Flat Appl) w (THead (Bind Abst) u t0)) +(pc1_pr0_u2 (THead (Flat Appl) v2 (THead (Bind b) u2 (lift (S O) O (THead +(Bind Abst) u t0)))) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) +(lift (S O) O (THead (Bind Abst) u t0)))) (pr0_upsilon b H16 v2 v2 (pr0_refl +v2) u2 u2 (pr0_refl u2) (lift (S O) O (THead (Bind Abst) u t0)) (lift (S O) O +(THead (Bind Abst) u t0)) (pr0_refl (lift (S O) O (THead (Bind Abst) u t0)))) +(THead (Flat Appl) w (THead (Bind Abst) u t0)) (pc1_head v2 w (pc1_pr0_x v2 w +H17) (THead (Bind b) u2 (lift (S O) O (THead (Bind Abst) u t0))) (THead (Bind +Abst) u t0) (pc1_pr0_r (THead (Bind b) u2 (lift (S O) O (THead (Bind Abst) u +t0))) (THead (Bind Abst) u t0) (pr0_zeta b H16 (THead (Bind Abst) u t0) +(THead (Bind Abst) u t0) (pr0_refl (THead (Bind Abst) u t0)) u2)) (Flat +Appl))) c2) (THead (Bind Abst) (lift (S O) O u) (lift (S O) (S O) t0)) +(lift_bind Abst u t0 (S O) O)))))))) (ty3_gen_bind g Abst (CHead c2 (Bind b) +u2) (lift (S O) O u) (lift (S O) (S O) t0) (lift (S O) O x) H31))))))))) +(ty3_gen_bind g b c2 u2 t4 (THead (Bind Abst) u t0) (H20 c2 H4 (THead (Bind +b) u2 t4) (pr0_comp u1 u2 H18 t3 t4 H19 (Bind b)))))))))) (ty3_gen_bind g +Abst c2 u t0 x H23))))) (ty3_correct g c2 (THead (Bind b) u2 t4) (THead (Bind +Abst) u t0) (H20 c2 H4 (THead (Bind b) u2 t4) (pr0_comp u1 u2 H18 t3 t4 H19 +(Bind b))))))))))) t2 H15)) v H14)) v1 (sym_eq T v1 w H13))) H12)) H11 H6 H7 +H8 H9))) | (pr0_delta u1 u2 H6 t3 t4 H7 w0 H8) \Rightarrow (\lambda (H9: (eq +T (THead (Bind Abbr) u1 t3) (THead (Flat Appl) w v))).(\lambda (H10: (eq T +(THead (Bind Abbr) u2 w0) t2)).((let H11 \def (eq_ind T (THead (Bind Abbr) u1 +t3) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort +_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) +\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) +\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) w v) +H9) in (False_ind ((eq T (THead (Bind Abbr) u2 w0) t2) \to ((pr0 u1 u2) \to +((pr0 t3 t4) \to ((subst0 O u2 t4 w0) \to (ty3 g c2 t2 (THead (Flat Appl) w +(THead (Bind Abst) u t0))))))) H11)) H10 H6 H7 H8))) | (pr0_zeta b H6 t3 t4 +H7 u0) \Rightarrow (\lambda (H8: (eq T (THead (Bind b) u0 (lift (S O) O t3)) +(THead (Flat Appl) w v))).(\lambda (H9: (eq T t4 t2)).((let H10 \def (eq_ind +T (THead (Bind b) u0 (lift (S O) O t3)) (\lambda (e: T).(match e in T return +(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda +(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow +False])])) I (THead (Flat Appl) w v) H8) in (False_ind ((eq T t4 t2) \to +((not (eq B b Abst)) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w +(THead (Bind Abst) u t0)))))) H10)) H9 H6 H7))) | (pr0_tau t3 t4 H6 u0) +\Rightarrow (\lambda (H7: (eq T (THead (Flat Cast) u0 t3) (THead (Flat Appl) +w v))).(\lambda (H8: (eq T t4 t2)).((let H9 \def (eq_ind T (THead (Flat Cast) +u0 t3) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with +[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) +\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) +\Rightarrow False | (Flat f) \Rightarrow (match f in F return (\lambda (_: +F).Prop) with [Appl \Rightarrow False | Cast \Rightarrow True])])])) I (THead +(Flat Appl) w v) H7) in (False_ind ((eq T t4 t2) \to ((pr0 t3 t4) \to (ty3 g +c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u t0))))) H9)) H8 H6)))]) in +(H6 (refl_equal T (THead (Flat Appl) w v)) (refl_equal T t2)))))))))))))))) +(\lambda (c: C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (ty3 g c t2 +t3)).(\lambda (H1: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t4: T).((pr0 +t2 t4) \to (ty3 g c2 t4 t3))))))).(\lambda (t0: T).(\lambda (_: (ty3 g c t3 +t0)).(\lambda (H3: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t4: T).((pr0 +t3 t4) \to (ty3 g c2 t4 t0))))))).(\lambda (c2: C).(\lambda (H4: (wcpr0 c +c2)).(\lambda (t4: T).(\lambda (H5: (pr0 (THead (Flat Cast) t3 t2) t4)).(let +H6 \def (match H5 in pr0 return (\lambda (t5: T).(\lambda (t6: T).(\lambda +(_: (pr0 t5 t6)).((eq T t5 (THead (Flat Cast) t3 t2)) \to ((eq T t6 t4) \to +(ty3 g c2 t4 (THead (Flat Cast) t0 t3))))))) with [(pr0_refl t5) \Rightarrow +(\lambda (H6: (eq T t5 (THead (Flat Cast) t3 t2))).(\lambda (H7: (eq T t5 +t4)).(eq_ind T (THead (Flat Cast) t3 t2) (\lambda (t6: T).((eq T t6 t4) \to +(ty3 g c2 t4 (THead (Flat Cast) t0 t3)))) (\lambda (H8: (eq T (THead (Flat +Cast) t3 t2) t4)).(eq_ind T (THead (Flat Cast) t3 t2) (\lambda (t6: T).(ty3 g +c2 t6 (THead (Flat Cast) t0 t3))) (ty3_cast g c2 t2 t3 (H1 c2 H4 t2 (pr0_refl +t2)) t0 (H3 c2 H4 t3 (pr0_refl t3))) t4 H8)) t5 (sym_eq T t5 (THead (Flat +Cast) t3 t2) H6) H7))) | (pr0_comp u1 u2 H6 t5 t6 H7 k) \Rightarrow (\lambda +(H8: (eq T (THead k u1 t5) (THead (Flat Cast) t3 t2))).(\lambda (H9: (eq T +(THead k u2 t6) t4)).((let H10 \def (f_equal T T (\lambda (e: T).(match e in +T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t5 | (TLRef _) +\Rightarrow t5 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t5) (THead +(Flat Cast) t3 t2) H8) in ((let H11 \def (f_equal T T (\lambda (e: T).(match +e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) +\Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t5) (THead +(Flat Cast) t3 t2) H8) in ((let H12 \def (f_equal T K (\lambda (e: T).(match +e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) +\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t5) (THead (Flat +Cast) t3 t2) H8) in (eq_ind K (Flat Cast) (\lambda (k0: K).((eq T u1 t3) \to +((eq T t5 t2) \to ((eq T (THead k0 u2 t6) t4) \to ((pr0 u1 u2) \to ((pr0 t5 +t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3)))))))) (\lambda (H13: (eq T u1 +t3)).(eq_ind T t3 (\lambda (t7: T).((eq T t5 t2) \to ((eq T (THead (Flat +Cast) u2 t6) t4) \to ((pr0 t7 u2) \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead +(Flat Cast) t0 t3))))))) (\lambda (H14: (eq T t5 t2)).(eq_ind T t2 (\lambda +(t7: T).((eq T (THead (Flat Cast) u2 t6) t4) \to ((pr0 t3 u2) \to ((pr0 t7 +t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3)))))) (\lambda (H15: (eq T +(THead (Flat Cast) u2 t6) t4)).(eq_ind T (THead (Flat Cast) u2 t6) (\lambda +(t7: T).((pr0 t3 u2) \to ((pr0 t2 t6) \to (ty3 g c2 t7 (THead (Flat Cast) t0 +t3))))) (\lambda (H16: (pr0 t3 u2)).(\lambda (H17: (pr0 t2 t6)).(ex_ind T +(\lambda (t7: T).(ty3 g c2 t0 t7)) (ty3 g c2 (THead (Flat Cast) u2 t6) (THead +(Flat Cast) t0 t3)) (\lambda (x: T).(\lambda (H18: (ty3 g c2 t0 x)).(ty3_conv +g c2 (THead (Flat Cast) t0 t3) (THead (Flat Cast) x t0) (ty3_cast g c2 t3 t0 +(H3 c2 H4 t3 (pr0_refl t3)) x H18) (THead (Flat Cast) u2 t6) (THead (Flat +Cast) t0 u2) (ty3_cast g c2 t6 u2 (ty3_conv g c2 u2 t0 (H3 c2 H4 u2 H16) t6 +t3 (H1 c2 H4 t6 H17) (pc3_pr2_r c2 t3 u2 (pr2_free c2 t3 u2 H16))) t0 (H3 c2 +H4 u2 H16)) (pc3_s c2 (THead (Flat Cast) t0 u2) (THead (Flat Cast) t0 t3) +(pc3_pr2_r c2 (THead (Flat Cast) t0 t3) (THead (Flat Cast) t0 u2) +(pr2_thin_dx c2 t3 u2 (pr2_free c2 t3 u2 H16) t0 Cast)))))) (ty3_correct g c2 +t3 t0 (H3 c2 H4 t3 (pr0_refl t3)))))) t4 H15)) t5 (sym_eq T t5 t2 H14))) u1 +(sym_eq T u1 t3 H13))) k (sym_eq K k (Flat Cast) H12))) H11)) H10)) H9 H6 +H7))) | (pr0_beta u v1 v2 H6 t5 t6 H7) \Rightarrow (\lambda (H8: (eq T (THead +(Flat Appl) v1 (THead (Bind Abst) u t5)) (THead (Flat Cast) t3 t2))).(\lambda +(H9: (eq T (THead (Bind Abbr) v2 t6) t4)).((let H10 \def (eq_ind T (THead +(Flat Appl) v1 (THead (Bind Abst) u t5)) (\lambda (e: T).(match e in T return +(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda +(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f +in F return (\lambda (_: F).Prop) with [Appl \Rightarrow True | Cast +\Rightarrow False])])])) I (THead (Flat Cast) t3 t2) H8) in (False_ind ((eq T +(THead (Bind Abbr) v2 t6) t4) \to ((pr0 v1 v2) \to ((pr0 t5 t6) \to (ty3 g c2 +t4 (THead (Flat Cast) t0 t3))))) H10)) H9 H6 H7))) | (pr0_upsilon b H6 v1 v2 +H7 u1 u2 H8 t5 t6 H9) \Rightarrow (\lambda (H10: (eq T (THead (Flat Appl) v1 +(THead (Bind b) u1 t5)) (THead (Flat Cast) t3 t2))).(\lambda (H11: (eq T +(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t6)) t4)).((let H12 +\def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t5)) (\lambda (e: +T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow +False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K +return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) +\Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl \Rightarrow +True | Cast \Rightarrow False])])])) I (THead (Flat Cast) t3 t2) H10) in +(False_ind ((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) +t6)) t4) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 +t5 t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3))))))) H12)) H11 H6 H7 H8 +H9))) | (pr0_delta u1 u2 H6 t5 t6 H7 w H8) \Rightarrow (\lambda (H9: (eq T +(THead (Bind Abbr) u1 t5) (THead (Flat Cast) t3 t2))).(\lambda (H10: (eq T +(THead (Bind Abbr) u2 w) t4)).((let H11 \def (eq_ind T (THead (Bind Abbr) u1 +t5) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort +_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) +\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) +\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Cast) t3 +t2) H9) in (False_ind ((eq T (THead (Bind Abbr) u2 w) t4) \to ((pr0 u1 u2) +\to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ty3 g c2 t4 (THead (Flat Cast) +t0 t3)))))) H11)) H10 H6 H7 H8))) | (pr0_zeta b H6 t5 t6 H7 u) \Rightarrow +(\lambda (H8: (eq T (THead (Bind b) u (lift (S O) O t5)) (THead (Flat Cast) +t3 t2))).(\lambda (H9: (eq T t6 t4)).((let H10 \def (eq_ind T (THead (Bind b) +u (lift (S O) O t5)) (\lambda (e: T).(match e in T return (\lambda (_: +T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | +(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with +[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat +Cast) t3 t2) H8) in (False_ind ((eq T t6 t4) \to ((not (eq B b Abst)) \to +((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3))))) H10)) H9 H6 H7))) +| (pr0_tau t5 t6 H6 u) \Rightarrow (\lambda (H7: (eq T (THead (Flat Cast) u +t5) (THead (Flat Cast) t3 t2))).(\lambda (H8: (eq T t6 t4)).((let H9 \def +(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with +[(TSort _) \Rightarrow t5 | (TLRef _) \Rightarrow t5 | (THead _ _ t7) +\Rightarrow t7])) (THead (Flat Cast) u t5) (THead (Flat Cast) t3 t2) H7) in +((let H10 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: +T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t7 +_) \Rightarrow t7])) (THead (Flat Cast) u t5) (THead (Flat Cast) t3 t2) H7) +in (eq_ind T t3 (\lambda (_: T).((eq T t5 t2) \to ((eq T t6 t4) \to ((pr0 t5 +t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3)))))) (\lambda (H11: (eq T t5 +t2)).(eq_ind T t2 (\lambda (t7: T).((eq T t6 t4) \to ((pr0 t7 t6) \to (ty3 g +c2 t4 (THead (Flat Cast) t0 t3))))) (\lambda (H12: (eq T t6 t4)).(eq_ind T t4 +(\lambda (t7: T).((pr0 t2 t7) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3)))) +(\lambda (H13: (pr0 t2 t4)).(ex_ind T (\lambda (t7: T).(ty3 g c2 t0 t7)) (ty3 +g c2 t4 (THead (Flat Cast) t0 t3)) (\lambda (x: T).(\lambda (H14: (ty3 g c2 +t0 x)).(ty3_conv g c2 (THead (Flat Cast) t0 t3) (THead (Flat Cast) x t0) +(ty3_cast g c2 t3 t0 (H3 c2 H4 t3 (pr0_refl t3)) x H14) t4 t3 (H1 c2 H4 t4 +H13) (pc3_pr2_x c2 t3 (THead (Flat Cast) t0 t3) (pr2_free c2 (THead (Flat +Cast) t0 t3) t3 (pr0_tau t3 t3 (pr0_refl t3) t0)))))) (ty3_correct g c2 t3 t0 +(H3 c2 H4 t3 (pr0_refl t3))))) t6 (sym_eq T t6 t4 H12))) t5 (sym_eq T t5 t2 +H11))) u (sym_eq T u t3 H10))) H9)) H8 H6)))]) in (H6 (refl_equal T (THead +(Flat Cast) t3 t2)) (refl_equal T t4))))))))))))))) c1 t1 t H))))). +(* COMMENTS +Initial nodes: 14710 +END *) + +theorem ty3_sred_pr0: + \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (g: G).(\forall +(c: C).(\forall (t: T).((ty3 g c t1 t) \to (ty3 g c t2 t))))))) +\def + \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(\lambda (g: +G).(\lambda (c: C).(\lambda (t: T).(\lambda (H0: (ty3 g c t1 +t)).(ty3_sred_wcpr0_pr0 g c t1 t H0 c (wcpr0_refl c) t2 H))))))). +(* COMMENTS +Initial nodes: 47 +END *) + +theorem ty3_sred_pr1: + \forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall (g: G).(\forall +(c: C).(\forall (t: T).((ty3 g c t1 t) \to (ty3 g c t2 t))))))) +\def + \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr1 t1 t2)).(pr1_ind (\lambda +(t: T).(\lambda (t0: T).(\forall (g: G).(\forall (c: C).(\forall (t3: +T).((ty3 g c t t3) \to (ty3 g c t0 t3))))))) (\lambda (t: T).(\lambda (g: +G).(\lambda (c: C).(\lambda (t0: T).(\lambda (H0: (ty3 g c t t0)).H0))))) +(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t4 t3)).(\lambda (t5: +T).(\lambda (_: (pr1 t3 t5)).(\lambda (H2: ((\forall (g: G).(\forall (c: +C).(\forall (t: T).((ty3 g c t3 t) \to (ty3 g c t5 t))))))).(\lambda (g: +G).(\lambda (c: C).(\lambda (t: T).(\lambda (H3: (ty3 g c t4 t)).(H2 g c t +(ty3_sred_pr0 t4 t3 H0 g c t H3)))))))))))) t1 t2 H))). +(* COMMENTS +Initial nodes: 151 +END *) + +theorem ty3_sred_pr2: + \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall +(g: G).(\forall (t: T).((ty3 g c t1 t) \to (ty3 g c t2 t))))))) +\def + \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 +t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (g: +G).(\forall (t3: T).((ty3 g c0 t t3) \to (ty3 g c0 t0 t3))))))) (\lambda (c0: +C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(\lambda (g: +G).(\lambda (t: T).(\lambda (H1: (ty3 g c0 t3 t)).(ty3_sred_wcpr0_pr0 g c0 t3 +t H1 c0 (wcpr0_refl c0) t4 H0)))))))) (\lambda (c0: C).(\lambda (d: +C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind +Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 +t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (g: +G).(\lambda (t0: T).(\lambda (H3: (ty3 g c0 t3 t0)).(ty3_subst0 g c0 t4 t0 +(ty3_sred_wcpr0_pr0 g c0 t3 t0 H3 c0 (wcpr0_refl c0) t4 H1) d u i H0 t +H2)))))))))))))) c t1 t2 H)))). +(* COMMENTS +Initial nodes: 205 +END *) + +theorem ty3_sred_pr3: + \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall +(g: G).(\forall (t: T).((ty3 g c t1 t) \to (ty3 g c t2 t))))))) +\def + \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 +t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (g: G).(\forall +(t3: T).((ty3 g c t t3) \to (ty3 g c t0 t3)))))) (\lambda (t: T).(\lambda (g: +G).(\lambda (t0: T).(\lambda (H0: (ty3 g c t t0)).H0)))) (\lambda (t3: +T).(\lambda (t4: T).(\lambda (H0: (pr2 c t4 t3)).(\lambda (t5: T).(\lambda +(_: (pr3 c t3 t5)).(\lambda (H2: ((\forall (g: G).(\forall (t: T).((ty3 g c +t3 t) \to (ty3 g c t5 t)))))).(\lambda (g: G).(\lambda (t: T).(\lambda (H3: +(ty3 g c t4 t)).(H2 g t (ty3_sred_pr2 c t4 t3 H0 g t H3))))))))))) t1 t2 +H)))). +(* COMMENTS +Initial nodes: 151 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/ty3/pr3_props.ma b/matita/matita/contribs/lambdadelta/basic_1/ty3/pr3_props.ma new file mode 100644 index 000000000..b5bb1fb69 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/ty3/pr3_props.ma @@ -0,0 +1,513 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/ty3/pr3.ma". + +theorem ty3_cred_pr2: + \forall (g: G).(\forall (c: C).(\forall (v1: T).(\forall (v2: T).((pr2 c v1 +v2) \to (\forall (b: B).(\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c +(Bind b) v1) t1 t2) \to (ty3 g (CHead c (Bind b) v2) t1 t2))))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda +(H: (pr2 c v1 v2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: +T).(\forall (b: B).(\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c0 (Bind +b) t) t1 t2) \to (ty3 g (CHead c0 (Bind b) t0) t1 t2)))))))) (\lambda (c0: +C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr0 t1 t2)).(\lambda (b: +B).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H1: (ty3 g (CHead c0 (Bind b) +t1) t0 t3)).(ty3_sred_wcpr0_pr0 g (CHead c0 (Bind b) t1) t0 t3 H1 (CHead c0 +(Bind b) t2) (wcpr0_comp c0 c0 (wcpr0_refl c0) t1 t2 H0 (Bind b)) t0 +(pr0_refl t0)))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: +T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr) +u))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H1: (pr0 t1 t2)).(\lambda +(t: T).(\lambda (H2: (subst0 i u t2 t)).(\lambda (b: B).(\lambda (t0: +T).(\lambda (t3: T).(\lambda (H3: (ty3 g (CHead c0 (Bind b) t1) t0 +t3)).(ty3_csubst0 g (CHead c0 (Bind b) t2) t0 t3 (ty3_sred_wcpr0_pr0 g (CHead +c0 (Bind b) t1) t0 t3 H3 (CHead c0 (Bind b) t2) (wcpr0_comp c0 c0 (wcpr0_refl +c0) t1 t2 H1 (Bind b)) t0 (pr0_refl t0)) d u (S i) (getl_clear_bind b (CHead +c0 (Bind b) t2) c0 t2 (clear_bind b c0 t2) (CHead d (Bind Abbr) u) i H0) +(CHead c0 (Bind b) t) (csubst0_snd_bind b i u t2 t H2 c0)))))))))))))))) c v1 +v2 H))))). +(* COMMENTS +Initial nodes: 383 +END *) + +theorem ty3_cred_pr3: + \forall (g: G).(\forall (c: C).(\forall (v1: T).(\forall (v2: T).((pr3 c v1 +v2) \to (\forall (b: B).(\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c +(Bind b) v1) t1 t2) \to (ty3 g (CHead c (Bind b) v2) t1 t2))))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda +(H: (pr3 c v1 v2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (b: +B).(\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c (Bind b) t) t1 t2) \to +(ty3 g (CHead c (Bind b) t0) t1 t2))))))) (\lambda (t: T).(\lambda (b: +B).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (ty3 g (CHead c (Bind b) +t) t1 t2)).H0))))) (\lambda (t2: T).(\lambda (t1: T).(\lambda (H0: (pr2 c t1 +t2)).(\lambda (t3: T).(\lambda (_: (pr3 c t2 t3)).(\lambda (H2: ((\forall (b: +B).(\forall (t4: T).(\forall (t5: T).((ty3 g (CHead c (Bind b) t2) t4 t5) \to +(ty3 g (CHead c (Bind b) t3) t4 t5))))))).(\lambda (b: B).(\lambda (t0: +T).(\lambda (t4: T).(\lambda (H3: (ty3 g (CHead c (Bind b) t1) t0 t4)).(H2 b +t0 t4 (ty3_cred_pr2 g c t1 t2 H0 b t0 t4 H3)))))))))))) v1 v2 H))))). +(* COMMENTS +Initial nodes: 215 +END *) + +theorem ty3_gen_lift: + \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (x: T).(\forall (h: +nat).(\forall (d: nat).((ty3 g c (lift h d t1) x) \to (\forall (e: C).((drop +h d c e) \to (ex2 T (\lambda (t2: T).(pc3 c (lift h d t2) x)) (\lambda (t2: +T).(ty3 g e t1 t2))))))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (x: T).(\lambda (h: +nat).(\lambda (d: nat).(\lambda (H: (ty3 g c (lift h d t1) x)).(insert_eq T +(lift h d t1) (\lambda (t: T).(ty3 g c t x)) (\lambda (_: T).(\forall (e: +C).((drop h d c e) \to (ex2 T (\lambda (t2: T).(pc3 c (lift h d t2) x)) +(\lambda (t2: T).(ty3 g e t1 t2)))))) (\lambda (y: T).(\lambda (H0: (ty3 g c +y x)).(unintro nat d (\lambda (n: nat).((eq T y (lift h n t1)) \to (\forall +(e: C).((drop h n c e) \to (ex2 T (\lambda (t2: T).(pc3 c (lift h n t2) x)) +(\lambda (t2: T).(ty3 g e t1 t2))))))) (unintro T t1 (\lambda (t: T).(\forall +(x0: nat).((eq T y (lift h x0 t)) \to (\forall (e: C).((drop h x0 c e) \to +(ex2 T (\lambda (t2: T).(pc3 c (lift h x0 t2) x)) (\lambda (t2: T).(ty3 g e t +t2)))))))) (ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: +T).(\forall (x0: T).(\forall (x1: nat).((eq T t (lift h x1 x0)) \to (\forall +(e: C).((drop h x1 c0 e) \to (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) +t0)) (\lambda (t2: T).(ty3 g e x0 t2))))))))))) (\lambda (c0: C).(\lambda +(t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda (_: ((\forall +(x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to (\forall (e: +C).((drop h x1 c0 e) \to (ex2 T (\lambda (t3: T).(pc3 c0 (lift h x1 t3) t)) +(\lambda (t3: T).(ty3 g e x0 t3)))))))))).(\lambda (u: T).(\lambda (t3: +T).(\lambda (H3: (ty3 g c0 u t3)).(\lambda (H4: ((\forall (x0: T).(\forall +(x1: nat).((eq T u (lift h x1 x0)) \to (\forall (e: C).((drop h x1 c0 e) \to +(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) t3)) (\lambda (t4: T).(ty3 g e +x0 t4)))))))))).(\lambda (H5: (pc3 c0 t3 t2)).(\lambda (x0: T).(\lambda (x1: +nat).(\lambda (H6: (eq T u (lift h x1 x0))).(\lambda (e: C).(\lambda (H7: +(drop h x1 c0 e)).(let H8 \def (eq_ind T u (\lambda (t0: T).(\forall (x2: +T).(\forall (x3: nat).((eq T t0 (lift h x3 x2)) \to (\forall (e0: C).((drop h +x3 c0 e0) \to (ex2 T (\lambda (t4: T).(pc3 c0 (lift h x3 t4) t3)) (\lambda +(t4: T).(ty3 g e0 x2 t4))))))))) H4 (lift h x1 x0) H6) in (let H9 \def +(eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 t3)) H3 (lift h x1 x0) H6) in (let +H10 \def (H8 x0 x1 (refl_equal T (lift h x1 x0)) e H7) in (ex2_ind T (\lambda +(t4: T).(pc3 c0 (lift h x1 t4) t3)) (\lambda (t4: T).(ty3 g e x0 t4)) (ex2 T +(\lambda (t4: T).(pc3 c0 (lift h x1 t4) t2)) (\lambda (t4: T).(ty3 g e x0 +t4))) (\lambda (x2: T).(\lambda (H11: (pc3 c0 (lift h x1 x2) t3)).(\lambda +(H12: (ty3 g e x0 x2)).(ex_intro2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) +t2)) (\lambda (t4: T).(ty3 g e x0 t4)) x2 (pc3_t t3 c0 (lift h x1 x2) H11 t2 +H5) H12)))) H10))))))))))))))))))) (\lambda (c0: C).(\lambda (m: +nat).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H1: (eq T (TSort m) (lift +h x1 x0))).(\lambda (e: C).(\lambda (_: (drop h x1 c0 e)).(eq_ind_r T (TSort +m) (\lambda (t: T).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (TSort +(next g m)))) (\lambda (t2: T).(ty3 g e t t2)))) (ex_intro2 T (\lambda (t2: +T).(pc3 c0 (lift h x1 t2) (TSort (next g m)))) (\lambda (t2: T).(ty3 g e +(TSort m) t2)) (TSort (next g m)) (eq_ind_r T (TSort (next g m)) (\lambda (t: +T).(pc3 c0 t (TSort (next g m)))) (pc3_refl c0 (TSort (next g m))) (lift h x1 +(TSort (next g m))) (lift_sort (next g m) h x1)) (ty3_sort g e m)) x0 +(lift_gen_sort h x1 m x0 H1))))))))) (\lambda (n: nat).(\lambda (c0: +C).(\lambda (d0: C).(\lambda (u: T).(\lambda (H1: (getl n c0 (CHead d0 (Bind +Abbr) u))).(\lambda (t: T).(\lambda (H2: (ty3 g d0 u t)).(\lambda (H3: +((\forall (x0: T).(\forall (x1: nat).((eq T u (lift h x1 x0)) \to (\forall +(e: C).((drop h x1 d0 e) \to (ex2 T (\lambda (t2: T).(pc3 d0 (lift h x1 t2) +t)) (\lambda (t2: T).(ty3 g e x0 t2)))))))))).(\lambda (x0: T).(\lambda (x1: +nat).(\lambda (H4: (eq T (TLRef n) (lift h x1 x0))).(\lambda (e: C).(\lambda +(H5: (drop h x1 c0 e)).(let H_x \def (lift_gen_lref x0 x1 h n H4) in (let H6 +\def H_x in (or_ind (land (lt n x1) (eq T x0 (TLRef n))) (land (le (plus x1 +h) n) (eq T x0 (TLRef (minus n h)))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h +x1 t2) (lift (S n) O t))) (\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (H7: +(land (lt n x1) (eq T x0 (TLRef n)))).(land_ind (lt n x1) (eq T x0 (TLRef n)) +(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t))) (\lambda +(t2: T).(ty3 g e x0 t2))) (\lambda (H8: (lt n x1)).(\lambda (H9: (eq T x0 +(TLRef n))).(eq_ind_r T (TLRef n) (\lambda (t0: T).(ex2 T (\lambda (t2: +T).(pc3 c0 (lift h x1 t2) (lift (S n) O t))) (\lambda (t2: T).(ty3 g e t0 +t2)))) (let H10 \def (eq_ind nat x1 (\lambda (n0: nat).(drop h n0 c0 e)) H5 +(S (plus n (minus x1 (S n)))) (lt_plus_minus n x1 H8)) in (ex3_2_ind T C +(\lambda (v: T).(\lambda (_: C).(eq T u (lift h (minus x1 (S n)) v)))) +(\lambda (v: T).(\lambda (e0: C).(getl n e (CHead e0 (Bind Abbr) v)))) +(\lambda (_: T).(\lambda (e0: C).(drop h (minus x1 (S n)) d0 e0))) (ex2 T +(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t))) (\lambda (t2: +T).(ty3 g e (TLRef n) t2))) (\lambda (x2: T).(\lambda (x3: C).(\lambda (H11: +(eq T u (lift h (minus x1 (S n)) x2))).(\lambda (H12: (getl n e (CHead x3 +(Bind Abbr) x2))).(\lambda (H13: (drop h (minus x1 (S n)) d0 x3)).(let H14 +\def (eq_ind T u (\lambda (t0: T).(\forall (x4: T).(\forall (x5: nat).((eq T +t0 (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 d0 e0) \to (ex2 T +(\lambda (t2: T).(pc3 d0 (lift h x5 t2) t)) (\lambda (t2: T).(ty3 g e0 x4 +t2))))))))) H3 (lift h (minus x1 (S n)) x2) H11) in (let H15 \def (eq_ind T u +(\lambda (t0: T).(ty3 g d0 t0 t)) H2 (lift h (minus x1 (S n)) x2) H11) in +(let H16 \def (H14 x2 (minus x1 (S n)) (refl_equal T (lift h (minus x1 (S n)) +x2)) x3 H13) in (ex2_ind T (\lambda (t2: T).(pc3 d0 (lift h (minus x1 (S n)) +t2) t)) (\lambda (t2: T).(ty3 g x3 x2 t2)) (ex2 T (\lambda (t2: T).(pc3 c0 +(lift h x1 t2) (lift (S n) O t))) (\lambda (t2: T).(ty3 g e (TLRef n) t2))) +(\lambda (x4: T).(\lambda (H17: (pc3 d0 (lift h (minus x1 (S n)) x4) +t)).(\lambda (H18: (ty3 g x3 x2 x4)).(eq_ind_r nat (plus (S n) (minus x1 (S +n))) (\lambda (n0: nat).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h n0 t2) (lift +(S n) O t))) (\lambda (t2: T).(ty3 g e (TLRef n) t2)))) (ex_intro2 T (\lambda +(t2: T).(pc3 c0 (lift h (plus (S n) (minus x1 (S n))) t2) (lift (S n) O t))) +(\lambda (t2: T).(ty3 g e (TLRef n) t2)) (lift (S n) O x4) (eq_ind_r T (lift +(S n) O (lift h (minus x1 (S n)) x4)) (\lambda (t0: T).(pc3 c0 t0 (lift (S n) +O t))) (pc3_lift c0 d0 (S n) O (getl_drop Abbr c0 d0 u n H1) (lift h (minus +x1 (S n)) x4) t H17) (lift h (plus (S n) (minus x1 (S n))) (lift (S n) O x4)) +(lift_d x4 h (S n) (minus x1 (S n)) O (le_O_n (minus x1 (S n))))) (ty3_abbr g +n e x3 x2 H12 x4 H18)) x1 (le_plus_minus (S n) x1 H8))))) H16))))))))) +(getl_drop_conf_lt Abbr c0 d0 u n H1 e h (minus x1 (S n)) H10))) x0 H9))) +H7)) (\lambda (H7: (land (le (plus x1 h) n) (eq T x0 (TLRef (minus n +h))))).(land_ind (le (plus x1 h) n) (eq T x0 (TLRef (minus n h))) (ex2 T +(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t))) (\lambda (t2: +T).(ty3 g e x0 t2))) (\lambda (H8: (le (plus x1 h) n)).(\lambda (H9: (eq T x0 +(TLRef (minus n h)))).(eq_ind_r T (TLRef (minus n h)) (\lambda (t0: T).(ex2 T +(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t))) (\lambda (t2: +T).(ty3 g e t0 t2)))) (ex_intro2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) +(lift (S n) O t))) (\lambda (t2: T).(ty3 g e (TLRef (minus n h)) t2)) (lift +(S (minus n h)) O t) (eq_ind_r T (lift (plus h (S (minus n h))) O t) (\lambda +(t0: T).(pc3 c0 t0 (lift (S n) O t))) (eq_ind nat (S (plus h (minus n h))) +(\lambda (n0: nat).(pc3 c0 (lift n0 O t) (lift (S n) O t))) (eq_ind nat n +(\lambda (n0: nat).(pc3 c0 (lift (S n0) O t) (lift (S n) O t))) (pc3_refl c0 +(lift (S n) O t)) (plus h (minus n h)) (le_plus_minus h n (le_trans h (plus +x1 h) n (le_plus_r x1 h) H8))) (plus h (S (minus n h))) (plus_n_Sm h (minus n +h))) (lift h x1 (lift (S (minus n h)) O t)) (lift_free t (S (minus n h)) h O +x1 (le_trans x1 (S (minus n h)) (plus O (S (minus n h))) (le_S_minus x1 h n +H8) (le_n (plus O (S (minus n h))))) (le_O_n x1))) (ty3_abbr g (minus n h) e +d0 u (getl_drop_conf_ge n (CHead d0 (Bind Abbr) u) c0 H1 e h x1 H5 H8) t H2)) +x0 H9))) H7)) H6)))))))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda +(d0: C).(\lambda (u: T).(\lambda (H1: (getl n c0 (CHead d0 (Bind Abst) +u))).(\lambda (t: T).(\lambda (H2: (ty3 g d0 u t)).(\lambda (H3: ((\forall +(x0: T).(\forall (x1: nat).((eq T u (lift h x1 x0)) \to (\forall (e: +C).((drop h x1 d0 e) \to (ex2 T (\lambda (t2: T).(pc3 d0 (lift h x1 t2) t)) +(\lambda (t2: T).(ty3 g e x0 t2)))))))))).(\lambda (x0: T).(\lambda (x1: +nat).(\lambda (H4: (eq T (TLRef n) (lift h x1 x0))).(\lambda (e: C).(\lambda +(H5: (drop h x1 c0 e)).(let H_x \def (lift_gen_lref x0 x1 h n H4) in (let H6 +\def H_x in (or_ind (land (lt n x1) (eq T x0 (TLRef n))) (land (le (plus x1 +h) n) (eq T x0 (TLRef (minus n h)))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h +x1 t2) (lift (S n) O u))) (\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (H7: +(land (lt n x1) (eq T x0 (TLRef n)))).(land_ind (lt n x1) (eq T x0 (TLRef n)) +(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O u))) (\lambda +(t2: T).(ty3 g e x0 t2))) (\lambda (H8: (lt n x1)).(\lambda (H9: (eq T x0 +(TLRef n))).(eq_ind_r T (TLRef n) (\lambda (t0: T).(ex2 T (\lambda (t2: +T).(pc3 c0 (lift h x1 t2) (lift (S n) O u))) (\lambda (t2: T).(ty3 g e t0 +t2)))) (let H10 \def (eq_ind nat x1 (\lambda (n0: nat).(drop h n0 c0 e)) H5 +(S (plus n (minus x1 (S n)))) (lt_plus_minus n x1 H8)) in (ex3_2_ind T C +(\lambda (v: T).(\lambda (_: C).(eq T u (lift h (minus x1 (S n)) v)))) +(\lambda (v: T).(\lambda (e0: C).(getl n e (CHead e0 (Bind Abst) v)))) +(\lambda (_: T).(\lambda (e0: C).(drop h (minus x1 (S n)) d0 e0))) (ex2 T +(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O u))) (\lambda (t2: +T).(ty3 g e (TLRef n) t2))) (\lambda (x2: T).(\lambda (x3: C).(\lambda (H11: +(eq T u (lift h (minus x1 (S n)) x2))).(\lambda (H12: (getl n e (CHead x3 +(Bind Abst) x2))).(\lambda (H13: (drop h (minus x1 (S n)) d0 x3)).(let H14 +\def (eq_ind T u (\lambda (t0: T).(\forall (x4: T).(\forall (x5: nat).((eq T +t0 (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 d0 e0) \to (ex2 T +(\lambda (t2: T).(pc3 d0 (lift h x5 t2) t)) (\lambda (t2: T).(ty3 g e0 x4 +t2))))))))) H3 (lift h (minus x1 (S n)) x2) H11) in (let H15 \def (eq_ind T u +(\lambda (t0: T).(ty3 g d0 t0 t)) H2 (lift h (minus x1 (S n)) x2) H11) in +(eq_ind_r T (lift h (minus x1 (S n)) x2) (\lambda (t0: T).(ex2 T (\lambda +(t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t0))) (\lambda (t2: T).(ty3 g e +(TLRef n) t2)))) (let H16 \def (H14 x2 (minus x1 (S n)) (refl_equal T (lift h +(minus x1 (S n)) x2)) x3 H13) in (ex2_ind T (\lambda (t2: T).(pc3 d0 (lift h +(minus x1 (S n)) t2) t)) (\lambda (t2: T).(ty3 g x3 x2 t2)) (ex2 T (\lambda +(t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O (lift h (minus x1 (S n)) x2)))) +(\lambda (t2: T).(ty3 g e (TLRef n) t2))) (\lambda (x4: T).(\lambda (_: (pc3 +d0 (lift h (minus x1 (S n)) x4) t)).(\lambda (H18: (ty3 g x3 x2 +x4)).(eq_ind_r nat (plus (S n) (minus x1 (S n))) (\lambda (n0: nat).(ex2 T +(\lambda (t2: T).(pc3 c0 (lift h n0 t2) (lift (S n) O (lift h (minus n0 (S +n)) x2)))) (\lambda (t2: T).(ty3 g e (TLRef n) t2)))) (ex_intro2 T (\lambda +(t2: T).(pc3 c0 (lift h (plus (S n) (minus x1 (S n))) t2) (lift (S n) O (lift +h (minus (plus (S n) (minus x1 (S n))) (S n)) x2)))) (\lambda (t2: T).(ty3 g +e (TLRef n) t2)) (lift (S n) O x2) (eq_ind_r T (lift (S n) O (lift h (minus +x1 (S n)) x2)) (\lambda (t0: T).(pc3 c0 t0 (lift (S n) O (lift h (minus (plus +(S n) (minus x1 (S n))) (S n)) x2)))) (eq_ind nat x1 (\lambda (n0: nat).(pc3 +c0 (lift (S n) O (lift h (minus x1 (S n)) x2)) (lift (S n) O (lift h (minus +n0 (S n)) x2)))) (pc3_refl c0 (lift (S n) O (lift h (minus x1 (S n)) x2))) +(plus (S n) (minus x1 (S n))) (le_plus_minus (S n) x1 H8)) (lift h (plus (S +n) (minus x1 (S n))) (lift (S n) O x2)) (lift_d x2 h (S n) (minus x1 (S n)) O +(le_O_n (minus x1 (S n))))) (ty3_abst g n e x3 x2 H12 x4 H18)) x1 +(le_plus_minus (S n) x1 H8))))) H16)) u H11)))))))) (getl_drop_conf_lt Abst +c0 d0 u n H1 e h (minus x1 (S n)) H10))) x0 H9))) H7)) (\lambda (H7: (land +(le (plus x1 h) n) (eq T x0 (TLRef (minus n h))))).(land_ind (le (plus x1 h) +n) (eq T x0 (TLRef (minus n h))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 +t2) (lift (S n) O u))) (\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (H8: (le +(plus x1 h) n)).(\lambda (H9: (eq T x0 (TLRef (minus n h)))).(eq_ind_r T +(TLRef (minus n h)) (\lambda (t0: T).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h +x1 t2) (lift (S n) O u))) (\lambda (t2: T).(ty3 g e t0 t2)))) (ex_intro2 T +(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O u))) (\lambda (t2: +T).(ty3 g e (TLRef (minus n h)) t2)) (lift (S (minus n h)) O u) (eq_ind_r T +(lift (plus h (S (minus n h))) O u) (\lambda (t0: T).(pc3 c0 t0 (lift (S n) O +u))) (eq_ind nat (S (plus h (minus n h))) (\lambda (n0: nat).(pc3 c0 (lift n0 +O u) (lift (S n) O u))) (eq_ind nat n (\lambda (n0: nat).(pc3 c0 (lift (S n0) +O u) (lift (S n) O u))) (pc3_refl c0 (lift (S n) O u)) (plus h (minus n h)) +(le_plus_minus h n (le_trans h (plus x1 h) n (le_plus_r x1 h) H8))) (plus h +(S (minus n h))) (plus_n_Sm h (minus n h))) (lift h x1 (lift (S (minus n h)) +O u)) (lift_free u (S (minus n h)) h O x1 (le_trans x1 (S (minus n h)) (plus +O (S (minus n h))) (le_S_minus x1 h n H8) (le_n (plus O (S (minus n h))))) +(le_O_n x1))) (ty3_abst g (minus n h) e d0 u (getl_drop_conf_ge n (CHead d0 +(Bind Abst) u) c0 H1 e h x1 H5 H8) t H2)) x0 H9))) H7)) H6)))))))))))))))) +(\lambda (c0: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H1: (ty3 g c0 u +t)).(\lambda (H2: ((\forall (x0: T).(\forall (x1: nat).((eq T u (lift h x1 +x0)) \to (\forall (e: C).((drop h x1 c0 e) \to (ex2 T (\lambda (t2: T).(pc3 +c0 (lift h x1 t2) t)) (\lambda (t2: T).(ty3 g e x0 t2)))))))))).(\lambda (b: +B).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H3: (ty3 g (CHead c0 (Bind b) +u) t2 t3)).(\lambda (H4: ((\forall (x0: T).(\forall (x1: nat).((eq T t2 (lift +h x1 x0)) \to (\forall (e: C).((drop h x1 (CHead c0 (Bind b) u) e) \to (ex2 T +(\lambda (t4: T).(pc3 (CHead c0 (Bind b) u) (lift h x1 t4) t3)) (\lambda (t4: +T).(ty3 g e x0 t4)))))))))).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H5: +(eq T (THead (Bind b) u t2) (lift h x1 x0))).(\lambda (e: C).(\lambda (H6: +(drop h x1 c0 e)).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 +(THead (Bind b) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u (lift h x1 +y0)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h (S x1) z)))) (ex2 T +(\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b) u t3))) (\lambda (t4: +T).(ty3 g e x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq T x0 +(THead (Bind b) x2 x3))).(\lambda (H8: (eq T u (lift h x1 x2))).(\lambda (H9: +(eq T t2 (lift h (S x1) x3))).(eq_ind_r T (THead (Bind b) x2 x3) (\lambda +(t0: T).(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b) u +t3))) (\lambda (t4: T).(ty3 g e t0 t4)))) (let H10 \def (eq_ind T t2 (\lambda +(t0: T).(\forall (x4: T).(\forall (x5: nat).((eq T t0 (lift h x5 x4)) \to +(\forall (e0: C).((drop h x5 (CHead c0 (Bind b) u) e0) \to (ex2 T (\lambda +(t4: T).(pc3 (CHead c0 (Bind b) u) (lift h x5 t4) t3)) (\lambda (t4: T).(ty3 +g e0 x4 t4))))))))) H4 (lift h (S x1) x3) H9) in (let H11 \def (eq_ind T t2 +(\lambda (t0: T).(ty3 g (CHead c0 (Bind b) u) t0 t3)) H3 (lift h (S x1) x3) +H9) in (let H12 \def (eq_ind T u (\lambda (t0: T).(ty3 g (CHead c0 (Bind b) +t0) (lift h (S x1) x3) t3)) H11 (lift h x1 x2) H8) in (let H13 \def (eq_ind T +u (\lambda (t0: T).(\forall (x4: T).(\forall (x5: nat).((eq T (lift h (S x1) +x3) (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 (CHead c0 (Bind b) t0) +e0) \to (ex2 T (\lambda (t4: T).(pc3 (CHead c0 (Bind b) t0) (lift h x5 t4) +t3)) (\lambda (t4: T).(ty3 g e0 x4 t4))))))))) H10 (lift h x1 x2) H8) in (let +H14 \def (eq_ind T u (\lambda (t0: T).(\forall (x4: T).(\forall (x5: +nat).((eq T t0 (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 c0 e0) \to +(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x5 t4) t)) (\lambda (t4: T).(ty3 g e0 +x4 t4))))))))) H2 (lift h x1 x2) H8) in (let H15 \def (eq_ind T u (\lambda +(t0: T).(ty3 g c0 t0 t)) H1 (lift h x1 x2) H8) in (eq_ind_r T (lift h x1 x2) +(\lambda (t0: T).(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind +b) t0 t3))) (\lambda (t4: T).(ty3 g e (THead (Bind b) x2 x3) t4)))) (let H16 +\def (H14 x2 x1 (refl_equal T (lift h x1 x2)) e H6) in (ex2_ind T (\lambda +(t4: T).(pc3 c0 (lift h x1 t4) t)) (\lambda (t4: T).(ty3 g e x2 t4)) (ex2 T +(\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b) (lift h x1 x2) t3))) +(\lambda (t4: T).(ty3 g e (THead (Bind b) x2 x3) t4))) (\lambda (x4: +T).(\lambda (_: (pc3 c0 (lift h x1 x4) t)).(\lambda (H18: (ty3 g e x2 +x4)).(let H19 \def (H13 x3 (S x1) (refl_equal T (lift h (S x1) x3)) (CHead e +(Bind b) x2) (drop_skip_bind h x1 c0 e H6 b x2)) in (ex2_ind T (\lambda (t4: +T).(pc3 (CHead c0 (Bind b) (lift h x1 x2)) (lift h (S x1) t4) t3)) (\lambda +(t4: T).(ty3 g (CHead e (Bind b) x2) x3 t4)) (ex2 T (\lambda (t4: T).(pc3 c0 +(lift h x1 t4) (THead (Bind b) (lift h x1 x2) t3))) (\lambda (t4: T).(ty3 g e +(THead (Bind b) x2 x3) t4))) (\lambda (x5: T).(\lambda (H20: (pc3 (CHead c0 +(Bind b) (lift h x1 x2)) (lift h (S x1) x5) t3)).(\lambda (H21: (ty3 g (CHead +e (Bind b) x2) x3 x5)).(ex_ind T (\lambda (t0: T).(ty3 g (CHead e (Bind b) +x2) x5 t0)) (ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b) +(lift h x1 x2) t3))) (\lambda (t4: T).(ty3 g e (THead (Bind b) x2 x3) t4))) +(\lambda (x6: T).(\lambda (_: (ty3 g (CHead e (Bind b) x2) x5 x6)).(ex_intro2 +T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b) (lift h x1 x2) +t3))) (\lambda (t4: T).(ty3 g e (THead (Bind b) x2 x3) t4)) (THead (Bind b) +x2 x5) (eq_ind_r T (THead (Bind b) (lift h x1 x2) (lift h (S x1) x5)) +(\lambda (t0: T).(pc3 c0 t0 (THead (Bind b) (lift h x1 x2) t3))) (pc3_head_2 +c0 (lift h x1 x2) (lift h (S x1) x5) t3 (Bind b) H20) (lift h x1 (THead (Bind +b) x2 x5)) (lift_bind b x2 x5 h x1)) (ty3_bind g e x2 x4 H18 b x3 x5 H21)))) +(ty3_correct g (CHead e (Bind b) x2) x3 x5 H21))))) H19))))) H16)) u +H8))))))) x0 H7)))))) (lift_gen_bind b u t2 x0 h x1 H5))))))))))))))))) +(\lambda (c0: C).(\lambda (w: T).(\lambda (u: T).(\lambda (H1: (ty3 g c0 w +u)).(\lambda (H2: ((\forall (x0: T).(\forall (x1: nat).((eq T w (lift h x1 +x0)) \to (\forall (e: C).((drop h x1 c0 e) \to (ex2 T (\lambda (t2: T).(pc3 +c0 (lift h x1 t2) u)) (\lambda (t2: T).(ty3 g e x0 t2)))))))))).(\lambda (v: +T).(\lambda (t: T).(\lambda (H3: (ty3 g c0 v (THead (Bind Abst) u +t))).(\lambda (H4: ((\forall (x0: T).(\forall (x1: nat).((eq T v (lift h x1 +x0)) \to (\forall (e: C).((drop h x1 c0 e) \to (ex2 T (\lambda (t2: T).(pc3 +c0 (lift h x1 t2) (THead (Bind Abst) u t))) (\lambda (t2: T).(ty3 g e x0 +t2)))))))))).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H5: (eq T (THead +(Flat Appl) w v) (lift h x1 x0))).(\lambda (e: C).(\lambda (H6: (drop h x1 c0 +e)).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Flat +Appl) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T w (lift h x1 y0)))) +(\lambda (_: T).(\lambda (z: T).(eq T v (lift h x1 z)))) (ex2 T (\lambda (t2: +T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) w (THead (Bind Abst) u t)))) +(\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (x2: T).(\lambda (x3: T).(\lambda +(H7: (eq T x0 (THead (Flat Appl) x2 x3))).(\lambda (H8: (eq T w (lift h x1 +x2))).(\lambda (H9: (eq T v (lift h x1 x3))).(eq_ind_r T (THead (Flat Appl) +x2 x3) (\lambda (t0: T).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead +(Flat Appl) w (THead (Bind Abst) u t)))) (\lambda (t2: T).(ty3 g e t0 t2)))) +(let H10 \def (eq_ind T v (\lambda (t0: T).(\forall (x4: T).(\forall (x5: +nat).((eq T t0 (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 c0 e0) \to +(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x5 t2) (THead (Bind Abst) u t))) +(\lambda (t2: T).(ty3 g e0 x4 t2))))))))) H4 (lift h x1 x3) H9) in (let H11 +\def (eq_ind T v (\lambda (t0: T).(ty3 g c0 t0 (THead (Bind Abst) u t))) H3 +(lift h x1 x3) H9) in (let H12 \def (eq_ind T w (\lambda (t0: T).(\forall +(x4: T).(\forall (x5: nat).((eq T t0 (lift h x5 x4)) \to (\forall (e0: +C).((drop h x5 c0 e0) \to (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x5 t2) u)) +(\lambda (t2: T).(ty3 g e0 x4 t2))))))))) H2 (lift h x1 x2) H8) in (let H13 +\def (eq_ind T w (\lambda (t0: T).(ty3 g c0 t0 u)) H1 (lift h x1 x2) H8) in +(eq_ind_r T (lift h x1 x2) (\lambda (t0: T).(ex2 T (\lambda (t2: T).(pc3 c0 +(lift h x1 t2) (THead (Flat Appl) t0 (THead (Bind Abst) u t)))) (\lambda (t2: +T).(ty3 g e (THead (Flat Appl) x2 x3) t2)))) (let H14 \def (H12 x2 x1 +(refl_equal T (lift h x1 x2)) e H6) in (ex2_ind T (\lambda (t2: T).(pc3 c0 +(lift h x1 t2) u)) (\lambda (t2: T).(ty3 g e x2 t2)) (ex2 T (\lambda (t2: +T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) (lift h x1 x2) (THead (Bind +Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead (Flat Appl) x2 x3) t2))) +(\lambda (x4: T).(\lambda (H15: (pc3 c0 (lift h x1 x4) u)).(\lambda (H16: +(ty3 g e x2 x4)).(let H17 \def (H10 x3 x1 (refl_equal T (lift h x1 x3)) e H6) +in (ex2_ind T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead (Bind Abst) u +t))) (\lambda (t2: T).(ty3 g e x3 t2)) (ex2 T (\lambda (t2: T).(pc3 c0 (lift +h x1 t2) (THead (Flat Appl) (lift h x1 x2) (THead (Bind Abst) u t)))) +(\lambda (t2: T).(ty3 g e (THead (Flat Appl) x2 x3) t2))) (\lambda (x5: +T).(\lambda (H18: (pc3 c0 (lift h x1 x5) (THead (Bind Abst) u t))).(\lambda +(H19: (ty3 g e x3 x5)).(ex3_2_ind T T (\lambda (u1: T).(\lambda (t2: T).(pr3 +e x5 (THead (Bind Abst) u1 t2)))) (\lambda (u1: T).(\lambda (_: T).(pr3 c0 u +(lift h x1 u1)))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall +(u0: T).(pr3 (CHead c0 (Bind b) u0) t (lift h (S x1) t2)))))) (ex2 T (\lambda +(t2: T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) (lift h x1 x2) (THead (Bind +Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead (Flat Appl) x2 x3) t2))) +(\lambda (x6: T).(\lambda (x7: T).(\lambda (H20: (pr3 e x5 (THead (Bind Abst) +x6 x7))).(\lambda (H21: (pr3 c0 u (lift h x1 x6))).(\lambda (H22: ((\forall +(b: B).(\forall (u0: T).(pr3 (CHead c0 (Bind b) u0) t (lift h (S x1) +x7)))))).(ex_ind T (\lambda (t0: T).(ty3 g e x5 t0)) (ex2 T (\lambda (t2: +T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) (lift h x1 x2) (THead (Bind +Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead (Flat Appl) x2 x3) t2))) +(\lambda (x8: T).(\lambda (H23: (ty3 g e x5 x8)).(let H_y \def (ty3_sred_pr3 +e x5 (THead (Bind Abst) x6 x7) H20 g x8 H23) in (ex3_2_ind T T (\lambda (t2: +T).(\lambda (_: T).(pc3 e (THead (Bind Abst) x6 t2) x8))) (\lambda (_: +T).(\lambda (t0: T).(ty3 g e x6 t0))) (\lambda (t2: T).(\lambda (_: T).(ty3 g +(CHead e (Bind Abst) x6) x7 t2))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 +t2) (THead (Flat Appl) (lift h x1 x2) (THead (Bind Abst) u t)))) (\lambda +(t2: T).(ty3 g e (THead (Flat Appl) x2 x3) t2))) (\lambda (x9: T).(\lambda +(x10: T).(\lambda (_: (pc3 e (THead (Bind Abst) x6 x9) x8)).(\lambda (H25: +(ty3 g e x6 x10)).(\lambda (H26: (ty3 g (CHead e (Bind Abst) x6) x7 +x9)).(ex_intro2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) +(lift h x1 x2) (THead (Bind Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead +(Flat Appl) x2 x3) t2)) (THead (Flat Appl) x2 (THead (Bind Abst) x6 x7)) +(eq_ind_r T (THead (Flat Appl) (lift h x1 x2) (lift h x1 (THead (Bind Abst) +x6 x7))) (\lambda (t0: T).(pc3 c0 t0 (THead (Flat Appl) (lift h x1 x2) (THead +(Bind Abst) u t)))) (pc3_thin_dx c0 (lift h x1 (THead (Bind Abst) x6 x7)) +(THead (Bind Abst) u t) (eq_ind_r T (THead (Bind Abst) (lift h x1 x6) (lift h +(S x1) x7)) (\lambda (t0: T).(pc3 c0 t0 (THead (Bind Abst) u t))) +(pc3_head_21 c0 (lift h x1 x6) u (pc3_pr3_x c0 (lift h x1 x6) u H21) (Bind +Abst) (lift h (S x1) x7) t (pc3_pr3_x (CHead c0 (Bind Abst) (lift h x1 x6)) +(lift h (S x1) x7) t (H22 Abst (lift h x1 x6)))) (lift h x1 (THead (Bind +Abst) x6 x7)) (lift_bind Abst x6 x7 h x1)) (lift h x1 x2) Appl) (lift h x1 +(THead (Flat Appl) x2 (THead (Bind Abst) x6 x7))) (lift_flat Appl x2 (THead +(Bind Abst) x6 x7) h x1)) (ty3_appl g e x2 x6 (ty3_conv g e x6 x10 H25 x2 x4 +H16 (pc3_gen_lift c0 x4 x6 h x1 (pc3_t u c0 (lift h x1 x4) H15 (lift h x1 x6) +(pc3_pr3_r c0 u (lift h x1 x6) H21)) e H6)) x3 x7 (ty3_conv g e (THead (Bind +Abst) x6 x7) (THead (Bind Abst) x6 x9) (ty3_bind g e x6 x10 H25 Abst x7 x9 +H26) x3 x5 H19 (pc3_pr3_r e x5 (THead (Bind Abst) x6 x7) H20))))))))) +(ty3_gen_bind g Abst e x6 x7 x8 H_y))))) (ty3_correct g e x3 x5 H19))))))) +(pc3_gen_lift_abst c0 x5 t u h x1 H18 e H6))))) H17))))) H14)) w H8))))) x0 +H7)))))) (lift_gen_flat Appl w v x0 h x1 H5)))))))))))))))) (\lambda (c0: +C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (ty3 g c0 t2 t3)).(\lambda +(H2: ((\forall (x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to +(\forall (e: C).((drop h x1 c0 e) \to (ex2 T (\lambda (t4: T).(pc3 c0 (lift h +x1 t4) t3)) (\lambda (t4: T).(ty3 g e x0 t4)))))))))).(\lambda (t0: +T).(\lambda (H3: (ty3 g c0 t3 t0)).(\lambda (H4: ((\forall (x0: T).(\forall +(x1: nat).((eq T t3 (lift h x1 x0)) \to (\forall (e: C).((drop h x1 c0 e) \to +(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) t0)) (\lambda (t4: T).(ty3 g e +x0 t4)))))))))).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H5: (eq T +(THead (Flat Cast) t3 t2) (lift h x1 x0))).(\lambda (e: C).(\lambda (H6: +(drop h x1 c0 e)).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 +(THead (Flat Cast) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T t3 (lift h +x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h x1 z)))) (ex2 T +(\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Flat Cast) t0 t3))) (\lambda +(t4: T).(ty3 g e x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq +T x0 (THead (Flat Cast) x2 x3))).(\lambda (H8: (eq T t3 (lift h x1 +x2))).(\lambda (H9: (eq T t2 (lift h x1 x3))).(eq_ind_r T (THead (Flat Cast) +x2 x3) (\lambda (t: T).(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead +(Flat Cast) t0 t3))) (\lambda (t4: T).(ty3 g e t t4)))) (let H10 \def (eq_ind +T t3 (\lambda (t: T).(\forall (x4: T).(\forall (x5: nat).((eq T t (lift h x5 +x4)) \to (\forall (e0: C).((drop h x5 c0 e0) \to (ex2 T (\lambda (t4: T).(pc3 +c0 (lift h x5 t4) t0)) (\lambda (t4: T).(ty3 g e0 x4 t4))))))))) H4 (lift h +x1 x2) H8) in (let H11 \def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 t t0)) H3 +(lift h x1 x2) H8) in (let H12 \def (eq_ind T t3 (\lambda (t: T).(\forall +(x4: T).(\forall (x5: nat).((eq T t2 (lift h x5 x4)) \to (\forall (e0: +C).((drop h x5 c0 e0) \to (ex2 T (\lambda (t4: T).(pc3 c0 (lift h x5 t4) t)) +(\lambda (t4: T).(ty3 g e0 x4 t4))))))))) H2 (lift h x1 x2) H8) in (let H13 +\def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 t2 t)) H1 (lift h x1 x2) H8) in +(eq_ind_r T (lift h x1 x2) (\lambda (t: T).(ex2 T (\lambda (t4: T).(pc3 c0 +(lift h x1 t4) (THead (Flat Cast) t0 t))) (\lambda (t4: T).(ty3 g e (THead +(Flat Cast) x2 x3) t4)))) (let H14 \def (eq_ind T t2 (\lambda (t: T).(ty3 g +c0 t (lift h x1 x2))) H13 (lift h x1 x3) H9) in (let H15 \def (eq_ind T t2 +(\lambda (t: T).(\forall (x4: T).(\forall (x5: nat).((eq T t (lift h x5 x4)) +\to (\forall (e0: C).((drop h x5 c0 e0) \to (ex2 T (\lambda (t4: T).(pc3 c0 +(lift h x5 t4) (lift h x1 x2))) (\lambda (t4: T).(ty3 g e0 x4 t4))))))))) H12 +(lift h x1 x3) H9) in (let H16 \def (H15 x3 x1 (refl_equal T (lift h x1 x3)) +e H6) in (ex2_ind T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (lift h x1 x2))) +(\lambda (t4: T).(ty3 g e x3 t4)) (ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 +t4) (THead (Flat Cast) t0 (lift h x1 x2)))) (\lambda (t4: T).(ty3 g e (THead +(Flat Cast) x2 x3) t4))) (\lambda (x4: T).(\lambda (H17: (pc3 c0 (lift h x1 +x4) (lift h x1 x2))).(\lambda (H18: (ty3 g e x3 x4)).(let H19 \def (H10 x2 x1 +(refl_equal T (lift h x1 x2)) e H6) in (ex2_ind T (\lambda (t4: T).(pc3 c0 +(lift h x1 t4) t0)) (\lambda (t4: T).(ty3 g e x2 t4)) (ex2 T (\lambda (t4: +T).(pc3 c0 (lift h x1 t4) (THead (Flat Cast) t0 (lift h x1 x2)))) (\lambda +(t4: T).(ty3 g e (THead (Flat Cast) x2 x3) t4))) (\lambda (x5: T).(\lambda +(H20: (pc3 c0 (lift h x1 x5) t0)).(\lambda (H21: (ty3 g e x2 x5)).(ex_intro2 +T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Flat Cast) t0 (lift h x1 +x2)))) (\lambda (t4: T).(ty3 g e (THead (Flat Cast) x2 x3) t4)) (THead (Flat +Cast) x5 x2) (eq_ind_r T (THead (Flat Cast) (lift h x1 x5) (lift h x1 x2)) +(\lambda (t: T).(pc3 c0 t (THead (Flat Cast) t0 (lift h x1 x2)))) (pc3_head_1 +c0 (lift h x1 x5) t0 H20 (Flat Cast) (lift h x1 x2)) (lift h x1 (THead (Flat +Cast) x5 x2)) (lift_flat Cast x5 x2 h x1)) (ty3_cast g e x3 x2 (ty3_conv g e +x2 x5 H21 x3 x4 H18 (pc3_gen_lift c0 x4 x2 h x1 H17 e H6)) x5 H21))))) +H19))))) H16)))) t3 H8))))) x0 H7)))))) (lift_gen_flat Cast t3 t2 x0 h x1 +H5))))))))))))))) c y x H0))))) H))))))). +(* COMMENTS +Initial nodes: 9781 +END *) + +theorem ty3_tred: + \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).((ty3 g c u +t1) \to (\forall (t2: T).((pr3 c t1 t2) \to (ty3 g c u t2))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (H: +(ty3 g c u t1)).(\lambda (t2: T).(\lambda (H0: (pr3 c t1 t2)).(ex_ind T +(\lambda (t: T).(ty3 g c t1 t)) (ty3 g c u t2) (\lambda (x: T).(\lambda (H1: +(ty3 g c t1 x)).(let H_y \def (ty3_sred_pr3 c t1 t2 H0 g x H1) in (ty3_conv g +c t2 x H_y u t1 H (pc3_pr3_r c t1 t2 H0))))) (ty3_correct g c u t1 H)))))))). +(* COMMENTS +Initial nodes: 121 +END *) + +theorem ty3_sconv_pc3: + \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (t1: T).((ty3 g c +u1 t1) \to (\forall (u2: T).(\forall (t2: T).((ty3 g c u2 t2) \to ((pc3 c u1 +u2) \to (pc3 c t1 t2))))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda +(H: (ty3 g c u1 t1)).(\lambda (u2: T).(\lambda (t2: T).(\lambda (H0: (ty3 g c +u2 t2)).(\lambda (H1: (pc3 c u1 u2)).(let H2 \def H1 in (ex2_ind T (\lambda +(t: T).(pr3 c u1 t)) (\lambda (t: T).(pr3 c u2 t)) (pc3 c t1 t2) (\lambda (x: +T).(\lambda (H3: (pr3 c u1 x)).(\lambda (H4: (pr3 c u2 x)).(let H_y \def +(ty3_sred_pr3 c u2 x H4 g t2 H0) in (let H_y0 \def (ty3_sred_pr3 c u1 x H3 g +t1 H) in (ty3_unique g c x t1 H_y0 t2 H_y)))))) H2)))))))))). +(* COMMENTS +Initial nodes: 141 +END *) + +theorem ty3_sred_back: + \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t0: T).((ty3 g c +t1 t0) \to (\forall (t2: T).((pr3 c t1 t2) \to (\forall (t: T).((ty3 g c t2 +t) \to (ty3 g c t1 t))))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t0: T).(\lambda +(H: (ty3 g c t1 t0)).(\lambda (t2: T).(\lambda (H0: (pr3 c t1 t2)).(\lambda +(t: T).(\lambda (H1: (ty3 g c t2 t)).(ex_ind T (\lambda (t3: T).(ty3 g c t +t3)) (ty3 g c t1 t) (\lambda (x: T).(\lambda (H2: (ty3 g c t x)).(ty3_conv g +c t x H2 t1 t0 H (ty3_unique g c t2 t0 (ty3_sred_pr3 c t1 t2 H0 g t0 H) t +H1)))) (ty3_correct g c t2 t H1)))))))))). +(* COMMENTS +Initial nodes: 137 +END *) + +theorem ty3_sconv: + \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (t1: T).((ty3 g c +u1 t1) \to (\forall (u2: T).(\forall (t2: T).((ty3 g c u2 t2) \to ((pc3 c u1 +u2) \to (ty3 g c u1 t2))))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda +(H: (ty3 g c u1 t1)).(\lambda (u2: T).(\lambda (t2: T).(\lambda (H0: (ty3 g c +u2 t2)).(\lambda (H1: (pc3 c u1 u2)).(let H2 \def H1 in (ex2_ind T (\lambda +(t: T).(pr3 c u1 t)) (\lambda (t: T).(pr3 c u2 t)) (ty3 g c u1 t2) (\lambda +(x: T).(\lambda (H3: (pr3 c u1 x)).(\lambda (H4: (pr3 c u2 x)).(ty3_sred_back +g c u1 t1 H x H3 t2 (ty3_sred_pr3 c u2 x H4 g t2 H0))))) H2)))))))))). +(* COMMENTS +Initial nodes: 129 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/ty3/props.ma b/matita/matita/contribs/lambdadelta/basic_1/ty3/props.ma new file mode 100644 index 000000000..579a8dd69 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/ty3/props.ma @@ -0,0 +1,691 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/ty3/fwd.ma". + +include "Basic-1/pc3/fwd.ma". + +theorem ty3_lift: + \forall (g: G).(\forall (e: C).(\forall (t1: T).(\forall (t2: T).((ty3 g e +t1 t2) \to (\forall (c: C).(\forall (d: nat).(\forall (h: nat).((drop h d c +e) \to (ty3 g c (lift h d t1) (lift h d t2)))))))))) +\def + \lambda (g: G).(\lambda (e: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda +(H: (ty3 g e t1 t2)).(ty3_ind g (\lambda (c: C).(\lambda (t: T).(\lambda (t0: +T).(\forall (c0: C).(\forall (d: nat).(\forall (h: nat).((drop h d c0 c) \to +(ty3 g c0 (lift h d t) (lift h d t0))))))))) (\lambda (c: C).(\lambda (t0: +T).(\lambda (t: T).(\lambda (_: (ty3 g c t0 t)).(\lambda (H1: ((\forall (c0: +C).(\forall (d: nat).(\forall (h: nat).((drop h d c0 c) \to (ty3 g c0 (lift h +d t0) (lift h d t)))))))).(\lambda (u: T).(\lambda (t3: T).(\lambda (_: (ty3 +g c u t3)).(\lambda (H3: ((\forall (c0: C).(\forall (d: nat).(\forall (h: +nat).((drop h d c0 c) \to (ty3 g c0 (lift h d u) (lift h d +t3)))))))).(\lambda (H4: (pc3 c t3 t0)).(\lambda (c0: C).(\lambda (d: +nat).(\lambda (h: nat).(\lambda (H5: (drop h d c0 c)).(ty3_conv g c0 (lift h +d t0) (lift h d t) (H1 c0 d h H5) (lift h d u) (lift h d t3) (H3 c0 d h H5) +(pc3_lift c0 c h d H5 t3 t0 H4)))))))))))))))) (\lambda (c: C).(\lambda (m: +nat).(\lambda (c0: C).(\lambda (d: nat).(\lambda (h: nat).(\lambda (_: (drop +h d c0 c)).(eq_ind_r T (TSort m) (\lambda (t: T).(ty3 g c0 t (lift h d (TSort +(next g m))))) (eq_ind_r T (TSort (next g m)) (\lambda (t: T).(ty3 g c0 +(TSort m) t)) (ty3_sort g c0 m) (lift h d (TSort (next g m))) (lift_sort +(next g m) h d)) (lift h d (TSort m)) (lift_sort m h d)))))))) (\lambda (n: +nat).(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c +(CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u +t)).(\lambda (H2: ((\forall (c0: C).(\forall (d0: nat).(\forall (h: +nat).((drop h d0 c0 d) \to (ty3 g c0 (lift h d0 u) (lift h d0 +t)))))))).(\lambda (c0: C).(\lambda (d0: nat).(\lambda (h: nat).(\lambda (H3: +(drop h d0 c0 c)).(lt_le_e n d0 (ty3 g c0 (lift h d0 (TLRef n)) (lift h d0 +(lift (S n) O t))) (\lambda (H4: (lt n d0)).(let H5 \def (drop_getl_trans_le +n d0 (le_S_n n d0 (le_S (S n) d0 H4)) c0 c h H3 (CHead d (Bind Abbr) u) H0) +in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop n O c0 e0))) +(\lambda (e0: C).(\lambda (e1: C).(drop h (minus d0 n) e0 e1))) (\lambda (_: +C).(\lambda (e1: C).(clear e1 (CHead d (Bind Abbr) u)))) (ty3 g c0 (lift h d0 +(TLRef n)) (lift h d0 (lift (S n) O t))) (\lambda (x0: C).(\lambda (x1: +C).(\lambda (H6: (drop n O c0 x0)).(\lambda (H7: (drop h (minus d0 n) x0 +x1)).(\lambda (H8: (clear x1 (CHead d (Bind Abbr) u))).(let H9 \def (eq_ind +nat (minus d0 n) (\lambda (n0: nat).(drop h n0 x0 x1)) H7 (S (minus d0 (S +n))) (minus_x_Sy d0 n H4)) in (let H10 \def (drop_clear_S x1 x0 h (minus d0 +(S n)) H9 Abbr d u H8) in (ex2_ind C (\lambda (c1: C).(clear x0 (CHead c1 +(Bind Abbr) (lift h (minus d0 (S n)) u)))) (\lambda (c1: C).(drop h (minus d0 +(S n)) c1 d)) (ty3 g c0 (lift h d0 (TLRef n)) (lift h d0 (lift (S n) O t))) +(\lambda (x: C).(\lambda (H11: (clear x0 (CHead x (Bind Abbr) (lift h (minus +d0 (S n)) u)))).(\lambda (H12: (drop h (minus d0 (S n)) x d)).(eq_ind_r T +(TLRef n) (\lambda (t0: T).(ty3 g c0 t0 (lift h d0 (lift (S n) O t)))) +(eq_ind nat (plus (S n) (minus d0 (S n))) (\lambda (n0: nat).(ty3 g c0 (TLRef +n) (lift h n0 (lift (S n) O t)))) (eq_ind_r T (lift (S n) O (lift h (minus d0 +(S n)) t)) (\lambda (t0: T).(ty3 g c0 (TLRef n) t0)) (eq_ind nat d0 (\lambda +(_: nat).(ty3 g c0 (TLRef n) (lift (S n) O (lift h (minus d0 (S n)) t)))) +(ty3_abbr g n c0 x (lift h (minus d0 (S n)) u) (getl_intro n c0 (CHead x +(Bind Abbr) (lift h (minus d0 (S n)) u)) x0 H6 H11) (lift h (minus d0 (S n)) +t) (H2 x (minus d0 (S n)) h H12)) (plus (S n) (minus d0 (S n))) +(le_plus_minus (S n) d0 H4)) (lift h (plus (S n) (minus d0 (S n))) (lift (S +n) O t)) (lift_d t h (S n) (minus d0 (S n)) O (le_O_n (minus d0 (S n))))) d0 +(le_plus_minus_r (S n) d0 H4)) (lift h d0 (TLRef n)) (lift_lref_lt n h d0 +H4))))) H10)))))))) H5))) (\lambda (H4: (le d0 n)).(eq_ind_r T (TLRef (plus n +h)) (\lambda (t0: T).(ty3 g c0 t0 (lift h d0 (lift (S n) O t)))) (eq_ind nat +(S n) (\lambda (_: nat).(ty3 g c0 (TLRef (plus n h)) (lift h d0 (lift (S n) O +t)))) (eq_ind_r T (lift (plus h (S n)) O t) (\lambda (t0: T).(ty3 g c0 (TLRef +(plus n h)) t0)) (eq_ind_r nat (plus (S n) h) (\lambda (n0: nat).(ty3 g c0 +(TLRef (plus n h)) (lift n0 O t))) (ty3_abbr g (plus n h) c0 d u +(drop_getl_trans_ge n c0 c d0 h H3 (CHead d (Bind Abbr) u) H0 H4) t H1) (plus +h (S n)) (plus_sym h (S n))) (lift h d0 (lift (S n) O t)) (lift_free t (S n) +h O d0 (le_S d0 n H4) (le_O_n d0))) (plus n (S O)) (eq_ind_r nat (plus (S O) +n) (\lambda (n0: nat).(eq nat (S n) n0)) (refl_equal nat (plus (S O) n)) +(plus n (S O)) (plus_sym n (S O)))) (lift h d0 (TLRef n)) (lift_lref_ge n h +d0 H4)))))))))))))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d: +C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind Abst) u))).(\lambda +(t: T).(\lambda (H1: (ty3 g d u t)).(\lambda (H2: ((\forall (c0: C).(\forall +(d0: nat).(\forall (h: nat).((drop h d0 c0 d) \to (ty3 g c0 (lift h d0 u) +(lift h d0 t)))))))).(\lambda (c0: C).(\lambda (d0: nat).(\lambda (h: +nat).(\lambda (H3: (drop h d0 c0 c)).(lt_le_e n d0 (ty3 g c0 (lift h d0 +(TLRef n)) (lift h d0 (lift (S n) O u))) (\lambda (H4: (lt n d0)).(let H5 +\def (drop_getl_trans_le n d0 (le_S_n n d0 (le_S (S n) d0 H4)) c0 c h H3 +(CHead d (Bind Abst) u) H0) in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: +C).(drop n O c0 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d0 n) +e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear e1 (CHead d (Bind Abst) +u)))) (ty3 g c0 (lift h d0 (TLRef n)) (lift h d0 (lift (S n) O u))) (\lambda +(x0: C).(\lambda (x1: C).(\lambda (H6: (drop n O c0 x0)).(\lambda (H7: (drop +h (minus d0 n) x0 x1)).(\lambda (H8: (clear x1 (CHead d (Bind Abst) u))).(let +H9 \def (eq_ind nat (minus d0 n) (\lambda (n0: nat).(drop h n0 x0 x1)) H7 (S +(minus d0 (S n))) (minus_x_Sy d0 n H4)) in (let H10 \def (drop_clear_S x1 x0 +h (minus d0 (S n)) H9 Abst d u H8) in (ex2_ind C (\lambda (c1: C).(clear x0 +(CHead c1 (Bind Abst) (lift h (minus d0 (S n)) u)))) (\lambda (c1: C).(drop h +(minus d0 (S n)) c1 d)) (ty3 g c0 (lift h d0 (TLRef n)) (lift h d0 (lift (S +n) O u))) (\lambda (x: C).(\lambda (H11: (clear x0 (CHead x (Bind Abst) (lift +h (minus d0 (S n)) u)))).(\lambda (H12: (drop h (minus d0 (S n)) x +d)).(eq_ind_r T (TLRef n) (\lambda (t0: T).(ty3 g c0 t0 (lift h d0 (lift (S +n) O u)))) (eq_ind nat (plus (S n) (minus d0 (S n))) (\lambda (n0: nat).(ty3 +g c0 (TLRef n) (lift h n0 (lift (S n) O u)))) (eq_ind_r T (lift (S n) O (lift +h (minus d0 (S n)) u)) (\lambda (t0: T).(ty3 g c0 (TLRef n) t0)) (eq_ind nat +d0 (\lambda (_: nat).(ty3 g c0 (TLRef n) (lift (S n) O (lift h (minus d0 (S +n)) u)))) (ty3_abst g n c0 x (lift h (minus d0 (S n)) u) (getl_intro n c0 +(CHead x (Bind Abst) (lift h (minus d0 (S n)) u)) x0 H6 H11) (lift h (minus +d0 (S n)) t) (H2 x (minus d0 (S n)) h H12)) (plus (S n) (minus d0 (S n))) +(le_plus_minus (S n) d0 H4)) (lift h (plus (S n) (minus d0 (S n))) (lift (S +n) O u)) (lift_d u h (S n) (minus d0 (S n)) O (le_O_n (minus d0 (S n))))) d0 +(le_plus_minus_r (S n) d0 H4)) (lift h d0 (TLRef n)) (lift_lref_lt n h d0 +H4))))) H10)))))))) H5))) (\lambda (H4: (le d0 n)).(eq_ind_r T (TLRef (plus n +h)) (\lambda (t0: T).(ty3 g c0 t0 (lift h d0 (lift (S n) O u)))) (eq_ind nat +(S n) (\lambda (_: nat).(ty3 g c0 (TLRef (plus n h)) (lift h d0 (lift (S n) O +u)))) (eq_ind_r T (lift (plus h (S n)) O u) (\lambda (t0: T).(ty3 g c0 (TLRef +(plus n h)) t0)) (eq_ind_r nat (plus (S n) h) (\lambda (n0: nat).(ty3 g c0 +(TLRef (plus n h)) (lift n0 O u))) (ty3_abst g (plus n h) c0 d u +(drop_getl_trans_ge n c0 c d0 h H3 (CHead d (Bind Abst) u) H0 H4) t H1) (plus +h (S n)) (plus_sym h (S n))) (lift h d0 (lift (S n) O u)) (lift_free u (S n) +h O d0 (le_S d0 n H4) (le_O_n d0))) (plus n (S O)) (eq_ind_r nat (plus (S O) +n) (\lambda (n0: nat).(eq nat (S n) n0)) (refl_equal nat (plus (S O) n)) +(plus n (S O)) (plus_sym n (S O)))) (lift h d0 (TLRef n)) (lift_lref_ge n h +d0 H4)))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (t: +T).(\lambda (_: (ty3 g c u t)).(\lambda (H1: ((\forall (c0: C).(\forall (d: +nat).(\forall (h: nat).((drop h d c0 c) \to (ty3 g c0 (lift h d u) (lift h d +t)))))))).(\lambda (b: B).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (ty3 +g (CHead c (Bind b) u) t0 t3)).(\lambda (H3: ((\forall (c0: C).(\forall (d: +nat).(\forall (h: nat).((drop h d c0 (CHead c (Bind b) u)) \to (ty3 g c0 +(lift h d t0) (lift h d t3)))))))).(\lambda (c0: C).(\lambda (d: +nat).(\lambda (h: nat).(\lambda (H4: (drop h d c0 c)).(eq_ind_r T (THead +(Bind b) (lift h d u) (lift h (s (Bind b) d) t0)) (\lambda (t4: T).(ty3 g c0 +t4 (lift h d (THead (Bind b) u t3)))) (eq_ind_r T (THead (Bind b) (lift h d +u) (lift h (s (Bind b) d) t3)) (\lambda (t4: T).(ty3 g c0 (THead (Bind b) +(lift h d u) (lift h (s (Bind b) d) t0)) t4)) (ty3_bind g c0 (lift h d u) +(lift h d t) (H1 c0 d h H4) b (lift h (S d) t0) (lift h (S d) t3) (H3 (CHead +c0 (Bind b) (lift h d u)) (S d) h (drop_skip_bind h d c0 c H4 b u))) (lift h +d (THead (Bind b) u t3)) (lift_head (Bind b) u t3 h d)) (lift h d (THead +(Bind b) u t0)) (lift_head (Bind b) u t0 h d)))))))))))))))) (\lambda (c: +C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c w u)).(\lambda (H1: +((\forall (c0: C).(\forall (d: nat).(\forall (h: nat).((drop h d c0 c) \to +(ty3 g c0 (lift h d w) (lift h d u)))))))).(\lambda (v: T).(\lambda (t: +T).(\lambda (_: (ty3 g c v (THead (Bind Abst) u t))).(\lambda (H3: ((\forall +(c0: C).(\forall (d: nat).(\forall (h: nat).((drop h d c0 c) \to (ty3 g c0 +(lift h d v) (lift h d (THead (Bind Abst) u t))))))))).(\lambda (c0: +C).(\lambda (d: nat).(\lambda (h: nat).(\lambda (H4: (drop h d c0 +c)).(eq_ind_r T (THead (Flat Appl) (lift h d w) (lift h (s (Flat Appl) d) v)) +(\lambda (t0: T).(ty3 g c0 t0 (lift h d (THead (Flat Appl) w (THead (Bind +Abst) u t))))) (eq_ind_r T (THead (Flat Appl) (lift h d w) (lift h (s (Flat +Appl) d) (THead (Bind Abst) u t))) (\lambda (t0: T).(ty3 g c0 (THead (Flat +Appl) (lift h d w) (lift h (s (Flat Appl) d) v)) t0)) (eq_ind_r T (THead +(Bind Abst) (lift h (s (Flat Appl) d) u) (lift h (s (Bind Abst) (s (Flat +Appl) d)) t)) (\lambda (t0: T).(ty3 g c0 (THead (Flat Appl) (lift h d w) +(lift h (s (Flat Appl) d) v)) (THead (Flat Appl) (lift h d w) t0))) (ty3_appl +g c0 (lift h d w) (lift h d u) (H1 c0 d h H4) (lift h d v) (lift h (S d) t) +(eq_ind T (lift h d (THead (Bind Abst) u t)) (\lambda (t0: T).(ty3 g c0 (lift +h d v) t0)) (H3 c0 d h H4) (THead (Bind Abst) (lift h d u) (lift h (S d) t)) +(lift_bind Abst u t h d))) (lift h (s (Flat Appl) d) (THead (Bind Abst) u t)) +(lift_head (Bind Abst) u t h (s (Flat Appl) d))) (lift h d (THead (Flat Appl) +w (THead (Bind Abst) u t))) (lift_head (Flat Appl) w (THead (Bind Abst) u t) +h d)) (lift h d (THead (Flat Appl) w v)) (lift_head (Flat Appl) w v h +d))))))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda (t3: T).(\lambda +(_: (ty3 g c t0 t3)).(\lambda (H1: ((\forall (c0: C).(\forall (d: +nat).(\forall (h: nat).((drop h d c0 c) \to (ty3 g c0 (lift h d t0) (lift h d +t3)))))))).(\lambda (t4: T).(\lambda (_: (ty3 g c t3 t4)).(\lambda (H3: +((\forall (c0: C).(\forall (d: nat).(\forall (h: nat).((drop h d c0 c) \to +(ty3 g c0 (lift h d t3) (lift h d t4)))))))).(\lambda (c0: C).(\lambda (d: +nat).(\lambda (h: nat).(\lambda (H4: (drop h d c0 c)).(eq_ind_r T (THead +(Flat Cast) (lift h d t3) (lift h (s (Flat Cast) d) t0)) (\lambda (t: T).(ty3 +g c0 t (lift h d (THead (Flat Cast) t4 t3)))) (eq_ind_r T (THead (Flat Cast) +(lift h d t4) (lift h (s (Flat Cast) d) t3)) (\lambda (t: T).(ty3 g c0 (THead +(Flat Cast) (lift h d t3) (lift h (s (Flat Cast) d) t0)) t)) (ty3_cast g c0 +(lift h (s (Flat Cast) d) t0) (lift h (s (Flat Cast) d) t3) (H1 c0 (s (Flat +Cast) d) h H4) (lift h d t4) (H3 c0 d h H4)) (lift h d (THead (Flat Cast) t4 +t3)) (lift_head (Flat Cast) t4 t3 h d)) (lift h d (THead (Flat Cast) t3 t0)) +(lift_head (Flat Cast) t3 t0 h d)))))))))))))) e t1 t2 H))))). +(* COMMENTS +Initial nodes: 4253 +END *) + +theorem ty3_correct: + \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c +t1 t2) \to (ex T (\lambda (t: T).(ty3 g c t2 t))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda +(H: (ty3 g c t1 t2)).(ty3_ind g (\lambda (c0: C).(\lambda (_: T).(\lambda +(t0: T).(ex T (\lambda (t3: T).(ty3 g c0 t0 t3)))))) (\lambda (c0: +C).(\lambda (t0: T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 t0 t)).(\lambda +(_: (ex T (\lambda (t3: T).(ty3 g c0 t t3)))).(\lambda (u: T).(\lambda (t3: +T).(\lambda (_: (ty3 g c0 u t3)).(\lambda (_: (ex T (\lambda (t4: T).(ty3 g +c0 t3 t4)))).(\lambda (_: (pc3 c0 t3 t0)).(ex_intro T (\lambda (t4: T).(ty3 g +c0 t0 t4)) t H0))))))))))) (\lambda (c0: C).(\lambda (m: nat).(ex_intro T +(\lambda (t: T).(ty3 g c0 (TSort (next g m)) t)) (TSort (next g (next g m))) +(ty3_sort g c0 (next g m))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: +C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abbr) +u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2: (ex T (\lambda +(t0: T).(ty3 g d t t0)))).(let H3 \def H2 in (ex_ind T (\lambda (t0: T).(ty3 +g d t t0)) (ex T (\lambda (t0: T).(ty3 g c0 (lift (S n) O t) t0))) (\lambda +(x: T).(\lambda (H4: (ty3 g d t x)).(ex_intro T (\lambda (t0: T).(ty3 g c0 +(lift (S n) O t) t0)) (lift (S n) O x) (ty3_lift g d t x H4 c0 O (S n) +(getl_drop Abbr c0 d u n H0))))) H3)))))))))) (\lambda (n: nat).(\lambda (c0: +C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind +Abst) u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u t)).(\lambda (_: (ex T +(\lambda (t0: T).(ty3 g d t t0)))).(ex_intro T (\lambda (t0: T).(ty3 g c0 +(lift (S n) O u) t0)) (lift (S n) O t) (ty3_lift g d u t H1 c0 O (S n) +(getl_drop Abst c0 d u n H0))))))))))) (\lambda (c0: C).(\lambda (u: +T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 u t)).(\lambda (_: (ex T (\lambda +(t0: T).(ty3 g c0 t t0)))).(\lambda (b: B).(\lambda (t0: T).(\lambda (t3: +T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t0 t3)).(\lambda (H3: (ex T +(\lambda (t4: T).(ty3 g (CHead c0 (Bind b) u) t3 t4)))).(let H4 \def H3 in +(ex_ind T (\lambda (t4: T).(ty3 g (CHead c0 (Bind b) u) t3 t4)) (ex T +(\lambda (t4: T).(ty3 g c0 (THead (Bind b) u t3) t4))) (\lambda (x: +T).(\lambda (H5: (ty3 g (CHead c0 (Bind b) u) t3 x)).(ex_intro T (\lambda +(t4: T).(ty3 g c0 (THead (Bind b) u t3) t4)) (THead (Bind b) u x) (ty3_bind g +c0 u t H0 b t3 x H5)))) H4)))))))))))) (\lambda (c0: C).(\lambda (w: +T).(\lambda (u: T).(\lambda (H0: (ty3 g c0 w u)).(\lambda (H1: (ex T (\lambda +(t: T).(ty3 g c0 u t)))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g +c0 v (THead (Bind Abst) u t))).(\lambda (H3: (ex T (\lambda (t0: T).(ty3 g c0 +(THead (Bind Abst) u t) t0)))).(let H4 \def H1 in (ex_ind T (\lambda (t0: +T).(ty3 g c0 u t0)) (ex T (\lambda (t0: T).(ty3 g c0 (THead (Flat Appl) w +(THead (Bind Abst) u t)) t0))) (\lambda (x: T).(\lambda (_: (ty3 g c0 u +x)).(let H6 \def H3 in (ex_ind T (\lambda (t0: T).(ty3 g c0 (THead (Bind +Abst) u t) t0)) (ex T (\lambda (t0: T).(ty3 g c0 (THead (Flat Appl) w (THead +(Bind Abst) u t)) t0))) (\lambda (x0: T).(\lambda (H7: (ty3 g c0 (THead (Bind +Abst) u t) x0)).(ex3_2_ind T T (\lambda (t3: T).(\lambda (_: T).(pc3 c0 +(THead (Bind Abst) u t3) x0))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u +t0))) (\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind Abst) u) t +t3))) (ex T (\lambda (t0: T).(ty3 g c0 (THead (Flat Appl) w (THead (Bind +Abst) u t)) t0))) (\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 c0 +(THead (Bind Abst) u x1) x0)).(\lambda (H9: (ty3 g c0 u x2)).(\lambda (H10: +(ty3 g (CHead c0 (Bind Abst) u) t x1)).(ex_intro T (\lambda (t0: T).(ty3 g c0 +(THead (Flat Appl) w (THead (Bind Abst) u t)) t0)) (THead (Flat Appl) w +(THead (Bind Abst) u x1)) (ty3_appl g c0 w u H0 (THead (Bind Abst) u t) x1 +(ty3_bind g c0 u x2 H9 Abst t x1 H10)))))))) (ty3_gen_bind g Abst c0 u t x0 +H7)))) H6)))) H4))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t3: +T).(\lambda (_: (ty3 g c0 t0 t3)).(\lambda (_: (ex T (\lambda (t: T).(ty3 g +c0 t3 t)))).(\lambda (t4: T).(\lambda (H2: (ty3 g c0 t3 t4)).(\lambda (H3: +(ex T (\lambda (t: T).(ty3 g c0 t4 t)))).(let H4 \def H3 in (ex_ind T +(\lambda (t: T).(ty3 g c0 t4 t)) (ex T (\lambda (t: T).(ty3 g c0 (THead (Flat +Cast) t4 t3) t))) (\lambda (x: T).(\lambda (H5: (ty3 g c0 t4 x)).(ex_intro T +(\lambda (t: T).(ty3 g c0 (THead (Flat Cast) t4 t3) t)) (THead (Flat Cast) x +t4) (ty3_cast g c0 t3 t4 H2 x H5)))) H4)))))))))) c t1 t2 H))))). +(* COMMENTS +Initial nodes: 1333 +END *) + +theorem ty3_unique: + \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).((ty3 g c u +t1) \to (\forall (t2: T).((ty3 g c u t2) \to (pc3 c t1 t2))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (H: +(ty3 g c u t1)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: +T).(\forall (t2: T).((ty3 g c0 t t2) \to (pc3 c0 t0 t2)))))) (\lambda (c0: +C).(\lambda (t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda +(_: ((\forall (t3: T).((ty3 g c0 t2 t3) \to (pc3 c0 t t3))))).(\lambda (u0: +T).(\lambda (t0: T).(\lambda (_: (ty3 g c0 u0 t0)).(\lambda (H3: ((\forall +(t3: T).((ty3 g c0 u0 t3) \to (pc3 c0 t0 t3))))).(\lambda (H4: (pc3 c0 t0 +t2)).(\lambda (t3: T).(\lambda (H5: (ty3 g c0 u0 t3)).(pc3_t t0 c0 t2 (pc3_s +c0 t2 t0 H4) t3 (H3 t3 H5)))))))))))))) (\lambda (c0: C).(\lambda (m: +nat).(\lambda (t2: T).(\lambda (H0: (ty3 g c0 (TSort m) t2)).(ty3_gen_sort g +c0 t2 m H0))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda +(u0: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abbr) u0))).(\lambda (t: +T).(\lambda (_: (ty3 g d u0 t)).(\lambda (H2: ((\forall (t2: T).((ty3 g d u0 +t2) \to (pc3 d t t2))))).(\lambda (t2: T).(\lambda (H3: (ty3 g c0 (TLRef n) +t2)).(or_ind (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: +T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda +(_: T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: C).(\lambda (u1: +T).(\lambda (t0: T).(ty3 g e u1 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda +(u1: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u1) t2)))) (\lambda (e: +C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u1))))) +(\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 t0))))) (pc3 c0 +(lift (S n) O t) t2) (\lambda (H4: (ex3_3 C T T (\lambda (_: C).(\lambda (_: +T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda (e: C).(\lambda +(u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: +C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 t0)))))).(ex3_3_ind C T T +(\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) +t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e +(Bind Abbr) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g +e u1 t0)))) (pc3 c0 (lift (S n) O t) t2) (\lambda (x0: C).(\lambda (x1: +T).(\lambda (x2: T).(\lambda (H5: (pc3 c0 (lift (S n) O x2) t2)).(\lambda +(H6: (getl n c0 (CHead x0 (Bind Abbr) x1))).(\lambda (H7: (ty3 g x0 x1 +x2)).(let H8 \def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda (c1: C).(getl n +c0 c1)) H0 (CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind Abbr) u0) n +H0 (CHead x0 (Bind Abbr) x1) H6)) in (let H9 \def (f_equal C C (\lambda (e: +C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | +(CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u0) (CHead x0 (Bind +Abbr) x1) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead x0 (Bind Abbr) +x1) H6)) in ((let H10 \def (f_equal C T (\lambda (e: C).(match e in C return +(\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t0) +\Rightarrow t0])) (CHead d (Bind Abbr) u0) (CHead x0 (Bind Abbr) x1) +(getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in +(\lambda (H11: (eq C d x0)).(let H12 \def (eq_ind_r T x1 (\lambda (t0: +T).(getl n c0 (CHead x0 (Bind Abbr) t0))) H8 u0 H10) in (let H13 \def +(eq_ind_r T x1 (\lambda (t0: T).(ty3 g x0 t0 x2)) H7 u0 H10) in (let H14 \def +(eq_ind_r C x0 (\lambda (c1: C).(getl n c0 (CHead c1 (Bind Abbr) u0))) H12 d +H11) in (let H15 \def (eq_ind_r C x0 (\lambda (c1: C).(ty3 g c1 u0 x2)) H13 d +H11) in (pc3_t (lift (S n) O x2) c0 (lift (S n) O t) (pc3_lift c0 d (S n) O +(getl_drop Abbr c0 d u0 n H14) t x2 (H2 x2 H15)) t2 H5))))))) H9))))))))) +H4)) (\lambda (H4: (ex3_3 C T T (\lambda (_: C).(\lambda (u1: T).(\lambda (_: +T).(pc3 c0 (lift (S n) O u1) t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda +(_: T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1: +T).(\lambda (t0: T).(ty3 g e u1 t0)))))).(ex3_3_ind C T T (\lambda (_: +C).(\lambda (u1: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u1) t2)))) (\lambda +(e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) +u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 t0)))) +(pc3 c0 (lift (S n) O t) t2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: +T).(\lambda (_: (pc3 c0 (lift (S n) O x1) t2)).(\lambda (H6: (getl n c0 +(CHead x0 (Bind Abst) x1))).(\lambda (_: (ty3 g x0 x1 x2)).(let H8 \def +(eq_ind C (CHead d (Bind Abbr) u0) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead +x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead x0 +(Bind Abst) x1) H6)) in (let H9 \def (eq_ind C (CHead d (Bind Abbr) u0) +(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) +\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda +(_: K).Prop) with [(Bind b) \Rightarrow (match b in B return (\lambda (_: +B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | Void +\Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead x0 (Bind Abst) +x1) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead x0 (Bind Abst) x1) +H6)) in (False_ind (pc3 c0 (lift (S n) O t) t2) H9))))))))) H4)) +(ty3_gen_lref g c0 t2 n H3)))))))))))) (\lambda (n: nat).(\lambda (c0: +C).(\lambda (d: C).(\lambda (u0: T).(\lambda (H0: (getl n c0 (CHead d (Bind +Abst) u0))).(\lambda (t: T).(\lambda (_: (ty3 g d u0 t)).(\lambda (_: +((\forall (t2: T).((ty3 g d u0 t2) \to (pc3 d t t2))))).(\lambda (t2: +T).(\lambda (H3: (ty3 g c0 (TLRef n) t2)).(or_ind (ex3_3 C T T (\lambda (_: +C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda +(e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) +u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 t0))))) +(ex3_3 C T T (\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(pc3 c0 (lift +(S n) O u1) t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n +c0 (CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda +(t0: T).(ty3 g e u1 t0))))) (pc3 c0 (lift (S n) O u0) t2) (\lambda (H4: +(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift +(S n) O t0) t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n +c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda +(t0: T).(ty3 g e u1 t0)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (_: +T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda (e: C).(\lambda +(u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: +C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 t0)))) (pc3 c0 (lift (S n) O +u0) t2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 +c0 (lift (S n) O x2) t2)).(\lambda (H6: (getl n c0 (CHead x0 (Bind Abbr) +x1))).(\lambda (_: (ty3 g x0 x1 x2)).(let H8 \def (eq_ind C (CHead d (Bind +Abst) u0) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead x0 (Bind Abbr) x1) +(getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in +(let H9 \def (eq_ind C (CHead d (Bind Abst) u0) (\lambda (ee: C).(match ee in +C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k +_) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b) +\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow +False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) +\Rightarrow False])])) I (CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d +(Bind Abst) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in (False_ind (pc3 c0 +(lift (S n) O u0) t2) H9))))))))) H4)) (\lambda (H4: (ex3_3 C T T (\lambda +(_: C).(\lambda (u1: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u1) t2)))) +(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind +Abst) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 +t0)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (u1: T).(\lambda (_: +T).(pc3 c0 (lift (S n) O u1) t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda +(_: T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1: +T).(\lambda (t0: T).(ty3 g e u1 t0)))) (pc3 c0 (lift (S n) O u0) t2) (\lambda +(x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H5: (pc3 c0 (lift (S n) O +x1) t2)).(\lambda (H6: (getl n c0 (CHead x0 (Bind Abst) x1))).(\lambda (H7: +(ty3 g x0 x1 x2)).(let H8 \def (eq_ind C (CHead d (Bind Abst) u0) (\lambda +(c1: C).(getl n c0 c1)) H0 (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d +(Bind Abst) u0) n H0 (CHead x0 (Bind Abst) x1) H6)) in (let H9 \def (f_equal +C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) +\Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abst) u0) +(CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead +x0 (Bind Abst) x1) H6)) in ((let H10 \def (f_equal C T (\lambda (e: C).(match +e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ +t0) \Rightarrow t0])) (CHead d (Bind Abst) u0) (CHead x0 (Bind Abst) x1) +(getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead x0 (Bind Abst) x1) H6)) in +(\lambda (H11: (eq C d x0)).(let H12 \def (eq_ind_r T x1 (\lambda (t0: +T).(getl n c0 (CHead x0 (Bind Abst) t0))) H8 u0 H10) in (let H13 \def +(eq_ind_r T x1 (\lambda (t0: T).(ty3 g x0 t0 x2)) H7 u0 H10) in (let H14 \def +(eq_ind_r T x1 (\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)) H5 u0 H10) in +(let H15 \def (eq_ind_r C x0 (\lambda (c1: C).(getl n c0 (CHead c1 (Bind +Abst) u0))) H12 d H11) in (let H16 \def (eq_ind_r C x0 (\lambda (c1: C).(ty3 +g c1 u0 x2)) H13 d H11) in H14))))))) H9))))))))) H4)) (ty3_gen_lref g c0 t2 +n H3)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (t: T).(\lambda +(_: (ty3 g c0 u0 t)).(\lambda (_: ((\forall (t2: T).((ty3 g c0 u0 t2) \to +(pc3 c0 t t2))))).(\lambda (b: B).(\lambda (t0: T).(\lambda (t2: T).(\lambda +(_: (ty3 g (CHead c0 (Bind b) u0) t0 t2)).(\lambda (H3: ((\forall (t3: +T).((ty3 g (CHead c0 (Bind b) u0) t0 t3) \to (pc3 (CHead c0 (Bind b) u0) t2 +t3))))).(\lambda (t3: T).(\lambda (H4: (ty3 g c0 (THead (Bind b) u0 t0) +t3)).(ex3_2_ind T T (\lambda (t4: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) +u0 t4) t3))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c0 u0 t5))) (\lambda +(t4: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u0) t0 t4))) (pc3 c0 (THead +(Bind b) u0 t2) t3) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (pc3 c0 +(THead (Bind b) u0 x0) t3)).(\lambda (_: (ty3 g c0 u0 x1)).(\lambda (H7: (ty3 +g (CHead c0 (Bind b) u0) t0 x0)).(pc3_t (THead (Bind b) u0 x0) c0 (THead +(Bind b) u0 t2) (pc3_head_2 c0 u0 t2 x0 (Bind b) (H3 x0 H7)) t3 H5)))))) +(ty3_gen_bind g b c0 u0 t0 t3 H4)))))))))))))) (\lambda (c0: C).(\lambda (w: +T).(\lambda (u0: T).(\lambda (_: (ty3 g c0 w u0)).(\lambda (_: ((\forall (t2: +T).((ty3 g c0 w t2) \to (pc3 c0 u0 t2))))).(\lambda (v: T).(\lambda (t: +T).(\lambda (_: (ty3 g c0 v (THead (Bind Abst) u0 t))).(\lambda (H3: +((\forall (t2: T).((ty3 g c0 v t2) \to (pc3 c0 (THead (Bind Abst) u0 t) +t2))))).(\lambda (t2: T).(\lambda (H4: (ty3 g c0 (THead (Flat Appl) w v) +t2)).(ex3_2_ind T T (\lambda (u1: T).(\lambda (t0: T).(pc3 c0 (THead (Flat +Appl) w (THead (Bind Abst) u1 t0)) t2))) (\lambda (u1: T).(\lambda (t0: +T).(ty3 g c0 v (THead (Bind Abst) u1 t0)))) (\lambda (u1: T).(\lambda (_: +T).(ty3 g c0 w u1))) (pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t)) +t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (pc3 c0 (THead (Flat +Appl) w (THead (Bind Abst) x0 x1)) t2)).(\lambda (H6: (ty3 g c0 v (THead +(Bind Abst) x0 x1))).(\lambda (_: (ty3 g c0 w x0)).(pc3_t (THead (Flat Appl) +w (THead (Bind Abst) x0 x1)) c0 (THead (Flat Appl) w (THead (Bind Abst) u0 +t)) (pc3_thin_dx c0 (THead (Bind Abst) u0 t) (THead (Bind Abst) x0 x1) (H3 +(THead (Bind Abst) x0 x1) H6) w Appl) t2 H5)))))) (ty3_gen_appl g c0 w v t2 +H4))))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: T).(\lambda +(_: (ty3 g c0 t0 t2)).(\lambda (_: ((\forall (t3: T).((ty3 g c0 t0 t3) \to +(pc3 c0 t2 t3))))).(\lambda (t3: T).(\lambda (_: (ty3 g c0 t2 t3)).(\lambda +(H3: ((\forall (t4: T).((ty3 g c0 t2 t4) \to (pc3 c0 t3 t4))))).(\lambda (t4: +T).(\lambda (H4: (ty3 g c0 (THead (Flat Cast) t2 t0) t4)).(ex3_ind T (\lambda +(t5: T).(pc3 c0 (THead (Flat Cast) t5 t2) t4)) (\lambda (_: T).(ty3 g c0 t0 +t2)) (\lambda (t5: T).(ty3 g c0 t2 t5)) (pc3 c0 (THead (Flat Cast) t3 t2) t4) +(\lambda (x0: T).(\lambda (H5: (pc3 c0 (THead (Flat Cast) x0 t2) +t4)).(\lambda (_: (ty3 g c0 t0 t2)).(\lambda (H7: (ty3 g c0 t2 x0)).(pc3_t +(THead (Flat Cast) x0 t2) c0 (THead (Flat Cast) t3 t2) (pc3_head_1 c0 t3 x0 +(H3 x0 H7) (Flat Cast) t2) t4 H5))))) (ty3_gen_cast g c0 t0 t2 t4 +H4)))))))))))) c u t1 H))))). +(* COMMENTS +Initial nodes: 3459 +END *) + +theorem ty3_gen_abst_abst: + \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall +(t2: T).((ty3 g c (THead (Bind Abst) u t1) (THead (Bind Abst) u t2)) \to (ex2 +T (\lambda (w: T).(ty3 g c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst) +u) t1 t2)))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda +(t2: T).(\lambda (H: (ty3 g c (THead (Bind Abst) u t1) (THead (Bind Abst) u +t2))).(ex_ind T (\lambda (t: T).(ty3 g c (THead (Bind Abst) u t2) t)) (ex2 T +(\lambda (w: T).(ty3 g c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) +t1 t2))) (\lambda (x: T).(\lambda (H0: (ty3 g c (THead (Bind Abst) u t2) +x)).(ex3_2_ind T T (\lambda (t3: T).(\lambda (_: T).(pc3 c (THead (Bind Abst) +u t3) x))) (\lambda (_: T).(\lambda (t: T).(ty3 g c u t))) (\lambda (t3: +T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t2 t3))) (ex2 T (\lambda +(w: T).(ty3 g c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t1 t2))) +(\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 c (THead (Bind Abst) u +x0) x)).(\lambda (_: (ty3 g c u x1)).(\lambda (H3: (ty3 g (CHead c (Bind +Abst) u) t2 x0)).(ex3_2_ind T T (\lambda (t3: T).(\lambda (_: T).(pc3 c +(THead (Bind Abst) u t3) (THead (Bind Abst) u t2)))) (\lambda (_: T).(\lambda +(t: T).(ty3 g c u t))) (\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c (Bind +Abst) u) t1 t3))) (ex2 T (\lambda (w: T).(ty3 g c u w)) (\lambda (_: T).(ty3 +g (CHead c (Bind Abst) u) t1 t2))) (\lambda (x2: T).(\lambda (x3: T).(\lambda +(H4: (pc3 c (THead (Bind Abst) u x2) (THead (Bind Abst) u t2))).(\lambda (H5: +(ty3 g c u x3)).(\lambda (H6: (ty3 g (CHead c (Bind Abst) u) t1 x2)).(let H_y +\def (pc3_gen_abst_shift c u x2 t2 H4) in (ex_intro2 T (\lambda (w: T).(ty3 g +c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t1 t2)) x3 H5 +(ty3_conv g (CHead c (Bind Abst) u) t2 x0 H3 t1 x2 H6 H_y)))))))) +(ty3_gen_bind g Abst c u t1 (THead (Bind Abst) u t2) H))))))) (ty3_gen_bind g +Abst c u t2 x H0)))) (ty3_correct g c (THead (Bind Abst) u t1) (THead (Bind +Abst) u t2) H))))))). +(* COMMENTS +Initial nodes: 571 +END *) + +theorem ty3_typecheck: + \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (v: T).((ty3 g c t +v) \to (ex T (\lambda (u: T).(ty3 g c (THead (Flat Cast) v t) u))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (v: T).(\lambda (H: +(ty3 g c t v)).(ex_ind T (\lambda (t0: T).(ty3 g c v t0)) (ex T (\lambda (u: +T).(ty3 g c (THead (Flat Cast) v t) u))) (\lambda (x: T).(\lambda (H0: (ty3 g +c v x)).(ex_intro T (\lambda (u: T).(ty3 g c (THead (Flat Cast) v t) u)) +(THead (Flat Cast) x v) (ty3_cast g c t v H x H0)))) (ty3_correct g c t v +H)))))). +(* COMMENTS +Initial nodes: 131 +END *) + +theorem ty3_getl_subst0: + \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t +u) \to (\forall (v0: T).(\forall (t0: T).(\forall (i: nat).((subst0 i v0 t +t0) \to (\forall (b: B).(\forall (d: C).(\forall (v: T).((getl i c (CHead d +(Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v w))))))))))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H: +(ty3 g c t u)).(ty3_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (_: +T).(\forall (v0: T).(\forall (t2: T).(\forall (i: nat).((subst0 i v0 t0 t2) +\to (\forall (b: B).(\forall (d: C).(\forall (v: T).((getl i c0 (CHead d +(Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v w)))))))))))))) (\lambda +(c0: C).(\lambda (t2: T).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t2 +t0)).(\lambda (_: ((\forall (v0: T).(\forall (t1: T).(\forall (i: +nat).((subst0 i v0 t2 t1) \to (\forall (b: B).(\forall (d: C).(\forall (v: +T).((getl i c0 (CHead d (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v +w))))))))))))).(\lambda (u0: T).(\lambda (t1: T).(\lambda (_: (ty3 g c0 u0 +t1)).(\lambda (H3: ((\forall (v0: T).(\forall (t3: T).(\forall (i: +nat).((subst0 i v0 u0 t3) \to (\forall (b: B).(\forall (d: C).(\forall (v: +T).((getl i c0 (CHead d (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v +w))))))))))))).(\lambda (_: (pc3 c0 t1 t2)).(\lambda (v0: T).(\lambda (t3: +T).(\lambda (i: nat).(\lambda (H5: (subst0 i v0 u0 t3)).(\lambda (b: +B).(\lambda (d: C).(\lambda (v: T).(\lambda (H6: (getl i c0 (CHead d (Bind b) +v))).(H3 v0 t3 i H5 b d v H6))))))))))))))))))) (\lambda (c0: C).(\lambda (m: +nat).(\lambda (v0: T).(\lambda (t0: T).(\lambda (i: nat).(\lambda (H0: +(subst0 i v0 (TSort m) t0)).(\lambda (b: B).(\lambda (d: C).(\lambda (v: +T).(\lambda (_: (getl i c0 (CHead d (Bind b) v))).(subst0_gen_sort v0 t0 i m +H0 (ex T (\lambda (w: T).(ty3 g d v w)))))))))))))) (\lambda (n: +nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (H0: (getl n +c0 (CHead d (Bind Abbr) u0))).(\lambda (t0: T).(\lambda (H1: (ty3 g d u0 +t0)).(\lambda (_: ((\forall (v0: T).(\forall (t1: T).(\forall (i: +nat).((subst0 i v0 u0 t1) \to (\forall (b: B).(\forall (d0: C).(\forall (v: +T).((getl i d (CHead d0 (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d0 v +w))))))))))))).(\lambda (v0: T).(\lambda (t1: T).(\lambda (i: nat).(\lambda +(H3: (subst0 i v0 (TLRef n) t1)).(\lambda (b: B).(\lambda (d0: C).(\lambda +(v: T).(\lambda (H4: (getl i c0 (CHead d0 (Bind b) v))).(land_ind (eq nat n +i) (eq T t1 (lift (S n) O v0)) (ex T (\lambda (w: T).(ty3 g d0 v w))) +(\lambda (H5: (eq nat n i)).(\lambda (_: (eq T t1 (lift (S n) O v0))).(let H7 +\def (eq_ind_r nat i (\lambda (n0: nat).(getl n0 c0 (CHead d0 (Bind b) v))) +H4 n H5) in (let H8 \def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda (c1: +C).(getl n c0 c1)) H0 (CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind +Abbr) u0) n H0 (CHead d0 (Bind b) v) H7)) in (let H9 \def (f_equal C C +(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) +\Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u0) +(CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead d0 +(Bind b) v) H7)) in ((let H10 \def (f_equal C B (\lambda (e: C).(match e in C +return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) +\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b0) +\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u0) +(CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead d0 +(Bind b) v) H7)) in ((let H11 \def (f_equal C T (\lambda (e: C).(match e in C +return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t2) +\Rightarrow t2])) (CHead d (Bind Abbr) u0) (CHead d0 (Bind b) v) (getl_mono +c0 (CHead d (Bind Abbr) u0) n H0 (CHead d0 (Bind b) v) H7)) in (\lambda (H12: +(eq B Abbr b)).(\lambda (H13: (eq C d d0)).(let H14 \def (eq_ind_r T v +(\lambda (t2: T).(getl n c0 (CHead d0 (Bind b) t2))) H8 u0 H11) in (eq_ind T +u0 (\lambda (t2: T).(ex T (\lambda (w: T).(ty3 g d0 t2 w)))) (let H15 \def +(eq_ind_r C d0 (\lambda (c1: C).(getl n c0 (CHead c1 (Bind b) u0))) H14 d +H13) in (eq_ind C d (\lambda (c1: C).(ex T (\lambda (w: T).(ty3 g c1 u0 w)))) +(let H16 \def (eq_ind_r B b (\lambda (b0: B).(getl n c0 (CHead d (Bind b0) +u0))) H15 Abbr H12) in (ex_intro T (\lambda (w: T).(ty3 g d u0 w)) t0 H1)) d0 +H13)) v H11))))) H10)) H9)))))) (subst0_gen_lref v0 t1 i n +H3)))))))))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: +C).(\lambda (u0: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abst) +u0))).(\lambda (t0: T).(\lambda (H1: (ty3 g d u0 t0)).(\lambda (_: ((\forall +(v0: T).(\forall (t1: T).(\forall (i: nat).((subst0 i v0 u0 t1) \to (\forall +(b: B).(\forall (d0: C).(\forall (v: T).((getl i d (CHead d0 (Bind b) v)) \to +(ex T (\lambda (w: T).(ty3 g d0 v w))))))))))))).(\lambda (v0: T).(\lambda +(t1: T).(\lambda (i: nat).(\lambda (H3: (subst0 i v0 (TLRef n) t1)).(\lambda +(b: B).(\lambda (d0: C).(\lambda (v: T).(\lambda (H4: (getl i c0 (CHead d0 +(Bind b) v))).(land_ind (eq nat n i) (eq T t1 (lift (S n) O v0)) (ex T +(\lambda (w: T).(ty3 g d0 v w))) (\lambda (H5: (eq nat n i)).(\lambda (_: (eq +T t1 (lift (S n) O v0))).(let H7 \def (eq_ind_r nat i (\lambda (n0: +nat).(getl n0 c0 (CHead d0 (Bind b) v))) H4 n H5) in (let H8 \def (eq_ind C +(CHead d (Bind Abst) u0) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead d0 (Bind +b) v) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead d0 (Bind b) v) H7)) +in (let H9 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda +(_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) +(CHead d (Bind Abst) u0) (CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind +Abst) u0) n H0 (CHead d0 (Bind b) v) H7)) in ((let H10 \def (f_equal C B +(\lambda (e: C).(match e in C return (\lambda (_: C).B) with [(CSort _) +\Rightarrow Abst | (CHead _ k _) \Rightarrow (match k in K return (\lambda +(_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abst])])) +(CHead d (Bind Abst) u0) (CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind +Abst) u0) n H0 (CHead d0 (Bind b) v) H7)) in ((let H11 \def (f_equal C T +(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) +\Rightarrow u0 | (CHead _ _ t2) \Rightarrow t2])) (CHead d (Bind Abst) u0) +(CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead d0 +(Bind b) v) H7)) in (\lambda (H12: (eq B Abst b)).(\lambda (H13: (eq C d +d0)).(let H14 \def (eq_ind_r T v (\lambda (t2: T).(getl n c0 (CHead d0 (Bind +b) t2))) H8 u0 H11) in (eq_ind T u0 (\lambda (t2: T).(ex T (\lambda (w: +T).(ty3 g d0 t2 w)))) (let H15 \def (eq_ind_r C d0 (\lambda (c1: C).(getl n +c0 (CHead c1 (Bind b) u0))) H14 d H13) in (eq_ind C d (\lambda (c1: C).(ex T +(\lambda (w: T).(ty3 g c1 u0 w)))) (let H16 \def (eq_ind_r B b (\lambda (b0: +B).(getl n c0 (CHead d (Bind b0) u0))) H15 Abst H12) in (ex_intro T (\lambda +(w: T).(ty3 g d u0 w)) t0 H1)) d0 H13)) v H11))))) H10)) H9)))))) +(subst0_gen_lref v0 t1 i n H3)))))))))))))))))) (\lambda (c0: C).(\lambda +(u0: T).(\lambda (t0: T).(\lambda (_: (ty3 g c0 u0 t0)).(\lambda (H1: +((\forall (v0: T).(\forall (t1: T).(\forall (i: nat).((subst0 i v0 u0 t1) \to +(\forall (b: B).(\forall (d: C).(\forall (v: T).((getl i c0 (CHead d (Bind b) +v)) \to (ex T (\lambda (w: T).(ty3 g d v w))))))))))))).(\lambda (b: +B).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) +u0) t1 t2)).(\lambda (H3: ((\forall (v0: T).(\forall (t3: T).(\forall (i: +nat).((subst0 i v0 t1 t3) \to (\forall (b0: B).(\forall (d: C).(\forall (v: +T).((getl i (CHead c0 (Bind b) u0) (CHead d (Bind b0) v)) \to (ex T (\lambda +(w: T).(ty3 g d v w))))))))))))).(\lambda (v0: T).(\lambda (t3: T).(\lambda +(i: nat).(\lambda (H4: (subst0 i v0 (THead (Bind b) u0 t1) t3)).(\lambda (b0: +B).(\lambda (d: C).(\lambda (v: T).(\lambda (H5: (getl i c0 (CHead d (Bind +b0) v))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t3 (THead (Bind b) u2 t1))) +(\lambda (u2: T).(subst0 i v0 u0 u2))) (ex2 T (\lambda (t4: T).(eq T t3 +(THead (Bind b) u0 t4))) (\lambda (t4: T).(subst0 (s (Bind b) i) v0 t1 t4))) +(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind b) u2 +t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 u0 u2))) (\lambda (_: +T).(\lambda (t4: T).(subst0 (s (Bind b) i) v0 t1 t4)))) (ex T (\lambda (w: +T).(ty3 g d v w))) (\lambda (H6: (ex2 T (\lambda (u2: T).(eq T t3 (THead +(Bind b) u2 t1))) (\lambda (u2: T).(subst0 i v0 u0 u2)))).(ex2_ind T (\lambda +(u2: T).(eq T t3 (THead (Bind b) u2 t1))) (\lambda (u2: T).(subst0 i v0 u0 +u2)) (ex T (\lambda (w: T).(ty3 g d v w))) (\lambda (x: T).(\lambda (_: (eq T +t3 (THead (Bind b) x t1))).(\lambda (H8: (subst0 i v0 u0 x)).(H1 v0 x i H8 b0 +d v H5)))) H6)) (\lambda (H6: (ex2 T (\lambda (t4: T).(eq T t3 (THead (Bind +b) u0 t4))) (\lambda (t4: T).(subst0 (s (Bind b) i) v0 t1 t4)))).(ex2_ind T +(\lambda (t4: T).(eq T t3 (THead (Bind b) u0 t4))) (\lambda (t4: T).(subst0 +(s (Bind b) i) v0 t1 t4)) (ex T (\lambda (w: T).(ty3 g d v w))) (\lambda (x: +T).(\lambda (_: (eq T t3 (THead (Bind b) u0 x))).(\lambda (H8: (subst0 (s +(Bind b) i) v0 t1 x)).(H3 v0 x (S i) H8 b0 d v (getl_head (Bind b) i c0 +(CHead d (Bind b0) v) H5 u0))))) H6)) (\lambda (H6: (ex3_2 T T (\lambda (u2: +T).(\lambda (t4: T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda (u2: +T).(\lambda (_: T).(subst0 i v0 u0 u2))) (\lambda (_: T).(\lambda (t4: +T).(subst0 (s (Bind b) i) v0 t1 t4))))).(ex3_2_ind T T (\lambda (u2: +T).(\lambda (t4: T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda (u2: +T).(\lambda (_: T).(subst0 i v0 u0 u2))) (\lambda (_: T).(\lambda (t4: +T).(subst0 (s (Bind b) i) v0 t1 t4))) (ex T (\lambda (w: T).(ty3 g d v w))) +(\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (eq T t3 (THead (Bind b) x0 +x1))).(\lambda (H8: (subst0 i v0 u0 x0)).(\lambda (_: (subst0 (s (Bind b) i) +v0 t1 x1)).(H1 v0 x0 i H8 b0 d v H5)))))) H6)) (subst0_gen_head (Bind b) v0 +u0 t1 t3 i H4)))))))))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda +(u0: T).(\lambda (_: (ty3 g c0 w u0)).(\lambda (H1: ((\forall (v0: +T).(\forall (t0: T).(\forall (i: nat).((subst0 i v0 w t0) \to (\forall (b: +B).(\forall (d: C).(\forall (v: T).((getl i c0 (CHead d (Bind b) v)) \to (ex +T (\lambda (w0: T).(ty3 g d v w0))))))))))))).(\lambda (v: T).(\lambda (t0: +T).(\lambda (_: (ty3 g c0 v (THead (Bind Abst) u0 t0))).(\lambda (H3: +((\forall (v0: T).(\forall (t1: T).(\forall (i: nat).((subst0 i v0 v t1) \to +(\forall (b: B).(\forall (d: C).(\forall (v1: T).((getl i c0 (CHead d (Bind +b) v1)) \to (ex T (\lambda (w0: T).(ty3 g d v1 w0))))))))))))).(\lambda (v0: +T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (H4: (subst0 i v0 (THead (Flat +Appl) w v) t1)).(\lambda (b: B).(\lambda (d: C).(\lambda (v1: T).(\lambda +(H5: (getl i c0 (CHead d (Bind b) v1))).(or3_ind (ex2 T (\lambda (u2: T).(eq +T t1 (THead (Flat Appl) u2 v))) (\lambda (u2: T).(subst0 i v0 w u2))) (ex2 T +(\lambda (t2: T).(eq T t1 (THead (Flat Appl) w t2))) (\lambda (t2: T).(subst0 +(s (Flat Appl) i) v0 v t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq +T t1 (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i +v0 w u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s (Flat Appl) i) v0 v +t2)))) (ex T (\lambda (w0: T).(ty3 g d v1 w0))) (\lambda (H6: (ex2 T (\lambda +(u2: T).(eq T t1 (THead (Flat Appl) u2 v))) (\lambda (u2: T).(subst0 i v0 w +u2)))).(ex2_ind T (\lambda (u2: T).(eq T t1 (THead (Flat Appl) u2 v))) +(\lambda (u2: T).(subst0 i v0 w u2)) (ex T (\lambda (w0: T).(ty3 g d v1 w0))) +(\lambda (x: T).(\lambda (_: (eq T t1 (THead (Flat Appl) x v))).(\lambda (H8: +(subst0 i v0 w x)).(H1 v0 x i H8 b d v1 H5)))) H6)) (\lambda (H6: (ex2 T +(\lambda (t2: T).(eq T t1 (THead (Flat Appl) w t2))) (\lambda (t2: T).(subst0 +(s (Flat Appl) i) v0 v t2)))).(ex2_ind T (\lambda (t2: T).(eq T t1 (THead +(Flat Appl) w t2))) (\lambda (t2: T).(subst0 (s (Flat Appl) i) v0 v t2)) (ex +T (\lambda (w0: T).(ty3 g d v1 w0))) (\lambda (x: T).(\lambda (_: (eq T t1 +(THead (Flat Appl) w x))).(\lambda (H8: (subst0 (s (Flat Appl) i) v0 v +x)).(H3 v0 x (s (Flat Appl) i) H8 b d v1 H5)))) H6)) (\lambda (H6: (ex3_2 T T +(\lambda (u2: T).(\lambda (t2: T).(eq T t1 (THead (Flat Appl) u2 t2)))) +(\lambda (u2: T).(\lambda (_: T).(subst0 i v0 w u2))) (\lambda (_: +T).(\lambda (t2: T).(subst0 (s (Flat Appl) i) v0 v t2))))).(ex3_2_ind T T +(\lambda (u2: T).(\lambda (t2: T).(eq T t1 (THead (Flat Appl) u2 t2)))) +(\lambda (u2: T).(\lambda (_: T).(subst0 i v0 w u2))) (\lambda (_: +T).(\lambda (t2: T).(subst0 (s (Flat Appl) i) v0 v t2))) (ex T (\lambda (w0: +T).(ty3 g d v1 w0))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (eq T t1 +(THead (Flat Appl) x0 x1))).(\lambda (_: (subst0 i v0 w x0)).(\lambda (H9: 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(THead (Flat Cast) u2 t1))) (\lambda (u2: T).(subst0 i v0 t2 u2))) (ex2 T +(\lambda (t4: T).(eq T t3 (THead (Flat Cast) t2 t4))) (\lambda (t4: +T).(subst0 (s (Flat Cast) i) v0 t1 t4))) (ex3_2 T T (\lambda (u2: T).(\lambda +(t4: T).(eq T t3 (THead (Flat Cast) u2 t4)))) (\lambda (u2: T).(\lambda (_: +T).(subst0 i v0 t2 u2))) (\lambda (_: T).(\lambda (t4: T).(subst0 (s (Flat +Cast) i) v0 t1 t4)))) (ex T (\lambda (w: T).(ty3 g d v w))) (\lambda (H6: +(ex2 T (\lambda (u2: T).(eq T t3 (THead (Flat Cast) u2 t1))) (\lambda (u2: +T).(subst0 i v0 t2 u2)))).(ex2_ind T (\lambda (u2: T).(eq T t3 (THead (Flat +Cast) u2 t1))) (\lambda (u2: T).(subst0 i v0 t2 u2)) (ex T (\lambda (w: +T).(ty3 g d v w))) (\lambda (x: T).(\lambda (_: (eq T t3 (THead (Flat Cast) x +t1))).(\lambda (H8: (subst0 i v0 t2 x)).(H3 v0 x i H8 b d v H5)))) H6)) +(\lambda (H6: (ex2 T (\lambda (t4: T).(eq T t3 (THead (Flat Cast) t2 t4))) +(\lambda (t4: T).(subst0 (s (Flat Cast) i) v0 t1 t4)))).(ex2_ind T (\lambda +(t4: T).(eq T t3 (THead (Flat Cast) t2 t4))) (\lambda (t4: T).(subst0 (s +(Flat Cast) i) v0 t1 t4)) (ex T (\lambda (w: T).(ty3 g d v w))) (\lambda (x: +T).(\lambda (_: (eq T t3 (THead (Flat Cast) t2 x))).(\lambda (H8: (subst0 (s +(Flat Cast) i) v0 t1 x)).(H1 v0 x (s (Flat Cast) i) H8 b d v H5)))) H6)) +(\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead +(Flat Cast) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 t2 u2))) +(\lambda (_: T).(\lambda (t4: T).(subst0 (s (Flat Cast) i) v0 t1 +t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead +(Flat Cast) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 t2 u2))) +(\lambda (_: T).(\lambda (t4: T).(subst0 (s (Flat Cast) i) v0 t1 t4))) (ex T +(\lambda (w: T).(ty3 g d v w))) (\lambda (x0: T).(\lambda (x1: T).(\lambda +(_: (eq T t3 (THead (Flat Cast) x0 x1))).(\lambda (H8: (subst0 i v0 t2 +x0)).(\lambda (_: (subst0 (s (Flat Cast) i) v0 t1 x1)).(H3 v0 x0 i H8 b d v +H5)))))) H6)) (subst0_gen_head (Flat Cast) v0 t2 t1 t3 i H4)))))))))))))))))) +c t u H))))). +(* COMMENTS +Initial nodes: 4343 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/ty3/sty0.ma b/matita/matita/contribs/lambdadelta/basic_1/ty3/sty0.ma new file mode 100644 index 000000000..3c630662c --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/ty3/sty0.ma @@ -0,0 +1,237 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/ty3/pr3_props.ma". + +include "Basic-1/sty0/fwd.ma". + +theorem ty3_sty0: + \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).((ty3 g c u +t1) \to (\forall (t2: T).((sty0 g c u t2) \to (ty3 g c u t2))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (H: +(ty3 g c u t1)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (_: +T).(\forall (t2: T).((sty0 g c0 t t2) \to (ty3 g c0 t t2)))))) (\lambda (c0: +C).(\lambda (t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda +(_: ((\forall (t3: T).((sty0 g c0 t2 t3) \to (ty3 g c0 t2 t3))))).(\lambda +(u0: T).(\lambda (t3: T).(\lambda (_: (ty3 g c0 u0 t3)).(\lambda (H3: +((\forall (t4: T).((sty0 g c0 u0 t4) \to (ty3 g c0 u0 t4))))).(\lambda (_: +(pc3 c0 t3 t2)).(\lambda (t0: T).(\lambda (H5: (sty0 g c0 u0 t0)).(H3 t0 +H5))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (t2: T).(\lambda +(H0: (sty0 g c0 (TSort m) t2)).(let H_y \def (sty0_gen_sort g c0 t2 m H0) in +(let H1 \def (f_equal T T (\lambda (e: T).e) t2 (TSort (next g m)) H_y) in +(eq_ind_r T (TSort (next g m)) (\lambda (t: T).(ty3 g c0 (TSort m) t)) +(ty3_sort g c0 m) t2 H1))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda +(d: C).(\lambda (u0: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abbr) +u0))).(\lambda (t: T).(\lambda (_: (ty3 g d u0 t)).(\lambda (H2: ((\forall +(t2: T).((sty0 g d u0 t2) \to (ty3 g d u0 t2))))).(\lambda (t2: T).(\lambda +(H3: (sty0 g c0 (TLRef n) t2)).(let H_x \def (sty0_gen_lref g c0 t2 n H3) in +(let H4 \def H_x in (or_ind (ex3_3 C T T (\lambda (e: C).(\lambda (u1: +T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: +C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_: +C).(\lambda (_: T).(\lambda (t0: T).(eq T t2 (lift (S n) O t0)))))) (ex3_3 C +T T (\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e +(Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g +e u1 t0)))) (\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq T t2 (lift +(S n) O u1)))))) (ty3 g c0 (TLRef n) t2) (\lambda (H5: (ex3_3 C T T (\lambda +(e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) +u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g e u1 t0)))) +(\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(eq T t2 (lift (S n) O +t0))))))).(ex3_3_ind C T T (\lambda (e: C).(\lambda (u1: T).(\lambda (_: +T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: C).(\lambda (u1: +T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_: C).(\lambda (_: +T).(\lambda (t0: T).(eq T t2 (lift (S n) O t0))))) (ty3 g c0 (TLRef n) t2) +(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H6: (getl n c0 +(CHead x0 (Bind Abbr) x1))).(\lambda (H7: (sty0 g x0 x1 x2)).(\lambda (H8: +(eq T t2 (lift (S n) O x2))).(let H9 \def (f_equal T T (\lambda (e: T).e) t2 +(lift (S n) O x2) H8) in (eq_ind_r T (lift (S n) O x2) (\lambda (t0: T).(ty3 +g c0 (TLRef n) t0)) (let H10 \def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda +(c1: C).(getl n c0 c1)) H0 (CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d +(Bind Abbr) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in (let H11 \def (f_equal +C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) +\Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u0) +(CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead +x0 (Bind Abbr) x1) H6)) in ((let H12 \def (f_equal C T (\lambda (e: C).(match +e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ +t0) \Rightarrow t0])) (CHead d (Bind Abbr) u0) (CHead x0 (Bind Abbr) x1) +(getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in +(\lambda (H13: (eq C d x0)).(let H14 \def (eq_ind_r T x1 (\lambda (t0: +T).(getl n c0 (CHead x0 (Bind Abbr) t0))) H10 u0 H12) in (let H15 \def +(eq_ind_r T x1 (\lambda (t0: T).(sty0 g x0 t0 x2)) H7 u0 H12) in (let H16 +\def (eq_ind_r C x0 (\lambda (c1: C).(getl n c0 (CHead c1 (Bind Abbr) u0))) +H14 d H13) in (let H17 \def (eq_ind_r C x0 (\lambda (c1: C).(sty0 g c1 u0 +x2)) H15 d H13) in (ty3_abbr g n c0 d u0 H16 x2 (H2 x2 H17)))))))) H11))) t2 +H9)))))))) H5)) (\lambda (H5: (ex3_3 C T T (\lambda (e: C).(\lambda (u1: +T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e: +C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_: +C).(\lambda (u1: T).(\lambda (_: T).(eq T t2 (lift (S n) O +u1))))))).(ex3_3_ind C T T (\lambda (e: C).(\lambda (u1: T).(\lambda (_: +T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1: +T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_: C).(\lambda (u1: +T).(\lambda (_: T).(eq T t2 (lift (S n) O u1))))) (ty3 g c0 (TLRef n) t2) +(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H6: (getl n c0 +(CHead x0 (Bind Abst) x1))).(\lambda (_: (sty0 g x0 x1 x2)).(\lambda (H8: (eq +T t2 (lift (S n) O x1))).(let H9 \def (f_equal T T (\lambda (e: T).e) t2 +(lift (S n) O x1) H8) in (eq_ind_r T (lift (S n) O x1) (\lambda (t0: T).(ty3 +g c0 (TLRef n) t0)) (let H10 \def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda +(c1: C).(getl n c0 c1)) H0 (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d +(Bind Abbr) u0) n H0 (CHead x0 (Bind Abst) x1) H6)) in (let H11 \def (eq_ind +C (CHead d (Bind Abbr) u0) (\lambda (ee: C).(match ee in C return (\lambda +(_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow +(match k in K return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match +b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst +\Rightarrow False | Void \Rightarrow False]) | (Flat _) \Rightarrow +False])])) I (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abbr) u0) +n H0 (CHead x0 (Bind Abst) x1) H6)) in (False_ind (ty3 g c0 (TLRef n) (lift +(S n) O x1)) H11))) t2 H9)))))))) H5)) H4))))))))))))) (\lambda (n: +nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (H0: (getl n +c0 (CHead d (Bind Abst) u0))).(\lambda (t: T).(\lambda (H1: (ty3 g d u0 +t)).(\lambda (_: ((\forall (t2: T).((sty0 g d u0 t2) \to (ty3 g d u0 +t2))))).(\lambda (t2: T).(\lambda (H3: (sty0 g c0 (TLRef n) t2)).(let H_x +\def (sty0_gen_lref g c0 t2 n H3) in (let H4 \def H_x in (or_ind (ex3_3 C T T +(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind +Abbr) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g e u1 +t0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(eq T t2 (lift (S n) +O t0)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u1: T).(\lambda (_: +T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1: +T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_: C).(\lambda (u1: +T).(\lambda (_: T).(eq T t2 (lift (S n) O u1)))))) (ty3 g c0 (TLRef n) t2) +(\lambda (H5: (ex3_3 C T T (\lambda (e: C).(\lambda (u1: T).(\lambda (_: +T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: C).(\lambda (u1: +T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_: C).(\lambda (_: +T).(\lambda (t0: T).(eq T t2 (lift (S n) O t0))))))).(ex3_3_ind C T T +(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind +Abbr) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g e u1 +t0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(eq T t2 (lift (S n) +O t0))))) (ty3 g c0 (TLRef n) t2) (\lambda (x0: C).(\lambda (x1: T).(\lambda +(x2: T).(\lambda (H6: (getl n c0 (CHead x0 (Bind Abbr) x1))).(\lambda (_: +(sty0 g x0 x1 x2)).(\lambda (H8: (eq T t2 (lift (S n) O x2))).(let H9 \def +(f_equal T T (\lambda (e: T).e) t2 (lift (S n) O x2) H8) in (eq_ind_r T (lift +(S n) O x2) (\lambda (t0: T).(ty3 g c0 (TLRef n) t0)) (let H10 \def (eq_ind C +(CHead d (Bind Abst) u0) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead x0 (Bind +Abbr) x1) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead x0 (Bind Abbr) +x1) H6)) in (let H11 \def (eq_ind C (CHead d (Bind Abst) u0) (\lambda (ee: +C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow +False | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) +with [(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with +[Abbr \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | +(Flat _) \Rightarrow False])])) I (CHead x0 (Bind Abbr) x1) (getl_mono c0 +(CHead d (Bind Abst) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in (False_ind +(ty3 g c0 (TLRef n) (lift (S n) O x2)) H11))) t2 H9)))))))) H5)) (\lambda +(H5: (ex3_3 C T T (\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 +(CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: +T).(sty0 g e u1 t0)))) (\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq T +t2 (lift (S n) O u1))))))).(ex3_3_ind C T T (\lambda (e: C).(\lambda (u1: +T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e: +C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_: +C).(\lambda (u1: T).(\lambda (_: T).(eq T t2 (lift (S n) O u1))))) (ty3 g c0 +(TLRef n) t2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda +(H6: (getl n c0 (CHead x0 (Bind Abst) x1))).(\lambda (H7: (sty0 g x0 x1 +x2)).(\lambda (H8: (eq T t2 (lift (S n) O x1))).(let H9 \def (f_equal T T +(\lambda (e: T).e) t2 (lift (S n) O x1) H8) in (eq_ind_r T (lift (S n) O x1) +(\lambda (t0: T).(ty3 g c0 (TLRef n) t0)) (let H10 \def (eq_ind C (CHead d +(Bind Abst) u0) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead x0 (Bind Abst) x1) +(getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead x0 (Bind Abst) x1) H6)) in +(let H11 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: +C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead +d (Bind Abst) u0) (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind +Abst) u0) n H0 (CHead x0 (Bind Abst) x1) H6)) in ((let H12 \def (f_equal C T +(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) +\Rightarrow u0 | (CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind Abst) u0) +(CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead +x0 (Bind Abst) x1) H6)) in (\lambda (H13: (eq C d x0)).(let H14 \def +(eq_ind_r T x1 (\lambda (t0: T).(getl n c0 (CHead x0 (Bind Abst) t0))) H10 u0 +H12) in (let H15 \def (eq_ind_r T x1 (\lambda (t0: T).(sty0 g x0 t0 x2)) H7 +u0 H12) in (eq_ind T u0 (\lambda (t0: T).(ty3 g c0 (TLRef n) (lift (S n) O +t0))) (let H16 \def (eq_ind_r C x0 (\lambda (c1: C).(getl n c0 (CHead c1 +(Bind Abst) u0))) H14 d H13) in (let H17 \def (eq_ind_r C x0 (\lambda (c1: +C).(sty0 g c1 u0 x2)) H15 d H13) in (ty3_abst g n c0 d u0 H16 t H1))) x1 +H12))))) H11))) t2 H9)))))))) H5)) H4))))))))))))) (\lambda (c0: C).(\lambda +(u0: T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 u0 t)).(\lambda (_: ((\forall +(t2: T).((sty0 g c0 u0 t2) \to (ty3 g c0 u0 t2))))).(\lambda (b: B).(\lambda +(t2: T).(\lambda (t3: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u0) t2 +t3)).(\lambda (H3: ((\forall (t4: T).((sty0 g (CHead c0 (Bind b) u0) t2 t4) +\to (ty3 g (CHead c0 (Bind b) u0) t2 t4))))).(\lambda (t0: T).(\lambda (H4: +(sty0 g c0 (THead (Bind b) u0 t2) t0)).(let H_x \def (sty0_gen_bind g b c0 u0 +t2 t0 H4) in (let H5 \def H_x in (ex2_ind T (\lambda (t4: T).(sty0 g (CHead +c0 (Bind b) u0) t2 t4)) (\lambda (t4: T).(eq T t0 (THead (Bind b) u0 t4))) +(ty3 g c0 (THead (Bind b) u0 t2) t0) (\lambda (x: T).(\lambda (H6: (sty0 g +(CHead c0 (Bind b) u0) t2 x)).(\lambda (H7: (eq T t0 (THead (Bind b) u0 +x))).(let H8 \def (f_equal T T (\lambda (e: T).e) t0 (THead (Bind b) u0 x) +H7) in (eq_ind_r T (THead (Bind b) u0 x) (\lambda (t4: T).(ty3 g c0 (THead +(Bind b) u0 t2) t4)) (ty3_bind g c0 u0 t H0 b t2 x (H3 x H6)) t0 H8))))) +H5))))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda (u0: T).(\lambda +(H0: (ty3 g c0 w u0)).(\lambda (_: ((\forall (t2: T).((sty0 g c0 w t2) \to +(ty3 g c0 w t2))))).(\lambda (v: T).(\lambda (t: T).(\lambda (H2: (ty3 g c0 v +(THead (Bind Abst) u0 t))).(\lambda (H3: ((\forall (t2: T).((sty0 g c0 v t2) +\to (ty3 g c0 v t2))))).(\lambda (t2: T).(\lambda (H4: (sty0 g c0 (THead +(Flat Appl) w v) t2)).(let H_x \def (sty0_gen_appl g c0 w v t2 H4) in (let H5 +\def H_x in (ex2_ind T (\lambda (t3: T).(sty0 g c0 v t3)) (\lambda (t3: +T).(eq T t2 (THead (Flat Appl) w t3))) (ty3 g c0 (THead (Flat Appl) w v) t2) +(\lambda (x: T).(\lambda (H6: (sty0 g c0 v x)).(\lambda (H7: (eq T t2 (THead +(Flat Appl) w x))).(let H8 \def (f_equal T T (\lambda (e: T).e) t2 (THead +(Flat Appl) w x) H7) in (eq_ind_r T (THead (Flat Appl) w x) (\lambda (t0: +T).(ty3 g c0 (THead (Flat Appl) w v) t0)) (let H_y \def (H3 x H6) in (let H9 +\def (ty3_unique g c0 v x H_y (THead (Bind Abst) u0 t) H2) in (ex_ind T +(\lambda (t0: T).(ty3 g c0 x t0)) (ty3 g c0 (THead (Flat Appl) w v) (THead +(Flat Appl) w x)) (\lambda (x0: T).(\lambda (H10: (ty3 g c0 x x0)).(ex_ind T +(\lambda (t0: T).(ty3 g c0 u0 t0)) (ty3 g c0 (THead (Flat Appl) w v) (THead +(Flat Appl) w x)) (\lambda (x1: T).(\lambda (_: (ty3 g c0 u0 x1)).(ex_ind T +(\lambda (t0: T).(ty3 g c0 (THead (Bind Abst) u0 t) t0)) (ty3 g c0 (THead +(Flat Appl) w v) (THead (Flat Appl) w x)) (\lambda (x2: T).(\lambda (H12: +(ty3 g c0 (THead (Bind Abst) u0 t) x2)).(ex3_2_ind T T (\lambda (t3: +T).(\lambda (_: T).(pc3 c0 (THead (Bind Abst) u0 t3) x2))) (\lambda (_: +T).(\lambda (t0: T).(ty3 g c0 u0 t0))) (\lambda (t3: T).(\lambda (_: T).(ty3 +g (CHead c0 (Bind Abst) u0) t t3))) (ty3 g c0 (THead (Flat Appl) w v) (THead +(Flat Appl) w x)) (\lambda (x3: T).(\lambda (x4: T).(\lambda (_: (pc3 c0 +(THead (Bind Abst) u0 x3) x2)).(\lambda (H14: (ty3 g c0 u0 x4)).(\lambda +(H15: (ty3 g (CHead c0 (Bind Abst) u0) t x3)).(ty3_conv g c0 (THead (Flat +Appl) w x) (THead (Flat Appl) w (THead (Bind Abst) u0 x3)) (ty3_appl g c0 w +u0 H0 x x3 (ty3_sconv g c0 x x0 H10 (THead (Bind Abst) u0 t) (THead (Bind +Abst) u0 x3) (ty3_bind g c0 u0 x4 H14 Abst t x3 H15) H9)) (THead (Flat Appl) +w v) (THead (Flat Appl) w (THead (Bind Abst) u0 t)) (ty3_appl g c0 w u0 H0 v +t H2) (pc3_thin_dx c0 (THead (Bind Abst) u0 t) x (ty3_unique g c0 v (THead +(Bind Abst) u0 t) H2 x H_y) w Appl))))))) (ty3_gen_bind g Abst c0 u0 t x2 +H12)))) (ty3_correct g c0 v (THead (Bind Abst) u0 t) H2)))) (ty3_correct g c0 +w u0 H0)))) (ty3_correct g c0 v x H_y)))) t2 H8))))) H5)))))))))))))) +(\lambda (c0: C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H0: (ty3 g c0 t2 +t3)).(\lambda (H1: ((\forall (t4: T).((sty0 g c0 t2 t4) \to (ty3 g c0 t2 +t4))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t3 t0)).(\lambda (H3: +((\forall (t4: T).((sty0 g c0 t3 t4) \to (ty3 g c0 t3 t4))))).(\lambda (t4: +T).(\lambda (H4: (sty0 g c0 (THead (Flat Cast) t3 t2) t4)).(let H_x \def +(sty0_gen_cast g c0 t3 t2 t4 H4) in (let H5 \def H_x in (ex3_2_ind T T +(\lambda (v2: T).(\lambda (_: T).(sty0 g c0 t3 v2))) (\lambda (_: T).(\lambda +(t5: T).(sty0 g c0 t2 t5))) (\lambda (v2: T).(\lambda (t5: T).(eq T t4 (THead +(Flat Cast) v2 t5)))) (ty3 g c0 (THead (Flat Cast) t3 t2) t4) (\lambda (x0: +T).(\lambda (x1: T).(\lambda (H6: (sty0 g c0 t3 x0)).(\lambda (H7: (sty0 g c0 +t2 x1)).(\lambda (H8: (eq T t4 (THead (Flat Cast) x0 x1))).(let H9 \def +(f_equal T T (\lambda (e: T).e) t4 (THead (Flat Cast) x0 x1) H8) in (eq_ind_r +T (THead (Flat Cast) x0 x1) (\lambda (t: T).(ty3 g c0 (THead (Flat Cast) t3 +t2) t)) (let H_y \def (H1 x1 H7) in (let H_y0 \def (H3 x0 H6) in (let H10 +\def (ty3_unique g c0 t2 x1 H_y t3 H0) in (ex_ind T (\lambda (t: T).(ty3 g c0 +x0 t)) (ty3 g c0 (THead (Flat Cast) t3 t2) (THead (Flat Cast) x0 x1)) +(\lambda (x: T).(\lambda (H11: (ty3 g c0 x0 x)).(ex_ind T (\lambda (t: +T).(ty3 g c0 x1 t)) (ty3 g c0 (THead (Flat Cast) t3 t2) (THead (Flat Cast) x0 +x1)) (\lambda (x2: T).(\lambda (H12: (ty3 g c0 x1 x2)).(ty3_conv g c0 (THead +(Flat Cast) x0 x1) (THead (Flat Cast) x x0) (ty3_cast g c0 x1 x0 (ty3_sconv g +c0 x1 x2 H12 t3 x0 H_y0 H10) x H11) (THead (Flat Cast) t3 t2) (THead (Flat +Cast) x0 t3) (ty3_cast g c0 t2 t3 H0 x0 H_y0) (pc3_thin_dx c0 t3 x1 +(ty3_unique g c0 t2 t3 H0 x1 H_y) x0 Cast)))) (ty3_correct g c0 t2 x1 H_y)))) +(ty3_correct g c0 t3 x0 H_y0))))) t4 H9))))))) H5))))))))))))) c u t1 H))))). +(* COMMENTS +Initial nodes: 4539 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/ty3/subst1.ma b/matita/matita/contribs/lambdadelta/basic_1/ty3/subst1.ma new file mode 100644 index 000000000..ca9516e58 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/ty3/subst1.ma @@ -0,0 +1,1102 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/ty3/props.ma". + +include "Basic-1/pc3/subst1.ma". + +include "Basic-1/getl/getl.ma". + +theorem ty3_gen_cabbr: + \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c +t1 t2) \to (\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c +(CHead e (Bind Abbr) u)) \to (\forall (a0: C).((csubst1 d u c a0) \to +(\forall (a: C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda +(_: T).(subst1 d u t1 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: +T).(subst1 d u t2 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 +g a y1 y2)))))))))))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda +(H: (ty3 g c t1 t2)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda +(t0: T).(\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead +e (Bind Abbr) u)) \to (\forall (a0: C).((csubst1 d u c0 a0) \to (\forall (a: +C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: +T).(subst1 d u t (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: +T).(subst1 d u t0 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 +g a y1 y2))))))))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t: +T).(\lambda (_: (ty3 g c0 t3 t)).(\lambda (H1: ((\forall (e: C).(\forall (u: +T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u)) \to (\forall (a0: +C).((csubst1 d u c0 a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (ex3_2 T +T (\lambda (y1: T).(\lambda (_: T).(subst1 d u t3 (lift (S O) d y1)))) +(\lambda (_: T).(\lambda (y2: T).(subst1 d u t (lift (S O) d y2)))) (\lambda +(y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))))).(\lambda (u: +T).(\lambda (t4: T).(\lambda (_: (ty3 g c0 u t4)).(\lambda (H3: ((\forall (e: +C).(\forall (u0: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u0)) +\to (\forall (a0: C).((csubst1 d u0 c0 a0) \to (\forall (a: C).((drop (S O) d +a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u (lift (S +O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t4 (lift (S O) d +y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 +y2)))))))))))))).(\lambda (H4: (pc3 c0 t4 t3)).(\lambda (e: C).(\lambda (u0: +T).(\lambda (d: nat).(\lambda (H5: (getl d c0 (CHead e (Bind Abbr) +u0))).(\lambda (a0: C).(\lambda (H6: (csubst1 d u0 c0 a0)).(\lambda (a: +C).(\lambda (H7: (drop (S O) d a0 a)).(let H8 \def (H3 e u0 d H5 a0 H6 a H7) +in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u (lift (S O) +d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t4 (lift (S O) d +y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T +(\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u (lift (S O) d y1)))) (\lambda +(_: T).(\lambda (y2: T).(subst1 d u0 t3 (lift (S O) d y2)))) (\lambda (y1: +T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1: +T).(\lambda (H9: (subst1 d u0 u (lift (S O) d x0))).(\lambda (H10: (subst1 d +u0 t4 (lift (S O) d x1))).(\lambda (H11: (ty3 g a x0 x1)).(let H12 \def (H1 e +u0 d H5 a0 H6 a H7) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: +T).(subst1 d u0 t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: +T).(subst1 d u0 t (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 +g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u (lift +(S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t3 (lift (S O) d +y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: +T).(\lambda (x3: T).(\lambda (H13: (subst1 d u0 t3 (lift (S O) d +x2))).(\lambda (_: (subst1 d u0 t (lift (S O) d x3))).(\lambda (H15: (ty3 g a +x2 x3)).(ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u +(lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t3 (lift +(S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) x0 x2 H9 +H13 (ty3_conv g a x2 x3 H15 x0 x1 H11 (pc3_gen_cabbr c0 t4 t3 H4 e u0 d H5 a0 +H6 a H7 x1 H10 x2 H13)))))))) H12))))))) H8)))))))))))))))))))) (\lambda (c0: +C).(\lambda (m: nat).(\lambda (e: C).(\lambda (u: T).(\lambda (d: +nat).(\lambda (_: (getl d c0 (CHead e (Bind Abbr) u))).(\lambda (a0: +C).(\lambda (_: (csubst1 d u c0 a0)).(\lambda (a: C).(\lambda (_: (drop (S O) +d a0 a)).(ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u (TSort +m) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u (TSort +(next g m)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a +y1 y2))) (TSort m) (TSort (next g m)) (eq_ind_r T (TSort m) (\lambda (t: +T).(subst1 d u (TSort m) t)) (subst1_refl d u (TSort m)) (lift (S O) d (TSort +m)) (lift_sort m (S O) d)) (eq_ind_r T (TSort (next g m)) (\lambda (t: +T).(subst1 d u (TSort (next g m)) t)) (subst1_refl d u (TSort (next g m))) +(lift (S O) d (TSort (next g m))) (lift_sort (next g m) (S O) d)) (ty3_sort g +a m)))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda +(u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abbr) u))).(\lambda (t: +T).(\lambda (H1: (ty3 g d u t)).(\lambda (H2: ((\forall (e: C).(\forall (u0: +T).(\forall (d0: nat).((getl d0 d (CHead e (Bind Abbr) u0)) \to (\forall (a0: +C).((csubst1 d0 u0 d a0) \to (\forall (a: C).((drop (S O) d0 a0 a) \to (ex3_2 +T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 u (lift (S O) d0 y1)))) +(\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 t (lift (S O) d0 y2)))) +(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))))).(\lambda (e: +C).(\lambda (u0: T).(\lambda (d0: nat).(\lambda (H3: (getl d0 c0 (CHead e +(Bind Abbr) u0))).(\lambda (a0: C).(\lambda (H4: (csubst1 d0 u0 c0 +a0)).(\lambda (a: C).(\lambda (H5: (drop (S O) d0 a0 a)).(lt_eq_gt_e n d0 +(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S +O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O t) +(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) +(\lambda (H6: (lt n d0)).(let H7 \def (eq_ind nat (minus d0 n) (\lambda (n0: +nat).(getl n0 (CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0))) +(getl_conf_le d0 (CHead e (Bind Abbr) u0) c0 H3 (CHead d (Bind Abbr) u) n H0 +(le_S_n n d0 (le_S (S n) d0 H6))) (S (minus d0 (S n))) (minus_x_Sy d0 n H6)) +in (ex2_ind C (\lambda (e2: C).(csubst1 (minus d0 n) u0 (CHead d (Bind Abbr) +u) e2)) (\lambda (e2: C).(getl n a0 e2)) (ex3_2 T T (\lambda (y1: T).(\lambda +(_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda +(y2: T).(subst1 d0 u0 (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: +T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x: C).(\lambda (H8: (csubst1 +(minus d0 n) u0 (CHead d (Bind Abbr) u) x)).(\lambda (H9: (getl n a0 x)).(let +H10 \def (eq_ind nat (minus d0 n) (\lambda (n0: nat).(csubst1 n0 u0 (CHead d +(Bind Abbr) u) x)) H8 (S (minus d0 (S n))) (minus_x_Sy d0 n H6)) in (let H11 +\def (csubst1_gen_head (Bind Abbr) d x u u0 (minus d0 (S n)) H10) in +(ex3_2_ind T C (\lambda (u2: T).(\lambda (c2: C).(eq C x (CHead c2 (Bind +Abbr) u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 (minus d0 (S n)) u0 u +u2))) (\lambda (_: T).(\lambda (c2: C).(csubst1 (minus d0 (S n)) u0 d c2))) +(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S +O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O t) +(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) +(\lambda (x0: T).(\lambda (x1: C).(\lambda (H12: (eq C x (CHead x1 (Bind +Abbr) x0))).(\lambda (H13: (subst1 (minus d0 (S n)) u0 u x0)).(\lambda (H14: +(csubst1 (minus d0 (S n)) u0 d x1)).(let H15 \def (eq_ind C x (\lambda (c1: +C).(getl n a0 c1)) H9 (CHead x1 (Bind Abbr) x0) H12) in (let H16 \def (eq_ind +nat d0 (\lambda (n0: nat).(drop (S O) n0 a0 a)) H5 (S (plus n (minus d0 (S +n)))) (lt_plus_minus n d0 H6)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: +C).(eq T x0 (lift (S O) (minus d0 (S n)) v)))) (\lambda (v: T).(\lambda (e0: +C).(getl n a (CHead e0 (Bind Abbr) v)))) (\lambda (_: T).(\lambda (e0: +C).(drop (S O) (minus d0 (S n)) x1 e0))) (ex3_2 T T (\lambda (y1: T).(\lambda +(_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda +(y2: T).(subst1 d0 u0 (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: +T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: T).(\lambda (x3: +C).(\lambda (H17: (eq T x0 (lift (S O) (minus d0 (S n)) x2))).(\lambda (H18: +(getl n a (CHead x3 (Bind Abbr) x2))).(\lambda (H19: (drop (S O) (minus d0 (S +n)) x1 x3)).(let H20 \def (eq_ind T x0 (\lambda (t0: T).(subst1 (minus d0 (S +n)) u0 u t0)) H13 (lift (S O) (minus d0 (S n)) x2) H17) in (let H21 \def (H2 +e u0 (minus d0 (S n)) (getl_gen_S (Bind Abbr) d (CHead e (Bind Abbr) u0) u +(minus d0 (S n)) H7) x1 H14 x3 H19) in (ex3_2_ind T T (\lambda (y1: +T).(\lambda (_: T).(subst1 (minus d0 (S n)) u0 u (lift (S O) (minus d0 (S n)) +y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 (minus d0 (S n)) u0 t (lift +(S O) (minus d0 (S n)) y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g x3 y1 +y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) +(lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S +n) O t) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 +y2)))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H22: (subst1 (minus d0 (S +n)) u0 u (lift (S O) (minus d0 (S n)) x4))).(\lambda (H23: (subst1 (minus d0 +(S n)) u0 t (lift (S O) (minus d0 (S n)) x5))).(\lambda (H24: (ty3 g x3 x4 +x5)).(let H25 \def (eq_ind T x4 (\lambda (t0: T).(ty3 g x3 t0 x5)) H24 x2 +(subst1_confluence_lift u x4 u0 (minus d0 (S n)) H22 x2 H20)) in (eq_ind_r +nat (plus (minus d0 (S n)) (S n)) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: +T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: +T).(\lambda (y2: T).(subst1 n0 u0 (lift (S n) O t) (lift (S O) d0 y2)))) +(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r nat (plus (S +n) (minus d0 (S n))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda +(_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda +(y2: T).(subst1 (plus (minus d0 (S n)) (S n)) u0 (lift (S n) O t) (lift (S O) +n0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro +T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 +y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 (plus (minus d0 (S n)) (S n)) +u0 (lift (S n) O t) (lift (S O) (plus (S n) (minus d0 (S n))) y2)))) (\lambda +(y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (TLRef n) (lift (S n) O x5) +(eq_ind_r T (TLRef n) (\lambda (t0: T).(subst1 d0 u0 (TLRef n) t0)) +(subst1_refl d0 u0 (TLRef n)) (lift (S O) d0 (TLRef n)) (lift_lref_lt n (S O) +d0 H6)) (eq_ind_r T (lift (S n) O (lift (S O) (minus d0 (S n)) x5)) (\lambda +(t0: T).(subst1 (plus (minus d0 (S n)) (S n)) u0 (lift (S n) O t) t0)) +(subst1_lift_ge t (lift (S O) (minus d0 (S n)) x5) u0 (minus d0 (S n)) (S n) +H23 O (le_O_n (minus d0 (S n)))) (lift (S O) (plus (S n) (minus d0 (S n))) +(lift (S n) O x5)) (lift_d x5 (S O) (S n) (minus d0 (S n)) O (le_O_n (minus +d0 (S n))))) (ty3_abbr g n a x3 x2 H18 x5 H25)) d0 (le_plus_minus (S n) d0 +H6)) d0 (le_plus_minus_sym (S n) d0 H6)))))))) H21)))))))) (getl_drop_conf_lt +Abbr a0 x1 x0 n H15 a (S O) (minus d0 (S n)) H16))))))))) H11)))))) +(csubst1_getl_lt d0 n H6 c0 a0 u0 H4 (CHead d (Bind Abbr) u) H0)))) (\lambda +(H6: (eq nat n d0)).(let H7 \def (eq_ind_r nat d0 (\lambda (n0: nat).(drop (S +O) n0 a0 a)) H5 n H6) in (let H8 \def (eq_ind_r nat d0 (\lambda (n0: +nat).(csubst1 n0 u0 c0 a0)) H4 n H6) in (let H9 \def (eq_ind_r nat d0 +(\lambda (n0: nat).(getl n0 c0 (CHead e (Bind Abbr) u0))) H3 n H6) in (eq_ind +nat n (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 +n0 u0 (TLRef n) (lift (S O) n0 y1)))) (\lambda (_: T).(\lambda (y2: +T).(subst1 n0 u0 (lift (S n) O t) (lift (S O) n0 y2)))) (\lambda (y1: +T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H10 \def (eq_ind C (CHead d +(Bind Abbr) u) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead e (Bind Abbr) u0) +(getl_mono c0 (CHead d (Bind Abbr) u) n H0 (CHead e (Bind Abbr) u0) H9)) in +(let H11 \def (f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda +(_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) +(CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d (Bind +Abbr) u) n H0 (CHead e (Bind Abbr) u0) H9)) in ((let H12 \def (f_equal C T +(\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) +\Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind Abbr) u) +(CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d (Bind Abbr) u) n H0 (CHead e +(Bind Abbr) u0) H9)) in (\lambda (H13: (eq C d e)).(let H14 \def (eq_ind_r T +u0 (\lambda (t0: T).(getl n c0 (CHead e (Bind Abbr) t0))) H10 u H12) in (let +H15 \def (eq_ind_r T u0 (\lambda (t0: T).(csubst1 n t0 c0 a0)) H8 u H12) in +(eq_ind T u (\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: +T).(subst1 n t0 (TLRef n) (lift (S O) n y1)))) (\lambda (_: T).(\lambda (y2: +T).(subst1 n t0 (lift (S n) O t) (lift (S O) n y2)))) (\lambda (y1: +T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H16 \def (eq_ind_r C e (\lambda +(c1: C).(getl n c0 (CHead c1 (Bind Abbr) u))) H14 d H13) in (ex3_2_intro T T +(\lambda (y1: T).(\lambda (_: T).(subst1 n u (TLRef n) (lift (S O) n y1)))) +(\lambda (_: T).(\lambda (y2: T).(subst1 n u (lift (S n) O t) (lift (S O) n +y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (lift n O u) (lift +n O t) (subst1_single n u (TLRef n) (lift (S O) n (lift n O u)) (eq_ind_r T +(lift (plus (S O) n) O u) (\lambda (t0: T).(subst0 n u (TLRef n) t0)) +(subst0_lref u n) (lift (S O) n (lift n O u)) (lift_free u n (S O) O n (le_n +(plus O n)) (le_O_n n)))) (eq_ind_r T (lift (plus (S O) n) O t) (\lambda (t0: +T).(subst1 n u (lift (S n) O t) t0)) (subst1_refl n u (lift (S n) O t)) (lift +(S O) n (lift n O t)) (lift_free t n (S O) O n (le_n (plus O n)) (le_O_n n))) +(ty3_lift g d u t H1 a O n (getl_conf_ge_drop Abbr a0 d u n (csubst1_getl_ge +n n (le_n n) c0 a0 u H15 (CHead d (Bind Abbr) u) H16) a H7)))) u0 H12))))) +H11))) d0 H6))))) (\lambda (H6: (lt d0 n)).(eq_ind_r nat (S (plus O (minus n +(S O)))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: +T).(subst1 d0 u0 (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda +(y2: T).(subst1 d0 u0 (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: +T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind nat (plus (S O) (minus n (S +O))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 +d0 u0 (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: +T).(subst1 d0 u0 (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: +T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r nat (plus (minus n (S O)) +(S O)) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 +d0 u0 (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: +T).(subst1 d0 u0 (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: +T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1: +T).(\lambda (_: T).(subst1 d0 u0 (TLRef (plus (minus n (S O)) (S O))) (lift +(S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O +t) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) +(TLRef (minus n (S O))) (lift n O t) (eq_ind_r T (TLRef (plus (minus n (S O)) +(S O))) (\lambda (t0: T).(subst1 d0 u0 (TLRef (plus (minus n (S O)) (S O))) +t0)) (subst1_refl d0 u0 (TLRef (plus (minus n (S O)) (S O)))) (lift (S O) d0 +(TLRef (minus n (S O)))) (lift_lref_ge (minus n (S O)) (S O) d0 (lt_le_minus +d0 n H6))) (eq_ind_r T (lift (plus (S O) n) O t) (\lambda (t0: T).(subst1 d0 +u0 (lift (S n) O t) t0)) (subst1_refl d0 u0 (lift (S n) O t)) (lift (S O) d0 +(lift n O t)) (lift_free t n (S O) O d0 (le_S_n d0 (plus O n) (le_S (S d0) +(plus O n) H6)) (le_O_n d0))) (eq_ind_r nat (S (minus n (S O))) (\lambda (n0: +nat).(ty3 g a (TLRef (minus n (S O))) (lift n0 O t))) (ty3_abbr g (minus n (S +O)) a d u (getl_drop_conf_ge n (CHead d (Bind Abbr) u) a0 (csubst1_getl_ge d0 +n (le_S_n d0 n (le_S (S d0) n H6)) c0 a0 u0 H4 (CHead d (Bind Abbr) u) H0) a +(S O) d0 H5 (eq_ind_r nat (plus (S O) d0) (\lambda (n0: nat).(le n0 n)) H6 +(plus d0 (S O)) (plus_sym d0 (S O)))) t H1) n (minus_x_SO n (le_lt_trans O d0 +n (le_O_n d0) H6)))) (plus (S O) (minus n (S O))) (plus_sym (S O) (minus n (S +O)))) (S (plus O (minus n (S O)))) (refl_equal nat (S (plus O (minus n (S +O)))))) n (lt_plus_minus O n (le_lt_trans O d0 n (le_O_n d0) +H6))))))))))))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: +C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abst) +u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u t)).(\lambda (H2: ((\forall (e: +C).(\forall (u0: T).(\forall (d0: nat).((getl d0 d (CHead e (Bind Abbr) u0)) +\to (\forall (a0: C).((csubst1 d0 u0 d a0) \to (\forall (a: C).((drop (S O) +d0 a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 u +(lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 t (lift +(S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 +y2)))))))))))))).(\lambda (e: C).(\lambda (u0: T).(\lambda (d0: nat).(\lambda +(H3: (getl d0 c0 (CHead e (Bind Abbr) u0))).(\lambda (a0: C).(\lambda (H4: +(csubst1 d0 u0 c0 a0)).(\lambda (a: C).(\lambda (H5: (drop (S O) d0 a0 +a)).(lt_eq_gt_e n d0 (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 +u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 +d0 u0 (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: +T).(ty3 g a y1 y2)))) (\lambda (H6: (lt n d0)).(let H7 \def (eq_ind nat +(minus d0 n) (\lambda (n0: nat).(getl n0 (CHead d (Bind Abst) u) (CHead e +(Bind Abbr) u0))) (getl_conf_le d0 (CHead e (Bind Abbr) u0) c0 H3 (CHead d +(Bind Abst) u) n H0 (le_S_n n d0 (le_S (S n) d0 H6))) (S (minus d0 (S n))) +(minus_x_Sy d0 n H6)) in (ex2_ind C (\lambda (e2: C).(csubst1 (minus d0 n) u0 +(CHead d (Bind Abst) u) e2)) (\lambda (e2: C).(getl n a0 e2)) (ex3_2 T T +(\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 +y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O u) (lift +(S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda +(x: C).(\lambda (H8: (csubst1 (minus d0 n) u0 (CHead d (Bind Abst) u) +x)).(\lambda (H9: (getl n a0 x)).(let H10 \def (eq_ind nat (minus d0 n) +(\lambda (n0: nat).(csubst1 n0 u0 (CHead d (Bind Abst) u) x)) H8 (S (minus d0 +(S n))) (minus_x_Sy d0 n H6)) in (let H11 \def (csubst1_gen_head (Bind Abst) +d x u u0 (minus d0 (S n)) H10) in (ex3_2_ind T C (\lambda (u2: T).(\lambda +(c2: C).(eq C x (CHead c2 (Bind Abst) u2)))) (\lambda (u2: T).(\lambda (_: +C).(subst1 (minus d0 (S n)) u0 u u2))) (\lambda (_: T).(\lambda (c2: +C).(csubst1 (minus d0 (S n)) u0 d c2))) (ex3_2 T T (\lambda (y1: T).(\lambda +(_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda +(y2: T).(subst1 d0 u0 (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: +T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1: +C).(\lambda (H12: (eq C x (CHead x1 (Bind Abst) x0))).(\lambda (H13: (subst1 +(minus d0 (S n)) u0 u x0)).(\lambda (H14: (csubst1 (minus d0 (S n)) u0 d +x1)).(let H15 \def (eq_ind C x (\lambda (c1: C).(getl n a0 c1)) H9 (CHead x1 +(Bind Abst) x0) H12) in (let H16 \def (eq_ind nat d0 (\lambda (n0: nat).(drop +(S O) n0 a0 a)) H5 (S (plus n (minus d0 (S n)))) (lt_plus_minus n d0 H6)) in +(ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T x0 (lift (S O) (minus d0 +(S n)) v)))) (\lambda (v: T).(\lambda (e0: C).(getl n a (CHead e0 (Bind Abst) +v)))) (\lambda (_: T).(\lambda (e0: C).(drop (S O) (minus d0 (S n)) x1 e0))) +(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S +O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O u) +(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) +(\lambda (x2: T).(\lambda (x3: C).(\lambda (H17: (eq T x0 (lift (S O) (minus +d0 (S n)) x2))).(\lambda (H18: (getl n a (CHead x3 (Bind Abst) x2))).(\lambda +(H19: (drop (S O) (minus d0 (S n)) x1 x3)).(let H20 \def (eq_ind T x0 +(\lambda (t0: T).(subst1 (minus d0 (S n)) u0 u t0)) H13 (lift (S O) (minus d0 +(S n)) x2) H17) in (let H21 \def (H2 e u0 (minus d0 (S n)) (getl_gen_S (Bind +Abst) d (CHead e (Bind Abbr) u0) u (minus d0 (S n)) H7) x1 H14 x3 H19) in +(ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(subst1 (minus d0 (S n)) u0 u +(lift (S O) (minus d0 (S n)) y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 +(minus d0 (S n)) u0 t (lift (S O) (minus d0 (S n)) y2)))) (\lambda (y1: +T).(\lambda (y2: T).(ty3 g x3 y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda +(_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda +(y2: T).(subst1 d0 u0 (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: +T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x4: T).(\lambda (x5: +T).(\lambda (H22: (subst1 (minus d0 (S n)) u0 u (lift (S O) (minus d0 (S n)) +x4))).(\lambda (_: (subst1 (minus d0 (S n)) u0 t (lift (S O) (minus d0 (S n)) +x5))).(\lambda (H24: (ty3 g x3 x4 x5)).(let H25 \def (eq_ind T x4 (\lambda +(t0: T).(ty3 g x3 t0 x5)) H24 x2 (subst1_confluence_lift u x4 u0 (minus d0 (S +n)) H22 x2 H20)) in (eq_ind_r nat (plus (minus d0 (S n)) (S n)) (\lambda (n0: +nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) +(lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 n0 u0 (lift (S +n) O u) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 +y2))))) (eq_ind_r nat (plus (S n) (minus d0 (S n))) (\lambda (n0: nat).(ex3_2 +T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 +y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 (plus (minus d0 (S n)) (S n)) +u0 (lift (S n) O u) (lift (S O) n0 y2)))) (\lambda (y1: T).(\lambda (y2: +T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: +T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda +(y2: T).(subst1 (plus (minus d0 (S n)) (S n)) u0 (lift (S n) O u) (lift (S O) +(plus (S n) (minus d0 (S n))) y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g +a y1 y2))) (TLRef n) (lift (S n) O x2) (eq_ind_r T (TLRef n) (\lambda (t0: +T).(subst1 d0 u0 (TLRef n) t0)) (subst1_refl d0 u0 (TLRef n)) (lift (S O) d0 +(TLRef n)) (lift_lref_lt n (S O) d0 H6)) (eq_ind_r T (lift (S n) O (lift (S +O) (minus d0 (S n)) x2)) (\lambda (t0: T).(subst1 (plus (minus d0 (S n)) (S +n)) u0 (lift (S n) O u) t0)) (subst1_lift_ge u (lift (S O) (minus d0 (S n)) +x2) u0 (minus d0 (S n)) (S n) H20 O (le_O_n (minus d0 (S n)))) (lift (S O) +(plus (S n) (minus d0 (S n))) (lift (S n) O x2)) (lift_d x2 (S O) (S n) +(minus d0 (S n)) O (le_O_n (minus d0 (S n))))) (ty3_abst g n a x3 x2 H18 x5 +H25)) d0 (le_plus_minus (S n) d0 H6)) d0 (le_plus_minus_sym (S n) d0 +H6)))))))) H21)))))))) (getl_drop_conf_lt Abst a0 x1 x0 n H15 a (S O) (minus +d0 (S n)) H16))))))))) H11)))))) (csubst1_getl_lt d0 n H6 c0 a0 u0 H4 (CHead +d (Bind Abst) u) H0)))) (\lambda (H6: (eq nat n d0)).(let H7 \def (eq_ind_r +nat d0 (\lambda (n0: nat).(drop (S O) n0 a0 a)) H5 n H6) in (let H8 \def +(eq_ind_r nat d0 (\lambda (n0: nat).(csubst1 n0 u0 c0 a0)) H4 n H6) in (let +H9 \def (eq_ind_r nat d0 (\lambda (n0: nat).(getl n0 c0 (CHead e (Bind Abbr) +u0))) H3 n H6) in (eq_ind nat n (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: +T).(\lambda (_: T).(subst1 n0 u0 (TLRef n) (lift (S O) n0 y1)))) (\lambda (_: +T).(\lambda (y2: T).(subst1 n0 u0 (lift (S n) O u) (lift (S O) n0 y2)))) +(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H10 \def (eq_ind C +(CHead d (Bind Abst) u) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead e (Bind +Abbr) u0) (getl_mono c0 (CHead d (Bind Abst) u) n H0 (CHead e (Bind Abbr) u0) +H9)) in (let H11 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (ee: +C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow +False | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) +with [(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with +[Abbr \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | +(Flat _) \Rightarrow False])])) I (CHead e (Bind Abbr) u0) (getl_mono c0 +(CHead d (Bind Abst) u) n H0 (CHead e (Bind Abbr) u0) H9)) in (False_ind +(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 n u0 (TLRef n) (lift (S +O) n y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 n u0 (lift (S n) O u) +(lift (S O) n y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) +H11))) d0 H6))))) (\lambda (H6: (lt d0 n)).(eq_ind_r nat (S (plus O (minus n +(S O)))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: +T).(subst1 d0 u0 (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda +(y2: T).(subst1 d0 u0 (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: +T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind nat (plus (S O) (minus n (S +O))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 +d0 u0 (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: +T).(subst1 d0 u0 (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: +T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r nat (plus (minus n (S O)) +(S O)) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 +d0 u0 (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: +T).(subst1 d0 u0 (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: +T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1: +T).(\lambda (_: T).(subst1 d0 u0 (TLRef (plus (minus n (S O)) (S O))) (lift +(S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O +u) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) +(TLRef (minus n (S O))) (lift n O u) (eq_ind_r T (TLRef (plus (minus n (S O)) +(S O))) (\lambda (t0: T).(subst1 d0 u0 (TLRef (plus (minus n (S O)) (S O))) +t0)) (subst1_refl d0 u0 (TLRef (plus (minus n (S O)) (S O)))) (lift (S O) d0 +(TLRef (minus n (S O)))) (lift_lref_ge (minus n (S O)) (S O) d0 (lt_le_minus +d0 n H6))) (eq_ind_r T (lift (plus (S O) n) O u) (\lambda (t0: T).(subst1 d0 +u0 (lift (S n) O u) t0)) (subst1_refl d0 u0 (lift (S n) O u)) (lift (S O) d0 +(lift n O u)) (lift_free u n (S O) O d0 (le_S_n d0 (plus O n) (le_S (S d0) +(plus O n) H6)) (le_O_n d0))) (eq_ind_r nat (S (minus n (S O))) (\lambda (n0: +nat).(ty3 g a (TLRef (minus n (S O))) (lift n0 O u))) (ty3_abst g (minus n (S +O)) a d u (getl_drop_conf_ge n (CHead d (Bind Abst) u) a0 (csubst1_getl_ge d0 +n (le_S_n d0 n (le_S (S d0) n H6)) c0 a0 u0 H4 (CHead d (Bind Abst) u) H0) a +(S O) d0 H5 (eq_ind_r nat (plus (S O) d0) (\lambda (n0: nat).(le n0 n)) H6 +(plus d0 (S O)) (plus_sym d0 (S O)))) t H1) n (minus_x_SO n (le_lt_trans O d0 +n (le_O_n d0) H6)))) (plus (S O) (minus n (S O))) (plus_sym (S O) (minus n (S +O)))) (S (plus O (minus n (S O)))) (refl_equal nat (S (plus O (minus n (S +O)))))) n (lt_plus_minus O n (le_lt_trans O d0 n (le_O_n d0) +H6))))))))))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (t: +T).(\lambda (_: (ty3 g c0 u t)).(\lambda (H1: ((\forall (e: C).(\forall (u0: +T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u0)) \to (\forall (a0: +C).((csubst1 d u0 c0 a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (ex3_2 +T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u (lift (S O) d y1)))) +(\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t (lift (S O) d y2)))) (\lambda +(y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))))).(\lambda (b: +B).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) +u) t3 t4)).(\lambda (H3: ((\forall (e: C).(\forall (u0: T).(\forall (d: +nat).((getl d (CHead c0 (Bind b) u) (CHead e (Bind Abbr) u0)) \to (\forall +(a0: C).((csubst1 d u0 (CHead c0 (Bind b) u) a0) \to (\forall (a: C).((drop +(S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 t3 +(lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t4 (lift +(S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 +y2)))))))))))))).(\lambda (e: C).(\lambda (u0: T).(\lambda (d: nat).(\lambda +(H4: (getl d c0 (CHead e (Bind Abbr) u0))).(\lambda (a0: C).(\lambda (H5: +(csubst1 d u0 c0 a0)).(\lambda (a: C).(\lambda (H6: (drop (S O) d a0 a)).(let +H7 \def (H1 e u0 d H4 a0 H5 a H6) in (ex3_2_ind T T (\lambda (y1: T).(\lambda +(_: T).(subst1 d u0 u (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: +T).(subst1 d u0 t (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 +g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 (THead +(Bind b) u t3) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 +d u0 (THead (Bind b) u t4) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda +(y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H8: +(subst1 d u0 u (lift (S O) d x0))).(\lambda (_: (subst1 d u0 t (lift (S O) d +x1))).(\lambda (H10: (ty3 g a x0 x1)).(let H11 \def (H3 e u0 (S d) (getl_head +(Bind b) d c0 (CHead e (Bind Abbr) u0) H4 u) (CHead a0 (Bind b) (lift (S O) d +x0)) (csubst1_bind b d u0 u (lift (S O) d x0) H8 c0 a0 H5) (CHead a (Bind b) +x0) (drop_skip_bind (S O) d a0 a H6 b x0)) in (ex3_2_ind T T (\lambda (y1: +T).(\lambda (_: T).(subst1 (S d) u0 t3 (lift (S O) (S d) y1)))) (\lambda (_: +T).(\lambda (y2: T).(subst1 (S d) u0 t4 (lift (S O) (S d) y2)))) (\lambda +(y1: T).(\lambda (y2: T).(ty3 g (CHead a (Bind b) x0) y1 y2))) (ex3_2 T T +(\lambda (y1: T).(\lambda (_: T).(subst1 d u0 (THead (Bind b) u t3) (lift (S +O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Bind b) u +t4) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) +(\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (subst1 (S d) u0 t3 (lift (S +O) (S d) x2))).(\lambda (H13: (subst1 (S d) u0 t4 (lift (S O) (S d) +x3))).(\lambda (H14: (ty3 g (CHead a (Bind b) x0) x2 x3)).(ex3_2_intro T T +(\lambda (y1: T).(\lambda (_: T).(subst1 d u0 (THead (Bind b) u t3) (lift (S +O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Bind b) u +t4) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) +(THead (Bind b) x0 x2) (THead (Bind b) x0 x3) (eq_ind_r T (THead (Bind b) +(lift (S O) d x0) (lift (S O) (S d) x2)) (\lambda (t0: T).(subst1 d u0 (THead +(Bind b) u t3) t0)) (subst1_head u0 u (lift (S O) d x0) d H8 (Bind b) t3 +(lift (S O) (S d) x2) H12) (lift (S O) d (THead (Bind b) x0 x2)) (lift_bind b +x0 x2 (S O) d)) (eq_ind_r T (THead (Bind b) (lift (S O) d x0) (lift (S O) (S +d) x3)) (\lambda (t0: T).(subst1 d u0 (THead (Bind b) u t4) t0)) (subst1_head +u0 u (lift (S O) d x0) d H8 (Bind b) t4 (lift (S O) (S d) x3) H13) (lift (S +O) d (THead (Bind b) x0 x3)) (lift_bind b x0 x3 (S O) d)) (ty3_bind g a x0 x1 +H10 b x2 x3 H14))))))) H11))))))) H7)))))))))))))))))))) (\lambda (c0: +C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda (H1: +((\forall (e: C).(\forall (u0: T).(\forall (d: nat).((getl d c0 (CHead e +(Bind Abbr) u0)) \to (\forall (a0: C).((csubst1 d u0 c0 a0) \to (\forall (a: +C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: +T).(subst1 d u0 w (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: +T).(subst1 d u0 u (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 +g a y1 y2)))))))))))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g +c0 v (THead (Bind Abst) u t))).(\lambda (H3: ((\forall (e: C).(\forall (u0: +T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u0)) \to (\forall (a0: +C).((csubst1 d u0 c0 a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (ex3_2 +T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 v (lift (S O) d y1)))) +(\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Bind Abst) u t) (lift +(S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 +y2)))))))))))))).(\lambda (e: C).(\lambda (u0: T).(\lambda (d: nat).(\lambda +(H4: (getl d c0 (CHead e (Bind Abbr) u0))).(\lambda (a0: C).(\lambda (H5: +(csubst1 d u0 c0 a0)).(\lambda (a: C).(\lambda (H6: (drop (S O) d a0 a)).(let +H7 \def (H3 e u0 d H4 a0 H5 a H6) in (ex3_2_ind T T (\lambda (y1: T).(\lambda +(_: T).(subst1 d u0 v (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: +T).(subst1 d u0 (THead (Bind Abst) u t) (lift (S O) d y2)))) (\lambda (y1: +T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda +(_: T).(subst1 d u0 (THead (Flat Appl) w v) (lift (S O) d y1)))) (\lambda (_: +T).(\lambda (y2: T).(subst1 d u0 (THead (Flat Appl) w (THead (Bind Abst) u +t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) +(\lambda (x0: T).(\lambda (x1: T).(\lambda (H8: (subst1 d u0 v (lift (S O) d +x0))).(\lambda (H9: (subst1 d u0 (THead (Bind Abst) u t) (lift (S O) d +x1))).(\lambda (H10: (ty3 g a x0 x1)).(let H11 \def (H1 e u0 d H4 a0 H5 a H6) +in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 w (lift (S O) +d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 u (lift (S O) d y2)))) +(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: +T).(\lambda (_: T).(subst1 d u0 (THead (Flat Appl) w v) (lift (S O) d y1)))) +(\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Flat Appl) w (THead +(Bind Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 +g a y1 y2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (subst1 d u0 w +(lift (S O) d x2))).(\lambda (H13: (subst1 d u0 u (lift (S O) d +x3))).(\lambda (H14: (ty3 g a x2 x3)).(let H_x \def (subst1_gen_head (Bind +Abst) u0 u t (lift (S O) d x1) d H9) in (let H15 \def H_x in (ex3_2_ind T T +(\lambda (u2: T).(\lambda (t3: T).(eq T (lift (S O) d x1) (THead (Bind Abst) +u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1 d u0 u u2))) (\lambda (_: +T).(\lambda (t3: T).(subst1 (S d) u0 t t3))) (ex3_2 T T (\lambda (y1: +T).(\lambda (_: T).(subst1 d u0 (THead (Flat Appl) w v) (lift (S O) d y1)))) +(\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Flat Appl) w (THead +(Bind Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 +g a y1 y2)))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H16: (eq T (lift (S +O) d x1) (THead (Bind Abst) x4 x5))).(\lambda (H17: (subst1 d u0 u +x4)).(\lambda (H18: (subst1 (S d) u0 t x5)).(let H19 \def (sym_eq T (lift (S +O) d x1) (THead (Bind Abst) x4 x5) H16) in (ex3_2_ind T T (\lambda (y: +T).(\lambda (z: T).(eq T x1 (THead (Bind Abst) y z)))) (\lambda (y: +T).(\lambda (_: T).(eq T x4 (lift (S O) d y)))) (\lambda (_: T).(\lambda (z: +T).(eq T x5 (lift (S O) (S d) z)))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: +T).(subst1 d u0 (THead (Flat Appl) w v) (lift (S O) d y1)))) (\lambda (_: +T).(\lambda (y2: T).(subst1 d u0 (THead (Flat Appl) w (THead (Bind Abst) u +t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) +(\lambda (x6: T).(\lambda (x7: T).(\lambda (H20: (eq T x1 (THead (Bind Abst) +x6 x7))).(\lambda (H21: (eq T x4 (lift (S O) d x6))).(\lambda (H22: (eq T x5 +(lift (S O) (S d) x7))).(let H23 \def (eq_ind T x5 (\lambda (t0: T).(subst1 +(S d) u0 t t0)) H18 (lift (S O) (S d) x7) H22) in (let H24 \def (eq_ind T x4 +(\lambda (t0: T).(subst1 d u0 u t0)) H17 (lift (S O) d x6) H21) in (let H25 +\def (eq_ind T x1 (\lambda (t0: T).(ty3 g a x0 t0)) H10 (THead (Bind Abst) x6 +x7) H20) in (let H26 \def (eq_ind T x6 (\lambda (t0: T).(ty3 g a x0 (THead +(Bind Abst) t0 x7))) H25 x3 (subst1_confluence_lift u x6 u0 d H24 x3 H13)) in +(ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 (THead (Flat +Appl) w v) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 +(THead (Flat Appl) w (THead (Bind Abst) u t)) (lift (S O) d y2)))) (\lambda +(y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (THead (Flat Appl) x2 x0) (THead +(Flat Appl) x2 (THead (Bind Abst) x3 x7)) (eq_ind_r T (THead (Flat Appl) +(lift (S O) d x2) (lift (S O) d x0)) (\lambda (t0: T).(subst1 d u0 (THead +(Flat Appl) w v) t0)) (subst1_head u0 w (lift (S O) d x2) d H12 (Flat Appl) v +(lift (S O) d x0) H8) (lift (S O) d (THead (Flat Appl) x2 x0)) (lift_flat +Appl x2 x0 (S O) d)) (eq_ind_r T (THead (Flat Appl) (lift (S O) d x2) (lift +(S O) d (THead (Bind Abst) x3 x7))) (\lambda (t0: T).(subst1 d u0 (THead +(Flat Appl) w (THead (Bind Abst) u t)) t0)) (subst1_head u0 w (lift (S O) d +x2) d H12 (Flat Appl) (THead (Bind Abst) u t) (lift (S O) d (THead (Bind +Abst) x3 x7)) (eq_ind_r T (THead (Bind Abst) (lift (S O) d x3) (lift (S O) (S +d) x7)) (\lambda (t0: T).(subst1 (s (Flat Appl) d) u0 (THead (Bind Abst) u t) +t0)) (subst1_head u0 u (lift (S O) d x3) (s (Flat Appl) d) H13 (Bind Abst) t +(lift (S O) (S d) x7) H23) (lift (S O) d (THead (Bind Abst) x3 x7)) +(lift_bind Abst x3 x7 (S O) d))) (lift (S O) d (THead (Flat Appl) x2 (THead +(Bind Abst) x3 x7))) (lift_flat Appl x2 (THead (Bind Abst) x3 x7) (S O) d)) +(ty3_appl g a x2 x3 H14 x0 x7 H26))))))))))) (lift_gen_bind Abst x4 x5 x1 (S +O) d H19)))))))) H15)))))))) H11))))))) H7))))))))))))))))))) (\lambda (c0: +C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (ty3 g c0 t3 t4)).(\lambda +(H1: ((\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead e +(Bind Abbr) u)) \to (\forall (a0: C).((csubst1 d u c0 a0) \to (\forall (a: +C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: +T).(subst1 d u t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: +T).(subst1 d u t4 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 +g a y1 y2)))))))))))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t4 +t0)).(\lambda (H3: ((\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl +d c0 (CHead e (Bind Abbr) u)) \to (\forall (a0: C).((csubst1 d u c0 a0) \to +(\forall (a: C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda +(_: T).(subst1 d u t4 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: +T).(subst1 d u t0 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 +g a y1 y2)))))))))))))).(\lambda (e: C).(\lambda (u: T).(\lambda (d: +nat).(\lambda (H4: (getl d c0 (CHead e (Bind Abbr) u))).(\lambda (a0: +C).(\lambda (H5: (csubst1 d u c0 a0)).(\lambda (a: C).(\lambda (H6: (drop (S +O) d a0 a)).(let H7 \def (H3 e u d H4 a0 H5 a H6) in (ex3_2_ind T T (\lambda +(y1: T).(\lambda (_: T).(subst1 d u t4 (lift (S O) d y1)))) (\lambda (_: +T).(\lambda (y2: T).(subst1 d u t0 (lift (S O) d y2)))) (\lambda (y1: +T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda +(_: T).(subst1 d u (THead (Flat Cast) t4 t3) (lift (S O) d y1)))) (\lambda +(_: T).(\lambda (y2: T).(subst1 d u (THead (Flat Cast) t0 t4) (lift (S O) d +y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: +T).(\lambda (x1: T).(\lambda (H8: (subst1 d u t4 (lift (S O) d x0))).(\lambda +(H9: (subst1 d u t0 (lift (S O) d x1))).(\lambda (H10: (ty3 g a x0 x1)).(let +H11 \def (H1 e u d H4 a0 H5 a H6) in (ex3_2_ind T T (\lambda (y1: T).(\lambda +(_: T).(subst1 d u t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: +T).(subst1 d u t4 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 +g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u (THead +(Flat Cast) t4 t3) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: +T).(subst1 d u (THead (Flat Cast) t0 t4) (lift (S O) d y2)))) (\lambda (y1: +T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: T).(\lambda (x3: +T).(\lambda (H12: (subst1 d u t3 (lift (S O) d x2))).(\lambda (H13: (subst1 d +u t4 (lift (S O) d x3))).(\lambda (H14: (ty3 g a x2 x3)).(let H15 \def +(eq_ind T x3 (\lambda (t: T).(ty3 g a x2 t)) H14 x0 (subst1_confluence_lift +t4 x3 u d H13 x0 H8)) in (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: +T).(subst1 d u (THead (Flat Cast) t4 t3) (lift (S O) d y1)))) (\lambda (_: +T).(\lambda (y2: T).(subst1 d u (THead (Flat Cast) t0 t4) (lift (S O) d +y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (THead (Flat Cast) +x0 x2) (THead (Flat Cast) x1 x0) (eq_ind_r T (THead (Flat Cast) (lift (S O) d +x0) (lift (S O) d x2)) (\lambda (t: T).(subst1 d u (THead (Flat Cast) t4 t3) +t)) (subst1_head u t4 (lift (S O) d x0) d H8 (Flat Cast) t3 (lift (S O) d x2) +H12) (lift (S O) d (THead (Flat Cast) x0 x2)) (lift_flat Cast x0 x2 (S O) d)) +(eq_ind_r T (THead (Flat Cast) (lift (S O) d x1) (lift (S O) d x0)) (\lambda +(t: T).(subst1 d u (THead (Flat Cast) t0 t4) t)) (subst1_head u t0 (lift (S +O) d x1) d H9 (Flat Cast) t4 (lift (S O) d x0) H8) (lift (S O) d (THead (Flat +Cast) x1 x0)) (lift_flat Cast x1 x0 (S O) d)) (ty3_cast g a x2 x0 H15 x1 +H10)))))))) H11))))))) H7)))))))))))))))))) c t1 t2 H))))). +(* COMMENTS +Initial nodes: 12848 +END *) + +theorem ty3_gen_cvoid: + \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c +t1 t2) \to (\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c +(CHead e (Bind Void) u)) \to (\forall (a: C).((drop (S O) d c a) \to (ex3_2 T +T (\lambda (y1: T).(\lambda (_: T).(eq T t1 (lift (S O) d y1)))) (\lambda (_: +T).(\lambda (y2: T).(eq T t2 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda +(y2: T).(ty3 g a y1 y2)))))))))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda +(H: (ty3 g c t1 t2)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda +(t0: T).(\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead +e (Bind Void) u)) \to (\forall (a: C).((drop (S O) d c0 a) \to (ex3_2 T T +(\lambda (y1: T).(\lambda (_: T).(eq T t (lift (S O) d y1)))) (\lambda (_: +T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda +(y2: T).(ty3 g a y1 y2))))))))))))) (\lambda (c0: C).(\lambda (t3: +T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 t3 t)).(\lambda (H1: ((\forall (e: +C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Void) u)) \to +(\forall (a: C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda +(_: T).(eq T t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t +(lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 +y2)))))))))))).(\lambda (u: T).(\lambda (t4: T).(\lambda (H2: (ty3 g c0 u +t4)).(\lambda (H3: ((\forall (e: C).(\forall (u0: T).(\forall (d: nat).((getl +d c0 (CHead e (Bind Void) u0)) \to (\forall (a: C).((drop (S O) d c0 a) \to +(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T u (lift (S O) d y1)))) +(\lambda (_: T).(\lambda (y2: T).(eq T t4 (lift (S O) d y2)))) (\lambda (y1: +T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (H4: (pc3 c0 t4 +t3)).(\lambda (e: C).(\lambda (u0: T).(\lambda (d: nat).(\lambda (H5: (getl d +c0 (CHead e (Bind Void) u0))).(\lambda (a: C).(\lambda (H6: (drop (S O) d c0 +a)).(let H7 \def (H3 e u0 d H5 a H6) in (ex3_2_ind T T (\lambda (y1: +T).(\lambda (_: T).(eq T u (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: +T).(eq T t4 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a +y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T u (lift (S O) d +y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t3 (lift (S O) d y2)))) +(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: +T).(\lambda (x1: T).(\lambda (H8: (eq T u (lift (S O) d x0))).(\lambda (H9: +(eq T t4 (lift (S O) d x1))).(\lambda (H10: (ty3 g a x0 x1)).(let H11 \def +(eq_ind T t4 (\lambda (t0: T).(pc3 c0 t0 t3)) H4 (lift (S O) d x1) H9) in +(let H12 \def (eq_ind T t4 (\lambda (t0: T).(ty3 g c0 u t0)) H2 (lift (S O) d +x1) H9) in (let H13 \def (eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 (lift (S +O) d x1))) H12 (lift (S O) d x0) H8) in (eq_ind_r T (lift (S O) d x0) +(\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t0 (lift +(S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t3 (lift (S O) d y2)))) +(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H14 \def (H1 e u0 +d H5 a H6) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T t3 (lift +(S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d y2)))) +(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: +T).(\lambda (_: T).(eq T (lift (S O) d x0) (lift (S O) d y1)))) (\lambda (_: +T).(\lambda (y2: T).(eq T t3 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda +(y2: T).(ty3 g a y1 y2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H15: +(eq T t3 (lift (S O) d x2))).(\lambda (H16: (eq T t (lift (S O) d +x3))).(\lambda (H17: (ty3 g a x2 x3)).(let H18 \def (eq_ind T t (\lambda (t0: +T).(ty3 g c0 t3 t0)) H0 (lift (S O) d x3) H16) in (let H19 \def (eq_ind T t3 +(\lambda (t0: T).(ty3 g c0 t0 (lift (S O) d x3))) H18 (lift (S O) d x2) H15) +in (let H20 \def (eq_ind T t3 (\lambda (t0: T).(pc3 c0 (lift (S O) d x1) t0)) +H11 (lift (S O) d x2) H15) in (eq_ind_r T (lift (S O) d x2) (\lambda (t0: +T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (lift (S O) d x0) (lift +(S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2)))) +(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T +(\lambda (y1: T).(\lambda (_: T).(eq T (lift (S O) d x0) (lift (S O) d y1)))) +(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d x2) (lift (S O) d y2)))) +(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) x0 x2 (refl_equal T (lift +(S O) d x0)) (refl_equal T (lift (S O) d x2)) (ty3_conv g a x2 x3 H17 x0 x1 +H10 (pc3_gen_lift c0 x1 x2 (S O) d H20 a H6))) t3 H15))))))))) H14)) u +H8))))))))) H7)))))))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda +(e: C).(\lambda (u: T).(\lambda (d: nat).(\lambda (_: (getl d c0 (CHead e +(Bind Void) u))).(\lambda (a: C).(\lambda (_: (drop (S O) d c0 +a)).(ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(eq T (TSort m) (lift +(S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (TSort (next g m)) +(lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) +(TSort m) (TSort (next g m)) (eq_ind_r T (TSort m) (\lambda (t: T).(eq T +(TSort m) t)) (refl_equal T (TSort m)) (lift (S O) d (TSort m)) (lift_sort m +(S O) d)) (eq_ind_r T (TSort (next g m)) (\lambda (t: T).(eq T (TSort (next g +m)) t)) (refl_equal T (TSort (next g m))) (lift (S O) d (TSort (next g m))) +(lift_sort (next g m) (S O) d)) (ty3_sort g a m)))))))))) (\lambda (n: +nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n +c0 (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u +t)).(\lambda (H2: ((\forall (e: C).(\forall (u0: T).(\forall (d0: nat).((getl +d0 d (CHead e (Bind Void) u0)) \to (\forall (a: C).((drop (S O) d0 d a) \to +(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T u (lift (S O) d0 y1)))) +(\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d0 y2)))) (\lambda (y1: +T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (e: C).(\lambda (u0: +T).(\lambda (d0: nat).(\lambda (H3: (getl d0 c0 (CHead e (Bind Void) +u0))).(\lambda (a: C).(\lambda (H4: (drop (S O) d0 c0 a)).(lt_eq_gt_e n d0 +(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 +y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O t) (lift (S O) d0 +y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (H5: (lt +n d0)).(let H6 \def (eq_ind nat (minus d0 n) (\lambda (n0: nat).(getl n0 +(CHead d (Bind Abbr) u) (CHead e (Bind Void) u0))) (getl_conf_le d0 (CHead e +(Bind Void) u0) c0 H3 (CHead d (Bind Abbr) u) n H0 (le_S_n n d0 (le_S (S n) +d0 H5))) (S (minus d0 (S n))) (minus_x_Sy d0 n H5)) in (let H7 \def (eq_ind +nat d0 (\lambda (n0: nat).(drop (S O) n0 c0 a)) H4 (S (plus n (minus d0 (S +n)))) (lt_plus_minus n d0 H5)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: +C).(eq T u (lift (S O) (minus d0 (S n)) v)))) (\lambda (v: T).(\lambda (e0: +C).(getl n a (CHead e0 (Bind Abbr) v)))) (\lambda (_: T).(\lambda (e0: +C).(drop (S O) (minus d0 (S n)) d e0))) (ex3_2 T T (\lambda (y1: T).(\lambda +(_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: +T).(eq T (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda +(y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1: C).(\lambda (H8: +(eq T u (lift (S O) (minus d0 (S n)) x0))).(\lambda (H9: (getl n a (CHead x1 +(Bind Abbr) x0))).(\lambda (H10: (drop (S O) (minus d0 (S n)) d x1)).(let H11 +\def (eq_ind T u (\lambda (t0: T).(\forall (e0: C).(\forall (u1: T).(\forall +(d1: nat).((getl d1 d (CHead e0 (Bind Void) u1)) \to (\forall (a0: C).((drop +(S O) d1 d a0) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t0 (lift +(S O) d1 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d1 +y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a0 y1 y2))))))))))) H2 (lift +(S O) (minus d0 (S n)) x0) H8) in (let H12 \def (eq_ind T u (\lambda (t0: +T).(ty3 g d t0 t)) H1 (lift (S O) (minus d0 (S n)) x0) H8) in (let H13 \def +(H11 e u0 (minus d0 (S n)) (getl_gen_S (Bind Abbr) d (CHead e (Bind Void) u0) +u (minus d0 (S n)) H6) x1 H10) in (ex3_2_ind T T (\lambda (y1: T).(\lambda +(_: T).(eq T (lift (S O) (minus d0 (S n)) x0) (lift (S O) (minus d0 (S n)) +y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) (minus d0 (S n)) +y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g x1 y1 y2))) (ex3_2 T T +(\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) +(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O t) (lift (S O) d0 y2)))) +(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: +T).(\lambda (x3: T).(\lambda (H14: (eq T (lift (S O) (minus d0 (S n)) x0) +(lift (S O) (minus d0 (S n)) x2))).(\lambda (H15: (eq T t (lift (S O) (minus +d0 (S n)) x3))).(\lambda (H16: (ty3 g x1 x2 x3)).(let H17 \def (eq_ind T t +(\lambda (t0: T).(ty3 g d (lift (S O) (minus d0 (S n)) x0) t0)) H12 (lift (S +O) (minus d0 (S n)) x3) H15) in (eq_ind_r T (lift (S O) (minus d0 (S n)) x3) +(\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) +(lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O +t0) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 +y2))))) (let H18 \def (eq_ind_r T x2 (\lambda (t0: T).(ty3 g x1 t0 x3)) H16 +x0 (lift_inj x0 x2 (S O) (minus d0 (S n)) H14)) in (eq_ind T (lift (S O) +(plus (S n) (minus d0 (S n))) (lift (S n) O x3)) (\lambda (t0: T).(ex3_2 T T +(\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) +(\lambda (_: T).(\lambda (y2: T).(eq T t0 (lift (S O) d0 y2)))) (\lambda (y1: +T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind nat d0 (\lambda (n0: +nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) +d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) n0 (lift (S n) O +x3)) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 +y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) +(lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d0 +(lift (S n) O x3)) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: +T).(ty3 g a y1 y2))) (TLRef n) (lift (S n) O x3) (eq_ind_r T (TLRef n) +(\lambda (t0: T).(eq T (TLRef n) t0)) (refl_equal T (TLRef n)) (lift (S O) d0 +(TLRef n)) (lift_lref_lt n (S O) d0 H5)) (refl_equal T (lift (S O) d0 (lift +(S n) O x3))) (ty3_abbr g n a x1 x0 H9 x3 H18)) (plus (S n) (minus d0 (S n))) +(le_plus_minus (S n) d0 H5)) (lift (S n) O (lift (S O) (minus d0 (S n)) x3)) +(lift_d x3 (S O) (S n) (minus d0 (S n)) O (le_O_n (minus d0 (S n)))))) t +H15))))))) H13))))))))) (getl_drop_conf_lt Abbr c0 d u n H0 a (S O) (minus d0 +(S n)) H7))))) (\lambda (H5: (eq nat n d0)).(let H6 \def (eq_ind_r nat d0 +(\lambda (n0: nat).(drop (S O) n0 c0 a)) H4 n H5) in (let H7 \def (eq_ind_r +nat d0 (\lambda (n0: nat).(getl n0 c0 (CHead e (Bind Void) u0))) H3 n H5) in +(eq_ind nat n (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: +T).(eq T (TLRef n) (lift (S O) n0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq +T (lift (S n) O t) (lift (S O) n0 y2)))) (\lambda (y1: T).(\lambda (y2: +T).(ty3 g a y1 y2))))) (let H8 \def (eq_ind C (CHead d (Bind Abbr) u) +(\lambda (c1: C).(getl n c0 c1)) H0 (CHead e (Bind Void) u0) (getl_mono c0 +(CHead d (Bind Abbr) u) n H0 (CHead e (Bind Void) u0) H7)) in (let H9 \def +(eq_ind C (CHead d (Bind Abbr) u) (\lambda (ee: C).(match ee in C return +(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) +\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b) +\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow +True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _) +\Rightarrow False])])) I (CHead e (Bind Void) u0) (getl_mono c0 (CHead d +(Bind Abbr) u) n H0 (CHead e (Bind Void) u0) H7)) in (False_ind (ex3_2 T T +(\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) n y1)))) +(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O t) (lift (S O) n y2)))) +(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) H9))) d0 H5)))) (\lambda +(H5: (lt d0 n)).(eq_ind_r nat (S (plus O (minus n (S O)))) (\lambda (n0: +nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n0) (lift (S O) +d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O t) (lift (S O) +d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind nat +(plus (S O) (minus n (S O))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: +T).(\lambda (_: T).(eq T (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: +T).(\lambda (y2: T).(eq T (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda +(y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r nat (plus (minus n (S +O)) (S O)) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq +T (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T +(lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 +g a y1 y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef +(plus (minus n (S O)) (S O))) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda +(y2: T).(eq T (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: +T).(\lambda (y2: T).(ty3 g a y1 y2))) (TLRef (minus n (S O))) (lift n O t) +(eq_ind_r T (TLRef (plus (minus n (S O)) (S O))) (\lambda (t0: T).(eq T +(TLRef (plus (minus n (S O)) (S O))) t0)) (refl_equal T (TLRef (plus (minus n +(S O)) (S O)))) (lift (S O) d0 (TLRef (minus n (S O)))) (lift_lref_ge (minus +n (S O)) (S O) d0 (lt_le_minus d0 n H5))) (eq_ind_r T (lift (plus (S O) n) O +t) (\lambda (t0: T).(eq T (lift (S n) O t) t0)) (refl_equal T (lift (S n) O +t)) (lift (S O) d0 (lift n O t)) (lift_free t n (S O) O d0 (le_S_n d0 (plus O +n) (le_S (S d0) (plus O n) H5)) (le_O_n d0))) (eq_ind_r nat (S (minus n (S +O))) (\lambda (n0: nat).(ty3 g a (TLRef (minus n (S O))) (lift n0 O t))) +(ty3_abbr g (minus n (S O)) a d u (getl_drop_conf_ge n (CHead d (Bind Abbr) +u) c0 H0 a (S O) d0 H4 (eq_ind_r nat (plus (S O) d0) (\lambda (n0: nat).(le +n0 n)) H5 (plus d0 (S O)) (plus_sym d0 (S O)))) t H1) n (minus_x_SO n +(le_lt_trans O d0 n (le_O_n d0) H5)))) (plus (S O) (minus n (S O))) (plus_sym +(S O) (minus n (S O)))) (S (plus O (minus n (S O)))) (refl_equal nat (S (plus +O (minus n (S O)))))) n (lt_plus_minus O n (le_lt_trans O d0 n (le_O_n d0) +H5))))))))))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: +C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abst) +u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u t)).(\lambda (H2: ((\forall (e: +C).(\forall (u0: T).(\forall (d0: nat).((getl d0 d (CHead e (Bind Void) u0)) +\to (\forall (a: C).((drop (S O) d0 d a) \to (ex3_2 T T (\lambda (y1: +T).(\lambda (_: T).(eq T u (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda +(y2: T).(eq T t (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 +g a y1 y2)))))))))))).(\lambda (e: C).(\lambda (u0: T).(\lambda (d0: +nat).(\lambda (H3: (getl d0 c0 (CHead e (Bind Void) u0))).(\lambda (a: +C).(\lambda (H4: (drop (S O) d0 c0 a)).(lt_eq_gt_e n d0 (ex3_2 T T (\lambda +(y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: +T).(\lambda (y2: T).(eq T (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda +(y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (H5: (lt n d0)).(let H6 +\def (eq_ind nat (minus d0 n) (\lambda (n0: nat).(getl n0 (CHead d (Bind +Abst) u) (CHead e (Bind Void) u0))) (getl_conf_le d0 (CHead e (Bind Void) u0) +c0 H3 (CHead d (Bind Abst) u) n H0 (le_S_n n d0 (le_S (S n) d0 H5))) (S +(minus d0 (S n))) (minus_x_Sy d0 n H5)) in (let H7 \def (eq_ind nat d0 +(\lambda (n0: nat).(drop (S O) n0 c0 a)) H4 (S (plus n (minus d0 (S n)))) +(lt_plus_minus n d0 H5)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: +C).(eq T u (lift (S O) (minus d0 (S n)) v)))) (\lambda (v: T).(\lambda (e0: +C).(getl n a (CHead e0 (Bind Abst) v)))) (\lambda (_: T).(\lambda (e0: +C).(drop (S O) (minus d0 (S n)) d e0))) (ex3_2 T T (\lambda (y1: T).(\lambda +(_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: +T).(eq T (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda +(y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1: C).(\lambda (H8: +(eq T u (lift (S O) (minus d0 (S n)) x0))).(\lambda (H9: (getl n a (CHead x1 +(Bind Abst) x0))).(\lambda (H10: (drop (S O) (minus d0 (S n)) d x1)).(let H11 +\def (eq_ind T u (\lambda (t0: T).(\forall (e0: C).(\forall (u1: T).(\forall +(d1: nat).((getl d1 d (CHead e0 (Bind Void) u1)) \to (\forall (a0: C).((drop +(S O) d1 d a0) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t0 (lift +(S O) d1 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d1 +y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a0 y1 y2))))))))))) H2 (lift +(S O) (minus d0 (S n)) x0) H8) in (let H12 \def (eq_ind T u (\lambda (t0: +T).(ty3 g d t0 t)) H1 (lift (S O) (minus d0 (S n)) x0) H8) in (eq_ind_r T +(lift (S O) (minus d0 (S n)) x0) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: +T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: +T).(\lambda (y2: T).(eq T (lift (S n) O t0) (lift (S O) d0 y2)))) (\lambda +(y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H13 \def (H11 e u0 (minus +d0 (S n)) (getl_gen_S (Bind Abst) d (CHead e (Bind Void) u0) u (minus d0 (S +n)) H6) x1 H10) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T +(lift (S O) (minus d0 (S n)) x0) (lift (S O) (minus d0 (S n)) y1)))) (\lambda +(_: T).(\lambda (y2: T).(eq T t (lift (S O) (minus d0 (S n)) y2)))) (\lambda +(y1: T).(\lambda (y2: T).(ty3 g x1 y1 y2))) (ex3_2 T T (\lambda (y1: +T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: +T).(\lambda (y2: T).(eq T (lift (S n) O (lift (S O) (minus d0 (S n)) x0)) +(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) +(\lambda (x2: T).(\lambda (x3: T).(\lambda (H14: (eq T (lift (S O) (minus d0 +(S n)) x0) (lift (S O) (minus d0 (S n)) x2))).(\lambda (H15: (eq T t (lift (S +O) (minus d0 (S n)) x3))).(\lambda (H16: (ty3 g x1 x2 x3)).(let H17 \def +(eq_ind T t (\lambda (t0: T).(ty3 g d (lift (S O) (minus d0 (S n)) x0) t0)) +H12 (lift (S O) (minus d0 (S n)) x3) H15) in (let H18 \def (eq_ind_r T x2 +(\lambda (t0: T).(ty3 g x1 t0 x3)) H16 x0 (lift_inj x0 x2 (S O) (minus d0 (S +n)) H14)) in (eq_ind T (lift (S O) (plus (S n) (minus d0 (S n))) (lift (S n) +O x0)) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T +(TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t0 +(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) +(eq_ind nat d0 (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: +T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq +T (lift (S O) n0 (lift (S n) O x0)) (lift (S O) d0 y2)))) (\lambda (y1: +T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1: +T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: +T).(\lambda (y2: T).(eq T (lift (S O) d0 (lift (S n) O x0)) (lift (S O) d0 +y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (TLRef n) (lift (S +n) O x0) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (TLRef n) t0)) +(refl_equal T (TLRef n)) (lift (S O) d0 (TLRef n)) (lift_lref_lt n (S O) d0 +H5)) (refl_equal T (lift (S O) d0 (lift (S n) O x0))) (ty3_abst g n a x1 x0 +H9 x3 H18)) (plus (S n) (minus d0 (S n))) (le_plus_minus (S n) d0 H5)) (lift +(S n) O (lift (S O) (minus d0 (S n)) x0)) (lift_d x0 (S O) (S n) (minus d0 (S +n)) O (le_O_n (minus d0 (S n)))))))))))) H13)) u H8)))))))) +(getl_drop_conf_lt Abst c0 d u n H0 a (S O) (minus d0 (S n)) H7))))) (\lambda +(H5: (eq nat n d0)).(let H6 \def (eq_ind_r nat d0 (\lambda (n0: nat).(drop (S +O) n0 c0 a)) H4 n H5) in (let H7 \def (eq_ind_r nat d0 (\lambda (n0: +nat).(getl n0 c0 (CHead e (Bind Void) u0))) H3 n H5) in (eq_ind nat n +(\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef +n) (lift (S O) n0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O +u) (lift (S O) n0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 +y2))))) (let H8 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (c1: C).(getl +n c0 c1)) H0 (CHead e (Bind Void) u0) (getl_mono c0 (CHead d (Bind Abst) u) n +H0 (CHead e (Bind Void) u0) H7)) in (let H9 \def (eq_ind C (CHead d (Bind +Abst) u) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with +[(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return +(\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B return +(\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow True | +Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead e (Bind +Void) u0) (getl_mono c0 (CHead d (Bind Abst) u) n H0 (CHead e (Bind Void) u0) +H7)) in (False_ind (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef +n) (lift (S O) n y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O +u) (lift (S O) n y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) +H9))) d0 H5)))) (\lambda (H5: (lt d0 n)).(eq_ind_r nat (S (plus O (minus n (S +O)))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T +(TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift +(S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a +y1 y2))))) (eq_ind nat (plus (S O) (minus n (S O))) (\lambda (n0: nat).(ex3_2 +T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n0) (lift (S O) d0 y1)))) +(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O u) (lift (S O) d0 y2)))) +(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r nat (plus +(minus n (S O)) (S O)) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: +T).(\lambda (_: T).(eq T (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: +T).(\lambda (y2: T).(eq T (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda +(y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1: +T).(\lambda (_: T).(eq T (TLRef (plus (minus n (S O)) (S O))) (lift (S O) d0 +y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O u) (lift (S O) d0 +y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (TLRef (minus n (S +O))) (lift n O u) (eq_ind_r T (TLRef (plus (minus n (S O)) (S O))) (\lambda +(t0: T).(eq T (TLRef (plus (minus n (S O)) (S O))) t0)) (refl_equal T (TLRef +(plus (minus n (S O)) (S O)))) (lift (S O) d0 (TLRef (minus n (S O)))) +(lift_lref_ge (minus n (S O)) (S O) d0 (lt_le_minus d0 n H5))) (eq_ind_r T +(lift (plus (S O) n) O u) (\lambda (t0: T).(eq T (lift (S n) O u) t0)) +(refl_equal T (lift (S n) O u)) (lift (S O) d0 (lift n O u)) (lift_free u n +(S O) O d0 (le_S_n d0 (plus O n) (le_S (S d0) (plus O n) H5)) (le_O_n d0))) +(eq_ind_r nat (S (minus n (S O))) (\lambda (n0: nat).(ty3 g a (TLRef (minus n +(S O))) (lift n0 O u))) (ty3_abst g (minus n (S O)) a d u (getl_drop_conf_ge +n (CHead d (Bind Abst) u) c0 H0 a (S O) d0 H4 (eq_ind_r nat (plus (S O) d0) +(\lambda (n0: nat).(le n0 n)) H5 (plus d0 (S O)) (plus_sym d0 (S O)))) t H1) +n (minus_x_SO n (le_lt_trans O d0 n (le_O_n d0) H5)))) (plus (S O) (minus n +(S O))) (plus_sym (S O) (minus n (S O)))) (S (plus O (minus n (S O)))) +(refl_equal nat (S (plus O (minus n (S O)))))) n (lt_plus_minus O n +(le_lt_trans O d0 n (le_O_n d0) H5))))))))))))))))))) (\lambda (c0: +C).(\lambda (u: T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 u t)).(\lambda +(H1: ((\forall (e: C).(\forall (u0: T).(\forall (d: nat).((getl d c0 (CHead e +(Bind Void) u0)) \to (\forall (a: C).((drop (S O) d c0 a) \to (ex3_2 T T +(\lambda (y1: T).(\lambda (_: T).(eq T u (lift (S O) d y1)))) (\lambda (_: +T).(\lambda (y2: T).(eq T t (lift (S O) d y2)))) (\lambda (y1: T).(\lambda +(y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (b: B).(\lambda (t3: T).(\lambda +(t4: T).(\lambda (H2: (ty3 g (CHead c0 (Bind b) u) t3 t4)).(\lambda (H3: +((\forall (e: C).(\forall (u0: T).(\forall (d: nat).((getl d (CHead c0 (Bind +b) u) (CHead e (Bind Void) u0)) \to (\forall (a: C).((drop (S O) d (CHead c0 +(Bind b) u) a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t3 (lift +(S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t4 (lift (S O) d y2)))) +(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (e: +C).(\lambda (u0: T).(\lambda (d: nat).(\lambda (H4: (getl d c0 (CHead e (Bind +Void) u0))).(\lambda (a: C).(\lambda (H5: (drop (S O) d c0 a)).(let H6 \def +(H1 e u0 d H4 a H5) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T +u (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d +y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T +(\lambda (y1: T).(\lambda (_: T).(eq T (THead (Bind b) u t3) (lift (S O) d +y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Bind b) u t4) (lift (S +O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda +(x0: T).(\lambda (x1: T).(\lambda (H7: (eq T u (lift (S O) d x0))).(\lambda +(H8: (eq T t (lift (S O) d x1))).(\lambda (H9: (ty3 g a x0 x1)).(let H10 \def +(eq_ind T t (\lambda (t0: T).(ty3 g c0 u t0)) H0 (lift (S O) d x1) H8) in +(let H11 \def (eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 (lift (S O) d x1))) +H10 (lift (S O) d x0) H7) in (let H12 \def (eq_ind T u (\lambda (t0: +T).(\forall (e0: C).(\forall (u1: T).(\forall (d0: nat).((getl d0 (CHead c0 +(Bind b) t0) (CHead e0 (Bind Void) u1)) \to (\forall (a0: C).((drop (S O) d0 +(CHead c0 (Bind b) t0) a0) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: +T).(eq T t3 (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t4 +(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a0 y1 +y2))))))))))) H3 (lift (S O) d x0) H7) in (let H13 \def (eq_ind T u (\lambda +(t0: T).(ty3 g (CHead c0 (Bind b) t0) t3 t4)) H2 (lift (S O) d x0) H7) in +(eq_ind_r T (lift (S O) d x0) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: +T).(\lambda (_: T).(eq T (THead (Bind b) t0 t3) (lift (S O) d y1)))) (\lambda +(_: T).(\lambda (y2: T).(eq T (THead (Bind b) t0 t4) (lift (S O) d y2)))) +(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H14 \def (H12 e u0 +(S d) (getl_head (Bind b) d c0 (CHead e (Bind Void) u0) H4 (lift (S O) d x0)) +(CHead a (Bind b) x0) (drop_skip_bind (S O) d c0 a H5 b x0)) in (ex3_2_ind T +T (\lambda (y1: T).(\lambda (_: T).(eq T t3 (lift (S O) (S d) y1)))) (\lambda +(_: T).(\lambda (y2: T).(eq T t4 (lift (S O) (S d) y2)))) (\lambda (y1: +T).(\lambda (y2: T).(ty3 g (CHead a (Bind b) x0) y1 y2))) (ex3_2 T T (\lambda +(y1: T).(\lambda (_: T).(eq T (THead (Bind b) (lift (S O) d x0) t3) (lift (S +O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Bind b) (lift (S +O) d x0) t4) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a +y1 y2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H15: (eq T t3 (lift (S +O) (S d) x2))).(\lambda (H16: (eq T t4 (lift (S O) (S d) x3))).(\lambda (H17: +(ty3 g (CHead a (Bind b) x0) x2 x3)).(eq_ind_r T (lift (S O) (S d) x3) +(\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead +(Bind b) (lift (S O) d x0) t3) (lift (S O) d y1)))) (\lambda (_: T).(\lambda +(y2: T).(eq T (THead (Bind b) (lift (S O) d x0) t0) (lift (S O) d y2)))) +(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r T (lift (S O) +(S d) x2) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T +(THead (Bind b) (lift (S O) d x0) t0) (lift (S O) d y1)))) (\lambda (_: +T).(\lambda (y2: T).(eq T (THead (Bind b) (lift (S O) d x0) (lift (S O) (S d) +x3)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 +y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Bind +b) (lift (S O) d x0) (lift (S O) (S d) x2)) (lift (S O) d y1)))) (\lambda (_: +T).(\lambda (y2: T).(eq T (THead (Bind b) (lift (S O) d x0) (lift (S O) (S d) +x3)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) +(THead (Bind b) x0 x2) (THead (Bind b) x0 x3) (sym_eq T (lift (S O) d (THead +(Bind b) x0 x2)) (THead (Bind b) (lift (S O) d x0) (lift (S O) (S d) x2)) +(lift_bind b x0 x2 (S O) d)) (sym_eq T (lift (S O) d (THead (Bind b) x0 x3)) +(THead (Bind b) (lift (S O) d x0) (lift (S O) (S d) x3)) (lift_bind b x0 x3 +(S O) d)) (ty3_bind g a x0 x1 H9 b x2 x3 H17)) t3 H15) t4 H16)))))) H14)) u +H7)))))))))) H6)))))))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda +(u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda (H1: ((\forall (e: C).(\forall +(u0: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Void) u0)) \to (\forall +(a: C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: +T).(eq T w (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T u +(lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 +y2)))))))))))).(\lambda (v: T).(\lambda (t: T).(\lambda (H2: (ty3 g c0 v +(THead (Bind Abst) u t))).(\lambda (H3: ((\forall (e: C).(\forall (u0: +T).(\forall (d: nat).((getl d c0 (CHead e (Bind Void) u0)) \to (\forall (a: +C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T +v (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Bind +Abst) u t) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 +y2)))))))))))).(\lambda (e: C).(\lambda (u0: T).(\lambda (d: nat).(\lambda +(H4: (getl d c0 (CHead e (Bind Void) u0))).(\lambda (a: C).(\lambda (H5: +(drop (S O) d c0 a)).(let H6 \def (H3 e u0 d H4 a H5) in (ex3_2_ind T T +(\lambda (y1: T).(\lambda (_: T).(eq T v (lift (S O) d y1)))) (\lambda (_: +T).(\lambda (y2: T).(eq T (THead (Bind Abst) u t) (lift (S O) d y2)))) +(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: +T).(\lambda (_: T).(eq T (THead (Flat Appl) w v) (lift (S O) d y1)))) +(\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) w (THead (Bind +Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a +y1 y2)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: (eq T v (lift (S O) +d x0))).(\lambda (H8: (eq T (THead (Bind Abst) u t) (lift (S O) d +x1))).(\lambda (H9: (ty3 g a x0 x1)).(let H10 \def (eq_ind T v (\lambda (t0: +T).(ty3 g c0 t0 (THead (Bind Abst) u t))) H2 (lift (S O) d x0) H7) in +(eq_ind_r T (lift (S O) d x0) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: +T).(\lambda (_: T).(eq T (THead (Flat Appl) w t0) (lift (S O) d y1)))) +(\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) w (THead (Bind +Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a +y1 y2))))) (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x1 (THead +(Bind Abst) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift (S O) d +y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift (S O) (S d) z)))) (ex3_2 +T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Appl) w (lift (S O) d +x0)) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat +Appl) w (THead (Bind Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: +T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: T).(\lambda (x3: +T).(\lambda (H11: (eq T x1 (THead (Bind Abst) x2 x3))).(\lambda (H12: (eq T u +(lift (S O) d x2))).(\lambda (H13: (eq T t (lift (S O) (S d) x3))).(let H14 +\def (eq_ind T x1 (\lambda (t0: T).(ty3 g a x0 t0)) H9 (THead (Bind Abst) x2 +x3) H11) in (eq_ind_r T (lift (S O) (S d) x3) (\lambda (t0: T).(ex3_2 T T +(\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Appl) w (lift (S O) d +x0)) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat +Appl) w (THead (Bind Abst) u t0)) (lift (S O) d y2)))) (\lambda (y1: +T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H15 \def (eq_ind T u (\lambda +(t0: T).(\forall (e0: C).(\forall (u1: T).(\forall (d0: nat).((getl d0 c0 +(CHead e0 (Bind Void) u1)) \to (\forall (a0: C).((drop (S O) d0 c0 a0) \to +(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T w (lift (S O) d0 y1)))) +(\lambda (_: T).(\lambda (y2: T).(eq T t0 (lift (S O) d0 y2)))) (\lambda (y1: +T).(\lambda (y2: T).(ty3 g a0 y1 y2))))))))))) H1 (lift (S O) d x2) H12) in +(eq_ind_r T (lift (S O) d x2) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: +T).(\lambda (_: T).(eq T (THead (Flat Appl) w (lift (S O) d x0)) (lift (S O) +d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) w (THead +(Bind Abst) t0 (lift (S O) (S d) x3))) (lift (S O) d y2)))) (\lambda (y1: +T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H16 \def (H15 e u0 d H4 a H5) in +(ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T w (lift (S O) d y1)))) +(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d x2) (lift (S O) d y2)))) +(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: +T).(\lambda (_: T).(eq T (THead (Flat Appl) w (lift (S O) d x0)) (lift (S O) +d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) w (THead +(Bind Abst) (lift (S O) d x2) (lift (S O) (S d) x3))) (lift (S O) d y2)))) +(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x4: +T).(\lambda (x5: T).(\lambda (H17: (eq T w (lift (S O) d x4))).(\lambda (H18: +(eq T (lift (S O) d x2) (lift (S O) d x5))).(\lambda (H19: (ty3 g a x4 +x5)).(eq_ind_r T (lift (S O) d x4) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: +T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (lift (S O) d x0)) (lift (S O) +d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) t0 (THead +(Bind Abst) (lift (S O) d x2) (lift (S O) (S d) x3))) (lift (S O) d y2)))) +(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H20 \def (eq_ind_r +T x5 (\lambda (t0: T).(ty3 g a x4 t0)) H19 x2 (lift_inj x2 x5 (S O) d H18)) +in (eq_ind T (lift (S O) d (THead (Bind Abst) x2 x3)) (\lambda (t0: T).(ex3_2 +T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Appl) (lift (S O) d +x4) (lift (S O) d x0)) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: +T).(eq T (THead (Flat Appl) (lift (S O) d x4) t0) (lift (S O) d y2)))) +(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind T (lift (S O) d +(THead (Flat Appl) x4 x0)) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: +T).(\lambda (_: T).(eq T t0 (lift (S O) d y1)))) (\lambda (_: T).(\lambda +(y2: T).(eq T (THead (Flat Appl) (lift (S O) d x4) (lift (S O) d (THead (Bind +Abst) x2 x3))) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g +a y1 y2))))) (eq_ind T (lift (S O) d (THead (Flat Appl) x4 (THead (Bind Abst) +x2 x3))) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T +(lift (S O) d (THead (Flat Appl) x4 x0)) (lift (S O) d y1)))) (\lambda (_: +T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda +(y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: +T).(eq T (lift (S O) d (THead (Flat Appl) x4 x0)) (lift (S O) d y1)))) +(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d (THead (Flat Appl) x4 +(THead (Bind Abst) x2 x3))) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda +(y2: T).(ty3 g a y1 y2))) (THead (Flat Appl) x4 x0) (THead (Flat Appl) x4 +(THead (Bind Abst) x2 x3)) (refl_equal T (lift (S O) d (THead (Flat Appl) x4 +x0))) (refl_equal T (lift (S O) d (THead (Flat Appl) x4 (THead (Bind Abst) x2 +x3)))) (ty3_appl g a x4 x2 H20 x0 x3 H14)) (THead (Flat Appl) (lift (S O) d +x4) (lift (S O) d (THead (Bind Abst) x2 x3))) (lift_flat Appl x4 (THead (Bind +Abst) x2 x3) (S O) d)) (THead (Flat Appl) (lift (S O) d x4) (lift (S O) d +x0)) (lift_flat Appl x4 x0 (S O) d)) (THead (Bind Abst) (lift (S O) d x2) +(lift (S O) (S d) x3)) (lift_bind Abst x2 x3 (S O) d))) w H17)))))) H16)) u +H12)) t H13))))))) (lift_gen_bind Abst u t x1 (S O) d H8)) v H7))))))) +H6))))))))))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: +T).(\lambda (H0: (ty3 g c0 t3 t4)).(\lambda (H1: ((\forall (e: C).(\forall +(u: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Void) u)) \to (\forall +(a: C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: +T).(eq T t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t4 +(lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 +y2)))))))))))).(\lambda (t0: T).(\lambda (H2: (ty3 g c0 t4 t0)).(\lambda (H3: +((\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead e (Bind +Void) u)) \to (\forall (a: C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda +(y1: T).(\lambda (_: T).(eq T t4 (lift (S O) d y1)))) (\lambda (_: +T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda +(y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (e: C).(\lambda (u: T).(\lambda +(d: nat).(\lambda (H4: (getl d c0 (CHead e (Bind Void) u))).(\lambda (a: +C).(\lambda (H5: (drop (S O) d c0 a)).(let H6 \def (H3 e u d H4 a H5) in +(ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T t4 (lift (S O) d y1)))) +(\lambda (_: T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2)))) (\lambda (y1: +T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda +(_: T).(eq T (THead (Flat Cast) t4 t3) (lift (S O) d y1)))) (\lambda (_: +T).(\lambda (y2: T).(eq T (THead (Flat Cast) t0 t4) (lift (S O) d y2)))) +(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: +T).(\lambda (x1: T).(\lambda (H7: (eq T t4 (lift (S O) d x0))).(\lambda (H8: +(eq T t0 (lift (S O) d x1))).(\lambda (H9: (ty3 g a x0 x1)).(let H10 \def +(eq_ind T t0 (\lambda (t: T).(ty3 g c0 t4 t)) H2 (lift (S O) d x1) H8) in +(eq_ind_r T (lift (S O) d x1) (\lambda (t: T).(ex3_2 T T (\lambda (y1: +T).(\lambda (_: T).(eq T (THead (Flat Cast) t4 t3) (lift (S O) d y1)))) +(\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Cast) t t4) (lift (S O) d +y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H11 \def +(eq_ind T t4 (\lambda (t: T).(ty3 g c0 t (lift (S O) d x1))) H10 (lift (S O) +d x0) H7) in (let H12 \def (eq_ind T t4 (\lambda (t: T).(\forall (e0: +C).(\forall (u0: T).(\forall (d0: nat).((getl d0 c0 (CHead e0 (Bind Void) +u0)) \to (\forall (a0: C).((drop (S O) d0 c0 a0) \to (ex3_2 T T (\lambda (y1: +T).(\lambda (_: T).(eq T t3 (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda +(y2: T).(eq T t (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 +g a0 y1 y2))))))))))) H1 (lift (S O) d x0) H7) in (let H13 \def (eq_ind T t4 +(\lambda (t: T).(ty3 g c0 t3 t)) H0 (lift (S O) d x0) H7) in (eq_ind_r T +(lift (S O) d x0) (\lambda (t: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: +T).(eq T (THead (Flat Cast) t t3) (lift (S O) d y1)))) (\lambda (_: +T).(\lambda (y2: T).(eq T (THead (Flat Cast) (lift (S O) d x1) t) (lift (S O) +d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H14 \def +(H12 e u d H4 a H5) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T +t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d +x0) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) +(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Cast) (lift (S +O) d x0) t3) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T +(THead (Flat Cast) (lift (S O) d x1) (lift (S O) d x0)) (lift (S O) d y2)))) +(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: +T).(\lambda (x3: T).(\lambda (H15: (eq T t3 (lift (S O) d x2))).(\lambda +(H16: (eq T (lift (S O) d x0) (lift (S O) d x3))).(\lambda (H17: (ty3 g a x2 +x3)).(let H18 \def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 t (lift (S O) d +x0))) H13 (lift (S O) d x2) H15) in (eq_ind_r T (lift (S O) d x2) (\lambda +(t: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Cast) +(lift (S O) d x0) t) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: +T).(eq T (THead (Flat Cast) (lift (S O) d x1) (lift (S O) d x0)) (lift (S O) +d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H19 \def +(eq_ind_r T x3 (\lambda (t: T).(ty3 g a x2 t)) H17 x0 (lift_inj x0 x3 (S O) d +H16)) in (eq_ind T (lift (S O) d (THead (Flat Cast) x0 x2)) (\lambda (t: +T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t (lift (S O) d y1)))) +(\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Cast) (lift (S O) d x1) +(lift (S O) d x0)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: +T).(ty3 g a y1 y2))))) (eq_ind T (lift (S O) d (THead (Flat Cast) x1 x0)) +(\lambda (t: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (lift (S O) +d (THead (Flat Cast) x0 x2)) (lift (S O) d y1)))) (\lambda (_: T).(\lambda +(y2: T).(eq T t (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g +a y1 y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(eq T (lift (S +O) d (THead (Flat Cast) x0 x2)) (lift (S O) d y1)))) (\lambda (_: T).(\lambda +(y2: T).(eq T (lift (S O) d (THead (Flat Cast) x1 x0)) (lift (S O) d y2)))) +(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (THead (Flat Cast) x0 x2) +(THead (Flat Cast) x1 x0) (refl_equal T (lift (S O) d (THead (Flat Cast) x0 +x2))) (refl_equal T (lift (S O) d (THead (Flat Cast) x1 x0))) (ty3_cast g a +x2 x0 H19 x1 H9)) (THead (Flat Cast) (lift (S O) d x1) (lift (S O) d x0)) +(lift_flat Cast x1 x0 (S O) d)) (THead (Flat Cast) (lift (S O) d x0) (lift (S +O) d x2)) (lift_flat Cast x0 x2 (S O) d))) t3 H15))))))) H14)) t4 H7)))) t0 +H8))))))) H6)))))))))))))))) c t1 t2 H))))). +(* COMMENTS +Initial nodes: 13105 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/wcpr0/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/wcpr0/defs.ma new file mode 100644 index 000000000..cdc9bf3a4 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/wcpr0/defs.ma @@ -0,0 +1,26 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/pr0/defs.ma". + +include "Basic-1/C/defs.ma". + +inductive wcpr0: C \to (C \to Prop) \def +| wcpr0_refl: \forall (c: C).(wcpr0 c c) +| wcpr0_comp: \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall +(u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (k: K).(wcpr0 (CHead c1 k +u1) (CHead c2 k u2)))))))). + diff --git a/matita/matita/contribs/lambdadelta/basic_1/wcpr0/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/wcpr0/fwd.ma new file mode 100644 index 000000000..2b0531a8a --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/wcpr0/fwd.ma @@ -0,0 +1,105 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/wcpr0/defs.ma". + +theorem wcpr0_gen_sort: + \forall (x: C).(\forall (n: nat).((wcpr0 (CSort n) x) \to (eq C x (CSort +n)))) +\def + \lambda (x: C).(\lambda (n: nat).(\lambda (H: (wcpr0 (CSort n) +x)).(insert_eq C (CSort n) (\lambda (c: C).(wcpr0 c x)) (\lambda (c: C).(eq C +x c)) (\lambda (y: C).(\lambda (H0: (wcpr0 y x)).(wcpr0_ind (\lambda (c: +C).(\lambda (c0: C).((eq C c (CSort n)) \to (eq C c0 c)))) (\lambda (c: +C).(\lambda (H1: (eq C c (CSort n))).(let H2 \def (f_equal C C (\lambda (e: +C).e) c (CSort n) H1) in (eq_ind_r C (CSort n) (\lambda (c0: C).(eq C c0 c0)) +(refl_equal C (CSort n)) c H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda +(_: (wcpr0 c1 c2)).(\lambda (_: (((eq C c1 (CSort n)) \to (eq C c2 +c1)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda +(k: K).(\lambda (H4: (eq C (CHead c1 k u1) (CSort n))).(let H5 \def (eq_ind C +(CHead c1 k u1) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) +with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I +(CSort n) H4) in (False_ind (eq C (CHead c2 k u2) (CHead c1 k u1)) +H5))))))))))) y x H0))) H))). +(* COMMENTS +Initial nodes: 249 +END *) + +theorem wcpr0_gen_head: + \forall (k: K).(\forall (c1: C).(\forall (x: C).(\forall (u1: T).((wcpr0 +(CHead c1 k u1) x) \to (or (eq C x (CHead c1 k u1)) (ex3_2 C T (\lambda (c2: +C).(\lambda (u2: T).(eq C x (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: +T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2))))))))) +\def + \lambda (k: K).(\lambda (c1: C).(\lambda (x: C).(\lambda (u1: T).(\lambda +(H: (wcpr0 (CHead c1 k u1) x)).(insert_eq C (CHead c1 k u1) (\lambda (c: +C).(wcpr0 c x)) (\lambda (c: C).(or (eq C x c) (ex3_2 C T (\lambda (c2: +C).(\lambda (u2: T).(eq C x (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: +T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))) (\lambda +(y: C).(\lambda (H0: (wcpr0 y x)).(wcpr0_ind (\lambda (c: C).(\lambda (c0: +C).((eq C c (CHead c1 k u1)) \to (or (eq C c0 c) (ex3_2 C T (\lambda (c2: +C).(\lambda (u2: T).(eq C c0 (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: +T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))))) +(\lambda (c: C).(\lambda (H1: (eq C c (CHead c1 k u1))).(let H2 \def (f_equal +C C (\lambda (e: C).e) c (CHead c1 k u1) H1) in (eq_ind_r C (CHead c1 k u1) +(\lambda (c0: C).(or (eq C c0 c0) (ex3_2 C T (\lambda (c2: C).(\lambda (u2: +T).(eq C c0 (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: T).(wcpr0 c1 +c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))) (or_introl (eq C +(CHead c1 k u1) (CHead c1 k u1)) (ex3_2 C T (\lambda (c2: C).(\lambda (u2: +T).(eq C (CHead c1 k u1) (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: +T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))) +(refl_equal C (CHead c1 k u1))) c H2)))) (\lambda (c0: C).(\lambda (c2: +C).(\lambda (H1: (wcpr0 c0 c2)).(\lambda (H2: (((eq C c0 (CHead c1 k u1)) \to +(or (eq C c2 c0) (ex3_2 C T (\lambda (c3: C).(\lambda (u2: T).(eq C c2 (CHead +c3 k u2)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: +C).(\lambda (u2: T).(pr0 u1 u2)))))))).(\lambda (u0: T).(\lambda (u2: +T).(\lambda (H3: (pr0 u0 u2)).(\lambda (k0: K).(\lambda (H4: (eq C (CHead c0 +k0 u0) (CHead c1 k u1))).(let H5 \def (f_equal C C (\lambda (e: C).(match e +in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) +\Rightarrow c])) (CHead c0 k0 u0) (CHead c1 k u1) H4) in ((let H6 \def +(f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with +[(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead c0 k0 u0) +(CHead c1 k u1) H4) in ((let H7 \def (f_equal C T (\lambda (e: C).(match e in +C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) +\Rightarrow t])) (CHead c0 k0 u0) (CHead c1 k u1) H4) in (\lambda (H8: (eq K +k0 k)).(\lambda (H9: (eq C c0 c1)).(eq_ind_r K k (\lambda (k1: K).(or (eq C +(CHead c2 k1 u2) (CHead c0 k1 u0)) (ex3_2 C T (\lambda (c3: C).(\lambda (u3: +T).(eq C (CHead c2 k1 u2) (CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_: +T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3)))))) (let H10 +\def (eq_ind T u0 (\lambda (t: T).(pr0 t u2)) H3 u1 H7) in (eq_ind_r T u1 +(\lambda (t: T).(or (eq C (CHead c2 k u2) (CHead c0 k t)) (ex3_2 C T (\lambda +(c3: C).(\lambda (u3: T).(eq C (CHead c2 k u2) (CHead c3 k u3)))) (\lambda +(c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 +u1 u3)))))) (let H11 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 k +u1)) \to (or (eq C c2 c) (ex3_2 C T (\lambda (c3: C).(\lambda (u3: T).(eq C +c2 (CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) +(\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))))))) H2 c1 H9) in (let H12 \def +(eq_ind C c0 (\lambda (c: C).(wcpr0 c c2)) H1 c1 H9) in (eq_ind_r C c1 +(\lambda (c: C).(or (eq C (CHead c2 k u2) (CHead c k u1)) (ex3_2 C T (\lambda +(c3: C).(\lambda (u3: T).(eq C (CHead c2 k u2) (CHead c3 k u3)))) (\lambda +(c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 +u1 u3)))))) (or_intror (eq C (CHead c2 k u2) (CHead c1 k u1)) (ex3_2 C T +(\lambda (c3: C).(\lambda (u3: T).(eq C (CHead c2 k u2) (CHead c3 k u3)))) +(\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda +(u3: T).(pr0 u1 u3)))) (ex3_2_intro C T (\lambda (c3: C).(\lambda (u3: T).(eq +C (CHead c2 k u2) (CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 +c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))) c2 u2 (refl_equal C +(CHead c2 k u2)) H12 H10)) c0 H9))) u0 H7)) k0 H8)))) H6)) H5))))))))))) y x +H0))) H))))). +(* COMMENTS +Initial nodes: 1133 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/wcpr0/getl.ma b/matita/matita/contribs/lambdadelta/basic_1/wcpr0/getl.ma new file mode 100644 index 000000000..d3a109e0d --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/wcpr0/getl.ma @@ -0,0 +1,464 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/wcpr0/defs.ma". + +include "Basic-1/getl/props.ma". + +theorem wcpr0_drop: + \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (h: +nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((drop h O c1 (CHead +e1 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c2 +(CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda +(_: C).(\lambda (u2: T).(pr0 u1 u2))))))))))) +\def + \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(wcpr0_ind +(\lambda (c: C).(\lambda (c0: C).(\forall (h: nat).(\forall (e1: C).(\forall +(u1: T).(\forall (k: K).((drop h O c (CHead e1 k u1)) \to (ex3_2 C T (\lambda +(e2: C).(\lambda (u2: T).(drop h O c0 (CHead e2 k u2)))) (\lambda (e2: +C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 +u2))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda +(u1: T).(\lambda (k: K).(\lambda (H0: (drop h O c (CHead e1 k +u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c (CHead +e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: +C).(\lambda (u2: T).(pr0 u1 u2))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl +u1)))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wcpr0 c3 +c4)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1: +T).(\forall (k: K).((drop h O c3 (CHead e1 k u1)) \to (ex3_2 C T (\lambda +(e2: C).(\lambda (u2: T).(drop h O c4 (CHead e2 k u2)))) (\lambda (e2: +C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 +u2))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 +u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall +(e1: C).(\forall (u3: T).(\forall (k0: K).((drop n O (CHead c3 k u1) (CHead +e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead +c4 k u2) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) +(\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))) (\lambda (e1: +C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (drop O O (CHead c3 k u1) +(CHead e1 k0 u0))).(let H4 \def (f_equal C C (\lambda (e: C).(match e in C +return (\lambda (_: C).C) with [(CSort _) \Rightarrow c3 | (CHead c _ _) +\Rightarrow c])) (CHead c3 k u1) (CHead e1 k0 u0) (drop_gen_refl (CHead c3 k +u1) (CHead e1 k0 u0) H3)) in ((let H5 \def (f_equal C K (\lambda (e: +C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | +(CHead _ k1 _) \Rightarrow k1])) (CHead c3 k u1) (CHead e1 k0 u0) +(drop_gen_refl (CHead c3 k u1) (CHead e1 k0 u0) H3)) in ((let H6 \def +(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with +[(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead c3 k u1) +(CHead e1 k0 u0) (drop_gen_refl (CHead c3 k u1) (CHead e1 k0 u0) H3)) in +(\lambda (H7: (eq K k k0)).(\lambda (H8: (eq C c3 e1)).(eq_ind K k (\lambda +(k1: K).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c4 k +u2) (CHead e2 k1 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) +(\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))))) (eq_ind T u1 (\lambda (t: +T).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c4 k u2) +(CHead e2 k u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda +(_: C).(\lambda (u3: T).(pr0 t u3))))) (eq_ind C c3 (\lambda (c: C).(ex3_2 C +T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c4 k u2) (CHead e2 k +u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 c e2))) (\lambda (_: +C).(\lambda (u3: T).(pr0 u1 u3))))) (ex3_2_intro C T (\lambda (e2: +C).(\lambda (u3: T).(drop O O (CHead c4 k u2) (CHead e2 k u3)))) (\lambda +(e2: C).(\lambda (_: T).(wcpr0 c3 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 +u1 u3))) c4 u2 (drop_refl (CHead c4 k u2)) H0 H2) e1 H8) u0 H6) k0 H7)))) +H5)) H4)))))) (K_ind (\lambda (k0: K).(\forall (n: nat).(((\forall (e1: +C).(\forall (u3: T).(\forall (k1: K).((drop n O (CHead c3 k0 u1) (CHead e1 k1 +u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c4 k0 +u2) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) +(\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))) \to (\forall (e1: +C).(\forall (u3: T).(\forall (k1: K).((drop (S n) O (CHead c3 k0 u1) (CHead +e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop (S n) O +(CHead c4 k0 u2) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 +e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))))) (\lambda (b: +B).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall +(k0: K).((drop n O (CHead c3 (Bind b) u1) (CHead e1 k0 u3)) \to (ex3_2 C T +(\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c4 (Bind b) u2) (CHead e2 +k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: +C).(\lambda (u4: T).(pr0 u3 u4)))))))))).(\lambda (e1: C).(\lambda (u0: +T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c3 (Bind b) u1) (CHead +e1 k0 u0))).(let H5 \def (H1 n e1 u0 k0 (drop_gen_drop (Bind b) c3 (CHead e1 +k0 u0) u1 n H4)) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop n +O c4 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) +(\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) (ex3_2 C T (\lambda (e2: +C).(\lambda (u3: T).(drop (S n) O (CHead c4 (Bind b) u2) (CHead e2 k0 u3)))) +(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda +(u3: T).(pr0 u0 u3)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (drop +n O c4 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 e1 x0)).(\lambda (H8: (pr0 u0 +x1)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead +c4 (Bind b) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 +e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) x0 x1 (drop_drop +(Bind b) n c4 (CHead x0 k0 x1) H6 u2) H7 H8)))))) H5))))))))) (\lambda (f: +F).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall +(k0: K).((drop n O (CHead c3 (Flat f) u1) (CHead e1 k0 u3)) \to (ex3_2 C T +(\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c4 (Flat f) u2) (CHead e2 +k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: +C).(\lambda (u4: T).(pr0 u3 u4)))))))))).(\lambda (e1: C).(\lambda (u0: +T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c3 (Flat f) u1) (CHead +e1 k0 u0))).(let H5 \def (H1 (S n) e1 u0 k0 (drop_gen_drop (Flat f) c3 (CHead +e1 k0 u0) u1 n H4)) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop +(S n) O c4 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 +e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) (ex3_2 C T (\lambda (e2: +C).(\lambda (u3: T).(drop (S n) O (CHead c4 (Flat f) u2) (CHead e2 k0 u3)))) +(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda +(u3: T).(pr0 u0 u3)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (drop +(S n) O c4 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 e1 x0)).(\lambda (H8: (pr0 +u0 x1)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O +(CHead c4 (Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: +T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) x0 x1 +(drop_drop (Flat f) n c4 (CHead x0 k0 x1) H6 u2) H7 H8)))))) H5))))))))) k) +h)))))))))) c1 c2 H))). +(* COMMENTS +Initial nodes: 1755 +END *) + +theorem wcpr0_drop_back: + \forall (c1: C).(\forall (c2: C).((wcpr0 c2 c1) \to (\forall (h: +nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((drop h O c1 (CHead +e1 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c2 +(CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda +(_: C).(\lambda (u2: T).(pr0 u2 u1))))))))))) +\def + \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c2 c1)).(wcpr0_ind +(\lambda (c: C).(\lambda (c0: C).(\forall (h: nat).(\forall (e1: C).(\forall +(u1: T).(\forall (k: K).((drop h O c0 (CHead e1 k u1)) \to (ex3_2 C T +(\lambda (e2: C).(\lambda (u2: T).(drop h O c (CHead e2 k u2)))) (\lambda +(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 +u2 u1))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda +(u1: T).(\lambda (k: K).(\lambda (H0: (drop h O c (CHead e1 k +u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c (CHead +e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: +C).(\lambda (u2: T).(pr0 u2 u1))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl +u1)))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wcpr0 c3 +c4)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1: +T).(\forall (k: K).((drop h O c4 (CHead e1 k u1)) \to (ex3_2 C T (\lambda +(e2: C).(\lambda (u2: T).(drop h O c3 (CHead e2 k u2)))) (\lambda (e2: +C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 +u1))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 +u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall +(e1: C).(\forall (u3: T).(\forall (k0: K).((drop n O (CHead c4 k u2) (CHead +e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead +c3 k u1) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) +(\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))) (\lambda (e1: +C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (drop O O (CHead c4 k u2) +(CHead e1 k0 u0))).(let H4 \def (f_equal C C (\lambda (e: C).(match e in C +return (\lambda (_: C).C) with [(CSort _) \Rightarrow c4 | (CHead c _ _) +\Rightarrow c])) (CHead c4 k u2) (CHead e1 k0 u0) (drop_gen_refl (CHead c4 k +u2) (CHead e1 k0 u0) H3)) in ((let H5 \def (f_equal C K (\lambda (e: +C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | +(CHead _ k1 _) \Rightarrow k1])) (CHead c4 k u2) (CHead e1 k0 u0) +(drop_gen_refl (CHead c4 k u2) (CHead e1 k0 u0) H3)) in ((let H6 \def +(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with +[(CSort _) \Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c4 k u2) +(CHead e1 k0 u0) (drop_gen_refl (CHead c4 k u2) (CHead e1 k0 u0) H3)) in +(\lambda (H7: (eq K k k0)).(\lambda (H8: (eq C c4 e1)).(eq_ind K k (\lambda +(k1: K).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c3 k +u1) (CHead e2 k1 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) +(\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))))) (eq_ind T u2 (\lambda (t: +T).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c3 k u1) +(CHead e2 k u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda +(_: C).(\lambda (u3: T).(pr0 u3 t))))) (eq_ind C c4 (\lambda (c: C).(ex3_2 C +T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c3 k u1) (CHead e2 k +u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 c))) (\lambda (_: +C).(\lambda (u3: T).(pr0 u3 u2))))) (ex3_2_intro C T (\lambda (e2: +C).(\lambda (u3: T).(drop O O (CHead c3 k u1) (CHead e2 k u3)))) (\lambda +(e2: C).(\lambda (_: T).(wcpr0 e2 c4))) (\lambda (_: C).(\lambda (u3: T).(pr0 +u3 u2))) c3 u1 (drop_refl (CHead c3 k u1)) H0 H2) e1 H8) u0 H6) k0 H7)))) +H5)) H4)))))) (K_ind (\lambda (k0: K).(\forall (n: nat).(((\forall (e1: +C).(\forall (u3: T).(\forall (k1: K).((drop n O (CHead c4 k0 u2) (CHead e1 k1 +u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c3 k0 +u1) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) +(\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))) \to (\forall (e1: +C).(\forall (u3: T).(\forall (k1: K).((drop (S n) O (CHead c4 k0 u2) (CHead +e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop (S n) O +(CHead c3 k0 u1) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 +e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))))) (\lambda (b: +B).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall +(k0: K).((drop n O (CHead c4 (Bind b) u2) (CHead e1 k0 u3)) \to (ex3_2 C T +(\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c3 (Bind b) u1) (CHead e2 +k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: +C).(\lambda (u4: T).(pr0 u4 u3)))))))))).(\lambda (e1: C).(\lambda (u0: +T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c4 (Bind b) u2) (CHead +e1 k0 u0))).(let H5 \def (H1 n e1 u0 k0 (drop_gen_drop (Bind b) c4 (CHead e1 +k0 u0) u2 n H4)) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop n +O c3 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) +(\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) (ex3_2 C T (\lambda (e2: +C).(\lambda (u3: T).(drop (S n) O (CHead c3 (Bind b) u1) (CHead e2 k0 u3)))) +(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda +(u3: T).(pr0 u3 u0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (drop +n O c3 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 x0 e1)).(\lambda (H8: (pr0 x1 +u0)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead +c3 (Bind b) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 +e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) x0 x1 (drop_drop +(Bind b) n c3 (CHead x0 k0 x1) H6 u1) H7 H8)))))) H5))))))))) (\lambda (f: +F).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall +(k0: K).((drop n O (CHead c4 (Flat f) u2) (CHead e1 k0 u3)) \to (ex3_2 C T +(\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c3 (Flat f) u1) (CHead e2 +k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: +C).(\lambda (u4: T).(pr0 u4 u3)))))))))).(\lambda (e1: C).(\lambda (u0: +T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c4 (Flat f) u2) (CHead +e1 k0 u0))).(let H5 \def (H1 (S n) e1 u0 k0 (drop_gen_drop (Flat f) c4 (CHead +e1 k0 u0) u2 n H4)) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop +(S n) O c3 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 +e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) (ex3_2 C T (\lambda (e2: +C).(\lambda (u3: T).(drop (S n) O (CHead c3 (Flat f) u1) (CHead e2 k0 u3)))) +(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda +(u3: T).(pr0 u3 u0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (drop +(S n) O c3 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 x0 e1)).(\lambda (H8: (pr0 +x1 u0)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O +(CHead c3 (Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: +T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) x0 x1 +(drop_drop (Flat f) n c3 (CHead x0 k0 x1) H6 u1) H7 H8)))))) H5))))))))) k) +h)))))))))) c2 c1 H))). +(* COMMENTS +Initial nodes: 1755 +END *) + +theorem wcpr0_getl: + \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (h: +nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((getl h c1 (CHead e1 +k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(getl h c2 (CHead e2 +k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: +C).(\lambda (u2: T).(pr0 u1 u2))))))))))) +\def + \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(wcpr0_ind +(\lambda (c: C).(\lambda (c0: C).(\forall (h: nat).(\forall (e1: C).(\forall +(u1: T).(\forall (k: K).((getl h c (CHead e1 k u1)) \to (ex3_2 C T (\lambda +(e2: C).(\lambda (u2: T).(getl h c0 (CHead e2 k u2)))) (\lambda (e2: +C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 +u2))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda +(u1: T).(\lambda (k: K).(\lambda (H0: (getl h c (CHead e1 k +u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(getl h c (CHead e2 +k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: +C).(\lambda (u2: T).(pr0 u1 u2))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl +u1)))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wcpr0 c3 +c4)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1: +T).(\forall (k: K).((getl h c3 (CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2: +C).(\lambda (u2: T).(getl h c4 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda +(_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 +u2))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 +u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall +(e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c3 k u1) (CHead e1 +k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c4 k +u2) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) +(\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))) (\lambda (e1: +C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (getl O (CHead c3 k u1) +(CHead e1 k0 u0))).(K_ind (\lambda (k1: K).((clear (CHead c3 k1 u1) (CHead e1 +k0 u0)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c4 k1 +u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) +(\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))))) (\lambda (b: B).(\lambda +(H4: (clear (CHead c3 (Bind b) u1) (CHead e1 k0 u0))).(let H5 \def (f_equal C +C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) +\Rightarrow e1 | (CHead c _ _) \Rightarrow c])) (CHead e1 k0 u0) (CHead c3 +(Bind b) u1) (clear_gen_bind b c3 (CHead e1 k0 u0) u1 H4)) in ((let H6 \def +(f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with +[(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead e1 k0 u0) +(CHead c3 (Bind b) u1) (clear_gen_bind b c3 (CHead e1 k0 u0) u1 H4)) in ((let +H7 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) +with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead e1 k0 +u0) (CHead c3 (Bind b) u1) (clear_gen_bind b c3 (CHead e1 k0 u0) u1 H4)) in +(\lambda (H8: (eq K k0 (Bind b))).(\lambda (H9: (eq C e1 c3)).(eq_ind_r K +(Bind b) (\lambda (k1: K).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl +O (CHead c4 (Bind b) u2) (CHead e2 k1 u3)))) (\lambda (e2: C).(\lambda (_: +T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))))) (eq_ind_r +T u1 (\lambda (t: T).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O +(CHead c4 (Bind b) u2) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda +(_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 t u3))))) +(eq_ind_r C c3 (\lambda (c: C).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: +T).(getl O (CHead c4 (Bind b) u2) (CHead e2 (Bind b) u3)))) (\lambda (e2: +C).(\lambda (_: T).(wcpr0 c e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 +u3))))) (ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c4 +(Bind b) u2) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda (_: +T).(wcpr0 c3 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))) c4 u2 +(getl_refl b c4 u2) H0 H2) e1 H9) u0 H7) k0 H8)))) H6)) H5)))) (\lambda (f: +F).(\lambda (H4: (clear (CHead c3 (Flat f) u1) (CHead e1 k0 u0))).(let H5 +\def (H1 O e1 u0 k0 (getl_intro O c3 (CHead e1 k0 u0) c3 (drop_refl c3) +(clear_gen_flat f c3 (CHead e1 k0 u0) u1 H4))) in (ex3_2_ind C T (\lambda +(e2: C).(\lambda (u3: T).(getl O c4 (CHead e2 k0 u3)))) (\lambda (e2: +C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 +u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c4 (Flat f) +u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) +(\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))) (\lambda (x0: C).(\lambda +(x1: T).(\lambda (H6: (getl O c4 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 e1 +x0)).(\lambda (H8: (pr0 u0 x1)).(ex3_2_intro C T (\lambda (e2: C).(\lambda +(u3: T).(getl O (CHead c4 (Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2: +C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 +u3))) x0 x1 (getl_flat c4 (CHead x0 k0 x1) O H6 f u2) H7 H8)))))) H5)))) k +(getl_gen_O (CHead c3 k u1) (CHead e1 k0 u0) H3)))))) (K_ind (\lambda (k0: +K).(\forall (n: nat).(((\forall (e1: C).(\forall (u3: T).(\forall (k1: +K).((getl n (CHead c3 k0 u1) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: +C).(\lambda (u4: T).(getl n (CHead c4 k0 u2) (CHead e2 k1 u4)))) (\lambda +(e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 +u3 u4))))))))) \to (\forall (e1: C).(\forall (u3: T).(\forall (k1: K).((getl +(S n) (CHead c3 k0 u1) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: +C).(\lambda (u4: T).(getl (S n) (CHead c4 k0 u2) (CHead e2 k1 u4)))) (\lambda +(e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 +u3 u4))))))))))) (\lambda (b: B).(\lambda (n: nat).(\lambda (_: ((\forall +(e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c3 (Bind b) u1) +(CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n +(CHead c4 (Bind b) u2) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: +T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 u3 +u4)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4: +(getl (S n) (CHead c3 (Bind b) u1) (CHead e1 k0 u0))).(let H5 \def (H1 n e1 +u0 k0 (getl_gen_S (Bind b) c3 (CHead e1 k0 u0) u1 n H4)) in (ex3_2_ind C T +(\lambda (e2: C).(\lambda (u3: T).(getl n c4 (CHead e2 k0 u3)))) (\lambda +(e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 +u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c4 +(Bind b) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 +e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))) (\lambda (x0: +C).(\lambda (x1: T).(\lambda (H6: (getl n c4 (CHead x0 k0 x1))).(\lambda (H7: +(wcpr0 e1 x0)).(\lambda (H8: (pr0 u0 x1)).(ex3_2_intro C T (\lambda (e2: +C).(\lambda (u3: T).(getl (S n) (CHead c4 (Bind b) u2) (CHead e2 k0 u3)))) +(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda +(u3: T).(pr0 u0 u3))) x0 x1 (getl_head (Bind b) n c4 (CHead x0 k0 x1) H6 u2) +H7 H8)))))) H5))))))))) (\lambda (f: F).(\lambda (n: nat).(\lambda (_: +((\forall (e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c3 (Flat +f) u1) (CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: +T).(getl n (CHead c4 (Flat f) u2) (CHead e2 k0 u4)))) (\lambda (e2: +C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 u3 +u4)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4: +(getl (S n) (CHead c3 (Flat f) u1) (CHead e1 k0 u0))).(let H5 \def (H1 (S n) +e1 u0 k0 (getl_gen_S (Flat f) c3 (CHead e1 k0 u0) u1 n H4)) in (ex3_2_ind C T +(\lambda (e2: C).(\lambda (u3: T).(getl (S n) c4 (CHead e2 k0 u3)))) (\lambda +(e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 +u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c4 +(Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 +e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))) (\lambda (x0: +C).(\lambda (x1: T).(\lambda (H6: (getl (S n) c4 (CHead x0 k0 x1))).(\lambda +(H7: (wcpr0 e1 x0)).(\lambda (H8: (pr0 u0 x1)).(ex3_2_intro C T (\lambda (e2: +C).(\lambda (u3: T).(getl (S n) (CHead c4 (Flat f) u2) (CHead e2 k0 u3)))) +(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda +(u3: T).(pr0 u0 u3))) x0 x1 (getl_head (Flat f) n c4 (CHead x0 k0 x1) H6 u2) +H7 H8)))))) H5))))))))) k) h)))))))))) c1 c2 H))). +(* COMMENTS +Initial nodes: 2103 +END *) + +theorem wcpr0_getl_back: + \forall (c1: C).(\forall (c2: C).((wcpr0 c2 c1) \to (\forall (h: +nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((getl h c1 (CHead e1 +k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(getl h c2 (CHead e2 +k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: +C).(\lambda (u2: T).(pr0 u2 u1))))))))))) +\def + \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c2 c1)).(wcpr0_ind +(\lambda (c: C).(\lambda (c0: C).(\forall (h: nat).(\forall (e1: C).(\forall +(u1: T).(\forall (k: K).((getl h c0 (CHead e1 k u1)) \to (ex3_2 C T (\lambda +(e2: C).(\lambda (u2: T).(getl h c (CHead e2 k u2)))) (\lambda (e2: +C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 +u1))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda +(u1: T).(\lambda (k: K).(\lambda (H0: (getl h c (CHead e1 k +u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(getl h c (CHead e2 +k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: +C).(\lambda (u2: T).(pr0 u2 u1))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl +u1)))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wcpr0 c3 +c4)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1: +T).(\forall (k: K).((getl h c4 (CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2: +C).(\lambda (u2: T).(getl h c3 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda +(_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 +u1))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 +u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall +(e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c4 k u2) (CHead e1 +k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c3 k +u1) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) +(\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))) (\lambda (e1: +C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (getl O (CHead c4 k u2) +(CHead e1 k0 u0))).(K_ind (\lambda (k1: K).((clear (CHead c4 k1 u2) (CHead e1 +k0 u0)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 k1 +u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) +(\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))))) (\lambda (b: B).(\lambda +(H4: (clear (CHead c4 (Bind b) u2) (CHead e1 k0 u0))).(let H5 \def (f_equal C +C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) +\Rightarrow e1 | (CHead c _ _) \Rightarrow c])) (CHead e1 k0 u0) (CHead c4 +(Bind b) u2) (clear_gen_bind b c4 (CHead e1 k0 u0) u2 H4)) in ((let H6 \def +(f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with +[(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead e1 k0 u0) +(CHead c4 (Bind b) u2) (clear_gen_bind b c4 (CHead e1 k0 u0) u2 H4)) in ((let +H7 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) +with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead e1 k0 +u0) (CHead c4 (Bind b) u2) (clear_gen_bind b c4 (CHead e1 k0 u0) u2 H4)) in +(\lambda (H8: (eq K k0 (Bind b))).(\lambda (H9: (eq C e1 c4)).(eq_ind_r K +(Bind b) (\lambda (k1: K).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl +O (CHead c3 (Bind b) u1) (CHead e2 k1 u3)))) (\lambda (e2: C).(\lambda (_: +T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))))) (eq_ind_r +T u2 (\lambda (t: T).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O +(CHead c3 (Bind b) u1) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda +(_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 t))))) +(eq_ind_r C c4 (\lambda (c: C).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: +T).(getl O (CHead c3 (Bind b) u1) (CHead e2 (Bind b) u3)))) (\lambda (e2: +C).(\lambda (_: T).(wcpr0 e2 c))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 +u2))))) (ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 +(Bind b) u1) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda (_: +T).(wcpr0 e2 c4))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u2))) c3 u1 +(getl_refl b c3 u1) H0 H2) e1 H9) u0 H7) k0 H8)))) H6)) H5)))) (\lambda (f: +F).(\lambda (H4: (clear (CHead c4 (Flat f) u2) (CHead e1 k0 u0))).(let H5 +\def (H1 O e1 u0 k0 (getl_intro O c4 (CHead e1 k0 u0) c4 (drop_refl c4) +(clear_gen_flat f c4 (CHead e1 k0 u0) u2 H4))) in (ex3_2_ind C T (\lambda +(e2: C).(\lambda (u3: T).(getl O c3 (CHead e2 k0 u3)))) (\lambda (e2: +C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 +u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 (Flat f) +u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) +(\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))) (\lambda (x0: C).(\lambda +(x1: T).(\lambda (H6: (getl O c3 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 x0 +e1)).(\lambda (H8: (pr0 x1 u0)).(ex3_2_intro C T (\lambda (e2: C).(\lambda +(u3: T).(getl O (CHead c3 (Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2: +C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 +u0))) x0 x1 (getl_flat c3 (CHead x0 k0 x1) O H6 f u1) H7 H8)))))) H5)))) k +(getl_gen_O (CHead c4 k u2) (CHead e1 k0 u0) H3)))))) (K_ind (\lambda (k0: +K).(\forall (n: nat).(((\forall (e1: C).(\forall (u3: T).(\forall (k1: +K).((getl n (CHead c4 k0 u2) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: +C).(\lambda (u4: T).(getl n (CHead c3 k0 u1) (CHead e2 k1 u4)))) (\lambda +(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 +u4 u3))))))))) \to (\forall (e1: C).(\forall (u3: T).(\forall (k1: K).((getl +(S n) (CHead c4 k0 u2) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: +C).(\lambda (u4: T).(getl (S n) (CHead c3 k0 u1) (CHead e2 k1 u4)))) (\lambda +(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 +u4 u3))))))))))) (\lambda (b: B).(\lambda (n: nat).(\lambda (_: ((\forall +(e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c4 (Bind b) u2) +(CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n +(CHead c3 (Bind b) u1) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: +T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 u4 +u3)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4: +(getl (S n) (CHead c4 (Bind b) u2) (CHead e1 k0 u0))).(let H5 \def (H1 n e1 +u0 k0 (getl_gen_S (Bind b) c4 (CHead e1 k0 u0) u2 n H4)) in (ex3_2_ind C T +(\lambda (e2: C).(\lambda (u3: T).(getl n c3 (CHead e2 k0 u3)))) (\lambda +(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 +u3 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c3 +(Bind b) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 +e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))) (\lambda (x0: +C).(\lambda (x1: T).(\lambda (H6: (getl n c3 (CHead x0 k0 x1))).(\lambda (H7: +(wcpr0 x0 e1)).(\lambda (H8: (pr0 x1 u0)).(ex3_2_intro C T (\lambda (e2: +C).(\lambda (u3: T).(getl (S n) (CHead c3 (Bind b) u1) (CHead e2 k0 u3)))) +(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda +(u3: T).(pr0 u3 u0))) x0 x1 (getl_head (Bind b) n c3 (CHead x0 k0 x1) H6 u1) +H7 H8)))))) H5))))))))) (\lambda (f: F).(\lambda (n: nat).(\lambda (_: +((\forall (e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c4 (Flat +f) u2) (CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: +T).(getl n (CHead c3 (Flat f) u1) (CHead e2 k0 u4)))) (\lambda (e2: +C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 u4 +u3)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4: +(getl (S n) (CHead c4 (Flat f) u2) (CHead e1 k0 u0))).(let H5 \def (H1 (S n) +e1 u0 k0 (getl_gen_S (Flat f) c4 (CHead e1 k0 u0) u2 n H4)) in (ex3_2_ind C T +(\lambda (e2: C).(\lambda (u3: T).(getl (S n) c3 (CHead e2 k0 u3)))) (\lambda +(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 +u3 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c3 +(Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 +e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))) (\lambda (x0: +C).(\lambda (x1: T).(\lambda (H6: (getl (S n) c3 (CHead x0 k0 x1))).(\lambda +(H7: (wcpr0 x0 e1)).(\lambda (H8: (pr0 x1 u0)).(ex3_2_intro C T (\lambda (e2: +C).(\lambda (u3: T).(getl (S n) (CHead c3 (Flat f) u1) (CHead e2 k0 u3)))) +(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda +(u3: T).(pr0 u3 u0))) x0 x1 (getl_head (Flat f) n c3 (CHead x0 k0 x1) H6 u1) +H7 H8)))))) H5))))))))) k) h)))))))))) c2 c1 H))). +(* COMMENTS +Initial nodes: 2103 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/wf3/clear.ma b/matita/matita/contribs/lambdadelta/basic_1/wf3/clear.ma new file mode 100644 index 000000000..b2ddd47af --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/wf3/clear.ma @@ -0,0 +1,91 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/wf3/fwd.ma". + +theorem wf3_clear_conf: + \forall (c1: C).(\forall (c: C).((clear c1 c) \to (\forall (g: G).(\forall +(c2: C).((wf3 g c1 c2) \to (wf3 g c c2)))))) +\def + \lambda (c1: C).(\lambda (c: C).(\lambda (H: (clear c1 c)).(clear_ind +(\lambda (c0: C).(\lambda (c2: C).(\forall (g: G).(\forall (c3: C).((wf3 g c0 +c3) \to (wf3 g c2 c3)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (u: +T).(\lambda (g: G).(\lambda (c2: C).(\lambda (H0: (wf3 g (CHead e (Bind b) u) +c2)).H0)))))) (\lambda (e: C).(\lambda (c0: C).(\lambda (_: (clear e +c0)).(\lambda (H1: ((\forall (g: G).(\forall (c2: C).((wf3 g e c2) \to (wf3 g +c0 c2)))))).(\lambda (f: F).(\lambda (u: T).(\lambda (g: G).(\lambda (c2: +C).(\lambda (H2: (wf3 g (CHead e (Flat f) u) c2)).(let H_y \def +(wf3_gen_flat1 g e c2 u f H2) in (H1 g c2 H_y))))))))))) c1 c H))). +(* COMMENTS +Initial nodes: 145 +END *) + +theorem clear_wf3_trans: + \forall (c1: C).(\forall (d1: C).((clear c1 d1) \to (\forall (g: G).(\forall +(d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda +(c2: C).(clear c2 d2)))))))) +\def + \lambda (c1: C).(\lambda (d1: C).(\lambda (H: (clear c1 d1)).(clear_ind +(\lambda (c: C).(\lambda (c0: C).(\forall (g: G).(\forall (d2: C).((wf3 g c0 +d2) \to (ex2 C (\lambda (c2: C).(wf3 g c c2)) (\lambda (c2: C).(clear c2 +d2)))))))) (\lambda (b: B).(\lambda (e: C).(\lambda (u: T).(\lambda (g: +G).(\lambda (d2: C).(\lambda (H0: (wf3 g (CHead e (Bind b) u) d2)).(let H_x +\def (wf3_gen_bind1 g e d2 u b H0) in (let H1 \def H_x in (or_ind (ex3_2 C T +(\lambda (c2: C).(\lambda (_: T).(eq C d2 (CHead c2 (Bind b) u)))) (\lambda +(c2: C).(\lambda (_: T).(wf3 g e c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g +e u w)))) (ex3 C (\lambda (c2: C).(eq C d2 (CHead c2 (Bind Void) (TSort O)))) +(\lambda (c2: C).(wf3 g e c2)) (\lambda (_: C).(\forall (w: T).((ty3 g e u w) +\to False)))) (ex2 C (\lambda (c2: C).(wf3 g (CHead e (Bind b) u) c2)) +(\lambda (c2: C).(clear c2 d2))) (\lambda (H2: (ex3_2 C T (\lambda (c2: +C).(\lambda (_: T).(eq C d2 (CHead c2 (Bind b) u)))) (\lambda (c2: +C).(\lambda (_: T).(wf3 g e c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g e u +w))))).(ex3_2_ind C T (\lambda (c2: C).(\lambda (_: T).(eq C d2 (CHead c2 +(Bind b) u)))) (\lambda (c2: C).(\lambda (_: T).(wf3 g e c2))) (\lambda (_: +C).(\lambda (w: T).(ty3 g e u w))) (ex2 C (\lambda (c2: C).(wf3 g (CHead e +(Bind b) u) c2)) (\lambda (c2: C).(clear c2 d2))) (\lambda (x0: C).(\lambda +(x1: T).(\lambda (H3: (eq C d2 (CHead x0 (Bind b) u))).(\lambda (H4: (wf3 g e +x0)).(\lambda (H5: (ty3 g e u x1)).(eq_ind_r C (CHead x0 (Bind b) u) (\lambda +(c: C).(ex2 C (\lambda (c2: C).(wf3 g (CHead e (Bind b) u) c2)) (\lambda (c2: +C).(clear c2 c)))) (ex_intro2 C (\lambda (c2: C).(wf3 g (CHead e (Bind b) u) +c2)) (\lambda (c2: C).(clear c2 (CHead x0 (Bind b) u))) (CHead x0 (Bind b) u) +(wf3_bind g e x0 H4 u x1 H5 b) (clear_bind b x0 u)) d2 H3)))))) H2)) (\lambda +(H2: (ex3 C (\lambda (c2: C).(eq C d2 (CHead c2 (Bind Void) (TSort O)))) +(\lambda (c2: C).(wf3 g e c2)) (\lambda (_: C).(\forall (w: T).((ty3 g e u w) +\to False))))).(ex3_ind C (\lambda (c2: C).(eq C d2 (CHead c2 (Bind Void) +(TSort O)))) (\lambda (c2: C).(wf3 g e c2)) (\lambda (_: C).(\forall (w: +T).((ty3 g e u w) \to False))) (ex2 C (\lambda (c2: C).(wf3 g (CHead e (Bind +b) u) c2)) (\lambda (c2: C).(clear c2 d2))) (\lambda (x0: C).(\lambda (H3: +(eq C d2 (CHead x0 (Bind Void) (TSort O)))).(\lambda (H4: (wf3 g e +x0)).(\lambda (H5: ((\forall (w: T).((ty3 g e u w) \to False)))).(eq_ind_r C +(CHead x0 (Bind Void) (TSort O)) (\lambda (c: C).(ex2 C (\lambda (c2: C).(wf3 +g (CHead e (Bind b) u) c2)) (\lambda (c2: C).(clear c2 c)))) (ex_intro2 C +(\lambda (c2: C).(wf3 g (CHead e (Bind b) u) c2)) (\lambda (c2: C).(clear c2 +(CHead x0 (Bind Void) (TSort O)))) (CHead x0 (Bind Void) (TSort O)) (wf3_void +g e x0 H4 u H5 b) (clear_bind Void x0 (TSort O))) d2 H3))))) H2)) H1))))))))) +(\lambda (e: C).(\lambda (c: C).(\lambda (_: (clear e c)).(\lambda (H1: +((\forall (g: G).(\forall (d2: C).((wf3 g c d2) \to (ex2 C (\lambda (c2: +C).(wf3 g e c2)) (\lambda (c2: C).(clear c2 d2)))))))).(\lambda (f: +F).(\lambda (u: T).(\lambda (g: G).(\lambda (d2: C).(\lambda (H2: (wf3 g c +d2)).(let H_x \def (H1 g d2 H2) in (let H3 \def H_x in (ex2_ind C (\lambda +(c2: C).(wf3 g e c2)) (\lambda (c2: C).(clear c2 d2)) (ex2 C (\lambda (c2: +C).(wf3 g (CHead e (Flat f) u) c2)) (\lambda (c2: C).(clear c2 d2))) (\lambda +(x: C).(\lambda (H4: (wf3 g e x)).(\lambda (H5: (clear x d2)).(ex_intro2 C +(\lambda (c2: C).(wf3 g (CHead e (Flat f) u) c2)) (\lambda (c2: C).(clear c2 +d2)) x (wf3_flat g e x H4 u f) H5)))) H3)))))))))))) c1 d1 H))). +(* COMMENTS +Initial nodes: 1023 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/wf3/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/wf3/defs.ma new file mode 100644 index 000000000..a99661941 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/wf3/defs.ma @@ -0,0 +1,29 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/ty3/defs.ma". + +inductive wf3 (g: G): C \to (C \to Prop) \def +| wf3_sort: \forall (m: nat).(wf3 g (CSort m) (CSort m)) +| wf3_bind: \forall (c1: C).(\forall (c2: C).((wf3 g c1 c2) \to (\forall (u: +T).(\forall (t: T).((ty3 g c1 u t) \to (\forall (b: B).(wf3 g (CHead c1 (Bind +b) u) (CHead c2 (Bind b) u)))))))) +| wf3_void: \forall (c1: C).(\forall (c2: C).((wf3 g c1 c2) \to (\forall (u: +T).(((\forall (t: T).((ty3 g c1 u t) \to False))) \to (\forall (b: B).(wf3 g +(CHead c1 (Bind b) u) (CHead c2 (Bind Void) (TSort O)))))))) +| wf3_flat: \forall (c1: C).(\forall (c2: C).((wf3 g c1 c2) \to (\forall (u: +T).(\forall (f: F).(wf3 g (CHead c1 (Flat f) u) c2))))). + diff --git a/matita/matita/contribs/lambdadelta/basic_1/wf3/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/wf3/fwd.ma new file mode 100644 index 000000000..71c903f06 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/wf3/fwd.ma @@ -0,0 +1,311 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/wf3/defs.ma". + +theorem wf3_gen_sort1: + \forall (g: G).(\forall (x: C).(\forall (m: nat).((wf3 g (CSort m) x) \to +(eq C x (CSort m))))) +\def + \lambda (g: G).(\lambda (x: C).(\lambda (m: nat).(\lambda (H: (wf3 g (CSort +m) x)).(insert_eq C (CSort m) (\lambda (c: C).(wf3 g c x)) (\lambda (c: +C).(eq C x c)) (\lambda (y: C).(\lambda (H0: (wf3 g y x)).(wf3_ind g (\lambda +(c: C).(\lambda (c0: C).((eq C c (CSort m)) \to (eq C c0 c)))) (\lambda (m0: +nat).(\lambda (H1: (eq C (CSort m0) (CSort m))).(let H2 \def (f_equal C nat +(\lambda (e: C).(match e in C return (\lambda (_: C).nat) with [(CSort n) +\Rightarrow n | (CHead _ _ _) \Rightarrow m0])) (CSort m0) (CSort m) H1) in +(eq_ind_r nat m (\lambda (n: nat).(eq C (CSort n) (CSort n))) (refl_equal C +(CSort m)) m0 H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (wf3 g c1 +c2)).(\lambda (_: (((eq C c1 (CSort m)) \to (eq C c2 c1)))).(\lambda (u: +T).(\lambda (t: T).(\lambda (_: (ty3 g c1 u t)).(\lambda (b: B).(\lambda (H4: +(eq C (CHead c1 (Bind b) u) (CSort m))).(let H5 \def (eq_ind C (CHead c1 +(Bind b) u) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with +[(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort m) +H4) in (False_ind (eq C (CHead c2 (Bind b) u) (CHead c1 (Bind b) u)) +H5))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (wf3 g c1 +c2)).(\lambda (_: (((eq C c1 (CSort m)) \to (eq C c2 c1)))).(\lambda (u: +T).(\lambda (_: ((\forall (t: T).((ty3 g c1 u t) \to False)))).(\lambda (b: +B).(\lambda (H4: (eq C (CHead c1 (Bind b) u) (CSort m))).(let H5 \def (eq_ind +C (CHead c1 (Bind b) u) (\lambda (ee: C).(match ee in C return (\lambda (_: +C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow +True])) I (CSort m) H4) in (False_ind (eq C (CHead c2 (Bind Void) (TSort O)) +(CHead c1 (Bind b) u)) H5)))))))))) (\lambda (c1: C).(\lambda (c2: +C).(\lambda (_: (wf3 g c1 c2)).(\lambda (_: (((eq C c1 (CSort m)) \to (eq C +c2 c1)))).(\lambda (u: T).(\lambda (f: F).(\lambda (H3: (eq C (CHead c1 (Flat +f) u) (CSort m))).(let H4 \def (eq_ind C (CHead c1 (Flat f) u) (\lambda (ee: +C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow +False | (CHead _ _ _) \Rightarrow True])) I (CSort m) H3) in (False_ind (eq C +c2 (CHead c1 (Flat f) u)) H4))))))))) y x H0))) H)))). +(* COMMENTS +Initial nodes: 523 +END *) + +theorem wf3_gen_bind1: + \forall (g: G).(\forall (c1: C).(\forall (x: C).(\forall (v: T).(\forall (b: +B).((wf3 g (CHead c1 (Bind b) v) x) \to (or (ex3_2 C T (\lambda (c2: +C).(\lambda (_: T).(eq C x (CHead c2 (Bind b) v)))) (\lambda (c2: C).(\lambda +(_: T).(wf3 g c1 c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 +C (\lambda (c2: C).(eq C x (CHead c2 (Bind Void) (TSort O)))) (\lambda (c2: +C).(wf3 g c1 c2)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to +False)))))))))) +\def + \lambda (g: G).(\lambda (c1: C).(\lambda (x: C).(\lambda (v: T).(\lambda (b: +B).(\lambda (H: (wf3 g (CHead c1 (Bind b) v) x)).(insert_eq C (CHead c1 (Bind +b) v) (\lambda (c: C).(wf3 g c x)) (\lambda (_: C).(or (ex3_2 C T (\lambda +(c2: C).(\lambda (_: T).(eq C x (CHead c2 (Bind b) v)))) (\lambda (c2: +C).(\lambda (_: T).(wf3 g c1 c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 +v w)))) (ex3 C (\lambda (c2: C).(eq C x (CHead c2 (Bind Void) (TSort O)))) +(\lambda (c2: C).(wf3 g c1 c2)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v +w) \to False)))))) (\lambda (y: C).(\lambda (H0: (wf3 g y x)).(wf3_ind g +(\lambda (c: C).(\lambda (c0: C).((eq C c (CHead c1 (Bind b) v)) \to (or +(ex3_2 C T (\lambda (c2: C).(\lambda (_: T).(eq C c0 (CHead c2 (Bind b) v)))) +(\lambda (c2: C).(\lambda (_: T).(wf3 g c1 c2))) (\lambda (_: C).(\lambda (w: +T).(ty3 g c1 v w)))) (ex3 C (\lambda (c2: C).(eq C c0 (CHead c2 (Bind Void) +(TSort O)))) (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (_: C).(\forall (w: +T).((ty3 g c1 v w) \to False)))))))) (\lambda (m: nat).(\lambda (H1: (eq C +(CSort m) (CHead c1 (Bind b) v))).(let H2 \def (eq_ind C (CSort m) (\lambda +(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) +\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead c1 (Bind b) v) +H1) in (False_ind (or (ex3_2 C T (\lambda (c2: C).(\lambda (_: T).(eq C +(CSort m) (CHead c2 (Bind b) v)))) (\lambda (c2: C).(\lambda (_: T).(wf3 g c1 +c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c2: +C).(eq C (CSort m) (CHead c2 (Bind Void) (TSort O)))) (\lambda (c2: C).(wf3 g +c1 c2)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to False))))) H2)))) +(\lambda (c0: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c0 c2)).(\lambda (H2: +(((eq C c0 (CHead c1 (Bind b) v)) \to (or (ex3_2 C T (\lambda (c3: +C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda (c3: +C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 +v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort O)))) +(\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v +w) \to False)))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H3: (ty3 g c0 +u t)).(\lambda (b0: B).(\lambda (H4: (eq C (CHead c0 (Bind b0) u) (CHead c1 +(Bind b) v))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return +(\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow +c])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) H4) in ((let H6 \def +(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with +[(CSort _) \Rightarrow b0 | (CHead _ k _) \Rightarrow (match k in K return +(\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow +b0])])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) H4) in ((let H7 \def +(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with +[(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead c0 (Bind +b0) u) (CHead c1 (Bind b) v) H4) in (\lambda (H8: (eq B b0 b)).(\lambda (H9: +(eq C c0 c1)).(eq_ind_r B b (\lambda (b1: B).(or (ex3_2 C T (\lambda (c3: +C).(\lambda (_: T).(eq C (CHead c2 (Bind b1) u) (CHead c3 (Bind b) v)))) +(\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: +T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C (CHead c2 (Bind b1) u) +(CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda +(_: C).(\forall (w: T).((ty3 g c1 v w) \to False)))))) (let H10 \def (eq_ind +T u (\lambda (t0: T).(ty3 g c0 t0 t)) H3 v H7) in (eq_ind_r T v (\lambda (t0: +T).(or (ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C (CHead c2 (Bind b) +t0) (CHead c3 (Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) +(\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq +C (CHead c2 (Bind b) t0) (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: +C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to +False)))))) (let H11 \def (eq_ind C c0 (\lambda (c: C).(ty3 g c v t)) H10 c1 +H9) in (let H12 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 (Bind b) +v)) \to (or (ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 +(Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: +C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead +c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: +C).(\forall (w: T).((ty3 g c1 v w) \to False))))))) H2 c1 H9) in (let H13 +\def (eq_ind C c0 (\lambda (c: C).(wf3 g c c2)) H1 c1 H9) in (or_introl +(ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C (CHead c2 (Bind b) v) +(CHead c3 (Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) +(\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq +C (CHead c2 (Bind b) v) (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: +C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to +False)))) (ex3_2_intro C T (\lambda (c3: C).(\lambda (_: T).(eq C (CHead c2 +(Bind b) v) (CHead c3 (Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g +c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w))) c2 t (refl_equal C +(CHead c2 (Bind b) v)) H13 H11))))) u H7)) b0 H8)))) H6)) H5))))))))))) +(\lambda (c0: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c0 c2)).(\lambda (H2: +(((eq C c0 (CHead c1 (Bind b) v)) \to (or (ex3_2 C T (\lambda (c3: +C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda (c3: +C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 +v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort O)))) +(\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v +w) \to False)))))))).(\lambda (u: T).(\lambda (H3: ((\forall (t: T).((ty3 g +c0 u t) \to False)))).(\lambda (b0: B).(\lambda (H4: (eq C (CHead c0 (Bind +b0) u) (CHead c1 (Bind b) v))).(let H5 \def (f_equal C C (\lambda (e: +C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | +(CHead c _ _) \Rightarrow c])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) +H4) in ((let H6 \def (f_equal C B (\lambda (e: C).(match e in C return +(\lambda (_: C).B) with [(CSort _) \Rightarrow b0 | (CHead _ k _) \Rightarrow +(match k in K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | +(Flat _) \Rightarrow b0])])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) H4) +in ((let H7 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda +(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) +(CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) H4) in (\lambda (_: (eq B b0 +b)).(\lambda (H9: (eq C c0 c1)).(let H10 \def (eq_ind T u (\lambda (t: +T).(\forall (t0: T).((ty3 g c0 t t0) \to False))) H3 v H7) in (let H11 \def +(eq_ind C c0 (\lambda (c: C).(\forall (t: T).((ty3 g c v t) \to False))) H10 +c1 H9) in (let H12 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 (Bind +b) v)) \to (or (ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 +(Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: +C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead +c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: +C).(\forall (w: T).((ty3 g c1 v w) \to False))))))) H2 c1 H9) in (let H13 +\def (eq_ind C c0 (\lambda (c: C).(wf3 g c c2)) H1 c1 H9) in (or_intror +(ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C (CHead c2 (Bind Void) +(TSort O)) (CHead c3 (Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g +c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda +(c3: C).(eq C (CHead c2 (Bind Void) (TSort O)) (CHead c3 (Bind Void) (TSort +O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g +c1 v w) \to False)))) (ex3_intro C (\lambda (c3: C).(eq C (CHead c2 (Bind +Void) (TSort O)) (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g +c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to False))) c2 +(refl_equal C (CHead c2 (Bind Void) (TSort O))) H13 H11))))))))) H6)) +H5)))))))))) (\lambda (c0: C).(\lambda (c2: C).(\lambda (_: (wf3 g c0 +c2)).(\lambda (_: (((eq C c0 (CHead c1 (Bind b) v)) \to (or (ex3_2 C T +(\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda +(c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 +g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort +O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g +c1 v w) \to False)))))))).(\lambda (u: T).(\lambda (f: F).(\lambda (H3: (eq C +(CHead c0 (Flat f) u) (CHead c1 (Bind b) v))).(let H4 \def (eq_ind C (CHead +c0 (Flat f) u) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) +with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K +return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) +\Rightarrow True])])) I (CHead c1 (Bind b) v) H3) in (False_ind (or (ex3_2 C +T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda +(c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 +g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort +O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g +c1 v w) \to False))))) H4))))))))) y x H0))) H)))))). +(* COMMENTS +Initial nodes: 2507 +END *) + +theorem wf3_gen_flat1: + \forall (g: G).(\forall (c1: C).(\forall (x: C).(\forall (v: T).(\forall (f: +F).((wf3 g (CHead c1 (Flat f) v) x) \to (wf3 g c1 x)))))) +\def + \lambda (g: G).(\lambda (c1: C).(\lambda (x: C).(\lambda (v: T).(\lambda (f: +F).(\lambda (H: (wf3 g (CHead c1 (Flat f) v) x)).(insert_eq C (CHead c1 (Flat +f) v) (\lambda (c: C).(wf3 g c x)) (\lambda (_: C).(wf3 g c1 x)) (\lambda (y: +C).(\lambda (H0: (wf3 g y x)).(wf3_ind g (\lambda (c: C).(\lambda (c0: +C).((eq C c (CHead c1 (Flat f) v)) \to (wf3 g c1 c0)))) (\lambda (m: +nat).(\lambda (H1: (eq C (CSort m) (CHead c1 (Flat f) v))).(let H2 \def +(eq_ind C (CSort m) (\lambda (ee: C).(match ee in C return (\lambda (_: +C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow +False])) I (CHead c1 (Flat f) v) H1) in (False_ind (wf3 g c1 (CSort m)) +H2)))) (\lambda (c0: C).(\lambda (c2: C).(\lambda (_: (wf3 g c0 c2)).(\lambda +(_: (((eq C c0 (CHead c1 (Flat f) v)) \to (wf3 g c1 c2)))).(\lambda (u: +T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (b: B).(\lambda (H4: +(eq C (CHead c0 (Bind b) u) (CHead c1 (Flat f) v))).(let H5 \def (eq_ind C +(CHead c0 (Bind b) u) (\lambda (ee: C).(match ee in C return (\lambda (_: +C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match +k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat +_) \Rightarrow False])])) I (CHead c1 (Flat f) v) H4) in (False_ind (wf3 g c1 +(CHead c2 (Bind b) u)) H5))))))))))) (\lambda (c0: C).(\lambda (c2: +C).(\lambda (_: (wf3 g c0 c2)).(\lambda (_: (((eq C c0 (CHead c1 (Flat f) v)) +\to (wf3 g c1 c2)))).(\lambda (u: T).(\lambda (_: ((\forall (t: T).((ty3 g c0 +u t) \to False)))).(\lambda (b: B).(\lambda (H4: (eq C (CHead c0 (Bind b) u) +(CHead c1 (Flat f) v))).(let H5 \def (eq_ind C (CHead c0 (Bind b) u) (\lambda +(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) +\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda +(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow +False])])) I (CHead c1 (Flat f) v) H4) in (False_ind (wf3 g c1 (CHead c2 +(Bind Void) (TSort O))) H5)))))))))) (\lambda (c0: C).(\lambda (c2: +C).(\lambda (H1: (wf3 g c0 c2)).(\lambda (H2: (((eq C c0 (CHead c1 (Flat f) +v)) \to (wf3 g c1 c2)))).(\lambda (u: T).(\lambda (f0: F).(\lambda (H3: (eq C +(CHead c0 (Flat f0) u) (CHead c1 (Flat f) v))).(let H4 \def (f_equal C C +(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) +\Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 (Flat f0) u) (CHead +c1 (Flat f) v) H3) in ((let H5 \def (f_equal C F (\lambda (e: C).(match e in +C return (\lambda (_: C).F) with [(CSort _) \Rightarrow f0 | (CHead _ k _) +\Rightarrow (match k in K return (\lambda (_: K).F) with [(Bind _) +\Rightarrow f0 | (Flat f1) \Rightarrow f1])])) (CHead c0 (Flat f0) u) (CHead +c1 (Flat f) v) H3) in ((let H6 \def (f_equal C T (\lambda (e: C).(match e in +C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) +\Rightarrow t])) (CHead c0 (Flat f0) u) (CHead c1 (Flat f) v) H3) in (\lambda +(_: (eq F f0 f)).(\lambda (H8: (eq C c0 c1)).(let H9 \def (eq_ind C c0 +(\lambda (c: C).((eq C c (CHead c1 (Flat f) v)) \to (wf3 g c1 c2))) H2 c1 H8) +in (let H10 \def (eq_ind C c0 (\lambda (c: C).(wf3 g c c2)) H1 c1 H8) in +H10))))) H5)) H4))))))))) y x H0))) H)))))). +(* COMMENTS +Initial nodes: 737 +END *) + +theorem wf3_gen_head2: + \forall (g: G).(\forall (x: C).(\forall (c: C).(\forall (v: T).(\forall (k: +K).((wf3 g x (CHead c k v)) \to (ex B (\lambda (b: B).(eq K k (Bind b))))))))) +\def + \lambda (g: G).(\lambda (x: C).(\lambda (c: C).(\lambda (v: T).(\lambda (k: +K).(\lambda (H: (wf3 g x (CHead c k v))).(insert_eq C (CHead c k v) (\lambda +(c0: C).(wf3 g x c0)) (\lambda (_: C).(ex B (\lambda (b: B).(eq K k (Bind +b))))) (\lambda (y: C).(\lambda (H0: (wf3 g x y)).(wf3_ind g (\lambda (_: +C).(\lambda (c1: C).((eq C c1 (CHead c k v)) \to (ex B (\lambda (b: B).(eq K +k (Bind b))))))) (\lambda (m: nat).(\lambda (H1: (eq C (CSort m) (CHead c k +v))).(let H2 \def (eq_ind C (CSort m) (\lambda (ee: C).(match ee in C return +(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) +\Rightarrow False])) I (CHead c k v) H1) in (False_ind (ex B (\lambda (b: +B).(eq K k (Bind b)))) H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: +(wf3 g c1 c2)).(\lambda (H2: (((eq C c2 (CHead c k v)) \to (ex B (\lambda (b: +B).(eq K k (Bind b))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H3: (ty3 +g c1 u t)).(\lambda (b: B).(\lambda (H4: (eq C (CHead c2 (Bind b) u) (CHead c +k v))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return +(\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ _) +\Rightarrow c0])) (CHead c2 (Bind b) u) (CHead c k v) H4) in ((let H6 \def +(f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with +[(CSort _) \Rightarrow (Bind b) | (CHead _ k0 _) \Rightarrow k0])) (CHead c2 +(Bind b) u) (CHead c k v) H4) in ((let H7 \def (f_equal C T (\lambda (e: +C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | +(CHead _ _ t0) \Rightarrow t0])) (CHead c2 (Bind b) u) (CHead c k v) H4) in +(\lambda (H8: (eq K (Bind b) k)).(\lambda (H9: (eq C c2 c)).(let H10 \def +(eq_ind T u (\lambda (t0: T).(ty3 g c1 t0 t)) H3 v H7) in (let H11 \def +(eq_ind C c2 (\lambda (c0: C).((eq C c0 (CHead c k v)) \to (ex B (\lambda +(b0: B).(eq K k (Bind b0)))))) H2 c H9) in (let H12 \def (eq_ind C c2 +(\lambda (c0: C).(wf3 g c1 c0)) H1 c H9) in (let H13 \def (eq_ind_r K k +(\lambda (k0: K).((eq C c (CHead c k0 v)) \to (ex B (\lambda (b0: B).(eq K k0 +(Bind b0)))))) H11 (Bind b) H8) in (eq_ind K (Bind b) (\lambda (k0: K).(ex B +(\lambda (b0: B).(eq K k0 (Bind b0))))) (ex_intro B (\lambda (b0: B).(eq K +(Bind b) (Bind b0))) b (refl_equal K (Bind b))) k H8)))))))) H6)) +H5))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c1 +c2)).(\lambda (H2: (((eq C c2 (CHead c k v)) \to (ex B (\lambda (b: B).(eq K +k (Bind b))))))).(\lambda (u: T).(\lambda (_: ((\forall (t: T).((ty3 g c1 u +t) \to False)))).(\lambda (_: B).(\lambda (H4: (eq C (CHead c2 (Bind Void) +(TSort O)) (CHead c k v))).(let H5 \def (f_equal C C (\lambda (e: C).(match e +in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ +_) \Rightarrow c0])) (CHead c2 (Bind Void) (TSort O)) (CHead c k v) H4) in +((let H6 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: +C).K) with [(CSort _) \Rightarrow (Bind Void) | (CHead _ k0 _) \Rightarrow +k0])) (CHead c2 (Bind Void) (TSort O)) (CHead c k v) H4) in ((let H7 \def +(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with +[(CSort _) \Rightarrow (TSort O) | (CHead _ _ t) \Rightarrow t])) (CHead c2 +(Bind Void) (TSort O)) (CHead c k v) H4) in (\lambda (H8: (eq K (Bind Void) +k)).(\lambda (H9: (eq C c2 c)).(let H10 \def (eq_ind C c2 (\lambda (c0: +C).((eq C c0 (CHead c k v)) \to (ex B (\lambda (b0: B).(eq K k (Bind b0)))))) +H2 c H9) in (let H11 \def (eq_ind C c2 (\lambda (c0: C).(wf3 g c1 c0)) H1 c +H9) in (let H12 \def (eq_ind_r K k (\lambda (k0: K).((eq C c (CHead c k0 v)) +\to (ex B (\lambda (b0: B).(eq K k0 (Bind b0)))))) H10 (Bind Void) H8) in +(eq_ind K (Bind Void) (\lambda (k0: K).(ex B (\lambda (b0: B).(eq K k0 (Bind +b0))))) (let H13 \def (eq_ind_r T v (\lambda (t: T).((eq C c (CHead c (Bind +Void) t)) \to (ex B (\lambda (b0: B).(eq K (Bind Void) (Bind b0)))))) H12 +(TSort O) H7) in (ex_intro B (\lambda (b0: B).(eq K (Bind Void) (Bind b0))) +Void (refl_equal K (Bind Void)))) k H8))))))) H6)) H5)))))))))) (\lambda (c1: +C).(\lambda (c2: C).(\lambda (H1: (wf3 g c1 c2)).(\lambda (H2: (((eq C c2 +(CHead c k v)) \to (ex B (\lambda (b: B).(eq K k (Bind b))))))).(\lambda (_: +T).(\lambda (_: F).(\lambda (H3: (eq C c2 (CHead c k v))).(let H4 \def +(f_equal C C (\lambda (e: C).e) c2 (CHead c k v) H3) in (let H5 \def (eq_ind +C c2 (\lambda (c0: C).((eq C c0 (CHead c k v)) \to (ex B (\lambda (b: B).(eq +K k (Bind b)))))) H2 (CHead c k v) H4) in (let H6 \def (eq_ind C c2 (\lambda +(c0: C).(wf3 g c1 c0)) H1 (CHead c k v) H4) in (H5 (refl_equal C (CHead c k +v))))))))))))) x y H0))) H)))))). +(* COMMENTS +Initial nodes: 1225 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/wf3/getl.ma b/matita/matita/contribs/lambdadelta/basic_1/wf3/getl.ma new file mode 100644 index 000000000..8a8f13780 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/wf3/getl.ma @@ -0,0 +1,205 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/wf3/clear.ma". + +include "Basic-1/ty3/dec.ma". + +theorem wf3_getl_conf: + \forall (b: B).(\forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall +(v: T).((getl i c1 (CHead d1 (Bind b) v)) \to (\forall (g: G).(\forall (c2: +C).((wf3 g c1 c2) \to (\forall (w: T).((ty3 g d1 v w) \to (ex2 C (\lambda +(d2: C).(getl i c2 (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 +d2))))))))))))) +\def + \lambda (b: B).(\lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: +C).(\forall (d1: C).(\forall (v: T).((getl n c1 (CHead d1 (Bind b) v)) \to +(\forall (g: G).(\forall (c2: C).((wf3 g c1 c2) \to (\forall (w: T).((ty3 g +d1 v w) \to (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind b) v))) +(\lambda (d2: C).(wf3 g d1 d2))))))))))))) (\lambda (c1: C).(\lambda (d1: +C).(\lambda (v: T).(\lambda (H: (getl O c1 (CHead d1 (Bind b) v))).(\lambda +(g: G).(\lambda (c2: C).(\lambda (H0: (wf3 g c1 c2)).(\lambda (w: T).(\lambda +(H1: (ty3 g d1 v w)).(let H_y \def (wf3_clear_conf c1 (CHead d1 (Bind b) v) +(getl_gen_O c1 (CHead d1 (Bind b) v) H) g c2 H0) in (let H_x \def +(wf3_gen_bind1 g d1 c2 v b H_y) in (let H2 \def H_x in (or_ind (ex3_2 C T +(\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda +(c3: C).(\lambda (_: T).(wf3 g d1 c3))) (\lambda (_: C).(\lambda (w0: T).(ty3 +g d1 v w0)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort +O)))) (\lambda (c3: C).(wf3 g d1 c3)) (\lambda (_: C).(\forall (w0: T).((ty3 +g d1 v w0) \to False)))) (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind +b) v))) (\lambda (d2: C).(wf3 g d1 d2))) (\lambda (H3: (ex3_2 C T (\lambda +(c3: C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda (c3: +C).(\lambda (_: T).(wf3 g d1 c3))) (\lambda (_: C).(\lambda (w0: T).(ty3 g d1 +v w0))))).(ex3_2_ind C T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 +(Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g d1 c3))) (\lambda (_: +C).(\lambda (w0: T).(ty3 g d1 v w0))) (ex2 C (\lambda (d2: C).(getl O c2 +(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2))) (\lambda (x0: +C).(\lambda (x1: T).(\lambda (H4: (eq C c2 (CHead x0 (Bind b) v))).(\lambda +(H5: (wf3 g d1 x0)).(\lambda (_: (ty3 g d1 v x1)).(eq_ind_r C (CHead x0 (Bind +b) v) (\lambda (c: C).(ex2 C (\lambda (d2: C).(getl O c (CHead d2 (Bind b) +v))) (\lambda (d2: C).(wf3 g d1 d2)))) (ex_intro2 C (\lambda (d2: C).(getl O +(CHead x0 (Bind b) v) (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)) +x0 (getl_refl b x0 v) H5) c2 H4)))))) H3)) (\lambda (H3: (ex3 C (\lambda (c3: +C).(eq C c2 (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g d1 +c3)) (\lambda (_: C).(\forall (w0: T).((ty3 g d1 v w0) \to +False))))).(ex3_ind C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort +O)))) (\lambda (c3: C).(wf3 g d1 c3)) (\lambda (_: C).(\forall (w0: T).((ty3 +g d1 v w0) \to False))) (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind b) +v))) (\lambda (d2: C).(wf3 g d1 d2))) (\lambda (x0: C).(\lambda (H4: (eq C c2 +(CHead x0 (Bind Void) (TSort O)))).(\lambda (_: (wf3 g d1 x0)).(\lambda (H6: +((\forall (w0: T).((ty3 g d1 v w0) \to False)))).(eq_ind_r C (CHead x0 (Bind +Void) (TSort O)) (\lambda (c: C).(ex2 C (\lambda (d2: C).(getl O c (CHead d2 +(Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)))) (let H_x0 \def (H6 w H1) in +(let H7 \def H_x0 in (False_ind (ex2 C (\lambda (d2: C).(getl O (CHead x0 +(Bind Void) (TSort O)) (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 +d2))) H7))) c2 H4))))) H3)) H2))))))))))))) (\lambda (n: nat).(\lambda (H: +((\forall (c1: C).(\forall (d1: C).(\forall (v: T).((getl n c1 (CHead d1 +(Bind b) v)) \to (\forall (g: G).(\forall (c2: C).((wf3 g c1 c2) \to (\forall +(w: T).((ty3 g d1 v w) \to (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind +b) v))) (\lambda (d2: C).(wf3 g d1 d2)))))))))))))).(\lambda (c1: C).(C_ind +(\lambda (c: C).(\forall (d1: C).(\forall (v: T).((getl (S n) c (CHead d1 +(Bind b) v)) \to (\forall (g: G).(\forall (c2: C).((wf3 g c c2) \to (\forall +(w: T).((ty3 g d1 v w) \to (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 +(Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)))))))))))) (\lambda (n0: +nat).(\lambda (d1: C).(\lambda (v: T).(\lambda (H0: (getl (S n) (CSort n0) +(CHead d1 (Bind b) v))).(\lambda (g: G).(\lambda (c2: C).(\lambda (_: (wf3 g +(CSort n0) c2)).(\lambda (w: T).(\lambda (_: (ty3 g d1 v w)).(getl_gen_sort +n0 (S n) (CHead d1 (Bind b) v) H0 (ex2 C (\lambda (d2: C).(getl (S n) c2 +(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2))))))))))))) (\lambda +(c: C).(\lambda (H0: ((\forall (d1: C).(\forall (v: T).((getl (S n) c (CHead +d1 (Bind b) v)) \to (\forall (g: G).(\forall (c2: C).((wf3 g c c2) \to +(\forall (w: T).((ty3 g d1 v w) \to (ex2 C (\lambda (d2: C).(getl (S n) c2 +(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2))))))))))))).(\lambda +(k: K).(\lambda (t: T).(\lambda (d1: C).(\lambda (v: T).(\lambda (H1: (getl +(S n) (CHead c k t) (CHead d1 (Bind b) v))).(\lambda (g: G).(\lambda (c2: +C).(\lambda (H2: (wf3 g (CHead c k t) c2)).(\lambda (w: T).(\lambda (H3: (ty3 +g d1 v w)).(K_ind (\lambda (k0: K).((wf3 g (CHead c k0 t) c2) \to ((getl (r +k0 n) c (CHead d1 (Bind b) v)) \to (ex2 C (\lambda (d2: C).(getl (S n) c2 +(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)))))) (\lambda (b0: +B).(\lambda (H4: (wf3 g (CHead c (Bind b0) t) c2)).(\lambda (H5: (getl (r +(Bind b0) n) c (CHead d1 (Bind b) v))).(let H_x \def (wf3_gen_bind1 g c c2 t +b0 H4) in (let H6 \def H_x in (or_ind (ex3_2 C T (\lambda (c3: C).(\lambda +(_: T).(eq C c2 (CHead c3 (Bind b0) t)))) (\lambda (c3: C).(\lambda (_: +T).(wf3 g c c3))) (\lambda (_: C).(\lambda (w0: T).(ty3 g c t w0)))) (ex3 C +(\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: +C).(wf3 g c c3)) (\lambda (_: C).(\forall (w0: T).((ty3 g c t w0) \to +False)))) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind b) v))) +(\lambda (d2: C).(wf3 g d1 d2))) (\lambda (H7: (ex3_2 C T (\lambda (c3: +C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b0) t)))) (\lambda (c3: +C).(\lambda (_: T).(wf3 g c c3))) (\lambda (_: C).(\lambda (w0: T).(ty3 g c t +w0))))).(ex3_2_ind C T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 +(Bind b0) t)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c c3))) (\lambda (_: +C).(\lambda (w0: T).(ty3 g c t w0))) (ex2 C (\lambda (d2: C).(getl (S n) c2 +(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2))) (\lambda (x0: +C).(\lambda (x1: T).(\lambda (H8: (eq C c2 (CHead x0 (Bind b0) t))).(\lambda +(H9: (wf3 g c x0)).(\lambda (_: (ty3 g c t x1)).(eq_ind_r C (CHead x0 (Bind +b0) t) (\lambda (c0: C).(ex2 C (\lambda (d2: C).(getl (S n) c0 (CHead d2 +(Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)))) (let H_x0 \def (H c d1 v H5 g +x0 H9 w H3) in (let H11 \def H_x0 in (ex2_ind C (\lambda (d2: C).(getl n x0 +(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)) (ex2 C (\lambda (d2: +C).(getl (S n) (CHead x0 (Bind b0) t) (CHead d2 (Bind b) v))) (\lambda (d2: +C).(wf3 g d1 d2))) (\lambda (x: C).(\lambda (H12: (getl n x0 (CHead x (Bind +b) v))).(\lambda (H13: (wf3 g d1 x)).(ex_intro2 C (\lambda (d2: C).(getl (S +n) (CHead x0 (Bind b0) t) (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 +d2)) x (getl_head (Bind b0) n x0 (CHead x (Bind b) v) H12 t) H13)))) H11))) +c2 H8)))))) H7)) (\lambda (H7: (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 +(Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c c3)) (\lambda (_: +C).(\forall (w0: T).((ty3 g c t w0) \to False))))).(ex3_ind C (\lambda (c3: +C).(eq C c2 (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c c3)) +(\lambda (_: C).(\forall (w0: T).((ty3 g c t w0) \to False))) (ex2 C (\lambda +(d2: C).(getl (S n) c2 (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 +d2))) (\lambda (x0: C).(\lambda (H8: (eq C c2 (CHead x0 (Bind Void) (TSort +O)))).(\lambda (H9: (wf3 g c x0)).(\lambda (_: ((\forall (w0: T).((ty3 g c t +w0) \to False)))).(eq_ind_r C (CHead x0 (Bind Void) (TSort O)) (\lambda (c0: +C).(ex2 C (\lambda (d2: C).(getl (S n) c0 (CHead d2 (Bind b) v))) (\lambda +(d2: C).(wf3 g d1 d2)))) (let H_x0 \def (H c d1 v H5 g x0 H9 w H3) in (let +H11 \def H_x0 in (ex2_ind C (\lambda (d2: C).(getl n x0 (CHead d2 (Bind b) +v))) (\lambda (d2: C).(wf3 g d1 d2)) (ex2 C (\lambda (d2: C).(getl (S n) +(CHead x0 (Bind Void) (TSort O)) (CHead d2 (Bind b) v))) (\lambda (d2: +C).(wf3 g d1 d2))) (\lambda (x: C).(\lambda (H12: (getl n x0 (CHead x (Bind +b) v))).(\lambda (H13: (wf3 g d1 x)).(ex_intro2 C (\lambda (d2: C).(getl (S +n) (CHead x0 (Bind Void) (TSort O)) (CHead d2 (Bind b) v))) (\lambda (d2: +C).(wf3 g d1 d2)) x (getl_head (Bind Void) n x0 (CHead x (Bind b) v) H12 +(TSort O)) H13)))) H11))) c2 H8))))) H7)) H6)))))) (\lambda (f: F).(\lambda +(H4: (wf3 g (CHead c (Flat f) t) c2)).(\lambda (H5: (getl (r (Flat f) n) c +(CHead d1 (Bind b) v))).(let H_y \def (wf3_gen_flat1 g c c2 t f H4) in (H0 d1 +v H5 g c2 H_y w H3))))) k H2 (getl_gen_S k c (CHead d1 (Bind b) v) t n +H1)))))))))))))) c1)))) i)). +(* COMMENTS +Initial nodes: 2531 +END *) + +theorem getl_wf3_trans: + \forall (i: nat).(\forall (c1: C).(\forall (d1: C).((getl i c1 d1) \to +(\forall (g: G).(\forall (d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda (c2: +C).(wf3 g c1 c2)) (\lambda (c2: C).(getl i c2 d2))))))))) +\def + \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (d1: +C).((getl n c1 d1) \to (\forall (g: G).(\forall (d2: C).((wf3 g d1 d2) \to +(ex2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(getl n c2 +d2)))))))))) (\lambda (c1: C).(\lambda (d1: C).(\lambda (H: (getl O c1 +d1)).(\lambda (g: G).(\lambda (d2: C).(\lambda (H0: (wf3 g d1 d2)).(let H_x +\def (clear_wf3_trans c1 d1 (getl_gen_O c1 d1 H) g d2 H0) in (let H1 \def H_x +in (ex2_ind C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(clear c2 d2)) +(ex2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(getl O c2 d2))) +(\lambda (x: C).(\lambda (H2: (wf3 g c1 x)).(\lambda (H3: (clear x +d2)).(ex_intro2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(getl O c2 +d2)) x H2 (getl_intro O x d2 x (drop_refl x) H3))))) H1))))))))) (\lambda (n: +nat).(\lambda (H: ((\forall (c1: C).(\forall (d1: C).((getl n c1 d1) \to +(\forall (g: G).(\forall (d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda (c2: +C).(wf3 g c1 c2)) (\lambda (c2: C).(getl n c2 d2))))))))))).(\lambda (c1: +C).(C_ind (\lambda (c: C).(\forall (d1: C).((getl (S n) c d1) \to (\forall +(g: G).(\forall (d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda (c2: C).(wf3 g c +c2)) (\lambda (c2: C).(getl (S n) c2 d2))))))))) (\lambda (n0: nat).(\lambda +(d1: C).(\lambda (H0: (getl (S n) (CSort n0) d1)).(\lambda (g: G).(\lambda +(d2: C).(\lambda (_: (wf3 g d1 d2)).(getl_gen_sort n0 (S n) d1 H0 (ex2 C +(\lambda (c2: C).(wf3 g (CSort n0) c2)) (\lambda (c2: C).(getl (S n) c2 +d2)))))))))) (\lambda (c: C).(\lambda (H0: ((\forall (d1: C).((getl (S n) c +d1) \to (\forall (g: G).(\forall (d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda +(c2: C).(wf3 g c c2)) (\lambda (c2: C).(getl (S n) c2 d2)))))))))).(\lambda +(k: K).(\lambda (t: T).(\lambda (d1: C).(\lambda (H1: (getl (S n) (CHead c k +t) d1)).(\lambda (g: G).(\lambda (d2: C).(\lambda (H2: (wf3 g d1 d2)).(K_ind +(\lambda (k0: K).((getl (r k0 n) c d1) \to (ex2 C (\lambda (c2: C).(wf3 g +(CHead c k0 t) c2)) (\lambda (c2: C).(getl (S n) c2 d2))))) (\lambda (b: +B).(\lambda (H3: (getl (r (Bind b) n) c d1)).(let H_x \def (H c d1 H3 g d2 +H2) in (let H4 \def H_x in (ex2_ind C (\lambda (c2: C).(wf3 g c c2)) (\lambda +(c2: C).(getl n c2 d2)) (ex2 C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) +c2)) (\lambda (c2: C).(getl (S n) c2 d2))) (\lambda (x: C).(\lambda (H5: (wf3 +g c x)).(\lambda (H6: (getl n x d2)).(let H_x0 \def (ty3_inference g c t) in +(let H7 \def H_x0 in (or_ind (ex T (\lambda (t2: T).(ty3 g c t t2))) (\forall +(t2: T).((ty3 g c t t2) \to False)) (ex2 C (\lambda (c2: C).(wf3 g (CHead c +(Bind b) t) c2)) (\lambda (c2: C).(getl (S n) c2 d2))) (\lambda (H8: (ex T +(\lambda (t2: T).(ty3 g c t t2)))).(ex_ind T (\lambda (t2: T).(ty3 g c t t2)) +(ex2 C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2)) (\lambda (c2: +C).(getl (S n) c2 d2))) (\lambda (x0: T).(\lambda (H9: (ty3 g c t +x0)).(ex_intro2 C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2)) (\lambda +(c2: C).(getl (S n) c2 d2)) (CHead x (Bind b) t) (wf3_bind g c x H5 t x0 H9 +b) (getl_head (Bind b) n x d2 H6 t)))) H8)) (\lambda (H8: ((\forall (t2: +T).((ty3 g c t t2) \to False)))).(ex_intro2 C (\lambda (c2: C).(wf3 g (CHead +c (Bind b) t) c2)) (\lambda (c2: C).(getl (S n) c2 d2)) (CHead x (Bind Void) +(TSort O)) (wf3_void g c x H5 t H8 b) (getl_head (Bind Void) n x d2 H6 (TSort +O)))) H7)))))) H4))))) (\lambda (f: F).(\lambda (H3: (getl (r (Flat f) n) c +d1)).(let H_x \def (H0 d1 H3 g d2 H2) in (let H4 \def H_x in (ex2_ind C +(\lambda (c2: C).(wf3 g c c2)) (\lambda (c2: C).(getl (S n) c2 d2)) (ex2 C +(\lambda (c2: C).(wf3 g (CHead c (Flat f) t) c2)) (\lambda (c2: C).(getl (S +n) c2 d2))) (\lambda (x: C).(\lambda (H5: (wf3 g c x)).(\lambda (H6: (getl (S +n) x d2)).(ex_intro2 C (\lambda (c2: C).(wf3 g (CHead c (Flat f) t) c2)) +(\lambda (c2: C).(getl (S n) c2 d2)) x (wf3_flat g c x H5 t f) H6)))) H4))))) +k (getl_gen_S k c d1 t n H1))))))))))) c1)))) i). +(* COMMENTS +Initial nodes: 1139 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/wf3/props.ma b/matita/matita/contribs/lambdadelta/basic_1/wf3/props.ma new file mode 100644 index 000000000..98a05c637 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/wf3/props.ma @@ -0,0 +1,248 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/wf3/ty3.ma". + +include "Basic-1/app/defs.ma". + +theorem wf3_mono: + \forall (g: G).(\forall (c: C).(\forall (c1: C).((wf3 g c c1) \to (\forall +(c2: C).((wf3 g c c2) \to (eq C c1 c2)))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (c1: C).(\lambda (H: (wf3 g c +c1)).(wf3_ind g (\lambda (c0: C).(\lambda (c2: C).(\forall (c3: C).((wf3 g c0 +c3) \to (eq C c2 c3))))) (\lambda (m: nat).(\lambda (c2: C).(\lambda (H0: +(wf3 g (CSort m) c2)).(let H_y \def (wf3_gen_sort1 g c2 m H0) in (eq_ind_r C +(CSort m) (\lambda (c0: C).(eq C (CSort m) c0)) (refl_equal C (CSort m)) c2 +H_y))))) (\lambda (c2: C).(\lambda (c3: C).(\lambda (_: (wf3 g c2 +c3)).(\lambda (H1: ((\forall (c4: C).((wf3 g c2 c4) \to (eq C c3 +c4))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H2: (ty3 g c2 u +t)).(\lambda (b: B).(\lambda (c0: C).(\lambda (H3: (wf3 g (CHead c2 (Bind b) +u) c0)).(let H_x \def (wf3_gen_bind1 g c2 c0 u b H3) in (let H4 \def H_x in +(or_ind (ex3_2 C T (\lambda (c4: C).(\lambda (_: T).(eq C c0 (CHead c4 (Bind +b) u)))) (\lambda (c4: C).(\lambda (_: T).(wf3 g c2 c4))) (\lambda (_: +C).(\lambda (w: T).(ty3 g c2 u w)))) (ex3 C (\lambda (c4: C).(eq C c0 (CHead +c4 (Bind Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda (_: +C).(\forall (w: T).((ty3 g c2 u w) \to False)))) (eq C (CHead c3 (Bind b) u) +c0) (\lambda (H5: (ex3_2 C T (\lambda (c4: C).(\lambda (_: T).(eq C c0 (CHead +c4 (Bind b) u)))) (\lambda (c4: C).(\lambda (_: T).(wf3 g c2 c4))) (\lambda +(_: C).(\lambda (w: T).(ty3 g c2 u w))))).(ex3_2_ind C T (\lambda (c4: +C).(\lambda (_: T).(eq C c0 (CHead c4 (Bind b) u)))) (\lambda (c4: +C).(\lambda (_: T).(wf3 g c2 c4))) (\lambda (_: C).(\lambda (w: T).(ty3 g c2 +u w))) (eq C (CHead c3 (Bind b) u) c0) (\lambda (x0: C).(\lambda (x1: +T).(\lambda (H6: (eq C c0 (CHead x0 (Bind b) u))).(\lambda (H7: (wf3 g c2 +x0)).(\lambda (_: (ty3 g c2 u x1)).(eq_ind_r C (CHead x0 (Bind b) u) (\lambda +(c4: C).(eq C (CHead c3 (Bind b) u) c4)) (f_equal3 C K T C CHead c3 x0 (Bind +b) (Bind b) u u (H1 x0 H7) (refl_equal K (Bind b)) (refl_equal T u)) c0 +H6)))))) H5)) (\lambda (H5: (ex3 C (\lambda (c4: C).(eq C c0 (CHead c4 (Bind +Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda (_: C).(\forall +(w: T).((ty3 g c2 u w) \to False))))).(ex3_ind C (\lambda (c4: C).(eq C c0 +(CHead c4 (Bind Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda +(_: C).(\forall (w: T).((ty3 g c2 u w) \to False))) (eq C (CHead c3 (Bind b) +u) c0) (\lambda (x0: C).(\lambda (H6: (eq C c0 (CHead x0 (Bind Void) (TSort +O)))).(\lambda (_: (wf3 g c2 x0)).(\lambda (H8: ((\forall (w: T).((ty3 g c2 u +w) \to False)))).(eq_ind_r C (CHead x0 (Bind Void) (TSort O)) (\lambda (c4: +C).(eq C (CHead c3 (Bind b) u) c4)) (let H_x0 \def (H8 t H2) in (let H9 \def +H_x0 in (False_ind (eq C (CHead c3 (Bind b) u) (CHead x0 (Bind Void) (TSort +O))) H9))) c0 H6))))) H5)) H4))))))))))))) (\lambda (c2: C).(\lambda (c3: +C).(\lambda (_: (wf3 g c2 c3)).(\lambda (H1: ((\forall (c4: C).((wf3 g c2 c4) +\to (eq C c3 c4))))).(\lambda (u: T).(\lambda (H2: ((\forall (t: T).((ty3 g +c2 u t) \to False)))).(\lambda (b: B).(\lambda (c0: C).(\lambda (H3: (wf3 g +(CHead c2 (Bind b) u) c0)).(let H_x \def (wf3_gen_bind1 g c2 c0 u b H3) in +(let H4 \def H_x in (or_ind (ex3_2 C T (\lambda (c4: C).(\lambda (_: T).(eq C +c0 (CHead c4 (Bind b) u)))) (\lambda (c4: C).(\lambda (_: T).(wf3 g c2 c4))) +(\lambda (_: C).(\lambda (w: T).(ty3 g c2 u w)))) (ex3 C (\lambda (c4: C).(eq +C c0 (CHead c4 (Bind Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) +(\lambda (_: C).(\forall (w: T).((ty3 g c2 u w) \to False)))) (eq C (CHead c3 +(Bind Void) (TSort O)) c0) (\lambda (H5: (ex3_2 C T (\lambda (c4: C).(\lambda +(_: T).(eq C c0 (CHead c4 (Bind b) u)))) (\lambda (c4: C).(\lambda (_: +T).(wf3 g c2 c4))) (\lambda (_: C).(\lambda (w: T).(ty3 g c2 u +w))))).(ex3_2_ind C T (\lambda (c4: C).(\lambda (_: T).(eq C c0 (CHead c4 +(Bind b) u)))) (\lambda (c4: C).(\lambda (_: T).(wf3 g c2 c4))) (\lambda (_: +C).(\lambda (w: T).(ty3 g c2 u w))) (eq C (CHead c3 (Bind Void) (TSort O)) +c0) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C c0 (CHead x0 (Bind +b) u))).(\lambda (_: (wf3 g c2 x0)).(\lambda (H8: (ty3 g c2 u x1)).(eq_ind_r +C (CHead x0 (Bind b) u) (\lambda (c4: C).(eq C (CHead c3 (Bind Void) (TSort +O)) c4)) (let H_x0 \def (H2 x1 H8) in (let H9 \def H_x0 in (False_ind (eq C +(CHead c3 (Bind Void) (TSort O)) (CHead x0 (Bind b) u)) H9))) c0 H6)))))) +H5)) (\lambda (H5: (ex3 C (\lambda (c4: C).(eq C c0 (CHead c4 (Bind Void) +(TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda (_: C).(\forall (w: +T).((ty3 g c2 u w) \to False))))).(ex3_ind C (\lambda (c4: C).(eq C c0 (CHead +c4 (Bind Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda (_: +C).(\forall (w: T).((ty3 g c2 u w) \to False))) (eq C (CHead c3 (Bind Void) +(TSort O)) c0) (\lambda (x0: C).(\lambda (H6: (eq C c0 (CHead x0 (Bind Void) +(TSort O)))).(\lambda (H7: (wf3 g c2 x0)).(\lambda (_: ((\forall (w: T).((ty3 +g c2 u w) \to False)))).(eq_ind_r C (CHead x0 (Bind Void) (TSort O)) (\lambda +(c4: C).(eq C (CHead c3 (Bind Void) (TSort O)) c4)) (f_equal3 C K T C CHead +c3 x0 (Bind Void) (Bind Void) (TSort O) (TSort O) (H1 x0 H7) (refl_equal K +(Bind Void)) (refl_equal T (TSort O))) c0 H6))))) H5)) H4)))))))))))) +(\lambda (c2: C).(\lambda (c3: C).(\lambda (_: (wf3 g c2 c3)).(\lambda (H1: +((\forall (c4: C).((wf3 g c2 c4) \to (eq C c3 c4))))).(\lambda (u: +T).(\lambda (f: F).(\lambda (c0: C).(\lambda (H2: (wf3 g (CHead c2 (Flat f) +u) c0)).(let H_y \def (wf3_gen_flat1 g c2 c0 u f H2) in (H1 c0 H_y)))))))))) +c c1 H)))). +(* COMMENTS +Initial nodes: 1555 +END *) + +theorem wf3_total: + \forall (g: G).(\forall (c1: C).(ex C (\lambda (c2: C).(wf3 g c1 c2)))) +\def + \lambda (g: G).(\lambda (c1: C).(C_ind (\lambda (c: C).(ex C (\lambda (c2: +C).(wf3 g c c2)))) (\lambda (n: nat).(ex_intro C (\lambda (c2: C).(wf3 g +(CSort n) c2)) (CSort n) (wf3_sort g n))) (\lambda (c: C).(\lambda (H: (ex C +(\lambda (c2: C).(wf3 g c c2)))).(\lambda (k: K).(\lambda (t: T).(let H0 \def +H in (ex_ind C (\lambda (c2: C).(wf3 g c c2)) (ex C (\lambda (c2: C).(wf3 g +(CHead c k t) c2))) (\lambda (x: C).(\lambda (H1: (wf3 g c x)).(K_ind +(\lambda (k0: K).(ex C (\lambda (c2: C).(wf3 g (CHead c k0 t) c2)))) (\lambda +(b: B).(let H_x \def (ty3_inference g c t) in (let H2 \def H_x in (or_ind (ex +T (\lambda (t2: T).(ty3 g c t t2))) (\forall (t2: T).((ty3 g c t t2) \to +False)) (ex C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2))) (\lambda +(H3: (ex T (\lambda (t2: T).(ty3 g c t t2)))).(ex_ind T (\lambda (t2: T).(ty3 +g c t t2)) (ex C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2))) (\lambda +(x0: T).(\lambda (H4: (ty3 g c t x0)).(ex_intro C (\lambda (c2: C).(wf3 g +(CHead c (Bind b) t) c2)) (CHead x (Bind b) t) (wf3_bind g c x H1 t x0 H4 +b)))) H3)) (\lambda (H3: ((\forall (t2: T).((ty3 g c t t2) \to +False)))).(ex_intro C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2)) +(CHead x (Bind Void) (TSort O)) (wf3_void g c x H1 t H3 b))) H2)))) (\lambda +(f: F).(ex_intro C (\lambda (c2: C).(wf3 g (CHead c (Flat f) t) c2)) x +(wf3_flat g c x H1 t f))) k))) H0)))))) c1)). +(* COMMENTS +Initial nodes: 435 +END *) + +theorem ty3_shift1: + \forall (g: G).(\forall (c: C).((wf3 g c c) \to (\forall (t1: T).(\forall +(t2: T).((ty3 g c t1 t2) \to (ty3 g (CSort (cbk c)) (app1 c t1) (app1 c +t2))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (H: (wf3 g c c)).(insert_eq C c +(\lambda (c0: C).(wf3 g c0 c)) (\lambda (c0: C).(\forall (t1: T).(\forall +(t2: T).((ty3 g c0 t1 t2) \to (ty3 g (CSort (cbk c0)) (app1 c0 t1) (app1 c0 +t2)))))) (\lambda (y: C).(\lambda (H0: (wf3 g y c)).(wf3_ind g (\lambda (c0: +C).(\lambda (c1: C).((eq C c0 c1) \to (\forall (t1: T).(\forall (t2: T).((ty3 +g c0 t1 t2) \to (ty3 g (CSort (cbk c0)) (app1 c0 t1) (app1 c0 t2)))))))) +(\lambda (m: nat).(\lambda (_: (eq C (CSort m) (CSort m))).(\lambda (t1: +T).(\lambda (t2: T).(\lambda (H2: (ty3 g (CSort m) t1 t2)).H2))))) (\lambda +(c1: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c1 c2)).(\lambda (H2: (((eq C +c1 c2) \to (\forall (t1: T).(\forall (t2: T).((ty3 g c1 t1 t2) \to (ty3 g +(CSort (cbk c1)) (app1 c1 t1) (app1 c1 t2)))))))).(\lambda (u: T).(\lambda +(t: T).(\lambda (H3: (ty3 g c1 u t)).(\lambda (b: B).(\lambda (H4: (eq C +(CHead c1 (Bind b) u) (CHead c2 (Bind b) u))).(\lambda (t1: T).(\lambda (t2: +T).(\lambda (H5: (ty3 g (CHead c1 (Bind b) u) t1 t2)).(let H6 \def (f_equal C +C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) +\Rightarrow c1 | (CHead c0 _ _) \Rightarrow c0])) (CHead c1 (Bind b) u) +(CHead c2 (Bind b) u) H4) in (let H7 \def (eq_ind_r C c2 (\lambda (c0: +C).((eq C c1 c0) \to (\forall (t3: T).(\forall (t4: T).((ty3 g c1 t3 t4) \to +(ty3 g (CSort (cbk c1)) (app1 c1 t3) (app1 c1 t4))))))) H2 c1 H6) in (let H8 +\def (eq_ind_r C c2 (\lambda (c0: C).(wf3 g c1 c0)) H1 c1 H6) in (ex_ind T +(\lambda (t0: T).(ty3 g (CHead c1 (Bind b) u) t2 t0)) (ty3 g (CSort (cbk c1)) +(app1 c1 (THead (Bind b) u t1)) (app1 c1 (THead (Bind b) u t2))) (\lambda (x: +T).(\lambda (_: (ty3 g (CHead c1 (Bind b) u) t2 x)).(H7 (refl_equal C c1) +(THead (Bind b) u t1) (THead (Bind b) u t2) (ty3_bind g c1 u t H3 b t1 t2 +H5)))) (ty3_correct g (CHead c1 (Bind b) u) t1 t2 H5))))))))))))))))) +(\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c1 c2)).(\lambda (H2: +(((eq C c1 c2) \to (\forall (t1: T).(\forall (t2: T).((ty3 g c1 t1 t2) \to +(ty3 g (CSort (cbk c1)) (app1 c1 t1) (app1 c1 t2)))))))).(\lambda (u: +T).(\lambda (H3: ((\forall (t: T).((ty3 g c1 u t) \to False)))).(\lambda (b: +B).(\lambda (H4: (eq C (CHead c1 (Bind b) u) (CHead c2 (Bind Void) (TSort +O)))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H5: (ty3 g (CHead c1 (Bind +b) u) t1 t2)).(let H6 \def (f_equal C C (\lambda (e: C).(match e in C return +(\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _ _) +\Rightarrow c0])) (CHead c1 (Bind b) u) (CHead c2 (Bind Void) (TSort O)) H4) +in ((let H7 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda +(_: C).B) with [(CSort _) \Rightarrow b | (CHead _ k _) \Rightarrow (match k +in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) +\Rightarrow b])])) (CHead c1 (Bind b) u) (CHead c2 (Bind Void) (TSort O)) H4) +in ((let H8 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda +(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) +(CHead c1 (Bind b) u) (CHead c2 (Bind Void) (TSort O)) H4) in (\lambda (H9: +(eq B b Void)).(\lambda (H10: (eq C c1 c2)).(let H11 \def (eq_ind B b +(\lambda (b0: B).(ty3 g (CHead c1 (Bind b0) u) t1 t2)) H5 Void H9) in +(eq_ind_r B Void (\lambda (b0: B).(ty3 g (CSort (cbk (CHead c1 (Bind b0) u))) +(app1 (CHead c1 (Bind b0) u) t1) (app1 (CHead c1 (Bind b0) u) t2))) (let H12 +\def (eq_ind T u (\lambda (t: T).(ty3 g (CHead c1 (Bind Void) t) t1 t2)) H11 +(TSort O) H8) in (let H13 \def (eq_ind T u (\lambda (t: T).(\forall (t0: +T).((ty3 g c1 t t0) \to False))) H3 (TSort O) H8) in (eq_ind_r T (TSort O) +(\lambda (t: T).(ty3 g (CSort (cbk (CHead c1 (Bind Void) t))) (app1 (CHead c1 +(Bind Void) t) t1) (app1 (CHead c1 (Bind Void) t) t2))) (let H14 \def +(eq_ind_r C c2 (\lambda (c0: C).((eq C c1 c0) \to (\forall (t3: T).(\forall +(t4: T).((ty3 g c1 t3 t4) \to (ty3 g (CSort (cbk c1)) (app1 c1 t3) (app1 c1 +t4))))))) H2 c1 H10) in (let H15 \def (eq_ind_r C c2 (\lambda (c0: C).(wf3 g +c1 c0)) H1 c1 H10) in (ex_ind T (\lambda (t: T).(ty3 g (CHead c1 (Bind Void) +(TSort O)) t2 t)) (ty3 g (CSort (cbk c1)) (app1 c1 (THead (Bind Void) (TSort +O) t1)) (app1 c1 (THead (Bind Void) (TSort O) t2))) (\lambda (x: T).(\lambda +(_: (ty3 g (CHead c1 (Bind Void) (TSort O)) t2 x)).(H14 (refl_equal C c1) +(THead (Bind Void) (TSort O) t1) (THead (Bind Void) (TSort O) t2) (ty3_bind g +c1 (TSort O) (TSort (next g O)) (ty3_sort g c1 O) Void t1 t2 H12)))) +(ty3_correct g (CHead c1 (Bind Void) (TSort O)) t1 t2 H12)))) u H8))) b +H9))))) H7)) H6))))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: +(wf3 g c1 c2)).(\lambda (H2: (((eq C c1 c2) \to (\forall (t1: T).(\forall +(t2: T).((ty3 g c1 t1 t2) \to (ty3 g (CSort (cbk c1)) (app1 c1 t1) (app1 c1 +t2)))))))).(\lambda (u: T).(\lambda (f: F).(\lambda (H3: (eq C (CHead c1 +(Flat f) u) c2)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g (CHead +c1 (Flat f) u) t1 t2)).(let H5 \def (f_equal C C (\lambda (e: C).e) (CHead c1 +(Flat f) u) c2 H3) in (let H6 \def (eq_ind_r C c2 (\lambda (c0: C).((eq C c1 +c0) \to (\forall (t3: T).(\forall (t4: T).((ty3 g c1 t3 t4) \to (ty3 g (CSort +(cbk c1)) (app1 c1 t3) (app1 c1 t4))))))) H2 (CHead c1 (Flat f) u) H5) in +(let H7 \def (eq_ind_r C c2 (\lambda (c0: C).(wf3 g c1 c0)) H1 (CHead c1 +(Flat f) u) H5) in (let H_x \def (wf3_gen_head2 g c1 c1 u (Flat f) H7) in +(let H8 \def H_x in (ex_ind B (\lambda (b: B).(eq K (Flat f) (Bind b))) (ty3 +g (CSort (cbk c1)) (app1 c1 (THead (Flat f) u t1)) (app1 c1 (THead (Flat f) u +t2))) (\lambda (x: B).(\lambda (H9: (eq K (Flat f) (Bind x))).(let H10 \def +(eq_ind K (Flat f) (\lambda (ee: K).(match ee in K return (\lambda (_: +K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])) I +(Bind x) H9) in (False_ind (ty3 g (CSort (cbk c1)) (app1 c1 (THead (Flat f) u +t1)) (app1 c1 (THead (Flat f) u t2))) H10)))) H8)))))))))))))))) y c H0))) +H))). +(* COMMENTS +Initial nodes: 1677 +END *) + +theorem wf3_idem: + \forall (g: G).(\forall (c1: C).(\forall (c2: C).((wf3 g c1 c2) \to (wf3 g +c2 c2)))) +\def + \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wf3 g c1 +c2)).(wf3_ind g (\lambda (_: C).(\lambda (c0: C).(wf3 g c0 c0))) (\lambda (m: +nat).(wf3_sort g m)) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wf3 g +c3 c4)).(\lambda (H1: (wf3 g c4 c4)).(\lambda (u: T).(\lambda (t: T).(\lambda +(H2: (ty3 g c3 u t)).(\lambda (b: B).(wf3_bind g c4 c4 H1 u t (wf3_ty3_conf g +c3 u t H2 c4 H0) b))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (_: +(wf3 g c3 c4)).(\lambda (H1: (wf3 g c4 c4)).(\lambda (u: T).(\lambda (_: +((\forall (t: T).((ty3 g c3 u t) \to False)))).(\lambda (_: B).(wf3_bind g c4 +c4 H1 (TSort O) (TSort (next g O)) (ty3_sort g c4 O) Void)))))))) (\lambda +(c3: C).(\lambda (c4: C).(\lambda (_: (wf3 g c3 c4)).(\lambda (H1: (wf3 g c4 +c4)).(\lambda (_: T).(\lambda (_: F).H1)))))) c1 c2 H)))). +(* COMMENTS +Initial nodes: 207 +END *) + +theorem wf3_ty3: + \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (u: T).((ty3 g c1 t +u) \to (ex2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(ty3 g c2 t +u))))))) +\def + \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H: +(ty3 g c1 t u)).(let H_x \def (wf3_total g c1) in (let H0 \def H_x in (ex_ind +C (\lambda (c2: C).(wf3 g c1 c2)) (ex2 C (\lambda (c2: C).(wf3 g c1 c2)) +(\lambda (c2: C).(ty3 g c2 t u))) (\lambda (x: C).(\lambda (H1: (wf3 g c1 +x)).(ex_intro2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(ty3 g c2 t +u)) x H1 (wf3_ty3_conf g c1 t u H x H1)))) H0))))))). +(* COMMENTS +Initial nodes: 123 +END *) + diff --git a/matita/matita/contribs/lambdadelta/basic_1/wf3/ty3.ma b/matita/matita/contribs/lambdadelta/basic_1/wf3/ty3.ma new file mode 100644 index 000000000..e47d4a2e0 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/wf3/ty3.ma @@ -0,0 +1,138 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Basic-1/wf3/getl.ma". + +theorem wf3_pr2_conf: + \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pr2 c1 +t1 t2) \to (\forall (c2: C).((wf3 g c1 c2) \to (\forall (u: T).((ty3 g c1 t1 +u) \to (pr2 c2 t1 t2))))))))) +\def + \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda +(H: (pr2 c1 t1 t2)).(pr2_ind (\lambda (c: C).(\lambda (t: T).(\lambda (t0: +T).(\forall (c2: C).((wf3 g c c2) \to (\forall (u: T).((ty3 g c t u) \to (pr2 +c2 t t0)))))))) (\lambda (c: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda +(H0: (pr0 t3 t4)).(\lambda (c2: C).(\lambda (_: (wf3 g c c2)).(\lambda (u: +T).(\lambda (_: (ty3 g c t3 u)).(pr2_free c2 t3 t4 H0))))))))) (\lambda (c: +C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c +(CHead d (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: +(pr0 t3 t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (c2: +C).(\lambda (H3: (wf3 g c c2)).(\lambda (u0: T).(\lambda (H4: (ty3 g c t3 +u0)).(let H_y \def (ty3_sred_pr0 t3 t4 H1 g c u0 H4) in (let H_x \def +(ty3_getl_subst0 g c t4 u0 H_y u t i H2 Abbr d u H0) in (let H5 \def H_x in +(ex_ind T (\lambda (w: T).(ty3 g d u w)) (pr2 c2 t3 t) (\lambda (x: +T).(\lambda (H6: (ty3 g d u x)).(let H_x0 \def (wf3_getl_conf Abbr i c d u H0 +g c2 H3 x H6) in (let H7 \def H_x0 in (ex2_ind C (\lambda (d2: C).(getl i c2 +(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(wf3 g d d2)) (pr2 c2 t3 t) +(\lambda (x0: C).(\lambda (H8: (getl i c2 (CHead x0 (Bind Abbr) u))).(\lambda +(_: (wf3 g d x0)).(pr2_delta c2 x0 u i H8 t3 t4 H1 t H2)))) H7))))) +H5)))))))))))))))))) c1 t1 t2 H))))). +(* COMMENTS +Initial nodes: 373 +END *) + +theorem wf3_pr3_conf: + \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pr3 c1 +t1 t2) \to (\forall (c2: C).((wf3 g c1 c2) \to (\forall (u: T).((ty3 g c1 t1 +u) \to (pr3 c2 t1 t2))))))))) +\def + \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda +(H: (pr3 c1 t1 t2)).(pr3_ind c1 (\lambda (t: T).(\lambda (t0: T).(\forall +(c2: C).((wf3 g c1 c2) \to (\forall (u: T).((ty3 g c1 t u) \to (pr3 c2 t +t0))))))) (\lambda (t: T).(\lambda (c2: C).(\lambda (_: (wf3 g c1 +c2)).(\lambda (u: T).(\lambda (_: (ty3 g c1 t u)).(pr3_refl c2 t)))))) +(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr2 c1 t4 t3)).(\lambda (t5: +T).(\lambda (_: (pr3 c1 t3 t5)).(\lambda (H2: ((\forall (c2: C).((wf3 g c1 +c2) \to (\forall (u: T).((ty3 g c1 t3 u) \to (pr3 c2 t3 t5))))))).(\lambda +(c2: C).(\lambda (H3: (wf3 g c1 c2)).(\lambda (u: T).(\lambda (H4: (ty3 g c1 +t4 u)).(pr3_sing c2 t3 t4 (wf3_pr2_conf g c1 t4 t3 H0 c2 H3 u H4) t5 (H2 c2 +H3 u (ty3_sred_pr2 c1 t4 t3 H0 g u H4))))))))))))) t1 t2 H))))). +(* COMMENTS +Initial nodes: 217 +END *) + +theorem wf3_pc3_conf: + \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pc3 c1 +t1 t2) \to (\forall (c2: C).((wf3 g c1 c2) \to (\forall (u1: T).((ty3 g c1 t1 +u1) \to (\forall (u2: T).((ty3 g c1 t2 u2) \to (pc3 c2 t1 t2))))))))))) +\def + \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda +(H: (pc3 c1 t1 t2)).(\lambda (c2: C).(\lambda (H0: (wf3 g c1 c2)).(\lambda +(u1: T).(\lambda (H1: (ty3 g c1 t1 u1)).(\lambda (u2: T).(\lambda (H2: (ty3 g +c1 t2 u2)).(let H3 \def H in (ex2_ind T (\lambda (t: T).(pr3 c1 t1 t)) +(\lambda (t: T).(pr3 c1 t2 t)) (pc3 c2 t1 t2) (\lambda (x: T).(\lambda (H4: +(pr3 c1 t1 x)).(\lambda (H5: (pr3 c1 t2 x)).(pc3_pr3_t c2 t1 x (wf3_pr3_conf +g c1 t1 x H4 c2 H0 u1 H1) t2 (wf3_pr3_conf g c1 t2 x H5 c2 H0 u2 H2))))) +H3)))))))))))). +(* COMMENTS +Initial nodes: 153 +END *) + +theorem wf3_ty3_conf: + \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c1 +t1 t2) \to (\forall (c2: C).((wf3 g c1 c2) \to (ty3 g c2 t1 t2))))))) +\def + \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda +(H: (ty3 g c1 t1 t2)).(ty3_ind g (\lambda (c: C).(\lambda (t: T).(\lambda +(t0: T).(\forall (c2: C).((wf3 g c c2) \to (ty3 g c2 t t0)))))) (\lambda (c: +C).(\lambda (t3: T).(\lambda (t: T).(\lambda (H0: (ty3 g c t3 t)).(\lambda +(H1: ((\forall (c2: C).((wf3 g c c2) \to (ty3 g c2 t3 t))))).(\lambda (u: +T).(\lambda (t4: T).(\lambda (H2: (ty3 g c u t4)).(\lambda (H3: ((\forall +(c2: C).((wf3 g c c2) \to (ty3 g c2 u t4))))).(\lambda (H4: (pc3 c t4 +t3)).(\lambda (c2: C).(\lambda (H5: (wf3 g c c2)).(ex_ind T (\lambda (t0: +T).(ty3 g c t4 t0)) (ty3 g c2 u t3) (\lambda (x: T).(\lambda (H6: (ty3 g c t4 +x)).(ty3_conv g c2 t3 t (H1 c2 H5) u t4 (H3 c2 H5) (wf3_pc3_conf g c t4 t3 H4 +c2 H5 x H6 t H0)))) (ty3_correct g c u t4 H2)))))))))))))) (\lambda (c: +C).(\lambda (m: nat).(\lambda (c2: C).(\lambda (_: (wf3 g c c2)).(ty3_sort g +c2 m))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d: C).(\lambda (u: +T).(\lambda (H0: (getl n c (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda +(H1: (ty3 g d u t)).(\lambda (H2: ((\forall (c2: C).((wf3 g d c2) \to (ty3 g +c2 u t))))).(\lambda (c2: C).(\lambda (H3: (wf3 g c c2)).(let H_x \def +(wf3_getl_conf Abbr n c d u H0 g c2 H3 t H1) in (let H4 \def H_x in (ex2_ind +C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: +C).(wf3 g d d2)) (ty3 g c2 (TLRef n) (lift (S n) O t)) (\lambda (x: +C).(\lambda (H5: (getl n c2 (CHead x (Bind Abbr) u))).(\lambda (H6: (wf3 g d +x)).(ty3_abbr g n c2 x u H5 t (H2 x H6))))) H4))))))))))))) (\lambda (n: +nat).(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c +(CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u +t)).(\lambda (H2: ((\forall (c2: C).((wf3 g d c2) \to (ty3 g c2 u +t))))).(\lambda (c2: C).(\lambda (H3: (wf3 g c c2)).(let H_x \def +(wf3_getl_conf Abst n c d u H0 g c2 H3 t H1) in (let H4 \def H_x in (ex2_ind +C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: +C).(wf3 g d d2)) (ty3 g c2 (TLRef n) (lift (S n) O u)) (\lambda (x: +C).(\lambda (H5: (getl n c2 (CHead x (Bind Abst) u))).(\lambda (H6: (wf3 g d +x)).(ty3_abst g n c2 x u H5 t (H2 x H6))))) H4))))))))))))) (\lambda (c: +C).(\lambda (u: T).(\lambda (t: T).(\lambda (H0: (ty3 g c u t)).(\lambda (H1: +((\forall (c2: C).((wf3 g c c2) \to (ty3 g c2 u t))))).(\lambda (b: +B).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (ty3 g (CHead c (Bind b) u) +t3 t4)).(\lambda (H3: ((\forall (c2: C).((wf3 g (CHead c (Bind b) u) c2) \to +(ty3 g c2 t3 t4))))).(\lambda (c2: C).(\lambda (H4: (wf3 g c c2)).(ty3_bind g +c2 u t (H1 c2 H4) b t3 t4 (H3 (CHead c2 (Bind b) u) (wf3_bind g c c2 H4 u t +H0 b))))))))))))))) (\lambda (c: C).(\lambda (w: T).(\lambda (u: T).(\lambda +(_: (ty3 g c w u)).(\lambda (H1: ((\forall (c2: C).((wf3 g c c2) \to (ty3 g +c2 w u))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c v (THead +(Bind Abst) u t))).(\lambda (H3: ((\forall (c2: C).((wf3 g c c2) \to (ty3 g +c2 v (THead (Bind Abst) u t)))))).(\lambda (c2: C).(\lambda (H4: (wf3 g c +c2)).(ty3_appl g c2 w u (H1 c2 H4) v t (H3 c2 H4))))))))))))) (\lambda (c: +C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (ty3 g c t3 t4)).(\lambda +(H1: ((\forall (c2: C).((wf3 g c c2) \to (ty3 g c2 t3 t4))))).(\lambda (t0: +T).(\lambda (_: (ty3 g c t4 t0)).(\lambda (H3: ((\forall (c2: C).((wf3 g c +c2) \to (ty3 g c2 t4 t0))))).(\lambda (c2: C).(\lambda (H4: (wf3 g c +c2)).(ty3_cast g c2 t3 t4 (H1 c2 H4) t0 (H3 c2 H4)))))))))))) c1 t1 t2 H))))). +(* COMMENTS +Initial nodes: 1027 +END *) + diff --git a/matita/matita/contribs/lambdadelta/ground_1/blt/defs.ma b/matita/matita/contribs/lambdadelta/ground_1/blt/defs.ma new file mode 100644 index 000000000..009627a3b --- /dev/null +++ b/matita/matita/contribs/lambdadelta/ground_1/blt/defs.ma @@ -0,0 +1,25 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Ground-1/preamble.ma". + +definition blt: + nat \to (nat \to bool) +\def + let rec blt (m: nat) (n: nat) on n: bool \def (match n with [O \Rightarrow +false | (S n0) \Rightarrow (match m with [O \Rightarrow true | (S m0) +\Rightarrow (blt m0 n0)])]) in blt. + diff --git a/matita/matita/contribs/lambdadelta/ground_1/blt/props.ma b/matita/matita/contribs/lambdadelta/ground_1/blt/props.ma new file mode 100644 index 000000000..e5b569925 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/ground_1/blt/props.ma @@ -0,0 +1,112 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Ground-1/blt/defs.ma". + +theorem lt_blt: + \forall (x: nat).(\forall (y: nat).((lt y x) \to (eq bool (blt y x) true))) +\def + \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((lt y n) \to +(eq bool (blt y n) true)))) (\lambda (y: nat).(\lambda (H: (lt y O)).(let H0 +\def (match H in le return (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat +n O) \to (eq bool (blt y O) true)))) with [le_n \Rightarrow (\lambda (H0: (eq +nat (S y) O)).(let H1 \def (eq_ind nat (S y) (\lambda (e: nat).(match e in +nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) +\Rightarrow True])) I O H0) in (False_ind (eq bool (blt y O) true) H1))) | +(le_S m H0) \Rightarrow (\lambda (H1: (eq nat (S m) O)).((let H2 \def (eq_ind +nat (S m) (\lambda (e: nat).(match e in nat return (\lambda (_: nat).Prop) +with [O \Rightarrow False | (S _) \Rightarrow True])) I O H1) in (False_ind +((le (S y) m) \to (eq bool (blt y O) true)) H2)) H0))]) in (H0 (refl_equal +nat O))))) (\lambda (n: nat).(\lambda (H: ((\forall (y: nat).((lt y n) \to +(eq bool (blt y n) true))))).(\lambda (y: nat).(nat_ind (\lambda (n0: +nat).((lt n0 (S n)) \to (eq bool (blt n0 (S n)) true))) (\lambda (_: (lt O (S +n))).(refl_equal bool true)) (\lambda (n0: nat).(\lambda (_: (((lt n0 (S n)) +\to (eq bool (match n0 with [O \Rightarrow true | (S m) \Rightarrow (blt m +n)]) true)))).(\lambda (H1: (lt (S n0) (S n))).(H n0 (le_S_n (S n0) n H1))))) +y)))) x). +(* COMMENTS +Initial nodes: 291 +END *) + +theorem le_bge: + \forall (x: nat).(\forall (y: nat).((le x y) \to (eq bool (blt y x) false))) +\def + \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((le n y) \to +(eq bool (blt y n) false)))) (\lambda (y: nat).(\lambda (_: (le O +y)).(refl_equal bool false))) (\lambda (n: nat).(\lambda (H: ((\forall (y: +nat).((le n y) \to (eq bool (blt y n) false))))).(\lambda (y: nat).(nat_ind +(\lambda (n0: nat).((le (S n) n0) \to (eq bool (blt n0 (S n)) false))) +(\lambda (H0: (le (S n) O)).(let H1 \def (match H0 in le return (\lambda (n0: +nat).(\lambda (_: (le ? n0)).((eq nat n0 O) \to (eq bool (blt O (S n)) +false)))) with [le_n \Rightarrow (\lambda (H1: (eq nat (S n) O)).(let H2 \def +(eq_ind nat (S n) (\lambda (e: nat).(match e in nat return (\lambda (_: +nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H1) in +(False_ind (eq bool (blt O (S n)) false) H2))) | (le_S m H1) \Rightarrow +(\lambda (H2: (eq nat (S m) O)).((let H3 \def (eq_ind nat (S m) (\lambda (e: +nat).(match e in nat return (\lambda (_: nat).Prop) with [O \Rightarrow False +| (S _) \Rightarrow True])) I O H2) in (False_ind ((le (S n) m) \to (eq bool +(blt O (S n)) false)) H3)) H1))]) in (H1 (refl_equal nat O)))) (\lambda (n0: +nat).(\lambda (_: (((le (S n) n0) \to (eq bool (blt n0 (S n)) +false)))).(\lambda (H1: (le (S n) (S n0))).(H n0 (le_S_n n n0 H1))))) y)))) +x). +(* COMMENTS +Initial nodes: 293 +END *) + +theorem blt_lt: + \forall (x: nat).(\forall (y: nat).((eq bool (blt y x) true) \to (lt y x))) +\def + \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((eq bool (blt +y n) true) \to (lt y n)))) (\lambda (y: nat).(\lambda (H: (eq bool (blt y O) +true)).(let H0 \def (match H in eq return (\lambda (b: bool).(\lambda (_: (eq +? ? b)).((eq bool b true) \to (lt y O)))) with [refl_equal \Rightarrow +(\lambda (H0: (eq bool (blt y O) true)).(let H1 \def (eq_ind bool (blt y O) +(\lambda (e: bool).(match e in bool return (\lambda (_: bool).Prop) with +[true \Rightarrow False | false \Rightarrow True])) I true H0) in (False_ind +(lt y O) H1)))]) in (H0 (refl_equal bool true))))) (\lambda (n: nat).(\lambda +(H: ((\forall (y: nat).((eq bool (blt y n) true) \to (lt y n))))).(\lambda +(y: nat).(nat_ind (\lambda (n0: nat).((eq bool (blt n0 (S n)) true) \to (lt +n0 (S n)))) (\lambda (_: (eq bool true true)).(le_S_n (S O) (S n) (le_n_S (S +O) (S n) (le_n_S O n (le_O_n n))))) (\lambda (n0: nat).(\lambda (_: (((eq +bool (match n0 with [O \Rightarrow true | (S m) \Rightarrow (blt m n)]) true) +\to (lt n0 (S n))))).(\lambda (H1: (eq bool (blt n0 n) true)).(lt_n_S n0 n (H +n0 H1))))) y)))) x). +(* COMMENTS +Initial nodes: 252 +END *) + +theorem bge_le: + \forall (x: nat).(\forall (y: nat).((eq bool (blt y x) false) \to (le x y))) +\def + \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((eq bool (blt +y n) false) \to (le n y)))) (\lambda (y: nat).(\lambda (_: (eq bool (blt y O) +false)).(le_O_n y))) (\lambda (n: nat).(\lambda (H: ((\forall (y: nat).((eq +bool (blt y n) false) \to (le n y))))).(\lambda (y: nat).(nat_ind (\lambda +(n0: nat).((eq bool (blt n0 (S n)) false) \to (le (S n) n0))) (\lambda (H0: +(eq bool (blt O (S n)) false)).(let H1 \def (match H0 in eq return (\lambda +(b: bool).(\lambda (_: (eq ? ? b)).((eq bool b false) \to (le (S n) O)))) +with [refl_equal \Rightarrow (\lambda (H1: (eq bool (blt O (S n)) +false)).(let H2 \def (eq_ind bool (blt O (S n)) (\lambda (e: bool).(match e +in bool return (\lambda (_: bool).Prop) with [true \Rightarrow True | false +\Rightarrow False])) I false H1) in (False_ind (le (S n) O) H2)))]) in (H1 +(refl_equal bool false)))) (\lambda (n0: nat).(\lambda (_: (((eq bool (blt n0 +(S n)) false) \to (le (S n) n0)))).(\lambda (H1: (eq bool (blt (S n0) (S n)) +false)).(le_S_n (S n) (S n0) (le_n_S (S n) (S n0) (le_n_S n n0 (H n0 +H1))))))) y)))) x). +(* COMMENTS +Initial nodes: 262 +END *) + diff --git a/matita/matita/contribs/lambdadelta/ground_1/definitions.ma b/matita/matita/contribs/lambdadelta/ground_1/definitions.ma new file mode 100644 index 000000000..df31468aa --- /dev/null +++ b/matita/matita/contribs/lambdadelta/ground_1/definitions.ma @@ -0,0 +1,22 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Ground-1/types/defs.ma". + +include "Ground-1/blt/defs.ma". + +include "Ground-1/plist/defs.ma". + diff --git a/matita/matita/contribs/lambdadelta/ground_1/ext/arith.ma b/matita/matita/contribs/lambdadelta/ground_1/ext/arith.ma new file mode 100644 index 000000000..f9796e7fd --- /dev/null +++ b/matita/matita/contribs/lambdadelta/ground_1/ext/arith.ma @@ -0,0 +1,737 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Ground-1/preamble.ma". + +theorem nat_dec: + \forall (n1: nat).(\forall (n2: nat).(or (eq nat n1 n2) ((eq nat n1 n2) \to +(\forall (P: Prop).P)))) +\def + \lambda (n1: nat).(nat_ind (\lambda (n: nat).(\forall (n2: nat).(or (eq nat +n n2) ((eq nat n n2) \to (\forall (P: Prop).P))))) (\lambda (n2: +nat).(nat_ind (\lambda (n: nat).(or (eq nat O n) ((eq nat O n) \to (\forall +(P: Prop).P)))) (or_introl (eq nat O O) ((eq nat O O) \to (\forall (P: +Prop).P)) (refl_equal nat O)) (\lambda (n: nat).(\lambda (_: (or (eq nat O n) +((eq nat O n) \to (\forall (P: Prop).P)))).(or_intror (eq nat O (S n)) ((eq +nat O (S n)) \to (\forall (P: Prop).P)) (\lambda (H0: (eq nat O (S +n))).(\lambda (P: Prop).(let H1 \def (eq_ind nat O (\lambda (ee: nat).(match +ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) +\Rightarrow False])) I (S n) H0) in (False_ind P H1))))))) n2)) (\lambda (n: +nat).(\lambda (H: ((\forall (n2: nat).(or (eq nat n n2) ((eq nat n n2) \to +(\forall (P: Prop).P)))))).(\lambda (n2: nat).(nat_ind (\lambda (n0: nat).(or +(eq nat (S n) n0) ((eq nat (S n) n0) \to (\forall (P: Prop).P)))) (or_intror +(eq nat (S n) O) ((eq nat (S n) O) \to (\forall (P: Prop).P)) (\lambda (H0: +(eq nat (S n) O)).(\lambda (P: Prop).(let H1 \def (eq_ind nat (S n) (\lambda +(ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow +False | (S _) \Rightarrow True])) I O H0) in (False_ind P H1))))) (\lambda +(n0: nat).(\lambda (H0: (or (eq nat (S n) n0) ((eq nat (S n) n0) \to (\forall +(P: Prop).P)))).(or_ind (eq nat n n0) ((eq nat n n0) \to (\forall (P: +Prop).P)) (or (eq nat (S n) (S n0)) ((eq nat (S n) (S n0)) \to (\forall (P: +Prop).P))) (\lambda (H1: (eq nat n n0)).(let H2 \def (eq_ind_r nat n0 +(\lambda (n3: nat).(or (eq nat (S n) n3) ((eq nat (S n) n3) \to (\forall (P: +Prop).P)))) H0 n H1) in (eq_ind nat n (\lambda (n3: nat).(or (eq nat (S n) (S +n3)) ((eq nat (S n) (S n3)) \to (\forall (P: Prop).P)))) (or_introl (eq nat +(S n) (S n)) ((eq nat (S n) (S n)) \to (\forall (P: Prop).P)) (refl_equal nat +(S n))) n0 H1))) (\lambda (H1: (((eq nat n n0) \to (\forall (P: +Prop).P)))).(or_intror (eq nat (S n) (S n0)) ((eq nat (S n) (S n0)) \to +(\forall (P: Prop).P)) (\lambda (H2: (eq nat (S n) (S n0))).(\lambda (P: +Prop).(let H3 \def (f_equal nat nat (\lambda (e: nat).(match e in nat return +(\lambda (_: nat).nat) with [O \Rightarrow n | (S n3) \Rightarrow n3])) (S n) +(S n0) H2) in (let H4 \def (eq_ind_r nat n0 (\lambda (n3: nat).((eq nat n n3) +\to (\forall (P0: Prop).P0))) H1 n H3) in (let H5 \def (eq_ind_r nat n0 +(\lambda (n3: nat).(or (eq nat (S n) n3) ((eq nat (S n) n3) \to (\forall (P0: +Prop).P0)))) H0 n H3) in (H4 (refl_equal nat n) P)))))))) (H n0)))) n2)))) +n1). +(* COMMENTS +Initial nodes: 676 +END *) + +theorem simpl_plus_r: + \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((eq nat (plus m n) +(plus p n)) \to (eq nat m p)))) +\def + \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (eq nat +(plus m n) (plus p n))).(simpl_plus_l n m p (eq_ind_r nat (plus m n) (\lambda +(n0: nat).(eq nat n0 (plus n p))) (eq_ind_r nat (plus p n) (\lambda (n0: +nat).(eq nat n0 (plus n p))) (sym_eq nat (plus n p) (plus p n) (plus_sym n +p)) (plus m n) H) (plus n m) (plus_sym n m)))))). +(* COMMENTS +Initial nodes: 119 +END *) + +theorem minus_Sx_Sy: + \forall (x: nat).(\forall (y: nat).(eq nat (minus (S x) (S y)) (minus x y))) +\def + \lambda (x: nat).(\lambda (y: nat).(refl_equal nat (minus x y))). +(* COMMENTS +Initial nodes: 13 +END *) + +theorem minus_plus_r: + \forall (m: nat).(\forall (n: nat).(eq nat (minus (plus m n) n) m)) +\def + \lambda (m: nat).(\lambda (n: nat).(eq_ind_r nat (plus n m) (\lambda (n0: +nat).(eq nat (minus n0 n) m)) (minus_plus n m) (plus m n) (plus_sym m n))). +(* COMMENTS +Initial nodes: 45 +END *) + +theorem plus_permute_2_in_3: + \forall (x: nat).(\forall (y: nat).(\forall (z: nat).(eq nat (plus (plus x +y) z) (plus (plus x z) y)))) +\def + \lambda (x: nat).(\lambda (y: nat).(\lambda (z: nat).(eq_ind_r nat (plus x +(plus y z)) (\lambda (n: nat).(eq nat n (plus (plus x z) y))) (eq_ind_r nat +(plus z y) (\lambda (n: nat).(eq nat (plus x n) (plus (plus x z) y))) (eq_ind +nat (plus (plus x z) y) (\lambda (n: nat).(eq nat n (plus (plus x z) y))) +(refl_equal nat (plus (plus x z) y)) (plus x (plus z y)) (plus_assoc_r x z +y)) (plus y z) (plus_sym y z)) (plus (plus x y) z) (plus_assoc_r x y z)))). +(* COMMENTS +Initial nodes: 163 +END *) + +theorem plus_permute_2_in_3_assoc: + \forall (n: nat).(\forall (h: nat).(\forall (k: nat).(eq nat (plus (plus n +h) k) (plus n (plus k h))))) +\def + \lambda (n: nat).(\lambda (h: nat).(\lambda (k: nat).(eq_ind_r nat (plus +(plus n k) h) (\lambda (n0: nat).(eq nat n0 (plus n (plus k h)))) (eq_ind_r +nat (plus (plus n k) h) (\lambda (n0: nat).(eq nat (plus (plus n k) h) n0)) +(refl_equal nat (plus (plus n k) h)) (plus n (plus k h)) (plus_assoc_l n k +h)) (plus (plus n h) k) (plus_permute_2_in_3 n h k)))). +(* COMMENTS +Initial nodes: 119 +END *) + +theorem plus_O: + \forall (x: nat).(\forall (y: nat).((eq nat (plus x y) O) \to (land (eq nat +x O) (eq nat y O)))) +\def + \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((eq nat (plus +n y) O) \to (land (eq nat n O) (eq nat y O))))) (\lambda (y: nat).(\lambda +(H: (eq nat (plus O y) O)).(conj (eq nat O O) (eq nat y O) (refl_equal nat O) +H))) (\lambda (n: nat).(\lambda (_: ((\forall (y: nat).((eq nat (plus n y) O) +\to (land (eq nat n O) (eq nat y O)))))).(\lambda (y: nat).(\lambda (H0: (eq +nat (plus (S n) y) O)).(let H1 \def (match H0 in eq return (\lambda (n0: +nat).(\lambda (_: (eq ? ? n0)).((eq nat n0 O) \to (land (eq nat (S n) O) (eq +nat y O))))) with [refl_equal \Rightarrow (\lambda (H1: (eq nat (plus (S n) +y) O)).(let H2 \def (eq_ind nat (plus (S n) y) (\lambda (e: nat).(match e in +nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) +\Rightarrow True])) I O H1) in (False_ind (land (eq nat (S n) O) (eq nat y +O)) H2)))]) in (H1 (refl_equal nat O))))))) x). +(* COMMENTS +Initial nodes: 233 +END *) + +theorem minus_Sx_SO: + \forall (x: nat).(eq nat (minus (S x) (S O)) x) +\def + \lambda (x: nat).(eq_ind nat x (\lambda (n: nat).(eq nat n x)) (refl_equal +nat x) (minus x O) (minus_n_O x)). +(* COMMENTS +Initial nodes: 33 +END *) + +theorem eq_nat_dec: + \forall (i: nat).(\forall (j: nat).(or (not (eq nat i j)) (eq nat i j))) +\def + \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (j: nat).(or (not (eq +nat n j)) (eq nat n j)))) (\lambda (j: nat).(nat_ind (\lambda (n: nat).(or +(not (eq nat O n)) (eq nat O n))) (or_intror (not (eq nat O O)) (eq nat O O) +(refl_equal nat O)) (\lambda (n: nat).(\lambda (_: (or (not (eq nat O n)) (eq +nat O n))).(or_introl (not (eq nat O (S n))) (eq nat O (S n)) (O_S n)))) j)) +(\lambda (n: nat).(\lambda (H: ((\forall (j: nat).(or (not (eq nat n j)) (eq +nat n j))))).(\lambda (j: nat).(nat_ind (\lambda (n0: nat).(or (not (eq nat +(S n) n0)) (eq nat (S n) n0))) (or_introl (not (eq nat (S n) O)) (eq nat (S +n) O) (sym_not_eq nat O (S n) (O_S n))) (\lambda (n0: nat).(\lambda (_: (or +(not (eq nat (S n) n0)) (eq nat (S n) n0))).(or_ind (not (eq nat n n0)) (eq +nat n n0) (or (not (eq nat (S n) (S n0))) (eq nat (S n) (S n0))) (\lambda +(H1: (not (eq nat n n0))).(or_introl (not (eq nat (S n) (S n0))) (eq nat (S +n) (S n0)) (not_eq_S n n0 H1))) (\lambda (H1: (eq nat n n0)).(or_intror (not +(eq nat (S n) (S n0))) (eq nat (S n) (S n0)) (f_equal nat nat S n n0 H1))) (H +n0)))) j)))) i). +(* COMMENTS +Initial nodes: 401 +END *) + +theorem neq_eq_e: + \forall (i: nat).(\forall (j: nat).(\forall (P: Prop).((((not (eq nat i j)) +\to P)) \to ((((eq nat i j) \to P)) \to P)))) +\def + \lambda (i: nat).(\lambda (j: nat).(\lambda (P: Prop).(\lambda (H: (((not +(eq nat i j)) \to P))).(\lambda (H0: (((eq nat i j) \to P))).(let o \def +(eq_nat_dec i j) in (or_ind (not (eq nat i j)) (eq nat i j) P H H0 o)))))). +(* COMMENTS +Initial nodes: 61 +END *) + +theorem le_false: + \forall (m: nat).(\forall (n: nat).(\forall (P: Prop).((le m n) \to ((le (S +n) m) \to P)))) +\def + \lambda (m: nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).(\forall (P: +Prop).((le n n0) \to ((le (S n0) n) \to P))))) (\lambda (n: nat).(\lambda (P: +Prop).(\lambda (_: (le O n)).(\lambda (H0: (le (S n) O)).(let H1 \def (match +H0 in le return (\lambda (n0: nat).(\lambda (_: (le ? n0)).((eq nat n0 O) \to +P))) with [le_n \Rightarrow (\lambda (H1: (eq nat (S n) O)).(let H2 \def +(eq_ind nat (S n) (\lambda (e: nat).(match e in nat return (\lambda (_: +nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H1) in +(False_ind P H2))) | (le_S m0 H1) \Rightarrow (\lambda (H2: (eq nat (S m0) +O)).((let H3 \def (eq_ind nat (S m0) (\lambda (e: nat).(match e in nat return +(\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) +I O H2) in (False_ind ((le (S n) m0) \to P) H3)) H1))]) in (H1 (refl_equal +nat O))))))) (\lambda (n: nat).(\lambda (H: ((\forall (n0: nat).(\forall (P: +Prop).((le n n0) \to ((le (S n0) n) \to P)))))).(\lambda (n0: nat).(nat_ind +(\lambda (n1: nat).(\forall (P: Prop).((le (S n) n1) \to ((le (S n1) (S n)) +\to P)))) (\lambda (P: Prop).(\lambda (H0: (le (S n) O)).(\lambda (_: (le (S +O) (S n))).(let H2 \def (match H0 in le return (\lambda (n1: nat).(\lambda +(_: (le ? n1)).((eq nat n1 O) \to P))) with [le_n \Rightarrow (\lambda (H2: +(eq nat (S n) O)).(let H3 \def (eq_ind nat (S n) (\lambda (e: nat).(match e +in nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) +\Rightarrow True])) I O H2) in (False_ind P H3))) | (le_S m0 H2) \Rightarrow +(\lambda (H3: (eq nat (S m0) O)).((let H4 \def (eq_ind nat (S m0) (\lambda +(e: nat).(match e in nat return (\lambda (_: nat).Prop) with [O \Rightarrow +False | (S _) \Rightarrow True])) I O H3) in (False_ind ((le (S n) m0) \to P) +H4)) H2))]) in (H2 (refl_equal nat O)))))) (\lambda (n1: nat).(\lambda (_: +((\forall (P: Prop).((le (S n) n1) \to ((le (S n1) (S n)) \to P))))).(\lambda +(P: Prop).(\lambda (H1: (le (S n) (S n1))).(\lambda (H2: (le (S (S n1)) (S +n))).(H n1 P (le_S_n n n1 H1) (le_S_n (S n1) n H2))))))) n0)))) m). +(* COMMENTS +Initial nodes: 409 +END *) + +theorem le_Sx_x: + \forall (x: nat).((le (S x) x) \to (\forall (P: Prop).P)) +\def + \lambda (x: nat).(\lambda (H: (le (S x) x)).(\lambda (P: Prop).(let H0 \def +le_Sn_n in (False_ind P (H0 x H))))). +(* COMMENTS +Initial nodes: 23 +END *) + +theorem le_n_pred: + \forall (n: nat).(\forall (m: nat).((le n m) \to (le (pred n) (pred m)))) +\def + \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(le_ind n (\lambda +(n0: nat).(le (pred n) (pred n0))) (le_n (pred n)) (\lambda (m0: +nat).(\lambda (_: (le n m0)).(\lambda (H1: (le (pred n) (pred m0))).(le_trans +(pred n) (pred m0) m0 H1 (le_pred_n m0))))) m H))). +(* COMMENTS +Initial nodes: 71 +END *) + +theorem minus_le: + \forall (x: nat).(\forall (y: nat).(le (minus x y) x)) +\def + \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).(le (minus n +y) n))) (\lambda (_: nat).(le_n O)) (\lambda (n: nat).(\lambda (H: ((\forall +(y: nat).(le (minus n y) n)))).(\lambda (y: nat).(nat_ind (\lambda (n0: +nat).(le (minus (S n) n0) (S n))) (le_n (S n)) (\lambda (n0: nat).(\lambda +(_: (le (match n0 with [O \Rightarrow (S n) | (S l) \Rightarrow (minus n l)]) +(S n))).(le_S (minus n n0) n (H n0)))) y)))) x). +(* COMMENTS +Initial nodes: 101 +END *) + +theorem le_plus_minus_sym: + \forall (n: nat).(\forall (m: nat).((le n m) \to (eq nat m (plus (minus m n) +n)))) +\def + \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(eq_ind_r nat +(plus n (minus m n)) (\lambda (n0: nat).(eq nat m n0)) (le_plus_minus n m H) +(plus (minus m n) n) (plus_sym (minus m n) n)))). +(* COMMENTS +Initial nodes: 61 +END *) + +theorem le_minus_minus: + \forall (x: nat).(\forall (y: nat).((le x y) \to (\forall (z: nat).((le y z) +\to (le (minus y x) (minus z x)))))) +\def + \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (le x y)).(\lambda (z: +nat).(\lambda (H0: (le y z)).(simpl_le_plus_l x (minus y x) (minus z x) +(eq_ind_r nat y (\lambda (n: nat).(le n (plus x (minus z x)))) (eq_ind_r nat +z (\lambda (n: nat).(le y n)) H0 (plus x (minus z x)) (le_plus_minus_r x z +(le_trans x y z H H0))) (plus x (minus y x)) (le_plus_minus_r x y H))))))). +(* COMMENTS +Initial nodes: 117 +END *) + +theorem le_minus_plus: + \forall (z: nat).(\forall (x: nat).((le z x) \to (\forall (y: nat).(eq nat +(minus (plus x y) z) (plus (minus x z) y))))) +\def + \lambda (z: nat).(nat_ind (\lambda (n: nat).(\forall (x: nat).((le n x) \to +(\forall (y: nat).(eq nat (minus (plus x y) n) (plus (minus x n) y)))))) +(\lambda (x: nat).(\lambda (H: (le O x)).(let H0 \def (match H in le return +(\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat n x) \to (\forall (y: +nat).(eq nat (minus (plus x y) O) (plus (minus x O) y)))))) with [le_n +\Rightarrow (\lambda (H0: (eq nat O x)).(eq_ind nat O (\lambda (n: +nat).(\forall (y: nat).(eq nat (minus (plus n y) O) (plus (minus n O) y)))) +(\lambda (y: nat).(sym_eq nat (plus (minus O O) y) (minus (plus O y) O) +(minus_n_O (plus O y)))) x H0)) | (le_S m H0) \Rightarrow (\lambda (H1: (eq +nat (S m) x)).(eq_ind nat (S m) (\lambda (n: nat).((le O m) \to (\forall (y: +nat).(eq nat (minus (plus n y) O) (plus (minus n O) y))))) (\lambda (_: (le O +m)).(\lambda (y: nat).(refl_equal nat (plus (minus (S m) O) y)))) x H1 H0))]) +in (H0 (refl_equal nat x))))) (\lambda (z0: nat).(\lambda (H: ((\forall (x: +nat).((le z0 x) \to (\forall (y: nat).(eq nat (minus (plus x y) z0) (plus +(minus x z0) y))))))).(\lambda (x: nat).(nat_ind (\lambda (n: nat).((le (S +z0) n) \to (\forall (y: nat).(eq nat (minus (plus n y) (S z0)) (plus (minus n +(S z0)) y))))) (\lambda (H0: (le (S z0) O)).(\lambda (y: nat).(let H1 \def +(match H0 in le return (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat n O) +\to (eq nat (minus (plus O y) (S z0)) (plus (minus O (S z0)) y))))) with +[le_n \Rightarrow (\lambda (H1: (eq nat (S z0) O)).(let H2 \def (eq_ind nat +(S z0) (\lambda (e: nat).(match e in nat return (\lambda (_: nat).Prop) with +[O \Rightarrow False | (S _) \Rightarrow True])) I O H1) in (False_ind (eq +nat (minus (plus O y) (S z0)) (plus (minus O (S z0)) y)) H2))) | (le_S m H1) +\Rightarrow (\lambda (H2: (eq nat (S m) O)).((let H3 \def (eq_ind nat (S m) +(\lambda (e: nat).(match e in nat return (\lambda (_: nat).Prop) with [O +\Rightarrow False | (S _) \Rightarrow True])) I O H2) in (False_ind ((le (S +z0) m) \to (eq nat (minus (plus O y) (S z0)) (plus (minus O (S z0)) y))) H3)) +H1))]) in (H1 (refl_equal nat O))))) (\lambda (n: nat).(\lambda (_: (((le (S +z0) n) \to (\forall (y: nat).(eq nat (minus (plus n y) (S z0)) (plus (minus n +(S z0)) y)))))).(\lambda (H1: (le (S z0) (S n))).(\lambda (y: nat).(H n +(le_S_n z0 n H1) y))))) x)))) z). +(* COMMENTS +Initial nodes: 603 +END *) + +theorem le_minus: + \forall (x: nat).(\forall (z: nat).(\forall (y: nat).((le (plus x y) z) \to +(le x (minus z y))))) +\def + \lambda (x: nat).(\lambda (z: nat).(\lambda (y: nat).(\lambda (H: (le (plus +x y) z)).(eq_ind nat (minus (plus x y) y) (\lambda (n: nat).(le n (minus z +y))) (le_minus_minus y (plus x y) (le_plus_r x y) z H) x (minus_plus_r x +y))))). +(* COMMENTS +Initial nodes: 69 +END *) + +theorem le_trans_plus_r: + \forall (x: nat).(\forall (y: nat).(\forall (z: nat).((le (plus x y) z) \to +(le y z)))) +\def + \lambda (x: nat).(\lambda (y: nat).(\lambda (z: nat).(\lambda (H: (le (plus +x y) z)).(le_trans y (plus x y) z (le_plus_r x y) H)))). +(* COMMENTS +Initial nodes: 35 +END *) + +theorem lt_x_O: + \forall (x: nat).((lt x O) \to (\forall (P: Prop).P)) +\def + \lambda (x: nat).(\lambda (H: (le (S x) O)).(\lambda (P: Prop).(let H_y \def +(le_n_O_eq (S x) H) in (let H0 \def (eq_ind nat O (\lambda (ee: nat).(match +ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) +\Rightarrow False])) I (S x) H_y) in (False_ind P H0))))). +(* COMMENTS +Initial nodes: 48 +END *) + +theorem le_gen_S: + \forall (m: nat).(\forall (x: nat).((le (S m) x) \to (ex2 nat (\lambda (n: +nat).(eq nat x (S n))) (\lambda (n: nat).(le m n))))) +\def + \lambda (m: nat).(\lambda (x: nat).(\lambda (H: (le (S m) x)).(let H0 \def +(match H in le return (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat n x) +\to (ex2 nat (\lambda (n0: nat).(eq nat x (S n0))) (\lambda (n0: nat).(le m +n0)))))) with [le_n \Rightarrow (\lambda (H0: (eq nat (S m) x)).(eq_ind nat +(S m) (\lambda (n: nat).(ex2 nat (\lambda (n0: nat).(eq nat n (S n0))) +(\lambda (n0: nat).(le m n0)))) (ex_intro2 nat (\lambda (n: nat).(eq nat (S +m) (S n))) (\lambda (n: nat).(le m n)) m (refl_equal nat (S m)) (le_n m)) x +H0)) | (le_S m0 H0) \Rightarrow (\lambda (H1: (eq nat (S m0) x)).(eq_ind nat +(S m0) (\lambda (n: nat).((le (S m) m0) \to (ex2 nat (\lambda (n0: nat).(eq +nat n (S n0))) (\lambda (n0: nat).(le m n0))))) (\lambda (H2: (le (S m) +m0)).(ex_intro2 nat (\lambda (n: nat).(eq nat (S m0) (S n))) (\lambda (n: +nat).(le m n)) m0 (refl_equal nat (S m0)) (le_S_n m m0 (le_S (S m) m0 H2)))) +x H1 H0))]) in (H0 (refl_equal nat x))))). +(* COMMENTS +Initial nodes: 261 +END *) + +theorem lt_x_plus_x_Sy: + \forall (x: nat).(\forall (y: nat).(lt x (plus x (S y)))) +\def + \lambda (x: nat).(\lambda (y: nat).(eq_ind_r nat (plus (S y) x) (\lambda (n: +nat).(lt x n)) (le_S_n (S x) (S (plus y x)) (le_n_S (S x) (S (plus y x)) +(le_n_S x (plus y x) (le_plus_r y x)))) (plus x (S y)) (plus_sym x (S y)))). +(* COMMENTS +Initial nodes: 83 +END *) + +theorem simpl_lt_plus_r: + \forall (p: nat).(\forall (n: nat).(\forall (m: nat).((lt (plus n p) (plus m +p)) \to (lt n m)))) +\def + \lambda (p: nat).(\lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt (plus +n p) (plus m p))).(simpl_lt_plus_l n m p (let H0 \def (eq_ind nat (plus n p) +(\lambda (n0: nat).(lt n0 (plus m p))) H (plus p n) (plus_sym n p)) in (let +H1 \def (eq_ind nat (plus m p) (\lambda (n0: nat).(lt (plus p n) n0)) H0 +(plus p m) (plus_sym m p)) in H1)))))). +(* COMMENTS +Initial nodes: 101 +END *) + +theorem minus_x_Sy: + \forall (x: nat).(\forall (y: nat).((lt y x) \to (eq nat (minus x y) (S +(minus x (S y)))))) +\def + \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((lt y n) \to +(eq nat (minus n y) (S (minus n (S y))))))) (\lambda (y: nat).(\lambda (H: +(lt y O)).(let H0 \def (match H in le return (\lambda (n: nat).(\lambda (_: +(le ? n)).((eq nat n O) \to (eq nat (minus O y) (S (minus O (S y))))))) with +[le_n \Rightarrow (\lambda (H0: (eq nat (S y) O)).(let H1 \def (eq_ind nat (S +y) (\lambda (e: nat).(match e in nat return (\lambda (_: nat).Prop) with [O +\Rightarrow False | (S _) \Rightarrow True])) I O H0) in (False_ind (eq nat +(minus O y) (S (minus O (S y)))) H1))) | (le_S m H0) \Rightarrow (\lambda +(H1: (eq nat (S m) O)).((let H2 \def (eq_ind nat (S m) (\lambda (e: +nat).(match e in nat return (\lambda (_: nat).Prop) with [O \Rightarrow False +| (S _) \Rightarrow True])) I O H1) in (False_ind ((le (S y) m) \to (eq nat +(minus O y) (S (minus O (S y))))) H2)) H0))]) in (H0 (refl_equal nat O))))) +(\lambda (n: nat).(\lambda (H: ((\forall (y: nat).((lt y n) \to (eq nat +(minus n y) (S (minus n (S y)))))))).(\lambda (y: nat).(nat_ind (\lambda (n0: +nat).((lt n0 (S n)) \to (eq nat (minus (S n) n0) (S (minus (S n) (S n0)))))) +(\lambda (_: (lt O (S n))).(eq_ind nat n (\lambda (n0: nat).(eq nat (S n) (S +n0))) (refl_equal nat (S n)) (minus n O) (minus_n_O n))) (\lambda (n0: +nat).(\lambda (_: (((lt n0 (S n)) \to (eq nat (minus (S n) n0) (S (minus (S +n) (S n0))))))).(\lambda (H1: (lt (S n0) (S n))).(let H2 \def (le_S_n (S n0) +n H1) in (H n0 H2))))) y)))) x). +(* COMMENTS +Initial nodes: 383 +END *) + +theorem lt_plus_minus: + \forall (x: nat).(\forall (y: nat).((lt x y) \to (eq nat y (S (plus x (minus +y (S x))))))) +\def + \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(le_plus_minus (S +x) y H))). +(* COMMENTS +Initial nodes: 19 +END *) + +theorem lt_plus_minus_r: + \forall (x: nat).(\forall (y: nat).((lt x y) \to (eq nat y (S (plus (minus y +(S x)) x))))) +\def + \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(eq_ind_r nat +(plus x (minus y (S x))) (\lambda (n: nat).(eq nat y (S n))) (lt_plus_minus x +y H) (plus (minus y (S x)) x) (plus_sym (minus y (S x)) x)))). +(* COMMENTS +Initial nodes: 69 +END *) + +theorem minus_x_SO: + \forall (x: nat).((lt O x) \to (eq nat x (S (minus x (S O))))) +\def + \lambda (x: nat).(\lambda (H: (lt O x)).(eq_ind nat (minus x O) (\lambda (n: +nat).(eq nat x n)) (eq_ind nat x (\lambda (n: nat).(eq nat x n)) (refl_equal +nat x) (minus x O) (minus_n_O x)) (S (minus x (S O))) (minus_x_Sy x O H))). +(* COMMENTS +Initial nodes: 77 +END *) + +theorem le_x_pred_y: + \forall (y: nat).(\forall (x: nat).((lt x y) \to (le x (pred y)))) +\def + \lambda (y: nat).(nat_ind (\lambda (n: nat).(\forall (x: nat).((lt x n) \to +(le x (pred n))))) (\lambda (x: nat).(\lambda (H: (lt x O)).(let H0 \def +(match H in le return (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat n O) +\to (le x O)))) with [le_n \Rightarrow (\lambda (H0: (eq nat (S x) O)).(let +H1 \def (eq_ind nat (S x) (\lambda (e: nat).(match e in nat return (\lambda +(_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H0) +in (False_ind (le x O) H1))) | (le_S m H0) \Rightarrow (\lambda (H1: (eq nat +(S m) O)).((let H2 \def (eq_ind nat (S m) (\lambda (e: nat).(match e in nat +return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow +True])) I O H1) in (False_ind ((le (S x) m) \to (le x O)) H2)) H0))]) in (H0 +(refl_equal nat O))))) (\lambda (n: nat).(\lambda (_: ((\forall (x: nat).((lt +x n) \to (le x (pred n)))))).(\lambda (x: nat).(\lambda (H0: (lt x (S +n))).(le_S_n x n H0))))) y). +(* COMMENTS +Initial nodes: 189 +END *) + +theorem lt_le_minus: + \forall (x: nat).(\forall (y: nat).((lt x y) \to (le x (minus y (S O))))) +\def + \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(le_minus x y (S +O) (eq_ind_r nat (plus (S O) x) (\lambda (n: nat).(le n y)) H (plus x (S O)) +(plus_sym x (S O)))))). +(* COMMENTS +Initial nodes: 57 +END *) + +theorem lt_le_e: + \forall (n: nat).(\forall (d: nat).(\forall (P: Prop).((((lt n d) \to P)) +\to ((((le d n) \to P)) \to P)))) +\def + \lambda (n: nat).(\lambda (d: nat).(\lambda (P: Prop).(\lambda (H: (((lt n +d) \to P))).(\lambda (H0: (((le d n) \to P))).(let H1 \def (le_or_lt d n) in +(or_ind (le d n) (lt n d) P H0 H H1)))))). +(* COMMENTS +Initial nodes: 49 +END *) + +theorem lt_eq_e: + \forall (x: nat).(\forall (y: nat).(\forall (P: Prop).((((lt x y) \to P)) +\to ((((eq nat x y) \to P)) \to ((le x y) \to P))))) +\def + \lambda (x: nat).(\lambda (y: nat).(\lambda (P: Prop).(\lambda (H: (((lt x +y) \to P))).(\lambda (H0: (((eq nat x y) \to P))).(\lambda (H1: (le x +y)).(or_ind (lt x y) (eq nat x y) P H H0 (le_lt_or_eq x y H1))))))). +(* COMMENTS +Initial nodes: 59 +END *) + +theorem lt_eq_gt_e: + \forall (x: nat).(\forall (y: nat).(\forall (P: Prop).((((lt x y) \to P)) +\to ((((eq nat x y) \to P)) \to ((((lt y x) \to P)) \to P))))) +\def + \lambda (x: nat).(\lambda (y: nat).(\lambda (P: Prop).(\lambda (H: (((lt x +y) \to P))).(\lambda (H0: (((eq nat x y) \to P))).(\lambda (H1: (((lt y x) +\to P))).(lt_le_e x y P H (\lambda (H2: (le y x)).(lt_eq_e y x P H1 (\lambda +(H3: (eq nat y x)).(H0 (sym_eq nat y x H3))) H2)))))))). +(* COMMENTS +Initial nodes: 79 +END *) + +theorem lt_gen_xS: + \forall (x: nat).(\forall (n: nat).((lt x (S n)) \to (or (eq nat x O) (ex2 +nat (\lambda (m: nat).(eq nat x (S m))) (\lambda (m: nat).(lt m n)))))) +\def + \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).((lt n (S +n0)) \to (or (eq nat n O) (ex2 nat (\lambda (m: nat).(eq nat n (S m))) +(\lambda (m: nat).(lt m n0))))))) (\lambda (n: nat).(\lambda (_: (lt O (S +n))).(or_introl (eq nat O O) (ex2 nat (\lambda (m: nat).(eq nat O (S m))) +(\lambda (m: nat).(lt m n))) (refl_equal nat O)))) (\lambda (n: nat).(\lambda +(_: ((\forall (n0: nat).((lt n (S n0)) \to (or (eq nat n O) (ex2 nat (\lambda +(m: nat).(eq nat n (S m))) (\lambda (m: nat).(lt m n0)))))))).(\lambda (n0: +nat).(\lambda (H0: (lt (S n) (S n0))).(or_intror (eq nat (S n) O) (ex2 nat +(\lambda (m: nat).(eq nat (S n) (S m))) (\lambda (m: nat).(lt m n0))) +(ex_intro2 nat (\lambda (m: nat).(eq nat (S n) (S m))) (\lambda (m: nat).(lt +m n0)) n (refl_equal nat (S n)) (le_S_n (S n) n0 H0))))))) x). +(* COMMENTS +Initial nodes: 243 +END *) + +theorem le_lt_false: + \forall (x: nat).(\forall (y: nat).((le x y) \to ((lt y x) \to (\forall (P: +Prop).P)))) +\def + \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (le x y)).(\lambda (H0: (lt +y x)).(\lambda (P: Prop).(False_ind P (le_not_lt x y H H0)))))). +(* COMMENTS +Initial nodes: 31 +END *) + +theorem lt_neq: + \forall (x: nat).(\forall (y: nat).((lt x y) \to (not (eq nat x y)))) +\def + \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(\lambda (H0: (eq +nat x y)).(let H1 \def (eq_ind nat x (\lambda (n: nat).(lt n y)) H y H0) in +(lt_n_n y H1))))). +(* COMMENTS +Initial nodes: 43 +END *) + +theorem arith0: + \forall (h2: nat).(\forall (d2: nat).(\forall (n: nat).((le (plus d2 h2) n) +\to (\forall (h1: nat).(le (plus d2 h1) (minus (plus n h1) h2)))))) +\def + \lambda (h2: nat).(\lambda (d2: nat).(\lambda (n: nat).(\lambda (H: (le +(plus d2 h2) n)).(\lambda (h1: nat).(eq_ind nat (minus (plus h2 (plus d2 h1)) +h2) (\lambda (n0: nat).(le n0 (minus (plus n h1) h2))) (le_minus_minus h2 +(plus h2 (plus d2 h1)) (le_plus_l h2 (plus d2 h1)) (plus n h1) (eq_ind_r nat +(plus (plus h2 d2) h1) (\lambda (n0: nat).(le n0 (plus n h1))) (eq_ind_r nat +(plus d2 h2) (\lambda (n0: nat).(le (plus n0 h1) (plus n h1))) (le_S_n (plus +(plus d2 h2) h1) (plus n h1) (le_n_S (plus (plus d2 h2) h1) (plus n h1) +(le_plus_plus (plus d2 h2) n h1 h1 H (le_n h1)))) (plus h2 d2) (plus_sym h2 +d2)) (plus h2 (plus d2 h1)) (plus_assoc_l h2 d2 h1))) (plus d2 h1) +(minus_plus h2 (plus d2 h1))))))). +(* COMMENTS +Initial nodes: 235 +END *) + +theorem O_minus: + \forall (x: nat).(\forall (y: nat).((le x y) \to (eq nat (minus x y) O))) +\def + \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((le n y) \to +(eq nat (minus n y) O)))) (\lambda (y: nat).(\lambda (_: (le O +y)).(refl_equal nat O))) (\lambda (x0: nat).(\lambda (H: ((\forall (y: +nat).((le x0 y) \to (eq nat (minus x0 y) O))))).(\lambda (y: nat).(nat_ind +(\lambda (n: nat).((le (S x0) n) \to (eq nat (match n with [O \Rightarrow (S +x0) | (S l) \Rightarrow (minus x0 l)]) O))) (\lambda (H0: (le (S x0) +O)).(ex2_ind nat (\lambda (n: nat).(eq nat O (S n))) (\lambda (n: nat).(le x0 +n)) (eq nat (S x0) O) (\lambda (x1: nat).(\lambda (H1: (eq nat O (S +x1))).(\lambda (_: (le x0 x1)).(let H3 \def (eq_ind nat O (\lambda (ee: +nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True +| (S _) \Rightarrow False])) I (S x1) H1) in (False_ind (eq nat (S x0) O) +H3))))) (le_gen_S x0 O H0))) (\lambda (n: nat).(\lambda (_: (((le (S x0) n) +\to (eq nat (match n with [O \Rightarrow (S x0) | (S l) \Rightarrow (minus x0 +l)]) O)))).(\lambda (H1: (le (S x0) (S n))).(H n (le_S_n x0 n H1))))) y)))) +x). +(* COMMENTS +Initial nodes: 252 +END *) + +theorem minus_minus: + \forall (z: nat).(\forall (x: nat).(\forall (y: nat).((le z x) \to ((le z y) +\to ((eq nat (minus x z) (minus y z)) \to (eq nat x y)))))) +\def + \lambda (z: nat).(nat_ind (\lambda (n: nat).(\forall (x: nat).(\forall (y: +nat).((le n x) \to ((le n y) \to ((eq nat (minus x n) (minus y n)) \to (eq +nat x y))))))) (\lambda (x: nat).(\lambda (y: nat).(\lambda (_: (le O +x)).(\lambda (_: (le O y)).(\lambda (H1: (eq nat (minus x O) (minus y +O))).(let H2 \def (eq_ind_r nat (minus x O) (\lambda (n: nat).(eq nat n +(minus y O))) H1 x (minus_n_O x)) in (let H3 \def (eq_ind_r nat (minus y O) +(\lambda (n: nat).(eq nat x n)) H2 y (minus_n_O y)) in H3))))))) (\lambda +(z0: nat).(\lambda (IH: ((\forall (x: nat).(\forall (y: nat).((le z0 x) \to +((le z0 y) \to ((eq nat (minus x z0) (minus y z0)) \to (eq nat x +y)))))))).(\lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((le +(S z0) n) \to ((le (S z0) y) \to ((eq nat (minus n (S z0)) (minus y (S z0))) +\to (eq nat n y)))))) (\lambda (y: nat).(\lambda (H: (le (S z0) O)).(\lambda +(_: (le (S z0) y)).(\lambda (_: (eq nat (minus O (S z0)) (minus y (S +z0)))).(ex2_ind nat (\lambda (n: nat).(eq nat O (S n))) (\lambda (n: nat).(le +z0 n)) (eq nat O y) (\lambda (x0: nat).(\lambda (H2: (eq nat O (S +x0))).(\lambda (_: (le z0 x0)).(let H4 \def (eq_ind nat O (\lambda (ee: +nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True +| (S _) \Rightarrow False])) I (S x0) H2) in (False_ind (eq nat O y) H4))))) +(le_gen_S z0 O H)))))) (\lambda (x0: nat).(\lambda (_: ((\forall (y: +nat).((le (S z0) x0) \to ((le (S z0) y) \to ((eq nat (minus x0 (S z0)) (minus +y (S z0))) \to (eq nat x0 y))))))).(\lambda (y: nat).(nat_ind (\lambda (n: +nat).((le (S z0) (S x0)) \to ((le (S z0) n) \to ((eq nat (minus (S x0) (S +z0)) (minus n (S z0))) \to (eq nat (S x0) n))))) (\lambda (H: (le (S z0) (S +x0))).(\lambda (H0: (le (S z0) O)).(\lambda (_: (eq nat (minus (S x0) (S z0)) +(minus O (S z0)))).(let H_y \def (le_S_n z0 x0 H) in (ex2_ind nat (\lambda +(n: nat).(eq nat O (S n))) (\lambda (n: nat).(le z0 n)) (eq nat (S x0) O) +(\lambda (x1: nat).(\lambda (H2: (eq nat O (S x1))).(\lambda (_: (le z0 +x1)).(let H4 \def (eq_ind nat O (\lambda (ee: nat).(match ee in nat return +(\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) +I (S x1) H2) in (False_ind (eq nat (S x0) O) H4))))) (le_gen_S z0 O H0)))))) +(\lambda (y0: nat).(\lambda (_: (((le (S z0) (S x0)) \to ((le (S z0) y0) \to +((eq nat (minus (S x0) (S z0)) (minus y0 (S z0))) \to (eq nat (S x0) +y0)))))).(\lambda (H: (le (S z0) (S x0))).(\lambda (H0: (le (S z0) (S +y0))).(\lambda (H1: (eq nat (minus (S x0) (S z0)) (minus (S y0) (S +z0)))).(f_equal nat nat S x0 y0 (IH x0 y0 (le_S_n z0 x0 H) (le_S_n z0 y0 H0) +H1))))))) y)))) x)))) z). +(* COMMENTS +Initial nodes: 751 +END *) + +theorem plus_plus: + \forall (z: nat).(\forall (x1: nat).(\forall (x2: nat).(\forall (y1: +nat).(\forall (y2: nat).((le x1 z) \to ((le x2 z) \to ((eq nat (plus (minus z +x1) y1) (plus (minus z x2) y2)) \to (eq nat (plus x1 y2) (plus x2 y1))))))))) +\def + \lambda (z: nat).(nat_ind (\lambda (n: nat).(\forall (x1: nat).(\forall (x2: +nat).(\forall (y1: nat).(\forall (y2: nat).((le x1 n) \to ((le x2 n) \to ((eq +nat (plus (minus n x1) y1) (plus (minus n x2) y2)) \to (eq nat (plus x1 y2) +(plus x2 y1)))))))))) (\lambda (x1: nat).(\lambda (x2: nat).(\lambda (y1: +nat).(\lambda (y2: nat).(\lambda (H: (le x1 O)).(\lambda (H0: (le x2 +O)).(\lambda (H1: (eq nat y1 y2)).(eq_ind nat y1 (\lambda (n: nat).(eq nat +(plus x1 n) (plus x2 y1))) (let H_y \def (le_n_O_eq x2 H0) in (eq_ind nat O +(\lambda (n: nat).(eq nat (plus x1 y1) (plus n y1))) (let H_y0 \def +(le_n_O_eq x1 H) in (eq_ind nat O (\lambda (n: nat).(eq nat (plus n y1) (plus +O y1))) (refl_equal nat (plus O y1)) x1 H_y0)) x2 H_y)) y2 H1)))))))) +(\lambda (z0: nat).(\lambda (IH: ((\forall (x1: nat).(\forall (x2: +nat).(\forall (y1: nat).(\forall (y2: nat).((le x1 z0) \to ((le x2 z0) \to +((eq nat (plus (minus z0 x1) y1) (plus (minus z0 x2) y2)) \to (eq nat (plus +x1 y2) (plus x2 y1))))))))))).(\lambda (x1: nat).(nat_ind (\lambda (n: +nat).(\forall (x2: nat).(\forall (y1: nat).(\forall (y2: nat).((le n (S z0)) +\to ((le x2 (S z0)) \to ((eq nat (plus (minus (S z0) n) y1) (plus (minus (S +z0) x2) y2)) \to (eq nat (plus n y2) (plus x2 y1))))))))) (\lambda (x2: +nat).(nat_ind (\lambda (n: nat).(\forall (y1: nat).(\forall (y2: nat).((le O +(S z0)) \to ((le n (S z0)) \to ((eq nat (plus (minus (S z0) O) y1) (plus +(minus (S z0) n) y2)) \to (eq nat (plus O y2) (plus n y1)))))))) (\lambda +(y1: nat).(\lambda (y2: nat).(\lambda (_: (le O (S z0))).(\lambda (_: (le O +(S z0))).(\lambda (H1: (eq nat (S (plus z0 y1)) (S (plus z0 y2)))).(let H_y +\def (IH O O) in (let H2 \def (eq_ind_r nat (minus z0 O) (\lambda (n: +nat).(\forall (y3: nat).(\forall (y4: nat).((le O z0) \to ((le O z0) \to ((eq +nat (plus n y3) (plus n y4)) \to (eq nat y4 y3))))))) H_y z0 (minus_n_O z0)) +in (H2 y1 y2 (le_O_n z0) (le_O_n z0) (eq_add_S (plus z0 y1) (plus z0 y2) +H1))))))))) (\lambda (x3: nat).(\lambda (_: ((\forall (y1: nat).(\forall (y2: +nat).((le O (S z0)) \to ((le x3 (S z0)) \to ((eq nat (S (plus z0 y1)) (plus +(match x3 with [O \Rightarrow (S z0) | (S l) \Rightarrow (minus z0 l)]) y2)) +\to (eq nat y2 (plus x3 y1))))))))).(\lambda (y1: nat).(\lambda (y2: +nat).(\lambda (_: (le O (S z0))).(\lambda (H0: (le (S x3) (S z0))).(\lambda +(H1: (eq nat (S (plus z0 y1)) (plus (minus z0 x3) y2))).(let H_y \def (IH O +x3 (S y1)) in (let H2 \def (eq_ind_r nat (minus z0 O) (\lambda (n: +nat).(\forall (y3: nat).((le O z0) \to ((le x3 z0) \to ((eq nat (plus n (S +y1)) (plus (minus z0 x3) y3)) \to (eq nat y3 (plus x3 (S y1)))))))) H_y z0 +(minus_n_O z0)) in (let H3 \def (eq_ind_r nat (plus z0 (S y1)) (\lambda (n: +nat).(\forall (y3: nat).((le O z0) \to ((le x3 z0) \to ((eq nat n (plus +(minus z0 x3) y3)) \to (eq nat y3 (plus x3 (S y1)))))))) H2 (S (plus z0 y1)) +(plus_n_Sm z0 y1)) in (let H4 \def (eq_ind_r nat (plus x3 (S y1)) (\lambda +(n: nat).(\forall (y3: nat).((le O z0) \to ((le x3 z0) \to ((eq nat (S (plus +z0 y1)) (plus (minus z0 x3) y3)) \to (eq nat y3 n)))))) H3 (S (plus x3 y1)) +(plus_n_Sm x3 y1)) in (H4 y2 (le_O_n z0) (le_S_n x3 z0 H0) H1)))))))))))) +x2)) (\lambda (x2: nat).(\lambda (_: ((\forall (x3: nat).(\forall (y1: +nat).(\forall (y2: nat).((le x2 (S z0)) \to ((le x3 (S z0)) \to ((eq nat +(plus (minus (S z0) x2) y1) (plus (minus (S z0) x3) y2)) \to (eq nat (plus x2 +y2) (plus x3 y1)))))))))).(\lambda (x3: nat).(nat_ind (\lambda (n: +nat).(\forall (y1: nat).(\forall (y2: nat).((le (S x2) (S z0)) \to ((le n (S +z0)) \to ((eq nat (plus (minus (S z0) (S x2)) y1) (plus (minus (S z0) n) y2)) +\to (eq nat (plus (S x2) y2) (plus n y1)))))))) (\lambda (y1: nat).(\lambda +(y2: nat).(\lambda (H: (le (S x2) (S z0))).(\lambda (_: (le O (S +z0))).(\lambda (H1: (eq nat (plus (minus z0 x2) y1) (S (plus z0 y2)))).(let +H_y \def (IH x2 O y1 (S y2)) in (let H2 \def (eq_ind_r nat (minus z0 O) +(\lambda (n: nat).((le x2 z0) \to ((le O z0) \to ((eq nat (plus (minus z0 x2) +y1) (plus n (S y2))) \to (eq nat (plus x2 (S y2)) y1))))) H_y z0 (minus_n_O +z0)) in (let H3 \def (eq_ind_r nat (plus z0 (S y2)) (\lambda (n: nat).((le x2 +z0) \to ((le O z0) \to ((eq nat (plus (minus z0 x2) y1) n) \to (eq nat (plus +x2 (S y2)) y1))))) H2 (S (plus z0 y2)) (plus_n_Sm z0 y2)) in (let H4 \def +(eq_ind_r nat (plus x2 (S y2)) (\lambda (n: nat).((le x2 z0) \to ((le O z0) +\to ((eq nat (plus (minus z0 x2) y1) (S (plus z0 y2))) \to (eq nat n y1))))) +H3 (S (plus x2 y2)) (plus_n_Sm x2 y2)) in (H4 (le_S_n x2 z0 H) (le_O_n z0) +H1)))))))))) (\lambda (x4: nat).(\lambda (_: ((\forall (y1: nat).(\forall +(y2: nat).((le (S x2) (S z0)) \to ((le x4 (S z0)) \to ((eq nat (plus (minus +z0 x2) y1) (plus (match x4 with [O \Rightarrow (S z0) | (S l) \Rightarrow +(minus z0 l)]) y2)) \to (eq nat (S (plus x2 y2)) (plus x4 +y1))))))))).(\lambda (y1: nat).(\lambda (y2: nat).(\lambda (H: (le (S x2) (S +z0))).(\lambda (H0: (le (S x4) (S z0))).(\lambda (H1: (eq nat (plus (minus z0 +x2) y1) (plus (minus z0 x4) y2))).(f_equal nat nat S (plus x2 y2) (plus x4 +y1) (IH x2 x4 y1 y2 (le_S_n x2 z0 H) (le_S_n x4 z0 H0) H1))))))))) x3)))) +x1)))) z). +(* COMMENTS +Initial nodes: 1495 +END *) + +theorem le_S_minus: + \forall (d: nat).(\forall (h: nat).(\forall (n: nat).((le (plus d h) n) \to +(le d (S (minus n h)))))) +\def + \lambda (d: nat).(\lambda (h: nat).(\lambda (n: nat).(\lambda (H: (le (plus +d h) n)).(let H0 \def (le_trans d (plus d h) n (le_plus_l d h) H) in (let H1 +\def (eq_ind nat n (\lambda (n0: nat).(le d n0)) H0 (plus (minus n h) h) +(le_plus_minus_sym h n (le_trans h (plus d h) n (le_plus_r d h) H))) in (le_S +d (minus n h) (le_minus d n h H))))))). +(* COMMENTS +Initial nodes: 107 +END *) + +theorem lt_x_pred_y: + \forall (x: nat).(\forall (y: nat).((lt x (pred y)) \to (lt (S x) y))) +\def + \lambda (x: nat).(\lambda (y: nat).(nat_ind (\lambda (n: nat).((lt x (pred +n)) \to (lt (S x) n))) (\lambda (H: (lt x O)).(lt_x_O x H (lt (S x) O))) +(\lambda (n: nat).(\lambda (_: (((lt x (pred n)) \to (lt (S x) n)))).(\lambda +(H0: (lt x n)).(le_S_n (S (S x)) (S n) (le_n_S (S (S x)) (S n) (le_n_S (S x) +n H0)))))) y)). +(* COMMENTS +Initial nodes: 103 +END *) + diff --git a/matita/matita/contribs/lambdadelta/ground_1/ext/tactics.ma b/matita/matita/contribs/lambdadelta/ground_1/ext/tactics.ma new file mode 100644 index 000000000..766db9f1f --- /dev/null +++ b/matita/matita/contribs/lambdadelta/ground_1/ext/tactics.ma @@ -0,0 +1,50 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Ground-1/preamble.ma". + +theorem insert_eq: + \forall (S: Set).(\forall (x: S).(\forall (P: ((S \to Prop))).(\forall (G: +((S \to Prop))).(((\forall (y: S).((P y) \to ((eq S y x) \to (G y))))) \to +((P x) \to (G x)))))) +\def + \lambda (S: Set).(\lambda (x: S).(\lambda (P: ((S \to Prop))).(\lambda (G: +((S \to Prop))).(\lambda (H: ((\forall (y: S).((P y) \to ((eq S y x) \to (G +y)))))).(\lambda (H0: (P x)).(H x H0 (refl_equal S x))))))). +(* COMMENTS +Initial nodes: 45 +END *) + +theorem unintro: + \forall (A: Set).(\forall (a: A).(\forall (P: ((A \to Prop))).(((\forall (x: +A).(P x))) \to (P a)))) +\def + \lambda (A: Set).(\lambda (a: A).(\lambda (P: ((A \to Prop))).(\lambda (H: +((\forall (x: A).(P x)))).(H a)))). +(* COMMENTS +Initial nodes: 17 +END *) + +theorem xinduction: + \forall (A: Set).(\forall (t: A).(\forall (P: ((A \to Prop))).(((\forall (x: +A).((eq A t x) \to (P x)))) \to (P t)))) +\def + \lambda (A: Set).(\lambda (t: A).(\lambda (P: ((A \to Prop))).(\lambda (H: +((\forall (x: A).((eq A t x) \to (P x))))).(H t (refl_equal A t))))). +(* COMMENTS +Initial nodes: 31 +END *) + diff --git a/matita/matita/contribs/lambdadelta/ground_1/plist/defs.ma b/matita/matita/contribs/lambdadelta/ground_1/plist/defs.ma new file mode 100644 index 000000000..8f9c1d3cc --- /dev/null +++ b/matita/matita/contribs/lambdadelta/ground_1/plist/defs.ma @@ -0,0 +1,43 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Ground-1/preamble.ma". + +inductive PList: Set \def +| PNil: PList +| PCons: nat \to (nat \to (PList \to PList)). + +definition PConsTail: + PList \to (nat \to (nat \to PList)) +\def + let rec PConsTail (hds: PList) on hds: (nat \to (nat \to PList)) \def +(\lambda (h0: nat).(\lambda (d0: nat).(match hds with [PNil \Rightarrow +(PCons h0 d0 PNil) | (PCons h d hds0) \Rightarrow (PCons h d (PConsTail hds0 +h0 d0))]))) in PConsTail. + +definition Ss: + PList \to PList +\def + let rec Ss (hds: PList) on hds: PList \def (match hds with [PNil \Rightarrow +PNil | (PCons h d hds0) \Rightarrow (PCons h (S d) (Ss hds0))]) in Ss. + +definition papp: + PList \to (PList \to PList) +\def + let rec papp (a: PList) on a: (PList \to PList) \def (\lambda (b: +PList).(match a with [PNil \Rightarrow b | (PCons h d a0) \Rightarrow (PCons +h d (papp a0 b))])) in papp. + diff --git a/matita/matita/contribs/lambdadelta/ground_1/plist/props.ma b/matita/matita/contribs/lambdadelta/ground_1/plist/props.ma new file mode 100644 index 000000000..990397229 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/ground_1/plist/props.ma @@ -0,0 +1,34 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Ground-1/plist/defs.ma". + +theorem papp_ss: + \forall (is1: PList).(\forall (is2: PList).(eq PList (papp (Ss is1) (Ss +is2)) (Ss (papp is1 is2)))) +\def + \lambda (is1: PList).(PList_ind (\lambda (p: PList).(\forall (is2: +PList).(eq PList (papp (Ss p) (Ss is2)) (Ss (papp p is2))))) (\lambda (is2: +PList).(refl_equal PList (Ss is2))) (\lambda (n: nat).(\lambda (n0: +nat).(\lambda (p: PList).(\lambda (H: ((\forall (is2: PList).(eq PList (papp +(Ss p) (Ss is2)) (Ss (papp p is2)))))).(\lambda (is2: PList).(eq_ind_r PList +(Ss (papp p is2)) (\lambda (p0: PList).(eq PList (PCons n (S n0) p0) (PCons n +(S n0) (Ss (papp p is2))))) (refl_equal PList (PCons n (S n0) (Ss (papp p +is2)))) (papp (Ss p) (Ss is2)) (H is2))))))) is1). +(* COMMENTS +Initial nodes: 151 +END *) + diff --git a/matita/matita/contribs/lambdadelta/ground_1/preamble.ma b/matita/matita/contribs/lambdadelta/ground_1/preamble.ma new file mode 100644 index 000000000..16ff2dc44 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/ground_1/preamble.ma @@ -0,0 +1,15 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "Legacy-1/theory.ma". diff --git a/matita/matita/contribs/lambdadelta/ground_1/spare.ma b/matita/matita/contribs/lambdadelta/ground_1/spare.ma new file mode 100644 index 000000000..a966c2ae2 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/ground_1/spare.ma @@ -0,0 +1,18 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Ground-1/theory.ma". + diff --git a/matita/matita/contribs/lambdadelta/ground_1/theory.ma b/matita/matita/contribs/lambdadelta/ground_1/theory.ma new file mode 100644 index 000000000..3b59dc6e8 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/ground_1/theory.ma @@ -0,0 +1,26 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Ground-1/ext/tactics.ma". + +include "Ground-1/ext/arith.ma". + +include "Ground-1/types/props.ma". + +include "Ground-1/blt/props.ma". + +include "Ground-1/plist/props.ma". + diff --git a/matita/matita/contribs/lambdadelta/ground_1/types/defs.ma b/matita/matita/contribs/lambdadelta/ground_1/types/defs.ma new file mode 100644 index 000000000..f94969d88 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/ground_1/types/defs.ma @@ -0,0 +1,172 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Ground-1/preamble.ma". + +inductive and3 (P0: Prop) (P1: Prop) (P2: Prop): Prop \def +| and3_intro: P0 \to (P1 \to (P2 \to (and3 P0 P1 P2))). + +inductive and4 (P0: Prop) (P1: Prop) (P2: Prop) (P3: Prop): Prop \def +| and4_intro: P0 \to (P1 \to (P2 \to (P3 \to (and4 P0 P1 P2 P3)))). + +inductive and5 (P0: Prop) (P1: Prop) (P2: Prop) (P3: Prop) (P4: Prop): Prop +\def +| and5_intro: P0 \to (P1 \to (P2 \to (P3 \to (P4 \to (and5 P0 P1 P2 P3 +P4))))). + +inductive or3 (P0: Prop) (P1: Prop) (P2: Prop): Prop \def +| or3_intro0: P0 \to (or3 P0 P1 P2) +| or3_intro1: P1 \to (or3 P0 P1 P2) +| or3_intro2: P2 \to (or3 P0 P1 P2). + +inductive or4 (P0: Prop) (P1: Prop) (P2: Prop) (P3: Prop): Prop \def +| or4_intro0: P0 \to (or4 P0 P1 P2 P3) +| or4_intro1: P1 \to (or4 P0 P1 P2 P3) +| or4_intro2: P2 \to (or4 P0 P1 P2 P3) +| or4_intro3: P3 \to (or4 P0 P1 P2 P3). + +inductive or5 (P0: Prop) (P1: Prop) (P2: Prop) (P3: Prop) (P4: Prop): Prop +\def +| or5_intro0: P0 \to (or5 P0 P1 P2 P3 P4) +| or5_intro1: P1 \to (or5 P0 P1 P2 P3 P4) +| or5_intro2: P2 \to (or5 P0 P1 P2 P3 P4) +| or5_intro3: P3 \to (or5 P0 P1 P2 P3 P4) +| or5_intro4: P4 \to (or5 P0 P1 P2 P3 P4). + +inductive ex3 (A0: Set) (P0: A0 \to Prop) (P1: A0 \to Prop) (P2: A0 \to +Prop): Prop \def +| ex3_intro: \forall (x0: A0).((P0 x0) \to ((P1 x0) \to ((P2 x0) \to (ex3 A0 +P0 P1 P2)))). + +inductive ex4 (A0: Set) (P0: A0 \to Prop) (P1: A0 \to Prop) (P2: A0 \to Prop) +(P3: A0 \to Prop): Prop \def +| ex4_intro: \forall (x0: A0).((P0 x0) \to ((P1 x0) \to ((P2 x0) \to ((P3 x0) +\to (ex4 A0 P0 P1 P2 P3))))). + +inductive ex_2 (A0: Set) (A1: Set) (P0: A0 \to (A1 \to Prop)): Prop \def +| ex_2_intro: \forall (x0: A0).(\forall (x1: A1).((P0 x0 x1) \to (ex_2 A0 A1 +P0))). + +inductive ex2_2 (A0: Set) (A1: Set) (P0: A0 \to (A1 \to Prop)) (P1: A0 \to +(A1 \to Prop)): Prop \def +| ex2_2_intro: \forall (x0: A0).(\forall (x1: A1).((P0 x0 x1) \to ((P1 x0 x1) +\to (ex2_2 A0 A1 P0 P1)))). + +inductive ex3_2 (A0: Set) (A1: Set) (P0: A0 \to (A1 \to Prop)) (P1: A0 \to +(A1 \to Prop)) (P2: A0 \to (A1 \to Prop)): Prop \def +| ex3_2_intro: \forall (x0: A0).(\forall (x1: A1).((P0 x0 x1) \to ((P1 x0 x1) +\to ((P2 x0 x1) \to (ex3_2 A0 A1 P0 P1 P2))))). + +inductive ex4_2 (A0: Set) (A1: Set) (P0: A0 \to (A1 \to Prop)) (P1: A0 \to +(A1 \to Prop)) (P2: A0 \to (A1 \to Prop)) (P3: A0 \to (A1 \to Prop)): Prop +\def +| ex4_2_intro: \forall (x0: A0).(\forall (x1: A1).((P0 x0 x1) \to ((P1 x0 x1) +\to ((P2 x0 x1) \to ((P3 x0 x1) \to (ex4_2 A0 A1 P0 P1 P2 P3)))))). + +inductive ex_3 (A0: Set) (A1: Set) (A2: Set) (P0: A0 \to (A1 \to (A2 \to +Prop))): Prop \def +| ex_3_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).((P0 x0 x1 +x2) \to (ex_3 A0 A1 A2 P0)))). + +inductive ex2_3 (A0: Set) (A1: Set) (A2: Set) (P0: A0 \to (A1 \to (A2 \to +Prop))) (P1: A0 \to (A1 \to (A2 \to Prop))): Prop \def +| ex2_3_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).((P0 x0 +x1 x2) \to ((P1 x0 x1 x2) \to (ex2_3 A0 A1 A2 P0 P1))))). + +inductive ex3_3 (A0: Set) (A1: Set) (A2: Set) (P0: A0 \to (A1 \to (A2 \to +Prop))) (P1: A0 \to (A1 \to (A2 \to Prop))) (P2: A0 \to (A1 \to (A2 \to +Prop))): Prop \def +| ex3_3_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).((P0 x0 +x1 x2) \to ((P1 x0 x1 x2) \to ((P2 x0 x1 x2) \to (ex3_3 A0 A1 A2 P0 P1 +P2)))))). + +inductive ex4_3 (A0: Set) (A1: Set) (A2: Set) (P0: A0 \to (A1 \to (A2 \to +Prop))) (P1: A0 \to (A1 \to (A2 \to Prop))) (P2: A0 \to (A1 \to (A2 \to +Prop))) (P3: A0 \to (A1 \to (A2 \to Prop))): Prop \def +| ex4_3_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).((P0 x0 +x1 x2) \to ((P1 x0 x1 x2) \to ((P2 x0 x1 x2) \to ((P3 x0 x1 x2) \to (ex4_3 A0 +A1 A2 P0 P1 P2 P3))))))). + +inductive ex5_3 (A0: Set) (A1: Set) (A2: Set) (P0: A0 \to (A1 \to (A2 \to +Prop))) (P1: A0 \to (A1 \to (A2 \to Prop))) (P2: A0 \to (A1 \to (A2 \to +Prop))) (P3: A0 \to (A1 \to (A2 \to Prop))) (P4: A0 \to (A1 \to (A2 \to +Prop))): Prop \def +| ex5_3_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).((P0 x0 +x1 x2) \to ((P1 x0 x1 x2) \to ((P2 x0 x1 x2) \to ((P3 x0 x1 x2) \to ((P4 x0 +x1 x2) \to (ex5_3 A0 A1 A2 P0 P1 P2 P3 P4)))))))). + +inductive ex3_4 (A0: Set) (A1: Set) (A2: Set) (A3: Set) (P0: A0 \to (A1 \to +(A2 \to (A3 \to Prop)))) (P1: A0 \to (A1 \to (A2 \to (A3 \to Prop)))) (P2: A0 +\to (A1 \to (A2 \to (A3 \to Prop)))): Prop \def +| ex3_4_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall +(x3: A3).((P0 x0 x1 x2 x3) \to ((P1 x0 x1 x2 x3) \to ((P2 x0 x1 x2 x3) \to +(ex3_4 A0 A1 A2 A3 P0 P1 P2))))))). + +inductive ex4_4 (A0: Set) (A1: Set) (A2: Set) (A3: Set) (P0: A0 \to (A1 \to +(A2 \to (A3 \to Prop)))) (P1: A0 \to (A1 \to (A2 \to (A3 \to Prop)))) (P2: A0 +\to (A1 \to (A2 \to (A3 \to Prop)))) (P3: A0 \to (A1 \to (A2 \to (A3 \to +Prop)))): Prop \def +| ex4_4_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall +(x3: A3).((P0 x0 x1 x2 x3) \to ((P1 x0 x1 x2 x3) \to ((P2 x0 x1 x2 x3) \to +((P3 x0 x1 x2 x3) \to (ex4_4 A0 A1 A2 A3 P0 P1 P2 P3)))))))). + +inductive ex4_5 (A0: Set) (A1: Set) (A2: Set) (A3: Set) (A4: Set) (P0: A0 \to +(A1 \to (A2 \to (A3 \to (A4 \to Prop))))) (P1: A0 \to (A1 \to (A2 \to (A3 \to +(A4 \to Prop))))) (P2: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to Prop))))) (P3: +A0 \to (A1 \to (A2 \to (A3 \to (A4 \to Prop))))): Prop \def +| ex4_5_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall +(x3: A3).(\forall (x4: A4).((P0 x0 x1 x2 x3 x4) \to ((P1 x0 x1 x2 x3 x4) \to +((P2 x0 x1 x2 x3 x4) \to ((P3 x0 x1 x2 x3 x4) \to (ex4_5 A0 A1 A2 A3 A4 P0 P1 +P2 P3))))))))). + +inductive ex5_5 (A0: Set) (A1: Set) (A2: Set) (A3: Set) (A4: Set) (P0: A0 \to +(A1 \to (A2 \to (A3 \to (A4 \to Prop))))) (P1: A0 \to (A1 \to (A2 \to (A3 \to +(A4 \to Prop))))) (P2: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to Prop))))) (P3: +A0 \to (A1 \to (A2 \to (A3 \to (A4 \to Prop))))) (P4: A0 \to (A1 \to (A2 \to +(A3 \to (A4 \to Prop))))): Prop \def +| ex5_5_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall +(x3: A3).(\forall (x4: A4).((P0 x0 x1 x2 x3 x4) \to ((P1 x0 x1 x2 x3 x4) \to +((P2 x0 x1 x2 x3 x4) \to ((P3 x0 x1 x2 x3 x4) \to ((P4 x0 x1 x2 x3 x4) \to +(ex5_5 A0 A1 A2 A3 A4 P0 P1 P2 P3 P4)))))))))). + +inductive ex6_6 (A0: Set) (A1: Set) (A2: Set) (A3: Set) (A4: Set) (A5: Set) +(P0: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to Prop)))))) (P1: A0 \to +(A1 \to (A2 \to (A3 \to (A4 \to (A5 \to Prop)))))) (P2: A0 \to (A1 \to (A2 +\to (A3 \to (A4 \to (A5 \to Prop)))))) (P3: A0 \to (A1 \to (A2 \to (A3 \to +(A4 \to (A5 \to Prop)))))) (P4: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 +\to Prop)))))) (P5: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to +Prop)))))): Prop \def +| ex6_6_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall +(x3: A3).(\forall (x4: A4).(\forall (x5: A5).((P0 x0 x1 x2 x3 x4 x5) \to ((P1 +x0 x1 x2 x3 x4 x5) \to ((P2 x0 x1 x2 x3 x4 x5) \to ((P3 x0 x1 x2 x3 x4 x5) +\to ((P4 x0 x1 x2 x3 x4 x5) \to ((P5 x0 x1 x2 x3 x4 x5) \to (ex6_6 A0 A1 A2 +A3 A4 A5 P0 P1 P2 P3 P4 P5)))))))))))). + +inductive ex6_7 (A0: Set) (A1: Set) (A2: Set) (A3: Set) (A4: Set) (A5: Set) +(A6: Set) (P0: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 \to +Prop))))))) (P1: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 \to +Prop))))))) (P2: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 \to +Prop))))))) (P3: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 \to +Prop))))))) (P4: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 \to +Prop))))))) (P5: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 \to +Prop))))))): Prop \def +| ex6_7_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall +(x3: A3).(\forall (x4: A4).(\forall (x5: A5).(\forall (x6: A6).((P0 x0 x1 x2 +x3 x4 x5 x6) \to ((P1 x0 x1 x2 x3 x4 x5 x6) \to ((P2 x0 x1 x2 x3 x4 x5 x6) +\to ((P3 x0 x1 x2 x3 x4 x5 x6) \to ((P4 x0 x1 x2 x3 x4 x5 x6) \to ((P5 x0 x1 +x2 x3 x4 x5 x6) \to (ex6_7 A0 A1 A2 A3 A4 A5 A6 P0 P1 P2 P3 P4 +P5))))))))))))). + diff --git a/matita/matita/contribs/lambdadelta/ground_1/types/props.ma b/matita/matita/contribs/lambdadelta/ground_1/types/props.ma new file mode 100644 index 000000000..9c326f44c --- /dev/null +++ b/matita/matita/contribs/lambdadelta/ground_1/types/props.ma @@ -0,0 +1,33 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Ground-1/types/defs.ma". + +theorem ex2_sym: + \forall (A: Set).(\forall (P: ((A \to Prop))).(\forall (Q: ((A \to +Prop))).((ex2 A (\lambda (x: A).(P x)) (\lambda (x: A).(Q x))) \to (ex2 A +(\lambda (x: A).(Q x)) (\lambda (x: A).(P x)))))) +\def + \lambda (A: Set).(\lambda (P: ((A \to Prop))).(\lambda (Q: ((A \to +Prop))).(\lambda (H: (ex2 A (\lambda (x: A).(P x)) (\lambda (x: A).(Q +x)))).(ex2_ind A (\lambda (x: A).(P x)) (\lambda (x: A).(Q x)) (ex2 A +(\lambda (x: A).(Q x)) (\lambda (x: A).(P x))) (\lambda (x: A).(\lambda (H0: +(P x)).(\lambda (H1: (Q x)).(ex_intro2 A (\lambda (x0: A).(Q x0)) (\lambda +(x0: A).(P x0)) x H1 H0)))) H)))). +(* COMMENTS +Initial nodes: 91 +END *) + diff --git a/matita/matita/contribs/lambdadelta/legacy_1/coq/defs.ma b/matita/matita/contribs/lambdadelta/legacy_1/coq/defs.ma new file mode 100644 index 000000000..7d4696229 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/legacy_1/coq/defs.ma @@ -0,0 +1,99 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Legacy-1/preamble.ma". + +inductive eq (A: Set) (x: A): A \to Prop \def +| refl_equal: eq A x x. + +inductive True: Prop \def +| I: True. + +inductive land (A: Prop) (B: Prop): Prop \def +| conj: A \to (B \to (land A B)). + +inductive or (A: Prop) (B: Prop): Prop \def +| or_introl: A \to (or A B) +| or_intror: B \to (or A B). + +inductive ex (A: Set) (P: A \to Prop): Prop \def +| ex_intro: \forall (x: A).((P x) \to (ex A P)). + +inductive ex2 (A: Set) (P: A \to Prop) (Q: A \to Prop): Prop \def +| ex_intro2: \forall (x: A).((P x) \to ((Q x) \to (ex2 A P Q))). + +definition not: + Prop \to Prop +\def + \lambda (A: Prop).(A \to False). + +inductive bool: Set \def +| true: bool +| false: bool. + +inductive nat: Set \def +| O: nat +| S: nat \to nat. + +inductive le (n: nat): nat \to Prop \def +| le_n: le n n +| le_S: \forall (m: nat).((le n m) \to (le n (S m))). + +definition lt: + nat \to (nat \to Prop) +\def + \lambda (n: nat).(\lambda (m: nat).(le (S n) m)). + +definition IsSucc: + nat \to Prop +\def + \lambda (n: nat).(match n with [O \Rightarrow False | (S _) \Rightarrow +True]). + +definition pred: + nat \to nat +\def + \lambda (n: nat).(match n with [O \Rightarrow O | (S u) \Rightarrow u]). + +definition plus: + nat \to (nat \to nat) +\def + let rec plus (n: nat) on n: (nat \to nat) \def (\lambda (m: nat).(match n +with [O \Rightarrow m | (S p) \Rightarrow (S (plus p m))])) in plus. + +definition minus: + nat \to (nat \to nat) +\def + let rec minus (n: nat) on n: (nat \to nat) \def (\lambda (m: nat).(match n +with [O \Rightarrow O | (S k) \Rightarrow (match m with [O \Rightarrow (S k) +| (S l) \Rightarrow (minus k l)])])) in minus. + +inductive Acc (A: Set) (R: A \to (A \to Prop)): A \to Prop \def +| Acc_intro: \forall (x: A).(((\forall (y: A).((R y x) \to (Acc A R y)))) \to +(Acc A R x)). + +definition well_founded: + \forall (A: Set).(((A \to (A \to Prop))) \to Prop) +\def + \lambda (A: Set).(\lambda (R: ((A \to (A \to Prop)))).(\forall (a: A).(Acc A +R a))). + +definition ltof: + \forall (A: Set).(((A \to nat)) \to (A \to (A \to Prop))) +\def + \lambda (A: Set).(\lambda (f: ((A \to nat))).(\lambda (a: A).(\lambda (b: +A).(lt (f a) (f b))))). + diff --git a/matita/matita/contribs/lambdadelta/legacy_1/coq/props.ma b/matita/matita/contribs/lambdadelta/legacy_1/coq/props.ma new file mode 100644 index 000000000..0b9d97b42 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/legacy_1/coq/props.ma @@ -0,0 +1,805 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Legacy-1/coq/defs.ma". + +theorem f_equal: + \forall (A: Set).(\forall (B: Set).(\forall (f: ((A \to B))).(\forall (x: +A).(\forall (y: A).((eq A x y) \to (eq B (f x) (f y))))))) +\def + \lambda (A: Set).(\lambda (B: Set).(\lambda (f: ((A \to B))).(\lambda (x: +A).(\lambda (y: A).(\lambda (H: (eq A x y)).(eq_ind A x (\lambda (a: A).(eq B +(f x) (f a))) (refl_equal B (f x)) y H)))))). +(* COMMENTS +Initial nodes: 51 +END *) + +theorem f_equal2: + \forall (A1: Set).(\forall (A2: Set).(\forall (B: Set).(\forall (f: ((A1 \to +(A2 \to B)))).(\forall (x1: A1).(\forall (y1: A1).(\forall (x2: A2).(\forall +(y2: A2).((eq A1 x1 y1) \to ((eq A2 x2 y2) \to (eq B (f x1 x2) (f y1 +y2))))))))))) +\def + \lambda (A1: Set).(\lambda (A2: Set).(\lambda (B: Set).(\lambda (f: ((A1 \to +(A2 \to B)))).(\lambda (x1: A1).(\lambda (y1: A1).(\lambda (x2: A2).(\lambda +(y2: A2).(\lambda (H: (eq A1 x1 y1)).(eq_ind A1 x1 (\lambda (a: A1).((eq A2 +x2 y2) \to (eq B (f x1 x2) (f a y2)))) (\lambda (H0: (eq A2 x2 y2)).(eq_ind +A2 x2 (\lambda (a: A2).(eq B (f x1 x2) (f x1 a))) (refl_equal B (f x1 x2)) y2 +H0)) y1 H))))))))). +(* COMMENTS +Initial nodes: 109 +END *) + +theorem f_equal3: + \forall (A1: Set).(\forall (A2: Set).(\forall (A3: Set).(\forall (B: +Set).(\forall (f: ((A1 \to (A2 \to (A3 \to B))))).(\forall (x1: A1).(\forall +(y1: A1).(\forall (x2: A2).(\forall (y2: A2).(\forall (x3: A3).(\forall (y3: +A3).((eq A1 x1 y1) \to ((eq A2 x2 y2) \to ((eq A3 x3 y3) \to (eq B (f x1 x2 +x3) (f y1 y2 y3))))))))))))))) +\def + \lambda (A1: Set).(\lambda (A2: Set).(\lambda (A3: Set).(\lambda (B: +Set).(\lambda (f: ((A1 \to (A2 \to (A3 \to B))))).(\lambda (x1: A1).(\lambda +(y1: A1).(\lambda (x2: A2).(\lambda (y2: A2).(\lambda (x3: A3).(\lambda (y3: +A3).(\lambda (H: (eq A1 x1 y1)).(eq_ind A1 x1 (\lambda (a: A1).((eq A2 x2 y2) +\to ((eq A3 x3 y3) \to (eq B (f x1 x2 x3) (f a y2 y3))))) (\lambda (H0: (eq +A2 x2 y2)).(eq_ind A2 x2 (\lambda (a: A2).((eq A3 x3 y3) \to (eq B (f x1 x2 +x3) (f x1 a y3)))) (\lambda (H1: (eq A3 x3 y3)).(eq_ind A3 x3 (\lambda (a: +A3).(eq B (f x1 x2 x3) (f x1 x2 a))) (refl_equal B (f x1 x2 x3)) y3 H1)) y2 +H0)) y1 H)))))))))))). +(* COMMENTS +Initial nodes: 183 +END *) + +theorem sym_eq: + \forall (A: Set).(\forall (x: A).(\forall (y: A).((eq A x y) \to (eq A y +x)))) +\def + \lambda (A: Set).(\lambda (x: A).(\lambda (y: A).(\lambda (H: (eq A x +y)).(eq_ind A x (\lambda (a: A).(eq A a x)) (refl_equal A x) y H)))). +(* COMMENTS +Initial nodes: 39 +END *) + +theorem eq_ind_r: + \forall (A: Set).(\forall (x: A).(\forall (P: ((A \to Prop))).((P x) \to +(\forall (y: A).((eq A y x) \to (P y)))))) +\def + \lambda (A: Set).(\lambda (x: A).(\lambda (P: ((A \to Prop))).(\lambda (H: +(P x)).(\lambda (y: A).(\lambda (H0: (eq A y x)).(match (sym_eq A y x H0) in +eq return (\lambda (a: A).(\lambda (_: (eq ? ? a)).(P a))) with [refl_equal +\Rightarrow H])))))). +(* COMMENTS +Initial nodes: 38 +END *) + +theorem trans_eq: + \forall (A: Set).(\forall (x: A).(\forall (y: A).(\forall (z: A).((eq A x y) +\to ((eq A y z) \to (eq A x z)))))) +\def + \lambda (A: Set).(\lambda (x: A).(\lambda (y: A).(\lambda (z: A).(\lambda +(H: (eq A x y)).(\lambda (H0: (eq A y z)).(eq_ind A y (\lambda (a: A).(eq A x +a)) H z H0)))))). +(* COMMENTS +Initial nodes: 45 +END *) + +theorem sym_not_eq: + \forall (A: Set).(\forall (x: A).(\forall (y: A).((not (eq A x y)) \to (not +(eq A y x))))) +\def + \lambda (A: Set).(\lambda (x: A).(\lambda (y: A).(\lambda (h1: (not (eq A x +y))).(\lambda (h2: (eq A y x)).(h1 (eq_ind A y (\lambda (a: A).(eq A a y)) +(refl_equal A y) x h2)))))). +(* COMMENTS +Initial nodes: 51 +END *) + +theorem nat_double_ind: + \forall (R: ((nat \to (nat \to Prop)))).(((\forall (n: nat).(R O n))) \to +(((\forall (n: nat).(R (S n) O))) \to (((\forall (n: nat).(\forall (m: +nat).((R n m) \to (R (S n) (S m)))))) \to (\forall (n: nat).(\forall (m: +nat).(R n m)))))) +\def + \lambda (R: ((nat \to (nat \to Prop)))).(\lambda (H: ((\forall (n: nat).(R O +n)))).(\lambda (H0: ((\forall (n: nat).(R (S n) O)))).(\lambda (H1: ((\forall +(n: nat).(\forall (m: nat).((R n m) \to (R (S n) (S m))))))).(\lambda (n: +nat).(nat_ind (\lambda (n0: nat).(\forall (m: nat).(R n0 m))) H (\lambda (n0: +nat).(\lambda (H2: ((\forall (m: nat).(R n0 m)))).(\lambda (m: nat).(nat_ind +(\lambda (n1: nat).(R (S n0) n1)) (H0 n0) (\lambda (n1: nat).(\lambda (_: (R +(S n0) n1)).(H1 n0 n1 (H2 n1)))) m)))) n))))). +(* COMMENTS +Initial nodes: 111 +END *) + +theorem eq_add_S: + \forall (n: nat).(\forall (m: nat).((eq nat (S n) (S m)) \to (eq nat n m))) +\def + \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (eq nat (S n) (S +m))).(f_equal nat nat pred (S n) (S m) H))). +(* COMMENTS +Initial nodes: 33 +END *) + +theorem O_S: + \forall (n: nat).(not (eq nat O (S n))) +\def + \lambda (n: nat).(\lambda (H: (eq nat O (S n))).(eq_ind nat (S n) (\lambda +(n0: nat).(IsSucc n0)) I O (sym_eq nat O (S n) H))). +(* COMMENTS +Initial nodes: 41 +END *) + +theorem not_eq_S: + \forall (n: nat).(\forall (m: nat).((not (eq nat n m)) \to (not (eq nat (S +n) (S m))))) +\def + \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (not (eq nat n m))).(\lambda +(H0: (eq nat (S n) (S m))).(H (eq_add_S n m H0))))). +(* COMMENTS +Initial nodes: 35 +END *) + +theorem pred_Sn: + \forall (m: nat).(eq nat m (pred (S m))) +\def + \lambda (m: nat).(refl_equal nat (pred (S m))). +(* COMMENTS +Initial nodes: 11 +END *) + +theorem S_pred: + \forall (n: nat).(\forall (m: nat).((lt m n) \to (eq nat n (S (pred n))))) +\def + \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt m n)).(le_ind (S m) +(\lambda (n0: nat).(eq nat n0 (S (pred n0)))) (refl_equal nat (S (pred (S +m)))) (\lambda (m0: nat).(\lambda (_: (le (S m) m0)).(\lambda (_: (eq nat m0 +(S (pred m0)))).(refl_equal nat (S (pred (S m0))))))) n H))). +(* COMMENTS +Initial nodes: 79 +END *) + +theorem le_trans: + \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((le n m) \to ((le m p) +\to (le n p))))) +\def + \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (le n +m)).(\lambda (H0: (le m p)).(le_ind m (\lambda (n0: nat).(le n n0)) H +(\lambda (m0: nat).(\lambda (_: (le m m0)).(\lambda (IHle: (le n m0)).(le_S n +m0 IHle)))) p H0))))). +(* COMMENTS +Initial nodes: 57 +END *) + +theorem le_trans_S: + \forall (n: nat).(\forall (m: nat).((le (S n) m) \to (le n m))) +\def + \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le (S n) m)).(le_trans n (S +n) m (le_S n n (le_n n)) H))). +(* COMMENTS +Initial nodes: 33 +END *) + +theorem le_n_S: + \forall (n: nat).(\forall (m: nat).((le n m) \to (le (S n) (S m)))) +\def + \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(le_ind n (\lambda +(n0: nat).(le (S n) (S n0))) (le_n (S n)) (\lambda (m0: nat).(\lambda (_: (le +n m0)).(\lambda (IHle: (le (S n) (S m0))).(le_S (S n) (S m0) IHle)))) m H))). +(* COMMENTS +Initial nodes: 65 +END *) + +theorem le_O_n: + \forall (n: nat).(le O n) +\def + \lambda (n: nat).(nat_ind (\lambda (n0: nat).(le O n0)) (le_n O) (\lambda +(n0: nat).(\lambda (IHn: (le O n0)).(le_S O n0 IHn))) n). +(* COMMENTS +Initial nodes: 33 +END *) + +theorem le_S_n: + \forall (n: nat).(\forall (m: nat).((le (S n) (S m)) \to (le n m))) +\def + \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le (S n) (S m))).(le_ind (S +n) (\lambda (n0: nat).(le (pred (S n)) (pred n0))) (le_n n) (\lambda (m0: +nat).(\lambda (H0: (le (S n) m0)).(\lambda (_: (le n (pred m0))).(le_trans_S +n m0 H0)))) (S m) H))). +(* COMMENTS +Initial nodes: 69 +END *) + +theorem le_Sn_O: + \forall (n: nat).(not (le (S n) O)) +\def + \lambda (n: nat).(\lambda (H: (le (S n) O)).(le_ind (S n) (\lambda (n0: +nat).(IsSucc n0)) I (\lambda (m: nat).(\lambda (_: (le (S n) m)).(\lambda (_: +(IsSucc m)).I))) O H)). +(* COMMENTS +Initial nodes: 43 +END *) + +theorem le_Sn_n: + \forall (n: nat).(not (le (S n) n)) +\def + \lambda (n: nat).(nat_ind (\lambda (n0: nat).(not (le (S n0) n0))) (le_Sn_O +O) (\lambda (n0: nat).(\lambda (IHn: (not (le (S n0) n0))).(\lambda (H: (le +(S (S n0)) (S n0))).(IHn (le_S_n (S n0) n0 H))))) n). +(* COMMENTS +Initial nodes: 57 +END *) + +theorem le_antisym: + \forall (n: nat).(\forall (m: nat).((le n m) \to ((le m n) \to (eq nat n +m)))) +\def + \lambda (n: nat).(\lambda (m: nat).(\lambda (h: (le n m)).(le_ind n (\lambda +(n0: nat).((le n0 n) \to (eq nat n n0))) (\lambda (_: (le n n)).(refl_equal +nat n)) (\lambda (m0: nat).(\lambda (H: (le n m0)).(\lambda (_: (((le m0 n) +\to (eq nat n m0)))).(\lambda (H1: (le (S m0) n)).(False_ind (eq nat n (S +m0)) (let H2 \def (le_trans (S m0) n m0 H1 H) in ((let H3 \def (le_Sn_n m0) +in (\lambda (H4: (le (S m0) m0)).(H3 H4))) H2))))))) m h))). +(* COMMENTS +Initial nodes: 119 +END *) + +theorem le_n_O_eq: + \forall (n: nat).((le n O) \to (eq nat O n)) +\def + \lambda (n: nat).(\lambda (H: (le n O)).(le_antisym O n (le_O_n n) H)). +(* COMMENTS +Initial nodes: 19 +END *) + +theorem le_elim_rel: + \forall (P: ((nat \to (nat \to Prop)))).(((\forall (p: nat).(P O p))) \to +(((\forall (p: nat).(\forall (q: nat).((le p q) \to ((P p q) \to (P (S p) (S +q))))))) \to (\forall (n: nat).(\forall (m: nat).((le n m) \to (P n m)))))) +\def + \lambda (P: ((nat \to (nat \to Prop)))).(\lambda (H: ((\forall (p: nat).(P O +p)))).(\lambda (H0: ((\forall (p: nat).(\forall (q: nat).((le p q) \to ((P p +q) \to (P (S p) (S q)))))))).(\lambda (n: nat).(nat_ind (\lambda (n0: +nat).(\forall (m: nat).((le n0 m) \to (P n0 m)))) (\lambda (m: nat).(\lambda +(_: (le O m)).(H m))) (\lambda (n0: nat).(\lambda (IHn: ((\forall (m: +nat).((le n0 m) \to (P n0 m))))).(\lambda (m: nat).(\lambda (Le: (le (S n0) +m)).(le_ind (S n0) (\lambda (n1: nat).(P (S n0) n1)) (H0 n0 n0 (le_n n0) (IHn +n0 (le_n n0))) (\lambda (m0: nat).(\lambda (H1: (le (S n0) m0)).(\lambda (_: +(P (S n0) m0)).(H0 n0 m0 (le_trans_S n0 m0 H1) (IHn m0 (le_trans_S n0 m0 +H1)))))) m Le))))) n)))). +(* COMMENTS +Initial nodes: 181 +END *) + +theorem lt_n_n: + \forall (n: nat).(not (lt n n)) +\def + le_Sn_n. +(* COMMENTS +Initial nodes: 1 +END *) + +theorem lt_n_S: + \forall (n: nat).(\forall (m: nat).((lt n m) \to (lt (S n) (S m)))) +\def + \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt n m)).(le_n_S (S n) m +H))). +(* COMMENTS +Initial nodes: 19 +END *) + +theorem lt_n_Sn: + \forall (n: nat).(lt n (S n)) +\def + \lambda (n: nat).(le_n (S n)). +(* COMMENTS +Initial nodes: 7 +END *) + +theorem lt_S_n: + \forall (n: nat).(\forall (m: nat).((lt (S n) (S m)) \to (lt n m))) +\def + \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt (S n) (S m))).(le_S_n (S +n) m H))). +(* COMMENTS +Initial nodes: 23 +END *) + +theorem lt_n_O: + \forall (n: nat).(not (lt n O)) +\def + le_Sn_O. +(* COMMENTS +Initial nodes: 1 +END *) + +theorem lt_trans: + \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((lt n m) \to ((lt m p) +\to (lt n p))))) +\def + \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (lt n +m)).(\lambda (H0: (lt m p)).(le_ind (S m) (\lambda (n0: nat).(lt n n0)) (le_S +(S n) m H) (\lambda (m0: nat).(\lambda (_: (le (S m) m0)).(\lambda (IHle: (lt +n m0)).(le_S (S n) m0 IHle)))) p H0))))). +(* COMMENTS +Initial nodes: 71 +END *) + +theorem lt_O_Sn: + \forall (n: nat).(lt O (S n)) +\def + \lambda (n: nat).(le_n_S O n (le_O_n n)). +(* COMMENTS +Initial nodes: 11 +END *) + +theorem lt_le_S: + \forall (n: nat).(\forall (p: nat).((lt n p) \to (le (S n) p))) +\def + \lambda (n: nat).(\lambda (p: nat).(\lambda (H: (lt n p)).H)). +(* COMMENTS +Initial nodes: 11 +END *) + +theorem le_not_lt: + \forall (n: nat).(\forall (m: nat).((le n m) \to (not (lt m n)))) +\def + \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(le_ind n (\lambda +(n0: nat).(not (lt n0 n))) (lt_n_n n) (\lambda (m0: nat).(\lambda (_: (le n +m0)).(\lambda (IHle: (not (lt m0 n))).(\lambda (H1: (lt (S m0) n)).(IHle +(le_trans_S (S m0) n H1)))))) m H))). +(* COMMENTS +Initial nodes: 67 +END *) + +theorem le_lt_n_Sm: + \forall (n: nat).(\forall (m: nat).((le n m) \to (lt n (S m)))) +\def + \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(le_n_S n m H))). +(* COMMENTS +Initial nodes: 17 +END *) + +theorem le_lt_trans: + \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((le n m) \to ((lt m p) +\to (lt n p))))) +\def + \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (le n +m)).(\lambda (H0: (lt m p)).(le_ind (S m) (\lambda (n0: nat).(lt n n0)) +(le_n_S n m H) (\lambda (m0: nat).(\lambda (_: (le (S m) m0)).(\lambda (IHle: +(lt n m0)).(le_S (S n) m0 IHle)))) p H0))))). +(* COMMENTS +Initial nodes: 69 +END *) + +theorem lt_le_trans: + \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((lt n m) \to ((le m p) +\to (lt n p))))) +\def + \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (lt n +m)).(\lambda (H0: (le m p)).(le_ind m (\lambda (n0: nat).(lt n n0)) H +(\lambda (m0: nat).(\lambda (_: (le m m0)).(\lambda (IHle: (lt n m0)).(le_S +(S n) m0 IHle)))) p H0))))). +(* COMMENTS +Initial nodes: 59 +END *) + +theorem lt_le_weak: + \forall (n: nat).(\forall (m: nat).((lt n m) \to (le n m))) +\def + \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt n m)).(le_trans_S n m +H))). +(* COMMENTS +Initial nodes: 17 +END *) + +theorem lt_n_Sm_le: + \forall (n: nat).(\forall (m: nat).((lt n (S m)) \to (le n m))) +\def + \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt n (S m))).(le_S_n n m +H))). +(* COMMENTS +Initial nodes: 19 +END *) + +theorem le_lt_or_eq: + \forall (n: nat).(\forall (m: nat).((le n m) \to (or (lt n m) (eq nat n m)))) +\def + \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(le_ind n (\lambda +(n0: nat).(or (lt n n0) (eq nat n n0))) (or_intror (lt n n) (eq nat n n) +(refl_equal nat n)) (\lambda (m0: nat).(\lambda (H0: (le n m0)).(\lambda (_: +(or (lt n m0) (eq nat n m0))).(or_introl (lt n (S m0)) (eq nat n (S m0)) +(le_n_S n m0 H0))))) m H))). +(* COMMENTS +Initial nodes: 109 +END *) + +theorem le_or_lt: + \forall (n: nat).(\forall (m: nat).(or (le n m) (lt m n))) +\def + \lambda (n: nat).(\lambda (m: nat).(nat_double_ind (\lambda (n0: +nat).(\lambda (n1: nat).(or (le n0 n1) (lt n1 n0)))) (\lambda (n0: +nat).(or_introl (le O n0) (lt n0 O) (le_O_n n0))) (\lambda (n0: +nat).(or_intror (le (S n0) O) (lt O (S n0)) (lt_le_S O (S n0) (lt_O_Sn n0)))) +(\lambda (n0: nat).(\lambda (m0: nat).(\lambda (H: (or (le n0 m0) (lt m0 +n0))).(or_ind (le n0 m0) (lt m0 n0) (or (le (S n0) (S m0)) (lt (S m0) (S +n0))) (\lambda (H0: (le n0 m0)).(or_introl (le (S n0) (S m0)) (lt (S m0) (S +n0)) (le_n_S n0 m0 H0))) (\lambda (H0: (lt m0 n0)).(or_intror (le (S n0) (S +m0)) (lt (S m0) (S n0)) (le_n_S (S m0) n0 H0))) H)))) n m)). +(* COMMENTS +Initial nodes: 209 +END *) + +theorem plus_n_O: + \forall (n: nat).(eq nat n (plus n O)) +\def + \lambda (n: nat).(nat_ind (\lambda (n0: nat).(eq nat n0 (plus n0 O))) +(refl_equal nat O) (\lambda (n0: nat).(\lambda (H: (eq nat n0 (plus n0 +O))).(f_equal nat nat S n0 (plus n0 O) H))) n). +(* COMMENTS +Initial nodes: 57 +END *) + +theorem plus_n_Sm: + \forall (n: nat).(\forall (m: nat).(eq nat (S (plus n m)) (plus n (S m)))) +\def + \lambda (m: nat).(\lambda (n: nat).(nat_ind (\lambda (n0: nat).(eq nat (S +(plus n0 n)) (plus n0 (S n)))) (refl_equal nat (S n)) (\lambda (n0: +nat).(\lambda (H: (eq nat (S (plus n0 n)) (plus n0 (S n)))).(f_equal nat nat +S (S (plus n0 n)) (plus n0 (S n)) H))) m)). +(* COMMENTS +Initial nodes: 85 +END *) + +theorem plus_sym: + \forall (n: nat).(\forall (m: nat).(eq nat (plus n m) (plus m n))) +\def + \lambda (n: nat).(\lambda (m: nat).(nat_ind (\lambda (n0: nat).(eq nat (plus +n0 m) (plus m n0))) (plus_n_O m) (\lambda (y: nat).(\lambda (H: (eq nat (plus +y m) (plus m y))).(eq_ind nat (S (plus m y)) (\lambda (n0: nat).(eq nat (S +(plus y m)) n0)) (f_equal nat nat S (plus y m) (plus m y) H) (plus m (S y)) +(plus_n_Sm m y)))) n)). +(* COMMENTS +Initial nodes: 111 +END *) + +theorem plus_Snm_nSm: + \forall (n: nat).(\forall (m: nat).(eq nat (plus (S n) m) (plus n (S m)))) +\def + \lambda (n: nat).(\lambda (m: nat).(eq_ind_r nat (plus m n) (\lambda (n0: +nat).(eq nat (S n0) (plus n (S m)))) (eq_ind_r nat (plus (S m) n) (\lambda +(n0: nat).(eq nat (S (plus m n)) n0)) (refl_equal nat (plus (S m) n)) (plus n +(S m)) (plus_sym n (S m))) (plus n m) (plus_sym n m))). +(* COMMENTS +Initial nodes: 99 +END *) + +theorem plus_assoc_l: + \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(eq nat (plus n (plus m +p)) (plus (plus n m) p)))) +\def + \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(nat_ind (\lambda (n0: +nat).(eq nat (plus n0 (plus m p)) (plus (plus n0 m) p))) (refl_equal nat +(plus m p)) (\lambda (n0: nat).(\lambda (H: (eq nat (plus n0 (plus m p)) +(plus (plus n0 m) p))).(f_equal nat nat S (plus n0 (plus m p)) (plus (plus n0 +m) p) H))) n))). +(* COMMENTS +Initial nodes: 101 +END *) + +theorem plus_assoc_r: + \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(eq nat (plus (plus n +m) p) (plus n (plus m p))))) +\def + \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(sym_eq nat (plus n +(plus m p)) (plus (plus n m) p) (plus_assoc_l n m p)))). +(* COMMENTS +Initial nodes: 37 +END *) + +theorem simpl_plus_l: + \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((eq nat (plus n m) +(plus n p)) \to (eq nat m p)))) +\def + \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (m: nat).(\forall (p: +nat).((eq nat (plus n0 m) (plus n0 p)) \to (eq nat m p))))) (\lambda (m: +nat).(\lambda (p: nat).(\lambda (H: (eq nat m p)).H))) (\lambda (n0: +nat).(\lambda (IHn: ((\forall (m: nat).(\forall (p: nat).((eq nat (plus n0 m) +(plus n0 p)) \to (eq nat m p)))))).(\lambda (m: nat).(\lambda (p: +nat).(\lambda (H: (eq nat (S (plus n0 m)) (S (plus n0 p)))).(IHn m p (IHn +(plus n0 m) (plus n0 p) (f_equal nat nat (plus n0) (plus n0 m) (plus n0 p) +(eq_add_S (plus n0 m) (plus n0 p) H))))))))) n). +(* COMMENTS +Initial nodes: 161 +END *) + +theorem minus_n_O: + \forall (n: nat).(eq nat n (minus n O)) +\def + \lambda (n: nat).(nat_ind (\lambda (n0: nat).(eq nat n0 (minus n0 O))) +(refl_equal nat O) (\lambda (n0: nat).(\lambda (_: (eq nat n0 (minus n0 +O))).(refl_equal nat (S n0)))) n). +(* COMMENTS +Initial nodes: 47 +END *) + +theorem minus_n_n: + \forall (n: nat).(eq nat O (minus n n)) +\def + \lambda (n: nat).(nat_ind (\lambda (n0: nat).(eq nat O (minus n0 n0))) +(refl_equal nat O) (\lambda (n0: nat).(\lambda (IHn: (eq nat O (minus n0 +n0))).IHn)) n). +(* COMMENTS +Initial nodes: 41 +END *) + +theorem minus_Sn_m: + \forall (n: nat).(\forall (m: nat).((le m n) \to (eq nat (S (minus n m)) +(minus (S n) m)))) +\def + \lambda (n: nat).(\lambda (m: nat).(\lambda (Le: (le m n)).(le_elim_rel +(\lambda (n0: nat).(\lambda (n1: nat).(eq nat (S (minus n1 n0)) (minus (S n1) +n0)))) (\lambda (p: nat).(f_equal nat nat S (minus p O) p (sym_eq nat p +(minus p O) (minus_n_O p)))) (\lambda (p: nat).(\lambda (q: nat).(\lambda (_: +(le p q)).(\lambda (H0: (eq nat (S (minus q p)) (match p with [O \Rightarrow +(S q) | (S l) \Rightarrow (minus q l)]))).H0)))) m n Le))). +(* COMMENTS +Initial nodes: 111 +END *) + +theorem plus_minus: + \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((eq nat n (plus m p)) +\to (eq nat p (minus n m))))) +\def + \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(nat_double_ind +(\lambda (n0: nat).(\lambda (n1: nat).((eq nat n1 (plus n0 p)) \to (eq nat p +(minus n1 n0))))) (\lambda (n0: nat).(\lambda (H: (eq nat n0 p)).(eq_ind nat +n0 (\lambda (n1: nat).(eq nat p n1)) (sym_eq nat n0 p H) (minus n0 O) +(minus_n_O n0)))) (\lambda (n0: nat).(\lambda (H: (eq nat O (S (plus n0 +p)))).(False_ind (eq nat p O) (let H0 \def H in ((let H1 \def (O_S (plus n0 +p)) in (\lambda (H2: (eq nat O (S (plus n0 p)))).(H1 H2))) H0))))) (\lambda +(n0: nat).(\lambda (m0: nat).(\lambda (H: (((eq nat m0 (plus n0 p)) \to (eq +nat p (minus m0 n0))))).(\lambda (H0: (eq nat (S m0) (S (plus n0 p)))).(H +(eq_add_S m0 (plus n0 p) H0)))))) m n))). +(* COMMENTS +Initial nodes: 199 +END *) + +theorem minus_plus: + \forall (n: nat).(\forall (m: nat).(eq nat (minus (plus n m) n) m)) +\def + \lambda (n: nat).(\lambda (m: nat).(sym_eq nat m (minus (plus n m) n) +(plus_minus (plus n m) n m (refl_equal nat (plus n m))))). +(* COMMENTS +Initial nodes: 41 +END *) + +theorem le_pred_n: + \forall (n: nat).(le (pred n) n) +\def + \lambda (n: nat).(nat_ind (\lambda (n0: nat).(le (pred n0) n0)) (le_n O) +(\lambda (n0: nat).(\lambda (_: (le (pred n0) n0)).(le_S (pred (S n0)) n0 +(le_n n0)))) n). +(* COMMENTS +Initial nodes: 43 +END *) + +theorem le_plus_l: + \forall (n: nat).(\forall (m: nat).(le n (plus n m))) +\def + \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (m: nat).(le n0 (plus +n0 m)))) (\lambda (m: nat).(le_O_n m)) (\lambda (n0: nat).(\lambda (IHn: +((\forall (m: nat).(le n0 (plus n0 m))))).(\lambda (m: nat).(le_n_S n0 (plus +n0 m) (IHn m))))) n). +(* COMMENTS +Initial nodes: 55 +END *) + +theorem le_plus_r: + \forall (n: nat).(\forall (m: nat).(le m (plus n m))) +\def + \lambda (n: nat).(\lambda (m: nat).(nat_ind (\lambda (n0: nat).(le m (plus +n0 m))) (le_n m) (\lambda (n0: nat).(\lambda (H: (le m (plus n0 m))).(le_S m +(plus n0 m) H))) n)). +(* COMMENTS +Initial nodes: 47 +END *) + +theorem simpl_le_plus_l: + \forall (p: nat).(\forall (n: nat).(\forall (m: nat).((le (plus p n) (plus p +m)) \to (le n m)))) +\def + \lambda (p: nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).(\forall (m: +nat).((le (plus n n0) (plus n m)) \to (le n0 m))))) (\lambda (n: +nat).(\lambda (m: nat).(\lambda (H: (le n m)).H))) (\lambda (p0: +nat).(\lambda (IHp: ((\forall (n: nat).(\forall (m: nat).((le (plus p0 n) +(plus p0 m)) \to (le n m)))))).(\lambda (n: nat).(\lambda (m: nat).(\lambda +(H: (le (S (plus p0 n)) (S (plus p0 m)))).(IHp n m (le_S_n (plus p0 n) (plus +p0 m) H))))))) p). +(* COMMENTS +Initial nodes: 113 +END *) + +theorem le_plus_trans: + \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((le n m) \to (le n +(plus m p))))) +\def + \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (le n +m)).(le_trans n m (plus m p) H (le_plus_l m p))))). +(* COMMENTS +Initial nodes: 31 +END *) + +theorem le_reg_l: + \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((le n m) \to (le (plus +p n) (plus p m))))) +\def + \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(nat_ind (\lambda (n0: +nat).((le n m) \to (le (plus n0 n) (plus n0 m)))) (\lambda (H: (le n m)).H) +(\lambda (p0: nat).(\lambda (IHp: (((le n m) \to (le (plus p0 n) (plus p0 +m))))).(\lambda (H: (le n m)).(le_n_S (plus p0 n) (plus p0 m) (IHp H))))) +p))). +(* COMMENTS +Initial nodes: 85 +END *) + +theorem le_plus_plus: + \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(\forall (q: nat).((le +n m) \to ((le p q) \to (le (plus n p) (plus m q))))))) +\def + \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (q: +nat).(\lambda (H: (le n m)).(\lambda (H0: (le p q)).(le_ind n (\lambda (n0: +nat).(le (plus n p) (plus n0 q))) (le_reg_l p q n H0) (\lambda (m0: +nat).(\lambda (_: (le n m0)).(\lambda (H2: (le (plus n p) (plus m0 q))).(le_S +(plus n p) (plus m0 q) H2)))) m H)))))). +(* COMMENTS +Initial nodes: 91 +END *) + +theorem le_plus_minus: + \forall (n: nat).(\forall (m: nat).((le n m) \to (eq nat m (plus n (minus m +n))))) +\def + \lambda (n: nat).(\lambda (m: nat).(\lambda (Le: (le n m)).(le_elim_rel +(\lambda (n0: nat).(\lambda (n1: nat).(eq nat n1 (plus n0 (minus n1 n0))))) +(\lambda (p: nat).(minus_n_O p)) (\lambda (p: nat).(\lambda (q: nat).(\lambda +(_: (le p q)).(\lambda (H0: (eq nat q (plus p (minus q p)))).(f_equal nat nat +S q (plus p (minus q p)) H0))))) n m Le))). +(* COMMENTS +Initial nodes: 91 +END *) + +theorem le_plus_minus_r: + \forall (n: nat).(\forall (m: nat).((le n m) \to (eq nat (plus n (minus m +n)) m))) +\def + \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(sym_eq nat m +(plus n (minus m n)) (le_plus_minus n m H)))). +(* COMMENTS +Initial nodes: 33 +END *) + +theorem simpl_lt_plus_l: + \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((lt (plus p n) (plus p +m)) \to (lt n m)))) +\def + \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(nat_ind (\lambda (n0: +nat).((lt (plus n0 n) (plus n0 m)) \to (lt n m))) (\lambda (H: (lt n m)).H) +(\lambda (p0: nat).(\lambda (IHp: (((lt (plus p0 n) (plus p0 m)) \to (lt n +m)))).(\lambda (H: (lt (S (plus p0 n)) (S (plus p0 m)))).(IHp (le_S_n (S +(plus p0 n)) (plus p0 m) H))))) p))). +(* COMMENTS +Initial nodes: 99 +END *) + +theorem lt_reg_l: + \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((lt n m) \to (lt (plus +p n) (plus p m))))) +\def + \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(nat_ind (\lambda (n0: +nat).((lt n m) \to (lt (plus n0 n) (plus n0 m)))) (\lambda (H: (lt n m)).H) +(\lambda (p0: nat).(\lambda (IHp: (((lt n m) \to (lt (plus p0 n) (plus p0 +m))))).(\lambda (H: (lt n m)).(lt_n_S (plus p0 n) (plus p0 m) (IHp H))))) +p))). +(* COMMENTS +Initial nodes: 85 +END *) + +theorem lt_reg_r: + \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((lt n m) \to (lt (plus +n p) (plus m p))))) +\def + \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (lt n +m)).(eq_ind_r nat (plus p n) (\lambda (n0: nat).(lt n0 (plus m p))) (eq_ind_r +nat (plus p m) (\lambda (n0: nat).(lt (plus p n) n0)) (nat_ind (\lambda (n0: +nat).(lt (plus n0 n) (plus n0 m))) H (\lambda (n0: nat).(\lambda (_: (lt +(plus n0 n) (plus n0 m))).(lt_reg_l n m (S n0) H))) p) (plus m p) (plus_sym m +p)) (plus n p) (plus_sym n p))))). +(* COMMENTS +Initial nodes: 129 +END *) + +theorem le_lt_plus_plus: + \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(\forall (q: nat).((le +n m) \to ((lt p q) \to (lt (plus n p) (plus m q))))))) +\def + \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (q: +nat).(\lambda (H: (le n m)).(\lambda (H0: (le (S p) q)).(eq_ind_r nat (plus n +(S p)) (\lambda (n0: nat).(le n0 (plus m q))) (le_plus_plus n m (S p) q H H0) +(plus (S n) p) (plus_Snm_nSm n p))))))). +(* COMMENTS +Initial nodes: 75 +END *) + +theorem lt_le_plus_plus: + \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(\forall (q: nat).((lt +n m) \to ((le p q) \to (lt (plus n p) (plus m q))))))) +\def + \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (q: +nat).(\lambda (H: (le (S n) m)).(\lambda (H0: (le p q)).(le_plus_plus (S n) m +p q H H0)))))). +(* COMMENTS +Initial nodes: 37 +END *) + +theorem lt_plus_plus: + \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(\forall (q: nat).((lt +n m) \to ((lt p q) \to (lt (plus n p) (plus m q))))))) +\def + \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (q: +nat).(\lambda (H: (lt n m)).(\lambda (H0: (lt p q)).(lt_le_plus_plus n m p q +H (lt_le_weak p q H0))))))). +(* COMMENTS +Initial nodes: 39 +END *) + +theorem well_founded_ltof: + \forall (A: Set).(\forall (f: ((A \to nat))).(well_founded A (ltof A f))) +\def + \lambda (A: Set).(\lambda (f: ((A \to nat))).(let H \def (\lambda (n: +nat).(nat_ind (\lambda (n0: nat).(\forall (a: A).((lt (f a) n0) \to (Acc A +(ltof A f) a)))) (\lambda (a: A).(\lambda (H: (lt (f a) O)).(False_ind (Acc A +(ltof A f) a) (let H0 \def H in ((let H1 \def (lt_n_O (f a)) in (\lambda (H2: +(lt (f a) O)).(H1 H2))) H0))))) (\lambda (n0: nat).(\lambda (IHn: ((\forall +(a: A).((lt (f a) n0) \to (Acc A (ltof A f) a))))).(\lambda (a: A).(\lambda +(ltSma: (lt (f a) (S n0))).(Acc_intro A (ltof A f) a (\lambda (b: A).(\lambda +(ltfafb: (lt (f b) (f a))).(IHn b (lt_le_trans (f b) (f a) n0 ltfafb +(lt_n_Sm_le (f a) n0 ltSma)))))))))) n)) in (\lambda (a: A).(H (S (f a)) a +(le_n (S (f a))))))). +(* COMMENTS +Initial nodes: 189 +END *) + +theorem lt_wf: + well_founded nat lt +\def + well_founded_ltof nat (\lambda (m: nat).m). +(* COMMENTS +Initial nodes: 7 +END *) + +theorem lt_wf_ind: + \forall (p: nat).(\forall (P: ((nat \to Prop))).(((\forall (n: +nat).(((\forall (m: nat).((lt m n) \to (P m)))) \to (P n)))) \to (P p))) +\def + \lambda (p: nat).(\lambda (P: ((nat \to Prop))).(\lambda (H: ((\forall (n: +nat).(((\forall (m: nat).((lt m n) \to (P m)))) \to (P n))))).(Acc_ind nat lt +(\lambda (n: nat).(P n)) (\lambda (x: nat).(\lambda (_: ((\forall (y: +nat).((lt y x) \to (Acc nat lt y))))).(\lambda (H1: ((\forall (y: nat).((lt y +x) \to (P y))))).(H x H1)))) p (lt_wf p)))). +(* COMMENTS +Initial nodes: 77 +END *) + diff --git a/matita/matita/contribs/lambdadelta/legacy_1/definitions.ma b/matita/matita/contribs/lambdadelta/legacy_1/definitions.ma new file mode 100644 index 000000000..63fc85890 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/legacy_1/definitions.ma @@ -0,0 +1,18 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Legacy-1/coq/defs.ma". + diff --git a/matita/matita/contribs/lambdadelta/legacy_1/preamble.ma b/matita/matita/contribs/lambdadelta/legacy_1/preamble.ma new file mode 100644 index 000000000..96c1bc1fa --- /dev/null +++ b/matita/matita/contribs/lambdadelta/legacy_1/preamble.ma @@ -0,0 +1,15 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +inductive False: Prop \def . diff --git a/matita/matita/contribs/lambdadelta/legacy_1/spare.ma b/matita/matita/contribs/lambdadelta/legacy_1/spare.ma new file mode 100644 index 000000000..77939a1b6 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/legacy_1/spare.ma @@ -0,0 +1,18 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Legacy-1/theory.ma". + diff --git a/matita/matita/contribs/lambdadelta/legacy_1/theory.ma b/matita/matita/contribs/lambdadelta/legacy_1/theory.ma new file mode 100644 index 000000000..4ee597e09 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/legacy_1/theory.ma @@ -0,0 +1,18 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "Legacy-1/coq/props.ma". + diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/A/defs.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/A/defs.ma deleted file mode 100644 index 2290c3de4..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/A/defs.ma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/preamble.ma". - -inductive A: Set \def -| ASort: nat \to (nat \to A) -| AHead: A \to (A \to A). - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/C/defs.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/C/defs.ma deleted file mode 100644 index cecc08267..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/C/defs.ma +++ /dev/null @@ -1,45 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/T/defs.ma". - -inductive C: Set \def -| CSort: nat \to C -| CHead: C \to (K \to (T \to C)). - -definition cweight: - C \to nat -\def - let rec cweight (c: C) on c: nat \def (match c with [(CSort _) \Rightarrow O -| (CHead c0 _ t) \Rightarrow (plus (cweight c0) (tweight t))]) in cweight. - -definition clt: - C \to (C \to Prop) -\def - \lambda (c1: C).(\lambda (c2: C).(lt (cweight c1) (cweight c2))). - -definition cle: - C \to (C \to Prop) -\def - \lambda (c1: C).(\lambda (c2: C).(le (cweight c1) (cweight c2))). - -definition CTail: - K \to (T \to (C \to C)) -\def - let rec CTail (k: K) (t: T) (c: C) on c: C \def (match c with [(CSort n) -\Rightarrow (CHead (CSort n) k t) | (CHead d h u) \Rightarrow (CHead (CTail k -t d) h u)]) in CTail. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/C/props.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/C/props.ma deleted file mode 100644 index 878abb3bc..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/C/props.ma +++ /dev/null @@ -1,139 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/C/defs.ma". - -include "Basic-1/T/props.ma". - -theorem clt_cong: - \forall (c: C).(\forall (d: C).((clt c d) \to (\forall (k: K).(\forall (t: -T).(clt (CHead c k t) (CHead d k t)))))) -\def - \lambda (c: C).(\lambda (d: C).(\lambda (H: (lt (cweight c) (cweight -d))).(\lambda (_: K).(\lambda (t: T).(lt_reg_r (cweight c) (cweight d) -(tweight t) H))))). -(* COMMENTS -Initial nodes: 33 -END *) - -theorem clt_head: - \forall (k: K).(\forall (c: C).(\forall (u: T).(clt c (CHead c k u)))) -\def - \lambda (_: K).(\lambda (c: C).(\lambda (u: T).(eq_ind_r nat (plus (cweight -c) O) (\lambda (n: nat).(lt n (plus (cweight c) (tweight u)))) -(le_lt_plus_plus (cweight c) (cweight c) O (tweight u) (le_n (cweight c)) -(tweight_lt u)) (cweight c) (plus_n_O (cweight c))))). -(* COMMENTS -Initial nodes: 69 -END *) - -theorem clt_wf__q_ind: - \forall (P: ((C \to Prop))).(((\forall (n: nat).((\lambda (P0: ((C \to -Prop))).(\lambda (n0: nat).(\forall (c: C).((eq nat (cweight c) n0) \to (P0 -c))))) P n))) \to (\forall (c: C).(P c))) -\def - let Q \def (\lambda (P: ((C \to Prop))).(\lambda (n: nat).(\forall (c: -C).((eq nat (cweight c) n) \to (P c))))) in (\lambda (P: ((C \to -Prop))).(\lambda (H: ((\forall (n: nat).(\forall (c: C).((eq nat (cweight c) -n) \to (P c)))))).(\lambda (c: C).(H (cweight c) c (refl_equal nat (cweight -c)))))). -(* COMMENTS -Initial nodes: 61 -END *) - -theorem clt_wf_ind: - \forall (P: ((C \to Prop))).(((\forall (c: C).(((\forall (d: C).((clt d c) -\to (P d)))) \to (P c)))) \to (\forall (c: C).(P c))) -\def - let Q \def (\lambda (P: ((C \to Prop))).(\lambda (n: nat).(\forall (c: -C).((eq nat (cweight c) n) \to (P c))))) in (\lambda (P: ((C \to -Prop))).(\lambda (H: ((\forall (c: C).(((\forall (d: C).((lt (cweight d) -(cweight c)) \to (P d)))) \to (P c))))).(\lambda (c: C).(clt_wf__q_ind -(\lambda (c0: C).(P c0)) (\lambda (n: nat).(lt_wf_ind n (Q (\lambda (c0: -C).(P c0))) (\lambda (n0: nat).(\lambda (H0: ((\forall (m: nat).((lt m n0) -\to (Q (\lambda (c0: C).(P c0)) m))))).(\lambda (c0: C).(\lambda (H1: (eq nat -(cweight c0) n0)).(let H2 \def (eq_ind_r nat n0 (\lambda (n1: nat).(\forall -(m: nat).((lt m n1) \to (\forall (c1: C).((eq nat (cweight c1) m) \to (P -c1)))))) H0 (cweight c0) H1) in (H c0 (\lambda (d: C).(\lambda (H3: (lt -(cweight d) (cweight c0))).(H2 (cweight d) H3 d (refl_equal nat (cweight -d))))))))))))) c)))). -(* COMMENTS -Initial nodes: 179 -END *) - -theorem chead_ctail: - \forall (c: C).(\forall (t: T).(\forall (k: K).(ex_3 K C T (\lambda (h: -K).(\lambda (d: C).(\lambda (u: T).(eq C (CHead c k t) (CTail h u d)))))))) -\def - \lambda (c: C).(C_ind (\lambda (c0: C).(\forall (t: T).(\forall (k: K).(ex_3 -K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C (CHead c0 k t) -(CTail h u d))))))))) (\lambda (n: nat).(\lambda (t: T).(\lambda (k: -K).(ex_3_intro K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C -(CHead (CSort n) k t) (CTail h u d))))) k (CSort n) t (refl_equal C (CHead -(CSort n) k t)))))) (\lambda (c0: C).(\lambda (H: ((\forall (t: T).(\forall -(k: K).(ex_3 K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C -(CHead c0 k t) (CTail h u d)))))))))).(\lambda (k: K).(\lambda (t: -T).(\lambda (t0: T).(\lambda (k0: K).(let H_x \def (H t k) in (let H0 \def -H_x in (ex_3_ind K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C -(CHead c0 k t) (CTail h u d))))) (ex_3 K C T (\lambda (h: K).(\lambda (d: -C).(\lambda (u: T).(eq C (CHead (CHead c0 k t) k0 t0) (CTail h u d)))))) -(\lambda (x0: K).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H1: (eq C (CHead -c0 k t) (CTail x0 x2 x1))).(eq_ind_r C (CTail x0 x2 x1) (\lambda (c1: -C).(ex_3 K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C (CHead -c1 k0 t0) (CTail h u d))))))) (ex_3_intro K C T (\lambda (h: K).(\lambda (d: -C).(\lambda (u: T).(eq C (CHead (CTail x0 x2 x1) k0 t0) (CTail h u d))))) x0 -(CHead x1 k0 t0) x2 (refl_equal C (CHead (CTail x0 x2 x1) k0 t0))) (CHead c0 -k t) H1))))) H0))))))))) c). -(* COMMENTS -Initial nodes: 395 -END *) - -theorem clt_thead: - \forall (k: K).(\forall (u: T).(\forall (c: C).(clt c (CTail k u c)))) -\def - \lambda (k: K).(\lambda (u: T).(\lambda (c: C).(C_ind (\lambda (c0: C).(clt -c0 (CTail k u c0))) (\lambda (n: nat).(clt_head k (CSort n) u)) (\lambda (c0: -C).(\lambda (H: (clt c0 (CTail k u c0))).(\lambda (k0: K).(\lambda (t: -T).(clt_cong c0 (CTail k u c0) H k0 t))))) c))). -(* COMMENTS -Initial nodes: 71 -END *) - -theorem c_tail_ind: - \forall (P: ((C \to Prop))).(((\forall (n: nat).(P (CSort n)))) \to -(((\forall (c: C).((P c) \to (\forall (k: K).(\forall (t: T).(P (CTail k t -c))))))) \to (\forall (c: C).(P c)))) -\def - \lambda (P: ((C \to Prop))).(\lambda (H: ((\forall (n: nat).(P (CSort -n))))).(\lambda (H0: ((\forall (c: C).((P c) \to (\forall (k: K).(\forall (t: -T).(P (CTail k t c)))))))).(\lambda (c: C).(clt_wf_ind (\lambda (c0: C).(P -c0)) (\lambda (c0: C).(C_ind (\lambda (c1: C).(((\forall (d: C).((clt d c1) -\to (P d)))) \to (P c1))) (\lambda (n: nat).(\lambda (_: ((\forall (d: -C).((clt d (CSort n)) \to (P d))))).(H n))) (\lambda (c1: C).(\lambda (_: -((((\forall (d: C).((clt d c1) \to (P d)))) \to (P c1)))).(\lambda (k: -K).(\lambda (t: T).(\lambda (H2: ((\forall (d: C).((clt d (CHead c1 k t)) \to -(P d))))).(let H_x \def (chead_ctail c1 t k) in (let H3 \def H_x in (ex_3_ind -K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C (CHead c1 k t) -(CTail h u d))))) (P (CHead c1 k t)) (\lambda (x0: K).(\lambda (x1: -C).(\lambda (x2: T).(\lambda (H4: (eq C (CHead c1 k t) (CTail x0 x2 -x1))).(eq_ind_r C (CTail x0 x2 x1) (\lambda (c2: C).(P c2)) (let H5 \def -(eq_ind C (CHead c1 k t) (\lambda (c2: C).(\forall (d: C).((clt d c2) \to (P -d)))) H2 (CTail x0 x2 x1) H4) in (H0 x1 (H5 x1 (clt_thead x0 x2 x1)) x0 x2)) -(CHead c1 k t) H4))))) H3)))))))) c0)) c)))). -(* COMMENTS -Initial nodes: 295 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/G/defs.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/G/defs.ma deleted file mode 100644 index 23edf5789..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/G/defs.ma +++ /dev/null @@ -1,23 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/preamble.ma". - -record G : Set \def { - next: (nat \to nat); - next_lt: (\forall (n: nat).(lt n (next n))) -}. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/T/dec.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/T/dec.ma deleted file mode 100644 index a088c40e3..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/T/dec.ma +++ /dev/null @@ -1,446 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/T/defs.ma". - -theorem terms_props__bind_dec: - \forall (b1: B).(\forall (b2: B).(or (eq B b1 b2) ((eq B b1 b2) \to (\forall -(P: Prop).P)))) -\def - \lambda (b1: B).(B_ind (\lambda (b: B).(\forall (b2: B).(or (eq B b b2) ((eq -B b b2) \to (\forall (P: Prop).P))))) (\lambda (b2: B).(B_ind (\lambda (b: -B).(or (eq B Abbr b) ((eq B Abbr b) \to (\forall (P: Prop).P)))) (or_introl -(eq B Abbr Abbr) ((eq B Abbr Abbr) \to (\forall (P: Prop).P)) (refl_equal B -Abbr)) (or_intror (eq B Abbr Abst) ((eq B Abbr Abst) \to (\forall (P: -Prop).P)) (\lambda (H: (eq B Abbr Abst)).(\lambda (P: Prop).(let H0 \def -(eq_ind B Abbr (\lambda (ee: B).(match ee in B return (\lambda (_: B).Prop) -with [Abbr \Rightarrow True | Abst \Rightarrow False | Void \Rightarrow -False])) I Abst H) in (False_ind P H0))))) (or_intror (eq B Abbr Void) ((eq B -Abbr Void) \to (\forall (P: Prop).P)) (\lambda (H: (eq B Abbr Void)).(\lambda -(P: Prop).(let H0 \def (eq_ind B Abbr (\lambda (ee: B).(match ee in B return -(\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | -Void \Rightarrow False])) I Void H) in (False_ind P H0))))) b2)) (\lambda -(b2: B).(B_ind (\lambda (b: B).(or (eq B Abst b) ((eq B Abst b) \to (\forall -(P: Prop).P)))) (or_intror (eq B Abst Abbr) ((eq B Abst Abbr) \to (\forall -(P: Prop).P)) (\lambda (H: (eq B Abst Abbr)).(\lambda (P: Prop).(let H0 \def -(eq_ind B Abst (\lambda (ee: B).(match ee in B return (\lambda (_: B).Prop) -with [Abbr \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow -False])) I Abbr H) in (False_ind P H0))))) (or_introl (eq B Abst Abst) ((eq B -Abst Abst) \to (\forall (P: Prop).P)) (refl_equal B Abst)) (or_intror (eq B -Abst Void) ((eq B Abst Void) \to (\forall (P: Prop).P)) (\lambda (H: (eq B -Abst Void)).(\lambda (P: Prop).(let H0 \def (eq_ind B Abst (\lambda (ee: -B).(match ee in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | -Abst \Rightarrow True | Void \Rightarrow False])) I Void H) in (False_ind P -H0))))) b2)) (\lambda (b2: B).(B_ind (\lambda (b: B).(or (eq B Void b) ((eq B -Void b) \to (\forall (P: Prop).P)))) (or_intror (eq B Void Abbr) ((eq B Void -Abbr) \to (\forall (P: Prop).P)) (\lambda (H: (eq B Void Abbr)).(\lambda (P: -Prop).(let H0 \def (eq_ind B Void (\lambda (ee: B).(match ee in B return -(\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False | -Void \Rightarrow True])) I Abbr H) in (False_ind P H0))))) (or_intror (eq B -Void Abst) ((eq B Void Abst) \to (\forall (P: Prop).P)) (\lambda (H: (eq B -Void Abst)).(\lambda (P: Prop).(let H0 \def (eq_ind B Void (\lambda (ee: -B).(match ee in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | -Abst \Rightarrow False | Void \Rightarrow True])) I Abst H) in (False_ind P -H0))))) (or_introl (eq B Void Void) ((eq B Void Void) \to (\forall (P: -Prop).P)) (refl_equal B Void)) b2)) b1). -(* COMMENTS -Initial nodes: 559 -END *) - -theorem bind_dec_not: - \forall (b1: B).(\forall (b2: B).(or (eq B b1 b2) (not (eq B b1 b2)))) -\def - \lambda (b1: B).(\lambda (b2: B).(let H_x \def (terms_props__bind_dec b1 b2) -in (let H \def H_x in (or_ind (eq B b1 b2) ((eq B b1 b2) \to (\forall (P: -Prop).P)) (or (eq B b1 b2) ((eq B b1 b2) \to False)) (\lambda (H0: (eq B b1 -b2)).(or_introl (eq B b1 b2) ((eq B b1 b2) \to False) H0)) (\lambda (H0: -(((eq B b1 b2) \to (\forall (P: Prop).P)))).(or_intror (eq B b1 b2) ((eq B b1 -b2) \to False) (\lambda (H1: (eq B b1 b2)).(H0 H1 False)))) H)))). -(* COMMENTS -Initial nodes: 131 -END *) - -theorem terms_props__flat_dec: - \forall (f1: F).(\forall (f2: F).(or (eq F f1 f2) ((eq F f1 f2) \to (\forall -(P: Prop).P)))) -\def - \lambda (f1: F).(F_ind (\lambda (f: F).(\forall (f2: F).(or (eq F f f2) ((eq -F f f2) \to (\forall (P: Prop).P))))) (\lambda (f2: F).(F_ind (\lambda (f: -F).(or (eq F Appl f) ((eq F Appl f) \to (\forall (P: Prop).P)))) (or_introl -(eq F Appl Appl) ((eq F Appl Appl) \to (\forall (P: Prop).P)) (refl_equal F -Appl)) (or_intror (eq F Appl Cast) ((eq F Appl Cast) \to (\forall (P: -Prop).P)) (\lambda (H: (eq F Appl Cast)).(\lambda (P: Prop).(let H0 \def -(eq_ind F Appl (\lambda (ee: F).(match ee in F return (\lambda (_: F).Prop) -with [Appl \Rightarrow True | Cast \Rightarrow False])) I Cast H) in -(False_ind P H0))))) f2)) (\lambda (f2: F).(F_ind (\lambda (f: F).(or (eq F -Cast f) ((eq F Cast f) \to (\forall (P: Prop).P)))) (or_intror (eq F Cast -Appl) ((eq F Cast Appl) \to (\forall (P: Prop).P)) (\lambda (H: (eq F Cast -Appl)).(\lambda (P: Prop).(let H0 \def (eq_ind F Cast (\lambda (ee: F).(match -ee in F return (\lambda (_: F).Prop) with [Appl \Rightarrow False | Cast -\Rightarrow True])) I Appl H) in (False_ind P H0))))) (or_introl (eq F Cast -Cast) ((eq F Cast Cast) \to (\forall (P: Prop).P)) (refl_equal F Cast)) f2)) -f1). -(* COMMENTS -Initial nodes: 263 -END *) - -theorem terms_props__kind_dec: - \forall (k1: K).(\forall (k2: K).(or (eq K k1 k2) ((eq K k1 k2) \to (\forall -(P: Prop).P)))) -\def - \lambda (k1: K).(K_ind (\lambda (k: K).(\forall (k2: K).(or (eq K k k2) ((eq -K k k2) \to (\forall (P: Prop).P))))) (\lambda (b: B).(\lambda (k2: K).(K_ind -(\lambda (k: K).(or (eq K (Bind b) k) ((eq K (Bind b) k) \to (\forall (P: -Prop).P)))) (\lambda (b0: B).(let H_x \def (terms_props__bind_dec b b0) in -(let H \def H_x in (or_ind (eq B b b0) ((eq B b b0) \to (\forall (P: -Prop).P)) (or (eq K (Bind b) (Bind b0)) ((eq K (Bind b) (Bind b0)) \to -(\forall (P: Prop).P))) (\lambda (H0: (eq B b b0)).(eq_ind B b (\lambda (b1: -B).(or (eq K (Bind b) (Bind b1)) ((eq K (Bind b) (Bind b1)) \to (\forall (P: -Prop).P)))) (or_introl (eq K (Bind b) (Bind b)) ((eq K (Bind b) (Bind b)) \to -(\forall (P: Prop).P)) (refl_equal K (Bind b))) b0 H0)) (\lambda (H0: (((eq B -b b0) \to (\forall (P: Prop).P)))).(or_intror (eq K (Bind b) (Bind b0)) ((eq -K (Bind b) (Bind b0)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq K (Bind b) -(Bind b0))).(\lambda (P: Prop).(let H2 \def (f_equal K B (\lambda (e: -K).(match e in K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | -(Flat _) \Rightarrow b])) (Bind b) (Bind b0) H1) in (let H3 \def (eq_ind_r B -b0 (\lambda (b1: B).((eq B b b1) \to (\forall (P0: Prop).P0))) H0 b H2) in -(H3 (refl_equal B b) P))))))) H)))) (\lambda (f: F).(or_intror (eq K (Bind b) -(Flat f)) ((eq K (Bind b) (Flat f)) \to (\forall (P: Prop).P)) (\lambda (H: -(eq K (Bind b) (Flat f))).(\lambda (P: Prop).(let H0 \def (eq_ind K (Bind b) -(\lambda (ee: K).(match ee in K return (\lambda (_: K).Prop) with [(Bind _) -\Rightarrow True | (Flat _) \Rightarrow False])) I (Flat f) H) in (False_ind -P H0)))))) k2))) (\lambda (f: F).(\lambda (k2: K).(K_ind (\lambda (k: K).(or -(eq K (Flat f) k) ((eq K (Flat f) k) \to (\forall (P: Prop).P)))) (\lambda -(b: B).(or_intror (eq K (Flat f) (Bind b)) ((eq K (Flat f) (Bind b)) \to -(\forall (P: Prop).P)) (\lambda (H: (eq K (Flat f) (Bind b))).(\lambda (P: -Prop).(let H0 \def (eq_ind K (Flat f) (\lambda (ee: K).(match ee in K return -(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])) I (Bind b) H) in (False_ind P H0)))))) (\lambda (f0: F).(let H_x \def -(terms_props__flat_dec f f0) in (let H \def H_x in (or_ind (eq F f f0) ((eq F -f f0) \to (\forall (P: Prop).P)) (or (eq K (Flat f) (Flat f0)) ((eq K (Flat -f) (Flat f0)) \to (\forall (P: Prop).P))) (\lambda (H0: (eq F f f0)).(eq_ind -F f (\lambda (f1: F).(or (eq K (Flat f) (Flat f1)) ((eq K (Flat f) (Flat f1)) -\to (\forall (P: Prop).P)))) (or_introl (eq K (Flat f) (Flat f)) ((eq K (Flat -f) (Flat f)) \to (\forall (P: Prop).P)) (refl_equal K (Flat f))) f0 H0)) -(\lambda (H0: (((eq F f f0) \to (\forall (P: Prop).P)))).(or_intror (eq K -(Flat f) (Flat f0)) ((eq K (Flat f) (Flat f0)) \to (\forall (P: Prop).P)) -(\lambda (H1: (eq K (Flat f) (Flat f0))).(\lambda (P: Prop).(let H2 \def -(f_equal K F (\lambda (e: K).(match e in K return (\lambda (_: K).F) with -[(Bind _) \Rightarrow f | (Flat f1) \Rightarrow f1])) (Flat f) (Flat f0) H1) -in (let H3 \def (eq_ind_r F f0 (\lambda (f1: F).((eq F f f1) \to (\forall -(P0: Prop).P0))) H0 f H2) in (H3 (refl_equal F f) P))))))) H)))) k2))) k1). -(* COMMENTS -Initial nodes: 799 -END *) - -theorem term_dec: - \forall (t1: T).(\forall (t2: T).(or (eq T t1 t2) ((eq T t1 t2) \to (\forall -(P: Prop).P)))) -\def - \lambda (t1: T).(T_ind (\lambda (t: T).(\forall (t2: T).(or (eq T t t2) ((eq -T t t2) \to (\forall (P: Prop).P))))) (\lambda (n: nat).(\lambda (t2: -T).(T_ind (\lambda (t: T).(or (eq T (TSort n) t) ((eq T (TSort n) t) \to -(\forall (P: Prop).P)))) (\lambda (n0: nat).(let H_x \def (nat_dec n n0) in -(let H \def H_x in (or_ind (eq nat n n0) ((eq nat n n0) \to (\forall (P: -Prop).P)) (or (eq T (TSort n) (TSort n0)) ((eq T (TSort n) (TSort n0)) \to -(\forall (P: Prop).P))) (\lambda (H0: (eq nat n n0)).(eq_ind nat n (\lambda -(n1: nat).(or (eq T (TSort n) (TSort n1)) ((eq T (TSort n) (TSort n1)) \to -(\forall (P: Prop).P)))) (or_introl (eq T (TSort n) (TSort n)) ((eq T (TSort -n) (TSort n)) \to (\forall (P: Prop).P)) (refl_equal T (TSort n))) n0 H0)) -(\lambda (H0: (((eq nat n n0) \to (\forall (P: Prop).P)))).(or_intror (eq T -(TSort n) (TSort n0)) ((eq T (TSort n) (TSort n0)) \to (\forall (P: Prop).P)) -(\lambda (H1: (eq T (TSort n) (TSort n0))).(\lambda (P: Prop).(let H2 \def -(f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with -[(TSort n1) \Rightarrow n1 | (TLRef _) \Rightarrow n | (THead _ _ _) -\Rightarrow n])) (TSort n) (TSort n0) H1) in (let H3 \def (eq_ind_r nat n0 -(\lambda (n1: nat).((eq nat n n1) \to (\forall (P0: Prop).P0))) H0 n H2) in -(H3 (refl_equal nat n) P))))))) H)))) (\lambda (n0: nat).(or_intror (eq T -(TSort n) (TLRef n0)) ((eq T (TSort n) (TLRef n0)) \to (\forall (P: Prop).P)) -(\lambda (H: (eq T (TSort n) (TLRef n0))).(\lambda (P: Prop).(let H0 \def -(eq_ind T (TSort n) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | -(THead _ _ _) \Rightarrow False])) I (TLRef n0) H) in (False_ind P H0)))))) -(\lambda (k: K).(\lambda (t: T).(\lambda (_: (or (eq T (TSort n) t) ((eq T -(TSort n) t) \to (\forall (P: Prop).P)))).(\lambda (t0: T).(\lambda (_: (or -(eq T (TSort n) t0) ((eq T (TSort n) t0) \to (\forall (P: -Prop).P)))).(or_intror (eq T (TSort n) (THead k t t0)) ((eq T (TSort n) -(THead k t t0)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq T (TSort n) -(THead k t t0))).(\lambda (P: Prop).(let H2 \def (eq_ind T (TSort n) (\lambda -(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow -False])) I (THead k t t0) H1) in (False_ind P H2)))))))))) t2))) (\lambda (n: -nat).(\lambda (t2: T).(T_ind (\lambda (t: T).(or (eq T (TLRef n) t) ((eq T -(TLRef n) t) \to (\forall (P: Prop).P)))) (\lambda (n0: nat).(or_intror (eq T -(TLRef n) (TSort n0)) ((eq T (TLRef n) (TSort n0)) \to (\forall (P: Prop).P)) -(\lambda (H: (eq T (TLRef n) (TSort n0))).(\lambda (P: Prop).(let H0 \def -(eq_ind T (TLRef n) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | -(THead _ _ _) \Rightarrow False])) I (TSort n0) H) in (False_ind P H0)))))) -(\lambda (n0: nat).(let H_x \def (nat_dec n n0) in (let H \def H_x in (or_ind -(eq nat n n0) ((eq nat n n0) \to (\forall (P: Prop).P)) (or (eq T (TLRef n) -(TLRef n0)) ((eq T (TLRef n) (TLRef n0)) \to (\forall (P: Prop).P))) (\lambda -(H0: (eq nat n n0)).(eq_ind nat n (\lambda (n1: nat).(or (eq T (TLRef n) -(TLRef n1)) ((eq T (TLRef n) (TLRef n1)) \to (\forall (P: Prop).P)))) -(or_introl (eq T (TLRef n) (TLRef n)) ((eq T (TLRef n) (TLRef n)) \to -(\forall (P: Prop).P)) (refl_equal T (TLRef n))) n0 H0)) (\lambda (H0: (((eq -nat n n0) \to (\forall (P: Prop).P)))).(or_intror (eq T (TLRef n) (TLRef n0)) -((eq T (TLRef n) (TLRef n0)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq T -(TLRef n) (TLRef n0))).(\lambda (P: Prop).(let H2 \def (f_equal T nat -(\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort _) -\Rightarrow n | (TLRef n1) \Rightarrow n1 | (THead _ _ _) \Rightarrow n])) -(TLRef n) (TLRef n0) H1) in (let H3 \def (eq_ind_r nat n0 (\lambda (n1: -nat).((eq nat n n1) \to (\forall (P0: Prop).P0))) H0 n H2) in (H3 (refl_equal -nat n) P))))))) H)))) (\lambda (k: K).(\lambda (t: T).(\lambda (_: (or (eq T -(TLRef n) t) ((eq T (TLRef n) t) \to (\forall (P: Prop).P)))).(\lambda (t0: -T).(\lambda (_: (or (eq T (TLRef n) t0) ((eq T (TLRef n) t0) \to (\forall (P: -Prop).P)))).(or_intror (eq T (TLRef n) (THead k t t0)) ((eq T (TLRef n) -(THead k t t0)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq T (TLRef n) -(THead k t t0))).(\lambda (P: Prop).(let H2 \def (eq_ind T (TLRef n) (\lambda -(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow -False])) I (THead k t t0) H1) in (False_ind P H2)))))))))) t2))) (\lambda (k: -K).(\lambda (t: T).(\lambda (H: ((\forall (t2: T).(or (eq T t t2) ((eq T t -t2) \to (\forall (P: Prop).P)))))).(\lambda (t0: T).(\lambda (H0: ((\forall -(t2: T).(or (eq T t0 t2) ((eq T t0 t2) \to (\forall (P: -Prop).P)))))).(\lambda (t2: T).(T_ind (\lambda (t3: T).(or (eq T (THead k t -t0) t3) ((eq T (THead k t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (n: -nat).(or_intror (eq T (THead k t t0) (TSort n)) ((eq T (THead k t t0) (TSort -n)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq T (THead k t t0) (TSort -n))).(\lambda (P: Prop).(let H2 \def (eq_ind T (THead k t t0) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I -(TSort n) H1) in (False_ind P H2)))))) (\lambda (n: nat).(or_intror (eq T -(THead k t t0) (TLRef n)) ((eq T (THead k t t0) (TLRef n)) \to (\forall (P: -Prop).P)) (\lambda (H1: (eq T (THead k t t0) (TLRef n))).(\lambda (P: -Prop).(let H2 \def (eq_ind T (THead k t t0) (\lambda (ee: T).(match ee in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H1) in -(False_ind P H2)))))) (\lambda (k0: K).(\lambda (t3: T).(\lambda (H1: (or (eq -T (THead k t t0) t3) ((eq T (THead k t t0) t3) \to (\forall (P: -Prop).P)))).(\lambda (t4: T).(\lambda (H2: (or (eq T (THead k t t0) t4) ((eq -T (THead k t t0) t4) \to (\forall (P: Prop).P)))).(let H_x \def (H t3) in -(let H3 \def H_x in (or_ind (eq T t t3) ((eq T t t3) \to (\forall (P: -Prop).P)) (or (eq T (THead k t t0) (THead k0 t3 t4)) ((eq T (THead k t t0) -(THead k0 t3 t4)) \to (\forall (P: Prop).P))) (\lambda (H4: (eq T t t3)).(let -H5 \def (eq_ind_r T t3 (\lambda (t5: T).(or (eq T (THead k t t0) t5) ((eq T -(THead k t t0) t5) \to (\forall (P: Prop).P)))) H1 t H4) in (eq_ind T t -(\lambda (t5: T).(or (eq T (THead k t t0) (THead k0 t5 t4)) ((eq T (THead k t -t0) (THead k0 t5 t4)) \to (\forall (P: Prop).P)))) (let H_x0 \def (H0 t4) in -(let H6 \def H_x0 in (or_ind (eq T t0 t4) ((eq T t0 t4) \to (\forall (P: -Prop).P)) (or (eq T (THead k t t0) (THead k0 t t4)) ((eq T (THead k t t0) -(THead k0 t t4)) \to (\forall (P: Prop).P))) (\lambda (H7: (eq T t0 t4)).(let -H8 \def (eq_ind_r T t4 (\lambda (t5: T).(or (eq T (THead k t t0) t5) ((eq T -(THead k t t0) t5) \to (\forall (P: Prop).P)))) H2 t0 H7) in (eq_ind T t0 -(\lambda (t5: T).(or (eq T (THead k t t0) (THead k0 t t5)) ((eq T (THead k t -t0) (THead k0 t t5)) \to (\forall (P: Prop).P)))) (let H_x1 \def -(terms_props__kind_dec k k0) in (let H9 \def H_x1 in (or_ind (eq K k k0) ((eq -K k k0) \to (\forall (P: Prop).P)) (or (eq T (THead k t t0) (THead k0 t t0)) -((eq T (THead k t t0) (THead k0 t t0)) \to (\forall (P: Prop).P))) (\lambda -(H10: (eq K k k0)).(eq_ind K k (\lambda (k1: K).(or (eq T (THead k t t0) -(THead k1 t t0)) ((eq T (THead k t t0) (THead k1 t t0)) \to (\forall (P: -Prop).P)))) (or_introl (eq T (THead k t t0) (THead k t t0)) ((eq T (THead k t -t0) (THead k t t0)) \to (\forall (P: Prop).P)) (refl_equal T (THead k t t0))) -k0 H10)) (\lambda (H10: (((eq K k k0) \to (\forall (P: Prop).P)))).(or_intror -(eq T (THead k t t0) (THead k0 t t0)) ((eq T (THead k t t0) (THead k0 t t0)) -\to (\forall (P: Prop).P)) (\lambda (H11: (eq T (THead k t t0) (THead k0 t -t0))).(\lambda (P: Prop).(let H12 \def (f_equal T K (\lambda (e: T).(match e -in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) -\Rightarrow k | (THead k1 _ _) \Rightarrow k1])) (THead k t t0) (THead k0 t -t0) H11) in (let H13 \def (eq_ind_r K k0 (\lambda (k1: K).((eq K k k1) \to -(\forall (P0: Prop).P0))) H10 k H12) in (H13 (refl_equal K k) P))))))) H9))) -t4 H7))) (\lambda (H7: (((eq T t0 t4) \to (\forall (P: Prop).P)))).(or_intror -(eq T (THead k t t0) (THead k0 t t4)) ((eq T (THead k t t0) (THead k0 t t4)) -\to (\forall (P: Prop).P)) (\lambda (H8: (eq T (THead k t t0) (THead k0 t -t4))).(\lambda (P: Prop).(let H9 \def (f_equal T K (\lambda (e: T).(match e -in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) -\Rightarrow k | (THead k1 _ _) \Rightarrow k1])) (THead k t t0) (THead k0 t -t4) H8) in ((let H10 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 -| (THead _ _ t5) \Rightarrow t5])) (THead k t t0) (THead k0 t t4) H8) in -(\lambda (_: (eq K k k0)).(let H12 \def (eq_ind_r T t4 (\lambda (t5: T).((eq -T t0 t5) \to (\forall (P0: Prop).P0))) H7 t0 H10) in (let H13 \def (eq_ind_r -T t4 (\lambda (t5: T).(or (eq T (THead k t t0) t5) ((eq T (THead k t t0) t5) -\to (\forall (P0: Prop).P0)))) H2 t0 H10) in (H12 (refl_equal T t0) P))))) -H9)))))) H6))) t3 H4))) (\lambda (H4: (((eq T t t3) \to (\forall (P: -Prop).P)))).(or_intror (eq T (THead k t t0) (THead k0 t3 t4)) ((eq T (THead k -t t0) (THead k0 t3 t4)) \to (\forall (P: Prop).P)) (\lambda (H5: (eq T (THead -k t t0) (THead k0 t3 t4))).(\lambda (P: Prop).(let H6 \def (f_equal T K -(\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) -\Rightarrow k | (TLRef _) \Rightarrow k | (THead k1 _ _) \Rightarrow k1])) -(THead k t t0) (THead k0 t3 t4) H5) in ((let H7 \def (f_equal T T (\lambda -(e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t -| (TLRef _) \Rightarrow t | (THead _ t5 _) \Rightarrow t5])) (THead k t t0) -(THead k0 t3 t4) H5) in ((let H8 \def (f_equal T T (\lambda (e: T).(match e -in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) -\Rightarrow t0 | (THead _ _ t5) \Rightarrow t5])) (THead k t t0) (THead k0 t3 -t4) H5) in (\lambda (H9: (eq T t t3)).(\lambda (_: (eq K k k0)).(let H11 \def -(eq_ind_r T t4 (\lambda (t5: T).(or (eq T (THead k t t0) t5) ((eq T (THead k -t t0) t5) \to (\forall (P0: Prop).P0)))) H2 t0 H8) in (let H12 \def (eq_ind_r -T t3 (\lambda (t5: T).((eq T t t5) \to (\forall (P0: Prop).P0))) H4 t H9) in -(let H13 \def (eq_ind_r T t3 (\lambda (t5: T).(or (eq T (THead k t t0) t5) -((eq T (THead k t t0) t5) \to (\forall (P0: Prop).P0)))) H1 t H9) in (H12 -(refl_equal T t) P))))))) H7)) H6)))))) H3)))))))) t2))))))) t1). -(* COMMENTS -Initial nodes: 2821 -END *) - -theorem binder_dec: - \forall (t: T).(or (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: -T).(eq T t (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall -(u: T).((eq T t (THead (Bind b) w u)) \to (\forall (P: Prop).P)))))) -\def - \lambda (t: T).(T_ind (\lambda (t0: T).(or (ex_3 B T T (\lambda (b: -B).(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w u)))))) -(\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t0 (THead (Bind b) w -u)) \to (\forall (P: Prop).P))))))) (\lambda (n: nat).(or_intror (ex_3 B T T -(\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T (TSort n) (THead (Bind -b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T (TSort n) -(THead (Bind b) w u)) \to (\forall (P: Prop).P))))) (\lambda (b: B).(\lambda -(w: T).(\lambda (u: T).(\lambda (H: (eq T (TSort n) (THead (Bind b) w -u))).(\lambda (P: Prop).(let H0 \def (eq_ind T (TSort n) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I -(THead (Bind b) w u) H) in (False_ind P H0))))))))) (\lambda (n: -nat).(or_intror (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: -T).(eq T (TLRef n) (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: -T).(\forall (u: T).((eq T (TLRef n) (THead (Bind b) w u)) \to (\forall (P: -Prop).P))))) (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(\lambda (H: (eq -T (TLRef n) (THead (Bind b) w u))).(\lambda (P: Prop).(let H0 \def (eq_ind T -(TLRef n) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) -\Rightarrow False])) I (THead (Bind b) w u) H) in (False_ind P H0))))))))) -(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t0: T).((or (ex_3 B T T -(\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w -u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t0 (THead -(Bind b) w u)) \to (\forall (P: Prop).P)))))) \to (\forall (t1: T).((or (ex_3 -B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t1 (THead (Bind -b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t1 (THead -(Bind b) w u)) \to (\forall (P: Prop).P)))))) \to (or (ex_3 B T T (\lambda -(b: B).(\lambda (w: T).(\lambda (u: T).(eq T (THead k0 t0 t1) (THead (Bind b) -w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T (THead k0 t0 -t1) (THead (Bind b) w u)) \to (\forall (P: Prop).P))))))))))) (\lambda (b: -B).(\lambda (t0: T).(\lambda (_: (or (ex_3 B T T (\lambda (b0: B).(\lambda -(w: T).(\lambda (u: T).(eq T t0 (THead (Bind b0) w u)))))) (\forall (b0: -B).(\forall (w: T).(\forall (u: T).((eq T t0 (THead (Bind b0) w u)) \to -(\forall (P: Prop).P))))))).(\lambda (t1: T).(\lambda (_: (or (ex_3 B T T -(\lambda (b0: B).(\lambda (w: T).(\lambda (u: T).(eq T t1 (THead (Bind b0) w -u)))))) (\forall (b0: B).(\forall (w: T).(\forall (u: T).((eq T t1 (THead -(Bind b0) w u)) \to (\forall (P: Prop).P))))))).(or_introl (ex_3 B T T -(\lambda (b0: B).(\lambda (w: T).(\lambda (u: T).(eq T (THead (Bind b) t0 t1) -(THead (Bind b0) w u)))))) (\forall (b0: B).(\forall (w: T).(\forall (u: -T).((eq T (THead (Bind b) t0 t1) (THead (Bind b0) w u)) \to (\forall (P: -Prop).P))))) (ex_3_intro B T T (\lambda (b0: B).(\lambda (w: T).(\lambda (u: -T).(eq T (THead (Bind b) t0 t1) (THead (Bind b0) w u))))) b t0 t1 (refl_equal -T (THead (Bind b) t0 t1))))))))) (\lambda (f: F).(\lambda (t0: T).(\lambda -(_: (or (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t0 -(THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: -T).((eq T t0 (THead (Bind b) w u)) \to (\forall (P: Prop).P))))))).(\lambda -(t1: T).(\lambda (_: (or (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda -(u: T).(eq T t1 (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: -T).(\forall (u: T).((eq T t1 (THead (Bind b) w u)) \to (\forall (P: -Prop).P))))))).(or_intror (ex_3 B T T (\lambda (b: B).(\lambda (w: -T).(\lambda (u: T).(eq T (THead (Flat f) t0 t1) (THead (Bind b) w u)))))) -(\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T (THead (Flat f) t0 t1) -(THead (Bind b) w u)) \to (\forall (P: Prop).P))))) (\lambda (b: B).(\lambda -(w: T).(\lambda (u: T).(\lambda (H1: (eq T (THead (Flat f) t0 t1) (THead -(Bind b) w u))).(\lambda (P: Prop).(let H2 \def (eq_ind T (THead (Flat f) t0 -t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort -_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _) -\Rightarrow (match k0 in K return (\lambda (_: K).Prop) with [(Bind _) -\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) w u) H1) -in (False_ind P H2))))))))))))) k)) t). -(* COMMENTS -Initial nodes: 1063 -END *) - -theorem abst_dec: - \forall (u: T).(\forall (v: T).(or (ex T (\lambda (t: T).(eq T u (THead -(Bind Abst) v t)))) (\forall (t: T).((eq T u (THead (Bind Abst) v t)) \to -(\forall (P: Prop).P))))) -\def - \lambda (u: T).(T_ind (\lambda (t: T).(\forall (v: T).(or (ex T (\lambda -(t0: T).(eq T t (THead (Bind Abst) v t0)))) (\forall (t0: T).((eq T t (THead -(Bind Abst) v t0)) \to (\forall (P: Prop).P)))))) (\lambda (n: nat).(\lambda -(v: T).(or_intror (ex T (\lambda (t: T).(eq T (TSort n) (THead (Bind Abst) v -t)))) (\forall (t: T).((eq T (TSort n) (THead (Bind Abst) v t)) \to (\forall -(P: Prop).P))) (\lambda (t: T).(\lambda (H: (eq T (TSort n) (THead (Bind -Abst) v t))).(\lambda (P: Prop).(let H0 \def (eq_ind T (TSort n) (\lambda -(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow -False])) I (THead (Bind Abst) v t) H) in (False_ind P H0)))))))) (\lambda (n: -nat).(\lambda (v: T).(or_intror (ex T (\lambda (t: T).(eq T (TLRef n) (THead -(Bind Abst) v t)))) (\forall (t: T).((eq T (TLRef n) (THead (Bind Abst) v t)) -\to (\forall (P: Prop).P))) (\lambda (t: T).(\lambda (H: (eq T (TLRef n) -(THead (Bind Abst) v t))).(\lambda (P: Prop).(let H0 \def (eq_ind T (TLRef n) -(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow -False])) I (THead (Bind Abst) v t) H) in (False_ind P H0)))))))) (\lambda (k: -K).(\lambda (t: T).(\lambda (_: ((\forall (v: T).(or (ex T (\lambda (t0: -T).(eq T t (THead (Bind Abst) v t0)))) (\forall (t0: T).((eq T t (THead (Bind -Abst) v t0)) \to (\forall (P: Prop).P))))))).(\lambda (t0: T).(\lambda (_: -((\forall (v: T).(or (ex T (\lambda (t1: T).(eq T t0 (THead (Bind Abst) v -t1)))) (\forall (t1: T).((eq T t0 (THead (Bind Abst) v t1)) \to (\forall (P: -Prop).P))))))).(\lambda (v: T).(let H_x \def (terms_props__kind_dec k (Bind -Abst)) in (let H1 \def H_x in (or_ind (eq K k (Bind Abst)) ((eq K k (Bind -Abst)) \to (\forall (P: Prop).P)) (or (ex T (\lambda (t1: T).(eq T (THead k t -t0) (THead (Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead k t t0) (THead -(Bind Abst) v t1)) \to (\forall (P: Prop).P)))) (\lambda (H2: (eq K k (Bind -Abst))).(eq_ind_r K (Bind Abst) (\lambda (k0: K).(or (ex T (\lambda (t1: -T).(eq T (THead k0 t t0) (THead (Bind Abst) v t1)))) (\forall (t1: T).((eq T -(THead k0 t t0) (THead (Bind Abst) v t1)) \to (\forall (P: Prop).P))))) (let -H_x0 \def (term_dec t v) in (let H3 \def H_x0 in (or_ind (eq T t v) ((eq T t -v) \to (\forall (P: Prop).P)) (or (ex T (\lambda (t1: T).(eq T (THead (Bind -Abst) t t0) (THead (Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead (Bind -Abst) t t0) (THead (Bind Abst) v t1)) \to (\forall (P: Prop).P)))) (\lambda -(H4: (eq T t v)).(eq_ind T t (\lambda (t1: T).(or (ex T (\lambda (t2: T).(eq -T (THead (Bind Abst) t t0) (THead (Bind Abst) t1 t2)))) (\forall (t2: T).((eq -T (THead (Bind Abst) t t0) (THead (Bind Abst) t1 t2)) \to (\forall (P: -Prop).P))))) (or_introl (ex T (\lambda (t1: T).(eq T (THead (Bind Abst) t t0) -(THead (Bind Abst) t t1)))) (\forall (t1: T).((eq T (THead (Bind Abst) t t0) -(THead (Bind Abst) t t1)) \to (\forall (P: Prop).P))) (ex_intro T (\lambda -(t1: T).(eq T (THead (Bind Abst) t t0) (THead (Bind Abst) t t1))) t0 -(refl_equal T (THead (Bind Abst) t t0)))) v H4)) (\lambda (H4: (((eq T t v) -\to (\forall (P: Prop).P)))).(or_intror (ex T (\lambda (t1: T).(eq T (THead -(Bind Abst) t t0) (THead (Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead -(Bind Abst) t t0) (THead (Bind Abst) v t1)) \to (\forall (P: Prop).P))) -(\lambda (t1: T).(\lambda (H5: (eq T (THead (Bind Abst) t t0) (THead (Bind -Abst) v t1))).(\lambda (P: Prop).(let H6 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t | -(TLRef _) \Rightarrow t | (THead _ t2 _) \Rightarrow t2])) (THead (Bind Abst) -t t0) (THead (Bind Abst) v t1) H5) in ((let H7 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | -(TLRef _) \Rightarrow t0 | (THead _ _ t2) \Rightarrow t2])) (THead (Bind -Abst) t t0) (THead (Bind Abst) v t1) H5) in (\lambda (H8: (eq T t v)).(H4 H8 -P))) H6))))))) H3))) k H2)) (\lambda (H2: (((eq K k (Bind Abst)) \to (\forall -(P: Prop).P)))).(or_intror (ex T (\lambda (t1: T).(eq T (THead k t t0) (THead -(Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead k t t0) (THead (Bind -Abst) v t1)) \to (\forall (P: Prop).P))) (\lambda (t1: T).(\lambda (H3: (eq T -(THead k t t0) (THead (Bind Abst) v t1))).(\lambda (P: Prop).(let H4 \def -(f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with -[(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) -\Rightarrow k0])) (THead k t t0) (THead (Bind Abst) v t1) H3) in ((let H5 -\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ t2 _) -\Rightarrow t2])) (THead k t t0) (THead (Bind Abst) v t1) H3) in ((let H6 -\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t2) -\Rightarrow t2])) (THead k t t0) (THead (Bind Abst) v t1) H3) in (\lambda (_: -(eq T t v)).(\lambda (H8: (eq K k (Bind Abst))).(H2 H8 P)))) H5)) H4))))))) -H1))))))))) u). -(* COMMENTS -Initial nodes: 1305 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/T/defs.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/T/defs.ma deleted file mode 100644 index 6ddbe6d0d..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/T/defs.ma +++ /dev/null @@ -1,43 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/preamble.ma". - -inductive B: Set \def -| Abbr: B -| Abst: B -| Void: B. - -inductive F: Set \def -| Appl: F -| Cast: F. - -inductive K: Set \def -| Bind: B \to K -| Flat: F \to K. - -inductive T: Set \def -| TSort: nat \to T -| TLRef: nat \to T -| THead: K \to (T \to (T \to T)). - -definition tweight: - T \to nat -\def - let rec tweight (t: T) on t: nat \def (match t with [(TSort _) \Rightarrow -(S O) | (TLRef _) \Rightarrow (S O) | (THead _ u t0) \Rightarrow (S (plus -(tweight u) (tweight t0)))]) in tweight. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/T/props.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/T/props.ma deleted file mode 100644 index faa9ed95d..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/T/props.ma +++ /dev/null @@ -1,111 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/T/defs.ma". - -theorem not_abbr_abst: - not (eq B Abbr Abst) -\def - \lambda (H: (eq B Abbr Abst)).(let H0 \def (eq_ind B Abbr (\lambda (ee: -B).(match ee in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow True | -Abst \Rightarrow False | Void \Rightarrow False])) I Abst H) in (False_ind -False H0)). -(* COMMENTS -Initial nodes: 34 -END *) - -theorem not_void_abst: - not (eq B Void Abst) -\def - \lambda (H: (eq B Void Abst)).(let H0 \def (eq_ind B Void (\lambda (ee: -B).(match ee in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | -Abst \Rightarrow False | Void \Rightarrow True])) I Abst H) in (False_ind -False H0)). -(* COMMENTS -Initial nodes: 34 -END *) - -theorem not_abbr_void: - not (eq B Abbr Void) -\def - \lambda (H: (eq B Abbr Void)).(let H0 \def (eq_ind B Abbr (\lambda (ee: -B).(match ee in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow True | -Abst \Rightarrow False | Void \Rightarrow False])) I Void H) in (False_ind -False H0)). -(* COMMENTS -Initial nodes: 34 -END *) - -theorem not_abst_void: - not (eq B Abst Void) -\def - \lambda (H: (eq B Abst Void)).(let H0 \def (eq_ind B Abst (\lambda (ee: -B).(match ee in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | -Abst \Rightarrow True | Void \Rightarrow False])) I Void H) in (False_ind -False H0)). -(* COMMENTS -Initial nodes: 34 -END *) - -theorem thead_x_y_y: - \forall (k: K).(\forall (v: T).(\forall (t: T).((eq T (THead k v t) t) \to -(\forall (P: Prop).P)))) -\def - \lambda (k: K).(\lambda (v: T).(\lambda (t: T).(T_ind (\lambda (t0: T).((eq -T (THead k v t0) t0) \to (\forall (P: Prop).P))) (\lambda (n: nat).(\lambda -(H: (eq T (THead k v (TSort n)) (TSort n))).(\lambda (P: Prop).(let H0 \def -(eq_ind T (THead k v (TSort n)) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H) in -(False_ind P H0))))) (\lambda (n: nat).(\lambda (H: (eq T (THead k v (TLRef -n)) (TLRef n))).(\lambda (P: Prop).(let H0 \def (eq_ind T (THead k v (TLRef -n)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort -_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) -\Rightarrow True])) I (TLRef n) H) in (False_ind P H0))))) (\lambda (k0: -K).(\lambda (t0: T).(\lambda (_: (((eq T (THead k v t0) t0) \to (\forall (P: -Prop).P)))).(\lambda (t1: T).(\lambda (H0: (((eq T (THead k v t1) t1) \to -(\forall (P: Prop).P)))).(\lambda (H1: (eq T (THead k v (THead k0 t0 t1)) -(THead k0 t0 t1))).(\lambda (P: Prop).(let H2 \def (f_equal T K (\lambda (e: -T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | -(TLRef _) \Rightarrow k | (THead k1 _ _) \Rightarrow k1])) (THead k v (THead -k0 t0 t1)) (THead k0 t0 t1) H1) in ((let H3 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v | -(TLRef _) \Rightarrow v | (THead _ t2 _) \Rightarrow t2])) (THead k v (THead -k0 t0 t1)) (THead k0 t0 t1) H1) in ((let H4 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow (THead -k0 t0 t1) | (TLRef _) \Rightarrow (THead k0 t0 t1) | (THead _ _ t2) -\Rightarrow t2])) (THead k v (THead k0 t0 t1)) (THead k0 t0 t1) H1) in -(\lambda (H5: (eq T v t0)).(\lambda (H6: (eq K k k0)).(let H7 \def (eq_ind T -v (\lambda (t2: T).((eq T (THead k t2 t1) t1) \to (\forall (P0: Prop).P0))) -H0 t0 H5) in (let H8 \def (eq_ind K k (\lambda (k1: K).((eq T (THead k1 t0 -t1) t1) \to (\forall (P0: Prop).P0))) H7 k0 H6) in (H8 H4 P)))))) H3)) -H2))))))))) t))). -(* COMMENTS -Initial nodes: 461 -END *) - -theorem tweight_lt: - \forall (t: T).(lt O (tweight t)) -\def - \lambda (t: T).(T_ind (\lambda (t0: T).(lt O (tweight t0))) (\lambda (_: -nat).(le_n (S O))) (\lambda (_: nat).(le_n (S O))) (\lambda (_: K).(\lambda -(t0: T).(\lambda (H: (lt O (tweight t0))).(\lambda (t1: T).(\lambda (_: (lt O -(tweight t1))).(le_S (S O) (plus (tweight t0) (tweight t1)) (le_plus_trans (S -O) (tweight t0) (tweight t1) H))))))) t). -(* COMMENTS -Initial nodes: 85 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/aplus/defs.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/aplus/defs.ma deleted file mode 100644 index 4095b163b..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/aplus/defs.ma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/asucc/defs.ma". - -definition aplus: - G \to (A \to (nat \to A)) -\def - let rec aplus (g: G) (a: A) (n: nat) on n: A \def (match n with [O -\Rightarrow a | (S n0) \Rightarrow (asucc g (aplus g a n0))]) in aplus. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/aplus/props.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/aplus/props.ma deleted file mode 100644 index 94bb9a069..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/aplus/props.ma +++ /dev/null @@ -1,282 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/aplus/defs.ma". - -include "Basic-1/next_plus/props.ma". - -theorem aplus_reg_r: - \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (h1: nat).(\forall -(h2: nat).((eq A (aplus g a1 h1) (aplus g a2 h2)) \to (\forall (h: nat).(eq A -(aplus g a1 (plus h h1)) (aplus g a2 (plus h h2))))))))) -\def - \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (h1: nat).(\lambda -(h2: nat).(\lambda (H: (eq A (aplus g a1 h1) (aplus g a2 h2))).(\lambda (h: -nat).(nat_ind (\lambda (n: nat).(eq A (aplus g a1 (plus n h1)) (aplus g a2 -(plus n h2)))) H (\lambda (n: nat).(\lambda (H0: (eq A (aplus g a1 (plus n -h1)) (aplus g a2 (plus n h2)))).(f_equal2 G A A asucc g g (aplus g a1 (plus n -h1)) (aplus g a2 (plus n h2)) (refl_equal G g) H0))) h))))))). -(* COMMENTS -Initial nodes: 143 -END *) - -theorem aplus_assoc: - \forall (g: G).(\forall (a: A).(\forall (h1: nat).(\forall (h2: nat).(eq A -(aplus g (aplus g a h1) h2) (aplus g a (plus h1 h2)))))) -\def - \lambda (g: G).(\lambda (a: A).(\lambda (h1: nat).(nat_ind (\lambda (n: -nat).(\forall (h2: nat).(eq A (aplus g (aplus g a n) h2) (aplus g a (plus n -h2))))) (\lambda (h2: nat).(refl_equal A (aplus g a h2))) (\lambda (n: -nat).(\lambda (_: ((\forall (h2: nat).(eq A (aplus g (aplus g a n) h2) (aplus -g a (plus n h2)))))).(\lambda (h2: nat).(nat_ind (\lambda (n0: nat).(eq A -(aplus g (asucc g (aplus g a n)) n0) (asucc g (aplus g a (plus n n0))))) -(eq_ind nat n (\lambda (n0: nat).(eq A (asucc g (aplus g a n)) (asucc g -(aplus g a n0)))) (refl_equal A (asucc g (aplus g a n))) (plus n O) (plus_n_O -n)) (\lambda (n0: nat).(\lambda (H0: (eq A (aplus g (asucc g (aplus g a n)) -n0) (asucc g (aplus g a (plus n n0))))).(eq_ind nat (S (plus n n0)) (\lambda -(n1: nat).(eq A (asucc g (aplus g (asucc g (aplus g a n)) n0)) (asucc g -(aplus g a n1)))) (f_equal2 G A A asucc g g (aplus g (asucc g (aplus g a n)) -n0) (asucc g (aplus g a (plus n n0))) (refl_equal G g) H0) (plus n (S n0)) -(plus_n_Sm n n0)))) h2)))) h1))). -(* COMMENTS -Initial nodes: 361 -END *) - -theorem aplus_asucc: - \forall (g: G).(\forall (h: nat).(\forall (a: A).(eq A (aplus g (asucc g a) -h) (asucc g (aplus g a h))))) -\def - \lambda (g: G).(\lambda (h: nat).(\lambda (a: A).(eq_ind_r A (aplus g a -(plus (S O) h)) (\lambda (a0: A).(eq A a0 (asucc g (aplus g a h)))) -(refl_equal A (asucc g (aplus g a h))) (aplus g (aplus g a (S O)) h) -(aplus_assoc g a (S O) h)))). -(* COMMENTS -Initial nodes: 87 -END *) - -theorem aplus_sort_O_S_simpl: - \forall (g: G).(\forall (n: nat).(\forall (k: nat).(eq A (aplus g (ASort O -n) (S k)) (aplus g (ASort O (next g n)) k)))) -\def - \lambda (g: G).(\lambda (n: nat).(\lambda (k: nat).(eq_ind A (aplus g (asucc -g (ASort O n)) k) (\lambda (a: A).(eq A a (aplus g (ASort O (next g n)) k))) -(refl_equal A (aplus g (ASort O (next g n)) k)) (asucc g (aplus g (ASort O n) -k)) (aplus_asucc g k (ASort O n))))). -(* COMMENTS -Initial nodes: 97 -END *) - -theorem aplus_sort_S_S_simpl: - \forall (g: G).(\forall (n: nat).(\forall (h: nat).(\forall (k: nat).(eq A -(aplus g (ASort (S h) n) (S k)) (aplus g (ASort h n) k))))) -\def - \lambda (g: G).(\lambda (n: nat).(\lambda (h: nat).(\lambda (k: nat).(eq_ind -A (aplus g (asucc g (ASort (S h) n)) k) (\lambda (a: A).(eq A a (aplus g -(ASort h n) k))) (refl_equal A (aplus g (ASort h n) k)) (asucc g (aplus g -(ASort (S h) n) k)) (aplus_asucc g k (ASort (S h) n)))))). -(* COMMENTS -Initial nodes: 97 -END *) - -theorem aplus_asort_O_simpl: - \forall (g: G).(\forall (h: nat).(\forall (n: nat).(eq A (aplus g (ASort O -n) h) (ASort O (next_plus g n h))))) -\def - \lambda (g: G).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall (n0: -nat).(eq A (aplus g (ASort O n0) n) (ASort O (next_plus g n0 n))))) (\lambda -(n: nat).(refl_equal A (ASort O n))) (\lambda (n: nat).(\lambda (H: ((\forall -(n0: nat).(eq A (aplus g (ASort O n0) n) (ASort O (next_plus g n0 -n)))))).(\lambda (n0: nat).(eq_ind A (aplus g (asucc g (ASort O n0)) n) -(\lambda (a: A).(eq A a (ASort O (next g (next_plus g n0 n))))) (eq_ind nat -(next_plus g (next g n0) n) (\lambda (n1: nat).(eq A (aplus g (ASort O (next -g n0)) n) (ASort O n1))) (H (next g n0)) (next g (next_plus g n0 n)) -(next_plus_next g n0 n)) (asucc g (aplus g (ASort O n0) n)) (aplus_asucc g n -(ASort O n0)))))) h)). -(* COMMENTS -Initial nodes: 229 -END *) - -theorem aplus_asort_le_simpl: - \forall (g: G).(\forall (h: nat).(\forall (k: nat).(\forall (n: nat).((le h -k) \to (eq A (aplus g (ASort k n) h) (ASort (minus k h) n)))))) -\def - \lambda (g: G).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall (k: -nat).(\forall (n0: nat).((le n k) \to (eq A (aplus g (ASort k n0) n) (ASort -(minus k n) n0)))))) (\lambda (k: nat).(\lambda (n: nat).(\lambda (_: (le O -k)).(eq_ind nat k (\lambda (n0: nat).(eq A (ASort k n) (ASort n0 n))) -(refl_equal A (ASort k n)) (minus k O) (minus_n_O k))))) (\lambda (h0: -nat).(\lambda (H: ((\forall (k: nat).(\forall (n: nat).((le h0 k) \to (eq A -(aplus g (ASort k n) h0) (ASort (minus k h0) n))))))).(\lambda (k: -nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).((le (S h0) n) \to (eq A -(asucc g (aplus g (ASort n n0) h0)) (ASort (minus n (S h0)) n0))))) (\lambda -(n: nat).(\lambda (H0: (le (S h0) O)).(ex2_ind nat (\lambda (n0: nat).(eq nat -O (S n0))) (\lambda (n0: nat).(le h0 n0)) (eq A (asucc g (aplus g (ASort O n) -h0)) (ASort (minus O (S h0)) n)) (\lambda (x: nat).(\lambda (H1: (eq nat O (S -x))).(\lambda (_: (le h0 x)).(let H3 \def (eq_ind nat O (\lambda (ee: -nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True -| (S _) \Rightarrow False])) I (S x) H1) in (False_ind (eq A (asucc g (aplus -g (ASort O n) h0)) (ASort (minus O (S h0)) n)) H3))))) (le_gen_S h0 O H0)))) -(\lambda (n: nat).(\lambda (_: ((\forall (n0: nat).((le (S h0) n) \to (eq A -(asucc g (aplus g (ASort n n0) h0)) (ASort (minus n (S h0)) n0)))))).(\lambda -(n0: nat).(\lambda (H1: (le (S h0) (S n))).(eq_ind A (aplus g (asucc g (ASort -(S n) n0)) h0) (\lambda (a: A).(eq A a (ASort (minus (S n) (S h0)) n0))) (H n -n0 (le_S_n h0 n H1)) (asucc g (aplus g (ASort (S n) n0) h0)) (aplus_asucc g -h0 (ASort (S n) n0))))))) k)))) h)). -(* COMMENTS -Initial nodes: 484 -END *) - -theorem aplus_asort_simpl: - \forall (g: G).(\forall (h: nat).(\forall (k: nat).(\forall (n: nat).(eq A -(aplus g (ASort k n) h) (ASort (minus k h) (next_plus g n (minus h k))))))) -\def - \lambda (g: G).(\lambda (h: nat).(\lambda (k: nat).(\lambda (n: -nat).(lt_le_e k h (eq A (aplus g (ASort k n) h) (ASort (minus k h) (next_plus -g n (minus h k)))) (\lambda (H: (lt k h)).(eq_ind_r nat (plus k (minus h k)) -(\lambda (n0: nat).(eq A (aplus g (ASort k n) n0) (ASort (minus k h) -(next_plus g n (minus h k))))) (eq_ind A (aplus g (aplus g (ASort k n) k) -(minus h k)) (\lambda (a: A).(eq A a (ASort (minus k h) (next_plus g n (minus -h k))))) (eq_ind_r A (ASort (minus k k) n) (\lambda (a: A).(eq A (aplus g a -(minus h k)) (ASort (minus k h) (next_plus g n (minus h k))))) (eq_ind nat O -(\lambda (n0: nat).(eq A (aplus g (ASort n0 n) (minus h k)) (ASort (minus k -h) (next_plus g n (minus h k))))) (eq_ind_r nat O (\lambda (n0: nat).(eq A -(aplus g (ASort O n) (minus h k)) (ASort n0 (next_plus g n (minus h k))))) -(aplus_asort_O_simpl g (minus h k) n) (minus k h) (O_minus k h (le_S_n k h -(le_S (S k) h H)))) (minus k k) (minus_n_n k)) (aplus g (ASort k n) k) -(aplus_asort_le_simpl g k k n (le_n k))) (aplus g (ASort k n) (plus k (minus -h k))) (aplus_assoc g (ASort k n) k (minus h k))) h (le_plus_minus k h -(le_S_n k h (le_S (S k) h H))))) (\lambda (H: (le h k)).(eq_ind_r A (ASort -(minus k h) n) (\lambda (a: A).(eq A a (ASort (minus k h) (next_plus g n -(minus h k))))) (eq_ind_r nat O (\lambda (n0: nat).(eq A (ASort (minus k h) -n) (ASort (minus k h) (next_plus g n n0)))) (refl_equal A (ASort (minus k h) -(next_plus g n O))) (minus h k) (O_minus h k H)) (aplus g (ASort k n) h) -(aplus_asort_le_simpl g h k n H))))))). -(* COMMENTS -Initial nodes: 587 -END *) - -theorem aplus_ahead_simpl: - \forall (g: G).(\forall (h: nat).(\forall (a1: A).(\forall (a2: A).(eq A -(aplus g (AHead a1 a2) h) (AHead a1 (aplus g a2 h)))))) -\def - \lambda (g: G).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall (a1: -A).(\forall (a2: A).(eq A (aplus g (AHead a1 a2) n) (AHead a1 (aplus g a2 -n)))))) (\lambda (a1: A).(\lambda (a2: A).(refl_equal A (AHead a1 a2)))) -(\lambda (n: nat).(\lambda (H: ((\forall (a1: A).(\forall (a2: A).(eq A -(aplus g (AHead a1 a2) n) (AHead a1 (aplus g a2 n))))))).(\lambda (a1: -A).(\lambda (a2: A).(eq_ind A (aplus g (asucc g (AHead a1 a2)) n) (\lambda -(a: A).(eq A a (AHead a1 (asucc g (aplus g a2 n))))) (eq_ind A (aplus g -(asucc g a2) n) (\lambda (a: A).(eq A (aplus g (asucc g (AHead a1 a2)) n) -(AHead a1 a))) (H a1 (asucc g a2)) (asucc g (aplus g a2 n)) (aplus_asucc g n -a2)) (asucc g (aplus g (AHead a1 a2) n)) (aplus_asucc g n (AHead a1 a2))))))) -h)). -(* COMMENTS -Initial nodes: 239 -END *) - -theorem aplus_asucc_false: - \forall (g: G).(\forall (a: A).(\forall (h: nat).((eq A (aplus g (asucc g a) -h) a) \to (\forall (P: Prop).P)))) -\def - \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(\forall (h: -nat).((eq A (aplus g (asucc g a0) h) a0) \to (\forall (P: Prop).P)))) -(\lambda (n: nat).(\lambda (n0: nat).(\lambda (h: nat).(\lambda (H: (eq A -(aplus g (match n with [O \Rightarrow (ASort O (next g n0)) | (S h0) -\Rightarrow (ASort h0 n0)]) h) (ASort n n0))).(\lambda (P: Prop).(nat_ind -(\lambda (n1: nat).((eq A (aplus g (match n1 with [O \Rightarrow (ASort O -(next g n0)) | (S h0) \Rightarrow (ASort h0 n0)]) h) (ASort n1 n0)) \to P)) -(\lambda (H0: (eq A (aplus g (ASort O (next g n0)) h) (ASort O n0))).(let H1 -\def (eq_ind A (aplus g (ASort O (next g n0)) h) (\lambda (a0: A).(eq A a0 -(ASort O n0))) H0 (ASort (minus O h) (next_plus g (next g n0) (minus h O))) -(aplus_asort_simpl g h O (next g n0))) in (let H2 \def (f_equal A nat -(\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort _ n1) -\Rightarrow n1 | (AHead _ _) \Rightarrow ((let rec next_plus (g0: G) (n1: -nat) (i: nat) on i: nat \def (match i with [O \Rightarrow n1 | (S i0) -\Rightarrow (next g0 (next_plus g0 n1 i0))]) in next_plus) g (next g n0) -(minus h O))])) (ASort (minus O h) (next_plus g (next g n0) (minus h O))) -(ASort O n0) H1) in (let H3 \def (eq_ind_r nat (minus h O) (\lambda (n1: -nat).(eq nat (next_plus g (next g n0) n1) n0)) H2 h (minus_n_O h)) in -(le_lt_false (next_plus g (next g n0) h) n0 (eq_ind nat (next_plus g (next g -n0) h) (\lambda (n1: nat).(le (next_plus g (next g n0) h) n1)) (le_n -(next_plus g (next g n0) h)) n0 H3) (next_plus_lt g h n0) P))))) (\lambda -(n1: nat).(\lambda (_: (((eq A (aplus g (match n1 with [O \Rightarrow (ASort -O (next g n0)) | (S h0) \Rightarrow (ASort h0 n0)]) h) (ASort n1 n0)) \to -P))).(\lambda (H0: (eq A (aplus g (ASort n1 n0) h) (ASort (S n1) n0))).(let -H1 \def (eq_ind A (aplus g (ASort n1 n0) h) (\lambda (a0: A).(eq A a0 (ASort -(S n1) n0))) H0 (ASort (minus n1 h) (next_plus g n0 (minus h n1))) -(aplus_asort_simpl g h n1 n0)) in (let H2 \def (f_equal A nat (\lambda (e: -A).(match e in A return (\lambda (_: A).nat) with [(ASort n2 _) \Rightarrow -n2 | (AHead _ _) \Rightarrow ((let rec minus (n2: nat) on n2: (nat \to nat) -\def (\lambda (m: nat).(match n2 with [O \Rightarrow O | (S k) \Rightarrow -(match m with [O \Rightarrow (S k) | (S l) \Rightarrow (minus k l)])])) in -minus) n1 h)])) (ASort (minus n1 h) (next_plus g n0 (minus h n1))) (ASort (S -n1) n0) H1) in ((let H3 \def (f_equal A nat (\lambda (e: A).(match e in A -return (\lambda (_: A).nat) with [(ASort _ n2) \Rightarrow n2 | (AHead _ _) -\Rightarrow ((let rec next_plus (g0: G) (n2: nat) (i: nat) on i: nat \def -(match i with [O \Rightarrow n2 | (S i0) \Rightarrow (next g0 (next_plus g0 -n2 i0))]) in next_plus) g n0 (minus h n1))])) (ASort (minus n1 h) (next_plus -g n0 (minus h n1))) (ASort (S n1) n0) H1) in (\lambda (H4: (eq nat (minus n1 -h) (S n1))).(le_Sx_x n1 (eq_ind nat (minus n1 h) (\lambda (n2: nat).(le n2 -n1)) (minus_le n1 h) (S n1) H4) P))) H2)))))) n H)))))) (\lambda (a0: -A).(\lambda (_: ((\forall (h: nat).((eq A (aplus g (asucc g a0) h) a0) \to -(\forall (P: Prop).P))))).(\lambda (a1: A).(\lambda (H0: ((\forall (h: -nat).((eq A (aplus g (asucc g a1) h) a1) \to (\forall (P: -Prop).P))))).(\lambda (h: nat).(\lambda (H1: (eq A (aplus g (AHead a0 (asucc -g a1)) h) (AHead a0 a1))).(\lambda (P: Prop).(let H2 \def (eq_ind A (aplus g -(AHead a0 (asucc g a1)) h) (\lambda (a2: A).(eq A a2 (AHead a0 a1))) H1 -(AHead a0 (aplus g (asucc g a1) h)) (aplus_ahead_simpl g h a0 (asucc g a1))) -in (let H3 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda -(_: A).A) with [(ASort _ _) \Rightarrow ((let rec aplus (g0: G) (a2: A) (n: -nat) on n: A \def (match n with [O \Rightarrow a2 | (S n0) \Rightarrow (asucc -g0 (aplus g0 a2 n0))]) in aplus) g (asucc g a1) h) | (AHead _ a2) \Rightarrow -a2])) (AHead a0 (aplus g (asucc g a1) h)) (AHead a0 a1) H2) in (H0 h H3 -P)))))))))) a)). -(* COMMENTS -Initial nodes: 977 -END *) - -theorem aplus_inj: - \forall (g: G).(\forall (h1: nat).(\forall (h2: nat).(\forall (a: A).((eq A -(aplus g a h1) (aplus g a h2)) \to (eq nat h1 h2))))) -\def - \lambda (g: G).(\lambda (h1: nat).(nat_ind (\lambda (n: nat).(\forall (h2: -nat).(\forall (a: A).((eq A (aplus g a n) (aplus g a h2)) \to (eq nat n -h2))))) (\lambda (h2: nat).(nat_ind (\lambda (n: nat).(\forall (a: A).((eq A -(aplus g a O) (aplus g a n)) \to (eq nat O n)))) (\lambda (a: A).(\lambda (_: -(eq A a a)).(refl_equal nat O))) (\lambda (n: nat).(\lambda (_: ((\forall (a: -A).((eq A a (aplus g a n)) \to (eq nat O n))))).(\lambda (a: A).(\lambda (H0: -(eq A a (asucc g (aplus g a n)))).(let H1 \def (eq_ind_r A (asucc g (aplus g -a n)) (\lambda (a0: A).(eq A a a0)) H0 (aplus g (asucc g a) n) (aplus_asucc g -n a)) in (aplus_asucc_false g a n (sym_eq A a (aplus g (asucc g a) n) H1) (eq -nat O (S n)))))))) h2)) (\lambda (n: nat).(\lambda (H: ((\forall (h2: -nat).(\forall (a: A).((eq A (aplus g a n) (aplus g a h2)) \to (eq nat n -h2)))))).(\lambda (h2: nat).(nat_ind (\lambda (n0: nat).(\forall (a: A).((eq -A (aplus g a (S n)) (aplus g a n0)) \to (eq nat (S n) n0)))) (\lambda (a: -A).(\lambda (H0: (eq A (asucc g (aplus g a n)) a)).(let H1 \def (eq_ind_r A -(asucc g (aplus g a n)) (\lambda (a0: A).(eq A a0 a)) H0 (aplus g (asucc g a) -n) (aplus_asucc g n a)) in (aplus_asucc_false g a n H1 (eq nat (S n) O))))) -(\lambda (n0: nat).(\lambda (_: ((\forall (a: A).((eq A (asucc g (aplus g a -n)) (aplus g a n0)) \to (eq nat (S n) n0))))).(\lambda (a: A).(\lambda (H1: -(eq A (asucc g (aplus g a n)) (asucc g (aplus g a n0)))).(let H2 \def -(eq_ind_r A (asucc g (aplus g a n)) (\lambda (a0: A).(eq A a0 (asucc g (aplus -g a n0)))) H1 (aplus g (asucc g a) n) (aplus_asucc g n a)) in (let H3 \def -(eq_ind_r A (asucc g (aplus g a n0)) (\lambda (a0: A).(eq A (aplus g (asucc g -a) n) a0)) H2 (aplus g (asucc g a) n0) (aplus_asucc g n0 a)) in (f_equal nat -nat S n n0 (H n0 (asucc g a) H3)))))))) h2)))) h1)). -(* COMMENTS -Initial nodes: 599 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/app/defs.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/app/defs.ma deleted file mode 100644 index 9cf1e37d1..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/app/defs.ma +++ /dev/null @@ -1,31 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/C/defs.ma". - -definition cbk: - C \to nat -\def - let rec cbk (c: C) on c: nat \def (match c with [(CSort m) \Rightarrow m | -(CHead c0 _ _) \Rightarrow (cbk c0)]) in cbk. - -definition app1: - C \to (T \to T) -\def - let rec app1 (c: C) on c: (T \to T) \def (\lambda (t: T).(match c with -[(CSort _) \Rightarrow t | (CHead c0 k u) \Rightarrow (app1 c0 (THead k u -t))])) in app1. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/aprem/defs.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/aprem/defs.ma deleted file mode 100644 index 78b49e920..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/aprem/defs.ma +++ /dev/null @@ -1,23 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/A/defs.ma". - -inductive aprem: nat \to (A \to (A \to Prop)) \def -| aprem_zero: \forall (a1: A).(\forall (a2: A).(aprem O (AHead a1 a2) a1)) -| aprem_succ: \forall (a2: A).(\forall (a: A).(\forall (i: nat).((aprem i a2 -a) \to (\forall (a1: A).(aprem (S i) (AHead a1 a2) a))))). - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/aprem/fwd.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/aprem/fwd.ma deleted file mode 100644 index ed48846a8..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/aprem/fwd.ma +++ /dev/null @@ -1,120 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/aprem/defs.ma". - -theorem aprem_gen_sort: - \forall (x: A).(\forall (i: nat).(\forall (h: nat).(\forall (n: nat).((aprem -i (ASort h n) x) \to False)))) -\def - \lambda (x: A).(\lambda (i: nat).(\lambda (h: nat).(\lambda (n: -nat).(\lambda (H: (aprem i (ASort h n) x)).(insert_eq A (ASort h n) (\lambda -(a: A).(aprem i a x)) (\lambda (_: A).False) (\lambda (y: A).(\lambda (H0: -(aprem i y x)).(aprem_ind (\lambda (_: nat).(\lambda (a: A).(\lambda (_: -A).((eq A a (ASort h n)) \to False)))) (\lambda (a1: A).(\lambda (a2: -A).(\lambda (H1: (eq A (AHead a1 a2) (ASort h n))).(let H2 \def (eq_ind A -(AHead a1 a2) (\lambda (ee: A).(match ee in A return (\lambda (_: A).Prop) -with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I -(ASort h n) H1) in (False_ind False H2))))) (\lambda (a2: A).(\lambda (a: -A).(\lambda (i0: nat).(\lambda (_: (aprem i0 a2 a)).(\lambda (_: (((eq A a2 -(ASort h n)) \to False))).(\lambda (a1: A).(\lambda (H3: (eq A (AHead a1 a2) -(ASort h n))).(let H4 \def (eq_ind A (AHead a1 a2) (\lambda (ee: A).(match ee -in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | -(AHead _ _) \Rightarrow True])) I (ASort h n) H3) in (False_ind False -H4))))))))) i y x H0))) H))))). -(* COMMENTS -Initial nodes: 227 -END *) - -theorem aprem_gen_head_O: - \forall (a1: A).(\forall (a2: A).(\forall (x: A).((aprem O (AHead a1 a2) x) -\to (eq A x a1)))) -\def - \lambda (a1: A).(\lambda (a2: A).(\lambda (x: A).(\lambda (H: (aprem O -(AHead a1 a2) x)).(insert_eq A (AHead a1 a2) (\lambda (a: A).(aprem O a x)) -(\lambda (_: A).(eq A x a1)) (\lambda (y: A).(\lambda (H0: (aprem O y -x)).(insert_eq nat O (\lambda (n: nat).(aprem n y x)) (\lambda (_: nat).((eq -A y (AHead a1 a2)) \to (eq A x a1))) (\lambda (y0: nat).(\lambda (H1: (aprem -y0 y x)).(aprem_ind (\lambda (n: nat).(\lambda (a: A).(\lambda (a0: A).((eq -nat n O) \to ((eq A a (AHead a1 a2)) \to (eq A a0 a1)))))) (\lambda (a0: -A).(\lambda (a3: A).(\lambda (_: (eq nat O O)).(\lambda (H3: (eq A (AHead a0 -a3) (AHead a1 a2))).(let H4 \def (f_equal A A (\lambda (e: A).(match e in A -return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead a _) -\Rightarrow a])) (AHead a0 a3) (AHead a1 a2) H3) in ((let H5 \def (f_equal A -A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) -\Rightarrow a3 | (AHead _ a) \Rightarrow a])) (AHead a0 a3) (AHead a1 a2) H3) -in (\lambda (H6: (eq A a0 a1)).H6)) H4)))))) (\lambda (a0: A).(\lambda (a: -A).(\lambda (i: nat).(\lambda (H2: (aprem i a0 a)).(\lambda (H3: (((eq nat i -O) \to ((eq A a0 (AHead a1 a2)) \to (eq A a a1))))).(\lambda (a3: A).(\lambda -(H4: (eq nat (S i) O)).(\lambda (H5: (eq A (AHead a3 a0) (AHead a1 a2))).(let -H6 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) -with [(ASort _ _) \Rightarrow a3 | (AHead a4 _) \Rightarrow a4])) (AHead a3 -a0) (AHead a1 a2) H5) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e -in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead _ -a4) \Rightarrow a4])) (AHead a3 a0) (AHead a1 a2) H5) in (\lambda (_: (eq A -a3 a1)).(let H9 \def (eq_ind A a0 (\lambda (a4: A).((eq nat i O) \to ((eq A -a4 (AHead a1 a2)) \to (eq A a a1)))) H3 a2 H7) in (let H10 \def (eq_ind A a0 -(\lambda (a4: A).(aprem i a4 a)) H2 a2 H7) in (let H11 \def (eq_ind nat (S i) -(\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O -\Rightarrow False | (S _) \Rightarrow True])) I O H4) in (False_ind (eq A a -a1) H11)))))) H6)))))))))) y0 y x H1))) H0))) H)))). -(* COMMENTS -Initial nodes: 500 -END *) - -theorem aprem_gen_head_S: - \forall (a1: A).(\forall (a2: A).(\forall (x: A).(\forall (i: nat).((aprem -(S i) (AHead a1 a2) x) \to (aprem i a2 x))))) -\def - \lambda (a1: A).(\lambda (a2: A).(\lambda (x: A).(\lambda (i: nat).(\lambda -(H: (aprem (S i) (AHead a1 a2) x)).(insert_eq A (AHead a1 a2) (\lambda (a: -A).(aprem (S i) a x)) (\lambda (_: A).(aprem i a2 x)) (\lambda (y: -A).(\lambda (H0: (aprem (S i) y x)).(insert_eq nat (S i) (\lambda (n: -nat).(aprem n y x)) (\lambda (_: nat).((eq A y (AHead a1 a2)) \to (aprem i a2 -x))) (\lambda (y0: nat).(\lambda (H1: (aprem y0 y x)).(aprem_ind (\lambda (n: -nat).(\lambda (a: A).(\lambda (a0: A).((eq nat n (S i)) \to ((eq A a (AHead -a1 a2)) \to (aprem i a2 a0)))))) (\lambda (a0: A).(\lambda (a3: A).(\lambda -(H2: (eq nat O (S i))).(\lambda (H3: (eq A (AHead a0 a3) (AHead a1 a2))).(let -H4 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) -with [(ASort _ _) \Rightarrow a0 | (AHead a _) \Rightarrow a])) (AHead a0 a3) -(AHead a1 a2) H3) in ((let H5 \def (f_equal A A (\lambda (e: A).(match e in A -return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a3 | (AHead _ a) -\Rightarrow a])) (AHead a0 a3) (AHead a1 a2) H3) in (\lambda (H6: (eq A a0 -a1)).(eq_ind_r A a1 (\lambda (a: A).(aprem i a2 a)) (let H7 \def (eq_ind nat -O (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O -\Rightarrow True | (S _) \Rightarrow False])) I (S i) H2) in (False_ind -(aprem i a2 a1) H7)) a0 H6))) H4)))))) (\lambda (a0: A).(\lambda (a: -A).(\lambda (i0: nat).(\lambda (H2: (aprem i0 a0 a)).(\lambda (H3: (((eq nat -i0 (S i)) \to ((eq A a0 (AHead a1 a2)) \to (aprem i a2 a))))).(\lambda (a3: -A).(\lambda (H4: (eq nat (S i0) (S i))).(\lambda (H5: (eq A (AHead a3 a0) -(AHead a1 a2))).(let H6 \def (f_equal A A (\lambda (e: A).(match e in A -return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a3 | (AHead a4 _) -\Rightarrow a4])) (AHead a3 a0) (AHead a1 a2) H5) in ((let H7 \def (f_equal A -A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) -\Rightarrow a0 | (AHead _ a4) \Rightarrow a4])) (AHead a3 a0) (AHead a1 a2) -H5) in (\lambda (_: (eq A a3 a1)).(let H9 \def (eq_ind A a0 (\lambda (a4: -A).((eq nat i0 (S i)) \to ((eq A a4 (AHead a1 a2)) \to (aprem i a2 a)))) H3 -a2 H7) in (let H10 \def (eq_ind A a0 (\lambda (a4: A).(aprem i0 a4 a)) H2 a2 -H7) in (let H11 \def (f_equal nat nat (\lambda (e: nat).(match e in nat -return (\lambda (_: nat).nat) with [O \Rightarrow i0 | (S n) \Rightarrow n])) -(S i0) (S i) H4) in (let H12 \def (eq_ind nat i0 (\lambda (n: nat).((eq nat n -(S i)) \to ((eq A a2 (AHead a1 a2)) \to (aprem i a2 a)))) H9 i H11) in (let -H13 \def (eq_ind nat i0 (\lambda (n: nat).(aprem n a2 a)) H10 i H11) in -H13))))))) H6)))))))))) y0 y x H1))) H0))) H))))). -(* COMMENTS -Initial nodes: 631 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/aprem/props.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/aprem/props.ma deleted file mode 100644 index fb8321062..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/aprem/props.ma +++ /dev/null @@ -1,76 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/aprem/fwd.ma". - -include "Basic-1/leq/defs.ma". - -theorem aprem_repl: - \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall -(i: nat).(\forall (b2: A).((aprem i a2 b2) \to (ex2 A (\lambda (b1: A).(leq g -b1 b2)) (\lambda (b1: A).(aprem i a1 b1))))))))) -\def - \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1 -a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(\forall (i: nat).(\forall -(b2: A).((aprem i a0 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda -(b1: A).(aprem i a b1)))))))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda -(n1: nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (_: (eq A (aplus g -(ASort h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (i: nat).(\lambda (b2: -A).(\lambda (H1: (aprem i (ASort h2 n2) b2)).(let H_x \def (aprem_gen_sort b2 -i h2 n2 H1) in (let H2 \def H_x in (False_ind (ex2 A (\lambda (b1: A).(leq g -b1 b2)) (\lambda (b1: A).(aprem i (ASort h1 n1) b1))) H2)))))))))))) (\lambda -(a0: A).(\lambda (a3: A).(\lambda (H0: (leq g a0 a3)).(\lambda (_: ((\forall -(i: nat).(\forall (b2: A).((aprem i a3 b2) \to (ex2 A (\lambda (b1: A).(leq g -b1 b2)) (\lambda (b1: A).(aprem i a0 b1)))))))).(\lambda (a4: A).(\lambda -(a5: A).(\lambda (_: (leq g a4 a5)).(\lambda (H3: ((\forall (i: nat).(\forall -(b2: A).((aprem i a5 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda -(b1: A).(aprem i a4 b1)))))))).(\lambda (i: nat).(\lambda (b2: A).(\lambda -(H4: (aprem i (AHead a3 a5) b2)).(nat_ind (\lambda (n: nat).((aprem n (AHead -a3 a5) b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem -n (AHead a0 a4) b1))))) (\lambda (H5: (aprem O (AHead a3 a5) b2)).(let H_y -\def (aprem_gen_head_O a3 a5 b2 H5) in (eq_ind_r A a3 (\lambda (a: A).(ex2 A -(\lambda (b1: A).(leq g b1 a)) (\lambda (b1: A).(aprem O (AHead a0 a4) b1)))) -(ex_intro2 A (\lambda (b1: A).(leq g b1 a3)) (\lambda (b1: A).(aprem O (AHead -a0 a4) b1)) a0 H0 (aprem_zero a0 a4)) b2 H_y))) (\lambda (i0: nat).(\lambda -(_: (((aprem i0 (AHead a3 a5) b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) -(\lambda (b1: A).(aprem i0 (AHead a0 a4) b1)))))).(\lambda (H5: (aprem (S i0) -(AHead a3 a5) b2)).(let H_y \def (aprem_gen_head_S a3 a5 b2 i0 H5) in (let -H_x \def (H3 i0 b2 H_y) in (let H6 \def H_x in (ex2_ind A (\lambda (b1: -A).(leq g b1 b2)) (\lambda (b1: A).(aprem i0 a4 b1)) (ex2 A (\lambda (b1: -A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S i0) (AHead a0 a4) b1))) (\lambda -(x: A).(\lambda (H7: (leq g x b2)).(\lambda (H8: (aprem i0 a4 x)).(ex_intro2 -A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S i0) (AHead a0 -a4) b1)) x H7 (aprem_succ a4 x i0 H8 a0))))) H6))))))) i H4)))))))))))) a1 a2 -H)))). -(* COMMENTS -Initial nodes: 621 -END *) - -theorem aprem_asucc: - \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (i: nat).((aprem i -a1 a2) \to (aprem i (asucc g a1) a2))))) -\def - \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (i: nat).(\lambda -(H: (aprem i a1 a2)).(aprem_ind (\lambda (n: nat).(\lambda (a: A).(\lambda -(a0: A).(aprem n (asucc g a) a0)))) (\lambda (a0: A).(\lambda (a3: -A).(aprem_zero a0 (asucc g a3)))) (\lambda (a0: A).(\lambda (a: A).(\lambda -(i0: nat).(\lambda (_: (aprem i0 a0 a)).(\lambda (H1: (aprem i0 (asucc g a0) -a)).(\lambda (a3: A).(aprem_succ (asucc g a0) a i0 H1 a3))))))) i a1 a2 -H))))). -(* COMMENTS -Initial nodes: 101 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/arity/aprem.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/arity/aprem.ma deleted file mode 100644 index 35e8f58e0..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/arity/aprem.ma +++ /dev/null @@ -1,260 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/arity/props.ma". - -include "Basic-1/arity/cimp.ma". - -include "Basic-1/aprem/props.ma". - -theorem arity_aprem: - \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t -a) \to (\forall (i: nat).(\forall (b: A).((aprem i a b) \to (ex2_3 C T nat -(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c)))) -(\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g -b))))))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: -(arity g c t a)).(arity_ind g (\lambda (c0: C).(\lambda (_: T).(\lambda (a0: -A).(\forall (i: nat).(\forall (b: A).((aprem i a0 b) \to (ex2_3 C T nat -(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) -(\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g -b)))))))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (i: nat).(\lambda -(b: A).(\lambda (H0: (aprem i (ASort O n) b)).(let H_x \def (aprem_gen_sort b -i O n H0) in (let H1 \def H_x in (False_ind (ex2_3 C T nat (\lambda (d: -C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: -C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b)))))) H1)))))))) -(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H0: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (a0: A).(\lambda (_: -(arity g d u a0)).(\lambda (H2: ((\forall (i0: nat).(\forall (b: A).((aprem -i0 a0 b) \to (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: -nat).(drop (plus i0 j) O d0 d)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda -(_: nat).(arity g d0 u0 (asucc g b))))))))))).(\lambda (i0: nat).(\lambda (b: -A).(\lambda (H3: (aprem i0 a0 b)).(let H_x \def (H2 i0 b H3) in (let H4 \def -H_x in (ex2_3_ind C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: -nat).(drop (plus i0 j) O d0 d)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda -(_: nat).(arity g d0 u0 (asucc g b))))) (ex2_3 C T nat (\lambda (d0: -C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d0 c0)))) (\lambda -(d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b)))))) -(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H5: (drop -(plus i0 x2) O x0 d)).(\lambda (H6: (arity g x0 x1 (asucc g b))).(let H_x0 -\def (getl_drop_conf_rev (plus i0 x2) x0 d H5 Abbr c0 u i H0) in (let H7 \def -H_x0 in (ex2_ind C (\lambda (c1: C).(drop (plus i0 x2) O c1 c0)) (\lambda -(c1: C).(drop (S i) (plus i0 x2) c1 x0)) (ex2_3 C T nat (\lambda (d0: -C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d0 c0)))) (\lambda -(d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b)))))) -(\lambda (x: C).(\lambda (H8: (drop (plus i0 x2) O x c0)).(\lambda (H9: (drop -(S i) (plus i0 x2) x x0)).(ex2_3_intro C T nat (\lambda (d0: C).(\lambda (_: -T).(\lambda (j: nat).(drop (plus i0 j) O d0 c0)))) (\lambda (d0: C).(\lambda -(u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b))))) x (lift (S i) (plus -i0 x2) x1) x2 H8 (arity_lift g x0 x1 (asucc g b) H6 x (S i) (plus i0 x2) -H9))))) H7)))))))) H4)))))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda -(u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abst) -u))).(\lambda (a0: A).(\lambda (_: (arity g d u (asucc g a0))).(\lambda (H2: -((\forall (i0: nat).(\forall (b: A).((aprem i0 (asucc g a0) b) \to (ex2_3 C T -nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d0 -d)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 -(asucc g b))))))))))).(\lambda (i0: nat).(\lambda (b: A).(\lambda (H3: (aprem -i0 a0 b)).(let H4 \def (H2 i0 b (aprem_asucc g a0 b i0 H3)) in (ex2_3_ind C T -nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d0 -d)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 -(asucc g b))))) (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: -nat).(drop (plus i0 j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda -(_: nat).(arity g d0 u0 (asucc g b)))))) (\lambda (x0: C).(\lambda (x1: -T).(\lambda (x2: nat).(\lambda (H5: (drop (plus i0 x2) O x0 d)).(\lambda (H6: -(arity g x0 x1 (asucc g b))).(let H_x \def (getl_drop_conf_rev (plus i0 x2) -x0 d H5 Abst c0 u i H0) in (let H7 \def H_x in (ex2_ind C (\lambda (c1: -C).(drop (plus i0 x2) O c1 c0)) (\lambda (c1: C).(drop (S i) (plus i0 x2) c1 -x0)) (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop -(plus i0 j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: -nat).(arity g d0 u0 (asucc g b)))))) (\lambda (x: C).(\lambda (H8: (drop -(plus i0 x2) O x c0)).(\lambda (H9: (drop (S i) (plus i0 x2) x -x0)).(ex2_3_intro C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: -nat).(drop (plus i0 j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda -(_: nat).(arity g d0 u0 (asucc g b))))) x (lift (S i) (plus i0 x2) x1) x2 H8 -(arity_lift g x0 x1 (asucc g b) H6 x (S i) (plus i0 x2) H9))))) H7)))))))) -H4))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda -(c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u -a1)).(\lambda (_: ((\forall (i: nat).(\forall (b0: A).((aprem i a1 b0) \to -(ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus -i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d -u0 (asucc g b0))))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: -(arity g (CHead c0 (Bind b) u) t0 a2)).(\lambda (H4: ((\forall (i: -nat).(\forall (b0: A).((aprem i a2 b0) \to (ex2_3 C T nat (\lambda (d: -C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d (CHead c0 (Bind b) -u))))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 -(asucc g b0))))))))))).(\lambda (i: nat).(\lambda (b0: A).(\lambda (H5: -(aprem i a2 b0)).(let H_x \def (H4 i b0 H5) in (let H6 \def H_x in (ex2_3_ind -C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O -d (CHead c0 (Bind b) u))))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: -nat).(arity g d u0 (asucc g b0))))) (ex2_3 C T nat (\lambda (d: C).(\lambda -(_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda -(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b0)))))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H7: (drop (plus i x2) O x0 -(CHead c0 (Bind b) u))).(\lambda (H8: (arity g x0 x1 (asucc g b0))).(let H9 -\def (eq_ind nat (S (plus i x2)) (\lambda (n: nat).(drop n O x0 c0)) (drop_S -b x0 c0 u (plus i x2) H7) (plus i (S x2)) (plus_n_Sm i x2)) in (ex2_3_intro C -T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d -c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 -(asucc g b0))))) x0 x1 (S x2) H9 H8))))))) H6))))))))))))))))) (\lambda (c0: -C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H0: (arity g c0 u (asucc g -a1))).(\lambda (_: ((\forall (i: nat).(\forall (b: A).((aprem i (asucc g a1) -b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop -(plus i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: -nat).(arity g d u0 (asucc g b))))))))))).(\lambda (t0: T).(\lambda (a2: -A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u) t0 a2)).(\lambda (H3: -((\forall (i: nat).(\forall (b: A).((aprem i a2 b) \to (ex2_3 C T nat -(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d (CHead -c0 (Bind Abst) u))))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: -nat).(arity g d u0 (asucc g b))))))))))).(\lambda (i: nat).(\lambda (b: -A).(\lambda (H4: (aprem i (AHead a1 a2) b)).(nat_ind (\lambda (n: -nat).((aprem n (AHead a1 a2) b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda -(_: T).(\lambda (j: nat).(drop (plus n j) O d c0)))) (\lambda (d: C).(\lambda -(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b)))))))) (\lambda (H5: -(aprem O (AHead a1 a2) b)).(let H_y \def (aprem_gen_head_O a1 a2 b H5) in -(eq_ind_r A a1 (\lambda (a0: A).(ex2_3 C T nat (\lambda (d: C).(\lambda (_: -T).(\lambda (j: nat).(drop (plus O j) O d c0)))) (\lambda (d: C).(\lambda -(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g a0))))))) (ex2_3_intro C T -nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus O j) O d -c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 -(asucc g a1))))) c0 u O (drop_refl c0) H0) b H_y))) (\lambda (i0: -nat).(\lambda (_: (((aprem i0 (AHead a1 a2) b) \to (ex2_3 C T nat (\lambda -(d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d c0)))) -(\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g -b))))))))).(\lambda (H5: (aprem (S i0) (AHead a1 a2) b)).(let H_y \def -(aprem_gen_head_S a1 a2 b i0 H5) in (let H_x \def (H3 i0 b H_y) in (let H6 -\def H_x in (ex2_3_ind C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: -nat).(drop (plus i0 j) O d (CHead c0 (Bind Abst) u))))) (\lambda (d: -C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))) (ex2_3 C -T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus (S i0) j) -O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 -(asucc g b)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: -nat).(\lambda (H7: (drop (plus i0 x2) O x0 (CHead c0 (Bind Abst) -u))).(\lambda (H8: (arity g x0 x1 (asucc g b))).(ex2_3_intro C T nat (\lambda -(d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus (S i0) j) O d c0)))) -(\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g -b))))) x0 x1 x2 (drop_S Abst x0 c0 u (plus i0 x2) H7) H8)))))) H6))))))) i -H4))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda -(_: (arity g c0 u a1)).(\lambda (_: ((\forall (i: nat).(\forall (b: -A).((aprem i a1 b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: -T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda -(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))))))))).(\lambda (t0: -T).(\lambda (a2: A).(\lambda (_: (arity g c0 t0 (AHead a1 a2))).(\lambda (H3: -((\forall (i: nat).(\forall (b: A).((aprem i (AHead a1 a2) b) \to (ex2_3 C T -nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d -c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 -(asucc g b))))))))))).(\lambda (i: nat).(\lambda (b: A).(\lambda (H4: (aprem -i a2 b)).(let H5 \def (H3 (S i) b (aprem_succ a2 b i H4 a1)) in (ex2_3_ind C -T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (S (plus i j)) -O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 -(asucc g b))))) (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: -nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda -(_: nat).(arity g d u0 (asucc g b)))))) (\lambda (x0: C).(\lambda (x1: -T).(\lambda (x2: nat).(\lambda (H6: (drop (S (plus i x2)) O x0 c0)).(\lambda -(H7: (arity g x0 x1 (asucc g b))).(C_ind (\lambda (c1: C).((drop (S (plus i -x2)) O c1 c0) \to ((arity g c1 x1 (asucc g b)) \to (ex2_3 C T nat (\lambda -(d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda -(d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))))))) -(\lambda (n: nat).(\lambda (H8: (drop (S (plus i x2)) O (CSort n) -c0)).(\lambda (_: (arity g (CSort n) x1 (asucc g b))).(and3_ind (eq C c0 -(CSort n)) (eq nat (S (plus i x2)) O) (eq nat O O) (ex2_3 C T nat (\lambda -(d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda -(d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b)))))) -(\lambda (_: (eq C c0 (CSort n))).(\lambda (H11: (eq nat (S (plus i x2)) -O)).(\lambda (_: (eq nat O O)).(let H13 \def (eq_ind nat (S (plus i x2)) -(\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O -\Rightarrow False | (S _) \Rightarrow True])) I O H11) in (False_ind (ex2_3 C -T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d -c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 -(asucc g b)))))) H13))))) (drop_gen_sort n (S (plus i x2)) O c0 H8))))) -(\lambda (d: C).(\lambda (IHd: (((drop (S (plus i x2)) O d c0) \to ((arity g -d x1 (asucc g b)) \to (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: -T).(\lambda (j: nat).(drop (plus i j) O d0 c0)))) (\lambda (d0: C).(\lambda -(u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b)))))))))).(\lambda (k: -K).(\lambda (t1: T).(\lambda (H8: (drop (S (plus i x2)) O (CHead d k t1) -c0)).(\lambda (H9: (arity g (CHead d k t1) x1 (asucc g b))).(K_ind (\lambda -(k0: K).((arity g (CHead d k0 t1) x1 (asucc g b)) \to ((drop (r k0 (plus i -x2)) O d c0) \to (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: -nat).(drop (plus i j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda -(_: nat).(arity g d0 u0 (asucc g b))))))))) (\lambda (b0: B).(\lambda (H10: -(arity g (CHead d (Bind b0) t1) x1 (asucc g b))).(\lambda (H11: (drop (r -(Bind b0) (plus i x2)) O d c0)).(ex2_3_intro C T nat (\lambda (d0: -C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d0 c0)))) (\lambda -(d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b))))) -(CHead d (Bind b0) t1) x1 (S x2) (eq_ind nat (S (plus i x2)) (\lambda (n: -nat).(drop n O (CHead d (Bind b0) t1) c0)) (drop_drop (Bind b0) (plus i x2) d -c0 H11 t1) (plus i (S x2)) (plus_n_Sm i x2)) H10)))) (\lambda (f: F).(\lambda -(H10: (arity g (CHead d (Flat f) t1) x1 (asucc g b))).(\lambda (H11: (drop (r -(Flat f) (plus i x2)) O d c0)).(let H12 \def (IHd H11 (arity_cimp_conf g -(CHead d (Flat f) t1) x1 (asucc g b) H10 d (cimp_flat_sx f d t1))) in -(ex2_3_ind C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop -(plus i j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: -nat).(arity g d0 u0 (asucc g b))))) (ex2_3 C T nat (\lambda (d0: C).(\lambda -(_: T).(\lambda (j: nat).(drop (plus i j) O d0 c0)))) (\lambda (d0: -C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b)))))) -(\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: nat).(\lambda (H13: (drop -(plus i x5) O x3 c0)).(\lambda (H14: (arity g x3 x4 (asucc g -b))).(ex2_3_intro C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: -nat).(drop (plus i j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda -(_: nat).(arity g d0 u0 (asucc g b))))) x3 x4 x5 H13 H14)))))) H12))))) k H9 -(drop_gen_drop k d c0 t1 (plus i x2) H8)))))))) x0 H6 H7)))))) -H5)))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a0: A).(\lambda -(_: (arity g c0 u (asucc g a0))).(\lambda (_: ((\forall (i: nat).(\forall (b: -A).((aprem i (asucc g a0) b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: -T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda -(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))))))))).(\lambda (t0: -T).(\lambda (_: (arity g c0 t0 a0)).(\lambda (H3: ((\forall (i: nat).(\forall -(b: A).((aprem i a0 b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: -T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda -(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))))))))).(\lambda (i: -nat).(\lambda (b: A).(\lambda (H4: (aprem i a0 b)).(let H_x \def (H3 i b H4) -in (let H5 \def H_x in (ex2_3_ind C T nat (\lambda (d: C).(\lambda (_: -T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda -(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))) (ex2_3 C T nat -(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) -(\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g -b)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H6: -(drop (plus i x2) O x0 c0)).(\lambda (H7: (arity g x0 x1 (asucc g -b))).(ex2_3_intro C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: -nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda -(_: nat).(arity g d u0 (asucc g b))))) x0 x1 x2 H6 H7)))))) H5)))))))))))))) -(\lambda (c0: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda (_: (arity g c0 -t0 a1)).(\lambda (H1: ((\forall (i: nat).(\forall (b: A).((aprem i a1 b) \to -(ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus -i j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d -u (asucc g b))))))))))).(\lambda (a2: A).(\lambda (H2: (leq g a1 -a2)).(\lambda (i: nat).(\lambda (b: A).(\lambda (H3: (aprem i a2 b)).(let H_x -\def (aprem_repl g a1 a2 H2 i b H3) in (let H4 \def H_x in (ex2_ind A -(\lambda (b1: A).(leq g b1 b)) (\lambda (b1: A).(aprem i a1 b1)) (ex2_3 C T -nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d -c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc -g b)))))) (\lambda (x: A).(\lambda (H5: (leq g x b)).(\lambda (H6: (aprem i -a1 x)).(let H_x0 \def (H1 i x H6) in (let H7 \def H_x0 in (ex2_3_ind C T nat -(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) -(\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g -x))))) (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop -(plus i j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: -nat).(arity g d u (asucc g b)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda -(x2: nat).(\lambda (H8: (drop (plus i x2) O x0 c0)).(\lambda (H9: (arity g x0 -x1 (asucc g x))).(ex2_3_intro C T nat (\lambda (d: C).(\lambda (_: -T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u: -T).(\lambda (_: nat).(arity g d u (asucc g b))))) x0 x1 x2 H8 (arity_repl g -x0 x1 (asucc g x) H9 (asucc g b) (asucc_repl g x b H5)))))))) H7)))))) -H4))))))))))))) c t a H))))). -(* COMMENTS -Initial nodes: 4526 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/arity/cimp.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/arity/cimp.ma deleted file mode 100644 index c6212ac15..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/arity/cimp.ma +++ /dev/null @@ -1,102 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/arity/defs.ma". - -include "Basic-1/cimp/props.ma". - -theorem arity_cimp_conf: - \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1 -t a) \to (\forall (c2: C).((cimp c1 c2) \to (arity g c2 t a))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: -(arity g c1 t a)).(arity_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda (a0: -A).(\forall (c2: C).((cimp c c2) \to (arity g c2 t0 a0)))))) (\lambda (c: -C).(\lambda (n: nat).(\lambda (c2: C).(\lambda (_: (cimp c c2)).(arity_sort g -c2 n))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: -nat).(\lambda (H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (a0: -A).(\lambda (_: (arity g d u a0)).(\lambda (H2: ((\forall (c2: C).((cimp d -c2) \to (arity g c2 u a0))))).(\lambda (c2: C).(\lambda (H3: (cimp c -c2)).(let H_x \def (H3 Abbr d u i H0) in (let H4 \def H_x in (ex_ind C -(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (arity g c2 (TLRef i) -a0) (\lambda (x: C).(\lambda (H5: (getl i c2 (CHead x (Bind Abbr) u))).(let -H_x0 \def (cimp_getl_conf c c2 H3 Abbr d u i H0) in (let H6 \def H_x0 in -(ex2_ind C (\lambda (d2: C).(cimp d d2)) (\lambda (d2: C).(getl i c2 (CHead -d2 (Bind Abbr) u))) (arity g c2 (TLRef i) a0) (\lambda (x0: C).(\lambda (H7: -(cimp d x0)).(\lambda (H8: (getl i c2 (CHead x0 (Bind Abbr) u))).(let H9 \def -(eq_ind C (CHead x (Bind Abbr) u) (\lambda (c0: C).(getl i c2 c0)) H5 (CHead -x0 (Bind Abbr) u) (getl_mono c2 (CHead x (Bind Abbr) u) i H5 (CHead x0 (Bind -Abbr) u) H8)) in (let H10 \def (f_equal C C (\lambda (e: C).(match e in C -return (\lambda (_: C).C) with [(CSort _) \Rightarrow x | (CHead c0 _ _) -\Rightarrow c0])) (CHead x (Bind Abbr) u) (CHead x0 (Bind Abbr) u) (getl_mono -c2 (CHead x (Bind Abbr) u) i H5 (CHead x0 (Bind Abbr) u) H8)) in (let H11 -\def (eq_ind_r C x0 (\lambda (c0: C).(getl i c2 (CHead c0 (Bind Abbr) u))) H9 -x H10) in (let H12 \def (eq_ind_r C x0 (\lambda (c0: C).(cimp d c0)) H7 x -H10) in (arity_abbr g c2 x u i H11 a0 (H2 x H12))))))))) H6))))) -H4))))))))))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: -nat).(\lambda (H0: (getl i c (CHead d (Bind Abst) u))).(\lambda (a0: -A).(\lambda (_: (arity g d u (asucc g a0))).(\lambda (H2: ((\forall (c2: -C).((cimp d c2) \to (arity g c2 u (asucc g a0)))))).(\lambda (c2: C).(\lambda -(H3: (cimp c c2)).(let H_x \def (H3 Abst d u i H0) in (let H4 \def H_x in -(ex_ind C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) (arity g c2 -(TLRef i) a0) (\lambda (x: C).(\lambda (H5: (getl i c2 (CHead x (Bind Abst) -u))).(let H_x0 \def (cimp_getl_conf c c2 H3 Abst d u i H0) in (let H6 \def -H_x0 in (ex2_ind C (\lambda (d2: C).(cimp d d2)) (\lambda (d2: C).(getl i c2 -(CHead d2 (Bind Abst) u))) (arity g c2 (TLRef i) a0) (\lambda (x0: -C).(\lambda (H7: (cimp d x0)).(\lambda (H8: (getl i c2 (CHead x0 (Bind Abst) -u))).(let H9 \def (eq_ind C (CHead x (Bind Abst) u) (\lambda (c0: C).(getl i -c2 c0)) H5 (CHead x0 (Bind Abst) u) (getl_mono c2 (CHead x (Bind Abst) u) i -H5 (CHead x0 (Bind Abst) u) H8)) in (let H10 \def (f_equal C C (\lambda (e: -C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow x | -(CHead c0 _ _) \Rightarrow c0])) (CHead x (Bind Abst) u) (CHead x0 (Bind -Abst) u) (getl_mono c2 (CHead x (Bind Abst) u) i H5 (CHead x0 (Bind Abst) u) -H8)) in (let H11 \def (eq_ind_r C x0 (\lambda (c0: C).(getl i c2 (CHead c0 -(Bind Abst) u))) H9 x H10) in (let H12 \def (eq_ind_r C x0 (\lambda (c0: -C).(cimp d c0)) H7 x H10) in (arity_abst g c2 x u i H11 a0 (H2 x H12))))))))) -H6))))) H4))))))))))))) (\lambda (b: B).(\lambda (H0: (not (eq B b -Abst))).(\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity -g c u a1)).(\lambda (H2: ((\forall (c2: C).((cimp c c2) \to (arity g c2 u -a1))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c -(Bind b) u) t0 a2)).(\lambda (H4: ((\forall (c2: C).((cimp (CHead c (Bind b) -u) c2) \to (arity g c2 t0 a2))))).(\lambda (c2: C).(\lambda (H5: (cimp c -c2)).(arity_bind g b H0 c2 u a1 (H2 c2 H5) t0 a2 (H4 (CHead c2 (Bind b) u) -(cimp_bind c c2 H5 b u)))))))))))))))) (\lambda (c: C).(\lambda (u: -T).(\lambda (a1: A).(\lambda (_: (arity g c u (asucc g a1))).(\lambda (H1: -((\forall (c2: C).((cimp c c2) \to (arity g c2 u (asucc g a1)))))).(\lambda -(t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c (Bind Abst) u) t0 -a2)).(\lambda (H3: ((\forall (c2: C).((cimp (CHead c (Bind Abst) u) c2) \to -(arity g c2 t0 a2))))).(\lambda (c2: C).(\lambda (H4: (cimp c -c2)).(arity_head g c2 u a1 (H1 c2 H4) t0 a2 (H3 (CHead c2 (Bind Abst) u) -(cimp_bind c c2 H4 Abst u)))))))))))))) (\lambda (c: C).(\lambda (u: -T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H1: ((\forall -(c2: C).((cimp c c2) \to (arity g c2 u a1))))).(\lambda (t0: T).(\lambda (a2: -A).(\lambda (_: (arity g c t0 (AHead a1 a2))).(\lambda (H3: ((\forall (c2: -C).((cimp c c2) \to (arity g c2 t0 (AHead a1 a2)))))).(\lambda (c2: -C).(\lambda (H4: (cimp c c2)).(arity_appl g c2 u a1 (H1 c2 H4) t0 a2 (H3 c2 -H4))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a0: A).(\lambda (_: -(arity g c u (asucc g a0))).(\lambda (H1: ((\forall (c2: C).((cimp c c2) \to -(arity g c2 u (asucc g a0)))))).(\lambda (t0: T).(\lambda (_: (arity g c t0 -a0)).(\lambda (H3: ((\forall (c2: C).((cimp c c2) \to (arity g c2 t0 -a0))))).(\lambda (c2: C).(\lambda (H4: (cimp c c2)).(arity_cast g c2 u a0 (H1 -c2 H4) t0 (H3 c2 H4)))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda -(a1: A).(\lambda (_: (arity g c t0 a1)).(\lambda (H1: ((\forall (c2: -C).((cimp c c2) \to (arity g c2 t0 a1))))).(\lambda (a2: A).(\lambda (H2: -(leq g a1 a2)).(\lambda (c2: C).(\lambda (H3: (cimp c c2)).(arity_repl g c2 -t0 a1 (H1 c2 H3) a2 H2)))))))))) c1 t a H))))). -(* COMMENTS -Initial nodes: 1505 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/arity/defs.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/arity/defs.ma deleted file mode 100644 index 99324dcbc..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/arity/defs.ma +++ /dev/null @@ -1,45 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/leq/defs.ma". - -include "Basic-1/getl/defs.ma". - -inductive arity (g: G): C \to (T \to (A \to Prop)) \def -| arity_sort: \forall (c: C).(\forall (n: nat).(arity g c (TSort n) (ASort O -n))) -| arity_abbr: \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: -nat).((getl i c (CHead d (Bind Abbr) u)) \to (\forall (a: A).((arity g d u a) -\to (arity g c (TLRef i) a))))))) -| arity_abst: \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: -nat).((getl i c (CHead d (Bind Abst) u)) \to (\forall (a: A).((arity g d u -(asucc g a)) \to (arity g c (TLRef i) a))))))) -| arity_bind: \forall (b: B).((not (eq B b Abst)) \to (\forall (c: -C).(\forall (u: T).(\forall (a1: A).((arity g c u a1) \to (\forall (t: -T).(\forall (a2: A).((arity g (CHead c (Bind b) u) t a2) \to (arity g c -(THead (Bind b) u t) a2))))))))) -| arity_head: \forall (c: C).(\forall (u: T).(\forall (a1: A).((arity g c u -(asucc g a1)) \to (\forall (t: T).(\forall (a2: A).((arity g (CHead c (Bind -Abst) u) t a2) \to (arity g c (THead (Bind Abst) u t) (AHead a1 a2)))))))) -| arity_appl: \forall (c: C).(\forall (u: T).(\forall (a1: A).((arity g c u -a1) \to (\forall (t: T).(\forall (a2: A).((arity g c t (AHead a1 a2)) \to -(arity g c (THead (Flat Appl) u t) a2))))))) -| arity_cast: \forall (c: C).(\forall (u: T).(\forall (a: A).((arity g c u -(asucc g a)) \to (\forall (t: T).((arity g c t a) \to (arity g c (THead (Flat -Cast) u t) a)))))) -| arity_repl: \forall (c: C).(\forall (t: T).(\forall (a1: A).((arity g c t -a1) \to (\forall (a2: A).((leq g a1 a2) \to (arity g c t a2)))))). - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/arity/fwd.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/arity/fwd.ma deleted file mode 100644 index 31fa35f3c..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/arity/fwd.ma +++ /dev/null @@ -1,1163 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/arity/defs.ma". - -include "Basic-1/leq/asucc.ma". - -include "Basic-1/getl/drop.ma". - -theorem arity_gen_sort: - \forall (g: G).(\forall (c: C).(\forall (n: nat).(\forall (a: A).((arity g c -(TSort n) a) \to (leq g a (ASort O n)))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (n: nat).(\lambda (a: A).(\lambda -(H: (arity g c (TSort n) a)).(insert_eq T (TSort n) (\lambda (t: T).(arity g -c t a)) (\lambda (_: T).(leq g a (ASort O n))) (\lambda (y: T).(\lambda (H0: -(arity g c y a)).(arity_ind g (\lambda (_: C).(\lambda (t: T).(\lambda (a0: -A).((eq T t (TSort n)) \to (leq g a0 (ASort O n)))))) (\lambda (_: -C).(\lambda (n0: nat).(\lambda (H1: (eq T (TSort n0) (TSort n))).(let H2 \def -(f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with -[(TSort n1) \Rightarrow n1 | (TLRef _) \Rightarrow n0 | (THead _ _ _) -\Rightarrow n0])) (TSort n0) (TSort n) H1) in (eq_ind_r nat n (\lambda (n1: -nat).(leq g (ASort O n1) (ASort O n))) (leq_refl g (ASort O n)) n0 H2))))) -(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda -(_: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (a0: A).(\lambda (_: (arity -g d u a0)).(\lambda (_: (((eq T u (TSort n)) \to (leq g a0 (ASort O -n))))).(\lambda (H4: (eq T (TLRef i) (TSort n))).(let H5 \def (eq_ind T -(TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) -\Rightarrow False])) I (TSort n) H4) in (False_ind (leq g a0 (ASort O n)) -H5))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: -nat).(\lambda (_: (getl i c0 (CHead d (Bind Abst) u))).(\lambda (a0: -A).(\lambda (_: (arity g d u (asucc g a0))).(\lambda (_: (((eq T u (TSort n)) -\to (leq g (asucc g a0) (ASort O n))))).(\lambda (H4: (eq T (TLRef i) (TSort -n))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (TSort n) H4) in -(False_ind (leq g a0 (ASort O n)) H5))))))))))) (\lambda (b: B).(\lambda (_: -(not (eq B b Abst))).(\lambda (c0: C).(\lambda (u: T).(\lambda (a1: -A).(\lambda (_: (arity g c0 u a1)).(\lambda (_: (((eq T u (TSort n)) \to (leq -g a1 (ASort O n))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (_: (arity g -(CHead c0 (Bind b) u) t a2)).(\lambda (_: (((eq T t (TSort n)) \to (leq g a2 -(ASort O n))))).(\lambda (H6: (eq T (THead (Bind b) u t) (TSort n))).(let H7 -\def (eq_ind T (THead (Bind b) u t) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H6) in -(False_ind (leq g a2 (ASort O n)) H7)))))))))))))) (\lambda (c0: C).(\lambda -(u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u (asucc g a1))).(\lambda -(_: (((eq T u (TSort n)) \to (leq g (asucc g a1) (ASort O n))))).(\lambda (t: -T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u) t -a2)).(\lambda (_: (((eq T t (TSort n)) \to (leq g a2 (ASort O n))))).(\lambda -(H5: (eq T (THead (Bind Abst) u t) (TSort n))).(let H6 \def (eq_ind T (THead -(Bind Abst) u t) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) -with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ -_) \Rightarrow True])) I (TSort n) H5) in (False_ind (leq g (AHead a1 a2) -(ASort O n)) H6)))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a1: -A).(\lambda (_: (arity g c0 u a1)).(\lambda (_: (((eq T u (TSort n)) \to (leq -g a1 (ASort O n))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (_: (arity g -c0 t (AHead a1 a2))).(\lambda (_: (((eq T t (TSort n)) \to (leq g (AHead a1 -a2) (ASort O n))))).(\lambda (H5: (eq T (THead (Flat Appl) u t) (TSort -n))).(let H6 \def (eq_ind T (THead (Flat Appl) u t) (\lambda (ee: T).(match -ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | -(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) -H5) in (False_ind (leq g a2 (ASort O n)) H6)))))))))))) (\lambda (c0: -C).(\lambda (u: T).(\lambda (a0: A).(\lambda (_: (arity g c0 u (asucc g -a0))).(\lambda (_: (((eq T u (TSort n)) \to (leq g (asucc g a0) (ASort O -n))))).(\lambda (t: T).(\lambda (_: (arity g c0 t a0)).(\lambda (_: (((eq T t -(TSort n)) \to (leq g a0 (ASort O n))))).(\lambda (H5: (eq T (THead (Flat -Cast) u t) (TSort n))).(let H6 \def (eq_ind T (THead (Flat Cast) u t) -(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow -True])) I (TSort n) H5) in (False_ind (leq g a0 (ASort O n)) H6))))))))))) -(\lambda (c0: C).(\lambda (t: T).(\lambda (a1: A).(\lambda (H1: (arity g c0 t -a1)).(\lambda (H2: (((eq T t (TSort n)) \to (leq g a1 (ASort O -n))))).(\lambda (a2: A).(\lambda (H3: (leq g a1 a2)).(\lambda (H4: (eq T t -(TSort n))).(let H5 \def (f_equal T T (\lambda (e: T).e) t (TSort n) H4) in -(let H6 \def (eq_ind T t (\lambda (t0: T).((eq T t0 (TSort n)) \to (leq g a1 -(ASort O n)))) H2 (TSort n) H5) in (let H7 \def (eq_ind T t (\lambda (t0: -T).(arity g c0 t0 a1)) H1 (TSort n) H5) in (leq_trans g a2 a1 (leq_sym g a1 -a2 H3) (ASort O n) (H6 (refl_equal T (TSort n))))))))))))))) c y a H0))) -H))))). -(* COMMENTS -Initial nodes: 1235 -END *) - -theorem arity_gen_lref: - \forall (g: G).(\forall (c: C).(\forall (i: nat).(\forall (a: A).((arity g c -(TLRef i) a) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c -(CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u a)))) -(ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c (CHead d (Bind Abst) -u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a)))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (i: nat).(\lambda (a: A).(\lambda -(H: (arity g c (TLRef i) a)).(insert_eq T (TLRef i) (\lambda (t: T).(arity g -c t a)) (\lambda (_: T).(or (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl -i c (CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u -a)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c (CHead d (Bind -Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a))))))) -(\lambda (y: T).(\lambda (H0: (arity g c y a)).(arity_ind g (\lambda (c0: -C).(\lambda (t: T).(\lambda (a0: A).((eq T t (TLRef i)) \to (or (ex2_2 C T -(\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u)))) -(\lambda (d: C).(\lambda (u: T).(arity g d u a0)))) (ex2_2 C T (\lambda (d: -C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d: -C).(\lambda (u: T).(arity g d u (asucc g a0)))))))))) (\lambda (c0: -C).(\lambda (n: nat).(\lambda (H1: (eq T (TSort n) (TLRef i))).(let H2 \def -(eq_ind T (TSort n) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | -(THead _ _ _) \Rightarrow False])) I (TLRef i) H1) in (False_ind (or (ex2_2 C -T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u)))) -(\lambda (d: C).(\lambda (u: T).(arity g d u (ASort O n))))) (ex2_2 C T -(\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) -(\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g (ASort O n))))))) -H2))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i0: -nat).(\lambda (H1: (getl i0 c0 (CHead d (Bind Abbr) u))).(\lambda (a0: -A).(\lambda (H2: (arity g d u a0)).(\lambda (_: (((eq T u (TLRef i)) \to (or -(ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl i d (CHead d0 (Bind Abbr) -u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a0)))) (ex2_2 C T -(\lambda (d0: C).(\lambda (u0: T).(getl i d (CHead d0 (Bind Abst) u0)))) -(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g -a0))))))))).(\lambda (H4: (eq T (TLRef i0) (TLRef i))).(let H5 \def (f_equal -T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort -_) \Rightarrow i0 | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow i0])) -(TLRef i0) (TLRef i) H4) in (let H6 \def (eq_ind nat i0 (\lambda (n: -nat).(getl n c0 (CHead d (Bind Abbr) u))) H1 i H5) in (or_introl (ex2_2 C T -(\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abbr) u0)))) -(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a0)))) (ex2_2 C T (\lambda -(d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0)))) (\lambda -(d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a0))))) (ex2_2_intro C T -(\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abbr) u0)))) -(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a0))) d u H6 H2))))))))))))) -(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i0: nat).(\lambda -(H1: (getl i0 c0 (CHead d (Bind Abst) u))).(\lambda (a0: A).(\lambda (H2: -(arity g d u (asucc g a0))).(\lambda (_: (((eq T u (TLRef i)) \to (or (ex2_2 -C T (\lambda (d0: C).(\lambda (u0: T).(getl i d (CHead d0 (Bind Abbr) u0)))) -(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a0))))) (ex2_2 C T -(\lambda (d0: C).(\lambda (u0: T).(getl i d (CHead d0 (Bind Abst) u0)))) -(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g (asucc g -a0)))))))))).(\lambda (H4: (eq T (TLRef i0) (TLRef i))).(let H5 \def (f_equal -T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort -_) \Rightarrow i0 | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow i0])) -(TLRef i0) (TLRef i) H4) in (let H6 \def (eq_ind nat i0 (\lambda (n: -nat).(getl n c0 (CHead d (Bind Abst) u))) H1 i H5) in (or_intror (ex2_2 C T -(\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abbr) u0)))) -(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a0)))) (ex2_2 C T (\lambda -(d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0)))) (\lambda -(d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a0))))) (ex2_2_intro C T -(\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0)))) -(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a0)))) d u H6 -H2))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda -(c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u -a1)).(\lambda (_: (((eq T u (TLRef i)) \to (or (ex2_2 C T (\lambda (d: -C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d: -C).(\lambda (u0: T).(arity g d u0 a1)))) (ex2_2 C T (\lambda (d: C).(\lambda -(u0: T).(getl i c0 (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0: -T).(arity g d u0 (asucc g a1))))))))).(\lambda (t: T).(\lambda (a2: -A).(\lambda (_: (arity g (CHead c0 (Bind b) u) t a2)).(\lambda (_: (((eq T t -(TLRef i)) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i (CHead -c0 (Bind b) u) (CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: -T).(arity g d u0 a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i -(CHead c0 (Bind b) u) (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda -(u0: T).(arity g d u0 (asucc g a2))))))))).(\lambda (H6: (eq T (THead (Bind -b) u t) (TLRef i))).(let H7 \def (eq_ind T (THead (Bind b) u t) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I -(TLRef i) H6) in (False_ind (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: -T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: -T).(arity g d u0 a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i -c0 (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 -(asucc g a2)))))) H7)))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda -(a1: A).(\lambda (_: (arity g c0 u (asucc g a1))).(\lambda (_: (((eq T u -(TLRef i)) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 -(CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 -(asucc g a1))))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 -(CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 -(asucc g (asucc g a1)))))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (_: -(arity g (CHead c0 (Bind Abst) u) t a2)).(\lambda (_: (((eq T t (TLRef i)) -\to (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i (CHead c0 (Bind -Abst) u) (CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity -g d u0 a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i (CHead c0 -(Bind Abst) u) (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0: -T).(arity g d u0 (asucc g a2))))))))).(\lambda (H5: (eq T (THead (Bind Abst) -u t) (TLRef i))).(let H6 \def (eq_ind T (THead (Bind Abst) u t) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I -(TLRef i) H5) in (False_ind (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: -T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: -T).(arity g d u0 (AHead a1 a2))))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: -T).(getl i c0 (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0: -T).(arity g d u0 (asucc g (AHead a1 a2))))))) H6)))))))))))) (\lambda (c0: -C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u a1)).(\lambda -(_: (((eq T u (TLRef i)) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: -T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: -T).(arity g d u0 a1)))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i -c0 (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 -(asucc g a1))))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (_: (arity g -c0 t (AHead a1 a2))).(\lambda (_: (((eq T t (TLRef i)) \to (or (ex2_2 C T -(\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abbr) u0)))) -(\lambda (d: C).(\lambda (u0: T).(arity g d u0 (AHead a1 a2))))) (ex2_2 C T -(\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abst) u0)))) -(\lambda (d: C).(\lambda (u0: T).(arity g d u0 (asucc g (AHead a1 -a2)))))))))).(\lambda (H5: (eq T (THead (Flat Appl) u t) (TLRef i))).(let H6 -\def (eq_ind T (THead (Flat Appl) u t) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef i) H5) in -(False_ind (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead -d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 a2)))) -(ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abst) -u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 (asucc g a2)))))) -H6)))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a0: A).(\lambda (_: -(arity g c0 u (asucc g a0))).(\lambda (_: (((eq T u (TLRef i)) \to (or (ex2_2 -C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abbr) u0)))) -(\lambda (d: C).(\lambda (u0: T).(arity g d u0 (asucc g a0))))) (ex2_2 C T -(\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abst) u0)))) -(\lambda (d: C).(\lambda (u0: T).(arity g d u0 (asucc g (asucc g -a0)))))))))).(\lambda (t: T).(\lambda (_: (arity g c0 t a0)).(\lambda (_: -(((eq T t (TLRef i)) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: -T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: -T).(arity g d u0 a0)))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i -c0 (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 -(asucc g a0))))))))).(\lambda (H5: (eq T (THead (Flat Cast) u t) (TLRef -i))).(let H6 \def (eq_ind T (THead (Flat Cast) u t) (\lambda (ee: T).(match -ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | -(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef i) -H5) in (False_ind (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 -(CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 -a0)))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind -Abst) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 (asucc g a0)))))) -H6))))))))))) (\lambda (c0: C).(\lambda (t: T).(\lambda (a1: A).(\lambda (H1: -(arity g c0 t a1)).(\lambda (H2: (((eq T t (TLRef i)) \to (or (ex2_2 C T -(\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u)))) -(\lambda (d: C).(\lambda (u: T).(arity g d u a1)))) (ex2_2 C T (\lambda (d: -C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d: -C).(\lambda (u: T).(arity g d u (asucc g a1))))))))).(\lambda (a2: -A).(\lambda (H3: (leq g a1 a2)).(\lambda (H4: (eq T t (TLRef i))).(let H5 -\def (f_equal T T (\lambda (e: T).e) t (TLRef i) H4) in (let H6 \def (eq_ind -T t (\lambda (t0: T).((eq T t0 (TLRef i)) \to (or (ex2_2 C T (\lambda (d: -C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u)))) (\lambda (d: -C).(\lambda (u: T).(arity g d u a1)))) (ex2_2 C T (\lambda (d: C).(\lambda -(u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u: -T).(arity g d u (asucc g a1)))))))) H2 (TLRef i) H5) in (let H7 \def (eq_ind -T t (\lambda (t0: T).(arity g c0 t0 a1)) H1 (TLRef i) H5) in (let H8 \def (H6 -(refl_equal T (TLRef i))) in (or_ind (ex2_2 C T (\lambda (d: C).(\lambda (u: -T).(getl i c0 (CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: -T).(arity g d u a1)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 -(CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u -(asucc g a1))))) (or (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 -(CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u -a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind -Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a2)))))) -(\lambda (H9: (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d -(Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u -a1))))).(ex2_2_ind C T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d -(Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u a1))) (or -(ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) -u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u a2)))) (ex2_2 C T (\lambda -(d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d: -C).(\lambda (u: T).(arity g d u (asucc g a2)))))) (\lambda (x0: C).(\lambda -(x1: T).(\lambda (H10: (getl i c0 (CHead x0 (Bind Abbr) x1))).(\lambda (H11: -(arity g x0 x1 a1)).(or_introl (ex2_2 C T (\lambda (d: C).(\lambda (u: -T).(getl i c0 (CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: -T).(arity g d u a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 -(CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u -(asucc g a2))))) (ex2_2_intro C T (\lambda (d: C).(\lambda (u: T).(getl i c0 -(CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u a2))) -x0 x1 H10 (arity_repl g x0 x1 a1 H11 a2 H3))))))) H9)) (\lambda (H9: (ex2_2 C -T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) -(\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a1)))))).(ex2_2_ind C T -(\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) -(\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a1)))) (or (ex2_2 C T -(\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u)))) -(\lambda (d: C).(\lambda (u: T).(arity g d u a2)))) (ex2_2 C T (\lambda (d: -C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d: -C).(\lambda (u: T).(arity g d u (asucc g a2)))))) (\lambda (x0: C).(\lambda -(x1: T).(\lambda (H10: (getl i c0 (CHead x0 (Bind Abst) x1))).(\lambda (H11: -(arity g x0 x1 (asucc g a1))).(or_intror (ex2_2 C T (\lambda (d: C).(\lambda -(u: T).(getl i c0 (CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: -T).(arity g d u a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 -(CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u -(asucc g a2))))) (ex2_2_intro C T (\lambda (d: C).(\lambda (u: T).(getl i c0 -(CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u -(asucc g a2)))) x0 x1 H10 (arity_repl g x0 x1 (asucc g a1) H11 (asucc g a2) -(asucc_repl g a1 a2 H3)))))))) H9)) H8))))))))))))) c y a H0))) H))))). -(* COMMENTS -Initial nodes: 3853 -END *) - -theorem arity_gen_bind: - \forall (b: B).((not (eq B b Abst)) \to (\forall (g: G).(\forall (c: -C).(\forall (u: T).(\forall (t: T).(\forall (a2: A).((arity g c (THead (Bind -b) u t) a2) \to (ex2 A (\lambda (a1: A).(arity g c u a1)) (\lambda (_: -A).(arity g (CHead c (Bind b) u) t a2)))))))))) -\def - \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (g: G).(\lambda -(c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (a2: A).(\lambda (H0: (arity -g c (THead (Bind b) u t) a2)).(insert_eq T (THead (Bind b) u t) (\lambda (t0: -T).(arity g c t0 a2)) (\lambda (_: T).(ex2 A (\lambda (a1: A).(arity g c u -a1)) (\lambda (_: A).(arity g (CHead c (Bind b) u) t a2)))) (\lambda (y: -T).(\lambda (H1: (arity g c y a2)).(arity_ind g (\lambda (c0: C).(\lambda -(t0: T).(\lambda (a: A).((eq T t0 (THead (Bind b) u t)) \to (ex2 A (\lambda -(a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t -a))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (H2: (eq T (TSort n) -(THead (Bind b) u t))).(let H3 \def (eq_ind T (TSort n) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I -(THead (Bind b) u t) H2) in (False_ind (ex2 A (\lambda (a1: A).(arity g c0 u -a1)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t (ASort O n)))) H3))))) -(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda -(_: (getl i c0 (CHead d (Bind Abbr) u0))).(\lambda (a: A).(\lambda (_: (arity -g d u0 a)).(\lambda (_: (((eq T u0 (THead (Bind b) u t)) \to (ex2 A (\lambda -(a1: A).(arity g d u a1)) (\lambda (_: A).(arity g (CHead d (Bind b) u) t -a)))))).(\lambda (H5: (eq T (TLRef i) (THead (Bind b) u t))).(let H6 \def -(eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | -(THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t) H5) in (False_ind -(ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g (CHead c0 -(Bind b) u) t a))) H6))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda -(u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind Abst) -u0))).(\lambda (a: A).(\lambda (_: (arity g d u0 (asucc g a))).(\lambda (_: -(((eq T u0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a1: A).(arity g d u -a1)) (\lambda (_: A).(arity g (CHead d (Bind b) u) t (asucc g -a))))))).(\lambda (H5: (eq T (TLRef i) (THead (Bind b) u t))).(let H6 \def -(eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | -(THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t) H5) in (False_ind -(ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g (CHead c0 -(Bind b) u) t a))) H6))))))))))) (\lambda (b0: B).(\lambda (H2: (not (eq B b0 -Abst))).(\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (H3: -(arity g c0 u0 a1)).(\lambda (H4: (((eq T u0 (THead (Bind b) u t)) \to (ex2 A -(\lambda (a3: A).(arity g c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind -b) u) t a1)))))).(\lambda (t0: T).(\lambda (a0: A).(\lambda (H5: (arity g -(CHead c0 (Bind b0) u0) t0 a0)).(\lambda (H6: (((eq T t0 (THead (Bind b) u -t)) \to (ex2 A (\lambda (a3: A).(arity g (CHead c0 (Bind b0) u0) u a3)) -(\lambda (_: A).(arity g (CHead (CHead c0 (Bind b0) u0) (Bind b) u) t -a0)))))).(\lambda (H7: (eq T (THead (Bind b0) u0 t0) (THead (Bind b) u -t))).(let H8 \def (f_equal T B (\lambda (e: T).(match e in T return (\lambda -(_: T).B) with [(TSort _) \Rightarrow b0 | (TLRef _) \Rightarrow b0 | (THead -k _ _) \Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b1) -\Rightarrow b1 | (Flat _) \Rightarrow b0])])) (THead (Bind b0) u0 t0) (THead -(Bind b) u t) H7) in ((let H9 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) -\Rightarrow u0 | (THead _ t1 _) \Rightarrow t1])) (THead (Bind b0) u0 t0) -(THead (Bind b) u t) H7) in ((let H10 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | -(TLRef _) \Rightarrow t0 | (THead _ _ t1) \Rightarrow t1])) (THead (Bind b0) -u0 t0) (THead (Bind b) u t) H7) in (\lambda (H11: (eq T u0 u)).(\lambda (H12: -(eq B b0 b)).(let H13 \def (eq_ind T t0 (\lambda (t1: T).((eq T t1 (THead -(Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g (CHead c0 (Bind b0) u0) u -a3)) (\lambda (_: A).(arity g (CHead (CHead c0 (Bind b0) u0) (Bind b) u) t -a0))))) H6 t H10) in (let H14 \def (eq_ind T t0 (\lambda (t1: T).(arity g -(CHead c0 (Bind b0) u0) t1 a0)) H5 t H10) in (let H15 \def (eq_ind T u0 -(\lambda (t1: T).((eq T t (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: -A).(arity g (CHead c0 (Bind b0) t1) u a3)) (\lambda (_: A).(arity g (CHead -(CHead c0 (Bind b0) t1) (Bind b) u) t a0))))) H13 u H11) in (let H16 \def -(eq_ind T u0 (\lambda (t1: T).(arity g (CHead c0 (Bind b0) t1) t a0)) H14 u -H11) in (let H17 \def (eq_ind T u0 (\lambda (t1: T).((eq T t1 (THead (Bind b) -u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (_: A).(arity g -(CHead c0 (Bind b) u) t a1))))) H4 u H11) in (let H18 \def (eq_ind T u0 -(\lambda (t1: T).(arity g c0 t1 a1)) H3 u H11) in (let H19 \def (eq_ind B b0 -(\lambda (b1: B).((eq T t (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: -A).(arity g (CHead c0 (Bind b1) u) u a3)) (\lambda (_: A).(arity g (CHead -(CHead c0 (Bind b1) u) (Bind b) u) t a0))))) H15 b H12) in (let H20 \def -(eq_ind B b0 (\lambda (b1: B).(arity g (CHead c0 (Bind b1) u) t a0)) H16 b -H12) in (let H21 \def (eq_ind B b0 (\lambda (b1: B).(not (eq B b1 Abst))) H2 -b H12) in (ex_intro2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (_: -A).(arity g (CHead c0 (Bind b) u) t a0)) a1 H18 H20))))))))))))) H9)) -H8)))))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: A).(\lambda -(H2: (arity g c0 u0 (asucc g a1))).(\lambda (H3: (((eq T u0 (THead (Bind b) u -t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (_: A).(arity g -(CHead c0 (Bind b) u) t (asucc g a1))))))).(\lambda (t0: T).(\lambda (a0: -A).(\lambda (H4: (arity g (CHead c0 (Bind Abst) u0) t0 a0)).(\lambda (H5: -(((eq T t0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g (CHead -c0 (Bind Abst) u0) u a3)) (\lambda (_: A).(arity g (CHead (CHead c0 (Bind -Abst) u0) (Bind b) u) t a0)))))).(\lambda (H6: (eq T (THead (Bind Abst) u0 -t0) (THead (Bind b) u t))).(let H7 \def (f_equal T B (\lambda (e: T).(match e -in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow Abst | (TLRef _) -\Rightarrow Abst | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abst])])) -(THead (Bind Abst) u0 t0) (THead (Bind b) u t) H6) in ((let H8 \def (f_equal -T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t1 _) \Rightarrow t1])) -(THead (Bind Abst) u0 t0) (THead (Bind b) u t) H6) in ((let H9 \def (f_equal -T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t1) \Rightarrow t1])) -(THead (Bind Abst) u0 t0) (THead (Bind b) u t) H6) in (\lambda (H10: (eq T u0 -u)).(\lambda (H11: (eq B Abst b)).(let H12 \def (eq_ind T t0 (\lambda (t1: -T).((eq T t1 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g -(CHead c0 (Bind Abst) u0) u a3)) (\lambda (_: A).(arity g (CHead (CHead c0 -(Bind Abst) u0) (Bind b) u) t a0))))) H5 t H9) in (let H13 \def (eq_ind T t0 -(\lambda (t1: T).(arity g (CHead c0 (Bind Abst) u0) t1 a0)) H4 t H9) in (let -H14 \def (eq_ind T u0 (\lambda (t1: T).((eq T t (THead (Bind b) u t)) \to -(ex2 A (\lambda (a3: A).(arity g (CHead c0 (Bind Abst) t1) u a3)) (\lambda -(_: A).(arity g (CHead (CHead c0 (Bind Abst) t1) (Bind b) u) t a0))))) H12 u -H10) in (let H15 \def (eq_ind T u0 (\lambda (t1: T).(arity g (CHead c0 (Bind -Abst) t1) t a0)) H13 u H10) in (let H16 \def (eq_ind T u0 (\lambda (t1: -T).((eq T t1 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u -a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t (asucc g a1)))))) H3 u -H10) in (let H17 \def (eq_ind T u0 (\lambda (t1: T).(arity g c0 t1 (asucc g -a1))) H2 u H10) in (let H18 \def (eq_ind_r B b (\lambda (b0: B).((eq T t -(THead (Bind b0) u t)) \to (ex2 A (\lambda (a3: A).(arity g (CHead c0 (Bind -Abst) u) u a3)) (\lambda (_: A).(arity g (CHead (CHead c0 (Bind Abst) u) -(Bind b0) u) t a0))))) H14 Abst H11) in (let H19 \def (eq_ind_r B b (\lambda -(b0: B).((eq T u (THead (Bind b0) u t)) \to (ex2 A (\lambda (a3: A).(arity g -c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b0) u) t (asucc g a1)))))) -H16 Abst H11) in (let H20 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 -Abst))) H Abst H11) in (eq_ind B Abst (\lambda (b0: B).(ex2 A (\lambda (a3: -A).(arity g c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b0) u) t -(AHead a1 a0))))) (let H21 \def (match (H20 (refl_equal B Abst)) in False -return (\lambda (_: False).(ex2 A (\lambda (a3: A).(arity g c0 u a3)) -(\lambda (_: A).(arity g (CHead c0 (Bind Abst) u) t (AHead a1 a0))))) with -[]) in H21) b H11))))))))))))) H8)) H7)))))))))))) (\lambda (c0: C).(\lambda -(u0: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u0 a1)).(\lambda (_: (((eq -T u0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) -(\lambda (_: A).(arity g (CHead c0 (Bind b) u) t a1)))))).(\lambda (t0: -T).(\lambda (a0: A).(\lambda (_: (arity g c0 t0 (AHead a1 a0))).(\lambda (_: -(((eq T t0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u -a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t (AHead a1 -a0))))))).(\lambda (H6: (eq T (THead (Flat Appl) u0 t0) (THead (Bind b) u -t))).(let H7 \def (eq_ind T (THead (Flat Appl) u0 t0) (\lambda (ee: T).(match -ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | -(TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return -(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind b) u t) H6) in (False_ind (ex2 A (\lambda (a3: -A).(arity g c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t a0))) -H7)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a: A).(\lambda (_: -(arity g c0 u0 (asucc g a))).(\lambda (_: (((eq T u0 (THead (Bind b) u t)) -\to (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g -(CHead c0 (Bind b) u) t (asucc g a))))))).(\lambda (t0: T).(\lambda (_: -(arity g c0 t0 a)).(\lambda (_: (((eq T t0 (THead (Bind b) u t)) \to (ex2 A -(\lambda (a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g (CHead c0 (Bind -b) u) t a)))))).(\lambda (H6: (eq T (THead (Flat Cast) u0 t0) (THead (Bind b) -u t))).(let H7 \def (eq_ind T (THead (Flat Cast) u0 t0) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K -return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) -\Rightarrow True])])) I (THead (Bind b) u t) H6) in (False_ind (ex2 A -(\lambda (a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g (CHead c0 (Bind -b) u) t a))) H7))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (a1: -A).(\lambda (H2: (arity g c0 t0 a1)).(\lambda (H3: (((eq T t0 (THead (Bind b) -u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (_: A).(arity g -(CHead c0 (Bind b) u) t a1)))))).(\lambda (a0: A).(\lambda (H4: (leq g a1 -a0)).(\lambda (H5: (eq T t0 (THead (Bind b) u t))).(let H6 \def (f_equal T T -(\lambda (e: T).e) t0 (THead (Bind b) u t) H5) in (let H7 \def (eq_ind T t0 -(\lambda (t1: T).((eq T t1 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: -A).(arity g c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t -a1))))) H3 (THead (Bind b) u t) H6) in (let H8 \def (eq_ind T t0 (\lambda -(t1: T).(arity g c0 t1 a1)) H2 (THead (Bind b) u t) H6) in (let H9 \def (H7 -(refl_equal T (THead (Bind b) u t))) in (ex2_ind A (\lambda (a3: A).(arity g -c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t a1)) (ex2 A -(\lambda (a3: A).(arity g c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind -b) u) t a0))) (\lambda (x: A).(\lambda (H10: (arity g c0 u x)).(\lambda (H11: -(arity g (CHead c0 (Bind b) u) t a1)).(ex_intro2 A (\lambda (a3: A).(arity g -c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t a0)) x H10 -(arity_repl g (CHead c0 (Bind b) u) t a1 H11 a0 H4))))) H9))))))))))))) c y -a2 H1))) H0)))))))). -(* COMMENTS -Initial nodes: 3365 -END *) - -theorem arity_gen_abst: - \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (a: -A).((arity g c (THead (Bind Abst) u t) a) \to (ex3_2 A A (\lambda (a1: -A).(\lambda (a2: A).(eq A a (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: -A).(arity g c u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g -(CHead c (Bind Abst) u) t a2))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (a: -A).(\lambda (H: (arity g c (THead (Bind Abst) u t) a)).(insert_eq T (THead -(Bind Abst) u t) (\lambda (t0: T).(arity g c t0 a)) (\lambda (_: T).(ex3_2 A -A (\lambda (a1: A).(\lambda (a2: A).(eq A a (AHead a1 a2)))) (\lambda (a1: -A).(\lambda (_: A).(arity g c u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: -A).(arity g (CHead c (Bind Abst) u) t a2))))) (\lambda (y: T).(\lambda (H0: -(arity g c y a)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (a0: -A).((eq T t0 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a1: -A).(\lambda (a2: A).(eq A a0 (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: -A).(arity g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g -(CHead c0 (Bind Abst) u) t a2)))))))) (\lambda (c0: C).(\lambda (n: -nat).(\lambda (H1: (eq T (TSort n) (THead (Bind Abst) u t))).(let H2 \def -(eq_ind T (TSort n) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | -(THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) u t) H1) in -(False_ind (ex3_2 A A (\lambda (a1: A).(\lambda (a2: A).(eq A (ASort O n) -(AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g c0 u (asucc g -a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g (CHead c0 (Bind Abst) u) t -a2)))) H2))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: -nat).(\lambda (_: (getl i c0 (CHead d (Bind Abbr) u0))).(\lambda (a0: -A).(\lambda (_: (arity g d u0 a0)).(\lambda (_: (((eq T u0 (THead (Bind Abst) -u t)) \to (ex3_2 A A (\lambda (a1: A).(\lambda (a2: A).(eq A a0 (AHead a1 -a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g d u (asucc g a1)))) (\lambda -(_: A).(\lambda (a2: A).(arity g (CHead d (Bind Abst) u) t a2))))))).(\lambda -(H4: (eq T (TLRef i) (THead (Bind Abst) u t))).(let H5 \def (eq_ind T (TLRef -i) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort -_) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow -False])) I (THead (Bind Abst) u t) H4) in (False_ind (ex3_2 A A (\lambda (a1: -A).(\lambda (a2: A).(eq A a0 (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: -A).(arity g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g -(CHead c0 (Bind Abst) u) t a2)))) H5))))))))))) (\lambda (c0: C).(\lambda (d: -C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind -Abst) u0))).(\lambda (a0: A).(\lambda (_: (arity g d u0 (asucc g -a0))).(\lambda (_: (((eq T u0 (THead (Bind Abst) u t)) \to (ex3_2 A A -(\lambda (a1: A).(\lambda (a2: A).(eq A (asucc g a0) (AHead a1 a2)))) -(\lambda (a1: A).(\lambda (_: A).(arity g d u (asucc g a1)))) (\lambda (_: -A).(\lambda (a2: A).(arity g (CHead d (Bind Abst) u) t a2))))))).(\lambda -(H4: (eq T (TLRef i) (THead (Bind Abst) u t))).(let H5 \def (eq_ind T (TLRef -i) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort -_) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow -False])) I (THead (Bind Abst) u t) H4) in (False_ind (ex3_2 A A (\lambda (a1: -A).(\lambda (a2: A).(eq A a0 (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: -A).(arity g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g -(CHead c0 (Bind Abst) u) t a2)))) H5))))))))))) (\lambda (b: B).(\lambda (H1: -(not (eq B b Abst))).(\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: -A).(\lambda (H2: (arity g c0 u0 a1)).(\lambda (H3: (((eq T u0 (THead (Bind -Abst) u t)) \to (ex3_2 A A (\lambda (a2: A).(\lambda (a3: A).(eq A a1 (AHead -a2 a3)))) (\lambda (a2: A).(\lambda (_: A).(arity g c0 u (asucc g a2)))) -(\lambda (_: A).(\lambda (a3: A).(arity g (CHead c0 (Bind Abst) u) t -a3))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (H4: (arity g (CHead c0 -(Bind b) u0) t0 a2)).(\lambda (H5: (((eq T t0 (THead (Bind Abst) u t)) \to -(ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) -(\lambda (a3: A).(\lambda (_: A).(arity g (CHead c0 (Bind b) u0) u (asucc g -a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead (CHead c0 (Bind b) -u0) (Bind Abst) u) t a4))))))).(\lambda (H6: (eq T (THead (Bind b) u0 t0) -(THead (Bind Abst) u t))).(let H7 \def (f_equal T B (\lambda (e: T).(match e -in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | (TLRef _) -\Rightarrow b | (THead k _ _) \Rightarrow (match k in K return (\lambda (_: -K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow b])])) (THead -(Bind b) u0 t0) (THead (Bind Abst) u t) H6) in ((let H8 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t1 _) \Rightarrow t1])) -(THead (Bind b) u0 t0) (THead (Bind Abst) u t) H6) in ((let H9 \def (f_equal -T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t1) \Rightarrow t1])) -(THead (Bind b) u0 t0) (THead (Bind Abst) u t) H6) in (\lambda (H10: (eq T u0 -u)).(\lambda (H11: (eq B b Abst)).(let H12 \def (eq_ind T t0 (\lambda (t1: -T).((eq T t1 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3: -A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: -A).(arity g (CHead c0 (Bind b) u0) u (asucc g a3)))) (\lambda (_: A).(\lambda -(a4: A).(arity g (CHead (CHead c0 (Bind b) u0) (Bind Abst) u) t a4)))))) H5 t -H9) in (let H13 \def (eq_ind T t0 (\lambda (t1: T).(arity g (CHead c0 (Bind -b) u0) t1 a2)) H4 t H9) in (let H14 \def (eq_ind T u0 (\lambda (t1: T).((eq T -t (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: -A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g (CHead -c0 (Bind b) t1) u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g -(CHead (CHead c0 (Bind b) t1) (Bind Abst) u) t a4)))))) H12 u H10) in (let -H15 \def (eq_ind T u0 (\lambda (t1: T).(arity g (CHead c0 (Bind b) t1) t a2)) -H13 u H10) in (let H16 \def (eq_ind T u0 (\lambda (t1: T).((eq T t1 (THead -(Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a1 -(AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g -a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t -a4)))))) H3 u H10) in (let H17 \def (eq_ind T u0 (\lambda (t1: T).(arity g c0 -t1 a1)) H2 u H10) in (let H18 \def (eq_ind B b (\lambda (b0: B).((eq T t -(THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq -A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g (CHead c0 -(Bind b0) u) u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g -(CHead (CHead c0 (Bind b0) u) (Bind Abst) u) t a4)))))) H14 Abst H11) in (let -H19 \def (eq_ind B b (\lambda (b0: B).(arity g (CHead c0 (Bind b0) u) t a2)) -H15 Abst H11) in (let H20 \def (eq_ind B b (\lambda (b0: B).(not (eq B b0 -Abst))) H1 Abst H11) in (let H21 \def (match (H20 (refl_equal B Abst)) in -False return (\lambda (_: False).(ex3_2 A A (\lambda (a3: A).(\lambda (a4: -A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u -(asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind -Abst) u) t a4))))) with []) in H21))))))))))))) H8)) H7)))))))))))))) -(\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (H1: (arity g c0 -u0 (asucc g a1))).(\lambda (H2: (((eq T u0 (THead (Bind Abst) u t)) \to -(ex3_2 A A (\lambda (a2: A).(\lambda (a3: A).(eq A (asucc g a1) (AHead a2 -a3)))) (\lambda (a2: A).(\lambda (_: A).(arity g c0 u (asucc g a2)))) -(\lambda (_: A).(\lambda (a3: A).(arity g (CHead c0 (Bind Abst) u) t -a3))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (H3: (arity g (CHead c0 -(Bind Abst) u0) t0 a2)).(\lambda (H4: (((eq T t0 (THead (Bind Abst) u t)) \to -(ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) -(\lambda (a3: A).(\lambda (_: A).(arity g (CHead c0 (Bind Abst) u0) u (asucc -g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead (CHead c0 (Bind -Abst) u0) (Bind Abst) u) t a4))))))).(\lambda (H5: (eq T (THead (Bind Abst) -u0 t0) (THead (Bind Abst) u t))).(let H6 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | -(TLRef _) \Rightarrow u0 | (THead _ t1 _) \Rightarrow t1])) (THead (Bind -Abst) u0 t0) (THead (Bind Abst) u t) H5) in ((let H7 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t1) \Rightarrow t1])) -(THead (Bind Abst) u0 t0) (THead (Bind Abst) u t) H5) in (\lambda (H8: (eq T -u0 u)).(let H9 \def (eq_ind T t0 (\lambda (t1: T).((eq T t1 (THead (Bind -Abst) u t)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead -a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g (CHead c0 (Bind Abst) u0) -u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead (CHead c0 -(Bind Abst) u0) (Bind Abst) u) t a4)))))) H4 t H7) in (let H10 \def (eq_ind T -t0 (\lambda (t1: T).(arity g (CHead c0 (Bind Abst) u0) t1 a2)) H3 t H7) in -(let H11 \def (eq_ind T u0 (\lambda (t1: T).((eq T t (THead (Bind Abst) u t)) -\to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) -(\lambda (a3: A).(\lambda (_: A).(arity g (CHead c0 (Bind Abst) t1) u (asucc -g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead (CHead c0 (Bind -Abst) t1) (Bind Abst) u) t a4)))))) H9 u H8) in (let H12 \def (eq_ind T u0 -(\lambda (t1: T).(arity g (CHead c0 (Bind Abst) t1) t a2)) H10 u H8) in (let -H13 \def (eq_ind T u0 (\lambda (t1: T).((eq T t1 (THead (Bind Abst) u t)) \to -(ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A (asucc g a1) (AHead a3 -a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g a3)))) -(\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t a4)))))) -H2 u H8) in (let H14 \def (eq_ind T u0 (\lambda (t1: T).(arity g c0 t1 (asucc -g a1))) H1 u H8) in (ex3_2_intro A A (\lambda (a3: A).(\lambda (a4: A).(eq A -(AHead a1 a2) (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u -(asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind -Abst) u) t a4))) a1 a2 (refl_equal A (AHead a1 a2)) H14 H12))))))))) -H6)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: A).(\lambda -(_: (arity g c0 u0 a1)).(\lambda (_: (((eq T u0 (THead (Bind Abst) u t)) \to -(ex3_2 A A (\lambda (a2: A).(\lambda (a3: A).(eq A a1 (AHead a2 a3)))) -(\lambda (a2: A).(\lambda (_: A).(arity g c0 u (asucc g a2)))) (\lambda (_: -A).(\lambda (a3: A).(arity g (CHead c0 (Bind Abst) u) t a3))))))).(\lambda -(t0: T).(\lambda (a2: A).(\lambda (_: (arity g c0 t0 (AHead a1 a2))).(\lambda -(_: (((eq T t0 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3: -A).(\lambda (a4: A).(eq A (AHead a1 a2) (AHead a3 a4)))) (\lambda (a3: -A).(\lambda (_: A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda -(a4: A).(arity g (CHead c0 (Bind Abst) u) t a4))))))).(\lambda (H5: (eq T -(THead (Flat Appl) u0 t0) (THead (Bind Abst) u t))).(let H6 \def (eq_ind T -(THead (Flat Appl) u0 t0) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind -Abst) u t) H5) in (False_ind (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq -A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g -a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t -a4)))) H6)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a0: -A).(\lambda (_: (arity g c0 u0 (asucc g a0))).(\lambda (_: (((eq T u0 (THead -(Bind Abst) u t)) \to (ex3_2 A A (\lambda (a1: A).(\lambda (a2: A).(eq A -(asucc g a0) (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g c0 u -(asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g (CHead c0 (Bind -Abst) u) t a2))))))).(\lambda (t0: T).(\lambda (_: (arity g c0 t0 -a0)).(\lambda (_: (((eq T t0 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda -(a1: A).(\lambda (a2: A).(eq A a0 (AHead a1 a2)))) (\lambda (a1: A).(\lambda -(_: A).(arity g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity -g (CHead c0 (Bind Abst) u) t a2))))))).(\lambda (H5: (eq T (THead (Flat Cast) -u0 t0) (THead (Bind Abst) u t))).(let H6 \def (eq_ind T (THead (Flat Cast) u0 -t0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort -_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) -\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u t) -H5) in (False_ind (ex3_2 A A (\lambda (a1: A).(\lambda (a2: A).(eq A a0 -(AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g c0 u (asucc g -a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g (CHead c0 (Bind Abst) u) t -a2)))) H6))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (a1: -A).(\lambda (H1: (arity g c0 t0 a1)).(\lambda (H2: (((eq T t0 (THead (Bind -Abst) u t)) \to (ex3_2 A A (\lambda (a2: A).(\lambda (a3: A).(eq A a1 (AHead -a2 a3)))) (\lambda (a2: A).(\lambda (_: A).(arity g c0 u (asucc g a2)))) -(\lambda (_: A).(\lambda (a3: A).(arity g (CHead c0 (Bind Abst) u) t -a3))))))).(\lambda (a2: A).(\lambda (H3: (leq g a1 a2)).(\lambda (H4: (eq T -t0 (THead (Bind Abst) u t))).(let H5 \def (f_equal T T (\lambda (e: T).e) t0 -(THead (Bind Abst) u t) H4) in (let H6 \def (eq_ind T t0 (\lambda (t1: -T).((eq T t1 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3: -A).(\lambda (a4: A).(eq A a1 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: -A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g -(CHead c0 (Bind Abst) u) t a4)))))) H2 (THead (Bind Abst) u t) H5) in (let H7 -\def (eq_ind T t0 (\lambda (t1: T).(arity g c0 t1 a1)) H1 (THead (Bind Abst) -u t) H5) in (let H8 \def (H6 (refl_equal T (THead (Bind Abst) u t))) in -(ex3_2_ind A A (\lambda (a3: A).(\lambda (a4: A).(eq A a1 (AHead a3 a4)))) -(\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g a3)))) (\lambda (_: -A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t a4))) (ex3_2 A A -(\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: -A).(\lambda (_: A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda -(a4: A).(arity g (CHead c0 (Bind Abst) u) t a4)))) (\lambda (x0: A).(\lambda -(x1: A).(\lambda (H9: (eq A a1 (AHead x0 x1))).(\lambda (H10: (arity g c0 u -(asucc g x0))).(\lambda (H11: (arity g (CHead c0 (Bind Abst) u) t x1)).(let -H12 \def (eq_ind A a1 (\lambda (a0: A).(leq g a0 a2)) H3 (AHead x0 x1) H9) in -(let H13 \def (eq_ind A a1 (\lambda (a0: A).(arity g c0 (THead (Bind Abst) u -t) a0)) H7 (AHead x0 x1) H9) in (let H_x \def (leq_gen_head1 g x0 x1 a2 H12) -in (let H14 \def H_x in (ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq -g x0 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g x1 a4))) (\lambda (a3: -A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) (ex3_2 A A (\lambda (a3: -A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: -A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g -(CHead c0 (Bind Abst) u) t a4)))) (\lambda (x2: A).(\lambda (x3: A).(\lambda -(H15: (leq g x0 x2)).(\lambda (H16: (leq g x1 x3)).(\lambda (H17: (eq A a2 -(AHead x2 x3))).(let H18 \def (f_equal A A (\lambda (e: A).e) a2 (AHead x2 -x3) H17) in (eq_ind_r A (AHead x2 x3) (\lambda (a0: A).(ex3_2 A A (\lambda -(a3: A).(\lambda (a4: A).(eq A a0 (AHead a3 a4)))) (\lambda (a3: A).(\lambda -(_: A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity -g (CHead c0 (Bind Abst) u) t a4))))) (ex3_2_intro A A (\lambda (a3: -A).(\lambda (a4: A).(eq A (AHead x2 x3) (AHead a3 a4)))) (\lambda (a3: -A).(\lambda (_: A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda -(a4: A).(arity g (CHead c0 (Bind Abst) u) t a4))) x2 x3 (refl_equal A (AHead -x2 x3)) (arity_repl g c0 u (asucc g x0) H10 (asucc g x2) (asucc_repl g x0 x2 -H15)) (arity_repl g (CHead c0 (Bind Abst) u) t x1 H11 x3 H16)) a2 H18))))))) -H14)))))))))) H8))))))))))))) c y a H0))) H)))))). -(* COMMENTS -Initial nodes: 4265 -END *) - -theorem arity_gen_appl: - \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (a2: -A).((arity g c (THead (Flat Appl) u t) a2) \to (ex2 A (\lambda (a1: A).(arity -g c u a1)) (\lambda (a1: A).(arity g c t (AHead a1 a2))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (a2: -A).(\lambda (H: (arity g c (THead (Flat Appl) u t) a2)).(insert_eq T (THead -(Flat Appl) u t) (\lambda (t0: T).(arity g c t0 a2)) (\lambda (_: T).(ex2 A -(\lambda (a1: A).(arity g c u a1)) (\lambda (a1: A).(arity g c t (AHead a1 -a2))))) (\lambda (y: T).(\lambda (H0: (arity g c y a2)).(arity_ind g (\lambda -(c0: C).(\lambda (t0: T).(\lambda (a: A).((eq T t0 (THead (Flat Appl) u t)) -\to (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t -(AHead a1 a)))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (H1: (eq T -(TSort n) (THead (Flat Appl) u t))).(let H2 \def (eq_ind T (TSort n) (\lambda -(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow -False])) I (THead (Flat Appl) u t) H1) in (False_ind (ex2 A (\lambda (a1: -A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t (AHead a1 (ASort O -n))))) H2))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: -nat).(\lambda (_: (getl i c0 (CHead d (Bind Abbr) u0))).(\lambda (a: -A).(\lambda (_: (arity g d u0 a)).(\lambda (_: (((eq T u0 (THead (Flat Appl) -u t)) \to (ex2 A (\lambda (a1: A).(arity g d u a1)) (\lambda (a1: A).(arity g -d t (AHead a1 a))))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat Appl) u -t))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) u -t) H4) in (False_ind (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: -A).(arity g c0 t (AHead a1 a)))) H5))))))))))) (\lambda (c0: C).(\lambda (d: -C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind -Abst) u0))).(\lambda (a: A).(\lambda (_: (arity g d u0 (asucc g a))).(\lambda -(_: (((eq T u0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda (a1: A).(arity g -d u a1)) (\lambda (a1: A).(arity g d t (AHead a1 (asucc g a)))))))).(\lambda -(H4: (eq T (TLRef i) (THead (Flat Appl) u t))).(let H5 \def (eq_ind T (TLRef -i) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort -_) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow -False])) I (THead (Flat Appl) u t) H4) in (False_ind (ex2 A (\lambda (a1: -A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t (AHead a1 a)))) -H5))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (c0: -C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u0 -a1)).(\lambda (_: (((eq T u0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda -(a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3 -a1))))))).(\lambda (t0: T).(\lambda (a0: A).(\lambda (_: (arity g (CHead c0 -(Bind b) u0) t0 a0)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u t)) \to -(ex2 A (\lambda (a3: A).(arity g (CHead c0 (Bind b) u0) u a3)) (\lambda (a3: -A).(arity g (CHead c0 (Bind b) u0) t (AHead a3 a0))))))).(\lambda (H6: (eq T -(THead (Bind b) u0 t0) (THead (Flat Appl) u t))).(let H7 \def (eq_ind T -(THead (Bind b) u0 t0) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat -Appl) u t) H6) in (False_ind (ex2 A (\lambda (a3: A).(arity g c0 u a3)) -(\lambda (a3: A).(arity g c0 t (AHead a3 a0)))) H7)))))))))))))) (\lambda -(c0: C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u0 (asucc -g a1))).(\lambda (_: (((eq T u0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda -(a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3 (asucc g -a1)))))))).(\lambda (t0: T).(\lambda (a0: A).(\lambda (_: (arity g (CHead c0 -(Bind Abst) u0) t0 a0)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u t)) \to -(ex2 A (\lambda (a3: A).(arity g (CHead c0 (Bind Abst) u0) u a3)) (\lambda -(a3: A).(arity g (CHead c0 (Bind Abst) u0) t (AHead a3 a0))))))).(\lambda -(H5: (eq T (THead (Bind Abst) u0 t0) (THead (Flat Appl) u t))).(let H6 \def -(eq_ind T (THead (Bind Abst) u0 t0) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow -False])])) I (THead (Flat Appl) u t) H5) in (False_ind (ex2 A (\lambda (a3: -A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3 (AHead a1 -a0))))) H6)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: -A).(\lambda (H1: (arity g c0 u0 a1)).(\lambda (H2: (((eq T u0 (THead (Flat -Appl) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3: -A).(arity g c0 t (AHead a3 a1))))))).(\lambda (t0: T).(\lambda (a0: -A).(\lambda (H3: (arity g c0 t0 (AHead a1 a0))).(\lambda (H4: (((eq T t0 -(THead (Flat Appl) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) -(\lambda (a3: A).(arity g c0 t (AHead a3 (AHead a1 a0)))))))).(\lambda (H5: -(eq T (THead (Flat Appl) u0 t0) (THead (Flat Appl) u t))).(let H6 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t1 _) -\Rightarrow t1])) (THead (Flat Appl) u0 t0) (THead (Flat Appl) u t) H5) in -((let H7 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: -T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ -t1) \Rightarrow t1])) (THead (Flat Appl) u0 t0) (THead (Flat Appl) u t) H5) -in (\lambda (H8: (eq T u0 u)).(let H9 \def (eq_ind T t0 (\lambda (t1: T).((eq -T t1 (THead (Flat Appl) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) -(\lambda (a3: A).(arity g c0 t (AHead a3 (AHead a1 a0))))))) H4 t H7) in (let -H10 \def (eq_ind T t0 (\lambda (t1: T).(arity g c0 t1 (AHead a1 a0))) H3 t -H7) in (let H11 \def (eq_ind T u0 (\lambda (t1: T).((eq T t1 (THead (Flat -Appl) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3: -A).(arity g c0 t (AHead a3 a1)))))) H2 u H8) in (let H12 \def (eq_ind T u0 -(\lambda (t1: T).(arity g c0 t1 a1)) H1 u H8) in (ex_intro2 A (\lambda (a3: -A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3 a0))) a1 H12 -H10))))))) H6)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a: -A).(\lambda (_: (arity g c0 u0 (asucc g a))).(\lambda (_: (((eq T u0 (THead -(Flat Appl) u t)) \to (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda -(a1: A).(arity g c0 t (AHead a1 (asucc g a)))))))).(\lambda (t0: T).(\lambda -(_: (arity g c0 t0 a)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u t)) \to -(ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t -(AHead a1 a))))))).(\lambda (H5: (eq T (THead (Flat Cast) u0 t0) (THead (Flat -Appl) u t))).(let H6 \def (eq_ind T (THead (Flat Cast) u0 t0) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K -return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) -\Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl \Rightarrow -False | Cast \Rightarrow True])])])) I (THead (Flat Appl) u t) H5) in -(False_ind (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity -g c0 t (AHead a1 a)))) H6))))))))))) (\lambda (c0: C).(\lambda (t0: -T).(\lambda (a1: A).(\lambda (H1: (arity g c0 t0 a1)).(\lambda (H2: (((eq T -t0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) -(\lambda (a3: A).(arity g c0 t (AHead a3 a1))))))).(\lambda (a0: A).(\lambda -(H3: (leq g a1 a0)).(\lambda (H4: (eq T t0 (THead (Flat Appl) u t))).(let H5 -\def (f_equal T T (\lambda (e: T).e) t0 (THead (Flat Appl) u t) H4) in (let -H6 \def (eq_ind T t0 (\lambda (t1: T).((eq T t1 (THead (Flat Appl) u t)) \to -(ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t -(AHead a3 a1)))))) H2 (THead (Flat Appl) u t) H5) in (let H7 \def (eq_ind T -t0 (\lambda (t1: T).(arity g c0 t1 a1)) H1 (THead (Flat Appl) u t) H5) in -(let H8 \def (H6 (refl_equal T (THead (Flat Appl) u t))) in (ex2_ind A -(\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3 -a1))) (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 -t (AHead a3 a0)))) (\lambda (x: A).(\lambda (H9: (arity g c0 u x)).(\lambda -(H10: (arity g c0 t (AHead x a1))).(ex_intro2 A (\lambda (a3: A).(arity g c0 -u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3 a0))) x H9 (arity_repl g c0 t -(AHead x a1) H10 (AHead x a0) (leq_head g x x (leq_refl g x) a1 a0 H3)))))) -H8))))))))))))) c y a2 H0))) H)))))). -(* COMMENTS -Initial nodes: 2277 -END *) - -theorem arity_gen_cast: - \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (a: -A).((arity g c (THead (Flat Cast) u t) a) \to (land (arity g c u (asucc g a)) -(arity g c t a))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (a: -A).(\lambda (H: (arity g c (THead (Flat Cast) u t) a)).(insert_eq T (THead -(Flat Cast) u t) (\lambda (t0: T).(arity g c t0 a)) (\lambda (_: T).(land -(arity g c u (asucc g a)) (arity g c t a))) (\lambda (y: T).(\lambda (H0: -(arity g c y a)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (a0: -A).((eq T t0 (THead (Flat Cast) u t)) \to (land (arity g c0 u (asucc g a0)) -(arity g c0 t a0)))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (H1: (eq T -(TSort n) (THead (Flat Cast) u t))).(let H2 \def (eq_ind T (TSort n) (\lambda -(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow -False])) I (THead (Flat Cast) u t) H1) in (False_ind (land (arity g c0 u -(asucc g (ASort O n))) (arity g c0 t (ASort O n))) H2))))) (\lambda (c0: -C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0 -(CHead d (Bind Abbr) u0))).(\lambda (a0: A).(\lambda (_: (arity g d u0 -a0)).(\lambda (_: (((eq T u0 (THead (Flat Cast) u t)) \to (land (arity g d u -(asucc g a0)) (arity g d t a0))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat -Cast) u t))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) u -t) H4) in (False_ind (land (arity g c0 u (asucc g a0)) (arity g c0 t a0)) -H5))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: -nat).(\lambda (_: (getl i c0 (CHead d (Bind Abst) u0))).(\lambda (a0: -A).(\lambda (_: (arity g d u0 (asucc g a0))).(\lambda (_: (((eq T u0 (THead -(Flat Cast) u t)) \to (land (arity g d u (asucc g (asucc g a0))) (arity g d t -(asucc g a0)))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat Cast) u -t))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) u -t) H4) in (False_ind (land (arity g c0 u (asucc g a0)) (arity g c0 t a0)) -H5))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (c0: -C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u0 -a1)).(\lambda (_: (((eq T u0 (THead (Flat Cast) u t)) \to (land (arity g c0 u -(asucc g a1)) (arity g c0 t a1))))).(\lambda (t0: T).(\lambda (a2: -A).(\lambda (_: (arity g (CHead c0 (Bind b) u0) t0 a2)).(\lambda (_: (((eq T -t0 (THead (Flat Cast) u t)) \to (land (arity g (CHead c0 (Bind b) u0) u -(asucc g a2)) (arity g (CHead c0 (Bind b) u0) t a2))))).(\lambda (H6: (eq T -(THead (Bind b) u0 t0) (THead (Flat Cast) u t))).(let H7 \def (eq_ind T -(THead (Bind b) u0 t0) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat -Cast) u t) H6) in (False_ind (land (arity g c0 u (asucc g a2)) (arity g c0 t -a2)) H7)))))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: -A).(\lambda (_: (arity g c0 u0 (asucc g a1))).(\lambda (_: (((eq T u0 (THead -(Flat Cast) u t)) \to (land (arity g c0 u (asucc g (asucc g a1))) (arity g c0 -t (asucc g a1)))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g -(CHead c0 (Bind Abst) u0) t0 a2)).(\lambda (_: (((eq T t0 (THead (Flat Cast) -u t)) \to (land (arity g (CHead c0 (Bind Abst) u0) u (asucc g a2)) (arity g -(CHead c0 (Bind Abst) u0) t a2))))).(\lambda (H5: (eq T (THead (Bind Abst) u0 -t0) (THead (Flat Cast) u t))).(let H6 \def (eq_ind T (THead (Bind Abst) u0 -t0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort -_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) -\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Cast) u t) -H5) in (False_ind (land (arity g c0 u (asucc g (AHead a1 a2))) (arity g c0 t -(AHead a1 a2))) H6)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda -(a1: A).(\lambda (_: (arity g c0 u0 a1)).(\lambda (_: (((eq T u0 (THead (Flat -Cast) u t)) \to (land (arity g c0 u (asucc g a1)) (arity g c0 t -a1))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g c0 t0 (AHead -a1 a2))).(\lambda (_: (((eq T t0 (THead (Flat Cast) u t)) \to (land (arity g -c0 u (asucc g (AHead a1 a2))) (arity g c0 t (AHead a1 a2)))))).(\lambda (H5: -(eq T (THead (Flat Appl) u0 t0) (THead (Flat Cast) u t))).(let H6 \def -(eq_ind T (THead (Flat Appl) u0 t0) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f -in F return (\lambda (_: F).Prop) with [Appl \Rightarrow True | Cast -\Rightarrow False])])])) I (THead (Flat Cast) u t) H5) in (False_ind (land -(arity g c0 u (asucc g a2)) (arity g c0 t a2)) H6)))))))))))) (\lambda (c0: -C).(\lambda (u0: T).(\lambda (a0: A).(\lambda (H1: (arity g c0 u0 (asucc g -a0))).(\lambda (H2: (((eq T u0 (THead (Flat Cast) u t)) \to (land (arity g c0 -u (asucc g (asucc g a0))) (arity g c0 t (asucc g a0)))))).(\lambda (t0: -T).(\lambda (H3: (arity g c0 t0 a0)).(\lambda (H4: (((eq T t0 (THead (Flat -Cast) u t)) \to (land (arity g c0 u (asucc g a0)) (arity g c0 t -a0))))).(\lambda (H5: (eq T (THead (Flat Cast) u0 t0) (THead (Flat Cast) u -t))).(let H6 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda -(_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead -_ t1 _) \Rightarrow t1])) (THead (Flat Cast) u0 t0) (THead (Flat Cast) u t) -H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 -| (THead _ _ t1) \Rightarrow t1])) (THead (Flat Cast) u0 t0) (THead (Flat -Cast) u t) H5) in (\lambda (H8: (eq T u0 u)).(let H9 \def (eq_ind T t0 -(\lambda (t1: T).((eq T t1 (THead (Flat Cast) u t)) \to (land (arity g c0 u -(asucc g a0)) (arity g c0 t a0)))) H4 t H7) in (let H10 \def (eq_ind T t0 -(\lambda (t1: T).(arity g c0 t1 a0)) H3 t H7) in (let H11 \def (eq_ind T u0 -(\lambda (t1: T).((eq T t1 (THead (Flat Cast) u t)) \to (land (arity g c0 u -(asucc g (asucc g a0))) (arity g c0 t (asucc g a0))))) H2 u H8) in (let H12 -\def (eq_ind T u0 (\lambda (t1: T).(arity g c0 t1 (asucc g a0))) H1 u H8) in -(conj (arity g c0 u (asucc g a0)) (arity g c0 t a0) H12 H10))))))) -H6))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda -(H1: (arity g c0 t0 a1)).(\lambda (H2: (((eq T t0 (THead (Flat Cast) u t)) -\to (land (arity g c0 u (asucc g a1)) (arity g c0 t a1))))).(\lambda (a2: -A).(\lambda (H3: (leq g a1 a2)).(\lambda (H4: (eq T t0 (THead (Flat Cast) u -t))).(let H5 \def (f_equal T T (\lambda (e: T).e) t0 (THead (Flat Cast) u t) -H4) in (let H6 \def (eq_ind T t0 (\lambda (t1: T).((eq T t1 (THead (Flat -Cast) u t)) \to (land (arity g c0 u (asucc g a1)) (arity g c0 t a1)))) H2 -(THead (Flat Cast) u t) H5) in (let H7 \def (eq_ind T t0 (\lambda (t1: -T).(arity g c0 t1 a1)) H1 (THead (Flat Cast) u t) H5) in (let H8 \def (H6 -(refl_equal T (THead (Flat Cast) u t))) in (land_ind (arity g c0 u (asucc g -a1)) (arity g c0 t a1) (land (arity g c0 u (asucc g a2)) (arity g c0 t a2)) -(\lambda (H9: (arity g c0 u (asucc g a1))).(\lambda (H10: (arity g c0 t -a1)).(conj (arity g c0 u (asucc g a2)) (arity g c0 t a2) (arity_repl g c0 u -(asucc g a1) H9 (asucc g a2) (asucc_repl g a1 a2 H3)) (arity_repl g c0 t a1 -H10 a2 H3)))) H8))))))))))))) c y a H0))) H)))))). -(* COMMENTS -Initial nodes: 2147 -END *) - -theorem arity_gen_appls: - \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (vs: TList).(\forall -(a2: A).((arity g c (THeads (Flat Appl) vs t) a2) \to (ex A (\lambda (a: -A).(arity g c t a)))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (vs: -TList).(TList_ind (\lambda (t0: TList).(\forall (a2: A).((arity g c (THeads -(Flat Appl) t0 t) a2) \to (ex A (\lambda (a: A).(arity g c t a)))))) (\lambda -(a2: A).(\lambda (H: (arity g c t a2)).(ex_intro A (\lambda (a: A).(arity g c -t a)) a2 H))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H: ((\forall -(a2: A).((arity g c (THeads (Flat Appl) t1 t) a2) \to (ex A (\lambda (a: -A).(arity g c t a))))))).(\lambda (a2: A).(\lambda (H0: (arity g c (THead -(Flat Appl) t0 (THeads (Flat Appl) t1 t)) a2)).(let H1 \def (arity_gen_appl g -c t0 (THeads (Flat Appl) t1 t) a2 H0) in (ex2_ind A (\lambda (a1: A).(arity g -c t0 a1)) (\lambda (a1: A).(arity g c (THeads (Flat Appl) t1 t) (AHead a1 -a2))) (ex A (\lambda (a: A).(arity g c t a))) (\lambda (x: A).(\lambda (_: -(arity g c t0 x)).(\lambda (H3: (arity g c (THeads (Flat Appl) t1 t) (AHead x -a2))).(let H_x \def (H (AHead x a2) H3) in (let H4 \def H_x in (ex_ind A -(\lambda (a: A).(arity g c t a)) (ex A (\lambda (a: A).(arity g c t a))) -(\lambda (x0: A).(\lambda (H5: (arity g c t x0)).(ex_intro A (\lambda (a: -A).(arity g c t a)) x0 H5))) H4)))))) H1))))))) vs)))). -(* COMMENTS -Initial nodes: 341 -END *) - -theorem arity_gen_lift: - \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).(\forall (h: -nat).(\forall (d: nat).((arity g c1 (lift h d t) a) \to (\forall (c2: -C).((drop h d c1 c2) \to (arity g c2 t a))))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (a: A).(\lambda (h: -nat).(\lambda (d: nat).(\lambda (H: (arity g c1 (lift h d t) a)).(insert_eq T -(lift h d t) (\lambda (t0: T).(arity g c1 t0 a)) (\lambda (_: T).(\forall -(c2: C).((drop h d c1 c2) \to (arity g c2 t a)))) (\lambda (y: T).(\lambda -(H0: (arity g c1 y a)).(unintro T t (\lambda (t0: T).((eq T y (lift h d t0)) -\to (\forall (c2: C).((drop h d c1 c2) \to (arity g c2 t0 a))))) (unintro nat -d (\lambda (n: nat).(\forall (x: T).((eq T y (lift h n x)) \to (\forall (c2: -C).((drop h n c1 c2) \to (arity g c2 x a)))))) (arity_ind g (\lambda (c: -C).(\lambda (t0: T).(\lambda (a0: A).(\forall (x: nat).(\forall (x0: T).((eq -T t0 (lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to (arity g c2 x0 -a0))))))))) (\lambda (c: C).(\lambda (n: nat).(\lambda (x: nat).(\lambda (x0: -T).(\lambda (H1: (eq T (TSort n) (lift h x x0))).(\lambda (c2: C).(\lambda -(_: (drop h x c c2)).(eq_ind_r T (TSort n) (\lambda (t0: T).(arity g c2 t0 -(ASort O n))) (arity_sort g c2 n) x0 (lift_gen_sort h x n x0 H1))))))))) -(\lambda (c: C).(\lambda (d0: C).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H1: (getl i c (CHead d0 (Bind Abbr) u))).(\lambda (a0: A).(\lambda (H2: -(arity g d0 u a0)).(\lambda (H3: ((\forall (x: nat).(\forall (x0: T).((eq T u -(lift h x x0)) \to (\forall (c2: C).((drop h x d0 c2) \to (arity g c2 x0 -a0)))))))).(\lambda (x: nat).(\lambda (x0: T).(\lambda (H4: (eq T (TLRef i) -(lift h x x0))).(\lambda (c2: C).(\lambda (H5: (drop h x c c2)).(let H_x \def -(lift_gen_lref x0 x h i H4) in (let H6 \def H_x in (or_ind (land (lt i x) (eq -T x0 (TLRef i))) (land (le (plus x h) i) (eq T x0 (TLRef (minus i h)))) -(arity g c2 x0 a0) (\lambda (H7: (land (lt i x) (eq T x0 (TLRef -i)))).(land_ind (lt i x) (eq T x0 (TLRef i)) (arity g c2 x0 a0) (\lambda (H8: -(lt i x)).(\lambda (H9: (eq T x0 (TLRef i))).(eq_ind_r T (TLRef i) (\lambda -(t0: T).(arity g c2 t0 a0)) (let H10 \def (eq_ind nat x (\lambda (n: -nat).(drop h n c c2)) H5 (S (plus i (minus x (S i)))) (lt_plus_minus i x H8)) -in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (minus x (S -i)) v)))) (\lambda (v: T).(\lambda (e0: C).(getl i c2 (CHead e0 (Bind Abbr) -v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (minus x (S i)) d0 e0))) -(arity g c2 (TLRef i) a0) (\lambda (x1: T).(\lambda (x2: C).(\lambda (H11: -(eq T u (lift h (minus x (S i)) x1))).(\lambda (H12: (getl i c2 (CHead x2 -(Bind Abbr) x1))).(\lambda (H13: (drop h (minus x (S i)) d0 x2)).(let H14 -\def (eq_ind T u (\lambda (t0: T).(\forall (x3: nat).(\forall (x4: T).((eq T -t0 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 d0 c3) \to (arity g c3 x4 -a0))))))) H3 (lift h (minus x (S i)) x1) H11) in (let H15 \def (eq_ind T u -(\lambda (t0: T).(arity g d0 t0 a0)) H2 (lift h (minus x (S i)) x1) H11) in -(arity_abbr g c2 x2 x1 i H12 a0 (H14 (minus x (S i)) x1 (refl_equal T (lift h -(minus x (S i)) x1)) x2 H13))))))))) (getl_drop_conf_lt Abbr c d0 u i H1 c2 h -(minus x (S i)) H10))) x0 H9))) H7)) (\lambda (H7: (land (le (plus x h) i) -(eq T x0 (TLRef (minus i h))))).(land_ind (le (plus x h) i) (eq T x0 (TLRef -(minus i h))) (arity g c2 x0 a0) (\lambda (H8: (le (plus x h) i)).(\lambda -(H9: (eq T x0 (TLRef (minus i h)))).(eq_ind_r T (TLRef (minus i h)) (\lambda -(t0: T).(arity g c2 t0 a0)) (arity_abbr g c2 d0 u (minus i h) -(getl_drop_conf_ge i (CHead d0 (Bind Abbr) u) c H1 c2 h x H5 H8) a0 H2) x0 -H9))) H7)) H6)))))))))))))))) (\lambda (c: C).(\lambda (d0: C).(\lambda (u: -T).(\lambda (i: nat).(\lambda (H1: (getl i c (CHead d0 (Bind Abst) -u))).(\lambda (a0: A).(\lambda (H2: (arity g d0 u (asucc g a0))).(\lambda -(H3: ((\forall (x: nat).(\forall (x0: T).((eq T u (lift h x x0)) \to (\forall -(c2: C).((drop h x d0 c2) \to (arity g c2 x0 (asucc g a0))))))))).(\lambda -(x: nat).(\lambda (x0: T).(\lambda (H4: (eq T (TLRef i) (lift h x -x0))).(\lambda (c2: C).(\lambda (H5: (drop h x c c2)).(let H_x \def -(lift_gen_lref x0 x h i H4) in (let H6 \def H_x in (or_ind (land (lt i x) (eq -T x0 (TLRef i))) (land (le (plus x h) i) (eq T x0 (TLRef (minus i h)))) -(arity g c2 x0 a0) (\lambda (H7: (land (lt i x) (eq T x0 (TLRef -i)))).(land_ind (lt i x) (eq T x0 (TLRef i)) (arity g c2 x0 a0) (\lambda (H8: -(lt i x)).(\lambda (H9: (eq T x0 (TLRef i))).(eq_ind_r T (TLRef i) (\lambda -(t0: T).(arity g c2 t0 a0)) (let H10 \def (eq_ind nat x (\lambda (n: -nat).(drop h n c c2)) H5 (S (plus i (minus x (S i)))) (lt_plus_minus i x H8)) -in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (minus x (S -i)) v)))) (\lambda (v: T).(\lambda (e0: C).(getl i c2 (CHead e0 (Bind Abst) -v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (minus x (S i)) d0 e0))) -(arity g c2 (TLRef i) a0) (\lambda (x1: T).(\lambda (x2: C).(\lambda (H11: -(eq T u (lift h (minus x (S i)) x1))).(\lambda (H12: (getl i c2 (CHead x2 -(Bind Abst) x1))).(\lambda (H13: (drop h (minus x (S i)) d0 x2)).(let H14 -\def (eq_ind T u (\lambda (t0: T).(\forall (x3: nat).(\forall (x4: T).((eq T -t0 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 d0 c3) \to (arity g c3 x4 -(asucc g a0)))))))) H3 (lift h (minus x (S i)) x1) H11) in (let H15 \def -(eq_ind T u (\lambda (t0: T).(arity g d0 t0 (asucc g a0))) H2 (lift h (minus -x (S i)) x1) H11) in (arity_abst g c2 x2 x1 i H12 a0 (H14 (minus x (S i)) x1 -(refl_equal T (lift h (minus x (S i)) x1)) x2 H13))))))))) (getl_drop_conf_lt -Abst c d0 u i H1 c2 h (minus x (S i)) H10))) x0 H9))) H7)) (\lambda (H7: -(land (le (plus x h) i) (eq T x0 (TLRef (minus i h))))).(land_ind (le (plus x -h) i) (eq T x0 (TLRef (minus i h))) (arity g c2 x0 a0) (\lambda (H8: (le -(plus x h) i)).(\lambda (H9: (eq T x0 (TLRef (minus i h)))).(eq_ind_r T -(TLRef (minus i h)) (\lambda (t0: T).(arity g c2 t0 a0)) (arity_abst g c2 d0 -u (minus i h) (getl_drop_conf_ge i (CHead d0 (Bind Abst) u) c H1 c2 h x H5 -H8) a0 H2) x0 H9))) H7)) H6)))))))))))))))) (\lambda (b: B).(\lambda (H1: -(not (eq B b Abst))).(\lambda (c: C).(\lambda (u: T).(\lambda (a1: -A).(\lambda (H2: (arity g c u a1)).(\lambda (H3: ((\forall (x: nat).(\forall -(x0: T).((eq T u (lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to -(arity g c2 x0 a1)))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (H4: -(arity g (CHead c (Bind b) u) t0 a2)).(\lambda (H5: ((\forall (x: -nat).(\forall (x0: T).((eq T t0 (lift h x x0)) \to (\forall (c2: C).((drop h -x (CHead c (Bind b) u) c2) \to (arity g c2 x0 a2)))))))).(\lambda (x: -nat).(\lambda (x0: T).(\lambda (H6: (eq T (THead (Bind b) u t0) (lift h x -x0))).(\lambda (c2: C).(\lambda (H7: (drop h x c c2)).(ex3_2_ind T T (\lambda -(y0: T).(\lambda (z: T).(eq T x0 (THead (Bind b) y0 z)))) (\lambda (y0: -T).(\lambda (_: T).(eq T u (lift h x y0)))) (\lambda (_: T).(\lambda (z: -T).(eq T t0 (lift h (S x) z)))) (arity g c2 x0 a2) (\lambda (x1: T).(\lambda -(x2: T).(\lambda (H8: (eq T x0 (THead (Bind b) x1 x2))).(\lambda (H9: (eq T u -(lift h x x1))).(\lambda (H10: (eq T t0 (lift h (S x) x2))).(eq_ind_r T -(THead (Bind b) x1 x2) (\lambda (t1: T).(arity g c2 t1 a2)) (let H11 \def -(eq_ind T t0 (\lambda (t1: T).(\forall (x3: nat).(\forall (x4: T).((eq T t1 -(lift h x3 x4)) \to (\forall (c3: C).((drop h x3 (CHead c (Bind b) u) c3) \to -(arity g c3 x4 a2))))))) H5 (lift h (S x) x2) H10) in (let H12 \def (eq_ind T -t0 (\lambda (t1: T).(arity g (CHead c (Bind b) u) t1 a2)) H4 (lift h (S x) -x2) H10) in (let H13 \def (eq_ind T u (\lambda (t1: T).(arity g (CHead c -(Bind b) t1) (lift h (S x) x2) a2)) H12 (lift h x x1) H9) in (let H14 \def -(eq_ind T u (\lambda (t1: T).(\forall (x3: nat).(\forall (x4: T).((eq T (lift -h (S x) x2) (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 (CHead c (Bind -b) t1) c3) \to (arity g c3 x4 a2))))))) H11 (lift h x x1) H9) in (let H15 -\def (eq_ind T u (\lambda (t1: T).(\forall (x3: nat).(\forall (x4: T).((eq T -t1 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 c c3) \to (arity g c3 x4 -a1))))))) H3 (lift h x x1) H9) in (let H16 \def (eq_ind T u (\lambda (t1: -T).(arity g c t1 a1)) H2 (lift h x x1) H9) in (arity_bind g b H1 c2 x1 a1 -(H15 x x1 (refl_equal T (lift h x x1)) c2 H7) x2 a2 (H14 (S x) x2 (refl_equal -T (lift h (S x) x2)) (CHead c2 (Bind b) x1) (drop_skip_bind h x c c2 H7 b -x1))))))))) x0 H8)))))) (lift_gen_bind b u t0 x0 h x H6)))))))))))))))))) -(\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H1: (arity g c u -(asucc g a1))).(\lambda (H2: ((\forall (x: nat).(\forall (x0: T).((eq T u -(lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to (arity g c2 x0 -(asucc g a1))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (H3: (arity g -(CHead c (Bind Abst) u) t0 a2)).(\lambda (H4: ((\forall (x: nat).(\forall -(x0: T).((eq T t0 (lift h x x0)) \to (\forall (c2: C).((drop h x (CHead c -(Bind Abst) u) c2) \to (arity g c2 x0 a2)))))))).(\lambda (x: nat).(\lambda -(x0: T).(\lambda (H5: (eq T (THead (Bind Abst) u t0) (lift h x x0))).(\lambda -(c2: C).(\lambda (H6: (drop h x c c2)).(ex3_2_ind T T (\lambda (y0: -T).(\lambda (z: T).(eq T x0 (THead (Bind Abst) y0 z)))) (\lambda (y0: -T).(\lambda (_: T).(eq T u (lift h x y0)))) (\lambda (_: T).(\lambda (z: -T).(eq T t0 (lift h (S x) z)))) (arity g c2 x0 (AHead a1 a2)) (\lambda (x1: -T).(\lambda (x2: T).(\lambda (H7: (eq T x0 (THead (Bind Abst) x1 -x2))).(\lambda (H8: (eq T u (lift h x x1))).(\lambda (H9: (eq T t0 (lift h (S -x) x2))).(eq_ind_r T (THead (Bind Abst) x1 x2) (\lambda (t1: T).(arity g c2 -t1 (AHead a1 a2))) (let H10 \def (eq_ind T t0 (\lambda (t1: T).(\forall (x3: -nat).(\forall (x4: T).((eq T t1 (lift h x3 x4)) \to (\forall (c3: C).((drop h -x3 (CHead c (Bind Abst) u) c3) \to (arity g c3 x4 a2))))))) H4 (lift h (S x) -x2) H9) in (let H11 \def (eq_ind T t0 (\lambda (t1: T).(arity g (CHead c -(Bind Abst) u) t1 a2)) H3 (lift h (S x) x2) H9) in (let H12 \def (eq_ind T u -(\lambda (t1: T).(arity g (CHead c (Bind Abst) t1) (lift h (S x) x2) a2)) H11 -(lift h x x1) H8) in (let H13 \def (eq_ind T u (\lambda (t1: T).(\forall (x3: -nat).(\forall (x4: T).((eq T (lift h (S x) x2) (lift h x3 x4)) \to (\forall -(c3: C).((drop h x3 (CHead c (Bind Abst) t1) c3) \to (arity g c3 x4 a2))))))) -H10 (lift h x x1) H8) in (let H14 \def (eq_ind T u (\lambda (t1: T).(\forall -(x3: nat).(\forall (x4: T).((eq T t1 (lift h x3 x4)) \to (\forall (c3: -C).((drop h x3 c c3) \to (arity g c3 x4 (asucc g a1)))))))) H2 (lift h x x1) -H8) in (let H15 \def (eq_ind T u (\lambda (t1: T).(arity g c t1 (asucc g -a1))) H1 (lift h x x1) H8) in (arity_head g c2 x1 a1 (H14 x x1 (refl_equal T -(lift h x x1)) c2 H6) x2 a2 (H13 (S x) x2 (refl_equal T (lift h (S x) x2)) -(CHead c2 (Bind Abst) x1) (drop_skip_bind h x c c2 H6 Abst x1))))))))) x0 -H7)))))) (lift_gen_bind Abst u t0 x0 h x H5)))))))))))))))) (\lambda (c: -C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H1: (arity g c u a1)).(\lambda -(H2: ((\forall (x: nat).(\forall (x0: T).((eq T u (lift h x x0)) \to (\forall -(c2: C).((drop h x c c2) \to (arity g c2 x0 a1)))))))).(\lambda (t0: -T).(\lambda (a2: A).(\lambda (H3: (arity g c t0 (AHead a1 a2))).(\lambda (H4: -((\forall (x: nat).(\forall (x0: T).((eq T t0 (lift h x x0)) \to (\forall -(c2: C).((drop h x c c2) \to (arity g c2 x0 (AHead a1 a2))))))))).(\lambda -(x: nat).(\lambda (x0: T).(\lambda (H5: (eq T (THead (Flat Appl) u t0) (lift -h x x0))).(\lambda (c2: C).(\lambda (H6: (drop h x c c2)).(ex3_2_ind T T -(\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Flat Appl) y0 z)))) -(\lambda (y0: T).(\lambda (_: T).(eq T u (lift h x y0)))) (\lambda (_: -T).(\lambda (z: T).(eq T t0 (lift h x z)))) (arity g c2 x0 a2) (\lambda (x1: -T).(\lambda (x2: T).(\lambda (H7: (eq T x0 (THead (Flat Appl) x1 -x2))).(\lambda (H8: (eq T u (lift h x x1))).(\lambda (H9: (eq T t0 (lift h x -x2))).(eq_ind_r T (THead (Flat Appl) x1 x2) (\lambda (t1: T).(arity g c2 t1 -a2)) (let H10 \def (eq_ind T t0 (\lambda (t1: T).(\forall (x3: nat).(\forall -(x4: T).((eq T t1 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 c c3) \to -(arity g c3 x4 (AHead a1 a2)))))))) H4 (lift h x x2) H9) in (let H11 \def -(eq_ind T t0 (\lambda (t1: T).(arity g c t1 (AHead a1 a2))) H3 (lift h x x2) -H9) in (let H12 \def (eq_ind T u (\lambda (t1: T).(\forall (x3: nat).(\forall -(x4: T).((eq T t1 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 c c3) \to -(arity g c3 x4 a1))))))) H2 (lift h x x1) H8) in (let H13 \def (eq_ind T u -(\lambda (t1: T).(arity g c t1 a1)) H1 (lift h x x1) H8) in (arity_appl g c2 -x1 a1 (H12 x x1 (refl_equal T (lift h x x1)) c2 H6) x2 a2 (H10 x x2 -(refl_equal T (lift h x x2)) c2 H6)))))) x0 H7)))))) (lift_gen_flat Appl u t0 -x0 h x H5)))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a0: -A).(\lambda (H1: (arity g c u (asucc g a0))).(\lambda (H2: ((\forall (x: -nat).(\forall (x0: T).((eq T u (lift h x x0)) \to (\forall (c2: C).((drop h x -c c2) \to (arity g c2 x0 (asucc g a0))))))))).(\lambda (t0: T).(\lambda (H3: -(arity g c t0 a0)).(\lambda (H4: ((\forall (x: nat).(\forall (x0: T).((eq T -t0 (lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to (arity g c2 x0 -a0)))))))).(\lambda (x: nat).(\lambda (x0: T).(\lambda (H5: (eq T (THead -(Flat Cast) u t0) (lift h x x0))).(\lambda (c2: C).(\lambda (H6: (drop h x c -c2)).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Flat -Cast) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u (lift h x y0)))) -(\lambda (_: T).(\lambda (z: T).(eq T t0 (lift h x z)))) (arity g c2 x0 a0) -(\lambda (x1: T).(\lambda (x2: T).(\lambda (H7: (eq T x0 (THead (Flat Cast) -x1 x2))).(\lambda (H8: (eq T u (lift h x x1))).(\lambda (H9: (eq T t0 (lift h -x x2))).(eq_ind_r T (THead (Flat Cast) x1 x2) (\lambda (t1: T).(arity g c2 t1 -a0)) (let H10 \def (eq_ind T t0 (\lambda (t1: T).(\forall (x3: nat).(\forall -(x4: T).((eq T t1 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 c c3) \to -(arity g c3 x4 a0))))))) H4 (lift h x x2) H9) in (let H11 \def (eq_ind T t0 -(\lambda (t1: T).(arity g c t1 a0)) H3 (lift h x x2) H9) in (let H12 \def -(eq_ind T u (\lambda (t1: T).(\forall (x3: nat).(\forall (x4: T).((eq T t1 -(lift h x3 x4)) \to (\forall (c3: C).((drop h x3 c c3) \to (arity g c3 x4 -(asucc g a0)))))))) H2 (lift h x x1) H8) in (let H13 \def (eq_ind T u -(\lambda (t1: T).(arity g c t1 (asucc g a0))) H1 (lift h x x1) H8) in -(arity_cast g c2 x1 a0 (H12 x x1 (refl_equal T (lift h x x1)) c2 H6) x2 (H10 -x x2 (refl_equal T (lift h x x2)) c2 H6)))))) x0 H7)))))) (lift_gen_flat Cast -u t0 x0 h x H5))))))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda (a1: -A).(\lambda (_: (arity g c t0 a1)).(\lambda (H2: ((\forall (x: nat).(\forall -(x0: T).((eq T t0 (lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to -(arity g c2 x0 a1)))))))).(\lambda (a2: A).(\lambda (H3: (leq g a1 -a2)).(\lambda (x: nat).(\lambda (x0: T).(\lambda (H4: (eq T t0 (lift h x -x0))).(\lambda (c2: C).(\lambda (H5: (drop h x c c2)).(arity_repl g c2 x0 a1 -(H2 x x0 H4 c2 H5) a2 H3))))))))))))) c1 y a H0))))) H))))))). -(* COMMENTS -Initial nodes: 4693 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/arity/lift1.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/arity/lift1.ma deleted file mode 100644 index 4077a8a1c..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/arity/lift1.ma +++ /dev/null @@ -1,44 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/arity/props.ma". - -include "Basic-1/drop1/fwd.ma". - -theorem arity_lift1: - \forall (g: G).(\forall (a: A).(\forall (c2: C).(\forall (hds: -PList).(\forall (c1: C).(\forall (t: T).((drop1 hds c1 c2) \to ((arity g c2 t -a) \to (arity g c1 (lift1 hds t) a)))))))) -\def - \lambda (g: G).(\lambda (a: A).(\lambda (c2: C).(\lambda (hds: -PList).(PList_ind (\lambda (p: PList).(\forall (c1: C).(\forall (t: -T).((drop1 p c1 c2) \to ((arity g c2 t a) \to (arity g c1 (lift1 p t) a)))))) -(\lambda (c1: C).(\lambda (t: T).(\lambda (H: (drop1 PNil c1 c2)).(\lambda -(H0: (arity g c2 t a)).(let H_y \def (drop1_gen_pnil c1 c2 H) in (eq_ind_r C -c2 (\lambda (c: C).(arity g c t a)) H0 c1 H_y)))))) (\lambda (n: -nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c1: -C).(\forall (t: T).((drop1 p c1 c2) \to ((arity g c2 t a) \to (arity g c1 -(lift1 p t) a))))))).(\lambda (c1: C).(\lambda (t: T).(\lambda (H0: (drop1 -(PCons n n0 p) c1 c2)).(\lambda (H1: (arity g c2 t a)).(let H_x \def -(drop1_gen_pcons c1 c2 p n n0 H0) in (let H2 \def H_x in (ex2_ind C (\lambda -(c3: C).(drop n n0 c1 c3)) (\lambda (c3: C).(drop1 p c3 c2)) (arity g c1 -(lift n n0 (lift1 p t)) a) (\lambda (x: C).(\lambda (H3: (drop n n0 c1 -x)).(\lambda (H4: (drop1 p x c2)).(arity_lift g x (lift1 p t) a (H x t H4 H1) -c1 n n0 H3)))) H2))))))))))) hds)))). -(* COMMENTS -Initial nodes: 289 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/arity/pr3.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/arity/pr3.ma deleted file mode 100644 index f8952088e..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/arity/pr3.ma +++ /dev/null @@ -1,635 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csuba/arity.ma". - -include "Basic-1/pr3/defs.ma". - -include "Basic-1/pr1/defs.ma". - -include "Basic-1/wcpr0/getl.ma". - -include "Basic-1/pr0/fwd.ma". - -include "Basic-1/arity/subst0.ma". - -theorem arity_sred_wcpr0_pr0: - \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (a: A).((arity g -c1 t1 a) \to (\forall (c2: C).((wcpr0 c1 c2) \to (\forall (t2: T).((pr0 t1 -t2) \to (arity g c2 t2 a))))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (a: A).(\lambda -(H: (arity g c1 t1 a)).(arity_ind g (\lambda (c: C).(\lambda (t: T).(\lambda -(a0: A).(\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 t t2) \to -(arity g c2 t2 a0)))))))) (\lambda (c: C).(\lambda (n: nat).(\lambda (c2: -C).(\lambda (_: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H1: (pr0 (TSort n) -t2)).(eq_ind_r T (TSort n) (\lambda (t: T).(arity g c2 t (ASort O n))) -(arity_sort g c2 n) t2 (pr0_gen_sort t2 n H1)))))))) (\lambda (c: C).(\lambda -(d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d -(Bind Abbr) u))).(\lambda (a0: A).(\lambda (_: (arity g d u a0)).(\lambda -(H2: ((\forall (c2: C).((wcpr0 d c2) \to (\forall (t2: T).((pr0 u t2) \to -(arity g c2 t2 a0))))))).(\lambda (c2: C).(\lambda (H3: (wcpr0 c -c2)).(\lambda (t2: T).(\lambda (H4: (pr0 (TLRef i) t2)).(eq_ind_r T (TLRef i) -(\lambda (t: T).(arity g c2 t a0)) (ex3_2_ind C T (\lambda (e2: C).(\lambda -(u2: T).(getl i c2 (CHead e2 (Bind Abbr) u2)))) (\lambda (e2: C).(\lambda (_: -T).(wcpr0 d e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u u2))) (arity g c2 -(TLRef i) a0) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: (getl i c2 -(CHead x0 (Bind Abbr) x1))).(\lambda (H6: (wcpr0 d x0)).(\lambda (H7: (pr0 u -x1)).(arity_abbr g c2 x0 x1 i H5 a0 (H2 x0 H6 x1 H7))))))) (wcpr0_getl c c2 -H3 i d u (Bind Abbr) H0)) t2 (pr0_gen_lref t2 i H4)))))))))))))) (\lambda (c: -C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c -(CHead d (Bind Abst) u))).(\lambda (a0: A).(\lambda (_: (arity g d u (asucc g -a0))).(\lambda (H2: ((\forall (c2: C).((wcpr0 d c2) \to (\forall (t2: -T).((pr0 u t2) \to (arity g c2 t2 (asucc g a0)))))))).(\lambda (c2: -C).(\lambda (H3: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H4: (pr0 (TLRef i) -t2)).(eq_ind_r T (TLRef i) (\lambda (t: T).(arity g c2 t a0)) (ex3_2_ind C T -(\lambda (e2: C).(\lambda (u2: T).(getl i c2 (CHead e2 (Bind Abst) u2)))) -(\lambda (e2: C).(\lambda (_: T).(wcpr0 d e2))) (\lambda (_: C).(\lambda (u2: -T).(pr0 u u2))) (arity g c2 (TLRef i) a0) (\lambda (x0: C).(\lambda (x1: -T).(\lambda (H5: (getl i c2 (CHead x0 (Bind Abst) x1))).(\lambda (H6: (wcpr0 -d x0)).(\lambda (H7: (pr0 u x1)).(arity_abst g c2 x0 x1 i H5 a0 (H2 x0 H6 x1 -H7))))))) (wcpr0_getl c c2 H3 i d u (Bind Abst) H0)) t2 (pr0_gen_lref t2 i -H4)))))))))))))) (\lambda (b: B).(\lambda (H0: (not (eq B b Abst))).(\lambda -(c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c u -a1)).(\lambda (H2: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 -u t2) \to (arity g c2 t2 a1))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda -(H3: (arity g (CHead c (Bind b) u) t a2)).(\lambda (H4: ((\forall (c2: -C).((wcpr0 (CHead c (Bind b) u) c2) \to (\forall (t2: T).((pr0 t t2) \to -(arity g c2 t2 a2))))))).(\lambda (c2: C).(\lambda (H5: (wcpr0 c -c2)).(\lambda (t2: T).(\lambda (H6: (pr0 (THead (Bind b) u t) t2)).(insert_eq -T (THead (Bind b) u t) (\lambda (t0: T).(pr0 t0 t2)) (\lambda (_: T).(arity g -c2 t2 a2)) (\lambda (y: T).(\lambda (H7: (pr0 y t2)).(pr0_ind (\lambda (t0: -T).(\lambda (t3: T).((eq T t0 (THead (Bind b) u t)) \to (arity g c2 t3 a2)))) -(\lambda (t0: T).(\lambda (H8: (eq T t0 (THead (Bind b) u t))).(let H9 \def -(f_equal T T (\lambda (e: T).e) t0 (THead (Bind b) u t) H8) in (eq_ind_r T -(THead (Bind b) u t) (\lambda (t3: T).(arity g c2 t3 a2)) (arity_bind g b H0 -c2 u a1 (H2 c2 H5 u (pr0_refl u)) t a2 (H4 (CHead c2 (Bind b) u) (wcpr0_comp -c c2 H5 u u (pr0_refl u) (Bind b)) t (pr0_refl t))) t0 H9)))) (\lambda (u1: -T).(\lambda (u2: T).(\lambda (H8: (pr0 u1 u2)).(\lambda (H9: (((eq T u1 -(THead (Bind b) u t)) \to (arity g c2 u2 a2)))).(\lambda (t3: T).(\lambda -(t4: T).(\lambda (H10: (pr0 t3 t4)).(\lambda (H11: (((eq T t3 (THead (Bind b) -u t)) \to (arity g c2 t4 a2)))).(\lambda (k: K).(\lambda (H12: (eq T (THead k -u1 t3) (THead (Bind b) u t))).(let H13 \def (f_equal T K (\lambda (e: -T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | -(TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) -(THead (Bind b) u t) H12) in ((let H14 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | -(TLRef _) \Rightarrow u1 | (THead _ t0 _) \Rightarrow t0])) (THead k u1 t3) -(THead (Bind b) u t) H12) in ((let H15 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | -(TLRef _) \Rightarrow t3 | (THead _ _ t0) \Rightarrow t0])) (THead k u1 t3) -(THead (Bind b) u t) H12) in (\lambda (H16: (eq T u1 u)).(\lambda (H17: (eq K -k (Bind b))).(eq_ind_r K (Bind b) (\lambda (k0: K).(arity g c2 (THead k0 u2 -t4) a2)) (let H18 \def (eq_ind T t3 (\lambda (t0: T).((eq T t0 (THead (Bind -b) u t)) \to (arity g c2 t4 a2))) H11 t H15) in (let H19 \def (eq_ind T t3 -(\lambda (t0: T).(pr0 t0 t4)) H10 t H15) in (let H20 \def (eq_ind T u1 -(\lambda (t0: T).((eq T t0 (THead (Bind b) u t)) \to (arity g c2 u2 a2))) H9 -u H16) in (let H21 \def (eq_ind T u1 (\lambda (t0: T).(pr0 t0 u2)) H8 u H16) -in (arity_bind g b H0 c2 u2 a1 (H2 c2 H5 u2 H21) t4 a2 (H4 (CHead c2 (Bind b) -u2) (wcpr0_comp c c2 H5 u u2 H21 (Bind b)) t4 H19)))))) k H17)))) H14)) -H13)))))))))))) (\lambda (u0: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda -(_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind b) u t)) \to (arity g -c2 v2 a2)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 -t4)).(\lambda (_: (((eq T t3 (THead (Bind b) u t)) \to (arity g c2 t4 -a2)))).(\lambda (H12: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) -(THead (Bind b) u t))).(let H13 \def (eq_ind T (THead (Flat Appl) v1 (THead -(Bind Abst) u0 t3)) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind -b) u t) H12) in (False_ind (arity g c2 (THead (Bind Abbr) v2 t4) a2) -H13)))))))))))) (\lambda (b0: B).(\lambda (_: (not (eq B b0 Abst))).(\lambda -(v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 -(THead (Bind b) u t)) \to (arity g c2 v2 a2)))).(\lambda (u1: T).(\lambda -(u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 (THead (Bind b) u -t)) \to (arity g c2 u2 a2)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: -(pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead (Bind b) u t)) \to (arity g c2 t4 -a2)))).(\lambda (H15: (eq T (THead (Flat Appl) v1 (THead (Bind b0) u1 t3)) -(THead (Bind b) u t))).(let H16 \def (eq_ind T (THead (Flat Appl) v1 (THead -(Bind b0) u1 t3)) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind -b) u t) H15) in (False_ind (arity g c2 (THead (Bind b0) u2 (THead (Flat Appl) -(lift (S O) O v2) t4)) a2) H16))))))))))))))))) (\lambda (u1: T).(\lambda -(u2: T).(\lambda (H8: (pr0 u1 u2)).(\lambda (H9: (((eq T u1 (THead (Bind b) u -t)) \to (arity g c2 u2 a2)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda -(H10: (pr0 t3 t4)).(\lambda (H11: (((eq T t3 (THead (Bind b) u t)) \to (arity -g c2 t4 a2)))).(\lambda (w: T).(\lambda (H12: (subst0 O u2 t4 w)).(\lambda -(H13: (eq T (THead (Bind Abbr) u1 t3) (THead (Bind b) u t))).(let H14 \def -(f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with -[(TSort _) \Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | (THead k _ _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b0) -\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (THead (Bind Abbr) u1 t3) -(THead (Bind b) u t) H13) in ((let H15 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | -(TLRef _) \Rightarrow u1 | (THead _ t0 _) \Rightarrow t0])) (THead (Bind -Abbr) u1 t3) (THead (Bind b) u t) H13) in ((let H16 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t0) \Rightarrow t0])) -(THead (Bind Abbr) u1 t3) (THead (Bind b) u t) H13) in (\lambda (H17: (eq T -u1 u)).(\lambda (H18: (eq B Abbr b)).(let H19 \def (eq_ind T t3 (\lambda (t0: -T).((eq T t0 (THead (Bind b) u t)) \to (arity g c2 t4 a2))) H11 t H16) in -(let H20 \def (eq_ind T t3 (\lambda (t0: T).(pr0 t0 t4)) H10 t H16) in (let -H21 \def (eq_ind T u1 (\lambda (t0: T).((eq T t0 (THead (Bind b) u t)) \to -(arity g c2 u2 a2))) H9 u H17) in (let H22 \def (eq_ind T u1 (\lambda (t0: -T).(pr0 t0 u2)) H8 u H17) in (let H23 \def (eq_ind_r B b (\lambda (b0: -B).((eq T t (THead (Bind b0) u t)) \to (arity g c2 t4 a2))) H19 Abbr H18) in -(let H24 \def (eq_ind_r B b (\lambda (b0: B).((eq T u (THead (Bind b0) u t)) -\to (arity g c2 u2 a2))) H21 Abbr H18) in (let H25 \def (eq_ind_r B b -(\lambda (b0: B).(\forall (c3: C).((wcpr0 (CHead c (Bind b0) u) c3) \to -(\forall (t5: T).((pr0 t t5) \to (arity g c3 t5 a2)))))) H4 Abbr H18) in (let -H26 \def (eq_ind_r B b (\lambda (b0: B).(arity g (CHead c (Bind b0) u) t a2)) -H3 Abbr H18) in (let H27 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 -Abst))) H0 Abbr H18) in (arity_bind g Abbr H27 c2 u2 a1 (H2 c2 H5 u2 H22) w -a2 (arity_subst0 g (CHead c2 (Bind Abbr) u2) t4 a2 (H25 (CHead c2 (Bind Abbr) -u2) (wcpr0_comp c c2 H5 u u2 H22 (Bind Abbr)) t4 H20) c2 u2 O (getl_refl Abbr -c2 u2) w H12)))))))))))))) H15)) H14))))))))))))) (\lambda (b0: B).(\lambda -(H8: (not (eq B b0 Abst))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H9: -(pr0 t3 t4)).(\lambda (H10: (((eq T t3 (THead (Bind b) u t)) \to (arity g c2 -t4 a2)))).(\lambda (u0: T).(\lambda (H11: (eq T (THead (Bind b0) u0 (lift (S -O) O t3)) (THead (Bind b) u t))).(let H12 \def (f_equal T B (\lambda (e: -T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b0 | -(TLRef _) \Rightarrow b0 | (THead k _ _) \Rightarrow (match k in K return -(\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow -b0])])) (THead (Bind b0) u0 (lift (S O) O t3)) (THead (Bind b) u t) H11) in -((let H13 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: -T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t0 -_) \Rightarrow t0])) (THead (Bind b0) u0 (lift (S O) O t3)) (THead (Bind b) u -t) H11) in ((let H14 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat -\to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n) -\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with -[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u1 t5) -\Rightarrow (THead k (lref_map f d u1) (lref_map f (s k d) t5))]) in -lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (TLRef _) \Rightarrow -((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match -t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef -(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | -(THead k u1 t5) \Rightarrow (THead k (lref_map f d u1) (lref_map f (s k d) -t5))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (THead _ _ t0) -\Rightarrow t0])) (THead (Bind b0) u0 (lift (S O) O t3)) (THead (Bind b) u t) -H11) in (\lambda (_: (eq T u0 u)).(\lambda (H16: (eq B b0 b)).(let H17 \def -(eq_ind B b0 (\lambda (b1: B).(not (eq B b1 Abst))) H8 b H16) in (let H18 -\def (eq_ind_r T t (\lambda (t0: T).((eq T t3 (THead (Bind b) u t0)) \to -(arity g c2 t4 a2))) H10 (lift (S O) O t3) H14) in (let H19 \def (eq_ind_r T -t (\lambda (t0: T).(\forall (c3: C).((wcpr0 (CHead c (Bind b) u) c3) \to -(\forall (t5: T).((pr0 t0 t5) \to (arity g c3 t5 a2)))))) H4 (lift (S O) O -t3) H14) in (let H20 \def (eq_ind_r T t (\lambda (t0: T).(arity g (CHead c -(Bind b) u) t0 a2)) H3 (lift (S O) O t3) H14) in (arity_gen_lift g (CHead c2 -(Bind b) u) t4 a2 (S O) O (H19 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H5 u u -(pr0_refl u) (Bind b)) (lift (S O) O t4) (pr0_lift t3 t4 H9 (S O) O)) c2 -(drop_drop (Bind b) O c2 c2 (drop_refl c2) u))))))))) H13)) H12)))))))))) -(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: -(((eq T t3 (THead (Bind b) u t)) \to (arity g c2 t4 a2)))).(\lambda (u0: -T).(\lambda (H10: (eq T (THead (Flat Cast) u0 t3) (THead (Bind b) u t))).(let -H11 \def (eq_ind T (THead (Flat Cast) u0 t3) (\lambda (ee: T).(match ee in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind b) u t) H10) in (False_ind (arity g c2 t4 a2) -H11)))))))) y t2 H7))) H6)))))))))))))))) (\lambda (c: C).(\lambda (u: -T).(\lambda (a1: A).(\lambda (_: (arity g c u (asucc g a1))).(\lambda (H1: -((\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 u t2) \to (arity g -c2 t2 (asucc g a1)))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (H2: -(arity g (CHead c (Bind Abst) u) t a2)).(\lambda (H3: ((\forall (c2: -C).((wcpr0 (CHead c (Bind Abst) u) c2) \to (\forall (t2: T).((pr0 t t2) \to -(arity g c2 t2 a2))))))).(\lambda (c2: C).(\lambda (H4: (wcpr0 c -c2)).(\lambda (t2: T).(\lambda (H5: (pr0 (THead (Bind Abst) u t) -t2)).(insert_eq T (THead (Bind Abst) u t) (\lambda (t0: T).(pr0 t0 t2)) -(\lambda (_: T).(arity g c2 t2 (AHead a1 a2))) (\lambda (y: T).(\lambda (H6: -(pr0 y t2)).(pr0_ind (\lambda (t0: T).(\lambda (t3: T).((eq T t0 (THead (Bind -Abst) u t)) \to (arity g c2 t3 (AHead a1 a2))))) (\lambda (t0: T).(\lambda -(H7: (eq T t0 (THead (Bind Abst) u t))).(let H8 \def (f_equal T T (\lambda -(e: T).e) t0 (THead (Bind Abst) u t) H7) in (eq_ind_r T (THead (Bind Abst) u -t) (\lambda (t3: T).(arity g c2 t3 (AHead a1 a2))) (arity_head g c2 u a1 (H1 -c2 H4 u (pr0_refl u)) t a2 (H3 (CHead c2 (Bind Abst) u) (wcpr0_comp c c2 H4 u -u (pr0_refl u) (Bind Abst)) t (pr0_refl t))) t0 H8)))) (\lambda (u1: -T).(\lambda (u2: T).(\lambda (H7: (pr0 u1 u2)).(\lambda (H8: (((eq T u1 -(THead (Bind Abst) u t)) \to (arity g c2 u2 (AHead a1 a2))))).(\lambda (t3: -T).(\lambda (t4: T).(\lambda (H9: (pr0 t3 t4)).(\lambda (H10: (((eq T t3 -(THead (Bind Abst) u t)) \to (arity g c2 t4 (AHead a1 a2))))).(\lambda (k: -K).(\lambda (H11: (eq T (THead k u1 t3) (THead (Bind Abst) u t))).(let H12 -\def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) -with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) -\Rightarrow k0])) (THead k u1 t3) (THead (Bind Abst) u t) H11) in ((let H13 -\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t0 _) -\Rightarrow t0])) (THead k u1 t3) (THead (Bind Abst) u t) H11) in ((let H14 -\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t0) -\Rightarrow t0])) (THead k u1 t3) (THead (Bind Abst) u t) H11) in (\lambda -(H15: (eq T u1 u)).(\lambda (H16: (eq K k (Bind Abst))).(eq_ind_r K (Bind -Abst) (\lambda (k0: K).(arity g c2 (THead k0 u2 t4) (AHead a1 a2))) (let H17 -\def (eq_ind T t3 (\lambda (t0: T).((eq T t0 (THead (Bind Abst) u t)) \to -(arity g c2 t4 (AHead a1 a2)))) H10 t H14) in (let H18 \def (eq_ind T t3 -(\lambda (t0: T).(pr0 t0 t4)) H9 t H14) in (let H19 \def (eq_ind T u1 -(\lambda (t0: T).((eq T t0 (THead (Bind Abst) u t)) \to (arity g c2 u2 (AHead -a1 a2)))) H8 u H15) in (let H20 \def (eq_ind T u1 (\lambda (t0: T).(pr0 t0 -u2)) H7 u H15) in (arity_head g c2 u2 a1 (H1 c2 H4 u2 H20) t4 a2 (H3 (CHead -c2 (Bind Abst) u2) (wcpr0_comp c c2 H4 u u2 H20 (Bind Abst)) t4 H18)))))) k -H16)))) H13)) H12)))))))))))) (\lambda (u0: T).(\lambda (v1: T).(\lambda (v2: -T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind Abst) u t)) -\to (arity g c2 v2 (AHead a1 a2))))).(\lambda (t3: T).(\lambda (t4: -T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead (Bind Abst) u t)) -\to (arity g c2 t4 (AHead a1 a2))))).(\lambda (H11: (eq T (THead (Flat Appl) -v1 (THead (Bind Abst) u0 t3)) (THead (Bind Abst) u t))).(let H12 \def (eq_ind -T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) (\lambda (ee: T).(match ee -in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef -_) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return -(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind Abst) u t) H11) in (False_ind (arity g c2 (THead -(Bind Abbr) v2 t4) (AHead a1 a2)) H12)))))))))))) (\lambda (b: B).(\lambda -(_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 -v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind Abst) u t)) \to (arity g c2 v2 -(AHead a1 a2))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 -u2)).(\lambda (_: (((eq T u1 (THead (Bind Abst) u t)) \to (arity g c2 u2 -(AHead a1 a2))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 -t4)).(\lambda (_: (((eq T t3 (THead (Bind Abst) u t)) \to (arity g c2 t4 -(AHead a1 a2))))).(\lambda (H14: (eq T (THead (Flat Appl) v1 (THead (Bind b) -u1 t3)) (THead (Bind Abst) u t))).(let H15 \def (eq_ind T (THead (Flat Appl) -v1 (THead (Bind b) u1 t3)) (\lambda (ee: T).(match ee in T return (\lambda -(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False -| (THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind -Abst) u t) H14) in (False_ind (arity g c2 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) (AHead a1 a2)) H15))))))))))))))))) (\lambda -(u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 -(THead (Bind Abst) u t)) \to (arity g c2 u2 (AHead a1 a2))))).(\lambda (t3: -T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead -(Bind Abst) u t)) \to (arity g c2 t4 (AHead a1 a2))))).(\lambda (w: -T).(\lambda (_: (subst0 O u2 t4 w)).(\lambda (H12: (eq T (THead (Bind Abbr) -u1 t3) (THead (Bind Abst) u t))).(let H13 \def (eq_ind T (THead (Bind Abbr) -u1 t3) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b) -\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow -True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _) -\Rightarrow False])])) I (THead (Bind Abst) u t) H12) in (False_ind (arity g -c2 (THead (Bind Abbr) u2 w) (AHead a1 a2)) H13))))))))))))) (\lambda (b: -B).(\lambda (H7: (not (eq B b Abst))).(\lambda (t3: T).(\lambda (t4: -T).(\lambda (_: (pr0 t3 t4)).(\lambda (H9: (((eq T t3 (THead (Bind Abst) u -t)) \to (arity g c2 t4 (AHead a1 a2))))).(\lambda (u0: T).(\lambda (H10: (eq -T (THead (Bind b) u0 (lift (S O) O t3)) (THead (Bind Abst) u t))).(let H11 -\def (f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) -with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b0) -\Rightarrow b0 | (Flat _) \Rightarrow b])])) (THead (Bind b) u0 (lift (S O) O -t3)) (THead (Bind Abst) u t) H10) in ((let H12 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | -(TLRef _) \Rightarrow u0 | (THead _ t0 _) \Rightarrow t0])) (THead (Bind b) -u0 (lift (S O) O t3)) (THead (Bind Abst) u t) H10) in ((let H13 \def (f_equal -T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T -\def (match t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow -(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) -| (THead k u1 t5) \Rightarrow (THead k (lref_map f d u1) (lref_map f (s k d) -t5))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (TLRef _) -\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T -\def (match t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow -(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) -| (THead k u1 t5) \Rightarrow (THead k (lref_map f d u1) (lref_map f (s k d) -t5))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (THead _ _ t0) -\Rightarrow t0])) (THead (Bind b) u0 (lift (S O) O t3)) (THead (Bind Abst) u -t) H10) in (\lambda (_: (eq T u0 u)).(\lambda (H15: (eq B b Abst)).(let H16 -\def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H7 Abst H15) in (let -H17 \def (eq_ind_r T t (\lambda (t0: T).((eq T t3 (THead (Bind Abst) u t0)) -\to (arity g c2 t4 (AHead a1 a2)))) H9 (lift (S O) O t3) H13) in (let H18 -\def (eq_ind_r T t (\lambda (t0: T).(\forall (c3: C).((wcpr0 (CHead c (Bind -Abst) u) c3) \to (\forall (t5: T).((pr0 t0 t5) \to (arity g c3 t5 a2)))))) H3 -(lift (S O) O t3) H13) in (let H19 \def (eq_ind_r T t (\lambda (t0: T).(arity -g (CHead c (Bind Abst) u) t0 a2)) H2 (lift (S O) O t3) H13) in (let H20 \def -(match (H16 (refl_equal B Abst)) in False return (\lambda (_: False).(arity g -c2 t4 (AHead a1 a2))) with []) in H20)))))))) H12)) H11)))))))))) (\lambda -(t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 -(THead (Bind Abst) u t)) \to (arity g c2 t4 (AHead a1 a2))))).(\lambda (u0: -T).(\lambda (H9: (eq T (THead (Flat Cast) u0 t3) (THead (Bind Abst) u -t))).(let H10 \def (eq_ind T (THead (Flat Cast) u0 t3) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K -return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) -\Rightarrow True])])) I (THead (Bind Abst) u t) H9) in (False_ind (arity g c2 -t4 (AHead a1 a2)) H10)))))))) y t2 H6))) H5)))))))))))))) (\lambda (c: -C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda -(H1: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 u t2) \to -(arity g c2 t2 a1))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (H2: -(arity g c t (AHead a1 a2))).(\lambda (H3: ((\forall (c2: C).((wcpr0 c c2) -\to (\forall (t2: T).((pr0 t t2) \to (arity g c2 t2 (AHead a1 -a2)))))))).(\lambda (c2: C).(\lambda (H4: (wcpr0 c c2)).(\lambda (t2: -T).(\lambda (H5: (pr0 (THead (Flat Appl) u t) t2)).(insert_eq T (THead (Flat -Appl) u t) (\lambda (t0: T).(pr0 t0 t2)) (\lambda (_: T).(arity g c2 t2 a2)) -(\lambda (y: T).(\lambda (H6: (pr0 y t2)).(pr0_ind (\lambda (t0: T).(\lambda -(t3: T).((eq T t0 (THead (Flat Appl) u t)) \to (arity g c2 t3 a2)))) (\lambda -(t0: T).(\lambda (H7: (eq T t0 (THead (Flat Appl) u t))).(let H8 \def -(f_equal T T (\lambda (e: T).e) t0 (THead (Flat Appl) u t) H7) in (eq_ind_r T -(THead (Flat Appl) u t) (\lambda (t3: T).(arity g c2 t3 a2)) (arity_appl g c2 -u a1 (H1 c2 H4 u (pr0_refl u)) t a2 (H3 c2 H4 t (pr0_refl t))) t0 H8)))) -(\lambda (u1: T).(\lambda (u2: T).(\lambda (H7: (pr0 u1 u2)).(\lambda (H8: -(((eq T u1 (THead (Flat Appl) u t)) \to (arity g c2 u2 a2)))).(\lambda (t3: -T).(\lambda (t4: T).(\lambda (H9: (pr0 t3 t4)).(\lambda (H10: (((eq T t3 -(THead (Flat Appl) u t)) \to (arity g c2 t4 a2)))).(\lambda (k: K).(\lambda -(H11: (eq T (THead k u1 t3) (THead (Flat Appl) u t))).(let H12 \def (f_equal -T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) -\Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) -(THead k u1 t3) (THead (Flat Appl) u t) H11) in ((let H13 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t0 _) \Rightarrow t0])) -(THead k u1 t3) (THead (Flat Appl) u t) H11) in ((let H14 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t0) \Rightarrow t0])) -(THead k u1 t3) (THead (Flat Appl) u t) H11) in (\lambda (H15: (eq T u1 -u)).(\lambda (H16: (eq K k (Flat Appl))).(eq_ind_r K (Flat Appl) (\lambda -(k0: K).(arity g c2 (THead k0 u2 t4) a2)) (let H17 \def (eq_ind T t3 (\lambda -(t0: T).((eq T t0 (THead (Flat Appl) u t)) \to (arity g c2 t4 a2))) H10 t -H14) in (let H18 \def (eq_ind T t3 (\lambda (t0: T).(pr0 t0 t4)) H9 t H14) in -(let H19 \def (eq_ind T u1 (\lambda (t0: T).((eq T t0 (THead (Flat Appl) u -t)) \to (arity g c2 u2 a2))) H8 u H15) in (let H20 \def (eq_ind T u1 (\lambda -(t0: T).(pr0 t0 u2)) H7 u H15) in (arity_appl g c2 u2 a1 (H1 c2 H4 u2 H20) t4 -a2 (H3 c2 H4 t4 H18)))))) k H16)))) H13)) H12)))))))))))) (\lambda (u0: -T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H7: (pr0 v1 v2)).(\lambda (H8: -(((eq T v1 (THead (Flat Appl) u t)) \to (arity g c2 v2 a2)))).(\lambda (t3: -T).(\lambda (t4: T).(\lambda (H9: (pr0 t3 t4)).(\lambda (H10: (((eq T t3 -(THead (Flat Appl) u t)) \to (arity g c2 t4 a2)))).(\lambda (H11: (eq T -(THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) (THead (Flat Appl) u -t))).(let H12 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda -(_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead -_ t0 _) \Rightarrow t0])) (THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) -(THead (Flat Appl) u t) H11) in ((let H13 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow (THead -(Bind Abst) u0 t3) | (TLRef _) \Rightarrow (THead (Bind Abst) u0 t3) | (THead -_ _ t0) \Rightarrow t0])) (THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) -(THead (Flat Appl) u t) H11) in (\lambda (H14: (eq T v1 u)).(let H15 \def -(eq_ind T v1 (\lambda (t0: T).((eq T t0 (THead (Flat Appl) u t)) \to (arity g -c2 v2 a2))) H8 u H14) in (let H16 \def (eq_ind T v1 (\lambda (t0: T).(pr0 t0 -v2)) H7 u H14) in (let H17 \def (eq_ind_r T t (\lambda (t0: T).((eq T t3 -(THead (Flat Appl) u t0)) \to (arity g c2 t4 a2))) H10 (THead (Bind Abst) u0 -t3) H13) in (let H18 \def (eq_ind_r T t (\lambda (t0: T).((eq T u (THead -(Flat Appl) u t0)) \to (arity g c2 v2 a2))) H15 (THead (Bind Abst) u0 t3) -H13) in (let H19 \def (eq_ind_r T t (\lambda (t0: T).(\forall (c3: C).((wcpr0 -c c3) \to (\forall (t5: T).((pr0 t0 t5) \to (arity g c3 t5 (AHead a1 -a2))))))) H3 (THead (Bind Abst) u0 t3) H13) in (let H20 \def (eq_ind_r T t -(\lambda (t0: T).(arity g c t0 (AHead a1 a2))) H2 (THead (Bind Abst) u0 t3) -H13) in (let H21 \def (H1 c2 H4 v2 H16) in (let H22 \def (H19 c2 H4 (THead -(Bind Abst) u0 t4) (pr0_comp u0 u0 (pr0_refl u0) t3 t4 H9 (Bind Abst))) in -(let H23 \def (arity_gen_abst g c2 u0 t4 (AHead a1 a2) H22) in (ex3_2_ind A A -(\lambda (a3: A).(\lambda (a4: A).(eq A (AHead a1 a2) (AHead a3 a4)))) -(\lambda (a3: A).(\lambda (_: A).(arity g c2 u0 (asucc g a3)))) (\lambda (_: -A).(\lambda (a4: A).(arity g (CHead c2 (Bind Abst) u0) t4 a4))) (arity g c2 -(THead (Bind Abbr) v2 t4) a2) (\lambda (x0: A).(\lambda (x1: A).(\lambda -(H24: (eq A (AHead a1 a2) (AHead x0 x1))).(\lambda (H25: (arity g c2 u0 -(asucc g x0))).(\lambda (H26: (arity g (CHead c2 (Bind Abst) u0) t4 x1)).(let -H27 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) -with [(ASort _ _) \Rightarrow a1 | (AHead a0 _) \Rightarrow a0])) (AHead a1 -a2) (AHead x0 x1) H24) in ((let H28 \def (f_equal A A (\lambda (e: A).(match -e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a2 | (AHead _ -a0) \Rightarrow a0])) (AHead a1 a2) (AHead x0 x1) H24) in (\lambda (H29: (eq -A a1 x0)).(let H30 \def (eq_ind_r A x1 (\lambda (a0: A).(arity g (CHead c2 -(Bind Abst) u0) t4 a0)) H26 a2 H28) in (let H31 \def (eq_ind_r A x0 (\lambda -(a0: A).(arity g c2 u0 (asucc g a0))) H25 a1 H29) in (arity_bind g Abbr -not_abbr_abst c2 v2 a1 H21 t4 a2 (csuba_arity g (CHead c2 (Bind Abst) u0) t4 -a2 H30 (CHead c2 (Bind Abbr) v2) (csuba_abst g c2 c2 (csuba_refl g c2) u0 a1 -H31 v2 H21))))))) H27))))))) H23)))))))))))) H12)))))))))))) (\lambda (b: -B).(\lambda (H7: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: -T).(\lambda (H8: (pr0 v1 v2)).(\lambda (H9: (((eq T v1 (THead (Flat Appl) u -t)) \to (arity g c2 v2 a2)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda -(H10: (pr0 u1 u2)).(\lambda (H11: (((eq T u1 (THead (Flat Appl) u t)) \to -(arity g c2 u2 a2)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H12: (pr0 -t3 t4)).(\lambda (H13: (((eq T t3 (THead (Flat Appl) u t)) \to (arity g c2 t4 -a2)))).(\lambda (H14: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) -(THead (Flat Appl) u t))).(let H15 \def (f_equal T T (\lambda (e: T).(match e -in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) -\Rightarrow v1 | (THead _ t0 _) \Rightarrow t0])) (THead (Flat Appl) v1 -(THead (Bind b) u1 t3)) (THead (Flat Appl) u t) H14) in ((let H16 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow (THead (Bind b) u1 t3) | (TLRef _) \Rightarrow (THead -(Bind b) u1 t3) | (THead _ _ t0) \Rightarrow t0])) (THead (Flat Appl) v1 -(THead (Bind b) u1 t3)) (THead (Flat Appl) u t) H14) in (\lambda (H17: (eq T -v1 u)).(let H18 \def (eq_ind T v1 (\lambda (t0: T).((eq T t0 (THead (Flat -Appl) u t)) \to (arity g c2 v2 a2))) H9 u H17) in (let H19 \def (eq_ind T v1 -(\lambda (t0: T).(pr0 t0 v2)) H8 u H17) in (let H20 \def (eq_ind_r T t -(\lambda (t0: T).((eq T t3 (THead (Flat Appl) u t0)) \to (arity g c2 t4 a2))) -H13 (THead (Bind b) u1 t3) H16) in (let H21 \def (eq_ind_r T t (\lambda (t0: -T).((eq T u1 (THead (Flat Appl) u t0)) \to (arity g c2 u2 a2))) H11 (THead -(Bind b) u1 t3) H16) in (let H22 \def (eq_ind_r T t (\lambda (t0: T).((eq T u -(THead (Flat Appl) u t0)) \to (arity g c2 v2 a2))) H18 (THead (Bind b) u1 t3) -H16) in (let H23 \def (eq_ind_r T t (\lambda (t0: T).(\forall (c3: C).((wcpr0 -c c3) \to (\forall (t5: T).((pr0 t0 t5) \to (arity g c3 t5 (AHead a1 -a2))))))) H3 (THead (Bind b) u1 t3) H16) in (let H24 \def (eq_ind_r T t -(\lambda (t0: T).(arity g c t0 (AHead a1 a2))) H2 (THead (Bind b) u1 t3) H16) -in (let H25 \def (H1 c2 H4 v2 H19) in (let H26 \def (H23 c2 H4 (THead (Bind -b) u2 t4) (pr0_comp u1 u2 H10 t3 t4 H12 (Bind b))) in (let H27 \def -(arity_gen_bind b H7 g c2 u2 t4 (AHead a1 a2) H26) in (ex2_ind A (\lambda -(a3: A).(arity g c2 u2 a3)) (\lambda (_: A).(arity g (CHead c2 (Bind b) u2) -t4 (AHead a1 a2))) (arity g c2 (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t4)) a2) (\lambda (x: A).(\lambda (H28: (arity g c2 u2 x)).(\lambda -(H29: (arity g (CHead c2 (Bind b) u2) t4 (AHead a1 a2))).(arity_bind g b H7 -c2 u2 x H28 (THead (Flat Appl) (lift (S O) O v2) t4) a2 (arity_appl g (CHead -c2 (Bind b) u2) (lift (S O) O v2) a1 (arity_lift g c2 v2 a1 H25 (CHead c2 -(Bind b) u2) (S O) O (drop_drop (Bind b) O c2 c2 (drop_refl c2) u2)) t4 a2 -H29))))) H27))))))))))))) H15))))))))))))))))) (\lambda (u1: T).(\lambda (u2: -T).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 (THead (Flat Appl) u t)) -\to (arity g c2 u2 a2)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 -t3 t4)).(\lambda (_: (((eq T t3 (THead (Flat Appl) u t)) \to (arity g c2 t4 -a2)))).(\lambda (w: T).(\lambda (_: (subst0 O u2 t4 w)).(\lambda (H12: (eq T -(THead (Bind Abbr) u1 t3) (THead (Flat Appl) u t))).(let H13 \def (eq_ind T -(THead (Bind Abbr) u1 t3) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat -Appl) u t) H12) in (False_ind (arity g c2 (THead (Bind Abbr) u2 w) a2) -H13))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda -(t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 -(THead (Flat Appl) u t)) \to (arity g c2 t4 a2)))).(\lambda (u0: T).(\lambda -(H10: (eq T (THead (Bind b) u0 (lift (S O) O t3)) (THead (Flat Appl) u -t))).(let H11 \def (eq_ind T (THead (Bind b) u0 (lift (S O) O t3)) (\lambda -(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow -(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | -(Flat _) \Rightarrow False])])) I (THead (Flat Appl) u t) H10) in (False_ind -(arity g c2 t4 a2) H11)))))))))) (\lambda (t3: T).(\lambda (t4: T).(\lambda -(_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead (Flat Appl) u t)) \to (arity -g c2 t4 a2)))).(\lambda (u0: T).(\lambda (H9: (eq T (THead (Flat Cast) u0 t3) -(THead (Flat Appl) u t))).(let H10 \def (eq_ind T (THead (Flat Cast) u0 t3) -(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow -(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | -(Flat f) \Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl -\Rightarrow False | Cast \Rightarrow True])])])) I (THead (Flat Appl) u t) -H9) in (False_ind (arity g c2 t4 a2) H10)))))))) y t2 H6))) H5)))))))))))))) -(\lambda (c: C).(\lambda (u: T).(\lambda (a0: A).(\lambda (_: (arity g c u -(asucc g a0))).(\lambda (H1: ((\forall (c2: C).((wcpr0 c c2) \to (\forall -(t2: T).((pr0 u t2) \to (arity g c2 t2 (asucc g a0)))))))).(\lambda (t: -T).(\lambda (_: (arity g c t a0)).(\lambda (H3: ((\forall (c2: C).((wcpr0 c -c2) \to (\forall (t2: T).((pr0 t t2) \to (arity g c2 t2 a0))))))).(\lambda -(c2: C).(\lambda (H4: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H5: (pr0 -(THead (Flat Cast) u t) t2)).(insert_eq T (THead (Flat Cast) u t) (\lambda -(t0: T).(pr0 t0 t2)) (\lambda (_: T).(arity g c2 t2 a0)) (\lambda (y: -T).(\lambda (H6: (pr0 y t2)).(pr0_ind (\lambda (t0: T).(\lambda (t3: T).((eq -T t0 (THead (Flat Cast) u t)) \to (arity g c2 t3 a0)))) (\lambda (t0: -T).(\lambda (H7: (eq T t0 (THead (Flat Cast) u t))).(let H8 \def (f_equal T T -(\lambda (e: T).e) t0 (THead (Flat Cast) u t) H7) in (eq_ind_r T (THead (Flat -Cast) u t) (\lambda (t3: T).(arity g c2 t3 a0)) (arity_cast g c2 u a0 (H1 c2 -H4 u (pr0_refl u)) t (H3 c2 H4 t (pr0_refl t))) t0 H8)))) (\lambda (u1: -T).(\lambda (u2: T).(\lambda (H7: (pr0 u1 u2)).(\lambda (H8: (((eq T u1 -(THead (Flat Cast) u t)) \to (arity g c2 u2 a0)))).(\lambda (t3: T).(\lambda -(t4: T).(\lambda (H9: (pr0 t3 t4)).(\lambda (H10: (((eq T t3 (THead (Flat -Cast) u t)) \to (arity g c2 t4 a0)))).(\lambda (k: K).(\lambda (H11: (eq T -(THead k u1 t3) (THead (Flat Cast) u t))).(let H12 \def (f_equal T K (\lambda -(e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k -| (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) -(THead (Flat Cast) u t) H11) in ((let H13 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | -(TLRef _) \Rightarrow u1 | (THead _ t0 _) \Rightarrow t0])) (THead k u1 t3) -(THead (Flat Cast) u t) H11) in ((let H14 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | -(TLRef _) \Rightarrow t3 | (THead _ _ t0) \Rightarrow t0])) (THead k u1 t3) -(THead (Flat Cast) u t) H11) in (\lambda (H15: (eq T u1 u)).(\lambda (H16: -(eq K k (Flat Cast))).(eq_ind_r K (Flat Cast) (\lambda (k0: K).(arity g c2 -(THead k0 u2 t4) a0)) (let H17 \def (eq_ind T t3 (\lambda (t0: T).((eq T t0 -(THead (Flat Cast) u t)) \to (arity g c2 t4 a0))) H10 t H14) in (let H18 \def -(eq_ind T t3 (\lambda (t0: T).(pr0 t0 t4)) H9 t H14) in (let H19 \def (eq_ind -T u1 (\lambda (t0: T).((eq T t0 (THead (Flat Cast) u t)) \to (arity g c2 u2 -a0))) H8 u H15) in (let H20 \def (eq_ind T u1 (\lambda (t0: T).(pr0 t0 u2)) -H7 u H15) in (arity_cast g c2 u2 a0 (H1 c2 H4 u2 H20) t4 (H3 c2 H4 t4 -H18)))))) k H16)))) H13)) H12)))))))))))) (\lambda (u0: T).(\lambda (v1: -T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (THead -(Flat Cast) u t)) \to (arity g c2 v2 a0)))).(\lambda (t3: T).(\lambda (t4: -T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead (Flat Cast) u t)) -\to (arity g c2 t4 a0)))).(\lambda (H11: (eq T (THead (Flat Appl) v1 (THead -(Bind Abst) u0 t3)) (THead (Flat Cast) u t))).(let H12 \def (eq_ind T (THead -(Flat Appl) v1 (THead (Bind Abst) u0 t3)) (\lambda (ee: T).(match ee in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f -in F return (\lambda (_: F).Prop) with [Appl \Rightarrow True | Cast -\Rightarrow False])])])) I (THead (Flat Cast) u t) H11) in (False_ind (arity -g c2 (THead (Bind Abbr) v2 t4) a0) H12)))))))))))) (\lambda (b: B).(\lambda -(_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 -v1 v2)).(\lambda (_: (((eq T v1 (THead (Flat Cast) u t)) \to (arity g c2 v2 -a0)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda -(_: (((eq T u1 (THead (Flat Cast) u t)) \to (arity g c2 u2 a0)))).(\lambda -(t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 -(THead (Flat Cast) u t)) \to (arity g c2 t4 a0)))).(\lambda (H14: (eq T -(THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Cast) u t))).(let -H15 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (\lambda -(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow -(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | -(Flat f) \Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl -\Rightarrow True | Cast \Rightarrow False])])])) I (THead (Flat Cast) u t) -H14) in (False_ind (arity g c2 (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t4)) a0) H15))))))))))))))))) (\lambda (u1: T).(\lambda (u2: -T).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 (THead (Flat Cast) u t)) -\to (arity g c2 u2 a0)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 -t3 t4)).(\lambda (_: (((eq T t3 (THead (Flat Cast) u t)) \to (arity g c2 t4 -a0)))).(\lambda (w: T).(\lambda (_: (subst0 O u2 t4 w)).(\lambda (H12: (eq T -(THead (Bind Abbr) u1 t3) (THead (Flat Cast) u t))).(let H13 \def (eq_ind T -(THead (Bind Abbr) u1 t3) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat -Cast) u t) H12) in (False_ind (arity g c2 (THead (Bind Abbr) u2 w) a0) -H13))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda -(t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 -(THead (Flat Cast) u t)) \to (arity g c2 t4 a0)))).(\lambda (u0: T).(\lambda -(H10: (eq T (THead (Bind b) u0 (lift (S O) O t3)) (THead (Flat Cast) u -t))).(let H11 \def (eq_ind T (THead (Bind b) u0 (lift (S O) O t3)) (\lambda -(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow -(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | -(Flat _) \Rightarrow False])])) I (THead (Flat Cast) u t) H10) in (False_ind -(arity g c2 t4 a0) H11)))))))))) (\lambda (t3: T).(\lambda (t4: T).(\lambda -(H7: (pr0 t3 t4)).(\lambda (H8: (((eq T t3 (THead (Flat Cast) u t)) \to -(arity g c2 t4 a0)))).(\lambda (u0: T).(\lambda (H9: (eq T (THead (Flat Cast) -u0 t3) (THead (Flat Cast) u t))).(let H10 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | -(TLRef _) \Rightarrow u0 | (THead _ t0 _) \Rightarrow t0])) (THead (Flat -Cast) u0 t3) (THead (Flat Cast) u t) H9) in ((let H11 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t0) \Rightarrow t0])) -(THead (Flat Cast) u0 t3) (THead (Flat Cast) u t) H9) in (\lambda (_: (eq T -u0 u)).(let H13 \def (eq_ind T t3 (\lambda (t0: T).((eq T t0 (THead (Flat -Cast) u t)) \to (arity g c2 t4 a0))) H8 t H11) in (let H14 \def (eq_ind T t3 -(\lambda (t0: T).(pr0 t0 t4)) H7 t H11) in (H3 c2 H4 t4 H14))))) H10)))))))) -y t2 H6))) H5))))))))))))) (\lambda (c: C).(\lambda (t: T).(\lambda (a1: -A).(\lambda (_: (arity g c t a1)).(\lambda (H1: ((\forall (c2: C).((wcpr0 c -c2) \to (\forall (t2: T).((pr0 t t2) \to (arity g c2 t2 a1))))))).(\lambda -(a2: A).(\lambda (H2: (leq g a1 a2)).(\lambda (c2: C).(\lambda (H3: (wcpr0 c -c2)).(\lambda (t2: T).(\lambda (H4: (pr0 t t2)).(arity_repl g c2 t2 a1 (H1 c2 -H3 t2 H4) a2 H2)))))))))))) c1 t1 a H))))). -(* COMMENTS -Initial nodes: 10246 -END *) - -theorem arity_sred_wcpr0_pr1: - \forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall (g: G).(\forall -(c1: C).(\forall (a: A).((arity g c1 t1 a) \to (\forall (c2: C).((wcpr0 c1 -c2) \to (arity g c2 t2 a))))))))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr1 t1 t2)).(pr1_ind (\lambda -(t: T).(\lambda (t0: T).(\forall (g: G).(\forall (c1: C).(\forall (a: -A).((arity g c1 t a) \to (\forall (c2: C).((wcpr0 c1 c2) \to (arity g c2 t0 -a))))))))) (\lambda (t: T).(\lambda (g: G).(\lambda (c1: C).(\lambda (a: -A).(\lambda (H0: (arity g c1 t a)).(\lambda (c2: C).(\lambda (H1: (wcpr0 c1 -c2)).(arity_sred_wcpr0_pr0 g c1 t a H0 c2 H1 t (pr0_refl t))))))))) (\lambda -(t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t4 t3)).(\lambda (t5: T).(\lambda -(_: (pr1 t3 t5)).(\lambda (H2: ((\forall (g: G).(\forall (c1: C).(\forall (a: -A).((arity g c1 t3 a) \to (\forall (c2: C).((wcpr0 c1 c2) \to (arity g c2 t5 -a))))))))).(\lambda (g: G).(\lambda (c1: C).(\lambda (a: A).(\lambda (H3: -(arity g c1 t4 a)).(\lambda (c2: C).(\lambda (H4: (wcpr0 c1 c2)).(H2 g c2 a -(arity_sred_wcpr0_pr0 g c1 t4 a H3 c2 H4 t3 H0) c2 (wcpr0_refl -c2)))))))))))))) t1 t2 H))). -(* COMMENTS -Initial nodes: 213 -END *) - -theorem arity_sred_pr2: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall -(g: G).(\forall (a: A).((arity g c t1 a) \to (arity g c t2 a))))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 -t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (g: -G).(\forall (a: A).((arity g c0 t a) \to (arity g c0 t0 a))))))) (\lambda -(c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(\lambda -(g: G).(\lambda (a: A).(\lambda (H1: (arity g c0 t3 a)).(arity_sred_wcpr0_pr0 -g c0 t3 a H1 c0 (wcpr0_refl c0) t4 H0)))))))) (\lambda (c0: C).(\lambda (d: -C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind -Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 -t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (g: -G).(\lambda (a: A).(\lambda (H3: (arity g c0 t3 a)).(arity_subst0 g c0 t4 a -(arity_sred_wcpr0_pr0 g c0 t3 a H3 c0 (wcpr0_refl c0) t4 H1) d u i H0 t -H2)))))))))))))) c t1 t2 H)))). -(* COMMENTS -Initial nodes: 205 -END *) - -theorem arity_sred_pr3: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall -(g: G).(\forall (a: A).((arity g c t1 a) \to (arity g c t2 a))))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 -t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (g: G).(\forall (a: -A).((arity g c t a) \to (arity g c t0 a)))))) (\lambda (t: T).(\lambda (g: -G).(\lambda (a: A).(\lambda (H0: (arity g c t a)).H0)))) (\lambda (t3: -T).(\lambda (t4: T).(\lambda (H0: (pr2 c t4 t3)).(\lambda (t5: T).(\lambda -(_: (pr3 c t3 t5)).(\lambda (H2: ((\forall (g: G).(\forall (a: A).((arity g c -t3 a) \to (arity g c t5 a)))))).(\lambda (g: G).(\lambda (a: A).(\lambda (H3: -(arity g c t4 a)).(H2 g a (arity_sred_pr2 c t4 t3 H0 g a H3))))))))))) t1 t2 -H)))). -(* COMMENTS -Initial nodes: 151 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/arity/props.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/arity/props.ma deleted file mode 100644 index 6f131df09..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/arity/props.ma +++ /dev/null @@ -1,434 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/arity/fwd.ma". - -theorem node_inh: - \forall (g: G).(\forall (n: nat).(\forall (k: nat).(ex_2 C T (\lambda (c: -C).(\lambda (t: T).(arity g c t (ASort k n))))))) -\def - \lambda (g: G).(\lambda (n: nat).(\lambda (k: nat).(nat_ind (\lambda (n0: -nat).(ex_2 C T (\lambda (c: C).(\lambda (t: T).(arity g c t (ASort n0 n)))))) -(ex_2_intro C T (\lambda (c: C).(\lambda (t: T).(arity g c t (ASort O n)))) -(CSort O) (TSort n) (arity_sort g (CSort O) n)) (\lambda (n0: nat).(\lambda -(H: (ex_2 C T (\lambda (c: C).(\lambda (t: T).(arity g c t (ASort n0 -n)))))).(let H0 \def H in (ex_2_ind C T (\lambda (c: C).(\lambda (t: -T).(arity g c t (ASort n0 n)))) (ex_2 C T (\lambda (c: C).(\lambda (t: -T).(arity g c t (ASort (S n0) n))))) (\lambda (x0: C).(\lambda (x1: -T).(\lambda (H1: (arity g x0 x1 (ASort n0 n))).(ex_2_intro C T (\lambda (c: -C).(\lambda (t: T).(arity g c t (ASort (S n0) n)))) (CHead x0 (Bind Abst) x1) -(TLRef O) (arity_abst g (CHead x0 (Bind Abst) x1) x0 x1 O (getl_refl Abst x0 -x1) (ASort (S n0) n) H1))))) H0)))) k))). -(* COMMENTS -Initial nodes: 253 -END *) - -theorem arity_lift: - \forall (g: G).(\forall (c2: C).(\forall (t: T).(\forall (a: A).((arity g c2 -t a) \to (\forall (c1: C).(\forall (h: nat).(\forall (d: nat).((drop h d c1 -c2) \to (arity g c1 (lift h d t) a))))))))) -\def - \lambda (g: G).(\lambda (c2: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: -(arity g c2 t a)).(arity_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda (a0: -A).(\forall (c1: C).(\forall (h: nat).(\forall (d: nat).((drop h d c1 c) \to -(arity g c1 (lift h d t0) a0)))))))) (\lambda (c: C).(\lambda (n: -nat).(\lambda (c1: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (_: (drop -h d c1 c)).(eq_ind_r T (TSort n) (\lambda (t0: T).(arity g c1 t0 (ASort O -n))) (arity_sort g c1 n) (lift h d (TSort n)) (lift_sort n h d)))))))) -(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (a0: A).(\lambda (H1: -(arity g d u a0)).(\lambda (H2: ((\forall (c1: C).(\forall (h: nat).(\forall -(d0: nat).((drop h d0 c1 d) \to (arity g c1 (lift h d0 u) a0))))))).(\lambda -(c1: C).(\lambda (h: nat).(\lambda (d0: nat).(\lambda (H3: (drop h d0 c1 -c)).(lt_le_e i d0 (arity g c1 (lift h d0 (TLRef i)) a0) (\lambda (H4: (lt i -d0)).(eq_ind_r T (TLRef i) (\lambda (t0: T).(arity g c1 t0 a0)) (let H5 \def -(drop_getl_trans_le i d0 (le_S_n i d0 (le_S (S i) d0 H4)) c1 c h H3 (CHead d -(Bind Abbr) u) H0) in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop i -O c1 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d0 i) e0 e1))) -(\lambda (_: C).(\lambda (e1: C).(clear e1 (CHead d (Bind Abbr) u)))) (arity -g c1 (TLRef i) a0) (\lambda (x0: C).(\lambda (x1: C).(\lambda (H6: (drop i O -c1 x0)).(\lambda (H7: (drop h (minus d0 i) x0 x1)).(\lambda (H8: (clear x1 -(CHead d (Bind Abbr) u))).(let H9 \def (eq_ind nat (minus d0 i) (\lambda (n: -nat).(drop h n x0 x1)) H7 (S (minus d0 (S i))) (minus_x_Sy d0 i H4)) in (let -H10 \def (drop_clear_S x1 x0 h (minus d0 (S i)) H9 Abbr d u H8) in (ex2_ind C -(\lambda (c3: C).(clear x0 (CHead c3 (Bind Abbr) (lift h (minus d0 (S i)) -u)))) (\lambda (c3: C).(drop h (minus d0 (S i)) c3 d)) (arity g c1 (TLRef i) -a0) (\lambda (x: C).(\lambda (H11: (clear x0 (CHead x (Bind Abbr) (lift h -(minus d0 (S i)) u)))).(\lambda (H12: (drop h (minus d0 (S i)) x -d)).(arity_abbr g c1 x (lift h (minus d0 (S i)) u) i (getl_intro i c1 (CHead -x (Bind Abbr) (lift h (minus d0 (S i)) u)) x0 H6 H11) a0 (H2 x h (minus d0 (S -i)) H12))))) H10)))))))) H5)) (lift h d0 (TLRef i)) (lift_lref_lt i h d0 -H4))) (\lambda (H4: (le d0 i)).(eq_ind_r T (TLRef (plus i h)) (\lambda (t0: -T).(arity g c1 t0 a0)) (arity_abbr g c1 d u (plus i h) (drop_getl_trans_ge i -c1 c d0 h H3 (CHead d (Bind Abbr) u) H0 H4) a0 H1) (lift h d0 (TLRef i)) -(lift_lref_ge i h d0 H4)))))))))))))))) (\lambda (c: C).(\lambda (d: -C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d (Bind -Abst) u))).(\lambda (a0: A).(\lambda (H1: (arity g d u (asucc g -a0))).(\lambda (H2: ((\forall (c1: C).(\forall (h: nat).(\forall (d0: -nat).((drop h d0 c1 d) \to (arity g c1 (lift h d0 u) (asucc g -a0)))))))).(\lambda (c1: C).(\lambda (h: nat).(\lambda (d0: nat).(\lambda -(H3: (drop h d0 c1 c)).(lt_le_e i d0 (arity g c1 (lift h d0 (TLRef i)) a0) -(\lambda (H4: (lt i d0)).(eq_ind_r T (TLRef i) (\lambda (t0: T).(arity g c1 -t0 a0)) (let H5 \def (drop_getl_trans_le i d0 (le_S_n i d0 (le_S (S i) d0 -H4)) c1 c h H3 (CHead d (Bind Abst) u) H0) in (ex3_2_ind C C (\lambda (e0: -C).(\lambda (_: C).(drop i O c1 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop -h (minus d0 i) e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear e1 (CHead d -(Bind Abst) u)))) (arity g c1 (TLRef i) a0) (\lambda (x0: C).(\lambda (x1: -C).(\lambda (H6: (drop i O c1 x0)).(\lambda (H7: (drop h (minus d0 i) x0 -x1)).(\lambda (H8: (clear x1 (CHead d (Bind Abst) u))).(let H9 \def (eq_ind -nat (minus d0 i) (\lambda (n: nat).(drop h n x0 x1)) H7 (S (minus d0 (S i))) -(minus_x_Sy d0 i H4)) in (let H10 \def (drop_clear_S x1 x0 h (minus d0 (S i)) -H9 Abst d u H8) in (ex2_ind C (\lambda (c3: C).(clear x0 (CHead c3 (Bind -Abst) (lift h (minus d0 (S i)) u)))) (\lambda (c3: C).(drop h (minus d0 (S -i)) c3 d)) (arity g c1 (TLRef i) a0) (\lambda (x: C).(\lambda (H11: (clear x0 -(CHead x (Bind Abst) (lift h (minus d0 (S i)) u)))).(\lambda (H12: (drop h -(minus d0 (S i)) x d)).(arity_abst g c1 x (lift h (minus d0 (S i)) u) i -(getl_intro i c1 (CHead x (Bind Abst) (lift h (minus d0 (S i)) u)) x0 H6 H11) -a0 (H2 x h (minus d0 (S i)) H12))))) H10)))))))) H5)) (lift h d0 (TLRef i)) -(lift_lref_lt i h d0 H4))) (\lambda (H4: (le d0 i)).(eq_ind_r T (TLRef (plus -i h)) (\lambda (t0: T).(arity g c1 t0 a0)) (arity_abst g c1 d u (plus i h) -(drop_getl_trans_ge i c1 c d0 h H3 (CHead d (Bind Abst) u) H0 H4) a0 H1) -(lift h d0 (TLRef i)) (lift_lref_ge i h d0 H4)))))))))))))))) (\lambda (b: -B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c: C).(\lambda (u: -T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H2: ((\forall -(c1: C).(\forall (h: nat).(\forall (d: nat).((drop h d c1 c) \to (arity g c1 -(lift h d u) a1))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity -g (CHead c (Bind b) u) t0 a2)).(\lambda (H4: ((\forall (c1: C).(\forall (h: -nat).(\forall (d: nat).((drop h d c1 (CHead c (Bind b) u)) \to (arity g c1 -(lift h d t0) a2))))))).(\lambda (c1: C).(\lambda (h: nat).(\lambda (d: -nat).(\lambda (H5: (drop h d c1 c)).(eq_ind_r T (THead (Bind b) (lift h d u) -(lift h (s (Bind b) d) t0)) (\lambda (t1: T).(arity g c1 t1 a2)) (arity_bind -g b H0 c1 (lift h d u) a1 (H2 c1 h d H5) (lift h (s (Bind b) d) t0) a2 (H4 -(CHead c1 (Bind b) (lift h d u)) h (s (Bind b) d) (drop_skip_bind h d c1 c H5 -b u))) (lift h d (THead (Bind b) u t0)) (lift_head (Bind b) u t0 h -d))))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda -(_: (arity g c u (asucc g a1))).(\lambda (H1: ((\forall (c1: C).(\forall (h: -nat).(\forall (d: nat).((drop h d c1 c) \to (arity g c1 (lift h d u) (asucc g -a1)))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c -(Bind Abst) u) t0 a2)).(\lambda (H3: ((\forall (c1: C).(\forall (h: -nat).(\forall (d: nat).((drop h d c1 (CHead c (Bind Abst) u)) \to (arity g c1 -(lift h d t0) a2))))))).(\lambda (c1: C).(\lambda (h: nat).(\lambda (d: -nat).(\lambda (H4: (drop h d c1 c)).(eq_ind_r T (THead (Bind Abst) (lift h d -u) (lift h (s (Bind Abst) d) t0)) (\lambda (t1: T).(arity g c1 t1 (AHead a1 -a2))) (arity_head g c1 (lift h d u) a1 (H1 c1 h d H4) (lift h (s (Bind Abst) -d) t0) a2 (H3 (CHead c1 (Bind Abst) (lift h d u)) h (s (Bind Abst) d) -(drop_skip_bind h d c1 c H4 Abst u))) (lift h d (THead (Bind Abst) u t0)) -(lift_head (Bind Abst) u t0 h d))))))))))))))) (\lambda (c: C).(\lambda (u: -T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H1: ((\forall -(c1: C).(\forall (h: nat).(\forall (d: nat).((drop h d c1 c) \to (arity g c1 -(lift h d u) a1))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity -g c t0 (AHead a1 a2))).(\lambda (H3: ((\forall (c1: C).(\forall (h: -nat).(\forall (d: nat).((drop h d c1 c) \to (arity g c1 (lift h d t0) (AHead -a1 a2)))))))).(\lambda (c1: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda -(H4: (drop h d c1 c)).(eq_ind_r T (THead (Flat Appl) (lift h d u) (lift h (s -(Flat Appl) d) t0)) (\lambda (t1: T).(arity g c1 t1 a2)) (arity_appl g c1 -(lift h d u) a1 (H1 c1 h d H4) (lift h (s (Flat Appl) d) t0) a2 (H3 c1 h (s -(Flat Appl) d) H4)) (lift h d (THead (Flat Appl) u t0)) (lift_head (Flat -Appl) u t0 h d))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a0: -A).(\lambda (_: (arity g c u (asucc g a0))).(\lambda (H1: ((\forall (c1: -C).(\forall (h: nat).(\forall (d: nat).((drop h d c1 c) \to (arity g c1 (lift -h d u) (asucc g a0)))))))).(\lambda (t0: T).(\lambda (_: (arity g c t0 -a0)).(\lambda (H3: ((\forall (c1: C).(\forall (h: nat).(\forall (d: -nat).((drop h d c1 c) \to (arity g c1 (lift h d t0) a0))))))).(\lambda (c1: -C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H4: (drop h d c1 -c)).(eq_ind_r T (THead (Flat Cast) (lift h d u) (lift h (s (Flat Cast) d) -t0)) (\lambda (t1: T).(arity g c1 t1 a0)) (arity_cast g c1 (lift h d u) a0 -(H1 c1 h d H4) (lift h (s (Flat Cast) d) t0) (H3 c1 h (s (Flat Cast) d) H4)) -(lift h d (THead (Flat Cast) u t0)) (lift_head (Flat Cast) u t0 h -d)))))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda -(_: (arity g c t0 a1)).(\lambda (H1: ((\forall (c1: C).(\forall (h: -nat).(\forall (d: nat).((drop h d c1 c) \to (arity g c1 (lift h d t0) -a1))))))).(\lambda (a2: A).(\lambda (H2: (leq g a1 a2)).(\lambda (c1: -C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H3: (drop h d c1 -c)).(arity_repl g c1 (lift h d t0) a1 (H1 c1 h d H3) a2 H2)))))))))))) c2 t a -H))))). -(* COMMENTS -Initial nodes: 2661 -END *) - -theorem arity_mono: - \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a1: A).((arity g c -t a1) \to (\forall (a2: A).((arity g c t a2) \to (leq g a1 a2))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a1: A).(\lambda (H: -(arity g c t a1)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (a: -A).(\forall (a2: A).((arity g c0 t0 a2) \to (leq g a a2)))))) (\lambda (c0: -C).(\lambda (n: nat).(\lambda (a2: A).(\lambda (H0: (arity g c0 (TSort n) -a2)).(leq_sym g a2 (ASort O n) (arity_gen_sort g c0 n a2 H0)))))) (\lambda -(c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl -i c0 (CHead d (Bind Abbr) u))).(\lambda (a: A).(\lambda (_: (arity g d u -a)).(\lambda (H2: ((\forall (a2: A).((arity g d u a2) \to (leq g a -a2))))).(\lambda (a2: A).(\lambda (H3: (arity g c0 (TLRef i) a2)).(let H4 -\def (arity_gen_lref g c0 i a2 H3) in (or_ind (ex2_2 C T (\lambda (d0: -C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abbr) u0)))) (\lambda (d0: -C).(\lambda (u0: T).(arity g d0 u0 a2)))) (ex2_2 C T (\lambda (d0: -C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0)))) (\lambda (d0: -C).(\lambda (u0: T).(arity g d0 u0 (asucc g a2))))) (leq g a a2) (\lambda -(H5: (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind -Abbr) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 -a2))))).(ex2_2_ind C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 -(Bind Abbr) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a2))) -(leq g a a2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl i c0 -(CHead x0 (Bind Abbr) x1))).(\lambda (H7: (arity g x0 x1 a2)).(let H8 \def -(eq_ind C (CHead d (Bind Abbr) u) (\lambda (c1: C).(getl i c0 c1)) H0 (CHead -x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead x0 (Bind -Abbr) x1) H6)) in (let H9 \def (f_equal C C (\lambda (e: C).(match e in C -return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) -\Rightarrow c1])) (CHead d (Bind Abbr) u) (CHead x0 (Bind Abbr) x1) -(getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead x0 (Bind Abbr) x1) H6)) in -((let H10 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: -C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead -d (Bind Abbr) u) (CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind Abbr) -u) i H0 (CHead x0 (Bind Abbr) x1) H6)) in (\lambda (H11: (eq C d x0)).(let -H12 \def (eq_ind_r T x1 (\lambda (t0: T).(getl i c0 (CHead x0 (Bind Abbr) -t0))) H8 u H10) in (let H13 \def (eq_ind_r T x1 (\lambda (t0: T).(arity g x0 -t0 a2)) H7 u H10) in (let H14 \def (eq_ind_r C x0 (\lambda (c1: C).(getl i c0 -(CHead c1 (Bind Abbr) u))) H12 d H11) in (let H15 \def (eq_ind_r C x0 -(\lambda (c1: C).(arity g c1 u a2)) H13 d H11) in (H2 a2 H15))))))) H9))))))) -H5)) (\lambda (H5: (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 -(CHead d0 (Bind Abst) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 -(asucc g a2)))))).(ex2_2_ind C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 -(CHead d0 (Bind Abst) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 -(asucc g a2)))) (leq g a a2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: -(getl i c0 (CHead x0 (Bind Abst) x1))).(\lambda (_: (arity g x0 x1 (asucc g -a2))).(let H8 \def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c1: C).(getl i -c0 c1)) H0 (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abbr) u) i -H0 (CHead x0 (Bind Abst) x1) H6)) in (let H9 \def (eq_ind C (CHead d (Bind -Abbr) u) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with -[(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return -(\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B return -(\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | -Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead x0 (Bind -Abst) x1) (getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead x0 (Bind Abst) -x1) H6)) in (False_ind (leq g a a2) H9))))))) H5)) H4)))))))))))) (\lambda -(c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl -i c0 (CHead d (Bind Abst) u))).(\lambda (a: A).(\lambda (_: (arity g d u -(asucc g a))).(\lambda (H2: ((\forall (a2: A).((arity g d u a2) \to (leq g -(asucc g a) a2))))).(\lambda (a2: A).(\lambda (H3: (arity g c0 (TLRef i) -a2)).(let H4 \def (arity_gen_lref g c0 i a2 H3) in (or_ind (ex2_2 C T -(\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abbr) u0)))) -(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a2)))) (ex2_2 C T (\lambda -(d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0)))) (\lambda -(d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a2))))) (leq g a a2) -(\lambda (H5: (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead -d0 (Bind Abbr) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 -a2))))).(ex2_2_ind C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 -(Bind Abbr) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a2))) -(leq g a a2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl i c0 -(CHead x0 (Bind Abbr) x1))).(\lambda (_: (arity g x0 x1 a2)).(let H8 \def -(eq_ind C (CHead d (Bind Abst) u) (\lambda (c1: C).(getl i c0 c1)) H0 (CHead -x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind Abst) u) i H0 (CHead x0 (Bind -Abbr) x1) H6)) in (let H9 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda -(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind b) \Rightarrow (match b in B return (\lambda (_: -B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow True | Void -\Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead x0 (Bind Abbr) -x1) (getl_mono c0 (CHead d (Bind Abst) u) i H0 (CHead x0 (Bind Abbr) x1) H6)) -in (False_ind (leq g a a2) H9))))))) H5)) (\lambda (H5: (ex2_2 C T (\lambda -(d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0)))) (\lambda -(d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a2)))))).(ex2_2_ind C T -(\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0)))) -(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a2)))) (leq g a a2) -(\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl i c0 (CHead x0 (Bind -Abst) x1))).(\lambda (H7: (arity g x0 x1 (asucc g a2))).(let H8 \def (eq_ind -C (CHead d (Bind Abst) u) (\lambda (c1: C).(getl i c0 c1)) H0 (CHead x0 (Bind -Abst) x1) (getl_mono c0 (CHead d (Bind Abst) u) i H0 (CHead x0 (Bind Abst) -x1) H6)) in (let H9 \def (f_equal C C (\lambda (e: C).(match e in C return -(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow -c1])) (CHead d (Bind Abst) u) (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead -d (Bind Abst) u) i H0 (CHead x0 (Bind Abst) x1) H6)) in ((let H10 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind -Abst) u) (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abst) u) i H0 -(CHead x0 (Bind Abst) x1) H6)) in (\lambda (H11: (eq C d x0)).(let H12 \def -(eq_ind_r T x1 (\lambda (t0: T).(getl i c0 (CHead x0 (Bind Abst) t0))) H8 u -H10) in (let H13 \def (eq_ind_r T x1 (\lambda (t0: T).(arity g x0 t0 (asucc g -a2))) H7 u H10) in (let H14 \def (eq_ind_r C x0 (\lambda (c1: C).(getl i c0 -(CHead c1 (Bind Abst) u))) H12 d H11) in (let H15 \def (eq_ind_r C x0 -(\lambda (c1: C).(arity g c1 u (asucc g a2))) H13 d H11) in (asucc_inj g a a2 -(H2 (asucc g a2) H15)))))))) H9))))))) H5)) H4)))))))))))) (\lambda (b: -B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c0: C).(\lambda (u: -T).(\lambda (a2: A).(\lambda (_: (arity g c0 u a2)).(\lambda (_: ((\forall -(a3: A).((arity g c0 u a3) \to (leq g a2 a3))))).(\lambda (t0: T).(\lambda -(a3: A).(\lambda (_: (arity g (CHead c0 (Bind b) u) t0 a3)).(\lambda (H4: -((\forall (a4: A).((arity g (CHead c0 (Bind b) u) t0 a4) \to (leq g a3 -a4))))).(\lambda (a0: A).(\lambda (H5: (arity g c0 (THead (Bind b) u t0) -a0)).(let H6 \def (arity_gen_bind b H0 g c0 u t0 a0 H5) in (ex2_ind A -(\lambda (a4: A).(arity g c0 u a4)) (\lambda (_: A).(arity g (CHead c0 (Bind -b) u) t0 a0)) (leq g a3 a0) (\lambda (x: A).(\lambda (_: (arity g c0 u -x)).(\lambda (H8: (arity g (CHead c0 (Bind b) u) t0 a0)).(H4 a0 H8)))) -H6))))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a2: A).(\lambda -(_: (arity g c0 u (asucc g a2))).(\lambda (H1: ((\forall (a3: A).((arity g c0 -u a3) \to (leq g (asucc g a2) a3))))).(\lambda (t0: T).(\lambda (a3: -A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u) t0 a3)).(\lambda (H3: -((\forall (a4: A).((arity g (CHead c0 (Bind Abst) u) t0 a4) \to (leq g a3 -a4))))).(\lambda (a0: A).(\lambda (H4: (arity g c0 (THead (Bind Abst) u t0) -a0)).(let H5 \def (arity_gen_abst g c0 u t0 a0 H4) in (ex3_2_ind A A (\lambda -(a4: A).(\lambda (a5: A).(eq A a0 (AHead a4 a5)))) (\lambda (a4: A).(\lambda -(_: A).(arity g c0 u (asucc g a4)))) (\lambda (_: A).(\lambda (a5: A).(arity -g (CHead c0 (Bind Abst) u) t0 a5))) (leq g (AHead a2 a3) a0) (\lambda (x0: -A).(\lambda (x1: A).(\lambda (H6: (eq A a0 (AHead x0 x1))).(\lambda (H7: -(arity g c0 u (asucc g x0))).(\lambda (H8: (arity g (CHead c0 (Bind Abst) u) -t0 x1)).(eq_ind_r A (AHead x0 x1) (\lambda (a: A).(leq g (AHead a2 a3) a)) -(leq_head g a2 x0 (asucc_inj g a2 x0 (H1 (asucc g x0) H7)) a3 x1 (H3 x1 H8)) -a0 H6)))))) H5))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a2: -A).(\lambda (_: (arity g c0 u a2)).(\lambda (_: ((\forall (a3: A).((arity g -c0 u a3) \to (leq g a2 a3))))).(\lambda (t0: T).(\lambda (a3: A).(\lambda (_: -(arity g c0 t0 (AHead a2 a3))).(\lambda (H3: ((\forall (a4: A).((arity g c0 -t0 a4) \to (leq g (AHead a2 a3) a4))))).(\lambda (a0: A).(\lambda (H4: (arity -g c0 (THead (Flat Appl) u t0) a0)).(let H5 \def (arity_gen_appl g c0 u t0 a0 -H4) in (ex2_ind A (\lambda (a4: A).(arity g c0 u a4)) (\lambda (a4: A).(arity -g c0 t0 (AHead a4 a0))) (leq g a3 a0) (\lambda (x: A).(\lambda (_: (arity g -c0 u x)).(\lambda (H7: (arity g c0 t0 (AHead x a0))).(ahead_inj_snd g a2 a3 x -a0 (H3 (AHead x a0) H7))))) H5))))))))))))) (\lambda (c0: C).(\lambda (u: -T).(\lambda (a: A).(\lambda (_: (arity g c0 u (asucc g a))).(\lambda (_: -((\forall (a2: A).((arity g c0 u a2) \to (leq g (asucc g a) a2))))).(\lambda -(t0: T).(\lambda (_: (arity g c0 t0 a)).(\lambda (H3: ((\forall (a2: -A).((arity g c0 t0 a2) \to (leq g a a2))))).(\lambda (a2: A).(\lambda (H4: -(arity g c0 (THead (Flat Cast) u t0) a2)).(let H5 \def (arity_gen_cast g c0 u -t0 a2 H4) in (land_ind (arity g c0 u (asucc g a2)) (arity g c0 t0 a2) (leq g -a a2) (\lambda (_: (arity g c0 u (asucc g a2))).(\lambda (H7: (arity g c0 t0 -a2)).(H3 a2 H7))) H5)))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda -(a2: A).(\lambda (_: (arity g c0 t0 a2)).(\lambda (H1: ((\forall (a3: -A).((arity g c0 t0 a3) \to (leq g a2 a3))))).(\lambda (a3: A).(\lambda (H2: -(leq g a2 a3)).(\lambda (a0: A).(\lambda (H3: (arity g c0 t0 a0)).(leq_trans -g a3 a2 (leq_sym g a2 a3 H2) a0 (H1 a0 H3))))))))))) c t a1 H))))). -(* COMMENTS -Initial nodes: 2947 -END *) - -theorem arity_repellent: - \forall (g: G).(\forall (c: C).(\forall (w: T).(\forall (t: T).(\forall (a1: -A).((arity g (CHead c (Bind Abst) w) t a1) \to (\forall (a2: A).((arity g c -(THead (Bind Abst) w t) a2) \to ((leq g a1 a2) \to (\forall (P: -Prop).P))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (w: T).(\lambda (t: T).(\lambda (a1: -A).(\lambda (H: (arity g (CHead c (Bind Abst) w) t a1)).(\lambda (a2: -A).(\lambda (H0: (arity g c (THead (Bind Abst) w t) a2)).(\lambda (H1: (leq g -a1 a2)).(\lambda (P: Prop).(let H_y \def (arity_repl g (CHead c (Bind Abst) -w) t a1 H a2 H1) in (let H2 \def (arity_gen_abst g c w t a2 H0) in (ex3_2_ind -A A (\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: -A).(\lambda (_: A).(arity g c w (asucc g a3)))) (\lambda (_: A).(\lambda (a4: -A).(arity g (CHead c (Bind Abst) w) t a4))) P (\lambda (x0: A).(\lambda (x1: -A).(\lambda (H3: (eq A a2 (AHead x0 x1))).(\lambda (_: (arity g c w (asucc g -x0))).(\lambda (H5: (arity g (CHead c (Bind Abst) w) t x1)).(let H6 \def -(eq_ind A a2 (\lambda (a: A).(arity g (CHead c (Bind Abst) w) t a)) H_y -(AHead x0 x1) H3) in (leq_ahead_false_2 g x1 x0 (arity_mono g (CHead c (Bind -Abst) w) t (AHead x0 x1) H6 x1 H5) P))))))) H2)))))))))))). -(* COMMENTS -Initial nodes: 283 -END *) - -theorem arity_appls_cast: - \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (vs: -TList).(\forall (a: A).((arity g c (THeads (Flat Appl) vs u) (asucc g a)) \to -((arity g c (THeads (Flat Appl) vs t) a) \to (arity g c (THeads (Flat Appl) -vs (THead (Flat Cast) u t)) a)))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (vs: -TList).(TList_ind (\lambda (t0: TList).(\forall (a: A).((arity g c (THeads -(Flat Appl) t0 u) (asucc g a)) \to ((arity g c (THeads (Flat Appl) t0 t) a) -\to (arity g c (THeads (Flat Appl) t0 (THead (Flat Cast) u t)) a))))) -(\lambda (a: A).(\lambda (H: (arity g c u (asucc g a))).(\lambda (H0: (arity -g c t a)).(arity_cast g c u a H t H0)))) (\lambda (t0: T).(\lambda (t1: -TList).(\lambda (H: ((\forall (a: A).((arity g c (THeads (Flat Appl) t1 u) -(asucc g a)) \to ((arity g c (THeads (Flat Appl) t1 t) a) \to (arity g c -(THeads (Flat Appl) t1 (THead (Flat Cast) u t)) a)))))).(\lambda (a: -A).(\lambda (H0: (arity g c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 u)) -(asucc g a))).(\lambda (H1: (arity g c (THead (Flat Appl) t0 (THeads (Flat -Appl) t1 t)) a)).(let H2 \def (arity_gen_appl g c t0 (THeads (Flat Appl) t1 -t) a H1) in (ex2_ind A (\lambda (a1: A).(arity g c t0 a1)) (\lambda (a1: -A).(arity g c (THeads (Flat Appl) t1 t) (AHead a1 a))) (arity g c (THead -(Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Flat Cast) u t))) a) (\lambda -(x: A).(\lambda (H3: (arity g c t0 x)).(\lambda (H4: (arity g c (THeads (Flat -Appl) t1 t) (AHead x a))).(let H5 \def (arity_gen_appl g c t0 (THeads (Flat -Appl) t1 u) (asucc g a) H0) in (ex2_ind A (\lambda (a1: A).(arity g c t0 a1)) -(\lambda (a1: A).(arity g c (THeads (Flat Appl) t1 u) (AHead a1 (asucc g -a)))) (arity g c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Flat -Cast) u t))) a) (\lambda (x0: A).(\lambda (H6: (arity g c t0 x0)).(\lambda -(H7: (arity g c (THeads (Flat Appl) t1 u) (AHead x0 (asucc g -a)))).(arity_appl g c t0 x H3 (THeads (Flat Appl) t1 (THead (Flat Cast) u t)) -a (H (AHead x a) (arity_repl g c (THeads (Flat Appl) t1 u) (AHead x (asucc g -a)) (arity_repl g c (THeads (Flat Appl) t1 u) (AHead x0 (asucc g a)) H7 -(AHead x (asucc g a)) (leq_head g x0 x (arity_mono g c t0 x0 H6 x H3) (asucc -g a) (asucc g a) (leq_refl g (asucc g a)))) (asucc g (AHead x a)) (leq_refl g -(asucc g (AHead x a)))) H4))))) H5))))) H2)))))))) vs))))). -(* COMMENTS -Initial nodes: 707 -END *) - -theorem arity_appls_abbr: - \forall (g: G).(\forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: -nat).((getl i c (CHead d (Bind Abbr) v)) \to (\forall (vs: TList).(\forall -(a: A).((arity g c (THeads (Flat Appl) vs (lift (S i) O v)) a) \to (arity g c -(THeads (Flat Appl) vs (TLRef i)) a))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: -nat).(\lambda (H: (getl i c (CHead d (Bind Abbr) v))).(\lambda (vs: -TList).(TList_ind (\lambda (t: TList).(\forall (a: A).((arity g c (THeads -(Flat Appl) t (lift (S i) O v)) a) \to (arity g c (THeads (Flat Appl) t -(TLRef i)) a)))) (\lambda (a: A).(\lambda (H0: (arity g c (lift (S i) O v) -a)).(arity_abbr g c d v i H a (arity_gen_lift g c v a (S i) O H0 d (getl_drop -Abbr c d v i H))))) (\lambda (t: T).(\lambda (t0: TList).(\lambda (H0: -((\forall (a: A).((arity g c (THeads (Flat Appl) t0 (lift (S i) O v)) a) \to -(arity g c (THeads (Flat Appl) t0 (TLRef i)) a))))).(\lambda (a: A).(\lambda -(H1: (arity g c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O -v))) a)).(let H2 \def (arity_gen_appl g c t (THeads (Flat Appl) t0 (lift (S -i) O v)) a H1) in (ex2_ind A (\lambda (a1: A).(arity g c t a1)) (\lambda (a1: -A).(arity g c (THeads (Flat Appl) t0 (lift (S i) O v)) (AHead a1 a))) (arity -g c (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) a) (\lambda (x: -A).(\lambda (H3: (arity g c t x)).(\lambda (H4: (arity g c (THeads (Flat -Appl) t0 (lift (S i) O v)) (AHead x a))).(arity_appl g c t x H3 (THeads (Flat -Appl) t0 (TLRef i)) a (H0 (AHead x a) H4))))) H2))))))) vs))))))). -(* COMMENTS -Initial nodes: 425 -END *) - -theorem arity_appls_bind: - \forall (g: G).(\forall (b: B).((not (eq B b Abst)) \to (\forall (c: -C).(\forall (v: T).(\forall (a1: A).((arity g c v a1) \to (\forall (t: -T).(\forall (vs: TList).(\forall (a2: A).((arity g (CHead c (Bind b) v) -(THeads (Flat Appl) (lifts (S O) O vs) t) a2) \to (arity g c (THeads (Flat -Appl) vs (THead (Bind b) v t)) a2))))))))))) -\def - \lambda (g: G).(\lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda -(c: C).(\lambda (v: T).(\lambda (a1: A).(\lambda (H0: (arity g c v -a1)).(\lambda (t: T).(\lambda (vs: TList).(TList_ind (\lambda (t0: -TList).(\forall (a2: A).((arity g (CHead c (Bind b) v) (THeads (Flat Appl) -(lifts (S O) O t0) t) a2) \to (arity g c (THeads (Flat Appl) t0 (THead (Bind -b) v t)) a2)))) (\lambda (a2: A).(\lambda (H1: (arity g (CHead c (Bind b) v) -t a2)).(arity_bind g b H c v a1 H0 t a2 H1))) (\lambda (t0: T).(\lambda (t1: -TList).(\lambda (H1: ((\forall (a2: A).((arity g (CHead c (Bind b) v) (THeads -(Flat Appl) (lifts (S O) O t1) t) a2) \to (arity g c (THeads (Flat Appl) t1 -(THead (Bind b) v t)) a2))))).(\lambda (a2: A).(\lambda (H2: (arity g (CHead -c (Bind b) v) (THead (Flat Appl) (lift (S O) O t0) (THeads (Flat Appl) (lifts -(S O) O t1) t)) a2)).(let H3 \def (arity_gen_appl g (CHead c (Bind b) v) -(lift (S O) O t0) (THeads (Flat Appl) (lifts (S O) O t1) t) a2 H2) in -(ex2_ind A (\lambda (a3: A).(arity g (CHead c (Bind b) v) (lift (S O) O t0) -a3)) (\lambda (a3: A).(arity g (CHead c (Bind b) v) (THeads (Flat Appl) -(lifts (S O) O t1) t) (AHead a3 a2))) (arity g c (THead (Flat Appl) t0 -(THeads (Flat Appl) t1 (THead (Bind b) v t))) a2) (\lambda (x: A).(\lambda -(H4: (arity g (CHead c (Bind b) v) (lift (S O) O t0) x)).(\lambda (H5: (arity -g (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O t1) t) (AHead x -a2))).(arity_appl g c t0 x (arity_gen_lift g (CHead c (Bind b) v) t0 x (S O) -O H4 c (drop_drop (Bind b) O c c (drop_refl c) v)) (THeads (Flat Appl) t1 -(THead (Bind b) v t)) a2 (H1 (AHead x a2) H5))))) H3))))))) vs))))))))). -(* COMMENTS -Initial nodes: 567 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/arity/subst0.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/arity/subst0.ma deleted file mode 100644 index 16046993b..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/arity/subst0.ma +++ /dev/null @@ -1,1137 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/arity/props.ma". - -include "Basic-1/fsubst0/fwd.ma". - -include "Basic-1/csubst0/getl.ma". - -include "Basic-1/subst0/dec.ma". - -include "Basic-1/subst0/fwd.ma". - -include "Basic-1/getl/getl.ma". - -theorem arity_gen_cvoid_subst0: - \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t -a) \to (\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d -(Bind Void) u)) \to (\forall (w: T).(\forall (v: T).((subst0 i w t v) \to -(\forall (P: Prop).P)))))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: -(arity g c t a)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (_: -A).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c0 (CHead d -(Bind Void) u)) \to (\forall (w: T).(\forall (v: T).((subst0 i w t0 v) \to -(\forall (P: Prop).P))))))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda -(d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d -(Bind Void) u))).(\lambda (w: T).(\lambda (v: T).(\lambda (H1: (subst0 i w -(TSort n) v)).(\lambda (P: Prop).(subst0_gen_sort w v i n H1 P))))))))))) -(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H0: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (a0: A).(\lambda (_: -(arity g d u a0)).(\lambda (_: ((\forall (d0: C).(\forall (u0: T).(\forall -(i0: nat).((getl i0 d (CHead d0 (Bind Void) u0)) \to (\forall (w: T).(\forall -(v: T).((subst0 i0 w u v) \to (\forall (P: Prop).P)))))))))).(\lambda (d0: -C).(\lambda (u0: T).(\lambda (i0: nat).(\lambda (H3: (getl i0 c0 (CHead d0 -(Bind Void) u0))).(\lambda (w: T).(\lambda (v: T).(\lambda (H4: (subst0 i0 w -(TLRef i) v)).(\lambda (P: Prop).(land_ind (eq nat i i0) (eq T v (lift (S i) -O w)) P (\lambda (H5: (eq nat i i0)).(\lambda (_: (eq T v (lift (S i) O -w))).(let H7 \def (eq_ind_r nat i0 (\lambda (n: nat).(getl n c0 (CHead d0 -(Bind Void) u0))) H3 i H5) in (let H8 \def (eq_ind C (CHead d (Bind Abbr) u) -(\lambda (c1: C).(getl i c0 c1)) H0 (CHead d0 (Bind Void) u0) (getl_mono c0 -(CHead d (Bind Abbr) u) i H0 (CHead d0 (Bind Void) u0) H7)) in (let H9 \def -(eq_ind C (CHead d (Bind Abbr) u) (\lambda (ee: C).(match ee in C return -(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b) -\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow -True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _) -\Rightarrow False])])) I (CHead d0 (Bind Void) u0) (getl_mono c0 (CHead d -(Bind Abbr) u) i H0 (CHead d0 (Bind Void) u0) H7)) in (False_ind P H9)))))) -(subst0_gen_lref w v i0 i H4)))))))))))))))))) (\lambda (c0: C).(\lambda (d: -C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind -Abst) u))).(\lambda (a0: A).(\lambda (_: (arity g d u (asucc g a0))).(\lambda -(_: ((\forall (d0: C).(\forall (u0: T).(\forall (i0: nat).((getl i0 d (CHead -d0 (Bind Void) u0)) \to (\forall (w: T).(\forall (v: T).((subst0 i0 w u v) -\to (\forall (P: Prop).P)))))))))).(\lambda (d0: C).(\lambda (u0: T).(\lambda -(i0: nat).(\lambda (H3: (getl i0 c0 (CHead d0 (Bind Void) u0))).(\lambda (w: -T).(\lambda (v: T).(\lambda (H4: (subst0 i0 w (TLRef i) v)).(\lambda (P: -Prop).(land_ind (eq nat i i0) (eq T v (lift (S i) O w)) P (\lambda (H5: (eq -nat i i0)).(\lambda (_: (eq T v (lift (S i) O w))).(let H7 \def (eq_ind_r nat -i0 (\lambda (n: nat).(getl n c0 (CHead d0 (Bind Void) u0))) H3 i H5) in (let -H8 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (c1: C).(getl i c0 c1)) H0 -(CHead d0 (Bind Void) u0) (getl_mono c0 (CHead d (Bind Abst) u) i H0 (CHead -d0 (Bind Void) u0) H7)) in (let H9 \def (eq_ind C (CHead d (Bind Abst) u) -(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind b) \Rightarrow (match b in B return (\lambda (_: -B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow True | Void -\Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead d0 (Bind Void) -u0) (getl_mono c0 (CHead d (Bind Abst) u) i H0 (CHead d0 (Bind Void) u0) H7)) -in (False_ind P H9)))))) (subst0_gen_lref w v i0 i H4)))))))))))))))))) -(\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (c0: C).(\lambda -(u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u a1)).(\lambda (H2: -((\forall (d: C).(\forall (u0: T).(\forall (i: nat).((getl i c0 (CHead d -(Bind Void) u0)) \to (\forall (w: T).(\forall (v: T).((subst0 i w u v) \to -(\forall (P: Prop).P)))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: -(arity g (CHead c0 (Bind b) u) t0 a2)).(\lambda (H4: ((\forall (d: -C).(\forall (u0: T).(\forall (i: nat).((getl i (CHead c0 (Bind b) u) (CHead d -(Bind Void) u0)) \to (\forall (w: T).(\forall (v: T).((subst0 i w t0 v) \to -(\forall (P: Prop).P)))))))))).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: -nat).(\lambda (H5: (getl i c0 (CHead d (Bind Void) u0))).(\lambda (w: -T).(\lambda (v: T).(\lambda (H6: (subst0 i w (THead (Bind b) u t0) -v)).(\lambda (P: Prop).(or3_ind (ex2 T (\lambda (u2: T).(eq T v (THead (Bind -b) u2 t0))) (\lambda (u2: T).(subst0 i w u u2))) (ex2 T (\lambda (t2: T).(eq -T v (THead (Bind b) u t2))) (\lambda (t2: T).(subst0 (s (Bind b) i) w t0 -t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Bind b) u2 -t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: -T).(\lambda (t2: T).(subst0 (s (Bind b) i) w t0 t2)))) P (\lambda (H7: (ex2 T -(\lambda (u2: T).(eq T v (THead (Bind b) u2 t0))) (\lambda (u2: T).(subst0 i -w u u2)))).(ex2_ind T (\lambda (u2: T).(eq T v (THead (Bind b) u2 t0))) -(\lambda (u2: T).(subst0 i w u u2)) P (\lambda (x: T).(\lambda (_: (eq T v -(THead (Bind b) x t0))).(\lambda (H9: (subst0 i w u x)).(H2 d u0 i H5 w x H9 -P)))) H7)) (\lambda (H7: (ex2 T (\lambda (t2: T).(eq T v (THead (Bind b) u -t2))) (\lambda (t2: T).(subst0 (s (Bind b) i) w t0 t2)))).(ex2_ind T (\lambda -(t2: T).(eq T v (THead (Bind b) u t2))) (\lambda (t2: T).(subst0 (s (Bind b) -i) w t0 t2)) P (\lambda (x: T).(\lambda (_: (eq T v (THead (Bind b) u -x))).(\lambda (H9: (subst0 (s (Bind b) i) w t0 x)).(H4 d u0 (S i) -(getl_clear_bind b (CHead c0 (Bind b) u) c0 u (clear_bind b c0 u) (CHead d -(Bind Void) u0) i H5) w x H9 P)))) H7)) (\lambda (H7: (ex3_2 T T (\lambda -(u2: T).(\lambda (t2: T).(eq T v (THead (Bind b) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda (t2: -T).(subst0 (s (Bind b) i) w t0 t2))))).(ex3_2_ind T T (\lambda (u2: -T).(\lambda (t2: T).(eq T v (THead (Bind b) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda (t2: -T).(subst0 (s (Bind b) i) w t0 t2))) P (\lambda (x0: T).(\lambda (x1: -T).(\lambda (_: (eq T v (THead (Bind b) x0 x1))).(\lambda (H9: (subst0 i w u -x0)).(\lambda (_: (subst0 (s (Bind b) i) w t0 x1)).(H2 d u0 i H5 w x0 H9 -P)))))) H7)) (subst0_gen_head (Bind b) w u t0 v i H6))))))))))))))))))))) -(\lambda (c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u -(asucc g a1))).(\lambda (H1: ((\forall (d: C).(\forall (u0: T).(\forall (i: -nat).((getl i c0 (CHead d (Bind Void) u0)) \to (\forall (w: T).(\forall (v: -T).((subst0 i w u v) \to (\forall (P: Prop).P)))))))))).(\lambda (t0: -T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u) t0 -a2)).(\lambda (H3: ((\forall (d: C).(\forall (u0: T).(\forall (i: nat).((getl -i (CHead c0 (Bind Abst) u) (CHead d (Bind Void) u0)) \to (\forall (w: -T).(\forall (v: T).((subst0 i w t0 v) \to (\forall (P: -Prop).P)))))))))).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda -(H4: (getl i c0 (CHead d (Bind Void) u0))).(\lambda (w: T).(\lambda (v: -T).(\lambda (H5: (subst0 i w (THead (Bind Abst) u t0) v)).(\lambda (P: -Prop).(or3_ind (ex2 T (\lambda (u2: T).(eq T v (THead (Bind Abst) u2 t0))) -(\lambda (u2: T).(subst0 i w u u2))) (ex2 T (\lambda (t2: T).(eq T v (THead -(Bind Abst) u t2))) (\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0 t2))) -(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Bind Abst) u2 -t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: -T).(\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0 t2)))) P (\lambda (H6: -(ex2 T (\lambda (u2: T).(eq T v (THead (Bind Abst) u2 t0))) (\lambda (u2: -T).(subst0 i w u u2)))).(ex2_ind T (\lambda (u2: T).(eq T v (THead (Bind -Abst) u2 t0))) (\lambda (u2: T).(subst0 i w u u2)) P (\lambda (x: T).(\lambda -(_: (eq T v (THead (Bind Abst) x t0))).(\lambda (H8: (subst0 i w u x)).(H1 d -u0 i H4 w x H8 P)))) H6)) (\lambda (H6: (ex2 T (\lambda (t2: T).(eq T v -(THead (Bind Abst) u t2))) (\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0 -t2)))).(ex2_ind T (\lambda (t2: T).(eq T v (THead (Bind Abst) u t2))) -(\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0 t2)) P (\lambda (x: -T).(\lambda (_: (eq T v (THead (Bind Abst) u x))).(\lambda (H8: (subst0 (s -(Bind Abst) i) w t0 x)).(H3 d u0 (S i) (getl_clear_bind Abst (CHead c0 (Bind -Abst) u) c0 u (clear_bind Abst c0 u) (CHead d (Bind Void) u0) i H4) w x H8 -P)))) H6)) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v -(THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w u -u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0 -t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Bind -Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda -(_: T).(\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0 t2))) P (\lambda (x0: -T).(\lambda (x1: T).(\lambda (_: (eq T v (THead (Bind Abst) x0 x1))).(\lambda -(H8: (subst0 i w u x0)).(\lambda (_: (subst0 (s (Bind Abst) i) w t0 x1)).(H1 -d u0 i H4 w x0 H8 P)))))) H6)) (subst0_gen_head (Bind Abst) w u t0 v i -H5))))))))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a1: -A).(\lambda (_: (arity g c0 u a1)).(\lambda (H1: ((\forall (d: C).(\forall -(u0: T).(\forall (i: nat).((getl i c0 (CHead d (Bind Void) u0)) \to (\forall -(w: T).(\forall (v: T).((subst0 i w u v) \to (\forall (P: -Prop).P)))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g c0 -t0 (AHead a1 a2))).(\lambda (H3: ((\forall (d: C).(\forall (u0: T).(\forall -(i: nat).((getl i c0 (CHead d (Bind Void) u0)) \to (\forall (w: T).(\forall -(v: T).((subst0 i w t0 v) \to (\forall (P: Prop).P)))))))))).(\lambda (d: -C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H4: (getl i c0 (CHead d (Bind -Void) u0))).(\lambda (w: T).(\lambda (v: T).(\lambda (H5: (subst0 i w (THead -(Flat Appl) u t0) v)).(\lambda (P: Prop).(or3_ind (ex2 T (\lambda (u2: T).(eq -T v (THead (Flat Appl) u2 t0))) (\lambda (u2: T).(subst0 i w u u2))) (ex2 T -(\lambda (t2: T).(eq T v (THead (Flat Appl) u t2))) (\lambda (t2: T).(subst0 -(s (Flat Appl) i) w t0 t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq -T v (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w -u u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s (Flat Appl) i) w t0 -t2)))) P (\lambda (H6: (ex2 T (\lambda (u2: T).(eq T v (THead (Flat Appl) u2 -t0))) (\lambda (u2: T).(subst0 i w u u2)))).(ex2_ind T (\lambda (u2: T).(eq T -v (THead (Flat Appl) u2 t0))) (\lambda (u2: T).(subst0 i w u u2)) P (\lambda -(x: T).(\lambda (_: (eq T v (THead (Flat Appl) x t0))).(\lambda (H8: (subst0 -i w u x)).(H1 d u0 i H4 w x H8 P)))) H6)) (\lambda (H6: (ex2 T (\lambda (t2: -T).(eq T v (THead (Flat Appl) u t2))) (\lambda (t2: T).(subst0 (s (Flat Appl) -i) w t0 t2)))).(ex2_ind T (\lambda (t2: T).(eq T v (THead (Flat Appl) u t2))) -(\lambda (t2: T).(subst0 (s (Flat Appl) i) w t0 t2)) P (\lambda (x: -T).(\lambda (_: (eq T v (THead (Flat Appl) u x))).(\lambda (H8: (subst0 (s -(Flat Appl) i) w t0 x)).(H3 d u0 i H4 w x H8 P)))) H6)) (\lambda (H6: (ex3_2 -T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Flat Appl) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda -(t2: T).(subst0 (s (Flat Appl) i) w t0 t2))))).(ex3_2_ind T T (\lambda (u2: -T).(\lambda (t2: T).(eq T v (THead (Flat Appl) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda (t2: -T).(subst0 (s (Flat Appl) i) w t0 t2))) P (\lambda (x0: T).(\lambda (x1: -T).(\lambda (_: (eq T v (THead (Flat Appl) x0 x1))).(\lambda (H8: (subst0 i w -u x0)).(\lambda (_: (subst0 (s (Flat Appl) i) w t0 x1)).(H1 d u0 i H4 w x0 H8 -P)))))) H6)) (subst0_gen_head (Flat Appl) w u t0 v i H5))))))))))))))))))) -(\lambda (c0: C).(\lambda (u: T).(\lambda (a0: A).(\lambda (_: (arity g c0 u -(asucc g a0))).(\lambda (H1: ((\forall (d: C).(\forall (u0: T).(\forall (i: -nat).((getl i c0 (CHead d (Bind Void) u0)) \to (\forall (w: T).(\forall (v: -T).((subst0 i w u v) \to (\forall (P: Prop).P)))))))))).(\lambda (t0: -T).(\lambda (_: (arity g c0 t0 a0)).(\lambda (H3: ((\forall (d: C).(\forall -(u0: T).(\forall (i: nat).((getl i c0 (CHead d (Bind Void) u0)) \to (\forall -(w: T).(\forall (v: T).((subst0 i w t0 v) \to (\forall (P: -Prop).P)))))))))).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda -(H4: (getl i c0 (CHead d (Bind Void) u0))).(\lambda (w: T).(\lambda (v: -T).(\lambda (H5: (subst0 i w (THead (Flat Cast) u t0) v)).(\lambda (P: -Prop).(or3_ind (ex2 T (\lambda (u2: T).(eq T v (THead (Flat Cast) u2 t0))) -(\lambda (u2: T).(subst0 i w u u2))) (ex2 T (\lambda (t2: T).(eq T v (THead -(Flat Cast) u t2))) (\lambda (t2: T).(subst0 (s (Flat Cast) i) w t0 t2))) -(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Flat Cast) u2 -t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: -T).(\lambda (t2: T).(subst0 (s (Flat Cast) i) w t0 t2)))) P (\lambda (H6: -(ex2 T (\lambda (u2: T).(eq T v (THead (Flat Cast) u2 t0))) (\lambda (u2: -T).(subst0 i w u u2)))).(ex2_ind T (\lambda (u2: T).(eq T v (THead (Flat -Cast) u2 t0))) (\lambda (u2: T).(subst0 i w u u2)) P (\lambda (x: T).(\lambda -(_: (eq T v (THead (Flat Cast) x t0))).(\lambda (H8: (subst0 i w u x)).(H1 d -u0 i H4 w x H8 P)))) H6)) (\lambda (H6: (ex2 T (\lambda (t2: T).(eq T v -(THead (Flat Cast) u t2))) (\lambda (t2: T).(subst0 (s (Flat Cast) i) w t0 -t2)))).(ex2_ind T (\lambda (t2: T).(eq T v (THead (Flat Cast) u t2))) -(\lambda (t2: T).(subst0 (s (Flat Cast) i) w t0 t2)) P (\lambda (x: -T).(\lambda (_: (eq T v (THead (Flat Cast) u x))).(\lambda (H8: (subst0 (s -(Flat Cast) i) w t0 x)).(H3 d u0 i H4 w x H8 P)))) H6)) (\lambda (H6: (ex3_2 -T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Flat Cast) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda -(t2: T).(subst0 (s (Flat Cast) i) w t0 t2))))).(ex3_2_ind T T (\lambda (u2: -T).(\lambda (t2: T).(eq T v (THead (Flat Cast) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda (t2: -T).(subst0 (s (Flat Cast) i) w t0 t2))) P (\lambda (x0: T).(\lambda (x1: -T).(\lambda (_: (eq T v (THead (Flat Cast) x0 x1))).(\lambda (H8: (subst0 i w -u x0)).(\lambda (_: (subst0 (s (Flat Cast) i) w t0 x1)).(H1 d u0 i H4 w x0 H8 -P)))))) H6)) (subst0_gen_head (Flat Cast) w u t0 v i H5)))))))))))))))))) -(\lambda (c0: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda (_: (arity g c0 -t0 a1)).(\lambda (H1: ((\forall (d: C).(\forall (u: T).(\forall (i: -nat).((getl i c0 (CHead d (Bind Void) u)) \to (\forall (w: T).(\forall (v: -T).((subst0 i w t0 v) \to (\forall (P: Prop).P)))))))))).(\lambda (a2: -A).(\lambda (_: (leq g a1 a2)).(\lambda (d: C).(\lambda (u: T).(\lambda (i: -nat).(\lambda (H3: (getl i c0 (CHead d (Bind Void) u))).(\lambda (w: -T).(\lambda (v: T).(\lambda (H4: (subst0 i w t0 v)).(\lambda (P: Prop).(H1 d -u i H3 w v H4 P)))))))))))))))) c t a H))))). -(* COMMENTS -Initial nodes: 4131 -END *) - -theorem arity_gen_cvoid: - \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t -a) \to (\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d -(Bind Void) u)) \to (ex T (\lambda (v: T).(eq T t (lift (S O) i v)))))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: -(arity g c t a)).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H0: (getl i c (CHead d (Bind Void) u))).(let H_x \def (dnf_dec u t i) in -(let H1 \def H_x in (ex_ind T (\lambda (v: T).(or (subst0 i u t (lift (S O) i -v)) (eq T t (lift (S O) i v)))) (ex T (\lambda (v: T).(eq T t (lift (S O) i -v)))) (\lambda (x: T).(\lambda (H2: (or (subst0 i u t (lift (S O) i x)) (eq T -t (lift (S O) i x)))).(or_ind (subst0 i u t (lift (S O) i x)) (eq T t (lift -(S O) i x)) (ex T (\lambda (v: T).(eq T t (lift (S O) i v)))) (\lambda (H3: -(subst0 i u t (lift (S O) i x))).(arity_gen_cvoid_subst0 g c t a H d u i H0 u -(lift (S O) i x) H3 (ex T (\lambda (v: T).(eq T t (lift (S O) i v)))))) -(\lambda (H3: (eq T t (lift (S O) i x))).(let H4 \def (eq_ind T t (\lambda -(t0: T).(arity g c t0 a)) H (lift (S O) i x) H3) in (eq_ind_r T (lift (S O) i -x) (\lambda (t0: T).(ex T (\lambda (v: T).(eq T t0 (lift (S O) i v))))) -(ex_intro T (\lambda (v: T).(eq T (lift (S O) i x) (lift (S O) i v))) x -(refl_equal T (lift (S O) i x))) t H3))) H2))) H1))))))))))). -(* COMMENTS -Initial nodes: 423 -END *) - -theorem arity_fsubst0: - \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (a: A).((arity g -c1 t1 a) \to (\forall (d1: C).(\forall (u: T).(\forall (i: nat).((getl i c1 -(CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u -c1 t1 c2 t2) \to (arity g c2 t2 a)))))))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (a: A).(\lambda -(H: (arity g c1 t1 a)).(arity_ind g (\lambda (c: C).(\lambda (t: T).(\lambda -(a0: A).(\forall (d1: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead -d1 (Bind Abbr) u)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u c t c2 -t2) \to (arity g c2 t2 a0))))))))))) (\lambda (c: C).(\lambda (n: -nat).(\lambda (d1: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (_: (getl i -c (CHead d1 (Bind Abbr) u))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H1: -(fsubst0 i u c (TSort n) c2 t2)).(let H_x \def (fsubst0_gen_base c c2 (TSort -n) t2 u i H1) in (let H2 \def H_x in (or3_ind (land (eq C c c2) (subst0 i u -(TSort n) t2)) (land (eq T (TSort n) t2) (csubst0 i u c c2)) (land (subst0 i -u (TSort n) t2) (csubst0 i u c c2)) (arity g c2 t2 (ASort O n)) (\lambda (H3: -(land (eq C c c2) (subst0 i u (TSort n) t2))).(land_ind (eq C c c2) (subst0 i -u (TSort n) t2) (arity g c2 t2 (ASort O n)) (\lambda (H4: (eq C c -c2)).(\lambda (H5: (subst0 i u (TSort n) t2)).(eq_ind C c (\lambda (c0: -C).(arity g c0 t2 (ASort O n))) (subst0_gen_sort u t2 i n H5 (arity g c t2 -(ASort O n))) c2 H4))) H3)) (\lambda (H3: (land (eq T (TSort n) t2) (csubst0 -i u c c2))).(land_ind (eq T (TSort n) t2) (csubst0 i u c c2) (arity g c2 t2 -(ASort O n)) (\lambda (H4: (eq T (TSort n) t2)).(\lambda (_: (csubst0 i u c -c2)).(eq_ind T (TSort n) (\lambda (t: T).(arity g c2 t (ASort O n))) -(arity_sort g c2 n) t2 H4))) H3)) (\lambda (H3: (land (subst0 i u (TSort n) -t2) (csubst0 i u c c2))).(land_ind (subst0 i u (TSort n) t2) (csubst0 i u c -c2) (arity g c2 t2 (ASort O n)) (\lambda (H4: (subst0 i u (TSort n) -t2)).(\lambda (_: (csubst0 i u c c2)).(subst0_gen_sort u t2 i n H4 (arity g -c2 t2 (ASort O n))))) H3)) H2)))))))))))) (\lambda (c: C).(\lambda (d: -C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d (Bind -Abbr) u))).(\lambda (a0: A).(\lambda (H1: (arity g d u a0)).(\lambda (H2: -((\forall (d1: C).(\forall (u0: T).(\forall (i0: nat).((getl i0 d (CHead d1 -(Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i0 u0 d u c2 -t2) \to (arity g c2 t2 a0)))))))))).(\lambda (d1: C).(\lambda (u0: -T).(\lambda (i0: nat).(\lambda (H3: (getl i0 c (CHead d1 (Bind Abbr) -u0))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H4: (fsubst0 i0 u0 c (TLRef -i) c2 t2)).(let H_x \def (fsubst0_gen_base c c2 (TLRef i) t2 u0 i0 H4) in -(let H5 \def H_x in (or3_ind (land (eq C c c2) (subst0 i0 u0 (TLRef i) t2)) -(land (eq T (TLRef i) t2) (csubst0 i0 u0 c c2)) (land (subst0 i0 u0 (TLRef i) -t2) (csubst0 i0 u0 c c2)) (arity g c2 t2 a0) (\lambda (H6: (land (eq C c c2) -(subst0 i0 u0 (TLRef i) t2))).(land_ind (eq C c c2) (subst0 i0 u0 (TLRef i) -t2) (arity g c2 t2 a0) (\lambda (H7: (eq C c c2)).(\lambda (H8: (subst0 i0 u0 -(TLRef i) t2)).(eq_ind C c (\lambda (c0: C).(arity g c0 t2 a0)) (land_ind (eq -nat i i0) (eq T t2 (lift (S i) O u0)) (arity g c t2 a0) (\lambda (H9: (eq nat -i i0)).(\lambda (H10: (eq T t2 (lift (S i) O u0))).(eq_ind_r T (lift (S i) O -u0) (\lambda (t: T).(arity g c t a0)) (let H11 \def (eq_ind_r nat i0 (\lambda -(n: nat).(getl n c (CHead d1 (Bind Abbr) u0))) H3 i H9) in (let H12 \def -(eq_ind C (CHead d (Bind Abbr) u) (\lambda (c0: C).(getl i c c0)) H0 (CHead -d1 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) i H0 (CHead d1 (Bind -Abbr) u0) H11)) in (let H13 \def (f_equal C C (\lambda (e: C).(match e in C -return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) -\Rightarrow c0])) (CHead d (Bind Abbr) u) (CHead d1 (Bind Abbr) u0) -(getl_mono c (CHead d (Bind Abbr) u) i H0 (CHead d1 (Bind Abbr) u0) H11)) in -((let H14 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: -C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d -(Bind Abbr) u) (CHead d1 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) -i H0 (CHead d1 (Bind Abbr) u0) H11)) in (\lambda (H15: (eq C d d1)).(let H16 -\def (eq_ind_r T u0 (\lambda (t: T).(getl i c (CHead d1 (Bind Abbr) t))) H12 -u H14) in (eq_ind T u (\lambda (t: T).(arity g c (lift (S i) O t) a0)) (let -H17 \def (eq_ind_r C d1 (\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr) u))) -H16 d H15) in (arity_lift g d u a0 H1 c (S i) O (getl_drop Abbr c d u i -H17))) u0 H14)))) H13)))) t2 H10))) (subst0_gen_lref u0 t2 i0 i H8)) c2 H7))) -H6)) (\lambda (H6: (land (eq T (TLRef i) t2) (csubst0 i0 u0 c c2))).(land_ind -(eq T (TLRef i) t2) (csubst0 i0 u0 c c2) (arity g c2 t2 a0) (\lambda (H7: (eq -T (TLRef i) t2)).(\lambda (H8: (csubst0 i0 u0 c c2)).(eq_ind T (TLRef i) -(\lambda (t: T).(arity g c2 t a0)) (lt_le_e i i0 (arity g c2 (TLRef i) a0) -(\lambda (H9: (lt i i0)).(let H10 \def (csubst0_getl_lt i0 i H9 c c2 u0 H8 -(CHead d (Bind Abbr) u) H0) in (or4_ind (getl i c2 (CHead d (Bind Abbr) u)) -(ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: -B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl i c2 (CHead e0 -(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda -(w: T).(subst0 (minus i0 (S i)) u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind -Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u1: T).(getl i c2 (CHead e2 (Bind b) u1)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S -i)) u0 e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead -e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(getl i c2 (CHead e2 (Bind b) w))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: -T).(subst0 (minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) -u0 e1 e2))))))) (arity g c2 (TLRef i) a0) (\lambda (H11: (getl i c2 (CHead d -(Bind Abbr) u))).(let H12 \def (eq_ind nat (minus i0 i) (\lambda (n: -nat).(getl n (CHead d (Bind Abbr) u) (CHead d1 (Bind Abbr) u0))) -(getl_conf_le i0 (CHead d1 (Bind Abbr) u0) c H3 (CHead d (Bind Abbr) u) i H0 -(le_S_n i i0 (le_S (S i) i0 H9))) (S (minus i0 (S i))) (minus_x_Sy i0 i H9)) -in (arity_abbr g c2 d u i H11 a0 H1))) (\lambda (H11: (ex3_4 B C T T (\lambda -(b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind -Abbr) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(getl i c2 (CHead e0 (Bind b) w)))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 -(minus i0 (S i)) u0 u1 w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda -(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead -e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(getl i c2 (CHead e0 (Bind b) w)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) -u0 u1 w))))) (arity g c2 (TLRef i) a0) (\lambda (x0: B).(\lambda (x1: -C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (eq C (CHead d (Bind -Abbr) u) (CHead x1 (Bind x0) x2))).(\lambda (H13: (getl i c2 (CHead x1 (Bind -x0) x3))).(\lambda (H14: (subst0 (minus i0 (S i)) u0 x2 x3)).(let H15 \def -(eq_ind nat (minus i0 i) (\lambda (n: nat).(getl n (CHead d (Bind Abbr) u) -(CHead d1 (Bind Abbr) u0))) (getl_conf_le i0 (CHead d1 (Bind Abbr) u0) c H3 -(CHead d (Bind Abbr) u) i H0 (le_S_n i i0 (le_S (S i) i0 H9))) (S (minus i0 -(S i))) (minus_x_Sy i0 i H9)) in (let H16 \def (f_equal C C (\lambda (e: -C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | -(CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) -x2) H12) in ((let H17 \def (f_equal C B (\lambda (e: C).(match e in C return -(\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) -\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead -x1 (Bind x0) x2) H12) in ((let H18 \def (f_equal C T (\lambda (e: C).(match e -in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) -\Rightarrow t])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H12) in -(\lambda (H19: (eq B Abbr x0)).(\lambda (H20: (eq C d x1)).(let H21 \def -(eq_ind_r T x2 (\lambda (t: T).(subst0 (minus i0 (S i)) u0 t x3)) H14 u H18) -in (let H22 \def (eq_ind_r C x1 (\lambda (c0: C).(getl i c2 (CHead c0 (Bind -x0) x3))) H13 d H20) in (let H23 \def (eq_ind_r B x0 (\lambda (b: B).(getl i -c2 (CHead d (Bind b) x3))) H22 Abbr H19) in (arity_abbr g c2 d x3 i H23 a0 -(H2 d1 u0 (r (Bind Abbr) (minus i0 (S i))) (getl_gen_S (Bind Abbr) d (CHead -d1 (Bind Abbr) u0) u (minus i0 (S i)) H15) d x3 (fsubst0_snd (r (Bind Abbr) -(minus i0 (S i))) u0 d u x3 H21))))))))) H17)) H16)))))))))) H11)) (\lambda -(H11: (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c2 -(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))).(ex3_4_ind B C C -T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C -(CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda -(_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c2 (CHead e2 (Bind b) -u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus i0 (S i)) u0 e1 e2))))) (arity g c2 (TLRef i) a0) (\lambda -(x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H12: (eq -C (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3))).(\lambda (H13: (getl i c2 -(CHead x2 (Bind x0) x3))).(\lambda (H14: (csubst0 (minus i0 (S i)) u0 x1 -x2)).(let H15 \def (eq_ind nat (minus i0 i) (\lambda (n: nat).(getl n (CHead -d (Bind Abbr) u) (CHead d1 (Bind Abbr) u0))) (getl_conf_le i0 (CHead d1 (Bind -Abbr) u0) c H3 (CHead d (Bind Abbr) u) i H0 (le_S_n i i0 (le_S (S i) i0 H9))) -(S (minus i0 (S i))) (minus_x_Sy i0 i H9)) in (let H16 \def (f_equal C C -(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abbr) u) -(CHead x1 (Bind x0) x3) H12) in ((let H17 \def (f_equal C B (\lambda (e: -C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | -(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind -b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) -(CHead x1 (Bind x0) x3) H12) in ((let H18 \def (f_equal C T (\lambda (e: -C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | -(CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) -x3) H12) in (\lambda (H19: (eq B Abbr x0)).(\lambda (H20: (eq C d x1)).(let -H21 \def (eq_ind_r T x3 (\lambda (t: T).(getl i c2 (CHead x2 (Bind x0) t))) -H13 u H18) in (let H22 \def (eq_ind_r C x1 (\lambda (c0: C).(csubst0 (minus -i0 (S i)) u0 c0 x2)) H14 d H20) in (let H23 \def (eq_ind_r B x0 (\lambda (b: -B).(getl i c2 (CHead x2 (Bind b) u))) H21 Abbr H19) in (arity_abbr g c2 x2 u -i H23 a0 (H2 d1 u0 (r (Bind Abbr) (minus i0 (S i))) (getl_gen_S (Bind Abbr) d -(CHead d1 (Bind Abbr) u0) u (minus i0 (S i)) H15) x2 u (fsubst0_fst (r (Bind -Abbr) (minus i0 (S i))) u0 d u x2 H22))))))))) H17)) H16)))))))))) H11)) -(\lambda (H11: (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 -(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(getl i c2 (CHead e2 (Bind b) w))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 -(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 -e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 -(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(getl i c2 (CHead e2 (Bind b) w))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 -(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) -(arity g c2 (TLRef i) a0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: -C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H12: (eq C (CHead d (Bind -Abbr) u) (CHead x1 (Bind x0) x3))).(\lambda (H13: (getl i c2 (CHead x2 (Bind -x0) x4))).(\lambda (H14: (subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H15: -(csubst0 (minus i0 (S i)) u0 x1 x2)).(let H16 \def (eq_ind nat (minus i0 i) -(\lambda (n: nat).(getl n (CHead d (Bind Abbr) u) (CHead d1 (Bind Abbr) u0))) -(getl_conf_le i0 (CHead d1 (Bind Abbr) u0) c H3 (CHead d (Bind Abbr) u) i H0 -(le_S_n i i0 (le_S (S i) i0 H9))) (S (minus i0 (S i))) (minus_x_Sy i0 i H9)) -in (let H17 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda -(_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) -(CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H12) in ((let H18 \def -(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with -[(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k in K return -(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow -Abbr])])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H12) in ((let H19 -\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind -Abbr) u) (CHead x1 (Bind x0) x3) H12) in (\lambda (H20: (eq B Abbr -x0)).(\lambda (H21: (eq C d x1)).(let H22 \def (eq_ind_r T x3 (\lambda (t: -T).(subst0 (minus i0 (S i)) u0 t x4)) H14 u H19) in (let H23 \def (eq_ind_r C -x1 (\lambda (c0: C).(csubst0 (minus i0 (S i)) u0 c0 x2)) H15 d H21) in (let -H24 \def (eq_ind_r B x0 (\lambda (b: B).(getl i c2 (CHead x2 (Bind b) x4))) -H13 Abbr H20) in (arity_abbr g c2 x2 x4 i H24 a0 (H2 d1 u0 (r (Bind Abbr) -(minus i0 (S i))) (getl_gen_S (Bind Abbr) d (CHead d1 (Bind Abbr) u0) u -(minus i0 (S i)) H16) x2 x4 (fsubst0_both (r (Bind Abbr) (minus i0 (S i))) u0 -d u x4 H22 x2 H23))))))))) H18)) H17)))))))))))) H11)) H10))) (\lambda (H9: -(le i0 i)).(arity_abbr g c2 d u i (csubst0_getl_ge i0 i H9 c c2 u0 H8 (CHead -d (Bind Abbr) u) H0) a0 H1))) t2 H7))) H6)) (\lambda (H6: (land (subst0 i0 u0 -(TLRef i) t2) (csubst0 i0 u0 c c2))).(land_ind (subst0 i0 u0 (TLRef i) t2) -(csubst0 i0 u0 c c2) (arity g c2 t2 a0) (\lambda (H7: (subst0 i0 u0 (TLRef i) -t2)).(\lambda (H8: (csubst0 i0 u0 c c2)).(land_ind (eq nat i i0) (eq T t2 -(lift (S i) O u0)) (arity g c2 t2 a0) (\lambda (H9: (eq nat i i0)).(\lambda -(H10: (eq T t2 (lift (S i) O u0))).(eq_ind_r T (lift (S i) O u0) (\lambda (t: -T).(arity g c2 t a0)) (let H11 \def (eq_ind_r nat i0 (\lambda (n: -nat).(csubst0 n u0 c c2)) H8 i H9) in (let H12 \def (eq_ind_r nat i0 (\lambda -(n: nat).(getl n c (CHead d1 (Bind Abbr) u0))) H3 i H9) in (let H13 \def -(eq_ind C (CHead d (Bind Abbr) u) (\lambda (c0: C).(getl i c c0)) H0 (CHead -d1 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) i H0 (CHead d1 (Bind -Abbr) u0) H12)) in (let H14 \def (f_equal C C (\lambda (e: C).(match e in C -return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) -\Rightarrow c0])) (CHead d (Bind Abbr) u) (CHead d1 (Bind Abbr) u0) -(getl_mono c (CHead d (Bind Abbr) u) i H0 (CHead d1 (Bind Abbr) u0) H12)) in -((let H15 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: -C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d -(Bind Abbr) u) (CHead d1 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) -i H0 (CHead d1 (Bind Abbr) u0) H12)) in (\lambda (H16: (eq C d d1)).(let H17 -\def (eq_ind_r T u0 (\lambda (t: T).(getl i c (CHead d1 (Bind Abbr) t))) H13 -u H15) in (let H18 \def (eq_ind_r T u0 (\lambda (t: T).(csubst0 i t c c2)) -H11 u H15) in (eq_ind T u (\lambda (t: T).(arity g c2 (lift (S i) O t) a0)) -(let H19 \def (eq_ind_r C d1 (\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr) -u))) H17 d H16) in (arity_lift g d u a0 H1 c2 (S i) O (getl_drop Abbr c2 d u -i (csubst0_getl_ge i i (le_n i) c c2 u H18 (CHead d (Bind Abbr) u) H19)))) u0 -H15))))) H14))))) t2 H10))) (subst0_gen_lref u0 t2 i0 i H7)))) H6)) -H5)))))))))))))))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda -(i: nat).(\lambda (H0: (getl i c (CHead d (Bind Abst) u))).(\lambda (a0: -A).(\lambda (H1: (arity g d u (asucc g a0))).(\lambda (H2: ((\forall (d1: -C).(\forall (u0: T).(\forall (i0: nat).((getl i0 d (CHead d1 (Bind Abbr) u0)) -\to (\forall (c2: C).(\forall (t2: T).((fsubst0 i0 u0 d u c2 t2) \to (arity g -c2 t2 (asucc g a0))))))))))).(\lambda (d1: C).(\lambda (u0: T).(\lambda (i0: -nat).(\lambda (H3: (getl i0 c (CHead d1 (Bind Abbr) u0))).(\lambda (c2: -C).(\lambda (t2: T).(\lambda (H4: (fsubst0 i0 u0 c (TLRef i) c2 t2)).(let H_x -\def (fsubst0_gen_base c c2 (TLRef i) t2 u0 i0 H4) in (let H5 \def H_x in -(or3_ind (land (eq C c c2) (subst0 i0 u0 (TLRef i) t2)) (land (eq T (TLRef i) -t2) (csubst0 i0 u0 c c2)) (land (subst0 i0 u0 (TLRef i) t2) (csubst0 i0 u0 c -c2)) (arity g c2 t2 a0) (\lambda (H6: (land (eq C c c2) (subst0 i0 u0 (TLRef -i) t2))).(land_ind (eq C c c2) (subst0 i0 u0 (TLRef i) t2) (arity g c2 t2 a0) -(\lambda (H7: (eq C c c2)).(\lambda (H8: (subst0 i0 u0 (TLRef i) t2)).(eq_ind -C c (\lambda (c0: C).(arity g c0 t2 a0)) (land_ind (eq nat i i0) (eq T t2 -(lift (S i) O u0)) (arity g c t2 a0) (\lambda (H9: (eq nat i i0)).(\lambda -(H10: (eq T t2 (lift (S i) O u0))).(eq_ind_r T (lift (S i) O u0) (\lambda (t: -T).(arity g c t a0)) (let H11 \def (eq_ind_r nat i0 (\lambda (n: nat).(getl n -c (CHead d1 (Bind Abbr) u0))) H3 i H9) in (let H12 \def (eq_ind C (CHead d -(Bind Abst) u) (\lambda (c0: C).(getl i c c0)) H0 (CHead d1 (Bind Abbr) u0) -(getl_mono c (CHead d (Bind Abst) u) i H0 (CHead d1 (Bind Abbr) u0) H11)) in -(let H13 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (ee: C).(match ee in -C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k -_) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b) -\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow -False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) -\Rightarrow False])])) I (CHead d1 (Bind Abbr) u0) (getl_mono c (CHead d -(Bind Abst) u) i H0 (CHead d1 (Bind Abbr) u0) H11)) in (False_ind (arity g c -(lift (S i) O u0) a0) H13)))) t2 H10))) (subst0_gen_lref u0 t2 i0 i H8)) c2 -H7))) H6)) (\lambda (H6: (land (eq T (TLRef i) t2) (csubst0 i0 u0 c -c2))).(land_ind (eq T (TLRef i) t2) (csubst0 i0 u0 c c2) (arity g c2 t2 a0) -(\lambda (H7: (eq T (TLRef i) t2)).(\lambda (H8: (csubst0 i0 u0 c -c2)).(eq_ind T (TLRef i) (\lambda (t: T).(arity g c2 t a0)) (lt_le_e i i0 -(arity g c2 (TLRef i) a0) (\lambda (H9: (lt i i0)).(let H10 \def -(csubst0_getl_lt i0 i H9 c c2 u0 H8 (CHead d (Bind Abst) u) H0) in (or4_ind -(getl i c2 (CHead d (Bind Abst) u)) (ex3_4 B C T T (\lambda (b: B).(\lambda -(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead -e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(getl i c2 (CHead e0 (Bind b) w)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) -u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1)))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c2 -(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) (ex4_5 B C C T T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1))))))) (\lambda -(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl -i c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))) (arity g c2 (TLRef i) a0) -(\lambda (H11: (getl i c2 (CHead d (Bind Abst) u))).(let H12 \def (eq_ind nat -(minus i0 i) (\lambda (n: nat).(getl n (CHead d (Bind Abst) u) (CHead d1 -(Bind Abbr) u0))) (getl_conf_le i0 (CHead d1 (Bind Abbr) u0) c H3 (CHead d -(Bind Abst) u) i H0 (le_S_n i i0 (le_S (S i) i0 H9))) (S (minus i0 (S i))) -(minus_x_Sy i0 i H9)) in (arity_abst g c2 d u i H11 a0 H1))) (\lambda (H11: -(ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(eq C (CHead d (Bind Abst) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: -B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl i c2 (CHead e0 -(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda -(w: T).(subst0 (minus i0 (S i)) u0 u1 w))))))).(ex3_4_ind B C T T (\lambda -(b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind -Abst) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(getl i c2 (CHead e0 (Bind b) w)))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 -(minus i0 (S i)) u0 u1 w))))) (arity g c2 (TLRef i) a0) (\lambda (x0: -B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (eq C -(CHead d (Bind Abst) u) (CHead x1 (Bind x0) x2))).(\lambda (H13: (getl i c2 -(CHead x1 (Bind x0) x3))).(\lambda (H14: (subst0 (minus i0 (S i)) u0 x2 -x3)).(let H15 \def (eq_ind nat (minus i0 i) (\lambda (n: nat).(getl n (CHead -d (Bind Abst) u) (CHead d1 (Bind Abbr) u0))) (getl_conf_le i0 (CHead d1 (Bind -Abbr) u0) c H3 (CHead d (Bind Abst) u) i H0 (le_S_n i i0 (le_S (S i) i0 H9))) -(S (minus i0 (S i))) (minus_x_Sy i0 i H9)) in (let H16 \def (f_equal C C -(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abst) u) -(CHead x1 (Bind x0) x2) H12) in ((let H17 \def (f_equal C B (\lambda (e: -C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abst | -(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind -b) \Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead d (Bind Abst) u) -(CHead x1 (Bind x0) x2) H12) in ((let H18 \def (f_equal C T (\lambda (e: -C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | -(CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) -x2) H12) in (\lambda (H19: (eq B Abst x0)).(\lambda (H20: (eq C d x1)).(let -H21 \def (eq_ind_r T x2 (\lambda (t: T).(subst0 (minus i0 (S i)) u0 t x3)) -H14 u H18) in (let H22 \def (eq_ind_r C x1 (\lambda (c0: C).(getl i c2 (CHead -c0 (Bind x0) x3))) H13 d H20) in (let H23 \def (eq_ind_r B x0 (\lambda (b: -B).(getl i c2 (CHead d (Bind b) x3))) H22 Abst H19) in (arity_abst g c2 d x3 -i H23 a0 (H2 d1 u0 (r (Bind Abst) (minus i0 (S i))) (getl_gen_S (Bind Abst) d -(CHead d1 (Bind Abbr) u0) u (minus i0 (S i)) H15) d x3 (fsubst0_snd (r (Bind -Abst) (minus i0 (S i))) u0 d u x3 H21))))))))) H17)) H16)))))))))) H11)) -(\lambda (H11: (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1)))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c2 -(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))).(ex3_4_ind B C C -T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C -(CHead d (Bind Abst) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda -(_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c2 (CHead e2 (Bind b) -u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus i0 (S i)) u0 e1 e2))))) (arity g c2 (TLRef i) a0) (\lambda -(x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H12: (eq -C (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3))).(\lambda (H13: (getl i c2 -(CHead x2 (Bind x0) x3))).(\lambda (H14: (csubst0 (minus i0 (S i)) u0 x1 -x2)).(let H15 \def (eq_ind nat (minus i0 i) (\lambda (n: nat).(getl n (CHead -d (Bind Abst) u) (CHead d1 (Bind Abbr) u0))) (getl_conf_le i0 (CHead d1 (Bind -Abbr) u0) c H3 (CHead d (Bind Abst) u) i H0 (le_S_n i i0 (le_S (S i) i0 H9))) -(S (minus i0 (S i))) (minus_x_Sy i0 i H9)) in (let H16 \def (f_equal C C -(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abst) u) -(CHead x1 (Bind x0) x3) H12) in ((let H17 \def (f_equal C B (\lambda (e: -C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abst | -(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind -b) \Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead d (Bind Abst) u) -(CHead x1 (Bind x0) x3) H12) in ((let H18 \def (f_equal C T (\lambda (e: -C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | -(CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) -x3) H12) in (\lambda (H19: (eq B Abst x0)).(\lambda (H20: (eq C d x1)).(let -H21 \def (eq_ind_r T x3 (\lambda (t: T).(getl i c2 (CHead x2 (Bind x0) t))) -H13 u H18) in (let H22 \def (eq_ind_r C x1 (\lambda (c0: C).(csubst0 (minus -i0 (S i)) u0 c0 x2)) H14 d H20) in (let H23 \def (eq_ind_r B x0 (\lambda (b: -B).(getl i c2 (CHead x2 (Bind b) u))) H21 Abst H19) in (arity_abst g c2 x2 u -i H23 a0 (H2 d1 u0 (r (Bind Abst) (minus i0 (S i))) (getl_gen_S (Bind Abst) d -(CHead d1 (Bind Abbr) u0) u (minus i0 (S i)) H15) x2 u (fsubst0_fst (r (Bind -Abst) (minus i0 (S i))) u0 d u x2 H22))))))))) H17)) H16)))))))))) H11)) -(\lambda (H11: (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead e1 -(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(getl i c2 (CHead e2 (Bind b) w))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 -(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 -e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead e1 -(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(getl i c2 (CHead e2 (Bind b) w))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 -(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) -(arity g c2 (TLRef i) a0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: -C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H12: (eq C (CHead d (Bind -Abst) u) (CHead x1 (Bind x0) x3))).(\lambda (H13: (getl i c2 (CHead x2 (Bind -x0) x4))).(\lambda (H14: (subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H15: -(csubst0 (minus i0 (S i)) u0 x1 x2)).(let H16 \def (eq_ind nat (minus i0 i) -(\lambda (n: nat).(getl n (CHead d (Bind Abst) u) (CHead d1 (Bind Abbr) u0))) -(getl_conf_le i0 (CHead d1 (Bind Abbr) u0) c H3 (CHead d (Bind Abst) u) i H0 -(le_S_n i i0 (le_S (S i) i0 H9))) (S (minus i0 (S i))) (minus_x_Sy i0 i H9)) -in (let H17 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda -(_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) -(CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3) H12) in ((let H18 \def -(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with -[(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match k in K return -(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow -Abst])])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3) H12) in ((let H19 -\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind -Abst) u) (CHead x1 (Bind x0) x3) H12) in (\lambda (H20: (eq B Abst -x0)).(\lambda (H21: (eq C d x1)).(let H22 \def (eq_ind_r T x3 (\lambda (t: -T).(subst0 (minus i0 (S i)) u0 t x4)) H14 u H19) in (let H23 \def (eq_ind_r C -x1 (\lambda (c0: C).(csubst0 (minus i0 (S i)) u0 c0 x2)) H15 d H21) in (let -H24 \def (eq_ind_r B x0 (\lambda (b: B).(getl i c2 (CHead x2 (Bind b) x4))) -H13 Abst H20) in (arity_abst g c2 x2 x4 i H24 a0 (H2 d1 u0 (r (Bind Abst) -(minus i0 (S i))) (getl_gen_S (Bind Abst) d (CHead d1 (Bind Abbr) u0) u -(minus i0 (S i)) H16) x2 x4 (fsubst0_both (r (Bind Abst) (minus i0 (S i))) u0 -d u x4 H22 x2 H23))))))))) H18)) H17)))))))))))) H11)) H10))) (\lambda (H9: -(le i0 i)).(arity_abst g c2 d u i (csubst0_getl_ge i0 i H9 c c2 u0 H8 (CHead -d (Bind Abst) u) H0) a0 H1))) t2 H7))) H6)) (\lambda (H6: (land (subst0 i0 u0 -(TLRef i) t2) (csubst0 i0 u0 c c2))).(land_ind (subst0 i0 u0 (TLRef i) t2) -(csubst0 i0 u0 c c2) (arity g c2 t2 a0) (\lambda (H7: (subst0 i0 u0 (TLRef i) -t2)).(\lambda (H8: (csubst0 i0 u0 c c2)).(land_ind (eq nat i i0) (eq T t2 -(lift (S i) O u0)) (arity g c2 t2 a0) (\lambda (H9: (eq nat i i0)).(\lambda -(H10: (eq T t2 (lift (S i) O u0))).(eq_ind_r T (lift (S i) O u0) (\lambda (t: -T).(arity g c2 t a0)) (let H11 \def (eq_ind_r nat i0 (\lambda (n: -nat).(csubst0 n u0 c c2)) H8 i H9) in (let H12 \def (eq_ind_r nat i0 (\lambda -(n: nat).(getl n c (CHead d1 (Bind Abbr) u0))) H3 i H9) in (let H13 \def -(eq_ind C (CHead d (Bind Abst) u) (\lambda (c0: C).(getl i c c0)) H0 (CHead -d1 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abst) u) i H0 (CHead d1 (Bind -Abbr) u0) H12)) in (let H14 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda -(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind b) \Rightarrow (match b in B return (\lambda (_: -B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow True | Void -\Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead d1 (Bind Abbr) -u0) (getl_mono c (CHead d (Bind Abst) u) i H0 (CHead d1 (Bind Abbr) u0) H12)) -in (False_ind (arity g c2 (lift (S i) O u0) a0) H14))))) t2 H10))) -(subst0_gen_lref u0 t2 i0 i H7)))) H6)) H5)))))))))))))))))) (\lambda (b: -B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c: C).(\lambda (u: -T).(\lambda (a1: A).(\lambda (H1: (arity g c u a1)).(\lambda (H2: ((\forall -(d1: C).(\forall (u0: T).(\forall (i: nat).((getl i c (CHead d1 (Bind Abbr) -u0)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 c u c2 t2) \to -(arity g c2 t2 a1)))))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (_: -(arity g (CHead c (Bind b) u) t a2)).(\lambda (H4: ((\forall (d1: C).(\forall -(u0: T).(\forall (i: nat).((getl i (CHead c (Bind b) u) (CHead d1 (Bind Abbr) -u0)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 (CHead c (Bind b) -u) t c2 t2) \to (arity g c2 t2 a2)))))))))).(\lambda (d1: C).(\lambda (u0: -T).(\lambda (i: nat).(\lambda (H5: (getl i c (CHead d1 (Bind Abbr) -u0))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H6: (fsubst0 i u0 c (THead -(Bind b) u t) c2 t2)).(let H_x \def (fsubst0_gen_base c c2 (THead (Bind b) u -t) t2 u0 i H6) in (let H7 \def H_x in (or3_ind (land (eq C c c2) (subst0 i u0 -(THead (Bind b) u t) t2)) (land (eq T (THead (Bind b) u t) t2) (csubst0 i u0 -c c2)) (land (subst0 i u0 (THead (Bind b) u t) t2) (csubst0 i u0 c c2)) -(arity g c2 t2 a2) (\lambda (H8: (land (eq C c c2) (subst0 i u0 (THead (Bind -b) u t) t2))).(land_ind (eq C c c2) (subst0 i u0 (THead (Bind b) u t) t2) -(arity g c2 t2 a2) (\lambda (H9: (eq C c c2)).(\lambda (H10: (subst0 i u0 -(THead (Bind b) u t) t2)).(eq_ind C c (\lambda (c0: C).(arity g c0 t2 a2)) -(or3_ind (ex2 T (\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t))) (\lambda -(u2: T).(subst0 i u0 u u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead (Bind b) -u t3))) (\lambda (t3: T).(subst0 (s (Bind b) i) u0 t t3))) (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) u2 t3)))) (\lambda -(u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: -T).(subst0 (s (Bind b) i) u0 t t3)))) (arity g c t2 a2) (\lambda (H11: (ex2 T -(\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t))) (\lambda (u2: T).(subst0 i -u0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t))) -(\lambda (u2: T).(subst0 i u0 u u2)) (arity g c t2 a2) (\lambda (x: -T).(\lambda (H12: (eq T t2 (THead (Bind b) x t))).(\lambda (H13: (subst0 i u0 -u x)).(eq_ind_r T (THead (Bind b) x t) (\lambda (t0: T).(arity g c t0 a2)) -(arity_bind g b H0 c x a1 (H2 d1 u0 i H5 c x (fsubst0_snd i u0 c u x H13)) t -a2 (H4 d1 u0 (S i) (getl_clear_bind b (CHead c (Bind b) u) c u (clear_bind b -c u) (CHead d1 (Bind Abbr) u0) i H5) (CHead c (Bind b) x) t (fsubst0_fst (S -i) u0 (CHead c (Bind b) u) t (CHead c (Bind b) x) (csubst0_snd_bind b i u0 u -x H13 c)))) t2 H12)))) H11)) (\lambda (H11: (ex2 T (\lambda (t3: T).(eq T t2 -(THead (Bind b) u t3))) (\lambda (t3: T).(subst0 (s (Bind b) i) u0 t -t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead (Bind b) u t3))) (\lambda -(t3: T).(subst0 (s (Bind b) i) u0 t t3)) (arity g c t2 a2) (\lambda (x: -T).(\lambda (H12: (eq T t2 (THead (Bind b) u x))).(\lambda (H13: (subst0 (s -(Bind b) i) u0 t x)).(eq_ind_r T (THead (Bind b) u x) (\lambda (t0: T).(arity -g c t0 a2)) (arity_bind g b H0 c u a1 H1 x a2 (H4 d1 u0 (S i) -(getl_clear_bind b (CHead c (Bind b) u) c u (clear_bind b c u) (CHead d1 -(Bind Abbr) u0) i H5) (CHead c (Bind b) u) x (fsubst0_snd (S i) u0 (CHead c -(Bind b) u) t x H13))) t2 H12)))) H11)) (\lambda (H11: (ex3_2 T T (\lambda -(u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: -T).(subst0 (s (Bind b) i) u0 t t3))))).(ex3_2_ind T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Bind b) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: -T).(subst0 (s (Bind b) i) u0 t t3))) (arity g c t2 a2) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H12: (eq T t2 (THead (Bind b) x0 x1))).(\lambda -(H13: (subst0 i u0 u x0)).(\lambda (H14: (subst0 (s (Bind b) i) u0 t -x1)).(eq_ind_r T (THead (Bind b) x0 x1) (\lambda (t0: T).(arity g c t0 a2)) -(arity_bind g b H0 c x0 a1 (H2 d1 u0 i H5 c x0 (fsubst0_snd i u0 c u x0 H13)) -x1 a2 (H4 d1 u0 (S i) (getl_clear_bind b (CHead c (Bind b) u) c u (clear_bind -b c u) (CHead d1 (Bind Abbr) u0) i H5) (CHead c (Bind b) x0) x1 (fsubst0_both -(S i) u0 (CHead c (Bind b) u) t x1 H14 (CHead c (Bind b) x0) -(csubst0_snd_bind b i u0 u x0 H13 c)))) t2 H12)))))) H11)) (subst0_gen_head -(Bind b) u0 u t t2 i H10)) c2 H9))) H8)) (\lambda (H8: (land (eq T (THead -(Bind b) u t) t2) (csubst0 i u0 c c2))).(land_ind (eq T (THead (Bind b) u t) -t2) (csubst0 i u0 c c2) (arity g c2 t2 a2) (\lambda (H9: (eq T (THead (Bind -b) u t) t2)).(\lambda (H10: (csubst0 i u0 c c2)).(eq_ind T (THead (Bind b) u -t) (\lambda (t0: T).(arity g c2 t0 a2)) (arity_bind g b H0 c2 u a1 (H2 d1 u0 -i H5 c2 u (fsubst0_fst i u0 c u c2 H10)) t a2 (H4 d1 u0 (S i) -(getl_clear_bind b (CHead c (Bind b) u) c u (clear_bind b c u) (CHead d1 -(Bind Abbr) u0) i H5) (CHead c2 (Bind b) u) t (fsubst0_fst (S i) u0 (CHead c -(Bind b) u) t (CHead c2 (Bind b) u) (csubst0_fst_bind b i c c2 u0 H10 u)))) -t2 H9))) H8)) (\lambda (H8: (land (subst0 i u0 (THead (Bind b) u t) t2) -(csubst0 i u0 c c2))).(land_ind (subst0 i u0 (THead (Bind b) u t) t2) -(csubst0 i u0 c c2) (arity g c2 t2 a2) (\lambda (H9: (subst0 i u0 (THead -(Bind b) u t) t2)).(\lambda (H10: (csubst0 i u0 c c2)).(or3_ind (ex2 T -(\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t))) (\lambda (u2: T).(subst0 i -u0 u u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead (Bind b) u t3))) (\lambda -(t3: T).(subst0 (s (Bind b) i) u0 t t3))) (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Bind b) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: -T).(subst0 (s (Bind b) i) u0 t t3)))) (arity g c2 t2 a2) (\lambda (H11: (ex2 -T (\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t))) (\lambda (u2: T).(subst0 -i u0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t))) -(\lambda (u2: T).(subst0 i u0 u u2)) (arity g c2 t2 a2) (\lambda (x: -T).(\lambda (H12: (eq T t2 (THead (Bind b) x t))).(\lambda (H13: (subst0 i u0 -u x)).(eq_ind_r T (THead (Bind b) x t) (\lambda (t0: T).(arity g c2 t0 a2)) -(arity_bind g b H0 c2 x a1 (H2 d1 u0 i H5 c2 x (fsubst0_both i u0 c u x H13 -c2 H10)) t a2 (H4 d1 u0 (S i) (getl_clear_bind b (CHead c (Bind b) u) c u -(clear_bind b c u) (CHead d1 (Bind Abbr) u0) i H5) (CHead c2 (Bind b) x) t -(fsubst0_fst (S i) u0 (CHead c (Bind b) u) t (CHead c2 (Bind b) x) -(csubst0_both_bind b i u0 u x H13 c c2 H10)))) t2 H12)))) H11)) (\lambda -(H11: (ex2 T (\lambda (t3: T).(eq T t2 (THead (Bind b) u t3))) (\lambda (t3: -T).(subst0 (s (Bind b) i) u0 t t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 -(THead (Bind b) u t3))) (\lambda (t3: T).(subst0 (s (Bind b) i) u0 t t3)) -(arity g c2 t2 a2) (\lambda (x: T).(\lambda (H12: (eq T t2 (THead (Bind b) u -x))).(\lambda (H13: (subst0 (s (Bind b) i) u0 t x)).(eq_ind_r T (THead (Bind -b) u x) (\lambda (t0: T).(arity g c2 t0 a2)) (arity_bind g b H0 c2 u a1 (H2 -d1 u0 i H5 c2 u (fsubst0_fst i u0 c u c2 H10)) x a2 (H4 d1 u0 (S i) -(getl_clear_bind b (CHead c (Bind b) u) c u (clear_bind b c u) (CHead d1 -(Bind Abbr) u0) i H5) (CHead c2 (Bind b) u) x (fsubst0_both (S i) u0 (CHead c -(Bind b) u) t x H13 (CHead c2 (Bind b) u) (csubst0_fst_bind b i c c2 u0 H10 -u)))) t2 H12)))) H11)) (\lambda (H11: (ex3_2 T T (\lambda (u2: T).(\lambda -(t3: T).(eq T t2 (THead (Bind b) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind b) -i) u0 t t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 -(THead (Bind b) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u -u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind b) i) u0 t t3))) -(arity g c2 t2 a2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H12: (eq T t2 -(THead (Bind b) x0 x1))).(\lambda (H13: (subst0 i u0 u x0)).(\lambda (H14: -(subst0 (s (Bind b) i) u0 t x1)).(eq_ind_r T (THead (Bind b) x0 x1) (\lambda -(t0: T).(arity g c2 t0 a2)) (arity_bind g b H0 c2 x0 a1 (H2 d1 u0 i H5 c2 x0 -(fsubst0_both i u0 c u x0 H13 c2 H10)) x1 a2 (H4 d1 u0 (S i) (getl_clear_bind -b (CHead c (Bind b) u) c u (clear_bind b c u) (CHead d1 (Bind Abbr) u0) i H5) -(CHead c2 (Bind b) x0) x1 (fsubst0_both (S i) u0 (CHead c (Bind b) u) t x1 -H14 (CHead c2 (Bind b) x0) (csubst0_both_bind b i u0 u x0 H13 c c2 H10)))) t2 -H12)))))) H11)) (subst0_gen_head (Bind b) u0 u t t2 i H9)))) H8)) -H7))))))))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a1: -A).(\lambda (H0: (arity g c u (asucc g a1))).(\lambda (H1: ((\forall (d1: -C).(\forall (u0: T).(\forall (i: nat).((getl i c (CHead d1 (Bind Abbr) u0)) -\to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 c u c2 t2) \to (arity g -c2 t2 (asucc g a1))))))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (_: -(arity g (CHead c (Bind Abst) u) t a2)).(\lambda (H3: ((\forall (d1: -C).(\forall (u0: T).(\forall (i: nat).((getl i (CHead c (Bind Abst) u) (CHead -d1 (Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 -(CHead c (Bind Abst) u) t c2 t2) \to (arity g c2 t2 a2)))))))))).(\lambda -(d1: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H4: (getl i c (CHead d1 -(Bind Abbr) u0))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H5: (fsubst0 i -u0 c (THead (Bind Abst) u t) c2 t2)).(let H_x \def (fsubst0_gen_base c c2 -(THead (Bind Abst) u t) t2 u0 i H5) in (let H6 \def H_x in (or3_ind (land (eq -C c c2) (subst0 i u0 (THead (Bind Abst) u t) t2)) (land (eq T (THead (Bind -Abst) u t) t2) (csubst0 i u0 c c2)) (land (subst0 i u0 (THead (Bind Abst) u -t) t2) (csubst0 i u0 c c2)) (arity g c2 t2 (AHead a1 a2)) (\lambda (H7: (land -(eq C c c2) (subst0 i u0 (THead (Bind Abst) u t) t2))).(land_ind (eq C c c2) -(subst0 i u0 (THead (Bind Abst) u t) t2) (arity g c2 t2 (AHead a1 a2)) -(\lambda (H8: (eq C c c2)).(\lambda (H9: (subst0 i u0 (THead (Bind Abst) u t) -t2)).(eq_ind C c (\lambda (c0: C).(arity g c0 t2 (AHead a1 a2))) (or3_ind -(ex2 T (\lambda (u2: T).(eq T t2 (THead (Bind Abst) u2 t))) (\lambda (u2: -T).(subst0 i u0 u u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead (Bind Abst) u -t3))) (\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3))) (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: -T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3)))) (arity g c t2 -(AHead a1 a2)) (\lambda (H10: (ex2 T (\lambda (u2: T).(eq T t2 (THead (Bind -Abst) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T (\lambda (u2: -T).(eq T t2 (THead (Bind Abst) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)) -(arity g c t2 (AHead a1 a2)) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead -(Bind Abst) x t))).(\lambda (H12: (subst0 i u0 u x)).(eq_ind_r T (THead (Bind -Abst) x t) (\lambda (t0: T).(arity g c t0 (AHead a1 a2))) (arity_head g c x -a1 (H1 d1 u0 i H4 c x (fsubst0_snd i u0 c u x H12)) t a2 (H3 d1 u0 (S i) -(getl_clear_bind Abst (CHead c (Bind Abst) u) c u (clear_bind Abst c u) -(CHead d1 (Bind Abbr) u0) i H4) (CHead c (Bind Abst) x) t (fsubst0_fst (S i) -u0 (CHead c (Bind Abst) u) t (CHead c (Bind Abst) x) (csubst0_snd_bind Abst i -u0 u x H12 c)))) t2 H11)))) H10)) (\lambda (H10: (ex2 T (\lambda (t3: T).(eq -T t2 (THead (Bind Abst) u t3))) (\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 -t t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead (Bind Abst) u t3))) -(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3)) (arity g c t2 (AHead a1 -a2)) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Bind Abst) u -x))).(\lambda (H12: (subst0 (s (Bind Abst) i) u0 t x)).(eq_ind_r T (THead -(Bind Abst) u x) (\lambda (t0: T).(arity g c t0 (AHead a1 a2))) (arity_head g -c u a1 H0 x a2 (H3 d1 u0 (S i) (getl_clear_bind Abst (CHead c (Bind Abst) u) -c u (clear_bind Abst c u) (CHead d1 (Bind Abbr) u0) i H4) (CHead c (Bind -Abst) u) x (fsubst0_snd (S i) u0 (CHead c (Bind Abst) u) t x H12))) t2 -H11)))) H10)) (\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq -T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i -u0 u u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t -t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead -(Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) -(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3))) (arity -g c t2 (AHead a1 a2)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq T -t2 (THead (Bind Abst) x0 x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda -(H13: (subst0 (s (Bind Abst) i) u0 t x1)).(eq_ind_r T (THead (Bind Abst) x0 -x1) (\lambda (t0: T).(arity g c t0 (AHead a1 a2))) (arity_head g c x0 a1 (H1 -d1 u0 i H4 c x0 (fsubst0_snd i u0 c u x0 H12)) x1 a2 (H3 d1 u0 (S i) -(getl_clear_bind Abst (CHead c (Bind Abst) u) c u (clear_bind Abst c u) -(CHead d1 (Bind Abbr) u0) i H4) (CHead c (Bind Abst) x0) x1 (fsubst0_both (S -i) u0 (CHead c (Bind Abst) u) t x1 H13 (CHead c (Bind Abst) x0) -(csubst0_snd_bind Abst i u0 u x0 H12 c)))) t2 H11)))))) H10)) -(subst0_gen_head (Bind Abst) u0 u t t2 i H9)) c2 H8))) H7)) (\lambda (H7: -(land (eq T (THead (Bind Abst) u t) t2) (csubst0 i u0 c c2))).(land_ind (eq T -(THead (Bind Abst) u t) t2) (csubst0 i u0 c c2) (arity g c2 t2 (AHead a1 a2)) -(\lambda (H8: (eq T (THead (Bind Abst) u t) t2)).(\lambda (H9: (csubst0 i u0 -c c2)).(eq_ind T (THead (Bind Abst) u t) (\lambda (t0: T).(arity g c2 t0 -(AHead a1 a2))) (arity_head g c2 u a1 (H1 d1 u0 i H4 c2 u (fsubst0_fst i u0 c -u c2 H9)) t a2 (H3 d1 u0 (S i) (getl_clear_bind Abst (CHead c (Bind Abst) u) -c u (clear_bind Abst c u) (CHead d1 (Bind Abbr) u0) i H4) (CHead c2 (Bind -Abst) u) t (fsubst0_fst (S i) u0 (CHead c (Bind Abst) u) t (CHead c2 (Bind -Abst) u) (csubst0_fst_bind Abst i c c2 u0 H9 u)))) t2 H8))) H7)) (\lambda -(H7: (land (subst0 i u0 (THead (Bind Abst) u t) t2) (csubst0 i u0 c -c2))).(land_ind (subst0 i u0 (THead (Bind Abst) u t) t2) (csubst0 i u0 c c2) -(arity g c2 t2 (AHead a1 a2)) (\lambda (H8: (subst0 i u0 (THead (Bind Abst) u -t) t2)).(\lambda (H9: (csubst0 i u0 c c2)).(or3_ind (ex2 T (\lambda (u2: -T).(eq T t2 (THead (Bind Abst) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2))) -(ex2 T (\lambda (t3: T).(eq T t2 (THead (Bind Abst) u t3))) (\lambda (t3: -T).(subst0 (s (Bind Abst) i) u0 t t3))) (ex3_2 T T (\lambda (u2: T).(\lambda -(t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind -Abst) i) u0 t t3)))) (arity g c2 t2 (AHead a1 a2)) (\lambda (H10: (ex2 T -(\lambda (u2: T).(eq T t2 (THead (Bind Abst) u2 t))) (\lambda (u2: T).(subst0 -i u0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T t2 (THead (Bind Abst) u2 t))) -(\lambda (u2: T).(subst0 i u0 u u2)) (arity g c2 t2 (AHead a1 a2)) (\lambda -(x: T).(\lambda (H11: (eq T t2 (THead (Bind Abst) x t))).(\lambda (H12: -(subst0 i u0 u x)).(eq_ind_r T (THead (Bind Abst) x t) (\lambda (t0: -T).(arity g c2 t0 (AHead a1 a2))) (arity_head g c2 x a1 (H1 d1 u0 i H4 c2 x -(fsubst0_both i u0 c u x H12 c2 H9)) t a2 (H3 d1 u0 (S i) (getl_clear_bind -Abst (CHead c (Bind Abst) u) c u (clear_bind Abst c u) (CHead d1 (Bind Abbr) -u0) i H4) (CHead c2 (Bind Abst) x) t (fsubst0_fst (S i) u0 (CHead c (Bind -Abst) u) t (CHead c2 (Bind Abst) x) (csubst0_both_bind Abst i u0 u x H12 c c2 -H9)))) t2 H11)))) H10)) (\lambda (H10: (ex2 T (\lambda (t3: T).(eq T t2 -(THead (Bind Abst) u t3))) (\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t -t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead (Bind Abst) u t3))) -(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3)) (arity g c2 t2 (AHead a1 -a2)) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Bind Abst) u -x))).(\lambda (H12: (subst0 (s (Bind Abst) i) u0 t x)).(eq_ind_r T (THead -(Bind Abst) u x) (\lambda (t0: T).(arity g c2 t0 (AHead a1 a2))) (arity_head -g c2 u a1 (H1 d1 u0 i H4 c2 u (fsubst0_fst i u0 c u c2 H9)) x a2 (H3 d1 u0 (S -i) (getl_clear_bind Abst (CHead c (Bind Abst) u) c u (clear_bind Abst c u) -(CHead d1 (Bind Abbr) u0) i H4) (CHead c2 (Bind Abst) u) x (fsubst0_both (S -i) u0 (CHead c (Bind Abst) u) t x H12 (CHead c2 (Bind Abst) u) -(csubst0_fst_bind Abst i c c2 u0 H9 u)))) t2 H11)))) H10)) (\lambda (H10: -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: -T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3))))).(ex3_2_ind T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: -T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3))) (arity g c2 t2 -(AHead a1 a2)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq T t2 -(THead (Bind Abst) x0 x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda (H13: -(subst0 (s (Bind Abst) i) u0 t x1)).(eq_ind_r T (THead (Bind Abst) x0 x1) -(\lambda (t0: T).(arity g c2 t0 (AHead a1 a2))) (arity_head g c2 x0 a1 (H1 d1 -u0 i H4 c2 x0 (fsubst0_both i u0 c u x0 H12 c2 H9)) x1 a2 (H3 d1 u0 (S i) -(getl_clear_bind Abst (CHead c (Bind Abst) u) c u (clear_bind Abst c u) -(CHead d1 (Bind Abbr) u0) i H4) (CHead c2 (Bind Abst) x0) x1 (fsubst0_both (S -i) u0 (CHead c (Bind Abst) u) t x1 H13 (CHead c2 (Bind Abst) x0) -(csubst0_both_bind Abst i u0 u x0 H12 c c2 H9)))) t2 H11)))))) H10)) -(subst0_gen_head (Bind Abst) u0 u t t2 i H8)))) H7)) H6))))))))))))))))))) -(\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H0: (arity g c u -a1)).(\lambda (H1: ((\forall (d1: C).(\forall (u0: T).(\forall (i: -nat).((getl i c (CHead d1 (Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2: -T).((fsubst0 i u0 c u c2 t2) \to (arity g c2 t2 a1)))))))))).(\lambda (t: -T).(\lambda (a2: A).(\lambda (H2: (arity g c t (AHead a1 a2))).(\lambda (H3: -((\forall (d1: C).(\forall (u0: T).(\forall (i: nat).((getl i c (CHead d1 -(Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 c t c2 -t2) \to (arity g c2 t2 (AHead a1 a2))))))))))).(\lambda (d1: C).(\lambda (u0: -T).(\lambda (i: nat).(\lambda (H4: (getl i c (CHead d1 (Bind Abbr) -u0))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H5: (fsubst0 i u0 c (THead -(Flat Appl) u t) c2 t2)).(let H_x \def (fsubst0_gen_base c c2 (THead (Flat -Appl) u t) t2 u0 i H5) in (let H6 \def H_x in (or3_ind (land (eq C c c2) -(subst0 i u0 (THead (Flat Appl) u t) t2)) (land (eq T (THead (Flat Appl) u t) -t2) (csubst0 i u0 c c2)) (land (subst0 i u0 (THead (Flat Appl) u t) t2) -(csubst0 i u0 c c2)) (arity g c2 t2 a2) (\lambda (H7: (land (eq C c c2) -(subst0 i u0 (THead (Flat Appl) u t) t2))).(land_ind (eq C c c2) (subst0 i u0 -(THead (Flat Appl) u t) t2) (arity g c2 t2 a2) (\lambda (H8: (eq C c -c2)).(\lambda (H9: (subst0 i u0 (THead (Flat Appl) u t) t2)).(eq_ind C c -(\lambda (c0: C).(arity g c0 t2 a2)) (or3_ind (ex2 T (\lambda (u2: T).(eq T -t2 (THead (Flat Appl) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2))) (ex2 T -(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u t3))) (\lambda (t3: T).(subst0 -(s (Flat Appl) i) u0 t t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq -T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i -u0 u u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 t -t3)))) (arity g c t2 a2) (\lambda (H10: (ex2 T (\lambda (u2: T).(eq T t2 -(THead (Flat Appl) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T -(\lambda (u2: T).(eq T t2 (THead (Flat Appl) u2 t))) (\lambda (u2: T).(subst0 -i u0 u u2)) (arity g c t2 a2) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead -(Flat Appl) x t))).(\lambda (H12: (subst0 i u0 u x)).(eq_ind_r T (THead (Flat -Appl) x t) (\lambda (t0: T).(arity g c t0 a2)) (arity_appl g c x a1 (H1 d1 u0 -i H4 c x (fsubst0_snd i u0 c u x H12)) t a2 H2) t2 H11)))) H10)) (\lambda -(H10: (ex2 T (\lambda (t3: T).(eq T t2 (THead (Flat Appl) u t3))) (\lambda -(t3: T).(subst0 (s (Flat Appl) i) u0 t t3)))).(ex2_ind T (\lambda (t3: T).(eq -T t2 (THead (Flat Appl) u t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 -t t3)) (arity g c t2 a2) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat -Appl) u x))).(\lambda (H12: (subst0 (s (Flat Appl) i) u0 t x)).(eq_ind_r T -(THead (Flat Appl) u x) (\lambda (t0: T).(arity g c t0 a2)) (arity_appl g c u -a1 H0 x a2 (H3 d1 u0 i H4 c x (fsubst0_snd i u0 c t x H12))) t2 H11)))) H10)) -(\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead -(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) -(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 t -t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead -(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) -(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 t t3))) (arity -g c t2 a2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq T t2 (THead -(Flat Appl) x0 x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda (H13: -(subst0 (s (Flat Appl) i) u0 t x1)).(eq_ind_r T (THead (Flat Appl) x0 x1) -(\lambda (t0: T).(arity g c t0 a2)) (arity_appl g c x0 a1 (H1 d1 u0 i H4 c x0 -(fsubst0_snd i u0 c u x0 H12)) x1 a2 (H3 d1 u0 i H4 c x1 (fsubst0_snd i u0 c -t x1 H13))) t2 H11)))))) H10)) (subst0_gen_head (Flat Appl) u0 u t t2 i H9)) -c2 H8))) H7)) (\lambda (H7: (land (eq T (THead (Flat Appl) u t) t2) (csubst0 -i u0 c c2))).(land_ind (eq T (THead (Flat Appl) u t) t2) (csubst0 i u0 c c2) -(arity g c2 t2 a2) (\lambda (H8: (eq T (THead (Flat Appl) u t) t2)).(\lambda -(H9: (csubst0 i u0 c c2)).(eq_ind T (THead (Flat Appl) u t) (\lambda (t0: -T).(arity g c2 t0 a2)) (arity_appl g c2 u a1 (H1 d1 u0 i H4 c2 u (fsubst0_fst -i u0 c u c2 H9)) t a2 (H3 d1 u0 i H4 c2 t (fsubst0_fst i u0 c t c2 H9))) t2 -H8))) H7)) (\lambda (H7: (land (subst0 i u0 (THead (Flat Appl) u t) t2) -(csubst0 i u0 c c2))).(land_ind (subst0 i u0 (THead (Flat Appl) u t) t2) -(csubst0 i u0 c c2) (arity g c2 t2 a2) (\lambda (H8: (subst0 i u0 (THead -(Flat Appl) u t) t2)).(\lambda (H9: (csubst0 i u0 c c2)).(or3_ind (ex2 T -(\lambda (u2: T).(eq T t2 (THead (Flat Appl) u2 t))) (\lambda (u2: T).(subst0 -i u0 u u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead (Flat Appl) u t3))) -(\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 t t3))) (ex3_2 T T (\lambda -(u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: -T).(subst0 (s (Flat Appl) i) u0 t t3)))) (arity g c2 t2 a2) (\lambda (H10: -(ex2 T (\lambda (u2: T).(eq T t2 (THead (Flat Appl) u2 t))) (\lambda (u2: -T).(subst0 i u0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T t2 (THead (Flat -Appl) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)) (arity g c2 t2 a2) -(\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat Appl) x t))).(\lambda -(H12: (subst0 i u0 u x)).(eq_ind_r T (THead (Flat Appl) x t) (\lambda (t0: -T).(arity g c2 t0 a2)) (arity_appl g c2 x a1 (H1 d1 u0 i H4 c2 x -(fsubst0_both i u0 c u x H12 c2 H9)) t a2 (H3 d1 u0 i H4 c2 t (fsubst0_fst i -u0 c t c2 H9))) t2 H11)))) H10)) (\lambda (H10: (ex2 T (\lambda (t3: T).(eq T -t2 (THead (Flat Appl) u t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 t -t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead (Flat Appl) u t3))) -(\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 t t3)) (arity g c2 t2 a2) -(\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat Appl) u x))).(\lambda -(H12: (subst0 (s (Flat Appl) i) u0 t x)).(eq_ind_r T (THead (Flat Appl) u x) -(\lambda (t0: T).(arity g c2 t0 a2)) (arity_appl g c2 u a1 (H1 d1 u0 i H4 c2 -u (fsubst0_fst i u0 c u c2 H9)) x a2 (H3 d1 u0 i H4 c2 x (fsubst0_both i u0 c -t x H12 c2 H9))) t2 H11)))) H10)) (\lambda (H10: (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: -T).(subst0 (s (Flat Appl) i) u0 t t3))))).(ex3_2_ind T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: -T).(subst0 (s (Flat Appl) i) u0 t t3))) (arity g c2 t2 a2) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H11: (eq T t2 (THead (Flat Appl) x0 -x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda (H13: (subst0 (s (Flat -Appl) i) u0 t x1)).(eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda (t0: -T).(arity g c2 t0 a2)) (arity_appl g c2 x0 a1 (H1 d1 u0 i H4 c2 x0 -(fsubst0_both i u0 c u x0 H12 c2 H9)) x1 a2 (H3 d1 u0 i H4 c2 x1 -(fsubst0_both i u0 c t x1 H13 c2 H9))) t2 H11)))))) H10)) (subst0_gen_head -(Flat Appl) u0 u t t2 i H8)))) H7)) H6))))))))))))))))))) (\lambda (c: -C).(\lambda (u: T).(\lambda (a0: A).(\lambda (H0: (arity g c u (asucc g -a0))).(\lambda (H1: ((\forall (d1: C).(\forall (u0: T).(\forall (i: -nat).((getl i c (CHead d1 (Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2: -T).((fsubst0 i u0 c u c2 t2) \to (arity g c2 t2 (asucc g -a0))))))))))).(\lambda (t: T).(\lambda (H2: (arity g c t a0)).(\lambda (H3: -((\forall (d1: C).(\forall (u0: T).(\forall (i: nat).((getl i c (CHead d1 -(Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 c t c2 -t2) \to (arity g c2 t2 a0)))))))))).(\lambda (d1: C).(\lambda (u0: -T).(\lambda (i: nat).(\lambda (H4: (getl i c (CHead d1 (Bind Abbr) -u0))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H5: (fsubst0 i u0 c (THead -(Flat Cast) u t) c2 t2)).(let H_x \def (fsubst0_gen_base c c2 (THead (Flat -Cast) u t) t2 u0 i H5) in (let H6 \def H_x in (or3_ind (land (eq C c c2) -(subst0 i u0 (THead (Flat Cast) u t) t2)) (land (eq T (THead (Flat Cast) u t) -t2) (csubst0 i u0 c c2)) (land (subst0 i u0 (THead (Flat Cast) u t) t2) -(csubst0 i u0 c c2)) (arity g c2 t2 a0) (\lambda (H7: (land (eq C c c2) -(subst0 i u0 (THead (Flat Cast) u t) t2))).(land_ind (eq C c c2) (subst0 i u0 -(THead (Flat Cast) u t) t2) (arity g c2 t2 a0) (\lambda (H8: (eq C c -c2)).(\lambda (H9: (subst0 i u0 (THead (Flat Cast) u t) t2)).(eq_ind C c -(\lambda (c0: C).(arity g c0 t2 a0)) (or3_ind (ex2 T (\lambda (u2: T).(eq T -t2 (THead (Flat Cast) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2))) (ex2 T -(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u t3))) (\lambda (t3: T).(subst0 -(s (Flat Cast) i) u0 t t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq -T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i -u0 u u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t -t3)))) (arity g c t2 a0) (\lambda (H10: (ex2 T (\lambda (u2: T).(eq T t2 -(THead (Flat Cast) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T -(\lambda (u2: T).(eq T t2 (THead (Flat Cast) u2 t))) (\lambda (u2: T).(subst0 -i u0 u u2)) (arity g c t2 a0) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead -(Flat Cast) x t))).(\lambda (H12: (subst0 i u0 u x)).(eq_ind_r T (THead (Flat -Cast) x t) (\lambda (t0: T).(arity g c t0 a0)) (arity_cast g c x a0 (H1 d1 u0 -i H4 c x (fsubst0_snd i u0 c u x H12)) t H2) t2 H11)))) H10)) (\lambda (H10: -(ex2 T (\lambda (t3: T).(eq T t2 (THead (Flat Cast) u t3))) (\lambda (t3: -T).(subst0 (s (Flat Cast) i) u0 t t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 -(THead (Flat Cast) u t3))) (\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t -t3)) (arity g c t2 a0) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat -Cast) u x))).(\lambda (H12: (subst0 (s (Flat Cast) i) u0 t x)).(eq_ind_r T -(THead (Flat Cast) u x) (\lambda (t0: T).(arity g c t0 a0)) (arity_cast g c u -a0 H0 x (H3 d1 u0 i H4 c x (fsubst0_snd i u0 c t x H12))) t2 H11)))) H10)) -(\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead -(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) -(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t -t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead -(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) -(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t t3))) (arity -g c t2 a0) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq T t2 (THead -(Flat Cast) x0 x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda (H13: -(subst0 (s (Flat Cast) i) u0 t x1)).(eq_ind_r T (THead (Flat Cast) x0 x1) -(\lambda (t0: T).(arity g c t0 a0)) (arity_cast g c x0 a0 (H1 d1 u0 i H4 c x0 -(fsubst0_snd i u0 c u x0 H12)) x1 (H3 d1 u0 i H4 c x1 (fsubst0_snd i u0 c t -x1 H13))) t2 H11)))))) H10)) (subst0_gen_head (Flat Cast) u0 u t t2 i H9)) c2 -H8))) H7)) (\lambda (H7: (land (eq T (THead (Flat Cast) u t) t2) (csubst0 i -u0 c c2))).(land_ind (eq T (THead (Flat Cast) u t) t2) (csubst0 i u0 c c2) -(arity g c2 t2 a0) (\lambda (H8: (eq T (THead (Flat Cast) u t) t2)).(\lambda -(H9: (csubst0 i u0 c c2)).(eq_ind T (THead (Flat Cast) u t) (\lambda (t0: -T).(arity g c2 t0 a0)) (arity_cast g c2 u a0 (H1 d1 u0 i H4 c2 u (fsubst0_fst -i u0 c u c2 H9)) t (H3 d1 u0 i H4 c2 t (fsubst0_fst i u0 c t c2 H9))) t2 -H8))) H7)) (\lambda (H7: (land (subst0 i u0 (THead (Flat Cast) u t) t2) -(csubst0 i u0 c c2))).(land_ind (subst0 i u0 (THead (Flat Cast) u t) t2) -(csubst0 i u0 c c2) (arity g c2 t2 a0) (\lambda (H8: (subst0 i u0 (THead -(Flat Cast) u t) t2)).(\lambda (H9: (csubst0 i u0 c c2)).(or3_ind (ex2 T -(\lambda (u2: T).(eq T t2 (THead (Flat Cast) u2 t))) (\lambda (u2: T).(subst0 -i u0 u u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead (Flat Cast) u t3))) -(\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t t3))) (ex3_2 T T (\lambda -(u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: -T).(subst0 (s (Flat Cast) i) u0 t t3)))) (arity g c2 t2 a0) (\lambda (H10: -(ex2 T (\lambda (u2: T).(eq T t2 (THead (Flat Cast) u2 t))) (\lambda (u2: -T).(subst0 i u0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T t2 (THead (Flat -Cast) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)) (arity g c2 t2 a0) -(\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat Cast) x t))).(\lambda -(H12: (subst0 i u0 u x)).(eq_ind_r T (THead (Flat Cast) x t) (\lambda (t0: -T).(arity g c2 t0 a0)) (arity_cast g c2 x a0 (H1 d1 u0 i H4 c2 x -(fsubst0_both i u0 c u x H12 c2 H9)) t (H3 d1 u0 i H4 c2 t (fsubst0_fst i u0 -c t c2 H9))) t2 H11)))) H10)) (\lambda (H10: (ex2 T (\lambda (t3: T).(eq T t2 -(THead (Flat Cast) u t3))) (\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t -t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead (Flat Cast) u t3))) -(\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t t3)) (arity g c2 t2 a0) -(\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat Cast) u x))).(\lambda -(H12: (subst0 (s (Flat Cast) i) u0 t x)).(eq_ind_r T (THead (Flat Cast) u x) -(\lambda (t0: T).(arity g c2 t0 a0)) (arity_cast g c2 u a0 (H1 d1 u0 i H4 c2 -u (fsubst0_fst i u0 c u c2 H9)) x (H3 d1 u0 i H4 c2 x (fsubst0_both i u0 c t -x H12 c2 H9))) t2 H11)))) H10)) (\lambda (H10: (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: -T).(subst0 (s (Flat Cast) i) u0 t t3))))).(ex3_2_ind T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: -T).(subst0 (s (Flat Cast) i) u0 t t3))) (arity g c2 t2 a0) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H11: (eq T t2 (THead (Flat Cast) x0 -x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda (H13: (subst0 (s (Flat -Cast) i) u0 t x1)).(eq_ind_r T (THead (Flat Cast) x0 x1) (\lambda (t0: -T).(arity g c2 t0 a0)) (arity_cast g c2 x0 a0 (H1 d1 u0 i H4 c2 x0 -(fsubst0_both i u0 c u x0 H12 c2 H9)) x1 (H3 d1 u0 i H4 c2 x1 (fsubst0_both i -u0 c t x1 H13 c2 H9))) t2 H11)))))) H10)) (subst0_gen_head (Flat Cast) u0 u t -t2 i H8)))) H7)) H6)))))))))))))))))) (\lambda (c: C).(\lambda (t: -T).(\lambda (a1: A).(\lambda (_: (arity g c t a1)).(\lambda (H1: ((\forall -(d1: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d1 (Bind Abbr) -u)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u c t c2 t2) \to (arity -g c2 t2 a1)))))))))).(\lambda (a2: A).(\lambda (H2: (leq g a1 a2)).(\lambda -(d1: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H3: (getl i c (CHead d1 -(Bind Abbr) u))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H4: (fsubst0 i u -c t c2 t2)).(let H_x \def (fsubst0_gen_base c c2 t t2 u i H4) in (let H5 \def -H_x in (or3_ind (land (eq C c c2) (subst0 i u t t2)) (land (eq T t t2) -(csubst0 i u c c2)) (land (subst0 i u t t2) (csubst0 i u c c2)) (arity g c2 -t2 a2) (\lambda (H6: (land (eq C c c2) (subst0 i u t t2))).(land_ind (eq C c -c2) (subst0 i u t t2) (arity g c2 t2 a2) (\lambda (H7: (eq C c c2)).(\lambda -(H8: (subst0 i u t t2)).(eq_ind C c (\lambda (c0: C).(arity g c0 t2 a2)) -(arity_repl g c t2 a1 (H1 d1 u i H3 c t2 (fsubst0_snd i u c t t2 H8)) a2 H2) -c2 H7))) H6)) (\lambda (H6: (land (eq T t t2) (csubst0 i u c c2))).(land_ind -(eq T t t2) (csubst0 i u c c2) (arity g c2 t2 a2) (\lambda (H7: (eq T t -t2)).(\lambda (H8: (csubst0 i u c c2)).(eq_ind T t (\lambda (t0: T).(arity g -c2 t0 a2)) (arity_repl g c2 t a1 (H1 d1 u i H3 c2 t (fsubst0_fst i u c t c2 -H8)) a2 H2) t2 H7))) H6)) (\lambda (H6: (land (subst0 i u t t2) (csubst0 i u -c c2))).(land_ind (subst0 i u t t2) (csubst0 i u c c2) (arity g c2 t2 a2) -(\lambda (H7: (subst0 i u t t2)).(\lambda (H8: (csubst0 i u c -c2)).(arity_repl g c2 t2 a1 (H1 d1 u i H3 c2 t2 (fsubst0_both i u c t t2 H7 -c2 H8)) a2 H2))) H6)) H5))))))))))))))))) c1 t1 a H))))). -(* COMMENTS -Initial nodes: 20387 -END *) - -theorem arity_subst0: - \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (a: A).((arity g c -t1 a) \to (\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead -d (Bind Abbr) u)) \to (\forall (t2: T).((subst0 i u t1 t2) \to (arity g c t2 -a))))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (a: A).(\lambda (H: -(arity g c t1 a)).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (t2: T).(\lambda (H1: -(subst0 i u t1 t2)).(arity_fsubst0 g c t1 a H d u i H0 c t2 (fsubst0_snd i u -c t1 t2 H1)))))))))))). -(* COMMENTS -Initial nodes: 89 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/asucc/defs.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/asucc/defs.ma deleted file mode 100644 index 2f1af0279..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/asucc/defs.ma +++ /dev/null @@ -1,28 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/A/defs.ma". - -include "Basic-1/G/defs.ma". - -definition asucc: - G \to (A \to A) -\def - let rec asucc (g: G) (l: A) on l: A \def (match l with [(ASort n0 n) -\Rightarrow (match n0 with [O \Rightarrow (ASort O (next g n)) | (S h) -\Rightarrow (ASort h n)]) | (AHead a1 a2) \Rightarrow (AHead a1 (asucc g -a2))]) in asucc. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/asucc/fwd.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/asucc/fwd.ma deleted file mode 100644 index 61fcb5799..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/asucc/fwd.ma +++ /dev/null @@ -1,99 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/asucc/defs.ma". - -theorem asucc_gen_sort: - \forall (g: G).(\forall (h: nat).(\forall (n: nat).(\forall (a: A).((eq A -(ASort h n) (asucc g a)) \to (ex_2 nat nat (\lambda (h0: nat).(\lambda (n0: -nat).(eq A a (ASort h0 n0))))))))) -\def - \lambda (g: G).(\lambda (h: nat).(\lambda (n: nat).(\lambda (a: A).(A_ind -(\lambda (a0: A).((eq A (ASort h n) (asucc g a0)) \to (ex_2 nat nat (\lambda -(h0: nat).(\lambda (n0: nat).(eq A a0 (ASort h0 n0))))))) (\lambda (n0: -nat).(\lambda (n1: nat).(\lambda (H: (eq A (ASort h n) (asucc g (ASort n0 -n1)))).(let H0 \def (f_equal A A (\lambda (e: A).e) (ASort h n) (match n0 -with [O \Rightarrow (ASort O (next g n1)) | (S h0) \Rightarrow (ASort h0 -n1)]) H) in (ex_2_intro nat nat (\lambda (h0: nat).(\lambda (n2: nat).(eq A -(ASort n0 n1) (ASort h0 n2)))) n0 n1 (refl_equal A (ASort n0 n1))))))) -(\lambda (a0: A).(\lambda (_: (((eq A (ASort h n) (asucc g a0)) \to (ex_2 nat -nat (\lambda (h0: nat).(\lambda (n0: nat).(eq A a0 (ASort h0 -n0)))))))).(\lambda (a1: A).(\lambda (_: (((eq A (ASort h n) (asucc g a1)) -\to (ex_2 nat nat (\lambda (h0: nat).(\lambda (n0: nat).(eq A a1 (ASort h0 -n0)))))))).(\lambda (H1: (eq A (ASort h n) (asucc g (AHead a0 a1)))).(let H2 -\def (eq_ind A (ASort h n) (\lambda (ee: A).(match ee in A return (\lambda -(_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow -False])) I (asucc g (AHead a0 a1)) H1) in (False_ind (ex_2 nat nat (\lambda -(h0: nat).(\lambda (n0: nat).(eq A (AHead a0 a1) (ASort h0 n0))))) H2))))))) -a)))). -(* COMMENTS -Initial nodes: 317 -END *) - -theorem asucc_gen_head: - \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a: A).((eq A -(AHead a1 a2) (asucc g a)) \to (ex2 A (\lambda (a0: A).(eq A a (AHead a1 -a0))) (\lambda (a0: A).(eq A a2 (asucc g a0)))))))) -\def - \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a: A).(A_ind -(\lambda (a0: A).((eq A (AHead a1 a2) (asucc g a0)) \to (ex2 A (\lambda (a3: -A).(eq A a0 (AHead a1 a3))) (\lambda (a3: A).(eq A a2 (asucc g a3)))))) -(\lambda (n: nat).(\lambda (n0: nat).(\lambda (H: (eq A (AHead a1 a2) (asucc -g (ASort n n0)))).(nat_ind (\lambda (n1: nat).((eq A (AHead a1 a2) (asucc g -(ASort n1 n0))) \to (ex2 A (\lambda (a0: A).(eq A (ASort n1 n0) (AHead a1 -a0))) (\lambda (a0: A).(eq A a2 (asucc g a0)))))) (\lambda (H0: (eq A (AHead -a1 a2) (asucc g (ASort O n0)))).(let H1 \def (eq_ind A (AHead a1 a2) (\lambda -(ee: A).(match ee in A return (\lambda (_: A).Prop) with [(ASort _ _) -\Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort O (next g n0)) -H0) in (False_ind (ex2 A (\lambda (a0: A).(eq A (ASort O n0) (AHead a1 a0))) -(\lambda (a0: A).(eq A a2 (asucc g a0)))) H1))) (\lambda (n1: nat).(\lambda -(_: (((eq A (AHead a1 a2) (asucc g (ASort n1 n0))) \to (ex2 A (\lambda (a0: -A).(eq A (ASort n1 n0) (AHead a1 a0))) (\lambda (a0: A).(eq A a2 (asucc g -a0))))))).(\lambda (H0: (eq A (AHead a1 a2) (asucc g (ASort (S n1) -n0)))).(let H1 \def (eq_ind A (AHead a1 a2) (\lambda (ee: A).(match ee in A -return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ -_) \Rightarrow True])) I (ASort n1 n0) H0) in (False_ind (ex2 A (\lambda (a0: -A).(eq A (ASort (S n1) n0) (AHead a1 a0))) (\lambda (a0: A).(eq A a2 (asucc g -a0)))) H1))))) n H)))) (\lambda (a0: A).(\lambda (H: (((eq A (AHead a1 a2) -(asucc g a0)) \to (ex2 A (\lambda (a3: A).(eq A a0 (AHead a1 a3))) (\lambda -(a3: A).(eq A a2 (asucc g a3))))))).(\lambda (a3: A).(\lambda (H0: (((eq A -(AHead a1 a2) (asucc g a3)) \to (ex2 A (\lambda (a4: A).(eq A a3 (AHead a1 -a4))) (\lambda (a4: A).(eq A a2 (asucc g a4))))))).(\lambda (H1: (eq A (AHead -a1 a2) (asucc g (AHead a0 a3)))).(let H2 \def (f_equal A A (\lambda (e: -A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a1 | -(AHead a4 _) \Rightarrow a4])) (AHead a1 a2) (AHead a0 (asucc g a3)) H1) in -((let H3 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: -A).A) with [(ASort _ _) \Rightarrow a2 | (AHead _ a4) \Rightarrow a4])) -(AHead a1 a2) (AHead a0 (asucc g a3)) H1) in (\lambda (H4: (eq A a1 a0)).(let -H5 \def (eq_ind_r A a0 (\lambda (a4: A).((eq A (AHead a1 a2) (asucc g a4)) -\to (ex2 A (\lambda (a5: A).(eq A a4 (AHead a1 a5))) (\lambda (a5: A).(eq A -a2 (asucc g a5)))))) H a1 H4) in (eq_ind A a1 (\lambda (a4: A).(ex2 A -(\lambda (a5: A).(eq A (AHead a4 a3) (AHead a1 a5))) (\lambda (a5: A).(eq A -a2 (asucc g a5))))) (let H6 \def (eq_ind A a2 (\lambda (a4: A).((eq A (AHead -a1 a4) (asucc g a3)) \to (ex2 A (\lambda (a5: A).(eq A a3 (AHead a1 a5))) -(\lambda (a5: A).(eq A a4 (asucc g a5)))))) H0 (asucc g a3) H3) in (let H7 -\def (eq_ind A a2 (\lambda (a4: A).((eq A (AHead a1 a4) (asucc g a1)) \to -(ex2 A (\lambda (a5: A).(eq A a1 (AHead a1 a5))) (\lambda (a5: A).(eq A a4 -(asucc g a5)))))) H5 (asucc g a3) H3) in (eq_ind_r A (asucc g a3) (\lambda -(a4: A).(ex2 A (\lambda (a5: A).(eq A (AHead a1 a3) (AHead a1 a5))) (\lambda -(a5: A).(eq A a4 (asucc g a5))))) (ex_intro2 A (\lambda (a4: A).(eq A (AHead -a1 a3) (AHead a1 a4))) (\lambda (a4: A).(eq A (asucc g a3) (asucc g a4))) a3 -(refl_equal A (AHead a1 a3)) (refl_equal A (asucc g a3))) a2 H3))) a0 H4)))) -H2))))))) a)))). -(* COMMENTS -Initial nodes: 957 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/cimp/defs.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/cimp/defs.ma deleted file mode 100644 index b45b64b59..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/cimp/defs.ma +++ /dev/null @@ -1,25 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/getl/defs.ma". - -definition cimp: - C \to (C \to Prop) -\def - \lambda (c1: C).(\lambda (c2: C).(\forall (b: B).(\forall (d1: C).(\forall -(w: T).(\forall (h: nat).((getl h c1 (CHead d1 (Bind b) w)) \to (ex C -(\lambda (d2: C).(getl h c2 (CHead d2 (Bind b) w)))))))))). - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/cimp/props.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/cimp/props.ma deleted file mode 100644 index 0834a7afe..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/cimp/props.ma +++ /dev/null @@ -1,139 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/cimp/defs.ma". - -include "Basic-1/getl/getl.ma". - -theorem cimp_flat_sx: - \forall (f: F).(\forall (c: C).(\forall (v: T).(cimp (CHead c (Flat f) v) -c))) -\def - \lambda (f: F).(\lambda (c: C).(\lambda (v: T).(\lambda (b: B).(\lambda (d1: -C).(\lambda (w: T).(\lambda (h: nat).(\lambda (H: (getl h (CHead c (Flat f) -v) (CHead d1 (Bind b) w))).(nat_ind (\lambda (n: nat).((getl n (CHead c (Flat -f) v) (CHead d1 (Bind b) w)) \to (ex C (\lambda (d2: C).(getl n c (CHead d2 -(Bind b) w)))))) (\lambda (H0: (getl O (CHead c (Flat f) v) (CHead d1 (Bind -b) w))).(ex_intro C (\lambda (d2: C).(getl O c (CHead d2 (Bind b) w))) d1 -(getl_intro O c (CHead d1 (Bind b) w) c (drop_refl c) (clear_gen_flat f c -(CHead d1 (Bind b) w) v (getl_gen_O (CHead c (Flat f) v) (CHead d1 (Bind b) -w) H0))))) (\lambda (h0: nat).(\lambda (_: (((getl h0 (CHead c (Flat f) v) -(CHead d1 (Bind b) w)) \to (ex C (\lambda (d2: C).(getl h0 c (CHead d2 (Bind -b) w))))))).(\lambda (H0: (getl (S h0) (CHead c (Flat f) v) (CHead d1 (Bind -b) w))).(ex_intro C (\lambda (d2: C).(getl (S h0) c (CHead d2 (Bind b) w))) -d1 (getl_gen_S (Flat f) c (CHead d1 (Bind b) w) v h0 H0))))) h H)))))))). -(* COMMENTS -Initial nodes: 327 -END *) - -theorem cimp_flat_dx: - \forall (f: F).(\forall (c: C).(\forall (v: T).(cimp c (CHead c (Flat f) -v)))) -\def - \lambda (f: F).(\lambda (c: C).(\lambda (v: T).(\lambda (b: B).(\lambda (d1: -C).(\lambda (w: T).(\lambda (h: nat).(\lambda (H: (getl h c (CHead d1 (Bind -b) w))).(ex_intro C (\lambda (d2: C).(getl h (CHead c (Flat f) v) (CHead d2 -(Bind b) w))) d1 (getl_flat c (CHead d1 (Bind b) w) h H f v))))))))). -(* COMMENTS -Initial nodes: 83 -END *) - -theorem cimp_bind: - \forall (c1: C).(\forall (c2: C).((cimp c1 c2) \to (\forall (b: B).(\forall -(v: T).(cimp (CHead c1 (Bind b) v) (CHead c2 (Bind b) v)))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (H: ((\forall (b: B).(\forall (d1: -C).(\forall (w: T).(\forall (h: nat).((getl h c1 (CHead d1 (Bind b) w)) \to -(ex C (\lambda (d2: C).(getl h c2 (CHead d2 (Bind b) w))))))))))).(\lambda -(b: B).(\lambda (v: T).(\lambda (b0: B).(\lambda (d1: C).(\lambda (w: -T).(\lambda (h: nat).(\lambda (H0: (getl h (CHead c1 (Bind b) v) (CHead d1 -(Bind b0) w))).(nat_ind (\lambda (n: nat).((getl n (CHead c1 (Bind b) v) -(CHead d1 (Bind b0) w)) \to (ex C (\lambda (d2: C).(getl n (CHead c2 (Bind b) -v) (CHead d2 (Bind b0) w)))))) (\lambda (H1: (getl O (CHead c1 (Bind b) v) -(CHead d1 (Bind b0) w))).(let H2 \def (f_equal C C (\lambda (e: C).(match e -in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d1 | (CHead c _ _) -\Rightarrow c])) (CHead d1 (Bind b0) w) (CHead c1 (Bind b) v) (clear_gen_bind -b c1 (CHead d1 (Bind b0) w) v (getl_gen_O (CHead c1 (Bind b) v) (CHead d1 -(Bind b0) w) H1))) in ((let H3 \def (f_equal C B (\lambda (e: C).(match e in -C return (\lambda (_: C).B) with [(CSort _) \Rightarrow b0 | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b1) -\Rightarrow b1 | (Flat _) \Rightarrow b0])])) (CHead d1 (Bind b0) w) (CHead -c1 (Bind b) v) (clear_gen_bind b c1 (CHead d1 (Bind b0) w) v (getl_gen_O -(CHead c1 (Bind b) v) (CHead d1 (Bind b0) w) H1))) in ((let H4 \def (f_equal -C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) -\Rightarrow w | (CHead _ _ t) \Rightarrow t])) (CHead d1 (Bind b0) w) (CHead -c1 (Bind b) v) (clear_gen_bind b c1 (CHead d1 (Bind b0) w) v (getl_gen_O -(CHead c1 (Bind b) v) (CHead d1 (Bind b0) w) H1))) in (\lambda (H5: (eq B b0 -b)).(\lambda (_: (eq C d1 c1)).(eq_ind_r T v (\lambda (t: T).(ex C (\lambda -(d2: C).(getl O (CHead c2 (Bind b) v) (CHead d2 (Bind b0) t))))) (eq_ind_r B -b (\lambda (b1: B).(ex C (\lambda (d2: C).(getl O (CHead c2 (Bind b) v) -(CHead d2 (Bind b1) v))))) (ex_intro C (\lambda (d2: C).(getl O (CHead c2 -(Bind b) v) (CHead d2 (Bind b) v))) c2 (getl_refl b c2 v)) b0 H5) w H4)))) -H3)) H2))) (\lambda (h0: nat).(\lambda (_: (((getl h0 (CHead c1 (Bind b) v) -(CHead d1 (Bind b0) w)) \to (ex C (\lambda (d2: C).(getl h0 (CHead c2 (Bind -b) v) (CHead d2 (Bind b0) w))))))).(\lambda (H1: (getl (S h0) (CHead c1 (Bind -b) v) (CHead d1 (Bind b0) w))).(let H_x \def (H b0 d1 w (r (Bind b) h0) -(getl_gen_S (Bind b) c1 (CHead d1 (Bind b0) w) v h0 H1)) in (let H2 \def H_x -in (ex_ind C (\lambda (d2: C).(getl h0 c2 (CHead d2 (Bind b0) w))) (ex C -(\lambda (d2: C).(getl (S h0) (CHead c2 (Bind b) v) (CHead d2 (Bind b0) w)))) -(\lambda (x: C).(\lambda (H3: (getl h0 c2 (CHead x (Bind b0) w))).(ex_intro C -(\lambda (d2: C).(getl (S h0) (CHead c2 (Bind b) v) (CHead d2 (Bind b0) w))) -x (getl_head (Bind b) h0 c2 (CHead x (Bind b0) w) H3 v)))) H2)))))) h -H0)))))))))). -(* COMMENTS -Initial nodes: 817 -END *) - -theorem cimp_getl_conf: - \forall (c1: C).(\forall (c2: C).((cimp c1 c2) \to (\forall (b: B).(\forall -(d1: C).(\forall (w: T).(\forall (i: nat).((getl i c1 (CHead d1 (Bind b) w)) -\to (ex2 C (\lambda (d2: C).(cimp d1 d2)) (\lambda (d2: C).(getl i c2 (CHead -d2 (Bind b) w))))))))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (H: ((\forall (b: B).(\forall (d1: -C).(\forall (w: T).(\forall (h: nat).((getl h c1 (CHead d1 (Bind b) w)) \to -(ex C (\lambda (d2: C).(getl h c2 (CHead d2 (Bind b) w))))))))))).(\lambda -(b: B).(\lambda (d1: C).(\lambda (w: T).(\lambda (i: nat).(\lambda (H0: (getl -i c1 (CHead d1 (Bind b) w))).(let H_x \def (H b d1 w i H0) in (let H1 \def -H_x in (ex_ind C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind b) w))) (ex2 C -(\lambda (d2: C).(\forall (b0: B).(\forall (d3: C).(\forall (w0: T).(\forall -(h: nat).((getl h d1 (CHead d3 (Bind b0) w0)) \to (ex C (\lambda (d4: -C).(getl h d2 (CHead d4 (Bind b0) w0)))))))))) (\lambda (d2: C).(getl i c2 -(CHead d2 (Bind b) w)))) (\lambda (x: C).(\lambda (H2: (getl i c2 (CHead x -(Bind b) w))).(ex_intro2 C (\lambda (d2: C).(\forall (b0: B).(\forall (d3: -C).(\forall (w0: T).(\forall (h: nat).((getl h d1 (CHead d3 (Bind b0) w0)) -\to (ex C (\lambda (d4: C).(getl h d2 (CHead d4 (Bind b0) w0)))))))))) -(\lambda (d2: C).(getl i c2 (CHead d2 (Bind b) w))) x (\lambda (b0: -B).(\lambda (d0: C).(\lambda (w0: T).(\lambda (h: nat).(\lambda (H3: (getl h -d1 (CHead d0 (Bind b0) w0))).(let H_y \def (getl_trans (S h) c1 (CHead d1 -(Bind b) w) i H0) in (let H_x0 \def (H b0 d0 w0 (plus (S h) i) (H_y (CHead d0 -(Bind b0) w0) (getl_head (Bind b) h d1 (CHead d0 (Bind b0) w0) H3 w))) in -(let H4 \def H_x0 in (ex_ind C (\lambda (d2: C).(getl (S (plus h i)) c2 -(CHead d2 (Bind b0) w0))) (ex C (\lambda (d2: C).(getl h x (CHead d2 (Bind -b0) w0)))) (\lambda (x0: C).(\lambda (H5: (getl (S (plus h i)) c2 (CHead x0 -(Bind b0) w0))).(let H_y0 \def (getl_conf_le (S (plus h i)) (CHead x0 (Bind -b0) w0) c2 H5 (CHead x (Bind b) w) i H2) in (let H6 \def (refl_equal nat -(plus (S h) i)) in (let H7 \def (eq_ind nat (S (plus h i)) (\lambda (n: -nat).(getl (minus n i) (CHead x (Bind b) w) (CHead x0 (Bind b0) w0))) (H_y0 -(le_S i (plus h i) (le_plus_r h i))) (plus (S h) i) H6) in (let H8 \def -(eq_ind nat (minus (plus (S h) i) i) (\lambda (n: nat).(getl n (CHead x (Bind -b) w) (CHead x0 (Bind b0) w0))) H7 (S h) (minus_plus_r (S h) i)) in (ex_intro -C (\lambda (d2: C).(getl h x (CHead d2 (Bind b0) w0))) x0 (getl_gen_S (Bind -b) x (CHead x0 (Bind b0) w0) w h H8)))))))) H4))))))))) H2))) H1)))))))))). -(* COMMENTS -Initial nodes: 673 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/clear/defs.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/clear/defs.ma deleted file mode 100644 index 30607a4fc..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/clear/defs.ma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/C/defs.ma". - -inductive clear: C \to (C \to Prop) \def -| clear_bind: \forall (b: B).(\forall (e: C).(\forall (u: T).(clear (CHead e -(Bind b) u) (CHead e (Bind b) u)))) -| clear_flat: \forall (e: C).(\forall (c: C).((clear e c) \to (\forall (f: -F).(\forall (u: T).(clear (CHead e (Flat f) u) c))))). - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/clear/drop.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/clear/drop.ma deleted file mode 100644 index aae3fedc0..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/clear/drop.ma +++ /dev/null @@ -1,181 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/clear/fwd.ma". - -include "Basic-1/drop/fwd.ma". - -theorem drop_clear: - \forall (c1: C).(\forall (c2: C).(\forall (i: nat).((drop (S i) O c1 c2) \to -(ex2_3 B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear c1 (CHead -e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e -c2)))))))) -\def - \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (i: -nat).((drop (S i) O c c2) \to (ex2_3 B C T (\lambda (b: B).(\lambda (e: -C).(\lambda (v: T).(clear c (CHead e (Bind b) v))))) (\lambda (_: B).(\lambda -(e: C).(\lambda (_: T).(drop i O e c2))))))))) (\lambda (n: nat).(\lambda -(c2: C).(\lambda (i: nat).(\lambda (H: (drop (S i) O (CSort n) c2)).(and3_ind -(eq C c2 (CSort n)) (eq nat (S i) O) (eq nat O O) (ex2_3 B C T (\lambda (b: -B).(\lambda (e: C).(\lambda (v: T).(clear (CSort n) (CHead e (Bind b) v))))) -(\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2))))) (\lambda -(_: (eq C c2 (CSort n))).(\lambda (H1: (eq nat (S i) O)).(\lambda (_: (eq nat -O O)).(let H3 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee in nat -return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow -True])) I O H1) in (False_ind (ex2_3 B C T (\lambda (b: B).(\lambda (e: -C).(\lambda (v: T).(clear (CSort n) (CHead e (Bind b) v))))) (\lambda (_: -B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2))))) H3))))) (drop_gen_sort -n (S i) O c2 H)))))) (\lambda (c: C).(\lambda (H: ((\forall (c2: C).(\forall -(i: nat).((drop (S i) O c c2) \to (ex2_3 B C T (\lambda (b: B).(\lambda (e: -C).(\lambda (v: T).(clear c (CHead e (Bind b) v))))) (\lambda (_: B).(\lambda -(e: C).(\lambda (_: T).(drop i O e c2)))))))))).(\lambda (k: K).(\lambda (t: -T).(\lambda (c2: C).(\lambda (i: nat).(\lambda (H0: (drop (S i) O (CHead c k -t) c2)).(K_ind (\lambda (k0: K).((drop (r k0 i) O c c2) \to (ex2_3 B C T -(\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear (CHead c k0 t) (CHead -e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e -c2))))))) (\lambda (b: B).(\lambda (H1: (drop (r (Bind b) i) O c -c2)).(ex2_3_intro B C T (\lambda (b0: B).(\lambda (e: C).(\lambda (v: -T).(clear (CHead c (Bind b) t) (CHead e (Bind b0) v))))) (\lambda (_: -B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2)))) b c t (clear_bind b c -t) H1))) (\lambda (f: F).(\lambda (H1: (drop (r (Flat f) i) O c c2)).(let H2 -\def (H c2 i H1) in (ex2_3_ind B C T (\lambda (b: B).(\lambda (e: C).(\lambda -(v: T).(clear c (CHead e (Bind b) v))))) (\lambda (_: B).(\lambda (e: -C).(\lambda (_: T).(drop i O e c2)))) (ex2_3 B C T (\lambda (b: B).(\lambda -(e: C).(\lambda (v: T).(clear (CHead c (Flat f) t) (CHead e (Bind b) v))))) -(\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2))))) (\lambda -(x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H3: (clear c (CHead x1 -(Bind x0) x2))).(\lambda (H4: (drop i O x1 c2)).(ex2_3_intro B C T (\lambda -(b: B).(\lambda (e: C).(\lambda (v: T).(clear (CHead c (Flat f) t) (CHead e -(Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e -c2)))) x0 x1 x2 (clear_flat c (CHead x1 (Bind x0) x2) H3 f t) H4)))))) H2)))) -k (drop_gen_drop k c c2 t i H0))))))))) c1). -(* COMMENTS -Initial nodes: 770 -END *) - -theorem drop_clear_O: - \forall (b: B).(\forall (c: C).(\forall (e1: C).(\forall (u: T).((clear c -(CHead e1 (Bind b) u)) \to (\forall (e2: C).(\forall (i: nat).((drop i O e1 -e2) \to (drop (S i) O c e2)))))))) -\def - \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e1: -C).(\forall (u: T).((clear c0 (CHead e1 (Bind b) u)) \to (\forall (e2: -C).(\forall (i: nat).((drop i O e1 e2) \to (drop (S i) O c0 e2)))))))) -(\lambda (n: nat).(\lambda (e1: C).(\lambda (u: T).(\lambda (H: (clear (CSort -n) (CHead e1 (Bind b) u))).(\lambda (e2: C).(\lambda (i: nat).(\lambda (_: -(drop i O e1 e2)).(clear_gen_sort (CHead e1 (Bind b) u) n H (drop (S i) O -(CSort n) e2))))))))) (\lambda (c0: C).(\lambda (H: ((\forall (e1: -C).(\forall (u: T).((clear c0 (CHead e1 (Bind b) u)) \to (\forall (e2: -C).(\forall (i: nat).((drop i O e1 e2) \to (drop (S i) O c0 -e2))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e1: C).(\lambda (u: -T).(\lambda (H0: (clear (CHead c0 k t) (CHead e1 (Bind b) u))).(\lambda (e2: -C).(\lambda (i: nat).(\lambda (H1: (drop i O e1 e2)).(K_ind (\lambda (k0: -K).((clear (CHead c0 k0 t) (CHead e1 (Bind b) u)) \to (drop (S i) O (CHead c0 -k0 t) e2))) (\lambda (b0: B).(\lambda (H2: (clear (CHead c0 (Bind b0) t) -(CHead e1 (Bind b) u))).(let H3 \def (f_equal C C (\lambda (e: C).(match e in -C return (\lambda (_: C).C) with [(CSort _) \Rightarrow e1 | (CHead c1 _ _) -\Rightarrow c1])) (CHead e1 (Bind b) u) (CHead c0 (Bind b0) t) -(clear_gen_bind b0 c0 (CHead e1 (Bind b) u) t H2)) in ((let H4 \def (f_equal -C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with [(CSort _) -\Rightarrow b | (CHead _ k0 _) \Rightarrow (match k0 in K return (\lambda (_: -K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow b])])) (CHead e1 -(Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e1 (Bind b) -u) t H2)) in ((let H5 \def (f_equal C T (\lambda (e: C).(match e in C return -(\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow -t0])) (CHead e1 (Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 -(CHead e1 (Bind b) u) t H2)) in (\lambda (H6: (eq B b b0)).(\lambda (H7: (eq -C e1 c0)).(let H8 \def (eq_ind C e1 (\lambda (c1: C).(drop i O c1 e2)) H1 c0 -H7) in (eq_ind B b (\lambda (b1: B).(drop (S i) O (CHead c0 (Bind b1) t) e2)) -(drop_drop (Bind b) i c0 e2 H8 t) b0 H6))))) H4)) H3)))) (\lambda (f: -F).(\lambda (H2: (clear (CHead c0 (Flat f) t) (CHead e1 (Bind b) -u))).(drop_drop (Flat f) i c0 e2 (H e1 u (clear_gen_flat f c0 (CHead e1 (Bind -b) u) t H2) e2 i H1) t))) k H0))))))))))) c)). -(* COMMENTS -Initial nodes: 619 -END *) - -theorem drop_clear_S: - \forall (x2: C).(\forall (x1: C).(\forall (h: nat).(\forall (d: nat).((drop -h (S d) x1 x2) \to (\forall (b: B).(\forall (c2: C).(\forall (u: T).((clear -x2 (CHead c2 (Bind b) u)) \to (ex2 C (\lambda (c1: C).(clear x1 (CHead c1 -(Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2))))))))))) -\def - \lambda (x2: C).(C_ind (\lambda (c: C).(\forall (x1: C).(\forall (h: -nat).(\forall (d: nat).((drop h (S d) x1 c) \to (\forall (b: B).(\forall (c2: -C).(\forall (u: T).((clear c (CHead c2 (Bind b) u)) \to (ex2 C (\lambda (c1: -C).(clear x1 (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 -c2)))))))))))) (\lambda (n: nat).(\lambda (x1: C).(\lambda (h: nat).(\lambda -(d: nat).(\lambda (_: (drop h (S d) x1 (CSort n))).(\lambda (b: B).(\lambda -(c2: C).(\lambda (u: T).(\lambda (H0: (clear (CSort n) (CHead c2 (Bind b) -u))).(clear_gen_sort (CHead c2 (Bind b) u) n H0 (ex2 C (\lambda (c1: -C).(clear x1 (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 -c2))))))))))))) (\lambda (c: C).(\lambda (H: ((\forall (x1: C).(\forall (h: -nat).(\forall (d: nat).((drop h (S d) x1 c) \to (\forall (b: B).(\forall (c2: -C).(\forall (u: T).((clear c (CHead c2 (Bind b) u)) \to (ex2 C (\lambda (c1: -C).(clear x1 (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 -c2))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (x1: C).(\lambda (h: -nat).(\lambda (d: nat).(\lambda (H0: (drop h (S d) x1 (CHead c k -t))).(\lambda (b: B).(\lambda (c2: C).(\lambda (u: T).(\lambda (H1: (clear -(CHead c k t) (CHead c2 (Bind b) u))).(ex2_ind C (\lambda (e: C).(eq C x1 -(CHead e k (lift h (r k d) t)))) (\lambda (e: C).(drop h (r k d) e c)) (ex2 C -(\lambda (c1: C).(clear x1 (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: -C).(drop h d c1 c2))) (\lambda (x: C).(\lambda (H2: (eq C x1 (CHead x k (lift -h (r k d) t)))).(\lambda (H3: (drop h (r k d) x c)).(eq_ind_r C (CHead x k -(lift h (r k d) t)) (\lambda (c0: C).(ex2 C (\lambda (c1: C).(clear c0 (CHead -c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2)))) (K_ind -(\lambda (k0: K).((clear (CHead c k0 t) (CHead c2 (Bind b) u)) \to ((drop h -(r k0 d) x c) \to (ex2 C (\lambda (c1: C).(clear (CHead x k0 (lift h (r k0 d) -t)) (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2)))))) -(\lambda (b0: B).(\lambda (H4: (clear (CHead c (Bind b0) t) (CHead c2 (Bind -b) u))).(\lambda (H5: (drop h (r (Bind b0) d) x c)).(let H6 \def (f_equal C C -(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) (CHead c2 (Bind b) u) -(CHead c (Bind b0) t) (clear_gen_bind b0 c (CHead c2 (Bind b) u) t H4)) in -((let H7 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: -C).B) with [(CSort _) \Rightarrow b | (CHead _ k0 _) \Rightarrow (match k0 in -K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) -\Rightarrow b])])) (CHead c2 (Bind b) u) (CHead c (Bind b0) t) -(clear_gen_bind b0 c (CHead c2 (Bind b) u) t H4)) in ((let H8 \def (f_equal C -T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) -\Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead c2 (Bind b) u) (CHead -c (Bind b0) t) (clear_gen_bind b0 c (CHead c2 (Bind b) u) t H4)) in (\lambda -(H9: (eq B b b0)).(\lambda (H10: (eq C c2 c)).(eq_ind_r T t (\lambda (t0: -T).(ex2 C (\lambda (c1: C).(clear (CHead x (Bind b0) (lift h (r (Bind b0) d) -t)) (CHead c1 (Bind b) (lift h d t0)))) (\lambda (c1: C).(drop h d c1 c2)))) -(eq_ind_r C c (\lambda (c0: C).(ex2 C (\lambda (c1: C).(clear (CHead x (Bind -b0) (lift h (r (Bind b0) d) t)) (CHead c1 (Bind b) (lift h d t)))) (\lambda -(c1: C).(drop h d c1 c0)))) (eq_ind_r B b0 (\lambda (b1: B).(ex2 C (\lambda -(c1: C).(clear (CHead x (Bind b0) (lift h (r (Bind b0) d) t)) (CHead c1 (Bind -b1) (lift h d t)))) (\lambda (c1: C).(drop h d c1 c)))) (ex_intro2 C (\lambda -(c1: C).(clear (CHead x (Bind b0) (lift h (r (Bind b0) d) t)) (CHead c1 (Bind -b0) (lift h d t)))) (\lambda (c1: C).(drop h d c1 c)) x (clear_bind b0 x -(lift h d t)) H5) b H9) c2 H10) u H8)))) H7)) H6))))) (\lambda (f: -F).(\lambda (H4: (clear (CHead c (Flat f) t) (CHead c2 (Bind b) u))).(\lambda -(H5: (drop h (r (Flat f) d) x c)).(let H6 \def (H x h d H5 b c2 u -(clear_gen_flat f c (CHead c2 (Bind b) u) t H4)) in (ex2_ind C (\lambda (c1: -C).(clear x (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 -c2)) (ex2 C (\lambda (c1: C).(clear (CHead x (Flat f) (lift h (r (Flat f) d) -t)) (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2))) -(\lambda (x0: C).(\lambda (H7: (clear x (CHead x0 (Bind b) (lift h d -u)))).(\lambda (H8: (drop h d x0 c2)).(ex_intro2 C (\lambda (c1: C).(clear -(CHead x (Flat f) (lift h (r (Flat f) d) t)) (CHead c1 (Bind b) (lift h d -u)))) (\lambda (c1: C).(drop h d c1 c2)) x0 (clear_flat x (CHead x0 (Bind b) -(lift h d u)) H7 f (lift h (r (Flat f) d) t)) H8)))) H6))))) k H1 H3) x1 -H2)))) (drop_gen_skip_r c x1 t h d k H0)))))))))))))) x2). -(* COMMENTS -Initial nodes: 1449 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/clear/fwd.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/clear/fwd.ma deleted file mode 100644 index d64ff77be..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/clear/fwd.ma +++ /dev/null @@ -1,164 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/clear/defs.ma". - -theorem clear_gen_sort: - \forall (x: C).(\forall (n: nat).((clear (CSort n) x) \to (\forall (P: -Prop).P))) -\def - \lambda (x: C).(\lambda (n: nat).(\lambda (H: (clear (CSort n) x)).(\lambda -(P: Prop).(insert_eq C (CSort n) (\lambda (c: C).(clear c x)) (\lambda (_: -C).P) (\lambda (y: C).(\lambda (H0: (clear y x)).(clear_ind (\lambda (c: -C).(\lambda (_: C).((eq C c (CSort n)) \to P))) (\lambda (b: B).(\lambda (e: -C).(\lambda (u: T).(\lambda (H1: (eq C (CHead e (Bind b) u) (CSort n))).(let -H2 \def (eq_ind C (CHead e (Bind b) u) (\lambda (ee: C).(match ee in C return -(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) -\Rightarrow True])) I (CSort n) H1) in (False_ind P H2)))))) (\lambda (e: -C).(\lambda (c: C).(\lambda (_: (clear e c)).(\lambda (_: (((eq C e (CSort -n)) \to P))).(\lambda (f: F).(\lambda (u: T).(\lambda (H3: (eq C (CHead e -(Flat f) u) (CSort n))).(let H4 \def (eq_ind C (CHead e (Flat f) u) (\lambda -(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H3) in -(False_ind P H4))))))))) y x H0))) H)))). -(* COMMENTS -Initial nodes: 215 -END *) - -theorem clear_gen_bind: - \forall (b: B).(\forall (e: C).(\forall (x: C).(\forall (u: T).((clear -(CHead e (Bind b) u) x) \to (eq C x (CHead e (Bind b) u)))))) -\def - \lambda (b: B).(\lambda (e: C).(\lambda (x: C).(\lambda (u: T).(\lambda (H: -(clear (CHead e (Bind b) u) x)).(insert_eq C (CHead e (Bind b) u) (\lambda -(c: C).(clear c x)) (\lambda (c: C).(eq C x c)) (\lambda (y: C).(\lambda (H0: -(clear y x)).(clear_ind (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead e -(Bind b) u)) \to (eq C c0 c)))) (\lambda (b0: B).(\lambda (e0: C).(\lambda -(u0: T).(\lambda (H1: (eq C (CHead e0 (Bind b0) u0) (CHead e (Bind b) -u))).(let H2 \def (f_equal C C (\lambda (e1: C).(match e1 in C return -(\lambda (_: C).C) with [(CSort _) \Rightarrow e0 | (CHead c _ _) \Rightarrow -c])) (CHead e0 (Bind b0) u0) (CHead e (Bind b) u) H1) in ((let H3 \def -(f_equal C B (\lambda (e1: C).(match e1 in C return (\lambda (_: C).B) with -[(CSort _) \Rightarrow b0 | (CHead _ k _) \Rightarrow (match k in K return -(\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow -b0])])) (CHead e0 (Bind b0) u0) (CHead e (Bind b) u) H1) in ((let H4 \def -(f_equal C T (\lambda (e1: C).(match e1 in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead e0 (Bind -b0) u0) (CHead e (Bind b) u) H1) in (\lambda (H5: (eq B b0 b)).(\lambda (H6: -(eq C e0 e)).(eq_ind_r T u (\lambda (t: T).(eq C (CHead e0 (Bind b0) t) -(CHead e0 (Bind b0) t))) (eq_ind_r C e (\lambda (c: C).(eq C (CHead c (Bind -b0) u) (CHead c (Bind b0) u))) (eq_ind_r B b (\lambda (b1: B).(eq C (CHead e -(Bind b1) u) (CHead e (Bind b1) u))) (refl_equal C (CHead e (Bind b) u)) b0 -H5) e0 H6) u0 H4)))) H3)) H2)))))) (\lambda (e0: C).(\lambda (c: C).(\lambda -(_: (clear e0 c)).(\lambda (_: (((eq C e0 (CHead e (Bind b) u)) \to (eq C c -e0)))).(\lambda (f: F).(\lambda (u0: T).(\lambda (H3: (eq C (CHead e0 (Flat -f) u0) (CHead e (Bind b) u))).(let H4 \def (eq_ind C (CHead e0 (Flat f) u0) -(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (CHead e (Bind b) u) H3) in (False_ind (eq C c (CHead e0 (Flat f) -u0)) H4))))))))) y x H0))) H))))). -(* COMMENTS -Initial nodes: 525 -END *) - -theorem clear_gen_flat: - \forall (f: F).(\forall (e: C).(\forall (x: C).(\forall (u: T).((clear -(CHead e (Flat f) u) x) \to (clear e x))))) -\def - \lambda (f: F).(\lambda (e: C).(\lambda (x: C).(\lambda (u: T).(\lambda (H: -(clear (CHead e (Flat f) u) x)).(insert_eq C (CHead e (Flat f) u) (\lambda -(c: C).(clear c x)) (\lambda (_: C).(clear e x)) (\lambda (y: C).(\lambda -(H0: (clear y x)).(clear_ind (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead -e (Flat f) u)) \to (clear e c0)))) (\lambda (b: B).(\lambda (e0: C).(\lambda -(u0: T).(\lambda (H1: (eq C (CHead e0 (Bind b) u0) (CHead e (Flat f) -u))).(let H2 \def (eq_ind C (CHead e0 (Bind b) u0) (\lambda (ee: C).(match ee -in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead -_ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) -\Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead e (Flat f) u) H1) -in (False_ind (clear e (CHead e0 (Bind b) u0)) H2)))))) (\lambda (e0: -C).(\lambda (c: C).(\lambda (H1: (clear e0 c)).(\lambda (H2: (((eq C e0 -(CHead e (Flat f) u)) \to (clear e c)))).(\lambda (f0: F).(\lambda (u0: -T).(\lambda (H3: (eq C (CHead e0 (Flat f0) u0) (CHead e (Flat f) u))).(let H4 -\def (f_equal C C (\lambda (e1: C).(match e1 in C return (\lambda (_: C).C) -with [(CSort _) \Rightarrow e0 | (CHead c0 _ _) \Rightarrow c0])) (CHead e0 -(Flat f0) u0) (CHead e (Flat f) u) H3) in ((let H5 \def (f_equal C F (\lambda -(e1: C).(match e1 in C return (\lambda (_: C).F) with [(CSort _) \Rightarrow -f0 | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).F) with -[(Bind _) \Rightarrow f0 | (Flat f1) \Rightarrow f1])])) (CHead e0 (Flat f0) -u0) (CHead e (Flat f) u) H3) in ((let H6 \def (f_equal C T (\lambda (e1: -C).(match e1 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | -(CHead _ _ t) \Rightarrow t])) (CHead e0 (Flat f0) u0) (CHead e (Flat f) u) -H3) in (\lambda (_: (eq F f0 f)).(\lambda (H8: (eq C e0 e)).(let H9 \def -(eq_ind C e0 (\lambda (c0: C).((eq C c0 (CHead e (Flat f) u)) \to (clear e -c))) H2 e H8) in (let H10 \def (eq_ind C e0 (\lambda (c0: C).(clear c0 c)) H1 -e H8) in H10))))) H5)) H4))))))))) y x H0))) H))))). -(* COMMENTS -Initial nodes: 453 -END *) - -theorem clear_gen_flat_r: - \forall (f: F).(\forall (x: C).(\forall (e: C).(\forall (u: T).((clear x -(CHead e (Flat f) u)) \to (\forall (P: Prop).P))))) -\def - \lambda (f: F).(\lambda (x: C).(\lambda (e: C).(\lambda (u: T).(\lambda (H: -(clear x (CHead e (Flat f) u))).(\lambda (P: Prop).(insert_eq C (CHead e -(Flat f) u) (\lambda (c: C).(clear x c)) (\lambda (_: C).P) (\lambda (y: -C).(\lambda (H0: (clear x y)).(clear_ind (\lambda (_: C).(\lambda (c0: -C).((eq C c0 (CHead e (Flat f) u)) \to P))) (\lambda (b: B).(\lambda (e0: -C).(\lambda (u0: T).(\lambda (H1: (eq C (CHead e0 (Bind b) u0) (CHead e (Flat -f) u))).(let H2 \def (eq_ind C (CHead e0 (Bind b) u0) (\lambda (ee: C).(match -ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | -(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead e (Flat -f) u) H1) in (False_ind P H2)))))) (\lambda (e0: C).(\lambda (c: C).(\lambda -(H1: (clear e0 c)).(\lambda (H2: (((eq C c (CHead e (Flat f) u)) \to -P))).(\lambda (_: F).(\lambda (_: T).(\lambda (H3: (eq C c (CHead e (Flat f) -u))).(let H4 \def (eq_ind C c (\lambda (c0: C).((eq C c0 (CHead e (Flat f) -u)) \to P)) H2 (CHead e (Flat f) u) H3) in (let H5 \def (eq_ind C c (\lambda -(c0: C).(clear e0 c0)) H1 (CHead e (Flat f) u) H3) in (H4 (refl_equal C -(CHead e (Flat f) u)))))))))))) x y H0))) H)))))). -(* COMMENTS -Initial nodes: 303 -END *) - -theorem clear_gen_all: - \forall (c1: C).(\forall (c2: C).((clear c1 c2) \to (ex_3 B C T (\lambda (b: -B).(\lambda (e: C).(\lambda (u: T).(eq C c2 (CHead e (Bind b) u)))))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (clear c1 c2)).(clear_ind -(\lambda (_: C).(\lambda (c0: C).(ex_3 B C T (\lambda (b: B).(\lambda (e: -C).(\lambda (u: T).(eq C c0 (CHead e (Bind b) u)))))))) (\lambda (b: -B).(\lambda (e: C).(\lambda (u: T).(ex_3_intro B C T (\lambda (b0: -B).(\lambda (e0: C).(\lambda (u0: T).(eq C (CHead e (Bind b) u) (CHead e0 -(Bind b0) u0))))) b e u (refl_equal C (CHead e (Bind b) u)))))) (\lambda (e: -C).(\lambda (c: C).(\lambda (H0: (clear e c)).(\lambda (H1: (ex_3 B C T -(\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(eq C c (CHead e0 (Bind b) -u))))))).(\lambda (_: F).(\lambda (_: T).(let H2 \def H1 in (ex_3_ind B C T -(\lambda (b: B).(\lambda (e0: C).(\lambda (u0: T).(eq C c (CHead e0 (Bind b) -u0))))) (ex_3 B C T (\lambda (b: B).(\lambda (e0: C).(\lambda (u0: T).(eq C c -(CHead e0 (Bind b) u0)))))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: -T).(\lambda (H3: (eq C c (CHead x1 (Bind x0) x2))).(let H4 \def (eq_ind C c -(\lambda (c0: C).(clear e c0)) H0 (CHead x1 (Bind x0) x2) H3) in (eq_ind_r C -(CHead x1 (Bind x0) x2) (\lambda (c0: C).(ex_3 B C T (\lambda (b: B).(\lambda -(e0: C).(\lambda (u0: T).(eq C c0 (CHead e0 (Bind b) u0))))))) (ex_3_intro B -C T (\lambda (b: B).(\lambda (e0: C).(\lambda (u0: T).(eq C (CHead x1 (Bind -x0) x2) (CHead e0 (Bind b) u0))))) x0 x1 x2 (refl_equal C (CHead x1 (Bind x0) -x2))) c H3)))))) H2)))))))) c1 c2 H))). -(* COMMENTS -Initial nodes: 381 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/clear/props.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/clear/props.ma deleted file mode 100644 index 68e250d76..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/clear/props.ma +++ /dev/null @@ -1,152 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/clear/fwd.ma". - -theorem clear_clear: - \forall (c1: C).(\forall (c2: C).((clear c1 c2) \to (clear c2 c2))) -\def - \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).((clear c c2) \to -(clear c2 c2)))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (H: (clear -(CSort n) c2)).(clear_gen_sort c2 n H (clear c2 c2))))) (\lambda (c: -C).(\lambda (H: ((\forall (c2: C).((clear c c2) \to (clear c2 -c2))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (H0: (clear -(CHead c k t) c2)).(K_ind (\lambda (k0: K).((clear (CHead c k0 t) c2) \to -(clear c2 c2))) (\lambda (b: B).(\lambda (H1: (clear (CHead c (Bind b) t) -c2)).(eq_ind_r C (CHead c (Bind b) t) (\lambda (c0: C).(clear c0 c0)) -(clear_bind b c t) c2 (clear_gen_bind b c c2 t H1)))) (\lambda (f: -F).(\lambda (H1: (clear (CHead c (Flat f) t) c2)).(H c2 (clear_gen_flat f c -c2 t H1)))) k H0))))))) c1). -(* COMMENTS -Initial nodes: 199 -END *) - -theorem clear_mono: - \forall (c: C).(\forall (c1: C).((clear c c1) \to (\forall (c2: C).((clear c -c2) \to (eq C c1 c2))))) -\def - \lambda (c: C).(C_ind (\lambda (c0: C).(\forall (c1: C).((clear c0 c1) \to -(\forall (c2: C).((clear c0 c2) \to (eq C c1 c2)))))) (\lambda (n: -nat).(\lambda (c1: C).(\lambda (_: (clear (CSort n) c1)).(\lambda (c2: -C).(\lambda (H0: (clear (CSort n) c2)).(clear_gen_sort c2 n H0 (eq C c1 -c2))))))) (\lambda (c0: C).(\lambda (H: ((\forall (c1: C).((clear c0 c1) \to -(\forall (c2: C).((clear c0 c2) \to (eq C c1 c2))))))).(\lambda (k: -K).(\lambda (t: T).(\lambda (c1: C).(\lambda (H0: (clear (CHead c0 k t) -c1)).(\lambda (c2: C).(\lambda (H1: (clear (CHead c0 k t) c2)).(K_ind -(\lambda (k0: K).((clear (CHead c0 k0 t) c1) \to ((clear (CHead c0 k0 t) c2) -\to (eq C c1 c2)))) (\lambda (b: B).(\lambda (H2: (clear (CHead c0 (Bind b) -t) c1)).(\lambda (H3: (clear (CHead c0 (Bind b) t) c2)).(eq_ind_r C (CHead c0 -(Bind b) t) (\lambda (c3: C).(eq C c1 c3)) (eq_ind_r C (CHead c0 (Bind b) t) -(\lambda (c3: C).(eq C c3 (CHead c0 (Bind b) t))) (refl_equal C (CHead c0 -(Bind b) t)) c1 (clear_gen_bind b c0 c1 t H2)) c2 (clear_gen_bind b c0 c2 t -H3))))) (\lambda (f: F).(\lambda (H2: (clear (CHead c0 (Flat f) t) -c1)).(\lambda (H3: (clear (CHead c0 (Flat f) t) c2)).(H c1 (clear_gen_flat f -c0 c1 t H2) c2 (clear_gen_flat f c0 c2 t H3))))) k H0 H1))))))))) c). -(* COMMENTS -Initial nodes: 357 -END *) - -theorem clear_trans: - \forall (c1: C).(\forall (c: C).((clear c1 c) \to (\forall (c2: C).((clear c -c2) \to (clear c1 c2))))) -\def - \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c0: C).((clear c c0) \to -(\forall (c2: C).((clear c0 c2) \to (clear c c2)))))) (\lambda (n: -nat).(\lambda (c: C).(\lambda (H: (clear (CSort n) c)).(\lambda (c2: -C).(\lambda (_: (clear c c2)).(clear_gen_sort c n H (clear (CSort n) -c2))))))) (\lambda (c: C).(\lambda (H: ((\forall (c0: C).((clear c c0) \to -(\forall (c2: C).((clear c0 c2) \to (clear c c2))))))).(\lambda (k: -K).(\lambda (t: T).(\lambda (c0: C).(\lambda (H0: (clear (CHead c k t) -c0)).(\lambda (c2: C).(\lambda (H1: (clear c0 c2)).(K_ind (\lambda (k0: -K).((clear (CHead c k0 t) c0) \to (clear (CHead c k0 t) c2))) (\lambda (b: -B).(\lambda (H2: (clear (CHead c (Bind b) t) c0)).(let H3 \def (eq_ind C c0 -(\lambda (c3: C).(clear c3 c2)) H1 (CHead c (Bind b) t) (clear_gen_bind b c -c0 t H2)) in (eq_ind_r C (CHead c (Bind b) t) (\lambda (c3: C).(clear (CHead -c (Bind b) t) c3)) (clear_bind b c t) c2 (clear_gen_bind b c c2 t H3))))) -(\lambda (f: F).(\lambda (H2: (clear (CHead c (Flat f) t) c0)).(clear_flat c -c2 (H c0 (clear_gen_flat f c c0 t H2) c2 H1) f t))) k H0))))))))) c1). -(* COMMENTS -Initial nodes: 299 -END *) - -theorem clear_ctail: - \forall (b: B).(\forall (c1: C).(\forall (c2: C).(\forall (u2: T).((clear c1 -(CHead c2 (Bind b) u2)) \to (\forall (k: K).(\forall (u1: T).(clear (CTail k -u1 c1) (CHead (CTail k u1 c2) (Bind b) u2)))))))) -\def - \lambda (b: B).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: -C).(\forall (u2: T).((clear c (CHead c2 (Bind b) u2)) \to (\forall (k: -K).(\forall (u1: T).(clear (CTail k u1 c) (CHead (CTail k u1 c2) (Bind b) -u2)))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (u2: T).(\lambda (H: -(clear (CSort n) (CHead c2 (Bind b) u2))).(\lambda (k: K).(\lambda (u1: -T).(K_ind (\lambda (k0: K).(clear (CHead (CSort n) k0 u1) (CHead (CTail k0 u1 -c2) (Bind b) u2))) (\lambda (b0: B).(clear_gen_sort (CHead c2 (Bind b) u2) n -H (clear (CHead (CSort n) (Bind b0) u1) (CHead (CTail (Bind b0) u1 c2) (Bind -b) u2)))) (\lambda (f: F).(clear_gen_sort (CHead c2 (Bind b) u2) n H (clear -(CHead (CSort n) (Flat f) u1) (CHead (CTail (Flat f) u1 c2) (Bind b) u2)))) -k))))))) (\lambda (c: C).(\lambda (H: ((\forall (c2: C).(\forall (u2: -T).((clear c (CHead c2 (Bind b) u2)) \to (\forall (k: K).(\forall (u1: -T).(clear (CTail k u1 c) (CHead (CTail k u1 c2) (Bind b) u2))))))))).(\lambda -(k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (u2: T).(\lambda (H0: (clear -(CHead c k t) (CHead c2 (Bind b) u2))).(\lambda (k0: K).(\lambda (u1: -T).(K_ind (\lambda (k1: K).((clear (CHead c k1 t) (CHead c2 (Bind b) u2)) \to -(clear (CHead (CTail k0 u1 c) k1 t) (CHead (CTail k0 u1 c2) (Bind b) u2)))) -(\lambda (b0: B).(\lambda (H1: (clear (CHead c (Bind b0) t) (CHead c2 (Bind -b) u2))).(let H2 \def (f_equal C C (\lambda (e: C).(match e in C return -(\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ _) -\Rightarrow c0])) (CHead c2 (Bind b) u2) (CHead c (Bind b0) t) -(clear_gen_bind b0 c (CHead c2 (Bind b) u2) t H1)) in ((let H3 \def (f_equal -C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with [(CSort _) -\Rightarrow b | (CHead _ k1 _) \Rightarrow (match k1 in K return (\lambda (_: -K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow b])])) (CHead c2 -(Bind b) u2) (CHead c (Bind b0) t) (clear_gen_bind b0 c (CHead c2 (Bind b) -u2) t H1)) in ((let H4 \def (f_equal C T (\lambda (e: C).(match e in C return -(\lambda (_: C).T) with [(CSort _) \Rightarrow u2 | (CHead _ _ t0) -\Rightarrow t0])) (CHead c2 (Bind b) u2) (CHead c (Bind b0) t) -(clear_gen_bind b0 c (CHead c2 (Bind b) u2) t H1)) in (\lambda (H5: (eq B b -b0)).(\lambda (H6: (eq C c2 c)).(eq_ind_r T t (\lambda (t0: T).(clear (CHead -(CTail k0 u1 c) (Bind b0) t) (CHead (CTail k0 u1 c2) (Bind b) t0))) (eq_ind_r -C c (\lambda (c0: C).(clear (CHead (CTail k0 u1 c) (Bind b0) t) (CHead (CTail -k0 u1 c0) (Bind b) t))) (eq_ind B b (\lambda (b1: B).(clear (CHead (CTail k0 -u1 c) (Bind b1) t) (CHead (CTail k0 u1 c) (Bind b) t))) (clear_bind b (CTail -k0 u1 c) t) b0 H5) c2 H6) u2 H4)))) H3)) H2)))) (\lambda (f: F).(\lambda (H1: -(clear (CHead c (Flat f) t) (CHead c2 (Bind b) u2))).(clear_flat (CTail k0 u1 -c) (CHead (CTail k0 u1 c2) (Bind b) u2) (H c2 u2 (clear_gen_flat f c (CHead -c2 (Bind b) u2) t H1) k0 u1) f t))) k H0)))))))))) c1)). -(* COMMENTS -Initial nodes: 819 -END *) - -theorem clear_cle: - \forall (c1: C).(\forall (c2: C).((clear c1 c2) \to (cle c2 c1))) -\def - \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).((clear c c2) \to -(le (cweight c2) (cweight c))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda -(H: (clear (CSort n) c2)).(clear_gen_sort c2 n H (le (cweight c2) O))))) -(\lambda (c: C).(\lambda (H: ((\forall (c2: C).((clear c c2) \to (le (cweight -c2) (cweight c)))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: -C).(\lambda (H0: (clear (CHead c k t) c2)).(K_ind (\lambda (k0: K).((clear -(CHead c k0 t) c2) \to (le (cweight c2) (plus (cweight c) (tweight t))))) -(\lambda (b: B).(\lambda (H1: (clear (CHead c (Bind b) t) c2)).(eq_ind_r C -(CHead c (Bind b) t) (\lambda (c0: C).(le (cweight c0) (plus (cweight c) -(tweight t)))) (le_n (plus (cweight c) (tweight t))) c2 (clear_gen_bind b c -c2 t H1)))) (\lambda (f: F).(\lambda (H1: (clear (CHead c (Flat f) t) -c2)).(le_plus_trans (cweight c2) (cweight c) (tweight t) (H c2 -(clear_gen_flat f c c2 t H1))))) k H0))))))) c1). -(* COMMENTS -Initial nodes: 247 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/clen/defs.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/clen/defs.ma deleted file mode 100644 index 07539ffad..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/clen/defs.ma +++ /dev/null @@ -1,26 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/C/defs.ma". - -include "Basic-1/s/defs.ma". - -definition clen: - C \to nat -\def - let rec clen (c: C) on c: nat \def (match c with [(CSort _) \Rightarrow O | -(CHead c0 k _) \Rightarrow (s k (clen c0))]) in clen. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/clen/getl.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/clen/getl.ma deleted file mode 100644 index af8a96cf5..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/clen/getl.ma +++ /dev/null @@ -1,361 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/clen/defs.ma". - -include "Basic-1/getl/props.ma". - -theorem getl_ctail_clen: - \forall (b: B).(\forall (t: T).(\forall (c: C).(ex nat (\lambda (n: -nat).(getl (clen c) (CTail (Bind b) t c) (CHead (CSort n) (Bind b) t)))))) -\def - \lambda (b: B).(\lambda (t: T).(\lambda (c: C).(C_ind (\lambda (c0: C).(ex -nat (\lambda (n: nat).(getl (clen c0) (CTail (Bind b) t c0) (CHead (CSort n) -(Bind b) t))))) (\lambda (n: nat).(ex_intro nat (\lambda (n0: nat).(getl O -(CHead (CSort n) (Bind b) t) (CHead (CSort n0) (Bind b) t))) n (getl_refl b -(CSort n) t))) (\lambda (c0: C).(\lambda (H: (ex nat (\lambda (n: nat).(getl -(clen c0) (CTail (Bind b) t c0) (CHead (CSort n) (Bind b) t))))).(\lambda (k: -K).(\lambda (t0: T).(let H0 \def H in (ex_ind nat (\lambda (n: nat).(getl -(clen c0) (CTail (Bind b) t c0) (CHead (CSort n) (Bind b) t))) (ex nat -(\lambda (n: nat).(getl (s k (clen c0)) (CHead (CTail (Bind b) t c0) k t0) -(CHead (CSort n) (Bind b) t)))) (\lambda (x: nat).(\lambda (H1: (getl (clen -c0) (CTail (Bind b) t c0) (CHead (CSort x) (Bind b) t))).(K_ind (\lambda (k0: -K).(ex nat (\lambda (n: nat).(getl (s k0 (clen c0)) (CHead (CTail (Bind b) t -c0) k0 t0) (CHead (CSort n) (Bind b) t))))) (\lambda (b0: B).(ex_intro nat -(\lambda (n: nat).(getl (S (clen c0)) (CHead (CTail (Bind b) t c0) (Bind b0) -t0) (CHead (CSort n) (Bind b) t))) x (getl_head (Bind b0) (clen c0) (CTail -(Bind b) t c0) (CHead (CSort x) (Bind b) t) H1 t0))) (\lambda (f: -F).(ex_intro nat (\lambda (n: nat).(getl (clen c0) (CHead (CTail (Bind b) t -c0) (Flat f) t0) (CHead (CSort n) (Bind b) t))) x (getl_flat (CTail (Bind b) -t c0) (CHead (CSort x) (Bind b) t) (clen c0) H1 f t0))) k))) H0)))))) c))). -(* COMMENTS -Initial nodes: 459 -END *) - -theorem getl_gen_tail: - \forall (k: K).(\forall (b: B).(\forall (u1: T).(\forall (u2: T).(\forall -(c2: C).(\forall (c1: C).(\forall (i: nat).((getl i (CTail k u1 c1) (CHead c2 -(Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) -(\lambda (e: C).(getl i c1 (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: -nat).(eq nat i (clen c1))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: -nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n)))))))))))) -\def - \lambda (k: K).(\lambda (b: B).(\lambda (u1: T).(\lambda (u2: T).(\lambda -(c2: C).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (i: nat).((getl i -(CTail k u1 c) (CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C -c2 (CTail k u1 e))) (\lambda (e: C).(getl i c (CHead e (Bind b) u2)))) (ex4 -nat (\lambda (_: nat).(eq nat i (clen c))) (\lambda (_: nat).(eq K k (Bind -b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort -n)))))))) (\lambda (n: nat).(\lambda (i: nat).(nat_ind (\lambda (n0: -nat).((getl n0 (CTail k u1 (CSort n)) (CHead c2 (Bind b) u2)) \to (or (ex2 C -(\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl n0 (CSort n) -(CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat n0 (clen (CSort -n)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) -(\lambda (n1: nat).(eq C c2 (CSort n1))))))) (\lambda (H: (getl O (CHead -(CSort n) k u1) (CHead c2 (Bind b) u2))).(K_ind (\lambda (k0: K).((clear -(CHead (CSort n) k0 u1) (CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda (e: -C).(eq C c2 (CTail k0 u1 e))) (\lambda (e: C).(getl O (CSort n) (CHead e -(Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O O)) (\lambda (_: -nat).(eq K k0 (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: -nat).(eq C c2 (CSort n0))))))) (\lambda (b0: B).(\lambda (H0: (clear (CHead -(CSort n) (Bind b0) u1) (CHead c2 (Bind b) u2))).(let H1 \def (f_equal C C -(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow c2 | (CHead c _ _) \Rightarrow c])) (CHead c2 (Bind b) u2) (CHead -(CSort n) (Bind b0) u1) (clear_gen_bind b0 (CSort n) (CHead c2 (Bind b) u2) -u1 H0)) in ((let H2 \def (f_equal C B (\lambda (e: C).(match e in C return -(\lambda (_: C).B) with [(CSort _) \Rightarrow b | (CHead _ k0 _) \Rightarrow -(match k0 in K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | -(Flat _) \Rightarrow b])])) (CHead c2 (Bind b) u2) (CHead (CSort n) (Bind b0) -u1) (clear_gen_bind b0 (CSort n) (CHead c2 (Bind b) u2) u1 H0)) in ((let H3 -\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c2 -(Bind b) u2) (CHead (CSort n) (Bind b0) u1) (clear_gen_bind b0 (CSort n) -(CHead c2 (Bind b) u2) u1 H0)) in (\lambda (H4: (eq B b b0)).(\lambda (H5: -(eq C c2 (CSort n))).(eq_ind_r C (CSort n) (\lambda (c: C).(or (ex2 C -(\lambda (e: C).(eq C c (CTail (Bind b0) u1 e))) (\lambda (e: C).(getl O -(CSort n) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O O)) -(\lambda (_: nat).(eq K (Bind b0) (Bind b))) (\lambda (_: nat).(eq T u1 u2)) -(\lambda (n0: nat).(eq C c (CSort n0)))))) (eq_ind_r T u1 (\lambda (t: T).(or -(ex2 C (\lambda (e: C).(eq C (CSort n) (CTail (Bind b0) u1 e))) (\lambda (e: -C).(getl O (CSort n) (CHead e (Bind b) t)))) (ex4 nat (\lambda (_: nat).(eq -nat O O)) (\lambda (_: nat).(eq K (Bind b0) (Bind b))) (\lambda (_: nat).(eq -T u1 t)) (\lambda (n0: nat).(eq C (CSort n) (CSort n0)))))) (eq_ind_r B b0 -(\lambda (b1: B).(or (ex2 C (\lambda (e: C).(eq C (CSort n) (CTail (Bind b0) -u1 e))) (\lambda (e: C).(getl O (CSort n) (CHead e (Bind b1) u1)))) (ex4 nat -(\lambda (_: nat).(eq nat O O)) (\lambda (_: nat).(eq K (Bind b0) (Bind b1))) -(\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: nat).(eq C (CSort n) (CSort -n0)))))) (or_intror (ex2 C (\lambda (e: C).(eq C (CSort n) (CTail (Bind b0) -u1 e))) (\lambda (e: C).(getl O (CSort n) (CHead e (Bind b0) u1)))) (ex4 nat -(\lambda (_: nat).(eq nat O O)) (\lambda (_: nat).(eq K (Bind b0) (Bind b0))) -(\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: nat).(eq C (CSort n) (CSort -n0)))) (ex4_intro nat (\lambda (_: nat).(eq nat O O)) (\lambda (_: nat).(eq K -(Bind b0) (Bind b0))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: nat).(eq -C (CSort n) (CSort n0))) n (refl_equal nat O) (refl_equal K (Bind b0)) -(refl_equal T u1) (refl_equal C (CSort n)))) b H4) u2 H3) c2 H5)))) H2)) -H1)))) (\lambda (f: F).(\lambda (H0: (clear (CHead (CSort n) (Flat f) u1) -(CHead c2 (Bind b) u2))).(clear_gen_sort (CHead c2 (Bind b) u2) n -(clear_gen_flat f (CSort n) (CHead c2 (Bind b) u2) u1 H0) (or (ex2 C (\lambda -(e: C).(eq C c2 (CTail (Flat f) u1 e))) (\lambda (e: C).(getl O (CSort n) -(CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O O)) (\lambda -(_: nat).(eq K (Flat f) (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda -(n0: nat).(eq C c2 (CSort n0)))))))) k (getl_gen_O (CHead (CSort n) k u1) -(CHead c2 (Bind b) u2) H))) (\lambda (n0: nat).(\lambda (_: (((getl n0 (CHead -(CSort n) k u1) (CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C -c2 (CTail k u1 e))) (\lambda (e: C).(getl n0 (CSort n) (CHead e (Bind b) -u2)))) (ex4 nat (\lambda (_: nat).(eq nat n0 O)) (\lambda (_: nat).(eq K k -(Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n1: nat).(eq C c2 (CSort -n1)))))))).(\lambda (H0: (getl (S n0) (CHead (CSort n) k u1) (CHead c2 (Bind -b) u2))).(getl_gen_sort n (r k n0) (CHead c2 (Bind b) u2) (getl_gen_S k -(CSort n) (CHead c2 (Bind b) u2) u1 n0 H0) (or (ex2 C (\lambda (e: C).(eq C -c2 (CTail k u1 e))) (\lambda (e: C).(getl (S n0) (CSort n) (CHead e (Bind b) -u2)))) (ex4 nat (\lambda (_: nat).(eq nat (S n0) O)) (\lambda (_: nat).(eq K -k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n1: nat).(eq C c2 -(CSort n1))))))))) i))) (\lambda (c: C).(\lambda (H: ((\forall (i: -nat).((getl i (CTail k u1 c) (CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda -(e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl i c (CHead e (Bind b) -u2)))) (ex4 nat (\lambda (_: nat).(eq nat i (clen c))) (\lambda (_: nat).(eq -K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 -(CSort n))))))))).(\lambda (k0: K).(\lambda (t: T).(\lambda (i: nat).(nat_ind -(\lambda (n: nat).((getl n (CTail k u1 (CHead c k0 t)) (CHead c2 (Bind b) -u2)) \to (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: -C).(getl n (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: -nat).(eq nat n (clen (CHead c k0 t)))) (\lambda (_: nat).(eq K k (Bind b))) -(\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0))))))) -(\lambda (H0: (getl O (CHead (CTail k u1 c) k0 t) (CHead c2 (Bind b) -u2))).(K_ind (\lambda (k1: K).((clear (CHead (CTail k u1 c) k1 t) (CHead c2 -(Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) -(\lambda (e: C).(getl O (CHead c k1 t) (CHead e (Bind b) u2)))) (ex4 nat -(\lambda (_: nat).(eq nat O (s k1 (clen c)))) (\lambda (_: nat).(eq K k (Bind -b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort -n))))))) (\lambda (b0: B).(\lambda (H1: (clear (CHead (CTail k u1 c) (Bind -b0) t) (CHead c2 (Bind b) u2))).(let H2 \def (f_equal C C (\lambda (e: -C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | -(CHead c0 _ _) \Rightarrow c0])) (CHead c2 (Bind b) u2) (CHead (CTail k u1 c) -(Bind b0) t) (clear_gen_bind b0 (CTail k u1 c) (CHead c2 (Bind b) u2) t H1)) -in ((let H3 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda -(_: C).B) with [(CSort _) \Rightarrow b | (CHead _ k1 _) \Rightarrow (match -k1 in K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) -\Rightarrow b])])) (CHead c2 (Bind b) u2) (CHead (CTail k u1 c) (Bind b0) t) -(clear_gen_bind b0 (CTail k u1 c) (CHead c2 (Bind b) u2) t H1)) in ((let H4 -\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow u2 | (CHead _ _ t0) \Rightarrow t0])) (CHead c2 -(Bind b) u2) (CHead (CTail k u1 c) (Bind b0) t) (clear_gen_bind b0 (CTail k -u1 c) (CHead c2 (Bind b) u2) t H1)) in (\lambda (H5: (eq B b b0)).(\lambda -(H6: (eq C c2 (CTail k u1 c))).(eq_ind T u2 (\lambda (t0: T).(or (ex2 C -(\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c -(Bind b0) t0) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O -(s (Bind b0) (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: -nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n)))))) (eq_ind B b -(\lambda (b1: B).(or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) -(\lambda (e: C).(getl O (CHead c (Bind b1) u2) (CHead e (Bind b) u2)))) (ex4 -nat (\lambda (_: nat).(eq nat O (s (Bind b1) (clen c)))) (\lambda (_: -nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq -C c2 (CSort n)))))) (let H7 \def (eq_ind C c2 (\lambda (c0: C).(\forall (i0: -nat).((getl i0 (CTail k u1 c) (CHead c0 (Bind b) u2)) \to (or (ex2 C (\lambda -(e: C).(eq C c0 (CTail k u1 e))) (\lambda (e: C).(getl i0 c (CHead e (Bind b) -u2)))) (ex4 nat (\lambda (_: nat).(eq nat i0 (clen c))) (\lambda (_: nat).(eq -K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c0 -(CSort n)))))))) H (CTail k u1 c) H6) in (eq_ind_r C (CTail k u1 c) (\lambda -(c0: C).(or (ex2 C (\lambda (e: C).(eq C c0 (CTail k u1 e))) (\lambda (e: -C).(getl O (CHead c (Bind b) u2) (CHead e (Bind b) u2)))) (ex4 nat (\lambda -(_: nat).(eq nat O (s (Bind b) (clen c)))) (\lambda (_: nat).(eq K k (Bind -b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c0 (CSort -n)))))) (or_introl (ex2 C (\lambda (e: C).(eq C (CTail k u1 c) (CTail k u1 -e))) (\lambda (e: C).(getl O (CHead c (Bind b) u2) (CHead e (Bind b) u2)))) -(ex4 nat (\lambda (_: nat).(eq nat O (s (Bind b) (clen c)))) (\lambda (_: -nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq -C (CTail k u1 c) (CSort n)))) (ex_intro2 C (\lambda (e: C).(eq C (CTail k u1 -c) (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Bind b) u2) (CHead e -(Bind b) u2))) c (refl_equal C (CTail k u1 c)) (getl_refl b c u2))) c2 H6)) -b0 H5) t H4)))) H3)) H2)))) (\lambda (f: F).(\lambda (H1: (clear (CHead -(CTail k u1 c) (Flat f) t) (CHead c2 (Bind b) u2))).(let H2 \def (H O -(getl_intro O (CTail k u1 c) (CHead c2 (Bind b) u2) (CTail k u1 c) (drop_refl -(CTail k u1 c)) (clear_gen_flat f (CTail k u1 c) (CHead c2 (Bind b) u2) t -H1))) in (or_ind (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda -(e: C).(getl O c (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat -O (clen c))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 -u2)) (\lambda (n: nat).(eq C c2 (CSort n)))) (or (ex2 C (\lambda (e: C).(eq C -c2 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e -(Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) -(\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda -(n: nat).(eq C c2 (CSort n))))) (\lambda (H3: (ex2 C (\lambda (e: C).(eq C c2 -(CTail k u1 e))) (\lambda (e: C).(getl O c (CHead e (Bind b) u2))))).(ex2_ind -C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl O c (CHead -e (Bind b) u2))) (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) -(\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind b) u2)))) (ex4 -nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq -K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 -(CSort n))))) (\lambda (x: C).(\lambda (H4: (eq C c2 (CTail k u1 -x))).(\lambda (H5: (getl O c (CHead x (Bind b) u2))).(eq_ind_r C (CTail k u1 -x) (\lambda (c0: C).(or (ex2 C (\lambda (e: C).(eq C c0 (CTail k u1 e))) -(\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind b) u2)))) (ex4 -nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq -K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c0 -(CSort n)))))) (or_introl (ex2 C (\lambda (e: C).(eq C (CTail k u1 x) (CTail -k u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind b) -u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) (\lambda -(_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: -nat).(eq C (CTail k u1 x) (CSort n)))) (ex_intro2 C (\lambda (e: C).(eq C -(CTail k u1 x) (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) -(CHead e (Bind b) u2))) x (refl_equal C (CTail k u1 x)) (getl_flat c (CHead x -(Bind b) u2) O H5 f t))) c2 H4)))) H3)) (\lambda (H3: (ex4 nat (\lambda (_: -nat).(eq nat O (clen c))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: -nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n))))).(ex4_ind nat -(\lambda (_: nat).(eq nat O (clen c))) (\lambda (_: nat).(eq K k (Bind b))) -(\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n))) (or -(ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl O -(CHead c (Flat f) t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq -nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda -(_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n))))) (\lambda (x0: -nat).(\lambda (H4: (eq nat O (clen c))).(\lambda (H5: (eq K k (Bind -b))).(\lambda (H6: (eq T u1 u2)).(\lambda (H7: (eq C c2 (CSort -x0))).(eq_ind_r C (CSort x0) (\lambda (c0: C).(or (ex2 C (\lambda (e: C).(eq -C c0 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e -(Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) -(\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda -(n: nat).(eq C c0 (CSort n)))))) (eq_ind T u1 (\lambda (t0: T).(or (ex2 C -(\lambda (e: C).(eq C (CSort x0) (CTail k u1 e))) (\lambda (e: C).(getl O -(CHead c (Flat f) t) (CHead e (Bind b) t0)))) (ex4 nat (\lambda (_: nat).(eq -nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda -(_: nat).(eq T u1 t0)) (\lambda (n: nat).(eq C (CSort x0) (CSort n)))))) -(eq_ind_r K (Bind b) (\lambda (k1: K).(or (ex2 C (\lambda (e: C).(eq C (CSort -x0) (CTail k1 u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e -(Bind b) u1)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) -(\lambda (_: nat).(eq K k1 (Bind b))) (\lambda (_: nat).(eq T u1 u1)) -(\lambda (n: nat).(eq C (CSort x0) (CSort n)))))) (or_intror (ex2 C (\lambda -(e: C).(eq C (CSort x0) (CTail (Bind b) u1 e))) (\lambda (e: C).(getl O -(CHead c (Flat f) t) (CHead e (Bind b) u1)))) (ex4 nat (\lambda (_: nat).(eq -nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq K (Bind b) (Bind b))) -(\lambda (_: nat).(eq T u1 u1)) (\lambda (n: nat).(eq C (CSort x0) (CSort -n)))) (ex4_intro nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) -(\lambda (_: nat).(eq K (Bind b) (Bind b))) (\lambda (_: nat).(eq T u1 u1)) -(\lambda (n: nat).(eq C (CSort x0) (CSort n))) x0 H4 (refl_equal K (Bind b)) -(refl_equal T u1) (refl_equal C (CSort x0)))) k H5) u2 H6) c2 H7)))))) H3)) -H2)))) k0 (getl_gen_O (CHead (CTail k u1 c) k0 t) (CHead c2 (Bind b) u2) -H0))) (\lambda (n: nat).(\lambda (H0: (((getl n (CHead (CTail k u1 c) k0 t) -(CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 -e))) (\lambda (e: C).(getl n (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat -(\lambda (_: nat).(eq nat n (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind -b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort -n0)))))))).(\lambda (H1: (getl (S n) (CHead (CTail k u1 c) k0 t) (CHead c2 -(Bind b) u2))).(let H_x \def (H (r k0 n) (getl_gen_S k0 (CTail k u1 c) (CHead -c2 (Bind b) u2) t n H1)) in (let H2 \def H_x in (or_ind (ex2 C (\lambda (e: -C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl (r k0 n) c (CHead e (Bind -b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat (r k0 n) (clen c))) (\lambda (_: -nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: -nat).(eq C c2 (CSort n0)))) (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 -e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 -nat (\lambda (_: nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K -k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 -(CSort n0))))) (\lambda (H3: (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) -(\lambda (e: C).(getl (r k0 n) c (CHead e (Bind b) u2))))).(ex2_ind C -(\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl (r k0 n) c -(CHead e (Bind b) u2))) (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) -(\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat -(\lambda (_: nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k -(Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort -n0))))) (\lambda (x: C).(\lambda (H4: (eq C c2 (CTail k u1 x))).(\lambda (H5: -(getl (r k0 n) c (CHead x (Bind b) u2))).(let H6 \def (eq_ind C c2 (\lambda -(c0: C).(getl (r k0 n) (CTail k u1 c) (CHead c0 (Bind b) u2))) (getl_gen_S k0 -(CTail k u1 c) (CHead c2 (Bind b) u2) t n H1) (CTail k u1 x) H4) in (let H7 -\def (eq_ind C c2 (\lambda (c0: C).((getl n (CHead (CTail k u1 c) k0 t) -(CHead c0 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c0 (CTail k u1 -e))) (\lambda (e: C).(getl n (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat -(\lambda (_: nat).(eq nat n (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind -b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c0 (CSort -n0))))))) H0 (CTail k u1 x) H4) in (eq_ind_r C (CTail k u1 x) (\lambda (c0: -C).(or (ex2 C (\lambda (e: C).(eq C c0 (CTail k u1 e))) (\lambda (e: C).(getl -(S n) (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq -nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: -nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c0 (CSort n0)))))) (or_introl -(ex2 C (\lambda (e: C).(eq C (CTail k u1 x) (CTail k u1 e))) (\lambda (e: -C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: -nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) -(\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C (CTail k u1 x) -(CSort n0)))) (ex_intro2 C (\lambda (e: C).(eq C (CTail k u1 x) (CTail k u1 -e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u2))) x -(refl_equal C (CTail k u1 x)) (getl_head k0 n c (CHead x (Bind b) u2) H5 t))) -c2 H4)))))) H3)) (\lambda (H3: (ex4 nat (\lambda (_: nat).(eq nat (r k0 n) -(clen c))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 -u2)) (\lambda (n0: nat).(eq C c2 (CSort n0))))).(ex4_ind nat (\lambda (_: -nat).(eq nat (r k0 n) (clen c))) (\lambda (_: nat).(eq K k (Bind b))) -(\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0))) (or -(ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl (S n) -(CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat (S -n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: -nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0))))) (\lambda (x0: -nat).(\lambda (H4: (eq nat (r k0 n) (clen c))).(\lambda (H5: (eq K k (Bind -b))).(\lambda (H6: (eq T u1 u2)).(\lambda (H7: (eq C c2 (CSort x0))).(let H8 -\def (eq_ind C c2 (\lambda (c0: C).(getl (r k0 n) (CTail k u1 c) (CHead c0 -(Bind b) u2))) (getl_gen_S k0 (CTail k u1 c) (CHead c2 (Bind b) u2) t n H1) -(CSort x0) H7) in (let H9 \def (eq_ind C c2 (\lambda (c0: C).((getl n (CHead -(CTail k u1 c) k0 t) (CHead c0 (Bind b) u2)) \to (or (ex2 C (\lambda (e: -C).(eq C c0 (CTail k u1 e))) (\lambda (e: C).(getl n (CHead c k0 t) (CHead e -(Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat n (s k0 (clen c)))) -(\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda -(n0: nat).(eq C c0 (CSort n0))))))) H0 (CSort x0) H7) in (eq_ind_r C (CSort -x0) (\lambda (c0: C).(or (ex2 C (\lambda (e: C).(eq C c0 (CTail k u1 e))) -(\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat -(\lambda (_: nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k -(Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c0 (CSort -n0)))))) (let H10 \def (eq_ind_r T u2 (\lambda (t0: T).((getl n (CHead (CTail -k u1 c) k0 t) (CHead (CSort x0) (Bind b) t0)) \to (or (ex2 C (\lambda (e: -C).(eq C (CSort x0) (CTail k u1 e))) (\lambda (e: C).(getl n (CHead c k0 t) -(CHead e (Bind b) t0)))) (ex4 nat (\lambda (_: nat).(eq nat n (s k0 (clen -c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 t0)) -(\lambda (n0: nat).(eq C (CSort x0) (CSort n0))))))) H9 u1 H6) in (let H11 -\def (eq_ind_r T u2 (\lambda (t0: T).(getl (r k0 n) (CTail k u1 c) (CHead -(CSort x0) (Bind b) t0))) H8 u1 H6) in (eq_ind T u1 (\lambda (t0: T).(or (ex2 -C (\lambda (e: C).(eq C (CSort x0) (CTail k u1 e))) (\lambda (e: C).(getl (S -n) (CHead c k0 t) (CHead e (Bind b) t0)))) (ex4 nat (\lambda (_: nat).(eq nat -(S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: -nat).(eq T u1 t0)) (\lambda (n0: nat).(eq C (CSort x0) (CSort n0)))))) (let -H12 \def (eq_ind K k (\lambda (k1: K).((getl n (CHead (CTail k1 u1 c) k0 t) -(CHead (CSort x0) (Bind b) u1)) \to (or (ex2 C (\lambda (e: C).(eq C (CSort -x0) (CTail k1 u1 e))) (\lambda (e: C).(getl n (CHead c k0 t) (CHead e (Bind -b) u1)))) (ex4 nat (\lambda (_: nat).(eq nat n (s k0 (clen c)))) (\lambda (_: -nat).(eq K k1 (Bind b))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: -nat).(eq C (CSort x0) (CSort n0))))))) H10 (Bind b) H5) in (let H13 \def -(eq_ind K k (\lambda (k1: K).(getl (r k0 n) (CTail k1 u1 c) (CHead (CSort x0) -(Bind b) u1))) H11 (Bind b) H5) in (eq_ind_r K (Bind b) (\lambda (k1: K).(or -(ex2 C (\lambda (e: C).(eq C (CSort x0) (CTail k1 u1 e))) (\lambda (e: -C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u1)))) (ex4 nat (\lambda (_: -nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k1 (Bind b))) -(\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: nat).(eq C (CSort x0) (CSort -n0)))))) (eq_ind nat (r k0 n) (\lambda (n0: nat).(or (ex2 C (\lambda (e: -C).(eq C (CSort x0) (CTail (Bind b) u1 e))) (\lambda (e: C).(getl (S n) -(CHead c k0 t) (CHead e (Bind b) u1)))) (ex4 nat (\lambda (_: nat).(eq nat (S -n) (s k0 n0))) (\lambda (_: nat).(eq K (Bind b) (Bind b))) (\lambda (_: -nat).(eq T u1 u1)) (\lambda (n1: nat).(eq C (CSort x0) (CSort n1)))))) -(eq_ind_r nat (S n) (\lambda (n0: nat).(or (ex2 C (\lambda (e: C).(eq C -(CSort x0) (CTail (Bind b) u1 e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) -(CHead e (Bind b) u1)))) (ex4 nat (\lambda (_: nat).(eq nat (S n) n0)) -(\lambda (_: nat).(eq K (Bind b) (Bind b))) (\lambda (_: nat).(eq T u1 u1)) -(\lambda (n1: nat).(eq C (CSort x0) (CSort n1)))))) (or_intror (ex2 C -(\lambda (e: C).(eq C (CSort x0) (CTail (Bind b) u1 e))) (\lambda (e: -C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u1)))) (ex4 nat (\lambda (_: -nat).(eq nat (S n) (S n))) (\lambda (_: nat).(eq K (Bind b) (Bind b))) -(\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: nat).(eq C (CSort x0) (CSort -n0)))) (ex4_intro nat (\lambda (_: nat).(eq nat (S n) (S n))) (\lambda (_: -nat).(eq K (Bind b) (Bind b))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: -nat).(eq C (CSort x0) (CSort n0))) x0 (refl_equal nat (S n)) (refl_equal K -(Bind b)) (refl_equal T u1) (refl_equal C (CSort x0)))) (s k0 (r k0 n)) (s_r -k0 n)) (clen c) H4) k H5))) u2 H6))) c2 H7)))))))) H3)) H2)))))) i)))))) -c1)))))). -(* COMMENTS -Initial nodes: 7489 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/cnt/defs.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/cnt/defs.ma deleted file mode 100644 index d6ab61d78..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/cnt/defs.ma +++ /dev/null @@ -1,23 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/T/defs.ma". - -inductive cnt: T \to Prop \def -| cnt_sort: \forall (n: nat).(cnt (TSort n)) -| cnt_head: \forall (t: T).((cnt t) \to (\forall (k: K).(\forall (v: T).(cnt -(THead k v t))))). - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/cnt/props.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/cnt/props.ma deleted file mode 100644 index 7fcd315eb..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/cnt/props.ma +++ /dev/null @@ -1,37 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/cnt/defs.ma". - -include "Basic-1/lift/fwd.ma". - -theorem cnt_lift: - \forall (t: T).((cnt t) \to (\forall (i: nat).(\forall (d: nat).(cnt (lift i -d t))))) -\def - \lambda (t: T).(\lambda (H: (cnt t)).(cnt_ind (\lambda (t0: T).(\forall (i: -nat).(\forall (d: nat).(cnt (lift i d t0))))) (\lambda (n: nat).(\lambda (i: -nat).(\lambda (d: nat).(eq_ind_r T (TSort n) (\lambda (t0: T).(cnt t0)) -(cnt_sort n) (lift i d (TSort n)) (lift_sort n i d))))) (\lambda (t0: -T).(\lambda (_: (cnt t0)).(\lambda (H1: ((\forall (i: nat).(\forall (d: -nat).(cnt (lift i d t0)))))).(\lambda (k: K).(\lambda (v: T).(\lambda (i: -nat).(\lambda (d: nat).(eq_ind_r T (THead k (lift i d v) (lift i (s k d) t0)) -(\lambda (t1: T).(cnt t1)) (cnt_head (lift i (s k d) t0) (H1 i (s k d)) k -(lift i d v)) (lift i d (THead k v t0)) (lift_head k v t0 i d))))))))) t H)). -(* COMMENTS -Initial nodes: 191 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/arity.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/arity.ma deleted file mode 100644 index fe021cd0f..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/arity.ma +++ /dev/null @@ -1,336 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csuba/getl.ma". - -include "Basic-1/csuba/props.ma". - -include "Basic-1/arity/props.ma". - -include "Basic-1/csubv/getl.ma". - -theorem csuba_arity: - \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1 -t a) \to (\forall (c2: C).((csuba g c1 c2) \to (arity g c2 t a))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: -(arity g c1 t a)).(arity_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda (a0: -A).(\forall (c2: C).((csuba g c c2) \to (arity g c2 t0 a0)))))) (\lambda (c: -C).(\lambda (n: nat).(\lambda (c2: C).(\lambda (_: (csuba g c -c2)).(arity_sort g c2 n))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: -T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d (Bind Abbr) -u))).(\lambda (a0: A).(\lambda (_: (arity g d u a0)).(\lambda (H2: ((\forall -(c2: C).((csuba g d c2) \to (arity g c2 u a0))))).(\lambda (c2: C).(\lambda -(H3: (csuba g c c2)).(let H4 \def (csuba_getl_abbr g c d u i H0 c2 H3) in -(ex2_ind C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda -(d2: C).(csuba g d d2)) (arity g c2 (TLRef i) a0) (\lambda (x: C).(\lambda -(H5: (getl i c2 (CHead x (Bind Abbr) u))).(\lambda (H6: (csuba g d -x)).(arity_abbr g c2 x u i H5 a0 (H2 x H6))))) H4)))))))))))) (\lambda (c: -C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c -(CHead d (Bind Abst) u))).(\lambda (a0: A).(\lambda (H1: (arity g d u (asucc -g a0))).(\lambda (H2: ((\forall (c2: C).((csuba g d c2) \to (arity g c2 u -(asucc g a0)))))).(\lambda (c2: C).(\lambda (H3: (csuba g c c2)).(let H4 \def -(csuba_getl_abst g c d u i H0 c2 H3) in (or_ind (ex2 C (\lambda (d2: C).(getl -i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d d2))) (ex4_3 C T -A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 -(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d u (asucc -g a1))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2 -a1))))) (arity g c2 (TLRef i) a0) (\lambda (H5: (ex2 C (\lambda (d2: C).(getl -i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d d2)))).(ex2_ind C -(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d d2)) (arity g c2 (TLRef i) a0) (\lambda (x: C).(\lambda (H6: -(getl i c2 (CHead x (Bind Abst) u))).(\lambda (H7: (csuba g d x)).(arity_abst -g c2 x u i H6 a0 (H2 x H7))))) H5)) (\lambda (H5: (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d d2)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d u (asucc g a1))))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2 -a1)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a1: A).(arity g d u (asucc g a1))))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a1: A).(arity g d2 u2 a1)))) (arity g c2 (TLRef i) a0) (\lambda -(x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H6: (getl i c2 (CHead x0 -(Bind Abbr) x1))).(\lambda (_: (csuba g d x0)).(\lambda (H8: (arity g d u -(asucc g x2))).(\lambda (H9: (arity g x0 x1 x2)).(arity_repl g c2 (TLRef i) -x2 (arity_abbr g c2 x0 x1 i H6 x2 H9) a0 (asucc_inj g x2 a0 (arity_mono g d u -(asucc g x2) H8 (asucc g a0) H1)))))))))) H5)) H4)))))))))))) (\lambda (b: -B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c: C).(\lambda (u: -T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H2: ((\forall -(c2: C).((csuba g c c2) \to (arity g c2 u a1))))).(\lambda (t0: T).(\lambda -(a2: A).(\lambda (_: (arity g (CHead c (Bind b) u) t0 a2)).(\lambda (H4: -((\forall (c2: C).((csuba g (CHead c (Bind b) u) c2) \to (arity g c2 t0 -a2))))).(\lambda (c2: C).(\lambda (H5: (csuba g c c2)).(arity_bind g b H0 c2 -u a1 (H2 c2 H5) t0 a2 (H4 (CHead c2 (Bind b) u) (csuba_head g c c2 H5 (Bind -b) u)))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a1: -A).(\lambda (_: (arity g c u (asucc g a1))).(\lambda (H1: ((\forall (c2: -C).((csuba g c c2) \to (arity g c2 u (asucc g a1)))))).(\lambda (t0: -T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c (Bind Abst) u) t0 -a2)).(\lambda (H3: ((\forall (c2: C).((csuba g (CHead c (Bind Abst) u) c2) -\to (arity g c2 t0 a2))))).(\lambda (c2: C).(\lambda (H4: (csuba g c -c2)).(arity_head g c2 u a1 (H1 c2 H4) t0 a2 (H3 (CHead c2 (Bind Abst) u) -(csuba_head g c c2 H4 (Bind Abst) u)))))))))))))) (\lambda (c: C).(\lambda -(u: T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H1: -((\forall (c2: C).((csuba g c c2) \to (arity g c2 u a1))))).(\lambda (t0: -T).(\lambda (a2: A).(\lambda (_: (arity g c t0 (AHead a1 a2))).(\lambda (H3: -((\forall (c2: C).((csuba g c c2) \to (arity g c2 t0 (AHead a1 -a2)))))).(\lambda (c2: C).(\lambda (H4: (csuba g c c2)).(arity_appl g c2 u a1 -(H1 c2 H4) t0 a2 (H3 c2 H4))))))))))))) (\lambda (c: C).(\lambda (u: -T).(\lambda (a0: A).(\lambda (_: (arity g c u (asucc g a0))).(\lambda (H1: -((\forall (c2: C).((csuba g c c2) \to (arity g c2 u (asucc g -a0)))))).(\lambda (t0: T).(\lambda (_: (arity g c t0 a0)).(\lambda (H3: -((\forall (c2: C).((csuba g c c2) \to (arity g c2 t0 a0))))).(\lambda (c2: -C).(\lambda (H4: (csuba g c c2)).(arity_cast g c2 u a0 (H1 c2 H4) t0 (H3 c2 -H4)))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda (_: -(arity g c t0 a1)).(\lambda (H1: ((\forall (c2: C).((csuba g c c2) \to (arity -g c2 t0 a1))))).(\lambda (a2: A).(\lambda (H2: (leq g a1 a2)).(\lambda (c2: -C).(\lambda (H3: (csuba g c c2)).(arity_repl g c2 t0 a1 (H1 c2 H3) a2 -H2)))))))))) c1 t a H))))). -(* COMMENTS -Initial nodes: 1505 -END *) - -theorem csuba_arity_rev: - \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1 -t a) \to (\forall (c2: C).((csuba g c2 c1) \to ((csubv c2 c1) \to (arity g c2 -t a)))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: -(arity g c1 t a)).(arity_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda (a0: -A).(\forall (c2: C).((csuba g c2 c) \to ((csubv c2 c) \to (arity g c2 t0 -a0))))))) (\lambda (c: C).(\lambda (n: nat).(\lambda (c2: C).(\lambda (_: -(csuba g c2 c)).(\lambda (_: (csubv c2 c)).(arity_sort g c2 n)))))) (\lambda -(c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl -i c (CHead d (Bind Abbr) u))).(\lambda (a0: A).(\lambda (H1: (arity g d u -a0)).(\lambda (H2: ((\forall (c2: C).((csuba g c2 d) \to ((csubv c2 d) \to -(arity g c2 u a0)))))).(\lambda (c2: C).(\lambda (H3: (csuba g c2 -c)).(\lambda (H4: (csubv c2 c)).(let H_x \def (csuba_getl_abbr_rev g c d u i -H0 c2 H3) in (let H5 \def H_x in (or3_ind (ex2 C (\lambda (d2: C).(getl i c2 -(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d2 d))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -d)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2 -(asucc g a1))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d -u a1))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d)))) (arity -g c2 (TLRef i) a0) (\lambda (H6: (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 -(Bind Abbr) u))) (\lambda (d2: C).(csuba g d2 d)))).(ex2_ind C (\lambda (d2: -C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d2 d)) -(arity g c2 (TLRef i) a0) (\lambda (x: C).(\lambda (H7: (getl i c2 (CHead x -(Bind Abbr) u))).(\lambda (H8: (csuba g x d)).(let H_x0 \def (csubv_getl_conf -c2 c H4 Abbr x u i H7) in (let H9 \def H_x0 in (ex2_3_ind B C T (\lambda (_: -B).(\lambda (d2: C).(\lambda (_: T).(csubv x d2)))) (\lambda (b2: B).(\lambda -(d2: C).(\lambda (v2: T).(getl i c (CHead d2 (Bind b2) v2))))) (arity g c2 -(TLRef i) a0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda -(H10: (csubv x x1)).(\lambda (H11: (getl i c (CHead x1 (Bind x0) x2))).(let -H12 \def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c0: C).(getl i c c0)) H0 -(CHead x1 (Bind x0) x2) (getl_mono c (CHead d (Bind Abbr) u) i H0 (CHead x1 -(Bind x0) x2) H11)) in (let H13 \def (f_equal C C (\lambda (e: C).(match e in -C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) -\Rightarrow c0])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) (getl_mono -c (CHead d (Bind Abbr) u) i H0 (CHead x1 (Bind x0) x2) H11)) in ((let H14 -\def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) -with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k in K -return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow -Abbr])])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) (getl_mono c (CHead -d (Bind Abbr) u) i H0 (CHead x1 (Bind x0) x2) H11)) in ((let H15 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind -Abbr) u) (CHead x1 (Bind x0) x2) (getl_mono c (CHead d (Bind Abbr) u) i H0 -(CHead x1 (Bind x0) x2) H11)) in (\lambda (H16: (eq B Abbr x0)).(\lambda -(H17: (eq C d x1)).(let H18 \def (eq_ind_r T x2 (\lambda (t0: T).(getl i c -(CHead x1 (Bind x0) t0))) H12 u H15) in (let H19 \def (eq_ind_r C x1 (\lambda -(c0: C).(getl i c (CHead c0 (Bind x0) u))) H18 d H17) in (let H20 \def -(eq_ind_r C x1 (\lambda (c0: C).(csubv x c0)) H10 d H17) in (let H21 \def -(eq_ind_r B x0 (\lambda (b: B).(getl i c (CHead d (Bind b) u))) H19 Abbr H16) -in (arity_abbr g c2 x u i H7 a0 (H2 x H8 H20))))))))) H14)) H13)))))))) -H9)))))) H6)) (\lambda (H6: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2 (asucc g a1))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d u a1)))))).(ex4_3_ind C T -A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 -(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d2 d)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2 -(asucc g a1))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d -u a1)))) (arity g c2 (TLRef i) a0) (\lambda (x0: C).(\lambda (x1: T).(\lambda -(x2: A).(\lambda (H7: (getl i c2 (CHead x0 (Bind Abst) x1))).(\lambda (_: -(csuba g x0 d)).(\lambda (H9: (arity g x0 x1 (asucc g x2))).(\lambda (H10: -(arity g d u x2)).(arity_repl g c2 (TLRef i) x2 (arity_abst g c2 x0 x1 i H7 -x2 H9) a0 (arity_mono g d u x2 H10 a0 H1))))))))) H6)) (\lambda (H6: (ex2_2 C -T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d))))).(ex2_2_ind C T (\lambda -(d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d))) (arity g c2 (TLRef i) a0) (\lambda -(x0: C).(\lambda (x1: T).(\lambda (H7: (getl i c2 (CHead x0 (Bind Void) -x1))).(\lambda (_: (csuba g x0 d)).(let H_x0 \def (csubv_getl_conf_void c2 c -H4 x0 x1 i H7) in (let H9 \def H_x0 in (ex2_2_ind C T (\lambda (d2: -C).(\lambda (_: T).(csubv x0 d2))) (\lambda (d2: C).(\lambda (v2: T).(getl i -c (CHead d2 (Bind Void) v2)))) (arity g c2 (TLRef i) a0) (\lambda (x2: -C).(\lambda (x3: T).(\lambda (_: (csubv x0 x2)).(\lambda (H11: (getl i c -(CHead x2 (Bind Void) x3))).(let H12 \def (eq_ind C (CHead d (Bind Abbr) u) -(\lambda (c0: C).(getl i c c0)) H0 (CHead x2 (Bind Void) x3) (getl_mono c -(CHead d (Bind Abbr) u) i H0 (CHead x2 (Bind Void) x3) H11)) in (let H13 \def -(eq_ind C (CHead d (Bind Abbr) u) (\lambda (ee: C).(match ee in C return -(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b) -\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow -True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _) -\Rightarrow False])])) I (CHead x2 (Bind Void) x3) (getl_mono c (CHead d -(Bind Abbr) u) i H0 (CHead x2 (Bind Void) x3) H11)) in (False_ind (arity g c2 -(TLRef i) a0) H13))))))) H9))))))) H6)) H5)))))))))))))) (\lambda (c: -C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c -(CHead d (Bind Abst) u))).(\lambda (a0: A).(\lambda (_: (arity g d u (asucc g -a0))).(\lambda (H2: ((\forall (c2: C).((csuba g c2 d) \to ((csubv c2 d) \to -(arity g c2 u (asucc g a0))))))).(\lambda (c2: C).(\lambda (H3: (csuba g c2 -c)).(\lambda (H4: (csubv c2 c)).(let H_x \def (csuba_getl_abst_rev g c d u i -H0 c2 H3) in (let H5 \def H_x in (or_ind (ex2 C (\lambda (d2: C).(getl i c2 -(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d)))) (arity g c2 (TLRef i) a0) -(\lambda (H6: (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d)))).(ex2_ind C (\lambda (d2: C).(getl i c2 -(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d)) (arity g c2 -(TLRef i) a0) (\lambda (x: C).(\lambda (H7: (getl i c2 (CHead x (Bind Abst) -u))).(\lambda (H8: (csuba g x d)).(let H_x0 \def (csubv_getl_conf c2 c H4 -Abst x u i H7) in (let H9 \def H_x0 in (ex2_3_ind B C T (\lambda (_: -B).(\lambda (d2: C).(\lambda (_: T).(csubv x d2)))) (\lambda (b2: B).(\lambda -(d2: C).(\lambda (v2: T).(getl i c (CHead d2 (Bind b2) v2))))) (arity g c2 -(TLRef i) a0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda -(H10: (csubv x x1)).(\lambda (H11: (getl i c (CHead x1 (Bind x0) x2))).(let -H12 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (c0: C).(getl i c c0)) H0 -(CHead x1 (Bind x0) x2) (getl_mono c (CHead d (Bind Abst) u) i H0 (CHead x1 -(Bind x0) x2) H11)) in (let H13 \def (f_equal C C (\lambda (e: C).(match e in -C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) -\Rightarrow c0])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x2) (getl_mono -c (CHead d (Bind Abst) u) i H0 (CHead x1 (Bind x0) x2) H11)) in ((let H14 -\def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) -with [(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match k in K -return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow -Abst])])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x2) (getl_mono c (CHead -d (Bind Abst) u) i H0 (CHead x1 (Bind x0) x2) H11)) in ((let H15 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind -Abst) u) (CHead x1 (Bind x0) x2) (getl_mono c (CHead d (Bind Abst) u) i H0 -(CHead x1 (Bind x0) x2) H11)) in (\lambda (H16: (eq B Abst x0)).(\lambda -(H17: (eq C d x1)).(let H18 \def (eq_ind_r T x2 (\lambda (t0: T).(getl i c -(CHead x1 (Bind x0) t0))) H12 u H15) in (let H19 \def (eq_ind_r C x1 (\lambda -(c0: C).(getl i c (CHead c0 (Bind x0) u))) H18 d H17) in (let H20 \def -(eq_ind_r C x1 (\lambda (c0: C).(csubv x c0)) H10 d H17) in (let H21 \def -(eq_ind_r B x0 (\lambda (b: B).(getl i c (CHead d (Bind b) u))) H19 Abst H16) -in (arity_abst g c2 x u i H7 a0 (H2 x H8 H20))))))))) H14)) H13)))))))) -H9)))))) H6)) (\lambda (H6: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(getl -i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g -d2 d))) (arity g c2 (TLRef i) a0) (\lambda (x0: C).(\lambda (x1: T).(\lambda -(H7: (getl i c2 (CHead x0 (Bind Void) x1))).(\lambda (_: (csuba g x0 d)).(let -H_x0 \def (csubv_getl_conf_void c2 c H4 x0 x1 i H7) in (let H9 \def H_x0 in -(ex2_2_ind C T (\lambda (d2: C).(\lambda (_: T).(csubv x0 d2))) (\lambda (d2: -C).(\lambda (v2: T).(getl i c (CHead d2 (Bind Void) v2)))) (arity g c2 (TLRef -i) a0) (\lambda (x2: C).(\lambda (x3: T).(\lambda (_: (csubv x0 x2)).(\lambda -(H11: (getl i c (CHead x2 (Bind Void) x3))).(let H12 \def (eq_ind C (CHead d -(Bind Abst) u) (\lambda (c0: C).(getl i c c0)) H0 (CHead x2 (Bind Void) x3) -(getl_mono c (CHead d (Bind Abst) u) i H0 (CHead x2 (Bind Void) x3) H11)) in -(let H13 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (ee: C).(match ee in -C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k -_) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b) -\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow -False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) -\Rightarrow False])])) I (CHead x2 (Bind Void) x3) (getl_mono c (CHead d -(Bind Abst) u) i H0 (CHead x2 (Bind Void) x3) H11)) in (False_ind (arity g c2 -(TLRef i) a0) H13))))))) H9))))))) H6)) H5)))))))))))))) (\lambda (b: -B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c: C).(\lambda (u: -T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H2: ((\forall -(c2: C).((csuba g c2 c) \to ((csubv c2 c) \to (arity g c2 u a1)))))).(\lambda -(t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c (Bind b) u) t0 -a2)).(\lambda (H4: ((\forall (c2: C).((csuba g c2 (CHead c (Bind b) u)) \to -((csubv c2 (CHead c (Bind b) u)) \to (arity g c2 t0 a2)))))).(\lambda (c2: -C).(\lambda (H5: (csuba g c2 c)).(\lambda (H6: (csubv c2 c)).(arity_bind g b -H0 c2 u a1 (H2 c2 H5 H6) t0 a2 (H4 (CHead c2 (Bind b) u) (csuba_head g c2 c -H5 (Bind b) u) (csubv_bind_same c2 c H6 b u u))))))))))))))))) (\lambda (c: -C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c u (asucc g -a1))).(\lambda (H1: ((\forall (c2: C).((csuba g c2 c) \to ((csubv c2 c) \to -(arity g c2 u (asucc g a1))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda -(_: (arity g (CHead c (Bind Abst) u) t0 a2)).(\lambda (H3: ((\forall (c2: -C).((csuba g c2 (CHead c (Bind Abst) u)) \to ((csubv c2 (CHead c (Bind Abst) -u)) \to (arity g c2 t0 a2)))))).(\lambda (c2: C).(\lambda (H4: (csuba g c2 -c)).(\lambda (H5: (csubv c2 c)).(arity_head g c2 u a1 (H1 c2 H4 H5) t0 a2 (H3 -(CHead c2 (Bind Abst) u) (csuba_head g c2 c H4 (Bind Abst) u) -(csubv_bind_same c2 c H5 Abst u u))))))))))))))) (\lambda (c: C).(\lambda (u: -T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H1: ((\forall -(c2: C).((csuba g c2 c) \to ((csubv c2 c) \to (arity g c2 u a1)))))).(\lambda -(t0: T).(\lambda (a2: A).(\lambda (_: (arity g c t0 (AHead a1 a2))).(\lambda -(H3: ((\forall (c2: C).((csuba g c2 c) \to ((csubv c2 c) \to (arity g c2 t0 -(AHead a1 a2))))))).(\lambda (c2: C).(\lambda (H4: (csuba g c2 c)).(\lambda -(H5: (csubv c2 c)).(arity_appl g c2 u a1 (H1 c2 H4 H5) t0 a2 (H3 c2 H4 -H5)))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a0: A).(\lambda -(_: (arity g c u (asucc g a0))).(\lambda (H1: ((\forall (c2: C).((csuba g c2 -c) \to ((csubv c2 c) \to (arity g c2 u (asucc g a0))))))).(\lambda (t0: -T).(\lambda (_: (arity g c t0 a0)).(\lambda (H3: ((\forall (c2: C).((csuba g -c2 c) \to ((csubv c2 c) \to (arity g c2 t0 a0)))))).(\lambda (c2: C).(\lambda -(H4: (csuba g c2 c)).(\lambda (H5: (csubv c2 c)).(arity_cast g c2 u a0 (H1 c2 -H4 H5) t0 (H3 c2 H4 H5))))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda -(a1: A).(\lambda (_: (arity g c t0 a1)).(\lambda (H1: ((\forall (c2: -C).((csuba g c2 c) \to ((csubv c2 c) \to (arity g c2 t0 a1)))))).(\lambda -(a2: A).(\lambda (H2: (leq g a1 a2)).(\lambda (c2: C).(\lambda (H3: (csuba g -c2 c)).(\lambda (H4: (csubv c2 c)).(arity_repl g c2 t0 a1 (H1 c2 H3 H4) a2 -H2))))))))))) c1 t a H))))). -(* COMMENTS -Initial nodes: 3597 -END *) - -theorem arity_appls_appl: - \forall (g: G).(\forall (c: C).(\forall (v: T).(\forall (a1: A).((arity g c -v a1) \to (\forall (u: T).((arity g c u (asucc g a1)) \to (\forall (t: -T).(\forall (vs: TList).(\forall (a2: A).((arity g c (THeads (Flat Appl) vs -(THead (Bind Abbr) v t)) a2) \to (arity g c (THeads (Flat Appl) vs (THead -(Flat Appl) v (THead (Bind Abst) u t))) a2))))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (v: T).(\lambda (a1: A).(\lambda (H: -(arity g c v a1)).(\lambda (u: T).(\lambda (H0: (arity g c u (asucc g -a1))).(\lambda (t: T).(\lambda (vs: TList).(TList_ind (\lambda (t0: -TList).(\forall (a2: A).((arity g c (THeads (Flat Appl) t0 (THead (Bind Abbr) -v t)) a2) \to (arity g c (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead -(Bind Abst) u t))) a2)))) (\lambda (a2: A).(\lambda (H1: (arity g c (THead -(Bind Abbr) v t) a2)).(let H_x \def (arity_gen_bind Abbr (\lambda (H2: (eq B -Abbr Abst)).(not_abbr_abst H2)) g c v t a2 H1) in (let H2 \def H_x in -(ex2_ind A (\lambda (a3: A).(arity g c v a3)) (\lambda (_: A).(arity g (CHead -c (Bind Abbr) v) t a2)) (arity g c (THead (Flat Appl) v (THead (Bind Abst) u -t)) a2) (\lambda (x: A).(\lambda (_: (arity g c v x)).(\lambda (H4: (arity g -(CHead c (Bind Abbr) v) t a2)).(arity_appl g c v a1 H (THead (Bind Abst) u t) -a2 (arity_head g c u a1 H0 t a2 (csuba_arity_rev g (CHead c (Bind Abbr) v) t -a2 H4 (CHead c (Bind Abst) u) (csuba_abst g c c (csuba_refl g c) u a1 H0 v H) -(csubv_bind c c (csubv_refl c) Abst (sym_not_eq B Void Abst not_void_abst) -Abbr u v))))))) H2))))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H1: -((\forall (a2: A).((arity g c (THeads (Flat Appl) t1 (THead (Bind Abbr) v t)) -a2) \to (arity g c (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind -Abst) u t))) a2))))).(\lambda (a2: A).(\lambda (H2: (arity g c (THead (Flat -Appl) t0 (THeads (Flat Appl) t1 (THead (Bind Abbr) v t))) a2)).(let H3 \def -(arity_gen_appl g c t0 (THeads (Flat Appl) t1 (THead (Bind Abbr) v t)) a2 H2) -in (ex2_ind A (\lambda (a3: A).(arity g c t0 a3)) (\lambda (a3: A).(arity g c -(THeads (Flat Appl) t1 (THead (Bind Abbr) v t)) (AHead a3 a2))) (arity g c -(THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead -(Bind Abst) u t)))) a2) (\lambda (x: A).(\lambda (H4: (arity g c t0 -x)).(\lambda (H5: (arity g c (THeads (Flat Appl) t1 (THead (Bind Abbr) v t)) -(AHead x a2))).(arity_appl g c t0 x H4 (THeads (Flat Appl) t1 (THead (Flat -Appl) v (THead (Bind Abst) u t))) a2 (H1 (AHead x a2) H5))))) H3))))))) -vs))))))))). -(* COMMENTS -Initial nodes: 687 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/clear.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/clear.ma deleted file mode 100644 index 2350fd1e8..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/clear.ma +++ /dev/null @@ -1,128 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csuba/defs.ma". - -include "Basic-1/clear/fwd.ma". - -theorem csuba_clear_conf: - \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csuba g c1 c2) \to -(\forall (e1: C).((clear c1 e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) -(\lambda (e2: C).(clear c2 e2)))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csuba g c1 -c2)).(csuba_ind g (\lambda (c: C).(\lambda (c0: C).(\forall (e1: C).((clear c -e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2: C).(clear c0 -e2))))))) (\lambda (n: nat).(\lambda (e1: C).(\lambda (H0: (clear (CSort n) -e1)).(clear_gen_sort e1 n H0 (ex2 C (\lambda (e2: C).(csuba g e1 e2)) -(\lambda (e2: C).(clear (CSort n) e2))))))) (\lambda (c3: C).(\lambda (c4: -C).(\lambda (H0: (csuba g c3 c4)).(\lambda (H1: ((\forall (e1: C).((clear c3 -e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2: C).(clear c4 -e2))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (e1: C).(\lambda (H2: -(clear (CHead c3 k u) e1)).(K_ind (\lambda (k0: K).((clear (CHead c3 k0 u) -e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2: C).(clear -(CHead c4 k0 u) e2))))) (\lambda (b: B).(\lambda (H3: (clear (CHead c3 (Bind -b) u) e1)).(eq_ind_r C (CHead c3 (Bind b) u) (\lambda (c: C).(ex2 C (\lambda -(e2: C).(csuba g c e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b) u) e2)))) -(ex_intro2 C (\lambda (e2: C).(csuba g (CHead c3 (Bind b) u) e2)) (\lambda -(e2: C).(clear (CHead c4 (Bind b) u) e2)) (CHead c4 (Bind b) u) (csuba_head g -c3 c4 H0 (Bind b) u) (clear_bind b c4 u)) e1 (clear_gen_bind b c3 e1 u H3)))) -(\lambda (f: F).(\lambda (H3: (clear (CHead c3 (Flat f) u) e1)).(let H4 \def -(H1 e1 (clear_gen_flat f c3 e1 u H3)) in (ex2_ind C (\lambda (e2: C).(csuba g -e1 e2)) (\lambda (e2: C).(clear c4 e2)) (ex2 C (\lambda (e2: C).(csuba g e1 -e2)) (\lambda (e2: C).(clear (CHead c4 (Flat f) u) e2))) (\lambda (x: -C).(\lambda (H5: (csuba g e1 x)).(\lambda (H6: (clear c4 x)).(ex_intro2 C -(\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2: C).(clear (CHead c4 (Flat f) -u) e2)) x H5 (clear_flat c4 x H6 f u))))) H4)))) k H2))))))))) (\lambda (c3: -C).(\lambda (c4: C).(\lambda (H0: (csuba g c3 c4)).(\lambda (_: ((\forall -(e1: C).((clear c3 e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) (\lambda -(e2: C).(clear c4 e2))))))).(\lambda (b: B).(\lambda (H2: (not (eq B b -Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (e1: C).(\lambda (H3: -(clear (CHead c3 (Bind Void) u1) e1)).(eq_ind_r C (CHead c3 (Bind Void) u1) -(\lambda (c: C).(ex2 C (\lambda (e2: C).(csuba g c e2)) (\lambda (e2: -C).(clear (CHead c4 (Bind b) u2) e2)))) (ex_intro2 C (\lambda (e2: C).(csuba -g (CHead c3 (Bind Void) u1) e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b) -u2) e2)) (CHead c4 (Bind b) u2) (csuba_void g c3 c4 H0 b H2 u1 u2) -(clear_bind b c4 u2)) e1 (clear_gen_bind Void c3 e1 u1 H3)))))))))))) -(\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csuba g c3 c4)).(\lambda (_: -((\forall (e1: C).((clear c3 e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) -(\lambda (e2: C).(clear c4 e2))))))).(\lambda (t: T).(\lambda (a: A).(\lambda -(H2: (arity g c3 t (asucc g a))).(\lambda (u: T).(\lambda (H3: (arity g c4 u -a)).(\lambda (e1: C).(\lambda (H4: (clear (CHead c3 (Bind Abst) t) -e1)).(eq_ind_r C (CHead c3 (Bind Abst) t) (\lambda (c: C).(ex2 C (\lambda -(e2: C).(csuba g c e2)) (\lambda (e2: C).(clear (CHead c4 (Bind Abbr) u) -e2)))) (ex_intro2 C (\lambda (e2: C).(csuba g (CHead c3 (Bind Abst) t) e2)) -(\lambda (e2: C).(clear (CHead c4 (Bind Abbr) u) e2)) (CHead c4 (Bind Abbr) -u) (csuba_abst g c3 c4 H0 t a H2 u H3) (clear_bind Abbr c4 u)) e1 -(clear_gen_bind Abst c3 e1 t H4))))))))))))) c1 c2 H)))). -(* COMMENTS -Initial nodes: 937 -END *) - -theorem csuba_clear_trans: - \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csuba g c2 c1) \to -(\forall (e1: C).((clear c1 e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) -(\lambda (e2: C).(clear c2 e2)))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csuba g c2 -c1)).(csuba_ind g (\lambda (c: C).(\lambda (c0: C).(\forall (e1: C).((clear -c0 e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) (\lambda (e2: C).(clear c -e2))))))) (\lambda (n: nat).(\lambda (e1: C).(\lambda (H0: (clear (CSort n) -e1)).(clear_gen_sort e1 n H0 (ex2 C (\lambda (e2: C).(csuba g e2 e1)) -(\lambda (e2: C).(clear (CSort n) e2))))))) (\lambda (c3: C).(\lambda (c4: -C).(\lambda (H0: (csuba g c3 c4)).(\lambda (H1: ((\forall (e1: C).((clear c4 -e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) (\lambda (e2: C).(clear c3 -e2))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (e1: C).(\lambda (H2: -(clear (CHead c4 k u) e1)).(K_ind (\lambda (k0: K).((clear (CHead c4 k0 u) -e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) (\lambda (e2: C).(clear -(CHead c3 k0 u) e2))))) (\lambda (b: B).(\lambda (H3: (clear (CHead c4 (Bind -b) u) e1)).(eq_ind_r C (CHead c4 (Bind b) u) (\lambda (c: C).(ex2 C (\lambda -(e2: C).(csuba g e2 c)) (\lambda (e2: C).(clear (CHead c3 (Bind b) u) e2)))) -(ex_intro2 C (\lambda (e2: C).(csuba g e2 (CHead c4 (Bind b) u))) (\lambda -(e2: C).(clear (CHead c3 (Bind b) u) e2)) (CHead c3 (Bind b) u) (csuba_head g -c3 c4 H0 (Bind b) u) (clear_bind b c3 u)) e1 (clear_gen_bind b c4 e1 u H3)))) -(\lambda (f: F).(\lambda (H3: (clear (CHead c4 (Flat f) u) e1)).(let H4 \def -(H1 e1 (clear_gen_flat f c4 e1 u H3)) in (ex2_ind C (\lambda (e2: C).(csuba g -e2 e1)) (\lambda (e2: C).(clear c3 e2)) (ex2 C (\lambda (e2: C).(csuba g e2 -e1)) (\lambda (e2: C).(clear (CHead c3 (Flat f) u) e2))) (\lambda (x: -C).(\lambda (H5: (csuba g x e1)).(\lambda (H6: (clear c3 x)).(ex_intro2 C -(\lambda (e2: C).(csuba g e2 e1)) (\lambda (e2: C).(clear (CHead c3 (Flat f) -u) e2)) x H5 (clear_flat c3 x H6 f u))))) H4)))) k H2))))))))) (\lambda (c3: -C).(\lambda (c4: C).(\lambda (H0: (csuba g c3 c4)).(\lambda (_: ((\forall -(e1: C).((clear c4 e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) (\lambda -(e2: C).(clear c3 e2))))))).(\lambda (b: B).(\lambda (H2: (not (eq B b -Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (e1: C).(\lambda (H3: -(clear (CHead c4 (Bind b) u2) e1)).(eq_ind_r C (CHead c4 (Bind b) u2) -(\lambda (c: C).(ex2 C (\lambda (e2: C).(csuba g e2 c)) (\lambda (e2: -C).(clear (CHead c3 (Bind Void) u1) e2)))) (ex_intro2 C (\lambda (e2: -C).(csuba g e2 (CHead c4 (Bind b) u2))) (\lambda (e2: C).(clear (CHead c3 -(Bind Void) u1) e2)) (CHead c3 (Bind Void) u1) (csuba_void g c3 c4 H0 b H2 u1 -u2) (clear_bind Void c3 u1)) e1 (clear_gen_bind b c4 e1 u2 H3)))))))))))) -(\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csuba g c3 c4)).(\lambda (_: -((\forall (e1: C).((clear c4 e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) -(\lambda (e2: C).(clear c3 e2))))))).(\lambda (t: T).(\lambda (a: A).(\lambda -(H2: (arity g c3 t (asucc g a))).(\lambda (u: T).(\lambda (H3: (arity g c4 u -a)).(\lambda (e1: C).(\lambda (H4: (clear (CHead c4 (Bind Abbr) u) -e1)).(eq_ind_r C (CHead c4 (Bind Abbr) u) (\lambda (c: C).(ex2 C (\lambda -(e2: C).(csuba g e2 c)) (\lambda (e2: C).(clear (CHead c3 (Bind Abst) t) -e2)))) (ex_intro2 C (\lambda (e2: C).(csuba g e2 (CHead c4 (Bind Abbr) u))) -(\lambda (e2: C).(clear (CHead c3 (Bind Abst) t) e2)) (CHead c3 (Bind Abst) -t) (csuba_abst g c3 c4 H0 t a H2 u H3) (clear_bind Abst c3 t)) e1 -(clear_gen_bind Abbr c4 e1 u H4))))))))))))) c2 c1 H)))). -(* COMMENTS -Initial nodes: 937 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/defs.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/defs.ma deleted file mode 100644 index cc6a46cd8..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/defs.ma +++ /dev/null @@ -1,30 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/arity/defs.ma". - -inductive csuba (g: G): C \to (C \to Prop) \def -| csuba_sort: \forall (n: nat).(csuba g (CSort n) (CSort n)) -| csuba_head: \forall (c1: C).(\forall (c2: C).((csuba g c1 c2) \to (\forall -(k: K).(\forall (u: T).(csuba g (CHead c1 k u) (CHead c2 k u)))))) -| csuba_void: \forall (c1: C).(\forall (c2: C).((csuba g c1 c2) \to (\forall -(b: B).((not (eq B b Void)) \to (\forall (u1: T).(\forall (u2: T).(csuba g -(CHead c1 (Bind Void) u1) (CHead c2 (Bind b) u2)))))))) -| csuba_abst: \forall (c1: C).(\forall (c2: C).((csuba g c1 c2) \to (\forall -(t: T).(\forall (a: A).((arity g c1 t (asucc g a)) \to (\forall (u: -T).((arity g c2 u a) \to (csuba g (CHead c1 (Bind Abst) t) (CHead c2 (Bind -Abbr) u))))))))). - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/drop.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/drop.ma deleted file mode 100644 index 1047ac13d..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/drop.ma +++ /dev/null @@ -1,2468 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csuba/fwd.ma". - -include "Basic-1/drop/fwd.ma". - -theorem csuba_drop_abbr: - \forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).((drop i -O c1 (CHead d1 (Bind Abbr) u)) \to (\forall (g: G).(\forall (c2: C).((csuba g -c1 c2) \to (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr) u))) -(\lambda (d2: C).(csuba g d1 d2)))))))))) -\def - \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (d1: -C).(\forall (u: T).((drop n O c1 (CHead d1 (Bind Abbr) u)) \to (\forall (g: -G).(\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(drop n O c2 -(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))))))))) -(\lambda (c1: C).(\lambda (d1: C).(\lambda (u: T).(\lambda (H: (drop O O c1 -(CHead d1 (Bind Abbr) u))).(\lambda (g: G).(\lambda (c2: C).(\lambda (H0: -(csuba g c1 c2)).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csuba g c c2)) H0 -(CHead d1 (Bind Abbr) u) (drop_gen_refl c1 (CHead d1 (Bind Abbr) u) H)) in -(let H_x \def (csuba_gen_abbr g d1 c2 u H1) in (let H2 \def H_x in (ex2_ind C -(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba -g d1 d2)) (ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abbr) u))) -(\lambda (d2: C).(csuba g d1 d2))) (\lambda (x: C).(\lambda (H3: (eq C c2 -(CHead x (Bind Abbr) u))).(\lambda (H4: (csuba g d1 x)).(eq_ind_r C (CHead x -(Bind Abbr) u) (\lambda (c: C).(ex2 C (\lambda (d2: C).(drop O O c (CHead d2 -(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (ex_intro2 C (\lambda -(d2: C).(drop O O (CHead x (Bind Abbr) u) (CHead d2 (Bind Abbr) u))) (\lambda -(d2: C).(csuba g d1 d2)) x (drop_refl (CHead x (Bind Abbr) u)) H4) c2 H3)))) -H2))))))))))) (\lambda (n: nat).(\lambda (H: ((\forall (c1: C).(\forall (d1: -C).(\forall (u: T).((drop n O c1 (CHead d1 (Bind Abbr) u)) \to (\forall (g: -G).(\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(drop n O c2 -(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 -d2)))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (d1: -C).(\forall (u: T).((drop (S n) O c (CHead d1 (Bind Abbr) u)) \to (\forall -(g: G).(\forall (c2: C).((csuba g c c2) \to (ex2 C (\lambda (d2: C).(drop (S -n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))))))) -(\lambda (n0: nat).(\lambda (d1: C).(\lambda (u: T).(\lambda (H0: (drop (S n) -O (CSort n0) (CHead d1 (Bind Abbr) u))).(\lambda (g: G).(\lambda (c2: -C).(\lambda (_: (csuba g (CSort n0) c2)).(and3_ind (eq C (CHead d1 (Bind -Abbr) u) (CSort n0)) (eq nat (S n) O) (eq nat O O) (ex2 C (\lambda (d2: -C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 -d2))) (\lambda (_: (eq C (CHead d1 (Bind Abbr) u) (CSort n0))).(\lambda (H3: -(eq nat (S n) O)).(\lambda (_: (eq nat O O)).(let H5 \def (eq_ind nat (S n) -(\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O -\Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind (ex2 C -(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: -C).(csuba g d1 d2))) H5))))) (drop_gen_sort n0 (S n) O (CHead d1 (Bind Abbr) -u) H0))))))))) (\lambda (c: C).(\lambda (H0: ((\forall (d1: C).(\forall (u: -T).((drop (S n) O c (CHead d1 (Bind Abbr) u)) \to (\forall (g: G).(\forall -(c2: C).((csuba g c c2) \to (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead -d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))))))))).(\lambda (k: -K).(\lambda (t: T).(\lambda (d1: C).(\lambda (u: T).(\lambda (H1: (drop (S n) -O (CHead c k t) (CHead d1 (Bind Abbr) u))).(\lambda (g: G).(\lambda (c2: -C).(\lambda (H2: (csuba g (CHead c k t) c2)).(K_ind (\lambda (k0: K).((csuba -g (CHead c k0 t) c2) \to ((drop (r k0 n) O c (CHead d1 (Bind Abbr) u)) \to -(ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda -(d2: C).(csuba g d1 d2)))))) (\lambda (b: B).(\lambda (H3: (csuba g (CHead c -(Bind b) t) c2)).(\lambda (H4: (drop (r (Bind b) n) O c (CHead d1 (Bind Abbr) -u))).(B_ind (\lambda (b0: B).((csuba g (CHead c (Bind b0) t) c2) \to ((drop -(r (Bind b0) n) O c (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: -C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 -d2)))))) (\lambda (H5: (csuba g (CHead c (Bind Abbr) t) c2)).(\lambda (H6: -(drop (r (Bind Abbr) n) O c (CHead d1 (Bind Abbr) u))).(let H_x \def -(csuba_gen_abbr g c c2 t H5) in (let H7 \def H_x in (ex2_ind C (\lambda (d2: -C).(eq C c2 (CHead d2 (Bind Abbr) t))) (\lambda (d2: C).(csuba g c d2)) (ex2 -C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: -C).(csuba g d1 d2))) (\lambda (x: C).(\lambda (H8: (eq C c2 (CHead x (Bind -Abbr) t))).(\lambda (H9: (csuba g c x)).(eq_ind_r C (CHead x (Bind Abbr) t) -(\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind -Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H10 \def (H c d1 u H6 g x -H9) in (ex2_ind C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u))) -(\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(drop (S n) O -(CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g -d1 d2))) (\lambda (x0: C).(\lambda (H11: (drop n O x (CHead x0 (Bind Abbr) -u))).(\lambda (H12: (csuba g d1 x0)).(let H13 \def (refl_equal nat (r (Bind -Abbr) n)) in (let H14 \def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x -(CHead x0 (Bind Abbr) u))) H11 (r (Bind Abbr) n) H13) in (ex_intro2 C -(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) -u))) (\lambda (d2: C).(csuba g d1 d2)) x0 (drop_drop (Bind Abbr) n x (CHead -x0 (Bind Abbr) u) H14 t) H12)))))) H10)) c2 H8)))) H7))))) (\lambda (H5: -(csuba g (CHead c (Bind Abst) t) c2)).(\lambda (H6: (drop (r (Bind Abst) n) O -c (CHead d1 (Bind Abbr) u))).(let H_x \def (csuba_gen_abst g c c2 t H5) in -(let H7 \def H_x in (or_ind (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind -Abst) t))) (\lambda (d2: C).(csuba g c d2))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g c d2)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex2 C -(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: -C).(csuba g d1 d2))) (\lambda (H8: (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 -(Bind Abst) t))) (\lambda (d2: C).(csuba g c d2)))).(ex2_ind C (\lambda (d2: -C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba g c d2)) (ex2 -C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: -C).(csuba g d1 d2))) (\lambda (x: C).(\lambda (H9: (eq C c2 (CHead x (Bind -Abst) t))).(\lambda (H10: (csuba g c x)).(eq_ind_r C (CHead x (Bind Abst) t) -(\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind -Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H11 \def (H c d1 u H6 g x -H10) in (ex2_ind C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u))) -(\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(drop (S n) O -(CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g -d1 d2))) (\lambda (x0: C).(\lambda (H12: (drop n O x (CHead x0 (Bind Abbr) -u))).(\lambda (H13: (csuba g d1 x0)).(let H14 \def (refl_equal nat (r (Bind -Abbr) n)) in (let H15 \def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x -(CHead x0 (Bind Abbr) u))) H12 (r (Bind Abbr) n) H14) in (ex_intro2 C -(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) -u))) (\lambda (d2: C).(csuba g d1 d2)) x0 (drop_drop (Bind Abst) n x (CHead -x0 (Bind Abbr) u) H15 t) H13)))))) H11)) c2 H9)))) H8)) (\lambda (H8: (ex4_3 -C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 -(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -c d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t (asucc -g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g c d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g c t (asucc g a))))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 a)))) (ex2 C (\lambda (d2: C).(drop (S n) O -c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda -(x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H9: (eq C c2 (CHead x0 -(Bind Abbr) x1))).(\lambda (H10: (csuba g c x0)).(\lambda (_: (arity g c t -(asucc g x2))).(\lambda (_: (arity g x0 x1 x2)).(eq_ind_r C (CHead x0 (Bind -Abbr) x1) (\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 -(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H13 \def (H c d1 u -H6 g x0 H10) in (ex2_ind C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind -Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(drop (S -n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abbr) u))) (\lambda (d2: -C).(csuba g d1 d2))) (\lambda (x: C).(\lambda (H14: (drop n O x0 (CHead x -(Bind Abbr) u))).(\lambda (H15: (csuba g d1 x)).(let H16 \def (refl_equal nat -(r (Bind Abbr) n)) in (let H17 \def (eq_ind nat n (\lambda (n0: nat).(drop n0 -O x0 (CHead x (Bind Abbr) u))) H14 (r (Bind Abbr) n) H16) in (ex_intro2 C -(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind -Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x (drop_drop (Bind Abbr) n x0 -(CHead x (Bind Abbr) u) H17 x1) H15)))))) H13)) c2 H9)))))))) H8)) H7))))) -(\lambda (H5: (csuba g (CHead c (Bind Void) t) c2)).(\lambda (H6: (drop (r -(Bind Void) n) O c (CHead d1 (Bind Abbr) u))).(let H_x \def (csuba_gen_void g -c c2 t H5) in (let H7 \def H_x in (ex2_3_ind B C T (\lambda (b0: B).(\lambda -(d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind b0) u2))))) (\lambda (_: -B).(\lambda (d2: C).(\lambda (_: T).(csuba g c d2)))) (ex2 C (\lambda (d2: -C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 -d2))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H8: (eq C -c2 (CHead x1 (Bind x0) x2))).(\lambda (H9: (csuba g c x1)).(eq_ind_r C (CHead -x1 (Bind x0) x2) (\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0 -(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H10 \def -(H c d1 u H6 g x1 H9) in (ex2_ind C (\lambda (d2: C).(drop n O x1 (CHead d2 -(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abbr) u))) (\lambda -(d2: C).(csuba g d1 d2))) (\lambda (x: C).(\lambda (H11: (drop n O x1 (CHead -x (Bind Abbr) u))).(\lambda (H12: (csuba g d1 x)).(let H13 \def (refl_equal -nat (r (Bind Abbr) n)) in (let H14 \def (eq_ind nat n (\lambda (n0: -nat).(drop n0 O x1 (CHead x (Bind Abbr) u))) H11 (r (Bind Abbr) n) H13) in -(ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2 -(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x (drop_drop (Bind x0) n -x1 (CHead x (Bind Abbr) u) H14 x2) H12)))))) H10)) c2 H8)))))) H7))))) b H3 -H4)))) (\lambda (f: F).(\lambda (H3: (csuba g (CHead c (Flat f) t) -c2)).(\lambda (H4: (drop (r (Flat f) n) O c (CHead d1 (Bind Abbr) u))).(let -H_x \def (csuba_gen_flat g c c2 t f H3) in (let H5 \def H_x in (ex2_2_ind C T -(\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Flat f) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g c d2))) (ex2 C (\lambda (d2: C).(drop (S n) -O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda -(x0: C).(\lambda (x1: T).(\lambda (H6: (eq C c2 (CHead x0 (Flat f) -x1))).(\lambda (H7: (csuba g c x0)).(eq_ind_r C (CHead x0 (Flat f) x1) -(\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind -Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H8 \def (H0 d1 u H4 g x0 -H7) in (ex2_ind C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind Abbr) -u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(drop (S n) O -(CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g -d1 d2))) (\lambda (x: C).(\lambda (H9: (drop (S n) O x0 (CHead x (Bind Abbr) -u))).(\lambda (H10: (csuba g d1 x)).(ex_intro2 C (\lambda (d2: C).(drop (S n) -O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g -d1 d2)) x (drop_drop (Flat f) n x0 (CHead x (Bind Abbr) u) H9 x1) H10)))) -H8)) c2 H6))))) H5)))))) k H2 (drop_gen_drop k c (CHead d1 (Bind Abbr) u) t n -H1)))))))))))) c1)))) i). -(* COMMENTS -Initial nodes: 3648 -END *) - -theorem csuba_drop_abst: - \forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u1: T).((drop i -O c1 (CHead d1 (Bind Abst) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba -g c1 c2) \to (or (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst) -u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a))))))))))))) -\def - \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (d1: -C).(\forall (u1: T).((drop n O c1 (CHead d1 (Bind Abst) u1)) \to (\forall (g: -G).(\forall (c2: C).((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(drop n -O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C -T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O c2 (CHead d2 -(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 -(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 -u2 a)))))))))))))) (\lambda (c1: C).(\lambda (d1: C).(\lambda (u1: -T).(\lambda (H: (drop O O c1 (CHead d1 (Bind Abst) u1))).(\lambda (g: -G).(\lambda (c2: C).(\lambda (H0: (csuba g c1 c2)).(let H1 \def (eq_ind C c1 -(\lambda (c: C).(csuba g c c2)) H0 (CHead d1 (Bind Abst) u1) (drop_gen_refl -c1 (CHead d1 (Bind Abst) u1) H)) in (let H_x \def (csuba_gen_abst g d1 c2 u1 -H1) in (let H2 \def H_x in (or_ind (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 -(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a))))) (or (ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) u1))) -(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(drop O O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (H3: -(ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: -C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind -Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: -C).(drop O O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 -d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O -O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: -A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda -(a: A).(arity g d2 u2 a)))))) (\lambda (x: C).(\lambda (H4: (eq C c2 (CHead x -(Bind Abst) u1))).(\lambda (H5: (csuba g d1 x)).(eq_ind_r C (CHead x (Bind -Abst) u1) (\lambda (c: C).(or (ex2 C (\lambda (d2: C).(drop O O c (CHead d2 -(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O c (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a))))))) (or_introl (ex2 C (\lambda (d2: C).(drop O O (CHead x (Bind Abst) -u1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T -A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O (CHead x (Bind -Abst) u1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: -C).(drop O O (CHead x (Bind Abst) u1) (CHead d2 (Bind Abst) u1))) (\lambda -(d2: C).(csuba g d1 d2)) x (drop_refl (CHead x (Bind Abst) u1)) H5)) c2 -H4)))) H3)) (\lambda (H3: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))).(ex4_3_ind C -T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 -(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 -(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 -u2 a)))) (or (ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) u1))) -(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(drop O O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H4: (eq C c2 (CHead x0 (Bind -Abbr) x1))).(\lambda (H5: (csuba g d1 x0)).(\lambda (H6: (arity g d1 u1 -(asucc g x2))).(\lambda (H7: (arity g x0 x1 x2)).(eq_ind_r C (CHead x0 (Bind -Abbr) x1) (\lambda (c: C).(or (ex2 C (\lambda (d2: C).(drop O O c (CHead d2 -(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O c (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a))))))) (or_intror (ex2 C (\lambda (d2: C).(drop O O (CHead x0 (Bind Abbr) -x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T -A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O (CHead x0 (Bind -Abbr) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O (CHead x0 (Bind Abbr) x1) -(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity -g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 a)))) x0 x1 x2 (drop_refl (CHead x0 (Bind Abbr) x1)) H5 H6 -H7)) c2 H4)))))))) H3)) H2))))))))))) (\lambda (n: nat).(\lambda (H: -((\forall (c1: C).(\forall (d1: C).(\forall (u1: T).((drop n O c1 (CHead d1 -(Bind Abst) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba g c1 c2) \to -(or (ex2 C (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abst) u1))) (\lambda -(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop n O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a))))))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (d1: -C).(\forall (u1: T).((drop (S n) O c (CHead d1 (Bind Abst) u1)) \to (\forall -(g: G).(\forall (c2: C).((csuba g c c2) \to (or (ex2 C (\lambda (d2: C).(drop -(S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O -c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: -A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda -(a: A).(arity g d2 u2 a))))))))))))) (\lambda (n0: nat).(\lambda (d1: -C).(\lambda (u1: T).(\lambda (H0: (drop (S n) O (CSort n0) (CHead d1 (Bind -Abst) u1))).(\lambda (g: G).(\lambda (c2: C).(\lambda (_: (csuba g (CSort n0) -c2)).(and3_ind (eq C (CHead d1 (Bind Abst) u1) (CSort n0)) (eq nat (S n) O) -(eq nat O O) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind -Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))) (\lambda (_: (eq C (CHead d1 (Bind Abst) u1) (CSort n0))).(\lambda -(H3: (eq nat (S n) O)).(\lambda (_: (eq nat O O)).(let H5 \def (eq_ind nat (S -n) (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O -\Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind (or (ex2 -C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: -C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) H5))))) -(drop_gen_sort n0 (S n) O (CHead d1 (Bind Abst) u1) H0))))))))) (\lambda (c: -C).(\lambda (H0: ((\forall (d1: C).(\forall (u1: T).((drop (S n) O c (CHead -d1 (Bind Abst) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba g c c2) \to -(or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) -(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (d1: C).(\lambda -(u1: T).(\lambda (H1: (drop (S n) O (CHead c k t) (CHead d1 (Bind Abst) -u1))).(\lambda (g: G).(\lambda (c2: C).(\lambda (H2: (csuba g (CHead c k t) -c2)).(K_ind (\lambda (k0: K).((csuba g (CHead c k0 t) c2) \to ((drop (r k0 n) -O c (CHead d1 (Bind Abst) u1)) \to (or (ex2 C (\lambda (d2: C).(drop (S n) O -c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T -A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead -d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity -g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 a))))))))) (\lambda (b: B).(\lambda (H3: (csuba g (CHead c -(Bind b) t) c2)).(\lambda (H4: (drop (r (Bind b) n) O c (CHead d1 (Bind Abst) -u1))).(B_ind (\lambda (b0: B).((csuba g (CHead c (Bind b0) t) c2) \to ((drop -(r (Bind b0) n) O c (CHead d1 (Bind Abst) u1)) \to (or (ex2 C (\lambda (d2: -C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 -d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S -n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))) (\lambda (H5: (csuba g -(CHead c (Bind Abbr) t) c2)).(\lambda (H6: (drop (r (Bind Abbr) n) O c (CHead -d1 (Bind Abst) u1))).(let H_x \def (csuba_gen_abbr g c c2 t H5) in (let H7 -\def H_x in (ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) -(\lambda (d2: C).(csuba g c d2)) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 -(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 -(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 -(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 -u2 a)))))) (\lambda (x: C).(\lambda (H8: (eq C c2 (CHead x (Bind Abbr) -t))).(\lambda (H9: (csuba g c x)).(eq_ind_r C (CHead x (Bind Abbr) t) -(\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind -Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a))))))) (let H10 \def (H c d1 u1 H6 g x H9) in (or_ind (ex2 C (\lambda (d2: -C).(drop n O x (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x -(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity -g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x -(Bind Abbr) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O -(CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda -(H11: (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) u1))) -(\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(drop n O x -(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C -(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) -u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t) -(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity -g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 a)))))) (\lambda (x0: C).(\lambda (H12: (drop n O x (CHead -x0 (Bind Abst) u1))).(\lambda (H13: (csuba g d1 x0)).(let H14 \def -(refl_equal nat (r (Bind Abbr) n)) in (let H15 \def (eq_ind nat n (\lambda -(n0: nat).(drop n0 O x (CHead x0 (Bind Abst) u1))) H12 (r (Bind Abbr) n) H14) -in (or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) -(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x -(Bind Abbr) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: -C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u1))) (\lambda -(d2: C).(csuba g d1 d2)) x0 (drop_drop (Bind Abbr) n x (CHead x0 (Bind Abst) -u1) H15 t) H13))))))) H11)) (\lambda (H11: (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abbr) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop n O x (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u1))) (\lambda -(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind -Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 -d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc -g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H12: -(drop n O x (CHead x0 (Bind Abbr) x1))).(\lambda (H13: (csuba g d1 -x0)).(\lambda (H14: (arity g d1 u1 (asucc g x2))).(\lambda (H15: (arity g x0 -x1 x2)).(let H16 \def (refl_equal nat (r (Bind Abbr) n)) in (let H17 \def -(eq_ind nat n (\lambda (n0: nat).(drop n0 O x (CHead x0 (Bind Abbr) x1))) H12 -(r (Bind Abbr) n) H16) in (or_intror (ex2 C (\lambda (d2: C).(drop (S n) O -(CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g -d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop -(S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C -T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x -(Bind Abbr) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x0 x1 x2 (drop_drop (Bind Abbr) -n x (CHead x0 (Bind Abbr) x1) H17 t) H13 H14 H15))))))))))) H11)) H10)) c2 -H8)))) H7))))) (\lambda (H5: (csuba g (CHead c (Bind Abst) t) c2)).(\lambda -(H6: (drop (r (Bind Abst) n) O c (CHead d1 (Bind Abst) u1))).(let H_x \def -(csuba_gen_abst g c c2 t H5) in (let H7 \def H_x in (or_ind (ex2 C (\lambda -(d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba g c d2))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 -(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g c d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g -c t (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity -g d2 u2 a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind -Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))) (\lambda (H8: (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) -t))) (\lambda (d2: C).(csuba g c d2)))).(ex2_ind C (\lambda (d2: C).(eq C c2 -(CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba g c d2)) (or (ex2 C -(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: -C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x: -C).(\lambda (H9: (eq C c2 (CHead x (Bind Abst) t))).(\lambda (H10: (csuba g c -x)).(eq_ind_r C (CHead x (Bind Abst) t) (\lambda (c0: C).(or (ex2 C (\lambda -(d2: C).(drop (S n) O c0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba -g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop (S n) O c0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda -(_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a))))))) (let H11 \def (H c d1 u1 H6 g -x H10) in (or_ind (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) -u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abbr) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (or -(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind -Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) -(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity -g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 a)))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x -(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C -(\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) u1))) (\lambda (d2: -C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind -Abst) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O -(CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x0: -C).(\lambda (H13: (drop n O x (CHead x0 (Bind Abst) u1))).(\lambda (H14: -(csuba g d1 x0)).(let H15 \def (refl_equal nat (r (Bind Abbr) n)) in (let H16 -\def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x (CHead x0 (Bind Abst) -u1))) H13 (r (Bind Abbr) n) H15) in (or_introl (ex2 C (\lambda (d2: C).(drop -(S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: -C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) -(ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 -(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) x0 (drop_drop (Bind Abst) -n x (CHead x0 (Bind Abst) u1) H16 t) H14))))))) H12)) (\lambda (H12: (ex4_3 C -T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 -(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 -(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 -u2 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop n O x (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u1))) (\lambda -(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind -Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 -d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc -g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H13: -(drop n O x (CHead x0 (Bind Abbr) x1))).(\lambda (H14: (csuba g d1 -x0)).(\lambda (H15: (arity g d1 u1 (asucc g x2))).(\lambda (H16: (arity g x0 -x1 x2)).(let H17 \def (refl_equal nat (r (Bind Abbr) n)) in (let H18 \def -(eq_ind nat n (\lambda (n0: nat).(drop n0 O x (CHead x0 (Bind Abbr) x1))) H13 -(r (Bind Abbr) n) H17) in (or_intror (ex2 C (\lambda (d2: C).(drop (S n) O -(CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g -d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop -(S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C -T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x -(Bind Abst) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x0 x1 x2 (drop_drop (Bind Abst) -n x (CHead x0 (Bind Abbr) x1) H18 t) H14 H15 H16))))))))))) H12)) H11)) c2 -H9)))) H8)) (\lambda (H8: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g c d2)))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g c t (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))).(ex4_3_ind C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g c d2)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C -(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: -C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H9: (eq C c2 (CHead x0 (Bind -Abbr) x1))).(\lambda (H10: (csuba g c x0)).(\lambda (_: (arity g c t (asucc g -x2))).(\lambda (_: (arity g x0 x1 x2)).(eq_ind_r C (CHead x0 (Bind Abbr) x1) -(\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind -Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a))))))) (let H13 \def (H c d1 u1 H6 g x0 H10) in (or_ind (ex2 C (\lambda -(d2: C).(drop n O x0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 -d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n -O x0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: -A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda -(a: A).(arity g d2 u2 a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead -x0 (Bind Abbr) x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 -d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S -n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda -(H14: (ex2 C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind Abst) u1))) -(\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(drop n O x0 -(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C -(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind -Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abbr) x1) -(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity -g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 a)))))) (\lambda (x: C).(\lambda (H15: (drop n O x0 (CHead -x (Bind Abst) u1))).(\lambda (H16: (csuba g d1 x)).(let H17 \def (refl_equal -nat (r (Bind Abbr) n)) in (let H18 \def (eq_ind nat n (\lambda (n0: -nat).(drop n0 O x0 (CHead x (Bind Abst) u1))) H15 (r (Bind Abbr) n) H17) in -(or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abbr) x1) -(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 -(Bind Abbr) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abst) u1))) -(\lambda (d2: C).(csuba g d1 d2)) x (drop_drop (Bind Abbr) n x0 (CHead x -(Bind Abst) u1) H18 x1) H16))))))) H14)) (\lambda (H14: (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x0 (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop n O x0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abst) u1))) -(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 -(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 -(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 -u2 a)))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: A).(\lambda (H15: -(drop n O x0 (CHead x3 (Bind Abbr) x4))).(\lambda (H16: (csuba g d1 -x3)).(\lambda (H17: (arity g d1 u1 (asucc g x5))).(\lambda (H18: (arity g x3 -x4 x5)).(let H19 \def (refl_equal nat (r (Bind Abbr) n)) in (let H20 \def -(eq_ind nat n (\lambda (n0: nat).(drop n0 O x0 (CHead x3 (Bind Abbr) x4))) -H15 (r (Bind Abbr) n) H19) in (or_intror (ex2 C (\lambda (d2: C).(drop (S n) -O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: -C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abbr) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) -(ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S -n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x3 x4 x5 -(drop_drop (Bind Abbr) n x0 (CHead x3 (Bind Abbr) x4) H20 x1) H16 H17 -H18))))))))))) H14)) H13)) c2 H9)))))))) H8)) H7))))) (\lambda (H5: (csuba g -(CHead c (Bind Void) t) c2)).(\lambda (H6: (drop (r (Bind Void) n) O c (CHead -d1 (Bind Abst) u1))).(let H_x \def (csuba_gen_void g c c2 t H5) in (let H7 -\def H_x in (ex2_3_ind B C T (\lambda (b0: B).(\lambda (d2: C).(\lambda (u2: -T).(eq C c2 (CHead d2 (Bind b0) u2))))) (\lambda (_: B).(\lambda (d2: -C).(\lambda (_: T).(csuba g c d2)))) (or (ex2 C (\lambda (d2: C).(drop (S n) -O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C -T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead -d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity -g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 a)))))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: -T).(\lambda (H8: (eq C c2 (CHead x1 (Bind x0) x2))).(\lambda (H9: (csuba g c -x1)).(eq_ind_r C (CHead x1 (Bind x0) x2) (\lambda (c0: C).(or (ex2 C (\lambda -(d2: C).(drop (S n) O c0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba -g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop (S n) O c0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda -(_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a))))))) (let H10 \def (H c d1 u1 H6 g -x1 H9) in (or_ind (ex2 C (\lambda (d2: C).(drop n O x1 (CHead d2 (Bind Abst) -u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop n O x1 (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x1 (Bind x0) x2) -(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x1 -(Bind x0) x2) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (H11: (ex2 C (\lambda -(d2: C).(drop n O x1 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 -d2)))).(ex2_ind C (\lambda (d2: C).(drop n O x1 (CHead d2 (Bind Abst) u1))) -(\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: C).(drop (S n) O -(CHead x1 (Bind x0) x2) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g -d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop -(S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x: -C).(\lambda (H12: (drop n O x1 (CHead x (Bind Abst) u1))).(\lambda (H13: -(csuba g d1 x)).(let H14 \def (refl_equal nat (r (Bind Abbr) n)) in (let H15 -\def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x1 (CHead x (Bind Abst) -u1))) H12 (r (Bind Abbr) n) H14) in (or_introl (ex2 C (\lambda (d2: C).(drop -(S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abst) u1))) (\lambda (d2: -C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abbr) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) -(ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2 -(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) x (drop_drop (Bind x0) n -x1 (CHead x (Bind Abst) u1) H15 x2) H13))))))) H11)) (\lambda (H11: (ex4_3 C -T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x1 (CHead d2 -(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 -(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 -u2 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop n O x1 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abst) u1))) (\lambda -(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind -Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 -d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc -g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: A).(\lambda (H12: -(drop n O x1 (CHead x3 (Bind Abbr) x4))).(\lambda (H13: (csuba g d1 -x3)).(\lambda (H14: (arity g d1 u1 (asucc g x5))).(\lambda (H15: (arity g x3 -x4 x5)).(let H16 \def (refl_equal nat (r (Bind Abbr) n)) in (let H17 \def -(eq_ind nat n (\lambda (n0: nat).(drop n0 O x1 (CHead x3 (Bind Abbr) x4))) -H12 (r (Bind Abbr) n) H16) in (or_intror (ex2 C (\lambda (d2: C).(drop (S n) -O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba -g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abbr) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) -(ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S -n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x3 x4 x5 -(drop_drop (Bind x0) n x1 (CHead x3 (Bind Abbr) x4) H17 x2) H13 H14 -H15))))))))))) H11)) H10)) c2 H8)))))) H7))))) b H3 H4)))) (\lambda (f: -F).(\lambda (H3: (csuba g (CHead c (Flat f) t) c2)).(\lambda (H4: (drop (r -(Flat f) n) O c (CHead d1 (Bind Abst) u1))).(let H_x \def (csuba_gen_flat g c -c2 t f H3) in (let H5 \def H_x in (ex2_2_ind C T (\lambda (d2: C).(\lambda -(u2: T).(eq C c2 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g c d2))) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 -(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind -Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 -d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc -g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C c2 (CHead x0 -(Flat f) x1))).(\lambda (H7: (csuba g c x0)).(eq_ind_r C (CHead x0 (Flat f) -x1) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 -(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind -Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 -d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc -g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a))))))) (let H8 \def (H0 d1 u1 H4 g x0 H7) in (or_ind (ex2 C (\lambda (d2: -C).(drop (S n) O x0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 -d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S -n) O x0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (or (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u1))) (\lambda -(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))) (\lambda (H9: (ex2 C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 -(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda -(d2: C).(drop (S n) O x0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba -g d1 d2)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) -(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 -(Flat f) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x: C).(\lambda (H10: -(drop (S n) O x0 (CHead x (Bind Abst) u1))).(\lambda (H11: (csuba g d1 -x)).(or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) -(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 -(Flat f) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u1))) (\lambda -(d2: C).(csuba g d1 d2)) x (drop_drop (Flat f) n x0 (CHead x (Bind Abst) u1) -H10 x1) H11))))) H9)) (\lambda (H9: (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(drop (S n) O x0 (CHead d2 (Bind Abbr) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop (S n) O x0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda -(_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u1))) (\lambda -(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))) (\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: A).(\lambda (H10: -(drop (S n) O x0 (CHead x2 (Bind Abbr) x3))).(\lambda (H11: (csuba g d1 -x2)).(\lambda (H12: (arity g d1 u1 (asucc g x4))).(\lambda (H13: (arity g x2 -x3 x4)).(or_intror (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) -x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T -A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 -(Flat f) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) -(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity -g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 a)))) x2 x3 x4 (drop_drop (Flat f) n x0 (CHead x2 (Bind -Abbr) x3) H10 x1) H11 H12 H13))))))))) H9)) H8)) c2 H6))))) H5)))))) k H2 -(drop_gen_drop k c (CHead d1 (Bind Abst) u1) t n H1)))))))))))) c1)))) i). -(* COMMENTS -Initial nodes: 12528 -END *) - -theorem csuba_drop_abst_rev: - \forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).((drop i -O c1 (CHead d1 (Bind Abst) u)) \to (\forall (g: G).(\forall (c2: C).((csuba g -c2 c1) \to (or (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst) -u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda -(u2: T).(drop i O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda -(_: T).(csuba g d2 d1)))))))))))) -\def - \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (d1: -C).(\forall (u: T).((drop n O c1 (CHead d1 (Bind Abst) u)) \to (\forall (g: -G).(\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: C).(drop n -O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop n O c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))))) (\lambda (c1: -C).(\lambda (d1: C).(\lambda (u: T).(\lambda (H: (drop O O c1 (CHead d1 (Bind -Abst) u))).(\lambda (g: G).(\lambda (c2: C).(\lambda (H0: (csuba g c2 -c1)).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csuba g c2 c)) H0 (CHead d1 -(Bind Abst) u) (drop_gen_refl c1 (CHead d1 (Bind Abst) u) H)) in (let H_x -\def (csuba_gen_abst_rev g d1 c2 u H1) in (let H2 \def H_x in (or_ind (ex2 C -(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba -g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or -(ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O -c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))) (\lambda (H3: (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) -u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(eq C c2 -(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C -(\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O -c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))) (\lambda (x: C).(\lambda (H4: (eq C c2 (CHead x (Bind Abst) -u))).(\lambda (H5: (csuba g x d1)).(eq_ind_r C (CHead x (Bind Abst) u) -(\lambda (c: C).(or (ex2 C (\lambda (d2: C).(drop O O c (CHead d2 (Bind Abst) -u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda -(u2: T).(drop O O c (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda -(_: T).(csuba g d2 d1)))))) (or_introl (ex2 C (\lambda (d2: C).(drop O O -(CHead x (Bind Abst) u) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g -d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O (CHead x -(Bind Abst) u) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop O O (CHead x (Bind -Abst) u) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x -(drop_refl (CHead x (Bind Abst) u)) H5)) c2 H4)))) H3)) (\lambda (H3: (ex2_2 -C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda -(d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1))) (or (ex2 C (\lambda (d2: C).(drop O O c2 -(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop O O c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (H4: (eq C c2 (CHead x0 (Bind Void) -x1))).(\lambda (H5: (csuba g x0 d1)).(eq_ind_r C (CHead x0 (Bind Void) x1) -(\lambda (c: C).(or (ex2 C (\lambda (d2: C).(drop O O c (CHead d2 (Bind Abst) -u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda -(u2: T).(drop O O c (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda -(_: T).(csuba g d2 d1)))))) (or_intror (ex2 C (\lambda (d2: C).(drop O O -(CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba -g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O (CHead x0 -(Bind Void) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: -T).(drop O O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1 (drop_refl (CHead x0 (Bind -Void) x1)) H5)) c2 H4))))) H3)) H2))))))))))) (\lambda (n: nat).(\lambda (H: -((\forall (c1: C).(\forall (d1: C).(\forall (u: T).((drop n O c1 (CHead d1 -(Bind Abst) u)) \to (\forall (g: G).(\forall (c2: C).((csuba g c2 c1) \to (or -(ex2 C (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O -c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (d1: -C).(\forall (u: T).((drop (S n) O c (CHead d1 (Bind Abst) u)) \to (\forall -(g: G).(\forall (c2: C).((csuba g c2 c) \to (or (ex2 C (\lambda (d2: C).(drop -(S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))))))) -(\lambda (n0: nat).(\lambda (d1: C).(\lambda (u: T).(\lambda (H0: (drop (S n) -O (CSort n0) (CHead d1 (Bind Abst) u))).(\lambda (g: G).(\lambda (c2: -C).(\lambda (_: (csuba g c2 (CSort n0))).(and3_ind (eq C (CHead d1 (Bind -Abst) u) (CSort n0)) (eq nat (S n) O) (eq nat O O) (or (ex2 C (\lambda (d2: -C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 -d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) -(\lambda (_: (eq C (CHead d1 (Bind Abst) u) (CSort n0))).(\lambda (H3: (eq -nat (S n) O)).(\lambda (_: (eq nat O O)).(let H5 \def (eq_ind nat (S n) -(\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O -\Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind (or (ex2 -C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) -O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g -d2 d1))))) H5))))) (drop_gen_sort n0 (S n) O (CHead d1 (Bind Abst) u) -H0))))))))) (\lambda (c: C).(\lambda (H0: ((\forall (d1: C).(\forall (u: -T).((drop (S n) O c (CHead d1 (Bind Abst) u)) \to (\forall (g: G).(\forall -(c2: C).((csuba g c2 c) \to (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 -(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))))).(\lambda -(k: K).(\lambda (t: T).(\lambda (d1: C).(\lambda (u: T).(\lambda (H1: (drop -(S n) O (CHead c k t) (CHead d1 (Bind Abst) u))).(\lambda (g: G).(\lambda -(c2: C).(\lambda (H2: (csuba g c2 (CHead c k t))).(K_ind (\lambda (k0: -K).((csuba g c2 (CHead c k0 t)) \to ((drop (r k0 n) O c (CHead d1 (Bind Abst) -u)) \to (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) -u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda -(u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))))))) (\lambda (b: B).(\lambda (H3: -(csuba g c2 (CHead c (Bind b) t))).(\lambda (H4: (drop (r (Bind b) n) O c -(CHead d1 (Bind Abst) u))).(B_ind (\lambda (b0: B).((csuba g c2 (CHead c -(Bind b0) t)) \to ((drop (r (Bind b0) n) O c (CHead d1 (Bind Abst) u)) \to -(or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda -(_: T).(csuba g d2 d1)))))))) (\lambda (H5: (csuba g c2 (CHead c (Bind Abbr) -t))).(\lambda (H6: (drop (r (Bind Abbr) n) O c (CHead d1 (Bind Abst) -u))).(let H_x \def (csuba_gen_abbr_rev g c c2 t H5) in (let H7 \def H_x in -(or3_ind (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) (\lambda -(d2: C).(csuba g d2 c))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 c)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 c)))) (or (ex2 C (\lambda (d2: -C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 -d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) -(\lambda (H8: (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) -(\lambda (d2: C).(csuba g d2 c)))).(ex2_ind C (\lambda (d2: C).(eq C c2 -(CHead d2 (Bind Abbr) t))) (\lambda (d2: C).(csuba g d2 c)) (or (ex2 C -(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) -O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g -d2 d1))))) (\lambda (x: C).(\lambda (H9: (eq C c2 (CHead x (Bind Abbr) -t))).(\lambda (H10: (csuba g x c)).(eq_ind_r C (CHead x (Bind Abbr) t) -(\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind -Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 (Bind Void) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H11 \def (H c d1 u H6 g x -H10) in (or_ind (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) -u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda -(u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda -(_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x -(Bind Abbr) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind -Abbr) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba -g d2 d1))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 -(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: -C).(drop n O x (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) -(or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 -(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda -(x0: C).(\lambda (H13: (drop n O x (CHead x0 (Bind Abst) u))).(\lambda (H14: -(csuba g x0 d1)).(or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x -(Bind Abbr) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind -Abbr) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba -g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) -t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x0 (drop_drop -(Bind Abbr) n x (CHead x0 (Bind Abst) u) H13 t) H14))))) H12)) (\lambda (H12: -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind -C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or (ex2 C (\lambda -(d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (H13: (drop n O x (CHead x0 (Bind Void) -x1))).(\lambda (H14: (csuba g x0 d1)).(or_intror (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u))) (\lambda -(d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop -(S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1 -(drop_drop (Bind Abbr) n x (CHead x0 (Bind Void) x1) H13 t) H14)))))) H12)) -H11)) c2 H9)))) H8)) (\lambda (H8: (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 c)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t a)))))).(ex4_3_ind C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -c)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc -g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t a)))) -(or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda -(_: T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: -A).(\lambda (H9: (eq C c2 (CHead x0 (Bind Abst) x1))).(\lambda (H10: (csuba g -x0 c)).(\lambda (_: (arity g x0 x1 (asucc g x2))).(\lambda (_: (arity g c t -x2)).(eq_ind_r C (CHead x0 (Bind Abst) x1) (\lambda (c0: C).(or (ex2 C -(\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) -O c0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g -d2 d1)))))) (let H13 \def (H c d1 u H6 g x0 H10) in (or_ind (ex2 C (\lambda -(d2: C).(drop n O x0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 -d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or -(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 -(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda -(H14: (ex2 C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop n O x0 -(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C -(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind -Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda -(x: C).(\lambda (H15: (drop n O x0 (CHead x (Bind Abst) u))).(\lambda (H16: -(csuba g x d1)).(or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 -(Bind Abst) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind -Abst) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x0 -(Bind Abst) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) -x (drop_drop (Bind Abst) n x0 (CHead x (Bind Abst) u) H15 x1) H16))))) H14)) -(\lambda (H14: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead -d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or -(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 -(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda -(x3: C).(\lambda (x4: T).(\lambda (H15: (drop n O x0 (CHead x3 (Bind Void) -x4))).(\lambda (H16: (csuba g x3 d1)).(or_intror (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda -(d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x3 x4 -(drop_drop (Bind Abst) n x0 (CHead x3 (Bind Void) x4) H15 x1) H16)))))) H14)) -H13)) c2 H9)))))))) H8)) (\lambda (H8: (ex2_2 C T (\lambda (d2: C).(\lambda -(u2: T).(eq C c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 c))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(eq C -c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -c))) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda -(_: T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H9: -(eq C c2 (CHead x0 (Bind Void) x1))).(\lambda (H10: (csuba g x0 c)).(eq_ind_r -C (CHead x0 (Bind Void) x1) (\lambda (c0: C).(or (ex2 C (\lambda (d2: -C).(drop (S n) O c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 -d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let -H11 \def (H c d1 u H6 g x0 H10) in (or_ind (ex2 C (\lambda (d2: C).(drop n O -x0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H12: (ex2 C -(\lambda (d2: C).(drop n O x0 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop n O x0 (CHead d2 -(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda -(H13: (drop n O x0 (CHead x (Bind Abst) u))).(\lambda (H14: (csuba g x -d1)).(or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) -x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) -(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop -(Bind Void) n x0 (CHead x (Bind Abst) u) H13 x1) H14))))) H12)) (\lambda -(H12: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead -d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or -(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 -(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda -(x2: C).(\lambda (x3: T).(\lambda (H13: (drop n O x0 (CHead x2 (Bind Void) -x3))).(\lambda (H14: (csuba g x2 d1)).(or_intror (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda -(d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x2 x3 -(drop_drop (Bind Void) n x0 (CHead x2 (Bind Void) x3) H13 x1) H14)))))) H12)) -H11)) c2 H9))))) H8)) H7))))) (\lambda (H5: (csuba g c2 (CHead c (Bind Abst) -t))).(\lambda (H6: (drop (r (Bind Abst) n) O c (CHead d1 (Bind Abst) -u))).(let H_x \def (csuba_gen_abst_rev g c c2 t H5) in (let H7 \def H_x in -(or_ind (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda -(d2: C).(csuba g d2 c))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C -c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -c)))) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda -(_: T).(csuba g d2 d1))))) (\lambda (H8: (ex2 C (\lambda (d2: C).(eq C c2 -(CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba g d2 c)))).(ex2_ind C -(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba -g d2 c)) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) -u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda -(u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda (H9: (eq C c2 -(CHead x (Bind Abst) t))).(\lambda (H10: (csuba g x c)).(eq_ind_r C (CHead x -(Bind Abst) t) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 -(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H11 \def (H -c d1 u H6 g x H10) in (or_ind (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 -(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: C).(drop (S n) -O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba -g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead -x (Bind Abst) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x -(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C -(\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind -Abst) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 -C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abst) t) -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))) (\lambda (x0: C).(\lambda (H13: (drop n O x (CHead x0 (Bind Abst) -u))).(\lambda (H14: (csuba g x0 d1)).(or_introl (ex2 C (\lambda (d2: C).(drop -(S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) -O (CHead x (Bind Abst) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S -n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1)) x0 (drop_drop (Bind Abst) n x (CHead x0 (Bind Abst) u) -H13 t) H14))))) H12)) (\lambda (H12: (ex2_2 C T (\lambda (d2: C).(\lambda -(u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda -(_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: -T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind -Abst) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 -C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abst) t) -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H13: (drop n O x (CHead -x0 (Bind Void) x1))).(\lambda (H14: (csuba g x0 d1)).(or_intror (ex2 C -(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) -u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda -(u2: T).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda -(d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1 -(drop_drop (Bind Abst) n x (CHead x0 (Bind Void) x1) H13 t) H14)))))) H12)) -H11)) c2 H9)))) H8)) (\lambda (H8: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(eq C c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 c))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(eq C -c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -c))) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda -(_: T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H9: -(eq C c2 (CHead x0 (Bind Void) x1))).(\lambda (H10: (csuba g x0 c)).(eq_ind_r -C (CHead x0 (Bind Void) x1) (\lambda (c0: C).(or (ex2 C (\lambda (d2: -C).(drop (S n) O c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 -d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let -H11 \def (H c d1 u H6 g x0 H10) in (or_ind (ex2 C (\lambda (d2: C).(drop n O -x0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H12: (ex2 C -(\lambda (d2: C).(drop n O x0 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop n O x0 (CHead d2 -(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda -(H13: (drop n O x0 (CHead x (Bind Abst) u))).(\lambda (H14: (csuba g x -d1)).(or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) -x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) -(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop -(Bind Void) n x0 (CHead x (Bind Abst) u) H13 x1) H14))))) H12)) (\lambda -(H12: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead -d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or -(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 -(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda -(x2: C).(\lambda (x3: T).(\lambda (H13: (drop n O x0 (CHead x2 (Bind Void) -x3))).(\lambda (H14: (csuba g x2 d1)).(or_intror (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda -(d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x2 x3 -(drop_drop (Bind Void) n x0 (CHead x2 (Bind Void) x3) H13 x1) H14)))))) H12)) -H11)) c2 H9))))) H8)) H7))))) (\lambda (H5: (csuba g c2 (CHead c (Bind Void) -t))).(\lambda (H6: (drop (r (Bind Void) n) O c (CHead d1 (Bind Abst) -u))).(let H_x \def (csuba_gen_void_rev g c c2 t H5) in (let H7 \def H_x in -(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Void) t))) (\lambda (d2: -C).(csuba g d2 c)) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 -(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda (H8: (eq -C c2 (CHead x (Bind Void) t))).(\lambda (H9: (csuba g x c)).(eq_ind_r C -(CHead x (Bind Void) t) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S -n) O c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 -C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H10 \def (H -c d1 u H6 g x H9) in (or_ind (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 -(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: C).(drop (S n) -O (CHead x (Bind Void) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba -g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead -x (Bind Void) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))) (\lambda (H11: (ex2 C (\lambda (d2: C).(drop n O x -(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C -(\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind -Void) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 -C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Void) t) -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))) (\lambda (x0: C).(\lambda (H12: (drop n O x (CHead x0 (Bind Abst) -u))).(\lambda (H13: (csuba g x0 d1)).(or_introl (ex2 C (\lambda (d2: C).(drop -(S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) -O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S -n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1)) x0 (drop_drop (Bind Void) n x (CHead x0 (Bind Abst) u) -H12 t) H13))))) H11)) (\lambda (H11: (ex2_2 C T (\lambda (d2: C).(\lambda -(u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda -(_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: -T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind -Void) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 -C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Void) t) -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H12: (drop n O x (CHead -x0 (Bind Void) x1))).(\lambda (H13: (csuba g x0 d1)).(or_intror (ex2 C -(\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abst) -u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda -(u2: T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda -(d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1 -(drop_drop (Bind Void) n x (CHead x0 (Bind Void) x1) H12 t) H13)))))) H11)) -H10)) c2 H8)))) H7))))) b H3 H4)))) (\lambda (f: F).(\lambda (H3: (csuba g c2 -(CHead c (Flat f) t))).(\lambda (H4: (drop (r (Flat f) n) O c (CHead d1 (Bind -Abst) u))).(let H_x \def (csuba_gen_flat_rev g c c2 t f H3) in (let H5 \def -H_x in (ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 -(Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 c))) (or (ex2 C -(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) -O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g -d2 d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C c2 (CHead x0 -(Flat f) x1))).(\lambda (H7: (csuba g x0 c)).(eq_ind_r C (CHead x0 (Flat f) -x1) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 -(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 (Bind Void) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H8 \def (H0 d1 u H4 g x0 -H7) in (or_ind (ex2 C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind Abst) -u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda -(u2: T).(drop (S n) O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: C).(drop (S n) -O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g -d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 -(Flat f) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))) (\lambda (H9: (ex2 C (\lambda (d2: C).(drop (S n) O x0 -(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C -(\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat -f) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C -T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f) x1) -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))) (\lambda (x: C).(\lambda (H10: (drop (S n) O x0 (CHead x (Bind Abst) -u))).(\lambda (H11: (csuba g x d1)).(or_introl (ex2 C (\lambda (d2: C).(drop -(S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) -O (CHead x0 (Flat f) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S -n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1)) x (drop_drop (Flat f) n x0 (CHead x (Bind Abst) u) H10 -x1) H11))))) H9)) (\lambda (H9: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda -(_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda -(_: T).(csuba g d2 d1))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 -(Flat f) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f) -x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))) (\lambda (x2: C).(\lambda (x3: T).(\lambda (H10: (drop (S n) O x0 -(CHead x2 (Bind Void) x3))).(\lambda (H11: (csuba g x2 d1)).(or_intror (ex2 C -(\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) -u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda -(u2: T).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda -(d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x2 x3 -(drop_drop (Flat f) n x0 (CHead x2 (Bind Void) x3) H10 x1) H11)))))) H9)) -H8)) c2 H6))))) H5)))))) k H2 (drop_gen_drop k c (CHead d1 (Bind Abst) u) t n -H1)))))))))))) c1)))) i). -(* COMMENTS -Initial nodes: 11438 -END *) - -theorem csuba_drop_abbr_rev: - \forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u1: T).((drop i -O c1 (CHead d1 (Bind Abbr) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba -g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop i O c2 (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))))))) -\def - \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (d1: -C).(\forall (u1: T).((drop n O c1 (CHead d1 (Bind Abbr) u1)) \to (\forall (g: -G).(\forall (c2: C).((csuba g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(drop n -O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C -T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O c2 (CHead d2 -(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O c2 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))))))))))) (\lambda (c1: C).(\lambda (d1: C).(\lambda (u1: T).(\lambda -(H: (drop O O c1 (CHead d1 (Bind Abbr) u1))).(\lambda (g: G).(\lambda (c2: -C).(\lambda (H0: (csuba g c2 c1)).(let H1 \def (eq_ind C c1 (\lambda (c: -C).(csuba g c2 c)) H0 (CHead d1 (Bind Abbr) u1) (drop_gen_refl c1 (CHead d1 -(Bind Abbr) u1) H)) in (let H_x \def (csuba_gen_abbr_rev g d1 c2 u1 H1) in -(let H2 \def H_x in (or3_ind (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 -C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: -C).(drop O O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O -O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop O O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))) (\lambda (H3: (ex2 C (\lambda (d2: C).(eq C c2 (CHead -d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda -(d2: C).(eq C c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1)) (or3 (ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abbr) u1))) -(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(drop O O c2 (CHead d2 (Bind Abst) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop O O c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda -(H4: (eq C c2 (CHead x (Bind Abbr) u1))).(\lambda (H5: (csuba g x -d1)).(eq_ind_r C (CHead x (Bind Abbr) u1) (\lambda (c: C).(or3 (ex2 C -(\lambda (d2: C).(drop O O c (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(drop O O c (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda -(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop O O c (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))))) (or3_intro0 (ex2 C (\lambda (d2: -C).(drop O O (CHead x (Bind Abbr) u1) (CHead d2 (Bind Abbr) u1))) (\lambda -(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop O O (CHead x (Bind Abbr) u1) (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O (CHead x (Bind Abbr) -u1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))) (ex_intro2 C (\lambda (d2: C).(drop O O (CHead x (Bind Abbr) u1) -(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_refl -(CHead x (Bind Abbr) u1)) H5)) c2 H4)))) H3)) (\lambda (H3: (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(eq C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C (\lambda (d2: -C).(drop O O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O -O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop O O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: -A).(\lambda (H4: (eq C c2 (CHead x0 (Bind Abst) x1))).(\lambda (H5: (csuba g -x0 d1)).(\lambda (H6: (arity g x0 x1 (asucc g x2))).(\lambda (H7: (arity g d1 -u1 x2)).(eq_ind_r C (CHead x0 (Bind Abst) x1) (\lambda (c: C).(or3 (ex2 C -(\lambda (d2: C).(drop O O c (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(drop O O c (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda -(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop O O c (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))))) (or3_intro1 (ex2 C (\lambda (d2: -C).(drop O O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abbr) u1))) (\lambda -(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop O O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O (CHead x0 (Bind Abst) -x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop O O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x0 x1 x2 -(drop_refl (CHead x0 (Bind Abst) x1)) H5 H6 H7)) c2 H4)))))))) H3)) (\lambda -(H3: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind -C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C (\lambda (d2: -C).(drop O O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O -O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop O O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H4: (eq C -c2 (CHead x0 (Bind Void) x1))).(\lambda (H5: (csuba g x0 d1)).(eq_ind_r C -(CHead x0 (Bind Void) x1) (\lambda (c: C).(or3 (ex2 C (\lambda (d2: C).(drop -O O c (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C -T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O c (CHead d2 -(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O c (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) -(or3_intro2 (ex2 C (\lambda (d2: C).(drop O O (CHead x0 (Bind Void) x1) -(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O (CHead x0 (Bind -Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop O O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T -(\lambda (d2: C).(\lambda (u2: T).(drop O O (CHead x0 (Bind Void) x1) (CHead -d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 -x1 (drop_refl (CHead x0 (Bind Void) x1)) H5)) c2 H4))))) H3)) H2))))))))))) -(\lambda (n: nat).(\lambda (H: ((\forall (c1: C).(\forall (d1: C).(\forall -(u1: T).((drop n O c1 (CHead d1 (Bind Abbr) u1)) \to (\forall (g: G).(\forall -(c2: C).((csuba g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(drop n O c2 (CHead -d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O c2 (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O c2 (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (d1: -C).(\forall (u1: T).((drop (S n) O c (CHead d1 (Bind Abbr) u1)) \to (\forall -(g: G).(\forall (c2: C).((csuba g c2 c) \to (or3 (ex2 C (\lambda (d2: -C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S -n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1)))))))))))) (\lambda (n0: -nat).(\lambda (d1: C).(\lambda (u1: T).(\lambda (H0: (drop (S n) O (CSort n0) -(CHead d1 (Bind Abbr) u1))).(\lambda (g: G).(\lambda (c2: C).(\lambda (_: -(csuba g c2 (CSort n0))).(and3_ind (eq C (CHead d1 (Bind Abbr) u1) (CSort -n0)) (eq nat (S n) O) (eq nat O O) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O -c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T -A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead -d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda -(_: T).(csuba g d2 d1))))) (\lambda (_: (eq C (CHead d1 (Bind Abbr) u1) -(CSort n0))).(\lambda (H3: (eq nat (S n) O)).(\lambda (_: (eq nat O O)).(let -H5 \def (eq_ind nat (S n) (\lambda (ee: nat).(match ee in nat return (\lambda -(_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H3) -in (False_ind (or3 (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) H5))))) -(drop_gen_sort n0 (S n) O (CHead d1 (Bind Abbr) u1) H0))))))))) (\lambda (c: -C).(\lambda (H0: ((\forall (d1: C).(\forall (u1: T).((drop (S n) O c (CHead -d1 (Bind Abbr) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba g c2 c) \to -(or3 (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) -(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 -C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))))).(\lambda -(k: K).(\lambda (t: T).(\lambda (d1: C).(\lambda (u1: T).(\lambda (H1: (drop -(S n) O (CHead c k t) (CHead d1 (Bind Abbr) u1))).(\lambda (g: G).(\lambda -(c2: C).(\lambda (H2: (csuba g c2 (CHead c k t))).(K_ind (\lambda (k0: -K).((csuba g c2 (CHead c k0 t)) \to ((drop (r k0 n) O c (CHead d1 (Bind Abbr) -u1)) \to (or3 (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))) (\lambda -(b: B).(\lambda (H3: (csuba g c2 (CHead c (Bind b) t))).(\lambda (H4: (drop -(r (Bind b) n) O c (CHead d1 (Bind Abbr) u1))).(B_ind (\lambda (b0: -B).((csuba g c2 (CHead c (Bind b0) t)) \to ((drop (r (Bind b0) n) O c (CHead -d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead -d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))))))) (\lambda (H5: (csuba g c2 (CHead c (Bind Abbr) t))).(\lambda (H6: -(drop (r (Bind Abbr) n) O c (CHead d1 (Bind Abbr) u1))).(let H_x \def -(csuba_gen_abbr_rev g c c2 t H5) in (let H7 \def H_x in (or3_ind (ex2 C -(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) (\lambda (d2: C).(csuba -g d2 c))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq -C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 c)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g c t a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C -c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -c)))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda -(H8: (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) (\lambda -(d2: C).(csuba g d2 c)))).(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 -(Bind Abbr) t))) (\lambda (d2: C).(csuba g d2 c)) (or3 (ex2 C (\lambda (d2: -C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S -n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda (H9: (eq -C c2 (CHead x (Bind Abbr) t))).(\lambda (H10: (csuba g x c)).(eq_ind_r C -(CHead x (Bind Abbr) t) (\lambda (c0: C).(or3 (ex2 C (\lambda (d2: C).(drop -(S n) O c0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O -c0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O c0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda -(_: T).(csuba g d2 d1)))))) (let H11 \def (H c d1 u1 H6 g x H10) in (or3_ind -(ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(drop n O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda -(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(drop (S -n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 -C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abbr) t) -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: -C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) -(or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 -(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) -t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: -C).(\lambda (H13: (drop n O x (CHead x0 (Bind Abbr) u1))).(\lambda (H14: -(csuba g x0 d1)).(or3_intro0 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x -(Bind Abbr) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O -(CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abbr) t) -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) -(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x0 (drop_drop -(Bind Abbr) n x (CHead x0 (Bind Abbr) u1) H13 t) H14))))) H12)) (\lambda -(H12: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n -O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C -(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t) -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H13: (drop n O x (CHead x0 -(Bind Abst) x1))).(\lambda (H14: (csuba g x0 d1)).(\lambda (H15: (arity g x0 -x1 (asucc g x2))).(\lambda (H16: (arity g d1 u1 x2)).(or3_intro1 (ex2 C -(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t) -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex4_3_intro C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x -(Bind Abbr) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a)))) x0 x1 x2 (drop_drop (Bind Abbr) n -x (CHead x0 (Bind Abst) x1) H13 t) H14 H15 H16))))))))) H12)) (\lambda (H12: -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind -C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C -(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t) -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (H13: (drop n O x (CHead x0 (Bind Void) -x1))).(\lambda (H14: (csuba g x0 d1)).(or3_intro2 (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u1))) (\lambda -(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x -(Bind Abbr) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1 (drop_drop (Bind -Abbr) n x (CHead x0 (Bind Void) x1) H13 t) H14)))))) H12)) H11)) c2 H9)))) -H8)) (\lambda (H8: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(eq C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 c)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g c t a)))))).(ex4_3_ind C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 c)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t a)))) (or3 (ex2 -C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H9: (eq C c2 (CHead x0 (Bind -Abst) x1))).(\lambda (H10: (csuba g x0 c)).(\lambda (_: (arity g x0 x1 (asucc -g x2))).(\lambda (_: (arity g c t x2)).(eq_ind_r C (CHead x0 (Bind Abst) x1) -(\lambda (c0: C).(or3 (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 -(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0 -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))))) (let H13 \def (H c d1 u1 H6 g x0 H10) in (or3_ind (ex2 C (\lambda -(d2: C).(drop n O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n -O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 -(Bind Abst) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S -n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))) (\lambda (H14: (ex2 C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: -C).(drop n O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) -(or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead -d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abst) -x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda -(H15: (drop n O x0 (CHead x (Bind Abbr) u1))).(\lambda (H16: (csuba g x -d1)).(or3_intro0 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) -x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T -A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 -(Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C -(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop (Bind Abst) n x0 -(CHead x (Bind Abbr) u1) H15 x1) H16))))) H14)) (\lambda (H14: (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x0 (CHead d2 -(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop n O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abbr) u1))) -(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 -(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead -x0 (Bind Abst) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: -A).(\lambda (H15: (drop n O x0 (CHead x3 (Bind Abst) x4))).(\lambda (H16: -(csuba g x3 d1)).(\lambda (H17: (arity g x3 x4 (asucc g x5))).(\lambda (H18: -(arity g d1 u1 x5)).(or3_intro1 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead -x0 (Bind Abst) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S -n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x3 x4 x5 -(drop_drop (Bind Abst) n x0 (CHead x3 (Bind Abst) x4) H15 x1) H16 H17 -H18))))))))) H14)) (\lambda (H14: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(drop -n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g -d2 d1))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) -(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 -(Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda -(x3: C).(\lambda (x4: T).(\lambda (H15: (drop n O x0 (CHead x3 (Bind Void) -x4))).(\lambda (H16: (csuba g x3 d1)).(or3_intro2 (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abbr) u1))) -(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 -(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead -x0 (Bind Abst) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x3 x4 (drop_drop (Bind -Abst) n x0 (CHead x3 (Bind Void) x4) H15 x1) H16)))))) H14)) H13)) c2 -H9)))))))) H8)) (\lambda (H8: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(eq C c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 c))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(eq C -c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -c))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda -(x0: C).(\lambda (x1: T).(\lambda (H9: (eq C c2 (CHead x0 (Bind Void) -x1))).(\lambda (H10: (csuba g x0 c)).(eq_ind_r C (CHead x0 (Bind Void) x1) -(\lambda (c0: C).(or3 (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 -(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0 -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))))) (let H11 \def (H c d1 u1 H6 g x0 H10) in (or3_ind (ex2 C (\lambda -(d2: C).(drop n O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n -O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 -(Bind Void) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S -n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: -C).(drop n O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) -(or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead -d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Void) -x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda -(H13: (drop n O x0 (CHead x (Bind Abbr) u1))).(\lambda (H14: (csuba g x -d1)).(or3_intro0 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) -x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T -A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 -(Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C -(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop (Bind Void) n x0 -(CHead x (Bind Abbr) u1) H13 x1) H14))))) H12)) (\lambda (H12: (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x0 (CHead d2 -(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop n O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abbr) u1))) -(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 -(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead -x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))) (\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: -A).(\lambda (H13: (drop n O x0 (CHead x2 (Bind Abst) x3))).(\lambda (H14: -(csuba g x2 d1)).(\lambda (H15: (arity g x2 x3 (asucc g x4))).(\lambda (H16: -(arity g d1 u1 x4)).(or3_intro1 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead -x0 (Bind Void) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S -n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x2 x3 x4 -(drop_drop (Bind Void) n x0 (CHead x2 (Bind Abst) x3) H13 x1) H14 H15 -H16))))))))) H12)) (\lambda (H12: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(drop -n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g -d2 d1))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) -(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 -(Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda -(x2: C).(\lambda (x3: T).(\lambda (H13: (drop n O x0 (CHead x2 (Bind Void) -x3))).(\lambda (H14: (csuba g x2 d1)).(or3_intro2 (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abbr) u1))) -(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 -(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead -x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x2 x3 (drop_drop (Bind -Void) n x0 (CHead x2 (Bind Void) x3) H13 x1) H14)))))) H12)) H11)) c2 H9))))) -H8)) H7))))) (\lambda (H5: (csuba g c2 (CHead c (Bind Abst) t))).(\lambda -(H6: (drop (r (Bind Abst) n) O c (CHead d1 (Bind Abbr) u1))).(let H_x \def -(csuba_gen_abst_rev g c c2 t H5) in (let H7 \def H_x in (or_ind (ex2 C -(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba -g d2 c))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 c)))) (or3 -(ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda -(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H8: -(ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2: -C).(csuba g d2 c)))).(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind -Abst) t))) (\lambda (d2: C).(csuba g d2 c)) (or3 (ex2 C (\lambda (d2: -C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S -n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda (H9: (eq -C c2 (CHead x (Bind Abst) t))).(\lambda (H10: (csuba g x c)).(eq_ind_r C -(CHead x (Bind Abst) t) (\lambda (c0: C).(or3 (ex2 C (\lambda (d2: C).(drop -(S n) O c0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O -c0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O c0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda -(_: T).(csuba g d2 d1)))))) (let H11 \def (H c d1 u1 H6 g x H10) in (or3_ind -(ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(drop n O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda -(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(drop (S -n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 -C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abst) t) -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: -C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) -(or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 -(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) -t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: -C).(\lambda (H13: (drop n O x (CHead x0 (Bind Abbr) u1))).(\lambda (H14: -(csuba g x0 d1)).(or3_intro0 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x -(Bind Abst) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O -(CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abst) t) -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) -(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x0 (drop_drop -(Bind Abst) n x (CHead x0 (Bind Abbr) u1) H13 t) H14))))) H12)) (\lambda -(H12: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n -O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C -(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H13: (drop n O x (CHead x0 -(Bind Abst) x1))).(\lambda (H14: (csuba g x0 d1)).(\lambda (H15: (arity g x0 -x1 (asucc g x2))).(\lambda (H16: (arity g d1 u1 x2)).(or3_intro1 (ex2 C -(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex4_3_intro C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x -(Bind Abst) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a)))) x0 x1 x2 (drop_drop (Bind Abst) n -x (CHead x0 (Bind Abst) x1) H13 t) H14 H15 H16))))))))) H12)) (\lambda (H12: -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind -C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C -(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (H13: (drop n O x (CHead x0 (Bind Void) -x1))).(\lambda (H14: (csuba g x0 d1)).(or3_intro2 (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u1))) (\lambda -(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x -(Bind Abst) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1 (drop_drop (Bind -Abst) n x (CHead x0 (Bind Void) x1) H13 t) H14)))))) H12)) H11)) c2 H9)))) -H8)) (\lambda (H8: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -c))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 c))) (or3 -(ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda -(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (H9: (eq C c2 (CHead x0 (Bind Void) -x1))).(\lambda (H10: (csuba g x0 c)).(eq_ind_r C (CHead x0 (Bind Void) x1) -(\lambda (c0: C).(or3 (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 -(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0 -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))))) (let H11 \def (H c d1 u1 H6 g x0 H10) in (or3_ind (ex2 C (\lambda -(d2: C).(drop n O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n -O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 -(Bind Void) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S -n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: -C).(drop n O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) -(or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead -d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Void) -x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda -(H13: (drop n O x0 (CHead x (Bind Abbr) u1))).(\lambda (H14: (csuba g x -d1)).(or3_intro0 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) -x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T -A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 -(Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C -(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop (Bind Void) n x0 -(CHead x (Bind Abbr) u1) H13 x1) H14))))) H12)) (\lambda (H12: (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x0 (CHead d2 -(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop n O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abbr) u1))) -(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 -(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead -x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))) (\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: -A).(\lambda (H13: (drop n O x0 (CHead x2 (Bind Abst) x3))).(\lambda (H14: -(csuba g x2 d1)).(\lambda (H15: (arity g x2 x3 (asucc g x4))).(\lambda (H16: -(arity g d1 u1 x4)).(or3_intro1 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead -x0 (Bind Void) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S -n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x2 x3 x4 -(drop_drop (Bind Void) n x0 (CHead x2 (Bind Abst) x3) H13 x1) H14 H15 -H16))))))))) H12)) (\lambda (H12: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(drop -n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g -d2 d1))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) -(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 -(Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda -(x2: C).(\lambda (x3: T).(\lambda (H13: (drop n O x0 (CHead x2 (Bind Void) -x3))).(\lambda (H14: (csuba g x2 d1)).(or3_intro2 (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abbr) u1))) -(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 -(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead -x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x2 x3 (drop_drop (Bind -Void) n x0 (CHead x2 (Bind Void) x3) H13 x1) H14)))))) H12)) H11)) c2 H9))))) -H8)) H7))))) (\lambda (H5: (csuba g c2 (CHead c (Bind Void) t))).(\lambda -(H6: (drop (r (Bind Void) n) O c (CHead d1 (Bind Abbr) u1))).(let H_x \def -(csuba_gen_void_rev g c c2 t H5) in (let H7 \def H_x in (ex2_ind C (\lambda -(d2: C).(eq C c2 (CHead d2 (Bind Void) t))) (\lambda (d2: C).(csuba g d2 c)) -(or3 (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) -(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 -C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: -C).(\lambda (H8: (eq C c2 (CHead x (Bind Void) t))).(\lambda (H9: (csuba g x -c)).(eq_ind_r C (CHead x (Bind Void) t) (\lambda (c0: C).(or3 (ex2 C (\lambda -(d2: C).(drop (S n) O c0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba -g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop (S n) O c0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda -(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 (Bind Void) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H10 \def (H c d1 u1 H6 g x -H9) in (or3_ind (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 -C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C -(\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H11: (ex2 C -(\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop n O x (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) (or3 (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) u1))) (\lambda -(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x -(Bind Void) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (H12: (drop n O x (CHead x0 -(Bind Abbr) u1))).(\lambda (H13: (csuba g x0 d1)).(or3_intro0 (ex2 C (\lambda -(d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) u1))) -(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x -(Bind Void) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x -(Bind Void) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) -x0 (drop_drop (Bind Void) n x (CHead x0 (Bind Abbr) u1) H12 t) H13))))) H11)) -(\lambda (H11: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop n O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 -(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H12: (drop n O x (CHead x0 -(Bind Abst) x1))).(\lambda (H13: (csuba g x0 d1)).(\lambda (H14: (arity g x0 -x1 (asucc g x2))).(\lambda (H15: (arity g d1 u1 x2)).(or3_intro1 (ex2 C -(\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex4_3_intro C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x -(Bind Void) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a)))) x0 x1 x2 (drop_drop (Bind Void) n -x (CHead x0 (Bind Abst) x1) H12 t) H13 H14 H15))))))))) H11)) (\lambda (H11: -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind -C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C -(\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (H12: (drop n O x (CHead x0 (Bind Void) -x1))).(\lambda (H13: (csuba g x0 d1)).(or3_intro2 (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) u1))) (\lambda -(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x -(Bind Void) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1 (drop_drop (Bind -Void) n x (CHead x0 (Bind Void) x1) H12 t) H13)))))) H11)) H10)) c2 H8)))) -H7))))) b H3 H4)))) (\lambda (f: F).(\lambda (H3: (csuba g c2 (CHead c (Flat -f) t))).(\lambda (H4: (drop (r (Flat f) n) O c (CHead d1 (Bind Abbr) -u1))).(let H_x \def (csuba_gen_flat_rev g c c2 t f H3) in (let H5 \def H_x in -(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Flat f) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 c))) (or3 (ex2 C (\lambda -(d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba -g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda -(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (x1: -T).(\lambda (H6: (eq C c2 (CHead x0 (Flat f) x1))).(\lambda (H7: (csuba g x0 -c)).(eq_ind_r C (CHead x0 (Flat f) x1) (\lambda (c0: C).(or3 (ex2 C (\lambda -(d2: C).(drop (S n) O c0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba -g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop (S n) O c0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda -(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 (Bind Void) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H8 \def (H0 d1 u1 H4 g x0 -H7) in (or3_ind (ex2 C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O x0 (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O x0 (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C -(\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Void) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H9: (ex2 C (\lambda -(d2: C).(drop (S n) O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba -g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) (or3 (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u1))) (\lambda -(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f) -x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))) (\lambda (x: C).(\lambda (H10: (drop (S n) O x0 (CHead x (Bind Abbr) -u1))).(\lambda (H11: (csuba g x d1)).(or3_intro0 (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u1))) (\lambda -(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f) -x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) -(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop -(Flat f) n x0 (CHead x (Bind Abbr) u1) H10 x1) H11))))) H9)) (\lambda (H9: -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O -x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop (S n) O x0 (CHead d2 (Bind Abst) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C -(\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Void) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x2: C).(\lambda (x3: -T).(\lambda (x4: A).(\lambda (H10: (drop (S n) O x0 (CHead x2 (Bind Abst) -x3))).(\lambda (H11: (csuba g x2 d1)).(\lambda (H12: (arity g x2 x3 (asucc g -x4))).(\lambda (H13: (arity g d1 u1 x4)).(or3_intro1 (ex2 C (\lambda (d2: -C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u1))) (\lambda -(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f) -x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x2 x3 x4 -(drop_drop (Flat f) n x0 (CHead x2 (Bind Abst) x3) H10 x1) H11 H12 -H13))))))))) H9)) (\lambda (H9: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda -(_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda -(_: T).(csuba g d2 d1))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 -(Flat f) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O -(CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f) x1) (CHead -d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) -(\lambda (x2: C).(\lambda (x3: T).(\lambda (H10: (drop (S n) O x0 (CHead x2 -(Bind Void) x3))).(\lambda (H11: (csuba g x2 d1)).(or3_intro2 (ex2 C (\lambda -(d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u1))) -(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead -x0 (Flat f) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: -T).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Void) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1))) x2 x3 (drop_drop (Flat f) n x0 -(CHead x2 (Bind Void) x3) H10 x1) H11)))))) H9)) H8)) c2 H6))))) H5)))))) k -H2 (drop_gen_drop k c (CHead d1 (Bind Abbr) u1) t n H1)))))))))))) c1)))) i). -(* COMMENTS -Initial nodes: 23852 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/fwd.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/fwd.ma deleted file mode 100644 index a618761fe..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/fwd.ma +++ /dev/null @@ -1,1083 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csuba/defs.ma". - -theorem csuba_gen_abbr: - \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g -(CHead d1 (Bind Abbr) u) c) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 -(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))))) -\def - \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u: T).(\lambda (H: -(csuba g (CHead d1 (Bind Abbr) u) c)).(insert_eq C (CHead d1 (Bind Abbr) u) -(\lambda (c0: C).(csuba g c0 c)) (\lambda (_: C).(ex2 C (\lambda (d2: C).(eq -C c (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (\lambda -(y: C).(\lambda (H0: (csuba g y c)).(csuba_ind g (\lambda (c0: C).(\lambda -(c1: C).((eq C c0 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(eq C -c1 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))) (\lambda -(n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Abbr) u))).(let H2 -\def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return (\lambda (_: -C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow -False])) I (CHead d1 (Bind Abbr) u) H1) in (False_ind (ex2 C (\lambda (d2: -C).(eq C (CSort n) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 -d2))) H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 -c2)).(\lambda (H2: (((eq C c1 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda -(d2: C).(eq C c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 -d2)))))).(\lambda (k: K).(\lambda (u0: T).(\lambda (H3: (eq C (CHead c1 k u0) -(CHead d1 (Bind Abbr) u))).(let H4 \def (f_equal C C (\lambda (e: C).(match e -in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _ -_) \Rightarrow c0])) (CHead c1 k u0) (CHead d1 (Bind Abbr) u) H3) in ((let H5 -\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) -with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c1 k -u0) (CHead d1 (Bind Abbr) u) H3) in ((let H6 \def (f_equal C T (\lambda (e: -C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | -(CHead _ _ t) \Rightarrow t])) (CHead c1 k u0) (CHead d1 (Bind Abbr) u) H3) -in (\lambda (H7: (eq K k (Bind Abbr))).(\lambda (H8: (eq C c1 d1)).(eq_ind_r -T u (\lambda (t: T).(ex2 C (\lambda (d2: C).(eq C (CHead c2 k t) (CHead d2 -(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (eq_ind_r K (Bind Abbr) -(\lambda (k0: K).(ex2 C (\lambda (d2: C).(eq C (CHead c2 k0 u) (CHead d2 -(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H9 \def (eq_ind C -c1 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda -(d2: C).(eq C c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 -d2))))) H2 d1 H8) in (let H10 \def (eq_ind C c1 (\lambda (c0: C).(csuba g c0 -c2)) H1 d1 H8) in (ex_intro2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Abbr) -u) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) c2 -(refl_equal C (CHead c2 (Bind Abbr) u)) H10))) k H7) u0 H6)))) H5)) -H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1 -c2)).(\lambda (_: (((eq C c1 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda -(d2: C).(eq C c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 -d2)))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: -T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u1) (CHead d1 -(Bind Abbr) u))).(let H5 \def (eq_ind C (CHead c1 (Bind Void) u1) (\lambda -(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind b0) \Rightarrow (match b0 in B return (\lambda (_: -B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False | Void -\Rightarrow True]) | (Flat _) \Rightarrow False])])) I (CHead d1 (Bind Abbr) -u) H4) in (False_ind (ex2 C (\lambda (d2: C).(eq C (CHead c2 (Bind b) u2) -(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) H5))))))))))) -(\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1 c2)).(\lambda (_: -(((eq C c1 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(eq C c2 -(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))).(\lambda (t: -T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g a))).(\lambda (u0: -T).(\lambda (_: (arity g c2 u0 a)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) -t) (CHead d1 (Bind Abbr) u))).(let H6 \def (eq_ind C (CHead c1 (Bind Abst) t) -(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind b) \Rightarrow (match b in B return (\lambda (_: -B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow True | Void -\Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead d1 (Bind Abbr) -u) H5) in (False_ind (ex2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Abbr) u0) -(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) H6)))))))))))) -y c H0))) H))))). -(* COMMENTS -Initial nodes: 1117 -END *) - -theorem csuba_gen_void: - \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).((csuba g -(CHead d1 (Bind Void) u1) c) \to (ex2_3 B C T (\lambda (b: B).(\lambda (d2: -C).(\lambda (u2: T).(eq C c (CHead d2 (Bind b) u2))))) (\lambda (_: -B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))))))))) -\def - \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda -(H: (csuba g (CHead d1 (Bind Void) u1) c)).(insert_eq C (CHead d1 (Bind Void) -u1) (\lambda (c0: C).(csuba g c0 c)) (\lambda (_: C).(ex2_3 B C T (\lambda -(b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Bind b) u2))))) -(\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))))) -(\lambda (y: C).(\lambda (H0: (csuba g y c)).(csuba_ind g (\lambda (c0: -C).(\lambda (c1: C).((eq C c0 (CHead d1 (Bind Void) u1)) \to (ex2_3 B C T -(\lambda (b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 (Bind b) -u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 -d2)))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind -Void) u1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C -return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) -\Rightarrow False])) I (CHead d1 (Bind Void) u1) H1) in (False_ind (ex2_3 B C -T (\lambda (b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C (CSort n) (CHead d2 -(Bind b) u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 -d2))))) H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 -c2)).(\lambda (H2: (((eq C c1 (CHead d1 (Bind Void) u1)) \to (ex2_3 B C T -(\lambda (b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind b) -u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 -d2)))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c1 k u) -(CHead d1 (Bind Void) u1))).(let H4 \def (f_equal C C (\lambda (e: C).(match -e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _ -_) \Rightarrow c0])) (CHead c1 k u) (CHead d1 (Bind Void) u1) H3) in ((let H5 -\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) -with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c1 k -u) (CHead d1 (Bind Void) u1) H3) in ((let H6 \def (f_equal C T (\lambda (e: -C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | -(CHead _ _ t) \Rightarrow t])) (CHead c1 k u) (CHead d1 (Bind Void) u1) H3) -in (\lambda (H7: (eq K k (Bind Void))).(\lambda (H8: (eq C c1 d1)).(eq_ind_r -T u1 (\lambda (t: T).(ex2_3 B C T (\lambda (b: B).(\lambda (d2: C).(\lambda -(u2: T).(eq C (CHead c2 k t) (CHead d2 (Bind b) u2))))) (\lambda (_: -B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))))) (eq_ind_r K (Bind -Void) (\lambda (k0: K).(ex2_3 B C T (\lambda (b: B).(\lambda (d2: C).(\lambda -(u2: T).(eq C (CHead c2 k0 u1) (CHead d2 (Bind b) u2))))) (\lambda (_: -B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))))) (let H9 \def (eq_ind -C c1 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Void) u1)) \to (ex2_3 B C T -(\lambda (b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind b) -u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 -d2))))))) H2 d1 H8) in (let H10 \def (eq_ind C c1 (\lambda (c0: C).(csuba g -c0 c2)) H1 d1 H8) in (ex2_3_intro B C T (\lambda (b: B).(\lambda (d2: -C).(\lambda (u2: T).(eq C (CHead c2 (Bind Void) u1) (CHead d2 (Bind b) -u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))) -Void c2 u1 (refl_equal C (CHead c2 (Bind Void) u1)) H10))) k H7) u H6)))) -H5)) H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 -c2)).(\lambda (H2: (((eq C c1 (CHead d1 (Bind Void) u1)) \to (ex2_3 B C T -(\lambda (b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind b) -u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 -d2)))))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u0: -T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u0) (CHead d1 -(Bind Void) u1))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C -return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _ _) -\Rightarrow c0])) (CHead c1 (Bind Void) u0) (CHead d1 (Bind Void) u1) H4) in -((let H6 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: -C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead -c1 (Bind Void) u0) (CHead d1 (Bind Void) u1) H4) in (\lambda (H7: (eq C c1 -d1)).(let H8 \def (eq_ind C c1 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind -Void) u1)) \to (ex2_3 B C T (\lambda (b0: B).(\lambda (d2: C).(\lambda (u3: -T).(eq C c2 (CHead d2 (Bind b0) u3))))) (\lambda (_: B).(\lambda (d2: -C).(\lambda (_: T).(csuba g d1 d2))))))) H2 d1 H7) in (let H9 \def (eq_ind C -c1 (\lambda (c0: C).(csuba g c0 c2)) H1 d1 H7) in (ex2_3_intro B C T (\lambda -(b0: B).(\lambda (d2: C).(\lambda (u3: T).(eq C (CHead c2 (Bind b) u2) (CHead -d2 (Bind b0) u3))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba -g d1 d2)))) b c2 u2 (refl_equal C (CHead c2 (Bind b) u2)) H9))))) -H5))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1 -c2)).(\lambda (_: (((eq C c1 (CHead d1 (Bind Void) u1)) \to (ex2_3 B C T -(\lambda (b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind b) -u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 -d2)))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc -g a))).(\lambda (u: T).(\lambda (_: (arity g c2 u a)).(\lambda (H5: (eq C -(CHead c1 (Bind Abst) t) (CHead d1 (Bind Void) u1))).(let H6 \def (eq_ind C -(CHead c1 (Bind Abst) t) (\lambda (ee: C).(match ee in C return (\lambda (_: -C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match -k in K return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B -return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow -True | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead d1 -(Bind Void) u1) H5) in (False_ind (ex2_3 B C T (\lambda (b: B).(\lambda (d2: -C).(\lambda (u2: T).(eq C (CHead c2 (Bind Abbr) u) (CHead d2 (Bind b) u2))))) -(\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))))) -H6)))))))))))) y c H0))) H))))). -(* COMMENTS -Initial nodes: 1418 -END *) - -theorem csuba_gen_abst: - \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).((csuba g -(CHead d1 (Bind Abst) u1) c) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead -d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))))))) -\def - \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda -(H: (csuba g (CHead d1 (Bind Abst) u1) c)).(insert_eq C (CHead d1 (Bind Abst) -u1) (\lambda (c0: C).(csuba g c0 c)) (\lambda (_: C).(or (ex2 C (\lambda (d2: -C).(eq C c (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead -d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity -g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 a))))))) (\lambda (y: C).(\lambda (H0: (csuba g y -c)).(csuba_ind g (\lambda (c0: C).(\lambda (c1: C).((eq C c0 (CHead d1 (Bind -Abst) u1)) \to (or (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abst) -u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind Abbr) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))) -(\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Abst) -u1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return -(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) -\Rightarrow False])) I (CHead d1 (Bind Abst) u1) H1) in (False_ind (or (ex2 C -(\lambda (d2: C).(eq C (CSort n) (CHead d2 (Bind Abst) u1))) (\lambda (d2: -C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(eq C (CSort n) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) H2)))) -(\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda -(H2: (((eq C c1 (CHead d1 (Bind Abst) u1)) \to (or (ex2 C (\lambda (d2: -C).(eq C c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 -(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity -g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 a))))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: -(eq C (CHead c1 k u) (CHead d1 (Bind Abst) u1))).(let H4 \def (f_equal C C -(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow c1 | (CHead c0 _ _) \Rightarrow c0])) (CHead c1 k u) (CHead d1 -(Bind Abst) u1) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e in -C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) -\Rightarrow k0])) (CHead c1 k u) (CHead d1 (Bind Abst) u1) H3) in ((let H6 -\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u) -(CHead d1 (Bind Abst) u1) H3) in (\lambda (H7: (eq K k (Bind Abst))).(\lambda -(H8: (eq C c1 d1)).(eq_ind_r T u1 (\lambda (t: T).(or (ex2 C (\lambda (d2: -C).(eq C (CHead c2 k t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g -d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C -(CHead c2 k t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a))))))) (eq_ind_r K (Bind Abst) -(\lambda (k0: K).(or (ex2 C (\lambda (d2: C).(eq C (CHead c2 k0 u1) (CHead d2 -(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c2 k0 u1) (CHead d2 -(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 -(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 -u2 a))))))) (let H9 \def (eq_ind C c1 (\lambda (c0: C).((eq C c0 (CHead d1 -(Bind Abst) u1)) \to (or (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind -Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))))) H2 -d1 H8) in (let H10 \def (eq_ind C c1 (\lambda (c0: C).(csuba g c0 c2)) H1 d1 -H8) in (or_introl (ex2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Abst) u1) -(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abst) -u1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: -A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda -(a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: C).(eq C (CHead c2 -(Bind Abst) u1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) -c2 (refl_equal C (CHead c2 (Bind Abst) u1)) H10)))) k H7) u H6)))) H5)) -H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1 -c2)).(\lambda (_: (((eq C c1 (CHead d1 (Bind Abst) u1)) \to (or (ex2 C -(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba -g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq -C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: -A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda -(a: A).(arity g d2 u2 a))))))))).(\lambda (b: B).(\lambda (_: (not (eq B b -Void))).(\lambda (u0: T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind -Void) u0) (CHead d1 (Bind Abst) u1))).(let H5 \def (eq_ind C (CHead c1 (Bind -Void) u0) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with -[(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return -(\lambda (_: K).Prop) with [(Bind b0) \Rightarrow (match b0 in B return -(\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False | -Void \Rightarrow True]) | (Flat _) \Rightarrow False])])) I (CHead d1 (Bind -Abst) u1) H4) in (False_ind (or (ex2 C (\lambda (d2: C).(eq C (CHead c2 (Bind -b) u2) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 -C T A (\lambda (d2: C).(\lambda (u3: T).(\lambda (_: A).(eq C (CHead c2 (Bind -b) u2) (CHead d2 (Bind Abbr) u3))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u3: T).(\lambda (a: A).(arity g d2 u3 a)))))) H5))))))))))) (\lambda (c1: -C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c1 -(CHead d1 (Bind Abst) u1)) \to (or (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 -(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a))))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (H3: (arity g c1 t (asucc -g a))).(\lambda (u: T).(\lambda (H4: (arity g c2 u a)).(\lambda (H5: (eq C -(CHead c1 (Bind Abst) t) (CHead d1 (Bind Abst) u1))).(let H6 \def (f_equal C -C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow c1 | (CHead c0 _ _) \Rightarrow c0])) (CHead c1 (Bind Abst) t) -(CHead d1 (Bind Abst) u1) H5) in ((let H7 \def (f_equal C T (\lambda (e: -C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow t | -(CHead _ _ t0) \Rightarrow t0])) (CHead c1 (Bind Abst) t) (CHead d1 (Bind -Abst) u1) H5) in (\lambda (H8: (eq C c1 d1)).(let H9 \def (eq_ind T t -(\lambda (t0: T).(arity g c1 t0 (asucc g a))) H3 u1 H7) in (let H10 \def -(eq_ind C c1 (\lambda (c0: C).(arity g c0 u1 (asucc g a))) H9 d1 H8) in (let -H11 \def (eq_ind C c1 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abst) u1)) -\to (or (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) u1))) (\lambda -(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 (asucc g a0))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 a0)))))))) H2 d1 H8) -in (let H12 \def (eq_ind C c1 (\lambda (c0: C).(csuba g c0 c2)) H1 d1 H8) in -(or_intror (ex2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Abbr) u) (CHead d2 -(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) u) -(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity -g d1 u1 (asucc g a0))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: -A).(arity g d2 u2 a0))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) u) (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 (asucc g -a0))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 -a0)))) c2 u a (refl_equal C (CHead c2 (Bind Abbr) u)) H12 H10 H4)))))))) -H6)))))))))))) y c H0))) H))))). -(* COMMENTS -Initial nodes: 2550 -END *) - -theorem csuba_gen_flat: - \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).(\forall -(f: F).((csuba g (CHead d1 (Flat f) u1) c) \to (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d1 d2))))))))) -\def - \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda -(f: F).(\lambda (H: (csuba g (CHead d1 (Flat f) u1) c)).(insert_eq C (CHead -d1 (Flat f) u1) (\lambda (c0: C).(csuba g c0 c)) (\lambda (_: C).(ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d1 d2))))) (\lambda (y: C).(\lambda (H0: -(csuba g y c)).(csuba_ind g (\lambda (c0: C).(\lambda (c1: C).((eq C c0 -(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq -C c1 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 -d2))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Flat f) -u1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return -(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) -\Rightarrow False])) I (CHead d1 (Flat f) u1) H1) in (False_ind (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(eq C (CSort n) (CHead d2 (Flat f) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))) H2)))) (\lambda (c1: -C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c1 -(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq -C c2 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 -d2))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c1 k u) -(CHead d1 (Flat f) u1))).(let H4 \def (f_equal C C (\lambda (e: C).(match e -in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _ -_) \Rightarrow c0])) (CHead c1 k u) (CHead d1 (Flat f) u1) H3) in ((let H5 -\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) -with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c1 k -u) (CHead d1 (Flat f) u1) H3) in ((let H6 \def (f_equal C T (\lambda (e: -C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | -(CHead _ _ t) \Rightarrow t])) (CHead c1 k u) (CHead d1 (Flat f) u1) H3) in -(\lambda (H7: (eq K k (Flat f))).(\lambda (H8: (eq C c1 d1)).(eq_ind_r T u1 -(\lambda (t: T).(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c2 -k t) (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 -d2))))) (eq_ind_r K (Flat f) (\lambda (k0: K).(ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(eq C (CHead c2 k0 u1) (CHead d2 (Flat f) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d1 d2))))) (let H9 \def (eq_ind C c1 -(\lambda (c0: C).((eq C c0 (CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda -(d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Flat f) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d1 d2)))))) H2 d1 H8) in (let H10 \def (eq_ind C -c1 (\lambda (c0: C).(csuba g c0 c2)) H1 d1 H8) in (ex2_2_intro C T (\lambda -(d2: C).(\lambda (u2: T).(eq C (CHead c2 (Flat f) u1) (CHead d2 (Flat f) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))) c2 u1 (refl_equal C -(CHead c2 (Flat f) u1)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: -C).(\lambda (c2: C).(\lambda (_: (csuba g c1 c2)).(\lambda (_: (((eq C c1 -(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq -C c2 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 -d2))))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u0: -T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u0) (CHead d1 -(Flat f) u1))).(let H5 \def (eq_ind C (CHead c1 (Bind Void) u0) (\lambda (ee: -C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow -False | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) -with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead d1 -(Flat f) u1) H4) in (False_ind (ex2_2 C T (\lambda (d2: C).(\lambda (u3: -T).(eq C (CHead c2 (Bind b) u2) (CHead d2 (Flat f) u3)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d1 d2)))) H5))))))))))) (\lambda (c1: C).(\lambda -(c2: C).(\lambda (_: (csuba g c1 c2)).(\lambda (_: (((eq C c1 (CHead d1 (Flat -f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 -(Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 -d2))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g -a))).(\lambda (u: T).(\lambda (_: (arity g c2 u a)).(\lambda (H5: (eq C -(CHead c1 (Bind Abst) t) (CHead d1 (Flat f) u1))).(let H6 \def (eq_ind C -(CHead c1 (Bind Abst) t) (\lambda (ee: C).(match ee in C return (\lambda (_: -C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match -k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat -_) \Rightarrow False])])) I (CHead d1 (Flat f) u1) H5) in (False_ind (ex2_2 C -T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c2 (Bind Abbr) u) (CHead d2 -(Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))) -H6)))))))))))) y c H0))) H)))))). -(* COMMENTS -Initial nodes: 1183 -END *) - -theorem csuba_gen_bind: - \forall (g: G).(\forall (b1: B).(\forall (e1: C).(\forall (c2: C).(\forall -(v1: T).((csuba g (CHead e1 (Bind b1) v1) c2) \to (ex2_3 B C T (\lambda (b2: -B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) -(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2)))))))))) -\def - \lambda (g: G).(\lambda (b1: B).(\lambda (e1: C).(\lambda (c2: C).(\lambda -(v1: T).(\lambda (H: (csuba g (CHead e1 (Bind b1) v1) c2)).(insert_eq C -(CHead e1 (Bind b1) v1) (\lambda (c: C).(csuba g c c2)) (\lambda (_: -C).(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 -(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: -T).(csuba g e1 e2)))))) (\lambda (y: C).(\lambda (H0: (csuba g y -c2)).(csuba_ind g (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead e1 (Bind -b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: -T).(eq C c0 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: -C).(\lambda (_: T).(csuba g e1 e2)))))))) (\lambda (n: nat).(\lambda (H1: (eq -C (CSort n) (CHead e1 (Bind b1) v1))).(let H2 \def (eq_ind C (CSort n) -(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 (Bind b1) -v1) H1) in (False_ind (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda -(v2: T).(eq C (CSort n) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda -(e2: C).(\lambda (_: T).(csuba g e1 e2))))) H2)))) (\lambda (c1: C).(\lambda -(c3: C).(\lambda (H1: (csuba g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 -(Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda -(v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: -C).(\lambda (_: T).(csuba g e1 e2)))))))).(\lambda (k: K).(\lambda (u: -T).(\lambda (H3: (eq C (CHead c1 k u) (CHead e1 (Bind b1) v1))).(let H4 \def -(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with -[(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u) -(CHead e1 (Bind b1) v1) H3) in ((let H5 \def (f_equal C K (\lambda (e: -C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | -(CHead _ k0 _) \Rightarrow k0])) (CHead c1 k u) (CHead e1 (Bind b1) v1) H3) -in ((let H6 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda -(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) -(CHead c1 k u) (CHead e1 (Bind b1) v1) H3) in (\lambda (H7: (eq K k (Bind -b1))).(\lambda (H8: (eq C c1 e1)).(eq_ind_r T v1 (\lambda (t: T).(ex2_3 B C T -(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 k t) -(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: -T).(csuba g e1 e2)))))) (eq_ind_r K (Bind b1) (\lambda (k0: K).(ex2_3 B C T -(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 k0 v1) -(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: -T).(csuba g e1 e2)))))) (let H9 \def (eq_ind C c1 (\lambda (c: C).((eq C c -(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: -C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: -B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2))))))) H2 e1 H8) in (let -H10 \def (eq_ind C c1 (\lambda (c: C).(csuba g c c3)) H1 e1 H8) in -(ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C -(CHead c3 (Bind b1) v1) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda -(e2: C).(\lambda (_: T).(csuba g e1 e2)))) b1 c3 v1 (refl_equal C (CHead c3 -(Bind b1) v1)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: -C).(\lambda (c3: C).(\lambda (H1: (csuba g c1 c3)).(\lambda (H2: (((eq C c1 -(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: -C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: -B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2)))))))).(\lambda (b: -B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: -T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) -v1))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda -(_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) -(CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) v1) H4) in ((let H6 \def -(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with -[(CSort _) \Rightarrow Void | (CHead _ k _) \Rightarrow (match k in K return -(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow -Void])])) (CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) v1) H4) in ((let H7 -\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead c1 -(Bind Void) u1) (CHead e1 (Bind b1) v1) H4) in (\lambda (H8: (eq B Void -b1)).(\lambda (H9: (eq C c1 e1)).(let H10 \def (eq_ind C c1 (\lambda (c: -C).((eq C c (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: -B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) -(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2))))))) H2 e1 -H9) in (let H11 \def (eq_ind C c1 (\lambda (c: C).(csuba g c c3)) H1 e1 H9) -in (let H12 \def (eq_ind_r B b1 (\lambda (b0: B).((eq C e1 (CHead e1 (Bind -b0) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: -T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: -C).(\lambda (_: T).(csuba g e1 e2))))))) H10 Void H8) in (ex2_3_intro B C T -(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind b) -u2) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: -T).(csuba g e1 e2)))) b c3 u2 (refl_equal C (CHead c3 (Bind b) u2)) -H11))))))) H6)) H5))))))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1: -(csuba g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind b1) v1)) \to (ex2_3 -B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 -(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g -e1 e2)))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (H3: (arity g c1 t -(asucc g a))).(\lambda (u: T).(\lambda (_: (arity g c3 u a)).(\lambda (H5: -(eq C (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1))).(let H6 \def -(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with -[(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 (Bind -Abst) t) (CHead e1 (Bind b1) v1) H5) in ((let H7 \def (f_equal C B (\lambda -(e: C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow -Abst | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with -[(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead c1 (Bind -Abst) t) (CHead e1 (Bind b1) v1) H5) in ((let H8 \def (f_equal C T (\lambda -(e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow t -| (CHead _ _ t0) \Rightarrow t0])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind -b1) v1) H5) in (\lambda (H9: (eq B Abst b1)).(\lambda (H10: (eq C c1 -e1)).(let H11 \def (eq_ind T t (\lambda (t0: T).(arity g c1 t0 (asucc g a))) -H3 v1 H8) in (let H12 \def (eq_ind C c1 (\lambda (c: C).(arity g c v1 (asucc -g a))) H11 e1 H10) in (let H13 \def (eq_ind C c1 (\lambda (c: C).((eq C c -(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: -C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: -B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2))))))) H2 e1 H10) in (let -H14 \def (eq_ind C c1 (\lambda (c: C).(csuba g c c3)) H1 e1 H10) in (let H15 -\def (eq_ind_r B b1 (\lambda (b: B).((eq C e1 (CHead e1 (Bind b) v1)) \to -(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 -(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: -T).(csuba g e1 e2))))))) H13 Abst H9) in (ex2_3_intro B C T (\lambda (b2: -B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 -(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g -e1 e2)))) Abbr c3 u (refl_equal C (CHead c3 (Bind Abbr) u)) H14))))))))) H7)) -H6)))))))))))) y c2 H0))) H)))))). -(* COMMENTS -Initial nodes: 1889 -END *) - -theorem csuba_gen_abst_rev: - \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g c -(CHead d1 (Bind Abst) u)) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead d2 -(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(eq C c (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1))))))))) -\def - \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u: T).(\lambda (H: -(csuba g c (CHead d1 (Bind Abst) u))).(insert_eq C (CHead d1 (Bind Abst) u) -(\lambda (c0: C).(csuba g c c0)) (\lambda (_: C).(or (ex2 C (\lambda (d2: -C).(eq C c (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (\lambda (y: -C).(\lambda (H0: (csuba g c y)).(csuba_ind g (\lambda (c0: C).(\lambda (c1: -C).((eq C c1 (CHead d1 (Bind Abst) u)) \to (or (ex2 C (\lambda (d2: C).(eq C -c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(eq C c0 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))) (\lambda (n: -nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Abst) u))).(let H2 \def -(eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return (\lambda (_: -C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow -False])) I (CHead d1 (Bind Abst) u) H1) in (False_ind (or (ex2 C (\lambda -(d2: C).(eq C (CSort n) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g -d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CSort n) (CHead -d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) -H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 -c2)).(\lambda (H2: (((eq C c2 (CHead d1 (Bind Abst) u)) \to (or (ex2 C -(\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba -g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))))))).(\lambda (k: K).(\lambda (u0: T).(\lambda (H3: (eq C (CHead c2 k -u0) (CHead d1 (Bind Abst) u))).(let H4 \def (f_equal C C (\lambda (e: -C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | -(CHead c0 _ _) \Rightarrow c0])) (CHead c2 k u0) (CHead d1 (Bind Abst) u) H3) -in ((let H5 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda -(_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) -(CHead c2 k u0) (CHead d1 (Bind Abst) u) H3) in ((let H6 \def (f_equal C T -(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) -\Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead c2 k u0) (CHead d1 -(Bind Abst) u) H3) in (\lambda (H7: (eq K k (Bind Abst))).(\lambda (H8: (eq C -c2 d1)).(eq_ind_r T u (\lambda (t: T).(or (ex2 C (\lambda (d2: C).(eq C -(CHead c1 k t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 k t) (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) -(eq_ind_r K (Bind Abst) (\lambda (k0: K).(or (ex2 C (\lambda (d2: C).(eq C -(CHead c1 k0 u) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 k0 u) (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let -H9 \def (eq_ind C c2 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abst) u)) \to -(or (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))))) H2 d1 H8) in (let H10 \def (eq_ind C c2 (\lambda (c0: C).(csuba g -c1 c0)) H1 d1 H8) in (or_introl (ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind -Abst) u) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 -C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 (Bind Abst) u) (CHead -d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) -(ex_intro2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abst) u) (CHead d2 (Bind -Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) c1 (refl_equal C (CHead c1 (Bind -Abst) u)) H10)))) k H7) u0 H6)))) H5)) H4))))))))) (\lambda (c1: C).(\lambda -(c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c2 (CHead d1 -(Bind Abst) u)) \to (or (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind -Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(eq C c1 (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))))))).(\lambda (b: B).(\lambda (H3: (not -(eq B b Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead -c2 (Bind b) u2) (CHead d1 (Bind Abst) u))).(let H5 \def (f_equal C C (\lambda -(e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 -| (CHead c0 _ _) \Rightarrow c0])) (CHead c2 (Bind b) u2) (CHead d1 (Bind -Abst) u) H4) in ((let H6 \def (f_equal C B (\lambda (e: C).(match e in C -return (\lambda (_: C).B) with [(CSort _) \Rightarrow b | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b0) -\Rightarrow b0 | (Flat _) \Rightarrow b])])) (CHead c2 (Bind b) u2) (CHead d1 -(Bind Abst) u) H4) in ((let H7 \def (f_equal C T (\lambda (e: C).(match e in -C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u2 | (CHead _ _ t) -\Rightarrow t])) (CHead c2 (Bind b) u2) (CHead d1 (Bind Abst) u) H4) in -(\lambda (H8: (eq B b Abst)).(\lambda (H9: (eq C c2 d1)).(let H10 \def -(eq_ind B b (\lambda (b0: B).(not (eq B b0 Void))) H3 Abst H8) in (let H11 -\def (eq_ind C c2 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abst) u)) \to -(or (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u3: T).(eq C c1 -(CHead d2 (Bind Void) u3)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))))) H2 d1 H9) in (let H12 \def (eq_ind C c2 (\lambda (c0: C).(csuba g -c1 c0)) H1 d1 H9) in (or_intror (ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind -Void) u1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u3: T).(eq C (CHead c1 (Bind Void) u1) -(CHead d2 (Bind Void) u3)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u3: T).(eq C (CHead c1 -(Bind Void) u1) (CHead d2 (Bind Void) u3)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))) c1 u1 (refl_equal C (CHead c1 (Bind Void) u1)) -H12)))))))) H6)) H5))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: -(csuba g c1 c2)).(\lambda (_: (((eq C c2 (CHead d1 (Bind Abst) u)) \to (or -(ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc -g a))).(\lambda (u0: T).(\lambda (_: (arity g c2 u0 a)).(\lambda (H5: (eq C -(CHead c2 (Bind Abbr) u0) (CHead d1 (Bind Abst) u))).(let H6 \def (eq_ind C -(CHead c2 (Bind Abbr) u0) (\lambda (ee: C).(match ee in C return (\lambda (_: -C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match -k in K return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B -return (\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow -False | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead -d1 (Bind Abst) u) H5) in (False_ind (or (ex2 C (\lambda (d2: C).(eq C (CHead -c1 (Bind Abst) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 -d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 (Bind -Abst) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba -g d2 d1))))) H6)))))))))))) c y H0))) H))))). -(* COMMENTS -Initial nodes: 1980 -END *) - -theorem csuba_gen_void_rev: - \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g c -(CHead d1 (Bind Void) u)) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind -Void) u))) (\lambda (d2: C).(csuba g d2 d1))))))) -\def - \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u: T).(\lambda (H: -(csuba g c (CHead d1 (Bind Void) u))).(insert_eq C (CHead d1 (Bind Void) u) -(\lambda (c0: C).(csuba g c c0)) (\lambda (_: C).(ex2 C (\lambda (d2: C).(eq -C c (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))) (\lambda -(y: C).(\lambda (H0: (csuba g c y)).(csuba_ind g (\lambda (c0: C).(\lambda -(c1: C).((eq C c1 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda (d2: C).(eq C -c0 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))))) (\lambda -(n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Void) u))).(let H2 -\def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return (\lambda (_: -C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow -False])) I (CHead d1 (Bind Void) u) H1) in (False_ind (ex2 C (\lambda (d2: -C).(eq C (CSort n) (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 -d1))) H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 -c2)).(\lambda (H2: (((eq C c2 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda -(d2: C).(eq C c1 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 -d1)))))).(\lambda (k: K).(\lambda (u0: T).(\lambda (H3: (eq C (CHead c2 k u0) -(CHead d1 (Bind Void) u))).(let H4 \def (f_equal C C (\lambda (e: C).(match e -in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ -_) \Rightarrow c0])) (CHead c2 k u0) (CHead d1 (Bind Void) u) H3) in ((let H5 -\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) -with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c2 k -u0) (CHead d1 (Bind Void) u) H3) in ((let H6 \def (f_equal C T (\lambda (e: -C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | -(CHead _ _ t) \Rightarrow t])) (CHead c2 k u0) (CHead d1 (Bind Void) u) H3) -in (\lambda (H7: (eq K k (Bind Void))).(\lambda (H8: (eq C c2 d1)).(eq_ind_r -T u (\lambda (t: T).(ex2 C (\lambda (d2: C).(eq C (CHead c1 k t) (CHead d2 -(Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))) (eq_ind_r K (Bind Void) -(\lambda (k0: K).(ex2 C (\lambda (d2: C).(eq C (CHead c1 k0 u) (CHead d2 -(Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))) (let H9 \def (eq_ind C -c2 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda -(d2: C).(eq C c1 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 -d1))))) H2 d1 H8) in (let H10 \def (eq_ind C c2 (\lambda (c0: C).(csuba g c1 -c0)) H1 d1 H8) in (ex_intro2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Void) -u) (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)) c1 -(refl_equal C (CHead c1 (Bind Void) u)) H10))) k H7) u0 H6)))) H5)) -H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 -c2)).(\lambda (H2: (((eq C c2 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda -(d2: C).(eq C c1 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 -d1)))))).(\lambda (b: B).(\lambda (H3: (not (eq B b Void))).(\lambda (u1: -T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c2 (Bind b) u2) (CHead d1 -(Bind Void) u))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C -return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ _) -\Rightarrow c0])) (CHead c2 (Bind b) u2) (CHead d1 (Bind Void) u) H4) in -((let H6 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: -C).B) with [(CSort _) \Rightarrow b | (CHead _ k _) \Rightarrow (match k in K -return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) -\Rightarrow b])])) (CHead c2 (Bind b) u2) (CHead d1 (Bind Void) u) H4) in -((let H7 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: -C).T) with [(CSort _) \Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead -c2 (Bind b) u2) (CHead d1 (Bind Void) u) H4) in (\lambda (H8: (eq B b -Void)).(\lambda (H9: (eq C c2 d1)).(let H10 \def (eq_ind B b (\lambda (b0: -B).(not (eq B b0 Void))) H3 Void H8) in (let H11 \def (eq_ind C c2 (\lambda -(c0: C).((eq C c0 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda (d2: C).(eq C -c1 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1))))) H2 d1 H9) -in (let H12 \def (eq_ind C c2 (\lambda (c0: C).(csuba g c1 c0)) H1 d1 H9) in -(let H13 \def (match (H10 (refl_equal B Void)) in False return (\lambda (_: -False).(ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Void) u1) (CHead d2 -(Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))) with []) in H13))))))) -H6)) H5))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1 -c2)).(\lambda (_: (((eq C c2 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda -(d2: C).(eq C c1 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 -d1)))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g -a))).(\lambda (u0: T).(\lambda (_: (arity g c2 u0 a)).(\lambda (H5: (eq C -(CHead c2 (Bind Abbr) u0) (CHead d1 (Bind Void) u))).(let H6 \def (eq_ind C -(CHead c2 (Bind Abbr) u0) (\lambda (ee: C).(match ee in C return (\lambda (_: -C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match -k in K return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B -return (\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow -False | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead -d1 (Bind Void) u) H5) in (False_ind (ex2 C (\lambda (d2: C).(eq C (CHead c1 -(Bind Abst) t) (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1))) -H6)))))))))))) c y H0))) H))))). -(* COMMENTS -Initial nodes: 1326 -END *) - -theorem csuba_gen_abbr_rev: - \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).((csuba g c -(CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(eq C c (CHead d2 -(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))) -\def - \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda -(H: (csuba g c (CHead d1 (Bind Abbr) u1))).(insert_eq C (CHead d1 (Bind Abbr) -u1) (\lambda (c0: C).(csuba g c c0)) (\lambda (_: C).(or3 (ex2 C (\lambda -(d2: C).(eq C c (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq -C c (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))))) (\lambda (y: C).(\lambda (H0: (csuba g c y)).(csuba_ind g (\lambda -(c0: C).(\lambda (c1: C).((eq C c1 (CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C -(\lambda (d2: C).(eq C c0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba -g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq -C c0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq -C c0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g -d2 d1)))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind -Abbr) u1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C -return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) -\Rightarrow False])) I (CHead d1 (Bind Abbr) u1) H1) in (False_ind (or3 (ex2 -C (\lambda (d2: C).(eq C (CSort n) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(eq C (CSort n) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(eq C (CSort n) (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) H2)))) (\lambda -(c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C -c2 (CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(eq C c1 -(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c2 k u) -(CHead d1 (Bind Abbr) u1))).(let H4 \def (f_equal C C (\lambda (e: C).(match -e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ -_) \Rightarrow c0])) (CHead c2 k u) (CHead d1 (Bind Abbr) u1) H3) in ((let H5 -\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) -with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c2 k -u) (CHead d1 (Bind Abbr) u1) H3) in ((let H6 \def (f_equal C T (\lambda (e: -C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | -(CHead _ _ t) \Rightarrow t])) (CHead c2 k u) (CHead d1 (Bind Abbr) u1) H3) -in (\lambda (H7: (eq K k (Bind Abbr))).(\lambda (H8: (eq C c2 d1)).(eq_ind_r -T u1 (\lambda (t: T).(or3 (ex2 C (\lambda (d2: C).(eq C (CHead c1 k t) (CHead -d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c1 k t) (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 k t) -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))))) (eq_ind_r K (Bind Abbr) (\lambda (k0: K).(or3 (ex2 C (\lambda (d2: -C).(eq C (CHead c1 k0 u1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba -g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq -C (CHead c1 k0 u1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda -(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(eq C (CHead c1 k0 u1) (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H9 \def (eq_ind C -c2 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C -(\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba -g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq -C c1 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq -C c1 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g -d2 d1))))))) H2 d1 H8) in (let H10 \def (eq_ind C c2 (\lambda (c0: C).(csuba -g c1 c0)) H1 d1 H8) in (or3_intro0 (ex2 C (\lambda (d2: C).(eq C (CHead c1 -(Bind Abbr) u1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C -(CHead c1 (Bind Abbr) u1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 (Bind Abbr) u1) (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) -(ex_intro2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abbr) u1) (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) c1 (refl_equal C (CHead c1 -(Bind Abbr) u1)) H10)))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: -C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c2 -(CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(eq C c1 (CHead -d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))).(\lambda (b: -B).(\lambda (H3: (not (eq B b Void))).(\lambda (u0: T).(\lambda (u2: -T).(\lambda (H4: (eq C (CHead c2 (Bind b) u2) (CHead d1 (Bind Abbr) -u1))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda -(_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) -(CHead c2 (Bind b) u2) (CHead d1 (Bind Abbr) u1) H4) in ((let H6 \def -(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with -[(CSort _) \Rightarrow b | (CHead _ k _) \Rightarrow (match k in K return -(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow -b])])) (CHead c2 (Bind b) u2) (CHead d1 (Bind Abbr) u1) H4) in ((let H7 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c2 (Bind b) -u2) (CHead d1 (Bind Abbr) u1) H4) in (\lambda (H8: (eq B b Abbr)).(\lambda -(H9: (eq C c2 d1)).(let H10 \def (eq_ind B b (\lambda (b0: B).(not (eq B b0 -Void))) H3 Abbr H8) in (let H11 \def (eq_ind C c2 (\lambda (c0: C).((eq C c0 -(CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(eq C c1 (CHead -d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u3: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind Abst) -u3))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u3: T).(\lambda (a: A).(arity g d2 u3 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u3: T).(eq C c1 (CHead d2 (Bind Void) -u3)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))) H2 d1 H9) in -(let H12 \def (eq_ind C c2 (\lambda (c0: C).(csuba g c1 c0)) H1 d1 H9) in -(or3_intro2 (ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Void) u0) (CHead d2 -(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u3: T).(\lambda (_: A).(eq C (CHead c1 (Bind Void) u0) -(CHead d2 (Bind Abst) u3))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u3: T).(\lambda (a: -A).(arity g d2 u3 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u3: T).(eq -C (CHead c1 (Bind Void) u0) (CHead d2 (Bind Void) u3)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: -C).(\lambda (u3: T).(eq C (CHead c1 (Bind Void) u0) (CHead d2 (Bind Void) -u3)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) c1 u0 (refl_equal C -(CHead c1 (Bind Void) u0)) H12)))))))) H6)) H5))))))))))) (\lambda (c1: -C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c2 -(CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(eq C c1 (CHead -d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))).(\lambda (t: -T).(\lambda (a: A).(\lambda (H3: (arity g c1 t (asucc g a))).(\lambda (u: -T).(\lambda (H4: (arity g c2 u a)).(\lambda (H5: (eq C (CHead c2 (Bind Abbr) -u) (CHead d1 (Bind Abbr) u1))).(let H6 \def (f_equal C C (\lambda (e: -C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | -(CHead c0 _ _) \Rightarrow c0])) (CHead c2 (Bind Abbr) u) (CHead d1 (Bind -Abbr) u1) H5) in ((let H7 \def (f_equal C T (\lambda (e: C).(match e in C -return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) -\Rightarrow t0])) (CHead c2 (Bind Abbr) u) (CHead d1 (Bind Abbr) u1) H5) in -(\lambda (H8: (eq C c2 d1)).(let H9 \def (eq_ind T u (\lambda (t0: T).(arity -g c2 t0 a)) H4 u1 H7) in (let H10 \def (eq_ind C c2 (\lambda (c0: C).(arity g -c0 u1 a)) H9 d1 H8) in (let H11 \def (eq_ind C c2 (\lambda (c0: C).((eq C c0 -(CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(eq C c1 (CHead -d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 (asucc g -a0))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 -a0))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))) H2 d1 H8) -in (let H12 \def (eq_ind C c2 (\lambda (c0: C).(csuba g c1 c0)) H1 d1 H8) in -(or3_intro1 (ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abst) t) (CHead d2 -(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) t) -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: -A).(arity g d2 u2 (asucc g a0))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a0: A).(arity g d1 u1 a0))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(eq C (CHead c1 (Bind Abst) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))) (ex4_3_intro C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) t) (CHead d2 -(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 -(asucc g a0))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 -u1 a0)))) c1 t a (refl_equal C (CHead c1 (Bind Abst) t)) H12 H3 H10)))))))) -H6)))))))))))) c y H0))) H))))). -(* COMMENTS -Initial nodes: 3459 -END *) - -theorem csuba_gen_flat_rev: - \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).(\forall -(f: F).((csuba g c (CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1))))))))) -\def - \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda -(f: F).(\lambda (H: (csuba g c (CHead d1 (Flat f) u1))).(insert_eq C (CHead -d1 (Flat f) u1) (\lambda (c0: C).(csuba g c c0)) (\lambda (_: C).(ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (y: C).(\lambda (H0: -(csuba g c y)).(csuba_ind g (\lambda (c0: C).(\lambda (c1: C).((eq C c1 -(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq -C c0 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Flat f) -u1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return -(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) -\Rightarrow False])) I (CHead d1 (Flat f) u1) H1) in (False_ind (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(eq C (CSort n) (CHead d2 (Flat f) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) H2)))) (\lambda (c1: -C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c2 -(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq -C c1 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c2 k u) -(CHead d1 (Flat f) u1))).(let H4 \def (f_equal C C (\lambda (e: C).(match e -in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ -_) \Rightarrow c0])) (CHead c2 k u) (CHead d1 (Flat f) u1) H3) in ((let H5 -\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) -with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c2 k -u) (CHead d1 (Flat f) u1) H3) in ((let H6 \def (f_equal C T (\lambda (e: -C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | -(CHead _ _ t) \Rightarrow t])) (CHead c2 k u) (CHead d1 (Flat f) u1) H3) in -(\lambda (H7: (eq K k (Flat f))).(\lambda (H8: (eq C c2 d1)).(eq_ind_r T u1 -(\lambda (t: T).(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 -k t) (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))) (eq_ind_r K (Flat f) (\lambda (k0: K).(ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(eq C (CHead c1 k0 u1) (CHead d2 (Flat f) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (let H9 \def (eq_ind C c2 -(\lambda (c0: C).((eq C c0 (CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda -(d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 (Flat f) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))))) H2 d1 H8) in (let H10 \def (eq_ind C -c2 (\lambda (c0: C).(csuba g c1 c0)) H1 d1 H8) in (ex2_2_intro C T (\lambda -(d2: C).(\lambda (u2: T).(eq C (CHead c1 (Flat f) u1) (CHead d2 (Flat f) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) c1 u1 (refl_equal C -(CHead c1 (Flat f) u1)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: -C).(\lambda (c2: C).(\lambda (_: (csuba g c1 c2)).(\lambda (_: (((eq C c2 -(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq -C c1 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u0: -T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c2 (Bind b) u2) (CHead d1 -(Flat f) u1))).(let H5 \def (eq_ind C (CHead c2 (Bind b) u2) (\lambda (ee: -C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow -False | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) -with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead d1 -(Flat f) u1) H4) in (False_ind (ex2_2 C T (\lambda (d2: C).(\lambda (u3: -T).(eq C (CHead c1 (Bind Void) u0) (CHead d2 (Flat f) u3)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))) H5))))))))))) (\lambda (c1: C).(\lambda -(c2: C).(\lambda (_: (csuba g c1 c2)).(\lambda (_: (((eq C c2 (CHead d1 (Flat -f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 -(Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g -a))).(\lambda (u: T).(\lambda (_: (arity g c2 u a)).(\lambda (H5: (eq C -(CHead c2 (Bind Abbr) u) (CHead d1 (Flat f) u1))).(let H6 \def (eq_ind C -(CHead c2 (Bind Abbr) u) (\lambda (ee: C).(match ee in C return (\lambda (_: -C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match -k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat -_) \Rightarrow False])])) I (CHead d1 (Flat f) u1) H5) in (False_ind (ex2_2 C -T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 (Bind Abst) t) (CHead d2 -(Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) -H6)))))))))))) c y H0))) H)))))). -(* COMMENTS -Initial nodes: 1183 -END *) - -theorem csuba_gen_bind_rev: - \forall (g: G).(\forall (b1: B).(\forall (e1: C).(\forall (c2: C).(\forall -(v1: T).((csuba g c2 (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: -B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) -(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1)))))))))) -\def - \lambda (g: G).(\lambda (b1: B).(\lambda (e1: C).(\lambda (c2: C).(\lambda -(v1: T).(\lambda (H: (csuba g c2 (CHead e1 (Bind b1) v1))).(insert_eq C -(CHead e1 (Bind b1) v1) (\lambda (c: C).(csuba g c2 c)) (\lambda (_: -C).(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 -(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: -T).(csuba g e2 e1)))))) (\lambda (y: C).(\lambda (H0: (csuba g c2 -y)).(csuba_ind g (\lambda (c: C).(\lambda (c0: C).((eq C c0 (CHead e1 (Bind -b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: -T).(eq C c (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: -C).(\lambda (_: T).(csuba g e2 e1)))))))) (\lambda (n: nat).(\lambda (H1: (eq -C (CSort n) (CHead e1 (Bind b1) v1))).(let H2 \def (eq_ind C (CSort n) -(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 (Bind b1) -v1) H1) in (False_ind (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda -(v2: T).(eq C (CSort n) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda -(e2: C).(\lambda (_: T).(csuba g e2 e1))))) H2)))) (\lambda (c1: C).(\lambda -(c3: C).(\lambda (H1: (csuba g c1 c3)).(\lambda (H2: (((eq C c3 (CHead e1 -(Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda -(v2: T).(eq C c1 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: -C).(\lambda (_: T).(csuba g e2 e1)))))))).(\lambda (k: K).(\lambda (u: -T).(\lambda (H3: (eq C (CHead c3 k u) (CHead e1 (Bind b1) v1))).(let H4 \def -(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with -[(CSort _) \Rightarrow c3 | (CHead c _ _) \Rightarrow c])) (CHead c3 k u) -(CHead e1 (Bind b1) v1) H3) in ((let H5 \def (f_equal C K (\lambda (e: -C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | -(CHead _ k0 _) \Rightarrow k0])) (CHead c3 k u) (CHead e1 (Bind b1) v1) H3) -in ((let H6 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda -(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) -(CHead c3 k u) (CHead e1 (Bind b1) v1) H3) in (\lambda (H7: (eq K k (Bind -b1))).(\lambda (H8: (eq C c3 e1)).(eq_ind_r T v1 (\lambda (t: T).(ex2_3 B C T -(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c1 k t) -(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: -T).(csuba g e2 e1)))))) (eq_ind_r K (Bind b1) (\lambda (k0: K).(ex2_3 B C T -(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c1 k0 v1) -(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: -T).(csuba g e2 e1)))))) (let H9 \def (eq_ind C c3 (\lambda (c: C).((eq C c -(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: -C).(\lambda (v2: T).(eq C c1 (CHead e2 (Bind b2) v2))))) (\lambda (_: -B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1))))))) H2 e1 H8) in (let -H10 \def (eq_ind C c3 (\lambda (c: C).(csuba g c1 c)) H1 e1 H8) in -(ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C -(CHead c1 (Bind b1) v1) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda -(e2: C).(\lambda (_: T).(csuba g e2 e1)))) b1 c1 v1 (refl_equal C (CHead c1 -(Bind b1) v1)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: -C).(\lambda (c3: C).(\lambda (H1: (csuba g c1 c3)).(\lambda (H2: (((eq C c3 -(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: -C).(\lambda (v2: T).(eq C c1 (CHead e2 (Bind b2) v2))))) (\lambda (_: -B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1)))))))).(\lambda (b: -B).(\lambda (H3: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: -T).(\lambda (H4: (eq C (CHead c3 (Bind b) u2) (CHead e1 (Bind b1) v1))).(let -H5 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) -with [(CSort _) \Rightarrow c3 | (CHead c _ _) \Rightarrow c])) (CHead c3 -(Bind b) u2) (CHead e1 (Bind b1) v1) H4) in ((let H6 \def (f_equal C B -(\lambda (e: C).(match e in C return (\lambda (_: C).B) with [(CSort _) -\Rightarrow b | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: -K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow b])])) (CHead c3 -(Bind b) u2) (CHead e1 (Bind b1) v1) H4) in ((let H7 \def (f_equal C T -(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) -\Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c3 (Bind b) u2) (CHead -e1 (Bind b1) v1) H4) in (\lambda (H8: (eq B b b1)).(\lambda (H9: (eq C c3 -e1)).(let H10 \def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Void))) H3 b1 -H8) in (let H11 \def (eq_ind C c3 (\lambda (c: C).((eq C c (CHead e1 (Bind -b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: -T).(eq C c1 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: -C).(\lambda (_: T).(csuba g e2 e1))))))) H2 e1 H9) in (let H12 \def (eq_ind C -c3 (\lambda (c: C).(csuba g c1 c)) H1 e1 H9) in (ex2_3_intro B C T (\lambda -(b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c1 (Bind Void) u1) -(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: -T).(csuba g e2 e1)))) Void c1 u1 (refl_equal C (CHead c1 (Bind Void) u1)) -H12))))))) H6)) H5))))))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1: -(csuba g c1 c3)).(\lambda (H2: (((eq C c3 (CHead e1 (Bind b1) v1)) \to (ex2_3 -B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c1 (CHead e2 -(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g -e2 e1)))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t -(asucc g a))).(\lambda (u: T).(\lambda (H4: (arity g c3 u a)).(\lambda (H5: -(eq C (CHead c3 (Bind Abbr) u) (CHead e1 (Bind b1) v1))).(let H6 \def -(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with -[(CSort _) \Rightarrow c3 | (CHead c _ _) \Rightarrow c])) (CHead c3 (Bind -Abbr) u) (CHead e1 (Bind b1) v1) H5) in ((let H7 \def (f_equal C B (\lambda -(e: C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow -Abbr | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with -[(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead c3 (Bind -Abbr) u) (CHead e1 (Bind b1) v1) H5) in ((let H8 \def (f_equal C T (\lambda -(e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u -| (CHead _ _ t0) \Rightarrow t0])) (CHead c3 (Bind Abbr) u) (CHead e1 (Bind -b1) v1) H5) in (\lambda (H9: (eq B Abbr b1)).(\lambda (H10: (eq C c3 -e1)).(let H11 \def (eq_ind T u (\lambda (t0: T).(arity g c3 t0 a)) H4 v1 H8) -in (let H12 \def (eq_ind C c3 (\lambda (c: C).(arity g c v1 a)) H11 e1 H10) -in (let H13 \def (eq_ind C c3 (\lambda (c: C).((eq C c (CHead e1 (Bind b1) -v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq -C c1 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda -(_: T).(csuba g e2 e1))))))) H2 e1 H10) in (let H14 \def (eq_ind C c3 -(\lambda (c: C).(csuba g c1 c)) H1 e1 H10) in (let H15 \def (eq_ind_r B b1 -(\lambda (b: B).((eq C e1 (CHead e1 (Bind b) v1)) \to (ex2_3 B C T (\lambda -(b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c1 (CHead e2 (Bind b2) -v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 -e1))))))) H13 Abbr H9) in (ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: -C).(\lambda (v2: T).(eq C (CHead c1 (Bind Abst) t) (CHead e2 (Bind b2) -v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1)))) -Abst c1 t (refl_equal C (CHead c1 (Bind Abst) t)) H14))))))))) H7)) -H6)))))))))))) c2 y H0))) H)))))). -(* COMMENTS -Initial nodes: 1831 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/getl.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/getl.ma deleted file mode 100644 index b49d9b7c2..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/getl.ma +++ /dev/null @@ -1,1178 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csuba/drop.ma". - -include "Basic-1/csuba/clear.ma". - -include "Basic-1/getl/clear.ma". - -theorem csuba_getl_abbr: - \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).(\forall -(i: nat).((getl i c1 (CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csuba g -c1 c2) \to (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) -(\lambda (d2: C).(csuba g d1 d2)))))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u: T).(\lambda -(i: nat).(\lambda (H: (getl i c1 (CHead d1 (Bind Abbr) u))).(let H0 \def -(getl_gen_all c1 (CHead d1 (Bind Abbr) u) i H) in (ex2_ind C (\lambda (e: -C).(drop i O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abbr) u))) -(\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(getl i c2 -(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))) (\lambda (x: -C).(\lambda (H1: (drop i O c1 x)).(\lambda (H2: (clear x (CHead d1 (Bind -Abbr) u))).(C_ind (\lambda (c: C).((drop i O c1 c) \to ((clear c (CHead d1 -(Bind Abbr) u)) \to (\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda -(d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 -d2)))))))) (\lambda (n: nat).(\lambda (_: (drop i O c1 (CSort n))).(\lambda -(H4: (clear (CSort n) (CHead d1 (Bind Abbr) u))).(clear_gen_sort (CHead d1 -(Bind Abbr) u) n H4 (\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda -(d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 -d2))))))))) (\lambda (x0: C).(\lambda (_: (((drop i O c1 x0) \to ((clear x0 -(CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csuba g c1 c2) \to (ex2 C -(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: -C).(csuba g d1 d2))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (H3: -(drop i O c1 (CHead x0 k t))).(\lambda (H4: (clear (CHead x0 k t) (CHead d1 -(Bind Abbr) u))).(K_ind (\lambda (k0: K).((drop i O c1 (CHead x0 k0 t)) \to -((clear (CHead x0 k0 t) (CHead d1 (Bind Abbr) u)) \to (\forall (c2: -C).((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind -Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))))) (\lambda (b: B).(\lambda -(H5: (drop i O c1 (CHead x0 (Bind b) t))).(\lambda (H6: (clear (CHead x0 -(Bind b) t) (CHead d1 (Bind Abbr) u))).(let H7 \def (f_equal C C (\lambda (e: -C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d1 | -(CHead c _ _) \Rightarrow c])) (CHead d1 (Bind Abbr) u) (CHead x0 (Bind b) t) -(clear_gen_bind b x0 (CHead d1 (Bind Abbr) u) t H6)) in ((let H8 \def -(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with -[(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow (match k0 in K -return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) -\Rightarrow Abbr])])) (CHead d1 (Bind Abbr) u) (CHead x0 (Bind b) t) -(clear_gen_bind b x0 (CHead d1 (Bind Abbr) u) t H6)) in ((let H9 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d1 (Bind -Abbr) u) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abbr) u) -t H6)) in (\lambda (H10: (eq B Abbr b)).(\lambda (H11: (eq C d1 x0)).(\lambda -(c2: C).(\lambda (H12: (csuba g c1 c2)).(let H13 \def (eq_ind_r T t (\lambda -(t0: T).(drop i O c1 (CHead x0 (Bind b) t0))) H5 u H9) in (let H14 \def -(eq_ind_r B b (\lambda (b0: B).(drop i O c1 (CHead x0 (Bind b0) u))) H13 Abbr -H10) in (let H15 \def (eq_ind_r C x0 (\lambda (c: C).(drop i O c1 (CHead c -(Bind Abbr) u))) H14 d1 H11) in (let H16 \def (csuba_drop_abbr i c1 d1 u H15 -g c2 H12) in (ex2_ind C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr) -u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(getl i c2 -(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x1: -C).(\lambda (H17: (drop i O c2 (CHead x1 (Bind Abbr) u))).(\lambda (H18: -(csuba g d1 x1)).(ex_intro2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind -Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x1 (getl_intro i c2 (CHead x1 -(Bind Abbr) u) (CHead x1 (Bind Abbr) u) H17 (clear_bind Abbr x1 u)) H18)))) -H16)))))))))) H8)) H7))))) (\lambda (f: F).(\lambda (H5: (drop i O c1 (CHead -x0 (Flat f) t))).(\lambda (H6: (clear (CHead x0 (Flat f) t) (CHead d1 (Bind -Abbr) u))).(let H7 \def H5 in (unintro C c1 (\lambda (c: C).((drop i O c -(CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g c c2) \to (ex2 C -(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: -C).(csuba g d1 d2))))))) (nat_ind (\lambda (n: nat).(\forall (x1: C).((drop n -O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g x1 c2) \to (ex2 C -(\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: -C).(csuba g d1 d2)))))))) (\lambda (x1: C).(\lambda (H8: (drop O O x1 (CHead -x0 (Flat f) t))).(\lambda (c2: C).(\lambda (H9: (csuba g x1 c2)).(let H10 -\def (eq_ind C x1 (\lambda (c: C).(csuba g c c2)) H9 (CHead x0 (Flat f) t) -(drop_gen_refl x1 (CHead x0 (Flat f) t) H8)) in (let H_y \def (clear_flat x0 -(CHead d1 (Bind Abbr) u) (clear_gen_flat f x0 (CHead d1 (Bind Abbr) u) t H6) -f t) in (let H11 \def (csuba_clear_conf g (CHead x0 (Flat f) t) c2 H10 (CHead -d1 (Bind Abbr) u) H_y) in (ex2_ind C (\lambda (e2: C).(csuba g (CHead d1 -(Bind Abbr) u) e2)) (\lambda (e2: C).(clear c2 e2)) (ex2 C (\lambda (d2: -C).(getl O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) -(\lambda (x2: C).(\lambda (H12: (csuba g (CHead d1 (Bind Abbr) u) -x2)).(\lambda (H13: (clear c2 x2)).(let H_x \def (csuba_gen_abbr g d1 x2 u -H12) in (let H14 \def H_x in (ex2_ind C (\lambda (d2: C).(eq C x2 (CHead d2 -(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: -C).(getl O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) -(\lambda (x3: C).(\lambda (H15: (eq C x2 (CHead x3 (Bind Abbr) u))).(\lambda -(H16: (csuba g d1 x3)).(let H17 \def (eq_ind C x2 (\lambda (c: C).(clear c2 -c)) H13 (CHead x3 (Bind Abbr) u) H15) in (ex_intro2 C (\lambda (d2: C).(getl -O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x3 -(getl_intro O c2 (CHead x3 (Bind Abbr) u) c2 (drop_refl c2) H17) H16))))) -H14)))))) H11)))))))) (\lambda (n: nat).(\lambda (H8: ((\forall (x1: -C).((drop n O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g x1 c2) -\to (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))) (\lambda -(d2: C).(csuba g d1 d2))))))))).(\lambda (x1: C).(\lambda (H9: (drop (S n) O -x1 (CHead x0 (Flat f) t))).(\lambda (c2: C).(\lambda (H10: (csuba g x1 -c2)).(let H11 \def (drop_clear x1 (CHead x0 (Flat f) t) n H9) in (ex2_3_ind B -C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear x1 (CHead e (Bind -b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop n O e (CHead -x0 (Flat f) t))))) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind -Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x2: B).(\lambda (x3: -C).(\lambda (x4: T).(\lambda (H12: (clear x1 (CHead x3 (Bind x2) -x4))).(\lambda (H13: (drop n O x3 (CHead x0 (Flat f) t))).(let H14 \def -(csuba_clear_conf g x1 c2 H10 (CHead x3 (Bind x2) x4) H12) in (ex2_ind C -(\lambda (e2: C).(csuba g (CHead x3 (Bind x2) x4) e2)) (\lambda (e2: -C).(clear c2 e2)) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind -Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x5: C).(\lambda (H15: -(csuba g (CHead x3 (Bind x2) x4) x5)).(\lambda (H16: (clear c2 x5)).(let H_x -\def (csuba_gen_bind g x2 x3 x5 x4 H15) in (let H17 \def H_x in (ex2_3_ind B -C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C x5 (CHead e2 -(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g -x3 e2)))) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) u))) -(\lambda (d2: C).(csuba g d1 d2))) (\lambda (x6: B).(\lambda (x7: C).(\lambda -(x8: T).(\lambda (H18: (eq C x5 (CHead x7 (Bind x6) x8))).(\lambda (H19: -(csuba g x3 x7)).(let H20 \def (eq_ind C x5 (\lambda (c: C).(clear c2 c)) H16 -(CHead x7 (Bind x6) x8) H18) in (let H21 \def (H8 x3 H13 x7 H19) in (ex2_ind -C (\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abbr) u))) (\lambda (d2: -C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind -Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x9: C).(\lambda (H22: -(getl n x7 (CHead x9 (Bind Abbr) u))).(\lambda (H23: (csuba g d1 -x9)).(ex_intro2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) u))) -(\lambda (d2: C).(csuba g d1 d2)) x9 (getl_clear_bind x6 c2 x7 x8 H20 (CHead -x9 (Bind Abbr) u) n H22) H23)))) H21)))))))) H17)))))) H14))))))) H11)))))))) -i) H7))))) k H3 H4))))))) x H1 H2)))) H0))))))). -(* COMMENTS -Initial nodes: 2319 -END *) - -theorem csuba_getl_abst: - \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u1: T).(\forall -(i: nat).((getl i c1 (CHead d1 (Bind Abst) u1)) \to (\forall (c2: C).((csuba -g c1 c2) \to (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) -u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a))))))))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u1: T).(\lambda -(i: nat).(\lambda (H: (getl i c1 (CHead d1 (Bind Abst) u1))).(let H0 \def -(getl_gen_all c1 (CHead d1 (Bind Abst) u1) i H) in (ex2_ind C (\lambda (e: -C).(drop i O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abst) u1))) -(\forall (c2: C).((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(getl i c2 -(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind -Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 -d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc -g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))))) (\lambda (x: C).(\lambda (H1: (drop i O c1 x)).(\lambda (H2: (clear -x (CHead d1 (Bind Abst) u1))).(C_ind (\lambda (c: C).((drop i O c1 c) \to -((clear c (CHead d1 (Bind Abst) u1)) \to (\forall (c2: C).((csuba g c1 c2) -\to (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) -(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))))) (\lambda -(n: nat).(\lambda (_: (drop i O c1 (CSort n))).(\lambda (H4: (clear (CSort n) -(CHead d1 (Bind Abst) u1))).(clear_gen_sort (CHead d1 (Bind Abst) u1) n H4 -(\forall (c2: C).((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(getl i c2 -(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind -Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 -d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc -g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))))))))) (\lambda (x0: C).(\lambda (_: (((drop i O c1 x0) \to ((clear x0 -(CHead d1 (Bind Abst) u1)) \to (\forall (c2: C).((csuba g c1 c2) \to (or (ex2 -C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: -C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))))))))).(\lambda (k: K).(\lambda -(t: T).(\lambda (H3: (drop i O c1 (CHead x0 k t))).(\lambda (H4: (clear -(CHead x0 k t) (CHead d1 (Bind Abst) u1))).(K_ind (\lambda (k0: K).((drop i O -c1 (CHead x0 k0 t)) \to ((clear (CHead x0 k0 t) (CHead d1 (Bind Abst) u1)) -\to (\forall (c2: C).((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(getl i -c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T -A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 -(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 -(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 -u2 a))))))))))) (\lambda (b: B).(\lambda (H5: (drop i O c1 (CHead x0 (Bind b) -t))).(\lambda (H6: (clear (CHead x0 (Bind b) t) (CHead d1 (Bind Abst) -u1))).(let H7 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda -(_: C).C) with [(CSort _) \Rightarrow d1 | (CHead c _ _) \Rightarrow c])) -(CHead d1 (Bind Abst) u1) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead -d1 (Bind Abst) u1) t H6)) in ((let H8 \def (f_equal C B (\lambda (e: -C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abst | -(CHead _ k0 _) \Rightarrow (match k0 in K return (\lambda (_: K).B) with -[(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abst])])) (CHead d1 (Bind -Abst) u1) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abst) -u1) t H6)) in ((let H9 \def (f_equal C T (\lambda (e: C).(match e in C return -(\lambda (_: C).T) with [(CSort _) \Rightarrow u1 | (CHead _ _ t0) -\Rightarrow t0])) (CHead d1 (Bind Abst) u1) (CHead x0 (Bind b) t) -(clear_gen_bind b x0 (CHead d1 (Bind Abst) u1) t H6)) in (\lambda (H10: (eq B -Abst b)).(\lambda (H11: (eq C d1 x0)).(\lambda (c2: C).(\lambda (H12: (csuba -g c1 c2)).(let H13 \def (eq_ind_r T t (\lambda (t0: T).(drop i O c1 (CHead x0 -(Bind b) t0))) H5 u1 H9) in (let H14 \def (eq_ind_r B b (\lambda (b0: -B).(drop i O c1 (CHead x0 (Bind b0) u1))) H13 Abst H10) in (let H15 \def -(eq_ind_r C x0 (\lambda (c: C).(drop i O c1 (CHead c (Bind Abst) u1))) H14 d1 -H11) in (let H16 \def (csuba_drop_abst i c1 d1 u1 H15 g c2 H12) in (or_ind -(ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst) u1))) (\lambda -(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (or (ex2 C -(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: -C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (H17: (ex2 C (\lambda -(d2: C).(drop i O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 -d2)))).(ex2_ind C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst) u1))) -(\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: C).(getl i c2 -(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind -Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 -d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc -g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))) (\lambda (x1: C).(\lambda (H18: (drop i O c2 (CHead x1 (Bind Abst) -u1))).(\lambda (H19: (csuba g d1 x1)).(or_introl (ex2 C (\lambda (d2: -C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 -(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity -g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: C).(getl i c2 (CHead d2 -(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) x1 (getl_intro i c2 -(CHead x1 (Bind Abst) u1) (CHead x1 (Bind Abst) u1) H18 (clear_bind Abst x1 -u1)) H19))))) H17)) (\lambda (H17: (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))).(ex4_3_ind C -T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 -(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 -(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 -u2 a)))) (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) -(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x1: -C).(\lambda (x2: T).(\lambda (x3: A).(\lambda (H18: (drop i O c2 (CHead x1 -(Bind Abbr) x2))).(\lambda (H19: (csuba g d1 x1)).(\lambda (H20: (arity g d1 -u1 (asucc g x3))).(\lambda (H21: (arity g x1 x2 x3)).(or_intror (ex2 C -(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: -C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) -x1 x2 x3 (getl_intro i c2 (CHead x1 (Bind Abbr) x2) (CHead x1 (Bind Abbr) x2) -H18 (clear_bind Abbr x1 x2)) H19 H20 H21))))))))) H17)) H16)))))))))) H8)) -H7))))) (\lambda (f: F).(\lambda (H5: (drop i O c1 (CHead x0 (Flat f) -t))).(\lambda (H6: (clear (CHead x0 (Flat f) t) (CHead d1 (Bind Abst) -u1))).(let H7 \def H5 in (unintro C c1 (\lambda (c: C).((drop i O c (CHead x0 -(Flat f) t)) \to (\forall (c2: C).((csuba g c c2) \to (or (ex2 C (\lambda -(d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 -d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i -c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: -A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda -(a: A).(arity g d2 u2 a)))))))))) (nat_ind (\lambda (n: nat).(\forall (x1: -C).((drop n O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g x1 c2) -\to (or (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) u1))) -(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(getl n c2 (CHead d2 (Bind Abbr) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))))) (\lambda -(x1: C).(\lambda (H8: (drop O O x1 (CHead x0 (Flat f) t))).(\lambda (c2: -C).(\lambda (H9: (csuba g x1 c2)).(let H10 \def (eq_ind C x1 (\lambda (c: -C).(csuba g c c2)) H9 (CHead x0 (Flat f) t) (drop_gen_refl x1 (CHead x0 (Flat -f) t) H8)) in (let H_y \def (clear_flat x0 (CHead d1 (Bind Abst) u1) -(clear_gen_flat f x0 (CHead d1 (Bind Abst) u1) t H6) f t) in (let H11 \def -(csuba_clear_conf g (CHead x0 (Flat f) t) c2 H10 (CHead d1 (Bind Abst) u1) -H_y) in (ex2_ind C (\lambda (e2: C).(csuba g (CHead d1 (Bind Abst) u1) e2)) -(\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(getl O c2 (CHead -d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))) (\lambda (x2: C).(\lambda (H12: (csuba g (CHead d1 (Bind Abst) u1) -x2)).(\lambda (H13: (clear c2 x2)).(let H_x \def (csuba_gen_abst g d1 x2 u1 -H12) in (let H14 \def H_x in (or_ind (ex2 C (\lambda (d2: C).(eq C x2 (CHead -d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C x2 (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a))))) (or (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u1))) -(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda -(H15: (ex2 C (\lambda (d2: C).(eq C x2 (CHead d2 (Bind Abst) u1))) (\lambda -(d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(eq C x2 (CHead d2 -(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: -C).(getl O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 -(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity -g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 a)))))) (\lambda (x3: C).(\lambda (H16: (eq C x2 (CHead x3 -(Bind Abst) u1))).(\lambda (H17: (csuba g d1 x3)).(let H18 \def (eq_ind C x2 -(\lambda (c: C).(clear c2 c)) H13 (CHead x3 (Bind Abst) u1) H16) in -(or_introl (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u1))) -(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C -(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: -C).(csuba g d1 d2)) x3 (getl_intro O c2 (CHead x3 (Bind Abst) u1) c2 -(drop_refl c2) H18) H17)))))) H15)) (\lambda (H15: (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(eq C x2 (CHead d2 (Bind Abbr) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(eq C x2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2: -C).(getl O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 -(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity -g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 a)))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: -A).(\lambda (H16: (eq C x2 (CHead x3 (Bind Abbr) x4))).(\lambda (H17: (csuba -g d1 x3)).(\lambda (H18: (arity g d1 u1 (asucc g x5))).(\lambda (H19: (arity -g x3 x4 x5)).(let H20 \def (eq_ind C x2 (\lambda (c: C).(clear c2 c)) H13 -(CHead x3 (Bind Abbr) x4) H16) in (or_intror (ex2 C (\lambda (d2: C).(getl O -c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T -A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 -(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 -(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 -u2 a))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(getl O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x3 x4 x5 (getl_intro O c2 (CHead -x3 (Bind Abbr) x4) c2 (drop_refl c2) H20) H17 H18 H19)))))))))) H15)) -H14)))))) H11)))))))) (\lambda (n: nat).(\lambda (H8: ((\forall (x1: -C).((drop n O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g x1 c2) -\to (or (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) u1))) -(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(getl n c2 (CHead d2 (Bind Abbr) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))))))))).(\lambda (x1: C).(\lambda (H9: (drop (S n) O x1 (CHead x0 (Flat -f) t))).(\lambda (c2: C).(\lambda (H10: (csuba g x1 c2)).(let H11 \def -(drop_clear x1 (CHead x0 (Flat f) t) n H9) in (ex2_3_ind B C T (\lambda (b: -B).(\lambda (e: C).(\lambda (v: T).(clear x1 (CHead e (Bind b) v))))) -(\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop n O e (CHead x0 (Flat -f) t))))) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) -u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: T).(\lambda (H12: -(clear x1 (CHead x3 (Bind x2) x4))).(\lambda (H13: (drop n O x3 (CHead x0 -(Flat f) t))).(let H14 \def (csuba_clear_conf g x1 c2 H10 (CHead x3 (Bind x2) -x4) H12) in (ex2_ind C (\lambda (e2: C).(csuba g (CHead x3 (Bind x2) x4) e2)) -(\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 -(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 -(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 -(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 -u2 a)))))) (\lambda (x5: C).(\lambda (H15: (csuba g (CHead x3 (Bind x2) x4) -x5)).(\lambda (H16: (clear c2 x5)).(let H_x \def (csuba_gen_bind g x2 x3 x5 -x4 H15) in (let H17 \def H_x in (ex2_3_ind B C T (\lambda (b2: B).(\lambda -(e2: C).(\lambda (v2: T).(eq C x5 (CHead e2 (Bind b2) v2))))) (\lambda (_: -B).(\lambda (e2: C).(\lambda (_: T).(csuba g x3 e2)))) (or (ex2 C (\lambda -(d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g -d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl -(S n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x6: B).(\lambda (x7: -C).(\lambda (x8: T).(\lambda (H18: (eq C x5 (CHead x7 (Bind x6) -x8))).(\lambda (H19: (csuba g x3 x7)).(let H20 \def (eq_ind C x5 (\lambda (c: -C).(clear c2 c)) H16 (CHead x7 (Bind x6) x8) H18) in (let H21 \def (H8 x3 H13 -x7 H19) in (or_ind (ex2 C (\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abst) -u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(getl n x7 (CHead d2 (Bind Abbr) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (or -(ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u1))) (\lambda -(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda -(H22: (ex2 C (\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abst) u1))) (\lambda -(d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(getl n x7 (CHead d2 -(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: -C).(getl (S n) c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 -d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S -n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: -A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda -(a: A).(arity g d2 u2 a)))))) (\lambda (x9: C).(\lambda (H23: (getl n x7 -(CHead x9 (Bind Abst) u1))).(\lambda (H24: (csuba g d1 x9)).(or_introl (ex2 C -(\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: -C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(getl (S n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda -(_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: -C).(getl (S n) c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 -d2)) x9 (getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Abst) u1) n H23) -H24))))) H22)) (\lambda (H22: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(getl n x7 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))).(ex4_3_ind C -T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl n x7 (CHead d2 -(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 -(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 -u2 a)))) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) -u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a)))))) (\lambda (x9: C).(\lambda (x10: T).(\lambda (x11: A).(\lambda (H23: -(getl n x7 (CHead x9 (Bind Abbr) x10))).(\lambda (H24: (csuba g d1 -x9)).(\lambda (H25: (arity g d1 u1 (asucc g x11))).(\lambda (H26: (arity g x9 -x10 x11)).(or_intror (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind -Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(getl (S n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda -(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x9 x10 x11 (getl_clear_bind x6 -c2 x7 x8 H20 (CHead x9 (Bind Abbr) x10) n H23) H24 H25 H26))))))))) H22)) -H21)))))))) H17)))))) H14))))))) H11)))))))) i) H7))))) k H3 H4))))))) x H1 -H2)))) H0))))))). -(* COMMENTS -Initial nodes: 6437 -END *) - -theorem csuba_getl_abst_rev: - \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).(\forall -(i: nat).((getl i c1 (CHead d1 (Bind Abst) u)) \to (\forall (c2: C).((csuba g -c2 c1) \to (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1)))))))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u: T).(\lambda -(i: nat).(\lambda (H: (getl i c1 (CHead d1 (Bind Abst) u))).(let H0 \def -(getl_gen_all c1 (CHead d1 (Bind Abst) u) i H) in (ex2_ind C (\lambda (e: -C).(drop i O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abst) u))) -(\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: C).(getl i c2 -(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))) (\lambda (x: -C).(\lambda (H1: (drop i O c1 x)).(\lambda (H2: (clear x (CHead d1 (Bind -Abst) u))).(C_ind (\lambda (c: C).((drop i O c1 c) \to ((clear c (CHead d1 -(Bind Abst) u)) \to (\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda -(d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 -d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))))) -(\lambda (n: nat).(\lambda (_: (drop i O c1 (CSort n))).(\lambda (H4: (clear -(CSort n) (CHead d1 (Bind Abst) u))).(clear_gen_sort (CHead d1 (Bind Abst) u) -n H4 (\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: C).(getl -i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))) (\lambda (x0: -C).(\lambda (_: (((drop i O c1 x0) \to ((clear x0 (CHead d1 (Bind Abst) u)) -\to (\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: C).(getl i -c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))).(\lambda (k: -K).(\lambda (t: T).(\lambda (H3: (drop i O c1 (CHead x0 k t))).(\lambda (H4: -(clear (CHead x0 k t) (CHead d1 (Bind Abst) u))).(K_ind (\lambda (k0: -K).((drop i O c1 (CHead x0 k0 t)) \to ((clear (CHead x0 k0 t) (CHead d1 (Bind -Abst) u)) \to (\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: -C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))))) (\lambda (b: -B).(\lambda (H5: (drop i O c1 (CHead x0 (Bind b) t))).(\lambda (H6: (clear -(CHead x0 (Bind b) t) (CHead d1 (Bind Abst) u))).(let H7 \def (f_equal C C -(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow d1 | (CHead c _ _) \Rightarrow c])) (CHead d1 (Bind Abst) u) -(CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abst) u) t H6)) in -((let H8 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: -C).B) with [(CSort _) \Rightarrow Abst | (CHead _ k0 _) \Rightarrow (match k0 -in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) -\Rightarrow Abst])])) (CHead d1 (Bind Abst) u) (CHead x0 (Bind b) t) -(clear_gen_bind b x0 (CHead d1 (Bind Abst) u) t H6)) in ((let H9 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d1 (Bind -Abst) u) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abst) u) -t H6)) in (\lambda (H10: (eq B Abst b)).(\lambda (H11: (eq C d1 x0)).(\lambda -(c2: C).(\lambda (H12: (csuba g c2 c1)).(let H13 \def (eq_ind_r T t (\lambda -(t0: T).(drop i O c1 (CHead x0 (Bind b) t0))) H5 u H9) in (let H14 \def -(eq_ind_r B b (\lambda (b0: B).(drop i O c1 (CHead x0 (Bind b0) u))) H13 Abst -H10) in (let H15 \def (eq_ind_r C x0 (\lambda (c: C).(drop i O c1 (CHead c -(Bind Abst) u))) H14 d1 H11) in (let H16 \def (csuba_drop_abst_rev i c1 d1 u -H15 g c2 H12) in (or_ind (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind -Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(drop i O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: C).(getl i c2 -(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H17: (ex2 C -(\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop i O c2 (CHead d2 -(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: -C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x1: -C).(\lambda (H18: (drop i O c2 (CHead x1 (Bind Abst) u))).(\lambda (H19: -(csuba g x1 d1)).(or_introl (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 -(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(getl i -c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x1 -(getl_intro i c2 (CHead x1 (Bind Abst) u) (CHead x1 (Bind Abst) u) H18 -(clear_bind Abst x1 u)) H19))))) H17)) (\lambda (H17: (ex2_2 C T (\lambda -(d2: C).(\lambda (u2: T).(drop i O c2 (CHead d2 (Bind Void) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: -C).(\lambda (u2: T).(drop i O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1))) (or (ex2 C (\lambda (d2: C).(getl i c2 -(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x1: -C).(\lambda (x2: T).(\lambda (H18: (drop i O c2 (CHead x1 (Bind Void) -x2))).(\lambda (H19: (csuba g x1 d1)).(or_intror (ex2 C (\lambda (d2: -C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T -(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x1 x2 (getl_intro i c2 -(CHead x1 (Bind Void) x2) (CHead x1 (Bind Void) x2) H18 (clear_bind Void x1 -x2)) H19)))))) H17)) H16)))))))))) H8)) H7))))) (\lambda (f: F).(\lambda (H5: -(drop i O c1 (CHead x0 (Flat f) t))).(\lambda (H6: (clear (CHead x0 (Flat f) -t) (CHead d1 (Bind Abst) u))).(let H7 \def H5 in (unintro C c1 (\lambda (c: -C).((drop i O c (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g c2 c) -\to (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))))))) (nat_ind (\lambda (n: nat).(\forall (x1: C).((drop -n O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g c2 x1) \to (or -(ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl n c2 -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))))))))) (\lambda (x1: C).(\lambda (H8: (drop O O x1 (CHead x0 (Flat f) -t))).(\lambda (c2: C).(\lambda (H9: (csuba g c2 x1)).(let H10 \def (eq_ind C -x1 (\lambda (c: C).(csuba g c2 c)) H9 (CHead x0 (Flat f) t) (drop_gen_refl x1 -(CHead x0 (Flat f) t) H8)) in (let H_y \def (clear_flat x0 (CHead d1 (Bind -Abst) u) (clear_gen_flat f x0 (CHead d1 (Bind Abst) u) t H6) f t) in (let H11 -\def (csuba_clear_trans g (CHead x0 (Flat f) t) c2 H10 (CHead d1 (Bind Abst) -u) H_y) in (ex2_ind C (\lambda (e2: C).(csuba g e2 (CHead d1 (Bind Abst) u))) -(\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(getl O c2 (CHead -d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda -(d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 (Bind Void) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x2: C).(\lambda (H12: -(csuba g x2 (CHead d1 (Bind Abst) u))).(\lambda (H13: (clear c2 x2)).(let H_x -\def (csuba_gen_abst_rev g d1 x2 u H12) in (let H14 \def H_x in (or_ind (ex2 -C (\lambda (d2: C).(eq C x2 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C x2 -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))) (or (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(getl O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))) (\lambda (H15: (ex2 C (\lambda (d2: C).(eq C x2 (CHead -d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda -(d2: C).(eq C x2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) -(or (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u))) (\lambda -(d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl -O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g -d2 d1))))) (\lambda (x3: C).(\lambda (H16: (eq C x2 (CHead x3 (Bind Abst) -u))).(\lambda (H17: (csuba g x3 d1)).(let H18 \def (eq_ind C x2 (\lambda (c: -C).(clear c2 c)) H13 (CHead x3 (Bind Abst) u) H16) in (or_introl (ex2 C -(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl O c2 -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))) (ex_intro2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1)) x3 (getl_intro O c2 (CHead x3 (Bind Abst) -u) c2 (drop_refl c2) H18) H17)))))) H15)) (\lambda (H15: (ex2_2 C T (\lambda -(d2: C).(\lambda (u2: T).(eq C x2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: -C).(\lambda (u2: T).(eq C x2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1))) (or (ex2 C (\lambda (d2: C).(getl O c2 -(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x3: -C).(\lambda (x4: T).(\lambda (H16: (eq C x2 (CHead x3 (Bind Void) -x4))).(\lambda (H17: (csuba g x3 d1)).(let H18 \def (eq_ind C x2 (\lambda (c: -C).(clear c2 c)) H13 (CHead x3 (Bind Void) x4) H16) in (or_intror (ex2 C -(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl O c2 -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead -d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x3 -x4 (getl_intro O c2 (CHead x3 (Bind Void) x4) c2 (drop_refl c2) H18) -H17))))))) H15)) H14)))))) H11)))))))) (\lambda (n: nat).(\lambda (H8: -((\forall (x1: C).((drop n O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: -C).((csuba g c2 x1) \to (or (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 -(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(getl n c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1))))))))))).(\lambda (x1: C).(\lambda (H9: -(drop (S n) O x1 (CHead x0 (Flat f) t))).(\lambda (c2: C).(\lambda (H10: -(csuba g c2 x1)).(let H11 \def (drop_clear x1 (CHead x0 (Flat f) t) n H9) in -(ex2_3_ind B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear x1 -(CHead e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: -T).(drop n O e (CHead x0 (Flat f) t))))) (or (ex2 C (\lambda (d2: C).(getl (S -n) c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C -T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x2: -B).(\lambda (x3: C).(\lambda (x4: T).(\lambda (H12: (clear x1 (CHead x3 (Bind -x2) x4))).(\lambda (H13: (drop n O x3 (CHead x0 (Flat f) t))).(let H14 \def -(csuba_clear_trans g x1 c2 H10 (CHead x3 (Bind x2) x4) H12) in (ex2_ind C -(\lambda (e2: C).(csuba g e2 (CHead x3 (Bind x2) x4))) (\lambda (e2: -C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind -Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x5: C).(\lambda (H15: (csuba -g x5 (CHead x3 (Bind x2) x4))).(\lambda (H16: (clear c2 x5)).(let H_x \def -(csuba_gen_bind_rev g x2 x3 x5 x4 H15) in (let H17 \def H_x in (ex2_3_ind B C -T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C x5 (CHead e2 (Bind -b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 -x3)))) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))) (\lambda (x6: B).(\lambda (x7: C).(\lambda (x8: -T).(\lambda (H18: (eq C x5 (CHead x7 (Bind x6) x8))).(\lambda (H19: (csuba g -x7 x3)).(let H20 \def (eq_ind C x5 (\lambda (c: C).(clear c2 c)) H16 (CHead -x7 (Bind x6) x8) H18) in (let H21 \def (H8 x3 H13 x7 H19) in (or_ind (ex2 C -(\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl n x7 -(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1)))) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))) (\lambda (H22: (ex2 C (\lambda (d2: C).(getl n x7 -(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C -(\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 -(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x9: C).(\lambda (H23: (getl -n x7 (CHead x9 (Bind Abst) u))).(\lambda (H24: (csuba g x9 d1)).(or_introl -(ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u))) (\lambda -(d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl -(S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba -g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind -Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x9 (getl_clear_bind x6 c2 x7 x8 -H20 (CHead x9 (Bind Abst) u) n H23) H24))))) H22)) (\lambda (H22: (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(getl n x7 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda -(d2: C).(\lambda (u2: T).(getl n x7 (CHead d2 (Bind Void) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1))) (or (ex2 C (\lambda (d2: C).(getl -(S n) c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 -C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x9: -C).(\lambda (x10: T).(\lambda (H23: (getl n x7 (CHead x9 (Bind Void) -x10))).(\lambda (H24: (csuba g x9 d1)).(or_intror (ex2 C (\lambda (d2: -C).(getl (S n) c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 -d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) -(ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x9 x10 -(getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Void) x10) n H23) H24)))))) -H22)) H21)))))))) H17)))))) H14))))))) H11)))))))) i) H7))))) k H3 H4))))))) -x H1 H2)))) H0))))))). -(* COMMENTS -Initial nodes: 4703 -END *) - -theorem csuba_getl_abbr_rev: - \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u1: T).(\forall -(i: nat).((getl i c1 (CHead d1 (Bind Abbr) u1)) \to (\forall (c2: C).((csuba -g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 -C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u1: T).(\lambda -(i: nat).(\lambda (H: (getl i c1 (CHead d1 (Bind Abbr) u1))).(let H0 \def -(getl_gen_all c1 (CHead d1 (Bind Abbr) u1) i H) in (ex2_ind C (\lambda (e: -C).(drop i O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abbr) u1))) -(\forall (c2: C).((csuba g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(getl i c2 -(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))) -(\lambda (x: C).(\lambda (H1: (drop i O c1 x)).(\lambda (H2: (clear x (CHead -d1 (Bind Abbr) u1))).(C_ind (\lambda (c: C).((drop i O c1 c) \to ((clear c -(CHead d1 (Bind Abbr) u1)) \to (\forall (c2: C).((csuba g c2 c1) \to (or3 -(ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))))))))) (\lambda (n: nat).(\lambda (_: -(drop i O c1 (CSort n))).(\lambda (H4: (clear (CSort n) (CHead d1 (Bind Abbr) -u1))).(clear_gen_sort (CHead d1 (Bind Abbr) u1) n H4 (\forall (c2: C).((csuba -g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 -C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))) (\lambda (x0: -C).(\lambda (_: (((drop i O c1 x0) \to ((clear x0 (CHead d1 (Bind Abbr) u1)) -\to (\forall (c2: C).((csuba g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(getl -i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C -T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 -(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 -d1))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (H3: (drop i O c1 -(CHead x0 k t))).(\lambda (H4: (clear (CHead x0 k t) (CHead d1 (Bind Abbr) -u1))).(K_ind (\lambda (k0: K).((drop i O c1 (CHead x0 k0 t)) \to ((clear -(CHead x0 k0 t) (CHead d1 (Bind Abbr) u1)) \to (\forall (c2: C).((csuba g c2 -c1) \to (or3 (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) -(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))))) (\lambda (b: -B).(\lambda (H5: (drop i O c1 (CHead x0 (Bind b) t))).(\lambda (H6: (clear -(CHead x0 (Bind b) t) (CHead d1 (Bind Abbr) u1))).(let H7 \def (f_equal C C -(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow d1 | (CHead c _ _) \Rightarrow c])) (CHead d1 (Bind Abbr) u1) -(CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abbr) u1) t H6)) -in ((let H8 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda -(_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow -(match k0 in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | -(Flat _) \Rightarrow Abbr])])) (CHead d1 (Bind Abbr) u1) (CHead x0 (Bind b) -t) (clear_gen_bind b x0 (CHead d1 (Bind Abbr) u1) t H6)) in ((let H9 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow u1 | (CHead _ _ t0) \Rightarrow t0])) (CHead d1 (Bind -Abbr) u1) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abbr) -u1) t H6)) in (\lambda (H10: (eq B Abbr b)).(\lambda (H11: (eq C d1 -x0)).(\lambda (c2: C).(\lambda (H12: (csuba g c2 c1)).(let H13 \def (eq_ind_r -T t (\lambda (t0: T).(drop i O c1 (CHead x0 (Bind b) t0))) H5 u1 H9) in (let -H14 \def (eq_ind_r B b (\lambda (b0: B).(drop i O c1 (CHead x0 (Bind b0) -u1))) H13 Abbr H10) in (let H15 \def (eq_ind_r C x0 (\lambda (c: C).(drop i O -c1 (CHead c (Bind Abbr) u1))) H14 d1 H11) in (let H16 \def -(csuba_drop_abbr_rev i c1 d1 u1 H15 g c2 H12) in (or3_ind (ex2 C (\lambda -(d2: C).(drop i O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop i -O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(drop i O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 -C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H17: (ex2 C -(\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop i O c2 (CHead d2 -(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) (or3 (ex2 C (\lambda (d2: -C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))) (\lambda (x1: C).(\lambda (H18: (drop i O c2 (CHead x1 -(Bind Abbr) u1))).(\lambda (H19: (csuba g x1 d1)).(or3_intro0 (ex2 C (\lambda -(d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i -c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(getl i c2 (CHead d2 -(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x1 (getl_intro i c2 -(CHead x1 (Bind Abbr) u1) (CHead x1 (Bind Abbr) u1) H18 (clear_bind Abbr x1 -u1)) H19))))) H17)) (\lambda (H17: (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abst) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T -A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 -(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a)))) (or3 (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) -(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x1: -C).(\lambda (x2: T).(\lambda (x3: A).(\lambda (H18: (drop i O c2 (CHead x1 -(Bind Abst) x2))).(\lambda (H19: (csuba g x1 d1)).(\lambda (H20: (arity g x1 -x2 (asucc g x3))).(\lambda (H21: (arity g d1 u1 x3)).(or3_intro1 (ex2 C -(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))) (ex4_3_intro C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x1 x2 x3 -(getl_intro i c2 (CHead x1 (Bind Abst) x2) (CHead x1 (Bind Abst) x2) H18 -(clear_bind Abst x1 x2)) H19 H20 H21))))))))) H17)) (\lambda (H17: (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(drop i O c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda -(d2: C).(\lambda (u2: T).(drop i O c2 (CHead d2 (Bind Void) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C (\lambda (d2: C).(getl -i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C -T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 -(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) -(\lambda (x1: C).(\lambda (x2: T).(\lambda (H18: (drop i O c2 (CHead x1 (Bind -Void) x2))).(\lambda (H19: (csuba g x1 d1)).(or3_intro2 (ex2 C (\lambda (d2: -C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: -T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))) x1 x2 (getl_intro i c2 (CHead x1 (Bind Void) x2) (CHead -x1 (Bind Void) x2) H18 (clear_bind Void x1 x2)) H19)))))) H17)) H16)))))))))) -H8)) H7))))) (\lambda (f: F).(\lambda (H5: (drop i O c1 (CHead x0 (Flat f) -t))).(\lambda (H6: (clear (CHead x0 (Flat f) t) (CHead d1 (Bind Abbr) -u1))).(let H7 \def H5 in (unintro C c1 (\lambda (c: C).((drop i O c (CHead x0 -(Flat f) t)) \to (\forall (c2: C).((csuba g c2 c) \to (or3 (ex2 C (\lambda -(d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i -c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))))))) (nat_ind (\lambda (n: nat).(\forall (x1: C).((drop -n O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g c2 x1) \to (or3 -(ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(getl n c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(getl n c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))))))))) (\lambda (x1: C).(\lambda (H8: -(drop O O x1 (CHead x0 (Flat f) t))).(\lambda (c2: C).(\lambda (H9: (csuba g -c2 x1)).(let H10 \def (eq_ind C x1 (\lambda (c: C).(csuba g c2 c)) H9 (CHead -x0 (Flat f) t) (drop_gen_refl x1 (CHead x0 (Flat f) t) H8)) in (let H_y \def -(clear_flat x0 (CHead d1 (Bind Abbr) u1) (clear_gen_flat f x0 (CHead d1 (Bind -Abbr) u1) t H6) f t) in (let H11 \def (csuba_clear_trans g (CHead x0 (Flat f) -t) c2 H10 (CHead d1 (Bind Abbr) u1) H_y) in (ex2_ind C (\lambda (e2: -C).(csuba g e2 (CHead d1 (Bind Abbr) u1))) (\lambda (e2: C).(clear c2 e2)) -(or3 (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u1))) (\lambda -(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x2: -C).(\lambda (H12: (csuba g x2 (CHead d1 (Bind Abbr) u1))).(\lambda (H13: -(clear c2 x2)).(let H_x \def (csuba_gen_abbr_rev g d1 x2 u1 H12) in (let H14 -\def H_x in (or3_ind (ex2 C (\lambda (d2: C).(eq C x2 (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(eq C x2 (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 -C T (\lambda (d2: C).(\lambda (u2: T).(eq C x2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: -C).(getl O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(getl O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))) (\lambda (H15: (ex2 C (\lambda (d2: C).(eq C x2 (CHead -d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda -(d2: C).(eq C x2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1)) (or3 (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u1))) -(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abst) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x3: -C).(\lambda (H16: (eq C x2 (CHead x3 (Bind Abbr) u1))).(\lambda (H17: (csuba -g x3 d1)).(let H18 \def (eq_ind C x2 (\lambda (c: C).(clear c2 c)) H13 (CHead -x3 (Bind Abbr) u1) H16) in (or3_intro0 (ex2 C (\lambda (d2: C).(getl O c2 -(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) -(ex_intro2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u1))) (\lambda -(d2: C).(csuba g d2 d1)) x3 (getl_intro O c2 (CHead x3 (Bind Abbr) u1) c2 -(drop_refl c2) H18) H17)))))) H15)) (\lambda (H15: (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(eq C x2 (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 -a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(eq C x2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C (\lambda (d2: -C).(getl O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) -(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(getl O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: -A).(\lambda (H16: (eq C x2 (CHead x3 (Bind Abst) x4))).(\lambda (H17: (csuba -g x3 d1)).(\lambda (H18: (arity g x3 x4 (asucc g x5))).(\lambda (H19: (arity -g d1 u1 x5)).(let H20 \def (eq_ind C x2 (\lambda (c: C).(clear c2 c)) H13 -(CHead x3 (Bind Abst) x4) H16) in (or3_intro1 (ex2 C (\lambda (d2: C).(getl O -c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T -A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 -(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) -(ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O -c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a)))) x3 x4 x5 (getl_intro O c2 (CHead x3 (Bind Abst) -x4) c2 (drop_refl c2) H20) H17 H18 H19)))))))))) H15)) (\lambda (H15: (ex2_2 -C T (\lambda (d2: C).(\lambda (u2: T).(eq C x2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda -(d2: C).(\lambda (u2: T).(eq C x2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C (\lambda (d2: C).(getl O c2 -(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind -Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 -d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 -(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) -(\lambda (x3: C).(\lambda (x4: T).(\lambda (H16: (eq C x2 (CHead x3 (Bind -Void) x4))).(\lambda (H17: (csuba g x3 d1)).(let H18 \def (eq_ind C x2 -(\lambda (c: C).(clear c2 c)) H13 (CHead x3 (Bind Void) x4) H16) in -(or3_intro2 (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u1))) -(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abst) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda -(d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 (Bind Void) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1))) x3 x4 (getl_intro O c2 (CHead x3 -(Bind Void) x4) c2 (drop_refl c2) H18) H17))))))) H15)) H14)))))) H11)))))))) -(\lambda (n: nat).(\lambda (H8: ((\forall (x1: C).((drop n O x1 (CHead x0 -(Flat f) t)) \to (\forall (c2: C).((csuba g c2 x1) \to (or3 (ex2 C (\lambda -(d2: C).(getl n c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl n -c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(getl n c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))))))))).(\lambda (x1: C).(\lambda (H9: (drop (S n) O x1 -(CHead x0 (Flat f) t))).(\lambda (c2: C).(\lambda (H10: (csuba g c2 x1)).(let -H11 \def (drop_clear x1 (CHead x0 (Flat f) t) n H9) in (ex2_3_ind B C T -(\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear x1 (CHead e (Bind b) -v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop n O e (CHead x0 -(Flat f) t))))) (or3 (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda -(x2: B).(\lambda (x3: C).(\lambda (x4: T).(\lambda (H12: (clear x1 (CHead x3 -(Bind x2) x4))).(\lambda (H13: (drop n O x3 (CHead x0 (Flat f) t))).(let H14 -\def (csuba_clear_trans g x1 c2 H10 (CHead x3 (Bind x2) x4) H12) in (ex2_ind -C (\lambda (e2: C).(csuba g e2 (CHead x3 (Bind x2) x4))) (\lambda (e2: -C).(clear c2 e2)) (or3 (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda -(x5: C).(\lambda (H15: (csuba g x5 (CHead x3 (Bind x2) x4))).(\lambda (H16: -(clear c2 x5)).(let H_x \def (csuba_gen_bind_rev g x2 x3 x5 x4 H15) in (let -H17 \def H_x in (ex2_3_ind B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda -(v2: T).(eq C x5 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: -C).(\lambda (_: T).(csuba g e2 x3)))) (or3 (ex2 C (\lambda (d2: C).(getl (S -n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C -T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead -d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))) (\lambda (x6: B).(\lambda (x7: C).(\lambda (x8: -T).(\lambda (H18: (eq C x5 (CHead x7 (Bind x6) x8))).(\lambda (H19: (csuba g -x7 x3)).(let H20 \def (eq_ind C x5 (\lambda (c: C).(clear c2 c)) H16 (CHead -x7 (Bind x6) x8) H18) in (let H21 \def (H8 x3 H13 x7 H19) in (or3_ind (ex2 C -(\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(getl n x7 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(getl n x7 (CHead d2 (Bind Void) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(getl (S -n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C -T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead -d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))) (\lambda (H22: (ex2 C (\lambda (d2: C).(getl n x7 -(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C -(\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1)) (or3 (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 -(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda -(x9: C).(\lambda (H23: (getl n x7 (CHead x9 (Bind Abbr) u1))).(\lambda (H24: -(csuba g x9 d1)).(or3_intro0 (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 -(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C -(\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1)) x9 (getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Abbr) -u1) n H23) H24))))) H22)) (\lambda (H22: (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(getl n x7 (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 -a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(getl n x7 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C (\lambda (d2: -C).(getl (S n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S -n) c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))) (\lambda (x9: C).(\lambda (x10: T).(\lambda (x11: -A).(\lambda (H23: (getl n x7 (CHead x9 (Bind Abst) x10))).(\lambda (H24: -(csuba g x9 d1)).(\lambda (H25: (arity g x9 x10 (asucc g x11))).(\lambda -(H26: (arity g d1 u1 x11)).(or3_intro1 (ex2 C (\lambda (d2: C).(getl (S n) c2 -(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 -(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g -d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 -u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead -d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) -(ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S -n) c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a)))) x9 x10 x11 (getl_clear_bind x6 c2 x7 x8 H20 -(CHead x9 (Bind Abst) x10) n H23) H24 H25 H26))))))))) H22)) (\lambda (H22: -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl n x7 (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind C T -(\lambda (d2: C).(\lambda (u2: T).(getl n x7 (CHead d2 (Bind Void) u2)))) -(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C (\lambda (d2: -C).(getl (S n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S -n) c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: -T).(csuba g d2 d1))))) (\lambda (x9: C).(\lambda (x10: T).(\lambda (H23: -(getl n x7 (CHead x9 (Bind Void) x10))).(\lambda (H24: (csuba g x9 -d1)).(or3_intro2 (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) -u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g -a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind -Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro -C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind Void) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x9 x10 -(getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Void) x10) n H23) H24)))))) -H22)) H21)))))))) H17)))))) H14))))))) H11)))))))) i) H7))))) k H3 H4))))))) -x H1 H2)))) H0))))))). -(* COMMENTS -Initial nodes: 9091 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/props.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/props.ma deleted file mode 100644 index 823e88825..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csuba/props.ma +++ /dev/null @@ -1,28 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csuba/defs.ma". - -theorem csuba_refl: - \forall (g: G).(\forall (c: C).(csuba g c c)) -\def - \lambda (g: G).(\lambda (c: C).(C_ind (\lambda (c0: C).(csuba g c0 c0)) -(\lambda (n: nat).(csuba_sort g n)) (\lambda (c0: C).(\lambda (H: (csuba g c0 -c0)).(\lambda (k: K).(\lambda (t: T).(csuba_head g c0 c0 H k t))))) c)). -(* COMMENTS -Initial nodes: 53 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/arity.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/arity.ma deleted file mode 100644 index 62922d311..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/arity.ma +++ /dev/null @@ -1,42 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubc/csuba.ma". - -theorem csubc_arity_conf: - \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to -(\forall (t: T).(\forall (a: A).((arity g c1 t a) \to (arity g c2 t a))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubc g c1 -c2)).(\lambda (t: T).(\lambda (a: A).(\lambda (H0: (arity g c1 t -a)).(csuba_arity g c1 t a H0 c2 (csubc_csuba g c1 c2 H)))))))). -(* COMMENTS -Initial nodes: 51 -END *) - -theorem csubc_arity_trans: - \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to -((csubv c1 c2) \to (\forall (t: T).(\forall (a: A).((arity g c2 t a) \to -(arity g c1 t a)))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubc g c1 -c2)).(\lambda (H0: (csubv c1 c2)).(\lambda (t: T).(\lambda (a: A).(\lambda -(H1: (arity g c2 t a)).(csuba_arity_rev g c2 t a H1 c1 (csubc_csuba g c1 c2 -H) H0)))))))). -(* COMMENTS -Initial nodes: 59 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/clear.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/clear.ma deleted file mode 100644 index 185bdff75..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/clear.ma +++ /dev/null @@ -1,170 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubc/fwd.ma". - -theorem csubc_clear_conf: - \forall (g: G).(\forall (c1: C).(\forall (e1: C).((clear c1 e1) \to (\forall -(c2: C).((csubc g c1 c2) \to (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda -(e2: C).(csubc g e1 e2)))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (e1: C).(\lambda (H: (clear c1 -e1)).(clear_ind (\lambda (c: C).(\lambda (c0: C).(\forall (c2: C).((csubc g c -c2) \to (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2: C).(csubc g c0 -e2))))))) (\lambda (b: B).(\lambda (e: C).(\lambda (u: T).(\lambda (c2: -C).(\lambda (H0: (csubc g (CHead e (Bind b) u) c2)).(let H_x \def -(csubc_gen_head_l g e c2 u (Bind b) H0) in (let H1 \def H_x in (or3_ind (ex2 -C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind b) u))) (\lambda (c3: C).(csubc g -e c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K -(Bind b) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq -C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda -(_: A).(csubc g e c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 -g (asucc g a) e u)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g -a c3 w))))) (ex4_3 B C T (\lambda (b0: B).(\lambda (c3: C).(\lambda (v2: -T).(eq C c2 (CHead c3 (Bind b0) v2))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: T).(eq K (Bind b) (Bind Void))))) (\lambda (b0: B).(\lambda -(_: C).(\lambda (_: T).(not (eq B b0 Void))))) (\lambda (_: B).(\lambda (c3: -C).(\lambda (_: T).(csubc g e c3))))) (ex2 C (\lambda (e2: C).(clear c2 e2)) -(\lambda (e2: C).(csubc g (CHead e (Bind b) u) e2))) (\lambda (H2: (ex2 C -(\lambda (c3: C).(eq C c2 (CHead c3 (Bind b) u))) (\lambda (c3: C).(csubc g e -c3)))).(ex2_ind C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind b) u))) (\lambda -(c3: C).(csubc g e c3)) (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2: -C).(csubc g (CHead e (Bind b) u) e2))) (\lambda (x: C).(\lambda (H3: (eq C c2 -(CHead x (Bind b) u))).(\lambda (H4: (csubc g e x)).(eq_ind_r C (CHead x -(Bind b) u) (\lambda (c: C).(ex2 C (\lambda (e2: C).(clear c e2)) (\lambda -(e2: C).(csubc g (CHead e (Bind b) u) e2)))) (ex_intro2 C (\lambda (e2: -C).(clear (CHead x (Bind b) u) e2)) (\lambda (e2: C).(csubc g (CHead e (Bind -b) u) e2)) (CHead x (Bind b) u) (clear_bind b x u) (csubc_head g e x H4 (Bind -b) u)) c2 H3)))) H2)) (\lambda (H2: (ex5_3 C T A (\lambda (_: C).(\lambda (_: -T).(\lambda (_: A).(eq K (Bind b) (Bind Abst))))) (\lambda (c3: C).(\lambda -(w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: -C).(\lambda (_: T).(\lambda (_: A).(csubc g e c3)))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(sc3 g (asucc g a) e u)))) (\lambda (c3: C).(\lambda -(w: T).(\lambda (a: A).(sc3 g a c3 w)))))).(ex5_3_ind C T A (\lambda (_: -C).(\lambda (_: T).(\lambda (_: A).(eq K (Bind b) (Bind Abst))))) (\lambda -(c3: C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) -(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g e c3)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) e u)))) (\lambda -(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w)))) (ex2 C (\lambda -(e2: C).(clear c2 e2)) (\lambda (e2: C).(csubc g (CHead e (Bind b) u) e2))) -(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H3: (eq K (Bind -b) (Bind Abst))).(\lambda (H4: (eq C c2 (CHead x0 (Bind Abbr) x1))).(\lambda -(H5: (csubc g e x0)).(\lambda (H6: (sc3 g (asucc g x2) e u)).(\lambda (H7: -(sc3 g x2 x0 x1)).(eq_ind_r C (CHead x0 (Bind Abbr) x1) (\lambda (c: C).(ex2 -C (\lambda (e2: C).(clear c e2)) (\lambda (e2: C).(csubc g (CHead e (Bind b) -u) e2)))) (let H8 \def (f_equal K B (\lambda (e0: K).(match e0 in K return -(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow b])) -(Bind b) (Bind Abst) H3) in (eq_ind_r B Abst (\lambda (b0: B).(ex2 C (\lambda -(e2: C).(clear (CHead x0 (Bind Abbr) x1) e2)) (\lambda (e2: C).(csubc g -(CHead e (Bind b0) u) e2)))) (ex_intro2 C (\lambda (e2: C).(clear (CHead x0 -(Bind Abbr) x1) e2)) (\lambda (e2: C).(csubc g (CHead e (Bind Abst) u) e2)) -(CHead x0 (Bind Abbr) x1) (clear_bind Abbr x0 x1) (csubc_abst g e x0 H5 u x2 -H6 x1 H7)) b H8)) c2 H4))))))))) H2)) (\lambda (H2: (ex4_3 B C T (\lambda -(b0: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b0) -v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K (Bind b) (Bind -Void))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b0 -Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g e -c3)))))).(ex4_3_ind B C T (\lambda (b0: B).(\lambda (c3: C).(\lambda (v2: -T).(eq C c2 (CHead c3 (Bind b0) v2))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: T).(eq K (Bind b) (Bind Void))))) (\lambda (b0: B).(\lambda -(_: C).(\lambda (_: T).(not (eq B b0 Void))))) (\lambda (_: B).(\lambda (c3: -C).(\lambda (_: T).(csubc g e c3)))) (ex2 C (\lambda (e2: C).(clear c2 e2)) -(\lambda (e2: C).(csubc g (CHead e (Bind b) u) e2))) (\lambda (x0: -B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H3: (eq C c2 (CHead x1 (Bind -x0) x2))).(\lambda (H4: (eq K (Bind b) (Bind Void))).(\lambda (H5: (not (eq B -x0 Void))).(\lambda (H6: (csubc g e x1)).(eq_ind_r C (CHead x1 (Bind x0) x2) -(\lambda (c: C).(ex2 C (\lambda (e2: C).(clear c e2)) (\lambda (e2: C).(csubc -g (CHead e (Bind b) u) e2)))) (let H7 \def (f_equal K B (\lambda (e0: -K).(match e0 in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | -(Flat _) \Rightarrow b])) (Bind b) (Bind Void) H4) in (eq_ind_r B Void -(\lambda (b0: B).(ex2 C (\lambda (e2: C).(clear (CHead x1 (Bind x0) x2) e2)) -(\lambda (e2: C).(csubc g (CHead e (Bind b0) u) e2)))) (ex_intro2 C (\lambda -(e2: C).(clear (CHead x1 (Bind x0) x2) e2)) (\lambda (e2: C).(csubc g (CHead -e (Bind Void) u) e2)) (CHead x1 (Bind x0) x2) (clear_bind x0 x1 x2) -(csubc_void g e x1 H6 x0 H5 u x2)) b H7)) c2 H3)))))))) H2)) H1)))))))) -(\lambda (e: C).(\lambda (c: C).(\lambda (_: (clear e c)).(\lambda (H1: -((\forall (c2: C).((csubc g e c2) \to (ex2 C (\lambda (e2: C).(clear c2 e2)) -(\lambda (e2: C).(csubc g c e2))))))).(\lambda (f: F).(\lambda (u: -T).(\lambda (c2: C).(\lambda (H2: (csubc g (CHead e (Flat f) u) c2)).(let H_x -\def (csubc_gen_head_l g e c2 u (Flat f) H2) in (let H3 \def H_x in (or3_ind -(ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 (Flat f) u))) (\lambda (c3: -C).(csubc g e c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: -A).(eq K (Flat f) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda -(_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: -T).(\lambda (_: A).(csubc g e c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(sc3 g (asucc g a) e u)))) (\lambda (c3: C).(\lambda (w: T).(\lambda -(a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: -C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: T).(eq K (Flat f) (Bind Void))))) (\lambda -(b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: -B).(\lambda (c3: C).(\lambda (_: T).(csubc g e c3))))) (ex2 C (\lambda (e2: -C).(clear c2 e2)) (\lambda (e2: C).(csubc g c e2))) (\lambda (H4: (ex2 C -(\lambda (c3: C).(eq C c2 (CHead c3 (Flat f) u))) (\lambda (c3: C).(csubc g e -c3)))).(ex2_ind C (\lambda (c3: C).(eq C c2 (CHead c3 (Flat f) u))) (\lambda -(c3: C).(csubc g e c3)) (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2: -C).(csubc g c e2))) (\lambda (x: C).(\lambda (H5: (eq C c2 (CHead x (Flat f) -u))).(\lambda (H6: (csubc g e x)).(eq_ind_r C (CHead x (Flat f) u) (\lambda -(c0: C).(ex2 C (\lambda (e2: C).(clear c0 e2)) (\lambda (e2: C).(csubc g c -e2)))) (let H_x0 \def (H1 x H6) in (let H7 \def H_x0 in (ex2_ind C (\lambda -(e2: C).(clear x e2)) (\lambda (e2: C).(csubc g c e2)) (ex2 C (\lambda (e2: -C).(clear (CHead x (Flat f) u) e2)) (\lambda (e2: C).(csubc g c e2))) -(\lambda (x0: C).(\lambda (H8: (clear x x0)).(\lambda (H9: (csubc g c -x0)).(ex_intro2 C (\lambda (e2: C).(clear (CHead x (Flat f) u) e2)) (\lambda -(e2: C).(csubc g c e2)) x0 (clear_flat x x0 H8 f u) H9)))) H7))) c2 H5)))) -H4)) (\lambda (H4: (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: -A).(eq K (Flat f) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda -(_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: -T).(\lambda (_: A).(csubc g e c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(sc3 g (asucc g a) e u)))) (\lambda (c3: C).(\lambda (w: T).(\lambda -(a: A).(sc3 g a c3 w)))))).(ex5_3_ind C T A (\lambda (_: C).(\lambda (_: -T).(\lambda (_: A).(eq K (Flat f) (Bind Abst))))) (\lambda (c3: C).(\lambda -(w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: -C).(\lambda (_: T).(\lambda (_: A).(csubc g e c3)))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(sc3 g (asucc g a) e u)))) (\lambda (c3: C).(\lambda -(w: T).(\lambda (a: A).(sc3 g a c3 w)))) (ex2 C (\lambda (e2: C).(clear c2 -e2)) (\lambda (e2: C).(csubc g c e2))) (\lambda (x0: C).(\lambda (x1: -T).(\lambda (x2: A).(\lambda (H5: (eq K (Flat f) (Bind Abst))).(\lambda (H6: -(eq C c2 (CHead x0 (Bind Abbr) x1))).(\lambda (_: (csubc g e x0)).(\lambda -(_: (sc3 g (asucc g x2) e u)).(\lambda (_: (sc3 g x2 x0 x1)).(eq_ind_r C -(CHead x0 (Bind Abbr) x1) (\lambda (c0: C).(ex2 C (\lambda (e2: C).(clear c0 -e2)) (\lambda (e2: C).(csubc g c e2)))) (let H10 \def (eq_ind K (Flat f) -(\lambda (ee: K).(match ee in K return (\lambda (_: K).Prop) with [(Bind _) -\Rightarrow False | (Flat _) \Rightarrow True])) I (Bind Abst) H5) in -(False_ind (ex2 C (\lambda (e2: C).(clear (CHead x0 (Bind Abbr) x1) e2)) -(\lambda (e2: C).(csubc g c e2))) H10)) c2 H6))))))))) H4)) (\lambda (H4: -(ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 -(CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -T).(eq K (Flat f) (Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda -(_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: -T).(csubc g e c3)))))).(ex4_3_ind B C T (\lambda (b: B).(\lambda (c3: -C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: T).(eq K (Flat f) (Bind Void))))) (\lambda -(b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: -B).(\lambda (c3: C).(\lambda (_: T).(csubc g e c3)))) (ex2 C (\lambda (e2: -C).(clear c2 e2)) (\lambda (e2: C).(csubc g c e2))) (\lambda (x0: B).(\lambda -(x1: C).(\lambda (x2: T).(\lambda (H5: (eq C c2 (CHead x1 (Bind x0) -x2))).(\lambda (H6: (eq K (Flat f) (Bind Void))).(\lambda (_: (not (eq B x0 -Void))).(\lambda (_: (csubc g e x1)).(eq_ind_r C (CHead x1 (Bind x0) x2) -(\lambda (c0: C).(ex2 C (\lambda (e2: C).(clear c0 e2)) (\lambda (e2: -C).(csubc g c e2)))) (let H9 \def (eq_ind K (Flat f) (\lambda (ee: K).(match -ee in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat -_) \Rightarrow True])) I (Bind Void) H6) in (False_ind (ex2 C (\lambda (e2: -C).(clear (CHead x1 (Bind x0) x2) e2)) (\lambda (e2: C).(csubc g c e2))) H9)) -c2 H5)))))))) H4)) H3))))))))))) c1 e1 H)))). -(* COMMENTS -Initial nodes: 2837 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/csuba.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/csuba.ma deleted file mode 100644 index 2abfe2a1f..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/csuba.ma +++ /dev/null @@ -1,40 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubc/defs.ma". - -include "Basic-1/sc3/props.ma". - -theorem csubc_csuba: - \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to (csuba -g c1 c2)))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubc g c1 -c2)).(csubc_ind g (\lambda (c: C).(\lambda (c0: C).(csuba g c c0))) (\lambda -(n: nat).(csuba_refl g (CSort n))) (\lambda (c3: C).(\lambda (c4: C).(\lambda -(_: (csubc g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (k: K).(\lambda -(v: T).(csuba_head g c3 c4 H1 k v))))))) (\lambda (c3: C).(\lambda (c4: -C).(\lambda (_: (csubc g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (b: -B).(\lambda (H2: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: -T).(csuba_void g c3 c4 H1 b H2 u1 u2))))))))) (\lambda (c3: C).(\lambda (c4: -C).(\lambda (_: (csubc g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (v: -T).(\lambda (a: A).(\lambda (H2: (sc3 g (asucc g a) c3 v)).(\lambda (w: -T).(\lambda (H3: (sc3 g a c4 w)).(csuba_abst g c3 c4 H1 v a (sc3_arity_gen g -c3 v (asucc g a) H2) w (sc3_arity_gen g c4 w a H3))))))))))) c1 c2 H)))). -(* COMMENTS -Initial nodes: 231 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/defs.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/defs.ma deleted file mode 100644 index 73a4c5625..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/defs.ma +++ /dev/null @@ -1,30 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/sc3/defs.ma". - -inductive csubc (g: G): C \to (C \to Prop) \def -| csubc_sort: \forall (n: nat).(csubc g (CSort n) (CSort n)) -| csubc_head: \forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to (\forall -(k: K).(\forall (v: T).(csubc g (CHead c1 k v) (CHead c2 k v)))))) -| csubc_void: \forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to (\forall -(b: B).((not (eq B b Void)) \to (\forall (u1: T).(\forall (u2: T).(csubc g -(CHead c1 (Bind Void) u1) (CHead c2 (Bind b) u2)))))))) -| csubc_abst: \forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to (\forall -(v: T).(\forall (a: A).((sc3 g (asucc g a) c1 v) \to (\forall (w: T).((sc3 g -a c2 w) \to (csubc g (CHead c1 (Bind Abst) v) (CHead c2 (Bind Abbr) -w))))))))). - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/drop.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/drop.ma deleted file mode 100644 index 195d36491..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/drop.ma +++ /dev/null @@ -1,475 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubc/fwd.ma". - -include "Basic-1/sc3/props.ma". - -theorem csubc_drop_conf_O: - \forall (g: G).(\forall (c1: C).(\forall (e1: C).(\forall (h: nat).((drop h -O c1 e1) \to (\forall (c2: C).((csubc g c1 c2) \to (ex2 C (\lambda (e2: -C).(drop h O c2 e2)) (\lambda (e2: C).(csubc g e1 e2))))))))) -\def - \lambda (g: G).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (e1: -C).(\forall (h: nat).((drop h O c e1) \to (\forall (c2: C).((csubc g c c2) -\to (ex2 C (\lambda (e2: C).(drop h O c2 e2)) (\lambda (e2: C).(csubc g e1 -e2))))))))) (\lambda (n: nat).(\lambda (e1: C).(\lambda (h: nat).(\lambda (H: -(drop h O (CSort n) e1)).(\lambda (c2: C).(\lambda (H0: (csubc g (CSort n) -c2)).(and3_ind (eq C e1 (CSort n)) (eq nat h O) (eq nat O O) (ex2 C (\lambda -(e2: C).(drop h O c2 e2)) (\lambda (e2: C).(csubc g e1 e2))) (\lambda (H1: -(eq C e1 (CSort n))).(\lambda (H2: (eq nat h O)).(\lambda (_: (eq nat O -O)).(eq_ind_r nat O (\lambda (n0: nat).(ex2 C (\lambda (e2: C).(drop n0 O c2 -e2)) (\lambda (e2: C).(csubc g e1 e2)))) (eq_ind_r C (CSort n) (\lambda (c: -C).(ex2 C (\lambda (e2: C).(drop O O c2 e2)) (\lambda (e2: C).(csubc g c -e2)))) (ex_intro2 C (\lambda (e2: C).(drop O O c2 e2)) (\lambda (e2: -C).(csubc g (CSort n) e2)) c2 (drop_refl c2) H0) e1 H1) h H2)))) -(drop_gen_sort n h O e1 H)))))))) (\lambda (c: C).(\lambda (H: ((\forall (e1: -C).(\forall (h: nat).((drop h O c e1) \to (\forall (c2: C).((csubc g c c2) -\to (ex2 C (\lambda (e2: C).(drop h O c2 e2)) (\lambda (e2: C).(csubc g e1 -e2)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e1: C).(\lambda (h: -nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c k t) e1) \to (\forall -(c2: C).((csubc g (CHead c k t) c2) \to (ex2 C (\lambda (e2: C).(drop n O c2 -e2)) (\lambda (e2: C).(csubc g e1 e2))))))) (\lambda (H0: (drop O O (CHead c -k t) e1)).(\lambda (c2: C).(\lambda (H1: (csubc g (CHead c k t) c2)).(eq_ind -C (CHead c k t) (\lambda (c0: C).(ex2 C (\lambda (e2: C).(drop O O c2 e2)) -(\lambda (e2: C).(csubc g c0 e2)))) (ex_intro2 C (\lambda (e2: C).(drop O O -c2 e2)) (\lambda (e2: C).(csubc g (CHead c k t) e2)) c2 (drop_refl c2) H1) e1 -(drop_gen_refl (CHead c k t) e1 H0))))) (\lambda (n: nat).(\lambda (H0: -(((drop n O (CHead c k t) e1) \to (\forall (c2: C).((csubc g (CHead c k t) -c2) \to (ex2 C (\lambda (e2: C).(drop n O c2 e2)) (\lambda (e2: C).(csubc g -e1 e2)))))))).(\lambda (H1: (drop (S n) O (CHead c k t) e1)).(\lambda (c2: -C).(\lambda (H2: (csubc g (CHead c k t) c2)).(let H_x \def (csubc_gen_head_l -g c c2 t k H2) in (let H3 \def H_x in (or3_ind (ex2 C (\lambda (c3: C).(eq C -c2 (CHead c3 k t))) (\lambda (c3: C).(csubc g c c3))) (ex5_3 C T A (\lambda -(_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3: -C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) -(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c c3)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c t)))) (\lambda -(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T -(\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b) -v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind -Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b -Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c c3))))) -(ex2 C (\lambda (e2: C).(drop (S n) O c2 e2)) (\lambda (e2: C).(csubc g e1 -e2))) (\lambda (H4: (ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 k t))) -(\lambda (c3: C).(csubc g c c3)))).(ex2_ind C (\lambda (c3: C).(eq C c2 -(CHead c3 k t))) (\lambda (c3: C).(csubc g c c3)) (ex2 C (\lambda (e2: -C).(drop (S n) O c2 e2)) (\lambda (e2: C).(csubc g e1 e2))) (\lambda (x: -C).(\lambda (H5: (eq C c2 (CHead x k t))).(\lambda (H6: (csubc g c -x)).(eq_ind_r C (CHead x k t) (\lambda (c0: C).(ex2 C (\lambda (e2: C).(drop -(S n) O c0 e2)) (\lambda (e2: C).(csubc g e1 e2)))) (let H_x0 \def (H e1 (r k -n) (drop_gen_drop k c e1 t n H1) x H6) in (let H7 \def H_x0 in (ex2_ind C -(\lambda (e2: C).(drop (r k n) O x e2)) (\lambda (e2: C).(csubc g e1 e2)) -(ex2 C (\lambda (e2: C).(drop (S n) O (CHead x k t) e2)) (\lambda (e2: -C).(csubc g e1 e2))) (\lambda (x0: C).(\lambda (H8: (drop (r k n) O x -x0)).(\lambda (H9: (csubc g e1 x0)).(ex_intro2 C (\lambda (e2: C).(drop (S n) -O (CHead x k t) e2)) (\lambda (e2: C).(csubc g e1 e2)) x0 (drop_drop k n x x0 -H8 t) H9)))) H7))) c2 H5)))) H4)) (\lambda (H4: (ex5_3 C T A (\lambda (_: -C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3: -C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) -(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c c3)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c t)))) (\lambda -(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w)))))).(ex5_3_ind C T A -(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) -(\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind -Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c -c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c -t)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w)))) (ex2 -C (\lambda (e2: C).(drop (S n) O c2 e2)) (\lambda (e2: C).(csubc g e1 e2))) -(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H5: (eq K k -(Bind Abst))).(\lambda (H6: (eq C c2 (CHead x0 (Bind Abbr) x1))).(\lambda -(H7: (csubc g c x0)).(\lambda (_: (sc3 g (asucc g x2) c t)).(\lambda (_: (sc3 -g x2 x0 x1)).(eq_ind_r C (CHead x0 (Bind Abbr) x1) (\lambda (c0: C).(ex2 C -(\lambda (e2: C).(drop (S n) O c0 e2)) (\lambda (e2: C).(csubc g e1 e2)))) -(let H10 \def (eq_ind K k (\lambda (k0: K).(drop (r k0 n) O c e1)) -(drop_gen_drop k c e1 t n H1) (Bind Abst) H5) in (let H11 \def (eq_ind K k -(\lambda (k0: K).((drop n O (CHead c k0 t) e1) \to (\forall (c3: C).((csubc g -(CHead c k0 t) c3) \to (ex2 C (\lambda (e2: C).(drop n O c3 e2)) (\lambda -(e2: C).(csubc g e1 e2))))))) H0 (Bind Abst) H5) in (let H_x0 \def (H e1 (r -(Bind Abst) n) H10 x0 H7) in (let H12 \def H_x0 in (ex2_ind C (\lambda (e2: -C).(drop n O x0 e2)) (\lambda (e2: C).(csubc g e1 e2)) (ex2 C (\lambda (e2: -C).(drop (S n) O (CHead x0 (Bind Abbr) x1) e2)) (\lambda (e2: C).(csubc g e1 -e2))) (\lambda (x: C).(\lambda (H13: (drop n O x0 x)).(\lambda (H14: (csubc g -e1 x)).(ex_intro2 C (\lambda (e2: C).(drop (S n) O (CHead x0 (Bind Abbr) x1) -e2)) (\lambda (e2: C).(csubc g e1 e2)) x (drop_drop (Bind Abbr) n x0 x H13 -x1) H14)))) H12))))) c2 H6))))))))) H4)) (\lambda (H4: (ex4_3 B C T (\lambda -(b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b) v2))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind Void))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) -(\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c -c3)))))).(ex4_3_ind B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: -T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: -C).(\lambda (_: T).(csubc g c c3)))) (ex2 C (\lambda (e2: C).(drop (S n) O c2 -e2)) (\lambda (e2: C).(csubc g e1 e2))) (\lambda (x0: B).(\lambda (x1: -C).(\lambda (x2: T).(\lambda (H5: (eq C c2 (CHead x1 (Bind x0) x2))).(\lambda -(H6: (eq K k (Bind Void))).(\lambda (_: (not (eq B x0 Void))).(\lambda (H8: -(csubc g c x1)).(eq_ind_r C (CHead x1 (Bind x0) x2) (\lambda (c0: C).(ex2 C -(\lambda (e2: C).(drop (S n) O c0 e2)) (\lambda (e2: C).(csubc g e1 e2)))) -(let H9 \def (eq_ind K k (\lambda (k0: K).(drop (r k0 n) O c e1)) -(drop_gen_drop k c e1 t n H1) (Bind Void) H6) in (let H10 \def (eq_ind K k -(\lambda (k0: K).((drop n O (CHead c k0 t) e1) \to (\forall (c3: C).((csubc g -(CHead c k0 t) c3) \to (ex2 C (\lambda (e2: C).(drop n O c3 e2)) (\lambda -(e2: C).(csubc g e1 e2))))))) H0 (Bind Void) H6) in (let H_x0 \def (H e1 (r -(Bind Void) n) H9 x1 H8) in (let H11 \def H_x0 in (ex2_ind C (\lambda (e2: -C).(drop n O x1 e2)) (\lambda (e2: C).(csubc g e1 e2)) (ex2 C (\lambda (e2: -C).(drop (S n) O (CHead x1 (Bind x0) x2) e2)) (\lambda (e2: C).(csubc g e1 -e2))) (\lambda (x: C).(\lambda (H12: (drop n O x1 x)).(\lambda (H13: (csubc g -e1 x)).(ex_intro2 C (\lambda (e2: C).(drop (S n) O (CHead x1 (Bind x0) x2) -e2)) (\lambda (e2: C).(csubc g e1 e2)) x (drop_drop (Bind x0) n x1 x H12 x2) -H13)))) H11))))) c2 H5)))))))) H4)) H3)))))))) h))))))) c1)). -(* COMMENTS -Initial nodes: 2389 -END *) - -theorem drop_csubc_trans: - \forall (g: G).(\forall (c2: C).(\forall (e2: C).(\forall (d: nat).(\forall -(h: nat).((drop h d c2 e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C -(\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c2 c1)))))))))) -\def - \lambda (g: G).(\lambda (c2: C).(C_ind (\lambda (c: C).(\forall (e2: -C).(\forall (d: nat).(\forall (h: nat).((drop h d c e2) \to (\forall (e1: -C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda -(c1: C).(csubc g c c1)))))))))) (\lambda (n: nat).(\lambda (e2: C).(\lambda -(d: nat).(\lambda (h: nat).(\lambda (H: (drop h d (CSort n) e2)).(\lambda -(e1: C).(\lambda (H0: (csubc g e2 e1)).(and3_ind (eq C e2 (CSort n)) (eq nat -h O) (eq nat d O) (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: -C).(csubc g (CSort n) c1))) (\lambda (H1: (eq C e2 (CSort n))).(\lambda (H2: -(eq nat h O)).(\lambda (H3: (eq nat d O)).(eq_ind_r nat O (\lambda (n0: -nat).(ex2 C (\lambda (c1: C).(drop n0 d c1 e1)) (\lambda (c1: C).(csubc g -(CSort n) c1)))) (eq_ind_r nat O (\lambda (n0: nat).(ex2 C (\lambda (c1: -C).(drop O n0 c1 e1)) (\lambda (c1: C).(csubc g (CSort n) c1)))) (let H4 \def -(eq_ind C e2 (\lambda (c: C).(csubc g c e1)) H0 (CSort n) H1) in (ex_intro2 C -(\lambda (c1: C).(drop O O c1 e1)) (\lambda (c1: C).(csubc g (CSort n) c1)) -e1 (drop_refl e1) H4)) d H3) h H2)))) (drop_gen_sort n h d e2 H))))))))) -(\lambda (c: C).(\lambda (H: ((\forall (e2: C).(\forall (d: nat).(\forall (h: -nat).((drop h d c e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C -(\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c -c1))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e2: C).(\lambda (d: -nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n (CHead c k t) -e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop h -n c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1)))))))) (\lambda (h: -nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c k t) e2) \to (\forall -(e1: C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop n O c1 e1)) -(\lambda (c1: C).(csubc g (CHead c k t) c1))))))) (\lambda (H0: (drop O O -(CHead c k t) e2)).(\lambda (e1: C).(\lambda (H1: (csubc g e2 e1)).(let H2 -\def (eq_ind_r C e2 (\lambda (c0: C).(csubc g c0 e1)) H1 (CHead c k t) -(drop_gen_refl (CHead c k t) e2 H0)) in (ex_intro2 C (\lambda (c1: C).(drop O -O c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1)) e1 (drop_refl e1) -H2))))) (\lambda (n: nat).(\lambda (_: (((drop n O (CHead c k t) e2) \to -(\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop n O c1 -e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1)))))))).(\lambda (H1: (drop -(S n) O (CHead c k t) e2)).(\lambda (e1: C).(\lambda (H2: (csubc g e2 -e1)).(let H_x \def (H e2 O (r k n) (drop_gen_drop k c e2 t n H1) e1 H2) in -(let H3 \def H_x in (ex2_ind C (\lambda (c1: C).(drop (r k n) O c1 e1)) -(\lambda (c1: C).(csubc g c c1)) (ex2 C (\lambda (c1: C).(drop (S n) O c1 -e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1))) (\lambda (x: C).(\lambda -(H4: (drop (r k n) O x e1)).(\lambda (H5: (csubc g c x)).(ex_intro2 C -(\lambda (c1: C).(drop (S n) O c1 e1)) (\lambda (c1: C).(csubc g (CHead c k -t) c1)) (CHead x k t) (drop_drop k n x e1 H4 t) (csubc_head g c x H5 k t))))) -H3)))))))) h)) (\lambda (n: nat).(\lambda (H0: ((\forall (h: nat).((drop h n -(CHead c k t) e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda -(c1: C).(drop h n c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t) -c1))))))))).(\lambda (h: nat).(\lambda (H1: (drop h (S n) (CHead c k t) -e2)).(\lambda (e1: C).(\lambda (H2: (csubc g e2 e1)).(ex3_2_ind C T (\lambda -(e: C).(\lambda (v: T).(eq C e2 (CHead e k v)))) (\lambda (_: C).(\lambda (v: -T).(eq T t (lift h (r k n) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r k -n) c e))) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1: -C).(csubc g (CHead c k t) c1))) (\lambda (x0: C).(\lambda (x1: T).(\lambda -(H3: (eq C e2 (CHead x0 k x1))).(\lambda (H4: (eq T t (lift h (r k n) -x1))).(\lambda (H5: (drop h (r k n) c x0)).(let H6 \def (eq_ind C e2 (\lambda -(c0: C).(csubc g c0 e1)) H2 (CHead x0 k x1) H3) in (let H7 \def (eq_ind C e2 -(\lambda (c0: C).(\forall (h0: nat).((drop h0 n (CHead c k t) c0) \to -(\forall (e3: C).((csubc g c0 e3) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 -e3)) (\lambda (c1: C).(csubc g (CHead c k t) c1)))))))) H0 (CHead x0 k x1) -H3) in (let H8 \def (eq_ind T t (\lambda (t0: T).(\forall (h0: nat).((drop h0 -n (CHead c k t0) (CHead x0 k x1)) \to (\forall (e3: C).((csubc g (CHead x0 k -x1) e3) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 e3)) (\lambda (c1: -C).(csubc g (CHead c k t0) c1)))))))) H7 (lift h (r k n) x1) H4) in (eq_ind_r -T (lift h (r k n) x1) (\lambda (t0: T).(ex2 C (\lambda (c1: C).(drop h (S n) -c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t0) c1)))) (let H_x \def -(csubc_gen_head_l g x0 e1 x1 k H6) in (let H9 \def H_x in (or3_ind (ex2 C -(\lambda (c3: C).(eq C e1 (CHead c3 k x1))) (\lambda (c3: C).(csubc g x0 -c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k -(Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C e1 -(CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: -A).(csubc g x0 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g -(asucc g a) x0 x1)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g -a c3 w))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: -T).(eq C e1 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: -C).(\lambda (_: T).(csubc g x0 c3))))) (ex2 C (\lambda (c1: C).(drop h (S n) -c1 e1)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1))) -(\lambda (H10: (ex2 C (\lambda (c3: C).(eq C e1 (CHead c3 k x1))) (\lambda -(c3: C).(csubc g x0 c3)))).(ex2_ind C (\lambda (c3: C).(eq C e1 (CHead c3 k -x1))) (\lambda (c3: C).(csubc g x0 c3)) (ex2 C (\lambda (c1: C).(drop h (S n) -c1 e1)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1))) -(\lambda (x: C).(\lambda (H11: (eq C e1 (CHead x k x1))).(\lambda (H12: -(csubc g x0 x)).(eq_ind_r C (CHead x k x1) (\lambda (c0: C).(ex2 C (\lambda -(c1: C).(drop h (S n) c1 c0)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r -k n) x1)) c1)))) (let H_x0 \def (H x0 (r k n) h H5 x H12) in (let H13 \def -H_x0 in (ex2_ind C (\lambda (c1: C).(drop h (r k n) c1 x)) (\lambda (c1: -C).(csubc g c c1)) (ex2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x k x1))) -(\lambda (c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1))) (\lambda (x2: -C).(\lambda (H14: (drop h (r k n) x2 x)).(\lambda (H15: (csubc g c -x2)).(ex_intro2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x k x1))) (\lambda -(c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1)) (CHead x2 k (lift h (r -k n) x1)) (drop_skip k h n x2 x H14 x1) (csubc_head g c x2 H15 k (lift h (r k -n) x1)))))) H13))) e1 H11)))) H10)) (\lambda (H10: (ex5_3 C T A (\lambda (_: -C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3: -C).(\lambda (w: T).(\lambda (_: A).(eq C e1 (CHead c3 (Bind Abbr) w))))) -(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g x0 c3)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) x0 x1)))) (\lambda -(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w)))))).(ex5_3_ind C T A -(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) -(\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C e1 (CHead c3 (Bind -Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g x0 -c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) x0 -x1)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w)))) -(ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1: C).(csubc g -(CHead c k (lift h (r k n) x1)) c1))) (\lambda (x2: C).(\lambda (x3: -T).(\lambda (x4: A).(\lambda (H11: (eq K k (Bind Abst))).(\lambda (H12: (eq C -e1 (CHead x2 (Bind Abbr) x3))).(\lambda (H13: (csubc g x0 x2)).(\lambda (H14: -(sc3 g (asucc g x4) x0 x1)).(\lambda (H15: (sc3 g x4 x2 x3)).(eq_ind_r C -(CHead x2 (Bind Abbr) x3) (\lambda (c0: C).(ex2 C (\lambda (c1: C).(drop h (S -n) c1 c0)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1)))) -(let H16 \def (eq_ind K k (\lambda (k0: K).(\forall (h0: nat).((drop h0 n -(CHead c k0 (lift h (r k0 n) x1)) (CHead x0 k0 x1)) \to (\forall (e3: -C).((csubc g (CHead x0 k0 x1) e3) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 -e3)) (\lambda (c1: C).(csubc g (CHead c k0 (lift h (r k0 n) x1)) c1)))))))) -H8 (Bind Abst) H11) in (let H17 \def (eq_ind K k (\lambda (k0: K).(drop h (r -k0 n) c x0)) H5 (Bind Abst) H11) in (eq_ind_r K (Bind Abst) (\lambda (k0: -K).(ex2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x2 (Bind Abbr) x3))) -(\lambda (c1: C).(csubc g (CHead c k0 (lift h (r k0 n) x1)) c1)))) (let H_x0 -\def (H x0 (r (Bind Abst) n) h H17 x2 H13) in (let H18 \def H_x0 in (ex2_ind -C (\lambda (c1: C).(drop h n c1 x2)) (\lambda (c1: C).(csubc g c c1)) (ex2 C -(\lambda (c1: C).(drop h (S n) c1 (CHead x2 (Bind Abbr) x3))) (\lambda (c1: -C).(csubc g (CHead c (Bind Abst) (lift h (r (Bind Abst) n) x1)) c1))) -(\lambda (x: C).(\lambda (H19: (drop h n x x2)).(\lambda (H20: (csubc g c -x)).(ex_intro2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x2 (Bind Abbr) -x3))) (\lambda (c1: C).(csubc g (CHead c (Bind Abst) (lift h (r (Bind Abst) -n) x1)) c1)) (CHead x (Bind Abbr) (lift h n x3)) (drop_skip_bind h n x x2 H19 -Abbr x3) (csubc_abst g c x H20 (lift h (r (Bind Abst) n) x1) x4 (sc3_lift g -(asucc g x4) x0 x1 H14 c h (r (Bind Abst) n) H17) (lift h n x3) (sc3_lift g -x4 x2 x3 H15 x h n H19)))))) H18))) k H11))) e1 H12))))))))) H10)) (\lambda -(H10: (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C e1 -(CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: -T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: -T).(csubc g x0 c3)))))).(ex4_3_ind B C T (\lambda (b: B).(\lambda (c3: -C).(\lambda (v2: T).(eq C e1 (CHead c3 (Bind b) v2))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: -B).(\lambda (c3: C).(\lambda (_: T).(csubc g x0 c3)))) (ex2 C (\lambda (c1: -C).(drop h (S n) c1 e1)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r k n) -x1)) c1))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: T).(\lambda (H11: -(eq C e1 (CHead x3 (Bind x2) x4))).(\lambda (H12: (eq K k (Bind -Void))).(\lambda (H13: (not (eq B x2 Void))).(\lambda (H14: (csubc g x0 -x3)).(eq_ind_r C (CHead x3 (Bind x2) x4) (\lambda (c0: C).(ex2 C (\lambda -(c1: C).(drop h (S n) c1 c0)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r -k n) x1)) c1)))) (let H15 \def (eq_ind K k (\lambda (k0: K).(\forall (h0: -nat).((drop h0 n (CHead c k0 (lift h (r k0 n) x1)) (CHead x0 k0 x1)) \to -(\forall (e3: C).((csubc g (CHead x0 k0 x1) e3) \to (ex2 C (\lambda (c1: -C).(drop h0 n c1 e3)) (\lambda (c1: C).(csubc g (CHead c k0 (lift h (r k0 n) -x1)) c1)))))))) H8 (Bind Void) H12) in (let H16 \def (eq_ind K k (\lambda -(k0: K).(drop h (r k0 n) c x0)) H5 (Bind Void) H12) in (eq_ind_r K (Bind -Void) (\lambda (k0: K).(ex2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x3 -(Bind x2) x4))) (\lambda (c1: C).(csubc g (CHead c k0 (lift h (r k0 n) x1)) -c1)))) (let H_x0 \def (H x0 (r (Bind Void) n) h H16 x3 H14) in (let H17 \def -H_x0 in (ex2_ind C (\lambda (c1: C).(drop h n c1 x3)) (\lambda (c1: C).(csubc -g c c1)) (ex2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x3 (Bind x2) x4))) -(\lambda (c1: C).(csubc g (CHead c (Bind Void) (lift h (r (Bind Void) n) x1)) -c1))) (\lambda (x: C).(\lambda (H18: (drop h n x x3)).(\lambda (H19: (csubc g -c x)).(ex_intro2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x3 (Bind x2) -x4))) (\lambda (c1: C).(csubc g (CHead c (Bind Void) (lift h (r (Bind Void) -n) x1)) c1)) (CHead x (Bind x2) (lift h n x4)) (drop_skip_bind h n x x3 H18 -x2 x4) (csubc_void g c x H19 x2 H13 (lift h (r (Bind Void) n) x1) (lift h n -x4)))))) H17))) k H12))) e1 H11)))))))) H10)) H9))) t H4))))))))) -(drop_gen_skip_l c e2 t h n k H1)))))))) d))))))) c2)). -(* COMMENTS -Initial nodes: 3747 -END *) - -theorem csubc_drop_conf_rev: - \forall (g: G).(\forall (c2: C).(\forall (e2: C).(\forall (d: nat).(\forall -(h: nat).((drop h d c2 e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C -(\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c1 c2)))))))))) -\def - \lambda (g: G).(\lambda (c2: C).(C_ind (\lambda (c: C).(\forall (e2: -C).(\forall (d: nat).(\forall (h: nat).((drop h d c e2) \to (\forall (e1: -C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda -(c1: C).(csubc g c1 c)))))))))) (\lambda (n: nat).(\lambda (e2: C).(\lambda -(d: nat).(\lambda (h: nat).(\lambda (H: (drop h d (CSort n) e2)).(\lambda -(e1: C).(\lambda (H0: (csubc g e1 e2)).(and3_ind (eq C e2 (CSort n)) (eq nat -h O) (eq nat d O) (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: -C).(csubc g c1 (CSort n)))) (\lambda (H1: (eq C e2 (CSort n))).(\lambda (H2: -(eq nat h O)).(\lambda (H3: (eq nat d O)).(eq_ind_r nat O (\lambda (n0: -nat).(ex2 C (\lambda (c1: C).(drop n0 d c1 e1)) (\lambda (c1: C).(csubc g c1 -(CSort n))))) (eq_ind_r nat O (\lambda (n0: nat).(ex2 C (\lambda (c1: -C).(drop O n0 c1 e1)) (\lambda (c1: C).(csubc g c1 (CSort n))))) (let H4 \def -(eq_ind C e2 (\lambda (c: C).(csubc g e1 c)) H0 (CSort n) H1) in (ex_intro2 C -(\lambda (c1: C).(drop O O c1 e1)) (\lambda (c1: C).(csubc g c1 (CSort n))) -e1 (drop_refl e1) H4)) d H3) h H2)))) (drop_gen_sort n h d e2 H))))))))) -(\lambda (c: C).(\lambda (H: ((\forall (e2: C).(\forall (d: nat).(\forall (h: -nat).((drop h d c e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C -(\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c1 -c))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e2: C).(\lambda (d: -nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n (CHead c k t) -e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop h -n c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t))))))))) (\lambda (h: -nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c k t) e2) \to (\forall -(e1: C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop n O c1 e1)) -(\lambda (c1: C).(csubc g c1 (CHead c k t)))))))) (\lambda (H0: (drop O O -(CHead c k t) e2)).(\lambda (e1: C).(\lambda (H1: (csubc g e1 e2)).(let H2 -\def (eq_ind_r C e2 (\lambda (c0: C).(csubc g e1 c0)) H1 (CHead c k t) -(drop_gen_refl (CHead c k t) e2 H0)) in (ex_intro2 C (\lambda (c1: C).(drop O -O c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t))) e1 (drop_refl e1) -H2))))) (\lambda (n: nat).(\lambda (_: (((drop n O (CHead c k t) e2) \to -(\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop n O c1 -e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t))))))))).(\lambda (H1: (drop -(S n) O (CHead c k t) e2)).(\lambda (e1: C).(\lambda (H2: (csubc g e1 -e2)).(let H_x \def (H e2 O (r k n) (drop_gen_drop k c e2 t n H1) e1 H2) in -(let H3 \def H_x in (ex2_ind C (\lambda (c1: C).(drop (r k n) O c1 e1)) -(\lambda (c1: C).(csubc g c1 c)) (ex2 C (\lambda (c1: C).(drop (S n) O c1 -e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t)))) (\lambda (x: C).(\lambda -(H4: (drop (r k n) O x e1)).(\lambda (H5: (csubc g x c)).(ex_intro2 C -(\lambda (c1: C).(drop (S n) O c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c -k t))) (CHead x k t) (drop_drop k n x e1 H4 t) (csubc_head g x c H5 k t))))) -H3)))))))) h)) (\lambda (n: nat).(\lambda (H0: ((\forall (h: nat).((drop h n -(CHead c k t) e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda -(c1: C).(drop h n c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k -t)))))))))).(\lambda (h: nat).(\lambda (H1: (drop h (S n) (CHead c k t) -e2)).(\lambda (e1: C).(\lambda (H2: (csubc g e1 e2)).(ex3_2_ind C T (\lambda -(e: C).(\lambda (v: T).(eq C e2 (CHead e k v)))) (\lambda (_: C).(\lambda (v: -T).(eq T t (lift h (r k n) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r k -n) c e))) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1: -C).(csubc g c1 (CHead c k t)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda -(H3: (eq C e2 (CHead x0 k x1))).(\lambda (H4: (eq T t (lift h (r k n) -x1))).(\lambda (H5: (drop h (r k n) c x0)).(let H6 \def (eq_ind C e2 (\lambda -(c0: C).(csubc g e1 c0)) H2 (CHead x0 k x1) H3) in (let H7 \def (eq_ind C e2 -(\lambda (c0: C).(\forall (h0: nat).((drop h0 n (CHead c k t) c0) \to -(\forall (e3: C).((csubc g e3 c0) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 -e3)) (\lambda (c1: C).(csubc g c1 (CHead c k t))))))))) H0 (CHead x0 k x1) -H3) in (let H8 \def (eq_ind T t (\lambda (t0: T).(\forall (h0: nat).((drop h0 -n (CHead c k t0) (CHead x0 k x1)) \to (\forall (e3: C).((csubc g e3 (CHead x0 -k x1)) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 e3)) (\lambda (c1: C).(csubc -g c1 (CHead c k t0))))))))) H7 (lift h (r k n) x1) H4) in (eq_ind_r T (lift h -(r k n) x1) (\lambda (t0: T).(ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) -(\lambda (c1: C).(csubc g c1 (CHead c k t0))))) (let H_x \def -(csubc_gen_head_r g x0 e1 x1 k H6) in (let H9 \def H_x in (or3_ind (ex2 C -(\lambda (c1: C).(eq C e1 (CHead c1 k x1))) (\lambda (c1: C).(csubc g c1 -x0))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k -(Bind Abbr))))) (\lambda (c1: C).(\lambda (v: T).(\lambda (_: A).(eq C e1 -(CHead c1 (Bind Abst) v))))) (\lambda (c1: C).(\lambda (_: T).(\lambda (_: -A).(csubc g c1 x0)))) (\lambda (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g -(asucc g a) c1 v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a -x0 x1))))) (ex4_3 B C T (\lambda (_: B).(\lambda (c1: C).(\lambda (v1: T).(eq -C e1 (CHead c1 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda -(_: T).(eq K k (Bind b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: -T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c1: C).(\lambda (_: -T).(csubc g c1 x0))))) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda -(c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1))))) (\lambda (H10: (ex2 C -(\lambda (c1: C).(eq C e1 (CHead c1 k x1))) (\lambda (c1: C).(csubc g c1 -x0)))).(ex2_ind C (\lambda (c1: C).(eq C e1 (CHead c1 k x1))) (\lambda (c1: -C).(csubc g c1 x0)) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda -(c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1))))) (\lambda (x: -C).(\lambda (H11: (eq C e1 (CHead x k x1))).(\lambda (H12: (csubc g x -x0)).(eq_ind_r C (CHead x k x1) (\lambda (c0: C).(ex2 C (\lambda (c1: -C).(drop h (S n) c1 c0)) (\lambda (c1: C).(csubc g c1 (CHead c k (lift h (r k -n) x1)))))) (let H_x0 \def (H x0 (r k n) h H5 x H12) in (let H13 \def H_x0 in -(ex2_ind C (\lambda (c1: C).(drop h (r k n) c1 x)) (\lambda (c1: C).(csubc g -c1 c)) (ex2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x k x1))) (\lambda -(c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1))))) (\lambda (x2: -C).(\lambda (H14: (drop h (r k n) x2 x)).(\lambda (H15: (csubc g x2 -c)).(ex_intro2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x k x1))) (\lambda -(c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1)))) (CHead x2 k (lift h (r -k n) x1)) (drop_skip k h n x2 x H14 x1) (csubc_head g x2 c H15 k (lift h (r k -n) x1)))))) H13))) e1 H11)))) H10)) (\lambda (H10: (ex5_3 C T A (\lambda (_: -C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c1: -C).(\lambda (v: T).(\lambda (_: A).(eq C e1 (CHead c1 (Bind Abst) v))))) -(\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 x0)))) (\lambda -(c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a x0 x1)))))).(ex5_3_ind C T A -(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) -(\lambda (c1: C).(\lambda (v: T).(\lambda (_: A).(eq C e1 (CHead c1 (Bind -Abst) v))))) (\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 -x0)))) (\lambda (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1 -v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a x0 x1)))) (ex2 -C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead -c k (lift h (r k n) x1))))) (\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: -A).(\lambda (H11: (eq K k (Bind Abbr))).(\lambda (H12: (eq C e1 (CHead x2 -(Bind Abst) x3))).(\lambda (H13: (csubc g x2 x0)).(\lambda (H14: (sc3 g -(asucc g x4) x2 x3)).(\lambda (H15: (sc3 g x4 x0 x1)).(eq_ind_r C (CHead x2 -(Bind Abst) x3) (\lambda (c0: C).(ex2 C (\lambda (c1: C).(drop h (S n) c1 -c0)) (\lambda (c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1)))))) (let -H16 \def (eq_ind K k (\lambda (k0: K).(\forall (h0: nat).((drop h0 n (CHead c -k0 (lift h (r k0 n) x1)) (CHead x0 k0 x1)) \to (\forall (e3: C).((csubc g e3 -(CHead x0 k0 x1)) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 e3)) (\lambda -(c1: C).(csubc g c1 (CHead c k0 (lift h (r k0 n) x1)))))))))) H8 (Bind Abbr) -H11) in (let H17 \def (eq_ind K k (\lambda (k0: K).(drop h (r k0 n) c x0)) H5 -(Bind Abbr) H11) in (eq_ind_r K (Bind Abbr) (\lambda (k0: K).(ex2 C (\lambda -(c1: C).(drop h (S n) c1 (CHead x2 (Bind Abst) x3))) (\lambda (c1: C).(csubc -g c1 (CHead c k0 (lift h (r k0 n) x1)))))) (let H_x0 \def (H x0 (r (Bind -Abbr) n) h H17 x2 H13) in (let H18 \def H_x0 in (ex2_ind C (\lambda (c1: -C).(drop h n c1 x2)) (\lambda (c1: C).(csubc g c1 c)) (ex2 C (\lambda (c1: -C).(drop h (S n) c1 (CHead x2 (Bind Abst) x3))) (\lambda (c1: C).(csubc g c1 -(CHead c (Bind Abbr) (lift h (r (Bind Abbr) n) x1))))) (\lambda (x: -C).(\lambda (H19: (drop h n x x2)).(\lambda (H20: (csubc g x c)).(ex_intro2 C -(\lambda (c1: C).(drop h (S n) c1 (CHead x2 (Bind Abst) x3))) (\lambda (c1: -C).(csubc g c1 (CHead c (Bind Abbr) (lift h (r (Bind Abbr) n) x1)))) (CHead x -(Bind Abst) (lift h n x3)) (drop_skip_bind h n x x2 H19 Abst x3) (csubc_abst -g x c H20 (lift h n x3) x4 (sc3_lift g (asucc g x4) x2 x3 H14 x h n H19) -(lift h (r (Bind Abbr) n) x1) (sc3_lift g x4 x0 x1 H15 c h (r (Bind Abbr) n) -H17)))))) H18))) k H11))) e1 H12))))))))) H10)) (\lambda (H10: (ex4_3 B C T -(\lambda (_: B).(\lambda (c1: C).(\lambda (v1: T).(eq C e1 (CHead c1 (Bind -Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind -b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) -(\lambda (_: B).(\lambda (c1: C).(\lambda (_: T).(csubc g c1 -x0)))))).(ex4_3_ind B C T (\lambda (_: B).(\lambda (c1: C).(\lambda (v1: -T).(eq C e1 (CHead c1 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c1: -C).(\lambda (_: T).(csubc g c1 x0)))) (ex2 C (\lambda (c1: C).(drop h (S n) -c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1))))) -(\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: T).(\lambda (H11: (eq C e1 -(CHead x3 (Bind Void) x4))).(\lambda (H12: (eq K k (Bind x2))).(\lambda (H13: -(not (eq B x2 Void))).(\lambda (H14: (csubc g x3 x0)).(eq_ind_r C (CHead x3 -(Bind Void) x4) (\lambda (c0: C).(ex2 C (\lambda (c1: C).(drop h (S n) c1 -c0)) (\lambda (c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1)))))) (let -H15 \def (eq_ind K k (\lambda (k0: K).(\forall (h0: nat).((drop h0 n (CHead c -k0 (lift h (r k0 n) x1)) (CHead x0 k0 x1)) \to (\forall (e3: C).((csubc g e3 -(CHead x0 k0 x1)) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 e3)) (\lambda -(c1: C).(csubc g c1 (CHead c k0 (lift h (r k0 n) x1)))))))))) H8 (Bind x2) -H12) in (let H16 \def (eq_ind K k (\lambda (k0: K).(drop h (r k0 n) c x0)) H5 -(Bind x2) H12) in (eq_ind_r K (Bind x2) (\lambda (k0: K).(ex2 C (\lambda (c1: -C).(drop h (S n) c1 (CHead x3 (Bind Void) x4))) (\lambda (c1: C).(csubc g c1 -(CHead c k0 (lift h (r k0 n) x1)))))) (let H_x0 \def (H x0 (r (Bind x2) n) h -H16 x3 H14) in (let H17 \def H_x0 in (ex2_ind C (\lambda (c1: C).(drop h n c1 -x3)) (\lambda (c1: C).(csubc g c1 c)) (ex2 C (\lambda (c1: C).(drop h (S n) -c1 (CHead x3 (Bind Void) x4))) (\lambda (c1: C).(csubc g c1 (CHead c (Bind -x2) (lift h (r (Bind x2) n) x1))))) (\lambda (x: C).(\lambda (H18: (drop h n -x x3)).(\lambda (H19: (csubc g x c)).(ex_intro2 C (\lambda (c1: C).(drop h (S -n) c1 (CHead x3 (Bind Void) x4))) (\lambda (c1: C).(csubc g c1 (CHead c (Bind -x2) (lift h (r (Bind x2) n) x1)))) (CHead x (Bind Void) (lift h n x4)) -(drop_skip_bind h n x x3 H18 Void x4) (csubc_void g x c H19 x2 H13 (lift h n -x4) (lift h (r (Bind x2) n) x1)))))) H17))) k H12))) e1 H11)))))))) H10)) -H9))) t H4))))))))) (drop_gen_skip_l c e2 t h n k H1)))))))) d))))))) c2)). -(* COMMENTS -Initial nodes: 3747 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/drop1.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/drop1.ma deleted file mode 100644 index 7c539cb4a..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/drop1.ma +++ /dev/null @@ -1,92 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubc/drop.ma". - -theorem drop1_csubc_trans: - \forall (g: G).(\forall (hds: PList).(\forall (c2: C).(\forall (e2: -C).((drop1 hds c2 e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C -(\lambda (c1: C).(drop1 hds c1 e1)) (\lambda (c1: C).(csubc g c2 c1))))))))) -\def - \lambda (g: G).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall -(c2: C).(\forall (e2: C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e2 -e1) \to (ex2 C (\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c2 -c1))))))))) (\lambda (c2: C).(\lambda (e2: C).(\lambda (H: (drop1 PNil c2 -e2)).(\lambda (e1: C).(\lambda (H0: (csubc g e2 e1)).(let H_y \def -(drop1_gen_pnil c2 e2 H) in (let H1 \def (eq_ind_r C e2 (\lambda (c: -C).(csubc g c e1)) H0 c2 H_y) in (ex_intro2 C (\lambda (c1: C).(drop1 PNil c1 -e1)) (\lambda (c1: C).(csubc g c2 c1)) e1 (drop1_nil e1) H1)))))))) (\lambda -(n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c2: -C).(\forall (e2: C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e2 e1) -\to (ex2 C (\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c2 -c1)))))))))).(\lambda (c2: C).(\lambda (e2: C).(\lambda (H0: (drop1 (PCons n -n0 p) c2 e2)).(\lambda (e1: C).(\lambda (H1: (csubc g e2 e1)).(let H_x \def -(drop1_gen_pcons c2 e2 p n n0 H0) in (let H2 \def H_x in (ex2_ind C (\lambda -(c3: C).(drop n n0 c2 c3)) (\lambda (c3: C).(drop1 p c3 e2)) (ex2 C (\lambda -(c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c2 c1))) -(\lambda (x: C).(\lambda (H3: (drop n n0 c2 x)).(\lambda (H4: (drop1 p x -e2)).(let H_x0 \def (H x e2 H4 e1 H1) in (let H5 \def H_x0 in (ex2_ind C -(\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g x c1)) (ex2 C -(\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c2 -c1))) (\lambda (x0: C).(\lambda (H6: (drop1 p x0 e1)).(\lambda (H7: (csubc g -x x0)).(let H_x1 \def (drop_csubc_trans g c2 x n0 n H3 x0 H7) in (let H8 \def -H_x1 in (ex2_ind C (\lambda (c1: C).(drop n n0 c1 x0)) (\lambda (c1: -C).(csubc g c2 c1)) (ex2 C (\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) -(\lambda (c1: C).(csubc g c2 c1))) (\lambda (x1: C).(\lambda (H9: (drop n n0 -x1 x0)).(\lambda (H10: (csubc g c2 x1)).(ex_intro2 C (\lambda (c1: C).(drop1 -(PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c2 c1)) x1 (drop1_cons x1 x0 -n n0 H9 e1 p H6) H10)))) H8)))))) H5)))))) H2)))))))))))) hds)). -(* COMMENTS -Initial nodes: 551 -END *) - -theorem csubc_drop1_conf_rev: - \forall (g: G).(\forall (hds: PList).(\forall (c2: C).(\forall (e2: -C).((drop1 hds c2 e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C -(\lambda (c1: C).(drop1 hds c1 e1)) (\lambda (c1: C).(csubc g c1 c2))))))))) -\def - \lambda (g: G).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall -(c2: C).(\forall (e2: C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e1 -e2) \to (ex2 C (\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c1 -c2))))))))) (\lambda (c2: C).(\lambda (e2: C).(\lambda (H: (drop1 PNil c2 -e2)).(\lambda (e1: C).(\lambda (H0: (csubc g e1 e2)).(let H_y \def -(drop1_gen_pnil c2 e2 H) in (let H1 \def (eq_ind_r C e2 (\lambda (c: -C).(csubc g e1 c)) H0 c2 H_y) in (ex_intro2 C (\lambda (c1: C).(drop1 PNil c1 -e1)) (\lambda (c1: C).(csubc g c1 c2)) e1 (drop1_nil e1) H1)))))))) (\lambda -(n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c2: -C).(\forall (e2: C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e1 e2) -\to (ex2 C (\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c1 -c2)))))))))).(\lambda (c2: C).(\lambda (e2: C).(\lambda (H0: (drop1 (PCons n -n0 p) c2 e2)).(\lambda (e1: C).(\lambda (H1: (csubc g e1 e2)).(let H_x \def -(drop1_gen_pcons c2 e2 p n n0 H0) in (let H2 \def H_x in (ex2_ind C (\lambda -(c3: C).(drop n n0 c2 c3)) (\lambda (c3: C).(drop1 p c3 e2)) (ex2 C (\lambda -(c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c1 c2))) -(\lambda (x: C).(\lambda (H3: (drop n n0 c2 x)).(\lambda (H4: (drop1 p x -e2)).(let H_x0 \def (H x e2 H4 e1 H1) in (let H5 \def H_x0 in (ex2_ind C -(\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c1 x)) (ex2 C -(\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c1 -c2))) (\lambda (x0: C).(\lambda (H6: (drop1 p x0 e1)).(\lambda (H7: (csubc g -x0 x)).(let H_x1 \def (csubc_drop_conf_rev g c2 x n0 n H3 x0 H7) in (let H8 -\def H_x1 in (ex2_ind C (\lambda (c1: C).(drop n n0 c1 x0)) (\lambda (c1: -C).(csubc g c1 c2)) (ex2 C (\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) -(\lambda (c1: C).(csubc g c1 c2))) (\lambda (x1: C).(\lambda (H9: (drop n n0 -x1 x0)).(\lambda (H10: (csubc g x1 c2)).(ex_intro2 C (\lambda (c1: C).(drop1 -(PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c1 c2)) x1 (drop1_cons x1 x0 -n n0 H9 e1 p H6) H10)))) H8)))))) H5)))))) H2)))))))))))) hds)). -(* COMMENTS -Initial nodes: 551 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/fwd.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/fwd.ma deleted file mode 100644 index fe04ddd45..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/fwd.ma +++ /dev/null @@ -1,673 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubc/defs.ma". - -theorem csubc_gen_sort_l: - \forall (g: G).(\forall (x: C).(\forall (n: nat).((csubc g (CSort n) x) \to -(eq C x (CSort n))))) -\def - \lambda (g: G).(\lambda (x: C).(\lambda (n: nat).(\lambda (H: (csubc g -(CSort n) x)).(insert_eq C (CSort n) (\lambda (c: C).(csubc g c x)) (\lambda -(c: C).(eq C x c)) (\lambda (y: C).(\lambda (H0: (csubc g y x)).(csubc_ind g -(\lambda (c: C).(\lambda (c0: C).((eq C c (CSort n)) \to (eq C c0 c)))) -(\lambda (n0: nat).(\lambda (H1: (eq C (CSort n0) (CSort n))).(let H2 \def -(f_equal C nat (\lambda (e: C).(match e in C return (\lambda (_: C).nat) with -[(CSort n1) \Rightarrow n1 | (CHead _ _ _) \Rightarrow n0])) (CSort n0) -(CSort n) H1) in (eq_ind_r nat n (\lambda (n1: nat).(eq C (CSort n1) (CSort -n1))) (refl_equal C (CSort n)) n0 H2)))) (\lambda (c1: C).(\lambda (c2: -C).(\lambda (_: (csubc g c1 c2)).(\lambda (_: (((eq C c1 (CSort n)) \to (eq C -c2 c1)))).(\lambda (k: K).(\lambda (v: T).(\lambda (H3: (eq C (CHead c1 k v) -(CSort n))).(let H4 \def (eq_ind C (CHead c1 k v) (\lambda (ee: C).(match ee -in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead -_ _ _) \Rightarrow True])) I (CSort n) H3) in (False_ind (eq C (CHead c2 k v) -(CHead c1 k v)) H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: -(csubc g c1 c2)).(\lambda (_: (((eq C c1 (CSort n)) \to (eq C c2 -c1)))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: -T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u1) (CSort -n))).(let H5 \def (eq_ind C (CHead c1 (Bind Void) u1) (\lambda (ee: C).(match -ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | -(CHead _ _ _) \Rightarrow True])) I (CSort n) H4) in (False_ind (eq C (CHead -c2 (Bind b) u2) (CHead c1 (Bind Void) u1)) H5))))))))))) (\lambda (c1: -C).(\lambda (c2: C).(\lambda (_: (csubc g c1 c2)).(\lambda (_: (((eq C c1 -(CSort n)) \to (eq C c2 c1)))).(\lambda (v: T).(\lambda (a: A).(\lambda (_: -(sc3 g (asucc g a) c1 v)).(\lambda (w: T).(\lambda (_: (sc3 g a c2 -w)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) v) (CSort n))).(let H6 \def -(eq_ind C (CHead c1 (Bind Abst) v) (\lambda (ee: C).(match ee in C return -(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) -\Rightarrow True])) I (CSort n) H5) in (False_ind (eq C (CHead c2 (Bind Abbr) -w) (CHead c1 (Bind Abst) v)) H6)))))))))))) y x H0))) H)))). -(* COMMENTS -Initial nodes: 533 -END *) - -theorem csubc_gen_head_l: - \forall (g: G).(\forall (c1: C).(\forall (x: C).(\forall (v: T).(\forall (k: -K).((csubc g (CHead c1 k v) x) \to (or3 (ex2 C (\lambda (c2: C).(eq C x -(CHead c2 k v))) (\lambda (c2: C).(csubc g c1 c2))) (ex5_3 C T A (\lambda (_: -C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c2: -C).(\lambda (w: T).(\lambda (_: A).(eq C x (CHead c2 (Bind Abbr) w))))) -(\lambda (c2: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c2)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda -(c2: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T -(\lambda (b: B).(\lambda (c2: C).(\lambda (v2: T).(eq C x (CHead c2 (Bind b) -v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind -Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b -Void))))) (\lambda (_: B).(\lambda (c2: C).(\lambda (_: T).(csubc g c1 -c2))))))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (x: C).(\lambda (v: T).(\lambda (k: -K).(\lambda (H: (csubc g (CHead c1 k v) x)).(insert_eq C (CHead c1 k v) -(\lambda (c: C).(csubc g c x)) (\lambda (_: C).(or3 (ex2 C (\lambda (c2: -C).(eq C x (CHead c2 k v))) (\lambda (c2: C).(csubc g c1 c2))) (ex5_3 C T A -(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) -(\lambda (c2: C).(\lambda (w: T).(\lambda (_: A).(eq C x (CHead c2 (Bind -Abbr) w))))) (\lambda (c2: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 -c2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 -v)))) (\lambda (c2: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c2 w))))) -(ex4_3 B C T (\lambda (b: B).(\lambda (c2: C).(\lambda (v2: T).(eq C x (CHead -c2 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k -(Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b -Void))))) (\lambda (_: B).(\lambda (c2: C).(\lambda (_: T).(csubc g c1 -c2))))))) (\lambda (y: C).(\lambda (H0: (csubc g y x)).(csubc_ind g (\lambda -(c: C).(\lambda (c0: C).((eq C c (CHead c1 k v)) \to (or3 (ex2 C (\lambda -(c2: C).(eq C c0 (CHead c2 k v))) (\lambda (c2: C).(csubc g c1 c2))) (ex5_3 C -T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) -(\lambda (c2: C).(\lambda (w: T).(\lambda (_: A).(eq C c0 (CHead c2 (Bind -Abbr) w))))) (\lambda (c2: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 -c2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 -v)))) (\lambda (c2: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c2 w))))) -(ex4_3 B C T (\lambda (b: B).(\lambda (c2: C).(\lambda (v2: T).(eq C c0 -(CHead c2 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: -T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c2: C).(\lambda (_: -T).(csubc g c1 c2))))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) -(CHead c1 k v))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee -in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ -_ _) \Rightarrow False])) I (CHead c1 k v) H1) in (False_ind (or3 (ex2 C -(\lambda (c2: C).(eq C (CSort n) (CHead c2 k v))) (\lambda (c2: C).(csubc g -c1 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k -(Bind Abst))))) (\lambda (c2: C).(\lambda (w: T).(\lambda (_: A).(eq C (CSort -n) (CHead c2 (Bind Abbr) w))))) (\lambda (c2: C).(\lambda (_: T).(\lambda (_: -A).(csubc g c1 c2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g -(asucc g a) c1 v)))) (\lambda (c2: C).(\lambda (w: T).(\lambda (a: A).(sc3 g -a c2 w))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c2: C).(\lambda (v2: -T).(eq C (CSort n) (CHead c2 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c2: -C).(\lambda (_: T).(csubc g c1 c2)))))) H2)))) (\lambda (c0: C).(\lambda (c2: -C).(\lambda (H1: (csubc g c0 c2)).(\lambda (H2: (((eq C c0 (CHead c1 k v)) -\to (or3 (ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 k v))) (\lambda (c3: -C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda -(_: A).(eq K k (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: -A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: -T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (c3: C).(\lambda (w: -T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda (b: B).(\lambda -(c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: -B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 c3))))))))).(\lambda (k0: -K).(\lambda (v0: T).(\lambda (H3: (eq C (CHead c0 k0 v0) (CHead c1 k -v))).(let H4 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda -(_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) -(CHead c0 k0 v0) (CHead c1 k v) H3) in ((let H5 \def (f_equal C K (\lambda -(e: C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k0 -| (CHead _ k1 _) \Rightarrow k1])) (CHead c0 k0 v0) (CHead c1 k v) H3) in -((let H6 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: -C).T) with [(CSort _) \Rightarrow v0 | (CHead _ _ t) \Rightarrow t])) (CHead -c0 k0 v0) (CHead c1 k v) H3) in (\lambda (H7: (eq K k0 k)).(\lambda (H8: (eq -C c0 c1)).(eq_ind_r T v (\lambda (t: T).(or3 (ex2 C (\lambda (c3: C).(eq C -(CHead c2 k0 t) (CHead c3 k v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C -T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) -(\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C (CHead c2 k0 t) (CHead -c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc -g c1 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g -a) c1 v)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 -w))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C -(CHead c2 k0 t) (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: -C).(\lambda (_: T).(csubc g c1 c3))))))) (eq_ind_r K k (\lambda (k1: K).(or3 -(ex2 C (\lambda (c3: C).(eq C (CHead c2 k1 v) (CHead c3 k v))) (\lambda (c3: -C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda -(_: A).(eq K k (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: -A).(eq C (CHead c2 k1 v) (CHead c3 (Bind Abbr) w))))) (\lambda (c3: -C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (c3: -C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda -(b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C (CHead c2 k1 v) (CHead c3 -(Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k -(Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b -Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 -c3))))))) (let H9 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 k v)) -\to (or3 (ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 k v))) (\lambda (c3: -C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda -(_: A).(eq K k (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: -A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: -T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (c3: C).(\lambda (w: -T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda (b: B).(\lambda -(c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: -B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 c3)))))))) H2 c1 H8) in (let -H10 \def (eq_ind C c0 (\lambda (c: C).(csubc g c c2)) H1 c1 H8) in -(or3_intro0 (ex2 C (\lambda (c3: C).(eq C (CHead c2 k v) (CHead c3 k v))) -(\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: -T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3: C).(\lambda (w: -T).(\lambda (_: A).(eq C (CHead c2 k v) (CHead c3 (Bind Abbr) w))))) (\lambda -(c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (c3: -C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda -(b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C (CHead c2 k v) (CHead c3 (Bind -b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind -Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b -Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 -c3))))) (ex_intro2 C (\lambda (c3: C).(eq C (CHead c2 k v) (CHead c3 k v))) -(\lambda (c3: C).(csubc g c1 c3)) c2 (refl_equal C (CHead c2 k v)) H10)))) k0 -H7) v0 H6)))) H5)) H4))))))))) (\lambda (c0: C).(\lambda (c2: C).(\lambda -(H1: (csubc g c0 c2)).(\lambda (H2: (((eq C c0 (CHead c1 k v)) \to (or3 (ex2 -C (\lambda (c3: C).(eq C c2 (CHead c3 k v))) (\lambda (c3: C).(csubc g c1 -c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k -(Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C c2 -(CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: -A).(csubc g c1 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g -(asucc g a) c1 v)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g -a c3 w))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: -T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: -C).(\lambda (_: T).(csubc g c1 c3))))))))).(\lambda (b: B).(\lambda (H3: (not -(eq B b Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead -c0 (Bind Void) u1) (CHead c1 k v))).(let H5 \def (f_equal C C (\lambda (e: -C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | -(CHead c _ _) \Rightarrow c])) (CHead c0 (Bind Void) u1) (CHead c1 k v) H4) -in ((let H6 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda -(_: C).K) with [(CSort _) \Rightarrow (Bind Void) | (CHead _ k0 _) -\Rightarrow k0])) (CHead c0 (Bind Void) u1) (CHead c1 k v) H4) in ((let H7 -\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead c0 -(Bind Void) u1) (CHead c1 k v) H4) in (\lambda (H8: (eq K (Bind Void) -k)).(\lambda (H9: (eq C c0 c1)).(let H10 \def (eq_ind C c0 (\lambda (c: -C).((eq C c (CHead c1 k v)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c2 (CHead -c3 k v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: -C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3: -C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) -(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda -(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T -(\lambda (b0: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind -b0) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind -Void))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b0 -Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 -c3)))))))) H2 c1 H9) in (let H11 \def (eq_ind C c0 (\lambda (c: C).(csubc g c -c2)) H1 c1 H9) in (let H12 \def (eq_ind_r K k (\lambda (k0: K).((eq C c1 -(CHead c1 k0 v)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 k0 v))) -(\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: -T).(\lambda (_: A).(eq K k0 (Bind Abst))))) (\lambda (c3: C).(\lambda (w: -T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: -C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (c3: -C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda -(b0: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b0) -v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k0 (Bind -Void))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b0 -Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 -c3)))))))) H10 (Bind Void) H8) in (eq_ind K (Bind Void) (\lambda (k0: K).(or3 -(ex2 C (\lambda (c3: C).(eq C (CHead c2 (Bind b) u2) (CHead c3 k0 v))) -(\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: -T).(\lambda (_: A).(eq K k0 (Bind Abst))))) (\lambda (c3: C).(\lambda (w: -T).(\lambda (_: A).(eq C (CHead c2 (Bind b) u2) (CHead c3 (Bind Abbr) w))))) -(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda -(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T -(\lambda (b0: B).(\lambda (c3: C).(\lambda (v2: T).(eq C (CHead c2 (Bind b) -u2) (CHead c3 (Bind b0) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -T).(eq K k0 (Bind Void))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: -T).(not (eq B b0 Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: -T).(csubc g c1 c3))))))) (or3_intro2 (ex2 C (\lambda (c3: C).(eq C (CHead c2 -(Bind b) u2) (CHead c3 (Bind Void) v))) (\lambda (c3: C).(csubc g c1 c3))) -(ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K (Bind -Void) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C -(CHead c2 (Bind b) u2) (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda -(_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (c3: C).(\lambda (w: -T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda (b0: B).(\lambda -(c3: C).(\lambda (v2: T).(eq C (CHead c2 (Bind b) u2) (CHead c3 (Bind b0) -v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K (Bind Void) -(Bind Void))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B -b0 Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 -c3))))) (ex4_3_intro B C T (\lambda (b0: B).(\lambda (c3: C).(\lambda (v2: -T).(eq C (CHead c2 (Bind b) u2) (CHead c3 (Bind b0) v2))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: T).(eq K (Bind Void) (Bind Void))))) (\lambda -(b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b0 Void))))) (\lambda (_: -B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 c3)))) b c2 u2 (refl_equal C -(CHead c2 (Bind b) u2)) (refl_equal K (Bind Void)) H3 H11)) k H8))))))) H6)) -H5))))))))))) (\lambda (c0: C).(\lambda (c2: C).(\lambda (H1: (csubc g c0 -c2)).(\lambda (H2: (((eq C c0 (CHead c1 k v)) \to (or3 (ex2 C (\lambda (c3: -C).(eq C c2 (CHead c3 k v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A -(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) -(\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind -Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 -c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 -v)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) -(ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 -(CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: -T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: -T).(csubc g c1 c3))))))))).(\lambda (v0: T).(\lambda (a: A).(\lambda (H3: -(sc3 g (asucc g a) c0 v0)).(\lambda (w: T).(\lambda (H4: (sc3 g a c2 -w)).(\lambda (H5: (eq C (CHead c0 (Bind Abst) v0) (CHead c1 k v))).(let H6 -\def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) -with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 -(Bind Abst) v0) (CHead c1 k v) H5) in ((let H7 \def (f_equal C K (\lambda (e: -C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow (Bind -Abst) | (CHead _ k0 _) \Rightarrow k0])) (CHead c0 (Bind Abst) v0) (CHead c1 -k v) H5) in ((let H8 \def (f_equal C T (\lambda (e: C).(match e in C return -(\lambda (_: C).T) with [(CSort _) \Rightarrow v0 | (CHead _ _ t) \Rightarrow -t])) (CHead c0 (Bind Abst) v0) (CHead c1 k v) H5) in (\lambda (H9: (eq K -(Bind Abst) k)).(\lambda (H10: (eq C c0 c1)).(let H11 \def (eq_ind T v0 -(\lambda (t: T).(sc3 g (asucc g a) c0 t)) H3 v H8) in (let H12 \def (eq_ind C -c0 (\lambda (c: C).(sc3 g (asucc g a) c v)) H11 c1 H10) in (let H13 \def -(eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 k v)) \to (or3 (ex2 C -(\lambda (c3: C).(eq C c2 (CHead c3 k v))) (\lambda (c3: C).(csubc g c1 c3))) -(ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind -Abst))))) (\lambda (c3: C).(\lambda (w0: T).(\lambda (_: A).(eq C c2 (CHead -c3 (Bind Abbr) w0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: -A).(csubc g c1 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g -(asucc g a0) c1 v)))) (\lambda (c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 -g a0 c3 w0))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: -T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: -C).(\lambda (_: T).(csubc g c1 c3)))))))) H2 c1 H10) in (let H14 \def (eq_ind -C c0 (\lambda (c: C).(csubc g c c2)) H1 c1 H10) in (let H15 \def (eq_ind_r K -k (\lambda (k0: K).((eq C c1 (CHead c1 k0 v)) \to (or3 (ex2 C (\lambda (c3: -C).(eq C c2 (CHead c3 k0 v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A -(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k0 (Bind Abst))))) -(\lambda (c3: C).(\lambda (w0: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind -Abbr) w0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 -c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g (asucc g a0) -c1 v)))) (\lambda (c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 g a0 c3 -w0))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C -c2 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -T).(eq K k0 (Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: -T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: -T).(csubc g c1 c3)))))))) H13 (Bind Abst) H9) in (eq_ind K (Bind Abst) -(\lambda (k0: K).(or3 (ex2 C (\lambda (c3: C).(eq C (CHead c2 (Bind Abbr) w) -(CHead c3 k0 v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda -(_: C).(\lambda (_: T).(\lambda (_: A).(eq K k0 (Bind Abst))))) (\lambda (c3: -C).(\lambda (w0: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) w) (CHead c3 -(Bind Abbr) w0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g -c1 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g (asucc g -a0) c1 v)))) (\lambda (c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 g a0 c3 -w0))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C -(CHead c2 (Bind Abbr) w) (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda -(_: C).(\lambda (_: T).(eq K k0 (Bind Void))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: -C).(\lambda (_: T).(csubc g c1 c3))))))) (or3_intro1 (ex2 C (\lambda (c3: -C).(eq C (CHead c2 (Bind Abbr) w) (CHead c3 (Bind Abst) v))) (\lambda (c3: -C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda -(_: A).(eq K (Bind Abst) (Bind Abst))))) (\lambda (c3: C).(\lambda (w0: -T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) w) (CHead c3 (Bind Abbr) -w0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g (asucc g a0) c1 v)))) -(\lambda (c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 g a0 c3 w0))))) -(ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C (CHead -c2 (Bind Abbr) w) (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: T).(eq K (Bind Abst) (Bind Void))))) (\lambda (b: B).(\lambda -(_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: -C).(\lambda (_: T).(csubc g c1 c3))))) (ex5_3_intro C T A (\lambda (_: -C).(\lambda (_: T).(\lambda (_: A).(eq K (Bind Abst) (Bind Abst))))) (\lambda -(c3: C).(\lambda (w0: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) w) -(CHead c3 (Bind Abbr) w0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: -A).(csubc g c1 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g -(asucc g a0) c1 v)))) (\lambda (c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 -g a0 c3 w0)))) c2 w a (refl_equal K (Bind Abst)) (refl_equal C (CHead c2 -(Bind Abbr) w)) H14 H12 H4)) k H9))))))))) H7)) H6)))))))))))) y x H0))) -H)))))). -(* COMMENTS -Initial nodes: 5205 -END *) - -theorem csubc_gen_sort_r: - \forall (g: G).(\forall (x: C).(\forall (n: nat).((csubc g x (CSort n)) \to -(eq C x (CSort n))))) -\def - \lambda (g: G).(\lambda (x: C).(\lambda (n: nat).(\lambda (H: (csubc g x -(CSort n))).(insert_eq C (CSort n) (\lambda (c: C).(csubc g x c)) (\lambda -(c: C).(eq C x c)) (\lambda (y: C).(\lambda (H0: (csubc g x y)).(csubc_ind g -(\lambda (c: C).(\lambda (c0: C).((eq C c0 (CSort n)) \to (eq C c c0)))) -(\lambda (n0: nat).(\lambda (H1: (eq C (CSort n0) (CSort n))).(let H2 \def -(f_equal C nat (\lambda (e: C).(match e in C return (\lambda (_: C).nat) with -[(CSort n1) \Rightarrow n1 | (CHead _ _ _) \Rightarrow n0])) (CSort n0) -(CSort n) H1) in (eq_ind_r nat n (\lambda (n1: nat).(eq C (CSort n1) (CSort -n1))) (refl_equal C (CSort n)) n0 H2)))) (\lambda (c1: C).(\lambda (c2: -C).(\lambda (_: (csubc g c1 c2)).(\lambda (_: (((eq C c2 (CSort n)) \to (eq C -c1 c2)))).(\lambda (k: K).(\lambda (v: T).(\lambda (H3: (eq C (CHead c2 k v) -(CSort n))).(let H4 \def (eq_ind C (CHead c2 k v) (\lambda (ee: C).(match ee -in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead -_ _ _) \Rightarrow True])) I (CSort n) H3) in (False_ind (eq C (CHead c1 k v) -(CHead c2 k v)) H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: -(csubc g c1 c2)).(\lambda (_: (((eq C c2 (CSort n)) \to (eq C c1 -c2)))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: -T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c2 (Bind b) u2) (CSort -n))).(let H5 \def (eq_ind C (CHead c2 (Bind b) u2) (\lambda (ee: C).(match ee -in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead -_ _ _) \Rightarrow True])) I (CSort n) H4) in (False_ind (eq C (CHead c1 -(Bind Void) u1) (CHead c2 (Bind b) u2)) H5))))))))))) (\lambda (c1: -C).(\lambda (c2: C).(\lambda (_: (csubc g c1 c2)).(\lambda (_: (((eq C c2 -(CSort n)) \to (eq C c1 c2)))).(\lambda (v: T).(\lambda (a: A).(\lambda (_: -(sc3 g (asucc g a) c1 v)).(\lambda (w: T).(\lambda (_: (sc3 g a c2 -w)).(\lambda (H5: (eq C (CHead c2 (Bind Abbr) w) (CSort n))).(let H6 \def -(eq_ind C (CHead c2 (Bind Abbr) w) (\lambda (ee: C).(match ee in C return -(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) -\Rightarrow True])) I (CSort n) H5) in (False_ind (eq C (CHead c1 (Bind Abst) -v) (CHead c2 (Bind Abbr) w)) H6)))))))))))) x y H0))) H)))). -(* COMMENTS -Initial nodes: 533 -END *) - -theorem csubc_gen_head_r: - \forall (g: G).(\forall (c2: C).(\forall (x: C).(\forall (w: T).(\forall (k: -K).((csubc g x (CHead c2 k w)) \to (or3 (ex2 C (\lambda (c1: C).(eq C x -(CHead c1 k w))) (\lambda (c1: C).(csubc g c1 c2))) (ex5_3 C T A (\lambda (_: -C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c1: -C).(\lambda (v: T).(\lambda (_: A).(eq C x (CHead c1 (Bind Abst) v))))) -(\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c2)))) (\lambda -(c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T -(\lambda (_: B).(\lambda (c1: C).(\lambda (v1: T).(eq C x (CHead c1 (Bind -Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind -b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) -(\lambda (_: B).(\lambda (c1: C).(\lambda (_: T).(csubc g c1 c2))))))))))) -\def - \lambda (g: G).(\lambda (c2: C).(\lambda (x: C).(\lambda (w: T).(\lambda (k: -K).(\lambda (H: (csubc g x (CHead c2 k w))).(insert_eq C (CHead c2 k w) -(\lambda (c: C).(csubc g x c)) (\lambda (_: C).(or3 (ex2 C (\lambda (c1: -C).(eq C x (CHead c1 k w))) (\lambda (c1: C).(csubc g c1 c2))) (ex5_3 C T A -(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) -(\lambda (c1: C).(\lambda (v: T).(\lambda (_: A).(eq C x (CHead c1 (Bind -Abst) v))))) (\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 -c2)))) (\lambda (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1 -v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) -(ex4_3 B C T (\lambda (_: B).(\lambda (c1: C).(\lambda (v1: T).(eq C x (CHead -c1 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(eq K -k (Bind b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b -Void))))) (\lambda (_: B).(\lambda (c1: C).(\lambda (_: T).(csubc g c1 -c2))))))) (\lambda (y: C).(\lambda (H0: (csubc g x y)).(csubc_ind g (\lambda -(c: C).(\lambda (c0: C).((eq C c0 (CHead c2 k w)) \to (or3 (ex2 C (\lambda -(c1: C).(eq C c (CHead c1 k w))) (\lambda (c1: C).(csubc g c1 c2))) (ex5_3 C -T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) -(\lambda (c1: C).(\lambda (v: T).(\lambda (_: A).(eq C c (CHead c1 (Bind -Abst) v))))) (\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 -c2)))) (\lambda (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1 -v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) -(ex4_3 B C T (\lambda (_: B).(\lambda (c1: C).(\lambda (v1: T).(eq C c (CHead -c1 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(eq K -k (Bind b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b -Void))))) (\lambda (_: B).(\lambda (c1: C).(\lambda (_: T).(csubc g c1 -c2))))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead c2 k -w))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return -(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) -\Rightarrow False])) I (CHead c2 k w) H1) in (False_ind (or3 (ex2 C (\lambda -(c1: C).(eq C (CSort n) (CHead c1 k w))) (\lambda (c1: C).(csubc g c1 c2))) -(ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind -Abbr))))) (\lambda (c1: C).(\lambda (v: T).(\lambda (_: A).(eq C (CSort n) -(CHead c1 (Bind Abst) v))))) (\lambda (c1: C).(\lambda (_: T).(\lambda (_: -A).(csubc g c1 c2)))) (\lambda (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g -(asucc g a) c1 v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a -c2 w))))) (ex4_3 B C T (\lambda (_: B).(\lambda (c1: C).(\lambda (v1: T).(eq -C (CSort n) (CHead c1 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c1: -C).(\lambda (_: T).(csubc g c1 c2)))))) H2)))) (\lambda (c1: C).(\lambda (c0: -C).(\lambda (H1: (csubc g c1 c0)).(\lambda (H2: (((eq C c0 (CHead c2 k w)) -\to (or3 (ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k w))) (\lambda (c3: -C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda -(_: A).(eq K k (Bind Abbr))))) (\lambda (c3: C).(\lambda (v: T).(\lambda (_: -A).(eq C c1 (CHead c3 (Bind Abst) v))))) (\lambda (c3: C).(\lambda (_: -T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v: -T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T (\lambda (_: B).(\lambda -(c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind Void) v1))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: -B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 c2))))))))).(\lambda (k0: -K).(\lambda (v: T).(\lambda (H3: (eq C (CHead c0 k0 v) (CHead c2 k w))).(let -H4 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) -with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k0 -v) (CHead c2 k w) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e -in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k1 -_) \Rightarrow k1])) (CHead c0 k0 v) (CHead c2 k w) H3) in ((let H6 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow v | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 v) -(CHead c2 k w) H3) in (\lambda (H7: (eq K k0 k)).(\lambda (H8: (eq C c0 -c2)).(eq_ind_r T w (\lambda (t: T).(or3 (ex2 C (\lambda (c3: C).(eq C (CHead -c1 k0 t) (CHead c3 k w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A -(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) -(\lambda (c3: C).(\lambda (v0: T).(\lambda (_: A).(eq C (CHead c1 k0 t) -(CHead c3 (Bind Abst) v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: -A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (a: A).(sc3 -g (asucc g a) c3 v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 -g a c2 w))))) (ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: -T).(eq C (CHead c1 k0 t) (CHead c3 (Bind Void) v1))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: -B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 c2))))))) (eq_ind_r K k -(\lambda (k1: K).(or3 (ex2 C (\lambda (c3: C).(eq C (CHead c1 k1 w) (CHead c3 -k w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: -C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c3: -C).(\lambda (v0: T).(\lambda (_: A).(eq C (CHead c1 k1 w) (CHead c3 (Bind -Abst) v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 -c2)))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (a: A).(sc3 g (asucc g a) -c3 v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) -(ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead -c1 k1 w) (CHead c3 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: -C).(\lambda (_: T).(csubc g c3 c2))))))) (let H9 \def (eq_ind C c0 (\lambda -(c: C).((eq C c (CHead c2 k w)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c1 -(CHead c3 k w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: -C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c3: -C).(\lambda (v0: T).(\lambda (_: A).(eq C c1 (CHead c3 (Bind Abst) v0))))) -(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda -(c3: C).(\lambda (v0: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v0)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C -T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind -Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind -b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) -(\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 c2)))))))) H2 c2 -H8) in (let H10 \def (eq_ind C c0 (\lambda (c: C).(csubc g c1 c)) H1 c2 H8) -in (or3_intro0 (ex2 C (\lambda (c3: C).(eq C (CHead c1 k w) (CHead c3 k w))) -(\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: -T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c3: C).(\lambda (v0: -T).(\lambda (_: A).(eq C (CHead c1 k w) (CHead c3 (Bind Abst) v0))))) -(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda -(c3: C).(\lambda (v0: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v0)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C -T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead c1 k w) -(CHead c3 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: -T).(eq K k (Bind b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not -(eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g -c3 c2))))) (ex_intro2 C (\lambda (c3: C).(eq C (CHead c1 k w) (CHead c3 k -w))) (\lambda (c3: C).(csubc g c3 c2)) c1 (refl_equal C (CHead c1 k w)) -H10)))) k0 H7) v H6)))) H5)) H4))))))))) (\lambda (c1: C).(\lambda (c0: -C).(\lambda (H1: (csubc g c1 c0)).(\lambda (H2: (((eq C c0 (CHead c2 k w)) -\to (or3 (ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k w))) (\lambda (c3: -C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda -(_: A).(eq K k (Bind Abbr))))) (\lambda (c3: C).(\lambda (v: T).(\lambda (_: -A).(eq C c1 (CHead c3 (Bind Abst) v))))) (\lambda (c3: C).(\lambda (_: -T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v: -T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T (\lambda (_: B).(\lambda -(c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind Void) v1))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: -B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 c2))))))))).(\lambda (b: -B).(\lambda (H3: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: -T).(\lambda (H4: (eq C (CHead c0 (Bind b) u2) (CHead c2 k w))).(let H5 \def -(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with -[(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 (Bind b) -u2) (CHead c2 k w) H4) in ((let H6 \def (f_equal C K (\lambda (e: C).(match e -in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow (Bind b) | (CHead -_ k0 _) \Rightarrow k0])) (CHead c0 (Bind b) u2) (CHead c2 k w) H4) in ((let -H7 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c0 -(Bind b) u2) (CHead c2 k w) H4) in (\lambda (H8: (eq K (Bind b) k)).(\lambda -(H9: (eq C c0 c2)).(let H10 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead -c2 k w)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k w))) (\lambda -(c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: -T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c3: C).(\lambda (v: -T).(\lambda (_: A).(eq C c1 (CHead c3 (Bind Abst) v))))) (\lambda (c3: -C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: -C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T (\lambda -(_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind Void) -v1))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind -b0))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b0 -Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 -c2)))))))) H2 c2 H9) in (let H11 \def (eq_ind C c0 (\lambda (c: C).(csubc g -c1 c)) H1 c2 H9) in (let H12 \def (eq_ind_r K k (\lambda (k0: K).((eq C c2 -(CHead c2 k0 w)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k0 w))) -(\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: -T).(\lambda (_: A).(eq K k0 (Bind Abbr))))) (\lambda (c3: C).(\lambda (v: -T).(\lambda (_: A).(eq C c1 (CHead c3 (Bind Abst) v))))) (\lambda (c3: -C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: -C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T (\lambda -(_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind Void) -v1))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(eq K k0 (Bind -b0))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b0 -Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 -c2)))))))) H10 (Bind b) H8) in (eq_ind K (Bind b) (\lambda (k0: K).(or3 (ex2 -C (\lambda (c3: C).(eq C (CHead c1 (Bind Void) u1) (CHead c3 k0 w))) (\lambda -(c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: -T).(\lambda (_: A).(eq K k0 (Bind Abbr))))) (\lambda (c3: C).(\lambda (v: -T).(\lambda (_: A).(eq C (CHead c1 (Bind Void) u1) (CHead c3 (Bind Abst) -v))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) -(\lambda (c3: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C -T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead c1 (Bind -Void) u1) (CHead c3 (Bind Void) v1))))) (\lambda (b0: B).(\lambda (_: -C).(\lambda (_: T).(eq K k0 (Bind b0))))) (\lambda (b0: B).(\lambda (_: -C).(\lambda (_: T).(not (eq B b0 Void))))) (\lambda (_: B).(\lambda (c3: -C).(\lambda (_: T).(csubc g c3 c2))))))) (or3_intro2 (ex2 C (\lambda (c3: -C).(eq C (CHead c1 (Bind Void) u1) (CHead c3 (Bind b) w))) (\lambda (c3: -C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda -(_: A).(eq K (Bind b) (Bind Abbr))))) (\lambda (c3: C).(\lambda (v: -T).(\lambda (_: A).(eq C (CHead c1 (Bind Void) u1) (CHead c3 (Bind Abst) -v))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) -(\lambda (c3: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C -T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead c1 (Bind -Void) u1) (CHead c3 (Bind Void) v1))))) (\lambda (b0: B).(\lambda (_: -C).(\lambda (_: T).(eq K (Bind b) (Bind b0))))) (\lambda (b0: B).(\lambda (_: -C).(\lambda (_: T).(not (eq B b0 Void))))) (\lambda (_: B).(\lambda (c3: -C).(\lambda (_: T).(csubc g c3 c2))))) (ex4_3_intro B C T (\lambda (_: -B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead c1 (Bind Void) u1) (CHead -c3 (Bind Void) v1))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(eq K -(Bind b) (Bind b0))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not -(eq B b0 Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g -c3 c2)))) b c1 u1 (refl_equal C (CHead c1 (Bind Void) u1)) (refl_equal K -(Bind b)) H3 H11)) k H8))))))) H6)) H5))))))))))) (\lambda (c1: C).(\lambda -(c0: C).(\lambda (H1: (csubc g c1 c0)).(\lambda (H2: (((eq C c0 (CHead c2 k -w)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k w))) (\lambda (c3: -C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda -(_: A).(eq K k (Bind Abbr))))) (\lambda (c3: C).(\lambda (v: T).(\lambda (_: -A).(eq C c1 (CHead c3 (Bind Abst) v))))) (\lambda (c3: C).(\lambda (_: -T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v: -T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T (\lambda (_: B).(\lambda -(c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind Void) v1))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: -B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 c2))))))))).(\lambda (v: -T).(\lambda (a: A).(\lambda (H3: (sc3 g (asucc g a) c1 v)).(\lambda (w0: -T).(\lambda (H4: (sc3 g a c0 w0)).(\lambda (H5: (eq C (CHead c0 (Bind Abbr) -w0) (CHead c2 k w))).(let H6 \def (f_equal C C (\lambda (e: C).(match e in C -return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) -\Rightarrow c])) (CHead c0 (Bind Abbr) w0) (CHead c2 k w) H5) in ((let H7 -\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) -with [(CSort _) \Rightarrow (Bind Abbr) | (CHead _ k0 _) \Rightarrow k0])) -(CHead c0 (Bind Abbr) w0) (CHead c2 k w) H5) in ((let H8 \def (f_equal C T -(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) -\Rightarrow w0 | (CHead _ _ t) \Rightarrow t])) (CHead c0 (Bind Abbr) w0) -(CHead c2 k w) H5) in (\lambda (H9: (eq K (Bind Abbr) k)).(\lambda (H10: (eq -C c0 c2)).(let H11 \def (eq_ind T w0 (\lambda (t: T).(sc3 g a c0 t)) H4 w H8) -in (let H12 \def (eq_ind C c0 (\lambda (c: C).(sc3 g a c w)) H11 c2 H10) in -(let H13 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c2 k w)) \to (or3 -(ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k w))) (\lambda (c3: C).(csubc g -c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k -(Bind Abbr))))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (_: A).(eq C c1 -(CHead c3 (Bind Abst) v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: -A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (a0: A).(sc3 -g (asucc g a0) c3 v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: -A).(sc3 g a0 c2 w))))) (ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda -(v1: T).(eq C c1 (CHead c3 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: -C).(\lambda (_: T).(csubc g c3 c2)))))))) H2 c2 H10) in (let H14 \def (eq_ind -C c0 (\lambda (c: C).(csubc g c1 c)) H1 c2 H10) in (let H15 \def (eq_ind_r K -k (\lambda (k0: K).((eq C c2 (CHead c2 k0 w)) \to (or3 (ex2 C (\lambda (c3: -C).(eq C c1 (CHead c3 k0 w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A -(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k0 (Bind Abbr))))) -(\lambda (c3: C).(\lambda (v0: T).(\lambda (_: A).(eq C c1 (CHead c3 (Bind -Abst) v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 -c2)))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (a0: A).(sc3 g (asucc g a0) -c3 v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g a0 c2 -w))))) (ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C -c1 (CHead c3 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: -T).(eq K k0 (Bind b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not -(eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g -c3 c2)))))))) H13 (Bind Abbr) H9) in (eq_ind K (Bind Abbr) (\lambda (k0: -K).(or3 (ex2 C (\lambda (c3: C).(eq C (CHead c1 (Bind Abst) v) (CHead c3 k0 -w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda -(_: T).(\lambda (_: A).(eq K k0 (Bind Abbr))))) (\lambda (c3: C).(\lambda -(v0: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) v) (CHead c3 (Bind Abst) -v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) -(\lambda (c3: C).(\lambda (v0: T).(\lambda (a0: A).(sc3 g (asucc g a0) c3 -v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g a0 c2 w))))) -(ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead -c1 (Bind Abst) v) (CHead c3 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (_: T).(eq K k0 (Bind b))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: -C).(\lambda (_: T).(csubc g c3 c2))))))) (or3_intro1 (ex2 C (\lambda (c3: -C).(eq C (CHead c1 (Bind Abst) v) (CHead c3 (Bind Abbr) w))) (\lambda (c3: -C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda -(_: A).(eq K (Bind Abbr) (Bind Abbr))))) (\lambda (c3: C).(\lambda (v0: -T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) v) (CHead c3 (Bind Abst) -v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) -(\lambda (c3: C).(\lambda (v0: T).(\lambda (a0: A).(sc3 g (asucc g a0) c3 -v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g a0 c2 w))))) -(ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead -c1 (Bind Abst) v) (CHead c3 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (_: T).(eq K (Bind Abbr) (Bind b))))) (\lambda (b: B).(\lambda -(_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: -C).(\lambda (_: T).(csubc g c3 c2))))) (ex5_3_intro C T A (\lambda (_: -C).(\lambda (_: T).(\lambda (_: A).(eq K (Bind Abbr) (Bind Abbr))))) (\lambda -(c3: C).(\lambda (v0: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) v) -(CHead c3 (Bind Abst) v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: -A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (a0: A).(sc3 -g (asucc g a0) c3 v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: -A).(sc3 g a0 c2 w)))) c1 v a (refl_equal K (Bind Abbr)) (refl_equal C (CHead -c1 (Bind Abst) v)) H14 H3 H12)) k H9))))))))) H7)) H6)))))))))))) x y H0))) -H)))))). -(* COMMENTS -Initial nodes: 5197 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/getl.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/getl.ma deleted file mode 100644 index 244c84927..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/getl.ma +++ /dev/null @@ -1,45 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubc/drop.ma". - -include "Basic-1/csubc/clear.ma". - -theorem csubc_getl_conf: - \forall (g: G).(\forall (c1: C).(\forall (e1: C).(\forall (i: nat).((getl i -c1 e1) \to (\forall (c2: C).((csubc g c1 c2) \to (ex2 C (\lambda (e2: -C).(getl i c2 e2)) (\lambda (e2: C).(csubc g e1 e2))))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (e1: C).(\lambda (i: nat).(\lambda -(H: (getl i c1 e1)).(\lambda (c2: C).(\lambda (H0: (csubc g c1 c2)).(let H1 -\def (getl_gen_all c1 e1 i H) in (ex2_ind C (\lambda (e: C).(drop i O c1 e)) -(\lambda (e: C).(clear e e1)) (ex2 C (\lambda (e2: C).(getl i c2 e2)) -(\lambda (e2: C).(csubc g e1 e2))) (\lambda (x: C).(\lambda (H2: (drop i O c1 -x)).(\lambda (H3: (clear x e1)).(let H_x \def (csubc_drop_conf_O g c1 x i H2 -c2 H0) in (let H4 \def H_x in (ex2_ind C (\lambda (e2: C).(drop i O c2 e2)) -(\lambda (e2: C).(csubc g x e2)) (ex2 C (\lambda (e2: C).(getl i c2 e2)) -(\lambda (e2: C).(csubc g e1 e2))) (\lambda (x0: C).(\lambda (H5: (drop i O -c2 x0)).(\lambda (H6: (csubc g x x0)).(let H_x0 \def (csubc_clear_conf g x e1 -H3 x0 H6) in (let H7 \def H_x0 in (ex2_ind C (\lambda (e2: C).(clear x0 e2)) -(\lambda (e2: C).(csubc g e1 e2)) (ex2 C (\lambda (e2: C).(getl i c2 e2)) -(\lambda (e2: C).(csubc g e1 e2))) (\lambda (x1: C).(\lambda (H8: (clear x0 -x1)).(\lambda (H9: (csubc g e1 x1)).(ex_intro2 C (\lambda (e2: C).(getl i c2 -e2)) (\lambda (e2: C).(csubc g e1 e2)) x1 (getl_intro i c2 x1 x0 H5 H8) -H9)))) H7)))))) H4)))))) H1)))))))). -(* COMMENTS -Initial nodes: 315 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/props.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/props.ma deleted file mode 100644 index d6399181c..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubc/props.ma +++ /dev/null @@ -1,30 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubc/defs.ma". - -include "Basic-1/sc3/props.ma". - -theorem csubc_refl: - \forall (g: G).(\forall (c: C).(csubc g c c)) -\def - \lambda (g: G).(\lambda (c: C).(C_ind (\lambda (c0: C).(csubc g c0 c0)) -(\lambda (n: nat).(csubc_sort g n)) (\lambda (c0: C).(\lambda (H: (csubc g c0 -c0)).(\lambda (k: K).(\lambda (t: T).(csubc_head g c0 c0 H k t))))) c)). -(* COMMENTS -Initial nodes: 53 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubst0/clear.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubst0/clear.ma deleted file mode 100644 index 0700d7d48..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubst0/clear.ma +++ /dev/null @@ -1,1139 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubst0/props.ma". - -include "Basic-1/csubst0/fwd.ma". - -include "Basic-1/clear/fwd.ma". - -theorem csubst0_clear_O: - \forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 O v c1 c2) \to -(\forall (c: C).((clear c1 c) \to (clear c2 c)))))) -\def - \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: -T).((csubst0 O v c c2) \to (\forall (c0: C).((clear c c0) \to (clear c2 -c0))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (H: -(csubst0 O v (CSort n) c2)).(\lambda (c: C).(\lambda (_: (clear (CSort n) -c)).(csubst0_gen_sort c2 v O n H (clear c2 c)))))))) (\lambda (c: C).(\lambda -(H: ((\forall (c2: C).(\forall (v: T).((csubst0 O v c c2) \to (\forall (c0: -C).((clear c c0) \to (clear c2 c0)))))))).(\lambda (k: K).(\lambda (t: -T).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 O v (CHead c k t) -c2)).(\lambda (c0: C).(\lambda (H1: (clear (CHead c k t) c0)).(or3_ind (ex3_2 -T nat (\lambda (_: T).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (u2: -T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: -nat).(subst0 j v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq -nat O (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k -t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (ex4_3 T C nat -(\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j))))) -(\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k -u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t -u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c -c3))))) (clear c2 c0) (\lambda (H2: (ex3_2 T nat (\lambda (_: T).(\lambda (j: -nat).(eq nat O (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead -c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t -u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: nat).(eq nat O (s k -j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda -(u2: T).(\lambda (j: nat).(subst0 j v t u2))) (clear c2 c0) (\lambda (x0: -T).(\lambda (x1: nat).(\lambda (H3: (eq nat O (s k x1))).(\lambda (H4: (eq C -c2 (CHead c k x0))).(\lambda (H5: (subst0 x1 v t x0)).(eq_ind_r C (CHead c k -x0) (\lambda (c3: C).(clear c3 c0)) (K_ind (\lambda (k0: K).((clear (CHead c -k0 t) c0) \to ((eq nat O (s k0 x1)) \to (clear (CHead c k0 x0) c0)))) -(\lambda (b: B).(\lambda (_: (clear (CHead c (Bind b) t) c0)).(\lambda (H7: -(eq nat O (s (Bind b) x1))).(let H8 \def (eq_ind nat O (\lambda (ee: -nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True -| (S _) \Rightarrow False])) I (S x1) H7) in (False_ind (clear (CHead c (Bind -b) x0) c0) H8))))) (\lambda (f: F).(\lambda (H6: (clear (CHead c (Flat f) t) -c0)).(\lambda (H7: (eq nat O (s (Flat f) x1))).(let H8 \def (eq_ind_r nat x1 -(\lambda (n: nat).(subst0 n v t x0)) H5 O H7) in (clear_flat c c0 -(clear_gen_flat f c c0 t H6) f x0))))) k H1 H3) c2 H4)))))) H2)) (\lambda -(H2: (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j)))) -(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: -C).(\lambda (j: nat).(csubst0 j v c c3))))).(ex3_2_ind C nat (\lambda (_: -C).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (c3: C).(\lambda (_: -nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j -v c c3))) (clear c2 c0) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H3: (eq -nat O (s k x1))).(\lambda (H4: (eq C c2 (CHead x0 k t))).(\lambda (H5: -(csubst0 x1 v c x0)).(eq_ind_r C (CHead x0 k t) (\lambda (c3: C).(clear c3 -c0)) (K_ind (\lambda (k0: K).((clear (CHead c k0 t) c0) \to ((eq nat O (s k0 -x1)) \to (clear (CHead x0 k0 t) c0)))) (\lambda (b: B).(\lambda (_: (clear -(CHead c (Bind b) t) c0)).(\lambda (H7: (eq nat O (s (Bind b) x1))).(let H8 -\def (eq_ind nat O (\lambda (ee: nat).(match ee in nat return (\lambda (_: -nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x1) H7) -in (False_ind (clear (CHead x0 (Bind b) t) c0) H8))))) (\lambda (f: -F).(\lambda (H6: (clear (CHead c (Flat f) t) c0)).(\lambda (H7: (eq nat O (s -(Flat f) x1))).(let H8 \def (eq_ind_r nat x1 (\lambda (n: nat).(csubst0 n v c -x0)) H5 O H7) in (clear_flat x0 c0 (H x0 v H8 c0 (clear_gen_flat f c c0 t -H6)) f t))))) k H1 H3) c2 H4)))))) H2)) (\lambda (H2: (ex4_3 T C nat (\lambda -(_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j))))) (\lambda (u2: -T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda -(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: -T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))))).(ex4_3_ind T C -nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j))))) -(\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k -u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t -u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c -c3)))) (clear c2 c0) (\lambda (x0: T).(\lambda (x1: C).(\lambda (x2: -nat).(\lambda (H3: (eq nat O (s k x2))).(\lambda (H4: (eq C c2 (CHead x1 k -x0))).(\lambda (H5: (subst0 x2 v t x0)).(\lambda (H6: (csubst0 x2 v c -x1)).(eq_ind_r C (CHead x1 k x0) (\lambda (c3: C).(clear c3 c0)) (K_ind -(\lambda (k0: K).((clear (CHead c k0 t) c0) \to ((eq nat O (s k0 x2)) \to -(clear (CHead x1 k0 x0) c0)))) (\lambda (b: B).(\lambda (_: (clear (CHead c -(Bind b) t) c0)).(\lambda (H8: (eq nat O (s (Bind b) x2))).(let H9 \def -(eq_ind nat O (\lambda (ee: nat).(match ee in nat return (\lambda (_: -nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x2) H8) -in (False_ind (clear (CHead x1 (Bind b) x0) c0) H9))))) (\lambda (f: -F).(\lambda (H7: (clear (CHead c (Flat f) t) c0)).(\lambda (H8: (eq nat O (s -(Flat f) x2))).(let H9 \def (eq_ind_r nat x2 (\lambda (n: nat).(csubst0 n v c -x1)) H6 O H8) in (let H10 \def (eq_ind_r nat x2 (\lambda (n: nat).(subst0 n v -t x0)) H5 O H8) in (clear_flat x1 c0 (H x1 v H9 c0 (clear_gen_flat f c c0 t -H7)) f x0)))))) k H1 H3) c2 H4)))))))) H2)) (csubst0_gen_head k c c2 t v O -H0))))))))))) c1). -(* COMMENTS -Initial nodes: 1582 -END *) - -theorem csubst0_clear_O_back: - \forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 O v c1 c2) \to -(\forall (c: C).((clear c2 c) \to (clear c1 c)))))) -\def - \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: -T).((csubst0 O v c c2) \to (\forall (c0: C).((clear c2 c0) \to (clear c -c0))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (H: -(csubst0 O v (CSort n) c2)).(\lambda (c: C).(\lambda (_: (clear c2 -c)).(csubst0_gen_sort c2 v O n H (clear (CSort n) c)))))))) (\lambda (c: -C).(\lambda (H: ((\forall (c2: C).(\forall (v: T).((csubst0 O v c c2) \to -(\forall (c0: C).((clear c2 c0) \to (clear c c0)))))))).(\lambda (k: -K).(\lambda (t: T).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 O -v (CHead c k t) c2)).(\lambda (c0: C).(\lambda (H1: (clear c2 c0)).(or3_ind -(ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda -(u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: -T).(\lambda (j: nat).(subst0 j v t u2)))) (ex3_2 C nat (\lambda (_: -C).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (c3: C).(\lambda (_: -nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j -v c c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: -nat).(eq nat O (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: -nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda -(j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: -nat).(csubst0 j v c c3))))) (clear (CHead c k t) c0) (\lambda (H2: (ex3_2 T -nat (\lambda (_: T).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (u2: -T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: -nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: -nat).(eq nat O (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead -c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))) (clear -(CHead c k t) c0) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H3: (eq nat O -(s k x1))).(\lambda (H4: (eq C c2 (CHead c k x0))).(\lambda (H5: (subst0 x1 v -t x0)).(let H6 \def (eq_ind C c2 (\lambda (c3: C).(clear c3 c0)) H1 (CHead c -k x0) H4) in (K_ind (\lambda (k0: K).((eq nat O (s k0 x1)) \to ((clear (CHead -c k0 x0) c0) \to (clear (CHead c k0 t) c0)))) (\lambda (b: B).(\lambda (H7: -(eq nat O (s (Bind b) x1))).(\lambda (_: (clear (CHead c (Bind b) x0) -c0)).(let H9 \def (eq_ind nat O (\lambda (ee: nat).(match ee in nat return -(\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) -I (S x1) H7) in (False_ind (clear (CHead c (Bind b) t) c0) H9))))) (\lambda -(f: F).(\lambda (H7: (eq nat O (s (Flat f) x1))).(\lambda (H8: (clear (CHead -c (Flat f) x0) c0)).(let H9 \def (eq_ind_r nat x1 (\lambda (n: nat).(subst0 n -v t x0)) H5 O H7) in (clear_flat c c0 (clear_gen_flat f c c0 x0 H8) f t))))) -k H3 H6))))))) H2)) (\lambda (H2: (ex3_2 C nat (\lambda (_: C).(\lambda (j: -nat).(eq nat O (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead -c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c -c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat O (s k -j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda -(c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (clear (CHead c k t) c0) -(\lambda (x0: C).(\lambda (x1: nat).(\lambda (H3: (eq nat O (s k -x1))).(\lambda (H4: (eq C c2 (CHead x0 k t))).(\lambda (H5: (csubst0 x1 v c -x0)).(let H6 \def (eq_ind C c2 (\lambda (c3: C).(clear c3 c0)) H1 (CHead x0 k -t) H4) in (K_ind (\lambda (k0: K).((eq nat O (s k0 x1)) \to ((clear (CHead x0 -k0 t) c0) \to (clear (CHead c k0 t) c0)))) (\lambda (b: B).(\lambda (H7: (eq -nat O (s (Bind b) x1))).(\lambda (_: (clear (CHead x0 (Bind b) t) c0)).(let -H9 \def (eq_ind nat O (\lambda (ee: nat).(match ee in nat return (\lambda (_: -nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x1) H7) -in (False_ind (clear (CHead c (Bind b) t) c0) H9))))) (\lambda (f: -F).(\lambda (H7: (eq nat O (s (Flat f) x1))).(\lambda (H8: (clear (CHead x0 -(Flat f) t) c0)).(let H9 \def (eq_ind_r nat x1 (\lambda (n: nat).(csubst0 n v -c x0)) H5 O H7) in (clear_flat c c0 (H x0 v H9 c0 (clear_gen_flat f x0 c0 t -H8)) f t))))) k H3 H6))))))) H2)) (\lambda (H2: (ex4_3 T C nat (\lambda (_: -T).(\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j))))) (\lambda (u2: -T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda -(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: -T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))))).(ex4_3_ind T C -nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j))))) -(\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k -u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t -u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c -c3)))) (clear (CHead c k t) c0) (\lambda (x0: T).(\lambda (x1: C).(\lambda -(x2: nat).(\lambda (H3: (eq nat O (s k x2))).(\lambda (H4: (eq C c2 (CHead x1 -k x0))).(\lambda (H5: (subst0 x2 v t x0)).(\lambda (H6: (csubst0 x2 v c -x1)).(let H7 \def (eq_ind C c2 (\lambda (c3: C).(clear c3 c0)) H1 (CHead x1 k -x0) H4) in (K_ind (\lambda (k0: K).((eq nat O (s k0 x2)) \to ((clear (CHead -x1 k0 x0) c0) \to (clear (CHead c k0 t) c0)))) (\lambda (b: B).(\lambda (H8: -(eq nat O (s (Bind b) x2))).(\lambda (_: (clear (CHead x1 (Bind b) x0) -c0)).(let H10 \def (eq_ind nat O (\lambda (ee: nat).(match ee in nat return -(\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) -I (S x2) H8) in (False_ind (clear (CHead c (Bind b) t) c0) H10))))) (\lambda -(f: F).(\lambda (H8: (eq nat O (s (Flat f) x2))).(\lambda (H9: (clear (CHead -x1 (Flat f) x0) c0)).(let H10 \def (eq_ind_r nat x2 (\lambda (n: -nat).(csubst0 n v c x1)) H6 O H8) in (let H11 \def (eq_ind_r nat x2 (\lambda -(n: nat).(subst0 n v t x0)) H5 O H8) in (clear_flat c c0 (H x1 v H10 c0 -(clear_gen_flat f x1 c0 x0 H9)) f t)))))) k H3 H7))))))))) H2)) -(csubst0_gen_head k c c2 t v O H0))))))))))) c1). -(* COMMENTS -Initial nodes: 1606 -END *) - -theorem csubst0_clear_S: - \forall (c1: C).(\forall (c2: C).(\forall (v: T).(\forall (i: nat).((csubst0 -(S i) v c1 c2) \to (\forall (c: C).((clear c1 c) \to (or4 (clear c2 c) (ex3_4 -B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq -C c (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: -T).(\lambda (u2: T).(clear c2 (CHead e (Bind b) u2)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) -(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(clear c2 (CHead e2 (Bind b) u)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b) u1))))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 i v e1 e2)))))))))))))) -\def - \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: -T).(\forall (i: nat).((csubst0 (S i) v c c2) \to (\forall (c0: C).((clear c -c0) \to (or4 (clear c2 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: -C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) -(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 -(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind -b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(clear c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e2 -(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 -e2))))))))))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda -(i: nat).(\lambda (H: (csubst0 (S i) v (CSort n) c2)).(\lambda (c: -C).(\lambda (_: (clear (CSort n) c)).(csubst0_gen_sort c2 v (S i) n H (or4 -(clear c2 c) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: -T).(\lambda (_: T).(eq C c (CHead e (Bind b) u1)))))) (\lambda (b: -B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e (Bind -b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind b) u)))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c2 -(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c -(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))))))))))) -(\lambda (c: C).(\lambda (H: ((\forall (c2: C).(\forall (v: T).(\forall (i: -nat).((csubst0 (S i) v c c2) \to (\forall (c0: C).((clear c c0) \to (or4 -(clear c2 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: -T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: -B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e (Bind -b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c2 -(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 -(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 -e2)))))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda -(v: T).(\lambda (i: nat).(\lambda (H0: (csubst0 (S i) v (CHead c k t) -c2)).(\lambda (c0: C).(\lambda (H1: (clear (CHead c k t) c0)).(or3_ind (ex3_2 -T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (S i) (s k j)))) (\lambda -(u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: -T).(\lambda (j: nat).(subst0 j v t u2)))) (ex3_2 C nat (\lambda (_: -C).(\lambda (j: nat).(eq nat (S i) (s k j)))) (\lambda (c3: C).(\lambda (_: -nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j -v c c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: -nat).(eq nat (S i) (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: -nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda -(j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: -nat).(csubst0 j v c c3))))) (or4 (clear c2 c0) (ex3_4 B C T T (\lambda (b: -B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind -b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: -T).(clear c2 (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 -(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(clear c2 (CHead e2 (Bind b) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) -(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear -c2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -i v e1 e2)))))))) (\lambda (H2: (ex3_2 T nat (\lambda (_: T).(\lambda (j: -nat).(eq nat (S i) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 -(CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t -u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (S i) (s k -j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda -(u2: T).(\lambda (j: nat).(subst0 j v t u2))) (or4 (clear c2 c0) (ex3_4 B C T -T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 -(CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: -T).(\lambda (u2: T).(clear c2 (CHead e (Bind b) u2)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) -(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(clear c2 (CHead e2 (Bind b) u)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 i v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: -nat).(\lambda (H3: (eq nat (S i) (s k x1))).(\lambda (H4: (eq C c2 (CHead c k -x0))).(\lambda (H5: (subst0 x1 v t x0)).(eq_ind_r C (CHead c k x0) (\lambda -(c3: C).(or4 (clear c3 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: -C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) -(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c3 -(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind -b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(clear c3 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear c3 (CHead e2 -(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 -e2))))))))) (K_ind (\lambda (k0: K).((clear (CHead c k0 t) c0) \to ((eq nat -(S i) (s k0 x1)) \to (or4 (clear (CHead c k0 x0) c0) (ex3_4 B C T T (\lambda -(b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e -(Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda -(u2: T).(clear (CHead c k0 x0) (CHead e (Bind b) u2)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) -(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead c k0 x0) (CHead e2 (Bind b) -u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind -b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (u2: T).(clear (CHead c k0 x0) (CHead e2 (Bind b) u2))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))))))) -(\lambda (b: B).(\lambda (H6: (clear (CHead c (Bind b) t) c0)).(\lambda (H7: -(eq nat (S i) (s (Bind b) x1))).(let H8 \def (f_equal nat nat (\lambda (e: -nat).(match e in nat return (\lambda (_: nat).nat) with [O \Rightarrow i | (S -n) \Rightarrow n])) (S i) (S x1) H7) in (let H9 \def (eq_ind_r nat x1 -(\lambda (n: nat).(subst0 n v t x0)) H5 i H8) in (eq_ind_r C (CHead c (Bind -b) t) (\lambda (c3: C).(or4 (clear (CHead c (Bind b) x0) c3) (ex3_4 B C T T -(\lambda (b0: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c3 -(CHead e (Bind b0) u1)))))) (\lambda (b0: B).(\lambda (e: C).(\lambda (_: -T).(\lambda (u2: T).(clear (CHead c (Bind b) x0) (CHead e (Bind b0) u2)))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v -u1 u2)))))) (ex3_4 B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C c3 (CHead e1 (Bind b0) u)))))) (\lambda (b0: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead c (Bind b) -x0) (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda -(b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq -C c3 (CHead e1 (Bind b0) u1))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead c (Bind b) x0) (CHead -e2 (Bind b0) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -i v e1 e2))))))))) (or4_intro1 (clear (CHead c (Bind b) x0) (CHead c (Bind b) -t)) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e: C).(\lambda (u1: T).(\lambda -(_: T).(eq C (CHead c (Bind b) t) (CHead e (Bind b0) u1)))))) (\lambda (b0: -B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead c (Bind b) -x0) (CHead e (Bind b0) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b0: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c (Bind b) -t) (CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(clear (CHead c (Bind b) x0) (CHead e2 (Bind b0) u)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e1 -(Bind b0) u1))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead c (Bind b) x0) (CHead e2 -(Bind b0) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 -e2))))))) (ex3_4_intro B C T T (\lambda (b0: B).(\lambda (e: C).(\lambda (u1: -T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e (Bind b0) u1)))))) -(\lambda (b0: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear -(CHead c (Bind b) x0) (CHead e (Bind b0) u2)))))) (\lambda (_: B).(\lambda -(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))) b c t x0 -(refl_equal C (CHead c (Bind b) t)) (clear_bind b c x0) H9)) c0 -(clear_gen_bind b c c0 t H6))))))) (\lambda (f: F).(\lambda (H6: (clear -(CHead c (Flat f) t) c0)).(\lambda (H7: (eq nat (S i) (s (Flat f) x1))).(let -H8 \def (f_equal nat nat (\lambda (e: nat).e) (S i) (s (Flat f) x1) H7) in -(let H9 \def (eq_ind_r nat x1 (\lambda (n: nat).(subst0 n v t x0)) H5 (S i) -H8) in (or4_intro0 (clear (CHead c (Flat f) x0) c0) (ex3_4 B C T T (\lambda -(b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e -(Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda -(u2: T).(clear (CHead c (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) -(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead c (Flat f) x0) (CHead e2 -(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 -(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (u2: T).(clear (CHead c (Flat f) x0) (CHead e2 (Bind b) -u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) -(clear_flat c c0 (clear_gen_flat f c c0 t H6) f x0))))))) k H1 H3) c2 -H4)))))) H2)) (\lambda (H2: (ex3_2 C nat (\lambda (_: C).(\lambda (j: -nat).(eq nat (S i) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 -(CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c -c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S i) (s k -j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda -(c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (or4 (clear c2 c0) (ex3_4 B C -T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 -(CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: -T).(\lambda (u2: T).(clear c2 (CHead e (Bind b) u2)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) -(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(clear c2 (CHead e2 (Bind b) u)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 i v e1 e2)))))))) (\lambda (x0: C).(\lambda (x1: -nat).(\lambda (H3: (eq nat (S i) (s k x1))).(\lambda (H4: (eq C c2 (CHead x0 -k t))).(\lambda (H5: (csubst0 x1 v c x0)).(eq_ind_r C (CHead x0 k t) (\lambda -(c3: C).(or4 (clear c3 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: -C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) -(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c3 -(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind -b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(clear c3 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear c3 (CHead e2 -(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 -e2))))))))) (K_ind (\lambda (k0: K).((clear (CHead c k0 t) c0) \to ((eq nat -(S i) (s k0 x1)) \to (or4 (clear (CHead x0 k0 t) c0) (ex3_4 B C T T (\lambda -(b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e -(Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda -(u2: T).(clear (CHead x0 k0 t) (CHead e (Bind b) u2)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) -(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 k0 t) (CHead e2 (Bind b) -u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind -b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (u2: T).(clear (CHead x0 k0 t) (CHead e2 (Bind b) u2))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))))))) -(\lambda (b: B).(\lambda (H6: (clear (CHead c (Bind b) t) c0)).(\lambda (H7: -(eq nat (S i) (s (Bind b) x1))).(let H8 \def (f_equal nat nat (\lambda (e: -nat).(match e in nat return (\lambda (_: nat).nat) with [O \Rightarrow i | (S -n) \Rightarrow n])) (S i) (S x1) H7) in (let H9 \def (eq_ind_r nat x1 -(\lambda (n: nat).(csubst0 n v c x0)) H5 i H8) in (eq_ind_r C (CHead c (Bind -b) t) (\lambda (c3: C).(or4 (clear (CHead x0 (Bind b) t) c3) (ex3_4 B C T T -(\lambda (b0: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c3 -(CHead e (Bind b0) u1)))))) (\lambda (b0: B).(\lambda (e: C).(\lambda (_: -T).(\lambda (u2: T).(clear (CHead x0 (Bind b) t) (CHead e (Bind b0) u2)))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v -u1 u2)))))) (ex3_4 B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C c3 (CHead e1 (Bind b0) u)))))) (\lambda (b0: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Bind b) -t) (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c3 -(CHead e1 (Bind b0) u1))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Bind b) t) (CHead e2 -(Bind b0) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 -e2))))))))) (or4_intro2 (clear (CHead x0 (Bind b) t) (CHead c (Bind b) t)) -(ex3_4 B C T T (\lambda (b0: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: -T).(eq C (CHead c (Bind b) t) (CHead e (Bind b0) u1)))))) (\lambda (b0: -B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Bind b) -t) (CHead e (Bind b0) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b0: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c (Bind b) -t) (CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(clear (CHead x0 (Bind b) t) (CHead e2 (Bind b0) u)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e1 -(Bind b0) u1))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Bind b) t) (CHead e2 -(Bind b0) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 -e2))))))) (ex3_4_intro B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C (CHead c (Bind b) t) (CHead e1 (Bind b0) u)))))) -(\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear -(CHead x0 (Bind b) t) (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2))))) b c x0 t -(refl_equal C (CHead c (Bind b) t)) (clear_bind b x0 t) H9)) c0 -(clear_gen_bind b c c0 t H6))))))) (\lambda (f: F).(\lambda (H6: (clear -(CHead c (Flat f) t) c0)).(\lambda (H7: (eq nat (S i) (s (Flat f) x1))).(let -H8 \def (f_equal nat nat (\lambda (e: nat).e) (S i) (s (Flat f) x1) H7) in -(let H9 \def (eq_ind_r nat x1 (\lambda (n: nat).(csubst0 n v c x0)) H5 (S i) -H8) in (let H10 \def (H x0 v i H9 c0 (clear_gen_flat f c c0 t H6)) in -(or4_ind (clear x0 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: -C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) -(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x0 -(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind -b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(clear x0 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear x0 (CHead e2 -(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 -e2))))))) (or4 (clear (CHead x0 (Flat f) t) c0) (ex3_4 B C T T (\lambda (b: -B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind -b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: -T).(clear (CHead x0 (Flat f) t) (CHead e (Bind b) u2)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) -(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Flat f) t) (CHead e2 -(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 -(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) -u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 -e2)))))))) (\lambda (H11: (clear x0 c0)).(or4_intro0 (clear (CHead x0 (Flat -f) t) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: -T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: -B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) -t) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind -b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) -(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear -(CHead x0 (Flat f) t) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda -(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 -u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (clear_flat x0 c0 H11 f t))) -(\lambda (H11: (ex3_4 B C T T (\lambda (b: B).(\lambda (e: 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x5)).(or4_intro1 (clear (CHead x0 (Flat f) t) c0) (ex3_4 B C T -T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 -(CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: -T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e (Bind b) u2)))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v -u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Flat f) -t) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 -(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 -(Bind b) u2))))))) 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B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x0 (CHead e2 (Bind -b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 i v e1 e2))))) (or4 (clear (CHead x0 (Flat f) t) c0) (ex3_4 B C T -T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 -(CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: -T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e (Bind b) u2)))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v -u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Flat f) -t) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 -(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 -(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 -e2)))))))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: -T).(\lambda (H12: (eq C c0 (CHead x3 (Bind x2) x5))).(\lambda (H13: (clear x0 -(CHead x4 (Bind x2) x5))).(\lambda (H14: (csubst0 i v x3 x4)).(or4_intro2 -(clear (CHead x0 (Flat f) t) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: -C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) -(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear -(CHead x0 (Flat f) t) (CHead e (Bind b) u2)))))) (\lambda (_: 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c0 -(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v e1 e2))))) x2 x3 x4 x5 H12 (clear_flat x0 (CHead x4 (Bind x2) x5) H13 f t) -H14))))))))) H11)) (\lambda (H11: (ex4_5 B C C T T (\lambda (b: B).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 -(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (u2: T).(clear x0 (CHead e2 (Bind b) u2))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 -i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))).(ex4_5_ind B C C T T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: 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e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 -(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) -u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 -e2)))))))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: -T).(\lambda (x6: T).(\lambda (H12: (eq C c0 (CHead x3 (Bind x2) -x5))).(\lambda (H13: (clear x0 (CHead x4 (Bind x2) x6))).(\lambda (H14: -(subst0 i v x5 x6)).(\lambda (H15: (csubst0 i v x3 x4)).(or4_intro3 (clear -(CHead x0 (Flat f) t) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: -C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) -(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: 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T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear -(CHead x0 (Flat f) t) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda -(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 -u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 i v e1 e2)))))) x2 x3 x4 x5 x6 H12 (clear_flat x0 -(CHead x4 (Bind x2) x6) H13 f t) H14 H15))))))))))) H11)) H10))))))) k H1 H3) -c2 H4)))))) H2)) (\lambda (H2: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: -C).(\lambda (j: nat).(eq nat (S i) (s k j))))) (\lambda (u2: T).(\lambda (c3: -C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda -(_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: -C).(\lambda (j: nat).(csubst0 j v c c3)))))).(ex4_3_ind T C nat (\lambda (_: -T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S i) (s k j))))) (\lambda (u2: -T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda -(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: -T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (or4 (clear c2 -c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda -(_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: -C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e (Bind b) u2)))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v -u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c2 (CHead e2 (Bind -b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind -b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 -i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))) (\lambda (x0: T).(\lambda -(x1: C).(\lambda (x2: nat).(\lambda (H3: (eq nat (S i) (s k x2))).(\lambda -(H4: (eq C c2 (CHead x1 k x0))).(\lambda (H5: (subst0 x2 v t x0)).(\lambda -(H6: (csubst0 x2 v c x1)).(eq_ind_r C (CHead x1 k x0) (\lambda (c3: C).(or4 -(clear c3 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: -T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: -B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c3 (CHead e (Bind -b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c3 -(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 -(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(clear c3 (CHead e2 (Bind b) u2))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))))) (K_ind -(\lambda (k0: K).((clear (CHead c k0 t) c0) \to ((eq nat (S i) (s k0 x2)) \to -(or4 (clear (CHead x1 k0 x0) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: -C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) -(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear -(CHead x1 k0 x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 -(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(clear (CHead x1 k0 x0) (CHead e2 (Bind b) u)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(clear (CHead x1 k0 x0) (CHead e2 (Bind b) u2))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 -i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))))))) (\lambda (b: B).(\lambda -(H7: (clear (CHead c (Bind b) t) c0)).(\lambda (H8: (eq nat (S i) (s (Bind b) -x2))).(let H9 \def (f_equal nat nat (\lambda (e: nat).(match e in nat return -(\lambda (_: nat).nat) with [O \Rightarrow i | (S n) \Rightarrow n])) (S i) -(S x2) H8) in (let H10 \def (eq_ind_r nat x2 (\lambda (n: nat).(csubst0 n v c -x1)) H6 i H9) in (let H11 \def (eq_ind_r nat x2 (\lambda (n: nat).(subst0 n v -t x0)) H5 i H9) in (eq_ind_r C (CHead c (Bind b) t) (\lambda (c3: C).(or4 -(clear (CHead x1 (Bind b) x0) c3) (ex3_4 B C T T (\lambda (b0: B).(\lambda -(e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c3 (CHead e (Bind b0) u1)))))) -(\lambda (b0: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear -(CHead x1 (Bind b) x0) (CHead e (Bind b0) u2)))))) (\lambda (_: B).(\lambda -(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C -T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3 -(CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(clear (CHead x1 (Bind b) x0) (CHead e2 (Bind b0) u)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C c3 (CHead e1 (Bind b0) u1))))))) -(\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(clear (CHead x1 (Bind b) x0) (CHead e2 (Bind b0) u2))))))) (\lambda -(_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))))) (or4_intro3 -(clear (CHead x1 (Bind b) x0) (CHead c (Bind b) t)) (ex3_4 B C T T (\lambda -(b0: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead c (Bind -b) t) (CHead e (Bind b0) u1)))))) (\lambda (b0: B).(\lambda (e: C).(\lambda -(_: T).(\lambda (u2: T).(clear (CHead x1 (Bind b) x0) (CHead e (Bind b0) -u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b0: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c (Bind b) t) (CHead e1 (Bind -b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(clear (CHead x1 (Bind b) x0) (CHead e2 (Bind b0) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) -(ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e1 (Bind b0) -u1))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (u2: T).(clear (CHead x1 (Bind b) x0) (CHead e2 (Bind b0) -u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) -(ex4_5_intro B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e1 -(Bind b0) u1))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Bind b) x0) (CHead e2 -(Bind b0) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 -e2)))))) b c x1 t x0 (refl_equal C (CHead c (Bind b) t)) (clear_bind b x1 x0) -H11 H10)) c0 (clear_gen_bind b c c0 t H7)))))))) (\lambda (f: F).(\lambda -(H7: (clear (CHead c (Flat f) t) c0)).(\lambda (H8: (eq nat (S i) (s (Flat f) -x2))).(let H9 \def (f_equal nat nat (\lambda (e: nat).e) (S i) (s (Flat f) -x2) H8) in (let H10 \def (eq_ind_r nat x2 (\lambda (n: nat).(csubst0 n v c -x1)) H6 (S i) H9) in (let H11 \def (eq_ind_r nat x2 (\lambda (n: nat).(subst0 -n v t x0)) H5 (S i) H9) in (let H12 \def (H x1 v i H10 c0 (clear_gen_flat f c -c0 t H7)) in (or4_ind (clear x1 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda -(e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) -(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x1 -(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind -b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(clear x1 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear x1 (CHead e2 -(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 -e2))))))) (or4 (clear (CHead x1 (Flat f) x0) c0) (ex3_4 B C T T (\lambda (b: -B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind -b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: -T).(clear (CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) -(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2 -(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 -(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) -u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 -e2)))))))) (\lambda (H13: (clear x1 c0)).(or4_intro0 (clear (CHead x1 (Flat -f) x0) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: -T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: -B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) -x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind -b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) -(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear -(CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda -(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 -u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (clear_flat x1 c0 H13 f x0))) -(\lambda (H13: (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: -T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: -B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x1 (CHead e (Bind -b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 i v u1 u2))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e: -C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) -(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x1 -(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 i v u1 u2))))) (or4 (clear (CHead x1 (Flat f) x0) -c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda -(_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: -C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e -(Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear -(CHead x1 (Flat f) x0) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C -C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear -(CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda -(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 -u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 i v e1 e2)))))))) (\lambda (x3: B).(\lambda (x4: -C).(\lambda (x5: T).(\lambda (x6: T).(\lambda (H14: (eq C c0 (CHead x4 (Bind -x3) x5))).(\lambda (H15: (clear x1 (CHead x4 (Bind x3) x6))).(\lambda (H16: -(subst0 i v x5 x6)).(or4_intro1 (clear (CHead x1 (Flat f) x0) c0) (ex3_4 B C -T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 -(CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: -T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v -u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x1 (Flat f) -x0) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 -(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 -(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 -e2))))))) (ex3_4_intro B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: -T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: -B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) -x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 i v u1 u2))))) x3 x4 x5 x6 H14 (clear_flat x1 -(CHead x4 (Bind x3) x6) H15 f x0) H16))))))))) H13)) (\lambda (H13: (ex3_4 B -C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C -c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(clear x1 (CHead e2 (Bind b) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 -e2))))))).(ex3_4_ind B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x1 (CHead e2 (Bind -b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 i v e1 e2))))) (or4 (clear (CHead x1 (Flat f) x0) c0) (ex3_4 B C -T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 -(CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: -T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v -u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x1 (Flat f) -x0) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 -(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 -(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 -e2)))))))) (\lambda (x3: B).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: -T).(\lambda (H14: (eq C c0 (CHead x4 (Bind x3) x6))).(\lambda (H15: (clear x1 -(CHead x5 (Bind x3) x6))).(\lambda (H16: (csubst0 i v x4 x5)).(or4_intro2 -(clear (CHead x1 (Flat f) x0) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: -C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) -(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear -(CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda -(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C -T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 -(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda -(_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (ex3_4_intro B C -C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 -(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v e1 e2))))) x3 x4 x5 x6 H14 (clear_flat x1 (CHead x5 (Bind x3) x6) H15 f x0) -H16))))))))) H13)) (\lambda (H13: (ex4_5 B C C T T (\lambda (b: B).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 -(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (u2: T).(clear x1 (CHead e2 (Bind b) u2))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 -i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))).(ex4_5_ind B C C T T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear x1 (CHead e2 -(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 -e2)))))) (or4 (clear (CHead x1 (Flat f) x0) c0) (ex3_4 B C T T (\lambda (b: -B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind -b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: -T).(clear (CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) -(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2 -(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 -(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) -u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 -e2)))))))) (\lambda (x3: B).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: -T).(\lambda (x7: T).(\lambda (H14: (eq C c0 (CHead x4 (Bind x3) -x6))).(\lambda (H15: (clear x1 (CHead x5 (Bind x3) x7))).(\lambda (H16: -(subst0 i v x6 x7)).(\lambda (H17: (csubst0 i v x4 x5)).(or4_intro3 (clear -(CHead x1 (Flat f) x0) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: -C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) -(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear -(CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda -(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C -T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 -(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda -(_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (ex4_5_intro B C -C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear -(CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda -(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 -u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 i v e1 e2)))))) x3 x4 x5 x6 x7 H14 (clear_flat x1 -(CHead x5 (Bind x3) x7) H15 f x0) H16 H17))))))))))) H13)) H12)))))))) k H1 -H3) c2 H4)))))))) H2)) (csubst0_gen_head k c c2 t v (S i) H0)))))))))))) c1). -(* COMMENTS -Initial nodes: 14968 -END *) - -theorem csubst0_clear_trans: - \forall (c1: C).(\forall (c2: C).(\forall (v: T).(\forall (i: nat).((csubst0 -i v c1 c2) \to (\forall (e2: C).((clear c2 e2) \to (or (clear c1 e2) (ex2 C -(\lambda (e1: C).(csubst0 i v e1 e2)) (\lambda (e1: C).(clear c1 e1)))))))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda (i: nat).(\lambda -(H: (csubst0 i v c1 c2)).(csubst0_ind (\lambda (n: nat).(\lambda (t: -T).(\lambda (c: C).(\lambda (c0: C).(\forall (e2: C).((clear c0 e2) \to (or -(clear c e2) (ex2 C (\lambda (e1: C).(csubst0 n t e1 e2)) (\lambda (e1: -C).(clear c e1)))))))))) (\lambda (k: K).(\lambda (i0: nat).(\lambda (v0: -T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H0: (subst0 i0 v0 u1 -u2)).(\lambda (c: C).(\lambda (e2: C).(\lambda (H1: (clear (CHead c k u2) -e2)).(K_ind (\lambda (k0: K).((clear (CHead c k0 u2) e2) \to (or (clear -(CHead c k0 u1) e2) (ex2 C (\lambda (e1: C).(csubst0 (s k0 i0) v0 e1 e2)) -(\lambda (e1: C).(clear (CHead c k0 u1) e1)))))) (\lambda (b: B).(\lambda -(H2: (clear (CHead c (Bind b) u2) e2)).(eq_ind_r C (CHead c (Bind b) u2) -(\lambda (c0: C).(or (clear (CHead c (Bind b) u1) c0) (ex2 C (\lambda (e1: -C).(csubst0 (s (Bind b) i0) v0 e1 c0)) (\lambda (e1: C).(clear (CHead c (Bind -b) u1) e1))))) (or_intror (clear (CHead c (Bind b) u1) (CHead c (Bind b) u2)) -(ex2 C (\lambda (e1: C).(csubst0 (S i0) v0 e1 (CHead c (Bind b) u2))) -(\lambda (e1: C).(clear (CHead c (Bind b) u1) e1))) (ex_intro2 C (\lambda -(e1: C).(csubst0 (S i0) v0 e1 (CHead c (Bind b) u2))) (\lambda (e1: C).(clear -(CHead c (Bind b) u1) e1)) (CHead c (Bind b) u1) (csubst0_snd_bind b i0 v0 u1 -u2 H0 c) (clear_bind b c u1))) e2 (clear_gen_bind b c e2 u2 H2)))) (\lambda -(f: F).(\lambda (H2: (clear (CHead c (Flat f) u2) e2)).(or_introl (clear -(CHead c (Flat f) u1) e2) (ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2)) -(\lambda (e1: C).(clear (CHead c (Flat f) u1) e1))) (clear_flat c e2 -(clear_gen_flat f c e2 u2 H2) f u1)))) k H1)))))))))) (\lambda (k: -K).(\lambda (i0: nat).(\lambda (c3: C).(\lambda (c4: C).(\lambda (v0: -T).(\lambda (H0: (csubst0 i0 v0 c3 c4)).(\lambda (H1: ((\forall (e2: -C).((clear c4 e2) \to (or (clear c3 e2) (ex2 C (\lambda (e1: C).(csubst0 i0 -v0 e1 e2)) (\lambda (e1: C).(clear c3 e1)))))))).(\lambda (u: T).(\lambda -(e2: C).(\lambda (H2: (clear (CHead c4 k u) e2)).(K_ind (\lambda (k0: -K).((clear (CHead c4 k0 u) e2) \to (or (clear (CHead c3 k0 u) e2) (ex2 C -(\lambda (e1: C).(csubst0 (s k0 i0) v0 e1 e2)) (\lambda (e1: C).(clear (CHead -c3 k0 u) e1)))))) (\lambda (b: B).(\lambda (H3: (clear (CHead c4 (Bind b) u) -e2)).(eq_ind_r C (CHead c4 (Bind b) u) (\lambda (c: C).(or (clear (CHead c3 -(Bind b) u) c) (ex2 C (\lambda (e1: C).(csubst0 (s (Bind b) i0) v0 e1 c)) -(\lambda (e1: C).(clear (CHead c3 (Bind b) u) e1))))) (or_intror (clear -(CHead c3 (Bind b) u) (CHead c4 (Bind b) u)) (ex2 C (\lambda (e1: C).(csubst0 -(S i0) v0 e1 (CHead c4 (Bind b) u))) (\lambda (e1: C).(clear (CHead c3 (Bind -b) u) e1))) (ex_intro2 C (\lambda (e1: C).(csubst0 (S i0) v0 e1 (CHead c4 -(Bind b) u))) (\lambda (e1: C).(clear (CHead c3 (Bind b) u) e1)) (CHead c3 -(Bind b) u) (csubst0_fst_bind b i0 c3 c4 v0 H0 u) (clear_bind b c3 u))) e2 -(clear_gen_bind b c4 e2 u H3)))) (\lambda (f: F).(\lambda (H3: (clear (CHead -c4 (Flat f) u) e2)).(let H_x \def (H1 e2 (clear_gen_flat f c4 e2 u H3)) in -(let H4 \def H_x in (or_ind (clear c3 e2) (ex2 C (\lambda (e1: C).(csubst0 i0 -v0 e1 e2)) (\lambda (e1: C).(clear c3 e1))) (or (clear (CHead c3 (Flat f) u) -e2) (ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear -(CHead c3 (Flat f) u) e1)))) (\lambda (H5: (clear c3 e2)).(or_introl (clear -(CHead c3 (Flat f) u) e2) (ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2)) -(\lambda (e1: C).(clear (CHead c3 (Flat f) u) e1))) (clear_flat c3 e2 H5 f -u))) (\lambda (H5: (ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda -(e1: C).(clear c3 e1)))).(ex2_ind C (\lambda (e1: C).(csubst0 i0 v0 e1 e2)) -(\lambda (e1: C).(clear c3 e1)) (or (clear (CHead c3 (Flat f) u) e2) (ex2 C -(\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear (CHead c3 -(Flat f) u) e1)))) (\lambda (x: C).(\lambda (H6: (csubst0 i0 v0 x -e2)).(\lambda (H7: (clear c3 x)).(or_intror (clear (CHead c3 (Flat f) u) e2) -(ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear (CHead -c3 (Flat f) u) e1))) (ex_intro2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2)) -(\lambda (e1: C).(clear (CHead c3 (Flat f) u) e1)) x H6 (clear_flat c3 x H7 f -u)))))) H5)) H4))))) k H2))))))))))) (\lambda (k: K).(\lambda (i0: -nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H0: (subst0 -i0 v0 u1 u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (H1: (csubst0 i0 v0 -c3 c4)).(\lambda (H2: ((\forall (e2: C).((clear c4 e2) \to (or (clear c3 e2) -(ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear c3 -e1)))))))).(\lambda (e2: C).(\lambda (H3: (clear (CHead c4 k u2) e2)).(K_ind -(\lambda (k0: K).((clear (CHead c4 k0 u2) e2) \to (or (clear (CHead c3 k0 u1) -e2) (ex2 C (\lambda (e1: C).(csubst0 (s k0 i0) v0 e1 e2)) (\lambda (e1: -C).(clear (CHead c3 k0 u1) e1)))))) (\lambda (b: B).(\lambda (H4: (clear -(CHead c4 (Bind b) u2) e2)).(eq_ind_r C (CHead c4 (Bind b) u2) (\lambda (c: -C).(or (clear (CHead c3 (Bind b) u1) c) (ex2 C (\lambda (e1: C).(csubst0 (s -(Bind b) i0) v0 e1 c)) (\lambda (e1: C).(clear (CHead c3 (Bind b) u1) e1))))) -(or_intror (clear (CHead c3 (Bind b) u1) (CHead c4 (Bind b) u2)) (ex2 C -(\lambda (e1: C).(csubst0 (S i0) v0 e1 (CHead c4 (Bind b) u2))) (\lambda (e1: -C).(clear (CHead c3 (Bind b) u1) e1))) (ex_intro2 C (\lambda (e1: C).(csubst0 -(S i0) v0 e1 (CHead c4 (Bind b) u2))) (\lambda (e1: C).(clear (CHead c3 (Bind -b) u1) e1)) (CHead c3 (Bind b) u1) (csubst0_both_bind b i0 v0 u1 u2 H0 c3 c4 -H1) (clear_bind b c3 u1))) e2 (clear_gen_bind b c4 e2 u2 H4)))) (\lambda (f: -F).(\lambda (H4: (clear (CHead c4 (Flat f) u2) e2)).(let H_x \def (H2 e2 -(clear_gen_flat f c4 e2 u2 H4)) in (let H5 \def H_x in (or_ind (clear c3 e2) -(ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear c3 -e1))) (or (clear (CHead c3 (Flat f) u1) e2) (ex2 C (\lambda (e1: C).(csubst0 -i0 v0 e1 e2)) (\lambda (e1: C).(clear (CHead c3 (Flat f) u1) e1)))) (\lambda -(H6: (clear c3 e2)).(or_introl (clear (CHead c3 (Flat f) u1) e2) (ex2 C -(\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear (CHead c3 -(Flat f) u1) e1))) (clear_flat c3 e2 H6 f u1))) (\lambda (H6: (ex2 C (\lambda -(e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear c3 e1)))).(ex2_ind C -(\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear c3 e1)) (or -(clear (CHead c3 (Flat f) u1) e2) (ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 -e2)) (\lambda (e1: C).(clear (CHead c3 (Flat f) u1) e1)))) (\lambda (x: -C).(\lambda (H7: (csubst0 i0 v0 x e2)).(\lambda (H8: (clear c3 x)).(or_intror -(clear (CHead c3 (Flat f) u1) e2) (ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 -e2)) (\lambda (e1: C).(clear (CHead c3 (Flat f) u1) e1))) (ex_intro2 C -(\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear (CHead c3 -(Flat f) u1) e1)) x H7 (clear_flat c3 x H8 f u1)))))) H6)) H5))))) k -H3))))))))))))) i v c1 c2 H))))). -(* COMMENTS -Initial nodes: 2085 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubst0/defs.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubst0/defs.ma deleted file mode 100644 index 0068e19f6..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubst0/defs.ma +++ /dev/null @@ -1,32 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/subst0/defs.ma". - -include "Basic-1/C/defs.ma". - -inductive csubst0: nat \to (T \to (C \to (C \to Prop))) \def -| csubst0_snd: \forall (k: K).(\forall (i: nat).(\forall (v: T).(\forall (u1: -T).(\forall (u2: T).((subst0 i v u1 u2) \to (\forall (c: C).(csubst0 (s k i) -v (CHead c k u1) (CHead c k u2)))))))) -| csubst0_fst: \forall (k: K).(\forall (i: nat).(\forall (c1: C).(\forall -(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (u: T).(csubst0 (s -k i) v (CHead c1 k u) (CHead c2 k u)))))))) -| csubst0_both: \forall (k: K).(\forall (i: nat).(\forall (v: T).(\forall -(u1: T).(\forall (u2: T).((subst0 i v u1 u2) \to (\forall (c1: C).(\forall -(c2: C).((csubst0 i v c1 c2) \to (csubst0 (s k i) v (CHead c1 k u1) (CHead c2 -k u2)))))))))). - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubst0/drop.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubst0/drop.ma deleted file mode 100644 index b6bc6cae7..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubst0/drop.ma +++ /dev/null @@ -1,6294 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubst0/fwd.ma". - -include "Basic-1/drop/fwd.ma". - -include "Basic-1/s/props.ma". - -theorem csubst0_drop_gt: - \forall (n: nat).(\forall (i: nat).((lt i n) \to (\forall (c1: C).(\forall -(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n O -c1 e) \to (drop n O c2 e))))))))) -\def - \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (i: nat).((lt i n0) -\to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) -\to (\forall (e: C).((drop n0 O c1 e) \to (drop n0 O c2 e)))))))))) (\lambda -(i: nat).(\lambda (H: (lt i O)).(\lambda (c1: C).(\lambda (c2: C).(\lambda -(v: T).(\lambda (_: (csubst0 i v c1 c2)).(\lambda (e: C).(\lambda (_: (drop O -O c1 e)).(lt_x_O i H (drop O O c2 e)))))))))) (\lambda (n0: nat).(\lambda (H: -((\forall (i: nat).((lt i n0) \to (\forall (c1: C).(\forall (c2: C).(\forall -(v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n0 O c1 e) \to (drop -n0 O c2 e))))))))))).(\lambda (i: nat).(\lambda (H0: (lt i (S n0))).(\lambda -(c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v -c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (drop (S n0) O c2 e))))))) -(\lambda (n1: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (_: (csubst0 i v -(CSort n1) c2)).(\lambda (e: C).(\lambda (H2: (drop (S n0) O (CSort n1) -e)).(and3_ind (eq C e (CSort n1)) (eq nat (S n0) O) (eq nat O O) (drop (S n0) -O c2 e) (\lambda (H3: (eq C e (CSort n1))).(\lambda (H4: (eq nat (S n0) -O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n1) (\lambda (c: C).(drop -(S n0) O c2 c)) (let H6 \def (eq_ind nat (S n0) (\lambda (ee: nat).(match ee -in nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) -\Rightarrow True])) I O H4) in (False_ind (drop (S n0) O c2 (CSort n1)) H6)) -e H3)))) (drop_gen_sort n1 (S n0) O e H2)))))))) (\lambda (c: C).(\lambda -(H1: ((\forall (c2: C).(\forall (v: T).((csubst0 i v c c2) \to (\forall (e: -C).((drop (S n0) O c e) \to (drop (S n0) O c2 e)))))))).(\lambda (k: -K).(\lambda (t: T).(\lambda (c2: C).(\lambda (v: T).(\lambda (H2: (csubst0 i -v (CHead c k t) c2)).(\lambda (e: C).(\lambda (H3: (drop (S n0) O (CHead c k -t) e)).(or3_ind (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k -j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda -(u2: T).(\lambda (j: nat).(subst0 j v t u2)))) (ex3_2 C nat (\lambda (_: -C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_: -nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j -v c c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: -nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: -nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda -(j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: -nat).(csubst0 j v c c3))))) (drop (S n0) O c2 e) (\lambda (H4: (ex3_2 T nat -(\lambda (_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: -T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: -nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: -nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead -c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))) (drop (S -n0) O c2 e) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H5: (eq nat i (s k -x1))).(\lambda (H6: (eq C c2 (CHead c k x0))).(\lambda (_: (subst0 x1 v t -x0)).(eq_ind_r C (CHead c k x0) (\lambda (c0: C).(drop (S n0) O c0 e)) (let -H8 \def (eq_ind nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0: -T).((csubst0 n1 v0 c c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (drop -(S n0) O c3 e0))))))) H1 (s k x1) H5) in (let H9 \def (eq_ind nat i (\lambda -(n1: nat).(lt n1 (S n0))) H0 (s k x1) H5) in (K_ind (\lambda (k0: K).((drop -(r k0 n0) O c e) \to (((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1) -v0 c c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3 -e0))))))) \to ((lt (s k0 x1) (S n0)) \to (drop (S n0) O (CHead c k0 x0) -e))))) (\lambda (b: B).(\lambda (H10: (drop (r (Bind b) n0) O c e)).(\lambda -(_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to -(\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3 -e0)))))))).(\lambda (_: (lt (s (Bind b) x1) (S n0))).(drop_drop (Bind b) n0 c -e H10 x0))))) (\lambda (f: F).(\lambda (H10: (drop (r (Flat f) n0) O c -e)).(\lambda (_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Flat f) x1) -v0 c c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3 -e0)))))))).(\lambda (H12: (lt (s (Flat f) x1) (S n0))).(or_ind (eq nat x1 O) -(ex2 nat (\lambda (m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0))) -(drop (S n0) O (CHead c (Flat f) x0) e) (\lambda (_: (eq nat x1 -O)).(drop_drop (Flat f) n0 c e H10 x0)) (\lambda (H13: (ex2 nat (\lambda (m: -nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda -(m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O -(CHead c (Flat f) x0) e) (\lambda (x: nat).(\lambda (_: (eq nat x1 (S -x))).(\lambda (_: (lt x n0)).(drop_drop (Flat f) n0 c e H10 x0)))) H13)) -(lt_gen_xS x1 n0 H12)))))) k (drop_gen_drop k c e t n0 H3) H8 H9))) c2 -H6)))))) H4)) (\lambda (H4: (ex3_2 C nat (\lambda (_: C).(\lambda (j: -nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead -c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c -c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k -j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda -(c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (drop (S n0) O c2 e) (\lambda -(x0: C).(\lambda (x1: nat).(\lambda (H5: (eq nat i (s k x1))).(\lambda (H6: -(eq C c2 (CHead x0 k t))).(\lambda (H7: (csubst0 x1 v c x0)).(eq_ind_r C -(CHead x0 k t) (\lambda (c0: C).(drop (S n0) O c0 e)) (let H8 \def (eq_ind -nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c -c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3 -e0))))))) H1 (s k x1) H5) in (let H9 \def (eq_ind nat i (\lambda (n1: -nat).(lt n1 (S n0))) H0 (s k x1) H5) in (K_ind (\lambda (k0: K).((drop (r k0 -n0) O c e) \to (((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c -c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3 -e0))))))) \to ((lt (s k0 x1) (S n0)) \to (drop (S n0) O (CHead x0 k0 t) -e))))) (\lambda (b: B).(\lambda (H10: (drop (r (Bind b) n0) O c e)).(\lambda -(_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to -(\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3 -e0)))))))).(\lambda (H12: (lt (s (Bind b) x1) (S n0))).(drop_drop (Bind b) n0 -x0 e (H x1 (lt_S_n x1 n0 H12) c x0 v H7 e H10) t))))) (\lambda (f: -F).(\lambda (H10: (drop (r (Flat f) n0) O c e)).(\lambda (H11: ((\forall (c3: -C).(\forall (v0: T).((csubst0 (s (Flat f) x1) v0 c c3) \to (\forall (e0: -C).((drop (S n0) O c e0) \to (drop (S n0) O c3 e0)))))))).(\lambda (H12: (lt -(s (Flat f) x1) (S n0))).(or_ind (eq nat x1 O) (ex2 nat (\lambda (m: nat).(eq -nat x1 (S m))) (\lambda (m: nat).(lt m n0))) (drop (S n0) O (CHead x0 (Flat -f) t) e) (\lambda (_: (eq nat x1 O)).(drop_drop (Flat f) n0 x0 e (H11 x0 v H7 -e H10) t)) (\lambda (H13: (ex2 nat (\lambda (m: nat).(eq nat x1 (S m))) -(\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda (m: nat).(eq nat x1 (S -m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O (CHead x0 (Flat f) t) e) -(\lambda (x: nat).(\lambda (_: (eq nat x1 (S x))).(\lambda (_: (lt x -n0)).(drop_drop (Flat f) n0 x0 e (H11 x0 v H7 e H10) t)))) H13)) (lt_gen_xS -x1 n0 H12)))))) k (drop_gen_drop k c e t n0 H3) H8 H9))) c2 H6)))))) H4)) -(\lambda (H4: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: -nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: -nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda -(j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: -nat).(csubst0 j v c c3)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_: -C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: -C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda -(_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: -C).(\lambda (j: nat).(csubst0 j v c c3)))) (drop (S n0) O c2 e) (\lambda (x0: -T).(\lambda (x1: C).(\lambda (x2: nat).(\lambda (H5: (eq nat i (s k -x2))).(\lambda (H6: (eq C c2 (CHead x1 k x0))).(\lambda (_: (subst0 x2 v t -x0)).(\lambda (H8: (csubst0 x2 v c x1)).(eq_ind_r C (CHead x1 k x0) (\lambda -(c0: C).(drop (S n0) O c0 e)) (let H9 \def (eq_ind nat i (\lambda (n1: -nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall -(e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3 e0))))))) H1 (s k x2) H5) -in (let H10 \def (eq_ind nat i (\lambda (n1: nat).(lt n1 (S n0))) H0 (s k x2) -H5) in (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to (((\forall (c3: -C).(\forall (v0: T).((csubst0 (s k0 x2) v0 c c3) \to (\forall (e0: C).((drop -(S n0) O c e0) \to (drop (S n0) O c3 e0))))))) \to ((lt (s k0 x2) (S n0)) \to -(drop (S n0) O (CHead x1 k0 x0) e))))) (\lambda (b: B).(\lambda (H11: (drop -(r (Bind b) n0) O c e)).(\lambda (_: ((\forall (c3: C).(\forall (v0: -T).((csubst0 (s (Bind b) x2) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c -e0) \to (drop (S n0) O c3 e0)))))))).(\lambda (H13: (lt (s (Bind b) x2) (S -n0))).(drop_drop (Bind b) n0 x1 e (H x2 (lt_S_n x2 n0 H13) c x1 v H8 e H11) -x0))))) (\lambda (f: F).(\lambda (H11: (drop (r (Flat f) n0) O c e)).(\lambda -(H12: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Flat f) x2) v0 c c3) -\to (\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3 -e0)))))))).(\lambda (H13: (lt (s (Flat f) x2) (S n0))).(or_ind (eq nat x2 O) -(ex2 nat (\lambda (m: nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m n0))) -(drop (S n0) O (CHead x1 (Flat f) x0) e) (\lambda (_: (eq nat x2 -O)).(drop_drop (Flat f) n0 x1 e (H12 x1 v H8 e H11) x0)) (\lambda (H14: (ex2 -nat (\lambda (m: nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m -n0)))).(ex2_ind nat (\lambda (m: nat).(eq nat x2 (S m))) (\lambda (m: -nat).(lt m n0)) (drop (S n0) O (CHead x1 (Flat f) x0) e) (\lambda (x: -nat).(\lambda (_: (eq nat x2 (S x))).(\lambda (_: (lt x n0)).(drop_drop (Flat -f) n0 x1 e (H12 x1 v H8 e H11) x0)))) H14)) (lt_gen_xS x2 n0 H13)))))) k -(drop_gen_drop k c e t n0 H3) H9 H10))) c2 H6)))))))) H4)) (csubst0_gen_head -k c c2 t v i H2))))))))))) c1)))))) n). -(* COMMENTS -Initial nodes: 3092 -END *) - -theorem csubst0_drop_gt_back: - \forall (n: nat).(\forall (i: nat).((lt i n) \to (\forall (c1: C).(\forall -(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n O -c2 e) \to (drop n O c1 e))))))))) -\def - \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (i: nat).((lt i n0) -\to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) -\to (\forall (e: C).((drop n0 O c2 e) \to (drop n0 O c1 e)))))))))) (\lambda -(i: nat).(\lambda (H: (lt i O)).(\lambda (c1: C).(\lambda (c2: C).(\lambda -(v: T).(\lambda (_: (csubst0 i v c1 c2)).(\lambda (e: C).(\lambda (_: (drop O -O c2 e)).(lt_x_O i H (drop O O c1 e)))))))))) (\lambda (n0: nat).(\lambda (H: -((\forall (i: nat).((lt i n0) \to (\forall (c1: C).(\forall (c2: C).(\forall -(v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n0 O c2 e) \to (drop -n0 O c1 e))))))))))).(\lambda (i: nat).(\lambda (H0: (lt i (S n0))).(\lambda -(c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v -c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c e))))))) -(\lambda (n1: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (H1: (csubst0 i -v (CSort n1) c2)).(\lambda (e: C).(\lambda (_: (drop (S n0) O c2 -e)).(csubst0_gen_sort c2 v i n1 H1 (drop (S n0) O (CSort n1) e)))))))) -(\lambda (c: C).(\lambda (H1: ((\forall (c2: C).(\forall (v: T).((csubst0 i v -c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c -e)))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (v: -T).(\lambda (H2: (csubst0 i v (CHead c k t) c2)).(\lambda (e: C).(\lambda -(H3: (drop (S n0) O c2 e)).(or3_ind (ex3_2 T nat (\lambda (_: T).(\lambda (j: -nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead -c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2)))) (ex3_2 C -nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: -C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: -nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: -C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: -C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda -(_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: -C).(\lambda (j: nat).(csubst0 j v c c3))))) (drop (S n0) O (CHead c k t) e) -(\lambda (H4: (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k -j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda -(u2: T).(\lambda (j: nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: -T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: -nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j -v t u2))) (drop (S n0) O (CHead c k t) e) (\lambda (x0: T).(\lambda (x1: -nat).(\lambda (H5: (eq nat i (s k x1))).(\lambda (H6: (eq C c2 (CHead c k -x0))).(\lambda (_: (subst0 x1 v t x0)).(let H8 \def (eq_ind C c2 (\lambda -(c0: C).(drop (S n0) O c0 e)) H3 (CHead c k x0) H6) in (let H9 \def (eq_ind -nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c -c3) \to (\forall (e0: C).((drop (S n0) O c3 e0) \to (drop (S n0) O c -e0))))))) H1 (s k x1) H5) in (let H10 \def (eq_ind nat i (\lambda (n1: -nat).(lt n1 (S n0))) H0 (s k x1) H5) in (K_ind (\lambda (k0: K).(((\forall -(c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c c3) \to (\forall (e0: -C).((drop (S n0) O c3 e0) \to (drop (S n0) O c e0))))))) \to ((lt (s k0 x1) -(S n0)) \to ((drop (r k0 n0) O c e) \to (drop (S n0) O (CHead c k0 t) e))))) -(\lambda (b: B).(\lambda (_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s -(Bind b) x1) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3 e0) \to (drop -(S n0) O c e0)))))))).(\lambda (_: (lt (s (Bind b) x1) (S n0))).(\lambda -(H13: (drop (r (Bind b) n0) O c e)).(drop_drop (Bind b) n0 c e H13 t))))) -(\lambda (f: F).(\lambda (_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s -(Flat f) x1) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3 e0) \to (drop -(S n0) O c e0)))))))).(\lambda (H12: (lt (s (Flat f) x1) (S n0))).(\lambda -(H13: (drop (r (Flat f) n0) O c e)).(or_ind (eq nat x1 O) (ex2 nat (\lambda -(m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0))) (drop (S n0) O -(CHead c (Flat f) t) e) (\lambda (_: (eq nat x1 O)).(drop_drop (Flat f) n0 c -e H13 t)) (\lambda (H14: (ex2 nat (\lambda (m: nat).(eq nat x1 (S m))) -(\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda (m: nat).(eq nat x1 (S -m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O (CHead c (Flat f) t) e) -(\lambda (x: nat).(\lambda (_: (eq nat x1 (S x))).(\lambda (_: (lt x -n0)).(drop_drop (Flat f) n0 c e H13 t)))) H14)) (lt_gen_xS x1 n0 H12)))))) k -H9 H10 (drop_gen_drop k c e x0 n0 H8)))))))))) H4)) (\lambda (H4: (ex3_2 C -nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: -C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: -nat).(csubst0 j v c c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: -nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead -c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (drop (S -n0) O (CHead c k t) e) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H5: (eq -nat i (s k x1))).(\lambda (H6: (eq C c2 (CHead x0 k t))).(\lambda (H7: -(csubst0 x1 v c x0)).(let H8 \def (eq_ind C c2 (\lambda (c0: C).(drop (S n0) -O c0 e)) H3 (CHead x0 k t) H6) in (let H9 \def (eq_ind nat i (\lambda (n1: -nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall -(e0: C).((drop (S n0) O c3 e0) \to (drop (S n0) O c e0))))))) H1 (s k x1) H5) -in (let H10 \def (eq_ind nat i (\lambda (n1: nat).(lt n1 (S n0))) H0 (s k x1) -H5) in (K_ind (\lambda (k0: K).(((\forall (c3: C).(\forall (v0: T).((csubst0 -(s k0 x1) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3 e0) \to (drop (S -n0) O c e0))))))) \to ((lt (s k0 x1) (S n0)) \to ((drop (r k0 n0) O x0 e) \to -(drop (S n0) O (CHead c k0 t) e))))) (\lambda (b: B).(\lambda (_: ((\forall -(c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to (\forall (e0: -C).((drop (S n0) O c3 e0) \to (drop (S n0) O c e0)))))))).(\lambda (H12: (lt -(s (Bind b) x1) (S n0))).(\lambda (H13: (drop (r (Bind b) n0) O x0 -e)).(drop_drop (Bind b) n0 c e (H x1 (lt_S_n x1 n0 H12) c x0 v H7 e H13) -t))))) (\lambda (f: F).(\lambda (H11: ((\forall (c3: C).(\forall (v0: -T).((csubst0 (s (Flat f) x1) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3 -e0) \to (drop (S n0) O c e0)))))))).(\lambda (H12: (lt (s (Flat f) x1) (S -n0))).(\lambda (H13: (drop (r (Flat f) n0) O x0 e)).(or_ind (eq nat x1 O) -(ex2 nat (\lambda (m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0))) -(drop (S n0) O (CHead c (Flat f) t) e) (\lambda (_: (eq nat x1 O)).(drop_drop -(Flat f) n0 c e (H11 x0 v H7 e H13) t)) (\lambda (H14: (ex2 nat (\lambda (m: -nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda -(m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O -(CHead c (Flat f) t) e) (\lambda (x: nat).(\lambda (_: (eq nat x1 (S -x))).(\lambda (_: (lt x n0)).(drop_drop (Flat f) n0 c e (H11 x0 v H7 e H13) -t)))) H14)) (lt_gen_xS x1 n0 H12)))))) k H9 H10 (drop_gen_drop k x0 e t n0 -H8)))))))))) H4)) (\lambda (H4: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: -C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: -C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda -(_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: -C).(\lambda (j: nat).(csubst0 j v c c3)))))).(ex4_3_ind T C nat (\lambda (_: -T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: -T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda -(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: -T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (drop (S n0) O -(CHead c k t) e) (\lambda (x0: T).(\lambda (x1: C).(\lambda (x2: -nat).(\lambda (H5: (eq nat i (s k x2))).(\lambda (H6: (eq C c2 (CHead x1 k -x0))).(\lambda (_: (subst0 x2 v t x0)).(\lambda (H8: (csubst0 x2 v c -x1)).(let H9 \def (eq_ind C c2 (\lambda (c0: C).(drop (S n0) O c0 e)) H3 -(CHead x1 k x0) H6) in (let H10 \def (eq_ind nat i (\lambda (n1: -nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall -(e0: C).((drop (S n0) O c3 e0) \to (drop (S n0) O c e0))))))) H1 (s k x2) H5) -in (let H11 \def (eq_ind nat i (\lambda (n1: nat).(lt n1 (S n0))) H0 (s k x2) -H5) in (K_ind (\lambda (k0: K).(((\forall (c3: C).(\forall (v0: T).((csubst0 -(s k0 x2) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3 e0) \to (drop (S -n0) O c e0))))))) \to ((lt (s k0 x2) (S n0)) \to ((drop (r k0 n0) O x1 e) \to -(drop (S n0) O (CHead c k0 t) e))))) (\lambda (b: B).(\lambda (_: ((\forall -(c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x2) v0 c c3) \to (\forall (e0: -C).((drop (S n0) O c3 e0) \to (drop (S n0) O c e0)))))))).(\lambda (H13: (lt -(s (Bind b) x2) (S n0))).(\lambda (H14: (drop (r (Bind b) n0) O x1 -e)).(drop_drop (Bind b) n0 c e (H x2 (lt_S_n x2 n0 H13) c x1 v H8 e H14) -t))))) (\lambda (f: F).(\lambda (H12: ((\forall (c3: C).(\forall (v0: -T).((csubst0 (s (Flat f) x2) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3 -e0) \to (drop (S n0) O c e0)))))))).(\lambda (H13: (lt (s (Flat f) x2) (S -n0))).(\lambda (H14: (drop (r (Flat f) n0) O x1 e)).(or_ind (eq nat x2 O) -(ex2 nat (\lambda (m: nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m n0))) -(drop (S n0) O (CHead c (Flat f) t) e) (\lambda (_: (eq nat x2 O)).(drop_drop -(Flat f) n0 c e (H12 x1 v H8 e H14) t)) (\lambda (H15: (ex2 nat (\lambda (m: -nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda -(m: nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O -(CHead c (Flat f) t) e) (\lambda (x: nat).(\lambda (_: (eq nat x2 (S -x))).(\lambda (_: (lt x n0)).(drop_drop (Flat f) n0 c e (H12 x1 v H8 e H14) -t)))) H15)) (lt_gen_xS x2 n0 H13)))))) k H10 H11 (drop_gen_drop k x1 e x0 n0 -H9)))))))))))) H4)) (csubst0_gen_head k c c2 t v i H2))))))))))) c1)))))) n). -(* COMMENTS -Initial nodes: 2989 -END *) - -theorem csubst0_drop_lt: - \forall (n: nat).(\forall (i: nat).((lt n i) \to (\forall (c1: C).(\forall -(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n O -c1 e) \to (or4 (drop n O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: -K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e0 k -w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (s k n)) v u w)))))) (ex3_4 K C C T (\lambda (k: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k -u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop n O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1 e2)))))) (ex4_5 K C C -T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda -(_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e2 k w))))))) -(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (s k n)) v u w)))))) (\lambda (k: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k -n)) v e1 e2)))))))))))))))) -\def - \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (i: nat).((lt n0 i) -\to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) -\to (\forall (e: C).((drop n0 O c1 e) \to (or4 (drop n0 O c2 e) (ex3_4 K C T -T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda -(w: T).(drop n0 O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n0)) v u w)))))) -(ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop n0 O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k n0)) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 -O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n0)) v u w)))))) -(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus i (s k n0)) v e1 e2))))))))))))))))) (\lambda (i: -nat).(\lambda (_: (lt O i)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (v: -T).(\lambda (H0: (csubst0 i v c1 c2)).(\lambda (e: C).(\lambda (H1: (drop O O -c1 e)).(eq_ind C c1 (\lambda (c: C).(or4 (drop O O c2 c) (ex3_4 K C T T -(\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c -(CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda -(w: T).(drop O O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k O)) v u w)))))) -(ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop O O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k O)) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 k u))))))) (\lambda (k: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O -c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k O)) v u w)))))) -(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus i (s k O)) v e1 e2))))))))) (csubst0_ind (\lambda (n0: -nat).(\lambda (t: T).(\lambda (c: C).(\lambda (c0: C).(or4 (drop O O c0 c) -(ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: -T).(eq C c (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop O O c0 (CHead e0 k w)))))) (\lambda (k: K).(\lambda -(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n0 (s k O)) t u w)))))) -(ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop O O c0 (CHead e2 k u)))))) (\lambda (k: K).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n0 (s k O)) t e1 -e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 k u))))))) (\lambda (k: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O -c0 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n0 (s k O)) t u w)))))) -(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus n0 (s k O)) t e1 e2)))))))))))) (\lambda (k: -K).(\lambda (i0: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: -T).(\lambda (H2: (subst0 i0 v0 u1 u2)).(\lambda (c: C).(let H3 \def (eq_ind_r -nat i0 (\lambda (n0: nat).(subst0 n0 v0 u1 u2)) H2 (minus (s k i0) (s k O)) -(s_arith0 k i0)) in (or4_intro1 (drop O O (CHead c k u2) (CHead c k u1)) -(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: -T).(eq C (CHead c k u1) (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c k u2) (CHead e0 k0 -w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus (s k i0) (s k0 O)) v0 u w)))))) (ex3_4 K C C T (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c k u1) -(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop O O (CHead c k u2) (CHead e2 k0 u)))))) (\lambda -(k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s -k i0) (s k0 O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c k u1) -(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c k u2) (CHead e2 k0 -w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 O)) v0 u w)))))) (\lambda -(k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: -T).(csubst0 (minus (s k i0) (s k0 O)) v0 e1 e2))))))) (ex3_4_intro K C T T -(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C -(CHead c k u1) (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c k u2) (CHead e0 k0 -w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus (s k i0) (s k0 O)) v0 u w))))) k c u1 u2 (refl_equal C -(CHead c k u1)) (drop_refl (CHead c k u2)) H3)))))))))) (\lambda (k: -K).(\lambda (i0: nat).(\lambda (c3: C).(\lambda (c4: C).(\lambda (v0: -T).(\lambda (H2: (csubst0 i0 v0 c3 c4)).(\lambda (H3: (or4 (drop O O c4 c3) -(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: -T).(eq C c3 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda -(_: T).(\lambda (w: T).(drop O O c4 (CHead e0 k0 w)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k0 -O)) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u: T).(eq C c3 (CHead e1 k0 u)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O c4 (CHead e2 k0 -u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus i0 (s k0 O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 -(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e2 k0 w))))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i0 (s k0 O)) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k0 -O)) v0 e1 e2))))))))).(\lambda (u: T).(let H4 \def (eq_ind_r nat i0 (\lambda -(n0: nat).(csubst0 n0 v0 c3 c4)) H2 (minus (s k i0) (s k O)) (s_arith0 k i0)) -in (let H5 \def (eq_ind_r nat i0 (\lambda (n0: nat).(or4 (drop O O c4 c3) -(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda -(_: T).(eq C c3 (CHead e0 k0 u0)))))) (\lambda (k0: K).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e0 k0 w)))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus n0 (s -k0 O)) v0 u0 w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u0: T).(eq C c3 (CHead e1 k0 u0)))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop O O c4 (CHead -e2 k0 u0)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 (minus n0 (s k0 O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq -C c3 (CHead e1 k0 u0))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e2 k0 w))))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda -(w: T).(subst0 (minus n0 (s k0 O)) v0 u0 w)))))) (\lambda (k0: K).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n0 -(s k0 O)) v0 e1 e2))))))))) H3 (minus (s k i0) (s k O)) (s_arith0 k i0)) in -(or4_intro2 (drop O O (CHead c4 k u) (CHead c3 k u)) (ex3_4 K C T T (\lambda -(k0: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 k -u) (CHead e0 k0 u0)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop O O (CHead c4 k u) (CHead e0 k0 w)))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k -i0) (s k0 O)) v0 u0 w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead c3 k u) (CHead e1 k0 -u0)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: -T).(drop O O (CHead c4 k u) (CHead e2 k0 u0)))))) (\lambda (k0: K).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 O)) -v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 k u) (CHead e1 k0 -u0))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop O O (CHead c4 k u) (CHead e2 k0 w))))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: -T).(subst0 (minus (s k i0) (s k0 O)) v0 u0 w)))))) (\lambda (k0: K).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k -i0) (s k0 O)) v0 e1 e2))))))) (ex3_4_intro K C C T (\lambda (k0: K).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead c3 k u) (CHead e1 k0 -u0)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: -T).(drop O O (CHead c4 k u) (CHead e2 k0 u0)))))) (\lambda (k0: K).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 O)) -v0 e1 e2))))) k c3 c4 u (refl_equal C (CHead c3 k u)) (drop_refl (CHead c4 k -u)) H4)))))))))))) (\lambda (k: K).(\lambda (i0: nat).(\lambda (v0: -T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 i0 v0 u1 -u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (H3: (csubst0 i0 v0 c3 -c4)).(\lambda (_: (or4 (drop O O c4 c3) (ex3_4 K C T T (\lambda (k0: -K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e0 k0 -u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(drop O O c4 (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k0 O)) v0 u w)))))) -(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c3 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u: T).(drop O O c4 (CHead e2 k0 u)))))) (\lambda (k0: -K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (s k0 -O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 k0 -u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop O O c4 (CHead e2 k0 w))))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus i0 (s k0 O)) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k0 O)) v0 e1 -e2))))))))).(let H5 \def (eq_ind_r nat i0 (\lambda (n0: nat).(subst0 n0 v0 u1 -u2)) H2 (minus (s k i0) (s k O)) (s_arith0 k i0)) in (let H6 \def (eq_ind_r -nat i0 (\lambda (n0: nat).(csubst0 n0 v0 c3 c4)) H3 (minus (s k i0) (s k O)) -(s_arith0 k i0)) in (or4_intro3 (drop O O (CHead c4 k u2) (CHead c3 k u1)) -(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: -T).(eq C (CHead c3 k u1) (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 k u2) (CHead e0 k0 -w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus (s k i0) (s k0 O)) v0 u w)))))) (ex3_4 K C C T (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c3 k -u1) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop O O (CHead c4 k u2) (CHead e2 k0 u)))))) (\lambda -(k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s -k i0) (s k0 O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 k u1) -(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 k u2) (CHead e2 k0 -w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 O)) v0 u w)))))) (\lambda -(k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: -T).(csubst0 (minus (s k i0) (s k0 O)) v0 e1 e2))))))) (ex4_5_intro K C C T T -(\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda -(_: T).(eq C (CHead c3 k u1) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 k -u2) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 O)) v0 u -w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 O)) v0 e1 e2)))))) k c3 c4 -u1 u2 (refl_equal C (CHead c3 k u1)) (drop_refl (CHead c4 k u2)) H5 -H6)))))))))))))) i v c1 c2 H0) e (drop_gen_refl c1 e H1)))))))))) (\lambda -(n0: nat).(\lambda (IHn: ((\forall (i: nat).((lt n0 i) \to (\forall (c1: -C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: -C).((drop n0 O c1 e) \to (or4 (drop n0 O c2 e) (ex3_4 K C T T (\lambda (k: -K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k -u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(drop n0 O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda -(u: T).(\lambda (w: T).(subst0 (minus i (s k n0)) v u w)))))) (ex3_4 K C C T -(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda -(u: T).(drop n0 O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k n0)) v e1 e2)))))) -(ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c2 (CHead -e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus i (s k n0)) v u w)))))) (\lambda (k: -K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -(minus i (s k n0)) v e1 e2)))))))))))))))))).(\lambda (i: nat).(\lambda (H: -(lt (S n0) i)).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: -C).(\forall (v: T).((csubst0 i v c c2) \to (\forall (e: C).((drop (S n0) O c -e) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: -K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead -e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k -u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w)))))) -(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus i (s k (S n0))) v e1 e2)))))))))))))) (\lambda (n1: -nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (_: (csubst0 i v (CSort n1) -c2)).(\lambda (e: C).(\lambda (H1: (drop (S n0) O (CSort n1) e)).(and3_ind -(eq C e (CSort n1)) (eq nat (S n0) O) (eq nat O O) (or4 (drop (S n0) O c2 e) -(ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: -T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: -K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S -n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda -(_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) -(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 -(minus i (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) -(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k (S -n0))) v e1 e2)))))))) (\lambda (H2: (eq C e (CSort n1))).(\lambda (H3: (eq -nat (S n0) O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n1) (\lambda (c: -C).(or4 (drop (S n0) O c2 c) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 k u)))))) (\lambda (k: -K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead -e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 k -u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 k u))))))) (\lambda (k: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w)))))) -(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus i (s k (S n0))) v e1 e2))))))))) (let H5 \def (eq_ind -nat (S n0) (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) -with [O \Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind -(or4 (drop (S n0) O c2 (CSort n1)) (ex3_4 K C T T (\lambda (k: K).(\lambda -(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CSort n1) (CHead e0 k u)))))) -(\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) -O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w)))))) (ex3_4 K C C T -(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CSort -n1) (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: -K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k -(S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CSort n1) (CHead e1 -k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k: -K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus i (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1 -e2)))))))) H5)) e H2)))) (drop_gen_sort n1 (S n0) O e H1)))))))) (\lambda (c: -C).(\lambda (H0: ((\forall (c2: C).(\forall (v: T).((csubst0 i v c c2) \to -(\forall (e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C -T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda -(w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w)))))) -(ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: -K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k -(S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k -u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k: -K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus i (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1 -e2))))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda -(v: T).(\lambda (H1: (csubst0 i v (CHead c k t) c2)).(\lambda (e: C).(\lambda -(H2: (drop (S n0) O (CHead c k t) e)).(or3_ind (ex3_2 T nat (\lambda (_: -T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: -nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j -v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k -j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda -(c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_: -T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: -T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda -(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: -T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))) (or4 (drop (S -n0) O c2 e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda -(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k0 -w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 -u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O c2 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k0 (S n0))) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O c2 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S -n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 (minus i (s k0 (S n0))) v e1 e2)))))))) -(\lambda (H3: (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k -j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda -(u2: T).(\lambda (j: nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: -T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: -nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j -v t u2))) (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k0: K).(\lambda -(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda -(k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 -(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (ex3_4 K C C T -(\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k0 u)))))) (\lambda (k0: -K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k0 -(S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 -u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k0 w))))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus i (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k0 -(S n0))) v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H4: -(eq nat i (s k x1))).(\lambda (H5: (eq C c2 (CHead c k x0))).(\lambda (_: -(subst0 x1 v t x0)).(eq_ind_r C (CHead c k x0) (\lambda (c0: C).(or4 (drop (S -n0) O c0 e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda -(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e0 k0 -w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 -u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O c0 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k0 (S n0))) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O c0 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S -n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 (minus i (s k0 (S n0))) v e1 e2))))))))) (let -H7 \def (eq_ind nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0: -T).((csubst0 n1 v0 c c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (or4 -(drop (S n0) O c3 e0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c3 -(CHead e1 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus n1 (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C -T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e0 -(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O c3 (CHead e2 k0 u)))))) (\lambda (k0: -K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n1 (s k0 -(S n0))) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 -u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus n1 (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 (s k0 -(S n0))) v0 e1 e2)))))))))))))) H0 (s k x1) H4) in (let H8 \def (eq_ind nat i -(\lambda (n1: nat).(lt (S n0) n1)) H (s k x1) H4) in (eq_ind_r nat (s k x1) -(\lambda (n1: nat).(or4 (drop (S n0) O (CHead c k x0) e) (ex3_4 K C T T -(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead c k x0) (CHead e0 k0 w)))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus n1 (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O -(CHead c k x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n1 (s k0 (S n0))) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead c k x0) (CHead e2 k0 w))))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus n1 (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 (s k0 -(S n0))) v e1 e2))))))))) (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to -(((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c c3) \to (\forall -(e0: C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T -(\lambda (k1: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 -(CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k1 w)))))) (\lambda (k1: -K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 x1) -(s k1 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k1 u)))))) (\lambda -(k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 -(CHead e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus (s k0 x1) (s k1 (S n0))) v0 e1 e2)))))) -(ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq C e0 (CHead e1 k1 u))))))) (\lambda (k1: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O c3 (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 x1) (s k1 (S n0))) v0 -u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (minus (s k0 x1) (s k1 (S n0))) v0 e1 -e2)))))))))))))) \to ((lt (S n0) (s k0 x1)) \to (or4 (drop (S n0) O (CHead c -k0 x0) e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda -(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c k0 x0) (CHead -e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus (s k0 x1) (s k1 (S n0))) v u w)))))) (ex3_4 K C C T -(\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O (CHead c k0 x0) (CHead e2 k1 u)))))) -(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 -(minus (s k0 x1) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c k0 x0) (CHead e2 -k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus (s k0 x1) (s k1 (S n0))) v u w)))))) -(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus (s k0 x1) (s k1 (S n0))) v e1 e2)))))))))))) (\lambda -(b: B).(\lambda (H9: (drop (r (Bind b) n0) O c e)).(\lambda (_: ((\forall -(c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to (\forall (e0: -C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T -(\lambda (k0: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 -(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) -x1) (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 -(CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v0 e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u))))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) -x1) (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 -(S n0))) v0 e1 e2))))))))))))))).(\lambda (_: (lt (S n0) (s (Bind b) -x1))).(or4_intro0 (drop (S n0) O (CHead c (Bind b) x0) e) (ex3_4 K C T T -(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead c (Bind b) x0) (CHead e0 k0 w)))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 -u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O (CHead c (Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0: -K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind -b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c (Bind b) x0) -(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S -n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1 -e2))))))) (drop_drop (Bind b) n0 c e H9 x0)))))) (\lambda (f: F).(\lambda -(H9: (drop (r (Flat f) n0) O c e)).(\lambda (_: ((\forall (c3: C).(\forall -(v0: T).((csubst0 (s (Flat f) x1) v0 c c3) \to (\forall (e0: C).((drop (S n0) -O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T (\lambda (k0: -K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 -u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S -n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 (CHead -e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v0 e1 e2)))))) (ex4_5 K C C -T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c3 -(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S -n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S -n0))) v0 e1 e2))))))))))))))).(\lambda (_: (lt (S n0) (s (Flat f) -x1))).(or4_intro0 (drop (S n0) O (CHead c (Flat f) x0) e) (ex3_4 K C T T -(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead c (Flat f) x0) (CHead e0 k0 w)))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 -u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O (CHead c (Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: -K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat -f) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c (Flat f) x0) -(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S -n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1 -e2))))))) (drop_drop (Flat f) n0 c e H9 x0)))))) k (drop_gen_drop k c e t n0 -H2) H7 H8) i H4))) c2 H5)))))) H3)) (\lambda (H3: (ex3_2 C nat (\lambda (_: -C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_: -nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j -v c c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k -j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda -(c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (or4 (drop (S n0) O c2 e) -(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: -T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda -(_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k0 w)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S -n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda -(_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k0 -u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus i (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq -C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k0 w))))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k0 -(S n0))) v e1 e2)))))))) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H4: -(eq nat i (s k x1))).(\lambda (H5: (eq C c2 (CHead x0 k t))).(\lambda (H6: -(csubst0 x1 v c x0)).(eq_ind_r C (CHead x0 k t) (\lambda (c0: C).(or4 (drop -(S n0) O c0 e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda -(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e0 k0 -w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 -u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O c0 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k0 (S n0))) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O c0 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S -n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 (minus i (s k0 (S n0))) v e1 e2))))))))) (let -H7 \def (eq_ind nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0: -T).((csubst0 n1 v0 c c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (or4 -(drop (S n0) O c3 e0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c3 -(CHead e1 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus n1 (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C -T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e0 -(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O c3 (CHead e2 k0 u)))))) (\lambda (k0: -K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n1 (s k0 -(S n0))) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 -u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus n1 (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 (s k0 -(S n0))) v0 e1 e2)))))))))))))) H0 (s k x1) H4) in (let H8 \def (eq_ind nat i -(\lambda (n1: nat).(lt (S n0) n1)) H (s k x1) H4) in (eq_ind_r nat (s k x1) -(\lambda (n1: nat).(or4 (drop (S n0) O (CHead x0 k t) e) (ex3_4 K C T T -(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x0 k t) (CHead e0 k0 w)))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus n1 (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O -(CHead x0 k t) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n1 (s k0 (S n0))) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x0 k t) (CHead e2 k0 w))))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus n1 (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 (s k0 -(S n0))) v e1 e2))))))))) (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to -(((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c c3) \to (\forall -(e0: C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T -(\lambda (k1: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 -(CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k1 w)))))) (\lambda (k1: -K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 x1) -(s k1 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k1 u)))))) (\lambda -(k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 -(CHead e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus (s k0 x1) (s k1 (S n0))) v0 e1 e2)))))) -(ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq C e0 (CHead e1 k1 u))))))) (\lambda (k1: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O c3 (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 x1) (s k1 (S n0))) v0 -u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (minus (s k0 x1) (s k1 (S n0))) v0 e1 -e2)))))))))))))) \to ((lt (S n0) (s k0 x1)) \to (or4 (drop (S n0) O (CHead x0 -k0 t) e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda -(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 k0 t) (CHead -e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus (s k0 x1) (s k1 (S n0))) v u w)))))) (ex3_4 K C C T -(\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O (CHead x0 k0 t) (CHead e2 k1 u)))))) -(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 -(minus (s k0 x1) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 k0 t) (CHead e2 -k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus (s k0 x1) (s k1 (S n0))) v u w)))))) -(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus (s k0 x1) (s k1 (S n0))) v e1 e2)))))))))))) (\lambda -(b: B).(\lambda (H9: (drop (r (Bind b) n0) O c e)).(\lambda (_: ((\forall -(c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to (\forall (e0: -C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T -(\lambda (k0: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 -(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) -x1) (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 -(CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v0 e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u))))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) -x1) (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 -(S n0))) v0 e1 e2))))))))))))))).(\lambda (H11: (lt (S n0) (s (Bind b) -x1))).(let H12 \def (IHn x1 (le_S_n (S n0) x1 H11) c x0 v H6 e H9) in -(or4_ind (drop n0 O x0 e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: -K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x0 (CHead e0 -k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus x1 (s k0 n0)) v u w)))))) (ex3_4 K C C T (\lambda (k0: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 -u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop n0 O x0 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus x1 (s k0 n0)) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop n0 O x0 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x1 (s k0 -n0)) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 (minus x1 (s k0 n0)) v e1 e2))))))) (or4 -(drop (S n0) O (CHead x0 (Bind b) t) e) (ex3_4 K C T T (\lambda (k0: -K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 -u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 k0 w)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) -x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 -(Bind b) t) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s -(Bind b) x1) (s k0 (S n0))) v e1 e2)))))))) (\lambda (H13: (drop n0 O x0 -e)).(or4_intro0 (drop (S n0) O (CHead x0 (Bind b) t) e) (ex3_4 K C T T -(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: 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(k0: K).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda -(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) -(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) -(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 -u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C -T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead -x0 (Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0: 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(\lambda (k0: K).(\lambda (_: C).(\lambda -(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 -(S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S -n0))) v e1 e2))))))) (ex3_4_intro K C T T (\lambda (k0: K).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x4) (CHead e0 k0 -u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 k0 w)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) -x1) (s k0 (S n0))) v u w))))) x2 x3 x4 x5 (refl_equal C (CHead x3 x2 x4)) -(drop_drop (Bind b) n0 x0 (CHead x3 x2 x5) H15 t) (eq_ind_r nat (S (s x2 n0)) -(\lambda (n1: nat).(subst0 (minus (s (Bind b) x1) n1) v x4 x5)) H16 (s x2 (S -n0)) (s_S x2 n0)))) e H14)))))))) H13)) (\lambda (H13: (ex3_4 K C C T -(\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 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T).(\lambda (H14: (eq C e (CHead x3 x2 x5))).(\lambda (H15: -(drop n0 O x0 (CHead x4 x2 x5))).(\lambda (H16: (csubst0 (minus x1 (s x2 n0)) -v x3 x4)).(eq_ind_r C (CHead x3 x2 x5) (\lambda (c0: C).(or4 (drop (S n0) O -(CHead x0 (Bind b) t) c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 k0 u)))))) (\lambda -(k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O -(CHead x0 (Bind b) t) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S -n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u: T).(eq C c0 (CHead e1 k0 u)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 -(Bind b) t) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 k0 u))))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s -(Bind b) x1) (s k0 (S n0))) v e1 e2))))))))) (or4_intro2 (drop (S n0) O -(CHead x0 (Bind b) t) (CHead x3 x2 x5)) (ex3_4 K C T T (\lambda (k0: -K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x5) -(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 k0 w)))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 x2 -x5) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 u)))))) -(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 -(minus (s (Bind b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq -C (CHead x3 x2 x5) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 -(Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda -(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 -(S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S -n0))) v e1 e2))))))) (ex3_4_intro K C C T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 x2 x5) (CHead e1 k0 -u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 u)))))) (\lambda (k0: -K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind -b) x1) (s k0 (S n0))) v e1 e2))))) x2 x3 x4 x5 (refl_equal C (CHead x3 x2 -x5)) (drop_drop (Bind b) n0 x0 (CHead x4 x2 x5) H15 t) (eq_ind_r nat (S (s x2 -n0)) (\lambda (n1: nat).(csubst0 (minus (s (Bind b) x1) n1) v x3 x4)) H16 (s -x2 (S n0)) (s_S x2 n0)))) e H14)))))))) H13)) (\lambda (H13: (ex4_5 K C C T T -(\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda -(_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x0 (CHead e2 -k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus x1 (s k0 n0)) v u w)))))) (\lambda (k0: -K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -(minus x1 (s k0 n0)) v e1 e2)))))))).(ex4_5_ind K C C T T (\lambda (k0: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x0 (CHead e2 k0 w))))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus x1 (s k0 n0)) v u w)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus x1 (s k0 -n0)) v e1 e2)))))) (or4 (drop (S n0) O (CHead x0 (Bind b) t) e) (ex3_4 K C T -T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 k0 w)))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus (s (Bind b) x1) (s k0 (S n0))) 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H12)))))) (\lambda (f: -F).(\lambda (H9: (drop (r (Flat f) n0) O c e)).(\lambda (H10: ((\forall (c3: -C).(\forall (v0: T).((csubst0 (s (Flat f) x1) v0 c c3) \to (\forall (e0: -C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T -(\lambda (k0: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 -(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) -x1) (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 -(CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v0 e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 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(CHead x3 x2 x4)) (drop_drop (Flat f) n0 x0 (CHead -x3 x2 x5) H15 t) H16)) e H14)))))))) H13)) (\lambda (H13: (ex3_4 K C C T -(\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O x0 (CHead e2 k0 u)))))) (\lambda (k0: -K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus x1 (s k0 -(S n0))) v e1 e2))))))).(ex3_4_ind K C C T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O x0 (CHead -e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus x1 (s k0 (S n0))) v e1 e2))))) (or4 (drop (S n0) O (CHead -x0 (Flat f) t) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda -(u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda -(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) -(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) -(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 -u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C -T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead -x0 (Flat f) t) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) -x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda 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(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S -n0))) v e1 e2))))))) (ex3_4_intro K C C T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 x2 x5) (CHead e1 k0 -u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 u)))))) (\lambda (k0: -K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat -f) x1) (s k0 (S n0))) v e1 e2))))) x2 x3 x4 x5 (refl_equal C (CHead x3 x2 -x5)) (drop_drop (Flat f) n0 x0 (CHead x4 x2 x5) H15 t) H16)) e H14)))))))) -H13)) (\lambda (H13: (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 -u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O x0 (CHead e2 k0 w))))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda 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(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S -n0))) v e1 e2))))))) (ex4_5_intro K C C T T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x5) -(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) -(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S -n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1 -e2)))))) x2 x3 x4 x5 x6 (refl_equal C (CHead x3 x2 x5)) (drop_drop (Flat f) -n0 x0 (CHead x4 x2 x6) H15 t) H16 H17)) e H14)))))))))) H13)) H12)))))) k -(drop_gen_drop k c e t n0 H2) H7 H8) i H4))) c2 H5)))))) H3)) (\lambda (H3: -(ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s -k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead -c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t -u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c -c3)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: -nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: -nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda -(j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: -nat).(csubst0 j v c c3)))) (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda -(k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 -u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O c2 (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S n0))) v u -w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k0 u)))))) -(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 -(minus i (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k0 w))))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k0 -(S n0))) v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: C).(\lambda (x2: -nat).(\lambda (H4: (eq nat i (s k x2))).(\lambda (H5: (eq C c2 (CHead x1 k -x0))).(\lambda (_: (subst0 x2 v t x0)).(\lambda (H7: (csubst0 x2 v c -x1)).(eq_ind_r C (CHead x1 k x0) (\lambda (c0: C).(or4 (drop (S n0) O c0 e) -(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: -T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda -(_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e0 k0 w)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S -n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda -(_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c0 (CHead e2 k0 -u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus i (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq -C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 k0 w))))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k0 -(S n0))) v e1 e2))))))))) (let H8 \def (eq_ind nat i (\lambda (n1: -nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall -(e0: C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T -(\lambda (k0: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 -(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n1 (s k0 (S -n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 (CHead -e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus n1 (s k0 (S n0))) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq -C e0 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c3 (CHead e2 k0 w))))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus n1 (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 -(s k0 (S n0))) v0 e1 e2)))))))))))))) H0 (s k x2) H4) in (let H9 \def (eq_ind -nat i (\lambda (n1: nat).(lt (S n0) n1)) H (s k x2) H4) in (eq_ind_r nat (s k -x2) (\lambda (n1: nat).(or4 (drop (S n0) O (CHead x1 k x0) e) (ex3_4 K C T T -(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 k x0) (CHead e0 k0 w)))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus n1 (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O -(CHead x1 k x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n1 (s k0 (S n0))) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x1 k x0) (CHead e2 k0 w))))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus n1 (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 (s k0 -(S n0))) v e1 e2))))))))) (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to -(((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x2) v0 c c3) \to (\forall -(e0: C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T -(\lambda (k1: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 -(CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k1 w)))))) (\lambda (k1: -K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 x2) -(s k1 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k1 u)))))) (\lambda -(k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 -(CHead e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus (s k0 x2) (s k1 (S n0))) v0 e1 e2)))))) -(ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq C e0 (CHead e1 k1 u))))))) (\lambda (k1: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O c3 (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 x2) (s k1 (S n0))) v0 -u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (minus (s k0 x2) (s k1 (S n0))) v0 e1 -e2)))))))))))))) \to ((lt (S n0) (s k0 x2)) \to (or4 (drop (S n0) O (CHead x1 -k0 x0) e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda -(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 k0 x0) -(CHead e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus (s k0 x2) (s k1 (S n0))) v u w)))))) (ex3_4 -K C C T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq -C e (CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O (CHead x1 k0 x0) (CHead e2 k1 u)))))) -(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 -(minus (s k0 x2) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 k0 x0) (CHead e2 -k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus (s k0 x2) (s k1 (S n0))) v u w)))))) -(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus (s k0 x2) (s k1 (S n0))) v e1 e2)))))))))))) (\lambda -(b: B).(\lambda (H10: (drop (r (Bind b) n0) O c e)).(\lambda (_: ((\forall -(c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x2) v0 c c3) \to (\forall (e0: -C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T -(\lambda (k0: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 -(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) -x2) (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 -(CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v0 e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u))))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) -x2) (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 -(S n0))) v0 e1 e2))))))))))))))).(\lambda (H12: (lt (S n0) (s (Bind b) -x2))).(let H13 \def (IHn x2 (le_S_n (S n0) x2 H12) c x1 v H7 e H10) in -(or4_ind (drop n0 O x1 e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: -K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x1 (CHead e0 -k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus x2 (s k0 n0)) v u w)))))) (ex3_4 K C C T (\lambda (k0: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 -u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop n0 O x1 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus x2 (s k0 n0)) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop n0 O x1 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x2 (s k0 -n0)) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 (minus x2 (s k0 n0)) v e1 e2))))))) (or4 -(drop (S n0) O (CHead x1 (Bind b) x0) e) (ex3_4 K C T T (\lambda (k0: -K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 -u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) -x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 -(Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s -(Bind b) x2) (s k0 (S n0))) v e1 e2)))))))) (\lambda (H14: (drop n0 O x1 -e)).(or4_intro0 (drop (S n0) O (CHead x1 (Bind b) x0) e) (ex3_4 K C T T -(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 -u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0: -K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind -b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) -(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S -n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 -e2))))))) (drop_drop (Bind b) n0 x1 e H14 x0))) (\lambda (H14: (ex3_4 K C T T -(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop n0 O x1 (CHead e0 k0 w)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x2 (s k0 -n0)) v u w))))))).(ex3_4_ind K C T T (\lambda (k0: K).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: -K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x1 (CHead e0 -k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus x2 (s k0 n0)) v u w))))) (or4 (drop (S n0) O (CHead x1 -(Bind b) x0) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda -(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) -(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) -(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 -u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C -T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead -x1 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) -x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 -(S n0))) v e1 e2)))))))) (\lambda (x3: K).(\lambda (x4: C).(\lambda (x5: -T).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x4 x3 x5))).(\lambda (H16: -(drop n0 O x1 (CHead x4 x3 x6))).(\lambda (H17: (subst0 (minus x2 (s x3 n0)) -v x5 x6)).(eq_ind_r C (CHead x4 x3 x5) (\lambda (c0: C).(or4 (drop (S n0) O -(CHead x1 (Bind b) x0) c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 k0 u)))))) (\lambda -(k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O -(CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S -n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u: T).(eq C c0 (CHead e1 k0 u)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 -(Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 k0 u))))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s -(Bind b) x2) (s k0 (S n0))) v e1 e2))))))))) (or4_intro1 (drop (S n0) O -(CHead x1 (Bind b) x0) (CHead x4 x3 x5)) (ex3_4 K C T T (\lambda (k0: -K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x5) -(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x4 x3 -x5) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 u)))))) -(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 -(minus (s (Bind b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq -C (CHead x4 x3 x5) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 -(Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda -(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 -(S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S -n0))) v e1 e2))))))) (ex3_4_intro K C T T (\lambda (k0: K).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x5) (CHead e0 k0 -u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) -x2) (s k0 (S n0))) v u w))))) x3 x4 x5 x6 (refl_equal C (CHead x4 x3 x5)) -(drop_drop (Bind b) n0 x1 (CHead x4 x3 x6) H16 x0) (eq_ind_r nat (S (s x3 -n0)) (\lambda (n1: nat).(subst0 (minus (s (Bind b) x2) n1) v x5 x6)) H17 (s -x3 (S n0)) (s_S x3 n0)))) e H15)))))))) H14)) (\lambda (H14: (ex3_4 K C C T -(\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop n0 O x1 (CHead e2 k0 u)))))) (\lambda (k0: -K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus x2 (s k0 -n0)) v e1 e2))))))).(ex3_4_ind K C C T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O x1 (CHead e2 -k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus x2 (s k0 n0)) v e1 e2))))) (or4 (drop (S n0) O (CHead x1 -(Bind b) x0) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda -(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) -(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) -(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 -u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C -T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead -x1 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) -x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 -(S n0))) v e1 e2)))))))) (\lambda (x3: K).(\lambda (x4: C).(\lambda (x5: -C).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x4 x3 x6))).(\lambda (H16: -(drop n0 O x1 (CHead x5 x3 x6))).(\lambda (H17: (csubst0 (minus x2 (s x3 n0)) -v x4 x5)).(eq_ind_r C (CHead x4 x3 x6) (\lambda (c0: C).(or4 (drop (S n0) O -(CHead x1 (Bind b) x0) c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 k0 u)))))) (\lambda -(k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O -(CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S -n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u: T).(eq C c0 (CHead e1 k0 u)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 -(Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 k0 u))))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s -(Bind b) x2) (s k0 (S n0))) v e1 e2))))))))) (or4_intro2 (drop (S n0) O -(CHead x1 (Bind b) x0) (CHead x4 x3 x6)) (ex3_4 K C T T (\lambda (k0: -K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6) -(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x4 x3 -x6) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 u)))))) -(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 -(minus (s (Bind b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq -C (CHead x4 x3 x6) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 -(Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda -(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 -(S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S -n0))) v e1 e2))))))) (ex3_4_intro K C C T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x4 x3 x6) (CHead e1 k0 -u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0: -K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind -b) x2) (s k0 (S n0))) v e1 e2))))) x3 x4 x5 x6 (refl_equal C (CHead x4 x3 -x6)) (drop_drop (Bind b) n0 x1 (CHead x5 x3 x6) H16 x0) (eq_ind_r nat (S (s -x3 n0)) (\lambda (n1: nat).(csubst0 (minus (s (Bind b) x2) n1) v x4 x5)) H17 -(s x3 (S n0)) (s_S x3 n0)))) e H15)))))))) H14)) (\lambda (H14: (ex4_5 K C C -T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x1 (CHead -e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus x2 (s k0 n0)) v u w)))))) (\lambda (k0: -K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -(minus x2 (s k0 n0)) v e1 e2)))))))).(ex4_5_ind K C C T T (\lambda (k0: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x1 (CHead e2 k0 w))))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus x2 (s k0 n0)) v u w)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus x2 (s k0 -n0)) v e1 e2)))))) (or4 (drop (S n0) O (CHead x1 (Bind b) x0) e) (ex3_4 K C T -T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 -u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0: -K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind -b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) -(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S -n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 -e2)))))))) (\lambda (x3: K).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: -T).(\lambda (x7: T).(\lambda (H15: (eq C e (CHead x4 x3 x6))).(\lambda (H16: -(drop n0 O x1 (CHead x5 x3 x7))).(\lambda (H17: (subst0 (minus x2 (s x3 n0)) -v x6 x7)).(\lambda (H18: (csubst0 (minus x2 (s x3 n0)) v x4 x5)).(eq_ind_r C -(CHead x4 x3 x6) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Bind b) x0) -c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda -(_: T).(eq C c0 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) -(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) -(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c0 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 -u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C -T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(eq C c0 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead -x1 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) -x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 -(S n0))) v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead x1 (Bind b) x0) -(CHead x4 x3 x6)) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda -(u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6) (CHead e0 k0 u)))))) (\lambda -(k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O -(CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S -n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u: T).(eq C (CHead x4 x3 x6) (CHead e1 k0 u)))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O -(CHead x1 (Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S -n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6) -(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) -(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S -n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 -e2))))))) (ex4_5_intro K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6) (CHead e1 k0 -u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 -w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) -(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 e2)))))) x3 x4 x5 -x6 x7 (refl_equal C (CHead x4 x3 x6)) (drop_drop (Bind b) n0 x1 (CHead x5 x3 -x7) H16 x0) (eq_ind_r nat (S (s x3 n0)) (\lambda (n1: nat).(subst0 (minus (s -(Bind b) x2) n1) v x6 x7)) H17 (s x3 (S n0)) (s_S x3 n0)) (eq_ind_r nat (S (s -x3 n0)) (\lambda (n1: nat).(csubst0 (minus (s (Bind b) x2) n1) v x4 x5)) H18 -(s x3 (S n0)) (s_S x3 n0)))) e H15)))))))))) H14)) H13)))))) (\lambda (f: -F).(\lambda (H10: (drop (r (Flat f) n0) O c e)).(\lambda (H11: ((\forall (c3: -C).(\forall (v0: T).((csubst0 (s (Flat f) x2) v0 c c3) \to (\forall (e0: -C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T -(\lambda (k0: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 -(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) -x2) (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 -(CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v0 e1 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w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda -(_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O x1 (CHead e2 k0 -u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus x2 (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq -C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x1 (CHead e2 k0 w))))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus x2 (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus x2 -(s k0 (S n0))) v e1 e2))))))) (or4 (drop (S n0) O (CHead x1 (Flat f) x0) e) -(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: -T).(eq C e 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(u: T).(eq C c0 (CHead e1 k0 u)))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O -(CHead x1 (Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S -n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 k0 -u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 -w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) -(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2))))))))) -(or4_intro2 (drop (S n0) O (CHead x1 (Flat f) x0) (CHead x4 x3 x6)) (ex3_4 K -C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C -(CHead x4 x3 x6) (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) -(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) -(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C (CHead x4 x3 x6) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0) -(CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6) (CHead e1 k0 -u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 -w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) -(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2))))))) -(ex3_4_intro K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C (CHead x4 x3 x6) (CHead e1 k0 u)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 -(Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 -e2))))) x3 x4 x5 x6 (refl_equal C (CHead x4 x3 x6)) (drop_drop (Flat f) n0 x1 -(CHead x5 x3 x6) H16 x0) H17)) e H15)))))))) H14)) (\lambda (H14: (ex4_5 K C -C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x1 -(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x2 (s k0 (S n0))) v u -w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (minus x2 (s k0 (S n0))) v e1 -e2)))))))).(ex4_5_ind K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O x1 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x2 (s k0 (S -n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 (minus x2 (s k0 (S n0))) v e1 e2)))))) (or4 -(drop (S n0) O (CHead x1 (Flat f) x0) e) (ex3_4 K C T T (\lambda (k0: -K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 -u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 k0 w)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) -x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 -(Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda -(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 w))))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s -(Flat f) x2) (s k0 (S n0))) v e1 e2)))))))) (\lambda (x3: K).(\lambda (x4: -C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (x7: T).(\lambda (H15: (eq C e -(CHead x4 x3 x6))).(\lambda (H16: (drop (S n0) O x1 (CHead x5 x3 -x7))).(\lambda (H17: (subst0 (minus x2 (s x3 (S n0))) v x6 x7)).(\lambda -(H18: (csubst0 (minus x2 (s x3 (S n0))) v x4 x5)).(eq_ind_r C (CHead x4 x3 -x6) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Flat f) x0) c0) (ex3_4 K -C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C -c0 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 k0 w)))))) -(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda -(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 -k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: -K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat -f) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 -(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) -(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S -n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 -e2))))))))) (or4_intro3 (drop (S n0) O (CHead x1 (Flat f) x0) (CHead x4 x3 -x6)) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C (CHead x4 x3 x6) (CHead e0 k0 u)))))) (\lambda (k0: -K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 -(Flat f) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda -(u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u -w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C (CHead x4 x3 x6) (CHead e1 k0 u)))))) (\lambda (k0: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 -(Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6) (CHead e1 k0 -u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 -w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) -(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2))))))) -(ex4_5_intro K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6) (CHead e1 k0 -u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 -w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) -(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2)))))) x3 x4 x5 -x6 x7 (refl_equal C (CHead x4 x3 x6)) (drop_drop (Flat f) n0 x1 (CHead x5 x3 -x7) H16 x0) H17 H18)) e H15)))))))))) H14)) H13)))))) k (drop_gen_drop k c e -t n0 H2) H8 H9) i H4))) c2 H5)))))))) H3)) (csubst0_gen_head k c c2 t v i -H1))))))))))) c1)))))) n). -(* COMMENTS -Initial nodes: 39886 -END *) - -theorem csubst0_drop_eq: - \forall (n: nat).(\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 -n v c1 c2) \to (\forall (e: C).((drop n O c1 e) \to (or4 (drop n O c2 e) -(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: -T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e0 (Flat f) w)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u -w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n O c2 (CHead e2 -(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 -(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(drop n O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O -v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))) -\def - \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (c1: C).(\forall (c2: -C).(\forall (v: T).((csubst0 n0 v c1 c2) \to (\forall (e: C).((drop n0 O c1 -e) \to (or4 (drop n0 O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O -c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat -f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop n0 O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: -T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c2 (CHead e2 -(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2))))))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda -(H: (csubst0 O v c1 c2)).(\lambda (e: C).(\lambda (H0: (drop O O c1 -e)).(eq_ind C c1 (\lambda (c: C).(or4 (drop O O c2 c) (ex3_4 F C T T (\lambda -(f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 -(Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda -(w: T).(drop O O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c -(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop O O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) -(ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq C c (CHead e1 (Flat f) u))))))) (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O -c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2))))))))) (insert_eq nat O (\lambda (n0: nat).(csubst0 n0 v c1 c2)) -(\lambda (n0: nat).(or4 (drop n0 n0 c2 c1) (ex3_4 F C T T (\lambda (f: -F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c1 (CHead e0 (Flat -f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(drop n0 n0 c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v u w)))))) (ex3_4 F C C T -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c1 -(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop n0 n0 c2 (CHead e2 (Flat f) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 v e1 -e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C c1 (CHead e1 (Flat f) u))))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop n0 n0 c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v u w)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 n0 v e1 e2))))))))) (\lambda (y: nat).(\lambda (H1: (csubst0 -y v c1 c2)).(csubst0_ind (\lambda (n0: nat).(\lambda (t: T).(\lambda (c: -C).(\lambda (c0: C).((eq nat n0 O) \to (or4 (drop n0 n0 c0 c) (ex3_4 F C T T -(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c -(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop n0 n0 c0 (CHead e0 (Flat f) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 t u w)))))) -(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop n0 n0 c0 (CHead e2 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 -t e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Flat f) u))))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop n0 n0 c0 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 t u w)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 n0 t e1 e2))))))))))))) (\lambda (k: K).(K_ind (\lambda (k0: -K).(\forall (i: nat).(\forall (v0: T).(\forall (u1: T).(\forall (u2: -T).((subst0 i v0 u1 u2) \to (\forall (c: C).((eq nat (s k0 i) O) \to (or4 -(drop (s k0 i) (s k0 i) (CHead c k0 u2) (CHead c k0 u1)) (ex3_4 F C T T -(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead -c k0 u1) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda -(_: T).(\lambda (w: T).(drop (s k0 i) (s k0 i) (CHead c k0 u2) (CHead e0 -(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (s k0 i) v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c k0 u1) (CHead e1 (Flat f) -u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (s k0 i) (s k0 i) (CHead c k0 u2) (CHead e2 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (s -k0 i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c k0 u1) -(CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) (s k0 i) (CHead c k0 u2) -(CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (s k0 i) v0 u w)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -(s k0 i) v0 e1 e2)))))))))))))))) (\lambda (b: B).(\lambda (i: nat).(\lambda -(v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0 i v0 u1 -u2)).(\lambda (c: C).(\lambda (H3: (eq nat (S i) O)).(let H4 \def (eq_ind nat -(S i) (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with -[O \Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind (or4 -(drop (S i) (S i) (CHead c (Bind b) u2) (CHead c (Bind b) u1)) (ex3_4 F C T T -(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead -c (Bind b) u1) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(drop (S i) (S i) (CHead c (Bind b) u2) -(CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (S i) v0 u w)))))) (ex3_4 F C C T (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c (Bind b) -u1) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S i) (S i) (CHead c (Bind b) u2) (CHead e2 (Flat f) -u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (S i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c (Bind -b) u1) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S i) (S i) (CHead c (Bind b) -u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (S i) v0 u w)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -(S i) v0 e1 e2)))))))) H4)))))))))) (\lambda (f: F).(\lambda (i: -nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 -i v0 u1 u2)).(\lambda (c: C).(\lambda (H3: (eq nat i O)).(let H4 \def (eq_ind -nat i (\lambda (n0: nat).(subst0 n0 v0 u1 u2)) H2 O H3) in (eq_ind_r nat O -(\lambda (n0: nat).(or4 (drop n0 n0 (CHead c (Flat f) u2) (CHead c (Flat f) -u1)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C (CHead c (Flat f) u1) (CHead e0 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0 -(CHead c (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda -(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v0 u w)))))) (ex3_4 F C C T -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C -(CHead c (Flat f) u1) (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda -(_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 n0 (CHead c (Flat f) u2) -(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C -(CHead c (Flat f) u1) (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0 (CHead c -(Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v0 u w)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 n0 v0 e1 e2))))))))) (or4_intro1 (drop O O (CHead c (Flat f) -u2) (CHead c (Flat f) u1)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c (Flat f) u1) (CHead e0 -(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda -(w: T).(drop O O (CHead c (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) -(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C (CHead c (Flat f) u1) (CHead e1 (Flat f0) u)))))) (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O (CHead c (Flat -f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda -(f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq -C (CHead c (Flat f) u1) (CHead e1 (Flat f0) u))))))) (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O -(CHead c (Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda -(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v0 e1 e2))))))) (ex3_4_intro F C T T (\lambda (f0: -F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c (Flat f) -u1) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda -(_: T).(\lambda (w: T).(drop O O (CHead c (Flat f) u2) (CHead e0 (Flat f0) -w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v0 u w))))) f c u1 u2 (refl_equal C (CHead c (Flat f) u1)) -(drop_refl (CHead c (Flat f) u2)) H4)) i H3)))))))))) k)) (\lambda (k: -K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (c3: C).(\forall (c4: -C).(\forall (v0: T).((csubst0 i v0 c3 c4) \to ((((eq nat i O) \to (or4 (drop -i i c4 c3) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C c3 (CHead e0 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop i i c4 (CHead e0 -(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 i v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 (Flat f) u)))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop i i c4 -(CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 i v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 -(CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop i i c4 (CHead e2 (Flat f) w))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 i v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v0 e1 e2)))))))))) \to (\forall -(u: T).((eq nat (s k0 i) O) \to (or4 (drop (s k0 i) (s k0 i) (CHead c4 k0 u) -(CHead c3 k0 u)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda -(u0: T).(\lambda (_: T).(eq C (CHead c3 k0 u) (CHead e0 (Flat f) u0)))))) -(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 -i) (s k0 i) (CHead c4 k0 u) (CHead e0 (Flat f) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (s k0 i) v0 u0 -w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u0: T).(eq C (CHead c3 k0 u) (CHead e1 (Flat f) u0)))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s k0 -i) (s k0 i) (CHead c4 k0 u) (CHead e2 (Flat f) u0)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (s k0 i) v0 e1 -e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 k0 u) (CHead e1 (Flat f) -u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (s k0 i) (s k0 i) (CHead c4 k0 u) (CHead e2 (Flat f) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: -T).(\lambda (w: T).(subst0 (s k0 i) v0 u0 w)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (s k0 i) v0 -e1 e2))))))))))))))))) (\lambda (b: B).(\lambda (i: nat).(\lambda (c3: -C).(\lambda (c4: C).(\lambda (v0: T).(\lambda (_: (csubst0 i v0 c3 -c4)).(\lambda (_: (((eq nat i O) \to (or4 (drop i i c4 c3) (ex3_4 F C T T -(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 -(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop i i c4 (CHead e0 (Flat f) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w)))))) -(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c3 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop i i c4 (CHead e2 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f) u))))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop i i c4 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (u: T).(\lambda (H4: (eq nat -(S i) O)).(let H5 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee in nat -return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow -True])) I O H4) in (False_ind (or4 (drop (S i) (S i) (CHead c4 (Bind b) u) -(CHead c3 (Bind b) u)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: -C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 (Bind b) u) (CHead e0 -(Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda -(w: T).(drop (S i) (S i) (CHead c4 (Bind b) u) (CHead e0 (Flat f) w)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (S -i) v0 u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u0: T).(eq C (CHead c3 (Bind b) u) (CHead e1 (Flat f) u0)))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S i) -(S i) (CHead c4 (Bind b) u) (CHead e2 (Flat f) u0)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (S i) v0 e1 -e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 (Bind b) u) (CHead e1 -(Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(drop (S i) (S i) (CHead c4 (Bind b) u) (CHead e2 -(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda -(u0: T).(\lambda (w: T).(subst0 (S i) v0 u0 w)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (S i) v0 e1 -e2)))))))) H5))))))))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (c3: -C).(\lambda (c4: C).(\lambda (v0: T).(\lambda (H2: (csubst0 i v0 c3 -c4)).(\lambda (H3: (((eq nat i O) \to (or4 (drop i i c4 c3) (ex3_4 F C T T -(\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 -(CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop i i c4 (CHead e0 (Flat f0) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w)))))) -(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c3 (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop i i c4 (CHead e2 (Flat f0) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f0) u))))))) -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(w: T).(drop i i c4 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (u: T).(\lambda (H4: (eq nat i -O)).(let H5 \def (eq_ind nat i (\lambda (n0: nat).((eq nat n0 O) \to (or4 -(drop n0 n0 c4 c3) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda -(u0: T).(\lambda (_: T).(eq C c3 (CHead e0 (Flat f0) u0)))))) (\lambda (f0: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0 c4 (CHead e0 -(Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda -(w: T).(subst0 n0 v0 u0 w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u0: T).(eq C c3 (CHead e1 (Flat f0) u0)))))) -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop n0 -n0 c4 (CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda -(f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq -C c3 (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0 c4 (CHead e2 (Flat f0) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: -T).(\lambda (w: T).(subst0 n0 v0 u0 w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 n0 v0 e1 -e2)))))))))) H3 O H4) in (let H6 \def (eq_ind nat i (\lambda (n0: -nat).(csubst0 n0 v0 c3 c4)) H2 O H4) in (eq_ind_r nat O (\lambda (n0: -nat).(or4 (drop n0 n0 (CHead c4 (Flat f) u) (CHead c3 (Flat f) u)) (ex3_4 F C -T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C -(CHead c3 (Flat f) u) (CHead e0 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda -(e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0 (CHead c4 (Flat f) u) -(CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: -T).(\lambda (w: T).(subst0 n0 v0 u0 w)))))) (ex3_4 F C C T (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead c3 (Flat f) -u) (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u0: T).(drop n0 n0 (CHead c4 (Flat f) u) (CHead e2 (Flat -f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 (Flat f) -u) (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0 (CHead c4 (Flat f) u) -(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u0: T).(\lambda (w: T).(subst0 n0 v0 u0 w)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -n0 v0 e1 e2))))))))) (or4_intro2 (drop O O (CHead c4 (Flat f) u) (CHead c3 -(Flat f) u)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: -T).(\lambda (_: T).(eq C (CHead c3 (Flat f) u) (CHead e0 (Flat f0) u0)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O -(CHead c4 (Flat f) u) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda -(_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v0 u0 w)))))) (ex3_4 F C C -T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C -(CHead c3 (Flat f) u) (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda -(_: C).(\lambda (e2: C).(\lambda (u0: T).(drop O O (CHead c4 (Flat f) u) -(CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C -(CHead c3 (Flat f) u) (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 -(Flat f) u) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v0 u0 w)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v0 e1 e2))))))) (ex3_4_intro F C C T (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead c3 (Flat f) -u) (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u0: T).(drop O O (CHead c4 (Flat f) u) (CHead e2 (Flat f0) -u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v0 e1 e2))))) f c3 c4 u (refl_equal C (CHead c3 (Flat f) u)) -(drop_refl (CHead c4 (Flat f) u)) H6)) i H4)))))))))))) k)) (\lambda (k: -K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (v0: T).(\forall (u1: -T).(\forall (u2: T).((subst0 i v0 u1 u2) \to (\forall (c3: C).(\forall (c4: -C).((csubst0 i v0 c3 c4) \to ((((eq nat i O) \to (or4 (drop i i c4 c3) (ex3_4 -F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq -C c3 (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop i i c4 (CHead e0 (Flat f) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w)))))) -(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c3 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop i i c4 (CHead e2 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f) u))))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop i i c4 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 i v0 e1 e2)))))))))) \to ((eq nat (s k0 i) O) \to (or4 (drop -(s k0 i) (s k0 i) (CHead c4 k0 u2) (CHead c3 k0 u1)) (ex3_4 F C T T (\lambda -(f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 k0 -u1) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (s k0 i) (s k0 i) (CHead c4 k0 u2) (CHead e0 (Flat -f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (s k0 i) v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c3 k0 u1) (CHead e1 (Flat f) -u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (s k0 i) (s k0 i) (CHead c4 k0 u2) (CHead e2 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (s -k0 i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 k0 u1) -(CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) (s k0 i) (CHead c4 k0 u2) -(CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (s k0 i) v0 u w)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -(s k0 i) v0 e1 e2))))))))))))))))))) (\lambda (b: B).(\lambda (i: -nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0 -i v0 u1 u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (_: (csubst0 i v0 c3 -c4)).(\lambda (_: (((eq nat i O) \to (or4 (drop i i c4 c3) (ex3_4 F C T T -(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 -(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop i i c4 (CHead e0 (Flat f) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w)))))) -(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c3 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop i i c4 (CHead e2 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f) u))))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop i i c4 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (H5: (eq nat (S i) O)).(let H6 -\def (eq_ind nat (S i) (\lambda (ee: nat).(match ee in nat return (\lambda -(_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H5) -in (False_ind (or4 (drop (S i) (S i) (CHead c4 (Bind b) u2) (CHead c3 (Bind -b) u1)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C (CHead c3 (Bind b) u1) (CHead e0 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S i) -(S i) (CHead c4 (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (S i) v0 u w)))))) -(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C (CHead c3 (Bind b) u1) (CHead e1 (Flat f) u)))))) (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S i) (S i) (CHead -c4 (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (S i) v0 e1 e2)))))) (ex4_5 F C -C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(eq C (CHead c3 (Bind b) u1) (CHead e1 (Flat f) u))))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S i) (S i) (CHead c4 (Bind b) u2) (CHead e2 (Flat f) w))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (S i) v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (S i) v0 e1 e2)))))))) -H6))))))))))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (v0: T).(\lambda -(u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 i v0 u1 u2)).(\lambda (c3: -C).(\lambda (c4: C).(\lambda (H3: (csubst0 i v0 c3 c4)).(\lambda (H4: (((eq -nat i O) \to (or4 (drop i i c4 c3) (ex3_4 F C T T (\lambda (f0: F).(\lambda -(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e0 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop i i -c4 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 i v0 u w)))))) (ex3_4 F C C T (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 (Flat -f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop i i c4 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v0 e1 e2)))))) (ex4_5 F C C T -T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda -(_: T).(eq C c3 (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop i i c4 (CHead e2 -(Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (w: T).(subst0 i v0 u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v0 e1 -e2))))))))))).(\lambda (H5: (eq nat i O)).(let H6 \def (eq_ind nat i (\lambda -(n0: nat).((eq nat n0 O) \to (or4 (drop n0 n0 c4 c3) (ex3_4 F C T T (\lambda -(f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e0 -(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda -(w: T).(drop n0 n0 c4 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda -(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v0 u w)))))) (ex3_4 F C C T -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3 -(CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop n0 n0 c4 (CHead e2 (Flat f0) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 v0 e1 -e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f0) u))))))) -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(w: T).(drop n0 n0 c4 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda -(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v0 u -w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))))))) H4 O H5) in (let H7 \def -(eq_ind nat i (\lambda (n0: nat).(csubst0 n0 v0 c3 c4)) H3 O H5) in (let H8 -\def (eq_ind nat i (\lambda (n0: nat).(subst0 n0 v0 u1 u2)) H2 O H5) in -(eq_ind_r nat O (\lambda (n0: nat).(or4 (drop n0 n0 (CHead c4 (Flat f) u2) -(CHead c3 (Flat f) u1)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 (Flat f) u1) (CHead e0 -(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda -(w: T).(drop n0 n0 (CHead c4 (Flat f) u2) (CHead e0 (Flat f0) w)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v0 -u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 n0 -(CHead c4 (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F -C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(eq C (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u))))))) -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(w: T).(drop n0 n0 (CHead c4 (Flat f) u2) (CHead e2 (Flat f0) w))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 n0 v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 n0 v0 e1 e2))))))))) (or4_intro3 -(drop O O (CHead c4 (Flat f) u2) (CHead c3 (Flat f) u1)) (ex3_4 F C T T -(\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C -(CHead c3 (Flat f) u1) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda -(e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 (Flat f) u2) -(CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v0 u w)))))) (ex3_4 F C C T (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c3 (Flat f) -u1) (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u: T).(drop O O (CHead c4 (Flat f) u2) (CHead e2 (Flat f0) -u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 (Flat f) -u1) (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 (Flat f) u2) -(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v0 e1 e2))))))) (ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 (Flat f) -u1) (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 (Flat f) u2) -(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v0 e1 e2)))))) f c3 c4 u1 u2 (refl_equal C (CHead c3 (Flat f) u1)) -(drop_refl (CHead c4 (Flat f) u2)) H8 H7)) i H5))))))))))))))) k)) y v c1 c2 -H1))) H) e (drop_gen_refl c1 e H0)))))))) (\lambda (n0: nat).(\lambda (IHn: -((\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 n0 v c1 c2) \to -(\forall (e: C).((drop n0 O c1 e) \to (or4 (drop n0 O c2 e) (ex3_4 F C T T -(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop n0 O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O c2 (CHead e2 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O -v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop n0 O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2)))))))))))))))).(\lambda (c1: C).(C_ind (\lambda -(c: C).(\forall (c2: C).(\forall (v: T).((csubst0 (S n0) v c c2) \to (\forall -(e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 F C T T -(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 (Flat f) -u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) -u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O -v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))) (\lambda (n1: -nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (_: (csubst0 (S n0) v (CSort -n1) c2)).(\lambda (e: C).(\lambda (H0: (drop (S n0) O (CSort n1) -e)).(and3_ind (eq C e (CSort n1)) (eq nat (S n0) O) (eq nat O O) (or4 (drop -(S n0) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead -e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda -(w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) -O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda -(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C -e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2)))))))) (\lambda (H1: (eq C e (CSort n1))).(\lambda (H2: (eq nat (S n0) -O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n1) (\lambda (c: C).(or4 -(drop (S n0) O c2 c) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda -(u: T).(\lambda (_: T).(eq C c (CHead e0 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead -e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda -(w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Flat f) u)))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) -O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda -(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C -c (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2))))))))) (let H4 \def (eq_ind nat (S n0) (\lambda (ee: nat).(match ee in -nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) -\Rightarrow True])) I O H2) in (False_ind (or4 (drop (S n0) O c2 (CSort n1)) -(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: -T).(eq C (CSort n1) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 (Flat f) -w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C (CSort n1) (CHead e1 (Flat f) -u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: -T).(eq C (CSort n1) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead -e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2)))))))) H4)) e H1)))) (drop_gen_sort n1 (S n0) O e H0)))))))) (\lambda (c: -C).(\lambda (H: ((\forall (c2: C).(\forall (v: T).((csubst0 (S n0) v c c2) -\to (\forall (e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 -F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq -C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 (Flat f) -u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) -u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O -v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))))))).(\lambda (k: K).(\lambda -(t: T).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 (S n0) v -(CHead c k t) c2)).(\lambda (e: C).(\lambda (H1: (drop (S n0) O (CHead c k t) -e)).(or3_ind (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (S n0) (s -k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) -(\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2)))) (ex3_2 C nat (\lambda -(_: C).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (c3: C).(\lambda -(_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: -nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: -C).(\lambda (j: nat).(eq nat (S n0) (s k j))))) (\lambda (u2: T).(\lambda -(c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: -T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: -T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))) (or4 (drop (S -n0) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead -e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda -(w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) -O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda -(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C -e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2)))))))) (\lambda (H2: (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq -nat (S n0) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k -u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))))).(ex3_2_ind T -nat (\lambda (_: T).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (u2: -T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: -nat).(subst0 j v t u2))) (or4 (drop (S n0) O c2 e) (ex3_4 F C T T (\lambda -(f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 -(Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda -(w: T).(drop (S n0) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda -(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) -(ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda -(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H3: (eq nat -(S n0) (s k x1))).(\lambda (H4: (eq C c2 (CHead c k x0))).(\lambda (H5: -(subst0 x1 v t x0)).(eq_ind_r C (CHead c k x0) (\lambda (c0: C).(or4 (drop (S -n0) O c0 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead -e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda -(w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) -O c0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda -(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C -e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 (Flat f) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2))))))))) (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to ((eq nat (S -n0) (s k0 x1)) \to (or4 (drop (S n0) O (CHead c k0 x0) e) (ex3_4 F C T T -(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead c k0 x0) (CHead e0 (Flat f) w)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u -w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c -k0 x0) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda -(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C -e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c k0 x0) (CHead e2 -(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2))))))))))) (\lambda (b: B).(\lambda (H6: (drop (r (Bind b) n0) O c -e)).(\lambda (H7: (eq nat (S n0) (s (Bind b) x1))).(let H8 \def (f_equal nat -nat (\lambda (e0: nat).(match e0 in nat return (\lambda (_: nat).nat) with [O -\Rightarrow n0 | (S n1) \Rightarrow n1])) (S n0) (S x1) H7) in (let H9 \def -(eq_ind_r nat x1 (\lambda (n1: nat).(subst0 n1 v t x0)) H5 n0 H8) in -(or4_intro0 (drop (S n0) O (CHead c (Bind b) x0) e) (ex3_4 F C T T (\lambda -(f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 -(Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda -(w: T).(drop (S n0) O (CHead c (Bind b) x0) (CHead e0 (Flat f) w)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u -w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c -(Bind b) x0) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: -T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c -(Bind b) x0) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2))))))) (drop_drop (Bind b) n0 c e H6 x0))))))) -(\lambda (f: F).(\lambda (H6: (drop (r (Flat f) n0) O c e)).(\lambda (H7: (eq -nat (S n0) (s (Flat f) x1))).(let H8 \def (f_equal nat nat (\lambda (e0: -nat).e0) (S n0) (s (Flat f) x1) H7) in (let H9 \def (eq_ind_r nat x1 (\lambda -(n1: nat).(subst0 n1 v t x0)) H5 (S n0) H8) in (or4_intro0 (drop (S n0) O -(CHead c (Flat f) x0) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O (CHead c (Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) x0) -(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c (Flat f) x0) -(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2))))))) (drop_drop (Flat f) n0 c e H6 x0))))))) k (drop_gen_drop k c -e t n0 H1) H3) c2 H4)))))) H2)) (\lambda (H2: (ex3_2 C nat (\lambda (_: -C).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (c3: C).(\lambda (_: -nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j -v c c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) -(s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) -(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (or4 (drop (S n0) O -c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead -e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda -(w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) -O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda -(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C -e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2)))))))) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H3: (eq nat (S n0) -(s k x1))).(\lambda (H4: (eq C c2 (CHead x0 k t))).(\lambda (H5: (csubst0 x1 -v c x0)).(eq_ind_r C (CHead x0 k t) (\lambda (c0: C).(or4 (drop (S n0) O c0 -e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda -(_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e0 (Flat f) -w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) -O c0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda -(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C -e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 (Flat f) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2))))))))) (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to ((eq nat (S -n0) (s k0 x1)) \to (or4 (drop (S n0) O (CHead x0 k0 t) e) (ex3_4 F C T T -(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x0 k0 t) (CHead e0 (Flat f) w)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u -w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 -k0 t) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda -(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C -e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 k0 t) (CHead e2 -(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2))))))))))) (\lambda (b: B).(\lambda (H6: (drop (r (Bind b) n0) O c -e)).(\lambda (H7: (eq nat (S n0) (s (Bind b) x1))).(let H8 \def (f_equal nat -nat (\lambda (e0: nat).(match e0 in nat return (\lambda (_: nat).nat) with [O -\Rightarrow n0 | (S n1) \Rightarrow n1])) (S n0) (S x1) H7) in (let H9 \def -(eq_ind_r nat x1 (\lambda (n1: nat).(csubst0 n1 v c x0)) H5 n0 H8) in (let -H10 \def (IHn c x0 v H9 e H6) in (or4_ind (drop n0 O x0 e) (ex3_4 F C T T -(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop n0 O x0 (CHead e0 (Flat f) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O x0 (CHead e2 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O -v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop n0 O x0 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead x0 (Bind b) t) e) -(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: -T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) -(CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat -f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) u)))))) (\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 -e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H11: -(drop n0 O x0 e)).(or4_intro0 (drop (S n0) O (CHead x0 (Bind b) t) e) (ex3_4 -F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq -C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f) -w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) -O (CHead x0 (Bind b) t) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C -C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O -v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Bind b) n0 x0 e H11 -t))) (\lambda (H11: (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda -(u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x0 (CHead e0 -(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w))))))).(ex3_4_ind F C T T (\lambda (f: F).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O -x0 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w))))) (or4 (drop (S n0) O (CHead x0 (Bind -b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 -(Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) -u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) -u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: -T).(\lambda (H12: (eq C e (CHead x3 (Flat x2) x4))).(\lambda (H13: (drop n0 O -x0 (CHead x3 (Flat x2) x5))).(\lambda (H14: (subst0 O v x4 x5)).(eq_ind_r C -(CHead x3 (Flat x2) x4) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x0 (Bind -b) t) c0) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 -(Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 -(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) -u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat -f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2))))))))) (or4_intro1 (drop (S n0) O (CHead x0 (Bind b) t) (CHead x3 (Flat -x2) x4)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x4) (CHead e0 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) -O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda -(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead -x3 (Flat x2) x4) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) -(CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C -(CHead x3 (Flat x2) x4) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead -x0 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C T T (\lambda (f: -F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) -x4) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f) -w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w))))) x2 x3 x4 x5 (refl_equal C (CHead x3 (Flat x2) x4)) -(drop_drop (Bind b) n0 x0 (CHead x3 (Flat x2) x5) H13 t) H14)) e H12)))))))) -H11)) (\lambda (H11: (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O x0 (CHead e2 -(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O -x0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2))))) (or4 (drop (S n0) O (CHead x0 -(Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 -(Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) -u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) -u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: -T).(\lambda (H12: (eq C e (CHead x3 (Flat x2) x5))).(\lambda (H13: (drop n0 O -x0 (CHead x4 (Flat x2) x5))).(\lambda (H14: (csubst0 O v x3 x4)).(eq_ind_r C -(CHead x3 (Flat x2) x5) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x0 (Bind -b) t) c0) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 -(Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 -(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) -u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat -f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2))))))))) (or4_intro2 (drop (S n0) O (CHead x0 (Bind b) t) (CHead x3 (Flat -x2) x5)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) -O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda -(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead -x3 (Flat x2) x5) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) -(CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C -(CHead x3 (Flat x2) x5) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead -x0 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C C T (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 (Flat x2) -x5) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) -u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v e1 e2))))) x2 x3 x4 x5 (refl_equal C (CHead x3 (Flat x2) x5)) -(drop_drop (Bind b) n0 x0 (CHead x4 (Flat x2) x5) H13 t) H14)) e H12)))))))) -H11)) (\lambda (H11: (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) -u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop n0 O x0 (CHead e2 (Flat f) w))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O -v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda -(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C -e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x0 (CHead e2 (Flat f) w))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) -O (CHead x0 (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) -O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda -(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) -u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) -u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: -T).(\lambda (x6: T).(\lambda (H12: (eq C e (CHead x3 (Flat x2) x5))).(\lambda -(H13: (drop n0 O x0 (CHead x4 (Flat x2) x6))).(\lambda (H14: (subst0 O v x5 -x6)).(\lambda (H15: (csubst0 O v x3 x4)).(eq_ind_r C (CHead x3 (Flat x2) x5) -(\lambda (c0: C).(or4 (drop (S n0) O (CHead x0 (Bind b) t) c0) (ex3_4 F C T T -(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 -(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f) -w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Flat f) u)))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) -O (CHead x0 (Bind b) t) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C -C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat f) u))))))) (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O -v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O -(CHead x0 (Bind b) t) (CHead x3 (Flat x2) x5)) (ex3_4 F C T T (\lambda (f: -F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) -x5) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f) -w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1 -(Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O -v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1 -(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat -f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) -(ex4_5_intro F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1 -(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat -f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) -x2 x3 x4 x5 x6 (refl_equal C (CHead x3 (Flat x2) x5)) (drop_drop (Bind b) n0 -x0 (CHead x4 (Flat x2) x6) H13 t) H14 H15)) e H12)))))))))) H11)) H10))))))) -(\lambda (f: F).(\lambda (H6: (drop (r (Flat f) n0) O c e)).(\lambda (H7: (eq -nat (S n0) (s (Flat f) x1))).(let H8 \def (f_equal nat nat (\lambda (e0: -nat).e0) (S n0) (s (Flat f) x1) H7) in (let H9 \def (eq_ind_r nat x1 (\lambda -(n1: nat).(csubst0 n1 v c x0)) H5 (S n0) H8) in (let H10 \def (H x0 v H9 e -H6) in (or4_ind (drop (S n0) O x0 e) (ex3_4 F C T T (\lambda (f0: F).(\lambda -(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O x0 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda -(u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat -f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O x0 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C -C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O x0 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead x0 (Flat f) t) e) -(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: -T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) -(CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat -f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u)))))) (\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 -e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H11: -(drop (S n0) O x0 e)).(or4_intro0 (drop (S n0) O (CHead x0 (Flat f) t) e) -(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: -T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) -(CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat -f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u)))))) (\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 -e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Flat -f) n0 x0 e H11 t))) (\lambda (H11: (ex3_4 F C T T (\lambda (f0: F).(\lambda -(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O x0 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda -(u: T).(\lambda (w: T).(subst0 O v u w))))))).(ex3_4_ind F C T T (\lambda -(f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 -(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda -(w: T).(drop (S n0) O x0 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda -(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))) (or4 (drop (S n0) -O (CHead x0 (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) -(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) -(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: T).(\lambda -(x5: T).(\lambda (H12: (eq C e (CHead x3 (Flat x2) x4))).(\lambda (H13: (drop -(S n0) O x0 (CHead x3 (Flat x2) x5))).(\lambda (H14: (subst0 O v x4 -x5)).(eq_ind_r C (CHead x3 (Flat x2) x4) (\lambda (c0: C).(or4 (drop (S n0) O -(CHead x0 (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c0 (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) -(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 -(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) -(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead x0 (Flat f) t) (CHead x3 -(Flat x2) x4)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x4) (CHead e0 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C (CHead x3 (Flat x2) x4) (CHead e1 (Flat f0) u)))))) (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 -(Flat f) t) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda -(_: T).(eq C (CHead x3 (Flat x2) x4) (CHead e1 (Flat f0) u))))))) (\lambda -(f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C -T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C -(CHead x3 (Flat x2) x4) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda -(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) -(CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w))))) x2 x3 x4 x5 (refl_equal C (CHead x3 -(Flat x2) x4)) (drop_drop (Flat f) n0 x0 (CHead x3 (Flat x2) x5) H13 t) H14)) -e H12)))))))) H11)) (\lambda (H11: (ex3_4 F C C T (\lambda (f0: F).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S -n0) O x0 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C -C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O x0 (CHead e2 (Flat f0) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) -(or4 (drop (S n0) O (CHead x0 (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: -F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat -f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) -(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) -(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda -(x5: T).(\lambda (H12: (eq C e (CHead x3 (Flat x2) x5))).(\lambda (H13: (drop -(S n0) O x0 (CHead x4 (Flat x2) x5))).(\lambda (H14: (csubst0 O v x3 -x4)).(eq_ind_r C (CHead x3 (Flat x2) x5) (\lambda (c0: C).(or4 (drop (S n0) O -(CHead x0 (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c0 (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) -(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 -(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) -(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead x0 (Flat f) t) (CHead x3 -(Flat x2) x5)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C (CHead x3 (Flat x2) x5) (CHead e1 (Flat f0) u)))))) (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 -(Flat f) t) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda -(_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1 (Flat f0) u))))))) (\lambda -(f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C -C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C -(CHead x3 (Flat x2) x5) (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda -(_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) -(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2))))) x2 x3 x4 x5 (refl_equal C (CHead -x3 (Flat x2) x5)) (drop_drop (Flat f) n0 x0 (CHead x4 (Flat x2) x5) H13 t) -H14)) e H12)))))))) H11)) (\lambda (H11: (ex4_5 F C C T T (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x0 (CHead e2 (Flat f0) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2)))))))).(ex4_5_ind F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(w: T).(drop (S n0) O x0 (CHead e2 (Flat f0) w))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O -v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) O (CHead x0 -(Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 -(Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) -u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat -f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: -T).(\lambda (x6: T).(\lambda (H12: (eq C e (CHead x3 (Flat x2) x5))).(\lambda -(H13: (drop (S n0) O x0 (CHead x4 (Flat x2) x6))).(\lambda (H14: (subst0 O v -x5 x6)).(\lambda (H15: (csubst0 O v x3 x4)).(eq_ind_r C (CHead x3 (Flat x2) -x5) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x0 (Flat f) t) c0) (ex3_4 F C -T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C -c0 (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) -w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S -n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) -(ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat f0) u))))))) (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O -v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O -(CHead x0 (Flat f) t) (CHead x3 (Flat x2) x5)) (ex3_4 F C T T (\lambda (f0: -F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) -x5) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda -(_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat -f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1 -(Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(u: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O -v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1 -(Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat -f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) -(ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1 -(Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat -f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) -x2 x3 x4 x5 x6 (refl_equal C (CHead x3 (Flat x2) x5)) (drop_drop (Flat f) n0 -x0 (CHead x4 (Flat x2) x6) H13 t) H14 H15)) e H12)))))))))) H11)) H10))))))) -k (drop_gen_drop k c e t n0 H1) H3) c2 H4)))))) H2)) (\lambda (H2: (ex4_3 T C -nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k -j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 -k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t -u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c -c3)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: -nat).(eq nat (S n0) (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda -(_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: -C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: -C).(\lambda (j: nat).(csubst0 j v c c3)))) (or4 (drop (S n0) O c2 e) (ex3_4 F -C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C -e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 (Flat f) -u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) -u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O -v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: -C).(\lambda (x2: nat).(\lambda (H3: (eq nat (S n0) (s k x2))).(\lambda (H4: -(eq C c2 (CHead x1 k x0))).(\lambda (H5: (subst0 x2 v t x0)).(\lambda (H6: -(csubst0 x2 v c x1)).(eq_ind_r C (CHead x1 k x0) (\lambda (c0: C).(or4 (drop -(S n0) O c0 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead -e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda -(w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) -O c0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda -(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C -e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 (Flat f) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2))))))))) (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to ((eq nat (S -n0) (s k0 x2)) \to (or4 (drop (S n0) O (CHead x1 k0 x0) e) (ex3_4 F C T T -(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 k0 x0) (CHead e0 (Flat f) w)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u -w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 -k0 x0) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda -(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C -e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 k0 x0) (CHead e2 -(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2))))))))))) (\lambda (b: B).(\lambda (H7: (drop (r (Bind b) n0) O c -e)).(\lambda (H8: (eq nat (S n0) (s (Bind b) x2))).(let H9 \def (f_equal nat -nat (\lambda (e0: nat).(match e0 in nat return (\lambda (_: nat).nat) with [O -\Rightarrow n0 | (S n1) \Rightarrow n1])) (S n0) (S x2) H8) in (let H10 \def -(eq_ind_r nat x2 (\lambda (n1: nat).(csubst0 n1 v c x1)) H6 n0 H9) in (let -H11 \def (eq_ind_r nat x2 (\lambda (n1: nat).(subst0 n1 v t x0)) H5 n0 H9) in -(let H12 \def (IHn c x1 v H10 e H7) in (or4_ind (drop n0 O x1 e) (ex3_4 F C T -T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop n0 O x1 (CHead e0 (Flat f) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O x1 (CHead e2 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O -v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop n0 O x1 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead x1 (Bind b) x0) -e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda -(_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) -(CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat -f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u)))))) (\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 -e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) w))))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H13: -(drop n0 O x1 e)).(or4_intro0 (drop (S n0) O (CHead x1 (Bind b) x0) e) (ex3_4 -F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq -C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) -w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) -O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C -C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) w))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O -v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Bind b) n0 x1 e H13 -x0))) (\lambda (H13: (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda -(u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x1 (CHead e0 -(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w))))))).(ex3_4_ind F C T T (\lambda (f: F).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O -x1 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w))))) (or4 (drop (S n0) O (CHead x1 (Bind -b) x0) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 -(Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) -u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) -u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: T).(\lambda (x6: -T).(\lambda (H14: (eq C e (CHead x4 (Flat x3) x5))).(\lambda (H15: (drop n0 O -x1 (CHead x4 (Flat x3) x6))).(\lambda (H16: (subst0 O v x5 x6)).(eq_ind_r C -(CHead x4 (Flat x3) x5) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Bind -b) x0) c0) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 -(Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 -(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) -u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat -f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2))))))))) (or4_intro1 (drop (S n0) O (CHead x1 (Bind b) x0) (CHead x4 (Flat -x3) x5)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x5) (CHead e0 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) -O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda -(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead -x4 (Flat x3) x5) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) -(CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C -(CHead x4 (Flat x3) x5) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead -x1 (Bind b) x0) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C T T (\lambda (f: -F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) -x5) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) -w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w))))) x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x3) x5)) -(drop_drop (Bind b) n0 x1 (CHead x4 (Flat x3) x6) H15 x0) H16)) e H14)))))))) -H13)) (\lambda (H13: (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O x1 (CHead e2 -(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O -x1 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2))))) (or4 (drop (S n0) O (CHead x1 -(Bind b) x0) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 -(Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) -u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) -u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: -T).(\lambda (H14: (eq C e (CHead x4 (Flat x3) x6))).(\lambda (H15: (drop n0 O -x1 (CHead x5 (Flat x3) x6))).(\lambda (H16: (csubst0 O v x4 x5)).(eq_ind_r C -(CHead x4 (Flat x3) x6) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Bind -b) x0) c0) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 -(Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 -(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) -u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat -f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2))))))))) (or4_intro2 (drop (S n0) O (CHead x1 (Bind b) x0) (CHead x4 (Flat -x3) x6)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) -O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda -(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead -x4 (Flat x3) x6) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) -(CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C -(CHead x4 (Flat x3) x6) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead -x1 (Bind b) x0) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C C T (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x4 (Flat x3) -x6) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) -u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v e1 e2))))) x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x3) x6)) -(drop_drop (Bind b) n0 x1 (CHead x5 (Flat x3) x6) H15 x0) H16)) e H14)))))))) -H13)) (\lambda (H13: (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) -u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop n0 O x1 (CHead e2 (Flat f) w))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O -v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda -(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C -e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x1 (CHead e2 (Flat f) w))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) -O (CHead x1 (Bind b) x0) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) -O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda -(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) -u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) -u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: -T).(\lambda (x7: T).(\lambda (H14: (eq C e (CHead x4 (Flat x3) x6))).(\lambda -(H15: (drop n0 O x1 (CHead x5 (Flat x3) x7))).(\lambda (H16: (subst0 O v x6 -x7)).(\lambda (H17: (csubst0 O v x4 x5)).(eq_ind_r C (CHead x4 (Flat x3) x6) -(\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Bind b) x0) c0) (ex3_4 F C T -T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 -(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) -w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Flat f) u)))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) -O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C -C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat f) u))))))) (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) w))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O -v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O -(CHead x1 (Bind b) x0) (CHead x4 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f: -F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) -x6) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) -w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 -(Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O -v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 -(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat -f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) -(ex4_5_intro F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 -(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat -f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) -x3 x4 x5 x6 x7 (refl_equal C (CHead x4 (Flat x3) x6)) (drop_drop (Bind b) n0 -x1 (CHead x5 (Flat x3) x7) H15 x0) H16 H17)) e H14)))))))))) H13)) -H12)))))))) (\lambda (f: F).(\lambda (H7: (drop (r (Flat f) n0) O c -e)).(\lambda (H8: (eq nat (S n0) (s (Flat f) x2))).(let H9 \def (f_equal nat -nat (\lambda (e0: nat).e0) (S n0) (s (Flat f) x2) H8) in (let H10 \def -(eq_ind_r nat x2 (\lambda (n1: nat).(csubst0 n1 v c x1)) H6 (S n0) H9) in -(let H11 \def (eq_ind_r nat x2 (\lambda (n1: nat).(subst0 n1 v t x0)) H5 (S -n0) H9) in (let H12 \def (H x1 v H10 e H7) in (or4_ind (drop (S n0) O x1 e) -(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: -T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x1 (CHead e0 (Flat f0) -w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S -n0) O x1 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda -(_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x1 (CHead -e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2))))))) (or4 (drop (S n0) O (CHead x1 (Flat f) x0) e) (ex3_4 F C T T -(\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) -w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S -n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) -(ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O -v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H13: (drop (S n0) O -x1 e)).(or4_intro0 (drop (S n0) O (CHead x1 (Flat f) x0) e) (ex3_4 F C T T -(\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) -w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S -n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) -(ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O -v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Flat f) n0 x1 e H13 -x0))) (\lambda (H13: (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O x1 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda -(u: T).(\lambda (w: T).(subst0 O v u w))))))).(ex3_4_ind F C T T (\lambda -(f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 -(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda -(w: T).(drop (S n0) O x1 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda -(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))) (or4 (drop (S n0) -O (CHead x1 (Flat f) x0) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0) -(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) -(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: T).(\lambda -(x6: T).(\lambda (H14: (eq C e (CHead x4 (Flat x3) x5))).(\lambda (H15: (drop -(S n0) O x1 (CHead x4 (Flat x3) x6))).(\lambda (H16: (subst0 O v x5 -x6)).(eq_ind_r C (CHead x4 (Flat x3) x5) (\lambda (c0: C).(or4 (drop (S n0) O -(CHead x1 (Flat f) x0) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C c0 (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0) -(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 -(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) -(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead x1 (Flat f) x0) (CHead -x4 (Flat x3) x5)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda -(u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x5) (CHead e0 (Flat f0) -u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C (CHead x4 (Flat x3) x5) (CHead e1 (Flat f0) u)))))) (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 -(Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda -(_: T).(eq C (CHead x4 (Flat x3) x5) (CHead e1 (Flat f0) u))))))) (\lambda -(f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C -T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C -(CHead x4 (Flat x3) x5) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda -(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) -(CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w))))) x3 x4 x5 x6 (refl_equal C (CHead x4 -(Flat x3) x5)) (drop_drop (Flat f) n0 x1 (CHead x4 (Flat x3) x6) H15 x0) -H16)) e H14)))))))) H13)) (\lambda (H13: (ex3_4 F C C T (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat -f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop (S n0) O x1 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind -F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq -C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u: T).(drop (S n0) O x1 (CHead e2 (Flat f0) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O -v e1 e2))))) (or4 (drop (S n0) O (CHead x1 (Flat f) x0) e) (ex3_4 F C T T -(\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) -w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S -n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) -(ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O -v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: -C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (H14: (eq C e (CHead x4 (Flat -x3) x6))).(\lambda (H15: (drop (S n0) O x1 (CHead x5 (Flat x3) x6))).(\lambda -(H16: (csubst0 O v x4 x5)).(eq_ind_r C (CHead x4 (Flat x3) x6) (\lambda (c0: -C).(or4 (drop (S n0) O (CHead x1 (Flat f) x0) c0) (ex3_4 F C T T (\lambda -(f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 -(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda -(w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u -w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C c0 (CHead e1 (Flat f0) u)))))) (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 -(Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda -(_: T).(eq C c0 (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 -(Flat f) x0) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead x1 (Flat -f) x0) (CHead x4 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0 -(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda -(w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u -w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S -n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) -(ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) -u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) -(ex3_4_intro F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S -n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) -x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x3) x6)) (drop_drop (Flat f) n0 x1 -(CHead x5 (Flat x3) x6) H15 x0) H16)) e H14)))))))) H13)) (\lambda (H13: -(ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O x1 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x1 (CHead e2 (Flat f0) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) -(or4 (drop (S n0) O (CHead x1 (Flat f) x0) e) (ex3_4 F C T T (\lambda (f0: -F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat -f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0) -(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) -(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda -(x6: T).(\lambda (x7: T).(\lambda (H14: (eq C e (CHead x4 (Flat x3) -x6))).(\lambda (H15: (drop (S n0) O x1 (CHead x5 (Flat x3) x7))).(\lambda -(H16: (subst0 O v x6 x7)).(\lambda (H17: (csubst0 O v x4 x5)).(eq_ind_r C -(CHead x4 (Flat x3) x6) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Flat -f) x0) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f0) u)))))) (\lambda (f0: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 -(Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 -(CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) -u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat -f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) -w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2))))))))) (or4_intro3 (drop (S n0) O (CHead x1 (Flat f) x0) (CHead x4 (Flat -x3) x6)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S -n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) -(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) u)))))) (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 -(Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda -(_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) u))))))) (\lambda -(f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex4_5_intro F C -C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) u))))))) -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) x3 x4 x5 x6 x7 -(refl_equal C (CHead x4 (Flat x3) x6)) (drop_drop (Flat f) n0 x1 (CHead x5 -(Flat x3) x7) H15 x0) H16 H17)) e H14)))))))))) H13)) H12)))))))) k -(drop_gen_drop k c e t n0 H1) H3) c2 H4)))))))) H2)) (csubst0_gen_head k c c2 -t v (S n0) H0))))))))))) c1)))) n). -(* COMMENTS -Initial nodes: 36162 -END *) - -theorem csubst0_drop_eq_back: - \forall (n: nat).(\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 -n v c1 c2) \to (\forall (e: C).((drop n O c2 e) \to (or4 (drop n O c1 e) -(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: -T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(drop n O c1 (CHead e0 (Flat f) u1)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v -u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n O c1 (CHead e1 -(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat -f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (_: T).(drop n O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 -O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))) -\def - \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (c1: C).(\forall (c2: -C).(\forall (v: T).((csubst0 n0 v c1 c2) \to (\forall (e: C).((drop n0 O c2 -e) \to (or4 (drop n0 O c1 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) -(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O -c1 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat -f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(drop n0 O c1 (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c1 (CHead e1 -(Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2))))))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda -(H: (csubst0 O v c1 c2)).(\lambda (e: C).(\lambda (H0: (drop O O c2 -e)).(eq_ind C c2 (\lambda (c: C).(or4 (drop O O c1 c) (ex3_4 F C T T (\lambda -(f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c (CHead e0 -(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda -(_: T).(drop O O c1 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c -(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(drop O O c1 (CHead e1 (Flat f) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) -(ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (u2: T).(eq C c (CHead e2 (Flat f) u2))))))) (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop O -O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda -(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2))))))))) (insert_eq nat O (\lambda (n0: nat).(csubst0 n0 v c1 c2)) -(\lambda (n0: nat).(or4 (drop n0 n0 c1 c2) (ex3_4 F C T T (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c2 (CHead e0 (Flat -f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(drop n0 n0 c1 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 n0 v u1 u2)))))) (ex3_4 F C C T -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c2 -(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(drop n0 n0 c1 (CHead e1 (Flat f) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 v e1 -e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C c2 (CHead e2 (Flat f) u2))))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop n0 n0 c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda -(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 n0 v u1 -u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 n0 v e1 e2))))))))) (\lambda (y: nat).(\lambda -(H1: (csubst0 y v c1 c2)).(csubst0_ind (\lambda (n0: nat).(\lambda (t: -T).(\lambda (c: C).(\lambda (c0: C).((eq nat n0 O) \to (or4 (drop n0 n0 c c0) -(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: -T).(eq C c0 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(drop n0 n0 c (CHead e0 (Flat f) u1)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 n0 -t u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 n0 c (CHead e1 -(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 n0 t e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e2 -(Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (_: T).(drop n0 n0 c (CHead e1 (Flat f) u1))))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 n0 t u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 n0 t e1 e2))))))))))))) (\lambda -(k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (v0: T).(\forall -(u1: T).(\forall (u2: T).((subst0 i v0 u1 u2) \to (\forall (c: C).((eq nat (s -k0 i) O) \to (or4 (drop (s k0 i) (s k0 i) (CHead c k0 u1) (CHead c k0 u2)) -(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: -T).(eq C (CHead c k0 u2) (CHead e0 (Flat f) u4)))))) (\lambda (f: F).(\lambda -(e0: C).(\lambda (u3: T).(\lambda (_: T).(drop (s k0 i) (s k0 i) (CHead c k0 -u1) (CHead e0 (Flat f) u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: -T).(\lambda (u4: T).(subst0 (s k0 i) v0 u3 u4)))))) (ex3_4 F C C T (\lambda -(f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c k0 u2) -(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(drop (s k0 i) (s k0 i) (CHead c k0 u1) (CHead e1 (Flat f) -u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (s k0 i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c k0 -u2) (CHead e2 (Flat f) u4))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u3: T).(\lambda (_: T).(drop (s k0 i) (s k0 i) (CHead c k0 -u1) (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u3: T).(\lambda (u4: T).(subst0 (s k0 i) v0 u3 u4)))))) (\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: -T).(csubst0 (s k0 i) v0 e1 e2)))))))))))))))) (\lambda (b: B).(\lambda (i: -nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0 -i v0 u1 u2)).(\lambda (c: C).(\lambda (H3: (eq nat (S i) O)).(let H4 \def -(eq_ind nat (S i) (\lambda (ee: nat).(match ee in nat return (\lambda (_: -nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H3) in -(False_ind (or4 (drop (S i) (S i) (CHead c (Bind b) u1) (CHead c (Bind b) -u2)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda -(u4: T).(eq C (CHead c (Bind b) u2) (CHead e0 (Flat f) u4)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop (S i) (S i) (CHead -c (Bind b) u1) (CHead e0 (Flat f) u3)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u3: T).(\lambda (u4: T).(subst0 (S i) v0 u3 u4)))))) (ex3_4 F C -C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C -(CHead c (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S i) (S i) (CHead c (Bind b) -u1) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (S i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda -(f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq -C (CHead c (Bind b) u2) (CHead e2 (Flat f) u4))))))) (\lambda (f: F).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (_: T).(drop (S i) (S i) -(CHead c (Bind b) u1) (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda -(_: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 (S i) v0 u3 -u4)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (S i) v0 e1 e2)))))))) H4)))))))))) (\lambda (f: -F).(\lambda (i: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: -T).(\lambda (H2: (subst0 i v0 u1 u2)).(\lambda (c: C).(\lambda (H3: (eq nat i -O)).(let H4 \def (eq_ind nat i (\lambda (n0: nat).(subst0 n0 v0 u1 u2)) H2 O -H3) in (eq_ind_r nat O (\lambda (n0: nat).(or4 (drop n0 n0 (CHead c (Flat f) -u1) (CHead c (Flat f) u2)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c (Flat f) u2) (CHead e0 -(Flat f0) u4)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u3: -T).(\lambda (_: T).(drop n0 n0 (CHead c (Flat f) u1) (CHead e0 (Flat f0) -u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: -T).(subst0 n0 v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c (Flat f) u2) (CHead e2 -(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(drop n0 n0 (CHead c (Flat f) u1) (CHead e1 (Flat f0) u)))))) (\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 v0 e1 -e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c (Flat f) u2) (CHead e2 -(Flat f0) u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u3: T).(\lambda (_: T).(drop n0 n0 (CHead c (Flat f) u1) (CHead -e1 (Flat f0) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u3: T).(\lambda (u4: T).(subst0 n0 v0 u3 u4)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -n0 v0 e1 e2))))))))) (or4_intro1 (drop O O (CHead c (Flat f) u1) (CHead c -(Flat f) u2)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (u4: T).(eq C (CHead c (Flat f) u2) (CHead e0 (Flat f0) u4)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop O O -(CHead c (Flat f) u1) (CHead e0 (Flat f0) u3)))))) (\lambda (_: F).(\lambda -(_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (ex3_4 F C -C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C -(CHead c (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u: T).(drop O O (CHead c (Flat f) u1) -(CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C -(CHead c (Flat f) u2) (CHead e2 (Flat f0) u4))))))) (\lambda (f0: F).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c -(Flat f) u1) (CHead e1 (Flat f0) u3))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v0 e1 e2))))))) (ex3_4_intro F C T T (\lambda (f0: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c (Flat f) -u2) (CHead e0 (Flat f0) u4)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda -(u3: T).(\lambda (_: T).(drop O O (CHead c (Flat f) u1) (CHead e0 (Flat f0) -u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: -T).(subst0 O v0 u3 u4))))) f c u1 u2 (refl_equal C (CHead c (Flat f) u2)) -(drop_refl (CHead c (Flat f) u1)) H4)) i H3)))))))))) k)) (\lambda (k: -K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (c3: C).(\forall (c4: -C).(\forall (v0: T).((csubst0 i v0 c3 c4) \to ((((eq nat i O) \to (or4 (drop -i i c3 c4) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (u2: T).(eq C c4 (CHead e0 (Flat f) u2)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop i i c3 (CHead e0 -(Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 i v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(eq C c4 (CHead e2 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop i i c3 -(CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 i v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 -(CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(drop i i c3 (CHead e1 (Flat f) u1))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 i v0 u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v0 e1 e2)))))))))) \to -(\forall (u: T).((eq nat (s k0 i) O) \to (or4 (drop (s k0 i) (s k0 i) (CHead -c3 k0 u) (CHead c4 k0 u)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 k0 u) (CHead e0 (Flat f) -u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(drop (s k0 i) (s k0 i) (CHead c3 k0 u) (CHead e0 (Flat f) u1)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (s -k0 i) v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u0: T).(eq C (CHead c4 k0 u) (CHead e2 (Flat f) u0)))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(drop (s k0 -i) (s k0 i) (CHead c3 k0 u) (CHead e1 (Flat f) u0)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (s k0 i) v0 e1 -e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 k0 u) (CHead e2 (Flat f) -u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (_: T).(drop (s k0 i) (s k0 i) (CHead c3 k0 u) (CHead e1 (Flat f) -u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 (s k0 i) v0 u1 u2)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (s k0 i) v0 -e1 e2))))))))))))))))) (\lambda (b: B).(\lambda (i: nat).(\lambda (c3: -C).(\lambda (c4: C).(\lambda (v0: T).(\lambda (_: (csubst0 i v0 c3 -c4)).(\lambda (_: (((eq nat i O) \to (or4 (drop i i c3 c4) (ex3_4 F C T T -(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 -(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(drop i i c3 (CHead e0 (Flat f) u1)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v0 u1 u2)))))) -(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(eq C c4 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(drop i i c3 (CHead e1 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e2 (Flat f) u2))))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop i i c3 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v0 u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (u: T).(\lambda (H4: (eq nat -(S i) O)).(let H5 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee in nat -return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow -True])) I O H4) in (False_ind (or4 (drop (S i) (S i) (CHead c3 (Bind b) u) -(CHead c4 (Bind b) u)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 (Bind b) u) (CHead e0 -(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S i) (S i) (CHead c3 (Bind b) u) (CHead e0 (Flat f) u1)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (S -i) v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u0: T).(eq C (CHead c4 (Bind b) u) (CHead e2 (Flat f) -u0)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: -T).(drop (S i) (S i) (CHead c3 (Bind b) u) (CHead e1 (Flat f) u0)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (S -i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 (Bind b) u) (CHead -e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S i) (S i) (CHead c3 (Bind b) u) -(CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (S i) v0 u1 u2)))))) (\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: -T).(csubst0 (S i) v0 e1 e2)))))))) H5))))))))))) (\lambda (f: F).(\lambda (i: -nat).(\lambda (c3: C).(\lambda (c4: C).(\lambda (v0: T).(\lambda (H2: -(csubst0 i v0 c3 c4)).(\lambda (H3: (((eq nat i O) \to (or4 (drop i i c3 c4) -(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda -(u2: T).(eq C c4 (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(drop i i c3 (CHead e0 (Flat f0) u1)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i -v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C c4 (CHead e2 (Flat f0) u)))))) (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop i i c3 (CHead e1 -(Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 i v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e2 -(Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(drop i i c3 (CHead e1 (Flat f0) -u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 i v0 u1 u2)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v0 e1 -e2))))))))))).(\lambda (u: T).(\lambda (H4: (eq nat i O)).(let H5 \def -(eq_ind nat i (\lambda (n0: nat).((eq nat n0 O) \to (or4 (drop n0 n0 c3 c4) -(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda -(u2: T).(eq C c4 (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(drop n0 n0 c3 (CHead e0 (Flat f0) -u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 n0 v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u0: T).(eq C c4 (CHead e2 (Flat f0) u0)))))) -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(drop n0 -n0 c3 (CHead e1 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda -(f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq -C c4 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 n0 c3 (CHead e1 (Flat f0) -u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 n0 v0 u1 u2)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 n0 v0 e1 -e2)))))))))) H3 O H4) in (let H6 \def (eq_ind nat i (\lambda (n0: -nat).(csubst0 n0 v0 c3 c4)) H2 O H4) in (eq_ind_r nat O (\lambda (n0: -nat).(or4 (drop n0 n0 (CHead c3 (Flat f) u) (CHead c4 (Flat f) u)) (ex3_4 F C -T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C -(CHead c4 (Flat f) u) (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda -(e0: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 n0 (CHead c3 (Flat f) u) -(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 n0 v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(eq C (CHead c4 (Flat f) -u) (CHead e2 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u0: T).(drop n0 n0 (CHead c3 (Flat f) u) (CHead e1 (Flat f0) -u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 (Flat f) -u) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 n0 (CHead c3 (Flat f) u) -(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 n0 v0 u1 u2)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -n0 v0 e1 e2))))))))) (or4_intro2 (drop O O (CHead c3 (Flat f) u) (CHead c4 -(Flat f) u)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (u2: T).(eq C (CHead c4 (Flat f) u) (CHead e0 (Flat f0) u2)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop O O -(CHead c3 (Flat f) u) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda -(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (ex3_4 F C -C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(eq C -(CHead c4 (Flat f) u) (CHead e2 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u0: T).(drop O O (CHead c3 (Flat f) u) -(CHead e1 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C -(CHead c4 (Flat f) u) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop O O (CHead c3 -(Flat f) u) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v0 e1 e2))))))) (ex3_4_intro F C C T (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(eq C (CHead c4 (Flat f) -u) (CHead e2 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u0: T).(drop O O (CHead c3 (Flat f) u) (CHead e1 (Flat f0) -u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v0 e1 e2))))) f c3 c4 u (refl_equal C (CHead c4 (Flat f) u)) -(drop_refl (CHead c3 (Flat f) u)) H6)) i H4)))))))))))) k)) (\lambda (k: -K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (v0: T).(\forall (u1: -T).(\forall (u2: T).((subst0 i v0 u1 u2) \to (\forall (c3: C).(\forall (c4: -C).((csubst0 i v0 c3 c4) \to ((((eq nat i O) \to (or4 (drop i i c3 c4) (ex3_4 -F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq -C c4 (CHead e0 (Flat f) u4)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda -(u3: T).(\lambda (_: T).(drop i i c3 (CHead e0 (Flat f) u3)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 i v0 u3 u4)))))) -(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(eq C c4 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(drop i i c3 (CHead e1 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u4: T).(eq C c4 (CHead e2 (Flat f) u4))))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda -(_: T).(drop i i c3 (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 i v0 u3 u4)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 i v0 e1 e2)))))))))) \to ((eq nat (s k0 i) O) \to (or4 (drop -(s k0 i) (s k0 i) (CHead c3 k0 u1) (CHead c4 k0 u2)) (ex3_4 F C T T (\lambda -(f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 k0 -u2) (CHead e0 (Flat f) u4)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda -(u3: T).(\lambda (_: T).(drop (s k0 i) (s k0 i) (CHead c3 k0 u1) (CHead e0 -(Flat f) u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda -(u4: T).(subst0 (s k0 i) v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c4 k0 u2) -(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(drop (s k0 i) (s k0 i) (CHead c3 k0 u1) (CHead e1 (Flat -f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (s k0 i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 k0 -u2) (CHead e2 (Flat f) u4))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u3: T).(\lambda (_: T).(drop (s k0 i) (s k0 i) (CHead c3 k0 -u1) (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u3: T).(\lambda (u4: T).(subst0 (s k0 i) v0 u3 u4)))))) (\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: -T).(csubst0 (s k0 i) v0 e1 e2))))))))))))))))))) (\lambda (b: B).(\lambda (i: -nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0 -i v0 u1 u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (_: (csubst0 i v0 c3 -c4)).(\lambda (_: (((eq nat i O) \to (or4 (drop i i c3 c4) (ex3_4 F C T T -(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C c4 -(CHead e0 (Flat f) u4)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u3: -T).(\lambda (_: T).(drop i i c3 (CHead e0 (Flat f) u3)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 i v0 u3 u4)))))) -(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(eq C c4 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(drop i i c3 (CHead e1 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u4: T).(eq C c4 (CHead e2 (Flat f) u4))))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda -(_: T).(drop i i c3 (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 i v0 u3 u4)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (H5: (eq nat (S i) O)).(let H6 -\def (eq_ind nat (S i) (\lambda (ee: nat).(match ee in nat return (\lambda -(_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H5) -in (False_ind (or4 (drop (S i) (S i) (CHead c3 (Bind b) u1) (CHead c4 (Bind -b) u2)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (u4: T).(eq C (CHead c4 (Bind b) u2) (CHead e0 (Flat f) u4)))))) -(\lambda (f: F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop (S i) -(S i) (CHead c3 (Bind b) u1) (CHead e0 (Flat f) u3)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 (S i) v0 u3 -u4)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C (CHead c4 (Bind b) u2) (CHead e2 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S i) -(S i) (CHead c3 (Bind b) u1) (CHead e1 (Flat f) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (S i) v0 e1 -e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 (Bind b) u2) (CHead e2 -(Flat f) u4))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u3: T).(\lambda (_: T).(drop (S i) (S i) (CHead c3 (Bind b) u1) (CHead e1 -(Flat f) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda -(u3: T).(\lambda (u4: T).(subst0 (S i) v0 u3 u4)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -(S i) v0 e1 e2)))))))) H6))))))))))))) (\lambda (f: F).(\lambda (i: -nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 -i v0 u1 u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (H3: (csubst0 i v0 c3 -c4)).(\lambda (H4: (((eq nat i O) \to (or4 (drop i i c3 c4) (ex3_4 F C T T -(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C c4 -(CHead e0 (Flat f0) u4)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u3: -T).(\lambda (_: T).(drop i i c3 (CHead e0 (Flat f0) u3)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 i v0 u3 u4)))))) -(ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(eq C c4 (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(drop i i c3 (CHead e1 (Flat f0) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i -v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C c4 (CHead e2 (Flat f0) -u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: -T).(\lambda (_: T).(drop i i c3 (CHead e1 (Flat f0) u3))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 -i v0 u3 u4)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (H5: (eq nat i -O)).(let H6 \def (eq_ind nat i (\lambda (n0: nat).((eq nat n0 O) \to (or4 -(drop n0 n0 c3 c4) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda -(_: T).(\lambda (u4: T).(eq C c4 (CHead e0 (Flat f0) u4)))))) (\lambda (f0: -F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop n0 n0 c3 (CHead e0 -(Flat f0) u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda -(u4: T).(subst0 n0 v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda -(_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c4 (CHead e2 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 n0 -c3 (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C c4 -(CHead e2 (Flat f0) u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u3: T).(\lambda (_: T).(drop n0 n0 c3 (CHead e1 (Flat f0) -u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u3: -T).(\lambda (u4: T).(subst0 n0 v0 u3 u4)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 n0 v0 e1 -e2)))))))))) H4 O H5) in (let H7 \def (eq_ind nat i (\lambda (n0: -nat).(csubst0 n0 v0 c3 c4)) H3 O H5) in (let H8 \def (eq_ind nat i (\lambda -(n0: nat).(subst0 n0 v0 u1 u2)) H2 O H5) in (eq_ind_r nat O (\lambda (n0: -nat).(or4 (drop n0 n0 (CHead c3 (Flat f) u1) (CHead c4 (Flat f) u2)) (ex3_4 F -C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq -C (CHead c4 (Flat f) u2) (CHead e0 (Flat f0) u4)))))) (\lambda (f0: -F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop n0 n0 (CHead c3 -(Flat f) u1) (CHead e0 (Flat f0) u3)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u3: T).(\lambda (u4: T).(subst0 n0 v0 u3 u4)))))) (ex3_4 F C C T -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C -(CHead c4 (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 n0 (CHead c3 (Flat f) u1) -(CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C -(CHead c4 (Flat f) u2) (CHead e2 (Flat f0) u4))))))) (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (_: T).(drop n0 -n0 (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u3))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 -n0 v0 u3 u4)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 n0 v0 e1 e2))))))))) (or4_intro3 (drop O O -(CHead c3 (Flat f) u1) (CHead c4 (Flat f) u2)) (ex3_4 F C T T (\lambda (f0: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 (Flat f) -u2) (CHead e0 (Flat f0) u4)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda -(u3: T).(\lambda (_: T).(drop O O (CHead c3 (Flat f) u1) (CHead e0 (Flat f0) -u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: -T).(subst0 O v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c4 (Flat f) u2) (CHead e2 -(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(drop O O (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u)))))) (\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 -e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 (Flat f) u2) (CHead e2 -(Flat f0) u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c3 (Flat f) u1) (CHead -e1 (Flat f0) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v0 e1 e2))))))) (ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 (Flat f) -u2) (CHead e2 (Flat f0) u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c3 (Flat f) u1) -(CHead e1 (Flat f0) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v0 e1 e2)))))) f c3 c4 u1 u2 (refl_equal C (CHead c4 (Flat f) u2)) -(drop_refl (CHead c3 (Flat f) u1)) H8 H7)) i H5))))))))))))))) k)) y v c1 c2 -H1))) H) e (drop_gen_refl c2 e H0)))))))) (\lambda (n0: nat).(\lambda (IHn: -((\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 n0 v c1 c2) \to -(\forall (e: C).((drop n0 O c2 e) \to (or4 (drop n0 O c1 e) (ex3_4 F C T T -(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e -(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(drop n0 O c1 (CHead e0 (Flat f) u1)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c1 (CHead e1 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O -v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop n0 O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda -(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 -u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))))).(\lambda (c1: C).(C_ind -(\lambda (c: C).(\forall (c2: C).(\forall (v: T).((csubst0 (S n0) v c c2) \to -(\forall (e: C).((drop (S n0) O c2 e) \to (or4 (drop (S n0) O c e) (ex3_4 F C -T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e -(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f) u1)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O c (CHead e1 (Flat f) -u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat -f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f) u1))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 -O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))) (\lambda (n1: -nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (H: (csubst0 (S n0) v (CSort -n1) c2)).(\lambda (e: C).(\lambda (_: (drop (S n0) O c2 e)).(csubst0_gen_sort -c2 v (S n0) n1 H (or4 (drop (S n0) O (CSort n1) e) (ex3_4 F C T T (\lambda -(f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 -(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S n0) O (CSort n1) (CHead e0 (Flat f) u1)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CSort n1) (CHead e1 (Flat -f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat -f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O (CSort n1) (CHead e1 (Flat f) u1))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))))))) -(\lambda (c: C).(\lambda (H: ((\forall (c2: C).(\forall (v: T).((csubst0 (S -n0) v c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (or4 (drop (S n0) O -c e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda -(u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f) -u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) -O c (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e -(CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f) -u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2))))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda -(v: T).(\lambda (H0: (csubst0 (S n0) v (CHead c k t) c2)).(\lambda (e: -C).(\lambda (H1: (drop (S n0) O c2 e)).(or3_ind (ex3_2 T nat (\lambda (_: -T).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (u2: T).(\lambda (_: -nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j -v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k -j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda -(c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_: -T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k j))))) (\lambda (u2: -T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda -(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: -T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))) (or4 (drop (S -n0) O (CHead c k t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) -(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O (CHead c k t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e -(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O -v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u1))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 -O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H2: (ex3_2 T nat -(\lambda (_: T).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (u2: -T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: -nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: -nat).(eq nat (S n0) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 -(CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))) (or4 -(drop (S n0) O (CHead c k t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) -(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O (CHead c k t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e -(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O -v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u1))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 -O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: T).(\lambda -(x1: nat).(\lambda (H3: (eq nat (S n0) (s k x1))).(\lambda (H4: (eq C c2 -(CHead c k x0))).(\lambda (H5: (subst0 x1 v t x0)).(let H6 \def (eq_ind C c2 -(\lambda (c0: C).(drop (S n0) O c0 e)) H1 (CHead c k x0) H4) in (K_ind -(\lambda (k0: K).((eq nat (S n0) (s k0 x1)) \to ((drop (r k0 n0) O c e) \to -(or4 (drop (S n0) O (CHead c k0 t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda -(e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) -(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O (CHead c k0 t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e -(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(drop (S n0) O (CHead c k0 t) (CHead e1 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O -v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S n0) O (CHead c k0 t) (CHead e1 (Flat f) u1))))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))) (\lambda (b: -B).(\lambda (H7: (eq nat (S n0) (s (Bind b) x1))).(\lambda (H8: (drop (r -(Bind b) n0) O c e)).(let H9 \def (f_equal nat nat (\lambda (e0: nat).(match -e0 in nat return (\lambda (_: nat).nat) with [O \Rightarrow n0 | (S n1) -\Rightarrow n1])) (S n0) (S x1) H7) in (let H10 \def (eq_ind_r nat x1 -(\lambda (n1: nat).(subst0 n1 v t x0)) H5 n0 H9) in (or4_intro0 (drop (S n0) -O (CHead c (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) -(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead -e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e -(CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) -(CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2))))))) (drop_drop (Bind b) n0 c e H8 t))))))) (\lambda (f: -F).(\lambda (H7: (eq nat (S n0) (s (Flat f) x1))).(\lambda (H8: (drop (r -(Flat f) n0) O c e)).(let H9 \def (f_equal nat nat (\lambda (e0: nat).e0) (S -n0) x1 H7) in (let H10 \def (eq_ind_r nat x1 (\lambda (n1: nat).(subst0 n1 v -t x0)) H5 (S n0) H9) in (or4_intro0 (drop (S n0) O (CHead c (Flat f) t) e) -(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda -(u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) -(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat -f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 -e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop -(Flat f) n0 c e H8 t))))))) k H3 (drop_gen_drop k c e x0 n0 H6)))))))) H2)) -(\lambda (H2: (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) -(s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) -(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))).(ex3_2_ind C nat -(\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (c3: -C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: -nat).(csubst0 j v c c3))) (or4 (drop (S n0) O (CHead c k t) e) (ex3_4 F C T T -(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e -(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O (CHead c k t) (CHead e0 (Flat f) u1)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v -u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c k -t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e -(CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k t) (CHead e1 -(Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2)))))))) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H3: (eq nat (S n0) -(s k x1))).(\lambda (H4: (eq C c2 (CHead x0 k t))).(\lambda (H5: (csubst0 x1 -v c x0)).(let H6 \def (eq_ind C c2 (\lambda (c0: C).(drop (S n0) O c0 e)) H1 -(CHead x0 k t) H4) in (K_ind (\lambda (k0: K).((eq nat (S n0) (s k0 x1)) \to -((drop (r k0 n0) O x0 e) \to (or4 (drop (S n0) O (CHead c k0 t) e) (ex3_4 F C -T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e -(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O (CHead c k0 t) (CHead e0 (Flat f) u1)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v -u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c -k0 t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda -(f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq -C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k0 t) (CHead -e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2))))))))))) (\lambda (b: B).(\lambda (H7: (eq nat (S n0) (s (Bind b) -x1))).(\lambda (H8: (drop (r (Bind b) n0) O x0 e)).(let H9 \def (f_equal nat -nat (\lambda (e0: nat).(match e0 in nat return (\lambda (_: nat).nat) with [O -\Rightarrow n0 | (S n1) \Rightarrow n1])) (S n0) (S x1) H7) in (let H10 \def -(eq_ind_r nat x1 (\lambda (n1: nat).(csubst0 n1 v c x0)) H5 n0 H9) in (let -H11 \def (IHn c x0 v H10 e H8) in (or4_ind (drop n0 O c e) (ex3_4 F C T T -(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e -(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O -v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop n0 O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e) -(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: -T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) -(CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat -f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 -e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda -(H12: (drop n0 O c e)).(or4_intro0 (drop (S n0) O (CHead c (Bind b) t) e) -(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: -T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) -(CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat -f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 -e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop -(Bind b) n0 c e H12 t))) (\lambda (H12: (ex3_4 F C T T (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat -f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))).(ex3_4_ind F C -T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e -(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))) -(or4 (drop (S n0) O (CHead c (Bind b) t) e) (ex3_4 F C T T (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat -f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 -u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c -(Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c -(Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda -(x4: T).(\lambda (x5: T).(\lambda (H13: (eq C e (CHead x3 (Flat x2) -x5))).(\lambda (H14: (drop n0 O c (CHead x3 (Flat x2) x4))).(\lambda (H15: -(subst0 O v x4 x5)).(eq_ind_r C (CHead x3 (Flat x2) x5) (\lambda (c0: C).(or4 -(drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat -f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 -u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c -(Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c -(Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead c (Bind -b) t) (CHead x3 (Flat x2) x5)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0 -(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v -u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C (CHead x3 (Flat x2) x5) (CHead e2 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) -O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C -C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e2 (Flat f) -u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) -u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) -(ex3_4_intro F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0 (Flat f) u2)))))) -(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))) -x2 x3 x4 x5 (refl_equal C (CHead x3 (Flat x2) x5)) (drop_drop (Bind b) n0 c -(CHead x3 (Flat x2) x4) H14 t) H15)) e H13)))))))) H12)) (\lambda (H12: -(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O -v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1 -(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 O v e1 e2))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e) -(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: -T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) -(CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat -f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 -e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda -(x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: T).(\lambda (H13: (eq -C e (CHead x4 (Flat x2) x5))).(\lambda (H14: (drop n0 O c (CHead x3 (Flat x2) -x5))).(\lambda (H15: (csubst0 O v x3 x4)).(eq_ind_r C (CHead x4 (Flat x2) x5) -(\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T -(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 -(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) -u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) -O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C -C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 -O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro2 (drop (S n0) O -(CHead c (Bind b) t) (CHead x4 (Flat x2) x5)) (ex3_4 F C T T (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat -x2) x5) (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda -(u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat -f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 -(Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O -v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 -(Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat -f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) -(ex3_4_intro F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) -O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) x2 x3 x4 x5 -(refl_equal C (CHead x4 (Flat x2) x5)) (drop_drop (Bind b) n0 c (CHead x3 -(Flat x2) x5) H14 t) H15)) e H13)))))))) H12)) (\lambda (H12: (ex4_5 F C C T -T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c (CHead e1 -(Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2)))))))).(ex4_5_ind F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop n0 O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e) -(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: -T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) -(CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat -f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 -e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda -(x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: T).(\lambda (x6: -T).(\lambda (H13: (eq C e (CHead x4 (Flat x2) x6))).(\lambda (H14: (drop n0 O -c (CHead x3 (Flat x2) x5))).(\lambda (H15: (subst0 O v x5 x6)).(\lambda (H16: -(csubst0 O v x3 x4)).(eq_ind_r C (CHead x4 (Flat x2) x6) (\lambda (c0: -C).(or4 (drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat -f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 -u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c -(Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c -(Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead c (Bind -b) t) (CHead x4 (Flat x2) x6)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e0 -(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v -u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) -O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C -C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 (Flat f) -u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) -u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) -(ex4_5_intro F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 -(Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat -f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) -x2 x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x2) x6)) (drop_drop (Bind b) n0 -c (CHead x3 (Flat x2) x5) H14 t) H15 H16)) e H13)))))))))) H12)) H11))))))) -(\lambda (f: F).(\lambda (H7: (eq nat (S n0) (s (Flat f) x1))).(\lambda (H8: -(drop (r (Flat f) n0) O x0 e)).(let H9 \def (f_equal nat nat (\lambda (e0: -nat).e0) (S n0) x1 H7) in (let H10 \def (eq_ind_r nat x1 (\lambda (n1: -nat).(csubst0 n1 v c x0)) H5 (S n0) H9) in (let H11 \def (H x0 v H10 e H8) in -(or4_ind (drop (S n0) O c e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O c (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat -f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(drop (S n0) O c (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead -e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2))))))) (or4 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T -(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e -(CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) -u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S -n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) -(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 -O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H12: (drop (S n0) -O c e)).(or4_intro0 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T -(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e -(CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) -u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S -n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) -(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 -O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Flat f) n0 c e -H12 t))) (\lambda (H12: (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O c (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))).(ex3_4_ind F C T T (\lambda -(f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 -(Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f0) u1)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))) -(or4 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat -f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 -u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c -(Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c -(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda -(x4: T).(\lambda (x5: T).(\lambda (H13: (eq C e (CHead x3 (Flat x2) -x5))).(\lambda (H14: (drop (S n0) O c (CHead x3 (Flat x2) x4))).(\lambda -(H15: (subst0 O v x4 x5)).(eq_ind_r C (CHead x3 (Flat x2) x5) (\lambda (c0: -C).(or4 (drop (S n0) O (CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat -f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 -u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c -(Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(eq C c0 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c -(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead c (Flat -f) t) (CHead x3 (Flat x2) x5)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0 -(Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) -u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x3 (Flat x2) x5) (CHead e2 -(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O -v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e2 -(Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) -(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2))))))) (ex3_4_intro F C T T (\lambda (f0: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0 -(Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) -u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 O v u1 u2))))) x2 x3 x4 x5 (refl_equal C (CHead x3 (Flat x2) x5)) -(drop_drop (Flat f) n0 c (CHead x3 (Flat x2) x4) H14 t) H15)) e H13)))))))) -H12)) (\lambda (H12: (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O c (CHead -e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat -f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(drop (S n0) O c (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) (or4 (drop (S n0) -O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead -e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e -(CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) -(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda -(x5: T).(\lambda (H13: (eq C e (CHead x4 (Flat x2) x5))).(\lambda (H14: (drop -(S n0) O c (CHead x3 (Flat x2) x5))).(\lambda (H15: (csubst0 O v x3 -x4)).(eq_ind_r C (CHead x4 (Flat x2) x5) (\lambda (c0: C).(or4 (drop (S n0) O -(CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat f0) u2)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead -e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 -(CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) -(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead c (Flat f) t) (CHead x4 -(Flat x2) x5)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x5) (CHead e0 (Flat f0) -u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 -u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S -n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) -(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 (Flat f0) -u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) -u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) -(ex3_4_intro F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S -n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) -x2 x3 x4 x5 (refl_equal C (CHead x4 (Flat x2) x5)) (drop_drop (Flat f) n0 c -(CHead x3 (Flat x2) x5) H14 t) H15)) e H13)))))))) H12)) (\lambda (H12: -(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O c (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e -(CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f0) -u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) -(or4 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat -f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 -u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c -(Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c -(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda -(x4: C).(\lambda (x5: T).(\lambda (x6: T).(\lambda (H13: (eq C e (CHead x4 -(Flat x2) x6))).(\lambda (H14: (drop (S n0) O c (CHead x3 (Flat x2) -x5))).(\lambda (H15: (subst0 O v x5 x6)).(\lambda (H16: (csubst0 O v x3 -x4)).(eq_ind_r C (CHead x4 (Flat x2) x6) (\lambda (c0: C).(or4 (drop (S n0) O -(CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat f0) u2)))))) -(\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead -e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 -(CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) -(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead c (Flat f) t) (CHead x4 -(Flat x2) x6)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e0 (Flat f0) -u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 -u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S -n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) -(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 (Flat f0) -u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) -u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) -(ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 -(Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) -(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2)))))) x2 x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x2) x6)) -(drop_drop (Flat f) n0 c (CHead x3 (Flat x2) x5) H14 t) H15 H16)) e -H13)))))))))) H12)) H11))))))) k H3 (drop_gen_drop k x0 e t n0 H6)))))))) -H2)) (\lambda (H2: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda -(j: nat).(eq nat (S n0) (s k j))))) (\lambda (u2: T).(\lambda (c3: -C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda -(_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: -C).(\lambda (j: nat).(csubst0 j v c c3)))))).(ex4_3_ind T C nat (\lambda (_: -T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k j))))) (\lambda (u2: -T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda -(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: -T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (or4 (drop (S n0) -O (CHead c k t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda -(_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c -k t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat -f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) -(ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O (CHead c k t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: C).(\lambda -(x2: nat).(\lambda (H3: (eq nat (S n0) (s k x2))).(\lambda (H4: (eq C c2 -(CHead x1 k x0))).(\lambda (H5: (subst0 x2 v t x0)).(\lambda (H6: (csubst0 x2 -v c x1)).(let H7 \def (eq_ind C c2 (\lambda (c0: C).(drop (S n0) O c0 e)) H1 -(CHead x1 k x0) H4) in (K_ind (\lambda (k0: K).((eq nat (S n0) (s k0 x2)) \to -((drop (r k0 n0) O x1 e) \to (or4 (drop (S n0) O (CHead c k0 t) e) (ex3_4 F C -T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e -(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O (CHead c k0 t) (CHead e0 (Flat f) u1)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v -u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c -k0 t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda -(f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq -C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k0 t) (CHead -e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2))))))))))) (\lambda (b: B).(\lambda (H8: (eq nat (S n0) (s (Bind b) -x2))).(\lambda (H9: (drop (r (Bind b) n0) O x1 e)).(let H10 \def (f_equal nat -nat (\lambda (e0: nat).(match e0 in nat return (\lambda (_: nat).nat) with [O -\Rightarrow n0 | (S n1) \Rightarrow n1])) (S n0) (S x2) H8) in (let H11 \def -(eq_ind_r nat x2 (\lambda (n1: nat).(csubst0 n1 v c x1)) H6 n0 H10) in (let -H12 \def (eq_ind_r nat x2 (\lambda (n1: nat).(subst0 n1 v t x0)) H5 n0 H10) -in (let H13 \def (IHn c x1 v H11 e H9) in (or4_ind (drop n0 O c e) (ex3_4 F C -T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e -(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O -v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop n0 O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e) -(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: -T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) -(CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat -f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 -e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda -(H14: (drop n0 O c e)).(or4_intro0 (drop (S n0) O (CHead c (Bind b) t) e) -(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: -T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) -(CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat -f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 -e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop -(Bind b) n0 c e H14 t))) (\lambda (H14: (ex3_4 F C T T (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat -f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))).(ex3_4_ind F C -T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e -(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))) -(or4 (drop (S n0) O (CHead c (Bind b) t) e) (ex3_4 F C T T (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat -f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 -u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c -(Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c -(Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda -(x5: T).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x4 (Flat x3) -x6))).(\lambda (H16: (drop n0 O c (CHead x4 (Flat x3) x5))).(\lambda (H17: -(subst0 O v x5 x6)).(eq_ind_r C (CHead x4 (Flat x3) x6) (\lambda (c0: C).(or4 -(drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat -f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 -u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c -(Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c -(Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead c (Bind -b) t) (CHead x4 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0 -(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v -u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C (CHead x4 (Flat x3) x6) (CHead e2 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) -O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C -C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (u2: T).(eq C (CHead x4 (Flat x3) x6) (CHead e2 (Flat f) -u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) -u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) -(ex3_4_intro F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (u2: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0 (Flat f) u2)))))) -(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))) -x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x3) x6)) (drop_drop (Bind b) n0 c -(CHead x4 (Flat x3) x5) H16 t) H17)) e H15)))))))) H14)) (\lambda (H14: -(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O -v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1 -(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 O v e1 e2))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e) -(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: -T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) -(CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat -f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 -e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda -(x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (H15: (eq -C e (CHead x5 (Flat x3) x6))).(\lambda (H16: (drop n0 O c (CHead x4 (Flat x3) -x6))).(\lambda (H17: (csubst0 O v x4 x5)).(eq_ind_r C (CHead x5 (Flat x3) x6) -(\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T -(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 -(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) -u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) -O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C -C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 -O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro2 (drop (S n0) O -(CHead c (Bind b) t) (CHead x5 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat -x3) x6) (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda -(u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat -f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2 -(Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O -v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2 -(Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat -f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) -(ex3_4_intro F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) -O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) x3 x4 x5 x6 -(refl_equal C (CHead x5 (Flat x3) x6)) (drop_drop (Bind b) n0 c (CHead x4 -(Flat x3) x6) H16 t) H17)) e H15)))))))) H14)) (\lambda (H14: (ex4_5 F C C T -T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c (CHead e1 -(Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 -e2)))))))).(ex4_5_ind F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop n0 O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e) -(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: -T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) -(CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat -f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 -e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda -(x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (x7: -T).(\lambda (H15: (eq C e (CHead x5 (Flat x3) x7))).(\lambda (H16: (drop n0 O -c (CHead x4 (Flat x3) x6))).(\lambda (H17: (subst0 O v x6 x7)).(\lambda (H18: -(csubst0 O v x4 x5)).(eq_ind_r C (CHead x5 (Flat x3) x7) (\lambda (c0: -C).(or4 (drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat -f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 -u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c -(Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c -(Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead c (Bind -b) t) (CHead x5 (Flat x3) x7)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e0 -(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v -u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) -O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C -C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2 (Flat f) -u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) -u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) -(ex4_5_intro F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2 -(Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat -f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) -x3 x4 x5 x6 x7 (refl_equal C (CHead x5 (Flat x3) x7)) (drop_drop (Bind b) n0 -c (CHead x4 (Flat x3) x6) H16 t) H17 H18)) e H15)))))))))) H14)) H13)))))))) -(\lambda (f: F).(\lambda (H8: (eq nat (S n0) (s (Flat f) x2))).(\lambda (H9: -(drop (r (Flat f) n0) O x1 e)).(let H10 \def (f_equal nat nat (\lambda (e0: -nat).e0) (S n0) x2 H8) in (let H11 \def (eq_ind_r nat x2 (\lambda (n1: -nat).(csubst0 n1 v c x1)) H6 (S n0) H10) in (let H12 \def (eq_ind_r nat x2 -(\lambda (n1: nat).(subst0 n1 v t x0)) H5 (S n0) H10) in (let H13 \def (H x1 -v H11 e H9) in (or4_ind (drop (S n0) O c e) (ex3_4 F C T T (\lambda (f0: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat -f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(drop (S n0) O c (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e -(CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(drop (S n0) O c (CHead e1 (Flat f0) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) -(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O c (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead c (Flat f) t) e) -(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda -(u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) -(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat -f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 -e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda -(H14: (drop (S n0) O c e)).(or4_intro0 (drop (S n0) O (CHead c (Flat f) t) e) -(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda -(u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) -(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat -f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 -e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop -(Flat f) n0 c e H14 t))) (\lambda (H14: (ex3_4 F C T T (\lambda (f0: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat -f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(drop (S n0) O c (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))).(ex3_4_ind F C -T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C -e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda -(u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f0) u1)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v -u1 u2))))) (or4 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T -(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e -(CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) -u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S -n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) -(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 -O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x3: F).(\lambda -(x4: C).(\lambda (x5: T).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x4 -(Flat x3) x6))).(\lambda (H16: (drop (S n0) O c (CHead x4 (Flat x3) -x5))).(\lambda (H17: (subst0 O v x5 x6)).(eq_ind_r C (CHead x4 (Flat x3) x6) -(\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Flat f) t) c0) (ex3_4 F C T T -(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 -(CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) -u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f0) u)))))) -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S -n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) -(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 -O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro1 (drop (S n0) O -(CHead c (Flat f) t) (CHead x4 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f0: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat -x3) x6) (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) -(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x4 (Flat x3) -x6) (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda -(_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat -f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat -x3) x6) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c -(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C T T (\lambda (f0: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat -x3) x6) (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) -(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2))))) x3 x4 x5 x6 (refl_equal C (CHead -x4 (Flat x3) x6)) (drop_drop (Flat f) n0 c (CHead x4 (Flat x3) x5) H16 t) -H17)) e H15)))))))) H14)) (\lambda (H14: (ex3_4 F C C T (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat -f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(drop (S n0) O c (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C -C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e -(CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(drop (S n0) O c (CHead e1 (Flat f0) u)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) -(or4 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat -f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 -u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c -(Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c -(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda -(x5: C).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x5 (Flat x3) -x6))).(\lambda (H16: (drop (S n0) O c (CHead x4 (Flat x3) x6))).(\lambda -(H17: (csubst0 O v x4 x5)).(eq_ind_r C (CHead x5 (Flat x3) x6) (\lambda (c0: -C).(or4 (drop (S n0) O (CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat -f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 -u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c -(Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(eq C c0 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c -(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead c (Flat -f) t) (CHead x5 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x6) (CHead e0 -(Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) -u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2 -(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O -v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2 -(Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) -(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2))))))) (ex3_4_intro F C C T (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2 -(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O -v e1 e2))))) x3 x4 x5 x6 (refl_equal C (CHead x5 (Flat x3) x6)) (drop_drop -(Flat f) n0 c (CHead x4 (Flat x3) x6) H16 t) H17)) e H15)))))))) H14)) -(\lambda (H14: (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) -u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f0) u1))))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C -C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S -n0) O c (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) O (CHead c (Flat f) t) e) -(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda -(u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) -(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat -f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda -(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 -e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) -(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda -(x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (x7: -T).(\lambda (H15: (eq C e (CHead x5 (Flat x3) x7))).(\lambda (H16: (drop (S -n0) O c (CHead x4 (Flat x3) x6))).(\lambda (H17: (subst0 O v x6 x7)).(\lambda -(H18: (csubst0 O v x4 x5)).(eq_ind_r C (CHead x5 (Flat x3) x7) (\lambda (c0: -C).(or4 (drop (S n0) O (CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat -f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: -T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda -(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 -u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c -(Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T -(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(u2: T).(eq C c0 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c -(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead c (Flat -f) t) (CHead x5 (Flat x3) x7)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e0 -(Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) -u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: -T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2 -(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O -v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2 -(Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) -(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2))))))) (ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat -x3) x7) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c -(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) -(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 O v e1 e2)))))) x3 x4 x5 x6 x7 (refl_equal C (CHead x5 (Flat -x3) x7)) (drop_drop (Flat f) n0 c (CHead x4 (Flat x3) x6) H16 t) H17 H18)) e -H15)))))))))) H14)) H13)))))))) k H3 (drop_gen_drop k x1 e x0 n0 H7)))))))))) -H2)) (csubst0_gen_head k c c2 t v (S n0) H0))))))))))) c1)))) n). -(* COMMENTS -Initial nodes: 34765 -END *) - -theorem csubst0_drop_lt_back: - \forall (n: nat).(\forall (i: nat).((lt n i) \to (\forall (c1: C).(\forall -(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e2: C).((drop n O -c2 e2) \to (or (drop n O c1 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) -v e1 e2)) (\lambda (e1: C).(drop n O c1 e1)))))))))))) -\def - \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (i: nat).((lt n0 i) -\to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) -\to (\forall (e2: C).((drop n0 O c2 e2) \to (or (drop n0 O c1 e2) (ex2 C -(\lambda (e1: C).(csubst0 (minus i n0) v e1 e2)) (\lambda (e1: C).(drop n0 O -c1 e1))))))))))))) (\lambda (i: nat).(\lambda (_: (lt O i)).(\lambda (c1: -C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 i v c1 -c2)).(\lambda (e2: C).(\lambda (H1: (drop O O c2 e2)).(eq_ind C c2 (\lambda -(c: C).(or (drop O O c1 c) (ex2 C (\lambda (e1: C).(csubst0 (minus i O) v e1 -c)) (\lambda (e1: C).(drop O O c1 e1))))) (eq_ind nat i (\lambda (n0: -nat).(or (drop O O c1 c2) (ex2 C (\lambda (e1: C).(csubst0 n0 v e1 c2)) -(\lambda (e1: C).(drop O O c1 e1))))) (or_intror (drop O O c1 c2) (ex2 C -(\lambda (e1: C).(csubst0 i v e1 c2)) (\lambda (e1: C).(drop O O c1 e1))) -(ex_intro2 C (\lambda (e1: C).(csubst0 i v e1 c2)) (\lambda (e1: C).(drop O O -c1 e1)) c1 H0 (drop_refl c1))) (minus i O) (minus_n_O i)) e2 (drop_gen_refl -c2 e2 H1)))))))))) (\lambda (n0: nat).(\lambda (IHn: ((\forall (i: nat).((lt -n0 i) \to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 -c2) \to (\forall (e2: C).((drop n0 O c2 e2) \to (or (drop n0 O c1 e2) (ex2 C -(\lambda (e1: C).(csubst0 (minus i n0) v e1 e2)) (\lambda (e1: C).(drop n0 O -c1 e1)))))))))))))).(\lambda (i: nat).(\lambda (H: (lt (S n0) i)).(\lambda -(c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v -c c2) \to (\forall (e2: C).((drop (S n0) O c2 e2) \to (or (drop (S n0) O c -e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i (S n0)) v e1 e2)) (\lambda (e1: -C).(drop (S n0) O c e1)))))))))) (\lambda (n1: nat).(\lambda (c2: C).(\lambda -(v: T).(\lambda (H0: (csubst0 i v (CSort n1) c2)).(\lambda (e2: C).(\lambda -(_: (drop (S n0) O c2 e2)).(csubst0_gen_sort c2 v i n1 H0 (or (drop (S n0) O -(CSort n1) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i (S n0)) v e1 e2)) -(\lambda (e1: C).(drop (S n0) O (CSort n1) e1))))))))))) (\lambda (c: -C).(\lambda (H0: ((\forall (c2: C).(\forall (v: T).((csubst0 i v c c2) \to -(\forall (e2: C).((drop (S n0) O c2 e2) \to (or (drop (S n0) O c e2) (ex2 C -(\lambda (e1: C).(csubst0 (minus i (S n0)) v e1 e2)) (\lambda (e1: C).(drop -(S n0) O c e1))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: -C).(\lambda (v: T).(\lambda (H1: (csubst0 i v (CHead c k t) c2)).(\lambda -(e2: C).(\lambda (H2: (drop (S n0) O c2 e2)).(or3_ind (ex3_2 T nat (\lambda -(_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: -nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j -v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k -j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda -(c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_: -T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: -T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda -(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: -T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))) (or (drop (S n0) -O (CHead c k t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i (S n0)) v e1 -e2)) (\lambda (e1: C).(drop (S n0) O (CHead c k t) e1)))) (\lambda (H3: -(ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda -(u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: -T).(\lambda (j: nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: -T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: -nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j -v t u2))) (or (drop (S n0) O (CHead c k t) e2) (ex2 C (\lambda (e1: -C).(csubst0 (minus i (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead -c k t) e1)))) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H4: (eq nat i (s -k x1))).(\lambda (H5: (eq C c2 (CHead c k x0))).(\lambda (_: (subst0 x1 v t -x0)).(let H7 \def (eq_ind C c2 (\lambda (c0: C).(drop (S n0) O c0 e2)) H2 -(CHead c k x0) H5) in (let H8 \def (eq_ind nat i (\lambda (n1: nat).(\forall -(c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall (e3: C).((drop (S -n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: C).(csubst0 -(minus n1 (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop (S n0) O c e1)))))))))) -H0 (s k x1) H4) in (let H9 \def (eq_ind nat i (\lambda (n1: nat).(lt (S n0) -n1)) H (s k x1) H4) in (eq_ind_r nat (s k x1) (\lambda (n1: nat).(or (drop (S -n0) O (CHead c k t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus n1 (S n0)) v -e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c k t) e1))))) (K_ind (\lambda -(k0: K).(((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c c3) \to -(\forall (e3: C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C -(\lambda (e1: C).(csubst0 (minus (s k0 x1) (S n0)) v0 e1 e3)) (\lambda (e1: -C).(drop (S n0) O c e1)))))))))) \to ((lt (S n0) (s k0 x1)) \to ((drop (r k0 -n0) O c e2) \to (or (drop (S n0) O (CHead c k0 t) e2) (ex2 C (\lambda (e1: -C).(csubst0 (minus (s k0 x1) (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) -O (CHead c k0 t) e1)))))))) (\lambda (b: B).(\lambda (_: ((\forall (c3: -C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to (\forall (e3: -C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: -C).(csubst0 (minus (s (Bind b) x1) (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop -(S n0) O c e1))))))))))).(\lambda (_: (lt (S n0) (s (Bind b) x1))).(\lambda -(H12: (drop (r (Bind b) n0) O c e2)).(or_introl (drop (S n0) O (CHead c (Bind -b) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x1 n0) v e1 e2)) (\lambda -(e1: C).(drop (S n0) O (CHead c (Bind b) t) e1))) (drop_drop (Bind b) n0 c e2 -H12 t)))))) (\lambda (f: F).(\lambda (_: ((\forall (c3: C).(\forall (v0: -T).((csubst0 (s (Flat f) x1) v0 c c3) \to (\forall (e3: C).((drop (S n0) O c3 -e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: C).(csubst0 (minus (s -(Flat f) x1) (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop (S n0) O c -e1))))))))))).(\lambda (_: (lt (S n0) (s (Flat f) x1))).(\lambda (H12: (drop -(r (Flat f) n0) O c e2)).(or_introl (drop (S n0) O (CHead c (Flat f) t) e2) -(ex2 C (\lambda (e1: C).(csubst0 (minus x1 (S n0)) v e1 e2)) (\lambda (e1: -C).(drop (S n0) O (CHead c (Flat f) t) e1))) (drop_drop (Flat f) n0 c e2 H12 -t)))))) k H8 H9 (drop_gen_drop k c e2 x0 n0 H7)) i H4))))))))) H3)) (\lambda -(H3: (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j)))) -(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: -C).(\lambda (j: nat).(csubst0 j v c c3))))).(ex3_2_ind C nat (\lambda (_: -C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_: -nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j -v c c3))) (or (drop (S n0) O (CHead c k t) e2) (ex2 C (\lambda (e1: -C).(csubst0 (minus i (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead -c k t) e1)))) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H4: (eq nat i (s -k x1))).(\lambda (H5: (eq C c2 (CHead x0 k t))).(\lambda (H6: (csubst0 x1 v c -x0)).(let H7 \def (eq_ind C c2 (\lambda (c0: C).(drop (S n0) O c0 e2)) H2 -(CHead x0 k t) H5) in (let H8 \def (eq_ind nat i (\lambda (n1: nat).(\forall -(c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall (e3: C).((drop (S -n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: C).(csubst0 -(minus n1 (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop (S n0) O c e1)))))))))) -H0 (s k x1) H4) in (let H9 \def (eq_ind nat i (\lambda (n1: nat).(lt (S n0) -n1)) H (s k x1) H4) in (eq_ind_r nat (s k x1) (\lambda (n1: nat).(or (drop (S -n0) O (CHead c k t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus n1 (S n0)) v -e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c k t) e1))))) (K_ind (\lambda -(k0: K).(((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c c3) \to -(\forall (e3: C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C -(\lambda (e1: C).(csubst0 (minus (s k0 x1) (S n0)) v0 e1 e3)) (\lambda (e1: -C).(drop (S n0) O c e1)))))))))) \to ((lt (S n0) (s k0 x1)) \to ((drop (r k0 -n0) O x0 e2) \to (or (drop (S n0) O (CHead c k0 t) e2) (ex2 C (\lambda (e1: -C).(csubst0 (minus (s k0 x1) (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) -O (CHead c k0 t) e1)))))))) (\lambda (b: B).(\lambda (_: ((\forall (c3: -C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to (\forall (e3: -C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: -C).(csubst0 (minus (s (Bind b) x1) (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop -(S n0) O c e1))))))))))).(\lambda (H11: (lt (S n0) (s (Bind b) x1))).(\lambda -(H12: (drop (r (Bind b) n0) O x0 e2)).(let H_x \def (IHn x1 (lt_S_n n0 x1 -H11) c x0 v H6 e2 H12) in (let H13 \def H_x in (or_ind (drop n0 O c e2) (ex2 -C (\lambda (e1: C).(csubst0 (minus x1 n0) v e1 e2)) (\lambda (e1: C).(drop n0 -O c e1))) (or (drop (S n0) O (CHead c (Bind b) t) e2) (ex2 C (\lambda (e1: -C).(csubst0 (minus x1 n0) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c -(Bind b) t) e1)))) (\lambda (H14: (drop n0 O c e2)).(or_introl (drop (S n0) O -(CHead c (Bind b) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x1 n0) v e1 -e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Bind b) t) e1))) (drop_drop -(Bind b) n0 c e2 H14 t))) (\lambda (H14: (ex2 C (\lambda (e1: C).(csubst0 -(minus x1 n0) v e1 e2)) (\lambda (e1: C).(drop n0 O c e1)))).(ex2_ind C -(\lambda (e1: C).(csubst0 (minus x1 n0) v e1 e2)) (\lambda (e1: C).(drop n0 O -c e1)) (or (drop (S n0) O (CHead c (Bind b) t) e2) (ex2 C (\lambda (e1: -C).(csubst0 (minus x1 n0) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c -(Bind b) t) e1)))) (\lambda (x: C).(\lambda (H15: (csubst0 (minus x1 n0) v x -e2)).(\lambda (H16: (drop n0 O c x)).(or_intror (drop (S n0) O (CHead c (Bind -b) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x1 n0) v e1 e2)) (\lambda -(e1: C).(drop (S n0) O (CHead c (Bind b) t) e1))) (ex_intro2 C (\lambda (e1: -C).(csubst0 (minus x1 n0) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c -(Bind b) t) e1)) x H15 (drop_drop (Bind b) n0 c x H16 t)))))) H14)) -H13))))))) (\lambda (f: F).(\lambda (H10: ((\forall (c3: C).(\forall (v0: -T).((csubst0 (s (Flat f) x1) v0 c c3) \to (\forall (e3: C).((drop (S n0) O c3 -e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: C).(csubst0 (minus (s -(Flat f) x1) (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop (S n0) O c -e1))))))))))).(\lambda (_: (lt (S n0) (s (Flat f) x1))).(\lambda (H12: (drop -(r (Flat f) n0) O x0 e2)).(let H_x \def (H10 x0 v H6 e2 H12) in (let H13 \def -H_x in (or_ind (drop (S n0) O c e2) (ex2 C (\lambda (e1: C).(csubst0 (minus -x1 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O c e1))) (or (drop (S n0) -O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x1 (S n0)) -v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Flat f) t) e1)))) -(\lambda (H14: (drop (S n0) O c e2)).(or_introl (drop (S n0) O (CHead c (Flat -f) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x1 (S n0)) v e1 e2)) -(\lambda (e1: C).(drop (S n0) O (CHead c (Flat f) t) e1))) (drop_drop (Flat -f) n0 c e2 H14 t))) (\lambda (H14: (ex2 C (\lambda (e1: C).(csubst0 (minus x1 -(S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O c e1)))).(ex2_ind C -(\lambda (e1: C).(csubst0 (minus x1 (S n0)) v e1 e2)) (\lambda (e1: C).(drop -(S n0) O c e1)) (or (drop (S n0) O (CHead c (Flat f) t) e2) (ex2 C (\lambda -(e1: C).(csubst0 (minus x1 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O -(CHead c (Flat f) t) e1)))) (\lambda (x: C).(\lambda (H15: (csubst0 (minus x1 -(S n0)) v x e2)).(\lambda (H16: (drop (S n0) O c x)).(or_intror (drop (S n0) -O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x1 (S n0)) -v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Flat f) t) e1))) -(ex_intro2 C (\lambda (e1: C).(csubst0 (minus x1 (S n0)) v e1 e2)) (\lambda -(e1: C).(drop (S n0) O (CHead c (Flat f) t) e1)) x H15 (drop_drop (Flat f) n0 -c x H16 t)))))) H14)) H13))))))) k H8 H9 (drop_gen_drop k x0 e2 t n0 H7)) i -H4))))))))) H3)) (\lambda (H3: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: -C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: -C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda -(_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: -C).(\lambda (j: nat).(csubst0 j v c c3)))))).(ex4_3_ind T C nat (\lambda (_: -T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: -T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda -(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: -T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (or (drop (S n0) -O (CHead c k t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i (S n0)) v e1 -e2)) (\lambda (e1: C).(drop (S n0) O (CHead c k t) e1)))) (\lambda (x0: -T).(\lambda (x1: C).(\lambda (x2: nat).(\lambda (H4: (eq nat i (s k -x2))).(\lambda (H5: (eq C c2 (CHead x1 k x0))).(\lambda (_: (subst0 x2 v t -x0)).(\lambda (H7: (csubst0 x2 v c x1)).(let H8 \def (eq_ind C c2 (\lambda -(c0: C).(drop (S n0) O c0 e2)) H2 (CHead x1 k x0) H5) in (let H9 \def (eq_ind -nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c -c3) \to (\forall (e3: C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) -(ex2 C (\lambda (e1: C).(csubst0 (minus n1 (S n0)) v0 e1 e3)) (\lambda (e1: -C).(drop (S n0) O c e1)))))))))) H0 (s k x2) H4) in (let H10 \def (eq_ind nat -i (\lambda (n1: nat).(lt (S n0) n1)) H (s k x2) H4) in (eq_ind_r nat (s k x2) -(\lambda (n1: nat).(or (drop (S n0) O (CHead c k t) e2) (ex2 C (\lambda (e1: -C).(csubst0 (minus n1 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O -(CHead c k t) e1))))) (K_ind (\lambda (k0: K).(((\forall (c3: C).(\forall -(v0: T).((csubst0 (s k0 x2) v0 c c3) \to (\forall (e3: C).((drop (S n0) O c3 -e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: C).(csubst0 (minus (s -k0 x2) (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop (S n0) O c e1)))))))))) \to -((lt (S n0) (s k0 x2)) \to ((drop (r k0 n0) O x1 e2) \to (or (drop (S n0) O -(CHead c k0 t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus (s k0 x2) (S n0)) -v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c k0 t) e1)))))))) (\lambda -(b: B).(\lambda (_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Bind b) -x2) v0 c c3) \to (\forall (e3: C).((drop (S n0) O c3 e3) \to (or (drop (S n0) -O c e3) (ex2 C (\lambda (e1: C).(csubst0 (minus (s (Bind b) x2) (S n0)) v0 e1 -e3)) (\lambda (e1: C).(drop (S n0) O c e1))))))))))).(\lambda (H12: (lt (S -n0) (s (Bind b) x2))).(\lambda (H13: (drop (r (Bind b) n0) O x1 e2)).(let H_x -\def (IHn x2 (lt_S_n n0 x2 H12) c x1 v H7 e2 H13) in (let H14 \def H_x in -(or_ind (drop n0 O c e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x2 n0) v e1 -e2)) (\lambda (e1: C).(drop n0 O c e1))) (or (drop (S n0) O (CHead c (Bind b) -t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x2 n0) v e1 e2)) (\lambda (e1: -C).(drop (S n0) O (CHead c (Bind b) t) e1)))) (\lambda (H15: (drop n0 O c -e2)).(or_introl (drop (S n0) O (CHead c (Bind b) t) e2) (ex2 C (\lambda (e1: -C).(csubst0 (minus x2 n0) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c -(Bind b) t) e1))) (drop_drop (Bind b) n0 c e2 H15 t))) (\lambda (H15: (ex2 C -(\lambda (e1: C).(csubst0 (minus x2 n0) v e1 e2)) (\lambda (e1: C).(drop n0 O -c e1)))).(ex2_ind C (\lambda (e1: C).(csubst0 (minus x2 n0) v e1 e2)) -(\lambda (e1: C).(drop n0 O c e1)) (or (drop (S n0) O (CHead c (Bind b) t) -e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x2 n0) v e1 e2)) (\lambda (e1: -C).(drop (S n0) O (CHead c (Bind b) t) e1)))) (\lambda (x: C).(\lambda (H16: -(csubst0 (minus x2 n0) v x e2)).(\lambda (H17: (drop n0 O c x)).(or_intror -(drop (S n0) O (CHead c (Bind b) t) e2) (ex2 C (\lambda (e1: C).(csubst0 -(minus x2 n0) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Bind b) t) -e1))) (ex_intro2 C (\lambda (e1: C).(csubst0 (minus x2 n0) v e1 e2)) (\lambda -(e1: C).(drop (S n0) O (CHead c (Bind b) t) e1)) x H16 (drop_drop (Bind b) n0 -c x H17 t)))))) H15)) H14))))))) (\lambda (f: F).(\lambda (H11: ((\forall -(c3: C).(\forall (v0: T).((csubst0 (s (Flat f) x2) v0 c c3) \to (\forall (e3: -C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: -C).(csubst0 (minus (s (Flat f) x2) (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop -(S n0) O c e1))))))))))).(\lambda (_: (lt (S n0) (s (Flat f) x2))).(\lambda -(H13: (drop (r (Flat f) n0) O x1 e2)).(let H_x \def (H11 x1 v H7 e2 H13) in -(let H14 \def H_x in (or_ind (drop (S n0) O c e2) (ex2 C (\lambda (e1: -C).(csubst0 (minus x2 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O c -e1))) (or (drop (S n0) O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1: -C).(csubst0 (minus x2 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O -(CHead c (Flat f) t) e1)))) (\lambda (H15: (drop (S n0) O c e2)).(or_introl -(drop (S n0) O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1: C).(csubst0 -(minus x2 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Flat f) -t) e1))) (drop_drop (Flat f) n0 c e2 H15 t))) (\lambda (H15: (ex2 C (\lambda -(e1: C).(csubst0 (minus x2 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O -c e1)))).(ex2_ind C (\lambda (e1: C).(csubst0 (minus x2 (S n0)) v e1 e2)) -(\lambda (e1: C).(drop (S n0) O c e1)) (or (drop (S n0) O (CHead c (Flat f) -t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x2 (S n0)) v e1 e2)) (\lambda -(e1: C).(drop (S n0) O (CHead c (Flat f) t) e1)))) (\lambda (x: C).(\lambda -(H16: (csubst0 (minus x2 (S n0)) v x e2)).(\lambda (H17: (drop (S n0) O c -x)).(or_intror (drop (S n0) O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1: -C).(csubst0 (minus x2 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O -(CHead c (Flat f) t) e1))) (ex_intro2 C (\lambda (e1: C).(csubst0 (minus x2 -(S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Flat f) t) e1)) x -H16 (drop_drop (Flat f) n0 c x H17 t)))))) H15)) H14))))))) k H9 H10 -(drop_gen_drop k x1 e2 x0 n0 H8)) i H4))))))))))) H3)) (csubst0_gen_head k c -c2 t v i H1))))))))))) c1)))))) n). -(* COMMENTS -Initial nodes: 5939 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubst0/fwd.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubst0/fwd.ma deleted file mode 100644 index 9b3de2983..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubst0/fwd.ma +++ /dev/null @@ -1,462 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubst0/defs.ma". - -theorem csubst0_gen_sort: - \forall (x: C).(\forall (v: T).(\forall (i: nat).(\forall (n: nat).((csubst0 -i v (CSort n) x) \to (\forall (P: Prop).P))))) -\def - \lambda (x: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda -(H: (csubst0 i v (CSort n) x)).(\lambda (P: Prop).(insert_eq C (CSort n) -(\lambda (c: C).(csubst0 i v c x)) (\lambda (_: C).P) (\lambda (y: -C).(\lambda (H0: (csubst0 i v y x)).(csubst0_ind (\lambda (_: nat).(\lambda -(_: T).(\lambda (c: C).(\lambda (_: C).((eq C c (CSort n)) \to P))))) -(\lambda (k: K).(\lambda (i0: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda -(u2: T).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (c: C).(\lambda (H2: (eq -C (CHead c k u1) (CSort n))).(let H3 \def (eq_ind C (CHead c k u1) (\lambda -(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H2) in -(False_ind P H3)))))))))) (\lambda (k: K).(\lambda (i0: nat).(\lambda (c1: -C).(\lambda (c2: C).(\lambda (v0: T).(\lambda (_: (csubst0 i0 v0 c1 -c2)).(\lambda (_: (((eq C c1 (CSort n)) \to P))).(\lambda (u: T).(\lambda -(H3: (eq C (CHead c1 k u) (CSort n))).(let H4 \def (eq_ind C (CHead c1 k u) -(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H3) in -(False_ind P H4))))))))))) (\lambda (k: K).(\lambda (i0: nat).(\lambda (v0: -T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0 i0 v0 u1 -u2)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csubst0 i0 v0 c1 -c2)).(\lambda (_: (((eq C c1 (CSort n)) \to P))).(\lambda (H4: (eq C (CHead -c1 k u1) (CSort n))).(let H5 \def (eq_ind C (CHead c1 k u1) (\lambda (ee: -C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow -False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H4) in (False_ind P -H5))))))))))))) i v y x H0))) H)))))). -(* COMMENTS -Initial nodes: 355 -END *) - -theorem csubst0_gen_head: - \forall (k: K).(\forall (c1: C).(\forall (x: C).(\forall (u1: T).(\forall -(v: T).(\forall (i: nat).((csubst0 i v (CHead c1 k u1) x) \to (or3 (ex3_2 T -nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: -T).(\lambda (_: nat).(eq C x (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: -nat).(subst0 j v u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq -nat i (s k j)))) (\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2 k -u1)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 c2)))) (ex4_3 T C -nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) -(\lambda (u2: T).(\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2 k -u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 -u2)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 -c2)))))))))))) -\def - \lambda (k: K).(\lambda (c1: C).(\lambda (x: C).(\lambda (u1: T).(\lambda -(v: T).(\lambda (i: nat).(\lambda (H: (csubst0 i v (CHead c1 k u1) -x)).(insert_eq C (CHead c1 k u1) (\lambda (c: C).(csubst0 i v c x)) (\lambda -(_: C).(or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k -j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C x (CHead c1 k u2)))) (\lambda -(u2: T).(\lambda (j: nat).(subst0 j v u1 u2)))) (ex3_2 C nat (\lambda (_: -C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c2: C).(\lambda (_: -nat).(eq C x (CHead c2 k u1)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j -v c1 c2)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: -nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c2: C).(\lambda (_: -nat).(eq C x (CHead c2 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda -(j: nat).(subst0 j v u1 u2)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: -nat).(csubst0 j v c1 c2))))))) (\lambda (y: C).(\lambda (H0: (csubst0 i v y -x)).(csubst0_ind (\lambda (n: nat).(\lambda (t: T).(\lambda (c: C).(\lambda -(c0: C).((eq C c (CHead c1 k u1)) \to (or3 (ex3_2 T nat (\lambda (_: -T).(\lambda (j: nat).(eq nat n (s k j)))) (\lambda (u2: T).(\lambda (_: -nat).(eq C c0 (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j -t u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat n (s k -j)))) (\lambda (c2: C).(\lambda (_: nat).(eq C c0 (CHead c2 k u1)))) (\lambda -(c2: C).(\lambda (j: nat).(csubst0 j t c1 c2)))) (ex4_3 T C nat (\lambda (_: -T).(\lambda (_: C).(\lambda (j: nat).(eq nat n (s k j))))) (\lambda (u2: -T).(\lambda (c2: C).(\lambda (_: nat).(eq C c0 (CHead c2 k u2))))) (\lambda -(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j t u1 u2)))) (\lambda (_: -T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j t c1 c2))))))))))) (\lambda -(k0: K).(\lambda (i0: nat).(\lambda (v0: T).(\lambda (u0: T).(\lambda (u2: -T).(\lambda (H1: (subst0 i0 v0 u0 u2)).(\lambda (c: C).(\lambda (H2: (eq C -(CHead c k0 u0) (CHead c1 k u1))).(let H3 \def (f_equal C C (\lambda (e: -C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c | -(CHead c0 _ _) \Rightarrow c0])) (CHead c k0 u0) (CHead c1 k u1) H2) in ((let -H4 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) -with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead c k0 -u0) (CHead c1 k u1) H2) in ((let H5 \def (f_equal C T (\lambda (e: C).(match -e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ -t) \Rightarrow t])) (CHead c k0 u0) (CHead c1 k u1) H2) in (\lambda (H6: (eq -K k0 k)).(\lambda (H7: (eq C c c1)).(eq_ind_r C c1 (\lambda (c0: C).(or3 -(ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) -(\lambda (u3: T).(\lambda (_: nat).(eq C (CHead c0 k0 u2) (CHead c1 k u3)))) -(\lambda (u3: T).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (ex3_2 C nat -(\lambda (_: C).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) (\lambda (c2: -C).(\lambda (_: nat).(eq C (CHead c0 k0 u2) (CHead c2 k u1)))) (\lambda (c2: -C).(\lambda (j: nat).(csubst0 j v0 c1 c2)))) (ex4_3 T C nat (\lambda (_: -T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k0 i0) (s k j))))) (\lambda -(u3: T).(\lambda (c2: C).(\lambda (_: nat).(eq C (CHead c0 k0 u2) (CHead c2 k -u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 -u3)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v0 c1 -c2))))))) (let H8 \def (eq_ind T u0 (\lambda (t: T).(subst0 i0 v0 t u2)) H1 -u1 H5) in (eq_ind_r K k (\lambda (k1: K).(or3 (ex3_2 T nat (\lambda (_: -T).(\lambda (j: nat).(eq nat (s k1 i0) (s k j)))) (\lambda (u3: T).(\lambda -(_: nat).(eq C (CHead c1 k1 u2) (CHead c1 k u3)))) (\lambda (u3: T).(\lambda -(j: nat).(subst0 j v0 u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: -nat).(eq nat (s k1 i0) (s k j)))) (\lambda (c2: C).(\lambda (_: nat).(eq C -(CHead c1 k1 u2) (CHead c2 k u1)))) (\lambda (c2: C).(\lambda (j: -nat).(csubst0 j v0 c1 c2)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: -C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j))))) (\lambda (u3: T).(\lambda -(c2: C).(\lambda (_: nat).(eq C (CHead c1 k1 u2) (CHead c2 k u3))))) (\lambda -(u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (\lambda (_: -T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v0 c1 c2))))))) (or3_intro0 -(ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) -(\lambda (u3: T).(\lambda (_: nat).(eq C (CHead c1 k u2) (CHead c1 k u3)))) -(\lambda (u3: T).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (ex3_2 C nat -(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (c2: -C).(\lambda (_: nat).(eq C (CHead c1 k u2) (CHead c2 k u1)))) (\lambda (c2: -C).(\lambda (j: nat).(csubst0 j v0 c1 c2)))) (ex4_3 T C nat (\lambda (_: -T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))) (\lambda -(u3: T).(\lambda (c2: C).(\lambda (_: nat).(eq C (CHead c1 k u2) (CHead c2 k -u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 -u3)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v0 c1 -c2))))) (ex3_2_intro T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i0) -(s k j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C (CHead c1 k u2) (CHead c1 -k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j v0 u1 u3))) u2 i0 -(refl_equal nat (s k i0)) (refl_equal C (CHead c1 k u2)) H8)) k0 H6)) c -H7)))) H4)) H3)))))))))) (\lambda (k0: K).(\lambda (i0: nat).(\lambda (c0: -C).(\lambda (c2: C).(\lambda (v0: T).(\lambda (H1: (csubst0 i0 v0 c0 -c2)).(\lambda (H2: (((eq C c0 (CHead c1 k u1)) \to (or3 (ex3_2 T nat (\lambda -(_: T).(\lambda (j: nat).(eq nat i0 (s k j)))) (\lambda (u2: T).(\lambda (_: -nat).(eq C c2 (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j -v0 u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i0 (s k -j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u1)))) (\lambda -(c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: -T).(\lambda (_: C).(\lambda (j: nat).(eq nat i0 (s k j))))) (\lambda (u2: -T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda -(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 u2)))) (\lambda (_: -T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3))))))))).(\lambda -(u: T).(\lambda (H3: (eq C (CHead c0 k0 u) (CHead c1 k u1))).(let H4 \def -(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with -[(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k0 u) -(CHead c1 k u1) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e in -C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) -\Rightarrow k1])) (CHead c0 k0 u) (CHead c1 k u1) H3) in ((let H6 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 u) -(CHead c1 k u1) H3) in (\lambda (H7: (eq K k0 k)).(\lambda (H8: (eq C c0 -c1)).(eq_ind_r T u1 (\lambda (t: T).(or3 (ex3_2 T nat (\lambda (_: -T).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) (\lambda (u2: T).(\lambda -(_: nat).(eq C (CHead c2 k0 t) (CHead c1 k u2)))) (\lambda (u2: T).(\lambda -(j: nat).(subst0 j v0 u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: -nat).(eq nat (s k0 i0) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C -(CHead c2 k0 t) (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: -nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: -C).(\lambda (j: nat).(eq nat (s k0 i0) (s k j))))) (\lambda (u2: T).(\lambda -(c3: C).(\lambda (_: nat).(eq C (CHead c2 k0 t) (CHead c3 k u2))))) (\lambda -(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 u2)))) (\lambda (_: -T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3))))))) (let H9 \def -(eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 k u1)) \to (or3 (ex3_2 T nat -(\lambda (_: T).(\lambda (j: nat).(eq nat i0 (s k j)))) (\lambda (u2: -T).(\lambda (_: nat).(eq C c2 (CHead c1 k u2)))) (\lambda (u2: T).(\lambda -(j: nat).(subst0 j v0 u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: -nat).(eq nat i0 (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead -c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 -T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat i0 (s k -j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 -k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 -u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 -c3)))))))) H2 c1 H8) in (let H10 \def (eq_ind C c0 (\lambda (c: C).(csubst0 -i0 v0 c c2)) H1 c1 H8) in (eq_ind_r K k (\lambda (k1: K).(or3 (ex3_2 T nat -(\lambda (_: T).(\lambda (j: nat).(eq nat (s k1 i0) (s k j)))) (\lambda (u2: -T).(\lambda (_: nat).(eq C (CHead c2 k1 u1) (CHead c1 k u2)))) (\lambda (u2: -T).(\lambda (j: nat).(subst0 j v0 u1 u2)))) (ex3_2 C nat (\lambda (_: -C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j)))) (\lambda (c3: C).(\lambda -(_: nat).(eq C (CHead c2 k1 u1) (CHead c3 k u1)))) (\lambda (c3: C).(\lambda -(j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: -C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j))))) (\lambda (u2: T).(\lambda -(c3: C).(\lambda (_: nat).(eq C (CHead c2 k1 u1) (CHead c3 k u2))))) (\lambda -(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 u2)))) (\lambda (_: -T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3))))))) (or3_intro1 -(ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) -(\lambda (u2: T).(\lambda (_: nat).(eq C (CHead c2 k u1) (CHead c1 k u2)))) -(\lambda (u2: T).(\lambda (j: nat).(subst0 j v0 u1 u2)))) (ex3_2 C nat -(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (c3: -C).(\lambda (_: nat).(eq C (CHead c2 k u1) (CHead c3 k u1)))) (\lambda (c3: -C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: -T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))) (\lambda -(u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k u1) (CHead c3 k -u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 -u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 -c3))))) (ex3_2_intro C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) -(s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k u1) (CHead c3 -k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3))) c2 i0 -(refl_equal nat (s k i0)) (refl_equal C (CHead c2 k u1)) H10)) k0 H7))) u -H6)))) H5)) H4))))))))))) (\lambda (k0: K).(\lambda (i0: nat).(\lambda (v0: -T).(\lambda (u0: T).(\lambda (u2: T).(\lambda (H1: (subst0 i0 v0 u0 -u2)).(\lambda (c0: C).(\lambda (c2: C).(\lambda (H2: (csubst0 i0 v0 c0 -c2)).(\lambda (H3: (((eq C c0 (CHead c1 k u1)) \to (or3 (ex3_2 T nat (\lambda -(_: T).(\lambda (j: nat).(eq nat i0 (s k j)))) (\lambda (u3: T).(\lambda (_: -nat).(eq C c2 (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j -v0 u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i0 (s k -j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u1)))) (\lambda -(c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: -T).(\lambda (_: C).(\lambda (j: nat).(eq nat i0 (s k j))))) (\lambda (u3: -T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u3))))) (\lambda -(u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (\lambda (_: -T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3))))))))).(\lambda -(H4: (eq C (CHead c0 k0 u0) (CHead c1 k u1))).(let H5 \def (f_equal C C -(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k0 u0) (CHead c1 k -u1) H4) in ((let H6 \def (f_equal C K (\lambda (e: C).(match e in C return -(\lambda (_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) -\Rightarrow k1])) (CHead c0 k0 u0) (CHead c1 k u1) H4) in ((let H7 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 u0) -(CHead c1 k u1) H4) in (\lambda (H8: (eq K k0 k)).(\lambda (H9: (eq C c0 -c1)).(let H10 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 k u1)) \to -(or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i0 (s k j)))) -(\lambda (u3: T).(\lambda (_: nat).(eq C c2 (CHead c1 k u3)))) (\lambda (u3: -T).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (ex3_2 C nat (\lambda (_: -C).(\lambda (j: nat).(eq nat i0 (s k j)))) (\lambda (c3: C).(\lambda (_: -nat).(eq C c2 (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 -j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: -nat).(eq nat i0 (s k j))))) (\lambda (u3: T).(\lambda (c3: C).(\lambda (_: -nat).(eq C c2 (CHead c3 k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda -(j: nat).(subst0 j v0 u1 u3)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: -nat).(csubst0 j v0 c1 c3)))))))) H3 c1 H9) in (let H11 \def (eq_ind C c0 -(\lambda (c: C).(csubst0 i0 v0 c c2)) H2 c1 H9) in (let H12 \def (eq_ind T u0 -(\lambda (t: T).(subst0 i0 v0 t u2)) H1 u1 H7) in (eq_ind_r K k (\lambda (k1: -K).(or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k1 i0) (s k -j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C (CHead c2 k1 u2) (CHead c1 k -u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (ex3_2 C nat -(\lambda (_: C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j)))) (\lambda (c3: -C).(\lambda (_: nat).(eq C (CHead c2 k1 u2) (CHead c3 k u1)))) (\lambda (c3: -C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: -T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j))))) (\lambda -(u3: T).(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k1 u2) (CHead c3 k -u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 -u3)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 -c3))))))) (or3_intro2 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat -(s k i0) (s k j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C (CHead c2 k u2) -(CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j v0 u1 u3)))) -(ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) -(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k u2) (CHead c3 k u1)))) -(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat -(\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k -j))))) (\lambda (u3: T).(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k -u2) (CHead c3 k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: -nat).(subst0 j v0 u1 u3)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: -nat).(csubst0 j v0 c1 c3))))) (ex4_3_intro T C nat (\lambda (_: T).(\lambda -(_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))) (\lambda (u3: -T).(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k u2) (CHead c3 k -u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 -u3)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 -c3)))) u2 c2 i0 (refl_equal nat (s k i0)) (refl_equal C (CHead c2 k u2)) H12 -H11)) k0 H8))))))) H6)) H5))))))))))))) i v y x H0))) H))))))). -(* COMMENTS -Initial nodes: 4039 -END *) - -theorem csubst0_gen_S_bind_2: - \forall (b: B).(\forall (x: C).(\forall (c2: C).(\forall (v: T).(\forall -(v2: T).(\forall (i: nat).((csubst0 (S i) v x (CHead c2 (Bind b) v2)) \to -(or3 (ex2 T (\lambda (v1: T).(subst0 i v v1 v2)) (\lambda (v1: T).(eq C x -(CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i v c1 c2)) -(\lambda (c1: C).(eq C x (CHead c1 (Bind b) v2)))) (ex3_2 C T (\lambda (_: -C).(\lambda (v1: T).(subst0 i v v1 v2))) (\lambda (c1: C).(\lambda (_: -T).(csubst0 i v c1 c2))) (\lambda (c1: C).(\lambda (v1: T).(eq C x (CHead c1 -(Bind b) v1)))))))))))) -\def - \lambda (b: B).(\lambda (x: C).(\lambda (c2: C).(\lambda (v: T).(\lambda -(v2: T).(\lambda (i: nat).(\lambda (H: (csubst0 (S i) v x (CHead c2 (Bind b) -v2))).(insert_eq C (CHead c2 (Bind b) v2) (\lambda (c: C).(csubst0 (S i) v x -c)) (\lambda (_: C).(or3 (ex2 T (\lambda (v1: T).(subst0 i v v1 v2)) (\lambda -(v1: T).(eq C x (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i -v c1 c2)) (\lambda (c1: C).(eq C x (CHead c1 (Bind b) v2)))) (ex3_2 C T -(\lambda (_: C).(\lambda (v1: T).(subst0 i v v1 v2))) (\lambda (c1: -C).(\lambda (_: T).(csubst0 i v c1 c2))) (\lambda (c1: C).(\lambda (v1: -T).(eq C x (CHead c1 (Bind b) v1))))))) (\lambda (y: C).(\lambda (H0: -(csubst0 (S i) v x y)).(insert_eq nat (S i) (\lambda (n: nat).(csubst0 n v x -y)) (\lambda (_: nat).((eq C y (CHead c2 (Bind b) v2)) \to (or3 (ex2 T -(\lambda (v1: T).(subst0 i v v1 v2)) (\lambda (v1: T).(eq C x (CHead c2 (Bind -b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i v c1 c2)) (\lambda (c1: C).(eq C -x (CHead c1 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: -T).(subst0 i v v1 v2))) (\lambda (c1: C).(\lambda (_: T).(csubst0 i v c1 -c2))) (\lambda (c1: C).(\lambda (v1: T).(eq C x (CHead c1 (Bind b) v1)))))))) -(\lambda (y0: nat).(\lambda (H1: (csubst0 y0 v x y)).(csubst0_ind (\lambda -(n: nat).(\lambda (t: T).(\lambda (c: C).(\lambda (c0: C).((eq nat n (S i)) -\to ((eq C c0 (CHead c2 (Bind b) v2)) \to (or3 (ex2 T (\lambda (v1: -T).(subst0 i t v1 v2)) (\lambda (v1: T).(eq C c (CHead c2 (Bind b) v1)))) -(ex2 C (\lambda (c1: C).(csubst0 i t c1 c2)) (\lambda (c1: C).(eq C c (CHead -c1 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i t v1 -v2))) (\lambda (c1: C).(\lambda (_: T).(csubst0 i t c1 c2))) (\lambda (c1: -C).(\lambda (v1: T).(eq C c (CHead c1 (Bind b) v1)))))))))))) (\lambda (k: -K).(\lambda (i0: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: -T).(\lambda (H2: (subst0 i0 v0 u1 u2)).(\lambda (c: C).(\lambda (H3: (eq nat -(s k i0) (S i))).(\lambda (H4: (eq C (CHead c k u2) (CHead c2 (Bind b) -v2))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda -(_: C).C) with [(CSort _) \Rightarrow c | (CHead c0 _ _) \Rightarrow c0])) -(CHead c k u2) (CHead c2 (Bind b) v2) H4) in ((let H6 \def (f_equal C K -(\lambda (e: C).(match e in C return (\lambda (_: C).K) with [(CSort _) -\Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c k u2) (CHead c2 -(Bind b) v2) H4) in ((let H7 \def (f_equal C T (\lambda (e: C).(match e in C -return (\lambda (_: C).T) with [(CSort _) \Rightarrow u2 | (CHead _ _ t) -\Rightarrow t])) (CHead c k u2) (CHead c2 (Bind b) v2) H4) in (\lambda (H8: -(eq K k (Bind b))).(\lambda (H9: (eq C c c2)).(eq_ind_r C c2 (\lambda (c0: -C).(or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C -(CHead c0 k u1) (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i -v0 c1 c2)) (\lambda (c1: C).(eq C (CHead c0 k u1) (CHead c1 (Bind b) v2)))) -(ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda -(c1: C).(\lambda (_: T).(csubst0 i v0 c1 c2))) (\lambda (c1: C).(\lambda (v1: -T).(eq C (CHead c0 k u1) (CHead c1 (Bind b) v1))))))) (let H10 \def (eq_ind T -u2 (\lambda (t: T).(subst0 i0 v0 u1 t)) H2 v2 H7) in (let H11 \def (eq_ind K -k (\lambda (k0: K).(eq nat (s k0 i0) (S i))) H3 (Bind b) H8) in (eq_ind_r K -(Bind b) (\lambda (k0: K).(or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) -(\lambda (v1: T).(eq C (CHead c2 k0 u1) (CHead c2 (Bind b) v1)))) (ex2 C -(\lambda (c1: C).(csubst0 i v0 c1 c2)) (\lambda (c1: C).(eq C (CHead c2 k0 -u1) (CHead c1 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: -T).(subst0 i v0 v1 v2))) (\lambda (c1: C).(\lambda (_: T).(csubst0 i v0 c1 -c2))) (\lambda (c1: C).(\lambda (v1: T).(eq C (CHead c2 k0 u1) (CHead c1 -(Bind b) v1))))))) (let H12 \def (f_equal nat nat (\lambda (e: nat).(match e -in nat return (\lambda (_: nat).nat) with [O \Rightarrow i0 | (S n) -\Rightarrow n])) (S i0) (S i) H11) in (let H13 \def (eq_ind nat i0 (\lambda -(n: nat).(subst0 n v0 u1 v2)) H10 i H12) in (or3_intro0 (ex2 T (\lambda (v1: -T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C (CHead c2 (Bind b) u1) (CHead -c2 (Bind b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i v0 c1 c2)) (\lambda -(c1: C).(eq C (CHead c2 (Bind b) u1) (CHead c1 (Bind b) v2)))) (ex3_2 C T -(\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda (c1: -C).(\lambda (_: T).(csubst0 i v0 c1 c2))) (\lambda (c1: C).(\lambda (v1: -T).(eq C (CHead c2 (Bind b) u1) (CHead c1 (Bind b) v1))))) (ex_intro2 T -(\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C (CHead c2 (Bind -b) u1) (CHead c2 (Bind b) v1))) u1 H13 (refl_equal C (CHead c2 (Bind b) -u1)))))) k H8))) c H9)))) H6)) H5))))))))))) (\lambda (k: K).(\lambda (i0: -nat).(\lambda (c1: C).(\lambda (c0: C).(\lambda (v0: T).(\lambda (H2: -(csubst0 i0 v0 c1 c0)).(\lambda (H3: (((eq nat i0 (S i)) \to ((eq C c0 (CHead -c2 (Bind b) v2)) \to (or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) -(\lambda (v1: T).(eq C c1 (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3: -C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C c1 (CHead c3 (Bind b) v2)))) -(ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda -(c3: C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1: -T).(eq C c1 (CHead c3 (Bind b) v1)))))))))).(\lambda (u: T).(\lambda (H4: (eq -nat (s k i0) (S i))).(\lambda (H5: (eq C (CHead c0 k u) (CHead c2 (Bind b) -v2))).(let H6 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda -(_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) -(CHead c0 k u) (CHead c2 (Bind b) v2) H5) in ((let H7 \def (f_equal C K -(\lambda (e: C).(match e in C return (\lambda (_: C).K) with [(CSort _) -\Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c0 k u) (CHead c2 -(Bind b) v2) H5) in ((let H8 \def (f_equal C T (\lambda (e: C).(match e in C -return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) -\Rightarrow t])) (CHead c0 k u) (CHead c2 (Bind b) v2) H5) in (\lambda (H9: -(eq K k (Bind b))).(\lambda (H10: (eq C c0 c2)).(eq_ind_r T v2 (\lambda (t: -T).(or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C -(CHead c1 k t) (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3: C).(csubst0 i -v0 c3 c2)) (\lambda (c3: C).(eq C (CHead c1 k t) (CHead c3 (Bind b) v2)))) -(ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda -(c3: C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1: -T).(eq C (CHead c1 k t) (CHead c3 (Bind b) v1))))))) (let H11 \def (eq_ind C -c0 (\lambda (c: C).((eq nat i0 (S i)) \to ((eq C c (CHead c2 (Bind b) v2)) -\to (or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C -c1 (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3: C).(csubst0 i v0 c3 c2)) -(\lambda (c3: C).(eq C c1 (CHead c3 (Bind b) v2)))) (ex3_2 C T (\lambda (_: -C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda (c3: C).(\lambda (_: -T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1: T).(eq C c1 (CHead -c3 (Bind b) v1))))))))) H3 c2 H10) in (let H12 \def (eq_ind C c0 (\lambda (c: -C).(csubst0 i0 v0 c1 c)) H2 c2 H10) in (let H13 \def (eq_ind K k (\lambda -(k0: K).(eq nat (s k0 i0) (S i))) H4 (Bind b) H9) in (eq_ind_r K (Bind b) -(\lambda (k0: K).(or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda -(v1: T).(eq C (CHead c1 k0 v2) (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3: -C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C (CHead c1 k0 v2) (CHead c3 -(Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 -v2))) (\lambda (c3: C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3: -C).(\lambda (v1: T).(eq C (CHead c1 k0 v2) (CHead c3 (Bind b) v1))))))) (let -H14 \def (f_equal nat nat (\lambda (e: nat).(match e in nat return (\lambda -(_: nat).nat) with [O \Rightarrow i0 | (S n) \Rightarrow n])) (S i0) (S i) -H13) in (let H15 \def (eq_ind nat i0 (\lambda (n: nat).((eq nat n (S i)) \to -((eq C c2 (CHead c2 (Bind b) v2)) \to (or3 (ex2 T (\lambda (v1: T).(subst0 i -v0 v1 v2)) (\lambda (v1: T).(eq C c1 (CHead c2 (Bind b) v1)))) (ex2 C -(\lambda (c3: C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C c1 (CHead c3 -(Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 -v2))) (\lambda (c3: C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3: -C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind b) v1))))))))) H11 i H14) in -(let H16 \def (eq_ind nat i0 (\lambda (n: nat).(csubst0 n v0 c1 c2)) H12 i -H14) in (or3_intro1 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda -(v1: T).(eq C (CHead c1 (Bind b) v2) (CHead c2 (Bind b) v1)))) (ex2 C -(\lambda (c3: C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C (CHead c1 (Bind -b) v2) (CHead c3 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: -T).(subst0 i v0 v1 v2))) (\lambda (c3: C).(\lambda (_: T).(csubst0 i v0 c3 -c2))) (\lambda (c3: C).(\lambda (v1: T).(eq C (CHead c1 (Bind b) v2) (CHead -c3 (Bind b) v1))))) (ex_intro2 C (\lambda (c3: C).(csubst0 i v0 c3 c2)) -(\lambda (c3: C).(eq C (CHead c1 (Bind b) v2) (CHead c3 (Bind b) v2))) c1 H16 -(refl_equal C (CHead c1 (Bind b) v2))))))) k H9)))) u H8)))) H7)) -H6)))))))))))) (\lambda (k: K).(\lambda (i0: nat).(\lambda (v0: T).(\lambda -(u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 i0 v0 u1 u2)).(\lambda (c1: -C).(\lambda (c0: C).(\lambda (H3: (csubst0 i0 v0 c1 c0)).(\lambda (H4: (((eq -nat i0 (S i)) \to ((eq C c0 (CHead c2 (Bind b) v2)) \to (or3 (ex2 T (\lambda -(v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C c1 (CHead c2 (Bind b) -v1)))) (ex2 C (\lambda (c3: C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C -c1 (CHead c3 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: -T).(subst0 i v0 v1 v2))) (\lambda (c3: C).(\lambda (_: T).(csubst0 i v0 c3 -c2))) (\lambda (c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind b) -v1)))))))))).(\lambda (H5: (eq nat (s k i0) (S i))).(\lambda (H6: (eq C -(CHead c0 k u2) (CHead c2 (Bind b) v2))).(let H7 \def (f_equal C C (\lambda -(e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 -| (CHead c _ _) \Rightarrow c])) (CHead c0 k u2) (CHead c2 (Bind b) v2) H6) -in ((let H8 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda -(_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) -(CHead c0 k u2) (CHead c2 (Bind b) v2) H6) in ((let H9 \def (f_equal C T -(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) -\Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c0 k u2) (CHead c2 -(Bind b) v2) H6) in (\lambda (H10: (eq K k (Bind b))).(\lambda (H11: (eq C c0 -c2)).(let H12 \def (eq_ind C c0 (\lambda (c: C).((eq nat i0 (S i)) \to ((eq C -c (CHead c2 (Bind b) v2)) \to (or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 -v2)) (\lambda (v1: T).(eq C c1 (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3: -C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C c1 (CHead c3 (Bind b) v2)))) -(ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda -(c3: C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1: -T).(eq C c1 (CHead c3 (Bind b) v1))))))))) H4 c2 H11) in (let H13 \def -(eq_ind C c0 (\lambda (c: C).(csubst0 i0 v0 c1 c)) H3 c2 H11) in (let H14 -\def (eq_ind T u2 (\lambda (t: T).(subst0 i0 v0 u1 t)) H2 v2 H9) in (let H15 -\def (eq_ind K k (\lambda (k0: K).(eq nat (s k0 i0) (S i))) H5 (Bind b) H10) -in (eq_ind_r K (Bind b) (\lambda (k0: K).(or3 (ex2 T (\lambda (v1: T).(subst0 -i v0 v1 v2)) (\lambda (v1: T).(eq C (CHead c1 k0 u1) (CHead c2 (Bind b) -v1)))) (ex2 C (\lambda (c3: C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C -(CHead c1 k0 u1) (CHead c3 (Bind b) v2)))) (ex3_2 C T (\lambda (_: -C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda (c3: C).(\lambda (_: -T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1: T).(eq C (CHead c1 -k0 u1) (CHead c3 (Bind b) v1))))))) (let H16 \def (f_equal nat nat (\lambda -(e: nat).(match e in nat return (\lambda (_: nat).nat) with [O \Rightarrow i0 -| (S n) \Rightarrow n])) (S i0) (S i) H15) in (let H17 \def (eq_ind nat i0 -(\lambda (n: nat).((eq nat n (S i)) \to ((eq C c2 (CHead c2 (Bind b) v2)) \to -(or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C c1 -(CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3: C).(csubst0 i v0 c3 c2)) -(\lambda (c3: C).(eq C c1 (CHead c3 (Bind b) v2)))) (ex3_2 C T (\lambda (_: -C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda (c3: C).(\lambda (_: -T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1: T).(eq C c1 (CHead -c3 (Bind b) v1))))))))) H12 i H16) in (let H18 \def (eq_ind nat i0 (\lambda -(n: nat).(csubst0 n v0 c1 c2)) H13 i H16) in (let H19 \def (eq_ind nat i0 -(\lambda (n: nat).(subst0 n v0 u1 v2)) H14 i H16) in (or3_intro2 (ex2 T -(\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C (CHead c1 (Bind -b) u1) (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3: C).(csubst0 i v0 c3 -c2)) (\lambda (c3: C).(eq C (CHead c1 (Bind b) u1) (CHead c3 (Bind b) v2)))) -(ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda -(c3: C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1: -T).(eq C (CHead c1 (Bind b) u1) (CHead c3 (Bind b) v1))))) (ex3_2_intro C T -(\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda (c3: -C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1: -T).(eq C (CHead c1 (Bind b) u1) (CHead c3 (Bind b) v1)))) c1 u1 H19 H18 -(refl_equal C (CHead c1 (Bind b) u1)))))))) k H10)))))))) H8)) -H7)))))))))))))) y0 v x y H1))) H0))) H))))))). -(* COMMENTS -Initial nodes: 3878 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubst0/getl.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubst0/getl.ma deleted file mode 100644 index 2701af000..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubst0/getl.ma +++ /dev/null @@ -1,1157 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubst0/clear.ma". - -include "Basic-1/csubst0/drop.ma". - -include "Basic-1/getl/fwd.ma". - -theorem csubst0_getl_ge: - \forall (i: nat).(\forall (n: nat).((le i n) \to (\forall (c1: C).(\forall -(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((getl n c1 -e) \to (getl n c2 e))))))))) -\def - \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (le i n)).(\lambda (c1: -C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 i v c1 -c2)).(\lambda (e: C).(\lambda (H1: (getl n c1 e)).(let H2 \def (getl_gen_all -c1 e n H1) in (ex2_ind C (\lambda (e0: C).(drop n O c1 e0)) (\lambda (e0: -C).(clear e0 e)) (getl n c2 e) (\lambda (x: C).(\lambda (H3: (drop n O c1 -x)).(\lambda (H4: (clear x e)).(lt_eq_gt_e i n (getl n c2 e) (\lambda (H5: -(lt i n)).(getl_intro n c2 e x (csubst0_drop_gt n i H5 c1 c2 v H0 x H3) H4)) -(\lambda (H5: (eq nat i n)).(let H6 \def (eq_ind_r nat n (\lambda (n0: -nat).(drop n0 O c1 x)) H3 i H5) in (let H7 \def (eq_ind_r nat n (\lambda (n0: -nat).(le i n0)) H i H5) in (eq_ind nat i (\lambda (n0: nat).(getl n0 c2 e)) -(let H8 \def (csubst0_drop_eq i c1 c2 v H0 x H6) in (or4_ind (drop i O c2 x) -(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: -T).(eq C x (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(drop i O c2 (CHead e0 (Flat f) w)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u -w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C x (CHead e1 (Flat f) u)))))) (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop i O c2 (CHead e2 -(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C x (CHead e1 -(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(drop i O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O -v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (getl i c2 e) (\lambda (H9: -(drop i O c2 x)).(getl_intro i c2 e x H9 H4)) (\lambda (H9: (ex3_4 F C T T -(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C x -(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop i O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u -w))))))).(ex3_4_ind F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C x (CHead e0 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop i O c2 (CHead e0 -(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 O v u w))))) (getl i c2 e) (\lambda (x0: F).(\lambda (x1: -C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H10: (eq C x (CHead x1 (Flat -x0) x2))).(\lambda (H11: (drop i O c2 (CHead x1 (Flat x0) x3))).(\lambda (_: -(subst0 O v x2 x3)).(let H13 \def (eq_ind C x (\lambda (c: C).(clear c e)) H4 -(CHead x1 (Flat x0) x2) H10) in (getl_intro i c2 e (CHead x1 (Flat x0) x3) -H11 (clear_flat x1 e (clear_gen_flat x0 x1 e x2 H13) x0 x3)))))))))) H9)) -(\lambda (H9: (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C x (CHead e1 (Flat f) u)))))) (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop i O c2 (CHead e2 -(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C x (CHead e1 (Flat f) u)))))) -(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop i O c2 -(CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2))))) (getl i c2 e) (\lambda (x0: -F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H10: (eq C x -(CHead x1 (Flat x0) x3))).(\lambda (H11: (drop i O c2 (CHead x2 (Flat x0) -x3))).(\lambda (H12: (csubst0 O v x1 x2)).(let H13 \def (eq_ind C x (\lambda -(c: C).(clear c e)) H4 (CHead x1 (Flat x0) x3) H10) in (getl_intro i c2 e -(CHead x2 (Flat x0) x3) H11 (clear_flat x2 e (csubst0_clear_O x1 x2 v H12 e -(clear_gen_flat x0 x1 e x3 H13)) x0 x3)))))))))) H9)) (\lambda (H9: (ex4_5 F -C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(eq C x (CHead e1 (Flat f) u))))))) (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop i O -c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda (f: F).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C x (CHead e1 (Flat f) -u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop i O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O -v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (getl i c2 e) (\lambda (x0: -F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: -T).(\lambda (H10: (eq C x (CHead x1 (Flat x0) x3))).(\lambda (H11: (drop i O -c2 (CHead x2 (Flat x0) x4))).(\lambda (_: (subst0 O v x3 x4)).(\lambda (H13: -(csubst0 O v x1 x2)).(let H14 \def (eq_ind C x (\lambda (c: C).(clear c e)) -H4 (CHead x1 (Flat x0) x3) H10) in (getl_intro i c2 e (CHead x2 (Flat x0) x4) -H11 (clear_flat x2 e (csubst0_clear_O x1 x2 v H13 e (clear_gen_flat x0 x1 e -x3 H14)) x0 x4)))))))))))) H9)) H8)) n H5)))) (\lambda (H5: (lt n -i)).(le_lt_false i n H H5 (getl n c2 e))))))) H2)))))))))). -(* COMMENTS -Initial nodes: 1525 -END *) - -theorem csubst0_getl_lt: - \forall (i: nat).(\forall (n: nat).((lt n i) \to (\forall (c1: C).(\forall -(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((getl n c1 -e) \to (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) -(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 -(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))))))))))) -\def - \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (lt n i)).(\lambda (c1: -C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 i v c1 -c2)).(\lambda (e: C).(\lambda (H1: (getl n c1 e)).(let H2 \def (getl_gen_all -c1 e n H1) in (ex2_ind C (\lambda (e0: C).(drop n O c1 e0)) (\lambda (e0: -C).(clear e0 e)) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda -(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) -(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 -(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x: -C).(\lambda (H3: (drop n O c1 x)).(\lambda (H4: (clear x e)).(let H5 \def -(csubst0_drop_lt n i H c1 c2 v H0 x H3) in (or4_ind (drop n O c2 x) (ex3_4 K -C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C -x (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop n O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda -(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n)) v u w)))))) -(ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(eq C x (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop n O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1 -e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C x (CHead e1 k u))))))) (\lambda (k: -K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n O -c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n)) v u w)))))) -(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus i (s k n)) v e1 e2))))))) (or4 (getl n c2 e) (ex3_4 B -C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C -e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 -(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))))) (\lambda (H6: (drop n O c2 x)).(or4_intro0 (getl n c2 e) -(ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: -T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 -(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: -T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 -(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -(minus i (S n)) v e1 e2))))))) (getl_intro n c2 e x H6 H4))) (\lambda (H6: -(ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: -T).(eq C x (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(drop n O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda -(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n)) v u -w))))))).(ex3_4_ind K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C x (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e0 k w)))))) (\lambda -(k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k -n)) v u w))))) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda -(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) -(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 -(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x0: -K).(\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq C x -(CHead x1 x0 x2))).(\lambda (H8: (drop n O c2 (CHead x1 x0 x3))).(\lambda -(H9: (subst0 (minus i (s x0 n)) v x2 x3)).(let H10 \def (eq_ind C x (\lambda -(c: C).(clear c e)) H4 (CHead x1 x0 x2) H7) in (K_ind (\lambda (k: K).((drop -n O c2 (CHead x1 k x3)) \to ((subst0 (minus i (s k n)) v x2 x3) \to ((clear -(CHead x1 k x2) e) \to (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: -B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind -b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 -B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq -C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))))))) (\lambda (b: -B).(\lambda (H11: (drop n O c2 (CHead x1 (Bind b) x3))).(\lambda (H12: -(subst0 (minus i (s (Bind b) n)) v x2 x3)).(\lambda (H13: (clear (CHead x1 -(Bind b) x2) e)).(eq_ind_r C (CHead x1 (Bind b) x2) (\lambda (c: C).(or4 -(getl n c2 c) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C c (CHead e0 (Bind b0) u)))))) (\lambda (b0: -B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 -(Bind b0) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda -(w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b0: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind -b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(getl n c2 (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) -(ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b0) u))))))) (\lambda (b0: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n -c2 (CHead e2 (Bind b0) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro1 (getl n c2 -(CHead x1 (Bind b) x2)) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Bind b) x2) (CHead e0 -(Bind b0) u)))))) (\lambda (b0: B).(\lambda (e0: C).(\lambda (_: T).(\lambda -(w: T).(getl n c2 (CHead e0 (Bind b0) w)))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 -B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq -C (CHead x1 (Bind b) x2) (CHead e1 (Bind b0) u)))))) (\lambda (b0: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 -(Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C -(CHead x1 (Bind b) x2) (CHead e1 (Bind b0) u))))))) (\lambda (b0: B).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 -(Bind b0) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -(minus i (S n)) v e1 e2))))))) (ex3_4_intro B C T T (\lambda (b0: B).(\lambda -(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Bind b) x2) (CHead -e0 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b0) w)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w))))) b x1 x2 x3 (refl_equal C (CHead x1 (Bind b) x2)) (getl_intro n c2 -(CHead x1 (Bind b) x3) (CHead x1 (Bind b) x3) H11 (clear_bind b x1 x3)) H12)) -e (clear_gen_bind b x1 e x2 H13)))))) (\lambda (f: F).(\lambda (H11: (drop n -O c2 (CHead x1 (Flat f) x3))).(\lambda (_: (subst0 (minus i (s (Flat f) n)) v -x2 x3)).(\lambda (H13: (clear (CHead x1 (Flat f) x2) e)).(or4_intro0 (getl n -c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: -B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 -(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind -b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) -(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n -c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (getl_intro n c2 e (CHead x1 -(Flat f) x3) H11 (clear_flat x1 e (clear_gen_flat f x1 e x2 H13) f x3))))))) -x0 H8 H9 H10))))))))) H6)) (\lambda (H6: (ex3_4 K C C T (\lambda (k: -K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C x (CHead e1 k -u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(drop n O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1 e2))))))).(ex3_4_ind -K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq -C x (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(drop n O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda -(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1 -e2))))) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) -(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 -(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x0: -K).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H7: (eq C x -(CHead x1 x0 x3))).(\lambda (H8: (drop n O c2 (CHead x2 x0 x3))).(\lambda -(H9: (csubst0 (minus i (s x0 n)) v x1 x2)).(let H10 \def (eq_ind C x (\lambda -(c: C).(clear c e)) H4 (CHead x1 x0 x3) H7) in (K_ind (\lambda (k: K).((drop -n O c2 (CHead x2 k x3)) \to ((csubst0 (minus i (s k n)) v x1 x2) \to ((clear -(CHead x1 k x3) e) \to (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: -B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind -b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 -B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq -C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))))))) (\lambda (b: -B).(\lambda (H11: (drop n O c2 (CHead x2 (Bind b) x3))).(\lambda (H12: -(csubst0 (minus i (s (Bind b) n)) v x1 x2)).(\lambda (H13: (clear (CHead x1 -(Bind b) x3) e)).(eq_ind_r C (CHead x1 (Bind b) x3) (\lambda (c: C).(or4 -(getl n c2 c) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C c (CHead e0 (Bind b0) u)))))) (\lambda (b0: -B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 -(Bind b0) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda -(w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b0: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind -b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(getl n c2 (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) -(ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b0) u))))))) (\lambda (b0: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n -c2 (CHead e2 (Bind b0) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro2 (getl n c2 -(CHead x1 (Bind b) x3)) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Bind b) x3) (CHead e0 -(Bind b0) u)))))) (\lambda (b0: B).(\lambda (e0: C).(\lambda (_: T).(\lambda -(w: T).(getl n c2 (CHead e0 (Bind b0) w)))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 -B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq -C (CHead x1 (Bind b) x3) (CHead e1 (Bind b0) u)))))) (\lambda (b0: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 -(Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C -(CHead x1 (Bind b) x3) (CHead e1 (Bind b0) u))))))) (\lambda (b0: B).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 -(Bind b0) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -(minus i (S n)) v e1 e2))))))) (ex3_4_intro B C C T (\lambda (b0: B).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x1 (Bind b) x3) (CHead -e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b0) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2))))) b x1 x2 x3 (refl_equal C (CHead x1 (Bind b) x3)) (getl_intro n -c2 (CHead x2 (Bind b) x3) (CHead x2 (Bind b) x3) H11 (clear_bind b x2 x3)) -H12)) e (clear_gen_bind b x1 e x3 H13)))))) (\lambda (f: F).(\lambda (H11: -(drop n O c2 (CHead x2 (Flat f) x3))).(\lambda (H12: (csubst0 (minus i (s -(Flat f) n)) v x1 x2)).(\lambda (H13: (clear (CHead x1 (Flat f) x3) e)).(let -H14 \def (eq_ind nat (minus i n) (\lambda (n0: nat).(csubst0 n0 v x1 x2)) H12 -(S (minus i (S n))) (minus_x_Sy i n H)) in (let H15 \def (csubst0_clear_S x1 -x2 v (minus i (S n)) H14 e (clear_gen_flat f x1 e x3 H13)) in (or4_ind (clear -x2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u1)))))) (\lambda (b: -B).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e0 -(Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 (minus i (S n)) v u1 u2)))))) (ex3_4 B C C T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind -b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(clear x2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) -(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u1))))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear -x2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (minus i (S n)) v u1 u2)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (or4 (getl n c2 e) (ex3_4 B C -T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 -(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))))) (\lambda (H16: (clear x2 e)).(or4_intro0 (getl n c2 e) (ex3_4 -B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq -C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 -(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2))))))) (getl_intro n c2 e (CHead x2 (Flat f) x3) H11 (clear_flat x2 e -H16 f x3)))) (\lambda (H16: (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u1)))))) -(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(clear x2 -(CHead e0 (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: -T).(\lambda (u2: T).(subst0 (minus i (S n)) v u1 u2))))))).(ex3_4_ind B C T T -(\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C e -(CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: -T).(\lambda (u2: T).(clear x2 (CHead e0 (Bind b) u2)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (minus i (S n)) -v u1 u2))))) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) -(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 -(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x4: -B).(\lambda (x5: C).(\lambda (x6: T).(\lambda (x7: T).(\lambda (H17: (eq C e -(CHead x5 (Bind x4) x6))).(\lambda (H18: (clear x2 (CHead x5 (Bind x4) -x7))).(\lambda (H19: (subst0 (minus i (S n)) v x6 x7)).(eq_ind_r C (CHead x5 -(Bind x4) x6) (\lambda (c: C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda (b: -B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Bind -b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 -B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq -C c (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b) u))))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro1 -(getl n c2 (CHead x5 (Bind x4) x6)) (ex3_4 B C T T (\lambda (b: B).(\lambda -(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) x6) (CHead -e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda -(w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 -B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq -C (CHead x5 (Bind x4) x6) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda -(_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 -(minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) -x6) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2))))))) (ex3_4_intro B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda -(u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) x6) (CHead e0 (Bind b) -u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w))))) x4 x5 -x6 x7 (refl_equal C (CHead x5 (Bind x4) x6)) (getl_intro n c2 (CHead x5 (Bind -x4) x7) (CHead x2 (Flat f) x3) H11 (clear_flat x2 (CHead x5 (Bind x4) x7) H18 -f x3)) H19)) e H17)))))))) H16)) (\lambda (H16: (ex3_4 B C C T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind -b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(clear x2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 -e2))))))).(ex3_4_ind B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x2 (CHead e2 (Bind -b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus i (S n)) v e1 e2))))) (or4 (getl n c2 e) (ex3_4 B C T T -(\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 -(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))))) (\lambda (x4: B).(\lambda (x5: C).(\lambda (x6: C).(\lambda -(x7: T).(\lambda (H17: (eq C e (CHead x5 (Bind x4) x7))).(\lambda (H18: -(clear x2 (CHead x6 (Bind x4) x7))).(\lambda (H19: (csubst0 (minus i (S n)) v -x5 x6)).(eq_ind_r C (CHead x5 (Bind x4) x7) (\lambda (c: C).(or4 (getl n c2 -c) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda -(_: T).(eq C c (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind b) u)))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 -(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: -T).(eq C c (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 -(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -(minus i (S n)) v e1 e2))))))))) (or4_intro2 (getl n c2 (CHead x5 (Bind x4) -x7)) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda -(_: T).(eq C (CHead x5 (Bind x4) x7) (CHead e0 (Bind b) u)))))) (\lambda (b: -B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 -(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x5 (Bind x4) -x7) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) x7) (CHead e1 -(Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))) -(ex3_4_intro B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C (CHead x5 (Bind x4) x7) (CHead e1 (Bind b) u)))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 -(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))) x4 x5 x6 x7 -(refl_equal C (CHead x5 (Bind x4) x7)) (getl_intro n c2 (CHead x6 (Bind x4) -x7) (CHead x2 (Flat f) x3) H11 (clear_flat x2 (CHead x6 (Bind x4) x7) H18 f -x3)) H19)) e H17)))))))) H16)) (\lambda (H16: (ex4_5 B C C T T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C e -(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e2 (Bind b) u2))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 (minus i (S n)) v u1 u2)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C e (CHead e1 (Bind -b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (u2: T).(clear x2 (CHead e2 (Bind b) u2))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 -(minus i (S n)) v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) -(or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda -(u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: -B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 -(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind -b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) -(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n -c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x4: B).(\lambda -(x5: C).(\lambda (x6: C).(\lambda (x7: T).(\lambda (x8: T).(\lambda (H17: (eq -C e (CHead x5 (Bind x4) x7))).(\lambda (H18: (clear x2 (CHead x6 (Bind x4) -x8))).(\lambda (H19: (subst0 (minus i (S n)) v x7 x8)).(\lambda (H20: -(csubst0 (minus i (S n)) v x5 x6)).(eq_ind_r C (CHead x5 (Bind x4) x7) -(\lambda (c: C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda (b: B).(\lambda -(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Bind b) u)))))) -(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 -(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c -(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b) u))))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro3 -(getl n c2 (CHead x5 (Bind x4) x7)) (ex3_4 B C T T (\lambda (b: B).(\lambda -(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) x7) (CHead -e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda -(w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 -B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq -C (CHead x5 (Bind x4) x7) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda -(_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 -(minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) -x7) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2))))))) (ex4_5_intro B C C T T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) -x7) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))) x4 x5 x6 x7 x8 (refl_equal C (CHead x5 (Bind x4) x7)) -(getl_intro n c2 (CHead x6 (Bind x4) x8) (CHead x2 (Flat f) x3) H11 -(clear_flat x2 (CHead x6 (Bind x4) x8) H18 f x3)) H19 H20)) e H17)))))))))) -H16)) H15))))))) x0 H8 H9 H10))))))))) H6)) (\lambda (H6: (ex4_5 K C C T T -(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: -T).(eq C x (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e2 k w))))))) (\lambda -(k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (s k n)) v u w)))))) (\lambda (k: K).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k -n)) v e1 e2)))))))).(ex4_5_ind K C C T T (\lambda (k: K).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C x (CHead e1 k -u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(drop n O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda -(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k -n)) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1 e2)))))) (or4 (getl n -c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: -B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 -(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind -b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) -(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n -c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x0: K).(\lambda -(x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H7: (eq -C x (CHead x1 x0 x3))).(\lambda (H8: (drop n O c2 (CHead x2 x0 x4))).(\lambda -(H9: (subst0 (minus i (s x0 n)) v x3 x4)).(\lambda (H10: (csubst0 (minus i (s -x0 n)) v x1 x2)).(let H11 \def (eq_ind C x (\lambda (c: C).(clear c e)) H4 -(CHead x1 x0 x3) H7) in (K_ind (\lambda (k: K).((drop n O c2 (CHead x2 k x4)) -\to ((subst0 (minus i (s k n)) v x3 x4) \to ((csubst0 (minus i (s k n)) v x1 -x2) \to ((clear (CHead x1 k x3) e) \to (or4 (getl n c2 e) (ex3_4 B C T T -(\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 -(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2))))))))))))) (\lambda (b: B).(\lambda (H12: (drop n O c2 (CHead x2 -(Bind b) x4))).(\lambda (H13: (subst0 (minus i (s (Bind b) n)) v x3 -x4)).(\lambda (H14: (csubst0 (minus i (s (Bind b) n)) v x1 x2)).(\lambda -(H15: (clear (CHead x1 (Bind b) x3) e)).(eq_ind_r C (CHead x1 (Bind b) x3) -(\lambda (c: C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda (b0: B).(\lambda -(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Bind b0) u)))))) -(\lambda (b0: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 -(CHead e0 (Bind b0) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T -(\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c -(CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b0) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b0) u))))))) -(\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(w: T).(getl n c2 (CHead e2 (Bind b0) w))))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro3 -(getl n c2 (CHead x1 (Bind b) x3)) (ex3_4 B C T T (\lambda (b0: B).(\lambda -(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Bind b) x3) (CHead -e0 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b0) w)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (ex3_4 B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C (CHead x1 (Bind b) x3) (CHead e1 (Bind b0) u)))))) -(\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 -(CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T -(\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda -(_: T).(eq C (CHead x1 (Bind b) x3) (CHead e1 (Bind b0) u))))))) (\lambda -(b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(getl n c2 (CHead e2 (Bind b0) w))))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (ex4_5_intro B C C -T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(eq C (CHead x1 (Bind b) x3) (CHead e1 (Bind b0) u))))))) -(\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(w: T).(getl n c2 (CHead e2 (Bind b0) w))))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) b x1 x2 x3 x4 -(refl_equal C (CHead x1 (Bind b) x3)) (getl_intro n c2 (CHead x2 (Bind b) x4) -(CHead x2 (Bind b) x4) H12 (clear_bind b x2 x4)) H13 H14)) e (clear_gen_bind -b x1 e x3 H15))))))) (\lambda (f: F).(\lambda (H12: (drop n O c2 (CHead x2 -(Flat f) x4))).(\lambda (_: (subst0 (minus i (s (Flat f) n)) v x3 -x4)).(\lambda (H14: (csubst0 (minus i (s (Flat f) n)) v x1 x2)).(\lambda -(H15: (clear (CHead x1 (Flat f) x3) e)).(let H16 \def (eq_ind nat (minus i n) -(\lambda (n0: nat).(csubst0 n0 v x1 x2)) H14 (S (minus i (S n))) (minus_x_Sy -i n H)) in (let H17 \def (csubst0_clear_S x1 x2 v (minus i (S n)) H16 e -(clear_gen_flat f x1 e x3 H15)) in (or4_ind (clear x2 e) (ex3_4 B C T T -(\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C e -(CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: -T).(\lambda (u2: T).(clear x2 (CHead e0 (Bind b) u2)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (minus i (S n)) -v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x2 (CHead e2 (Bind -b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C e -(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e2 (Bind b) u2))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 (minus i (S n)) v u1 u2)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2))))))) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda -(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) -(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 -(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e -(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (H18: -(clear x2 e)).(or4_intro0 (getl n c2 e) (ex3_4 B C T T (\lambda (b: -B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind -b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 -B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq -C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (\lambda (_: 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(\lambda (b: B).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 -(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: -T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 -(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -(minus i (S n)) v e1 e2)))))))) (\lambda (x5: B).(\lambda (x6: C).(\lambda -(x7: T).(\lambda (x8: T).(\lambda (H19: (eq C e (CHead x6 (Bind x5) -x7))).(\lambda (H20: (clear x2 (CHead x6 (Bind x5) x8))).(\lambda (H21: -(subst0 (minus i (S n)) v x7 x8)).(eq_ind_r C (CHead x6 (Bind x5) x7) -(\lambda (c: C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda (b: B).(\lambda -(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Bind b) u)))))) -(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 -(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c -(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b) u))))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro1 -(getl n c2 (CHead x6 (Bind x5) x7)) (ex3_4 B C T T (\lambda (b: B).(\lambda -(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) x7) (CHead -e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda -(w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 -B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq -C (CHead x6 (Bind x5) x7) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda -(_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 -(minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) -x7) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2))))))) (ex3_4_intro B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda -(u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) x7) (CHead e0 (Bind b) -u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w))))) x5 x6 -x7 x8 (refl_equal C (CHead x6 (Bind x5) x7)) (getl_intro n c2 (CHead x6 (Bind -x5) x8) (CHead x2 (Flat f) x4) H12 (clear_flat x2 (CHead x6 (Bind x5) x8) H20 -f x4)) H21)) e H19)))))))) H18)) (\lambda (H18: (ex3_4 B C C T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind -b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(clear x2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 -e2))))))).(ex3_4_ind B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x2 (CHead e2 (Bind -b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus i (S n)) v e1 e2))))) (or4 (getl n c2 e) (ex3_4 B C T T -(\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 -(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e -(CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))))) (\lambda (x5: B).(\lambda (x6: C).(\lambda (x7: C).(\lambda -(x8: T).(\lambda (H19: (eq C e (CHead x6 (Bind x5) x8))).(\lambda (H20: -(clear x2 (CHead x7 (Bind x5) x8))).(\lambda (H21: (csubst0 (minus i (S n)) v -x6 x7)).(eq_ind_r C (CHead x6 (Bind x5) x8) (\lambda (c: C).(or4 (getl n c2 -c) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda -(_: T).(eq C c (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind b) u)))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 -(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: -T).(eq C c (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 -(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -(minus i (S n)) v e1 e2))))))))) (or4_intro2 (getl n c2 (CHead x6 (Bind x5) -x8)) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda -(_: T).(eq C (CHead x6 (Bind x5) x8) (CHead e0 (Bind b) u)))))) (\lambda (b: -B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 -(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x6 (Bind x5) -x8) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) x8) (CHead e1 -(Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))) -(ex3_4_intro B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(eq C (CHead x6 (Bind x5) x8) (CHead e1 (Bind b) u)))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 -(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))) x5 x6 x7 x8 -(refl_equal C (CHead x6 (Bind x5) x8)) (getl_intro n c2 (CHead x7 (Bind x5) -x8) (CHead x2 (Flat f) x4) H12 (clear_flat x2 (CHead x7 (Bind x5) x8) H20 f -x4)) H21)) e H19)))))))) H18)) (\lambda (H18: (ex4_5 B C C T T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C e -(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e2 (Bind b) u2))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 (minus i (S n)) v u1 u2)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C e (CHead e1 (Bind -b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (u2: T).(clear x2 (CHead e2 (Bind b) u2))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 -(minus i (S n)) v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) -(or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda -(u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: -B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 -(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind -b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: -T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) -(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n -c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x5: B).(\lambda -(x6: C).(\lambda (x7: C).(\lambda (x8: T).(\lambda (x9: T).(\lambda (H19: (eq -C e (CHead x6 (Bind x5) x8))).(\lambda (H20: (clear x2 (CHead x7 (Bind x5) -x9))).(\lambda (H21: (subst0 (minus i (S n)) v x8 x9)).(\lambda (H22: -(csubst0 (minus i (S n)) v x6 x7)).(eq_ind_r C (CHead x6 (Bind x5) x8) -(\lambda (c: C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda (b: B).(\lambda -(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Bind b) u)))))) -(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 -(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: -T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c -(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b) u))))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: -T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v -u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro3 -(getl n c2 (CHead x6 (Bind x5) x8)) (ex3_4 B C T T (\lambda (b: B).(\lambda -(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) x8) (CHead -e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda -(w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 -B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq -C (CHead x6 (Bind x5) x8) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda -(_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 -(minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) -x8) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2))))))) (ex4_5_intro B C C T T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) -x8) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) -v e1 e2)))))) x5 x6 x7 x8 x9 (refl_equal C (CHead x6 (Bind x5) x8)) -(getl_intro n c2 (CHead x7 (Bind x5) x9) (CHead x2 (Flat f) x4) H12 -(clear_flat x2 (CHead x7 (Bind x5) x9) H20 f x4)) H21 H22)) e H19)))))))))) -H18)) H17)))))))) x0 H8 H9 H10 H11))))))))))) H6)) H5))))) H2)))))))))). -(* COMMENTS -Initial nodes: 17179 -END *) - -theorem csubst0_getl_ge_back: - \forall (i: nat).(\forall (n: nat).((le i n) \to (\forall (c1: C).(\forall -(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((getl n c2 -e) \to (getl n c1 e))))))))) -\def - \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (le i n)).(\lambda (c1: -C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 i v c1 -c2)).(\lambda (e: C).(\lambda (H1: (getl n c2 e)).(let H2 \def (getl_gen_all -c2 e n H1) in (ex2_ind C (\lambda (e0: C).(drop n O c2 e0)) (\lambda (e0: -C).(clear e0 e)) (getl n c1 e) (\lambda (x: C).(\lambda (H3: (drop n O c2 -x)).(\lambda (H4: (clear x e)).(lt_eq_gt_e i n (getl n c1 e) (\lambda (H5: -(lt i n)).(getl_intro n c1 e x (csubst0_drop_gt_back n i H5 c1 c2 v H0 x H3) -H4)) (\lambda (H5: (eq nat i n)).(let H6 \def (eq_ind_r nat n (\lambda (n0: -nat).(drop n0 O c2 x)) H3 i H5) in (let H7 \def (eq_ind_r nat n (\lambda (n0: -nat).(le i n0)) H i H5) in (eq_ind nat i (\lambda (n0: nat).(getl n0 c1 e)) -(let H8 \def (csubst0_drop_eq_back i c1 c2 v H0 x H6) in (or4_ind (drop i O -c1 x) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (u2: T).(eq C x (CHead e0 (Flat f) u2)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop i O c1 (CHead e0 -(Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (u: T).(eq C x (CHead e2 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop i O c1 -(CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: -F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C x -(CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(drop i O c1 (CHead e1 (Flat f) u1))))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda -(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (getl i c1 -e) (\lambda (H9: (drop i O c1 x)).(getl_intro i c1 e x H9 H4)) (\lambda (H9: -(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: -T).(eq C x (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(drop i O c1 (CHead e0 (Flat f) u1)))))) -(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v -u1 u2))))))).(ex3_4_ind F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: -T).(\lambda (u2: T).(eq C x (CHead e0 (Flat f) u2)))))) (\lambda (f: -F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop i O c1 (CHead e0 -(Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda -(u2: T).(subst0 O v u1 u2))))) (getl i c1 e) (\lambda (x0: F).(\lambda (x1: -C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H10: (eq C x (CHead x1 (Flat -x0) x3))).(\lambda (H11: (drop i O c1 (CHead x1 (Flat x0) x2))).(\lambda (_: -(subst0 O v x2 x3)).(let H13 \def (eq_ind C x (\lambda (c: C).(clear c e)) H4 -(CHead x1 (Flat x0) x3) H10) in (getl_intro i c1 e (CHead x1 (Flat x0) x2) -H11 (clear_flat x1 e (clear_gen_flat x0 x1 e x3 H13) x0 x2)))))))))) H9)) -(\lambda (H9: (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: -C).(\lambda (u: T).(eq C x (CHead e2 (Flat f) u)))))) (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop i O c1 (CHead e1 -(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda -(_: C).(\lambda (e2: C).(\lambda (u: T).(eq C x (CHead e2 (Flat f) u)))))) -(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop i O c1 -(CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 O v e1 e2))))) (getl i c1 e) (\lambda (x0: -F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H10: (eq C x -(CHead x2 (Flat x0) x3))).(\lambda (H11: (drop i O c1 (CHead x1 (Flat x0) -x3))).(\lambda (H12: (csubst0 O v x1 x2)).(let H13 \def (eq_ind C x (\lambda -(c: C).(clear c e)) H4 (CHead x2 (Flat x0) x3) H10) in (getl_intro i c1 e -(CHead x1 (Flat x0) x3) H11 (clear_flat x1 e (csubst0_clear_O_back x1 x2 v -H12 e (clear_gen_flat x0 x2 e x3 H13)) x0 x3)))))))))) H9)) (\lambda (H9: -(ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (u2: T).(eq C x (CHead e2 (Flat f) u2))))))) (\lambda (f: -F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop i -O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda -(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: -F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda (f: F).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C x (CHead e2 (Flat -f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: -T).(\lambda (_: T).(drop i O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: -F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 -O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (getl i c1 e) (\lambda (x0: -F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: -T).(\lambda (H10: (eq C x (CHead x2 (Flat x0) x4))).(\lambda (H11: (drop i O -c1 (CHead x1 (Flat x0) x3))).(\lambda (_: (subst0 O v x3 x4)).(\lambda (H13: -(csubst0 O v x1 x2)).(let H14 \def (eq_ind C x (\lambda (c: C).(clear c e)) -H4 (CHead x2 (Flat x0) x4) H10) in (getl_intro i c1 e (CHead x1 (Flat x0) x3) -H11 (clear_flat x1 e (csubst0_clear_O_back x1 x2 v H13 e (clear_gen_flat x0 -x2 e x4 H14)) x0 x3)))))))))))) H9)) H8)) n H5)))) (\lambda (H5: (lt n -i)).(le_lt_false i n H H5 (getl n c1 e))))))) H2)))))))))). -(* COMMENTS -Initial nodes: 1525 -END *) - -theorem csubst0_getl_lt_back: - \forall (n: nat).(\forall (i: nat).((lt n i) \to (\forall (c1: C).(\forall -(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e2: C).((getl n c2 -e2) \to (or (getl n c1 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) v e1 -e2)) (\lambda (e1: C).(getl n c1 e1)))))))))))) -\def - \lambda (n: nat).(\lambda (i: nat).(\lambda (H: (lt n i)).(\lambda (c1: -C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 i v c1 -c2)).(\lambda (e2: C).(\lambda (H1: (getl n c2 e2)).(let H2 \def -(getl_gen_all c2 e2 n H1) in (ex2_ind C (\lambda (e: C).(drop n O c2 e)) -(\lambda (e: C).(clear e e2)) (or (getl n c1 e2) (ex2 C (\lambda (e1: -C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(getl n c1 e1)))) (\lambda -(x: C).(\lambda (H3: (drop n O c2 x)).(\lambda (H4: (clear x e2)).(let H_x -\def (csubst0_drop_lt_back n i H c1 c2 v H0 x H3) in (let H5 \def H_x in -(or_ind (drop n O c1 x) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) v e1 x)) -(\lambda (e1: C).(drop n O c1 e1))) (or (getl n c1 e2) (ex2 C (\lambda (e1: -C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(getl n c1 e1)))) (\lambda -(H6: (drop n O c1 x)).(or_introl (getl n c1 e2) (ex2 C (\lambda (e1: -C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(getl n c1 e1))) -(getl_intro n c1 e2 x H6 H4))) (\lambda (H6: (ex2 C (\lambda (e1: C).(csubst0 -(minus i n) v e1 x)) (\lambda (e1: C).(drop n O c1 e1)))).(ex2_ind C (\lambda -(e1: C).(csubst0 (minus i n) v e1 x)) (\lambda (e1: C).(drop n O c1 e1)) (or -(getl n c1 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) v e1 e2)) -(\lambda (e1: C).(getl n c1 e1)))) (\lambda (x0: C).(\lambda (H7: (csubst0 -(minus i n) v x0 x)).(\lambda (H8: (drop n O c1 x0)).(let H_x0 \def -(csubst0_clear_trans x0 x v (minus i n) H7 e2 H4) in (let H9 \def H_x0 in -(or_ind (clear x0 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) v e1 e2)) -(\lambda (e1: C).(clear x0 e1))) (or (getl n c1 e2) (ex2 C (\lambda (e1: -C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(getl n c1 e1)))) (\lambda -(H10: (clear x0 e2)).(or_introl (getl n c1 e2) (ex2 C (\lambda (e1: -C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(getl n c1 e1))) -(getl_intro n c1 e2 x0 H8 H10))) (\lambda (H10: (ex2 C (\lambda (e1: -C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(clear x0 e1)))).(ex2_ind -C (\lambda (e1: C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(clear x0 -e1)) (or (getl n c1 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) v e1 -e2)) (\lambda (e1: C).(getl n c1 e1)))) (\lambda (x1: C).(\lambda (H11: -(csubst0 (minus i n) v x1 e2)).(\lambda (H12: (clear x0 x1)).(or_intror (getl -n c1 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: -C).(getl n c1 e1))) (ex_intro2 C (\lambda (e1: C).(csubst0 (minus i n) v e1 -e2)) (\lambda (e1: C).(getl n c1 e1)) x1 H11 (getl_intro n c1 x1 x0 H8 -H12)))))) H10)) H9)))))) H6)) H5)))))) H2)))))))))). -(* COMMENTS -Initial nodes: 801 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubst0/props.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubst0/props.ma deleted file mode 100644 index 28b75ec37..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubst0/props.ma +++ /dev/null @@ -1,61 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubst0/defs.ma". - -theorem csubst0_snd_bind: - \forall (b: B).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall -(u2: T).((subst0 i v u1 u2) \to (\forall (c: C).(csubst0 (S i) v (CHead c -(Bind b) u1) (CHead c (Bind b) u2)))))))) -\def - \lambda (b: B).(\lambda (i: nat).(\lambda (v: T).(\lambda (u1: T).(\lambda -(u2: T).(\lambda (H: (subst0 i v u1 u2)).(\lambda (c: C).(eq_ind nat (s (Bind -b) i) (\lambda (n: nat).(csubst0 n v (CHead c (Bind b) u1) (CHead c (Bind b) -u2))) (csubst0_snd (Bind b) i v u1 u2 H c) (S i) (refl_equal nat (S -i))))))))). -(* COMMENTS -Initial nodes: 91 -END *) - -theorem csubst0_fst_bind: - \forall (b: B).(\forall (i: nat).(\forall (c1: C).(\forall (c2: C).(\forall -(v: T).((csubst0 i v c1 c2) \to (\forall (u: T).(csubst0 (S i) v (CHead c1 -(Bind b) u) (CHead c2 (Bind b) u)))))))) -\def - \lambda (b: B).(\lambda (i: nat).(\lambda (c1: C).(\lambda (c2: C).(\lambda -(v: T).(\lambda (H: (csubst0 i v c1 c2)).(\lambda (u: T).(eq_ind nat (s (Bind -b) i) (\lambda (n: nat).(csubst0 n v (CHead c1 (Bind b) u) (CHead c2 (Bind b) -u))) (csubst0_fst (Bind b) i c1 c2 v H u) (S i) (refl_equal nat (S i))))))))). -(* COMMENTS -Initial nodes: 91 -END *) - -theorem csubst0_both_bind: - \forall (b: B).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall -(u2: T).((subst0 i v u1 u2) \to (\forall (c1: C).(\forall (c2: C).((csubst0 i -v c1 c2) \to (csubst0 (S i) v (CHead c1 (Bind b) u1) (CHead c2 (Bind b) -u2)))))))))) -\def - \lambda (b: B).(\lambda (i: nat).(\lambda (v: T).(\lambda (u1: T).(\lambda -(u2: T).(\lambda (H: (subst0 i v u1 u2)).(\lambda (c1: C).(\lambda (c2: -C).(\lambda (H0: (csubst0 i v c1 c2)).(eq_ind nat (s (Bind b) i) (\lambda (n: -nat).(csubst0 n v (CHead c1 (Bind b) u1) (CHead c2 (Bind b) u2))) -(csubst0_both (Bind b) i v u1 u2 H c1 c2 H0) (S i) (refl_equal nat (S -i))))))))))). -(* COMMENTS -Initial nodes: 107 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubst1/defs.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubst1/defs.ma deleted file mode 100644 index d6ba0a942..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubst1/defs.ma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubst0/defs.ma". - -inductive csubst1 (i: nat) (v: T) (c1: C): C \to Prop \def -| csubst1_refl: csubst1 i v c1 c1 -| csubst1_sing: \forall (c2: C).((csubst0 i v c1 c2) \to (csubst1 i v c1 c2)). - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubst1/fwd.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubst1/fwd.ma deleted file mode 100644 index fe71a56fd..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubst1/fwd.ma +++ /dev/null @@ -1,117 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubst1/defs.ma". - -include "Basic-1/csubst0/fwd.ma". - -include "Basic-1/subst1/props.ma". - -theorem csubst1_gen_head: - \forall (k: K).(\forall (c1: C).(\forall (x: C).(\forall (u1: T).(\forall -(v: T).(\forall (i: nat).((csubst1 (s k i) v (CHead c1 k u1) x) \to (ex3_2 T -C (\lambda (u2: T).(\lambda (c2: C).(eq C x (CHead c2 k u2)))) (\lambda (u2: -T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c2: -C).(csubst1 i v c1 c2)))))))))) -\def - \lambda (k: K).(\lambda (c1: C).(\lambda (x: C).(\lambda (u1: T).(\lambda -(v: T).(\lambda (i: nat).(\lambda (H: (csubst1 (s k i) v (CHead c1 k u1) -x)).(csubst1_ind (s k i) v (CHead c1 k u1) (\lambda (c: C).(ex3_2 T C -(\lambda (u2: T).(\lambda (c2: C).(eq C c (CHead c2 k u2)))) (\lambda (u2: -T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c2: -C).(csubst1 i v c1 c2))))) (ex3_2_intro T C (\lambda (u2: T).(\lambda (c2: -C).(eq C (CHead c1 k u1) (CHead c2 k u2)))) (\lambda (u2: T).(\lambda (_: -C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c2: C).(csubst1 i v c1 -c2))) u1 c1 (refl_equal C (CHead c1 k u1)) (subst1_refl i v u1) (csubst1_refl -i v c1)) (\lambda (c2: C).(\lambda (H0: (csubst0 (s k i) v (CHead c1 k u1) -c2)).(or3_ind (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i) -(s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c1 k u2)))) -(\lambda (u2: T).(\lambda (j: nat).(subst0 j v u1 u2)))) (ex3_2 C nat -(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i) (s k j)))) (\lambda (c3: -C).(\lambda (_: nat).(eq C c2 (CHead c3 k u1)))) (\lambda (c3: C).(\lambda -(j: nat).(csubst0 j v c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: -C).(\lambda (j: nat).(eq nat (s k i) (s k j))))) (\lambda (u2: T).(\lambda -(c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: -T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u2)))) (\lambda (_: -T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3))))) (ex3_2 T C -(\lambda (u2: T).(\lambda (c3: C).(eq C c2 (CHead c3 k u2)))) (\lambda (u2: -T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3: -C).(csubst1 i v c1 c3)))) (\lambda (H1: (ex3_2 T nat (\lambda (_: T).(\lambda -(j: nat).(eq nat (s k i) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C -c2 (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v u1 -u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i) (s -k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c1 k u2)))) -(\lambda (u2: T).(\lambda (j: nat).(subst0 j v u1 u2))) (ex3_2 T C (\lambda -(u2: T).(\lambda (c3: C).(eq C c2 (CHead c3 k u2)))) (\lambda (u2: -T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3: -C).(csubst1 i v c1 c3)))) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H2: -(eq nat (s k i) (s k x1))).(\lambda (H3: (eq C c2 (CHead c1 k x0))).(\lambda -(H4: (subst0 x1 v u1 x0)).(eq_ind_r C (CHead c1 k x0) (\lambda (c: C).(ex3_2 -T C (\lambda (u2: T).(\lambda (c3: C).(eq C c (CHead c3 k u2)))) (\lambda -(u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3: -C).(csubst1 i v c1 c3))))) (let H5 \def (eq_ind_r nat x1 (\lambda (n: -nat).(subst0 n v u1 x0)) H4 i (s_inj k i x1 H2)) in (ex3_2_intro T C (\lambda -(u2: T).(\lambda (c3: C).(eq C (CHead c1 k x0) (CHead c3 k u2)))) (\lambda -(u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3: -C).(csubst1 i v c1 c3))) x0 c1 (refl_equal C (CHead c1 k x0)) (subst1_single -i v u1 x0 H5) (csubst1_refl i v c1))) c2 H3)))))) H1)) (\lambda (H1: (ex3_2 C -nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i) (s k j)))) (\lambda -(c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u1)))) (\lambda (c3: -C).(\lambda (j: nat).(csubst0 j v c1 c3))))).(ex3_2_ind C nat (\lambda (_: -C).(\lambda (j: nat).(eq nat (s k i) (s k j)))) (\lambda (c3: C).(\lambda (_: -nat).(eq C c2 (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 -j v c1 c3))) (ex3_2 T C (\lambda (u2: T).(\lambda (c3: C).(eq C c2 (CHead c3 -k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: -T).(\lambda (c3: C).(csubst1 i v c1 c3)))) (\lambda (x0: C).(\lambda (x1: -nat).(\lambda (H2: (eq nat (s k i) (s k x1))).(\lambda (H3: (eq C c2 (CHead -x0 k u1))).(\lambda (H4: (csubst0 x1 v c1 x0)).(eq_ind_r C (CHead x0 k u1) -(\lambda (c: C).(ex3_2 T C (\lambda (u2: T).(\lambda (c3: C).(eq C c (CHead -c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda -(_: T).(\lambda (c3: C).(csubst1 i v c1 c3))))) (let H5 \def (eq_ind_r nat x1 -(\lambda (n: nat).(csubst0 n v c1 x0)) H4 i (s_inj k i x1 H2)) in -(ex3_2_intro T C (\lambda (u2: T).(\lambda (c3: C).(eq C (CHead x0 k u1) -(CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) -(\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1 c3))) u1 x0 (refl_equal C -(CHead x0 k u1)) (subst1_refl i v u1) (csubst1_sing i v c1 x0 H5))) c2 -H3)))))) H1)) (\lambda (H1: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: -C).(\lambda (j: nat).(eq nat (s k i) (s k j))))) (\lambda (u2: T).(\lambda -(c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: -T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u2)))) (\lambda (_: -T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))))).(ex4_3_ind T C -nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i) (s k -j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 -k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 -u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 -c3)))) (ex3_2 T C (\lambda (u2: T).(\lambda (c3: C).(eq C c2 (CHead c3 k -u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: -T).(\lambda (c3: C).(csubst1 i v c1 c3)))) (\lambda (x0: T).(\lambda (x1: -C).(\lambda (x2: nat).(\lambda (H2: (eq nat (s k i) (s k x2))).(\lambda (H3: -(eq C c2 (CHead x1 k x0))).(\lambda (H4: (subst0 x2 v u1 x0)).(\lambda (H5: -(csubst0 x2 v c1 x1)).(eq_ind_r C (CHead x1 k x0) (\lambda (c: C).(ex3_2 T C -(\lambda (u2: T).(\lambda (c3: C).(eq C c (CHead c3 k u2)))) (\lambda (u2: -T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3: -C).(csubst1 i v c1 c3))))) (let H6 \def (eq_ind_r nat x2 (\lambda (n: -nat).(csubst0 n v c1 x1)) H5 i (s_inj k i x2 H2)) in (let H7 \def (eq_ind_r -nat x2 (\lambda (n: nat).(subst0 n v u1 x0)) H4 i (s_inj k i x2 H2)) in -(ex3_2_intro T C (\lambda (u2: T).(\lambda (c3: C).(eq C (CHead x1 k x0) -(CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) -(\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1 c3))) x0 x1 (refl_equal C -(CHead x1 k x0)) (subst1_single i v u1 x0 H7) (csubst1_sing i v c1 x1 H6)))) -c2 H3)))))))) H1)) (csubst0_gen_head k c1 c2 u1 v (s k i) H0)))) x H))))))). -(* COMMENTS -Initial nodes: 1817 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubst1/getl.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubst1/getl.ma deleted file mode 100644 index e24ba9533..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubst1/getl.ma +++ /dev/null @@ -1,283 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubst1/props.ma". - -include "Basic-1/csubst0/getl.ma". - -include "Basic-1/subst1/props.ma". - -include "Basic-1/drop/props.ma". - -theorem csubst1_getl_ge: - \forall (i: nat).(\forall (n: nat).((le i n) \to (\forall (c1: C).(\forall -(c2: C).(\forall (v: T).((csubst1 i v c1 c2) \to (\forall (e: C).((getl n c1 -e) \to (getl n c2 e))))))))) -\def - \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (le i n)).(\lambda (c1: -C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst1 i v c1 -c2)).(csubst1_ind i v c1 (\lambda (c: C).(\forall (e: C).((getl n c1 e) \to -(getl n c e)))) (\lambda (e: C).(\lambda (H1: (getl n c1 e)).H1)) (\lambda -(c3: C).(\lambda (H1: (csubst0 i v c1 c3)).(\lambda (e: C).(\lambda (H2: -(getl n c1 e)).(csubst0_getl_ge i n H c1 c3 v H1 e H2))))) c2 H0))))))). -(* COMMENTS -Initial nodes: 111 -END *) - -theorem csubst1_getl_lt: - \forall (i: nat).(\forall (n: nat).((lt n i) \to (\forall (c1: C).(\forall -(c2: C).(\forall (v: T).((csubst1 i v c1 c2) \to (\forall (e1: C).((getl n c1 -e1) \to (ex2 C (\lambda (e2: C).(csubst1 (minus i n) v e1 e2)) (\lambda (e2: -C).(getl n c2 e2))))))))))) -\def - \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (lt n i)).(\lambda (c1: -C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst1 i v c1 -c2)).(csubst1_ind i v c1 (\lambda (c: C).(\forall (e1: C).((getl n c1 e1) \to -(ex2 C (\lambda (e2: C).(csubst1 (minus i n) v e1 e2)) (\lambda (e2: C).(getl -n c e2)))))) (\lambda (e1: C).(\lambda (H1: (getl n c1 e1)).(eq_ind_r nat (S -(minus i (S n))) (\lambda (n0: nat).(ex2 C (\lambda (e2: C).(csubst1 n0 v e1 -e2)) (\lambda (e2: C).(getl n c1 e2)))) (ex_intro2 C (\lambda (e2: -C).(csubst1 (S (minus i (S n))) v e1 e2)) (\lambda (e2: C).(getl n c1 e2)) e1 -(csubst1_refl (S (minus i (S n))) v e1) H1) (minus i n) (minus_x_Sy i n H)))) -(\lambda (c3: C).(\lambda (H1: (csubst0 i v c1 c3)).(\lambda (e1: C).(\lambda -(H2: (getl n c1 e1)).(eq_ind_r nat (S (minus i (S n))) (\lambda (n0: -nat).(ex2 C (\lambda (e2: C).(csubst1 n0 v e1 e2)) (\lambda (e2: C).(getl n -c3 e2)))) (let H3 \def (csubst0_getl_lt i n H c1 c3 v H1 e1 H2) in (or4_ind -(getl n c3 e1) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e1 (CHead e0 (Bind b) u)))))) (\lambda (b: -B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e0 -(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: -B).(\lambda (e2: C).(\lambda (_: C).(\lambda (u: T).(eq C e1 (CHead e2 (Bind -b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e3: C).(\lambda (u: -T).(getl n c3 (CHead e3 (Bind b) u)))))) (\lambda (_: B).(\lambda (e2: -C).(\lambda (e3: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e2 e3)))))) -(ex4_5 B C C T T (\lambda (b: B).(\lambda (e2: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq C e1 (CHead e2 (Bind b) u))))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e3: C).(\lambda (_: T).(\lambda (w: T).(getl n -c3 (CHead e3 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) -(\lambda (_: B).(\lambda (e2: C).(\lambda (e3: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus i (S n)) v e2 e3))))))) (ex2 C (\lambda (e2: -C).(csubst1 (S (minus i (S n))) v e1 e2)) (\lambda (e2: C).(getl n c3 e2))) -(\lambda (H4: (getl n c3 e1)).(ex_intro2 C (\lambda (e2: C).(csubst1 (S -(minus i (S n))) v e1 e2)) (\lambda (e2: C).(getl n c3 e2)) e1 (csubst1_refl -(S (minus i (S n))) v e1) H4)) (\lambda (H4: (ex3_4 B C T T (\lambda (b: -B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e1 (CHead e0 (Bind -b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(getl n c3 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u -w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: -T).(\lambda (_: T).(eq C e1 (CHead e0 (Bind b) u)))))) (\lambda (b: -B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e0 -(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: -T).(subst0 (minus i (S n)) v u w))))) (ex2 C (\lambda (e2: C).(csubst1 (S -(minus i (S n))) v e1 e2)) (\lambda (e2: C).(getl n c3 e2))) (\lambda (x0: -B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H5: (eq C e1 -(CHead x1 (Bind x0) x2))).(\lambda (H6: (getl n c3 (CHead x1 (Bind x0) -x3))).(\lambda (H7: (subst0 (minus i (S n)) v x2 x3)).(eq_ind_r C (CHead x1 -(Bind x0) x2) (\lambda (c: C).(ex2 C (\lambda (e2: C).(csubst1 (S (minus i (S -n))) v c e2)) (\lambda (e2: C).(getl n c3 e2)))) (ex_intro2 C (\lambda (e2: -C).(csubst1 (S (minus i (S n))) v (CHead x1 (Bind x0) x2) e2)) (\lambda (e2: -C).(getl n c3 e2)) (CHead x1 (Bind x0) x3) (csubst1_sing (S (minus i (S n))) -v (CHead x1 (Bind x0) x2) (CHead x1 (Bind x0) x3) (csubst0_snd_bind x0 (minus -i (S n)) v x2 x3 H7 x1)) H6) e1 H5)))))))) H4)) (\lambda (H4: (ex3_4 B C C T -(\lambda (b: B).(\lambda (e2: C).(\lambda (_: C).(\lambda (u: T).(eq C e1 -(CHead e2 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e3: -C).(\lambda (u: T).(getl n c3 (CHead e3 (Bind b) u)))))) (\lambda (_: -B).(\lambda (e2: C).(\lambda (e3: C).(\lambda (_: T).(csubst0 (minus i (S n)) -v e2 e3))))))).(ex3_4_ind B C C T (\lambda (b: B).(\lambda (e2: C).(\lambda -(_: C).(\lambda (u: T).(eq C e1 (CHead e2 (Bind b) u)))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e3: C).(\lambda (u: T).(getl n c3 (CHead e3 -(Bind b) u)))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (e3: C).(\lambda -(_: T).(csubst0 (minus i (S n)) v e2 e3))))) (ex2 C (\lambda (e2: C).(csubst1 -(S (minus i (S n))) v e1 e2)) (\lambda (e2: C).(getl n c3 e2))) (\lambda (x0: -B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H5: (eq C e1 -(CHead x1 (Bind x0) x3))).(\lambda (H6: (getl n c3 (CHead x2 (Bind x0) -x3))).(\lambda (H7: (csubst0 (minus i (S n)) v x1 x2)).(eq_ind_r C (CHead x1 -(Bind x0) x3) (\lambda (c: C).(ex2 C (\lambda (e2: C).(csubst1 (S (minus i (S -n))) v c e2)) (\lambda (e2: C).(getl n c3 e2)))) (ex_intro2 C (\lambda (e2: -C).(csubst1 (S (minus i (S n))) v (CHead x1 (Bind x0) x3) e2)) (\lambda (e2: -C).(getl n c3 e2)) (CHead x2 (Bind x0) x3) (csubst1_sing (S (minus i (S n))) -v (CHead x1 (Bind x0) x3) (CHead x2 (Bind x0) x3) (csubst0_fst_bind x0 (minus -i (S n)) x1 x2 v H7 x3)) H6) e1 H5)))))))) H4)) (\lambda (H4: (ex4_5 B C C T -T (\lambda (b: B).(\lambda (e2: C).(\lambda (_: C).(\lambda (u: T).(\lambda -(_: T).(eq C e1 (CHead e2 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e3: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e3 -(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: -B).(\lambda (e2: C).(\lambda (e3: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -(minus i (S n)) v e2 e3)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda -(e2: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e1 (CHead e2 -(Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e3: C).(\lambda -(_: T).(\lambda (w: T).(getl n c3 (CHead e3 (Bind b) w))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 -(minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (e3: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e2 e3)))))) -(ex2 C (\lambda (e2: C).(csubst1 (S (minus i (S n))) v e1 e2)) (\lambda (e2: -C).(getl n c3 e2))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: -C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H5: (eq C e1 (CHead x1 (Bind -x0) x3))).(\lambda (H6: (getl n c3 (CHead x2 (Bind x0) x4))).(\lambda (H7: -(subst0 (minus i (S n)) v x3 x4)).(\lambda (H8: (csubst0 (minus i (S n)) v x1 -x2)).(eq_ind_r C (CHead x1 (Bind x0) x3) (\lambda (c: C).(ex2 C (\lambda (e2: -C).(csubst1 (S (minus i (S n))) v c e2)) (\lambda (e2: C).(getl n c3 e2)))) -(ex_intro2 C (\lambda (e2: C).(csubst1 (S (minus i (S n))) v (CHead x1 (Bind -x0) x3) e2)) (\lambda (e2: C).(getl n c3 e2)) (CHead x2 (Bind x0) x4) -(csubst1_sing (S (minus i (S n))) v (CHead x1 (Bind x0) x3) (CHead x2 (Bind -x0) x4) (csubst0_both_bind x0 (minus i (S n)) v x3 x4 H7 x1 x2 H8)) H6) e1 -H5)))))))))) H4)) H3)) (minus i n) (minus_x_Sy i n H)))))) c2 H0))))))). -(* COMMENTS -Initial nodes: 2035 -END *) - -theorem csubst1_getl_ge_back: - \forall (i: nat).(\forall (n: nat).((le i n) \to (\forall (c1: C).(\forall -(c2: C).(\forall (v: T).((csubst1 i v c1 c2) \to (\forall (e: C).((getl n c2 -e) \to (getl n c1 e))))))))) -\def - \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (le i n)).(\lambda (c1: -C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst1 i v c1 -c2)).(csubst1_ind i v c1 (\lambda (c: C).(\forall (e: C).((getl n c e) \to -(getl n c1 e)))) (\lambda (e: C).(\lambda (H1: (getl n c1 e)).H1)) (\lambda -(c3: C).(\lambda (H1: (csubst0 i v c1 c3)).(\lambda (e: C).(\lambda (H2: -(getl n c3 e)).(csubst0_getl_ge_back i n H c1 c3 v H1 e H2))))) c2 H0))))))). -(* COMMENTS -Initial nodes: 111 -END *) - -theorem getl_csubst1: - \forall (d: nat).(\forall (c: C).(\forall (e: C).(\forall (u: T).((getl d c -(CHead e (Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0: C).(\lambda (_: -C).(csubst1 d u c a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) d a0 -a)))))))) -\def - \lambda (d: nat).(nat_ind (\lambda (n: nat).(\forall (c: C).(\forall (e: -C).(\forall (u: T).((getl n c (CHead e (Bind Abbr) u)) \to (ex2_2 C C -(\lambda (a0: C).(\lambda (_: C).(csubst1 n u c a0))) (\lambda (a0: -C).(\lambda (a: C).(drop (S O) n a0 a))))))))) (\lambda (c: C).(C_ind -(\lambda (c0: C).(\forall (e: C).(\forall (u: T).((getl O c0 (CHead e (Bind -Abbr) u)) \to (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O u c0 -a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0 a)))))))) (\lambda -(n: nat).(\lambda (e: C).(\lambda (u: T).(\lambda (H: (getl O (CSort n) -(CHead e (Bind Abbr) u))).(getl_gen_sort n O (CHead e (Bind Abbr) u) H (ex2_2 -C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O u (CSort n) a0))) (\lambda -(a0: C).(\lambda (a: C).(drop (S O) O a0 a))))))))) (\lambda (c0: C).(\lambda -(H: ((\forall (e: C).(\forall (u: T).((getl O c0 (CHead e (Bind Abbr) u)) \to -(ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O u c0 a0))) (\lambda -(a0: C).(\lambda (a: C).(drop (S O) O a0 a))))))))).(\lambda (k: K).(K_ind -(\lambda (k0: K).(\forall (t: T).(\forall (e: C).(\forall (u: T).((getl O -(CHead c0 k0 t) (CHead e (Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0: -C).(\lambda (_: C).(csubst1 O u (CHead c0 k0 t) a0))) (\lambda (a0: -C).(\lambda (a: C).(drop (S O) O a0 a))))))))) (\lambda (b: B).(\lambda (t: -T).(\lambda (e: C).(\lambda (u: T).(\lambda (H0: (getl O (CHead c0 (Bind b) -t) (CHead e (Bind Abbr) u))).(let H1 \def (f_equal C C (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow e | -(CHead c1 _ _) \Rightarrow c1])) (CHead e (Bind Abbr) u) (CHead c0 (Bind b) -t) (clear_gen_bind b c0 (CHead e (Bind Abbr) u) t (getl_gen_O (CHead c0 (Bind -b) t) (CHead e (Bind Abbr) u) H0))) in ((let H2 \def (f_equal C B (\lambda -(e0: C).(match e0 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow -Abbr | (CHead _ k0 _) \Rightarrow (match k0 in K return (\lambda (_: K).B) -with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead e -(Bind Abbr) u) (CHead c0 (Bind b) t) (clear_gen_bind b c0 (CHead e (Bind -Abbr) u) t (getl_gen_O (CHead c0 (Bind b) t) (CHead e (Bind Abbr) u) H0))) in -((let H3 \def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda -(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) -(CHead e (Bind Abbr) u) (CHead c0 (Bind b) t) (clear_gen_bind b c0 (CHead e -(Bind Abbr) u) t (getl_gen_O (CHead c0 (Bind b) t) (CHead e (Bind Abbr) u) -H0))) in (\lambda (H4: (eq B Abbr b)).(\lambda (_: (eq C e c0)).(eq_ind_r T t -(\lambda (t0: T).(ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O t0 -(CHead c0 (Bind b) t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0 -a))))) (eq_ind B Abbr (\lambda (b0: B).(ex2_2 C C (\lambda (a0: C).(\lambda -(_: C).(csubst1 O t (CHead c0 (Bind b0) t) a0))) (\lambda (a0: C).(\lambda -(a: C).(drop (S O) O a0 a))))) (ex2_2_intro C C (\lambda (a0: C).(\lambda (_: -C).(csubst1 O t (CHead c0 (Bind Abbr) t) a0))) (\lambda (a0: C).(\lambda (a: -C).(drop (S O) O a0 a))) (CHead c0 (Bind Abbr) t) c0 (csubst1_refl O t (CHead -c0 (Bind Abbr) t)) (drop_drop (Bind Abbr) O c0 c0 (drop_refl c0) t)) b H4) u -H3)))) H2)) H1))))))) (\lambda (f: F).(\lambda (t: T).(\lambda (e: -C).(\lambda (u: T).(\lambda (H0: (getl O (CHead c0 (Flat f) t) (CHead e (Bind -Abbr) u))).(let H_x \def (subst1_ex u t O) in (let H1 \def H_x in (ex_ind T -(\lambda (t2: T).(subst1 O u t (lift (S O) O t2))) (ex2_2 C C (\lambda (a0: -C).(\lambda (_: C).(csubst1 O u (CHead c0 (Flat f) t) a0))) (\lambda (a0: -C).(\lambda (a: C).(drop (S O) O a0 a)))) (\lambda (x: T).(\lambda (H2: -(subst1 O u t (lift (S O) O x))).(let H3 \def (H e u (getl_intro O c0 (CHead -e (Bind Abbr) u) c0 (drop_refl c0) (clear_gen_flat f c0 (CHead e (Bind Abbr) -u) t (getl_gen_O (CHead c0 (Flat f) t) (CHead e (Bind Abbr) u) H0)))) in -(ex2_2_ind C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O u c0 a0))) -(\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0 a))) (ex2_2 C C (\lambda -(a0: C).(\lambda (_: C).(csubst1 O u (CHead c0 (Flat f) t) a0))) (\lambda -(a0: C).(\lambda (a: C).(drop (S O) O a0 a)))) (\lambda (x0: C).(\lambda (x1: -C).(\lambda (H4: (csubst1 O u c0 x0)).(\lambda (H5: (drop (S O) O x0 -x1)).(ex2_2_intro C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O u (CHead c0 -(Flat f) t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0 a))) -(CHead x0 (Flat f) (lift (S O) O x)) x1 (csubst1_flat f O u t (lift (S O) O -x) H2 c0 x0 H4) (drop_drop (Flat f) O x0 x1 H5 (lift (S O) O x))))))) H3)))) -H1)))))))) k)))) c)) (\lambda (n: nat).(\lambda (H: ((\forall (c: C).(\forall -(e: C).(\forall (u: T).((getl n c (CHead e (Bind Abbr) u)) \to (ex2_2 C C -(\lambda (a0: C).(\lambda (_: C).(csubst1 n u c a0))) (\lambda (a0: -C).(\lambda (a: C).(drop (S O) n a0 a)))))))))).(\lambda (c: C).(C_ind -(\lambda (c0: C).(\forall (e: C).(\forall (u: T).((getl (S n) c0 (CHead e -(Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S -n) u c0 a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a)))))))) -(\lambda (n0: nat).(\lambda (e: C).(\lambda (u: T).(\lambda (H0: (getl (S n) -(CSort n0) (CHead e (Bind Abbr) u))).(getl_gen_sort n0 (S n) (CHead e (Bind -Abbr) u) H0 (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u -(CSort n0) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 -a))))))))) (\lambda (c0: C).(\lambda (H0: ((\forall (e: C).(\forall (u: -T).((getl (S n) c0 (CHead e (Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0: -C).(\lambda (_: C).(csubst1 (S n) u c0 a0))) (\lambda (a0: C).(\lambda (a: -C).(drop (S O) (S n) a0 a))))))))).(\lambda (k: K).(K_ind (\lambda (k0: -K).(\forall (t: T).(\forall (e: C).(\forall (u: T).((getl (S n) (CHead c0 k0 -t) (CHead e (Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0: C).(\lambda (_: -C).(csubst1 (S n) u (CHead c0 k0 t) a0))) (\lambda (a0: C).(\lambda (a: -C).(drop (S O) (S n) a0 a))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda -(e: C).(\lambda (u: T).(\lambda (H1: (getl (S n) (CHead c0 (Bind b) t) (CHead -e (Bind Abbr) u))).(let H_x \def (subst1_ex u t n) in (let H2 \def H_x in -(ex_ind T (\lambda (t2: T).(subst1 n u t (lift (S O) n t2))) (ex2_2 C C -(\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u (CHead c0 (Bind b) t) a0))) -(\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a)))) (\lambda (x: -T).(\lambda (H3: (subst1 n u t (lift (S O) n x))).(let H4 \def (H c0 e u -(getl_gen_S (Bind b) c0 (CHead e (Bind Abbr) u) t n H1)) in (ex2_2_ind C C -(\lambda (a0: C).(\lambda (_: C).(csubst1 n u c0 a0))) (\lambda (a0: -C).(\lambda (a: C).(drop (S O) n a0 a))) (ex2_2 C C (\lambda (a0: C).(\lambda -(_: C).(csubst1 (S n) u (CHead c0 (Bind b) t) a0))) (\lambda (a0: C).(\lambda -(a: C).(drop (S O) (S n) a0 a)))) (\lambda (x0: C).(\lambda (x1: C).(\lambda -(H5: (csubst1 n u c0 x0)).(\lambda (H6: (drop (S O) n x0 x1)).(ex2_2_intro C -C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u (CHead c0 (Bind b) t) -a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a))) (CHead x0 -(Bind b) (lift (S O) n x)) (CHead x1 (Bind b) x) (csubst1_bind b n u t (lift -(S O) n x) H3 c0 x0 H5) (drop_skip_bind (S O) n x0 x1 H6 b x)))))) H4)))) -H2)))))))) (\lambda (f: F).(\lambda (t: T).(\lambda (e: C).(\lambda (u: -T).(\lambda (H1: (getl (S n) (CHead c0 (Flat f) t) (CHead e (Bind Abbr) -u))).(let H_x \def (subst1_ex u t (S n)) in (let H2 \def H_x in (ex_ind T -(\lambda (t2: T).(subst1 (S n) u t (lift (S O) (S n) t2))) (ex2_2 C C -(\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u (CHead c0 (Flat f) t) a0))) -(\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a)))) (\lambda (x: -T).(\lambda (H3: (subst1 (S n) u t (lift (S O) (S n) x))).(let H4 \def (H0 e -u (getl_gen_S (Flat f) c0 (CHead e (Bind Abbr) u) t n H1)) in (ex2_2_ind C C -(\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u c0 a0))) (\lambda (a0: -C).(\lambda (a: C).(drop (S O) (S n) a0 a))) (ex2_2 C C (\lambda (a0: -C).(\lambda (_: C).(csubst1 (S n) u (CHead c0 (Flat f) t) a0))) (\lambda (a0: -C).(\lambda (a: C).(drop (S O) (S n) a0 a)))) (\lambda (x0: C).(\lambda (x1: -C).(\lambda (H5: (csubst1 (S n) u c0 x0)).(\lambda (H6: (drop (S O) (S n) x0 -x1)).(ex2_2_intro C C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u -(CHead c0 (Flat f) t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S -n) a0 a))) (CHead x0 (Flat f) (lift (S O) (S n) x)) (CHead x1 (Flat f) x) -(csubst1_flat f (S n) u t (lift (S O) (S n) x) H3 c0 x0 H5) (drop_skip_flat -(S O) n x0 x1 H6 f x)))))) H4)))) H2)))))))) k)))) c)))) d). -(* COMMENTS -Initial nodes: 2467 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubst1/props.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubst1/props.ma deleted file mode 100644 index 518a86564..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubst1/props.ma +++ /dev/null @@ -1,75 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubst1/defs.ma". - -include "Basic-1/subst1/defs.ma". - -theorem csubst1_head: - \forall (k: K).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall -(u2: T).((subst1 i v u1 u2) \to (\forall (c1: C).(\forall (c2: C).((csubst1 i -v c1 c2) \to (csubst1 (s k i) v (CHead c1 k u1) (CHead c2 k u2)))))))))) -\def - \lambda (k: K).(\lambda (i: nat).(\lambda (v: T).(\lambda (u1: T).(\lambda -(u2: T).(\lambda (H: (subst1 i v u1 u2)).(subst1_ind i v u1 (\lambda (t: -T).(\forall (c1: C).(\forall (c2: C).((csubst1 i v c1 c2) \to (csubst1 (s k -i) v (CHead c1 k u1) (CHead c2 k t)))))) (\lambda (c1: C).(\lambda (c2: -C).(\lambda (H0: (csubst1 i v c1 c2)).(csubst1_ind i v c1 (\lambda (c: -C).(csubst1 (s k i) v (CHead c1 k u1) (CHead c k u1))) (csubst1_refl (s k i) -v (CHead c1 k u1)) (\lambda (c3: C).(\lambda (H1: (csubst0 i v c1 -c3)).(csubst1_sing (s k i) v (CHead c1 k u1) (CHead c3 k u1) (csubst0_fst k i -c1 c3 v H1 u1)))) c2 H0)))) (\lambda (t2: T).(\lambda (H0: (subst0 i v u1 -t2)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csubst1 i v c1 -c2)).(csubst1_ind i v c1 (\lambda (c: C).(csubst1 (s k i) v (CHead c1 k u1) -(CHead c k t2))) (csubst1_sing (s k i) v (CHead c1 k u1) (CHead c1 k t2) -(csubst0_snd k i v u1 t2 H0 c1)) (\lambda (c3: C).(\lambda (H2: (csubst0 i v -c1 c3)).(csubst1_sing (s k i) v (CHead c1 k u1) (CHead c3 k t2) (csubst0_both -k i v u1 t2 H0 c1 c3 H2)))) c2 H1)))))) u2 H)))))). -(* COMMENTS -Initial nodes: 365 -END *) - -theorem csubst1_bind: - \forall (b: B).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall -(u2: T).((subst1 i v u1 u2) \to (\forall (c1: C).(\forall (c2: C).((csubst1 i -v c1 c2) \to (csubst1 (S i) v (CHead c1 (Bind b) u1) (CHead c2 (Bind b) -u2)))))))))) -\def - \lambda (b: B).(\lambda (i: nat).(\lambda (v: T).(\lambda (u1: T).(\lambda -(u2: T).(\lambda (H: (subst1 i v u1 u2)).(\lambda (c1: C).(\lambda (c2: -C).(\lambda (H0: (csubst1 i v c1 c2)).(eq_ind nat (s (Bind b) i) (\lambda (n: -nat).(csubst1 n v (CHead c1 (Bind b) u1) (CHead c2 (Bind b) u2))) -(csubst1_head (Bind b) i v u1 u2 H c1 c2 H0) (S i) (refl_equal nat (S -i))))))))))). -(* COMMENTS -Initial nodes: 107 -END *) - -theorem csubst1_flat: - \forall (f: F).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall -(u2: T).((subst1 i v u1 u2) \to (\forall (c1: C).(\forall (c2: C).((csubst1 i -v c1 c2) \to (csubst1 i v (CHead c1 (Flat f) u1) (CHead c2 (Flat f) -u2)))))))))) -\def - \lambda (f: F).(\lambda (i: nat).(\lambda (v: T).(\lambda (u1: T).(\lambda -(u2: T).(\lambda (H: (subst1 i v u1 u2)).(\lambda (c1: C).(\lambda (c2: -C).(\lambda (H0: (csubst1 i v c1 c2)).(eq_ind nat (s (Flat f) i) (\lambda (n: -nat).(csubst1 n v (CHead c1 (Flat f) u1) (CHead c2 (Flat f) u2))) -(csubst1_head (Flat f) i v u1 u2 H c1 c2 H0) i (refl_equal nat i)))))))))). -(* COMMENTS -Initial nodes: 103 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/clear.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/clear.ma deleted file mode 100644 index c60700147..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/clear.ma +++ /dev/null @@ -1,74 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubt/defs.ma". - -include "Basic-1/clear/fwd.ma". - -theorem csubt_clear_conf: - \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to -(\forall (e1: C).((clear c1 e1) \to (ex2 C (\lambda (e2: C).(csubt g e1 e2)) -(\lambda (e2: C).(clear c2 e2)))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubt g c1 -c2)).(csubt_ind g (\lambda (c: C).(\lambda (c0: C).(\forall (e1: C).((clear c -e1) \to (ex2 C (\lambda (e2: C).(csubt g e1 e2)) (\lambda (e2: C).(clear c0 -e2))))))) (\lambda (n: nat).(\lambda (e1: C).(\lambda (H0: (clear (CSort n) -e1)).(clear_gen_sort e1 n H0 (ex2 C (\lambda (e2: C).(csubt g e1 e2)) -(\lambda (e2: C).(clear (CSort n) e2))))))) (\lambda (c3: C).(\lambda (c4: -C).(\lambda (H0: (csubt g c3 c4)).(\lambda (H1: ((\forall (e1: C).((clear c3 -e1) \to (ex2 C (\lambda (e2: C).(csubt g e1 e2)) (\lambda (e2: C).(clear c4 -e2))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (e1: C).(\lambda (H2: -(clear (CHead c3 k u) e1)).(K_ind (\lambda (k0: K).((clear (CHead c3 k0 u) -e1) \to (ex2 C (\lambda (e2: C).(csubt g e1 e2)) (\lambda (e2: C).(clear -(CHead c4 k0 u) e2))))) (\lambda (b: B).(\lambda (H3: (clear (CHead c3 (Bind -b) u) e1)).(eq_ind_r C (CHead c3 (Bind b) u) (\lambda (c: C).(ex2 C (\lambda -(e2: C).(csubt g c e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b) u) e2)))) -(ex_intro2 C (\lambda (e2: C).(csubt g (CHead c3 (Bind b) u) e2)) (\lambda -(e2: C).(clear (CHead c4 (Bind b) u) e2)) (CHead c4 (Bind b) u) (csubt_head g -c3 c4 H0 (Bind b) u) (clear_bind b c4 u)) e1 (clear_gen_bind b c3 e1 u H3)))) -(\lambda (f: F).(\lambda (H3: (clear (CHead c3 (Flat f) u) e1)).(let H4 \def -(H1 e1 (clear_gen_flat f c3 e1 u H3)) in (ex2_ind C (\lambda (e2: C).(csubt g -e1 e2)) (\lambda (e2: C).(clear c4 e2)) (ex2 C (\lambda (e2: C).(csubt g e1 -e2)) (\lambda (e2: C).(clear (CHead c4 (Flat f) u) e2))) (\lambda (x: -C).(\lambda (H5: (csubt g e1 x)).(\lambda (H6: (clear c4 x)).(ex_intro2 C -(\lambda (e2: C).(csubt g e1 e2)) (\lambda (e2: C).(clear (CHead c4 (Flat f) -u) e2)) x H5 (clear_flat c4 x H6 f u))))) H4)))) k H2))))))))) (\lambda (c3: -C).(\lambda (c4: C).(\lambda (H0: (csubt g c3 c4)).(\lambda (_: ((\forall -(e1: C).((clear c3 e1) \to (ex2 C (\lambda (e2: C).(csubt g e1 e2)) (\lambda -(e2: C).(clear c4 e2))))))).(\lambda (b: B).(\lambda (H2: (not (eq B b -Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (e1: C).(\lambda (H3: -(clear (CHead c3 (Bind Void) u1) e1)).(eq_ind_r C (CHead c3 (Bind Void) u1) -(\lambda (c: C).(ex2 C (\lambda (e2: C).(csubt g c e2)) (\lambda (e2: -C).(clear (CHead c4 (Bind b) u2) e2)))) (ex_intro2 C (\lambda (e2: C).(csubt -g (CHead c3 (Bind Void) u1) e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b) -u2) e2)) (CHead c4 (Bind b) u2) (csubt_void g c3 c4 H0 b H2 u1 u2) -(clear_bind b c4 u2)) e1 (clear_gen_bind Void c3 e1 u1 H3)))))))))))) -(\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csubt g c3 c4)).(\lambda (_: -((\forall (e1: C).((clear c3 e1) \to (ex2 C (\lambda (e2: C).(csubt g e1 e2)) -(\lambda (e2: C).(clear c4 e2))))))).(\lambda (u: T).(\lambda (t: T).(\lambda -(H2: (ty3 g c3 u t)).(\lambda (H3: (ty3 g c4 u t)).(\lambda (e1: C).(\lambda -(H4: (clear (CHead c3 (Bind Abst) t) e1)).(eq_ind_r C (CHead c3 (Bind Abst) -t) (\lambda (c: C).(ex2 C (\lambda (e2: C).(csubt g c e2)) (\lambda (e2: -C).(clear (CHead c4 (Bind Abbr) u) e2)))) (ex_intro2 C (\lambda (e2: -C).(csubt g (CHead c3 (Bind Abst) t) e2)) (\lambda (e2: C).(clear (CHead c4 -(Bind Abbr) u) e2)) (CHead c4 (Bind Abbr) u) (csubt_abst g c3 c4 H0 u t H2 -H3) (clear_bind Abbr c4 u)) e1 (clear_gen_bind Abst c3 e1 t H4)))))))))))) c1 -c2 H)))). -(* COMMENTS -Initial nodes: 929 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/csuba.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/csuba.ma deleted file mode 100644 index b2cf1183f..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/csuba.ma +++ /dev/null @@ -1,42 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/ty3/arity.ma". - -theorem csubt_csuba: - \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to (csuba -g c1 c2)))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubt g c1 -c2)).(csubt_ind g (\lambda (c: C).(\lambda (c0: C).(csuba g c c0))) (\lambda -(n: nat).(csuba_refl g (CSort n))) (\lambda (c3: C).(\lambda (c4: C).(\lambda -(_: (csubt g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (k: K).(\lambda -(u: T).(csuba_head g c3 c4 H1 k u))))))) (\lambda (c3: C).(\lambda (c4: -C).(\lambda (_: (csubt g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (b: -B).(\lambda (H2: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: -T).(csuba_void g c3 c4 H1 b H2 u1 u2))))))))) (\lambda (c3: C).(\lambda (c4: -C).(\lambda (_: (csubt g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (u: -T).(\lambda (t: T).(\lambda (H2: (ty3 g c3 u t)).(\lambda (_: (ty3 g c4 u -t)).(let H_x \def (ty3_arity g c3 u t H2) in (let H4 \def H_x in (ex2_ind A -(\lambda (a1: A).(arity g c3 u a1)) (\lambda (a1: A).(arity g c3 t (asucc g -a1))) (csuba g (CHead c3 (Bind Abst) t) (CHead c4 (Bind Abbr) u)) (\lambda -(x: A).(\lambda (H5: (arity g c3 u x)).(\lambda (H6: (arity g c3 t (asucc g -x))).(csuba_abst g c3 c4 H1 t x H6 u (csuba_arity g c3 u x H5 c4 H1))))) -H4))))))))))) c1 c2 H)))). -(* COMMENTS -Initial nodes: 313 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/defs.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/defs.ma deleted file mode 100644 index a12c9f829..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/defs.ma +++ /dev/null @@ -1,29 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/ty3/defs.ma". - -inductive csubt (g: G): C \to (C \to Prop) \def -| csubt_sort: \forall (n: nat).(csubt g (CSort n) (CSort n)) -| csubt_head: \forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to (\forall -(k: K).(\forall (u: T).(csubt g (CHead c1 k u) (CHead c2 k u)))))) -| csubt_void: \forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to (\forall -(b: B).((not (eq B b Void)) \to (\forall (u1: T).(\forall (u2: T).(csubt g -(CHead c1 (Bind Void) u1) (CHead c2 (Bind b) u2)))))))) -| csubt_abst: \forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to (\forall -(u: T).(\forall (t: T).((ty3 g c1 u t) \to ((ty3 g c2 u t) \to (csubt g -(CHead c1 (Bind Abst) t) (CHead c2 (Bind Abbr) u)))))))). - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/drop.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/drop.ma deleted file mode 100644 index adaedcc9a..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/drop.ma +++ /dev/null @@ -1,590 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubt/fwd.ma". - -include "Basic-1/drop/fwd.ma". - -theorem csubt_drop_flat: - \forall (g: G).(\forall (f: F).(\forall (n: nat).(\forall (c1: C).(\forall -(c2: C).((csubt g c1 c2) \to (\forall (d1: C).(\forall (u: T).((drop n O c1 -(CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda -(d2: C).(drop n O c2 (CHead d2 (Flat f) u)))))))))))) -\def - \lambda (g: G).(\lambda (f: F).(\lambda (n: nat).(nat_ind (\lambda (n0: -nat).(\forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to (\forall (d1: -C).(\forall (u: T).((drop n0 O c1 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda -(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 (Flat f) -u))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubt g c1 -c2)).(\lambda (d1: C).(\lambda (u: T).(\lambda (H0: (drop O O c1 (CHead d1 -(Flat f) u))).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c2)) H -(CHead d1 (Flat f) u) (drop_gen_refl c1 (CHead d1 (Flat f) u) H0)) in (let -H_x \def (csubt_gen_flat g d1 c2 u f H1) in (let H2 \def H_x in (ex2_ind C -(\lambda (e2: C).(eq C c2 (CHead e2 (Flat f) u))) (\lambda (e2: C).(csubt g -d1 e2)) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O -c2 (CHead d2 (Flat f) u)))) (\lambda (x: C).(\lambda (H3: (eq C c2 (CHead x -(Flat f) u))).(\lambda (H4: (csubt g d1 x)).(eq_ind_r C (CHead x (Flat f) u) -(\lambda (c: C).(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(drop O O c (CHead d2 (Flat f) u))))) (ex_intro2 C (\lambda (d2: C).(csubt -g d1 d2)) (\lambda (d2: C).(drop O O (CHead x (Flat f) u) (CHead d2 (Flat f) -u))) x H4 (drop_refl (CHead x (Flat f) u))) c2 H3)))) H2)))))))))) (\lambda -(n0: nat).(\lambda (H: ((\forall (c1: C).(\forall (c2: C).((csubt g c1 c2) -\to (\forall (d1: C).(\forall (u: T).((drop n0 O c1 (CHead d1 (Flat f) u)) -\to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c2 -(CHead d2 (Flat f) u)))))))))))).(\lambda (c1: C).(\lambda (c2: C).(\lambda -(H0: (csubt g c1 c2)).(csubt_ind g (\lambda (c: C).(\lambda (c0: C).(\forall -(d1: C).(\forall (u: T).((drop (S n0) O c (CHead d1 (Flat f) u)) \to (ex2 C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O c0 (CHead -d2 (Flat f) u))))))))) (\lambda (n1: nat).(\lambda (d1: C).(\lambda (u: -T).(\lambda (H1: (drop (S n0) O (CSort n1) (CHead d1 (Flat f) u))).(and3_ind -(eq C (CHead d1 (Flat f) u) (CSort n1)) (eq nat (S n0) O) (eq nat O O) (ex2 C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) -(CHead d2 (Flat f) u)))) (\lambda (_: (eq C (CHead d1 (Flat f) u) (CSort -n1))).(\lambda (H3: (eq nat (S n0) O)).(\lambda (_: (eq nat O O)).(let H5 -\def (eq_ind nat (S n0) (\lambda (ee: nat).(match ee in nat return (\lambda -(_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H3) -in (False_ind (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop -(S n0) O (CSort n1) (CHead d2 (Flat f) u)))) H5))))) (drop_gen_sort n1 (S n0) -O (CHead d1 (Flat f) u) H1)))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda -(H1: (csubt g c0 c3)).(\lambda (H2: ((\forall (d1: C).(\forall (u: T).((drop -(S n0) O c0 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 -d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Flat f) -u))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (u: T).(\forall -(d1: C).(\forall (u0: T).((drop (S n0) O (CHead c0 k0 u) (CHead d1 (Flat f) -u0)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S -n0) O (CHead c3 k0 u) (CHead d2 (Flat f) u0))))))))) (\lambda (b: B).(\lambda -(u: T).(\lambda (d1: C).(\lambda (u0: T).(\lambda (H3: (drop (S n0) O (CHead -c0 (Bind b) u) (CHead d1 (Flat f) u0))).(ex2_ind C (\lambda (d2: C).(csubt g -d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Flat f) u0))) (ex2 C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 -(Bind b) u) (CHead d2 (Flat f) u0)))) (\lambda (x: C).(\lambda (H4: (csubt g -d1 x)).(\lambda (H5: (drop n0 O c3 (CHead x (Flat f) u0))).(ex_intro2 C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 -(Bind b) u) (CHead d2 (Flat f) u0))) x H4 (drop_drop (Bind b) n0 c3 (CHead x -(Flat f) u0) H5 u))))) (H c0 c3 H1 d1 u0 (drop_gen_drop (Bind b) c0 (CHead d1 -(Flat f) u0) u n0 H3)))))))) (\lambda (f0: F).(\lambda (u: T).(\lambda (d1: -C).(\lambda (u0: T).(\lambda (H3: (drop (S n0) O (CHead c0 (Flat f0) u) -(CHead d1 (Flat f) u0))).(ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Flat f) u0))) (ex2 C (\lambda -(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f0) -u) (CHead d2 (Flat f) u0)))) (\lambda (x: C).(\lambda (H4: (csubt g d1 -x)).(\lambda (H5: (drop (S n0) O c3 (CHead x (Flat f) u0))).(ex_intro2 C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 -(Flat f0) u) (CHead d2 (Flat f) u0))) x H4 (drop_drop (Flat f0) n0 c3 (CHead -x (Flat f) u0) H5 u))))) (H2 d1 u0 (drop_gen_drop (Flat f0) c0 (CHead d1 -(Flat f) u0) u n0 H3)))))))) k)))))) (\lambda (c0: C).(\lambda (c3: -C).(\lambda (H1: (csubt g c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (u: -T).((drop (S n0) O c0 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: -C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Flat f) -u))))))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: -T).(\lambda (u2: T).(\lambda (d1: C).(\lambda (u: T).(\lambda (H4: (drop (S -n0) O (CHead c0 (Bind Void) u1) (CHead d1 (Flat f) u))).(ex2_ind C (\lambda -(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Flat f) -u))) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O -(CHead c3 (Bind b) u2) (CHead d2 (Flat f) u)))) (\lambda (x: C).(\lambda (H5: -(csubt g d1 x)).(\lambda (H6: (drop n0 O c3 (CHead x (Flat f) u))).(ex_intro2 -C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 -(Bind b) u2) (CHead d2 (Flat f) u))) x H5 (drop_drop (Bind b) n0 c3 (CHead x -(Flat f) u) H6 u2))))) (H c0 c3 H1 d1 u (drop_gen_drop (Bind Void) c0 (CHead -d1 (Flat f) u) u1 n0 H4)))))))))))))) (\lambda (c0: C).(\lambda (c3: -C).(\lambda (H1: (csubt g c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (u: -T).((drop (S n0) O c0 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: -C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Flat f) -u))))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u -t)).(\lambda (_: (ty3 g c3 u t)).(\lambda (d1: C).(\lambda (u0: T).(\lambda -(H5: (drop (S n0) O (CHead c0 (Bind Abst) t) (CHead d1 (Flat f) -u0))).(ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 -O c3 (CHead d2 (Flat f) u0))) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Flat f) -u0)))) (\lambda (x: C).(\lambda (H6: (csubt g d1 x)).(\lambda (H7: (drop n0 O -c3 (CHead x (Flat f) u0))).(ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Flat f) -u0))) x H6 (drop_drop (Bind Abbr) n0 c3 (CHead x (Flat f) u0) H7 u))))) (H c0 -c3 H1 d1 u0 (drop_gen_drop (Bind Abst) c0 (CHead d1 (Flat f) u0) t n0 -H5)))))))))))))) c1 c2 H0)))))) n))). -(* COMMENTS -Initial nodes: 2090 -END *) - -theorem csubt_drop_abbr: - \forall (g: G).(\forall (n: nat).(\forall (c1: C).(\forall (c2: C).((csubt g -c1 c2) \to (\forall (d1: C).(\forall (u: T).((drop n O c1 (CHead d1 (Bind -Abbr) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop -n O c2 (CHead d2 (Bind Abbr) u))))))))))) -\def - \lambda (g: G).(\lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (c1: -C).(\forall (c2: C).((csubt g c1 c2) \to (\forall (d1: C).(\forall (u: -T).((drop n0 O c1 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: -C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 (Bind Abbr) -u))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubt g c1 -c2)).(\lambda (d1: C).(\lambda (u: T).(\lambda (H0: (drop O O c1 (CHead d1 -(Bind Abbr) u))).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c2)) H -(CHead d1 (Bind Abbr) u) (drop_gen_refl c1 (CHead d1 (Bind Abbr) u) H0)) in -(let H2 \def (csubt_gen_abbr g d1 c2 u H1) in (ex2_ind C (\lambda (e2: C).(eq -C c2 (CHead e2 (Bind Abbr) u))) (\lambda (e2: C).(csubt g d1 e2)) (ex2 C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 -(Bind Abbr) u)))) (\lambda (x: C).(\lambda (H3: (eq C c2 (CHead x (Bind Abbr) -u))).(\lambda (H4: (csubt g d1 x)).(eq_ind_r C (CHead x (Bind Abbr) u) -(\lambda (c: C).(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(drop O O c (CHead d2 (Bind Abbr) u))))) (ex_intro2 C (\lambda (d2: -C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O (CHead x (Bind Abbr) u) (CHead -d2 (Bind Abbr) u))) x H4 (drop_refl (CHead x (Bind Abbr) u))) c2 H3)))) -H2))))))))) (\lambda (n0: nat).(\lambda (H: ((\forall (c1: C).(\forall (c2: -C).((csubt g c1 c2) \to (\forall (d1: C).(\forall (u: T).((drop n0 O c1 -(CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(drop n0 O c2 (CHead d2 (Bind Abbr) u)))))))))))).(\lambda -(c1: C).(\lambda (c2: C).(\lambda (H0: (csubt g c1 c2)).(csubt_ind g (\lambda -(c: C).(\lambda (c0: C).(\forall (d1: C).(\forall (u: T).((drop (S n0) O c -(CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(drop (S n0) O c0 (CHead d2 (Bind Abbr) u))))))))) (\lambda -(n1: nat).(\lambda (d1: C).(\lambda (u: T).(\lambda (H1: (drop (S n0) O -(CSort n1) (CHead d1 (Bind Abbr) u))).(and3_ind (eq C (CHead d1 (Bind Abbr) -u) (CSort n1)) (eq nat (S n0) O) (eq nat O O) (ex2 C (\lambda (d2: C).(csubt -g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) -u)))) (\lambda (_: (eq C (CHead d1 (Bind Abbr) u) (CSort n1))).(\lambda (H3: -(eq nat (S n0) O)).(\lambda (_: (eq nat O O)).(let H5 \def (eq_ind nat (S n0) -(\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O -\Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind (ex2 C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) -(CHead d2 (Bind Abbr) u)))) H5))))) (drop_gen_sort n1 (S n0) O (CHead d1 -(Bind Abbr) u) H1)))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H1: -(csubt g c0 c3)).(\lambda (H2: ((\forall (d1: C).(\forall (u: T).((drop (S -n0) O c0 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 -d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abbr) -u))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (u: T).(\forall -(d1: C).(\forall (u0: T).((drop (S n0) O (CHead c0 k0 u) (CHead d1 (Bind -Abbr) u0)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(drop (S n0) O (CHead c3 k0 u) (CHead d2 (Bind Abbr) u0))))))))) (\lambda -(b: B).(\lambda (u: T).(\lambda (d1: C).(\lambda (u0: T).(\lambda (H3: (drop -(S n0) O (CHead c0 (Bind b) u) (CHead d1 (Bind Abbr) u0))).(ex2_ind C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 -(Bind Abbr) u0))) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abbr) u0)))) (\lambda -(x: C).(\lambda (H4: (csubt g d1 x)).(\lambda (H5: (drop n0 O c3 (CHead x -(Bind Abbr) u0))).(ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda -(d2: C).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abbr) u0))) x H4 -(drop_drop (Bind b) n0 c3 (CHead x (Bind Abbr) u0) H5 u))))) (H c0 c3 H1 d1 -u0 (drop_gen_drop (Bind b) c0 (CHead d1 (Bind Abbr) u0) u n0 H3)))))))) -(\lambda (f: F).(\lambda (u: T).(\lambda (d1: C).(\lambda (u0: T).(\lambda -(H3: (drop (S n0) O (CHead c0 (Flat f) u) (CHead d1 (Bind Abbr) -u0))).(ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S -n0) O c3 (CHead d2 (Bind Abbr) u0))) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind Abbr) -u0)))) (\lambda (x: C).(\lambda (H4: (csubt g d1 x)).(\lambda (H5: (drop (S -n0) O c3 (CHead x (Bind Abbr) u0))).(ex_intro2 C (\lambda (d2: C).(csubt g d1 -d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind -Abbr) u0))) x H4 (drop_drop (Flat f) n0 c3 (CHead x (Bind Abbr) u0) H5 u))))) -(H2 d1 u0 (drop_gen_drop (Flat f) c0 (CHead d1 (Bind Abbr) u0) u n0 -H3)))))))) k)))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H1: (csubt g -c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (u: T).((drop (S n0) O c0 -(CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u))))))))).(\lambda -(b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: -T).(\lambda (d1: C).(\lambda (u: T).(\lambda (H4: (drop (S n0) O (CHead c0 -(Bind Void) u1) (CHead d1 (Bind Abbr) u))).(ex2_ind C (\lambda (d2: C).(csubt -g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abbr) u))) (ex2 C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 -(Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda (x: C).(\lambda (H5: (csubt -g d1 x)).(\lambda (H6: (drop n0 O c3 (CHead x (Bind Abbr) u))).(ex_intro2 C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 -(Bind b) u2) (CHead d2 (Bind Abbr) u))) x H5 (drop_drop (Bind b) n0 c3 (CHead -x (Bind Abbr) u) H6 u2))))) (H c0 c3 H1 d1 u (drop_gen_drop (Bind Void) c0 -(CHead d1 (Bind Abbr) u) u1 n0 H4)))))))))))))) (\lambda (c0: C).(\lambda -(c3: C).(\lambda (H1: (csubt g c0 c3)).(\lambda (_: ((\forall (d1: -C).(\forall (u: T).((drop (S n0) O c0 (CHead d1 (Bind Abbr) u)) \to (ex2 C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead -d2 (Bind Abbr) u))))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (_: (ty3 g -c0 u t)).(\lambda (_: (ty3 g c3 u t)).(\lambda (d1: C).(\lambda (u0: -T).(\lambda (H5: (drop (S n0) O (CHead c0 (Bind Abst) t) (CHead d1 (Bind -Abbr) u0))).(ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(drop n0 O c3 (CHead d2 (Bind Abbr) u0))) (ex2 C (\lambda (d2: C).(csubt g -d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 -(Bind Abbr) u0)))) (\lambda (x: C).(\lambda (H6: (csubt g d1 x)).(\lambda -(H7: (drop n0 O c3 (CHead x (Bind Abbr) u0))).(ex_intro2 C (\lambda (d2: -C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) -(CHead d2 (Bind Abbr) u0))) x H6 (drop_drop (Bind Abbr) n0 c3 (CHead x (Bind -Abbr) u0) H7 u))))) (H c0 c3 H1 d1 u0 (drop_gen_drop (Bind Abst) c0 (CHead d1 -(Bind Abbr) u0) t n0 H5)))))))))))))) c1 c2 H0)))))) n)). -(* COMMENTS -Initial nodes: 2084 -END *) - -theorem csubt_drop_abst: - \forall (g: G).(\forall (n: nat).(\forall (c1: C).(\forall (c2: C).((csubt g -c1 c2) \to (\forall (d1: C).(\forall (t: T).((drop n O c1 (CHead d1 (Bind -Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(drop n O c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: -C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop n -O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u -t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))))) -\def - \lambda (g: G).(\lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (c1: -C).(\forall (c2: C).((csubt g c1 c2) \to (\forall (d1: C).(\forall (t: -T).((drop n0 O c1 (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: -C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 (Bind Abst) -t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda -(d2: C).(\lambda (u: T).(drop n0 O c2 (CHead d2 (Bind Abbr) u)))) (\lambda -(_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 -g d2 u t)))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubt g -c1 c2)).(\lambda (d1: C).(\lambda (t: T).(\lambda (H0: (drop O O c1 (CHead d1 -(Bind Abst) t))).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c2)) H -(CHead d1 (Bind Abst) t) (drop_gen_refl c1 (CHead d1 (Bind Abst) t) H0)) in -(let H2 \def (csubt_gen_abst g d1 c2 t H1) in (or_ind (ex2 C (\lambda (e2: -C).(eq C c2 (CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csubt g d1 e2))) -(ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) -v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g d1 e2))) (\lambda (_: -C).(\lambda (v2: T).(ty3 g d1 v2 t))) (\lambda (e2: C).(\lambda (v2: T).(ty3 -g e2 v2 t)))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(drop O O c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: -C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop O -O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u -t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (H3: (ex2 C -(\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csubt -g d1 e2)))).(ex2_ind C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abst) t))) -(\lambda (e2: C).(csubt g d1 e2)) (or (ex2 C (\lambda (d2: C).(csubt g d1 -d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T -(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda -(u: T).(drop O O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: -T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) -(\lambda (x: C).(\lambda (H4: (eq C c2 (CHead x (Bind Abst) t))).(\lambda -(H5: (csubt g d1 x)).(eq_ind_r C (CHead x (Bind Abst) t) (\lambda (c: C).(or -(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O c (CHead -d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 -d2))) (\lambda (d2: C).(\lambda (u: T).(drop O O c (CHead d2 (Bind Abbr) -u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: -C).(\lambda (u: T).(ty3 g d2 u t)))))) (or_introl (ex2 C (\lambda (d2: -C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O (CHead x (Bind Abst) t) (CHead -d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 -d2))) (\lambda (d2: C).(\lambda (u: T).(drop O O (CHead x (Bind Abst) t) -(CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) -(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: -C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O (CHead x (Bind Abst) t) (CHead -d2 (Bind Abst) t))) x H5 (drop_refl (CHead x (Bind Abst) t)))) c2 H4)))) H3)) -(\lambda (H3: (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 -(Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g d1 e2))) -(\lambda (_: C).(\lambda (v2: T).(ty3 g d1 v2 t))) (\lambda (e2: C).(\lambda -(v2: T).(ty3 g e2 v2 t))))).(ex4_2_ind C T (\lambda (e2: C).(\lambda (v2: -T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: -T).(csubt g d1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g d1 v2 t))) -(\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 t))) (or (ex2 C (\lambda (d2: -C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) t)))) -(ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: -C).(\lambda (u: T).(drop O O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: -C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g -d2 u t))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H4: (eq C c2 (CHead -x0 (Bind Abbr) x1))).(\lambda (H5: (csubt g d1 x0)).(\lambda (H6: (ty3 g d1 -x1 t)).(\lambda (H7: (ty3 g x0 x1 t)).(eq_ind_r C (CHead x0 (Bind Abbr) x1) -(\lambda (c: C).(or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(drop O O c (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: -C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop O -O c (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u -t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))) (or_intror (ex2 C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O (CHead x0 (Bind -Abbr) x1) (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda -(_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop O O (CHead x0 -(Bind Abbr) x1) (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: -T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) -(ex4_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda -(d2: C).(\lambda (u: T).(drop O O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind -Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: -C).(\lambda (u: T).(ty3 g d2 u t))) x0 x1 H5 (drop_refl (CHead x0 (Bind Abbr) -x1)) H6 H7)) c2 H4))))))) H3)) H2))))))))) (\lambda (n0: nat).(\lambda (H: -((\forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to (\forall (d1: -C).(\forall (t: T).((drop n0 O c1 (CHead d1 (Bind Abst) t)) \to (or (ex2 C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 -(Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 -d2))) (\lambda (d2: C).(\lambda (u: T).(drop n0 O c2 (CHead d2 (Bind Abbr) -u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: -C).(\lambda (u: T).(ty3 g d2 u t))))))))))))).(\lambda (c1: C).(\lambda (c2: -C).(\lambda (H0: (csubt g c1 c2)).(csubt_ind g (\lambda (c: C).(\lambda (c0: -C).(\forall (d1: C).(\forall (t: T).((drop (S n0) O c (CHead d1 (Bind Abst) -t)) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop -(S n0) O c0 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda -(_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O c0 -(CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) -(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))) (\lambda (n1: -nat).(\lambda (d1: C).(\lambda (t: T).(\lambda (H1: (drop (S n0) O (CSort n1) -(CHead d1 (Bind Abst) t))).(and3_ind (eq C (CHead d1 (Bind Abst) t) (CSort -n1)) (eq nat (S n0) O) (eq nat O O) (or (ex2 C (\lambda (d2: C).(csubt g d1 -d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abst) t)))) -(ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: -C).(\lambda (u: T).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) u)))) -(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda -(u: T).(ty3 g d2 u t))))) (\lambda (_: (eq C (CHead d1 (Bind Abst) t) (CSort -n1))).(\lambda (H3: (eq nat (S n0) O)).(\lambda (_: (eq nat O O)).(let H5 -\def (eq_ind nat (S n0) (\lambda (ee: nat).(match ee in nat return (\lambda -(_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H3) -in (False_ind (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda -(d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: -T).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) u)))) (\lambda (_: -C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g -d2 u t))))) H5))))) (drop_gen_sort n1 (S n0) O (CHead d1 (Bind Abst) t) -H1)))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H1: (csubt g c0 -c3)).(\lambda (H2: ((\forall (d1: C).(\forall (t: T).((drop (S n0) O c0 -(CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abst) t)))) (ex4_2 C T -(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda -(u: T).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda -(u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u -t)))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (u: T).(\forall -(d1: C).(\forall (t: T).((drop (S n0) O (CHead c0 k0 u) (CHead d1 (Bind Abst) -t)) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop -(S n0) O (CHead c3 k0 u) (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: -C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop -(S n0) O (CHead c3 k0 u) (CHead d2 (Bind Abbr) u0)))) (\lambda (_: -C).(\lambda (u0: T).(ty3 g d1 u0 t))) (\lambda (d2: C).(\lambda (u0: T).(ty3 -g d2 u0 t)))))))))) (\lambda (b: B).(\lambda (u: T).(\lambda (d1: C).(\lambda -(t: T).(\lambda (H3: (drop (S n0) O (CHead c0 (Bind b) u) (CHead d1 (Bind -Abst) t))).(or_ind (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(drop n0 O c3 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: -C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop -n0 O c3 (CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g -d1 u0 t))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))) (or (ex2 C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 -(Bind b) u) (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda -(_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O -(CHead c3 (Bind b) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda -(u0: T).(ty3 g d1 u0 t))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 -t))))) (\lambda (H4: (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(drop n0 O c3 (CHead d2 (Bind Abst) t))))).(ex2_ind C (\lambda (d2: -C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abst) t))) -(or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O -(CHead c3 (Bind b) u) (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: -C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop -(S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (_: -C).(\lambda (u0: T).(ty3 g d1 u0 t))) (\lambda (d2: C).(\lambda (u0: T).(ty3 -g d2 u0 t))))) (\lambda (x: C).(\lambda (H5: (csubt g d1 x)).(\lambda (H6: -(drop n0 O c3 (CHead x (Bind Abst) t))).(or_introl (ex2 C (\lambda (d2: -C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u) -(CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: -T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead -c3 (Bind b) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: -T).(ty3 g d1 u0 t))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))) -(ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) -O (CHead c3 (Bind b) u) (CHead d2 (Bind Abst) t))) x H5 (drop_drop (Bind b) -n0 c3 (CHead x (Bind Abst) t) H6 u)))))) H4)) (\lambda (H4: (ex4_2 C T -(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda -(u0: T).(drop n0 O c3 (CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda -(u0: T).(ty3 g d1 u0 t))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 -t))))).(ex4_2_ind C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) -(\lambda (d2: C).(\lambda (u0: T).(drop n0 O c3 (CHead d2 (Bind Abbr) u0)))) -(\lambda (_: C).(\lambda (u0: T).(ty3 g d1 u0 t))) (\lambda (d2: C).(\lambda -(u0: T).(ty3 g d2 u0 t))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abst) -t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda -(d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind -Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g d1 u0 t))) (\lambda (d2: -C).(\lambda (u0: T).(ty3 g d2 u0 t))))) (\lambda (x0: C).(\lambda (x1: -T).(\lambda (H5: (csubt g d1 x0)).(\lambda (H6: (drop n0 O c3 (CHead x0 (Bind -Abbr) x1))).(\lambda (H7: (ty3 g d1 x1 t)).(\lambda (H8: (ty3 g x0 x1 -t)).(or_intror (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(drop (S n0) O (CHead c3 (Bind b) u) 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(d2: -C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abst) t0)))) -(ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: -C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind -Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g d1 u0 t0))) (\lambda -(d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0)))) (ex_intro2 C (\lambda (d2: -C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) -(CHead d2 (Bind Abst) t0))) x H7 (drop_drop (Bind Abbr) n0 c3 (CHead x (Bind -Abst) t0) H8 u)))))) H6)) (\lambda (H6: (ex4_2 C T (\lambda (d2: C).(\lambda -(_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop n0 O c3 -(CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g d1 u0 -t0))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0))))).(ex4_2_ind C T -(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda -(u0: T).(drop n0 O c3 (CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda -(u0: T).(ty3 g d1 u0 t0))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 -t0))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S -n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abst) t0)))) (ex4_2 C T -(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda -(u0: T).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) -(\lambda (_: C).(\lambda (u0: T).(ty3 g d1 u0 t0))) (\lambda (d2: C).(\lambda -(u0: T).(ty3 g d2 u0 t0))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H7: -(csubt g d1 x0)).(\lambda (H8: (drop n0 O c3 (CHead x0 (Bind Abbr) -x1))).(\lambda (H9: (ty3 g d1 x1 t0)).(\lambda (H10: (ty3 g x0 x1 -t0)).(or_intror (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abst) t0)))) -(ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: -C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind -Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g d1 u0 t0))) (\lambda -(d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0)))) (ex4_2_intro C T (\lambda (d2: -C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop -(S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (_: -C).(\lambda (u0: T).(ty3 g d1 u0 t0))) (\lambda (d2: C).(\lambda (u0: T).(ty3 -g d2 u0 t0))) x0 x1 H7 (drop_drop (Bind Abbr) n0 c3 (CHead x0 (Bind Abbr) x1) -H8 u) H9 H10)))))))) H6)) (H c0 c3 H1 d1 t0 (drop_gen_drop (Bind Abst) c0 -(CHead d1 (Bind Abst) t0) t n0 H5)))))))))))))) c1 c2 H0)))))) n)). -(* COMMENTS -Initial nodes: 7940 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/fwd.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/fwd.ma deleted file mode 100644 index 63a3eca47..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/fwd.ma +++ /dev/null @@ -1,398 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubt/defs.ma". - -theorem csubt_gen_abbr: - \forall (g: G).(\forall (e1: C).(\forall (c2: C).(\forall (v: T).((csubt g -(CHead e1 (Bind Abbr) v) c2) \to (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 -(Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2))))))) -\def - \lambda (g: G).(\lambda (e1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda -(H: (csubt g (CHead e1 (Bind Abbr) v) c2)).(insert_eq C (CHead e1 (Bind Abbr) -v) (\lambda (c: C).(csubt g c c2)) (\lambda (_: C).(ex2 C (\lambda (e2: -C).(eq C c2 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))) -(\lambda (y: C).(\lambda (H0: (csubt g y c2)).(csubt_ind g (\lambda (c: -C).(\lambda (c0: C).((eq C c (CHead e1 (Bind Abbr) v)) \to (ex2 C (\lambda -(e2: C).(eq C c0 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 -e2)))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead e1 (Bind -Abbr) v))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C -return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) -\Rightarrow False])) I (CHead e1 (Bind Abbr) v) H1) in (False_ind (ex2 C -(\lambda (e2: C).(eq C (CSort n) (CHead e2 (Bind Abbr) v))) (\lambda (e2: -C).(csubt g e1 e2))) H2)))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1: -(csubt g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind Abbr) v)) \to (ex2 -C (\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abbr) v))) (\lambda (e2: -C).(csubt g e1 e2)))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C -(CHead c1 k u) (CHead e1 (Bind Abbr) v))).(let H4 \def (f_equal C C (\lambda -(e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 -| (CHead c _ _) \Rightarrow c])) (CHead c1 k u) (CHead e1 (Bind Abbr) v) H3) -in ((let H5 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda -(_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) -(CHead c1 k u) (CHead e1 (Bind Abbr) v) H3) in ((let H6 \def (f_equal C T -(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) -\Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u) (CHead e1 (Bind -Abbr) v) H3) in (\lambda (H7: (eq K k (Bind Abbr))).(\lambda (H8: (eq C c1 -e1)).(eq_ind_r T v (\lambda (t: T).(ex2 C (\lambda (e2: C).(eq C (CHead c3 k -t) (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))) (eq_ind_r K -(Bind Abbr) (\lambda (k0: K).(ex2 C (\lambda (e2: C).(eq C (CHead c3 k0 v) -(CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))) (let H9 \def -(eq_ind C c1 (\lambda (c: C).((eq C c (CHead e1 (Bind Abbr) v)) \to (ex2 C -(\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt -g e1 e2))))) H2 e1 H8) in (let H10 \def (eq_ind C c1 (\lambda (c: C).(csubt g -c c3)) H1 e1 H8) in (ex_intro2 C (\lambda (e2: C).(eq C (CHead c3 (Bind Abbr) -v) (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)) c3 -(refl_equal C (CHead c3 (Bind Abbr) v)) H10))) k H7) u H6)))) H5)) -H4))))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (_: (csubt g c1 -c3)).(\lambda (_: (((eq C c1 (CHead e1 (Bind Abbr) v)) \to (ex2 C (\lambda -(e2: C).(eq C c3 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 -e2)))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: -T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u1) (CHead e1 -(Bind Abbr) v))).(let H5 \def (eq_ind C (CHead c1 (Bind Void) u1) (\lambda -(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind b0) \Rightarrow (match b0 in B return (\lambda (_: -B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False | Void -\Rightarrow True]) | (Flat _) \Rightarrow False])])) I (CHead e1 (Bind Abbr) -v) H4) in (False_ind (ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind b) u2) -(CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2))) H5))))))))))) -(\lambda (c1: C).(\lambda (c3: C).(\lambda (_: (csubt g c1 c3)).(\lambda (_: -(((eq C c1 (CHead e1 (Bind Abbr) v)) \to (ex2 C (\lambda (e2: C).(eq C c3 -(CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))))).(\lambda (u: -T).(\lambda (t: T).(\lambda (_: (ty3 g c1 u t)).(\lambda (_: (ty3 g c3 u -t)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) t) (CHead e1 (Bind Abbr) -v))).(let H6 \def (eq_ind C (CHead c1 (Bind Abst) t) (\lambda (ee: C).(match -ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | -(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr -\Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat -_) \Rightarrow False])])) I (CHead e1 (Bind Abbr) v) H5) in (False_ind (ex2 C -(\lambda (e2: C).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 (Bind Abbr) v))) -(\lambda (e2: C).(csubt g e1 e2))) H6))))))))))) y c2 H0))) H))))). -(* COMMENTS -Initial nodes: 1111 -END *) - -theorem csubt_gen_abst: - \forall (g: G).(\forall (e1: C).(\forall (c2: C).(\forall (v1: T).((csubt g -(CHead e1 (Bind Abst) v1) c2) \to (or (ex2 C (\lambda (e2: C).(eq C c2 (CHead -e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda -(e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: -C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g -e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))))))))) -\def - \lambda (g: G).(\lambda (e1: C).(\lambda (c2: C).(\lambda (v1: T).(\lambda -(H: (csubt g (CHead e1 (Bind Abst) v1) c2)).(insert_eq C (CHead e1 (Bind -Abst) v1) (\lambda (c: C).(csubt g c c2)) (\lambda (_: C).(or (ex2 C (\lambda -(e2: C).(eq C c2 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 -e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind -Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: -C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 -g e2 v2 v1)))))) (\lambda (y: C).(\lambda (H0: (csubt g y c2)).(csubt_ind g -(\lambda (c: C).(\lambda (c0: C).((eq C c (CHead e1 (Bind Abst) v1)) \to (or -(ex2 C (\lambda (e2: C).(eq C c0 (CHead e2 (Bind Abst) v1))) (\lambda (e2: -C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c0 -(CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 -e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: -C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))))) (\lambda (n: nat).(\lambda (H1: -(eq C (CSort n) (CHead e1 (Bind Abst) v1))).(let H2 \def (eq_ind C (CSort n) -(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 (Bind Abst) -v1) H1) in (False_ind (or (ex2 C (\lambda (e2: C).(eq C (CSort n) (CHead e2 -(Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: -C).(\lambda (v2: T).(eq C (CSort n) (CHead e2 (Bind Abbr) v2)))) (\lambda -(e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: -T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))))) -H2)))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1: (csubt g c1 -c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind Abst) v1)) \to (or (ex2 C -(\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt -g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 -(Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) -(\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda -(v2: T).(ty3 g e2 v2 v1)))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: -(eq C (CHead c1 k u) (CHead e1 (Bind Abst) v1))).(let H4 \def (f_equal C C -(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u) (CHead e1 -(Bind Abst) v1) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e in -C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) -\Rightarrow k0])) (CHead c1 k u) (CHead e1 (Bind Abst) v1) H3) in ((let H6 -\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u) -(CHead e1 (Bind Abst) v1) H3) in (\lambda (H7: (eq K k (Bind Abst))).(\lambda -(H8: (eq C c1 e1)).(eq_ind_r T v1 (\lambda (t: T).(or (ex2 C (\lambda (e2: -C).(eq C (CHead c3 k t) (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g -e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 k t) -(CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 -e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: -C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))) (eq_ind_r K (Bind Abst) (\lambda -(k0: K).(or (ex2 C (\lambda (e2: C).(eq C (CHead c3 k0 v1) (CHead e2 (Bind -Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: -C).(\lambda (v2: T).(eq C (CHead c3 k0 v1) (CHead e2 (Bind Abbr) v2)))) -(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda -(v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 -v1)))))) (let H9 \def (eq_ind C c1 (\lambda (c: C).((eq C c (CHead e1 (Bind -Abst) v1)) \to (or (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abst) -v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda -(v2: T).(eq C c3 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: -T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) -(\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))))))) H2 e1 H8) in (let -H10 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H8) in (or_introl -(ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind Abst) v1) (CHead e2 (Bind Abst) -v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda -(v2: T).(eq C (CHead c3 (Bind Abst) v1) (CHead e2 (Bind Abbr) v2)))) (\lambda -(e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: -T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))) -(ex_intro2 C (\lambda (e2: C).(eq C (CHead c3 (Bind Abst) v1) (CHead e2 (Bind -Abst) v1))) (\lambda (e2: C).(csubt g e1 e2)) c3 (refl_equal C (CHead c3 -(Bind Abst) v1)) H10)))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: -C).(\lambda (c3: C).(\lambda (_: (csubt g c1 c3)).(\lambda (_: (((eq C c1 -(CHead e1 (Bind Abst) v1)) \to (or (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 -(Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: -C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: -C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g -e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 -v1)))))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: -T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u1) (CHead e1 -(Bind Abst) v1))).(let H5 \def (eq_ind C (CHead c1 (Bind Void) u1) (\lambda -(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind b0) \Rightarrow (match b0 in B return (\lambda (_: -B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False | Void -\Rightarrow True]) | (Flat _) \Rightarrow False])])) I (CHead e1 (Bind Abst) -v1) H4) in (False_ind (or (ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind b) -u2) (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T -(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind b) u2) (CHead e2 -(Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) -(\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda -(v2: T).(ty3 g e2 v2 v1))))) H5))))))))))) (\lambda (c1: C).(\lambda (c3: -C).(\lambda (H1: (csubt g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind -Abst) v1)) \to (or (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abst) -v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda -(v2: T).(eq C c3 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: -T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) -(\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))))).(\lambda (u: -T).(\lambda (t: T).(\lambda (H3: (ty3 g c1 u t)).(\lambda (H4: (ty3 g c3 u -t)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) t) (CHead e1 (Bind Abst) -v1))).(let H6 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda -(_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) -(CHead c1 (Bind Abst) t) (CHead e1 (Bind Abst) v1) H5) in ((let H7 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow t | (CHead _ _ t0) \Rightarrow t0])) (CHead c1 (Bind -Abst) t) (CHead e1 (Bind Abst) v1) H5) in (\lambda (H8: (eq C c1 e1)).(let H9 -\def (eq_ind T t (\lambda (t0: T).(ty3 g c3 u t0)) H4 v1 H7) in (let H10 \def -(eq_ind T t (\lambda (t0: T).(ty3 g c1 u t0)) H3 v1 H7) in (let H11 \def -(eq_ind C c1 (\lambda (c: C).(ty3 g c u v1)) H10 e1 H8) in (let H12 \def -(eq_ind C c1 (\lambda (c: C).((eq C c (CHead e1 (Bind Abst) v1)) \to (or (ex2 -C (\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abst) v1))) (\lambda (e2: -C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c3 -(CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 -e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: -C).(\lambda (v2: T).(ty3 g e2 v2 v1))))))) H2 e1 H8) in (let H13 \def (eq_ind -C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H8) in (or_intror (ex2 C (\lambda -(e2: C).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 (Bind Abst) v1))) (\lambda -(e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C -(CHead c3 (Bind Abbr) u) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: -C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g -e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))) -(ex4_2_intro C T (\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind -Abbr) u) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt -g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: -C).(\lambda (v2: T).(ty3 g e2 v2 v1))) c3 u (refl_equal C (CHead c3 (Bind -Abbr) u)) H13 H11 H9))))))))) H6))))))))))) y c2 H0))) H))))). -(* COMMENTS -Initial nodes: 2362 -END *) - -theorem csubt_gen_flat: - \forall (g: G).(\forall (e1: C).(\forall (c2: C).(\forall (v: T).(\forall -(f: F).((csubt g (CHead e1 (Flat f) v) c2) \to (ex2 C (\lambda (e2: C).(eq C -c2 (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g e1 e2)))))))) -\def - \lambda (g: G).(\lambda (e1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda -(f: F).(\lambda (H: (csubt g (CHead e1 (Flat f) v) c2)).(insert_eq C (CHead -e1 (Flat f) v) (\lambda (c: C).(csubt g c c2)) (\lambda (_: C).(ex2 C -(\lambda (e2: C).(eq C c2 (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g -e1 e2)))) (\lambda (y: C).(\lambda (H0: (csubt g y c2)).(csubt_ind g (\lambda -(c: C).(\lambda (c0: C).((eq C c (CHead e1 (Flat f) v)) \to (ex2 C (\lambda -(e2: C).(eq C c0 (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g e1 -e2)))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead e1 (Flat f) -v))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return -(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) -\Rightarrow False])) I (CHead e1 (Flat f) v) H1) in (False_ind (ex2 C -(\lambda (e2: C).(eq C (CSort n) (CHead e2 (Flat f) v))) (\lambda (e2: -C).(csubt g e1 e2))) H2)))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1: -(csubt g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Flat f) v)) \to (ex2 C -(\lambda (e2: C).(eq C c3 (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g -e1 e2)))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c1 k -u) (CHead e1 (Flat f) v))).(let H4 \def (f_equal C C (\lambda (e: C).(match e -in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) -\Rightarrow c])) (CHead c1 k u) (CHead e1 (Flat f) v) H3) in ((let H5 \def -(f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with -[(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c1 k u) -(CHead e1 (Flat f) v) H3) in ((let H6 \def (f_equal C T (\lambda (e: -C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | -(CHead _ _ t) \Rightarrow t])) (CHead c1 k u) (CHead e1 (Flat f) v) H3) in -(\lambda (H7: (eq K k (Flat f))).(\lambda (H8: (eq C c1 e1)).(eq_ind_r T v -(\lambda (t: T).(ex2 C (\lambda (e2: C).(eq C (CHead c3 k t) (CHead e2 (Flat -f) v))) (\lambda (e2: C).(csubt g e1 e2)))) (eq_ind_r K (Flat f) (\lambda -(k0: K).(ex2 C (\lambda (e2: C).(eq C (CHead c3 k0 v) (CHead e2 (Flat f) v))) -(\lambda (e2: C).(csubt g e1 e2)))) (let H9 \def (eq_ind C c1 (\lambda (c: -C).((eq C c (CHead e1 (Flat f) v)) \to (ex2 C (\lambda (e2: C).(eq C c3 -(CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g e1 e2))))) H2 e1 H8) in -(let H10 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H8) in -(ex_intro2 C (\lambda (e2: C).(eq C (CHead c3 (Flat f) v) (CHead e2 (Flat f) -v))) (\lambda (e2: C).(csubt g e1 e2)) c3 (refl_equal C (CHead c3 (Flat f) -v)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: C).(\lambda (c3: -C).(\lambda (_: (csubt g c1 c3)).(\lambda (_: (((eq C c1 (CHead e1 (Flat f) -v)) \to (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 (Flat f) v))) (\lambda -(e2: C).(csubt g e1 e2)))))).(\lambda (b: B).(\lambda (_: (not (eq B b -Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind -Void) u1) (CHead e1 (Flat f) v))).(let H5 \def (eq_ind C (CHead c1 (Bind -Void) u1) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with -[(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return -(\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow -False])])) I (CHead e1 (Flat f) v) H4) in (False_ind (ex2 C (\lambda (e2: -C).(eq C (CHead c3 (Bind b) u2) (CHead e2 (Flat f) v))) (\lambda (e2: -C).(csubt g e1 e2))) H5))))))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda -(_: (csubt g c1 c3)).(\lambda (_: (((eq C c1 (CHead e1 (Flat f) v)) \to (ex2 -C (\lambda (e2: C).(eq C c3 (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g -e1 e2)))))).(\lambda (u: T).(\lambda (t: T).(\lambda (_: (ty3 g c1 u -t)).(\lambda (_: (ty3 g c3 u t)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) t) -(CHead e1 (Flat f) v))).(let H6 \def (eq_ind C (CHead c1 (Bind Abst) t) -(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow -False])])) I (CHead e1 (Flat f) v) H5) in (False_ind (ex2 C (\lambda (e2: -C).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 (Flat f) v))) (\lambda (e2: -C).(csubt g e1 e2))) H6))))))))))) y c2 H0))) H)))))). -(* COMMENTS -Initial nodes: 1103 -END *) - -theorem csubt_gen_bind: - \forall (g: G).(\forall (b1: B).(\forall (e1: C).(\forall (c2: C).(\forall -(v1: T).((csubt g (CHead e1 (Bind b1) v1) c2) \to (ex2_3 B C T (\lambda (b2: -B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) -(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2)))))))))) -\def - \lambda (g: G).(\lambda (b1: B).(\lambda (e1: C).(\lambda (c2: C).(\lambda -(v1: T).(\lambda (H: (csubt g (CHead e1 (Bind b1) v1) c2)).(insert_eq C -(CHead e1 (Bind b1) v1) (\lambda (c: C).(csubt g c c2)) (\lambda (_: -C).(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 -(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: -T).(csubt g e1 e2)))))) (\lambda (y: C).(\lambda (H0: (csubt g y -c2)).(csubt_ind g (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead e1 (Bind -b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: -T).(eq C c0 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: -C).(\lambda (_: T).(csubt g e1 e2)))))))) (\lambda (n: nat).(\lambda (H1: (eq -C (CSort n) (CHead e1 (Bind b1) v1))).(let H2 \def (eq_ind C (CSort n) -(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 (Bind b1) -v1) H1) in (False_ind (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda -(v2: T).(eq C (CSort n) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda -(e2: C).(\lambda (_: T).(csubt g e1 e2))))) H2)))) (\lambda (c1: C).(\lambda -(c3: C).(\lambda (H1: (csubt g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 -(Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda -(v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: -C).(\lambda (_: T).(csubt g e1 e2)))))))).(\lambda (k: K).(\lambda (u: -T).(\lambda (H3: (eq C (CHead c1 k u) (CHead e1 (Bind b1) v1))).(let H4 \def -(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with -[(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u) -(CHead e1 (Bind b1) v1) H3) in ((let H5 \def (f_equal C K (\lambda (e: -C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | -(CHead _ k0 _) \Rightarrow k0])) (CHead c1 k u) (CHead e1 (Bind b1) v1) H3) -in ((let H6 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda -(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) -(CHead c1 k u) (CHead e1 (Bind b1) v1) H3) in (\lambda (H7: (eq K k (Bind -b1))).(\lambda (H8: (eq C c1 e1)).(eq_ind_r T v1 (\lambda (t: T).(ex2_3 B C T -(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 k t) -(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: -T).(csubt g e1 e2)))))) (eq_ind_r K (Bind b1) (\lambda (k0: K).(ex2_3 B C T -(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 k0 v1) -(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: -T).(csubt g e1 e2)))))) (let H9 \def (eq_ind C c1 (\lambda (c: C).((eq C c -(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: -C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: -B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))))))) H2 e1 H8) in (let -H10 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H8) in -(ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C -(CHead c3 (Bind b1) v1) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda -(e2: C).(\lambda (_: T).(csubt g e1 e2)))) b1 c3 v1 (refl_equal C (CHead c3 -(Bind b1) v1)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: -C).(\lambda (c3: C).(\lambda (H1: (csubt g c1 c3)).(\lambda (H2: (((eq C c1 -(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: -C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: -B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2)))))))).(\lambda (b: -B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: -T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) -v1))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda -(_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) -(CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) v1) H4) in ((let H6 \def -(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with -[(CSort _) \Rightarrow Void | (CHead _ k _) \Rightarrow (match k in K return -(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow -Void])])) (CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) v1) H4) in ((let H7 -\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead c1 -(Bind Void) u1) (CHead e1 (Bind b1) v1) H4) in (\lambda (H8: (eq B Void -b1)).(\lambda (H9: (eq C c1 e1)).(let H10 \def (eq_ind C c1 (\lambda (c: -C).((eq C c (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: -B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) -(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))))))) H2 e1 -H9) in (let H11 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H9) -in (let H12 \def (eq_ind_r B b1 (\lambda (b0: B).((eq C e1 (CHead e1 (Bind -b0) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: -T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: -C).(\lambda (_: T).(csubt g e1 e2))))))) H10 Void H8) in (ex2_3_intro B C T -(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind b) -u2) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: -T).(csubt g e1 e2)))) b c3 u2 (refl_equal C (CHead c3 (Bind b) u2)) -H11))))))) H6)) H5))))))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1: -(csubt g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind b1) v1)) \to (ex2_3 -B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 -(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g -e1 e2)))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H3: (ty3 g c1 u -t)).(\lambda (H4: (ty3 g c3 u t)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) -t) (CHead e1 (Bind b1) v1))).(let H6 \def (f_equal C C (\lambda (e: C).(match -e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ -_) \Rightarrow c])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H5) in -((let H7 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: -C).B) with [(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match k -in K return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) -\Rightarrow Abst])])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H5) in -((let H8 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: -C).T) with [(CSort _) \Rightarrow t | (CHead _ _ t0) \Rightarrow t0])) (CHead -c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H5) in (\lambda (H9: (eq B Abst -b1)).(\lambda (H10: (eq C c1 e1)).(let H11 \def (eq_ind T t (\lambda (t0: -T).(ty3 g c3 u t0)) H4 v1 H8) in (let H12 \def (eq_ind T t (\lambda (t0: -T).(ty3 g c1 u t0)) H3 v1 H8) in (let H13 \def (eq_ind C c1 (\lambda (c: -C).(ty3 g c u v1)) H12 e1 H10) in (let H14 \def (eq_ind C c1 (\lambda (c: -C).((eq C c (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: -B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) -(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))))))) H2 e1 -H10) in (let H15 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H10) -in (let H16 \def (eq_ind_r B b1 (\lambda (b: B).((eq C e1 (CHead e1 (Bind b) -v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq -C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda -(_: T).(csubt g e1 e2))))))) H14 Abst H9) in (ex2_3_intro B C T (\lambda (b2: -B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 -(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g -e1 e2)))) Abbr c3 u (refl_equal C (CHead c3 (Bind Abbr) u)) H15)))))))))) -H7)) H6))))))))))) y c2 H0))) H)))))). -(* COMMENTS -Initial nodes: 1899 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/getl.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/getl.ma deleted file mode 100644 index df14528ed..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/getl.ma +++ /dev/null @@ -1,426 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubt/clear.ma". - -include "Basic-1/csubt/drop.ma". - -include "Basic-1/getl/clear.ma". - -theorem csubt_getl_abbr: - \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).(\forall -(n: nat).((getl n c1 (CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csubt g -c1 c2) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n -c2 (CHead d2 (Bind Abbr) u))))))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u: T).(\lambda -(n: nat).(\lambda (H: (getl n c1 (CHead d1 (Bind Abbr) u))).(let H0 \def -(getl_gen_all c1 (CHead d1 (Bind Abbr) u) n H) in (ex2_ind C (\lambda (e: -C).(drop n O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abbr) u))) -(\forall (c2: C).((csubt g c1 c2) \to (ex2 C (\lambda (d2: C).(csubt g d1 -d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u)))))) (\lambda (x: -C).(\lambda (H1: (drop n O c1 x)).(\lambda (H2: (clear x (CHead d1 (Bind -Abbr) u))).(C_ind (\lambda (c: C).((drop n O c1 c) \to ((clear c (CHead d1 -(Bind Abbr) u)) \to (\forall (c2: C).((csubt g c1 c2) \to (ex2 C (\lambda -(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) -u))))))))) (\lambda (n0: nat).(\lambda (_: (drop n O c1 (CSort n0))).(\lambda -(H4: (clear (CSort n0) (CHead d1 (Bind Abbr) u))).(clear_gen_sort (CHead d1 -(Bind Abbr) u) n0 H4 (\forall (c2: C).((csubt g c1 c2) \to (ex2 C (\lambda -(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) -u)))))))))) (\lambda (x0: C).(\lambda (_: (((drop n O c1 x0) \to ((clear x0 -(CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csubt g c1 c2) \to (ex2 C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind -Abbr) u)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (H3: (drop n O c1 -(CHead x0 k t))).(\lambda (H4: (clear (CHead x0 k t) (CHead d1 (Bind Abbr) -u))).(K_ind (\lambda (k0: K).((drop n O c1 (CHead x0 k0 t)) \to ((clear -(CHead x0 k0 t) (CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csubt g c1 -c2) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 -(CHead d2 (Bind Abbr) u))))))))) (\lambda (b: B).(\lambda (H5: (drop n O c1 -(CHead x0 (Bind b) t))).(\lambda (H6: (clear (CHead x0 (Bind b) t) (CHead d1 -(Bind Abbr) u))).(let H7 \def (f_equal C C (\lambda (e: C).(match e in C -return (\lambda (_: C).C) with [(CSort _) \Rightarrow d1 | (CHead c _ _) -\Rightarrow c])) (CHead d1 (Bind Abbr) u) (CHead x0 (Bind b) t) -(clear_gen_bind b x0 (CHead d1 (Bind Abbr) u) t H6)) in ((let H8 \def -(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with -[(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow (match k0 in K -return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) -\Rightarrow Abbr])])) (CHead d1 (Bind Abbr) u) (CHead x0 (Bind b) t) -(clear_gen_bind b x0 (CHead d1 (Bind Abbr) u) t H6)) in ((let H9 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d1 (Bind -Abbr) u) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abbr) u) -t H6)) in (\lambda (H10: (eq B Abbr b)).(\lambda (H11: (eq C d1 x0)).(\lambda -(c2: C).(\lambda (H12: (csubt g c1 c2)).(let H13 \def (eq_ind_r T t (\lambda -(t0: T).(drop n O c1 (CHead x0 (Bind b) t0))) H5 u H9) in (let H14 \def -(eq_ind_r B b (\lambda (b0: B).(drop n O c1 (CHead x0 (Bind b0) u))) H13 Abbr -H10) in (let H15 \def (eq_ind_r C x0 (\lambda (c: C).(drop n O c1 (CHead c -(Bind Abbr) u))) H14 d1 H11) in (ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abbr) u))) (ex2 C (\lambda (d2: -C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u)))) -(\lambda (x1: C).(\lambda (H16: (csubt g d1 x1)).(\lambda (H17: (drop n O c2 -(CHead x1 (Bind Abbr) u))).(ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))) x1 H16 (getl_intro n -c2 (CHead x1 (Bind Abbr) u) (CHead x1 (Bind Abbr) u) H17 (clear_bind Abbr x1 -u)))))) (csubt_drop_abbr g n c1 c2 H12 d1 u H15)))))))))) H8)) H7))))) -(\lambda (f: F).(\lambda (H5: (drop n O c1 (CHead x0 (Flat f) t))).(\lambda -(H6: (clear (CHead x0 (Flat f) t) (CHead d1 (Bind Abbr) u))).(let H7 \def H5 -in (unintro C c1 (\lambda (c: C).((drop n O c (CHead x0 (Flat f) t)) \to -(\forall (c2: C).((csubt g c c2) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u)))))))) (nat_ind (\lambda -(n0: nat).(\forall (x1: C).((drop n0 O x1 (CHead x0 (Flat f) t)) \to (\forall -(c2: C).((csubt g x1 c2) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(getl n0 c2 (CHead d2 (Bind Abbr) u))))))))) (\lambda (x1: -C).(\lambda (H8: (drop O O x1 (CHead x0 (Flat f) t))).(\lambda (c2: -C).(\lambda (H9: (csubt g x1 c2)).(let H10 \def (eq_ind C x1 (\lambda (c: -C).(csubt g c c2)) H9 (CHead x0 (Flat f) t) (drop_gen_refl x1 (CHead x0 (Flat -f) t) H8)) in (let H_y \def (clear_flat x0 (CHead d1 (Bind Abbr) u) -(clear_gen_flat f x0 (CHead d1 (Bind Abbr) u) t H6) f t) in (let H11 \def -(csubt_clear_conf g (CHead x0 (Flat f) t) c2 H10 (CHead d1 (Bind Abbr) u) -H_y) in (ex2_ind C (\lambda (e2: C).(csubt g (CHead d1 (Bind Abbr) u) e2)) -(\lambda (e2: C).(clear c2 e2)) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (x2: -C).(\lambda (H12: (csubt g (CHead d1 (Bind Abbr) u) x2)).(\lambda (H13: -(clear c2 x2)).(let H14 \def (csubt_gen_abbr g d1 x2 u H12) in (ex2_ind C -(\lambda (e2: C).(eq C x2 (CHead e2 (Bind Abbr) u))) (\lambda (e2: C).(csubt -g d1 e2)) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl O -c2 (CHead d2 (Bind Abbr) u)))) (\lambda (x3: C).(\lambda (H15: (eq C x2 -(CHead x3 (Bind Abbr) u))).(\lambda (H16: (csubt g d1 x3)).(let H17 \def -(eq_ind C x2 (\lambda (c: C).(clear c2 c)) H13 (CHead x3 (Bind Abbr) u) H15) -in (ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl O c2 -(CHead d2 (Bind Abbr) u))) x3 H16 (getl_intro O c2 (CHead x3 (Bind Abbr) u) -c2 (drop_refl c2) H17)))))) H14))))) H11)))))))) (\lambda (n0: nat).(\lambda -(H8: ((\forall (x1: C).((drop n0 O x1 (CHead x0 (Flat f) t)) \to (\forall -(c2: C).((csubt g x1 c2) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(getl n0 c2 (CHead d2 (Bind Abbr) u)))))))))).(\lambda (x1: -C).(\lambda (H9: (drop (S n0) O x1 (CHead x0 (Flat f) t))).(\lambda (c2: -C).(\lambda (H10: (csubt g x1 c2)).(let H11 \def (drop_clear x1 (CHead x0 -(Flat f) t) n0 H9) in (ex2_3_ind B C T (\lambda (b: B).(\lambda (e: -C).(\lambda (v: T).(clear x1 (CHead e (Bind b) v))))) (\lambda (_: -B).(\lambda (e: C).(\lambda (_: T).(drop n0 O e (CHead x0 (Flat f) t))))) -(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 -(CHead d2 (Bind Abbr) u)))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: -T).(\lambda (H12: (clear x1 (CHead x3 (Bind x2) x4))).(\lambda (H13: (drop n0 -O x3 (CHead x0 (Flat f) t))).(let H14 \def (csubt_clear_conf g x1 c2 H10 -(CHead x3 (Bind x2) x4) H12) in (ex2_ind C (\lambda (e2: C).(csubt g (CHead -x3 (Bind x2) x4) e2)) (\lambda (e2: C).(clear c2 e2)) (ex2 C (\lambda (d2: -C).(csubt g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr) -u)))) (\lambda (x5: C).(\lambda (H15: (csubt g (CHead x3 (Bind x2) x4) -x5)).(\lambda (H16: (clear c2 x5)).(let H17 \def (csubt_gen_bind g x2 x3 x5 -x4 H15) in (ex2_3_ind B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: -T).(eq C x5 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: -C).(\lambda (_: T).(csubt g x3 e2)))) (ex2 C (\lambda (d2: C).(csubt g d1 -d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda -(x6: B).(\lambda (x7: C).(\lambda (x8: T).(\lambda (H18: (eq C x5 (CHead x7 -(Bind x6) x8))).(\lambda (H19: (csubt g x3 x7)).(let H20 \def (eq_ind C x5 -(\lambda (c: C).(clear c2 c)) H16 (CHead x7 (Bind x6) x8) H18) in (let H21 -\def (H8 x3 H13 x7 H19) in (ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(getl n0 x7 (CHead d2 (Bind Abbr) u))) (ex2 C (\lambda (d2: -C).(csubt g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr) -u)))) (\lambda (x9: C).(\lambda (H22: (csubt g d1 x9)).(\lambda (H23: (getl -n0 x7 (CHead x9 (Bind Abbr) u))).(ex_intro2 C (\lambda (d2: C).(csubt g d1 -d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr) u))) x9 H22 -(getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Abbr) u) n0 H23))))) -H21)))))))) H17))))) H14))))))) H11)))))))) n) H7))))) k H3 H4))))))) x H1 -H2)))) H0))))))). -(* COMMENTS -Initial nodes: 2313 -END *) - -theorem csubt_getl_abst: - \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (t: T).(\forall -(n: nat).((getl n c1 (CHead d1 (Bind Abst) t)) \to (\forall (c2: C).((csubt g -c1 c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: -C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n -c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u -t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (t: T).(\lambda -(n: nat).(\lambda (H: (getl n c1 (CHead d1 (Bind Abst) t))).(let H0 \def -(getl_gen_all c1 (CHead d1 (Bind Abst) t) n H) in (ex2_ind C (\lambda (e: -C).(drop n O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abst) t))) -(\forall (c2: C).((csubt g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 -d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T -(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda -(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: -T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))))) -(\lambda (x: C).(\lambda (H1: (drop n O c1 x)).(\lambda (H2: (clear x (CHead -d1 (Bind Abst) t))).(C_ind (\lambda (c: C).((drop n O c1 c) \to ((clear c -(CHead d1 (Bind Abst) t)) \to (\forall (c2: C).((csubt g c1 c2) \to (or (ex2 -C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 -(Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 -d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) -(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda -(u: T).(ty3 g d2 u t)))))))))) (\lambda (n0: nat).(\lambda (_: (drop n O c1 -(CSort n0))).(\lambda (H4: (clear (CSort n0) (CHead d1 (Bind Abst) -t))).(clear_gen_sort (CHead d1 (Bind Abst) t) n0 H4 (\forall (c2: C).((csubt -g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: -C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n -c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u -t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))))))))) (\lambda (x0: -C).(\lambda (_: (((drop n O c1 x0) \to ((clear x0 (CHead d1 (Bind Abst) t)) -\to (\forall (c2: C).((csubt g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csubt -g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T -(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda -(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: -T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u -t))))))))))).(\lambda (k: K).(\lambda (t0: T).(\lambda (H3: (drop n O c1 -(CHead x0 k t0))).(\lambda (H4: (clear (CHead x0 k t0) (CHead d1 (Bind Abst) -t))).(K_ind (\lambda (k0: K).((drop n O c1 (CHead x0 k0 t0)) \to ((clear -(CHead x0 k0 t0) (CHead d1 (Bind Abst) t)) \to (\forall (c2: C).((csubt g c1 -c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n -c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: -T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 -(Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda -(d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))) (\lambda (b: B).(\lambda (H5: -(drop n O c1 (CHead x0 (Bind b) t0))).(\lambda (H6: (clear (CHead x0 (Bind b) -t0) (CHead d1 (Bind Abst) t))).(let H7 \def (f_equal C C (\lambda (e: -C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d1 | -(CHead c _ _) \Rightarrow c])) (CHead d1 (Bind Abst) t) (CHead x0 (Bind b) -t0) (clear_gen_bind b x0 (CHead d1 (Bind Abst) t) t0 H6)) in ((let H8 \def -(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with -[(CSort _) \Rightarrow Abst | (CHead _ k0 _) \Rightarrow (match k0 in K -return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) -\Rightarrow Abst])])) (CHead d1 (Bind Abst) t) (CHead x0 (Bind b) t0) -(clear_gen_bind b x0 (CHead d1 (Bind Abst) t) t0 H6)) in ((let H9 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow t | (CHead _ _ t1) \Rightarrow t1])) (CHead d1 (Bind -Abst) t) (CHead x0 (Bind b) t0) (clear_gen_bind b x0 (CHead d1 (Bind Abst) t) -t0 H6)) in (\lambda (H10: (eq B Abst b)).(\lambda (H11: (eq C d1 -x0)).(\lambda (c2: C).(\lambda (H12: (csubt g c1 c2)).(let H13 \def (eq_ind_r -T t0 (\lambda (t1: T).(drop n O c1 (CHead x0 (Bind b) t1))) H5 t H9) in (let -H14 \def (eq_ind_r B b (\lambda (b0: B).(drop n O c1 (CHead x0 (Bind b0) t))) -H13 Abst H10) in (let H15 \def (eq_ind_r C x0 (\lambda (c: C).(drop n O c1 -(CHead c (Bind Abst) t))) H14 d1 H11) in (or_ind (ex2 C (\lambda (d2: -C).(csubt g d1 d2)) (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abst) t)))) -(ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: -C).(\lambda (u: T).(drop n O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: -C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g -d2 u t)))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: -C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n -c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u -t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (H16: (ex2 -C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n O c2 (CHead d2 -(Bind Abst) t))))).(ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(drop n O c2 (CHead d2 (Bind Abst) t))) (or (ex2 C (\lambda (d2: C).(csubt -g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T -(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda -(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: -T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) -(\lambda (x1: C).(\lambda (H17: (csubt g d1 x1)).(\lambda (H18: (drop n O c2 -(CHead x1 (Bind Abst) t))).(or_introl (ex2 C (\lambda (d2: C).(csubt g d1 -d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T -(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda -(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: -T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) -(ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 -(CHead d2 (Bind Abst) t))) x1 H17 (getl_intro n c2 (CHead x1 (Bind Abst) t) -(CHead x1 (Bind Abst) t) H18 (clear_bind Abst x1 t))))))) H16)) (\lambda -(H16: (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda -(d2: C).(\lambda (u: T).(drop n O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: -C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g -d2 u t))))).(ex4_2_ind C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) -(\lambda (d2: C).(\lambda (u: T).(drop n O c2 (CHead d2 (Bind Abbr) u)))) -(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda -(u: T).(ty3 g d2 u t))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda -(d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: -C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n -c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u -t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x1: -C).(\lambda (x2: T).(\lambda (H17: (csubt g d1 x1)).(\lambda (H18: (drop n O -c2 (CHead x1 (Bind Abbr) x2))).(\lambda (H19: (ty3 g d1 x2 t)).(\lambda (H20: -(ty3 g x1 x2 t)).(or_intror (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda -(d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: -C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n -c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u -t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (ex4_2_intro C T -(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda -(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: -T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))) x1 x2 -H17 (getl_intro n c2 (CHead x1 (Bind Abbr) x2) (CHead x1 (Bind Abbr) x2) H18 -(clear_bind Abbr x1 x2)) H19 H20)))))))) H16)) (csubt_drop_abst g n c1 c2 H12 -d1 t H15)))))))))) H8)) H7))))) (\lambda (f: F).(\lambda (H5: (drop n O c1 -(CHead x0 (Flat f) t0))).(\lambda (H6: (clear (CHead x0 (Flat f) t0) (CHead -d1 (Bind Abst) t))).(let H7 \def H5 in (unintro C c1 (\lambda (c: C).((drop n -O c (CHead x0 (Flat f) t0)) \to (\forall (c2: C).((csubt g c c2) \to (or (ex2 -C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 -(Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 -d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) -(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda -(u: T).(ty3 g d2 u t))))))))) (nat_ind (\lambda (n0: nat).(\forall (x1: -C).((drop n0 O x1 (CHead x0 (Flat f) t0)) \to (\forall (c2: C).((csubt g x1 -c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl -n0 c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: -T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n0 c2 (CHead d2 -(Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda -(d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))) (\lambda (x1: C).(\lambda -(H8: (drop O O x1 (CHead x0 (Flat f) t0))).(\lambda (c2: C).(\lambda (H9: -(csubt g x1 c2)).(let H10 \def (eq_ind C x1 (\lambda (c: C).(csubt g c c2)) -H9 (CHead x0 (Flat f) t0) (drop_gen_refl x1 (CHead x0 (Flat f) t0) H8)) in -(let H_y \def (clear_flat x0 (CHead d1 (Bind Abst) t) (clear_gen_flat f x0 -(CHead d1 (Bind Abst) t) t0 H6) f t0) in (let H11 \def (csubt_clear_conf g -(CHead x0 (Flat f) t0) c2 H10 (CHead d1 (Bind Abst) t) H_y) in (ex2_ind C -(\lambda (e2: C).(csubt g (CHead d1 (Bind Abst) t) e2)) (\lambda (e2: -C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: -C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl O -c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u -t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x2: -C).(\lambda (H12: (csubt g (CHead d1 (Bind Abst) t) x2)).(\lambda (H13: -(clear c2 x2)).(let H14 \def (csubt_gen_abst g d1 x2 t H12) in (or_ind (ex2 C -(\lambda (e2: C).(eq C x2 (CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csubt -g d1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C x2 (CHead e2 -(Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g d1 e2))) -(\lambda (_: C).(\lambda (v2: T).(ty3 g d1 v2 t))) (\lambda (e2: C).(\lambda -(v2: T).(ty3 g e2 v2 t)))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda -(d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: -T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: -T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) -(\lambda (H15: (ex2 C (\lambda (e2: C).(eq C x2 (CHead e2 (Bind Abst) t))) -(\lambda (e2: C).(csubt g d1 e2)))).(ex2_ind C (\lambda (e2: C).(eq C x2 -(CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csubt g d1 e2)) (or (ex2 C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind -Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) -(\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr) u)))) -(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda -(u: T).(ty3 g d2 u t))))) (\lambda (x3: C).(\lambda (H16: (eq C x2 (CHead x3 -(Bind Abst) t))).(\lambda (H17: (csubt g d1 x3)).(let H18 \def (eq_ind C x2 -(\lambda (c: C).(clear c2 c)) H13 (CHead x3 (Bind Abst) t) H16) in (or_introl -(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead -d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 -d2))) (\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr) u)))) -(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda -(u: T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t))) x3 H17 (getl_intro O -c2 (CHead x3 (Bind Abst) t) c2 (drop_refl c2) H18))))))) H15)) (\lambda (H15: -(ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C x2 (CHead e2 (Bind Abbr) -v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g d1 e2))) (\lambda (_: -C).(\lambda (v2: T).(ty3 g d1 v2 t))) (\lambda (e2: C).(\lambda (v2: T).(ty3 -g e2 v2 t))))).(ex4_2_ind C T (\lambda (e2: C).(\lambda (v2: T).(eq C x2 -(CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g d1 -e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g d1 v2 t))) (\lambda (e2: -C).(\lambda (v2: T).(ty3 g e2 v2 t))) (or (ex2 C (\lambda (d2: C).(csubt g d1 -d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T -(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda -(u: T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: -T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) -(\lambda (x3: C).(\lambda (x4: T).(\lambda (H16: (eq C x2 (CHead x3 (Bind -Abbr) x4))).(\lambda (H17: (csubt g d1 x3)).(\lambda (H18: (ty3 g d1 x4 -t)).(\lambda (H19: (ty3 g x3 x4 t)).(let H20 \def (eq_ind C x2 (\lambda (c: -C).(clear c2 c)) H13 (CHead x3 (Bind Abbr) x4) H16) in (or_intror (ex2 C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind -Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) -(\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr) u)))) -(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda -(u: T).(ty3 g d2 u t)))) (ex4_2_intro C T (\lambda (d2: C).(\lambda (_: -T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 -(Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda -(d2: C).(\lambda (u: T).(ty3 g d2 u t))) x3 x4 H17 (getl_intro O c2 (CHead x3 -(Bind Abbr) x4) c2 (drop_refl c2) H20) H18 H19))))))))) H15)) H14))))) -H11)))))))) (\lambda (n0: nat).(\lambda (H8: ((\forall (x1: C).((drop n0 O x1 -(CHead x0 (Flat f) t0)) \to (\forall (c2: C).((csubt g x1 c2) \to (or (ex2 C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n0 c2 (CHead d2 -(Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 -d2))) (\lambda (d2: C).(\lambda (u: T).(getl n0 c2 (CHead d2 (Bind Abbr) -u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: -C).(\lambda (u: T).(ty3 g d2 u t))))))))))).(\lambda (x1: C).(\lambda (H9: -(drop (S n0) O x1 (CHead x0 (Flat f) t0))).(\lambda (c2: C).(\lambda (H10: -(csubt g x1 c2)).(let H11 \def (drop_clear x1 (CHead x0 (Flat f) t0) n0 H9) -in (ex2_3_ind B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear x1 -(CHead e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: -T).(drop n0 O e (CHead x0 (Flat f) t0))))) (or (ex2 C (\lambda (d2: C).(csubt -g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex4_2 -C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: -C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: -C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g -d2 u t))))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: T).(\lambda (H12: -(clear x1 (CHead x3 (Bind x2) x4))).(\lambda (H13: (drop n0 O x3 (CHead x0 -(Flat f) t0))).(let H14 \def (csubt_clear_conf g x1 c2 H10 (CHead x3 (Bind -x2) x4) H12) in (ex2_ind C (\lambda (e2: C).(csubt g (CHead x3 (Bind x2) x4) -e2)) (\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(csubt g d1 -d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T -(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda -(u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda -(u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) -(\lambda (x5: C).(\lambda (H15: (csubt g (CHead x3 (Bind x2) x4) -x5)).(\lambda (H16: (clear c2 x5)).(let H17 \def (csubt_gen_bind g x2 x3 x5 -x4 H15) in (ex2_3_ind B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: -T).(eq C x5 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: -C).(\lambda (_: T).(csubt g x3 e2)))) (or (ex2 C (\lambda (d2: C).(csubt g d1 -d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T -(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda -(u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda -(u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) -(\lambda (x6: B).(\lambda (x7: C).(\lambda (x8: T).(\lambda (H18: (eq C x5 -(CHead x7 (Bind x6) x8))).(\lambda (H19: (csubt g x3 x7)).(let H20 \def -(eq_ind C x5 (\lambda (c: C).(clear c2 c)) H16 (CHead x7 (Bind x6) x8) H18) -in (let H21 \def (H8 x3 H13 x7 H19) in (or_ind (ex2 C (\lambda (d2: C).(csubt -g d1 d2)) (\lambda (d2: C).(getl n0 x7 (CHead d2 (Bind Abst) t)))) (ex4_2 C T -(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda -(u: T).(getl n0 x7 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: -T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (or -(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 -(CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: -T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead -d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) -(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (H22: (ex2 C -(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n0 x7 (CHead d2 -(Bind Abst) t))))).(ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(getl n0 x7 (CHead d2 (Bind Abst) t))) (or (ex2 C (\lambda (d2: C).(csubt -g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex4_2 -C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: -C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: -C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g -d2 u t))))) (\lambda (x9: C).(\lambda (H23: (csubt g d1 x9)).(\lambda (H24: -(getl n0 x7 (CHead x9 (Bind Abst) t))).(or_introl (ex2 C (\lambda (d2: -C).(csubt g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) -t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda -(d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda -(_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 -g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: -C).(getl (S n0) c2 (CHead d2 (Bind Abst) t))) x9 H23 (getl_clear_bind x6 c2 -x7 x8 H20 (CHead x9 (Bind Abst) t) n0 H24)))))) H22)) (\lambda (H22: (ex4_2 C -T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: -C).(\lambda (u: T).(getl n0 x7 (CHead d2 (Bind Abbr) u)))) (\lambda (_: -C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g -d2 u t))))).(ex4_2_ind C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) -(\lambda (d2: C).(\lambda (u: T).(getl n0 x7 (CHead d2 (Bind Abbr) u)))) -(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda -(u: T).(ty3 g d2 u t))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda -(d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: -C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl -(S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g -d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x9: -C).(\lambda (x10: T).(\lambda (H23: (csubt g d1 x9)).(\lambda (H24: (getl n0 -x7 (CHead x9 (Bind Abbr) x10))).(\lambda (H25: (ty3 g d1 x10 t)).(\lambda -(H26: (ty3 g x9 x10 t)).(or_intror (ex2 C (\lambda (d2: C).(csubt g d1 d2)) -(\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T -(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda -(u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda -(u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) -(ex4_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda -(d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda -(_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 -g d2 u t))) x9 x10 H23 (getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Abbr) -x10) n0 H24) H25 H26)))))))) H22)) H21)))))))) H17))))) H14))))))) -H11)))))))) n) H7))))) k H3 H4))))))) x H1 H2)))) H0))))))). -(* COMMENTS -Initial nodes: 5861 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/pc3.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/pc3.ma deleted file mode 100644 index 900ba49e9..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/pc3.ma +++ /dev/null @@ -1,62 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubt/getl.ma". - -include "Basic-1/pc3/left.ma". - -theorem csubt_pr2: - \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pr2 c1 -t1 t2) \to (\forall (c2: C).((csubt g c1 c2) \to (pr2 c2 t1 t2))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H: (pr2 c1 t1 t2)).(pr2_ind (\lambda (c: C).(\lambda (t: T).(\lambda (t0: -T).(\forall (c2: C).((csubt g c c2) \to (pr2 c2 t t0)))))) (\lambda (c: -C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(\lambda (c2: -C).(\lambda (_: (csubt g c c2)).(pr2_free c2 t3 t4 H0))))))) (\lambda (c: -C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c -(CHead d (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: -(pr0 t3 t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (c2: -C).(\lambda (H3: (csubt g c c2)).(let H4 \def (csubt_getl_abbr g c d u i H0 -c2 H3) in (ex2_ind C (\lambda (d2: C).(csubt g d d2)) (\lambda (d2: C).(getl -i c2 (CHead d2 (Bind Abbr) u))) (pr2 c2 t3 t) (\lambda (x: C).(\lambda (_: -(csubt g d x)).(\lambda (H6: (getl i c2 (CHead x (Bind Abbr) u))).(pr2_delta -c2 x u i H6 t3 t4 H1 t H2)))) H4)))))))))))))) c1 t1 t2 H))))). -(* COMMENTS -Initial nodes: 245 -END *) - -theorem csubt_pc3: - \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pc3 c1 -t1 t2) \to (\forall (c2: C).((csubt g c1 c2) \to (pc3 c2 t1 t2))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H: (pc3 c1 t1 t2)).(pc3_ind_left c1 (\lambda (t: T).(\lambda (t0: -T).(\forall (c2: C).((csubt g c1 c2) \to (pc3 c2 t t0))))) (\lambda (t: -T).(\lambda (c2: C).(\lambda (_: (csubt g c1 c2)).(pc3_refl c2 t)))) (\lambda -(t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c1 t0 t3)).(\lambda (t4: -T).(\lambda (_: (pc3 c1 t3 t4)).(\lambda (H2: ((\forall (c2: C).((csubt g c1 -c2) \to (pc3 c2 t3 t4))))).(\lambda (c2: C).(\lambda (H3: (csubt g c1 -c2)).(pc3_t t3 c2 t0 (pc3_pr2_r c2 t0 t3 (csubt_pr2 g c1 t0 t3 H0 c2 H3)) t4 -(H2 c2 H3)))))))))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c1 -t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3 c1 t0 t4)).(\lambda (H2: ((\forall -(c2: C).((csubt g c1 c2) \to (pc3 c2 t0 t4))))).(\lambda (c2: C).(\lambda -(H3: (csubt g c1 c2)).(pc3_t t0 c2 t3 (pc3_pr2_x c2 t3 t0 (csubt_pr2 g c1 t0 -t3 H0 c2 H3)) t4 (H2 c2 H3)))))))))) t1 t2 H))))). -(* COMMENTS -Initial nodes: 245 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/props.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/props.ma deleted file mode 100644 index 2efc87ccb..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/props.ma +++ /dev/null @@ -1,28 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubt/defs.ma". - -theorem csubt_refl: - \forall (g: G).(\forall (c: C).(csubt g c c)) -\def - \lambda (g: G).(\lambda (c: C).(C_ind (\lambda (c0: C).(csubt g c0 c0)) -(\lambda (n: nat).(csubt_sort g n)) (\lambda (c0: C).(\lambda (H: (csubt g c0 -c0)).(\lambda (k: K).(\lambda (t: T).(csubt_head g c0 c0 H k t))))) c)). -(* COMMENTS -Initial nodes: 53 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/ty3.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/ty3.ma deleted file mode 100644 index e6199ac15..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubt/ty3.ma +++ /dev/null @@ -1,102 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubt/pc3.ma". - -include "Basic-1/csubt/props.ma". - -theorem csubt_ty3: - \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c1 -t1 t2) \to (\forall (c2: C).((csubt g c1 c2) \to (ty3 g c2 t1 t2))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H: (ty3 g c1 t1 t2)).(ty3_ind g (\lambda (c: C).(\lambda (t: T).(\lambda -(t0: T).(\forall (c2: C).((csubt g c c2) \to (ty3 g c2 t t0)))))) (\lambda -(c: C).(\lambda (t0: T).(\lambda (t: T).(\lambda (_: (ty3 g c t0 t)).(\lambda -(H1: ((\forall (c2: C).((csubt g c c2) \to (ty3 g c2 t0 t))))).(\lambda (u: -T).(\lambda (t3: T).(\lambda (_: (ty3 g c u t3)).(\lambda (H3: ((\forall (c2: -C).((csubt g c c2) \to (ty3 g c2 u t3))))).(\lambda (H4: (pc3 c t3 -t0)).(\lambda (c2: C).(\lambda (H5: (csubt g c c2)).(ty3_conv g c2 t0 t (H1 -c2 H5) u t3 (H3 c2 H5) (csubt_pc3 g c t3 t0 H4 c2 H5)))))))))))))) (\lambda -(c: C).(\lambda (m: nat).(\lambda (c2: C).(\lambda (_: (csubt g c -c2)).(ty3_sort g c2 m))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d: -C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind Abbr) u))).(\lambda -(t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2: ((\forall (c2: C).((csubt g -d c2) \to (ty3 g c2 u t))))).(\lambda (c2: C).(\lambda (H3: (csubt g c -c2)).(let H4 \def (csubt_getl_abbr g c d u n H0 c2 H3) in (ex2_ind C (\lambda -(d2: C).(csubt g d d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) -u))) (ty3 g c2 (TLRef n) (lift (S n) O t)) (\lambda (x: C).(\lambda (H5: -(csubt g d x)).(\lambda (H6: (getl n c2 (CHead x (Bind Abbr) u))).(ty3_abbr g -n c2 x u H6 t (H2 x H5))))) H4)))))))))))) (\lambda (n: nat).(\lambda (c: -C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind -Abst) u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2: -((\forall (c2: C).((csubt g d c2) \to (ty3 g c2 u t))))).(\lambda (c2: -C).(\lambda (H3: (csubt g c c2)).(let H4 \def (csubt_getl_abst g c d u n H0 -c2 H3) in (or_ind (ex2 C (\lambda (d2: C).(csubt g d d2)) (\lambda (d2: -C).(getl n c2 (CHead d2 (Bind Abst) u)))) (ex4_2 C T (\lambda (d2: -C).(\lambda (_: T).(csubt g d d2))) (\lambda (d2: C).(\lambda (u0: T).(getl n -c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g d u0 -u))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 u)))) (ty3 g c2 (TLRef n) -(lift (S n) O u)) (\lambda (H5: (ex2 C (\lambda (d2: C).(csubt g d d2)) -(\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) u))))).(ex2_ind C (\lambda -(d2: C).(csubt g d d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) -u))) (ty3 g c2 (TLRef n) (lift (S n) O u)) (\lambda (x: C).(\lambda (H6: -(csubt g d x)).(\lambda (H7: (getl n c2 (CHead x (Bind Abst) u))).(ty3_abst g -n c2 x u H7 t (H2 x H6))))) H5)) (\lambda (H5: (ex4_2 C T (\lambda (d2: -C).(\lambda (_: T).(csubt g d d2))) (\lambda (d2: C).(\lambda (u0: T).(getl n -c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g d u0 -u))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 u))))).(ex4_2_ind C T -(\lambda (d2: C).(\lambda (_: T).(csubt g d d2))) (\lambda (d2: C).(\lambda -(u0: T).(getl n c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: -T).(ty3 g d u0 u))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 u))) (ty3 -g c2 (TLRef n) (lift (S n) O u)) (\lambda (x0: C).(\lambda (x1: T).(\lambda -(_: (csubt g d x0)).(\lambda (H7: (getl n c2 (CHead x0 (Bind Abbr) -x1))).(\lambda (_: (ty3 g d x1 u)).(\lambda (H9: (ty3 g x0 x1 u)).(ty3_abbr g -n c2 x0 x1 H7 u H9))))))) H5)) H4)))))))))))) (\lambda (c: C).(\lambda (u: -T).(\lambda (t: T).(\lambda (_: (ty3 g c u t)).(\lambda (H1: ((\forall (c2: -C).((csubt g c c2) \to (ty3 g c2 u t))))).(\lambda (b: B).(\lambda (t0: -T).(\lambda (t3: T).(\lambda (_: (ty3 g (CHead c (Bind b) u) t0 t3)).(\lambda -(H3: ((\forall (c2: C).((csubt g (CHead c (Bind b) u) c2) \to (ty3 g c2 t0 -t3))))).(\lambda (c2: C).(\lambda (H4: (csubt g c c2)).(ty3_bind g c2 u t (H1 -c2 H4) b t0 t3 (H3 (CHead c2 (Bind b) u) (csubt_head g c c2 H4 (Bind b) -u))))))))))))))) (\lambda (c: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: -(ty3 g c w u)).(\lambda (H1: ((\forall (c2: C).((csubt g c c2) \to (ty3 g c2 -w u))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c v (THead (Bind -Abst) u t))).(\lambda (H3: ((\forall (c2: C).((csubt g c c2) \to (ty3 g c2 v -(THead (Bind Abst) u t)))))).(\lambda (c2: C).(\lambda (H4: (csubt g c -c2)).(ty3_appl g c2 w u (H1 c2 H4) v t (H3 c2 H4))))))))))))) (\lambda (c: -C).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (ty3 g c t0 t3)).(\lambda -(H1: ((\forall (c2: C).((csubt g c c2) \to (ty3 g c2 t0 t3))))).(\lambda (t4: -T).(\lambda (_: (ty3 g c t3 t4)).(\lambda (H3: ((\forall (c2: C).((csubt g c -c2) \to (ty3 g c2 t3 t4))))).(\lambda (c2: C).(\lambda (H4: (csubt g c -c2)).(ty3_cast g c2 t0 t3 (H1 c2 H4) t4 (H3 c2 H4)))))))))))) c1 t1 t2 H))))). -(* COMMENTS -Initial nodes: 1325 -END *) - -theorem csubt_ty3_ld: - \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (v: T).((ty3 g c u -v) \to (\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c (Bind Abst) v) t1 -t2) \to (ty3 g (CHead c (Bind Abbr) u) t1 t2)))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (v: T).(\lambda (H: -(ty3 g c u v)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (ty3 g (CHead -c (Bind Abst) v) t1 t2)).(csubt_ty3 g (CHead c (Bind Abst) v) t1 t2 H0 (CHead -c (Bind Abbr) u) (csubt_abst g c c (csubt_refl g c) u v H H))))))))). -(* COMMENTS -Initial nodes: 91 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubv/clear.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubv/clear.ma deleted file mode 100644 index 5c54f5b9a..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubv/clear.ma +++ /dev/null @@ -1,194 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubv/defs.ma". - -include "Basic-1/clear/fwd.ma". - -theorem csubv_clear_conf: - \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (b1: -B).(\forall (d1: C).(\forall (v1: T).((clear c1 (CHead d1 (Bind b1) v1)) \to -(ex2_3 B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 -d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda (v2: T).(clear c2 (CHead d2 -(Bind b2) v2)))))))))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubv c1 c2)).(csubv_ind -(\lambda (c: C).(\lambda (c0: C).(\forall (b1: B).(\forall (d1: C).(\forall -(v1: T).((clear c (CHead d1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (_: -B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2: -B).(\lambda (d2: C).(\lambda (v2: T).(clear c0 (CHead d2 (Bind b2) -v2)))))))))))) (\lambda (n: nat).(\lambda (b1: B).(\lambda (d1: C).(\lambda -(v1: T).(\lambda (H0: (clear (CSort n) (CHead d1 (Bind b1) -v1))).(clear_gen_sort (CHead d1 (Bind b1) v1) n H0 (ex2_3 B C T (\lambda (_: -B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2: -B).(\lambda (d2: C).(\lambda (v2: T).(clear (CSort n) (CHead d2 (Bind b2) -v2)))))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csubv c3 -c4)).(\lambda (_: ((\forall (b1: B).(\forall (d1: C).(\forall (v1: T).((clear -c3 (CHead d1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (_: B).(\lambda (d2: -C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2: B).(\lambda (d2: -C).(\lambda (v2: T).(clear c4 (CHead d2 (Bind b2) v2)))))))))))).(\lambda -(v1: T).(\lambda (v2: T).(\lambda (b1: B).(\lambda (d1: C).(\lambda (v0: -T).(\lambda (H2: (clear (CHead c3 (Bind Void) v1) (CHead d1 (Bind b1) -v0))).(let H3 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda -(_: C).C) with [(CSort _) \Rightarrow d1 | (CHead c _ _) \Rightarrow c])) -(CHead d1 (Bind b1) v0) (CHead c3 (Bind Void) v1) (clear_gen_bind Void c3 -(CHead d1 (Bind b1) v0) v1 H2)) in ((let H4 \def (f_equal C B (\lambda (e: -C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow b1 | -(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind -b) \Rightarrow b | (Flat _) \Rightarrow b1])])) (CHead d1 (Bind b1) v0) -(CHead c3 (Bind Void) v1) (clear_gen_bind Void c3 (CHead d1 (Bind b1) v0) v1 -H2)) in ((let H5 \def (f_equal C T (\lambda (e: C).(match e in C return -(\lambda (_: C).T) with [(CSort _) \Rightarrow v0 | (CHead _ _ t) \Rightarrow -t])) (CHead d1 (Bind b1) v0) (CHead c3 (Bind Void) v1) (clear_gen_bind Void -c3 (CHead d1 (Bind b1) v0) v1 H2)) in (\lambda (_: (eq B b1 Void)).(\lambda -(H7: (eq C d1 c3)).(eq_ind_r C c3 (\lambda (c: C).(ex2_3 B C T (\lambda (_: -B).(\lambda (d2: C).(\lambda (_: T).(csubv c d2)))) (\lambda (b2: B).(\lambda -(d2: C).(\lambda (v3: T).(clear (CHead c4 (Bind Void) v2) (CHead d2 (Bind b2) -v3))))))) (ex2_3_intro B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: -T).(csubv c3 d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda (v3: T).(clear -(CHead c4 (Bind Void) v2) (CHead d2 (Bind b2) v3))))) Void c4 v2 H0 -(clear_bind Void c4 v2)) d1 H7)))) H4)) H3)))))))))))) (\lambda (c3: -C).(\lambda (c4: C).(\lambda (H0: (csubv c3 c4)).(\lambda (_: ((\forall (b1: -B).(\forall (d1: C).(\forall (v1: T).((clear c3 (CHead d1 (Bind b1) v1)) \to -(ex2_3 B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 -d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda (v2: T).(clear c4 (CHead d2 -(Bind b2) v2)))))))))))).(\lambda (b1: B).(\lambda (_: (not (eq B b1 -Void))).(\lambda (b2: B).(\lambda (v1: T).(\lambda (v2: T).(\lambda (b0: -B).(\lambda (d1: C).(\lambda (v0: T).(\lambda (H3: (clear (CHead c3 (Bind b1) -v1) (CHead d1 (Bind b0) v0))).(let H4 \def (f_equal C C (\lambda (e: -C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d1 | -(CHead c _ _) \Rightarrow c])) (CHead d1 (Bind b0) v0) (CHead c3 (Bind b1) -v1) (clear_gen_bind b1 c3 (CHead d1 (Bind b0) v0) v1 H3)) in ((let H5 \def -(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with -[(CSort _) \Rightarrow b0 | (CHead _ k _) \Rightarrow (match k in K return -(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow -b0])])) (CHead d1 (Bind b0) v0) (CHead c3 (Bind b1) v1) (clear_gen_bind b1 c3 -(CHead d1 (Bind b0) v0) v1 H3)) in ((let H6 \def (f_equal C T (\lambda (e: -C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow v0 | -(CHead _ _ t) \Rightarrow t])) (CHead d1 (Bind b0) v0) (CHead c3 (Bind b1) -v1) (clear_gen_bind b1 c3 (CHead d1 (Bind b0) v0) v1 H3)) in (\lambda (_: (eq -B b0 b1)).(\lambda (H8: (eq C d1 c3)).(eq_ind_r C c3 (\lambda (c: C).(ex2_3 B -C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv c d2)))) (\lambda -(b3: B).(\lambda (d2: C).(\lambda (v3: T).(clear (CHead c4 (Bind b2) v2) -(CHead d2 (Bind b3) v3))))))) (ex2_3_intro B C T (\lambda (_: B).(\lambda -(d2: C).(\lambda (_: T).(csubv c3 d2)))) (\lambda (b3: B).(\lambda (d2: -C).(\lambda (v3: T).(clear (CHead c4 (Bind b2) v2) (CHead d2 (Bind b3) -v3))))) b2 c4 v2 H0 (clear_bind b2 c4 v2)) d1 H8)))) H5)) H4))))))))))))))) -(\lambda (c3: C).(\lambda (c4: C).(\lambda (_: (csubv c3 c4)).(\lambda (H1: -((\forall (b1: B).(\forall (d1: C).(\forall (v1: T).((clear c3 (CHead d1 -(Bind b1) v1)) \to (ex2_3 B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: -T).(csubv d1 d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda (v2: T).(clear -c4 (CHead d2 (Bind b2) v2)))))))))))).(\lambda (f1: F).(\lambda (f2: -F).(\lambda (v1: T).(\lambda (v2: T).(\lambda (b1: B).(\lambda (d1: -C).(\lambda (v0: T).(\lambda (H2: (clear (CHead c3 (Flat f1) v1) (CHead d1 -(Bind b1) v0))).(let H_x \def (H1 b1 d1 v0 (clear_gen_flat f1 c3 (CHead d1 -(Bind b1) v0) v1 H2)) in (let H3 \def H_x in (ex2_3_ind B C T (\lambda (_: -B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2: -B).(\lambda (d2: C).(\lambda (v3: T).(clear c4 (CHead d2 (Bind b2) v3))))) -(ex2_3 B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 -d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda (v3: T).(clear (CHead c4 -(Flat f2) v2) (CHead d2 (Bind b2) v3)))))) (\lambda (x0: B).(\lambda (x1: -C).(\lambda (x2: T).(\lambda (H4: (csubv d1 x1)).(\lambda (H5: (clear c4 -(CHead x1 (Bind x0) x2))).(ex2_3_intro B C T (\lambda (_: B).(\lambda (d2: -C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2: B).(\lambda (d2: -C).(\lambda (v3: T).(clear (CHead c4 (Flat f2) v2) (CHead d2 (Bind b2) -v3))))) x0 x1 x2 H4 (clear_flat c4 (CHead x1 (Bind x0) x2) H5 f2 v2))))))) -H3))))))))))))))) c1 c2 H))). -(* COMMENTS -Initial nodes: 1357 -END *) - -theorem csubv_clear_conf_void: - \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (d1: -C).(\forall (v1: T).((clear c1 (CHead d1 (Bind Void) v1)) \to (ex2_2 C T -(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2))) (\lambda (d2: C).(\lambda -(v2: T).(clear c2 (CHead d2 (Bind Void) v2)))))))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubv c1 c2)).(csubv_ind -(\lambda (c: C).(\lambda (c0: C).(\forall (d1: C).(\forall (v1: T).((clear c -(CHead d1 (Bind Void) v1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (_: -T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(clear c0 (CHead d2 -(Bind Void) v2)))))))))) (\lambda (n: nat).(\lambda (d1: C).(\lambda (v1: -T).(\lambda (H0: (clear (CSort n) (CHead d1 (Bind Void) v1))).(clear_gen_sort -(CHead d1 (Bind Void) v1) n H0 (ex2_2 C T (\lambda (d2: C).(\lambda (_: -T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(clear (CSort n) (CHead -d2 (Bind Void) v2)))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: -(csubv c3 c4)).(\lambda (_: ((\forall (d1: C).(\forall (v1: T).((clear c3 -(CHead d1 (Bind Void) v1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (_: -T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(clear c4 (CHead d2 -(Bind Void) v2)))))))))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (d1: -C).(\lambda (v0: T).(\lambda (H2: (clear (CHead c3 (Bind Void) v1) (CHead d1 -(Bind Void) v0))).(let H3 \def (f_equal C C (\lambda (e: C).(match e in C -return (\lambda (_: C).C) with [(CSort _) \Rightarrow d1 | (CHead c _ _) -\Rightarrow c])) (CHead d1 (Bind Void) v0) (CHead c3 (Bind Void) v1) -(clear_gen_bind Void c3 (CHead d1 (Bind Void) v0) v1 H2)) in ((let H4 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow v0 | (CHead _ _ t) \Rightarrow t])) (CHead d1 (Bind -Void) v0) (CHead c3 (Bind Void) v1) (clear_gen_bind Void c3 (CHead d1 (Bind -Void) v0) v1 H2)) in (\lambda (H5: (eq C d1 c3)).(eq_ind_r C c3 (\lambda (c: -C).(ex2_2 C T (\lambda (d2: C).(\lambda (_: T).(csubv c d2))) (\lambda (d2: -C).(\lambda (v3: T).(clear (CHead c4 (Bind Void) v2) (CHead d2 (Bind Void) -v3)))))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csubv c3 d2))) -(\lambda (d2: C).(\lambda (v3: T).(clear (CHead c4 (Bind Void) v2) (CHead d2 -(Bind Void) v3)))) c4 v2 H0 (clear_bind Void c4 v2)) d1 H5))) H3))))))))))) -(\lambda (c3: C).(\lambda (c4: C).(\lambda (_: (csubv c3 c4)).(\lambda (_: -((\forall (d1: C).(\forall (v1: T).((clear c3 (CHead d1 (Bind Void) v1)) \to -(ex2_2 C T (\lambda (d2: C).(\lambda (_: T).(csubv d1 d2))) (\lambda (d2: -C).(\lambda (v2: T).(clear c4 (CHead d2 (Bind Void) v2)))))))))).(\lambda -(b1: B).(\lambda (H2: (not (eq B b1 Void))).(\lambda (b2: B).(\lambda (v1: -T).(\lambda (v2: T).(\lambda (d1: C).(\lambda (v0: T).(\lambda (H3: (clear -(CHead c3 (Bind b1) v1) (CHead d1 (Bind Void) v0))).(let H4 \def (f_equal C C -(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow d1 | (CHead c _ _) \Rightarrow c])) (CHead d1 (Bind Void) v0) -(CHead c3 (Bind b1) v1) (clear_gen_bind b1 c3 (CHead d1 (Bind Void) v0) v1 -H3)) in ((let H5 \def (f_equal C B (\lambda (e: C).(match e in C return -(\lambda (_: C).B) with [(CSort _) \Rightarrow Void | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) -\Rightarrow b | (Flat _) \Rightarrow Void])])) (CHead d1 (Bind Void) v0) -(CHead c3 (Bind b1) v1) (clear_gen_bind b1 c3 (CHead d1 (Bind Void) v0) v1 -H3)) in ((let H6 \def (f_equal C T (\lambda (e: C).(match e in C return -(\lambda (_: C).T) with [(CSort _) \Rightarrow v0 | (CHead _ _ t) \Rightarrow -t])) (CHead d1 (Bind Void) v0) (CHead c3 (Bind b1) v1) (clear_gen_bind b1 c3 -(CHead d1 (Bind Void) v0) v1 H3)) in (\lambda (H7: (eq B Void b1)).(\lambda -(H8: (eq C d1 c3)).(eq_ind_r C c3 (\lambda (c: C).(ex2_2 C T (\lambda (d2: -C).(\lambda (_: T).(csubv c d2))) (\lambda (d2: C).(\lambda (v3: T).(clear -(CHead c4 (Bind b2) v2) (CHead d2 (Bind Void) v3)))))) (let H9 \def (eq_ind_r -B b1 (\lambda (b: B).(not (eq B b Void))) H2 Void H7) in (let H10 \def (match -(H9 (refl_equal B Void)) in False return (\lambda (_: False).(ex2_2 C T -(\lambda (d2: C).(\lambda (_: T).(csubv c3 d2))) (\lambda (d2: C).(\lambda -(v3: T).(clear (CHead c4 (Bind b2) v2) (CHead d2 (Bind Void) v3)))))) with -[]) in H10)) d1 H8)))) H5)) H4)))))))))))))) (\lambda (c3: C).(\lambda (c4: -C).(\lambda (_: (csubv c3 c4)).(\lambda (H1: ((\forall (d1: C).(\forall (v1: -T).((clear c3 (CHead d1 (Bind Void) v1)) \to (ex2_2 C T (\lambda (d2: -C).(\lambda (_: T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(clear -c4 (CHead d2 (Bind Void) v2)))))))))).(\lambda (f1: F).(\lambda (f2: -F).(\lambda (v1: T).(\lambda (v2: T).(\lambda (d1: C).(\lambda (v0: -T).(\lambda (H2: (clear (CHead c3 (Flat f1) v1) (CHead d1 (Bind Void) -v0))).(let H_x \def (H1 d1 v0 (clear_gen_flat f1 c3 (CHead d1 (Bind Void) v0) -v1 H2)) in (let H3 \def H_x in (ex2_2_ind C T (\lambda (d2: C).(\lambda (_: -T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v3: T).(clear c4 (CHead d2 -(Bind Void) v3)))) (ex2_2 C T (\lambda (d2: C).(\lambda (_: T).(csubv d1 -d2))) (\lambda (d2: C).(\lambda (v3: T).(clear (CHead c4 (Flat f2) v2) (CHead -d2 (Bind Void) v3))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H4: (csubv -d1 x0)).(\lambda (H5: (clear c4 (CHead x0 (Bind Void) x1))).(ex2_2_intro C T -(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2))) (\lambda (d2: C).(\lambda -(v3: T).(clear (CHead c4 (Flat f2) v2) (CHead d2 (Bind Void) v3)))) x0 x1 H4 -(clear_flat c4 (CHead x0 (Bind Void) x1) H5 f2 v2)))))) H3)))))))))))))) c1 -c2 H))). -(* COMMENTS -Initial nodes: 1205 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubv/defs.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubv/defs.ma deleted file mode 100644 index 6ffb63608..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubv/defs.ma +++ /dev/null @@ -1,30 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/C/defs.ma". - -inductive csubv: C \to (C \to Prop) \def -| csubv_sort: \forall (n: nat).(csubv (CSort n) (CSort n)) -| csubv_void: \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall -(v1: T).(\forall (v2: T).(csubv (CHead c1 (Bind Void) v1) (CHead c2 (Bind -Void) v2)))))) -| csubv_bind: \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall -(b1: B).((not (eq B b1 Void)) \to (\forall (b2: B).(\forall (v1: T).(\forall -(v2: T).(csubv (CHead c1 (Bind b1) v1) (CHead c2 (Bind b2) v2))))))))) -| csubv_flat: \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall -(f1: F).(\forall (f2: F).(\forall (v1: T).(\forall (v2: T).(csubv (CHead c1 -(Flat f1) v1) (CHead c2 (Flat f2) v2)))))))). - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubv/drop.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubv/drop.ma deleted file mode 100644 index 007670dbb..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubv/drop.ma +++ /dev/null @@ -1,115 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubv/props.ma". - -include "Basic-1/drop/fwd.ma". - -theorem csubv_drop_conf: - \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (e1: -C).(\forall (h: nat).((drop h O c1 e1) \to (ex2 C (\lambda (e2: C).(csubv e1 -e2)) (\lambda (e2: C).(drop h O c2 e2)))))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubv c1 c2)).(csubv_ind -(\lambda (c: C).(\lambda (c0: C).(\forall (e1: C).(\forall (h: nat).((drop h -O c e1) \to (ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop h O -c0 e2)))))))) (\lambda (n: nat).(\lambda (e1: C).(\lambda (h: nat).(\lambda -(H0: (drop h O (CSort n) e1)).(and3_ind (eq C e1 (CSort n)) (eq nat h O) (eq -nat O O) (ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop h O -(CSort n) e2))) (\lambda (H1: (eq C e1 (CSort n))).(\lambda (H2: (eq nat h -O)).(\lambda (_: (eq nat O O)).(eq_ind_r nat O (\lambda (n0: nat).(ex2 C -(\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop n0 O (CSort n) e2)))) -(eq_ind_r C (CSort n) (\lambda (c: C).(ex2 C (\lambda (e2: C).(csubv c e2)) -(\lambda (e2: C).(drop O O (CSort n) e2)))) (ex_intro2 C (\lambda (e2: -C).(csubv (CSort n) e2)) (\lambda (e2: C).(drop O O (CSort n) e2)) (CSort n) -(csubv_refl (CSort n)) (drop_refl (CSort n))) e1 H1) h H2)))) (drop_gen_sort -n h O e1 H0)))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csubv c3 -c4)).(\lambda (H1: ((\forall (e1: C).(\forall (h: nat).((drop h O c3 e1) \to -(ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop h O c4 -e2)))))))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (e1: C).(\lambda (h: -nat).(\lambda (H2: (drop h O (CHead c3 (Bind Void) v1) e1)).(nat_ind (\lambda -(n: nat).((drop n O (CHead c3 (Bind Void) v1) e1) \to (ex2 C (\lambda (e2: -C).(csubv e1 e2)) (\lambda (e2: C).(drop n O (CHead c4 (Bind Void) v2) -e2))))) (\lambda (H3: (drop O O (CHead c3 (Bind Void) v1) e1)).(eq_ind C -(CHead c3 (Bind Void) v1) (\lambda (c: C).(ex2 C (\lambda (e2: C).(csubv c -e2)) (\lambda (e2: C).(drop O O (CHead c4 (Bind Void) v2) e2)))) (ex_intro2 C -(\lambda (e2: C).(csubv (CHead c3 (Bind Void) v1) e2)) (\lambda (e2: C).(drop -O O (CHead c4 (Bind Void) v2) e2)) (CHead c4 (Bind Void) v2) (csubv_bind_same -c3 c4 H0 Void v1 v2) (drop_refl (CHead c4 (Bind Void) v2))) e1 (drop_gen_refl -(CHead c3 (Bind Void) v1) e1 H3))) (\lambda (h0: nat).(\lambda (_: (((drop h0 -O (CHead c3 (Bind Void) v1) e1) \to (ex2 C (\lambda (e2: C).(csubv e1 e2)) -(\lambda (e2: C).(drop h0 O (CHead c4 (Bind Void) v2) e2)))))).(\lambda (H3: -(drop (S h0) O (CHead c3 (Bind Void) v1) e1)).(let H_x \def (H1 e1 (r (Bind -Void) h0) (drop_gen_drop (Bind Void) c3 e1 v1 h0 H3)) in (let H4 \def H_x in -(ex2_ind C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop h0 O c4 -e2)) (ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop (S h0) O -(CHead c4 (Bind Void) v2) e2))) (\lambda (x: C).(\lambda (H5: (csubv e1 -x)).(\lambda (H6: (drop h0 O c4 x)).(ex_intro2 C (\lambda (e2: C).(csubv e1 -e2)) (\lambda (e2: C).(drop (S h0) O (CHead c4 (Bind Void) v2) e2)) x H5 -(drop_drop (Bind Void) h0 c4 x H6 v2))))) H4)))))) h H2)))))))))) (\lambda -(c3: C).(\lambda (c4: C).(\lambda (H0: (csubv c3 c4)).(\lambda (H1: ((\forall -(e1: C).(\forall (h: nat).((drop h O c3 e1) \to (ex2 C (\lambda (e2: -C).(csubv e1 e2)) (\lambda (e2: C).(drop h O c4 e2)))))))).(\lambda (b1: -B).(\lambda (H2: (not (eq B b1 Void))).(\lambda (b2: B).(\lambda (v1: -T).(\lambda (v2: T).(\lambda (e1: C).(\lambda (h: nat).(\lambda (H3: (drop h -O (CHead c3 (Bind b1) v1) e1)).(nat_ind (\lambda (n: nat).((drop n O (CHead -c3 (Bind b1) v1) e1) \to (ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: -C).(drop n O (CHead c4 (Bind b2) v2) e2))))) (\lambda (H4: (drop O O (CHead -c3 (Bind b1) v1) e1)).(eq_ind C (CHead c3 (Bind b1) v1) (\lambda (c: C).(ex2 -C (\lambda (e2: C).(csubv c e2)) (\lambda (e2: C).(drop O O (CHead c4 (Bind -b2) v2) e2)))) (ex_intro2 C (\lambda (e2: C).(csubv (CHead c3 (Bind b1) v1) -e2)) (\lambda (e2: C).(drop O O (CHead c4 (Bind b2) v2) e2)) (CHead c4 (Bind -b2) v2) (csubv_bind c3 c4 H0 b1 H2 b2 v1 v2) (drop_refl (CHead c4 (Bind b2) -v2))) e1 (drop_gen_refl (CHead c3 (Bind b1) v1) e1 H4))) (\lambda (h0: -nat).(\lambda (_: (((drop h0 O (CHead c3 (Bind b1) v1) e1) \to (ex2 C -(\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop h0 O (CHead c4 (Bind -b2) v2) e2)))))).(\lambda (H4: (drop (S h0) O (CHead c3 (Bind b1) v1) -e1)).(let H_x \def (H1 e1 (r (Bind b1) h0) (drop_gen_drop (Bind b1) c3 e1 v1 -h0 H4)) in (let H5 \def H_x in (ex2_ind C (\lambda (e2: C).(csubv e1 e2)) -(\lambda (e2: C).(drop h0 O c4 e2)) (ex2 C (\lambda (e2: C).(csubv e1 e2)) -(\lambda (e2: C).(drop (S h0) O (CHead c4 (Bind b2) v2) e2))) (\lambda (x: -C).(\lambda (H6: (csubv e1 x)).(\lambda (H7: (drop h0 O c4 x)).(ex_intro2 C -(\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop (S h0) O (CHead c4 -(Bind b2) v2) e2)) x H6 (drop_drop (Bind b2) h0 c4 x H7 v2))))) H5)))))) h -H3))))))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csubv c3 -c4)).(\lambda (H1: ((\forall (e1: C).(\forall (h: nat).((drop h O c3 e1) \to -(ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop h O c4 -e2)))))))).(\lambda (f1: F).(\lambda (f2: F).(\lambda (v1: T).(\lambda (v2: -T).(\lambda (e1: C).(\lambda (h: nat).(\lambda (H2: (drop h O (CHead c3 (Flat -f1) v1) e1)).(nat_ind (\lambda (n: nat).((drop n O (CHead c3 (Flat f1) v1) -e1) \to (ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop n O -(CHead c4 (Flat f2) v2) e2))))) (\lambda (H3: (drop O O (CHead c3 (Flat f1) -v1) e1)).(eq_ind C (CHead c3 (Flat f1) v1) (\lambda (c: C).(ex2 C (\lambda -(e2: C).(csubv c e2)) (\lambda (e2: C).(drop O O (CHead c4 (Flat f2) v2) -e2)))) (ex_intro2 C (\lambda (e2: C).(csubv (CHead c3 (Flat f1) v1) e2)) -(\lambda (e2: C).(drop O O (CHead c4 (Flat f2) v2) e2)) (CHead c4 (Flat f2) -v2) (csubv_flat c3 c4 H0 f1 f2 v1 v2) (drop_refl (CHead c4 (Flat f2) v2))) e1 -(drop_gen_refl (CHead c3 (Flat f1) v1) e1 H3))) (\lambda (h0: nat).(\lambda -(_: (((drop h0 O (CHead c3 (Flat f1) v1) e1) \to (ex2 C (\lambda (e2: -C).(csubv e1 e2)) (\lambda (e2: C).(drop h0 O (CHead c4 (Flat f2) v2) -e2)))))).(\lambda (H3: (drop (S h0) O (CHead c3 (Flat f1) v1) e1)).(let H_x -\def (H1 e1 (r (Flat f1) h0) (drop_gen_drop (Flat f1) c3 e1 v1 h0 H3)) in -(let H4 \def H_x in (ex2_ind C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: -C).(drop (S h0) O c4 e2)) (ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda -(e2: C).(drop (S h0) O (CHead c4 (Flat f2) v2) e2))) (\lambda (x: C).(\lambda -(H5: (csubv e1 x)).(\lambda (H6: (drop (S h0) O c4 x)).(ex_intro2 C (\lambda -(e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop (S h0) O (CHead c4 (Flat f2) -v2) e2)) x H5 (drop_drop (Flat f2) h0 c4 x H6 v2))))) H4)))))) h -H2)))))))))))) c1 c2 H))). -(* COMMENTS -Initial nodes: 1897 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubv/getl.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubv/getl.ma deleted file mode 100644 index bbba95084..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubv/getl.ma +++ /dev/null @@ -1,90 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubv/clear.ma". - -include "Basic-1/csubv/drop.ma". - -include "Basic-1/getl/fwd.ma". - -theorem csubv_getl_conf: - \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (b1: -B).(\forall (d1: C).(\forall (v1: T).(\forall (i: nat).((getl i c1 (CHead d1 -(Bind b1) v1)) \to (ex2_3 B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: -T).(csubv d1 d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda (v2: T).(getl -i c2 (CHead d2 (Bind b2) v2))))))))))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubv c1 c2)).(\lambda (b1: -B).(\lambda (d1: C).(\lambda (v1: T).(\lambda (i: nat).(\lambda (H0: (getl i -c1 (CHead d1 (Bind b1) v1))).(let H1 \def (getl_gen_all c1 (CHead d1 (Bind -b1) v1) i H0) in (ex2_ind C (\lambda (e: C).(drop i O c1 e)) (\lambda (e: -C).(clear e (CHead d1 (Bind b1) v1))) (ex2_3 B C T (\lambda (_: B).(\lambda -(d2: C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2: B).(\lambda (d2: -C).(\lambda (v2: T).(getl i c2 (CHead d2 (Bind b2) v2)))))) (\lambda (x: -C).(\lambda (H2: (drop i O c1 x)).(\lambda (H3: (clear x (CHead d1 (Bind b1) -v1))).(let H_x \def (csubv_drop_conf c1 c2 H x i H2) in (let H4 \def H_x in -(ex2_ind C (\lambda (e2: C).(csubv x e2)) (\lambda (e2: C).(drop i O c2 e2)) -(ex2_3 B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 -d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda (v2: T).(getl i c2 (CHead -d2 (Bind b2) v2)))))) (\lambda (x0: C).(\lambda (H5: (csubv x x0)).(\lambda -(H6: (drop i O c2 x0)).(let H_x0 \def (csubv_clear_conf x x0 H5 b1 d1 v1 H3) -in (let H7 \def H_x0 in (ex2_3_ind B C T (\lambda (_: B).(\lambda (d2: -C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2: B).(\lambda (d2: -C).(\lambda (v2: T).(clear x0 (CHead d2 (Bind b2) v2))))) (ex2_3 B C T -(\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2)))) (\lambda -(b2: B).(\lambda (d2: C).(\lambda (v2: T).(getl i c2 (CHead d2 (Bind b2) -v2)))))) (\lambda (x1: B).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H8: -(csubv d1 x2)).(\lambda (H9: (clear x0 (CHead x2 (Bind x1) x3))).(ex2_3_intro -B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2)))) -(\lambda (b2: B).(\lambda (d2: C).(\lambda (v2: T).(getl i c2 (CHead d2 (Bind -b2) v2))))) x1 x2 x3 H8 (getl_intro i c2 (CHead x2 (Bind x1) x3) x0 H6 -H9))))))) H7)))))) H4)))))) H1))))))))). -(* COMMENTS -Initial nodes: 455 -END *) - -theorem csubv_getl_conf_void: - \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (d1: -C).(\forall (v1: T).(\forall (i: nat).((getl i c1 (CHead d1 (Bind Void) v1)) -\to (ex2_2 C T (\lambda (d2: C).(\lambda (_: T).(csubv d1 d2))) (\lambda (d2: -C).(\lambda (v2: T).(getl i c2 (CHead d2 (Bind Void) v2))))))))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubv c1 c2)).(\lambda (d1: -C).(\lambda (v1: T).(\lambda (i: nat).(\lambda (H0: (getl i c1 (CHead d1 -(Bind Void) v1))).(let H1 \def (getl_gen_all c1 (CHead d1 (Bind Void) v1) i -H0) in (ex2_ind C (\lambda (e: C).(drop i O c1 e)) (\lambda (e: C).(clear e -(CHead d1 (Bind Void) v1))) (ex2_2 C T (\lambda (d2: C).(\lambda (_: -T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(getl i c2 (CHead d2 -(Bind Void) v2))))) (\lambda (x: C).(\lambda (H2: (drop i O c1 x)).(\lambda -(H3: (clear x (CHead d1 (Bind Void) v1))).(let H_x \def (csubv_drop_conf c1 -c2 H x i H2) in (let H4 \def H_x in (ex2_ind C (\lambda (e2: C).(csubv x e2)) -(\lambda (e2: C).(drop i O c2 e2)) (ex2_2 C T (\lambda (d2: C).(\lambda (_: -T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(getl i c2 (CHead d2 -(Bind Void) v2))))) (\lambda (x0: C).(\lambda (H5: (csubv x x0)).(\lambda -(H6: (drop i O c2 x0)).(let H_x0 \def (csubv_clear_conf_void x x0 H5 d1 v1 -H3) in (let H7 \def H_x0 in (ex2_2_ind C T (\lambda (d2: C).(\lambda (_: -T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(clear x0 (CHead d2 -(Bind Void) v2)))) (ex2_2 C T (\lambda (d2: C).(\lambda (_: T).(csubv d1 -d2))) (\lambda (d2: C).(\lambda (v2: T).(getl i c2 (CHead d2 (Bind Void) -v2))))) (\lambda (x1: C).(\lambda (x2: T).(\lambda (H8: (csubv d1 -x1)).(\lambda (H9: (clear x0 (CHead x1 (Bind Void) x2))).(ex2_2_intro C T -(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2))) (\lambda (d2: C).(\lambda -(v2: T).(getl i c2 (CHead d2 (Bind Void) v2)))) x1 x2 H8 (getl_intro i c2 -(CHead x1 (Bind Void) x2) x0 H6 H9)))))) H7)))))) H4)))))) H1)))))))). -(* COMMENTS -Initial nodes: 417 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubv/props.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubv/props.ma deleted file mode 100644 index 716922824..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/csubv/props.ma +++ /dev/null @@ -1,48 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubv/defs.ma". - -include "Basic-1/T/props.ma". - -theorem csubv_bind_same: - \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (b: B).(\forall -(v1: T).(\forall (v2: T).(csubv (CHead c1 (Bind b) v1) (CHead c2 (Bind b) -v2))))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubv c1 c2)).(\lambda (b: -B).(B_ind (\lambda (b0: B).(\forall (v1: T).(\forall (v2: T).(csubv (CHead c1 -(Bind b0) v1) (CHead c2 (Bind b0) v2))))) (\lambda (v1: T).(\lambda (v2: -T).(csubv_bind c1 c2 H Abbr (\lambda (H0: (eq B Abbr Void)).(not_abbr_void -H0)) Abbr v1 v2))) (\lambda (v1: T).(\lambda (v2: T).(csubv_bind c1 c2 H Abst -(sym_not_eq B Void Abst not_void_abst) Abst v1 v2))) (\lambda (v1: -T).(\lambda (v2: T).(csubv_void c1 c2 H v1 v2))) b)))). -(* COMMENTS -Initial nodes: 121 -END *) - -theorem csubv_refl: - \forall (c: C).(csubv c c) -\def - \lambda (c: C).(C_ind (\lambda (c0: C).(csubv c0 c0)) (\lambda (n: -nat).(csubv_sort n)) (\lambda (c0: C).(\lambda (H: (csubv c0 c0)).(\lambda -(k: K).(K_ind (\lambda (k0: K).(\forall (t: T).(csubv (CHead c0 k0 t) (CHead -c0 k0 t)))) (\lambda (b: B).(\lambda (t: T).(csubv_bind_same c0 c0 H b t t))) -(\lambda (f: F).(\lambda (t: T).(csubv_flat c0 c0 H f f t t))) k)))) c). -(* COMMENTS -Initial nodes: 93 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/definitions.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/definitions.ma deleted file mode 100644 index 5863b4243..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/definitions.ma +++ /dev/null @@ -1,68 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/tlt/defs.ma". - -include "Basic-1/iso/defs.ma". - -include "Basic-1/clen/defs.ma". - -include "Basic-1/flt/defs.ma". - -include "Basic-1/app/defs.ma". - -include "Basic-1/cnt/defs.ma". - -include "Basic-1/cimp/defs.ma". - -include "Basic-1/csubv/defs.ma". - -include "Basic-1/subst/defs.ma". - -include "Basic-1/subst1/defs.ma". - -include "Basic-1/csubst1/defs.ma". - -include "Basic-1/fsubst0/defs.ma". - -include "Basic-1/next_plus/defs.ma". - -include "Basic-1/sty1/defs.ma". - -include "Basic-1/llt/defs.ma". - -include "Basic-1/aprem/defs.ma". - -include "Basic-1/ex0/defs.ma". - -include "Basic-1/wcpr0/defs.ma". - -include "Basic-1/csuba/defs.ma". - -include "Basic-1/nf2/defs.ma". - -include "Basic-1/ex2/defs.ma". - -include "Basic-1/csubc/defs.ma". - -include "Basic-1/pc1/defs.ma". - -include "Basic-1/ex1/defs.ma". - -include "Basic-1/csubt/defs.ma". - -include "Basic-1/wf3/defs.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/drop/defs.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/drop/defs.ma deleted file mode 100644 index 987ae0ceb..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/drop/defs.ma +++ /dev/null @@ -1,31 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/C/defs.ma". - -include "Basic-1/lift/defs.ma". - -include "Basic-1/r/defs.ma". - -inductive drop: nat \to (nat \to (C \to (C \to Prop))) \def -| drop_refl: \forall (c: C).(drop O O c c) -| drop_drop: \forall (k: K).(\forall (h: nat).(\forall (c: C).(\forall (e: -C).((drop (r k h) O c e) \to (\forall (u: T).(drop (S h) O (CHead c k u) -e)))))) -| drop_skip: \forall (k: K).(\forall (h: nat).(\forall (d: nat).(\forall (c: -C).(\forall (e: C).((drop h (r k d) c e) \to (\forall (u: T).(drop h (S d) -(CHead c k (lift h (r k d) u)) (CHead e k u)))))))). - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/drop/fwd.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/drop/fwd.ma deleted file mode 100644 index 48495c148..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/drop/fwd.ma +++ /dev/null @@ -1,384 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/drop/defs.ma". - -theorem drop_gen_sort: - \forall (n: nat).(\forall (h: nat).(\forall (d: nat).(\forall (x: C).((drop -h d (CSort n) x) \to (and3 (eq C x (CSort n)) (eq nat h O) (eq nat d O)))))) -\def - \lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (x: -C).(\lambda (H: (drop h d (CSort n) x)).(insert_eq C (CSort n) (\lambda (c: -C).(drop h d c x)) (\lambda (c: C).(and3 (eq C x c) (eq nat h O) (eq nat d -O))) (\lambda (y: C).(\lambda (H0: (drop h d y x)).(drop_ind (\lambda (n0: -nat).(\lambda (n1: nat).(\lambda (c: C).(\lambda (c0: C).((eq C c (CSort n)) -\to (and3 (eq C c0 c) (eq nat n0 O) (eq nat n1 O))))))) (\lambda (c: -C).(\lambda (H1: (eq C c (CSort n))).(let H2 \def (f_equal C C (\lambda (e: -C).e) c (CSort n) H1) in (eq_ind_r C (CSort n) (\lambda (c0: C).(and3 (eq C -c0 c0) (eq nat O O) (eq nat O O))) (and3_intro (eq C (CSort n) (CSort n)) (eq -nat O O) (eq nat O O) (refl_equal C (CSort n)) (refl_equal nat O) (refl_equal -nat O)) c H2)))) (\lambda (k: K).(\lambda (h0: nat).(\lambda (c: C).(\lambda -(e: C).(\lambda (_: (drop (r k h0) O c e)).(\lambda (_: (((eq C c (CSort n)) -\to (and3 (eq C e c) (eq nat (r k h0) O) (eq nat O O))))).(\lambda (u: -T).(\lambda (H3: (eq C (CHead c k u) (CSort n))).(let H4 \def (eq_ind C -(CHead c k u) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) -with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I -(CSort n) H3) in (False_ind (and3 (eq C e (CHead c k u)) (eq nat (S h0) O) -(eq nat O O)) H4)))))))))) (\lambda (k: K).(\lambda (h0: nat).(\lambda (d0: -nat).(\lambda (c: C).(\lambda (e: C).(\lambda (_: (drop h0 (r k d0) c -e)).(\lambda (_: (((eq C c (CSort n)) \to (and3 (eq C e c) (eq nat h0 O) (eq -nat (r k d0) O))))).(\lambda (u: T).(\lambda (H3: (eq C (CHead c k (lift h0 -(r k d0) u)) (CSort n))).(let H4 \def (eq_ind C (CHead c k (lift h0 (r k d0) -u)) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort -_) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H3) in -(False_ind (and3 (eq C (CHead e k u) (CHead c k (lift h0 (r k d0) u))) (eq -nat h0 O) (eq nat (S d0) O)) H4))))))))))) h d y x H0))) H))))). -(* COMMENTS -Initial nodes: 595 -END *) - -theorem drop_gen_refl: - \forall (x: C).(\forall (e: C).((drop O O x e) \to (eq C x e))) -\def - \lambda (x: C).(\lambda (e: C).(\lambda (H: (drop O O x e)).(insert_eq nat O -(\lambda (n: nat).(drop n O x e)) (\lambda (_: nat).(eq C x e)) (\lambda (y: -nat).(\lambda (H0: (drop y O x e)).(insert_eq nat O (\lambda (n: nat).(drop y -n x e)) (\lambda (n: nat).((eq nat y n) \to (eq C x e))) (\lambda (y0: -nat).(\lambda (H1: (drop y y0 x e)).(drop_ind (\lambda (n: nat).(\lambda (n0: -nat).(\lambda (c: C).(\lambda (c0: C).((eq nat n0 O) \to ((eq nat n n0) \to -(eq C c c0))))))) (\lambda (c: C).(\lambda (_: (eq nat O O)).(\lambda (_: (eq -nat O O)).(refl_equal C c)))) (\lambda (k: K).(\lambda (h: nat).(\lambda (c: -C).(\lambda (e0: C).(\lambda (_: (drop (r k h) O c e0)).(\lambda (_: (((eq -nat O O) \to ((eq nat (r k h) O) \to (eq C c e0))))).(\lambda (u: T).(\lambda -(_: (eq nat O O)).(\lambda (H5: (eq nat (S h) O)).(let H6 \def (eq_ind nat (S -h) (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O -\Rightarrow False | (S _) \Rightarrow True])) I O H5) in (False_ind (eq C -(CHead c k u) e0) H6))))))))))) (\lambda (k: K).(\lambda (h: nat).(\lambda -(d: nat).(\lambda (c: C).(\lambda (e0: C).(\lambda (H2: (drop h (r k d) c -e0)).(\lambda (H3: (((eq nat (r k d) O) \to ((eq nat h (r k d)) \to (eq C c -e0))))).(\lambda (u: T).(\lambda (H4: (eq nat (S d) O)).(\lambda (H5: (eq nat -h (S d))).(let H6 \def (f_equal nat nat (\lambda (e1: nat).e1) h (S d) H5) in -(let H7 \def (eq_ind nat h (\lambda (n: nat).((eq nat (r k d) O) \to ((eq nat -n (r k d)) \to (eq C c e0)))) H3 (S d) H6) in (let H8 \def (eq_ind nat h -(\lambda (n: nat).(drop n (r k d) c e0)) H2 (S d) H6) in (eq_ind_r nat (S d) -(\lambda (n: nat).(eq C (CHead c k (lift n (r k d) u)) (CHead e0 k u))) (let -H9 \def (eq_ind nat (S d) (\lambda (ee: nat).(match ee in nat return (\lambda -(_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H4) -in (False_ind (eq C (CHead c k (lift (S d) (r k d) u)) (CHead e0 k u)) H9)) h -H6)))))))))))))) y y0 x e H1))) H0))) H))). -(* COMMENTS -Initial nodes: 561 -END *) - -theorem drop_gen_drop: - \forall (k: K).(\forall (c: C).(\forall (x: C).(\forall (u: T).(\forall (h: -nat).((drop (S h) O (CHead c k u) x) \to (drop (r k h) O c x)))))) -\def - \lambda (k: K).(\lambda (c: C).(\lambda (x: C).(\lambda (u: T).(\lambda (h: -nat).(\lambda (H: (drop (S h) O (CHead c k u) x)).(insert_eq C (CHead c k u) -(\lambda (c0: C).(drop (S h) O c0 x)) (\lambda (_: C).(drop (r k h) O c x)) -(\lambda (y: C).(\lambda (H0: (drop (S h) O y x)).(insert_eq nat O (\lambda -(n: nat).(drop (S h) n y x)) (\lambda (n: nat).((eq C y (CHead c k u)) \to -(drop (r k h) n c x))) (\lambda (y0: nat).(\lambda (H1: (drop (S h) y0 y -x)).(insert_eq nat (S h) (\lambda (n: nat).(drop n y0 y x)) (\lambda (_: -nat).((eq nat y0 O) \to ((eq C y (CHead c k u)) \to (drop (r k h) y0 c x)))) -(\lambda (y1: nat).(\lambda (H2: (drop y1 y0 y x)).(drop_ind (\lambda (n: -nat).(\lambda (n0: nat).(\lambda (c0: C).(\lambda (c1: C).((eq nat n (S h)) -\to ((eq nat n0 O) \to ((eq C c0 (CHead c k u)) \to (drop (r k h) n0 c -c1)))))))) (\lambda (c0: C).(\lambda (H3: (eq nat O (S h))).(\lambda (_: (eq -nat O O)).(\lambda (H5: (eq C c0 (CHead c k u))).(eq_ind_r C (CHead c k u) -(\lambda (c1: C).(drop (r k h) O c c1)) (let H6 \def (eq_ind nat O (\lambda -(ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow -True | (S _) \Rightarrow False])) I (S h) H3) in (False_ind (drop (r k h) O c -(CHead c k u)) H6)) c0 H5))))) (\lambda (k0: K).(\lambda (h0: nat).(\lambda -(c0: C).(\lambda (e: C).(\lambda (H3: (drop (r k0 h0) O c0 e)).(\lambda (H4: -(((eq nat (r k0 h0) (S h)) \to ((eq nat O O) \to ((eq C c0 (CHead c k u)) \to -(drop (r k h) O c e)))))).(\lambda (u0: T).(\lambda (H5: (eq nat (S h0) (S -h))).(\lambda (_: (eq nat O O)).(\lambda (H7: (eq C (CHead c0 k0 u0) (CHead c -k u))).(let H8 \def (f_equal C C (\lambda (e0: C).(match e0 in C return -(\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c1 _ _) -\Rightarrow c1])) (CHead c0 k0 u0) (CHead c k u) H7) in ((let H9 \def -(f_equal C K (\lambda (e0: C).(match e0 in C return (\lambda (_: C).K) with -[(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead c0 k0 u0) -(CHead c k u) H7) in ((let H10 \def (f_equal C T (\lambda (e0: C).(match e0 -in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) -\Rightarrow t])) (CHead c0 k0 u0) (CHead c k u) H7) in (\lambda (H11: (eq K -k0 k)).(\lambda (H12: (eq C c0 c)).(let H13 \def (eq_ind C c0 (\lambda (c1: -C).((eq nat (r k0 h0) (S h)) \to ((eq nat O O) \to ((eq C c1 (CHead c k u)) -\to (drop (r k h) O c e))))) H4 c H12) in (let H14 \def (eq_ind C c0 (\lambda -(c1: C).(drop (r k0 h0) O c1 e)) H3 c H12) in (let H15 \def (eq_ind K k0 -(\lambda (k1: K).((eq nat (r k1 h0) (S h)) \to ((eq nat O O) \to ((eq C c -(CHead c k u)) \to (drop (r k h) O c e))))) H13 k H11) in (let H16 \def -(eq_ind K k0 (\lambda (k1: K).(drop (r k1 h0) O c e)) H14 k H11) in (let H17 -\def (f_equal nat nat (\lambda (e0: nat).(match e0 in nat return (\lambda (_: -nat).nat) with [O \Rightarrow h0 | (S n) \Rightarrow n])) (S h0) (S h) H5) in -(let H18 \def (eq_ind nat h0 (\lambda (n: nat).((eq nat (r k n) (S h)) \to -((eq nat O O) \to ((eq C c (CHead c k u)) \to (drop (r k h) O c e))))) H15 h -H17) in (let H19 \def (eq_ind nat h0 (\lambda (n: nat).(drop (r k n) O c e)) -H16 h H17) in H19)))))))))) H9)) H8)))))))))))) (\lambda (k0: K).(\lambda -(h0: nat).(\lambda (d: nat).(\lambda (c0: C).(\lambda (e: C).(\lambda (H3: -(drop h0 (r k0 d) c0 e)).(\lambda (H4: (((eq nat h0 (S h)) \to ((eq nat (r k0 -d) O) \to ((eq C c0 (CHead c k u)) \to (drop (r k h) (r k0 d) c -e)))))).(\lambda (u0: T).(\lambda (H5: (eq nat h0 (S h))).(\lambda (H6: (eq -nat (S d) O)).(\lambda (H7: (eq C (CHead c0 k0 (lift h0 (r k0 d) u0)) (CHead -c k u))).(let H8 \def (eq_ind nat h0 (\lambda (n: nat).(eq C (CHead c0 k0 -(lift n (r k0 d) u0)) (CHead c k u))) H7 (S h) H5) in (let H9 \def (eq_ind -nat h0 (\lambda (n: nat).((eq nat n (S h)) \to ((eq nat (r k0 d) O) \to ((eq -C c0 (CHead c k u)) \to (drop (r k h) (r k0 d) c e))))) H4 (S h) H5) in (let -H10 \def (eq_ind nat h0 (\lambda (n: nat).(drop n (r k0 d) c0 e)) H3 (S h) -H5) in (let H11 \def (f_equal C C (\lambda (e0: C).(match e0 in C return -(\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c1 _ _) -\Rightarrow c1])) (CHead c0 k0 (lift (S h) (r k0 d) u0)) (CHead c k u) H8) in -((let H12 \def (f_equal C K (\lambda (e0: C).(match e0 in C return (\lambda -(_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) -(CHead c0 k0 (lift (S h) (r k0 d) u0)) (CHead c k u) H8) in ((let H13 \def -(f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d0: nat) (t: -T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) -\Rightarrow (TLRef (match (blt i d0) with [true \Rightarrow i | false -\Rightarrow (f i)])) | (THead k1 u1 t0) \Rightarrow (THead k1 (lref_map f d0 -u1) (lref_map f (s k1 d0) t0))]) in lref_map) (\lambda (x0: nat).(plus x0 (S -h))) (r k0 d) u0) | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 (lift (S h) -(r k0 d) u0)) (CHead c k u) H8) in (\lambda (H14: (eq K k0 k)).(\lambda (H15: -(eq C c0 c)).(let H16 \def (eq_ind C c0 (\lambda (c1: C).((eq nat (S h) (S -h)) \to ((eq nat (r k0 d) O) \to ((eq C c1 (CHead c k u)) \to (drop (r k h) -(r k0 d) c e))))) H9 c H15) in (let H17 \def (eq_ind C c0 (\lambda (c1: -C).(drop (S h) (r k0 d) c1 e)) H10 c H15) in (let H18 \def (eq_ind K k0 -(\lambda (k1: K).(eq T (lift (S h) (r k1 d) u0) u)) H13 k H14) in (let H19 -\def (eq_ind K k0 (\lambda (k1: K).((eq nat (S h) (S h)) \to ((eq nat (r k1 -d) O) \to ((eq C c (CHead c k u)) \to (drop (r k h) (r k1 d) c e))))) H16 k -H14) in (let H20 \def (eq_ind K k0 (\lambda (k1: K).(drop (S h) (r k1 d) c -e)) H17 k H14) in (eq_ind_r K k (\lambda (k1: K).(drop (r k h) (S d) c (CHead -e k1 u0))) (let H21 \def (eq_ind_r T u (\lambda (t: T).((eq nat (S h) (S h)) -\to ((eq nat (r k d) O) \to ((eq C c (CHead c k t)) \to (drop (r k h) (r k d) -c e))))) H19 (lift (S h) (r k d) u0) H18) in (let H22 \def (eq_ind nat (S d) -(\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O -\Rightarrow False | (S _) \Rightarrow True])) I O H6) in (False_ind (drop (r -k h) (S d) c (CHead e k u0)) H22))) k0 H14))))))))) H12)) H11)))))))))))))))) -y1 y0 y x H2))) H1))) H0))) H)))))). -(* COMMENTS -Initial nodes: 1856 -END *) - -theorem drop_gen_skip_r: - \forall (c: C).(\forall (x: C).(\forall (u: T).(\forall (h: nat).(\forall -(d: nat).(\forall (k: K).((drop h (S d) x (CHead c k u)) \to (ex2 C (\lambda -(e: C).(eq C x (CHead e k (lift h (r k d) u)))) (\lambda (e: C).(drop h (r k -d) e c))))))))) -\def - \lambda (c: C).(\lambda (x: C).(\lambda (u: T).(\lambda (h: nat).(\lambda -(d: nat).(\lambda (k: K).(\lambda (H: (drop h (S d) x (CHead c k -u))).(insert_eq C (CHead c k u) (\lambda (c0: C).(drop h (S d) x c0)) -(\lambda (_: C).(ex2 C (\lambda (e: C).(eq C x (CHead e k (lift h (r k d) -u)))) (\lambda (e: C).(drop h (r k d) e c)))) (\lambda (y: C).(\lambda (H0: -(drop h (S d) x y)).(insert_eq nat (S d) (\lambda (n: nat).(drop h n x y)) -(\lambda (_: nat).((eq C y (CHead c k u)) \to (ex2 C (\lambda (e: C).(eq C x -(CHead e k (lift h (r k d) u)))) (\lambda (e: C).(drop h (r k d) e c))))) -(\lambda (y0: nat).(\lambda (H1: (drop h y0 x y)).(drop_ind (\lambda (n: -nat).(\lambda (n0: nat).(\lambda (c0: C).(\lambda (c1: C).((eq nat n0 (S d)) -\to ((eq C c1 (CHead c k u)) \to (ex2 C (\lambda (e: C).(eq C c0 (CHead e k -(lift n (r k d) u)))) (\lambda (e: C).(drop n (r k d) e c))))))))) (\lambda -(c0: C).(\lambda (H2: (eq nat O (S d))).(\lambda (H3: (eq C c0 (CHead c k -u))).(eq_ind_r C (CHead c k u) (\lambda (c1: C).(ex2 C (\lambda (e: C).(eq C -c1 (CHead e k (lift O (r k d) u)))) (\lambda (e: C).(drop O (r k d) e c)))) -(let H4 \def (eq_ind nat O (\lambda (ee: nat).(match ee in nat return -(\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) -I (S d) H2) in (False_ind (ex2 C (\lambda (e: C).(eq C (CHead c k u) (CHead e -k (lift O (r k d) u)))) (\lambda (e: C).(drop O (r k d) e c))) H4)) c0 H3)))) -(\lambda (k0: K).(\lambda (h0: nat).(\lambda (c0: C).(\lambda (e: C).(\lambda -(H2: (drop (r k0 h0) O c0 e)).(\lambda (H3: (((eq nat O (S d)) \to ((eq C e -(CHead c k u)) \to (ex2 C (\lambda (e0: C).(eq C c0 (CHead e0 k (lift (r k0 -h0) (r k d) u)))) (\lambda (e0: C).(drop (r k0 h0) (r k d) e0 -c))))))).(\lambda (u0: T).(\lambda (H4: (eq nat O (S d))).(\lambda (H5: (eq C -e (CHead c k u))).(let H6 \def (eq_ind C e (\lambda (c1: C).((eq nat O (S d)) -\to ((eq C c1 (CHead c k u)) \to (ex2 C (\lambda (e0: C).(eq C c0 (CHead e0 k -(lift (r k0 h0) (r k d) u)))) (\lambda (e0: C).(drop (r k0 h0) (r k d) e0 -c)))))) H3 (CHead c k u) H5) in (let H7 \def (eq_ind C e (\lambda (c1: -C).(drop (r k0 h0) O c0 c1)) H2 (CHead c k u) H5) in (let H8 \def (eq_ind nat -O (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O -\Rightarrow True | (S _) \Rightarrow False])) I (S d) H4) in (False_ind (ex2 -C (\lambda (e0: C).(eq C (CHead c0 k0 u0) (CHead e0 k (lift (S h0) (r k d) -u)))) (\lambda (e0: C).(drop (S h0) (r k d) e0 c))) H8))))))))))))) (\lambda -(k0: K).(\lambda (h0: nat).(\lambda (d0: nat).(\lambda (c0: C).(\lambda (e: -C).(\lambda (H2: (drop h0 (r k0 d0) c0 e)).(\lambda (H3: (((eq nat (r k0 d0) -(S d)) \to ((eq C e (CHead c k u)) \to (ex2 C (\lambda (e0: C).(eq C c0 -(CHead e0 k (lift h0 (r k d) u)))) (\lambda (e0: C).(drop h0 (r k d) e0 -c))))))).(\lambda (u0: T).(\lambda (H4: (eq nat (S d0) (S d))).(\lambda (H5: -(eq C (CHead e k0 u0) (CHead c k u))).(let H6 \def (f_equal C C (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow e | -(CHead c1 _ _) \Rightarrow c1])) (CHead e k0 u0) (CHead c k u) H5) in ((let -H7 \def (f_equal C K (\lambda (e0: C).(match e0 in C return (\lambda (_: -C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) -(CHead e k0 u0) (CHead c k u) H5) in ((let H8 \def (f_equal C T (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | -(CHead _ _ t) \Rightarrow t])) (CHead e k0 u0) (CHead c k u) H5) in (\lambda -(H9: (eq K k0 k)).(\lambda (H10: (eq C e c)).(eq_ind_r T u (\lambda (t: -T).(ex2 C (\lambda (e0: C).(eq C (CHead c0 k0 (lift h0 (r k0 d0) t)) (CHead -e0 k (lift h0 (r k d) u)))) (\lambda (e0: C).(drop h0 (r k d) e0 c)))) (let -H11 \def (eq_ind C e (\lambda (c1: C).((eq nat (r k0 d0) (S d)) \to ((eq C c1 -(CHead c k u)) \to (ex2 C (\lambda (e0: C).(eq C c0 (CHead e0 k (lift h0 (r k -d) u)))) (\lambda (e0: C).(drop h0 (r k d) e0 c)))))) H3 c H10) in (let H12 -\def (eq_ind C e (\lambda (c1: C).(drop h0 (r k0 d0) c0 c1)) H2 c H10) in -(let H13 \def (eq_ind K k0 (\lambda (k1: K).((eq nat (r k1 d0) (S d)) \to -((eq C c (CHead c k u)) \to (ex2 C (\lambda (e0: C).(eq C c0 (CHead e0 k -(lift h0 (r k d) u)))) (\lambda (e0: C).(drop h0 (r k d) e0 c)))))) H11 k H9) -in (let H14 \def (eq_ind K k0 (\lambda (k1: K).(drop h0 (r k1 d0) c0 c)) H12 -k H9) in (eq_ind_r K k (\lambda (k1: K).(ex2 C (\lambda (e0: C).(eq C (CHead -c0 k1 (lift h0 (r k1 d0) u)) (CHead e0 k (lift h0 (r k d) u)))) (\lambda (e0: -C).(drop h0 (r k d) e0 c)))) (let H15 \def (f_equal nat nat (\lambda (e0: -nat).(match e0 in nat return (\lambda (_: nat).nat) with [O \Rightarrow d0 | -(S n) \Rightarrow n])) (S d0) (S d) H4) in (let H16 \def (eq_ind nat d0 -(\lambda (n: nat).((eq nat (r k n) (S d)) \to ((eq C c (CHead c k u)) \to -(ex2 C (\lambda (e0: C).(eq C c0 (CHead e0 k (lift h0 (r k d) u)))) (\lambda -(e0: C).(drop h0 (r k d) e0 c)))))) H13 d H15) in (let H17 \def (eq_ind nat -d0 (\lambda (n: nat).(drop h0 (r k n) c0 c)) H14 d H15) in (eq_ind_r nat d -(\lambda (n: nat).(ex2 C (\lambda (e0: C).(eq C (CHead c0 k (lift h0 (r k n) -u)) (CHead e0 k (lift h0 (r k d) u)))) (\lambda (e0: C).(drop h0 (r k d) e0 -c)))) (ex_intro2 C (\lambda (e0: C).(eq C (CHead c0 k (lift h0 (r k d) u)) -(CHead e0 k (lift h0 (r k d) u)))) (\lambda (e0: C).(drop h0 (r k d) e0 c)) -c0 (refl_equal C (CHead c0 k (lift h0 (r k d) u))) H17) d0 H15)))) k0 H9))))) -u0 H8)))) H7)) H6)))))))))))) h y0 x y H1))) H0))) H))))))). -(* COMMENTS -Initial nodes: 1758 -END *) - -theorem drop_gen_skip_l: - \forall (c: C).(\forall (x: C).(\forall (u: T).(\forall (h: nat).(\forall -(d: nat).(\forall (k: K).((drop h (S d) (CHead c k u) x) \to (ex3_2 C T -(\lambda (e: C).(\lambda (v: T).(eq C x (CHead e k v)))) (\lambda (_: -C).(\lambda (v: T).(eq T u (lift h (r k d) v)))) (\lambda (e: C).(\lambda (_: -T).(drop h (r k d) c e)))))))))) -\def - \lambda (c: C).(\lambda (x: C).(\lambda (u: T).(\lambda (h: nat).(\lambda -(d: nat).(\lambda (k: K).(\lambda (H: (drop h (S d) (CHead c k u) -x)).(insert_eq C (CHead c k u) (\lambda (c0: C).(drop h (S d) c0 x)) (\lambda -(_: C).(ex3_2 C T (\lambda (e: C).(\lambda (v: T).(eq C x (CHead e k v)))) -(\lambda (_: C).(\lambda (v: T).(eq T u (lift h (r k d) v)))) (\lambda (e: -C).(\lambda (_: T).(drop h (r k d) c e))))) (\lambda (y: C).(\lambda (H0: -(drop h (S d) y x)).(insert_eq nat (S d) (\lambda (n: nat).(drop h n y x)) -(\lambda (_: nat).((eq C y (CHead c k u)) \to (ex3_2 C T (\lambda (e: -C).(\lambda (v: T).(eq C x (CHead e k v)))) (\lambda (_: C).(\lambda (v: -T).(eq T u (lift h (r k d) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r k -d) c e)))))) (\lambda (y0: nat).(\lambda (H1: (drop h y0 y x)).(drop_ind -(\lambda (n: nat).(\lambda (n0: nat).(\lambda (c0: C).(\lambda (c1: C).((eq -nat n0 (S d)) \to ((eq C c0 (CHead c k u)) \to (ex3_2 C T (\lambda (e: -C).(\lambda (v: T).(eq C c1 (CHead e k v)))) (\lambda (_: C).(\lambda (v: -T).(eq T u (lift n (r k d) v)))) (\lambda (e: C).(\lambda (_: T).(drop n (r k -d) c e)))))))))) (\lambda (c0: C).(\lambda (H2: (eq nat O (S d))).(\lambda -(H3: (eq C c0 (CHead c k u))).(eq_ind_r C (CHead c k u) (\lambda (c1: -C).(ex3_2 C T (\lambda (e: C).(\lambda (v: T).(eq C c1 (CHead e k v)))) -(\lambda (_: C).(\lambda (v: T).(eq T u (lift O (r k d) v)))) (\lambda (e: -C).(\lambda (_: T).(drop O (r k d) c e))))) (let H4 \def (eq_ind nat O -(\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O -\Rightarrow True | (S _) \Rightarrow False])) I (S d) H2) in (False_ind -(ex3_2 C T (\lambda (e: C).(\lambda (v: T).(eq C (CHead c k u) (CHead e k -v)))) (\lambda (_: C).(\lambda (v: T).(eq T u (lift O (r k d) v)))) (\lambda -(e: C).(\lambda (_: T).(drop O (r k d) c e)))) H4)) c0 H3)))) (\lambda (k0: -K).(\lambda (h0: nat).(\lambda (c0: C).(\lambda (e: C).(\lambda (H2: (drop (r -k0 h0) O c0 e)).(\lambda (H3: (((eq nat O (S d)) \to ((eq C c0 (CHead c k u)) -\to (ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 k v)))) -(\lambda (_: C).(\lambda (v: T).(eq T u (lift (r k0 h0) (r k d) v)))) -(\lambda (e0: C).(\lambda (_: T).(drop (r k0 h0) (r k d) c -e0)))))))).(\lambda (u0: T).(\lambda (H4: (eq nat O (S d))).(\lambda (H5: (eq -C (CHead c0 k0 u0) (CHead c k u))).(let H6 \def (f_equal C C (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | -(CHead c1 _ _) \Rightarrow c1])) (CHead c0 k0 u0) (CHead c k u) H5) in ((let -H7 \def (f_equal C K (\lambda (e0: C).(match e0 in C return (\lambda (_: -C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) -(CHead c0 k0 u0) (CHead c k u) H5) in ((let H8 \def (f_equal C T (\lambda -(e0: C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow -u0 | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 u0) (CHead c k u) H5) in -(\lambda (H9: (eq K k0 k)).(\lambda (H10: (eq C c0 c)).(let H11 \def (eq_ind -C c0 (\lambda (c1: C).((eq nat O (S d)) \to ((eq C c1 (CHead c k u)) \to -(ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 k v)))) -(\lambda (_: C).(\lambda (v: T).(eq T u (lift (r k0 h0) (r k d) v)))) -(\lambda (e0: C).(\lambda (_: T).(drop (r k0 h0) (r k d) c e0))))))) H3 c -H10) in (let H12 \def (eq_ind C c0 (\lambda (c1: C).(drop (r k0 h0) O c1 e)) -H2 c H10) in (let H13 \def (eq_ind K k0 (\lambda (k1: K).((eq nat O (S d)) -\to ((eq C c (CHead c k u)) \to (ex3_2 C T (\lambda (e0: C).(\lambda (v: -T).(eq C e (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T u (lift (r -k1 h0) (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop (r k1 h0) (r k d) -c e0))))))) H11 k H9) in (let H14 \def (eq_ind K k0 (\lambda (k1: K).(drop (r -k1 h0) O c e)) H12 k H9) in (let H15 \def (eq_ind nat O (\lambda (ee: -nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True -| (S _) \Rightarrow False])) I (S d) H4) in (False_ind (ex3_2 C T (\lambda -(e0: C).(\lambda (v: T).(eq C e (CHead e0 k v)))) (\lambda (_: C).(\lambda -(v: T).(eq T u (lift (S h0) (r k d) v)))) (\lambda (e0: C).(\lambda (_: -T).(drop (S h0) (r k d) c e0)))) H15))))))))) H7)) H6))))))))))) (\lambda -(k0: K).(\lambda (h0: nat).(\lambda (d0: nat).(\lambda (c0: C).(\lambda (e: -C).(\lambda (H2: (drop h0 (r k0 d0) c0 e)).(\lambda (H3: (((eq nat (r k0 d0) -(S d)) \to ((eq C c0 (CHead c k u)) \to (ex3_2 C T (\lambda (e0: C).(\lambda -(v: T).(eq C e (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T u -(lift h0 (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h0 (r k d) c -e0)))))))).(\lambda (u0: T).(\lambda (H4: (eq nat (S d0) (S d))).(\lambda -(H5: (eq C (CHead c0 k0 (lift h0 (r k0 d0) u0)) (CHead c k u))).(let H6 \def -(f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with -[(CSort _) \Rightarrow c0 | (CHead c1 _ _) \Rightarrow c1])) (CHead c0 k0 -(lift h0 (r k0 d0) u0)) (CHead c k u) H5) in ((let H7 \def (f_equal C K -(\lambda (e0: C).(match e0 in C return (\lambda (_: C).K) with [(CSort _) -\Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead c0 k0 (lift h0 (r k0 -d0) u0)) (CHead c k u) H5) in ((let H8 \def (f_equal C T (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow ((let -rec lref_map (f: ((nat \to nat))) (d1: nat) (t: T) on t: T \def (match t with -[(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i -d1) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k1 u1 t0) -\Rightarrow (THead k1 (lref_map f d1 u1) (lref_map f (s k1 d1) t0))]) in -lref_map) (\lambda (x0: nat).(plus x0 h0)) (r k0 d0) u0) | (CHead _ _ t) -\Rightarrow t])) (CHead c0 k0 (lift h0 (r k0 d0) u0)) (CHead c k u) H5) in -(\lambda (H9: (eq K k0 k)).(\lambda (H10: (eq C c0 c)).(let H11 \def (eq_ind -C c0 (\lambda (c1: C).((eq nat (r k0 d0) (S d)) \to ((eq C c1 (CHead c k u)) -\to (ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 k v)))) -(\lambda (_: C).(\lambda (v: T).(eq T u (lift h0 (r k d) v)))) (\lambda (e0: -C).(\lambda (_: T).(drop h0 (r k d) c e0))))))) H3 c H10) in (let H12 \def -(eq_ind C c0 (\lambda (c1: C).(drop h0 (r k0 d0) c1 e)) H2 c H10) in (let H13 -\def (eq_ind K k0 (\lambda (k1: K).(eq T (lift h0 (r k1 d0) u0) u)) H8 k H9) -in (let H14 \def (eq_ind K k0 (\lambda (k1: K).((eq nat (r k1 d0) (S d)) \to -((eq C c (CHead c k u)) \to (ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C -e (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T u (lift h0 (r k d) -v)))) (\lambda (e0: C).(\lambda (_: T).(drop h0 (r k d) c e0))))))) H11 k H9) -in (let H15 \def (eq_ind K k0 (\lambda (k1: K).(drop h0 (r k1 d0) c e)) H12 k -H9) in (eq_ind_r K k (\lambda (k1: K).(ex3_2 C T (\lambda (e0: C).(\lambda -(v: T).(eq C (CHead e k1 u0) (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: -T).(eq T u (lift h0 (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h0 -(r k d) c e0))))) (let H16 \def (eq_ind_r T u (\lambda (t: T).((eq nat (r k -d0) (S d)) \to ((eq C c (CHead c k t)) \to (ex3_2 C T (\lambda (e0: -C).(\lambda (v: T).(eq C e (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: -T).(eq T t (lift h0 (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h0 -(r k d) c e0))))))) H14 (lift h0 (r k d0) u0) H13) in (eq_ind T (lift h0 (r k -d0) u0) (\lambda (t: T).(ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C -(CHead e k u0) (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T t -(lift h0 (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h0 (r k d) c -e0))))) (let H17 \def (f_equal nat nat (\lambda (e0: nat).(match e0 in nat -return (\lambda (_: nat).nat) with [O \Rightarrow d0 | (S n) \Rightarrow n])) -(S d0) (S d) H4) in (let H18 \def (eq_ind nat d0 (\lambda (n: nat).((eq nat -(r k n) (S d)) \to ((eq C c (CHead c k (lift h0 (r k n) u0))) \to (ex3_2 C T -(\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 k v)))) (\lambda (_: -C).(\lambda (v: T).(eq T (lift h0 (r k n) u0) (lift h0 (r k d) v)))) (\lambda -(e0: C).(\lambda (_: T).(drop h0 (r k d) c e0))))))) H16 d H17) in (let H19 -\def (eq_ind nat d0 (\lambda (n: nat).(drop h0 (r k n) c e)) H15 d H17) in -(eq_ind_r nat d (\lambda (n: nat).(ex3_2 C T (\lambda (e0: C).(\lambda (v: -T).(eq C (CHead e k u0) (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq -T (lift h0 (r k n) u0) (lift h0 (r k d) v)))) (\lambda (e0: C).(\lambda (_: -T).(drop h0 (r k d) c e0))))) (ex3_2_intro C T (\lambda (e0: C).(\lambda (v: -T).(eq C (CHead e k u0) (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq -T (lift h0 (r k d) u0) (lift h0 (r k d) v)))) (\lambda (e0: C).(\lambda (_: -T).(drop h0 (r k d) c e0))) e u0 (refl_equal C (CHead e k u0)) (refl_equal T -(lift h0 (r k d) u0)) H19) d0 H17)))) u H13)) k0 H9))))))))) H7)) -H6)))))))))))) h y0 y x H1))) H0))) H))))))). -(* COMMENTS -Initial nodes: 2574 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/drop/props.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/drop/props.ma deleted file mode 100644 index ac802b105..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/drop/props.ma +++ /dev/null @@ -1,737 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/drop/fwd.ma". - -include "Basic-1/lift/props.ma". - -include "Basic-1/r/props.ma". - -theorem drop_skip_bind: - \forall (h: nat).(\forall (d: nat).(\forall (c: C).(\forall (e: C).((drop h -d c e) \to (\forall (b: B).(\forall (u: T).(drop h (S d) (CHead c (Bind b) -(lift h d u)) (CHead e (Bind b) u)))))))) -\def - \lambda (h: nat).(\lambda (d: nat).(\lambda (c: C).(\lambda (e: C).(\lambda -(H: (drop h d c e)).(\lambda (b: B).(\lambda (u: T).(eq_ind nat (r (Bind b) -d) (\lambda (n: nat).(drop h (S d) (CHead c (Bind b) (lift h n u)) (CHead e -(Bind b) u))) (drop_skip (Bind b) h d c e H u) d (refl_equal nat d)))))))). -(* COMMENTS -Initial nodes: 95 -END *) - -theorem drop_skip_flat: - \forall (h: nat).(\forall (d: nat).(\forall (c: C).(\forall (e: C).((drop h -(S d) c e) \to (\forall (f: F).(\forall (u: T).(drop h (S d) (CHead c (Flat -f) (lift h (S d) u)) (CHead e (Flat f) u)))))))) -\def - \lambda (h: nat).(\lambda (d: nat).(\lambda (c: C).(\lambda (e: C).(\lambda -(H: (drop h (S d) c e)).(\lambda (f: F).(\lambda (u: T).(eq_ind nat (r (Flat -f) d) (\lambda (n: nat).(drop h (S d) (CHead c (Flat f) (lift h n u)) (CHead -e (Flat f) u))) (drop_skip (Flat f) h d c e H u) (S d) (refl_equal nat (S -d))))))))). -(* COMMENTS -Initial nodes: 101 -END *) - -theorem drop_S: - \forall (b: B).(\forall (c: C).(\forall (e: C).(\forall (u: T).(\forall (h: -nat).((drop h O c (CHead e (Bind b) u)) \to (drop (S h) O c e)))))) -\def - \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e: -C).(\forall (u: T).(\forall (h: nat).((drop h O c0 (CHead e (Bind b) u)) \to -(drop (S h) O c0 e)))))) (\lambda (n: nat).(\lambda (e: C).(\lambda (u: -T).(\lambda (h: nat).(\lambda (H: (drop h O (CSort n) (CHead e (Bind b) -u))).(and3_ind (eq C (CHead e (Bind b) u) (CSort n)) (eq nat h O) (eq nat O -O) (drop (S h) O (CSort n) e) (\lambda (H0: (eq C (CHead e (Bind b) u) (CSort -n))).(\lambda (H1: (eq nat h O)).(\lambda (_: (eq nat O O)).(eq_ind_r nat O -(\lambda (n0: nat).(drop (S n0) O (CSort n) e)) (let H3 \def (eq_ind C (CHead -e (Bind b) u) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) -with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I -(CSort n) H0) in (False_ind (drop (S O) O (CSort n) e) H3)) h H1)))) -(drop_gen_sort n h O (CHead e (Bind b) u) H))))))) (\lambda (c0: C).(\lambda -(H: ((\forall (e: C).(\forall (u: T).(\forall (h: nat).((drop h O c0 (CHead e -(Bind b) u)) \to (drop (S h) O c0 e))))))).(\lambda (k: K).(\lambda (t: -T).(\lambda (e: C).(\lambda (u: T).(\lambda (h: nat).(nat_ind (\lambda (n: -nat).((drop n O (CHead c0 k t) (CHead e (Bind b) u)) \to (drop (S n) O (CHead -c0 k t) e))) (\lambda (H0: (drop O O (CHead c0 k t) (CHead e (Bind b) -u))).(let H1 \def (f_equal C C (\lambda (e0: C).(match e0 in C return -(\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c1 _ _) -\Rightarrow c1])) (CHead c0 k t) (CHead e (Bind b) u) (drop_gen_refl (CHead -c0 k t) (CHead e (Bind b) u) H0)) in ((let H2 \def (f_equal C K (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | -(CHead _ k0 _) \Rightarrow k0])) (CHead c0 k t) (CHead e (Bind b) u) -(drop_gen_refl (CHead c0 k t) (CHead e (Bind b) u) H0)) in ((let H3 \def -(f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow t | (CHead _ _ t0) \Rightarrow t0])) (CHead c0 k t) -(CHead e (Bind b) u) (drop_gen_refl (CHead c0 k t) (CHead e (Bind b) u) H0)) -in (\lambda (H4: (eq K k (Bind b))).(\lambda (H5: (eq C c0 e)).(eq_ind C c0 -(\lambda (c1: C).(drop (S O) O (CHead c0 k t) c1)) (eq_ind_r K (Bind b) -(\lambda (k0: K).(drop (S O) O (CHead c0 k0 t) c0)) (drop_drop (Bind b) O c0 -c0 (drop_refl c0) t) k H4) e H5)))) H2)) H1))) (\lambda (n: nat).(\lambda (_: -(((drop n O (CHead c0 k t) (CHead e (Bind b) u)) \to (drop (S n) O (CHead c0 -k t) e)))).(\lambda (H1: (drop (S n) O (CHead c0 k t) (CHead e (Bind b) -u))).(drop_drop k (S n) c0 e (eq_ind_r nat (S (r k n)) (\lambda (n0: -nat).(drop n0 O c0 e)) (H e u (r k n) (drop_gen_drop k c0 (CHead e (Bind b) -u) t n H1)) (r k (S n)) (r_S k n)) t)))) h)))))))) c)). -(* COMMENTS -Initial nodes: 807 -END *) - -theorem drop_ctail: - \forall (c1: C).(\forall (c2: C).(\forall (d: nat).(\forall (h: nat).((drop -h d c1 c2) \to (\forall (k: K).(\forall (u: T).(drop h d (CTail k u c1) -(CTail k u c2)))))))) -\def - \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (d: -nat).(\forall (h: nat).((drop h d c c2) \to (\forall (k: K).(\forall (u: -T).(drop h d (CTail k u c) (CTail k u c2))))))))) (\lambda (n: nat).(\lambda -(c2: C).(\lambda (d: nat).(\lambda (h: nat).(\lambda (H: (drop h d (CSort n) -c2)).(\lambda (k: K).(\lambda (u: T).(and3_ind (eq C c2 (CSort n)) (eq nat h -O) (eq nat d O) (drop h d (CTail k u (CSort n)) (CTail k u c2)) (\lambda (H0: -(eq C c2 (CSort n))).(\lambda (H1: (eq nat h O)).(\lambda (H2: (eq nat d -O)).(eq_ind_r nat O (\lambda (n0: nat).(drop n0 d (CTail k u (CSort n)) -(CTail k u c2))) (eq_ind_r nat O (\lambda (n0: nat).(drop O n0 (CTail k u -(CSort n)) (CTail k u c2))) (eq_ind_r C (CSort n) (\lambda (c: C).(drop O O -(CTail k u (CSort n)) (CTail k u c))) (drop_refl (CTail k u (CSort n))) c2 -H0) d H2) h H1)))) (drop_gen_sort n h d c2 H))))))))) (\lambda (c2: -C).(\lambda (IHc: ((\forall (c3: C).(\forall (d: nat).(\forall (h: -nat).((drop h d c2 c3) \to (\forall (k: K).(\forall (u: T).(drop h d (CTail k -u c2) (CTail k u c3)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c3: -C).(\lambda (d: nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n -(CHead c2 k t) c3) \to (\forall (k0: K).(\forall (u: T).(drop h n (CTail k0 u -(CHead c2 k t)) (CTail k0 u c3))))))) (\lambda (h: nat).(nat_ind (\lambda (n: -nat).((drop n O (CHead c2 k t) c3) \to (\forall (k0: K).(\forall (u: T).(drop -n O (CTail k0 u (CHead c2 k t)) (CTail k0 u c3)))))) (\lambda (H: (drop O O -(CHead c2 k t) c3)).(\lambda (k0: K).(\lambda (u: T).(eq_ind C (CHead c2 k t) -(\lambda (c: C).(drop O O (CTail k0 u (CHead c2 k t)) (CTail k0 u c))) -(drop_refl (CTail k0 u (CHead c2 k t))) c3 (drop_gen_refl (CHead c2 k t) c3 -H))))) (\lambda (n: nat).(\lambda (_: (((drop n O (CHead c2 k t) c3) \to -(\forall (k0: K).(\forall (u: T).(drop n O (CTail k0 u (CHead c2 k t)) (CTail -k0 u c3))))))).(\lambda (H0: (drop (S n) O (CHead c2 k t) c3)).(\lambda (k0: -K).(\lambda (u: T).(drop_drop k n (CTail k0 u c2) (CTail k0 u c3) (IHc c3 O -(r k n) (drop_gen_drop k c2 c3 t n H0) k0 u) t)))))) h)) (\lambda (n: -nat).(\lambda (H: ((\forall (h: nat).((drop h n (CHead c2 k t) c3) \to -(\forall (k0: K).(\forall (u: T).(drop h n (CTail k0 u (CHead c2 k t)) (CTail -k0 u c3)))))))).(\lambda (h: nat).(\lambda (H0: (drop h (S n) (CHead c2 k t) -c3)).(\lambda (k0: K).(\lambda (u: T).(ex3_2_ind C T (\lambda (e: C).(\lambda -(v: T).(eq C c3 (CHead e k v)))) (\lambda (_: C).(\lambda (v: T).(eq T t -(lift h (r k n) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r k n) c2 e))) -(drop h (S n) (CTail k0 u (CHead c2 k t)) (CTail k0 u c3)) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (H1: (eq C c3 (CHead x0 k x1))).(\lambda (H2: -(eq T t (lift h (r k n) x1))).(\lambda (H3: (drop h (r k n) c2 x0)).(let H4 -\def (eq_ind C c3 (\lambda (c: C).(\forall (h0: nat).((drop h0 n (CHead c2 k -t) c) \to (\forall (k1: K).(\forall (u0: T).(drop h0 n (CTail k1 u0 (CHead c2 -k t)) (CTail k1 u0 c))))))) H (CHead x0 k x1) H1) in (eq_ind_r C (CHead x0 k -x1) (\lambda (c: C).(drop h (S n) (CTail k0 u (CHead c2 k t)) (CTail k0 u -c))) (let H5 \def (eq_ind T t (\lambda (t0: T).(\forall (h0: nat).((drop h0 n -(CHead c2 k t0) (CHead x0 k x1)) \to (\forall (k1: K).(\forall (u0: T).(drop -h0 n (CTail k1 u0 (CHead c2 k t0)) (CTail k1 u0 (CHead x0 k x1)))))))) H4 -(lift h (r k n) x1) H2) in (eq_ind_r T (lift h (r k n) x1) (\lambda (t0: -T).(drop h (S n) (CTail k0 u (CHead c2 k t0)) (CTail k0 u (CHead x0 k x1)))) -(drop_skip k h n (CTail k0 u c2) (CTail k0 u x0) (IHc x0 (r k n) h H3 k0 u) -x1) t H2)) c3 H1))))))) (drop_gen_skip_l c2 c3 t h n k H0)))))))) d))))))) -c1). -(* COMMENTS -Initial nodes: 1211 -END *) - -theorem drop_mono: - \forall (c: C).(\forall (x1: C).(\forall (d: nat).(\forall (h: nat).((drop h -d c x1) \to (\forall (x2: C).((drop h d c x2) \to (eq C x1 x2))))))) -\def - \lambda (c: C).(C_ind (\lambda (c0: C).(\forall (x1: C).(\forall (d: -nat).(\forall (h: nat).((drop h d c0 x1) \to (\forall (x2: C).((drop h d c0 -x2) \to (eq C x1 x2)))))))) (\lambda (n: nat).(\lambda (x1: C).(\lambda (d: -nat).(\lambda (h: nat).(\lambda (H: (drop h d (CSort n) x1)).(\lambda (x2: -C).(\lambda (H0: (drop h d (CSort n) x2)).(and3_ind (eq C x2 (CSort n)) (eq -nat h O) (eq nat d O) (eq C x1 x2) (\lambda (H1: (eq C x2 (CSort -n))).(\lambda (H2: (eq nat h O)).(\lambda (H3: (eq nat d O)).(and3_ind (eq C -x1 (CSort n)) (eq nat h O) (eq nat d O) (eq C x1 x2) (\lambda (H4: (eq C x1 -(CSort n))).(\lambda (H5: (eq nat h O)).(\lambda (H6: (eq nat d O)).(eq_ind_r -C (CSort n) (\lambda (c0: C).(eq C x1 c0)) (let H7 \def (eq_ind nat h -(\lambda (n0: nat).(eq nat n0 O)) H2 O H5) in (let H8 \def (eq_ind nat d -(\lambda (n0: nat).(eq nat n0 O)) H3 O H6) in (eq_ind_r C (CSort n) (\lambda -(c0: C).(eq C c0 (CSort n))) (refl_equal C (CSort n)) x1 H4))) x2 H1)))) -(drop_gen_sort n h d x1 H))))) (drop_gen_sort n h d x2 H0))))))))) (\lambda -(c0: C).(\lambda (H: ((\forall (x1: C).(\forall (d: nat).(\forall (h: -nat).((drop h d c0 x1) \to (\forall (x2: C).((drop h d c0 x2) \to (eq C x1 -x2))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (x1: C).(\lambda (d: -nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n (CHead c0 k t) -x1) \to (\forall (x2: C).((drop h n (CHead c0 k t) x2) \to (eq C x1 x2)))))) -(\lambda (h: nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c0 k t) x1) -\to (\forall (x2: C).((drop n O (CHead c0 k t) x2) \to (eq C x1 x2))))) -(\lambda (H0: (drop O O (CHead c0 k t) x1)).(\lambda (x2: C).(\lambda (H1: -(drop O O (CHead c0 k t) x2)).(eq_ind C (CHead c0 k t) (\lambda (c1: C).(eq C -x1 c1)) (eq_ind C (CHead c0 k t) (\lambda (c1: C).(eq C c1 (CHead c0 k t))) -(refl_equal C (CHead c0 k t)) x1 (drop_gen_refl (CHead c0 k t) x1 H0)) x2 -(drop_gen_refl (CHead c0 k t) x2 H1))))) (\lambda (n: nat).(\lambda (_: -(((drop n O (CHead c0 k t) x1) \to (\forall (x2: C).((drop n O (CHead c0 k t) -x2) \to (eq C x1 x2)))))).(\lambda (H1: (drop (S n) O (CHead c0 k t) -x1)).(\lambda (x2: C).(\lambda (H2: (drop (S n) O (CHead c0 k t) x2)).(H x1 O -(r k n) (drop_gen_drop k c0 x1 t n H1) x2 (drop_gen_drop k c0 x2 t n -H2))))))) h)) (\lambda (n: nat).(\lambda (H0: ((\forall (h: nat).((drop h n -(CHead c0 k t) x1) \to (\forall (x2: C).((drop h n (CHead c0 k t) x2) \to (eq -C x1 x2))))))).(\lambda (h: nat).(\lambda (H1: (drop h (S n) (CHead c0 k t) -x1)).(\lambda (x2: C).(\lambda (H2: (drop h (S n) (CHead c0 k t) -x2)).(ex3_2_ind C T (\lambda (e: C).(\lambda (v: T).(eq C x2 (CHead e k v)))) -(\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r k n) v)))) (\lambda (e: -C).(\lambda (_: T).(drop h (r k n) c0 e))) (eq C x1 x2) (\lambda (x0: -C).(\lambda (x3: T).(\lambda (H3: (eq C x2 (CHead x0 k x3))).(\lambda (H4: -(eq T t (lift h (r k n) x3))).(\lambda (H5: (drop h (r k n) c0 -x0)).(ex3_2_ind C T (\lambda (e: C).(\lambda (v: T).(eq C x1 (CHead e k v)))) -(\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r k n) v)))) (\lambda (e: -C).(\lambda (_: T).(drop h (r k n) c0 e))) (eq C x1 x2) (\lambda (x4: -C).(\lambda (x5: T).(\lambda (H6: (eq C x1 (CHead x4 k x5))).(\lambda (H7: -(eq T t (lift h (r k n) x5))).(\lambda (H8: (drop h (r k n) c0 x4)).(eq_ind_r -C (CHead x0 k x3) (\lambda (c1: C).(eq C x1 c1)) (let H9 \def (eq_ind C x1 -(\lambda (c1: C).(\forall (h0: nat).((drop h0 n (CHead c0 k t) c1) \to -(\forall (x6: C).((drop h0 n (CHead c0 k t) x6) \to (eq C c1 x6)))))) H0 -(CHead x4 k x5) H6) in (eq_ind_r C (CHead x4 k x5) (\lambda (c1: C).(eq C c1 -(CHead x0 k x3))) (let H10 \def (eq_ind T t (\lambda (t0: T).(\forall (h0: -nat).((drop h0 n (CHead c0 k t0) (CHead x4 k x5)) \to (\forall (x6: C).((drop -h0 n (CHead c0 k t0) x6) \to (eq C (CHead x4 k x5) x6)))))) H9 (lift h (r k -n) x5) H7) in (let H11 \def (eq_ind T t (\lambda (t0: T).(eq T t0 (lift h (r -k n) x3))) H4 (lift h (r k n) x5) H7) in (let H12 \def (eq_ind T x5 (\lambda -(t0: T).(\forall (h0: nat).((drop h0 n (CHead c0 k (lift h (r k n) t0)) -(CHead x4 k t0)) \to (\forall (x6: C).((drop h0 n (CHead c0 k (lift h (r k n) -t0)) x6) \to (eq C (CHead x4 k t0) x6)))))) H10 x3 (lift_inj x5 x3 h (r k n) -H11)) in (eq_ind_r T x3 (\lambda (t0: T).(eq C (CHead x4 k t0) (CHead x0 k -x3))) (f_equal3 C K T C CHead x4 x0 k k x3 x3 (sym_eq C x0 x4 (H x0 (r k n) h -H5 x4 H8)) (refl_equal K k) (refl_equal T x3)) x5 (lift_inj x5 x3 h (r k n) -H11))))) x1 H6)) x2 H3)))))) (drop_gen_skip_l c0 x1 t h n k H1))))))) -(drop_gen_skip_l c0 x2 t h n k H2)))))))) d))))))) c). -(* COMMENTS -Initial nodes: 1539 -END *) - -theorem drop_conf_lt: - \forall (k: K).(\forall (i: nat).(\forall (u: T).(\forall (c0: C).(\forall -(c: C).((drop i O c (CHead c0 k u)) \to (\forall (e: C).(\forall (h: -nat).(\forall (d: nat).((drop h (S (plus i d)) c e) \to (ex3_2 T C (\lambda -(v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda -(e0: C).(drop i O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop -h (r k d) c0 e0))))))))))))) -\def - \lambda (k: K).(\lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (u: -T).(\forall (c0: C).(\forall (c: C).((drop n O c (CHead c0 k u)) \to (\forall -(e: C).(\forall (h: nat).(\forall (d: nat).((drop h (S (plus n d)) c e) \to -(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) -(\lambda (v: T).(\lambda (e0: C).(drop n O e (CHead e0 k v)))) (\lambda (_: -T).(\lambda (e0: C).(drop h (r k d) c0 e0))))))))))))) (\lambda (u: -T).(\lambda (c0: C).(\lambda (c: C).(\lambda (H: (drop O O c (CHead c0 k -u))).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H0: (drop -h (S (plus O d)) c e)).(let H1 \def (eq_ind C c (\lambda (c1: C).(drop h (S -(plus O d)) c1 e)) H0 (CHead c0 k u) (drop_gen_refl c (CHead c0 k u) H)) in -(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 k v)))) -(\lambda (_: C).(\lambda (v: T).(eq T u (lift h (r k (plus O d)) v)))) -(\lambda (e0: C).(\lambda (_: T).(drop h (r k (plus O d)) c0 e0))) (ex3_2 T C -(\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: -T).(\lambda (e0: C).(drop O O e (CHead e0 k v)))) (\lambda (_: T).(\lambda -(e0: C).(drop h (r k d) c0 e0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda -(H2: (eq C e (CHead x0 k x1))).(\lambda (H3: (eq T u (lift h (r k (plus O d)) -x1))).(\lambda (H4: (drop h (r k (plus O d)) c0 x0)).(eq_ind_r C (CHead x0 k -x1) (\lambda (c1: C).(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift -h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop O O c1 (CHead e0 k -v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0))))) (eq_ind_r T -(lift h (r k (plus O d)) x1) (\lambda (t: T).(ex3_2 T C (\lambda (v: -T).(\lambda (_: C).(eq T t (lift h (r k d) v)))) (\lambda (v: T).(\lambda -(e0: C).(drop O O (CHead x0 k x1) (CHead e0 k v)))) (\lambda (_: T).(\lambda -(e0: C).(drop h (r k d) c0 e0))))) (ex3_2_intro T C (\lambda (v: T).(\lambda -(_: C).(eq T (lift h (r k (plus O d)) x1) (lift h (r k d) v)))) (\lambda (v: -T).(\lambda (e0: C).(drop O O (CHead x0 k x1) (CHead e0 k v)))) (\lambda (_: -T).(\lambda (e0: C).(drop h (r k d) c0 e0))) x1 x0 (refl_equal T (lift h (r k -d) x1)) (drop_refl (CHead x0 k x1)) H4) u H3) e H2)))))) (drop_gen_skip_l c0 -e u h (plus O d) k H1))))))))))) (\lambda (i0: nat).(\lambda (H: ((\forall -(u: T).(\forall (c0: C).(\forall (c: C).((drop i0 O c (CHead c0 k u)) \to -(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h (S (plus i0 d)) -c e) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) -v)))) (\lambda (v: T).(\lambda (e0: C).(drop i0 O e (CHead e0 k v)))) -(\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0)))))))))))))).(\lambda -(u: T).(\lambda (c0: C).(\lambda (c: C).(C_ind (\lambda (c1: C).((drop (S i0) -O c1 (CHead c0 k u)) \to (\forall (e: C).(\forall (h: nat).(\forall (d: -nat).((drop h (S (plus (S i0) d)) c1 e) \to (ex3_2 T C (\lambda (v: -T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda -(e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: -C).(drop h (r k d) c0 e0)))))))))) (\lambda (n: nat).(\lambda (_: (drop (S -i0) O (CSort n) (CHead c0 k u))).(\lambda (e: C).(\lambda (h: nat).(\lambda -(d: nat).(\lambda (H1: (drop h (S (plus (S i0) d)) (CSort n) e)).(and3_ind -(eq C e (CSort n)) (eq nat h O) (eq nat (S (plus (S i0) d)) O) (ex3_2 T C -(\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: -T).(\lambda (e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda (_: -T).(\lambda (e0: C).(drop h (r k d) c0 e0)))) (\lambda (_: (eq C e (CSort -n))).(\lambda (_: (eq nat h O)).(\lambda (H4: (eq nat (S (plus (S i0) d)) -O)).(let H5 \def (eq_ind nat (S (plus (S i0) d)) (\lambda (ee: nat).(match ee -in nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) -\Rightarrow True])) I O H4) in (False_ind (ex3_2 T C (\lambda (v: T).(\lambda -(_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop -(S i0) O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) -c0 e0)))) H5))))) (drop_gen_sort n h (S (plus (S i0) d)) e H1)))))))) -(\lambda (c1: C).(\lambda (H0: (((drop (S i0) O c1 (CHead c0 k u)) \to -(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h (S (plus (S i0) -d)) c1 e) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k -d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S i0) O e (CHead e0 k v)))) -(\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0))))))))))).(\lambda -(k0: K).(K_ind (\lambda (k1: K).(\forall (t: T).((drop (S i0) O (CHead c1 k1 -t) (CHead c0 k u)) \to (\forall (e: C).(\forall (h: nat).(\forall (d: -nat).((drop h (S (plus (S i0) d)) (CHead c1 k1 t) e) \to (ex3_2 T C (\lambda -(v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda -(e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: -C).(drop h (r k d) c0 e0))))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda -(H1: (drop (S i0) O (CHead c1 (Bind b) t) (CHead c0 k u))).(\lambda (e: -C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2: (drop h (S (plus (S i0) -d)) (CHead c1 (Bind b) t) e)).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: -T).(eq C e (CHead e0 (Bind b) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t -(lift h (r (Bind b) (plus (S i0) d)) v)))) (\lambda (e0: C).(\lambda (_: -T).(drop h (r (Bind b) (plus (S i0) d)) c1 e0))) (ex3_2 T C (\lambda (v: -T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda -(e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: -C).(drop h (r k d) c0 e0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H3: -(eq C e (CHead x0 (Bind b) x1))).(\lambda (_: (eq T t (lift h (r (Bind b) -(plus (S i0) d)) x1))).(\lambda (H5: (drop h (r (Bind b) (plus (S i0) d)) c1 -x0)).(eq_ind_r C (CHead x0 (Bind b) x1) (\lambda (c2: C).(ex3_2 T C (\lambda -(v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda -(e0: C).(drop (S i0) O c2 (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: -C).(drop h (r k d) c0 e0))))) (let H6 \def (H u c0 c1 (drop_gen_drop (Bind b) -c1 (CHead c0 k u) t i0 H1) x0 h d H5) in (ex3_2_ind T C (\lambda (v: -T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda -(e0: C).(drop i0 O x0 (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: -C).(drop h (r k d) c0 e0))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T -u (lift h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S i0) O -(CHead x0 (Bind b) x1) (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: -C).(drop h (r k d) c0 e0)))) (\lambda (x2: T).(\lambda (x3: C).(\lambda (H7: -(eq T u (lift h (r k d) x2))).(\lambda (H8: (drop i0 O x0 (CHead x3 k -x2))).(\lambda (H9: (drop h (r k d) c0 x3)).(ex3_2_intro T C (\lambda (v: -T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda -(e0: C).(drop (S i0) O (CHead x0 (Bind b) x1) (CHead e0 k v)))) (\lambda (_: -T).(\lambda (e0: C).(drop h (r k d) c0 e0))) x2 x3 H7 (drop_drop (Bind b) i0 -x0 (CHead x3 k x2) H8 x1) H9)))))) H6)) e H3)))))) (drop_gen_skip_l c1 e t h -(plus (S i0) d) (Bind b) H2))))))))) (\lambda (f: F).(\lambda (t: T).(\lambda -(H1: (drop (S i0) O (CHead c1 (Flat f) t) (CHead c0 k u))).(\lambda (e: -C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2: (drop h (S (plus (S i0) -d)) (CHead c1 (Flat f) t) e)).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: -T).(eq C e (CHead e0 (Flat f) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t -(lift h (r (Flat f) (plus (S i0) d)) v)))) (\lambda (e0: C).(\lambda (_: -T).(drop h (r (Flat f) (plus (S i0) d)) c1 e0))) (ex3_2 T C (\lambda (v: -T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda -(e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: -C).(drop h (r k d) c0 e0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H3: -(eq C e (CHead x0 (Flat f) x1))).(\lambda (_: (eq T t (lift h (r (Flat f) -(plus (S i0) d)) x1))).(\lambda (H5: (drop h (r (Flat f) (plus (S i0) d)) c1 -x0)).(eq_ind_r C (CHead x0 (Flat f) x1) (\lambda (c2: C).(ex3_2 T C (\lambda -(v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda -(e0: C).(drop (S i0) O c2 (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: -C).(drop h (r k d) c0 e0))))) (ex3_2_ind T C (\lambda (v: T).(\lambda (_: -C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S -i0) O x0 (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) -c0 e0))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) -v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S i0) O (CHead x0 (Flat f) x1) -(CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0)))) -(\lambda (x2: T).(\lambda (x3: C).(\lambda (H6: (eq T u (lift h (r k d) -x2))).(\lambda (H7: (drop (S i0) O x0 (CHead x3 k x2))).(\lambda (H8: (drop h -(r k d) c0 x3)).(ex3_2_intro T C (\lambda (v: T).(\lambda (_: C).(eq T u -(lift h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S i0) O (CHead -x0 (Flat f) x1) (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r -k d) c0 e0))) x2 x3 H6 (drop_drop (Flat f) i0 x0 (CHead x3 k x2) H7 x1) -H8)))))) (H0 (drop_gen_drop (Flat f) c1 (CHead c0 k u) t i0 H1) x0 h d H5)) e -H3)))))) (drop_gen_skip_l c1 e t h (plus (S i0) d) (Flat f) H2))))))))) -k0)))) c)))))) i)). -(* COMMENTS -Initial nodes: 2972 -END *) - -theorem drop_conf_ge: - \forall (i: nat).(\forall (a: C).(\forall (c: C).((drop i O c a) \to -(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to ((le -(plus d h) i) \to (drop (minus i h) O e a))))))))) -\def - \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (a: C).(\forall (c: -C).((drop n O c a) \to (\forall (e: C).(\forall (h: nat).(\forall (d: -nat).((drop h d c e) \to ((le (plus d h) n) \to (drop (minus n h) O e -a)))))))))) (\lambda (a: C).(\lambda (c: C).(\lambda (H: (drop O O c -a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H0: (drop h -d c e)).(\lambda (H1: (le (plus d h) O)).(let H2 \def (eq_ind C c (\lambda -(c0: C).(drop h d c0 e)) H0 a (drop_gen_refl c a H)) in (let H_y \def -(le_n_O_eq (plus d h) H1) in (land_ind (eq nat d O) (eq nat h O) (drop (minus -O h) O e a) (\lambda (H3: (eq nat d O)).(\lambda (H4: (eq nat h O)).(let H5 -\def (eq_ind nat d (\lambda (n: nat).(drop h n a e)) H2 O H3) in (let H6 \def -(eq_ind nat h (\lambda (n: nat).(drop n O a e)) H5 O H4) in (eq_ind_r nat O -(\lambda (n: nat).(drop (minus O n) O e a)) (eq_ind C a (\lambda (c0: -C).(drop (minus O O) O c0 a)) (drop_refl a) e (drop_gen_refl a e H6)) h -H4))))) (plus_O d h (sym_eq nat O (plus d h) H_y))))))))))))) (\lambda (i0: -nat).(\lambda (H: ((\forall (a: C).(\forall (c: C).((drop i0 O c a) \to -(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to ((le -(plus d h) i0) \to (drop (minus i0 h) O e a))))))))))).(\lambda (a: -C).(\lambda (c: C).(C_ind (\lambda (c0: C).((drop (S i0) O c0 a) \to (\forall -(e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c0 e) \to ((le (plus d -h) (S i0)) \to (drop (minus (S i0) h) O e a)))))))) (\lambda (n: -nat).(\lambda (H0: (drop (S i0) O (CSort n) a)).(\lambda (e: C).(\lambda (h: -nat).(\lambda (d: nat).(\lambda (H1: (drop h d (CSort n) e)).(\lambda (H2: -(le (plus d h) (S i0))).(and3_ind (eq C e (CSort n)) (eq nat h O) (eq nat d -O) (drop (minus (S i0) h) O e a) (\lambda (H3: (eq C e (CSort n))).(\lambda -(H4: (eq nat h O)).(\lambda (H5: (eq nat d O)).(and3_ind (eq C a (CSort n)) -(eq nat (S i0) O) (eq nat O O) (drop (minus (S i0) h) O e a) (\lambda (H6: -(eq C a (CSort n))).(\lambda (H7: (eq nat (S i0) O)).(\lambda (_: (eq nat O -O)).(let H9 \def (eq_ind nat d (\lambda (n0: nat).(le (plus n0 h) (S i0))) H2 -O H5) in (let H10 \def (eq_ind nat h (\lambda (n0: nat).(le (plus O n0) (S -i0))) H9 O H4) in (eq_ind_r nat O (\lambda (n0: nat).(drop (minus (S i0) n0) -O e a)) (eq_ind_r C (CSort n) (\lambda (c0: C).(drop (minus (S i0) O) O c0 -a)) (eq_ind_r C (CSort n) (\lambda (c0: C).(drop (minus (S i0) O) O (CSort n) -c0)) (let H11 \def (eq_ind nat (S i0) (\lambda (ee: nat).(match ee in nat -return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow -True])) I O H7) in (False_ind (drop (minus (S i0) O) O (CSort n) (CSort n)) -H11)) a H6) e H3) h H4)))))) (drop_gen_sort n (S i0) O a H0))))) -(drop_gen_sort n h d e H1))))))))) (\lambda (c0: C).(\lambda (H0: (((drop (S -i0) O c0 a) \to (\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h -d c0 e) \to ((le (plus d h) (S i0)) \to (drop (minus (S i0) h) O e -a))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t: T).((drop (S -i0) O (CHead c0 k0 t) a) \to (\forall (e: C).(\forall (h: nat).(\forall (d: -nat).((drop h d (CHead c0 k0 t) e) \to ((le (plus d h) (S i0)) \to (drop -(minus (S i0) h) O e a))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (H1: -(drop (S i0) O (CHead c0 (Bind b) t) a)).(\lambda (e: C).(\lambda (h: -nat).(\lambda (d: nat).(\lambda (H2: (drop h d (CHead c0 (Bind b) t) -e)).(\lambda (H3: (le (plus d h) (S i0))).(nat_ind (\lambda (n: nat).((drop h -n (CHead c0 (Bind b) t) e) \to ((le (plus n h) (S i0)) \to (drop (minus (S -i0) h) O e a)))) (\lambda (H4: (drop h O (CHead c0 (Bind b) t) e)).(\lambda -(H5: (le (plus O h) (S i0))).(nat_ind (\lambda (n: nat).((drop n O (CHead c0 -(Bind b) t) e) \to ((le (plus O n) (S i0)) \to (drop (minus (S i0) n) O e -a)))) (\lambda (H6: (drop O O (CHead c0 (Bind b) t) e)).(\lambda (_: (le -(plus O O) (S i0))).(eq_ind C (CHead c0 (Bind b) t) (\lambda (c1: C).(drop -(minus (S i0) O) O c1 a)) (drop_drop (Bind b) i0 c0 a (drop_gen_drop (Bind b) -c0 a t i0 H1) t) e (drop_gen_refl (CHead c0 (Bind b) t) e H6)))) (\lambda -(h0: nat).(\lambda (_: (((drop h0 O (CHead c0 (Bind b) t) e) \to ((le (plus O -h0) (S i0)) \to (drop (minus (S i0) h0) O e a))))).(\lambda (H6: (drop (S h0) -O (CHead c0 (Bind b) t) e)).(\lambda (H7: (le (plus O (S h0)) (S i0))).(H a -c0 (drop_gen_drop (Bind b) c0 a t i0 H1) e h0 O (drop_gen_drop (Bind b) c0 e -t h0 H6) (le_S_n (plus O h0) i0 H7)))))) h H4 H5))) (\lambda (d0: -nat).(\lambda (_: (((drop h d0 (CHead c0 (Bind b) t) e) \to ((le (plus d0 h) -(S i0)) \to (drop (minus (S i0) h) O e a))))).(\lambda (H4: (drop h (S d0) -(CHead c0 (Bind b) t) e)).(\lambda (H5: (le (plus (S d0) h) (S -i0))).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 (Bind -b) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r (Bind b) d0) -v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r (Bind b) d0) c0 e0))) (drop -(minus (S i0) h) O e a) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C -e (CHead x0 (Bind b) x1))).(\lambda (_: (eq T t (lift h (r (Bind b) d0) -x1))).(\lambda (H8: (drop h (r (Bind b) d0) c0 x0)).(eq_ind_r C (CHead x0 -(Bind b) x1) (\lambda (c1: C).(drop (minus (S i0) h) O c1 a)) (eq_ind nat (S -(minus i0 h)) (\lambda (n: nat).(drop n O (CHead x0 (Bind b) x1) a)) -(drop_drop (Bind b) (minus i0 h) x0 a (H a c0 (drop_gen_drop (Bind b) c0 a t -i0 H1) x0 h d0 H8 (le_S_n (plus d0 h) i0 H5)) x1) (minus (S i0) h) -(minus_Sn_m i0 h (le_trans_plus_r d0 h i0 (le_S_n (plus d0 h) i0 H5)))) e -H6)))))) (drop_gen_skip_l c0 e t h d0 (Bind b) H4)))))) d H2 H3))))))))) -(\lambda (f: F).(\lambda (t: T).(\lambda (H1: (drop (S i0) O (CHead c0 (Flat -f) t) a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2: -(drop h d (CHead c0 (Flat f) t) e)).(\lambda (H3: (le (plus d h) (S -i0))).(nat_ind (\lambda (n: nat).((drop h n (CHead c0 (Flat f) t) e) \to ((le -(plus n h) (S i0)) \to (drop (minus (S i0) h) O e a)))) (\lambda (H4: (drop h -O (CHead c0 (Flat f) t) e)).(\lambda (H5: (le (plus O h) (S i0))).(nat_ind -(\lambda (n: nat).((drop n O (CHead c0 (Flat f) t) e) \to ((le (plus O n) (S -i0)) \to (drop (minus (S i0) n) O e a)))) (\lambda (H6: (drop O O (CHead c0 -(Flat f) t) e)).(\lambda (_: (le (plus O O) (S i0))).(eq_ind C (CHead c0 -(Flat f) t) (\lambda (c1: C).(drop (minus (S i0) O) O c1 a)) (drop_drop (Flat -f) i0 c0 a (drop_gen_drop (Flat f) c0 a t i0 H1) t) e (drop_gen_refl (CHead -c0 (Flat f) t) e H6)))) (\lambda (h0: nat).(\lambda (_: (((drop h0 O (CHead -c0 (Flat f) t) e) \to ((le (plus O h0) (S i0)) \to (drop (minus (S i0) h0) O -e a))))).(\lambda (H6: (drop (S h0) O (CHead c0 (Flat f) t) e)).(\lambda (H7: -(le (plus O (S h0)) (S i0))).(H0 (drop_gen_drop (Flat f) c0 a t i0 H1) e (S -h0) O (drop_gen_drop (Flat f) c0 e t h0 H6) H7))))) h H4 H5))) (\lambda (d0: -nat).(\lambda (_: (((drop h d0 (CHead c0 (Flat f) t) e) \to ((le (plus d0 h) -(S i0)) \to (drop (minus (S i0) h) O e a))))).(\lambda (H4: (drop h (S d0) -(CHead c0 (Flat f) t) e)).(\lambda (H5: (le (plus (S d0) h) (S -i0))).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 (Flat -f) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r (Flat f) d0) -v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r (Flat f) d0) c0 e0))) (drop -(minus (S i0) h) O e a) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C -e (CHead x0 (Flat f) x1))).(\lambda (_: (eq T t (lift h (r (Flat f) d0) -x1))).(\lambda (H8: (drop h (r (Flat f) d0) c0 x0)).(eq_ind_r C (CHead x0 -(Flat f) x1) (\lambda (c1: C).(drop (minus (S i0) h) O c1 a)) (let H9 \def -(eq_ind_r nat (minus (S i0) h) (\lambda (n: nat).(drop n O x0 a)) (H0 -(drop_gen_drop (Flat f) c0 a t i0 H1) x0 h (S d0) H8 H5) (S (minus i0 h)) -(minus_Sn_m i0 h (le_trans_plus_r d0 h i0 (le_S_n (plus d0 h) i0 H5)))) in -(eq_ind nat (S (minus i0 h)) (\lambda (n: nat).(drop n O (CHead x0 (Flat f) -x1) a)) (drop_drop (Flat f) (minus i0 h) x0 a H9 x1) (minus (S i0) h) -(minus_Sn_m i0 h (le_trans_plus_r d0 h i0 (le_S_n (plus d0 h) i0 H5))))) e -H6)))))) (drop_gen_skip_l c0 e t h d0 (Flat f) H4)))))) d H2 H3))))))))) -k)))) c))))) i). -(* COMMENTS -Initial nodes: 2726 -END *) - -theorem drop_conf_rev: - \forall (j: nat).(\forall (e1: C).(\forall (e2: C).((drop j O e1 e2) \to -(\forall (c2: C).(\forall (i: nat).((drop i O c2 e2) \to (ex2 C (\lambda (c1: -C).(drop j O c1 c2)) (\lambda (c1: C).(drop i j c1 e1))))))))) -\def - \lambda (j: nat).(nat_ind (\lambda (n: nat).(\forall (e1: C).(\forall (e2: -C).((drop n O e1 e2) \to (\forall (c2: C).(\forall (i: nat).((drop i O c2 e2) -\to (ex2 C (\lambda (c1: C).(drop n O c1 c2)) (\lambda (c1: C).(drop i n c1 -e1)))))))))) (\lambda (e1: C).(\lambda (e2: C).(\lambda (H: (drop O O e1 -e2)).(\lambda (c2: C).(\lambda (i: nat).(\lambda (H0: (drop i O c2 e2)).(let -H1 \def (eq_ind_r C e2 (\lambda (c: C).(drop i O c2 c)) H0 e1 (drop_gen_refl -e1 e2 H)) in (ex_intro2 C (\lambda (c1: C).(drop O O c1 c2)) (\lambda (c1: -C).(drop i O c1 e1)) c2 (drop_refl c2) H1)))))))) (\lambda (j0: nat).(\lambda -(IHj: ((\forall (e1: C).(\forall (e2: C).((drop j0 O e1 e2) \to (\forall (c2: -C).(\forall (i: nat).((drop i O c2 e2) \to (ex2 C (\lambda (c1: C).(drop j0 O -c1 c2)) (\lambda (c1: C).(drop i j0 c1 e1))))))))))).(\lambda (e1: C).(C_ind -(\lambda (c: C).(\forall (e2: C).((drop (S j0) O c e2) \to (\forall (c2: -C).(\forall (i: nat).((drop i O c2 e2) \to (ex2 C (\lambda (c1: C).(drop (S -j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 c))))))))) (\lambda (n: -nat).(\lambda (e2: C).(\lambda (H: (drop (S j0) O (CSort n) e2)).(\lambda -(c2: C).(\lambda (i: nat).(\lambda (H0: (drop i O c2 e2)).(and3_ind (eq C e2 -(CSort n)) (eq nat (S j0) O) (eq nat O O) (ex2 C (\lambda (c1: C).(drop (S -j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 (CSort n)))) (\lambda (H1: -(eq C e2 (CSort n))).(\lambda (H2: (eq nat (S j0) O)).(\lambda (_: (eq nat O -O)).(let H4 \def (eq_ind C e2 (\lambda (c: C).(drop i O c2 c)) H0 (CSort n) -H1) in (let H5 \def (eq_ind nat (S j0) (\lambda (ee: nat).(match ee in nat -return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow -True])) I O H2) in (False_ind (ex2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) -(\lambda (c1: C).(drop i (S j0) c1 (CSort n)))) H5)))))) (drop_gen_sort n (S -j0) O e2 H)))))))) (\lambda (e2: C).(\lambda (IHe1: ((\forall (e3: C).((drop -(S j0) O e2 e3) \to (\forall (c2: C).(\forall (i: nat).((drop i O c2 e3) \to -(ex2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S -j0) c1 e2)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e3: C).(\lambda -(H: (drop (S j0) O (CHead e2 k t) e3)).(\lambda (c2: C).(\lambda (i: -nat).(\lambda (H0: (drop i O c2 e3)).(K_ind (\lambda (k0: K).((drop (r k0 j0) -O e2 e3) \to (ex2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: -C).(drop i (S j0) c1 (CHead e2 k0 t)))))) (\lambda (b: B).(\lambda (H1: (drop -(r (Bind b) j0) O e2 e3)).(let H_x \def (IHj e2 e3 H1 c2 i H0) in (let H2 -\def H_x in (ex2_ind C (\lambda (c1: C).(drop j0 O c1 c2)) (\lambda (c1: -C).(drop i j0 c1 e2)) (ex2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda -(c1: C).(drop i (S j0) c1 (CHead e2 (Bind b) t)))) (\lambda (x: C).(\lambda -(H3: (drop j0 O x c2)).(\lambda (H4: (drop i j0 x e2)).(ex_intro2 C (\lambda -(c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 (CHead e2 -(Bind b) t))) (CHead x (Bind b) (lift i (r (Bind b) j0) t)) (drop_drop (Bind -b) j0 x c2 H3 (lift i (r (Bind b) j0) t)) (drop_skip (Bind b) i j0 x e2 H4 -t))))) H2))))) (\lambda (f: F).(\lambda (H1: (drop (r (Flat f) j0) O e2 -e3)).(let H_x \def (IHe1 e3 H1 c2 i H0) in (let H2 \def H_x in (ex2_ind C -(\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 -e2)) (ex2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i -(S j0) c1 (CHead e2 (Flat f) t)))) (\lambda (x: C).(\lambda (H3: (drop (S j0) -O x c2)).(\lambda (H4: (drop i (S j0) x e2)).(ex_intro2 C (\lambda (c1: -C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 (CHead e2 (Flat -f) t))) (CHead x (Flat f) (lift i (r (Flat f) j0) t)) (drop_drop (Flat f) j0 -x c2 H3 (lift i (r (Flat f) j0) t)) (drop_skip (Flat f) i j0 x e2 H4 t))))) -H2))))) k (drop_gen_drop k e2 e3 t j0 H))))))))))) e1)))) j). -(* COMMENTS -Initial nodes: 1154 -END *) - -theorem drop_trans_le: - \forall (i: nat).(\forall (d: nat).((le i d) \to (\forall (c1: C).(\forall -(c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((drop i O -c2 e2) \to (ex2 C (\lambda (e1: C).(drop i O c1 e1)) (\lambda (e1: C).(drop h -(minus d i) e1 e2))))))))))) -\def - \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (d: nat).((le n d) \to -(\forall (c1: C).(\forall (c2: C).(\forall (h: nat).((drop h d c1 c2) \to -(\forall (e2: C).((drop n O c2 e2) \to (ex2 C (\lambda (e1: C).(drop n O c1 -e1)) (\lambda (e1: C).(drop h (minus d n) e1 e2)))))))))))) (\lambda (d: -nat).(\lambda (_: (le O d)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (h: -nat).(\lambda (H0: (drop h d c1 c2)).(\lambda (e2: C).(\lambda (H1: (drop O O -c2 e2)).(let H2 \def (eq_ind C c2 (\lambda (c: C).(drop h d c1 c)) H0 e2 -(drop_gen_refl c2 e2 H1)) in (eq_ind nat d (\lambda (n: nat).(ex2 C (\lambda -(e1: C).(drop O O c1 e1)) (\lambda (e1: C).(drop h n e1 e2)))) (ex_intro2 C -(\lambda (e1: C).(drop O O c1 e1)) (\lambda (e1: C).(drop h d e1 e2)) c1 -(drop_refl c1) H2) (minus d O) (minus_n_O d))))))))))) (\lambda (i0: -nat).(\lambda (IHi: ((\forall (d: nat).((le i0 d) \to (\forall (c1: -C).(\forall (c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: -C).((drop i0 O c2 e2) \to (ex2 C (\lambda (e1: C).(drop i0 O c1 e1)) (\lambda -(e1: C).(drop h (minus d i0) e1 e2))))))))))))).(\lambda (d: nat).(nat_ind -(\lambda (n: nat).((le (S i0) n) \to (\forall (c1: C).(\forall (c2: -C).(\forall (h: nat).((drop h n c1 c2) \to (\forall (e2: C).((drop (S i0) O -c2 e2) \to (ex2 C (\lambda (e1: C).(drop (S i0) O c1 e1)) (\lambda (e1: -C).(drop h (minus n (S i0)) e1 e2))))))))))) (\lambda (H: (le (S i0) -O)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (h: nat).(\lambda (_: (drop h -O c1 c2)).(\lambda (e2: C).(\lambda (_: (drop (S i0) O c2 e2)).(ex2_ind nat -(\lambda (n: nat).(eq nat O (S n))) (\lambda (n: nat).(le i0 n)) (ex2 C -(\lambda (e1: C).(drop (S i0) O c1 e1)) (\lambda (e1: C).(drop h (minus O (S -i0)) e1 e2))) (\lambda (x: nat).(\lambda (H2: (eq nat O (S x))).(\lambda (_: -(le i0 x)).(let H4 \def (eq_ind nat O (\lambda (ee: nat).(match ee in nat -return (\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow -False])) I (S x) H2) in (False_ind (ex2 C (\lambda (e1: C).(drop (S i0) O c1 -e1)) (\lambda (e1: C).(drop h (minus O (S i0)) e1 e2))) H4))))) (le_gen_S i0 -O H))))))))) (\lambda (d0: nat).(\lambda (_: (((le (S i0) d0) \to (\forall -(c1: C).(\forall (c2: C).(\forall (h: nat).((drop h d0 c1 c2) \to (\forall -(e2: C).((drop (S i0) O c2 e2) \to (ex2 C (\lambda (e1: C).(drop (S i0) O c1 -e1)) (\lambda (e1: C).(drop h (minus d0 (S i0)) e1 e2)))))))))))).(\lambda -(H: (le (S i0) (S d0))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: -C).(\forall (h: nat).((drop h (S d0) c c2) \to (\forall (e2: C).((drop (S i0) -O c2 e2) \to (ex2 C (\lambda (e1: C).(drop (S i0) O c e1)) (\lambda (e1: -C).(drop h (minus (S d0) (S i0)) e1 e2))))))))) (\lambda (n: nat).(\lambda -(c2: C).(\lambda (h: nat).(\lambda (H0: (drop h (S d0) (CSort n) -c2)).(\lambda (e2: C).(\lambda (H1: (drop (S i0) O c2 e2)).(and3_ind (eq C c2 -(CSort n)) (eq nat h O) (eq nat (S d0) O) (ex2 C (\lambda (e1: C).(drop (S -i0) O (CSort n) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2))) -(\lambda (H2: (eq C c2 (CSort n))).(\lambda (_: (eq nat h O)).(\lambda (_: -(eq nat (S d0) O)).(let H5 \def (eq_ind C c2 (\lambda (c: C).(drop (S i0) O c -e2)) H1 (CSort n) H2) in (and3_ind (eq C e2 (CSort n)) (eq nat (S i0) O) (eq -nat O O) (ex2 C (\lambda (e1: C).(drop (S i0) O (CSort n) e1)) (\lambda (e1: -C).(drop h (minus (S d0) (S i0)) e1 e2))) (\lambda (H6: (eq C e2 (CSort -n))).(\lambda (H7: (eq nat (S i0) O)).(\lambda (_: (eq nat O O)).(eq_ind_r C -(CSort n) (\lambda (c: C).(ex2 C (\lambda (e1: C).(drop (S i0) O (CSort n) -e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 c)))) (let H9 \def -(eq_ind nat (S i0) (\lambda (ee: nat).(match ee in nat return (\lambda (_: -nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H7) in -(False_ind (ex2 C (\lambda (e1: C).(drop (S i0) O (CSort n) e1)) (\lambda -(e1: C).(drop h (minus (S d0) (S i0)) e1 (CSort n)))) H9)) e2 H6)))) -(drop_gen_sort n (S i0) O e2 H5)))))) (drop_gen_sort n h (S d0) c2 H0)))))))) -(\lambda (c2: C).(\lambda (IHc: ((\forall (c3: C).(\forall (h: nat).((drop h -(S d0) c2 c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to (ex2 C (\lambda -(e1: C).(drop (S i0) O c2 e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) -e1 e2)))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t: -T).(\forall (c3: C).(\forall (h: nat).((drop h (S d0) (CHead c2 k0 t) c3) \to -(\forall (e2: C).((drop (S i0) O c3 e2) \to (ex2 C (\lambda (e1: C).(drop (S -i0) O (CHead c2 k0 t) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 -e2)))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (c3: C).(\lambda (h: -nat).(\lambda (H0: (drop h (S d0) (CHead c2 (Bind b) t) c3)).(\lambda (e2: -C).(\lambda (H1: (drop (S i0) O c3 e2)).(ex3_2_ind C T (\lambda (e: -C).(\lambda (v: T).(eq C c3 (CHead e (Bind b) v)))) (\lambda (_: C).(\lambda -(v: T).(eq T t (lift h (r (Bind b) d0) v)))) (\lambda (e: C).(\lambda (_: -T).(drop h (r (Bind b) d0) c2 e))) (ex2 C (\lambda (e1: C).(drop (S i0) O -(CHead c2 (Bind b) t) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 -e2))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H2: (eq C c3 (CHead x0 -(Bind b) x1))).(\lambda (H3: (eq T t (lift h (r (Bind b) d0) x1))).(\lambda -(H4: (drop h (r (Bind b) d0) c2 x0)).(let H5 \def (eq_ind C c3 (\lambda (c: -C).(drop (S i0) O c e2)) H1 (CHead x0 (Bind b) x1) H2) in (eq_ind_r T (lift h -(r (Bind b) d0) x1) (\lambda (t0: T).(ex2 C (\lambda (e1: C).(drop (S i0) O -(CHead c2 (Bind b) t0) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 -e2)))) (ex2_ind C (\lambda (e1: C).(drop i0 O c2 e1)) (\lambda (e1: C).(drop -h (minus d0 i0) e1 e2)) (ex2 C (\lambda (e1: C).(drop (S i0) O (CHead c2 -(Bind b) (lift h (r (Bind b) d0) x1)) e1)) (\lambda (e1: C).(drop h (minus (S -d0) (S i0)) e1 e2))) (\lambda (x: C).(\lambda (H6: (drop i0 O c2 x)).(\lambda -(H7: (drop h (minus d0 i0) x e2)).(ex_intro2 C (\lambda (e1: C).(drop (S i0) -O (CHead c2 (Bind b) (lift h (r (Bind b) d0) x1)) e1)) (\lambda (e1: C).(drop -h (minus (S d0) (S i0)) e1 e2)) x (drop_drop (Bind b) i0 c2 x H6 (lift h (r -(Bind b) d0) x1)) H7)))) (IHi d0 (le_S_n i0 d0 H) c2 x0 h H4 e2 -(drop_gen_drop (Bind b) x0 e2 x1 i0 H5))) t H3))))))) (drop_gen_skip_l c2 c3 -t h d0 (Bind b) H0))))))))) (\lambda (f: F).(\lambda (t: T).(\lambda (c3: -C).(\lambda (h: nat).(\lambda (H0: (drop h (S d0) (CHead c2 (Flat f) t) -c3)).(\lambda (e2: C).(\lambda (H1: (drop (S i0) O c3 e2)).(ex3_2_ind C T -(\lambda (e: C).(\lambda (v: T).(eq C c3 (CHead e (Flat f) v)))) (\lambda (_: -C).(\lambda (v: T).(eq T t (lift h (r (Flat f) d0) v)))) (\lambda (e: -C).(\lambda (_: T).(drop h (r (Flat f) d0) c2 e))) (ex2 C (\lambda (e1: -C).(drop (S i0) O (CHead c2 (Flat f) t) e1)) (\lambda (e1: C).(drop h (minus -(S d0) (S i0)) e1 e2))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H2: (eq C -c3 (CHead x0 (Flat f) x1))).(\lambda (H3: (eq T t (lift h (r (Flat f) d0) -x1))).(\lambda (H4: (drop h (r (Flat f) d0) c2 x0)).(let H5 \def (eq_ind C c3 -(\lambda (c: C).(drop (S i0) O c e2)) H1 (CHead x0 (Flat f) x1) H2) in -(eq_ind_r T (lift h (r (Flat f) d0) x1) (\lambda (t0: T).(ex2 C (\lambda (e1: -C).(drop (S i0) O (CHead c2 (Flat f) t0) e1)) (\lambda (e1: C).(drop h (minus -(S d0) (S i0)) e1 e2)))) (ex2_ind C (\lambda (e1: C).(drop (S i0) O c2 e1)) -(\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2)) (ex2 C (\lambda (e1: -C).(drop (S i0) O (CHead c2 (Flat f) (lift h (r (Flat f) d0) x1)) e1)) -(\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2))) (\lambda (x: -C).(\lambda (H6: (drop (S i0) O c2 x)).(\lambda (H7: (drop h (minus (S d0) (S -i0)) x e2)).(ex_intro2 C (\lambda (e1: C).(drop (S i0) O (CHead c2 (Flat f) -(lift h (r (Flat f) d0) x1)) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S -i0)) e1 e2)) x (drop_drop (Flat f) i0 c2 x H6 (lift h (r (Flat f) d0) x1)) -H7)))) (IHc x0 h H4 e2 (drop_gen_drop (Flat f) x0 e2 x1 i0 H5))) t H3))))))) -(drop_gen_skip_l c2 c3 t h d0 (Flat f) H0))))))))) k)))) c1))))) d)))) i). -(* COMMENTS -Initial nodes: 2453 -END *) - -theorem drop_trans_ge: - \forall (i: nat).(\forall (c1: C).(\forall (c2: C).(\forall (d: -nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((drop i O c2 -e2) \to ((le d i) \to (drop (plus i h) O c1 e2))))))))) -\def - \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (c2: -C).(\forall (d: nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: -C).((drop n O c2 e2) \to ((le d n) \to (drop (plus n h) O c1 e2)))))))))) -(\lambda (c1: C).(\lambda (c2: C).(\lambda (d: nat).(\lambda (h: -nat).(\lambda (H: (drop h d c1 c2)).(\lambda (e2: C).(\lambda (H0: (drop O O -c2 e2)).(\lambda (H1: (le d O)).(eq_ind C c2 (\lambda (c: C).(drop (plus O h) -O c1 c)) (let H_y \def (le_n_O_eq d H1) in (let H2 \def (eq_ind_r nat d -(\lambda (n: nat).(drop h n c1 c2)) H O H_y) in H2)) e2 (drop_gen_refl c2 e2 -H0)))))))))) (\lambda (i0: nat).(\lambda (IHi: ((\forall (c1: C).(\forall -(c2: C).(\forall (d: nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall -(e2: C).((drop i0 O c2 e2) \to ((le d i0) \to (drop (plus i0 h) O c1 -e2))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: -C).(\forall (d: nat).(\forall (h: nat).((drop h d c c2) \to (\forall (e2: -C).((drop (S i0) O c2 e2) \to ((le d (S i0)) \to (drop (plus (S i0) h) O c -e2))))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (d: nat).(\lambda (h: -nat).(\lambda (H: (drop h d (CSort n) c2)).(\lambda (e2: C).(\lambda (H0: -(drop (S i0) O c2 e2)).(\lambda (H1: (le d (S i0))).(and3_ind (eq C c2 (CSort -n)) (eq nat h O) (eq nat d O) (drop (S (plus i0 h)) O (CSort n) e2) (\lambda -(H2: (eq C c2 (CSort n))).(\lambda (H3: (eq nat h O)).(\lambda (H4: (eq nat d -O)).(eq_ind_r nat O (\lambda (n0: nat).(drop (S (plus i0 n0)) O (CSort n) -e2)) (let H5 \def (eq_ind nat d (\lambda (n0: nat).(le n0 (S i0))) H1 O H4) -in (let H6 \def (eq_ind C c2 (\lambda (c: C).(drop (S i0) O c e2)) H0 (CSort -n) H2) in (and3_ind (eq C e2 (CSort n)) (eq nat (S i0) O) (eq nat O O) (drop -(S (plus i0 O)) O (CSort n) e2) (\lambda (H7: (eq C e2 (CSort n))).(\lambda -(H8: (eq nat (S i0) O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n) -(\lambda (c: C).(drop (S (plus i0 O)) O (CSort n) c)) (let H10 \def (eq_ind -nat (S i0) (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) -with [O \Rightarrow False | (S _) \Rightarrow True])) I O H8) in (False_ind -(drop (S (plus i0 O)) O (CSort n) (CSort n)) H10)) e2 H7)))) (drop_gen_sort n -(S i0) O e2 H6)))) h H3)))) (drop_gen_sort n h d c2 H)))))))))) (\lambda (c2: -C).(\lambda (IHc: ((\forall (c3: C).(\forall (d: nat).(\forall (h: -nat).((drop h d c2 c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to ((le d -(S i0)) \to (drop (S (plus i0 h)) O c2 e2)))))))))).(\lambda (k: K).(\lambda -(t: T).(\lambda (c3: C).(\lambda (d: nat).(nat_ind (\lambda (n: nat).(\forall -(h: nat).((drop h n (CHead c2 k t) c3) \to (\forall (e2: C).((drop (S i0) O -c3 e2) \to ((le n (S i0)) \to (drop (S (plus i0 h)) O (CHead c2 k t) -e2))))))) (\lambda (h: nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c2 k -t) c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to ((le O (S i0)) \to -(drop (S (plus i0 n)) O (CHead c2 k t) e2)))))) (\lambda (H: (drop O O (CHead -c2 k t) c3)).(\lambda (e2: C).(\lambda (H0: (drop (S i0) O c3 e2)).(\lambda -(_: (le O (S i0))).(let H2 \def (eq_ind_r C c3 (\lambda (c: C).(drop (S i0) O -c e2)) H0 (CHead c2 k t) (drop_gen_refl (CHead c2 k t) c3 H)) in (eq_ind nat -i0 (\lambda (n: nat).(drop (S n) O (CHead c2 k t) e2)) (drop_drop k i0 c2 e2 -(drop_gen_drop k c2 e2 t i0 H2) t) (plus i0 O) (plus_n_O i0))))))) (\lambda -(n: nat).(\lambda (_: (((drop n O (CHead c2 k t) c3) \to (\forall (e2: -C).((drop (S i0) O c3 e2) \to ((le O (S i0)) \to (drop (S (plus i0 n)) O -(CHead c2 k t) e2))))))).(\lambda (H0: (drop (S n) O (CHead c2 k t) -c3)).(\lambda (e2: C).(\lambda (H1: (drop (S i0) O c3 e2)).(\lambda (H2: (le -O (S i0))).(eq_ind nat (S (plus i0 n)) (\lambda (n0: nat).(drop (S n0) O -(CHead c2 k t) e2)) (drop_drop k (S (plus i0 n)) c2 e2 (eq_ind_r nat (S (r k -(plus i0 n))) (\lambda (n0: nat).(drop n0 O c2 e2)) (eq_ind_r nat (plus i0 (r -k n)) (\lambda (n0: nat).(drop (S n0) O c2 e2)) (IHc c3 O (r k n) -(drop_gen_drop k c2 c3 t n H0) e2 H1 H2) (r k (plus i0 n)) (r_plus_sym k i0 -n)) (r k (S (plus i0 n))) (r_S k (plus i0 n))) t) (plus i0 (S n)) (plus_n_Sm -i0 n)))))))) h)) (\lambda (d0: nat).(\lambda (IHd: ((\forall (h: nat).((drop -h d0 (CHead c2 k t) c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to ((le -d0 (S i0)) \to (drop (S (plus i0 h)) O (CHead c2 k t) e2)))))))).(\lambda (h: -nat).(\lambda (H: (drop h (S d0) (CHead c2 k t) c3)).(\lambda (e2: -C).(\lambda (H0: (drop (S i0) O c3 e2)).(\lambda (H1: (le (S d0) (S -i0))).(ex3_2_ind C T (\lambda (e: C).(\lambda (v: T).(eq C c3 (CHead e k -v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r k d0) v)))) (\lambda -(e: C).(\lambda (_: T).(drop h (r k d0) c2 e))) (drop (S (plus i0 h)) O -(CHead c2 k t) e2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H2: (eq C c3 -(CHead x0 k x1))).(\lambda (H3: (eq T t (lift h (r k d0) x1))).(\lambda (H4: -(drop h (r k d0) c2 x0)).(let H5 \def (eq_ind C c3 (\lambda (c: C).(\forall -(h0: nat).((drop h0 d0 (CHead c2 k t) c) \to (\forall (e3: C).((drop (S i0) O -c e3) \to ((le d0 (S i0)) \to (drop (S (plus i0 h0)) O (CHead c2 k t) -e3))))))) IHd (CHead x0 k x1) H2) in (let H6 \def (eq_ind C c3 (\lambda (c: -C).(drop (S i0) O c e2)) H0 (CHead x0 k x1) H2) in (let H7 \def (eq_ind T t -(\lambda (t0: T).(\forall (h0: nat).((drop h0 d0 (CHead c2 k t0) (CHead x0 k -x1)) \to (\forall (e3: C).((drop (S i0) O (CHead x0 k x1) e3) \to ((le d0 (S -i0)) \to (drop (S (plus i0 h0)) O (CHead c2 k t0) e3))))))) H5 (lift h (r k -d0) x1) H3) in (eq_ind_r T (lift h (r k d0) x1) (\lambda (t0: T).(drop (S -(plus i0 h)) O (CHead c2 k t0) e2)) (drop_drop k (plus i0 h) c2 e2 (K_ind -(\lambda (k0: K).((drop h (r k0 d0) c2 x0) \to ((drop (r k0 i0) O x0 e2) \to -(drop (r k0 (plus i0 h)) O c2 e2)))) (\lambda (b: B).(\lambda (H8: (drop h (r -(Bind b) d0) c2 x0)).(\lambda (H9: (drop (r (Bind b) i0) O x0 e2)).(IHi c2 x0 -(r (Bind b) d0) h H8 e2 H9 (le_S_n (r (Bind b) d0) i0 H1))))) (\lambda (f: -F).(\lambda (H8: (drop h (r (Flat f) d0) c2 x0)).(\lambda (H9: (drop (r (Flat -f) i0) O x0 e2)).(IHc x0 (r (Flat f) d0) h H8 e2 H9 H1)))) k H4 -(drop_gen_drop k x0 e2 x1 i0 H6)) (lift h (r k d0) x1)) t H3))))))))) -(drop_gen_skip_l c2 c3 t h d0 k H))))))))) d))))))) c1)))) i). -(* COMMENTS -Initial nodes: 2020 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/drop1/defs.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/drop1/defs.ma deleted file mode 100644 index c0e14f438..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/drop1/defs.ma +++ /dev/null @@ -1,35 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/drop/defs.ma". - -include "Basic-1/lift1/defs.ma". - -inductive drop1: PList \to (C \to (C \to Prop)) \def -| drop1_nil: \forall (c: C).(drop1 PNil c c) -| drop1_cons: \forall (c1: C).(\forall (c2: C).(\forall (h: nat).(\forall (d: -nat).((drop h d c1 c2) \to (\forall (c3: C).(\forall (hds: PList).((drop1 hds -c2 c3) \to (drop1 (PCons h d hds) c1 c3)))))))). - -definition ptrans: - PList \to (nat \to PList) -\def - let rec ptrans (hds: PList) on hds: (nat \to PList) \def (\lambda (i: -nat).(match hds with [PNil \Rightarrow PNil | (PCons h d hds0) \Rightarrow -(let j \def (trans hds0 i) in (let q \def (ptrans hds0 i) in (match (blt j d) -with [true \Rightarrow (PCons h (minus d (S j)) q) | false \Rightarrow -q])))])) in ptrans. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/drop1/fwd.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/drop1/fwd.ma deleted file mode 100644 index 6e4d1789e..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/drop1/fwd.ma +++ /dev/null @@ -1,81 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/drop1/defs.ma". - -theorem drop1_gen_pnil: - \forall (c1: C).(\forall (c2: C).((drop1 PNil c1 c2) \to (eq C c1 c2))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (drop1 PNil c1 c2)).(insert_eq -PList PNil (\lambda (p: PList).(drop1 p c1 c2)) (\lambda (_: PList).(eq C c1 -c2)) (\lambda (y: PList).(\lambda (H0: (drop1 y c1 c2)).(drop1_ind (\lambda -(p: PList).(\lambda (c: C).(\lambda (c0: C).((eq PList p PNil) \to (eq C c -c0))))) (\lambda (c: C).(\lambda (_: (eq PList PNil PNil)).(refl_equal C c))) -(\lambda (c3: C).(\lambda (c4: C).(\lambda (h: nat).(\lambda (d: -nat).(\lambda (_: (drop h d c3 c4)).(\lambda (c5: C).(\lambda (hds: -PList).(\lambda (_: (drop1 hds c4 c5)).(\lambda (_: (((eq PList hds PNil) \to -(eq C c4 c5)))).(\lambda (H4: (eq PList (PCons h d hds) PNil)).(let H5 \def -(eq_ind PList (PCons h d hds) (\lambda (ee: PList).(match ee in PList return -(\lambda (_: PList).Prop) with [PNil \Rightarrow False | (PCons _ _ _) -\Rightarrow True])) I PNil H4) in (False_ind (eq C c3 c5) H5)))))))))))) y c1 -c2 H0))) H))). -(* COMMENTS -Initial nodes: 198 -END *) - -theorem drop1_gen_pcons: - \forall (c1: C).(\forall (c3: C).(\forall (hds: PList).(\forall (h: -nat).(\forall (d: nat).((drop1 (PCons h d hds) c1 c3) \to (ex2 C (\lambda -(c2: C).(drop h d c1 c2)) (\lambda (c2: C).(drop1 hds c2 c3)))))))) -\def - \lambda (c1: C).(\lambda (c3: C).(\lambda (hds: PList).(\lambda (h: -nat).(\lambda (d: nat).(\lambda (H: (drop1 (PCons h d hds) c1 c3)).(insert_eq -PList (PCons h d hds) (\lambda (p: PList).(drop1 p c1 c3)) (\lambda (_: -PList).(ex2 C (\lambda (c2: C).(drop h d c1 c2)) (\lambda (c2: C).(drop1 hds -c2 c3)))) (\lambda (y: PList).(\lambda (H0: (drop1 y c1 c3)).(drop1_ind -(\lambda (p: PList).(\lambda (c: C).(\lambda (c0: C).((eq PList p (PCons h d -hds)) \to (ex2 C (\lambda (c2: C).(drop h d c c2)) (\lambda (c2: C).(drop1 -hds c2 c0))))))) (\lambda (c: C).(\lambda (H1: (eq PList PNil (PCons h d -hds))).(let H2 \def (eq_ind PList PNil (\lambda (ee: PList).(match ee in -PList return (\lambda (_: PList).Prop) with [PNil \Rightarrow True | (PCons _ -_ _) \Rightarrow False])) I (PCons h d hds) H1) in (False_ind (ex2 C (\lambda -(c2: C).(drop h d c c2)) (\lambda (c2: C).(drop1 hds c2 c))) H2)))) (\lambda -(c2: C).(\lambda (c4: C).(\lambda (h0: nat).(\lambda (d0: nat).(\lambda (H1: -(drop h0 d0 c2 c4)).(\lambda (c5: C).(\lambda (hds0: PList).(\lambda (H2: -(drop1 hds0 c4 c5)).(\lambda (H3: (((eq PList hds0 (PCons h d hds)) \to (ex2 -C (\lambda (c6: C).(drop h d c4 c6)) (\lambda (c6: C).(drop1 hds c6 -c5)))))).(\lambda (H4: (eq PList (PCons h0 d0 hds0) (PCons h d hds))).(let H5 -\def (f_equal PList nat (\lambda (e: PList).(match e in PList return (\lambda -(_: PList).nat) with [PNil \Rightarrow h0 | (PCons n _ _) \Rightarrow n])) -(PCons h0 d0 hds0) (PCons h d hds) H4) in ((let H6 \def (f_equal PList nat -(\lambda (e: PList).(match e in PList return (\lambda (_: PList).nat) with -[PNil \Rightarrow d0 | (PCons _ n _) \Rightarrow n])) (PCons h0 d0 hds0) -(PCons h d hds) H4) in ((let H7 \def (f_equal PList PList (\lambda (e: -PList).(match e in PList return (\lambda (_: PList).PList) with [PNil -\Rightarrow hds0 | (PCons _ _ p) \Rightarrow p])) (PCons h0 d0 hds0) (PCons h -d hds) H4) in (\lambda (H8: (eq nat d0 d)).(\lambda (H9: (eq nat h0 h)).(let -H10 \def (eq_ind PList hds0 (\lambda (p: PList).((eq PList p (PCons h d hds)) -\to (ex2 C (\lambda (c6: C).(drop h d c4 c6)) (\lambda (c6: C).(drop1 hds c6 -c5))))) H3 hds H7) in (let H11 \def (eq_ind PList hds0 (\lambda (p: -PList).(drop1 p c4 c5)) H2 hds H7) in (let H12 \def (eq_ind nat d0 (\lambda -(n: nat).(drop h0 n c2 c4)) H1 d H8) in (let H13 \def (eq_ind nat h0 (\lambda -(n: nat).(drop n d c2 c4)) H12 h H9) in (ex_intro2 C (\lambda (c6: C).(drop h -d c2 c6)) (\lambda (c6: C).(drop1 hds c6 c5)) c4 H13 H11)))))))) H6)) -H5)))))))))))) y c1 c3 H0))) H)))))). -(* COMMENTS -Initial nodes: 587 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/drop1/getl.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/drop1/getl.ma deleted file mode 100644 index dbee79850..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/drop1/getl.ma +++ /dev/null @@ -1,110 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/drop1/fwd.ma". - -include "Basic-1/getl/drop.ma". - -theorem drop1_getl_trans: - \forall (hds: PList).(\forall (c1: C).(\forall (c2: C).((drop1 hds c2 c1) -\to (\forall (b: B).(\forall (e1: C).(\forall (v: T).(\forall (i: nat).((getl -i c1 (CHead e1 (Bind b) v)) \to (ex2 C (\lambda (e2: C).(drop1 (ptrans hds i) -e2 e1)) (\lambda (e2: C).(getl (trans hds i) c2 (CHead e2 (Bind b) (lift1 -(ptrans hds i) v))))))))))))) -\def - \lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (c1: -C).(\forall (c2: C).((drop1 p c2 c1) \to (\forall (b: B).(\forall (e1: -C).(\forall (v: T).(\forall (i: nat).((getl i c1 (CHead e1 (Bind b) v)) \to -(ex2 C (\lambda (e2: C).(drop1 (ptrans p i) e2 e1)) (\lambda (e2: C).(getl -(trans p i) c2 (CHead e2 (Bind b) (lift1 (ptrans p i) v)))))))))))))) -(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (drop1 PNil c2 c1)).(\lambda -(b: B).(\lambda (e1: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H0: (getl -i c1 (CHead e1 (Bind b) v))).(let H_y \def (drop1_gen_pnil c2 c1 H) in -(eq_ind_r C c1 (\lambda (c: C).(ex2 C (\lambda (e2: C).(drop1 PNil e2 e1)) -(\lambda (e2: C).(getl i c (CHead e2 (Bind b) v))))) (ex_intro2 C (\lambda -(e2: C).(drop1 PNil e2 e1)) (\lambda (e2: C).(getl i c1 (CHead e2 (Bind b) -v))) e1 (drop1_nil e1) H0) c2 H_y)))))))))) (\lambda (h: nat).(\lambda (d: -nat).(\lambda (hds0: PList).(\lambda (H: ((\forall (c1: C).(\forall (c2: -C).((drop1 hds0 c2 c1) \to (\forall (b: B).(\forall (e1: C).(\forall (v: -T).(\forall (i: nat).((getl i c1 (CHead e1 (Bind b) v)) \to (ex2 C (\lambda -(e2: C).(drop1 (ptrans hds0 i) e2 e1)) (\lambda (e2: C).(getl (trans hds0 i) -c2 (CHead e2 (Bind b) (lift1 (ptrans hds0 i) v))))))))))))))).(\lambda (c1: -C).(\lambda (c2: C).(\lambda (H0: (drop1 (PCons h d hds0) c2 c1)).(\lambda -(b: B).(\lambda (e1: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H1: (getl -i c1 (CHead e1 (Bind b) v))).(let H_x \def (drop1_gen_pcons c2 c1 hds0 h d -H0) in (let H2 \def H_x in (ex2_ind C (\lambda (c3: C).(drop h d c2 c3)) -(\lambda (c3: C).(drop1 hds0 c3 c1)) (ex2 C (\lambda (e2: C).(drop1 (match -(blt (trans hds0 i) d) with [true \Rightarrow (PCons h (minus d (S (trans -hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)]) e2 e1)) -(\lambda (e2: C).(getl (match (blt (trans hds0 i) d) with [true \Rightarrow -(trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 -(Bind b) (lift1 (match (blt (trans hds0 i) d) with [true \Rightarrow (PCons h -(minus d (S (trans hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans -hds0 i)]) v))))) (\lambda (x: C).(\lambda (H3: (drop h d c2 x)).(\lambda (H4: -(drop1 hds0 x c1)).(xinduction bool (blt (trans hds0 i) d) (\lambda (b0: -bool).(ex2 C (\lambda (e2: C).(drop1 (match b0 with [true \Rightarrow (PCons -h (minus d (S (trans hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans -hds0 i)]) e2 e1)) (\lambda (e2: C).(getl (match b0 with [true \Rightarrow -(trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 -(Bind b) (lift1 (match b0 with [true \Rightarrow (PCons h (minus d (S (trans -hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)]) v)))))) -(\lambda (x_x: bool).(bool_ind (\lambda (b0: bool).((eq bool (blt (trans hds0 -i) d) b0) \to (ex2 C (\lambda (e2: C).(drop1 (match b0 with [true \Rightarrow -(PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) | false \Rightarrow -(ptrans hds0 i)]) e2 e1)) (\lambda (e2: C).(getl (match b0 with [true -\Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2 -(CHead e2 (Bind b) (lift1 (match b0 with [true \Rightarrow (PCons h (minus d -(S (trans hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)]) -v))))))) (\lambda (H5: (eq bool (blt (trans hds0 i) d) true)).(let H_x0 \def -(H c1 x H4 b e1 v i H1) in (let H6 \def H_x0 in (ex2_ind C (\lambda (e2: -C).(drop1 (ptrans hds0 i) e2 e1)) (\lambda (e2: C).(getl (trans hds0 i) x -(CHead e2 (Bind b) (lift1 (ptrans hds0 i) v)))) (ex2 C (\lambda (e2: -C).(drop1 (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) e2 e1)) -(\lambda (e2: C).(getl (trans hds0 i) c2 (CHead e2 (Bind b) (lift1 (PCons h -(minus d (S (trans hds0 i))) (ptrans hds0 i)) v))))) (\lambda (x0: -C).(\lambda (H7: (drop1 (ptrans hds0 i) x0 e1)).(\lambda (H8: (getl (trans -hds0 i) x (CHead x0 (Bind b) (lift1 (ptrans hds0 i) v)))).(let H_x1 \def -(drop_getl_trans_lt (trans hds0 i) d (blt_lt d (trans hds0 i) H5) c2 x h H3 b -x0 (lift1 (ptrans hds0 i) v) H8) in (let H9 \def H_x1 in (ex2_ind C (\lambda -(e2: C).(getl (trans hds0 i) c2 (CHead e2 (Bind b) (lift h (minus d (S (trans -hds0 i))) (lift1 (ptrans hds0 i) v))))) (\lambda (e2: C).(drop h (minus d (S -(trans hds0 i))) e2 x0)) (ex2 C (\lambda (e2: C).(drop1 (PCons h (minus d (S -(trans hds0 i))) (ptrans hds0 i)) e2 e1)) (\lambda (e2: C).(getl (trans hds0 -i) c2 (CHead e2 (Bind b) (lift1 (PCons h (minus d (S (trans hds0 i))) (ptrans -hds0 i)) v))))) (\lambda (x1: C).(\lambda (H10: (getl (trans hds0 i) c2 -(CHead x1 (Bind b) (lift h (minus d (S (trans hds0 i))) (lift1 (ptrans hds0 -i) v))))).(\lambda (H11: (drop h (minus d (S (trans hds0 i))) x1 -x0)).(ex_intro2 C (\lambda (e2: C).(drop1 (PCons h (minus d (S (trans hds0 -i))) (ptrans hds0 i)) e2 e1)) (\lambda (e2: C).(getl (trans hds0 i) c2 (CHead -e2 (Bind b) (lift1 (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) -v)))) x1 (drop1_cons x1 x0 h (minus d (S (trans hds0 i))) H11 e1 (ptrans hds0 -i) H7) H10)))) H9)))))) H6)))) (\lambda (H5: (eq bool (blt (trans hds0 i) d) -false)).(let H_x0 \def (H c1 x H4 b e1 v i H1) in (let H6 \def H_x0 in -(ex2_ind C (\lambda (e2: C).(drop1 (ptrans hds0 i) e2 e1)) (\lambda (e2: -C).(getl (trans hds0 i) x (CHead e2 (Bind b) (lift1 (ptrans hds0 i) v)))) -(ex2 C (\lambda (e2: C).(drop1 (ptrans hds0 i) e2 e1)) (\lambda (e2: C).(getl -(plus (trans hds0 i) h) c2 (CHead e2 (Bind b) (lift1 (ptrans hds0 i) v))))) -(\lambda (x0: C).(\lambda (H7: (drop1 (ptrans hds0 i) x0 e1)).(\lambda (H8: -(getl (trans hds0 i) x (CHead x0 (Bind b) (lift1 (ptrans hds0 i) v)))).(let -H9 \def (drop_getl_trans_ge (trans hds0 i) c2 x d h H3 (CHead x0 (Bind b) -(lift1 (ptrans hds0 i) v)) H8) in (ex_intro2 C (\lambda (e2: C).(drop1 -(ptrans hds0 i) e2 e1)) (\lambda (e2: C).(getl (plus (trans hds0 i) h) c2 -(CHead e2 (Bind b) (lift1 (ptrans hds0 i) v)))) x0 H7 (H9 (bge_le d (trans -hds0 i) H5))))))) H6)))) x_x)))))) H2))))))))))))))) hds). -(* COMMENTS -Initial nodes: 1674 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/drop1/props.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/drop1/props.ma deleted file mode 100644 index 240219b95..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/drop1/props.ma +++ /dev/null @@ -1,97 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/drop1/fwd.ma". - -include "Basic-1/drop/props.ma". - -include "Basic-1/getl/defs.ma". - -theorem drop1_skip_bind: - \forall (b: B).(\forall (e: C).(\forall (hds: PList).(\forall (c: -C).(\forall (u: T).((drop1 hds c e) \to (drop1 (Ss hds) (CHead c (Bind b) -(lift1 hds u)) (CHead e (Bind b) u))))))) -\def - \lambda (b: B).(\lambda (e: C).(\lambda (hds: PList).(PList_ind (\lambda (p: -PList).(\forall (c: C).(\forall (u: T).((drop1 p c e) \to (drop1 (Ss p) -(CHead c (Bind b) (lift1 p u)) (CHead e (Bind b) u)))))) (\lambda (c: -C).(\lambda (u: T).(\lambda (H: (drop1 PNil c e)).(let H_y \def -(drop1_gen_pnil c e H) in (eq_ind_r C e (\lambda (c0: C).(drop1 PNil (CHead -c0 (Bind b) u) (CHead e (Bind b) u))) (drop1_nil (CHead e (Bind b) u)) c -H_y))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda -(H: ((\forall (c: C).(\forall (u: T).((drop1 p c e) \to (drop1 (Ss p) (CHead -c (Bind b) (lift1 p u)) (CHead e (Bind b) u))))))).(\lambda (c: C).(\lambda -(u: T).(\lambda (H0: (drop1 (PCons n n0 p) c e)).(let H_x \def -(drop1_gen_pcons c e p n n0 H0) in (let H1 \def H_x in (ex2_ind C (\lambda -(c2: C).(drop n n0 c c2)) (\lambda (c2: C).(drop1 p c2 e)) (drop1 (PCons n (S -n0) (Ss p)) (CHead c (Bind b) (lift n n0 (lift1 p u))) (CHead e (Bind b) u)) -(\lambda (x: C).(\lambda (H2: (drop n n0 c x)).(\lambda (H3: (drop1 p x -e)).(drop1_cons (CHead c (Bind b) (lift n n0 (lift1 p u))) (CHead x (Bind b) -(lift1 p u)) n (S n0) (drop_skip_bind n n0 c x H2 b (lift1 p u)) (CHead e -(Bind b) u) (Ss p) (H x u H3))))) H1)))))))))) hds))). -(* COMMENTS -Initial nodes: 379 -END *) - -theorem drop1_cons_tail: - \forall (c2: C).(\forall (c3: C).(\forall (h: nat).(\forall (d: nat).((drop -h d c2 c3) \to (\forall (hds: PList).(\forall (c1: C).((drop1 hds c1 c2) \to -(drop1 (PConsTail hds h d) c1 c3)))))))) -\def - \lambda (c2: C).(\lambda (c3: C).(\lambda (h: nat).(\lambda (d: -nat).(\lambda (H: (drop h d c2 c3)).(\lambda (hds: PList).(PList_ind (\lambda -(p: PList).(\forall (c1: C).((drop1 p c1 c2) \to (drop1 (PConsTail p h d) c1 -c3)))) (\lambda (c1: C).(\lambda (H0: (drop1 PNil c1 c2)).(let H_y \def -(drop1_gen_pnil c1 c2 H0) in (eq_ind_r C c2 (\lambda (c: C).(drop1 (PCons h d -PNil) c c3)) (drop1_cons c2 c3 h d H c3 PNil (drop1_nil c3)) c1 H_y)))) -(\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H0: -((\forall (c1: C).((drop1 p c1 c2) \to (drop1 (PConsTail p h d) c1 -c3))))).(\lambda (c1: C).(\lambda (H1: (drop1 (PCons n n0 p) c1 c2)).(let H_x -\def (drop1_gen_pcons c1 c2 p n n0 H1) in (let H2 \def H_x in (ex2_ind C -(\lambda (c4: C).(drop n n0 c1 c4)) (\lambda (c4: C).(drop1 p c4 c2)) (drop1 -(PCons n n0 (PConsTail p h d)) c1 c3) (\lambda (x: C).(\lambda (H3: (drop n -n0 c1 x)).(\lambda (H4: (drop1 p x c2)).(drop1_cons c1 x n n0 H3 c3 -(PConsTail p h d) (H0 x H4))))) H2))))))))) hds)))))). -(* COMMENTS -Initial nodes: 271 -END *) - -theorem drop1_trans: - \forall (is1: PList).(\forall (c1: C).(\forall (c0: C).((drop1 is1 c1 c0) -\to (\forall (is2: PList).(\forall (c2: C).((drop1 is2 c0 c2) \to (drop1 -(papp is1 is2) c1 c2))))))) -\def - \lambda (is1: PList).(PList_ind (\lambda (p: PList).(\forall (c1: -C).(\forall (c0: C).((drop1 p c1 c0) \to (\forall (is2: PList).(\forall (c2: -C).((drop1 is2 c0 c2) \to (drop1 (papp p is2) c1 c2)))))))) (\lambda (c1: -C).(\lambda (c0: C).(\lambda (H: (drop1 PNil c1 c0)).(\lambda (is2: -PList).(\lambda (c2: C).(\lambda (H0: (drop1 is2 c0 c2)).(let H_y \def -(drop1_gen_pnil c1 c0 H) in (let H1 \def (eq_ind_r C c0 (\lambda (c: -C).(drop1 is2 c c2)) H0 c1 H_y) in H1)))))))) (\lambda (n: nat).(\lambda (n0: -nat).(\lambda (p: PList).(\lambda (H: ((\forall (c1: C).(\forall (c0: -C).((drop1 p c1 c0) \to (\forall (is2: PList).(\forall (c2: C).((drop1 is2 c0 -c2) \to (drop1 (papp p is2) c1 c2))))))))).(\lambda (c1: C).(\lambda (c0: -C).(\lambda (H0: (drop1 (PCons n n0 p) c1 c0)).(\lambda (is2: PList).(\lambda -(c2: C).(\lambda (H1: (drop1 is2 c0 c2)).(let H_x \def (drop1_gen_pcons c1 c0 -p n n0 H0) in (let H2 \def H_x in (ex2_ind C (\lambda (c3: C).(drop n n0 c1 -c3)) (\lambda (c3: C).(drop1 p c3 c0)) (drop1 (PCons n n0 (papp p is2)) c1 -c2) (\lambda (x: C).(\lambda (H3: (drop n n0 c1 x)).(\lambda (H4: (drop1 p x -c0)).(drop1_cons c1 x n n0 H3 c2 (papp p is2) (H x c0 H4 is2 c2 H1))))) -H2))))))))))))) is1). -(* COMMENTS -Initial nodes: 287 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/ex0/defs.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/ex0/defs.ma deleted file mode 100644 index 6ab98b1af..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/ex0/defs.ma +++ /dev/null @@ -1,32 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/A/defs.ma". - -include "Basic-1/G/defs.ma". - -definition gz: - G -\def - mk_G S lt_n_Sn. - -inductive leqz: A \to (A \to Prop) \def -| leqz_sort: \forall (h1: nat).(\forall (h2: nat).(\forall (n1: nat).(\forall -(n2: nat).((eq nat (plus h1 n2) (plus h2 n1)) \to (leqz (ASort h1 n1) (ASort -h2 n2)))))) -| leqz_head: \forall (a1: A).(\forall (a2: A).((leqz a1 a2) \to (\forall (a3: -A).(\forall (a4: A).((leqz a3 a4) \to (leqz (AHead a1 a3) (AHead a2 a4))))))). - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/ex0/props.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/ex0/props.ma deleted file mode 100644 index 96dd77da3..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/ex0/props.ma +++ /dev/null @@ -1,207 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/ex0/defs.ma". - -include "Basic-1/leq/defs.ma". - -include "Basic-1/aplus/props.ma". - -theorem aplus_gz_le: - \forall (k: nat).(\forall (h: nat).(\forall (n: nat).((le h k) \to (eq A -(aplus gz (ASort h n) k) (ASort O (plus (minus k h) n)))))) -\def - \lambda (k: nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).(\forall (n0: -nat).((le h n) \to (eq A (aplus gz (ASort h n0) n) (ASort O (plus (minus n h) -n0))))))) (\lambda (h: nat).(\lambda (n: nat).(\lambda (H: (le h O)).(let H_y -\def (le_n_O_eq h H) in (eq_ind nat O (\lambda (n0: nat).(eq A (ASort n0 n) -(ASort O n))) (refl_equal A (ASort O n)) h H_y))))) (\lambda (k0: -nat).(\lambda (IH: ((\forall (h: nat).(\forall (n: nat).((le h k0) \to (eq A -(aplus gz (ASort h n) k0) (ASort O (plus (minus k0 h) n)))))))).(\lambda (h: -nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).((le n (S k0)) \to (eq A -(asucc gz (aplus gz (ASort n n0) k0)) (ASort O (plus (match n with [O -\Rightarrow (S k0) | (S l) \Rightarrow (minus k0 l)]) n0)))))) (\lambda (n: -nat).(\lambda (_: (le O (S k0))).(eq_ind A (aplus gz (asucc gz (ASort O n)) -k0) (\lambda (a: A).(eq A a (ASort O (S (plus k0 n))))) (eq_ind_r A (ASort O -(plus (minus k0 O) (S n))) (\lambda (a: A).(eq A a (ASort O (S (plus k0 -n))))) (eq_ind nat k0 (\lambda (n0: nat).(eq A (ASort O (plus n0 (S n))) -(ASort O (S (plus k0 n))))) (eq_ind nat (S (plus k0 n)) (\lambda (n0: -nat).(eq A (ASort O n0) (ASort O (S (plus k0 n))))) (refl_equal A (ASort O (S -(plus k0 n)))) (plus k0 (S n)) (plus_n_Sm k0 n)) (minus k0 O) (minus_n_O k0)) -(aplus gz (ASort O (S n)) k0) (IH O (S n) (le_O_n k0))) (asucc gz (aplus gz -(ASort O n) k0)) (aplus_asucc gz k0 (ASort O n))))) (\lambda (n: -nat).(\lambda (_: ((\forall (n0: nat).((le n (S k0)) \to (eq A (asucc gz -(aplus gz (ASort n n0) k0)) (ASort O (plus (match n with [O \Rightarrow (S -k0) | (S l) \Rightarrow (minus k0 l)]) n0))))))).(\lambda (n0: nat).(\lambda -(H0: (le (S n) (S k0))).(let H_y \def (le_S_n n k0 H0) in (eq_ind A (aplus gz -(ASort n n0) k0) (\lambda (a: A).(eq A (asucc gz (aplus gz (ASort (S n) n0) -k0)) a)) (eq_ind A (aplus gz (asucc gz (ASort (S n) n0)) k0) (\lambda (a: -A).(eq A a (aplus gz (ASort n n0) k0))) (refl_equal A (aplus gz (ASort n n0) -k0)) (asucc gz (aplus gz (ASort (S n) n0) k0)) (aplus_asucc gz k0 (ASort (S -n) n0))) (ASort O (plus (minus k0 n) n0)) (IH n n0 H_y))))))) h)))) k). -(* COMMENTS -Initial nodes: 683 -END *) - -theorem aplus_gz_ge: - \forall (n: nat).(\forall (k: nat).(\forall (h: nat).((le k h) \to (eq A -(aplus gz (ASort h n) k) (ASort (minus h k) n))))) -\def - \lambda (n: nat).(\lambda (k: nat).(nat_ind (\lambda (n0: nat).(\forall (h: -nat).((le n0 h) \to (eq A (aplus gz (ASort h n) n0) (ASort (minus h n0) -n))))) (\lambda (h: nat).(\lambda (_: (le O h)).(eq_ind nat h (\lambda (n0: -nat).(eq A (ASort h n) (ASort n0 n))) (refl_equal A (ASort h n)) (minus h O) -(minus_n_O h)))) (\lambda (k0: nat).(\lambda (IH: ((\forall (h: nat).((le k0 -h) \to (eq A (aplus gz (ASort h n) k0) (ASort (minus h k0) n)))))).(\lambda -(h: nat).(nat_ind (\lambda (n0: nat).((le (S k0) n0) \to (eq A (asucc gz -(aplus gz (ASort n0 n) k0)) (ASort (minus n0 (S k0)) n)))) (\lambda (H: (le -(S k0) O)).(ex2_ind nat (\lambda (n0: nat).(eq nat O (S n0))) (\lambda (n0: -nat).(le k0 n0)) (eq A (asucc gz (aplus gz (ASort O n) k0)) (ASort O n)) -(\lambda (x: nat).(\lambda (H0: (eq nat O (S x))).(\lambda (_: (le k0 -x)).(let H2 \def (eq_ind nat O (\lambda (ee: nat).(match ee in nat return -(\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) -I (S x) H0) in (False_ind (eq A (asucc gz (aplus gz (ASort O n) k0)) (ASort O -n)) H2))))) (le_gen_S k0 O H))) (\lambda (n0: nat).(\lambda (_: (((le (S k0) -n0) \to (eq A (asucc gz (aplus gz (ASort n0 n) k0)) (ASort (minus n0 (S k0)) -n))))).(\lambda (H0: (le (S k0) (S n0))).(let H_y \def (le_S_n k0 n0 H0) in -(eq_ind A (aplus gz (ASort n0 n) k0) (\lambda (a: A).(eq A (asucc gz (aplus -gz (ASort (S n0) n) k0)) a)) (eq_ind A (aplus gz (asucc gz (ASort (S n0) n)) -k0) (\lambda (a: A).(eq A a (aplus gz (ASort n0 n) k0))) (refl_equal A (aplus -gz (ASort n0 n) k0)) (asucc gz (aplus gz (ASort (S n0) n) k0)) (aplus_asucc -gz k0 (ASort (S n0) n))) (ASort (minus n0 k0) n) (IH n0 H_y)))))) h)))) k)). -(* COMMENTS -Initial nodes: 524 -END *) - -theorem next_plus_gz: - \forall (n: nat).(\forall (h: nat).(eq nat (next_plus gz n h) (plus h n))) -\def - \lambda (n: nat).(\lambda (h: nat).(nat_ind (\lambda (n0: nat).(eq nat -(next_plus gz n n0) (plus n0 n))) (refl_equal nat n) (\lambda (n0: -nat).(\lambda (H: (eq nat (next_plus gz n n0) (plus n0 n))).(f_equal nat nat -S (next_plus gz n n0) (plus n0 n) H))) h)). -(* COMMENTS -Initial nodes: 77 -END *) - -theorem leqz_leq: - \forall (a1: A).(\forall (a2: A).((leq gz a1 a2) \to (leqz a1 a2))) -\def - \lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq gz a1 a2)).(leq_ind gz -(\lambda (a: A).(\lambda (a0: A).(leqz a a0))) (\lambda (h1: nat).(\lambda -(h2: nat).(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda -(H0: (eq A (aplus gz (ASort h1 n1) k) (aplus gz (ASort h2 n2) k))).(lt_le_e k -h1 (leqz (ASort h1 n1) (ASort h2 n2)) (\lambda (H1: (lt k h1)).(lt_le_e k h2 -(leqz (ASort h1 n1) (ASort h2 n2)) (\lambda (H2: (lt k h2)).(let H3 \def -(eq_ind A (aplus gz (ASort h1 n1) k) (\lambda (a: A).(eq A a (aplus gz (ASort -h2 n2) k))) H0 (ASort (minus h1 k) n1) (aplus_gz_ge n1 k h1 (le_S_n k h1 -(le_S (S k) h1 H1)))) in (let H4 \def (eq_ind A (aplus gz (ASort h2 n2) k) -(\lambda (a: A).(eq A (ASort (minus h1 k) n1) a)) H3 (ASort (minus h2 k) n2) -(aplus_gz_ge n2 k h2 (le_S_n k h2 (le_S (S k) h2 H2)))) in (let H5 \def -(f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with -[(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow ((let rec minus (n: nat) -on n: (nat \to nat) \def (\lambda (m: nat).(match n with [O \Rightarrow O | -(S k0) \Rightarrow (match m with [O \Rightarrow (S k0) | (S l) \Rightarrow -(minus k0 l)])])) in minus) h1 k)])) (ASort (minus h1 k) n1) (ASort (minus h2 -k) n2) H4) in ((let H6 \def (f_equal A nat (\lambda (e: A).(match e in A -return (\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) -\Rightarrow n1])) (ASort (minus h1 k) n1) (ASort (minus h2 k) n2) H4) in -(\lambda (H7: (eq nat (minus h1 k) (minus h2 k))).(eq_ind nat n1 (\lambda (n: -nat).(leqz (ASort h1 n1) (ASort h2 n))) (eq_ind nat h1 (\lambda (n: -nat).(leqz (ASort h1 n1) (ASort n n1))) (leqz_sort h1 h1 n1 n1 (refl_equal -nat (plus h1 n1))) h2 (minus_minus k h1 h2 (le_S_n k h1 (le_S (S k) h1 H1)) -(le_S_n k h2 (le_S (S k) h2 H2)) H7)) n2 H6))) H5))))) (\lambda (H2: (le h2 -k)).(let H3 \def (eq_ind A (aplus gz (ASort h1 n1) k) (\lambda (a: A).(eq A a -(aplus gz (ASort h2 n2) k))) H0 (ASort (minus h1 k) n1) (aplus_gz_ge n1 k h1 -(le_S_n k h1 (le_S (S k) h1 H1)))) in (let H4 \def (eq_ind A (aplus gz (ASort -h2 n2) k) (\lambda (a: A).(eq A (ASort (minus h1 k) n1) a)) H3 (ASort O (plus -(minus k h2) n2)) (aplus_gz_le k h2 n2 H2)) in (let H5 \def (eq_ind nat -(minus h1 k) (\lambda (n: nat).(eq A (ASort n n1) (ASort O (plus (minus k h2) -n2)))) H4 (S (minus h1 (S k))) (minus_x_Sy h1 k H1)) in (let H6 \def (eq_ind -A (ASort (S (minus h1 (S k))) n1) (\lambda (ee: A).(match ee in A return -(\lambda (_: A).Prop) with [(ASort n _) \Rightarrow (match n in nat return -(\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True]) -| (AHead _ _) \Rightarrow False])) I (ASort O (plus (minus k h2) n2)) H5) in -(False_ind (leqz (ASort h1 n1) (ASort h2 n2)) H6)))))))) (\lambda (H1: (le h1 -k)).(lt_le_e k h2 (leqz (ASort h1 n1) (ASort h2 n2)) (\lambda (H2: (lt k -h2)).(let H3 \def (eq_ind A (aplus gz (ASort h1 n1) k) (\lambda (a: A).(eq A -a (aplus gz (ASort h2 n2) k))) H0 (ASort O (plus (minus k h1) n1)) -(aplus_gz_le k h1 n1 H1)) in (let H4 \def (eq_ind A (aplus gz (ASort h2 n2) -k) (\lambda (a: A).(eq A (ASort O (plus (minus k h1) n1)) a)) H3 (ASort -(minus h2 k) n2) (aplus_gz_ge n2 k h2 (le_S_n k h2 (le_S (S k) h2 H2)))) in -(let H5 \def (sym_eq A (ASort O (plus (minus k h1) n1)) (ASort (minus h2 k) -n2) H4) in (let H6 \def (eq_ind nat (minus h2 k) (\lambda (n: nat).(eq A -(ASort n n2) (ASort O (plus (minus k h1) n1)))) H5 (S (minus h2 (S k))) -(minus_x_Sy h2 k H2)) in (let H7 \def (eq_ind A (ASort (S (minus h2 (S k))) -n2) (\lambda (ee: A).(match ee in A return (\lambda (_: A).Prop) with [(ASort -n _) \Rightarrow (match n in nat return (\lambda (_: nat).Prop) with [O -\Rightarrow False | (S _) \Rightarrow True]) | (AHead _ _) \Rightarrow -False])) I (ASort O (plus (minus k h1) n1)) H6) in (False_ind (leqz (ASort h1 -n1) (ASort h2 n2)) H7))))))) (\lambda (H2: (le h2 k)).(let H3 \def (eq_ind A -(aplus gz (ASort h1 n1) k) (\lambda (a: A).(eq A a (aplus gz (ASort h2 n2) -k))) H0 (ASort O (plus (minus k h1) n1)) (aplus_gz_le k h1 n1 H1)) in (let H4 -\def (eq_ind A (aplus gz (ASort h2 n2) k) (\lambda (a: A).(eq A (ASort O -(plus (minus k h1) n1)) a)) H3 (ASort O (plus (minus k h2) n2)) (aplus_gz_le -k h2 n2 H2)) in (let H5 \def (f_equal A nat (\lambda (e: A).(match e in A -return (\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) -\Rightarrow ((let rec plus (n: nat) on n: (nat \to nat) \def (\lambda (m: -nat).(match n with [O \Rightarrow m | (S p) \Rightarrow (S (plus p m))])) in -plus) (minus k h1) n1)])) (ASort O (plus (minus k h1) n1)) (ASort O (plus -(minus k h2) n2)) H4) in (let H_y \def (plus_plus k h1 h2 n1 n2 H1 H2 H5) in -(leqz_sort h1 h2 n1 n2 H_y))))))))))))))) (\lambda (a0: A).(\lambda (a3: -A).(\lambda (_: (leq gz a0 a3)).(\lambda (H1: (leqz a0 a3)).(\lambda (a4: -A).(\lambda (a5: A).(\lambda (_: (leq gz a4 a5)).(\lambda (H3: (leqz a4 -a5)).(leqz_head a0 a3 H1 a4 a5 H3))))))))) a1 a2 H))). -(* COMMENTS -Initial nodes: 1375 -END *) - -theorem leq_leqz: - \forall (a1: A).(\forall (a2: A).((leqz a1 a2) \to (leq gz a1 a2))) -\def - \lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leqz a1 a2)).(leqz_ind -(\lambda (a: A).(\lambda (a0: A).(leq gz a a0))) (\lambda (h1: nat).(\lambda -(h2: nat).(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (H0: (eq nat (plus -h1 n2) (plus h2 n1))).(leq_sort gz h1 h2 n1 n2 (plus h1 h2) (eq_ind_r A -(ASort (minus h1 (plus h1 h2)) (next_plus gz n1 (minus (plus h1 h2) h1))) -(\lambda (a: A).(eq A a (aplus gz (ASort h2 n2) (plus h1 h2)))) (eq_ind_r A -(ASort (minus h2 (plus h1 h2)) (next_plus gz n2 (minus (plus h1 h2) h2))) -(\lambda (a: A).(eq A (ASort (minus h1 (plus h1 h2)) (next_plus gz n1 (minus -(plus h1 h2) h1))) a)) (eq_ind_r nat h2 (\lambda (n: nat).(eq A (ASort (minus -h1 (plus h1 h2)) (next_plus gz n1 n)) (ASort (minus h2 (plus h1 h2)) -(next_plus gz n2 (minus (plus h1 h2) h2))))) (eq_ind_r nat h1 (\lambda (n: -nat).(eq A (ASort (minus h1 (plus h1 h2)) (next_plus gz n1 h2)) (ASort (minus -h2 (plus h1 h2)) (next_plus gz n2 n)))) (eq_ind_r nat O (\lambda (n: nat).(eq -A (ASort n (next_plus gz n1 h2)) (ASort (minus h2 (plus h1 h2)) (next_plus gz -n2 h1)))) (eq_ind_r nat O (\lambda (n: nat).(eq A (ASort O (next_plus gz n1 -h2)) (ASort n (next_plus gz n2 h1)))) (eq_ind_r nat (plus h2 n1) (\lambda (n: -nat).(eq A (ASort O n) (ASort O (next_plus gz n2 h1)))) (eq_ind_r nat (plus -h1 n2) (\lambda (n: nat).(eq A (ASort O (plus h2 n1)) (ASort O n))) (f_equal -nat A (ASort O) (plus h2 n1) (plus h1 n2) (sym_eq nat (plus h1 n2) (plus h2 -n1) H0)) (next_plus gz n2 h1) (next_plus_gz n2 h1)) (next_plus gz n1 h2) -(next_plus_gz n1 h2)) (minus h2 (plus h1 h2)) (O_minus h2 (plus h1 h2) -(le_plus_r h1 h2))) (minus h1 (plus h1 h2)) (O_minus h1 (plus h1 h2) -(le_plus_l h1 h2))) (minus (plus h1 h2) h2) (minus_plus_r h1 h2)) (minus -(plus h1 h2) h1) (minus_plus h1 h2)) (aplus gz (ASort h2 n2) (plus h1 h2)) -(aplus_asort_simpl gz (plus h1 h2) h2 n2)) (aplus gz (ASort h1 n1) (plus h1 -h2)) (aplus_asort_simpl gz (plus h1 h2) h1 n1)))))))) (\lambda (a0: -A).(\lambda (a3: A).(\lambda (_: (leqz a0 a3)).(\lambda (H1: (leq gz a0 -a3)).(\lambda (a4: A).(\lambda (a5: A).(\lambda (_: (leqz a4 a5)).(\lambda -(H3: (leq gz a4 a5)).(leq_head gz a0 a3 H1 a4 a5 H3))))))))) a1 a2 H))). -(* COMMENTS -Initial nodes: 717 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/ex1/defs.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/ex1/defs.ma deleted file mode 100644 index dc18f6fc2..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/ex1/defs.ma +++ /dev/null @@ -1,29 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/C/defs.ma". - -definition ex1_c: - C -\def - CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) -(Bind Abst) (TLRef O). - -definition ex1_t: - T -\def - THead (Flat Appl) (TLRef O) (THead (Bind Abst) (TLRef (S (S O))) (TSort O)). - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/ex1/props.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/ex1/props.ma deleted file mode 100644 index 5c442f5bb..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/ex1/props.ma +++ /dev/null @@ -1,536 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/ex1/defs.ma". - -include "Basic-1/ty3/fwd.ma". - -include "Basic-1/pc3/fwd.ma". - -include "Basic-1/nf2/pr3.ma". - -include "Basic-1/nf2/props.ma". - -include "Basic-1/arity/defs.ma". - -include "Basic-1/leq/props.ma". - -theorem ex1__leq_sort_SS: - \forall (g: G).(\forall (k: nat).(\forall (n: nat).(leq g (ASort k n) (asucc -g (asucc g (ASort (S (S k)) n)))))) -\def - \lambda (g: G).(\lambda (k: nat).(\lambda (n: nat).(leq_refl g (asucc g -(asucc g (ASort (S (S k)) n)))))). -(* COMMENTS -Initial nodes: 27 -END *) - -theorem ex1_arity: - \forall (g: G).(arity g ex1_c ex1_t (ASort O O)) -\def - \lambda (g: G).(arity_appl g (CHead (CHead (CHead (CSort O) (Bind Abst) -(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef O) (ASort (S -(S O)) O) (arity_abst g (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) -(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) (TLRef O) O (getl_refl Abst (CHead -(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (TLRef O)) -(ASort (S (S O)) O) (arity_abst g (CHead (CHead (CSort O) (Bind Abst) (TSort -O)) (Bind Abst) (TSort O)) (CHead (CSort O) (Bind Abst) (TSort O)) (TSort O) -O (getl_refl Abst (CHead (CSort O) (Bind Abst) (TSort O)) (TSort O)) (asucc g -(ASort (S (S O)) O)) (arity_repl g (CHead (CSort O) (Bind Abst) (TSort O)) -(TSort O) (ASort O O) (arity_sort g (CHead (CSort O) (Bind Abst) (TSort O)) -O) (asucc g (asucc g (ASort (S (S O)) O))) (ex1__leq_sort_SS g O O)))) (THead -(Bind Abst) (TLRef (S (S O))) (TSort O)) (ASort O O) (arity_head g (CHead -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind -Abst) (TLRef O)) (TLRef (S (S O))) (ASort (S (S O)) O) (arity_abst g (CHead -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind -Abst) (TLRef O)) (CSort O) (TSort O) (S (S O)) (getl_clear_bind Abst (CHead -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind -Abst) (TLRef O)) (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) -(TSort O)) (TLRef O) (clear_bind Abst (CHead (CHead (CSort O) (Bind Abst) -(TSort O)) (Bind Abst) (TSort O)) (TLRef O)) (CHead (CSort O) (Bind Abst) -(TSort O)) (S O) (getl_head (Bind Abst) O (CHead (CSort O) (Bind Abst) (TSort -O)) (CHead (CSort O) (Bind Abst) (TSort O)) (getl_refl Abst (CSort O) (TSort -O)) (TSort O))) (asucc g (ASort (S (S O)) O)) (arity_repl g (CSort O) (TSort -O) (ASort O O) (arity_sort g (CSort O) O) (asucc g (asucc g (ASort (S (S O)) -O))) (ex1__leq_sort_SS g O O))) (TSort O) (ASort O O) (arity_sort g (CHead -(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) -(Bind Abst) (TLRef O)) (Bind Abst) (TLRef (S (S O)))) O))). -(* COMMENTS -Initial nodes: 753 -END *) - -theorem ex1_ty3: - \forall (g: G).(\forall (u: T).((ty3 g ex1_c ex1_t u) \to (\forall (P: -Prop).P))) -\def - \lambda (g: G).(\lambda (u: T).(\lambda (H: (ty3 g (CHead (CHead (CHead -(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef -O)) (THead (Flat Appl) (TLRef O) (THead (Bind Abst) (TLRef (S (S O))) (TSort -O))) u)).(\lambda (P: Prop).(ex3_2_ind T T (\lambda (u0: T).(\lambda (t: -T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) -(TSort O)) (Bind Abst) (TLRef O)) (THead (Flat Appl) (TLRef O) (THead (Bind -Abst) u0 t)) u))) (\lambda (u0: T).(\lambda (t: T).(ty3 g (CHead (CHead -(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) -(TLRef O)) (THead (Bind Abst) (TLRef (S (S O))) (TSort O)) (THead (Bind Abst) -u0 t)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g (CHead (CHead (CHead (CSort -O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) -(TLRef O) u0))) P (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 (CHead -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind -Abst) (TLRef O)) (THead (Flat Appl) (TLRef O) (THead (Bind Abst) x0 x1)) -u)).(\lambda (H1: (ty3 g (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort -O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (THead (Bind Abst) (TLRef -(S (S O))) (TSort O)) (THead (Bind Abst) x0 x1))).(\lambda (H2: (ty3 g (CHead -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind -Abst) (TLRef O)) (TLRef O) x0)).(or_ind (ex3_3 C T T (\lambda (_: C).(\lambda -(_: T).(\lambda (t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort -O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S O) O t) x0)))) -(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl O (CHead (CHead (CHead -(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef -O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(t: T).(ty3 g e u0 t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: -T).(\lambda (_: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) -(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S O) O u0) x0)))) -(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl O (CHead (CHead (CHead -(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef -O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(t: T).(ty3 g e u0 t))))) P (\lambda (H3: (ex3_3 C T T (\lambda (_: -C).(\lambda (_: T).(\lambda (t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S O) O -t) x0)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl O (CHead -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind -Abst) (TLRef O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (t: T).(ty3 g e u0 t)))))).(ex3_3_ind C T T (\lambda (_: -C).(\lambda (_: T).(\lambda (t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S O) O -t) x0)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl O (CHead -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind -Abst) (TLRef O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (t: T).(ty3 g e u0 t)))) P (\lambda (x2: C).(\lambda (x3: -T).(\lambda (x4: T).(\lambda (_: (pc3 (CHead (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S O) O -x4) x0)).(\lambda (H5: (getl O (CHead (CHead (CHead (CSort O) (Bind Abst) -(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead x2 (Bind -Abbr) x3))).(\lambda (_: (ty3 g x2 x3 x4)).(ex3_2_ind T T (\lambda (t2: -T).(\lambda (_: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) -(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (THead (Bind Abst) (TLRef (S (S -O))) t2) (THead (Bind Abst) x0 x1)))) (\lambda (_: T).(\lambda (t: T).(ty3 g -(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) -(Bind Abst) (TLRef O)) (TLRef (S (S O))) t))) (\lambda (t2: T).(\lambda (_: -T).(ty3 g (CHead (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind -Abst) (TSort O)) (Bind Abst) (TLRef O)) (Bind Abst) (TLRef (S (S O)))) (TSort -O) t2))) P (\lambda (x5: T).(\lambda (x6: T).(\lambda (_: (pc3 (CHead (CHead -(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) -(TLRef O)) (THead (Bind Abst) (TLRef (S (S O))) x5) (THead (Bind Abst) x0 -x1))).(\lambda (H8: (ty3 g (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort -O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) -x6)).(\lambda (_: (ty3 g (CHead (CHead (CHead (CHead (CSort O) (Bind Abst) -(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (Bind Abst) (TLRef -(S (S O)))) (TSort O) x5)).(or_ind (ex3_3 C T T (\lambda (_: C).(\lambda (_: -T).(\lambda (t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) -(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O t) x6)))) -(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) (CHead -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind -Abst) (TLRef O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (t: T).(ty3 g e u0 t))))) (ex3_3 C T T (\lambda (_: C).(\lambda -(u0: T).(\lambda (_: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) -(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) -O u0) x6)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) -(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) -(Bind Abst) (TLRef O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda -(u0: T).(\lambda (t: T).(ty3 g e u0 t))))) P (\lambda (H10: (ex3_3 C T T -(\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 (CHead (CHead (CHead -(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef -O)) (lift (S (S (S O))) O t) x6)))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(_: T).(getl (S (S O)) (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) -(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead e (Bind Abbr) u0))))) -(\lambda (e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 -t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 -(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) -(Bind Abst) (TLRef O)) (lift (S (S (S O))) O t) x6)))) (\lambda (e: -C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) (CHead (CHead (CHead -(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef -O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(t: T).(ty3 g e u0 t)))) P (\lambda (x7: C).(\lambda (x8: T).(\lambda (x9: -T).(\lambda (_: (pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) -(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O x9) -x6)).(\lambda (H12: (getl (S (S O)) (CHead (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead x7 -(Bind Abbr) x8))).(\lambda (_: (ty3 g x7 x8 x9)).(let H14 \def (getl_gen_all -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead -x7 (Bind Abbr) x8) (r (Bind Abst) (S O)) (getl_gen_S (Bind Abst) (CHead -(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead x7 -(Bind Abbr) x8) (TLRef O) (S O) H12)) in (ex2_ind C (\lambda (e: C).(drop (S -O) O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) -e)) (\lambda (e: C).(clear e (CHead x7 (Bind Abbr) x8))) P (\lambda (x: -C).(\lambda (_: (drop (S O) O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) -(Bind Abst) (TSort O)) x)).(\lambda (_: (clear x (CHead x7 (Bind Abbr) -x8))).(let H17 \def (eq_ind C (CHead x2 (Bind Abbr) x3) (\lambda (ee: -C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow -False | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) -with [(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with -[Abbr \Rightarrow True | Abst \Rightarrow False | Void \Rightarrow False]) | -(Flat _) \Rightarrow False])])) I (CHead (CHead (CHead (CSort O) (Bind Abst) -(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (clear_gen_bind Abst -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead -x2 (Bind Abbr) x3) (TLRef O) (getl_gen_O (CHead (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead x2 -(Bind Abbr) x3) H5))) in (False_ind P H17))))) H14)))))))) H10)) (\lambda -(H10: (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 -(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) -(Bind Abst) (TLRef O)) (lift (S (S (S O))) O u0) x6)))) (\lambda (e: -C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) (CHead (CHead (CHead -(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef -O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(t: T).(ty3 g e u0 t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (u0: -T).(\lambda (_: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) -(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O u0) -x6)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) -(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) -(Bind Abst) (TLRef O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda -(u0: T).(\lambda (t: T).(ty3 g e u0 t)))) P (\lambda (x7: C).(\lambda (x8: -T).(\lambda (x9: T).(\lambda (_: (pc3 (CHead (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S -O))) O x8) x6)).(\lambda (H12: (getl (S (S O)) (CHead (CHead (CHead (CSort O) -(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead -x7 (Bind Abst) x8))).(\lambda (_: (ty3 g x7 x8 x9)).(let H14 \def -(getl_gen_all (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) -(TSort O)) (CHead x7 (Bind Abst) x8) (r (Bind Abst) (S O)) (getl_gen_S (Bind -Abst) (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) -(CHead x7 (Bind Abst) x8) (TLRef O) (S O) H12)) in (ex2_ind C (\lambda (e: -C).(drop (S O) O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) -(TSort O)) e)) (\lambda (e: C).(clear e (CHead x7 (Bind Abst) x8))) P -(\lambda (x: C).(\lambda (_: (drop (S O) O (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) x)).(\lambda (_: (clear x (CHead x7 -(Bind Abst) x8))).(let H17 \def (eq_ind C (CHead x2 (Bind Abbr) x3) (\lambda -(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind b) \Rightarrow (match b in B return (\lambda (_: -B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | Void -\Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead (CHead (CHead -(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef -O)) (clear_gen_bind Abst (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind -Abst) (TSort O)) (CHead x2 (Bind Abbr) x3) (TLRef O) (getl_gen_O (CHead -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind -Abst) (TLRef O)) (CHead x2 (Bind Abbr) x3) H5))) in (False_ind P H17))))) -H14)))))))) H10)) (ty3_gen_lref g (CHead (CHead (CHead (CSort O) (Bind Abst) -(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) x6 (S (S O)) -H8))))))) (ty3_gen_bind g Abst (CHead (CHead (CHead (CSort O) (Bind Abst) -(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) -(TSort O) (THead (Bind Abst) x0 x1) H1)))))))) H3)) (\lambda (H3: (ex3_3 C T -T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 (CHead (CHead (CHead -(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef -O)) (lift (S O) O u0) x0)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: -T).(getl O (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) -(TSort O)) (Bind Abst) (TLRef O)) (CHead e (Bind Abst) u0))))) (\lambda (e: -C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t)))))).(ex3_3_ind C T T -(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 (CHead (CHead (CHead -(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef -O)) (lift (S O) O u0) x0)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: -T).(getl O (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) -(TSort O)) (Bind Abst) (TLRef O)) (CHead e (Bind Abst) u0))))) (\lambda (e: -C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t)))) P (\lambda (x2: -C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H4: (pc3 (CHead (CHead (CHead -(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef -O)) (lift (S O) O x3) x0)).(\lambda (H5: (getl O (CHead (CHead (CHead (CSort -O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) -(CHead x2 (Bind Abst) x3))).(\lambda (H6: (ty3 g x2 x3 x4)).(ex3_2_ind T T -(\lambda (t2: T).(\lambda (_: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (THead (Bind -Abst) (TLRef (S (S O))) t2) (THead (Bind Abst) x0 x1)))) (\lambda (_: -T).(\lambda (t: T).(ty3 g (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort -O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) t))) -(\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead (CHead (CHead (CHead (CSort O) -(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (Bind -Abst) (TLRef (S (S O)))) (TSort O) t2))) P (\lambda (x5: T).(\lambda (x6: -T).(\lambda (H7: (pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) -(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (THead (Bind Abst) (TLRef (S (S -O))) x5) (THead (Bind Abst) x0 x1))).(\lambda (H8: (ty3 g (CHead (CHead -(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) -(TLRef O)) (TLRef (S (S O))) x6)).(\lambda (_: (ty3 g (CHead (CHead (CHead -(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) -(TLRef O)) (Bind Abst) (TLRef (S (S O)))) (TSort O) x5)).(or_ind (ex3_3 C T T -(\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 (CHead (CHead (CHead -(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef -O)) (lift (S (S (S O))) O t) x6)))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(_: T).(getl (S (S O)) (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) -(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead e (Bind Abbr) u0))))) -(\lambda (e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t))))) (ex3_3 C -T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 (CHead (CHead -(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) -(TLRef O)) (lift (S (S (S O))) O u0) x6)))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (_: T).(getl (S (S O)) (CHead (CHead (CHead (CSort O) (Bind Abst) -(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead e (Bind Abst) -u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t))))) P -(\lambda (H10: (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: -T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) -(TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O t) x6)))) (\lambda -(e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) (CHead (CHead (CHead -(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef -O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(t: T).(ty3 g e u0 t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (_: -T).(\lambda (t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) -(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O t) x6)))) -(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) (CHead -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind -Abst) (TLRef O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (t: T).(ty3 g e u0 t)))) P (\lambda (x7: C).(\lambda (x8: -T).(\lambda (x9: T).(\lambda (_: (pc3 (CHead (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S -O))) O x9) x6)).(\lambda (H12: (getl (S (S O)) (CHead (CHead (CHead (CSort O) -(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead -x7 (Bind Abbr) x8))).(\lambda (_: (ty3 g x7 x8 x9)).(let H14 \def -(getl_gen_all (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) -(TSort O)) (CHead x7 (Bind Abbr) x8) (r (Bind Abst) (S O)) (getl_gen_S (Bind -Abst) (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) -(CHead x7 (Bind Abbr) x8) (TLRef O) (S O) H12)) in (ex2_ind C (\lambda (e: -C).(drop (S O) O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) -(TSort O)) e)) (\lambda (e: C).(clear e (CHead x7 (Bind Abbr) x8))) P -(\lambda (x: C).(\lambda (H15: (drop (S O) O (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) x)).(\lambda (H16: (clear x (CHead x7 -(Bind Abbr) x8))).(let H17 \def (f_equal C C (\lambda (e: C).(match e in C -return (\lambda (_: C).C) with [(CSort _) \Rightarrow x2 | (CHead c _ _) -\Rightarrow c])) (CHead x2 (Bind Abst) x3) (CHead (CHead (CHead (CSort O) -(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) -(clear_gen_bind Abst (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind -Abst) (TSort O)) (CHead x2 (Bind Abst) x3) (TLRef O) (getl_gen_O (CHead -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind -Abst) (TLRef O)) (CHead x2 (Bind Abst) x3) H5))) in ((let H18 \def (f_equal C -T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) -\Rightarrow x3 | (CHead _ _ t) \Rightarrow t])) (CHead x2 (Bind Abst) x3) -(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) -(Bind Abst) (TLRef O)) (clear_gen_bind Abst (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead x2 (Bind Abst) x3) (TLRef O) -(getl_gen_O (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) -(TSort O)) (Bind Abst) (TLRef O)) (CHead x2 (Bind Abst) x3) H5))) in (\lambda -(H19: (eq C x2 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) -(TSort O)))).(let H20 \def (eq_ind T x3 (\lambda (t: T).(ty3 g x2 t x4)) H6 -(TLRef O) H18) in (let H21 \def (eq_ind T x3 (\lambda (t: T).(pc3 (CHead -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind -Abst) (TLRef O)) (lift (S O) O t) x0)) H4 (TLRef O) H18) in (let H22 \def -(eq_ind C x2 (\lambda (c: C).(ty3 g c (TLRef O) x4)) H20 (CHead (CHead (CSort -O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) H19) in (let H23 \def -(eq_ind_r C x (\lambda (c: C).(clear c (CHead x7 (Bind Abbr) x8))) H16 (CHead -(CSort O) (Bind Abst) (TSort O)) (drop_gen_refl (CHead (CSort O) (Bind Abst) -(TSort O)) x (drop_gen_drop (Bind Abst) (CHead (CSort O) (Bind Abst) (TSort -O)) x (TSort O) O H15))) in (let H24 \def (eq_ind C (CHead x7 (Bind Abbr) x8) -(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind b) \Rightarrow (match b in B return (\lambda (_: -B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | Void -\Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead (CSort O) -(Bind Abst) (TSort O)) (clear_gen_bind Abst (CSort O) (CHead x7 (Bind Abbr) -x8) (TSort O) H23)) in (False_ind P H24)))))))) H17))))) H14)))))))) H10)) -(\lambda (H10: (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: -T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) -(TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O u0) x6)))) (\lambda -(e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) (CHead (CHead (CHead -(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef -O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(t: T).(ty3 g e u0 t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (u0: -T).(\lambda (_: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) -(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O u0) -x6)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) -(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) -(Bind Abst) (TLRef O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda -(u0: T).(\lambda (t: T).(ty3 g e u0 t)))) P (\lambda (x7: C).(\lambda (x8: -T).(\lambda (x9: T).(\lambda (H11: (pc3 (CHead (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S -O))) O x8) x6)).(\lambda (H12: (getl (S (S O)) (CHead (CHead (CHead (CSort O) -(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead -x7 (Bind Abst) x8))).(\lambda (H13: (ty3 g x7 x8 x9)).(let H14 \def -(getl_gen_all (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) -(TSort O)) (CHead x7 (Bind Abst) x8) (r (Bind Abst) (S O)) (getl_gen_S (Bind -Abst) (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) -(CHead x7 (Bind Abst) x8) (TLRef O) (S O) H12)) in (ex2_ind C (\lambda (e: -C).(drop (S O) O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) -(TSort O)) e)) (\lambda (e: C).(clear e (CHead x7 (Bind Abst) x8))) P -(\lambda (x: C).(\lambda (H15: (drop (S O) O (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) x)).(\lambda (H16: (clear x (CHead x7 -(Bind Abst) x8))).(let H17 \def (f_equal C C (\lambda (e: C).(match e in C -return (\lambda (_: C).C) with [(CSort _) \Rightarrow x2 | (CHead c _ _) -\Rightarrow c])) (CHead x2 (Bind Abst) x3) (CHead (CHead (CHead (CSort O) -(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) -(clear_gen_bind Abst (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind -Abst) (TSort O)) (CHead x2 (Bind Abst) x3) (TLRef O) (getl_gen_O (CHead -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind -Abst) (TLRef O)) (CHead x2 (Bind Abst) x3) H5))) in ((let H18 \def (f_equal C -T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) -\Rightarrow x3 | (CHead _ _ t) \Rightarrow t])) (CHead x2 (Bind Abst) x3) -(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) -(Bind Abst) (TLRef O)) (clear_gen_bind Abst (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead x2 (Bind Abst) x3) (TLRef O) -(getl_gen_O (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) -(TSort O)) (Bind Abst) (TLRef O)) (CHead x2 (Bind Abst) x3) H5))) in (\lambda -(H19: (eq C x2 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) -(TSort O)))).(let H20 \def (eq_ind T x3 (\lambda (t: T).(ty3 g x2 t x4)) H6 -(TLRef O) H18) in (let H21 \def (eq_ind T x3 (\lambda (t: T).(pc3 (CHead -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind -Abst) (TLRef O)) (lift (S O) O t) x0)) H4 (TLRef O) H18) in (let H22 \def -(eq_ind C x2 (\lambda (c: C).(ty3 g c (TLRef O) x4)) H20 (CHead (CHead (CSort -O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) H19) in (let H23 \def -(eq_ind_r C x (\lambda (c: C).(clear c (CHead x7 (Bind Abst) x8))) H16 (CHead -(CSort O) (Bind Abst) (TSort O)) (drop_gen_refl (CHead (CSort O) (Bind Abst) -(TSort O)) x (drop_gen_drop (Bind Abst) (CHead (CSort O) (Bind Abst) (TSort -O)) x (TSort O) O H15))) in (let H24 \def (f_equal C C (\lambda (e: C).(match -e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow x7 | (CHead c _ -_) \Rightarrow c])) (CHead x7 (Bind Abst) x8) (CHead (CSort O) (Bind Abst) -(TSort O)) (clear_gen_bind Abst (CSort O) (CHead x7 (Bind Abst) x8) (TSort O) -H23)) in ((let H25 \def (f_equal C T (\lambda (e: C).(match e in C return -(\lambda (_: C).T) with [(CSort _) \Rightarrow x8 | (CHead _ _ t) \Rightarrow -t])) (CHead x7 (Bind Abst) x8) (CHead (CSort O) (Bind Abst) (TSort O)) -(clear_gen_bind Abst (CSort O) (CHead x7 (Bind Abst) x8) (TSort O) H23)) in -(\lambda (H26: (eq C x7 (CSort O))).(let H27 \def (eq_ind T x8 (\lambda (t: -T).(ty3 g x7 t x9)) H13 (TSort O) H25) in (let H28 \def (eq_ind T x8 (\lambda -(t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) -(TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O t) x6)) H11 (TSort O) -H25) in (let H29 \def (eq_ind C x7 (\lambda (c: C).(ty3 g c (TSort O) x9)) -H27 (CSort O) H26) in (or_ind (ex3_3 C T T (\lambda (_: C).(\lambda (_: -T).(\lambda (t: T).(pc3 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind -Abst) (TSort O)) (lift (S O) O t) x4)))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (_: T).(getl O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) -(Bind Abst) (TSort O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda -(u0: T).(\lambda (t: T).(ty3 g e u0 t))))) (ex3_3 C T T (\lambda (_: -C).(\lambda (u0: T).(\lambda (_: T).(pc3 (CHead (CHead (CSort O) (Bind Abst) -(TSort O)) (Bind Abst) (TSort O)) (lift (S O) O u0) x4)))) (\lambda (e: -C).(\lambda (u0: T).(\lambda (_: T).(getl O (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead e (Bind Abst) u0))))) (\lambda -(e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t))))) P (\lambda (H30: -(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 (CHead -(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (lift (S O) O -t) x4)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl O (CHead -(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead e (Bind -Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 -t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (lift -(S O) O t) x4)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl O -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead -e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g -e u0 t)))) P (\lambda (x10: C).(\lambda (x11: T).(\lambda (x12: T).(\lambda -(_: (pc3 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort -O)) (lift (S O) O x12) x4)).(\lambda (H32: (getl O (CHead (CHead (CSort O) -(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead x10 (Bind Abbr) -x11))).(\lambda (_: (ty3 g x10 x11 x12)).(let H34 \def (eq_ind C (CHead x10 -(Bind Abbr) x11) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) -with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K -return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B return -(\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | -Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead (CHead -(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (clear_gen_bind Abst -(CHead (CSort O) (Bind Abst) (TSort O)) (CHead x10 (Bind Abbr) x11) (TSort O) -(getl_gen_O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort -O)) (CHead x10 (Bind Abbr) x11) H32))) in (False_ind P H34)))))))) H30)) -(\lambda (H30: (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: -T).(pc3 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) -(lift (S O) O u0) x4)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: -T).(getl O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort -O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(t: T).(ty3 g e u0 t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (u0: -T).(\lambda (_: T).(pc3 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind -Abst) (TSort O)) (lift (S O) O u0) x4)))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (_: T).(getl O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) -(Bind Abst) (TSort O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda -(u0: T).(\lambda (t: T).(ty3 g e u0 t)))) P (\lambda (x10: C).(\lambda (x11: -T).(\lambda (x12: T).(\lambda (H31: (pc3 (CHead (CHead (CSort O) (Bind Abst) -(TSort O)) (Bind Abst) (TSort O)) (lift (S O) O x11) x4)).(\lambda (H32: -(getl O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) -(CHead x10 (Bind Abst) x11))).(\lambda (H33: (ty3 g x10 x11 x12)).(let H34 -\def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) -with [(CSort _) \Rightarrow x10 | (CHead c _ _) \Rightarrow c])) (CHead x10 -(Bind Abst) x11) (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) -(TSort O)) (clear_gen_bind Abst (CHead (CSort O) (Bind Abst) (TSort O)) -(CHead x10 (Bind Abst) x11) (TSort O) (getl_gen_O (CHead (CHead (CSort O) -(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead x10 (Bind Abst) x11) -H32))) in ((let H35 \def (f_equal C T (\lambda (e: C).(match e in C return -(\lambda (_: C).T) with [(CSort _) \Rightarrow x11 | (CHead _ _ t) -\Rightarrow t])) (CHead x10 (Bind Abst) x11) (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) (clear_gen_bind Abst (CHead (CSort O) -(Bind Abst) (TSort O)) (CHead x10 (Bind Abst) x11) (TSort O) (getl_gen_O -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead -x10 (Bind Abst) x11) H32))) in (\lambda (H36: (eq C x10 (CHead (CSort O) -(Bind Abst) (TSort O)))).(let H37 \def (eq_ind T x11 (\lambda (t: T).(ty3 g -x10 t x12)) H33 (TSort O) H35) in (let H38 \def (eq_ind T x11 (\lambda (t: -T).(pc3 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) -(lift (S O) O t) x4)) H31 (TSort O) H35) in (let H39 \def (eq_ind C x10 -(\lambda (c: C).(ty3 g c (TSort O) x12)) H37 (CHead (CSort O) (Bind Abst) -(TSort O)) H36) in (land_ind (pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) -(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) -x0) (\forall (b: B).(\forall (u0: T).(pc3 (CHead (CHead (CHead (CHead (CSort -O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (Bind -b) u0) x5 x1))) P (\lambda (H40: (pc3 (CHead (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S -O))) x0)).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pc3 (CHead (CHead -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind -Abst) (TLRef O)) (Bind b) u0) x5 x1))))).(let H42 \def (eq_ind T (lift (S O) -O (TLRef O)) (\lambda (t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) -(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) -t)) (pc3_t x0 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind -Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) H40 (lift (S O) O -(TLRef O)) (ex2_sym T (pr3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort -O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S O) O (TLRef O))) -(pr3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort -O)) (Bind Abst) (TLRef O)) x0) H21)) (TLRef (plus O (S O))) (lift_lref_ge O -(S O) O (le_n O))) in (let H43 \def H42 in (ex2_ind T (\lambda (t: T).(pr3 -(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) -(Bind Abst) (TLRef O)) (TLRef (S (S O))) t)) (\lambda (t: T).(pr3 (CHead -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind -Abst) (TLRef O)) (TLRef (S O)) t)) P (\lambda (x13: T).(\lambda (H44: (pr3 -(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) -(Bind Abst) (TLRef O)) (TLRef (S (S O))) x13)).(\lambda (H45: (pr3 (CHead -(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind -Abst) (TLRef O)) (TLRef (S O)) x13)).(let H46 \def (eq_ind_r T x13 (\lambda -(t: T).(eq T (TLRef (S (S O))) t)) (nf2_pr3_unfold (CHead (CHead (CHead -(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef -O)) (TLRef (S (S O))) x13 H44 (nf2_lref_abst (CHead (CHead (CHead (CSort O) -(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CSort -O) (TSort O) (S (S O)) (getl_clear_bind Abst (CHead (CHead (CHead (CSort O) -(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead -(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (TLRef O) -(clear_bind Abst (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) -(TSort O)) (TLRef O)) (CHead (CSort O) (Bind Abst) (TSort O)) (S O) -(getl_head (Bind Abst) O (CHead (CSort O) (Bind Abst) (TSort O)) (CHead -(CSort O) (Bind Abst) (TSort O)) (getl_refl Abst (CSort O) (TSort O)) (TSort -O))))) (TLRef (S O)) (nf2_pr3_unfold (CHead (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S O)) -x13 H45 (nf2_lref_abst (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) -(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead (CSort O) (Bind Abst) -(TSort O)) (TSort O) (S O) (getl_head (Bind Abst) O (CHead (CHead (CSort O) -(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead (CHead (CSort O) (Bind -Abst) (TSort O)) (Bind Abst) (TSort O)) (getl_refl Abst (CHead (CSort O) -(Bind Abst) (TSort O)) (TSort O)) (TLRef O))))) in (let H47 \def (eq_ind T -(TLRef (S (S O))) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef n) \Rightarrow (match n -in nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S n0) -\Rightarrow (match n0 in nat return (\lambda (_: nat).Prop) with [O -\Rightarrow False | (S _) \Rightarrow True])]) | (THead _ _ _) \Rightarrow -False])) I (TLRef (S O)) H46) in (False_ind P H47)))))) H43))))) -(pc3_gen_abst (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind -Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) x0 x5 x1 H7))))))) -H34)))))))) H30)) (ty3_gen_lref g (CHead (CHead (CSort O) (Bind Abst) (TSort -O)) (Bind Abst) (TSort O)) x4 O H22))))))) H24)))))))) H17))))) H14)))))))) -H10)) (ty3_gen_lref g (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) -(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) x6 (S (S O)) H8))))))) -(ty3_gen_bind g Abst (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) -(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) (TSort O) -(THead (Bind Abst) x0 x1) H1)))))))) H3)) (ty3_gen_lref g (CHead (CHead -(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) -(TLRef O)) x0 O H2))))))) (ty3_gen_appl g (CHead (CHead (CHead (CSort O) -(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef -O) (THead (Bind Abst) (TLRef (S (S O))) (TSort O)) u H))))). -(* COMMENTS -Initial nodes: 9973 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/ex2/defs.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/ex2/defs.ma deleted file mode 100644 index 35b5df73e..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/ex2/defs.ma +++ /dev/null @@ -1,28 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/C/defs.ma". - -definition ex2_c: - C -\def - CSort O. - -definition ex2_t: - T -\def - THead (Flat Appl) (TSort O) (TSort O). - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/ex2/props.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/ex2/props.ma deleted file mode 100644 index 4dfa6a582..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/ex2/props.ma +++ /dev/null @@ -1,159 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/ex2/defs.ma". - -include "Basic-1/nf2/defs.ma". - -include "Basic-1/pr2/fwd.ma". - -include "Basic-1/arity/fwd.ma". - -theorem ex2_nf2: - nf2 ex2_c ex2_t -\def - \lambda (t2: T).(\lambda (H: (pr2 (CSort O) (THead (Flat Appl) (TSort O) -(TSort O)) t2)).(let H0 \def (pr2_gen_appl (CSort O) (TSort O) (TSort O) t2 -H) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead -(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 (CSort O) (TSort -O) u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 (CSort O) (TSort O) t3)))) -(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(eq T (TSort O) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda -(_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 -t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: -T).(pr2 (CSort O) (TSort O) u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead (CSort O) -(Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B -b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (TSort O) (THead (Bind b) y1 -z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: -T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat -Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 (CSort O) (TSort -O) u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CSort O) y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead (CSort O) (Bind b) y2) z1 z2)))))))) (eq T (THead (Flat -Appl) (TSort O) (TSort O)) t2) (\lambda (H1: (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 (CSort O) (TSort O) u2))) (\lambda (_: T).(\lambda -(t3: T).(pr2 (CSort O) (TSort O) t3))))).(ex3_2_ind T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 (CSort O) (TSort O) u2))) (\lambda (_: T).(\lambda -(t3: T).(pr2 (CSort O) (TSort O) t3))) (eq T (THead (Flat Appl) (TSort O) -(TSort O)) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H2: (eq T t2 -(THead (Flat Appl) x0 x1))).(\lambda (H3: (pr2 (CSort O) (TSort O) -x0)).(\lambda (H4: (pr2 (CSort O) (TSort O) x1)).(let H5 \def (eq_ind T x1 -(\lambda (t: T).(eq T t2 (THead (Flat Appl) x0 t))) H2 (TSort O) -(pr2_gen_sort (CSort O) x1 O H4)) in (let H6 \def (eq_ind T x0 (\lambda (t: -T).(eq T t2 (THead (Flat Appl) t (TSort O)))) H5 (TSort O) (pr2_gen_sort -(CSort O) x0 O H3)) in (eq_ind_r T (THead (Flat Appl) (TSort O) (TSort O)) -(\lambda (t: T).(eq T (THead (Flat Appl) (TSort O) (TSort O)) t)) (refl_equal -T (THead (Flat Appl) (TSort O) (TSort O))) t2 H6)))))))) H1)) (\lambda (H1: -(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(eq T (TSort O) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda -(_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 -t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: -T).(pr2 (CSort O) (TSort O) u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead (CSort O) -(Bind b) u) z1 t3))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T (TSort O) (THead (Bind Abst) y1 -z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: -T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 (CSort O) (TSort O) u2))))) (\lambda -(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: -B).(\forall (u: T).(pr2 (CHead (CSort O) (Bind b) u) z1 t3))))))) (eq T -(THead (Flat Appl) (TSort O) (TSort O)) t2) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H2: (eq T (TSort O) (THead -(Bind Abst) x0 x1))).(\lambda (H3: (eq T t2 (THead (Bind Abbr) x2 -x3))).(\lambda (H4: (pr2 (CSort O) (TSort O) x2)).(\lambda (_: ((\forall (b: -B).(\forall (u: T).(pr2 (CHead (CSort O) (Bind b) u) x1 x3))))).(let H6 \def -(eq_ind T x2 (\lambda (t: T).(eq T t2 (THead (Bind Abbr) t x3))) H3 (TSort O) -(pr2_gen_sort (CSort O) x2 O H4)) in (eq_ind_r T (THead (Bind Abbr) (TSort O) -x3) (\lambda (t: T).(eq T (THead (Flat Appl) (TSort O) (TSort O)) t)) (let H7 -\def (eq_ind T (TSort O) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | -(THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) x0 x1) H2) in -(False_ind (eq T (THead (Flat Appl) (TSort O) (TSort O)) (THead (Bind Abbr) -(TSort O) x3)) H7)) t2 H6)))))))))) H1)) (\lambda (H1: (ex6_6 B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T -(TSort O) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T -t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 (CSort O) (TSort O) u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CSort O) y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda -(z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead (CSort -O) (Bind b) y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (TSort O) -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 (CSort O) (TSort O) u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 -(CSort O) y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead (CSort O) -(Bind b) y2) z1 z2))))))) (eq T (THead (Flat Appl) (TSort O) (TSort O)) t2) -(\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda -(x4: T).(\lambda (x5: T).(\lambda (_: (not (eq B x0 Abst))).(\lambda (H3: (eq -T (TSort O) (THead (Bind x0) x1 x2))).(\lambda (H4: (eq T t2 (THead (Bind x0) -x5 (THead (Flat Appl) (lift (S O) O x4) x3)))).(\lambda (H5: (pr2 (CSort O) -(TSort O) x4)).(\lambda (H6: (pr2 (CSort O) x1 x5)).(\lambda (_: (pr2 (CHead -(CSort O) (Bind x0) x5) x2 x3)).(let H_y \def (pr2_gen_csort x1 x5 O H6) in -(let H8 \def (eq_ind T x4 (\lambda (t: T).(eq T t2 (THead (Bind x0) x5 (THead -(Flat Appl) (lift (S O) O t) x3)))) H4 (TSort O) (pr2_gen_sort (CSort O) x4 O -H5)) in (eq_ind_r T (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O -(TSort O)) x3)) (\lambda (t: T).(eq T (THead (Flat Appl) (TSort O) (TSort O)) -t)) (let H9 \def (eq_ind T (TSort O) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Bind x0) x1 -x2) H3) in (False_ind (eq T (THead (Flat Appl) (TSort O) (TSort O)) (THead -(Bind x0) x5 (THead (Flat Appl) (lift (S O) O (TSort O)) x3))) H9)) t2 -H8))))))))))))))) H1)) H0))). -(* COMMENTS -Initial nodes: 1939 -END *) - -theorem ex2_arity: - \forall (g: G).(\forall (a: A).((arity g ex2_c ex2_t a) \to (\forall (P: -Prop).P))) -\def - \lambda (g: G).(\lambda (a: A).(\lambda (H: (arity g (CSort O) (THead (Flat -Appl) (TSort O) (TSort O)) a)).(\lambda (P: Prop).(let H0 \def -(arity_gen_appl g (CSort O) (TSort O) (TSort O) a H) in (ex2_ind A (\lambda -(a1: A).(arity g (CSort O) (TSort O) a1)) (\lambda (a1: A).(arity g (CSort O) -(TSort O) (AHead a1 a))) P (\lambda (x: A).(\lambda (_: (arity g (CSort O) -(TSort O) x)).(\lambda (H2: (arity g (CSort O) (TSort O) (AHead x a))).(let -H_x \def (leq_gen_head1 g x a (ASort O O) (arity_gen_sort g (CSort O) O -(AHead x a) H2)) in (let H3 \def H_x in (ex3_2_ind A A (\lambda (a3: -A).(\lambda (_: A).(leq g x a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a -a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort O O) (AHead a3 a4)))) P -(\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g x x0)).(\lambda (_: -(leq g a x1)).(\lambda (H6: (eq A (ASort O O) (AHead x0 x1))).(let H7 \def -(eq_ind A (ASort O O) (\lambda (ee: A).(match ee in A return (\lambda (_: -A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow -False])) I (AHead x0 x1) H6) in (False_ind P H7))))))) H3)))))) H0))))). -(* COMMENTS -Initial nodes: 289 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/flt/defs.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/flt/defs.ma deleted file mode 100644 index 3191e1e38..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/flt/defs.ma +++ /dev/null @@ -1,29 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/C/defs.ma". - -definition fweight: - C \to (T \to nat) -\def - \lambda (c: C).(\lambda (t: T).(plus (cweight c) (tweight t))). - -definition flt: - C \to (T \to (C \to (T \to Prop))) -\def - \lambda (c1: C).(\lambda (t1: T).(\lambda (c2: C).(\lambda (t2: T).(lt -(fweight c1 t1) (fweight c2 t2))))). - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/flt/props.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/flt/props.ma deleted file mode 100644 index 57df41528..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/flt/props.ma +++ /dev/null @@ -1,154 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/flt/defs.ma". - -include "Basic-1/C/props.ma". - -theorem flt_thead_sx: - \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).(flt c u c -(THead k u t))))) -\def - \lambda (_: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t: -T).(le_lt_plus_plus (cweight c) (cweight c) (tweight u) (S (plus (tweight u) -(tweight t))) (le_n (cweight c)) (le_n_S (tweight u) (plus (tweight u) -(tweight t)) (le_plus_l (tweight u) (tweight t))))))). -(* COMMENTS -Initial nodes: 65 -END *) - -theorem flt_thead_dx: - \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).(flt c t c -(THead k u t))))) -\def - \lambda (_: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t: -T).(le_lt_plus_plus (cweight c) (cweight c) (tweight t) (S (plus (tweight u) -(tweight t))) (le_n (cweight c)) (le_n_S (tweight t) (plus (tweight u) -(tweight t)) (le_plus_r (tweight u) (tweight t))))))). -(* COMMENTS -Initial nodes: 65 -END *) - -theorem flt_shift: - \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).(flt (CHead c -k u) t c (THead k u t))))) -\def - \lambda (_: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(eq_ind nat -(S (plus (cweight c) (plus (tweight u) (tweight t)))) (\lambda (n: nat).(lt -(plus (plus (cweight c) (tweight u)) (tweight t)) n)) (eq_ind_r nat (plus -(plus (cweight c) (tweight u)) (tweight t)) (\lambda (n: nat).(lt (plus (plus -(cweight c) (tweight u)) (tweight t)) (S n))) (le_n (S (plus (plus (cweight -c) (tweight u)) (tweight t)))) (plus (cweight c) (plus (tweight u) (tweight -t))) (plus_assoc_l (cweight c) (tweight u) (tweight t))) (plus (cweight c) (S -(plus (tweight u) (tweight t)))) (plus_n_Sm (cweight c) (plus (tweight u) -(tweight t))))))). -(* COMMENTS -Initial nodes: 179 -END *) - -theorem flt_arith0: - \forall (k: K).(\forall (c: C).(\forall (t: T).(\forall (i: nat).(flt c t -(CHead c k t) (TLRef i))))) -\def - \lambda (_: K).(\lambda (c: C).(\lambda (t: T).(\lambda (_: -nat).(lt_x_plus_x_Sy (plus (cweight c) (tweight t)) O)))). -(* COMMENTS -Initial nodes: 21 -END *) - -theorem flt_arith1: - \forall (k1: K).(\forall (c1: C).(\forall (c2: C).(\forall (t1: T).((cle -(CHead c1 k1 t1) c2) \to (\forall (k2: K).(\forall (t2: T).(\forall (i: -nat).(flt c1 t1 (CHead c2 k2 t2) (TLRef i))))))))) -\def - \lambda (_: K).(\lambda (c1: C).(\lambda (c2: C).(\lambda (t1: T).(\lambda -(H: (le (plus (cweight c1) (tweight t1)) (cweight c2))).(\lambda (_: -K).(\lambda (t2: T).(\lambda (_: nat).(le_lt_trans (plus (cweight c1) -(tweight t1)) (cweight c2) (plus (plus (cweight c2) (tweight t2)) (S O)) H -(eq_ind_r nat (plus (S O) (plus (cweight c2) (tweight t2))) (\lambda (n: -nat).(lt (cweight c2) n)) (le_lt_n_Sm (cweight c2) (plus (cweight c2) -(tweight t2)) (le_plus_l (cweight c2) (tweight t2))) (plus (plus (cweight c2) -(tweight t2)) (S O)) (plus_sym (plus (cweight c2) (tweight t2)) (S -O))))))))))). -(* COMMENTS -Initial nodes: 151 -END *) - -theorem flt_arith2: - \forall (c1: C).(\forall (c2: C).(\forall (t1: T).(\forall (i: nat).((flt c1 -t1 c2 (TLRef i)) \to (\forall (k2: K).(\forall (t2: T).(\forall (j: nat).(flt -c1 t1 (CHead c2 k2 t2) (TLRef j))))))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (t1: T).(\lambda (_: nat).(\lambda -(H: (lt (plus (cweight c1) (tweight t1)) (plus (cweight c2) (S O)))).(\lambda -(_: K).(\lambda (t2: T).(\lambda (_: nat).(lt_le_trans (plus (cweight c1) -(tweight t1)) (plus (cweight c2) (S O)) (plus (plus (cweight c2) (tweight -t2)) (S O)) H (le_plus_plus (cweight c2) (plus (cweight c2) (tweight t2)) (S -O) (S O) (le_plus_l (cweight c2) (tweight t2)) (le_n (S O))))))))))). -(* COMMENTS -Initial nodes: 115 -END *) - -theorem flt_trans: - \forall (c1: C).(\forall (c2: C).(\forall (t1: T).(\forall (t2: T).((flt c1 -t1 c2 t2) \to (\forall (c3: C).(\forall (t3: T).((flt c2 t2 c3 t3) \to (flt -c1 t1 c3 t3)))))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H: (lt (fweight c1 t1) (fweight c2 t2))).(\lambda (c3: C).(\lambda (t3: -T).(\lambda (H0: (lt (fweight c2 t2) (fweight c3 t3))).(lt_trans (fweight c1 -t1) (fweight c2 t2) (fweight c3 t3) H H0)))))))). -(* COMMENTS -Initial nodes: 63 -END *) - -theorem flt_wf__q_ind: - \forall (P: ((C \to (T \to Prop)))).(((\forall (n: nat).((\lambda (P0: ((C -\to (T \to Prop)))).(\lambda (n0: nat).(\forall (c: C).(\forall (t: T).((eq -nat (fweight c t) n0) \to (P0 c t)))))) P n))) \to (\forall (c: C).(\forall -(t: T).(P c t)))) -\def - let Q \def (\lambda (P: ((C \to (T \to Prop)))).(\lambda (n: nat).(\forall -(c: C).(\forall (t: T).((eq nat (fweight c t) n) \to (P c t)))))) in (\lambda -(P: ((C \to (T \to Prop)))).(\lambda (H: ((\forall (n: nat).(\forall (c: -C).(\forall (t: T).((eq nat (fweight c t) n) \to (P c t))))))).(\lambda (c: -C).(\lambda (t: T).(H (fweight c t) c t (refl_equal nat (fweight c t))))))). -(* COMMENTS -Initial nodes: 85 -END *) - -theorem flt_wf_ind: - \forall (P: ((C \to (T \to Prop)))).(((\forall (c2: C).(\forall (t2: -T).(((\forall (c1: C).(\forall (t1: T).((flt c1 t1 c2 t2) \to (P c1 t1))))) -\to (P c2 t2))))) \to (\forall (c: C).(\forall (t: T).(P c t)))) -\def - let Q \def (\lambda (P: ((C \to (T \to Prop)))).(\lambda (n: nat).(\forall -(c: C).(\forall (t: T).((eq nat (fweight c t) n) \to (P c t)))))) in (\lambda -(P: ((C \to (T \to Prop)))).(\lambda (H: ((\forall (c2: C).(\forall (t2: -T).(((\forall (c1: C).(\forall (t1: T).((flt c1 t1 c2 t2) \to (P c1 t1))))) -\to (P c2 t2)))))).(\lambda (c: C).(\lambda (t: T).(flt_wf__q_ind P (\lambda -(n: nat).(lt_wf_ind n (Q P) (\lambda (n0: nat).(\lambda (H0: ((\forall (m: -nat).((lt m n0) \to (Q P m))))).(\lambda (c0: C).(\lambda (t0: T).(\lambda -(H1: (eq nat (fweight c0 t0) n0)).(let H2 \def (eq_ind_r nat n0 (\lambda (n1: -nat).(\forall (m: nat).((lt m n1) \to (\forall (c1: C).(\forall (t1: T).((eq -nat (fweight c1 t1) m) \to (P c1 t1))))))) H0 (fweight c0 t0) H1) in (H c0 t0 -(\lambda (c1: C).(\lambda (t1: T).(\lambda (H3: (flt c1 t1 c0 t0)).(H2 -(fweight c1 t1) H3 c1 t1 (refl_equal nat (fweight c1 t1))))))))))))))) c -t))))). -(* COMMENTS -Initial nodes: 211 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/fsubst0/defs.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/fsubst0/defs.ma deleted file mode 100644 index f39baebf7..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/fsubst0/defs.ma +++ /dev/null @@ -1,26 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubst0/defs.ma". - -inductive fsubst0 (i: nat) (v: T) (c1: C) (t1: T): C \to (T \to Prop) \def -| fsubst0_snd: \forall (t2: T).((subst0 i v t1 t2) \to (fsubst0 i v c1 t1 c1 -t2)) -| fsubst0_fst: \forall (c2: C).((csubst0 i v c1 c2) \to (fsubst0 i v c1 t1 c2 -t1)) -| fsubst0_both: \forall (t2: T).((subst0 i v t1 t2) \to (\forall (c2: -C).((csubst0 i v c1 c2) \to (fsubst0 i v c1 t1 c2 t2)))). - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/fsubst0/fwd.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/fsubst0/fwd.ma deleted file mode 100644 index 3c1212510..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/fsubst0/fwd.ma +++ /dev/null @@ -1,43 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/fsubst0/defs.ma". - -theorem fsubst0_gen_base: - \forall (c1: C).(\forall (c2: C).(\forall (t1: T).(\forall (t2: T).(\forall -(v: T).(\forall (i: nat).((fsubst0 i v c1 t1 c2 t2) \to (or3 (land (eq C c1 -c2) (subst0 i v t1 t2)) (land (eq T t1 t2) (csubst0 i v c1 c2)) (land (subst0 -i v t1 t2) (csubst0 i v c1 c2))))))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(v: T).(\lambda (i: nat).(\lambda (H: (fsubst0 i v c1 t1 c2 t2)).(fsubst0_ind -i v c1 t1 (\lambda (c: C).(\lambda (t: T).(or3 (land (eq C c1 c) (subst0 i v -t1 t)) (land (eq T t1 t) (csubst0 i v c1 c)) (land (subst0 i v t1 t) (csubst0 -i v c1 c))))) (\lambda (t0: T).(\lambda (H0: (subst0 i v t1 t0)).(or3_intro0 -(land (eq C c1 c1) (subst0 i v t1 t0)) (land (eq T t1 t0) (csubst0 i v c1 -c1)) (land (subst0 i v t1 t0) (csubst0 i v c1 c1)) (conj (eq C c1 c1) (subst0 -i v t1 t0) (refl_equal C c1) H0)))) (\lambda (c0: C).(\lambda (H0: (csubst0 i -v c1 c0)).(or3_intro1 (land (eq C c1 c0) (subst0 i v t1 t1)) (land (eq T t1 -t1) (csubst0 i v c1 c0)) (land (subst0 i v t1 t1) (csubst0 i v c1 c0)) (conj -(eq T t1 t1) (csubst0 i v c1 c0) (refl_equal T t1) H0)))) (\lambda (t0: -T).(\lambda (H0: (subst0 i v t1 t0)).(\lambda (c0: C).(\lambda (H1: (csubst0 -i v c1 c0)).(or3_intro2 (land (eq C c1 c0) (subst0 i v t1 t0)) (land (eq T t1 -t0) (csubst0 i v c1 c0)) (land (subst0 i v t1 t0) (csubst0 i v c1 c0)) (conj -(subst0 i v t1 t0) (csubst0 i v c1 c0) H0 H1)))))) c2 t2 H))))))). -(* COMMENTS -Initial nodes: 431 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/getl/clear.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/getl/clear.ma deleted file mode 100644 index de0a85bd3..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/getl/clear.ma +++ /dev/null @@ -1,153 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/getl/props.ma". - -include "Basic-1/clear/drop.ma". - -theorem clear_getl_trans: - \forall (i: nat).(\forall (c2: C).(\forall (c3: C).((getl i c2 c3) \to -(\forall (c1: C).((clear c1 c2) \to (getl i c1 c3)))))) -\def - \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c2: C).(\forall (c3: -C).((getl n c2 c3) \to (\forall (c1: C).((clear c1 c2) \to (getl n c1 -c3))))))) (\lambda (c2: C).(\lambda (c3: C).(\lambda (H: (getl O c2 -c3)).(\lambda (c1: C).(\lambda (H0: (clear c1 c2)).(getl_intro O c1 c3 c1 -(drop_refl c1) (clear_trans c1 c2 H0 c3 (getl_gen_O c2 c3 H)))))))) (\lambda -(n: nat).(\lambda (_: ((\forall (c2: C).(\forall (c3: C).((getl n c2 c3) \to -(\forall (c1: C).((clear c1 c2) \to (getl n c1 c3)))))))).(\lambda (c2: -C).(C_ind (\lambda (c: C).(\forall (c3: C).((getl (S n) c c3) \to (\forall -(c1: C).((clear c1 c) \to (getl (S n) c1 c3)))))) (\lambda (n0: nat).(\lambda -(c3: C).(\lambda (H0: (getl (S n) (CSort n0) c3)).(\lambda (c1: C).(\lambda -(_: (clear c1 (CSort n0))).(getl_gen_sort n0 (S n) c3 H0 (getl (S n) c1 -c3))))))) (\lambda (c: C).(\lambda (_: ((\forall (c3: C).((getl (S n) c c3) -\to (\forall (c1: C).((clear c1 c) \to (getl (S n) c1 c3))))))).(\lambda (k: -K).(\lambda (t: T).(\lambda (c3: C).(\lambda (H1: (getl (S n) (CHead c k t) -c3)).(\lambda (c1: C).(\lambda (H2: (clear c1 (CHead c k t))).(K_ind (\lambda -(k0: K).((getl (S n) (CHead c k0 t) c3) \to ((clear c1 (CHead c k0 t)) \to -(getl (S n) c1 c3)))) (\lambda (b: B).(\lambda (H3: (getl (S n) (CHead c -(Bind b) t) c3)).(\lambda (H4: (clear c1 (CHead c (Bind b) t))).(let H5 \def -(getl_gen_all c c3 (r (Bind b) n) (getl_gen_S (Bind b) c c3 t n H3)) in -(ex2_ind C (\lambda (e: C).(drop n O c e)) (\lambda (e: C).(clear e c3)) -(getl (S n) c1 c3) (\lambda (x: C).(\lambda (H6: (drop n O c x)).(\lambda -(H7: (clear x c3)).(getl_intro (S n) c1 c3 x (drop_clear_O b c1 c t H4 x n -H6) H7)))) H5))))) (\lambda (f: F).(\lambda (_: (getl (S n) (CHead c (Flat f) -t) c3)).(\lambda (H4: (clear c1 (CHead c (Flat f) t))).(clear_gen_flat_r f c1 -c t H4 (getl (S n) c1 c3))))) k H1 H2))))))))) c2)))) i). -(* COMMENTS -Initial nodes: 525 -END *) - -theorem getl_clear_trans: - \forall (i: nat).(\forall (c1: C).(\forall (c2: C).((getl i c1 c2) \to -(\forall (c3: C).((clear c2 c3) \to (getl i c1 c3)))))) -\def - \lambda (i: nat).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (getl i c1 -c2)).(\lambda (c3: C).(\lambda (H0: (clear c2 c3)).(let H1 \def (getl_gen_all -c1 c2 i H) in (ex2_ind C (\lambda (e: C).(drop i O c1 e)) (\lambda (e: -C).(clear e c2)) (getl i c1 c3) (\lambda (x: C).(\lambda (H2: (drop i O c1 -x)).(\lambda (H3: (clear x c2)).(let H4 \def (clear_gen_all x c2 H3) in -(ex_3_ind B C T (\lambda (b: B).(\lambda (e: C).(\lambda (u: T).(eq C c2 -(CHead e (Bind b) u))))) (getl i c1 c3) (\lambda (x0: B).(\lambda (x1: -C).(\lambda (x2: T).(\lambda (H5: (eq C c2 (CHead x1 (Bind x0) x2))).(let H6 -\def (eq_ind C c2 (\lambda (c: C).(clear x c)) H3 (CHead x1 (Bind x0) x2) H5) -in (let H7 \def (eq_ind C c2 (\lambda (c: C).(clear c c3)) H0 (CHead x1 (Bind -x0) x2) H5) in (eq_ind_r C (CHead x1 (Bind x0) x2) (\lambda (c: C).(getl i c1 -c)) (getl_intro i c1 (CHead x1 (Bind x0) x2) x H2 H6) c3 (clear_gen_bind x0 -x1 c3 x2 H7)))))))) H4))))) H1))))))). -(* COMMENTS -Initial nodes: 269 -END *) - -theorem getl_clear_bind: - \forall (b: B).(\forall (c: C).(\forall (e1: C).(\forall (v: T).((clear c -(CHead e1 (Bind b) v)) \to (\forall (e2: C).(\forall (n: nat).((getl n e1 e2) -\to (getl (S n) c e2)))))))) -\def - \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e1: -C).(\forall (v: T).((clear c0 (CHead e1 (Bind b) v)) \to (\forall (e2: -C).(\forall (n: nat).((getl n e1 e2) \to (getl (S n) c0 e2)))))))) (\lambda -(n: nat).(\lambda (e1: C).(\lambda (v: T).(\lambda (H: (clear (CSort n) -(CHead e1 (Bind b) v))).(\lambda (e2: C).(\lambda (n0: nat).(\lambda (_: -(getl n0 e1 e2)).(clear_gen_sort (CHead e1 (Bind b) v) n H (getl (S n0) -(CSort n) e2))))))))) (\lambda (c0: C).(\lambda (H: ((\forall (e1: -C).(\forall (v: T).((clear c0 (CHead e1 (Bind b) v)) \to (\forall (e2: -C).(\forall (n: nat).((getl n e1 e2) \to (getl (S n) c0 e2))))))))).(\lambda -(k: K).(\lambda (t: T).(\lambda (e1: C).(\lambda (v: T).(\lambda (H0: (clear -(CHead c0 k t) (CHead e1 (Bind b) v))).(\lambda (e2: C).(\lambda (n: -nat).(\lambda (H1: (getl n e1 e2)).(K_ind (\lambda (k0: K).((clear (CHead c0 -k0 t) (CHead e1 (Bind b) v)) \to (getl (S n) (CHead c0 k0 t) e2))) (\lambda -(b0: B).(\lambda (H2: (clear (CHead c0 (Bind b0) t) (CHead e1 (Bind b) -v))).(let H3 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda -(_: C).C) with [(CSort _) \Rightarrow e1 | (CHead c1 _ _) \Rightarrow c1])) -(CHead e1 (Bind b) v) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e1 -(Bind b) v) t H2)) in ((let H4 \def (f_equal C B (\lambda (e: C).(match e in -C return (\lambda (_: C).B) with [(CSort _) \Rightarrow b | (CHead _ k0 _) -\Rightarrow (match k0 in K return (\lambda (_: K).B) with [(Bind b1) -\Rightarrow b1 | (Flat _) \Rightarrow b])])) (CHead e1 (Bind b) v) (CHead c0 -(Bind b0) t) (clear_gen_bind b0 c0 (CHead e1 (Bind b) v) t H2)) in ((let H5 -\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow v | (CHead _ _ t0) \Rightarrow t0])) (CHead e1 -(Bind b) v) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e1 (Bind b) -v) t H2)) in (\lambda (H6: (eq B b b0)).(\lambda (H7: (eq C e1 c0)).(let H8 -\def (eq_ind C e1 (\lambda (c1: C).(getl n c1 e2)) H1 c0 H7) in (eq_ind B b -(\lambda (b1: B).(getl (S n) (CHead c0 (Bind b1) t) e2)) (getl_head (Bind b) -n c0 e2 H8 t) b0 H6))))) H4)) H3)))) (\lambda (f: F).(\lambda (H2: (clear -(CHead c0 (Flat f) t) (CHead e1 (Bind b) v))).(getl_flat c0 e2 (S n) (H e1 v -(clear_gen_flat f c0 (CHead e1 (Bind b) v) t H2) e2 n H1) f t))) k -H0))))))))))) c)). -(* COMMENTS -Initial nodes: 599 -END *) - -theorem getl_clear_conf: - \forall (i: nat).(\forall (c1: C).(\forall (c3: C).((getl i c1 c3) \to -(\forall (c2: C).((clear c1 c2) \to (getl i c2 c3)))))) -\def - \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (c3: -C).((getl n c1 c3) \to (\forall (c2: C).((clear c1 c2) \to (getl n c2 -c3))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H: (getl O c1 -c3)).(\lambda (c2: C).(\lambda (H0: (clear c1 c2)).(eq_ind C c3 (\lambda (c: -C).(getl O c c3)) (let H1 \def (clear_gen_all c1 c3 (getl_gen_O c1 c3 H)) in -(ex_3_ind B C T (\lambda (b: B).(\lambda (e: C).(\lambda (u: T).(eq C c3 -(CHead e (Bind b) u))))) (getl O c3 c3) (\lambda (x0: B).(\lambda (x1: -C).(\lambda (x2: T).(\lambda (H2: (eq C c3 (CHead x1 (Bind x0) x2))).(let H3 -\def (eq_ind C c3 (\lambda (c: C).(clear c1 c)) (getl_gen_O c1 c3 H) (CHead -x1 (Bind x0) x2) H2) in (eq_ind_r C (CHead x1 (Bind x0) x2) (\lambda (c: -C).(getl O c c)) (getl_refl x0 x1 x2) c3 H2)))))) H1)) c2 (clear_mono c1 c3 -(getl_gen_O c1 c3 H) c2 H0))))))) (\lambda (n: nat).(\lambda (_: ((\forall -(c1: C).(\forall (c3: C).((getl n c1 c3) \to (\forall (c2: C).((clear c1 c2) -\to (getl n c2 c3)))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall -(c3: C).((getl (S n) c c3) \to (\forall (c2: C).((clear c c2) \to (getl (S n) -c2 c3)))))) (\lambda (n0: nat).(\lambda (c3: C).(\lambda (H0: (getl (S n) -(CSort n0) c3)).(\lambda (c2: C).(\lambda (_: (clear (CSort n0) -c2)).(getl_gen_sort n0 (S n) c3 H0 (getl (S n) c2 c3))))))) (\lambda (c: -C).(\lambda (H0: ((\forall (c3: C).((getl (S n) c c3) \to (\forall (c2: -C).((clear c c2) \to (getl (S n) c2 c3))))))).(\lambda (k: K).(\lambda (t: -T).(\lambda (c3: C).(\lambda (H1: (getl (S n) (CHead c k t) c3)).(\lambda -(c2: C).(\lambda (H2: (clear (CHead c k t) c2)).(K_ind (\lambda (k0: -K).((getl (S n) (CHead c k0 t) c3) \to ((clear (CHead c k0 t) c2) \to (getl -(S n) c2 c3)))) (\lambda (b: B).(\lambda (H3: (getl (S n) (CHead c (Bind b) -t) c3)).(\lambda (H4: (clear (CHead c (Bind b) t) c2)).(eq_ind_r C (CHead c -(Bind b) t) (\lambda (c0: C).(getl (S n) c0 c3)) (getl_head (Bind b) n c c3 -(getl_gen_S (Bind b) c c3 t n H3) t) c2 (clear_gen_bind b c c2 t H4))))) -(\lambda (f: F).(\lambda (H3: (getl (S n) (CHead c (Flat f) t) c3)).(\lambda -(H4: (clear (CHead c (Flat f) t) c2)).(H0 c3 (getl_gen_S (Flat f) c c3 t n -H3) c2 (clear_gen_flat f c c2 t H4))))) k H1 H2))))))))) c1)))) i). -(* COMMENTS -Initial nodes: 641 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/getl/dec.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/getl/dec.ma deleted file mode 100644 index 278362590..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/getl/dec.ma +++ /dev/null @@ -1,100 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/getl/props.ma". - -theorem getl_dec: - \forall (c: C).(\forall (i: nat).(or (ex_3 C B T (\lambda (e: C).(\lambda -(b: B).(\lambda (v: T).(getl i c (CHead e (Bind b) v)))))) (\forall (d: -C).((getl i c d) \to (\forall (P: Prop).P))))) -\def - \lambda (c: C).(C_ind (\lambda (c0: C).(\forall (i: nat).(or (ex_3 C B T -(\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl i c0 (CHead e (Bind b) -v)))))) (\forall (d: C).((getl i c0 d) \to (\forall (P: Prop).P)))))) -(\lambda (n: nat).(\lambda (i: nat).(or_intror (ex_3 C B T (\lambda (e: -C).(\lambda (b: B).(\lambda (v: T).(getl i (CSort n) (CHead e (Bind b) -v)))))) (\forall (d: C).((getl i (CSort n) d) \to (\forall (P: Prop).P))) -(\lambda (d: C).(\lambda (H: (getl i (CSort n) d)).(\lambda (P: -Prop).(getl_gen_sort n i d H P))))))) (\lambda (c0: C).(\lambda (H: ((\forall -(i: nat).(or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: -T).(getl i c0 (CHead e (Bind b) v)))))) (\forall (d: C).((getl i c0 d) \to -(\forall (P: Prop).P))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (i: -nat).(nat_ind (\lambda (n: nat).(or (ex_3 C B T (\lambda (e: C).(\lambda (b: -B).(\lambda (v: T).(getl n (CHead c0 k t) (CHead e (Bind b) v)))))) (\forall -(d: C).((getl n (CHead c0 k t) d) \to (\forall (P: Prop).P))))) (K_ind -(\lambda (k0: K).(or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: -T).(getl O (CHead c0 k0 t) (CHead e (Bind b) v)))))) (\forall (d: C).((getl O -(CHead c0 k0 t) d) \to (\forall (P: Prop).P))))) (\lambda (b: B).(or_introl -(ex_3 C B T (\lambda (e: C).(\lambda (b0: B).(\lambda (v: T).(getl O (CHead -c0 (Bind b) t) (CHead e (Bind b0) v)))))) (\forall (d: C).((getl O (CHead c0 -(Bind b) t) d) \to (\forall (P: Prop).P))) (ex_3_intro C B T (\lambda (e: -C).(\lambda (b0: B).(\lambda (v: T).(getl O (CHead c0 (Bind b) t) (CHead e -(Bind b0) v))))) c0 b t (getl_refl b c0 t)))) (\lambda (f: F).(let H_x \def -(H O) in (let H0 \def H_x in (or_ind (ex_3 C B T (\lambda (e: C).(\lambda (b: -B).(\lambda (v: T).(getl O c0 (CHead e (Bind b) v)))))) (\forall (d: -C).((getl O c0 d) \to (\forall (P: Prop).P))) (or (ex_3 C B T (\lambda (e: -C).(\lambda (b: B).(\lambda (v: T).(getl O (CHead c0 (Flat f) t) (CHead e -(Bind b) v)))))) (\forall (d: C).((getl O (CHead c0 (Flat f) t) d) \to -(\forall (P: Prop).P)))) (\lambda (H1: (ex_3 C B T (\lambda (e: C).(\lambda -(b: B).(\lambda (v: T).(getl O c0 (CHead e (Bind b) v))))))).(ex_3_ind C B T -(\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl O c0 (CHead e (Bind b) -v))))) (or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl -O (CHead c0 (Flat f) t) (CHead e (Bind b) v)))))) (\forall (d: C).((getl O -(CHead c0 (Flat f) t) d) \to (\forall (P: Prop).P)))) (\lambda (x0: -C).(\lambda (x1: B).(\lambda (x2: T).(\lambda (H2: (getl O c0 (CHead x0 (Bind -x1) x2))).(or_introl (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: -T).(getl O (CHead c0 (Flat f) t) (CHead e (Bind b) v)))))) (\forall (d: -C).((getl O (CHead c0 (Flat f) t) d) \to (\forall (P: Prop).P))) (ex_3_intro -C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl O (CHead c0 (Flat -f) t) (CHead e (Bind b) v))))) x0 x1 x2 (getl_flat c0 (CHead x0 (Bind x1) x2) -O H2 f t))))))) H1)) (\lambda (H1: ((\forall (d: C).((getl O c0 d) \to -(\forall (P: Prop).P))))).(or_intror (ex_3 C B T (\lambda (e: C).(\lambda (b: -B).(\lambda (v: T).(getl O (CHead c0 (Flat f) t) (CHead e (Bind b) v)))))) -(\forall (d: C).((getl O (CHead c0 (Flat f) t) d) \to (\forall (P: Prop).P))) -(\lambda (d: C).(\lambda (H2: (getl O (CHead c0 (Flat f) t) d)).(\lambda (P: -Prop).(H1 d (getl_intro O c0 d c0 (drop_refl c0) (clear_gen_flat f c0 d t -(getl_gen_O (CHead c0 (Flat f) t) d H2))) P)))))) H0)))) k) (\lambda (n: -nat).(\lambda (_: (or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda -(v: T).(getl n (CHead c0 k t) (CHead e (Bind b) v)))))) (\forall (d: -C).((getl n (CHead c0 k t) d) \to (\forall (P: Prop).P))))).(let H_x \def (H -(r k n)) in (let H1 \def H_x in (or_ind (ex_3 C B T (\lambda (e: C).(\lambda -(b: B).(\lambda (v: T).(getl (r k n) c0 (CHead e (Bind b) v)))))) (\forall -(d: C).((getl (r k n) c0 d) \to (\forall (P: Prop).P))) (or (ex_3 C B T -(\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl (S n) (CHead c0 k t) -(CHead e (Bind b) v)))))) (\forall (d: C).((getl (S n) (CHead c0 k t) d) \to -(\forall (P: Prop).P)))) (\lambda (H2: (ex_3 C B T (\lambda (e: C).(\lambda -(b: B).(\lambda (v: T).(getl (r k n) c0 (CHead e (Bind b) v))))))).(ex_3_ind -C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl (r k n) c0 (CHead -e (Bind b) v))))) (or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda -(v: T).(getl (S n) (CHead c0 k t) (CHead e (Bind b) v)))))) (\forall (d: -C).((getl (S n) (CHead c0 k t) d) \to (\forall (P: Prop).P)))) (\lambda (x0: -C).(\lambda (x1: B).(\lambda (x2: T).(\lambda (H3: (getl (r k n) c0 (CHead x0 -(Bind x1) x2))).(or_introl (ex_3 C B T (\lambda (e: C).(\lambda (b: -B).(\lambda (v: T).(getl (S n) (CHead c0 k t) (CHead e (Bind b) v)))))) -(\forall (d: C).((getl (S n) (CHead c0 k t) d) \to (\forall (P: Prop).P))) -(ex_3_intro C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl (S n) -(CHead c0 k t) (CHead e (Bind b) v))))) x0 x1 x2 (getl_head k n c0 (CHead x0 -(Bind x1) x2) H3 t))))))) H2)) (\lambda (H2: ((\forall (d: C).((getl (r k n) -c0 d) \to (\forall (P: Prop).P))))).(or_intror (ex_3 C B T (\lambda (e: -C).(\lambda (b: B).(\lambda (v: T).(getl (S n) (CHead c0 k t) (CHead e (Bind -b) v)))))) (\forall (d: C).((getl (S n) (CHead c0 k t) d) \to (\forall (P: -Prop).P))) (\lambda (d: C).(\lambda (H3: (getl (S n) (CHead c0 k t) -d)).(\lambda (P: Prop).(H2 d (getl_gen_S k c0 d t n H3) P)))))) H1))))) -i)))))) c). -(* COMMENTS -Initial nodes: 1563 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/getl/defs.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/getl/defs.ma deleted file mode 100644 index a6efd2eee..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/getl/defs.ma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/drop/defs.ma". - -include "Basic-1/clear/defs.ma". - -inductive getl (h: nat) (c1: C) (c2: C): Prop \def -| getl_intro: \forall (e: C).((drop h O c1 e) \to ((clear e c2) \to (getl h -c1 c2))). - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/getl/drop.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/getl/drop.ma deleted file mode 100644 index e4404e137..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/getl/drop.ma +++ /dev/null @@ -1,514 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/getl/props.ma". - -include "Basic-1/clear/drop.ma". - -theorem getl_drop: - \forall (b: B).(\forall (c: C).(\forall (e: C).(\forall (u: T).(\forall (h: -nat).((getl h c (CHead e (Bind b) u)) \to (drop (S h) O c e)))))) -\def - \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e: -C).(\forall (u: T).(\forall (h: nat).((getl h c0 (CHead e (Bind b) u)) \to -(drop (S h) O c0 e)))))) (\lambda (n: nat).(\lambda (e: C).(\lambda (u: -T).(\lambda (h: nat).(\lambda (H: (getl h (CSort n) (CHead e (Bind b) -u))).(getl_gen_sort n h (CHead e (Bind b) u) H (drop (S h) O (CSort n) -e))))))) (\lambda (c0: C).(\lambda (H: ((\forall (e: C).(\forall (u: -T).(\forall (h: nat).((getl h c0 (CHead e (Bind b) u)) \to (drop (S h) O c0 -e))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e: C).(\lambda (u: -T).(\lambda (h: nat).(nat_ind (\lambda (n: nat).((getl n (CHead c0 k t) -(CHead e (Bind b) u)) \to (drop (S n) O (CHead c0 k t) e))) (\lambda (H0: -(getl O (CHead c0 k t) (CHead e (Bind b) u))).(K_ind (\lambda (k0: K).((clear -(CHead c0 k0 t) (CHead e (Bind b) u)) \to (drop (S O) O (CHead c0 k0 t) e))) -(\lambda (b0: B).(\lambda (H1: (clear (CHead c0 (Bind b0) t) (CHead e (Bind -b) u))).(let H2 \def (f_equal C C (\lambda (e0: C).(match e0 in C return -(\lambda (_: C).C) with [(CSort _) \Rightarrow e | (CHead c1 _ _) \Rightarrow -c1])) (CHead e (Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 -(CHead e (Bind b) u) t H1)) in ((let H3 \def (f_equal C B (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow b | -(CHead _ k0 _) \Rightarrow (match k0 in K return (\lambda (_: K).B) with -[(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow b])])) (CHead e (Bind b) u) -(CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e (Bind b) u) t H1)) in -((let H4 \def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda -(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) -(CHead e (Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e -(Bind b) u) t H1)) in (\lambda (H5: (eq B b b0)).(\lambda (H6: (eq C e -c0)).(eq_ind_r C c0 (\lambda (c1: C).(drop (S O) O (CHead c0 (Bind b0) t) -c1)) (eq_ind B b (\lambda (b1: B).(drop (S O) O (CHead c0 (Bind b1) t) c0)) -(drop_drop (Bind b) O c0 c0 (drop_refl c0) t) b0 H5) e H6)))) H3)) H2)))) -(\lambda (f: F).(\lambda (H1: (clear (CHead c0 (Flat f) t) (CHead e (Bind b) -u))).(drop_clear_O b (CHead c0 (Flat f) t) e u (clear_flat c0 (CHead e (Bind -b) u) (clear_gen_flat f c0 (CHead e (Bind b) u) t H1) f t) e O (drop_refl -e)))) k (getl_gen_O (CHead c0 k t) (CHead e (Bind b) u) H0))) (\lambda (n: -nat).(\lambda (_: (((getl n (CHead c0 k t) (CHead e (Bind b) u)) \to (drop (S -n) O (CHead c0 k t) e)))).(\lambda (H1: (getl (S n) (CHead c0 k t) (CHead e -(Bind b) u))).(drop_drop k (S n) c0 e (eq_ind_r nat (S (r k n)) (\lambda (n0: -nat).(drop n0 O c0 e)) (H e u (r k n) (getl_gen_S k c0 (CHead e (Bind b) u) t -n H1)) (r k (S n)) (r_S k n)) t)))) h)))))))) c)). -(* COMMENTS -Initial nodes: 827 -END *) - -theorem getl_drop_conf_lt: - \forall (b: B).(\forall (c: C).(\forall (c0: C).(\forall (u: T).(\forall (i: -nat).((getl i c (CHead c0 (Bind b) u)) \to (\forall (e: C).(\forall (h: -nat).(\forall (d: nat).((drop h (S (plus i d)) c e) \to (ex3_2 T C (\lambda -(v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: -C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop -h d c0 e0))))))))))))) -\def - \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (c1: -C).(\forall (u: T).(\forall (i: nat).((getl i c0 (CHead c1 (Bind b) u)) \to -(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h (S (plus i d)) -c0 e) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) -(\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda -(_: T).(\lambda (e0: C).(drop h d c1 e0))))))))))))) (\lambda (n: -nat).(\lambda (c0: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H: (getl i -(CSort n) (CHead c0 (Bind b) u))).(\lambda (e: C).(\lambda (h: nat).(\lambda -(d: nat).(\lambda (_: (drop h (S (plus i d)) (CSort n) e)).(getl_gen_sort n i -(CHead c0 (Bind b) u) H (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u -(lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) -v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c0 e0)))))))))))))) (\lambda -(c0: C).(\lambda (H: ((\forall (c1: C).(\forall (u: T).(\forall (i: -nat).((getl i c0 (CHead c1 (Bind b) u)) \to (\forall (e: C).(\forall (h: -nat).(\forall (d: nat).((drop h (S (plus i d)) c0 e) \to (ex3_2 T C (\lambda -(v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: -C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop -h d c1 e0)))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c1: -C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i (CHead c0 k t) -(CHead c1 (Bind b) u))).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: -nat).(\lambda (H1: (drop h (S (plus i d)) (CHead c0 k t) e)).(let H2 \def -(getl_gen_all (CHead c0 k t) (CHead c1 (Bind b) u) i H0) in (ex2_ind C -(\lambda (e0: C).(drop i O (CHead c0 k t) e0)) (\lambda (e0: C).(clear e0 -(CHead c1 (Bind b) u))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u -(lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) -v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x: -C).(\lambda (H3: (drop i O (CHead c0 k t) x)).(\lambda (H4: (clear x (CHead -c1 (Bind b) u))).(C_ind (\lambda (c2: C).((drop i O (CHead c0 k t) c2) \to -((clear c2 (CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: -C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead -e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) -(\lambda (n: nat).(\lambda (_: (drop i O (CHead c0 k t) (CSort n))).(\lambda -(H6: (clear (CSort n) (CHead c1 (Bind b) u))).(clear_gen_sort (CHead c1 (Bind -b) u) n H6 (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) -(\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda -(_: T).(\lambda (e0: C).(drop h d c1 e0)))))))) (\lambda (x0: C).(\lambda -(IHx: (((drop i O (CHead c0 k t) x0) \to ((clear x0 (CHead c1 (Bind b) u)) -\to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) -(\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda -(_: T).(\lambda (e0: C).(drop h d c1 e0)))))))).(\lambda (k0: K).(\lambda -(t0: T).(\lambda (H5: (drop i O (CHead c0 k t) (CHead x0 k0 t0))).(\lambda -(H6: (clear (CHead x0 k0 t0) (CHead c1 (Bind b) u))).(K_ind (\lambda (k1: -K).((drop i O (CHead c0 k t) (CHead x0 k1 t0)) \to ((clear (CHead x0 k1 t0) -(CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u -(lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) -v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) (\lambda (b0: -B).(\lambda (H7: (drop i O (CHead c0 k t) (CHead x0 (Bind b0) t0))).(\lambda -(H8: (clear (CHead x0 (Bind b0) t0) (CHead c1 (Bind b) u))).(let H9 \def -(f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with -[(CSort _) \Rightarrow c1 | (CHead c2 _ _) \Rightarrow c2])) (CHead c1 (Bind -b) u) (CHead x0 (Bind b0) t0) (clear_gen_bind b0 x0 (CHead c1 (Bind b) u) t0 -H8)) in ((let H10 \def (f_equal C B (\lambda (e0: C).(match e0 in C return -(\lambda (_: C).B) with [(CSort _) \Rightarrow b | (CHead _ k1 _) \Rightarrow -(match k1 in K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | -(Flat _) \Rightarrow b])])) (CHead c1 (Bind b) u) (CHead x0 (Bind b0) t0) -(clear_gen_bind b0 x0 (CHead c1 (Bind b) u) t0 H8)) in ((let H11 \def -(f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow u | (CHead _ _ t1) \Rightarrow t1])) (CHead c1 (Bind -b) u) (CHead x0 (Bind b0) t0) (clear_gen_bind b0 x0 (CHead c1 (Bind b) u) t0 -H8)) in (\lambda (H12: (eq B b b0)).(\lambda (H13: (eq C c1 x0)).(let H14 -\def (eq_ind_r T t0 (\lambda (t1: T).(drop i O (CHead c0 k t) (CHead x0 (Bind -b0) t1))) H7 u H11) in (let H15 \def (eq_ind_r B b0 (\lambda (b1: B).(drop i -O (CHead c0 k t) (CHead x0 (Bind b1) u))) H14 b H12) in (let H16 \def -(eq_ind_r C x0 (\lambda (c2: C).((drop i O (CHead c0 k t) c2) \to ((clear c2 -(CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u -(lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) -v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) IHx c1 H13) in -(let H17 \def (eq_ind_r C x0 (\lambda (c2: C).(drop i O (CHead c0 k t) (CHead -c2 (Bind b) u))) H15 c1 H13) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: -C).(eq T u (lift h (r (Bind b) d) v)))) (\lambda (v: T).(\lambda (e0: -C).(drop i O e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: -C).(drop h (r (Bind b) d) c1 e0))) (ex3_2 T C (\lambda (v: T).(\lambda (_: -C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead -e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))) -(\lambda (x1: T).(\lambda (x2: C).(\lambda (H18: (eq T u (lift h (r (Bind b) -d) x1))).(\lambda (H19: (drop i O e (CHead x2 (Bind b) x1))).(\lambda (H20: -(drop h (r (Bind b) d) c1 x2)).(let H21 \def (eq_ind T u (\lambda (t1: -T).((drop i O (CHead c0 k t) c1) \to ((clear c1 (CHead c1 (Bind b) t1)) \to -(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T t1 (lift h d v)))) (\lambda -(v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: -T).(\lambda (e0: C).(drop h d c1 e0))))))) H16 (lift h (r (Bind b) d) x1) -H18) in (eq_ind_r T (lift h (r (Bind b) d) x1) (\lambda (t1: T).(ex3_2 T C -(\lambda (v: T).(\lambda (_: C).(eq T t1 (lift h d v)))) (\lambda (v: -T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: -T).(\lambda (e0: C).(drop h d c1 e0))))) (ex3_2_intro T C (\lambda (v: -T).(\lambda (_: C).(eq T (lift h (r (Bind b) d) x1) (lift h d v)))) (\lambda -(v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: -T).(\lambda (e0: C).(drop h d c1 e0))) x1 x2 (refl_equal T (lift h d x1)) -(getl_intro i e (CHead x2 (Bind b) x1) (CHead x2 (Bind b) x1) H19 (clear_bind -b x2 x1)) H20) u H18))))))) (drop_conf_lt (Bind b) i u c1 (CHead c0 k t) H17 -e h d H1))))))))) H10)) H9))))) (\lambda (f: F).(\lambda (H7: (drop i O -(CHead c0 k t) (CHead x0 (Flat f) t0))).(\lambda (H8: (clear (CHead x0 (Flat -f) t0) (CHead c1 (Bind b) u))).(nat_ind (\lambda (n: nat).((drop h (S (plus n -d)) (CHead c0 k t) e) \to ((drop n O (CHead c0 k t) (CHead x0 (Flat f) t0)) -\to ((((drop n O (CHead c0 k t) x0) \to ((clear x0 (CHead c1 (Bind b) u)) \to -(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda -(v: T).(\lambda (e0: C).(getl n e (CHead e0 (Bind b) v)))) (\lambda (_: -T).(\lambda (e0: C).(drop h d c1 e0))))))) \to (ex3_2 T C (\lambda (v: -T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: -C).(getl n e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop -h d c1 e0)))))))) (\lambda (H9: (drop h (S (plus O d)) (CHead c0 k t) -e)).(\lambda (H10: (drop O O (CHead c0 k t) (CHead x0 (Flat f) t0))).(\lambda -(IHx0: (((drop O O (CHead c0 k t) x0) \to ((clear x0 (CHead c1 (Bind b) u)) -\to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) -(\lambda (v: T).(\lambda (e0: C).(getl O e (CHead e0 (Bind b) v)))) (\lambda -(_: T).(\lambda (e0: C).(drop h d c1 e0)))))))).(let H11 \def (f_equal C C -(\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow c0 | (CHead c2 _ _) \Rightarrow c2])) (CHead c0 k t) (CHead x0 -(Flat f) t0) (drop_gen_refl (CHead c0 k t) (CHead x0 (Flat f) t0) H10)) in -((let H12 \def (f_equal C K (\lambda (e0: C).(match e0 in C return (\lambda -(_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k1 _) \Rightarrow k1])) -(CHead c0 k t) (CHead x0 (Flat f) t0) (drop_gen_refl (CHead c0 k t) (CHead x0 -(Flat f) t0) H10)) in ((let H13 \def (f_equal C T (\lambda (e0: C).(match e0 -in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow t | (CHead _ _ t1) -\Rightarrow t1])) (CHead c0 k t) (CHead x0 (Flat f) t0) (drop_gen_refl (CHead -c0 k t) (CHead x0 (Flat f) t0) H10)) in (\lambda (H14: (eq K k (Flat -f))).(\lambda (H15: (eq C c0 x0)).(let H16 \def (eq_ind_r C x0 (\lambda (c2: -C).(clear c2 (CHead c1 (Bind b) u))) (clear_gen_flat f x0 (CHead c1 (Bind b) -u) t0 H8) c0 H15) in (let H17 \def (eq_ind_r C x0 (\lambda (c2: C).((drop O O -(CHead c0 k t) c2) \to ((clear c2 (CHead c1 (Bind b) u)) \to (ex3_2 T C -(\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: -T).(\lambda (e0: C).(getl O e (CHead e0 (Bind b) v)))) (\lambda (_: -T).(\lambda (e0: C).(drop h d c1 e0))))))) IHx0 c0 H15) in (let H18 \def -(eq_ind K k (\lambda (k1: K).((drop O O (CHead c0 k1 t) c0) \to ((clear c0 -(CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u -(lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl O e (CHead e0 (Bind b) -v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) H17 (Flat f) -H14) in (let H19 \def (eq_ind K k (\lambda (k1: K).(drop h (S (plus O d)) -(CHead c0 k1 t) e)) H9 (Flat f) H14) in (ex3_2_ind C T (\lambda (e0: -C).(\lambda (v: T).(eq C e (CHead e0 (Flat f) v)))) (\lambda (_: C).(\lambda -(v: T).(eq T t (lift h (r (Flat f) (plus O d)) v)))) (\lambda (e0: -C).(\lambda (_: T).(drop h (r (Flat f) (plus O d)) c0 e0))) (ex3_2 T C -(\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: -T).(\lambda (e0: C).(getl O e (CHead e0 (Bind b) v)))) (\lambda (_: -T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x1: C).(\lambda (x2: -T).(\lambda (H20: (eq C e (CHead x1 (Flat f) x2))).(\lambda (H21: (eq T t -(lift h (r (Flat f) (plus O d)) x2))).(\lambda (H22: (drop h (r (Flat f) -(plus O d)) c0 x1)).(let H23 \def (f_equal T T (\lambda (e0: T).e0) t (lift h -(r (Flat f) (plus O d)) x2) H21) in (let H24 \def (eq_ind C e (\lambda (c2: -C).((drop O O (CHead c0 (Flat f) t) c0) \to ((clear c0 (CHead c1 (Bind b) u)) -\to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) -(\lambda (v: T).(\lambda (e0: C).(getl O c2 (CHead e0 (Bind b) v)))) (\lambda -(_: T).(\lambda (e0: C).(drop h d c1 e0))))))) H18 (CHead x1 (Flat f) x2) -H20) in (eq_ind_r C (CHead x1 (Flat f) x2) (\lambda (c2: C).(ex3_2 T C -(\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: -T).(\lambda (e0: C).(getl O c2 (CHead e0 (Bind b) v)))) (\lambda (_: -T).(\lambda (e0: C).(drop h d c1 e0))))) (let H25 \def (eq_ind T t (\lambda -(t1: T).((drop O O (CHead c0 (Flat f) t1) c0) \to ((clear c0 (CHead c1 (Bind -b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) -(\lambda (v: T).(\lambda (e0: C).(getl O (CHead x1 (Flat f) x2) (CHead e0 -(Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) H24 -(lift h (S d) x2) H23) in (let H26 \def (H c1 u O (getl_intro O c0 (CHead c1 -(Bind b) u) c0 (drop_refl c0) H16) x1 h d H22) in (ex3_2_ind T C (\lambda (v: -T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: -C).(getl O x1 (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop -h d c1 e0))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d -v)))) (\lambda (v: T).(\lambda (e0: C).(getl O (CHead x1 (Flat f) x2) (CHead -e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))) -(\lambda (x3: T).(\lambda (x4: C).(\lambda (H27: (eq T u (lift h d -x3))).(\lambda (H28: (getl O x1 (CHead x4 (Bind b) x3))).(\lambda (H29: (drop -h d c1 x4)).(let H30 \def (eq_ind T u (\lambda (t1: T).((drop O O (CHead c0 -(Flat f) (lift h (S d) x2)) c0) \to ((clear c0 (CHead c1 (Bind b) t1)) \to -(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T t1 (lift h d v)))) (\lambda -(v: T).(\lambda (e0: C).(getl O (CHead x1 (Flat f) x2) (CHead e0 (Bind b) -v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) H25 (lift h d -x3) H27) in (let H31 \def (eq_ind T u (\lambda (t1: T).(clear c0 (CHead c1 -(Bind b) t1))) H16 (lift h d x3) H27) in (eq_ind_r T (lift h d x3) (\lambda -(t1: T).(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T t1 (lift h d v)))) -(\lambda (v: T).(\lambda (e0: C).(getl O (CHead x1 (Flat f) x2) (CHead e0 -(Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))) -(ex3_2_intro T C (\lambda (v: T).(\lambda (_: C).(eq T (lift h d x3) (lift h -d v)))) (\lambda (v: T).(\lambda (e0: C).(getl O (CHead x1 (Flat f) x2) -(CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))) -x3 x4 (refl_equal T (lift h d x3)) (getl_flat x1 (CHead x4 (Bind b) x3) O H28 -f x2) H29) u H27)))))))) H26))) e H20)))))))) (drop_gen_skip_l c0 e t h (plus -O d) (Flat f) H19))))))))) H12)) H11))))) (\lambda (i0: nat).(\lambda (IHi: -(((drop h (S (plus i0 d)) (CHead c0 k t) e) \to ((drop i0 O (CHead c0 k t) -(CHead x0 (Flat f) t0)) \to ((((drop i0 O (CHead c0 k t) x0) \to ((clear x0 -(CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u -(lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i0 e (CHead e0 (Bind -b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) \to (ex3_2 T -C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: -T).(\lambda (e0: C).(getl i0 e (CHead e0 (Bind b) v)))) (\lambda (_: -T).(\lambda (e0: C).(drop h d c1 e0))))))))).(\lambda (H9: (drop h (S (plus -(S i0) d)) (CHead c0 k t) e)).(\lambda (H10: (drop (S i0) O (CHead c0 k t) -(CHead x0 (Flat f) t0))).(\lambda (IHx0: (((drop (S i0) O (CHead c0 k t) x0) -\to ((clear x0 (CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda -(_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl (S i0) -e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 -e0)))))))).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 -k v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r k (plus (S i0) d)) -v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r k (plus (S i0) d)) c0 e0))) -(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda -(v: T).(\lambda (e0: C).(getl (S i0) e (CHead e0 (Bind b) v)))) (\lambda (_: -T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x1: C).(\lambda (x2: -T).(\lambda (H11: (eq C e (CHead x1 k x2))).(\lambda (H12: (eq T t (lift h (r -k (plus (S i0) d)) x2))).(\lambda (H13: (drop h (r k (plus (S i0) d)) c0 -x1)).(let H14 \def (f_equal T T (\lambda (e0: T).e0) t (lift h (r k (plus (S -i0) d)) x2) H12) in (let H15 \def (eq_ind C e (\lambda (c2: C).((drop h (S -(plus i0 d)) (CHead c0 k t) c2) \to ((drop i0 O (CHead c0 k t) (CHead x0 -(Flat f) t0)) \to ((((drop i0 O (CHead c0 k t) x0) \to ((clear x0 (CHead c1 -(Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d -v)))) (\lambda (v: T).(\lambda (e0: C).(getl i0 c2 (CHead e0 (Bind b) v)))) -(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) \to (ex3_2 T C -(\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: -T).(\lambda (e0: C).(getl i0 c2 (CHead e0 (Bind b) v)))) (\lambda (_: -T).(\lambda (e0: C).(drop h d c1 e0)))))))) IHi (CHead x1 k x2) H11) in (let -H16 \def (eq_ind C e (\lambda (c2: C).((drop (S i0) O (CHead c0 k t) x0) \to -((clear x0 (CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: -C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl (S i0) c2 -(CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 -e0))))))) IHx0 (CHead x1 k x2) H11) in (eq_ind_r C (CHead x1 k x2) (\lambda -(c2: C).(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) -(\lambda (v: T).(\lambda (e0: C).(getl (S i0) c2 (CHead e0 (Bind b) v)))) -(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))) (let H17 \def (eq_ind T -t (\lambda (t1: T).((drop (S i0) O (CHead c0 k t1) x0) \to ((clear x0 (CHead -c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift -h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl (S i0) (CHead x1 k x2) -(CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 -e0))))))) H16 (lift h (r k (S (plus i0 d))) x2) H14) in (let H18 \def (eq_ind -T t (\lambda (t1: T).((drop h (S (plus i0 d)) (CHead c0 k t1) (CHead x1 k -x2)) \to ((drop i0 O (CHead c0 k t1) (CHead x0 (Flat f) t0)) \to ((((drop i0 -O (CHead c0 k t1) x0) \to ((clear x0 (CHead c1 (Bind b) u)) \to (ex3_2 T C -(\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: -T).(\lambda (e0: C).(getl i0 (CHead x1 k x2) (CHead e0 (Bind b) v)))) -(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) \to (ex3_2 T C -(\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: -T).(\lambda (e0: C).(getl i0 (CHead x1 k x2) (CHead e0 (Bind b) v)))) -(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))))))) H15 (lift h (r k (S -(plus i0 d))) x2) H14) in (let H19 \def (eq_ind nat (r k (plus (S i0) d)) -(\lambda (n: nat).(drop h n c0 x1)) H13 (plus (r k (S i0)) d) (r_plus k (S -i0) d)) in (let H20 \def (eq_ind nat (r k (S i0)) (\lambda (n: nat).(drop h -(plus n d) c0 x1)) H19 (S (r k i0)) (r_S k i0)) in (let H21 \def (H c1 u (r k -i0) (getl_intro (r k i0) c0 (CHead c1 (Bind b) u) (CHead x0 (Flat f) t0) -(drop_gen_drop k c0 (CHead x0 (Flat f) t0) t i0 H10) (clear_flat x0 (CHead c1 -(Bind b) u) (clear_gen_flat f x0 (CHead c1 (Bind b) u) t0 H8) f t0)) x1 h d -H20) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d -v)))) (\lambda (v: T).(\lambda (e0: C).(getl (r k i0) x1 (CHead e0 (Bind b) -v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))) (ex3_2 T C (\lambda -(v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: -C).(getl (S i0) (CHead x1 k x2) (CHead e0 (Bind b) v)))) (\lambda (_: -T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x3: T).(\lambda (x4: -C).(\lambda (H22: (eq T u (lift h d x3))).(\lambda (H23: (getl (r k i0) x1 -(CHead x4 (Bind b) x3))).(\lambda (H24: (drop h d c1 x4)).(let H25 \def -(eq_ind T u (\lambda (t1: T).((drop (S i0) O (CHead c0 k (lift h (r k (S -(plus i0 d))) x2)) x0) \to ((clear x0 (CHead c1 (Bind b) t1)) \to (ex3_2 T C -(\lambda (v: T).(\lambda (_: C).(eq T t1 (lift h d v)))) (\lambda (v: -T).(\lambda (e0: C).(getl (S i0) (CHead x1 k x2) (CHead e0 (Bind b) v)))) -(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) H17 (lift h d x3) -H22) in (let H26 \def (eq_ind T u (\lambda (t1: T).(clear x0 (CHead c1 (Bind -b) t1))) (clear_gen_flat f x0 (CHead c1 (Bind b) u) t0 H8) (lift h d x3) H22) -in (eq_ind_r T (lift h d x3) (\lambda (t1: T).(ex3_2 T C (\lambda (v: -T).(\lambda (_: C).(eq T t1 (lift h d v)))) (\lambda (v: T).(\lambda (e0: -C).(getl (S i0) (CHead x1 k x2) (CHead e0 (Bind b) v)))) (\lambda (_: -T).(\lambda (e0: C).(drop h d c1 e0))))) (ex3_2_intro T C (\lambda (v: -T).(\lambda (_: C).(eq T (lift h d x3) (lift h d v)))) (\lambda (v: -T).(\lambda (e0: C).(getl (S i0) (CHead x1 k x2) (CHead e0 (Bind b) v)))) -(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))) x3 x4 (refl_equal T (lift -h d x3)) (getl_head k i0 x1 (CHead x4 (Bind b) x3) H23 x2) H24) u H22)))))))) -H21)))))) e H11))))))))) (drop_gen_skip_l c0 e t h (plus (S i0) d) k -H9))))))) i H1 H7 IHx)))) k0 H5 H6))))))) x H3 H4)))) H2)))))))))))))) c)). -(* COMMENTS -Initial nodes: 6137 -END *) - -theorem getl_drop_conf_ge: - \forall (i: nat).(\forall (a: C).(\forall (c: C).((getl i c a) \to (\forall -(e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to ((le (plus d -h) i) \to (getl (minus i h) e a))))))))) -\def - \lambda (i: nat).(\lambda (a: C).(\lambda (c: C).(\lambda (H: (getl i c -a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H0: (drop h -d c e)).(\lambda (H1: (le (plus d h) i)).(let H2 \def (getl_gen_all c a i H) -in (ex2_ind C (\lambda (e0: C).(drop i O c e0)) (\lambda (e0: C).(clear e0 -a)) (getl (minus i h) e a) (\lambda (x: C).(\lambda (H3: (drop i O c -x)).(\lambda (H4: (clear x a)).(getl_intro (minus i h) e a x (drop_conf_ge i -x c H3 e h d H0 H1) H4)))) H2)))))))))). -(* COMMENTS -Initial nodes: 141 -END *) - -theorem getl_conf_ge_drop: - \forall (b: B).(\forall (c1: C).(\forall (e: C).(\forall (u: T).(\forall (i: -nat).((getl i c1 (CHead e (Bind b) u)) \to (\forall (c2: C).((drop (S O) i c1 -c2) \to (drop i O c2 e)))))))) -\def - \lambda (b: B).(\lambda (c1: C).(\lambda (e: C).(\lambda (u: T).(\lambda (i: -nat).(\lambda (H: (getl i c1 (CHead e (Bind b) u))).(\lambda (c2: C).(\lambda -(H0: (drop (S O) i c1 c2)).(let H3 \def (eq_ind nat (minus (S i) (S O)) -(\lambda (n: nat).(drop n O c2 e)) (drop_conf_ge (S i) e c1 (getl_drop b c1 e -u i H) c2 (S O) i H0 (eq_ind_r nat (plus (S O) i) (\lambda (n: nat).(le n (S -i))) (le_n (S i)) (plus i (S O)) (plus_sym i (S O)))) i (minus_Sx_SO i)) in -H3)))))))). -(* COMMENTS -Initial nodes: 151 -END *) - -theorem getl_drop_conf_rev: - \forall (j: nat).(\forall (e1: C).(\forall (e2: C).((drop j O e1 e2) \to -(\forall (b: B).(\forall (c2: C).(\forall (v2: T).(\forall (i: nat).((getl i -c2 (CHead e2 (Bind b) v2)) \to (ex2 C (\lambda (c1: C).(drop j O c1 c2)) -(\lambda (c1: C).(drop (S i) j c1 e1))))))))))) -\def - \lambda (j: nat).(\lambda (e1: C).(\lambda (e2: C).(\lambda (H: (drop j O e1 -e2)).(\lambda (b: B).(\lambda (c2: C).(\lambda (v2: T).(\lambda (i: -nat).(\lambda (H0: (getl i c2 (CHead e2 (Bind b) v2))).(drop_conf_rev j e1 e2 -H c2 (S i) (getl_drop b c2 e2 v2 i H0)))))))))). -(* COMMENTS -Initial nodes: 69 -END *) - -theorem drop_getl_trans_lt: - \forall (i: nat).(\forall (d: nat).((lt i d) \to (\forall (c1: C).(\forall -(c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (b: B).(\forall (e2: -C).(\forall (v: T).((getl i c2 (CHead e2 (Bind b) v)) \to (ex2 C (\lambda -(e1: C).(getl i c1 (CHead e1 (Bind b) (lift h (minus d (S i)) v)))) (\lambda -(e1: C).(drop h (minus d (S i)) e1 e2))))))))))))) -\def - \lambda (i: nat).(\lambda (d: nat).(\lambda (H: (lt i d)).(\lambda (c1: -C).(\lambda (c2: C).(\lambda (h: nat).(\lambda (H0: (drop h d c1 -c2)).(\lambda (b: B).(\lambda (e2: C).(\lambda (v: T).(\lambda (H1: (getl i -c2 (CHead e2 (Bind b) v))).(let H2 \def (getl_gen_all c2 (CHead e2 (Bind b) -v) i H1) in (ex2_ind C (\lambda (e: C).(drop i O c2 e)) (\lambda (e: -C).(clear e (CHead e2 (Bind b) v))) (ex2 C (\lambda (e1: C).(getl i c1 (CHead -e1 (Bind b) (lift h (minus d (S i)) v)))) (\lambda (e1: C).(drop h (minus d -(S i)) e1 e2))) (\lambda (x: C).(\lambda (H3: (drop i O c2 x)).(\lambda (H4: -(clear x (CHead e2 (Bind b) v))).(ex2_ind C (\lambda (e1: C).(drop i O c1 -e1)) (\lambda (e1: C).(drop h (minus d i) e1 x)) (ex2 C (\lambda (e1: -C).(getl i c1 (CHead e1 (Bind b) (lift h (minus d (S i)) v)))) (\lambda (e1: -C).(drop h (minus d (S i)) e1 e2))) (\lambda (x0: C).(\lambda (H5: (drop i O -c1 x0)).(\lambda (H6: (drop h (minus d i) x0 x)).(let H7 \def (eq_ind nat -(minus d i) (\lambda (n: nat).(drop h n x0 x)) H6 (S (minus d (S i))) -(minus_x_Sy d i H)) in (let H8 \def (drop_clear_S x x0 h (minus d (S i)) H7 b -e2 v H4) in (ex2_ind C (\lambda (c3: C).(clear x0 (CHead c3 (Bind b) (lift h -(minus d (S i)) v)))) (\lambda (c3: C).(drop h (minus d (S i)) c3 e2)) (ex2 C -(\lambda (e1: C).(getl i c1 (CHead e1 (Bind b) (lift h (minus d (S i)) v)))) -(\lambda (e1: C).(drop h (minus d (S i)) e1 e2))) (\lambda (x1: C).(\lambda -(H9: (clear x0 (CHead x1 (Bind b) (lift h (minus d (S i)) v)))).(\lambda -(H10: (drop h (minus d (S i)) x1 e2)).(ex_intro2 C (\lambda (e1: C).(getl i -c1 (CHead e1 (Bind b) (lift h (minus d (S i)) v)))) (\lambda (e1: C).(drop h -(minus d (S i)) e1 e2)) x1 (getl_intro i c1 (CHead x1 (Bind b) (lift h (minus -d (S i)) v)) x0 H5 H9) H10)))) H8)))))) (drop_trans_le i d (le_S_n i d (le_S -(S i) d H)) c1 c2 h H0 x H3))))) H2)))))))))))). -(* COMMENTS -Initial nodes: 627 -END *) - -theorem drop_getl_trans_le: - \forall (i: nat).(\forall (d: nat).((le i d) \to (\forall (c1: C).(\forall -(c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((getl i c2 -e2) \to (ex3_2 C C (\lambda (e0: C).(\lambda (_: C).(drop i O c1 e0))) -(\lambda (e0: C).(\lambda (e1: C).(drop h (minus d i) e0 e1))) (\lambda (_: -C).(\lambda (e1: C).(clear e1 e2)))))))))))) -\def - \lambda (i: nat).(\lambda (d: nat).(\lambda (H: (le i d)).(\lambda (c1: -C).(\lambda (c2: C).(\lambda (h: nat).(\lambda (H0: (drop h d c1 -c2)).(\lambda (e2: C).(\lambda (H1: (getl i c2 e2)).(let H2 \def -(getl_gen_all c2 e2 i H1) in (ex2_ind C (\lambda (e: C).(drop i O c2 e)) -(\lambda (e: C).(clear e e2)) (ex3_2 C C (\lambda (e0: C).(\lambda (_: -C).(drop i O c1 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d i) -e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear e1 e2)))) (\lambda (x: -C).(\lambda (H3: (drop i O c2 x)).(\lambda (H4: (clear x e2)).(let H5 \def -(drop_trans_le i d H c1 c2 h H0 x H3) in (ex2_ind C (\lambda (e1: C).(drop i -O c1 e1)) (\lambda (e1: C).(drop h (minus d i) e1 x)) (ex3_2 C C (\lambda -(e0: C).(\lambda (_: C).(drop i O c1 e0))) (\lambda (e0: C).(\lambda (e1: -C).(drop h (minus d i) e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear e1 -e2)))) (\lambda (x0: C).(\lambda (H6: (drop i O c1 x0)).(\lambda (H7: (drop h -(minus d i) x0 x)).(ex3_2_intro C C (\lambda (e0: C).(\lambda (_: C).(drop i -O c1 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d i) e0 e1))) -(\lambda (_: C).(\lambda (e1: C).(clear e1 e2))) x0 x H6 H7 H4)))) H5))))) -H2)))))))))). -(* COMMENTS -Initial nodes: 323 -END *) - -theorem drop_getl_trans_ge: - \forall (i: nat).(\forall (c1: C).(\forall (c2: C).(\forall (d: -nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((getl i c2 e2) -\to ((le d i) \to (getl (plus i h) c1 e2))))))))) -\def - \lambda (i: nat).(\lambda (c1: C).(\lambda (c2: C).(\lambda (d: -nat).(\lambda (h: nat).(\lambda (H: (drop h d c1 c2)).(\lambda (e2: -C).(\lambda (H0: (getl i c2 e2)).(\lambda (H1: (le d i)).(let H2 \def -(getl_gen_all c2 e2 i H0) in (ex2_ind C (\lambda (e: C).(drop i O c2 e)) -(\lambda (e: C).(clear e e2)) (getl (plus i h) c1 e2) (\lambda (x: -C).(\lambda (H3: (drop i O c2 x)).(\lambda (H4: (clear x e2)).(getl_intro -(plus i h) c1 e2 x (drop_trans_ge i c1 c2 d h H x H3 H1) H4)))) H2)))))))))). -(* COMMENTS -Initial nodes: 137 -END *) - -theorem getl_drop_trans: - \forall (c1: C).(\forall (c2: C).(\forall (h: nat).((getl h c1 c2) \to -(\forall (e2: C).(\forall (i: nat).((drop (S i) O c2 e2) \to (drop (S (plus i -h)) O c1 e2))))))) -\def - \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (h: -nat).((getl h c c2) \to (\forall (e2: C).(\forall (i: nat).((drop (S i) O c2 -e2) \to (drop (S (plus i h)) O c e2)))))))) (\lambda (n: nat).(\lambda (c2: -C).(\lambda (h: nat).(\lambda (H: (getl h (CSort n) c2)).(\lambda (e2: -C).(\lambda (i: nat).(\lambda (_: (drop (S i) O c2 e2)).(getl_gen_sort n h c2 -H (drop (S (plus i h)) O (CSort n) e2))))))))) (\lambda (c2: C).(\lambda -(IHc: ((\forall (c3: C).(\forall (h: nat).((getl h c2 c3) \to (\forall (e2: -C).(\forall (i: nat).((drop (S i) O c3 e2) \to (drop (S (plus i h)) O c2 -e2))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t: T).(\forall -(c3: C).(\forall (h: nat).((getl h (CHead c2 k0 t) c3) \to (\forall (e2: -C).(\forall (i: nat).((drop (S i) O c3 e2) \to (drop (S (plus i h)) O (CHead -c2 k0 t) e2))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (c3: -C).(\lambda (h: nat).(nat_ind (\lambda (n: nat).((getl n (CHead c2 (Bind b) -t) c3) \to (\forall (e2: C).(\forall (i: nat).((drop (S i) O c3 e2) \to (drop -(S (plus i n)) O (CHead c2 (Bind b) t) e2)))))) (\lambda (H: (getl O (CHead -c2 (Bind b) t) c3)).(\lambda (e2: C).(\lambda (i: nat).(\lambda (H0: (drop (S -i) O c3 e2)).(let H1 \def (eq_ind C c3 (\lambda (c: C).(drop (S i) O c e2)) -H0 (CHead c2 (Bind b) t) (clear_gen_bind b c2 c3 t (getl_gen_O (CHead c2 -(Bind b) t) c3 H))) in (eq_ind nat i (\lambda (n: nat).(drop (S n) O (CHead -c2 (Bind b) t) e2)) (drop_drop (Bind b) i c2 e2 (drop_gen_drop (Bind b) c2 e2 -t i H1) t) (plus i O) (plus_n_O i))))))) (\lambda (n: nat).(\lambda (_: -(((getl n (CHead c2 (Bind b) t) c3) \to (\forall (e2: C).(\forall (i: -nat).((drop (S i) O c3 e2) \to (drop (S (plus i n)) O (CHead c2 (Bind b) t) -e2))))))).(\lambda (H0: (getl (S n) (CHead c2 (Bind b) t) c3)).(\lambda (e2: -C).(\lambda (i: nat).(\lambda (H1: (drop (S i) O c3 e2)).(eq_ind nat (plus (S -i) n) (\lambda (n0: nat).(drop (S n0) O (CHead c2 (Bind b) t) e2)) (drop_drop -(Bind b) (plus (S i) n) c2 e2 (IHc c3 n (getl_gen_S (Bind b) c2 c3 t n H0) e2 -i H1) t) (plus i (S n)) (plus_Snm_nSm i n)))))))) h))))) (\lambda (f: -F).(\lambda (t: T).(\lambda (c3: C).(\lambda (h: nat).(nat_ind (\lambda (n: -nat).((getl n (CHead c2 (Flat f) t) c3) \to (\forall (e2: C).(\forall (i: -nat).((drop (S i) O c3 e2) \to (drop (S (plus i n)) O (CHead c2 (Flat f) t) -e2)))))) (\lambda (H: (getl O (CHead c2 (Flat f) t) c3)).(\lambda (e2: -C).(\lambda (i: nat).(\lambda (H0: (drop (S i) O c3 e2)).(drop_drop (Flat f) -(plus i O) c2 e2 (IHc c3 O (getl_intro O c2 c3 c2 (drop_refl c2) -(clear_gen_flat f c2 c3 t (getl_gen_O (CHead c2 (Flat f) t) c3 H))) e2 i H0) -t))))) (\lambda (n: nat).(\lambda (_: (((getl n (CHead c2 (Flat f) t) c3) \to -(\forall (e2: C).(\forall (i: nat).((drop (S i) O c3 e2) \to (drop (S (plus i -n)) O (CHead c2 (Flat f) t) e2))))))).(\lambda (H0: (getl (S n) (CHead c2 -(Flat f) t) c3)).(\lambda (e2: C).(\lambda (i: nat).(\lambda (H1: (drop (S i) -O c3 e2)).(drop_drop (Flat f) (plus i (S n)) c2 e2 (IHc c3 (S n) (getl_gen_S -(Flat f) c2 c3 t n H0) e2 i H1) t))))))) h))))) k)))) c1). -(* COMMENTS -Initial nodes: 953 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/getl/flt.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/getl/flt.ma deleted file mode 100644 index 3d6c90f73..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/getl/flt.ma +++ /dev/null @@ -1,67 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/getl/fwd.ma". - -include "Basic-1/clear/props.ma". - -include "Basic-1/flt/props.ma". - -theorem getl_flt: - \forall (b: B).(\forall (c: C).(\forall (e: C).(\forall (u: T).(\forall (i: -nat).((getl i c (CHead e (Bind b) u)) \to (flt e u c (TLRef i))))))) -\def - \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e: -C).(\forall (u: T).(\forall (i: nat).((getl i c0 (CHead e (Bind b) u)) \to -(flt e u c0 (TLRef i))))))) (\lambda (n: nat).(\lambda (e: C).(\lambda (u: -T).(\lambda (i: nat).(\lambda (H: (getl i (CSort n) (CHead e (Bind b) -u))).(getl_gen_sort n i (CHead e (Bind b) u) H (flt e u (CSort n) (TLRef -i)))))))) (\lambda (c0: C).(\lambda (H: ((\forall (e: C).(\forall (u: -T).(\forall (i: nat).((getl i c0 (CHead e (Bind b) u)) \to (flt e u c0 (TLRef -i)))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e: C).(\lambda (u: -T).(\lambda (i: nat).(nat_ind (\lambda (n: nat).((getl n (CHead c0 k t) -(CHead e (Bind b) u)) \to (flt e u (CHead c0 k t) (TLRef n)))) (\lambda (H0: -(getl O (CHead c0 k t) (CHead e (Bind b) u))).(K_ind (\lambda (k0: K).((clear -(CHead c0 k0 t) (CHead e (Bind b) u)) \to (flt e u (CHead c0 k0 t) (TLRef -O)))) (\lambda (b0: B).(\lambda (H1: (clear (CHead c0 (Bind b0) t) (CHead e -(Bind b) u))).(let H2 \def (f_equal C C (\lambda (e0: C).(match e0 in C -return (\lambda (_: C).C) with [(CSort _) \Rightarrow e | (CHead c1 _ _) -\Rightarrow c1])) (CHead e (Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind -b0 c0 (CHead e (Bind b) u) t H1)) in ((let H3 \def (f_equal C B (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow b | -(CHead _ k0 _) \Rightarrow (match k0 in K return (\lambda (_: K).B) with -[(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow b])])) (CHead e (Bind b) u) -(CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e (Bind b) u) t H1)) in -((let H4 \def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda -(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) -(CHead e (Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e -(Bind b) u) t H1)) in (\lambda (H5: (eq B b b0)).(\lambda (H6: (eq C e -c0)).(eq_ind_r T t (\lambda (t0: T).(flt e t0 (CHead c0 (Bind b0) t) (TLRef -O))) (eq_ind_r C c0 (\lambda (c1: C).(flt c1 t (CHead c0 (Bind b0) t) (TLRef -O))) (eq_ind B b (\lambda (b1: B).(flt c0 t (CHead c0 (Bind b1) t) (TLRef -O))) (flt_arith0 (Bind b) c0 t O) b0 H5) e H6) u H4)))) H3)) H2)))) (\lambda -(f: F).(\lambda (H1: (clear (CHead c0 (Flat f) t) (CHead e (Bind b) -u))).(flt_arith1 (Bind b) e c0 u (clear_cle c0 (CHead e (Bind b) u) -(clear_gen_flat f c0 (CHead e (Bind b) u) t H1)) (Flat f) t O))) k -(getl_gen_O (CHead c0 k t) (CHead e (Bind b) u) H0))) (\lambda (n: -nat).(\lambda (_: (((getl n (CHead c0 k t) (CHead e (Bind b) u)) \to (flt e u -(CHead c0 k t) (TLRef n))))).(\lambda (H1: (getl (S n) (CHead c0 k t) (CHead -e (Bind b) u))).(let H_y \def (H e u (r k n) (getl_gen_S k c0 (CHead e (Bind -b) u) t n H1)) in (flt_arith2 e c0 u (r k n) H_y k t (S n)))))) i)))))))) c)). -(* COMMENTS -Initial nodes: 815 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/getl/fwd.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/getl/fwd.ma deleted file mode 100644 index b378f61fc..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/getl/fwd.ma +++ /dev/null @@ -1,148 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/getl/defs.ma". - -include "Basic-1/drop/fwd.ma". - -include "Basic-1/clear/fwd.ma". - -theorem getl_gen_all: - \forall (c1: C).(\forall (c2: C).(\forall (i: nat).((getl i c1 c2) \to (ex2 -C (\lambda (e: C).(drop i O c1 e)) (\lambda (e: C).(clear e c2)))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (i: nat).(\lambda (H: (getl i c1 -c2)).(getl_ind i c1 c2 (ex2 C (\lambda (e: C).(drop i O c1 e)) (\lambda (e: -C).(clear e c2))) (\lambda (e: C).(\lambda (H0: (drop i O c1 e)).(\lambda -(H1: (clear e c2)).(ex_intro2 C (\lambda (e0: C).(drop i O c1 e0)) (\lambda -(e0: C).(clear e0 c2)) e H0 H1)))) H)))). -(* COMMENTS -Initial nodes: 95 -END *) - -theorem getl_gen_sort: - \forall (n: nat).(\forall (h: nat).(\forall (x: C).((getl h (CSort n) x) \to -(\forall (P: Prop).P)))) -\def - \lambda (n: nat).(\lambda (h: nat).(\lambda (x: C).(\lambda (H: (getl h -(CSort n) x)).(\lambda (P: Prop).(let H0 \def (getl_gen_all (CSort n) x h H) -in (ex2_ind C (\lambda (e: C).(drop h O (CSort n) e)) (\lambda (e: C).(clear -e x)) P (\lambda (x0: C).(\lambda (H1: (drop h O (CSort n) x0)).(\lambda (H2: -(clear x0 x)).(and3_ind (eq C x0 (CSort n)) (eq nat h O) (eq nat O O) P -(\lambda (H3: (eq C x0 (CSort n))).(\lambda (_: (eq nat h O)).(\lambda (_: -(eq nat O O)).(let H6 \def (eq_ind C x0 (\lambda (c: C).(clear c x)) H2 -(CSort n) H3) in (clear_gen_sort x n H6 P))))) (drop_gen_sort n h O x0 -H1))))) H0)))))). -(* COMMENTS -Initial nodes: 179 -END *) - -theorem getl_gen_O: - \forall (e: C).(\forall (x: C).((getl O e x) \to (clear e x))) -\def - \lambda (e: C).(\lambda (x: C).(\lambda (H: (getl O e x)).(let H0 \def -(getl_gen_all e x O H) in (ex2_ind C (\lambda (e0: C).(drop O O e e0)) -(\lambda (e0: C).(clear e0 x)) (clear e x) (\lambda (x0: C).(\lambda (H1: -(drop O O e x0)).(\lambda (H2: (clear x0 x)).(let H3 \def (eq_ind_r C x0 -(\lambda (c: C).(clear c x)) H2 e (drop_gen_refl e x0 H1)) in H3)))) H0)))). -(* COMMENTS -Initial nodes: 99 -END *) - -theorem getl_gen_S: - \forall (k: K).(\forall (c: C).(\forall (x: C).(\forall (u: T).(\forall (h: -nat).((getl (S h) (CHead c k u) x) \to (getl (r k h) c x)))))) -\def - \lambda (k: K).(\lambda (c: C).(\lambda (x: C).(\lambda (u: T).(\lambda (h: -nat).(\lambda (H: (getl (S h) (CHead c k u) x)).(let H0 \def (getl_gen_all -(CHead c k u) x (S h) H) in (ex2_ind C (\lambda (e: C).(drop (S h) O (CHead c -k u) e)) (\lambda (e: C).(clear e x)) (getl (r k h) c x) (\lambda (x0: -C).(\lambda (H1: (drop (S h) O (CHead c k u) x0)).(\lambda (H2: (clear x0 -x)).(getl_intro (r k h) c x x0 (drop_gen_drop k c x0 u h H1) H2)))) H0))))))). -(* COMMENTS -Initial nodes: 145 -END *) - -theorem getl_gen_2: - \forall (c1: C).(\forall (c2: C).(\forall (i: nat).((getl i c1 c2) \to (ex_3 -B C T (\lambda (b: B).(\lambda (c: C).(\lambda (v: T).(eq C c2 (CHead c (Bind -b) v))))))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (i: nat).(\lambda (H: (getl i c1 -c2)).(let H0 \def (getl_gen_all c1 c2 i H) in (ex2_ind C (\lambda (e: -C).(drop i O c1 e)) (\lambda (e: C).(clear e c2)) (ex_3 B C T (\lambda (b: -B).(\lambda (c: C).(\lambda (v: T).(eq C c2 (CHead c (Bind b) v)))))) -(\lambda (x: C).(\lambda (_: (drop i O c1 x)).(\lambda (H2: (clear x -c2)).(let H3 \def (clear_gen_all x c2 H2) in (ex_3_ind B C T (\lambda (b: -B).(\lambda (e: C).(\lambda (u: T).(eq C c2 (CHead e (Bind b) u))))) (ex_3 B -C T (\lambda (b: B).(\lambda (c: C).(\lambda (v: T).(eq C c2 (CHead c (Bind -b) v)))))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H4: -(eq C c2 (CHead x1 (Bind x0) x2))).(let H5 \def (eq_ind C c2 (\lambda (c: -C).(clear x c)) H2 (CHead x1 (Bind x0) x2) H4) in (eq_ind_r C (CHead x1 (Bind -x0) x2) (\lambda (c: C).(ex_3 B C T (\lambda (b: B).(\lambda (c0: C).(\lambda -(v: T).(eq C c (CHead c0 (Bind b) v))))))) (ex_3_intro B C T (\lambda (b: -B).(\lambda (c: C).(\lambda (v: T).(eq C (CHead x1 (Bind x0) x2) (CHead c -(Bind b) v))))) x0 x1 x2 (refl_equal C (CHead x1 (Bind x0) x2))) c2 H4)))))) -H3))))) H0))))). -(* COMMENTS -Initial nodes: 325 -END *) - -theorem getl_gen_flat: - \forall (f: F).(\forall (e: C).(\forall (d: C).(\forall (v: T).(\forall (i: -nat).((getl i (CHead e (Flat f) v) d) \to (getl i e d)))))) -\def - \lambda (f: F).(\lambda (e: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: -nat).(nat_ind (\lambda (n: nat).((getl n (CHead e (Flat f) v) d) \to (getl n -e d))) (\lambda (H: (getl O (CHead e (Flat f) v) d)).(getl_intro O e d e -(drop_refl e) (clear_gen_flat f e d v (getl_gen_O (CHead e (Flat f) v) d -H)))) (\lambda (n: nat).(\lambda (_: (((getl n (CHead e (Flat f) v) d) \to -(getl n e d)))).(\lambda (H0: (getl (S n) (CHead e (Flat f) v) -d)).(getl_gen_S (Flat f) e d v n H0)))) i))))). -(* COMMENTS -Initial nodes: 155 -END *) - -theorem getl_gen_bind: - \forall (b: B).(\forall (e: C).(\forall (d: C).(\forall (v: T).(\forall (i: -nat).((getl i (CHead e (Bind b) v) d) \to (or (land (eq nat i O) (eq C d -(CHead e (Bind b) v))) (ex2 nat (\lambda (j: nat).(eq nat i (S j))) (\lambda -(j: nat).(getl j e d))))))))) -\def - \lambda (b: B).(\lambda (e: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: -nat).(nat_ind (\lambda (n: nat).((getl n (CHead e (Bind b) v) d) \to (or -(land (eq nat n O) (eq C d (CHead e (Bind b) v))) (ex2 nat (\lambda (j: -nat).(eq nat n (S j))) (\lambda (j: nat).(getl j e d)))))) (\lambda (H: (getl -O (CHead e (Bind b) v) d)).(eq_ind_r C (CHead e (Bind b) v) (\lambda (c: -C).(or (land (eq nat O O) (eq C c (CHead e (Bind b) v))) (ex2 nat (\lambda -(j: nat).(eq nat O (S j))) (\lambda (j: nat).(getl j e c))))) (or_introl -(land (eq nat O O) (eq C (CHead e (Bind b) v) (CHead e (Bind b) v))) (ex2 nat -(\lambda (j: nat).(eq nat O (S j))) (\lambda (j: nat).(getl j e (CHead e -(Bind b) v)))) (conj (eq nat O O) (eq C (CHead e (Bind b) v) (CHead e (Bind -b) v)) (refl_equal nat O) (refl_equal C (CHead e (Bind b) v)))) d -(clear_gen_bind b e d v (getl_gen_O (CHead e (Bind b) v) d H)))) (\lambda (n: -nat).(\lambda (_: (((getl n (CHead e (Bind b) v) d) \to (or (land (eq nat n -O) (eq C d (CHead e (Bind b) v))) (ex2 nat (\lambda (j: nat).(eq nat n (S -j))) (\lambda (j: nat).(getl j e d))))))).(\lambda (H0: (getl (S n) (CHead e -(Bind b) v) d)).(or_intror (land (eq nat (S n) O) (eq C d (CHead e (Bind b) -v))) (ex2 nat (\lambda (j: nat).(eq nat (S n) (S j))) (\lambda (j: nat).(getl -j e d))) (ex_intro2 nat (\lambda (j: nat).(eq nat (S n) (S j))) (\lambda (j: -nat).(getl j e d)) n (refl_equal nat (S n)) (getl_gen_S (Bind b) e d v n -H0)))))) i))))). -(* COMMENTS -Initial nodes: 525 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/getl/getl.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/getl/getl.ma deleted file mode 100644 index 71327becf..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/getl/getl.ma +++ /dev/null @@ -1,57 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/getl/drop.ma". - -include "Basic-1/getl/clear.ma". - -theorem getl_conf_le: - \forall (i: nat).(\forall (a: C).(\forall (c: C).((getl i c a) \to (\forall -(e: C).(\forall (h: nat).((getl h c e) \to ((le h i) \to (getl (minus i h) e -a)))))))) -\def - \lambda (i: nat).(\lambda (a: C).(\lambda (c: C).(\lambda (H: (getl i c -a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (H0: (getl h c e)).(\lambda -(H1: (le h i)).(let H2 \def (getl_gen_all c e h H0) in (ex2_ind C (\lambda -(e0: C).(drop h O c e0)) (\lambda (e0: C).(clear e0 e)) (getl (minus i h) e -a) (\lambda (x: C).(\lambda (H3: (drop h O c x)).(\lambda (H4: (clear x -e)).(getl_clear_conf (minus i h) x a (getl_drop_conf_ge i a c H x h O H3 H1) -e H4)))) H2))))))))). -(* COMMENTS -Initial nodes: 133 -END *) - -theorem getl_trans: - \forall (i: nat).(\forall (c1: C).(\forall (c2: C).(\forall (h: nat).((getl -h c1 c2) \to (\forall (e2: C).((getl i c2 e2) \to (getl (plus i h) c1 -e2))))))) -\def - \lambda (i: nat).(\lambda (c1: C).(\lambda (c2: C).(\lambda (h: -nat).(\lambda (H: (getl h c1 c2)).(\lambda (e2: C).(\lambda (H0: (getl i c2 -e2)).(let H1 \def (getl_gen_all c2 e2 i H0) in (ex2_ind C (\lambda (e: -C).(drop i O c2 e)) (\lambda (e: C).(clear e e2)) (getl (plus i h) c1 e2) -(\lambda (x: C).(\lambda (H2: (drop i O c2 x)).(\lambda (H3: (clear x -e2)).(nat_ind (\lambda (n: nat).((drop n O c2 x) \to (getl (plus n h) c1 -e2))) (\lambda (H4: (drop O O c2 x)).(let H5 \def (eq_ind_r C x (\lambda (c: -C).(clear c e2)) H3 c2 (drop_gen_refl c2 x H4)) in (getl_clear_trans (plus O -h) c1 c2 H e2 H5))) (\lambda (i0: nat).(\lambda (_: (((drop i0 O c2 x) \to -(getl (plus i0 h) c1 e2)))).(\lambda (H4: (drop (S i0) O c2 x)).(let H_y \def -(getl_drop_trans c1 c2 h H x i0 H4) in (getl_intro (plus (S i0) h) c1 e2 x -H_y H3))))) i H2)))) H1)))))))). -(* COMMENTS -Initial nodes: 247 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/getl/props.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/getl/props.ma deleted file mode 100644 index 9399ad4ab..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/getl/props.ma +++ /dev/null @@ -1,104 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/getl/fwd.ma". - -include "Basic-1/drop/props.ma". - -include "Basic-1/clear/props.ma". - -theorem getl_refl: - \forall (b: B).(\forall (c: C).(\forall (u: T).(getl O (CHead c (Bind b) u) -(CHead c (Bind b) u)))) -\def - \lambda (b: B).(\lambda (c: C).(\lambda (u: T).(getl_intro O (CHead c (Bind -b) u) (CHead c (Bind b) u) (CHead c (Bind b) u) (drop_refl (CHead c (Bind b) -u)) (clear_bind b c u)))). -(* COMMENTS -Initial nodes: 59 -END *) - -theorem getl_head: - \forall (k: K).(\forall (h: nat).(\forall (c: C).(\forall (e: C).((getl (r k -h) c e) \to (\forall (u: T).(getl (S h) (CHead c k u) e)))))) -\def - \lambda (k: K).(\lambda (h: nat).(\lambda (c: C).(\lambda (e: C).(\lambda -(H: (getl (r k h) c e)).(\lambda (u: T).(let H0 \def (getl_gen_all c e (r k -h) H) in (ex2_ind C (\lambda (e0: C).(drop (r k h) O c e0)) (\lambda (e0: -C).(clear e0 e)) (getl (S h) (CHead c k u) e) (\lambda (x: C).(\lambda (H1: -(drop (r k h) O c x)).(\lambda (H2: (clear x e)).(getl_intro (S h) (CHead c k -u) e x (drop_drop k h c x H1 u) H2)))) H0))))))). -(* COMMENTS -Initial nodes: 137 -END *) - -theorem getl_flat: - \forall (c: C).(\forall (e: C).(\forall (h: nat).((getl h c e) \to (\forall -(f: F).(\forall (u: T).(getl h (CHead c (Flat f) u) e)))))) -\def - \lambda (c: C).(\lambda (e: C).(\lambda (h: nat).(\lambda (H: (getl h c -e)).(\lambda (f: F).(\lambda (u: T).(let H0 \def (getl_gen_all c e h H) in -(ex2_ind C (\lambda (e0: C).(drop h O c e0)) (\lambda (e0: C).(clear e0 e)) -(getl h (CHead c (Flat f) u) e) (\lambda (x: C).(\lambda (H1: (drop h O c -x)).(\lambda (H2: (clear x e)).(nat_ind (\lambda (n: nat).((drop n O c x) \to -(getl n (CHead c (Flat f) u) e))) (\lambda (H3: (drop O O c x)).(let H4 \def -(eq_ind_r C x (\lambda (c0: C).(clear c0 e)) H2 c (drop_gen_refl c x H3)) in -(getl_intro O (CHead c (Flat f) u) e (CHead c (Flat f) u) (drop_refl (CHead c -(Flat f) u)) (clear_flat c e H4 f u)))) (\lambda (h0: nat).(\lambda (_: -(((drop h0 O c x) \to (getl h0 (CHead c (Flat f) u) e)))).(\lambda (H3: (drop -(S h0) O c x)).(getl_intro (S h0) (CHead c (Flat f) u) e x (drop_drop (Flat -f) h0 c x H3 u) H2)))) h H1)))) H0))))))). -(* COMMENTS -Initial nodes: 285 -END *) - -theorem getl_ctail: - \forall (b: B).(\forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: -nat).((getl i c (CHead d (Bind b) u)) \to (\forall (k: K).(\forall (v: -T).(getl i (CTail k v c) (CHead (CTail k v d) (Bind b) u))))))))) -\def - \lambda (b: B).(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: -nat).(\lambda (H: (getl i c (CHead d (Bind b) u))).(\lambda (k: K).(\lambda -(v: T).(let H0 \def (getl_gen_all c (CHead d (Bind b) u) i H) in (ex2_ind C -(\lambda (e: C).(drop i O c e)) (\lambda (e: C).(clear e (CHead d (Bind b) -u))) (getl i (CTail k v c) (CHead (CTail k v d) (Bind b) u)) (\lambda (x: -C).(\lambda (H1: (drop i O c x)).(\lambda (H2: (clear x (CHead d (Bind b) -u))).(getl_intro i (CTail k v c) (CHead (CTail k v d) (Bind b) u) (CTail k v -x) (drop_ctail c x O i H1 k v) (clear_ctail b x d u H2 k v))))) H0))))))))). -(* COMMENTS -Initial nodes: 203 -END *) - -theorem getl_mono: - \forall (c: C).(\forall (x1: C).(\forall (h: nat).((getl h c x1) \to -(\forall (x2: C).((getl h c x2) \to (eq C x1 x2)))))) -\def - \lambda (c: C).(\lambda (x1: C).(\lambda (h: nat).(\lambda (H: (getl h c -x1)).(\lambda (x2: C).(\lambda (H0: (getl h c x2)).(let H1 \def (getl_gen_all -c x2 h H0) in (ex2_ind C (\lambda (e: C).(drop h O c e)) (\lambda (e: -C).(clear e x2)) (eq C x1 x2) (\lambda (x: C).(\lambda (H2: (drop h O c -x)).(\lambda (H3: (clear x x2)).(let H4 \def (getl_gen_all c x1 h H) in -(ex2_ind C (\lambda (e: C).(drop h O c e)) (\lambda (e: C).(clear e x1)) (eq -C x1 x2) (\lambda (x0: C).(\lambda (H5: (drop h O c x0)).(\lambda (H6: (clear -x0 x1)).(let H7 \def (eq_ind C x (\lambda (c0: C).(drop h O c c0)) H2 x0 -(drop_mono c x O h H2 x0 H5)) in (let H8 \def (eq_ind_r C x0 (\lambda (c0: -C).(drop h O c c0)) H7 x (drop_mono c x O h H2 x0 H5)) in (let H9 \def -(eq_ind_r C x0 (\lambda (c0: C).(clear c0 x1)) H6 x (drop_mono c x O h H2 x0 -H5)) in (clear_mono x x1 H9 x2 H3))))))) H4))))) H1))))))). -(* COMMENTS -Initial nodes: 269 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/iso/defs.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/iso/defs.ma deleted file mode 100644 index 202b8a300..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/iso/defs.ma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/T/defs.ma". - -inductive iso: T \to (T \to Prop) \def -| iso_sort: \forall (n1: nat).(\forall (n2: nat).(iso (TSort n1) (TSort n2))) -| iso_lref: \forall (i1: nat).(\forall (i2: nat).(iso (TLRef i1) (TLRef i2))) -| iso_head: \forall (v1: T).(\forall (v2: T).(\forall (t1: T).(\forall (t2: -T).(\forall (k: K).(iso (THead k v1 t1) (THead k v2 t2)))))). - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/iso/fwd.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/iso/fwd.ma deleted file mode 100644 index 761e982f7..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/iso/fwd.ma +++ /dev/null @@ -1,191 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/iso/defs.ma". - -include "Basic-1/tlist/defs.ma". - -theorem iso_gen_sort: - \forall (u2: T).(\forall (n1: nat).((iso (TSort n1) u2) \to (ex nat (\lambda -(n2: nat).(eq T u2 (TSort n2)))))) -\def - \lambda (u2: T).(\lambda (n1: nat).(\lambda (H: (iso (TSort n1) -u2)).(insert_eq T (TSort n1) (\lambda (t: T).(iso t u2)) (\lambda (_: T).(ex -nat (\lambda (n2: nat).(eq T u2 (TSort n2))))) (\lambda (y: T).(\lambda (H0: -(iso y u2)).(iso_ind (\lambda (t: T).(\lambda (t0: T).((eq T t (TSort n1)) -\to (ex nat (\lambda (n2: nat).(eq T t0 (TSort n2))))))) (\lambda (n0: -nat).(\lambda (n2: nat).(\lambda (H1: (eq T (TSort n0) (TSort n1))).(let H2 -\def (f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) -with [(TSort n) \Rightarrow n | (TLRef _) \Rightarrow n0 | (THead _ _ _) -\Rightarrow n0])) (TSort n0) (TSort n1) H1) in (ex_intro nat (\lambda (n3: -nat).(eq T (TSort n2) (TSort n3))) n2 (refl_equal T (TSort n2))))))) (\lambda -(i1: nat).(\lambda (i2: nat).(\lambda (H1: (eq T (TLRef i1) (TSort n1))).(let -H2 \def (eq_ind T (TLRef i1) (\lambda (ee: T).(match ee in T return (\lambda -(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | -(THead _ _ _) \Rightarrow False])) I (TSort n1) H1) in (False_ind (ex nat -(\lambda (n2: nat).(eq T (TLRef i2) (TSort n2)))) H2))))) (\lambda (v1: -T).(\lambda (v2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k: -K).(\lambda (H1: (eq T (THead k v1 t1) (TSort n1))).(let H2 \def (eq_ind T -(THead k v1 t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) -with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ -_) \Rightarrow True])) I (TSort n1) H1) in (False_ind (ex nat (\lambda (n2: -nat).(eq T (THead k v2 t2) (TSort n2)))) H2)))))))) y u2 H0))) H))). -(* COMMENTS -Initial nodes: 321 -END *) - -theorem iso_gen_lref: - \forall (u2: T).(\forall (n1: nat).((iso (TLRef n1) u2) \to (ex nat (\lambda -(n2: nat).(eq T u2 (TLRef n2)))))) -\def - \lambda (u2: T).(\lambda (n1: nat).(\lambda (H: (iso (TLRef n1) -u2)).(insert_eq T (TLRef n1) (\lambda (t: T).(iso t u2)) (\lambda (_: T).(ex -nat (\lambda (n2: nat).(eq T u2 (TLRef n2))))) (\lambda (y: T).(\lambda (H0: -(iso y u2)).(iso_ind (\lambda (t: T).(\lambda (t0: T).((eq T t (TLRef n1)) -\to (ex nat (\lambda (n2: nat).(eq T t0 (TLRef n2))))))) (\lambda (n0: -nat).(\lambda (n2: nat).(\lambda (H1: (eq T (TSort n0) (TLRef n1))).(let H2 -\def (eq_ind T (TSort n0) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | -(THead _ _ _) \Rightarrow False])) I (TLRef n1) H1) in (False_ind (ex nat -(\lambda (n3: nat).(eq T (TSort n2) (TLRef n3)))) H2))))) (\lambda (i1: -nat).(\lambda (i2: nat).(\lambda (H1: (eq T (TLRef i1) (TLRef n1))).(let H2 -\def (f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) -with [(TSort _) \Rightarrow i1 | (TLRef n) \Rightarrow n | (THead _ _ _) -\Rightarrow i1])) (TLRef i1) (TLRef n1) H1) in (ex_intro nat (\lambda (n2: -nat).(eq T (TLRef i2) (TLRef n2))) i2 (refl_equal T (TLRef i2))))))) (\lambda -(v1: T).(\lambda (v2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k: -K).(\lambda (H1: (eq T (THead k v1 t1) (TLRef n1))).(let H2 \def (eq_ind T -(THead k v1 t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) -with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ -_) \Rightarrow True])) I (TLRef n1) H1) in (False_ind (ex nat (\lambda (n2: -nat).(eq T (THead k v2 t2) (TLRef n2)))) H2)))))))) y u2 H0))) H))). -(* COMMENTS -Initial nodes: 321 -END *) - -theorem iso_gen_head: - \forall (k: K).(\forall (v1: T).(\forall (t1: T).(\forall (u2: T).((iso -(THead k v1 t1) u2) \to (ex_2 T T (\lambda (v2: T).(\lambda (t2: T).(eq T u2 -(THead k v2 t2))))))))) -\def - \lambda (k: K).(\lambda (v1: T).(\lambda (t1: T).(\lambda (u2: T).(\lambda -(H: (iso (THead k v1 t1) u2)).(insert_eq T (THead k v1 t1) (\lambda (t: -T).(iso t u2)) (\lambda (_: T).(ex_2 T T (\lambda (v2: T).(\lambda (t2: -T).(eq T u2 (THead k v2 t2)))))) (\lambda (y: T).(\lambda (H0: (iso y -u2)).(iso_ind (\lambda (t: T).(\lambda (t0: T).((eq T t (THead k v1 t1)) \to -(ex_2 T T (\lambda (v2: T).(\lambda (t2: T).(eq T t0 (THead k v2 t2)))))))) -(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (H1: (eq T (TSort n1) (THead k -v1 t1))).(let H2 \def (eq_ind T (TSort n1) (\lambda (ee: T).(match ee in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead k v1 t1) H1) -in (False_ind (ex_2 T T (\lambda (v2: T).(\lambda (t2: T).(eq T (TSort n2) -(THead k v2 t2))))) H2))))) (\lambda (i1: nat).(\lambda (i2: nat).(\lambda -(H1: (eq T (TLRef i1) (THead k v1 t1))).(let H2 \def (eq_ind T (TLRef i1) -(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow -False])) I (THead k v1 t1) H1) in (False_ind (ex_2 T T (\lambda (v2: -T).(\lambda (t2: T).(eq T (TLRef i2) (THead k v2 t2))))) H2))))) (\lambda -(v0: T).(\lambda (v2: T).(\lambda (t0: T).(\lambda (t2: T).(\lambda (k0: -K).(\lambda (H1: (eq T (THead k0 v0 t0) (THead k v1 t1))).(let H2 \def -(f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with -[(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _ _) -\Rightarrow k1])) (THead k0 v0 t0) (THead k v1 t1) H1) in ((let H3 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow v0 | (TLRef _) \Rightarrow v0 | (THead _ t _) -\Rightarrow t])) (THead k0 v0 t0) (THead k v1 t1) H1) in ((let H4 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) -\Rightarrow t])) (THead k0 v0 t0) (THead k v1 t1) H1) in (\lambda (_: (eq T -v0 v1)).(\lambda (H6: (eq K k0 k)).(eq_ind_r K k (\lambda (k1: K).(ex_2 T T -(\lambda (v3: T).(\lambda (t3: T).(eq T (THead k1 v2 t2) (THead k v3 t3)))))) -(ex_2_intro T T (\lambda (v3: T).(\lambda (t3: T).(eq T (THead k v2 t2) -(THead k v3 t3)))) v2 t2 (refl_equal T (THead k v2 t2))) k0 H6)))) H3)) -H2)))))))) y u2 H0))) H))))). -(* COMMENTS -Initial nodes: 545 -END *) - -theorem iso_flats_lref_bind_false: - \forall (f: F).(\forall (b: B).(\forall (i: nat).(\forall (v: T).(\forall -(t: T).(\forall (vs: TList).((iso (THeads (Flat f) vs (TLRef i)) (THead (Bind -b) v t)) \to (\forall (P: Prop).P))))))) -\def - \lambda (f: F).(\lambda (b: B).(\lambda (i: nat).(\lambda (v: T).(\lambda -(t: T).(\lambda (vs: TList).(TList_ind (\lambda (t0: TList).((iso (THeads -(Flat f) t0 (TLRef i)) (THead (Bind b) v t)) \to (\forall (P: Prop).P))) -(\lambda (H: (iso (TLRef i) (THead (Bind b) v t))).(\lambda (P: Prop).(let -H_x \def (iso_gen_lref (THead (Bind b) v t) i H) in (let H0 \def H_x in -(ex_ind nat (\lambda (n2: nat).(eq T (THead (Bind b) v t) (TLRef n2))) P -(\lambda (x: nat).(\lambda (H1: (eq T (THead (Bind b) v t) (TLRef x))).(let -H2 \def (eq_ind T (THead (Bind b) v t) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef x) H1) in -(False_ind P H2)))) H0))))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda -(_: (((iso (THeads (Flat f) t1 (TLRef i)) (THead (Bind b) v t)) \to (\forall -(P: Prop).P)))).(\lambda (H0: (iso (THead (Flat f) t0 (THeads (Flat f) t1 -(TLRef i))) (THead (Bind b) v t))).(\lambda (P: Prop).(let H_x \def -(iso_gen_head (Flat f) t0 (THeads (Flat f) t1 (TLRef i)) (THead (Bind b) v t) -H0) in (let H1 \def H_x in (ex_2_ind T T (\lambda (v2: T).(\lambda (t2: -T).(eq T (THead (Bind b) v t) (THead (Flat f) v2 t2)))) P (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H2: (eq T (THead (Bind b) v t) (THead (Flat f) -x0 x1))).(let H3 \def (eq_ind T (THead (Bind b) v t) (\lambda (ee: T).(match -ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | -(TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return -(\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow -False])])) I (THead (Flat f) x0 x1) H2) in (False_ind P H3))))) H1)))))))) -vs)))))). -(* COMMENTS -Initial nodes: 391 -END *) - -theorem iso_flats_flat_bind_false: - \forall (f1: F).(\forall (f2: F).(\forall (b: B).(\forall (v: T).(\forall -(v2: T).(\forall (t: T).(\forall (t2: T).(\forall (vs: TList).((iso (THeads -(Flat f1) vs (THead (Flat f2) v2 t2)) (THead (Bind b) v t)) \to (\forall (P: -Prop).P))))))))) -\def - \lambda (f1: F).(\lambda (f2: F).(\lambda (b: B).(\lambda (v: T).(\lambda -(v2: T).(\lambda (t: T).(\lambda (t2: T).(\lambda (vs: TList).(TList_ind -(\lambda (t0: TList).((iso (THeads (Flat f1) t0 (THead (Flat f2) v2 t2)) -(THead (Bind b) v t)) \to (\forall (P: Prop).P))) (\lambda (H: (iso (THead -(Flat f2) v2 t2) (THead (Bind b) v t))).(\lambda (P: Prop).(let H_x \def -(iso_gen_head (Flat f2) v2 t2 (THead (Bind b) v t) H) in (let H0 \def H_x in -(ex_2_ind T T (\lambda (v3: T).(\lambda (t3: T).(eq T (THead (Bind b) v t) -(THead (Flat f2) v3 t3)))) P (\lambda (x0: T).(\lambda (x1: T).(\lambda (H1: -(eq T (THead (Bind b) v t) (THead (Flat f2) x0 x1))).(let H2 \def (eq_ind T -(THead (Bind b) v t) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat -f2) x0 x1) H1) in (False_ind P H2))))) H0))))) (\lambda (t0: T).(\lambda (t1: -TList).(\lambda (_: (((iso (THeads (Flat f1) t1 (THead (Flat f2) v2 t2)) -(THead (Bind b) v t)) \to (\forall (P: Prop).P)))).(\lambda (H0: (iso (THead -(Flat f1) t0 (THeads (Flat f1) t1 (THead (Flat f2) v2 t2))) (THead (Bind b) v -t))).(\lambda (P: Prop).(let H_x \def (iso_gen_head (Flat f1) t0 (THeads -(Flat f1) t1 (THead (Flat f2) v2 t2)) (THead (Bind b) v t) H0) in (let H1 -\def H_x in (ex_2_ind T T (\lambda (v3: T).(\lambda (t3: T).(eq T (THead -(Bind b) v t) (THead (Flat f1) v3 t3)))) P (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H2: (eq T (THead (Bind b) v t) (THead (Flat f1) x0 x1))).(let H3 -\def (eq_ind T (THead (Bind b) v t) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow -False])])) I (THead (Flat f1) x0 x1) H2) in (False_ind P H3))))) H1)))))))) -vs)))))))). -(* COMMENTS -Initial nodes: 461 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/iso/props.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/iso/props.ma deleted file mode 100644 index af521d073..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/iso/props.ma +++ /dev/null @@ -1,56 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/iso/fwd.ma". - -theorem iso_refl: - \forall (t: T).(iso t t) -\def - \lambda (t: T).(T_ind (\lambda (t0: T).(iso t0 t0)) (\lambda (n: -nat).(iso_sort n n)) (\lambda (n: nat).(iso_lref n n)) (\lambda (k: -K).(\lambda (t0: T).(\lambda (_: (iso t0 t0)).(\lambda (t1: T).(\lambda (_: -(iso t1 t1)).(iso_head t0 t0 t1 t1 k)))))) t). -(* COMMENTS -Initial nodes: 59 -END *) - -theorem iso_trans: - \forall (t1: T).(\forall (t2: T).((iso t1 t2) \to (\forall (t3: T).((iso t2 -t3) \to (iso t1 t3))))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (iso t1 t2)).(iso_ind (\lambda -(t: T).(\lambda (t0: T).(\forall (t3: T).((iso t0 t3) \to (iso t t3))))) -(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (t3: T).(\lambda (H0: (iso -(TSort n2) t3)).(let H_x \def (iso_gen_sort t3 n2 H0) in (let H1 \def H_x in -(ex_ind nat (\lambda (n3: nat).(eq T t3 (TSort n3))) (iso (TSort n1) t3) -(\lambda (x: nat).(\lambda (H2: (eq T t3 (TSort x))).(eq_ind_r T (TSort x) -(\lambda (t: T).(iso (TSort n1) t)) (iso_sort n1 x) t3 H2))) H1))))))) -(\lambda (i1: nat).(\lambda (i2: nat).(\lambda (t3: T).(\lambda (H0: (iso -(TLRef i2) t3)).(let H_x \def (iso_gen_lref t3 i2 H0) in (let H1 \def H_x in -(ex_ind nat (\lambda (n2: nat).(eq T t3 (TLRef n2))) (iso (TLRef i1) t3) -(\lambda (x: nat).(\lambda (H2: (eq T t3 (TLRef x))).(eq_ind_r T (TLRef x) -(\lambda (t: T).(iso (TLRef i1) t)) (iso_lref i1 x) t3 H2))) H1))))))) -(\lambda (v1: T).(\lambda (v2: T).(\lambda (t3: T).(\lambda (t4: T).(\lambda -(k: K).(\lambda (t5: T).(\lambda (H0: (iso (THead k v2 t4) t5)).(let H_x \def -(iso_gen_head k v2 t4 t5 H0) in (let H1 \def H_x in (ex_2_ind T T (\lambda -(v3: T).(\lambda (t6: T).(eq T t5 (THead k v3 t6)))) (iso (THead k v1 t3) t5) -(\lambda (x0: T).(\lambda (x1: T).(\lambda (H2: (eq T t5 (THead k x0 -x1))).(eq_ind_r T (THead k x0 x1) (\lambda (t: T).(iso (THead k v1 t3) t)) -(iso_head v1 x0 t3 x1 k) t5 H2)))) H1)))))))))) t1 t2 H))). -(* COMMENTS -Initial nodes: 351 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/leq/asucc.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/leq/asucc.ma deleted file mode 100644 index fd9e7c1d3..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/leq/asucc.ma +++ /dev/null @@ -1,479 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/leq/props.ma". - -theorem asucc_repl: - \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (leq g -(asucc g a1) (asucc g a2))))) -\def - \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1 -a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(leq g (asucc g a) (asucc g -a0)))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: nat).(\lambda (n2: -nat).(\lambda (k: nat).(\lambda (H0: (eq A (aplus g (ASort h1 n1) k) (aplus g -(ASort h2 n2) k))).(nat_ind (\lambda (n: nat).((eq A (aplus g (ASort n n1) k) -(aplus g (ASort h2 n2) k)) \to (leq g (match n with [O \Rightarrow (ASort O -(next g n1)) | (S h) \Rightarrow (ASort h n1)]) (match h2 with [O \Rightarrow -(ASort O (next g n2)) | (S h) \Rightarrow (ASort h n2)])))) (\lambda (H1: (eq -A (aplus g (ASort O n1) k) (aplus g (ASort h2 n2) k))).(nat_ind (\lambda (n: -nat).((eq A (aplus g (ASort O n1) k) (aplus g (ASort n n2) k)) \to (leq g -(ASort O (next g n1)) (match n with [O \Rightarrow (ASort O (next g n2)) | (S -h) \Rightarrow (ASort h n2)])))) (\lambda (H2: (eq A (aplus g (ASort O n1) k) -(aplus g (ASort O n2) k))).(leq_sort g O O (next g n1) (next g n2) k (eq_ind -A (aplus g (ASort O n1) (S k)) (\lambda (a: A).(eq A a (aplus g (ASort O -(next g n2)) k))) (eq_ind A (aplus g (ASort O n2) (S k)) (\lambda (a: A).(eq -A (aplus g (ASort O n1) (S k)) a)) (eq_ind_r A (aplus g (ASort O n2) k) -(\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g (ASort O n2) k)))) -(refl_equal A (asucc g (aplus g (ASort O n2) k))) (aplus g (ASort O n1) k) -H2) (aplus g (ASort O (next g n2)) k) (aplus_sort_O_S_simpl g n2 k)) (aplus g -(ASort O (next g n1)) k) (aplus_sort_O_S_simpl g n1 k)))) (\lambda (h3: -nat).(\lambda (_: (((eq A (aplus g (ASort O n1) k) (aplus g (ASort h3 n2) k)) -\to (leq g (ASort O (next g n1)) (match h3 with [O \Rightarrow (ASort O (next -g n2)) | (S h) \Rightarrow (ASort h n2)]))))).(\lambda (H2: (eq A (aplus g -(ASort O n1) k) (aplus g (ASort (S h3) n2) k))).(leq_sort g O h3 (next g n1) -n2 k (eq_ind A (aplus g (ASort O n1) (S k)) (\lambda (a: A).(eq A a (aplus g -(ASort h3 n2) k))) (eq_ind A (aplus g (ASort (S h3) n2) (S k)) (\lambda (a: -A).(eq A (aplus g (ASort O n1) (S k)) a)) (eq_ind_r A (aplus g (ASort (S h3) -n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g (ASort (S h3) n2) -k)))) (refl_equal A (asucc g (aplus g (ASort (S h3) n2) k))) (aplus g (ASort -O n1) k) H2) (aplus g (ASort h3 n2) k) (aplus_sort_S_S_simpl g n2 h3 k)) -(aplus g (ASort O (next g n1)) k) (aplus_sort_O_S_simpl g n1 k)))))) h2 H1)) -(\lambda (h3: nat).(\lambda (IHh1: (((eq A (aplus g (ASort h3 n1) k) (aplus g -(ASort h2 n2) k)) \to (leq g (match h3 with [O \Rightarrow (ASort O (next g -n1)) | (S h) \Rightarrow (ASort h n1)]) (match h2 with [O \Rightarrow (ASort -O (next g n2)) | (S h) \Rightarrow (ASort h n2)]))))).(\lambda (H1: (eq A -(aplus g (ASort (S h3) n1) k) (aplus g (ASort h2 n2) k))).(nat_ind (\lambda -(n: nat).((eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort n n2) k)) \to -((((eq A (aplus g (ASort h3 n1) k) (aplus g (ASort n n2) k)) \to (leq g -(match h3 with [O \Rightarrow (ASort O (next g n1)) | (S h) \Rightarrow -(ASort h n1)]) (match n with [O \Rightarrow (ASort O (next g n2)) | (S h) -\Rightarrow (ASort h n2)])))) \to (leq g (ASort h3 n1) (match n with [O -\Rightarrow (ASort O (next g n2)) | (S h) \Rightarrow (ASort h n2)]))))) -(\lambda (H2: (eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort O n2) -k))).(\lambda (_: (((eq A (aplus g (ASort h3 n1) k) (aplus g (ASort O n2) k)) -\to (leq g (match h3 with [O \Rightarrow (ASort O (next g n1)) | (S h) -\Rightarrow (ASort h n1)]) (ASort O (next g n2)))))).(leq_sort g h3 O n1 -(next g n2) k (eq_ind A (aplus g (ASort O n2) (S k)) (\lambda (a: A).(eq A -(aplus g (ASort h3 n1) k) a)) (eq_ind A (aplus g (ASort (S h3) n1) (S k)) -(\lambda (a: A).(eq A a (aplus g (ASort O n2) (S k)))) (eq_ind_r A (aplus g -(ASort O n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g (ASort O -n2) k)))) (refl_equal A (asucc g (aplus g (ASort O n2) k))) (aplus g (ASort -(S h3) n1) k) H2) (aplus g (ASort h3 n1) k) (aplus_sort_S_S_simpl g n1 h3 k)) -(aplus g (ASort O (next g n2)) k) (aplus_sort_O_S_simpl g n2 k))))) (\lambda -(h4: nat).(\lambda (_: (((eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort -h4 n2) k)) \to ((((eq A (aplus g (ASort h3 n1) k) (aplus g (ASort h4 n2) k)) -\to (leq g (match h3 with [O \Rightarrow (ASort O (next g n1)) | (S h) -\Rightarrow (ASort h n1)]) (match h4 with [O \Rightarrow (ASort O (next g -n2)) | (S h) \Rightarrow (ASort h n2)])))) \to (leq g (ASort h3 n1) (match h4 -with [O \Rightarrow (ASort O (next g n2)) | (S h) \Rightarrow (ASort h -n2)])))))).(\lambda (H2: (eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort -(S h4) n2) k))).(\lambda (_: (((eq A (aplus g (ASort h3 n1) k) (aplus g -(ASort (S h4) n2) k)) \to (leq g (match h3 with [O \Rightarrow (ASort O (next -g n1)) | (S h) \Rightarrow (ASort h n1)]) (ASort h4 n2))))).(leq_sort g h3 h4 -n1 n2 k (eq_ind A (aplus g (ASort (S h3) n1) (S k)) (\lambda (a: A).(eq A a -(aplus g (ASort h4 n2) k))) (eq_ind A (aplus g (ASort (S h4) n2) (S k)) -(\lambda (a: A).(eq A (aplus g (ASort (S h3) n1) (S k)) a)) (eq_ind_r A -(aplus g (ASort (S h4) n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g -(aplus g (ASort (S h4) n2) k)))) (refl_equal A (asucc g (aplus g (ASort (S -h4) n2) k))) (aplus g (ASort (S h3) n1) k) H2) (aplus g (ASort h4 n2) k) -(aplus_sort_S_S_simpl g n2 h4 k)) (aplus g (ASort h3 n1) k) -(aplus_sort_S_S_simpl g n1 h3 k))))))) h2 H1 IHh1)))) h1 H0))))))) (\lambda -(a3: A).(\lambda (a4: A).(\lambda (H0: (leq g a3 a4)).(\lambda (_: (leq g -(asucc g a3) (asucc g a4))).(\lambda (a5: A).(\lambda (a6: A).(\lambda (_: -(leq g a5 a6)).(\lambda (H3: (leq g (asucc g a5) (asucc g a6))).(leq_head g -a3 a4 H0 (asucc g a5) (asucc g a6) H3))))))))) a1 a2 H)))). -(* COMMENTS -Initial nodes: 1907 -END *) - -theorem asucc_inj: - \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (asucc g a1) (asucc -g a2)) \to (leq g a1 a2)))) -\def - \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2: -A).((leq g (asucc g a) (asucc g a2)) \to (leq g a a2)))) (\lambda (n: -nat).(\lambda (n0: nat).(\lambda (a2: A).(A_ind (\lambda (a: A).((leq g -(asucc g (ASort n n0)) (asucc g a)) \to (leq g (ASort n n0) a))) (\lambda -(n1: nat).(\lambda (n2: nat).(\lambda (H: (leq g (asucc g (ASort n n0)) -(asucc g (ASort n1 n2)))).(nat_ind (\lambda (n3: nat).((leq g (asucc g (ASort -n3 n0)) (asucc g (ASort n1 n2))) \to (leq g (ASort n3 n0) (ASort n1 n2)))) -(\lambda (H0: (leq g (asucc g (ASort O n0)) (asucc g (ASort n1 -n2)))).(nat_ind (\lambda (n3: nat).((leq g (asucc g (ASort O n0)) (asucc g -(ASort n3 n2))) \to (leq g (ASort O n0) (ASort n3 n2)))) (\lambda (H1: (leq g -(asucc g (ASort O n0)) (asucc g (ASort O n2)))).(let H_x \def (leq_gen_sort1 -g O (next g n0) (ASort O (next g n2)) H1) in (let H2 \def H_x in (ex2_3_ind -nat nat nat (\lambda (n3: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A -(aplus g (ASort O (next g n0)) k) (aplus g (ASort h2 n3) k))))) (\lambda (n3: -nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A (ASort O (next g n2)) (ASort -h2 n3))))) (leq g (ASort O n0) (ASort O n2)) (\lambda (x0: nat).(\lambda (x1: -nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort O (next g n0)) -x2) (aplus g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort O (next g n2)) -(ASort x1 x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e in A -return (\lambda (_: A).nat) with [(ASort n3 _) \Rightarrow n3 | (AHead _ _) -\Rightarrow O])) (ASort O (next g n2)) (ASort x1 x0) H4) in ((let H6 \def -(f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with -[(ASort _ n3) \Rightarrow n3 | (AHead _ _) \Rightarrow ((match g with [(mk_G -next _) \Rightarrow next]) n2)])) (ASort O (next g n2)) (ASort x1 x0) H4) in -(\lambda (H7: (eq nat O x1)).(let H8 \def (eq_ind_r nat x1 (\lambda (n3: -nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort n3 x0) x2))) H3 -O H7) in (let H9 \def (eq_ind_r nat x0 (\lambda (n3: nat).(eq A (aplus g -(ASort O (next g n0)) x2) (aplus g (ASort O n3) x2))) H8 (next g n2) H6) in -(let H10 \def (eq_ind_r A (aplus g (ASort O (next g n0)) x2) (\lambda (a: -A).(eq A a (aplus g (ASort O (next g n2)) x2))) H9 (aplus g (ASort O n0) (S -x2)) (aplus_sort_O_S_simpl g n0 x2)) in (let H11 \def (eq_ind_r A (aplus g -(ASort O (next g n2)) x2) (\lambda (a: A).(eq A (aplus g (ASort O n0) (S x2)) -a)) H10 (aplus g (ASort O n2) (S x2)) (aplus_sort_O_S_simpl g n2 x2)) in -(leq_sort g O O n0 n2 (S x2) H11))))))) H5))))))) H2)))) (\lambda (n3: -nat).(\lambda (_: (((leq g (asucc g (ASort O n0)) (asucc g (ASort n3 n2))) -\to (leq g (ASort O n0) (ASort n3 n2))))).(\lambda (H1: (leq g (asucc g -(ASort O n0)) (asucc g (ASort (S n3) n2)))).(let H_x \def (leq_gen_sort1 g O -(next g n0) (ASort n3 n2) H1) in (let H2 \def H_x in (ex2_3_ind nat nat nat -(\lambda (n4: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort -O (next g n0)) k) (aplus g (ASort h2 n4) k))))) (\lambda (n4: nat).(\lambda -(h2: nat).(\lambda (_: nat).(eq A (ASort n3 n2) (ASort h2 n4))))) (leq g -(ASort O n0) (ASort (S n3) n2)) (\lambda (x0: nat).(\lambda (x1: -nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort O (next g n0)) -x2) (aplus g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort n3 n2) (ASort x1 -x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e in A return -(\lambda (_: A).nat) with [(ASort n4 _) \Rightarrow n4 | (AHead _ _) -\Rightarrow n3])) (ASort n3 n2) (ASort x1 x0) H4) in ((let H6 \def (f_equal A -nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort _ -n4) \Rightarrow n4 | (AHead _ _) \Rightarrow n2])) (ASort n3 n2) (ASort x1 -x0) H4) in (\lambda (H7: (eq nat n3 x1)).(let H8 \def (eq_ind_r nat x1 -(\lambda (n4: nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort -n4 x0) x2))) H3 n3 H7) in (let H9 \def (eq_ind_r nat x0 (\lambda (n4: -nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort n3 n4) x2))) H8 -n2 H6) in (let H10 \def (eq_ind_r A (aplus g (ASort O (next g n0)) x2) -(\lambda (a: A).(eq A a (aplus g (ASort n3 n2) x2))) H9 (aplus g (ASort O n0) -(S x2)) (aplus_sort_O_S_simpl g n0 x2)) in (let H11 \def (eq_ind_r A (aplus g -(ASort n3 n2) x2) (\lambda (a: A).(eq A (aplus g (ASort O n0) (S x2)) a)) H10 -(aplus g (ASort (S n3) n2) (S x2)) (aplus_sort_S_S_simpl g n2 n3 x2)) in -(leq_sort g O (S n3) n0 n2 (S x2) H11))))))) H5))))))) H2)))))) n1 H0)) -(\lambda (n3: nat).(\lambda (IHn: (((leq g (asucc g (ASort n3 n0)) (asucc g -(ASort n1 n2))) \to (leq g (ASort n3 n0) (ASort n1 n2))))).(\lambda (H0: (leq -g (asucc g (ASort (S n3) n0)) (asucc g (ASort n1 n2)))).(nat_ind (\lambda -(n4: nat).((leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort n4 n2))) \to -((((leq g (asucc g (ASort n3 n0)) (asucc g (ASort n4 n2))) \to (leq g (ASort -n3 n0) (ASort n4 n2)))) \to (leq g (ASort (S n3) n0) (ASort n4 n2))))) -(\lambda (H1: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort O -n2)))).(\lambda (_: (((leq g (asucc g (ASort n3 n0)) (asucc g (ASort O n2))) -\to (leq g (ASort n3 n0) (ASort O n2))))).(let H_x \def (leq_gen_sort1 g n3 -n0 (ASort O (next g n2)) H1) in (let H2 \def H_x in (ex2_3_ind nat nat nat -(\lambda (n4: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort -n3 n0) k) (aplus g (ASort h2 n4) k))))) (\lambda (n4: nat).(\lambda (h2: -nat).(\lambda (_: nat).(eq A (ASort O (next g n2)) (ASort h2 n4))))) (leq g -(ASort (S n3) n0) (ASort O n2)) (\lambda (x0: nat).(\lambda (x1: -nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort n3 n0) x2) (aplus -g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort O (next g n2)) (ASort x1 -x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e in A return -(\lambda (_: A).nat) with [(ASort n4 _) \Rightarrow n4 | (AHead _ _) -\Rightarrow O])) (ASort O (next g n2)) (ASort x1 x0) H4) in ((let H6 \def -(f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with -[(ASort _ n4) \Rightarrow n4 | (AHead _ _) \Rightarrow ((match g with [(mk_G -next _) \Rightarrow next]) n2)])) (ASort O (next g n2)) (ASort x1 x0) H4) in -(\lambda (H7: (eq nat O x1)).(let H8 \def (eq_ind_r nat x1 (\lambda (n4: -nat).(eq A (aplus g (ASort n3 n0) x2) (aplus g (ASort n4 x0) x2))) H3 O H7) -in (let H9 \def (eq_ind_r nat x0 (\lambda (n4: nat).(eq A (aplus g (ASort n3 -n0) x2) (aplus g (ASort O n4) x2))) H8 (next g n2) H6) in (let H10 \def -(eq_ind_r A (aplus g (ASort n3 n0) x2) (\lambda (a: A).(eq A a (aplus g -(ASort O (next g n2)) x2))) H9 (aplus g (ASort (S n3) n0) (S x2)) -(aplus_sort_S_S_simpl g n0 n3 x2)) in (let H11 \def (eq_ind_r A (aplus g -(ASort O (next g n2)) x2) (\lambda (a: A).(eq A (aplus g (ASort (S n3) n0) (S -x2)) a)) H10 (aplus g (ASort O n2) (S x2)) (aplus_sort_O_S_simpl g n2 x2)) in -(leq_sort g (S n3) O n0 n2 (S x2) H11))))))) H5))))))) H2))))) (\lambda (n4: -nat).(\lambda (_: (((leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort n4 -n2))) \to ((((leq g (asucc g (ASort n3 n0)) (asucc g (ASort n4 n2))) \to (leq -g (ASort n3 n0) (ASort n4 n2)))) \to (leq g (ASort (S n3) n0) (ASort n4 -n2)))))).(\lambda (H1: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort (S -n4) n2)))).(\lambda (_: (((leq g (asucc g (ASort n3 n0)) (asucc g (ASort (S -n4) n2))) \to (leq g (ASort n3 n0) (ASort (S n4) n2))))).(let H_x \def -(leq_gen_sort1 g n3 n0 (ASort n4 n2) H1) in (let H2 \def H_x in (ex2_3_ind -nat nat nat (\lambda (n5: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A -(aplus g (ASort n3 n0) k) (aplus g (ASort h2 n5) k))))) (\lambda (n5: -nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A (ASort n4 n2) (ASort h2 -n5))))) (leq g (ASort (S n3) n0) (ASort (S n4) n2)) (\lambda (x0: -nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g -(ASort n3 n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort n4 -n2) (ASort x1 x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e in A -return (\lambda (_: A).nat) with [(ASort n5 _) \Rightarrow n5 | (AHead _ _) -\Rightarrow n4])) (ASort n4 n2) (ASort x1 x0) H4) in ((let H6 \def (f_equal A -nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort _ -n5) \Rightarrow n5 | (AHead _ _) \Rightarrow n2])) (ASort n4 n2) (ASort x1 -x0) H4) in (\lambda (H7: (eq nat n4 x1)).(let H8 \def (eq_ind_r nat x1 -(\lambda (n5: nat).(eq A (aplus g (ASort n3 n0) x2) (aplus g (ASort n5 x0) -x2))) H3 n4 H7) in (let H9 \def (eq_ind_r nat x0 (\lambda (n5: nat).(eq A -(aplus g (ASort n3 n0) x2) (aplus g (ASort n4 n5) x2))) H8 n2 H6) in (let H10 -\def (eq_ind_r A (aplus g (ASort n3 n0) x2) (\lambda (a: A).(eq A a (aplus g -(ASort n4 n2) x2))) H9 (aplus g (ASort (S n3) n0) (S x2)) -(aplus_sort_S_S_simpl g n0 n3 x2)) in (let H11 \def (eq_ind_r A (aplus g -(ASort n4 n2) x2) (\lambda (a: A).(eq A (aplus g (ASort (S n3) n0) (S x2)) -a)) H10 (aplus g (ASort (S n4) n2) (S x2)) (aplus_sort_S_S_simpl g n2 n4 x2)) -in (leq_sort g (S n3) (S n4) n0 n2 (S x2) H11))))))) H5))))))) H2))))))) n1 -H0 IHn)))) n H)))) (\lambda (a: A).(\lambda (H: (((leq g (asucc g (ASort n -n0)) (asucc g a)) \to (leq g (ASort n n0) a)))).(\lambda (a0: A).(\lambda -(H0: (((leq g (asucc g (ASort n n0)) (asucc g a0)) \to (leq g (ASort n n0) -a0)))).(\lambda (H1: (leq g (asucc g (ASort n n0)) (asucc g (AHead a -a0)))).(nat_ind (\lambda (n1: nat).((((leq g (asucc g (ASort n1 n0)) (asucc g -a)) \to (leq g (ASort n1 n0) a))) \to ((((leq g (asucc g (ASort n1 n0)) -(asucc g a0)) \to (leq g (ASort n1 n0) a0))) \to ((leq g (asucc g (ASort n1 -n0)) (asucc g (AHead a a0))) \to (leq g (ASort n1 n0) (AHead a a0)))))) -(\lambda (_: (((leq g (asucc g (ASort O n0)) (asucc g a)) \to (leq g (ASort O -n0) a)))).(\lambda (_: (((leq g (asucc g (ASort O n0)) (asucc g a0)) \to (leq -g (ASort O n0) a0)))).(\lambda (H4: (leq g (asucc g (ASort O n0)) (asucc g -(AHead a a0)))).(let H_x \def (leq_gen_sort1 g O (next g n0) (AHead a (asucc -g a0)) H4) in (let H5 \def H_x in (ex2_3_ind nat nat nat (\lambda (n2: -nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort O (next g -n0)) k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: -nat).(\lambda (_: nat).(eq A (AHead a (asucc g a0)) (ASort h2 n2))))) (leq g -(ASort O n0) (AHead a a0)) (\lambda (x0: nat).(\lambda (x1: nat).(\lambda -(x2: nat).(\lambda (_: (eq A (aplus g (ASort O (next g n0)) x2) (aplus g -(ASort x1 x0) x2))).(\lambda (H7: (eq A (AHead a (asucc g a0)) (ASort x1 -x0))).(let H8 \def (eq_ind A (AHead a (asucc g a0)) (\lambda (ee: A).(match -ee in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | -(AHead _ _) \Rightarrow True])) I (ASort x1 x0) H7) in (False_ind (leq g -(ASort O n0) (AHead a a0)) H8))))))) H5)))))) (\lambda (n1: nat).(\lambda (_: -(((((leq g (asucc g (ASort n1 n0)) (asucc g a)) \to (leq g (ASort n1 n0) a))) -\to ((((leq g (asucc g (ASort n1 n0)) (asucc g a0)) \to (leq g (ASort n1 n0) -a0))) \to ((leq g (asucc g (ASort n1 n0)) (asucc g (AHead a a0))) \to (leq g -(ASort n1 n0) (AHead a a0))))))).(\lambda (_: (((leq g (asucc g (ASort (S n1) -n0)) (asucc g a)) \to (leq g (ASort (S n1) n0) a)))).(\lambda (_: (((leq g -(asucc g (ASort (S n1) n0)) (asucc g a0)) \to (leq g (ASort (S n1) n0) -a0)))).(\lambda (H4: (leq g (asucc g (ASort (S n1) n0)) (asucc g (AHead a -a0)))).(let H_x \def (leq_gen_sort1 g n1 n0 (AHead a (asucc g a0)) H4) in -(let H5 \def H_x in (ex2_3_ind nat nat nat (\lambda (n2: nat).(\lambda (h2: -nat).(\lambda (k: nat).(eq A (aplus g (ASort n1 n0) k) (aplus g (ASort h2 n2) -k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A (AHead a -(asucc g a0)) (ASort h2 n2))))) (leq g (ASort (S n1) n0) (AHead a a0)) -(\lambda (x0: nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (_: (eq A -(aplus g (ASort n1 n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H7: (eq A -(AHead a (asucc g a0)) (ASort x1 x0))).(let H8 \def (eq_ind A (AHead a (asucc -g a0)) (\lambda (ee: A).(match ee in A return (\lambda (_: A).Prop) with -[(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort x1 -x0) H7) in (False_ind (leq g (ASort (S n1) n0) (AHead a a0)) H8))))))) -H5)))))))) n H H0 H1)))))) a2)))) (\lambda (a: A).(\lambda (_: ((\forall (a2: -A).((leq g (asucc g a) (asucc g a2)) \to (leq g a a2))))).(\lambda (a0: -A).(\lambda (H0: ((\forall (a2: A).((leq g (asucc g a0) (asucc g a2)) \to -(leq g a0 a2))))).(\lambda (a2: A).(A_ind (\lambda (a3: A).((leq g (asucc g -(AHead a a0)) (asucc g a3)) \to (leq g (AHead a a0) a3))) (\lambda (n: -nat).(\lambda (n0: nat).(\lambda (H1: (leq g (asucc g (AHead a a0)) (asucc g -(ASort n n0)))).(nat_ind (\lambda (n1: nat).((leq g (asucc g (AHead a a0)) -(asucc g (ASort n1 n0))) \to (leq g (AHead a a0) (ASort n1 n0)))) (\lambda -(H2: (leq g (asucc g (AHead a a0)) (asucc g (ASort O n0)))).(let H_x \def -(leq_gen_head1 g a (asucc g a0) (ASort O (next g n0)) H2) in (let H3 \def H_x -in (ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq g a a3))) (\lambda -(_: A).(\lambda (a4: A).(leq g (asucc g a0) a4))) (\lambda (a3: A).(\lambda -(a4: A).(eq A (ASort O (next g n0)) (AHead a3 a4)))) (leq g (AHead a a0) -(ASort O n0)) (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g a -x0)).(\lambda (_: (leq g (asucc g a0) x1)).(\lambda (H6: (eq A (ASort O (next -g n0)) (AHead x0 x1))).(let H7 \def (eq_ind A (ASort O (next g n0)) (\lambda -(ee: A).(match ee in A return (\lambda (_: A).Prop) with [(ASort _ _) -\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H6) in -(False_ind (leq g (AHead a a0) (ASort O n0)) H7))))))) H3)))) (\lambda (n1: -nat).(\lambda (_: (((leq g (asucc g (AHead a a0)) (asucc g (ASort n1 n0))) -\to (leq g (AHead a a0) (ASort n1 n0))))).(\lambda (H2: (leq g (asucc g -(AHead a a0)) (asucc g (ASort (S n1) n0)))).(let H_x \def (leq_gen_head1 g a -(asucc g a0) (ASort n1 n0) H2) in (let H3 \def H_x in (ex3_2_ind A A (\lambda -(a3: A).(\lambda (_: A).(leq g a a3))) (\lambda (_: A).(\lambda (a4: A).(leq -g (asucc g a0) a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort n1 n0) -(AHead a3 a4)))) (leq g (AHead a a0) (ASort (S n1) n0)) (\lambda (x0: -A).(\lambda (x1: A).(\lambda (_: (leq g a x0)).(\lambda (_: (leq g (asucc g -a0) x1)).(\lambda (H6: (eq A (ASort n1 n0) (AHead x0 x1))).(let H7 \def -(eq_ind A (ASort n1 n0) (\lambda (ee: A).(match ee in A return (\lambda (_: -A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow -False])) I (AHead x0 x1) H6) in (False_ind (leq g (AHead a a0) (ASort (S n1) -n0)) H7))))))) H3)))))) n H1)))) (\lambda (a3: A).(\lambda (_: (((leq g -(asucc g (AHead a a0)) (asucc g a3)) \to (leq g (AHead a a0) a3)))).(\lambda -(a4: A).(\lambda (_: (((leq g (asucc g (AHead a a0)) (asucc g a4)) \to (leq g -(AHead a a0) a4)))).(\lambda (H3: (leq g (asucc g (AHead a a0)) (asucc g -(AHead a3 a4)))).(let H_x \def (leq_gen_head1 g a (asucc g a0) (AHead a3 -(asucc g a4)) H3) in (let H4 \def H_x in (ex3_2_ind A A (\lambda (a5: -A).(\lambda (_: A).(leq g a a5))) (\lambda (_: A).(\lambda (a6: A).(leq g -(asucc g a0) a6))) (\lambda (a5: A).(\lambda (a6: A).(eq A (AHead a3 (asucc g -a4)) (AHead a5 a6)))) (leq g (AHead a a0) (AHead a3 a4)) (\lambda (x0: -A).(\lambda (x1: A).(\lambda (H5: (leq g a x0)).(\lambda (H6: (leq g (asucc g -a0) x1)).(\lambda (H7: (eq A (AHead a3 (asucc g a4)) (AHead x0 x1))).(let H8 -\def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) -with [(ASort _ _) \Rightarrow a3 | (AHead a5 _) \Rightarrow a5])) (AHead a3 -(asucc g a4)) (AHead x0 x1) H7) in ((let H9 \def (f_equal A A (\lambda (e: -A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow -((let rec asucc (g0: G) (l: A) on l: A \def (match l with [(ASort n0 n) -\Rightarrow (match n0 with [O \Rightarrow (ASort O (next g0 n)) | (S h) -\Rightarrow (ASort h n)]) | (AHead a5 a6) \Rightarrow (AHead a5 (asucc g0 -a6))]) in asucc) g a4) | (AHead _ a5) \Rightarrow a5])) (AHead a3 (asucc g -a4)) (AHead x0 x1) H7) in (\lambda (H10: (eq A a3 x0)).(let H11 \def -(eq_ind_r A x1 (\lambda (a5: A).(leq g (asucc g a0) a5)) H6 (asucc g a4) H9) -in (let H12 \def (eq_ind_r A x0 (\lambda (a5: A).(leq g a a5)) H5 a3 H10) in -(leq_head g a a3 H12 a0 a4 (H0 a4 H11)))))) H8))))))) H4)))))))) a2)))))) -a1)). -(* COMMENTS -Initial nodes: 4697 -END *) - -theorem leq_asucc: - \forall (g: G).(\forall (a: A).(ex A (\lambda (a0: A).(leq g a (asucc g -a0))))) -\def - \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(ex A (\lambda (a1: -A).(leq g a0 (asucc g a1))))) (\lambda (n: nat).(\lambda (n0: nat).(ex_intro -A (\lambda (a0: A).(leq g (ASort n n0) (asucc g a0))) (ASort (S n) n0) -(leq_refl g (ASort n n0))))) (\lambda (a0: A).(\lambda (_: (ex A (\lambda -(a1: A).(leq g a0 (asucc g a1))))).(\lambda (a1: A).(\lambda (H0: (ex A -(\lambda (a2: A).(leq g a1 (asucc g a2))))).(let H1 \def H0 in (ex_ind A -(\lambda (a2: A).(leq g a1 (asucc g a2))) (ex A (\lambda (a2: A).(leq g -(AHead a0 a1) (asucc g a2)))) (\lambda (x: A).(\lambda (H2: (leq g a1 (asucc -g x))).(ex_intro A (\lambda (a2: A).(leq g (AHead a0 a1) (asucc g a2))) -(AHead a0 x) (leq_head g a0 a0 (leq_refl g a0) a1 (asucc g x) H2)))) H1)))))) -a)). -(* COMMENTS -Initial nodes: 221 -END *) - -theorem leq_ahead_asucc_false: - \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (AHead a1 a2) -(asucc g a1)) \to (\forall (P: Prop).P)))) -\def - \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2: -A).((leq g (AHead a a2) (asucc g a)) \to (\forall (P: Prop).P)))) (\lambda -(n: nat).(\lambda (n0: nat).(\lambda (a2: A).(\lambda (H: (leq g (AHead -(ASort n n0) a2) (match n with [O \Rightarrow (ASort O (next g n0)) | (S h) -\Rightarrow (ASort h n0)]))).(\lambda (P: Prop).(nat_ind (\lambda (n1: -nat).((leq g (AHead (ASort n1 n0) a2) (match n1 with [O \Rightarrow (ASort O -(next g n0)) | (S h) \Rightarrow (ASort h n0)])) \to P)) (\lambda (H0: (leq g -(AHead (ASort O n0) a2) (ASort O (next g n0)))).(let H_x \def (leq_gen_head1 -g (ASort O n0) a2 (ASort O (next g n0)) H0) in (let H1 \def H_x in (ex3_2_ind -A A (\lambda (a3: A).(\lambda (_: A).(leq g (ASort O n0) a3))) (\lambda (_: -A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A -(ASort O (next g n0)) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: -A).(\lambda (_: (leq g (ASort O n0) x0)).(\lambda (_: (leq g a2 x1)).(\lambda -(H4: (eq A (ASort O (next g n0)) (AHead x0 x1))).(let H5 \def (eq_ind A -(ASort O (next g n0)) (\lambda (ee: A).(match ee in A return (\lambda (_: -A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow -False])) I (AHead x0 x1) H4) in (False_ind P H5))))))) H1)))) (\lambda (n1: -nat).(\lambda (_: (((leq g (AHead (ASort n1 n0) a2) (match n1 with [O -\Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)])) \to -P))).(\lambda (H0: (leq g (AHead (ASort (S n1) n0) a2) (ASort n1 n0))).(let -H_x \def (leq_gen_head1 g (ASort (S n1) n0) a2 (ASort n1 n0) H0) in (let H1 -\def H_x in (ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq g (ASort (S -n1) n0) a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: -A).(\lambda (a4: A).(eq A (ASort n1 n0) (AHead a3 a4)))) P (\lambda (x0: -A).(\lambda (x1: A).(\lambda (_: (leq g (ASort (S n1) n0) x0)).(\lambda (_: -(leq g a2 x1)).(\lambda (H4: (eq A (ASort n1 n0) (AHead x0 x1))).(let H5 \def -(eq_ind A (ASort n1 n0) (\lambda (ee: A).(match ee in A return (\lambda (_: -A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow -False])) I (AHead x0 x1) H4) in (False_ind P H5))))))) H1)))))) n H)))))) -(\lambda (a: A).(\lambda (_: ((\forall (a2: A).((leq g (AHead a a2) (asucc g -a)) \to (\forall (P: Prop).P))))).(\lambda (a0: A).(\lambda (_: ((\forall -(a2: A).((leq g (AHead a0 a2) (asucc g a0)) \to (\forall (P: -Prop).P))))).(\lambda (a2: A).(\lambda (H1: (leq g (AHead (AHead a a0) a2) -(AHead a (asucc g a0)))).(\lambda (P: Prop).(let H_x \def (leq_gen_head1 g -(AHead a a0) a2 (AHead a (asucc g a0)) H1) in (let H2 \def H_x in (ex3_2_ind -A A (\lambda (a3: A).(\lambda (_: A).(leq g (AHead a a0) a3))) (\lambda (_: -A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A -(AHead a (asucc g a0)) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: -A).(\lambda (H3: (leq g (AHead a a0) x0)).(\lambda (H4: (leq g a2 -x1)).(\lambda (H5: (eq A (AHead a (asucc g a0)) (AHead x0 x1))).(let H6 \def -(f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with -[(ASort _ _) \Rightarrow a | (AHead a3 _) \Rightarrow a3])) (AHead a (asucc g -a0)) (AHead x0 x1) H5) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e -in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow ((let rec asucc -(g0: G) (l: A) on l: A \def (match l with [(ASort n0 n) \Rightarrow (match n0 -with [O \Rightarrow (ASort O (next g0 n)) | (S h) \Rightarrow (ASort h n)]) | -(AHead a3 a4) \Rightarrow (AHead a3 (asucc g0 a4))]) in asucc) g a0) | (AHead -_ a3) \Rightarrow a3])) (AHead a (asucc g a0)) (AHead x0 x1) H5) in (\lambda -(H8: (eq A a x0)).(let H9 \def (eq_ind_r A x1 (\lambda (a3: A).(leq g a2 a3)) -H4 (asucc g a0) H7) in (let H10 \def (eq_ind_r A x0 (\lambda (a3: A).(leq g -(AHead a a0) a3)) H3 a H8) in (leq_ahead_false_1 g a a0 H10 P))))) H6))))))) -H2)))))))))) a1)). -(* COMMENTS -Initial nodes: 927 -END *) - -theorem leq_asucc_false: - \forall (g: G).(\forall (a: A).((leq g (asucc g a) a) \to (\forall (P: -Prop).P))) -\def - \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).((leq g (asucc g a0) -a0) \to (\forall (P: Prop).P))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda -(H: (leq g (match n with [O \Rightarrow (ASort O (next g n0)) | (S h) -\Rightarrow (ASort h n0)]) (ASort n n0))).(\lambda (P: Prop).(nat_ind -(\lambda (n1: nat).((leq g (match n1 with [O \Rightarrow (ASort O (next g -n0)) | (S h) \Rightarrow (ASort h n0)]) (ASort n1 n0)) \to P)) (\lambda (H0: -(leq g (ASort O (next g n0)) (ASort O n0))).(let H_x \def (leq_gen_sort1 g O -(next g n0) (ASort O n0) H0) in (let H1 \def H_x in (ex2_3_ind nat nat nat -(\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort -O (next g n0)) k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda -(h2: nat).(\lambda (_: nat).(eq A (ASort O n0) (ASort h2 n2))))) P (\lambda -(x0: nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (H2: (eq A (aplus g -(ASort O (next g n0)) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H3: (eq A -(ASort O n0) (ASort x1 x0))).(let H4 \def (f_equal A nat (\lambda (e: -A).(match e in A return (\lambda (_: A).nat) with [(ASort n1 _) \Rightarrow -n1 | (AHead _ _) \Rightarrow O])) (ASort O n0) (ASort x1 x0) H3) in ((let H5 -\def (f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) -with [(ASort _ n1) \Rightarrow n1 | (AHead _ _) \Rightarrow n0])) (ASort O -n0) (ASort x1 x0) H3) in (\lambda (H6: (eq nat O x1)).(let H7 \def (eq_ind_r -nat x1 (\lambda (n1: nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g -(ASort n1 x0) x2))) H2 O H6) in (let H8 \def (eq_ind_r nat x0 (\lambda (n1: -nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort O n1) x2))) H7 -n0 H5) in (let H9 \def (eq_ind_r A (aplus g (ASort O (next g n0)) x2) -(\lambda (a0: A).(eq A a0 (aplus g (ASort O n0) x2))) H8 (aplus g (ASort O -n0) (S x2)) (aplus_sort_O_S_simpl g n0 x2)) in (let H_y \def (aplus_inj g (S -x2) x2 (ASort O n0) H9) in (le_Sx_x x2 (eq_ind_r nat x2 (\lambda (n1: -nat).(le n1 x2)) (le_n x2) (S x2) H_y) P))))))) H4))))))) H1)))) (\lambda -(n1: nat).(\lambda (_: (((leq g (match n1 with [O \Rightarrow (ASort O (next -g n0)) | (S h) \Rightarrow (ASort h n0)]) (ASort n1 n0)) \to P))).(\lambda -(H0: (leq g (ASort n1 n0) (ASort (S n1) n0))).(let H_x \def (leq_gen_sort1 g -n1 n0 (ASort (S n1) n0) H0) in (let H1 \def H_x in (ex2_3_ind nat nat nat -(\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort -n1 n0) k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: -nat).(\lambda (_: nat).(eq A (ASort (S n1) n0) (ASort h2 n2))))) P (\lambda -(x0: nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (H2: (eq A (aplus g -(ASort n1 n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H3: (eq A (ASort (S -n1) n0) (ASort x1 x0))).(let H4 \def (f_equal A nat (\lambda (e: A).(match e -in A return (\lambda (_: A).nat) with [(ASort n2 _) \Rightarrow n2 | (AHead _ -_) \Rightarrow (S n1)])) (ASort (S n1) n0) (ASort x1 x0) H3) in ((let H5 \def -(f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with -[(ASort _ n2) \Rightarrow n2 | (AHead _ _) \Rightarrow n0])) (ASort (S n1) -n0) (ASort x1 x0) H3) in (\lambda (H6: (eq nat (S n1) x1)).(let H7 \def -(eq_ind_r nat x1 (\lambda (n2: nat).(eq A (aplus g (ASort n1 n0) x2) (aplus g -(ASort n2 x0) x2))) H2 (S n1) H6) in (let H8 \def (eq_ind_r nat x0 (\lambda -(n2: nat).(eq A (aplus g (ASort n1 n0) x2) (aplus g (ASort (S n1) n2) x2))) -H7 n0 H5) in (let H9 \def (eq_ind_r A (aplus g (ASort n1 n0) x2) (\lambda -(a0: A).(eq A a0 (aplus g (ASort (S n1) n0) x2))) H8 (aplus g (ASort (S n1) -n0) (S x2)) (aplus_sort_S_S_simpl g n0 n1 x2)) in (let H_y \def (aplus_inj g -(S x2) x2 (ASort (S n1) n0) H9) in (le_Sx_x x2 (eq_ind_r nat x2 (\lambda (n2: -nat).(le n2 x2)) (le_n x2) (S x2) H_y) P))))))) H4))))))) H1)))))) n H))))) -(\lambda (a0: A).(\lambda (_: (((leq g (asucc g a0) a0) \to (\forall (P: -Prop).P)))).(\lambda (a1: A).(\lambda (H0: (((leq g (asucc g a1) a1) \to -(\forall (P: Prop).P)))).(\lambda (H1: (leq g (AHead a0 (asucc g a1)) (AHead -a0 a1))).(\lambda (P: Prop).(let H_x \def (leq_gen_head1 g a0 (asucc g a1) -(AHead a0 a1) H1) in (let H2 \def H_x in (ex3_2_ind A A (\lambda (a3: -A).(\lambda (_: A).(leq g a0 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g -(asucc g a1) a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (AHead a0 a1) -(AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (H3: (leq g a0 -x0)).(\lambda (H4: (leq g (asucc g a1) x1)).(\lambda (H5: (eq A (AHead a0 a1) -(AHead x0 x1))).(let H6 \def (f_equal A A (\lambda (e: A).(match e in A -return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead a2 _) -\Rightarrow a2])) (AHead a0 a1) (AHead x0 x1) H5) in ((let H7 \def (f_equal A -A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) -\Rightarrow a1 | (AHead _ a2) \Rightarrow a2])) (AHead a0 a1) (AHead x0 x1) -H5) in (\lambda (H8: (eq A a0 x0)).(let H9 \def (eq_ind_r A x1 (\lambda (a2: -A).(leq g (asucc g a1) a2)) H4 a1 H7) in (let H10 \def (eq_ind_r A x0 -(\lambda (a2: A).(leq g a0 a2)) H3 a0 H8) in (H0 H9 P))))) H6))))))) -H2))))))))) a)). -(* COMMENTS -Initial nodes: 1327 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/leq/defs.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/leq/defs.ma deleted file mode 100644 index 5a5308280..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/leq/defs.ma +++ /dev/null @@ -1,26 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/aplus/defs.ma". - -inductive leq (g: G): A \to (A \to Prop) \def -| leq_sort: \forall (h1: nat).(\forall (h2: nat).(\forall (n1: nat).(\forall -(n2: nat).(\forall (k: nat).((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort -h2 n2) k)) \to (leq g (ASort h1 n1) (ASort h2 n2))))))) -| leq_head: \forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall (a3: -A).(\forall (a4: A).((leq g a3 a4) \to (leq g (AHead a1 a3) (AHead a2 -a4))))))). - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/leq/fwd.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/leq/fwd.ma deleted file mode 100644 index e259fa19e..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/leq/fwd.ma +++ /dev/null @@ -1,244 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/leq/defs.ma". - -theorem leq_gen_sort1: - \forall (g: G).(\forall (h1: nat).(\forall (n1: nat).(\forall (a2: A).((leq -g (ASort h1 n1) a2) \to (ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: -nat).(\lambda (k: nat).(eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) -k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a2 -(ASort h2 n2)))))))))) -\def - \lambda (g: G).(\lambda (h1: nat).(\lambda (n1: nat).(\lambda (a2: -A).(\lambda (H: (leq g (ASort h1 n1) a2)).(insert_eq A (ASort h1 n1) (\lambda -(a: A).(leq g a a2)) (\lambda (a: A).(ex2_3 nat nat nat (\lambda (n2: -nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g a k) (aplus g (ASort -h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A -a2 (ASort h2 n2))))))) (\lambda (y: A).(\lambda (H0: (leq g y a2)).(leq_ind g -(\lambda (a: A).(\lambda (a0: A).((eq A a (ASort h1 n1)) \to (ex2_3 nat nat -nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g a -k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: -nat).(\lambda (_: nat).(eq A a0 (ASort h2 n2))))))))) (\lambda (h0: -nat).(\lambda (h2: nat).(\lambda (n0: nat).(\lambda (n2: nat).(\lambda (k: -nat).(\lambda (H1: (eq A (aplus g (ASort h0 n0) k) (aplus g (ASort h2 n2) -k))).(\lambda (H2: (eq A (ASort h0 n0) (ASort h1 n1))).(let H3 \def (f_equal -A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort -n _) \Rightarrow n | (AHead _ _) \Rightarrow h0])) (ASort h0 n0) (ASort h1 -n1) H2) in ((let H4 \def (f_equal A nat (\lambda (e: A).(match e in A return -(\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) -\Rightarrow n0])) (ASort h0 n0) (ASort h1 n1) H2) in (\lambda (H5: (eq nat h0 -h1)).(let H6 \def (eq_ind nat n0 (\lambda (n: nat).(eq A (aplus g (ASort h0 -n) k) (aplus g (ASort h2 n2) k))) H1 n1 H4) in (eq_ind_r nat n1 (\lambda (n: -nat).(ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (k0: -nat).(eq A (aplus g (ASort h0 n) k0) (aplus g (ASort h3 n3) k0))))) (\lambda -(n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A (ASort h2 n2) (ASort h3 -n3))))))) (let H7 \def (eq_ind nat h0 (\lambda (n: nat).(eq A (aplus g (ASort -n n1) k) (aplus g (ASort h2 n2) k))) H6 h1 H5) in (eq_ind_r nat h1 (\lambda -(n: nat).(ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda -(k0: nat).(eq A (aplus g (ASort n n1) k0) (aplus g (ASort h3 n3) k0))))) -(\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A (ASort h2 n2) -(ASort h3 n3))))))) (ex2_3_intro nat nat nat (\lambda (n3: nat).(\lambda (h3: -nat).(\lambda (k0: nat).(eq A (aplus g (ASort h1 n1) k0) (aplus g (ASort h3 -n3) k0))))) (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A -(ASort h2 n2) (ASort h3 n3))))) n2 h2 k H7 (refl_equal A (ASort h2 n2))) h0 -H5)) n0 H4)))) H3))))))))) (\lambda (a1: A).(\lambda (a3: A).(\lambda (_: -(leq g a1 a3)).(\lambda (_: (((eq A a1 (ASort h1 n1)) \to (ex2_3 nat nat nat -(\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g a1 k) -(aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda -(_: nat).(eq A a3 (ASort h2 n2))))))))).(\lambda (a4: A).(\lambda (a5: -A).(\lambda (_: (leq g a4 a5)).(\lambda (_: (((eq A a4 (ASort h1 n1)) \to -(ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: -nat).(eq A (aplus g a4 k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: -nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a5 (ASort h2 -n2))))))))).(\lambda (H5: (eq A (AHead a1 a4) (ASort h1 n1))).(let H6 \def -(eq_ind A (AHead a1 a4) (\lambda (ee: A).(match ee in A return (\lambda (_: -A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow -True])) I (ASort h1 n1) H5) in (False_ind (ex2_3 nat nat nat (\lambda (n2: -nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (AHead a1 a4) k) -(aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda -(_: nat).(eq A (AHead a3 a5) (ASort h2 n2)))))) H6))))))))))) y a2 H0))) -H))))). -(* COMMENTS -Initial nodes: 913 -END *) - -theorem leq_gen_head1: - \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a: A).((leq g -(AHead a1 a2) a) \to (ex3_2 A A (\lambda (a3: A).(\lambda (_: A).(leq g a1 -a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: -A).(\lambda (a4: A).(eq A a (AHead a3 a4))))))))) -\def - \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a: A).(\lambda -(H: (leq g (AHead a1 a2) a)).(insert_eq A (AHead a1 a2) (\lambda (a0: A).(leq -g a0 a)) (\lambda (_: A).(ex3_2 A A (\lambda (a3: A).(\lambda (_: A).(leq g -a1 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: -A).(\lambda (a4: A).(eq A a (AHead a3 a4)))))) (\lambda (y: A).(\lambda (H0: -(leq g y a)).(leq_ind g (\lambda (a0: A).(\lambda (a3: A).((eq A a0 (AHead a1 -a2)) \to (ex3_2 A A (\lambda (a4: A).(\lambda (_: A).(leq g a1 a4))) (\lambda -(_: A).(\lambda (a5: A).(leq g a2 a5))) (\lambda (a4: A).(\lambda (a5: A).(eq -A a3 (AHead a4 a5)))))))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: -nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (_: (eq A (aplus g (ASort -h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (H2: (eq A (ASort h1 n1) -(AHead a1 a2))).(let H3 \def (eq_ind A (ASort h1 n1) (\lambda (ee: A).(match -ee in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | -(AHead _ _) \Rightarrow False])) I (AHead a1 a2) H2) in (False_ind (ex3_2 A A -(\lambda (a3: A).(\lambda (_: A).(leq g a1 a3))) (\lambda (_: A).(\lambda -(a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort h2 n2) -(AHead a3 a4))))) H3))))))))) (\lambda (a0: A).(\lambda (a3: A).(\lambda (H1: -(leq g a0 a3)).(\lambda (H2: (((eq A a0 (AHead a1 a2)) \to (ex3_2 A A -(\lambda (a4: A).(\lambda (_: A).(leq g a1 a4))) (\lambda (_: A).(\lambda -(a5: A).(leq g a2 a5))) (\lambda (a4: A).(\lambda (a5: A).(eq A a3 (AHead a4 -a5)))))))).(\lambda (a4: A).(\lambda (a5: A).(\lambda (H3: (leq g a4 -a5)).(\lambda (H4: (((eq A a4 (AHead a1 a2)) \to (ex3_2 A A (\lambda (a6: -A).(\lambda (_: A).(leq g a1 a6))) (\lambda (_: A).(\lambda (a7: A).(leq g a2 -a7))) (\lambda (a6: A).(\lambda (a7: A).(eq A a5 (AHead a6 -a7)))))))).(\lambda (H5: (eq A (AHead a0 a4) (AHead a1 a2))).(let H6 \def -(f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with -[(ASort _ _) \Rightarrow a0 | (AHead a6 _) \Rightarrow a6])) (AHead a0 a4) -(AHead a1 a2) H5) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e in A -return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a4 | (AHead _ a6) -\Rightarrow a6])) (AHead a0 a4) (AHead a1 a2) H5) in (\lambda (H8: (eq A a0 -a1)).(let H9 \def (eq_ind A a4 (\lambda (a6: A).((eq A a6 (AHead a1 a2)) \to -(ex3_2 A A (\lambda (a7: A).(\lambda (_: A).(leq g a1 a7))) (\lambda (_: -A).(\lambda (a8: A).(leq g a2 a8))) (\lambda (a7: A).(\lambda (a8: A).(eq A -a5 (AHead a7 a8))))))) H4 a2 H7) in (let H10 \def (eq_ind A a4 (\lambda (a6: -A).(leq g a6 a5)) H3 a2 H7) in (let H11 \def (eq_ind A a0 (\lambda (a6: -A).((eq A a6 (AHead a1 a2)) \to (ex3_2 A A (\lambda (a7: A).(\lambda (_: -A).(leq g a1 a7))) (\lambda (_: A).(\lambda (a8: A).(leq g a2 a8))) (\lambda -(a7: A).(\lambda (a8: A).(eq A a3 (AHead a7 a8))))))) H2 a1 H8) in (let H12 -\def (eq_ind A a0 (\lambda (a6: A).(leq g a6 a3)) H1 a1 H8) in (ex3_2_intro A -A (\lambda (a6: A).(\lambda (_: A).(leq g a1 a6))) (\lambda (_: A).(\lambda -(a7: A).(leq g a2 a7))) (\lambda (a6: A).(\lambda (a7: A).(eq A (AHead a3 a5) -(AHead a6 a7)))) a3 a5 H12 H10 (refl_equal A (AHead a3 a5))))))))) -H6))))))))))) y a H0))) H))))). -(* COMMENTS -Initial nodes: 797 -END *) - -theorem leq_gen_sort2: - \forall (g: G).(\forall (h1: nat).(\forall (n1: nat).(\forall (a2: A).((leq -g a2 (ASort h1 n1)) \to (ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: -nat).(\lambda (k: nat).(eq A (aplus g (ASort h2 n2) k) (aplus g (ASort h1 n1) -k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a2 -(ASort h2 n2)))))))))) -\def - \lambda (g: G).(\lambda (h1: nat).(\lambda (n1: nat).(\lambda (a2: -A).(\lambda (H: (leq g a2 (ASort h1 n1))).(insert_eq A (ASort h1 n1) (\lambda -(a: A).(leq g a2 a)) (\lambda (a: A).(ex2_3 nat nat nat (\lambda (n2: -nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort h2 n2) k) -(aplus g a k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq -A a2 (ASort h2 n2))))))) (\lambda (y: A).(\lambda (H0: (leq g a2 y)).(leq_ind -g (\lambda (a: A).(\lambda (a0: A).((eq A a0 (ASort h1 n1)) \to (ex2_3 nat -nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus -g (ASort h2 n2) k) (aplus g a0 k))))) (\lambda (n2: nat).(\lambda (h2: -nat).(\lambda (_: nat).(eq A a (ASort h2 n2))))))))) (\lambda (h0: -nat).(\lambda (h2: nat).(\lambda (n0: nat).(\lambda (n2: nat).(\lambda (k: -nat).(\lambda (H1: (eq A (aplus g (ASort h0 n0) k) (aplus g (ASort h2 n2) -k))).(\lambda (H2: (eq A (ASort h2 n2) (ASort h1 n1))).(let H3 \def (f_equal -A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort -n _) \Rightarrow n | (AHead _ _) \Rightarrow h2])) (ASort h2 n2) (ASort h1 -n1) H2) in ((let H4 \def (f_equal A nat (\lambda (e: A).(match e in A return -(\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) -\Rightarrow n2])) (ASort h2 n2) (ASort h1 n1) H2) in (\lambda (H5: (eq nat h2 -h1)).(let H6 \def (eq_ind nat n2 (\lambda (n: nat).(eq A (aplus g (ASort h0 -n0) k) (aplus g (ASort h2 n) k))) H1 n1 H4) in (eq_ind_r nat n1 (\lambda (n: -nat).(ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (k0: -nat).(eq A (aplus g (ASort h3 n3) k0) (aplus g (ASort h2 n) k0))))) (\lambda -(n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A (ASort h0 n0) (ASort h3 -n3))))))) (let H7 \def (eq_ind nat h2 (\lambda (n: nat).(eq A (aplus g (ASort -h0 n0) k) (aplus g (ASort n n1) k))) H6 h1 H5) in (eq_ind_r nat h1 (\lambda -(n: nat).(ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda -(k0: nat).(eq A (aplus g (ASort h3 n3) k0) (aplus g (ASort n n1) k0))))) -(\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A (ASort h0 n0) -(ASort h3 n3))))))) (ex2_3_intro nat nat nat (\lambda (n3: nat).(\lambda (h3: -nat).(\lambda (k0: nat).(eq A (aplus g (ASort h3 n3) k0) (aplus g (ASort h1 -n1) k0))))) (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A -(ASort h0 n0) (ASort h3 n3))))) n0 h0 k H7 (refl_equal A (ASort h0 n0))) h2 -H5)) n2 H4)))) H3))))))))) (\lambda (a1: A).(\lambda (a3: A).(\lambda (_: -(leq g a1 a3)).(\lambda (_: (((eq A a3 (ASort h1 n1)) \to (ex2_3 nat nat nat -(\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort -h2 n2) k) (aplus g a3 k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda -(_: nat).(eq A a1 (ASort h2 n2))))))))).(\lambda (a4: A).(\lambda (a5: -A).(\lambda (_: (leq g a4 a5)).(\lambda (_: (((eq A a5 (ASort h1 n1)) \to -(ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: -nat).(eq A (aplus g (ASort h2 n2) k) (aplus g a5 k))))) (\lambda (n2: -nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a4 (ASort h2 -n2))))))))).(\lambda (H5: (eq A (AHead a3 a5) (ASort h1 n1))).(let H6 \def -(eq_ind A (AHead a3 a5) (\lambda (ee: A).(match ee in A return (\lambda (_: -A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow -True])) I (ASort h1 n1) H5) in (False_ind (ex2_3 nat nat nat (\lambda (n2: -nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort h2 n2) k) -(aplus g (AHead a3 a5) k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda -(_: nat).(eq A (AHead a1 a4) (ASort h2 n2)))))) H6))))))))))) a2 y H0))) -H))))). -(* COMMENTS -Initial nodes: 913 -END *) - -theorem leq_gen_head2: - \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a: A).((leq g a -(AHead a1 a2)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (_: A).(leq g a3 -a1))) (\lambda (_: A).(\lambda (a4: A).(leq g a4 a2))) (\lambda (a3: -A).(\lambda (a4: A).(eq A a (AHead a3 a4))))))))) -\def - \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a: A).(\lambda -(H: (leq g a (AHead a1 a2))).(insert_eq A (AHead a1 a2) (\lambda (a0: A).(leq -g a a0)) (\lambda (_: A).(ex3_2 A A (\lambda (a3: A).(\lambda (_: A).(leq g -a3 a1))) (\lambda (_: A).(\lambda (a4: A).(leq g a4 a2))) (\lambda (a3: -A).(\lambda (a4: A).(eq A a (AHead a3 a4)))))) (\lambda (y: A).(\lambda (H0: -(leq g a y)).(leq_ind g (\lambda (a0: A).(\lambda (a3: A).((eq A a3 (AHead a1 -a2)) \to (ex3_2 A A (\lambda (a4: A).(\lambda (_: A).(leq g a4 a1))) (\lambda -(_: A).(\lambda (a5: A).(leq g a5 a2))) (\lambda (a4: A).(\lambda (a5: A).(eq -A a0 (AHead a4 a5)))))))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: -nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (_: (eq A (aplus g (ASort -h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (H2: (eq A (ASort h2 n2) -(AHead a1 a2))).(let H3 \def (eq_ind A (ASort h2 n2) (\lambda (ee: A).(match -ee in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | -(AHead _ _) \Rightarrow False])) I (AHead a1 a2) H2) in (False_ind (ex3_2 A A -(\lambda (a3: A).(\lambda (_: A).(leq g a3 a1))) (\lambda (_: A).(\lambda -(a4: A).(leq g a4 a2))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort h1 n1) -(AHead a3 a4))))) H3))))))))) (\lambda (a0: A).(\lambda (a3: A).(\lambda (H1: -(leq g a0 a3)).(\lambda (H2: (((eq A a3 (AHead a1 a2)) \to (ex3_2 A A -(\lambda (a4: A).(\lambda (_: A).(leq g a4 a1))) (\lambda (_: A).(\lambda -(a5: A).(leq g a5 a2))) (\lambda (a4: A).(\lambda (a5: A).(eq A a0 (AHead a4 -a5)))))))).(\lambda (a4: A).(\lambda (a5: A).(\lambda (H3: (leq g a4 -a5)).(\lambda (H4: (((eq A a5 (AHead a1 a2)) \to (ex3_2 A A (\lambda (a6: -A).(\lambda (_: A).(leq g a6 a1))) (\lambda (_: A).(\lambda (a7: A).(leq g a7 -a2))) (\lambda (a6: A).(\lambda (a7: A).(eq A a4 (AHead a6 -a7)))))))).(\lambda (H5: (eq A (AHead a3 a5) (AHead a1 a2))).(let H6 \def -(f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with -[(ASort _ _) \Rightarrow a3 | (AHead a6 _) \Rightarrow a6])) (AHead a3 a5) -(AHead a1 a2) H5) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e in A -return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a5 | (AHead _ a6) -\Rightarrow a6])) (AHead a3 a5) (AHead a1 a2) H5) in (\lambda (H8: (eq A a3 -a1)).(let H9 \def (eq_ind A a5 (\lambda (a6: A).((eq A a6 (AHead a1 a2)) \to -(ex3_2 A A (\lambda (a7: A).(\lambda (_: A).(leq g a7 a1))) (\lambda (_: -A).(\lambda (a8: A).(leq g a8 a2))) (\lambda (a7: A).(\lambda (a8: A).(eq A -a4 (AHead a7 a8))))))) H4 a2 H7) in (let H10 \def (eq_ind A a5 (\lambda (a6: -A).(leq g a4 a6)) H3 a2 H7) in (let H11 \def (eq_ind A a3 (\lambda (a6: -A).((eq A a6 (AHead a1 a2)) \to (ex3_2 A A (\lambda (a7: A).(\lambda (_: -A).(leq g a7 a1))) (\lambda (_: A).(\lambda (a8: A).(leq g a8 a2))) (\lambda -(a7: A).(\lambda (a8: A).(eq A a0 (AHead a7 a8))))))) H2 a1 H8) in (let H12 -\def (eq_ind A a3 (\lambda (a6: A).(leq g a0 a6)) H1 a1 H8) in (ex3_2_intro A -A (\lambda (a6: A).(\lambda (_: A).(leq g a6 a1))) (\lambda (_: A).(\lambda -(a7: A).(leq g a7 a2))) (\lambda (a6: A).(\lambda (a7: A).(eq A (AHead a0 a4) -(AHead a6 a7)))) a0 a4 H12 H10 (refl_equal A (AHead a0 a4))))))))) -H6))))))))))) a y H0))) H))))). -(* COMMENTS -Initial nodes: 797 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/leq/props.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/leq/props.ma deleted file mode 100644 index b83fc503e..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/leq/props.ma +++ /dev/null @@ -1,233 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/leq/fwd.ma". - -include "Basic-1/aplus/props.ma". - -theorem ahead_inj_snd: - \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a3: A).(\forall -(a4: A).((leq g (AHead a1 a2) (AHead a3 a4)) \to (leq g a2 a4)))))) -\def - \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a3: A).(\lambda -(a4: A).(\lambda (H: (leq g (AHead a1 a2) (AHead a3 a4))).(let H_x \def -(leq_gen_head1 g a1 a2 (AHead a3 a4) H) in (let H0 \def H_x in (ex3_2_ind A A -(\lambda (a5: A).(\lambda (_: A).(leq g a1 a5))) (\lambda (_: A).(\lambda -(a6: A).(leq g a2 a6))) (\lambda (a5: A).(\lambda (a6: A).(eq A (AHead a3 a4) -(AHead a5 a6)))) (leq g a2 a4) (\lambda (x0: A).(\lambda (x1: A).(\lambda -(H1: (leq g a1 x0)).(\lambda (H2: (leq g a2 x1)).(\lambda (H3: (eq A (AHead -a3 a4) (AHead x0 x1))).(let H4 \def (f_equal A A (\lambda (e: A).(match e in -A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a3 | (AHead a _) -\Rightarrow a])) (AHead a3 a4) (AHead x0 x1) H3) in ((let H5 \def (f_equal A -A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) -\Rightarrow a4 | (AHead _ a) \Rightarrow a])) (AHead a3 a4) (AHead x0 x1) H3) -in (\lambda (H6: (eq A a3 x0)).(let H7 \def (eq_ind_r A x1 (\lambda (a: -A).(leq g a2 a)) H2 a4 H5) in (let H8 \def (eq_ind_r A x0 (\lambda (a: -A).(leq g a1 a)) H1 a3 H6) in H7)))) H4))))))) H0)))))))). -(* COMMENTS -Initial nodes: 259 -END *) - -theorem leq_refl: - \forall (g: G).(\forall (a: A).(leq g a a)) -\def - \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(leq g a0 a0)) -(\lambda (n: nat).(\lambda (n0: nat).(leq_sort g n n n0 n0 O (refl_equal A -(aplus g (ASort n n0) O))))) (\lambda (a0: A).(\lambda (H: (leq g a0 -a0)).(\lambda (a1: A).(\lambda (H0: (leq g a1 a1)).(leq_head g a0 a0 H a1 a1 -H0))))) a)). -(* COMMENTS -Initial nodes: 87 -END *) - -theorem leq_eq: - \forall (g: G).(\forall (a1: A).(\forall (a2: A).((eq A a1 a2) \to (leq g a1 -a2)))) -\def - \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (eq A a1 -a2)).(eq_ind A a1 (\lambda (a: A).(leq g a1 a)) (leq_refl g a1) a2 H)))). -(* COMMENTS -Initial nodes: 39 -END *) - -theorem leq_sym: - \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (leq g -a2 a1)))) -\def - \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1 -a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(leq g a0 a))) (\lambda (h1: -nat).(\lambda (h2: nat).(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (k: -nat).(\lambda (H0: (eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) -k))).(leq_sort g h2 h1 n2 n1 k (sym_eq A (aplus g (ASort h1 n1) k) (aplus g -(ASort h2 n2) k) H0)))))))) (\lambda (a3: A).(\lambda (a4: A).(\lambda (_: -(leq g a3 a4)).(\lambda (H1: (leq g a4 a3)).(\lambda (a5: A).(\lambda (a6: -A).(\lambda (_: (leq g a5 a6)).(\lambda (H3: (leq g a6 a5)).(leq_head g a4 a3 -H1 a6 a5 H3))))))))) a1 a2 H)))). -(* COMMENTS -Initial nodes: 173 -END *) - -theorem leq_trans: - \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall -(a3: A).((leq g a2 a3) \to (leq g a1 a3)))))) -\def - \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1 -a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(\forall (a3: A).((leq g a0 -a3) \to (leq g a a3))))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: -nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (H0: (eq A (aplus g (ASort -h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (a3: A).(\lambda (H1: (leq g -(ASort h2 n2) a3)).(let H_x \def (leq_gen_sort1 g h2 n2 a3 H1) in (let H2 -\def H_x in (ex2_3_ind nat nat nat (\lambda (n3: nat).(\lambda (h3: -nat).(\lambda (k0: nat).(eq A (aplus g (ASort h2 n2) k0) (aplus g (ASort h3 -n3) k0))))) (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A a3 -(ASort h3 n3))))) (leq g (ASort h1 n1) a3) (\lambda (x0: nat).(\lambda (x1: -nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort h2 n2) x2) (aplus -g (ASort x1 x0) x2))).(\lambda (H4: (eq A a3 (ASort x1 x0))).(let H5 \def -(f_equal A A (\lambda (e: A).e) a3 (ASort x1 x0) H4) in (eq_ind_r A (ASort x1 -x0) (\lambda (a: A).(leq g (ASort h1 n1) a)) (lt_le_e k x2 (leq g (ASort h1 -n1) (ASort x1 x0)) (\lambda (H6: (lt k x2)).(let H_y \def (aplus_reg_r g -(ASort h1 n1) (ASort h2 n2) k k H0 (minus x2 k)) in (let H7 \def (eq_ind_r -nat (plus (minus x2 k) k) (\lambda (n: nat).(eq A (aplus g (ASort h1 n1) n) -(aplus g (ASort h2 n2) n))) H_y x2 (le_plus_minus_sym k x2 (le_trans k (S k) -x2 (le_S k k (le_n k)) H6))) in (leq_sort g h1 x1 n1 x0 x2 (trans_eq A (aplus -g (ASort h1 n1) x2) (aplus g (ASort h2 n2) x2) (aplus g (ASort x1 x0) x2) H7 -H3))))) (\lambda (H6: (le x2 k)).(let H_y \def (aplus_reg_r g (ASort h2 n2) -(ASort x1 x0) x2 x2 H3 (minus k x2)) in (let H7 \def (eq_ind_r nat (plus -(minus k x2) x2) (\lambda (n: nat).(eq A (aplus g (ASort h2 n2) n) (aplus g -(ASort x1 x0) n))) H_y k (le_plus_minus_sym x2 k H6)) in (leq_sort g h1 x1 n1 -x0 k (trans_eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k) (aplus g -(ASort x1 x0) k) H0 H7)))))) a3 H5))))))) H2))))))))))) (\lambda (a3: -A).(\lambda (a4: A).(\lambda (_: (leq g a3 a4)).(\lambda (H1: ((\forall (a5: -A).((leq g a4 a5) \to (leq g a3 a5))))).(\lambda (a5: A).(\lambda (a6: -A).(\lambda (_: (leq g a5 a6)).(\lambda (H3: ((\forall (a7: A).((leq g a6 a7) -\to (leq g a5 a7))))).(\lambda (a0: A).(\lambda (H4: (leq g (AHead a4 a6) -a0)).(let H_x \def (leq_gen_head1 g a4 a6 a0 H4) in (let H5 \def H_x in -(ex3_2_ind A A (\lambda (a7: A).(\lambda (_: A).(leq g a4 a7))) (\lambda (_: -A).(\lambda (a8: A).(leq g a6 a8))) (\lambda (a7: A).(\lambda (a8: A).(eq A -a0 (AHead a7 a8)))) (leq g (AHead a3 a5) a0) (\lambda (x0: A).(\lambda (x1: -A).(\lambda (H6: (leq g a4 x0)).(\lambda (H7: (leq g a6 x1)).(\lambda (H8: -(eq A a0 (AHead x0 x1))).(let H9 \def (f_equal A A (\lambda (e: A).e) a0 -(AHead x0 x1) H8) in (eq_ind_r A (AHead x0 x1) (\lambda (a: A).(leq g (AHead -a3 a5) a)) (leq_head g a3 x0 (H1 x0 H6) a5 x1 (H3 x1 H7)) a0 H9))))))) -H5))))))))))))) a1 a2 H)))). -(* COMMENTS -Initial nodes: 869 -END *) - -theorem leq_ahead_false_1: - \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (AHead a1 a2) a1) -\to (\forall (P: Prop).P)))) -\def - \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2: -A).((leq g (AHead a a2) a) \to (\forall (P: Prop).P)))) (\lambda (n: -nat).(\lambda (n0: nat).(\lambda (a2: A).(\lambda (H: (leq g (AHead (ASort n -n0) a2) (ASort n n0))).(\lambda (P: Prop).(nat_ind (\lambda (n1: nat).((leq g -(AHead (ASort n1 n0) a2) (ASort n1 n0)) \to P)) (\lambda (H0: (leq g (AHead -(ASort O n0) a2) (ASort O n0))).(let H_x \def (leq_gen_head1 g (ASort O n0) -a2 (ASort O n0) H0) in (let H1 \def H_x in (ex3_2_ind A A (\lambda (a3: -A).(\lambda (_: A).(leq g (ASort O n0) a3))) (\lambda (_: A).(\lambda (a4: -A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort O n0) -(AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g -(ASort O n0) x0)).(\lambda (_: (leq g a2 x1)).(\lambda (H4: (eq A (ASort O -n0) (AHead x0 x1))).(let H5 \def (eq_ind A (ASort O n0) (\lambda (ee: -A).(match ee in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow -True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H4) in (False_ind P -H5))))))) H1)))) (\lambda (n1: nat).(\lambda (_: (((leq g (AHead (ASort n1 -n0) a2) (ASort n1 n0)) \to P))).(\lambda (H0: (leq g (AHead (ASort (S n1) n0) -a2) (ASort (S n1) n0))).(let H_x \def (leq_gen_head1 g (ASort (S n1) n0) a2 -(ASort (S n1) n0) H0) in (let H1 \def H_x in (ex3_2_ind A A (\lambda (a3: -A).(\lambda (_: A).(leq g (ASort (S n1) n0) a3))) (\lambda (_: A).(\lambda -(a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort (S n1) -n0) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g -(ASort (S n1) n0) x0)).(\lambda (_: (leq g a2 x1)).(\lambda (H4: (eq A (ASort -(S n1) n0) (AHead x0 x1))).(let H5 \def (eq_ind A (ASort (S n1) n0) (\lambda -(ee: A).(match ee in A return (\lambda (_: A).Prop) with [(ASort _ _) -\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H4) in -(False_ind P H5))))))) H1)))))) n H)))))) (\lambda (a: A).(\lambda (H: -((\forall (a2: A).((leq g (AHead a a2) a) \to (\forall (P: -Prop).P))))).(\lambda (a0: A).(\lambda (_: ((\forall (a2: A).((leq g (AHead -a0 a2) a0) \to (\forall (P: Prop).P))))).(\lambda (a2: A).(\lambda (H1: (leq -g (AHead (AHead a a0) a2) (AHead a a0))).(\lambda (P: Prop).(let H_x \def -(leq_gen_head1 g (AHead a a0) a2 (AHead a a0) H1) in (let H2 \def H_x in -(ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq g (AHead a a0) a3))) -(\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda -(a4: A).(eq A (AHead a a0) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: -A).(\lambda (H3: (leq g (AHead a a0) x0)).(\lambda (H4: (leq g a2 -x1)).(\lambda (H5: (eq A (AHead a a0) (AHead x0 x1))).(let H6 \def (f_equal A -A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) -\Rightarrow a | (AHead a3 _) \Rightarrow a3])) (AHead a a0) (AHead x0 x1) H5) -in ((let H7 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda -(_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead _ a3) \Rightarrow a3])) -(AHead a a0) (AHead x0 x1) H5) in (\lambda (H8: (eq A a x0)).(let H9 \def -(eq_ind_r A x1 (\lambda (a3: A).(leq g a2 a3)) H4 a0 H7) in (let H10 \def -(eq_ind_r A x0 (\lambda (a3: A).(leq g (AHead a a0) a3)) H3 a H8) in (H a0 -H10 P))))) H6))))))) H2)))))))))) a1)). -(* COMMENTS -Initial nodes: 797 -END *) - -theorem leq_ahead_false_2: - \forall (g: G).(\forall (a2: A).(\forall (a1: A).((leq g (AHead a1 a2) a2) -\to (\forall (P: Prop).P)))) -\def - \lambda (g: G).(\lambda (a2: A).(A_ind (\lambda (a: A).(\forall (a1: -A).((leq g (AHead a1 a) a) \to (\forall (P: Prop).P)))) (\lambda (n: -nat).(\lambda (n0: nat).(\lambda (a1: A).(\lambda (H: (leq g (AHead a1 (ASort -n n0)) (ASort n n0))).(\lambda (P: Prop).(nat_ind (\lambda (n1: nat).((leq g -(AHead a1 (ASort n1 n0)) (ASort n1 n0)) \to P)) (\lambda (H0: (leq g (AHead -a1 (ASort O n0)) (ASort O n0))).(let H_x \def (leq_gen_head1 g a1 (ASort O -n0) (ASort O n0) H0) in (let H1 \def H_x in (ex3_2_ind A A (\lambda (a3: -A).(\lambda (_: A).(leq g a1 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g -(ASort O n0) a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort O n0) -(AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g a1 -x0)).(\lambda (_: (leq g (ASort O n0) x1)).(\lambda (H4: (eq A (ASort O n0) -(AHead x0 x1))).(let H5 \def (eq_ind A (ASort O n0) (\lambda (ee: A).(match -ee in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | -(AHead _ _) \Rightarrow False])) I (AHead x0 x1) H4) in (False_ind P -H5))))))) H1)))) (\lambda (n1: nat).(\lambda (_: (((leq g (AHead a1 (ASort n1 -n0)) (ASort n1 n0)) \to P))).(\lambda (H0: (leq g (AHead a1 (ASort (S n1) -n0)) (ASort (S n1) n0))).(let H_x \def (leq_gen_head1 g a1 (ASort (S n1) n0) -(ASort (S n1) n0) H0) in (let H1 \def H_x in (ex3_2_ind A A (\lambda (a3: -A).(\lambda (_: A).(leq g a1 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g -(ASort (S n1) n0) a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort (S n1) -n0) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g -a1 x0)).(\lambda (_: (leq g (ASort (S n1) n0) x1)).(\lambda (H4: (eq A (ASort -(S n1) n0) (AHead x0 x1))).(let H5 \def (eq_ind A (ASort (S n1) n0) (\lambda -(ee: A).(match ee in A return (\lambda (_: A).Prop) with [(ASort _ _) -\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H4) in -(False_ind P H5))))))) H1)))))) n H)))))) (\lambda (a: A).(\lambda (_: -((\forall (a1: A).((leq g (AHead a1 a) a) \to (\forall (P: -Prop).P))))).(\lambda (a0: A).(\lambda (H0: ((\forall (a1: A).((leq g (AHead -a1 a0) a0) \to (\forall (P: Prop).P))))).(\lambda (a1: A).(\lambda (H1: (leq -g (AHead a1 (AHead a a0)) (AHead a a0))).(\lambda (P: Prop).(let H_x \def -(leq_gen_head1 g a1 (AHead a a0) (AHead a a0) H1) in (let H2 \def H_x in -(ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq g a1 a3))) (\lambda (_: -A).(\lambda (a4: A).(leq g (AHead a a0) a4))) (\lambda (a3: A).(\lambda (a4: -A).(eq A (AHead a a0) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: -A).(\lambda (H3: (leq g a1 x0)).(\lambda (H4: (leq g (AHead a a0) -x1)).(\lambda (H5: (eq A (AHead a a0) (AHead x0 x1))).(let H6 \def (f_equal A -A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) -\Rightarrow a | (AHead a3 _) \Rightarrow a3])) (AHead a a0) (AHead x0 x1) H5) -in ((let H7 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda -(_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead _ a3) \Rightarrow a3])) -(AHead a a0) (AHead x0 x1) H5) in (\lambda (H8: (eq A a x0)).(let H9 \def -(eq_ind_r A x1 (\lambda (a3: A).(leq g (AHead a a0) a3)) H4 a0 H7) in (let -H10 \def (eq_ind_r A x0 (\lambda (a3: A).(leq g a1 a3)) H3 a H8) in (H0 a H9 -P))))) H6))))))) H2)))))))))) a2)). -(* COMMENTS -Initial nodes: 797 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/lift/defs.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/lift/defs.ma deleted file mode 100644 index 046506672..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/lift/defs.ma +++ /dev/null @@ -1,42 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/tlist/defs.ma". - -include "Basic-1/s/defs.ma". - -definition lref_map: - ((nat \to nat)) \to (nat \to (T \to T)) -\def - let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t -with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match -(blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u -t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in -lref_map. - -definition lift: - nat \to (nat \to (T \to T)) -\def - \lambda (h: nat).(\lambda (i: nat).(\lambda (t: T).(lref_map (\lambda (x: -nat).(plus x h)) i t))). - -definition lifts: - nat \to (nat \to (TList \to TList)) -\def - let rec lifts (h: nat) (d: nat) (ts: TList) on ts: TList \def (match ts with -[TNil \Rightarrow TNil | (TCons t ts0) \Rightarrow (TCons (lift h d t) (lifts -h d ts0))]) in lifts. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/lift/fwd.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/lift/fwd.ma deleted file mode 100644 index 324fed2fb..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/lift/fwd.ma +++ /dev/null @@ -1,434 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/lift/defs.ma". - -theorem lift_sort: - \forall (n: nat).(\forall (h: nat).(\forall (d: nat).(eq T (lift h d (TSort -n)) (TSort n)))) -\def - \lambda (n: nat).(\lambda (_: nat).(\lambda (_: nat).(refl_equal T (TSort -n)))). -(* COMMENTS -Initial nodes: 13 -END *) - -theorem lift_lref_lt: - \forall (n: nat).(\forall (h: nat).(\forall (d: nat).((lt n d) \to (eq T -(lift h d (TLRef n)) (TLRef n))))) -\def - \lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (lt n -d)).(eq_ind bool true (\lambda (b: bool).(eq T (TLRef (match b with [true -\Rightarrow n | false \Rightarrow (plus n h)])) (TLRef n))) (refl_equal T -(TLRef n)) (blt n d) (sym_eq bool (blt n d) true (lt_blt d n H)))))). -(* COMMENTS -Initial nodes: 72 -END *) - -theorem lift_lref_ge: - \forall (n: nat).(\forall (h: nat).(\forall (d: nat).((le d n) \to (eq T -(lift h d (TLRef n)) (TLRef (plus n h)))))) -\def - \lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (le d -n)).(eq_ind bool false (\lambda (b: bool).(eq T (TLRef (match b with [true -\Rightarrow n | false \Rightarrow (plus n h)])) (TLRef (plus n h)))) -(refl_equal T (TLRef (plus n h))) (blt n d) (sym_eq bool (blt n d) false -(le_bge d n H)))))). -(* COMMENTS -Initial nodes: 80 -END *) - -theorem lift_head: - \forall (k: K).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall -(d: nat).(eq T (lift h d (THead k u t)) (THead k (lift h d u) (lift h (s k d) -t))))))) -\def - \lambda (k: K).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda -(d: nat).(refl_equal T (THead k (lift h d u) (lift h (s k d) t))))))). -(* COMMENTS -Initial nodes: 37 -END *) - -theorem lift_bind: - \forall (b: B).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall -(d: nat).(eq T (lift h d (THead (Bind b) u t)) (THead (Bind b) (lift h d u) -(lift h (S d) t))))))) -\def - \lambda (b: B).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda -(d: nat).(refl_equal T (THead (Bind b) (lift h d u) (lift h (S d) t))))))). -(* COMMENTS -Initial nodes: 37 -END *) - -theorem lift_flat: - \forall (f: F).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall -(d: nat).(eq T (lift h d (THead (Flat f) u t)) (THead (Flat f) (lift h d u) -(lift h d t))))))) -\def - \lambda (f: F).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda -(d: nat).(refl_equal T (THead (Flat f) (lift h d u) (lift h d t))))))). -(* COMMENTS -Initial nodes: 35 -END *) - -theorem lift_gen_sort: - \forall (h: nat).(\forall (d: nat).(\forall (n: nat).(\forall (t: T).((eq T -(TSort n) (lift h d t)) \to (eq T t (TSort n)))))) -\def - \lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (t: T).(T_ind -(\lambda (t0: T).((eq T (TSort n) (lift h d t0)) \to (eq T t0 (TSort n)))) -(\lambda (n0: nat).(\lambda (H: (eq T (TSort n) (lift h d (TSort -n0)))).(sym_eq T (TSort n) (TSort n0) H))) (\lambda (n0: nat).(\lambda (H: -(eq T (TSort n) (lift h d (TLRef n0)))).(lt_le_e n0 d (eq T (TLRef n0) (TSort -n)) (\lambda (_: (lt n0 d)).(let H1 \def (eq_ind T (lift h d (TLRef n0)) -(\lambda (t0: T).(eq T (TSort n) t0)) H (TLRef n0) (lift_lref_lt n0 h d (let -H1 \def (eq_ind T (TSort n) (\lambda (ee: T).(match ee in T return (\lambda -(_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | -(THead _ _ _) \Rightarrow False])) I (lift h d (TLRef n0)) H) in (False_ind -(lt n0 d) H1)))) in (let H2 \def (eq_ind T (TSort n) (\lambda (ee: T).(match -ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | -(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef n0) -H1) in (False_ind (eq T (TLRef n0) (TSort n)) H2)))) (\lambda (_: (le d -n0)).(let H1 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t0: T).(eq T -(TSort n) t0)) H (TLRef (plus n0 h)) (lift_lref_ge n0 h d (let H1 \def -(eq_ind T (TSort n) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | -(THead _ _ _) \Rightarrow False])) I (lift h d (TLRef n0)) H) in (False_ind -(le d n0) H1)))) in (let H2 \def (eq_ind T (TSort n) (\lambda (ee: T).(match -ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | -(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef -(plus n0 h)) H1) in (False_ind (eq T (TLRef n0) (TSort n)) H2))))))) (\lambda -(k: K).(\lambda (t0: T).(\lambda (_: (((eq T (TSort n) (lift h d t0)) \to (eq -T t0 (TSort n))))).(\lambda (t1: T).(\lambda (_: (((eq T (TSort n) (lift h d -t1)) \to (eq T t1 (TSort n))))).(\lambda (H1: (eq T (TSort n) (lift h d -(THead k t0 t1)))).(let H2 \def (eq_ind T (lift h d (THead k t0 t1)) (\lambda -(t2: T).(eq T (TSort n) t2)) H1 (THead k (lift h d t0) (lift h (s k d) t1)) -(lift_head k t0 t1 h d)) in (let H3 \def (eq_ind T (TSort n) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I -(THead k (lift h d t0) (lift h (s k d) t1)) H2) in (False_ind (eq T (THead k -t0 t1) (TSort n)) H3))))))))) t)))). -(* COMMENTS -Initial nodes: 613 -END *) - -theorem lift_gen_lref: - \forall (t: T).(\forall (d: nat).(\forall (h: nat).(\forall (i: nat).((eq T -(TLRef i) (lift h d t)) \to (or (land (lt i d) (eq T t (TLRef i))) (land (le -(plus d h) i) (eq T t (TLRef (minus i h))))))))) -\def - \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d: nat).(\forall (h: -nat).(\forall (i: nat).((eq T (TLRef i) (lift h d t0)) \to (or (land (lt i d) -(eq T t0 (TLRef i))) (land (le (plus d h) i) (eq T t0 (TLRef (minus i -h)))))))))) (\lambda (n: nat).(\lambda (d: nat).(\lambda (h: nat).(\lambda -(i: nat).(\lambda (H: (eq T (TLRef i) (lift h d (TSort n)))).(let H0 \def -(eq_ind T (lift h d (TSort n)) (\lambda (t0: T).(eq T (TLRef i) t0)) H (TSort -n) (lift_sort n h d)) in (let H1 \def (eq_ind T (TLRef i) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I -(TSort n) H0) in (False_ind (or (land (lt i d) (eq T (TSort n) (TLRef i))) -(land (le (plus d h) i) (eq T (TSort n) (TLRef (minus i h))))) H1)))))))) -(\lambda (n: nat).(\lambda (d: nat).(\lambda (h: nat).(\lambda (i: -nat).(\lambda (H: (eq T (TLRef i) (lift h d (TLRef n)))).(lt_le_e n d (or -(land (lt i d) (eq T (TLRef n) (TLRef i))) (land (le (plus d h) i) (eq T -(TLRef n) (TLRef (minus i h))))) (\lambda (H0: (lt n d)).(let H1 \def (eq_ind -T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (TLRef i) t0)) H (TLRef n) -(lift_lref_lt n h d H0)) in (let H2 \def (f_equal T nat (\lambda (e: -T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow i | -(TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow i])) (TLRef i) (TLRef -n) H1) in (eq_ind_r nat n (\lambda (n0: nat).(or (land (lt n0 d) (eq T (TLRef -n) (TLRef n0))) (land (le (plus d h) n0) (eq T (TLRef n) (TLRef (minus n0 -h)))))) (or_introl (land (lt n d) (eq T (TLRef n) (TLRef n))) (land (le (plus -d h) n) (eq T (TLRef n) (TLRef (minus n h)))) (conj (lt n d) (eq T (TLRef n) -(TLRef n)) H0 (refl_equal T (TLRef n)))) i H2)))) (\lambda (H0: (le d -n)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (TLRef -i) t0)) H (TLRef (plus n h)) (lift_lref_ge n h d H0)) in (let H2 \def -(f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with -[(TSort _) \Rightarrow i | (TLRef n0) \Rightarrow n0 | (THead _ _ _) -\Rightarrow i])) (TLRef i) (TLRef (plus n h)) H1) in (eq_ind_r nat (plus n h) -(\lambda (n0: nat).(or (land (lt n0 d) (eq T (TLRef n) (TLRef n0))) (land (le -(plus d h) n0) (eq T (TLRef n) (TLRef (minus n0 h)))))) (eq_ind_r nat n -(\lambda (n0: nat).(or (land (lt (plus n h) d) (eq T (TLRef n) (TLRef (plus n -h)))) (land (le (plus d h) (plus n h)) (eq T (TLRef n) (TLRef n0))))) -(or_intror (land (lt (plus n h) d) (eq T (TLRef n) (TLRef (plus n h)))) (land -(le (plus d h) (plus n h)) (eq T (TLRef n) (TLRef n))) (conj (le (plus d h) -(plus n h)) (eq T (TLRef n) (TLRef n)) (le_plus_plus d n h h H0 (le_n h)) -(refl_equal T (TLRef n)))) (minus (plus n h) h) (minus_plus_r n h)) i -H2)))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (_: ((\forall (d: -nat).(\forall (h: nat).(\forall (i: nat).((eq T (TLRef i) (lift h d t0)) \to -(or (land (lt i d) (eq T t0 (TLRef i))) (land (le (plus d h) i) (eq T t0 -(TLRef (minus i h))))))))))).(\lambda (t1: T).(\lambda (_: ((\forall (d: -nat).(\forall (h: nat).(\forall (i: nat).((eq T (TLRef i) (lift h d t1)) \to -(or (land (lt i d) (eq T t1 (TLRef i))) (land (le (plus d h) i) (eq T t1 -(TLRef (minus i h))))))))))).(\lambda (d: nat).(\lambda (h: nat).(\lambda (i: -nat).(\lambda (H1: (eq T (TLRef i) (lift h d (THead k t0 t1)))).(let H2 \def -(eq_ind T (lift h d (THead k t0 t1)) (\lambda (t2: T).(eq T (TLRef i) t2)) H1 -(THead k (lift h d t0) (lift h (s k d) t1)) (lift_head k t0 t1 h d)) in (let -H3 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda -(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | -(THead _ _ _) \Rightarrow False])) I (THead k (lift h d t0) (lift h (s k d) -t1)) H2) in (False_ind (or (land (lt i d) (eq T (THead k t0 t1) (TLRef i))) -(land (le (plus d h) i) (eq T (THead k t0 t1) (TLRef (minus i h))))) -H3)))))))))))) t). -(* COMMENTS -Initial nodes: 1221 -END *) - -theorem lift_gen_lref_lt: - \forall (h: nat).(\forall (d: nat).(\forall (n: nat).((lt n d) \to (\forall -(t: T).((eq T (TLRef n) (lift h d t)) \to (eq T t (TLRef n))))))) -\def - \lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (H: (lt n -d)).(\lambda (t: T).(\lambda (H0: (eq T (TLRef n) (lift h d t))).(let H_x -\def (lift_gen_lref t d h n H0) in (let H1 \def H_x in (or_ind (land (lt n d) -(eq T t (TLRef n))) (land (le (plus d h) n) (eq T t (TLRef (minus n h)))) (eq -T t (TLRef n)) (\lambda (H2: (land (lt n d) (eq T t (TLRef n)))).(land_ind -(lt n d) (eq T t (TLRef n)) (eq T t (TLRef n)) (\lambda (_: (lt n -d)).(\lambda (H4: (eq T t (TLRef n))).(eq_ind_r T (TLRef n) (\lambda (t0: -T).(eq T t0 (TLRef n))) (refl_equal T (TLRef n)) t H4))) H2)) (\lambda (H2: -(land (le (plus d h) n) (eq T t (TLRef (minus n h))))).(land_ind (le (plus d -h) n) (eq T t (TLRef (minus n h))) (eq T t (TLRef n)) (\lambda (H3: (le (plus -d h) n)).(\lambda (H4: (eq T t (TLRef (minus n h)))).(eq_ind_r T (TLRef -(minus n h)) (\lambda (t0: T).(eq T t0 (TLRef n))) (le_false (plus d h) n (eq -T (TLRef (minus n h)) (TLRef n)) H3 (lt_le_S n (plus d h) (le_plus_trans (S -n) d h H))) t H4))) H2)) H1)))))))). -(* COMMENTS -Initial nodes: 363 -END *) - -theorem lift_gen_lref_false: - \forall (h: nat).(\forall (d: nat).(\forall (n: nat).((le d n) \to ((lt n -(plus d h)) \to (\forall (t: T).((eq T (TLRef n) (lift h d t)) \to (\forall -(P: Prop).P))))))) -\def - \lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (H: (le d -n)).(\lambda (H0: (lt n (plus d h))).(\lambda (t: T).(\lambda (H1: (eq T -(TLRef n) (lift h d t))).(\lambda (P: Prop).(let H_x \def (lift_gen_lref t d -h n H1) in (let H2 \def H_x in (or_ind (land (lt n d) (eq T t (TLRef n))) -(land (le (plus d h) n) (eq T t (TLRef (minus n h)))) P (\lambda (H3: (land -(lt n d) (eq T t (TLRef n)))).(land_ind (lt n d) (eq T t (TLRef n)) P -(\lambda (H4: (lt n d)).(\lambda (_: (eq T t (TLRef n))).(le_false d n P H -H4))) H3)) (\lambda (H3: (land (le (plus d h) n) (eq T t (TLRef (minus n -h))))).(land_ind (le (plus d h) n) (eq T t (TLRef (minus n h))) P (\lambda -(H4: (le (plus d h) n)).(\lambda (_: (eq T t (TLRef (minus n h)))).(le_false -(plus d h) n P H4 H0))) H3)) H2)))))))))). -(* COMMENTS -Initial nodes: 269 -END *) - -theorem lift_gen_lref_ge: - \forall (h: nat).(\forall (d: nat).(\forall (n: nat).((le d n) \to (\forall -(t: T).((eq T (TLRef (plus n h)) (lift h d t)) \to (eq T t (TLRef n))))))) -\def - \lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (H: (le d -n)).(\lambda (t: T).(\lambda (H0: (eq T (TLRef (plus n h)) (lift h d -t))).(let H_x \def (lift_gen_lref t d h (plus n h) H0) in (let H1 \def H_x in -(or_ind (land (lt (plus n h) d) (eq T t (TLRef (plus n h)))) (land (le (plus -d h) (plus n h)) (eq T t (TLRef (minus (plus n h) h)))) (eq T t (TLRef n)) -(\lambda (H2: (land (lt (plus n h) d) (eq T t (TLRef (plus n h))))).(land_ind -(lt (plus n h) d) (eq T t (TLRef (plus n h))) (eq T t (TLRef n)) (\lambda -(H3: (lt (plus n h) d)).(\lambda (H4: (eq T t (TLRef (plus n h)))).(eq_ind_r -T (TLRef (plus n h)) (\lambda (t0: T).(eq T t0 (TLRef n))) (le_false d n (eq -T (TLRef (plus n h)) (TLRef n)) H (lt_le_S n d (simpl_lt_plus_r h n d -(lt_le_trans (plus n h) d (plus d h) H3 (le_plus_l d h))))) t H4))) H2)) -(\lambda (H2: (land (le (plus d h) (plus n h)) (eq T t (TLRef (minus (plus n -h) h))))).(land_ind (le (plus d h) (plus n h)) (eq T t (TLRef (minus (plus n -h) h))) (eq T t (TLRef n)) (\lambda (_: (le (plus d h) (plus n h))).(\lambda -(H4: (eq T t (TLRef (minus (plus n h) h)))).(eq_ind_r T (TLRef (minus (plus n -h) h)) (\lambda (t0: T).(eq T t0 (TLRef n))) (f_equal nat T TLRef (minus -(plus n h) h) n (minus_plus_r n h)) t H4))) H2)) H1)))))))). -(* COMMENTS -Initial nodes: 473 -END *) - -theorem lift_gen_head: - \forall (k: K).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h: -nat).(\forall (d: nat).((eq T (THead k u t) (lift h d x)) \to (ex3_2 T T -(\lambda (y: T).(\lambda (z: T).(eq T x (THead k y z)))) (\lambda (y: -T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: -T).(eq T t (lift h (s k d) z))))))))))) -\def - \lambda (k: K).(\lambda (u: T).(\lambda (t: T).(\lambda (x: T).(T_ind -(\lambda (t0: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k u t) -(lift h d t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead -k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda -(_: T).(\lambda (z: T).(eq T t (lift h (s k d) z))))))))) (\lambda (n: -nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (eq T (THead k u t) -(lift h d (TSort n)))).(let H0 \def (eq_ind T (lift h d (TSort n)) (\lambda -(t0: T).(eq T (THead k u t) t0)) H (TSort n) (lift_sort n h d)) in (let H1 -\def (eq_ind T (THead k u t) (\lambda (ee: T).(match ee in T return (\lambda -(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False -| (THead _ _ _) \Rightarrow True])) I (TSort n) H0) in (False_ind (ex3_2 T T -(\lambda (y: T).(\lambda (z: T).(eq T (TSort n) (THead k y z)))) (\lambda (y: -T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: -T).(eq T t (lift h (s k d) z))))) H1))))))) (\lambda (n: nat).(\lambda (h: -nat).(\lambda (d: nat).(\lambda (H: (eq T (THead k u t) (lift h d (TLRef -n)))).(lt_le_e n d (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) -(THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) -(\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k d) z))))) (\lambda (H0: -(lt n d)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T -(THead k u t) t0)) H (TLRef n) (lift_lref_lt n h d H0)) in (let H2 \def -(eq_ind T (THead k u t) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead _ _ _) \Rightarrow True])) I (TLRef n) H1) in (False_ind (ex3_2 T T -(\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead k y z)))) (\lambda (y: -T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: -T).(eq T t (lift h (s k d) z))))) H2)))) (\lambda (H0: (le d n)).(let H1 \def -(eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (THead k u t) t0)) H -(TLRef (plus n h)) (lift_lref_ge n h d H0)) in (let H2 \def (eq_ind T (THead -k u t) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) -\Rightarrow True])) I (TLRef (plus n h)) H1) in (False_ind (ex3_2 T T -(\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead k y z)))) (\lambda (y: -T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: -T).(eq T t (lift h (s k d) z))))) H2))))))))) (\lambda (k0: K).(\lambda (t0: -T).(\lambda (H: ((\forall (h: nat).(\forall (d: nat).((eq T (THead k u t) -(lift h d t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead -k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda -(_: T).(\lambda (z: T).(eq T t (lift h (s k d) z)))))))))).(\lambda (t1: -T).(\lambda (H0: ((\forall (h: nat).(\forall (d: nat).((eq T (THead k u t) -(lift h d t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t1 (THead -k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda -(_: T).(\lambda (z: T).(eq T t (lift h (s k d) z)))))))))).(\lambda (h: -nat).(\lambda (d: nat).(\lambda (H1: (eq T (THead k u t) (lift h d (THead k0 -t0 t1)))).(let H2 \def (eq_ind T (lift h d (THead k0 t0 t1)) (\lambda (t2: -T).(eq T (THead k u t) t2)) H1 (THead k0 (lift h d t0) (lift h (s k0 d) t1)) -(lift_head k0 t0 t1 h d)) in (let H3 \def (f_equal T K (\lambda (e: T).(match -e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) -\Rightarrow k | (THead k1 _ _) \Rightarrow k1])) (THead k u t) (THead k0 -(lift h d t0) (lift h (s k0 d) t1)) H2) in ((let H4 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t2 _) \Rightarrow t2])) -(THead k u t) (THead k0 (lift h d t0) (lift h (s k0 d) t1)) H2) in ((let H5 -\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t2) -\Rightarrow t2])) (THead k u t) (THead k0 (lift h d t0) (lift h (s k0 d) t1)) -H2) in (\lambda (H6: (eq T u (lift h d t0))).(\lambda (H7: (eq K k k0)).(let -H8 \def (eq_ind_r K k0 (\lambda (k1: K).(eq T t (lift h (s k1 d) t1))) H5 k -H7) in (eq_ind K k (\lambda (k1: K).(ex3_2 T T (\lambda (y: T).(\lambda (z: -T).(eq T (THead k1 t0 t1) (THead k y z)))) (\lambda (y: T).(\lambda (_: -T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s -k d) z)))))) (let H9 \def (eq_ind T t (\lambda (t2: T).(\forall (h0: -nat).(\forall (d0: nat).((eq T (THead k u t2) (lift h0 d0 t1)) \to (ex3_2 T T -(\lambda (y: T).(\lambda (z: T).(eq T t1 (THead k y z)))) (\lambda (y: -T).(\lambda (_: T).(eq T u (lift h0 d0 y)))) (\lambda (_: T).(\lambda (z: -T).(eq T t2 (lift h0 (s k d0) z))))))))) H0 (lift h (s k d) t1) H8) in (let -H10 \def (eq_ind T t (\lambda (t2: T).(\forall (h0: nat).(\forall (d0: -nat).((eq T (THead k u t2) (lift h0 d0 t0)) \to (ex3_2 T T (\lambda (y: -T).(\lambda (z: T).(eq T t0 (THead k y z)))) (\lambda (y: T).(\lambda (_: -T).(eq T u (lift h0 d0 y)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift -h0 (s k d0) z))))))))) H (lift h (s k d) t1) H8) in (eq_ind_r T (lift h (s k -d) t1) (\lambda (t2: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T -(THead k t0 t1) (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u -(lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h (s k d) -z)))))) (let H11 \def (eq_ind T u (\lambda (t2: T).(\forall (h0: -nat).(\forall (d0: nat).((eq T (THead k t2 (lift h (s k d) t1)) (lift h0 d0 -t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead k y z)))) -(\lambda (y: T).(\lambda (_: T).(eq T t2 (lift h0 d0 y)))) (\lambda (_: -T).(\lambda (z: T).(eq T (lift h (s k d) t1) (lift h0 (s k d0) z))))))))) H10 -(lift h d t0) H6) in (let H12 \def (eq_ind T u (\lambda (t2: T).(\forall (h0: -nat).(\forall (d0: nat).((eq T (THead k t2 (lift h (s k d) t1)) (lift h0 d0 -t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t1 (THead k y z)))) -(\lambda (y: T).(\lambda (_: T).(eq T t2 (lift h0 d0 y)))) (\lambda (_: -T).(\lambda (z: T).(eq T (lift h (s k d) t1) (lift h0 (s k d0) z))))))))) H9 -(lift h d t0) H6) in (eq_ind_r T (lift h d t0) (\lambda (t2: T).(ex3_2 T T -(\lambda (y: T).(\lambda (z: T).(eq T (THead k t0 t1) (THead k y z)))) -(\lambda (y: T).(\lambda (_: T).(eq T t2 (lift h d y)))) (\lambda (_: -T).(\lambda (z: T).(eq T (lift h (s k d) t1) (lift h (s k d) z)))))) -(ex3_2_intro T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k t0 t1) (THead -k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t0) (lift h d y)))) -(\lambda (_: T).(\lambda (z: T).(eq T (lift h (s k d) t1) (lift h (s k d) -z)))) t0 t1 (refl_equal T (THead k t0 t1)) (refl_equal T (lift h d t0)) -(refl_equal T (lift h (s k d) t1))) u H6))) t H8))) k0 H7))))) H4)) -H3))))))))))) x)))). -(* COMMENTS -Initial nodes: 2083 -END *) - -theorem lift_gen_bind: - \forall (b: B).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h: -nat).(\forall (d: nat).((eq T (THead (Bind b) u t) (lift h d x)) \to (ex3_2 T -T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Bind b) y z)))) (\lambda -(y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: -T).(eq T t (lift h (S d) z))))))))))) -\def - \lambda (b: B).(\lambda (u: T).(\lambda (t: T).(\lambda (x: T).(\lambda (h: -nat).(\lambda (d: nat).(\lambda (H: (eq T (THead (Bind b) u t) (lift h d -x))).(let H_x \def (lift_gen_head (Bind b) u t x h d H) in (let H0 \def H_x -in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Bind b) y -z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: -T).(\lambda (z: T).(eq T t (lift h (S d) z)))) (ex3_2 T T (\lambda (y: -T).(\lambda (z: T).(eq T x (THead (Bind b) y z)))) (\lambda (y: T).(\lambda -(_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift -h (S d) z))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H1: (eq T x (THead -(Bind b) x0 x1))).(\lambda (H2: (eq T u (lift h d x0))).(\lambda (H3: (eq T t -(lift h (S d) x1))).(eq_ind_r T (THead (Bind b) x0 x1) (\lambda (t0: -T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead (Bind b) y -z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: -T).(\lambda (z: T).(eq T t (lift h (S d) z)))))) (eq_ind_r T (lift h (S d) -x1) (\lambda (t0: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead -(Bind b) x0 x1) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T -u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t0 (lift h (S d) -z)))))) (eq_ind_r T (lift h d x0) (\lambda (t0: T).(ex3_2 T T (\lambda (y: -T).(\lambda (z: T).(eq T (THead (Bind b) x0 x1) (THead (Bind b) y z)))) -(\lambda (y: T).(\lambda (_: T).(eq T t0 (lift h d y)))) (\lambda (_: -T).(\lambda (z: T).(eq T (lift h (S d) x1) (lift h (S d) z)))))) (ex3_2_intro -T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Bind b) x0 x1) (THead (Bind -b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d x0) (lift h d -y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h (S d) x1) (lift h (S d) -z)))) x0 x1 (refl_equal T (THead (Bind b) x0 x1)) (refl_equal T (lift h d -x0)) (refl_equal T (lift h (S d) x1))) u H2) t H3) x H1)))))) H0))))))))). -(* COMMENTS -Initial nodes: 637 -END *) - -theorem lift_gen_flat: - \forall (f: F).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h: -nat).(\forall (d: nat).((eq T (THead (Flat f) u t) (lift h d x)) \to (ex3_2 T -T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Flat f) y z)))) (\lambda -(y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: -T).(eq T t (lift h d z))))))))))) -\def - \lambda (f: F).(\lambda (u: T).(\lambda (t: T).(\lambda (x: T).(\lambda (h: -nat).(\lambda (d: nat).(\lambda (H: (eq T (THead (Flat f) u t) (lift h d -x))).(let H_x \def (lift_gen_head (Flat f) u t x h d H) in (let H0 \def H_x -in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Flat f) y -z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: -T).(\lambda (z: T).(eq T t (lift h d z)))) (ex3_2 T T (\lambda (y: -T).(\lambda (z: T).(eq T x (THead (Flat f) y z)))) (\lambda (y: T).(\lambda -(_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift -h d z))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H1: (eq T x (THead -(Flat f) x0 x1))).(\lambda (H2: (eq T u (lift h d x0))).(\lambda (H3: (eq T t -(lift h d x1))).(eq_ind_r T (THead (Flat f) x0 x1) (\lambda (t0: T).(ex3_2 T -T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead (Flat f) y z)))) (\lambda -(y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: -T).(eq T t (lift h d z)))))) (eq_ind_r T (lift h d x1) (\lambda (t0: -T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Flat f) x0 x1) -(THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d -y)))) (\lambda (_: T).(\lambda (z: T).(eq T t0 (lift h d z)))))) (eq_ind_r T -(lift h d x0) (\lambda (t0: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq -T (THead (Flat f) x0 x1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: -T).(eq T t0 (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h d -x1) (lift h d z)))))) (ex3_2_intro T T (\lambda (y: T).(\lambda (z: T).(eq T -(THead (Flat f) x0 x1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: -T).(eq T (lift h d x0) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T -(lift h d x1) (lift h d z)))) x0 x1 (refl_equal T (THead (Flat f) x0 x1)) -(refl_equal T (lift h d x0)) (refl_equal T (lift h d x1))) u H2) t H3) x -H1)))))) H0))))))))). -(* COMMENTS -Initial nodes: 615 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/lift/props.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/lift/props.ma deleted file mode 100644 index f0ed22451..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/lift/props.ma +++ /dev/null @@ -1,592 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/lift/fwd.ma". - -include "Basic-1/s/props.ma". - -theorem thead_x_lift_y_y: - \forall (k: K).(\forall (t: T).(\forall (v: T).(\forall (h: nat).(\forall -(d: nat).((eq T (THead k v (lift h d t)) t) \to (\forall (P: Prop).P)))))) -\def - \lambda (k: K).(\lambda (t: T).(T_ind (\lambda (t0: T).(\forall (v: -T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k v (lift h d t0)) t0) -\to (\forall (P: Prop).P)))))) (\lambda (n: nat).(\lambda (v: T).(\lambda (h: -nat).(\lambda (d: nat).(\lambda (H: (eq T (THead k v (lift h d (TSort n))) -(TSort n))).(\lambda (P: Prop).(let H0 \def (eq_ind T (THead k v (lift h d -(TSort n))) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) -\Rightarrow True])) I (TSort n) H) in (False_ind P H0)))))))) (\lambda (n: -nat).(\lambda (v: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (eq T -(THead k v (lift h d (TLRef n))) (TLRef n))).(\lambda (P: Prop).(let H0 \def -(eq_ind T (THead k v (lift h d (TLRef n))) (\lambda (ee: T).(match ee in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H) in -(False_ind P H0)))))))) (\lambda (k0: K).(\lambda (t0: T).(\lambda (_: -((\forall (v: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k v (lift -h d t0)) t0) \to (\forall (P: Prop).P))))))).(\lambda (t1: T).(\lambda (H0: -((\forall (v: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k v (lift -h d t1)) t1) \to (\forall (P: Prop).P))))))).(\lambda (v: T).(\lambda (h: -nat).(\lambda (d: nat).(\lambda (H1: (eq T (THead k v (lift h d (THead k0 t0 -t1))) (THead k0 t0 t1))).(\lambda (P: Prop).(let H2 \def (f_equal T K -(\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) -\Rightarrow k | (TLRef _) \Rightarrow k | (THead k1 _ _) \Rightarrow k1])) -(THead k v (lift h d (THead k0 t0 t1))) (THead k0 t0 t1) H1) in ((let H3 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow v | (TLRef _) \Rightarrow v | (THead _ t2 _) -\Rightarrow t2])) (THead k v (lift h d (THead k0 t0 t1))) (THead k0 t0 t1) -H1) in ((let H4 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow (THead k0 ((let rec lref_map -(f: ((nat \to nat))) (d0: nat) (t2: T) on t2: T \def (match t2 with [(TSort -n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) -with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k1 u t3) -\Rightarrow (THead k1 (lref_map f d0 u) (lref_map f (s k1 d0) t3))]) in -lref_map) (\lambda (x: nat).(plus x h)) d t0) ((let rec lref_map (f: ((nat -\to nat))) (d0: nat) (t2: T) on t2: T \def (match t2 with [(TSort n) -\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) with -[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k1 u t3) -\Rightarrow (THead k1 (lref_map f d0 u) (lref_map f (s k1 d0) t3))]) in -lref_map) (\lambda (x: nat).(plus x h)) (s k0 d) t1)) | (TLRef _) \Rightarrow -(THead k0 ((let rec lref_map (f: ((nat \to nat))) (d0: nat) (t2: T) on t2: T -\def (match t2 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow -(TLRef (match (blt i d0) with [true \Rightarrow i | false \Rightarrow (f -i)])) | (THead k1 u t3) \Rightarrow (THead k1 (lref_map f d0 u) (lref_map f -(s k1 d0) t3))]) in lref_map) (\lambda (x: nat).(plus x h)) d t0) ((let rec -lref_map (f: ((nat \to nat))) (d0: nat) (t2: T) on t2: T \def (match t2 with -[(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i -d0) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k1 u t3) -\Rightarrow (THead k1 (lref_map f d0 u) (lref_map f (s k1 d0) t3))]) in -lref_map) (\lambda (x: nat).(plus x h)) (s k0 d) t1)) | (THead _ _ t2) -\Rightarrow t2])) (THead k v (lift h d (THead k0 t0 t1))) (THead k0 t0 t1) -H1) in (\lambda (_: (eq T v t0)).(\lambda (H6: (eq K k k0)).(let H7 \def -(eq_ind K k (\lambda (k1: K).(\forall (v0: T).(\forall (h0: nat).(\forall -(d0: nat).((eq T (THead k1 v0 (lift h0 d0 t1)) t1) \to (\forall (P0: -Prop).P0)))))) H0 k0 H6) in (let H8 \def (eq_ind T (lift h d (THead k0 t0 -t1)) (\lambda (t2: T).(eq T t2 t1)) H4 (THead k0 (lift h d t0) (lift h (s k0 -d) t1)) (lift_head k0 t0 t1 h d)) in (H7 (lift h d t0) h (s k0 d) H8 P)))))) -H3)) H2)))))))))))) t)). -(* COMMENTS -Initial nodes: 887 -END *) - -theorem lift_r: - \forall (t: T).(\forall (d: nat).(eq T (lift O d t) t)) -\def - \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d: nat).(eq T (lift O d t0) -t0))) (\lambda (n: nat).(\lambda (_: nat).(refl_equal T (TSort n)))) (\lambda -(n: nat).(\lambda (d: nat).(lt_le_e n d (eq T (lift O d (TLRef n)) (TLRef n)) -(\lambda (H: (lt n d)).(eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 (TLRef -n))) (refl_equal T (TLRef n)) (lift O d (TLRef n)) (lift_lref_lt n O d H))) -(\lambda (H: (le d n)).(eq_ind_r T (TLRef (plus n O)) (\lambda (t0: T).(eq T -t0 (TLRef n))) (f_equal nat T TLRef (plus n O) n (sym_eq nat n (plus n O) -(plus_n_O n))) (lift O d (TLRef n)) (lift_lref_ge n O d H)))))) (\lambda (k: -K).(\lambda (t0: T).(\lambda (H: ((\forall (d: nat).(eq T (lift O d t0) -t0)))).(\lambda (t1: T).(\lambda (H0: ((\forall (d: nat).(eq T (lift O d t1) -t1)))).(\lambda (d: nat).(eq_ind_r T (THead k (lift O d t0) (lift O (s k d) -t1)) (\lambda (t2: T).(eq T t2 (THead k t0 t1))) (f_equal3 K T T T THead k k -(lift O d t0) t0 (lift O (s k d) t1) t1 (refl_equal K k) (H d) (H0 (s k d))) -(lift O d (THead k t0 t1)) (lift_head k t0 t1 O d)))))))) t). -(* COMMENTS -Initial nodes: 367 -END *) - -theorem lift_lref_gt: - \forall (d: nat).(\forall (n: nat).((lt d n) \to (eq T (lift (S O) d (TLRef -(pred n))) (TLRef n)))) -\def - \lambda (d: nat).(\lambda (n: nat).(\lambda (H: (lt d n)).(eq_ind_r T (TLRef -(plus (pred n) (S O))) (\lambda (t: T).(eq T t (TLRef n))) (eq_ind nat (plus -(S O) (pred n)) (\lambda (n0: nat).(eq T (TLRef n0) (TLRef n))) (eq_ind nat n -(\lambda (n0: nat).(eq T (TLRef n0) (TLRef n))) (refl_equal T (TLRef n)) (S -(pred n)) (S_pred n d H)) (plus (pred n) (S O)) (plus_sym (S O) (pred n))) -(lift (S O) d (TLRef (pred n))) (lift_lref_ge (pred n) (S O) d (le_S_n d -(pred n) (eq_ind nat n (\lambda (n0: nat).(le (S d) n0)) H (S (pred n)) -(S_pred n d H))))))). -(* COMMENTS -Initial nodes: 193 -END *) - -theorem lifts_tapp: - \forall (h: nat).(\forall (d: nat).(\forall (v: T).(\forall (vs: TList).(eq -TList (lifts h d (TApp vs v)) (TApp (lifts h d vs) (lift h d v)))))) -\def - \lambda (h: nat).(\lambda (d: nat).(\lambda (v: T).(\lambda (vs: -TList).(TList_ind (\lambda (t: TList).(eq TList (lifts h d (TApp t v)) (TApp -(lifts h d t) (lift h d v)))) (refl_equal TList (TCons (lift h d v) TNil)) -(\lambda (t: T).(\lambda (t0: TList).(\lambda (H: (eq TList (lifts h d (TApp -t0 v)) (TApp (lifts h d t0) (lift h d v)))).(eq_ind_r TList (TApp (lifts h d -t0) (lift h d v)) (\lambda (t1: TList).(eq TList (TCons (lift h d t) t1) -(TCons (lift h d t) (TApp (lifts h d t0) (lift h d v))))) (refl_equal TList -(TCons (lift h d t) (TApp (lifts h d t0) (lift h d v)))) (lifts h d (TApp t0 -v)) H)))) vs)))). -(* COMMENTS -Initial nodes: 215 -END *) - -theorem lift_inj: - \forall (x: T).(\forall (t: T).(\forall (h: nat).(\forall (d: nat).((eq T -(lift h d x) (lift h d t)) \to (eq T x t))))) -\def - \lambda (x: T).(T_ind (\lambda (t: T).(\forall (t0: T).(\forall (h: -nat).(\forall (d: nat).((eq T (lift h d t) (lift h d t0)) \to (eq T t -t0)))))) (\lambda (n: nat).(\lambda (t: T).(\lambda (h: nat).(\lambda (d: -nat).(\lambda (H: (eq T (lift h d (TSort n)) (lift h d t))).(let H0 \def -(eq_ind T (lift h d (TSort n)) (\lambda (t0: T).(eq T t0 (lift h d t))) H -(TSort n) (lift_sort n h d)) in (sym_eq T t (TSort n) (lift_gen_sort h d n t -H0)))))))) (\lambda (n: nat).(\lambda (t: T).(\lambda (h: nat).(\lambda (d: -nat).(\lambda (H: (eq T (lift h d (TLRef n)) (lift h d t))).(lt_le_e n d (eq -T (TLRef n) t) (\lambda (H0: (lt n d)).(let H1 \def (eq_ind T (lift h d -(TLRef n)) (\lambda (t0: T).(eq T t0 (lift h d t))) H (TLRef n) (lift_lref_lt -n h d H0)) in (sym_eq T t (TLRef n) (lift_gen_lref_lt h d n (lt_le_trans n d -d H0 (le_n d)) t H1)))) (\lambda (H0: (le d n)).(let H1 \def (eq_ind T (lift -h d (TLRef n)) (\lambda (t0: T).(eq T t0 (lift h d t))) H (TLRef (plus n h)) -(lift_lref_ge n h d H0)) in (sym_eq T t (TLRef n) (lift_gen_lref_ge h d n H0 -t H1)))))))))) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t: -T).(((\forall (t0: T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t) -(lift h d t0)) \to (eq T t t0)))))) \to (\forall (t0: T).(((\forall (t1: -T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t0) (lift h d t1)) -\to (eq T t0 t1)))))) \to (\forall (t1: T).(\forall (h: nat).(\forall (d: -nat).((eq T (lift h d (THead k0 t t0)) (lift h d t1)) \to (eq T (THead k0 t -t0) t1)))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (H: ((\forall (t0: -T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t) (lift h d t0)) \to -(eq T t t0))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (t1: T).(\forall -(h: nat).(\forall (d: nat).((eq T (lift h d t0) (lift h d t1)) \to (eq T t0 -t1))))))).(\lambda (t1: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: -(eq T (lift h d (THead (Bind b) t t0)) (lift h d t1))).(let H2 \def (eq_ind T -(lift h d (THead (Bind b) t t0)) (\lambda (t2: T).(eq T t2 (lift h d t1))) H1 -(THead (Bind b) (lift h d t) (lift h (S d) t0)) (lift_bind b t t0 h d)) in -(ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T t1 (THead (Bind b) y -z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t) (lift h d y)))) -(\lambda (_: T).(\lambda (z: T).(eq T (lift h (S d) t0) (lift h (S d) z)))) -(eq T (THead (Bind b) t t0) t1) (\lambda (x0: T).(\lambda (x1: T).(\lambda -(H3: (eq T t1 (THead (Bind b) x0 x1))).(\lambda (H4: (eq T (lift h d t) (lift -h d x0))).(\lambda (H5: (eq T (lift h (S d) t0) (lift h (S d) x1))).(eq_ind_r -T (THead (Bind b) x0 x1) (\lambda (t2: T).(eq T (THead (Bind b) t t0) t2)) -(f_equal3 K T T T THead (Bind b) (Bind b) t x0 t0 x1 (refl_equal K (Bind b)) -(H x0 h d H4) (H0 x1 h (S d) H5)) t1 H3)))))) (lift_gen_bind b (lift h d t) -(lift h (S d) t0) t1 h d H2)))))))))))) (\lambda (f: F).(\lambda (t: -T).(\lambda (H: ((\forall (t0: T).(\forall (h: nat).(\forall (d: nat).((eq T -(lift h d t) (lift h d t0)) \to (eq T t t0))))))).(\lambda (t0: T).(\lambda -(H0: ((\forall (t1: T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d -t0) (lift h d t1)) \to (eq T t0 t1))))))).(\lambda (t1: T).(\lambda (h: -nat).(\lambda (d: nat).(\lambda (H1: (eq T (lift h d (THead (Flat f) t t0)) -(lift h d t1))).(let H2 \def (eq_ind T (lift h d (THead (Flat f) t t0)) -(\lambda (t2: T).(eq T t2 (lift h d t1))) H1 (THead (Flat f) (lift h d t) -(lift h d t0)) (lift_flat f t t0 h d)) in (ex3_2_ind T T (\lambda (y: -T).(\lambda (z: T).(eq T t1 (THead (Flat f) y z)))) (\lambda (y: T).(\lambda -(_: T).(eq T (lift h d t) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq -T (lift h d t0) (lift h d z)))) (eq T (THead (Flat f) t t0) t1) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H3: (eq T t1 (THead (Flat f) x0 x1))).(\lambda -(H4: (eq T (lift h d t) (lift h d x0))).(\lambda (H5: (eq T (lift h d t0) -(lift h d x1))).(eq_ind_r T (THead (Flat f) x0 x1) (\lambda (t2: T).(eq T -(THead (Flat f) t t0) t2)) (f_equal3 K T T T THead (Flat f) (Flat f) t x0 t0 -x1 (refl_equal K (Flat f)) (H x0 h d H4) (H0 x1 h d H5)) t1 H3)))))) -(lift_gen_flat f (lift h d t) (lift h d t0) t1 h d H2)))))))))))) k)) x). -(* COMMENTS -Initial nodes: 1391 -END *) - -theorem lift_gen_lift: - \forall (t1: T).(\forall (x: T).(\forall (h1: nat).(\forall (h2: -nat).(\forall (d1: nat).(\forall (d2: nat).((le d1 d2) \to ((eq T (lift h1 d1 -t1) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1 -t2))) (\lambda (t2: T).(eq T t1 (lift h2 d2 t2))))))))))) -\def - \lambda (t1: T).(T_ind (\lambda (t: T).(\forall (x: T).(\forall (h1: -nat).(\forall (h2: nat).(\forall (d1: nat).(\forall (d2: nat).((le d1 d2) \to -((eq T (lift h1 d1 t) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2: -T).(eq T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T t (lift h2 d2 -t2)))))))))))) (\lambda (n: nat).(\lambda (x: T).(\lambda (h1: nat).(\lambda -(h2: nat).(\lambda (d1: nat).(\lambda (d2: nat).(\lambda (_: (le d1 -d2)).(\lambda (H0: (eq T (lift h1 d1 (TSort n)) (lift h2 (plus d2 h1) -x))).(let H1 \def (eq_ind T (lift h1 d1 (TSort n)) (\lambda (t: T).(eq T t -(lift h2 (plus d2 h1) x))) H0 (TSort n) (lift_sort n h1 d1)) in (eq_ind_r T -(TSort n) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h1 d1 t2))) -(\lambda (t2: T).(eq T (TSort n) (lift h2 d2 t2))))) (ex_intro2 T (\lambda -(t2: T).(eq T (TSort n) (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TSort n) -(lift h2 d2 t2))) (TSort n) (eq_ind_r T (TSort n) (\lambda (t: T).(eq T -(TSort n) t)) (refl_equal T (TSort n)) (lift h1 d1 (TSort n)) (lift_sort n h1 -d1)) (eq_ind_r T (TSort n) (\lambda (t: T).(eq T (TSort n) t)) (refl_equal T -(TSort n)) (lift h2 d2 (TSort n)) (lift_sort n h2 d2))) x (lift_gen_sort h2 -(plus d2 h1) n x H1))))))))))) (\lambda (n: nat).(\lambda (x: T).(\lambda -(h1: nat).(\lambda (h2: nat).(\lambda (d1: nat).(\lambda (d2: nat).(\lambda -(H: (le d1 d2)).(\lambda (H0: (eq T (lift h1 d1 (TLRef n)) (lift h2 (plus d2 -h1) x))).(lt_le_e n d1 (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1 t2))) -(\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2)))) (\lambda (H1: (lt n -d1)).(let H2 \def (eq_ind T (lift h1 d1 (TLRef n)) (\lambda (t: T).(eq T t -(lift h2 (plus d2 h1) x))) H0 (TLRef n) (lift_lref_lt n h1 d1 H1)) in -(eq_ind_r T (TLRef n) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift -h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2))))) (ex_intro2 T -(\lambda (t2: T).(eq T (TLRef n) (lift h1 d1 t2))) (\lambda (t2: T).(eq T -(TLRef n) (lift h2 d2 t2))) (TLRef n) (eq_ind_r T (TLRef n) (\lambda (t: -T).(eq T (TLRef n) t)) (refl_equal T (TLRef n)) (lift h1 d1 (TLRef n)) -(lift_lref_lt n h1 d1 H1)) (eq_ind_r T (TLRef n) (\lambda (t: T).(eq T (TLRef -n) t)) (refl_equal T (TLRef n)) (lift h2 d2 (TLRef n)) (lift_lref_lt n h2 d2 -(lt_le_trans n d1 d2 H1 H)))) x (lift_gen_lref_lt h2 (plus d2 h1) n -(lt_le_trans n d1 (plus d2 h1) H1 (le_plus_trans d1 d2 h1 H)) x H2)))) -(\lambda (H1: (le d1 n)).(let H2 \def (eq_ind T (lift h1 d1 (TLRef n)) -(\lambda (t: T).(eq T t (lift h2 (plus d2 h1) x))) H0 (TLRef (plus n h1)) -(lift_lref_ge n h1 d1 H1)) in (lt_le_e n d2 (ex2 T (\lambda (t2: T).(eq T x -(lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2)))) -(\lambda (H3: (lt n d2)).(eq_ind_r T (TLRef (plus n h1)) (\lambda (t: T).(ex2 -T (\lambda (t2: T).(eq T t (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) -(lift h2 d2 t2))))) (ex_intro2 T (\lambda (t2: T).(eq T (TLRef (plus n h1)) -(lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2))) (TLRef -n) (eq_ind_r T (TLRef (plus n h1)) (\lambda (t: T).(eq T (TLRef (plus n h1)) -t)) (refl_equal T (TLRef (plus n h1))) (lift h1 d1 (TLRef n)) (lift_lref_ge n -h1 d1 H1)) (eq_ind_r T (TLRef n) (\lambda (t: T).(eq T (TLRef n) t)) -(refl_equal T (TLRef n)) (lift h2 d2 (TLRef n)) (lift_lref_lt n h2 d2 H3))) x -(lift_gen_lref_lt h2 (plus d2 h1) (plus n h1) (lt_reg_r n d2 h1 H3) x H2))) -(\lambda (H3: (le d2 n)).(lt_le_e n (plus d2 h2) (ex2 T (\lambda (t2: T).(eq -T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2)))) -(\lambda (H4: (lt n (plus d2 h2))).(lift_gen_lref_false h2 (plus d2 h1) (plus -n h1) (le_plus_plus d2 n h1 h1 H3 (le_n h1)) (eq_ind_r nat (plus (plus d2 h2) -h1) (\lambda (n0: nat).(lt (plus n h1) n0)) (lt_reg_r n (plus d2 h2) h1 H4) -(plus (plus d2 h1) h2) (plus_permute_2_in_3 d2 h1 h2)) x H2 (ex2 T (\lambda -(t2: T).(eq T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 -d2 t2)))))) (\lambda (H4: (le (plus d2 h2) n)).(let H5 \def (eq_ind nat (plus -n h1) (\lambda (n0: nat).(eq T (TLRef n0) (lift h2 (plus d2 h1) x))) H2 (plus -(minus (plus n h1) h2) h2) (le_plus_minus_sym h2 (plus n h1) (le_plus_trans -h2 n h1 (le_trans h2 (plus d2 h2) n (le_plus_r d2 h2) H4)))) in (eq_ind_r T -(TLRef (minus (plus n h1) h2)) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T -t (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2))))) -(ex_intro2 T (\lambda (t2: T).(eq T (TLRef (minus (plus n h1) h2)) (lift h1 -d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2))) (TLRef (minus n -h2)) (eq_ind_r nat (plus (minus n h2) h1) (\lambda (n0: nat).(eq T (TLRef n0) -(lift h1 d1 (TLRef (minus n h2))))) (eq_ind_r T (TLRef (plus (minus n h2) -h1)) (\lambda (t: T).(eq T (TLRef (plus (minus n h2) h1)) t)) (refl_equal T -(TLRef (plus (minus n h2) h1))) (lift h1 d1 (TLRef (minus n h2))) -(lift_lref_ge (minus n h2) h1 d1 (le_trans d1 d2 (minus n h2) H (le_minus d2 -n h2 H4)))) (minus (plus n h1) h2) (le_minus_plus h2 n (le_trans h2 (plus d2 -h2) n (le_plus_r d2 h2) H4) h1)) (eq_ind_r nat (plus (minus n h2) h2) -(\lambda (n0: nat).(eq T (TLRef n0) (lift h2 d2 (TLRef (minus n0 h2))))) -(eq_ind_r T (TLRef (plus (minus (plus (minus n h2) h2) h2) h2)) (\lambda (t: -T).(eq T (TLRef (plus (minus n h2) h2)) t)) (f_equal nat T TLRef (plus (minus -n h2) h2) (plus (minus (plus (minus n h2) h2) h2) h2) (f_equal2 nat nat nat -plus (minus n h2) (minus (plus (minus n h2) h2) h2) h2 h2 (sym_eq nat (minus -(plus (minus n h2) h2) h2) (minus n h2) (minus_plus_r (minus n h2) h2)) -(refl_equal nat h2))) (lift h2 d2 (TLRef (minus (plus (minus n h2) h2) h2))) -(lift_lref_ge (minus (plus (minus n h2) h2) h2) h2 d2 (le_minus d2 (plus -(minus n h2) h2) h2 (le_plus_plus d2 (minus n h2) h2 h2 (le_minus d2 n h2 H4) -(le_n h2))))) n (le_plus_minus_sym h2 n (le_trans h2 (plus d2 h2) n -(le_plus_r d2 h2) H4)))) x (lift_gen_lref_ge h2 (plus d2 h1) (minus (plus n -h1) h2) (arith0 h2 d2 n H4 h1) x H5)))))))))))))))))) (\lambda (k: -K).(\lambda (t: T).(\lambda (H: ((\forall (x: T).(\forall (h1: nat).(\forall -(h2: nat).(\forall (d1: nat).(\forall (d2: nat).((le d1 d2) \to ((eq T (lift -h1 d1 t) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2: T).(eq T x (lift -h1 d1 t2))) (\lambda (t2: T).(eq T t (lift h2 d2 t2))))))))))))).(\lambda -(t0: T).(\lambda (H0: ((\forall (x: T).(\forall (h1: nat).(\forall (h2: -nat).(\forall (d1: nat).(\forall (d2: nat).((le d1 d2) \to ((eq T (lift h1 d1 -t0) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1 -t2))) (\lambda (t2: T).(eq T t0 (lift h2 d2 t2))))))))))))).(\lambda (x: -T).(\lambda (h1: nat).(\lambda (h2: nat).(\lambda (d1: nat).(\lambda (d2: -nat).(\lambda (H1: (le d1 d2)).(\lambda (H2: (eq T (lift h1 d1 (THead k t -t0)) (lift h2 (plus d2 h1) x))).(K_ind (\lambda (k0: K).((eq T (lift h1 d1 -(THead k0 t t0)) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2: T).(eq T -x (lift h1 d1 t2))) (\lambda (t2: T).(eq T (THead k0 t t0) (lift h2 d2 -t2)))))) (\lambda (b: B).(\lambda (H3: (eq T (lift h1 d1 (THead (Bind b) t -t0)) (lift h2 (plus d2 h1) x))).(let H4 \def (eq_ind T (lift h1 d1 (THead -(Bind b) t t0)) (\lambda (t2: T).(eq T t2 (lift h2 (plus d2 h1) x))) H3 -(THead (Bind b) (lift h1 d1 t) (lift h1 (S d1) t0)) (lift_bind b t t0 h1 d1)) -in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Bind b) y -z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h1 d1 t) (lift h2 (plus d2 -h1) y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h1 (S d1) t0) (lift h2 -(S (plus d2 h1)) z)))) (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1 t2))) -(\lambda (t2: T).(eq T (THead (Bind b) t t0) (lift h2 d2 t2)))) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H5: (eq T x (THead (Bind b) x0 x1))).(\lambda -(H6: (eq T (lift h1 d1 t) (lift h2 (plus d2 h1) x0))).(\lambda (H7: (eq T -(lift h1 (S d1) t0) (lift h2 (S (plus d2 h1)) x1))).(eq_ind_r T (THead (Bind -b) x0 x1) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h1 d1 t3))) -(\lambda (t3: T).(eq T (THead (Bind b) t t0) (lift h2 d2 t3))))) (ex2_ind T -(\lambda (t2: T).(eq T x0 (lift h1 d1 t2))) (\lambda (t2: T).(eq T t (lift h2 -d2 t2))) (ex2 T (\lambda (t2: T).(eq T (THead (Bind b) x0 x1) (lift h1 d1 -t2))) (\lambda (t2: T).(eq T (THead (Bind b) t t0) (lift h2 d2 t2)))) -(\lambda (x2: T).(\lambda (H8: (eq T x0 (lift h1 d1 x2))).(\lambda (H9: (eq T -t (lift h2 d2 x2))).(eq_ind_r T (lift h1 d1 x2) (\lambda (t2: T).(ex2 T -(\lambda (t3: T).(eq T (THead (Bind b) t2 x1) (lift h1 d1 t3))) (\lambda (t3: -T).(eq T (THead (Bind b) t t0) (lift h2 d2 t3))))) (eq_ind_r T (lift h2 d2 -x2) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead (Bind b) (lift h1 -d1 x2) x1) (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead (Bind b) t2 t0) -(lift h2 d2 t3))))) (let H10 \def (refl_equal nat (plus (S d2) h1)) in (let -H11 \def (eq_ind nat (S (plus d2 h1)) (\lambda (n: nat).(eq T (lift h1 (S d1) -t0) (lift h2 n x1))) H7 (plus (S d2) h1) H10) in (ex2_ind T (\lambda (t2: -T).(eq T x1 (lift h1 (S d1) t2))) (\lambda (t2: T).(eq T t0 (lift h2 (S d2) -t2))) (ex2 T (\lambda (t2: T).(eq T (THead (Bind b) (lift h1 d1 x2) x1) (lift -h1 d1 t2))) (\lambda (t2: T).(eq T (THead (Bind b) (lift h2 d2 x2) t0) (lift -h2 d2 t2)))) (\lambda (x3: T).(\lambda (H12: (eq T x1 (lift h1 (S d1) -x3))).(\lambda (H13: (eq T t0 (lift h2 (S d2) x3))).(eq_ind_r T (lift h1 (S -d1) x3) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead (Bind b) (lift -h1 d1 x2) t2) (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead (Bind b) (lift -h2 d2 x2) t0) (lift h2 d2 t3))))) (eq_ind_r T (lift h2 (S d2) x3) (\lambda -(t2: T).(ex2 T (\lambda (t3: T).(eq T (THead (Bind b) (lift h1 d1 x2) (lift -h1 (S d1) x3)) (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead (Bind b) (lift -h2 d2 x2) t2) (lift h2 d2 t3))))) (ex_intro2 T (\lambda (t2: T).(eq T (THead -(Bind b) (lift h1 d1 x2) (lift h1 (S d1) x3)) (lift h1 d1 t2))) (\lambda (t2: -T).(eq T (THead (Bind b) (lift h2 d2 x2) (lift h2 (S d2) x3)) (lift h2 d2 -t2))) (THead (Bind b) x2 x3) (eq_ind_r T (THead (Bind b) (lift h1 d1 x2) -(lift h1 (S d1) x3)) (\lambda (t2: T).(eq T (THead (Bind b) (lift h1 d1 x2) -(lift h1 (S d1) x3)) t2)) (refl_equal T (THead (Bind b) (lift h1 d1 x2) (lift -h1 (S d1) x3))) (lift h1 d1 (THead (Bind b) x2 x3)) (lift_bind b x2 x3 h1 -d1)) (eq_ind_r T (THead (Bind b) (lift h2 d2 x2) (lift h2 (S d2) x3)) -(\lambda (t2: T).(eq T (THead (Bind b) (lift h2 d2 x2) (lift h2 (S d2) x3)) -t2)) (refl_equal T (THead (Bind b) (lift h2 d2 x2) (lift h2 (S d2) x3))) -(lift h2 d2 (THead (Bind b) x2 x3)) (lift_bind b x2 x3 h2 d2))) t0 H13) x1 -H12)))) (H0 x1 h1 h2 (S d1) (S d2) (le_n_S d1 d2 H1) H11)))) t H9) x0 H8)))) -(H x0 h1 h2 d1 d2 H1 H6)) x H5)))))) (lift_gen_bind b (lift h1 d1 t) (lift h1 -(S d1) t0) x h2 (plus d2 h1) H4))))) (\lambda (f: F).(\lambda (H3: (eq T -(lift h1 d1 (THead (Flat f) t t0)) (lift h2 (plus d2 h1) x))).(let H4 \def -(eq_ind T (lift h1 d1 (THead (Flat f) t t0)) (\lambda (t2: T).(eq T t2 (lift -h2 (plus d2 h1) x))) H3 (THead (Flat f) (lift h1 d1 t) (lift h1 d1 t0)) -(lift_flat f t t0 h1 d1)) in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: -T).(eq T x (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T -(lift h1 d1 t) (lift h2 (plus d2 h1) y)))) (\lambda (_: T).(\lambda (z: -T).(eq T (lift h1 d1 t0) (lift h2 (plus d2 h1) z)))) (ex2 T (\lambda (t2: -T).(eq T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T (THead (Flat f) t t0) -(lift h2 d2 t2)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T x -(THead (Flat f) x0 x1))).(\lambda (H6: (eq T (lift h1 d1 t) (lift h2 (plus d2 -h1) x0))).(\lambda (H7: (eq T (lift h1 d1 t0) (lift h2 (plus d2 h1) -x1))).(eq_ind_r T (THead (Flat f) x0 x1) (\lambda (t2: T).(ex2 T (\lambda -(t3: T).(eq T t2 (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead (Flat f) t -t0) (lift h2 d2 t3))))) (ex2_ind T (\lambda (t2: T).(eq T x0 (lift h1 d1 -t2))) (\lambda (t2: T).(eq T t (lift h2 d2 t2))) (ex2 T (\lambda (t2: T).(eq -T (THead (Flat f) x0 x1) (lift h1 d1 t2))) (\lambda (t2: T).(eq T (THead -(Flat f) t t0) (lift h2 d2 t2)))) (\lambda (x2: T).(\lambda (H8: (eq T x0 -(lift h1 d1 x2))).(\lambda (H9: (eq T t (lift h2 d2 x2))).(eq_ind_r T (lift -h1 d1 x2) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead (Flat f) t2 -x1) (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead (Flat f) t t0) (lift h2 -d2 t3))))) (eq_ind_r T (lift h2 d2 x2) (\lambda (t2: T).(ex2 T (\lambda (t3: -T).(eq T (THead (Flat f) (lift h1 d1 x2) x1) (lift h1 d1 t3))) (\lambda (t3: -T).(eq T (THead (Flat f) t2 t0) (lift h2 d2 t3))))) (ex2_ind T (\lambda (t2: -T).(eq T x1 (lift h1 d1 t2))) (\lambda (t2: T).(eq T t0 (lift h2 d2 t2))) -(ex2 T (\lambda (t2: T).(eq T (THead (Flat f) (lift h1 d1 x2) x1) (lift h1 d1 -t2))) (\lambda (t2: T).(eq T (THead (Flat f) (lift h2 d2 x2) t0) (lift h2 d2 -t2)))) (\lambda (x3: T).(\lambda (H10: (eq T x1 (lift h1 d1 x3))).(\lambda -(H11: (eq T t0 (lift h2 d2 x3))).(eq_ind_r T (lift h1 d1 x3) (\lambda (t2: -T).(ex2 T (\lambda (t3: T).(eq T (THead (Flat f) (lift h1 d1 x2) t2) (lift h1 -d1 t3))) (\lambda (t3: T).(eq T (THead (Flat f) (lift h2 d2 x2) t0) (lift h2 -d2 t3))))) (eq_ind_r T (lift h2 d2 x3) (\lambda (t2: T).(ex2 T (\lambda (t3: -T).(eq T (THead (Flat f) (lift h1 d1 x2) (lift h1 d1 x3)) (lift h1 d1 t3))) -(\lambda (t3: T).(eq T (THead (Flat f) (lift h2 d2 x2) t2) (lift h2 d2 -t3))))) (ex_intro2 T (\lambda (t2: T).(eq T (THead (Flat f) (lift h1 d1 x2) -(lift h1 d1 x3)) (lift h1 d1 t2))) (\lambda (t2: T).(eq T (THead (Flat f) -(lift h2 d2 x2) (lift h2 d2 x3)) (lift h2 d2 t2))) (THead (Flat f) x2 x3) -(eq_ind_r T (THead (Flat f) (lift h1 d1 x2) (lift h1 d1 x3)) (\lambda (t2: -T).(eq T (THead (Flat f) (lift h1 d1 x2) (lift h1 d1 x3)) t2)) (refl_equal T -(THead (Flat f) (lift h1 d1 x2) (lift h1 d1 x3))) (lift h1 d1 (THead (Flat f) -x2 x3)) (lift_flat f x2 x3 h1 d1)) (eq_ind_r T (THead (Flat f) (lift h2 d2 -x2) (lift h2 d2 x3)) (\lambda (t2: T).(eq T (THead (Flat f) (lift h2 d2 x2) -(lift h2 d2 x3)) t2)) (refl_equal T (THead (Flat f) (lift h2 d2 x2) (lift h2 -d2 x3))) (lift h2 d2 (THead (Flat f) x2 x3)) (lift_flat f x2 x3 h2 d2))) t0 -H11) x1 H10)))) (H0 x1 h1 h2 d1 d2 H1 H7)) t H9) x0 H8)))) (H x0 h1 h2 d1 d2 -H1 H6)) x H5)))))) (lift_gen_flat f (lift h1 d1 t) (lift h1 d1 t0) x h2 (plus -d2 h1) H4))))) k H2))))))))))))) t1). -(* COMMENTS -Initial nodes: 5037 -END *) - -theorem lifts_inj: - \forall (xs: TList).(\forall (ts: TList).(\forall (h: nat).(\forall (d: -nat).((eq TList (lifts h d xs) (lifts h d ts)) \to (eq TList xs ts))))) -\def - \lambda (xs: TList).(TList_ind (\lambda (t: TList).(\forall (ts: -TList).(\forall (h: nat).(\forall (d: nat).((eq TList (lifts h d t) (lifts h -d ts)) \to (eq TList t ts)))))) (\lambda (ts: TList).(TList_ind (\lambda (t: -TList).(\forall (h: nat).(\forall (d: nat).((eq TList (lifts h d TNil) (lifts -h d t)) \to (eq TList TNil t))))) (\lambda (_: nat).(\lambda (_: -nat).(\lambda (_: (eq TList TNil TNil)).(refl_equal TList TNil)))) (\lambda -(t: T).(\lambda (t0: TList).(\lambda (_: ((\forall (h: nat).(\forall (d: -nat).((eq TList TNil (lifts h d t0)) \to (eq TList TNil t0)))))).(\lambda (h: -nat).(\lambda (d: nat).(\lambda (H0: (eq TList TNil (TCons (lift h d t) -(lifts h d t0)))).(let H1 \def (eq_ind TList TNil (\lambda (ee: TList).(match -ee in TList return (\lambda (_: TList).Prop) with [TNil \Rightarrow True | -(TCons _ _) \Rightarrow False])) I (TCons (lift h d t) (lifts h d t0)) H0) in -(False_ind (eq TList TNil (TCons t t0)) H1)))))))) ts)) (\lambda (t: -T).(\lambda (t0: TList).(\lambda (H: ((\forall (ts: TList).(\forall (h: -nat).(\forall (d: nat).((eq TList (lifts h d t0) (lifts h d ts)) \to (eq -TList t0 ts))))))).(\lambda (ts: TList).(TList_ind (\lambda (t1: -TList).(\forall (h: nat).(\forall (d: nat).((eq TList (lifts h d (TCons t -t0)) (lifts h d t1)) \to (eq TList (TCons t t0) t1))))) (\lambda (h: -nat).(\lambda (d: nat).(\lambda (H0: (eq TList (TCons (lift h d t) (lifts h d -t0)) TNil)).(let H1 \def (eq_ind TList (TCons (lift h d t) (lifts h d t0)) -(\lambda (ee: TList).(match ee in TList return (\lambda (_: TList).Prop) with -[TNil \Rightarrow False | (TCons _ _) \Rightarrow True])) I TNil H0) in -(False_ind (eq TList (TCons t t0) TNil) H1))))) (\lambda (t1: T).(\lambda -(t2: TList).(\lambda (_: ((\forall (h: nat).(\forall (d: nat).((eq TList -(TCons (lift h d t) (lifts h d t0)) (lifts h d t2)) \to (eq TList (TCons t -t0) t2)))))).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: (eq TList -(TCons (lift h d t) (lifts h d t0)) (TCons (lift h d t1) (lifts h d -t2)))).(let H2 \def (f_equal TList T (\lambda (e: TList).(match e in TList -return (\lambda (_: TList).T) with [TNil \Rightarrow ((let rec lref_map (f: -((nat \to nat))) (d0: nat) (t3: T) on t3: T \def (match t3 with [(TSort n) -\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) with -[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t4) \Rightarrow -(THead k (lref_map f d0 u) (lref_map f (s k d0) t4))]) in lref_map) (\lambda -(x: nat).(plus x h)) d t) | (TCons t3 _) \Rightarrow t3])) (TCons (lift h d -t) (lifts h d t0)) (TCons (lift h d t1) (lifts h d t2)) H1) in ((let H3 \def -(f_equal TList TList (\lambda (e: TList).(match e in TList return (\lambda -(_: TList).TList) with [TNil \Rightarrow ((let rec lifts (h0: nat) (d0: nat) -(ts0: TList) on ts0: TList \def (match ts0 with [TNil \Rightarrow TNil | -(TCons t3 ts1) \Rightarrow (TCons (lift h0 d0 t3) (lifts h0 d0 ts1))]) in -lifts) h d t0) | (TCons _ t3) \Rightarrow t3])) (TCons (lift h d t) (lifts h -d t0)) (TCons (lift h d t1) (lifts h d t2)) H1) in (\lambda (H4: (eq T (lift -h d t) (lift h d t1))).(eq_ind T t (\lambda (t3: T).(eq TList (TCons t t0) -(TCons t3 t2))) (f_equal2 T TList TList TCons t t t0 t2 (refl_equal T t) (H -t2 h d H3)) t1 (lift_inj t t1 h d H4)))) H2)))))))) ts))))) xs). -(* COMMENTS -Initial nodes: 772 -END *) - -theorem lift_free: - \forall (t: T).(\forall (h: nat).(\forall (k: nat).(\forall (d: -nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to (eq T (lift k e -(lift h d t)) (lift (plus k h) d t)))))))) -\def - \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (h: nat).(\forall (k: -nat).(\forall (d: nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to -(eq T (lift k e (lift h d t0)) (lift (plus k h) d t0))))))))) (\lambda (n: -nat).(\lambda (h: nat).(\lambda (k: nat).(\lambda (d: nat).(\lambda (e: -nat).(\lambda (_: (le e (plus d h))).(\lambda (_: (le d e)).(eq_ind_r T -(TSort n) (\lambda (t0: T).(eq T (lift k e t0) (lift (plus k h) d (TSort -n)))) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq T t0 (lift (plus k h) d -(TSort n)))) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq T (TSort n) t0)) -(refl_equal T (TSort n)) (lift (plus k h) d (TSort n)) (lift_sort n (plus k -h) d)) (lift k e (TSort n)) (lift_sort n k e)) (lift h d (TSort n)) -(lift_sort n h d))))))))) (\lambda (n: nat).(\lambda (h: nat).(\lambda (k: -nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (H: (le e (plus d -h))).(\lambda (H0: (le d e)).(lt_le_e n d (eq T (lift k e (lift h d (TLRef -n))) (lift (plus k h) d (TLRef n))) (\lambda (H1: (lt n d)).(eq_ind_r T -(TLRef n) (\lambda (t0: T).(eq T (lift k e t0) (lift (plus k h) d (TLRef -n)))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 (lift (plus k h) d -(TLRef n)))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (TLRef n) t0)) -(refl_equal T (TLRef n)) (lift (plus k h) d (TLRef n)) (lift_lref_lt n (plus -k h) d H1)) (lift k e (TLRef n)) (lift_lref_lt n k e (lt_le_trans n d e H1 -H0))) (lift h d (TLRef n)) (lift_lref_lt n h d H1))) (\lambda (H1: (le d -n)).(eq_ind_r T (TLRef (plus n h)) (\lambda (t0: T).(eq T (lift k e t0) (lift -(plus k h) d (TLRef n)))) (eq_ind_r T (TLRef (plus (plus n h) k)) (\lambda -(t0: T).(eq T t0 (lift (plus k h) d (TLRef n)))) (eq_ind_r T (TLRef (plus n -(plus k h))) (\lambda (t0: T).(eq T (TLRef (plus (plus n h) k)) t0)) (f_equal -nat T TLRef (plus (plus n h) k) (plus n (plus k h)) -(plus_permute_2_in_3_assoc n h k)) (lift (plus k h) d (TLRef n)) -(lift_lref_ge n (plus k h) d H1)) (lift k e (TLRef (plus n h))) (lift_lref_ge -(plus n h) k e (le_trans e (plus d h) (plus n h) H (le_plus_plus d n h h H1 -(le_n h))))) (lift h d (TLRef n)) (lift_lref_ge n h d H1))))))))))) (\lambda -(k: K).(\lambda (t0: T).(\lambda (H: ((\forall (h: nat).(\forall (k0: -nat).(\forall (d: nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to -(eq T (lift k0 e (lift h d t0)) (lift (plus k0 h) d t0)))))))))).(\lambda -(t1: T).(\lambda (H0: ((\forall (h: nat).(\forall (k0: nat).(\forall (d: -nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to (eq T (lift k0 e -(lift h d t1)) (lift (plus k0 h) d t1)))))))))).(\lambda (h: nat).(\lambda -(k0: nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (H1: (le e (plus d -h))).(\lambda (H2: (le d e)).(eq_ind_r T (THead k (lift h d t0) (lift h (s k -d) t1)) (\lambda (t2: T).(eq T (lift k0 e t2) (lift (plus k0 h) d (THead k t0 -t1)))) (eq_ind_r T (THead k (lift k0 e (lift h d t0)) (lift k0 (s k e) (lift -h (s k d) t1))) (\lambda (t2: T).(eq T t2 (lift (plus k0 h) d (THead k t0 -t1)))) (eq_ind_r T (THead k (lift (plus k0 h) d t0) (lift (plus k0 h) (s k d) -t1)) (\lambda (t2: T).(eq T (THead k (lift k0 e (lift h d t0)) (lift k0 (s k -e) (lift h (s k d) t1))) t2)) (f_equal3 K T T T THead k k (lift k0 e (lift h -d t0)) (lift (plus k0 h) d t0) (lift k0 (s k e) (lift h (s k d) t1)) (lift -(plus k0 h) (s k d) t1) (refl_equal K k) (H h k0 d e H1 H2) (H0 h k0 (s k d) -(s k e) (eq_ind nat (s k (plus d h)) (\lambda (n: nat).(le (s k e) n)) (s_le -k e (plus d h) H1) (plus (s k d) h) (s_plus k d h)) (s_le k d e H2))) (lift -(plus k0 h) d (THead k t0 t1)) (lift_head k t0 t1 (plus k0 h) d)) (lift k0 e -(THead k (lift h d t0) (lift h (s k d) t1))) (lift_head k (lift h d t0) (lift -h (s k d) t1) k0 e)) (lift h d (THead k t0 t1)) (lift_head k t0 t1 h -d))))))))))))) t). -(* COMMENTS -Initial nodes: 1407 -END *) - -theorem lift_d: - \forall (t: T).(\forall (h: nat).(\forall (k: nat).(\forall (d: -nat).(\forall (e: nat).((le e d) \to (eq T (lift h (plus k d) (lift k e t)) -(lift k e (lift h d t)))))))) -\def - \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (h: nat).(\forall (k: -nat).(\forall (d: nat).(\forall (e: nat).((le e d) \to (eq T (lift h (plus k -d) (lift k e t0)) (lift k e (lift h d t0))))))))) (\lambda (n: nat).(\lambda -(h: nat).(\lambda (k: nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (_: -(le e d)).(eq_ind_r T (TSort n) (\lambda (t0: T).(eq T (lift h (plus k d) t0) -(lift k e (lift h d (TSort n))))) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq -T t0 (lift k e (lift h d (TSort n))))) (eq_ind_r T (TSort n) (\lambda (t0: -T).(eq T (TSort n) (lift k e t0))) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq -T (TSort n) t0)) (refl_equal T (TSort n)) (lift k e (TSort n)) (lift_sort n k -e)) (lift h d (TSort n)) (lift_sort n h d)) (lift h (plus k d) (TSort n)) -(lift_sort n h (plus k d))) (lift k e (TSort n)) (lift_sort n k e)))))))) -(\lambda (n: nat).(\lambda (h: nat).(\lambda (k: nat).(\lambda (d: -nat).(\lambda (e: nat).(\lambda (H: (le e d)).(lt_le_e n e (eq T (lift h -(plus k d) (lift k e (TLRef n))) (lift k e (lift h d (TLRef n)))) (\lambda -(H0: (lt n e)).(let H1 \def (lt_le_trans n e d H0 H) in (eq_ind_r T (TLRef n) -(\lambda (t0: T).(eq T (lift h (plus k d) t0) (lift k e (lift h d (TLRef -n))))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 (lift k e (lift h d -(TLRef n))))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (TLRef n) (lift k -e t0))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (TLRef n) t0)) -(refl_equal T (TLRef n)) (lift k e (TLRef n)) (lift_lref_lt n k e H0)) (lift -h d (TLRef n)) (lift_lref_lt n h d H1)) (lift h (plus k d) (TLRef n)) -(lift_lref_lt n h (plus k d) (lt_le_trans n d (plus k d) H1 (le_plus_r k -d)))) (lift k e (TLRef n)) (lift_lref_lt n k e H0)))) (\lambda (H0: (le e -n)).(eq_ind_r T (TLRef (plus n k)) (\lambda (t0: T).(eq T (lift h (plus k d) -t0) (lift k e (lift h d (TLRef n))))) (eq_ind_r nat (plus d k) (\lambda (n0: -nat).(eq T (lift h n0 (TLRef (plus n k))) (lift k e (lift h d (TLRef n))))) -(lt_le_e n d (eq T (lift h (plus d k) (TLRef (plus n k))) (lift k e (lift h d -(TLRef n)))) (\lambda (H1: (lt n d)).(eq_ind_r T (TLRef (plus n k)) (\lambda -(t0: T).(eq T t0 (lift k e (lift h d (TLRef n))))) (eq_ind_r T (TLRef n) -(\lambda (t0: T).(eq T (TLRef (plus n k)) (lift k e t0))) (eq_ind_r T (TLRef -(plus n k)) (\lambda (t0: T).(eq T (TLRef (plus n k)) t0)) (refl_equal T -(TLRef (plus n k))) (lift k e (TLRef n)) (lift_lref_ge n k e H0)) (lift h d -(TLRef n)) (lift_lref_lt n h d H1)) (lift h (plus d k) (TLRef (plus n k))) -(lift_lref_lt (plus n k) h (plus d k) (lt_reg_r n d k H1)))) (\lambda (H1: -(le d n)).(eq_ind_r T (TLRef (plus (plus n k) h)) (\lambda (t0: T).(eq T t0 -(lift k e (lift h d (TLRef n))))) (eq_ind_r T (TLRef (plus n h)) (\lambda -(t0: T).(eq T (TLRef (plus (plus n k) h)) (lift k e t0))) (eq_ind_r T (TLRef -(plus (plus n h) k)) (\lambda (t0: T).(eq T (TLRef (plus (plus n k) h)) t0)) -(f_equal nat T TLRef (plus (plus n k) h) (plus (plus n h) k) (sym_eq nat -(plus (plus n h) k) (plus (plus n k) h) (plus_permute_2_in_3 n h k))) (lift k -e (TLRef (plus n h))) (lift_lref_ge (plus n h) k e (le_plus_trans e n h H0))) -(lift h d (TLRef n)) (lift_lref_ge n h d H1)) (lift h (plus d k) (TLRef (plus -n k))) (lift_lref_ge (plus n k) h (plus d k) (le_plus_plus d n k k H1 (le_n -k)))))) (plus k d) (plus_sym k d)) (lift k e (TLRef n)) (lift_lref_ge n k e -H0)))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (H: ((\forall (h: -nat).(\forall (k0: nat).(\forall (d: nat).(\forall (e: nat).((le e d) \to (eq -T (lift h (plus k0 d) (lift k0 e t0)) (lift k0 e (lift h d -t0)))))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (h: nat).(\forall (k0: -nat).(\forall (d: nat).(\forall (e: nat).((le e d) \to (eq T (lift h (plus k0 -d) (lift k0 e t1)) (lift k0 e (lift h d t1)))))))))).(\lambda (h: -nat).(\lambda (k0: nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (H1: (le -e d)).(eq_ind_r T (THead k (lift k0 e t0) (lift k0 (s k e) t1)) (\lambda (t2: -T).(eq T (lift h (plus k0 d) t2) (lift k0 e (lift h d (THead k t0 t1))))) -(eq_ind_r T (THead k (lift h (plus k0 d) (lift k0 e t0)) (lift h (s k (plus -k0 d)) (lift k0 (s k e) t1))) (\lambda (t2: T).(eq T t2 (lift k0 e (lift h d -(THead k t0 t1))))) (eq_ind_r T (THead k (lift h d t0) (lift h (s k d) t1)) -(\lambda (t2: T).(eq T (THead k (lift h (plus k0 d) (lift k0 e t0)) (lift h -(s k (plus k0 d)) (lift k0 (s k e) t1))) (lift k0 e t2))) (eq_ind_r T (THead -k (lift k0 e (lift h d t0)) (lift k0 (s k e) (lift h (s k d) t1))) (\lambda -(t2: T).(eq T (THead k (lift h (plus k0 d) (lift k0 e t0)) (lift h (s k (plus -k0 d)) (lift k0 (s k e) t1))) t2)) (eq_ind_r nat (plus k0 (s k d)) (\lambda -(n: nat).(eq T (THead k (lift h (plus k0 d) (lift k0 e t0)) (lift h n (lift -k0 (s k e) t1))) (THead k (lift k0 e (lift h d t0)) (lift k0 (s k e) (lift h -(s k d) t1))))) (f_equal3 K T T T THead k k (lift h (plus k0 d) (lift k0 e -t0)) (lift k0 e (lift h d t0)) (lift h (plus k0 (s k d)) (lift k0 (s k e) -t1)) (lift k0 (s k e) (lift h (s k d) t1)) (refl_equal K k) (H h k0 d e H1) -(H0 h k0 (s k d) (s k e) (s_le k e d H1))) (s k (plus k0 d)) (s_plus_sym k k0 -d)) (lift k0 e (THead k (lift h d t0) (lift h (s k d) t1))) (lift_head k -(lift h d t0) (lift h (s k d) t1) k0 e)) (lift h d (THead k t0 t1)) -(lift_head k t0 t1 h d)) (lift h (plus k0 d) (THead k (lift k0 e t0) (lift k0 -(s k e) t1))) (lift_head k (lift k0 e t0) (lift k0 (s k e) t1) h (plus k0 -d))) (lift k0 e (THead k t0 t1)) (lift_head k t0 t1 k0 e)))))))))))) t). -(* COMMENTS -Initial nodes: 2143 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/lift/tlt.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/lift/tlt.ma deleted file mode 100644 index 1d8edc7df..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/lift/tlt.ma +++ /dev/null @@ -1,299 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/lift/fwd.ma". - -include "Basic-1/tlt/props.ma". - -theorem lift_weight_map: - \forall (t: T).(\forall (h: nat).(\forall (d: nat).(\forall (f: ((nat \to -nat))).(((\forall (m: nat).((le d m) \to (eq nat (f m) O)))) \to (eq nat -(weight_map f (lift h d t)) (weight_map f t)))))) -\def - \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (h: nat).(\forall (d: -nat).(\forall (f: ((nat \to nat))).(((\forall (m: nat).((le d m) \to (eq nat -(f m) O)))) \to (eq nat (weight_map f (lift h d t0)) (weight_map f t0))))))) -(\lambda (n: nat).(\lambda (_: nat).(\lambda (d: nat).(\lambda (f: ((nat \to -nat))).(\lambda (_: ((\forall (m: nat).((le d m) \to (eq nat (f m) -O))))).(refl_equal nat (weight_map f (TSort n)))))))) (\lambda (n: -nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (f: ((nat \to -nat))).(\lambda (H: ((\forall (m: nat).((le d m) \to (eq nat (f m) -O))))).(lt_le_e n d (eq nat (weight_map f (lift h d (TLRef n))) (weight_map f -(TLRef n))) (\lambda (H0: (lt n d)).(eq_ind_r T (TLRef n) (\lambda (t0: -T).(eq nat (weight_map f t0) (weight_map f (TLRef n)))) (refl_equal nat -(weight_map f (TLRef n))) (lift h d (TLRef n)) (lift_lref_lt n h d H0))) -(\lambda (H0: (le d n)).(eq_ind_r T (TLRef (plus n h)) (\lambda (t0: T).(eq -nat (weight_map f t0) (weight_map f (TLRef n)))) (eq_ind_r nat O (\lambda -(n0: nat).(eq nat (f (plus n h)) n0)) (H (plus n h) (le_plus_trans d n h H0)) -(f n) (H n H0)) (lift h d (TLRef n)) (lift_lref_ge n h d H0))))))))) (\lambda -(k: K).(\lambda (t0: T).(\lambda (H: ((\forall (h: nat).(\forall (d: -nat).(\forall (f: ((nat \to nat))).(((\forall (m: nat).((le d m) \to (eq nat -(f m) O)))) \to (eq nat (weight_map f (lift h d t0)) (weight_map f -t0)))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (h: nat).(\forall (d: -nat).(\forall (f: ((nat \to nat))).(((\forall (m: nat).((le d m) \to (eq nat -(f m) O)))) \to (eq nat (weight_map f (lift h d t1)) (weight_map f -t1)))))))).(\lambda (h: nat).(\lambda (d: nat).(\lambda (f: ((nat \to -nat))).(\lambda (H1: ((\forall (m: nat).((le d m) \to (eq nat (f m) -O))))).(K_ind (\lambda (k0: K).(eq nat (weight_map f (lift h d (THead k0 t0 -t1))) (weight_map f (THead k0 t0 t1)))) (\lambda (b: B).(eq_ind_r T (THead -(Bind b) (lift h d t0) (lift h (s (Bind b) d) t1)) (\lambda (t2: T).(eq nat -(weight_map f t2) (weight_map f (THead (Bind b) t0 t1)))) (B_ind (\lambda -(b0: B).(eq nat (match b0 with [Abbr \Rightarrow (S (plus (weight_map f (lift -h d t0)) (weight_map (wadd f (S (weight_map f (lift h d t0)))) (lift h (S d) -t1)))) | Abst \Rightarrow (S (plus (weight_map f (lift h d t0)) (weight_map -(wadd f O) (lift h (S d) t1)))) | Void \Rightarrow (S (plus (weight_map f -(lift h d t0)) (weight_map (wadd f O) (lift h (S d) t1))))]) (match b0 with -[Abbr \Rightarrow (S (plus (weight_map f t0) (weight_map (wadd f (S -(weight_map f t0))) t1))) | Abst \Rightarrow (S (plus (weight_map f t0) -(weight_map (wadd f O) t1))) | Void \Rightarrow (S (plus (weight_map f t0) -(weight_map (wadd f O) t1)))]))) (eq_ind_r nat (weight_map f t0) (\lambda (n: -nat).(eq nat (S (plus n (weight_map (wadd f (S n)) (lift h (S d) t1)))) (S -(plus (weight_map f t0) (weight_map (wadd f (S (weight_map f t0))) t1))))) -(eq_ind_r nat (weight_map (wadd f (S (weight_map f t0))) t1) (\lambda (n: -nat).(eq nat (S (plus (weight_map f t0) n)) (S (plus (weight_map f t0) -(weight_map (wadd f (S (weight_map f t0))) t1))))) (refl_equal nat (S (plus -(weight_map f t0) (weight_map (wadd f (S (weight_map f t0))) t1)))) -(weight_map (wadd f (S (weight_map f t0))) (lift h (S d) t1)) (H0 h (S d) -(wadd f (S (weight_map f t0))) (\lambda (m: nat).(\lambda (H2: (le (S d) -m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S n))) (\lambda (n: nat).(le d -n)) (eq nat (wadd f (S (weight_map f t0)) m) O) (\lambda (x: nat).(\lambda -(H3: (eq nat m (S x))).(\lambda (H4: (le d x)).(eq_ind_r nat (S x) (\lambda -(n: nat).(eq nat (wadd f (S (weight_map f t0)) n) O)) (H1 x H4) m H3)))) -(le_gen_S d m H2)))))) (weight_map f (lift h d t0)) (H h d f H1)) (eq_ind_r -nat (weight_map (wadd f O) t1) (\lambda (n: nat).(eq nat (S (plus (weight_map -f (lift h d t0)) n)) (S (plus (weight_map f t0) (weight_map (wadd f O) -t1))))) (f_equal nat nat S (plus (weight_map f (lift h d t0)) (weight_map -(wadd f O) t1)) (plus (weight_map f t0) (weight_map (wadd f O) t1)) (f_equal2 -nat nat nat plus (weight_map f (lift h d t0)) (weight_map f t0) (weight_map -(wadd f O) t1) (weight_map (wadd f O) t1) (H h d f H1) (refl_equal nat -(weight_map (wadd f O) t1)))) (weight_map (wadd f O) (lift h (S d) t1)) (H0 h -(S d) (wadd f O) (\lambda (m: nat).(\lambda (H2: (le (S d) m)).(ex2_ind nat -(\lambda (n: nat).(eq nat m (S n))) (\lambda (n: nat).(le d n)) (eq nat (wadd -f O m) O) (\lambda (x: nat).(\lambda (H3: (eq nat m (S x))).(\lambda (H4: (le -d x)).(eq_ind_r nat (S x) (\lambda (n: nat).(eq nat (wadd f O n) O)) (H1 x -H4) m H3)))) (le_gen_S d m H2)))))) (eq_ind_r nat (weight_map (wadd f O) t1) -(\lambda (n: nat).(eq nat (S (plus (weight_map f (lift h d t0)) n)) (S (plus -(weight_map f t0) (weight_map (wadd f O) t1))))) (f_equal nat nat S (plus -(weight_map f (lift h d t0)) (weight_map (wadd f O) t1)) (plus (weight_map f -t0) (weight_map (wadd f O) t1)) (f_equal2 nat nat nat plus (weight_map f -(lift h d t0)) (weight_map f t0) (weight_map (wadd f O) t1) (weight_map (wadd -f O) t1) (H h d f H1) (refl_equal nat (weight_map (wadd f O) t1)))) -(weight_map (wadd f O) (lift h (S d) t1)) (H0 h (S d) (wadd f O) (\lambda (m: -nat).(\lambda (H2: (le (S d) m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S -n))) (\lambda (n: nat).(le d n)) (eq nat (wadd f O m) O) (\lambda (x: -nat).(\lambda (H3: (eq nat m (S x))).(\lambda (H4: (le d x)).(eq_ind_r nat (S -x) (\lambda (n: nat).(eq nat (wadd f O n) O)) (H1 x H4) m H3)))) (le_gen_S d -m H2)))))) b) (lift h d (THead (Bind b) t0 t1)) (lift_head (Bind b) t0 t1 h -d))) (\lambda (f0: F).(eq_ind_r T (THead (Flat f0) (lift h d t0) (lift h (s -(Flat f0) d) t1)) (\lambda (t2: T).(eq nat (weight_map f t2) (weight_map f -(THead (Flat f0) t0 t1)))) (f_equal nat nat S (plus (weight_map f (lift h d -t0)) (weight_map f (lift h d t1))) (plus (weight_map f t0) (weight_map f t1)) -(f_equal2 nat nat nat plus (weight_map f (lift h d t0)) (weight_map f t0) -(weight_map f (lift h d t1)) (weight_map f t1) (H h d f H1) (H0 h d f H1))) -(lift h d (THead (Flat f0) t0 t1)) (lift_head (Flat f0) t0 t1 h d))) -k)))))))))) t). -(* COMMENTS -Initial nodes: 1969 -END *) - -theorem lift_weight: - \forall (t: T).(\forall (h: nat).(\forall (d: nat).(eq nat (weight (lift h d -t)) (weight t)))) -\def - \lambda (t: T).(\lambda (h: nat).(\lambda (d: nat).(lift_weight_map t h d -(\lambda (_: nat).O) (\lambda (m: nat).(\lambda (_: (le d m)).(refl_equal nat -O)))))). -(* COMMENTS -Initial nodes: 31 -END *) - -theorem lift_weight_add: - \forall (w: nat).(\forall (t: T).(\forall (h: nat).(\forall (d: -nat).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall -(m: nat).((lt m d) \to (eq nat (g m) (f m))))) \to ((eq nat (g d) w) \to -(((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f m))))) \to (eq nat -(weight_map f (lift h d t)) (weight_map g (lift (S h) d t))))))))))) -\def - \lambda (w: nat).(\lambda (t: T).(T_ind (\lambda (t0: T).(\forall (h: -nat).(\forall (d: nat).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (m: nat).((lt m d) \to (eq nat (g m) (f m))))) \to ((eq nat -(g d) w) \to (((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f m))))) -\to (eq nat (weight_map f (lift h d t0)) (weight_map g (lift (S h) d -t0))))))))))) (\lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda -(f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (_: ((\forall (m: -nat).((lt m d) \to (eq nat (g m) (f m)))))).(\lambda (_: (eq nat (g d) -w)).(\lambda (_: ((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f -m)))))).(refl_equal nat (weight_map g (lift (S h) d (TSort n)))))))))))) -(\lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (f: ((nat \to -nat))).(\lambda (g: ((nat \to nat))).(\lambda (H: ((\forall (m: nat).((lt m -d) \to (eq nat (g m) (f m)))))).(\lambda (_: (eq nat (g d) w)).(\lambda (H1: -((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f m)))))).(lt_le_e n d -(eq nat (weight_map f (lift h d (TLRef n))) (weight_map g (lift (S h) d -(TLRef n)))) (\lambda (H2: (lt n d)).(eq_ind_r T (TLRef n) (\lambda (t0: -T).(eq nat (weight_map f t0) (weight_map g (lift (S h) d (TLRef n))))) -(eq_ind_r T (TLRef n) (\lambda (t0: T).(eq nat (weight_map f (TLRef n)) -(weight_map g t0))) (sym_eq nat (g n) (f n) (H n H2)) (lift (S h) d (TLRef -n)) (lift_lref_lt n (S h) d H2)) (lift h d (TLRef n)) (lift_lref_lt n h d -H2))) (\lambda (H2: (le d n)).(eq_ind_r T (TLRef (plus n h)) (\lambda (t0: -T).(eq nat (weight_map f t0) (weight_map g (lift (S h) d (TLRef n))))) -(eq_ind_r T (TLRef (plus n (S h))) (\lambda (t0: T).(eq nat (weight_map f -(TLRef (plus n h))) (weight_map g t0))) (eq_ind nat (S (plus n h)) (\lambda -(n0: nat).(eq nat (f (plus n h)) (g n0))) (sym_eq nat (g (S (plus n h))) (f -(plus n h)) (H1 (plus n h) (le_plus_trans d n h H2))) (plus n (S h)) -(plus_n_Sm n h)) (lift (S h) d (TLRef n)) (lift_lref_ge n (S h) d H2)) (lift -h d (TLRef n)) (lift_lref_ge n h d H2)))))))))))) (\lambda (k: K).(\lambda -(t0: T).(\lambda (H: ((\forall (h: nat).(\forall (d: nat).(\forall (f: ((nat -\to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).((lt m d) \to -(eq nat (g m) (f m))))) \to ((eq nat (g d) w) \to (((\forall (m: nat).((le d -m) \to (eq nat (g (S m)) (f m))))) \to (eq nat (weight_map f (lift h d t0)) -(weight_map g (lift (S h) d t0)))))))))))).(\lambda (t1: T).(\lambda (H0: -((\forall (h: nat).(\forall (d: nat).(\forall (f: ((nat \to nat))).(\forall -(g: ((nat \to nat))).(((\forall (m: nat).((lt m d) \to (eq nat (g m) (f -m))))) \to ((eq nat (g d) w) \to (((\forall (m: nat).((le d m) \to (eq nat (g -(S m)) (f m))))) \to (eq nat (weight_map f (lift h d t1)) (weight_map g (lift -(S h) d t1)))))))))))).(\lambda (h: nat).(\lambda (d: nat).(\lambda (f: ((nat -\to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H1: ((\forall (m: -nat).((lt m d) \to (eq nat (g m) (f m)))))).(\lambda (H2: (eq nat (g d) -w)).(\lambda (H3: ((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f -m)))))).(K_ind (\lambda (k0: K).(eq nat (weight_map f (lift h d (THead k0 t0 -t1))) (weight_map g (lift (S h) d (THead k0 t0 t1))))) (\lambda (b: -B).(eq_ind_r T (THead (Bind b) (lift h d t0) (lift h (s (Bind b) d) t1)) -(\lambda (t2: T).(eq nat (weight_map f t2) (weight_map g (lift (S h) d (THead -(Bind b) t0 t1))))) (eq_ind_r T (THead (Bind b) (lift (S h) d t0) (lift (S h) -(s (Bind b) d) t1)) (\lambda (t2: T).(eq nat (weight_map f (THead (Bind b) -(lift h d t0) (lift h (s (Bind b) d) t1))) (weight_map g t2))) (B_ind -(\lambda (b0: B).(eq nat (match b0 with [Abbr \Rightarrow (S (plus -(weight_map f (lift h d t0)) (weight_map (wadd f (S (weight_map f (lift h d -t0)))) (lift h (S d) t1)))) | Abst \Rightarrow (S (plus (weight_map f (lift h -d t0)) (weight_map (wadd f O) (lift h (S d) t1)))) | Void \Rightarrow (S -(plus (weight_map f (lift h d t0)) (weight_map (wadd f O) (lift h (S d) -t1))))]) (match b0 with [Abbr \Rightarrow (S (plus (weight_map g (lift (S h) -d t0)) (weight_map (wadd g (S (weight_map g (lift (S h) d t0)))) (lift (S h) -(S d) t1)))) | Abst \Rightarrow (S (plus (weight_map g (lift (S h) d t0)) -(weight_map (wadd g O) (lift (S h) (S d) t1)))) | Void \Rightarrow (S (plus -(weight_map g (lift (S h) d t0)) (weight_map (wadd g O) (lift (S h) (S d) -t1))))]))) (f_equal nat nat S (plus (weight_map f (lift h d t0)) (weight_map -(wadd f (S (weight_map f (lift h d t0)))) (lift h (S d) t1))) (plus -(weight_map g (lift (S h) d t0)) (weight_map (wadd g (S (weight_map g (lift -(S h) d t0)))) (lift (S h) (S d) t1))) (f_equal2 nat nat nat plus (weight_map -f (lift h d t0)) (weight_map g (lift (S h) d t0)) (weight_map (wadd f (S -(weight_map f (lift h d t0)))) (lift h (S d) t1)) (weight_map (wadd g (S -(weight_map g (lift (S h) d t0)))) (lift (S h) (S d) t1)) (H h d f g H1 H2 -H3) (H0 h (S d) (wadd f (S (weight_map f (lift h d t0)))) (wadd g (S -(weight_map g (lift (S h) d t0)))) (\lambda (m: nat).(\lambda (H4: (lt m (S -d))).(or_ind (eq nat m O) (ex2 nat (\lambda (m0: nat).(eq nat m (S m0))) -(\lambda (m0: nat).(lt m0 d))) (eq nat (wadd g (S (weight_map g (lift (S h) d -t0))) m) (wadd f (S (weight_map f (lift h d t0))) m)) (\lambda (H5: (eq nat m -O)).(eq_ind_r nat O (\lambda (n: nat).(eq nat (wadd g (S (weight_map g (lift -(S h) d t0))) n) (wadd f (S (weight_map f (lift h d t0))) n))) (f_equal nat -nat S (weight_map g (lift (S h) d t0)) (weight_map f (lift h d t0)) (sym_eq -nat (weight_map f (lift h d t0)) (weight_map g (lift (S h) d t0)) (H h d f g -H1 H2 H3))) m H5)) (\lambda (H5: (ex2 nat (\lambda (m0: nat).(eq nat m (S -m0))) (\lambda (m0: nat).(lt m0 d)))).(ex2_ind nat (\lambda (m0: nat).(eq nat -m (S m0))) (\lambda (m0: nat).(lt m0 d)) (eq nat (wadd g (S (weight_map g -(lift (S h) d t0))) m) (wadd f (S (weight_map f (lift h d t0))) m)) (\lambda -(x: nat).(\lambda (H6: (eq nat m (S x))).(\lambda (H7: (lt x d)).(eq_ind_r -nat (S x) (\lambda (n: nat).(eq nat (wadd g (S (weight_map g (lift (S h) d -t0))) n) (wadd f (S (weight_map f (lift h d t0))) n))) (H1 x H7) m H6)))) -H5)) (lt_gen_xS m d H4)))) H2 (\lambda (m: nat).(\lambda (H4: (le (S d) -m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S n))) (\lambda (n: nat).(le d -n)) (eq nat (g m) (wadd f (S (weight_map f (lift h d t0))) m)) (\lambda (x: -nat).(\lambda (H5: (eq nat m (S x))).(\lambda (H6: (le d x)).(eq_ind_r nat (S -x) (\lambda (n: nat).(eq nat (g n) (wadd f (S (weight_map f (lift h d t0))) -n))) (H3 x H6) m H5)))) (le_gen_S d m H4))))))) (f_equal nat nat S (plus -(weight_map f (lift h d t0)) (weight_map (wadd f O) (lift h (S d) t1))) (plus -(weight_map g (lift (S h) d t0)) (weight_map (wadd g O) (lift (S h) (S d) -t1))) (f_equal2 nat nat nat plus (weight_map f (lift h d t0)) (weight_map g -(lift (S h) d t0)) (weight_map (wadd f O) (lift h (S d) t1)) (weight_map -(wadd g O) (lift (S h) (S d) t1)) (H h d f g H1 H2 H3) (H0 h (S d) (wadd f O) -(wadd g O) (\lambda (m: nat).(\lambda (H4: (lt m (S d))).(or_ind (eq nat m O) -(ex2 nat (\lambda (m0: nat).(eq nat m (S m0))) (\lambda (m0: nat).(lt m0 d))) -(eq nat (wadd g O m) (wadd f O m)) (\lambda (H5: (eq nat m O)).(eq_ind_r nat -O (\lambda (n: nat).(eq nat (wadd g O n) (wadd f O n))) (refl_equal nat O) m -H5)) (\lambda (H5: (ex2 nat (\lambda (m0: nat).(eq nat m (S m0))) (\lambda -(m0: nat).(lt m0 d)))).(ex2_ind nat (\lambda (m0: nat).(eq nat m (S m0))) -(\lambda (m0: nat).(lt m0 d)) (eq nat (wadd g O m) (wadd f O m)) (\lambda (x: -nat).(\lambda (H6: (eq nat m (S x))).(\lambda (H7: (lt x d)).(eq_ind_r nat (S -x) (\lambda (n: nat).(eq nat (wadd g O n) (wadd f O n))) (H1 x H7) m H6)))) -H5)) (lt_gen_xS m d H4)))) H2 (\lambda (m: nat).(\lambda (H4: (le (S d) -m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S n))) (\lambda (n: nat).(le d -n)) (eq nat (g m) (wadd f O m)) (\lambda (x: nat).(\lambda (H5: (eq nat m (S -x))).(\lambda (H6: (le d x)).(eq_ind_r nat (S x) (\lambda (n: nat).(eq nat (g -n) (wadd f O n))) (H3 x H6) m H5)))) (le_gen_S d m H4))))))) (f_equal nat nat -S (plus (weight_map f (lift h d t0)) (weight_map (wadd f O) (lift h (S d) -t1))) (plus (weight_map g (lift (S h) d t0)) (weight_map (wadd g O) (lift (S -h) (S d) t1))) (f_equal2 nat nat nat plus (weight_map f (lift h d t0)) -(weight_map g (lift (S h) d t0)) (weight_map (wadd f O) (lift h (S d) t1)) -(weight_map (wadd g O) (lift (S h) (S d) t1)) (H h d f g H1 H2 H3) (H0 h (S -d) (wadd f O) (wadd g O) (\lambda (m: nat).(\lambda (H4: (lt m (S -d))).(or_ind (eq nat m O) (ex2 nat (\lambda (m0: nat).(eq nat m (S m0))) -(\lambda (m0: nat).(lt m0 d))) (eq nat (wadd g O m) (wadd f O m)) (\lambda -(H5: (eq nat m O)).(eq_ind_r nat O (\lambda (n: nat).(eq nat (wadd g O n) -(wadd f O n))) (refl_equal nat O) m H5)) (\lambda (H5: (ex2 nat (\lambda (m0: -nat).(eq nat m (S m0))) (\lambda (m0: nat).(lt m0 d)))).(ex2_ind nat (\lambda -(m0: nat).(eq nat m (S m0))) (\lambda (m0: nat).(lt m0 d)) (eq nat (wadd g O -m) (wadd f O m)) (\lambda (x: nat).(\lambda (H6: (eq nat m (S x))).(\lambda -(H7: (lt x d)).(eq_ind_r nat (S x) (\lambda (n: nat).(eq nat (wadd g O n) -(wadd f O n))) (H1 x H7) m H6)))) H5)) (lt_gen_xS m d H4)))) H2 (\lambda (m: -nat).(\lambda (H4: (le (S d) m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S -n))) (\lambda (n: nat).(le d n)) (eq nat (g m) (wadd f O m)) (\lambda (x: -nat).(\lambda (H5: (eq nat m (S x))).(\lambda (H6: (le d x)).(eq_ind_r nat (S -x) (\lambda (n: nat).(eq nat (g n) (wadd f O n))) (H3 x H6) m H5)))) -(le_gen_S d m H4))))))) b) (lift (S h) d (THead (Bind b) t0 t1)) (lift_head -(Bind b) t0 t1 (S h) d)) (lift h d (THead (Bind b) t0 t1)) (lift_head (Bind -b) t0 t1 h d))) (\lambda (f0: F).(eq_ind_r T (THead (Flat f0) (lift h d t0) -(lift h (s (Flat f0) d) t1)) (\lambda (t2: T).(eq nat (weight_map f t2) -(weight_map g (lift (S h) d (THead (Flat f0) t0 t1))))) (eq_ind_r T (THead -(Flat f0) (lift (S h) d t0) (lift (S h) (s (Flat f0) d) t1)) (\lambda (t2: -T).(eq nat (weight_map f (THead (Flat f0) (lift h d t0) (lift h (s (Flat f0) -d) t1))) (weight_map g t2))) (f_equal nat nat S (plus (weight_map f (lift h d -t0)) (weight_map f (lift h d t1))) (plus (weight_map g (lift (S h) d t0)) -(weight_map g (lift (S h) d t1))) (f_equal2 nat nat nat plus (weight_map f -(lift h d t0)) (weight_map g (lift (S h) d t0)) (weight_map f (lift h d t1)) -(weight_map g (lift (S h) d t1)) (H h d f g H1 H2 H3) (H0 h d f g H1 H2 H3))) -(lift (S h) d (THead (Flat f0) t0 t1)) (lift_head (Flat f0) t0 t1 (S h) d)) -(lift h d (THead (Flat f0) t0 t1)) (lift_head (Flat f0) t0 t1 h d))) -k))))))))))))) t)). -(* COMMENTS -Initial nodes: 3697 -END *) - -theorem lift_weight_add_O: - \forall (w: nat).(\forall (t: T).(\forall (h: nat).(\forall (f: ((nat \to -nat))).(eq nat (weight_map f (lift h O t)) (weight_map (wadd f w) (lift (S h) -O t)))))) -\def - \lambda (w: nat).(\lambda (t: T).(\lambda (h: nat).(\lambda (f: ((nat \to -nat))).(lift_weight_add (plus (wadd f w O) O) t h O f (wadd f w) (\lambda (m: -nat).(\lambda (H: (lt m O)).(lt_x_O m H (eq nat (wadd f w m) (f m))))) -(plus_n_O (wadd f w O)) (\lambda (m: nat).(\lambda (_: (le O m)).(refl_equal -nat (f m)))))))). -(* COMMENTS -Initial nodes: 93 -END *) - -theorem lift_tlt_dx: - \forall (k: K).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall -(d: nat).(tlt t (THead k u (lift h d t))))))) -\def - \lambda (k: K).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda -(d: nat).(eq_ind nat (weight (lift h d t)) (\lambda (n: nat).(lt n (weight -(THead k u (lift h d t))))) (tlt_head_dx k u (lift h d t)) (weight t) -(lift_weight t h d)))))). -(* COMMENTS -Initial nodes: 71 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/lift1/defs.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/lift1/defs.ma deleted file mode 100644 index 1f473cbce..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/lift1/defs.ma +++ /dev/null @@ -1,40 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/lift/defs.ma". - -definition trans: - PList \to (nat \to nat) -\def - let rec trans (hds: PList) on hds: (nat \to nat) \def (\lambda (i: -nat).(match hds with [PNil \Rightarrow i | (PCons h d hds0) \Rightarrow (let -j \def (trans hds0 i) in (match (blt j d) with [true \Rightarrow j | false -\Rightarrow (plus j h)]))])) in trans. - -definition lift1: - PList \to (T \to T) -\def - let rec lift1 (hds: PList) on hds: (T \to T) \def (\lambda (t: T).(match hds -with [PNil \Rightarrow t | (PCons h d hds0) \Rightarrow (lift h d (lift1 hds0 -t))])) in lift1. - -definition lifts1: - PList \to (TList \to TList) -\def - let rec lifts1 (hds: PList) (ts: TList) on ts: TList \def (match ts with -[TNil \Rightarrow TNil | (TCons t ts0) \Rightarrow (TCons (lift1 hds t) -(lifts1 hds ts0))]) in lifts1. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/lift1/fwd.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/lift1/fwd.ma deleted file mode 100644 index e9ae2d11c..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/lift1/fwd.ma +++ /dev/null @@ -1,164 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/lift1/defs.ma". - -include "Basic-1/lift/fwd.ma". - -theorem lift1_sort: - \forall (n: nat).(\forall (is: PList).(eq T (lift1 is (TSort n)) (TSort n))) -\def - \lambda (n: nat).(\lambda (is: PList).(PList_ind (\lambda (p: PList).(eq T -(lift1 p (TSort n)) (TSort n))) (refl_equal T (TSort n)) (\lambda (n0: -nat).(\lambda (n1: nat).(\lambda (p: PList).(\lambda (H: (eq T (lift1 p -(TSort n)) (TSort n))).(eq_ind_r T (TSort n) (\lambda (t: T).(eq T (lift n0 -n1 t) (TSort n))) (refl_equal T (TSort n)) (lift1 p (TSort n)) H))))) is)). -(* COMMENTS -Initial nodes: 99 -END *) - -theorem lift1_lref: - \forall (hds: PList).(\forall (i: nat).(eq T (lift1 hds (TLRef i)) (TLRef -(trans hds i)))) -\def - \lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (i: nat).(eq T -(lift1 p (TLRef i)) (TLRef (trans p i))))) (\lambda (i: nat).(refl_equal T -(TLRef i))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda -(H: ((\forall (i: nat).(eq T (lift1 p (TLRef i)) (TLRef (trans p -i)))))).(\lambda (i: nat).(eq_ind_r T (TLRef (trans p i)) (\lambda (t: T).(eq -T (lift n n0 t) (TLRef (match (blt (trans p i) n0) with [true \Rightarrow -(trans p i) | false \Rightarrow (plus (trans p i) n)])))) (refl_equal T -(TLRef (match (blt (trans p i) n0) with [true \Rightarrow (trans p i) | false -\Rightarrow (plus (trans p i) n)]))) (lift1 p (TLRef i)) (H i))))))) hds). -(* COMMENTS -Initial nodes: 165 -END *) - -theorem lift1_bind: - \forall (b: B).(\forall (hds: PList).(\forall (u: T).(\forall (t: T).(eq T -(lift1 hds (THead (Bind b) u t)) (THead (Bind b) (lift1 hds u) (lift1 (Ss -hds) t)))))) -\def - \lambda (b: B).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall -(u: T).(\forall (t: T).(eq T (lift1 p (THead (Bind b) u t)) (THead (Bind b) -(lift1 p u) (lift1 (Ss p) t)))))) (\lambda (u: T).(\lambda (t: T).(refl_equal -T (THead (Bind b) u t)))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: -PList).(\lambda (H: ((\forall (u: T).(\forall (t: T).(eq T (lift1 p (THead -(Bind b) u t)) (THead (Bind b) (lift1 p u) (lift1 (Ss p) t))))))).(\lambda -(u: T).(\lambda (t: T).(eq_ind_r T (THead (Bind b) (lift1 p u) (lift1 (Ss p) -t)) (\lambda (t0: T).(eq T (lift n n0 t0) (THead (Bind b) (lift n n0 (lift1 p -u)) (lift n (S n0) (lift1 (Ss p) t))))) (eq_ind_r T (THead (Bind b) (lift n -n0 (lift1 p u)) (lift n (S n0) (lift1 (Ss p) t))) (\lambda (t0: T).(eq T t0 -(THead (Bind b) (lift n n0 (lift1 p u)) (lift n (S n0) (lift1 (Ss p) t))))) -(refl_equal T (THead (Bind b) (lift n n0 (lift1 p u)) (lift n (S n0) (lift1 -(Ss p) t)))) (lift n n0 (THead (Bind b) (lift1 p u) (lift1 (Ss p) t))) -(lift_bind b (lift1 p u) (lift1 (Ss p) t) n n0)) (lift1 p (THead (Bind b) u -t)) (H u t)))))))) hds)). -(* COMMENTS -Initial nodes: 379 -END *) - -theorem lift1_flat: - \forall (f: F).(\forall (hds: PList).(\forall (u: T).(\forall (t: T).(eq T -(lift1 hds (THead (Flat f) u t)) (THead (Flat f) (lift1 hds u) (lift1 hds -t)))))) -\def - \lambda (f: F).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall -(u: T).(\forall (t: T).(eq T (lift1 p (THead (Flat f) u t)) (THead (Flat f) -(lift1 p u) (lift1 p t)))))) (\lambda (u: T).(\lambda (t: T).(refl_equal T -(THead (Flat f) u t)))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: -PList).(\lambda (H: ((\forall (u: T).(\forall (t: T).(eq T (lift1 p (THead -(Flat f) u t)) (THead (Flat f) (lift1 p u) (lift1 p t))))))).(\lambda (u: -T).(\lambda (t: T).(eq_ind_r T (THead (Flat f) (lift1 p u) (lift1 p t)) -(\lambda (t0: T).(eq T (lift n n0 t0) (THead (Flat f) (lift n n0 (lift1 p u)) -(lift n n0 (lift1 p t))))) (eq_ind_r T (THead (Flat f) (lift n n0 (lift1 p -u)) (lift n n0 (lift1 p t))) (\lambda (t0: T).(eq T t0 (THead (Flat f) (lift -n n0 (lift1 p u)) (lift n n0 (lift1 p t))))) (refl_equal T (THead (Flat f) -(lift n n0 (lift1 p u)) (lift n n0 (lift1 p t)))) (lift n n0 (THead (Flat f) -(lift1 p u) (lift1 p t))) (lift_flat f (lift1 p u) (lift1 p t) n n0)) (lift1 -p (THead (Flat f) u t)) (H u t)))))))) hds)). -(* COMMENTS -Initial nodes: 353 -END *) - -theorem lift1_cons_tail: - \forall (t: T).(\forall (h: nat).(\forall (d: nat).(\forall (hds: PList).(eq -T (lift1 (PConsTail hds h d) t) (lift1 hds (lift h d t)))))) -\def - \lambda (t: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (hds: -PList).(PList_ind (\lambda (p: PList).(eq T (lift1 (PConsTail p h d) t) -(lift1 p (lift h d t)))) (refl_equal T (lift h d t)) (\lambda (n: -nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: (eq T (lift1 -(PConsTail p h d) t) (lift1 p (lift h d t)))).(eq_ind_r T (lift1 p (lift h d -t)) (\lambda (t0: T).(eq T (lift n n0 t0) (lift n n0 (lift1 p (lift h d -t))))) (refl_equal T (lift n n0 (lift1 p (lift h d t)))) (lift1 (PConsTail p -h d) t) H))))) hds)))). -(* COMMENTS -Initial nodes: 171 -END *) - -theorem lifts1_flat: - \forall (f: F).(\forall (hds: PList).(\forall (t: T).(\forall (ts: -TList).(eq T (lift1 hds (THeads (Flat f) ts t)) (THeads (Flat f) (lifts1 hds -ts) (lift1 hds t)))))) -\def - \lambda (f: F).(\lambda (hds: PList).(\lambda (t: T).(\lambda (ts: -TList).(TList_ind (\lambda (t0: TList).(eq T (lift1 hds (THeads (Flat f) t0 -t)) (THeads (Flat f) (lifts1 hds t0) (lift1 hds t)))) (refl_equal T (lift1 -hds t)) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H: (eq T (lift1 hds -(THeads (Flat f) t1 t)) (THeads (Flat f) (lifts1 hds t1) (lift1 hds -t)))).(eq_ind_r T (THead (Flat f) (lift1 hds t0) (lift1 hds (THeads (Flat f) -t1 t))) (\lambda (t2: T).(eq T t2 (THead (Flat f) (lift1 hds t0) (THeads -(Flat f) (lifts1 hds t1) (lift1 hds t))))) (eq_ind_r T (THeads (Flat f) -(lifts1 hds t1) (lift1 hds t)) (\lambda (t2: T).(eq T (THead (Flat f) (lift1 -hds t0) t2) (THead (Flat f) (lift1 hds t0) (THeads (Flat f) (lifts1 hds t1) -(lift1 hds t))))) (refl_equal T (THead (Flat f) (lift1 hds t0) (THeads (Flat -f) (lifts1 hds t1) (lift1 hds t)))) (lift1 hds (THeads (Flat f) t1 t)) H) -(lift1 hds (THead (Flat f) t0 (THeads (Flat f) t1 t))) (lift1_flat f hds t0 -(THeads (Flat f) t1 t)))))) ts)))). -(* COMMENTS -Initial nodes: 329 -END *) - -theorem lifts1_nil: - \forall (ts: TList).(eq TList (lifts1 PNil ts) ts) -\def - \lambda (ts: TList).(TList_ind (\lambda (t: TList).(eq TList (lifts1 PNil t) -t)) (refl_equal TList TNil) (\lambda (t: T).(\lambda (t0: TList).(\lambda (H: -(eq TList (lifts1 PNil t0) t0)).(eq_ind_r TList t0 (\lambda (t1: TList).(eq -TList (TCons t t1) (TCons t t0))) (refl_equal TList (TCons t t0)) (lifts1 -PNil t0) H)))) ts). -(* COMMENTS -Initial nodes: 83 -END *) - -theorem lifts1_cons: - \forall (h: nat).(\forall (d: nat).(\forall (hds: PList).(\forall (ts: -TList).(eq TList (lifts1 (PCons h d hds) ts) (lifts h d (lifts1 hds ts)))))) -\def - \lambda (h: nat).(\lambda (d: nat).(\lambda (hds: PList).(\lambda (ts: -TList).(TList_ind (\lambda (t: TList).(eq TList (lifts1 (PCons h d hds) t) -(lifts h d (lifts1 hds t)))) (refl_equal TList TNil) (\lambda (t: T).(\lambda -(t0: TList).(\lambda (H: (eq TList (lifts1 (PCons h d hds) t0) (lifts h d -(lifts1 hds t0)))).(eq_ind_r TList (lifts h d (lifts1 hds t0)) (\lambda (t1: -TList).(eq TList (TCons (lift h d (lift1 hds t)) t1) (TCons (lift h d (lift1 -hds t)) (lifts h d (lifts1 hds t0))))) (refl_equal TList (TCons (lift h d -(lift1 hds t)) (lifts h d (lifts1 hds t0)))) (lifts1 (PCons h d hds) t0) -H)))) ts)))). -(* COMMENTS -Initial nodes: 187 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/lift1/props.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/lift1/props.ma deleted file mode 100644 index ebda0267b..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/lift1/props.ma +++ /dev/null @@ -1,139 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/lift/props.ma". - -include "Basic-1/drop1/defs.ma". - -theorem lift1_lift1: - \forall (is1: PList).(\forall (is2: PList).(\forall (t: T).(eq T (lift1 is1 -(lift1 is2 t)) (lift1 (papp is1 is2) t)))) -\def - \lambda (is1: PList).(PList_ind (\lambda (p: PList).(\forall (is2: -PList).(\forall (t: T).(eq T (lift1 p (lift1 is2 t)) (lift1 (papp p is2) -t))))) (\lambda (is2: PList).(\lambda (t: T).(refl_equal T (lift1 is2 t)))) -(\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: -((\forall (is2: PList).(\forall (t: T).(eq T (lift1 p (lift1 is2 t)) (lift1 -(papp p is2) t)))))).(\lambda (is2: PList).(\lambda (t: T).(f_equal3 nat nat -T T lift n n n0 n0 (lift1 p (lift1 is2 t)) (lift1 (papp p is2) t) (refl_equal -nat n) (refl_equal nat n0) (H is2 t)))))))) is1). -(* COMMENTS -Initial nodes: 145 -END *) - -theorem lift1_xhg: - \forall (hds: PList).(\forall (t: T).(eq T (lift1 (Ss hds) (lift (S O) O t)) -(lift (S O) O (lift1 hds t)))) -\def - \lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (t: T).(eq T -(lift1 (Ss p) (lift (S O) O t)) (lift (S O) O (lift1 p t))))) (\lambda (t: -T).(refl_equal T (lift (S O) O t))) (\lambda (h: nat).(\lambda (d: -nat).(\lambda (p: PList).(\lambda (H: ((\forall (t: T).(eq T (lift1 (Ss p) -(lift (S O) O t)) (lift (S O) O (lift1 p t)))))).(\lambda (t: T).(eq_ind_r T -(lift (S O) O (lift1 p t)) (\lambda (t0: T).(eq T (lift h (S d) t0) (lift (S -O) O (lift h d (lift1 p t))))) (eq_ind nat (plus (S O) d) (\lambda (n: -nat).(eq T (lift h n (lift (S O) O (lift1 p t))) (lift (S O) O (lift h d -(lift1 p t))))) (eq_ind_r T (lift (S O) O (lift h d (lift1 p t))) (\lambda -(t0: T).(eq T t0 (lift (S O) O (lift h d (lift1 p t))))) (refl_equal T (lift -(S O) O (lift h d (lift1 p t)))) (lift h (plus (S O) d) (lift (S O) O (lift1 -p t))) (lift_d (lift1 p t) h (S O) d O (le_O_n d))) (S d) (refl_equal nat (S -d))) (lift1 (Ss p) (lift (S O) O t)) (H t))))))) hds). -(* COMMENTS -Initial nodes: 371 -END *) - -theorem lifts1_xhg: - \forall (hds: PList).(\forall (ts: TList).(eq TList (lifts1 (Ss hds) (lifts -(S O) O ts)) (lifts (S O) O (lifts1 hds ts)))) -\def - \lambda (hds: PList).(\lambda (ts: TList).(TList_ind (\lambda (t: TList).(eq -TList (lifts1 (Ss hds) (lifts (S O) O t)) (lifts (S O) O (lifts1 hds t)))) -(refl_equal TList TNil) (\lambda (t: T).(\lambda (t0: TList).(\lambda (H: (eq -TList (lifts1 (Ss hds) (lifts (S O) O t0)) (lifts (S O) O (lifts1 hds -t0)))).(eq_ind_r T (lift (S O) O (lift1 hds t)) (\lambda (t1: T).(eq TList -(TCons t1 (lifts1 (Ss hds) (lifts (S O) O t0))) (TCons (lift (S O) O (lift1 -hds t)) (lifts (S O) O (lifts1 hds t0))))) (eq_ind_r TList (lifts (S O) O -(lifts1 hds t0)) (\lambda (t1: TList).(eq TList (TCons (lift (S O) O (lift1 -hds t)) t1) (TCons (lift (S O) O (lift1 hds t)) (lifts (S O) O (lifts1 hds -t0))))) (refl_equal TList (TCons (lift (S O) O (lift1 hds t)) (lifts (S O) O -(lifts1 hds t0)))) (lifts1 (Ss hds) (lifts (S O) O t0)) H) (lift1 (Ss hds) -(lift (S O) O t)) (lift1_xhg hds t))))) ts)). -(* COMMENTS -Initial nodes: 307 -END *) - -theorem lift1_free: - \forall (hds: PList).(\forall (i: nat).(\forall (t: T).(eq T (lift1 hds -(lift (S i) O t)) (lift (S (trans hds i)) O (lift1 (ptrans hds i) t))))) -\def - \lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (i: -nat).(\forall (t: T).(eq T (lift1 p (lift (S i) O t)) (lift (S (trans p i)) O -(lift1 (ptrans p i) t)))))) (\lambda (i: nat).(\lambda (t: T).(refl_equal T -(lift (S i) O t)))) (\lambda (h: nat).(\lambda (d: nat).(\lambda (hds0: -PList).(\lambda (H: ((\forall (i: nat).(\forall (t: T).(eq T (lift1 hds0 -(lift (S i) O t)) (lift (S (trans hds0 i)) O (lift1 (ptrans hds0 i) -t))))))).(\lambda (i: nat).(\lambda (t: T).(eq_ind_r T (lift (S (trans hds0 -i)) O (lift1 (ptrans hds0 i) t)) (\lambda (t0: T).(eq T (lift h d t0) (lift -(S (match (blt (trans hds0 i) d) with [true \Rightarrow (trans hds0 i) | -false \Rightarrow (plus (trans hds0 i) h)])) O (lift1 (match (blt (trans hds0 -i) d) with [true \Rightarrow (PCons h (minus d (S (trans hds0 i))) (ptrans -hds0 i)) | false \Rightarrow (ptrans hds0 i)]) t)))) (xinduction bool (blt -(trans hds0 i) d) (\lambda (b: bool).(eq T (lift h d (lift (S (trans hds0 i)) -O (lift1 (ptrans hds0 i) t))) (lift (S (match b with [true \Rightarrow (trans -hds0 i) | false \Rightarrow (plus (trans hds0 i) h)])) O (lift1 (match b with -[true \Rightarrow (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) | -false \Rightarrow (ptrans hds0 i)]) t)))) (\lambda (x_x: bool).(bool_ind -(\lambda (b: bool).((eq bool (blt (trans hds0 i) d) b) \to (eq T (lift h d -(lift (S (trans hds0 i)) O (lift1 (ptrans hds0 i) t))) (lift (S (match b with -[true \Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i) -h)])) O (lift1 (match b with [true \Rightarrow (PCons h (minus d (S (trans -hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)]) t))))) -(\lambda (H0: (eq bool (blt (trans hds0 i) d) true)).(eq_ind_r nat (plus (S -(trans hds0 i)) (minus d (S (trans hds0 i)))) (\lambda (n: nat).(eq T (lift h -n (lift (S (trans hds0 i)) O (lift1 (ptrans hds0 i) t))) (lift (S (trans hds0 -i)) O (lift1 (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) t)))) -(eq_ind_r T (lift (S (trans hds0 i)) O (lift h (minus d (S (trans hds0 i))) -(lift1 (ptrans hds0 i) t))) (\lambda (t0: T).(eq T t0 (lift (S (trans hds0 -i)) O (lift1 (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) t)))) -(refl_equal T (lift (S (trans hds0 i)) O (lift1 (PCons h (minus d (S (trans -hds0 i))) (ptrans hds0 i)) t))) (lift h (plus (S (trans hds0 i)) (minus d (S -(trans hds0 i)))) (lift (S (trans hds0 i)) O (lift1 (ptrans hds0 i) t))) -(lift_d (lift1 (ptrans hds0 i) t) h (S (trans hds0 i)) (minus d (S (trans -hds0 i))) O (le_O_n (minus d (S (trans hds0 i)))))) d (le_plus_minus (S -(trans hds0 i)) d (bge_le (S (trans hds0 i)) d (le_bge (S (trans hds0 i)) d -(lt_le_S (trans hds0 i) d (blt_lt d (trans hds0 i) H0))))))) (\lambda (H0: -(eq bool (blt (trans hds0 i) d) false)).(eq_ind_r T (lift (plus h (S (trans -hds0 i))) O (lift1 (ptrans hds0 i) t)) (\lambda (t0: T).(eq T t0 (lift (S -(plus (trans hds0 i) h)) O (lift1 (ptrans hds0 i) t)))) (eq_ind nat (S (plus -h (trans hds0 i))) (\lambda (n: nat).(eq T (lift n O (lift1 (ptrans hds0 i) -t)) (lift (S (plus (trans hds0 i) h)) O (lift1 (ptrans hds0 i) t)))) -(eq_ind_r nat (plus (trans hds0 i) h) (\lambda (n: nat).(eq T (lift (S n) O -(lift1 (ptrans hds0 i) t)) (lift (S (plus (trans hds0 i) h)) O (lift1 (ptrans -hds0 i) t)))) (refl_equal T (lift (S (plus (trans hds0 i) h)) O (lift1 -(ptrans hds0 i) t))) (plus h (trans hds0 i)) (plus_sym h (trans hds0 i))) -(plus h (S (trans hds0 i))) (plus_n_Sm h (trans hds0 i))) (lift h d (lift (S -(trans hds0 i)) O (lift1 (ptrans hds0 i) t))) (lift_free (lift1 (ptrans hds0 -i) t) (S (trans hds0 i)) h O d (eq_ind nat (S (plus O (trans hds0 i))) -(\lambda (n: nat).(le d n)) (eq_ind_r nat (plus (trans hds0 i) O) (\lambda -(n: nat).(le d (S n))) (le_S d (plus (trans hds0 i) O) (le_plus_trans d -(trans hds0 i) O (bge_le d (trans hds0 i) H0))) (plus O (trans hds0 i)) -(plus_sym O (trans hds0 i))) (plus O (S (trans hds0 i))) (plus_n_Sm O (trans -hds0 i))) (le_O_n d)))) x_x))) (lift1 hds0 (lift (S i) O t)) (H i t)))))))) -hds). -(* COMMENTS -Initial nodes: 1339 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/llt/defs.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/llt/defs.ma deleted file mode 100644 index 96d869935..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/llt/defs.ma +++ /dev/null @@ -1,30 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/A/defs.ma". - -definition lweight: - A \to nat -\def - let rec lweight (a: A) on a: nat \def (match a with [(ASort _ _) \Rightarrow -O | (AHead a1 a2) \Rightarrow (S (plus (lweight a1) (lweight a2)))]) in -lweight. - -definition llt: - A \to (A \to Prop) -\def - \lambda (a1: A).(\lambda (a2: A).(lt (lweight a1) (lweight a2))). - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/llt/props.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/llt/props.ma deleted file mode 100644 index ef1f15a96..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/llt/props.ma +++ /dev/null @@ -1,114 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/llt/defs.ma". - -include "Basic-1/leq/defs.ma". - -theorem lweight_repl: - \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (eq nat -(lweight a1) (lweight a2))))) -\def - \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1 -a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(eq nat (lweight a) (lweight -a0)))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: nat).(\lambda (n2: -nat).(\lambda (k: nat).(\lambda (_: (eq A (aplus g (ASort h1 n1) k) (aplus g -(ASort h2 n2) k))).(refl_equal nat O))))))) (\lambda (a0: A).(\lambda (a3: -A).(\lambda (_: (leq g a0 a3)).(\lambda (H1: (eq nat (lweight a0) (lweight -a3))).(\lambda (a4: A).(\lambda (a5: A).(\lambda (_: (leq g a4 a5)).(\lambda -(H3: (eq nat (lweight a4) (lweight a5))).(f_equal nat nat S (plus (lweight -a0) (lweight a4)) (plus (lweight a3) (lweight a5)) (f_equal2 nat nat nat plus -(lweight a0) (lweight a3) (lweight a4) (lweight a5) H1 H3)))))))))) a1 a2 -H)))). -(* COMMENTS -Initial nodes: 189 -END *) - -theorem llt_repl: - \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall -(a3: A).((llt a1 a3) \to (llt a2 a3)))))) -\def - \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1 -a2)).(\lambda (a3: A).(\lambda (H0: (lt (lweight a1) (lweight a3))).(let H1 -\def (eq_ind nat (lweight a1) (\lambda (n: nat).(lt n (lweight a3))) H0 -(lweight a2) (lweight_repl g a1 a2 H)) in H1)))))). -(* COMMENTS -Initial nodes: 61 -END *) - -theorem llt_trans: - \forall (a1: A).(\forall (a2: A).(\forall (a3: A).((llt a1 a2) \to ((llt a2 -a3) \to (llt a1 a3))))) -\def - \lambda (a1: A).(\lambda (a2: A).(\lambda (a3: A).(\lambda (H: (lt (lweight -a1) (lweight a2))).(\lambda (H0: (lt (lweight a2) (lweight a3))).(lt_trans -(lweight a1) (lweight a2) (lweight a3) H H0))))). -(* COMMENTS -Initial nodes: 43 -END *) - -theorem llt_head_sx: - \forall (a1: A).(\forall (a2: A).(llt a1 (AHead a1 a2))) -\def - \lambda (a1: A).(\lambda (a2: A).(le_n_S (lweight a1) (plus (lweight a1) -(lweight a2)) (le_plus_l (lweight a1) (lweight a2)))). -(* COMMENTS -Initial nodes: 29 -END *) - -theorem llt_head_dx: - \forall (a1: A).(\forall (a2: A).(llt a2 (AHead a1 a2))) -\def - \lambda (a1: A).(\lambda (a2: A).(le_n_S (lweight a2) (plus (lweight a1) -(lweight a2)) (le_plus_r (lweight a1) (lweight a2)))). -(* COMMENTS -Initial nodes: 29 -END *) - -theorem llt_wf__q_ind: - \forall (P: ((A \to Prop))).(((\forall (n: nat).((\lambda (P0: ((A \to -Prop))).(\lambda (n0: nat).(\forall (a: A).((eq nat (lweight a) n0) \to (P0 -a))))) P n))) \to (\forall (a: A).(P a))) -\def - let Q \def (\lambda (P: ((A \to Prop))).(\lambda (n: nat).(\forall (a: -A).((eq nat (lweight a) n) \to (P a))))) in (\lambda (P: ((A \to -Prop))).(\lambda (H: ((\forall (n: nat).(\forall (a: A).((eq nat (lweight a) -n) \to (P a)))))).(\lambda (a: A).(H (lweight a) a (refl_equal nat (lweight -a)))))). -(* COMMENTS -Initial nodes: 61 -END *) - -theorem llt_wf_ind: - \forall (P: ((A \to Prop))).(((\forall (a2: A).(((\forall (a1: A).((llt a1 -a2) \to (P a1)))) \to (P a2)))) \to (\forall (a: A).(P a))) -\def - let Q \def (\lambda (P: ((A \to Prop))).(\lambda (n: nat).(\forall (a: -A).((eq nat (lweight a) n) \to (P a))))) in (\lambda (P: ((A \to -Prop))).(\lambda (H: ((\forall (a2: A).(((\forall (a1: A).((lt (lweight a1) -(lweight a2)) \to (P a1)))) \to (P a2))))).(\lambda (a: A).(llt_wf__q_ind -(\lambda (a0: A).(P a0)) (\lambda (n: nat).(lt_wf_ind n (Q (\lambda (a0: -A).(P a0))) (\lambda (n0: nat).(\lambda (H0: ((\forall (m: nat).((lt m n0) -\to (Q (\lambda (a0: A).(P a0)) m))))).(\lambda (a0: A).(\lambda (H1: (eq nat -(lweight a0) n0)).(let H2 \def (eq_ind_r nat n0 (\lambda (n1: nat).(\forall -(m: nat).((lt m n1) \to (\forall (a1: A).((eq nat (lweight a1) m) \to (P -a1)))))) H0 (lweight a0) H1) in (H a0 (\lambda (a1: A).(\lambda (H3: (lt -(lweight a1) (lweight a0))).(H2 (lweight a1) H3 a1 (refl_equal nat (lweight -a1))))))))))))) a)))). -(* COMMENTS -Initial nodes: 179 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/next_plus/defs.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/next_plus/defs.ma deleted file mode 100644 index 1ff7ecdff..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/next_plus/defs.ma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/G/defs.ma". - -definition next_plus: - G \to (nat \to (nat \to nat)) -\def - let rec next_plus (g: G) (n: nat) (i: nat) on i: nat \def (match i with [O -\Rightarrow n | (S i0) \Rightarrow (next g (next_plus g n i0))]) in next_plus. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/next_plus/props.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/next_plus/props.ma deleted file mode 100644 index 258c3a711..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/next_plus/props.ma +++ /dev/null @@ -1,68 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/next_plus/defs.ma". - -theorem next_plus_assoc: - \forall (g: G).(\forall (n: nat).(\forall (h1: nat).(\forall (h2: nat).(eq -nat (next_plus g (next_plus g n h1) h2) (next_plus g n (plus h1 h2)))))) -\def - \lambda (g: G).(\lambda (n: nat).(\lambda (h1: nat).(nat_ind (\lambda (n0: -nat).(\forall (h2: nat).(eq nat (next_plus g (next_plus g n n0) h2) -(next_plus g n (plus n0 h2))))) (\lambda (h2: nat).(refl_equal nat (next_plus -g n h2))) (\lambda (n0: nat).(\lambda (_: ((\forall (h2: nat).(eq nat -(next_plus g (next_plus g n n0) h2) (next_plus g n (plus n0 h2)))))).(\lambda -(h2: nat).(nat_ind (\lambda (n1: nat).(eq nat (next_plus g (next g (next_plus -g n n0)) n1) (next g (next_plus g n (plus n0 n1))))) (eq_ind nat n0 (\lambda -(n1: nat).(eq nat (next g (next_plus g n n0)) (next g (next_plus g n n1)))) -(refl_equal nat (next g (next_plus g n n0))) (plus n0 O) (plus_n_O n0)) -(\lambda (n1: nat).(\lambda (H0: (eq nat (next_plus g (next g (next_plus g n -n0)) n1) (next g (next_plus g n (plus n0 n1))))).(eq_ind nat (S (plus n0 n1)) -(\lambda (n2: nat).(eq nat (next g (next_plus g (next g (next_plus g n n0)) -n1)) (next g (next_plus g n n2)))) (f_equal nat nat (next g) (next_plus g -(next g (next_plus g n n0)) n1) (next g (next_plus g n (plus n0 n1))) H0) -(plus n0 (S n1)) (plus_n_Sm n0 n1)))) h2)))) h1))). -(* COMMENTS -Initial nodes: 351 -END *) - -theorem next_plus_next: - \forall (g: G).(\forall (n: nat).(\forall (h: nat).(eq nat (next_plus g -(next g n) h) (next g (next_plus g n h))))) -\def - \lambda (g: G).(\lambda (n: nat).(\lambda (h: nat).(eq_ind_r nat (next_plus -g n (plus (S O) h)) (\lambda (n0: nat).(eq nat n0 (next g (next_plus g n -h)))) (refl_equal nat (next g (next_plus g n h))) (next_plus g (next_plus g n -(S O)) h) (next_plus_assoc g n (S O) h)))). -(* COMMENTS -Initial nodes: 87 -END *) - -theorem next_plus_lt: - \forall (g: G).(\forall (h: nat).(\forall (n: nat).(lt n (next_plus g (next -g n) h)))) -\def - \lambda (g: G).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall (n0: -nat).(lt n0 (next_plus g (next g n0) n)))) (\lambda (n: nat).(next_lt g n)) -(\lambda (n: nat).(\lambda (H: ((\forall (n0: nat).(lt n0 (next_plus g (next -g n0) n))))).(\lambda (n0: nat).(eq_ind nat (next_plus g (next g (next g n0)) -n) (\lambda (n1: nat).(lt n0 n1)) (lt_trans n0 (next g n0) (next_plus g (next -g (next g n0)) n) (next_lt g n0) (H (next g n0))) (next g (next_plus g (next -g n0) n)) (next_plus_next g (next g n0) n))))) h)). -(* COMMENTS -Initial nodes: 153 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/arity.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/arity.ma deleted file mode 100644 index 98770d9e9..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/arity.ma +++ /dev/null @@ -1,496 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/nf2/fwd.ma". - -include "Basic-1/arity/subst0.ma". - -theorem arity_nf2_inv_all: - \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t -a) \to ((nf2 c t) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t -(THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c w))) -(\lambda (w: T).(\lambda (u: T).(nf2 (CHead c (Bind Abst) w) u)))) (ex nat -(\lambda (n: nat).(eq T t (TSort n)))) (ex3_2 TList nat (\lambda (ws: -TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i))))) -(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c (TLRef i))))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: -(arity g c t a)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (_: -A).((nf2 c0 t0) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t0 -(THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) -(\lambda (w: T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat -(\lambda (n: nat).(eq T t0 (TSort n)))) (ex3_2 TList nat (\lambda (ws: -TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i))))) -(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))))))) (\lambda (c0: C).(\lambda -(n: nat).(\lambda (_: (nf2 c0 (TSort n))).(or3_intro1 (ex3_2 T T (\lambda (w: -T).(\lambda (u: T).(eq T (TSort n) (THead (Bind Abst) w u)))) (\lambda (w: -T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead -c0 (Bind Abst) w) u)))) (ex nat (\lambda (n0: nat).(eq T (TSort n) (TSort -n0)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (TSort -n) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef -i))))) (ex_intro nat (\lambda (n0: nat).(eq T (TSort n) (TSort n0))) n -(refl_equal T (TSort n))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: -T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr) -u))).(\lambda (a0: A).(\lambda (_: (arity g d u a0)).(\lambda (_: (((nf2 d u) -\to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u (THead (Bind -Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 d w))) (\lambda (w: -T).(\lambda (u0: T).(nf2 (CHead d (Bind Abst) w) u0)))) (ex nat (\lambda (n: -nat).(eq T u (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i0: -nat).(eq T u (THeads (Flat Appl) ws (TLRef i0))))) (\lambda (ws: -TList).(\lambda (_: nat).(nfs2 d ws))) (\lambda (_: TList).(\lambda (i0: -nat).(nf2 d (TLRef i0))))))))).(\lambda (H3: (nf2 c0 (TLRef -i))).(nf2_gen_lref c0 d u i H0 H3 (or3 (ex3_2 T T (\lambda (w: T).(\lambda -(u0: T).(eq T (TLRef i) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda -(_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind -Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (TLRef i) (TSort n)))) (ex3_2 -TList nat (\lambda (ws: TList).(\lambda (i0: nat).(eq T (TLRef i) (THeads -(Flat Appl) ws (TLRef i0))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 -ws))) (\lambda (_: TList).(\lambda (i0: nat).(nf2 c0 (TLRef -i0)))))))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda -(i: nat).(\lambda (_: (getl i c0 (CHead d (Bind Abst) u))).(\lambda (a0: -A).(\lambda (_: (arity g d u (asucc g a0))).(\lambda (_: (((nf2 d u) \to (or3 -(ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u (THead (Bind Abst) w -u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 d w))) (\lambda (w: T).(\lambda -(u0: T).(nf2 (CHead d (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T u -(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i0: nat).(eq T u -(THeads (Flat Appl) ws (TLRef i0))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 d ws))) (\lambda (_: TList).(\lambda (i0: nat).(nf2 d (TLRef -i0))))))))).(\lambda (H3: (nf2 c0 (TLRef i))).(or3_intro2 (ex3_2 T T (\lambda -(w: T).(\lambda (u0: T).(eq T (TLRef i) (THead (Bind Abst) w u0)))) (\lambda -(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 -(CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (TLRef i) -(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i0: nat).(eq T -(TLRef i) (THeads (Flat Appl) ws (TLRef i0))))) (\lambda (ws: TList).(\lambda -(_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i0: nat).(nf2 c0 (TLRef -i0))))) (ex3_2_intro TList nat (\lambda (ws: TList).(\lambda (i0: nat).(eq T -(TLRef i) (THeads (Flat Appl) ws (TLRef i0))))) (\lambda (ws: TList).(\lambda -(_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i0: nat).(nf2 c0 (TLRef -i0)))) TNil i (refl_equal T (TLRef i)) I H3))))))))))) (\lambda (b: -B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c0: C).(\lambda (u: -T).(\lambda (a1: A).(\lambda (_: (arity g c0 u a1)).(\lambda (_: (((nf2 c0 u) -\to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u (THead (Bind -Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: -T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: -nat).(eq T u (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: -nat).(eq T u (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: -TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: -nat).(nf2 c0 (TLRef i))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda -(H3: (arity g (CHead c0 (Bind b) u) t0 a2)).(\lambda (_: (((nf2 (CHead c0 -(Bind b) u) t0) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T t0 -(THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 (CHead c0 -(Bind b) u) w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead (CHead c0 (Bind -b) u) (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort n)))) -(ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads -(Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 -(CHead c0 (Bind b) u) ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 (CHead -c0 (Bind b) u) (TLRef i))))))))).(\lambda (H5: (nf2 c0 (THead (Bind b) u -t0))).(B_ind (\lambda (b0: B).((not (eq B b0 Abst)) \to ((arity g (CHead c0 -(Bind b0) u) t0 a2) \to ((nf2 c0 (THead (Bind b0) u t0)) \to (or3 (ex3_2 T T -(\lambda (w: T).(\lambda (u0: T).(eq T (THead (Bind b0) u t0) (THead (Bind -Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: -T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: -nat).(eq T (THead (Bind b0) u t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws: -TList).(\lambda (i: nat).(eq T (THead (Bind b0) u t0) (THeads (Flat Appl) ws -(TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda -(_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))))))) (\lambda (_: (not (eq -B Abbr Abst))).(\lambda (_: (arity g (CHead c0 (Bind Abbr) u) t0 -a2)).(\lambda (H8: (nf2 c0 (THead (Bind Abbr) u t0))).(nf2_gen_abbr c0 u t0 -H8 (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Bind Abbr) -u t0) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 -w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) -(ex nat (\lambda (n: nat).(eq T (THead (Bind Abbr) u t0) (TSort n)))) (ex3_2 -TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Bind Abbr) u -t0) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef -i)))))))))) (\lambda (H6: (not (eq B Abst Abst))).(\lambda (_: (arity g -(CHead c0 (Bind Abst) u) t0 a2)).(\lambda (_: (nf2 c0 (THead (Bind Abst) u -t0))).(let H9 \def (match (H6 (refl_equal B Abst)) in False return (\lambda -(_: False).(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead -(Bind Abst) u t0) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: -T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) -w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind Abst) u t0) (TSort -n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead -(Bind Abst) u t0) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: -TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: -nat).(nf2 c0 (TLRef i))))))) with []) in H9)))) (\lambda (_: (not (eq B Void -Abst))).(\lambda (H7: (arity g (CHead c0 (Bind Void) u) t0 a2)).(\lambda (H8: -(nf2 c0 (THead (Bind Void) u t0))).(let H9 \def (arity_gen_cvoid g (CHead c0 -(Bind Void) u) t0 a2 H7 c0 u O (getl_refl Void c0 u)) in (ex_ind T (\lambda -(v: T).(eq T t0 (lift (S O) O v))) (or3 (ex3_2 T T (\lambda (w: T).(\lambda -(u0: T).(eq T (THead (Bind Void) u t0) (THead (Bind Abst) w u0)))) (\lambda -(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 -(CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind -Void) u t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: -nat).(eq T (THead (Bind Void) u t0) (THeads (Flat Appl) ws (TLRef i))))) -(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))) (\lambda (x: T).(\lambda -(H10: (eq T t0 (lift (S O) O x))).(let H11 \def (eq_ind T t0 (\lambda (t1: -T).(nf2 c0 (THead (Bind Void) u t1))) H8 (lift (S O) O x) H10) in (eq_ind_r T -(lift (S O) O x) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (w: T).(\lambda -(u0: T).(eq T (THead (Bind Void) u t1) (THead (Bind Abst) w u0)))) (\lambda -(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 -(CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind -Void) u t1) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: -nat).(eq T (THead (Bind Void) u t1) (THeads (Flat Appl) ws (TLRef i))))) -(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))) (nf2_gen_void c0 u x H11 -(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Bind Void) u -(lift (S O) O x)) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: -T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) -w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind Void) u (lift (S O) O -x)) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq -T (THead (Bind Void) u (lift (S O) O x)) (THeads (Flat Appl) ws (TLRef i))))) -(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))) t0 H10)))) H9))))) b H0 H3 -H5))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda -(_: (arity g c0 u (asucc g a1))).(\lambda (_: (((nf2 c0 u) \to (or3 (ex3_2 T -T (\lambda (w: T).(\lambda (u0: T).(eq T u (THead (Bind Abst) w u0)))) -(\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: -T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T u -(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T u -(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef -i))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c0 -(Bind Abst) u) t0 a2)).(\lambda (_: (((nf2 (CHead c0 (Bind Abst) u) t0) \to -(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) -w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 (CHead c0 (Bind Abst) u) w))) -(\lambda (w: T).(\lambda (u0: T).(nf2 (CHead (CHead c0 (Bind Abst) u) (Bind -Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort n)))) (ex3_2 TList -nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws -(TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 (CHead c0 (Bind -Abst) u) ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 (CHead c0 (Bind -Abst) u) (TLRef i))))))))).(\lambda (H4: (nf2 c0 (THead (Bind Abst) u -t0))).(let H5 \def (nf2_gen_abst c0 u t0 H4) in (land_ind (nf2 c0 u) (nf2 -(CHead c0 (Bind Abst) u) t0) (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: -T).(eq T (THead (Bind Abst) u t0) (THead (Bind Abst) w u0)))) (\lambda (w: -T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead -c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind Abst) u -t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq -T (THead (Bind Abst) u t0) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: -TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: -nat).(nf2 c0 (TLRef i)))))) (\lambda (H6: (nf2 c0 u)).(\lambda (H7: (nf2 -(CHead c0 (Bind Abst) u) t0)).(or3_intro0 (ex3_2 T T (\lambda (w: T).(\lambda -(u0: T).(eq T (THead (Bind Abst) u t0) (THead (Bind Abst) w u0)))) (\lambda -(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 -(CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind -Abst) u t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: -nat).(eq T (THead (Bind Abst) u t0) (THeads (Flat Appl) ws (TLRef i))))) -(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))) (ex3_2_intro T T (\lambda (w: -T).(\lambda (u0: T).(eq T (THead (Bind Abst) u t0) (THead (Bind Abst) w -u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda -(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0))) u t0 (refl_equal T (THead (Bind -Abst) u t0)) H6 H7)))) H5)))))))))))) (\lambda (c0: C).(\lambda (u: -T).(\lambda (a1: A).(\lambda (_: (arity g c0 u a1)).(\lambda (_: (((nf2 c0 u) -\to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u (THead (Bind -Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: -T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: -nat).(eq T u (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: -nat).(eq T u (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: -TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: -nat).(nf2 c0 (TLRef i))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda -(H2: (arity g c0 t0 (AHead a1 a2))).(\lambda (H3: (((nf2 c0 t0) \to (or3 -(ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w -u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda -(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T -t0 (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T -t0 (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef -i))))))))).(\lambda (H4: (nf2 c0 (THead (Flat Appl) u t0))).(let H5 \def -(nf2_gen_flat Appl c0 u t0 H4) in (land_ind (nf2 c0 u) (nf2 c0 t0) (or3 -(ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t0) -(THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) -(\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat -(\lambda (n: nat).(eq T (THead (Flat Appl) u t0) (TSort n)))) (ex3_2 TList -nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u t0) -(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef -i)))))) (\lambda (H6: (nf2 c0 u)).(\lambda (H7: (nf2 c0 t0)).(let H_x \def -(H3 H7) in (let H8 \def H_x in (or3_ind (ex3_2 T T (\lambda (w: T).(\lambda -(u0: T).(eq T t0 (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: -T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) -w) u0)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort n)))) (ex3_2 TList nat -(\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef -i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))) (or3 (ex3_2 T T (\lambda (w: -T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t0) (THead (Bind Abst) w -u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda -(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T -(THead (Flat Appl) u t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws: -TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u t0) (THeads (Flat Appl) -ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) -(\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))) (\lambda (H9: -(ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w -u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda -(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0))))).(ex3_2_ind T T (\lambda (w: -T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w u0)))) (\lambda (w: -T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead -c0 (Bind Abst) w) u0))) (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq -T (THead (Flat Appl) u t0) (THead (Bind Abst) w u0)))) (\lambda (w: -T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead -c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u -t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq -T (THead (Flat Appl) u t0) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: -TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: -nat).(nf2 c0 (TLRef i)))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H10: -(eq T t0 (THead (Bind Abst) x0 x1))).(\lambda (_: (nf2 c0 x0)).(\lambda (_: -(nf2 (CHead c0 (Bind Abst) x0) x1)).(let H13 \def (eq_ind T t0 (\lambda (t1: -T).(nf2 c0 (THead (Flat Appl) u t1))) H4 (THead (Bind Abst) x0 x1) H10) in -(let H14 \def (eq_ind T t0 (\lambda (t1: T).(arity g c0 t1 (AHead a1 a2))) H2 -(THead (Bind Abst) x0 x1) H10) in (eq_ind_r T (THead (Bind Abst) x0 x1) -(\lambda (t1: T).(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T -(THead (Flat Appl) u t1) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda -(_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind -Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u t1) -(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T -(THead (Flat Appl) u t1) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: -TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: -nat).(nf2 c0 (TLRef i))))))) (nf2_gen_beta c0 u x0 x1 H13 (or3 (ex3_2 T T -(\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl) u (THead (Bind -Abst) x0 x1)) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: -T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) -w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u (THead (Bind -Abst) x0 x1)) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: -nat).(eq T (THead (Flat Appl) u (THead (Bind Abst) x0 x1)) (THeads (Flat -Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) -(\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))) t0 H10)))))))) -H9)) (\lambda (H9: (ex nat (\lambda (n: nat).(eq T t0 (TSort n))))).(ex_ind -nat (\lambda (n: nat).(eq T t0 (TSort n))) (or3 (ex3_2 T T (\lambda (w: -T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t0) (THead (Bind Abst) w -u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda -(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T -(THead (Flat Appl) u t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws: -TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u t0) (THeads (Flat Appl) -ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) -(\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))) (\lambda (x: -nat).(\lambda (H10: (eq T t0 (TSort x))).(let H11 \def (eq_ind T t0 (\lambda -(t1: T).(nf2 c0 (THead (Flat Appl) u t1))) H4 (TSort x) H10) in (let H12 \def -(eq_ind T t0 (\lambda (t1: T).(arity g c0 t1 (AHead a1 a2))) H2 (TSort x) -H10) in (eq_ind_r T (TSort x) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (w: -T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t1) (THead (Bind Abst) w -u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda -(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T -(THead (Flat Appl) u t1) (TSort n)))) (ex3_2 TList nat (\lambda (ws: -TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u t1) (THeads (Flat Appl) -ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) -(\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))) (let H_x0 \def -(leq_gen_head1 g a1 a2 (ASort O x) (arity_gen_sort g c0 x (AHead a1 a2) H12)) -in (let H13 \def H_x0 in (ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq -g a1 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: -A).(\lambda (a4: A).(eq A (ASort O x) (AHead a3 a4)))) (or3 (ex3_2 T T -(\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl) u (TSort x)) (THead -(Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda -(w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda -(n: nat).(eq T (THead (Flat Appl) u (TSort x)) (TSort n)))) (ex3_2 TList nat -(\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u (TSort x)) -(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef -i)))))) (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g a1 -x0)).(\lambda (_: (leq g a2 x1)).(\lambda (H16: (eq A (ASort O x) (AHead x0 -x1))).(let H17 \def (eq_ind A (ASort O x) (\lambda (ee: A).(match ee in A -return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) -\Rightarrow False])) I (AHead x0 x1) H16) in (False_ind (or3 (ex3_2 T T -(\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl) u (TSort x)) (THead -(Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda -(w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda -(n: nat).(eq T (THead (Flat Appl) u (TSort x)) (TSort n)))) (ex3_2 TList nat -(\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u (TSort x)) -(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef -i)))))) H17))))))) H13))) t0 H10))))) H9)) (\lambda (H9: (ex3_2 TList nat -(\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef -i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))).(ex3_2_ind TList nat (\lambda -(ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i))))) -(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))) (or3 (ex3_2 T T (\lambda (w: -T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t0) (THead (Bind Abst) w -u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda -(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T -(THead (Flat Appl) u t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws: -TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u t0) (THeads (Flat Appl) -ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) -(\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))) (\lambda (x0: -TList).(\lambda (x1: nat).(\lambda (H10: (eq T t0 (THeads (Flat Appl) x0 -(TLRef x1)))).(\lambda (H11: (nfs2 c0 x0)).(\lambda (H12: (nf2 c0 (TLRef -x1))).(let H13 \def (eq_ind T t0 (\lambda (t1: T).(nf2 c0 (THead (Flat Appl) -u t1))) H4 (THeads (Flat Appl) x0 (TLRef x1)) H10) in (let H14 \def (eq_ind T -t0 (\lambda (t1: T).(arity g c0 t1 (AHead a1 a2))) H2 (THeads (Flat Appl) x0 -(TLRef x1)) H10) in (eq_ind_r T (THeads (Flat Appl) x0 (TLRef x1)) (\lambda -(t1: T).(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat -Appl) u t1) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 -c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) -(ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u t1) (TSort n)))) (ex3_2 -TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u -t1) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef -i))))))) (or3_intro2 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead -(Flat Appl) u (THeads (Flat Appl) x0 (TLRef x1))) (THead (Bind Abst) w u0)))) -(\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: -T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T -(THead (Flat Appl) u (THeads (Flat Appl) x0 (TLRef x1))) (TSort n)))) (ex3_2 -TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u -(THeads (Flat Appl) x0 (TLRef x1))) (THeads (Flat Appl) ws (TLRef i))))) -(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))) (ex3_2_intro TList nat -(\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u (THeads -(Flat Appl) x0 (TLRef x1))) (THeads (Flat Appl) ws (TLRef i))))) (\lambda -(ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: -nat).(nf2 c0 (TLRef i)))) (TCons u x0) x1 (refl_equal T (THead (Flat Appl) u -(THeads (Flat Appl) x0 (TLRef x1)))) (conj (nf2 c0 u) (nfs2 c0 x0) H6 H11) -H12)) t0 H10)))))))) H9)) H8))))) H5)))))))))))) (\lambda (c0: C).(\lambda -(u: T).(\lambda (a0: A).(\lambda (_: (arity g c0 u (asucc g a0))).(\lambda -(_: (((nf2 c0 u) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u -(THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) -(\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat -(\lambda (n: nat).(eq T u (TSort n)))) (ex3_2 TList nat (\lambda (ws: -TList).(\lambda (i: nat).(eq T u (THeads (Flat Appl) ws (TLRef i))))) -(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))))).(\lambda (t0: T).(\lambda -(_: (arity g c0 t0 a0)).(\lambda (_: (((nf2 c0 t0) \to (or3 (ex3_2 T T -(\lambda (w: T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w u0)))) -(\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: -T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T t0 -(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 -(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef -i))))))))).(\lambda (H4: (nf2 c0 (THead (Flat Cast) u t0))).(nf2_gen_cast c0 -u t0 H4 (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat -Cast) u t0) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 -c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) -(ex nat (\lambda (n: nat).(eq T (THead (Flat Cast) u t0) (TSort n)))) (ex3_2 -TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Cast) u -t0) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef -i)))))))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda -(_: (arity g c0 t0 a1)).(\lambda (H1: (((nf2 c0 t0) \to (or3 (ex3_2 T T -(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda -(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 -(CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort -n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 -(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef -i))))))))).(\lambda (a2: A).(\lambda (_: (leq g a1 a2)).(\lambda (H3: (nf2 c0 -t0)).(let H_x \def (H1 H3) in (let H4 \def H_x in (or3_ind (ex3_2 T T -(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda -(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 -(CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort -n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 -(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef -i))))) (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind -Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: -T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: -nat).(eq T t0 (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: -nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: -TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: -nat).(nf2 c0 (TLRef i)))))) (\lambda (H5: (ex3_2 T T (\lambda (w: T).(\lambda -(u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: -T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) -u))))).(ex3_2_ind T T (\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind -Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: -T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u))) (or3 (ex3_2 T T -(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda -(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 -(CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort -n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 -(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef -i)))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H6: (eq T t0 (THead (Bind -Abst) x0 x1))).(\lambda (H7: (nf2 c0 x0)).(\lambda (H8: (nf2 (CHead c0 (Bind -Abst) x0) x1)).(eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t1: T).(or3 -(ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t1 (THead (Bind Abst) w -u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda -(u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t1 -(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t1 -(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef -i))))))) (or3_intro0 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T (THead -(Bind Abst) x0 x1) (THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: -T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) -u)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind Abst) x0 x1) (TSort n)))) -(ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Bind -Abst) x0 x1) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: -TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: -nat).(nf2 c0 (TLRef i))))) (ex3_2_intro T T (\lambda (w: T).(\lambda (u: -T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) w u)))) (\lambda (w: -T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead -c0 (Bind Abst) w) u))) x0 x1 (refl_equal T (THead (Bind Abst) x0 x1)) H7 H8)) -t0 H6)))))) H5)) (\lambda (H5: (ex nat (\lambda (n: nat).(eq T t0 (TSort -n))))).(ex_ind nat (\lambda (n: nat).(eq T t0 (TSort n))) (or3 (ex3_2 T T -(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda -(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 -(CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort -n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 -(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef -i)))))) (\lambda (x: nat).(\lambda (H6: (eq T t0 (TSort x))).(eq_ind_r T -(TSort x) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u: -T).(eq T t1 (THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 -c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) -(ex nat (\lambda (n: nat).(eq T t1 (TSort n)))) (ex3_2 TList nat (\lambda -(ws: TList).(\lambda (i: nat).(eq T t1 (THeads (Flat Appl) ws (TLRef i))))) -(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))) (or3_intro1 (ex3_2 T T -(\lambda (w: T).(\lambda (u: T).(eq T (TSort x) (THead (Bind Abst) w u)))) -(\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: -T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T (TSort -x) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T -(TSort x) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda -(_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef -i))))) (ex_intro nat (\lambda (n: nat).(eq T (TSort x) (TSort n))) x -(refl_equal T (TSort x)))) t0 H6))) H5)) (\lambda (H5: (ex3_2 TList nat -(\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef -i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))).(ex3_2_ind TList nat (\lambda -(ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i))))) -(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))) (or3 (ex3_2 T T (\lambda (w: -T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda (w: -T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead -c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort n)))) -(ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads -(Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 -ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))) (\lambda -(x0: TList).(\lambda (x1: nat).(\lambda (H6: (eq T t0 (THeads (Flat Appl) x0 -(TLRef x1)))).(\lambda (H7: (nfs2 c0 x0)).(\lambda (H8: (nf2 c0 (TLRef -x1))).(eq_ind_r T (THeads (Flat Appl) x0 (TLRef x1)) (\lambda (t1: T).(or3 -(ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t1 (THead (Bind Abst) w -u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda -(u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t1 -(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t1 -(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef -i))))))) (or3_intro2 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T (THeads -(Flat Appl) x0 (TLRef x1)) (THead (Bind Abst) w u)))) (\lambda (w: -T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead -c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T (THeads (Flat Appl) -x0 (TLRef x1)) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda -(i: nat).(eq T (THeads (Flat Appl) x0 (TLRef x1)) (THeads (Flat Appl) ws -(TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda -(_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))) (ex3_2_intro TList nat -(\lambda (ws: TList).(\lambda (i: nat).(eq T (THeads (Flat Appl) x0 (TLRef -x1)) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef -i)))) x0 x1 (refl_equal T (THeads (Flat Appl) x0 (TLRef x1))) H7 H8)) t0 -H6)))))) H5)) H4))))))))))) c t a H))))). -(* COMMENTS -Initial nodes: 9193 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/dec.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/dec.ma deleted file mode 100644 index 33b652baf..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/dec.ma +++ /dev/null @@ -1,200 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/nf2/defs.ma". - -include "Basic-1/pr2/clen.ma". - -include "Basic-1/pr2/fwd.ma". - -include "Basic-1/pr0/dec.ma". - -include "Basic-1/C/props.ma". - -theorem nf2_dec: - \forall (c: C).(\forall (t1: T).(or (nf2 c t1) (ex2 T (\lambda (t2: T).((eq -T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c t1 t2))))) -\def - \lambda (c: C).(c_tail_ind (\lambda (c0: C).(\forall (t1: T).(or (\forall -(t2: T).((pr2 c0 t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 -t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c0 t1 t2)))))) (\lambda -(n: nat).(\lambda (t1: T).(let H_x \def (nf0_dec t1) in (let H \def H_x in -(or_ind (\forall (t2: T).((pr0 t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: -T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t1 t2))) -(or (\forall (t2: T).((pr2 (CSort n) t1 t2) \to (eq T t1 t2))) (ex2 T -(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: -T).(pr2 (CSort n) t1 t2)))) (\lambda (H0: ((\forall (t2: T).((pr0 t1 t2) \to -(eq T t1 t2))))).(or_introl (\forall (t2: T).((pr2 (CSort n) t1 t2) \to (eq T -t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) -(\lambda (t2: T).(pr2 (CSort n) t1 t2))) (\lambda (t2: T).(\lambda (H1: (pr2 -(CSort n) t1 t2)).(let H_y \def (pr2_gen_csort t1 t2 n H1) in (H0 t2 -H_y)))))) (\lambda (H0: (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall -(P: Prop).P))) (\lambda (t2: T).(pr0 t1 t2)))).(ex2_ind T (\lambda (t2: -T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t1 t2)) -(or (\forall (t2: T).((pr2 (CSort n) t1 t2) \to (eq T t1 t2))) (ex2 T -(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: -T).(pr2 (CSort n) t1 t2)))) (\lambda (x: T).(\lambda (H1: (((eq T t1 x) \to -(\forall (P: Prop).P)))).(\lambda (H2: (pr0 t1 x)).(or_intror (\forall (t2: -T).((pr2 (CSort n) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T -t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CSort n) t1 t2))) -(ex_intro2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) -(\lambda (t2: T).(pr2 (CSort n) t1 t2)) x H1 (pr2_free (CSort n) t1 x -H2)))))) H0)) H))))) (\lambda (c0: C).(\lambda (H: ((\forall (t1: T).(or -(\forall (t2: T).((pr2 c0 t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: -T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c0 t1 -t2))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (t1: T).(let H_x \def (H -t1) in (let H0 \def H_x in (or_ind (\forall (t2: T).((pr2 c0 t1 t2) \to (eq T -t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) -(\lambda (t2: T).(pr2 c0 t1 t2))) (or (\forall (t2: T).((pr2 (CTail k t c0) -t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall -(P: Prop).P))) (\lambda (t2: T).(pr2 (CTail k t c0) t1 t2)))) (\lambda (H1: -((\forall (t2: T).((pr2 c0 t1 t2) \to (eq T t1 t2))))).(K_ind (\lambda (k0: -K).(or (\forall (t2: T).((pr2 (CTail k0 t c0) t1 t2) \to (eq T t1 t2))) (ex2 -T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: -T).(pr2 (CTail k0 t c0) t1 t2))))) (\lambda (b: B).(B_ind (\lambda (b0: -B).(or (\forall (t2: T).((pr2 (CTail (Bind b0) t c0) t1 t2) \to (eq T t1 -t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) -(\lambda (t2: T).(pr2 (CTail (Bind b0) t c0) t1 t2))))) (let H_x0 \def -(dnf_dec t t1 (clen c0)) in (let H2 \def H_x0 in (ex_ind T (\lambda (v: -T).(or (subst0 (clen c0) t t1 (lift (S O) (clen c0) v)) (eq T t1 (lift (S O) -(clen c0) v)))) (or (\forall (t2: T).((pr2 (CTail (Bind Abbr) t c0) t1 t2) -\to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) t1 t2)))) (\lambda -(x: T).(\lambda (H3: (or (subst0 (clen c0) t t1 (lift (S O) (clen c0) x)) (eq -T t1 (lift (S O) (clen c0) x)))).(or_ind (subst0 (clen c0) t t1 (lift (S O) -(clen c0) x)) (eq T t1 (lift (S O) (clen c0) x)) (or (\forall (t2: T).((pr2 -(CTail (Bind Abbr) t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: -T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail -(Bind Abbr) t c0) t1 t2)))) (\lambda (H4: (subst0 (clen c0) t t1 (lift (S O) -(clen c0) x))).(let H_x1 \def (getl_ctail_clen Abbr t c0) in (let H5 \def -H_x1 in (ex_ind nat (\lambda (n: nat).(getl (clen c0) (CTail (Bind Abbr) t -c0) (CHead (CSort n) (Bind Abbr) t))) (or (\forall (t2: T).((pr2 (CTail (Bind -Abbr) t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) -\to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) t1 -t2)))) (\lambda (x0: nat).(\lambda (H6: (getl (clen c0) (CTail (Bind Abbr) t -c0) (CHead (CSort x0) (Bind Abbr) t))).(or_intror (\forall (t2: T).((pr2 -(CTail (Bind Abbr) t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: -T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail -(Bind Abbr) t c0) t1 t2))) (ex_intro2 T (\lambda (t2: T).((eq T t1 t2) \to -(\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) t1 -t2)) (lift (S O) (clen c0) x) (\lambda (H7: (eq T t1 (lift (S O) (clen c0) -x))).(\lambda (P: Prop).(let H8 \def (eq_ind T t1 (\lambda (t0: T).(subst0 -(clen c0) t t0 (lift (S O) (clen c0) x))) H4 (lift (S O) (clen c0) x) H7) in -(subst0_gen_lift_false x t (lift (S O) (clen c0) x) (S O) (clen c0) (clen c0) -(le_n (clen c0)) (eq_ind_r nat (plus (S O) (clen c0)) (\lambda (n: nat).(lt -(clen c0) n)) (le_n (plus (S O) (clen c0))) (plus (clen c0) (S O)) (plus_sym -(clen c0) (S O))) H8 P)))) (pr2_delta (CTail (Bind Abbr) t c0) (CSort x0) t -(clen c0) H6 t1 t1 (pr0_refl t1) (lift (S O) (clen c0) x) H4))))) H5)))) -(\lambda (H4: (eq T t1 (lift (S O) (clen c0) x))).(let H5 \def (eq_ind T t1 -(\lambda (t0: T).(\forall (t2: T).((pr2 c0 t0 t2) \to (eq T t0 t2)))) H1 -(lift (S O) (clen c0) x) H4) in (eq_ind_r T (lift (S O) (clen c0) x) (\lambda -(t0: T).(or (\forall (t2: T).((pr2 (CTail (Bind Abbr) t c0) t0 t2) \to (eq T -t0 t2))) (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: Prop).P))) -(\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) t0 t2))))) (or_introl (\forall -(t2: T).((pr2 (CTail (Bind Abbr) t c0) (lift (S O) (clen c0) x) t2) \to (eq T -(lift (S O) (clen c0) x) t2))) (ex2 T (\lambda (t2: T).((eq T (lift (S O) -(clen c0) x) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail -(Bind Abbr) t c0) (lift (S O) (clen c0) x) t2))) (\lambda (t2: T).(\lambda -(H6: (pr2 (CTail (Bind Abbr) t c0) (lift (S O) (clen c0) x) t2)).(let H_x1 -\def (pr2_gen_ctail (Bind Abbr) c0 t (lift (S O) (clen c0) x) t2 H6) in (let -H7 \def H_x1 in (or_ind (pr2 c0 (lift (S O) (clen c0) x) t2) (ex3 T (\lambda -(_: T).(eq K (Bind Abbr) (Bind Abbr))) (\lambda (t0: T).(pr0 (lift (S O) -(clen c0) x) t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2))) (eq T (lift -(S O) (clen c0) x) t2) (\lambda (H8: (pr2 c0 (lift (S O) (clen c0) x) -t2)).(H5 t2 H8)) (\lambda (H8: (ex3 T (\lambda (_: T).(eq K (Bind Abbr) (Bind -Abbr))) (\lambda (t0: T).(pr0 (lift (S O) (clen c0) x) t0)) (\lambda (t0: -T).(subst0 (clen c0) t t0 t2)))).(ex3_ind T (\lambda (_: T).(eq K (Bind Abbr) -(Bind Abbr))) (\lambda (t0: T).(pr0 (lift (S O) (clen c0) x) t0)) (\lambda -(t0: T).(subst0 (clen c0) t t0 t2)) (eq T (lift (S O) (clen c0) x) t2) -(\lambda (x0: T).(\lambda (_: (eq K (Bind Abbr) (Bind Abbr))).(\lambda (H10: -(pr0 (lift (S O) (clen c0) x) x0)).(\lambda (H11: (subst0 (clen c0) t x0 -t2)).(ex2_ind T (\lambda (t3: T).(eq T x0 (lift (S O) (clen c0) t3))) -(\lambda (t3: T).(pr0 x t3)) (eq T (lift (S O) (clen c0) x) t2) (\lambda (x1: -T).(\lambda (H12: (eq T x0 (lift (S O) (clen c0) x1))).(\lambda (_: (pr0 x -x1)).(let H14 \def (eq_ind T x0 (\lambda (t0: T).(subst0 (clen c0) t t0 t2)) -H11 (lift (S O) (clen c0) x1) H12) in (subst0_gen_lift_false x1 t t2 (S O) -(clen c0) (clen c0) (le_n (clen c0)) (eq_ind_r nat (plus (S O) (clen c0)) -(\lambda (n: nat).(lt (clen c0) n)) (le_n (plus (S O) (clen c0))) (plus (clen -c0) (S O)) (plus_sym (clen c0) (S O))) H14 (eq T (lift (S O) (clen c0) x) -t2)))))) (pr0_gen_lift x x0 (S O) (clen c0) H10)))))) H8)) H7)))))) t1 H4))) -H3))) H2))) (or_introl (\forall (t2: T).((pr2 (CTail (Bind Abst) t c0) t1 t2) -\to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abst) t c0) t1 t2))) (\lambda -(t2: T).(\lambda (H2: (pr2 (CTail (Bind Abst) t c0) t1 t2)).(let H_x0 \def -(pr2_gen_ctail (Bind Abst) c0 t t1 t2 H2) in (let H3 \def H_x0 in (or_ind -(pr2 c0 t1 t2) (ex3 T (\lambda (_: T).(eq K (Bind Abst) (Bind Abbr))) -(\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2))) -(eq T t1 t2) (\lambda (H4: (pr2 c0 t1 t2)).(H1 t2 H4)) (\lambda (H4: (ex3 T -(\lambda (_: T).(eq K (Bind Abst) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) -(\lambda (t0: T).(subst0 (clen c0) t t0 t2)))).(ex3_ind T (\lambda (_: T).(eq -K (Bind Abst) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: -T).(subst0 (clen c0) t t0 t2)) (eq T t1 t2) (\lambda (x0: T).(\lambda (H5: -(eq K (Bind Abst) (Bind Abbr))).(\lambda (_: (pr0 t1 x0)).(\lambda (_: -(subst0 (clen c0) t x0 t2)).(let H8 \def (eq_ind K (Bind Abst) (\lambda (ee: -K).(match ee in K return (\lambda (_: K).Prop) with [(Bind b0) \Rightarrow -(match b0 in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | -Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) \Rightarrow -False])) I (Bind Abbr) H5) in (False_ind (eq T t1 t2) H8)))))) H4)) H3)))))) -(or_introl (\forall (t2: T).((pr2 (CTail (Bind Void) t c0) t1 t2) \to (eq T -t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) -(\lambda (t2: T).(pr2 (CTail (Bind Void) t c0) t1 t2))) (\lambda (t2: -T).(\lambda (H2: (pr2 (CTail (Bind Void) t c0) t1 t2)).(let H_x0 \def -(pr2_gen_ctail (Bind Void) c0 t t1 t2 H2) in (let H3 \def H_x0 in (or_ind -(pr2 c0 t1 t2) (ex3 T (\lambda (_: T).(eq K (Bind Void) (Bind Abbr))) -(\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2))) -(eq T t1 t2) (\lambda (H4: (pr2 c0 t1 t2)).(H1 t2 H4)) (\lambda (H4: (ex3 T -(\lambda (_: T).(eq K (Bind Void) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) -(\lambda (t0: T).(subst0 (clen c0) t t0 t2)))).(ex3_ind T (\lambda (_: T).(eq -K (Bind Void) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: -T).(subst0 (clen c0) t t0 t2)) (eq T t1 t2) (\lambda (x0: T).(\lambda (H5: -(eq K (Bind Void) (Bind Abbr))).(\lambda (_: (pr0 t1 x0)).(\lambda (_: -(subst0 (clen c0) t x0 t2)).(let H8 \def (eq_ind K (Bind Void) (\lambda (ee: -K).(match ee in K return (\lambda (_: K).Prop) with [(Bind b0) \Rightarrow -(match b0 in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | -Abst \Rightarrow False | Void \Rightarrow True]) | (Flat _) \Rightarrow -False])) I (Bind Abbr) H5) in (False_ind (eq T t1 t2) H8)))))) H4)) H3)))))) -b)) (\lambda (f: F).(or_introl (\forall (t2: T).((pr2 (CTail (Flat f) t c0) -t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall -(P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Flat f) t c0) t1 t2))) (\lambda -(t2: T).(\lambda (H2: (pr2 (CTail (Flat f) t c0) t1 t2)).(let H_x0 \def -(pr2_gen_ctail (Flat f) c0 t t1 t2 H2) in (let H3 \def H_x0 in (or_ind (pr2 -c0 t1 t2) (ex3 T (\lambda (_: T).(eq K (Flat f) (Bind Abbr))) (\lambda (t0: -T).(pr0 t1 t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2))) (eq T t1 t2) -(\lambda (H4: (pr2 c0 t1 t2)).(H1 t2 H4)) (\lambda (H4: (ex3 T (\lambda (_: -T).(eq K (Flat f) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: -T).(subst0 (clen c0) t t0 t2)))).(ex3_ind T (\lambda (_: T).(eq K (Flat f) -(Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(subst0 (clen -c0) t t0 t2)) (eq T t1 t2) (\lambda (x0: T).(\lambda (H5: (eq K (Flat f) -(Bind Abbr))).(\lambda (_: (pr0 t1 x0)).(\lambda (_: (subst0 (clen c0) t x0 -t2)).(let H8 \def (eq_ind K (Flat f) (\lambda (ee: K).(match ee in K return -(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])) I (Bind Abbr) H5) in (False_ind (eq T t1 t2) H8)))))) H4)) H3))))))) -k)) (\lambda (H1: (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr2 c0 t1 t2)))).(ex2_ind T (\lambda (t2: -T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c0 t1 t2)) -(or (\forall (t2: T).((pr2 (CTail k t c0) t1 t2) \to (eq T t1 t2))) (ex2 T -(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: -T).(pr2 (CTail k t c0) t1 t2)))) (\lambda (x: T).(\lambda (H2: (((eq T t1 x) -\to (\forall (P: Prop).P)))).(\lambda (H3: (pr2 c0 t1 x)).(or_intror (\forall -(t2: T).((pr2 (CTail k t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: -T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail k t -c0) t1 t2))) (ex_intro2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr2 (CTail k t c0) t1 t2)) x H2 (pr2_ctail c0 t1 -x H3 k t)))))) H1)) H0)))))))) c). -(* COMMENTS -Initial nodes: 3653 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/defs.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/defs.ma deleted file mode 100644 index 98e931c0b..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/defs.ma +++ /dev/null @@ -1,31 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr2/defs.ma". - -definition nf2: - C \to (T \to Prop) -\def - \lambda (c: C).(\lambda (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (eq T t1 -t2)))). - -definition nfs2: - C \to (TList \to Prop) -\def - let rec nfs2 (c: C) (ts: TList) on ts: Prop \def (match ts with [TNil -\Rightarrow True | (TCons t ts0) \Rightarrow (land (nf2 c t) (nfs2 c ts0))]) -in nfs2. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/fwd.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/fwd.ma deleted file mode 100644 index 9138ff2fa..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/fwd.ma +++ /dev/null @@ -1,220 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/nf2/defs.ma". - -include "Basic-1/pr2/clen.ma". - -include "Basic-1/subst0/dec.ma". - -include "Basic-1/T/props.ma". - -theorem nf2_gen_lref: - \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c -(CHead d (Bind Abbr) u)) \to ((nf2 c (TLRef i)) \to (\forall (P: Prop).P)))))) -\def - \lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H0: ((\forall (t2: T).((pr2 -c (TLRef i) t2) \to (eq T (TLRef i) t2))))).(\lambda (P: -Prop).(lift_gen_lref_false (S i) O i (le_O_n i) (le_n (plus O (S i))) u (H0 -(lift (S i) O u) (pr2_delta c d u i H (TLRef i) (TLRef i) (pr0_refl (TLRef -i)) (lift (S i) O u) (subst0_lref u i))) P))))))). -(* COMMENTS -Initial nodes: 129 -END *) - -theorem nf2_gen_abst: - \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Bind Abst) u -t)) \to (land (nf2 c u) (nf2 (CHead c (Bind Abst) u) t))))) -\def - \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: ((\forall (t2: -T).((pr2 c (THead (Bind Abst) u t) t2) \to (eq T (THead (Bind Abst) u t) -t2))))).(conj (\forall (t2: T).((pr2 c u t2) \to (eq T u t2))) (\forall (t2: -T).((pr2 (CHead c (Bind Abst) u) t t2) \to (eq T t t2))) (\lambda (t2: -T).(\lambda (H0: (pr2 c u t2)).(let H1 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | -(TLRef _) \Rightarrow u | (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abst) -u t) (THead (Bind Abst) t2 t) (H (THead (Bind Abst) t2 t) (pr2_head_1 c u t2 -H0 (Bind Abst) t))) in (let H2 \def (eq_ind_r T t2 (\lambda (t0: T).(pr2 c u -t0)) H0 u H1) in (eq_ind T u (\lambda (t0: T).(eq T u t0)) (refl_equal T u) -t2 H1))))) (\lambda (t2: T).(\lambda (H0: (pr2 (CHead c (Bind Abst) u) t -t2)).(let H1 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda -(_: T).T) with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ -_ t0) \Rightarrow t0])) (THead (Bind Abst) u t) (THead (Bind Abst) u t2) (H -(THead (Bind Abst) u t2) (let H_y \def (pr2_gen_cbind Abst c u t t2 H0) in -H_y))) in (let H2 \def (eq_ind_r T t2 (\lambda (t0: T).(pr2 (CHead c (Bind -Abst) u) t t0)) H0 t H1) in (eq_ind T t (\lambda (t0: T).(eq T t t0)) -(refl_equal T t) t2 H1))))))))). -(* COMMENTS -Initial nodes: 353 -END *) - -theorem nf2_gen_cast: - \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Flat Cast) u -t)) \to (\forall (P: Prop).P)))) -\def - \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (nf2 c (THead -(Flat Cast) u t))).(\lambda (P: Prop).(thead_x_y_y (Flat Cast) u t (H t -(pr2_free c (THead (Flat Cast) u t) t (pr0_tau t t (pr0_refl t) u))) P))))). -(* COMMENTS -Initial nodes: 65 -END *) - -theorem nf2_gen_beta: - \forall (c: C).(\forall (u: T).(\forall (v: T).(\forall (t: T).((nf2 c -(THead (Flat Appl) u (THead (Bind Abst) v t))) \to (\forall (P: Prop).P))))) -\def - \lambda (c: C).(\lambda (u: T).(\lambda (v: T).(\lambda (t: T).(\lambda (H: -((\forall (t2: T).((pr2 c (THead (Flat Appl) u (THead (Bind Abst) v t)) t2) -\to (eq T (THead (Flat Appl) u (THead (Bind Abst) v t)) t2))))).(\lambda (P: -Prop).(let H0 \def (eq_ind T (THead (Flat Appl) u (THead (Bind Abst) v t)) -(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow -(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | -(Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u t) (H (THead (Bind -Abbr) u t) (pr2_free c (THead (Flat Appl) u (THead (Bind Abst) v t)) (THead -(Bind Abbr) u t) (pr0_beta v u u (pr0_refl u) t t (pr0_refl t))))) in -(False_ind P H0))))))). -(* COMMENTS -Initial nodes: 183 -END *) - -theorem nf2_gen_flat: - \forall (f: F).(\forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c -(THead (Flat f) u t)) \to (land (nf2 c u) (nf2 c t)))))) -\def - \lambda (f: F).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: -((\forall (t2: T).((pr2 c (THead (Flat f) u t) t2) \to (eq T (THead (Flat f) -u t) t2))))).(conj (\forall (t2: T).((pr2 c u t2) \to (eq T u t2))) (\forall -(t2: T).((pr2 c t t2) \to (eq T t t2))) (\lambda (t2: T).(\lambda (H0: (pr2 c -u t2)).(let H1 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | -(THead _ t0 _) \Rightarrow t0])) (THead (Flat f) u t) (THead (Flat f) t2 t) -(H (THead (Flat f) t2 t) (pr2_head_1 c u t2 H0 (Flat f) t))) in H1))) -(\lambda (t2: T).(\lambda (H0: (pr2 c t t2)).(let H1 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t0) \Rightarrow t0])) -(THead (Flat f) u t) (THead (Flat f) u t2) (H (THead (Flat f) u t2) -(pr2_head_2 c u t t2 (Flat f) (pr2_cflat c t t2 H0 f u)))) in H1)))))))). -(* COMMENTS -Initial nodes: 251 -END *) - -theorem nf2_gen__nf2_gen_aux: - \forall (b: B).(\forall (x: T).(\forall (u: T).(\forall (d: nat).((eq T -(THead (Bind b) u (lift (S O) d x)) x) \to (\forall (P: Prop).P))))) -\def - \lambda (b: B).(\lambda (x: T).(T_ind (\lambda (t: T).(\forall (u: -T).(\forall (d: nat).((eq T (THead (Bind b) u (lift (S O) d t)) t) \to -(\forall (P: Prop).P))))) (\lambda (n: nat).(\lambda (u: T).(\lambda (d: -nat).(\lambda (H: (eq T (THead (Bind b) u (lift (S O) d (TSort n))) (TSort -n))).(\lambda (P: Prop).(let H0 \def (eq_ind T (THead (Bind b) u (lift (S O) -d (TSort n))) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) -with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ -_) \Rightarrow True])) I (TSort n) H) in (False_ind P H0))))))) (\lambda (n: -nat).(\lambda (u: T).(\lambda (d: nat).(\lambda (H: (eq T (THead (Bind b) u -(lift (S O) d (TLRef n))) (TLRef n))).(\lambda (P: Prop).(let H0 \def (eq_ind -T (THead (Bind b) u (lift (S O) d (TLRef n))) (\lambda (ee: T).(match ee in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H) in -(False_ind P H0))))))) (\lambda (k: K).(\lambda (t: T).(\lambda (_: ((\forall -(u: T).(\forall (d: nat).((eq T (THead (Bind b) u (lift (S O) d t)) t) \to -(\forall (P: Prop).P)))))).(\lambda (t0: T).(\lambda (H0: ((\forall (u: -T).(\forall (d: nat).((eq T (THead (Bind b) u (lift (S O) d t0)) t0) \to -(\forall (P: Prop).P)))))).(\lambda (u: T).(\lambda (d: nat).(\lambda (H1: -(eq T (THead (Bind b) u (lift (S O) d (THead k t t0))) (THead k t -t0))).(\lambda (P: Prop).(let H2 \def (f_equal T K (\lambda (e: T).(match e -in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow (Bind b) | (TLRef -_) \Rightarrow (Bind b) | (THead k0 _ _) \Rightarrow k0])) (THead (Bind b) u -(lift (S O) d (THead k t t0))) (THead k t t0) H1) in ((let H3 \def (f_equal T -T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t1 _) \Rightarrow t1])) -(THead (Bind b) u (lift (S O) d (THead k t t0))) (THead k t t0) H1) in ((let -H4 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow (THead k ((let rec lref_map (f: ((nat \to nat))) -(d0: nat) (t1: T) on t1: T \def (match t1 with [(TSort n) \Rightarrow (TSort -n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) with [true \Rightarrow i -| false \Rightarrow (f i)])) | (THead k0 u0 t2) \Rightarrow (THead k0 -(lref_map f d0 u0) (lref_map f (s k0 d0) t2))]) in lref_map) (\lambda (x0: -nat).(plus x0 (S O))) d t) ((let rec lref_map (f: ((nat \to nat))) (d0: nat) -(t1: T) on t1: T \def (match t1 with [(TSort n) \Rightarrow (TSort n) | -(TLRef i) \Rightarrow (TLRef (match (blt i d0) with [true \Rightarrow i | -false \Rightarrow (f i)])) | (THead k0 u0 t2) \Rightarrow (THead k0 (lref_map -f d0 u0) (lref_map f (s k0 d0) t2))]) in lref_map) (\lambda (x0: nat).(plus -x0 (S O))) (s k d) t0)) | (TLRef _) \Rightarrow (THead k ((let rec lref_map -(f: ((nat \to nat))) (d0: nat) (t1: T) on t1: T \def (match t1 with [(TSort -n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) -with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k0 u0 t2) -\Rightarrow (THead k0 (lref_map f d0 u0) (lref_map f (s k0 d0) t2))]) in -lref_map) (\lambda (x0: nat).(plus x0 (S O))) d t) ((let rec lref_map (f: -((nat \to nat))) (d0: nat) (t1: T) on t1: T \def (match t1 with [(TSort n) -\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) with -[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k0 u0 t2) -\Rightarrow (THead k0 (lref_map f d0 u0) (lref_map f (s k0 d0) t2))]) in -lref_map) (\lambda (x0: nat).(plus x0 (S O))) (s k d) t0)) | (THead _ _ t1) -\Rightarrow t1])) (THead (Bind b) u (lift (S O) d (THead k t t0))) (THead k t -t0) H1) in (\lambda (_: (eq T u t)).(\lambda (H6: (eq K (Bind b) k)).(let H7 -\def (eq_ind_r K k (\lambda (k0: K).(eq T (lift (S O) d (THead k0 t t0)) t0)) -H4 (Bind b) H6) in (let H8 \def (eq_ind T (lift (S O) d (THead (Bind b) t -t0)) (\lambda (t1: T).(eq T t1 t0)) H7 (THead (Bind b) (lift (S O) d t) (lift -(S O) (S d) t0)) (lift_bind b t t0 (S O) d)) in (H0 (lift (S O) d t) (S d) H8 -P)))))) H3)) H2))))))))))) x)). -(* COMMENTS -Initial nodes: 935 -END *) - -theorem nf2_gen_abbr: - \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Bind Abbr) u -t)) \to (\forall (P: Prop).P)))) -\def - \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: ((\forall (t2: -T).((pr2 c (THead (Bind Abbr) u t) t2) \to (eq T (THead (Bind Abbr) u t) -t2))))).(\lambda (P: Prop).(let H_x \def (dnf_dec u t O) in (let H0 \def H_x -in (ex_ind T (\lambda (v: T).(or (subst0 O u t (lift (S O) O v)) (eq T t -(lift (S O) O v)))) P (\lambda (x: T).(\lambda (H1: (or (subst0 O u t (lift -(S O) O x)) (eq T t (lift (S O) O x)))).(or_ind (subst0 O u t (lift (S O) O -x)) (eq T t (lift (S O) O x)) P (\lambda (H2: (subst0 O u t (lift (S O) O -x))).(let H3 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda -(_: T).T) with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ -_ t0) \Rightarrow t0])) (THead (Bind Abbr) u t) (THead (Bind Abbr) u (lift (S -O) O x)) (H (THead (Bind Abbr) u (lift (S O) O x)) (pr2_free c (THead (Bind -Abbr) u t) (THead (Bind Abbr) u (lift (S O) O x)) (pr0_delta u u (pr0_refl u) -t t (pr0_refl t) (lift (S O) O x) H2)))) in (let H4 \def (eq_ind T t (\lambda -(t0: T).(subst0 O u t0 (lift (S O) O x))) H2 (lift (S O) O x) H3) in -(subst0_refl u (lift (S O) O x) O H4 P)))) (\lambda (H2: (eq T t (lift (S O) -O x))).(let H3 \def (eq_ind T t (\lambda (t0: T).(\forall (t2: T).((pr2 c -(THead (Bind Abbr) u t0) t2) \to (eq T (THead (Bind Abbr) u t0) t2)))) H -(lift (S O) O x) H2) in (nf2_gen__nf2_gen_aux Abbr x u O (H3 x (pr2_free c -(THead (Bind Abbr) u (lift (S O) O x)) x (pr0_zeta Abbr not_abbr_abst x x -(pr0_refl x) u))) P))) H1))) H0))))))). -(* COMMENTS -Initial nodes: 511 -END *) - -theorem nf2_gen_void: - \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Bind Void) u -(lift (S O) O t))) \to (\forall (P: Prop).P)))) -\def - \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: ((\forall (t2: -T).((pr2 c (THead (Bind Void) u (lift (S O) O t)) t2) \to (eq T (THead (Bind -Void) u (lift (S O) O t)) t2))))).(\lambda (P: Prop).(nf2_gen__nf2_gen_aux -Void t u O (H t (pr2_free c (THead (Bind Void) u (lift (S O) O t)) t -(pr0_zeta Void (sym_not_eq B Abst Void not_abst_void) t t (pr0_refl t) u))) -P))))). -(* COMMENTS -Initial nodes: 121 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/iso.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/iso.ma deleted file mode 100644 index 6a2ce00f8..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/iso.ma +++ /dev/null @@ -1,130 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/nf2/pr3.ma". - -include "Basic-1/pr3/fwd.ma". - -include "Basic-1/iso/props.ma". - -theorem nf2_iso_appls_lref: - \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (vs: -TList).(\forall (u: T).((pr3 c (THeads (Flat Appl) vs (TLRef i)) u) \to (iso -(THeads (Flat Appl) vs (TLRef i)) u)))))) -\def - \lambda (c: C).(\lambda (i: nat).(\lambda (H: (nf2 c (TLRef i))).(\lambda -(vs: TList).(TList_ind (\lambda (t: TList).(\forall (u: T).((pr3 c (THeads -(Flat Appl) t (TLRef i)) u) \to (iso (THeads (Flat Appl) t (TLRef i)) u)))) -(\lambda (u: T).(\lambda (H0: (pr3 c (TLRef i) u)).(let H_y \def -(nf2_pr3_unfold c (TLRef i) u H0 H) in (let H1 \def (eq_ind_r T u (\lambda -(t: T).(pr3 c (TLRef i) t)) H0 (TLRef i) H_y) in (eq_ind T (TLRef i) (\lambda -(t: T).(iso (TLRef i) t)) (iso_refl (TLRef i)) u H_y))))) (\lambda (t: -T).(\lambda (t0: TList).(\lambda (H0: ((\forall (u: T).((pr3 c (THeads (Flat -Appl) t0 (TLRef i)) u) \to (iso (THeads (Flat Appl) t0 (TLRef i)) -u))))).(\lambda (u: T).(\lambda (H1: (pr3 c (THead (Flat Appl) t (THeads -(Flat Appl) t0 (TLRef i))) u)).(let H2 \def (pr3_gen_appl c t (THeads (Flat -Appl) t0 (TLRef i)) u H1) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda -(t2: T).(eq T u (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c t u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) -t0 (TLRef i)) t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) u))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))) (\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat -Appl) t0 (TLRef i)) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u0: -T).(pr3 (CHead c (Bind b) u0) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat -Appl) t0 (TLRef i)) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda -(_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: -T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) -u))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) -y2) z1 z2)))))))) (iso (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef -i))) u) (\lambda (H3: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T u -(THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) -(\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) -t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T u (THead (Flat -Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_: -T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) t2))) (iso -(THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) u) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H4: (eq T u (THead (Flat Appl) x0 -x1))).(\lambda (_: (pr3 c t x0)).(\lambda (_: (pr3 c (THeads (Flat Appl) t0 -(TLRef i)) x1)).(eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda (t1: T).(iso -(THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) t1)) (iso_head t x0 -(THeads (Flat Appl) t0 (TLRef i)) x1 (Flat Appl)) u H4)))))) H3)) (\lambda -(H3: (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t2: T).(pr3 c (THead (Bind Abbr) u2 t2) u))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))) (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat -Appl) t0 (TLRef i)) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u0: -T).(pr3 (CHead c (Bind b) u0) z1 t2))))))))).(ex4_4_ind T T T T (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind -Abbr) u2 t2) u))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c t u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(t2: T).(\forall (b: B).(\forall (u0: T).(pr3 (CHead c (Bind b) u0) z1 -t2))))))) (iso (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) u) -(\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda -(_: (pr3 c (THead (Bind Abbr) x2 x3) u)).(\lambda (_: (pr3 c t x2)).(\lambda -(H6: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind Abst) x0 -x1))).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pr3 (CHead c (Bind b) -u0) x1 x3))))).(let H_y \def (H0 (THead (Bind Abst) x0 x1) H6) in -(iso_flats_lref_bind_false Appl Abst i x0 x1 t0 H_y (iso (THead (Flat Appl) t -(THeads (Flat Appl) t0 (TLRef i))) u))))))))))) H3)) (\lambda (H3: (ex6_6 B T -T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind b) y1 z1)))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda -(u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift -(S O) O u2) z2)) u))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 -(CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat -Appl) t0 (TLRef i)) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda -(_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: -T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) -u))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) -y2) z1 z2))))))) (iso (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) -u) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: -T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (not (eq B x0 -Abst))).(\lambda (H5: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind -x0) x1 x2))).(\lambda (_: (pr3 c (THead (Bind x0) x5 (THead (Flat Appl) (lift -(S O) O x4) x3)) u)).(\lambda (_: (pr3 c t x4)).(\lambda (_: (pr3 c x1 -x5)).(\lambda (_: (pr3 (CHead c (Bind x0) x5) x2 x3)).(let H_y \def (H0 -(THead (Bind x0) x1 x2) H5) in (iso_flats_lref_bind_false Appl x0 i x1 x2 t0 -H_y (iso (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) -u))))))))))))))) H3)) H2))))))) vs)))). -(* COMMENTS -Initial nodes: 1817 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/lift1.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/lift1.ma deleted file mode 100644 index d50790336..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/lift1.ma +++ /dev/null @@ -1,41 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/nf2/props.ma". - -include "Basic-1/drop1/fwd.ma". - -theorem nf2_lift1: - \forall (e: C).(\forall (hds: PList).(\forall (c: C).(\forall (t: T).((drop1 -hds c e) \to ((nf2 e t) \to (nf2 c (lift1 hds t))))))) -\def - \lambda (e: C).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall -(c: C).(\forall (t: T).((drop1 p c e) \to ((nf2 e t) \to (nf2 c (lift1 p -t))))))) (\lambda (c: C).(\lambda (t: T).(\lambda (H: (drop1 PNil c -e)).(\lambda (H0: (nf2 e t)).(let H_y \def (drop1_gen_pnil c e H) in -(eq_ind_r C e (\lambda (c0: C).(nf2 c0 t)) H0 c H_y)))))) (\lambda (n: -nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c: -C).(\forall (t: T).((drop1 p c e) \to ((nf2 e t) \to (nf2 c (lift1 p -t)))))))).(\lambda (c: C).(\lambda (t: T).(\lambda (H0: (drop1 (PCons n n0 p) -c e)).(\lambda (H1: (nf2 e t)).(let H_x \def (drop1_gen_pcons c e p n n0 H0) -in (let H2 \def H_x in (ex2_ind C (\lambda (c2: C).(drop n n0 c c2)) (\lambda -(c2: C).(drop1 p c2 e)) (nf2 c (lift n n0 (lift1 p t))) (\lambda (x: -C).(\lambda (H3: (drop n n0 c x)).(\lambda (H4: (drop1 p x e)).(nf2_lift x -(lift1 p t) (H x t H4 H1) c n n0 H3)))) H2))))))))))) hds)). -(* COMMENTS -Initial nodes: 249 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/pr3.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/pr3.ma deleted file mode 100644 index 3db24223e..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/pr3.ma +++ /dev/null @@ -1,56 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/nf2/defs.ma". - -include "Basic-1/pr3/pr3.ma". - -theorem nf2_pr3_unfold: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to ((nf2 c -t1) \to (eq T t1 t2))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 -t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).((nf2 c t) \to (eq T t -t0)))) (\lambda (t: T).(\lambda (H0: (nf2 c t)).(H0 t (pr2_free c t t -(pr0_refl t))))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c t3 -t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2: (((nf2 c t0) -\to (eq T t0 t4)))).(\lambda (H3: (nf2 c t3)).(let H4 \def H3 in (let H5 \def -(eq_ind T t3 (\lambda (t: T).(nf2 c t)) H3 t0 (H4 t0 H0)) in (let H6 \def -(eq_ind T t3 (\lambda (t: T).(pr2 c t t0)) H0 t0 (H4 t0 H0)) in (eq_ind_r T -t0 (\lambda (t: T).(eq T t t4)) (H2 H5) t3 (H4 t0 H0)))))))))))) t1 t2 H)))). -(* COMMENTS -Initial nodes: 187 -END *) - -theorem nf2_pr3_confluence: - \forall (c: C).(\forall (t1: T).((nf2 c t1) \to (\forall (t2: T).((nf2 c t2) -\to (\forall (t: T).((pr3 c t t1) \to ((pr3 c t t2) \to (eq T t1 t2)))))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (H: (nf2 c t1)).(\lambda (t2: -T).(\lambda (H0: (nf2 c t2)).(\lambda (t: T).(\lambda (H1: (pr3 c t -t1)).(\lambda (H2: (pr3 c t t2)).(ex2_ind T (\lambda (t0: T).(pr3 c t2 t0)) -(\lambda (t0: T).(pr3 c t1 t0)) (eq T t1 t2) (\lambda (x: T).(\lambda (H3: -(pr3 c t2 x)).(\lambda (H4: (pr3 c t1 x)).(let H_y \def (nf2_pr3_unfold c t1 -x H4 H) in (let H5 \def (eq_ind_r T x (\lambda (t0: T).(pr3 c t1 t0)) H4 t1 -H_y) in (let H6 \def (eq_ind_r T x (\lambda (t0: T).(pr3 c t2 t0)) H3 t1 H_y) -in (let H_y0 \def (nf2_pr3_unfold c t2 t1 H6 H0) in (let H7 \def (eq_ind T t2 -(\lambda (t0: T).(pr3 c t0 t1)) H6 t1 H_y0) in (eq_ind_r T t1 (\lambda (t0: -T).(eq T t1 t0)) (refl_equal T t1) t2 H_y0))))))))) (pr3_confluence c t t2 H2 -t1 H1))))))))). -(* COMMENTS -Initial nodes: 215 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/props.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/props.ma deleted file mode 100644 index 2f0f092f2..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/nf2/props.ma +++ /dev/null @@ -1,341 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/nf2/defs.ma". - -include "Basic-1/pr2/fwd.ma". - -theorem nf2_sort: - \forall (c: C).(\forall (n: nat).(nf2 c (TSort n))) -\def - \lambda (c: C).(\lambda (n: nat).(\lambda (t2: T).(\lambda (H: (pr2 c (TSort -n) t2)).(eq_ind_r T (TSort n) (\lambda (t: T).(eq T (TSort n) t)) (refl_equal -T (TSort n)) t2 (pr2_gen_sort c t2 n H))))). -(* COMMENTS -Initial nodes: 55 -END *) - -theorem nf2_csort_lref: - \forall (n: nat).(\forall (i: nat).(nf2 (CSort n) (TLRef i))) -\def - \lambda (n: nat).(\lambda (i: nat).(\lambda (t2: T).(\lambda (H: (pr2 (CSort -n) (TLRef i) t2)).(let H0 \def (pr2_gen_lref (CSort n) t2 i H) in (or_ind (eq -T t2 (TLRef i)) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i (CSort n) -(CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T t2 (lift (S -i) O u))))) (eq T (TLRef i) t2) (\lambda (H1: (eq T t2 (TLRef i))).(eq_ind_r -T (TLRef i) (\lambda (t: T).(eq T (TLRef i) t)) (refl_equal T (TLRef i)) t2 -H1)) (\lambda (H1: (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i (CSort -n) (CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T t2 (lift -(S i) O u)))))).(ex2_2_ind C T (\lambda (d: C).(\lambda (u: T).(getl i (CSort -n) (CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T t2 (lift -(S i) O u)))) (eq T (TLRef i) t2) (\lambda (x0: C).(\lambda (x1: T).(\lambda -(H2: (getl i (CSort n) (CHead x0 (Bind Abbr) x1))).(\lambda (H3: (eq T t2 -(lift (S i) O x1))).(eq_ind_r T (lift (S i) O x1) (\lambda (t: T).(eq T -(TLRef i) t)) (getl_gen_sort n i (CHead x0 (Bind Abbr) x1) H2 (eq T (TLRef i) -(lift (S i) O x1))) t2 H3))))) H1)) H0))))). -(* COMMENTS -Initial nodes: 355 -END *) - -theorem nf2_abst: - \forall (c: C).(\forall (u: T).((nf2 c u) \to (\forall (b: B).(\forall (v: -T).(\forall (t: T).((nf2 (CHead c (Bind b) v) t) \to (nf2 c (THead (Bind -Abst) u t)))))))) -\def - \lambda (c: C).(\lambda (u: T).(\lambda (H: ((\forall (t2: T).((pr2 c u t2) -\to (eq T u t2))))).(\lambda (b: B).(\lambda (v: T).(\lambda (t: T).(\lambda -(H0: ((\forall (t2: T).((pr2 (CHead c (Bind b) v) t t2) \to (eq T t -t2))))).(\lambda (t2: T).(\lambda (H1: (pr2 c (THead (Bind Abst) u t) -t2)).(let H2 \def (pr2_gen_abst c u t t2 H1) in (ex3_2_ind T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c u u2))) (\lambda (_: T).(\lambda (t3: T).(\forall -(b0: B).(\forall (u0: T).(pr2 (CHead c (Bind b0) u0) t t3))))) (eq T (THead -(Bind Abst) u t) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H3: (eq T t2 -(THead (Bind Abst) x0 x1))).(\lambda (H4: (pr2 c u x0)).(\lambda (H5: -((\forall (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind b0) u0) t -x1))))).(eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t0: T).(eq T (THead -(Bind Abst) u t) t0)) (f_equal3 K T T T THead (Bind Abst) (Bind Abst) u x0 t -x1 (refl_equal K (Bind Abst)) (H x0 H4) (H0 x1 (H5 b v))) t2 H3)))))) -H2)))))))))). -(* COMMENTS -Initial nodes: 299 -END *) - -theorem nf2_abst_shift: - \forall (c: C).(\forall (u: T).((nf2 c u) \to (\forall (t: T).((nf2 (CHead c -(Bind Abst) u) t) \to (nf2 c (THead (Bind Abst) u t)))))) -\def - \lambda (c: C).(\lambda (u: T).(\lambda (H: ((\forall (t2: T).((pr2 c u t2) -\to (eq T u t2))))).(\lambda (t: T).(\lambda (H0: ((\forall (t2: T).((pr2 -(CHead c (Bind Abst) u) t t2) \to (eq T t t2))))).(\lambda (t2: T).(\lambda -(H1: (pr2 c (THead (Bind Abst) u t) t2)).(let H2 \def (pr2_gen_abst c u t t2 -H1) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u u2))) (\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) -u0) t t3))))) (eq T (THead (Bind Abst) u t) t2) (\lambda (x0: T).(\lambda -(x1: T).(\lambda (H3: (eq T t2 (THead (Bind Abst) x0 x1))).(\lambda (H4: (pr2 -c u x0)).(\lambda (H5: ((\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind -b) u0) t x1))))).(eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t0: T).(eq T -(THead (Bind Abst) u t) t0)) (f_equal3 K T T T THead (Bind Abst) (Bind Abst) -u x0 t x1 (refl_equal K (Bind Abst)) (H x0 H4) (H0 x1 (H5 Abst u))) t2 -H3)))))) H2)))))))). -(* COMMENTS -Initial nodes: 295 -END *) - -theorem nfs2_tapp: - \forall (c: C).(\forall (t: T).(\forall (ts: TList).((nfs2 c (TApp ts t)) -\to (land (nfs2 c ts) (nf2 c t))))) -\def - \lambda (c: C).(\lambda (t: T).(\lambda (ts: TList).(TList_ind (\lambda (t0: -TList).((nfs2 c (TApp t0 t)) \to (land (nfs2 c t0) (nf2 c t)))) (\lambda (H: -(land (nf2 c t) True)).(let H0 \def H in (land_ind (nf2 c t) True (land True -(nf2 c t)) (\lambda (H1: (nf2 c t)).(\lambda (_: True).(conj True (nf2 c t) I -H1))) H0))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H: (((nfs2 c -(TApp t1 t)) \to (land (nfs2 c t1) (nf2 c t))))).(\lambda (H0: (land (nf2 c -t0) (nfs2 c (TApp t1 t)))).(let H1 \def H0 in (land_ind (nf2 c t0) (nfs2 c -(TApp t1 t)) (land (land (nf2 c t0) (nfs2 c t1)) (nf2 c t)) (\lambda (H2: -(nf2 c t0)).(\lambda (H3: (nfs2 c (TApp t1 t))).(let H_x \def (H H3) in (let -H4 \def H_x in (land_ind (nfs2 c t1) (nf2 c t) (land (land (nf2 c t0) (nfs2 c -t1)) (nf2 c t)) (\lambda (H5: (nfs2 c t1)).(\lambda (H6: (nf2 c t)).(conj -(land (nf2 c t0) (nfs2 c t1)) (nf2 c t) (conj (nf2 c t0) (nfs2 c t1) H2 H5) -H6))) H4))))) H1)))))) ts))). -(* COMMENTS -Initial nodes: 295 -END *) - -theorem nf2_appls_lref: - \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (vs: -TList).((nfs2 c vs) \to (nf2 c (THeads (Flat Appl) vs (TLRef i))))))) -\def - \lambda (c: C).(\lambda (i: nat).(\lambda (H: (nf2 c (TLRef i))).(\lambda -(vs: TList).(TList_ind (\lambda (t: TList).((nfs2 c t) \to (nf2 c (THeads -(Flat Appl) t (TLRef i))))) (\lambda (_: True).H) (\lambda (t: T).(\lambda -(t0: TList).(\lambda (H0: (((nfs2 c t0) \to (nf2 c (THeads (Flat Appl) t0 -(TLRef i)))))).(\lambda (H1: (land (nf2 c t) (nfs2 c t0))).(let H2 \def H1 in -(land_ind (nf2 c t) (nfs2 c t0) (nf2 c (THead (Flat Appl) t (THeads (Flat -Appl) t0 (TLRef i)))) (\lambda (H3: (nf2 c t)).(\lambda (H4: (nfs2 c -t0)).(let H_y \def (H0 H4) in (\lambda (t2: T).(\lambda (H5: (pr2 c (THead -(Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) t2)).(let H6 \def -(pr2_gen_appl c t (THeads (Flat Appl) t0 (TLRef i)) t2 H5) in (or3_ind (ex3_2 -T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c t u2))) (\lambda (_: T).(\lambda (t3: -T).(pr2 c (THeads (Flat Appl) t0 (TLRef i)) t3)))) (ex4_4 T T T T (\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THeads (Flat -Appl) t0 (TLRef i)) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda -(_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 -t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: -T).(pr2 c t u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 -t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) -(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(eq T (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead -(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t -u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (eq T (THead (Flat Appl) t -(THeads (Flat Appl) t0 (TLRef i))) t2) (\lambda (H7: (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c t u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c -(THeads (Flat Appl) t0 (TLRef i)) t3))))).(ex3_2_ind T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c t u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c -(THeads (Flat Appl) t0 (TLRef i)) t3))) (eq T (THead (Flat Appl) t (THeads -(Flat Appl) t0 (TLRef i))) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda -(H8: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H9: (pr2 c t -x0)).(\lambda (H10: (pr2 c (THeads (Flat Appl) t0 (TLRef i)) x1)).(eq_ind_r T -(THead (Flat Appl) x0 x1) (\lambda (t1: T).(eq T (THead (Flat Appl) t (THeads -(Flat Appl) t0 (TLRef i))) t1)) (let H11 \def (eq_ind_r T x1 (\lambda (t1: -T).(pr2 c (THeads (Flat Appl) t0 (TLRef i)) t1)) H10 (THeads (Flat Appl) t0 -(TLRef i)) (H_y x1 H10)) in (eq_ind T (THeads (Flat Appl) t0 (TLRef i)) -(\lambda (t1: T).(eq T (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef -i))) (THead (Flat Appl) x0 t1))) (let H12 \def (eq_ind_r T x0 (\lambda (t1: -T).(pr2 c t t1)) H9 t (H3 x0 H9)) in (eq_ind T t (\lambda (t1: T).(eq T -(THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) (THead (Flat Appl) t1 -(THeads (Flat Appl) t0 (TLRef i))))) (refl_equal T (THead (Flat Appl) t -(THeads (Flat Appl) t0 (TLRef i)))) x0 (H3 x0 H9))) x1 (H_y x1 H10))) t2 -H8)))))) H7)) (\lambda (H7: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THeads (Flat Appl) t0 (TLRef i)) -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t u2))))) (\lambda -(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: -B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3))))))))).(ex4_4_ind T T T -T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THeads (Flat Appl) t0 (TLRef i)) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c t u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) -z1 t3))))))) (eq T (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) -t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: -T).(\lambda (H8: (eq T (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind Abst) -x0 x1))).(\lambda (H9: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (_: (pr2 -c t x2)).(\lambda (_: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) -u) x1 x3))))).(eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda (t1: T).(eq T -(THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) t1)) (TList_ind -(\lambda (t1: TList).((nf2 c (THeads (Flat Appl) t1 (TLRef i))) \to ((eq T -(THeads (Flat Appl) t1 (TLRef i)) (THead (Bind Abst) x0 x1)) \to (eq T (THead -(Flat Appl) t (THeads (Flat Appl) t1 (TLRef i))) (THead (Bind Abbr) x2 -x3))))) (\lambda (_: (nf2 c (THeads (Flat Appl) TNil (TLRef i)))).(\lambda -(H13: (eq T (THeads (Flat Appl) TNil (TLRef i)) (THead (Bind Abst) x0 -x1))).(let H14 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) x0 -x1) H13) in (False_ind (eq T (THead (Flat Appl) t (THeads (Flat Appl) TNil -(TLRef i))) (THead (Bind Abbr) x2 x3)) H14)))) (\lambda (t1: T).(\lambda (t3: -TList).(\lambda (_: (((nf2 c (THeads (Flat Appl) t3 (TLRef i))) \to ((eq T -(THeads (Flat Appl) t3 (TLRef i)) (THead (Bind Abst) x0 x1)) \to (eq T (THead -(Flat Appl) t (THeads (Flat Appl) t3 (TLRef i))) (THead (Bind Abbr) x2 -x3)))))).(\lambda (_: (nf2 c (THeads (Flat Appl) (TCons t1 t3) (TLRef -i)))).(\lambda (H13: (eq T (THeads (Flat Appl) (TCons t1 t3) (TLRef i)) -(THead (Bind Abst) x0 x1))).(let H14 \def (eq_ind T (THead (Flat Appl) t1 -(THeads (Flat Appl) t3 (TLRef i))) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind Abst) x0 x1) H13) in (False_ind (eq T (THead (Flat -Appl) t (THeads (Flat Appl) (TCons t1 t3) (TLRef i))) (THead (Bind Abbr) x2 -x3)) H14))))))) t0 H_y H8) t2 H9))))))))) H7)) (\lambda (H7: (ex6_6 B T T T T -T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq -T (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind b) y1 z1)))))))) (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: -T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S -O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) -y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B -b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THeads (Flat Appl) t0 (TLRef i)) -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c t u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) (eq T (THead -(Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) t2) (\lambda (x0: -B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: -T).(\lambda (x5: T).(\lambda (_: (not (eq B x0 Abst))).(\lambda (H9: (eq T -(THeads (Flat Appl) t0 (TLRef i)) (THead (Bind x0) x1 x2))).(\lambda (H10: -(eq T t2 (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) -x3)))).(\lambda (_: (pr2 c t x4)).(\lambda (_: (pr2 c x1 x5)).(\lambda (_: -(pr2 (CHead c (Bind x0) x5) x2 x3)).(eq_ind_r T (THead (Bind x0) x5 (THead -(Flat Appl) (lift (S O) O x4) x3)) (\lambda (t1: T).(eq T (THead (Flat Appl) -t (THeads (Flat Appl) t0 (TLRef i))) t1)) (TList_ind (\lambda (t1: -TList).((nf2 c (THeads (Flat Appl) t1 (TLRef i))) \to ((eq T (THeads (Flat -Appl) t1 (TLRef i)) (THead (Bind x0) x1 x2)) \to (eq T (THead (Flat Appl) t -(THeads (Flat Appl) t1 (TLRef i))) (THead (Bind x0) x5 (THead (Flat Appl) -(lift (S O) O x4) x3)))))) (\lambda (_: (nf2 c (THeads (Flat Appl) TNil -(TLRef i)))).(\lambda (H15: (eq T (THeads (Flat Appl) TNil (TLRef i)) (THead -(Bind x0) x1 x2))).(let H16 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match -ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | -(TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead -(Bind x0) x1 x2) H15) in (False_ind (eq T (THead (Flat Appl) t (THeads (Flat -Appl) TNil (TLRef i))) (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O -x4) x3))) H16)))) (\lambda (t1: T).(\lambda (t3: TList).(\lambda (_: (((nf2 c -(THeads (Flat Appl) t3 (TLRef i))) \to ((eq T (THeads (Flat Appl) t3 (TLRef -i)) (THead (Bind x0) x1 x2)) \to (eq T (THead (Flat Appl) t (THeads (Flat -Appl) t3 (TLRef i))) (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) -x3))))))).(\lambda (_: (nf2 c (THeads (Flat Appl) (TCons t1 t3) (TLRef -i)))).(\lambda (H15: (eq T (THeads (Flat Appl) (TCons t1 t3) (TLRef i)) -(THead (Bind x0) x1 x2))).(let H16 \def (eq_ind T (THead (Flat Appl) t1 -(THeads (Flat Appl) t3 (TLRef i))) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind x0) x1 x2) H15) in (False_ind (eq T (THead (Flat -Appl) t (THeads (Flat Appl) (TCons t1 t3) (TLRef i))) (THead (Bind x0) x5 -(THead (Flat Appl) (lift (S O) O x4) x3))) H16))))))) t0 H_y H9) t2 -H10))))))))))))) H7)) H6))))))) H2)))))) vs)))). -(* COMMENTS -Initial nodes: 2915 -END *) - -theorem nf2_appl_lref: - \forall (c: C).(\forall (u: T).((nf2 c u) \to (\forall (i: nat).((nf2 c -(TLRef i)) \to (nf2 c (THead (Flat Appl) u (TLRef i))))))) -\def - \lambda (c: C).(\lambda (u: T).(\lambda (H: (nf2 c u)).(\lambda (i: -nat).(\lambda (H0: (nf2 c (TLRef i))).(let H_y \def (nf2_appls_lref c i H0 -(TCons u TNil)) in (H_y (conj (nf2 c u) True H I))))))). -(* COMMENTS -Initial nodes: 49 -END *) - -theorem nf2_lref_abst: - \forall (c: C).(\forall (e: C).(\forall (u: T).(\forall (i: nat).((getl i c -(CHead e (Bind Abst) u)) \to (nf2 c (TLRef i)))))) -\def - \lambda (c: C).(\lambda (e: C).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H: (getl i c (CHead e (Bind Abst) u))).(\lambda (t2: T).(\lambda (H0: (pr2 c -(TLRef i) t2)).(let H1 \def (pr2_gen_lref c t2 i H0) in (or_ind (eq T t2 -(TLRef i)) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c (CHead d -(Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T t2 (lift (S i) O -u0))))) (eq T (TLRef i) t2) (\lambda (H2: (eq T t2 (TLRef i))).(eq_ind_r T -(TLRef i) (\lambda (t: T).(eq T (TLRef i) t)) (refl_equal T (TLRef i)) t2 -H2)) (\lambda (H2: (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c -(CHead d (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T t2 (lift -(S i) O u0)))))).(ex2_2_ind C T (\lambda (d: C).(\lambda (u0: T).(getl i c -(CHead d (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T t2 (lift -(S i) O u0)))) (eq T (TLRef i) t2) (\lambda (x0: C).(\lambda (x1: T).(\lambda -(H3: (getl i c (CHead x0 (Bind Abbr) x1))).(\lambda (H4: (eq T t2 (lift (S i) -O x1))).(eq_ind_r T (lift (S i) O x1) (\lambda (t: T).(eq T (TLRef i) t)) -(let H5 \def (eq_ind C (CHead e (Bind Abst) u) (\lambda (c0: C).(getl i c -c0)) H (CHead x0 (Bind Abbr) x1) (getl_mono c (CHead e (Bind Abst) u) i H -(CHead x0 (Bind Abbr) x1) H3)) in (let H6 \def (eq_ind C (CHead e (Bind Abst) -u) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort -_) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return -(\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B return -(\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow True | -Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead x0 (Bind -Abbr) x1) (getl_mono c (CHead e (Bind Abst) u) i H (CHead x0 (Bind Abbr) x1) -H3)) in (False_ind (eq T (TLRef i) (lift (S i) O x1)) H6))) t2 H4))))) H2)) -H1)))))))). -(* COMMENTS -Initial nodes: 494 -END *) - -theorem nf2_lift: - \forall (d: C).(\forall (t: T).((nf2 d t) \to (\forall (c: C).(\forall (h: -nat).(\forall (i: nat).((drop h i c d) \to (nf2 c (lift h i t)))))))) -\def - \lambda (d: C).(\lambda (t: T).(\lambda (H: ((\forall (t2: T).((pr2 d t t2) -\to (eq T t t2))))).(\lambda (c: C).(\lambda (h: nat).(\lambda (i: -nat).(\lambda (H0: (drop h i c d)).(\lambda (t2: T).(\lambda (H1: (pr2 c -(lift h i t) t2)).(let H2 \def (pr2_gen_lift c t t2 h i H1 d H0) in (ex2_ind -T (\lambda (t3: T).(eq T t2 (lift h i t3))) (\lambda (t3: T).(pr2 d t t3)) -(eq T (lift h i t) t2) (\lambda (x: T).(\lambda (H3: (eq T t2 (lift h i -x))).(\lambda (H4: (pr2 d t x)).(eq_ind_r T (lift h i x) (\lambda (t0: T).(eq -T (lift h i t) t0)) (let H_y \def (H x H4) in (let H5 \def (eq_ind_r T x -(\lambda (t0: T).(pr2 d t t0)) H4 t H_y) in (eq_ind T t (\lambda (t0: T).(eq -T (lift h i t) (lift h i t0))) (refl_equal T (lift h i t)) x H_y))) t2 H3)))) -H2)))))))))). -(* COMMENTS -Initial nodes: 245 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pc1/defs.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pc1/defs.ma deleted file mode 100644 index 80726a188..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pc1/defs.ma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr1/defs.ma". - -definition pc1: - T \to (T \to Prop) -\def - \lambda (t1: T).(\lambda (t2: T).(ex2 T (\lambda (t: T).(pr1 t1 t)) (\lambda -(t: T).(pr1 t2 t)))). - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pc1/props.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pc1/props.ma deleted file mode 100644 index 0da1a5aca..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pc1/props.ma +++ /dev/null @@ -1,146 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pc1/defs.ma". - -include "Basic-1/pr1/pr1.ma". - -theorem pc1_pr0_r: - \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (pc1 t1 t2))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(ex_intro2 T -(\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)) t2 (pr1_pr0 t1 t2 H) -(pr1_refl t2)))). -(* COMMENTS -Initial nodes: 43 -END *) - -theorem pc1_pr0_x: - \forall (t1: T).(\forall (t2: T).((pr0 t2 t1) \to (pc1 t1 t2))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t2 t1)).(ex_intro2 T -(\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)) t1 (pr1_refl t1) -(pr1_pr0 t2 t1 H)))). -(* COMMENTS -Initial nodes: 43 -END *) - -theorem pc1_refl: - \forall (t: T).(pc1 t t) -\def - \lambda (t: T).(ex_intro2 T (\lambda (t0: T).(pr1 t t0)) (\lambda (t0: -T).(pr1 t t0)) t (pr1_refl t) (pr1_refl t)). -(* COMMENTS -Initial nodes: 31 -END *) - -theorem pc1_pr0_u: - \forall (t2: T).(\forall (t1: T).((pr0 t1 t2) \to (\forall (t3: T).((pc1 t2 -t3) \to (pc1 t1 t3))))) -\def - \lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pr0 t1 t2)).(\lambda (t3: -T).(\lambda (H0: (pc1 t2 t3)).(let H1 \def H0 in (ex2_ind T (\lambda (t: -T).(pr1 t2 t)) (\lambda (t: T).(pr1 t3 t)) (pc1 t1 t3) (\lambda (x: -T).(\lambda (H2: (pr1 t2 x)).(\lambda (H3: (pr1 t3 x)).(ex_intro2 T (\lambda -(t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t3 t)) x (pr1_sing t2 t1 H x H2) -H3)))) H1)))))). -(* COMMENTS -Initial nodes: 97 -END *) - -theorem pc1_s: - \forall (t2: T).(\forall (t1: T).((pc1 t1 t2) \to (pc1 t2 t1))) -\def - \lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pc1 t1 t2)).(let H0 \def H in -(ex2_ind T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)) (pc1 t2 -t1) (\lambda (x: T).(\lambda (H1: (pr1 t1 x)).(\lambda (H2: (pr1 t2 -x)).(ex_intro2 T (\lambda (t: T).(pr1 t2 t)) (\lambda (t: T).(pr1 t1 t)) x H2 -H1)))) H0)))). -(* COMMENTS -Initial nodes: 79 -END *) - -theorem pc1_head_1: - \forall (u1: T).(\forall (u2: T).((pc1 u1 u2) \to (\forall (t: T).(\forall -(k: K).(pc1 (THead k u1 t) (THead k u2 t)))))) -\def - \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pc1 u1 u2)).(\lambda (t: -T).(\lambda (k: K).(let H0 \def H in (ex2_ind T (\lambda (t0: T).(pr1 u1 t0)) -(\lambda (t0: T).(pr1 u2 t0)) (pc1 (THead k u1 t) (THead k u2 t)) (\lambda -(x: T).(\lambda (H1: (pr1 u1 x)).(\lambda (H2: (pr1 u2 x)).(ex_intro2 T -(\lambda (t0: T).(pr1 (THead k u1 t) t0)) (\lambda (t0: T).(pr1 (THead k u2 -t) t0)) (THead k x t) (pr1_head_1 u1 x H1 t k) (pr1_head_1 u2 x H2 t k))))) -H0)))))). -(* COMMENTS -Initial nodes: 133 -END *) - -theorem pc1_head_2: - \forall (t1: T).(\forall (t2: T).((pc1 t1 t2) \to (\forall (u: T).(\forall -(k: K).(pc1 (THead k u t1) (THead k u t2)))))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc1 t1 t2)).(\lambda (u: -T).(\lambda (k: K).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr1 t1 t)) -(\lambda (t: T).(pr1 t2 t)) (pc1 (THead k u t1) (THead k u t2)) (\lambda (x: -T).(\lambda (H1: (pr1 t1 x)).(\lambda (H2: (pr1 t2 x)).(ex_intro2 T (\lambda -(t: T).(pr1 (THead k u t1) t)) (\lambda (t: T).(pr1 (THead k u t2) t)) (THead -k u x) (pr1_head_2 t1 x H1 u k) (pr1_head_2 t2 x H2 u k))))) H0)))))). -(* COMMENTS -Initial nodes: 133 -END *) - -theorem pc1_t: - \forall (t2: T).(\forall (t1: T).((pc1 t1 t2) \to (\forall (t3: T).((pc1 t2 -t3) \to (pc1 t1 t3))))) -\def - \lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pc1 t1 t2)).(\lambda (t3: -T).(\lambda (H0: (pc1 t2 t3)).(let H1 \def H0 in (ex2_ind T (\lambda (t: -T).(pr1 t2 t)) (\lambda (t: T).(pr1 t3 t)) (pc1 t1 t3) (\lambda (x: -T).(\lambda (H2: (pr1 t2 x)).(\lambda (H3: (pr1 t3 x)).(let H4 \def H in -(ex2_ind T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)) (pc1 t1 -t3) (\lambda (x0: T).(\lambda (H5: (pr1 t1 x0)).(\lambda (H6: (pr1 t2 -x0)).(ex2_ind T (\lambda (t: T).(pr1 x0 t)) (\lambda (t: T).(pr1 x t)) (pc1 -t1 t3) (\lambda (x1: T).(\lambda (H7: (pr1 x0 x1)).(\lambda (H8: (pr1 x -x1)).(ex_intro2 T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t3 t)) x1 -(pr1_t x0 t1 H5 x1 H7) (pr1_t x t3 H3 x1 H8))))) (pr1_confluence t2 x0 H6 x -H2))))) H4))))) H1)))))). -(* COMMENTS -Initial nodes: 203 -END *) - -theorem pc1_pr0_u2: - \forall (t0: T).(\forall (t1: T).((pr0 t0 t1) \to (\forall (t2: T).((pc1 t0 -t2) \to (pc1 t1 t2))))) -\def - \lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr0 t0 t1)).(\lambda (t2: -T).(\lambda (H0: (pc1 t0 t2)).(pc1_t t0 t1 (pc1_pr0_x t1 t0 H) t2 H0))))). -(* COMMENTS -Initial nodes: 35 -END *) - -theorem pc1_head: - \forall (u1: T).(\forall (u2: T).((pc1 u1 u2) \to (\forall (t1: T).(\forall -(t2: T).((pc1 t1 t2) \to (\forall (k: K).(pc1 (THead k u1 t1) (THead k u2 -t2)))))))) -\def - \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pc1 u1 u2)).(\lambda (t1: -T).(\lambda (t2: T).(\lambda (H0: (pc1 t1 t2)).(\lambda (k: K).(pc1_t (THead -k u2 t1) (THead k u1 t1) (pc1_head_1 u1 u2 H t1 k) (THead k u2 t2) -(pc1_head_2 t1 t2 H0 u2 k)))))))). -(* COMMENTS -Initial nodes: 71 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/dec.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/dec.ma deleted file mode 100644 index aa7a4d89e..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/dec.ma +++ /dev/null @@ -1,152 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/ty3/arity_props.ma". - -include "Basic-1/nf2/fwd.ma". - -theorem pc3_dec: - \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (t1: T).((ty3 g c -u1 t1) \to (\forall (u2: T).(\forall (t2: T).((ty3 g c u2 t2) \to (or (pc3 c -u1 u2) ((pc3 c u1 u2) \to False))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda -(H: (ty3 g c u1 t1)).(\lambda (u2: T).(\lambda (t2: T).(\lambda (H0: (ty3 g c -u2 t2)).(let H_y \def (ty3_sn3 g c u1 t1 H) in (let H_y0 \def (ty3_sn3 g c u2 -t2 H0) in (let H_x \def (nf2_sn3 c u1 H_y) in (let H1 \def H_x in (ex2_ind T -(\lambda (u: T).(pr3 c u1 u)) (\lambda (u: T).(nf2 c u)) (or (pc3 c u1 u2) -((pc3 c u1 u2) \to False)) (\lambda (x: T).(\lambda (H2: (pr3 c u1 -x)).(\lambda (H3: (nf2 c x)).(let H_x0 \def (nf2_sn3 c u2 H_y0) in (let H4 -\def H_x0 in (ex2_ind T (\lambda (u: T).(pr3 c u2 u)) (\lambda (u: T).(nf2 c -u)) (or (pc3 c u1 u2) ((pc3 c u1 u2) \to False)) (\lambda (x0: T).(\lambda -(H5: (pr3 c u2 x0)).(\lambda (H6: (nf2 c x0)).(let H_x1 \def (term_dec x x0) -in (let H7 \def H_x1 in (or_ind (eq T x x0) ((eq T x x0) \to (\forall (P: -Prop).P)) (or (pc3 c u1 u2) ((pc3 c u1 u2) \to False)) (\lambda (H8: (eq T x -x0)).(let H9 \def (eq_ind_r T x0 (\lambda (t: T).(nf2 c t)) H6 x H8) in (let -H10 \def (eq_ind_r T x0 (\lambda (t: T).(pr3 c u2 t)) H5 x H8) in (or_introl -(pc3 c u1 u2) ((pc3 c u1 u2) \to False) (pc3_pr3_t c u1 x H2 u2 H10))))) -(\lambda (H8: (((eq T x x0) \to (\forall (P: Prop).P)))).(or_intror (pc3 c u1 -u2) ((pc3 c u1 u2) \to False) (\lambda (H9: (pc3 c u1 u2)).(let H10 \def H9 -in (ex2_ind T (\lambda (t: T).(pr3 c u1 t)) (\lambda (t: T).(pr3 c u2 t)) -False (\lambda (x1: T).(\lambda (H11: (pr3 c u1 x1)).(\lambda (H12: (pr3 c u2 -x1)).(let H_x2 \def (pr3_confluence c u2 x0 H5 x1 H12) in (let H13 \def H_x2 -in (ex2_ind T (\lambda (t: T).(pr3 c x0 t)) (\lambda (t: T).(pr3 c x1 t)) -False (\lambda (x2: T).(\lambda (H14: (pr3 c x0 x2)).(\lambda (H15: (pr3 c x1 -x2)).(let H_y1 \def (nf2_pr3_unfold c x0 x2 H14 H6) in (let H16 \def -(eq_ind_r T x2 (\lambda (t: T).(pr3 c x1 t)) H15 x0 H_y1) in (let H17 \def -(nf2_pr3_confluence c x H3 x0 H6 u1 H2) in (H8 (H17 (pr3_t x1 u1 c H11 x0 -H16)) False))))))) H13)))))) H10))))) H7)))))) H4)))))) H1)))))))))))). -(* COMMENTS -Initial nodes: 551 -END *) - -theorem pc3_abst_dec: - \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (t1: T).((ty3 g c -u1 t1) \to (\forall (u2: T).(\forall (t2: T).((ty3 g c u2 t2) \to (or (ex4_2 -T T (\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst) u2 u)))) -(\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u) t1))) -(\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_: T).(\lambda -(v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind Abst) u2 u)) -\to False)))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda -(H: (ty3 g c u1 t1)).(\lambda (u2: T).(\lambda (t2: T).(\lambda (H0: (ty3 g c -u2 t2)).(let H1 \def (ty3_sn3 g c u1 t1 H) in (let H2 \def (ty3_sn3 g c u2 t2 -H0) in (let H_x \def (nf2_sn3 c u1 H1) in (let H3 \def H_x in (ex2_ind T -(\lambda (u: T).(pr3 c u1 u)) (\lambda (u: T).(nf2 c u)) (or (ex4_2 T T -(\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst) u2 u)))) -(\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u) t1))) -(\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_: T).(\lambda -(v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind Abst) u2 u)) -\to False))) (\lambda (x: T).(\lambda (H4: (pr3 c u1 x)).(\lambda (H5: (nf2 c -x)).(let H_x0 \def (nf2_sn3 c u2 H2) in (let H6 \def H_x0 in (ex2_ind T -(\lambda (u: T).(pr3 c u2 u)) (\lambda (u: T).(nf2 c u)) (or (ex4_2 T T -(\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst) u2 u)))) -(\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u) t1))) -(\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_: T).(\lambda -(v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind Abst) u2 u)) -\to False))) (\lambda (x0: T).(\lambda (H7: (pr3 c u2 x0)).(\lambda (H8: (nf2 -c x0)).(let H_x1 \def (abst_dec x x0) in (let H9 \def H_x1 in (or_ind (ex T -(\lambda (t: T).(eq T x (THead (Bind Abst) x0 t)))) (\forall (t: T).((eq T x -(THead (Bind Abst) x0 t)) \to (\forall (P: Prop).P))) (or (ex4_2 T T (\lambda -(u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst) u2 u)))) (\lambda (u: -T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u) t1))) (\lambda (_: -T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_: T).(\lambda (v2: T).(nf2 c -v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind Abst) u2 u)) \to False))) -(\lambda (H10: (ex T (\lambda (t: T).(eq T x (THead (Bind Abst) x0 -t))))).(ex_ind T (\lambda (t: T).(eq T x (THead (Bind Abst) x0 t))) (or -(ex4_2 T T (\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst) u2 -u)))) (\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u) -t1))) (\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_: -T).(\lambda (v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind -Abst) u2 u)) \to False))) (\lambda (x1: T).(\lambda (H11: (eq T x (THead -(Bind Abst) x0 x1))).(let H12 \def (eq_ind T x (\lambda (t: T).(nf2 c t)) H5 -(THead (Bind Abst) x0 x1) H11) in (let H13 \def (eq_ind T x (\lambda (t: -T).(pr3 c u1 t)) H4 (THead (Bind Abst) x0 x1) H11) in (let H_y \def -(ty3_sred_pr3 c u1 (THead (Bind Abst) x0 x1) H13 g t1 H) in (or_introl (ex4_2 -T T (\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst) u2 u)))) -(\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u) t1))) -(\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_: T).(\lambda -(v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind Abst) u2 u)) -\to False)) (ex4_2_intro T T (\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead -(Bind Abst) u2 u)))) (\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind -Abst) v2 u) t1))) (\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda -(_: T).(\lambda (v2: T).(nf2 c v2))) x1 x0 (pc3_pr3_t c u1 (THead (Bind Abst) -x0 x1) H13 (THead (Bind Abst) u2 x1) (pr3_head_12 c u2 x0 H7 (Bind Abst) x1 -x1 (pr3_refl (CHead c (Bind Abst) x0) x1))) H_y H7 H8))))))) H10)) (\lambda -(H10: ((\forall (t: T).((eq T x (THead (Bind Abst) x0 t)) \to (\forall (P: -Prop).P))))).(or_intror (ex4_2 T T (\lambda (u: T).(\lambda (_: T).(pc3 c u1 -(THead (Bind Abst) u2 u)))) (\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead -(Bind Abst) v2 u) t1))) (\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) -(\lambda (_: T).(\lambda (v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1 -(THead (Bind Abst) u2 u)) \to False)) (\lambda (u: T).(\lambda (H11: (pc3 c -u1 (THead (Bind Abst) u2 u))).(let H12 \def H11 in (ex2_ind T (\lambda (t: -T).(pr3 c u1 t)) (\lambda (t: T).(pr3 c (THead (Bind Abst) u2 u) t)) False -(\lambda (x1: T).(\lambda (H13: (pr3 c u1 x1)).(\lambda (H14: (pr3 c (THead -(Bind Abst) u2 u) x1)).(ex2_ind T (\lambda (t: T).(pr3 c x1 t)) (\lambda (t: -T).(pr3 c x t)) False (\lambda (x2: T).(\lambda (H15: (pr3 c x1 x2)).(\lambda -(H16: (pr3 c x x2)).(let H_y \def (nf2_pr3_unfold c x x2 H16 H5) in (let H17 -\def (eq_ind_r T x2 (\lambda (t: T).(pr3 c x1 t)) H15 x H_y) in (let H18 \def -(pr3_gen_abst c u2 u x1 H14) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: -T).(eq T x1 (THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: -T).(pr3 c u2 u3))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall -(u0: T).(pr3 (CHead c (Bind b) u0) u t3))))) False (\lambda (x3: T).(\lambda -(x4: T).(\lambda (H19: (eq T x1 (THead (Bind Abst) x3 x4))).(\lambda (H20: -(pr3 c u2 x3)).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pr3 (CHead c -(Bind b) u0) u x4))))).(let H22 \def (eq_ind T x1 (\lambda (t: T).(pr3 c t -x)) H17 (THead (Bind Abst) x3 x4) H19) in (let H23 \def (pr3_gen_abst c x3 x4 -x H22) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead -(Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c x3 u3))) -(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr3 (CHead -c (Bind b) u0) x4 t3))))) False (\lambda (x5: T).(\lambda (x6: T).(\lambda -(H24: (eq T x (THead (Bind Abst) x5 x6))).(\lambda (H25: (pr3 c x3 -x5)).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pr3 (CHead c (Bind b) -u0) x4 x6))))).(let H27 \def (eq_ind T x (\lambda (t: T).(\forall (t0: -T).((eq T t (THead (Bind Abst) x0 t0)) \to (\forall (P: Prop).P)))) H10 -(THead (Bind Abst) x5 x6) H24) in (let H28 \def (eq_ind T x (\lambda (t: -T).(nf2 c t)) H5 (THead (Bind Abst) x5 x6) H24) in (let H29 \def -(nf2_gen_abst c x5 x6 H28) in (land_ind (nf2 c x5) (nf2 (CHead c (Bind Abst) -x5) x6) False (\lambda (H30: (nf2 c x5)).(\lambda (_: (nf2 (CHead c (Bind -Abst) x5) x6)).(let H32 \def (nf2_pr3_confluence c x0 H8 x5 H30 u2 H7) in -(H27 x6 (sym_eq T (THead (Bind Abst) x0 x6) (THead (Bind Abst) x5 x6) -(f_equal3 K T T T THead (Bind Abst) (Bind Abst) x0 x5 x6 x6 (refl_equal K -(Bind Abst)) (H32 (pr3_t x3 u2 c H20 x5 H25)) (refl_equal T x6))) False)))) -H29))))))))) H23)))))))) H18))))))) (pr3_confluence c u1 x1 H13 x H4))))) -H12)))))) H9)))))) H6)))))) H3)))))))))))). -(* COMMENTS -Initial nodes: 1759 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/defs.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/defs.ma deleted file mode 100644 index e7ea2b24f..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/defs.ma +++ /dev/null @@ -1,31 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr3/defs.ma". - -definition pc3: - C \to (T \to (T \to Prop)) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(ex2 T (\lambda (t: T).(pr3 -c t1 t)) (\lambda (t: T).(pr3 c t2 t))))). - -inductive pc3_left (c: C): T \to (T \to Prop) \def -| pc3_left_r: \forall (t: T).(pc3_left c t t) -| pc3_left_ur: \forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall -(t3: T).((pc3_left c t2 t3) \to (pc3_left c t1 t3))))) -| pc3_left_ux: \forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall -(t3: T).((pc3_left c t1 t3) \to (pc3_left c t2 t3))))). - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/fsubst0.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/fsubst0.ma deleted file mode 100644 index c563ca397..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/fsubst0.ma +++ /dev/null @@ -1,726 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pc3/left.ma". - -include "Basic-1/fsubst0/defs.ma". - -include "Basic-1/csubst0/getl.ma". - -theorem pc3_pr2_fsubst0: - \forall (c1: C).(\forall (t1: T).(\forall (t: T).((pr2 c1 t1 t) \to (\forall -(i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t2: T).((fsubst0 i u c1 -t1 c2 t2) \to (\forall (e: C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3 -c2 t2 t))))))))))) -\def - \lambda (c1: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H: (pr2 c1 t1 -t)).(pr2_ind (\lambda (c: C).(\lambda (t0: T).(\lambda (t2: T).(\forall (i: -nat).(\forall (u: T).(\forall (c2: C).(\forall (t3: T).((fsubst0 i u c t0 c2 -t3) \to (\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) \to (pc3 c2 t3 -t2))))))))))) (\lambda (c: C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H0: -(pr0 t2 t3)).(\lambda (i: nat).(\lambda (u: T).(\lambda (c2: C).(\lambda (t0: -T).(\lambda (H1: (fsubst0 i u c t2 c2 t0)).(fsubst0_ind i u c t2 (\lambda -(c0: C).(\lambda (t4: T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) -\to (pc3 c0 t4 t3))))) (\lambda (t4: T).(\lambda (H2: (subst0 i u t2 -t4)).(\lambda (e: C).(\lambda (H3: (getl i c (CHead e (Bind Abbr) -u))).(or_ind (pr0 t4 t3) (ex2 T (\lambda (w2: T).(pr0 t4 w2)) (\lambda (w2: -T).(subst0 i u t3 w2))) (pc3 c t4 t3) (\lambda (H4: (pr0 t4 t3)).(pc3_pr2_r c -t4 t3 (pr2_free c t4 t3 H4))) (\lambda (H4: (ex2 T (\lambda (w2: T).(pr0 t4 -w2)) (\lambda (w2: T).(subst0 i u t3 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 -t4 w2)) (\lambda (w2: T).(subst0 i u t3 w2)) (pc3 c t4 t3) (\lambda (x: -T).(\lambda (H5: (pr0 t4 x)).(\lambda (H6: (subst0 i u t3 x)).(pc3_pr2_u c x -t4 (pr2_free c t4 x H5) t3 (pc3_pr2_x c x t3 (pr2_delta c e u i H3 t3 t3 -(pr0_refl t3) x H6)))))) H4)) (pr0_subst0 t2 t3 H0 u t4 i H2 u (pr0_refl -u))))))) (\lambda (c0: C).(\lambda (_: (csubst0 i u c c0)).(\lambda (e: -C).(\lambda (_: (getl i c (CHead e (Bind Abbr) u))).(pc3_pr2_r c0 t2 t3 -(pr2_free c0 t2 t3 H0)))))) (\lambda (t4: T).(\lambda (H2: (subst0 i u t2 -t4)).(\lambda (c0: C).(\lambda (H3: (csubst0 i u c c0)).(\lambda (e: -C).(\lambda (H4: (getl i c (CHead e (Bind Abbr) u))).(or_ind (pr0 t4 t3) (ex2 -T (\lambda (w2: T).(pr0 t4 w2)) (\lambda (w2: T).(subst0 i u t3 w2))) (pc3 c0 -t4 t3) (\lambda (H5: (pr0 t4 t3)).(pc3_pr2_r c0 t4 t3 (pr2_free c0 t4 t3 -H5))) (\lambda (H5: (ex2 T (\lambda (w2: T).(pr0 t4 w2)) (\lambda (w2: -T).(subst0 i u t3 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t4 w2)) (\lambda -(w2: T).(subst0 i u t3 w2)) (pc3 c0 t4 t3) (\lambda (x: T).(\lambda (H6: (pr0 -t4 x)).(\lambda (H7: (subst0 i u t3 x)).(pc3_pr2_u c0 x t4 (pr2_free c0 t4 x -H6) t3 (pc3_pr2_x c0 x t3 (pr2_delta c0 e u i (csubst0_getl_ge i i (le_n i) c -c0 u H3 (CHead e (Bind Abbr) u) H4) t3 t3 (pr0_refl t3) x H7)))))) H5)) -(pr0_subst0 t2 t3 H0 u t4 i H2 u (pr0_refl u))))))))) c2 t0 H1)))))))))) -(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (t2: T).(\lambda (t3: -T).(\lambda (H1: (pr0 t2 t3)).(\lambda (t0: T).(\lambda (H2: (subst0 i u t3 -t0)).(\lambda (i0: nat).(\lambda (u0: T).(\lambda (c2: C).(\lambda (t4: -T).(\lambda (H3: (fsubst0 i0 u0 c t2 c2 t4)).(fsubst0_ind i0 u0 c t2 (\lambda -(c0: C).(\lambda (t5: T).(\forall (e: C).((getl i0 c (CHead e (Bind Abbr) -u0)) \to (pc3 c0 t5 t0))))) (\lambda (t5: T).(\lambda (H4: (subst0 i0 u0 t2 -t5)).(\lambda (e: C).(\lambda (H5: (getl i0 c (CHead e (Bind Abbr) -u0))).(pc3_t t2 c t5 (pc3_s c t5 t2 (pc3_pr2_r c t2 t5 (pr2_delta c e u0 i0 -H5 t2 t2 (pr0_refl t2) t5 H4))) t0 (pc3_pr2_r c t2 t0 (pr2_delta c d u i H0 -t2 t3 H1 t0 H2))))))) (\lambda (c0: C).(\lambda (H4: (csubst0 i0 u0 c -c0)).(\lambda (e: C).(\lambda (H5: (getl i0 c (CHead e (Bind Abbr) -u0))).(lt_le_e i i0 (pc3 c0 t2 t0) (\lambda (H6: (lt i i0)).(let H7 \def -(csubst0_getl_lt i0 i H6 c c0 u0 H4 (CHead d (Bind Abbr) u) H0) in (or4_ind -(getl i c0 (CHead d (Bind Abbr) u)) (ex3_4 B C T T (\lambda (b: B).(\lambda -(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead -e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) -u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c0 -(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) (ex4_5 B C C T T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda -(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl -i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))) (pc3 c0 t2 t0) (\lambda (H8: -(getl i c0 (CHead d (Bind Abbr) u))).(pc3_pr2_r c0 t2 t0 (pr2_delta c0 d u i -H8 t2 t3 H1 t0 H2))) (\lambda (H8: (ex3_4 B C T T (\lambda (b: B).(\lambda -(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead -e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) -u0 u1 w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda -(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b) -u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w))))) -(pc3 c0 t2 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda -(x3: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) -x2))).(\lambda (H10: (getl i c0 (CHead x1 (Bind x0) x3))).(\lambda (H11: -(subst0 (minus i0 (S i)) u0 x2 x3)).(let H12 \def (f_equal C C (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | -(CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) -x2) H9) in ((let H13 \def (f_equal C B (\lambda (e0: C).(match e0 in C return -(\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) -\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead -x1 (Bind x0) x2) H9) in ((let H14 \def (f_equal C T (\lambda (e0: C).(match -e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ -t5) \Rightarrow t5])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H9) in -(\lambda (H15: (eq B Abbr x0)).(\lambda (H16: (eq C d x1)).(let H17 \def -(eq_ind_r T x2 (\lambda (t5: T).(subst0 (minus i0 (S i)) u0 t5 x3)) H11 u -H14) in (let H18 \def (eq_ind_r C x1 (\lambda (c3: C).(getl i c0 (CHead c3 -(Bind x0) x3))) H10 d H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b: -B).(getl i c0 (CHead d (Bind b) x3))) H18 Abbr H15) in (ex2_ind T (\lambda -(t5: T).(subst0 i x3 t3 t5)) (\lambda (t5: T).(subst0 (S (plus (minus i0 (S -i)) i)) u0 t0 t5)) (pc3 c0 t2 t0) (\lambda (x: T).(\lambda (H20: (subst0 i x3 -t3 x)).(\lambda (H21: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let -H22 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n: -nat).(subst0 n u0 t0 x)) H21 i0 (lt_plus_minus_r i i0 H6)) in (pc3_pr2_u c0 x -t2 (pr2_delta c0 d x3 i H19 t2 t3 H1 x H20) t0 (pc3_pr2_x c0 x t0 (pr2_delta -c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H4 (CHead e (Bind Abbr) -u0) H5) t0 t0 (pr0_refl t0) x H22))))))) (subst0_subst0_back t3 t0 u i H2 x3 -u0 (minus i0 (S i)) H17)))))))) H13)) H12))))))))) H8)) (\lambda (H8: (ex3_4 -B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq -C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 -(Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))).(ex3_4_ind B C C T (\lambda -(b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind -Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 (Bind b) u1)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S -i)) u0 e1 e2))))) (pc3 c0 t2 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda -(x2: C).(\lambda (x3: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead -x1 (Bind x0) x3))).(\lambda (H10: (getl i c0 (CHead x2 (Bind x0) -x3))).(\lambda (H11: (csubst0 (minus i0 (S i)) u0 x1 x2)).(let H12 \def -(f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with -[(CSort _) \Rightarrow d | (CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind -Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H13 \def (f_equal C B (\lambda -(e0: C).(match e0 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow -Abbr | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with -[(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) -u) (CHead x1 (Bind x0) x3) H9) in ((let H14 \def (f_equal C T (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | -(CHead _ _ t5) \Rightarrow t5])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) -x3) H9) in (\lambda (H15: (eq B Abbr x0)).(\lambda (H16: (eq C d x1)).(let -H17 \def (eq_ind_r T x3 (\lambda (t5: T).(getl i c0 (CHead x2 (Bind x0) t5))) -H10 u H14) in (let H18 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus -i0 (S i)) u0 c3 x2)) H11 d H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b: -B).(getl i c0 (CHead x2 (Bind b) u))) H17 Abbr H15) in (pc3_pr2_r c0 t2 t0 -(pr2_delta c0 x2 u i H19 t2 t3 H1 t0 H2)))))))) H13)) H12))))))))) H8)) -(\lambda (H8: (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 -(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 -(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 -e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 -(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 -(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) -(pc3 c0 t2 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda -(x3: T).(\lambda (x4: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead -x1 (Bind x0) x3))).(\lambda (H10: (getl i c0 (CHead x2 (Bind x0) -x4))).(\lambda (H11: (subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H12: -(csubst0 (minus i0 (S i)) u0 x1 x2)).(let H13 \def (f_equal C C (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | -(CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) -x3) H9) in ((let H14 \def (f_equal C B (\lambda (e0: C).(match e0 in C return -(\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) -\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead -x1 (Bind x0) x3) H9) in ((let H15 \def (f_equal C T (\lambda (e0: C).(match -e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ -t5) \Rightarrow t5])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in -(\lambda (H16: (eq B Abbr x0)).(\lambda (H17: (eq C d x1)).(let H18 \def -(eq_ind_r T x3 (\lambda (t5: T).(subst0 (minus i0 (S i)) u0 t5 x4)) H11 u -H15) in (let H19 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus i0 (S -i)) u0 c3 x2)) H12 d H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b: -B).(getl i c0 (CHead x2 (Bind b) x4))) H10 Abbr H16) in (ex2_ind T (\lambda -(t5: T).(subst0 i x4 t3 t5)) (\lambda (t5: T).(subst0 (S (plus (minus i0 (S -i)) i)) u0 t0 t5)) (pc3 c0 t2 t0) (\lambda (x: T).(\lambda (H21: (subst0 i x4 -t3 x)).(\lambda (H22: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let -H23 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n: -nat).(subst0 n u0 t0 x)) H22 i0 (lt_plus_minus_r i i0 H6)) in (pc3_pr2_u c0 x -t2 (pr2_delta c0 x2 x4 i H20 t2 t3 H1 x H21) t0 (pc3_pr2_x c0 x t0 (pr2_delta -c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H4 (CHead e (Bind Abbr) -u0) H5) t0 t0 (pr0_refl t0) x H23))))))) (subst0_subst0_back t3 t0 u i H2 x4 -u0 (minus i0 (S i)) H18)))))))) H14)) H13))))))))))) H8)) H7))) (\lambda (H6: -(le i0 i)).(pc3_pr2_r c0 t2 t0 (pr2_delta c0 d u i (csubst0_getl_ge i0 i H6 c -c0 u0 H4 (CHead d (Bind Abbr) u) H0) t2 t3 H1 t0 H2)))))))) (\lambda (t5: -T).(\lambda (H4: (subst0 i0 u0 t2 t5)).(\lambda (c0: C).(\lambda (H5: -(csubst0 i0 u0 c c0)).(\lambda (e: C).(\lambda (H6: (getl i0 c (CHead e (Bind -Abbr) u0))).(lt_le_e i i0 (pc3 c0 t5 t0) (\lambda (H7: (lt i i0)).(let H8 -\def (csubst0_getl_lt i0 i H7 c c0 u0 H5 (CHead d (Bind Abbr) u) H0) in -(or4_ind (getl i c0 (CHead d (Bind Abbr) u)) (ex3_4 B C T T (\lambda (b: -B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind -Abbr) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 -(minus i0 (S i)) u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 -(Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(u1: T).(getl i c0 (CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) -(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) -u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 -(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))) -(pc3 c0 t5 t0) (\lambda (H9: (getl i c0 (CHead d (Bind Abbr) u))).(pc3_pr2_u2 -c0 t2 t5 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 -(CHead e (Bind Abbr) u0) H6) t2 t2 (pr0_refl t2) t5 H4) t0 (pc3_pr2_r c0 t2 -t0 (pr2_delta c0 d u i H9 t2 t3 H1 t0 H2)))) (\lambda (H9: (ex3_4 B C T T -(\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C -(CHead d (Bind Abbr) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda -(e0: C).(\lambda (_: T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 -(minus i0 (S i)) u0 u1 w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda -(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead -e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) -u0 u1 w))))) (pc3 c0 t5 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: -T).(\lambda (x3: T).(\lambda (H10: (eq C (CHead d (Bind Abbr) u) (CHead x1 -(Bind x0) x2))).(\lambda (H11: (getl i c0 (CHead x1 (Bind x0) x3))).(\lambda -(H12: (subst0 (minus i0 (S i)) u0 x2 x3)).(let H13 \def (f_equal C C (\lambda -(e0: C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow -d | (CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind -x0) x2) H10) in ((let H14 \def (f_equal C B (\lambda (e0: C).(match e0 in C -return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) -\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead -x1 (Bind x0) x2) H10) in ((let H15 \def (f_equal C T (\lambda (e0: C).(match -e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ -t6) \Rightarrow t6])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H10) in -(\lambda (H16: (eq B Abbr x0)).(\lambda (H17: (eq C d x1)).(let H18 \def -(eq_ind_r T x2 (\lambda (t6: T).(subst0 (minus i0 (S i)) u0 t6 x3)) H12 u -H15) in (let H19 \def (eq_ind_r C x1 (\lambda (c3: C).(getl i c0 (CHead c3 -(Bind x0) x3))) H11 d H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b: -B).(getl i c0 (CHead d (Bind b) x3))) H19 Abbr H16) in (ex2_ind T (\lambda -(t6: T).(subst0 i x3 t3 t6)) (\lambda (t6: T).(subst0 (S (plus (minus i0 (S -i)) i)) u0 t0 t6)) (pc3 c0 t5 t0) (\lambda (x: T).(\lambda (H21: (subst0 i x3 -t3 x)).(\lambda (H22: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let -H23 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n: -nat).(subst0 n u0 t0 x)) H22 i0 (lt_plus_minus_r i i0 H7)) in (pc3_pr2_u2 c0 -t2 t5 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 -(CHead e (Bind Abbr) u0) H6) t2 t2 (pr0_refl t2) t5 H4) t0 (pc3_pr2_u c0 x t2 -(pr2_delta c0 d x3 i H20 t2 t3 H1 x H21) t0 (pc3_pr2_x c0 x t0 (pr2_delta c0 -e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e (Bind Abbr) u0) -H6) t0 t0 (pr0_refl t0) x H23)))))))) (subst0_subst0_back t3 t0 u i H2 x3 u0 -(minus i0 (S i)) H18)))))))) H14)) H13))))))))) H9)) (\lambda (H9: (ex3_4 B C -C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C -(CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda -(_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 (Bind b) -u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))).(ex3_4_ind B C C T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind -Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 (Bind b) u1)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S -i)) u0 e1 e2))))) (pc3 c0 t5 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda -(x2: C).(\lambda (x3: T).(\lambda (H10: (eq C (CHead d (Bind Abbr) u) (CHead -x1 (Bind x0) x3))).(\lambda (H11: (getl i c0 (CHead x2 (Bind x0) -x3))).(\lambda (H12: (csubst0 (minus i0 (S i)) u0 x1 x2)).(let H13 \def -(f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with -[(CSort _) \Rightarrow d | (CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind -Abbr) u) (CHead x1 (Bind x0) x3) H10) in ((let H14 \def (f_equal C B (\lambda -(e0: C).(match e0 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow -Abbr | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with -[(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) -u) (CHead x1 (Bind x0) x3) H10) in ((let H15 \def (f_equal C T (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | -(CHead _ _ t6) \Rightarrow t6])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) -x3) H10) in (\lambda (H16: (eq B Abbr x0)).(\lambda (H17: (eq C d x1)).(let -H18 \def (eq_ind_r T x3 (\lambda (t6: T).(getl i c0 (CHead x2 (Bind x0) t6))) -H11 u H15) in (let H19 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus -i0 (S i)) u0 c3 x2)) H12 d H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b: -B).(getl i c0 (CHead x2 (Bind b) u))) H18 Abbr H16) in (pc3_pr2_u2 c0 t2 t5 -(pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e -(Bind Abbr) u0) H6) t2 t2 (pr0_refl t2) t5 H4) t0 (pc3_pr2_r c0 t2 t0 -(pr2_delta c0 x2 u i H20 t2 t3 H1 t0 H2))))))))) H14)) H13))))))))) H9)) -(\lambda (H9: (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 -(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 -(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 -e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 -(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 -(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) -(pc3 c0 t5 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda -(x3: T).(\lambda (x4: T).(\lambda (H10: (eq C (CHead d (Bind Abbr) u) (CHead -x1 (Bind x0) x3))).(\lambda (H11: (getl i c0 (CHead x2 (Bind x0) -x4))).(\lambda (H12: (subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H13: -(csubst0 (minus i0 (S i)) u0 x1 x2)).(let H14 \def (f_equal C C (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | -(CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) -x3) H10) in ((let H15 \def (f_equal C B (\lambda (e0: C).(match e0 in C -return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) -\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead -x1 (Bind x0) x3) H10) in ((let H16 \def (f_equal C T (\lambda (e0: C).(match -e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ -t6) \Rightarrow t6])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H10) in -(\lambda (H17: (eq B Abbr x0)).(\lambda (H18: (eq C d x1)).(let H19 \def -(eq_ind_r T x3 (\lambda (t6: T).(subst0 (minus i0 (S i)) u0 t6 x4)) H12 u -H16) in (let H20 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus i0 (S -i)) u0 c3 x2)) H13 d H18) in (let H21 \def (eq_ind_r B x0 (\lambda (b: -B).(getl i c0 (CHead x2 (Bind b) x4))) H11 Abbr H17) in (ex2_ind T (\lambda -(t6: T).(subst0 i x4 t3 t6)) (\lambda (t6: T).(subst0 (S (plus (minus i0 (S -i)) i)) u0 t0 t6)) (pc3 c0 t5 t0) (\lambda (x: T).(\lambda (H22: (subst0 i x4 -t3 x)).(\lambda (H23: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let -H24 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n: -nat).(subst0 n u0 t0 x)) H23 i0 (lt_plus_minus_r i i0 H7)) in (pc3_pr2_u2 c0 -t2 t5 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 -(CHead e (Bind Abbr) u0) H6) t2 t2 (pr0_refl t2) t5 H4) t0 (pc3_pr2_u c0 x t2 -(pr2_delta c0 x2 x4 i H21 t2 t3 H1 x H22) t0 (pc3_pr2_x c0 x t0 (pr2_delta c0 -e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e (Bind Abbr) u0) -H6) t0 t0 (pr0_refl t0) x H24)))))))) (subst0_subst0_back t3 t0 u i H2 x4 u0 -(minus i0 (S i)) H19)))))))) H15)) H14))))))))))) H9)) H8))) (\lambda (H7: -(le i0 i)).(pc3_pr2_u2 c0 t2 t5 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 -(le_n i0) c c0 u0 H5 (CHead e (Bind Abbr) u0) H6) t2 t2 (pr0_refl t2) t5 H4) -t0 (pc3_pr2_r c0 t2 t0 (pr2_delta c0 d u i (csubst0_getl_ge i0 i H7 c c0 u0 -H5 (CHead d (Bind Abbr) u) H0) t2 t3 H1 t0 H2))))))))))) c2 t4 -H3)))))))))))))))) c1 t1 t H)))). -(* COMMENTS -Initial nodes: 6455 -END *) - -theorem pc3_pr2_fsubst0_back: - \forall (c1: C).(\forall (t: T).(\forall (t1: T).((pr2 c1 t t1) \to (\forall -(i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t2: T).((fsubst0 i u c1 -t1 c2 t2) \to (\forall (e: C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3 -c2 t t2))))))))))) -\def - \lambda (c1: C).(\lambda (t: T).(\lambda (t1: T).(\lambda (H: (pr2 c1 t -t1)).(pr2_ind (\lambda (c: C).(\lambda (t0: T).(\lambda (t2: T).(\forall (i: -nat).(\forall (u: T).(\forall (c2: C).(\forall (t3: T).((fsubst0 i u c t2 c2 -t3) \to (\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) \to (pc3 c2 t0 -t3))))))))))) (\lambda (c: C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H0: -(pr0 t2 t3)).(\lambda (i: nat).(\lambda (u: T).(\lambda (c2: C).(\lambda (t0: -T).(\lambda (H1: (fsubst0 i u c t3 c2 t0)).(fsubst0_ind i u c t3 (\lambda -(c0: C).(\lambda (t4: T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) -\to (pc3 c0 t2 t4))))) (\lambda (t4: T).(\lambda (H2: (subst0 i u t3 -t4)).(\lambda (e: C).(\lambda (H3: (getl i c (CHead e (Bind Abbr) -u))).(pc3_pr2_u c t3 t2 (pr2_free c t2 t3 H0) t4 (pc3_pr2_r c t3 t4 -(pr2_delta c e u i H3 t3 t3 (pr0_refl t3) t4 H2))))))) (\lambda (c0: -C).(\lambda (_: (csubst0 i u c c0)).(\lambda (e: C).(\lambda (_: (getl i c -(CHead e (Bind Abbr) u))).(pc3_pr2_r c0 t2 t3 (pr2_free c0 t2 t3 H0)))))) -(\lambda (t4: T).(\lambda (H2: (subst0 i u t3 t4)).(\lambda (c0: C).(\lambda -(H3: (csubst0 i u c c0)).(\lambda (e: C).(\lambda (H4: (getl i c (CHead e -(Bind Abbr) u))).(pc3_pr2_u c0 t3 t2 (pr2_free c0 t2 t3 H0) t4 (pc3_pr2_r c0 -t3 t4 (pr2_delta c0 e u i (csubst0_getl_ge i i (le_n i) c c0 u H3 (CHead e -(Bind Abbr) u) H4) t3 t3 (pr0_refl t3) t4 H2))))))))) c2 t0 H1)))))))))) -(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (t2: T).(\lambda (t3: -T).(\lambda (H1: (pr0 t2 t3)).(\lambda (t0: T).(\lambda (H2: (subst0 i u t3 -t0)).(\lambda (i0: nat).(\lambda (u0: T).(\lambda (c2: C).(\lambda (t4: -T).(\lambda (H3: (fsubst0 i0 u0 c t0 c2 t4)).(fsubst0_ind i0 u0 c t0 (\lambda -(c0: C).(\lambda (t5: T).(\forall (e: C).((getl i0 c (CHead e (Bind Abbr) -u0)) \to (pc3 c0 t2 t5))))) (\lambda (t5: T).(\lambda (H4: (subst0 i0 u0 t0 -t5)).(\lambda (e: C).(\lambda (H5: (getl i0 c (CHead e (Bind Abbr) -u0))).(pc3_t t3 c t2 (pc3_pr3_r c t2 t3 (pr3_pr2 c t2 t3 (pr2_free c t2 t3 -H1))) t5 (pc3_pr3_r c t3 t5 (pr3_sing c t0 t3 (pr2_delta c d u i H0 t3 t3 -(pr0_refl t3) t0 H2) t5 (pr3_pr2 c t0 t5 (pr2_delta c e u0 i0 H5 t0 t0 -(pr0_refl t0) t5 H4))))))))) (\lambda (c0: C).(\lambda (H4: (csubst0 i0 u0 c -c0)).(\lambda (e: C).(\lambda (H5: (getl i0 c (CHead e (Bind Abbr) -u0))).(lt_le_e i i0 (pc3 c0 t2 t0) (\lambda (H6: (lt i i0)).(let H7 \def -(csubst0_getl_lt i0 i H6 c c0 u0 H4 (CHead d (Bind Abbr) u) H0) in (or4_ind -(getl i c0 (CHead d (Bind Abbr) u)) (ex3_4 B C T T (\lambda (b: B).(\lambda -(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead -e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) -u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c0 -(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) (ex4_5 B C C T T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda -(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl -i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))) (pc3 c0 t2 t0) (\lambda (H8: -(getl i c0 (CHead d (Bind Abbr) u))).(pc3_pr2_r c0 t2 t0 (pr2_delta c0 d u i -H8 t2 t3 H1 t0 H2))) (\lambda (H8: (ex3_4 B C T T (\lambda (b: B).(\lambda -(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead -e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) -u0 u1 w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda -(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b) -u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w))))) -(pc3 c0 t2 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda -(x3: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) -x2))).(\lambda (H10: (getl i c0 (CHead x1 (Bind x0) x3))).(\lambda (H11: -(subst0 (minus i0 (S i)) u0 x2 x3)).(let H12 \def (f_equal C C (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | -(CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) -x2) H9) in ((let H13 \def (f_equal C B (\lambda (e0: C).(match e0 in C return -(\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) -\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead -x1 (Bind x0) x2) H9) in ((let H14 \def (f_equal C T (\lambda (e0: C).(match -e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ -t5) \Rightarrow t5])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H9) in -(\lambda (H15: (eq B Abbr x0)).(\lambda (H16: (eq C d x1)).(let H17 \def -(eq_ind_r T x2 (\lambda (t5: T).(subst0 (minus i0 (S i)) u0 t5 x3)) H11 u -H14) in (let H18 \def (eq_ind_r C x1 (\lambda (c3: C).(getl i c0 (CHead c3 -(Bind x0) x3))) H10 d H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b: -B).(getl i c0 (CHead d (Bind b) x3))) H18 Abbr H15) in (ex2_ind T (\lambda -(t5: T).(subst0 i x3 t3 t5)) (\lambda (t5: T).(subst0 (S (plus (minus i0 (S -i)) i)) u0 t0 t5)) (pc3 c0 t2 t0) (\lambda (x: T).(\lambda (H20: (subst0 i x3 -t3 x)).(\lambda (H21: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let -H22 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n: -nat).(subst0 n u0 t0 x)) H21 i0 (lt_plus_minus_r i i0 H6)) in (pc3_pr2_u c0 x -t2 (pr2_delta c0 d x3 i H19 t2 t3 H1 x H20) t0 (pc3_pr2_x c0 x t0 (pr2_delta -c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H4 (CHead e (Bind Abbr) -u0) H5) t0 t0 (pr0_refl t0) x H22))))))) (subst0_subst0_back t3 t0 u i H2 x3 -u0 (minus i0 (S i)) H17)))))))) H13)) H12))))))))) H8)) (\lambda (H8: (ex3_4 -B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq -C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 -(Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))).(ex3_4_ind B C C T (\lambda -(b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind -Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 (Bind b) u1)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S -i)) u0 e1 e2))))) (pc3 c0 t2 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda -(x2: C).(\lambda (x3: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead -x1 (Bind x0) x3))).(\lambda (H10: (getl i c0 (CHead x2 (Bind x0) -x3))).(\lambda (H11: (csubst0 (minus i0 (S i)) u0 x1 x2)).(let H12 \def -(f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with -[(CSort _) \Rightarrow d | (CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind -Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H13 \def (f_equal C B (\lambda -(e0: C).(match e0 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow -Abbr | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with -[(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) -u) (CHead x1 (Bind x0) x3) H9) in ((let H14 \def (f_equal C T (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | -(CHead _ _ t5) \Rightarrow t5])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) -x3) H9) in (\lambda (H15: (eq B Abbr x0)).(\lambda (H16: (eq C d x1)).(let -H17 \def (eq_ind_r T x3 (\lambda (t5: T).(getl i c0 (CHead x2 (Bind x0) t5))) -H10 u H14) in (let H18 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus -i0 (S i)) u0 c3 x2)) H11 d H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b: -B).(getl i c0 (CHead x2 (Bind b) u))) H17 Abbr H15) in (pc3_pr2_r c0 t2 t0 -(pr2_delta c0 x2 u i H19 t2 t3 H1 t0 H2)))))))) H13)) H12))))))))) H8)) -(\lambda (H8: (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 -(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 -(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 -e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 -(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 -(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) -(pc3 c0 t2 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda -(x3: T).(\lambda (x4: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead -x1 (Bind x0) x3))).(\lambda (H10: (getl i c0 (CHead x2 (Bind x0) -x4))).(\lambda (H11: (subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H12: -(csubst0 (minus i0 (S i)) u0 x1 x2)).(let H13 \def (f_equal C C (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | -(CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) -x3) H9) in ((let H14 \def (f_equal C B (\lambda (e0: C).(match e0 in C return -(\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) -\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead -x1 (Bind x0) x3) H9) in ((let H15 \def (f_equal C T (\lambda (e0: C).(match -e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ -t5) \Rightarrow t5])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in -(\lambda (H16: (eq B Abbr x0)).(\lambda (H17: (eq C d x1)).(let H18 \def -(eq_ind_r T x3 (\lambda (t5: T).(subst0 (minus i0 (S i)) u0 t5 x4)) H11 u -H15) in (let H19 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus i0 (S -i)) u0 c3 x2)) H12 d H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b: -B).(getl i c0 (CHead x2 (Bind b) x4))) H10 Abbr H16) in (ex2_ind T (\lambda -(t5: T).(subst0 i x4 t3 t5)) (\lambda (t5: T).(subst0 (S (plus (minus i0 (S -i)) i)) u0 t0 t5)) (pc3 c0 t2 t0) (\lambda (x: T).(\lambda (H21: (subst0 i x4 -t3 x)).(\lambda (H22: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let -H23 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n: -nat).(subst0 n u0 t0 x)) H22 i0 (lt_plus_minus_r i i0 H6)) in (pc3_pr2_u c0 x -t2 (pr2_delta c0 x2 x4 i H20 t2 t3 H1 x H21) t0 (pc3_pr2_x c0 x t0 (pr2_delta -c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H4 (CHead e (Bind Abbr) -u0) H5) t0 t0 (pr0_refl t0) x H23))))))) (subst0_subst0_back t3 t0 u i H2 x4 -u0 (minus i0 (S i)) H18)))))))) H14)) H13))))))))))) H8)) H7))) (\lambda (H6: -(le i0 i)).(pc3_pr2_r c0 t2 t0 (pr2_delta c0 d u i (csubst0_getl_ge i0 i H6 c -c0 u0 H4 (CHead d (Bind Abbr) u) H0) t2 t3 H1 t0 H2)))))))) (\lambda (t5: -T).(\lambda (H4: (subst0 i0 u0 t0 t5)).(\lambda (c0: C).(\lambda (H5: -(csubst0 i0 u0 c c0)).(\lambda (e: C).(\lambda (H6: (getl i0 c (CHead e (Bind -Abbr) u0))).(lt_le_e i i0 (pc3 c0 t2 t5) (\lambda (H7: (lt i i0)).(let H8 -\def (csubst0_getl_lt i0 i H7 c c0 u0 H5 (CHead d (Bind Abbr) u) H0) in -(or4_ind (getl i c0 (CHead d (Bind Abbr) u)) (ex3_4 B C T T (\lambda (b: -B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind -Abbr) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: -C).(\lambda (_: T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 -(minus i0 (S i)) u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: -C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 -(Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(u1: T).(getl i c0 (CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: -C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) -(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) -u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 -(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))) -(pc3 c0 t2 t5) (\lambda (H9: (getl i c0 (CHead d (Bind Abbr) u))).(pc3_pr2_u -c0 t3 t2 (pr2_free c0 t2 t3 H1) t5 (pc3_pr3_r c0 t3 t5 (pr3_sing c0 t0 t3 -(pr2_delta c0 d u i H9 t3 t3 (pr0_refl t3) t0 H2) t5 (pr3_pr2 c0 t0 t5 -(pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e -(Bind Abbr) u0) H6) t0 t0 (pr0_refl t0) t5 H4)))))) (\lambda (H9: (ex3_4 B C -T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C -(CHead d (Bind Abbr) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda -(e0: C).(\lambda (_: T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 -(minus i0 (S i)) u0 u1 w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda -(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead -e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) -u0 u1 w))))) (pc3 c0 t2 t5) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: -T).(\lambda (x3: T).(\lambda (H10: (eq C (CHead d (Bind Abbr) u) (CHead x1 -(Bind x0) x2))).(\lambda (H11: (getl i c0 (CHead x1 (Bind x0) x3))).(\lambda -(H12: (subst0 (minus i0 (S i)) u0 x2 x3)).(let H13 \def (f_equal C C (\lambda -(e0: C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow -d | (CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind -x0) x2) H10) in ((let H14 \def (f_equal C B (\lambda (e0: C).(match e0 in C -return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) -\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead -x1 (Bind x0) x2) H10) in ((let H15 \def (f_equal C T (\lambda (e0: C).(match -e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ -t6) \Rightarrow t6])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H10) in -(\lambda (H16: (eq B Abbr x0)).(\lambda (H17: (eq C d x1)).(let H18 \def -(eq_ind_r T x2 (\lambda (t6: T).(subst0 (minus i0 (S i)) u0 t6 x3)) H12 u -H15) in (let H19 \def (eq_ind_r C x1 (\lambda (c3: C).(getl i c0 (CHead c3 -(Bind x0) x3))) H11 d H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b: -B).(getl i c0 (CHead d (Bind b) x3))) H19 Abbr H16) in (ex2_ind T (\lambda -(t6: T).(subst0 i x3 t3 t6)) (\lambda (t6: T).(subst0 (S (plus (minus i0 (S -i)) i)) u0 t0 t6)) (pc3 c0 t2 t5) (\lambda (x: T).(\lambda (H21: (subst0 i x3 -t3 x)).(\lambda (H22: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let -H23 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n: -nat).(subst0 n u0 t0 x)) H22 i0 (lt_plus_minus_r i i0 H7)) in (pc3_pr2_u c0 x -t2 (pr2_delta c0 d x3 i H20 t2 t3 H1 x H21) t5 (pc3_pr2_u2 c0 t0 x (pr2_delta -c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e (Bind Abbr) -u0) H6) t0 t0 (pr0_refl t0) x H23) t5 (pc3_pr2_r c0 t0 t5 (pr2_delta c0 e u0 -i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e (Bind Abbr) u0) H6) -t0 t0 (pr0_refl t0) t5 H4)))))))) (subst0_subst0_back t3 t0 u i H2 x3 u0 -(minus i0 (S i)) H18)))))))) H14)) H13))))))))) H9)) (\lambda (H9: (ex3_4 B C -C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C -(CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda -(_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 (Bind b) -u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))).(ex3_4_ind B C C T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind -Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 (Bind b) u1)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S -i)) u0 e1 e2))))) (pc3 c0 t2 t5) (\lambda (x0: B).(\lambda (x1: C).(\lambda -(x2: C).(\lambda (x3: T).(\lambda (H10: (eq C (CHead d (Bind Abbr) u) (CHead -x1 (Bind x0) x3))).(\lambda (H11: (getl i c0 (CHead x2 (Bind x0) -x3))).(\lambda (H12: (csubst0 (minus i0 (S i)) u0 x1 x2)).(let H13 \def -(f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with -[(CSort _) \Rightarrow d | (CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind -Abbr) u) (CHead x1 (Bind x0) x3) H10) in ((let H14 \def (f_equal C B (\lambda -(e0: C).(match e0 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow -Abbr | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with -[(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) -u) (CHead x1 (Bind x0) x3) H10) in ((let H15 \def (f_equal C T (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | -(CHead _ _ t6) \Rightarrow t6])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) -x3) H10) in (\lambda (H16: (eq B Abbr x0)).(\lambda (H17: (eq C d x1)).(let -H18 \def (eq_ind_r T x3 (\lambda (t6: T).(getl i c0 (CHead x2 (Bind x0) t6))) -H11 u H15) in (let H19 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus -i0 (S i)) u0 c3 x2)) H12 d H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b: -B).(getl i c0 (CHead x2 (Bind b) u))) H18 Abbr H16) in (pc3_pr2_u c0 t0 t2 -(pr2_delta c0 x2 u i H20 t2 t3 H1 t0 H2) t5 (pc3_pr2_r c0 t0 t5 (pr2_delta c0 -e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e (Bind Abbr) u0) -H6) t0 t0 (pr0_refl t0) t5 H4))))))))) H14)) H13))))))))) H9)) (\lambda (H9: -(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) -u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: -T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 -(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 -e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 -(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda -(_: T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 -(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) -(pc3 c0 t2 t5) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda -(x3: T).(\lambda (x4: T).(\lambda (H10: (eq C (CHead d (Bind Abbr) u) (CHead -x1 (Bind x0) x3))).(\lambda (H11: (getl i c0 (CHead x2 (Bind x0) -x4))).(\lambda (H12: (subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H13: -(csubst0 (minus i0 (S i)) u0 x1 x2)).(let H14 \def (f_equal C C (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | -(CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) -x3) H10) in ((let H15 \def (f_equal C B (\lambda (e0: C).(match e0 in C -return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) -\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead -x1 (Bind x0) x3) H10) in ((let H16 \def (f_equal C T (\lambda (e0: C).(match -e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ -t6) \Rightarrow t6])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H10) in -(\lambda (H17: (eq B Abbr x0)).(\lambda (H18: (eq C d x1)).(let H19 \def -(eq_ind_r T x3 (\lambda (t6: T).(subst0 (minus i0 (S i)) u0 t6 x4)) H12 u -H16) in (let H20 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus i0 (S -i)) u0 c3 x2)) H13 d H18) in (let H21 \def (eq_ind_r B x0 (\lambda (b: -B).(getl i c0 (CHead x2 (Bind b) x4))) H11 Abbr H17) in (ex2_ind T (\lambda -(t6: T).(subst0 i x4 t3 t6)) (\lambda (t6: T).(subst0 (S (plus (minus i0 (S -i)) i)) u0 t0 t6)) (pc3 c0 t2 t5) (\lambda (x: T).(\lambda (H22: (subst0 i x4 -t3 x)).(\lambda (H23: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let -H24 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n: -nat).(subst0 n u0 t0 x)) H23 i0 (lt_plus_minus_r i i0 H7)) in (pc3_pr2_u c0 x -t2 (pr2_delta c0 x2 x4 i H21 t2 t3 H1 x H22) t5 (pc3_pr2_u2 c0 t0 x -(pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e -(Bind Abbr) u0) H6) t0 t0 (pr0_refl t0) x H24) t5 (pc3_pr2_r c0 t0 t5 -(pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e -(Bind Abbr) u0) H6) t0 t0 (pr0_refl t0) t5 H4)))))))) (subst0_subst0_back t3 -t0 u i H2 x4 u0 (minus i0 (S i)) H19)))))))) H15)) H14))))))))))) H9)) H8))) -(\lambda (H7: (le i0 i)).(pc3_pr2_u c0 t0 t2 (pr2_delta c0 d u i -(csubst0_getl_ge i0 i H7 c c0 u0 H5 (CHead d (Bind Abbr) u) H0) t2 t3 H1 t0 -H2) t5 (pc3_pr2_r c0 t0 t5 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n -i0) c c0 u0 H5 (CHead e (Bind Abbr) u0) H6) t0 t0 (pr0_refl t0) t5 -H4))))))))))) c2 t4 H3)))))))))))))))) c1 t t1 H)))). -(* COMMENTS -Initial nodes: 6191 -END *) - -theorem pc3_fsubst0: - \forall (c1: C).(\forall (t1: T).(\forall (t: T).((pc3 c1 t1 t) \to (\forall -(i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t2: T).((fsubst0 i u c1 -t1 c2 t2) \to (\forall (e: C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3 -c2 t2 t))))))))))) -\def - \lambda (c1: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H: (pc3 c1 t1 -t)).(pc3_ind_left c1 (\lambda (t0: T).(\lambda (t2: T).(\forall (i: -nat).(\forall (u: T).(\forall (c2: C).(\forall (t3: T).((fsubst0 i u c1 t0 c2 -t3) \to (\forall (e: C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3 c2 t3 -t2)))))))))) (\lambda (t0: T).(\lambda (i: nat).(\lambda (u: T).(\lambda (c2: -C).(\lambda (t2: T).(\lambda (H0: (fsubst0 i u c1 t0 c2 t2)).(fsubst0_ind i u -c1 t0 (\lambda (c: C).(\lambda (t3: T).(\forall (e: C).((getl i c1 (CHead e -(Bind Abbr) u)) \to (pc3 c t3 t0))))) (\lambda (t3: T).(\lambda (H1: (subst0 -i u t0 t3)).(\lambda (e: C).(\lambda (H2: (getl i c1 (CHead e (Bind Abbr) -u))).(pc3_pr2_x c1 t3 t0 (pr2_delta c1 e u i H2 t0 t0 (pr0_refl t0) t3 -H1)))))) (\lambda (c0: C).(\lambda (_: (csubst0 i u c1 c0)).(\lambda (e: -C).(\lambda (_: (getl i c1 (CHead e (Bind Abbr) u))).(pc3_refl c0 t0))))) -(\lambda (t3: T).(\lambda (H1: (subst0 i u t0 t3)).(\lambda (c0: C).(\lambda -(H2: (csubst0 i u c1 c0)).(\lambda (e: C).(\lambda (H3: (getl i c1 (CHead e -(Bind Abbr) u))).(pc3_pr2_x c0 t3 t0 (pr2_delta c0 e u i (csubst0_getl_ge i i -(le_n i) c1 c0 u H2 (CHead e (Bind Abbr) u) H3) t0 t0 (pr0_refl t0) t3 -H1)))))))) c2 t2 H0))))))) (\lambda (t0: T).(\lambda (t2: T).(\lambda (H0: -(pr2 c1 t0 t2)).(\lambda (t3: T).(\lambda (H1: (pc3 c1 t2 t3)).(\lambda (H2: -((\forall (i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t4: -T).((fsubst0 i u c1 t2 c2 t4) \to (\forall (e: C).((getl i c1 (CHead e (Bind -Abbr) u)) \to (pc3 c2 t4 t3)))))))))).(\lambda (i: nat).(\lambda (u: -T).(\lambda (c2: C).(\lambda (t4: T).(\lambda (H3: (fsubst0 i u c1 t0 c2 -t4)).(fsubst0_ind i u c1 t0 (\lambda (c: C).(\lambda (t5: T).(\forall (e: -C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3 c t5 t3))))) (\lambda (t5: -T).(\lambda (H4: (subst0 i u t0 t5)).(\lambda (e: C).(\lambda (H5: (getl i c1 -(CHead e (Bind Abbr) u))).(pc3_t t2 c1 t5 (pc3_pr2_fsubst0 c1 t0 t2 H0 i u c1 -t5 (fsubst0_snd i u c1 t0 t5 H4) e H5) t3 H1))))) (\lambda (c0: C).(\lambda -(H4: (csubst0 i u c1 c0)).(\lambda (e: C).(\lambda (H5: (getl i c1 (CHead e -(Bind Abbr) u))).(pc3_t t2 c0 t0 (pc3_pr2_fsubst0 c1 t0 t2 H0 i u c0 t0 -(fsubst0_fst i u c1 t0 c0 H4) e H5) t3 (H2 i u c0 t2 (fsubst0_fst i u c1 t2 -c0 H4) e H5)))))) (\lambda (t5: T).(\lambda (H4: (subst0 i u t0 t5)).(\lambda -(c0: C).(\lambda (H5: (csubst0 i u c1 c0)).(\lambda (e: C).(\lambda (H6: -(getl i c1 (CHead e (Bind Abbr) u))).(pc3_t t2 c0 t5 (pc3_pr2_fsubst0 c1 t0 -t2 H0 i u c0 t5 (fsubst0_both i u c1 t0 t5 H4 c0 H5) e H6) t3 (H2 i u c0 t2 -(fsubst0_fst i u c1 t2 c0 H5) e H6)))))))) c2 t4 H3)))))))))))) (\lambda (t0: -T).(\lambda (t2: T).(\lambda (H0: (pr2 c1 t0 t2)).(\lambda (t3: T).(\lambda -(H1: (pc3 c1 t0 t3)).(\lambda (H2: ((\forall (i: nat).(\forall (u: -T).(\forall (c2: C).(\forall (t4: T).((fsubst0 i u c1 t0 c2 t4) \to (\forall -(e: C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3 c2 t4 -t3)))))))))).(\lambda (i: nat).(\lambda (u: T).(\lambda (c2: C).(\lambda (t4: -T).(\lambda (H3: (fsubst0 i u c1 t2 c2 t4)).(fsubst0_ind i u c1 t2 (\lambda -(c: C).(\lambda (t5: T).(\forall (e: C).((getl i c1 (CHead e (Bind Abbr) u)) -\to (pc3 c t5 t3))))) (\lambda (t5: T).(\lambda (H4: (subst0 i u t2 -t5)).(\lambda (e: C).(\lambda (H5: (getl i c1 (CHead e (Bind Abbr) -u))).(pc3_t t0 c1 t5 (pc3_s c1 t5 t0 (pc3_pr2_fsubst0_back c1 t0 t2 H0 i u c1 -t5 (fsubst0_snd i u c1 t2 t5 H4) e H5)) t3 H1))))) (\lambda (c0: C).(\lambda -(H4: (csubst0 i u c1 c0)).(\lambda (e: C).(\lambda (H5: (getl i c1 (CHead e -(Bind Abbr) u))).(pc3_t t0 c0 t2 (pc3_s c0 t2 t0 (pc3_pr2_fsubst0_back c1 t0 -t2 H0 i u c0 t2 (fsubst0_fst i u c1 t2 c0 H4) e H5)) t3 (H2 i u c0 t0 -(fsubst0_fst i u c1 t0 c0 H4) e H5)))))) (\lambda (t5: T).(\lambda (H4: -(subst0 i u t2 t5)).(\lambda (c0: C).(\lambda (H5: (csubst0 i u c1 -c0)).(\lambda (e: C).(\lambda (H6: (getl i c1 (CHead e (Bind Abbr) -u))).(pc3_t t0 c0 t5 (pc3_s c0 t5 t0 (pc3_pr2_fsubst0_back c1 t0 t2 H0 i u c0 -t5 (fsubst0_both i u c1 t2 t5 H4 c0 H5) e H6)) t3 (H2 i u c0 t0 (fsubst0_fst -i u c1 t0 c0 H5) e H6)))))))) c2 t4 H3)))))))))))) t1 t H)))). -(* COMMENTS -Initial nodes: 1249 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/fwd.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/fwd.ma deleted file mode 100644 index c89e4563c..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/fwd.ma +++ /dev/null @@ -1,333 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pc3/props.ma". - -include "Basic-1/pr3/fwd.ma". - -theorem pc3_gen_sort: - \forall (c: C).(\forall (m: nat).(\forall (n: nat).((pc3 c (TSort m) (TSort -n)) \to (eq nat m n)))) -\def - \lambda (c: C).(\lambda (m: nat).(\lambda (n: nat).(\lambda (H: (pc3 c -(TSort m) (TSort n))).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr3 c -(TSort m) t)) (\lambda (t: T).(pr3 c (TSort n) t)) (eq nat m n) (\lambda (x: -T).(\lambda (H1: (pr3 c (TSort m) x)).(\lambda (H2: (pr3 c (TSort n) x)).(let -H3 \def (eq_ind T x (\lambda (t: T).(eq T t (TSort n))) (pr3_gen_sort c x n -H2) (TSort m) (pr3_gen_sort c x m H1)) in (let H4 \def (f_equal T nat -(\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort n0) -\Rightarrow n0 | (TLRef _) \Rightarrow m | (THead _ _ _) \Rightarrow m])) -(TSort m) (TSort n) H3) in H4))))) H0))))). -(* COMMENTS -Initial nodes: 153 -END *) - -theorem pc3_gen_abst: - \forall (c: C).(\forall (u1: T).(\forall (u2: T).(\forall (t1: T).(\forall -(t2: T).((pc3 c (THead (Bind Abst) u1 t1) (THead (Bind Abst) u2 t2)) \to -(land (pc3 c u1 u2) (\forall (b: B).(\forall (u: T).(pc3 (CHead c (Bind b) u) -t1 t2))))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (t1: T).(\lambda -(t2: T).(\lambda (H: (pc3 c (THead (Bind Abst) u1 t1) (THead (Bind Abst) u2 -t2))).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr3 c (THead (Bind Abst) -u1 t1) t)) (\lambda (t: T).(pr3 c (THead (Bind Abst) u2 t2) t)) (land (pc3 c -u1 u2) (\forall (b: B).(\forall (u: T).(pc3 (CHead c (Bind b) u) t1 t2)))) -(\lambda (x: T).(\lambda (H1: (pr3 c (THead (Bind Abst) u1 t1) x)).(\lambda -(H2: (pr3 c (THead (Bind Abst) u2 t2) x)).(let H3 \def (pr3_gen_abst c u2 t2 -x H2) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead -(Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c u2 u3))) -(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead -c (Bind b) u) t2 t3))))) (land (pc3 c u1 u2) (\forall (b: B).(\forall (u: -T).(pc3 (CHead c (Bind b) u) t1 t2)))) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H4: (eq T x (THead (Bind Abst) x0 x1))).(\lambda (H5: (pr3 c u2 -x0)).(\lambda (H6: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -t2 x1))))).(let H7 \def (pr3_gen_abst c u1 t1 x H1) in (ex3_2_ind T T -(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u3 t3)))) -(\lambda (u3: T).(\lambda (_: T).(pr3 c u1 u3))) (\lambda (_: T).(\lambda -(t3: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t1 t3))))) -(land (pc3 c u1 u2) (\forall (b: B).(\forall (u: T).(pc3 (CHead c (Bind b) u) -t1 t2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H8: (eq T x (THead -(Bind Abst) x2 x3))).(\lambda (H9: (pr3 c u1 x2)).(\lambda (H10: ((\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t1 x3))))).(let H11 \def -(eq_ind T x (\lambda (t: T).(eq T t (THead (Bind Abst) x0 x1))) H4 (THead -(Bind Abst) x2 x3) H8) in (let H12 \def (f_equal T T (\lambda (e: T).(match e -in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x2 | (TLRef _) -\Rightarrow x2 | (THead _ t _) \Rightarrow t])) (THead (Bind Abst) x2 x3) -(THead (Bind Abst) x0 x1) H11) in ((let H13 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x3 | -(TLRef _) \Rightarrow x3 | (THead _ _ t) \Rightarrow t])) (THead (Bind Abst) -x2 x3) (THead (Bind Abst) x0 x1) H11) in (\lambda (H14: (eq T x2 x0)).(let -H15 \def (eq_ind T x3 (\lambda (t: T).(\forall (b: B).(\forall (u: T).(pr3 -(CHead c (Bind b) u) t1 t)))) H10 x1 H13) in (let H16 \def (eq_ind T x2 -(\lambda (t: T).(pr3 c u1 t)) H9 x0 H14) in (conj (pc3 c u1 u2) (\forall (b: -B).(\forall (u: T).(pc3 (CHead c (Bind b) u) t1 t2))) (pc3_pr3_t c u1 x0 H16 -u2 H5) (\lambda (b: B).(\lambda (u: T).(pc3_pr3_t (CHead c (Bind b) u) t1 x1 -(H15 b u) t2 (H6 b u))))))))) H12)))))))) H7))))))) H3))))) H0))))))). -(* COMMENTS -Initial nodes: 715 -END *) - -theorem pc3_gen_abst_shift: - \forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).((pc3 c -(THead (Bind Abst) u t1) (THead (Bind Abst) u t2)) \to (pc3 (CHead c (Bind -Abst) u) t1 t2))))) -\def - \lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H: (pc3 c (THead (Bind Abst) u t1) (THead (Bind Abst) u t2))).(let H_x \def -(pc3_gen_abst c u u t1 t2 H) in (let H0 \def H_x in (land_ind (pc3 c u u) -(\forall (b: B).(\forall (u0: T).(pc3 (CHead c (Bind b) u0) t1 t2))) (pc3 -(CHead c (Bind Abst) u) t1 t2) (\lambda (_: (pc3 c u u)).(\lambda (H2: -((\forall (b: B).(\forall (u0: T).(pc3 (CHead c (Bind b) u0) t1 t2))))).(H2 -Abst u))) H0))))))). -(* COMMENTS -Initial nodes: 129 -END *) - -theorem pc3_gen_lift: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).(\forall (h: nat).(\forall -(d: nat).((pc3 c (lift h d t1) (lift h d t2)) \to (\forall (e: C).((drop h d -c e) \to (pc3 e t1 t2)))))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (h: nat).(\lambda -(d: nat).(\lambda (H: (pc3 c (lift h d t1) (lift h d t2))).(\lambda (e: -C).(\lambda (H0: (drop h d c e)).(let H1 \def H in (ex2_ind T (\lambda (t: -T).(pr3 c (lift h d t1) t)) (\lambda (t: T).(pr3 c (lift h d t2) t)) (pc3 e -t1 t2) (\lambda (x: T).(\lambda (H2: (pr3 c (lift h d t1) x)).(\lambda (H3: -(pr3 c (lift h d t2) x)).(let H4 \def (pr3_gen_lift c t2 x h d H3 e H0) in -(ex2_ind T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr3 e -t2 t3)) (pc3 e t1 t2) (\lambda (x0: T).(\lambda (H5: (eq T x (lift h d -x0))).(\lambda (H6: (pr3 e t2 x0)).(let H7 \def (pr3_gen_lift c t1 x h d H2 e -H0) in (ex2_ind T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: -T).(pr3 e t1 t3)) (pc3 e t1 t2) (\lambda (x1: T).(\lambda (H8: (eq T x (lift -h d x1))).(\lambda (H9: (pr3 e t1 x1)).(let H10 \def (eq_ind T x (\lambda (t: -T).(eq T t (lift h d x0))) H5 (lift h d x1) H8) in (let H11 \def (eq_ind T x1 -(\lambda (t: T).(pr3 e t1 t)) H9 x0 (lift_inj x1 x0 h d H10)) in (pc3_pr3_t e -t1 x0 H11 t2 H6)))))) H7))))) H4))))) H1))))))))). -(* COMMENTS -Initial nodes: 363 -END *) - -theorem pc3_gen_not_abst: - \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (t1: -T).(\forall (t2: T).(\forall (u1: T).(\forall (u2: T).((pc3 c (THead (Bind b) -u1 t1) (THead (Bind Abst) u2 t2)) \to (pc3 (CHead c (Bind b) u1) t1 (lift (S -O) O (THead (Bind Abst) u2 t2)))))))))) -\def - \lambda (b: B).(B_ind (\lambda (b0: B).((not (eq B b0 Abst)) \to (\forall -(c: C).(\forall (t1: T).(\forall (t2: T).(\forall (u1: T).(\forall (u2: -T).((pc3 c (THead (Bind b0) u1 t1) (THead (Bind Abst) u2 t2)) \to (pc3 (CHead -c (Bind b0) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2))))))))))) (\lambda -(_: (not (eq B Abbr Abst))).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H0: (pc3 c (THead (Bind Abbr) -u1 t1) (THead (Bind Abst) u2 t2))).(let H1 \def H0 in (ex2_ind T (\lambda (t: -T).(pr3 c (THead (Bind Abbr) u1 t1) t)) (\lambda (t: T).(pr3 c (THead (Bind -Abst) u2 t2) t)) (pc3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O (THead (Bind -Abst) u2 t2))) (\lambda (x: T).(\lambda (H2: (pr3 c (THead (Bind Abbr) u1 t1) -x)).(\lambda (H3: (pr3 c (THead (Bind Abst) u2 t2) x)).(let H4 \def -(pr3_gen_abbr c u1 t1 x H2) in (or_ind (ex3_2 T T (\lambda (u3: T).(\lambda -(t3: T).(eq T x (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: -T).(pr3 c u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr3 (CHead c (Bind Abbr) -u1) t1 t3)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O x)) (pc3 (CHead -c (Bind Abbr) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2))) (\lambda (H5: -(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3 -t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c u1 u3))) (\lambda (_: -T).(\lambda (t3: T).(pr3 (CHead c (Bind Abbr) u1) t1 t3))))).(ex3_2_ind T T -(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3 t3)))) -(\lambda (u3: T).(\lambda (_: T).(pr3 c u1 u3))) (\lambda (_: T).(\lambda -(t3: T).(pr3 (CHead c (Bind Abbr) u1) t1 t3))) (pc3 (CHead c (Bind Abbr) u1) -t1 (lift (S O) O (THead (Bind Abst) u2 t2))) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H6: (eq T x (THead (Bind Abbr) x0 x1))).(\lambda (_: (pr3 c u1 -x0)).(\lambda (_: (pr3 (CHead c (Bind Abbr) u1) t1 x1)).(let H9 \def -(pr3_gen_abst c u2 t2 x H3) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: -T).(eq T x (THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 -c u2 u3))) (\lambda (_: T).(\lambda (t3: T).(\forall (b0: B).(\forall (u: -T).(pr3 (CHead c (Bind b0) u) t2 t3))))) (pc3 (CHead c (Bind Abbr) u1) t1 -(lift (S O) O (THead (Bind Abst) u2 t2))) (\lambda (x2: T).(\lambda (x3: -T).(\lambda (H10: (eq T x (THead (Bind Abst) x2 x3))).(\lambda (_: (pr3 c u2 -x2)).(\lambda (_: ((\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) -u) t2 x3))))).(let H13 \def (eq_ind T x (\lambda (t: T).(eq T t (THead (Bind -Abbr) x0 x1))) H6 (THead (Bind Abst) x2 x3) H10) in (let H14 \def (eq_ind T -(THead (Bind Abst) x2 x3) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind b0) \Rightarrow (match b0 in B return (\lambda (_: B).Prop) with [Abbr -\Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat -_) \Rightarrow False])])) I (THead (Bind Abbr) x0 x1) H13) in (False_ind (pc3 -(CHead c (Bind Abbr) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2))) -H14)))))))) H9))))))) H5)) (\lambda (H5: (pr3 (CHead c (Bind Abbr) u1) t1 -(lift (S O) O x))).(let H6 \def (pr3_gen_abst c u2 t2 x H3) in (ex3_2_ind T T -(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u3 t3)))) -(\lambda (u3: T).(\lambda (_: T).(pr3 c u2 u3))) (\lambda (_: T).(\lambda -(t3: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t2 -t3))))) (pc3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O (THead (Bind Abst) u2 -t2))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: (eq T x (THead (Bind -Abst) x0 x1))).(\lambda (H8: (pr3 c u2 x0)).(\lambda (H9: ((\forall (b0: -B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t2 x1))))).(let H10 \def -(eq_ind T x (\lambda (t: T).(pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O -t))) H5 (THead (Bind Abst) x0 x1) H7) in (pc3_pr3_t (CHead c (Bind Abbr) u1) -t1 (lift (S O) O (THead (Bind Abst) x0 x1)) H10 (lift (S O) O (THead (Bind -Abst) u2 t2)) (pr3_lift (CHead c (Bind Abbr) u1) c (S O) O (drop_drop (Bind -Abbr) O c c (drop_refl c) u1) (THead (Bind Abst) u2 t2) (THead (Bind Abst) x0 -x1) (pr3_head_12 c u2 x0 H8 (Bind Abst) t2 x1 (H9 Abst x0)))))))))) H6))) -H4))))) H1))))))))) (\lambda (H: (not (eq B Abst Abst))).(\lambda (c: -C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (u1: T).(\lambda (u2: -T).(\lambda (_: (pc3 c (THead (Bind Abst) u1 t1) (THead (Bind Abst) u2 -t2))).(let H1 \def (match (H (refl_equal B Abst)) in False return (\lambda -(_: False).(pc3 (CHead c (Bind Abst) u1) t1 (lift (S O) O (THead (Bind Abst) -u2 t2)))) with []) in H1)))))))) (\lambda (_: (not (eq B Void -Abst))).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (u1: -T).(\lambda (u2: T).(\lambda (H0: (pc3 c (THead (Bind Void) u1 t1) (THead -(Bind Abst) u2 t2))).(let H1 \def H0 in (ex2_ind T (\lambda (t: T).(pr3 c -(THead (Bind Void) u1 t1) t)) (\lambda (t: T).(pr3 c (THead (Bind Abst) u2 -t2) t)) (pc3 (CHead c (Bind Void) u1) t1 (lift (S O) O (THead (Bind Abst) u2 -t2))) (\lambda (x: T).(\lambda (H2: (pr3 c (THead (Bind Void) u1 t1) -x)).(\lambda (H3: (pr3 c (THead (Bind Abst) u2 t2) x)).(let H4 \def -(pr3_gen_void c u1 t1 x H2) in (or_ind (ex3_2 T T (\lambda (u3: T).(\lambda -(t3: T).(eq T x (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: -T).(pr3 c u1 u3))) (\lambda (_: T).(\lambda (t3: T).(\forall (b0: B).(\forall -(u: T).(pr3 (CHead c (Bind b0) u) t1 t3)))))) (pr3 (CHead c (Bind Void) u1) -t1 (lift (S O) O x)) (pc3 (CHead c (Bind Void) u1) t1 (lift (S O) O (THead -(Bind Abst) u2 t2))) (\lambda (H5: (ex3_2 T T (\lambda (u3: T).(\lambda (t3: -T).(eq T x (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 -c u1 u3))) (\lambda (_: T).(\lambda (t3: T).(\forall (b0: B).(\forall (u: -T).(pr3 (CHead c (Bind b0) u) t1 t3))))))).(ex3_2_ind T T (\lambda (u3: -T).(\lambda (t3: T).(eq T x (THead (Bind Void) u3 t3)))) (\lambda (u3: -T).(\lambda (_: T).(pr3 c u1 u3))) (\lambda (_: T).(\lambda (t3: T).(\forall -(b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t1 t3))))) (pc3 (CHead c -(Bind Void) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2))) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H6: (eq T x (THead (Bind Void) x0 -x1))).(\lambda (_: (pr3 c u1 x0)).(\lambda (_: ((\forall (b0: B).(\forall (u: -T).(pr3 (CHead c (Bind b0) u) t1 x1))))).(let H9 \def (pr3_gen_abst c u2 t2 x -H3) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind -Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c u2 u3))) (\lambda (_: -T).(\lambda (t3: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) -u) t2 t3))))) (pc3 (CHead c (Bind Void) u1) t1 (lift (S O) O (THead (Bind -Abst) u2 t2))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H10: (eq T x -(THead (Bind Abst) x2 x3))).(\lambda (_: (pr3 c u2 x2)).(\lambda (_: -((\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t2 x3))))).(let -H13 \def (eq_ind T x (\lambda (t: T).(eq T t (THead (Bind Void) x0 x1))) H6 -(THead (Bind Abst) x2 x3) H10) in (let H14 \def (eq_ind T (THead (Bind Abst) -x2 x3) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b0) -\Rightarrow (match b0 in B return (\lambda (_: B).Prop) with [Abbr -\Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat -_) \Rightarrow False])])) I (THead (Bind Void) x0 x1) H13) in (False_ind (pc3 -(CHead c (Bind Void) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2))) -H14)))))))) H9))))))) H5)) (\lambda (H5: (pr3 (CHead c (Bind Void) u1) t1 -(lift (S O) O x))).(let H6 \def (pr3_gen_abst c u2 t2 x H3) in (ex3_2_ind T T -(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u3 t3)))) -(\lambda (u3: T).(\lambda (_: T).(pr3 c u2 u3))) (\lambda (_: T).(\lambda -(t3: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t2 -t3))))) (pc3 (CHead c (Bind Void) u1) t1 (lift (S O) O (THead (Bind Abst) u2 -t2))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: (eq T x (THead (Bind -Abst) x0 x1))).(\lambda (H8: (pr3 c u2 x0)).(\lambda (H9: ((\forall (b0: -B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t2 x1))))).(let H10 \def -(eq_ind T x (\lambda (t: T).(pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O -t))) H5 (THead (Bind Abst) x0 x1) H7) in (pc3_pr3_t (CHead c (Bind Void) u1) -t1 (lift (S O) O (THead (Bind Abst) x0 x1)) H10 (lift (S O) O (THead (Bind -Abst) u2 t2)) (pr3_lift (CHead c (Bind Void) u1) c (S O) O (drop_drop (Bind -Void) O c c (drop_refl c) u1) (THead (Bind Abst) u2 t2) (THead (Bind Abst) x0 -x1) (pr3_head_12 c u2 x0 H8 (Bind Abst) t2 x1 (H9 Abst x0)))))))))) H6))) -H4))))) H1))))))))) b). -(* COMMENTS -Initial nodes: 2427 -END *) - -theorem pc3_gen_lift_abst: - \forall (c: C).(\forall (t: T).(\forall (t2: T).(\forall (u2: T).(\forall -(h: nat).(\forall (d: nat).((pc3 c (lift h d t) (THead (Bind Abst) u2 t2)) -\to (\forall (e: C).((drop h d c e) \to (ex3_2 T T (\lambda (u1: T).(\lambda -(t1: T).(pr3 e t (THead (Bind Abst) u1 t1)))) (\lambda (u1: T).(\lambda (_: -T).(pr3 c u2 (lift h d u1)))) (\lambda (_: T).(\lambda (t1: T).(\forall (b: -B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t2 (lift h (S d) -t1))))))))))))))) -\def - \lambda (c: C).(\lambda (t: T).(\lambda (t2: T).(\lambda (u2: T).(\lambda -(h: nat).(\lambda (d: nat).(\lambda (H: (pc3 c (lift h d t) (THead (Bind -Abst) u2 t2))).(\lambda (e: C).(\lambda (H0: (drop h d c e)).(let H1 \def H -in (ex2_ind T (\lambda (t0: T).(pr3 c (lift h d t) t0)) (\lambda (t0: T).(pr3 -c (THead (Bind Abst) u2 t2) t0)) (ex3_2 T T (\lambda (u1: T).(\lambda (t1: -T).(pr3 e t (THead (Bind Abst) u1 t1)))) (\lambda (u1: T).(\lambda (_: -T).(pr3 c u2 (lift h d u1)))) (\lambda (_: T).(\lambda (t1: T).(\forall (b: -B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t2 (lift h (S d) t1))))))) -(\lambda (x: T).(\lambda (H2: (pr3 c (lift h d t) x)).(\lambda (H3: (pr3 c -(THead (Bind Abst) u2 t2) x)).(let H4 \def (pr3_gen_lift c t x h d H2 e H0) -in (ex2_ind T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr3 -e t t3)) (ex3_2 T T (\lambda (u1: T).(\lambda (t1: T).(pr3 e t (THead (Bind -Abst) u1 t1)))) (\lambda (u1: T).(\lambda (_: T).(pr3 c u2 (lift h d u1)))) -(\lambda (_: T).(\lambda (t1: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead -c (Bind b) u) t2 (lift h (S d) t1))))))) (\lambda (x0: T).(\lambda (H5: (eq T -x (lift h d x0))).(\lambda (H6: (pr3 e t x0)).(let H7 \def (pr3_gen_abst c u2 -t2 x H3) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead -(Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c u2 u3))) -(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead -c (Bind b) u) t2 t3))))) (ex3_2 T T (\lambda (u1: T).(\lambda (t1: T).(pr3 e -t (THead (Bind Abst) u1 t1)))) (\lambda (u1: T).(\lambda (_: T).(pr3 c u2 -(lift h d u1)))) (\lambda (_: T).(\lambda (t1: T).(\forall (b: B).(\forall -(u: T).(pr3 (CHead c (Bind b) u) t2 (lift h (S d) t1))))))) (\lambda (x1: -T).(\lambda (x2: T).(\lambda (H8: (eq T x (THead (Bind Abst) x1 -x2))).(\lambda (H9: (pr3 c u2 x1)).(\lambda (H10: ((\forall (b: B).(\forall -(u: T).(pr3 (CHead c (Bind b) u) t2 x2))))).(let H11 \def (eq_ind T x -(\lambda (t0: T).(eq T t0 (lift h d x0))) H5 (THead (Bind Abst) x1 x2) H8) in -(ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x0 (THead (Bind Abst) y -z)))) (\lambda (y: T).(\lambda (_: T).(eq T x1 (lift h d y)))) (\lambda (_: -T).(\lambda (z: T).(eq T x2 (lift h (S d) z)))) (ex3_2 T T (\lambda (u1: -T).(\lambda (t1: T).(pr3 e t (THead (Bind Abst) u1 t1)))) (\lambda (u1: -T).(\lambda (_: T).(pr3 c u2 (lift h d u1)))) (\lambda (_: T).(\lambda (t1: -T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t2 (lift h (S d) -t1))))))) (\lambda (x3: T).(\lambda (x4: T).(\lambda (H12: (eq T x0 (THead -(Bind Abst) x3 x4))).(\lambda (H13: (eq T x1 (lift h d x3))).(\lambda (H14: -(eq T x2 (lift h (S d) x4))).(let H15 \def (eq_ind T x2 (\lambda (t0: -T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t2 t0)))) H10 -(lift h (S d) x4) H14) in (let H16 \def (eq_ind T x1 (\lambda (t0: T).(pr3 c -u2 t0)) H9 (lift h d x3) H13) in (let H17 \def (eq_ind T x0 (\lambda (t0: -T).(pr3 e t t0)) H6 (THead (Bind Abst) x3 x4) H12) in (ex3_2_intro T T -(\lambda (u1: T).(\lambda (t1: T).(pr3 e t (THead (Bind Abst) u1 t1)))) -(\lambda (u1: T).(\lambda (_: T).(pr3 c u2 (lift h d u1)))) (\lambda (_: -T).(\lambda (t1: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -t2 (lift h (S d) t1)))))) x3 x4 H17 H16 H15))))))))) (lift_gen_bind Abst x1 -x2 x0 h d H11)))))))) H7))))) H4))))) H1)))))))))). -(* COMMENTS -Initial nodes: 973 -END *) - -theorem pc3_gen_sort_abst: - \forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (n: nat).((pc3 c -(TSort n) (THead (Bind Abst) u t)) \to (\forall (P: Prop).P))))) -\def - \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (n: nat).(\lambda -(H: (pc3 c (TSort n) (THead (Bind Abst) u t))).(\lambda (P: Prop).(let H0 -\def H in (ex2_ind T (\lambda (t0: T).(pr3 c (TSort n) t0)) (\lambda (t0: -T).(pr3 c (THead (Bind Abst) u t) t0)) P (\lambda (x: T).(\lambda (H1: (pr3 c -(TSort n) x)).(\lambda (H2: (pr3 c (THead (Bind Abst) u t) x)).(let H3 \def -(pr3_gen_abst c u t x H2) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: -T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 -c u u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u0: -T).(pr3 (CHead c (Bind b) u0) t t2))))) P (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H4: (eq T x (THead (Bind Abst) x0 x1))).(\lambda (_: (pr3 c u -x0)).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pr3 (CHead c (Bind b) -u0) t x1))))).(let H7 \def (eq_ind T x (\lambda (t0: T).(eq T t0 (TSort n))) -(pr3_gen_sort c x n H1) (THead (Bind Abst) x0 x1) H4) in (let H8 \def (eq_ind -T (THead (Bind Abst) x0 x1) (\lambda (ee: T).(match ee in T return (\lambda -(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False -| (THead _ _ _) \Rightarrow True])) I (TSort n) H7) in (False_ind P -H8)))))))) H3))))) H0))))))). -(* COMMENTS -Initial nodes: 303 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/left.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/left.ma deleted file mode 100644 index 12d22d10c..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/left.ma +++ /dev/null @@ -1,125 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pc3/props.ma". - -theorem pc3_ind_left__pc3_left_pr3: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to -(pc3_left c t1 t2)))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 -t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(pc3_left c t t0))) (\lambda -(t: T).(pc3_left_r c t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 -c t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2: -(pc3_left c t0 t4)).(pc3_left_ur c t3 t0 H0 t4 H2))))))) t1 t2 H)))). -(* COMMENTS -Initial nodes: 87 -END *) - -theorem pc3_ind_left__pc3_left_trans: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3_left c t1 t2) \to -(\forall (t3: T).((pc3_left c t2 t3) \to (pc3_left c t1 t3)))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3_left c t1 -t2)).(pc3_left_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (t3: -T).((pc3_left c t0 t3) \to (pc3_left c t t3))))) (\lambda (t: T).(\lambda -(t3: T).(\lambda (H0: (pc3_left c t t3)).H0))) (\lambda (t0: T).(\lambda (t3: -T).(\lambda (H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t3 -t4)).(\lambda (H2: ((\forall (t5: T).((pc3_left c t4 t5) \to (pc3_left c t3 -t5))))).(\lambda (t5: T).(\lambda (H3: (pc3_left c t4 t5)).(pc3_left_ur c t0 -t3 H0 t5 (H2 t5 H3)))))))))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: -(pr2 c t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t0 t4)).(\lambda -(H2: ((\forall (t5: T).((pc3_left c t4 t5) \to (pc3_left c t0 -t5))))).(\lambda (t5: T).(\lambda (H3: (pc3_left c t4 t5)).(pc3_left_ux c t0 -t3 H0 t5 (H2 t5 H3)))))))))) t1 t2 H)))). -(* COMMENTS -Initial nodes: 195 -END *) - -theorem pc3_ind_left__pc3_left_sym: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3_left c t1 t2) \to -(pc3_left c t2 t1)))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3_left c t1 -t2)).(pc3_left_ind c (\lambda (t: T).(\lambda (t0: T).(pc3_left c t0 t))) -(\lambda (t: T).(pc3_left_r c t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda -(H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t3 -t4)).(\lambda (H2: (pc3_left c t4 t3)).(pc3_ind_left__pc3_left_trans c t4 t3 -H2 t0 (pc3_left_ux c t0 t3 H0 t0 (pc3_left_r c t0))))))))) (\lambda (t0: -T).(\lambda (t3: T).(\lambda (H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda -(_: (pc3_left c t0 t4)).(\lambda (H2: (pc3_left c t4 -t0)).(pc3_ind_left__pc3_left_trans c t4 t0 H2 t3 (pc3_left_ur c t0 t3 H0 t3 -(pc3_left_r c t3))))))))) t1 t2 H)))). -(* COMMENTS -Initial nodes: 163 -END *) - -theorem pc3_ind_left__pc3_left_pc3: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to -(pc3_left c t1 t2)))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c t1 -t2)).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: -T).(pr3 c t2 t)) (pc3_left c t1 t2) (\lambda (x: T).(\lambda (H1: (pr3 c t1 -x)).(\lambda (H2: (pr3 c t2 x)).(pc3_ind_left__pc3_left_trans c t1 x -(pc3_ind_left__pc3_left_pr3 c t1 x H1) t2 (pc3_ind_left__pc3_left_sym c t2 x -(pc3_ind_left__pc3_left_pr3 c t2 x H2)))))) H0))))). -(* COMMENTS -Initial nodes: 105 -END *) - -theorem pc3_ind_left__pc3_pc3_left: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3_left c t1 t2) \to -(pc3 c t1 t2)))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3_left c t1 -t2)).(pc3_left_ind c (\lambda (t: T).(\lambda (t0: T).(pc3 c t t0))) (\lambda -(t: T).(pc3_refl c t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c -t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t3 t4)).(\lambda (H2: (pc3 -c t3 t4)).(pc3_t t3 c t0 (pc3_pr2_r c t0 t3 H0) t4 H2))))))) (\lambda (t0: -T).(\lambda (t3: T).(\lambda (H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda -(_: (pc3_left c t0 t4)).(\lambda (H2: (pc3 c t0 t4)).(pc3_t t0 c t3 -(pc3_pr2_x c t3 t0 H0) t4 H2))))))) t1 t2 H)))). -(* COMMENTS -Initial nodes: 147 -END *) - -theorem pc3_ind_left: - \forall (c: C).(\forall (P: ((T \to (T \to Prop)))).(((\forall (t: T).(P t -t))) \to (((\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall (t3: -T).((pc3 c t2 t3) \to ((P t2 t3) \to (P t1 t3)))))))) \to (((\forall (t1: -T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall (t3: T).((pc3 c t1 t3) \to -((P t1 t3) \to (P t2 t3)))))))) \to (\forall (t: T).(\forall (t0: T).((pc3 c -t t0) \to (P t t0)))))))) -\def - \lambda (c: C).(\lambda (P: ((T \to (T \to Prop)))).(\lambda (H: ((\forall -(t: T).(P t t)))).(\lambda (H0: ((\forall (t1: T).(\forall (t2: T).((pr2 c t1 -t2) \to (\forall (t3: T).((pc3 c t2 t3) \to ((P t2 t3) \to (P t1 -t3))))))))).(\lambda (H1: ((\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) -\to (\forall (t3: T).((pc3 c t1 t3) \to ((P t1 t3) \to (P t2 -t3))))))))).(\lambda (t: T).(\lambda (t0: T).(\lambda (H2: (pc3 c t -t0)).(pc3_left_ind c (\lambda (t1: T).(\lambda (t2: T).(P t1 t2))) H (\lambda -(t1: T).(\lambda (t2: T).(\lambda (H3: (pr2 c t1 t2)).(\lambda (t3: -T).(\lambda (H4: (pc3_left c t2 t3)).(\lambda (H5: (P t2 t3)).(H0 t1 t2 H3 t3 -(pc3_ind_left__pc3_pc3_left c t2 t3 H4) H5))))))) (\lambda (t1: T).(\lambda -(t2: T).(\lambda (H3: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda (H4: (pc3_left -c t1 t3)).(\lambda (H5: (P t1 t3)).(H1 t1 t2 H3 t3 -(pc3_ind_left__pc3_pc3_left c t1 t3 H4) H5))))))) t t0 -(pc3_ind_left__pc3_left_pc3 c t t0 H2))))))))). -(* COMMENTS -Initial nodes: 225 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/nf2.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/nf2.ma deleted file mode 100644 index ed930b85e..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/nf2.ma +++ /dev/null @@ -1,52 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pc3/defs.ma". - -include "Basic-1/nf2/pr3.ma". - -theorem pc3_nf2: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to ((nf2 c -t1) \to ((nf2 c t2) \to (eq T t1 t2)))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c t1 -t2)).(\lambda (H0: (nf2 c t1)).(\lambda (H1: (nf2 c t2)).(let H2 \def H in -(ex2_ind T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) (eq T -t1 t2) (\lambda (x: T).(\lambda (H3: (pr3 c t1 x)).(\lambda (H4: (pr3 c t2 -x)).(let H_y \def (nf2_pr3_unfold c t1 x H3 H0) in (let H5 \def (eq_ind_r T x -(\lambda (t: T).(pr3 c t2 t)) H4 t1 H_y) in (let H6 \def (eq_ind_r T x -(\lambda (t: T).(pr3 c t1 t)) H3 t1 H_y) in (let H_y0 \def (nf2_pr3_unfold c -t2 t1 H5 H1) in (let H7 \def (eq_ind T t2 (\lambda (t: T).(pr3 c t t1)) H5 t1 -H_y0) in (eq_ind_r T t1 (\lambda (t: T).(eq T t1 t)) (refl_equal T t1) t2 -H_y0))))))))) H2))))))). -(* COMMENTS -Initial nodes: 195 -END *) - -theorem pc3_nf2_unfold: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to ((nf2 c -t2) \to (pr3 c t1 t2))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c t1 -t2)).(\lambda (H0: (nf2 c t2)).(let H1 \def H in (ex2_ind T (\lambda (t: -T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) (pr3 c t1 t2) (\lambda (x: -T).(\lambda (H2: (pr3 c t1 x)).(\lambda (H3: (pr3 c t2 x)).(let H_y \def -(nf2_pr3_unfold c t2 x H3 H0) in (let H4 \def (eq_ind_r T x (\lambda (t: -T).(pr3 c t1 t)) H2 t2 H_y) in H4))))) H1)))))). -(* COMMENTS -Initial nodes: 109 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/pc1.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/pc1.ma deleted file mode 100644 index dd89f951d..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/pc1.ma +++ /dev/null @@ -1,36 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pc3/defs.ma". - -include "Basic-1/pc1/defs.ma". - -include "Basic-1/pr3/pr1.ma". - -theorem pc3_pc1: - \forall (t1: T).(\forall (t2: T).((pc1 t1 t2) \to (\forall (c: C).(pc3 c t1 -t2)))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc1 t1 t2)).(\lambda (c: -C).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: -T).(pr1 t2 t)) (pc3 c t1 t2) (\lambda (x: T).(\lambda (H1: (pr1 t1 -x)).(\lambda (H2: (pr1 t2 x)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) -(\lambda (t: T).(pr3 c t2 t)) x (pr3_pr1 t1 x H1 c) (pr3_pr1 t2 x H2 c))))) -H0))))). -(* COMMENTS -Initial nodes: 103 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/props.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/props.ma deleted file mode 100644 index b0a9a2f50..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/props.ma +++ /dev/null @@ -1,483 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pc3/defs.ma". - -include "Basic-1/pr3/pr3.ma". - -theorem clear_pc3_trans: - \forall (c2: C).(\forall (t1: T).(\forall (t2: T).((pc3 c2 t1 t2) \to -(\forall (c1: C).((clear c1 c2) \to (pc3 c1 t1 t2)))))) -\def - \lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c2 t1 -t2)).(\lambda (c1: C).(\lambda (H0: (clear c1 c2)).(let H1 \def H in (ex2_ind -T (\lambda (t: T).(pr3 c2 t1 t)) (\lambda (t: T).(pr3 c2 t2 t)) (pc3 c1 t1 -t2) (\lambda (x: T).(\lambda (H2: (pr3 c2 t1 x)).(\lambda (H3: (pr3 c2 t2 -x)).(ex_intro2 T (\lambda (t: T).(pr3 c1 t1 t)) (\lambda (t: T).(pr3 c1 t2 -t)) x (clear_pr3_trans c2 t1 x H2 c1 H0) (clear_pr3_trans c2 t2 x H3 c1 -H0))))) H1))))))). -(* COMMENTS -Initial nodes: 129 -END *) - -theorem pc3_pr2_r: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (pc3 c -t1 t2)))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 -t2)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) -t2 (pr3_pr2 c t1 t2 H) (pr3_refl c t2))))). -(* COMMENTS -Initial nodes: 55 -END *) - -theorem pc3_pr2_x: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t2 t1) \to (pc3 c -t1 t2)))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t2 -t1)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) -t1 (pr3_refl c t1) (pr3_pr2 c t2 t1 H))))). -(* COMMENTS -Initial nodes: 55 -END *) - -theorem pc3_pr3_r: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (pc3 c -t1 t2)))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 -t2)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) -t2 H (pr3_refl c t2))))). -(* COMMENTS -Initial nodes: 47 -END *) - -theorem pc3_pr3_x: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t2 t1) \to (pc3 c -t1 t2)))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t2 -t1)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) -t1 (pr3_refl c t1) H)))). -(* COMMENTS -Initial nodes: 47 -END *) - -theorem pc3_pr3_t: - \forall (c: C).(\forall (t1: T).(\forall (t0: T).((pr3 c t1 t0) \to (\forall -(t2: T).((pr3 c t2 t0) \to (pc3 c t1 t2)))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t0: T).(\lambda (H: (pr3 c t1 -t0)).(\lambda (t2: T).(\lambda (H0: (pr3 c t2 t0)).(ex_intro2 T (\lambda (t: -T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) t0 H H0)))))). -(* COMMENTS -Initial nodes: 53 -END *) - -theorem pc3_refl: - \forall (c: C).(\forall (t: T).(pc3 c t t)) -\def - \lambda (c: C).(\lambda (t: T).(ex_intro2 T (\lambda (t0: T).(pr3 c t t0)) -(\lambda (t0: T).(pr3 c t t0)) t (pr3_refl c t) (pr3_refl c t))). -(* COMMENTS -Initial nodes: 41 -END *) - -theorem pc3_s: - \forall (c: C).(\forall (t2: T).(\forall (t1: T).((pc3 c t1 t2) \to (pc3 c -t2 t1)))) -\def - \lambda (c: C).(\lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pc3 c t1 -t2)).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: -T).(pr3 c t2 t)) (pc3 c t2 t1) (\lambda (x: T).(\lambda (H1: (pr3 c t1 -x)).(\lambda (H2: (pr3 c t2 x)).(ex_intro2 T (\lambda (t: T).(pr3 c t2 t)) -(\lambda (t: T).(pr3 c t1 t)) x H2 H1)))) H0))))). -(* COMMENTS -Initial nodes: 97 -END *) - -theorem pc3_thin_dx: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to (\forall -(u: T).(\forall (f: F).(pc3 c (THead (Flat f) u t1) (THead (Flat f) u -t2))))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c t1 -t2)).(\lambda (u: T).(\lambda (f: F).(let H0 \def H in (ex2_ind T (\lambda -(t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) (pc3 c (THead (Flat f) u -t1) (THead (Flat f) u t2)) (\lambda (x: T).(\lambda (H1: (pr3 c t1 -x)).(\lambda (H2: (pr3 c t2 x)).(ex_intro2 T (\lambda (t: T).(pr3 c (THead -(Flat f) u t1) t)) (\lambda (t: T).(pr3 c (THead (Flat f) u t2) t)) (THead -(Flat f) u x) (pr3_thin_dx c t1 x H1 u f) (pr3_thin_dx c t2 x H2 u f))))) -H0))))))). -(* COMMENTS -Initial nodes: 165 -END *) - -theorem pc3_head_1: - \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pc3 c u1 u2) \to (\forall -(k: K).(\forall (t: T).(pc3 c (THead k u1 t) (THead k u2 t))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pc3 c u1 -u2)).(\lambda (k: K).(\lambda (t: T).(let H0 \def H in (ex2_ind T (\lambda -(t0: T).(pr3 c u1 t0)) (\lambda (t0: T).(pr3 c u2 t0)) (pc3 c (THead k u1 t) -(THead k u2 t)) (\lambda (x: T).(\lambda (H1: (pr3 c u1 x)).(\lambda (H2: -(pr3 c u2 x)).(ex_intro2 T (\lambda (t0: T).(pr3 c (THead k u1 t) t0)) -(\lambda (t0: T).(pr3 c (THead k u2 t) t0)) (THead k x t) (pr3_head_12 c u1 x -H1 k t t (pr3_refl (CHead c k x) t)) (pr3_head_12 c u2 x H2 k t t (pr3_refl -(CHead c k x) t)))))) H0))))))). -(* COMMENTS -Initial nodes: 183 -END *) - -theorem pc3_head_2: - \forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).(\forall -(k: K).((pc3 (CHead c k u) t1 t2) \to (pc3 c (THead k u t1) (THead k u -t2))))))) -\def - \lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(k: K).(\lambda (H: (pc3 (CHead c k u) t1 t2)).(let H0 \def H in (ex2_ind T -(\lambda (t: T).(pr3 (CHead c k u) t1 t)) (\lambda (t: T).(pr3 (CHead c k u) -t2 t)) (pc3 c (THead k u t1) (THead k u t2)) (\lambda (x: T).(\lambda (H1: -(pr3 (CHead c k u) t1 x)).(\lambda (H2: (pr3 (CHead c k u) t2 x)).(ex_intro2 -T (\lambda (t: T).(pr3 c (THead k u t1) t)) (\lambda (t: T).(pr3 c (THead k u -t2) t)) (THead k u x) (pr3_head_12 c u u (pr3_refl c u) k t1 x H1) -(pr3_head_12 c u u (pr3_refl c u) k t2 x H2))))) H0))))))). -(* COMMENTS -Initial nodes: 201 -END *) - -theorem pc3_pr2_u: - \forall (c: C).(\forall (t2: T).(\forall (t1: T).((pr2 c t1 t2) \to (\forall -(t3: T).((pc3 c t2 t3) \to (pc3 c t1 t3)))))) -\def - \lambda (c: C).(\lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pr2 c t1 -t2)).(\lambda (t3: T).(\lambda (H0: (pc3 c t2 t3)).(let H1 \def H0 in -(ex2_ind T (\lambda (t: T).(pr3 c t2 t)) (\lambda (t: T).(pr3 c t3 t)) (pc3 c -t1 t3) (\lambda (x: T).(\lambda (H2: (pr3 c t2 x)).(\lambda (H3: (pr3 c t3 -x)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t3 t)) -x (pr3_sing c t2 t1 H x H2) H3)))) H1))))))). -(* COMMENTS -Initial nodes: 119 -END *) - -theorem pc3_t: - \forall (t2: T).(\forall (c: C).(\forall (t1: T).((pc3 c t1 t2) \to (\forall -(t3: T).((pc3 c t2 t3) \to (pc3 c t1 t3)))))) -\def - \lambda (t2: T).(\lambda (c: C).(\lambda (t1: T).(\lambda (H: (pc3 c t1 -t2)).(\lambda (t3: T).(\lambda (H0: (pc3 c t2 t3)).(let H1 \def H0 in -(ex2_ind T (\lambda (t: T).(pr3 c t2 t)) (\lambda (t: T).(pr3 c t3 t)) (pc3 c -t1 t3) (\lambda (x: T).(\lambda (H2: (pr3 c t2 x)).(\lambda (H3: (pr3 c t3 -x)).(let H4 \def H in (ex2_ind T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: -T).(pr3 c t2 t)) (pc3 c t1 t3) (\lambda (x0: T).(\lambda (H5: (pr3 c t1 -x0)).(\lambda (H6: (pr3 c t2 x0)).(ex2_ind T (\lambda (t: T).(pr3 c x0 t)) -(\lambda (t: T).(pr3 c x t)) (pc3 c t1 t3) (\lambda (x1: T).(\lambda (H7: -(pr3 c x0 x1)).(\lambda (H8: (pr3 c x x1)).(pc3_pr3_t c t1 x1 (pr3_t x0 t1 c -H5 x1 H7) t3 (pr3_t x t3 c H3 x1 H8))))) (pr3_confluence c t2 x0 H6 x H2))))) -H4))))) H1))))))). -(* COMMENTS -Initial nodes: 233 -END *) - -theorem pc3_pr2_u2: - \forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr2 c t0 t1) \to (\forall -(t2: T).((pc3 c t0 t2) \to (pc3 c t1 t2)))))) -\def - \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr2 c t0 -t1)).(\lambda (t2: T).(\lambda (H0: (pc3 c t0 t2)).(pc3_t t0 c t1 (pc3_pr2_x -c t1 t0 H) t2 H0)))))). -(* COMMENTS -Initial nodes: 45 -END *) - -theorem pc3_pr3_conf: - \forall (c: C).(\forall (t: T).(\forall (t1: T).((pc3 c t t1) \to (\forall -(t2: T).((pr3 c t t2) \to (pc3 c t2 t1)))))) -\def - \lambda (c: C).(\lambda (t: T).(\lambda (t1: T).(\lambda (H: (pc3 c t -t1)).(\lambda (t2: T).(\lambda (H0: (pr3 c t t2)).(pc3_t t c t2 (pc3_pr3_x c -t2 t H0) t1 H)))))). -(* COMMENTS -Initial nodes: 45 -END *) - -theorem pc3_head_12: - \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pc3 c u1 u2) \to (\forall -(k: K).(\forall (t1: T).(\forall (t2: T).((pc3 (CHead c k u2) t1 t2) \to (pc3 -c (THead k u1 t1) (THead k u2 t2))))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pc3 c u1 -u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pc3 -(CHead c k u2) t1 t2)).(pc3_t (THead k u2 t1) c (THead k u1 t1) (pc3_head_1 c -u1 u2 H k t1) (THead k u2 t2) (pc3_head_2 c u2 t1 t2 k H0))))))))). -(* COMMENTS -Initial nodes: 89 -END *) - -theorem pc3_head_21: - \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pc3 c u1 u2) \to (\forall -(k: K).(\forall (t1: T).(\forall (t2: T).((pc3 (CHead c k u1) t1 t2) \to (pc3 -c (THead k u1 t1) (THead k u2 t2))))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pc3 c u1 -u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pc3 -(CHead c k u1) t1 t2)).(pc3_t (THead k u1 t2) c (THead k u1 t1) (pc3_head_2 c -u1 t1 t2 k H0) (THead k u2 t2) (pc3_head_1 c u1 u2 H k t2))))))))). -(* COMMENTS -Initial nodes: 89 -END *) - -theorem pc3_pr0_pr2_t: - \forall (u1: T).(\forall (u2: T).((pr0 u2 u1) \to (\forall (c: C).(\forall -(t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pc3 -(CHead c k u1) t1 t2)))))))) -\def - \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr0 u2 u1)).(\lambda (c: -C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k: K).(\lambda (H0: (pr2 -(CHead c k u2) t1 t2)).(insert_eq C (CHead c k u2) (\lambda (c0: C).(pr2 c0 -t1 t2)) (\lambda (_: C).(pc3 (CHead c k u1) t1 t2)) (\lambda (y: C).(\lambda -(H1: (pr2 y t1 t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: -T).((eq C c0 (CHead c k u2)) \to (pc3 (CHead c k u1) t t0))))) (\lambda (c0: -C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (H3: -(eq C c0 (CHead c k u2))).(let H4 \def (f_equal C C (\lambda (e: C).e) c0 -(CHead c k u2) H3) in (pc3_pr2_r (CHead c k u1) t3 t4 (pr2_free (CHead c k -u1) t3 t4 H2)))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: -T).(\lambda (i: nat).(\lambda (H2: (getl i c0 (CHead d (Bind Abbr) -u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda -(t: T).(\lambda (H4: (subst0 i u t4 t)).(\lambda (H5: (eq C c0 (CHead c k -u2))).(let H6 \def (f_equal C C (\lambda (e: C).e) c0 (CHead c k u2) H5) in -(let H7 \def (eq_ind C c0 (\lambda (c1: C).(getl i c1 (CHead d (Bind Abbr) -u))) H2 (CHead c k u2) H6) in (nat_ind (\lambda (n: nat).((getl n (CHead c k -u2) (CHead d (Bind Abbr) u)) \to ((subst0 n u t4 t) \to (pc3 (CHead c k u1) -t3 t)))) (\lambda (H8: (getl O (CHead c k u2) (CHead d (Bind Abbr) -u))).(\lambda (H9: (subst0 O u t4 t)).(K_ind (\lambda (k0: K).((clear (CHead -c k0 u2) (CHead d (Bind Abbr) u)) \to (pc3 (CHead c k0 u1) t3 t))) (\lambda -(b: B).(\lambda (H10: (clear (CHead c (Bind b) u2) (CHead d (Bind Abbr) -u))).(let H11 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda -(_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) -(CHead d (Bind Abbr) u) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d -(Bind Abbr) u) u2 H10)) in ((let H12 \def (f_equal C B (\lambda (e: C).(match -e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ -k0 _) \Rightarrow (match k0 in K return (\lambda (_: K).B) with [(Bind b0) -\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) -(CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 H10)) in -((let H13 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: -C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead -d (Bind Abbr) u) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind -Abbr) u) u2 H10)) in (\lambda (H14: (eq B Abbr b)).(\lambda (_: (eq C d -c)).(let H16 \def (eq_ind T u (\lambda (t0: T).(subst0 O t0 t4 t)) H9 u2 H13) -in (eq_ind B Abbr (\lambda (b0: B).(pc3 (CHead c (Bind b0) u1) t3 t)) -(ex2_ind T (\lambda (t0: T).(subst0 O u1 t4 t0)) (\lambda (t0: T).(pr0 t t0)) -(pc3 (CHead c (Bind Abbr) u1) t3 t) (\lambda (x: T).(\lambda (H17: (subst0 O -u1 t4 x)).(\lambda (H18: (pr0 t x)).(pc3_pr3_t (CHead c (Bind Abbr) u1) t3 x -(pr3_pr2 (CHead c (Bind Abbr) u1) t3 x (pr2_delta (CHead c (Bind Abbr) u1) c -u1 O (getl_refl Abbr c u1) t3 t4 H3 x H17)) t (pr3_pr2 (CHead c (Bind Abbr) -u1) t x (pr2_free (CHead c (Bind Abbr) u1) t x H18)))))) (pr0_subst0_fwd u2 -t4 t O H16 u1 H)) b H14))))) H12)) H11)))) (\lambda (f: F).(\lambda (H10: -(clear (CHead c (Flat f) u2) (CHead d (Bind Abbr) u))).(clear_pc3_trans -(CHead d (Bind Abbr) u) t3 t (pc3_pr2_r (CHead d (Bind Abbr) u) t3 t -(pr2_delta (CHead d (Bind Abbr) u) d u O (getl_refl Abbr d u) t3 t4 H3 t H9)) -(CHead c (Flat f) u1) (clear_flat c (CHead d (Bind Abbr) u) (clear_gen_flat f -c (CHead d (Bind Abbr) u) u2 H10) f u1)))) k (getl_gen_O (CHead c k u2) -(CHead d (Bind Abbr) u) H8)))) (\lambda (i0: nat).(\lambda (IHi: (((getl i0 -(CHead c k u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t4 t) \to (pc3 -(CHead c k u1) t3 t))))).(\lambda (H8: (getl (S i0) (CHead c k u2) (CHead d -(Bind Abbr) u))).(\lambda (H9: (subst0 (S i0) u t4 t)).(K_ind (\lambda (k0: -K).((((getl i0 (CHead c k0 u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t4 -t) \to (pc3 (CHead c k0 u1) t3 t)))) \to ((getl (r k0 i0) c (CHead d (Bind -Abbr) u)) \to (pc3 (CHead c k0 u1) t3 t)))) (\lambda (b: B).(\lambda (_: -(((getl i0 (CHead c (Bind b) u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u -t4 t) \to (pc3 (CHead c (Bind b) u1) t3 t))))).(\lambda (H10: (getl (r (Bind -b) i0) c (CHead d (Bind Abbr) u))).(pc3_pr2_r (CHead c (Bind b) u1) t3 t -(pr2_delta (CHead c (Bind b) u1) d u (S i0) (getl_head (Bind b) i0 c (CHead d -(Bind Abbr) u) H10 u1) t3 t4 H3 t H9))))) (\lambda (f: F).(\lambda (_: -(((getl i0 (CHead c (Flat f) u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u -t4 t) \to (pc3 (CHead c (Flat f) u1) t3 t))))).(\lambda (H10: (getl (r (Flat -f) i0) c (CHead d (Bind Abbr) u))).(pc3_pr2_r (CHead c (Flat f) u1) t3 t -(pr2_cflat c t3 t (pr2_delta c d u (r (Flat f) i0) H10 t3 t4 H3 t H9) f -u1))))) k IHi (getl_gen_S k c (CHead d (Bind Abbr) u) u2 i0 H8)))))) i H7 -H4)))))))))))))) y t1 t2 H1))) H0)))))))). -(* COMMENTS -Initial nodes: 1533 -END *) - -theorem pc3_pr2_pr2_t: - \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr2 c u2 u1) \to (\forall -(t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pc3 -(CHead c k u1) t1 t2)))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr2 c u2 -u1)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (t1: -T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c0 k t) t1 t2) \to (pc3 -(CHead c0 k t0) t1 t2)))))))) (\lambda (c0: C).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (H0: (pr0 t1 t2)).(\lambda (t0: T).(\lambda (t3: T).(\lambda (k: -K).(\lambda (H1: (pr2 (CHead c0 k t1) t0 t3)).(pc3_pr0_pr2_t t2 t1 H0 c0 t0 -t3 k H1))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda -(i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t1: -T).(\lambda (t2: T).(\lambda (H1: (pr0 t1 t2)).(\lambda (t: T).(\lambda (H2: -(subst0 i u t2 t)).(\lambda (t0: T).(\lambda (t3: T).(\lambda (k: K).(\lambda -(H3: (pr2 (CHead c0 k t1) t0 t3)).(insert_eq C (CHead c0 k t1) (\lambda (c1: -C).(pr2 c1 t0 t3)) (\lambda (_: C).(pc3 (CHead c0 k t) t0 t3)) (\lambda (y: -C).(\lambda (H4: (pr2 y t0 t3)).(pr2_ind (\lambda (c1: C).(\lambda (t4: -T).(\lambda (t5: T).((eq C c1 (CHead c0 k t1)) \to (pc3 (CHead c0 k t) t4 -t5))))) (\lambda (c1: C).(\lambda (t4: T).(\lambda (t5: T).(\lambda (H5: (pr0 -t4 t5)).(\lambda (_: (eq C c1 (CHead c0 k t1))).(pc3_pr2_r (CHead c0 k t) t4 -t5 (pr2_free (CHead c0 k t) t4 t5 H5))))))) (\lambda (c1: C).(\lambda (d0: -C).(\lambda (u0: T).(\lambda (i0: nat).(\lambda (H5: (getl i0 c1 (CHead d0 -(Bind Abbr) u0))).(\lambda (t4: T).(\lambda (t5: T).(\lambda (H6: (pr0 t4 -t5)).(\lambda (t6: T).(\lambda (H7: (subst0 i0 u0 t5 t6)).(\lambda (H8: (eq C -c1 (CHead c0 k t1))).(let H9 \def (eq_ind C c1 (\lambda (c2: C).(getl i0 c2 -(CHead d0 (Bind Abbr) u0))) H5 (CHead c0 k t1) H8) in (nat_ind (\lambda (n: -nat).((getl n (CHead c0 k t1) (CHead d0 (Bind Abbr) u0)) \to ((subst0 n u0 t5 -t6) \to (pc3 (CHead c0 k t) t4 t6)))) (\lambda (H10: (getl O (CHead c0 k t1) -(CHead d0 (Bind Abbr) u0))).(\lambda (H11: (subst0 O u0 t5 t6)).(K_ind -(\lambda (k0: K).((clear (CHead c0 k0 t1) (CHead d0 (Bind Abbr) u0)) \to (pc3 -(CHead c0 k0 t) t4 t6))) (\lambda (b: B).(\lambda (H12: (clear (CHead c0 -(Bind b) t1) (CHead d0 (Bind Abbr) u0))).(let H13 \def (f_equal C C (\lambda -(e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d0 -| (CHead c2 _ _) \Rightarrow c2])) (CHead d0 (Bind Abbr) u0) (CHead c0 (Bind -b) t1) (clear_gen_bind b c0 (CHead d0 (Bind Abbr) u0) t1 H12)) in ((let H14 -\def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) -with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow (match k0 in K -return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) -\Rightarrow Abbr])])) (CHead d0 (Bind Abbr) u0) (CHead c0 (Bind b) t1) -(clear_gen_bind b c0 (CHead d0 (Bind Abbr) u0) t1 H12)) in ((let H15 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow u0 | (CHead _ _ t7) \Rightarrow t7])) (CHead d0 (Bind -Abbr) u0) (CHead c0 (Bind b) t1) (clear_gen_bind b c0 (CHead d0 (Bind Abbr) -u0) t1 H12)) in (\lambda (H16: (eq B Abbr b)).(\lambda (_: (eq C d0 c0)).(let -H18 \def (eq_ind T u0 (\lambda (t7: T).(subst0 O t7 t5 t6)) H11 t1 H15) in -(eq_ind B Abbr (\lambda (b0: B).(pc3 (CHead c0 (Bind b0) t) t4 t6)) (ex2_ind -T (\lambda (t7: T).(subst0 O t2 t5 t7)) (\lambda (t7: T).(pr0 t6 t7)) (pc3 -(CHead c0 (Bind Abbr) t) t4 t6) (\lambda (x: T).(\lambda (H19: (subst0 O t2 -t5 x)).(\lambda (H20: (pr0 t6 x)).(ex2_ind T (\lambda (t7: T).(subst0 O t t5 -t7)) (\lambda (t7: T).(subst0 (S (plus i O)) u x t7)) (pc3 (CHead c0 (Bind -Abbr) t) t4 t6) (\lambda (x0: T).(\lambda (H21: (subst0 O t t5 x0)).(\lambda -(H22: (subst0 (S (plus i O)) u x x0)).(let H23 \def (f_equal nat nat S (plus -i O) i (sym_eq nat i (plus i O) (plus_n_O i))) in (let H24 \def (eq_ind nat -(S (plus i O)) (\lambda (n: nat).(subst0 n u x x0)) H22 (S i) H23) in -(pc3_pr2_u (CHead c0 (Bind Abbr) t) x0 t4 (pr2_delta (CHead c0 (Bind Abbr) t) -c0 t O (getl_refl Abbr c0 t) t4 t5 H6 x0 H21) t6 (pc3_pr2_x (CHead c0 (Bind -Abbr) t) x0 t6 (pr2_delta (CHead c0 (Bind Abbr) t) d u (S i) (getl_head (Bind -Abbr) i c0 (CHead d (Bind Abbr) u) H0 t) t6 x H20 x0 H24)))))))) -(subst0_subst0_back t5 x t2 O H19 t u i H2))))) (pr0_subst0_fwd t1 t5 t6 O -H18 t2 H1)) b H16))))) H14)) H13)))) (\lambda (f: F).(\lambda (H12: (clear -(CHead c0 (Flat f) t1) (CHead d0 (Bind Abbr) u0))).(clear_pc3_trans (CHead d0 -(Bind Abbr) u0) t4 t6 (pc3_pr2_r (CHead d0 (Bind Abbr) u0) t4 t6 (pr2_delta -(CHead d0 (Bind Abbr) u0) d0 u0 O (getl_refl Abbr d0 u0) t4 t5 H6 t6 H11)) -(CHead c0 (Flat f) t) (clear_flat c0 (CHead d0 (Bind Abbr) u0) -(clear_gen_flat f c0 (CHead d0 (Bind Abbr) u0) t1 H12) f t)))) k (getl_gen_O -(CHead c0 k t1) (CHead d0 (Bind Abbr) u0) H10)))) (\lambda (i1: nat).(\lambda -(_: (((getl i1 (CHead c0 k t1) (CHead d0 (Bind Abbr) u0)) \to ((subst0 i1 u0 -t5 t6) \to (pc3 (CHead c0 k t) t4 t6))))).(\lambda (H10: (getl (S i1) (CHead -c0 k t1) (CHead d0 (Bind Abbr) u0))).(\lambda (H11: (subst0 (S i1) u0 t5 -t6)).(K_ind (\lambda (k0: K).((getl (r k0 i1) c0 (CHead d0 (Bind Abbr) u0)) -\to (pc3 (CHead c0 k0 t) t4 t6))) (\lambda (b: B).(\lambda (H12: (getl (r -(Bind b) i1) c0 (CHead d0 (Bind Abbr) u0))).(pc3_pr2_r (CHead c0 (Bind b) t) -t4 t6 (pr2_delta (CHead c0 (Bind b) t) d0 u0 (S i1) (getl_head (Bind b) i1 c0 -(CHead d0 (Bind Abbr) u0) H12 t) t4 t5 H6 t6 H11)))) (\lambda (f: F).(\lambda -(H12: (getl (r (Flat f) i1) c0 (CHead d0 (Bind Abbr) u0))).(pc3_pr2_r (CHead -c0 (Flat f) t) t4 t6 (pr2_cflat c0 t4 t6 (pr2_delta c0 d0 u0 (r (Flat f) i1) -H12 t4 t5 H6 t6 H11) f t)))) k (getl_gen_S k c0 (CHead d0 (Bind Abbr) u0) t1 -i1 H10)))))) i0 H9 H7))))))))))))) y t0 t3 H4))) H3))))))))))))))) c u2 u1 -H)))). -(* COMMENTS -Initial nodes: 1671 -END *) - -theorem pc3_pr2_pr3_t: - \forall (c: C).(\forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall -(k: K).((pr3 (CHead c k u2) t1 t2) \to (\forall (u1: T).((pr2 c u2 u1) \to -(pc3 (CHead c k u1) t1 t2)))))))) -\def - \lambda (c: C).(\lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(k: K).(\lambda (H: (pr3 (CHead c k u2) t1 t2)).(pr3_ind (CHead c k u2) -(\lambda (t: T).(\lambda (t0: T).(\forall (u1: T).((pr2 c u2 u1) \to (pc3 -(CHead c k u1) t t0))))) (\lambda (t: T).(\lambda (u1: T).(\lambda (_: (pr2 c -u2 u1)).(pc3_refl (CHead c k u1) t)))) (\lambda (t0: T).(\lambda (t3: -T).(\lambda (H0: (pr2 (CHead c k u2) t3 t0)).(\lambda (t4: T).(\lambda (_: -(pr3 (CHead c k u2) t0 t4)).(\lambda (H2: ((\forall (u1: T).((pr2 c u2 u1) -\to (pc3 (CHead c k u1) t0 t4))))).(\lambda (u1: T).(\lambda (H3: (pr2 c u2 -u1)).(pc3_t t0 (CHead c k u1) t3 (pc3_pr2_pr2_t c u1 u2 H3 t3 t0 k H0) t4 (H2 -u1 H3)))))))))) t1 t2 H)))))). -(* COMMENTS -Initial nodes: 199 -END *) - -theorem pc3_pr3_pc3_t: - \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u2 u1) \to (\forall -(t1: T).(\forall (t2: T).(\forall (k: K).((pc3 (CHead c k u2) t1 t2) \to (pc3 -(CHead c k u1) t1 t2)))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u2 -u1)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (t1: T).(\forall -(t2: T).(\forall (k: K).((pc3 (CHead c k t) t1 t2) \to (pc3 (CHead c k t0) t1 -t2))))))) (\lambda (t: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k: -K).(\lambda (H0: (pc3 (CHead c k t) t1 t2)).H0))))) (\lambda (t2: T).(\lambda -(t1: T).(\lambda (H0: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda (_: (pr3 c t2 -t3)).(\lambda (H2: ((\forall (t4: T).(\forall (t5: T).(\forall (k: K).((pc3 -(CHead c k t2) t4 t5) \to (pc3 (CHead c k t3) t4 t5))))))).(\lambda (t0: -T).(\lambda (t4: T).(\lambda (k: K).(\lambda (H3: (pc3 (CHead c k t1) t0 -t4)).(H2 t0 t4 k (let H4 \def H3 in (ex2_ind T (\lambda (t: T).(pr3 (CHead c -k t1) t0 t)) (\lambda (t: T).(pr3 (CHead c k t1) t4 t)) (pc3 (CHead c k t2) -t0 t4) (\lambda (x: T).(\lambda (H5: (pr3 (CHead c k t1) t0 x)).(\lambda (H6: -(pr3 (CHead c k t1) t4 x)).(pc3_t x (CHead c k t2) t0 (pc3_pr2_pr3_t c t1 t0 -x k H5 t2 H0) t4 (pc3_s (CHead c k t2) x t4 (pc3_pr2_pr3_t c t1 t4 x k H6 t2 -H0)))))) H4))))))))))))) u2 u1 H)))). -(* COMMENTS -Initial nodes: 319 -END *) - -theorem pc3_lift: - \forall (c: C).(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h -d c e) \to (\forall (t1: T).(\forall (t2: T).((pc3 e t1 t2) \to (pc3 c (lift -h d t1) (lift h d t2))))))))) -\def - \lambda (c: C).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda -(H: (drop h d c e)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pc3 e t1 -t2)).(let H1 \def H0 in (ex2_ind T (\lambda (t: T).(pr3 e t1 t)) (\lambda (t: -T).(pr3 e t2 t)) (pc3 c (lift h d t1) (lift h d t2)) (\lambda (x: T).(\lambda -(H2: (pr3 e t1 x)).(\lambda (H3: (pr3 e t2 x)).(pc3_pr3_t c (lift h d t1) -(lift h d x) (pr3_lift c e h d H t1 x H2) (lift h d t2) (pr3_lift c e h d H -t2 x H3))))) H1))))))))). -(* COMMENTS -Initial nodes: 159 -END *) - -theorem pc3_eta: - \forall (c: C).(\forall (t: T).(\forall (w: T).(\forall (u: T).((pc3 c t -(THead (Bind Abst) w u)) \to (\forall (v: T).((pc3 c v w) \to (pc3 c (THead -(Bind Abst) v (THead (Flat Appl) (TLRef O) (lift (S O) O t))) t))))))) -\def - \lambda (c: C).(\lambda (t: T).(\lambda (w: T).(\lambda (u: T).(\lambda (H: -(pc3 c t (THead (Bind Abst) w u))).(\lambda (v: T).(\lambda (H0: (pc3 c v -w)).(pc3_t (THead (Bind Abst) w (THead (Flat Appl) (TLRef O) (lift (S O) O -(THead (Bind Abst) w u)))) c (THead (Bind Abst) v (THead (Flat Appl) (TLRef -O) (lift (S O) O t))) (pc3_head_21 c v w H0 (Bind Abst) (THead (Flat Appl) -(TLRef O) (lift (S O) O t)) (THead (Flat Appl) (TLRef O) (lift (S O) O (THead -(Bind Abst) w u))) (pc3_thin_dx (CHead c (Bind Abst) v) (lift (S O) O t) -(lift (S O) O (THead (Bind Abst) w u)) (pc3_lift (CHead c (Bind Abst) v) c (S -O) O (drop_drop (Bind Abst) O c c (drop_refl c) v) t (THead (Bind Abst) w u) -H) (TLRef O) Appl)) t (pc3_t (THead (Bind Abst) w u) c (THead (Bind Abst) w -(THead (Flat Appl) (TLRef O) (lift (S O) O (THead (Bind Abst) w u)))) -(pc3_pr3_r c (THead (Bind Abst) w (THead (Flat Appl) (TLRef O) (lift (S O) O -(THead (Bind Abst) w u)))) (THead (Bind Abst) w u) (pr3_eta c w u w (pr3_refl -c w))) t (pc3_s c (THead (Bind Abst) w u) t H))))))))). -(* COMMENTS -Initial nodes: 399 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/subst1.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/subst1.ma deleted file mode 100644 index 3af245b3a..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/subst1.ma +++ /dev/null @@ -1,48 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pc3/props.ma". - -include "Basic-1/pr3/subst1.ma". - -theorem pc3_gen_cabbr: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to (\forall -(e: C).(\forall (u: T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) -\to (\forall (a0: C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d -a0 a) \to (\forall (x1: T).((subst1 d u t1 (lift (S O) d x1)) \to (\forall -(x2: T).((subst1 d u t2 (lift (S O) d x2)) \to (pc3 a x1 x2)))))))))))))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c t1 -t2)).(\lambda (e: C).(\lambda (u: T).(\lambda (d: nat).(\lambda (H0: (getl d -c (CHead e (Bind Abbr) u))).(\lambda (a0: C).(\lambda (H1: (csubst1 d u c -a0)).(\lambda (a: C).(\lambda (H2: (drop (S O) d a0 a)).(\lambda (x1: -T).(\lambda (H3: (subst1 d u t1 (lift (S O) d x1))).(\lambda (x2: T).(\lambda -(H4: (subst1 d u t2 (lift (S O) d x2))).(let H5 \def H in (ex2_ind T (\lambda -(t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) (pc3 a x1 x2) (\lambda (x: -T).(\lambda (H6: (pr3 c t1 x)).(\lambda (H7: (pr3 c t2 x)).(ex2_ind T -(\lambda (x3: T).(subst1 d u x (lift (S O) d x3))) (\lambda (x3: T).(pr3 a x2 -x3)) (pc3 a x1 x2) (\lambda (x0: T).(\lambda (H8: (subst1 d u x (lift (S O) d -x0))).(\lambda (H9: (pr3 a x2 x0)).(ex2_ind T (\lambda (x3: T).(subst1 d u x -(lift (S O) d x3))) (\lambda (x3: T).(pr3 a x1 x3)) (pc3 a x1 x2) (\lambda -(x3: T).(\lambda (H10: (subst1 d u x (lift (S O) d x3))).(\lambda (H11: (pr3 -a x1 x3)).(let H12 \def (eq_ind T x3 (\lambda (t: T).(pr3 a x1 t)) H11 x0 -(subst1_confluence_lift x x3 u d H10 x0 H8)) in (pc3_pr3_t a x1 x0 H12 x2 -H9))))) (pr3_gen_cabbr c t1 x H6 e u d H0 a0 H1 a H2 x1 H3))))) -(pr3_gen_cabbr c t2 x H7 e u d H0 a0 H1 a H2 x2 H4))))) H5))))))))))))))))). -(* COMMENTS -Initial nodes: 405 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/wcpr0.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/wcpr0.ma deleted file mode 100644 index 2b51edf62..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pc3/wcpr0.ma +++ /dev/null @@ -1,96 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pc3/props.ma". - -include "Basic-1/wcpr0/getl.ma". - -theorem pc3_wcpr0__pc3_wcpr0_t_aux: - \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (k: K).(\forall -(u: T).(\forall (t1: T).(\forall (t2: T).((pr3 (CHead c1 k u) t1 t2) \to (pc3 -(CHead c2 k u) t1 t2)))))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(\lambda (k: -K).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 -(CHead c1 k u) t1 t2)).(pr3_ind (CHead c1 k u) (\lambda (t: T).(\lambda (t0: -T).(pc3 (CHead c2 k u) t t0))) (\lambda (t: T).(pc3_refl (CHead c2 k u) t)) -(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr2 (CHead c1 k u) t4 -t3)).(\lambda (t5: T).(\lambda (_: (pr3 (CHead c1 k u) t3 t5)).(\lambda (H3: -(pc3 (CHead c2 k u) t3 t5)).(pc3_t t3 (CHead c2 k u) t4 (insert_eq C (CHead -c1 k u) (\lambda (c: C).(pr2 c t4 t3)) (\lambda (_: C).(pc3 (CHead c2 k u) t4 -t3)) (\lambda (y: C).(\lambda (H4: (pr2 y t4 t3)).(pr2_ind (\lambda (c: -C).(\lambda (t: T).(\lambda (t0: T).((eq C c (CHead c1 k u)) \to (pc3 (CHead -c2 k u) t t0))))) (\lambda (c: C).(\lambda (t6: T).(\lambda (t0: T).(\lambda -(H5: (pr0 t6 t0)).(\lambda (_: (eq C c (CHead c1 k u))).(pc3_pr2_r (CHead c2 -k u) t6 t0 (pr2_free (CHead c2 k u) t6 t0 H5))))))) (\lambda (c: C).(\lambda -(d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H5: (getl i c (CHead d -(Bind Abbr) u0))).(\lambda (t6: T).(\lambda (t0: T).(\lambda (H6: (pr0 t6 -t0)).(\lambda (t: T).(\lambda (H7: (subst0 i u0 t0 t)).(\lambda (H8: (eq C c -(CHead c1 k u))).(let H9 \def (eq_ind C c (\lambda (c0: C).(getl i c0 (CHead -d (Bind Abbr) u0))) H5 (CHead c1 k u) H8) in (ex3_2_ind C T (\lambda (e2: -C).(\lambda (u2: T).(getl i (CHead c2 k u) (CHead e2 (Bind Abbr) u2)))) -(\lambda (e2: C).(\lambda (_: T).(wcpr0 d e2))) (\lambda (_: C).(\lambda (u2: -T).(pr0 u0 u2))) (pc3 (CHead c2 k u) t6 t) (\lambda (x0: C).(\lambda (x1: -T).(\lambda (H10: (getl i (CHead c2 k u) (CHead x0 (Bind Abbr) x1))).(\lambda -(_: (wcpr0 d x0)).(\lambda (H12: (pr0 u0 x1)).(ex2_ind T (\lambda (t7: -T).(subst0 i x1 t0 t7)) (\lambda (t7: T).(pr0 t t7)) (pc3 (CHead c2 k u) t6 -t) (\lambda (x: T).(\lambda (H13: (subst0 i x1 t0 x)).(\lambda (H14: (pr0 t -x)).(pc3_pr2_u (CHead c2 k u) x t6 (pr2_delta (CHead c2 k u) x0 x1 i H10 t6 -t0 H6 x H13) t (pc3_pr2_x (CHead c2 k u) x t (pr2_free (CHead c2 k u) t x -H14)))))) (pr0_subst0_fwd u0 t0 t i H7 x1 H12))))))) (wcpr0_getl (CHead c1 k -u) (CHead c2 k u) (wcpr0_comp c1 c2 H u u (pr0_refl u) k) i d u0 (Bind Abbr) -H9)))))))))))))) y t4 t3 H4))) H1) t5 H3))))))) t1 t2 H0)))))))). -(* COMMENTS -Initial nodes: 689 -END *) - -theorem pc3_wcpr0_t: - \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (t1: -T).(\forall (t2: T).((pr3 c1 t1 t2) \to (pc3 c2 t1 t2)))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(wcpr0_ind -(\lambda (c: C).(\lambda (c0: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 -t2) \to (pc3 c0 t1 t2)))))) (\lambda (c: C).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (H0: (pr3 c t1 t2)).(pc3_pr3_r c t1 t2 H0))))) (\lambda (c0: -C).(\lambda (c3: C).(\lambda (H0: (wcpr0 c0 c3)).(\lambda (_: ((\forall (t1: -T).(\forall (t2: T).((pr3 c0 t1 t2) \to (pc3 c3 t1 t2)))))).(\lambda (u1: -T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 u2)).(\lambda (k: K).(\lambda (t1: -T).(\lambda (t2: T).(\lambda (H3: (pr3 (CHead c0 k u1) t1 t2)).(let H4 \def -(pc3_pr2_pr3_t c0 u1 t1 t2 k H3 u2 (pr2_free c0 u1 u2 H2)) in (ex2_ind T -(\lambda (t: T).(pr3 (CHead c0 k u2) t1 t)) (\lambda (t: T).(pr3 (CHead c0 k -u2) t2 t)) (pc3 (CHead c3 k u2) t1 t2) (\lambda (x: T).(\lambda (H5: (pr3 -(CHead c0 k u2) t1 x)).(\lambda (H6: (pr3 (CHead c0 k u2) t2 x)).(pc3_t x -(CHead c3 k u2) t1 (pc3_wcpr0__pc3_wcpr0_t_aux c0 c3 H0 k u2 t1 x H5) t2 -(pc3_s (CHead c3 k u2) x t2 (pc3_wcpr0__pc3_wcpr0_t_aux c0 c3 H0 k u2 t2 x -H6)))))) H4))))))))))))) c1 c2 H))). -(* COMMENTS -Initial nodes: 299 -END *) - -theorem pc3_wcpr0: - \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (t1: -T).(\forall (t2: T).((pc3 c1 t1 t2) \to (pc3 c2 t1 t2)))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(\lambda (t1: -T).(\lambda (t2: T).(\lambda (H0: (pc3 c1 t1 t2)).(let H1 \def H0 in (ex2_ind -T (\lambda (t: T).(pr3 c1 t1 t)) (\lambda (t: T).(pr3 c1 t2 t)) (pc3 c2 t1 -t2) (\lambda (x: T).(\lambda (H2: (pr3 c1 t1 x)).(\lambda (H3: (pr3 c1 t2 -x)).(pc3_t x c2 t1 (pc3_wcpr0_t c1 c2 H t1 x H2) t2 (pc3_s c2 x t2 -(pc3_wcpr0_t c1 c2 H t2 x H3)))))) H1))))))). -(* COMMENTS -Initial nodes: 121 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr0/dec.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr0/dec.ma deleted file mode 100644 index c28504bee..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr0/dec.ma +++ /dev/null @@ -1,529 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr0/fwd.ma". - -include "Basic-1/subst0/dec.ma". - -include "Basic-1/T/dec.ma". - -include "Basic-1/T/props.ma". - -theorem nf0_dec: - \forall (t1: T).(or (\forall (t2: T).((pr0 t1 t2) \to (eq T t1 t2))) (ex2 T -(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: -T).(pr0 t1 t2)))) -\def - \lambda (t1: T).(T_ind (\lambda (t: T).(or (\forall (t2: T).((pr0 t t2) \to -(eq T t t2))) (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 t t2))))) (\lambda (n: nat).(or_introl -(\forall (t2: T).((pr0 (TSort n) t2) \to (eq T (TSort n) t2))) (ex2 T -(\lambda (t2: T).((eq T (TSort n) t2) \to (\forall (P: Prop).P))) (\lambda -(t2: T).(pr0 (TSort n) t2))) (\lambda (t2: T).(\lambda (H: (pr0 (TSort n) -t2)).(eq_ind_r T (TSort n) (\lambda (t: T).(eq T (TSort n) t)) (refl_equal T -(TSort n)) t2 (pr0_gen_sort t2 n H)))))) (\lambda (n: nat).(or_introl -(\forall (t2: T).((pr0 (TLRef n) t2) \to (eq T (TLRef n) t2))) (ex2 T -(\lambda (t2: T).((eq T (TLRef n) t2) \to (\forall (P: Prop).P))) (\lambda -(t2: T).(pr0 (TLRef n) t2))) (\lambda (t2: T).(\lambda (H: (pr0 (TLRef n) -t2)).(eq_ind_r T (TLRef n) (\lambda (t: T).(eq T (TLRef n) t)) (refl_equal T -(TLRef n)) t2 (pr0_gen_lref t2 n H)))))) (\lambda (k: K).(\lambda (t: -T).(\lambda (H: (or (\forall (t2: T).((pr0 t t2) \to (eq T t t2))) (ex2 T -(\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: -T).(pr0 t t2))))).(\lambda (t0: T).(\lambda (H0: (or (\forall (t2: T).((pr0 -t0 t2) \to (eq T t0 t2))) (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall -(P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2))))).(K_ind (\lambda (k0: K).(or -(\forall (t2: T).((pr0 (THead k0 t t0) t2) \to (eq T (THead k0 t t0) t2))) -(ex2 T (\lambda (t2: T).((eq T (THead k0 t t0) t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 (THead k0 t t0) t2))))) (\lambda (b: -B).(B_ind (\lambda (b0: B).(or (\forall (t2: T).((pr0 (THead (Bind b0) t t0) -t2) \to (eq T (THead (Bind b0) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T -(THead (Bind b0) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 -(THead (Bind b0) t t0) t2))))) (or_intror (\forall (t2: T).((pr0 (THead (Bind -Abbr) t t0) t2) \to (eq T (THead (Bind Abbr) t t0) t2))) (ex2 T (\lambda (t2: -T).((eq T (THead (Bind Abbr) t t0) t2) \to (\forall (P: Prop).P))) (\lambda -(t2: T).(pr0 (THead (Bind Abbr) t t0) t2))) (let H_x \def (dnf_dec t t0 O) in -(let H1 \def H_x in (ex_ind T (\lambda (v: T).(or (subst0 O t t0 (lift (S O) -O v)) (eq T t0 (lift (S O) O v)))) (ex2 T (\lambda (t2: T).((eq T (THead -(Bind Abbr) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 -(THead (Bind Abbr) t t0) t2))) (\lambda (x: T).(\lambda (H2: (or (subst0 O t -t0 (lift (S O) O x)) (eq T t0 (lift (S O) O x)))).(or_ind (subst0 O t t0 -(lift (S O) O x)) (eq T t0 (lift (S O) O x)) (ex2 T (\lambda (t2: T).((eq T -(THead (Bind Abbr) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: -T).(pr0 (THead (Bind Abbr) t t0) t2))) (\lambda (H3: (subst0 O t t0 (lift (S -O) O x))).(ex_intro2 T (\lambda (t2: T).((eq T (THead (Bind Abbr) t t0) t2) -\to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abbr) t t0) -t2)) (THead (Bind Abbr) t (lift (S O) O x)) (\lambda (H4: (eq T (THead (Bind -Abbr) t t0) (THead (Bind Abbr) t (lift (S O) O x)))).(\lambda (P: Prop).(let -H5 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t2) -\Rightarrow t2])) (THead (Bind Abbr) t t0) (THead (Bind Abbr) t (lift (S O) O -x)) H4) in (let H6 \def (eq_ind T t0 (\lambda (t2: T).(subst0 O t t2 (lift (S -O) O x))) H3 (lift (S O) O x) H5) in (subst0_refl t (lift (S O) O x) O H6 -P))))) (pr0_delta t t (pr0_refl t) t0 t0 (pr0_refl t0) (lift (S O) O x) H3))) -(\lambda (H3: (eq T t0 (lift (S O) O x))).(eq_ind_r T (lift (S O) O x) -(\lambda (t2: T).(ex2 T (\lambda (t3: T).((eq T (THead (Bind Abbr) t t2) t3) -\to (\forall (P: Prop).P))) (\lambda (t3: T).(pr0 (THead (Bind Abbr) t t2) -t3)))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Bind Abbr) t (lift (S O) -O x)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind -Abbr) t (lift (S O) O x)) t2)) x (\lambda (H4: (eq T (THead (Bind Abbr) t -(lift (S O) O x)) x)).(\lambda (P: Prop).(thead_x_lift_y_y (Bind Abbr) x t (S -O) O H4 P))) (pr0_zeta Abbr not_abbr_abst x x (pr0_refl x) t)) t0 H3)) H2))) -H1)))) (let H1 \def H in (or_ind (\forall (t2: T).((pr0 t t2) \to (eq T t -t2))) (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) -(\lambda (t2: T).(pr0 t t2))) (or (\forall (t2: T).((pr0 (THead (Bind Abst) t -t0) t2) \to (eq T (THead (Bind Abst) t t0) t2))) (ex2 T (\lambda (t2: T).((eq -T (THead (Bind Abst) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: -T).(pr0 (THead (Bind Abst) t t0) t2)))) (\lambda (H2: ((\forall (t2: T).((pr0 -t t2) \to (eq T t t2))))).(let H3 \def H0 in (or_ind (\forall (t2: T).((pr0 -t0 t2) \to (eq T t0 t2))) (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall -(P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2))) (or (\forall (t2: T).((pr0 -(THead (Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0) t2))) (ex2 T -(\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2)))) (\lambda -(H4: ((\forall (t2: T).((pr0 t0 t2) \to (eq T t0 t2))))).(or_introl (\forall -(t2: T).((pr0 (THead (Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0) -t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to -(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2))) -(\lambda (t2: T).(\lambda (H5: (pr0 (THead (Bind Abst) t t0) t2)).(ex3_2_ind -T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t0 t3))) (eq T (THead (Bind Abst) t t0) t2) (\lambda (x0: T).(\lambda -(x1: T).(\lambda (H6: (eq T t2 (THead (Bind Abst) x0 x1))).(\lambda (H7: (pr0 -t x0)).(\lambda (H8: (pr0 t0 x1)).(let H_y \def (H4 x1 H8) in (let H_y0 \def -(H2 x0 H7) in (let H9 \def (eq_ind_r T x1 (\lambda (t3: T).(pr0 t0 t3)) H8 t0 -H_y) in (let H10 \def (eq_ind_r T x1 (\lambda (t3: T).(eq T t2 (THead (Bind -Abst) x0 t3))) H6 t0 H_y) in (let H11 \def (eq_ind_r T x0 (\lambda (t3: -T).(pr0 t t3)) H7 t H_y0) in (let H12 \def (eq_ind_r T x0 (\lambda (t3: -T).(eq T t2 (THead (Bind Abst) t3 t0))) H10 t H_y0) in (eq_ind_r T (THead -(Bind Abst) t t0) (\lambda (t3: T).(eq T (THead (Bind Abst) t t0) t3)) -(refl_equal T (THead (Bind Abst) t t0)) t2 H12)))))))))))) (pr0_gen_abst t t0 -t2 H5)))))) (\lambda (H4: (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall -(P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2)))).(ex2_ind T (\lambda (t2: -T).((eq T t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2)) -(or (\forall (t2: T).((pr0 (THead (Bind Abst) t t0) t2) \to (eq T (THead -(Bind Abst) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abst) t -t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) -t t0) t2)))) (\lambda (x: T).(\lambda (H5: (((eq T t0 x) \to (\forall (P: -Prop).P)))).(\lambda (H6: (pr0 t0 x)).(or_intror (\forall (t2: T).((pr0 -(THead (Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0) t2))) (ex2 T -(\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2))) (ex_intro2 T -(\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2)) (THead (Bind -Abst) t x) (\lambda (H7: (eq T (THead (Bind Abst) t t0) (THead (Bind Abst) t -x))).(\lambda (P: Prop).(let H8 \def (f_equal T T (\lambda (e: T).(match e in -T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) -\Rightarrow t0 | (THead _ _ t2) \Rightarrow t2])) (THead (Bind Abst) t t0) -(THead (Bind Abst) t x) H7) in (let H9 \def (eq_ind_r T x (\lambda (t2: -T).(pr0 t0 t2)) H6 t0 H8) in (let H10 \def (eq_ind_r T x (\lambda (t2: -T).((eq T t0 t2) \to (\forall (P0: Prop).P0))) H5 t0 H8) in (H10 (refl_equal -T t0) P)))))) (pr0_comp t t (pr0_refl t) t0 x H6 (Bind Abst))))))) H4)) H3))) -(\lambda (H2: (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 t t2)))).(ex2_ind T (\lambda (t2: T).((eq T -t t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2)) (or (\forall -(t2: T).((pr0 (THead (Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0) -t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to -(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2)))) -(\lambda (x: T).(\lambda (H3: (((eq T t x) \to (\forall (P: -Prop).P)))).(\lambda (H4: (pr0 t x)).(or_intror (\forall (t2: T).((pr0 (THead -(Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0) t2))) (ex2 T -(\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2))) (ex_intro2 T -(\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2)) (THead (Bind -Abst) x t0) (\lambda (H5: (eq T (THead (Bind Abst) t t0) (THead (Bind Abst) x -t0))).(\lambda (P: Prop).(let H6 \def (f_equal T T (\lambda (e: T).(match e -in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t | (TLRef _) -\Rightarrow t | (THead _ t2 _) \Rightarrow t2])) (THead (Bind Abst) t t0) -(THead (Bind Abst) x t0) H5) in (let H7 \def (eq_ind_r T x (\lambda (t2: -T).(pr0 t t2)) H4 t H6) in (let H8 \def (eq_ind_r T x (\lambda (t2: T).((eq T -t t2) \to (\forall (P0: Prop).P0))) H3 t H6) in (H8 (refl_equal T t) P)))))) -(pr0_comp t x H4 t0 t0 (pr0_refl t0) (Bind Abst))))))) H2)) H1)) (let H_x -\def (dnf_dec t t0 O) in (let H1 \def H_x in (ex_ind T (\lambda (v: T).(or -(subst0 O t t0 (lift (S O) O v)) (eq T t0 (lift (S O) O v)))) (or (\forall -(t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) -t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to -(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)))) -(\lambda (x: T).(\lambda (H2: (or (subst0 O t t0 (lift (S O) O x)) (eq T t0 -(lift (S O) O x)))).(or_ind (subst0 O t t0 (lift (S O) O x)) (eq T t0 (lift -(S O) O x)) (or (\forall (t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T -(THead (Bind Void) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind -Void) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead -(Bind Void) t t0) t2)))) (\lambda (H3: (subst0 O t t0 (lift (S O) O x))).(let -H4 \def H in (or_ind (\forall (t2: T).((pr0 t t2) \to (eq T t t2))) (ex2 T -(\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: -T).(pr0 t t2))) (or (\forall (t2: T).((pr0 (THead (Bind Void) t t0) t2) \to -(eq T (THead (Bind Void) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead -(Bind Void) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 -(THead (Bind Void) t t0) t2)))) (\lambda (H5: ((\forall (t2: T).((pr0 t t2) -\to (eq T t t2))))).(let H6 \def H0 in (or_ind (\forall (t2: T).((pr0 t0 t2) -\to (eq T t0 t2))) (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 t0 t2))) (or (\forall (t2: T).((pr0 (THead -(Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) t2))) (ex2 T -(\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)))) (\lambda -(H7: ((\forall (t2: T).((pr0 t0 t2) \to (eq T t0 t2))))).(or_introl (\forall -(t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) -t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to -(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2))) -(\lambda (t2: T).(\lambda (H8: (pr0 (THead (Bind Void) t t0) t2)).(or_ind -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda -(t3: T).(pr0 t0 t3)))) (pr0 t0 (lift (S O) O t2)) (eq T (THead (Bind Void) t -t0) t2) (\lambda (H9: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 -(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) -(\lambda (_: T).(\lambda (t3: T).(pr0 t0 t3))))).(ex3_2_ind T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t0 -t3))) (eq T (THead (Bind Void) t t0) t2) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H10: (eq T t2 (THead (Bind Void) x0 x1))).(\lambda (H11: (pr0 t -x0)).(\lambda (H12: (pr0 t0 x1)).(let H_y \def (H7 x1 H12) in (let H_y0 \def -(H5 x0 H11) in (let H13 \def (eq_ind_r T x1 (\lambda (t3: T).(pr0 t0 t3)) H12 -t0 H_y) in (let H14 \def (eq_ind_r T x1 (\lambda (t3: T).(eq T t2 (THead -(Bind Void) x0 t3))) H10 t0 H_y) in (let H15 \def (eq_ind_r T x0 (\lambda -(t3: T).(pr0 t t3)) H11 t H_y0) in (let H16 \def (eq_ind_r T x0 (\lambda (t3: -T).(eq T t2 (THead (Bind Void) t3 t0))) H14 t H_y0) in (eq_ind_r T (THead -(Bind Void) t t0) (\lambda (t3: T).(eq T (THead (Bind Void) t t0) t3)) -(refl_equal T (THead (Bind Void) t t0)) t2 H16)))))))))))) H9)) (\lambda (H9: -(pr0 t0 (lift (S O) O t2))).(let H_y \def (H7 (lift (S O) O t2) H9) in (let -H10 \def (eq_ind T t0 (\lambda (t3: T).(subst0 O t t3 (lift (S O) O x))) H3 -(lift (S O) O t2) H_y) in (eq_ind_r T (lift (S O) O t2) (\lambda (t3: T).(eq -T (THead (Bind Void) t t3) t2)) (subst0_gen_lift_false t2 t (lift (S O) O x) -(S O) O O (le_n O) (eq_ind_r nat (plus (S O) O) (\lambda (n: nat).(lt O n)) -(le_n (plus (S O) O)) (plus O (S O)) (plus_sym O (S O))) H10 (eq T (THead -(Bind Void) t (lift (S O) O t2)) t2)) t0 H_y)))) (pr0_gen_void t t0 t2 -H8)))))) (\lambda (H7: (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 t0 t2)))).(ex2_ind T (\lambda (t2: T).((eq T -t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2)) (or (\forall -(t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) -t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to -(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)))) -(\lambda (x0: T).(\lambda (H8: (((eq T t0 x0) \to (\forall (P: -Prop).P)))).(\lambda (H9: (pr0 t0 x0)).(or_intror (\forall (t2: T).((pr0 -(THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) t2))) (ex2 T -(\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2))) (ex_intro2 T -(\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)) (THead (Bind -Void) t x0) (\lambda (H10: (eq T (THead (Bind Void) t t0) (THead (Bind Void) -t x0))).(\lambda (P: Prop).(let H11 \def (f_equal T T (\lambda (e: T).(match -e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) -\Rightarrow t0 | (THead _ _ t2) \Rightarrow t2])) (THead (Bind Void) t t0) -(THead (Bind Void) t x0) H10) in (let H12 \def (eq_ind_r T x0 (\lambda (t2: -T).(pr0 t0 t2)) H9 t0 H11) in (let H13 \def (eq_ind_r T x0 (\lambda (t2: -T).((eq T t0 t2) \to (\forall (P0: Prop).P0))) H8 t0 H11) in (H13 (refl_equal -T t0) P)))))) (pr0_comp t t (pr0_refl t) t0 x0 H9 (Bind Void))))))) H7)) -H6))) (\lambda (H5: (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 t t2)))).(ex2_ind T (\lambda (t2: T).((eq T -t t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2)) (or (\forall -(t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) -t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to -(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)))) -(\lambda (x0: T).(\lambda (H6: (((eq T t x0) \to (\forall (P: -Prop).P)))).(\lambda (H7: (pr0 t x0)).(or_intror (\forall (t2: T).((pr0 -(THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) t2))) (ex2 T -(\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2))) (ex_intro2 T -(\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)) (THead (Bind -Void) x0 t0) (\lambda (H8: (eq T (THead (Bind Void) t t0) (THead (Bind Void) -x0 t0))).(\lambda (P: Prop).(let H9 \def (f_equal T T (\lambda (e: T).(match -e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t | (TLRef _) -\Rightarrow t | (THead _ t2 _) \Rightarrow t2])) (THead (Bind Void) t t0) -(THead (Bind Void) x0 t0) H8) in (let H10 \def (eq_ind_r T x0 (\lambda (t2: -T).(pr0 t t2)) H7 t H9) in (let H11 \def (eq_ind_r T x0 (\lambda (t2: T).((eq -T t t2) \to (\forall (P0: Prop).P0))) H6 t H9) in (H11 (refl_equal T t) -P)))))) (pr0_comp t x0 H7 t0 t0 (pr0_refl t0) (Bind Void))))))) H5)) H4))) -(\lambda (H3: (eq T t0 (lift (S O) O x))).(let H4 \def (eq_ind T t0 (\lambda -(t2: T).(or (\forall (t3: T).((pr0 t2 t3) \to (eq T t2 t3))) (ex2 T (\lambda -(t3: T).((eq T t2 t3) \to (\forall (P: Prop).P))) (\lambda (t3: T).(pr0 t2 -t3))))) H0 (lift (S O) O x) H3) in (eq_ind_r T (lift (S O) O x) (\lambda (t2: -T).(or (\forall (t3: T).((pr0 (THead (Bind Void) t t2) t3) \to (eq T (THead -(Bind Void) t t2) t3))) (ex2 T (\lambda (t3: T).((eq T (THead (Bind Void) t -t2) t3) \to (\forall (P: Prop).P))) (\lambda (t3: T).(pr0 (THead (Bind Void) -t t2) t3))))) (or_intror (\forall (t2: T).((pr0 (THead (Bind Void) t (lift (S -O) O x)) t2) \to (eq T (THead (Bind Void) t (lift (S O) O x)) t2))) (ex2 T -(\lambda (t2: T).((eq T (THead (Bind Void) t (lift (S O) O x)) t2) \to -(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t (lift (S -O) O x)) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Bind Void) t -(lift (S O) O x)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 -(THead (Bind Void) t (lift (S O) O x)) t2)) x (\lambda (H5: (eq T (THead -(Bind Void) t (lift (S O) O x)) x)).(\lambda (P: Prop).(thead_x_lift_y_y -(Bind Void) x t (S O) O H5 P))) (pr0_zeta Void (sym_not_eq B Abst Void -not_abst_void) x x (pr0_refl x) t))) t0 H3))) H2))) H1))) b)) (\lambda (f: -F).(F_ind (\lambda (f0: F).(or (\forall (t2: T).((pr0 (THead (Flat f0) t t0) -t2) \to (eq T (THead (Flat f0) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T -(THead (Flat f0) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 -(THead (Flat f0) t t0) t2))))) (let H_x \def (binder_dec t0) in (let H1 \def -H_x in (or_ind (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: -T).(eq T t0 (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: -T).(\forall (u: T).((eq T t0 (THead (Bind b) w u)) \to (\forall (P: -Prop).P))))) (or (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq -T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat -Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead -(Flat Appl) t t0) t2)))) (\lambda (H2: (ex_3 B T T (\lambda (b: B).(\lambda -(w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w u))))))).(ex_3_ind B T T -(\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w -u))))) (or (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T -(THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat -Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead -(Flat Appl) t t0) t2)))) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: -T).(\lambda (H3: (eq T t0 (THead (Bind x0) x1 x2))).(let H4 \def (eq_ind T t0 -(\lambda (t2: T).(or (\forall (t3: T).((pr0 t2 t3) \to (eq T t2 t3))) (ex2 T -(\lambda (t3: T).((eq T t2 t3) \to (\forall (P: Prop).P))) (\lambda (t3: -T).(pr0 t2 t3))))) H0 (THead (Bind x0) x1 x2) H3) in (eq_ind_r T (THead (Bind -x0) x1 x2) (\lambda (t2: T).(or (\forall (t3: T).((pr0 (THead (Flat Appl) t -t2) t3) \to (eq T (THead (Flat Appl) t t2) t3))) (ex2 T (\lambda (t3: T).((eq -T (THead (Flat Appl) t t2) t3) \to (\forall (P: Prop).P))) (\lambda (t3: -T).(pr0 (THead (Flat Appl) t t2) t3))))) (B_ind (\lambda (b: B).((or (\forall -(t2: T).((pr0 (THead (Bind b) x1 x2) t2) \to (eq T (THead (Bind b) x1 x2) -t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind b) x1 x2) t2) \to (\forall -(P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind b) x1 x2) t2)))) \to (or -(\forall (t2: T).((pr0 (THead (Flat Appl) t (THead (Bind b) x1 x2)) t2) \to -(eq T (THead (Flat Appl) t (THead (Bind b) x1 x2)) t2))) (ex2 T (\lambda (t2: -T).((eq T (THead (Flat Appl) t (THead (Bind b) x1 x2)) t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead (Bind b) x1 x2)) -t2)))))) (\lambda (_: (or (\forall (t2: T).((pr0 (THead (Bind Abbr) x1 x2) -t2) \to (eq T (THead (Bind Abbr) x1 x2) t2))) (ex2 T (\lambda (t2: T).((eq T -(THead (Bind Abbr) x1 x2) t2) \to (\forall (P: Prop).P))) (\lambda (t2: -T).(pr0 (THead (Bind Abbr) x1 x2) t2))))).(or_intror (\forall (t2: T).((pr0 -(THead (Flat Appl) t (THead (Bind Abbr) x1 x2)) t2) \to (eq T (THead (Flat -Appl) t (THead (Bind Abbr) x1 x2)) t2))) (ex2 T (\lambda (t2: T).((eq T -(THead (Flat Appl) t (THead (Bind Abbr) x1 x2)) t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead (Bind Abbr) x1 -x2)) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t (THead -(Bind Abbr) x1 x2)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 -(THead (Flat Appl) t (THead (Bind Abbr) x1 x2)) t2)) (THead (Bind Abbr) x1 -(THead (Flat Appl) (lift (S O) O t) x2)) (\lambda (H6: (eq T (THead (Flat -Appl) t (THead (Bind Abbr) x1 x2)) (THead (Bind Abbr) x1 (THead (Flat Appl) -(lift (S O) O t) x2)))).(\lambda (P: Prop).(let H7 \def (eq_ind T (THead -(Flat Appl) t (THead (Bind Abbr) x1 x2)) (\lambda (ee: T).(match ee in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead _ _ t2) \Rightarrow (match t2 in T return (\lambda -(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False -| (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])])) I (THead (Bind -Abbr) x1 (THead (Flat Appl) (lift (S O) O t) x2)) H6) in (False_ind P H7)))) -(pr0_upsilon Abbr not_abbr_abst t t (pr0_refl t) x1 x1 (pr0_refl x1) x2 x2 -(pr0_refl x2))))) (\lambda (_: (or (\forall (t2: T).((pr0 (THead (Bind Abst) -x1 x2) t2) \to (eq T (THead (Bind Abst) x1 x2) t2))) (ex2 T (\lambda (t2: -T).((eq T (THead (Bind Abst) x1 x2) t2) \to (\forall (P: Prop).P))) (\lambda -(t2: T).(pr0 (THead (Bind Abst) x1 x2) t2))))).(or_intror (\forall (t2: -T).((pr0 (THead (Flat Appl) t (THead (Bind Abst) x1 x2)) t2) \to (eq T (THead -(Flat Appl) t (THead (Bind Abst) x1 x2)) t2))) (ex2 T (\lambda (t2: T).((eq T -(THead (Flat Appl) t (THead (Bind Abst) x1 x2)) t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead (Bind Abst) x1 -x2)) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t (THead -(Bind Abst) x1 x2)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 -(THead (Flat Appl) t (THead (Bind Abst) x1 x2)) t2)) (THead (Bind Abbr) t x2) -(\lambda (H6: (eq T (THead (Flat Appl) t (THead (Bind Abst) x1 x2)) (THead -(Bind Abbr) t x2))).(\lambda (P: Prop).(let H7 \def (eq_ind T (THead (Flat -Appl) t (THead (Bind Abst) x1 x2)) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind Abbr) t x2) H6) in (False_ind P H7)))) (pr0_beta x1 -t t (pr0_refl t) x2 x2 (pr0_refl x2))))) (\lambda (_: (or (\forall (t2: -T).((pr0 (THead (Bind Void) x1 x2) t2) \to (eq T (THead (Bind Void) x1 x2) -t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) x1 x2) t2) \to -(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) x1 x2) -t2))))).(or_intror (\forall (t2: T).((pr0 (THead (Flat Appl) t (THead (Bind -Void) x1 x2)) t2) \to (eq T (THead (Flat Appl) t (THead (Bind Void) x1 x2)) -t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t (THead (Bind Void) -x1 x2)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat -Appl) t (THead (Bind Void) x1 x2)) t2))) (ex_intro2 T (\lambda (t2: T).((eq T -(THead (Flat Appl) t (THead (Bind Void) x1 x2)) t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead (Bind Void) x1 -x2)) t2)) (THead (Bind Void) x1 (THead (Flat Appl) (lift (S O) O t) x2)) -(\lambda (H6: (eq T (THead (Flat Appl) t (THead (Bind Void) x1 x2)) (THead -(Bind Void) x1 (THead (Flat Appl) (lift (S O) O t) x2)))).(\lambda (P: -Prop).(let H7 \def (eq_ind T (THead (Flat Appl) t (THead (Bind Void) x1 x2)) -(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ t2) \Rightarrow -(match t2 in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False -| (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K -return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) -\Rightarrow False])])])) I (THead (Bind Void) x1 (THead (Flat Appl) (lift (S -O) O t) x2)) H6) in (False_ind P H7)))) (pr0_upsilon Void (sym_not_eq B Abst -Void not_abst_void) t t (pr0_refl t) x1 x1 (pr0_refl x1) x2 x2 (pr0_refl -x2))))) x0 H4) t0 H3)))))) H2)) (\lambda (H2: ((\forall (b: B).(\forall (w: -T).(\forall (u: T).((eq T t0 (THead (Bind b) w u)) \to (\forall (P: -Prop).P))))))).(let H3 \def H in (or_ind (\forall (t2: T).((pr0 t t2) \to (eq -T t t2))) (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) -(\lambda (t2: T).(pr0 t t2))) (or (\forall (t2: T).((pr0 (THead (Flat Appl) t -t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq -T (THead (Flat Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: -T).(pr0 (THead (Flat Appl) t t0) t2)))) (\lambda (H4: ((\forall (t2: T).((pr0 -t t2) \to (eq T t t2))))).(let H5 \def H0 in (or_ind (\forall (t2: T).((pr0 -t0 t2) \to (eq T t0 t2))) (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall -(P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2))) (or (\forall (t2: T).((pr0 -(THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T -(\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)))) (\lambda -(H6: ((\forall (t2: T).((pr0 t0 t2) \to (eq T t0 t2))))).(or_introl (\forall -(t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) -t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to -(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2))) -(\lambda (t2: T).(\lambda (H7: (pr0 (THead (Flat Appl) t t0) t2)).(or3_ind -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda -(t3: T).(pr0 t0 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T t0 (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 -(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr0 t u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t0 (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead -(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 t -u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(t3: T).(pr0 z1 t3)))))))) (eq T (THead (Flat Appl) t t0) t2) (\lambda (H8: -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda -(t3: T).(pr0 t0 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq -T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t -u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t0 t3))) (eq T (THead (Flat Appl) -t t0) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H9: (eq T t2 (THead -(Flat Appl) x0 x1))).(\lambda (H10: (pr0 t x0)).(\lambda (H11: (pr0 t0 -x1)).(let H_y \def (H6 x1 H11) in (let H_y0 \def (H4 x0 H10) in (let H12 \def -(eq_ind_r T x1 (\lambda (t3: T).(pr0 t0 t3)) H11 t0 H_y) in (let H13 \def -(eq_ind_r T x1 (\lambda (t3: T).(eq T t2 (THead (Flat Appl) x0 t3))) H9 t0 -H_y) in (let H14 \def (eq_ind_r T x0 (\lambda (t3: T).(pr0 t t3)) H10 t H_y0) -in (let H15 \def (eq_ind_r T x0 (\lambda (t3: T).(eq T t2 (THead (Flat Appl) -t3 t0))) H13 t H_y0) in (eq_ind_r T (THead (Flat Appl) t t0) (\lambda (t3: -T).(eq T (THead (Flat Appl) t t0) t3)) (refl_equal T (THead (Flat Appl) t -t0)) t2 H15)))))))))))) H8)) (\lambda (H8: (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t0 (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr0 t u2))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))).(ex4_4_ind T T T T -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t0 -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 t u2))))) (\lambda -(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))) (eq -T (THead (Flat Appl) t t0) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda -(x2: T).(\lambda (x3: T).(\lambda (H9: (eq T t0 (THead (Bind Abst) x0 -x1))).(\lambda (H10: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (_: (pr0 t -x2)).(\lambda (_: (pr0 x1 x3)).(eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda -(t3: T).(eq T (THead (Flat Appl) t t0) t3)) (let H13 \def (eq_ind T t0 -(\lambda (t3: T).(\forall (t4: T).((pr0 t3 t4) \to (eq T t3 t4)))) H6 (THead -(Bind Abst) x0 x1) H9) in (let H14 \def (eq_ind T t0 (\lambda (t3: -T).(\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t3 (THead (Bind b) -w u)) \to (\forall (P: Prop).P)))))) H2 (THead (Bind Abst) x0 x1) H9) in -(eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t3: T).(eq T (THead (Flat -Appl) t t3) (THead (Bind Abbr) x2 x3))) (H14 Abst x0 x1 (H13 (THead (Bind -Abst) x0 x1) (pr0_refl (THead (Bind Abst) x0 x1))) (eq T (THead (Flat Appl) t -(THead (Bind Abst) x0 x1)) (THead (Bind Abbr) x2 x3))) t0 H9))) t2 -H10))))))))) H8)) (\lambda (H8: (ex6_6 B T T T T T (\lambda (b: B).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not -(eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t0 (THead (Bind b) -y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead (Flat -Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 t u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 -t3))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T t0 (THead (Bind b) y1 z1)))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead (Flat Appl) (lift -(S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 t u2))))))) (\lambda -(_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: -T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))) -(eq T (THead (Flat Appl) t t0) t2) (\lambda (x0: B).(\lambda (x1: T).(\lambda -(x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (not -(eq B x0 Abst))).(\lambda (H10: (eq T t0 (THead (Bind x0) x1 x2))).(\lambda -(H11: (eq T t2 (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) -x5)))).(\lambda (_: (pr0 t x3)).(\lambda (_: (pr0 x1 x4)).(\lambda (_: (pr0 -x2 x5)).(eq_ind_r T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) -x5)) (\lambda (t3: T).(eq T (THead (Flat Appl) t t0) t3)) (let H15 \def -(eq_ind T t0 (\lambda (t3: T).(\forall (t4: T).((pr0 t3 t4) \to (eq T t3 -t4)))) H6 (THead (Bind x0) x1 x2) H10) in (let H16 \def (eq_ind T t0 (\lambda -(t3: T).(\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t3 (THead -(Bind b) w u)) \to (\forall (P: Prop).P)))))) H2 (THead (Bind x0) x1 x2) H10) -in (eq_ind_r T (THead (Bind x0) x1 x2) (\lambda (t3: T).(eq T (THead (Flat -Appl) t t3) (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)))) -(H16 x0 x1 x2 (H15 (THead (Bind x0) x1 x2) (pr0_refl (THead (Bind x0) x1 -x2))) (eq T (THead (Flat Appl) t (THead (Bind x0) x1 x2)) (THead (Bind x0) x4 -(THead (Flat Appl) (lift (S O) O x3) x5)))) t0 H10))) t2 H11))))))))))))) -H8)) (pr0_gen_appl t t0 t2 H7)))))) (\lambda (H6: (ex2 T (\lambda (t2: -T).((eq T t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t0 -t2)))).(ex2_ind T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: Prop).P))) -(\lambda (t2: T).(pr0 t0 t2)) (or (\forall (t2: T).((pr0 (THead (Flat Appl) t -t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq -T (THead (Flat Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: -T).(pr0 (THead (Flat Appl) t t0) t2)))) (\lambda (x: T).(\lambda (H7: (((eq T -t0 x) \to (\forall (P: Prop).P)))).(\lambda (H8: (pr0 t0 x)).(or_intror -(\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat -Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) -\to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) -t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to -(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)) -(THead (Flat Appl) t x) (\lambda (H9: (eq T (THead (Flat Appl) t t0) (THead -(Flat Appl) t x))).(\lambda (P: Prop).(let H10 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | -(TLRef _) \Rightarrow t0 | (THead _ _ t2) \Rightarrow t2])) (THead (Flat -Appl) t t0) (THead (Flat Appl) t x) H9) in (let H11 \def (eq_ind_r T x -(\lambda (t2: T).(pr0 t0 t2)) H8 t0 H10) in (let H12 \def (eq_ind_r T x -(\lambda (t2: T).((eq T t0 t2) \to (\forall (P0: Prop).P0))) H7 t0 H10) in -(H12 (refl_equal T t0) P)))))) (pr0_comp t t (pr0_refl t) t0 x H8 (Flat -Appl))))))) H6)) H5))) (\lambda (H4: (ex2 T (\lambda (t2: T).((eq T t t2) \to -(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2)))).(ex2_ind T (\lambda -(t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2)) -(or (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead -(Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t -t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) -t t0) t2)))) (\lambda (x: T).(\lambda (H5: (((eq T t x) \to (\forall (P: -Prop).P)))).(\lambda (H6: (pr0 t x)).(or_intror (\forall (t2: T).((pr0 (THead -(Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T -(\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2))) (ex_intro2 T -(\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P: -Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)) (THead (Flat -Appl) x t0) (\lambda (H7: (eq T (THead (Flat Appl) t t0) (THead (Flat Appl) x -t0))).(\lambda (P: Prop).(let H8 \def (f_equal T T (\lambda (e: T).(match e -in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t | (TLRef _) -\Rightarrow t | (THead _ t2 _) \Rightarrow t2])) (THead (Flat Appl) t t0) -(THead (Flat Appl) x t0) H7) in (let H9 \def (eq_ind_r T x (\lambda (t2: -T).(pr0 t t2)) H6 t H8) in (let H10 \def (eq_ind_r T x (\lambda (t2: T).((eq -T t t2) \to (\forall (P0: Prop).P0))) H5 t H8) in (H10 (refl_equal T t) -P)))))) (pr0_comp t x H6 t0 t0 (pr0_refl t0) (Flat Appl))))))) H4)) H3))) -H1))) (or_intror (\forall (t2: T).((pr0 (THead (Flat Cast) t t0) t2) \to (eq -T (THead (Flat Cast) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat -Cast) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead -(Flat Cast) t t0) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Flat -Cast) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead -(Flat Cast) t t0) t2)) t0 (\lambda (H1: (eq T (THead (Flat Cast) t t0) -t0)).(\lambda (P: Prop).(thead_x_y_y (Flat Cast) t t0 H1 P))) (pr0_tau t0 t0 -(pr0_refl t0) t))) f)) k)))))) t1). -(* COMMENTS -Initial nodes: 10459 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr0/defs.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr0/defs.ma deleted file mode 100644 index 0568e070c..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr0/defs.ma +++ /dev/null @@ -1,40 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/subst0/defs.ma". - -inductive pr0: T \to (T \to Prop) \def -| pr0_refl: \forall (t: T).(pr0 t t) -| pr0_comp: \forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (t1: -T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (k: K).(pr0 (THead k u1 t1) -(THead k u2 t2)))))))) -| pr0_beta: \forall (u: T).(\forall (v1: T).(\forall (v2: T).((pr0 v1 v2) \to -(\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (pr0 (THead (Flat Appl) v1 -(THead (Bind Abst) u t1)) (THead (Bind Abbr) v2 t2)))))))) -| pr0_upsilon: \forall (b: B).((not (eq B b Abst)) \to (\forall (v1: -T).(\forall (v2: T).((pr0 v1 v2) \to (\forall (u1: T).(\forall (u2: T).((pr0 -u1 u2) \to (\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (pr0 (THead -(Flat Appl) v1 (THead (Bind b) u1 t1)) (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t2))))))))))))) -| pr0_delta: \forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (t1: -T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (w: T).((subst0 O u2 t2 w) \to -(pr0 (THead (Bind Abbr) u1 t1) (THead (Bind Abbr) u2 w))))))))) -| pr0_zeta: \forall (b: B).((not (eq B b Abst)) \to (\forall (t1: T).(\forall -(t2: T).((pr0 t1 t2) \to (\forall (u: T).(pr0 (THead (Bind b) u (lift (S O) O -t1)) t2)))))) -| pr0_tau: \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (u: -T).(pr0 (THead (Flat Cast) u t1) t2)))). - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr0/fwd.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr0/fwd.ma deleted file mode 100644 index 46caceab4..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr0/fwd.ma +++ /dev/null @@ -1,2018 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr0/props.ma". - -theorem pr0_gen_sort: - \forall (x: T).(\forall (n: nat).((pr0 (TSort n) x) \to (eq T x (TSort n)))) -\def - \lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr0 (TSort n) x)).(insert_eq -T (TSort n) (\lambda (t: T).(pr0 t x)) (\lambda (t: T).(eq T x t)) (\lambda -(y: T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: T).(\lambda (t0: -T).((eq T t (TSort n)) \to (eq T t0 t)))) (\lambda (t: T).(\lambda (H1: (eq T -t (TSort n))).(let H2 \def (f_equal T T (\lambda (e: T).e) t (TSort n) H1) in -(eq_ind_r T (TSort n) (\lambda (t0: T).(eq T t0 t0)) (refl_equal T (TSort n)) -t H2)))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda -(_: (((eq T u1 (TSort n)) \to (eq T u2 u1)))).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2 -t1)))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u1 t1) (TSort n))).(let -H6 \def (eq_ind T (THead k u1 t1) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H5) in -(False_ind (eq T (THead k u2 t2) (THead k u1 t1)) H6)))))))))))) (\lambda (u: -T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: -(((eq T v1 (TSort n)) \to (eq T v2 v1)))).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2 -t1)))).(\lambda (H5: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t1)) -(TSort n))).(let H6 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u -t1)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) -\Rightarrow True])) I (TSort n) H5) in (False_ind (eq T (THead (Bind Abbr) v2 -t2) (THead (Flat Appl) v1 (THead (Bind Abst) u t1))) H6)))))))))))) (\lambda -(b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: -T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (TSort n)) \to (eq T v2 -v1)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda -(_: (((eq T u1 (TSort n)) \to (eq T u2 u1)))).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2 -t1)))).(\lambda (H8: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t1)) -(TSort n))).(let H9 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 -t1)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) -\Rightarrow True])) I (TSort n) H8) in (False_ind (eq T (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Flat Appl) v1 (THead (Bind -b) u1 t1))) H9))))))))))))))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda -(_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 (TSort n)) \to (eq T u2 -u1)))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda -(_: (((eq T t1 (TSort n)) \to (eq T t2 t1)))).(\lambda (w: T).(\lambda (_: -(subst0 O u2 t2 w)).(\lambda (H6: (eq T (THead (Bind Abbr) u1 t1) (TSort -n))).(let H7 \def (eq_ind T (THead (Bind Abbr) u1 t1) (\lambda (ee: T).(match -ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | -(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) -H6) in (False_ind (eq T (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u1 t1)) -H7))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda -(t1: T).(\lambda (t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 -(TSort n)) \to (eq T t2 t1)))).(\lambda (u: T).(\lambda (H4: (eq T (THead -(Bind b) u (lift (S O) O t1)) (TSort n))).(let H5 \def (eq_ind T (THead (Bind -b) u (lift (S O) O t1)) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead _ _ _) \Rightarrow True])) I (TSort n) H4) in (False_ind (eq T t2 -(THead (Bind b) u (lift (S O) O t1))) H5)))))))))) (\lambda (t1: T).(\lambda -(t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq -T t2 t1)))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u t1) -(TSort n))).(let H4 \def (eq_ind T (THead (Flat Cast) u t1) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I -(TSort n) H3) in (False_ind (eq T t2 (THead (Flat Cast) u t1)) H4)))))))) y x -H0))) H))). -(* COMMENTS -Initial nodes: 1045 -END *) - -theorem pr0_gen_lref: - \forall (x: T).(\forall (n: nat).((pr0 (TLRef n) x) \to (eq T x (TLRef n)))) -\def - \lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr0 (TLRef n) x)).(insert_eq -T (TLRef n) (\lambda (t: T).(pr0 t x)) (\lambda (t: T).(eq T x t)) (\lambda -(y: T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: T).(\lambda (t0: -T).((eq T t (TLRef n)) \to (eq T t0 t)))) (\lambda (t: T).(\lambda (H1: (eq T -t (TLRef n))).(let H2 \def (f_equal T T (\lambda (e: T).e) t (TLRef n) H1) in -(eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 t0)) (refl_equal T (TLRef n)) -t H2)))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda -(_: (((eq T u1 (TLRef n)) \to (eq T u2 u1)))).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (eq T t2 -t1)))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u1 t1) (TLRef n))).(let -H6 \def (eq_ind T (THead k u1 t1) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H5) in -(False_ind (eq T (THead k u2 t2) (THead k u1 t1)) H6)))))))))))) (\lambda (u: -T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: -(((eq T v1 (TLRef n)) \to (eq T v2 v1)))).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (eq T t2 -t1)))).(\lambda (H5: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t1)) -(TLRef n))).(let H6 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u -t1)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) -\Rightarrow True])) I (TLRef n) H5) in (False_ind (eq T (THead (Bind Abbr) v2 -t2) (THead (Flat Appl) v1 (THead (Bind Abst) u t1))) H6)))))))))))) (\lambda -(b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: -T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (TLRef n)) \to (eq T v2 -v1)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda -(_: (((eq T u1 (TLRef n)) \to (eq T u2 u1)))).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (eq T t2 -t1)))).(\lambda (H8: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t1)) -(TLRef n))).(let H9 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 -t1)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) -\Rightarrow True])) I (TLRef n) H8) in (False_ind (eq T (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Flat Appl) v1 (THead (Bind -b) u1 t1))) H9))))))))))))))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda -(_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 (TLRef n)) \to (eq T u2 -u1)))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda -(_: (((eq T t1 (TLRef n)) \to (eq T t2 t1)))).(\lambda (w: T).(\lambda (_: -(subst0 O u2 t2 w)).(\lambda (H6: (eq T (THead (Bind Abbr) u1 t1) (TLRef -n))).(let H7 \def (eq_ind T (THead (Bind Abbr) u1 t1) (\lambda (ee: T).(match -ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | -(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) -H6) in (False_ind (eq T (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u1 t1)) -H7))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda -(t1: T).(\lambda (t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 -(TLRef n)) \to (eq T t2 t1)))).(\lambda (u: T).(\lambda (H4: (eq T (THead -(Bind b) u (lift (S O) O t1)) (TLRef n))).(let H5 \def (eq_ind T (THead (Bind -b) u (lift (S O) O t1)) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead _ _ _) \Rightarrow True])) I (TLRef n) H4) in (False_ind (eq T t2 -(THead (Bind b) u (lift (S O) O t1))) H5)))))))))) (\lambda (t1: T).(\lambda -(t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (eq -T t2 t1)))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u t1) -(TLRef n))).(let H4 \def (eq_ind T (THead (Flat Cast) u t1) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I -(TLRef n) H3) in (False_ind (eq T t2 (THead (Flat Cast) u t1)) H4)))))))) y x -H0))) H))). -(* COMMENTS -Initial nodes: 1045 -END *) - -theorem pr0_gen_abst: - \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Bind Abst) u1 -t1) x) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind -Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: -T).(\lambda (t2: T).(pr0 t1 t2))))))) -\def - \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead -(Bind Abst) u1 t1) x)).(insert_eq T (THead (Bind Abst) u1 t1) (\lambda (t: -T).(pr0 t x)) (\lambda (_: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 -u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))))) (\lambda (y: -T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: T).(\lambda (t0: T).((eq T -t (THead (Bind Abst) u1 t1)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T t0 (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))))))) (\lambda -(t: T).(\lambda (H1: (eq T t (THead (Bind Abst) u1 t1))).(let H2 \def -(f_equal T T (\lambda (e: T).e) t (THead (Bind Abst) u1 t1) H1) in (eq_ind_r -T (THead (Bind Abst) u1 t1) (\lambda (t0: T).(ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T t0 (THead (Bind Abst) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 -t2))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Bind -Abst) u1 t1) (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))) u1 t1 -(refl_equal T (THead (Bind Abst) u1 t1)) (pr0_refl u1) (pr0_refl t1)) t -H2)))) (\lambda (u0: T).(\lambda (u2: T).(\lambda (H1: (pr0 u0 u2)).(\lambda -(H2: (((eq T u0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T (\lambda (u3: -T).(\lambda (t2: T).(eq T u2 (THead (Bind Abst) u3 t2)))) (\lambda (u3: -T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 -t2))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H3: (pr0 t0 -t2)).(\lambda (H4: (((eq T t0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T -(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u3 t3)))) -(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3))))))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u0 t0) -(THead (Bind Abst) u1 t1))).(let H6 \def (f_equal T K (\lambda (e: T).(match -e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) -\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u0 t0) (THead (Bind -Abst) u1 t1) H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) -\Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead k u0 t0) (THead (Bind -Abst) u1 t1) H5) in ((let H8 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) -\Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k u0 t0) (THead (Bind -Abst) u1 t1) H5) in (\lambda (H9: (eq T u0 u1)).(\lambda (H10: (eq K k (Bind -Abst))).(eq_ind_r K (Bind Abst) (\lambda (k0: K).(ex3_2 T T (\lambda (u3: -T).(\lambda (t3: T).(eq T (THead k0 u2 t2) (THead (Bind Abst) u3 t3)))) -(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3))))) (let H11 \def (eq_ind T t0 (\lambda (t: T).((eq T t (THead -(Bind Abst) u1 t1)) \to (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 -(THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) -(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))))) H4 t1 H8) in (let H12 \def -(eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H3 t1 H8) in (let H13 \def (eq_ind T -u0 (\lambda (t: T).((eq T t (THead (Bind Abst) u1 t1)) \to (ex3_2 T T -(\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead (Bind Abst) u3 t3)))) -(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3)))))) H2 u1 H9) in (let H14 \def (eq_ind T u0 (\lambda (t: -T).(pr0 t u2)) H1 u1 H9) in (ex3_2_intro T T (\lambda (u3: T).(\lambda (t3: -T).(eq T (THead (Bind Abst) u2 t2) (THead (Bind Abst) u3 t3)))) (\lambda (u3: -T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 -t3))) u2 t2 (refl_equal T (THead (Bind Abst) u2 t2)) H14 H12))))) k H10)))) -H7)) H6)))))))))))) (\lambda (u: T).(\lambda (v1: T).(\lambda (v2: -T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind Abst) u1 -t1)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind -Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: -T).(\lambda (t2: T).(pr0 t1 t2))))))).(\lambda (t0: T).(\lambda (t2: -T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Abst) u1 -t1)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t1 t3))))))).(\lambda (H5: (eq T (THead (Flat Appl) -v1 (THead (Bind Abst) u t0)) (THead (Bind Abst) u1 t1))).(let H6 \def (eq_ind -T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (ee: T).(match ee -in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef -_) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return -(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind Abst) u1 t1) H5) in (False_ind (ex3_2 T T (\lambda -(u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) (THead (Bind Abst) -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t1 t3)))) H6)))))))))))) (\lambda (b: B).(\lambda -(_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 -v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T -(\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind Abst) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: -T).(pr0 t1 t2))))))).(\lambda (u0: T).(\lambda (u2: T).(\lambda (_: (pr0 u0 -u2)).(\lambda (_: (((eq T u0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T -(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Bind Abst) u3 t2)))) -(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: -T).(pr0 t1 t2))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 -t2)).(\lambda (_: (((eq T t0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T -(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u3 t3)))) -(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3))))))).(\lambda (H8: (eq T (THead (Flat Appl) v1 (THead (Bind -b) u0 t0)) (THead (Bind Abst) u1 t1))).(let H9 \def (eq_ind T (THead (Flat -Appl) v1 (THead (Bind b) u0 t0)) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind Abst) u1 t1) H8) in (False_ind (ex3_2 T T (\lambda -(u3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t2)) (THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: -T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) -H9))))))))))))))))) (\lambda (u0: T).(\lambda (u2: T).(\lambda (_: (pr0 u0 -u2)).(\lambda (_: (((eq T u0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T -(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Bind Abst) u3 t2)))) -(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: -T).(pr0 t1 t2))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 -t2)).(\lambda (_: (((eq T t0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T -(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u3 t3)))) -(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3))))))).(\lambda (w: T).(\lambda (_: (subst0 O u2 t2 -w)).(\lambda (H6: (eq T (THead (Bind Abbr) u0 t0) (THead (Bind Abst) u1 -t1))).(let H7 \def (eq_ind T (THead (Bind Abbr) u0 t0) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K -return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B return -(\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | -Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (THead (Bind -Abst) u1 t1) H6) in (False_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t3: -T).(eq T (THead (Bind Abbr) u2 w) (THead (Bind Abst) u3 t3)))) (\lambda (u3: -T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 -t3)))) H7))))))))))))) (\lambda (b: B).(\lambda (H1: (not (eq B b -Abst))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda -(H3: (((eq T t0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 -t3))))))).(\lambda (u: T).(\lambda (H4: (eq T (THead (Bind b) u (lift (S O) O -t0)) (THead (Bind Abst) u1 t1))).(let H5 \def (f_equal T B (\lambda (e: -T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | -(TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k in K return -(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow -b])])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abst) u1 t1) H4) in -((let H6 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: -T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) -\Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abst) u1 -t1) H4) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat -\to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow -(TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true -\Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t3) \Rightarrow -(THead k (lref_map f d u0) (lref_map f (s k d) t3))]) in lref_map) (\lambda -(x0: nat).(plus x0 (S O))) O t0) | (TLRef _) \Rightarrow ((let rec lref_map -(f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) -\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with -[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t3) -\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t3))]) in -lref_map) (\lambda (x0: nat).(plus x0 (S O))) O t0) | (THead _ _ t) -\Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abst) u1 -t1) H4) in (\lambda (_: (eq T u u1)).(\lambda (H9: (eq B b Abst)).(let H10 -\def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H1 Abst H9) in (let -H11 \def (eq_ind_r T t1 (\lambda (t: T).((eq T t0 (THead (Bind Abst) u1 t)) -\to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t t3)))))) H3 (lift (S O) O t0) H7) in (eq_ind T -(lift (S O) O t0) (\lambda (t: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t3: -T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t t3))))) (let H12 -\def (match (H10 (refl_equal B Abst)) in False return (\lambda (_: -False).(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr0 (lift (S O) O t0) t3))))) with []) in H12) t1 -H7)))))) H6)) H5)))))))))) (\lambda (t0: T).(\lambda (t2: T).(\lambda (_: -(pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T -T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3))))))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u -t0) (THead (Bind Abst) u1 t1))).(let H4 \def (eq_ind T (THead (Flat Cast) u -t0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort -_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) -\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u1 -t1) H3) in (False_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 -(THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) -(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) H4)))))))) y x H0))) H)))). -(* COMMENTS -Initial nodes: 2838 -END *) - -theorem pr0_gen_appl: - \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Flat Appl) u1 -t1) x) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead -(Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda -(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T x (THead (Bind b) -v2 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: B).(\lambda -(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 -u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(t2: T).(pr0 z1 t2)))))))))))) -\def - \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead -(Flat Appl) u1 t1) x)).(insert_eq T (THead (Flat Appl) u1 t1) (\lambda (t: -T).(pr0 t x)) (\lambda (_: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 -u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda -(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) -(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T x -(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda -(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: -T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 -v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))))))) (\lambda (y: -T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: T).(\lambda (t0: T).((eq T -t (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda -(t2: T).(eq T t0 (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T -T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda -(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) -(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T -t0 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: -T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 -v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))))))))) (\lambda (t: -T).(\lambda (H1: (eq T t (THead (Flat Appl) u1 t1))).(let H2 \def (f_equal T -T (\lambda (e: T).e) t (THead (Flat Appl) u1 t1) H1) in (eq_ind_r T (THead -(Flat Appl) u1 t1) (\lambda (t0: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda -(t2: T).(eq T t0 (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T -T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda -(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) -(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T -t0 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: -T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 -v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))))))) (or3_intro0 (ex3_2 T -T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Appl) u1 t1) (THead -(Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda -(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t2: T).(eq T (THead (Flat Appl) u1 t1) (THead (Bind Abbr) u2 t2)))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 -u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: -T).(pr0 z1 t2)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B -b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(v2: T).(\lambda (t2: T).(eq T (THead (Flat Appl) u1 t1) (THead (Bind b) v2 -(THead (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 -u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(t2: T).(pr0 z1 t2)))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: -T).(eq T (THead (Flat Appl) u1 t1) (THead (Flat Appl) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 -t2))) u1 t1 (refl_equal T (THead (Flat Appl) u1 t1)) (pr0_refl u1) (pr0_refl -t1))) t H2)))) (\lambda (u0: T).(\lambda (u2: T).(\lambda (H1: (pr0 u0 -u2)).(\lambda (H2: (((eq T u0 (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T -(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) -(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: -T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Bind -Abbr) u3 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda -(_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(u3: T).(\lambda (v2: T).(\lambda (t2: T).(eq T u2 (THead (Bind b) v2 (THead -(Flat Appl) (lift (S O) O u3) t2))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 -u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(t2: T).(pr0 z1 t2)))))))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda -(H3: (pr0 t0 t2)).(\lambda (H4: (((eq T t0 (THead (Flat Appl) u1 t1)) \to -(or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) -u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 -z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: -T).(eq T t2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -b) v2 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda -(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3)))))))))))).(\lambda (k: K).(\lambda (H5: (eq -T (THead k u0 t0) (THead (Flat Appl) u1 t1))).(let H6 \def (f_equal T K -(\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) -\Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) -(THead k u0 t0) (THead (Flat Appl) u1 t1) H5) in ((let H7 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) -(THead k u0 t0) (THead (Flat Appl) u1 t1) H5) in ((let H8 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) -(THead k u0 t0) (THead (Flat Appl) u1 t1) H5) in (\lambda (H9: (eq T u0 -u1)).(\lambda (H10: (eq K k (Flat Appl))).(eq_ind_r K (Flat Appl) (\lambda -(k0: K).(or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead k0 u2 -t2) (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 -u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead -(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: -T).(\lambda (t3: T).(eq T (THead k0 u2 t2) (THead (Bind Abbr) u3 t3)))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 -u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: -T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B -b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda -(v2: T).(\lambda (t3: T).(eq T (THead k0 u2 t2) (THead (Bind b) v2 (THead -(Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 -u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(t3: T).(pr0 z1 t3)))))))))) (let H11 \def (eq_ind T t0 (\lambda (t: T).((eq -T t (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda -(t3: T).(eq T t2 (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: -T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T -T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: -T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda -(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) -(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T -t2 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: -T).(\lambda (_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 -v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))))))) H4 t1 H8) in (let -H12 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H3 t1 H8) in (let H13 \def -(eq_ind T u0 (\lambda (t: T).((eq T t (THead (Flat Appl) u1 t1)) \to (or3 -(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead (Flat Appl) u3 -t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 -z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: -T).(eq T u2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T u2 (THead (Bind -b) v2 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda -(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3))))))))))) H2 u1 H9) in (let H14 \def (eq_ind -T u0 (\lambda (t: T).(pr0 t u2)) H1 u1 H9) in (or3_intro0 (ex3_2 T T (\lambda -(u3: T).(\lambda (t3: T).(eq T (THead (Flat Appl) u2 t2) (THead (Flat Appl) -u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 -z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: -T).(eq T (THead (Flat Appl) u2 t2) (THead (Bind Abbr) u3 t3)))))) (\lambda -(_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 -t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) -(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (v2: T).(\lambda -(t3: T).(eq T (THead (Flat Appl) u2 t2) (THead (Bind b) v2 (THead (Flat Appl) -(lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 -t3)))))))) (ex3_2_intro T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead -(Flat Appl) u2 t2) (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: -T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) u2 t2 -(refl_equal T (THead (Flat Appl) u2 t2)) H14 H12)))))) k H10)))) H7)) -H6)))))))))))) (\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda -(H1: (pr0 v1 v2)).(\lambda (H2: (((eq T v1 (THead (Flat Appl) u1 t1)) \to -(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Flat Appl) -u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: -T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 -z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: -T).(eq T v2 (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (v3: T).(\lambda (t2: T).(eq T v2 (THead (Bind -b) v3 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda -(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (t2: T).(pr0 z1 t2)))))))))))).(\lambda (t0: T).(\lambda (t2: -T).(\lambda (H3: (pr0 t0 t2)).(\lambda (H4: (((eq T t0 (THead (Flat Appl) u1 -t1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead -(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda -(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T t2 (THead (Bind -b) v3 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda -(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3)))))))))))).(\lambda (H5: (eq T (THead (Flat -Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat Appl) u1 t1))).(let H6 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t _) -\Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat -Appl) u1 t1) H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow (THead (Bind Abst) u -t0) | (TLRef _) \Rightarrow (THead (Bind Abst) u t0) | (THead _ _ t) -\Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat -Appl) u1 t1) H5) in (\lambda (H8: (eq T v1 u1)).(let H9 \def (eq_ind T v1 -(\lambda (t: T).((eq T t (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T v2 (THead (Flat Appl) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T v2 (THead (Bind -Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T v2 (THead (Bind b) v3 (THead -(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 -u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(t3: T).(pr0 z1 t3))))))))))) H2 u1 H8) in (let H10 \def (eq_ind T v1 -(\lambda (t: T).(pr0 t v2)) H1 u1 H8) in (let H11 \def (eq_ind_r T t1 -(\lambda (t: T).((eq T t0 (THead (Flat Appl) u1 t)) \to (or3 (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v3 (THead -(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 -u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(t3: T).(pr0 z1 t3))))))))))) H4 (THead (Bind Abst) u t0) H7) in (let H12 -\def (eq_ind_r T t1 (\lambda (t: T).((eq T u1 (THead (Flat Appl) u1 t)) \to -(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T v2 (THead (Flat Appl) -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 -z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: -T).(eq T v2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T v2 (THead (Bind -b) v3 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda -(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3))))))))))) H9 (THead (Bind Abst) u t0) H7) in -(eq_ind T (THead (Bind Abst) u t0) (\lambda (t: T).(or3 (ex3_2 T T (\lambda -(u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) (THead (Flat Appl) -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 -z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: -T).(eq T (THead (Bind Abbr) v2 t2) (THead (Bind Abbr) u2 t3)))))) (\lambda -(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 -t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) -(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(eq T t (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v3: T).(\lambda -(t3: T).(eq T (THead (Bind Abbr) v2 t2) (THead (Bind b) v3 (THead (Flat Appl) -(lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 -t3)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T -(THead (Bind Abbr) v2 t2) (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 (THead -(Bind Abst) u t0) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) u t0) (THead -(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) (THead (Bind Abbr) u2 -t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: -T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B -b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) u t0) (THead -(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T (THead (Bind -Abbr) v2 t2) (THead (Bind b) v3 (THead (Flat Appl) (lift (S O) O u2) -t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda -(y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 -y1 v3))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))))) (ex4_4_intro T T T T -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind Abst) u t0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) -v2 t2) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))) u t0 v2 t2 -(refl_equal T (THead (Bind Abst) u t0)) (refl_equal T (THead (Bind Abbr) v2 -t2)) H10 H3)) t1 H7))))))) H6)))))))))))) (\lambda (b: B).(\lambda (H1: (not -(eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H2: (pr0 v1 -v2)).(\lambda (H3: (((eq T v1 (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T -(\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Flat Appl) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: -T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind -Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T (\lambda (b0: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (v3: T).(\lambda (t2: T).(eq T v2 (THead (Bind b0) v3 (THead -(Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 -u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(t2: T).(pr0 z1 t2)))))))))))).(\lambda (u0: T).(\lambda (u2: T).(\lambda -(H4: (pr0 u0 u2)).(\lambda (H5: (((eq T u0 (THead (Flat Appl) u1 t1)) \to -(or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) -u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: -T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 -z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: -T).(eq T u2 (THead (Bind Abbr) u3 t2)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T -(\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t2: T).(eq T u2 (THead (Bind -b0) v3 (THead (Flat Appl) (lift (S O) O u3) t2))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda -(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (t2: T).(pr0 z1 t2)))))))))))).(\lambda (t0: T).(\lambda (t2: -T).(\lambda (H6: (pr0 t0 t2)).(\lambda (H7: (((eq T t0 (THead (Flat Appl) u1 -t1)) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead -(Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda -(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda -(t3: T).(eq T t2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T -(\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T t2 (THead (Bind -b0) v3 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda -(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3)))))))))))).(\lambda (H8: (eq T (THead (Flat -Appl) v1 (THead (Bind b) u0 t0)) (THead (Flat Appl) u1 t1))).(let H9 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t _) -\Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead (Flat -Appl) u1 t1) H8) in ((let H10 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow (THead (Bind b) u0 t0) -| (TLRef _) \Rightarrow (THead (Bind b) u0 t0) | (THead _ _ t) \Rightarrow -t])) (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead (Flat Appl) u1 t1) -H8) in (\lambda (H11: (eq T v1 u1)).(let H12 \def (eq_ind T v1 (\lambda (t: -T).((eq T t (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T (\lambda (u3: -T).(\lambda (t3: T).(eq T v2 (THead (Flat Appl) u3 t3)))) (\lambda (u3: -T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 -t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T v2 (THead (Bind -Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda -(_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b0: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T v2 (THead (Bind b0) v3 (THead -(Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 -u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(t3: T).(pr0 z1 t3))))))))))) H3 u1 H11) in (let H13 \def (eq_ind T v1 -(\lambda (t: T).(pr0 t v2)) H2 u1 H11) in (let H14 \def (eq_ind_r T t1 -(\lambda (t: T).((eq T t0 (THead (Flat Appl) u1 t)) \to (or3 (ex3_2 T T -(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u3 t3)))) -(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind -Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda -(_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b0: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind -b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T t2 (THead (Bind b0) v3 (THead -(Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 -u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(t3: T).(pr0 z1 t3))))))))))) H7 (THead (Bind b) u0 t0) H10) in (let H15 \def -(eq_ind_r T t1 (\lambda (t: T).((eq T u0 (THead (Flat Appl) u1 t)) \to (or3 -(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead (Flat Appl) u3 -t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 -z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: -T).(eq T u2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T -(\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t -(THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T u2 (THead (Bind -b0) v3 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda -(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3))))))))))) H5 (THead (Bind b) u0 t0) H10) in -(let H16 \def (eq_ind_r T t1 (\lambda (t: T).((eq T u1 (THead (Flat Appl) u1 -t)) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T v2 (THead -(Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda -(_: T).(\lambda (t3: T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda -(t3: T).(eq T v2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T -(\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t -(THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T v2 (THead (Bind -b0) v3 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda -(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3))))))))))) H12 (THead (Bind b) u0 t0) H10) in -(eq_ind T (THead (Bind b) u0 t0) (\lambda (t: T).(or3 (ex3_2 T T (\lambda -(u3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t2)) (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: -T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t t3)))) (ex4_4 T T T -T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: -T).(\lambda (t3: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O -v2) t2)) (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T -(\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t -(THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Bind b0) v3 (THead (Flat -Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 -t3)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T -(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Flat -Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: -T).(\lambda (t3: T).(pr0 (THead (Bind b) u0 t0) t3)))) (ex4_4 T T T T -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind b) u0 t0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Bind Abbr) u3 t3)))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 -u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: -T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B -b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) u0 t0) (THead -(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Bind b0) v3 (THead (Flat -Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 -t3)))))))) (ex6_6_intro B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 -Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) u0 t0) (THead (Bind -b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t2)) (THead (Bind b0) v3 (THead (Flat Appl) -(lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 -t3))))))) b u0 t0 v2 u2 t2 H1 (refl_equal T (THead (Bind b) u0 t0)) -(refl_equal T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2))) -H13 H4 H6)) t1 H10)))))))) H9))))))))))))))))) (\lambda (u0: T).(\lambda (u2: -T).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (((eq T u0 (THead (Flat Appl) u1 -t1)) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead -(Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda -(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda -(t2: T).(eq T u2 (THead (Bind Abbr) u3 t2)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t2: T).(eq T u2 (THead (Bind -b) v2 (THead (Flat Appl) (lift (S O) O u3) t2))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda -(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (t2: T).(pr0 z1 t2)))))))))))).(\lambda (t0: T).(\lambda (t2: -T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u1 -t1)) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead -(Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda -(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda -(t3: T).(eq T t2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -b) v2 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda -(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3)))))))))))).(\lambda (w: T).(\lambda (_: -(subst0 O u2 t2 w)).(\lambda (H6: (eq T (THead (Bind Abbr) u0 t0) (THead -(Flat Appl) u1 t1))).(let H7 \def (eq_ind T (THead (Bind Abbr) u0 t0) -(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow -(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | -(Flat _) \Rightarrow False])])) I (THead (Flat Appl) u1 t1) H6) in (False_ind -(or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 -w) (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 -u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead -(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: -T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u3 -t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: -T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B -b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda -(v2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 w) (THead (Bind b) v2 -(THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 -u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(t3: T).(pr0 z1 t3))))))))) H7))))))))))))) (\lambda (b: B).(\lambda (_: (not -(eq B b Abst))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 -t2)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b0: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b0) v2 (THead -(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 -u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(t3: T).(pr0 z1 t3)))))))))))).(\lambda (u: T).(\lambda (H4: (eq T (THead -(Bind b) u (lift (S O) O t0)) (THead (Flat Appl) u1 t1))).(let H5 \def -(eq_ind T (THead (Bind b) u (lift (S O) O t0)) (\lambda (ee: T).(match ee in -T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow -False])])) I (THead (Flat Appl) u1 t1) H4) in (False_ind (or3 (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b0: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b0) v2 (THead -(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 -u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(t3: T).(pr0 z1 t3))))))))) H5)))))))))) (\lambda (t0: T).(\lambda (t2: -T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u1 -t1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead -(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda -(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda -(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3)))))))))))).(\lambda (u: T).(\lambda (H3: (eq -T (THead (Flat Cast) u t0) (THead (Flat Appl) u1 t1))).(let H4 \def (eq_ind T -(THead (Flat Cast) u t0) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f in F return -(\lambda (_: F).Prop) with [Appl \Rightarrow False | Cast \Rightarrow -True])])])) I (THead (Flat Appl) u1 t1) H3) in (False_ind (or3 (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead -(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 -u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(t3: T).(pr0 z1 t3))))))))) H4)))))))) y x H0))) H)))). -(* COMMENTS -Initial nodes: 12299 -END *) - -theorem pr0_gen_cast: - \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Flat Cast) u1 -t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead -(Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda -(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 x))))) -\def - \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead -(Flat Cast) u1 t1) x)).(insert_eq T (THead (Flat Cast) u1 t1) (\lambda (t: -T).(pr0 t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 -u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 x))) -(\lambda (y: T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: T).(\lambda -(t0: T).((eq T t (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T t0 (THead (Flat Cast) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 -t2)))) (pr0 t1 t0))))) (\lambda (t: T).(\lambda (H1: (eq T t (THead (Flat -Cast) u1 t1))).(let H2 \def (f_equal T T (\lambda (e: T).e) t (THead (Flat -Cast) u1 t1) H1) in (eq_ind_r T (THead (Flat Cast) u1 t1) (\lambda (t0: -T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Flat -Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: -T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 t0))) (or_introl (ex3_2 T T -(\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Cast) u1 t1) (THead -(Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda -(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (THead (Flat Cast) u1 t1)) -(ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Cast) -u1 t1) (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 -u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))) u1 t1 (refl_equal T -(THead (Flat Cast) u1 t1)) (pr0_refl u1) (pr0_refl t1))) t H2)))) (\lambda -(u0: T).(\lambda (u2: T).(\lambda (H1: (pr0 u0 u2)).(\lambda (H2: (((eq T u0 -(THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: -T).(eq T u2 (THead (Flat Cast) u3 t2)))) (\lambda (u3: T).(\lambda (_: -T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 -u2))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H3: (pr0 t0 t2)).(\lambda -(H4: (((eq T t0 (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: -T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u3 t3)))) (\lambda (u3: -T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 -t3)))) (pr0 t1 t2))))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u0 t0) -(THead (Flat Cast) u1 t1))).(let H6 \def (f_equal T K (\lambda (e: T).(match -e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) -\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u0 t0) (THead (Flat -Cast) u1 t1) H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) -\Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead k u0 t0) (THead (Flat -Cast) u1 t1) H5) in ((let H8 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) -\Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k u0 t0) (THead (Flat -Cast) u1 t1) H5) in (\lambda (H9: (eq T u0 u1)).(\lambda (H10: (eq K k (Flat -Cast))).(eq_ind_r K (Flat Cast) (\lambda (k0: K).(or (ex3_2 T T (\lambda (u3: -T).(\lambda (t3: T).(eq T (THead k0 u2 t2) (THead (Flat Cast) u3 t3)))) -(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3)))) (pr0 t1 (THead k0 u2 t2)))) (let H11 \def (eq_ind T t0 -(\lambda (t: T).((eq T t (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T -(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u3 t3)))) -(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3)))) (pr0 t1 t2)))) H4 t1 H8) in (let H12 \def (eq_ind T t0 -(\lambda (t: T).(pr0 t t2)) H3 t1 H8) in (let H13 \def (eq_ind T u0 (\lambda -(t: T).((eq T t (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: -T).(\lambda (t3: T).(eq T u2 (THead (Flat Cast) u3 t3)))) (\lambda (u3: -T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 -t3)))) (pr0 t1 u2)))) H2 u1 H9) in (let H14 \def (eq_ind T u0 (\lambda (t: -T).(pr0 t u2)) H1 u1 H9) in (or_introl (ex3_2 T T (\lambda (u3: T).(\lambda -(t3: T).(eq T (THead (Flat Cast) u2 t2) (THead (Flat Cast) u3 t3)))) (\lambda -(u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 -t1 t3)))) (pr0 t1 (THead (Flat Cast) u2 t2)) (ex3_2_intro T T (\lambda (u3: -T).(\lambda (t3: T).(eq T (THead (Flat Cast) u2 t2) (THead (Flat Cast) u3 -t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t1 t3))) u2 t2 (refl_equal T (THead (Flat Cast) u2 -t2)) H14 H12)))))) k H10)))) H7)) H6)))))))))))) (\lambda (u: T).(\lambda -(v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 -(THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T v2 (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 -v2))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda -(_: (((eq T t0 (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 -t3)))) (pr0 t1 t2))))).(\lambda (H5: (eq T (THead (Flat Appl) v1 (THead (Bind -Abst) u t0)) (THead (Flat Cast) u1 t1))).(let H6 \def (eq_ind T (THead (Flat -Appl) v1 (THead (Bind Abst) u t0)) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f -in F return (\lambda (_: F).Prop) with [Appl \Rightarrow True | Cast -\Rightarrow False])])])) I (THead (Flat Cast) u1 t1) H5) in (False_ind (or -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) -(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) -(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (THead (Bind Abbr) v2 -t2))) H6)))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b -Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda -(_: (((eq T v1 (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T v2 (THead (Flat Cast) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 -t2)))) (pr0 t1 v2))))).(\lambda (u0: T).(\lambda (u2: T).(\lambda (_: (pr0 u0 -u2)).(\lambda (_: (((eq T u0 (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T -(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Cast) u3 t2)))) -(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: -T).(pr0 t1 t2)))) (pr0 t1 u2))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda -(_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Flat Cast) u1 t1)) \to (or -(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u3 -t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2))))).(\lambda (H8: (eq T (THead -(Flat Appl) v1 (THead (Bind b) u0 t0)) (THead (Flat Cast) u1 t1))).(let H9 -\def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K -return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) -\Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl \Rightarrow -True | Cast \Rightarrow False])])])) I (THead (Flat Cast) u1 t1) H8) in -(False_ind (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Flat Cast) u3 -t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t2)))) H9))))))))))))))))) (\lambda (u0: T).(\lambda -(u2: T).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (((eq T u0 (THead (Flat Cast) -u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead -(Flat Cast) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda -(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 u2))))).(\lambda (t0: -T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead -(Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq -T t2 (THead (Flat Cast) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 -u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 -t2))))).(\lambda (w: T).(\lambda (_: (subst0 O u2 t2 w)).(\lambda (H6: (eq T -(THead (Bind Abbr) u0 t0) (THead (Flat Cast) u1 t1))).(let H7 \def (eq_ind T -(THead (Bind Abbr) u0 t0) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat -Cast) u1 t1) H6) in (False_ind (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: -T).(eq T (THead (Bind Abbr) u2 w) (THead (Flat Cast) u3 t3)))) (\lambda (u3: -T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 -t3)))) (pr0 t1 (THead (Bind Abbr) u2 w))) H7))))))))))))) (\lambda (b: -B).(\lambda (_: (not (eq B b Abst))).(\lambda (t0: T).(\lambda (t2: -T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Flat Cast) u1 -t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead -(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda -(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2))))).(\lambda (u: -T).(\lambda (H4: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead (Flat -Cast) u1 t1))).(let H5 \def (eq_ind T (THead (Bind b) u (lift (S O) O t0)) -(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow -(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | -(Flat _) \Rightarrow False])])) I (THead (Flat Cast) u1 t1) H4) in (False_ind -(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2)) H5)))))))))) (\lambda (t0: -T).(\lambda (t2: T).(\lambda (H1: (pr0 t0 t2)).(\lambda (H2: (((eq T t0 -(THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: -T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 -t2))))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u t0) (THead -(Flat Cast) u1 t1))).(let H4 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) -\Rightarrow u | (THead _ t _) \Rightarrow t])) (THead (Flat Cast) u t0) -(THead (Flat Cast) u1 t1) H3) in ((let H5 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | -(TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Flat Cast) -u t0) (THead (Flat Cast) u1 t1) H3) in (\lambda (_: (eq T u u1)).(let H7 \def -(eq_ind T t0 (\lambda (t: T).((eq T t (THead (Flat Cast) u1 t1)) \to (or -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2)))) H2 t1 H5) in (let H8 \def -(eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H1 t1 H5) in (or_intror (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3)))) (pr0 t1 t2) H8))))) H4)))))))) y x H0))) H)))). -(* COMMENTS -Initial nodes: 2911 -END *) - -theorem pr0_gen_abbr: - \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Bind Abbr) u1 -t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead -(Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda -(u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y: T).(pr0 t1 y)) -(\lambda (y: T).(subst0 O u2 y t2))))))) (pr0 t1 (lift (S O) O x)))))) -\def - \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead -(Bind Abbr) u1 t1) x)).(insert_eq T (THead (Bind Abbr) u1 t1) (\lambda (t: -T).(pr0 t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 -u1 u2))) (\lambda (u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda -(y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y t2))))))) (pr0 t1 (lift (S -O) O x)))) (\lambda (y: T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: -T).(\lambda (t0: T).((eq T t (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T -(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: -T).(or (pr0 t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: -T).(subst0 O u2 y0 t2))))))) (pr0 t1 (lift (S O) O t0)))))) (\lambda (t: -T).(\lambda (H1: (eq T t (THead (Bind Abbr) u1 t1))).(let H2 \def (f_equal T -T (\lambda (e: T).e) t (THead (Bind Abbr) u1 t1) H1) in (eq_ind_r T (THead -(Bind Abbr) u1 t1) (\lambda (t0: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda -(t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T -(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 t2))))))) (pr0 -t1 (lift (S O) O t0)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T (THead (Bind Abbr) u1 t1) (THead (Bind Abbr) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: T).(or (pr0 -t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 -t2))))))) (pr0 t1 (lift (S O) O (THead (Bind Abbr) u1 t1))) (ex3_2_intro T T -(\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Bind Abbr) u1 t1) (THead -(Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda -(u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) -(\lambda (y0: T).(subst0 O u2 y0 t2)))))) u1 t1 (refl_equal T (THead (Bind -Abbr) u1 t1)) (pr0_refl u1) (or_introl (pr0 t1 t1) (ex2 T (\lambda (y0: -T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u1 y0 t1))) (pr0_refl t1)))) t -H2)))) (\lambda (u0: T).(\lambda (u2: T).(\lambda (H1: (pr0 u0 u2)).(\lambda -(H2: (((eq T u0 (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: -T).(\lambda (t2: T).(eq T u2 (THead (Bind Abbr) u3 t2)))) (\lambda (u3: -T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t2: T).(or (pr0 -t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 -t2))))))) (pr0 t1 (lift (S O) O u2)))))).(\lambda (t0: T).(\lambda (t2: -T).(\lambda (H3: (pr0 t0 t2)).(\lambda (H4: (((eq T t0 (THead (Bind Abbr) u1 -t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead -(Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda -(u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) -(\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O -t2)))))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u0 t0) (THead (Bind -Abbr) u1 t1))).(let H6 \def (f_equal T K (\lambda (e: T).(match e in T return -(\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | -(THead k0 _ _) \Rightarrow k0])) (THead k u0 t0) (THead (Bind Abbr) u1 t1) -H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 -| (THead _ t _) \Rightarrow t])) (THead k u0 t0) (THead (Bind Abbr) u1 t1) -H5) in ((let H8 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 -| (THead _ _ t) \Rightarrow t])) (THead k u0 t0) (THead (Bind Abbr) u1 t1) -H5) in (\lambda (H9: (eq T u0 u1)).(\lambda (H10: (eq K k (Bind -Abbr))).(eq_ind_r K (Bind Abbr) (\lambda (k0: K).(or (ex3_2 T T (\lambda (u3: -T).(\lambda (t3: T).(eq T (THead k0 u2 t2) (THead (Bind Abbr) u3 t3)))) -(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: -T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: -T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O (THead k0 u2 t2))))) (let -H11 \def (eq_ind T t0 (\lambda (t: T).((eq T t (THead (Bind Abbr) u1 t1)) \to -(or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) -u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: -T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) -(\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O t2))))) H4 -t1 H8) in (let H12 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H3 t1 H8) in -(let H13 \def (eq_ind T u0 (\lambda (t: T).((eq T t (THead (Bind Abbr) u1 -t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead -(Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda -(u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) -(\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O u2))))) H2 -u1 H9) in (let H14 \def (eq_ind T u0 (\lambda (t: T).(pr0 t u2)) H1 u1 H9) in -(or_introl (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind -Abbr) u2 t2) (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: -T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T -(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 -t1 (lift (S O) O (THead (Bind Abbr) u2 t2))) (ex3_2_intro T T (\lambda (u3: -T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 t2) (THead (Bind Abbr) u3 -t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: -T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) -(\lambda (y0: T).(subst0 O u3 y0 t3)))))) u2 t2 (refl_equal T (THead (Bind -Abbr) u2 t2)) H14 (or_introl (pr0 t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) -(\lambda (y0: T).(subst0 O u2 y0 t2))) H12))))))) k H10)))) H7)) -H6)))))))))))) (\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: -(pr0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind Abbr) u1 t1)) \to (or -(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind Abbr) u2 -t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: -T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) -(\lambda (y0: T).(subst0 O u2 y0 t2))))))) (pr0 t1 (lift (S O) O -v2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda -(_: (((eq T t0 (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0 -t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 -t3))))))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (H5: (eq T (THead (Flat -Appl) v1 (THead (Bind Abst) u t0)) (THead (Bind Abbr) u1 t1))).(let H6 \def -(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K -return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) -\Rightarrow True])])) I (THead (Bind Abbr) u1 t1) H5) in (False_ind (or -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) -(THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) -(\lambda (u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: -T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S -O) O (THead (Bind Abbr) v2 t2)))) H6)))))))))))) (\lambda (b: B).(\lambda (_: -(not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 -v2)).(\lambda (_: (((eq T v1 (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T -(\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind Abbr) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: -T).(or (pr0 t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: -T).(subst0 O u2 y0 t2))))))) (pr0 t1 (lift (S O) O v2)))))).(\lambda (u0: -T).(\lambda (u2: T).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (((eq T u0 (THead -(Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq -T u2 (THead (Bind Abbr) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 -u3))) (\lambda (u3: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y0: -T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 t2))))))) (pr0 t1 (lift (S -O) O u2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 -t2)).(\lambda (_: (((eq T t0 (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T -(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u3 t3)))) -(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: -T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: -T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (H8: (eq -T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead (Bind Abbr) u1 -t1))).(let H9 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) -(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow -(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | -(Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u1 t1) H8) in (False_ind -(or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Bind Abbr) u3 t3)))) -(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: -T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: -T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t2))))) H9))))))))))))))))) (\lambda (u0: -T).(\lambda (u2: T).(\lambda (H1: (pr0 u0 u2)).(\lambda (H2: (((eq T u0 -(THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: -T).(eq T u2 (THead (Bind Abbr) u3 t2)))) (\lambda (u3: T).(\lambda (_: -T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T -(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 t2))))))) (pr0 -t1 (lift (S O) O u2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H3: -(pr0 t0 t2)).(\lambda (H4: (((eq T t0 (THead (Bind Abbr) u1 t1)) \to (or -(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u3 -t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: -T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) -(\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O -t2)))))).(\lambda (w: T).(\lambda (H5: (subst0 O u2 t2 w)).(\lambda (H6: (eq -T (THead (Bind Abbr) u0 t0) (THead (Bind Abbr) u1 t1))).(let H7 \def (f_equal -T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) -(THead (Bind Abbr) u0 t0) (THead (Bind Abbr) u1 t1) H6) in ((let H8 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) -\Rightarrow t])) (THead (Bind Abbr) u0 t0) (THead (Bind Abbr) u1 t1) H6) in -(\lambda (H9: (eq T u0 u1)).(let H10 \def (eq_ind T t0 (\lambda (t: T).((eq T -t (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda -(t3: T).(eq T t2 (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: -T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T -(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 -t1 (lift (S O) O t2))))) H4 t1 H8) in (let H11 \def (eq_ind T t0 (\lambda (t: -T).(pr0 t t2)) H3 t1 H8) in (let H12 \def (eq_ind T u0 (\lambda (t: T).((eq T -t (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda -(t3: T).(eq T u2 (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: -T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T -(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 -t1 (lift (S O) O u2))))) H2 u1 H9) in (let H13 \def (eq_ind T u0 (\lambda (t: -T).(pr0 t u2)) H1 u1 H9) in (or_introl (ex3_2 T T (\lambda (u3: T).(\lambda -(t3: T).(eq T (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u3 t3)))) (\lambda -(u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or -(pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O -u3 y0 t3))))))) (pr0 t1 (lift (S O) O (THead (Bind Abbr) u2 w))) (ex3_2_intro -T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 w) (THead -(Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda -(u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) -(\lambda (y0: T).(subst0 O u3 y0 t3)))))) u2 w (refl_equal T (THead (Bind -Abbr) u2 w)) H13 (or_intror (pr0 t1 w) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) -(\lambda (y0: T).(subst0 O u2 y0 w))) (ex_intro2 T (\lambda (y0: T).(pr0 t1 -y0)) (\lambda (y0: T).(subst0 O u2 y0 w)) t2 H11 H5)))))))))) H7))))))))))))) -(\lambda (b: B).(\lambda (H1: (not (eq B b Abst))).(\lambda (t0: T).(\lambda -(t2: T).(\lambda (H2: (pr0 t0 t2)).(\lambda (H3: (((eq T t0 (THead (Bind -Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 -(THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) -(\lambda (u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: -T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S -O) O t2)))))).(\lambda (u: T).(\lambda (H4: (eq T (THead (Bind b) u (lift (S -O) O t0)) (THead (Bind Abbr) u1 t1))).(let H5 \def (f_equal T B (\lambda (e: -T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | -(TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k in K return -(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow -b])])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abbr) u1 t1) H4) in -((let H6 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: -T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) -\Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abbr) u1 -t1) H4) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat -\to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow -(TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true -\Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t3) \Rightarrow -(THead k (lref_map f d u0) (lref_map f (s k d) t3))]) in lref_map) (\lambda -(x0: nat).(plus x0 (S O))) O t0) | (TLRef _) \Rightarrow ((let rec lref_map -(f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) -\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with -[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t3) -\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t3))]) in -lref_map) (\lambda (x0: nat).(plus x0 (S O))) O t0) | (THead _ _ t) -\Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abbr) u1 -t1) H4) in (\lambda (_: (eq T u u1)).(\lambda (H9: (eq B b Abbr)).(let H10 -\def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H1 Abbr H9) in (let -H11 \def (eq_ind_r T t1 (\lambda (t: T).((eq T t0 (THead (Bind Abbr) u1 t)) -\to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: -T).(\lambda (t3: T).(or (pr0 t t3) (ex2 T (\lambda (y0: T).(pr0 t y0)) -(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t (lift (S O) O t2))))) H3 -(lift (S O) O t0) H7) in (eq_ind T (lift (S O) O t0) (\lambda (t: T).(or -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: -T).(\lambda (t3: T).(or (pr0 t t3) (ex2 T (\lambda (y0: T).(pr0 t y0)) -(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t (lift (S O) O t2)))) -(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: -T).(\lambda (t3: T).(or (pr0 (lift (S O) O t0) t3) (ex2 T (\lambda (y0: -T).(pr0 (lift (S O) O t0) y0)) (\lambda (y0: T).(subst0 O u2 y0 t3))))))) -(pr0 (lift (S O) O t0) (lift (S O) O t2)) (pr0_lift t0 t2 H2 (S O) O)) t1 -H7)))))) H6)) H5)))))))))) (\lambda (t0: T).(\lambda (t2: T).(\lambda (_: -(pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Abbr) u1 t1)) \to (or -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: -T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) -(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S O) O -t2)))))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u t0) (THead -(Bind Abbr) u1 t1))).(let H4 \def (eq_ind T (THead (Flat Cast) u t0) (\lambda -(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow -(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | -(Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u1 t1) H3) in (False_ind -(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: -T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) -(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S O) O t2))) -H4)))))))) y x H0))) H)))). -(* COMMENTS -Initial nodes: 4711 -END *) - -theorem pr0_gen_void: - \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Bind Void) u1 -t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead -(Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda -(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O x)))))) -\def - \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead -(Bind Void) u1 t1) x)).(insert_eq T (THead (Bind Void) u1 t1) (\lambda (t: -T).(pr0 t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 -u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) -O x)))) (\lambda (y: T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: -T).(\lambda (t0: T).((eq T t (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T -(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Void) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: -T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O t0)))))) (\lambda (t: T).(\lambda -(H1: (eq T t (THead (Bind Void) u1 t1))).(let H2 \def (f_equal T T (\lambda -(e: T).e) t (THead (Bind Void) u1 t1) H1) in (eq_ind_r T (THead (Bind Void) -u1 t1) (\lambda (t0: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq -T t0 (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 -u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O -t0)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead -(Bind Void) u1 t1) (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 -(lift (S O) O (THead (Bind Void) u1 t1))) (ex3_2_intro T T (\lambda (u2: -T).(\lambda (t2: T).(eq T (THead (Bind Void) u1 t1) (THead (Bind Void) u2 -t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: -T).(\lambda (t2: T).(pr0 t1 t2))) u1 t1 (refl_equal T (THead (Bind Void) u1 -t1)) (pr0_refl u1) (pr0_refl t1))) t H2)))) (\lambda (u0: T).(\lambda (u2: -T).(\lambda (H1: (pr0 u0 u2)).(\lambda (H2: (((eq T u0 (THead (Bind Void) u1 -t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead -(Bind Void) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda -(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O -u2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H3: (pr0 t0 -t2)).(\lambda (H4: (((eq T t0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T -(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u3 t3)))) -(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (k: K).(\lambda -(H5: (eq T (THead k u0 t0) (THead (Bind Void) u1 t1))).(let H6 \def (f_equal -T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) -\Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) -(THead k u0 t0) (THead (Bind Void) u1 t1) H5) in ((let H7 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) -(THead k u0 t0) (THead (Bind Void) u1 t1) H5) in ((let H8 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) -(THead k u0 t0) (THead (Bind Void) u1 t1) H5) in (\lambda (H9: (eq T u0 -u1)).(\lambda (H10: (eq K k (Bind Void))).(eq_ind_r K (Bind Void) (\lambda -(k0: K).(or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead k0 u2 -t2) (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 -u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O -(THead k0 u2 t2))))) (let H11 \def (eq_ind T t0 (\lambda (t: T).((eq T t -(THead (Bind Void) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: -T).(eq T t2 (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: -T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 -(lift (S O) O t2))))) H4 t1 H8) in (let H12 \def (eq_ind T t0 (\lambda (t: -T).(pr0 t t2)) H3 t1 H8) in (let H13 \def (eq_ind T u0 (\lambda (t: T).((eq T -t (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda -(t3: T).(eq T u2 (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: -T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 -(lift (S O) O u2))))) H2 u1 H9) in (let H14 \def (eq_ind T u0 (\lambda (t: -T).(pr0 t u2)) H1 u1 H9) in (or_introl (ex3_2 T T (\lambda (u3: T).(\lambda -(t3: T).(eq T (THead (Bind Void) u2 t2) (THead (Bind Void) u3 t3)))) (\lambda -(u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 -t1 t3)))) (pr0 t1 (lift (S O) O (THead (Bind Void) u2 t2))) (ex3_2_intro T T -(\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Void) u2 t2) (THead -(Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda -(_: T).(\lambda (t3: T).(pr0 t1 t3))) u2 t2 (refl_equal T (THead (Bind Void) -u2 t2)) H14 H12)))))) k H10)))) H7)) H6)))))))))))) (\lambda (u: T).(\lambda -(v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 -(THead (Bind Void) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T v2 (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 -(lift (S O) O v2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 -t2)).(\lambda (_: (((eq T t0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (H5: (eq T (THead -(Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Bind Void) u1 t1))).(let H6 -\def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K -return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) -\Rightarrow True])])) I (THead (Bind Void) u1 t1) H5) in (False_ind (or -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) -(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) -(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O (THead -(Bind Abbr) v2 t2)))) H6)))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B -b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 -v2)).(\lambda (_: (((eq T v1 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T -(\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind Void) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: -T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O v2)))))).(\lambda (u0: T).(\lambda -(u2: T).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (((eq T u0 (THead (Bind Void) -u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead -(Bind Void) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda -(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O -u2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda -(_: (((eq T t0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: -T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u3 t3)))) (\lambda (u3: -T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 -t3)))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (H8: (eq T (THead (Flat Appl) -v1 (THead (Bind b) u0 t0)) (THead (Bind Void) u1 t1))).(let H9 \def (eq_ind T -(THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (\lambda (ee: T).(match ee in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind Void) u1 t1) H8) in (False_ind (or (ex3_2 T T -(\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t2)) (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda -(_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 -(lift (S O) O (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2))))) -H9))))))))))))))))) (\lambda (u0: T).(\lambda (u2: T).(\lambda (_: (pr0 u0 -u2)).(\lambda (_: (((eq T u0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T -(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Bind Void) u3 t2)))) -(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: -T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O u2)))))).(\lambda (t0: T).(\lambda -(t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Void) -u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead -(Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda -(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O -t2)))))).(\lambda (w: T).(\lambda (_: (subst0 O u2 t2 w)).(\lambda (H6: (eq T -(THead (Bind Abbr) u0 t0) (THead (Bind Void) u1 t1))).(let H7 \def (eq_ind T -(THead (Bind Abbr) u0 t0) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr -\Rightarrow True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat -_) \Rightarrow False])])) I (THead (Bind Void) u1 t1) H6) in (False_ind (or -(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 w) -(THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) -(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O (THead -(Bind Abbr) u2 w)))) H7))))))))))))) (\lambda (b: B).(\lambda (H1: (not (eq B -b Abst))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H2: (pr0 t0 -t2)).(\lambda (H3: (((eq T t0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (u: T).(\lambda -(H4: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Void) u1 -t1))).(let H5 \def (f_equal T B (\lambda (e: T).(match e in T return (\lambda -(_: T).B) with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead k -_ _) \Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b0) -\Rightarrow b0 | (Flat _) \Rightarrow b])])) (THead (Bind b) u (lift (S O) O -t0)) (THead (Bind Void) u1 t1) H4) in ((let H6 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | -(TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t])) (THead (Bind b) u -(lift (S O) O t0)) (THead (Bind Void) u1 t1) H4) in ((let H7 \def (f_equal T -T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T -\def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow -(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) -| (THead k u0 t3) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) -t3))]) in lref_map) (\lambda (x0: nat).(plus x0 (S O))) O t0) | (TLRef _) -\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T -\def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow -(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) -| (THead k u0 t3) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) -t3))]) in lref_map) (\lambda (x0: nat).(plus x0 (S O))) O t0) | (THead _ _ t) -\Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Void) u1 -t1) H4) in (\lambda (_: (eq T u u1)).(\lambda (H9: (eq B b Void)).(let H10 -\def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H1 Void H9) in (let -H11 \def (eq_ind_r T t1 (\lambda (t: T).((eq T t0 (THead (Bind Void) u1 t)) -\to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t t3)))) (pr0 t (lift (S O) O t2))))) H3 (lift (S O) -O t0) H7) in (eq_ind T (lift (S O) O t0) (\lambda (t: T).(or (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t t3)))) (pr0 t (lift (S O) O t2)))) (or_intror (ex3_2 T T (\lambda -(u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 (lift -(S O) O t0) t3)))) (pr0 (lift (S O) O t0) (lift (S O) O t2)) (pr0_lift t0 t2 -H2 (S O) O)) t1 H7)))))) H6)) H5)))))))))) (\lambda (t0: T).(\lambda (t2: -T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Void) u1 -t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead -(Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda -(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O -t2)))))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u t0) (THead -(Bind Void) u1 t1))).(let H4 \def (eq_ind T (THead (Flat Cast) u t0) (\lambda -(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow -(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | -(Flat _) \Rightarrow True])])) I (THead (Bind Void) u1 t1) H3) in (False_ind -(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2))) H4)))))))) y x -H0))) H)))). -(* COMMENTS -Initial nodes: 3436 -END *) - -theorem pr0_gen_lift: - \forall (t1: T).(\forall (x: T).(\forall (h: nat).(\forall (d: nat).((pr0 -(lift h d t1) x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda -(t2: T).(pr0 t1 t2))))))) -\def - \lambda (t1: T).(\lambda (x: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda -(H: (pr0 (lift h d t1) x)).(insert_eq T (lift h d t1) (\lambda (t: T).(pr0 t -x)) (\lambda (_: T).(ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda -(t2: T).(pr0 t1 t2)))) (\lambda (y: T).(\lambda (H0: (pr0 y x)).(unintro nat -d (\lambda (n: nat).((eq T y (lift h n t1)) \to (ex2 T (\lambda (t2: T).(eq T -x (lift h n t2))) (\lambda (t2: T).(pr0 t1 t2))))) (unintro T t1 (\lambda (t: -T).(\forall (x0: nat).((eq T y (lift h x0 t)) \to (ex2 T (\lambda (t2: T).(eq -T x (lift h x0 t2))) (\lambda (t2: T).(pr0 t t2)))))) (pr0_ind (\lambda (t: -T).(\lambda (t0: T).(\forall (x0: T).(\forall (x1: nat).((eq T t (lift h x1 -x0)) \to (ex2 T (\lambda (t2: T).(eq T t0 (lift h x1 t2))) (\lambda (t2: -T).(pr0 x0 t2)))))))) (\lambda (t: T).(\lambda (x0: T).(\lambda (x1: -nat).(\lambda (H1: (eq T t (lift h x1 x0))).(ex_intro2 T (\lambda (t2: T).(eq -T t (lift h x1 t2))) (\lambda (t2: T).(pr0 x0 t2)) x0 H1 (pr0_refl x0)))))) -(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda (H2: -((\forall (x0: T).(\forall (x1: nat).((eq T u1 (lift h x1 x0)) \to (ex2 T -(\lambda (t2: T).(eq T u2 (lift h x1 t2))) (\lambda (t2: T).(pr0 x0 -t2)))))))).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2 -t3)).(\lambda (H4: ((\forall (x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 -x0)) \to (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: -T).(pr0 x0 t4)))))))).(\lambda (k: K).(\lambda (x0: T).(\lambda (x1: -nat).(\lambda (H5: (eq T (THead k u1 t2) (lift h x1 x0))).(K_ind (\lambda -(k0: K).((eq T (THead k0 u1 t2) (lift h x1 x0)) \to (ex2 T (\lambda (t4: -T).(eq T (THead k0 u2 t3) (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4))))) -(\lambda (b: B).(\lambda (H6: (eq T (THead (Bind b) u1 t2) (lift h x1 -x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Bind -b) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u1 (lift h x1 y0)))) -(\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h (S x1) z)))) (ex2 T (\lambda -(t4: T).(eq T (THead (Bind b) u2 t3) (lift h x1 t4))) (\lambda (t4: T).(pr0 -x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq T x0 (THead -(Bind b) x2 x3))).(\lambda (H8: (eq T u1 (lift h x1 x2))).(\lambda (H9: (eq T -t2 (lift h (S x1) x3))).(eq_ind_r T (THead (Bind b) x2 x3) (\lambda (t: -T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 t3) (lift h x1 t4))) -(\lambda (t4: T).(pr0 t t4)))) (ex2_ind T (\lambda (t4: T).(eq T t3 (lift h -(S x1) t4))) (\lambda (t4: T).(pr0 x3 t4)) (ex2 T (\lambda (t4: T).(eq T -(THead (Bind b) u2 t3) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind b) -x2 x3) t4))) (\lambda (x4: T).(\lambda (H_x: (eq T t3 (lift h (S x1) -x4))).(\lambda (H10: (pr0 x3 x4)).(eq_ind_r T (lift h (S x1) x4) (\lambda (t: -T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 t) (lift h x1 t4))) -(\lambda (t4: T).(pr0 (THead (Bind b) x2 x3) t4)))) (ex2_ind T (\lambda (t4: -T).(eq T u2 (lift h x1 t4))) (\lambda (t4: T).(pr0 x2 t4)) (ex2 T (\lambda -(t4: T).(eq T (THead (Bind b) u2 (lift h (S x1) x4)) (lift h x1 t4))) -(\lambda (t4: T).(pr0 (THead (Bind b) x2 x3) t4))) (\lambda (x5: T).(\lambda -(H_x0: (eq T u2 (lift h x1 x5))).(\lambda (H11: (pr0 x2 x5)).(eq_ind_r T -(lift h x1 x5) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) -t (lift h (S x1) x4)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind b) -x2 x3) t4)))) (ex_intro2 T (\lambda (t4: T).(eq T (THead (Bind b) (lift h x1 -x5) (lift h (S x1) x4)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind -b) x2 x3) t4)) (THead (Bind b) x5 x4) (sym_eq T (lift h x1 (THead (Bind b) x5 -x4)) (THead (Bind b) (lift h x1 x5) (lift h (S x1) x4)) (lift_bind b x5 x4 h -x1)) (pr0_comp x2 x5 H11 x3 x4 H10 (Bind b))) u2 H_x0)))) (H2 x2 x1 H8)) t3 -H_x)))) (H4 x3 (S x1) H9)) x0 H7)))))) (lift_gen_bind b u1 t2 x0 h x1 H6)))) -(\lambda (f: F).(\lambda (H6: (eq T (THead (Flat f) u1 t2) (lift h x1 -x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Flat -f) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u1 (lift h x1 y0)))) -(\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h x1 z)))) (ex2 T (\lambda -(t4: T).(eq T (THead (Flat f) u2 t3) (lift h x1 t4))) (\lambda (t4: T).(pr0 -x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq T x0 (THead -(Flat f) x2 x3))).(\lambda (H8: (eq T u1 (lift h x1 x2))).(\lambda (H9: (eq T -t2 (lift h x1 x3))).(eq_ind_r T (THead (Flat f) x2 x3) (\lambda (t: T).(ex2 T -(\lambda (t4: T).(eq T (THead (Flat f) u2 t3) (lift h x1 t4))) (\lambda (t4: -T).(pr0 t t4)))) (ex2_ind T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) -(\lambda (t4: T).(pr0 x3 t4)) (ex2 T (\lambda (t4: T).(eq T (THead (Flat f) -u2 t3) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat f) x2 x3) t4))) -(\lambda (x4: T).(\lambda (H_x: (eq T t3 (lift h x1 x4))).(\lambda (H10: (pr0 -x3 x4)).(eq_ind_r T (lift h x1 x4) (\lambda (t: T).(ex2 T (\lambda (t4: -T).(eq T (THead (Flat f) u2 t) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead -(Flat f) x2 x3) t4)))) (ex2_ind T (\lambda (t4: T).(eq T u2 (lift h x1 t4))) -(\lambda (t4: T).(pr0 x2 t4)) (ex2 T (\lambda (t4: T).(eq T (THead (Flat f) -u2 (lift h x1 x4)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat f) x2 -x3) t4))) (\lambda (x5: T).(\lambda (H_x0: (eq T u2 (lift h x1 x5))).(\lambda -(H11: (pr0 x2 x5)).(eq_ind_r T (lift h x1 x5) (\lambda (t: T).(ex2 T (\lambda -(t4: T).(eq T (THead (Flat f) t (lift h x1 x4)) (lift h x1 t4))) (\lambda -(t4: T).(pr0 (THead (Flat f) x2 x3) t4)))) (ex_intro2 T (\lambda (t4: T).(eq -T (THead (Flat f) (lift h x1 x5) (lift h x1 x4)) (lift h x1 t4))) (\lambda -(t4: T).(pr0 (THead (Flat f) x2 x3) t4)) (THead (Flat f) x5 x4) (sym_eq T -(lift h x1 (THead (Flat f) x5 x4)) (THead (Flat f) (lift h x1 x5) (lift h x1 -x4)) (lift_flat f x5 x4 h x1)) (pr0_comp x2 x5 H11 x3 x4 H10 (Flat f))) u2 -H_x0)))) (H2 x2 x1 H8)) t3 H_x)))) (H4 x3 x1 H9)) x0 H7)))))) (lift_gen_flat -f u1 t2 x0 h x1 H6)))) k H5))))))))))))) (\lambda (u: T).(\lambda (v1: -T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (H2: ((\forall (x0: -T).(\forall (x1: nat).((eq T v1 (lift h x1 x0)) \to (ex2 T (\lambda (t2: -T).(eq T v2 (lift h x1 t2))) (\lambda (t2: T).(pr0 x0 t2)))))))).(\lambda -(t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2 t3)).(\lambda (H4: ((\forall -(x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to (ex2 T (\lambda (t4: -T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4)))))))).(\lambda -(x0: T).(\lambda (x1: nat).(\lambda (H5: (eq T (THead (Flat Appl) v1 (THead -(Bind Abst) u t2)) (lift h x1 x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda -(z: T).(eq T x0 (THead (Flat Appl) y0 z)))) (\lambda (y0: T).(\lambda (_: -T).(eq T v1 (lift h x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T (THead -(Bind Abst) u t2) (lift h x1 z)))) (ex2 T (\lambda (t4: T).(eq T (THead (Bind -Abbr) v2 t3) (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4))) (\lambda (x2: -T).(\lambda (x3: T).(\lambda (H6: (eq T x0 (THead (Flat Appl) x2 -x3))).(\lambda (H7: (eq T v1 (lift h x1 x2))).(\lambda (H8: (eq T (THead -(Bind Abst) u t2) (lift h x1 x3))).(eq_ind_r T (THead (Flat Appl) x2 x3) -(\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) v2 t3) (lift -h x1 t4))) (\lambda (t4: T).(pr0 t t4)))) (ex3_2_ind T T (\lambda (y0: -T).(\lambda (z: T).(eq T x3 (THead (Bind Abst) y0 z)))) (\lambda (y0: -T).(\lambda (_: T).(eq T u (lift h x1 y0)))) (\lambda (_: T).(\lambda (z: -T).(eq T t2 (lift h (S x1) z)))) (ex2 T (\lambda (t4: T).(eq T (THead (Bind -Abbr) v2 t3) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 x3) -t4))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H9: (eq T x3 (THead (Bind -Abst) x4 x5))).(\lambda (_: (eq T u (lift h x1 x4))).(\lambda (H11: (eq T t2 -(lift h (S x1) x5))).(eq_ind_r T (THead (Bind Abst) x4 x5) (\lambda (t: -T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) v2 t3) (lift h x1 t4))) -(\lambda (t4: T).(pr0 (THead (Flat Appl) x2 t) t4)))) (ex2_ind T (\lambda -(t4: T).(eq T t3 (lift h (S x1) t4))) (\lambda (t4: T).(pr0 x5 t4)) (ex2 T -(\lambda (t4: T).(eq T (THead (Bind Abbr) v2 t3) (lift h x1 t4))) (\lambda -(t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind Abst) x4 x5)) t4))) (\lambda -(x6: T).(\lambda (H_x: (eq T t3 (lift h (S x1) x6))).(\lambda (H12: (pr0 x5 -x6)).(eq_ind_r T (lift h (S x1) x6) (\lambda (t: T).(ex2 T (\lambda (t4: -T).(eq T (THead (Bind Abbr) v2 t) (lift h x1 t4))) (\lambda (t4: T).(pr0 -(THead (Flat Appl) x2 (THead (Bind Abst) x4 x5)) t4)))) (ex2_ind T (\lambda -(t4: T).(eq T v2 (lift h x1 t4))) (\lambda (t4: T).(pr0 x2 t4)) (ex2 T -(\lambda (t4: 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h x1 H5)))))))))))))) (\lambda (b: B).(\lambda -(H1: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 -v1 v2)).(\lambda (H3: ((\forall (x0: T).(\forall (x1: nat).((eq T v1 (lift h -x1 x0)) \to (ex2 T (\lambda (t2: T).(eq T v2 (lift h x1 t2))) (\lambda (t2: -T).(pr0 x0 t2)))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 -u2)).(\lambda (H5: ((\forall (x0: T).(\forall (x1: nat).((eq T u1 (lift h x1 -x0)) \to (ex2 T (\lambda (t2: T).(eq T u2 (lift h x1 t2))) (\lambda (t2: -T).(pr0 x0 t2)))))))).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2 -t3)).(\lambda (H7: ((\forall (x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 -x0)) \to (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: -T).(pr0 x0 t4)))))))).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H8: (eq T -(THead (Flat Appl) v1 (THead (Bind b) u1 t2)) (lift h x1 x0))).(ex3_2_ind T T -(\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Flat Appl) y0 z)))) -(\lambda (y0: T).(\lambda (_: T).(eq T v1 (lift h x1 y0)))) (\lambda (_: -T).(\lambda (z: T).(eq T (THead (Bind b) u1 t2) (lift h x1 z)))) (ex2 T -(\lambda (t4: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O -v2) t3)) (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4))) (\lambda (x2: -T).(\lambda (x3: T).(\lambda (H9: (eq T x0 (THead (Flat Appl) x2 -x3))).(\lambda (H10: (eq T v1 (lift h x1 x2))).(\lambda (H11: (eq T (THead -(Bind b) u1 t2) (lift h x1 x3))).(eq_ind_r T (THead (Flat Appl) x2 x3) -(\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t3)) (lift h x1 t4))) (\lambda (t4: T).(pr0 t t4)))) -(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x3 (THead (Bind b) y0 -z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u1 (lift h x1 y0)))) (\lambda -(_: T).(\lambda (z: T).(eq T t2 (lift h (S x1) z)))) (ex2 T (\lambda (t4: -T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) (lift h -x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 x3) t4))) (\lambda (x4: -T).(\lambda (x5: T).(\lambda (H12: (eq T x3 (THead (Bind b) x4 x5))).(\lambda -(H13: (eq T u1 (lift h x1 x4))).(\lambda (H14: (eq T t2 (lift h (S x1) -x5))).(eq_ind_r T (THead (Bind b) x4 x5) (\lambda (t: T).(ex2 T (\lambda (t4: -T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) (lift h -x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 t) t4)))) (ex2_ind T -(\lambda (t4: T).(eq T t3 (lift h (S x1) t4))) (\lambda (t4: T).(pr0 x5 t4)) -(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t3)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 -(THead (Bind b) x4 x5)) t4))) (\lambda (x6: T).(\lambda (H_x: (eq T t3 (lift -h (S x1) x6))).(\lambda (H15: (pr0 x5 x6)).(eq_ind_r T (lift h (S x1) x6) -(\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead -(Flat Appl) x2 (THead (Bind b) x4 x5)) t4)))) (ex2_ind T (\lambda (t4: T).(eq -T u2 (lift h x1 t4))) (\lambda (t4: T).(pr0 x4 t4)) (ex2 T (\lambda (t4: -T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (lift h (S -x1) x6))) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead -(Bind b) x4 x5)) t4))) (\lambda (x7: T).(\lambda (H_x0: (eq T u2 (lift h x1 -x7))).(\lambda (H16: (pr0 x4 x7)).(eq_ind_r T (lift h x1 x7) (\lambda (t: -T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) t (THead (Flat Appl) (lift -(S O) O v2) (lift h (S x1) x6))) (lift h x1 t4))) (\lambda (t4: T).(pr0 -(THead (Flat Appl) x2 (THead (Bind b) x4 x5)) t4)))) (ex2_ind T (\lambda (t4: -T).(eq T v2 (lift h x1 t4))) (\lambda (t4: T).(pr0 x2 t4)) (ex2 T (\lambda -(t4: T).(eq T (THead (Bind b) (lift h x1 x7) (THead (Flat Appl) (lift (S O) O -v2) (lift h (S x1) x6))) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat -Appl) x2 (THead (Bind b) x4 x5)) t4))) (\lambda (x8: T).(\lambda (H_x1: (eq T -v2 (lift h x1 x8))).(\lambda (H17: (pr0 x2 x8)).(eq_ind_r T (lift h x1 x8) -(\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) (lift h x1 x7) -(THead (Flat Appl) (lift (S O) O t) (lift h (S x1) x6))) (lift h x1 t4))) -(\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind b) x4 x5)) t4)))) -(eq_ind T (lift h (plus (S O) x1) (lift (S O) O x8)) (\lambda (t: T).(ex2 T -(\lambda (t4: T).(eq T (THead (Bind b) (lift h x1 x7) (THead (Flat Appl) t -(lift h (S x1) x6))) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat -Appl) x2 (THead (Bind b) x4 x5)) t4)))) (eq_ind T (lift h (S x1) (THead (Flat -Appl) (lift (S O) O x8) x6)) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T -(THead (Bind b) (lift h x1 x7) t) (lift h x1 t4))) (\lambda (t4: T).(pr0 -(THead (Flat Appl) x2 (THead (Bind b) x4 x5)) t4)))) (ex_intro2 T (\lambda -(t4: T).(eq T (THead (Bind b) (lift h x1 x7) (lift h (S x1) (THead (Flat -Appl) (lift (S O) O x8) x6))) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead -(Flat Appl) x2 (THead (Bind b) x4 x5)) t4)) (THead (Bind b) x7 (THead (Flat -Appl) (lift (S O) O x8) x6)) (sym_eq T (lift h x1 (THead (Bind b) x7 (THead -(Flat Appl) (lift (S O) O x8) x6))) (THead (Bind b) (lift h x1 x7) (lift h (S -x1) (THead (Flat Appl) (lift (S O) O x8) x6))) (lift_bind b x7 (THead (Flat -Appl) (lift (S O) O x8) x6) h x1)) (pr0_upsilon b H1 x2 x8 H17 x4 x7 H16 x5 -x6 H15)) (THead (Flat Appl) (lift h (S x1) (lift (S O) O x8)) (lift h (S x1) -x6)) (lift_flat Appl (lift (S O) O x8) x6 h (S x1))) (lift (S O) O (lift h x1 -x8)) (lift_d x8 h (S O) x1 O (le_O_n x1))) v2 H_x1)))) (H3 x2 x1 H10)) u2 -H_x0)))) (H5 x4 x1 H13)) t3 H_x)))) (H7 x5 (S x1) H14)) x3 H12)))))) -(lift_gen_bind b u1 t2 x3 h x1 H11)) x0 H9)))))) (lift_gen_flat Appl v1 -(THead (Bind b) u1 t2) x0 h x1 H8))))))))))))))))))) (\lambda (u1: -T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda (H2: ((\forall (x0: -T).(\forall (x1: nat).((eq T u1 (lift h x1 x0)) \to (ex2 T (\lambda (t2: -T).(eq T u2 (lift h x1 t2))) (\lambda (t2: T).(pr0 x0 t2)))))))).(\lambda -(t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2 t3)).(\lambda (H4: ((\forall -(x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to (ex2 T (\lambda (t4: -T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4)))))))).(\lambda (w: -T).(\lambda (H5: (subst0 O u2 t3 w)).(\lambda (x0: T).(\lambda (x1: -nat).(\lambda (H6: (eq T (THead (Bind Abbr) u1 t2) (lift h x1 -x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Bind -Abbr) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u1 (lift h x1 y0)))) -(\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h (S x1) z)))) (ex2 T (\lambda -(t4: T).(eq T (THead (Bind Abbr) u2 w) (lift h x1 t4))) (\lambda (t4: T).(pr0 -x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq T x0 (THead -(Bind Abbr) x2 x3))).(\lambda (H8: (eq T u1 (lift h x1 x2))).(\lambda (H9: -(eq T t2 (lift h (S x1) x3))).(eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda -(t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) u2 w) (lift h x1 -t4))) (\lambda (t4: T).(pr0 t t4)))) (ex2_ind T (\lambda (t4: T).(eq T t3 -(lift h (S x1) t4))) (\lambda (t4: T).(pr0 x3 t4)) (ex2 T (\lambda (t4: -T).(eq T (THead (Bind Abbr) u2 w) (lift h x1 t4))) (\lambda (t4: T).(pr0 -(THead (Bind Abbr) x2 x3) t4))) (\lambda (x4: T).(\lambda (H_x: (eq T t3 -(lift h (S x1) x4))).(\lambda (H10: (pr0 x3 x4)).(let H11 \def (eq_ind T t3 -(\lambda (t: T).(subst0 O u2 t w)) H5 (lift h (S x1) x4) H_x) in (ex2_ind T -(\lambda (t4: T).(eq T u2 (lift h x1 t4))) (\lambda (t4: T).(pr0 x2 t4)) (ex2 -T (\lambda (t4: T).(eq T (THead (Bind Abbr) u2 w) (lift h x1 t4))) (\lambda -(t4: T).(pr0 (THead (Bind Abbr) x2 x3) t4))) (\lambda (x5: T).(\lambda (H_x0: -(eq T u2 (lift h x1 x5))).(\lambda (H12: (pr0 x2 x5)).(eq_ind_r T (lift h x1 -x5) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) t w) -(lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind Abbr) x2 x3) t4)))) (let -H13 \def (eq_ind T u2 (\lambda (t: T).(subst0 O t (lift h (S x1) x4) w)) H11 -(lift h x1 x5) H_x0) in (let H14 \def (refl_equal nat (S (plus O x1))) in -(let H15 \def (eq_ind nat (S x1) (\lambda (n: nat).(subst0 O (lift h x1 x5) -(lift h n x4) w)) H13 (S (plus O x1)) H14) in (ex2_ind T (\lambda (t4: T).(eq -T w (lift h (S (plus O x1)) t4))) (\lambda (t4: T).(subst0 O x5 x4 t4)) (ex2 -T (\lambda (t4: T).(eq T (THead (Bind Abbr) (lift h x1 x5) w) (lift h x1 -t4))) (\lambda (t4: T).(pr0 (THead (Bind Abbr) x2 x3) t4))) (\lambda (x6: -T).(\lambda (H16: (eq T w (lift h (S (plus O x1)) x6))).(\lambda (H17: -(subst0 O x5 x4 x6)).(eq_ind_r T (lift h (S (plus O x1)) x6) (\lambda (t: -T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) (lift h x1 x5) t) (lift h -x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind Abbr) x2 x3) t4)))) (ex_intro2 T -(\lambda (t4: T).(eq T (THead (Bind Abbr) (lift h x1 x5) (lift h (S (plus O -x1)) x6)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind Abbr) x2 x3) -t4)) (THead (Bind Abbr) x5 x6) (sym_eq T (lift h x1 (THead (Bind Abbr) x5 -x6)) (THead (Bind Abbr) (lift h x1 x5) (lift h (S (plus O x1)) x6)) -(lift_bind Abbr x5 x6 h (plus O x1))) (pr0_delta x2 x5 H12 x3 x4 H10 x6 H17)) -w H16)))) (subst0_gen_lift_lt x5 x4 w O h x1 H15))))) u2 H_x0)))) (H2 x2 x1 -H8)))))) (H4 x3 (S x1) H9)) x0 H7)))))) (lift_gen_bind Abbr u1 t2 x0 h x1 -H6))))))))))))))) (\lambda (b: B).(\lambda (H1: (not (eq B b Abst))).(\lambda -(t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2 t3)).(\lambda (H3: ((\forall -(x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to (ex2 T (\lambda (t4: -T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4)))))))).(\lambda (u: -T).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H4: (eq T (THead (Bind b) u -(lift (S O) O t2)) (lift h x1 x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda -(z: T).(eq T x0 (THead (Bind b) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq -T u (lift h x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift (S O) O t2) -(lift h (S x1) z)))) (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) -(\lambda (t4: T).(pr0 x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda -(H5: (eq T x0 (THead (Bind b) x2 x3))).(\lambda (_: (eq T u (lift h x1 -x2))).(\lambda (H7: (eq T (lift (S O) O t2) (lift h (S x1) x3))).(eq_ind_r T -(THead (Bind b) x2 x3) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T t3 (lift -h x1 t4))) (\lambda (t4: T).(pr0 t t4)))) (let H8 \def (eq_ind_r nat (plus (S -O) x1) (\lambda (n: nat).(eq nat (S x1) n)) (refl_equal nat (plus (S O) x1)) -(plus x1 (S O)) (plus_sym x1 (S O))) in (let H9 \def (eq_ind nat (S x1) -(\lambda (n: nat).(eq T (lift (S O) O t2) (lift h n x3))) H7 (plus x1 (S O)) -H8) in (ex2_ind T (\lambda (t4: T).(eq T x3 (lift (S O) O t4))) (\lambda (t4: -T).(eq T t2 (lift h x1 t4))) (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 -t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 x3) t4))) (\lambda (x4: -T).(\lambda (H10: (eq T x3 (lift (S O) O x4))).(\lambda (H11: (eq T t2 (lift -h x1 x4))).(eq_ind_r T (lift (S O) O x4) (\lambda (t: T).(ex2 T (\lambda (t4: -T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 t) -t4)))) (ex2_ind T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: -T).(pr0 x4 t4)) (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda -(t4: T).(pr0 (THead (Bind b) x2 (lift (S O) O x4)) t4))) (\lambda (x5: -T).(\lambda (H_x: (eq T t3 (lift h x1 x5))).(\lambda (H12: (pr0 x4 -x5)).(eq_ind_r T (lift h x1 x5) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T -t (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 (lift (S O) O -x4)) t4)))) (ex_intro2 T (\lambda (t4: T).(eq T (lift h x1 x5) (lift h x1 -t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 (lift (S O) O x4)) t4)) x5 -(refl_equal T (lift h x1 x5)) (pr0_zeta b H1 x4 x5 H12 x2)) t3 H_x)))) (H3 x4 -x1 H11)) x3 H10)))) (lift_gen_lift t2 x3 (S O) h O x1 (le_O_n x1) H9)))) x0 -H5)))))) (lift_gen_bind b u (lift (S O) O t2) x0 h x1 H4)))))))))))) (\lambda -(t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2 t3)).(\lambda (H2: ((\forall -(x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to (ex2 T (\lambda (t4: -T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4)))))))).(\lambda (u: -T).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H3: (eq T (THead (Flat Cast) -u t2) (lift h x1 x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T -x0 (THead (Flat Cast) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u (lift -h x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h x1 z)))) (ex2 T -(\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4))) -(\lambda (x2: T).(\lambda (x3: T).(\lambda (H4: (eq T x0 (THead (Flat Cast) -x2 x3))).(\lambda (_: (eq T u (lift h x1 x2))).(\lambda (H6: (eq T t2 (lift h -x1 x3))).(eq_ind_r T (THead (Flat Cast) x2 x3) (\lambda (t: T).(ex2 T -(\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 t t4)))) -(ex2_ind T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 -x3 t4)) (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: -T).(pr0 (THead (Flat Cast) x2 x3) t4))) (\lambda (x4: T).(\lambda (H_x: (eq T -t3 (lift h x1 x4))).(\lambda (H7: (pr0 x3 x4)).(eq_ind_r T (lift h x1 x4) -(\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T t (lift h x1 t4))) (\lambda -(t4: T).(pr0 (THead (Flat Cast) x2 x3) t4)))) (ex_intro2 T (\lambda (t4: -T).(eq T (lift h x1 x4) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat -Cast) x2 x3) t4)) x4 (refl_equal T (lift h x1 x4)) (pr0_tau x3 x4 H7 x2)) t3 -H_x)))) (H2 x3 x1 H6)) x0 H4)))))) (lift_gen_flat Cast u t2 x0 h x1 -H3)))))))))) y x H0))))) H))))). -(* COMMENTS -Initial nodes: 7569 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr0/pr0.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr0/pr0.ma deleted file mode 100644 index 9a3b397fe..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr0/pr0.ma +++ /dev/null @@ -1,2507 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr0/fwd.ma". - -include "Basic-1/lift/tlt.ma". - -theorem pr0_confluence__pr0_cong_upsilon_refl: - \forall (b: B).((not (eq B b Abst)) \to (\forall (u0: T).(\forall (u3: -T).((pr0 u0 u3) \to (\forall (t4: T).(\forall (t5: T).((pr0 t4 t5) \to -(\forall (u2: T).(\forall (v2: T).(\forall (x: T).((pr0 u2 x) \to ((pr0 v2 x) -\to (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u0 t4)) -t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O -v2) t5)) t))))))))))))))) -\def - \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (u0: T).(\lambda -(u3: T).(\lambda (H0: (pr0 u0 u3)).(\lambda (t4: T).(\lambda (t5: T).(\lambda -(H1: (pr0 t4 t5)).(\lambda (u2: T).(\lambda (v2: T).(\lambda (x: T).(\lambda -(H2: (pr0 u2 x)).(\lambda (H3: (pr0 v2 x)).(ex_intro2 T (\lambda (t: T).(pr0 -(THead (Flat Appl) u2 (THead (Bind b) u0 t4)) t)) (\lambda (t: T).(pr0 (THead -(Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t)) (THead (Bind b) u3 -(THead (Flat Appl) (lift (S O) O x) t5)) (pr0_upsilon b H u2 x H2 u0 u3 H0 t4 -t5 H1) (pr0_comp u3 u3 (pr0_refl u3) (THead (Flat Appl) (lift (S O) O v2) t5) -(THead (Flat Appl) (lift (S O) O x) t5) (pr0_comp (lift (S O) O v2) (lift (S -O) O x) (pr0_lift v2 x H3 (S O) O) t5 t5 (pr0_refl t5) (Flat Appl)) (Bind -b))))))))))))))). -(* COMMENTS -Initial nodes: 257 -END *) - -theorem pr0_confluence__pr0_cong_upsilon_cong: - \forall (b: B).((not (eq B b Abst)) \to (\forall (u2: T).(\forall (v2: -T).(\forall (x: T).((pr0 u2 x) \to ((pr0 v2 x) \to (\forall (t2: T).(\forall -(t5: T).(\forall (x0: T).((pr0 t2 x0) \to ((pr0 t5 x0) \to (\forall (u5: -T).(\forall (u3: T).(\forall (x1: T).((pr0 u5 x1) \to ((pr0 u3 x1) \to (ex2 T -(\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 t2)) t)) -(\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) -t5)) t))))))))))))))))))) -\def - \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (u2: T).(\lambda -(v2: T).(\lambda (x: T).(\lambda (H0: (pr0 u2 x)).(\lambda (H1: (pr0 v2 -x)).(\lambda (t2: T).(\lambda (t5: T).(\lambda (x0: T).(\lambda (H2: (pr0 t2 -x0)).(\lambda (H3: (pr0 t5 x0)).(\lambda (u5: T).(\lambda (u3: T).(\lambda -(x1: T).(\lambda (H4: (pr0 u5 x1)).(\lambda (H5: (pr0 u3 x1)).(ex_intro2 T -(\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 t2)) t)) -(\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) -t5)) t)) (THead (Bind b) x1 (THead (Flat Appl) (lift (S O) O x) x0)) -(pr0_upsilon b H u2 x H0 u5 x1 H4 t2 x0 H2) (pr0_comp u3 x1 H5 (THead (Flat -Appl) (lift (S O) O v2) t5) (THead (Flat Appl) (lift (S O) O x) x0) (pr0_comp -(lift (S O) O v2) (lift (S O) O x) (pr0_lift v2 x H1 (S O) O) t5 x0 H3 (Flat -Appl)) (Bind b))))))))))))))))))). -(* COMMENTS -Initial nodes: 269 -END *) - -theorem pr0_confluence__pr0_cong_upsilon_delta: - (not (eq B Abbr Abst)) \to (\forall (u5: T).(\forall (t2: T).(\forall (w: -T).((subst0 O u5 t2 w) \to (\forall (u2: T).(\forall (v2: T).(\forall (x: -T).((pr0 u2 x) \to ((pr0 v2 x) \to (\forall (t5: T).(\forall (x0: T).((pr0 t2 -x0) \to ((pr0 t5 x0) \to (\forall (u3: T).(\forall (x1: T).((pr0 u5 x1) \to -((pr0 u3 x1) \to (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead -(Bind Abbr) u5 w)) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 (THead -(Flat Appl) (lift (S O) O v2) t5)) t)))))))))))))))))))) -\def - \lambda (H: (not (eq B Abbr Abst))).(\lambda (u5: T).(\lambda (t2: -T).(\lambda (w: T).(\lambda (H0: (subst0 O u5 t2 w)).(\lambda (u2: -T).(\lambda (v2: T).(\lambda (x: T).(\lambda (H1: (pr0 u2 x)).(\lambda (H2: -(pr0 v2 x)).(\lambda (t5: T).(\lambda (x0: T).(\lambda (H3: (pr0 t2 -x0)).(\lambda (H4: (pr0 t5 x0)).(\lambda (u3: T).(\lambda (x1: T).(\lambda -(H5: (pr0 u5 x1)).(\lambda (H6: (pr0 u3 x1)).(or_ind (pr0 w x0) (ex2 T -(\lambda (w2: T).(pr0 w w2)) (\lambda (w2: T).(subst0 O x1 x0 w2))) (ex2 T -(\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abbr) u5 w)) t)) -(\lambda (t: T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O -v2) t5)) t))) (\lambda (H7: (pr0 w x0)).(ex_intro2 T (\lambda (t: T).(pr0 -(THead (Flat Appl) u2 (THead (Bind Abbr) u5 w)) t)) (\lambda (t: T).(pr0 -(THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t)) (THead -(Bind Abbr) x1 (THead (Flat Appl) (lift (S O) O x) x0)) (pr0_upsilon Abbr H -u2 x H1 u5 x1 H5 w x0 H7) (pr0_comp u3 x1 H6 (THead (Flat Appl) (lift (S O) O -v2) t5) (THead (Flat Appl) (lift (S O) O x) x0) (pr0_comp (lift (S O) O v2) -(lift (S O) O x) (pr0_lift v2 x H2 (S O) O) t5 x0 H4 (Flat Appl)) (Bind -Abbr)))) (\lambda (H7: (ex2 T (\lambda (w2: T).(pr0 w w2)) (\lambda (w2: -T).(subst0 O x1 x0 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 w w2)) (\lambda -(w2: T).(subst0 O x1 x0 w2)) (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) -u2 (THead (Bind Abbr) u5 w)) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 -(THead (Flat Appl) (lift (S O) O v2) t5)) t))) (\lambda (x2: T).(\lambda (H8: -(pr0 w x2)).(\lambda (H9: (subst0 O x1 x0 x2)).(ex_intro2 T (\lambda (t: -T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abbr) u5 w)) t)) (\lambda (t: -T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t)) -(THead (Bind Abbr) x1 (THead (Flat Appl) (lift (S O) O x) x2)) (pr0_upsilon -Abbr H u2 x H1 u5 x1 H5 w x2 H8) (pr0_delta u3 x1 H6 (THead (Flat Appl) (lift -(S O) O v2) t5) (THead (Flat Appl) (lift (S O) O x) x0) (pr0_comp (lift (S O) -O v2) (lift (S O) O x) (pr0_lift v2 x H2 (S O) O) t5 x0 H4 (Flat Appl)) -(THead (Flat Appl) (lift (S O) O x) x2) (subst0_snd (Flat Appl) x1 x2 x0 O H9 -(lift (S O) O x))))))) H7)) (pr0_subst0 t2 x0 H3 u5 w O H0 x1 -H5))))))))))))))))))). -(* COMMENTS -Initial nodes: 769 -END *) - -theorem pr0_confluence__pr0_cong_upsilon_zeta: - \forall (b: B).((not (eq B b Abst)) \to (\forall (u0: T).(\forall (u3: -T).((pr0 u0 u3) \to (\forall (u2: T).(\forall (v2: T).(\forall (x0: T).((pr0 -u2 x0) \to ((pr0 v2 x0) \to (\forall (x: T).(\forall (t3: T).(\forall (x1: -T).((pr0 x x1) \to ((pr0 t3 x1) \to (ex2 T (\lambda (t: T).(pr0 (THead (Flat -Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) -(lift (S O) O v2) (lift (S O) O x))) t))))))))))))))))) -\def - \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (u0: T).(\lambda -(u3: T).(\lambda (_: (pr0 u0 u3)).(\lambda (u2: T).(\lambda (v2: T).(\lambda -(x0: T).(\lambda (H1: (pr0 u2 x0)).(\lambda (H2: (pr0 v2 x0)).(\lambda (x: -T).(\lambda (t3: T).(\lambda (x1: T).(\lambda (H3: (pr0 x x1)).(\lambda (H4: -(pr0 t3 x1)).(eq_ind T (lift (S O) O (THead (Flat Appl) v2 x)) (\lambda (t: -T).(ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3) t0)) (\lambda (t0: -T).(pr0 (THead (Bind b) u3 t) t0)))) (ex_intro2 T (\lambda (t: T).(pr0 (THead -(Flat Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (lift (S O) O -(THead (Flat Appl) v2 x))) t)) (THead (Flat Appl) x0 x1) (pr0_comp u2 x0 H1 -t3 x1 H4 (Flat Appl)) (pr0_zeta b H (THead (Flat Appl) v2 x) (THead (Flat -Appl) x0 x1) (pr0_comp v2 x0 H2 x x1 H3 (Flat Appl)) u3)) (THead (Flat Appl) -(lift (S O) O v2) (lift (S O) O x)) (lift_flat Appl v2 x (S O) -O)))))))))))))))). -(* COMMENTS -Initial nodes: 283 -END *) - -theorem pr0_confluence__pr0_cong_delta: - \forall (u3: T).(\forall (t5: T).(\forall (w: T).((subst0 O u3 t5 w) \to -(\forall (u2: T).(\forall (x: T).((pr0 u2 x) \to ((pr0 u3 x) \to (\forall -(t3: T).(\forall (x0: T).((pr0 t3 x0) \to ((pr0 t5 x0) \to (ex2 T (\lambda -(t: T).(pr0 (THead (Bind Abbr) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind -Abbr) u3 w) t)))))))))))))) -\def - \lambda (u3: T).(\lambda (t5: T).(\lambda (w: T).(\lambda (H: (subst0 O u3 -t5 w)).(\lambda (u2: T).(\lambda (x: T).(\lambda (H0: (pr0 u2 x)).(\lambda -(H1: (pr0 u3 x)).(\lambda (t3: T).(\lambda (x0: T).(\lambda (H2: (pr0 t3 -x0)).(\lambda (H3: (pr0 t5 x0)).(or_ind (pr0 w x0) (ex2 T (\lambda (w2: -T).(pr0 w w2)) (\lambda (w2: T).(subst0 O x x0 w2))) (ex2 T (\lambda (t: -T).(pr0 (THead (Bind Abbr) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) -u3 w) t))) (\lambda (H4: (pr0 w x0)).(ex_intro2 T (\lambda (t: T).(pr0 (THead -(Bind Abbr) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w) t)) -(THead (Bind Abbr) x x0) (pr0_comp u2 x H0 t3 x0 H2 (Bind Abbr)) (pr0_comp u3 -x H1 w x0 H4 (Bind Abbr)))) (\lambda (H4: (ex2 T (\lambda (w2: T).(pr0 w w2)) -(\lambda (w2: T).(subst0 O x x0 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 w -w2)) (\lambda (w2: T).(subst0 O x x0 w2)) (ex2 T (\lambda (t: T).(pr0 (THead -(Bind Abbr) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w) t))) -(\lambda (x1: T).(\lambda (H5: (pr0 w x1)).(\lambda (H6: (subst0 O x x0 -x1)).(ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 t3) t)) (\lambda -(t: T).(pr0 (THead (Bind Abbr) u3 w) t)) (THead (Bind Abbr) x x1) (pr0_delta -u2 x H0 t3 x0 H2 x1 H6) (pr0_comp u3 x H1 w x1 H5 (Bind Abbr)))))) H4)) -(pr0_subst0 t5 x0 H3 u3 w O H x H1))))))))))))). -(* COMMENTS -Initial nodes: 409 -END *) - -theorem pr0_confluence__pr0_upsilon_upsilon: - \forall (b: B).((not (eq B b Abst)) \to (\forall (v1: T).(\forall (v2: -T).(\forall (x0: T).((pr0 v1 x0) \to ((pr0 v2 x0) \to (\forall (u1: -T).(\forall (u2: T).(\forall (x1: T).((pr0 u1 x1) \to ((pr0 u2 x1) \to -(\forall (t1: T).(\forall (t2: T).(\forall (x2: T).((pr0 t1 x2) \to ((pr0 t2 -x2) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u1 (THead (Flat Appl) -(lift (S O) O v1) t1)) t)) (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t2)) t))))))))))))))))))) -\def - \lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda -(v2: T).(\lambda (x0: T).(\lambda (H0: (pr0 v1 x0)).(\lambda (H1: (pr0 v2 -x0)).(\lambda (u1: T).(\lambda (u2: T).(\lambda (x1: T).(\lambda (H2: (pr0 u1 -x1)).(\lambda (H3: (pr0 u2 x1)).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(x2: T).(\lambda (H4: (pr0 t1 x2)).(\lambda (H5: (pr0 t2 x2)).(ex_intro2 T -(\lambda (t: T).(pr0 (THead (Bind b) u1 (THead (Flat Appl) (lift (S O) O v1) -t1)) t)) (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t2)) t)) (THead (Bind b) x1 (THead (Flat Appl) (lift (S O) O x0) -x2)) (pr0_comp u1 x1 H2 (THead (Flat Appl) (lift (S O) O v1) t1) (THead (Flat -Appl) (lift (S O) O x0) x2) (pr0_comp (lift (S O) O v1) (lift (S O) O x0) -(pr0_lift v1 x0 H0 (S O) O) t1 x2 H4 (Flat Appl)) (Bind b)) (pr0_comp u2 x1 -H3 (THead (Flat Appl) (lift (S O) O v2) t2) (THead (Flat Appl) (lift (S O) O -x0) x2) (pr0_comp (lift (S O) O v2) (lift (S O) O x0) (pr0_lift v2 x0 H1 (S -O) O) t2 x2 H5 (Flat Appl)) (Bind b))))))))))))))))))). -(* COMMENTS -Initial nodes: 347 -END *) - -theorem pr0_confluence__pr0_delta_delta: - \forall (u2: T).(\forall (t3: T).(\forall (w: T).((subst0 O u2 t3 w) \to -(\forall (u3: T).(\forall (t5: T).(\forall (w0: T).((subst0 O u3 t5 w0) \to -(\forall (x: T).((pr0 u2 x) \to ((pr0 u3 x) \to (\forall (x0: T).((pr0 t3 x0) -\to ((pr0 t5 x0) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) -(\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t)))))))))))))))) -\def - \lambda (u2: T).(\lambda (t3: T).(\lambda (w: T).(\lambda (H: (subst0 O u2 -t3 w)).(\lambda (u3: T).(\lambda (t5: T).(\lambda (w0: T).(\lambda (H0: -(subst0 O u3 t5 w0)).(\lambda (x: T).(\lambda (H1: (pr0 u2 x)).(\lambda (H2: -(pr0 u3 x)).(\lambda (x0: T).(\lambda (H3: (pr0 t3 x0)).(\lambda (H4: (pr0 t5 -x0)).(or_ind (pr0 w0 x0) (ex2 T (\lambda (w2: T).(pr0 w0 w2)) (\lambda (w2: -T).(subst0 O x x0 w2))) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) -t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (H5: (pr0 w0 -x0)).(or_ind (pr0 w x0) (ex2 T (\lambda (w2: T).(pr0 w w2)) (\lambda (w2: -T).(subst0 O x x0 w2))) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) -t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (H6: (pr0 w -x0)).(ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda -(t: T).(pr0 (THead (Bind Abbr) u3 w0) t)) (THead (Bind Abbr) x x0) (pr0_comp -u2 x H1 w x0 H6 (Bind Abbr)) (pr0_comp u3 x H2 w0 x0 H5 (Bind Abbr)))) -(\lambda (H6: (ex2 T (\lambda (w2: T).(pr0 w w2)) (\lambda (w2: T).(subst0 O -x x0 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 w w2)) (\lambda (w2: T).(subst0 -O x x0 w2)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda -(t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (x1: T).(\lambda (H7: -(pr0 w x1)).(\lambda (H8: (subst0 O x x0 x1)).(ex_intro2 T (\lambda (t: -T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) -u3 w0) t)) (THead (Bind Abbr) x x1) (pr0_comp u2 x H1 w x1 H7 (Bind Abbr)) -(pr0_delta u3 x H2 w0 x0 H5 x1 H8))))) H6)) (pr0_subst0 t3 x0 H3 u2 w O H x -H1))) (\lambda (H5: (ex2 T (\lambda (w2: T).(pr0 w0 w2)) (\lambda (w2: -T).(subst0 O x x0 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 w0 w2)) (\lambda -(w2: T).(subst0 O x x0 w2)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 -w) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (x1: -T).(\lambda (H6: (pr0 w0 x1)).(\lambda (H7: (subst0 O x x0 x1)).(or_ind (pr0 -w x0) (ex2 T (\lambda (w2: T).(pr0 w w2)) (\lambda (w2: T).(subst0 O x x0 -w2))) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: -T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (H8: (pr0 w x0)).(ex_intro2 T -(\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead -(Bind Abbr) u3 w0) t)) (THead (Bind Abbr) x x1) (pr0_delta u2 x H1 w x0 H8 x1 -H7) (pr0_comp u3 x H2 w0 x1 H6 (Bind Abbr)))) (\lambda (H8: (ex2 T (\lambda -(w2: T).(pr0 w w2)) (\lambda (w2: T).(subst0 O x x0 w2)))).(ex2_ind T -(\lambda (w2: T).(pr0 w w2)) (\lambda (w2: T).(subst0 O x x0 w2)) (ex2 T -(\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead -(Bind Abbr) u3 w0) t))) (\lambda (x2: T).(\lambda (H9: (pr0 w x2)).(\lambda -(H10: (subst0 O x x0 x2)).(or4_ind (eq T x2 x1) (ex2 T (\lambda (t: -T).(subst0 O x x2 t)) (\lambda (t: T).(subst0 O x x1 t))) (subst0 O x x2 x1) -(subst0 O x x1 x2) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) -(\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (H11: (eq T x2 -x1)).(let H12 \def (eq_ind T x2 (\lambda (t: T).(pr0 w t)) H9 x1 H11) in -(ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: -T).(pr0 (THead (Bind Abbr) u3 w0) t)) (THead (Bind Abbr) x x1) (pr0_comp u2 x -H1 w x1 H12 (Bind Abbr)) (pr0_comp u3 x H2 w0 x1 H6 (Bind Abbr))))) (\lambda -(H11: (ex2 T (\lambda (t: T).(subst0 O x x2 t)) (\lambda (t: T).(subst0 O x -x1 t)))).(ex2_ind T (\lambda (t: T).(subst0 O x x2 t)) (\lambda (t: -T).(subst0 O x x1 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) -t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (x3: -T).(\lambda (H12: (subst0 O x x2 x3)).(\lambda (H13: (subst0 O x x1 -x3)).(ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda -(t: T).(pr0 (THead (Bind Abbr) u3 w0) t)) (THead (Bind Abbr) x x3) (pr0_delta -u2 x H1 w x2 H9 x3 H12) (pr0_delta u3 x H2 w0 x1 H6 x3 H13))))) H11)) -(\lambda (H11: (subst0 O x x2 x1)).(ex_intro2 T (\lambda (t: T).(pr0 (THead -(Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t)) -(THead (Bind Abbr) x x1) (pr0_delta u2 x H1 w x2 H9 x1 H11) (pr0_comp u3 x H2 -w0 x1 H6 (Bind Abbr)))) (\lambda (H11: (subst0 O x x1 x2)).(ex_intro2 T -(\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead -(Bind Abbr) u3 w0) t)) (THead (Bind Abbr) x x2) (pr0_comp u2 x H1 w x2 H9 -(Bind Abbr)) (pr0_delta u3 x H2 w0 x1 H6 x2 H11))) (subst0_confluence_eq x0 -x2 x O H10 x1 H7))))) H8)) (pr0_subst0 t3 x0 H3 u2 w O H x H1))))) H5)) -(pr0_subst0 t5 x0 H4 u3 w0 O H0 x H2))))))))))))))). -(* COMMENTS -Initial nodes: 1501 -END *) - -theorem pr0_confluence__pr0_delta_tau: - \forall (u2: T).(\forall (t3: T).(\forall (w: T).((subst0 O u2 t3 w) \to -(\forall (t4: T).((pr0 (lift (S O) O t4) t3) \to (\forall (t2: T).(ex2 T -(\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 t2 -t))))))))) -\def - \lambda (u2: T).(\lambda (t3: T).(\lambda (w: T).(\lambda (H: (subst0 O u2 -t3 w)).(\lambda (t4: T).(\lambda (H0: (pr0 (lift (S O) O t4) t3)).(\lambda -(t2: T).(ex2_ind T (\lambda (t5: T).(eq T t3 (lift (S O) O t5))) (\lambda -(t5: T).(pr0 t4 t5)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) -(\lambda (t: T).(pr0 t2 t))) (\lambda (x: T).(\lambda (H1: (eq T t3 (lift (S -O) O x))).(\lambda (_: (pr0 t4 x)).(let H3 \def (eq_ind T t3 (\lambda (t: -T).(subst0 O u2 t w)) H (lift (S O) O x) H1) in (subst0_gen_lift_false x u2 w -(S O) O O (le_n O) (eq_ind_r nat (plus (S O) O) (\lambda (n: nat).(lt O n)) -(le_n (plus (S O) O)) (plus O (S O)) (plus_sym O (S O))) H3 (ex2 T (\lambda -(t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 t2 t)))))))) -(pr0_gen_lift t4 t3 (S O) O H0)))))))). -(* COMMENTS -Initial nodes: 257 -END *) - -theorem pr0_confluence: - \forall (t0: T).(\forall (t1: T).((pr0 t0 t1) \to (\forall (t2: T).((pr0 t0 -t2) \to (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t))))))) -\def - \lambda (t0: T).(tlt_wf_ind (\lambda (t: T).(\forall (t1: T).((pr0 t t1) \to -(\forall (t2: T).((pr0 t t2) \to (ex2 T (\lambda (t3: T).(pr0 t1 t3)) -(\lambda (t3: T).(pr0 t2 t3)))))))) (\lambda (t: T).(\lambda (H: ((\forall -(v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 -v t2) \to (ex2 T (\lambda (t3: T).(pr0 t1 t3)) (\lambda (t3: T).(pr0 t2 -t3))))))))))).(\lambda (t1: T).(\lambda (H0: (pr0 t t1)).(\lambda (t2: -T).(\lambda (H1: (pr0 t t2)).(let H2 \def (match H0 in pr0 return (\lambda -(t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).((eq T t3 t) \to ((eq T t4 -t1) \to (ex2 T (\lambda (t5: T).(pr0 t1 t5)) (\lambda (t5: T).(pr0 t2 -t5)))))))) with [(pr0_refl t3) \Rightarrow (\lambda (H2: (eq T t3 -t)).(\lambda (H3: (eq T t3 t1)).(eq_ind T t (\lambda (t4: T).((eq T t4 t1) -\to (ex2 T (\lambda (t5: T).(pr0 t1 t5)) (\lambda (t5: T).(pr0 t2 t5))))) -(\lambda (H4: (eq T t t1)).(eq_ind T t1 (\lambda (_: T).(ex2 T (\lambda (t5: -T).(pr0 t1 t5)) (\lambda (t5: T).(pr0 t2 t5)))) (let H5 \def (match H1 in pr0 -return (\lambda (t4: T).(\lambda (t5: T).(\lambda (_: (pr0 t4 t5)).((eq T t4 -t) \to ((eq T t5 t2) \to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: -T).(pr0 t2 t6)))))))) with [(pr0_refl t4) \Rightarrow (\lambda (H5: (eq T t4 -t)).(\lambda (H6: (eq T t4 t2)).(eq_ind T t (\lambda (t5: T).((eq T t5 t2) -\to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6))))) -(\lambda (H7: (eq T t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t6: -T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6)))) (let H8 \def (eq_ind T t -(\lambda (t5: T).(eq T t4 t5)) H5 t2 H7) in (let H9 \def (eq_ind T t (\lambda -(t5: T).(eq T t5 t1)) H4 t2 H7) in (let H10 \def (eq_ind T t (\lambda (t5: -T).(eq T t3 t5)) H2 t2 H7) in (let H11 \def (eq_ind T t (\lambda (t5: -T).(\forall (v: T).((tlt v t5) \to (\forall (t6: T).((pr0 v t6) \to (\forall -(t7: T).((pr0 v t7) \to (ex2 T (\lambda (t8: T).(pr0 t6 t8)) (\lambda (t8: -T).(pr0 t7 t8)))))))))) H t2 H7) in (let H12 \def (eq_ind T t2 (\lambda (t5: -T).(\forall (v: T).((tlt v t5) \to (\forall (t6: T).((pr0 v t6) \to (\forall -(t7: T).((pr0 v t7) \to (ex2 T (\lambda (t8: T).(pr0 t6 t8)) (\lambda (t8: -T).(pr0 t7 t8)))))))))) H11 t1 H9) in (eq_ind_r T t1 (\lambda (t5: T).(ex2 T -(\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t5 t6)))) (let H13 \def -(eq_ind T t2 (\lambda (t5: T).(eq T t3 t5)) H10 t1 H9) in (ex_intro2 T -(\lambda (t5: T).(pr0 t1 t5)) (\lambda (t5: T).(pr0 t1 t5)) t1 (pr0_refl t1) -(pr0_refl t1))) t2 H9)))))) t (sym_eq T t t2 H7))) t4 (sym_eq T t4 t H5) -H6))) | (pr0_comp u1 u2 H5 t4 t5 H6 k) \Rightarrow (\lambda (H7: (eq T (THead -k u1 t4) t)).(\lambda (H8: (eq T (THead k u2 t5) t2)).(eq_ind T (THead k u1 -t4) (\lambda (_: T).((eq T (THead k u2 t5) t2) \to ((pr0 u1 u2) \to ((pr0 t4 -t5) \to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 -t7))))))) (\lambda (H9: (eq T (THead k u2 t5) t2)).(eq_ind T (THead k u2 t5) -(\lambda (t6: T).((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t7: -T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t6 t7)))))) (\lambda (H10: (pr0 u1 -u2)).(\lambda (H11: (pr0 t4 t5)).(let H12 \def (eq_ind_r T t (\lambda (t6: -T).(eq T t6 t1)) H4 (THead k u1 t4) H7) in (eq_ind T (THead k u1 t4) (\lambda -(t6: T).(ex2 T (\lambda (t7: T).(pr0 t6 t7)) (\lambda (t7: T).(pr0 (THead k -u2 t5) t7)))) (let H13 \def (eq_ind_r T t (\lambda (t6: T).(eq T t3 t6)) H2 -(THead k u1 t4) H7) in (let H14 \def (eq_ind_r T t (\lambda (t6: T).(\forall -(v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall (t8: -T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 -t8 t9)))))))))) H (THead k u1 t4) H7) in (ex_intro2 T (\lambda (t6: T).(pr0 -(THead k u1 t4) t6)) (\lambda (t6: T).(pr0 (THead k u2 t5) t6)) (THead k u2 -t5) (pr0_comp u1 u2 H10 t4 t5 H11 k) (pr0_refl (THead k u2 t5))))) t1 H12)))) -t2 H9)) t H7 H8 H5 H6))) | (pr0_beta u v1 v2 H5 t4 t5 H6) \Rightarrow -(\lambda (H7: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t4)) -t)).(\lambda (H8: (eq T (THead (Bind Abbr) v2 t5) t2)).(eq_ind T (THead (Flat -Appl) v1 (THead (Bind Abst) u t4)) (\lambda (_: T).((eq T (THead (Bind Abbr) -v2 t5) t2) \to ((pr0 v1 v2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t7: T).(pr0 -t1 t7)) (\lambda (t7: T).(pr0 t2 t7))))))) (\lambda (H9: (eq T (THead (Bind -Abbr) v2 t5) t2)).(eq_ind T (THead (Bind Abbr) v2 t5) (\lambda (t6: T).((pr0 -v1 v2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda -(t7: T).(pr0 t6 t7)))))) (\lambda (H10: (pr0 v1 v2)).(\lambda (H11: (pr0 t4 -t5)).(let H12 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 t1)) H4 (THead -(Flat Appl) v1 (THead (Bind Abst) u t4)) H7) in (eq_ind T (THead (Flat Appl) -v1 (THead (Bind Abst) u t4)) (\lambda (t6: T).(ex2 T (\lambda (t7: T).(pr0 t6 -t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t5) t7)))) (let H13 \def -(eq_ind_r T t (\lambda (t6: T).(eq T t3 t6)) H2 (THead (Flat Appl) v1 (THead -(Bind Abst) u t4)) H7) in (let H14 \def (eq_ind_r T t (\lambda (t6: -T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall -(t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: -T).(pr0 t8 t9)))))))))) H (THead (Flat Appl) v1 (THead (Bind Abst) u t4)) H7) -in (ex_intro2 T (\lambda (t6: T).(pr0 (THead (Flat Appl) v1 (THead (Bind -Abst) u t4)) t6)) (\lambda (t6: T).(pr0 (THead (Bind Abbr) v2 t5) t6)) (THead -(Bind Abbr) v2 t5) (pr0_beta u v1 v2 H10 t4 t5 H11) (pr0_refl (THead (Bind -Abbr) v2 t5))))) t1 H12)))) t2 H9)) t H7 H8 H5 H6))) | (pr0_upsilon b H5 v1 -v2 H6 u1 u2 H7 t4 t5 H8) \Rightarrow (\lambda (H9: (eq T (THead (Flat Appl) -v1 (THead (Bind b) u1 t4)) t)).(\lambda (H10: (eq T (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t5)) t2)).(eq_ind T (THead (Flat Appl) v1 -(THead (Bind b) u1 t4)) (\lambda (_: T).((eq T (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t5)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1 -v2) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t7: T).(pr0 t1 -t7)) (\lambda (t7: T).(pr0 t2 t7))))))))) (\lambda (H11: (eq T (THead (Bind -b) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) t2)).(eq_ind T (THead (Bind -b) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) (\lambda (t6: T).((not (eq B -b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T -(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t6 t7)))))))) (\lambda -(H12: (not (eq B b Abst))).(\lambda (H13: (pr0 v1 v2)).(\lambda (H14: (pr0 u1 -u2)).(\lambda (H15: (pr0 t4 t5)).(let H16 \def (eq_ind_r T t (\lambda (t6: -T).(eq T t6 t1)) H4 (THead (Flat Appl) v1 (THead (Bind b) u1 t4)) H9) in -(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t4)) (\lambda (t6: T).(ex2 -T (\lambda (t7: T).(pr0 t6 t7)) (\lambda (t7: T).(pr0 (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t5)) t7)))) (let H17 \def (eq_ind_r T t -(\lambda (t6: T).(eq T t3 t6)) H2 (THead (Flat Appl) v1 (THead (Bind b) u1 -t4)) H9) in (let H18 \def (eq_ind_r T t (\lambda (t6: T).(\forall (v: -T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall (t8: T).((pr0 v -t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8 -t9)))))))))) H (THead (Flat Appl) v1 (THead (Bind b) u1 t4)) H9) in -(pr0_confluence__pr0_cong_upsilon_refl b H12 u1 u2 H14 t4 t5 H15 v1 v2 v2 H13 -(pr0_refl v2)))) t1 H16)))))) t2 H11)) t H9 H10 H5 H6 H7 H8))) | (pr0_delta -u1 u2 H5 t4 t5 H6 w H7) \Rightarrow (\lambda (H8: (eq T (THead (Bind Abbr) u1 -t4) t)).(\lambda (H9: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead -(Bind Abbr) u1 t4) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t2) \to -((pr0 u1 u2) \to ((pr0 t4 t5) \to ((subst0 O u2 t5 w) \to (ex2 T (\lambda -(t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)))))))) (\lambda (H10: (eq T -(THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda -(t6: T).((pr0 u1 u2) \to ((pr0 t4 t5) \to ((subst0 O u2 t5 w) \to (ex2 T -(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t6 t7))))))) (\lambda -(H11: (pr0 u1 u2)).(\lambda (H12: (pr0 t4 t5)).(\lambda (H13: (subst0 O u2 t5 -w)).(let H14 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 t1)) H4 (THead -(Bind Abbr) u1 t4) H8) in (eq_ind T (THead (Bind Abbr) u1 t4) (\lambda (t6: -T).(ex2 T (\lambda (t7: T).(pr0 t6 t7)) (\lambda (t7: T).(pr0 (THead (Bind -Abbr) u2 w) t7)))) (let H15 \def (eq_ind_r T t (\lambda (t6: T).(eq T t3 t6)) -H2 (THead (Bind Abbr) u1 t4) H8) in (let H16 \def (eq_ind_r T t (\lambda (t6: -T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall -(t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: -T).(pr0 t8 t9)))))))))) H (THead (Bind Abbr) u1 t4) H8) in (ex_intro2 T -(\lambda (t6: T).(pr0 (THead (Bind Abbr) u1 t4) t6)) (\lambda (t6: T).(pr0 -(THead (Bind Abbr) u2 w) t6)) (THead (Bind Abbr) u2 w) (pr0_delta u1 u2 H11 -t4 t5 H12 w H13) (pr0_refl (THead (Bind Abbr) u2 w))))) t1 H14))))) t2 H10)) -t H8 H9 H5 H6 H7))) | (pr0_zeta b H5 t4 t5 H6 u) \Rightarrow (\lambda (H7: -(eq T (THead (Bind b) u (lift (S O) O t4)) t)).(\lambda (H8: (eq T t5 -t2)).(eq_ind T (THead (Bind b) u (lift (S O) O t4)) (\lambda (_: T).((eq T t5 -t2) \to ((not (eq B b Abst)) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t7: -T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))))))) (\lambda (H9: (eq T t5 -t2)).(eq_ind T t2 (\lambda (t6: T).((not (eq B b Abst)) \to ((pr0 t4 t6) \to -(ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)))))) -(\lambda (H10: (not (eq B b Abst))).(\lambda (H11: (pr0 t4 t2)).(let H12 \def -(eq_ind_r T t (\lambda (t6: T).(eq T t6 t1)) H4 (THead (Bind b) u (lift (S O) -O t4)) H7) in (eq_ind T (THead (Bind b) u (lift (S O) O t4)) (\lambda (t6: -T).(ex2 T (\lambda (t7: T).(pr0 t6 t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let -H13 \def (eq_ind_r T t (\lambda (t6: T).(eq T t3 t6)) H2 (THead (Bind b) u -(lift (S O) O t4)) H7) in (let H14 \def (eq_ind_r T t (\lambda (t6: -T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall -(t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: -T).(pr0 t8 t9)))))))))) H (THead (Bind b) u (lift (S O) O t4)) H7) in -(ex_intro2 T (\lambda (t6: T).(pr0 (THead (Bind b) u (lift (S O) O t4)) t6)) -(\lambda (t6: T).(pr0 t2 t6)) t2 (pr0_zeta b H10 t4 t2 H11 u) (pr0_refl -t2)))) t1 H12)))) t5 (sym_eq T t5 t2 H9))) t H7 H8 H5 H6))) | (pr0_tau t4 t5 -H5 u) \Rightarrow (\lambda (H6: (eq T (THead (Flat Cast) u t4) t)).(\lambda -(H7: (eq T t5 t2)).(eq_ind T (THead (Flat Cast) u t4) (\lambda (_: T).((eq T -t5 t2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda -(t7: T).(pr0 t2 t7)))))) (\lambda (H8: (eq T t5 t2)).(eq_ind T t2 (\lambda -(t6: T).((pr0 t4 t6) \to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: -T).(pr0 t2 t7))))) (\lambda (H9: (pr0 t4 t2)).(let H10 \def (eq_ind_r T t -(\lambda (t6: T).(eq T t6 t1)) H4 (THead (Flat Cast) u t4) H6) in (eq_ind T -(THead (Flat Cast) u t4) (\lambda (t6: T).(ex2 T (\lambda (t7: T).(pr0 t6 -t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let H11 \def (eq_ind_r T t (\lambda -(t6: T).(eq T t3 t6)) H2 (THead (Flat Cast) u t4) H6) in (let H12 \def -(eq_ind_r T t (\lambda (t6: T).(\forall (v: T).((tlt v t6) \to (\forall (t7: -T).((pr0 v t7) \to (\forall (t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: -T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead (Flat Cast) u -t4) H6) in (ex_intro2 T (\lambda (t6: T).(pr0 (THead (Flat Cast) u t4) t6)) -(\lambda (t6: T).(pr0 t2 t6)) t2 (pr0_tau t4 t2 H9 u) (pr0_refl t2)))) t1 -H10))) t5 (sym_eq T t5 t2 H8))) t H6 H7 H5)))]) in (H5 (refl_equal T t) -(refl_equal T t2))) t (sym_eq T t t1 H4))) t3 (sym_eq T t3 t H2) H3))) | -(pr0_comp u1 u2 H2 t3 t4 H3 k) \Rightarrow (\lambda (H4: (eq T (THead k u1 -t3) t)).(\lambda (H5: (eq T (THead k u2 t4) t1)).(eq_ind T (THead k u1 t3) -(\lambda (_: T).((eq T (THead k u2 t4) t1) \to ((pr0 u1 u2) \to ((pr0 t3 t4) -\to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6))))))) -(\lambda (H6: (eq T (THead k u2 t4) t1)).(eq_ind T (THead k u2 t4) (\lambda -(t5: T).((pr0 u1 u2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t5 -t6)) (\lambda (t6: T).(pr0 t2 t6)))))) (\lambda (H7: (pr0 u1 u2)).(\lambda -(H8: (pr0 t3 t4)).(let H9 \def (match H1 in pr0 return (\lambda (t5: -T).(\lambda (t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 t) \to ((eq T t6 t2) -\to (ex2 T (\lambda (t7: T).(pr0 (THead k u2 t4) t7)) (\lambda (t7: T).(pr0 -t2 t7)))))))) with [(pr0_refl t5) \Rightarrow (\lambda (H9: (eq T t5 -t)).(\lambda (H10: (eq T t5 t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2) -\to (ex2 T (\lambda (t7: T).(pr0 (THead k u2 t4) t7)) (\lambda (t7: T).(pr0 -t2 t7))))) (\lambda (H11: (eq T t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T -(\lambda (t7: T).(pr0 (THead k u2 t4) t7)) (\lambda (t7: T).(pr0 t2 t7)))) -(let H12 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 t2)) H11 (THead k u1 -t3) H4) in (eq_ind T (THead k u1 t3) (\lambda (t6: T).(ex2 T (\lambda (t7: -T).(pr0 (THead k u2 t4) t7)) (\lambda (t7: T).(pr0 t6 t7)))) (let H13 \def -(eq_ind_r T t (\lambda (t6: T).(eq T t5 t6)) H9 (THead k u1 t3) H4) in (let -H14 \def (eq_ind_r T t (\lambda (t6: T).(\forall (v: T).((tlt v t6) \to -(\forall (t7: T).((pr0 v t7) \to (\forall (t8: T).((pr0 v t8) \to (ex2 T -(\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead -k u1 t3) H4) in (ex_intro2 T (\lambda (t6: T).(pr0 (THead k u2 t4) t6)) -(\lambda (t6: T).(pr0 (THead k u1 t3) t6)) (THead k u2 t4) (pr0_refl (THead k -u2 t4)) (pr0_comp u1 u2 H7 t3 t4 H8 k)))) t2 H12)) t (sym_eq T t t2 H11))) t5 -(sym_eq T t5 t H9) H10))) | (pr0_comp u0 u3 H9 t5 t6 H10 k0) \Rightarrow -(\lambda (H11: (eq T (THead k0 u0 t5) t)).(\lambda (H12: (eq T (THead k0 u3 -t6) t2)).(eq_ind T (THead k0 u0 t5) (\lambda (_: T).((eq T (THead k0 u3 t6) -t2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead -k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H13: (eq T (THead -k0 u3 t6) t2)).(eq_ind T (THead k0 u3 t6) (\lambda (t7: T).((pr0 u0 u3) \to -((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda -(t8: T).(pr0 t7 t8)))))) (\lambda (H14: (pr0 u0 u3)).(\lambda (H15: (pr0 t5 -t6)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead k u1 t3) t7)) -H4 (THead k0 u0 t5) H11) in (let H17 \def (f_equal T K (\lambda (e: T).(match -e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) -\Rightarrow k | (THead k1 _ _) \Rightarrow k1])) (THead k u1 t3) (THead k0 u0 -t5) H16) in ((let H18 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 -| (THead _ t7 _) \Rightarrow t7])) (THead k u1 t3) (THead k0 u0 t5) H16) in -((let H19 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: -T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ -t7) \Rightarrow t7])) (THead k u1 t3) (THead k0 u0 t5) H16) in (\lambda (H20: -(eq T u1 u0)).(\lambda (H21: (eq K k k0)).(let H22 \def (eq_ind_r T t -(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) -\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) -(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead k0 u0 t5) H11) in (eq_ind_r -K k0 (\lambda (k1: K).(ex2 T (\lambda (t7: T).(pr0 (THead k1 u2 t4) t7)) -(\lambda (t7: T).(pr0 (THead k0 u3 t6) t7)))) (let H23 \def (eq_ind T u1 -(\lambda (t7: T).(pr0 t7 u2)) H7 u0 H20) in (let H24 \def (eq_ind T t3 -(\lambda (t7: T).(pr0 t7 t4)) H8 t5 H19) in (ex2_ind T (\lambda (t7: T).(pr0 -t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead k0 -u2 t4) t7)) (\lambda (t7: T).(pr0 (THead k0 u3 t6) t7))) (\lambda (x: -T).(\lambda (H25: (pr0 t4 x)).(\lambda (H26: (pr0 t6 x)).(ex2_ind T (\lambda -(t7: T).(pr0 u2 t7)) (\lambda (t7: T).(pr0 u3 t7)) (ex2 T (\lambda (t7: -T).(pr0 (THead k0 u2 t4) t7)) (\lambda (t7: T).(pr0 (THead k0 u3 t6) t7))) -(\lambda (x0: T).(\lambda (H27: (pr0 u2 x0)).(\lambda (H28: (pr0 u3 -x0)).(ex_intro2 T (\lambda (t7: T).(pr0 (THead k0 u2 t4) t7)) (\lambda (t7: -T).(pr0 (THead k0 u3 t6) t7)) (THead k0 x0 x) (pr0_comp u2 x0 H27 t4 x H25 -k0) (pr0_comp u3 x0 H28 t6 x H26 k0))))) (H22 u0 (tlt_head_sx k0 u0 t5) u2 -H23 u3 H14))))) (H22 t5 (tlt_head_dx k0 u0 t5) t4 H24 t6 H15)))) k H21))))) -H18)) H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_beta u v1 v2 H9 t5 t6 H10) -\Rightarrow (\lambda (H11: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u -t5)) t)).(\lambda (H12: (eq T (THead (Bind Abbr) v2 t6) t2)).(eq_ind T (THead -(Flat Appl) v1 (THead (Bind Abst) u t5)) (\lambda (_: T).((eq T (THead (Bind -Abbr) v2 t6) t2) \to ((pr0 v1 v2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: -T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda -(H13: (eq T (THead (Bind Abbr) v2 t6) t2)).(eq_ind T (THead (Bind Abbr) v2 -t6) (\lambda (t7: T).((pr0 v1 v2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: -T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (H14: -(pr0 v1 v2)).(\lambda (H15: (pr0 t5 t6)).(let H16 \def (eq_ind_r T t (\lambda -(t7: T).(eq T (THead k u1 t3) t7)) H4 (THead (Flat Appl) v1 (THead (Bind -Abst) u t5)) H11) in (let H17 \def (f_equal T K (\lambda (e: T).(match e in T -return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) -\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) (THead (Flat -Appl) v1 (THead (Bind Abst) u t5)) H16) in ((let H18 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) -(THead k u1 t3) (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) H16) in ((let -H19 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7) -\Rightarrow t7])) (THead k u1 t3) (THead (Flat Appl) v1 (THead (Bind Abst) u -t5)) H16) in (\lambda (H20: (eq T u1 v1)).(\lambda (H21: (eq K k (Flat -Appl))).(let H22 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v -t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to -(ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 -t10)))))))))) H (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) H11) in -(eq_ind_r K (Flat Appl) (\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 (THead -k0 u2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t6) t7)))) (let -H23 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H7 v1 H20) in (let H24 -\def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H8 (THead (Bind Abst) u t5) -H19) in (let H25 \def (match H24 in pr0 return (\lambda (t7: T).(\lambda (t8: -T).(\lambda (_: (pr0 t7 t8)).((eq T t7 (THead (Bind Abst) u t5)) \to ((eq T -t8 t4) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) -(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9)))))))) with [(pr0_refl -t7) \Rightarrow (\lambda (H25: (eq T t7 (THead (Bind Abst) u t5))).(\lambda -(H26: (eq T t7 t4)).(eq_ind T (THead (Bind Abst) u t5) (\lambda (t8: T).((eq -T t8 t4) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) -(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9))))) (\lambda (H27: (eq T -(THead (Bind Abst) u t5) t4)).(eq_ind T (THead (Bind Abst) u t5) (\lambda -(t8: T).(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t8) t9)) (\lambda -(t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9)))) (ex2_ind T (\lambda (t8: -T).(pr0 u2 t8)) (\lambda (t8: T).(pr0 v2 t8)) (ex2 T (\lambda (t8: T).(pr0 -(THead (Flat Appl) u2 (THead (Bind Abst) u t5)) t8)) (\lambda (t8: T).(pr0 -(THead (Bind Abbr) v2 t6) t8))) (\lambda (x: T).(\lambda (H28: (pr0 u2 -x)).(\lambda (H29: (pr0 v2 x)).(ex_intro2 T (\lambda (t8: T).(pr0 (THead -(Flat Appl) u2 (THead (Bind Abst) u t5)) t8)) (\lambda (t8: T).(pr0 (THead -(Bind Abbr) v2 t6) t8)) (THead (Bind Abbr) x t6) (pr0_beta u u2 x H28 t5 t6 -H15) (pr0_comp v2 x H29 t6 t6 (pr0_refl t6) (Bind Abbr)))))) (H22 v1 -(tlt_head_sx (Flat Appl) v1 (THead (Bind Abst) u t5)) u2 H23 v2 H14)) t4 -H27)) t7 (sym_eq T t7 (THead (Bind Abst) u t5) H25) H26))) | (pr0_comp u0 u3 -H25 t7 t8 H26 k0) \Rightarrow (\lambda (H27: (eq T (THead k0 u0 t7) (THead -(Bind Abst) u t5))).(\lambda (H28: (eq T (THead k0 u3 t8) t4)).((let H29 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow t7 | (TLRef _) \Rightarrow t7 | (THead _ _ t9) -\Rightarrow t9])) (THead k0 u0 t7) (THead (Bind Abst) u t5) H27) in ((let H30 -\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t9 _) -\Rightarrow t9])) (THead k0 u0 t7) (THead (Bind Abst) u t5) H27) in ((let H31 -\def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) -with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _ _) -\Rightarrow k1])) (THead k0 u0 t7) (THead (Bind Abst) u t5) H27) in (eq_ind K -(Bind Abst) (\lambda (k1: K).((eq T u0 u) \to ((eq T t7 t5) \to ((eq T (THead -k1 u3 t8) t4) \to ((pr0 u0 u3) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: -T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind -Abbr) v2 t6) t9))))))))) (\lambda (H32: (eq T u0 u)).(eq_ind T u (\lambda -(t9: T).((eq T t7 t5) \to ((eq T (THead (Bind Abst) u3 t8) t4) \to ((pr0 t9 -u3) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 -t4) t10)) (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t6) t10)))))))) -(\lambda (H33: (eq T t7 t5)).(eq_ind T t5 (\lambda (t9: T).((eq T (THead -(Bind Abst) u3 t8) t4) \to ((pr0 u u3) \to ((pr0 t9 t8) \to (ex2 T (\lambda -(t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda (t10: T).(pr0 (THead -(Bind Abbr) v2 t6) t10))))))) (\lambda (H34: (eq T (THead (Bind Abst) u3 t8) -t4)).(eq_ind T (THead (Bind Abst) u3 t8) (\lambda (t9: T).((pr0 u u3) \to -((pr0 t5 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t9) -t10)) (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t6) t10)))))) (\lambda (_: -(pr0 u u3)).(\lambda (H36: (pr0 t5 t8)).(ex2_ind T (\lambda (t9: T).(pr0 t8 -t9)) (\lambda (t9: T).(pr0 t6 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat -Appl) u2 (THead (Bind Abst) u3 t8)) t9)) (\lambda (t9: T).(pr0 (THead (Bind -Abbr) v2 t6) t9))) (\lambda (x: T).(\lambda (H37: (pr0 t8 x)).(\lambda (H38: -(pr0 t6 x)).(ex2_ind T (\lambda (t9: T).(pr0 u2 t9)) (\lambda (t9: T).(pr0 v2 -t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u3 -t8)) t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9))) (\lambda (x0: -T).(\lambda (H39: (pr0 u2 x0)).(\lambda (H40: (pr0 v2 x0)).(ex_intro2 T -(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u3 t8)) t9)) -(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9)) (THead (Bind Abbr) x0 x) -(pr0_beta u3 u2 x0 H39 t8 x H37) (pr0_comp v2 x0 H40 t6 x H38 (Bind -Abbr)))))) (H22 v1 (tlt_head_sx (Flat Appl) v1 (THead (Bind Abst) u t5)) u2 -H23 v2 H14))))) (H22 t5 (tlt_trans (THead (Bind Abst) u t5) t5 (THead (Flat -Appl) v1 (THead (Bind Abst) u t5)) (tlt_head_dx (Bind Abst) u t5) -(tlt_head_dx (Flat Appl) v1 (THead (Bind Abst) u t5))) t8 H36 t6 H15)))) t4 -H34)) t7 (sym_eq T t7 t5 H33))) u0 (sym_eq T u0 u H32))) k0 (sym_eq K k0 -(Bind Abst) H31))) H30)) H29)) H28 H25 H26))) | (pr0_beta u0 v0 v3 H25 t7 t8 -H26) \Rightarrow (\lambda (H27: (eq T (THead (Flat Appl) v0 (THead (Bind -Abst) u0 t7)) (THead (Bind Abst) u t5))).(\lambda (H28: (eq T (THead (Bind -Abbr) v3 t8) t4)).((let H29 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind -Abst) u0 t7)) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _) -\Rightarrow (match k0 in K return (\lambda (_: K).Prop) with [(Bind _) -\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u t5) -H27) in (False_ind ((eq T (THead (Bind Abbr) v3 t8) t4) \to ((pr0 v0 v3) \to -((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) -(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9)))))) H29)) H28 H25 H26))) -| (pr0_upsilon b H25 v0 v3 H26 u0 u3 H27 t7 t8 H28) \Rightarrow (\lambda -(H29: (eq T (THead (Flat Appl) v0 (THead (Bind b) u0 t7)) (THead (Bind Abst) -u t5))).(\lambda (H30: (eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S -O) O v3) t8)) t4)).((let H31 \def (eq_ind T (THead (Flat Appl) v0 (THead -(Bind b) u0 t7)) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) -with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ -_) \Rightarrow (match k0 in K return (\lambda (_: K).Prop) with [(Bind _) -\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u t5) -H29) in (False_ind ((eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O -v3) t8)) t4) \to ((not (eq B b Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3) \to -((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) -(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9)))))))) H31)) H30 H25 H26 -H27 H28))) | (pr0_delta u0 u3 H25 t7 t8 H26 w H27) \Rightarrow (\lambda (H28: -(eq T (THead (Bind Abbr) u0 t7) (THead (Bind Abst) u t5))).(\lambda (H29: (eq -T (THead (Bind Abbr) u3 w) t4)).((let H30 \def (eq_ind T (THead (Bind Abbr) -u0 t7) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _) -\Rightarrow (match k0 in K return (\lambda (_: K).Prop) with [(Bind b) -\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow -True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _) -\Rightarrow False])])) I (THead (Bind Abst) u t5) H28) in (False_ind ((eq T -(THead (Bind Abbr) u3 w) t4) \to ((pr0 u0 u3) \to ((pr0 t7 t8) \to ((subst0 O -u3 t8 w) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) -(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9))))))) H30)) H29 H25 H26 -H27))) | (pr0_zeta b H25 t7 t8 H26 u0) \Rightarrow (\lambda (H27: (eq T -(THead (Bind b) u0 (lift (S O) O t7)) (THead (Bind Abst) u t5))).(\lambda -(H28: (eq T t8 t4)).((let H29 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: -((nat \to nat))) (d: nat) (t9: T) on t9: T \def (match t9 with [(TSort n) -\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with -[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k0 u3 t10) -\Rightarrow (THead k0 (lref_map f d u3) (lref_map f (s k0 d) t10))]) in -lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (TLRef _) \Rightarrow -((let rec lref_map (f: ((nat \to nat))) (d: nat) (t9: T) on t9: T \def (match -t9 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef -(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | -(THead k0 u3 t10) \Rightarrow (THead k0 (lref_map f d u3) (lref_map f (s k0 -d) t10))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (THead _ _ -t9) \Rightarrow t9])) (THead (Bind b) u0 (lift (S O) O t7)) (THead (Bind -Abst) u t5) H27) in ((let H30 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) -\Rightarrow u0 | (THead _ t9 _) \Rightarrow t9])) (THead (Bind b) u0 (lift (S -O) O t7)) (THead (Bind Abst) u t5) H27) in ((let H31 \def (f_equal T B -(\lambda (e: T).(match e in T return (\lambda (_: T).B) with [(TSort _) -\Rightarrow b | (TLRef _) \Rightarrow b | (THead k0 _ _) \Rightarrow (match -k0 in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) -\Rightarrow b])])) (THead (Bind b) u0 (lift (S O) O t7)) (THead (Bind Abst) u -t5) H27) in (eq_ind B Abst (\lambda (b0: B).((eq T u0 u) \to ((eq T (lift (S -O) O t7) t5) \to ((eq T t8 t4) \to ((not (eq B b0 Abst)) \to ((pr0 t7 t8) \to -(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: -T).(pr0 (THead (Bind Abbr) v2 t6) t9))))))))) (\lambda (H32: (eq T u0 -u)).(eq_ind T u (\lambda (_: T).((eq T (lift (S O) O t7) t5) \to ((eq T t8 -t4) \to ((not (eq B Abst Abst)) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: -T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda (t10: T).(pr0 (THead (Bind -Abbr) v2 t6) t10)))))))) (\lambda (H33: (eq T (lift (S O) O t7) t5)).(eq_ind -T (lift (S O) O t7) (\lambda (_: T).((eq T t8 t4) \to ((not (eq B Abst Abst)) -\to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) -t10)) (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t6) t10))))))) (\lambda -(H34: (eq T t8 t4)).(eq_ind T t4 (\lambda (t9: T).((not (eq B Abst Abst)) \to -((pr0 t7 t9) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) -t10)) (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t6) t10)))))) (\lambda -(H35: (not (eq B Abst Abst))).(\lambda (_: (pr0 t7 t4)).(let H37 \def (match -(H35 (refl_equal B Abst)) in False return (\lambda (_: False).(ex2 T (\lambda -(t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead -(Bind Abbr) v2 t6) t9)))) with []) in H37))) t8 (sym_eq T t8 t4 H34))) t5 -H33)) u0 (sym_eq T u0 u H32))) b (sym_eq B b Abst H31))) H30)) H29)) H28 H25 -H26))) | (pr0_tau t7 t8 H25 u0) \Rightarrow (\lambda (H26: (eq T (THead (Flat -Cast) u0 t7) (THead (Bind Abst) u t5))).(\lambda (H27: (eq T t8 t4)).((let -H28 \def (eq_ind T (THead (Flat Cast) u0 t7) (\lambda (e: T).(match e in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind Abst) u t5) H26) in (False_ind ((eq T t8 t4) \to -((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) -(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9))))) H28)) H27 H25)))]) in -(H25 (refl_equal T (THead (Bind Abst) u t5)) (refl_equal T t4))))) k H21))))) -H18)) H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_upsilon b H9 v1 v2 H10 u0 -u3 H11 t5 t6 H12) \Rightarrow (\lambda (H13: (eq T (THead (Flat Appl) v1 -(THead (Bind b) u0 t5)) t)).(\lambda (H14: (eq T (THead (Bind b) u3 (THead -(Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Flat Appl) v1 -(THead (Bind b) u0 t5)) (\lambda (_: T).((eq T (THead (Bind b) u3 (THead -(Flat Appl) (lift (S O) O v2) t6)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1 -v2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead -k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H15: (eq T -(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T -(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) (\lambda (t7: -T).((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) -\to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 -t7 t8)))))))) (\lambda (H16: (not (eq B b Abst))).(\lambda (H17: (pr0 v1 -v2)).(\lambda (H18: (pr0 u0 u3)).(\lambda (H19: (pr0 t5 t6)).(let H20 \def -(eq_ind_r T t (\lambda (t7: T).(eq T (THead k u1 t3) t7)) H4 (THead (Flat -Appl) v1 (THead (Bind b) u0 t5)) H13) in (let H21 \def (f_equal T K (\lambda -(e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k -| (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) -(THead (Flat Appl) v1 (THead (Bind b) u0 t5)) H20) in ((let H22 \def (f_equal -T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) -(THead k u1 t3) (THead (Flat Appl) v1 (THead (Bind b) u0 t5)) H20) in ((let -H23 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7) -\Rightarrow t7])) (THead k u1 t3) (THead (Flat Appl) v1 (THead (Bind b) u0 -t5)) H20) in (\lambda (H24: (eq T u1 v1)).(\lambda (H25: (eq K k (Flat -Appl))).(let H26 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v -t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to -(ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 -t10)))))))))) H (THead (Flat Appl) v1 (THead (Bind b) u0 t5)) H13) in -(eq_ind_r K (Flat Appl) (\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 (THead -k0 u2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) -(lift (S O) O v2) t6)) t7)))) (let H27 \def (eq_ind T u1 (\lambda (t7: -T).(pr0 t7 u2)) H7 v1 H24) in (let H28 \def (eq_ind T t3 (\lambda (t7: -T).(pr0 t7 t4)) H8 (THead (Bind b) u0 t5) H23) in (let H29 \def (match H28 in -pr0 return (\lambda (t7: T).(\lambda (t8: T).(\lambda (_: (pr0 t7 t8)).((eq T -t7 (THead (Bind b) u0 t5)) \to ((eq T t8 t4) \to (ex2 T (\lambda (t9: T).(pr0 -(THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 -(THead (Flat Appl) (lift (S O) O v2) t6)) t9)))))))) with [(pr0_refl t7) -\Rightarrow (\lambda (H29: (eq T t7 (THead (Bind b) u0 t5))).(\lambda (H30: -(eq T t7 t4)).(eq_ind T (THead (Bind b) u0 t5) (\lambda (t8: T).((eq T t8 t4) -\to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: -T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9))))) -(\lambda (H31: (eq T (THead (Bind b) u0 t5) t4)).(eq_ind T (THead (Bind b) u0 -t5) (\lambda (t8: T).(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t8) -t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) -O v2) t6)) t9)))) (ex2_ind T (\lambda (t8: T).(pr0 u2 t8)) (\lambda (t8: -T).(pr0 v2 t8)) (ex2 T (\lambda (t8: T).(pr0 (THead (Flat Appl) u2 (THead -(Bind b) u0 t5)) t8)) (\lambda (t8: T).(pr0 (THead (Bind b) u3 (THead (Flat -Appl) (lift (S O) O v2) t6)) t8))) (\lambda (x: T).(\lambda (H32: (pr0 u2 -x)).(\lambda (H33: (pr0 v2 x)).(pr0_confluence__pr0_cong_upsilon_refl b H16 -u0 u3 H18 t5 t6 H19 u2 v2 x H32 H33)))) (H26 v1 (tlt_head_sx (Flat Appl) v1 -(THead (Bind b) u0 t5)) u2 H27 v2 H17)) t4 H31)) t7 (sym_eq T t7 (THead (Bind -b) u0 t5) H29) H30))) | (pr0_comp u4 u5 H29 t7 t8 H30 k0) \Rightarrow -(\lambda (H31: (eq T (THead k0 u4 t7) (THead (Bind b) u0 t5))).(\lambda (H32: -(eq T (THead k0 u5 t8) t4)).((let H33 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t7 | -(TLRef _) \Rightarrow t7 | (THead _ _ t9) \Rightarrow t9])) (THead k0 u4 t7) -(THead (Bind b) u0 t5) H31) in ((let H34 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u4 | -(TLRef _) \Rightarrow u4 | (THead _ t9 _) \Rightarrow t9])) (THead k0 u4 t7) -(THead (Bind b) u0 t5) H31) in ((let H35 \def (f_equal T K (\lambda (e: -T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k0 | -(TLRef _) \Rightarrow k0 | (THead k1 _ _) \Rightarrow k1])) (THead k0 u4 t7) -(THead (Bind b) u0 t5) H31) in (eq_ind K (Bind b) (\lambda (k1: K).((eq T u4 -u0) \to ((eq T t7 t5) \to ((eq T (THead k1 u5 t8) t4) \to ((pr0 u4 u5) \to -((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) -(\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) -t6)) t9))))))))) (\lambda (H36: (eq T u4 u0)).(eq_ind T u0 (\lambda (t9: -T).((eq T t7 t5) \to ((eq T (THead (Bind b) u5 t8) t4) \to ((pr0 t9 u5) \to -((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) -t10)) (\lambda (t10: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S -O) O v2) t6)) t10)))))))) (\lambda (H37: (eq T t7 t5)).(eq_ind T t5 (\lambda -(t9: T).((eq T (THead (Bind b) u5 t8) t4) \to ((pr0 u0 u5) \to ((pr0 t9 t8) -\to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda -(t10: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) -t10))))))) (\lambda (H38: (eq T (THead (Bind b) u5 t8) t4)).(eq_ind T (THead -(Bind b) u5 t8) (\lambda (t9: T).((pr0 u0 u5) \to ((pr0 t5 t8) \to (ex2 T -(\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t9) t10)) (\lambda (t10: T).(pr0 -(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t10)))))) -(\lambda (H39: (pr0 u0 u5)).(\lambda (H40: (pr0 t5 t8)).(ex2_ind T (\lambda -(t9: T).(pr0 t8 t9)) (\lambda (t9: T).(pr0 t6 t9)) (ex2 T (\lambda (t9: -T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 t8)) t9)) (\lambda (t9: -T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9))) -(\lambda (x: T).(\lambda (H41: (pr0 t8 x)).(\lambda (H42: (pr0 t6 -x)).(ex2_ind T (\lambda (t9: T).(pr0 u5 t9)) (\lambda (t9: T).(pr0 u3 t9)) -(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 t8)) -t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) -O v2) t6)) t9))) (\lambda (x0: T).(\lambda (H43: (pr0 u5 x0)).(\lambda (H44: -(pr0 u3 x0)).(ex2_ind T (\lambda (t9: T).(pr0 u2 t9)) (\lambda (t9: T).(pr0 -v2 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 -t8)) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift -(S O) O v2) t6)) t9))) (\lambda (x1: T).(\lambda (H45: (pr0 u2 x1)).(\lambda -(H46: (pr0 v2 x1)).(pr0_confluence__pr0_cong_upsilon_cong b H16 u2 v2 x1 H45 -H46 t8 t6 x H41 H42 u5 u3 x0 H43 H44)))) (H26 v1 (tlt_head_sx (Flat Appl) v1 -(THead (Bind b) u0 t5)) u2 H27 v2 H17))))) (H26 u0 (tlt_trans (THead (Bind b) -u0 t5) u0 (THead (Flat Appl) v1 (THead (Bind b) u0 t5)) (tlt_head_sx (Bind b) -u0 t5) (tlt_head_dx (Flat Appl) v1 (THead (Bind b) u0 t5))) u5 H39 u3 -H18))))) (H26 t5 (tlt_trans (THead (Bind b) u0 t5) t5 (THead (Flat Appl) v1 -(THead (Bind b) u0 t5)) (tlt_head_dx (Bind b) u0 t5) (tlt_head_dx (Flat Appl) -v1 (THead (Bind b) u0 t5))) t8 H40 t6 H19)))) t4 H38)) t7 (sym_eq T t7 t5 -H37))) u4 (sym_eq T u4 u0 H36))) k0 (sym_eq K k0 (Bind b) H35))) H34)) H33)) -H32 H29 H30))) | (pr0_beta u v0 v3 H29 t7 t8 H30) \Rightarrow (\lambda (H31: -(eq T (THead (Flat Appl) v0 (THead (Bind Abst) u t7)) (THead (Bind b) u0 -t5))).(\lambda (H32: (eq T (THead (Bind Abbr) v3 t8) t4)).((let H33 \def -(eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u t7)) (\lambda (e: -T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in -K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) -\Rightarrow True])])) I (THead (Bind b) u0 t5) H31) in (False_ind ((eq T -(THead (Bind Abbr) v3 t8) t4) \to ((pr0 v0 v3) \to ((pr0 t7 t8) \to (ex2 T -(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 -(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9)))))) H33)) -H32 H29 H30))) | (pr0_upsilon b0 H29 v0 v3 H30 u4 u5 H31 t7 t8 H32) -\Rightarrow (\lambda (H33: (eq T (THead (Flat Appl) v0 (THead (Bind b0) u4 -t7)) (THead (Bind b) u0 t5))).(\lambda (H34: (eq T (THead (Bind b0) u5 (THead -(Flat Appl) (lift (S O) O v3) t8)) t4)).((let H35 \def (eq_ind T (THead (Flat -Appl) v0 (THead (Bind b0) u4 t7)) (\lambda (e: T).(match e in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind b) u0 t5) H33) in (False_ind ((eq T (THead (Bind b0) -u5 (THead (Flat Appl) (lift (S O) O v3) t8)) t4) \to ((not (eq B b0 Abst)) -\to ((pr0 v0 v3) \to ((pr0 u4 u5) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: -T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) -u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9)))))))) H35)) H34 H29 H30 H31 -H32))) | (pr0_delta u4 u5 H29 t7 t8 H30 w H31) \Rightarrow (\lambda (H32: (eq -T (THead (Bind Abbr) u4 t7) (THead (Bind b) u0 t5))).(\lambda (H33: (eq T -(THead (Bind Abbr) u5 w) t4)).((let H34 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t7 | -(TLRef _) \Rightarrow t7 | (THead _ _ t9) \Rightarrow t9])) (THead (Bind -Abbr) u4 t7) (THead (Bind b) u0 t5) H32) in ((let H35 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow u4 | (TLRef _) \Rightarrow u4 | (THead _ t9 _) \Rightarrow t9])) -(THead (Bind Abbr) u4 t7) (THead (Bind b) u0 t5) H32) in ((let H36 \def -(f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with -[(TSort _) \Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | (THead k0 _ _) -\Rightarrow (match k0 in K return (\lambda (_: K).B) with [(Bind b0) -\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (THead (Bind Abbr) u4 t7) -(THead (Bind b) u0 t5) H32) in (eq_ind B Abbr (\lambda (b0: B).((eq T u4 u0) -\to ((eq T t7 t5) \to ((eq T (THead (Bind Abbr) u5 w) t4) \to ((pr0 u4 u5) -\to ((pr0 t7 t8) \to ((subst0 O u5 t8 w) \to (ex2 T (\lambda (t9: T).(pr0 -(THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind b0) u3 -(THead (Flat Appl) (lift (S O) O v2) t6)) t9)))))))))) (\lambda (H37: (eq T -u4 u0)).(eq_ind T u0 (\lambda (t9: T).((eq T t7 t5) \to ((eq T (THead (Bind -Abbr) u5 w) t4) \to ((pr0 t9 u5) \to ((pr0 t7 t8) \to ((subst0 O u5 t8 w) \to -(ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda (t10: -T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) -t10))))))))) (\lambda (H38: (eq T t7 t5)).(eq_ind T t5 (\lambda (t9: T).((eq -T (THead (Bind Abbr) u5 w) t4) \to ((pr0 u0 u5) \to ((pr0 t9 t8) \to ((subst0 -O u5 t8 w) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) -(\lambda (t10: T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O -v2) t6)) t10)))))))) (\lambda (H39: (eq T (THead (Bind Abbr) u5 w) -t4)).(eq_ind T (THead (Bind Abbr) u5 w) (\lambda (t9: T).((pr0 u0 u5) \to -((pr0 t5 t8) \to ((subst0 O u5 t8 w) \to (ex2 T (\lambda (t10: T).(pr0 (THead -(Flat Appl) u2 t9) t10)) (\lambda (t10: T).(pr0 (THead (Bind Abbr) u3 (THead -(Flat Appl) (lift (S O) O v2) t6)) t10))))))) (\lambda (H40: (pr0 u0 -u5)).(\lambda (H41: (pr0 t5 t8)).(\lambda (H42: (subst0 O u5 t8 w)).(let H43 -\def (eq_ind_r B b (\lambda (b0: B).(\forall (v: T).((tlt v (THead (Flat -Appl) v1 (THead (Bind b0) u0 t5))) \to (\forall (t9: T).((pr0 v t9) \to -(\forall (t10: T).((pr0 v t10) \to (ex2 T (\lambda (t11: T).(pr0 t9 t11)) -(\lambda (t11: T).(pr0 t10 t11)))))))))) H26 Abbr H36) in (let H44 \def -(eq_ind_r B b (\lambda (b0: B).(eq T t3 (THead (Bind b0) u0 t5))) H23 Abbr -H36) in (let H45 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 Abst))) -H16 Abbr H36) in (ex2_ind T (\lambda (t9: T).(pr0 t8 t9)) (\lambda (t9: -T).(pr0 t6 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead -(Bind Abbr) u5 w)) t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) u3 (THead -(Flat Appl) (lift (S O) O v2) t6)) t9))) (\lambda (x: T).(\lambda (H46: (pr0 -t8 x)).(\lambda (H47: (pr0 t6 x)).(ex2_ind T (\lambda (t9: T).(pr0 u5 t9)) -(\lambda (t9: T).(pr0 u3 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) -u2 (THead (Bind Abbr) u5 w)) t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) u3 -(THead (Flat Appl) (lift (S O) O v2) t6)) t9))) (\lambda (x0: T).(\lambda -(H48: (pr0 u5 x0)).(\lambda (H49: (pr0 u3 x0)).(ex2_ind T (\lambda (t9: -T).(pr0 u2 t9)) (\lambda (t9: T).(pr0 v2 t9)) (ex2 T (\lambda (t9: T).(pr0 -(THead (Flat Appl) u2 (THead (Bind Abbr) u5 w)) t9)) (\lambda (t9: T).(pr0 -(THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9))) -(\lambda (x1: T).(\lambda (H50: (pr0 u2 x1)).(\lambda (H51: (pr0 v2 -x1)).(pr0_confluence__pr0_cong_upsilon_delta H45 u5 t8 w H42 u2 v2 x1 H50 H51 -t6 x H46 H47 u3 x0 H48 H49)))) (H43 v1 (tlt_head_sx (Flat Appl) v1 (THead -(Bind Abbr) u0 t5)) u2 H27 v2 H17))))) (H43 u0 (tlt_trans (THead (Bind Abbr) -u0 t5) u0 (THead (Flat Appl) v1 (THead (Bind Abbr) u0 t5)) (tlt_head_sx (Bind -Abbr) u0 t5) (tlt_head_dx (Flat Appl) v1 (THead (Bind Abbr) u0 t5))) u5 H40 -u3 H18))))) (H43 t5 (tlt_trans (THead (Bind Abbr) u0 t5) t5 (THead (Flat -Appl) v1 (THead (Bind Abbr) u0 t5)) (tlt_head_dx (Bind Abbr) u0 t5) -(tlt_head_dx (Flat Appl) v1 (THead (Bind Abbr) u0 t5))) t8 H41 t6 H19)))))))) -t4 H39)) t7 (sym_eq T t7 t5 H38))) u4 (sym_eq T u4 u0 H37))) b H36)) H35)) -H34)) H33 H29 H30 H31))) | (pr0_zeta b0 H29 t7 t8 H30 u) \Rightarrow (\lambda -(H31: (eq T (THead (Bind b0) u (lift (S O) O t7)) (THead (Bind b) u0 -t5))).(\lambda (H32: (eq T t8 t4)).((let H33 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let -rec lref_map (f: ((nat \to nat))) (d: nat) (t9: T) on t9: T \def (match t9 -with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match -(blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k0 -u4 t10) \Rightarrow (THead k0 (lref_map f d u4) (lref_map f (s k0 d) t10))]) -in lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (TLRef _) \Rightarrow -((let rec lref_map (f: ((nat \to nat))) (d: nat) (t9: T) on t9: T \def (match -t9 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef -(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | -(THead k0 u4 t10) \Rightarrow (THead k0 (lref_map f d u4) (lref_map f (s k0 -d) t10))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (THead _ _ -t9) \Rightarrow t9])) (THead (Bind b0) u (lift (S O) O t7)) (THead (Bind b) -u0 t5) H31) in ((let H34 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) -\Rightarrow u | (THead _ t9 _) \Rightarrow t9])) (THead (Bind b0) u (lift (S -O) O t7)) (THead (Bind b) u0 t5) H31) in ((let H35 \def (f_equal T B (\lambda -(e: T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b0 -| (TLRef _) \Rightarrow b0 | (THead k0 _ _) \Rightarrow (match k0 in K return -(\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow -b0])])) (THead (Bind b0) u (lift (S O) O t7)) (THead (Bind b) u0 t5) H31) in -(eq_ind B b (\lambda (b1: B).((eq T u u0) \to ((eq T (lift (S O) O t7) t5) -\to ((eq T t8 t4) \to ((not (eq B b1 Abst)) \to ((pr0 t7 t8) \to (ex2 T -(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 -(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9))))))))) -(\lambda (H36: (eq T u u0)).(eq_ind T u0 (\lambda (_: T).((eq T (lift (S O) O -t7) t5) \to ((eq T t8 t4) \to ((not (eq B b Abst)) \to ((pr0 t7 t8) \to (ex2 -T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda (t10: -T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) -t10)))))))) (\lambda (H37: (eq T (lift (S O) O t7) t5)).(eq_ind T (lift (S O) -O t7) (\lambda (_: T).((eq T t8 t4) \to ((not (eq B b Abst)) \to ((pr0 t7 t8) -\to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda -(t10: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) -t10))))))) (\lambda (H38: (eq T t8 t4)).(eq_ind T t4 (\lambda (t9: T).((not -(eq B b Abst)) \to ((pr0 t7 t9) \to (ex2 T (\lambda (t10: T).(pr0 (THead -(Flat Appl) u2 t4) t10)) (\lambda (t10: T).(pr0 (THead (Bind b) u3 (THead -(Flat Appl) (lift (S O) O v2) t6)) t10)))))) (\lambda (H39: (not (eq B b -Abst))).(\lambda (H40: (pr0 t7 t4)).(let H41 \def (eq_ind_r T t5 (\lambda -(t9: T).(\forall (v: T).((tlt v (THead (Flat Appl) v1 (THead (Bind b) u0 -t9))) \to (\forall (t10: T).((pr0 v t10) \to (\forall (t11: T).((pr0 v t11) -\to (ex2 T (\lambda (t12: T).(pr0 t10 t12)) (\lambda (t12: T).(pr0 t11 -t12)))))))))) H26 (lift (S O) O t7) H37) in (let H42 \def (eq_ind_r T t5 -(\lambda (t9: T).(eq T t3 (THead (Bind b) u0 t9))) H23 (lift (S O) O t7) H37) -in (let H43 \def (eq_ind_r T t5 (\lambda (t9: T).(pr0 t9 t6)) H19 (lift (S O) -O t7) H37) in (ex2_ind T (\lambda (t9: T).(eq T t6 (lift (S O) O t9))) -(\lambda (t9: T).(pr0 t7 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) -u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift -(S O) O v2) t6)) t9))) (\lambda (x: T).(\lambda (H44: (eq T t6 (lift (S O) O -x))).(\lambda (H45: (pr0 t7 x)).(eq_ind_r T (lift (S O) O x) (\lambda (t9: -T).(ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda -(t10: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t9)) -t10)))) (ex2_ind T (\lambda (t9: T).(pr0 x t9)) (\lambda (t9: T).(pr0 t4 t9)) -(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: -T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) (lift (S O) O -x))) t9))) (\lambda (x0: T).(\lambda (H46: (pr0 x x0)).(\lambda (H47: (pr0 t4 -x0)).(ex2_ind T (\lambda (t9: T).(pr0 u2 t9)) (\lambda (t9: T).(pr0 v2 t9)) -(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: -T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) (lift (S O) O -x))) t9))) (\lambda (x1: T).(\lambda (H48: (pr0 u2 x1)).(\lambda (H49: (pr0 -v2 x1)).(pr0_confluence__pr0_cong_upsilon_zeta b H39 u0 u3 H18 u2 v2 x1 H48 -H49 x t4 x0 H46 H47)))) (H41 v1 (tlt_head_sx (Flat Appl) v1 (THead (Bind b) -u0 (lift (S O) O t7))) u2 H27 v2 H17))))) (H41 t7 (tlt_trans (THead (Bind b) -u0 (lift (S O) O t7)) t7 (THead (Flat Appl) v1 (THead (Bind b) u0 (lift (S O) -O t7))) (lift_tlt_dx (Bind b) u0 t7 (S O) O) (tlt_head_dx (Flat Appl) v1 -(THead (Bind b) u0 (lift (S O) O t7)))) x H45 t4 H40)) t6 H44)))) -(pr0_gen_lift t7 t6 (S O) O H43))))))) t8 (sym_eq T t8 t4 H38))) t5 H37)) u -(sym_eq T u u0 H36))) b0 (sym_eq B b0 b H35))) H34)) H33)) H32 H29 H30))) | -(pr0_tau t7 t8 H29 u) \Rightarrow (\lambda (H30: (eq T (THead (Flat Cast) u -t7) (THead (Bind b) u0 t5))).(\lambda (H31: (eq T t8 t4)).((let H32 \def -(eq_ind T (THead (Flat Cast) u t7) (\lambda (e: T).(match e in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind b) u0 t5) H30) in (False_ind ((eq T t8 t4) \to ((pr0 -t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) -(\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) -t6)) t9))))) H32)) H31 H29)))]) in (H29 (refl_equal T (THead (Bind b) u0 t5)) -(refl_equal T t4))))) k H25))))) H22)) H21))))))) t2 H15)) t H13 H14 H9 H10 -H11 H12))) | (pr0_delta u0 u3 H9 t5 t6 H10 w H11) \Rightarrow (\lambda (H12: -(eq T (THead (Bind Abbr) u0 t5) t)).(\lambda (H13: (eq T (THead (Bind Abbr) -u3 w) t2)).(eq_ind T (THead (Bind Abbr) u0 t5) (\lambda (_: T).((eq T (THead -(Bind Abbr) u3 w) t2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to ((subst0 O u3 t6 -w) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: -T).(pr0 t2 t8)))))))) (\lambda (H14: (eq T (THead (Bind Abbr) u3 w) -t2)).(eq_ind T (THead (Bind Abbr) u3 w) (\lambda (t7: T).((pr0 u0 u3) \to -((pr0 t5 t6) \to ((subst0 O u3 t6 w) \to (ex2 T (\lambda (t8: T).(pr0 (THead -k u2 t4) t8)) (\lambda (t8: T).(pr0 t7 t8))))))) (\lambda (H15: (pr0 u0 -u3)).(\lambda (H16: (pr0 t5 t6)).(\lambda (H17: (subst0 O u3 t6 w)).(let H18 -\def (eq_ind_r T t (\lambda (t7: T).(eq T (THead k u1 t3) t7)) H4 (THead -(Bind Abbr) u0 t5) H12) in (let H19 \def (f_equal T K (\lambda (e: T).(match -e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) -\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) (THead (Bind -Abbr) u0 t5) H18) in ((let H20 \def (f_equal T T (\lambda (e: T).(match e in -T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) -\Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t3) (THead -(Bind Abbr) u0 t5) H18) in ((let H21 \def (f_equal T T (\lambda (e: T).(match -e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) -\Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t3) (THead -(Bind Abbr) u0 t5) H18) in (\lambda (H22: (eq T u1 u0)).(\lambda (H23: (eq K -k (Bind Abbr))).(let H24 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: -T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v -t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 -t10)))))))))) H (THead (Bind Abbr) u0 t5) H12) in (eq_ind_r K (Bind Abbr) -(\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 (THead k0 u2 t4) t7)) (\lambda -(t7: T).(pr0 (THead (Bind Abbr) u3 w) t7)))) (let H25 \def (eq_ind T u1 -(\lambda (t7: T).(pr0 t7 u2)) H7 u0 H22) in (let H26 \def (eq_ind T t3 -(\lambda (t7: T).(pr0 t7 t4)) H8 t5 H21) in (ex2_ind T (\lambda (t7: T).(pr0 -t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead -(Bind Abbr) u2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u3 w) t7))) -(\lambda (x: T).(\lambda (H27: (pr0 t4 x)).(\lambda (H28: (pr0 t6 -x)).(ex2_ind T (\lambda (t7: T).(pr0 u2 t7)) (\lambda (t7: T).(pr0 u3 t7)) -(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 t4) t7)) (\lambda (t7: -T).(pr0 (THead (Bind Abbr) u3 w) t7))) (\lambda (x0: T).(\lambda (H29: (pr0 -u2 x0)).(\lambda (H30: (pr0 u3 x0)).(pr0_confluence__pr0_cong_delta u3 t6 w -H17 u2 x0 H29 H30 t4 x H27 H28)))) (H24 u0 (tlt_head_sx (Bind Abbr) u0 t5) u2 -H25 u3 H15))))) (H24 t5 (tlt_head_dx (Bind Abbr) u0 t5) t4 H26 t6 H16)))) k -H23))))) H20)) H19)))))) t2 H14)) t H12 H13 H9 H10 H11))) | (pr0_zeta b H9 t5 -t6 H10 u) \Rightarrow (\lambda (H11: (eq T (THead (Bind b) u (lift (S O) O -t5)) t)).(\lambda (H12: (eq T t6 t2)).(eq_ind T (THead (Bind b) u (lift (S O) -O t5)) (\lambda (_: T).((eq T t6 t2) \to ((not (eq B b Abst)) \to ((pr0 t5 -t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: -T).(pr0 t2 t8))))))) (\lambda (H13: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: -T).((not (eq B b Abst)) \to ((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 -(THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H14: (not -(eq B b Abst))).(\lambda (H15: (pr0 t5 t2)).(let H16 \def (eq_ind_r T t -(\lambda (t7: T).(eq T (THead k u1 t3) t7)) H4 (THead (Bind b) u (lift (S O) -O t5)) H11) in (let H17 \def (f_equal T K (\lambda (e: T).(match e in T -return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) -\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) (THead (Bind -b) u (lift (S O) O t5)) H16) in ((let H18 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | -(TLRef _) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t3) -(THead (Bind b) u (lift (S O) O t5)) H16) in ((let H19 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) -(THead k u1 t3) (THead (Bind b) u (lift (S O) O t5)) H16) in (\lambda (H20: -(eq T u1 u)).(\lambda (H21: (eq K k (Bind b))).(let H22 \def (eq_ind_r T t -(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) -\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) -(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Bind b) u (lift (S O) O -t5)) H11) in (eq_ind_r K (Bind b) (\lambda (k0: K).(ex2 T (\lambda (t7: -T).(pr0 (THead k0 u2 t4) t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let H23 \def -(eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H7 u H20) in (let H24 \def (eq_ind -T t3 (\lambda (t7: T).(pr0 t7 t4)) H8 (lift (S O) O t5) H19) in (ex2_ind T -(\lambda (t7: T).(eq T t4 (lift (S O) O t7))) (\lambda (t7: T).(pr0 t5 t7)) -(ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 t4) t7)) (\lambda (t7: -T).(pr0 t2 t7))) (\lambda (x: T).(\lambda (H25: (eq T t4 (lift (S O) O -x))).(\lambda (H26: (pr0 t5 x)).(eq_ind_r T (lift (S O) O x) (\lambda (t7: -T).(ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 t7) t8)) (\lambda (t8: -T).(pr0 t2 t8)))) (ex2_ind T (\lambda (t7: T).(pr0 x t7)) (\lambda (t7: -T).(pr0 t2 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (lift (S O) O -x)) t7)) (\lambda (t7: T).(pr0 t2 t7))) (\lambda (x0: T).(\lambda (H27: (pr0 -x x0)).(\lambda (H28: (pr0 t2 x0)).(ex_intro2 T (\lambda (t7: T).(pr0 (THead -(Bind b) u2 (lift (S O) O x)) t7)) (\lambda (t7: T).(pr0 t2 t7)) x0 (pr0_zeta -b H14 x x0 H27 u2) H28)))) (H22 t5 (lift_tlt_dx (Bind b) u t5 (S O) O) x H26 -t2 H15)) t4 H25)))) (pr0_gen_lift t5 t4 (S O) O H24)))) k H21))))) H18)) -H17))))) t6 (sym_eq T t6 t2 H13))) t H11 H12 H9 H10))) | (pr0_tau t5 t6 H9 u) -\Rightarrow (\lambda (H10: (eq T (THead (Flat Cast) u t5) t)).(\lambda (H11: -(eq T t6 t2)).(eq_ind T (THead (Flat Cast) u t5) (\lambda (_: T).((eq T t6 -t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) -(\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H12: (eq T t6 t2)).(eq_ind T t2 -(\lambda (t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 -t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))) (\lambda (H13: (pr0 t5 t2)).(let -H14 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead k u1 t3) t7)) H4 (THead -(Flat Cast) u t5) H10) in (let H15 \def (f_equal T K (\lambda (e: T).(match e -in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) -\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) (THead (Flat -Cast) u t5) H14) in ((let H16 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) -\Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t3) (THead -(Flat Cast) u t5) H14) in ((let H17 \def (f_equal T T (\lambda (e: T).(match -e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) -\Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t3) (THead -(Flat Cast) u t5) H14) in (\lambda (H18: (eq T u1 u)).(\lambda (H19: (eq K k -(Flat Cast))).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: -T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v -t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 -t10)))))))))) H (THead (Flat Cast) u t5) H10) in (eq_ind_r K (Flat Cast) -(\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 (THead k0 u2 t4) t7)) (\lambda -(t7: T).(pr0 t2 t7)))) (let H21 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 -u2)) H7 u H18) in (let H22 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H8 -t5 H17) in (ex2_ind T (\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t2 -t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Flat Cast) u2 t4) t7)) (\lambda -(t7: T).(pr0 t2 t7))) (\lambda (x: T).(\lambda (H23: (pr0 t4 x)).(\lambda -(H24: (pr0 t2 x)).(ex_intro2 T (\lambda (t7: T).(pr0 (THead (Flat Cast) u2 -t4) t7)) (\lambda (t7: T).(pr0 t2 t7)) x (pr0_tau t4 x H23 u2) H24)))) (H20 -t5 (tlt_head_dx (Flat Cast) u t5) t4 H22 t2 H13)))) k H19))))) H16)) H15)))) -t6 (sym_eq T t6 t2 H12))) t H10 H11 H9)))]) in (H9 (refl_equal T t) -(refl_equal T t2))))) t1 H6)) t H4 H5 H2 H3))) | (pr0_beta u v1 v2 H2 t3 t4 -H3) \Rightarrow (\lambda (H4: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) -u t3)) t)).(\lambda (H5: (eq T (THead (Bind Abbr) v2 t4) t1)).(eq_ind T -(THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (\lambda (_: T).((eq T (THead -(Bind Abbr) v2 t4) t1) \to ((pr0 v1 v2) \to ((pr0 t3 t4) \to (ex2 T (\lambda -(t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6))))))) (\lambda (H6: (eq T -(THead (Bind Abbr) v2 t4) t1)).(eq_ind T (THead (Bind Abbr) v2 t4) (\lambda -(t5: T).((pr0 v1 v2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t5 -t6)) (\lambda (t6: T).(pr0 t2 t6)))))) (\lambda (H7: (pr0 v1 v2)).(\lambda -(H8: (pr0 t3 t4)).(let H9 \def (match H1 in pr0 return (\lambda (t5: -T).(\lambda (t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 t) \to ((eq T t6 t2) -\to (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: -T).(pr0 t2 t7)))))))) with [(pr0_refl t5) \Rightarrow (\lambda (H9: (eq T t5 -t)).(\lambda (H10: (eq T t5 t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2) -\to (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: -T).(pr0 t2 t7))))) (\lambda (H11: (eq T t t2)).(eq_ind T t2 (\lambda (_: -T).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: -T).(pr0 t2 t7)))) (let H12 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 t2)) -H11 (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) H4) in (eq_ind T (THead -(Flat Appl) v1 (THead (Bind Abst) u t3)) (\lambda (t6: T).(ex2 T (\lambda -(t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 t6 t7)))) -(let H13 \def (eq_ind_r T t (\lambda (t6: T).(eq T t5 t6)) H9 (THead (Flat -Appl) v1 (THead (Bind Abst) u t3)) H4) in (let H14 \def (eq_ind_r T t -(\lambda (t6: T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) -\to (\forall (t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) -(\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead (Flat Appl) v1 (THead (Bind -Abst) u t3)) H4) in (ex_intro2 T (\lambda (t6: T).(pr0 (THead (Bind Abbr) v2 -t4) t6)) (\lambda (t6: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u -t3)) t6)) (THead (Bind Abbr) v2 t4) (pr0_refl (THead (Bind Abbr) v2 t4)) -(pr0_beta u v1 v2 H7 t3 t4 H8)))) t2 H12)) t (sym_eq T t t2 H11))) t5 (sym_eq -T t5 t H9) H10))) | (pr0_comp u1 u2 H9 t5 t6 H10 k) \Rightarrow (\lambda -(H11: (eq T (THead k u1 t5) t)).(\lambda (H12: (eq T (THead k u2 t6) -t2)).(eq_ind T (THead k u1 t5) (\lambda (_: T).((eq T (THead k u2 t6) t2) \to -((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind -Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H13: (eq T -(THead k u2 t6) t2)).(eq_ind T (THead k u2 t6) (\lambda (t7: T).((pr0 u1 u2) -\to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) -t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (H14: (pr0 u1 u2)).(\lambda -(H15: (pr0 t5 t6)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead -(Flat Appl) v1 (THead (Bind Abst) u t3)) t7)) H4 (THead k u1 t5) H11) in (let -H17 \def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) -with [(TSort _) \Rightarrow (Flat Appl) | (TLRef _) \Rightarrow (Flat Appl) | -(THead k0 _ _) \Rightarrow k0])) (THead (Flat Appl) v1 (THead (Bind Abst) u -t3)) (THead k u1 t5) H16) in ((let H18 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | -(TLRef _) \Rightarrow v1 | (THead _ t7 _) \Rightarrow t7])) (THead (Flat -Appl) v1 (THead (Bind Abst) u t3)) (THead k u1 t5) H16) in ((let H19 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow (THead (Bind Abst) u t3) | (TLRef _) \Rightarrow -(THead (Bind Abst) u t3) | (THead _ _ t7) \Rightarrow t7])) (THead (Flat -Appl) v1 (THead (Bind Abst) u t3)) (THead k u1 t5) H16) in (\lambda (H20: (eq -T v1 u1)).(\lambda (H21: (eq K (Flat Appl) k)).(eq_ind K (Flat Appl) (\lambda -(k0: K).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda -(t7: T).(pr0 (THead k0 u2 t6) t7)))) (let H22 \def (eq_ind_r K k (\lambda -(k0: K).(eq T (THead k0 u1 t5) t)) H11 (Flat Appl) H21) in (let H23 \def -(eq_ind_r T t5 (\lambda (t7: T).(pr0 t7 t6)) H15 (THead (Bind Abst) u t3) -H19) in (let H24 \def (match H23 in pr0 return (\lambda (t7: T).(\lambda (t8: -T).(\lambda (_: (pr0 t7 t8)).((eq T t7 (THead (Bind Abst) u t3)) \to ((eq T -t8 t6) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) -(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t6) t9)))))))) with [(pr0_refl -t7) \Rightarrow (\lambda (H24: (eq T t7 (THead (Bind Abst) u t3))).(\lambda -(H25: (eq T t7 t6)).(eq_ind T (THead (Bind Abst) u t3) (\lambda (t8: T).((eq -T t8 t6) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) -(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t6) t9))))) (\lambda (H26: (eq T -(THead (Bind Abst) u t3) t6)).(eq_ind T (THead (Bind Abst) u t3) (\lambda -(t8: T).(ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda -(t9: T).(pr0 (THead (Flat Appl) u2 t8) t9)))) (let H27 \def (eq_ind_r T t5 -(\lambda (t8: T).(eq T (THead (Flat Appl) u1 t8) t)) H22 (THead (Bind Abst) u -t3) H19) in (let H28 \def (eq_ind_r T t (\lambda (t8: T).(\forall (v: -T).((tlt v t8) \to (\forall (t9: T).((pr0 v t9) \to (\forall (t10: T).((pr0 v -t10) \to (ex2 T (\lambda (t11: T).(pr0 t9 t11)) (\lambda (t11: T).(pr0 t10 -t11)))))))))) H (THead (Flat Appl) u1 (THead (Bind Abst) u t3)) H27) in (let -H29 \def (eq_ind T v1 (\lambda (t8: T).(pr0 t8 v2)) H7 u1 H20) in (ex2_ind T -(\lambda (t8: T).(pr0 v2 t8)) (\lambda (t8: T).(pr0 u2 t8)) (ex2 T (\lambda -(t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 (THead -(Flat Appl) u2 (THead (Bind Abst) u t3)) t8))) (\lambda (x: T).(\lambda (H30: -(pr0 v2 x)).(\lambda (H31: (pr0 u2 x)).(ex_intro2 T (\lambda (t8: T).(pr0 -(THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 (THead (Flat Appl) u2 -(THead (Bind Abst) u t3)) t8)) (THead (Bind Abbr) x t4) (pr0_comp v2 x H30 t4 -t4 (pr0_refl t4) (Bind Abbr)) (pr0_beta u u2 x H31 t3 t4 H8))))) (H28 u1 -(tlt_head_sx (Flat Appl) u1 (THead (Bind Abst) u t3)) v2 H29 u2 H14))))) t6 -H26)) t7 (sym_eq T t7 (THead (Bind Abst) u t3) H24) H25))) | (pr0_comp u0 u3 -H24 t7 t8 H25 k0) \Rightarrow (\lambda (H26: (eq T (THead k0 u0 t7) (THead -(Bind Abst) u t3))).(\lambda (H27: (eq T (THead k0 u3 t8) t6)).((let H28 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow t7 | (TLRef _) \Rightarrow t7 | (THead _ _ t9) -\Rightarrow t9])) (THead k0 u0 t7) (THead (Bind Abst) u t3) H26) in ((let H29 -\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t9 _) -\Rightarrow t9])) (THead k0 u0 t7) (THead (Bind Abst) u t3) H26) in ((let H30 -\def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) -with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _ _) -\Rightarrow k1])) (THead k0 u0 t7) (THead (Bind Abst) u t3) H26) in (eq_ind K -(Bind Abst) (\lambda (k1: K).((eq T u0 u) \to ((eq T t7 t3) \to ((eq T (THead -k1 u3 t8) t6) \to ((pr0 u0 u3) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: -T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Flat -Appl) u2 t6) t9))))))))) (\lambda (H31: (eq T u0 u)).(eq_ind T u (\lambda -(t9: T).((eq T t7 t3) \to ((eq T (THead (Bind Abst) u3 t8) t6) \to ((pr0 t9 -u3) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 -t4) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t6) t10)))))))) -(\lambda (H32: (eq T t7 t3)).(eq_ind T t3 (\lambda (t9: T).((eq T (THead -(Bind Abst) u3 t8) t6) \to ((pr0 u u3) \to ((pr0 t9 t8) \to (ex2 T (\lambda -(t10: T).(pr0 (THead (Bind Abbr) v2 t4) t10)) (\lambda (t10: T).(pr0 (THead -(Flat Appl) u2 t6) t10))))))) (\lambda (H33: (eq T (THead (Bind Abst) u3 t8) -t6)).(eq_ind T (THead (Bind Abst) u3 t8) (\lambda (t9: T).((pr0 u u3) \to -((pr0 t3 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t4) -t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t9) t10)))))) (\lambda (_: -(pr0 u u3)).(\lambda (H35: (pr0 t3 t8)).(let H36 \def (eq_ind_r T t5 (\lambda -(t9: T).(eq T (THead (Flat Appl) u1 t9) t)) H22 (THead (Bind Abst) u t3) H19) -in (let H37 \def (eq_ind_r T t (\lambda (t9: T).(\forall (v: T).((tlt v t9) -\to (\forall (t10: T).((pr0 v t10) \to (\forall (t11: T).((pr0 v t11) \to -(ex2 T (\lambda (t12: T).(pr0 t10 t12)) (\lambda (t12: T).(pr0 t11 -t12)))))))))) H (THead (Flat Appl) u1 (THead (Bind Abst) u t3)) H36) in (let -H38 \def (eq_ind T v1 (\lambda (t9: T).(pr0 t9 v2)) H7 u1 H20) in (ex2_ind T -(\lambda (t9: T).(pr0 v2 t9)) (\lambda (t9: T).(pr0 u2 t9)) (ex2 T (\lambda -(t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead -(Flat Appl) u2 (THead (Bind Abst) u3 t8)) t9))) (\lambda (x: T).(\lambda -(H39: (pr0 v2 x)).(\lambda (H40: (pr0 u2 x)).(ex2_ind T (\lambda (t9: T).(pr0 -t8 t9)) (\lambda (t9: T).(pr0 t4 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead -(Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead -(Bind Abst) u3 t8)) t9))) (\lambda (x0: T).(\lambda (H41: (pr0 t8 -x0)).(\lambda (H42: (pr0 t4 x0)).(ex_intro2 T (\lambda (t9: T).(pr0 (THead -(Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead -(Bind Abst) u3 t8)) t9)) (THead (Bind Abbr) x x0) (pr0_comp v2 x H39 t4 x0 -H42 (Bind Abbr)) (pr0_beta u3 u2 x H40 t8 x0 H41))))) (H37 t3 (tlt_trans -(THead (Bind Abst) u t3) t3 (THead (Flat Appl) u1 (THead (Bind Abst) u t3)) -(tlt_head_dx (Bind Abst) u t3) (tlt_head_dx (Flat Appl) u1 (THead (Bind Abst) -u t3))) t8 H35 t4 H8))))) (H37 u1 (tlt_head_sx (Flat Appl) u1 (THead (Bind -Abst) u t3)) v2 H38 u2 H14))))))) t6 H33)) t7 (sym_eq T t7 t3 H32))) u0 -(sym_eq T u0 u H31))) k0 (sym_eq K k0 (Bind Abst) H30))) H29)) H28)) H27 H24 -H25))) | (pr0_beta u0 v0 v3 H24 t7 t8 H25) \Rightarrow (\lambda (H26: (eq T -(THead (Flat Appl) v0 (THead (Bind Abst) u0 t7)) (THead (Bind Abst) u -t3))).(\lambda (H27: (eq T (THead (Bind Abbr) v3 t8) t6)).((let H28 \def -(eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u0 t7)) (\lambda (e: -T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in -K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) -\Rightarrow True])])) I (THead (Bind Abst) u t3) H26) in (False_ind ((eq T -(THead (Bind Abbr) v3 t8) t6) \to ((pr0 v0 v3) \to ((pr0 t7 t8) \to (ex2 T -(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 -(THead (Flat Appl) u2 t6) t9)))))) H28)) H27 H24 H25))) | (pr0_upsilon b H24 -v0 v3 H25 u0 u3 H26 t7 t8 H27) \Rightarrow (\lambda (H28: (eq T (THead (Flat -Appl) v0 (THead (Bind b) u0 t7)) (THead (Bind Abst) u t3))).(\lambda (H29: -(eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v3) t8)) t6)).((let -H30 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind b) u0 t7)) (\lambda (e: -T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in -K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) -\Rightarrow True])])) I (THead (Bind Abst) u t3) H28) in (False_ind ((eq T -(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v3) t8)) t6) \to ((not -(eq B b Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3) \to ((pr0 t7 t8) \to (ex2 T -(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 -(THead (Flat Appl) u2 t6) t9)))))))) H30)) H29 H24 H25 H26 H27))) | -(pr0_delta u0 u3 H24 t7 t8 H25 w H26) \Rightarrow (\lambda (H27: (eq T (THead -(Bind Abbr) u0 t7) (THead (Bind Abst) u t3))).(\lambda (H28: (eq T (THead -(Bind Abbr) u3 w) t6)).((let H29 \def (eq_ind T (THead (Bind Abbr) u0 t7) -(\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow -(match k0 in K return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match -b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst -\Rightarrow False | Void \Rightarrow False]) | (Flat _) \Rightarrow -False])])) I (THead (Bind Abst) u t3) H27) in (False_ind ((eq T (THead (Bind -Abbr) u3 w) t6) \to ((pr0 u0 u3) \to ((pr0 t7 t8) \to ((subst0 O u3 t8 w) \to -(ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: -T).(pr0 (THead (Flat Appl) u2 t6) t9))))))) H29)) H28 H24 H25 H26))) | -(pr0_zeta b H24 t7 t8 H25 u0) \Rightarrow (\lambda (H26: (eq T (THead (Bind -b) u0 (lift (S O) O t7)) (THead (Bind Abst) u t3))).(\lambda (H27: (eq T t8 -t6)).((let H28 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat -\to nat))) (d: nat) (t9: T) on t9: T \def (match t9 with [(TSort n) -\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with -[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k0 u3 t10) -\Rightarrow (THead k0 (lref_map f d u3) (lref_map f (s k0 d) t10))]) in -lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (TLRef _) \Rightarrow -((let rec lref_map (f: ((nat \to nat))) (d: nat) (t9: T) on t9: T \def (match -t9 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef -(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | -(THead k0 u3 t10) \Rightarrow (THead k0 (lref_map f d u3) (lref_map f (s k0 -d) t10))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (THead _ _ -t9) \Rightarrow t9])) (THead (Bind b) u0 (lift (S O) O t7)) (THead (Bind -Abst) u t3) H26) in ((let H29 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) -\Rightarrow u0 | (THead _ t9 _) \Rightarrow t9])) (THead (Bind b) u0 (lift (S -O) O t7)) (THead (Bind Abst) u t3) H26) in ((let H30 \def (f_equal T B -(\lambda (e: T).(match e in T return (\lambda (_: T).B) with [(TSort _) -\Rightarrow b | (TLRef _) \Rightarrow b | (THead k0 _ _) \Rightarrow (match -k0 in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) -\Rightarrow b])])) (THead (Bind b) u0 (lift (S O) O t7)) (THead (Bind Abst) u -t3) H26) in (eq_ind B Abst (\lambda (b0: B).((eq T u0 u) \to ((eq T (lift (S -O) O t7) t3) \to ((eq T t8 t6) \to ((not (eq B b0 Abst)) \to ((pr0 t7 t8) \to -(ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: -T).(pr0 (THead (Flat Appl) u2 t6) t9))))))))) (\lambda (H31: (eq T u0 -u)).(eq_ind T u (\lambda (_: T).((eq T (lift (S O) O t7) t3) \to ((eq T t8 -t6) \to ((not (eq B Abst Abst)) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: -T).(pr0 (THead (Bind Abbr) v2 t4) t10)) (\lambda (t10: T).(pr0 (THead (Flat -Appl) u2 t6) t10)))))))) (\lambda (H32: (eq T (lift (S O) O t7) t3)).(eq_ind -T (lift (S O) O t7) (\lambda (_: T).((eq T t8 t6) \to ((not (eq B Abst Abst)) -\to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t4) -t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t6) t10))))))) (\lambda -(H33: (eq T t8 t6)).(eq_ind T t6 (\lambda (t9: T).((not (eq B Abst Abst)) \to -((pr0 t7 t9) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t4) -t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t6) t10)))))) (\lambda -(H34: (not (eq B Abst Abst))).(\lambda (_: (pr0 t7 t6)).(let H36 \def (match -(H34 (refl_equal B Abst)) in False return (\lambda (_: False).(ex2 T (\lambda -(t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead -(Flat Appl) u2 t6) t9)))) with []) in H36))) t8 (sym_eq T t8 t6 H33))) t3 -H32)) u0 (sym_eq T u0 u H31))) b (sym_eq B b Abst H30))) H29)) H28)) H27 H24 -H25))) | (pr0_tau t7 t8 H24 u0) \Rightarrow (\lambda (H25: (eq T (THead (Flat -Cast) u0 t7) (THead (Bind Abst) u t3))).(\lambda (H26: (eq T t8 t6)).((let -H27 \def (eq_ind T (THead (Flat Cast) u0 t7) (\lambda (e: T).(match e in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind Abst) u t3) H25) in (False_ind ((eq T t8 t6) \to -((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) -(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t6) t9))))) H27)) H26 H24)))]) in -(H24 (refl_equal T (THead (Bind Abst) u t3)) (refl_equal T t6))))) k H21)))) -H18)) H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_beta u0 v0 v3 H9 t5 t6 -H10) \Rightarrow (\lambda (H11: (eq T (THead (Flat Appl) v0 (THead (Bind -Abst) u0 t5)) t)).(\lambda (H12: (eq T (THead (Bind Abbr) v3 t6) t2)).(eq_ind -T (THead (Flat Appl) v0 (THead (Bind Abst) u0 t5)) (\lambda (_: T).((eq T -(THead (Bind Abbr) v3 t6) t2) \to ((pr0 v0 v3) \to ((pr0 t5 t6) \to (ex2 T -(\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 -t8))))))) (\lambda (H13: (eq T (THead (Bind Abbr) v3 t6) t2)).(eq_ind T -(THead (Bind Abbr) v3 t6) (\lambda (t7: T).((pr0 v0 v3) \to ((pr0 t5 t6) \to -(ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: -T).(pr0 t7 t8)))))) (\lambda (H14: (pr0 v0 v3)).(\lambda (H15: (pr0 t5 -t6)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1 -(THead (Bind Abst) u t3)) t7)) H4 (THead (Flat Appl) v0 (THead (Bind Abst) u0 -t5)) H11) in (let H17 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 -| (THead _ t7 _) \Rightarrow t7])) (THead (Flat Appl) v1 (THead (Bind Abst) u -t3)) (THead (Flat Appl) v0 (THead (Bind Abst) u0 t5)) H16) in ((let H18 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ _ t7) -\Rightarrow (match t7 in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t8 _) \Rightarrow t8])])) -(THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (THead (Flat Appl) v0 (THead -(Bind Abst) u0 t5)) H16) in ((let H19 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | -(TLRef _) \Rightarrow t3 | (THead _ _ t7) \Rightarrow (match t7 in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 -| (THead _ _ t8) \Rightarrow t8])])) (THead (Flat Appl) v1 (THead (Bind Abst) -u t3)) (THead (Flat Appl) v0 (THead (Bind Abst) u0 t5)) H16) in (\lambda (_: -(eq T u u0)).(\lambda (H21: (eq T v1 v0)).(let H22 \def (eq_ind_r T t -(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) -\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) -(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Appl) v0 (THead (Bind -Abst) u0 t5)) H11) in (let H23 \def (eq_ind T v1 (\lambda (t7: T).(pr0 t7 -v2)) H7 v0 H21) in (let H24 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) -H8 t5 H19) in (ex2_ind T (\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 -t6 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda -(t7: T).(pr0 (THead (Bind Abbr) v3 t6) t7))) (\lambda (x: T).(\lambda (H25: -(pr0 t4 x)).(\lambda (H26: (pr0 t6 x)).(ex2_ind T (\lambda (t7: T).(pr0 v2 -t7)) (\lambda (t7: T).(pr0 v3 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind -Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v3 t6) t7))) -(\lambda (x0: T).(\lambda (H27: (pr0 v2 x0)).(\lambda (H28: (pr0 v3 -x0)).(ex_intro2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) -(\lambda (t7: T).(pr0 (THead (Bind Abbr) v3 t6) t7)) (THead (Bind Abbr) x0 x) -(pr0_comp v2 x0 H27 t4 x H25 (Bind Abbr)) (pr0_comp v3 x0 H28 t6 x H26 (Bind -Abbr)))))) (H22 v0 (tlt_head_sx (Flat Appl) v0 (THead (Bind Abst) u0 t5)) v2 -H23 v3 H14))))) (H22 t5 (tlt_trans (THead (Bind Abst) u0 t5) t5 (THead (Flat -Appl) v0 (THead (Bind Abst) u0 t5)) (tlt_head_dx (Bind Abst) u0 t5) -(tlt_head_dx (Flat Appl) v0 (THead (Bind Abst) u0 t5))) t4 H24 t6 H15)))))))) -H18)) H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_upsilon b H9 v0 v3 H10 u1 -u2 H11 t5 t6 H12) \Rightarrow (\lambda (H13: (eq T (THead (Flat Appl) v0 -(THead (Bind b) u1 t5)) t)).(\lambda (H14: (eq T (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v3) t6)) t2)).(eq_ind T (THead (Flat Appl) v0 -(THead (Bind b) u1 t5)) (\lambda (_: T).((eq T (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v3) t6)) t2) \to ((not (eq B b Abst)) \to ((pr0 v0 -v3) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead -(Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H15: -(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v3) t6)) -t2)).(eq_ind T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v3) t6)) -(\lambda (t7: T).((not (eq B b Abst)) \to ((pr0 v0 v3) \to ((pr0 u1 u2) \to -((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) -(\lambda (t8: T).(pr0 t7 t8)))))))) (\lambda (H16: (not (eq B b -Abst))).(\lambda (_: (pr0 v0 v3)).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (pr0 -t5 t6)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) -v1 (THead (Bind Abst) u t3)) t7)) H4 (THead (Flat Appl) v0 (THead (Bind b) u1 -t5)) H13) in (let H21 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 -| (THead _ t7 _) \Rightarrow t7])) (THead (Flat Appl) v1 (THead (Bind Abst) u -t3)) (THead (Flat Appl) v0 (THead (Bind b) u1 t5)) H20) in ((let H22 \def -(f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with -[(TSort _) \Rightarrow Abst | (TLRef _) \Rightarrow Abst | (THead _ _ t7) -\Rightarrow (match t7 in T return (\lambda (_: T).B) with [(TSort _) -\Rightarrow Abst | (TLRef _) \Rightarrow Abst | (THead k _ _) \Rightarrow -(match k in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | -(Flat _) \Rightarrow Abst])])])) (THead (Flat Appl) v1 (THead (Bind Abst) u -t3)) (THead (Flat Appl) v0 (THead (Bind b) u1 t5)) H20) in ((let H23 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ _ t7) -\Rightarrow (match t7 in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t8 _) \Rightarrow t8])])) -(THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (THead (Flat Appl) v0 (THead -(Bind b) u1 t5)) H20) in ((let H24 \def (f_equal T T (\lambda (e: T).(match e -in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) -\Rightarrow t3 | (THead _ _ t7) \Rightarrow (match t7 in T return (\lambda -(_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead -_ _ t8) \Rightarrow t8])])) (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) -(THead (Flat Appl) v0 (THead (Bind b) u1 t5)) H20) in (\lambda (_: (eq T u -u1)).(\lambda (H26: (eq B Abst b)).(\lambda (_: (eq T v1 v0)).(eq_ind B Abst -(\lambda (b0: B).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) -(\lambda (t7: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O -v3) t6)) t7)))) (let H28 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 -Abst))) H16 Abst H26) in (let H29 \def (match (H28 (refl_equal B Abst)) in -False return (\lambda (_: False).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind -Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abst) u2 (THead (Flat -Appl) (lift (S O) O v3) t6)) t7)))) with []) in H29)) b H26))))) H23)) H22)) -H21))))))) t2 H15)) t H13 H14 H9 H10 H11 H12))) | (pr0_delta u1 u2 H9 t5 t6 -H10 w H11) \Rightarrow (\lambda (H12: (eq T (THead (Bind Abbr) u1 t5) -t)).(\lambda (H13: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind -Abbr) u1 t5) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t2) \to ((pr0 u1 -u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T (\lambda (t8: T).(pr0 -(THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8)))))))) (\lambda -(H14: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u2 w) -(\lambda (t7: T).((pr0 u1 u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to -(ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: -T).(pr0 t7 t8))))))) (\lambda (_: (pr0 u1 u2)).(\lambda (_: (pr0 t5 -t6)).(\lambda (_: (subst0 O u2 t6 w)).(let H18 \def (eq_ind_r T t (\lambda -(t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) t7)) H4 (THead -(Bind Abbr) u1 t5) H12) in (let H19 \def (eq_ind T (THead (Flat Appl) v1 -(THead (Bind Abst) u t3)) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind -Abbr) u1 t5) H18) in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind -Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7))) -H19)))))) t2 H14)) t H12 H13 H9 H10 H11))) | (pr0_zeta b H9 t5 t6 H10 u0) -\Rightarrow (\lambda (H11: (eq T (THead (Bind b) u0 (lift (S O) O t5)) -t)).(\lambda (H12: (eq T t6 t2)).(eq_ind T (THead (Bind b) u0 (lift (S O) O -t5)) (\lambda (_: T).((eq T t6 t2) \to ((not (eq B b Abst)) \to ((pr0 t5 t6) -\to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: -T).(pr0 t2 t8))))))) (\lambda (H13: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: -T).((not (eq B b Abst)) \to ((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 -(THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (_: -(not (eq B b Abst))).(\lambda (_: (pr0 t5 t2)).(let H16 \def (eq_ind_r T t -(\lambda (t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) t7)) -H4 (THead (Bind b) u0 (lift (S O) O t5)) H11) in (let H17 \def (eq_ind T -(THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (\lambda (ee: T).(match ee in -T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind b) u0 (lift (S O) O t5)) H16) in (False_ind (ex2 T -(\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 t2 -t7))) H17))))) t6 (sym_eq T t6 t2 H13))) t H11 H12 H9 H10))) | (pr0_tau t5 t6 -H9 u0) \Rightarrow (\lambda (H10: (eq T (THead (Flat Cast) u0 t5) -t)).(\lambda (H11: (eq T t6 t2)).(eq_ind T (THead (Flat Cast) u0 t5) (\lambda -(_: T).((eq T t6 t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead -(Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H12: (eq -T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8: -T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))) -(\lambda (_: (pr0 t5 t2)).(let H14 \def (eq_ind_r T t (\lambda (t7: T).(eq T -(THead (Flat Appl) v1 (THead (Bind Abst) u t3)) t7)) H4 (THead (Flat Cast) u0 -t5) H10) in (let H15 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) -u t3)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) -\Rightarrow False | (Flat f) \Rightarrow (match f in F return (\lambda (_: -F).Prop) with [Appl \Rightarrow True | Cast \Rightarrow False])])])) I (THead -(Flat Cast) u0 t5) H14) in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead -(Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 t2 t7))) H15)))) t6 (sym_eq T -t6 t2 H12))) t H10 H11 H9)))]) in (H9 (refl_equal T t) (refl_equal T t2))))) -t1 H6)) t H4 H5 H2 H3))) | (pr0_upsilon b H2 v1 v2 H3 u1 u2 H4 t3 t4 H5) -\Rightarrow (\lambda (H6: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) -t)).(\lambda (H7: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O -v2) t4)) t1)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) -(\lambda (_: T).((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O -v2) t4)) t1) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to -((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 -t2 t6))))))))) (\lambda (H8: (eq T (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4)) t1)).(eq_ind T (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4)) (\lambda (t5: T).((not (eq B b Abst)) \to ((pr0 v1 v2) -\to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t5 t6)) -(\lambda (t6: T).(pr0 t2 t6)))))))) (\lambda (H9: (not (eq B b -Abst))).(\lambda (H10: (pr0 v1 v2)).(\lambda (H11: (pr0 u1 u2)).(\lambda -(H12: (pr0 t3 t4)).(let H13 \def (match H1 in pr0 return (\lambda (t5: -T).(\lambda (t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 t) \to ((eq T t6 t2) -\to (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift -(S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 t2 t7)))))))) with [(pr0_refl t5) -\Rightarrow (\lambda (H13: (eq T t5 t)).(\lambda (H14: (eq T t5 t2)).(eq_ind -T t (\lambda (t6: T).((eq T t6 t2) \to (ex2 T (\lambda (t7: T).(pr0 (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: -T).(pr0 t2 t7))))) (\lambda (H15: (eq T t t2)).(eq_ind T t2 (\lambda (_: -T).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift -(S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let H16 \def (eq_ind_r -T t (\lambda (t6: T).(eq T t6 t2)) H15 (THead (Flat Appl) v1 (THead (Bind b) -u1 t3)) H6) in (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) -(\lambda (t6: T).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 t6 t7)))) (let H17 -\def (eq_ind_r T t (\lambda (t6: T).(eq T t5 t6)) H13 (THead (Flat Appl) v1 -(THead (Bind b) u1 t3)) H6) in (let H18 \def (eq_ind_r T t (\lambda (t6: -T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall -(t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: -T).(pr0 t8 t9)))))))))) H (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) H6) -in (ex2_sym T (pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 t3))) (pr0 (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) -(pr0_confluence__pr0_cong_upsilon_refl b H9 u1 u2 H11 t3 t4 H12 v1 v2 v2 H10 -(pr0_refl v2))))) t2 H16)) t (sym_eq T t t2 H15))) t5 (sym_eq T t5 t H13) -H14))) | (pr0_comp u0 u3 H13 t5 t6 H14 k) \Rightarrow (\lambda (H15: (eq T -(THead k u0 t5) t)).(\lambda (H16: (eq T (THead k u3 t6) t2)).(eq_ind T -(THead k u0 t5) (\lambda (_: T).((eq T (THead k u3 t6) t2) \to ((pr0 u0 u3) -\to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) -(\lambda (H17: (eq T (THead k u3 t6) t2)).(eq_ind T (THead k u3 t6) (\lambda -(t7: T).((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: -T).(pr0 t7 t8)))))) (\lambda (H18: (pr0 u0 u3)).(\lambda (H19: (pr0 t5 -t6)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1 -(THead (Bind b) u1 t3)) t7)) H6 (THead k u0 t5) H15) in (let H21 \def -(f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with -[(TSort _) \Rightarrow (Flat Appl) | (TLRef _) \Rightarrow (Flat Appl) | -(THead k0 _ _) \Rightarrow k0])) (THead (Flat Appl) v1 (THead (Bind b) u1 -t3)) (THead k u0 t5) H20) in ((let H22 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | -(TLRef _) \Rightarrow v1 | (THead _ t7 _) \Rightarrow t7])) (THead (Flat -Appl) v1 (THead (Bind b) u1 t3)) (THead k u0 t5) H20) in ((let H23 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow (THead (Bind b) u1 t3) | (TLRef _) \Rightarrow (THead -(Bind b) u1 t3) | (THead _ _ t7) \Rightarrow t7])) (THead (Flat Appl) v1 -(THead (Bind b) u1 t3)) (THead k u0 t5) H20) in (\lambda (H24: (eq T v1 -u0)).(\lambda (H25: (eq K (Flat Appl) k)).(eq_ind K (Flat Appl) (\lambda (k0: -K).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift -(S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 (THead k0 u3 t6) t7)))) (let H26 -\def (eq_ind_r K k (\lambda (k0: K).(eq T (THead k0 u0 t5) t)) H15 (Flat -Appl) H25) in (let H27 \def (eq_ind_r T t5 (\lambda (t7: T).(pr0 t7 t6)) H19 -(THead (Bind b) u1 t3) H23) in (let H28 \def (match H27 in pr0 return -(\lambda (t7: T).(\lambda (t8: T).(\lambda (_: (pr0 t7 t8)).((eq T t7 (THead -(Bind b) u1 t3)) \to ((eq T t8 t6) \to (ex2 T (\lambda (t9: T).(pr0 (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: -T).(pr0 (THead (Flat Appl) u3 t6) t9)))))))) with [(pr0_refl t7) \Rightarrow -(\lambda (H28: (eq T t7 (THead (Bind b) u1 t3))).(\lambda (H29: (eq T t7 -t6)).(eq_ind T (THead (Bind b) u1 t3) (\lambda (t8: T).((eq T t8 t6) \to (ex2 -T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O -v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9))))) -(\lambda (H30: (eq T (THead (Bind b) u1 t3) t6)).(eq_ind T (THead (Bind b) u1 -t3) (\lambda (t8: T).(ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat -Appl) u3 t8) t9)))) (let H31 \def (eq_ind_r T t5 (\lambda (t8: T).(eq T -(THead (Flat Appl) u0 t8) t)) H26 (THead (Bind b) u1 t3) H23) in (let H32 -\def (eq_ind_r T t (\lambda (t8: T).(\forall (v: T).((tlt v t8) \to (\forall -(t9: T).((pr0 v t9) \to (\forall (t10: T).((pr0 v t10) \to (ex2 T (\lambda -(t11: T).(pr0 t9 t11)) (\lambda (t11: T).(pr0 t10 t11)))))))))) H (THead -(Flat Appl) u0 (THead (Bind b) u1 t3)) H31) in (let H33 \def (eq_ind T v1 -(\lambda (t8: T).(pr0 t8 v2)) H10 u0 H24) in (ex2_ind T (\lambda (t8: T).(pr0 -v2 t8)) (\lambda (t8: T).(pr0 u3 t8)) (ex2 T (\lambda (t8: T).(pr0 (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: -T).(pr0 (THead (Flat Appl) u3 (THead (Bind b) u1 t3)) t8))) (\lambda (x: -T).(\lambda (H34: (pr0 v2 x)).(\lambda (H35: (pr0 u3 x)).(ex2_sym T (pr0 -(THead (Flat Appl) u3 (THead (Bind b) u1 t3))) (pr0 (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t4))) (pr0_confluence__pr0_cong_upsilon_refl b -H9 u1 u2 H11 t3 t4 H12 u3 v2 x H35 H34))))) (H32 u0 (tlt_head_sx (Flat Appl) -u0 (THead (Bind b) u1 t3)) v2 H33 u3 H18))))) t6 H30)) t7 (sym_eq T t7 (THead -(Bind b) u1 t3) H28) H29))) | (pr0_comp u4 u5 H28 t7 t8 H29 k0) \Rightarrow -(\lambda (H30: (eq T (THead k0 u4 t7) (THead (Bind b) u1 t3))).(\lambda (H31: -(eq T (THead k0 u5 t8) t6)).((let H32 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t7 | -(TLRef _) \Rightarrow t7 | (THead _ _ t9) \Rightarrow t9])) (THead k0 u4 t7) -(THead (Bind b) u1 t3) H30) in ((let H33 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u4 | -(TLRef _) \Rightarrow u4 | (THead _ t9 _) \Rightarrow t9])) (THead k0 u4 t7) -(THead (Bind b) u1 t3) H30) in ((let H34 \def (f_equal T K (\lambda (e: -T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k0 | -(TLRef _) \Rightarrow k0 | (THead k1 _ _) \Rightarrow k1])) (THead k0 u4 t7) -(THead (Bind b) u1 t3) H30) in (eq_ind K (Bind b) (\lambda (k1: K).((eq T u4 -u1) \to ((eq T t7 t3) \to ((eq T (THead k1 u5 t8) t6) \to ((pr0 u4 u5) \to -((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 -t6) t9))))))))) (\lambda (H35: (eq T u4 u1)).(eq_ind T u1 (\lambda (t9: -T).((eq T t7 t3) \to ((eq T (THead (Bind b) u5 t8) t6) \to ((pr0 t9 u5) \to -((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat -Appl) u3 t6) t10)))))))) (\lambda (H36: (eq T t7 t3)).(eq_ind T t3 (\lambda -(t9: T).((eq T (THead (Bind b) u5 t8) t6) \to ((pr0 u1 u5) \to ((pr0 t9 t8) -\to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift -(S O) O v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6) -t10))))))) (\lambda (H37: (eq T (THead (Bind b) u5 t8) t6)).(eq_ind T (THead -(Bind b) u5 t8) (\lambda (t9: T).((pr0 u1 u5) \to ((pr0 t3 t8) \to (ex2 T -(\lambda (t10: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O -v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t9) t10)))))) -(\lambda (H38: (pr0 u1 u5)).(\lambda (H39: (pr0 t3 t8)).(let H40 \def -(eq_ind_r T t5 (\lambda (t9: T).(eq T (THead (Flat Appl) u0 t9) t)) H26 -(THead (Bind b) u1 t3) H23) in (let H41 \def (eq_ind_r T t (\lambda (t9: -T).(\forall (v: T).((tlt v t9) \to (\forall (t10: T).((pr0 v t10) \to -(\forall (t11: T).((pr0 v t11) \to (ex2 T (\lambda (t12: T).(pr0 t10 t12)) -(\lambda (t12: T).(pr0 t11 t12)))))))))) H (THead (Flat Appl) u0 (THead (Bind -b) u1 t3)) H40) in (let H42 \def (eq_ind T v1 (\lambda (t9: T).(pr0 t9 v2)) -H10 u0 H24) in (ex2_ind T (\lambda (t9: T).(pr0 v2 t9)) (\lambda (t9: T).(pr0 -u3 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 -(THead (Bind b) u5 t8)) t9))) (\lambda (x: T).(\lambda (H43: (pr0 v2 -x)).(\lambda (H44: (pr0 u3 x)).(ex2_ind T (\lambda (t9: T).(pr0 t8 t9)) -(\lambda (t9: T).(pr0 t4 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead -(Flat Appl) u3 (THead (Bind b) u5 t8)) t9))) (\lambda (x0: T).(\lambda (H45: -(pr0 t8 x0)).(\lambda (H46: (pr0 t4 x0)).(ex2_ind T (\lambda (t9: T).(pr0 u5 -t9)) (\lambda (t9: T).(pr0 u2 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Bind -b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 -(THead (Flat Appl) u3 (THead (Bind b) u5 t8)) t9))) (\lambda (x1: T).(\lambda -(H47: (pr0 u5 x1)).(\lambda (H48: (pr0 u2 x1)).(ex2_sym T (pr0 (THead (Flat -Appl) u3 (THead (Bind b) u5 t8))) (pr0 (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4))) (pr0_confluence__pr0_cong_upsilon_cong b H9 u3 v2 x -H44 H43 t8 t4 x0 H45 H46 u5 u2 x1 H47 H48))))) (H41 u1 (tlt_trans (THead -(Bind b) u1 t3) u1 (THead (Flat Appl) u0 (THead (Bind b) u1 t3)) (tlt_head_sx -(Bind b) u1 t3) (tlt_head_dx (Flat Appl) u0 (THead (Bind b) u1 t3))) u5 H38 -u2 H11))))) (H41 t3 (tlt_trans (THead (Bind b) u1 t3) t3 (THead (Flat Appl) -u0 (THead (Bind b) u1 t3)) (tlt_head_dx (Bind b) u1 t3) (tlt_head_dx (Flat -Appl) u0 (THead (Bind b) u1 t3))) t8 H39 t4 H12))))) (H41 u0 (tlt_head_sx -(Flat Appl) u0 (THead (Bind b) u1 t3)) v2 H42 u3 H18))))))) t6 H37)) t7 -(sym_eq T t7 t3 H36))) u4 (sym_eq T u4 u1 H35))) k0 (sym_eq K k0 (Bind b) -H34))) H33)) H32)) H31 H28 H29))) | (pr0_beta u v0 v3 H28 t7 t8 H29) -\Rightarrow (\lambda (H30: (eq T (THead (Flat Appl) v0 (THead (Bind Abst) u -t7)) (THead (Bind b) u1 t3))).(\lambda (H31: (eq T (THead (Bind Abbr) v3 t8) -t6)).((let H32 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u t7)) -(\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow -(match k0 in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False -| (Flat _) \Rightarrow True])])) I (THead (Bind b) u1 t3) H30) in (False_ind -((eq T (THead (Bind Abbr) v3 t8) t6) \to ((pr0 v0 v3) \to ((pr0 t7 t8) \to -(ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9)))))) -H32)) H31 H28 H29))) | (pr0_upsilon b0 H28 v0 v3 H29 u4 u5 H30 t7 t8 H31) -\Rightarrow (\lambda (H32: (eq T (THead (Flat Appl) v0 (THead (Bind b0) u4 -t7)) (THead (Bind b) u1 t3))).(\lambda (H33: (eq T (THead (Bind b0) u5 (THead -(Flat Appl) (lift (S O) O v3) t8)) t6)).((let H34 \def (eq_ind T (THead (Flat -Appl) v0 (THead (Bind b0) u4 t7)) (\lambda (e: T).(match e in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind b) u1 t3) H32) in (False_ind ((eq T (THead (Bind b0) -u5 (THead (Flat Appl) (lift (S O) O v3) t8)) t6) \to ((not (eq B b0 Abst)) -\to ((pr0 v0 v3) \to ((pr0 u4 u5) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: -T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) -(\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9)))))))) H34)) H33 H28 H29 -H30 H31))) | (pr0_delta u4 u5 H28 t7 t8 H29 w H30) \Rightarrow (\lambda (H31: -(eq T (THead (Bind Abbr) u4 t7) (THead (Bind b) u1 t3))).(\lambda (H32: (eq T -(THead (Bind Abbr) u5 w) t6)).((let H33 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t7 | -(TLRef _) \Rightarrow t7 | (THead _ _ t9) \Rightarrow t9])) (THead (Bind -Abbr) u4 t7) (THead (Bind b) u1 t3) H31) in ((let H34 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow u4 | (TLRef _) \Rightarrow u4 | (THead _ t9 _) \Rightarrow t9])) -(THead (Bind Abbr) u4 t7) (THead (Bind b) u1 t3) H31) in ((let H35 \def -(f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with -[(TSort _) \Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | (THead k0 _ _) -\Rightarrow (match k0 in K return (\lambda (_: K).B) with [(Bind b0) -\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (THead (Bind Abbr) u4 t7) -(THead (Bind b) u1 t3) H31) in (eq_ind B Abbr (\lambda (b0: B).((eq T u4 u1) -\to ((eq T t7 t3) \to ((eq T (THead (Bind Abbr) u5 w) t6) \to ((pr0 u4 u5) -\to ((pr0 t7 t8) \to ((subst0 O u5 t8 w) \to (ex2 T (\lambda (t9: T).(pr0 -(THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda -(t9: T).(pr0 (THead (Flat Appl) u3 t6) t9)))))))))) (\lambda (H36: (eq T u4 -u1)).(eq_ind T u1 (\lambda (t9: T).((eq T t7 t3) \to ((eq T (THead (Bind -Abbr) u5 w) t6) \to ((pr0 t9 u5) \to ((pr0 t7 t8) \to ((subst0 O u5 t8 w) \to -(ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift -(S O) O v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6) -t10))))))))) (\lambda (H37: (eq T t7 t3)).(eq_ind T t3 (\lambda (t9: T).((eq -T (THead (Bind Abbr) u5 w) t6) \to ((pr0 u1 u5) \to ((pr0 t9 t8) \to ((subst0 -O u5 t8 w) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) u2 (THead -(Flat Appl) (lift (S O) O v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat -Appl) u3 t6) t10)))))))) (\lambda (H38: (eq T (THead (Bind Abbr) u5 w) -t6)).(eq_ind T (THead (Bind Abbr) u5 w) (\lambda (t9: T).((pr0 u1 u5) \to -((pr0 t3 t8) \to ((subst0 O u5 t8 w) \to (ex2 T (\lambda (t10: T).(pr0 (THead -(Bind Abbr) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t10)) (\lambda (t10: -T).(pr0 (THead (Flat Appl) u3 t9) t10))))))) (\lambda (H39: (pr0 u1 -u5)).(\lambda (H40: (pr0 t3 t8)).(\lambda (H41: (subst0 O u5 t8 w)).(let H42 -\def (eq_ind_r B b (\lambda (b0: B).(eq T (THead (Bind b0) u1 t3) t5)) H23 -Abbr H35) in (let H43 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 -Abst))) H9 Abbr H35) in (let H44 \def (eq_ind_r B b (\lambda (b0: B).(eq T -(THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t1)) H8 Abbr -H35) in (let H45 \def (eq_ind_r T t5 (\lambda (t9: T).(eq T (THead (Flat -Appl) u0 t9) t)) H26 (THead (Bind Abbr) u1 t3) H42) in (let H46 \def -(eq_ind_r T t (\lambda (t9: T).(\forall (v: T).((tlt v t9) \to (\forall (t10: -T).((pr0 v t10) \to (\forall (t11: T).((pr0 v t11) \to (ex2 T (\lambda (t12: -T).(pr0 t10 t12)) (\lambda (t12: T).(pr0 t11 t12)))))))))) H (THead (Flat -Appl) u0 (THead (Bind Abbr) u1 t3)) H45) in (let H47 \def (eq_ind T v1 -(\lambda (t9: T).(pr0 t9 v2)) H10 u0 H24) in (ex2_ind T (\lambda (t9: T).(pr0 -v2 t9)) (\lambda (t9: T).(pr0 u3 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead -(Bind Abbr) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: -T).(pr0 (THead (Flat Appl) u3 (THead (Bind Abbr) u5 w)) t9))) (\lambda (x: -T).(\lambda (H48: (pr0 v2 x)).(\lambda (H49: (pr0 u3 x)).(ex2_ind T (\lambda -(t9: T).(pr0 t8 t9)) (\lambda (t9: T).(pr0 t4 t9)) (ex2 T (\lambda (t9: -T).(pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) -(\lambda (t9: T).(pr0 (THead (Flat Appl) u3 (THead (Bind Abbr) u5 w)) t9))) -(\lambda (x0: T).(\lambda (H50: (pr0 t8 x0)).(\lambda (H51: (pr0 t4 -x0)).(ex2_ind T (\lambda (t9: T).(pr0 u5 t9)) (\lambda (t9: T).(pr0 u2 t9)) -(ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift -(S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 (THead -(Bind Abbr) u5 w)) t9))) (\lambda (x1: T).(\lambda (H52: (pr0 u5 -x1)).(\lambda (H53: (pr0 u2 x1)).(ex2_sym T (pr0 (THead (Flat Appl) u3 (THead -(Bind Abbr) u5 w))) (pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift (S O) -O v2) t4))) (pr0_confluence__pr0_cong_upsilon_delta H43 u5 t8 w H41 u3 v2 x -H49 H48 t4 x0 H50 H51 u2 x1 H52 H53))))) (H46 u1 (tlt_trans (THead (Bind -Abbr) u1 t3) u1 (THead (Flat Appl) u0 (THead (Bind Abbr) u1 t3)) (tlt_head_sx -(Bind Abbr) u1 t3) (tlt_head_dx (Flat Appl) u0 (THead (Bind Abbr) u1 t3))) u5 -H39 u2 H11))))) (H46 t3 (tlt_trans (THead (Bind Abbr) u1 t3) t3 (THead (Flat -Appl) u0 (THead (Bind Abbr) u1 t3)) (tlt_head_dx (Bind Abbr) u1 t3) -(tlt_head_dx (Flat Appl) u0 (THead (Bind Abbr) u1 t3))) t8 H40 t4 H12))))) -(H46 u0 (tlt_head_sx (Flat Appl) u0 (THead (Bind Abbr) u1 t3)) v2 H47 u3 -H18))))))))))) t6 H38)) t7 (sym_eq T t7 t3 H37))) u4 (sym_eq T u4 u1 H36))) b -H35)) H34)) H33)) H32 H28 H29 H30))) | (pr0_zeta b0 H28 t7 t8 H29 u) -\Rightarrow (\lambda (H30: (eq T (THead (Bind b0) u (lift (S O) O t7)) (THead -(Bind b) u1 t3))).(\lambda (H31: (eq T t8 t6)).((let H32 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t9: T) on t9: T -\def (match t9 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow -(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) -| (THead k0 u4 t10) \Rightarrow (THead k0 (lref_map f d u4) (lref_map f (s k0 -d) t10))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (TLRef _) -\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t9: T) on t9: T -\def (match t9 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow -(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) -| (THead k0 u4 t10) \Rightarrow (THead k0 (lref_map f d u4) (lref_map f (s k0 -d) t10))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (THead _ _ -t9) \Rightarrow t9])) (THead (Bind b0) u (lift (S O) O t7)) (THead (Bind b) -u1 t3) H30) in ((let H33 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) -\Rightarrow u | (THead _ t9 _) \Rightarrow t9])) (THead (Bind b0) u (lift (S -O) O t7)) (THead (Bind b) u1 t3) H30) in ((let H34 \def (f_equal T B (\lambda -(e: T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b0 -| (TLRef _) \Rightarrow b0 | (THead k0 _ _) \Rightarrow (match k0 in K return -(\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow -b0])])) (THead (Bind b0) u (lift (S O) O t7)) (THead (Bind b) u1 t3) H30) in -(eq_ind B b (\lambda (b1: B).((eq T u u1) \to ((eq T (lift (S O) O t7) t3) -\to ((eq T t8 t6) \to ((not (eq B b1 Abst)) \to ((pr0 t7 t8) \to (ex2 T -(\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) -t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9))))))))) -(\lambda (H35: (eq T u u1)).(eq_ind T u1 (\lambda (_: T).((eq T (lift (S O) O -t7) t3) \to ((eq T t8 t6) \to ((not (eq B b Abst)) \to ((pr0 t7 t8) \to (ex2 -T (\lambda (t10: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O -v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6) t10)))))))) -(\lambda (H36: (eq T (lift (S O) O t7) t3)).(eq_ind T (lift (S O) O t7) -(\lambda (_: T).((eq T t8 t6) \to ((not (eq B b Abst)) \to ((pr0 t7 t8) \to -(ex2 T (\lambda (t10: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6) -t10))))))) (\lambda (H37: (eq T t8 t6)).(eq_ind T t6 (\lambda (t9: T).((not -(eq B b Abst)) \to ((pr0 t7 t9) \to (ex2 T (\lambda (t10: T).(pr0 (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t10)) (\lambda (t10: -T).(pr0 (THead (Flat Appl) u3 t6) t10)))))) (\lambda (H38: (not (eq B b -Abst))).(\lambda (H39: (pr0 t7 t6)).(let H40 \def (eq_ind_r T t3 (\lambda -(t9: T).(eq T (THead (Bind b) u1 t9) t5)) H23 (lift (S O) O t7) H36) in (let -H41 \def (eq_ind_r T t5 (\lambda (t9: T).(eq T (THead (Flat Appl) u0 t9) t)) -H26 (THead (Bind b) u1 (lift (S O) O t7)) H40) in (let H42 \def (eq_ind_r T t -(\lambda (t9: T).(\forall (v: T).((tlt v t9) \to (\forall (t10: T).((pr0 v -t10) \to (\forall (t11: T).((pr0 v t11) \to (ex2 T (\lambda (t12: T).(pr0 t10 -t12)) (\lambda (t12: T).(pr0 t11 t12)))))))))) H (THead (Flat Appl) u0 (THead -(Bind b) u1 (lift (S O) O t7))) H41) in (let H43 \def (eq_ind_r T t3 (\lambda -(t9: T).(pr0 t9 t4)) H12 (lift (S O) O t7) H36) in (ex2_ind T (\lambda (t9: -T).(eq T t4 (lift (S O) O t9))) (\lambda (t9: T).(pr0 t7 t9)) (ex2 T (\lambda -(t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9))) (\lambda (x: -T).(\lambda (H44: (eq T t4 (lift (S O) O x))).(\lambda (H45: (pr0 t7 -x)).(eq_ind_r T (lift (S O) O x) (\lambda (t9: T).(ex2 T (\lambda (t10: -T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t9)) t10)) -(\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6) t10)))) (let H46 \def -(eq_ind T v1 (\lambda (t9: T).(pr0 t9 v2)) H10 u0 H24) in (ex2_ind T (\lambda -(t9: T).(pr0 v2 t9)) (\lambda (t9: T).(pr0 u3 t9)) (ex2 T (\lambda (t9: -T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (lift (S O) O -x))) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9))) (\lambda (x0: -T).(\lambda (H47: (pr0 v2 x0)).(\lambda (H48: (pr0 u3 x0)).(ex2_ind T -(\lambda (t9: T).(pr0 x t9)) (\lambda (t9: T).(pr0 t6 t9)) (ex2 T (\lambda -(t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (lift (S -O) O x))) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9))) (\lambda -(x1: T).(\lambda (H49: (pr0 x x1)).(\lambda (H50: (pr0 t6 x1)).(ex2_sym T -(pr0 (THead (Flat Appl) u3 t6)) (pr0 (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) (lift (S O) O x)))) (pr0_confluence__pr0_cong_upsilon_zeta -b H38 u1 u2 H11 u3 v2 x0 H48 H47 x t6 x1 H49 H50))))) (H42 t7 (tlt_trans -(THead (Bind b) u1 (lift (S O) O t7)) t7 (THead (Flat Appl) u0 (THead (Bind -b) u1 (lift (S O) O t7))) (lift_tlt_dx (Bind b) u1 t7 (S O) O) (tlt_head_dx -(Flat Appl) u0 (THead (Bind b) u1 (lift (S O) O t7)))) x H45 t6 H39))))) (H42 -u0 (tlt_head_sx (Flat Appl) u0 (THead (Bind b) u1 (lift (S O) O t7))) v2 H46 -u3 H18))) t4 H44)))) (pr0_gen_lift t7 t4 (S O) O H43)))))))) t8 (sym_eq T t8 -t6 H37))) t3 H36)) u (sym_eq T u u1 H35))) b0 (sym_eq B b0 b H34))) H33)) -H32)) H31 H28 H29))) | (pr0_tau t7 t8 H28 u) \Rightarrow (\lambda (H29: (eq T -(THead (Flat Cast) u t7) (THead (Bind b) u1 t3))).(\lambda (H30: (eq T t8 -t6)).((let H31 \def (eq_ind T (THead (Flat Cast) u t7) (\lambda (e: T).(match -e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | -(TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K -return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) -\Rightarrow True])])) I (THead (Bind b) u1 t3) H29) in (False_ind ((eq T t8 -t6) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead -(Flat Appl) u3 t6) t9))))) H31)) H30 H28)))]) in (H28 (refl_equal T (THead -(Bind b) u1 t3)) (refl_equal T t6))))) k H25)))) H22)) H21))))) t2 H17)) t -H15 H16 H13 H14))) | (pr0_beta u v0 v3 H13 t5 t6 H14) \Rightarrow (\lambda -(H15: (eq T (THead (Flat Appl) v0 (THead (Bind Abst) u t5)) t)).(\lambda -(H16: (eq T (THead (Bind Abbr) v3 t6) t2)).(eq_ind T (THead (Flat Appl) v0 -(THead (Bind Abst) u t5)) (\lambda (_: T).((eq T (THead (Bind Abbr) v3 t6) -t2) \to ((pr0 v0 v3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: -T).(pr0 t2 t8))))))) (\lambda (H17: (eq T (THead (Bind Abbr) v3 t6) -t2)).(eq_ind T (THead (Bind Abbr) v3 t6) (\lambda (t7: T).((pr0 v0 v3) \to -((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda -(_: (pr0 v0 v3)).(\lambda (_: (pr0 t5 t6)).(let H20 \def (eq_ind_r T t -(\lambda (t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) t7)) H6 -(THead (Flat Appl) v0 (THead (Bind Abst) u t5)) H15) in (let H21 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t7 _) -\Rightarrow t7])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat -Appl) v0 (THead (Bind Abst) u t5)) H20) in ((let H22 \def (f_equal T B -(\lambda (e: T).(match e in T return (\lambda (_: T).B) with [(TSort _) -\Rightarrow b | (TLRef _) \Rightarrow b | (THead _ _ t7) \Rightarrow (match -t7 in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | (TLRef _) -\Rightarrow b | (THead k _ _) \Rightarrow (match k in K return (\lambda (_: -K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow b])])])) (THead -(Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) v0 (THead (Bind -Abst) u t5)) H20) in ((let H23 \def (f_equal T T (\lambda (e: T).(match e in -T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) -\Rightarrow u1 | (THead _ _ t7) \Rightarrow (match t7 in T return (\lambda -(_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead -_ t8 _) \Rightarrow t8])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) -(THead (Flat Appl) v0 (THead (Bind Abst) u t5)) H20) in ((let H24 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7) -\Rightarrow (match t7 in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t8) \Rightarrow -t8])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) v0 -(THead (Bind Abst) u t5)) H20) in (\lambda (_: (eq T u1 u)).(\lambda (H26: -(eq B b Abst)).(\lambda (H27: (eq T v1 v0)).(let H28 \def (eq_ind_r T t -(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) -\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) -(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Appl) v0 (THead (Bind -Abst) u t5)) H15) in (let H29 \def (eq_ind T v1 (\lambda (t7: T).(pr0 t7 v2)) -H10 v0 H27) in (eq_ind_r B Abst (\lambda (b0: B).(ex2 T (\lambda (t7: T).(pr0 -(THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda -(t7: T).(pr0 (THead (Bind Abbr) v3 t6) t7)))) (let H30 \def (eq_ind B b -(\lambda (b0: B).(not (eq B b0 Abst))) H9 Abst H26) in (let H31 \def (match -(H30 (refl_equal B Abst)) in False return (\lambda (_: False).(ex2 T (\lambda -(t7: T).(pr0 (THead (Bind Abst) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v3 t6) t7)))) with []) in H31)) -b H26))))))) H23)) H22)) H21))))) t2 H17)) t H15 H16 H13 H14))) | -(pr0_upsilon b0 H13 v0 v3 H14 u0 u3 H15 t5 t6 H16) \Rightarrow (\lambda (H17: -(eq T (THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) t)).(\lambda (H18: (eq T -(THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) t2)).(eq_ind T -(THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) (\lambda (_: T).((eq T (THead -(Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) t2) \to ((not (eq B b0 -Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda -(t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -t8)) (\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H19: (eq T (THead (Bind -b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) t2)).(eq_ind T (THead (Bind -b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) (\lambda (t7: T).((not (eq B -b0 Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T -(\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) -t4)) t8)) (\lambda (t8: T).(pr0 t7 t8)))))))) (\lambda (_: (not (eq B b0 -Abst))).(\lambda (H21: (pr0 v0 v3)).(\lambda (H22: (pr0 u0 u3)).(\lambda -(H23: (pr0 t5 t6)).(let H24 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead -(Flat Appl) v1 (THead (Bind b) u1 t3)) t7)) H6 (THead (Flat Appl) v0 (THead -(Bind b0) u0 t5)) H17) in (let H25 \def (f_equal T T (\lambda (e: T).(match e -in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) -\Rightarrow v1 | (THead _ t7 _) \Rightarrow t7])) (THead (Flat Appl) v1 -(THead (Bind b) u1 t3)) (THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) H24) -in ((let H26 \def (f_equal T B (\lambda (e: T).(match e in T return (\lambda -(_: T).B) with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead _ -_ t7) \Rightarrow (match t7 in T return (\lambda (_: T).B) with [(TSort _) -\Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k -in K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) -\Rightarrow b])])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead -(Flat Appl) v0 (THead (Bind b0) u0 t5)) H24) in ((let H27 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ _ t7) \Rightarrow (match -t7 in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) -\Rightarrow u1 | (THead _ t8 _) \Rightarrow t8])])) (THead (Flat Appl) v1 -(THead (Bind b) u1 t3)) (THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) H24) -in ((let H28 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda -(_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead -_ _ t7) \Rightarrow (match t7 in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t8) \Rightarrow -t8])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) v0 -(THead (Bind b0) u0 t5)) H24) in (\lambda (H29: (eq T u1 u0)).(\lambda (H30: -(eq B b b0)).(\lambda (H31: (eq T v1 v0)).(let H32 \def (eq_ind_r T t -(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) -\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) -(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Appl) v0 (THead (Bind -b0) u0 t5)) H17) in (let H33 \def (eq_ind T v1 (\lambda (t7: T).(pr0 t7 v2)) -H10 v0 H31) in (eq_ind_r B b0 (\lambda (b1: B).(ex2 T (\lambda (t7: T).(pr0 -(THead (Bind b1) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda -(t7: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) -t7)))) (let H34 \def (eq_ind B b (\lambda (b1: B).(not (eq B b1 Abst))) H9 b0 -H30) in (let H35 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H11 u0 H29) -in (let H36 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H12 t5 H28) in -(ex2_ind T (\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T -(\lambda (t7: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O -v2) t4)) t7)) (\lambda (t7: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) -(lift (S O) O v3) t6)) t7))) (\lambda (x: T).(\lambda (H37: (pr0 t4 -x)).(\lambda (H38: (pr0 t6 x)).(ex2_ind T (\lambda (t7: T).(pr0 u2 t7)) -(\lambda (t7: T).(pr0 u3 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b0) -u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 -(THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) t7))) (\lambda -(x0: T).(\lambda (H39: (pr0 u2 x0)).(\lambda (H40: (pr0 u3 x0)).(ex2_ind T -(\lambda (t7: T).(pr0 v2 t7)) (\lambda (t7: T).(pr0 v3 t7)) (ex2 T (\lambda -(t7: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -t7)) (\lambda (t7: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) -O v3) t6)) t7))) (\lambda (x1: T).(\lambda (H41: (pr0 v2 x1)).(\lambda (H42: -(pr0 v3 x1)).(pr0_confluence__pr0_upsilon_upsilon b0 H34 v2 v3 x1 H41 H42 u2 -u3 x0 H39 H40 t4 t6 x H37 H38)))) (H32 v0 (tlt_head_sx (Flat Appl) v0 (THead -(Bind b0) u0 t5)) v2 H33 v3 H21))))) (H32 u0 (tlt_trans (THead (Bind b0) u0 -t5) u0 (THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) (tlt_head_sx (Bind b0) -u0 t5) (tlt_head_dx (Flat Appl) v0 (THead (Bind b0) u0 t5))) u2 H35 u3 -H22))))) (H32 t5 (tlt_trans (THead (Bind b0) u0 t5) t5 (THead (Flat Appl) v0 -(THead (Bind b0) u0 t5)) (tlt_head_dx (Bind b0) u0 t5) (tlt_head_dx (Flat -Appl) v0 (THead (Bind b0) u0 t5))) t4 H36 t6 H23))))) b H30))))))) H27)) -H26)) H25))))))) t2 H19)) t H17 H18 H13 H14 H15 H16))) | (pr0_delta u0 u3 H13 -t5 t6 H14 w H15) \Rightarrow (\lambda (H16: (eq T (THead (Bind Abbr) u0 t5) -t)).(\lambda (H17: (eq T (THead (Bind Abbr) u3 w) t2)).(eq_ind T (THead (Bind -Abbr) u0 t5) (\lambda (_: T).((eq T (THead (Bind Abbr) u3 w) t2) \to ((pr0 u0 -u3) \to ((pr0 t5 t6) \to ((subst0 O u3 t6 w) \to (ex2 T (\lambda (t8: T).(pr0 -(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda -(t8: T).(pr0 t2 t8)))))))) (\lambda (H18: (eq T (THead (Bind Abbr) u3 w) -t2)).(eq_ind T (THead (Bind Abbr) u3 w) (\lambda (t7: T).((pr0 u0 u3) \to -((pr0 t5 t6) \to ((subst0 O u3 t6 w) \to (ex2 T (\lambda (t8: T).(pr0 (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: -T).(pr0 t7 t8))))))) (\lambda (_: (pr0 u0 u3)).(\lambda (_: (pr0 t5 -t6)).(\lambda (_: (subst0 O u3 t6 w)).(let H22 \def (eq_ind_r T t (\lambda -(t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) t7)) H6 (THead -(Bind Abbr) u0 t5) H16) in (let H23 \def (eq_ind T (THead (Flat Appl) v1 -(THead (Bind b) u1 t3)) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind -Abbr) u0 t5) H22) in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 -(THead (Bind Abbr) u3 w) t7))) H23)))))) t2 H18)) t H16 H17 H13 H14 H15))) | -(pr0_zeta b0 H13 t5 t6 H14 u) \Rightarrow (\lambda (H15: (eq T (THead (Bind -b0) u (lift (S O) O t5)) t)).(\lambda (H16: (eq T t6 t2)).(eq_ind T (THead -(Bind b0) u (lift (S O) O t5)) (\lambda (_: T).((eq T t6 t2) \to ((not (eq B -b0 Abst)) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t2 -t8))))))) (\lambda (H17: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((not -(eq B b0 Abst)) \to ((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: -T).(pr0 t2 t8)))))) (\lambda (_: (not (eq B b0 Abst))).(\lambda (_: (pr0 t5 -t2)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1 -(THead (Bind b) u1 t3)) t7)) H6 (THead (Bind b0) u (lift (S O) O t5)) H15) in -(let H21 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) -(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow -(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | -(Flat _) \Rightarrow True])])) I (THead (Bind b0) u (lift (S O) O t5)) H20) -in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 t2 t7))) H21))))) t6 -(sym_eq T t6 t2 H17))) t H15 H16 H13 H14))) | (pr0_tau t5 t6 H13 u) -\Rightarrow (\lambda (H14: (eq T (THead (Flat Cast) u t5) t)).(\lambda (H15: -(eq T t6 t2)).(eq_ind T (THead (Flat Cast) u t5) (\lambda (_: T).((eq T t6 -t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t2 -t8)))))) (\lambda (H16: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5 -t7) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t2 t8))))) (\lambda (_: -(pr0 t5 t2)).(let H18 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat -Appl) v1 (THead (Bind b) u1 t3)) t7)) H6 (THead (Flat Cast) u t5) H14) in -(let H19 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) -(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow -(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | -(Flat f) \Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl -\Rightarrow True | Cast \Rightarrow False])])])) I (THead (Flat Cast) u t5) -H18) in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 t2 t7))) -H19)))) t6 (sym_eq T t6 t2 H16))) t H14 H15 H13)))]) in (H13 (refl_equal T t) -(refl_equal T t2))))))) t1 H8)) t H6 H7 H2 H3 H4 H5))) | (pr0_delta u1 u2 H2 -t3 t4 H3 w H4) \Rightarrow (\lambda (H5: (eq T (THead (Bind Abbr) u1 t3) -t)).(\lambda (H6: (eq T (THead (Bind Abbr) u2 w) t1)).(eq_ind T (THead (Bind -Abbr) u1 t3) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t1) \to ((pr0 u1 -u2) \to ((pr0 t3 t4) \to ((subst0 O u2 t4 w) \to (ex2 T (\lambda (t6: T).(pr0 -t1 t6)) (\lambda (t6: T).(pr0 t2 t6)))))))) (\lambda (H7: (eq T (THead (Bind -Abbr) u2 w) t1)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t5: T).((pr0 u1 -u2) \to ((pr0 t3 t4) \to ((subst0 O u2 t4 w) \to (ex2 T (\lambda (t6: T).(pr0 -t5 t6)) (\lambda (t6: T).(pr0 t2 t6))))))) (\lambda (H8: (pr0 u1 -u2)).(\lambda (H9: (pr0 t3 t4)).(\lambda (H10: (subst0 O u2 t4 w)).(let H11 -\def (match H1 in pr0 return (\lambda (t5: T).(\lambda (t6: T).(\lambda (_: -(pr0 t5 t6)).((eq T t5 t) \to ((eq T t6 t2) \to (ex2 T (\lambda (t7: T).(pr0 -(THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7)))))))) with -[(pr0_refl t5) \Rightarrow (\lambda (H11: (eq T t5 t)).(\lambda (H12: (eq T -t5 t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2) \to (ex2 T (\lambda (t7: -T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7))))) -(\lambda (H13: (eq T t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t7: -T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let -H14 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 t2)) H13 (THead (Bind Abbr) -u1 t3) H5) in (eq_ind T (THead (Bind Abbr) u1 t3) (\lambda (t6: T).(ex2 T -(\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t6 -t7)))) (let H15 \def (eq_ind_r T t (\lambda (t6: T).(eq T t5 t6)) H11 (THead -(Bind Abbr) u1 t3) H5) in (let H16 \def (eq_ind_r T t (\lambda (t6: -T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall -(t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: -T).(pr0 t8 t9)))))))))) H (THead (Bind Abbr) u1 t3) H5) in (ex_intro2 T -(\lambda (t6: T).(pr0 (THead (Bind Abbr) u2 w) t6)) (\lambda (t6: T).(pr0 -(THead (Bind Abbr) u1 t3) t6)) (THead (Bind Abbr) u2 w) (pr0_refl (THead -(Bind Abbr) u2 w)) (pr0_delta u1 u2 H8 t3 t4 H9 w H10)))) t2 H14)) t (sym_eq -T t t2 H13))) t5 (sym_eq T t5 t H11) H12))) | (pr0_comp u0 u3 H11 t5 t6 H12 -k) \Rightarrow (\lambda (H13: (eq T (THead k u0 t5) t)).(\lambda (H14: (eq T -(THead k u3 t6) t2)).(eq_ind T (THead k u0 t5) (\lambda (_: T).((eq T (THead -k u3 t6) t2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: -T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) -(\lambda (H15: (eq T (THead k u3 t6) t2)).(eq_ind T (THead k u3 t6) (\lambda -(t7: T).((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead -(Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (H16: (pr0 -u0 u3)).(\lambda (H17: (pr0 t5 t6)).(let H18 \def (eq_ind_r T t (\lambda (t7: -T).(eq T (THead (Bind Abbr) u1 t3) t7)) H5 (THead k u0 t5) H13) in (let H19 -\def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) -with [(TSort _) \Rightarrow (Bind Abbr) | (TLRef _) \Rightarrow (Bind Abbr) | -(THead k0 _ _) \Rightarrow k0])) (THead (Bind Abbr) u1 t3) (THead k u0 t5) -H18) in ((let H20 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 -| (THead _ t7 _) \Rightarrow t7])) (THead (Bind Abbr) u1 t3) (THead k u0 t5) -H18) in ((let H21 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 -| (THead _ _ t7) \Rightarrow t7])) (THead (Bind Abbr) u1 t3) (THead k u0 t5) -H18) in (\lambda (H22: (eq T u1 u0)).(\lambda (H23: (eq K (Bind Abbr) -k)).(eq_ind K (Bind Abbr) (\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 -(THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 (THead k0 u3 t6) t7)))) -(let H24 \def (eq_ind_r K k (\lambda (k0: K).(eq T (THead k0 u0 t5) t)) H13 -(Bind Abbr) H23) in (let H25 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: -T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v -t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 -t10)))))))))) H (THead (Bind Abbr) u0 t5) H24) in (let H26 \def (eq_ind T u1 -(\lambda (t7: T).(pr0 t7 u2)) H8 u0 H22) in (let H27 \def (eq_ind T t3 -(\lambda (t7: T).(pr0 t7 t4)) H9 t5 H21) in (ex2_ind T (\lambda (t7: T).(pr0 -t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead -(Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u3 t6) t7))) -(\lambda (x: T).(\lambda (H28: (pr0 t4 x)).(\lambda (H29: (pr0 t6 -x)).(ex2_ind T (\lambda (t7: T).(pr0 u2 t7)) (\lambda (t7: T).(pr0 u3 t7)) -(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: -T).(pr0 (THead (Bind Abbr) u3 t6) t7))) (\lambda (x0: T).(\lambda (H30: (pr0 -u2 x0)).(\lambda (H31: (pr0 u3 x0)).(ex2_sym T (pr0 (THead (Bind Abbr) u3 -t6)) (pr0 (THead (Bind Abbr) u2 w)) (pr0_confluence__pr0_cong_delta u2 t4 w -H10 u3 x0 H31 H30 t6 x H29 H28))))) (H25 u0 (tlt_head_sx (Bind Abbr) u0 t5) -u2 H26 u3 H16))))) (H25 t5 (tlt_head_dx (Bind Abbr) u0 t5) t4 H27 t6 -H17)))))) k H23)))) H20)) H19))))) t2 H15)) t H13 H14 H11 H12))) | (pr0_beta -u v1 v2 H11 t5 t6 H12) \Rightarrow (\lambda (H13: (eq T (THead (Flat Appl) v1 -(THead (Bind Abst) u t5)) t)).(\lambda (H14: (eq T (THead (Bind Abbr) v2 t6) -t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) (\lambda (_: -T).((eq T (THead (Bind Abbr) v2 t6) t2) \to ((pr0 v1 v2) \to ((pr0 t5 t6) \to -(ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: -T).(pr0 t2 t8))))))) (\lambda (H15: (eq T (THead (Bind Abbr) v2 t6) -t2)).(eq_ind T (THead (Bind Abbr) v2 t6) (\lambda (t7: T).((pr0 v1 v2) \to -((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) -(\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 -t5 t6)).(let H18 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind Abbr) -u1 t3) t7)) H5 (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) H13) in (let -H19 \def (eq_ind T (THead (Bind Abbr) u1 t3) (\lambda (ee: T).(match ee in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow -False])])) I (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) H18) in -(False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) -(\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t6) t7))) H19))))) t2 H15)) t H13 -H14 H11 H12))) | (pr0_upsilon b H11 v1 v2 H12 u0 u3 H13 t5 t6 H14) -\Rightarrow (\lambda (H15: (eq T (THead (Flat Appl) v1 (THead (Bind b) u0 -t5)) t)).(\lambda (H16: (eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S -O) O v2) t6)) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u0 t5)) -(\lambda (_: T).((eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O -v2) t6)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u3) \to -((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) -(\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H17: (eq T (THead (Bind b) u3 -(THead (Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Bind b) u3 -(THead (Flat Appl) (lift (S O) O v2) t6)) (\lambda (t7: T).((not (eq B b -Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda -(t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t7 -t8)))))))) (\lambda (_: (not (eq B b Abst))).(\lambda (_: (pr0 v1 -v2)).(\lambda (_: (pr0 u0 u3)).(\lambda (_: (pr0 t5 t6)).(let H22 \def -(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind Abbr) u1 t3) t7)) H5 (THead -(Flat Appl) v1 (THead (Bind b) u0 t5)) H15) in (let H23 \def (eq_ind T (THead -(Bind Abbr) u1 t3) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat -Appl) v1 (THead (Bind b) u0 t5)) H22) in (False_ind (ex2 T (\lambda (t7: -T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 (THead (Bind b) -u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t7))) H23))))))) t2 H17)) t H15 -H16 H11 H12 H13 H14))) | (pr0_delta u0 u3 H11 t5 t6 H12 w0 H13) \Rightarrow -(\lambda (H14: (eq T (THead (Bind Abbr) u0 t5) t)).(\lambda (H15: (eq T -(THead (Bind Abbr) u3 w0) t2)).(eq_ind T (THead (Bind Abbr) u0 t5) (\lambda -(_: T).((eq T (THead (Bind Abbr) u3 w0) t2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) -\to ((subst0 O u3 t6 w0) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) -u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8)))))))) (\lambda (H16: (eq T (THead -(Bind Abbr) u3 w0) t2)).(eq_ind T (THead (Bind Abbr) u3 w0) (\lambda (t7: -T).((pr0 u0 u3) \to ((pr0 t5 t6) \to ((subst0 O u3 t6 w0) \to (ex2 T (\lambda -(t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t7 t8))))))) -(\lambda (H17: (pr0 u0 u3)).(\lambda (H18: (pr0 t5 t6)).(\lambda (H19: -(subst0 O u3 t6 w0)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T -(THead (Bind Abbr) u1 t3) t7)) H5 (THead (Bind Abbr) u0 t5) H14) in (let H21 -\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t7 _) -\Rightarrow t7])) (THead (Bind Abbr) u1 t3) (THead (Bind Abbr) u0 t5) H20) in -((let H22 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: -T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ -t7) \Rightarrow t7])) (THead (Bind Abbr) u1 t3) (THead (Bind Abbr) u0 t5) -H20) in (\lambda (H23: (eq T u1 u0)).(let H24 \def (eq_ind_r T t (\lambda -(t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to -(\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) -(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Bind Abbr) u0 t5) H14) in -(let H25 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H8 u0 H23) in (let -H26 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H9 t5 H22) in (ex2_ind T -(\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda -(t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 (THead (Bind -Abbr) u3 w0) t7))) (\lambda (x: T).(\lambda (H27: (pr0 t4 x)).(\lambda (H28: -(pr0 t6 x)).(ex2_ind T (\lambda (t7: T).(pr0 u2 t7)) (\lambda (t7: T).(pr0 u3 -t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: -T).(pr0 (THead (Bind Abbr) u3 w0) t7))) (\lambda (x0: T).(\lambda (H29: (pr0 -u2 x0)).(\lambda (H30: (pr0 u3 x0)).(pr0_confluence__pr0_delta_delta u2 t4 w -H10 u3 t6 w0 H19 x0 H29 H30 x H27 H28)))) (H24 u0 (tlt_head_sx (Bind Abbr) u0 -t5) u2 H25 u3 H17))))) (H24 t5 (tlt_head_dx (Bind Abbr) u0 t5) t4 H26 t6 -H18))))))) H21)))))) t2 H16)) t H14 H15 H11 H12 H13))) | (pr0_zeta b H11 t5 -t6 H12 u) \Rightarrow (\lambda (H13: (eq T (THead (Bind b) u (lift (S O) O -t5)) t)).(\lambda (H14: (eq T t6 t2)).(eq_ind T (THead (Bind b) u (lift (S O) -O t5)) (\lambda (_: T).((eq T t6 t2) \to ((not (eq B b Abst)) \to ((pr0 t5 -t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda -(t8: T).(pr0 t2 t8))))))) (\lambda (H15: (eq T t6 t2)).(eq_ind T t2 (\lambda -(t7: T).((not (eq B b Abst)) \to ((pr0 t5 t7) \to (ex2 T (\lambda (t8: -T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) -(\lambda (H16: (not (eq B b Abst))).(\lambda (H17: (pr0 t5 t2)).(let H18 \def -(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind Abbr) u1 t3) t7)) H5 (THead -(Bind b) u (lift (S O) O t5)) H13) in (let H19 \def (f_equal T B (\lambda (e: -T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow Abbr | -(TLRef _) \Rightarrow Abbr | (THead k _ _) \Rightarrow (match k in K return -(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow -Abbr])])) (THead (Bind Abbr) u1 t3) (THead (Bind b) u (lift (S O) O t5)) H18) -in ((let H20 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda -(_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead -_ t7 _) \Rightarrow t7])) (THead (Bind Abbr) u1 t3) (THead (Bind b) u (lift -(S O) O t5)) H18) in ((let H21 \def (f_equal T T (\lambda (e: T).(match e in -T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) -\Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead (Bind Abbr) u1 t3) -(THead (Bind b) u (lift (S O) O t5)) H18) in (\lambda (H22: (eq T u1 -u)).(\lambda (H23: (eq B Abbr b)).(let H24 \def (eq_ind_r B b (\lambda (b0: -B).(not (eq B b0 Abst))) H16 Abbr H23) in (let H25 \def (eq_ind_r B b -(\lambda (b0: B).(eq T (THead (Bind b0) u (lift (S O) O t5)) t)) H13 Abbr -H23) in (let H26 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v -t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to -(ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 -t10)))))))))) H (THead (Bind Abbr) u (lift (S O) O t5)) H25) in (let H27 \def -(eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H8 u H22) in (let H28 \def (eq_ind -T t3 (\lambda (t7: T).(pr0 t7 t4)) H9 (lift (S O) O t5) H21) in (ex2_ind T -(\lambda (t7: T).(eq T t4 (lift (S O) O t7))) (\lambda (t7: T).(pr0 t5 t7)) -(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: -T).(pr0 t2 t7))) (\lambda (x: T).(\lambda (H29: (eq T t4 (lift (S O) O -x))).(\lambda (H30: (pr0 t5 x)).(let H31 \def (eq_ind T t4 (\lambda (t7: -T).(subst0 O u2 t7 w)) H10 (lift (S O) O x) H29) in (ex2_ind T (\lambda (t7: -T).(pr0 x t7)) (\lambda (t7: T).(pr0 t2 t7)) (ex2 T (\lambda (t7: T).(pr0 -(THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7))) (\lambda (x0: -T).(\lambda (_: (pr0 x x0)).(\lambda (_: (pr0 t2 -x0)).(pr0_confluence__pr0_delta_tau u2 (lift (S O) O x) w H31 x (pr0_refl -(lift (S O) O x)) t2)))) (H26 t5 (lift_tlt_dx (Bind Abbr) u t5 (S O) O) x H30 -t2 H17)))))) (pr0_gen_lift t5 t4 (S O) O H28)))))))))) H20)) H19))))) t6 -(sym_eq T t6 t2 H15))) t H13 H14 H11 H12))) | (pr0_tau t5 t6 H11 u) -\Rightarrow (\lambda (H12: (eq T (THead (Flat Cast) u t5) t)).(\lambda (H13: -(eq T t6 t2)).(eq_ind T (THead (Flat Cast) u t5) (\lambda (_: T).((eq T t6 -t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 -w) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H14: (eq T t6 -t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8: -T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8))))) -(\lambda (_: (pr0 t5 t2)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T -(THead (Bind Abbr) u1 t3) t7)) H5 (THead (Flat Cast) u t5) H12) in (let H17 -\def (eq_ind T (THead (Bind Abbr) u1 t3) (\lambda (ee: T).(match ee in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow -False])])) I (THead (Flat Cast) u t5) H16) in (False_ind (ex2 T (\lambda (t7: -T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7))) H17)))) -t6 (sym_eq T t6 t2 H14))) t H12 H13 H11)))]) in (H11 (refl_equal T t) -(refl_equal T t2)))))) t1 H7)) t H5 H6 H2 H3 H4))) | (pr0_zeta b H2 t3 t4 H3 -u) \Rightarrow (\lambda (H4: (eq T (THead (Bind b) u (lift (S O) O t3)) -t)).(\lambda (H5: (eq T t4 t1)).(eq_ind T (THead (Bind b) u (lift (S O) O -t3)) (\lambda (_: T).((eq T t4 t1) \to ((not (eq B b Abst)) \to ((pr0 t3 t4) -\to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6))))))) -(\lambda (H6: (eq T t4 t1)).(eq_ind T t1 (\lambda (t5: T).((not (eq B b -Abst)) \to ((pr0 t3 t5) \to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda -(t6: T).(pr0 t2 t6)))))) (\lambda (H7: (not (eq B b Abst))).(\lambda (H8: -(pr0 t3 t1)).(let H9 \def (match H1 in pr0 return (\lambda (t5: T).(\lambda -(t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 t) \to ((eq T t6 t2) \to (ex2 T -(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)))))))) with -[(pr0_refl t5) \Rightarrow (\lambda (H9: (eq T t5 t)).(\lambda (H10: (eq T t5 -t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2) \to (ex2 T (\lambda (t7: -T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))))) (\lambda (H11: (eq T t -t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t7: T).(pr0 t1 t7)) -(\lambda (t7: T).(pr0 t2 t7)))) (let H12 \def (eq_ind_r T t (\lambda (t6: -T).(eq T t6 t2)) H11 (THead (Bind b) u (lift (S O) O t3)) H4) in (eq_ind T -(THead (Bind b) u (lift (S O) O t3)) (\lambda (t6: T).(ex2 T (\lambda (t7: -T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t6 t7)))) (let H13 \def (eq_ind_r T t -(\lambda (t6: T).(eq T t5 t6)) H9 (THead (Bind b) u (lift (S O) O t3)) H4) in -(let H14 \def (eq_ind_r T t (\lambda (t6: T).(\forall (v: T).((tlt v t6) \to -(\forall (t7: T).((pr0 v t7) \to (\forall (t8: T).((pr0 v t8) \to (ex2 T -(\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead -(Bind b) u (lift (S O) O t3)) H4) in (ex_intro2 T (\lambda (t6: T).(pr0 t1 -t6)) (\lambda (t6: T).(pr0 (THead (Bind b) u (lift (S O) O t3)) t6)) t1 -(pr0_refl t1) (pr0_zeta b H7 t3 t1 H8 u)))) t2 H12)) t (sym_eq T t t2 H11))) -t5 (sym_eq T t5 t H9) H10))) | (pr0_comp u1 u2 H9 t5 t6 H10 k) \Rightarrow -(\lambda (H11: (eq T (THead k u1 t5) t)).(\lambda (H12: (eq T (THead k u2 t6) -t2)).(eq_ind T (THead k u1 t5) (\lambda (_: T).((eq T (THead k u2 t6) t2) \to -((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) -(\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H13: (eq T (THead k u2 t6) -t2)).(eq_ind T (THead k u2 t6) (\lambda (t7: T).((pr0 u1 u2) \to ((pr0 t5 t6) -\to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 t8)))))) -(\lambda (_: (pr0 u1 u2)).(\lambda (H15: (pr0 t5 t6)).(let H16 \def (eq_ind_r -T t (\lambda (t7: T).(eq T (THead (Bind b) u (lift (S O) O t3)) t7)) H4 -(THead k u1 t5) H11) in (let H17 \def (f_equal T K (\lambda (e: T).(match e -in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow (Bind b) | (TLRef -_) \Rightarrow (Bind b) | (THead k0 _ _) \Rightarrow k0])) (THead (Bind b) u -(lift (S O) O t3)) (THead k u1 t5) H16) in ((let H18 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t7 _) \Rightarrow t7])) -(THead (Bind b) u (lift (S O) O t3)) (THead k u1 t5) H16) in ((let H19 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t7: -T) on t7: T \def (match t7 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) -\Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false -\Rightarrow (f i)])) | (THead k0 u0 t8) \Rightarrow (THead k0 (lref_map f d -u0) (lref_map f (s k0 d) t8))]) in lref_map) (\lambda (x: nat).(plus x (S -O))) O t3) | (TLRef _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) -(d: nat) (t7: T) on t7: T \def (match t7 with [(TSort n) \Rightarrow (TSort -n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | -false \Rightarrow (f i)])) | (THead k0 u0 t8) \Rightarrow (THead k0 (lref_map -f d u0) (lref_map f (s k0 d) t8))]) in lref_map) (\lambda (x: nat).(plus x (S -O))) O t3) | (THead _ _ t7) \Rightarrow t7])) (THead (Bind b) u (lift (S O) O -t3)) (THead k u1 t5) H16) in (\lambda (_: (eq T u u1)).(\lambda (H21: (eq K -(Bind b) k)).(eq_ind K (Bind b) (\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 -t1 t7)) (\lambda (t7: T).(pr0 (THead k0 u2 t6) t7)))) (let H22 \def (eq_ind_r -K k (\lambda (k0: K).(eq T (THead k0 u1 t5) t)) H11 (Bind b) H21) in (let H23 -\def (eq_ind_r T t5 (\lambda (t7: T).(pr0 t7 t6)) H15 (lift (S O) O t3) H19) -in (ex2_ind T (\lambda (t7: T).(eq T t6 (lift (S O) O t7))) (\lambda (t7: -T).(pr0 t3 t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 -(THead (Bind b) u2 t6) t7))) (\lambda (x: T).(\lambda (H24: (eq T t6 (lift (S -O) O x))).(\lambda (H25: (pr0 t3 x)).(let H26 \def (eq_ind_r T t5 (\lambda -(t7: T).(eq T (THead (Bind b) u1 t7) t)) H22 (lift (S O) O t3) H19) in (let -H27 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to -(\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T -(\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H -(THead (Bind b) u1 (lift (S O) O t3)) H26) in (eq_ind_r T (lift (S O) O x) -(\lambda (t7: T).(ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 -(THead (Bind b) u2 t7) t8)))) (ex2_ind T (\lambda (t7: T).(pr0 x t7)) -(\lambda (t7: T).(pr0 t1 t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda -(t7: T).(pr0 (THead (Bind b) u2 (lift (S O) O x)) t7))) (\lambda (x0: -T).(\lambda (H28: (pr0 x x0)).(\lambda (H29: (pr0 t1 x0)).(ex_intro2 T -(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Bind b) u2 (lift -(S O) O x)) t7)) x0 H29 (pr0_zeta b H7 x x0 H28 u2))))) (H27 t3 (lift_tlt_dx -(Bind b) u1 t3 (S O) O) x H25 t1 H8)) t6 H24)))))) (pr0_gen_lift t3 t6 (S O) -O H23)))) k H21)))) H18)) H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_beta -u0 v1 v2 H9 t5 t6 H10) \Rightarrow (\lambda (H11: (eq T (THead (Flat Appl) v1 -(THead (Bind Abst) u0 t5)) t)).(\lambda (H12: (eq T (THead (Bind Abbr) v2 t6) -t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t5)) (\lambda (_: -T).((eq T (THead (Bind Abbr) v2 t6) t2) \to ((pr0 v1 v2) \to ((pr0 t5 t6) \to -(ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))))) -(\lambda (H13: (eq T (THead (Bind Abbr) v2 t6) t2)).(eq_ind T (THead (Bind -Abbr) v2 t6) (\lambda (t7: T).((pr0 v1 v2) \to ((pr0 t5 t6) \to (ex2 T -(\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (_: -(pr0 v1 v2)).(\lambda (_: (pr0 t5 t6)).(let H16 \def (eq_ind_r T t (\lambda -(t7: T).(eq T (THead (Bind b) u (lift (S O) O t3)) t7)) H4 (THead (Flat Appl) -v1 (THead (Bind Abst) u0 t5)) H11) in (let H17 \def (eq_ind T (THead (Bind b) -u (lift (S O) O t3)) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat -Appl) v1 (THead (Bind Abst) u0 t5)) H16) in (False_ind (ex2 T (\lambda (t7: -T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t6) t7))) -H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_upsilon b0 H9 v1 v2 H10 u1 u2 -H11 t5 t6 H12) \Rightarrow (\lambda (H13: (eq T (THead (Flat Appl) v1 (THead -(Bind b0) u1 t5)) t)).(\lambda (H14: (eq T (THead (Bind b0) u2 (THead (Flat -Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Flat Appl) v1 (THead -(Bind b0) u1 t5)) (\lambda (_: T).((eq T (THead (Bind b0) u2 (THead (Flat -Appl) (lift (S O) O v2) t6)) t2) \to ((not (eq B b0 Abst)) \to ((pr0 v1 v2) -\to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) -(\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H15: (eq T (THead (Bind b0) u2 -(THead (Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Bind b0) u2 -(THead (Flat Appl) (lift (S O) O v2) t6)) (\lambda (t7: T).((not (eq B b0 -Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda -(t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 t8)))))))) (\lambda (_: (not -(eq B b0 Abst))).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 u1 u2)).(\lambda -(_: (pr0 t5 t6)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead -(Bind b) u (lift (S O) O t3)) t7)) H4 (THead (Flat Appl) v1 (THead (Bind b0) -u1 t5)) H13) in (let H21 \def (eq_ind T (THead (Bind b) u (lift (S O) O t3)) -(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow -(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | -(Flat _) \Rightarrow False])])) I (THead (Flat Appl) v1 (THead (Bind b0) u1 -t5)) H20) in (False_ind (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: -T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t6)) t7))) -H21))))))) t2 H15)) t H13 H14 H9 H10 H11 H12))) | (pr0_delta u1 u2 H9 t5 t6 -H10 w H11) \Rightarrow (\lambda (H12: (eq T (THead (Bind Abbr) u1 t5) -t)).(\lambda (H13: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind -Abbr) u1 t5) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t2) \to ((pr0 u1 -u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T (\lambda (t8: T).(pr0 -t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))))) (\lambda (H14: (eq T (THead (Bind -Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t7: T).((pr0 u1 -u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T (\lambda (t8: T).(pr0 -t1 t8)) (\lambda (t8: T).(pr0 t7 t8))))))) (\lambda (_: (pr0 u1 u2)).(\lambda -(H16: (pr0 t5 t6)).(\lambda (H17: (subst0 O u2 t6 w)).(let H18 \def (eq_ind_r -T t (\lambda (t7: T).(eq T (THead (Bind b) u (lift (S O) O t3)) t7)) H4 -(THead (Bind Abbr) u1 t5) H12) in (let H19 \def (f_equal T B (\lambda (e: -T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | -(TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k in K return -(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow -b])])) (THead (Bind b) u (lift (S O) O t3)) (THead (Bind Abbr) u1 t5) H18) in -((let H20 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: -T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t7 -_) \Rightarrow t7])) (THead (Bind b) u (lift (S O) O t3)) (THead (Bind Abbr) -u1 t5) H18) in ((let H21 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: -((nat \to nat))) (d: nat) (t7: T) on t7: T \def (match t7 with [(TSort n) -\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with -[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t8) -\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t8))]) in -lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (TLRef _) \Rightarrow -((let rec lref_map (f: ((nat \to nat))) (d: nat) (t7: T) on t7: T \def (match -t7 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef -(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | -(THead k u0 t8) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) -t8))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (THead _ _ t7) -\Rightarrow t7])) (THead (Bind b) u (lift (S O) O t3)) (THead (Bind Abbr) u1 -t5) H18) in (\lambda (_: (eq T u u1)).(\lambda (H23: (eq B b Abbr)).(let H24 -\def (eq_ind_r T t5 (\lambda (t7: T).(pr0 t7 t6)) H16 (lift (S O) O t3) H21) -in (ex2_ind T (\lambda (t7: T).(eq T t6 (lift (S O) O t7))) (\lambda (t7: -T).(pr0 t3 t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 -(THead (Bind Abbr) u2 w) t7))) (\lambda (x: T).(\lambda (H25: (eq T t6 (lift -(S O) O x))).(\lambda (H26: (pr0 t3 x)).(let H27 \def (eq_ind_r T t5 (\lambda -(t7: T).(eq T (THead (Bind Abbr) u1 t7) t)) H12 (lift (S O) O t3) H21) in -(let H28 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to -(\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T -(\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H -(THead (Bind Abbr) u1 (lift (S O) O t3)) H27) in (let H29 \def (eq_ind T t6 -(\lambda (t7: T).(subst0 O u2 t7 w)) H17 (lift (S O) O x) H25) in (let H30 -\def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H7 Abbr H23) in -(ex2_ind T (\lambda (t7: T).(pr0 x t7)) (\lambda (t7: T).(pr0 t1 t7)) (ex2 T -(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) -t7))) (\lambda (x0: T).(\lambda (_: (pr0 x x0)).(\lambda (_: (pr0 t1 -x0)).(ex2_sym T (pr0 (THead (Bind Abbr) u2 w)) (pr0 t1) -(pr0_confluence__pr0_delta_tau u2 (lift (S O) O x) w H29 x (pr0_refl (lift (S -O) O x)) t1))))) (H28 t3 (lift_tlt_dx (Bind Abbr) u1 t3 (S O) O) x H26 t1 -H8))))))))) (pr0_gen_lift t3 t6 (S O) O H24)))))) H20)) H19)))))) t2 H14)) t -H12 H13 H9 H10 H11))) | (pr0_zeta b0 H9 t5 t6 H10 u0) \Rightarrow (\lambda -(H11: (eq T (THead (Bind b0) u0 (lift (S O) O t5)) t)).(\lambda (H12: (eq T -t6 t2)).(eq_ind T (THead (Bind b0) u0 (lift (S O) O t5)) (\lambda (_: T).((eq -T t6 t2) \to ((not (eq B b0 Abst)) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: -T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H13: (eq T t6 -t2)).(eq_ind T t2 (\lambda (t7: T).((not (eq B b0 Abst)) \to ((pr0 t5 t7) \to -(ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))) -(\lambda (_: (not (eq B b0 Abst))).(\lambda (H15: (pr0 t5 t2)).(let H16 \def -(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind b) u (lift (S O) O t3)) -t7)) H4 (THead (Bind b0) u0 (lift (S O) O t5)) H11) in (let H17 \def (f_equal -T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with [(TSort _) -\Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k -in K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) -\Rightarrow b])])) (THead (Bind b) u (lift (S O) O t3)) (THead (Bind b0) u0 -(lift (S O) O t5)) H16) in ((let H18 \def (f_equal T T (\lambda (e: T).(match -e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) -\Rightarrow u | (THead _ t7 _) \Rightarrow t7])) (THead (Bind b) u (lift (S -O) O t3)) (THead (Bind b0) u0 (lift (S O) O t5)) H16) in ((let H19 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t7: -T) on t7: T \def (match t7 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) -\Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false -\Rightarrow (f i)])) | (THead k u1 t8) \Rightarrow (THead k (lref_map f d u1) -(lref_map f (s k d) t8))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O -t3) | (TLRef _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) -(t7: T) on t7: T \def (match t7 with [(TSort n) \Rightarrow (TSort n) | -(TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | -false \Rightarrow (f i)])) | (THead k u1 t8) \Rightarrow (THead k (lref_map f -d u1) (lref_map f (s k d) t8))]) in lref_map) (\lambda (x: nat).(plus x (S -O))) O t3) | (THead _ _ t7) \Rightarrow t7])) (THead (Bind b) u (lift (S O) O -t3)) (THead (Bind b0) u0 (lift (S O) O t5)) H16) in (\lambda (_: (eq T u -u0)).(\lambda (H21: (eq B b b0)).(let H22 \def (eq_ind_r T t (\lambda (t7: -T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall -(t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: -T).(pr0 t9 t10)))))))))) H (THead (Bind b0) u0 (lift (S O) O t5)) H11) in -(let H23 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t1)) H8 t5 (lift_inj t3 -t5 (S O) O H19)) in (let H24 \def (eq_ind B b (\lambda (b1: B).(not (eq B b1 -Abst))) H7 b0 H21) in (ex2_ind T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: -T).(pr0 t2 t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 -t7))) (\lambda (x: T).(\lambda (H25: (pr0 t1 x)).(\lambda (H26: (pr0 t2 -x)).(ex_intro2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)) -x H25 H26)))) (H22 t5 (lift_tlt_dx (Bind b0) u0 t5 (S O) O) t1 H23 t2 -H15)))))))) H18)) H17))))) t6 (sym_eq T t6 t2 H13))) t H11 H12 H9 H10))) | -(pr0_tau t5 t6 H9 u0) \Rightarrow (\lambda (H10: (eq T (THead (Flat Cast) u0 -t5) t)).(\lambda (H11: (eq T t6 t2)).(eq_ind T (THead (Flat Cast) u0 t5) -(\lambda (_: T).((eq T t6 t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: -T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H12: (eq T t6 -t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8: -T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))) (\lambda (_: (pr0 t5 -t2)).(let H14 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind b) u -(lift (S O) O t3)) t7)) H4 (THead (Flat Cast) u0 t5) H10) in (let H15 \def -(eq_ind T (THead (Bind b) u (lift (S O) O t3)) (\lambda (ee: T).(match ee in -T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow -False])])) I (THead (Flat Cast) u0 t5) H14) in (False_ind (ex2 T (\lambda -(t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))) H15)))) t6 (sym_eq T t6 -t2 H12))) t H10 H11 H9)))]) in (H9 (refl_equal T t) (refl_equal T t2))))) t4 -(sym_eq T t4 t1 H6))) t H4 H5 H2 H3))) | (pr0_tau t3 t4 H2 u) \Rightarrow -(\lambda (H3: (eq T (THead (Flat Cast) u t3) t)).(\lambda (H4: (eq T t4 -t1)).(eq_ind T (THead (Flat Cast) u t3) (\lambda (_: T).((eq T t4 t1) \to -((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 -t2 t6)))))) (\lambda (H5: (eq T t4 t1)).(eq_ind T t1 (\lambda (t5: T).((pr0 -t3 t5) \to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 -t6))))) (\lambda (H6: (pr0 t3 t1)).(let H7 \def (match H1 in pr0 return -(\lambda (t5: T).(\lambda (t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 t) \to -((eq T t6 t2) \to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 -t2 t7)))))))) with [(pr0_refl t5) \Rightarrow (\lambda (H7: (eq T t5 -t)).(\lambda (H8: (eq T t5 t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2) -\to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))))) -(\lambda (H9: (eq T t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t7: -T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let H10 \def (eq_ind_r T t -(\lambda (t6: T).(eq T t6 t2)) H9 (THead (Flat Cast) u t3) H3) in (eq_ind T -(THead (Flat Cast) u t3) (\lambda (t6: T).(ex2 T (\lambda (t7: T).(pr0 t1 -t7)) (\lambda (t7: T).(pr0 t6 t7)))) (let H11 \def (eq_ind_r T t (\lambda -(t6: T).(eq T t5 t6)) H7 (THead (Flat Cast) u t3) H3) in (let H12 \def -(eq_ind_r T t (\lambda (t6: T).(\forall (v: T).((tlt v t6) \to (\forall (t7: -T).((pr0 v t7) \to (\forall (t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: -T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead (Flat Cast) u -t3) H3) in (ex_intro2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 -(THead (Flat Cast) u t3) t6)) t1 (pr0_refl t1) (pr0_tau t3 t1 H6 u)))) t2 -H10)) t (sym_eq T t t2 H9))) t5 (sym_eq T t5 t H7) H8))) | (pr0_comp u1 u2 H7 -t5 t6 H8 k) \Rightarrow (\lambda (H9: (eq T (THead k u1 t5) t)).(\lambda -(H10: (eq T (THead k u2 t6) t2)).(eq_ind T (THead k u1 t5) (\lambda (_: -T).((eq T (THead k u2 t6) t2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T -(\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda -(H11: (eq T (THead k u2 t6) t2)).(eq_ind T (THead k u2 t6) (\lambda (t7: -T).((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) -(\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (_: (pr0 u1 u2)).(\lambda (H13: -(pr0 t5 t6)).(let H14 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat -Cast) u t3) t7)) H3 (THead k u1 t5) H9) in (let H15 \def (f_equal T K -(\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) -\Rightarrow (Flat Cast) | (TLRef _) \Rightarrow (Flat Cast) | (THead k0 _ _) -\Rightarrow k0])) (THead (Flat Cast) u t3) (THead k u1 t5) H14) in ((let H16 -\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t7 _) -\Rightarrow t7])) (THead (Flat Cast) u t3) (THead k u1 t5) H14) in ((let H17 -\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7) -\Rightarrow t7])) (THead (Flat Cast) u t3) (THead k u1 t5) H14) in (\lambda -(_: (eq T u u1)).(\lambda (H19: (eq K (Flat Cast) k)).(eq_ind K (Flat Cast) -(\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 -(THead k0 u2 t6) t7)))) (let H20 \def (eq_ind_r K k (\lambda (k0: K).(eq T -(THead k0 u1 t5) t)) H9 (Flat Cast) H19) in (let H21 \def (eq_ind_r T t -(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) -\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) -(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Cast) u1 t5) H20) in -(let H22 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t1)) H6 t5 H17) in -(ex2_ind T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T -(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Flat Cast) u2 t6) -t7))) (\lambda (x: T).(\lambda (H23: (pr0 t1 x)).(\lambda (H24: (pr0 t6 -x)).(ex_intro2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead -(Flat Cast) u2 t6) t7)) x H23 (pr0_tau t6 x H24 u2))))) (H21 t5 (tlt_head_dx -(Flat Cast) u1 t5) t1 H22 t6 H13))))) k H19)))) H16)) H15))))) t2 H11)) t H9 -H10 H7 H8))) | (pr0_beta u0 v1 v2 H7 t5 t6 H8) \Rightarrow (\lambda (H9: (eq -T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t5)) t)).(\lambda (H10: (eq T -(THead (Bind Abbr) v2 t6) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind -Abst) u0 t5)) (\lambda (_: T).((eq T (THead (Bind Abbr) v2 t6) t2) \to ((pr0 -v1 v2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda -(t8: T).(pr0 t2 t8))))))) (\lambda (H11: (eq T (THead (Bind Abbr) v2 t6) -t2)).(eq_ind T (THead (Bind Abbr) v2 t6) (\lambda (t7: T).((pr0 v1 v2) \to -((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 -t7 t8)))))) (\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 t5 t6)).(let H14 \def -(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Cast) u t3) t7)) H3 (THead -(Flat Appl) v1 (THead (Bind Abst) u0 t5)) H9) in (let H15 \def (eq_ind T -(THead (Flat Cast) u t3) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f in F return -(\lambda (_: F).Prop) with [Appl \Rightarrow False | Cast \Rightarrow -True])])])) I (THead (Flat Appl) v1 (THead (Bind Abst) u0 t5)) H14) in -(False_ind (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead -(Bind Abbr) v2 t6) t7))) H15))))) t2 H11)) t H9 H10 H7 H8))) | (pr0_upsilon b -H7 v1 v2 H8 u1 u2 H9 t5 t6 H10) \Rightarrow (\lambda (H11: (eq T (THead (Flat -Appl) v1 (THead (Bind b) u1 t5)) t)).(\lambda (H12: (eq T (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Flat Appl) -v1 (THead (Bind b) u1 t5)) (\lambda (_: T).((eq T (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t6)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1 -v2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 -t8)) (\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H13: (eq T (THead (Bind -b) u2 (THead (Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Bind -b) u2 (THead (Flat Appl) (lift (S O) O v2) t6)) (\lambda (t7: T).((not (eq B -b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T -(\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 t8)))))))) (\lambda -(_: (not (eq B b Abst))).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 u1 -u2)).(\lambda (_: (pr0 t5 t6)).(let H18 \def (eq_ind_r T t (\lambda (t7: -T).(eq T (THead (Flat Cast) u t3) t7)) H3 (THead (Flat Appl) v1 (THead (Bind -b) u1 t5)) H11) in (let H19 \def (eq_ind T (THead (Flat Cast) u t3) (\lambda -(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow -(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | -(Flat f) \Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl -\Rightarrow False | Cast \Rightarrow True])])])) I (THead (Flat Appl) v1 -(THead (Bind b) u1 t5)) H18) in (False_ind (ex2 T (\lambda (t7: T).(pr0 t1 -t7)) (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) -O v2) t6)) t7))) H19))))))) t2 H13)) t H11 H12 H7 H8 H9 H10))) | (pr0_delta -u1 u2 H7 t5 t6 H8 w H9) \Rightarrow (\lambda (H10: (eq T (THead (Bind Abbr) -u1 t5) t)).(\lambda (H11: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T -(THead (Bind Abbr) u1 t5) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t2) -\to ((pr0 u1 u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T (\lambda -(t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))))) (\lambda (H12: (eq T -(THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda -(t7: T).((pr0 u1 u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T -(\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 t8))))))) (\lambda (_: -(pr0 u1 u2)).(\lambda (_: (pr0 t5 t6)).(\lambda (_: (subst0 O u2 t6 w)).(let -H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Cast) u t3) t7)) -H3 (THead (Bind Abbr) u1 t5) H10) in (let H17 \def (eq_ind T (THead (Flat -Cast) u t3) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) -\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u1 -t5) H16) in (False_ind (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: -T).(pr0 (THead (Bind Abbr) u2 w) t7))) H17)))))) t2 H12)) t H10 H11 H7 H8 -H9))) | (pr0_zeta b H7 t5 t6 H8 u0) \Rightarrow (\lambda (H9: (eq T (THead -(Bind b) u0 (lift (S O) O t5)) t)).(\lambda (H10: (eq T t6 t2)).(eq_ind T -(THead (Bind b) u0 (lift (S O) O t5)) (\lambda (_: T).((eq T t6 t2) \to ((not -(eq B b Abst)) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) -(\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H11: (eq T t6 t2)).(eq_ind T t2 -(\lambda (t7: T).((not (eq B b Abst)) \to ((pr0 t5 t7) \to (ex2 T (\lambda -(t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (_: (not (eq -B b Abst))).(\lambda (_: (pr0 t5 t2)).(let H14 \def (eq_ind_r T t (\lambda -(t7: T).(eq T (THead (Flat Cast) u t3) t7)) H3 (THead (Bind b) u0 (lift (S O) -O t5)) H9) in (let H15 \def (eq_ind T (THead (Flat Cast) u t3) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K -return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) -\Rightarrow True])])) I (THead (Bind b) u0 (lift (S O) O t5)) H14) in -(False_ind (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 -t7))) H15))))) t6 (sym_eq T t6 t2 H11))) t H9 H10 H7 H8))) | (pr0_tau t5 t6 -H7 u0) \Rightarrow (\lambda (H8: (eq T (THead (Flat Cast) u0 t5) t)).(\lambda -(H9: (eq T t6 t2)).(eq_ind T (THead (Flat Cast) u0 t5) (\lambda (_: T).((eq T -t6 t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda -(t8: T).(pr0 t2 t8)))))) (\lambda (H10: (eq T t6 t2)).(eq_ind T t2 (\lambda -(t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: -T).(pr0 t2 t8))))) (\lambda (H11: (pr0 t5 t2)).(let H12 \def (eq_ind_r T t -(\lambda (t7: T).(eq T (THead (Flat Cast) u t3) t7)) H3 (THead (Flat Cast) u0 -t5) H8) in (let H13 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | -(THead _ t7 _) \Rightarrow t7])) (THead (Flat Cast) u t3) (THead (Flat Cast) -u0 t5) H12) in ((let H14 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) -\Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead (Flat Cast) u t3) -(THead (Flat Cast) u0 t5) H12) in (\lambda (_: (eq T u u0)).(let H16 \def -(eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: -T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: -T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Cast) -u0 t5) H8) in (let H17 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t1)) H6 t5 -H14) in (ex2_ind T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 -t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))) -(\lambda (x: T).(\lambda (H18: (pr0 t1 x)).(\lambda (H19: (pr0 t2 -x)).(ex_intro2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)) -x H18 H19)))) (H16 t5 (tlt_head_dx (Flat Cast) u0 t5) t1 H17 t2 H11)))))) -H13)))) t6 (sym_eq T t6 t2 H10))) t H8 H9 H7)))]) in (H7 (refl_equal T t) -(refl_equal T t2)))) t4 (sym_eq T t4 t1 H5))) t H3 H4 H2)))]) in (H2 -(refl_equal T t) (refl_equal T t1))))))))) t0). -(* COMMENTS -Initial nodes: 46103 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr0/props.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr0/props.ma deleted file mode 100644 index d7f69d691..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr0/props.ma +++ /dev/null @@ -1,1758 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr0/defs.ma". - -include "Basic-1/subst0/subst0.ma". - -theorem pr0_lift: - \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (h: nat).(\forall -(d: nat).(pr0 (lift h d t1) (lift h d t2)))))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(pr0_ind (\lambda -(t: T).(\lambda (t0: T).(\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t) -(lift h d t0)))))) (\lambda (t: T).(\lambda (h: nat).(\lambda (d: -nat).(pr0_refl (lift h d t))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda -(_: (pr0 u1 u2)).(\lambda (H1: ((\forall (h: nat).(\forall (d: nat).(pr0 -(lift h d u1) (lift h d u2)))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda -(_: (pr0 t3 t4)).(\lambda (H3: ((\forall (h: nat).(\forall (d: nat).(pr0 -(lift h d t3) (lift h d t4)))))).(\lambda (k: K).(\lambda (h: nat).(\lambda -(d: nat).(eq_ind_r T (THead k (lift h d u1) (lift h (s k d) t3)) (\lambda (t: -T).(pr0 t (lift h d (THead k u2 t4)))) (eq_ind_r T (THead k (lift h d u2) -(lift h (s k d) t4)) (\lambda (t: T).(pr0 (THead k (lift h d u1) (lift h (s k -d) t3)) t)) (pr0_comp (lift h d u1) (lift h d u2) (H1 h d) (lift h (s k d) -t3) (lift h (s k d) t4) (H3 h (s k d)) k) (lift h d (THead k u2 t4)) -(lift_head k u2 t4 h d)) (lift h d (THead k u1 t3)) (lift_head k u1 t3 h -d))))))))))))) (\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: -(pr0 v1 v2)).(\lambda (H1: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h -d v1) (lift h d v2)))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 -t3 t4)).(\lambda (H3: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t3) -(lift h d t4)))))).(\lambda (h: nat).(\lambda (d: nat).(eq_ind_r T (THead -(Flat Appl) (lift h d v1) (lift h (s (Flat Appl) d) (THead (Bind Abst) u -t3))) (\lambda (t: T).(pr0 t (lift h d (THead (Bind Abbr) v2 t4)))) (eq_ind_r -T (THead (Bind Abst) (lift h (s (Flat Appl) d) u) (lift h (s (Bind Abst) (s -(Flat Appl) d)) t3)) (\lambda (t: T).(pr0 (THead (Flat Appl) (lift h d v1) t) -(lift h d (THead (Bind Abbr) v2 t4)))) (eq_ind_r T (THead (Bind Abbr) (lift h -d v2) (lift h (s (Bind Abbr) d) t4)) (\lambda (t: T).(pr0 (THead (Flat Appl) -(lift h d v1) (THead (Bind Abst) (lift h (s (Flat Appl) d) u) (lift h (s -(Bind Abst) (s (Flat Appl) d)) t3))) t)) (pr0_beta (lift h (s (Flat Appl) d) -u) (lift h d v1) (lift h d v2) (H1 h d) (lift h (s (Bind Abst) (s (Flat Appl) -d)) t3) (lift h (s (Bind Abbr) d) t4) (H3 h (s (Bind Abbr) d))) (lift h d -(THead (Bind Abbr) v2 t4)) (lift_head (Bind Abbr) v2 t4 h d)) (lift h (s -(Flat Appl) d) (THead (Bind Abst) u t3)) (lift_head (Bind Abst) u t3 h (s -(Flat Appl) d))) (lift h d (THead (Flat Appl) v1 (THead (Bind Abst) u t3))) -(lift_head (Flat Appl) v1 (THead (Bind Abst) u t3) h d))))))))))))) (\lambda -(b: B).(\lambda (H0: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: -T).(\lambda (_: (pr0 v1 v2)).(\lambda (H2: ((\forall (h: nat).(\forall (d: -nat).(pr0 (lift h d v1) (lift h d v2)))))).(\lambda (u1: T).(\lambda (u2: -T).(\lambda (_: (pr0 u1 u2)).(\lambda (H4: ((\forall (h: nat).(\forall (d: -nat).(pr0 (lift h d u1) (lift h d u2)))))).(\lambda (t3: T).(\lambda (t4: -T).(\lambda (_: (pr0 t3 t4)).(\lambda (H6: ((\forall (h: nat).(\forall (d: -nat).(pr0 (lift h d t3) (lift h d t4)))))).(\lambda (h: nat).(\lambda (d: -nat).(eq_ind_r T (THead (Flat Appl) (lift h d v1) (lift h (s (Flat Appl) d) -(THead (Bind b) u1 t3))) (\lambda (t: T).(pr0 t (lift h d (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t4))))) (eq_ind_r T (THead (Bind b) -(lift h (s (Flat Appl) d) u1) (lift h (s (Bind b) (s (Flat Appl) d)) t3)) -(\lambda (t: T).(pr0 (THead (Flat Appl) (lift h d v1) t) (lift h d (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))))) (eq_ind_r T (THead -(Bind b) (lift h d u2) (lift h (s (Bind b) d) (THead (Flat Appl) (lift (S O) -O v2) t4))) (\lambda (t: T).(pr0 (THead (Flat Appl) (lift h d v1) (THead -(Bind b) (lift h (s (Flat Appl) d) u1) (lift h (s (Bind b) (s (Flat Appl) d)) -t3))) t)) (eq_ind_r T (THead (Flat Appl) (lift h (s (Bind b) d) (lift (S O) O -v2)) (lift h (s (Flat Appl) (s (Bind b) d)) t4)) (\lambda (t: T).(pr0 (THead -(Flat Appl) (lift h d v1) (THead (Bind b) (lift h (s (Flat Appl) d) u1) (lift -h (s (Bind b) (s (Flat Appl) d)) t3))) (THead (Bind b) (lift h d u2) t))) -(eq_ind nat (plus (S O) d) (\lambda (n: nat).(pr0 (THead (Flat Appl) (lift h -d v1) (THead (Bind b) (lift h d u1) (lift h n t3))) (THead (Bind b) (lift h d -u2) (THead (Flat Appl) (lift h n (lift (S O) O v2)) (lift h n t4))))) -(eq_ind_r T (lift (S O) O (lift h d v2)) (\lambda (t: T).(pr0 (THead (Flat -Appl) (lift h d v1) (THead (Bind b) (lift h d u1) (lift h (plus (S O) d) -t3))) (THead (Bind b) (lift h d u2) (THead (Flat Appl) t (lift h (plus (S O) -d) t4))))) (pr0_upsilon b H0 (lift h d v1) (lift h d v2) (H2 h d) (lift h d -u1) (lift h d u2) (H4 h d) (lift h (plus (S O) d) t3) (lift h (plus (S O) d) -t4) (H6 h (plus (S O) d))) (lift h (plus (S O) d) (lift (S O) O v2)) (lift_d -v2 h (S O) d O (le_O_n d))) (S d) (refl_equal nat (S d))) (lift h (s (Bind b) -d) (THead (Flat Appl) (lift (S O) O v2) t4)) (lift_head (Flat Appl) (lift (S -O) O v2) t4 h (s (Bind b) d))) (lift h d (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4))) (lift_head (Bind b) u2 (THead (Flat Appl) (lift -(S O) O v2) t4) h d)) (lift h (s (Flat Appl) d) (THead (Bind b) u1 t3)) -(lift_head (Bind b) u1 t3 h (s (Flat Appl) d))) (lift h d (THead (Flat Appl) -v1 (THead (Bind b) u1 t3))) (lift_head (Flat Appl) v1 (THead (Bind b) u1 t3) -h d)))))))))))))))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 -u2)).(\lambda (H1: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d u1) -(lift h d u2)))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 -t4)).(\lambda (H3: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t3) -(lift h d t4)))))).(\lambda (w: T).(\lambda (H4: (subst0 O u2 t4 w)).(\lambda -(h: nat).(\lambda (d: nat).(eq_ind_r T (THead (Bind Abbr) (lift h d u1) (lift -h (s (Bind Abbr) d) t3)) (\lambda (t: T).(pr0 t (lift h d (THead (Bind Abbr) -u2 w)))) (eq_ind_r T (THead (Bind Abbr) (lift h d u2) (lift h (s (Bind Abbr) -d) w)) (\lambda (t: T).(pr0 (THead (Bind Abbr) (lift h d u1) (lift h (s (Bind -Abbr) d) t3)) t)) (pr0_delta (lift h d u1) (lift h d u2) (H1 h d) (lift h (S -d) t3) (lift h (S d) t4) (H3 h (S d)) (lift h (S d) w) (let d' \def (S d) in -(eq_ind nat (minus (S d) (S O)) (\lambda (n: nat).(subst0 O (lift h n u2) -(lift h d' t4) (lift h d' w))) (subst0_lift_lt t4 w u2 O H4 (S d) (le_n_S O d -(le_O_n d)) h) d (eq_ind nat d (\lambda (n: nat).(eq nat n d)) (refl_equal -nat d) (minus d O) (minus_n_O d))))) (lift h d (THead (Bind Abbr) u2 w)) -(lift_head (Bind Abbr) u2 w h d)) (lift h d (THead (Bind Abbr) u1 t3)) -(lift_head (Bind Abbr) u1 t3 h d)))))))))))))) (\lambda (b: B).(\lambda (H0: -(not (eq B b Abst))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 -t4)).(\lambda (H2: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t3) -(lift h d t4)))))).(\lambda (u: T).(\lambda (h: nat).(\lambda (d: -nat).(eq_ind_r T (THead (Bind b) (lift h d u) (lift h (s (Bind b) d) (lift (S -O) O t3))) (\lambda (t: T).(pr0 t (lift h d t4))) (eq_ind nat (plus (S O) d) -(\lambda (n: nat).(pr0 (THead (Bind b) (lift h d u) (lift h n (lift (S O) O -t3))) (lift h d t4))) (eq_ind_r T (lift (S O) O (lift h d t3)) (\lambda (t: -T).(pr0 (THead (Bind b) (lift h d u) t) (lift h d t4))) (pr0_zeta b H0 (lift -h d t3) (lift h d t4) (H2 h d) (lift h d u)) (lift h (plus (S O) d) (lift (S -O) O t3)) (lift_d t3 h (S O) d O (le_O_n d))) (S d) (refl_equal nat (S d))) -(lift h d (THead (Bind b) u (lift (S O) O t3))) (lift_head (Bind b) u (lift -(S O) O t3) h d))))))))))) (\lambda (t3: T).(\lambda (t4: T).(\lambda (_: -(pr0 t3 t4)).(\lambda (H1: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h -d t3) (lift h d t4)))))).(\lambda (u: T).(\lambda (h: nat).(\lambda (d: -nat).(eq_ind_r T (THead (Flat Cast) (lift h d u) (lift h (s (Flat Cast) d) -t3)) (\lambda (t: T).(pr0 t (lift h d t4))) (pr0_tau (lift h (s (Flat Cast) -d) t3) (lift h d t4) (H1 h d) (lift h d u)) (lift h d (THead (Flat Cast) u -t3)) (lift_head (Flat Cast) u t3 h d))))))))) t1 t2 H))). -(* COMMENTS -Initial nodes: 2845 -END *) - -theorem pr0_subst0_back: - \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst0 -i u2 t1 t2) \to (\forall (u1: T).((pr0 u1 u2) \to (ex2 T (\lambda (t: -T).(subst0 i u1 t1 t)) (\lambda (t: T).(pr0 t t2))))))))) -\def - \lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (i: nat).(\lambda -(H: (subst0 i u2 t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t: -T).(\lambda (t0: T).(\lambda (t3: T).(\forall (u1: T).((pr0 u1 t) \to (ex2 T -(\lambda (t4: T).(subst0 n u1 t0 t4)) (\lambda (t4: T).(pr0 t4 t3))))))))) -(\lambda (v: T).(\lambda (i0: nat).(\lambda (u1: T).(\lambda (H0: (pr0 u1 -v)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1 (TLRef i0) t)) (\lambda (t: -T).(pr0 t (lift (S i0) O v))) (lift (S i0) O u1) (subst0_lref u1 i0) -(pr0_lift u1 v H0 (S i0) O)))))) (\lambda (v: T).(\lambda (u3: T).(\lambda -(u1: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v u1 u3)).(\lambda (H1: -((\forall (u4: T).((pr0 u4 v) \to (ex2 T (\lambda (t: T).(subst0 i0 u4 u1 t)) -(\lambda (t: T).(pr0 t u3))))))).(\lambda (t: T).(\lambda (k: K).(\lambda -(u0: T).(\lambda (H2: (pr0 u0 v)).(ex2_ind T (\lambda (t0: T).(subst0 i0 u0 -u1 t0)) (\lambda (t0: T).(pr0 t0 u3)) (ex2 T (\lambda (t0: T).(subst0 i0 u0 -(THead k u1 t) t0)) (\lambda (t0: T).(pr0 t0 (THead k u3 t)))) (\lambda (x: -T).(\lambda (H3: (subst0 i0 u0 u1 x)).(\lambda (H4: (pr0 x u3)).(ex_intro2 T -(\lambda (t0: T).(subst0 i0 u0 (THead k u1 t) t0)) (\lambda (t0: T).(pr0 t0 -(THead k u3 t))) (THead k x t) (subst0_fst u0 x u1 i0 H3 t k) (pr0_comp x u3 -H4 t t (pr0_refl t) k))))) (H1 u0 H2)))))))))))) (\lambda (k: K).(\lambda (v: -T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (i0: nat).(\lambda (_: (subst0 -(s k i0) v t4 t3)).(\lambda (H1: ((\forall (u1: T).((pr0 u1 v) \to (ex2 T -(\lambda (t: T).(subst0 (s k i0) u1 t4 t)) (\lambda (t: T).(pr0 t -t3))))))).(\lambda (u: T).(\lambda (u1: T).(\lambda (H2: (pr0 u1 v)).(ex2_ind -T (\lambda (t: T).(subst0 (s k i0) u1 t4 t)) (\lambda (t: T).(pr0 t t3)) (ex2 -T (\lambda (t: T).(subst0 i0 u1 (THead k u t4) t)) (\lambda (t: T).(pr0 t -(THead k u t3)))) (\lambda (x: T).(\lambda (H3: (subst0 (s k i0) u1 t4 -x)).(\lambda (H4: (pr0 x t3)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1 -(THead k u t4) t)) (\lambda (t: T).(pr0 t (THead k u t3))) (THead k u x) -(subst0_snd k u1 x t4 i0 H3 u) (pr0_comp u u (pr0_refl u) x t3 H4 k))))) (H1 -u1 H2)))))))))))) (\lambda (v: T).(\lambda (u1: T).(\lambda (u3: T).(\lambda -(i0: nat).(\lambda (_: (subst0 i0 v u1 u3)).(\lambda (H1: ((\forall (u4: -T).((pr0 u4 v) \to (ex2 T (\lambda (t: T).(subst0 i0 u4 u1 t)) (\lambda (t: -T).(pr0 t u3))))))).(\lambda (k: K).(\lambda (t3: T).(\lambda (t4: -T).(\lambda (_: (subst0 (s k i0) v t3 t4)).(\lambda (H3: ((\forall (u4: -T).((pr0 u4 v) \to (ex2 T (\lambda (t: T).(subst0 (s k i0) u4 t3 t)) (\lambda -(t: T).(pr0 t t4))))))).(\lambda (u0: T).(\lambda (H4: (pr0 u0 v)).(ex2_ind T -(\lambda (t: T).(subst0 (s k i0) u0 t3 t)) (\lambda (t: T).(pr0 t t4)) (ex2 T -(\lambda (t: T).(subst0 i0 u0 (THead k u1 t3) t)) (\lambda (t: T).(pr0 t -(THead k u3 t4)))) (\lambda (x: T).(\lambda (H5: (subst0 (s k i0) u0 t3 -x)).(\lambda (H6: (pr0 x t4)).(ex2_ind T (\lambda (t: T).(subst0 i0 u0 u1 t)) -(\lambda (t: T).(pr0 t u3)) (ex2 T (\lambda (t: T).(subst0 i0 u0 (THead k u1 -t3) t)) (\lambda (t: T).(pr0 t (THead k u3 t4)))) (\lambda (x0: T).(\lambda -(H7: (subst0 i0 u0 u1 x0)).(\lambda (H8: (pr0 x0 u3)).(ex_intro2 T (\lambda -(t: T).(subst0 i0 u0 (THead k u1 t3) t)) (\lambda (t: T).(pr0 t (THead k u3 -t4))) (THead k x0 x) (subst0_both u0 u1 x0 i0 H7 k t3 x H5) (pr0_comp x0 u3 -H8 x t4 H6 k))))) (H1 u0 H4))))) (H3 u0 H4))))))))))))))) i u2 t1 t2 H))))). -(* COMMENTS -Initial nodes: 979 -END *) - -theorem pr0_subst0_fwd: - \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst0 -i u2 t1 t2) \to (\forall (u1: T).((pr0 u2 u1) \to (ex2 T (\lambda (t: -T).(subst0 i u1 t1 t)) (\lambda (t: T).(pr0 t2 t))))))))) -\def - \lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (i: nat).(\lambda -(H: (subst0 i u2 t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t: -T).(\lambda (t0: T).(\lambda (t3: T).(\forall (u1: T).((pr0 t u1) \to (ex2 T -(\lambda (t4: T).(subst0 n u1 t0 t4)) (\lambda (t4: T).(pr0 t3 t4))))))))) -(\lambda (v: T).(\lambda (i0: nat).(\lambda (u1: T).(\lambda (H0: (pr0 v -u1)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1 (TLRef i0) t)) (\lambda (t: -T).(pr0 (lift (S i0) O v) t)) (lift (S i0) O u1) (subst0_lref u1 i0) -(pr0_lift v u1 H0 (S i0) O)))))) (\lambda (v: T).(\lambda (u3: T).(\lambda -(u1: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v u1 u3)).(\lambda (H1: -((\forall (u4: T).((pr0 v u4) \to (ex2 T (\lambda (t: T).(subst0 i0 u4 u1 t)) -(\lambda (t: T).(pr0 u3 t))))))).(\lambda (t: T).(\lambda (k: K).(\lambda -(u0: T).(\lambda (H2: (pr0 v u0)).(ex2_ind T (\lambda (t0: T).(subst0 i0 u0 -u1 t0)) (\lambda (t0: T).(pr0 u3 t0)) (ex2 T (\lambda (t0: T).(subst0 i0 u0 -(THead k u1 t) t0)) (\lambda (t0: T).(pr0 (THead k u3 t) t0))) (\lambda (x: -T).(\lambda (H3: (subst0 i0 u0 u1 x)).(\lambda (H4: (pr0 u3 x)).(ex_intro2 T -(\lambda (t0: T).(subst0 i0 u0 (THead k u1 t) t0)) (\lambda (t0: T).(pr0 -(THead k u3 t) t0)) (THead k x t) (subst0_fst u0 x u1 i0 H3 t k) (pr0_comp u3 -x H4 t t (pr0_refl t) k))))) (H1 u0 H2)))))))))))) (\lambda (k: K).(\lambda -(v: T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (i0: nat).(\lambda (_: -(subst0 (s k i0) v t4 t3)).(\lambda (H1: ((\forall (u1: T).((pr0 v u1) \to -(ex2 T (\lambda (t: T).(subst0 (s k i0) u1 t4 t)) (\lambda (t: T).(pr0 t3 -t))))))).(\lambda (u: T).(\lambda (u1: T).(\lambda (H2: (pr0 v u1)).(ex2_ind -T (\lambda (t: T).(subst0 (s k i0) u1 t4 t)) (\lambda (t: T).(pr0 t3 t)) (ex2 -T (\lambda (t: T).(subst0 i0 u1 (THead k u t4) t)) (\lambda (t: T).(pr0 -(THead k u t3) t))) (\lambda (x: T).(\lambda (H3: (subst0 (s k i0) u1 t4 -x)).(\lambda (H4: (pr0 t3 x)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1 -(THead k u t4) t)) (\lambda (t: T).(pr0 (THead k u t3) t)) (THead k u x) -(subst0_snd k u1 x t4 i0 H3 u) (pr0_comp u u (pr0_refl u) t3 x H4 k))))) (H1 -u1 H2)))))))))))) (\lambda (v: T).(\lambda (u1: T).(\lambda (u3: T).(\lambda -(i0: nat).(\lambda (_: (subst0 i0 v u1 u3)).(\lambda (H1: ((\forall (u4: -T).((pr0 v u4) \to (ex2 T (\lambda (t: T).(subst0 i0 u4 u1 t)) (\lambda (t: -T).(pr0 u3 t))))))).(\lambda (k: K).(\lambda (t3: T).(\lambda (t4: -T).(\lambda (_: (subst0 (s k i0) v t3 t4)).(\lambda (H3: ((\forall (u4: -T).((pr0 v u4) \to (ex2 T (\lambda (t: T).(subst0 (s k i0) u4 t3 t)) (\lambda -(t: T).(pr0 t4 t))))))).(\lambda (u0: T).(\lambda (H4: (pr0 v u0)).(ex2_ind T -(\lambda (t: T).(subst0 (s k i0) u0 t3 t)) (\lambda (t: T).(pr0 t4 t)) (ex2 T -(\lambda (t: T).(subst0 i0 u0 (THead k u1 t3) t)) (\lambda (t: T).(pr0 (THead -k u3 t4) t))) (\lambda (x: T).(\lambda (H5: (subst0 (s k i0) u0 t3 -x)).(\lambda (H6: (pr0 t4 x)).(ex2_ind T (\lambda (t: T).(subst0 i0 u0 u1 t)) -(\lambda (t: T).(pr0 u3 t)) (ex2 T (\lambda (t: T).(subst0 i0 u0 (THead k u1 -t3) t)) (\lambda (t: T).(pr0 (THead k u3 t4) t))) (\lambda (x0: T).(\lambda -(H7: (subst0 i0 u0 u1 x0)).(\lambda (H8: (pr0 u3 x0)).(ex_intro2 T (\lambda -(t: T).(subst0 i0 u0 (THead k u1 t3) t)) (\lambda (t: T).(pr0 (THead k u3 t4) -t)) (THead k x0 x) (subst0_both u0 u1 x0 i0 H7 k t3 x H5) (pr0_comp u3 x0 H8 -t4 x H6 k))))) (H1 u0 H4))))) (H3 u0 H4))))))))))))))) i u2 t1 t2 H))))). -(* COMMENTS -Initial nodes: 979 -END *) - -theorem pr0_subst0: - \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (v1: T).(\forall -(w1: T).(\forall (i: nat).((subst0 i v1 t1 w1) \to (\forall (v2: T).((pr0 v1 -v2) \to (or (pr0 w1 t2) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: -T).(subst0 i v2 t2 w2)))))))))))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(pr0_ind (\lambda -(t: T).(\lambda (t0: T).(\forall (v1: T).(\forall (w1: T).(\forall (i: -nat).((subst0 i v1 t w1) \to (\forall (v2: T).((pr0 v1 v2) \to (or (pr0 w1 -t0) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t0 -w2)))))))))))) (\lambda (t: T).(\lambda (v1: T).(\lambda (w1: T).(\lambda (i: -nat).(\lambda (H0: (subst0 i v1 t w1)).(\lambda (v2: T).(\lambda (H1: (pr0 v1 -v2)).(or_intror (pr0 w1 t) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: -T).(subst0 i v2 t w2))) (ex2_sym T (subst0 i v2 t) (pr0 w1) (pr0_subst0_fwd -v1 t w1 i H0 v2 H1)))))))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (H0: -(pr0 u1 u2)).(\lambda (H1: ((\forall (v1: T).(\forall (w1: T).(\forall (i: -nat).((subst0 i v1 u1 w1) \to (\forall (v2: T).((pr0 v1 v2) \to (or (pr0 w1 -u2) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 u2 -w2)))))))))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 -t4)).(\lambda (H3: ((\forall (v1: T).(\forall (w1: T).(\forall (i: -nat).((subst0 i v1 t3 w1) \to (\forall (v2: T).((pr0 v1 v2) \to (or (pr0 w1 -t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 -w2)))))))))))).(\lambda (k: K).(\lambda (v1: T).(\lambda (w1: T).(\lambda (i: -nat).(\lambda (H4: (subst0 i v1 (THead k u1 t3) w1)).(\lambda (v2: -T).(\lambda (H5: (pr0 v1 v2)).(or3_ind (ex2 T (\lambda (u3: T).(eq T w1 -(THead k u3 t3))) (\lambda (u3: T).(subst0 i v1 u1 u3))) (ex2 T (\lambda (t5: -T).(eq T w1 (THead k u1 t5))) (\lambda (t5: T).(subst0 (s k i) v1 t3 t5))) -(ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T w1 (THead k u3 t5)))) -(\lambda (u3: T).(\lambda (_: T).(subst0 i v1 u1 u3))) (\lambda (_: -T).(\lambda (t5: T).(subst0 (s k i) v1 t3 t5)))) (or (pr0 w1 (THead k u2 t4)) -(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 (THead k -u2 t4) w2)))) (\lambda (H6: (ex2 T (\lambda (u3: T).(eq T w1 (THead k u3 -t3))) (\lambda (u3: T).(subst0 i v1 u1 u3)))).(ex2_ind T (\lambda (u3: T).(eq -T w1 (THead k u3 t3))) (\lambda (u3: T).(subst0 i v1 u1 u3)) (or (pr0 w1 -(THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: -T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x: T).(\lambda (H7: (eq T w1 -(THead k x t3))).(\lambda (H8: (subst0 i v1 u1 x)).(eq_ind_r T (THead k x t3) -(\lambda (t: T).(or (pr0 t (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 t -w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))))) (or_ind (pr0 x u2) -(ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 u2 w2))) -(or (pr0 (THead k x t3) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead -k x t3) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda -(H9: (pr0 x u2)).(or_introl (pr0 (THead k x t3) (THead k u2 t4)) (ex2 T -(\lambda (w2: T).(pr0 (THead k x t3) w2)) (\lambda (w2: T).(subst0 i v2 -(THead k u2 t4) w2))) (pr0_comp x u2 H9 t3 t4 H2 k))) (\lambda (H9: (ex2 T -(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 u2 w2)))).(ex2_ind -T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 u2 w2)) (or (pr0 -(THead k x t3) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x t3) -w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x0: -T).(\lambda (H10: (pr0 x x0)).(\lambda (H11: (subst0 i v2 u2 x0)).(or_intror -(pr0 (THead k x t3) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x -t3) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))) (ex_intro2 T -(\lambda (w2: T).(pr0 (THead k x t3) w2)) (\lambda (w2: T).(subst0 i v2 -(THead k u2 t4) w2)) (THead k x0 t4) (pr0_comp x x0 H10 t3 t4 H2 k) -(subst0_fst v2 x0 u2 i H11 t4 k)))))) H9)) (H1 v1 x i H8 v2 H5)) w1 H7)))) -H6)) (\lambda (H6: (ex2 T (\lambda (t5: T).(eq T w1 (THead k u1 t5))) -(\lambda (t5: T).(subst0 (s k i) v1 t3 t5)))).(ex2_ind T (\lambda (t5: T).(eq -T w1 (THead k u1 t5))) (\lambda (t5: T).(subst0 (s k i) v1 t3 t5)) (or (pr0 -w1 (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: -T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x: T).(\lambda (H7: (eq T w1 -(THead k u1 x))).(\lambda (H8: (subst0 (s k i) v1 t3 x)).(eq_ind_r T (THead k -u1 x) (\lambda (t: T).(or (pr0 t (THead k u2 t4)) (ex2 T (\lambda (w2: -T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))))) (or_ind -(pr0 x t4) (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s k -i) v2 t4 w2))) (or (pr0 (THead k u1 x) (THead k u2 t4)) (ex2 T (\lambda (w2: -T).(pr0 (THead k u1 x) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) -w2)))) (\lambda (H9: (pr0 x t4)).(or_introl (pr0 (THead k u1 x) (THead k u2 -t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k u1 x) w2)) (\lambda (w2: -T).(subst0 i v2 (THead k u2 t4) w2))) (pr0_comp u1 u2 H0 x t4 H9 k))) -(\lambda (H9: (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s -k i) v2 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: -T).(subst0 (s k i) v2 t4 w2)) (or (pr0 (THead k u1 x) (THead k u2 t4)) (ex2 T -(\lambda (w2: T).(pr0 (THead k u1 x) w2)) (\lambda (w2: T).(subst0 i v2 -(THead k u2 t4) w2)))) (\lambda (x0: T).(\lambda (H10: (pr0 x x0)).(\lambda -(H11: (subst0 (s k i) v2 t4 x0)).(or_intror (pr0 (THead k u1 x) (THead k u2 -t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k u1 x) w2)) (\lambda (w2: -T).(subst0 i v2 (THead k u2 t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 -(THead k u1 x) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)) (THead -k u2 x0) (pr0_comp u1 u2 H0 x x0 H10 k) (subst0_snd k v2 x0 t4 i H11 u2)))))) -H9)) (H3 v1 x (s k i) H8 v2 H5)) w1 H7)))) H6)) (\lambda (H6: (ex3_2 T T -(\lambda (u3: T).(\lambda (t5: T).(eq T w1 (THead k u3 t5)))) (\lambda (u3: -T).(\lambda (_: T).(subst0 i v1 u1 u3))) (\lambda (_: T).(\lambda (t5: -T).(subst0 (s k i) v1 t3 t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda -(t5: T).(eq T w1 (THead k u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 -i v1 u1 u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s k i) v1 t3 t5))) -(or (pr0 w1 (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda -(w2: T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H7: (eq T w1 (THead k x0 x1))).(\lambda (H8: (subst0 i v1 u1 -x0)).(\lambda (H9: (subst0 (s k i) v1 t3 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w2)))).(ex2_ind T (\lambda -(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 u2 w2)) (or (pr0 (THead k -x0 x1) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x0 x1) w2)) -(\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x: T).(\lambda -(H12: (pr0 x0 x)).(\lambda (H13: (subst0 i v2 u2 x)).(or_intror (pr0 (THead k -x0 x1) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x0 x1) w2)) -(\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))) (ex_intro2 T (\lambda -(w2: T).(pr0 (THead k x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 -t4) w2)) (THead k x t4) (pr0_comp x0 x H12 x1 t4 H10 k) (subst0_fst v2 x u2 i -H13 t4 k)))))) H11)) (H1 v1 x0 i H8 v2 H5))) (\lambda (H10: (ex2 T (\lambda -(w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s k i) v2 t4 w2)))).(ex2_ind T -(\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s k i) v2 t4 w2)) (or -(pr0 (THead k x0 x1) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k -x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x: -T).(\lambda (H11: (pr0 x1 x)).(\lambda (H12: (subst0 (s k i) v2 t4 -x)).(or_ind (pr0 x0 u2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: -T).(subst0 i v2 u2 w2))) (or (pr0 (THead k x0 x1) (THead k u2 t4)) (ex2 T -(\lambda (w2: T).(pr0 (THead k x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 -(THead k u2 t4) w2)))) (\lambda (H13: (pr0 x0 u2)).(or_intror (pr0 (THead k -x0 x1) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x0 x1) w2)) -(\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))) (ex_intro2 T (\lambda -(w2: T).(pr0 (THead k x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 -t4) w2)) (THead k u2 x) (pr0_comp x0 u2 H13 x1 x H11 k) (subst0_snd k v2 x t4 -i H12 u2)))) (\lambda (H13: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda -(w2: T).(subst0 i v2 u2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) -(\lambda (w2: T).(subst0 i v2 u2 w2)) (or (pr0 (THead k x0 x1) (THead k u2 -t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x0 x1) w2)) (\lambda (w2: -T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x2: T).(\lambda (H14: (pr0 -x0 x2)).(\lambda (H15: (subst0 i v2 u2 x2)).(or_intror (pr0 (THead k x0 x1) -(THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x0 x1) w2)) (\lambda -(w2: T).(subst0 i v2 (THead k u2 t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 -(THead k x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)) -(THead k x2 x) (pr0_comp x0 x2 H14 x1 x H11 k) (subst0_both v2 u2 x2 i H15 k -t4 x H12)))))) H13)) (H1 v1 x0 i H8 v2 H5))))) H10)) (H3 v1 x1 (s k i) H9 v2 -H5)) w1 H7)))))) H6)) (subst0_gen_head k v1 u1 t3 w1 i H4))))))))))))))))) -(\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H0: (pr0 v1 -v2)).(\lambda (H1: ((\forall (v3: T).(\forall (w1: T).(\forall (i: -nat).((subst0 i v3 v1 w1) \to (\forall (v4: T).((pr0 v3 v4) \to (or (pr0 w1 -v2) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v4 v2 -w2)))))))))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 -t4)).(\lambda 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T).(pr0 (THead (Flat -Appl) x (THead (Bind Abst) u t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead -(Bind Abbr) v2 t4) w2)))) (\lambda (H9: (pr0 x v2)).(or_introl (pr0 (THead -(Flat Appl) x (THead (Bind Abst) u t3)) (THead (Bind Abbr) v2 t4)) (ex2 T -(\lambda (w2: T).(pr0 (THead (Flat Appl) x (THead (Bind Abst) u t3)) w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (pr0_beta u x -v2 H9 t3 t4 H2))) (\lambda (H9: (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda -(w2: T).(subst0 i v3 v2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) -(\lambda (w2: T).(subst0 i v3 v2 w2)) (or (pr0 (THead (Flat Appl) x (THead -(Bind Abst) u t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 -(THead (Flat Appl) x (THead (Bind Abst) u t3)) w2)) (\lambda (w2: T).(subst0 -i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (x0: T).(\lambda (H10: (pr0 x -x0)).(\lambda (H11: (subst0 i v3 v2 x0)).(or_intror (pr0 (THead (Flat Appl) x -(THead (Bind Abst) u t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x (THead (Bind Abst) u t3)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x (THead (Bind Abst) u t3)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)) (THead (Bind Abbr) x0 t4) -(pr0_beta u x x0 H10 t3 t4 H2) (subst0_fst v3 x0 v2 i H11 t4 (Bind -Abbr))))))) H9)) (H1 v0 x i H8 v3 H5)) w1 H7)))) H6)) (\lambda (H6: (ex2 T -(\lambda (t5: T).(eq T w1 (THead (Flat Appl) v1 t5))) (\lambda (t5: -T).(subst0 (s (Flat Appl) i) v0 (THead (Bind Abst) u t3) t5)))).(ex2_ind T -(\lambda (t5: T).(eq T w1 (THead (Flat Appl) v1 t5))) (\lambda (t5: -T).(subst0 (s (Flat Appl) i) v0 (THead (Bind Abst) u t3) t5)) (or (pr0 w1 -(THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (x: T).(\lambda -(H7: (eq T w1 (THead (Flat Appl) v1 x))).(\lambda (H8: (subst0 (s (Flat Appl) -i) v0 (THead (Bind Abst) u t3) x)).(or3_ind (ex2 T (\lambda (u2: T).(eq T x -(THead (Bind Abst) u2 t3))) (\lambda (u2: T).(subst0 (s (Flat Appl) i) v0 u -u2))) (ex2 T (\lambda (t5: T).(eq T x (THead (Bind Abst) u t5))) (\lambda -(t5: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5))) (ex3_2 T T -(\lambda (u2: T).(\lambda (t5: T).(eq T x (THead (Bind Abst) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u u2))) -(\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 -t3 t5)))) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 -w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) -(\lambda (H9: (ex2 T (\lambda (u2: T).(eq T x (THead (Bind Abst) u2 t3))) -(\lambda (u2: T).(subst0 (s (Flat Appl) i) v0 u u2)))).(ex2_ind T (\lambda -(u2: T).(eq T x (THead (Bind Abst) u2 t3))) (\lambda (u2: T).(subst0 (s (Flat -Appl) i) v0 u u2)) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda -(w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) -w2)))) (\lambda (x0: T).(\lambda (H10: (eq T x (THead (Bind Abst) x0 -t3))).(\lambda (_: (subst0 (s (Flat Appl) i) v0 u x0)).(let H12 \def (eq_ind -T x (\lambda (t: T).(eq T w1 (THead (Flat Appl) v1 t))) H7 (THead (Bind Abst) -x0 t3) H10) in (eq_ind_r T (THead (Flat Appl) v1 (THead (Bind Abst) x0 t3)) -(\lambda (t: T).(or (pr0 t (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: -T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) -w2))))) (or_introl (pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 t3)) -(THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 -(THead (Bind Abst) x0 t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind -Abbr) v2 t4) w2))) (pr0_beta x0 v1 v2 H0 t3 t4 H2)) w1 H12))))) H9)) (\lambda -(H9: (ex2 T (\lambda (t5: T).(eq T x (THead (Bind Abst) u t5))) (\lambda (t5: -T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5)))).(ex2_ind T (\lambda -(t5: T).(eq T x (THead (Bind Abst) u t5))) (\lambda (t5: T).(subst0 (s (Bind -Abst) (s (Flat Appl) i)) v0 t3 t5)) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) -(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead -(Bind Abbr) v2 t4) w2)))) (\lambda (x0: T).(\lambda (H10: (eq T x (THead -(Bind Abst) u x0))).(\lambda (H11: (subst0 (s (Bind Abst) (s (Flat Appl) i)) -v0 t3 x0)).(let H12 \def (eq_ind T x (\lambda (t: T).(eq T w1 (THead (Flat -Appl) v1 t))) H7 (THead (Bind Abst) u x0) H10) in (eq_ind_r T (THead (Flat -Appl) v1 (THead (Bind Abst) u x0)) (\lambda (t: T).(or (pr0 t (THead (Bind -Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i -v3 (THead (Bind Abbr) v2 t4) w2))))) (or_ind (pr0 x0 t4) (ex2 T (\lambda (w2: -T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 -t4 w2))) (or (pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u x0)) (THead -(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead -(Bind Abst) u x0)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 -t4) w2)))) (\lambda (H13: (pr0 x0 t4)).(or_introl (pr0 (THead (Flat Appl) v1 -(THead (Bind Abst) u x0)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u x0)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (pr0_beta u v1 v2 H0 x0 t4 -H13))) (\lambda (H13: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: -T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda -(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) -i)) v3 t4 w2)) (or (pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u x0)) -(THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 -(THead (Bind Abst) u x0)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind -Abbr) v2 t4) w2)))) (\lambda (x1: T).(\lambda (H14: (pr0 x0 x1)).(\lambda -(H15: (subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 x1)).(or_intror (pr0 -(THead (Flat Appl) v1 (THead (Bind Abst) u x0)) (THead (Bind Abbr) v2 t4)) -(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u x0)) -w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (ex_intro2 -T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u x0)) w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)) (THead (Bind -Abbr) v2 x1) (pr0_beta u v1 v2 H0 x0 x1 H14) (subst0_snd (Bind Abbr) v3 x1 t4 -i H15 v2)))))) H13)) (H3 v0 x0 (s (Bind Abst) (s (Flat Appl) i)) H11 v3 H5)) -w1 H12))))) H9)) (\lambda (H9: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: -T).(eq T x (THead (Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 (s (Flat Appl) i) v0 u u2))) (\lambda (_: T).(\lambda (t5: -T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5))))).(ex3_2_ind T T -(\lambda (u2: T).(\lambda (t5: T).(eq T x (THead (Bind Abst) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u u2))) -(\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 -t3 t5))) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 -w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) -(\lambda (x0: T).(\lambda (x1: T).(\lambda (H10: (eq T x (THead (Bind Abst) -x0 x1))).(\lambda (_: (subst0 (s (Flat Appl) i) v0 u x0)).(\lambda (H12: -(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 x1)).(let H13 \def (eq_ind T -x (\lambda (t: T).(eq T w1 (THead (Flat Appl) v1 t))) H7 (THead (Bind Abst) -x0 x1) H10) in (eq_ind_r T (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) -(\lambda (t: T).(or (pr0 t (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: -T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) -w2))))) (or_ind (pr0 x1 t4) (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda -(w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2))) (or (pr0 (THead -(Flat Appl) v1 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) v2 t4)) (ex2 T -(\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H14: -(pr0 x1 t4)).(or_introl (pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) -(THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 -(THead (Bind Abst) x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind -Abbr) v2 t4) w2))) (pr0_beta x0 v1 v2 H0 x1 t4 H14))) (\lambda (H14: (ex2 T -(\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s -(Flat Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x1 w2)) -(\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2)) (or -(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) v2 -t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 -x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) -(\lambda (x2: T).(\lambda (H15: (pr0 x1 x2)).(\lambda (H16: (subst0 (s (Bind -Abst) (s (Flat Appl) i)) v3 t4 x2)).(or_intror (pr0 (THead (Flat Appl) v1 -(THead (Bind Abst) x0 x1)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)) (THead (Bind Abbr) v2 x2) -(pr0_beta x0 v1 v2 H0 x1 x2 H15) (subst0_snd (Bind Abbr) v3 x2 t4 i H16 -v2)))))) H14)) (H3 v0 x1 (s (Bind Abst) (s (Flat Appl) i)) H12 v3 H5)) w1 -H13))))))) H9)) (subst0_gen_head (Bind Abst) v0 u t3 x (s (Flat Appl) i) -H8))))) H6)) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T -w1 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 -v1 u2))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Flat Appl) i) v0 (THead -(Bind Abst) u t3) t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: -T).(eq T w1 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 i v0 v1 u2))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Flat -Appl) i) v0 (THead (Bind Abst) u t3) t5))) (or (pr0 w1 (THead (Bind Abbr) v2 -t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 -(THead (Bind Abbr) v2 t4) w2)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda -(H7: (eq T w1 (THead (Flat Appl) x0 x1))).(\lambda (H8: (subst0 i v0 v1 -x0)).(\lambda (H9: (subst0 (s (Flat Appl) i) v0 (THead (Bind Abst) u t3) -x1)).(or3_ind (ex2 T (\lambda (u2: T).(eq T x1 (THead (Bind Abst) u2 t3))) -(\lambda (u2: T).(subst0 (s (Flat Appl) i) v0 u u2))) (ex2 T (\lambda (t5: -T).(eq T x1 (THead (Bind Abst) u t5))) (\lambda (t5: T).(subst0 (s (Bind -Abst) (s (Flat Appl) i)) v0 t3 t5))) (ex3_2 T T (\lambda (u2: T).(\lambda -(t5: T).(eq T x1 (THead (Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 (s (Flat Appl) i) v0 u u2))) (\lambda (_: T).(\lambda (t5: -T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5)))) (or (pr0 w1 (THead -(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H10: (ex2 T -(\lambda (u2: T).(eq T x1 (THead (Bind Abst) u2 t3))) (\lambda (u2: -T).(subst0 (s (Flat Appl) i) v0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T x1 -(THead (Bind Abst) u2 t3))) (\lambda (u2: T).(subst0 (s (Flat Appl) i) v0 u -u2)) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 -w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda -(x: T).(\lambda (H11: (eq T x1 (THead (Bind Abst) x t3))).(\lambda (_: -(subst0 (s (Flat Appl) i) v0 u x)).(let H13 \def (eq_ind T x1 (\lambda (t: -T).(eq T w1 (THead (Flat Appl) x0 t))) H7 (THead (Bind Abst) x t3) H11) in -(eq_ind_r T (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) (\lambda (t: -T).(or (pr0 t (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 t w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))))) (or_ind (pr0 -x0 v2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 -w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) (THead (Bind -Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind -Abst) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) -w2)))) (\lambda (H14: (pr0 x0 v2)).(or_introl (pr0 (THead (Flat Appl) x0 -(THead (Bind Abst) x t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (pr0_beta x x0 v2 H14 t3 t4 -H2))) (\lambda (H14: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: -T).(subst0 i v3 v2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda -(w2: T).(subst0 i v3 v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind -Abst) x t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead -(Flat Appl) x0 (THead (Bind Abst) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 -(THead (Bind Abbr) v2 t4) w2)))) (\lambda (x2: T).(\lambda (H15: (pr0 x0 -x2)).(\lambda (H16: (subst0 i v3 v2 x2)).(or_intror (pr0 (THead (Flat Appl) -x0 (THead (Bind Abst) x t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)) (THead (Bind Abbr) x2 t4) -(pr0_beta x x0 x2 H15 t3 t4 H2) (subst0_fst v3 x2 v2 i H16 t4 (Bind -Abbr))))))) H14)) (H1 v0 x0 i H8 v3 H5)) w1 H13))))) H10)) (\lambda (H10: -(ex2 T (\lambda (t5: T).(eq T x1 (THead (Bind Abst) u t5))) (\lambda (t5: -T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5)))).(ex2_ind T (\lambda -(t5: T).(eq T x1 (THead (Bind Abst) u t5))) (\lambda (t5: T).(subst0 (s (Bind -Abst) (s (Flat Appl) i)) v0 t3 t5)) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) -(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead -(Bind Abbr) v2 t4) w2)))) (\lambda (x: T).(\lambda (H11: (eq T x1 (THead -(Bind Abst) u x))).(\lambda (H12: (subst0 (s (Bind Abst) (s (Flat Appl) i)) -v0 t3 x)).(let H13 \def (eq_ind T x1 (\lambda (t: T).(eq T w1 (THead (Flat -Appl) x0 t))) H7 (THead (Bind Abst) u x) H11) in (eq_ind_r T (THead (Flat -Appl) x0 (THead (Bind Abst) u x)) (\lambda (t: T).(or (pr0 t (THead (Bind -Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i -v3 (THead (Bind Abbr) v2 t4) w2))))) (or_ind (pr0 x t4) (ex2 T (\lambda (w2: -T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 -t4 w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind -Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind -Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) -w2)))) (\lambda (H14: (pr0 x t4)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2: -T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat -Appl) x0 (THead (Bind Abst) u x)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda -(w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) w2)) (\lambda -(w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H15: (pr0 x0 -v2)).(or_introl (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead -(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead -(Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 -t4) w2))) (pr0_beta u x0 v2 H15 x t4 H14))) (\lambda (H15: (ex2 T (\lambda -(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)))).(ex2_ind T -(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)) (or (pr0 -(THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind Abbr) v2 t4)) -(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) -w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda -(x2: T).(\lambda (H16: (pr0 x0 x2)).(\lambda (H17: (subst0 i v3 v2 -x2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead -(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead -(Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 -t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead -(Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 -t4) w2)) (THead (Bind Abbr) x2 t4) (pr0_beta u x0 x2 H16 x t4 H14) -(subst0_fst v3 x2 v2 i H17 t4 (Bind Abbr))))))) H15)) (H1 v0 x0 i H8 v3 H5))) -(\lambda (H14: (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 -(s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda (w2: -T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 -t4 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind -Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind -Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) -w2)))) (\lambda (x2: T).(\lambda (H15: (pr0 x x2)).(\lambda (H16: (subst0 (s -(Bind Abst) (s (Flat Appl) i)) v3 t4 x2)).(or_ind (pr0 x0 v2) (ex2 T (\lambda -(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead -(Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind Abbr) v2 t4)) (ex2 T -(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H17: -(pr0 x0 v2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) -(THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 -(THead (Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind -Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 -(THead (Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind -Abbr) v2 t4) w2)) (THead (Bind Abbr) v2 x2) (pr0_beta u x0 v2 H17 x x2 H15) -(subst0_snd (Bind Abbr) v3 x2 t4 i H16 v2)))) (\lambda (H17: (ex2 T (\lambda -(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)))).(ex2_ind T -(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)) (or (pr0 -(THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind Abbr) v2 t4)) -(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) -w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda -(x3: T).(\lambda (H18: (pr0 x0 x3)).(\lambda (H19: (subst0 i v3 v2 -x3)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead -(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead -(Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 -t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead -(Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 -t4) w2)) (THead (Bind Abbr) x3 x2) (pr0_beta u x0 x3 H18 x x2 H15) -(subst0_both v3 v2 x3 i H19 (Bind Abbr) t4 x2 H16)))))) H17)) (H1 v0 x0 i H8 -v3 H5))))) H14)) (H3 v0 x (s (Bind Abst) (s (Flat Appl) i)) H12 v3 H5)) w1 -H13))))) H10)) (\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: -T).(eq T x1 (THead (Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 (s (Flat Appl) i) v0 u u2))) (\lambda (_: T).(\lambda (t5: -T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5))))).(ex3_2_ind T T -(\lambda (u2: T).(\lambda (t5: T).(eq T x1 (THead (Bind Abst) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u u2))) -(\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 -t3 t5))) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 -w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) -(\lambda (x2: T).(\lambda (x3: T).(\lambda (H11: (eq T x1 (THead (Bind Abst) -x2 x3))).(\lambda (_: (subst0 (s (Flat Appl) i) v0 u x2)).(\lambda (H13: -(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 x3)).(let H14 \def (eq_ind T -x1 (\lambda (t: T).(eq T w1 (THead (Flat Appl) x0 t))) H7 (THead (Bind Abst) -x2 x3) H11) in (eq_ind_r T (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) -(\lambda (t: T).(or (pr0 t (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: -T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) -w2))))) (or_ind (pr0 x3 t4) (ex2 T (\lambda (w2: T).(pr0 x3 w2)) (\lambda -(w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2))) (or (pr0 (THead -(Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead (Bind Abbr) v2 t4)) (ex2 T -(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H15: -(pr0 x3 t4)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) -(\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat Appl) x0 (THead -(Bind Abst) x2 x3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 -(THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H16: (pr0 x0 -v2)).(or_introl (pr0 (THead 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(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead -(Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 -t4) w2)) (THead (Bind Abbr) x t4) (pr0_beta x2 x0 x H17 x3 t4 H15) -(subst0_fst v3 x v2 i H18 t4 (Bind Abbr))))))) H16)) (H1 v0 x0 i H8 v3 H5))) -(\lambda (H15: (ex2 T (\lambda (w2: T).(pr0 x3 w2)) (\lambda (w2: T).(subst0 -(s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda (w2: -T).(pr0 x3 w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 -t4 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead -(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead -(Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 -t4) w2)))) (\lambda (x: T).(\lambda (H16: (pr0 x3 x)).(\lambda (H17: (subst0 -(s (Bind Abst) (s (Flat Appl) i)) v3 t4 x)).(or_ind (pr0 x0 v2) (ex2 T -(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 -(THead (Flat Appl) x0 (THead (Bind 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T).(pr0 (THead (Flat Appl) x0 (THead -(Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 -t4) w2)))) (\lambda (x4: T).(\lambda (H19: (pr0 x0 x4)).(\lambda (H20: -(subst0 i v3 v2 x4)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) -x2 x3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat -Appl) x0 (THead (Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead -(Bind Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat -Appl) x0 (THead (Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead -(Bind Abbr) v2 t4) w2)) (THead (Bind Abbr) x4 x) (pr0_beta x2 x0 x4 H19 x3 x -H16) (subst0_both v3 v2 x4 i H20 (Bind Abbr) t4 x H17)))))) H18)) (H1 v0 x0 i -H8 v3 H5))))) H15)) (H3 v0 x3 (s (Bind Abst) (s (Flat Appl) i)) H13 v3 H5)) -w1 H14))))))) H10)) (subst0_gen_head (Bind Abst) v0 u t3 x1 (s (Flat Appl) i) -H9))))))) H6)) (subst0_gen_head (Flat Appl) v0 v1 (THead (Bind Abst) u t3) w1 -i H4))))))))))))))))) (\lambda (b: B).(\lambda (H0: (not (eq B b -Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H1: (pr0 v1 v2)).(\lambda -(H2: ((\forall (v3: T).(\forall (w1: T).(\forall (i: nat).((subst0 i v3 v1 -w1) \to (\forall (v4: T).((pr0 v3 v4) \to (or (pr0 w1 v2) (ex2 T (\lambda -(w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v4 v2 w2)))))))))))).(\lambda -(u1: T).(\lambda (u2: T).(\lambda (H3: (pr0 u1 u2)).(\lambda (H4: ((\forall -(v3: T).(\forall (w1: T).(\forall (i: nat).((subst0 i v3 u1 w1) \to (\forall -(v4: T).((pr0 v3 v4) \to (or (pr0 w1 u2) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) -(\lambda (w2: T).(subst0 i v4 u2 w2)))))))))))).(\lambda (t3: T).(\lambda -(t4: T).(\lambda (H5: (pr0 t3 t4)).(\lambda (H6: ((\forall (v3: T).(\forall -(w1: T).(\forall (i: nat).((subst0 i v3 t3 w1) \to (\forall (v4: T).((pr0 v3 -v4) \to (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: -T).(subst0 i v4 t4 w2)))))))))))).(\lambda (v0: T).(\lambda (w1: T).(\lambda -(i: nat).(\lambda (H7: (subst0 i v0 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T).(subst0 i -v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) -(\lambda (H12: (pr0 x v2)).(or_introl (pr0 (THead (Flat Appl) x (THead (Bind -b) u1 t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 -T (\lambda (w2: T).(pr0 (THead (Flat Appl) x (THead (Bind b) u1 t3)) w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t4)) w2))) (pr0_upsilon b H0 x v2 H12 u1 u2 H3 t3 t4 H5))) (\lambda -(H12: (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v3 v2 -w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v3 -v2 w2)) (or (pr0 (THead (Flat Appl) x (THead (Bind b) u1 t3)) (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 -(THead (Flat Appl) x (THead (Bind b) u1 t3)) w2)) (\lambda (w2: T).(subst0 i -v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) -(\lambda (x0: T).(\lambda (H13: (pr0 x x0)).(\lambda (H14: (subst0 i v3 v2 -x0)).(or_intror (pr0 (THead (Flat Appl) x (THead (Bind b) u1 t3)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x (THead (Bind b) u1 t3)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x (THead (Bind b) -u1 t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 (THead (Flat Appl) (lift -(S O) O x0) t4)) (pr0_upsilon b H0 x x0 H13 u1 u2 H3 t3 t4 H5) (subst0_snd -(Bind b) v3 (THead (Flat Appl) (lift (S O) O x0) t4) (THead (Flat Appl) (lift -(S O) O v2) t4) i (subst0_fst v3 (lift (S O) O x0) (lift (S O) O v2) (s (Bind -b) i) (subst0_lift_ge_s v2 x0 v3 i H14 O (le_O_n i) b) t4 (Flat Appl)) -u2)))))) H12)) (H2 v0 x i H11 v3 H8)) w1 H10)))) H9)) (\lambda (H9: (ex2 T -(\lambda (t5: T).(eq T w1 (THead (Flat Appl) v1 t5))) (\lambda (t5: -T).(subst0 (s (Flat Appl) i) v0 (THead (Bind b) u1 t3) t5)))).(ex2_ind T -(\lambda (t5: T).(eq T w1 (THead (Flat Appl) v1 t5))) (\lambda (t5: -T).(subst0 (s (Flat Appl) i) v0 (THead (Bind b) u1 t3) t5)) (or (pr0 w1 -(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda -(w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x: T).(\lambda (H10: (eq -T w1 (THead (Flat Appl) v1 x))).(\lambda (H11: (subst0 (s (Flat Appl) i) v0 -(THead (Bind b) u1 t3) x)).(or3_ind (ex2 T (\lambda (u3: T).(eq T x (THead -(Bind b) u3 t3))) (\lambda (u3: T).(subst0 (s (Flat Appl) i) v0 u1 u3))) (ex2 -T (\lambda (t5: T).(eq T x (THead (Bind b) u1 t5))) (\lambda (t5: T).(subst0 -(s (Bind b) (s (Flat Appl) i)) v0 t3 t5))) (ex3_2 T T (\lambda (u3: -T).(\lambda (t5: T).(eq T x (THead (Bind b) u3 t5)))) (\lambda (u3: -T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u1 u3))) (\lambda (_: -T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 t5)))) (or -(pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T -(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H12: (ex2 T -(\lambda (u3: T).(eq T x (THead (Bind b) u3 t3))) (\lambda (u3: T).(subst0 (s -(Flat Appl) i) v0 u1 u3)))).(ex2_ind T (\lambda (u3: T).(eq T x (THead (Bind -b) u3 t3))) (\lambda (u3: T).(subst0 (s (Flat Appl) i) v0 u1 u3)) (or (pr0 w1 -(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda -(w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x0: T).(\lambda (H13: (eq -T x (THead (Bind b) x0 t3))).(\lambda (H14: (subst0 (s (Flat Appl) i) v0 u1 -x0)).(let H15 \def (eq_ind T x (\lambda (t: T).(eq T w1 (THead (Flat Appl) v1 -t))) H10 (THead (Bind b) x0 t3) H13) in (eq_ind_r T (THead (Flat Appl) v1 -(THead (Bind b) x0 t3)) (\lambda (t: T).(or (pr0 t (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 t w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t4)) w2))))) (or_ind (pr0 x0 u2) (ex2 T (\lambda (w2: T).(pr0 x0 -w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 u2 w2))) (or (pr0 (THead -(Flat Appl) v1 (THead (Bind b) x0 t3)) (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 -(THead (Bind b) x0 t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H16: (pr0 x0 -u2)).(or_introl (pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 t3)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 t3)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2))) (pr0_upsilon b H0 v1 v2 H1 x0 u2 H16 t3 t4 H5))) (\lambda (H16: (ex2 T -(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 -u2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 -(s (Flat Appl) i) v3 u2 w2)) (or (pr0 (THead (Flat Appl) v1 (THead (Bind b) -x0 t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T -(\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 t3)) w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t4)) w2)))) (\lambda (x1: T).(\lambda (H17: (pr0 x0 x1)).(\lambda -(H18: (subst0 (s (Flat Appl) i) v3 u2 x1)).(or_intror (pr0 (THead (Flat Appl) -v1 (THead (Bind b) x0 t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) -O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind -b) x0 t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead -(Flat Appl) v1 (THead (Bind b) x0 t3)) w2)) (\lambda (w2: T).(subst0 i v3 -(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)) (THead -(Bind b) x1 (THead (Flat Appl) (lift (S O) O v2) t4)) (pr0_upsilon b H0 v1 v2 -H1 x0 x1 H17 t3 t4 H5) (subst0_fst v3 x1 u2 i H18 (THead (Flat Appl) (lift (S -O) O v2) t4) (Bind b))))))) H16)) (H4 v0 x0 (s (Flat Appl) i) H14 v3 H8)) w1 -H15))))) H12)) (\lambda (H12: (ex2 T (\lambda (t5: T).(eq T x (THead (Bind b) -u1 t5))) (\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 -t5)))).(ex2_ind T (\lambda (t5: T).(eq T x (THead (Bind b) u1 t5))) (\lambda -(t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 t5)) (or (pr0 w1 (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x0: T).(\lambda (H13: (eq T x -(THead (Bind b) u1 x0))).(\lambda (H14: (subst0 (s (Bind b) (s (Flat Appl) -i)) v0 t3 x0)).(let H15 \def (eq_ind T x (\lambda (t: T).(eq T w1 (THead -(Flat Appl) v1 t))) H10 (THead (Bind b) u1 x0) H13) in (eq_ind_r T (THead -(Flat Appl) v1 (THead (Bind b) u1 x0)) (\lambda (t: T).(or (pr0 t (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) w2))))) (or_ind (pr0 x0 t4) (ex2 T (\lambda (w2: -T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 -w2))) (or (pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 x0)) (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 -(THead (Flat Appl) v1 (THead (Bind b) u1 x0)) w2)) (\lambda (w2: T).(subst0 i -v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) -(\lambda (H16: (pr0 x0 t4)).(or_introl (pr0 (THead (Flat Appl) v1 (THead -(Bind b) u1 x0)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) -t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 -x0)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4)) w2))) (pr0_upsilon b H0 v1 v2 H1 u1 u2 H3 x0 t4 H16))) -(\lambda (H16: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 -(s (Bind b) (s (Flat Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 -x0 w2)) (\lambda (w2: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 w2)) -(or (pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 x0)) (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 -(THead (Flat Appl) v1 (THead (Bind b) u1 x0)) w2)) (\lambda (w2: T).(subst0 i -v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) -(\lambda (x1: T).(\lambda (H17: (pr0 x0 x1)).(\lambda (H18: (subst0 (s (Bind -b) (s (Flat Appl) i)) v3 t4 x1)).(or_intror (pr0 (THead (Flat Appl) v1 (THead -(Bind b) u1 x0)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) -t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 -x0)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat -Appl) v1 (THead (Bind b) u1 x0)) w2)) (\lambda (w2: T).(subst0 i v3 (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) x1)) (pr0_upsilon b H0 v1 v2 H1 u1 u2 H3 -x0 x1 H17) (subst0_snd (Bind b) v3 (THead (Flat Appl) (lift (S O) O v2) x1) -(THead (Flat Appl) (lift (S O) O v2) t4) i (subst0_snd (Flat Appl) v3 x1 t4 -(s (Bind b) i) H18 (lift (S O) O v2)) u2)))))) H16)) (H6 v0 x0 (s (Bind b) (s -(Flat Appl) i)) H14 v3 H8)) w1 H15))))) H12)) (\lambda (H12: (ex3_2 T T -(\lambda (u3: T).(\lambda (t5: T).(eq T x (THead (Bind b) u3 t5)))) (\lambda -(u3: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u1 u3))) (\lambda (_: -T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 -t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t5: T).(eq T x (THead (Bind -b) u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u1 -u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) -v0 t3 t5))) (or (pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O -v2) t4))) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 -(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda -(x0: T).(\lambda (x1: T).(\lambda (H13: (eq T x (THead (Bind b) x0 -x1))).(\lambda (H14: (subst0 (s (Flat Appl) i) v0 u1 x0)).(\lambda (H15: -(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 x1)).(let H16 \def (eq_ind T x -(\lambda (t: T).(eq T w1 (THead (Flat Appl) v1 t))) H10 (THead (Bind b) x0 -x1) H13) in (eq_ind_r T (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) -(\lambda (t: T).(or (pr0 t (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) -O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 -(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2))))) (or_ind -(pr0 x1 t4) (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s -(Bind b) (s (Flat Appl) i)) v3 t4 w2))) (or (pr0 (THead (Flat Appl) v1 (THead -(Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) -t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 -x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4)) w2)))) (\lambda (H17: (pr0 x1 t4)).(or_ind (pr0 x0 u2) -(ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) -i) v3 u2 w2))) (or (pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2)))) (\lambda (H18: (pr0 x0 u2)).(or_introl (pr0 (THead (Flat Appl) v1 -(THead (Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O -v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) -x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) w2))) (pr0_upsilon b H0 v1 v2 H1 x0 u2 H18 x1 t4 -H17))) (\lambda (H18: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: -T).(subst0 (s (Flat Appl) i) v3 u2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 -w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 u2 w2)) (or (pr0 (THead -(Flat Appl) v1 (THead (Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 -(THead (Bind b) x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x2: T).(\lambda -(H19: (pr0 x0 x2)).(\lambda (H20: (subst0 (s (Flat Appl) i) v3 u2 -x2)).(or_intror (pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind -b) x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) x2 (THead (Flat Appl) (lift -(S O) O v2) t4)) (pr0_upsilon b H0 v1 v2 H1 x0 x2 H19 x1 t4 H17) (subst0_fst -v3 x2 u2 i H20 (THead (Flat Appl) (lift (S O) O v2) t4) (Bind b))))))) H18)) -(H4 v0 x0 (s (Flat Appl) i) H14 v3 H8))) (\lambda (H17: (ex2 T (\lambda (w2: -T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 -w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s -(Bind b) (s (Flat Appl) i)) v3 t4 w2)) (or (pr0 (THead (Flat Appl) v1 (THead -(Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) -t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 -x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4)) w2)))) (\lambda (x2: T).(\lambda (H18: (pr0 x1 -x2)).(\lambda (H19: (subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 x2)).(or_ind -(pr0 x0 u2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s -(Flat Appl) i) v3 u2 w2))) (or (pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 -x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T -(\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t4)) w2)))) (\lambda (H20: (pr0 x0 u2)).(or_intror (pr0 (THead (Flat -Appl) v1 (THead (Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift -(S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead -(Bind b) x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 -(THead (Flat Appl) v1 (THead (Bind b) x0 x1)) w2)) (\lambda (w2: T).(subst0 i -v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) x2)) (pr0_upsilon b H0 v1 v2 -H1 x0 u2 H20 x1 x2 H18) (subst0_snd (Bind b) v3 (THead (Flat Appl) (lift (S -O) O v2) x2) (THead (Flat Appl) (lift (S O) O v2) t4) i (subst0_snd (Flat -Appl) v3 x2 t4 (s (Bind b) i) H19 (lift (S O) O v2)) u2)))) (\lambda (H20: -(ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) -i) v3 u2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: -T).(subst0 (s (Flat Appl) i) v3 u2 w2)) (or (pr0 (THead (Flat Appl) v1 (THead -(Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) -t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 -x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4)) w2)))) (\lambda (x3: T).(\lambda (H21: (pr0 x0 -x3)).(\lambda (H22: (subst0 (s (Flat Appl) i) v3 u2 x3)).(or_intror (pr0 -(THead (Flat Appl) v1 (THead (Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) -v1 (THead (Bind b) x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2)) (THead (Bind b) x3 (THead (Flat Appl) (lift (S O) O v2) x2)) -(pr0_upsilon b H0 v1 v2 H1 x0 x3 H21 x1 x2 H18) (subst0_both v3 u2 x3 i H22 -(Bind b) (THead (Flat Appl) (lift (S O) O v2) t4) (THead (Flat Appl) (lift (S -O) O v2) x2) (subst0_snd (Flat Appl) v3 x2 t4 (s (Bind b) i) H19 (lift (S O) -O v2)))))))) H20)) (H4 v0 x0 (s (Flat Appl) i) H14 v3 H8))))) H17)) (H6 v0 x1 -(s (Bind b) (s (Flat Appl) i)) H15 v3 H8)) w1 H16))))))) H12)) -(subst0_gen_head (Bind b) v0 u1 t3 x (s (Flat Appl) i) H11))))) H9)) (\lambda -(H9: (ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T w1 (THead (Flat Appl) -u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v0 v1 u3))) (\lambda (_: -T).(\lambda (t5: T).(subst0 (s (Flat Appl) i) v0 (THead (Bind b) u1 t3) -t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t5: T).(eq T w1 (THead -(Flat Appl) u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v0 v1 u3))) -(\lambda (_: T).(\lambda (t5: T).(subst0 (s (Flat Appl) i) v0 (THead (Bind b) -u1 t3) t5))) (or (pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O -v2) t4))) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 -(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda -(x0: T).(\lambda (x1: T).(\lambda (H10: (eq T w1 (THead (Flat Appl) x0 -x1))).(\lambda (H11: (subst0 i v0 v1 x0)).(\lambda (H12: (subst0 (s (Flat -Appl) i) v0 (THead (Bind b) u1 t3) x1)).(or3_ind (ex2 T (\lambda (u3: T).(eq -T x1 (THead (Bind b) u3 t3))) (\lambda (u3: T).(subst0 (s (Flat Appl) i) v0 -u1 u3))) (ex2 T (\lambda (t5: T).(eq T x1 (THead (Bind b) u1 t5))) (\lambda -(t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 t5))) (ex3_2 T T -(\lambda (u3: T).(\lambda (t5: T).(eq T x1 (THead (Bind b) u3 t5)))) (\lambda -(u3: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u1 u3))) (\lambda (_: -T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 t5)))) (or -(pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T -(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H13: (ex2 T -(\lambda (u3: T).(eq T x1 (THead (Bind b) u3 t3))) (\lambda (u3: T).(subst0 -(s (Flat Appl) i) v0 u1 u3)))).(ex2_ind T (\lambda (u3: T).(eq T x1 (THead -(Bind b) u3 t3))) (\lambda (u3: T).(subst0 (s (Flat Appl) i) v0 u1 u3)) (or -(pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T -(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x: T).(\lambda -(H14: (eq T x1 (THead (Bind b) x t3))).(\lambda (H15: (subst0 (s (Flat Appl) -i) v0 u1 x)).(let H16 \def (eq_ind T x1 (\lambda (t: T).(eq T w1 (THead (Flat -Appl) x0 t))) H10 (THead (Bind b) x t3) H14) in (eq_ind_r T (THead (Flat -Appl) x0 (THead (Bind b) x t3)) (\lambda (t: T).(or (pr0 t (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 t -w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4)) w2))))) (or_ind (pr0 x u2) (ex2 T (\lambda (w2: -T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 u2 w2))) (or -(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat -Appl) x0 (THead (Bind b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H17: -(pr0 x u2)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda -(w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) -x t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T -(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t4)) w2)))) (\lambda (H18: (pr0 x0 v2)).(or_introl (pr0 (THead (Flat -Appl) x0 (THead (Bind b) x t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift -(S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead -(Bind b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t4)) w2))) (pr0_upsilon b H0 x0 v2 H18 x u2 H17 -t3 t4 H5))) (\lambda (H18: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: -T).(subst0 i v3 v2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda -(w2: T).(subst0 i v3 v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x -t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T -(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t4)) w2)))) (\lambda (x2: T).(\lambda (H19: (pr0 x0 x2)).(\lambda -(H20: (subst0 i v3 v2 x2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind -b) x t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 -T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 -(THead (Bind b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O x2) t4)) (pr0_upsilon b H0 x0 x2 H19 x u2 H17 t3 t4 -H5) (subst0_snd (Bind b) v3 (THead (Flat Appl) (lift (S O) O x2) t4) (THead -(Flat Appl) (lift (S O) O v2) t4) i (subst0_fst v3 (lift (S O) O x2) (lift (S -O) O v2) (s (Bind b) i) (subst0_lift_ge_s v2 x2 v3 i H20 O (le_O_n i) b) t4 -(Flat Appl)) u2)))))) H18)) (H2 v0 x0 i H11 v3 H8))) (\lambda (H17: (ex2 T -(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 u2 -w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s -(Flat Appl) i) v3 u2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x -t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T -(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) -(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S -O) O v2) t4)) w2)))) (\lambda (x2: T).(\lambda (H18: (pr0 x x2)).(\lambda -(H19: (subst0 (s (Flat Appl) i) v3 u2 x2)).(or_ind (pr0 x0 v2) (ex2 T -(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 -(THead (Flat Appl) x0 (THead (Bind b) x t3)) (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) -x0 (THead (Bind b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H20: (pr0 x0 -v2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind -b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) x2 (THead (Flat Appl) (lift -(S O) O v2) t4)) (pr0_upsilon b H0 x0 v2 H20 x x2 H18 t3 t4 H5) (subst0_fst -v3 x2 u2 i H19 (THead (Flat Appl) (lift (S O) O v2) t4) (Bind b))))) (\lambda -(H20: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 -w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 -v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 -(THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) (\lambda (w2: T).(subst0 i -v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) -(\lambda (x3: T).(\lambda (H21: (pr0 x0 x3)).(\lambda (H22: (subst0 i v3 v2 -x3)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind -b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) x2 (THead (Flat Appl) (lift -(S O) O x3) t4)) (pr0_upsilon b H0 x0 x3 H21 x x2 H18 t3 t4 H5) (subst0_both -v3 u2 x2 i H19 (Bind b) (THead (Flat Appl) (lift (S O) O v2) t4) (THead (Flat -Appl) (lift (S O) O x3) t4) (subst0_fst v3 (lift (S O) O x3) (lift (S O) O -v2) (s (Bind b) i) (subst0_lift_ge_s v2 x3 v3 i H22 O (le_O_n i) b) t4 (Flat -Appl)))))))) H20)) (H2 v0 x0 i H11 v3 H8))))) H17)) (H4 v0 x (s (Flat Appl) -i) H15 v3 H8)) w1 H16))))) H13)) (\lambda (H13: (ex2 T (\lambda (t5: T).(eq T -x1 (THead (Bind b) u1 t5))) (\lambda (t5: T).(subst0 (s (Bind b) (s (Flat -Appl) i)) v0 t3 t5)))).(ex2_ind T (\lambda (t5: T).(eq T x1 (THead (Bind b) -u1 t5))) (\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 t5)) -(or (pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) -(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x: -T).(\lambda (H14: (eq T x1 (THead (Bind b) u1 x))).(\lambda (H15: (subst0 (s -(Bind b) (s (Flat Appl) i)) v0 t3 x)).(let H16 \def (eq_ind T x1 (\lambda (t: -T).(eq T w1 (THead (Flat Appl) x0 t))) H10 (THead (Bind b) u1 x) H14) in -(eq_ind_r T (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (\lambda (t: T).(or -(pr0 t (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T -(\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t4)) w2))))) (or_ind (pr0 x t4) (ex2 T -(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind b) (s (Flat -Appl) i)) v3 t4 w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) -(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda -(w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2)))) (\lambda (H17: (pr0 x t4)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2: -T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat -Appl) x0 (THead (Bind b) u1 x)) (THead (Bind b) u2 (THead (Flat Appl) (lift -(S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead -(Bind b) u1 x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H18: (pr0 x0 -v2)).(or_introl (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2))) (pr0_upsilon b H0 x0 v2 H18 u1 u2 H3 x t4 H17))) (\lambda (H18: (ex2 T -(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 -w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 -v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 -(THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: T).(subst0 i -v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) -(\lambda (x2: T).(\lambda (H19: (pr0 x0 x2)).(\lambda (H20: (subst0 i v3 v2 -x2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind -b) u1 x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 (THead (Flat Appl) (lift -(S O) O x2) t4)) (pr0_upsilon b H0 x0 x2 H19 u1 u2 H3 x t4 H17) (subst0_snd -(Bind b) v3 (THead (Flat Appl) (lift (S O) O x2) t4) (THead (Flat Appl) (lift -(S O) O v2) t4) i (subst0_fst v3 (lift (S O) O x2) (lift (S O) O v2) (s (Bind -b) i) (subst0_lift_ge_s v2 x2 v3 i H20 O (le_O_n i) b) t4 (Flat Appl)) -u2)))))) H18)) (H2 v0 x0 i H11 v3 H8))) (\lambda (H17: (ex2 T (\lambda (w2: -T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 -w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s -(Bind b) (s (Flat Appl) i)) v3 t4 w2)) (or (pr0 (THead (Flat Appl) x0 (THead -(Bind b) u1 x)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) -(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) -w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4)) w2)))) (\lambda (x2: T).(\lambda (H18: (pr0 x -x2)).(\lambda (H19: (subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 x2)).(or_ind -(pr0 x0 v2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i -v3 v2 w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead 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t4 (s (Bind b) i) H19 (lift (S O) O v2)) u2)))) (\lambda (H20: (ex2 T -(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 -w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 -v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 -(THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: T).(subst0 i -v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) -(\lambda (x3: T).(\lambda (H21: (pr0 x0 x3)).(\lambda (H22: (subst0 i v3 v2 -x3)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind -b) u1 x)) w2)) (\lambda 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T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat -Appl) i)) v0 t3 t5))) (or (pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift -(S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H14: (eq T x1 (THead (Bind -b) x2 x3))).(\lambda (H15: (subst0 (s (Flat Appl) i) v0 u1 x2)).(\lambda -(H16: (subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 x3)).(let H17 \def (eq_ind -T x1 (\lambda (t: T).(eq T w1 (THead (Flat Appl) x0 t))) H10 (THead (Bind b) -x2 x3) H14) in (eq_ind_r T (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) -(\lambda (t: T).(or (pr0 t (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) -O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 -(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2))))) (or_ind -(pr0 x3 t4) (ex2 T (\lambda (w2: T).(pr0 x3 w2)) (\lambda (w2: T).(subst0 (s -(Bind b) (s (Flat Appl) 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(\lambda (w2: T).(subst0 (s (Flat Appl) -i) v3 u2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x2 w2)) (\lambda (w2: -T).(subst0 (s (Flat Appl) i) v3 u2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead -(Bind b) x2 x3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) -t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 -x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4)) w2)))) (\lambda (x4: T).(\lambda (H22: (pr0 x2 -x4)).(\lambda (H23: (subst0 (s (Flat Appl) i) v3 u2 x4)).(or_ind (pr0 x0 v2) -(ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) -(or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 -(THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i -v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) -(\lambda (H24: (pr0 x0 v2)).(or_intror (pr0 (THead 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(THead (Bind -b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 -(THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i -v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) -(\lambda (x5: T).(\lambda (H25: (pr0 x0 x5)).(\lambda (H26: (subst0 i v3 v2 -x5)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: -T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) -w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind -b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) x4 (THead (Flat Appl) (lift -(S O) O x5) x)) (pr0_upsilon b H0 x0 x5 H25 x2 x4 H22 x3 x H19) (subst0_both -v3 u2 x4 i H23 (Bind b) (THead (Flat Appl) (lift (S O) O v2) t4) (THead (Flat -Appl) (lift (S O) O x5) x) (subst0_both v3 (lift (S O) O v2) (lift (S O) O -x5) (s (Bind b) i) (subst0_lift_ge_s v2 x5 v3 i H26 O (le_O_n i) b) (Flat -Appl) t4 x H20))))))) H24)) (H2 v0 x0 i H11 v3 H8))))) H21)) (H4 v0 x2 (s -(Flat Appl) i) H15 v3 H8))))) H18)) (H6 v0 x3 (s (Bind b) (s (Flat Appl) i)) -H16 v3 H8)) w1 H17))))))) H13)) (subst0_gen_head (Bind b) v0 u1 t3 x1 (s -(Flat Appl) i) H12))))))) H9)) (subst0_gen_head (Flat Appl) v0 v1 (THead -(Bind b) u1 t3) w1 i H7)))))))))))))))))))))) (\lambda (u1: T).(\lambda (u2: -T).(\lambda (H0: (pr0 u1 u2)).(\lambda (H1: ((\forall (v1: T).(\forall (w1: -T).(\forall (i: nat).((subst0 i v1 u1 w1) \to (\forall (v2: T).((pr0 v1 v2) -\to (or (pr0 w1 u2) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: -T).(subst0 i v2 u2 w2)))))))))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda -(H2: (pr0 t3 t4)).(\lambda (H3: ((\forall (v1: T).(\forall (w1: T).(\forall -(i: nat).((subst0 i v1 t3 w1) \to (\forall (v2: T).((pr0 v1 v2) \to (or (pr0 -w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 -w2)))))))))))).(\lambda (w: T).(\lambda (H4: (subst0 O u2 t4 w)).(\lambda -(v1: T).(\lambda (w1: T).(\lambda (i: nat).(\lambda (H5: (subst0 i v1 (THead -(Bind Abbr) u1 t3) w1)).(\lambda (v2: T).(\lambda (H6: (pr0 v1 v2)).(or3_ind -(ex2 T (\lambda (u3: T).(eq T w1 (THead (Bind Abbr) u3 t3))) (\lambda (u3: -T).(subst0 i v1 u1 u3))) (ex2 T (\lambda (t5: T).(eq T w1 (THead (Bind Abbr) -u1 t5))) (\lambda (t5: T).(subst0 (s (Bind Abbr) i) v1 t3 t5))) (ex3_2 T T -(\lambda (u3: T).(\lambda (t5: T).(eq T w1 (THead (Bind Abbr) u3 t5)))) -(\lambda (u3: T).(\lambda (_: T).(subst0 i v1 u1 u3))) (\lambda (_: -T).(\lambda (t5: T).(subst0 (s (Bind Abbr) i) v1 t3 t5)))) (or (pr0 w1 (THead -(Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: -T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (H7: (ex2 T (\lambda -(u3: T).(eq T w1 (THead (Bind Abbr) u3 t3))) (\lambda (u3: T).(subst0 i v1 u1 -u3)))).(ex2_ind T (\lambda (u3: T).(eq T w1 (THead (Bind Abbr) u3 t3))) -(\lambda (u3: T).(subst0 i v1 u1 u3)) (or (pr0 w1 (THead (Bind Abbr) u2 w)) -(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 (THead -(Bind Abbr) u2 w) w2)))) (\lambda (x: T).(\lambda (H8: (eq T w1 (THead (Bind -Abbr) x t3))).(\lambda (H9: (subst0 i v1 u1 x)).(eq_ind_r T (THead (Bind -Abbr) x t3) (\lambda (t: T).(or (pr0 t (THead (Bind Abbr) u2 w)) (ex2 T -(\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) -u2 w) w2))))) (or_ind (pr0 x u2) (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda -(w2: T).(subst0 i v2 u2 w2))) (or (pr0 (THead (Bind Abbr) x t3) (THead (Bind -Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x t3) w2)) -(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (H10: -(pr0 x u2)).(or_introl (pr0 (THead (Bind Abbr) x t3) (THead (Bind Abbr) u2 -w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x t3) w2)) (\lambda (w2: -T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))) (pr0_delta x u2 H10 t3 t4 H2 w -H4))) (\lambda (H10: (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: -T).(subst0 i v2 u2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda -(w2: T).(subst0 i v2 u2 w2)) (or (pr0 (THead (Bind Abbr) x t3) (THead (Bind -Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x t3) w2)) -(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x0: -T).(\lambda (H11: (pr0 x x0)).(\lambda (H12: (subst0 i v2 u2 x0)).(ex2_ind T -(\lambda (t: T).(subst0 O x0 t4 t)) (\lambda (t: T).(subst0 (S (plus i O)) v2 -w t)) (or (pr0 (THead (Bind Abbr) x t3) (THead (Bind Abbr) u2 w)) (ex2 T -(\lambda (w2: T).(pr0 (THead (Bind Abbr) x t3) w2)) (\lambda (w2: T).(subst0 -i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x1: T).(\lambda (H13: (subst0 -O x0 t4 x1)).(\lambda (H14: (subst0 (S (plus i O)) v2 w x1)).(let H15 \def -(f_equal nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O i))) in -(let H16 \def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n v2 w -x1)) H14 (S i) H15) in (or_intror (pr0 (THead (Bind Abbr) x t3) (THead (Bind -Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x t3) w2)) -(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))) (ex_intro2 T -(\lambda (w2: T).(pr0 (THead (Bind Abbr) x t3) w2)) (\lambda (w2: T).(subst0 -i v2 (THead (Bind Abbr) u2 w) w2)) (THead (Bind Abbr) x0 x1) (pr0_delta x x0 -H11 t3 t4 H2 x1 H13) (subst0_both v2 u2 x0 i H12 (Bind Abbr) w x1 H16)))))))) -(subst0_subst0_back t4 w u2 O H4 x0 v2 i H12))))) H10)) (H1 v1 x i H9 v2 H6)) -w1 H8)))) H7)) (\lambda (H7: (ex2 T (\lambda (t5: T).(eq T w1 (THead (Bind -Abbr) u1 t5))) (\lambda (t5: T).(subst0 (s (Bind Abbr) i) v1 t3 -t5)))).(ex2_ind T (\lambda (t5: T).(eq T w1 (THead (Bind Abbr) u1 t5))) -(\lambda (t5: T).(subst0 (s (Bind Abbr) i) v1 t3 t5)) (or (pr0 w1 (THead -(Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: -T).(subst0 i v2 (THead (Bind 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(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: -T).(subst0 (s (Bind Abbr) i) v2 t4 w2)) (or (pr0 (THead (Bind Abbr) u1 x) -(THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) u1 -x) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) -(\lambda (x0: T).(\lambda (H11: (pr0 x x0)).(\lambda (H12: (subst0 (s (Bind -Abbr) i) v2 t4 x0)).(ex2_ind T (\lambda (t: T).(subst0 O u2 x0 t)) (\lambda -(t: T).(subst0 (s (Bind Abbr) i) v2 w t)) (or (pr0 (THead (Bind Abbr) u1 x) -(THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) u1 -x) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) -(\lambda (x1: T).(\lambda (H13: (subst0 O u2 x0 x1)).(\lambda (H14: (subst0 -(s (Bind Abbr) i) v2 w x1)).(or_intror (pr0 (THead (Bind Abbr) u1 x) (THead -(Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) u1 x) w2)) -(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))) (ex_intro2 T -(\lambda (w2: T).(pr0 (THead (Bind Abbr) u1 x) w2)) (\lambda (w2: T).(subst0 -i v2 (THead (Bind Abbr) u2 w) w2)) (THead (Bind Abbr) u2 x1) (pr0_delta u1 u2 -H0 x x0 H11 x1 H13) (subst0_snd (Bind Abbr) v2 x1 w i H14 u2)))))) -(subst0_confluence_neq t4 x0 v2 (s (Bind Abbr) i) H12 w u2 O H4 (sym_not_eq -nat O (S i) (O_S i))))))) H10)) (H3 v1 x (s (Bind Abbr) i) H9 v2 H6)) w1 -H8)))) H7)) (\lambda (H7: (ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T -w1 (THead (Bind Abbr) u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v1 -u1 u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind Abbr) i) v1 t3 -t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t5: T).(eq T w1 (THead -(Bind Abbr) u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v1 u1 u3))) -(\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind Abbr) i) v1 t3 t5))) (or -(pr0 w1 (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) -(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H8: (eq T w1 (THead (Bind 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-w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: -T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))) (pr0_delta x0 u2 H12 x1 t4 H11 -w H4))) (\lambda (H12: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: -T).(subst0 i v2 u2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda -(w2: T).(subst0 i v2 u2 w2)) (or (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind -Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) -(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x: -T).(\lambda (H13: (pr0 x0 x)).(\lambda (H14: (subst0 i v2 u2 x)).(ex2_ind T -(\lambda (t: T).(subst0 O x t4 t)) (\lambda (t: T).(subst0 (S (plus i O)) v2 -w t)) (or (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T -(\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 -i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x2: T).(\lambda (H15: (subst0 -O x t4 x2)).(\lambda (H16: (subst0 (S (plus i O)) v2 w x2)).(let H17 \def -(f_equal nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O i))) in -(let H18 \def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n v2 w -x2)) H16 (S i) H17) in (or_intror (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind -Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) -(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))) (ex_intro2 T -(\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 -i v2 (THead (Bind Abbr) u2 w) w2)) (THead (Bind Abbr) x x2) (pr0_delta x0 x -H13 x1 t4 H11 x2 H15) (subst0_both v2 u2 x i H14 (Bind Abbr) w x2 H18)))))))) -(subst0_subst0_back t4 w u2 O H4 x v2 i H14))))) H12)) (H1 v1 x0 i H9 v2 -H6))) (\lambda (H11: (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: -T).(subst0 (s (Bind Abbr) i) v2 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x1 -w2)) (\lambda (w2: T).(subst0 (s (Bind Abbr) i) v2 t4 w2)) (or (pr0 (THead -(Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 -(THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind -Abbr) u2 w) w2)))) (\lambda (x: T).(\lambda (H12: (pr0 x1 x)).(\lambda (H13: -(subst0 (s (Bind Abbr) i) v2 t4 x)).(or_ind (pr0 x0 u2) (ex2 T (\lambda (w2: -T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 u2 w2))) (or (pr0 (THead (Bind -Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead -(Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 -w) w2)))) (\lambda (H14: (pr0 x0 u2)).(ex2_ind T (\lambda (t: T).(subst0 O u2 -x t)) (\lambda (t: T).(subst0 (s (Bind Abbr) i) v2 w t)) (or (pr0 (THead -(Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 -(THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind -Abbr) u2 w) w2)))) (\lambda (x2: T).(\lambda (H15: (subst0 O u2 x -x2)).(\lambda (H16: (subst0 (s (Bind Abbr) i) v2 w x2)).(or_intror (pr0 -(THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: -T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead -(Bind Abbr) u2 w) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) -x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)) -(THead (Bind Abbr) u2 x2) (pr0_delta x0 u2 H14 x1 x H12 x2 H15) (subst0_snd -(Bind Abbr) v2 x2 w i H16 u2)))))) (subst0_confluence_neq t4 x v2 (s (Bind -Abbr) i) H13 w u2 O H4 (sym_not_eq nat O (S i) (O_S i))))) (\lambda (H14: -(ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 u2 -w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 -u2 w2)) (or (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T -(\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 -i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x2: T).(\lambda (H15: (pr0 x0 -x2)).(\lambda (H16: (subst0 i v2 u2 x2)).(ex2_ind T (\lambda (t: T).(subst0 O -x2 t4 t)) (\lambda (t: T).(subst0 (S (plus i O)) v2 w t)) (or (pr0 (THead -(Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 -(THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind -Abbr) u2 w) w2)))) (\lambda (x3: T).(\lambda (H17: (subst0 O x2 t4 -x3)).(\lambda (H18: (subst0 (S (plus i O)) v2 w x3)).(let H19 \def (f_equal -nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O i))) in (let H20 -\def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n v2 w x3)) H18 (S -i) H19) in (ex2_ind T (\lambda (t: T).(subst0 (s (Bind Abbr) i) v2 x3 t)) -(\lambda (t: T).(subst0 O x2 x t)) (or (pr0 (THead (Bind Abbr) x0 x1) (THead -(Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) -w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda -(x4: T).(\lambda (H21: (subst0 (s (Bind Abbr) i) v2 x3 x4)).(\lambda (H22: -(subst0 O x2 x x4)).(or_intror (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind -Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) -(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))) (ex_intro2 T -(\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 -i v2 (THead (Bind Abbr) u2 w) w2)) (THead (Bind Abbr) x2 x4) (pr0_delta x0 x2 -H15 x1 x H12 x4 H22) (subst0_both v2 u2 x2 i H16 (Bind Abbr) w x4 -(subst0_trans x3 w v2 (s (Bind Abbr) i) H20 x4 H21))))))) -(subst0_confluence_neq t4 x3 x2 O H17 x v2 (s (Bind Abbr) i) H13 (O_S -i)))))))) (subst0_subst0_back t4 w u2 O H4 x2 v2 i H16))))) H14)) (H1 v1 x0 i -H9 v2 H6))))) H11)) (H3 v1 x1 (s (Bind Abbr) i) H10 v2 H6)) w1 H8)))))) H7)) -(subst0_gen_head (Bind Abbr) v1 u1 t3 w1 i H5)))))))))))))))))) (\lambda (b: -B).(\lambda (H0: (not (eq B b Abst))).(\lambda (t3: T).(\lambda (t4: -T).(\lambda (H1: (pr0 t3 t4)).(\lambda (H2: ((\forall (v1: T).(\forall (w1: -T).(\forall (i: nat).((subst0 i v1 t3 w1) \to (\forall (v2: T).((pr0 v1 v2) -\to (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: -T).(subst0 i v2 t4 w2)))))))))))).(\lambda (u: T).(\lambda (v1: T).(\lambda -(w1: T).(\lambda (i: nat).(\lambda (H3: (subst0 i v1 (THead (Bind b) u (lift -(S O) O t3)) w1)).(\lambda (v2: T).(\lambda (H4: (pr0 v1 v2)).(or3_ind (ex2 T -(\lambda (u2: T).(eq T w1 (THead (Bind b) u2 (lift (S O) O t3)))) (\lambda -(u2: T).(subst0 i v1 u u2))) (ex2 T (\lambda (t5: T).(eq T w1 (THead (Bind b) -u t5))) (\lambda (t5: T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5))) -(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T w1 (THead (Bind b) u2 -t5)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: -T).(\lambda (t5: T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5)))) (or -(pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i -v2 t4 w2)))) (\lambda (H5: (ex2 T (\lambda (u2: T).(eq T w1 (THead (Bind b) -u2 (lift (S O) O t3)))) (\lambda (u2: T).(subst0 i v1 u u2)))).(ex2_ind T -(\lambda (u2: T).(eq T w1 (THead (Bind b) u2 (lift (S O) O t3)))) (\lambda -(u2: T).(subst0 i v1 u u2)) (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 -w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H6: -(eq T w1 (THead (Bind b) x (lift (S O) O t3)))).(\lambda (_: (subst0 i v1 u -x)).(eq_ind_r T (THead (Bind b) x (lift (S O) O t3)) (\lambda (t: T).(or (pr0 -t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4 -w2))))) (or_introl (pr0 (THead (Bind b) x (lift (S O) O t3)) t4) (ex2 T -(\lambda (w2: T).(pr0 (THead (Bind b) x (lift (S O) O t3)) w2)) (\lambda (w2: -T).(subst0 i v2 t4 w2))) (pr0_zeta b H0 t3 t4 H1 x)) w1 H6)))) H5)) (\lambda -(H5: (ex2 T (\lambda (t5: T).(eq T w1 (THead (Bind b) u t5))) (\lambda (t5: -T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5)))).(ex2_ind T (\lambda -(t5: T).(eq T w1 (THead (Bind b) u t5))) (\lambda (t5: T).(subst0 (s (Bind b) -i) v1 (lift (S O) O t3) t5)) (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 -w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H6: -(eq T w1 (THead (Bind b) u x))).(\lambda (H7: (subst0 (s (Bind b) i) v1 (lift -(S O) O t3) x)).(ex2_ind T (\lambda (t5: T).(eq T x (lift (S O) O t5))) -(\lambda (t5: T).(subst0 (minus (s (Bind b) i) (S O)) v1 t3 t5)) (or (pr0 w1 -t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 -w2)))) (\lambda (x0: T).(\lambda (H8: (eq T x (lift (S O) O x0))).(\lambda -(H9: (subst0 (minus (s (Bind b) i) (S O)) v1 t3 x0)).(let H10 \def (eq_ind T -x (\lambda (t: T).(eq T w1 (THead (Bind b) u t))) H6 (lift (S O) O x0) H8) in -(eq_ind_r T (THead (Bind b) u (lift (S O) O x0)) (\lambda (t: T).(or (pr0 t -t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4 -w2))))) (let H11 \def (eq_ind_r nat (minus i O) (\lambda (n: nat).(subst0 n -v1 t3 x0)) H9 i (minus_n_O i)) in (or_ind (pr0 x0 t4) (ex2 T (\lambda (w2: -T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (or (pr0 (THead (Bind -b) u (lift (S O) O x0)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) u -(lift (S O) O x0)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda -(H12: (pr0 x0 t4)).(or_introl (pr0 (THead (Bind b) u (lift (S O) O x0)) t4) -(ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) u (lift (S O) O x0)) w2)) -(\lambda (w2: T).(subst0 i v2 t4 w2))) (pr0_zeta b H0 x0 t4 H12 u))) (\lambda -(H12: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 t4 -w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 -t4 w2)) (or (pr0 (THead (Bind b) u (lift (S O) O x0)) t4) (ex2 T (\lambda -(w2: T).(pr0 (THead (Bind b) u (lift (S O) O x0)) w2)) (\lambda (w2: -T).(subst0 i v2 t4 w2)))) (\lambda (x1: T).(\lambda (H13: (pr0 x0 -x1)).(\lambda (H14: (subst0 i v2 t4 x1)).(or_intror (pr0 (THead (Bind b) u -(lift (S O) O x0)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) u (lift -(S O) O x0)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (ex_intro2 T -(\lambda (w2: T).(pr0 (THead (Bind b) u (lift (S O) O x0)) w2)) (\lambda (w2: -T).(subst0 i v2 t4 w2)) x1 (pr0_zeta b H0 x0 x1 H13 u) H14))))) H12)) (H2 v1 -x0 i H11 v2 H4))) w1 H10))))) (subst0_gen_lift_ge v1 t3 x (s (Bind b) i) (S -O) O H7 (le_n_S O i (le_O_n i))))))) H5)) (\lambda (H5: (ex3_2 T T (\lambda -(u2: T).(\lambda (t5: T).(eq T w1 (THead (Bind b) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: -T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5))))).(ex3_2_ind T T -(\lambda (u2: T).(\lambda (t5: T).(eq T w1 (THead (Bind b) u2 t5)))) (\lambda -(u2: T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: -T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5))) (or (pr0 w1 t4) (ex2 T -(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) -(\lambda (x0: T).(\lambda (x1: T).(\lambda (H6: (eq T w1 (THead (Bind b) x0 -x1))).(\lambda (_: (subst0 i v1 u x0)).(\lambda (H8: (subst0 (s (Bind b) i) -v1 (lift (S O) O t3) x1)).(ex2_ind T (\lambda (t5: T).(eq T x1 (lift (S O) O -t5))) (\lambda (t5: T).(subst0 (minus (s (Bind b) i) (S O)) v1 t3 t5)) (or -(pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i -v2 t4 w2)))) (\lambda (x: T).(\lambda (H9: (eq T x1 (lift (S O) O -x))).(\lambda (H10: (subst0 (minus (s (Bind b) i) (S O)) v1 t3 x)).(let H11 -\def (eq_ind T x1 (\lambda (t: T).(eq T w1 (THead (Bind b) x0 t))) H6 (lift -(S O) O x) H9) in (eq_ind_r T (THead (Bind b) x0 (lift (S O) O x)) (\lambda -(t: T).(or (pr0 t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: -T).(subst0 i v2 t4 w2))))) (let H12 \def (eq_ind_r nat (minus i O) (\lambda -(n: nat).(subst0 n v1 t3 x)) H10 i (minus_n_O i)) in (or_ind (pr0 x t4) (ex2 -T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (or -(pr0 (THead (Bind b) x0 (lift (S O) O x)) t4) (ex2 T (\lambda (w2: T).(pr0 -(THead (Bind b) x0 (lift (S O) O x)) w2)) (\lambda (w2: T).(subst0 i v2 t4 -w2)))) (\lambda (H13: (pr0 x t4)).(or_introl (pr0 (THead (Bind b) x0 (lift (S -O) O x)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) x0 (lift (S O) O -x)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (pr0_zeta b H0 x t4 H13 x0))) -(\lambda (H13: (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i -v2 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 -i v2 t4 w2)) (or (pr0 (THead (Bind b) x0 (lift (S O) O x)) t4) (ex2 T -(\lambda (w2: T).(pr0 (THead (Bind b) x0 (lift (S O) O x)) w2)) (\lambda (w2: -T).(subst0 i v2 t4 w2)))) (\lambda (x2: T).(\lambda (H14: (pr0 x -x2)).(\lambda (H15: (subst0 i v2 t4 x2)).(or_intror (pr0 (THead (Bind b) x0 -(lift (S O) O x)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) x0 (lift -(S O) O x)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (ex_intro2 T (\lambda -(w2: T).(pr0 (THead (Bind b) x0 (lift (S O) O x)) w2)) (\lambda (w2: -T).(subst0 i v2 t4 w2)) x2 (pr0_zeta b H0 x x2 H14 x0) H15))))) H13)) (H2 v1 -x i H12 v2 H4))) w1 H11))))) (subst0_gen_lift_ge v1 t3 x1 (s (Bind b) i) (S -O) O H8 (le_n_S O i (le_O_n i))))))))) H5)) (subst0_gen_head (Bind b) v1 u -(lift (S O) O t3) w1 i H3))))))))))))))) (\lambda (t3: T).(\lambda (t4: -T).(\lambda (H0: (pr0 t3 t4)).(\lambda (H1: ((\forall (v1: T).(\forall (w1: -T).(\forall (i: nat).((subst0 i v1 t3 w1) \to (\forall (v2: T).((pr0 v1 v2) -\to (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: -T).(subst0 i v2 t4 w2)))))))))))).(\lambda (u: T).(\lambda (v1: T).(\lambda -(w1: T).(\lambda (i: nat).(\lambda (H2: (subst0 i v1 (THead (Flat Cast) u t3) -w1)).(\lambda (v2: T).(\lambda (H3: (pr0 v1 v2)).(or3_ind (ex2 T (\lambda -(u2: T).(eq T w1 (THead (Flat Cast) u2 t3))) (\lambda (u2: T).(subst0 i v1 u -u2))) (ex2 T (\lambda (t5: T).(eq T w1 (THead (Flat Cast) u t5))) (\lambda -(t5: T).(subst0 (s (Flat Cast) i) v1 t3 t5))) (ex3_2 T T (\lambda (u2: -T).(\lambda (t5: T).(eq T w1 (THead (Flat Cast) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: -T).(subst0 (s (Flat Cast) i) v1 t3 t5)))) (or (pr0 w1 t4) (ex2 T (\lambda -(w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (H4: -(ex2 T (\lambda (u2: T).(eq T w1 (THead (Flat Cast) u2 t3))) (\lambda (u2: -T).(subst0 i v1 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T w1 (THead (Flat -Cast) u2 t3))) (\lambda (u2: T).(subst0 i v1 u u2)) (or (pr0 w1 t4) (ex2 T -(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) -(\lambda (x: T).(\lambda (H5: (eq T w1 (THead (Flat Cast) x t3))).(\lambda -(_: (subst0 i v1 u x)).(eq_ind_r T (THead (Flat Cast) x t3) (\lambda (t: -T).(or (pr0 t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: -T).(subst0 i v2 t4 w2))))) (or_introl (pr0 (THead (Flat Cast) x t3) t4) (ex2 -T (\lambda (w2: T).(pr0 (THead (Flat Cast) x t3) w2)) (\lambda (w2: -T).(subst0 i v2 t4 w2))) (pr0_tau t3 t4 H0 x)) w1 H5)))) H4)) (\lambda (H4: -(ex2 T (\lambda (t5: T).(eq T w1 (THead (Flat Cast) u t5))) (\lambda (t5: -T).(subst0 (s (Flat Cast) i) v1 t3 t5)))).(ex2_ind T (\lambda (t5: T).(eq T -w1 (THead (Flat Cast) u t5))) (\lambda (t5: T).(subst0 (s (Flat Cast) i) v1 -t3 t5)) (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: -T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H5: (eq T w1 (THead (Flat -Cast) u x))).(\lambda (H6: (subst0 (s (Flat Cast) i) v1 t3 x)).(eq_ind_r T -(THead (Flat Cast) u x) (\lambda (t: T).(or (pr0 t t4) (ex2 T (\lambda (w2: -T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))))) (or_ind (pr0 x t4) -(ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Flat Cast) -i) v2 t4 w2))) (or (pr0 (THead (Flat Cast) u x) t4) (ex2 T (\lambda (w2: -T).(pr0 (THead (Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) -(\lambda (H7: (pr0 x t4)).(or_introl (pr0 (THead (Flat Cast) u x) t4) (ex2 T -(\lambda (w2: T).(pr0 (THead (Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i -v2 t4 w2))) (pr0_tau x t4 H7 u))) (\lambda (H7: (ex2 T (\lambda (w2: T).(pr0 -x w2)) (\lambda (w2: T).(subst0 (s (Flat Cast) i) v2 t4 w2)))).(ex2_ind T -(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Flat Cast) i) v2 t4 -w2)) (or (pr0 (THead (Flat Cast) u x) t4) (ex2 T (\lambda (w2: T).(pr0 (THead -(Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x0: -T).(\lambda (H8: (pr0 x x0)).(\lambda (H9: (subst0 (s (Flat Cast) i) v2 t4 -x0)).(or_intror (pr0 (THead (Flat Cast) u x) t4) (ex2 T (\lambda (w2: T).(pr0 -(THead (Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) -(ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) u x) w2)) (\lambda (w2: -T).(subst0 i v2 t4 w2)) x0 (pr0_tau x x0 H8 u) H9))))) H7)) (H1 v1 x (s (Flat -Cast) i) H6 v2 H3)) w1 H5)))) H4)) (\lambda (H4: (ex3_2 T T (\lambda (u2: -T).(\lambda (t5: T).(eq T w1 (THead (Flat Cast) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: -T).(subst0 (s (Flat Cast) i) v1 t3 t5))))).(ex3_2_ind T T (\lambda (u2: -T).(\lambda (t5: T).(eq T w1 (THead (Flat Cast) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: -T).(subst0 (s (Flat Cast) i) v1 t3 t5))) (or (pr0 w1 t4) (ex2 T (\lambda (w2: -T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H5: (eq T w1 (THead (Flat Cast) x0 -x1))).(\lambda (_: (subst0 i v1 u x0)).(\lambda (H7: (subst0 (s (Flat Cast) -i) v1 t3 x1)).(eq_ind_r T (THead (Flat Cast) x0 x1) (\lambda (t: T).(or (pr0 -t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4 -w2))))) (or_ind (pr0 x1 t4) (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda -(w2: T).(subst0 (s (Flat Cast) i) v2 t4 w2))) (or (pr0 (THead (Flat Cast) x0 -x1) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) x0 x1) w2)) (\lambda -(w2: T).(subst0 i v2 t4 w2)))) (\lambda (H8: (pr0 x1 t4)).(or_introl (pr0 -(THead (Flat Cast) x0 x1) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) -x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (pr0_tau x1 t4 H8 x0))) -(\lambda (H8: (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 -(s (Flat Cast) i) v2 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x1 w2)) -(\lambda (w2: T).(subst0 (s (Flat Cast) i) v2 t4 w2)) (or (pr0 (THead (Flat -Cast) x0 x1) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) x0 x1) w2)) -(\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H9: (pr0 x1 -x)).(\lambda (H10: (subst0 (s (Flat Cast) i) v2 t4 x)).(or_intror (pr0 (THead -(Flat Cast) x0 x1) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) x0 x1) -w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (ex_intro2 T (\lambda (w2: -T).(pr0 (THead (Flat Cast) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)) -x (pr0_tau x1 x H9 x0) H10))))) H8)) (H1 v1 x1 (s (Flat Cast) i) H7 v2 H3)) -w1 H5)))))) H4)) (subst0_gen_head (Flat Cast) v1 u t3 w1 i H2))))))))))))) t1 -t2 H))). -(* COMMENTS -Initial nodes: 38857 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr0/subst1.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr0/subst1.ma deleted file mode 100644 index 877f87f01..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr0/subst1.ma +++ /dev/null @@ -1,105 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr0/props.ma". - -include "Basic-1/subst1/defs.ma". - -theorem pr0_delta1: - \forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (t1: T).(\forall -(t2: T).((pr0 t1 t2) \to (\forall (w: T).((subst1 O u2 t2 w) \to (pr0 (THead -(Bind Abbr) u1 t1) (THead (Bind Abbr) u2 w))))))))) -\def - \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr0 u1 u2)).(\lambda (t1: -T).(\lambda (t2: T).(\lambda (H0: (pr0 t1 t2)).(\lambda (w: T).(\lambda (H1: -(subst1 O u2 t2 w)).(subst1_ind O u2 t2 (\lambda (t: T).(pr0 (THead (Bind -Abbr) u1 t1) (THead (Bind Abbr) u2 t))) (pr0_comp u1 u2 H t1 t2 H0 (Bind -Abbr)) (\lambda (t0: T).(\lambda (H2: (subst0 O u2 t2 t0)).(pr0_delta u1 u2 H -t1 t2 H0 t0 H2))) w H1)))))))). -(* COMMENTS -Initial nodes: 115 -END *) - -theorem pr0_subst1_back: - \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst1 -i u2 t1 t2) \to (\forall (u1: T).((pr0 u1 u2) \to (ex2 T (\lambda (t: -T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t t2))))))))) -\def - \lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (i: nat).(\lambda -(H: (subst1 i u2 t1 t2)).(subst1_ind i u2 t1 (\lambda (t: T).(\forall (u1: -T).((pr0 u1 u2) \to (ex2 T (\lambda (t0: T).(subst1 i u1 t1 t0)) (\lambda -(t0: T).(pr0 t0 t)))))) (\lambda (u1: T).(\lambda (_: (pr0 u1 u2)).(ex_intro2 -T (\lambda (t: T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t t1)) t1 -(subst1_refl i u1 t1) (pr0_refl t1)))) (\lambda (t0: T).(\lambda (H0: (subst0 -i u2 t1 t0)).(\lambda (u1: T).(\lambda (H1: (pr0 u1 u2)).(ex2_ind T (\lambda -(t: T).(subst0 i u1 t1 t)) (\lambda (t: T).(pr0 t t0)) (ex2 T (\lambda (t: -T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t t0))) (\lambda (x: T).(\lambda -(H2: (subst0 i u1 t1 x)).(\lambda (H3: (pr0 x t0)).(ex_intro2 T (\lambda (t: -T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t t0)) x (subst1_single i u1 t1 x -H2) H3)))) (pr0_subst0_back u2 t1 t0 i H0 u1 H1)))))) t2 H))))). -(* COMMENTS -Initial nodes: 251 -END *) - -theorem pr0_subst1_fwd: - \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst1 -i u2 t1 t2) \to (\forall (u1: T).((pr0 u2 u1) \to (ex2 T (\lambda (t: -T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t2 t))))))))) -\def - \lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (i: nat).(\lambda -(H: (subst1 i u2 t1 t2)).(subst1_ind i u2 t1 (\lambda (t: T).(\forall (u1: -T).((pr0 u2 u1) \to (ex2 T (\lambda (t0: T).(subst1 i u1 t1 t0)) (\lambda -(t0: T).(pr0 t t0)))))) (\lambda (u1: T).(\lambda (_: (pr0 u2 u1)).(ex_intro2 -T (\lambda (t: T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t1 t)) t1 -(subst1_refl i u1 t1) (pr0_refl t1)))) (\lambda (t0: T).(\lambda (H0: (subst0 -i u2 t1 t0)).(\lambda (u1: T).(\lambda (H1: (pr0 u2 u1)).(ex2_ind T (\lambda -(t: T).(subst0 i u1 t1 t)) (\lambda (t: T).(pr0 t0 t)) (ex2 T (\lambda (t: -T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t0 t))) (\lambda (x: T).(\lambda -(H2: (subst0 i u1 t1 x)).(\lambda (H3: (pr0 t0 x)).(ex_intro2 T (\lambda (t: -T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t0 t)) x (subst1_single i u1 t1 x -H2) H3)))) (pr0_subst0_fwd u2 t1 t0 i H0 u1 H1)))))) t2 H))))). -(* COMMENTS -Initial nodes: 251 -END *) - -theorem pr0_subst1: - \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (v1: T).(\forall -(w1: T).(\forall (i: nat).((subst1 i v1 t1 w1) \to (\forall (v2: T).((pr0 v1 -v2) \to (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst1 i v2 t2 -w2))))))))))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(\lambda (v1: -T).(\lambda (w1: T).(\lambda (i: nat).(\lambda (H0: (subst1 i v1 t1 -w1)).(subst1_ind i v1 t1 (\lambda (t: T).(\forall (v2: T).((pr0 v1 v2) \to -(ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst1 i v2 t2 w2)))))) -(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(ex_intro2 T (\lambda (w2: T).(pr0 -t1 w2)) (\lambda (w2: T).(subst1 i v2 t2 w2)) t2 H (subst1_refl i v2 t2)))) -(\lambda (t0: T).(\lambda (H1: (subst0 i v1 t1 t0)).(\lambda (v2: T).(\lambda -(H2: (pr0 v1 v2)).(or_ind (pr0 t0 t2) (ex2 T (\lambda (w2: T).(pr0 t0 w2)) -(\lambda (w2: T).(subst0 i v2 t2 w2))) (ex2 T (\lambda (w2: T).(pr0 t0 w2)) -(\lambda (w2: T).(subst1 i v2 t2 w2))) (\lambda (H3: (pr0 t0 t2)).(ex_intro2 -T (\lambda (w2: T).(pr0 t0 w2)) (\lambda (w2: T).(subst1 i v2 t2 w2)) t2 H3 -(subst1_refl i v2 t2))) (\lambda (H3: (ex2 T (\lambda (w2: T).(pr0 t0 w2)) -(\lambda (w2: T).(subst0 i v2 t2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t0 -w2)) (\lambda (w2: T).(subst0 i v2 t2 w2)) (ex2 T (\lambda (w2: T).(pr0 t0 -w2)) (\lambda (w2: T).(subst1 i v2 t2 w2))) (\lambda (x: T).(\lambda (H4: -(pr0 t0 x)).(\lambda (H5: (subst0 i v2 t2 x)).(ex_intro2 T (\lambda (w2: -T).(pr0 t0 w2)) (\lambda (w2: T).(subst1 i v2 t2 w2)) x H4 (subst1_single i -v2 t2 x H5))))) H3)) (pr0_subst0 t1 t2 H v1 t0 i H1 v2 H2)))))) w1 H0))))))). -(* COMMENTS -Initial nodes: 385 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr1/defs.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr1/defs.ma deleted file mode 100644 index 84db71acb..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr1/defs.ma +++ /dev/null @@ -1,23 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr0/defs.ma". - -inductive pr1: T \to (T \to Prop) \def -| pr1_refl: \forall (t: T).(pr1 t t) -| pr1_sing: \forall (t2: T).(\forall (t1: T).((pr0 t1 t2) \to (\forall (t3: -T).((pr1 t2 t3) \to (pr1 t1 t3))))). - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr1/pr1.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr1/pr1.ma deleted file mode 100644 index ec469da50..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr1/pr1.ma +++ /dev/null @@ -1,70 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr1/props.ma". - -include "Basic-1/pr0/pr0.ma". - -theorem pr1_strip: - \forall (t0: T).(\forall (t1: T).((pr1 t0 t1) \to (\forall (t2: T).((pr0 t0 -t2) \to (ex2 T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t))))))) -\def - \lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr1 t0 t1)).(pr1_ind (\lambda -(t: T).(\lambda (t2: T).(\forall (t3: T).((pr0 t t3) \to (ex2 T (\lambda (t4: -T).(pr1 t2 t4)) (\lambda (t4: T).(pr1 t3 t4))))))) (\lambda (t: T).(\lambda -(t2: T).(\lambda (H0: (pr0 t t2)).(ex_intro2 T (\lambda (t3: T).(pr1 t t3)) -(\lambda (t3: T).(pr1 t2 t3)) t2 (pr1_pr0 t t2 H0) (pr1_refl t2))))) (\lambda -(t2: T).(\lambda (t3: T).(\lambda (H0: (pr0 t3 t2)).(\lambda (t4: T).(\lambda -(_: (pr1 t2 t4)).(\lambda (H2: ((\forall (t5: T).((pr0 t2 t5) \to (ex2 T -(\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t))))))).(\lambda (t5: -T).(\lambda (H3: (pr0 t3 t5)).(ex2_ind T (\lambda (t: T).(pr0 t5 t)) (\lambda -(t: T).(pr0 t2 t)) (ex2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 -t))) (\lambda (x: T).(\lambda (H4: (pr0 t5 x)).(\lambda (H5: (pr0 t2 x)).(let -H6 \def (H2 x H5) in (ex2_ind T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: -T).(pr1 x t)) (ex2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t))) -(\lambda (x0: T).(\lambda (H7: (pr1 t4 x0)).(\lambda (H8: (pr1 x -x0)).(ex_intro2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t)) x0 -H7 (pr1_t x t5 (pr1_pr0 t5 x H4) x0 H8))))) H6))))) (pr0_confluence t3 t5 H3 -t2 H0)))))))))) t0 t1 H))). -(* COMMENTS -Initial nodes: 317 -END *) - -theorem pr1_confluence: - \forall (t0: T).(\forall (t1: T).((pr1 t0 t1) \to (\forall (t2: T).((pr1 t0 -t2) \to (ex2 T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t))))))) -\def - \lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr1 t0 t1)).(pr1_ind (\lambda -(t: T).(\lambda (t2: T).(\forall (t3: T).((pr1 t t3) \to (ex2 T (\lambda (t4: -T).(pr1 t2 t4)) (\lambda (t4: T).(pr1 t3 t4))))))) (\lambda (t: T).(\lambda -(t2: T).(\lambda (H0: (pr1 t t2)).(ex_intro2 T (\lambda (t3: T).(pr1 t t3)) -(\lambda (t3: T).(pr1 t2 t3)) t2 H0 (pr1_refl t2))))) (\lambda (t2: -T).(\lambda (t3: T).(\lambda (H0: (pr0 t3 t2)).(\lambda (t4: T).(\lambda (_: -(pr1 t2 t4)).(\lambda (H2: ((\forall (t5: T).((pr1 t2 t5) \to (ex2 T (\lambda -(t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t))))))).(\lambda (t5: T).(\lambda -(H3: (pr1 t3 t5)).(let H_x \def (pr1_strip t3 t5 H3 t2 H0) in (let H4 \def -H_x in (ex2_ind T (\lambda (t: T).(pr1 t5 t)) (\lambda (t: T).(pr1 t2 t)) -(ex2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t))) (\lambda (x: -T).(\lambda (H5: (pr1 t5 x)).(\lambda (H6: (pr1 t2 x)).(let H_x0 \def (H2 x -H6) in (let H7 \def H_x0 in (ex2_ind T (\lambda (t: T).(pr1 t4 t)) (\lambda -(t: T).(pr1 x t)) (ex2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 -t))) (\lambda (x0: T).(\lambda (H8: (pr1 t4 x0)).(\lambda (H9: (pr1 x -x0)).(ex_intro2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t)) x0 -H8 (pr1_t x t5 H5 x0 H9))))) H7)))))) H4))))))))))) t0 t1 H))). -(* COMMENTS -Initial nodes: 311 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr1/props.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr1/props.ma deleted file mode 100644 index 0615bed96..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr1/props.ma +++ /dev/null @@ -1,126 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr1/defs.ma". - -include "Basic-1/pr0/subst1.ma". - -include "Basic-1/subst1/props.ma". - -include "Basic-1/T/props.ma". - -theorem pr1_pr0: - \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (pr1 t1 t2))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(pr1_sing t2 t1 H -t2 (pr1_refl t2)))). -(* COMMENTS -Initial nodes: 23 -END *) - -theorem pr1_t: - \forall (t2: T).(\forall (t1: T).((pr1 t1 t2) \to (\forall (t3: T).((pr1 t2 -t3) \to (pr1 t1 t3))))) -\def - \lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pr1 t1 t2)).(pr1_ind (\lambda -(t: T).(\lambda (t0: T).(\forall (t3: T).((pr1 t0 t3) \to (pr1 t t3))))) -(\lambda (t: T).(\lambda (t3: T).(\lambda (H0: (pr1 t t3)).H0))) (\lambda -(t0: T).(\lambda (t3: T).(\lambda (H0: (pr0 t3 t0)).(\lambda (t4: T).(\lambda -(_: (pr1 t0 t4)).(\lambda (H2: ((\forall (t5: T).((pr1 t4 t5) \to (pr1 t0 -t5))))).(\lambda (t5: T).(\lambda (H3: (pr1 t4 t5)).(pr1_sing t0 t3 H0 t5 (H2 -t5 H3)))))))))) t1 t2 H))). -(* COMMENTS -Initial nodes: 103 -END *) - -theorem pr1_head_1: - \forall (u1: T).(\forall (u2: T).((pr1 u1 u2) \to (\forall (t: T).(\forall -(k: K).(pr1 (THead k u1 t) (THead k u2 t)))))) -\def - \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr1 u1 u2)).(\lambda (t: -T).(\lambda (k: K).(pr1_ind (\lambda (t0: T).(\lambda (t1: T).(pr1 (THead k -t0 t) (THead k t1 t)))) (\lambda (t0: T).(pr1_refl (THead k t0 t))) (\lambda -(t2: T).(\lambda (t1: T).(\lambda (H0: (pr0 t1 t2)).(\lambda (t3: T).(\lambda -(_: (pr1 t2 t3)).(\lambda (H2: (pr1 (THead k t2 t) (THead k t3 t))).(pr1_sing -(THead k t2 t) (THead k t1 t) (pr0_comp t1 t2 H0 t t (pr0_refl t) k) (THead k -t3 t) H2))))))) u1 u2 H))))). -(* COMMENTS -Initial nodes: 137 -END *) - -theorem pr1_head_2: - \forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall (u: T).(\forall -(k: K).(pr1 (THead k u t1) (THead k u t2)))))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr1 t1 t2)).(\lambda (u: -T).(\lambda (k: K).(pr1_ind (\lambda (t: T).(\lambda (t0: T).(pr1 (THead k u -t) (THead k u t0)))) (\lambda (t: T).(pr1_refl (THead k u t))) (\lambda (t0: -T).(\lambda (t3: T).(\lambda (H0: (pr0 t3 t0)).(\lambda (t4: T).(\lambda (_: -(pr1 t0 t4)).(\lambda (H2: (pr1 (THead k u t0) (THead k u t4))).(pr1_sing -(THead k u t0) (THead k u t3) (pr0_comp u u (pr0_refl u) t3 t0 H0 k) (THead k -u t4) H2))))))) t1 t2 H))))). -(* COMMENTS -Initial nodes: 137 -END *) - -theorem pr1_comp: - \forall (v: T).(\forall (w: T).((pr1 v w) \to (\forall (t: T).(\forall (u: -T).((pr1 t u) \to (\forall (k: K).(pr1 (THead k v t) (THead k w u)))))))) -\def - \lambda (v: T).(\lambda (w: T).(\lambda (H: (pr1 v w)).(pr1_ind (\lambda (t: -T).(\lambda (t0: T).(\forall (t1: T).(\forall (u: T).((pr1 t1 u) \to (\forall -(k: K).(pr1 (THead k t t1) (THead k t0 u)))))))) (\lambda (t: T).(\lambda -(t0: T).(\lambda (u: T).(\lambda (H0: (pr1 t0 u)).(\lambda (k: K).(pr1_head_2 -t0 u H0 t k)))))) (\lambda (t2: T).(\lambda (t1: T).(\lambda (H0: (pr0 t1 -t2)).(\lambda (t3: T).(\lambda (H1: (pr1 t2 t3)).(\lambda (_: ((\forall (t: -T).(\forall (u: T).((pr1 t u) \to (\forall (k: K).(pr1 (THead k t2 t) (THead -k t3 u)))))))).(\lambda (t: T).(\lambda (u: T).(\lambda (H3: (pr1 t -u)).(\lambda (k: K).(pr1_ind (\lambda (t0: T).(\lambda (t4: T).(pr1 (THead k -t1 t0) (THead k t3 t4)))) (\lambda (t0: T).(pr1_head_1 t1 t3 (pr1_sing t2 t1 -H0 t3 H1) t0 k)) (\lambda (t0: T).(\lambda (t4: T).(\lambda (H4: (pr0 t4 -t0)).(\lambda (t5: T).(\lambda (_: (pr1 t0 t5)).(\lambda (H6: (pr1 (THead k -t1 t0) (THead k t3 t5))).(pr1_sing (THead k t1 t0) (THead k t1 t4) (pr0_comp -t1 t1 (pr0_refl t1) t4 t0 H4 k) (THead k t3 t5) H6))))))) t u H3))))))))))) v -w H))). -(* COMMENTS -Initial nodes: 273 -END *) - -theorem pr1_eta: - \forall (w: T).(\forall (u: T).(let t \def (THead (Bind Abst) w u) in -(\forall (v: T).((pr1 v w) \to (pr1 (THead (Bind Abst) v (THead (Flat Appl) -(TLRef O) (lift (S O) O t))) t))))) -\def - \lambda (w: T).(\lambda (u: T).(let t \def (THead (Bind Abst) w u) in -(\lambda (v: T).(\lambda (H: (pr1 v w)).(eq_ind_r T (THead (Bind Abst) (lift -(S O) O w) (lift (S O) (S O) u)) (\lambda (t0: T).(pr1 (THead (Bind Abst) v -(THead (Flat Appl) (TLRef O) t0)) (THead (Bind Abst) w u))) (pr1_comp v w H -(THead (Flat Appl) (TLRef O) (THead (Bind Abst) (lift (S O) O w) (lift (S O) -(S O) u))) u (pr1_sing (THead (Bind Abbr) (TLRef O) (lift (S O) (S O) u)) -(THead (Flat Appl) (TLRef O) (THead (Bind Abst) (lift (S O) O w) (lift (S O) -(S O) u))) (pr0_beta (lift (S O) O w) (TLRef O) (TLRef O) (pr0_refl (TLRef -O)) (lift (S O) (S O) u) (lift (S O) (S O) u) (pr0_refl (lift (S O) (S O) -u))) u (pr1_sing (THead (Bind Abbr) (TLRef O) (lift (S O) O u)) (THead (Bind -Abbr) (TLRef O) (lift (S O) (S O) u)) (pr0_delta1 (TLRef O) (TLRef O) -(pr0_refl (TLRef O)) (lift (S O) (S O) u) (lift (S O) (S O) u) (pr0_refl -(lift (S O) (S O) u)) (lift (S O) O u) (subst1_lift_S u O O (le_n O))) u -(pr1_pr0 (THead (Bind Abbr) (TLRef O) (lift (S O) O u)) u (pr0_zeta Abbr -not_abbr_abst u u (pr0_refl u) (TLRef O))))) (Bind Abst)) (lift (S O) O -(THead (Bind Abst) w u)) (lift_bind Abst w u (S O) O)))))). -(* COMMENTS -Initial nodes: 463 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr2/clen.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr2/clen.ma deleted file mode 100644 index df9ea1fd9..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr2/clen.ma +++ /dev/null @@ -1,161 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr2/props.ma". - -include "Basic-1/clen/getl.ma". - -theorem pr2_gen_ctail: - \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall -(t2: T).((pr2 (CTail k u c) t1 t2) \to (or (pr2 c t1 t2) (ex3 T (\lambda (_: -T).(eq K k (Bind Abbr))) (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(subst0 -(clen c) u t t2))))))))) -\def - \lambda (k: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda -(t2: T).(\lambda (H: (pr2 (CTail k u c) t1 t2)).(insert_eq C (CTail k u c) -(\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(or (pr2 c t1 t2) (ex3 T -(\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t: T).(pr0 t1 t)) (\lambda -(t: T).(subst0 (clen c) u t t2))))) (\lambda (y: C).(\lambda (H0: (pr2 y t1 -t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 -(CTail k u c)) \to (or (pr2 c t t0) (ex3 T (\lambda (_: T).(eq K k (Bind -Abbr))) (\lambda (t3: T).(pr0 t t3)) (\lambda (t3: T).(subst0 (clen c) u t3 -t0)))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: -(pr0 t3 t4)).(\lambda (_: (eq C c0 (CTail k u c))).(or_introl (pr2 c t3 t4) -(ex3 T (\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t: T).(pr0 t3 t)) -(\lambda (t: T).(subst0 (clen c) u t t4))) (pr2_free c t3 t4 H1))))))) -(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda -(H1: (getl i c0 (CHead d (Bind Abbr) u0))).(\lambda (t3: T).(\lambda (t4: -T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: (subst0 i u0 t4 -t)).(\lambda (H4: (eq C c0 (CTail k u c))).(let H5 \def (eq_ind C c0 (\lambda -(c1: C).(getl i c1 (CHead d (Bind Abbr) u0))) H1 (CTail k u c) H4) in (let -H_x \def (getl_gen_tail k Abbr u u0 d c i H5) in (let H6 \def H_x in (or_ind -(ex2 C (\lambda (e: C).(eq C d (CTail k u e))) (\lambda (e: C).(getl i c -(CHead e (Bind Abbr) u0)))) (ex4 nat (\lambda (_: nat).(eq nat i (clen c))) -(\lambda (_: nat).(eq K k (Bind Abbr))) (\lambda (_: nat).(eq T u u0)) -(\lambda (n: nat).(eq C d (CSort n)))) (or (pr2 c t3 t) (ex3 T (\lambda (_: -T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0: -T).(subst0 (clen c) u t0 t)))) (\lambda (H7: (ex2 C (\lambda (e: C).(eq C d -(CTail k u e))) (\lambda (e: C).(getl i c (CHead e (Bind Abbr) -u0))))).(ex2_ind C (\lambda (e: C).(eq C d (CTail k u e))) (\lambda (e: -C).(getl i c (CHead e (Bind Abbr) u0))) (or (pr2 c t3 t) (ex3 T (\lambda (_: -T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0: -T).(subst0 (clen c) u t0 t)))) (\lambda (x: C).(\lambda (_: (eq C d (CTail k -u x))).(\lambda (H9: (getl i c (CHead x (Bind Abbr) u0))).(or_introl (pr2 c -t3 t) (ex3 T (\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3 -t0)) (\lambda (t0: T).(subst0 (clen c) u t0 t))) (pr2_delta c x u0 i H9 t3 t4 -H2 t H3))))) H7)) (\lambda (H7: (ex4 nat (\lambda (_: nat).(eq nat i (clen -c))) (\lambda (_: nat).(eq K k (Bind Abbr))) (\lambda (_: nat).(eq T u u0)) -(\lambda (n: nat).(eq C d (CSort n))))).(ex4_ind nat (\lambda (_: nat).(eq -nat i (clen c))) (\lambda (_: nat).(eq K k (Bind Abbr))) (\lambda (_: -nat).(eq T u u0)) (\lambda (n: nat).(eq C d (CSort n))) (or (pr2 c t3 t) (ex3 -T (\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3 t0)) -(\lambda (t0: T).(subst0 (clen c) u t0 t)))) (\lambda (x0: nat).(\lambda (H8: -(eq nat i (clen c))).(\lambda (H9: (eq K k (Bind Abbr))).(\lambda (H10: (eq T -u u0)).(\lambda (_: (eq C d (CSort x0))).(let H12 \def (eq_ind nat i (\lambda -(n: nat).(subst0 n u0 t4 t)) H3 (clen c) H8) in (let H13 \def (eq_ind_r T u0 -(\lambda (t0: T).(subst0 (clen c) t0 t4 t)) H12 u H10) in (eq_ind_r K (Bind -Abbr) (\lambda (k0: K).(or (pr2 c t3 t) (ex3 T (\lambda (_: T).(eq K k0 (Bind -Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0 -t))))) (or_intror (pr2 c t3 t) (ex3 T (\lambda (_: T).(eq K (Bind Abbr) (Bind -Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0 -t))) (ex3_intro T (\lambda (_: T).(eq K (Bind Abbr) (Bind Abbr))) (\lambda -(t0: T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0 t)) t4 -(refl_equal K (Bind Abbr)) H2 H13)) k H9)))))))) H7)) H6))))))))))))))) y t1 -t2 H0))) H)))))). -(* COMMENTS -Initial nodes: 1161 -END *) - -theorem pr2_gen_cbind: - \forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall -(t2: T).((pr2 (CHead c (Bind b) v) t1 t2) \to (pr2 c (THead (Bind b) v t1) -(THead (Bind b) v t2))))))) -\def - \lambda (b: B).(\lambda (c: C).(\lambda (v: T).(\lambda (t1: T).(\lambda -(t2: T).(\lambda (H: (pr2 (CHead c (Bind b) v) t1 t2)).(insert_eq C (CHead c -(Bind b) v) (\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 c (THead -(Bind b) v t1) (THead (Bind b) v t2))) (\lambda (y: C).(\lambda (H0: (pr2 y -t1 t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 -(CHead c (Bind b) v)) \to (pr2 c (THead (Bind b) v t) (THead (Bind b) v -t0)))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: -(pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c (Bind b) v))).(pr2_free c (THead -(Bind b) v t3) (THead (Bind b) v t4) (pr0_comp v v (pr0_refl v) t3 t4 H1 -(Bind b)))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: -nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t3: -T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: -(subst0 i u t4 t)).(\lambda (H4: (eq C c0 (CHead c (Bind b) v))).(let H5 \def -(eq_ind C c0 (\lambda (c1: C).(getl i c1 (CHead d (Bind Abbr) u))) H1 (CHead -c (Bind b) v) H4) in (let H_x \def (getl_gen_bind b c (CHead d (Bind Abbr) u) -v i H5) in (let H6 \def H_x in (or_ind (land (eq nat i O) (eq C (CHead d -(Bind Abbr) u) (CHead c (Bind b) v))) (ex2 nat (\lambda (j: nat).(eq nat i (S -j))) (\lambda (j: nat).(getl j c (CHead d (Bind Abbr) u)))) (pr2 c (THead -(Bind b) v t3) (THead (Bind b) v t)) (\lambda (H7: (land (eq nat i O) (eq C -(CHead d (Bind Abbr) u) (CHead c (Bind b) v)))).(land_ind (eq nat i O) (eq C -(CHead d (Bind Abbr) u) (CHead c (Bind b) v)) (pr2 c (THead (Bind b) v t3) -(THead (Bind b) v t)) (\lambda (H8: (eq nat i O)).(\lambda (H9: (eq C (CHead -d (Bind Abbr) u) (CHead c (Bind b) v))).(let H10 \def (f_equal C C (\lambda -(e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d -| (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u) (CHead c (Bind b) -v) H9) in ((let H11 \def (f_equal C B (\lambda (e: C).(match e in C return -(\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b0) -\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) -(CHead c (Bind b) v) H9) in ((let H12 \def (f_equal C T (\lambda (e: -C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | -(CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind Abbr) u) (CHead c (Bind b) v) -H9) in (\lambda (H13: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H15 \def -(eq_ind nat i (\lambda (n: nat).(subst0 n u t4 t)) H3 O H8) in (let H16 \def -(eq_ind T u (\lambda (t0: T).(subst0 O t0 t4 t)) H15 v H12) in (eq_ind B Abbr -(\lambda (b0: B).(pr2 c (THead (Bind b0) v t3) (THead (Bind b0) v t))) -(pr2_free c (THead (Bind Abbr) v t3) (THead (Bind Abbr) v t) (pr0_delta v v -(pr0_refl v) t3 t4 H2 t H16)) b H13)))))) H11)) H10)))) H7)) (\lambda (H7: -(ex2 nat (\lambda (j: nat).(eq nat i (S j))) (\lambda (j: nat).(getl j c -(CHead d (Bind Abbr) u))))).(ex2_ind nat (\lambda (j: nat).(eq nat i (S j))) -(\lambda (j: nat).(getl j c (CHead d (Bind Abbr) u))) (pr2 c (THead (Bind b) -v t3) (THead (Bind b) v t)) (\lambda (x: nat).(\lambda (H8: (eq nat i (S -x))).(\lambda (H9: (getl x c (CHead d (Bind Abbr) u))).(let H10 \def (f_equal -nat nat (\lambda (e: nat).e) i (S x) H8) in (let H11 \def (eq_ind nat i -(\lambda (n: nat).(subst0 n u t4 t)) H3 (S x) H10) in (pr2_head_2 c v t3 t -(Bind b) (pr2_delta (CHead c (Bind b) v) d u (S x) (getl_clear_bind b (CHead -c (Bind b) v) c v (clear_bind b c v) (CHead d (Bind Abbr) u) x H9) t3 t4 H2 t -H11))))))) H7)) H6))))))))))))))) y t1 t2 H0))) H)))))). -(* COMMENTS -Initial nodes: 1085 -END *) - -theorem pr2_gen_cflat: - \forall (f: F).(\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall -(t2: T).((pr2 (CHead c (Flat f) v) t1 t2) \to (pr2 c t1 t2)))))) -\def - \lambda (f: F).(\lambda (c: C).(\lambda (v: T).(\lambda (t1: T).(\lambda -(t2: T).(\lambda (H: (pr2 (CHead c (Flat f) v) t1 t2)).(insert_eq C (CHead c -(Flat f) v) (\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 c t1 t2)) -(\lambda (y: C).(\lambda (H0: (pr2 y t1 t2)).(pr2_ind (\lambda (c0: -C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CHead c (Flat f) v)) \to (pr2 -c t t0))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: -(pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c (Flat f) v))).(pr2_free c t3 t4 -H1)))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: -nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t3: -T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: -(subst0 i u t4 t)).(\lambda (H4: (eq C c0 (CHead c (Flat f) v))).(let H5 \def -(eq_ind C c0 (\lambda (c1: C).(getl i c1 (CHead d (Bind Abbr) u))) H1 (CHead -c (Flat f) v) H4) in (let H_y \def (getl_gen_flat f c (CHead d (Bind Abbr) u) -v i H5) in (pr2_delta c d u i H_y t3 t4 H2 t H3)))))))))))))) y t1 t2 H0))) -H)))))). -(* COMMENTS -Initial nodes: 293 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr2/defs.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr2/defs.ma deleted file mode 100644 index e21568232..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr2/defs.ma +++ /dev/null @@ -1,28 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr0/defs.ma". - -include "Basic-1/getl/defs.ma". - -inductive pr2: C \to (T \to (T \to Prop)) \def -| pr2_free: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to -(pr2 c t1 t2)))) -| pr2_delta: \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: -nat).((getl i c (CHead d (Bind Abbr) u)) \to (\forall (t1: T).(\forall (t2: -T).((pr0 t1 t2) \to (\forall (t: T).((subst0 i u t2 t) \to (pr2 c t1 -t)))))))))). - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr2/fwd.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr2/fwd.ma deleted file mode 100644 index 1a18f7dbc..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr2/fwd.ma +++ /dev/null @@ -1,3343 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr2/defs.ma". - -include "Basic-1/pr0/fwd.ma". - -include "Basic-1/getl/drop.ma". - -include "Basic-1/getl/clear.ma". - -theorem pr2_gen_sort: - \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr2 c (TSort n) x) \to -(eq T x (TSort n))))) -\def - \lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr2 c (TSort -n) x)).(insert_eq T (TSort n) (\lambda (t: T).(pr2 c t x)) (\lambda (t: -T).(eq T x t)) (\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda -(_: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (TSort n)) \to (eq T t0 -t))))) (\lambda (_: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H1: (pr0 -t1 t2)).(\lambda (H2: (eq T t1 (TSort n))).(let H3 \def (eq_ind T t1 (\lambda -(t: T).(pr0 t t2)) H1 (TSort n) H2) in (eq_ind_r T (TSort n) (\lambda (t: -T).(eq T t2 t)) (eq_ind_r T (TSort n) (\lambda (t: T).(eq T t (TSort n))) -(refl_equal T (TSort n)) t2 (pr0_gen_sort t2 n H3)) t1 H2))))))) (\lambda -(c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (_: (getl -i c0 (CHead d (Bind Abbr) u))).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H2: (pr0 t1 t2)).(\lambda (t: T).(\lambda (H3: (subst0 i u t2 t)).(\lambda -(H4: (eq T t1 (TSort n))).(let H5 \def (eq_ind T t1 (\lambda (t0: T).(pr0 t0 -t2)) H2 (TSort n) H4) in (eq_ind_r T (TSort n) (\lambda (t0: T).(eq T t t0)) -(let H6 \def (eq_ind T t2 (\lambda (t0: T).(subst0 i u t0 t)) H3 (TSort n) -(pr0_gen_sort t2 n H5)) in (subst0_gen_sort u t i n H6 (eq T t (TSort n)))) -t1 H4))))))))))))) c y x H0))) H)))). -(* COMMENTS -Initial nodes: 347 -END *) - -theorem pr2_gen_lref: - \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr2 c (TLRef n) x) \to -(or (eq T x (TLRef n)) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c -(CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T x (lift (S -n) O u))))))))) -\def - \lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr2 c (TLRef -n) x)).(insert_eq T (TLRef n) (\lambda (t: T).(pr2 c t x)) (\lambda (t: -T).(or (eq T x t) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c (CHead -d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T x (lift (S n) O -u))))))) (\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0: -C).(\lambda (t: T).(\lambda (t0: T).((eq T t (TLRef n)) \to (or (eq T t0 t) -(ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c0 (CHead d (Bind Abbr) -u)))) (\lambda (_: C).(\lambda (u: T).(eq T t0 (lift (S n) O u)))))))))) -(\lambda (c0: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H1: (pr0 t1 -t2)).(\lambda (H2: (eq T t1 (TLRef n))).(let H3 \def (eq_ind T t1 (\lambda -(t: T).(pr0 t t2)) H1 (TLRef n) H2) in (eq_ind_r T (TLRef n) (\lambda (t: -T).(or (eq T t2 t) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c0 -(CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T t2 (lift (S -n) O u))))))) (eq_ind_r T (TLRef n) (\lambda (t: T).(or (eq T t (TLRef n)) -(ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c0 (CHead d (Bind Abbr) -u)))) (\lambda (_: C).(\lambda (u: T).(eq T t (lift (S n) O u))))))) -(or_introl (eq T (TLRef n) (TLRef n)) (ex2_2 C T (\lambda (d: C).(\lambda (u: -T).(getl n c0 (CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq -T (TLRef n) (lift (S n) O u))))) (refl_equal T (TLRef n))) t2 (pr0_gen_lref -t2 n H3)) t1 H2))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: -T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr) -u))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H2: (pr0 t1 t2)).(\lambda -(t: T).(\lambda (H3: (subst0 i u t2 t)).(\lambda (H4: (eq T t1 (TLRef -n))).(let H5 \def (eq_ind T t1 (\lambda (t0: T).(pr0 t0 t2)) H2 (TLRef n) H4) -in (eq_ind_r T (TLRef n) (\lambda (t0: T).(or (eq T t t0) (ex2_2 C T (\lambda -(d0: C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind Abbr) u0)))) (\lambda (_: -C).(\lambda (u0: T).(eq T t (lift (S n) O u0))))))) (let H6 \def (eq_ind T t2 -(\lambda (t0: T).(subst0 i u t0 t)) H3 (TLRef n) (pr0_gen_lref t2 n H5)) in -(land_ind (eq nat n i) (eq T t (lift (S n) O u)) (or (eq T t (TLRef n)) -(ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind Abbr) -u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T t (lift (S n) O u0)))))) -(\lambda (H7: (eq nat n i)).(\lambda (H8: (eq T t (lift (S n) O -u))).(eq_ind_r T (lift (S n) O u) (\lambda (t0: T).(or (eq T t0 (TLRef n)) -(ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind Abbr) -u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T t0 (lift (S n) O u0))))))) (let -H9 \def (eq_ind_r nat i (\lambda (n0: nat).(getl n0 c0 (CHead d (Bind Abbr) -u))) H1 n H7) in (or_intror (eq T (lift (S n) O u) (TLRef n)) (ex2_2 C T -(\lambda (d0: C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind Abbr) u0)))) -(\lambda (_: C).(\lambda (u0: T).(eq T (lift (S n) O u) (lift (S n) O u0))))) -(ex2_2_intro C T (\lambda (d0: C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind -Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T (lift (S n) O u) (lift (S -n) O u0)))) d u H9 (refl_equal T (lift (S n) O u))))) t H8))) -(subst0_gen_lref u t i n H6))) t1 H4))))))))))))) c y x H0))) H)))). -(* COMMENTS -Initial nodes: 1003 -END *) - -theorem pr2_gen_abst: - \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c -(THead (Bind Abst) u1 t1) x) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 -c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: -T).(pr2 (CHead c (Bind b) u) t1 t2)))))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda -(H: (pr2 c (THead (Bind Abst) u1 t1) x)).(insert_eq T (THead (Bind Abst) u1 -t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall -(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2))))))) (\lambda (y: -T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0: C).(\lambda (t: -T).(\lambda (t0: T).((eq T t (THead (Bind Abst) u1 t1)) \to (ex3_2 T T -(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abst) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 -t2)))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H1: -(pr0 t0 t2)).(\lambda (H2: (eq T t0 (THead (Bind Abst) u1 t1))).(let H3 \def -(eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Bind Abst) u1 t1) H2) in -(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: -T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall -(u: T).(pr2 (CHead c0 (Bind b) u) t1 t3)))))) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H4: (eq T t2 (THead (Bind Abst) x0 x1))).(\lambda (H5: (pr0 u1 -x0)).(\lambda (H6: (pr0 t1 x1)).(eq_ind_r T (THead (Bind Abst) x0 x1) -(\lambda (t: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead -(Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) -(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead -c0 (Bind b) u) t1 t3))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: -T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall -(b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 t3))))) x0 x1 -(refl_equal T (THead (Bind Abst) x0 x1)) (pr2_free c0 u1 x0 H5) (\lambda (b: -B).(\lambda (u: T).(pr2_free (CHead c0 (Bind b) u) t1 x1 H6)))) t2 H4)))))) -(pr0_gen_abst u1 t1 t2 H3)))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda -(u: T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr) -u))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H2: (pr0 t0 t2)).(\lambda -(t: T).(\lambda (H3: (subst0 i u t2 t)).(\lambda (H4: (eq T t0 (THead (Bind -Abst) u1 t1))).(let H5 \def (eq_ind T t0 (\lambda (t3: T).(pr0 t3 t2)) H2 -(THead (Bind Abst) u1 t1) H4) in (ex3_2_ind T T (\lambda (u2: T).(\lambda -(t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 -t3)))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H6: (eq T t2 (THead -(Bind Abst) x0 x1))).(\lambda (H7: (pr0 u1 x0)).(\lambda (H8: (pr0 t1 -x1)).(let H9 \def (eq_ind T t2 (\lambda (t3: T).(subst0 i u t3 t)) H3 (THead -(Bind Abst) x0 x1) H6) in (or3_ind (ex2 T (\lambda (u2: T).(eq T t (THead -(Bind Abst) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2))) (ex2 T (\lambda -(t3: T).(eq T t (THead (Bind Abst) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind -Abst) i) u x1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t -(THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 -u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u x1 t3)))) -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) t1 t3)))))) (\lambda (H10: (ex2 T (\lambda (u2: T).(eq T t (THead (Bind -Abst) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2)))).(ex2_ind T (\lambda -(u2: T).(eq T t (THead (Bind Abst) u2 x1))) (\lambda (u2: T).(subst0 i u x0 -u2)) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst) -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) t1 t3)))))) (\lambda (x2: T).(\lambda (H11: (eq T t (THead (Bind Abst) x2 -x1))).(\lambda (H12: (subst0 i u x0 x2)).(eq_ind_r T (THead (Bind Abst) x2 -x1) (\lambda (t3: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 -(THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 -(CHead c0 (Bind b) u0) t1 t4))))))) (ex3_2_intro T T (\lambda (u2: -T).(\lambda (t3: T).(eq T (THead (Bind Abst) x2 x1) (THead (Bind Abst) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) t1 t3))))) x2 x1 (refl_equal T (THead (Bind Abst) x2 x1)) (pr2_delta c0 d -u i H1 u1 x0 H7 x2 H12) (\lambda (b: B).(\lambda (u0: T).(pr2_free (CHead c0 -(Bind b) u0) t1 x1 H8)))) t H11)))) H10)) (\lambda (H10: (ex2 T (\lambda (t3: -T).(eq T t (THead (Bind Abst) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind -Abst) i) u x1 t3)))).(ex2_ind T (\lambda (t3: T).(eq T t (THead (Bind Abst) -x0 t3))) (\lambda (t3: T).(subst0 (s (Bind Abst) i) u x1 t3)) (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 -t3)))))) (\lambda (x2: T).(\lambda (H11: (eq T t (THead (Bind Abst) x0 -x2))).(\lambda (H12: (subst0 (s (Bind Abst) i) u x1 x2)).(eq_ind_r T (THead -(Bind Abst) x0 x2) (\lambda (t3: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t4: -T).(eq T t3 (THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall -(u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t4))))))) (ex3_2_intro T T (\lambda -(u2: T).(\lambda (t3: T).(eq T (THead (Bind Abst) x0 x2) (THead (Bind Abst) -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) t1 t3))))) x0 x2 (refl_equal T (THead (Bind Abst) x0 x2)) (pr2_free c0 u1 -x0 H7) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u -(S i) (getl_head (Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 x1 H8 x2 -H12)))) t H11)))) H10)) (\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda -(t3: T).(eq T t (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind -Abst) i) u x1 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T -t (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u -x0 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u x1 -t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst) -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) t1 t3)))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H11: (eq T t -(THead (Bind Abst) x2 x3))).(\lambda (H12: (subst0 i u x0 x2)).(\lambda (H13: -(subst0 (s (Bind Abst) i) u x1 x3)).(eq_ind_r T (THead (Bind Abst) x2 x3) -(\lambda (t3: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead -(Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) -(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead -c0 (Bind b) u0) t1 t4))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: -T).(eq T (THead (Bind Abst) x2 x3) (THead (Bind Abst) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall -(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))))) x2 x3 -(refl_equal T (THead (Bind Abst) x2 x3)) (pr2_delta c0 d u i H1 u1 x0 H7 x2 -H12) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u -(S i) (getl_head (Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 x1 H8 x3 -H13)))) t H11)))))) H10)) (subst0_gen_head (Bind Abst) u x0 x1 t i H9)))))))) -(pr0_gen_abst u1 t1 t2 H5)))))))))))))) c y x H0))) H))))). -(* COMMENTS -Initial nodes: 2383 -END *) - -theorem pr2_gen_cast: - \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c -(THead (Flat Cast) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda -(t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c t1 t2)))) (pr2 c -t1 x)))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda -(H: (pr2 c (THead (Flat Cast) u1 t1) x)).(insert_eq T (THead (Flat Cast) u1 -t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c t1 -t2)))) (pr2 c t1 x))) (\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind -(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Flat Cast) -u1 t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead -(Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) -(\lambda (_: T).(\lambda (t2: T).(pr2 c0 t1 t2)))) (pr2 c0 t1 t0)))))) -(\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H1: (pr0 t0 -t2)).(\lambda (H2: (eq T t0 (THead (Flat Cast) u1 t1))).(let H3 \def (eq_ind -T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Flat Cast) u1 t1) H2) in (or_ind -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2) (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -t1 t3)))) (pr2 c0 t1 t2)) (\lambda (H4: (ex3_2 T T (\lambda (u2: T).(\lambda -(t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind -T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 -(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t2)) -(\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T t2 (THead (Flat Cast) -x0 x1))).(\lambda (H6: (pr0 u1 x0)).(\lambda (H7: (pr0 t1 x1)).(eq_ind_r T -(THead (Flat Cast) x0 x1) (\lambda (t: T).(or (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -t1 t3)))) (pr2 c0 t1 t))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda -(t3: T).(eq T (THead (Flat Cast) x0 x1) (THead (Flat Cast) u2 t3)))) (\lambda -(u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: -T).(pr2 c0 t1 t3)))) (pr2 c0 t1 (THead (Flat Cast) x0 x1)) (ex3_2_intro T T -(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Cast) x0 x1) (THead -(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) -(\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3))) x0 x1 (refl_equal T (THead -(Flat Cast) x0 x1)) (pr2_free c0 u1 x0 H6) (pr2_free c0 t1 x1 H7))) t2 -H5)))))) H4)) (\lambda (H4: (pr0 t1 t2)).(or_intror (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -t1 t3)))) (pr2 c0 t1 t2) (pr2_free c0 t1 t2 H4))) (pr0_gen_cast u1 t1 t2 -H3)))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: -nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t0: -T).(\lambda (t2: T).(\lambda (H2: (pr0 t0 t2)).(\lambda (t: T).(\lambda (H3: -(subst0 i u t2 t)).(\lambda (H4: (eq T t0 (THead (Flat Cast) u1 t1))).(let H5 -\def (eq_ind T t0 (\lambda (t3: T).(pr0 t3 t2)) H2 (THead (Flat Cast) u1 t1) -H4) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead -(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda -(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2) (or (ex3_2 T T (\lambda -(u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -t1 t3)))) (pr2 c0 t1 t)) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda -(t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind -T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t -(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t)) -(\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: (eq T t2 (THead (Flat Cast) -x0 x1))).(\lambda (H8: (pr0 u1 x0)).(\lambda (H9: (pr0 t1 x1)).(let H10 \def -(eq_ind T t2 (\lambda (t3: T).(subst0 i u t3 t)) H3 (THead (Flat Cast) x0 x1) -H7) in (or3_ind (ex2 T (\lambda (u2: T).(eq T t (THead (Flat Cast) u2 x1))) -(\lambda (u2: T).(subst0 i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T t (THead -(Flat Cast) x0 t3))) (\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3))) -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: -T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3)))) (or (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t)) (\lambda (H11: (ex2 T (\lambda (u2: -T).(eq T t (THead (Flat Cast) u2 x1))) (\lambda (u2: T).(subst0 i u x0 -u2)))).(ex2_ind T (\lambda (u2: T).(eq T t (THead (Flat Cast) u2 x1))) -(\lambda (u2: T).(subst0 i u x0 u2)) (or (ex3_2 T T (\lambda (u2: T).(\lambda -(t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 -c0 t1 t)) (\lambda (x2: T).(\lambda (H12: (eq T t (THead (Flat Cast) x2 -x1))).(\lambda (H13: (subst0 i u x0 x2)).(or_introl (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -t1 t3)))) (pr2 c0 t1 t) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: -T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 -c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3))) x2 x1 H12 -(pr2_delta c0 d u i H1 u1 x0 H8 x2 H13) (pr2_free c0 t1 x1 H9)))))) H11)) -(\lambda (H11: (ex2 T (\lambda (t3: T).(eq T t (THead (Flat Cast) x0 t3))) -(\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3)))).(ex2_ind T (\lambda -(t3: T).(eq T t (THead (Flat Cast) x0 t3))) (\lambda (t3: T).(subst0 (s (Flat -Cast) i) u x1 t3)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t -(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t)) -(\lambda (x2: T).(\lambda (H12: (eq T t (THead (Flat Cast) x0 x2))).(\lambda -(H13: (subst0 (s (Flat Cast) i) u x1 x2)).(or_introl (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -t1 t3)))) (pr2 c0 t1 t) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: -T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 -c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3))) x0 x2 H12 -(pr2_free c0 u1 x0 H8) (pr2_delta c0 d u i H1 t1 x1 H9 x2 H13)))))) H11)) -(\lambda (H11: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead -(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) -(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 -t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat -Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) -(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3))) (or -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t)) (\lambda (x2: -T).(\lambda (x3: T).(\lambda (H12: (eq T t (THead (Flat Cast) x2 -x3))).(\lambda (H13: (subst0 i u x0 x2)).(\lambda (H14: (subst0 (s (Flat -Cast) i) u x1 x3)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: -T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 -c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t) -(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast) -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr2 c0 t1 t3))) x2 x3 H12 (pr2_delta c0 d u i H1 u1 x0 -H8 x2 H13) (pr2_delta c0 d u i H1 t1 x1 H9 x3 H14)))))))) H11)) -(subst0_gen_head (Flat Cast) u x0 x1 t i H10)))))))) H6)) (\lambda (H6: (pr0 -t1 t2)).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t -(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t) -(pr2_delta c0 d u i H1 t1 t2 H6 t H3))) (pr0_gen_cast u1 t1 t2 -H5)))))))))))))) c y x H0))) H))))). -(* COMMENTS -Initial nodes: 2659 -END *) - -theorem pr2_gen_csort: - \forall (t1: T).(\forall (t2: T).(\forall (n: nat).((pr2 (CSort n) t1 t2) -\to (pr0 t1 t2)))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (n: nat).(\lambda (H: (pr2 (CSort -n) t1 t2)).(insert_eq C (CSort n) (\lambda (c: C).(pr2 c t1 t2)) (\lambda (_: -C).(pr0 t1 t2)) (\lambda (y: C).(\lambda (H0: (pr2 y t1 t2)).(pr2_ind -(\lambda (c: C).(\lambda (t: T).(\lambda (t0: T).((eq C c (CSort n)) \to (pr0 -t t0))))) (\lambda (c: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: -(pr0 t3 t4)).(\lambda (_: (eq C c (CSort n))).H1))))) (\lambda (c: -C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H1: (getl i c -(CHead d (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 -t3 t4)).(\lambda (t: T).(\lambda (_: (subst0 i u t4 t)).(\lambda (H4: (eq C c -(CSort n))).(let H5 \def (eq_ind C c (\lambda (c0: C).(getl i c0 (CHead d -(Bind Abbr) u))) H1 (CSort n) H4) in (getl_gen_sort n i (CHead d (Bind Abbr) -u) H5 (pr0 t3 t)))))))))))))) y t1 t2 H0))) H)))). -(* COMMENTS -Initial nodes: 221 -END *) - -theorem pr2_gen_appl: - \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c -(THead (Flat Appl) u1 t1) x) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda -(t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c t1 t2)))) (ex4_4 T -T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T -t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall -(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T -T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq -T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead -(Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda -(H: (pr2 c (THead (Flat Appl) u1 t1) x)).(insert_eq T (THead (Flat Appl) u1 -t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(or3 (ex3_2 T T (\lambda -(u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c t1 -t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind -Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) -z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) -(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) -z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) -y2) z1 z2)))))))))) (\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind -(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Flat Appl) -u1 t1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead -(Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) -(\lambda (_: T).(\lambda (t2: T).(pr2 c0 t1 t2)))) (ex4_4 T T T T (\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead -(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall -(b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) z1 t2)))))))) (ex6_6 B T T -T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T -t0 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 -y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 -z2))))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: T).(\lambda -(H1: (pr0 t0 t2)).(\lambda (H2: (eq T t0 (THead (Flat Appl) u1 t1))).(let H3 -\def (eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Flat Appl) u1 t1) -H2) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead -(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda -(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda -(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3)))))))) (or3 (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) -u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda -(u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift -(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 -(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (H4: (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 -t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead -(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda -(_: T).(\lambda (t3: T).(pr0 t1 t3))) (or3 (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) -u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda -(u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift -(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 -(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H5: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H6: (pr0 u1 -x0)).(\lambda (H7: (pr0 t1 x1)).(eq_ind_r T (THead (Flat Appl) x0 x1) -(\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t -(THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall -(b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T -T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t -(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda -(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 -y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 -z2)))))))))) (or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T -(THead (Flat Appl) x0 x1) (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) -x0 x1) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall -(u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Flat -Appl) x0 x1) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) -z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2)))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T -(THead (Flat Appl) x0 x1) (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -t1 t3))) x0 x1 (refl_equal T (THead (Flat Appl) x0 x1)) (pr2_free c0 u1 x0 -H6) (pr2_free c0 t1 x1 H7))) t2 H5)))))) H4)) (\lambda (H4: (ex4_4 T T T T -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda -(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 -t3))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda -(t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 -T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq -T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall -(b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T -T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T -t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 -y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 -z2))))))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: -T).(\lambda (H5: (eq T t1 (THead (Bind Abst) x0 x1))).(\lambda (H6: (eq T t2 -(THead (Bind Abbr) x2 x3))).(\lambda (H7: (pr0 u1 x2)).(\lambda (H8: (pr0 x1 -x3)).(eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda (t: T).(or3 (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t -(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 -(CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead -(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 -u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (eq_ind_r T (THead (Bind -Abst) x0 x1) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: -T).(eq T (THead (Bind Abbr) x2 x3) (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) -x2 x3) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall -(u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind -Abbr) x2 x3) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) -z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: -T).(eq T (THead (Bind Abbr) x2 x3) (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) u2 -t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: -T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) -u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead -(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind -Abbr) x2 x3) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) -z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead -(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) u2 -t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: -T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) -u) z1 t3))))))) x0 x1 x2 x3 (refl_equal T (THead (Bind Abst) x0 x1)) -(refl_equal T (THead (Bind Abbr) x2 x3)) (pr2_free c0 u1 x2 H7) (\lambda (b: -B).(\lambda (u: T).(pr2_free (CHead c0 (Bind b) u) x1 x3 H8))))) t1 H5) t2 -H6))))))))) H4)) (\lambda (H4: (ex6_6 B T T T T T (\lambda (b: B).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not -(eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) -y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead (Flat -Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 -t3))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead (Flat Appl) (lift -(S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 -t3))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead -(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) -(\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead -(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall -(b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T -T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T -t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 -y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 -z2))))))))) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: -T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H5: (not (eq B x0 -Abst))).(\lambda (H6: (eq T t1 (THead (Bind x0) x1 x2))).(\lambda (H7: (eq T -t2 (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)))).(\lambda -(H8: (pr0 u1 x3)).(\lambda (H9: (pr0 x1 x4)).(\lambda (H10: (pr0 x2 -x5)).(eq_ind_r T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) -x5)) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T -t (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall -(b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T -T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t -(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda -(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 -y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 -z2)))))))))) (eq_ind_r T (THead (Bind x0) x1 x2) (\lambda (t: T).(or3 (ex3_2 -T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat -Appl) (lift (S O) O x3) x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) -x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind Abbr) u2 t3)))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: -T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) -(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T t (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T -(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind b) -y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 -c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat -Appl) (lift (S O) O x3) x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) -(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 -(CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind -x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T -(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind b) -y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 -c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) -(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda -(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 -y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))) -x0 x1 x2 x5 x3 x4 H5 (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T -(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5))) (pr2_free c0 -u1 x3 H8) (pr2_free c0 x1 x4 H9) (pr2_free (CHead c0 (Bind x0) x4) x2 x5 -H10))) t1 H6) t2 H7))))))))))))) H4)) (pr0_gen_appl u1 t1 t2 H3)))))))) -(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t0: T).(\lambda (t2: -T).(\lambda (H2: (pr0 t0 t2)).(\lambda (t: T).(\lambda (H3: (subst0 i u t2 -t)).(\lambda (H4: (eq T t0 (THead (Flat Appl) u1 t1))).(let H5 \def (eq_ind T -t0 (\lambda (t3: T).(pr0 t3 t2)) H2 (THead (Flat Appl) u1 t1) H4) in (or3_ind -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 -z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: -T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda -(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3)))))))) (or3 (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind -Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda -(u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift -(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 -(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (H6: (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 -t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead -(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda -(_: T).(\lambda (t3: T).(pr0 t1 t3))) (or3 (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind -Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda -(u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift -(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 -(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H7: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H8: (pr0 u1 -x0)).(\lambda (H9: (pr0 t1 x1)).(let H10 \def (eq_ind T t2 (\lambda (t3: -T).(subst0 i u t3 t)) H3 (THead (Flat Appl) x0 x1) H7) in (or3_ind (ex2 T -(\lambda (u2: T).(eq T t (THead (Flat Appl) u2 x1))) (\lambda (u2: T).(subst0 -i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T t (THead (Flat Appl) x0 t3))) -(\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 t3))) (ex3_2 T T (\lambda -(u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3: -T).(subst0 (s (Flat Appl) i) u x1 t3)))) (or3 (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind -Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda -(u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift -(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 -(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (H11: (ex2 T (\lambda (u2: -T).(eq T t (THead (Flat Appl) u2 x1))) (\lambda (u2: T).(subst0 i u x0 -u2)))).(ex2_ind T (\lambda (u2: T).(eq T t (THead (Flat Appl) u2 x1))) -(\lambda (u2: T).(subst0 i u x0 u2)) (or3 (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind -Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda -(u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift -(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 -(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x2: T).(\lambda (H12: (eq T t -(THead (Flat Appl) x2 x1))).(\lambda (H13: (subst0 i u x0 x2)).(eq_ind_r T -(THead (Flat Appl) x2 x1) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: -T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 -t1 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind -Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda -(u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift -(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 -(CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro0 (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x1) (THead (Flat Appl) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t3: T).(eq T (THead (Flat Appl) x2 x1) (THead (Bind Abbr) u2 t3)))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: -T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) -(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T -(THead (Flat Appl) x2 x1) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O -u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2)))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T -(THead (Flat Appl) x2 x1) (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -t1 t3))) x2 x1 (refl_equal T (THead (Flat Appl) x2 x1)) (pr2_delta c0 d u i -H1 u1 x0 H8 x2 H13) (pr2_free c0 t1 x1 H9))) t H12)))) H11)) (\lambda (H11: -(ex2 T (\lambda (t3: T).(eq T t (THead (Flat Appl) x0 t3))) (\lambda (t3: -T).(subst0 (s (Flat Appl) i) u x1 t3)))).(ex2_ind T (\lambda (t3: T).(eq T t -(THead (Flat Appl) x0 t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 -t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat -Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda -(_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall -(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) -y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 -c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x2: T).(\lambda -(H12: (eq T t (THead (Flat Appl) x0 x2))).(\lambda (H13: (subst0 (s (Flat -Appl) i) u x1 x2)).(eq_ind_r T (THead (Flat Appl) x0 x2) (\lambda (t3: -T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat -Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda -(_: T).(\lambda (t4: T).(pr2 c0 t1 t4)))) (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall -(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) -(or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat -Appl) x0 x2) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 -T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq -T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x0 x2) (THead (Bind Abbr) -u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: -T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda -(u2: T).(\lambda (y2: T).(eq T (THead (Flat Appl) x0 x2) (THead (Bind b) y2 -(THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 -u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) (ex3_2_intro T T (\lambda -(u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x0 x2) (THead (Flat Appl) -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr2 c0 t1 t3))) x0 x2 (refl_equal T (THead (Flat Appl) -x0 x2)) (pr2_free c0 u1 x0 H8) (pr2_delta c0 d u i H1 t1 x1 H9 x2 H13))) t -H12)))) H11)) (\lambda (H11: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq -T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u -x0 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 -t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat -Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) -(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 t3))) (or3 -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall -(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) -y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 -c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x2: T).(\lambda -(x3: T).(\lambda (H12: (eq T t (THead (Flat Appl) x2 x3))).(\lambda (H13: -(subst0 i u x0 x2)).(\lambda (H14: (subst0 (s (Flat Appl) i) u x1 -x3)).(eq_ind_r T (THead (Flat Appl) x2 x3) (\lambda (t3: T).(or3 (ex3_2 T T -(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t4: T).(pr2 c0 t1 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 -(THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 -(CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead -(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 -u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro0 (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x3) (THead -(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) -(\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead -(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x3) (THead (Bind Abbr) u2 -t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: -T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda -(u2: T).(\lambda (y2: T).(eq T (THead (Flat Appl) x2 x3) (THead (Bind b) y2 -(THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 -u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) (ex3_2_intro T T (\lambda -(u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x3) (THead (Flat Appl) -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr2 c0 t1 t3))) x2 x3 (refl_equal T (THead (Flat Appl) -x2 x3)) (pr2_delta c0 d u i H1 u1 x0 H8 x2 H13) (pr2_delta c0 d u i H1 t1 x1 -H9 x3 H14))) t H12)))))) H11)) (subst0_gen_head (Flat Appl) u x0 x1 t i -H10)))))))) H6)) (\lambda (H6: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 -(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (t3: T).(pr0 z1 t3))))))).(ex4_4_ind T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))) (or3 (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t -(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 -(CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead -(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 -u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x0: T).(\lambda -(x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq T t1 (THead (Bind -Abst) x0 x1))).(\lambda (H8: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda -(H9: (pr0 u1 x2)).(\lambda (H10: (pr0 x1 x3)).(let H11 \def (eq_ind T t2 -(\lambda (t3: T).(subst0 i u t3 t)) H3 (THead (Bind Abbr) x2 x3) H8) in -(eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t3: T).(or3 (ex3_2 T T -(\lambda (u2: T).(\lambda (t4: T).(eq T t (THead (Flat Appl) u2 t4)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t4: T).(pr2 c0 t3 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T t3 (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t -(THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 -(CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t3 (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead -(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 -u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_ind (ex2 T (\lambda -(u2: T).(eq T t (THead (Bind Abbr) u2 x3))) (\lambda (u2: T).(subst0 i u x2 -u2))) (ex2 T (\lambda (t3: T).(eq T t (THead (Bind Abbr) x2 t3))) (\lambda -(t3: T).(subst0 (s (Bind Abbr) i) u x3 t3))) (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i u x2 u2))) (\lambda (_: T).(\lambda (t3: -T).(subst0 (s (Bind Abbr) i) u x3 t3)))) (or3 (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall -(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T -T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: -T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) -O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2))))))))) (\lambda (H12: (ex2 T (\lambda (u2: T).(eq T t (THead -(Bind Abbr) u2 x3))) (\lambda (u2: T).(subst0 i u x2 u2)))).(ex2_ind T -(\lambda (u2: T).(eq T t (THead (Bind Abbr) u2 x3))) (\lambda (u2: T).(subst0 -i u x2 u2)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead -(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) -(\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind Abst) x0 x1) t3)))) -(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda -(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind -Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead -(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) -y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 -c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x4: T).(\lambda -(H13: (eq T t (THead (Bind Abbr) x4 x3))).(\lambda (H14: (subst0 i u x2 -x4)).(eq_ind_r T (THead (Bind Abbr) x4 x3) (\lambda (t3: T).(or3 (ex3_2 T T -(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t4: T).(pr2 c0 (THead (Bind Abst) x0 x1) t4)))) (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) -x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: -T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) -(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: -T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S -O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: -T).(eq T (THead (Bind Abbr) x4 x3) (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x3) (THead (Bind Abbr) u2 -t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: -T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead -(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind -Abbr) x4 x3) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) -z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: 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(\lambda (t3: T).(eq T t (THead (Bind Abbr) x2 t3))) -(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x3 t3)) (or3 (ex3_2 T T (\lambda -(u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall -(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T -T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: 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(_: T).(\lambda (t4: T).(pr2 c0 -(THead (Bind Abst) x0 x1) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall -(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T -T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: -T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) 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T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead -(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x4) (THead (Bind Abbr) u2 -t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: -T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) z1 t3))))))) x0 x1 x2 x4 (refl_equal T (THead (Bind Abst) x0 x1)) -(refl_equal T (THead (Bind Abbr) x2 x4)) (pr2_free c0 u1 x2 H9) (\lambda (b: -B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) -(getl_clear_bind b (CHead c0 (Bind b) u0) c0 u0 (clear_bind b c0 u0) (CHead d -(Bind Abbr) u) i H1) x1 x3 H10 x4 H14))))) t H13)))) H12)) (\lambda (H12: -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 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T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) -(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: -T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) -O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2))))))))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H13: (eq T t -(THead (Bind Abbr) x4 x5))).(\lambda (H14: (subst0 i u x2 x4)).(\lambda (H15: -(subst0 (s (Bind Abbr) i) u x3 x5)).(eq_ind_r T (THead (Bind Abbr) x4 x5) -(\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 -(THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind Abst) x0 x1) -t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1 -z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: -T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall -(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) -z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: -T).(eq T (THead (Bind Abbr) x4 x5) (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2 -t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: -T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead -(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind -Abbr) x4 x5) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) -z2))))))))) (\lambda (_: B).(\lambda (_: 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(\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u -(S i) (getl_clear_bind b (CHead c0 (Bind b) u0) c0 u0 (clear_bind b c0 u0) -(CHead d (Bind Abbr) u) i H1) x1 x3 H10 x5 H15))))) t H13)))))) H12)) -(subst0_gen_head (Bind Abbr) u x2 x3 t i H11)) t1 H7)))))))))) H6)) (\lambda -(H6: (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) -(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda -(t3: T).(eq T t2 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) -t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda -(y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 -y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))))).(ex6_6_ind B T T T T -T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq -T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead -(Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda -(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (t3: T).(pr0 z1 t3))))))) (or3 (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind -Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda -(u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead 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T).(\lambda (y2: T).(eq T t (THead (Bind b) -y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 -c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_ind (ex2 T (\lambda -(u2: T).(eq T t (THead (Bind x0) u2 (THead (Flat Appl) (lift (S O) O x3) -x5)))) (\lambda (u2: T).(subst0 i u x4 u2))) (ex2 T (\lambda (t3: T).(eq T t -(THead (Bind x0) x4 t3))) (\lambda (t3: T).(subst0 (s (Bind x0) i) u (THead -(Flat Appl) (lift (S O) O x3) x5) t3))) (ex3_2 T T (\lambda (u2: T).(\lambda -(t3: T).(eq T t (THead (Bind x0) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 i u x4 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind x0) -i) u (THead (Flat Appl) (lift (S O) O x3) x5) t3)))) (or3 (ex3_2 T T (\lambda -(u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall -(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) -z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2))))))))) (\lambda (H14: (ex2 T (\lambda (u2: T).(eq T t (THead -(Bind x0) u2 (THead (Flat Appl) (lift (S O) O x3) x5)))) (\lambda (u2: -T).(subst0 i u x4 u2)))).(ex2_ind T (\lambda (u2: T).(eq T t (THead (Bind x0) -u2 (THead (Flat Appl) (lift (S O) O x3) x5)))) (\lambda (u2: T).(subst0 i u -x4 u2)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead -(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) -(\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) -(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind -Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead -(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 -u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x6: T).(\lambda -(H15: (eq T t (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) -x5)))).(\lambda (H16: (subst0 i u x4 x6)).(eq_ind_r T (THead (Bind x0) x6 -(THead (Flat Appl) (lift (S O) O x3) x5)) (\lambda (t3: T).(or3 (ex3_2 T T -(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 -x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall -(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T -T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: -T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S -O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: -T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead -(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) -(\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) -(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) -x6 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind Abbr) u2 t3)))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: -T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) -(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: -T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) -O x3) x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) -z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B -b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead -(Flat Appl) (lift (S O) O x3) x5)) (THead (Bind b) y2 (THead (Flat Appl) -(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 -(CHead c0 (Bind b) y2) z1 z2))))))) x0 x1 x2 x5 x3 x6 H7 (refl_equal T (THead -(Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x6 (THead (Flat Appl) (lift -(S O) O x3) x5))) (pr2_free c0 u1 x3 H10) (pr2_delta c0 d u i H1 x1 x4 H11 x6 -H16) (pr2_free (CHead c0 (Bind x0) x6) x2 x5 H12))) t H15)))) H14)) (\lambda -(H14: (ex2 T (\lambda (t3: T).(eq T t (THead (Bind x0) x4 t3))) (\lambda (t3: -T).(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) O x3) x5) -t3)))).(ex2_ind T (\lambda (t3: T).(eq T t (THead (Bind x0) x4 t3))) (\lambda -(t3: T).(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) O x3) x5) -t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat -Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda -(_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind -Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead -(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 -u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x6: T).(\lambda -(H15: (eq T t (THead (Bind x0) x4 x6))).(\lambda (H16: (subst0 (s (Bind x0) -i) u (THead (Flat Appl) (lift (S O) O x3) x5) x6)).(eq_ind_r T (THead (Bind -x0) x4 x6) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: -T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) -x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 -z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: -T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall -(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) -z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2)))))))))) (or3_ind (ex2 T (\lambda (u2: T).(eq T x6 (THead (Flat -Appl) u2 x5))) (\lambda (u2: T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) -u2))) (ex2 T (\lambda (t3: T).(eq T x6 (THead (Flat Appl) 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-(\lambda (u2: T).(eq T x6 (THead (Flat Appl) u2 x5))) (\lambda (u2: -T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) u2)))).(ex2_ind T (\lambda -(u2: T).(eq T x6 (THead (Flat Appl) u2 x5))) (\lambda (u2: T).(subst0 (s -(Bind x0) i) u (lift (S O) O x3) u2)) (or3 (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 x6) (THead (Flat Appl) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) -x4 x6) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall -(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind b) y2 (THead (Flat Appl) -(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 -(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x7: T).(\lambda (H18: (eq T -x6 (THead (Flat Appl) x7 x5))).(\lambda (H19: (subst0 (s (Bind x0) i) u (lift -(S 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u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) x7 x5)) (THead (Bind -Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(z2: T).(\lambda (u2: T).(\lambda 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(\lambda (_: T).(\lambda (t4: T).(pr2 c0 -(THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t4: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) t3 x5)) (THead (Bind -Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(z2: T).(\lambda (u2: T).(\lambda (y2: 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(_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B -b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead -(Flat Appl) (lift (S O) O x8) x5)) (THead (Bind b) y2 (THead (Flat Appl) -(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 -(CHead c0 (Bind b) y2) z1 z2))))))) x0 x1 x2 x5 x8 x4 H7 (refl_equal T (THead -(Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x4 (THead (Flat Appl) (lift -(S O) O x8) x5))) (pr2_delta c0 d u i H1 u1 x3 H10 x8 H22) (pr2_free c0 x1 x4 -H11) (pr2_free (CHead c0 (Bind x0) x4) x2 x5 H12))) x7 H20))))) -(subst0_gen_lift_ge u x3 x7 (s (Bind x0) i) (S O) O H19 (le_n_S O i (le_O_n -i)))) x6 H18)))) H17)) (\lambda (H17: (ex2 T (\lambda (t3: T).(eq T x6 (THead -(Flat Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) -(s (Bind x0) i)) u x5 t3)))).(ex2_ind T (\lambda (t3: T).(eq T x6 (THead -(Flat Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) -(s (Bind x0) i)) u x5 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: -T).(eq T (THead (Bind x0) x4 x6) (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: 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z2))))))) -x0 x1 x2 x7 x3 x4 H7 (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T -(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x7))) (pr2_free c0 -u1 x3 H10) (pr2_free c0 x1 x4 H11) (pr2_delta (CHead c0 (Bind x0) x4) d u (S -i) (getl_clear_bind x0 (CHead c0 (Bind x0) x4) c0 x4 (clear_bind x0 c0 x4) -(CHead d (Bind Abbr) u) i H1) x2 x5 H12 x7 H19))) x6 H18)))) H17)) (\lambda -(H17: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x6 (THead (Flat -Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0) i) u -(lift (S O) O x3) u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat -Appl) (s (Bind x0) i)) u x5 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda -(t3: T).(eq T x6 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) u2))) (\lambda (_: T).(\lambda -(t3: T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3))) (or3 (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 x6) (THead 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(THead -(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 -u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (ex2_ind T (\lambda (t3: -T).(eq T x7 (lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus (s (Bind x0) -i) (S O)) u x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T -(THead (Bind x0) x4 (THead (Flat Appl) x7 x8)) (THead (Flat Appl) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 -x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda 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B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x9: T).(\lambda -(H21: (eq T x7 (lift (S O) O x9))).(\lambda (H22: (subst0 (minus (s (Bind x0) -i) (S O)) u x3 x9)).(let H23 \def (eq_ind nat (minus (s (Bind x0) i) (S O)) -(\lambda (n: nat).(subst0 n u x3 x9)) H22 i (s_arith1 x0 i)) in (eq_ind_r T -(lift (S O) O x9) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda -(t4: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) t3 x8)) (THead (Flat -Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda -(_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: 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z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B -b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead -(Flat Appl) (lift (S O) O x9) x8)) (THead (Bind b) y2 (THead (Flat Appl) -(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 -(CHead c0 (Bind b) y2) z1 z2))))))) x0 x1 x2 x8 x9 x4 H7 (refl_equal T (THead -(Bind x0) x1 x2)) 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x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) -x6 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall -(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(eq T (THead (Bind x0) x6 t3) (THead (Bind b) y2 (THead (Flat Appl) -(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 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-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2))))))))) (\lambda (x9: T).(\lambda (H21: (eq T x8 (lift (S O) O -x9))).(\lambda (H22: (subst0 (minus (s (Bind x0) i) (S O)) u x3 x9)).(let H23 -\def (eq_ind nat (minus (s (Bind x0) i) (S O)) (\lambda (n: nat).(subst0 n u -x3 x9)) H22 i (s_arith1 x0 i)) in (eq_ind_r T (lift (S O) O x9) (\lambda (t3: -T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) -x6 (THead (Flat Appl) t3 x5)) (THead (Flat Appl) u2 t4)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 -(THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t4: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t3 x5)) (THead (Bind -Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 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-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: -T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x9) x5)) (THead -(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) -(\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) -(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) -x6 (THead (Flat Appl) (lift (S O) O x9) x5)) (THead (Bind Abbr) u2 t3)))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: -T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) -(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: -T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) -O x9) x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) -z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B -b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead -(Flat Appl) (lift (S O) O x9) x5)) (THead (Bind b) y2 (THead (Flat Appl) -(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 -(CHead c0 (Bind b) y2) z1 z2))))))) x0 x1 x2 x5 x9 x6 H7 (refl_equal T (THead -(Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x6 (THead (Flat Appl) (lift -(S O) O x9) x5))) (pr2_delta c0 d u i H1 u1 x3 H10 x9 H23) (pr2_delta c0 d u -i H1 x1 x4 H11 x6 H16) (pr2_free (CHead c0 (Bind x0) x6) x2 x5 H12))) x8 -H21))))) (subst0_gen_lift_ge u x3 x8 (s (Bind x0) i) (S O) O H20 (le_n_S O i -(le_O_n i)))) x7 H19)))) H18)) (\lambda (H18: (ex2 T (\lambda (t3: T).(eq T -x7 (THead (Flat Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s -(Flat Appl) (s (Bind x0) i)) u x5 t3)))).(ex2_ind T (\lambda (t3: T).(eq T x7 -(THead (Flat Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat -Appl) (s (Bind x0) i)) u x5 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda -(t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Flat Appl) u2 t3)))) (\lambda -(u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: -T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 -x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind Abbr) u2 -t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: -T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 x7) -(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda -(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 -y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 -z2))))))))) (\lambda (x8: T).(\lambda (H19: (eq T x7 (THead (Flat Appl) (lift -(S O) O x3) x8))).(\lambda (H20: (subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 -x8)).(eq_ind_r T (THead (Flat Appl) (lift (S O) O x3) x8) (\lambda (t3: -T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) -x6 t3) (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 -u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) -t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 -z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: -T).(eq T (THead (Bind x0) x6 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall -(b: 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(t3: T).(eq T (THead (Bind x0) x6 (THead (Flat -Appl) (lift (S O) O x3) x8)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 -(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8)) -(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 -(CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind -x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T -(THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8)) (THead (Bind b) -y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 -c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8)) -(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda -(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 -y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))) -x0 x1 x2 x8 x3 x6 H7 (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T -(THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8))) (pr2_free c0 -u1 x3 H10) (pr2_delta c0 d u i H1 x1 x4 H11 x6 H16) (pr2_delta (CHead c0 -(Bind x0) x6) d u (S i) (getl_clear_bind x0 (CHead c0 (Bind x0) x6) c0 x6 -(clear_bind x0 c0 x6) (CHead d (Bind Abbr) u) i H1) x2 x5 H12 x8 H20))) x7 -H19)))) H18)) (\lambda (H18: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq -T x7 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s -(Bind x0) i) u (lift (S O) O x3) u2))) (\lambda (_: T).(\lambda (t3: -T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3))))).(ex3_2_ind T T -(\lambda (u2: T).(\lambda (t3: T).(eq T x7 (THead (Flat Appl) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) -u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind x0) -i)) u x5 t3))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead -(Bind x0) x6 x7) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) -x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 -z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: -T).(eq T (THead (Bind x0) x6 x7) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall -(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T -T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: -T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind b) y2 (THead -(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 -u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x8: T).(\lambda -(x9: T).(\lambda (H19: (eq T x7 (THead (Flat Appl) x8 x9))).(\lambda (H20: -(subst0 (s (Bind x0) i) u (lift (S O) O x3) x8)).(\lambda (H21: (subst0 (s -(Flat Appl) (s (Bind x0) i)) u x5 x9)).(eq_ind_r T (THead (Flat Appl) x8 x9) -(\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T -(THead (Bind x0) x6 t3) (THead (Flat Appl) u2 t4)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 -(THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(t4: T).(eq T (THead (Bind x0) x6 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda -(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall -(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T -T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: -T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 t3) (THead (Bind b) y2 (THead -(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 -u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (ex2_ind T (\lambda (t3: -T).(eq T x8 (lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus (s (Bind x0) -i) (S O)) u x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T -(THead (Bind x0) x6 (THead (Flat Appl) x8 x9)) (THead (Flat Appl) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 -x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x8 x9)) -(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 -(CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind -x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T -(THead (Bind x0) x6 (THead (Flat Appl) x8 x9)) (THead (Bind b) y2 (THead -(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 -u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x10: T).(\lambda -(H22: (eq T x8 (lift (S O) O x10))).(\lambda (H23: (subst0 (minus (s (Bind -x0) i) (S O)) u x3 x10)).(let H24 \def (eq_ind nat (minus (s (Bind x0) i) (S -O)) (\lambda (n: nat).(subst0 n u x3 x10)) H23 i (s_arith1 x0 i)) in -(eq_ind_r T (lift (S O) O x10) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: -T).(\lambda (t4: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t3 x9)) -(THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) -(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) -x6 (THead (Flat Appl) t3 x9)) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall -(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T -T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: -T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t3 x9)) -(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda -(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 -y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 -z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T -(THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x10) x9)) (THead (Flat -Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda -(_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) -x6 (THead (Flat Appl) (lift (S O) O x10) x9)) (THead (Bind Abbr) u2 t3)))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: -T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) -(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: -T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) -O x10) x9)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) -z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) -y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B -b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead -(Flat Appl) (lift (S O) O x10) x9)) (THead (Bind b) y2 (THead (Flat Appl) -(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 -(CHead c0 (Bind b) y2) z1 z2))))))) x0 x1 x2 x9 x10 x6 H7 (refl_equal T -(THead (Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x6 (THead (Flat Appl) -(lift (S O) O x10) x9))) (pr2_delta c0 d u i H1 u1 x3 H10 x10 H24) (pr2_delta -c0 d u i H1 x1 x4 H11 x6 H16) (pr2_delta (CHead c0 (Bind x0) x6) d u (S i) -(getl_clear_bind x0 (CHead c0 (Bind x0) x6) c0 x6 (clear_bind x0 c0 x6) -(CHead d (Bind Abbr) u) i H1) x2 x5 H12 x9 H21))) x8 H22))))) -(subst0_gen_lift_ge u x3 x8 (s (Bind x0) i) (S O) O H20 (le_n_S O i (le_O_n -i)))) x7 H19)))))) H18)) (subst0_gen_head (Flat Appl) u (lift (S O) O x3) x5 -x7 (s (Bind x0) i) H17)) t H15)))))) H14)) (subst0_gen_head (Bind x0) u x4 -(THead (Flat Appl) (lift (S O) O x3) x5) t i H13)) t1 H8)))))))))))))) H6)) -(pr0_gen_appl u1 t1 t2 H5)))))))))))))) c y x H0))) H))))). -(* COMMENTS -Initial nodes: 38859 -END *) - -theorem pr2_gen_abbr: - \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c -(THead (Bind Abbr) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda -(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(or3 (\forall (b: -B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2))) (ex2 T (\lambda (u: -T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t2))) (ex3_2 T -T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) -(\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: -T).(pr2 (CHead c (Bind Abbr) u1) z t2)))))))) (\forall (b: B).(\forall (u: -T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda -(H: (pr2 c (THead (Bind Abbr) u1 t1) x)).(insert_eq T (THead (Bind Abbr) u1 -t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(or3 -(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2))) (ex2 T -(\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 -t2))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) -u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: -T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t2)))))))) (\forall (b: -B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))) -(\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0: C).(\lambda -(t: T).(\lambda (t0: T).((eq T t (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T -T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t2: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 -t2))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind -Abbr) u) t1 t2))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 -(Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) -(\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t2)))))))) -(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O -t0))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H1: -(pr0 t0 t2)).(\lambda (H2: (eq T t0 (THead (Bind Abbr) u1 t1))).(let H3 \def -(eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Bind Abbr) u1 t1) H2) in -(or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: -T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) -(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S O) O t2)) (or -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind -b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead -c0 (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 -(CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 -z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z -t3)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 -(lift (S O) O t2))))) (\lambda (H4: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: -T).(eq T t2 (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T -(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 -t3)))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead -(Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda -(u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) -(\lambda (y0: T).(subst0 O u2 y0 t3)))))) (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 -(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 t3))) (ex2 T -(\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind Abbr) u) t1 -t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) -u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: -T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: -B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O t2))))) -(\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T t2 (THead (Bind Abbr) -x0 x1))).(\lambda (H6: (pr0 u1 x0)).(\lambda (H_x: (or (pr0 t1 x1) (ex2 T -(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O x0 y0 -x1))))).(or_ind (pr0 t1 x1) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda -(y0: T).(subst0 O x0 y0 x1))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: -T).(eq T t2 (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: -B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 t3))) (ex2 T (\lambda (u: -T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind Abbr) u) t1 t3))) (ex3_2 -T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) -(\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: -T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u: -T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O t2))))) (\lambda (H7: (pr0 t1 -x1)).(eq_ind_r T (THead (Bind Abbr) x0 x1) (\lambda (t: T).(or (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 -t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind -Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 -(Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) -(\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) -(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O -t)))))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead -(Bind Abbr) x0 x1) (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: -B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 t3))) (ex2 T (\lambda (u: -T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind Abbr) u) t1 t3))) (ex3_2 -T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) -(\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: -T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u: -T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O (THead (Bind Abbr) x0 x1))))) -(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) -x0 x1) (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 -u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u: -T).(pr2 (CHead c0 (Bind b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) -(\lambda (u: T).(pr2 (CHead c0 (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda -(y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: -T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead -c0 (Bind Abbr) u1) z t3))))))) x0 x1 (refl_equal T (THead (Bind Abbr) x0 x1)) -(pr2_free c0 u1 x0 H6) (or3_intro0 (\forall (b: B).(\forall (u: T).(pr2 -(CHead c0 (Bind b) u) t1 x1))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda -(u: T).(pr2 (CHead c0 (Bind Abbr) u) t1 x1))) (ex3_2 T T (\lambda (y0: -T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: -T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead -c0 (Bind Abbr) u1) z x1)))) (\lambda (b: B).(\lambda (u: T).(pr2_free (CHead -c0 (Bind b) u) t1 x1 H7)))))) t2 H5)) (\lambda (H_x0: (ex2 T (\lambda (y0: -T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O x0 y0 x1)))).(ex2_ind T (\lambda -(y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O x0 y0 x1)) (or (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 -t3))) (ex2 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T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead -c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead -c0 (Bind b) u0) t1 (lift (S O) O t)))) (ex3_2_intro T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 -(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T -(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) -t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind -Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: -T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3))))))) x0 x3 H13 -(pr2_free c0 u1 x0 H8) (or3_intro2 (\forall (b: B).(\forall (u0: T).(pr2 -(CHead c0 (Bind b) u0) t1 x3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda -(u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 x3))) (ex3_2 T T (\lambda (y0: -T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: -T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead -c0 (Bind Abbr) u1) z x3)))) (ex3_2_intro T T (\lambda (y0: T).(\lambda (_: -T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: -T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) -u1) z x3))) x4 x1 (pr2_delta (CHead c0 (Bind Abbr) u1) c0 u1 O (getl_refl -Abbr c0 u1) t1 x2 H9 x4 H15) H16 (pr2_delta (CHead c0 (Bind Abbr) u1) d u (S -i) (getl_head (Bind Abbr) i c0 (CHead d (Bind Abbr) u) H1 u1) x1 x1 (pr0_refl -x1) x3 H14)))))))) (pr0_subst0_back x0 x2 x1 O H10 u1 H8))))) H12)) (\lambda -(H12: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: -T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x1 t3))))).(ex3_2_ind T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: -T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x1 t3))) (or (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 -t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 -(Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 -(CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 -z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z -t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 -(lift (S O) O t))))) (\lambda (x3: T).(\lambda (x4: T).(\lambda (H13: (eq T t -(THead (Bind Abbr) x3 x4))).(\lambda (H14: (subst0 i u x0 x3)).(\lambda (H15: -(subst0 (s (Bind Abbr) i) u x1 x4)).(ex2_ind T (\lambda (t3: T).(subst0 O u1 -x2 t3)) (\lambda (t3: T).(pr0 t3 x1)) (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 -(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T -(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) -t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind -Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: -T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: -B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))) -(\lambda (x5: T).(\lambda (H16: (subst0 O u1 x2 x5)).(\lambda (H17: (pr0 x5 -x1)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead -(Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) -(\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 -(CHead c0 (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda -(u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y0: -T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: -T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead -c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead -c0 (Bind b) u0) t1 (lift (S O) O t)))) (ex3_2_intro T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 -(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T -(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) -t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind -Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: -T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3))))))) x3 x4 H13 -(pr2_delta c0 d u i H1 u1 x0 H8 x3 H14) (or3_intro2 (\forall (b: B).(\forall -(u0: T).(pr2 (CHead c0 (Bind b) u0) t1 x4))) (ex2 T (\lambda (u0: T).(pr0 u1 -u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 x4))) (ex3_2 T T -(\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) -(\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: -T).(pr2 (CHead c0 (Bind Abbr) u1) z x4)))) (ex3_2_intro T T (\lambda (y0: -T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: -T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead -c0 (Bind Abbr) u1) z x4))) x5 x1 (pr2_delta (CHead c0 (Bind Abbr) u1) c0 u1 O -(getl_refl Abbr c0 u1) t1 x2 H9 x5 H16) H17 (pr2_delta (CHead c0 (Bind Abbr) -u1) d u (S i) (getl_head (Bind Abbr) i c0 (CHead d (Bind Abbr) u) H1 u1) x1 -x1 (pr0_refl x1) x4 H15)))))))) (pr0_subst0_back x0 x2 x1 O H10 u1 H8))))))) -H12)) (subst0_gen_head (Bind Abbr) u x0 x1 t i H11)))))) H_x0)) H_x)))))) -H6)) (\lambda (H6: (pr0 t1 (lift (S O) O t2))).(or_intror (ex3_2 T T (\lambda -(u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 -(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T -(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) -t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind -Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: -T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: -B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))) -(\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) -(getl_head (Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 (lift (S O) O t2) -H6 (lift (S O) O t) (subst0_lift_ge_S t2 t u i H3 O (le_O_n i))))))) -(pr0_gen_abbr u1 t1 t2 H5)))))))))))))) c y x H0))) H))))). -(* COMMENTS -Initial nodes: 11671 -END *) - -theorem pr2_gen_void: - \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c -(THead (Bind Void) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda -(t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall -(u: T).(pr2 (CHead c (Bind b) u) t1 t2)))))) (\forall (b: B).(\forall (u: -T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda -(H: (pr2 c (THead (Bind Void) u1 t1) x)).(insert_eq T (THead (Bind Void) u1 -t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall -(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2)))))) (\forall (b: -B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))) -(\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0: C).(\lambda -(t: T).(\lambda (t0: T).((eq T t (THead (Bind Void) u1 t1)) \to (or (ex3_2 T -T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Void) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 -t2)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 (lift -(S O) O t0))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: -T).(\lambda (H1: (pr0 t0 t2)).(\lambda (H2: (eq T t0 (THead (Bind Void) u1 -t1))).(let H3 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Bind -Void) u1 t1) H2) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq -T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 -u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O -t2)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda -(_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind -b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) -t1 (lift (S O) O t2))))) (\lambda (H4: (ex3_2 T T (\lambda (u2: T).(\lambda -(t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind -T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 -(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 -(CHead c0 (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead -c0 (Bind b) u) t1 (lift (S O) O t2))))) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H5: (eq T t2 (THead (Bind Void) x0 x1))).(\lambda (H6: (pr0 u1 -x0)).(\lambda (H7: (pr0 t1 x1)).(eq_ind_r T (THead (Bind Void) x0 x1) -(\lambda (t: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t -(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 -(CHead c0 (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead -c0 (Bind b) u) t1 (lift (S O) O t)))))) (or_introl (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T (THead (Bind Void) x0 x1) (THead (Bind Void) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) -u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 -(lift (S O) O (THead (Bind Void) x0 x1))))) (ex3_2_intro T T (\lambda (u2: -T).(\lambda (t3: T).(eq T (THead (Bind Void) x0 x1) (THead (Bind Void) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) -u) t1 t3))))) x0 x1 (refl_equal T (THead (Bind Void) x0 x1)) (pr2_free c0 u1 -x0 H6) (\lambda (b: B).(\lambda (u: T).(pr2_free (CHead c0 (Bind b) u) t1 x1 -H7))))) t2 H5)))))) H4)) (\lambda (H4: (pr0 t1 (lift (S O) O t2))).(or_intror -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) -u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 -(lift (S O) O t2)))) (\lambda (b: B).(\lambda (u: T).(pr2_free (CHead c0 -(Bind b) u) t1 (lift (S O) O t2) H4))))) (pr0_gen_void u1 t1 t2 H3)))))))) -(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t0: T).(\lambda (t2: -T).(\lambda (H2: (pr0 t0 t2)).(\lambda (t: T).(\lambda (H3: (subst0 i u t2 -t)).(\lambda (H4: (eq T t0 (THead (Bind Void) u1 t1))).(let H5 \def (eq_ind T -t0 (\lambda (t3: T).(pr0 t3 t2)) H2 (THead (Bind Void) u1 t1) H4) in (or_ind -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2)) (or (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 -t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 -(lift (S O) O t))))) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: -T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind -T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: -T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t -(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 -(CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 -(CHead c0 (Bind b) u0) t1 (lift (S O) O t))))) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H7: (eq T t2 (THead (Bind Void) x0 x1))).(\lambda (H8: (pr0 u1 -x0)).(\lambda (H9: (pr0 t1 x1)).(let H10 \def (eq_ind T t2 (\lambda (t3: -T).(subst0 i u t3 t)) H3 (THead (Bind Void) x0 x1) H7) in (or3_ind (ex2 T -(\lambda (u2: T).(eq T t (THead (Bind Void) u2 x1))) (\lambda (u2: T).(subst0 -i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T t (THead (Bind Void) x0 t3))) -(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3))) (ex3_2 T T (\lambda -(u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3: -T).(subst0 (s (Bind Void) i) u x1 t3)))) (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall -(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: -B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))) -(\lambda (H11: (ex2 T (\lambda (u2: T).(eq T t (THead (Bind Void) u2 x1))) -(\lambda (u2: T).(subst0 i u x0 u2)))).(ex2_ind T (\lambda (u2: T).(eq T t -(THead (Bind Void) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2)) (or (ex3_2 T -T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 -t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 -(lift (S O) O t))))) (\lambda (x2: T).(\lambda (H12: (eq T t (THead (Bind -Void) x2 x1))).(\lambda (H13: (subst0 i u x0 x2)).(or_introl (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 -t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 -(lift (S O) O t)))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T -t (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 -u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 -(CHead c0 (Bind b) u0) t1 t3))))) x2 x1 H12 (pr2_delta c0 d u i H1 u1 x0 H8 -x2 H13) (\lambda (b: B).(\lambda (u0: T).(pr2_free (CHead c0 (Bind b) u0) t1 -x1 H9)))))))) H11)) (\lambda (H11: (ex2 T (\lambda (t3: T).(eq T t (THead -(Bind Void) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 -t3)))).(ex2_ind T (\lambda (t3: T).(eq T t (THead (Bind Void) x0 t3))) -(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3)) (or (ex3_2 T T (\lambda -(u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall -(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: -B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))) -(\lambda (x2: T).(\lambda (H12: (eq T t (THead (Bind Void) x0 x2))).(\lambda -(H13: (subst0 (s (Bind Void) i) u x1 x2)).(or_introl (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall -(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: -B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))) -(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) t1 t3))))) x0 x2 H12 (pr2_free c0 u1 x0 H8) (\lambda (b: B).(\lambda (u0: -T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) (getl_head (Bind b) i c0 -(CHead d (Bind Abbr) u) H1 u0) t1 x1 H9 x2 H13)))))))) H11)) (\lambda (H11: -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: -T).(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3))))).(ex3_2_ind T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: -T).(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3))) (or (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda -(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 -t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 -(lift (S O) O t))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (eq T t -(THead (Bind Void) x2 x3))).(\lambda (H13: (subst0 i u x0 x2)).(\lambda (H14: -(subst0 (s (Bind Void) i) u x1 x3)).(or_introl (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall -(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: -B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))) -(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) -u0) t1 t3))))) x2 x3 H12 (pr2_delta c0 d u i H1 u1 x0 H8 x2 H13) (\lambda (b: -B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) (getl_head -(Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 x1 H9 x3 H14)))))))))) H11)) -(subst0_gen_head (Bind Void) u x0 x1 t i H10)))))))) H6)) (\lambda (H6: (pr0 -t1 (lift (S O) O t2))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: -T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 -c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: -T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0: -T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))) (\lambda (b: -B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) (getl_head -(Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 (lift (S O) O t2) H6 (lift (S -O) O t) (subst0_lift_ge_S t2 t u i H3 O (le_O_n i))))))) (pr0_gen_void u1 t1 -t2 H5)))))))))))))) c y x H0))) H))))). -(* COMMENTS -Initial nodes: 3467 -END *) - -theorem pr2_gen_lift: - \forall (c: C).(\forall (t1: T).(\forall (x: T).(\forall (h: nat).(\forall -(d: nat).((pr2 c (lift h d t1) x) \to (\forall (e: C).((drop h d c e) \to -(ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr2 e t1 -t2)))))))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (x: T).(\lambda (h: nat).(\lambda -(d: nat).(\lambda (H: (pr2 c (lift h d t1) x)).(insert_eq T (lift h d t1) -(\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(\forall (e: C).((drop h d c e) -\to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr2 e -t1 t2)))))) (\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0: -C).(\lambda (t: T).(\lambda (t0: T).((eq T t (lift h d t1)) \to (\forall (e: -C).((drop h d c0 e) \to (ex2 T (\lambda (t2: T).(eq T t0 (lift h d t2))) -(\lambda (t2: T).(pr2 e t1 t2))))))))) (\lambda (c0: C).(\lambda (t0: -T).(\lambda (t2: T).(\lambda (H1: (pr0 t0 t2)).(\lambda (H2: (eq T t0 (lift h -d t1))).(\lambda (e: C).(\lambda (_: (drop h d c0 e)).(let H4 \def (eq_ind T -t0 (\lambda (t: T).(pr0 t t2)) H1 (lift h d t1) H2) in (ex2_ind T (\lambda -(t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(pr0 t1 t3)) (ex2 T -(\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))) -(\lambda (x0: T).(\lambda (H5: (eq T t2 (lift h d x0))).(\lambda (H6: (pr0 t1 -x0)).(eq_ind_r T (lift h d x0) (\lambda (t: T).(ex2 T (\lambda (t3: T).(eq T -t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)))) (ex_intro2 T (\lambda -(t3: T).(eq T (lift h d x0) (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)) -x0 (refl_equal T (lift h d x0)) (pr2_free e t1 x0 H6)) t2 H5)))) -(pr0_gen_lift t1 t2 h d H4)))))))))) (\lambda (c0: C).(\lambda (d0: -C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d0 (Bind -Abbr) u))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H2: (pr0 t0 -t2)).(\lambda (t: T).(\lambda (H3: (subst0 i u t2 t)).(\lambda (H4: (eq T t0 -(lift h d t1))).(\lambda (e: C).(\lambda (H5: (drop h d c0 e)).(let H6 \def -(eq_ind T t0 (\lambda (t3: T).(pr0 t3 t2)) H2 (lift h d t1) H4) in (ex2_ind T -(\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(pr0 t1 t3)) (ex2 -T (\lambda (t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))) -(\lambda (x0: T).(\lambda (H7: (eq T t2 (lift h d x0))).(\lambda (H8: (pr0 t1 -x0)).(let H9 \def (eq_ind T t2 (\lambda (t3: T).(subst0 i u t3 t)) H3 (lift h -d x0) H7) in (lt_le_e i d (ex2 T (\lambda (t3: T).(eq T t (lift h d t3))) -(\lambda (t3: T).(pr2 e t1 t3))) (\lambda (H10: (lt i d)).(let H11 \def -(eq_ind nat d (\lambda (n: nat).(subst0 i u (lift h n x0) t)) H9 (S (plus i -(minus d (S i)))) (lt_plus_minus i d H10)) in (let H12 \def (eq_ind nat d -(\lambda (n: nat).(drop h n c0 e)) H5 (S (plus i (minus d (S i)))) -(lt_plus_minus i d H10)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: -C).(eq T u (lift h (minus d (S i)) v)))) (\lambda (v: T).(\lambda (e0: -C).(getl i e (CHead e0 (Bind Abbr) v)))) (\lambda (_: T).(\lambda (e0: -C).(drop h (minus d (S i)) d0 e0))) (ex2 T (\lambda (t3: T).(eq T t (lift h d -t3))) (\lambda (t3: T).(pr2 e t1 t3))) (\lambda (x1: T).(\lambda (x2: -C).(\lambda (H13: (eq T u (lift h (minus d (S i)) x1))).(\lambda (H14: (getl -i e (CHead x2 (Bind Abbr) x1))).(\lambda (_: (drop h (minus d (S i)) d0 -x2)).(let H16 \def (eq_ind T u (\lambda (t3: T).(subst0 i t3 (lift h (S (plus -i (minus d (S i)))) x0) t)) H11 (lift h (minus d (S i)) x1) H13) in (ex2_ind -T (\lambda (t3: T).(eq T t (lift h (S (plus i (minus d (S i)))) t3))) -(\lambda (t3: T).(subst0 i x1 x0 t3)) (ex2 T (\lambda (t3: T).(eq T t (lift h -d t3))) (\lambda (t3: T).(pr2 e t1 t3))) (\lambda (x3: T).(\lambda (H17: (eq -T t (lift h (S (plus i (minus d (S i)))) x3))).(\lambda (H18: (subst0 i x1 x0 -x3)).(let H19 \def (eq_ind_r nat (S (plus i (minus d (S i)))) (\lambda (n: -nat).(eq T t (lift h n x3))) H17 d (lt_plus_minus i d H10)) in (ex_intro2 T -(\lambda (t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)) x3 -H19 (pr2_delta e x2 x1 i H14 t1 x0 H8 x3 H18)))))) (subst0_gen_lift_lt x1 x0 -t i h (minus d (S i)) H16)))))))) (getl_drop_conf_lt Abbr c0 d0 u i H1 e h -(minus d (S i)) H12))))) (\lambda (H10: (le d i)).(lt_le_e i (plus d h) (ex2 -T (\lambda (t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))) -(\lambda (H11: (lt i (plus d h))).(subst0_gen_lift_false x0 u t h d i H10 H11 -H9 (ex2 T (\lambda (t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 -t3))))) (\lambda (H11: (le (plus d h) i)).(ex2_ind T (\lambda (t3: T).(eq T t -(lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u x0 t3)) (ex2 T -(\lambda (t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))) -(\lambda (x1: T).(\lambda (H12: (eq T t (lift h d x1))).(\lambda (H13: -(subst0 (minus i h) u x0 x1)).(ex_intro2 T (\lambda (t3: T).(eq T t (lift h d -t3))) (\lambda (t3: T).(pr2 e t1 t3)) x1 H12 (pr2_delta e d0 u (minus i h) -(getl_drop_conf_ge i (CHead d0 (Bind Abbr) u) c0 H1 e h d H5 H11) t1 x0 H8 x1 -H13))))) (subst0_gen_lift_ge u x0 t i h d H9 H11)))))))))) (pr0_gen_lift t1 -t2 h d H6)))))))))))))))) c y x H0))) H)))))). -(* COMMENTS -Initial nodes: 1579 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr2/pr2.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr2/pr2.ma deleted file mode 100644 index f8df4e9e2..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr2/pr2.ma +++ /dev/null @@ -1,258 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr2/defs.ma". - -include "Basic-1/pr0/pr0.ma". - -include "Basic-1/getl/props.ma". - -theorem pr2_confluence__pr2_free_free: - \forall (c: C).(\forall (t0: T).(\forall (t1: T).(\forall (t2: T).((pr0 t0 -t1) \to ((pr0 t0 t2) \to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: -T).(pr2 c t2 t)))))))) -\def - \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H: (pr0 t0 t1)).(\lambda (H0: (pr0 t0 t2)).(ex2_ind T (\lambda (t: T).(pr0 -t2 t)) (\lambda (t: T).(pr0 t1 t)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) -(\lambda (t: T).(pr2 c t2 t))) (\lambda (x: T).(\lambda (H1: (pr0 t2 -x)).(\lambda (H2: (pr0 t1 x)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) -(\lambda (t: T).(pr2 c t2 t)) x (pr2_free c t1 x H2) (pr2_free c t2 x H1))))) -(pr0_confluence t0 t2 H0 t1 H))))))). -(* COMMENTS -Initial nodes: 135 -END *) - -theorem pr2_confluence__pr2_free_delta: - \forall (c: C).(\forall (d: C).(\forall (t0: T).(\forall (t1: T).(\forall -(t2: T).(\forall (t4: T).(\forall (u: T).(\forall (i: nat).((pr0 t0 t1) \to -((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t4) \to ((subst0 i u t4 t2) -\to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 -t)))))))))))))) -\def - \lambda (c: C).(\lambda (d: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda -(t2: T).(\lambda (t4: T).(\lambda (u: T).(\lambda (i: nat).(\lambda (H: (pr0 -t0 t1)).(\lambda (H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H1: (pr0 -t0 t4)).(\lambda (H2: (subst0 i u t4 t2)).(ex2_ind T (\lambda (t: T).(pr0 t4 -t)) (\lambda (t: T).(pr0 t1 t)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda -(t: T).(pr2 c t2 t))) (\lambda (x: T).(\lambda (H3: (pr0 t4 x)).(\lambda (H4: -(pr0 t1 x)).(or_ind (pr0 t2 x) (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda -(w2: T).(subst0 i u x w2))) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: -T).(pr2 c t2 t))) (\lambda (H5: (pr0 t2 x)).(ex_intro2 T (\lambda (t: T).(pr2 -c t1 t)) (\lambda (t: T).(pr2 c t2 t)) x (pr2_free c t1 x H4) (pr2_free c t2 -x H5))) (\lambda (H5: (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: -T).(subst0 i u x w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t2 w2)) (\lambda -(w2: T).(subst0 i u x w2)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: -T).(pr2 c t2 t))) (\lambda (x0: T).(\lambda (H6: (pr0 t2 x0)).(\lambda (H7: -(subst0 i u x x0)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: -T).(pr2 c t2 t)) x0 (pr2_delta c d u i H0 t1 x H4 x0 H7) (pr2_free c t2 x0 -H6))))) H5)) (pr0_subst0 t4 x H3 u t2 i H2 u (pr0_refl u)))))) -(pr0_confluence t0 t4 H1 t1 H))))))))))))). -(* COMMENTS -Initial nodes: 403 -END *) - -theorem pr2_confluence__pr2_delta_delta: - \forall (c: C).(\forall (d: C).(\forall (d0: C).(\forall (t0: T).(\forall -(t1: T).(\forall (t2: T).(\forall (t3: T).(\forall (t4: T).(\forall (u: -T).(\forall (u0: T).(\forall (i: nat).(\forall (i0: nat).((getl i c (CHead d -(Bind Abbr) u)) \to ((pr0 t0 t3) \to ((subst0 i u t3 t1) \to ((getl i0 c -(CHead d0 (Bind Abbr) u0)) \to ((pr0 t0 t4) \to ((subst0 i0 u0 t4 t2) \to -(ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 -t)))))))))))))))))))) -\def - \lambda (c: C).(\lambda (d: C).(\lambda (d0: C).(\lambda (t0: T).(\lambda -(t1: T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (u: -T).(\lambda (u0: T).(\lambda (i: nat).(\lambda (i0: nat).(\lambda (H: (getl i -c (CHead d (Bind Abbr) u))).(\lambda (H0: (pr0 t0 t3)).(\lambda (H1: (subst0 -i u t3 t1)).(\lambda (H2: (getl i0 c (CHead d0 (Bind Abbr) u0))).(\lambda -(H3: (pr0 t0 t4)).(\lambda (H4: (subst0 i0 u0 t4 t2)).(ex2_ind T (\lambda (t: -T).(pr0 t4 t)) (\lambda (t: T).(pr0 t3 t)) (ex2 T (\lambda (t: T).(pr2 c t1 -t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (x: T).(\lambda (H5: (pr0 t4 -x)).(\lambda (H6: (pr0 t3 x)).(or_ind (pr0 t1 x) (ex2 T (\lambda (w2: T).(pr0 -t1 w2)) (\lambda (w2: T).(subst0 i u x w2))) (ex2 T (\lambda (t: T).(pr2 c t1 -t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (H7: (pr0 t1 x)).(or_ind (pr0 t2 -x) (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: T).(subst0 i0 u0 x -w2))) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) -(\lambda (H8: (pr0 t2 x)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda -(t: T).(pr2 c t2 t)) x (pr2_free c t1 x H7) (pr2_free c t2 x H8))) (\lambda -(H8: (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: T).(subst0 i0 u0 x -w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: T).(subst0 i0 -u0 x w2)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) -(\lambda (x0: T).(\lambda (H9: (pr0 t2 x0)).(\lambda (H10: (subst0 i0 u0 x -x0)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t)) -x0 (pr2_delta c d0 u0 i0 H2 t1 x H7 x0 H10) (pr2_free c t2 x0 H9))))) H8)) -(pr0_subst0 t4 x H5 u0 t2 i0 H4 u0 (pr0_refl u0)))) (\lambda (H7: (ex2 T -(\lambda (w2: T).(pr0 t1 w2)) (\lambda (w2: T).(subst0 i u x w2)))).(ex2_ind -T (\lambda (w2: T).(pr0 t1 w2)) (\lambda (w2: T).(subst0 i u x w2)) (ex2 T -(\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (x0: -T).(\lambda (H8: (pr0 t1 x0)).(\lambda (H9: (subst0 i u x x0)).(or_ind (pr0 -t2 x) (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: T).(subst0 i0 u0 x -w2))) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) -(\lambda (H10: (pr0 t2 x)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) -(\lambda (t: T).(pr2 c t2 t)) x0 (pr2_free c t1 x0 H8) (pr2_delta c d u i H -t2 x H10 x0 H9))) (\lambda (H10: (ex2 T (\lambda (w2: T).(pr0 t2 w2)) -(\lambda (w2: T).(subst0 i0 u0 x w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t2 -w2)) (\lambda (w2: T).(subst0 i0 u0 x w2)) (ex2 T (\lambda (t: T).(pr2 c t1 -t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (x1: T).(\lambda (H11: (pr0 t2 -x1)).(\lambda (H12: (subst0 i0 u0 x x1)).(neq_eq_e i i0 (ex2 T (\lambda (t: -T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (H13: (not (eq nat i -i0))).(ex2_ind T (\lambda (t: T).(subst0 i u x1 t)) (\lambda (t: T).(subst0 -i0 u0 x0 t)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 -t))) (\lambda (x2: T).(\lambda (H14: (subst0 i u x1 x2)).(\lambda (H15: -(subst0 i0 u0 x0 x2)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: -T).(pr2 c t2 t)) x2 (pr2_delta c d0 u0 i0 H2 t1 x0 H8 x2 H15) (pr2_delta c d -u i H t2 x1 H11 x2 H14))))) (subst0_confluence_neq x x1 u0 i0 H12 x0 u i H9 -(sym_not_eq nat i i0 H13)))) (\lambda (H13: (eq nat i i0)).(let H14 \def -(eq_ind_r nat i0 (\lambda (n: nat).(subst0 n u0 x x1)) H12 i H13) in (let H15 -\def (eq_ind_r nat i0 (\lambda (n: nat).(getl n c (CHead d0 (Bind Abbr) u0))) -H2 i H13) in (let H16 \def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c0: -C).(getl i c c0)) H (CHead d0 (Bind Abbr) u0) (getl_mono c (CHead d (Bind -Abbr) u) i H (CHead d0 (Bind Abbr) u0) H15)) in (let H17 \def (f_equal C C -(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abbr) u) -(CHead d0 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) i H (CHead d0 -(Bind Abbr) u0) H15)) in ((let H18 \def (f_equal C T (\lambda (e: C).(match e -in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) -\Rightarrow t])) (CHead d (Bind Abbr) u) (CHead d0 (Bind Abbr) u0) (getl_mono -c (CHead d (Bind Abbr) u) i H (CHead d0 (Bind Abbr) u0) H15)) in (\lambda -(H19: (eq C d d0)).(let H20 \def (eq_ind_r T u0 (\lambda (t: T).(subst0 i t x -x1)) H14 u H18) in (let H21 \def (eq_ind_r T u0 (\lambda (t: T).(getl i c -(CHead d0 (Bind Abbr) t))) H16 u H18) in (let H22 \def (eq_ind_r C d0 -(\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr) u))) H21 d H19) in (or4_ind -(eq T x1 x0) (ex2 T (\lambda (t: T).(subst0 i u x1 t)) (\lambda (t: -T).(subst0 i u x0 t))) (subst0 i u x1 x0) (subst0 i u x0 x1) (ex2 T (\lambda -(t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (H23: (eq T x1 -x0)).(let H24 \def (eq_ind T x1 (\lambda (t: T).(pr0 t2 t)) H11 x0 H23) in -(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t)) x0 -(pr2_free c t1 x0 H8) (pr2_free c t2 x0 H24)))) (\lambda (H23: (ex2 T -(\lambda (t: T).(subst0 i u x1 t)) (\lambda (t: T).(subst0 i u x0 -t)))).(ex2_ind T (\lambda (t: T).(subst0 i u x1 t)) (\lambda (t: T).(subst0 i -u x0 t)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) -(\lambda (x2: T).(\lambda (H24: (subst0 i u x1 x2)).(\lambda (H25: (subst0 i -u x0 x2)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c -t2 t)) x2 (pr2_delta c d u i H22 t1 x0 H8 x2 H25) (pr2_delta c d u i H22 t2 -x1 H11 x2 H24))))) H23)) (\lambda (H23: (subst0 i u x1 x0)).(ex_intro2 T -(\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t)) x0 (pr2_free c t1 -x0 H8) (pr2_delta c d u i H22 t2 x1 H11 x0 H23))) (\lambda (H23: (subst0 i u -x0 x1)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 -t)) x1 (pr2_delta c d u i H22 t1 x0 H8 x1 H23) (pr2_free c t2 x1 H11))) -(subst0_confluence_eq x x1 u i H20 x0 H9))))))) H17)))))))))) H10)) -(pr0_subst0 t4 x H5 u0 t2 i0 H4 u0 (pr0_refl u0)))))) H7)) (pr0_subst0 t3 x -H6 u t1 i H1 u (pr0_refl u)))))) (pr0_confluence t0 t4 H3 t3 -H0))))))))))))))))))). -(* COMMENTS -Initial nodes: 1901 -END *) - -theorem pr2_confluence: - \forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr2 c t0 t1) \to (\forall -(t2: T).((pr2 c t0 t2) \to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: -T).(pr2 c t2 t)))))))) -\def - \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr2 c t0 -t1)).(\lambda (t2: T).(\lambda (H0: (pr2 c t0 t2)).(let H1 \def (match H in -pr2 return (\lambda (c0: C).(\lambda (t: T).(\lambda (t3: T).(\lambda (_: -(pr2 c0 t t3)).((eq C c0 c) \to ((eq T t t0) \to ((eq T t3 t1) \to (ex2 T -(\lambda (t4: T).(pr2 c t1 t4)) (\lambda (t4: T).(pr2 c t2 t4)))))))))) with -[(pr2_free c0 t3 t4 H1) \Rightarrow (\lambda (H2: (eq C c0 c)).(\lambda (H3: -(eq T t3 t0)).(\lambda (H4: (eq T t4 t1)).(eq_ind C c (\lambda (_: C).((eq T -t3 t0) \to ((eq T t4 t1) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t: T).(pr2 c -t1 t)) (\lambda (t: T).(pr2 c t2 t))))))) (\lambda (H5: (eq T t3 t0)).(eq_ind -T t0 (\lambda (t: T).((eq T t4 t1) \to ((pr0 t t4) \to (ex2 T (\lambda (t5: -T).(pr2 c t1 t5)) (\lambda (t5: T).(pr2 c t2 t5)))))) (\lambda (H6: (eq T t4 -t1)).(eq_ind T t1 (\lambda (t: T).((pr0 t0 t) \to (ex2 T (\lambda (t5: -T).(pr2 c t1 t5)) (\lambda (t5: T).(pr2 c t2 t5))))) (\lambda (H7: (pr0 t0 -t1)).(let H8 \def (match H0 in pr2 return (\lambda (c1: C).(\lambda (t: -T).(\lambda (t5: T).(\lambda (_: (pr2 c1 t t5)).((eq C c1 c) \to ((eq T t t0) -\to ((eq T t5 t2) \to (ex2 T (\lambda (t6: T).(pr2 c t1 t6)) (\lambda (t6: -T).(pr2 c t2 t6)))))))))) with [(pr2_free c1 t5 t6 H8) \Rightarrow (\lambda -(H9: (eq C c1 c)).(\lambda (H10: (eq T t5 t0)).(\lambda (H11: (eq T t6 -t2)).(eq_ind C c (\lambda (_: C).((eq T t5 t0) \to ((eq T t6 t2) \to ((pr0 t5 -t6) \to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 -t))))))) (\lambda (H12: (eq T t5 t0)).(eq_ind T t0 (\lambda (t: T).((eq T t6 -t2) \to ((pr0 t t6) \to (ex2 T (\lambda (t7: T).(pr2 c t1 t7)) (\lambda (t7: -T).(pr2 c t2 t7)))))) (\lambda (H13: (eq T t6 t2)).(eq_ind T t2 (\lambda (t: -T).((pr0 t0 t) \to (ex2 T (\lambda (t7: T).(pr2 c t1 t7)) (\lambda (t7: -T).(pr2 c t2 t7))))) (\lambda (H14: (pr0 t0 -t2)).(pr2_confluence__pr2_free_free c t0 t1 t2 H7 H14)) t6 (sym_eq T t6 t2 -H13))) t5 (sym_eq T t5 t0 H12))) c1 (sym_eq C c1 c H9) H10 H11 H8)))) | -(pr2_delta c1 d u i H8 t5 t6 H9 t H10) \Rightarrow (\lambda (H11: (eq C c1 -c)).(\lambda (H12: (eq T t5 t0)).(\lambda (H13: (eq T t t2)).(eq_ind C c -(\lambda (c2: C).((eq T t5 t0) \to ((eq T t t2) \to ((getl i c2 (CHead d -(Bind Abbr) u)) \to ((pr0 t5 t6) \to ((subst0 i u t6 t) \to (ex2 T (\lambda -(t7: T).(pr2 c t1 t7)) (\lambda (t7: T).(pr2 c t2 t7))))))))) (\lambda (H14: -(eq T t5 t0)).(eq_ind T t0 (\lambda (t7: T).((eq T t t2) \to ((getl i c -(CHead d (Bind Abbr) u)) \to ((pr0 t7 t6) \to ((subst0 i u t6 t) \to (ex2 T -(\lambda (t8: T).(pr2 c t1 t8)) (\lambda (t8: T).(pr2 c t2 t8)))))))) -(\lambda (H15: (eq T t t2)).(eq_ind T t2 (\lambda (t7: T).((getl i c (CHead d -(Bind Abbr) u)) \to ((pr0 t0 t6) \to ((subst0 i u t6 t7) \to (ex2 T (\lambda -(t8: T).(pr2 c t1 t8)) (\lambda (t8: T).(pr2 c t2 t8))))))) (\lambda (H16: -(getl i c (CHead d (Bind Abbr) u))).(\lambda (H17: (pr0 t0 t6)).(\lambda -(H18: (subst0 i u t6 t2)).(pr2_confluence__pr2_free_delta c d t0 t1 t2 t6 u i -H7 H16 H17 H18)))) t (sym_eq T t t2 H15))) t5 (sym_eq T t5 t0 H14))) c1 -(sym_eq C c1 c H11) H12 H13 H8 H9 H10))))]) in (H8 (refl_equal C c) -(refl_equal T t0) (refl_equal T t2)))) t4 (sym_eq T t4 t1 H6))) t3 (sym_eq T -t3 t0 H5))) c0 (sym_eq C c0 c H2) H3 H4 H1)))) | (pr2_delta c0 d u i H1 t3 t4 -H2 t H3) \Rightarrow (\lambda (H4: (eq C c0 c)).(\lambda (H5: (eq T t3 -t0)).(\lambda (H6: (eq T t t1)).(eq_ind C c (\lambda (c1: C).((eq T t3 t0) -\to ((eq T t t1) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t3 t4) -\to ((subst0 i u t4 t) \to (ex2 T (\lambda (t5: T).(pr2 c t1 t5)) (\lambda -(t5: T).(pr2 c t2 t5))))))))) (\lambda (H7: (eq T t3 t0)).(eq_ind T t0 -(\lambda (t5: T).((eq T t t1) \to ((getl i c (CHead d (Bind Abbr) u)) \to -((pr0 t5 t4) \to ((subst0 i u t4 t) \to (ex2 T (\lambda (t6: T).(pr2 c t1 -t6)) (\lambda (t6: T).(pr2 c t2 t6)))))))) (\lambda (H8: (eq T t t1)).(eq_ind -T t1 (\lambda (t5: T).((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t4) -\to ((subst0 i u t4 t5) \to (ex2 T (\lambda (t6: T).(pr2 c t1 t6)) (\lambda -(t6: T).(pr2 c t2 t6))))))) (\lambda (H9: (getl i c (CHead d (Bind Abbr) -u))).(\lambda (H10: (pr0 t0 t4)).(\lambda (H11: (subst0 i u t4 t1)).(let H12 -\def (match H0 in pr2 return (\lambda (c1: C).(\lambda (t5: T).(\lambda (t6: -T).(\lambda (_: (pr2 c1 t5 t6)).((eq C c1 c) \to ((eq T t5 t0) \to ((eq T t6 -t2) \to (ex2 T (\lambda (t7: T).(pr2 c t1 t7)) (\lambda (t7: T).(pr2 c t2 -t7)))))))))) with [(pr2_free c1 t5 t6 H12) \Rightarrow (\lambda (H13: (eq C -c1 c)).(\lambda (H14: (eq T t5 t0)).(\lambda (H15: (eq T t6 t2)).(eq_ind C c -(\lambda (_: C).((eq T t5 t0) \to ((eq T t6 t2) \to ((pr0 t5 t6) \to (ex2 T -(\lambda (t7: T).(pr2 c t1 t7)) (\lambda (t7: T).(pr2 c t2 t7))))))) (\lambda -(H16: (eq T t5 t0)).(eq_ind T t0 (\lambda (t7: T).((eq T t6 t2) \to ((pr0 t7 -t6) \to (ex2 T (\lambda (t8: T).(pr2 c t1 t8)) (\lambda (t8: T).(pr2 c t2 -t8)))))) (\lambda (H17: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t0 -t7) \to (ex2 T (\lambda (t8: T).(pr2 c t1 t8)) (\lambda (t8: T).(pr2 c t2 -t8))))) (\lambda (H18: (pr0 t0 t2)).(ex2_sym T (pr2 c t2) (pr2 c t1) -(pr2_confluence__pr2_free_delta c d t0 t2 t1 t4 u i H18 H9 H10 H11))) t6 -(sym_eq T t6 t2 H17))) t5 (sym_eq T t5 t0 H16))) c1 (sym_eq C c1 c H13) H14 -H15 H12)))) | (pr2_delta c1 d0 u0 i0 H12 t5 t6 H13 t7 H14) \Rightarrow -(\lambda (H15: (eq C c1 c)).(\lambda (H16: (eq T t5 t0)).(\lambda (H17: (eq T -t7 t2)).(eq_ind C c (\lambda (c2: C).((eq T t5 t0) \to ((eq T t7 t2) \to -((getl i0 c2 (CHead d0 (Bind Abbr) u0)) \to ((pr0 t5 t6) \to ((subst0 i0 u0 -t6 t7) \to (ex2 T (\lambda (t8: T).(pr2 c t1 t8)) (\lambda (t8: T).(pr2 c t2 -t8))))))))) (\lambda (H18: (eq T t5 t0)).(eq_ind T t0 (\lambda (t8: T).((eq T -t7 t2) \to ((getl i0 c (CHead d0 (Bind Abbr) u0)) \to ((pr0 t8 t6) \to -((subst0 i0 u0 t6 t7) \to (ex2 T (\lambda (t9: T).(pr2 c t1 t9)) (\lambda -(t9: T).(pr2 c t2 t9)))))))) (\lambda (H19: (eq T t7 t2)).(eq_ind T t2 -(\lambda (t8: T).((getl i0 c (CHead d0 (Bind Abbr) u0)) \to ((pr0 t0 t6) \to -((subst0 i0 u0 t6 t8) \to (ex2 T (\lambda (t9: T).(pr2 c t1 t9)) (\lambda -(t9: T).(pr2 c t2 t9))))))) (\lambda (H20: (getl i0 c (CHead d0 (Bind Abbr) -u0))).(\lambda (H21: (pr0 t0 t6)).(\lambda (H22: (subst0 i0 u0 t6 -t2)).(pr2_confluence__pr2_delta_delta c d d0 t0 t1 t2 t4 t6 u u0 i i0 H9 H10 -H11 H20 H21 H22)))) t7 (sym_eq T t7 t2 H19))) t5 (sym_eq T t5 t0 H18))) c1 -(sym_eq C c1 c H15) H16 H17 H12 H13 H14))))]) in (H12 (refl_equal C c) -(refl_equal T t0) (refl_equal T t2)))))) t (sym_eq T t t1 H8))) t3 (sym_eq T -t3 t0 H7))) c0 (sym_eq C c0 c H4) H5 H6 H1 H2 H3))))]) in (H1 (refl_equal C -c) (refl_equal T t0) (refl_equal T t1)))))))). -(* COMMENTS -Initial nodes: 2087 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr2/props.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr2/props.ma deleted file mode 100644 index 2faeb6ebd..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr2/props.ma +++ /dev/null @@ -1,307 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr2/defs.ma". - -include "Basic-1/pr0/props.ma". - -include "Basic-1/getl/drop.ma". - -include "Basic-1/getl/clear.ma". - -theorem pr2_thin_dx: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall -(u: T).(\forall (f: F).(pr2 c (THead (Flat f) u t1) (THead (Flat f) u -t2))))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 -t2)).(\lambda (u: T).(\lambda (f: F).(pr2_ind (\lambda (c0: C).(\lambda (t: -T).(\lambda (t0: T).(pr2 c0 (THead (Flat f) u t) (THead (Flat f) u t0))))) -(\lambda (c0: C).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr0 t0 -t3)).(pr2_free c0 (THead (Flat f) u t0) (THead (Flat f) u t3) (pr0_comp u u -(pr0_refl u) t0 t3 H0 (Flat f))))))) (\lambda (c0: C).(\lambda (d: -C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind -Abbr) u0))).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H1: (pr0 t0 -t3)).(\lambda (t: T).(\lambda (H2: (subst0 i u0 t3 t)).(pr2_delta c0 d u0 i -H0 (THead (Flat f) u t0) (THead (Flat f) u t3) (pr0_comp u u (pr0_refl u) t0 -t3 H1 (Flat f)) (THead (Flat f) u t) (subst0_snd (Flat f) u0 t t3 i H2 -u)))))))))))) c t1 t2 H)))))). -(* COMMENTS -Initial nodes: 239 -END *) - -theorem pr2_head_1: - \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr2 c u1 u2) \to (\forall -(k: K).(\forall (t: T).(pr2 c (THead k u1 t) (THead k u2 t))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr2 c u1 -u2)).(\lambda (k: K).(\lambda (t: T).(pr2_ind (\lambda (c0: C).(\lambda (t0: -T).(\lambda (t1: T).(pr2 c0 (THead k t0 t) (THead k t1 t))))) (\lambda (c0: -C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr0 t1 t2)).(pr2_free c0 -(THead k t1 t) (THead k t2 t) (pr0_comp t1 t2 H0 t t (pr0_refl t) k)))))) -(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H0: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (H1: (pr0 t1 t2)).(\lambda (t0: T).(\lambda (H2: (subst0 i u t2 -t0)).(pr2_delta c0 d u i H0 (THead k t1 t) (THead k t2 t) (pr0_comp t1 t2 H1 -t t (pr0_refl t) k) (THead k t0 t) (subst0_fst u t0 t2 i H2 t k)))))))))))) c -u1 u2 H)))))). -(* COMMENTS -Initial nodes: 219 -END *) - -theorem pr2_head_2: - \forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).(\forall -(k: K).((pr2 (CHead c k u) t1 t2) \to (pr2 c (THead k u t1) (THead k u -t2))))))) -\def - \lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(k: K).(\lambda (H: (pr2 (CHead c k u) t1 t2)).(insert_eq C (CHead c k u) -(\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 c (THead k u t1) (THead -k u t2))) (\lambda (y: C).(\lambda (H0: (pr2 y t1 t2)).(pr2_ind (\lambda (c0: -C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CHead c k u)) \to (pr2 c -(THead k u t) (THead k u t0)))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda -(t4: T).(\lambda (H1: (pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c k -u))).(pr2_free c (THead k u t3) (THead k u t4) (pr0_comp u u (pr0_refl u) t3 -t4 H1 k))))))) (K_ind (\lambda (k0: K).(\forall (c0: C).(\forall (d: -C).(\forall (u0: T).(\forall (i: nat).((getl i c0 (CHead d (Bind Abbr) u0)) -\to (\forall (t3: T).(\forall (t4: T).((pr0 t3 t4) \to (\forall (t: -T).((subst0 i u0 t4 t) \to ((eq C c0 (CHead c k0 u)) \to (pr2 c (THead k0 u -t3) (THead k0 u t)))))))))))))) (\lambda (b: B).(\lambda (c0: C).(\lambda (d: -C).(\lambda (u0: T).(\lambda (i: nat).(nat_ind (\lambda (n: nat).((getl n c0 -(CHead d (Bind Abbr) u0)) \to (\forall (t3: T).(\forall (t4: T).((pr0 t3 t4) -\to (\forall (t: T).((subst0 n u0 t4 t) \to ((eq C c0 (CHead c (Bind b) u)) -\to (pr2 c (THead (Bind b) u t3) (THead (Bind b) u t)))))))))) (\lambda (H1: -(getl O c0 (CHead d (Bind Abbr) u0))).(\lambda (t3: T).(\lambda (t4: -T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: (subst0 O u0 t4 -t)).(\lambda (H4: (eq C c0 (CHead c (Bind b) u))).(let H5 \def (eq_ind C c0 -(\lambda (c1: C).(getl O c1 (CHead d (Bind Abbr) u0))) H1 (CHead c (Bind b) -u) H4) in (let H6 \def (f_equal C C (\lambda (e: C).(match e in C return -(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow -c1])) (CHead d (Bind Abbr) u0) (CHead c (Bind b) u) (clear_gen_bind b c -(CHead d (Bind Abbr) u0) u (getl_gen_O (CHead c (Bind b) u) (CHead d (Bind -Abbr) u0) H5))) in ((let H7 \def (f_equal C B (\lambda (e: C).(match e in C -return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _) -\Rightarrow (match k0 in K return (\lambda (_: K).B) with [(Bind b0) -\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u0) -(CHead c (Bind b) u) (clear_gen_bind b c (CHead d (Bind Abbr) u0) u -(getl_gen_O (CHead c (Bind b) u) (CHead d (Bind Abbr) u0) H5))) in ((let H8 -\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow u0 | (CHead _ _ t0) \Rightarrow t0])) (CHead d -(Bind Abbr) u0) (CHead c (Bind b) u) (clear_gen_bind b c (CHead d (Bind Abbr) -u0) u (getl_gen_O (CHead c (Bind b) u) (CHead d (Bind Abbr) u0) H5))) in -(\lambda (H9: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H11 \def (eq_ind T -u0 (\lambda (t0: T).(subst0 O t0 t4 t)) H3 u H8) in (eq_ind B Abbr (\lambda -(b0: B).(pr2 c (THead (Bind b0) u t3) (THead (Bind b0) u t))) (pr2_free c -(THead (Bind Abbr) u t3) (THead (Bind Abbr) u t) (pr0_delta u u (pr0_refl u) -t3 t4 H2 t H11)) b H9))))) H7)) H6)))))))))) (\lambda (n: nat).(\lambda (H1: -(((getl n c0 (CHead d (Bind Abbr) u0)) \to (\forall (t3: T).(\forall (t4: -T).((pr0 t3 t4) \to (\forall (t: T).((subst0 n u0 t4 t) \to ((eq C c0 (CHead -c (Bind b) u)) \to (pr2 c (THead (Bind b) u t3) (THead (Bind b) u -t))))))))))).(\lambda (H2: (getl (S n) c0 (CHead d (Bind Abbr) u0))).(\lambda -(t3: T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda (t: T).(\lambda -(H4: (subst0 (S n) u0 t4 t)).(\lambda (H5: (eq C c0 (CHead c (Bind b) -u))).(let H6 \def (eq_ind C c0 (\lambda (c1: C).(getl (S n) c1 (CHead d (Bind -Abbr) u0))) H2 (CHead c (Bind b) u) H5) in (let H7 \def (eq_ind C c0 (\lambda -(c1: C).((getl n c1 (CHead d (Bind Abbr) u0)) \to (\forall (t5: T).(\forall -(t6: T).((pr0 t5 t6) \to (\forall (t0: T).((subst0 n u0 t6 t0) \to ((eq C c1 -(CHead c (Bind b) u)) \to (pr2 c (THead (Bind b) u t5) (THead (Bind b) u -t0)))))))))) H1 (CHead c (Bind b) u) H5) in (pr2_delta c d u0 (r (Bind b) n) -(getl_gen_S (Bind b) c (CHead d (Bind Abbr) u0) u n H6) (THead (Bind b) u t3) -(THead (Bind b) u t4) (pr0_comp u u (pr0_refl u) t3 t4 H3 (Bind b)) (THead -(Bind b) u t) (subst0_snd (Bind b) u0 t t4 (r (Bind b) n) H4 u))))))))))))) -i)))))) (\lambda (f: F).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: -T).(\lambda (i: nat).(nat_ind (\lambda (n: nat).((getl n c0 (CHead d (Bind -Abbr) u0)) \to (\forall (t3: T).(\forall (t4: T).((pr0 t3 t4) \to (\forall -(t: T).((subst0 n u0 t4 t) \to ((eq C c0 (CHead c (Flat f) u)) \to (pr2 c -(THead (Flat f) u t3) (THead (Flat f) u t)))))))))) (\lambda (H1: (getl O c0 -(CHead d (Bind Abbr) u0))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: -(pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: (subst0 O u0 t4 t)).(\lambda (H4: -(eq C c0 (CHead c (Flat f) u))).(let H5 \def (eq_ind C c0 (\lambda (c1: -C).(getl O c1 (CHead d (Bind Abbr) u0))) H1 (CHead c (Flat f) u) H4) in -(pr2_delta c d u0 O (getl_intro O c (CHead d (Bind Abbr) u0) c (drop_refl c) -(clear_gen_flat f c (CHead d (Bind Abbr) u0) u (getl_gen_O (CHead c (Flat f) -u) (CHead d (Bind Abbr) u0) H5))) (THead (Flat f) u t3) (THead (Flat f) u t4) -(pr0_comp u u (pr0_refl u) t3 t4 H2 (Flat f)) (THead (Flat f) u t) -(subst0_snd (Flat f) u0 t t4 O H3 u)))))))))) (\lambda (n: nat).(\lambda (H1: -(((getl n c0 (CHead d (Bind Abbr) u0)) \to (\forall (t3: T).(\forall (t4: -T).((pr0 t3 t4) \to (\forall (t: T).((subst0 n u0 t4 t) \to ((eq C c0 (CHead -c (Flat f) u)) \to (pr2 c (THead (Flat f) u t3) (THead (Flat f) u -t))))))))))).(\lambda (H2: (getl (S n) c0 (CHead d (Bind Abbr) u0))).(\lambda -(t3: T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda (t: T).(\lambda -(H4: (subst0 (S n) u0 t4 t)).(\lambda (H5: (eq C c0 (CHead c (Flat f) -u))).(let H6 \def (eq_ind C c0 (\lambda (c1: C).(getl (S n) c1 (CHead d (Bind -Abbr) u0))) H2 (CHead c (Flat f) u) H5) in (let H7 \def (eq_ind C c0 (\lambda -(c1: C).((getl n c1 (CHead d (Bind Abbr) u0)) \to (\forall (t5: T).(\forall -(t6: T).((pr0 t5 t6) \to (\forall (t0: T).((subst0 n u0 t6 t0) \to ((eq C c1 -(CHead c (Flat f) u)) \to (pr2 c (THead (Flat f) u t5) (THead (Flat f) u -t0)))))))))) H1 (CHead c (Flat f) u) H5) in (pr2_delta c d u0 (r (Flat f) n) -(getl_gen_S (Flat f) c (CHead d (Bind Abbr) u0) u n H6) (THead (Flat f) u t3) -(THead (Flat f) u t4) (pr0_comp u u (pr0_refl u) t3 t4 H3 (Flat f)) (THead -(Flat f) u t) (subst0_snd (Flat f) u0 t t4 (r (Flat f) n) H4 u))))))))))))) -i)))))) k) y t1 t2 H0))) H)))))). -(* COMMENTS -Initial nodes: 1947 -END *) - -theorem clear_pr2_trans: - \forall (c2: C).(\forall (t1: T).(\forall (t2: T).((pr2 c2 t1 t2) \to -(\forall (c1: C).((clear c1 c2) \to (pr2 c1 t1 t2)))))) -\def - \lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c2 t1 -t2)).(pr2_ind (\lambda (c: C).(\lambda (t: T).(\lambda (t0: T).(\forall (c1: -C).((clear c1 c) \to (pr2 c1 t t0)))))) (\lambda (c: C).(\lambda (t3: -T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(\lambda (c1: C).(\lambda (_: -(clear c1 c)).(pr2_free c1 t3 t4 H0))))))) (\lambda (c: C).(\lambda (d: -C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d (Bind -Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 -t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (c1: -C).(\lambda (H3: (clear c1 c)).(pr2_delta c1 d u i (clear_getl_trans i c -(CHead d (Bind Abbr) u) H0 c1 H3) t3 t4 H1 t H2))))))))))))) c2 t1 t2 H)))). -(* COMMENTS -Initial nodes: 171 -END *) - -theorem pr2_cflat: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall -(f: F).(\forall (v: T).(pr2 (CHead c (Flat f) v) t1 t2)))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 -t2)).(\lambda (f: F).(\lambda (v: T).(pr2_ind (\lambda (c0: C).(\lambda (t: -T).(\lambda (t0: T).(pr2 (CHead c0 (Flat f) v) t t0)))) (\lambda (c0: -C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(pr2_free -(CHead c0 (Flat f) v) t3 t4 H0))))) (\lambda (c0: C).(\lambda (d: C).(\lambda -(u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr) -u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 t4)).(\lambda -(t: T).(\lambda (H2: (subst0 i u t4 t)).(pr2_delta (CHead c0 (Flat f) v) d u -i (getl_flat c0 (CHead d (Bind Abbr) u) i H0 f v) t3 t4 H1 t H2))))))))))) c -t1 t2 H)))))). -(* COMMENTS -Initial nodes: 175 -END *) - -theorem pr2_ctail: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall -(k: K).(\forall (u: T).(pr2 (CTail k u c) t1 t2)))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 -t2)).(\lambda (k: K).(\lambda (u: T).(pr2_ind (\lambda (c0: C).(\lambda (t: -T).(\lambda (t0: T).(pr2 (CTail k u c0) t t0)))) (\lambda (c0: C).(\lambda -(t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(pr2_free (CTail k u c0) -t3 t4 H0))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: -nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr) u0))).(\lambda (t3: -T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H2: -(subst0 i u0 t4 t)).(pr2_delta (CTail k u c0) (CTail k u d) u0 i (getl_ctail -Abbr c0 d u0 i H0 k u) t3 t4 H1 t H2))))))))))) c t1 t2 H)))))). -(* COMMENTS -Initial nodes: 171 -END *) - -theorem pr2_change: - \forall (b: B).((not (eq B b Abbr)) \to (\forall (c: C).(\forall (v1: -T).(\forall (t1: T).(\forall (t2: T).((pr2 (CHead c (Bind b) v1) t1 t2) \to -(\forall (v2: T).(pr2 (CHead c (Bind b) v2) t1 t2)))))))) -\def - \lambda (b: B).(\lambda (H: (not (eq B b Abbr))).(\lambda (c: C).(\lambda -(v1: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr2 (CHead c (Bind -b) v1) t1 t2)).(\lambda (v2: T).(insert_eq C (CHead c (Bind b) v1) (\lambda -(c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 (CHead c (Bind b) v2) t1 t2)) -(\lambda (y: C).(\lambda (H1: (pr2 y t1 t2)).(pr2_ind (\lambda (c0: -C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CHead c (Bind b) v1)) \to (pr2 -(CHead c (Bind b) v2) t t0))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda -(t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c (Bind b) -v1))).(pr2_free (CHead c (Bind b) v2) t3 t4 H2)))))) (\lambda (c0: -C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H2: (getl i c0 -(CHead d (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H3: -(pr0 t3 t4)).(\lambda (t: T).(\lambda (H4: (subst0 i u t4 t)).(\lambda (H5: -(eq C c0 (CHead c (Bind b) v1))).(let H6 \def (eq_ind C c0 (\lambda (c1: -C).(getl i c1 (CHead d (Bind Abbr) u))) H2 (CHead c (Bind b) v1) H5) in -(nat_ind (\lambda (n: nat).((getl n (CHead c (Bind b) v1) (CHead d (Bind -Abbr) u)) \to ((subst0 n u t4 t) \to (pr2 (CHead c (Bind b) v2) t3 t)))) -(\lambda (H7: (getl O (CHead c (Bind b) v1) (CHead d (Bind Abbr) -u))).(\lambda (H8: (subst0 O u t4 t)).(let H9 \def (f_equal C C (\lambda (e: -C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | -(CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u) (CHead c (Bind b) -v1) (clear_gen_bind b c (CHead d (Bind Abbr) u) v1 (getl_gen_O (CHead c (Bind -b) v1) (CHead d (Bind Abbr) u) H7))) in ((let H10 \def (f_equal C B (\lambda -(e: C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow -Abbr | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with -[(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind -Abbr) u) (CHead c (Bind b) v1) (clear_gen_bind b c (CHead d (Bind Abbr) u) v1 -(getl_gen_O (CHead c (Bind b) v1) (CHead d (Bind Abbr) u) H7))) in ((let H11 -\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d -(Bind Abbr) u) (CHead c (Bind b) v1) (clear_gen_bind b c (CHead d (Bind Abbr) -u) v1 (getl_gen_O (CHead c (Bind b) v1) (CHead d (Bind Abbr) u) H7))) in -(\lambda (H12: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H14 \def (eq_ind -T u (\lambda (t0: T).(subst0 O t0 t4 t)) H8 v1 H11) in (let H15 \def -(eq_ind_r B b (\lambda (b0: B).(not (eq B b0 Abbr))) H Abbr H12) in (eq_ind B -Abbr (\lambda (b0: B).(pr2 (CHead c (Bind b0) v2) t3 t)) (let H16 \def (match -(H15 (refl_equal B Abbr)) in False return (\lambda (_: False).(pr2 (CHead c -(Bind Abbr) v2) t3 t)) with []) in H16) b H12)))))) H10)) H9)))) (\lambda -(i0: nat).(\lambda (_: (((getl i0 (CHead c (Bind b) v1) (CHead d (Bind Abbr) -u)) \to ((subst0 i0 u t4 t) \to (pr2 (CHead c (Bind b) v2) t3 t))))).(\lambda -(H7: (getl (S i0) (CHead c (Bind b) v1) (CHead d (Bind Abbr) u))).(\lambda -(H8: (subst0 (S i0) u t4 t)).(pr2_delta (CHead c (Bind b) v2) d u (S i0) -(getl_head (Bind b) i0 c (CHead d (Bind Abbr) u) (getl_gen_S (Bind b) c -(CHead d (Bind Abbr) u) v1 i0 H7) v2) t3 t4 H3 t H8))))) i H6 H4))))))))))))) -y t1 t2 H1))) H0)))))))). -(* COMMENTS -Initial nodes: 913 -END *) - -theorem pr2_lift: - \forall (c: C).(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h -d c e) \to (\forall (t1: T).(\forall (t2: T).((pr2 e t1 t2) \to (pr2 c (lift -h d t1) (lift h d t2))))))))) -\def - \lambda (c: C).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda -(H: (drop h d c e)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr2 e t1 -t2)).(insert_eq C e (\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 c -(lift h d t1) (lift h d t2))) (\lambda (y: C).(\lambda (H1: (pr2 y t1 -t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 e) -\to (pr2 c (lift h d t) (lift h d t0)))))) (\lambda (c0: C).(\lambda (t3: -T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (_: (eq C c0 -e)).(pr2_free c (lift h d t3) (lift h d t4) (pr0_lift t3 t4 H2 h d))))))) -(\lambda (c0: C).(\lambda (d0: C).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H2: (getl i c0 (CHead d0 (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: -T).(\lambda (H3: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H4: (subst0 i u t4 -t)).(\lambda (H5: (eq C c0 e)).(let H6 \def (eq_ind C c0 (\lambda (c1: -C).(getl i c1 (CHead d0 (Bind Abbr) u))) H2 e H5) in (lt_le_e i d (pr2 c -(lift h d t3) (lift h d t)) (\lambda (H7: (lt i d)).(let H8 \def -(drop_getl_trans_le i d (le_S_n i d (le_S (S i) d H7)) c e h H (CHead d0 -(Bind Abbr) u) H6) in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop i -O c e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d i) e0 e1))) -(\lambda (_: C).(\lambda (e1: C).(clear e1 (CHead d0 (Bind Abbr) u)))) (pr2 c -(lift h d t3) (lift h d t)) (\lambda (x0: C).(\lambda (x1: C).(\lambda (H9: -(drop i O c x0)).(\lambda (H10: (drop h (minus d i) x0 x1)).(\lambda (H11: -(clear x1 (CHead d0 (Bind Abbr) u))).(let H12 \def (eq_ind nat (minus d i) -(\lambda (n: nat).(drop h n x0 x1)) H10 (S (minus d (S i))) (minus_x_Sy d i -H7)) in (let H13 \def (drop_clear_S x1 x0 h (minus d (S i)) H12 Abbr d0 u -H11) in (ex2_ind C (\lambda (c1: C).(clear x0 (CHead c1 (Bind Abbr) (lift h -(minus d (S i)) u)))) (\lambda (c1: C).(drop h (minus d (S i)) c1 d0)) (pr2 c -(lift h d t3) (lift h d t)) (\lambda (x: C).(\lambda (H14: (clear x0 (CHead x -(Bind Abbr) (lift h (minus d (S i)) u)))).(\lambda (_: (drop h (minus d (S -i)) x d0)).(pr2_delta c x (lift h (minus d (S i)) u) i (getl_intro i c (CHead -x (Bind Abbr) (lift h (minus d (S i)) u)) x0 H9 H14) (lift h d t3) (lift h d -t4) (pr0_lift t3 t4 H3 h d) (lift h d t) (subst0_lift_lt t4 t u i H4 d H7 -h))))) H13)))))))) H8))) (\lambda (H7: (le d i)).(pr2_delta c d0 u (plus i h) -(drop_getl_trans_ge i c e d h H (CHead d0 (Bind Abbr) u) H6 H7) (lift h d t3) -(lift h d t4) (pr0_lift t3 t4 H3 h d) (lift h d t) (subst0_lift_ge t4 t u i h -H4 d H7)))))))))))))))) y t1 t2 H1))) H0)))))))). -(* COMMENTS -Initial nodes: 849 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr2/subst1.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr2/subst1.ma deleted file mode 100644 index 47704dd52..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr2/subst1.ma +++ /dev/null @@ -1,281 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr2/defs.ma". - -include "Basic-1/pr0/subst1.ma". - -include "Basic-1/pr0/fwd.ma". - -include "Basic-1/csubst1/getl.ma". - -include "Basic-1/csubst1/fwd.ma". - -include "Basic-1/subst1/subst1.ma". - -include "Basic-1/getl/drop.ma". - -theorem pr2_delta1: - \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c -(CHead d (Bind Abbr) u)) \to (\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) -\to (\forall (t: T).((subst1 i u t2 t) \to (pr2 c t1 t)))))))))) -\def - \lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H: (getl i c (CHead d (Bind Abbr) u))).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (H0: (pr0 t1 t2)).(\lambda (t: T).(\lambda (H1: (subst1 i u t2 -t)).(subst1_ind i u t2 (\lambda (t0: T).(pr2 c t1 t0)) (pr2_free c t1 t2 H0) -(\lambda (t0: T).(\lambda (H2: (subst0 i u t2 t0)).(pr2_delta c d u i H t1 t2 -H0 t0 H2))) t H1)))))))))). -(* COMMENTS -Initial nodes: 111 -END *) - -theorem pr2_subst1: - \forall (c: C).(\forall (e: C).(\forall (v: T).(\forall (i: nat).((getl i c -(CHead e (Bind Abbr) v)) \to (\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) -\to (\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T (\lambda (w2: T).(pr2 c -w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2)))))))))))) -\def - \lambda (c: C).(\lambda (e: C).(\lambda (v: T).(\lambda (i: nat).(\lambda -(H: (getl i c (CHead e (Bind Abbr) v))).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (H0: (pr2 c t1 t2)).(insert_eq C c (\lambda (c0: C).(pr2 c0 t1 -t2)) (\lambda (c0: C).(\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T -(\lambda (w2: T).(pr2 c0 w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2)))))) -(\lambda (y: C).(\lambda (H1: (pr2 y t1 t2)).(pr2_ind (\lambda (c0: -C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 c) \to (\forall (w1: -T).((subst1 i v t w1) \to (ex2 T (\lambda (w2: T).(pr2 c0 w1 w2)) (\lambda -(w2: T).(subst1 i v t0 w2))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda -(t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (H3: (eq C c0 c)).(\lambda (w1: -T).(\lambda (H4: (subst1 i v t3 w1)).(eq_ind_r C c (\lambda (c1: C).(ex2 T -(\lambda (w2: T).(pr2 c1 w1 w2)) (\lambda (w2: T).(subst1 i v t4 w2)))) -(ex2_ind T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst1 i v t4 w2)) -(ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t4 w2))) -(\lambda (x: T).(\lambda (H5: (pr0 w1 x)).(\lambda (H6: (subst1 i v t4 -x)).(ex_intro2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v -t4 w2)) x (pr2_free c w1 x H5) H6)))) (pr0_subst1 t3 t4 H2 v w1 i H4 v -(pr0_refl v))) c0 H3)))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: -T).(\lambda (i0: nat).(\lambda (H2: (getl i0 c0 (CHead d (Bind Abbr) -u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda -(t: T).(\lambda (H4: (subst0 i0 u t4 t)).(\lambda (H5: (eq C c0 c)).(\lambda -(w1: T).(\lambda (H6: (subst1 i v t3 w1)).(let H7 \def (eq_ind C c0 (\lambda -(c1: C).(getl i0 c1 (CHead d (Bind Abbr) u))) H2 c H5) in (eq_ind_r C c -(\lambda (c1: C).(ex2 T (\lambda (w2: T).(pr2 c1 w1 w2)) (\lambda (w2: -T).(subst1 i v t w2)))) (ex2_ind T (\lambda (w2: T).(pr0 w1 w2)) (\lambda -(w2: T).(subst1 i v t4 w2)) (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda -(w2: T).(subst1 i v t w2))) (\lambda (x: T).(\lambda (H8: (pr0 w1 -x)).(\lambda (H9: (subst1 i v t4 x)).(neq_eq_e i i0 (ex2 T (\lambda (w2: -T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t w2))) (\lambda (H10: (not -(eq nat i i0))).(ex2_ind T (\lambda (t0: T).(subst1 i v t t0)) (\lambda (t0: -T).(subst1 i0 u x t0)) (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: -T).(subst1 i v t w2))) (\lambda (x0: T).(\lambda (H11: (subst1 i v t -x0)).(\lambda (H12: (subst1 i0 u x x0)).(ex_intro2 T (\lambda (w2: T).(pr2 c -w1 w2)) (\lambda (w2: T).(subst1 i v t w2)) x0 (pr2_delta1 c d u i0 H7 w1 x -H8 x0 H12) H11)))) (subst1_confluence_neq t4 t u i0 (subst1_single i0 u t4 t -H4) x v i H9 (sym_not_eq nat i i0 H10)))) (\lambda (H10: (eq nat i i0)).(let -H11 \def (eq_ind_r nat i0 (\lambda (n: nat).(subst0 n u t4 t)) H4 i H10) in -(let H12 \def (eq_ind_r nat i0 (\lambda (n: nat).(getl n c (CHead d (Bind -Abbr) u))) H7 i H10) in (let H13 \def (eq_ind C (CHead e (Bind Abbr) v) -(\lambda (c1: C).(getl i c c1)) H (CHead d (Bind Abbr) u) (getl_mono c (CHead -e (Bind Abbr) v) i H (CHead d (Bind Abbr) u) H12)) in (let H14 \def (f_equal -C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow e | (CHead c1 _ _) \Rightarrow c1])) (CHead e (Bind Abbr) v) -(CHead d (Bind Abbr) u) (getl_mono c (CHead e (Bind Abbr) v) i H (CHead d -(Bind Abbr) u) H12)) in ((let H15 \def (f_equal C T (\lambda (e0: C).(match -e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow v | (CHead _ _ -t0) \Rightarrow t0])) (CHead e (Bind Abbr) v) (CHead d (Bind Abbr) u) -(getl_mono c (CHead e (Bind Abbr) v) i H (CHead d (Bind Abbr) u) H12)) in -(\lambda (H16: (eq C e d)).(let H17 \def (eq_ind_r T u (\lambda (t0: T).(getl -i c (CHead d (Bind Abbr) t0))) H13 v H15) in (let H18 \def (eq_ind_r T u -(\lambda (t0: T).(subst0 i t0 t4 t)) H11 v H15) in (let H19 \def (eq_ind_r C -d (\lambda (c1: C).(getl i c (CHead c1 (Bind Abbr) v))) H17 e H16) in -(ex2_ind T (\lambda (t0: T).(subst1 i v t t0)) (\lambda (t0: T).(subst1 i v x -t0)) (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t -w2))) (\lambda (x0: T).(\lambda (H20: (subst1 i v t x0)).(\lambda (H21: -(subst1 i v x x0)).(ex_intro2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: -T).(subst1 i v t w2)) x0 (pr2_delta1 c e v i H19 w1 x H8 x0 H21) H20)))) -(subst1_confluence_eq t4 t v i (subst1_single i v t4 t H18) x H9))))))) -H14)))))))))) (pr0_subst1 t3 t4 H3 v w1 i H6 v (pr0_refl v))) c0 -H5))))))))))))))) y t1 t2 H1))) H0)))))))). -(* COMMENTS -Initial nodes: 1311 -END *) - -theorem pr2_gen_cabbr: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall -(e: C).(\forall (u: T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) -\to (\forall (a0: C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d -a0 a) \to (\forall (x1: T).((subst1 d u t1 (lift (S O) d x1)) \to (ex2 T -(\lambda (x2: T).(subst1 d u t2 (lift (S O) d x2))) (\lambda (x2: T).(pr2 a -x1 x2)))))))))))))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 -t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (e: -C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u)) \to -(\forall (a0: C).((csubst1 d u c0 a0) \to (\forall (a: C).((drop (S O) d a0 -a) \to (\forall (x1: T).((subst1 d u t (lift (S O) d x1)) \to (ex2 T (\lambda -(x2: T).(subst1 d u t0 (lift (S O) d x2))) (\lambda (x2: T).(pr2 a x1 -x2)))))))))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: -T).(\lambda (H0: (pr0 t3 t4)).(\lambda (e: C).(\lambda (u: T).(\lambda (d: -nat).(\lambda (_: (getl d c0 (CHead e (Bind Abbr) u))).(\lambda (a0: -C).(\lambda (_: (csubst1 d u c0 a0)).(\lambda (a: C).(\lambda (_: (drop (S O) -d a0 a)).(\lambda (x1: T).(\lambda (H4: (subst1 d u t3 (lift (S O) d -x1))).(ex2_ind T (\lambda (w2: T).(pr0 (lift (S O) d x1) w2)) (\lambda (w2: -T).(subst1 d u t4 w2)) (ex2 T (\lambda (x2: T).(subst1 d u t4 (lift (S O) d -x2))) (\lambda (x2: T).(pr2 a x1 x2))) (\lambda (x: T).(\lambda (H5: (pr0 -(lift (S O) d x1) x)).(\lambda (H6: (subst1 d u t4 x)).(ex2_ind T (\lambda -(t5: T).(eq T x (lift (S O) d t5))) (\lambda (t5: T).(pr0 x1 t5)) (ex2 T -(\lambda (x2: T).(subst1 d u t4 (lift (S O) d x2))) (\lambda (x2: T).(pr2 a -x1 x2))) (\lambda (x0: T).(\lambda (H7: (eq T x (lift (S O) d x0))).(\lambda -(H8: (pr0 x1 x0)).(let H9 \def (eq_ind T x (\lambda (t: T).(subst1 d u t4 t)) -H6 (lift (S O) d x0) H7) in (ex_intro2 T (\lambda (x2: T).(subst1 d u t4 -(lift (S O) d x2))) (\lambda (x2: T).(pr2 a x1 x2)) x0 H9 (pr2_free a x1 x0 -H8)))))) (pr0_gen_lift x1 x (S O) d H5))))) (pr0_subst1 t3 t4 H0 u (lift (S -O) d x1) d H4 u (pr0_refl u))))))))))))))))) (\lambda (c0: C).(\lambda (d: -C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind -Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 -t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (e: -C).(\lambda (u0: T).(\lambda (d0: nat).(\lambda (H3: (getl d0 c0 (CHead e -(Bind Abbr) u0))).(\lambda (a0: C).(\lambda (H4: (csubst1 d0 u0 c0 -a0)).(\lambda (a: C).(\lambda (H5: (drop (S O) d0 a0 a)).(\lambda (x1: -T).(\lambda (H6: (subst1 d0 u0 t3 (lift (S O) d0 x1))).(ex2_ind T (\lambda -(w2: T).(pr0 (lift (S O) d0 x1) w2)) (\lambda (w2: T).(subst1 d0 u0 t4 w2)) -(ex2 T (\lambda (x2: T).(subst1 d0 u0 t (lift (S O) d0 x2))) (\lambda (x2: -T).(pr2 a x1 x2))) (\lambda (x: T).(\lambda (H7: (pr0 (lift (S O) d0 x1) -x)).(\lambda (H8: (subst1 d0 u0 t4 x)).(ex2_ind T (\lambda (t5: T).(eq T x -(lift (S O) d0 t5))) (\lambda (t5: T).(pr0 x1 t5)) (ex2 T (\lambda (x2: -T).(subst1 d0 u0 t (lift (S O) d0 x2))) (\lambda (x2: T).(pr2 a x1 x2))) -(\lambda (x0: T).(\lambda (H9: (eq T x (lift (S O) d0 x0))).(\lambda (H10: -(pr0 x1 x0)).(let H11 \def (eq_ind T x (\lambda (t0: T).(subst1 d0 u0 t4 t0)) -H8 (lift (S O) d0 x0) H9) in (lt_eq_gt_e i d0 (ex2 T (\lambda (x2: T).(subst1 -d0 u0 t (lift (S O) d0 x2))) (\lambda (x2: T).(pr2 a x1 x2))) (\lambda (H12: -(lt i d0)).(ex2_ind T (\lambda (t0: T).(subst1 d0 u0 t t0)) (\lambda (t0: -T).(subst1 i u (lift (S O) d0 x0) t0)) (ex2 T (\lambda (x2: T).(subst1 d0 u0 -t (lift (S O) d0 x2))) (\lambda (x2: T).(pr2 a x1 x2))) (\lambda (x2: -T).(\lambda (H13: (subst1 d0 u0 t x2)).(\lambda (H14: (subst1 i u (lift (S O) -d0 x0) x2)).(ex2_ind C (\lambda (e2: C).(csubst1 (minus d0 i) u0 (CHead d -(Bind Abbr) u) e2)) (\lambda (e2: C).(getl i a0 e2)) (ex2 T (\lambda (x3: -T).(subst1 d0 u0 t (lift (S O) d0 x3))) (\lambda (x3: T).(pr2 a x1 x3))) -(\lambda (x3: C).(\lambda (H15: (csubst1 (minus d0 i) u0 (CHead d (Bind Abbr) -u) x3)).(\lambda (H16: (getl i a0 x3)).(let H17 \def (eq_ind nat (minus d0 i) -(\lambda (n: nat).(csubst1 n u0 (CHead d (Bind Abbr) u) x3)) H15 (S (minus d0 -(S i))) (minus_x_Sy d0 i H12)) in (let H18 \def (csubst1_gen_head (Bind Abbr) -d x3 u u0 (minus d0 (S i)) H17) in (ex3_2_ind T C (\lambda (u2: T).(\lambda -(c2: C).(eq C x3 (CHead c2 (Bind Abbr) u2)))) (\lambda (u2: T).(\lambda (_: -C).(subst1 (minus d0 (S i)) u0 u u2))) (\lambda (_: T).(\lambda (c2: -C).(csubst1 (minus d0 (S i)) u0 d c2))) (ex2 T (\lambda (x4: T).(subst1 d0 u0 -t (lift (S O) d0 x4))) (\lambda (x4: T).(pr2 a x1 x4))) (\lambda (x4: -T).(\lambda (x5: C).(\lambda (H19: (eq C x3 (CHead x5 (Bind Abbr) -x4))).(\lambda (H20: (subst1 (minus d0 (S i)) u0 u x4)).(\lambda (_: (csubst1 -(minus d0 (S i)) u0 d x5)).(let H22 \def (eq_ind C x3 (\lambda (c1: C).(getl -i a0 c1)) H16 (CHead x5 (Bind Abbr) x4) H19) in (let H23 \def (eq_ind nat d0 -(\lambda (n: nat).(drop (S O) n a0 a)) H5 (S (plus i (minus d0 (S i)))) -(lt_plus_minus i d0 H12)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: -C).(eq T x4 (lift (S O) (minus d0 (S i)) v)))) (\lambda (v: T).(\lambda (e0: -C).(getl i a (CHead e0 (Bind Abbr) v)))) (\lambda (_: T).(\lambda (e0: -C).(drop (S O) (minus d0 (S i)) x5 e0))) (ex2 T (\lambda (x6: T).(subst1 d0 -u0 t (lift (S O) d0 x6))) (\lambda (x6: T).(pr2 a x1 x6))) (\lambda (x6: -T).(\lambda (x7: C).(\lambda (H24: (eq T x4 (lift (S O) (minus d0 (S i)) -x6))).(\lambda (H25: (getl i a (CHead x7 (Bind Abbr) x6))).(\lambda (_: (drop -(S O) (minus d0 (S i)) x5 x7)).(let H27 \def (eq_ind T x4 (\lambda (t0: -T).(subst1 (minus d0 (S i)) u0 u t0)) H20 (lift (S O) (minus d0 (S i)) x6) -H24) in (ex2_ind T (\lambda (t0: T).(subst1 i (lift (S O) (minus d0 (S i)) -x6) (lift (S O) d0 x0) t0)) (\lambda (t0: T).(subst1 (S (plus (minus d0 (S -i)) i)) u0 x2 t0)) (ex2 T (\lambda (x8: T).(subst1 d0 u0 t (lift (S O) d0 -x8))) (\lambda (x8: T).(pr2 a x1 x8))) (\lambda (x8: T).(\lambda (H28: -(subst1 i (lift (S O) (minus d0 (S i)) x6) (lift (S O) d0 x0) x8)).(\lambda -(H29: (subst1 (S (plus (minus d0 (S i)) i)) u0 x2 x8)).(let H30 \def (eq_ind -nat d0 (\lambda (n: nat).(subst1 i (lift (S O) (minus d0 (S i)) x6) (lift (S -O) n x0) x8)) H28 (S (plus i (minus d0 (S i)))) (lt_plus_minus i d0 H12)) in -(ex2_ind T (\lambda (t5: T).(eq T x8 (lift (S O) (S (plus i (minus d0 (S -i)))) t5))) (\lambda (t5: T).(subst1 i x6 x0 t5)) (ex2 T (\lambda (x9: -T).(subst1 d0 u0 t (lift (S O) d0 x9))) (\lambda (x9: T).(pr2 a x1 x9))) -(\lambda (x9: T).(\lambda (H31: (eq T x8 (lift (S O) (S (plus i (minus d0 (S -i)))) x9))).(\lambda (H32: (subst1 i x6 x0 x9)).(let H33 \def (eq_ind T x8 -(\lambda (t0: T).(subst1 (S (plus (minus d0 (S i)) i)) u0 x2 t0)) H29 (lift -(S O) (S (plus i (minus d0 (S i)))) x9) H31) in (let H34 \def (eq_ind_r nat -(S (plus i (minus d0 (S i)))) (\lambda (n: nat).(subst1 (S (plus (minus d0 (S -i)) i)) u0 x2 (lift (S O) n x9))) H33 d0 (lt_plus_minus i d0 H12)) in (let -H35 \def (eq_ind_r nat (S (plus (minus d0 (S i)) i)) (\lambda (n: -nat).(subst1 n u0 x2 (lift (S O) d0 x9))) H34 d0 (lt_plus_minus_r i d0 H12)) -in (ex_intro2 T (\lambda (x10: T).(subst1 d0 u0 t (lift (S O) d0 x10))) -(\lambda (x10: T).(pr2 a x1 x10)) x9 (subst1_trans x2 t u0 d0 H13 (lift (S O) -d0 x9) H35) (pr2_delta1 a x7 x6 i H25 x1 x0 H10 x9 H32)))))))) -(subst1_gen_lift_lt x6 x0 x8 i (S O) (minus d0 (S i)) H30)))))) -(subst1_subst1_back (lift (S O) d0 x0) x2 u i H14 (lift (S O) (minus d0 (S -i)) x6) u0 (minus d0 (S i)) H27)))))))) (getl_drop_conf_lt Abbr a0 x5 x4 i -H22 a (S O) (minus d0 (S i)) H23))))))))) H18)))))) (csubst1_getl_lt d0 i H12 -c0 a0 u0 H4 (CHead d (Bind Abbr) u) H0))))) (subst1_confluence_neq t4 t u i -(subst1_single i u t4 t H2) (lift (S O) d0 x0) u0 d0 H11 (lt_neq i d0 H12)))) -(\lambda (H12: (eq nat i d0)).(let H13 \def (eq_ind_r nat d0 (\lambda (n: -nat).(subst1 n u0 t4 (lift (S O) n x0))) H11 i H12) in (let H14 \def -(eq_ind_r nat d0 (\lambda (n: nat).(drop (S O) n a0 a)) H5 i H12) in (let H15 -\def (eq_ind_r nat d0 (\lambda (n: nat).(csubst1 n u0 c0 a0)) H4 i H12) in -(let H16 \def (eq_ind_r nat d0 (\lambda (n: nat).(getl n c0 (CHead e (Bind -Abbr) u0))) H3 i H12) in (eq_ind nat i (\lambda (n: nat).(ex2 T (\lambda (x2: -T).(subst1 n u0 t (lift (S O) n x2))) (\lambda (x2: T).(pr2 a x1 x2)))) (let -H17 \def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c1: C).(getl i c0 c1)) -H0 (CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead -e (Bind Abbr) u0) H16)) in (let H18 \def (f_equal C C (\lambda (e0: C).(match -e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ -_) \Rightarrow c1])) (CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0) -(getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead e (Bind Abbr) u0) H16)) in -((let H19 \def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda -(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) -(CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d (Bind -Abbr) u) i H0 (CHead e (Bind Abbr) u0) H16)) in (\lambda (H20: (eq C d -e)).(let H21 \def (eq_ind_r T u0 (\lambda (t0: T).(getl i c0 (CHead e (Bind -Abbr) t0))) H17 u H19) in (let H22 \def (eq_ind_r T u0 (\lambda (t0: -T).(subst1 i t0 t4 (lift (S O) i x0))) H13 u H19) in (let H23 \def (eq_ind_r -T u0 (\lambda (t0: T).(csubst1 i t0 c0 a0)) H15 u H19) in (eq_ind T u -(\lambda (t0: T).(ex2 T (\lambda (x2: T).(subst1 i t0 t (lift (S O) i x2))) -(\lambda (x2: T).(pr2 a x1 x2)))) (let H24 \def (eq_ind_r C e (\lambda (c1: -C).(getl i c0 (CHead c1 (Bind Abbr) u))) H21 d H20) in (ex2_ind T (\lambda -(t0: T).(subst1 i u t t0)) (\lambda (t0: T).(subst1 i u (lift (S O) i x0) -t0)) (ex2 T (\lambda (x2: T).(subst1 i u t (lift (S O) i x2))) (\lambda (x2: -T).(pr2 a x1 x2))) (\lambda (x2: T).(\lambda (H25: (subst1 i u t -x2)).(\lambda (H26: (subst1 i u (lift (S O) i x0) x2)).(let H27 \def (eq_ind -T x2 (\lambda (t0: T).(subst1 i u t t0)) H25 (lift (S O) i x0) -(subst1_gen_lift_eq x0 u x2 (S O) i i (le_n i) (eq_ind_r nat (plus (S O) i) -(\lambda (n: nat).(lt i n)) (le_n (plus (S O) i)) (plus i (S O)) (plus_sym i -(S O))) H26)) in (ex_intro2 T (\lambda (x3: T).(subst1 i u t (lift (S O) i -x3))) (\lambda (x3: T).(pr2 a x1 x3)) x0 H27 (pr2_free a x1 x0 H10)))))) -(subst1_confluence_eq t4 t u i (subst1_single i u t4 t H2) (lift (S O) i x0) -H22))) u0 H19)))))) H18))) d0 H12)))))) (\lambda (H12: (lt d0 i)).(ex2_ind T -(\lambda (t0: T).(subst1 d0 u0 t t0)) (\lambda (t0: T).(subst1 i u (lift (S -O) d0 x0) t0)) (ex2 T (\lambda (x2: T).(subst1 d0 u0 t (lift (S O) d0 x2))) -(\lambda (x2: T).(pr2 a x1 x2))) (\lambda (x2: T).(\lambda (H13: (subst1 d0 -u0 t x2)).(\lambda (H14: (subst1 i u (lift (S O) d0 x0) x2)).(ex2_ind T -(\lambda (t5: T).(eq T x2 (lift (S O) d0 t5))) (\lambda (t5: T).(subst1 -(minus i (S O)) u x0 t5)) (ex2 T (\lambda (x3: T).(subst1 d0 u0 t (lift (S O) -d0 x3))) (\lambda (x3: T).(pr2 a x1 x3))) (\lambda (x3: T).(\lambda (H15: (eq -T x2 (lift (S O) d0 x3))).(\lambda (H16: (subst1 (minus i (S O)) u x0 -x3)).(let H17 \def (eq_ind T x2 (\lambda (t0: T).(subst1 d0 u0 t t0)) H13 -(lift (S O) d0 x3) H15) in (ex_intro2 T (\lambda (x4: T).(subst1 d0 u0 t -(lift (S O) d0 x4))) (\lambda (x4: T).(pr2 a x1 x4)) x3 H17 (pr2_delta1 a d u -(minus i (S O)) (getl_drop_conf_ge i (CHead d (Bind Abbr) u) a0 -(csubst1_getl_ge d0 i (le_S_n d0 i (le_S (S d0) i H12)) c0 a0 u0 H4 (CHead d -(Bind Abbr) u) H0) a (S O) d0 H5 (eq_ind_r nat (plus (S O) d0) (\lambda (n: -nat).(le n i)) H12 (plus d0 (S O)) (plus_sym d0 (S O)))) x1 x0 H10 x3 -H16)))))) (subst1_gen_lift_ge u x0 x2 i (S O) d0 H14 (eq_ind_r nat (plus (S -O) d0) (\lambda (n: nat).(le n i)) H12 (plus d0 (S O)) (plus_sym d0 (S -O)))))))) (subst1_confluence_neq t4 t u i (subst1_single i u t4 t H2) (lift -(S O) d0 x0) u0 d0 H11 (sym_not_eq nat d0 i (lt_neq d0 i H12)))))))))) -(pr0_gen_lift x1 x (S O) d0 H7))))) (pr0_subst1 t3 t4 H1 u0 (lift (S O) d0 -x1) d0 H6 u0 (pr0_refl u0))))))))))))))))))))))) c t1 t2 H)))). -(* COMMENTS -Initial nodes: 3757 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/defs.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/defs.ma deleted file mode 100644 index 9619ee1d7..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/defs.ma +++ /dev/null @@ -1,23 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr2/defs.ma". - -inductive pr3 (c: C): T \to (T \to Prop) \def -| pr3_refl: \forall (t: T).(pr3 c t t) -| pr3_sing: \forall (t2: T).(\forall (t1: T).((pr2 c t1 t2) \to (\forall (t3: -T).((pr3 c t2 t3) \to (pr3 c t1 t3))))). - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/fwd.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/fwd.ma deleted file mode 100644 index 726c420f0..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/fwd.ma +++ /dev/null @@ -1,1604 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr3/props.ma". - -include "Basic-1/pr2/fwd.ma". - -theorem pr3_gen_sort: - \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr3 c (TSort n) x) \to -(eq T x (TSort n))))) -\def - \lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr3 c (TSort -n) x)).(insert_eq T (TSort n) (\lambda (t: T).(pr3 c t x)) (\lambda (t: -T).(eq T x t)) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(pr3_ind c (\lambda -(t: T).(\lambda (t0: T).((eq T t (TSort n)) \to (eq T t0 t)))) (\lambda (t: -T).(\lambda (_: (eq T t (TSort n))).(refl_equal T t))) (\lambda (t2: -T).(\lambda (t1: T).(\lambda (H1: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda -(_: (pr3 c t2 t3)).(\lambda (H3: (((eq T t2 (TSort n)) \to (eq T t3 -t2)))).(\lambda (H4: (eq T t1 (TSort n))).(let H5 \def (eq_ind T t1 (\lambda -(t: T).(pr2 c t t2)) H1 (TSort n) H4) in (eq_ind_r T (TSort n) (\lambda (t: -T).(eq T t3 t)) (let H6 \def (eq_ind T t2 (\lambda (t: T).((eq T t (TSort n)) -\to (eq T t3 t))) H3 (TSort n) (pr2_gen_sort c t2 n H5)) in (H6 (refl_equal T -(TSort n)))) t1 H4))))))))) y x H0))) H)))). -(* COMMENTS -Initial nodes: 253 -END *) - -theorem pr3_gen_abst: - \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c -(THead (Bind Abst) u1 t1) x) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 -c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: -T).(pr3 (CHead c (Bind b) u) t1 t2)))))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda -(H: (pr3 c (THead (Bind Abst) u1 t1) x)).(insert_eq T (THead (Bind Abst) u1 -t1) (\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t1 t2))))))) (\lambda (y: -T).(\lambda (H0: (pr3 c y x)).(unintro T t1 (\lambda (t: T).((eq T y (THead -(Bind Abst) u1 t)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x -(THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) -(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead -c (Bind b) u) t t2)))))))) (unintro T u1 (\lambda (t: T).(\forall (x0: -T).((eq T y (THead (Bind Abst) t x0)) \to (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_: T).(\lambda (t2: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x0 t2))))))))) (pr3_ind c -(\lambda (t: T).(\lambda (t0: T).(\forall (x0: T).(\forall (x1: T).((eq T t -(THead (Bind Abst) x0 x1)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T t0 (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall -(u: T).(pr3 (CHead c (Bind b) u) x1 t2))))))))))) (\lambda (t: T).(\lambda -(x0: T).(\lambda (x1: T).(\lambda (H1: (eq T t (THead (Bind Abst) x0 -x1))).(ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T t (THead (Bind -Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: -T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -x1 t2))))) x0 x1 H1 (pr3_refl c x0) (\lambda (b: B).(\lambda (u: T).(pr3_refl -(CHead c (Bind b) u) x1)))))))) (\lambda (t2: T).(\lambda (t3: T).(\lambda -(H1: (pr2 c t3 t2)).(\lambda (t4: T).(\lambda (_: (pr3 c t2 t4)).(\lambda -(H3: ((\forall (x0: T).(\forall (x1: T).((eq T t2 (THead (Bind Abst) x0 x1)) -\to (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst) -u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: -T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -x1 t5))))))))))).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t3 -(THead (Bind Abst) x0 x1))).(let H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c -t t2)) H1 (THead (Bind Abst) x0 x1) H4) in (let H6 \def (pr2_gen_abst c x0 x1 -t2 H5) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead -(Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead -c (Bind b) u) x1 t5))))) (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T -t4 (THead (Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 -u2))) (\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 -(CHead c (Bind b) u) x1 t5)))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda -(H7: (eq T t2 (THead (Bind Abst) x2 x3))).(\lambda (H8: (pr2 c x0 -x2)).(\lambda (H9: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) -x1 x3))))).(let H10 \def (eq_ind T t2 (\lambda (t: T).(\forall (x4: -T).(\forall (x5: T).((eq T t (THead (Bind Abst) x4 x5)) \to (ex3_2 T T -(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x4 u2))) (\lambda (_: T).(\lambda -(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x5 -t5)))))))))) H3 (THead (Bind Abst) x2 x3) H7) in (let H11 \def (H10 x2 x3 -(refl_equal T (THead (Bind Abst) x2 x3))) in (ex3_2_ind T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 t5))))) (ex3_2 T T -(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) -(\lambda (x4: T).(\lambda (x5: T).(\lambda (H12: (eq T t4 (THead (Bind Abst) -x4 x5))).(\lambda (H13: (pr3 c x2 x4)).(\lambda (H14: ((\forall (b: -B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 x5))))).(ex3_2_intro T T -(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5))))) -x4 x5 H12 (pr3_sing c x2 x0 H8 x4 H13) (\lambda (b: B).(\lambda (u: -T).(pr3_sing (CHead c (Bind b) u) x3 x1 (H9 b u) x5 (H14 b u)))))))))) -H11)))))))) H6)))))))))))) y x H0))))) H))))). -(* COMMENTS -Initial nodes: 1261 -END *) - -theorem pr3_gen_cast: - \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c -(THead (Flat Cast) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda -(t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t1 t2)))) (pr3 c -t1 x)))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda -(H: (pr3 c (THead (Flat Cast) u1 t1) x)).(insert_eq T (THead (Flat Cast) u1 -t1) (\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t1 -t2)))) (pr3 c t1 x))) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(unintro T -t1 (\lambda (t: T).((eq T y (THead (Flat Cast) u1 t)) \to (or (ex3_2 T T -(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda -(t2: T).(pr3 c t t2)))) (pr3 c t x)))) (unintro T u1 (\lambda (t: T).(\forall -(x0: T).((eq T y (THead (Flat Cast) t x0)) \to (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c x0 -t2)))) (pr3 c x0 x))))) (pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall -(x0: T).(\forall (x1: T).((eq T t (THead (Flat Cast) x0 x1)) \to (or (ex3_2 T -T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Flat Cast) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t2: T).(pr3 c x1 t2)))) (pr3 c x1 t0))))))) (\lambda (t: T).(\lambda (x0: -T).(\lambda (x1: T).(\lambda (H1: (eq T t (THead (Flat Cast) x0 -x1))).(eq_ind_r T (THead (Flat Cast) x0 x1) (\lambda (t0: T).(or (ex3_2 T T -(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Flat Cast) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t2: T).(pr3 c x1 t2)))) (pr3 c x1 t0))) (or_introl (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T (THead (Flat Cast) x0 x1) (THead (Flat Cast) u2 -t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: -T).(\lambda (t2: T).(pr3 c x1 t2)))) (pr3 c x1 (THead (Flat Cast) x0 x1)) -(ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Cast) -x0 x1) (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c -x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c x1 t2))) x0 x1 (refl_equal T -(THead (Flat Cast) x0 x1)) (pr3_refl c x0) (pr3_refl c x1))) t H1))))) -(\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c t3 t2)).(\lambda (t4: -T).(\lambda (H2: (pr3 c t2 t4)).(\lambda (H3: ((\forall (x0: T).(\forall (x1: -T).((eq T t2 (THead (Flat Cast) x0 x1)) \to (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 -t5)))) (pr3 c x1 t4))))))).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: -(eq T t3 (THead (Flat Cast) x0 x1))).(let H5 \def (eq_ind T t3 (\lambda (t: -T).(pr2 c t t2)) H1 (THead (Flat Cast) x0 x1) H4) in (let H6 \def -(pr2_gen_cast c x0 x1 t2 H5) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda -(t5: T).(eq T t2 (THead (Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr2 c x1 t5)))) (pr2 c -x1 t2) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat -Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: -T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4)) (\lambda (H7: (ex3_2 T T -(\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Flat Cast) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda -(t5: T).(pr2 c x1 t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: -T).(eq T t2 (THead (Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr2 c x1 t5))) (or -(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 -t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: -T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4)) (\lambda (x2: T).(\lambda -(x3: T).(\lambda (H8: (eq T t2 (THead (Flat Cast) x2 x3))).(\lambda (H9: (pr2 -c x0 x2)).(\lambda (H10: (pr2 c x1 x3)).(let H11 \def (eq_ind T t2 (\lambda -(t: T).(\forall (x4: T).(\forall (x5: T).((eq T t (THead (Flat Cast) x4 x5)) -\to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat -Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x4 u2))) (\lambda (_: -T).(\lambda (t5: T).(pr3 c x5 t5)))) (pr3 c x5 t4)))))) H3 (THead (Flat Cast) -x2 x3) H8) in (let H12 \def (H11 x2 x3 (refl_equal T (THead (Flat Cast) x2 -x3))) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead -(Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr3 c x3 t5)))) (pr3 c x3 t4) (or (ex3_2 T -T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4)) (\lambda (H13: (ex3_2 T T (\lambda -(u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x3 -t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead -(Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr3 c x3 t5))) (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 -t5)))) (pr3 c x1 t4)) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H14: (eq T -t4 (THead (Flat Cast) x4 x5))).(\lambda (H15: (pr3 c x2 x4)).(\lambda (H16: -(pr3 c x3 x5)).(eq_ind_r T (THead (Flat Cast) x4 x5) (\lambda (t: T).(or -(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t (THead (Flat Cast) u2 -t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: -T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t))) (or_introl (ex3_2 T T -(\lambda (u2: T).(\lambda (t5: T).(eq T (THead (Flat Cast) x4 x5) (THead -(Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 (THead (Flat -Cast) x4 x5)) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t5: T).(eq T (THead -(Flat Cast) x4 x5) (THead (Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5))) x4 x5 -(refl_equal T (THead (Flat Cast) x4 x5)) (pr3_sing c x2 x0 H9 x4 H15) -(pr3_sing c x3 x1 H10 x5 H16))) t4 H14)))))) H13)) (\lambda (H13: (pr3 c x3 -t4)).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead -(Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4) (pr3_sing c -x3 x1 H10 t4 H13))) H12)))))))) H7)) (\lambda (H7: (pr2 c x1 t2)).(or_intror -(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 -t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: -T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4) (pr3_sing c t2 x1 H7 t4 -H2))) H6)))))))))))) y x H0))))) H))))). -(* COMMENTS -Initial nodes: 2001 -END *) - -theorem pr3_gen_lift: - \forall (c: C).(\forall (t1: T).(\forall (x: T).(\forall (h: nat).(\forall -(d: nat).((pr3 c (lift h d t1) x) \to (\forall (e: C).((drop h d c e) \to -(ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr3 e t1 -t2)))))))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (x: T).(\lambda (h: nat).(\lambda -(d: nat).(\lambda (H: (pr3 c (lift h d t1) x)).(insert_eq T (lift h d t1) -(\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(\forall (e: C).((drop h d c e) -\to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr3 e -t1 t2)))))) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(unintro T t1 (\lambda -(t: T).((eq T y (lift h d t)) \to (\forall (e: C).((drop h d c e) \to (ex2 T -(\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr3 e t t2))))))) -(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (x0: T).((eq T t (lift h -d x0)) \to (\forall (e: C).((drop h d c e) \to (ex2 T (\lambda (t2: T).(eq T -t0 (lift h d t2))) (\lambda (t2: T).(pr3 e x0 t2))))))))) (\lambda (t: -T).(\lambda (x0: T).(\lambda (H1: (eq T t (lift h d x0))).(\lambda (e: -C).(\lambda (_: (drop h d c e)).(ex_intro2 T (\lambda (t2: T).(eq T t (lift h -d t2))) (\lambda (t2: T).(pr3 e x0 t2)) x0 H1 (pr3_refl e x0))))))) (\lambda -(t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c t3 t2)).(\lambda (t4: -T).(\lambda (_: (pr3 c t2 t4)).(\lambda (H3: ((\forall (x0: T).((eq T t2 -(lift h d x0)) \to (\forall (e: C).((drop h d c e) \to (ex2 T (\lambda (t5: -T).(eq T t4 (lift h d t5))) (\lambda (t5: T).(pr3 e x0 t5))))))))).(\lambda -(x0: T).(\lambda (H4: (eq T t3 (lift h d x0))).(\lambda (e: C).(\lambda (H5: -(drop h d c e)).(let H6 \def (eq_ind T t3 (\lambda (t: T).(pr2 c t t2)) H1 -(lift h d x0) H4) in (let H7 \def (pr2_gen_lift c x0 t2 h d H6 e H5) in -(ex2_ind T (\lambda (t5: T).(eq T t2 (lift h d t5))) (\lambda (t5: T).(pr2 e -x0 t5)) (ex2 T (\lambda (t5: T).(eq T t4 (lift h d t5))) (\lambda (t5: -T).(pr3 e x0 t5))) (\lambda (x1: T).(\lambda (H8: (eq T t2 (lift h d -x1))).(\lambda (H9: (pr2 e x0 x1)).(ex2_ind T (\lambda (t5: T).(eq T t4 (lift -h d t5))) (\lambda (t5: T).(pr3 e x1 t5)) (ex2 T (\lambda (t5: T).(eq T t4 -(lift h d t5))) (\lambda (t5: T).(pr3 e x0 t5))) (\lambda (x2: T).(\lambda -(H10: (eq T t4 (lift h d x2))).(\lambda (H11: (pr3 e x1 x2)).(ex_intro2 T -(\lambda (t5: T).(eq T t4 (lift h d t5))) (\lambda (t5: T).(pr3 e x0 t5)) x2 -H10 (pr3_sing e x1 x0 H9 x2 H11))))) (H3 x1 H8 e H5))))) H7))))))))))))) y x -H0)))) H)))))). -(* COMMENTS -Initial nodes: 689 -END *) - -theorem pr3_gen_lref: - \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr3 c (TLRef n) x) \to -(or (eq T x (TLRef n)) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda -(_: T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: -T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(v: T).(eq T x (lift (S n) O v)))))))))) -\def - \lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr3 c (TLRef -n) x)).(insert_eq T (TLRef n) (\lambda (t: T).(pr3 c t x)) (\lambda (t: -T).(or (eq T x t) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: -T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: -T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(v: T).(eq T x (lift (S n) O v)))))))) (\lambda (y: T).(\lambda (H0: (pr3 c y -x)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).((eq T t (TLRef n)) \to (or -(eq T t0 t) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: -T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: -T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(v: T).(eq T t0 (lift (S n) O v)))))))))) (\lambda (t: T).(\lambda (_: (eq T -t (TLRef n))).(or_introl (eq T t t) (ex3_3 C T T (\lambda (d: C).(\lambda (u: -T).(\lambda (_: T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: -C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda -(_: T).(\lambda (v: T).(eq T t (lift (S n) O v)))))) (refl_equal T t)))) -(\lambda (t2: T).(\lambda (t1: T).(\lambda (H1: (pr2 c t1 t2)).(\lambda (t3: -T).(\lambda (H2: (pr3 c t2 t3)).(\lambda (H3: (((eq T t2 (TLRef n)) \to (or -(eq T t3 t2) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: -T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: -T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(v: T).(eq T t3 (lift (S n) O v)))))))))).(\lambda (H4: (eq T t1 (TLRef -n))).(let H5 \def (eq_ind T t1 (\lambda (t: T).(pr2 c t t2)) H1 (TLRef n) H4) -in (eq_ind_r T (TLRef n) (\lambda (t: T).(or (eq T t3 t) (ex3_3 C T T -(\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead d (Bind -Abbr) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T t3 (lift (S n) O -v)))))))) (let H6 \def (pr2_gen_lref c t2 n H5) in (or_ind (eq T t2 (TLRef -n)) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) -u)))) (\lambda (_: C).(\lambda (u: T).(eq T t2 (lift (S n) O u))))) (or (eq T -t3 (TLRef n)) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: -T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: -T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(v: T).(eq T t3 (lift (S n) O v))))))) (\lambda (H7: (eq T t2 (TLRef -n))).(let H8 \def (eq_ind T t2 (\lambda (t: T).((eq T t (TLRef n)) \to (or -(eq T t3 t) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: -T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: -T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(v: T).(eq T t3 (lift (S n) O v))))))))) H3 (TLRef n) H7) in (let H9 \def -(eq_ind T t2 (\lambda (t: T).(pr3 c t t3)) H2 (TLRef n) H7) in (H8 -(refl_equal T (TLRef n)))))) (\lambda (H7: (ex2_2 C T (\lambda (d: -C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda (_: -C).(\lambda (u: T).(eq T t2 (lift (S n) O u)))))).(ex2_2_ind C T (\lambda (d: -C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda (_: -C).(\lambda (u: T).(eq T t2 (lift (S n) O u)))) (or (eq T t3 (TLRef n)) -(ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead -d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (v: T).(pr3 d u -v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T t3 (lift (S n) O -v))))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H8: (getl n c (CHead x0 -(Bind Abbr) x1))).(\lambda (H9: (eq T t2 (lift (S n) O x1))).(let H10 \def -(eq_ind T t2 (\lambda (t: T).((eq T t (TLRef n)) \to (or (eq T t3 t) (ex3_3 C -T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead d (Bind -Abbr) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T t3 (lift (S n) O -v))))))))) H3 (lift (S n) O x1) H9) in (let H11 \def (eq_ind T t2 (\lambda -(t: T).(pr3 c t t3)) H2 (lift (S n) O x1) H9) in (let H12 \def (pr3_gen_lift -c x1 t3 (S n) O H11 x0 (getl_drop Abbr c x0 x1 n H8)) in (ex2_ind T (\lambda -(t4: T).(eq T t3 (lift (S n) O t4))) (\lambda (t4: T).(pr3 x0 x1 t4)) (or (eq -T t3 (TLRef n)) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: -T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: -T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(v: T).(eq T t3 (lift (S n) O v))))))) (\lambda (x2: T).(\lambda (H13: (eq T -t3 (lift (S n) O x2))).(\lambda (H14: (pr3 x0 x1 x2)).(or_intror (eq T t3 -(TLRef n)) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(getl -n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (v: -T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T t3 -(lift (S n) O v)))))) (ex3_3_intro C T T (\lambda (d: C).(\lambda (u: -T).(\lambda (_: T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: -C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda -(_: T).(\lambda (v: T).(eq T t3 (lift (S n) O v))))) x0 x1 x2 H8 H14 H13))))) -H12)))))))) H7)) H6)) t1 H4))))))))) y x H0))) H)))). -(* COMMENTS -Initial nodes: 1515 -END *) - -theorem pr3_gen_void: - \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c -(THead (Bind Void) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda -(t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall -(u: T).(pr3 (CHead c (Bind b) u) t1 t2)))))) (pr3 (CHead c (Bind Void) u1) t1 -(lift (S O) O x))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda -(H: (pr3 c (THead (Bind Void) u1 t1) x)).(insert_eq T (THead (Bind Void) u1 -t1) (\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t1 t2)))))) (pr3 (CHead c -(Bind Void) u1) t1 (lift (S O) O x)))) (\lambda (y: T).(\lambda (H0: (pr3 c y -x)).(unintro T t1 (\lambda (t: T).((eq T y (THead (Bind Void) u1 t)) \to (or -(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 -t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: -T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -t t2)))))) (pr3 (CHead c (Bind Void) u1) t (lift (S O) O x))))) (unintro T u1 -(\lambda (t: T).(\forall (x0: T).((eq T y (THead (Bind Void) t x0)) \to (or -(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 -t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_: -T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -x0 t2)))))) (pr3 (CHead c (Bind Void) t) x0 (lift (S O) O x)))))) (pr3_ind c -(\lambda (t: T).(\lambda (t0: T).(\forall (x0: T).(\forall (x1: T).((eq T t -(THead (Bind Void) x0 x1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T t0 (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall -(u: T).(pr3 (CHead c (Bind b) u) x1 t2)))))) (pr3 (CHead c (Bind Void) x0) x1 -(lift (S O) O t0)))))))) (\lambda (t: T).(\lambda (x0: T).(\lambda (x1: -T).(\lambda (H1: (eq T t (THead (Bind Void) x0 x1))).(eq_ind_r T (THead (Bind -Void) x0 x1) (\lambda (t0: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T t0 (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall -(u: T).(pr3 (CHead c (Bind b) u) x1 t2)))))) (pr3 (CHead c (Bind Void) x0) x1 -(lift (S O) O t0)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T (THead (Bind Void) x0 x1) (THead (Bind Void) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t2)))))) (pr3 (CHead c -(Bind Void) x0) x1 (lift (S O) O (THead (Bind Void) x0 x1))) (ex3_2_intro T T -(\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Bind Void) x0 x1) (THead -(Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) -(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead -c (Bind b) u) x1 t2))))) x0 x1 (refl_equal T (THead (Bind Void) x0 x1)) -(pr3_refl c x0) (\lambda (b: B).(\lambda (u: T).(pr3_refl (CHead c (Bind b) -u) x1))))) t H1))))) (\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c -t3 t2)).(\lambda (t4: T).(\lambda (H2: (pr3 c t2 t4)).(\lambda (H3: ((\forall -(x0: T).(\forall (x1: T).((eq T t2 (THead (Bind Void) x0 x1)) \to (or (ex3_2 -T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) -(pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4)))))))).(\lambda (x0: -T).(\lambda (x1: T).(\lambda (H4: (eq T t3 (THead (Bind Void) x0 x1))).(let -H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c t t2)) H1 (THead (Bind Void) x0 -x1) H4) in (let H6 \def (pr2_gen_void c x0 x1 t2 H5) in (or_ind (ex3_2 T T -(\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind Void) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda -(t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 t5)))))) -(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 (lift (S O) O -t2)))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind -Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: -T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4))) (\lambda -(H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind Void) -u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: -T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) -x1 t5))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead -(Bind Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead -c (Bind b) u) x1 t5))))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq -T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 -u2))) (\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 -(CHead c (Bind b) u) x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) -O t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H8: (eq T t2 (THead (Bind -Void) x2 x3))).(\lambda (H9: (pr2 c x0 x2)).(\lambda (H10: ((\forall (b: -B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 x3))))).(let H11 \def (eq_ind -T t2 (\lambda (t: T).(\forall (x4: T).(\forall (x5: T).((eq T t (THead (Bind -Void) x4 x5)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 -(THead (Bind Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x4 u2))) -(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead -c (Bind b) u) x5 t5)))))) (pr3 (CHead c (Bind Void) x4) x5 (lift (S O) O -t4))))))) H3 (THead (Bind Void) x2 x3) H8) in (let H12 \def (H11 x2 x3 -(refl_equal T (THead (Bind Void) x2 x3))) in (or_ind (ex3_2 T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 t5)))))) (pr3 (CHead c -(Bind Void) x2) x3 (lift (S O) O t4)) (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) (pr3 (CHead c -(Bind Void) x0) x1 (lift (S O) O t4))) (\lambda (H13: (ex3_2 T T (\lambda -(u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 t5))))))).(ex3_2_ind T T -(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda -(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 t5))))) -(or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) -u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: -T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4))) (\lambda -(x4: T).(\lambda (x5: T).(\lambda (H14: (eq T t4 (THead (Bind Void) x4 -x5))).(\lambda (H15: (pr3 c x2 x4)).(\lambda (H16: ((\forall (b: B).(\forall -(u: T).(pr3 (CHead c (Bind b) u) x3 x5))))).(or_introl (ex3_2 T T (\lambda -(u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) (pr3 (CHead c -(Bind Void) x0) x1 (lift (S O) O t4)) (ex3_2_intro T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5))))) x4 x5 H14 -(pr3_sing c x2 x0 H9 x4 H15) (\lambda (b: B).(\lambda (u: T).(pr3_sing (CHead -c (Bind b) u) x3 x1 (H10 b u) x5 (H16 b u))))))))))) H13)) (\lambda (H13: -(pr3 (CHead c (Bind Void) x2) x3 (lift (S O) O t4))).(or_intror (ex3_2 T T -(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) -(pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4)) (pr3_sing (CHead c (Bind -Void) x0) x3 x1 (H10 Void x0) (lift (S O) O t4) (pr3_pr2_pr3_t c x2 x3 (lift -(S O) O t4) (Bind Void) H13 x0 H9)))) H12)))))))) H7)) (\lambda (H7: -((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 (lift (S O) O -t2)))))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 -(THead (Bind Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead -c (Bind b) u) x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4)) -(pr3_sing (CHead c (Bind Void) x0) (lift (S O) O t2) x1 (H7 Void x0) (lift (S -O) O t4) (pr3_lift (CHead c (Bind Void) x0) c (S O) O (drop_drop (Bind Void) -O c c (drop_refl c) x0) t2 t4 H2)))) H6)))))))))))) y x H0))))) H))))). -(* COMMENTS -Initial nodes: 2645 -END *) - -theorem pr3_gen_abbr: - \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c -(THead (Bind Abbr) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda -(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) -u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O x))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda -(H: (pr3 c (THead (Bind Abbr) u1 t1) x)).(insert_eq T (THead (Bind Abbr) u1 -t1) (\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 -(CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S -O) O x)))) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(unintro T t1 (\lambda -(t: T).((eq T y (THead (Bind Abbr) u1 t)) \to (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 -(CHead c (Bind Abbr) u1) t t2)))) (pr3 (CHead c (Bind Abbr) u1) t (lift (S O) -O x))))) (unintro T u1 (\lambda (t: T).(\forall (x0: T).((eq T y (THead (Bind -Abbr) t x0)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x -(THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) -(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) t) x0 t2)))) (pr3 -(CHead c (Bind Abbr) t) x0 (lift (S O) O x)))))) (pr3_ind c (\lambda (t: -T).(\lambda (t0: T).(\forall (x0: T).(\forall (x1: T).((eq T t (THead (Bind -Abbr) x0 x1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 -(THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) -(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) x0) x1 t2)))) (pr3 -(CHead c (Bind Abbr) x0) x1 (lift (S O) O t0)))))))) (\lambda (t: T).(\lambda -(x0: T).(\lambda (x1: T).(\lambda (H1: (eq T t (THead (Bind Abbr) x0 -x1))).(eq_ind_r T (THead (Bind Abbr) x0 x1) (\lambda (t0: T).(or (ex3_2 T T -(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t2: T).(pr3 (CHead c (Bind Abbr) x0) x1 t2)))) (pr3 (CHead c (Bind Abbr) x0) -x1 (lift (S O) O t0)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 -(CHead c (Bind Abbr) x0) x1 t2)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S -O) O (THead (Bind Abbr) x0 x1))) (ex3_2_intro T T (\lambda (u2: T).(\lambda -(t2: T).(eq T (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 t2)))) (\lambda -(u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 -(CHead c (Bind Abbr) x0) x1 t2))) x0 x1 (refl_equal T (THead (Bind Abbr) x0 -x1)) (pr3_refl c x0) (pr3_refl (CHead c (Bind Abbr) x0) x1))) t H1))))) -(\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c t3 t2)).(\lambda (t4: -T).(\lambda (H2: (pr3 c t2 t4)).(\lambda (H3: ((\forall (x0: T).(\forall (x1: -T).((eq T t2 (THead (Bind Abbr) x0 x1)) \to (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S -O) O t4)))))))).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t3 -(THead (Bind Abbr) x0 x1))).(let H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c -t t2)) H1 (THead (Bind Abbr) x0 x1) H4) in (let H6 \def (pr2_gen_abbr c x0 x1 -t2 H5) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 -(THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 -(CHead c (Bind b) u) x1 t5))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: -T).(pr2 (CHead c (Bind Abbr) u) x1 t5))) (ex3_2 T T (\lambda (y0: T).(\lambda -(_: T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: -T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) -z t5)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 -(lift (S O) O t2)))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T -t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 -u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 -t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (H7: -(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind Abbr) u2 -t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: -T).(\lambda (t5: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind -b) u) x1 t5))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead -c (Bind Abbr) u) x1 t5))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 -(CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 -z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) z -t5))))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead -(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 -(CHead c (Bind b) u) x1 t5))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: -T).(pr2 (CHead c (Bind Abbr) u) x1 t5))) (ex3_2 T T (\lambda (y0: T).(\lambda -(_: T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: -T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) -z t5))))))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead -(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 -(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (x2: T).(\lambda -(x3: T).(\lambda (H8: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (H9: (pr2 -c x0 x2)).(\lambda (H10: (or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c -(Bind b) u) x1 x3))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 -(CHead c (Bind Abbr) u) x1 x3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: -T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: -T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) -z x3)))))).(or3_ind (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) -x1 x3))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead c -(Bind Abbr) u) x1 x3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 -(CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 -z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) z x3)))) -(or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) -u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: -T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c -(Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (H11: ((\forall (b: -B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 x3))))).(let H12 \def (eq_ind -T t2 (\lambda (t: T).(\forall (x4: T).(\forall (x5: T).((eq T t (THead (Bind -Abbr) x4 x5)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 -(THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x4 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x4) x5 t5)))) (pr3 -(CHead c (Bind Abbr) x4) x5 (lift (S O) O t4))))))) H3 (THead (Bind Abbr) x2 -x3) H8) in (let H13 \def (H12 x2 x3 (refl_equal T (THead (Bind Abbr) x2 x3))) -in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind -Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: -T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 t5)))) (pr3 (CHead c -(Bind Abbr) x2) x3 (lift (S O) O t4)) (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S -O) O t4))) (\lambda (H14: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T -t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 -u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 -t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead -(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 t5))) (or -(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 -t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: -T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c -(Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (x4: T).(\lambda (x5: -T).(\lambda (H15: (eq T t4 (THead (Bind Abbr) x4 x5))).(\lambda (H16: (pr3 c -x2 x4)).(\lambda (H17: (pr3 (CHead c (Bind Abbr) x2) x3 x5)).(eq_ind_r T -(THead (Bind Abbr) x4 x5) (\lambda (t: T).(or (ex3_2 T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t (THead (Bind Abbr) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S -O) O t)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T -(THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S -O) O (THead (Bind Abbr) x4 x5))) (ex3_2_intro T T (\lambda (u2: T).(\lambda -(t5: T).(eq T (THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2 t5)))) (\lambda -(u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x0) x1 t5))) x4 x5 (refl_equal T (THead (Bind Abbr) x4 -x5)) (pr3_sing c x2 x0 H9 x4 H16) (pr3_sing (CHead c (Bind Abbr) x0) x3 x1 -(H11 Abbr x0) x5 (pr3_pr2_pr3_t c x2 x3 x5 (Bind Abbr) H17 x0 H9)))) t4 -H15)))))) H14)) (\lambda (H14: (pr3 (CHead c (Bind Abbr) x2) x3 (lift (S O) O -t4))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead -(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 -(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4)) (pr3_sing (CHead c (Bind Abbr) -x0) x3 x1 (H11 Abbr x0) (lift (S O) O t4) (pr3_pr2_pr3_t c x2 x3 (lift (S O) -O t4) (Bind Abbr) H14 x0 H9)))) H13)))) (\lambda (H11: (ex2 T (\lambda (u: -T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) x1 -x3)))).(ex2_ind T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead c -(Bind Abbr) u) x1 x3)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T -t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 -u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 -t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (x4: -T).(\lambda (H12: (pr0 x0 x4)).(\lambda (H13: (pr2 (CHead c (Bind Abbr) x4) -x1 x3)).(let H14 \def (eq_ind T t2 (\lambda (t: T).(\forall (x5: T).(\forall -(x6: T).((eq T t (THead (Bind Abbr) x5 x6)) \to (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x5 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x5) x6 t5)))) (pr3 (CHead c (Bind Abbr) x5) x6 (lift (S -O) O t4))))))) H3 (THead (Bind Abbr) x2 x3) H8) in (let H15 \def (H14 x2 x3 -(refl_equal T (THead (Bind Abbr) x2 x3))) in (or_ind (ex3_2 T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x2) x3 t5)))) (pr3 (CHead c (Bind Abbr) x2) x3 (lift (S -O) O t4)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead -(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 -(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (H16: (ex3_2 T T -(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda -(t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 t5))))).(ex3_2_ind T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x2) x3 t5))) (or (ex3_2 T T (\lambda (u2: T).(\lambda -(t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) -x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda -(x5: T).(\lambda (x6: T).(\lambda (H17: (eq T t4 (THead (Bind Abbr) x5 -x6))).(\lambda (H18: (pr3 c x2 x5)).(\lambda (H19: (pr3 (CHead c (Bind Abbr) -x2) x3 x6)).(eq_ind_r T (THead (Bind Abbr) x5 x6) (\lambda (t: T).(or (ex3_2 -T T (\lambda (u2: T).(\lambda (t5: T).(eq T t (THead (Bind Abbr) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) -x1 (lift (S O) O t)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t5: -T).(eq T (THead (Bind Abbr) x5 x6) (THead (Bind Abbr) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S -O) O (THead (Bind Abbr) x5 x6))) (ex3_2_intro T T (\lambda (u2: T).(\lambda -(t5: T).(eq T (THead (Bind Abbr) x5 x6) (THead (Bind Abbr) u2 t5)))) (\lambda -(u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x0) x1 t5))) x5 x6 (refl_equal T (THead (Bind Abbr) x5 -x6)) (pr3_sing c x2 x0 H9 x5 H18) (pr3_t x3 x1 (CHead c (Bind Abbr) x0) -(pr3_pr0_pr2_t x0 x4 H12 c x1 x3 (Bind Abbr) H13) x6 (pr3_pr2_pr3_t c x2 x3 -x6 (Bind Abbr) H19 x0 H9)))) t4 H17)))))) H16)) (\lambda (H16: (pr3 (CHead c -(Bind Abbr) x2) x3 (lift (S O) O t4))).(or_intror (ex3_2 T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S -O) O t4)) (pr3_t x3 x1 (CHead c (Bind Abbr) x0) (pr3_pr0_pr2_t x0 x4 H12 c x1 -x3 (Bind Abbr) H13) (lift (S O) O t4) (pr3_pr2_pr3_t c x2 x3 (lift (S O) O -t4) (Bind Abbr) H16 x0 H9)))) H15)))))) H11)) (\lambda (H11: (ex3_2 T T -(\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) -(\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: -T).(pr2 (CHead c (Bind Abbr) x0) z x3))))).(ex3_2_ind T T (\lambda (y0: -T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: -T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c -(Bind Abbr) x0) z x3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq -T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 -u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 -t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (x4: -T).(\lambda (x5: T).(\lambda (H12: (pr2 (CHead c (Bind Abbr) x0) x1 -x4)).(\lambda (H13: (pr0 x4 x5)).(\lambda (H14: (pr2 (CHead c (Bind Abbr) x0) -x5 x3)).(let H15 \def (eq_ind T t2 (\lambda (t: T).(\forall (x6: T).(\forall -(x7: T).((eq T t (THead (Bind Abbr) x6 x7)) \to (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x6 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x6) x7 t5)))) (pr3 (CHead c (Bind Abbr) x6) x7 (lift (S -O) O t4))))))) H3 (THead (Bind Abbr) x2 x3) H8) in (let H16 \def (H15 x2 x3 -(refl_equal T (THead (Bind Abbr) x2 x3))) in (or_ind (ex3_2 T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x2) x3 t5)))) (pr3 (CHead c (Bind Abbr) x2) x3 (lift (S -O) O t4)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead -(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 -(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (H17: (ex3_2 T T -(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda -(t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 t5))))).(ex3_2_ind T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x2) x3 t5))) (or (ex3_2 T T (\lambda (u2: T).(\lambda -(t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) -x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda -(x6: T).(\lambda (x7: T).(\lambda (H18: (eq T t4 (THead (Bind Abbr) x6 -x7))).(\lambda (H19: (pr3 c x2 x6)).(\lambda (H20: (pr3 (CHead c (Bind Abbr) -x2) x3 x7)).(eq_ind_r T (THead (Bind Abbr) x6 x7) (\lambda (t: T).(or (ex3_2 -T T (\lambda (u2: T).(\lambda (t5: T).(eq T t (THead (Bind Abbr) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) -x1 (lift (S O) O t)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t5: -T).(eq T (THead (Bind Abbr) x6 x7) (THead (Bind Abbr) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S -O) O (THead (Bind Abbr) x6 x7))) (ex3_2_intro T T (\lambda (u2: T).(\lambda -(t5: T).(eq T (THead (Bind Abbr) x6 x7) (THead (Bind Abbr) u2 t5)))) (\lambda -(u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x0) x1 t5))) x6 x7 (refl_equal T (THead (Bind Abbr) x6 -x7)) (pr3_sing c x2 x0 H9 x6 H19) (pr3_sing (CHead c (Bind Abbr) x0) x4 x1 -H12 x7 (pr3_sing (CHead c (Bind Abbr) x0) x5 x4 (pr2_free (CHead c (Bind -Abbr) x0) x4 x5 H13) x7 (pr3_sing (CHead c (Bind Abbr) x0) x3 x5 H14 x7 -(pr3_pr2_pr3_t c x2 x3 x7 (Bind Abbr) H20 x0 H9)))))) t4 H18)))))) H17)) -(\lambda (H17: (pr3 (CHead c (Bind Abbr) x2) x3 (lift (S O) O -t4))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead -(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 -(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4)) (pr3_sing (CHead c (Bind Abbr) -x0) x4 x1 H12 (lift (S O) O t4) (pr3_sing (CHead c (Bind Abbr) x0) x5 x4 -(pr2_free (CHead c (Bind Abbr) x0) x4 x5 H13) (lift (S O) O t4) (pr3_sing -(CHead c (Bind Abbr) x0) x3 x5 H14 (lift (S O) O t4) (pr3_pr2_pr3_t c x2 x3 -(lift (S O) O t4) (Bind Abbr) H17 x0 H9)))))) H16)))))))) H11)) H10)))))) -H7)) (\lambda (H7: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) -x1 (lift (S O) O t2)))))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda -(t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) -x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4)) (pr3_sing -(CHead c (Bind Abbr) x0) (lift (S O) O t2) x1 (H7 Abbr x0) (lift (S O) O t4) -(pr3_lift (CHead c (Bind Abbr) x0) c (S O) O (drop_drop (Bind Abbr) O c c -(drop_refl c) x0) t2 t4 H2)))) H6)))))))))))) y x H0))))) H))))). -(* COMMENTS -Initial nodes: 5983 -END *) - -theorem pr3_gen_appl: - \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c -(THead (Flat Appl) u1 t1) x) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda -(t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t1 t2)))) (ex4_4 T -T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 -c (THead (Bind Abbr) u2 t2) x))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (_: T).(pr3 c u1 u2))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c t1 (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T -T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 -c t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) x))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda -(H: (pr3 c (THead (Flat Appl) u1 t1) x)).(insert_eq T (THead (Flat Appl) u1 -t1) (\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(or3 (ex3_2 T T (\lambda -(u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t1 -t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) x))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))))) -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c t1 -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) -(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(pr3 c t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) -x))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) -y2) z1 z2)))))))))) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(unintro T t1 -(\lambda (t: T).((eq T y (THead (Flat Appl) u1 t)) \to (or3 (ex3_2 T T -(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda -(t2: T).(pr3 c t t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) x))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 -u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(pr3 c t (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 -(CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c t (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead -(Flat Appl) (lift (S O) O u2) z2)) x))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 -u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))) (unintro T u1 (\lambda -(t: T).(\forall (x0: T).((eq T y (THead (Flat Appl) t x0)) \to (or3 (ex3_2 T -T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_: T).(\lambda (t2: -T).(pr3 c x0 t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) x))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))) (\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x0 (THead -(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) -(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(pr3 c x0 (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) -x))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) -y2) z1 z2)))))))))))) (pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall -(x0: T).(\forall (x1: T).((eq T t (THead (Flat Appl) x0 x1)) \to (or3 (ex3_2 -T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Flat Appl) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t2: T).(pr3 c x1 t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) t0))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 -u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 -(CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead -(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) -y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t0))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))))))) -(\lambda (t: T).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H1: (eq T t -(THead (Flat Appl) x0 x1))).(eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda -(t0: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead -(Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) -(\lambda (_: T).(\lambda (t2: T).(pr3 c x1 t2)))) (ex4_4 T T T T (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind -Abbr) u2 t2) t0))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T -T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 -c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t0))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))) -(or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat -Appl) x0 x1) (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c x1 t2)))) (ex4_4 T -T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 -c (THead (Bind Abbr) u2 t2) (THead (Flat Appl) x0 x1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))) -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) -(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) -(THead (Flat Appl) x0 x1)))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 -(CHead c (Bind b) y2) z1 z2)))))))) (ex3_2_intro T T (\lambda (u2: -T).(\lambda (t2: T).(eq T (THead (Flat Appl) x0 x1) (THead (Flat Appl) u2 -t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: -T).(\lambda (t2: T).(pr3 c x1 t2))) x0 x1 (refl_equal T (THead (Flat Appl) x0 -x1)) (pr3_refl c x0) (pr3_refl c x1))) t H1))))) (\lambda (t2: T).(\lambda -(t3: T).(\lambda (H1: (pr2 c t3 t2)).(\lambda (t4: T).(\lambda (H2: (pr3 c t2 -t4)).(\lambda (H3: ((\forall (x0: T).(\forall (x1: T).((eq T t2 (THead (Flat -Appl) x0 x1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 -(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind -Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T -T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 -c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 -z2)))))))))))))).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t3 -(THead (Flat Appl) x0 x1))).(let H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c -t t2)) H1 (THead (Flat Appl) x0 x1) H4) in (let H6 \def (pr2_gen_appl c x0 x1 -t2 H5) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 -(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr2 c x1 t5)))) (ex4_4 T T T T (\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x1 (THead -(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t5: T).(eq T t2 (THead (Bind Abbr) u2 t5)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T -T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq -T x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead -(Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (or3 (ex3_2 -T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 -u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 -(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead -(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) -y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda -(H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Flat Appl) -u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: -T).(\lambda (t5: T).(pr2 c x1 t5))))).(ex3_2_ind T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t2 (THead (Flat Appl) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr2 c x1 -t5))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat -Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: -T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda -(_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) -t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 -c x0 u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 -(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead -(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) -y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda -(x2: T).(\lambda (x3: T).(\lambda (H8: (eq T t2 (THead (Flat Appl) x2 -x3))).(\lambda (H9: (pr2 c x0 x2)).(\lambda (H10: (pr2 c x1 x3)).(let H11 -\def (eq_ind T t2 (\lambda (t: T).(\forall (x4: T).(\forall (x5: T).((eq T t -(THead (Flat Appl) x4 x5)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: -T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c x4 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x5 t5)))) (ex4_4 T -T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 -c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (_: T).(pr3 c x4 u2))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c x5 (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T -T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 -c x5 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c x4 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))))) H3 -(THead (Flat Appl) x2 x3) H8) in (let H12 \def (eq_ind T t2 (\lambda (t: -T).(pr3 c t t4)) H2 (THead (Flat Appl) x2 x3) H8) in (let H13 \def (H11 x2 x3 -(refl_equal T (THead (Flat Appl) x2 x3))) in (or3_ind (ex3_2 T T (\lambda -(u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x3 -t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))))) -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x3 -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) -(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(pr3 c x3 (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) -t4))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 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B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 -c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda -(H14: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat -Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: -T).(\lambda (t5: T).(pr3 c x3 t5))))).(ex3_2_ind T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x3 -t5))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat -Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: -T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda -(_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) -t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 -c x0 u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 -(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead -(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) -y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda -(x4: T).(\lambda (x5: T).(\lambda (H15: (eq T t4 (THead (Flat Appl) x4 -x5))).(\lambda (H16: (pr3 c x2 x4)).(\lambda (H17: (pr3 c x3 x5)).(eq_ind_r T -(THead (Flat Appl) x4 x5) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t (THead (Flat Appl) u2 t5)))) (\lambda (u2: 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-t))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) -y2) z1 z2)))))))))) (or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: -T).(eq T (THead (Flat Appl) x4 x5) (THead (Flat Appl) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 -t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) (THead (Flat Appl) x4 -x5)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: -T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (z1: T).(\lambda (_: 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T).(\lambda (t5: T).(eq T (THead (Flat Appl) -x4 x5) (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c -x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5))) x4 x5 (refl_equal T -(THead (Flat Appl) x4 x5)) (pr3_sing c x2 x0 H9 x4 H16) (pr3_sing c x3 x1 H10 -x5 H17))) t4 H15)))))) H14)) (\lambda (H14: (ex4_4 T T T T (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind -Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr3 c x2 u2))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c x3 (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5))))))))).(ex4_4_ind T -T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 -c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (_: T).(pr3 c x2 u2))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c x3 (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5))))))) (or3 (ex3_2 T T -(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 -u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 -(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead -(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) -y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda -(x4: T).(\lambda (x5: T).(\lambda (x6: T).(\lambda (x7: T).(\lambda (H15: -(pr3 c (THead (Bind Abbr) x6 x7) t4)).(\lambda (H16: (pr3 c x2 x6)).(\lambda -(H17: (pr3 c x3 (THead (Bind Abst) x4 x5))).(\lambda (H18: ((\forall (b: -B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x5 x7))))).(or3_intro1 (ex3_2 T -T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 -u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 -(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead -(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) -y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (ex4_4_intro -T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: -T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 -t5))))))) x4 x5 x6 x7 H15 (pr3_sing c x2 x0 H9 x6 H16) (pr3_sing c x3 x1 H10 -(THead (Bind Abst) x4 x5) H17) H18)))))))))) H14)) (\lambda (H14: (ex6_6 B T -T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(pr3 c x3 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c -(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr3 c x2 u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 -y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 -z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c x3 (THead (Bind b) y1 z1)))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda -(u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift -(S O) O u2) z2)) t4))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 -(CHead c (Bind b) y2) z1 z2))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda -(t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T -T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 -c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T -T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 -c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda -(x4: B).(\lambda (x5: T).(\lambda (x6: T).(\lambda (x7: T).(\lambda (x8: -T).(\lambda (x9: T).(\lambda (H15: (not (eq B x4 Abst))).(\lambda (H16: (pr3 -c x3 (THead (Bind x4) x5 x6))).(\lambda (H17: (pr3 c (THead (Bind x4) x9 -(THead (Flat Appl) (lift (S O) O x8) x7)) t4)).(\lambda (H18: (pr3 c x2 -x8)).(\lambda (H19: (pr3 c x5 x9)).(\lambda (H20: (pr3 (CHead c (Bind x4) x9) -x6 x7)).(or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 -(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind -Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T -T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 -c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (ex6_6_intro -B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(pr3 c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c -(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 -y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))) -x4 x5 x6 x7 x8 x9 H15 (pr3_sing c x3 x1 H10 (THead (Bind x4) x5 x6) H16) H17 -(pr3_sing c x2 x0 H9 x8 H18) H19 H20)))))))))))))) H14)) H13))))))))) H7)) -(\lambda (H7: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind -Abbr) u2 t5)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c x0 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) -z1 t5))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (_: T).(eq T x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind -Abbr) u2 t5)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c x0 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) -z1 t5))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 -(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind -Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T -T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 -c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda -(x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H8: (eq -T x1 (THead (Bind Abst) x2 x3))).(\lambda (H9: (eq T t2 (THead (Bind Abbr) x4 -x5))).(\lambda (H10: (pr2 c x0 x4)).(\lambda (H11: ((\forall (b: B).(\forall -(u: T).(pr2 (CHead c (Bind b) u) x3 x5))))).(eq_ind_r T (THead (Bind Abst) x2 -x3) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T -t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 -u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c t t5)))) (ex4_4 T T T T -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c -(THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c t (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T -T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 -c t (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))) (let H12 -\def (eq_ind T t2 (\lambda (t: T).(\forall (x6: T).(\forall (x7: T).((eq T t -(THead (Flat Appl) x6 x7)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: -T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c x6 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x7 t5)))) (ex4_4 T -T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 -c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (_: T).(pr3 c x6 u2))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c x7 (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T -T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 -c x7 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c x6 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))))) H3 -(THead (Bind Abbr) x4 x5) H9) in (let H13 \def (eq_ind T t2 (\lambda (t: -T).(pr3 c t t4)) H2 (THead (Bind Abbr) x4 x5) H9) in (or3_intro1 (ex3_2 T T -(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t5: T).(pr3 c (THead (Bind Abst) x2 x3) t5)))) (ex4_4 T T T T (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind -Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) x2 x3) (THead -(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) -(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) x2 x3) (THead (Bind b) y1 -z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: -T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat -Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 -u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (ex4_4_intro T T T T -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c -(THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) x2 x3) (THead -(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -z1 t5))))))) x2 x3 x4 x5 H13 (pr3_pr2 c x0 x4 H10) (pr3_refl c (THead (Bind -Abst) x2 x3)) (\lambda (b: B).(\lambda (u: T).(pr3_pr2 (CHead c (Bind b) u) -x3 x5 (H11 b u)))))))) x1 H8))))))))) H7)) (\lambda (H7: (ex6_6 B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T x1 -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind -B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T -t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 -y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) -(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) -u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: -T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda -(_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) -t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 -c x0 u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 -(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead -(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) -y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda -(x2: B).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (x6: -T).(\lambda (x7: T).(\lambda (H8: (not (eq B x2 Abst))).(\lambda (H9: (eq T -x1 (THead (Bind x2) x3 x4))).(\lambda (H10: (eq T t2 (THead (Bind x2) x7 -(THead (Flat Appl) (lift (S O) O x6) x5)))).(\lambda (H11: (pr2 c x0 -x6)).(\lambda (H12: (pr2 c x3 x7)).(\lambda (H13: (pr2 (CHead c (Bind x2) x7) -x4 x5)).(eq_ind_r T (THead (Bind x2) x3 x4) (\lambda (t: T).(or3 (ex3_2 T T -(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t5: T).(pr3 c t t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 -u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(pr3 c t (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 -(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c t (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead -(Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 -u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))) (let H14 \def (eq_ind T t2 -(\lambda (t: T).(\forall (x8: T).(\forall (x9: T).((eq T t (THead (Flat Appl) -x8 x9)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead -(Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x8 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr3 c x9 t5)))) (ex4_4 T T T T (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind -Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr3 c x8 u2))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c x9 (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T -T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 -c x9 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c x8 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))))) H3 -(THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) O x6) x5)) H10) in (let -H15 \def (eq_ind T t2 (\lambda (t: T).(pr3 c t t4)) H2 (THead (Bind x2) x7 -(THead (Flat Appl) (lift (S O) O x6) x5)) H10) in (or3_intro2 (ex3_2 T T -(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t5: T).(pr3 c (THead (Bind x2) x3 x4) t5)))) (ex4_4 T T T T (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind -Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind x2) x3 x4) (THead -(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) -(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(pr3 c (THead (Bind x2) x3 x4) (THead (Bind b) y1 -z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: -T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat -Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 -u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c -(THead (Bind x2) x3 x4) (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) -t4))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) -y2) z1 z2))))))) x2 x3 x4 x5 x6 x7 H8 (pr3_refl c (THead (Bind x2) x3 x4)) -H15 (pr3_pr2 c x0 x6 H11) (pr3_pr2 c x3 x7 H12) (pr3_pr2 (CHead c (Bind x2) -x7) x4 x5 H13))))) x1 H9))))))))))))) H7)) H6)))))))))))) y x H0))))) H))))). -(* COMMENTS -Initial nodes: 12691 -END *) - -theorem pr3_gen_bind: - \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (u1: -T).(\forall (t1: T).(\forall (x: T).((pr3 c (THead (Bind b) u1 t1) x) \to (or -(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind b) u2 -t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: -T).(\lambda (t2: T).(pr3 (CHead c (Bind b) u1) t1 t2)))) (pr3 (CHead c (Bind -b) u1) t1 (lift (S O) O x))))))))) -\def - \lambda (b: B).(B_ind (\lambda (b0: B).((not (eq B b0 Abst)) \to (\forall -(c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c (THead (Bind -b0) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x -(THead (Bind b0) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) -(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind b0) u1) t1 t2)))) (pr3 -(CHead c (Bind b0) u1) t1 (lift (S O) O x)))))))))) (\lambda (_: (not (eq B -Abbr Abst))).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: -T).(\lambda (H0: (pr3 c (THead (Bind Abbr) u1 t1) x)).(let H1 \def -(pr3_gen_abbr c u1 t1 x H0) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda -(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) -u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O x)) (or (ex3_2 T -T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda -(t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) -t1 (lift (S O) O x))) (\lambda (H2: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 -c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 -t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind -Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: -T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2))) (or (ex3_2 T T -(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda -(t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) -t1 (lift (S O) O x))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H3: (eq T x -(THead (Bind Abbr) x0 x1))).(\lambda (H4: (pr3 c u1 x0)).(\lambda (H5: (pr3 -(CHead c (Bind Abbr) u1) t1 x1)).(or_introl (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 -(CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S -O) O x)) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead -(Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) -(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2))) x0 x1 -H3 H4 H5))))))) H2)) (\lambda (H2: (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S -O) O x))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x -(THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) -(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2)))) (pr3 -(CHead c (Bind Abbr) u1) t1 (lift (S O) O x)) H2)) H1)))))))) (\lambda (H: -(not (eq B Abst Abst))).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: -T).(\lambda (x: T).(\lambda (_: (pr3 c (THead (Bind Abst) u1 t1) x)).(let H1 -\def (match (H (refl_equal B Abst)) in False return (\lambda (_: False).(or -(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 -t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: -T).(\lambda (t2: T).(pr3 (CHead c (Bind Abst) u1) t1 t2)))) (pr3 (CHead c -(Bind Abst) u1) t1 (lift (S O) O x)))) with []) in H1))))))) (\lambda (_: -(not (eq B Void Abst))).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: -T).(\lambda (x: T).(\lambda (H0: (pr3 c (THead (Bind Void) u1 t1) x)).(let H1 -\def (pr3_gen_void c u1 t1 x H0) in (or_ind (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall -(b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t1 t2)))))) (pr3 (CHead c -(Bind Void) u1) t1 (lift (S O) O x)) (or (ex3_2 T T (\lambda (u2: T).(\lambda -(t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) -u1) t1 t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O x))) (\lambda -(H2: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) -u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: -T).(\lambda (t2: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) -u) t1 t2))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T x -(THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) -(\lambda (_: T).(\lambda (t2: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead -c (Bind b0) u) t1 t2))))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 -c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) u1) t1 -t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O x))) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H3: (eq T x (THead (Bind Void) x0 -x1))).(\lambda (H4: (pr3 c u1 x0)).(\lambda (H5: ((\forall (b0: B).(\forall -(u: T).(pr3 (CHead c (Bind b0) u) t1 x1))))).(or_introl (ex3_2 T T (\lambda -(u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 -(CHead c (Bind Void) u1) t1 t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S -O) O x)) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead -(Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) -(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) u1) t1 t2))) x0 x1 -H3 H4 (H5 Void u1)))))))) H2)) (\lambda (H2: (pr3 (CHead c (Bind Void) u1) t1 -(lift (S O) O x))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 -c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) u1) t1 -t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O x)) H2)) H1)))))))) b). -(* COMMENTS -Initial nodes: 1721 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/iso.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/iso.ma deleted file mode 100644 index c7b2fb7aa..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/iso.ma +++ /dev/null @@ -1,1155 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr3/fwd.ma". - -include "Basic-1/iso/props.ma". - -include "Basic-1/tlist/props.ma". - -theorem pr3_iso_appls_abbr: - \forall (c: C).(\forall (d: C).(\forall (w: T).(\forall (i: nat).((getl i c -(CHead d (Bind Abbr) w)) \to (\forall (vs: TList).(let u1 \def (THeads (Flat -Appl) vs (TLRef i)) in (\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to -(\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) vs (lift (S i) O w)) -u2)))))))))) -\def - \lambda (c: C).(\lambda (d: C).(\lambda (w: T).(\lambda (i: nat).(\lambda -(H: (getl i c (CHead d (Bind Abbr) w))).(\lambda (vs: TList).(TList_ind -(\lambda (t: TList).(let u1 \def (THeads (Flat Appl) t (TLRef i)) in (\forall -(u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to -(pr3 c (THeads (Flat Appl) t (lift (S i) O w)) u2)))))) (\lambda (u2: -T).(\lambda (H0: (pr3 c (TLRef i) u2)).(\lambda (H1: (((iso (TLRef i) u2) \to -(\forall (P: Prop).P)))).(let H2 \def (pr3_gen_lref c u2 i H0) in (or_ind (eq -T u2 (TLRef i)) (ex3_3 C T T (\lambda (d0: C).(\lambda (u: T).(\lambda (_: -T).(getl i c (CHead d0 (Bind Abbr) u))))) (\lambda (d0: C).(\lambda (u: -T).(\lambda (v: T).(pr3 d0 u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(v: T).(eq T u2 (lift (S i) O v)))))) (pr3 c (lift (S i) O w) u2) (\lambda -(H3: (eq T u2 (TLRef i))).(let H4 \def (eq_ind T u2 (\lambda (t: T).((iso -(TLRef i) t) \to (\forall (P: Prop).P))) H1 (TLRef i) H3) in (eq_ind_r T -(TLRef i) (\lambda (t: T).(pr3 c (lift (S i) O w) t)) (H4 (iso_refl (TLRef -i)) (pr3 c (lift (S i) O w) (TLRef i))) u2 H3))) (\lambda (H3: (ex3_3 C T T -(\lambda (d0: C).(\lambda (u: T).(\lambda (_: T).(getl i c (CHead d0 (Bind -Abbr) u))))) (\lambda (d0: C).(\lambda (u: T).(\lambda (v: T).(pr3 d0 u v)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T u2 (lift (S i) O -v))))))).(ex3_3_ind C T T (\lambda (d0: C).(\lambda (u: T).(\lambda (_: -T).(getl i c (CHead d0 (Bind Abbr) u))))) (\lambda (d0: C).(\lambda (u: -T).(\lambda (v: T).(pr3 d0 u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(v: T).(eq T u2 (lift (S i) O v))))) (pr3 c (lift (S i) O w) u2) (\lambda -(x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H4: (getl i c (CHead x0 -(Bind Abbr) x1))).(\lambda (H5: (pr3 x0 x1 x2)).(\lambda (H6: (eq T u2 (lift -(S i) O x2))).(let H7 \def (eq_ind T u2 (\lambda (t: T).((iso (TLRef i) t) -\to (\forall (P: Prop).P))) H1 (lift (S i) O x2) H6) in (eq_ind_r T (lift (S -i) O x2) (\lambda (t: T).(pr3 c (lift (S i) O w) t)) (let H8 \def (eq_ind C -(CHead d (Bind Abbr) w) (\lambda (c0: C).(getl i c c0)) H (CHead x0 (Bind -Abbr) x1) (getl_mono c (CHead d (Bind Abbr) w) i H (CHead x0 (Bind Abbr) x1) -H4)) in (let H9 \def (f_equal C C (\lambda (e: C).(match e in C return -(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow -c0])) (CHead d (Bind Abbr) w) (CHead x0 (Bind Abbr) x1) (getl_mono c (CHead d -(Bind Abbr) w) i H (CHead x0 (Bind Abbr) x1) H4)) in ((let H10 \def (f_equal -C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) -\Rightarrow w | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abbr) w) (CHead -x0 (Bind Abbr) x1) (getl_mono c (CHead d (Bind Abbr) w) i H (CHead x0 (Bind -Abbr) x1) H4)) in (\lambda (H11: (eq C d x0)).(let H12 \def (eq_ind_r T x1 -(\lambda (t: T).(getl i c (CHead x0 (Bind Abbr) t))) H8 w H10) in (let H13 -\def (eq_ind_r T x1 (\lambda (t: T).(pr3 x0 t x2)) H5 w H10) in (let H14 \def -(eq_ind_r C x0 (\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr) w))) H12 d -H11) in (let H15 \def (eq_ind_r C x0 (\lambda (c0: C).(pr3 c0 w x2)) H13 d -H11) in (pr3_lift c d (S i) O (getl_drop Abbr c d w i H14) w x2 H15))))))) -H9))) u2 H6)))))))) H3)) H2))))) (\lambda (t: T).(\lambda (t0: -TList).(\lambda (H0: ((\forall (u2: T).((pr3 c (THeads (Flat Appl) t0 (TLRef -i)) u2) \to ((((iso (THeads (Flat Appl) t0 (TLRef i)) u2) \to (\forall (P: -Prop).P))) \to (pr3 c (THeads (Flat Appl) t0 (lift (S i) O w)) -u2)))))).(\lambda (u2: T).(\lambda (H1: (pr3 c (THead (Flat Appl) t (THeads -(Flat Appl) t0 (TLRef i))) u2)).(\lambda (H2: (((iso (THead (Flat Appl) t -(THeads (Flat Appl) t0 (TLRef i))) u2) \to (\forall (P: Prop).P)))).(let H3 -\def (pr3_gen_appl c t (THeads (Flat Appl) t0 (TLRef i)) u2 H1) in (or3_ind -(ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 -t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t u3))) (\lambda (_: -T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) t2)))) (ex4_4 T -T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 -c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u3: T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) -(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead -(Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t -u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (pr3 c (THead (Flat Appl) t -(THeads (Flat Appl) t0 (lift (S i) O w))) u2) (\lambda (H4: (ex3_2 T T -(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) -(\lambda (u3: T).(\lambda (_: T).(pr3 c t u3))) (\lambda (_: T).(\lambda (t2: -T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) t2))))).(ex3_2_ind T T (\lambda -(u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: -T).(\lambda (_: T).(pr3 c t u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c -(THeads (Flat Appl) t0 (TLRef i)) t2))) (pr3 c (THead (Flat Appl) t (THeads -(Flat Appl) t0 (lift (S i) O w))) u2) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H5: (eq T u2 (THead (Flat Appl) x0 x1))).(\lambda (_: (pr3 c t -x0)).(\lambda (_: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) x1)).(let H8 \def -(eq_ind T u2 (\lambda (t1: T).((iso (THead (Flat Appl) t (THeads (Flat Appl) -t0 (TLRef i))) t1) \to (\forall (P: Prop).P))) H2 (THead (Flat Appl) x0 x1) -H5) in (eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda (t1: T).(pr3 c (THead -(Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O w))) t1)) (H8 (iso_head t -x0 (THeads (Flat Appl) t0 (TLRef i)) x1 (Flat Appl)) (pr3 c (THead (Flat -Appl) t (THeads (Flat Appl) t0 (lift (S i) O w))) (THead (Flat Appl) x0 x1))) -u2 H5))))))) H4)) (\lambda (H4: (ex4_4 T T T T (\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t -u3))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2))))))))).(ex4_4_ind T -T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 -c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u3: T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -z1 t2))))))) (pr3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O -w))) u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: -T).(\lambda (H5: (pr3 c (THead (Bind Abbr) x2 x3) u2)).(\lambda (H6: (pr3 c t -x2)).(\lambda (H7: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind -Abst) x0 x1))).(\lambda (H8: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c -(Bind b) u) x1 x3))))).(pr3_t (THead (Bind Abbr) t x1) (THead (Flat Appl) t -(THeads (Flat Appl) t0 (lift (S i) O w))) c (pr3_t (THead (Flat Appl) t -(THead (Bind Abst) x0 x1)) (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift -(S i) O w))) c (pr3_thin_dx c (THeads (Flat Appl) t0 (lift (S i) O w)) (THead -(Bind Abst) x0 x1) (H0 (THead (Bind Abst) x0 x1) H7 (\lambda (H9: (iso -(THeads (Flat Appl) t0 (TLRef i)) (THead (Bind Abst) x0 x1))).(\lambda (P: -Prop).(iso_flats_lref_bind_false Appl Abst i x0 x1 t0 H9 P)))) t Appl) (THead -(Bind Abbr) t x1) (pr3_pr2 c (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) -(THead (Bind Abbr) t x1) (pr2_free c (THead (Flat Appl) t (THead (Bind Abst) -x0 x1)) (THead (Bind Abbr) t x1) (pr0_beta x0 t t (pr0_refl t) x1 x1 -(pr0_refl x1))))) u2 (pr3_t (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) t -x1) c (pr3_head_12 c t x2 H6 (Bind Abbr) x1 x3 (H8 Abbr x2)) u2 H5)))))))))) -H4)) (\lambda (H4: (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B -b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) -y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda -(_: T).(pr3 c t u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind -B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind b) y1 z1)))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda -(u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift -(S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t u3))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 -(CHead c (Bind b) y2) z1 z2))))))) (pr3 c (THead (Flat Appl) t (THeads (Flat -Appl) t0 (lift (S i) O w))) u2) (\lambda (x0: B).(\lambda (x1: T).(\lambda -(x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H5: (not -(eq B x0 Abst))).(\lambda (H6: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) -(THead (Bind x0) x1 x2))).(\lambda (H7: (pr3 c (THead (Bind x0) x5 (THead -(Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda (H8: (pr3 c t x4)).(\lambda -(H9: (pr3 c x1 x5)).(\lambda (H10: (pr3 (CHead c (Bind x0) x5) x2 x3)).(pr3_t -(THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) (THead (Flat -Appl) t (THeads (Flat Appl) t0 (lift (S i) O w))) c (pr3_t (THead (Bind x0) -x1 (THead (Flat Appl) (lift (S O) O t) x2)) (THead (Flat Appl) t (THeads -(Flat Appl) t0 (lift (S i) O w))) c (pr3_t (THead (Flat Appl) t (THead (Bind -x0) x1 x2)) (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O w))) c -(pr3_thin_dx c (THeads (Flat Appl) t0 (lift (S i) O w)) (THead (Bind x0) x1 -x2) (H0 (THead (Bind x0) x1 x2) H6 (\lambda (H11: (iso (THeads (Flat Appl) t0 -(TLRef i)) (THead (Bind x0) x1 x2))).(\lambda (P: -Prop).(iso_flats_lref_bind_false Appl x0 i x1 x2 t0 H11 P)))) t Appl) (THead -(Bind x0) x1 (THead (Flat Appl) (lift (S O) O t) x2)) (pr3_pr2 c (THead (Flat -Appl) t (THead (Bind x0) x1 x2)) (THead (Bind x0) x1 (THead (Flat Appl) (lift -(S O) O t) x2)) (pr2_free c (THead (Flat Appl) t (THead (Bind x0) x1 x2)) -(THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O t) x2)) (pr0_upsilon x0 -H5 t t (pr0_refl t) x1 x1 (pr0_refl x1) x2 x2 (pr0_refl x2))))) (THead (Bind -x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) (pr3_head_12 c x1 x1 -(pr3_refl c x1) (Bind x0) (THead (Flat Appl) (lift (S O) O t) x2) (THead -(Flat Appl) (lift (S O) O x4) x2) (pr3_head_12 (CHead c (Bind x0) x1) (lift -(S O) O t) (lift (S O) O x4) (pr3_lift (CHead c (Bind x0) x1) c (S O) O -(drop_drop (Bind x0) O c c (drop_refl c) x1) t x4 H8) (Flat Appl) x2 x2 -(pr3_refl (CHead (CHead c (Bind x0) x1) (Flat Appl) (lift (S O) O x4)) x2)))) -u2 (pr3_t (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) -(THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) c (pr3_head_12 -c x1 x5 H9 (Bind x0) (THead (Flat Appl) (lift (S O) O x4) x2) (THead (Flat -Appl) (lift (S O) O x4) x3) (pr3_thin_dx (CHead c (Bind x0) x5) x2 x3 H10 -(lift (S O) O x4) Appl)) u2 H7)))))))))))))) H4)) H3)))))))) vs)))))). -(* COMMENTS -Initial nodes: 3759 -END *) - -theorem pr3_iso_appls_cast: - \forall (c: C).(\forall (v: T).(\forall (t: T).(\forall (vs: TList).(let u1 -\def (THeads (Flat Appl) vs (THead (Flat Cast) v t)) in (\forall (u2: -T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to (pr3 c -(THeads (Flat Appl) vs t) u2)))))))) -\def - \lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (vs: -TList).(TList_ind (\lambda (t0: TList).(let u1 \def (THeads (Flat Appl) t0 -(THead (Flat Cast) v t)) in (\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 -u2) \to (\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) t0 t) u2)))))) -(\lambda (u2: T).(\lambda (H: (pr3 c (THead (Flat Cast) v t) u2)).(\lambda -(H0: (((iso (THead (Flat Cast) v t) u2) \to (\forall (P: Prop).P)))).(let H1 -\def (pr3_gen_cast c v t u2 H) in (or_ind (ex3_2 T T (\lambda (u3: -T).(\lambda (t2: T).(eq T u2 (THead (Flat Cast) u3 t2)))) (\lambda (u3: -T).(\lambda (_: T).(pr3 c v u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t -t2)))) (pr3 c t u2) (pr3 c t u2) (\lambda (H2: (ex3_2 T T (\lambda (u3: -T).(\lambda (t2: T).(eq T u2 (THead (Flat Cast) u3 t2)))) (\lambda (u3: -T).(\lambda (_: T).(pr3 c v u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t -t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead -(Flat Cast) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c v u3))) -(\lambda (_: T).(\lambda (t2: T).(pr3 c t t2))) (pr3 c t u2) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H3: (eq T u2 (THead (Flat Cast) x0 -x1))).(\lambda (_: (pr3 c v x0)).(\lambda (_: (pr3 c t x1)).(let H6 \def -(eq_ind T u2 (\lambda (t0: T).((iso (THead (Flat Cast) v t) t0) \to (\forall -(P: Prop).P))) H0 (THead (Flat Cast) x0 x1) H3) in (eq_ind_r T (THead (Flat -Cast) x0 x1) (\lambda (t0: T).(pr3 c t t0)) (H6 (iso_head v x0 t x1 (Flat -Cast)) (pr3 c t (THead (Flat Cast) x0 x1))) u2 H3))))))) H2)) (\lambda (H2: -(pr3 c t u2)).H2) H1))))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H: -((\forall (u2: T).((pr3 c (THeads (Flat Appl) t1 (THead (Flat Cast) v t)) u2) -\to ((((iso (THeads (Flat Appl) t1 (THead (Flat Cast) v t)) u2) \to (\forall -(P: Prop).P))) \to (pr3 c (THeads (Flat Appl) t1 t) u2)))))).(\lambda (u2: -T).(\lambda (H0: (pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead -(Flat Cast) v t))) u2)).(\lambda (H1: (((iso (THead (Flat Appl) t0 (THeads -(Flat Appl) t1 (THead (Flat Cast) v t))) u2) \to (\forall (P: -Prop).P)))).(let H2 \def (pr3_gen_appl c t0 (THeads (Flat Appl) t1 (THead -(Flat Cast) v t)) u2 H0) in (or3_ind (ex3_2 T T (\lambda (u3: T).(\lambda -(t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: -T).(pr3 c t0 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat -Appl) t1 (THead (Flat Cast) v t)) t2)))) (ex4_4 T T T T (\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind -Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: -T).(\lambda (_: T).(pr3 c t0 u3))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat -Cast) v t)) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 -(CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat -Appl) t1 (THead (Flat Cast) v t)) (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u3: T).(\lambda -(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u3) z2)) -u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) -y2) z1 z2)))))))) (pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 t)) u2) -(\lambda (H3: (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead -(Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3))) -(\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat -Cast) v t)) t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 -(THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3))) -(\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat -Cast) v t)) t2))) (pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 t)) u2) -(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T u2 (THead (Flat Appl) -x0 x1))).(\lambda (_: (pr3 c t0 x0)).(\lambda (_: (pr3 c (THeads (Flat Appl) -t1 (THead (Flat Cast) v t)) x1)).(let H7 \def (eq_ind T u2 (\lambda (t2: -T).((iso (THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Flat Cast) v -t))) t2) \to (\forall (P: Prop).P))) H1 (THead (Flat Appl) x0 x1) H4) in -(eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda (t2: T).(pr3 c (THead (Flat -Appl) t0 (THeads (Flat Appl) t1 t)) t2)) (H7 (iso_head t0 x0 (THeads (Flat -Appl) t1 (THead (Flat Cast) v t)) x1 (Flat Appl)) (pr3 c (THead (Flat Appl) -t0 (THeads (Flat Appl) t1 t)) (THead (Flat Appl) x0 x1))) u2 H4))))))) H3)) -(\lambda (H3: (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: -T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3))))) -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c -(THeads (Flat Appl) t1 (THead (Flat Cast) v t)) (THead (Bind Abst) y1 -z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: -T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 -t2))))))))).(ex4_4_ind T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: -T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3))))) -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c -(THeads (Flat Appl) t1 (THead (Flat Cast) v t)) (THead (Bind Abst) y1 -z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: -T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2))))))) -(pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 t)) u2) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H4: (pr3 c -(THead (Bind Abbr) x2 x3) u2)).(\lambda (H5: (pr3 c t0 x2)).(\lambda (H6: -(pr3 c (THeads (Flat Appl) t1 (THead (Flat Cast) v t)) (THead (Bind Abst) x0 -x1))).(\lambda (H7: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) -u) x1 x3))))).(pr3_t (THead (Bind Abbr) t0 x1) (THead (Flat Appl) t0 (THeads -(Flat Appl) t1 t)) c (pr3_t (THead (Flat Appl) t0 (THead (Bind Abst) x0 x1)) -(THead (Flat Appl) t0 (THeads (Flat Appl) t1 t)) c (pr3_thin_dx c (THeads -(Flat Appl) t1 t) (THead (Bind Abst) x0 x1) (H (THead (Bind Abst) x0 x1) H6 -(\lambda (H8: (iso (THeads (Flat Appl) t1 (THead (Flat Cast) v t)) (THead -(Bind Abst) x0 x1))).(\lambda (P: Prop).(iso_flats_flat_bind_false Appl Cast -Abst x0 v x1 t t1 H8 P)))) t0 Appl) (THead (Bind Abbr) t0 x1) (pr3_pr2 c -(THead (Flat Appl) t0 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) t0 x1) -(pr2_free c (THead (Flat Appl) t0 (THead (Bind Abst) x0 x1)) (THead (Bind -Abbr) t0 x1) (pr0_beta x0 t0 t0 (pr0_refl t0) x1 x1 (pr0_refl x1))))) u2 -(pr3_t (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) t0 x1) c (pr3_head_12 c -t0 x2 H5 (Bind Abbr) x1 x3 (H7 Abbr x2)) u2 H4)))))))))) H3)) (\lambda (H3: -(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat Cast) v t)) (THead (Bind b) -y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: -T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat -Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t0 -u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c -(THeads (Flat Appl) t1 (THead (Flat Cast) v t)) (THead (Bind b) y1 z1)))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda -(u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift -(S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 -(CHead c (Bind b) y2) z1 z2))))))) (pr3 c (THead (Flat Appl) t0 (THeads (Flat -Appl) t1 t)) u2) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda -(x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H4: (not (eq B x0 -Abst))).(\lambda (H5: (pr3 c (THeads (Flat Appl) t1 (THead (Flat Cast) v t)) -(THead (Bind x0) x1 x2))).(\lambda (H6: (pr3 c (THead (Bind x0) x5 (THead -(Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda (H7: (pr3 c t0 x4)).(\lambda -(H8: (pr3 c x1 x5)).(\lambda (H9: (pr3 (CHead c (Bind x0) x5) x2 x3)).(pr3_t -(THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) (THead (Flat -Appl) t0 (THeads (Flat Appl) t1 t)) c (pr3_t (THead (Bind x0) x1 (THead (Flat -Appl) (lift (S O) O t0) x2)) (THead (Flat Appl) t0 (THeads (Flat Appl) t1 t)) -c (pr3_t (THead (Flat Appl) t0 (THead (Bind x0) x1 x2)) (THead (Flat Appl) t0 -(THeads (Flat Appl) t1 t)) c (pr3_thin_dx c (THeads (Flat Appl) t1 t) (THead -(Bind x0) x1 x2) (H (THead (Bind x0) x1 x2) H5 (\lambda (H10: (iso (THeads -(Flat Appl) t1 (THead (Flat Cast) v t)) (THead (Bind x0) x1 x2))).(\lambda -(P: Prop).(iso_flats_flat_bind_false Appl Cast x0 x1 v x2 t t1 H10 P)))) t0 -Appl) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O t0) x2)) (pr3_pr2 -c (THead (Flat Appl) t0 (THead (Bind x0) x1 x2)) (THead (Bind x0) x1 (THead -(Flat Appl) (lift (S O) O t0) x2)) (pr2_free c (THead (Flat Appl) t0 (THead -(Bind x0) x1 x2)) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O t0) -x2)) (pr0_upsilon x0 H4 t0 t0 (pr0_refl t0) x1 x1 (pr0_refl x1) x2 x2 -(pr0_refl x2))))) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) -x2)) (pr3_head_12 c x1 x1 (pr3_refl c x1) (Bind x0) (THead (Flat Appl) (lift -(S O) O t0) x2) (THead (Flat Appl) (lift (S O) O x4) x2) (pr3_head_12 (CHead -c (Bind x0) x1) (lift (S O) O t0) (lift (S O) O x4) (pr3_lift (CHead c (Bind -x0) x1) c (S O) O (drop_drop (Bind x0) O c c (drop_refl c) x1) t0 x4 H7) -(Flat Appl) x2 x2 (pr3_refl (CHead (CHead c (Bind x0) x1) (Flat Appl) (lift -(S O) O x4)) x2)))) u2 (pr3_t (THead (Bind x0) x5 (THead (Flat Appl) (lift (S -O) O x4) x3)) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) c -(pr3_head_12 c x1 x5 H8 (Bind x0) (THead (Flat Appl) (lift (S O) O x4) x2) -(THead (Flat Appl) (lift (S O) O x4) x3) (pr3_thin_dx (CHead c (Bind x0) x5) -x2 x3 H9 (lift (S O) O x4) Appl)) u2 H6)))))))))))))) H3)) H2)))))))) vs)))). -(* COMMENTS -Initial nodes: 3297 -END *) - -theorem pr3_iso_appl_bind: - \forall (b: B).((not (eq B b Abst)) \to (\forall (v1: T).(\forall (v2: -T).(\forall (t: T).(let u1 \def (THead (Flat Appl) v1 (THead (Bind b) v2 t)) -in (\forall (c: C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to -(\forall (P: Prop).P))) \to (pr3 c (THead (Bind b) v2 (THead (Flat Appl) -(lift (S O) O v1) t)) u2)))))))))) -\def - \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (v1: T).(\lambda -(v2: T).(\lambda (t: T).(\lambda (c: C).(\lambda (u2: T).(\lambda (H0: (pr3 c -(THead (Flat Appl) v1 (THead (Bind b) v2 t)) u2)).(\lambda (H1: (((iso (THead -(Flat Appl) v1 (THead (Bind b) v2 t)) u2) \to (\forall (P: Prop).P)))).(let -H2 \def (pr3_gen_appl c v1 (THead (Bind b) v2 t) u2 H0) in (or3_ind (ex3_2 T -T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) -(\lambda (u3: T).(\lambda (_: T).(pr3 c v1 u3))) (\lambda (_: T).(\lambda -(t2: T).(pr3 c (THead (Bind b) v2 t) t2)))) (ex4_4 T T T T (\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind -Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: -T).(\lambda (_: T).(pr3 c v1 u3))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind b) v2 t) (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(t2: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) z1 -t2)))))))) (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) -(\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(pr3 c (THead (Bind b) v2 t) (THead (Bind b0) y1 -z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: -T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b0) y2 (THead (Flat -Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v1 -u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b0: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2)))))))) (pr3 c (THead (Bind b) v2 -(THead (Flat Appl) (lift (S O) O v1) t)) u2) (\lambda (H3: (ex3_2 T T -(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) -(\lambda (u3: T).(\lambda (_: T).(pr3 c v1 u3))) (\lambda (_: T).(\lambda -(t2: T).(pr3 c (THead (Bind b) v2 t) t2))))).(ex3_2_ind T T (\lambda (u3: -T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: -T).(\lambda (_: T).(pr3 c v1 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c -(THead (Bind b) v2 t) t2))) (pr3 c (THead (Bind b) v2 (THead (Flat Appl) -(lift (S O) O v1) t)) u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq -T u2 (THead (Flat Appl) x0 x1))).(\lambda (_: (pr3 c v1 x0)).(\lambda (_: -(pr3 c (THead (Bind b) v2 t) x1)).(let H7 \def (eq_ind T u2 (\lambda (t0: -T).((iso (THead (Flat Appl) v1 (THead (Bind b) v2 t)) t0) \to (\forall (P: -Prop).P))) H1 (THead (Flat Appl) x0 x1) H4) in (eq_ind_r T (THead (Flat Appl) -x0 x1) (\lambda (t0: T).(pr3 c (THead (Bind b) v2 (THead (Flat Appl) (lift (S -O) O v1) t)) t0)) (H7 (iso_head v1 x0 (THead (Bind b) v2 t) x1 (Flat Appl)) -(pr3 c (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) t)) (THead -(Flat Appl) x0 x1))) u2 H4))))))) H3)) (\lambda (H3: (ex4_4 T T T T (\lambda -(_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind -Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: -T).(\lambda (_: T).(pr3 c v1 u3))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind b) v2 t) (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(t2: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) z1 -t2))))))))).(ex4_4_ind T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: -T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v1 u3))))) -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c -(THead (Bind b) v2 t) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b0: B).(\forall (u: -T).(pr3 (CHead c (Bind b0) u) z1 t2))))))) (pr3 c (THead (Bind b) v2 (THead -(Flat Appl) (lift (S O) O v1) t)) u2) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H4: (pr3 c (THead (Bind Abbr) -x2 x3) u2)).(\lambda (H5: (pr3 c v1 x2)).(\lambda (H6: (pr3 c (THead (Bind b) -v2 t) (THead (Bind Abst) x0 x1))).(\lambda (H7: ((\forall (b0: B).(\forall -(u: T).(pr3 (CHead c (Bind b0) u) x1 x3))))).(pr3_t (THead (Bind Abbr) x2 x3) -(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) t)) c (let H_x \def -(pr3_gen_bind b H c v2 t (THead (Bind Abst) x0 x1) H6) in (let H8 \def H_x in -(or_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind Abst) -x0 x1) (THead (Bind b) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c v2 -u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind b) v2) t t2)))) -(pr3 (CHead c (Bind b) v2) t (lift (S O) O (THead (Bind Abst) x0 x1))) (pr3 c -(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) t)) (THead (Bind -Abbr) x2 x3)) (\lambda (H9: (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq -T (THead (Bind Abst) x0 x1) (THead (Bind b) u3 t2)))) (\lambda (u3: -T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 -(CHead c (Bind b) v2) t t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: -T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) u3 t2)))) (\lambda (u3: -T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 -(CHead c (Bind b) v2) t t2))) (pr3 c (THead (Bind b) v2 (THead (Flat Appl) -(lift (S O) O v1) t)) (THead (Bind Abbr) x2 x3)) (\lambda (x4: T).(\lambda -(x5: T).(\lambda (H10: (eq T (THead (Bind Abst) x0 x1) (THead (Bind b) x4 -x5))).(\lambda (H11: (pr3 c v2 x4)).(\lambda (H12: (pr3 (CHead c (Bind b) v2) -t x5)).(let H13 \def (f_equal T B (\lambda (e: T).(match e in T return -(\lambda (_: T).B) with [(TSort _) \Rightarrow Abst | (TLRef _) \Rightarrow -Abst | (THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).B) with -[(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abst])])) (THead (Bind Abst) -x0 x1) (THead (Bind b) x4 x5) H10) in ((let H14 \def (f_equal T T (\lambda -(e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 -| (TLRef _) \Rightarrow x0 | (THead _ t0 _) \Rightarrow t0])) (THead (Bind -Abst) x0 x1) (THead (Bind b) x4 x5) H10) in ((let H15 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow x1 | (TLRef _) \Rightarrow x1 | (THead _ _ t0) \Rightarrow t0])) -(THead (Bind Abst) x0 x1) (THead (Bind b) x4 x5) H10) in (\lambda (H16: (eq T -x0 x4)).(\lambda (H17: (eq B Abst b)).(let H18 \def (eq_ind_r T x5 (\lambda -(t0: T).(pr3 (CHead c (Bind b) v2) t t0)) H12 x1 H15) in (let H19 \def -(eq_ind_r T x4 (\lambda (t0: T).(pr3 c v2 t0)) H11 x0 H16) in (let H20 \def -(eq_ind_r B b (\lambda (b0: B).(pr3 (CHead c (Bind b0) v2) t x1)) H18 Abst -H17) in (let H21 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 Abst))) H -Abst H17) in (eq_ind B Abst (\lambda (b0: B).(pr3 c (THead (Bind b0) v2 -(THead (Flat Appl) (lift (S O) O v1) t)) (THead (Bind Abbr) x2 x3))) (let H22 -\def (match (H21 (refl_equal B Abst)) in False return (\lambda (_: -False).(pr3 c (THead (Bind Abst) v2 (THead (Flat Appl) (lift (S O) O v1) t)) -(THead (Bind Abbr) x2 x3))) with []) in H22) b H17)))))))) H14)) H13))))))) -H9)) (\lambda (H9: (pr3 (CHead c (Bind b) v2) t (lift (S O) O (THead (Bind -Abst) x0 x1)))).(pr3_t (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O -x2) (lift (S O) O (THead (Bind Abst) x0 x1)))) (THead (Bind b) v2 (THead -(Flat Appl) (lift (S O) O v1) t)) c (pr3_head_2 c v2 (THead (Flat Appl) (lift -(S O) O v1) t) (THead (Flat Appl) (lift (S O) O x2) (lift (S O) O (THead -(Bind Abst) x0 x1))) (Bind b) (pr3_flat (CHead c (Bind b) v2) (lift (S O) O -v1) (lift (S O) O x2) (pr3_lift (CHead c (Bind b) v2) c (S O) O (drop_drop -(Bind b) O c c (drop_refl c) v2) v1 x2 H5) t (lift (S O) O (THead (Bind Abst) -x0 x1)) H9 Appl)) (THead (Bind Abbr) x2 x3) (eq_ind T (lift (S O) O (THead -(Flat Appl) x2 (THead (Bind Abst) x0 x1))) (\lambda (t0: T).(pr3 c (THead -(Bind b) v2 t0) (THead (Bind Abbr) x2 x3))) (pr3_sing c (THead (Bind Abbr) x2 -x1) (THead (Bind b) v2 (lift (S O) O (THead (Flat Appl) x2 (THead (Bind Abst) -x0 x1)))) (pr2_free c (THead (Bind b) v2 (lift (S O) O (THead (Flat Appl) x2 -(THead (Bind Abst) x0 x1)))) (THead (Bind Abbr) x2 x1) (pr0_zeta b H (THead -(Flat Appl) x2 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) x2 x1) (pr0_beta -x0 x2 x2 (pr0_refl x2) x1 x1 (pr0_refl x1)) v2)) (THead (Bind Abbr) x2 x3) -(pr3_head_12 c x2 x2 (pr3_refl c x2) (Bind Abbr) x1 x3 (H7 Abbr x2))) (THead -(Flat Appl) (lift (S O) O x2) (lift (S O) O (THead (Bind Abst) x0 x1))) -(lift_flat Appl x2 (THead (Bind Abst) x0 x1) (S O) O)))) H8))) u2 H4))))))))) -H3)) (\lambda (H3: (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B -b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind b) v2 t) (THead -(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b0) -y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda -(_: T).(pr3 c v1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2))))))))).(ex6_6_ind -B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(pr3 c (THead (Bind b) v2 t) (THead (Bind b0) y1 z1)))))))) (\lambda -(b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u3: -T).(\lambda (y2: T).(pr3 c (THead (Bind b0) y2 (THead (Flat Appl) (lift (S O) -O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v1 u3))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 c y1 y2))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) -y2) z1 z2))))))) (pr3 c (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O -v1) t)) u2) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: -T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H4: (not (eq B x0 -Abst))).(\lambda (H5: (pr3 c (THead (Bind b) v2 t) (THead (Bind x0) x1 -x2))).(\lambda (H6: (pr3 c (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) -O x4) x3)) u2)).(\lambda (H7: (pr3 c v1 x4)).(\lambda (H8: (pr3 c x1 -x5)).(\lambda (H9: (pr3 (CHead c (Bind x0) x5) x2 x3)).(pr3_t (THead (Bind -x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) (THead (Bind b) v2 (THead -(Flat Appl) (lift (S O) O v1) t)) c (let H_x \def (pr3_gen_bind b H c v2 t -(THead (Bind x0) x1 x2) H5) in (let H10 \def H_x in (or_ind (ex3_2 T T -(\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind -b) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: -T).(\lambda (t2: T).(pr3 (CHead c (Bind b) v2) t t2)))) (pr3 (CHead c (Bind -b) v2) t (lift (S O) O (THead (Bind x0) x1 x2))) (pr3 c (THead (Bind b) v2 -(THead (Flat Appl) (lift (S O) O v1) t)) (THead (Bind x0) x5 (THead (Flat -Appl) (lift (S O) O x4) x3))) (\lambda (H11: (ex3_2 T T (\lambda (u3: -T).(\lambda (t2: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) u3 t2)))) -(\lambda (u3: T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: T).(\lambda -(t2: T).(pr3 (CHead c (Bind b) v2) t t2))))).(ex3_2_ind T T (\lambda (u3: -T).(\lambda (t2: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) u3 t2)))) -(\lambda (u3: T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: T).(\lambda -(t2: T).(pr3 (CHead c (Bind b) v2) t t2))) (pr3 c (THead (Bind b) v2 (THead -(Flat Appl) (lift (S O) O v1) t)) (THead (Bind x0) x5 (THead (Flat Appl) -(lift (S O) O x4) x3))) (\lambda (x6: T).(\lambda (x7: T).(\lambda (H12: (eq -T (THead (Bind x0) x1 x2) (THead (Bind b) x6 x7))).(\lambda (H13: (pr3 c v2 -x6)).(\lambda (H14: (pr3 (CHead c (Bind b) v2) t x7)).(let H15 \def (f_equal -T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with [(TSort _) -\Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead k _ _) \Rightarrow (match -k in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) -\Rightarrow x0])])) (THead (Bind x0) x1 x2) (THead (Bind b) x6 x7) H12) in -((let H16 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: -T).T) with [(TSort _) \Rightarrow x1 | (TLRef _) \Rightarrow x1 | (THead _ t0 -_) \Rightarrow t0])) (THead (Bind x0) x1 x2) (THead (Bind b) x6 x7) H12) in -((let H17 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: -T).T) with [(TSort _) \Rightarrow x2 | (TLRef _) \Rightarrow x2 | (THead _ _ -t0) \Rightarrow t0])) (THead (Bind x0) x1 x2) (THead (Bind b) x6 x7) H12) in -(\lambda (H18: (eq T x1 x6)).(\lambda (H19: (eq B x0 b)).(let H20 \def -(eq_ind_r T x7 (\lambda (t0: T).(pr3 (CHead c (Bind b) v2) t t0)) H14 x2 H17) -in (let H21 \def (eq_ind_r T x6 (\lambda (t0: T).(pr3 c v2 t0)) H13 x1 H18) -in (let H22 \def (eq_ind B x0 (\lambda (b0: B).(pr3 (CHead c (Bind b0) x5) x2 -x3)) H9 b H19) in (let H23 \def (eq_ind B x0 (\lambda (b0: B).(not (eq B b0 -Abst))) H4 b H19) in (eq_ind_r B b (\lambda (b0: B).(pr3 c (THead (Bind b) v2 -(THead (Flat Appl) (lift (S O) O v1) t)) (THead (Bind b0) x5 (THead (Flat -Appl) (lift (S O) O x4) x3)))) (pr3_head_21 c v2 x5 (pr3_t x1 v2 c H21 x5 H8) -(Bind b) (THead (Flat Appl) (lift (S O) O v1) t) (THead (Flat Appl) (lift (S -O) O x4) x3) (pr3_flat (CHead c (Bind b) v2) (lift (S O) O v1) (lift (S O) O -x4) (pr3_lift (CHead c (Bind b) v2) c (S O) O (drop_drop (Bind b) O c c -(drop_refl c) v2) v1 x4 H7) t x3 (pr3_t x2 t (CHead c (Bind b) v2) H20 x3 -(pr3_pr3_pr3_t c v2 x1 H21 x2 x3 (Bind b) (pr3_pr3_pr3_t c x1 x5 H8 x2 x3 -(Bind b) H22))) Appl)) x0 H19)))))))) H16)) H15))))))) H11)) (\lambda (H11: -(pr3 (CHead c (Bind b) v2) t (lift (S O) O (THead (Bind x0) x1 x2)))).(pr3_t -(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O x4) (lift (S O) O (THead -(Bind x0) x1 x2)))) (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) -t)) c (pr3_head_2 c v2 (THead (Flat Appl) (lift (S O) O v1) t) (THead (Flat -Appl) (lift (S O) O x4) (lift (S O) O (THead (Bind x0) x1 x2))) (Bind b) -(pr3_flat (CHead c (Bind b) v2) (lift (S O) O v1) (lift (S O) O x4) (pr3_lift -(CHead c (Bind b) v2) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) v2) -v1 x4 H7) t (lift (S O) O (THead (Bind x0) x1 x2)) H11 Appl)) (THead (Bind -x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) (eq_ind T (lift (S O) O -(THead (Flat Appl) x4 (THead (Bind x0) x1 x2))) (\lambda (t0: T).(pr3 c -(THead (Bind b) v2 t0) (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O -x4) x3)))) (pr3_sing c (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O -x4) x2)) (THead (Bind b) v2 (lift (S O) O (THead (Flat Appl) x4 (THead (Bind -x0) x1 x2)))) (pr2_free c (THead (Bind b) v2 (lift (S O) O (THead (Flat Appl) -x4 (THead (Bind x0) x1 x2)))) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S -O) O x4) x2)) (pr0_zeta b H (THead (Flat Appl) x4 (THead (Bind x0) x1 x2)) -(THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) (pr0_upsilon x0 -H4 x4 x4 (pr0_refl x4) x1 x1 (pr0_refl x1) x2 x2 (pr0_refl x2)) v2)) (THead -(Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) (pr3_head_12 c x1 x5 -H8 (Bind x0) (THead (Flat Appl) (lift (S O) O x4) x2) (THead (Flat Appl) -(lift (S O) O x4) x3) (pr3_thin_dx (CHead c (Bind x0) x5) x2 x3 H9 (lift (S -O) O x4) Appl))) (THead (Flat Appl) (lift (S O) O x4) (lift (S O) O (THead -(Bind x0) x1 x2))) (lift_flat Appl x4 (THead (Bind x0) x1 x2) (S O) O)))) -H10))) u2 H6))))))))))))) H3)) H2)))))))))). -(* COMMENTS -Initial nodes: 4805 -END *) - -theorem pr3_iso_appls_appl_bind: - \forall (b: B).((not (eq B b Abst)) \to (\forall (v: T).(\forall (u: -T).(\forall (t: T).(\forall (vs: TList).(let u1 \def (THeads (Flat Appl) vs -(THead (Flat Appl) v (THead (Bind b) u t))) in (\forall (c: C).(\forall (u2: -T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to (pr3 c -(THeads (Flat Appl) vs (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) -t))) u2))))))))))) -\def - \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (v: T).(\lambda -(u: T).(\lambda (t: T).(\lambda (vs: TList).(TList_ind (\lambda (t0: -TList).(let u1 \def (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind -b) u t))) in (\forall (c: C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 -u2) \to (\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) t0 (THead -(Bind b) u (THead (Flat Appl) (lift (S O) O v) t))) u2))))))) (\lambda (c: -C).(\lambda (u2: T).(\lambda (H0: (pr3 c (THead (Flat Appl) v (THead (Bind b) -u t)) u2)).(\lambda (H1: (((iso (THead (Flat Appl) v (THead (Bind b) u t)) -u2) \to (\forall (P: Prop).P)))).(pr3_iso_appl_bind b H v u t c u2 H0 H1))))) -(\lambda (t0: T).(\lambda (t1: TList).(\lambda (H0: ((\forall (c: C).(\forall -(u2: T).((pr3 c (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u -t))) u2) \to ((((iso (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind -b) u t))) u2) \to (\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) t1 -(THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t))) u2))))))).(\lambda -(c: C).(\lambda (u2: T).(\lambda (H1: (pr3 c (THead (Flat Appl) t0 (THeads -(Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t)))) u2)).(\lambda -(H2: (((iso (THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Flat Appl) v -(THead (Bind b) u t)))) u2) \to (\forall (P: Prop).P)))).(let H3 \def -(pr3_gen_appl c t0 (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind -b) u t))) u2 H1) in (or3_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq -T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t0 -u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t1 (THead -(Flat Appl) v (THead (Bind b) u t))) t2)))) (ex4_4 T T T T (\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind -Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: -T).(\lambda (_: T).(pr3 c t0 u3))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat -Appl) v (THead (Bind b) u t))) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b0: -B).(\forall (u0: T).(pr3 (CHead c (Bind b0) u0) z1 t2)))))))) (ex6_6 B T T T -T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 -c (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t))) (THead -(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b0) -y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda -(_: T).(pr3 c t0 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2)))))))) (pr3 c -(THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat -Appl) (lift (S O) O v) t)))) u2) (\lambda (H4: (ex3_2 T T (\lambda (u3: -T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: -T).(\lambda (_: T).(pr3 c t0 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c -(THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t))) -t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead -(Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3))) -(\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat -Appl) v (THead (Bind b) u t))) t2))) (pr3 c (THead (Flat Appl) t0 (THeads -(Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) -u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T u2 (THead (Flat -Appl) x0 x1))).(\lambda (_: (pr3 c t0 x0)).(\lambda (_: (pr3 c (THeads (Flat -Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t))) x1)).(let H8 \def -(eq_ind T u2 (\lambda (t2: T).((iso (THead (Flat Appl) t0 (THeads (Flat Appl) -t1 (THead (Flat Appl) v (THead (Bind b) u t)))) t2) \to (\forall (P: -Prop).P))) H2 (THead (Flat Appl) x0 x1) H5) in (eq_ind_r T (THead (Flat Appl) -x0 x1) (\lambda (t2: T).(pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 -(THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) t2)) (H8 -(iso_head t0 x0 (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u -t))) x1 (Flat Appl)) (pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 -(THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) (THead (Flat -Appl) x0 x1))) u2 H5))))))) H4)) (\lambda (H4: (ex4_4 T T T T (\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind -Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: -T).(\lambda (_: T).(pr3 c t0 u3))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat -Appl) v (THead (Bind b) u t))) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b0: -B).(\forall (u0: T).(pr3 (CHead c (Bind b0) u0) z1 t2))))))))).(ex4_4_ind T T -T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c -(THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u3: T).(\lambda (_: T).(pr3 c t0 u3))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat -Appl) v (THead (Bind b) u t))) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b0: -B).(\forall (u0: T).(pr3 (CHead c (Bind b0) u0) z1 t2))))))) (pr3 c (THead -(Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) -(lift (S O) O v) t)))) u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: -T).(\lambda (x3: T).(\lambda (H5: (pr3 c (THead (Bind Abbr) x2 x3) -u2)).(\lambda (H6: (pr3 c t0 x2)).(\lambda (H7: (pr3 c (THeads (Flat Appl) t1 -(THead (Flat Appl) v (THead (Bind b) u t))) (THead (Bind Abst) x0 -x1))).(\lambda (H8: ((\forall (b0: B).(\forall (u0: T).(pr3 (CHead c (Bind -b0) u0) x1 x3))))).(pr3_t (THead (Bind Abbr) t0 x1) (THead (Flat Appl) t0 -(THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) -t)))) c (pr3_t (THead (Flat Appl) t0 (THead (Bind Abst) x0 x1)) (THead (Flat -Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) (lift (S -O) O v) t)))) c (pr3_thin_dx c (THeads (Flat Appl) t1 (THead (Bind b) u -(THead (Flat Appl) (lift (S O) O v) t))) (THead (Bind Abst) x0 x1) (H0 c -(THead (Bind Abst) x0 x1) H7 (\lambda (H9: (iso (THeads (Flat Appl) t1 (THead -(Flat Appl) v (THead (Bind b) u t))) (THead (Bind Abst) x0 x1))).(\lambda (P: -Prop).(iso_flats_flat_bind_false Appl Appl Abst x0 v x1 (THead (Bind b) u t) -t1 H9 P)))) t0 Appl) (THead (Bind Abbr) t0 x1) (pr3_pr2 c (THead (Flat Appl) -t0 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) t0 x1) (pr2_free c (THead -(Flat Appl) t0 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) t0 x1) (pr0_beta -x0 t0 t0 (pr0_refl t0) x1 x1 (pr0_refl x1))))) u2 (pr3_t (THead (Bind Abbr) -x2 x3) (THead (Bind Abbr) t0 x1) c (pr3_head_12 c t0 x2 H6 (Bind Abbr) x1 x3 -(H8 Abbr x2)) u2 H5)))))))))) H4)) (\lambda (H4: (ex6_6 B T T T T T (\lambda -(b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c -(THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t))) (THead -(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b0) -y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda -(_: T).(pr3 c t0 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2))))))))).(ex6_6_ind -B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u -t))) (THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind -b0) y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda -(_: T).(pr3 c t0 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2))))))) (pr3 c -(THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat -Appl) (lift (S O) O v) t)))) u2) (\lambda (x0: B).(\lambda (x1: T).(\lambda -(x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H5: (not -(eq B x0 Abst))).(\lambda (H6: (pr3 c (THeads (Flat Appl) t1 (THead (Flat -Appl) v (THead (Bind b) u t))) (THead (Bind x0) x1 x2))).(\lambda (H7: (pr3 c -(THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda -(H8: (pr3 c t0 x4)).(\lambda (H9: (pr3 c x1 x5)).(\lambda (H10: (pr3 (CHead c -(Bind x0) x5) x2 x3)).(pr3_t (THead (Bind x0) x1 (THead (Flat Appl) (lift (S -O) O x4) x2)) (THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u -(THead (Flat Appl) (lift (S O) O v) t)))) c (pr3_t (THead (Bind x0) x1 (THead -(Flat Appl) (lift (S O) O t0) x2)) (THead (Flat Appl) t0 (THeads (Flat Appl) -t1 (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) c (pr3_t -(THead (Flat Appl) t0 (THead (Bind x0) x1 x2)) (THead (Flat Appl) t0 (THeads -(Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) c -(pr3_thin_dx c (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) -(lift (S O) O v) t))) (THead (Bind x0) x1 x2) (H0 c (THead (Bind x0) x1 x2) -H6 (\lambda (H11: (iso (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead -(Bind b) u t))) (THead (Bind x0) x1 x2))).(\lambda (P: -Prop).(iso_flats_flat_bind_false Appl Appl x0 x1 v x2 (THead (Bind b) u t) t1 -H11 P)))) t0 Appl) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O t0) -x2)) (pr3_pr2 c (THead (Flat Appl) t0 (THead (Bind x0) x1 x2)) (THead (Bind -x0) x1 (THead (Flat Appl) (lift (S O) O t0) x2)) (pr2_free c (THead (Flat -Appl) t0 (THead (Bind x0) x1 x2)) (THead (Bind x0) x1 (THead (Flat Appl) -(lift (S O) O t0) x2)) (pr0_upsilon x0 H5 t0 t0 (pr0_refl t0) x1 x1 (pr0_refl -x1) x2 x2 (pr0_refl x2))))) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S -O) O x4) x2)) (pr3_head_12 c x1 x1 (pr3_refl c x1) (Bind x0) (THead (Flat -Appl) (lift (S O) O t0) x2) (THead (Flat Appl) (lift (S O) O x4) x2) -(pr3_head_12 (CHead c (Bind x0) x1) (lift (S O) O t0) (lift (S O) O x4) -(pr3_lift (CHead c (Bind x0) x1) c (S O) O (drop_drop (Bind x0) O c c -(drop_refl c) x1) t0 x4 H8) (Flat Appl) x2 x2 (pr3_refl (CHead (CHead c (Bind -x0) x1) (Flat Appl) (lift (S O) O x4)) x2)))) u2 (pr3_t (THead (Bind x0) x5 -(THead (Flat Appl) (lift (S O) O x4) x3)) (THead (Bind x0) x1 (THead (Flat -Appl) (lift (S O) O x4) x2)) c (pr3_head_12 c x1 x5 H9 (Bind x0) (THead (Flat -Appl) (lift (S O) O x4) x2) (THead (Flat Appl) (lift (S O) O x4) x3) -(pr3_thin_dx (CHead c (Bind x0) x5) x2 x3 H10 (lift (S O) O x4) Appl)) u2 -H7)))))))))))))) H4)) H3))))))))) vs)))))). -(* COMMENTS -Initial nodes: 3571 -END *) - -theorem pr3_iso_appls_bind: - \forall (b: B).((not (eq B b Abst)) \to (\forall (vs: TList).(\forall (u: -T).(\forall (t: T).(let u1 \def (THeads (Flat Appl) vs (THead (Bind b) u t)) -in (\forall (c: C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to -(\forall (P: Prop).P))) \to (pr3 c (THead (Bind b) u (THeads (Flat Appl) -(lifts (S O) O vs) t)) u2)))))))))) -\def - \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (vs: -TList).(tlist_ind_rev (\lambda (t: TList).(\forall (u: T).(\forall (t0: -T).(let u1 \def (THeads (Flat Appl) t (THead (Bind b) u t0)) in (\forall (c: -C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: -Prop).P))) \to (pr3 c (THead (Bind b) u (THeads (Flat Appl) (lifts (S O) O t) -t0)) u2))))))))) (\lambda (u: T).(\lambda (t: T).(\lambda (c: C).(\lambda -(u2: T).(\lambda (H0: (pr3 c (THead (Bind b) u t) u2)).(\lambda (_: (((iso -(THead (Bind b) u t) u2) \to (\forall (P: Prop).P)))).H0)))))) (\lambda (ts: -TList).(\lambda (t: T).(\lambda (H0: ((\forall (u: T).(\forall (t0: -T).(\forall (c: C).(\forall (u2: T).((pr3 c (THeads (Flat Appl) ts (THead -(Bind b) u t0)) u2) \to ((((iso (THeads (Flat Appl) ts (THead (Bind b) u t0)) -u2) \to (\forall (P: Prop).P))) \to (pr3 c (THead (Bind b) u (THeads (Flat -Appl) (lifts (S O) O ts) t0)) u2))))))))).(\lambda (u: T).(\lambda (t0: -T).(\lambda (c: C).(\lambda (u2: T).(\lambda (H1: (pr3 c (THeads (Flat Appl) -(TApp ts t) (THead (Bind b) u t0)) u2)).(\lambda (H2: (((iso (THeads (Flat -Appl) (TApp ts t) (THead (Bind b) u t0)) u2) \to (\forall (P: -Prop).P)))).(eq_ind_r TList (TApp (lifts (S O) O ts) (lift (S O) O t)) -(\lambda (t1: TList).(pr3 c (THead (Bind b) u (THeads (Flat Appl) t1 t0)) -u2)) (eq_ind_r T (THeads (Flat Appl) (lifts (S O) O ts) (THead (Flat Appl) -(lift (S O) O t) t0)) (\lambda (t1: T).(pr3 c (THead (Bind b) u t1) u2)) (let -H3 \def (eq_ind T (THeads (Flat Appl) (TApp ts t) (THead (Bind b) u t0)) -(\lambda (t1: T).(pr3 c t1 u2)) H1 (THeads (Flat Appl) ts (THead (Flat Appl) -t (THead (Bind b) u t0))) (theads_tapp (Flat Appl) t (THead (Bind b) u t0) -ts)) in (let H4 \def (eq_ind T (THeads (Flat Appl) (TApp ts t) (THead (Bind -b) u t0)) (\lambda (t1: T).((iso t1 u2) \to (\forall (P: Prop).P))) H2 -(THeads (Flat Appl) ts (THead (Flat Appl) t (THead (Bind b) u t0))) -(theads_tapp (Flat Appl) t (THead (Bind b) u t0) ts)) in (TList_ind (\lambda -(t1: TList).(((\forall (u0: T).(\forall (t2: T).(\forall (c0: C).(\forall -(u3: T).((pr3 c0 (THeads (Flat Appl) t1 (THead (Bind b) u0 t2)) u3) \to -((((iso (THeads (Flat Appl) t1 (THead (Bind b) u0 t2)) u3) \to (\forall (P: -Prop).P))) \to (pr3 c0 (THead (Bind b) u0 (THeads (Flat Appl) (lifts (S O) O -t1) t2)) u3)))))))) \to ((pr3 c (THeads (Flat Appl) t1 (THead (Flat Appl) t -(THead (Bind b) u t0))) u2) \to ((((iso (THeads (Flat Appl) t1 (THead (Flat -Appl) t (THead (Bind b) u t0))) u2) \to (\forall (P: Prop).P))) \to (pr3 c -(THead (Bind b) u (THeads (Flat Appl) (lifts (S O) O t1) (THead (Flat Appl) -(lift (S O) O t) t0))) u2))))) (\lambda (_: ((\forall (u0: T).(\forall (t1: -T).(\forall (c0: C).(\forall (u3: T).((pr3 c0 (THeads (Flat Appl) TNil (THead -(Bind b) u0 t1)) u3) \to ((((iso (THeads (Flat Appl) TNil (THead (Bind b) u0 -t1)) u3) \to (\forall (P: Prop).P))) \to (pr3 c0 (THead (Bind b) u0 (THeads -(Flat Appl) (lifts (S O) O TNil) t1)) u3))))))))).(\lambda (H6: (pr3 c -(THeads (Flat Appl) TNil (THead (Flat Appl) t (THead (Bind b) u t0))) -u2)).(\lambda (H7: (((iso (THeads (Flat Appl) TNil (THead (Flat Appl) t -(THead (Bind b) u t0))) u2) \to (\forall (P: Prop).P)))).(pr3_iso_appl_bind b -H t u t0 c u2 H6 H7)))) (\lambda (t1: T).(\lambda (ts0: TList).(\lambda (_: -((((\forall (u0: T).(\forall (t2: T).(\forall (c0: C).(\forall (u3: T).((pr3 -c0 (THeads (Flat Appl) ts0 (THead (Bind b) u0 t2)) u3) \to ((((iso (THeads -(Flat Appl) ts0 (THead (Bind b) u0 t2)) u3) \to (\forall (P: Prop).P))) \to -(pr3 c0 (THead (Bind b) u0 (THeads (Flat Appl) (lifts (S O) O ts0) t2)) -u3)))))))) \to ((pr3 c (THeads (Flat Appl) ts0 (THead (Flat Appl) t (THead -(Bind b) u t0))) u2) \to ((((iso (THeads (Flat Appl) ts0 (THead (Flat Appl) t -(THead (Bind b) u t0))) u2) \to (\forall (P: Prop).P))) \to (pr3 c (THead -(Bind b) u (THeads (Flat Appl) (lifts (S O) O ts0) (THead (Flat Appl) (lift -(S O) O t) t0))) u2)))))).(\lambda (H5: ((\forall (u0: T).(\forall (t2: -T).(\forall (c0: C).(\forall (u3: T).((pr3 c0 (THeads (Flat Appl) (TCons t1 -ts0) (THead (Bind b) u0 t2)) u3) \to ((((iso (THeads (Flat Appl) (TCons t1 -ts0) (THead (Bind b) u0 t2)) u3) \to (\forall (P: Prop).P))) \to (pr3 c0 -(THead (Bind b) u0 (THeads (Flat Appl) (lifts (S O) O (TCons t1 ts0)) t2)) -u3))))))))).(\lambda (H6: (pr3 c (THeads (Flat Appl) (TCons t1 ts0) (THead -(Flat Appl) t (THead (Bind b) u t0))) u2)).(\lambda (H7: (((iso (THeads (Flat -Appl) (TCons t1 ts0) (THead (Flat Appl) t (THead (Bind b) u t0))) u2) \to -(\forall (P: Prop).P)))).(H5 u (THead (Flat Appl) (lift (S O) O t) t0) c u2 -(pr3_iso_appls_appl_bind b H t u t0 (TCons t1 ts0) c u2 H6 H7) (\lambda (H8: -(iso (THeads (Flat Appl) (TCons t1 ts0) (THead (Bind b) u (THead (Flat Appl) -(lift (S O) O t) t0))) u2)).(\lambda (P: Prop).(H7 (iso_trans (THeads (Flat -Appl) (TCons t1 ts0) (THead (Flat Appl) t (THead (Bind b) u t0))) (THeads -(Flat Appl) (TCons t1 ts0) (THead (Bind b) u (THead (Flat Appl) (lift (S O) O -t) t0))) (iso_head t1 t1 (THeads (Flat Appl) ts0 (THead (Flat Appl) t (THead -(Bind b) u t0))) (THeads (Flat Appl) ts0 (THead (Bind b) u (THead (Flat Appl) -(lift (S O) O t) t0))) (Flat Appl)) u2 H8) P)))))))))) ts H0 H3 H4))) (THeads -(Flat Appl) (TApp (lifts (S O) O ts) (lift (S O) O t)) t0) (theads_tapp (Flat -Appl) (lift (S O) O t) t0 (lifts (S O) O ts))) (lifts (S O) O (TApp ts t)) -(lifts_tapp (S O) O t ts))))))))))) vs))). -(* COMMENTS -Initial nodes: 1681 -END *) - -theorem pr3_iso_beta: - \forall (v: T).(\forall (w: T).(\forall (t: T).(let u1 \def (THead (Flat -Appl) v (THead (Bind Abst) w t)) in (\forall (c: C).(\forall (u2: T).((pr3 c -u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to (pr3 c (THead (Bind -Abbr) v t) u2)))))))) -\def - \lambda (v: T).(\lambda (w: T).(\lambda (t: T).(\lambda (c: C).(\lambda (u2: -T).(\lambda (H: (pr3 c (THead (Flat Appl) v (THead (Bind Abst) w t)) -u2)).(\lambda (H0: (((iso (THead (Flat Appl) v (THead (Bind Abst) w t)) u2) -\to (\forall (P: Prop).P)))).(let H1 \def (pr3_gen_appl c v (THead (Bind -Abst) w t) u2 H) in (or3_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq -T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c v -u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THead (Bind Abst) w t) t2)))) -(ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: -T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v u3))))) (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) -w t) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind -b) u) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B -b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) w t) (THead -(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) -y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda -(_: T).(pr3 c v u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (pr3 c -(THead (Bind Abbr) v t) u2) (\lambda (H2: (ex3_2 T T (\lambda (u3: -T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: -T).(\lambda (_: T).(pr3 c v u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c -(THead (Bind Abst) w t) t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: -T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: -T).(pr3 c v u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THead (Bind Abst) -w t) t2))) (pr3 c (THead (Bind Abbr) v t) u2) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H3: (eq T u2 (THead (Flat Appl) x0 x1))).(\lambda (_: (pr3 c v -x0)).(\lambda (_: (pr3 c (THead (Bind Abst) w t) x1)).(let H6 \def (eq_ind T -u2 (\lambda (t0: T).((iso (THead (Flat Appl) v (THead (Bind Abst) w t)) t0) -\to (\forall (P: Prop).P))) H0 (THead (Flat Appl) x0 x1) H3) in (eq_ind_r T -(THead (Flat Appl) x0 x1) (\lambda (t0: T).(pr3 c (THead (Bind Abbr) v t) -t0)) (H6 (iso_head v x0 (THead (Bind Abst) w t) x1 (Flat Appl)) (pr3 c (THead -(Bind Abbr) v t) (THead (Flat Appl) x0 x1))) u2 H3))))))) H2)) (\lambda (H2: -(ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: -T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v u3))))) (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) -w t) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind -b) u) z1 t2))))))))).(ex4_4_ind T T T T (\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v -u3))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(pr3 c (THead (Bind Abst) w t) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall -(u: T).(pr3 (CHead c (Bind b) u) z1 t2))))))) (pr3 c (THead (Bind Abbr) v t) -u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: -T).(\lambda (H3: (pr3 c (THead (Bind Abbr) x2 x3) u2)).(\lambda (H4: (pr3 c v -x2)).(\lambda (H5: (pr3 c (THead (Bind Abst) w t) (THead (Bind Abst) x0 -x1))).(\lambda (H6: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) -u) x1 x3))))).(let H7 \def (pr3_gen_abst c w t (THead (Bind Abst) x0 x1) H5) -in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind Abst) -x0 x1) (THead (Bind Abst) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c w -u3))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 -(CHead c (Bind b) u) t t2))))) (pr3 c (THead (Bind Abbr) v t) u2) (\lambda -(x4: T).(\lambda (x5: T).(\lambda (H8: (eq T (THead (Bind Abst) x0 x1) (THead -(Bind Abst) x4 x5))).(\lambda (H9: (pr3 c w x4)).(\lambda (H10: ((\forall (b: -B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t x5))))).(let H11 \def (f_equal -T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ t0 _) \Rightarrow t0])) -(THead (Bind Abst) x0 x1) (THead (Bind Abst) x4 x5) H8) in ((let H12 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow x1 | (TLRef _) \Rightarrow x1 | (THead _ _ t0) -\Rightarrow t0])) (THead (Bind Abst) x0 x1) (THead (Bind Abst) x4 x5) H8) in -(\lambda (H13: (eq T x0 x4)).(let H14 \def (eq_ind_r T x5 (\lambda (t0: -T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t t0)))) H10 x1 -H12) in (let H15 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 c w t0)) H9 x0 -H13) in (pr3_t (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) v t) c -(pr3_head_12 c v x2 H4 (Bind Abbr) t x3 (pr3_t x1 t (CHead c (Bind Abbr) x2) -(H14 Abbr x2) x3 (H6 Abbr x2))) u2 H3))))) H11))))))) H7)))))))))) H2)) -(\lambda (H2: (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) -(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) w t) (THead (Bind b) y1 -z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: -T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat -Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v -u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c -(THead (Bind Abst) w t) (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u3: T).(\lambda -(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u3) z2)) -u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v u3))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) -y2) z1 z2))))))) (pr3 c (THead (Bind Abbr) v t) u2) (\lambda (x0: B).(\lambda -(x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: -T).(\lambda (H3: (not (eq B x0 Abst))).(\lambda (H4: (pr3 c (THead (Bind -Abst) w t) (THead (Bind x0) x1 x2))).(\lambda (H5: (pr3 c (THead (Bind x0) x5 -(THead (Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda (_: (pr3 c v -x4)).(\lambda (_: (pr3 c x1 x5)).(\lambda (H8: (pr3 (CHead c (Bind x0) x5) x2 -x3)).(let H9 \def (pr3_gen_abst c w t (THead (Bind x0) x1 x2) H4) in -(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind x0) x1 -x2) (THead (Bind Abst) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c w -u3))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 -(CHead c (Bind b) u) t t2))))) (pr3 c (THead (Bind Abbr) v t) u2) (\lambda -(x6: T).(\lambda (x7: T).(\lambda (H10: (eq T (THead (Bind x0) x1 x2) (THead -(Bind Abst) x6 x7))).(\lambda (H11: (pr3 c w x6)).(\lambda (H12: ((\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t x7))))).(let H13 \def -(f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with -[(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead k _ _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) -\Rightarrow b | (Flat _) \Rightarrow x0])])) (THead (Bind x0) x1 x2) (THead -(Bind Abst) x6 x7) H10) in ((let H14 \def (f_equal T T (\lambda (e: T).(match -e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x1 | (TLRef _) -\Rightarrow x1 | (THead _ t0 _) \Rightarrow t0])) (THead (Bind x0) x1 x2) -(THead (Bind Abst) x6 x7) H10) in ((let H15 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x2 | -(TLRef _) \Rightarrow x2 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind x0) -x1 x2) (THead (Bind Abst) x6 x7) H10) in (\lambda (H16: (eq T x1 -x6)).(\lambda (H17: (eq B x0 Abst)).(let H18 \def (eq_ind_r T x7 (\lambda -(t0: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t t0)))) -H12 x2 H15) in (let H19 \def (eq_ind_r T x6 (\lambda (t0: T).(pr3 c w t0)) -H11 x1 H16) in (let H20 \def (eq_ind B x0 (\lambda (b: B).(pr3 (CHead c (Bind -b) x5) x2 x3)) H8 Abst H17) in (let H21 \def (eq_ind B x0 (\lambda (b: -B).(pr3 c (THead (Bind b) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) u2)) -H5 Abst H17) in (let H22 \def (eq_ind B x0 (\lambda (b: B).(not (eq B b -Abst))) H3 Abst H17) in (let H23 \def (match (H22 (refl_equal B Abst)) in -False return (\lambda (_: False).(pr3 c (THead (Bind Abbr) v t) u2)) with []) -in H23))))))))) H14)) H13))))))) H9)))))))))))))) H2)) H1)))))))). -(* COMMENTS -Initial nodes: 2459 -END *) - -theorem pr3_iso_appls_beta: - \forall (us: TList).(\forall (v: T).(\forall (w: T).(\forall (t: T).(let u1 -\def (THeads (Flat Appl) us (THead (Flat Appl) v (THead (Bind Abst) w t))) in -(\forall (c: C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to -(\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) us (THead (Bind Abbr) -v t)) u2))))))))) -\def - \lambda (us: TList).(TList_ind (\lambda (t: TList).(\forall (v: T).(\forall -(w: T).(\forall (t0: T).(let u1 \def (THeads (Flat Appl) t (THead (Flat Appl) -v (THead (Bind Abst) w t0))) in (\forall (c: C).(\forall (u2: T).((pr3 c u1 -u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to (pr3 c (THeads (Flat -Appl) t (THead (Bind Abbr) v t0)) u2)))))))))) (\lambda (v: T).(\lambda (w: -T).(\lambda (t: T).(\lambda (c: C).(\lambda (u2: T).(\lambda (H: (pr3 c -(THead (Flat Appl) v (THead (Bind Abst) w t)) u2)).(\lambda (H0: (((iso -(THead (Flat Appl) v (THead (Bind Abst) w t)) u2) \to (\forall (P: -Prop).P)))).(pr3_iso_beta v w t c u2 H H0)))))))) (\lambda (t: T).(\lambda -(t0: TList).(\lambda (H: ((\forall (v: T).(\forall (w: T).(\forall (t1: -T).(\forall (c: C).(\forall (u2: T).((pr3 c (THeads (Flat Appl) t0 (THead -(Flat Appl) v (THead (Bind Abst) w t1))) u2) \to ((((iso (THeads (Flat Appl) -t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) u2) \to (\forall (P: -Prop).P))) \to (pr3 c (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1)) -u2)))))))))).(\lambda (v: T).(\lambda (w: T).(\lambda (t1: T).(\lambda (c: -C).(\lambda (u2: T).(\lambda (H0: (pr3 c (THead (Flat Appl) t (THeads (Flat -Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1)))) u2)).(\lambda (H1: -(((iso (THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Flat Appl) v -(THead (Bind Abst) w t1)))) u2) \to (\forall (P: Prop).P)))).(let H2 \def -(pr3_gen_appl c t (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind -Abst) w t1))) u2 H0) in (or3_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t2: -T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: -T).(pr3 c t u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) -t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) t2)))) (ex4_4 T T T T -(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c -(THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u3: T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat -Appl) v (THead (Bind Abst) w t1))) (THead (Bind Abst) y1 z1)))))) (\lambda -(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: -B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c -(THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) (THead -(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) -y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda -(_: T).(pr3 c t u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (pr3 c -(THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) u2) -(\lambda (H3: (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead -(Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t u3))) -(\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat -Appl) v (THead (Bind Abst) w t1))) t2))))).(ex3_2_ind T T (\lambda (u3: -T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: -T).(\lambda (_: T).(pr3 c t u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c -(THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) t2))) -(pr3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) -u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T u2 (THead (Flat -Appl) x0 x1))).(\lambda (_: (pr3 c t x0)).(\lambda (_: (pr3 c (THeads (Flat -Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) x1)).(let H7 \def -(eq_ind T u2 (\lambda (t2: T).((iso (THead (Flat Appl) t (THeads (Flat Appl) -t0 (THead (Flat Appl) v (THead (Bind Abst) w t1)))) t2) \to (\forall (P: -Prop).P))) H1 (THead (Flat Appl) x0 x1) H4) in (eq_ind_r T (THead (Flat Appl) -x0 x1) (\lambda (t2: T).(pr3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 -(THead (Bind Abbr) v t1))) t2)) (H7 (iso_head t x0 (THeads (Flat Appl) t0 -(THead (Flat Appl) v (THead (Bind Abst) w t1))) x1 (Flat Appl)) (pr3 c (THead -(Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) (THead (Flat -Appl) x0 x1))) u2 H4))))))) H3)) (\lambda (H3: (ex4_4 T T T T (\lambda (_: -T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind -Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: -T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat -Appl) v (THead (Bind Abst) w t1))) (THead (Bind Abst) y1 z1)))))) (\lambda -(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: -B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2))))))))).(ex4_4_ind T T T -T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c -(THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u3: T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat -Appl) v (THead (Bind Abst) w t1))) (THead (Bind Abst) y1 z1)))))) (\lambda -(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: -B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2))))))) (pr3 c (THead (Flat -Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) u2) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H4: (pr3 c -(THead (Bind Abbr) x2 x3) u2)).(\lambda (H5: (pr3 c t x2)).(\lambda (H6: (pr3 -c (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) -(THead (Bind Abst) x0 x1))).(\lambda (H7: ((\forall (b: B).(\forall (u: -T).(pr3 (CHead c (Bind b) u) x1 x3))))).(pr3_t (THead (Bind Abbr) t x1) -(THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) c -(pr3_t (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) (THead (Flat Appl) t -(THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) c (pr3_thin_dx c (THeads -(Flat Appl) t0 (THead (Bind Abbr) v t1)) (THead (Bind Abst) x0 x1) (H v w t1 -c (THead (Bind Abst) x0 x1) H6 (\lambda (H8: (iso (THeads (Flat Appl) t0 -(THead (Flat Appl) v (THead (Bind Abst) w t1))) (THead (Bind Abst) x0 -x1))).(\lambda (P: Prop).(iso_flats_flat_bind_false Appl Appl Abst x0 v x1 -(THead (Bind Abst) w t1) t0 H8 P)))) t Appl) (THead (Bind Abbr) t x1) -(pr3_pr2 c (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) -t x1) (pr2_free c (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) (THead -(Bind Abbr) t x1) (pr0_beta x0 t t (pr0_refl t) x1 x1 (pr0_refl x1))))) u2 -(pr3_t (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) t x1) c (pr3_head_12 c t -x2 H5 (Bind Abbr) x1 x3 (H7 Abbr x2)) u2 H4)))))))))) H3)) (\lambda (H3: -(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) -w t1))) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c -(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: -T).(\lambda (_: T).(pr3 c t u3))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 -y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 -z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat -Appl) v (THead (Bind Abst) w t1))) (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u3: T).(\lambda -(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u3) z2)) -u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t u3))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) -y2) z1 z2))))))) (pr3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (THead -(Bind Abbr) v t1))) u2) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: -T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H4: (not (eq -B x0 Abst))).(\lambda (H5: (pr3 c (THeads (Flat Appl) t0 (THead (Flat Appl) v -(THead (Bind Abst) w t1))) (THead (Bind x0) x1 x2))).(\lambda (H6: (pr3 c -(THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda -(H7: (pr3 c t x4)).(\lambda (H8: (pr3 c x1 x5)).(\lambda (H9: (pr3 (CHead c -(Bind x0) x5) x2 x3)).(pr3_t (THead (Bind x0) x1 (THead (Flat Appl) (lift (S -O) O x4) x2)) (THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) -v t1))) c (pr3_t (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O t) x2)) -(THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) c -(pr3_t (THead (Flat Appl) t (THead (Bind x0) x1 x2)) (THead (Flat Appl) t -(THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) c (pr3_thin_dx c (THeads -(Flat Appl) t0 (THead (Bind Abbr) v t1)) (THead (Bind x0) x1 x2) (H v w t1 c -(THead (Bind x0) x1 x2) H5 (\lambda (H10: (iso (THeads (Flat Appl) t0 (THead -(Flat Appl) v (THead (Bind Abst) w t1))) (THead (Bind x0) x1 x2))).(\lambda -(P: Prop).(iso_flats_flat_bind_false Appl Appl x0 x1 v x2 (THead (Bind Abst) -w t1) t0 H10 P)))) t Appl) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) -O t) x2)) (pr3_pr2 c (THead (Flat Appl) t (THead (Bind x0) x1 x2)) (THead -(Bind x0) x1 (THead (Flat Appl) (lift (S O) O t) x2)) (pr2_free c (THead -(Flat Appl) t (THead (Bind x0) x1 x2)) (THead (Bind x0) x1 (THead (Flat Appl) -(lift (S O) O t) x2)) (pr0_upsilon x0 H4 t t (pr0_refl t) x1 x1 (pr0_refl x1) -x2 x2 (pr0_refl x2))))) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O -x4) x2)) (pr3_head_12 c x1 x1 (pr3_refl c x1) (Bind x0) (THead (Flat Appl) -(lift (S O) O t) x2) (THead (Flat Appl) (lift (S O) O x4) x2) (pr3_head_12 -(CHead c (Bind x0) x1) (lift (S O) O t) (lift (S O) O x4) (pr3_lift (CHead c -(Bind x0) x1) c (S O) O (drop_drop (Bind x0) O c c (drop_refl c) x1) t x4 H7) -(Flat Appl) x2 x2 (pr3_refl (CHead (CHead c (Bind x0) x1) (Flat Appl) (lift -(S O) O x4)) x2)))) u2 (pr3_t (THead (Bind x0) x5 (THead (Flat Appl) (lift (S -O) O x4) x3)) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) c -(pr3_head_12 c x1 x5 H8 (Bind x0) (THead (Flat Appl) (lift (S O) O x4) x2) -(THead (Flat Appl) (lift (S O) O x4) x3) (pr3_thin_dx (CHead c (Bind x0) x5) -x2 x3 H9 (lift (S O) O x4) Appl)) u2 H6)))))))))))))) H3)) H2)))))))))))) us). -(* COMMENTS -Initial nodes: 3345 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/pr1.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/pr1.ma deleted file mode 100644 index e428daf37..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/pr1.ma +++ /dev/null @@ -1,34 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr3/defs.ma". - -include "Basic-1/pr1/defs.ma". - -theorem pr3_pr1: - \forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall (c: C).(pr3 c t1 -t2)))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr1 t1 t2)).(pr1_ind (\lambda -(t: T).(\lambda (t0: T).(\forall (c: C).(pr3 c t t0)))) (\lambda (t: -T).(\lambda (c: C).(pr3_refl c t))) (\lambda (t0: T).(\lambda (t3: -T).(\lambda (H0: (pr0 t3 t0)).(\lambda (t4: T).(\lambda (_: (pr1 t0 -t4)).(\lambda (H2: ((\forall (c: C).(pr3 c t0 t4)))).(\lambda (c: -C).(pr3_sing c t0 t3 (pr2_free c t3 t0 H0) t4 (H2 c))))))))) t1 t2 H))). -(* COMMENTS -Initial nodes: 95 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/pr3.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/pr3.ma deleted file mode 100644 index 935850b1c..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/pr3.ma +++ /dev/null @@ -1,74 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr3/props.ma". - -include "Basic-1/pr2/pr2.ma". - -theorem pr3_strip: - \forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr3 c t0 t1) \to (\forall -(t2: T).((pr2 c t0 t2) \to (ex2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: -T).(pr3 c t2 t)))))))) -\def - \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr3 c t0 -t1)).(pr3_ind c (\lambda (t: T).(\lambda (t2: T).(\forall (t3: T).((pr2 c t -t3) \to (ex2 T (\lambda (t4: T).(pr3 c t2 t4)) (\lambda (t4: T).(pr3 c t3 -t4))))))) (\lambda (t: T).(\lambda (t2: T).(\lambda (H0: (pr2 c t -t2)).(ex_intro2 T (\lambda (t3: T).(pr3 c t t3)) (\lambda (t3: T).(pr3 c t2 -t3)) t2 (pr3_pr2 c t t2 H0) (pr3_refl c t2))))) (\lambda (t2: T).(\lambda -(t3: T).(\lambda (H0: (pr2 c t3 t2)).(\lambda (t4: T).(\lambda (_: (pr3 c t2 -t4)).(\lambda (H2: ((\forall (t5: T).((pr2 c t2 t5) \to (ex2 T (\lambda (t: -T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t))))))).(\lambda (t5: T).(\lambda -(H3: (pr2 c t3 t5)).(ex2_ind T (\lambda (t: T).(pr2 c t5 t)) (\lambda (t: -T).(pr2 c t2 t)) (ex2 T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c -t5 t))) (\lambda (x: T).(\lambda (H4: (pr2 c t5 x)).(\lambda (H5: (pr2 c t2 -x)).(ex2_ind T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c x t)) -(ex2 T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t))) (\lambda -(x0: T).(\lambda (H6: (pr3 c t4 x0)).(\lambda (H7: (pr3 c x x0)).(ex_intro2 T -(\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t)) x0 H6 (pr3_sing c -x t5 H4 x0 H7))))) (H2 x H5))))) (pr2_confluence c t3 t5 H3 t2 H0)))))))))) -t0 t1 H)))). -(* COMMENTS -Initial nodes: 375 -END *) - -theorem pr3_confluence: - \forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr3 c t0 t1) \to (\forall -(t2: T).((pr3 c t0 t2) \to (ex2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: -T).(pr3 c t2 t)))))))) -\def - \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr3 c t0 -t1)).(pr3_ind c (\lambda (t: T).(\lambda (t2: T).(\forall (t3: T).((pr3 c t -t3) \to (ex2 T (\lambda (t4: T).(pr3 c t2 t4)) (\lambda (t4: T).(pr3 c t3 -t4))))))) (\lambda (t: T).(\lambda (t2: T).(\lambda (H0: (pr3 c t -t2)).(ex_intro2 T (\lambda (t3: T).(pr3 c t t3)) (\lambda (t3: T).(pr3 c t2 -t3)) t2 H0 (pr3_refl c t2))))) (\lambda (t2: T).(\lambda (t3: T).(\lambda -(H0: (pr2 c t3 t2)).(\lambda (t4: T).(\lambda (_: (pr3 c t2 t4)).(\lambda -(H2: ((\forall (t5: T).((pr3 c t2 t5) \to (ex2 T (\lambda (t: T).(pr3 c t4 -t)) (\lambda (t: T).(pr3 c t5 t))))))).(\lambda (t5: T).(\lambda (H3: (pr3 c -t3 t5)).(ex2_ind T (\lambda (t: T).(pr3 c t5 t)) (\lambda (t: T).(pr3 c t2 -t)) (ex2 T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t))) -(\lambda (x: T).(\lambda (H4: (pr3 c t5 x)).(\lambda (H5: (pr3 c t2 -x)).(ex2_ind T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c x t)) -(ex2 T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t))) (\lambda -(x0: T).(\lambda (H6: (pr3 c t4 x0)).(\lambda (H7: (pr3 c x x0)).(ex_intro2 T -(\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t)) x0 H6 (pr3_t x t5 -c H4 x0 H7))))) (H2 x H5))))) (pr3_strip c t3 t5 H3 t2 H0)))))))))) t0 t1 -H)))). -(* COMMENTS -Initial nodes: 367 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/props.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/props.ma deleted file mode 100644 index 777f5d85d..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/props.ma +++ /dev/null @@ -1,407 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr3/pr1.ma". - -include "Basic-1/pr2/props.ma". - -include "Basic-1/pr1/props.ma". - -theorem clear_pr3_trans: - \forall (c2: C).(\forall (t1: T).(\forall (t2: T).((pr3 c2 t1 t2) \to -(\forall (c1: C).((clear c1 c2) \to (pr3 c1 t1 t2)))))) -\def - \lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c2 t1 -t2)).(\lambda (c1: C).(\lambda (H0: (clear c1 c2)).(pr3_ind c2 (\lambda (t: -T).(\lambda (t0: T).(pr3 c1 t t0))) (\lambda (t: T).(pr3_refl c1 t)) (\lambda -(t3: T).(\lambda (t4: T).(\lambda (H1: (pr2 c2 t4 t3)).(\lambda (t5: -T).(\lambda (_: (pr3 c2 t3 t5)).(\lambda (H3: (pr3 c1 t3 t5)).(pr3_sing c1 t3 -t4 (clear_pr2_trans c2 t4 t3 H1 c1 H0) t5 H3))))))) t1 t2 H)))))). -(* COMMENTS -Initial nodes: 107 -END *) - -theorem pr3_pr2: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (pr3 c -t1 t2)))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 -t2)).(pr3_sing c t2 t1 H t2 (pr3_refl c t2))))). -(* COMMENTS -Initial nodes: 31 -END *) - -theorem pr3_t: - \forall (t2: T).(\forall (t1: T).(\forall (c: C).((pr3 c t1 t2) \to (\forall -(t3: T).((pr3 c t2 t3) \to (pr3 c t1 t3)))))) -\def - \lambda (t2: T).(\lambda (t1: T).(\lambda (c: C).(\lambda (H: (pr3 c t1 -t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (t3: T).((pr3 c t0 -t3) \to (pr3 c t t3))))) (\lambda (t: T).(\lambda (t3: T).(\lambda (H0: (pr3 -c t t3)).H0))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c t3 -t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2: ((\forall -(t5: T).((pr3 c t4 t5) \to (pr3 c t0 t5))))).(\lambda (t5: T).(\lambda (H3: -(pr3 c t4 t5)).(pr3_sing c t0 t3 H0 t5 (H2 t5 H3)))))))))) t1 t2 H)))). -(* COMMENTS -Initial nodes: 127 -END *) - -theorem pr3_thin_dx: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall -(u: T).(\forall (f: F).(pr3 c (THead (Flat f) u t1) (THead (Flat f) u -t2))))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 -t2)).(\lambda (u: T).(\lambda (f: F).(pr3_ind c (\lambda (t: T).(\lambda (t0: -T).(pr3 c (THead (Flat f) u t) (THead (Flat f) u t0)))) (\lambda (t: -T).(pr3_refl c (THead (Flat f) u t))) (\lambda (t0: T).(\lambda (t3: -T).(\lambda (H0: (pr2 c t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0 -t4)).(\lambda (H2: (pr3 c (THead (Flat f) u t0) (THead (Flat f) u -t4))).(pr3_sing c (THead (Flat f) u t0) (THead (Flat f) u t3) (pr2_thin_dx c -t3 t0 H0 u f) (THead (Flat f) u t4) H2))))))) t1 t2 H)))))). -(* COMMENTS -Initial nodes: 167 -END *) - -theorem pr3_head_1: - \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall -(k: K).(\forall (t: T).(pr3 c (THead k u1 t) (THead k u2 t))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1 -u2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (k: K).(\forall -(t1: T).(pr3 c (THead k t t1) (THead k t0 t1)))))) (\lambda (t: T).(\lambda -(k: K).(\lambda (t0: T).(pr3_refl c (THead k t t0))))) (\lambda (t2: -T).(\lambda (t1: T).(\lambda (H0: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda -(_: (pr3 c t2 t3)).(\lambda (H2: ((\forall (k: K).(\forall (t: T).(pr3 c -(THead k t2 t) (THead k t3 t)))))).(\lambda (k: K).(\lambda (t: T).(pr3_sing -c (THead k t2 t) (THead k t1 t) (pr2_head_1 c t1 t2 H0 k t) (THead k t3 t) -(H2 k t)))))))))) u1 u2 H)))). -(* COMMENTS -Initial nodes: 167 -END *) - -theorem pr3_head_2: - \forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).(\forall -(k: K).((pr3 (CHead c k u) t1 t2) \to (pr3 c (THead k u t1) (THead k u -t2))))))) -\def - \lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(k: K).(\lambda (H: (pr3 (CHead c k u) t1 t2)).(pr3_ind (CHead c k u) -(\lambda (t: T).(\lambda (t0: T).(pr3 c (THead k u t) (THead k u t0)))) -(\lambda (t: T).(pr3_refl c (THead k u t))) (\lambda (t0: T).(\lambda (t3: -T).(\lambda (H0: (pr2 (CHead c k u) t3 t0)).(\lambda (t4: T).(\lambda (_: -(pr3 (CHead c k u) t0 t4)).(\lambda (H2: (pr3 c (THead k u t0) (THead k u -t4))).(pr3_sing c (THead k u t0) (THead k u t3) (pr2_head_2 c u t3 t0 k H0) -(THead k u t4) H2))))))) t1 t2 H)))))). -(* COMMENTS -Initial nodes: 175 -END *) - -theorem pr3_head_21: - \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall -(k: K).(\forall (t1: T).(\forall (t2: T).((pr3 (CHead c k u1) t1 t2) \to (pr3 -c (THead k u1 t1) (THead k u2 t2))))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1 -u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 -(CHead c k u1) t1 t2)).(pr3_t (THead k u1 t2) (THead k u1 t1) c (pr3_head_2 c -u1 t1 t2 k H0) (THead k u2 t2) (pr3_head_1 c u1 u2 H k t2))))))))). -(* COMMENTS -Initial nodes: 89 -END *) - -theorem pr3_head_12: - \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall -(k: K).(\forall (t1: T).(\forall (t2: T).((pr3 (CHead c k u2) t1 t2) \to (pr3 -c (THead k u1 t1) (THead k u2 t2))))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1 -u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 -(CHead c k u2) t1 t2)).(pr3_t (THead k u2 t1) (THead k u1 t1) c (pr3_head_1 c -u1 u2 H k t1) (THead k u2 t2) (pr3_head_2 c u2 t1 t2 k H0))))))))). -(* COMMENTS -Initial nodes: 89 -END *) - -theorem pr3_cflat: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall -(f: F).(\forall (v: T).(pr3 (CHead c (Flat f) v) t1 t2)))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 -t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (f: F).(\forall (v: -T).(pr3 (CHead c (Flat f) v) t t0))))) (\lambda (t: T).(\lambda (f: -F).(\lambda (v: T).(pr3_refl (CHead c (Flat f) v) t)))) (\lambda (t3: -T).(\lambda (t4: T).(\lambda (H0: (pr2 c t4 t3)).(\lambda (t5: T).(\lambda -(_: (pr3 c t3 t5)).(\lambda (H2: ((\forall (f: F).(\forall (v: T).(pr3 (CHead -c (Flat f) v) t3 t5))))).(\lambda (f: F).(\lambda (v: T).(pr3_sing (CHead c -(Flat f) v) t3 t4 (pr2_cflat c t4 t3 H0 f v) t5 (H2 f v)))))))))) t1 t2 H)))). -(* COMMENTS -Initial nodes: 151 -END *) - -theorem pr3_flat: - \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall -(t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall (f: F).(pr3 c (THead -(Flat f) u1 t1) (THead (Flat f) u2 t2))))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1 -u2)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 c t1 t2)).(\lambda -(f: F).(pr3_head_12 c u1 u2 H (Flat f) t1 t2 (pr3_cflat c t1 t2 H0 f -u2))))))))). -(* COMMENTS -Initial nodes: 59 -END *) - -theorem pr3_pr0_pr2_t: - \forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (c: C).(\forall -(t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pr3 -(CHead c k u1) t1 t2)))))))) -\def - \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr0 u1 u2)).(\lambda (c: -C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k: K).(\lambda (H0: (pr2 -(CHead c k u2) t1 t2)).(insert_eq C (CHead c k u2) (\lambda (c0: C).(pr2 c0 -t1 t2)) (\lambda (_: C).(pr3 (CHead c k u1) t1 t2)) (\lambda (y: C).(\lambda -(H1: (pr2 y t1 t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: -T).((eq C c0 (CHead c k u2)) \to (pr3 (CHead c k u1) t t0))))) (\lambda (c0: -C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (_: -(eq C c0 (CHead c k u2))).(pr3_pr2 (CHead c k u1) t3 t4 (pr2_free (CHead c k -u1) t3 t4 H2))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda -(i: nat).(\lambda (H2: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t3: -T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H4: -(subst0 i u t4 t)).(\lambda (H5: (eq C c0 (CHead c k u2))).(let H6 \def -(eq_ind C c0 (\lambda (c1: C).(getl i c1 (CHead d (Bind Abbr) u))) H2 (CHead -c k u2) H5) in (nat_ind (\lambda (n: nat).((getl n (CHead c k u2) (CHead d -(Bind Abbr) u)) \to ((subst0 n u t4 t) \to (pr3 (CHead c k u1) t3 t)))) -(\lambda (H7: (getl O (CHead c k u2) (CHead d (Bind Abbr) u))).(\lambda (H8: -(subst0 O u t4 t)).(K_ind (\lambda (k0: K).((getl O (CHead c k0 u2) (CHead d -(Bind Abbr) u)) \to (pr3 (CHead c k0 u1) t3 t))) (\lambda (b: B).(\lambda -(H9: (getl O (CHead c (Bind b) u2) (CHead d (Bind Abbr) u))).(let H10 \def -(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with -[(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind -Abbr) u) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 -(getl_gen_O (CHead c (Bind b) u2) (CHead d (Bind Abbr) u) H9))) in ((let H11 -\def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) -with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow (match k0 in K -return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) -\Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead c (Bind b) u2) -(clear_gen_bind b c (CHead d (Bind Abbr) u) u2 (getl_gen_O (CHead c (Bind b) -u2) (CHead d (Bind Abbr) u) H9))) in ((let H12 \def (f_equal C T (\lambda (e: -C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | -(CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind Abbr) u) (CHead c (Bind b) -u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 (getl_gen_O (CHead c (Bind -b) u2) (CHead d (Bind Abbr) u) H9))) in (\lambda (H13: (eq B Abbr -b)).(\lambda (_: (eq C d c)).(let H15 \def (eq_ind T u (\lambda (t0: -T).(subst0 O t0 t4 t)) H8 u2 H12) in (eq_ind B Abbr (\lambda (b0: B).(pr3 -(CHead c (Bind b0) u1) t3 t)) (ex2_ind T (\lambda (t0: T).(subst0 O u1 t4 -t0)) (\lambda (t0: T).(pr0 t0 t)) (pr3 (CHead c (Bind Abbr) u1) t3 t) -(\lambda (x: T).(\lambda (H16: (subst0 O u1 t4 x)).(\lambda (H17: (pr0 x -t)).(pr3_sing (CHead c (Bind Abbr) u1) x t3 (pr2_delta (CHead c (Bind Abbr) -u1) c u1 O (getl_refl Abbr c u1) t3 t4 H3 x H16) t (pr3_pr2 (CHead c (Bind -Abbr) u1) x t (pr2_free (CHead c (Bind Abbr) u1) x t H17)))))) -(pr0_subst0_back u2 t4 t O H15 u1 H)) b H13))))) H11)) H10)))) (\lambda (f: -F).(\lambda (H9: (getl O (CHead c (Flat f) u2) (CHead d (Bind Abbr) -u))).(pr3_pr2 (CHead c (Flat f) u1) t3 t (pr2_cflat c t3 t (pr2_delta c d u O -(getl_intro O c (CHead d (Bind Abbr) u) c (drop_refl c) (clear_gen_flat f c -(CHead d (Bind Abbr) u) u2 (getl_gen_O (CHead c (Flat f) u2) (CHead d (Bind -Abbr) u) H9))) t3 t4 H3 t H8) f u1)))) k H7))) (\lambda (i0: nat).(\lambda -(IHi: (((getl i0 (CHead c k u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t4 -t) \to (pr3 (CHead c k u1) t3 t))))).(\lambda (H7: (getl (S i0) (CHead c k -u2) (CHead d (Bind Abbr) u))).(\lambda (H8: (subst0 (S i0) u t4 t)).(K_ind -(\lambda (k0: K).((getl (S i0) (CHead c k0 u2) (CHead d (Bind Abbr) u)) \to -((((getl i0 (CHead c k0 u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t4 t) -\to (pr3 (CHead c k0 u1) t3 t)))) \to (pr3 (CHead c k0 u1) t3 t)))) (\lambda -(b: B).(\lambda (H9: (getl (S i0) (CHead c (Bind b) u2) (CHead d (Bind Abbr) -u))).(\lambda (_: (((getl i0 (CHead c (Bind b) u2) (CHead d (Bind Abbr) u)) -\to ((subst0 i0 u t4 t) \to (pr3 (CHead c (Bind b) u1) t3 t))))).(pr3_pr2 -(CHead c (Bind b) u1) t3 t (pr2_delta (CHead c (Bind b) u1) d u (S i0) -(getl_head (Bind b) i0 c (CHead d (Bind Abbr) u) (getl_gen_S (Bind b) c -(CHead d (Bind Abbr) u) u2 i0 H9) u1) t3 t4 H3 t H8))))) (\lambda (f: -F).(\lambda (H9: (getl (S i0) (CHead c (Flat f) u2) (CHead d (Bind Abbr) -u))).(\lambda (_: (((getl i0 (CHead c (Flat f) u2) (CHead d (Bind Abbr) u)) -\to ((subst0 i0 u t4 t) \to (pr3 (CHead c (Flat f) u1) t3 t))))).(pr3_pr2 -(CHead c (Flat f) u1) t3 t (pr2_cflat c t3 t (pr2_delta c d u (r (Flat f) i0) -(getl_gen_S (Flat f) c (CHead d (Bind Abbr) u) u2 i0 H9) t3 t4 H3 t H8) f -u1))))) k H7 IHi))))) i H6 H4))))))))))))) y t1 t2 H1))) H0)))))))). -(* COMMENTS -Initial nodes: 1557 -END *) - -theorem pr3_pr2_pr2_t: - \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr2 c u1 u2) \to (\forall -(t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pr3 -(CHead c k u1) t1 t2)))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr2 c u1 -u2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (t1: -T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c0 k t0) t1 t2) \to (pr3 -(CHead c0 k t) t1 t2)))))))) (\lambda (c0: C).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (H0: (pr0 t1 t2)).(\lambda (t0: T).(\lambda (t3: T).(\lambda (k: -K).(\lambda (H1: (pr2 (CHead c0 k t2) t0 t3)).(pr3_pr0_pr2_t t1 t2 H0 c0 t0 -t3 k H1))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda -(i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t1: -T).(\lambda (t2: T).(\lambda (H1: (pr0 t1 t2)).(\lambda (t: T).(\lambda (H2: -(subst0 i u t2 t)).(\lambda (t0: T).(\lambda (t3: T).(\lambda (k: K).(\lambda -(H3: (pr2 (CHead c0 k t) t0 t3)).(insert_eq C (CHead c0 k t) (\lambda (c1: -C).(pr2 c1 t0 t3)) (\lambda (_: C).(pr3 (CHead c0 k t1) t0 t3)) (\lambda (y: -C).(\lambda (H4: (pr2 y t0 t3)).(pr2_ind (\lambda (c1: C).(\lambda (t4: -T).(\lambda (t5: T).((eq C c1 (CHead c0 k t)) \to (pr3 (CHead c0 k t1) t4 -t5))))) (\lambda (c1: C).(\lambda (t4: T).(\lambda (t5: T).(\lambda (H5: (pr0 -t4 t5)).(\lambda (_: (eq C c1 (CHead c0 k t))).(pr3_pr2 (CHead c0 k t1) t4 t5 -(pr2_free (CHead c0 k t1) t4 t5 H5))))))) (\lambda (c1: C).(\lambda (d0: -C).(\lambda (u0: T).(\lambda (i0: nat).(\lambda (H5: (getl i0 c1 (CHead d0 -(Bind Abbr) u0))).(\lambda (t4: T).(\lambda (t5: T).(\lambda (H6: (pr0 t4 -t5)).(\lambda (t6: T).(\lambda (H7: (subst0 i0 u0 t5 t6)).(\lambda (H8: (eq C -c1 (CHead c0 k t))).(let H9 \def (eq_ind C c1 (\lambda (c2: C).(getl i0 c2 -(CHead d0 (Bind Abbr) u0))) H5 (CHead c0 k t) H8) in (nat_ind (\lambda (n: -nat).((getl n (CHead c0 k t) (CHead d0 (Bind Abbr) u0)) \to ((subst0 n u0 t5 -t6) \to (pr3 (CHead c0 k t1) t4 t6)))) (\lambda (H10: (getl O (CHead c0 k t) -(CHead d0 (Bind Abbr) u0))).(\lambda (H11: (subst0 O u0 t5 t6)).(K_ind -(\lambda (k0: K).((clear (CHead c0 k0 t) (CHead d0 (Bind Abbr) u0)) \to (pr3 -(CHead c0 k0 t1) t4 t6))) (\lambda (b: B).(\lambda (H12: (clear (CHead c0 -(Bind b) t) (CHead d0 (Bind Abbr) u0))).(let H13 \def (f_equal C C (\lambda -(e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d0 -| (CHead c2 _ _) \Rightarrow c2])) (CHead d0 (Bind Abbr) u0) (CHead c0 (Bind -b) t) (clear_gen_bind b c0 (CHead d0 (Bind Abbr) u0) t H12)) in ((let H14 -\def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) -with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow (match k0 in K -return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) -\Rightarrow Abbr])])) (CHead d0 (Bind Abbr) u0) (CHead c0 (Bind b) t) -(clear_gen_bind b c0 (CHead d0 (Bind Abbr) u0) t H12)) in ((let H15 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow u0 | (CHead _ _ t7) \Rightarrow t7])) (CHead d0 (Bind -Abbr) u0) (CHead c0 (Bind b) t) (clear_gen_bind b c0 (CHead d0 (Bind Abbr) -u0) t H12)) in (\lambda (H16: (eq B Abbr b)).(\lambda (_: (eq C d0 c0)).(let -H18 \def (eq_ind T u0 (\lambda (t7: T).(subst0 O t7 t5 t6)) H11 t H15) in -(eq_ind B Abbr (\lambda (b0: B).(pr3 (CHead c0 (Bind b0) t1) t4 t6)) (ex2_ind -T (\lambda (t7: T).(subst0 O t2 t5 t7)) (\lambda (t7: T).(subst0 (S (plus i -O)) u t7 t6)) (pr3 (CHead c0 (Bind Abbr) t1) t4 t6) (\lambda (x: T).(\lambda -(H19: (subst0 O t2 t5 x)).(\lambda (H20: (subst0 (S (plus i O)) u x t6)).(let -H21 \def (f_equal nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O -i))) in (let H22 \def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n -u x t6)) H20 (S i) H21) in (ex2_ind T (\lambda (t7: T).(subst0 O t1 t5 t7)) -(\lambda (t7: T).(pr0 t7 x)) (pr3 (CHead c0 (Bind Abbr) t1) t4 t6) (\lambda -(x0: T).(\lambda (H23: (subst0 O t1 t5 x0)).(\lambda (H24: (pr0 x0 -x)).(pr3_sing (CHead c0 (Bind Abbr) t1) x0 t4 (pr2_delta (CHead c0 (Bind -Abbr) t1) c0 t1 O (getl_refl Abbr c0 t1) t4 t5 H6 x0 H23) t6 (pr3_pr2 (CHead -c0 (Bind Abbr) t1) x0 t6 (pr2_delta (CHead c0 (Bind Abbr) t1) d u (S i) -(getl_clear_bind Abbr (CHead c0 (Bind Abbr) t1) c0 t1 (clear_bind Abbr c0 t1) -(CHead d (Bind Abbr) u) i H0) x0 x H24 t6 H22)))))) (pr0_subst0_back t2 t5 x -O H19 t1 H1))))))) (subst0_subst0 t5 t6 t O H18 t2 u i H2)) b H16))))) H14)) -H13)))) (\lambda (f: F).(\lambda (H12: (clear (CHead c0 (Flat f) t) (CHead d0 -(Bind Abbr) u0))).(pr3_pr2 (CHead c0 (Flat f) t1) t4 t6 (pr2_cflat c0 t4 t6 -(pr2_delta c0 d0 u0 O (getl_intro O c0 (CHead d0 (Bind Abbr) u0) c0 -(drop_refl c0) (clear_gen_flat f c0 (CHead d0 (Bind Abbr) u0) t H12)) t4 t5 -H6 t6 H11) f t1)))) k (getl_gen_O (CHead c0 k t) (CHead d0 (Bind Abbr) u0) -H10)))) (\lambda (i1: nat).(\lambda (_: (((getl i1 (CHead c0 k t) (CHead d0 -(Bind Abbr) u0)) \to ((subst0 i1 u0 t5 t6) \to (pr3 (CHead c0 k t1) t4 -t6))))).(\lambda (H10: (getl (S i1) (CHead c0 k t) (CHead d0 (Bind Abbr) -u0))).(\lambda (H11: (subst0 (S i1) u0 t5 t6)).(K_ind (\lambda (k0: K).((getl -(S i1) (CHead c0 k0 t) (CHead d0 (Bind Abbr) u0)) \to (pr3 (CHead c0 k0 t1) -t4 t6))) (\lambda (b: B).(\lambda (H12: (getl (S i1) (CHead c0 (Bind b) t) -(CHead d0 (Bind Abbr) u0))).(pr3_pr2 (CHead c0 (Bind b) t1) t4 t6 (pr2_delta -(CHead c0 (Bind b) t1) d0 u0 (S i1) (getl_head (Bind b) i1 c0 (CHead d0 (Bind -Abbr) u0) (getl_gen_S (Bind b) c0 (CHead d0 (Bind Abbr) u0) t i1 H12) t1) t4 -t5 H6 t6 H11)))) (\lambda (f: F).(\lambda (H12: (getl (S i1) (CHead c0 (Flat -f) t) (CHead d0 (Bind Abbr) u0))).(pr3_pr2 (CHead c0 (Flat f) t1) t4 t6 -(pr2_cflat c0 t4 t6 (pr2_delta c0 d0 u0 (r (Flat f) i1) (getl_gen_S (Flat f) -c0 (CHead d0 (Bind Abbr) u0) t i1 H12) t4 t5 H6 t6 H11) f t1)))) k H10))))) -i0 H9 H7))))))))))))) y t0 t3 H4))) H3))))))))))))))) c u1 u2 H)))). -(* COMMENTS -Initial nodes: 1697 -END *) - -theorem pr3_pr2_pr3_t: - \forall (c: C).(\forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall -(k: K).((pr3 (CHead c k u2) t1 t2) \to (\forall (u1: T).((pr2 c u1 u2) \to -(pr3 (CHead c k u1) t1 t2)))))))) -\def - \lambda (c: C).(\lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(k: K).(\lambda (H: (pr3 (CHead c k u2) t1 t2)).(pr3_ind (CHead c k u2) -(\lambda (t: T).(\lambda (t0: T).(\forall (u1: T).((pr2 c u1 u2) \to (pr3 -(CHead c k u1) t t0))))) (\lambda (t: T).(\lambda (u1: T).(\lambda (_: (pr2 c -u1 u2)).(pr3_refl (CHead c k u1) t)))) (\lambda (t0: T).(\lambda (t3: -T).(\lambda (H0: (pr2 (CHead c k u2) t3 t0)).(\lambda (t4: T).(\lambda (_: -(pr3 (CHead c k u2) t0 t4)).(\lambda (H2: ((\forall (u1: T).((pr2 c u1 u2) -\to (pr3 (CHead c k u1) t0 t4))))).(\lambda (u1: T).(\lambda (H3: (pr2 c u1 -u2)).(pr3_t t0 t3 (CHead c k u1) (pr3_pr2_pr2_t c u1 u2 H3 t3 t0 k H0) t4 (H2 -u1 H3)))))))))) t1 t2 H)))))). -(* COMMENTS -Initial nodes: 199 -END *) - -theorem pr3_pr3_pr3_t: - \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall -(t1: T).(\forall (t2: T).(\forall (k: K).((pr3 (CHead c k u2) t1 t2) \to (pr3 -(CHead c k u1) t1 t2)))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1 -u2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (t1: T).(\forall -(t2: T).(\forall (k: K).((pr3 (CHead c k t0) t1 t2) \to (pr3 (CHead c k t) t1 -t2))))))) (\lambda (t: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k: -K).(\lambda (H0: (pr3 (CHead c k t) t1 t2)).H0))))) (\lambda (t2: T).(\lambda -(t1: T).(\lambda (H0: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda (_: (pr3 c t2 -t3)).(\lambda (H2: ((\forall (t4: T).(\forall (t5: T).(\forall (k: K).((pr3 -(CHead c k t3) t4 t5) \to (pr3 (CHead c k t2) t4 t5))))))).(\lambda (t0: -T).(\lambda (t4: T).(\lambda (k: K).(\lambda (H3: (pr3 (CHead c k t3) t0 -t4)).(pr3_pr2_pr3_t c t2 t0 t4 k (H2 t0 t4 k H3) t1 H0))))))))))) u1 u2 H)))). -(* COMMENTS -Initial nodes: 187 -END *) - -theorem pr3_lift: - \forall (c: C).(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h -d c e) \to (\forall (t1: T).(\forall (t2: T).((pr3 e t1 t2) \to (pr3 c (lift -h d t1) (lift h d t2))))))))) -\def - \lambda (c: C).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda -(H: (drop h d c e)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 e t1 -t2)).(pr3_ind e (\lambda (t: T).(\lambda (t0: T).(pr3 c (lift h d t) (lift h -d t0)))) (\lambda (t: T).(pr3_refl c (lift h d t))) (\lambda (t0: T).(\lambda -(t3: T).(\lambda (H1: (pr2 e t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 e t0 -t4)).(\lambda (H3: (pr3 c (lift h d t0) (lift h d t4))).(pr3_sing c (lift h d -t0) (lift h d t3) (pr2_lift c e h d H t3 t0 H1) (lift h d t4) H3))))))) t1 t2 -H0)))))))). -(* COMMENTS -Initial nodes: 167 -END *) - -theorem pr3_eta: - \forall (c: C).(\forall (w: T).(\forall (u: T).(let t \def (THead (Bind -Abst) w u) in (\forall (v: T).((pr3 c v w) \to (pr3 c (THead (Bind Abst) v -(THead (Flat Appl) (TLRef O) (lift (S O) O t))) t)))))) -\def - \lambda (c: C).(\lambda (w: T).(\lambda (u: T).(let t \def (THead (Bind -Abst) w u) in (\lambda (v: T).(\lambda (H: (pr3 c v w)).(eq_ind_r T (THead -(Bind Abst) (lift (S O) O w) (lift (S O) (S O) u)) (\lambda (t0: T).(pr3 c -(THead (Bind Abst) v (THead (Flat Appl) (TLRef O) t0)) (THead (Bind Abst) w -u))) (pr3_head_12 c v w H (Bind Abst) (THead (Flat Appl) (TLRef O) (THead -(Bind Abst) (lift (S O) O w) (lift (S O) (S O) u))) u (pr3_pr1 (THead (Flat -Appl) (TLRef O) (THead (Bind Abst) (lift (S O) O w) (lift (S O) (S O) u))) u -(pr1_sing (THead (Bind Abbr) (TLRef O) (lift (S O) (S O) u)) (THead (Flat -Appl) (TLRef O) (THead (Bind Abst) (lift (S O) O w) (lift (S O) (S O) u))) -(pr0_beta (lift (S O) O w) (TLRef O) (TLRef O) (pr0_refl (TLRef O)) (lift (S -O) (S O) u) (lift (S O) (S O) u) (pr0_refl (lift (S O) (S O) u))) u (pr1_sing -(THead (Bind Abbr) (TLRef O) (lift (S O) O u)) (THead (Bind Abbr) (TLRef O) -(lift (S O) (S O) u)) (pr0_delta1 (TLRef O) (TLRef O) (pr0_refl (TLRef O)) -(lift (S O) (S O) u) (lift (S O) (S O) u) (pr0_refl (lift (S O) (S O) u)) -(lift (S O) O u) (subst1_lift_S u O O (le_n O))) u (pr1_pr0 (THead (Bind -Abbr) (TLRef O) (lift (S O) O u)) u (pr0_zeta Abbr not_abbr_abst u u -(pr0_refl u) (TLRef O))))) (CHead c (Bind Abst) w))) (lift (S O) O (THead -(Bind Abst) w u)) (lift_bind Abst w u (S O) O))))))). -(* COMMENTS -Initial nodes: 523 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/subst1.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/subst1.ma deleted file mode 100644 index 3bcbd4965..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/subst1.ma +++ /dev/null @@ -1,95 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr3/defs.ma". - -include "Basic-1/pr2/subst1.ma". - -theorem pr3_subst1: - \forall (c: C).(\forall (e: C).(\forall (v: T).(\forall (i: nat).((getl i c -(CHead e (Bind Abbr) v)) \to (\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) -\to (\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T (\lambda (w2: T).(pr3 c -w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2)))))))))))) -\def - \lambda (c: C).(\lambda (e: C).(\lambda (v: T).(\lambda (i: nat).(\lambda -(H: (getl i c (CHead e (Bind Abbr) v))).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (H0: (pr3 c t1 t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: -T).(\forall (w1: T).((subst1 i v t w1) \to (ex2 T (\lambda (w2: T).(pr3 c w1 -w2)) (\lambda (w2: T).(subst1 i v t0 w2))))))) (\lambda (t: T).(\lambda (w1: -T).(\lambda (H1: (subst1 i v t w1)).(ex_intro2 T (\lambda (w2: T).(pr3 c w1 -w2)) (\lambda (w2: T).(subst1 i v t w2)) w1 (pr3_refl c w1) H1)))) (\lambda -(t3: T).(\lambda (t4: T).(\lambda (H1: (pr2 c t4 t3)).(\lambda (t5: -T).(\lambda (_: (pr3 c t3 t5)).(\lambda (H3: ((\forall (w1: T).((subst1 i v -t3 w1) \to (ex2 T (\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1 i -v t5 w2))))))).(\lambda (w1: T).(\lambda (H4: (subst1 i v t4 w1)).(ex2_ind T -(\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t3 w2)) (ex2 T -(\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1 i v t5 w2))) -(\lambda (x: T).(\lambda (H5: (pr2 c w1 x)).(\lambda (H6: (subst1 i v t3 -x)).(ex2_ind T (\lambda (w2: T).(pr3 c x w2)) (\lambda (w2: T).(subst1 i v t5 -w2)) (ex2 T (\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1 i v t5 -w2))) (\lambda (x0: T).(\lambda (H7: (pr3 c x x0)).(\lambda (H8: (subst1 i v -t5 x0)).(ex_intro2 T (\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1 -i v t5 w2)) x0 (pr3_sing c x w1 H5 x0 H7) H8)))) (H3 x H6))))) (pr2_subst1 c -e v i H t4 t3 H1 w1 H4)))))))))) t1 t2 H0)))))))). -(* COMMENTS -Initial nodes: 425 -END *) - -theorem pr3_gen_cabbr: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall -(e: C).(\forall (u: T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) -\to (\forall (a0: C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d -a0 a) \to (\forall (x1: T).((subst1 d u t1 (lift (S O) d x1)) \to (ex2 T -(\lambda (x2: T).(subst1 d u t2 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a -x1 x2)))))))))))))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 -t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (e: C).(\forall (u: -T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) \to (\forall (a0: -C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (\forall -(x1: T).((subst1 d u t (lift (S O) d x1)) \to (ex2 T (\lambda (x2: T).(subst1 -d u t0 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x1 x2))))))))))))))) -(\lambda (t: T).(\lambda (e: C).(\lambda (u: T).(\lambda (d: nat).(\lambda -(_: (getl d c (CHead e (Bind Abbr) u))).(\lambda (a0: C).(\lambda (_: -(csubst1 d u c a0)).(\lambda (a: C).(\lambda (_: (drop (S O) d a0 -a)).(\lambda (x1: T).(\lambda (H3: (subst1 d u t (lift (S O) d -x1))).(ex_intro2 T (\lambda (x2: T).(subst1 d u t (lift (S O) d x2))) -(\lambda (x2: T).(pr3 a x1 x2)) x1 H3 (pr3_refl a x1))))))))))))) (\lambda -(t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c t3 t0)).(\lambda (t4: -T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2: ((\forall (e: C).(\forall (u: -T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) \to (\forall (a0: -C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (\forall -(x1: T).((subst1 d u t0 (lift (S O) d x1)) \to (ex2 T (\lambda (x2: -T).(subst1 d u t4 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x1 -x2))))))))))))))).(\lambda (e: C).(\lambda (u: T).(\lambda (d: nat).(\lambda -(H3: (getl d c (CHead e (Bind Abbr) u))).(\lambda (a0: C).(\lambda (H4: -(csubst1 d u c a0)).(\lambda (a: C).(\lambda (H5: (drop (S O) d a0 -a)).(\lambda (x1: T).(\lambda (H6: (subst1 d u t3 (lift (S O) d -x1))).(ex2_ind T (\lambda (x2: T).(subst1 d u t0 (lift (S O) d x2))) (\lambda -(x2: T).(pr2 a x1 x2)) (ex2 T (\lambda (x2: T).(subst1 d u t4 (lift (S O) d -x2))) (\lambda (x2: T).(pr3 a x1 x2))) (\lambda (x: T).(\lambda (H7: (subst1 -d u t0 (lift (S O) d x))).(\lambda (H8: (pr2 a x1 x)).(ex2_ind T (\lambda -(x2: T).(subst1 d u t4 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x x2)) -(ex2 T (\lambda (x2: T).(subst1 d u t4 (lift (S O) d x2))) (\lambda (x2: -T).(pr3 a x1 x2))) (\lambda (x0: T).(\lambda (H9: (subst1 d u t4 (lift (S O) -d x0))).(\lambda (H10: (pr3 a x x0)).(ex_intro2 T (\lambda (x2: T).(subst1 d -u t4 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x1 x2)) x0 H9 (pr3_sing a x -x1 H8 x0 H10))))) (H2 e u d H3 a0 H4 a H5 x H7))))) (pr2_gen_cabbr c t3 t0 H0 -e u d H3 a0 H4 a H5 x1 H6)))))))))))))))))) t1 t2 H)))). -(* COMMENTS -Initial nodes: 731 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/wcpr0.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/wcpr0.ma deleted file mode 100644 index 052f4e5b2..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/pr3/wcpr0.ma +++ /dev/null @@ -1,66 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr3/props.ma". - -include "Basic-1/wcpr0/getl.ma". - -theorem pr3_wcpr0_t: - \forall (c1: C).(\forall (c2: C).((wcpr0 c2 c1) \to (\forall (t1: -T).(\forall (t2: T).((pr3 c1 t1 t2) \to (pr3 c2 t1 t2)))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c2 c1)).(wcpr0_ind -(\lambda (c: C).(\lambda (c0: C).(\forall (t1: T).(\forall (t2: T).((pr3 c0 -t1 t2) \to (pr3 c t1 t2)))))) (\lambda (c: C).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (H0: (pr3 c t1 t2)).H0)))) (\lambda (c0: C).(\lambda (c3: -C).(\lambda (H0: (wcpr0 c0 c3)).(\lambda (_: ((\forall (t1: T).(\forall (t2: -T).((pr3 c3 t1 t2) \to (pr3 c0 t1 t2)))))).(\lambda (u1: T).(\lambda (u2: -T).(\lambda (H2: (pr0 u1 u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (H3: (pr3 (CHead c3 k u2) t1 t2)).(pr3_ind (CHead c3 k u1) -(\lambda (t: T).(\lambda (t0: T).(pr3 (CHead c0 k u1) t t0))) (\lambda (t: -T).(pr3_refl (CHead c0 k u1) t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda -(H4: (pr2 (CHead c3 k u1) t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 (CHead -c3 k u1) t0 t4)).(\lambda (H6: (pr3 (CHead c0 k u1) t0 t4)).(pr3_t t0 t3 -(CHead c0 k u1) (insert_eq C (CHead c3 k u1) (\lambda (c: C).(pr2 c t3 t0)) -(\lambda (_: C).(pr3 (CHead c0 k u1) t3 t0)) (\lambda (y: C).(\lambda (H7: -(pr2 y t3 t0)).(pr2_ind (\lambda (c: C).(\lambda (t: T).(\lambda (t5: T).((eq -C c (CHead c3 k u1)) \to (pr3 (CHead c0 k u1) t t5))))) (\lambda (c: -C).(\lambda (t5: T).(\lambda (t6: T).(\lambda (H8: (pr0 t5 t6)).(\lambda (_: -(eq C c (CHead c3 k u1))).(pr3_pr2 (CHead c0 k u1) t5 t6 (pr2_free (CHead c0 -k u1) t5 t6 H8))))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: -T).(\lambda (i: nat).(\lambda (H8: (getl i c (CHead d (Bind Abbr) -u))).(\lambda (t5: T).(\lambda (t6: T).(\lambda (H9: (pr0 t5 t6)).(\lambda -(t: T).(\lambda (H10: (subst0 i u t6 t)).(\lambda (H11: (eq C c (CHead c3 k -u1))).(let H12 \def (eq_ind C c (\lambda (c4: C).(getl i c4 (CHead d (Bind -Abbr) u))) H8 (CHead c3 k u1) H11) in (ex3_2_ind C T (\lambda (e2: -C).(\lambda (u3: T).(getl i (CHead c0 k u1) (CHead e2 (Bind Abbr) u3)))) -(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 d))) (\lambda (_: C).(\lambda (u3: -T).(pr0 u3 u))) (pr3 (CHead c0 k u1) t5 t) (\lambda (x0: C).(\lambda (x1: -T).(\lambda (H13: (getl i (CHead c0 k u1) (CHead x0 (Bind Abbr) -x1))).(\lambda (_: (wcpr0 x0 d)).(\lambda (H15: (pr0 x1 u)).(ex2_ind T -(\lambda (t7: T).(subst0 i x1 t6 t7)) (\lambda (t7: T).(pr0 t7 t)) (pr3 -(CHead c0 k u1) t5 t) (\lambda (x: T).(\lambda (H16: (subst0 i x1 t6 -x)).(\lambda (H17: (pr0 x t)).(pr3_sing (CHead c0 k u1) x t5 (pr2_delta -(CHead c0 k u1) x0 x1 i H13 t5 t6 H9 x H16) t (pr3_pr2 (CHead c0 k u1) x t -(pr2_free (CHead c0 k u1) x t H17)))))) (pr0_subst0_back u t6 t i H10 x1 -H15))))))) (wcpr0_getl_back (CHead c3 k u1) (CHead c0 k u1) (wcpr0_comp c0 c3 -H0 u1 u1 (pr0_refl u1) k) i d u (Bind Abbr) H12)))))))))))))) y t3 t0 H7))) -H4) t4 H6))))))) t1 t2 (pr3_pr2_pr3_t c3 u2 t1 t2 k H3 u1 (pr2_free c3 u1 u2 -H2)))))))))))))) c2 c1 H))). -(* COMMENTS -Initial nodes: 799 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/preamble.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/preamble.ma deleted file mode 100644 index c9d6a172a..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/preamble.ma +++ /dev/null @@ -1,15 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "Ground-1/theory.ma". diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/r/defs.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/r/defs.ma deleted file mode 100644 index cde40fdf8..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/r/defs.ma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/T/defs.ma". - -definition r: - K \to (nat \to nat) -\def - \lambda (k: K).(\lambda (i: nat).(match k with [(Bind _) \Rightarrow i | -(Flat _) \Rightarrow (S i)])). - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/r/props.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/r/props.ma deleted file mode 100644 index 0815aaf5e..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/r/props.ma +++ /dev/null @@ -1,117 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/r/defs.ma". - -include "Basic-1/s/defs.ma". - -theorem r_S: - \forall (k: K).(\forall (i: nat).(eq nat (r k (S i)) (S (r k i)))) -\def - \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(eq nat (r k0 (S -i)) (S (r k0 i))))) (\lambda (b: B).(\lambda (i: nat).(refl_equal nat (S (r -(Bind b) i))))) (\lambda (f: F).(\lambda (i: nat).(refl_equal nat (S (r (Flat -f) i))))) k). -(* COMMENTS -Initial nodes: 65 -END *) - -theorem r_plus: - \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (r k (plus i j)) -(plus (r k i) j)))) -\def - \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: -nat).(eq nat (r k0 (plus i j)) (plus (r k0 i) j))))) (\lambda (b: B).(\lambda -(i: nat).(\lambda (j: nat).(refl_equal nat (plus (r (Bind b) i) j))))) -(\lambda (f: F).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (plus (r -(Flat f) i) j))))) k). -(* COMMENTS -Initial nodes: 79 -END *) - -theorem r_plus_sym: - \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (r k (plus i j)) -(plus i (r k j))))) -\def - \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: -nat).(eq nat (r k0 (plus i j)) (plus i (r k0 j)))))) (\lambda (_: B).(\lambda -(i: nat).(\lambda (j: nat).(refl_equal nat (plus i j))))) (\lambda (_: -F).(\lambda (i: nat).(\lambda (j: nat).(plus_n_Sm i j)))) k). -(* COMMENTS -Initial nodes: 63 -END *) - -theorem r_minus: - \forall (i: nat).(\forall (n: nat).((lt n i) \to (\forall (k: K).(eq nat -(minus (r k i) (S n)) (r k (minus i (S n))))))) -\def - \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (lt n i)).(\lambda (k: -K).(K_ind (\lambda (k0: K).(eq nat (minus (r k0 i) (S n)) (r k0 (minus i (S -n))))) (\lambda (_: B).(refl_equal nat (minus i (S n)))) (\lambda (_: -F).(minus_x_Sy i n H)) k)))). -(* COMMENTS -Initial nodes: 69 -END *) - -theorem r_dis: - \forall (k: K).(\forall (P: Prop).(((((\forall (i: nat).(eq nat (r k i) i))) -\to P)) \to (((((\forall (i: nat).(eq nat (r k i) (S i)))) \to P)) \to P))) -\def - \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (P: Prop).(((((\forall (i: -nat).(eq nat (r k0 i) i))) \to P)) \to (((((\forall (i: nat).(eq nat (r k0 i) -(S i)))) \to P)) \to P)))) (\lambda (b: B).(\lambda (P: Prop).(\lambda (H: -((((\forall (i: nat).(eq nat (r (Bind b) i) i))) \to P))).(\lambda (_: -((((\forall (i: nat).(eq nat (r (Bind b) i) (S i)))) \to P))).(H (\lambda (i: -nat).(refl_equal nat i))))))) (\lambda (f: F).(\lambda (P: Prop).(\lambda (_: -((((\forall (i: nat).(eq nat (r (Flat f) i) i))) \to P))).(\lambda (H0: -((((\forall (i: nat).(eq nat (r (Flat f) i) (S i)))) \to P))).(H0 (\lambda -(i: nat).(refl_equal nat (S i)))))))) k). -(* COMMENTS -Initial nodes: 151 -END *) - -theorem s_r: - \forall (k: K).(\forall (i: nat).(eq nat (s k (r k i)) (S i))) -\def - \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(eq nat (s k0 (r k0 -i)) (S i)))) (\lambda (_: B).(\lambda (i: nat).(refl_equal nat (S i)))) -(\lambda (_: F).(\lambda (i: nat).(refl_equal nat (S i)))) k). -(* COMMENTS -Initial nodes: 51 -END *) - -theorem r_arith0: - \forall (k: K).(\forall (i: nat).(eq nat (minus (r k (S i)) (S O)) (r k i))) -\def - \lambda (k: K).(\lambda (i: nat).(eq_ind_r nat (S (r k i)) (\lambda (n: -nat).(eq nat (minus n (S O)) (r k i))) (eq_ind_r nat (r k i) (\lambda (n: -nat).(eq nat n (r k i))) (refl_equal nat (r k i)) (minus (S (r k i)) (S O)) -(minus_Sx_SO (r k i))) (r k (S i)) (r_S k i))). -(* COMMENTS -Initial nodes: 105 -END *) - -theorem r_arith1: - \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (minus (r k (S -i)) (S j)) (minus (r k i) j)))) -\def - \lambda (k: K).(\lambda (i: nat).(\lambda (j: nat).(eq_ind_r nat (S (r k i)) -(\lambda (n: nat).(eq nat (minus n (S j)) (minus (r k i) j))) (refl_equal nat -(minus (r k i) j)) (r k (S i)) (r_S k i)))). -(* COMMENTS -Initial nodes: 69 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/s/defs.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/s/defs.ma deleted file mode 100644 index 0d1fb7914..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/s/defs.ma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/T/defs.ma". - -definition s: - K \to (nat \to nat) -\def - \lambda (k: K).(\lambda (i: nat).(match k with [(Bind _) \Rightarrow (S i) | -(Flat _) \Rightarrow i])). - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/s/props.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/s/props.ma deleted file mode 100644 index 3cb4fbd74..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/s/props.ma +++ /dev/null @@ -1,151 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/s/defs.ma". - -theorem s_S: - \forall (k: K).(\forall (i: nat).(eq nat (s k (S i)) (S (s k i)))) -\def - \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(eq nat (s k0 (S -i)) (S (s k0 i))))) (\lambda (b: B).(\lambda (i: nat).(refl_equal nat (S (s -(Bind b) i))))) (\lambda (f: F).(\lambda (i: nat).(refl_equal nat (S (s (Flat -f) i))))) k). -(* COMMENTS -Initial nodes: 65 -END *) - -theorem s_plus: - \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (s k (plus i j)) -(plus (s k i) j)))) -\def - \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: -nat).(eq nat (s k0 (plus i j)) (plus (s k0 i) j))))) (\lambda (b: B).(\lambda -(i: nat).(\lambda (j: nat).(refl_equal nat (plus (s (Bind b) i) j))))) -(\lambda (f: F).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (plus (s -(Flat f) i) j))))) k). -(* COMMENTS -Initial nodes: 79 -END *) - -theorem s_plus_sym: - \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (s k (plus i j)) -(plus i (s k j))))) -\def - \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: -nat).(eq nat (s k0 (plus i j)) (plus i (s k0 j)))))) (\lambda (_: B).(\lambda -(i: nat).(\lambda (j: nat).(eq_ind_r nat (plus i (S j)) (\lambda (n: nat).(eq -nat n (plus i (S j)))) (refl_equal nat (plus i (S j))) (S (plus i j)) -(plus_n_Sm i j))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (j: -nat).(refl_equal nat (plus i (s (Flat f) j)))))) k). -(* COMMENTS -Initial nodes: 117 -END *) - -theorem s_minus: - \forall (k: K).(\forall (i: nat).(\forall (j: nat).((le j i) \to (eq nat (s -k (minus i j)) (minus (s k i) j))))) -\def - \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: -nat).((le j i) \to (eq nat (s k0 (minus i j)) (minus (s k0 i) j)))))) -(\lambda (_: B).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (le j -i)).(eq_ind_r nat (minus (S i) j) (\lambda (n: nat).(eq nat n (minus (S i) -j))) (refl_equal nat (minus (S i) j)) (S (minus i j)) (minus_Sn_m i j H)))))) -(\lambda (f: F).(\lambda (i: nat).(\lambda (j: nat).(\lambda (_: (le j -i)).(refl_equal nat (minus (s (Flat f) i) j)))))) k). -(* COMMENTS -Initial nodes: 137 -END *) - -theorem minus_s_s: - \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (minus (s k i) (s -k j)) (minus i j)))) -\def - \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: -nat).(eq nat (minus (s k0 i) (s k0 j)) (minus i j))))) (\lambda (_: -B).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (minus i j))))) -(\lambda (_: F).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (minus i -j))))) k). -(* COMMENTS -Initial nodes: 67 -END *) - -theorem s_le: - \forall (k: K).(\forall (i: nat).(\forall (j: nat).((le i j) \to (le (s k i) -(s k j))))) -\def - \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: -nat).((le i j) \to (le (s k0 i) (s k0 j)))))) (\lambda (_: B).(\lambda (i: -nat).(\lambda (j: nat).(\lambda (H: (le i j)).(le_n_S i j H))))) (\lambda (_: -F).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (le i j)).H)))) k). -(* COMMENTS -Initial nodes: 65 -END *) - -theorem s_lt: - \forall (k: K).(\forall (i: nat).(\forall (j: nat).((lt i j) \to (lt (s k i) -(s k j))))) -\def - \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: -nat).((lt i j) \to (lt (s k0 i) (s k0 j)))))) (\lambda (_: B).(\lambda (i: -nat).(\lambda (j: nat).(\lambda (H: (lt i j)).(le_n_S (S i) j H))))) (\lambda -(_: F).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (lt i j)).H)))) k). -(* COMMENTS -Initial nodes: 67 -END *) - -theorem s_inj: - \forall (k: K).(\forall (i: nat).(\forall (j: nat).((eq nat (s k i) (s k j)) -\to (eq nat i j)))) -\def - \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: -nat).((eq nat (s k0 i) (s k0 j)) \to (eq nat i j))))) (\lambda (b: -B).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (eq nat (s (Bind b) i) (s -(Bind b) j))).(eq_add_S i j H))))) (\lambda (f: F).(\lambda (i: nat).(\lambda -(j: nat).(\lambda (H: (eq nat (s (Flat f) i) (s (Flat f) j))).H)))) k). -(* COMMENTS -Initial nodes: 97 -END *) - -theorem s_inc: - \forall (k: K).(\forall (i: nat).(le i (s k i))) -\def - \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(le i (s k0 i)))) -(\lambda (b: B).(\lambda (i: nat).(le_S_n i (s (Bind b) i) (le_S (S i) (s -(Bind b) i) (le_n (s (Bind b) i)))))) (\lambda (f: F).(\lambda (i: nat).(le_n -(s (Flat f) i)))) k). -(* COMMENTS -Initial nodes: 73 -END *) - -theorem s_arith0: - \forall (k: K).(\forall (i: nat).(eq nat (minus (s k i) (s k O)) i)) -\def - \lambda (k: K).(\lambda (i: nat).(eq_ind_r nat (minus i O) (\lambda (n: -nat).(eq nat n i)) (eq_ind nat i (\lambda (n: nat).(eq nat n i)) (refl_equal -nat i) (minus i O) (minus_n_O i)) (minus (s k i) (s k O)) (minus_s_s k i O))). -(* COMMENTS -Initial nodes: 77 -END *) - -theorem s_arith1: - \forall (b: B).(\forall (i: nat).(eq nat (minus (s (Bind b) i) (S O)) i)) -\def - \lambda (_: B).(\lambda (i: nat).(eq_ind nat i (\lambda (n: nat).(eq nat n -i)) (refl_equal nat i) (minus i O) (minus_n_O i))). -(* COMMENTS -Initial nodes: 35 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/sc3/arity.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/sc3/arity.ma deleted file mode 100644 index 651321580..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/sc3/arity.ma +++ /dev/null @@ -1,322 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubc/arity.ma". - -include "Basic-1/csubc/getl.ma". - -include "Basic-1/csubc/drop1.ma". - -include "Basic-1/csubc/props.ma". - -theorem sc3_arity_csubc: - \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1 -t a) \to (\forall (d1: C).(\forall (is: PList).((drop1 is d1 c1) \to (\forall -(c2: C).((csubc g d1 c2) \to (sc3 g a c2 (lift1 is t))))))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: -(arity g c1 t a)).(arity_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda (a0: -A).(\forall (d1: C).(\forall (is: PList).((drop1 is d1 c) \to (\forall (c2: -C).((csubc g d1 c2) \to (sc3 g a0 c2 (lift1 is t0)))))))))) (\lambda (c: -C).(\lambda (n: nat).(\lambda (d1: C).(\lambda (is: PList).(\lambda (_: -(drop1 is d1 c)).(\lambda (c2: C).(\lambda (_: (csubc g d1 c2)).(eq_ind_r T -(TSort n) (\lambda (t0: T).(land (arity g c2 t0 (ASort O n)) (sn3 c2 t0))) -(conj (arity g c2 (TSort n) (ASort O n)) (sn3 c2 (TSort n)) (arity_sort g c2 -n) (sn3_nf2 c2 (TSort n) (nf2_sort c2 n))) (lift1 is (TSort n)) (lift1_sort n -is))))))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: -nat).(\lambda (H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (a0: -A).(\lambda (_: (arity g d u a0)).(\lambda (H2: ((\forall (d1: C).(\forall -(is: PList).((drop1 is d1 d) \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g -a0 c2 (lift1 is u))))))))).(\lambda (d1: C).(\lambda (is: PList).(\lambda -(H3: (drop1 is d1 c)).(\lambda (c2: C).(\lambda (H4: (csubc g d1 c2)).(let -H_x \def (drop1_getl_trans is c d1 H3 Abbr d u i H0) in (let H5 \def H_x in -(ex2_ind C (\lambda (e2: C).(drop1 (ptrans is i) e2 d)) (\lambda (e2: -C).(getl (trans is i) d1 (CHead e2 (Bind Abbr) (lift1 (ptrans is i) u)))) -(sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x: C).(\lambda (_: (drop1 -(ptrans is i) x d)).(\lambda (H7: (getl (trans is i) d1 (CHead x (Bind Abbr) -(lift1 (ptrans is i) u)))).(let H_x0 \def (csubc_getl_conf g d1 (CHead x -(Bind Abbr) (lift1 (ptrans is i) u)) (trans is i) H7 c2 H4) in (let H8 \def -H_x0 in (ex2_ind C (\lambda (e2: C).(getl (trans is i) c2 e2)) (\lambda (e2: -C).(csubc g (CHead x (Bind Abbr) (lift1 (ptrans is i) u)) e2)) (sc3 g a0 c2 -(lift1 is (TLRef i))) (\lambda (x0: C).(\lambda (H9: (getl (trans is i) c2 -x0)).(\lambda (H10: (csubc g (CHead x (Bind Abbr) (lift1 (ptrans is i) u)) -x0)).(let H_x1 \def (csubc_gen_head_l g x x0 (lift1 (ptrans is i) u) (Bind -Abbr) H10) in (let H11 \def H_x1 in (or3_ind (ex2 C (\lambda (c3: C).(eq C x0 -(CHead c3 (Bind Abbr) (lift1 (ptrans is i) u)))) (\lambda (c3: C).(csubc g x -c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K -(Bind Abbr) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: -A).(eq C x0 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: -T).(\lambda (_: A).(csubc g x c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans is i) u))))) (\lambda (c3: -C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3 w))))) (ex4_3 B C T (\lambda -(b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C x0 (CHead c3 (Bind b) v2))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K (Bind Abbr) (Bind -Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b -Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g x c3))))) -(sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (H12: (ex2 C (\lambda (c3: C).(eq -C x0 (CHead c3 (Bind Abbr) (lift1 (ptrans is i) u)))) (\lambda (c3: C).(csubc -g x c3)))).(ex2_ind C (\lambda (c3: C).(eq C x0 (CHead c3 (Bind Abbr) (lift1 -(ptrans is i) u)))) (\lambda (c3: C).(csubc g x c3)) (sc3 g a0 c2 (lift1 is -(TLRef i))) (\lambda (x1: C).(\lambda (H13: (eq C x0 (CHead x1 (Bind Abbr) -(lift1 (ptrans is i) u)))).(\lambda (_: (csubc g x x1)).(let H15 \def (eq_ind -C x0 (\lambda (c0: C).(getl (trans is i) c2 c0)) H9 (CHead x1 (Bind Abbr) -(lift1 (ptrans is i) u)) H13) in (let H_y \def (sc3_abbr g a0 TNil) in -(eq_ind_r T (TLRef (trans is i)) (\lambda (t0: T).(sc3 g a0 c2 t0)) (H_y -(trans is i) x1 (lift1 (ptrans is i) u) c2 (eq_ind T (lift1 is (lift (S i) O -u)) (\lambda (t0: T).(sc3 g a0 c2 t0)) (eq_ind T (lift1 (PConsTail is (S i) -O) u) (\lambda (t0: T).(sc3 g a0 c2 t0)) (H2 d1 (PConsTail is (S i) O) -(drop1_cons_tail c d (S i) O (getl_drop Abbr c d u i H0) is d1 H3) c2 H4) -(lift1 is (lift (S i) O u)) (lift1_cons_tail u (S i) O is)) (lift (S (trans -is i)) O (lift1 (ptrans is i) u)) (lift1_free is i u)) H15) (lift1 is (TLRef -i)) (lift1_lref is i))))))) H12)) (\lambda (H12: (ex5_3 C T A (\lambda (_: -C).(\lambda (_: T).(\lambda (_: A).(eq K (Bind Abbr) (Bind Abst))))) (\lambda -(c3: C).(\lambda (w: T).(\lambda (_: A).(eq C x0 (CHead c3 (Bind Abbr) w))))) -(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g x c3)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans -is i) u))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3 -w)))))).(ex5_3_ind C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq -K (Bind Abbr) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: -A).(eq C x0 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: -T).(\lambda (_: A).(csubc g x c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans is i) u))))) (\lambda (c3: -C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3 w)))) (sc3 g a0 c2 (lift1 is -(TLRef i))) (\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: A).(\lambda (H13: -(eq K (Bind Abbr) (Bind Abst))).(\lambda (H14: (eq C x0 (CHead x1 (Bind Abbr) -x2))).(\lambda (_: (csubc g x x1)).(\lambda (_: (sc3 g (asucc g x3) x (lift1 -(ptrans is i) u))).(\lambda (_: (sc3 g x3 x1 x2)).(let H18 \def (eq_ind C x0 -(\lambda (c0: C).(getl (trans is i) c2 c0)) H9 (CHead x1 (Bind Abbr) x2) H14) -in (let H19 \def (eq_ind K (Bind Abbr) (\lambda (ee: K).(match ee in K return -(\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B return -(\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | -Void \Rightarrow False]) | (Flat _) \Rightarrow False])) I (Bind Abst) H13) -in (False_ind (sc3 g a0 c2 (lift1 is (TLRef i))) H19))))))))))) H12)) -(\lambda (H12: (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: -T).(eq C x0 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: T).(eq K (Bind Abbr) (Bind Void))))) (\lambda (b: B).(\lambda -(_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: -C).(\lambda (_: T).(csubc g x c3)))))).(ex4_3_ind B C T (\lambda (b: -B).(\lambda (c3: C).(\lambda (v2: T).(eq C x0 (CHead c3 (Bind b) v2))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K (Bind Abbr) (Bind -Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b -Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g x c3)))) -(sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x1: B).(\lambda (x2: C).(\lambda -(x3: T).(\lambda (H13: (eq C x0 (CHead x2 (Bind x1) x3))).(\lambda (H14: (eq -K (Bind Abbr) (Bind Void))).(\lambda (_: (not (eq B x1 Void))).(\lambda (_: -(csubc g x x2)).(let H17 \def (eq_ind C x0 (\lambda (c0: C).(getl (trans is -i) c2 c0)) H9 (CHead x2 (Bind x1) x3) H13) in (let H18 \def (eq_ind K (Bind -Abbr) (\lambda (ee: K).(match ee in K return (\lambda (_: K).Prop) with -[(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr -\Rightarrow True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat -_) \Rightarrow False])) I (Bind Void) H14) in (False_ind (sc3 g a0 c2 (lift1 -is (TLRef i))) H18)))))))))) H12)) H11)))))) H8)))))) H5)))))))))))))))) -(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H0: (getl i c (CHead d (Bind Abst) u))).(\lambda (a0: A).(\lambda (H1: -(arity g d u (asucc g a0))).(\lambda (_: ((\forall (d1: C).(\forall (is: -PList).((drop1 is d1 d) \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g -(asucc g a0) c2 (lift1 is u))))))))).(\lambda (d1: C).(\lambda (is: -PList).(\lambda (H3: (drop1 is d1 c)).(\lambda (c2: C).(\lambda (H4: (csubc g -d1 c2)).(let H5 \def H0 in (let H_x \def (drop1_getl_trans is c d1 H3 Abst d -u i H5) in (let H6 \def H_x in (ex2_ind C (\lambda (e2: C).(drop1 (ptrans is -i) e2 d)) (\lambda (e2: C).(getl (trans is i) d1 (CHead e2 (Bind Abst) (lift1 -(ptrans is i) u)))) (sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x: -C).(\lambda (H7: (drop1 (ptrans is i) x d)).(\lambda (H8: (getl (trans is i) -d1 (CHead x (Bind Abst) (lift1 (ptrans is i) u)))).(let H_x0 \def -(csubc_getl_conf g d1 (CHead x (Bind Abst) (lift1 (ptrans is i) u)) (trans is -i) H8 c2 H4) in (let H9 \def H_x0 in (ex2_ind C (\lambda (e2: C).(getl (trans -is i) c2 e2)) (\lambda (e2: C).(csubc g (CHead x (Bind Abst) (lift1 (ptrans -is i) u)) e2)) (sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x0: C).(\lambda -(H10: (getl (trans is i) c2 x0)).(\lambda (H11: (csubc g (CHead x (Bind Abst) -(lift1 (ptrans is i) u)) x0)).(let H_x1 \def (csubc_gen_head_l g x x0 (lift1 -(ptrans is i) u) (Bind Abst) H11) in (let H12 \def H_x1 in (or3_ind (ex2 C -(\lambda (c3: C).(eq C x0 (CHead c3 (Bind Abst) (lift1 (ptrans is i) u)))) -(\lambda (c3: C).(csubc g x c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: -T).(\lambda (_: A).(eq K (Bind Abst) (Bind Abst))))) (\lambda (c3: -C).(\lambda (w: T).(\lambda (_: A).(eq C x0 (CHead c3 (Bind Abbr) w))))) -(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g x c3)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans -is i) u))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3 -w))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C -x0 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -T).(eq K (Bind Abst) (Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda -(_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: -T).(csubc g x c3))))) (sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (H13: (ex2 -C (\lambda (c3: C).(eq C x0 (CHead c3 (Bind Abst) (lift1 (ptrans is i) u)))) -(\lambda (c3: C).(csubc g x c3)))).(ex2_ind C (\lambda (c3: C).(eq C x0 -(CHead c3 (Bind Abst) (lift1 (ptrans is i) u)))) (\lambda (c3: C).(csubc g x -c3)) (sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x1: C).(\lambda (H14: (eq C -x0 (CHead x1 (Bind Abst) (lift1 (ptrans is i) u)))).(\lambda (_: (csubc g x -x1)).(let H16 \def (eq_ind C x0 (\lambda (c0: C).(getl (trans is i) c2 c0)) -H10 (CHead x1 (Bind Abst) (lift1 (ptrans is i) u)) H14) in (let H_y \def -(sc3_abst g a0 TNil) in (eq_ind_r T (TLRef (trans is i)) (\lambda (t0: -T).(sc3 g a0 c2 t0)) (H_y c2 (trans is i) (csubc_arity_conf g d1 c2 H4 (TLRef -(trans is i)) a0 (eq_ind T (lift1 is (TLRef i)) (\lambda (t0: T).(arity g d1 -t0 a0)) (arity_lift1 g a0 c is d1 (TLRef i) H3 (arity_abst g c d u i H0 a0 -H1)) (TLRef (trans is i)) (lift1_lref is i))) (nf2_lref_abst c2 x1 (lift1 -(ptrans is i) u) (trans is i) H16) I) (lift1 is (TLRef i)) (lift1_lref is -i))))))) H13)) (\lambda (H13: (ex5_3 C T A (\lambda (_: C).(\lambda (_: -T).(\lambda (_: A).(eq K (Bind Abst) (Bind Abst))))) (\lambda (c3: -C).(\lambda (w: T).(\lambda (_: A).(eq C x0 (CHead c3 (Bind Abbr) w))))) -(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g x c3)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans -is i) u))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3 -w)))))).(ex5_3_ind C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq -K (Bind Abst) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: -A).(eq C x0 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: -T).(\lambda (_: A).(csubc g x c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans is i) u))))) (\lambda (c3: -C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3 w)))) (sc3 g a0 c2 (lift1 is -(TLRef i))) (\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: A).(\lambda (_: -(eq K (Bind Abst) (Bind Abst))).(\lambda (H15: (eq C x0 (CHead x1 (Bind Abbr) -x2))).(\lambda (_: (csubc g x x1)).(\lambda (H17: (sc3 g (asucc g x3) x -(lift1 (ptrans is i) u))).(\lambda (H18: (sc3 g x3 x1 x2)).(let H19 \def -(eq_ind C x0 (\lambda (c0: C).(getl (trans is i) c2 c0)) H10 (CHead x1 (Bind -Abbr) x2) H15) in (let H_y \def (sc3_abbr g a0 TNil) in (eq_ind_r T (TLRef -(trans is i)) (\lambda (t0: T).(sc3 g a0 c2 t0)) (H_y (trans is i) x1 x2 c2 -(let H_y0 \def (arity_lift1 g (asucc g a0) d (ptrans is i) x u H7 H1) in (let -H_y1 \def (sc3_arity_gen g x (lift1 (ptrans is i) u) (asucc g x3) H17) in -(sc3_repl g x3 c2 (lift (S (trans is i)) O x2) (sc3_lift g x3 x1 x2 H18 c2 (S -(trans is i)) O (getl_drop Abbr c2 x1 x2 (trans is i) H19)) a0 (asucc_inj g -x3 a0 (arity_mono g x (lift1 (ptrans is i) u) (asucc g x3) H_y1 (asucc g a0) -H_y0))))) H19) (lift1 is (TLRef i)) (lift1_lref is i)))))))))))) H13)) -(\lambda (H13: (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: -T).(eq C x0 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (_: T).(eq K (Bind Abst) (Bind Void))))) (\lambda (b: B).(\lambda -(_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: -C).(\lambda (_: T).(csubc g x c3)))))).(ex4_3_ind B C T (\lambda (b: -B).(\lambda (c3: C).(\lambda (v2: T).(eq C x0 (CHead c3 (Bind b) v2))))) -(\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K (Bind Abst) (Bind -Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b -Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g x c3)))) -(sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x1: B).(\lambda (x2: C).(\lambda -(x3: T).(\lambda (H14: (eq C x0 (CHead x2 (Bind x1) x3))).(\lambda (H15: (eq -K (Bind Abst) (Bind Void))).(\lambda (_: (not (eq B x1 Void))).(\lambda (_: -(csubc g x x2)).(let H18 \def (eq_ind C x0 (\lambda (c0: C).(getl (trans is -i) c2 c0)) H10 (CHead x2 (Bind x1) x3) H14) in (let H19 \def (eq_ind K (Bind -Abst) (\lambda (ee: K).(match ee in K return (\lambda (_: K).Prop) with -[(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr -\Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat -_) \Rightarrow False])) I (Bind Void) H15) in (False_ind (sc3 g a0 c2 (lift1 -is (TLRef i))) H19)))))))))) H13)) H12)))))) H9)))))) H6))))))))))))))))) -(\lambda (b: B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c: C).(\lambda -(u: T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H2: -((\forall (d1: C).(\forall (is: PList).((drop1 is d1 c) \to (\forall (c2: -C).((csubc g d1 c2) \to (sc3 g a1 c2 (lift1 is u))))))))).(\lambda (t0: -T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c (Bind b) u) t0 -a2)).(\lambda (H4: ((\forall (d1: C).(\forall (is: PList).((drop1 is d1 -(CHead c (Bind b) u)) \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g a2 c2 -(lift1 is t0))))))))).(\lambda (d1: C).(\lambda (is: PList).(\lambda (H5: -(drop1 is d1 c)).(\lambda (c2: C).(\lambda (H6: (csubc g d1 c2)).(let H_y -\def (sc3_bind g b H0 a1 a2 TNil) in (eq_ind_r T (THead (Bind b) (lift1 is u) -(lift1 (Ss is) t0)) (\lambda (t1: T).(sc3 g a2 c2 t1)) (H_y c2 (lift1 is u) -(lift1 (Ss is) t0) (H4 (CHead d1 (Bind b) (lift1 is u)) (Ss is) -(drop1_skip_bind b c is d1 u H5) (CHead c2 (Bind b) (lift1 is u)) (csubc_head -g d1 c2 H6 (Bind b) (lift1 is u))) (H2 d1 is H5 c2 H6)) (lift1 is (THead -(Bind b) u t0)) (lift1_bind b is u t0))))))))))))))))))) (\lambda (c: -C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H0: (arity g c u (asucc g -a1))).(\lambda (H1: ((\forall (d1: C).(\forall (is: PList).((drop1 is d1 c) -\to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g (asucc g a1) c2 (lift1 is -u))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (H2: (arity g (CHead c -(Bind Abst) u) t0 a2)).(\lambda (H3: ((\forall (d1: C).(\forall (is: -PList).((drop1 is d1 (CHead c (Bind Abst) u)) \to (\forall (c2: C).((csubc g -d1 c2) \to (sc3 g a2 c2 (lift1 is t0))))))))).(\lambda (d1: C).(\lambda (is: -PList).(\lambda (H4: (drop1 is d1 c)).(\lambda (c2: C).(\lambda (H5: (csubc g -d1 c2)).(eq_ind_r T (THead (Bind Abst) (lift1 is u) (lift1 (Ss is) t0)) -(\lambda (t1: T).(land (arity g c2 t1 (AHead a1 a2)) (\forall (d: C).(\forall -(w: T).((sc3 g a1 d w) \to (\forall (is0: PList).((drop1 is0 d c2) \to (sc3 g -a2 d (THead (Flat Appl) w (lift1 is0 t1)))))))))) (conj (arity g c2 (THead -(Bind Abst) (lift1 is u) (lift1 (Ss is) t0)) (AHead a1 a2)) (\forall (d: -C).(\forall (w: T).((sc3 g a1 d w) \to (\forall (is0: PList).((drop1 is0 d -c2) \to (sc3 g a2 d (THead (Flat Appl) w (lift1 is0 (THead (Bind Abst) (lift1 -is u) (lift1 (Ss is) t0)))))))))) (csubc_arity_conf g d1 c2 H5 (THead (Bind -Abst) (lift1 is u) (lift1 (Ss is) t0)) (AHead a1 a2) (arity_head g d1 (lift1 -is u) a1 (arity_lift1 g (asucc g a1) c is d1 u H4 H0) (lift1 (Ss is) t0) a2 -(arity_lift1 g a2 (CHead c (Bind Abst) u) (Ss is) (CHead d1 (Bind Abst) -(lift1 is u)) t0 (drop1_skip_bind Abst c is d1 u H4) H2))) (\lambda (d: -C).(\lambda (w: T).(\lambda (H6: (sc3 g a1 d w)).(\lambda (is0: -PList).(\lambda (H7: (drop1 is0 d c2)).(eq_ind_r T (THead (Bind Abst) (lift1 -is0 (lift1 is u)) (lift1 (Ss is0) (lift1 (Ss is) t0))) (\lambda (t1: T).(sc3 -g a2 d (THead (Flat Appl) w t1))) (let H8 \def (sc3_appl g a1 a2 TNil) in (H8 -d w (lift1 (Ss is0) (lift1 (Ss is) t0)) (let H_y \def (sc3_bind g Abbr -(\lambda (H9: (eq B Abbr Abst)).(not_abbr_abst H9)) a1 a2 TNil) in (H_y d w -(lift1 (Ss is0) (lift1 (Ss is) t0)) (let H_x \def (csubc_drop1_conf_rev g is0 -d c2 H7 d1 H5) in (let H9 \def H_x in (ex2_ind C (\lambda (c3: C).(drop1 is0 -c3 d1)) (\lambda (c3: C).(csubc g c3 d)) (sc3 g a2 (CHead d (Bind Abbr) w) -(lift1 (Ss is0) (lift1 (Ss is) t0))) (\lambda (x: C).(\lambda (H10: (drop1 -is0 x d1)).(\lambda (H11: (csubc g x d)).(eq_ind_r T (lift1 (papp (Ss is0) -(Ss is)) t0) (\lambda (t1: T).(sc3 g a2 (CHead d (Bind Abbr) w) t1)) -(eq_ind_r PList (Ss (papp is0 is)) (\lambda (p: PList).(sc3 g a2 (CHead d -(Bind Abbr) w) (lift1 p t0))) (H3 (CHead x (Bind Abst) (lift1 (papp is0 is) -u)) (Ss (papp is0 is)) (drop1_skip_bind Abst c (papp is0 is) x u (drop1_trans -is0 x d1 H10 is c H4)) (CHead d (Bind Abbr) w) (csubc_abst g x d H11 (lift1 -(papp is0 is) u) a1 (H1 x (papp is0 is) (drop1_trans is0 x d1 H10 is c H4) x -(csubc_refl g x)) w H6)) (papp (Ss is0) (Ss is)) (papp_ss is0 is)) (lift1 (Ss -is0) (lift1 (Ss is) t0)) (lift1_lift1 (Ss is0) (Ss is) t0))))) H9))) H6)) H6 -(lift1 is0 (lift1 is u)) (sc3_lift1 g c2 (asucc g a1) is0 d (lift1 is u) (H1 -d1 is H4 c2 H5) H7))) (lift1 is0 (THead (Bind Abst) (lift1 is u) (lift1 (Ss -is) t0))) (lift1_bind Abst is0 (lift1 is u) (lift1 (Ss is) t0))))))))) (lift1 -is (THead (Bind Abst) u t0)) (lift1_bind Abst is u t0)))))))))))))))) -(\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c u -a1)).(\lambda (H1: ((\forall (d1: C).(\forall (is: PList).((drop1 is d1 c) -\to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g a1 c2 (lift1 is -u))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g c t0 -(AHead a1 a2))).(\lambda (H3: ((\forall (d1: C).(\forall (is: PList).((drop1 -is d1 c) \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g (AHead a1 a2) c2 -(lift1 is t0))))))))).(\lambda (d1: C).(\lambda (is: PList).(\lambda (H4: -(drop1 is d1 c)).(\lambda (c2: C).(\lambda (H5: (csubc g d1 c2)).(let H_y -\def (H1 d1 is H4 c2 H5) in (let H_y0 \def (H3 d1 is H4 c2 H5) in (let H6 -\def H_y0 in (land_ind (arity g c2 (lift1 is t0) (AHead a1 a2)) (\forall (d: -C).(\forall (w: T).((sc3 g a1 d w) \to (\forall (is0: PList).((drop1 is0 d -c2) \to (sc3 g a2 d (THead (Flat Appl) w (lift1 is0 (lift1 is t0))))))))) -(sc3 g a2 c2 (lift1 is (THead (Flat Appl) u t0))) (\lambda (_: (arity g c2 -(lift1 is t0) (AHead a1 a2))).(\lambda (H8: ((\forall (d: C).(\forall (w: -T).((sc3 g a1 d w) \to (\forall (is0: PList).((drop1 is0 d c2) \to (sc3 g a2 -d (THead (Flat Appl) w (lift1 is0 (lift1 is t0))))))))))).(let H_y1 \def (H8 -c2 (lift1 is u) H_y PNil) in (eq_ind_r T (THead (Flat Appl) (lift1 is u) -(lift1 is t0)) (\lambda (t1: T).(sc3 g a2 c2 t1)) (H_y1 (drop1_nil c2)) -(lift1 is (THead (Flat Appl) u t0)) (lift1_flat Appl is u t0))))) -H6)))))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a0: -A).(\lambda (_: (arity g c u (asucc g a0))).(\lambda (H1: ((\forall (d1: -C).(\forall (is: PList).((drop1 is d1 c) \to (\forall (c2: C).((csubc g d1 -c2) \to (sc3 g (asucc g a0) c2 (lift1 is u))))))))).(\lambda (t0: T).(\lambda -(_: (arity g c t0 a0)).(\lambda (H3: ((\forall (d1: C).(\forall (is: -PList).((drop1 is d1 c) \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g a0 -c2 (lift1 is t0))))))))).(\lambda (d1: C).(\lambda (is: PList).(\lambda (H4: -(drop1 is d1 c)).(\lambda (c2: C).(\lambda (H5: (csubc g d1 c2)).(let H_y -\def (sc3_cast g a0 TNil) in (eq_ind_r T (THead (Flat Cast) (lift1 is u) -(lift1 is t0)) (\lambda (t1: T).(sc3 g a0 c2 t1)) (H_y c2 (lift1 is u) (H1 d1 -is H4 c2 H5) (lift1 is t0) (H3 d1 is H4 c2 H5)) (lift1 is (THead (Flat Cast) -u t0)) (lift1_flat Cast is u t0)))))))))))))))) (\lambda (c: C).(\lambda (t0: -T).(\lambda (a1: A).(\lambda (_: (arity g c t0 a1)).(\lambda (H1: ((\forall -(d1: C).(\forall (is: PList).((drop1 is d1 c) \to (\forall (c2: C).((csubc g -d1 c2) \to (sc3 g a1 c2 (lift1 is t0))))))))).(\lambda (a2: A).(\lambda (H2: -(leq g a1 a2)).(\lambda (d1: C).(\lambda (is: PList).(\lambda (H3: (drop1 is -d1 c)).(\lambda (c2: C).(\lambda (H4: (csubc g d1 c2)).(sc3_repl g a1 c2 -(lift1 is t0) (H1 d1 is H3 c2 H4) a2 H2))))))))))))) c1 t a H))))). -(* COMMENTS -Initial nodes: 5940 -END *) - -theorem sc3_arity: - \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t -a) \to (sc3 g a c t))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: -(arity g c t a)).(let H_y \def (sc3_arity_csubc g c t a H c PNil) in (H_y -(drop1_nil c) c (csubc_refl g c))))))). -(* COMMENTS -Initial nodes: 47 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/sc3/defs.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/sc3/defs.ma deleted file mode 100644 index 32fdfda84..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/sc3/defs.ma +++ /dev/null @@ -1,32 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/sn3/defs.ma". - -include "Basic-1/arity/defs.ma". - -include "Basic-1/drop1/defs.ma". - -definition sc3: - G \to (A \to (C \to (T \to Prop))) -\def - let rec sc3 (g: G) (a: A) on a: (C \to (T \to Prop)) \def (\lambda (c: -C).(\lambda (t: T).(match a with [(ASort h n) \Rightarrow (land (arity g c t -(ASort h n)) (sn3 c t)) | (AHead a1 a2) \Rightarrow (land (arity g c t (AHead -a1 a2)) (\forall (d: C).(\forall (w: T).((sc3 g a1 d w) \to (\forall (is: -PList).((drop1 is d c) \to (sc3 g a2 d (THead (Flat Appl) w (lift1 is -t)))))))))]))) in sc3. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/sc3/props.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/sc3/props.ma deleted file mode 100644 index e1d909251..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/sc3/props.ma +++ /dev/null @@ -1,728 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/sc3/defs.ma". - -include "Basic-1/sn3/lift1.ma". - -include "Basic-1/nf2/lift1.ma". - -include "Basic-1/csuba/arity.ma". - -include "Basic-1/arity/lift1.ma". - -include "Basic-1/arity/aprem.ma". - -include "Basic-1/llt/props.ma". - -include "Basic-1/drop1/getl.ma". - -include "Basic-1/drop1/props.ma". - -include "Basic-1/lift1/props.ma". - -theorem sc3_arity_gen: - \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((sc3 g a c -t) \to (arity g c t a))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(A_ind -(\lambda (a0: A).((sc3 g a0 c t) \to (arity g c t a0))) (\lambda (n: -nat).(\lambda (n0: nat).(\lambda (H: (land (arity g c t (ASort n n0)) (sn3 c -t))).(let H0 \def H in (land_ind (arity g c t (ASort n n0)) (sn3 c t) (arity -g c t (ASort n n0)) (\lambda (H1: (arity g c t (ASort n n0))).(\lambda (_: -(sn3 c t)).H1)) H0))))) (\lambda (a0: A).(\lambda (_: (((sc3 g a0 c t) \to -(arity g c t a0)))).(\lambda (a1: A).(\lambda (_: (((sc3 g a1 c t) \to (arity -g c t a1)))).(\lambda (H1: (land (arity g c t (AHead a0 a1)) (\forall (d: -C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) -\to (sc3 g a1 d (THead (Flat Appl) w (lift1 is t)))))))))).(let H2 \def H1 in -(land_ind (arity g c t (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g -a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat -Appl) w (lift1 is t)))))))) (arity g c t (AHead a0 a1)) (\lambda (H3: (arity -g c t (AHead a0 a1))).(\lambda (_: ((\forall (d: C).(\forall (w: T).((sc3 g -a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat -Appl) w (lift1 is t)))))))))).H3)) H2))))))) a)))). -(* COMMENTS -Initial nodes: 369 -END *) - -theorem sc3_repl: - \forall (g: G).(\forall (a1: A).(\forall (c: C).(\forall (t: T).((sc3 g a1 c -t) \to (\forall (a2: A).((leq g a1 a2) \to (sc3 g a2 c t))))))) -\def - \lambda (g: G).(\lambda (a1: A).(llt_wf_ind (\lambda (a: A).(\forall (c: -C).(\forall (t: T).((sc3 g a c t) \to (\forall (a2: A).((leq g a a2) \to (sc3 -g a2 c t))))))) (\lambda (a2: A).(A_ind (\lambda (a: A).(((\forall (a3: -A).((llt a3 a) \to (\forall (c: C).(\forall (t: T).((sc3 g a3 c t) \to -(\forall (a4: A).((leq g a3 a4) \to (sc3 g a4 c t))))))))) \to (\forall (c: -C).(\forall (t: T).((sc3 g a c t) \to (\forall (a3: A).((leq g a a3) \to (sc3 -g a3 c t)))))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (_: ((\forall -(a3: A).((llt a3 (ASort n n0)) \to (\forall (c: C).(\forall (t: T).((sc3 g a3 -c t) \to (\forall (a4: A).((leq g a3 a4) \to (sc3 g a4 c t)))))))))).(\lambda -(c: C).(\lambda (t: T).(\lambda (H0: (land (arity g c t (ASort n n0)) (sn3 c -t))).(\lambda (a3: A).(\lambda (H1: (leq g (ASort n n0) a3)).(let H2 \def H0 -in (land_ind (arity g c t (ASort n n0)) (sn3 c t) (sc3 g a3 c t) (\lambda -(H3: (arity g c t (ASort n n0))).(\lambda (H4: (sn3 c t)).(let H_y \def -(arity_repl g c t (ASort n n0) H3 a3 H1) in (let H_x \def (leq_gen_sort1 g n -n0 a3 H1) in (let H5 \def H_x in (ex2_3_ind nat nat nat (\lambda (n2: -nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort n n0) k) -(aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda -(_: nat).(eq A a3 (ASort h2 n2))))) (sc3 g a3 c t) (\lambda (x0: -nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (_: (eq A (aplus g (ASort -n n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H7: (eq A a3 (ASort x1 -x0))).(let H8 \def (f_equal A A (\lambda (e: A).e) a3 (ASort x1 x0) H7) in -(let H9 \def (eq_ind A a3 (\lambda (a: A).(arity g c t a)) H_y (ASort x1 x0) -H8) in (eq_ind_r A (ASort x1 x0) (\lambda (a: A).(sc3 g a c t)) (conj (arity -g c t (ASort x1 x0)) (sn3 c t) H9 H4) a3 H8)))))))) H5)))))) H2)))))))))) -(\lambda (a: A).(\lambda (_: ((((\forall (a3: A).((llt a3 a) \to (\forall (c: -C).(\forall (t: T).((sc3 g a3 c t) \to (\forall (a4: A).((leq g a3 a4) \to -(sc3 g a4 c t))))))))) \to (\forall (c: C).(\forall (t: T).((sc3 g a c t) \to -(\forall (a3: A).((leq g a a3) \to (sc3 g a3 c t))))))))).(\lambda (a0: -A).(\lambda (H0: ((((\forall (a3: A).((llt a3 a0) \to (\forall (c: -C).(\forall (t: T).((sc3 g a3 c t) \to (\forall (a4: A).((leq g a3 a4) \to -(sc3 g a4 c t))))))))) \to (\forall (c: C).(\forall (t: T).((sc3 g a0 c t) -\to (\forall (a3: A).((leq g a0 a3) \to (sc3 g a3 c t))))))))).(\lambda (H1: -((\forall (a3: A).((llt a3 (AHead a a0)) \to (\forall (c: C).(\forall (t: -T).((sc3 g a3 c t) \to (\forall (a4: A).((leq g a3 a4) \to (sc3 g a4 c -t)))))))))).(\lambda (c: C).(\lambda (t: T).(\lambda (H2: (land (arity g c t -(AHead a a0)) (\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall -(is: PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w (lift1 is -t)))))))))).(\lambda (a3: A).(\lambda (H3: (leq g (AHead a a0) a3)).(let H4 -\def H2 in (land_ind (arity g c t (AHead a a0)) (\forall (d: C).(\forall (w: -T).((sc3 g a d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a0 d -(THead (Flat Appl) w (lift1 is t)))))))) (sc3 g a3 c t) (\lambda (H5: (arity -g c t (AHead a a0))).(\lambda (H6: ((\forall (d: C).(\forall (w: T).((sc3 g a -d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat -Appl) w (lift1 is t)))))))))).(let H_x \def (leq_gen_head1 g a a0 a3 H3) in -(let H7 \def H_x in (ex3_2_ind A A (\lambda (a4: A).(\lambda (_: A).(leq g a -a4))) (\lambda (_: A).(\lambda (a5: A).(leq g a0 a5))) (\lambda (a4: -A).(\lambda (a5: A).(eq A a3 (AHead a4 a5)))) (sc3 g a3 c t) (\lambda (x0: -A).(\lambda (x1: A).(\lambda (H8: (leq g a x0)).(\lambda (H9: (leq g a0 -x1)).(\lambda (H10: (eq A a3 (AHead x0 x1))).(let H11 \def (f_equal A A -(\lambda (e: A).e) a3 (AHead x0 x1) H10) in (eq_ind_r A (AHead x0 x1) -(\lambda (a4: A).(sc3 g a4 c t)) (conj (arity g c t (AHead x0 x1)) (\forall -(d: C).(\forall (w: T).((sc3 g x0 d w) \to (\forall (is: PList).((drop1 is d -c) \to (sc3 g x1 d (THead (Flat Appl) w (lift1 is t)))))))) (arity_repl g c t -(AHead a a0) H5 (AHead x0 x1) (leq_head g a x0 H8 a0 x1 H9)) (\lambda (d: -C).(\lambda (w: T).(\lambda (H12: (sc3 g x0 d w)).(\lambda (is: -PList).(\lambda (H13: (drop1 is d c)).(H0 (\lambda (a4: A).(\lambda (H14: -(llt a4 a0)).(\lambda (c0: C).(\lambda (t0: T).(\lambda (H15: (sc3 g a4 c0 -t0)).(\lambda (a5: A).(\lambda (H16: (leq g a4 a5)).(H1 a4 (llt_trans a4 a0 -(AHead a a0) H14 (llt_head_dx a a0)) c0 t0 H15 a5 H16)))))))) d (THead (Flat -Appl) w (lift1 is t)) (H6 d w (H1 x0 (llt_repl g a x0 H8 (AHead a a0) -(llt_head_sx a a0)) d w H12 a (leq_sym g a x0 H8)) is H13) x1 H9))))))) a3 -H11))))))) H7))))) H4)))))))))))) a2)) a1)). -(* COMMENTS -Initial nodes: 1359 -END *) - -theorem sc3_lift: - \forall (g: G).(\forall (a: A).(\forall (e: C).(\forall (t: T).((sc3 g a e -t) \to (\forall (c: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) -\to (sc3 g a c (lift h d t)))))))))) -\def - \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(\forall (e: -C).(\forall (t: T).((sc3 g a0 e t) \to (\forall (c: C).(\forall (h: -nat).(\forall (d: nat).((drop h d c e) \to (sc3 g a0 c (lift h d t)))))))))) -(\lambda (n: nat).(\lambda (n0: nat).(\lambda (e: C).(\lambda (t: T).(\lambda -(H: (land (arity g e t (ASort n n0)) (sn3 e t))).(\lambda (c: C).(\lambda (h: -nat).(\lambda (d: nat).(\lambda (H0: (drop h d c e)).(let H1 \def H in -(land_ind (arity g e t (ASort n n0)) (sn3 e t) (land (arity g c (lift h d t) -(ASort n n0)) (sn3 c (lift h d t))) (\lambda (H2: (arity g e t (ASort n -n0))).(\lambda (H3: (sn3 e t)).(conj (arity g c (lift h d t) (ASort n n0)) -(sn3 c (lift h d t)) (arity_lift g e t (ASort n n0) H2 c h d H0) (sn3_lift e -t H3 c h d H0)))) H1))))))))))) (\lambda (a0: A).(\lambda (_: ((\forall (e: -C).(\forall (t: T).((sc3 g a0 e t) \to (\forall (c: C).(\forall (h: -nat).(\forall (d: nat).((drop h d c e) \to (sc3 g a0 c (lift h d -t))))))))))).(\lambda (a1: A).(\lambda (_: ((\forall (e: C).(\forall (t: -T).((sc3 g a1 e t) \to (\forall (c: C).(\forall (h: nat).(\forall (d: -nat).((drop h d c e) \to (sc3 g a1 c (lift h d t))))))))))).(\lambda (e: -C).(\lambda (t: T).(\lambda (H1: (land (arity g e t (AHead a0 a1)) (\forall -(d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d -e) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is t)))))))))).(\lambda (c: -C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2: (drop h d c e)).(let H3 -\def H1 in (land_ind (arity g e t (AHead a0 a1)) (\forall (d0: C).(\forall -(w: T).((sc3 g a0 d0 w) \to (\forall (is: PList).((drop1 is d0 e) \to (sc3 g -a1 d0 (THead (Flat Appl) w (lift1 is t)))))))) (land (arity g c (lift h d t) -(AHead a0 a1)) (\forall (d0: C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall -(is: PList).((drop1 is d0 c) \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is -(lift h d t)))))))))) (\lambda (H4: (arity g e t (AHead a0 a1))).(\lambda -(H5: ((\forall (d0: C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: -PList).((drop1 is d0 e) \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is -t)))))))))).(conj (arity g c (lift h d t) (AHead a0 a1)) (\forall (d0: -C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: PList).((drop1 is d0 c) -\to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is (lift h d t))))))))) -(arity_lift g e t (AHead a0 a1) H4 c h d H2) (\lambda (d0: C).(\lambda (w: -T).(\lambda (H6: (sc3 g a0 d0 w)).(\lambda (is: PList).(\lambda (H7: (drop1 -is d0 c)).(let H_y \def (H5 d0 w H6 (PConsTail is h d)) in (eq_ind T (lift1 -(PConsTail is h d) t) (\lambda (t0: T).(sc3 g a1 d0 (THead (Flat Appl) w -t0))) (H_y (drop1_cons_tail c e h d H2 is d0 H7)) (lift1 is (lift h d t)) -(lift1_cons_tail t h d is))))))))))) H3))))))))))))) a)). -(* COMMENTS -Initial nodes: 849 -END *) - -theorem sc3_lift1: - \forall (g: G).(\forall (e: C).(\forall (a: A).(\forall (hds: -PList).(\forall (c: C).(\forall (t: T).((sc3 g a e t) \to ((drop1 hds c e) -\to (sc3 g a c (lift1 hds t))))))))) -\def - \lambda (g: G).(\lambda (e: C).(\lambda (a: A).(\lambda (hds: -PList).(PList_ind (\lambda (p: PList).(\forall (c: C).(\forall (t: T).((sc3 g -a e t) \to ((drop1 p c e) \to (sc3 g a c (lift1 p t))))))) (\lambda (c: -C).(\lambda (t: T).(\lambda (H: (sc3 g a e t)).(\lambda (H0: (drop1 PNil c -e)).(let H_y \def (drop1_gen_pnil c e H0) in (eq_ind_r C e (\lambda (c0: -C).(sc3 g a c0 t)) H c H_y)))))) (\lambda (n: nat).(\lambda (n0: -nat).(\lambda (p: PList).(\lambda (H: ((\forall (c: C).(\forall (t: T).((sc3 -g a e t) \to ((drop1 p c e) \to (sc3 g a c (lift1 p t)))))))).(\lambda (c: -C).(\lambda (t: T).(\lambda (H0: (sc3 g a e t)).(\lambda (H1: (drop1 (PCons n -n0 p) c e)).(let H_x \def (drop1_gen_pcons c e p n n0 H1) in (let H2 \def H_x -in (ex2_ind C (\lambda (c2: C).(drop n n0 c c2)) (\lambda (c2: C).(drop1 p c2 -e)) (sc3 g a c (lift n n0 (lift1 p t))) (\lambda (x: C).(\lambda (H3: (drop n -n0 c x)).(\lambda (H4: (drop1 p x e)).(sc3_lift g a x (lift1 p t) (H x t H0 -H4) c n n0 H3)))) H2))))))))))) hds)))). -(* COMMENTS -Initial nodes: 289 -END *) - -theorem sc3_abbr: - \forall (g: G).(\forall (a: A).(\forall (vs: TList).(\forall (i: -nat).(\forall (d: C).(\forall (v: T).(\forall (c: C).((sc3 g a c (THeads -(Flat Appl) vs (lift (S i) O v))) \to ((getl i c (CHead d (Bind Abbr) v)) \to -(sc3 g a c (THeads (Flat Appl) vs (TLRef i))))))))))) -\def - \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(\forall (vs: -TList).(\forall (i: nat).(\forall (d: C).(\forall (v: T).(\forall (c: -C).((sc3 g a0 c (THeads (Flat Appl) vs (lift (S i) O v))) \to ((getl i c -(CHead d (Bind Abbr) v)) \to (sc3 g a0 c (THeads (Flat Appl) vs (TLRef -i))))))))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (vs: -TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(\lambda (c: -C).(\lambda (H: (land (arity g c (THeads (Flat Appl) vs (lift (S i) O v)) -(ASort n n0)) (sn3 c (THeads (Flat Appl) vs (lift (S i) O v))))).(\lambda -(H0: (getl i c (CHead d (Bind Abbr) v))).(let H1 \def H in (land_ind (arity g -c (THeads (Flat Appl) vs (lift (S i) O v)) (ASort n n0)) (sn3 c (THeads (Flat -Appl) vs (lift (S i) O v))) (land (arity g c (THeads (Flat Appl) vs (TLRef -i)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (TLRef i)))) (\lambda (H2: -(arity g c (THeads (Flat Appl) vs (lift (S i) O v)) (ASort n n0))).(\lambda -(H3: (sn3 c (THeads (Flat Appl) vs (lift (S i) O v)))).(conj (arity g c -(THeads (Flat Appl) vs (TLRef i)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs -(TLRef i))) (arity_appls_abbr g c d v i H0 vs (ASort n n0) H2) -(sn3_appls_abbr c d v i H0 vs H3)))) H1))))))))))) (\lambda (a0: A).(\lambda -(_: ((\forall (vs: TList).(\forall (i: nat).(\forall (d: C).(\forall (v: -T).(\forall (c: C).((sc3 g a0 c (THeads (Flat Appl) vs (lift (S i) O v))) \to -((getl i c (CHead d (Bind Abbr) v)) \to (sc3 g a0 c (THeads (Flat Appl) vs -(TLRef i)))))))))))).(\lambda (a1: A).(\lambda (H0: ((\forall (vs: -TList).(\forall (i: nat).(\forall (d: C).(\forall (v: T).(\forall (c: -C).((sc3 g a1 c (THeads (Flat Appl) vs (lift (S i) O v))) \to ((getl i c -(CHead d (Bind Abbr) v)) \to (sc3 g a1 c (THeads (Flat Appl) vs (TLRef -i)))))))))))).(\lambda (vs: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda -(v: T).(\lambda (c: C).(\lambda (H1: (land (arity g c (THeads (Flat Appl) vs -(lift (S i) O v)) (AHead a0 a1)) (\forall (d0: C).(\forall (w: T).((sc3 g a0 -d0 w) \to (\forall (is: PList).((drop1 is d0 c) \to (sc3 g a1 d0 (THead (Flat -Appl) w (lift1 is (THeads (Flat Appl) vs (lift (S i) O v)))))))))))).(\lambda -(H2: (getl i c (CHead d (Bind Abbr) v))).(let H3 \def H1 in (land_ind (arity -g c (THeads (Flat Appl) vs (lift (S i) O v)) (AHead a0 a1)) (\forall (d0: -C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: PList).((drop1 is d0 c) -\to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs (lift -(S i) O v)))))))))) (land (arity g c (THeads (Flat Appl) vs (TLRef i)) (AHead -a0 a1)) (\forall (d0: C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: -PList).((drop1 is d0 c) \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is -(THeads (Flat Appl) vs (TLRef i))))))))))) (\lambda (H4: (arity g c (THeads -(Flat Appl) vs (lift (S i) O v)) (AHead a0 a1))).(\lambda (H5: ((\forall (d0: -C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: PList).((drop1 is d0 c) -\to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs (lift -(S i) O v)))))))))))).(conj (arity g c (THeads (Flat Appl) vs (TLRef i)) -(AHead a0 a1)) (\forall (d0: C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall -(is: PList).((drop1 is d0 c) \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is -(THeads (Flat Appl) vs (TLRef i)))))))))) (arity_appls_abbr g c d v i H2 vs -(AHead a0 a1) H4) (\lambda (d0: C).(\lambda (w: T).(\lambda (H6: (sc3 g a0 d0 -w)).(\lambda (is: PList).(\lambda (H7: (drop1 is d0 c)).(let H_x \def -(drop1_getl_trans is c d0 H7 Abbr d v i H2) in (let H8 \def H_x in (ex2_ind C -(\lambda (e2: C).(drop1 (ptrans is i) e2 d)) (\lambda (e2: C).(getl (trans is -i) d0 (CHead e2 (Bind Abbr) (lift1 (ptrans is i) v)))) (sc3 g a1 d0 (THead -(Flat Appl) w (lift1 is (THeads (Flat Appl) vs (TLRef i))))) (\lambda (x: -C).(\lambda (_: (drop1 (ptrans is i) x d)).(\lambda (H10: (getl (trans is i) -d0 (CHead x (Bind Abbr) (lift1 (ptrans is i) v)))).(let H_y \def (H0 (TCons w -(lifts1 is vs))) in (eq_ind_r T (THeads (Flat Appl) (lifts1 is vs) (lift1 is -(TLRef i))) (\lambda (t: T).(sc3 g a1 d0 (THead (Flat Appl) w t))) (eq_ind_r -T (TLRef (trans is i)) (\lambda (t: T).(sc3 g a1 d0 (THead (Flat Appl) w -(THeads (Flat Appl) (lifts1 is vs) t)))) (H_y (trans is i) x (lift1 (ptrans -is i) v) d0 (eq_ind T (lift1 is (lift (S i) O v)) (\lambda (t: T).(sc3 g a1 -d0 (THead (Flat Appl) w (THeads (Flat Appl) (lifts1 is vs) t)))) (eq_ind T -(lift1 is (THeads (Flat Appl) vs (lift (S i) O v))) (\lambda (t: T).(sc3 g a1 -d0 (THead (Flat Appl) w t))) (H5 d0 w H6 is H7) (THeads (Flat Appl) (lifts1 -is vs) (lift1 is (lift (S i) O v))) (lifts1_flat Appl is (lift (S i) O v) -vs)) (lift (S (trans is i)) O (lift1 (ptrans is i) v)) (lift1_free is i v)) -H10) (lift1 is (TLRef i)) (lift1_lref is i)) (lift1 is (THeads (Flat Appl) vs -(TLRef i))) (lifts1_flat Appl is (TLRef i) vs)))))) H8))))))))))) -H3))))))))))))) a)). -(* COMMENTS -Initial nodes: 1563 -END *) - -theorem sc3_cast: - \forall (g: G).(\forall (a: A).(\forall (vs: TList).(\forall (c: C).(\forall -(u: T).((sc3 g (asucc g a) c (THeads (Flat Appl) vs u)) \to (\forall (t: -T).((sc3 g a c (THeads (Flat Appl) vs t)) \to (sc3 g a c (THeads (Flat Appl) -vs (THead (Flat Cast) u t)))))))))) -\def - \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(\forall (vs: -TList).(\forall (c: C).(\forall (u: T).((sc3 g (asucc g a0) c (THeads (Flat -Appl) vs u)) \to (\forall (t: T).((sc3 g a0 c (THeads (Flat Appl) vs t)) \to -(sc3 g a0 c (THeads (Flat Appl) vs (THead (Flat Cast) u t)))))))))) (\lambda -(n: nat).(\lambda (n0: nat).(\lambda (vs: TList).(\lambda (c: C).(\lambda (u: -T).(\lambda (H: (sc3 g (match n with [O \Rightarrow (ASort O (next g n0)) | -(S h) \Rightarrow (ASort h n0)]) c (THeads (Flat Appl) vs u))).(\lambda (t: -T).(\lambda (H0: (land (arity g c (THeads (Flat Appl) vs t) (ASort n n0)) -(sn3 c (THeads (Flat Appl) vs t)))).(nat_ind (\lambda (n1: nat).((sc3 g -(match n1 with [O \Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow -(ASort h n0)]) c (THeads (Flat Appl) vs u)) \to ((land (arity g c (THeads -(Flat Appl) vs t) (ASort n1 n0)) (sn3 c (THeads (Flat Appl) vs t))) \to (land -(arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort n1 n0)) -(sn3 c (THeads (Flat Appl) vs (THead (Flat Cast) u t))))))) (\lambda (H1: -(sc3 g (ASort O (next g n0)) c (THeads (Flat Appl) vs u))).(\lambda (H2: -(land (arity g c (THeads (Flat Appl) vs t) (ASort O n0)) (sn3 c (THeads (Flat -Appl) vs t)))).(let H3 \def H1 in (land_ind (arity g c (THeads (Flat Appl) vs -u) (ASort O (next g n0))) (sn3 c (THeads (Flat Appl) vs u)) (land (arity g c -(THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort O n0)) (sn3 c (THeads -(Flat Appl) vs (THead (Flat Cast) u t)))) (\lambda (H4: (arity g c (THeads -(Flat Appl) vs u) (ASort O (next g n0)))).(\lambda (H5: (sn3 c (THeads (Flat -Appl) vs u))).(let H6 \def H2 in (land_ind (arity g c (THeads (Flat Appl) vs -t) (ASort O n0)) (sn3 c (THeads (Flat Appl) vs t)) (land (arity g c (THeads -(Flat Appl) vs (THead (Flat Cast) u t)) (ASort O n0)) (sn3 c (THeads (Flat -Appl) vs (THead (Flat Cast) u t)))) (\lambda (H7: (arity g c (THeads (Flat -Appl) vs t) (ASort O n0))).(\lambda (H8: (sn3 c (THeads (Flat Appl) vs -t))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort -O n0)) (sn3 c (THeads (Flat Appl) vs (THead (Flat Cast) u t))) -(arity_appls_cast g c u t vs (ASort O n0) H4 H7) (sn3_appls_cast c vs u H5 t -H8)))) H6)))) H3)))) (\lambda (n1: nat).(\lambda (_: (((sc3 g (match n1 with -[O \Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)]) c -(THeads (Flat Appl) vs u)) \to ((land (arity g c (THeads (Flat Appl) vs t) -(ASort n1 n0)) (sn3 c (THeads (Flat Appl) vs t))) \to (land (arity g c -(THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort n1 n0)) (sn3 c (THeads -(Flat Appl) vs (THead (Flat Cast) u t)))))))).(\lambda (H1: (sc3 g (ASort n1 -n0) c (THeads (Flat Appl) vs u))).(\lambda (H2: (land (arity g c (THeads -(Flat Appl) vs t) (ASort (S n1) n0)) (sn3 c (THeads (Flat Appl) vs t)))).(let -H3 \def H1 in (land_ind (arity g c (THeads (Flat Appl) vs u) (ASort n1 n0)) -(sn3 c (THeads (Flat Appl) vs u)) (land (arity g c (THeads (Flat Appl) vs -(THead (Flat Cast) u t)) (ASort (S n1) n0)) (sn3 c (THeads (Flat Appl) vs -(THead (Flat Cast) u t)))) (\lambda (H4: (arity g c (THeads (Flat Appl) vs u) -(ASort n1 n0))).(\lambda (H5: (sn3 c (THeads (Flat Appl) vs u))).(let H6 \def -H2 in (land_ind (arity g c (THeads (Flat Appl) vs t) (ASort (S n1) n0)) (sn3 -c (THeads (Flat Appl) vs t)) (land (arity g c (THeads (Flat Appl) vs (THead -(Flat Cast) u t)) (ASort (S n1) n0)) (sn3 c (THeads (Flat Appl) vs (THead -(Flat Cast) u t)))) (\lambda (H7: (arity g c (THeads (Flat Appl) vs t) (ASort -(S n1) n0))).(\lambda (H8: (sn3 c (THeads (Flat Appl) vs t))).(conj (arity g -c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort (S n1) n0)) (sn3 c -(THeads (Flat Appl) vs (THead (Flat Cast) u t))) (arity_appls_cast g c u t vs -(ASort (S n1) n0) H4 H7) (sn3_appls_cast c vs u H5 t H8)))) H6)))) H3)))))) n -H H0))))))))) (\lambda (a0: A).(\lambda (_: ((\forall (vs: TList).(\forall -(c: C).(\forall (u: T).((sc3 g (asucc g a0) c (THeads (Flat Appl) vs u)) \to -(\forall (t: T).((sc3 g a0 c (THeads (Flat Appl) vs t)) \to (sc3 g a0 c -(THeads (Flat Appl) vs (THead (Flat Cast) u t))))))))))).(\lambda (a1: -A).(\lambda (H0: ((\forall (vs: TList).(\forall (c: C).(\forall (u: T).((sc3 -g (asucc g a1) c (THeads (Flat Appl) vs u)) \to (\forall (t: T).((sc3 g a1 c -(THeads (Flat Appl) vs t)) \to (sc3 g a1 c (THeads (Flat Appl) vs (THead -(Flat Cast) u t))))))))))).(\lambda (vs: TList).(\lambda (c: C).(\lambda (u: -T).(\lambda (H1: (land (arity g c (THeads (Flat Appl) vs u) (AHead a0 (asucc -g a1))) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: -PList).((drop1 is d c) \to (sc3 g (asucc g a1) d (THead (Flat Appl) w (lift1 -is (THeads (Flat Appl) vs u))))))))))).(\lambda (t: T).(\lambda (H2: (land -(arity g c (THeads (Flat Appl) vs t) (AHead a0 a1)) (\forall (d: C).(\forall -(w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 -d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs t))))))))))).(let H3 -\def H1 in (land_ind (arity g c (THeads (Flat Appl) vs u) (AHead a0 (asucc g -a1))) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: -PList).((drop1 is d c) \to (sc3 g (asucc g a1) d (THead (Flat Appl) w (lift1 -is (THeads (Flat Appl) vs u))))))))) (land (arity g c (THeads (Flat Appl) vs -(THead (Flat Cast) u t)) (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 -g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead -(Flat Appl) w (lift1 is (THeads (Flat Appl) vs (THead (Flat Cast) u -t))))))))))) (\lambda (H4: (arity g c (THeads (Flat Appl) vs u) (AHead a0 -(asucc g a1)))).(\lambda (H5: ((\forall (d: C).(\forall (w: T).((sc3 g a0 d -w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g (asucc g a1) d (THead -(Flat Appl) w (lift1 is (THeads (Flat Appl) vs u))))))))))).(let H6 \def H2 -in (land_ind (arity g c (THeads (Flat Appl) vs t) (AHead a0 a1)) (\forall (d: -C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) -\to (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs -t))))))))) (land (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) -(AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall -(is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is -(THeads (Flat Appl) vs (THead (Flat Cast) u t))))))))))) (\lambda (H7: (arity -g c (THeads (Flat Appl) vs t) (AHead a0 a1))).(\lambda (H8: ((\forall (d: -C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) -\to (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs -t))))))))))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) -(AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall -(is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is -(THeads (Flat Appl) vs (THead (Flat Cast) u t)))))))))) (arity_appls_cast g c -u t vs (AHead a0 a1) H4 H7) (\lambda (d: C).(\lambda (w: T).(\lambda (H9: -(sc3 g a0 d w)).(\lambda (is: PList).(\lambda (H10: (drop1 is d c)).(let H_y -\def (H0 (TCons w (lifts1 is vs))) in (eq_ind_r T (THeads (Flat Appl) (lifts1 -is vs) (lift1 is (THead (Flat Cast) u t))) (\lambda (t0: T).(sc3 g a1 d -(THead (Flat Appl) w t0))) (eq_ind_r T (THead (Flat Cast) (lift1 is u) (lift1 -is t)) (\lambda (t0: T).(sc3 g a1 d (THead (Flat Appl) w (THeads (Flat Appl) -(lifts1 is vs) t0)))) (H_y d (lift1 is u) (eq_ind T (lift1 is (THeads (Flat -Appl) vs u)) (\lambda (t0: T).(sc3 g (asucc g a1) d (THead (Flat Appl) w -t0))) (H5 d w H9 is H10) (THeads (Flat Appl) (lifts1 is vs) (lift1 is u)) -(lifts1_flat Appl is u vs)) (lift1 is t) (eq_ind T (lift1 is (THeads (Flat -Appl) vs t)) (\lambda (t0: T).(sc3 g a1 d (THead (Flat Appl) w t0))) (H8 d w -H9 is H10) (THeads (Flat Appl) (lifts1 is vs) (lift1 is t)) (lifts1_flat Appl -is t vs))) (lift1 is (THead (Flat Cast) u t)) (lift1_flat Cast is u t)) -(lift1 is (THeads (Flat Appl) vs (THead (Flat Cast) u t))) (lifts1_flat Appl -is (THead (Flat Cast) u t) vs))))))))))) H6)))) H3)))))))))))) a)). -(* COMMENTS -Initial nodes: 2625 -END *) - -theorem sc3_props__sc3_sn3_abst: - \forall (g: G).(\forall (a: A).(land (\forall (c: C).(\forall (t: T).((sc3 g -a c t) \to (sn3 c t)))) (\forall (vs: TList).(\forall (i: nat).(let t \def -(THeads (Flat Appl) vs (TLRef i)) in (\forall (c: C).((arity g c t a) \to -((nf2 c (TLRef i)) \to ((sns3 c vs) \to (sc3 g a c t)))))))))) -\def - \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(land (\forall (c: -C).(\forall (t: T).((sc3 g a0 c t) \to (sn3 c t)))) (\forall (vs: -TList).(\forall (i: nat).(let t \def (THeads (Flat Appl) vs (TLRef i)) in -(\forall (c: C).((arity g c t a0) \to ((nf2 c (TLRef i)) \to ((sns3 c vs) \to -(sc3 g a0 c t)))))))))) (\lambda (n: nat).(\lambda (n0: nat).(conj (\forall -(c: C).(\forall (t: T).((land (arity g c t (ASort n n0)) (sn3 c t)) \to (sn3 -c t)))) (\forall (vs: TList).(\forall (i: nat).(\forall (c: C).((arity g c -(THeads (Flat Appl) vs (TLRef i)) (ASort n n0)) \to ((nf2 c (TLRef i)) \to -((sns3 c vs) \to (land (arity g c (THeads (Flat Appl) vs (TLRef i)) (ASort n -n0)) (sn3 c (THeads (Flat Appl) vs (TLRef i)))))))))) (\lambda (c: -C).(\lambda (t: T).(\lambda (H: (land (arity g c t (ASort n n0)) (sn3 c -t))).(let H0 \def H in (land_ind (arity g c t (ASort n n0)) (sn3 c t) (sn3 c -t) (\lambda (_: (arity g c t (ASort n n0))).(\lambda (H2: (sn3 c t)).H2)) -H0))))) (\lambda (vs: TList).(\lambda (i: nat).(\lambda (c: C).(\lambda (H: -(arity g c (THeads (Flat Appl) vs (TLRef i)) (ASort n n0))).(\lambda (H0: -(nf2 c (TLRef i))).(\lambda (H1: (sns3 c vs)).(conj (arity g c (THeads (Flat -Appl) vs (TLRef i)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (TLRef i))) H -(sn3_appls_lref c i H0 vs H1))))))))))) (\lambda (a0: A).(\lambda (H: (land -(\forall (c: C).(\forall (t: T).((sc3 g a0 c t) \to (sn3 c t)))) (\forall -(vs: TList).(\forall (i: nat).(\forall (c: C).((arity g c (THeads (Flat Appl) -vs (TLRef i)) a0) \to ((nf2 c (TLRef i)) \to ((sns3 c vs) \to (sc3 g a0 c -(THeads (Flat Appl) vs (TLRef i))))))))))).(\lambda (a1: A).(\lambda (H0: -(land (\forall (c: C).(\forall (t: T).((sc3 g a1 c t) \to (sn3 c t)))) -(\forall (vs: TList).(\forall (i: nat).(\forall (c: C).((arity g c (THeads -(Flat Appl) vs (TLRef i)) a1) \to ((nf2 c (TLRef i)) \to ((sns3 c vs) \to -(sc3 g a1 c (THeads (Flat Appl) vs (TLRef i))))))))))).(conj (\forall (c: -C).(\forall (t: T).((land (arity g c t (AHead a0 a1)) (\forall (d: -C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) -\to (sc3 g a1 d (THead (Flat Appl) w (lift1 is t))))))))) \to (sn3 c t)))) -(\forall (vs: TList).(\forall (i: nat).(\forall (c: C).((arity g c (THeads -(Flat Appl) vs (TLRef i)) (AHead a0 a1)) \to ((nf2 c (TLRef i)) \to ((sns3 c -vs) \to (land (arity g c (THeads (Flat Appl) vs (TLRef i)) (AHead a0 a1)) -(\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: -PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads -(Flat Appl) vs (TLRef i))))))))))))))))) (\lambda (c: C).(\lambda (t: -T).(\lambda (H1: (land (arity g c t (AHead a0 a1)) (\forall (d: C).(\forall -(w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 -d (THead (Flat Appl) w (lift1 is t)))))))))).(let H2 \def H in (land_ind -(\forall (c0: C).(\forall (t0: T).((sc3 g a0 c0 t0) \to (sn3 c0 t0)))) -(\forall (vs: TList).(\forall (i: nat).(\forall (c0: C).((arity g c0 (THeads -(Flat Appl) vs (TLRef i)) a0) \to ((nf2 c0 (TLRef i)) \to ((sns3 c0 vs) \to -(sc3 g a0 c0 (THeads (Flat Appl) vs (TLRef i))))))))) (sn3 c t) (\lambda (_: -((\forall (c0: C).(\forall (t0: T).((sc3 g a0 c0 t0) \to (sn3 c0 -t0)))))).(\lambda (H4: ((\forall (vs: TList).(\forall (i: nat).(\forall (c0: -C).((arity g c0 (THeads (Flat Appl) vs (TLRef i)) a0) \to ((nf2 c0 (TLRef i)) -\to ((sns3 c0 vs) \to (sc3 g a0 c0 (THeads (Flat Appl) vs (TLRef -i))))))))))).(let H5 \def H0 in (land_ind (\forall (c0: C).(\forall (t0: -T).((sc3 g a1 c0 t0) \to (sn3 c0 t0)))) (\forall (vs: TList).(\forall (i: -nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) vs (TLRef i)) a1) \to -((nf2 c0 (TLRef i)) \to ((sns3 c0 vs) \to (sc3 g a1 c0 (THeads (Flat Appl) vs -(TLRef i))))))))) (sn3 c t) (\lambda (H6: ((\forall (c0: C).(\forall (t0: -T).((sc3 g a1 c0 t0) \to (sn3 c0 t0)))))).(\lambda (_: ((\forall (vs: -TList).(\forall (i: nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) vs -(TLRef i)) a1) \to ((nf2 c0 (TLRef i)) \to ((sns3 c0 vs) \to (sc3 g a1 c0 -(THeads (Flat Appl) vs (TLRef i))))))))))).(let H8 \def H1 in (land_ind -(arity g c t (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) -\to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w -(lift1 is t)))))))) (sn3 c t) (\lambda (H9: (arity g c t (AHead a0 -a1))).(\lambda (H10: ((\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to -(\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w -(lift1 is t)))))))))).(let H_y \def (arity_aprem g c t (AHead a0 a1) H9 O a0) -in (let H11 \def (H_y (aprem_zero a0 a1)) in (ex2_3_ind C T nat (\lambda (d: -C).(\lambda (_: T).(\lambda (j: nat).(drop j O d c)))) (\lambda (d: -C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g a0))))) (sn3 c t) -(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H12: (drop x2 -O x0 c)).(\lambda (H13: (arity g x0 x1 (asucc g a0))).(let H_y0 \def (H10 -(CHead x0 (Bind Abst) x1) (TLRef O) (H4 TNil O (CHead x0 (Bind Abst) x1) -(arity_abst g (CHead x0 (Bind Abst) x1) x0 x1 O (getl_refl Abst x0 x1) a0 -H13) (nf2_lref_abst (CHead x0 (Bind Abst) x1) x0 x1 O (getl_refl Abst x0 x1)) -I) (PCons (S x2) O PNil)) in (let H_y1 \def (H6 (CHead x0 (Bind Abst) x1) -(THead (Flat Appl) (TLRef O) (lift (S x2) O t)) (H_y0 (drop1_cons (CHead x0 -(Bind Abst) x1) c (S x2) O (drop_drop (Bind Abst) x2 x0 c H12 x1) c PNil -(drop1_nil c)))) in (let H_x \def (sn3_gen_flat Appl (CHead x0 (Bind Abst) -x1) (TLRef O) (lift (S x2) O t) H_y1) in (let H14 \def H_x in (land_ind (sn3 -(CHead x0 (Bind Abst) x1) (TLRef O)) (sn3 (CHead x0 (Bind Abst) x1) (lift (S -x2) O t)) (sn3 c t) (\lambda (_: (sn3 (CHead x0 (Bind Abst) x1) (TLRef -O))).(\lambda (H16: (sn3 (CHead x0 (Bind Abst) x1) (lift (S x2) O -t))).(sn3_gen_lift (CHead x0 (Bind Abst) x1) t (S x2) O H16 c (drop_drop -(Bind Abst) x2 x0 c H12 x1)))) H14)))))))))) H11))))) H8)))) H5)))) H2))))) -(\lambda (vs: TList).(\lambda (i: nat).(\lambda (c: C).(\lambda (H1: (arity g -c (THeads (Flat Appl) vs (TLRef i)) (AHead a0 a1))).(\lambda (H2: (nf2 c -(TLRef i))).(\lambda (H3: (sns3 c vs)).(conj (arity g c (THeads (Flat Appl) -vs (TLRef i)) (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) -\to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w -(lift1 is (THeads (Flat Appl) vs (TLRef i)))))))))) H1 (\lambda (d: -C).(\lambda (w: T).(\lambda (H4: (sc3 g a0 d w)).(\lambda (is: -PList).(\lambda (H5: (drop1 is d c)).(let H6 \def H in (land_ind (\forall -(c0: C).(\forall (t: T).((sc3 g a0 c0 t) \to (sn3 c0 t)))) (\forall (vs0: -TList).(\forall (i0: nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) -vs0 (TLRef i0)) a0) \to ((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a0 -c0 (THeads (Flat Appl) vs0 (TLRef i0))))))))) (sc3 g a1 d (THead (Flat Appl) -w (lift1 is (THeads (Flat Appl) vs (TLRef i))))) (\lambda (H7: ((\forall (c0: -C).(\forall (t: T).((sc3 g a0 c0 t) \to (sn3 c0 t)))))).(\lambda (_: -((\forall (vs0: TList).(\forall (i0: nat).(\forall (c0: C).((arity g c0 -(THeads (Flat Appl) vs0 (TLRef i0)) a0) \to ((nf2 c0 (TLRef i0)) \to ((sns3 -c0 vs0) \to (sc3 g a0 c0 (THeads (Flat Appl) vs0 (TLRef i0))))))))))).(let H9 -\def H0 in (land_ind (\forall (c0: C).(\forall (t: T).((sc3 g a1 c0 t) \to -(sn3 c0 t)))) (\forall (vs0: TList).(\forall (i0: nat).(\forall (c0: -C).((arity g c0 (THeads (Flat Appl) vs0 (TLRef i0)) a1) \to ((nf2 c0 (TLRef -i0)) \to ((sns3 c0 vs0) \to (sc3 g a1 c0 (THeads (Flat Appl) vs0 (TLRef -i0))))))))) (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs -(TLRef i))))) (\lambda (_: ((\forall (c0: C).(\forall (t: T).((sc3 g a1 c0 t) -\to (sn3 c0 t)))))).(\lambda (H11: ((\forall (vs0: TList).(\forall (i0: -nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) vs0 (TLRef i0)) a1) -\to ((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a1 c0 (THeads (Flat -Appl) vs0 (TLRef i0))))))))))).(let H_y \def (H11 (TCons w (lifts1 is vs))) -in (eq_ind_r T (THeads (Flat Appl) (lifts1 is vs) (lift1 is (TLRef i))) -(\lambda (t: T).(sc3 g a1 d (THead (Flat Appl) w t))) (eq_ind_r T (TLRef -(trans is i)) (\lambda (t: T).(sc3 g a1 d (THead (Flat Appl) w (THeads (Flat -Appl) (lifts1 is vs) t)))) (H_y (trans is i) d (eq_ind T (lift1 is (TLRef i)) -(\lambda (t: T).(arity g d (THead (Flat Appl) w (THeads (Flat Appl) (lifts1 -is vs) t)) a1)) (eq_ind T (lift1 is (THeads (Flat Appl) vs (TLRef i))) -(\lambda (t: T).(arity g d (THead (Flat Appl) w t) a1)) (arity_appl g d w a0 -(sc3_arity_gen g d w a0 H4) (lift1 is (THeads (Flat Appl) vs (TLRef i))) a1 -(arity_lift1 g (AHead a0 a1) c is d (THeads (Flat Appl) vs (TLRef i)) H5 H1)) -(THeads (Flat Appl) (lifts1 is vs) (lift1 is (TLRef i))) (lifts1_flat Appl is -(TLRef i) vs)) (TLRef (trans is i)) (lift1_lref is i)) (eq_ind T (lift1 is -(TLRef i)) (\lambda (t: T).(nf2 d t)) (nf2_lift1 c is d (TLRef i) H5 H2) -(TLRef (trans is i)) (lift1_lref is i)) (conj (sn3 d w) (sns3 d (lifts1 is -vs)) (H7 d w H4) (sns3_lifts1 c is d H5 vs H3))) (lift1 is (TLRef i)) -(lift1_lref is i)) (lift1 is (THeads (Flat Appl) vs (TLRef i))) (lifts1_flat -Appl is (TLRef i) vs))))) H9)))) H6))))))))))))))))))) a)). -(* COMMENTS -Initial nodes: 2737 -END *) - -theorem sc3_sn3: - \forall (g: G).(\forall (a: A).(\forall (c: C).(\forall (t: T).((sc3 g a c -t) \to (sn3 c t))))) -\def - \lambda (g: G).(\lambda (a: A).(\lambda (c: C).(\lambda (t: T).(\lambda (H: -(sc3 g a c t)).(let H_x \def (sc3_props__sc3_sn3_abst g a) in (let H0 \def -H_x in (land_ind (\forall (c0: C).(\forall (t0: T).((sc3 g a c0 t0) \to (sn3 -c0 t0)))) (\forall (vs: TList).(\forall (i: nat).(\forall (c0: C).((arity g -c0 (THeads (Flat Appl) vs (TLRef i)) a) \to ((nf2 c0 (TLRef i)) \to ((sns3 c0 -vs) \to (sc3 g a c0 (THeads (Flat Appl) vs (TLRef i))))))))) (sn3 c t) -(\lambda (H1: ((\forall (c0: C).(\forall (t0: T).((sc3 g a c0 t0) \to (sn3 c0 -t0)))))).(\lambda (_: ((\forall (vs: TList).(\forall (i: nat).(\forall (c0: -C).((arity g c0 (THeads (Flat Appl) vs (TLRef i)) a) \to ((nf2 c0 (TLRef i)) -\to ((sns3 c0 vs) \to (sc3 g a c0 (THeads (Flat Appl) vs (TLRef -i))))))))))).(H1 c t H))) H0))))))). -(* COMMENTS -Initial nodes: 203 -END *) - -theorem sc3_abst: - \forall (g: G).(\forall (a: A).(\forall (vs: TList).(\forall (c: C).(\forall -(i: nat).((arity g c (THeads (Flat Appl) vs (TLRef i)) a) \to ((nf2 c (TLRef -i)) \to ((sns3 c vs) \to (sc3 g a c (THeads (Flat Appl) vs (TLRef i)))))))))) -\def - \lambda (g: G).(\lambda (a: A).(\lambda (vs: TList).(\lambda (c: C).(\lambda -(i: nat).(\lambda (H: (arity g c (THeads (Flat Appl) vs (TLRef i)) -a)).(\lambda (H0: (nf2 c (TLRef i))).(\lambda (H1: (sns3 c vs)).(let H_x \def -(sc3_props__sc3_sn3_abst g a) in (let H2 \def H_x in (land_ind (\forall (c0: -C).(\forall (t: T).((sc3 g a c0 t) \to (sn3 c0 t)))) (\forall (vs0: -TList).(\forall (i0: nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) -vs0 (TLRef i0)) a) \to ((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a -c0 (THeads (Flat Appl) vs0 (TLRef i0))))))))) (sc3 g a c (THeads (Flat Appl) -vs (TLRef i))) (\lambda (_: ((\forall (c0: C).(\forall (t: T).((sc3 g a c0 t) -\to (sn3 c0 t)))))).(\lambda (H4: ((\forall (vs0: TList).(\forall (i0: -nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) vs0 (TLRef i0)) a) \to -((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a c0 (THeads (Flat Appl) -vs0 (TLRef i0))))))))))).(H4 vs i c H H0 H1))) H2)))))))))). -(* COMMENTS -Initial nodes: 249 -END *) - -theorem sc3_bind: - \forall (g: G).(\forall (b: B).((not (eq B b Abst)) \to (\forall (a1: -A).(\forall (a2: A).(\forall (vs: TList).(\forall (c: C).(\forall (v: -T).(\forall (t: T).((sc3 g a2 (CHead c (Bind b) v) (THeads (Flat Appl) (lifts -(S O) O vs) t)) \to ((sc3 g a1 c v) \to (sc3 g a2 c (THeads (Flat Appl) vs -(THead (Bind b) v t))))))))))))) -\def - \lambda (g: G).(\lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda -(a1: A).(\lambda (a2: A).(A_ind (\lambda (a: A).(\forall (vs: TList).(\forall -(c: C).(\forall (v: T).(\forall (t: T).((sc3 g a (CHead c (Bind b) v) (THeads -(Flat Appl) (lifts (S O) O vs) t)) \to ((sc3 g a1 c v) \to (sc3 g a c (THeads -(Flat Appl) vs (THead (Bind b) v t)))))))))) (\lambda (n: nat).(\lambda (n0: -nat).(\lambda (vs: TList).(\lambda (c: C).(\lambda (v: T).(\lambda (t: -T).(\lambda (H0: (land (arity g (CHead c (Bind b) v) (THeads (Flat Appl) -(lifts (S O) O vs) t) (ASort n n0)) (sn3 (CHead c (Bind b) v) (THeads (Flat -Appl) (lifts (S O) O vs) t)))).(\lambda (H1: (sc3 g a1 c v)).(let H2 \def H0 -in (land_ind (arity g (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O -vs) t) (ASort n n0)) (sn3 (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S -O) O vs) t)) (land (arity g c (THeads (Flat Appl) vs (THead (Bind b) v t)) -(ASort n n0)) (sn3 c (THeads (Flat Appl) vs (THead (Bind b) v t)))) (\lambda -(H3: (arity g (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O vs) t) -(ASort n n0))).(\lambda (H4: (sn3 (CHead c (Bind b) v) (THeads (Flat Appl) -(lifts (S O) O vs) t))).(conj (arity g c (THeads (Flat Appl) vs (THead (Bind -b) v t)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (THead (Bind b) v t))) -(arity_appls_bind g b H c v a1 (sc3_arity_gen g c v a1 H1) t vs (ASort n n0) -H3) (sn3_appls_bind b H c v (sc3_sn3 g a1 c v H1) vs t H4)))) H2)))))))))) -(\lambda (a: A).(\lambda (_: ((\forall (vs: TList).(\forall (c: C).(\forall -(v: T).(\forall (t: T).((sc3 g a (CHead c (Bind b) v) (THeads (Flat Appl) -(lifts (S O) O vs) t)) \to ((sc3 g a1 c v) \to (sc3 g a c (THeads (Flat Appl) -vs (THead (Bind b) v t))))))))))).(\lambda (a0: A).(\lambda (H1: ((\forall -(vs: TList).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sc3 g a0 (CHead -c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O vs) t)) \to ((sc3 g a1 c v) -\to (sc3 g a0 c (THeads (Flat Appl) vs (THead (Bind b) v -t))))))))))).(\lambda (vs: TList).(\lambda (c: C).(\lambda (v: T).(\lambda -(t: T).(\lambda (H2: (land (arity g (CHead c (Bind b) v) (THeads (Flat Appl) -(lifts (S O) O vs) t) (AHead a a0)) (\forall (d: C).(\forall (w: T).((sc3 g a -d w) \to (\forall (is: PList).((drop1 is d (CHead c (Bind b) v)) \to (sc3 g -a0 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) (lifts (S O) O vs) -t))))))))))).(\lambda (H3: (sc3 g a1 c v)).(let H4 \def H2 in (land_ind -(arity g (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O vs) t) -(AHead a a0)) (\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall -(is: PList).((drop1 is d (CHead c (Bind b) v)) \to (sc3 g a0 d (THead (Flat -Appl) w (lift1 is (THeads (Flat Appl) (lifts (S O) O vs) t))))))))) (land -(arity g c (THeads (Flat Appl) vs (THead (Bind b) v t)) (AHead a a0)) -(\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall (is: -PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w (lift1 is (THeads -(Flat Appl) vs (THead (Bind b) v t))))))))))) (\lambda (H5: (arity g (CHead c -(Bind b) v) (THeads (Flat Appl) (lifts (S O) O vs) t) (AHead a a0))).(\lambda -(H6: ((\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall (is: -PList).((drop1 is d (CHead c (Bind b) v)) \to (sc3 g a0 d (THead (Flat Appl) -w (lift1 is (THeads (Flat Appl) (lifts (S O) O vs) t))))))))))).(conj (arity -g c (THeads (Flat Appl) vs (THead (Bind b) v t)) (AHead a a0)) (\forall (d: -C).(\forall (w: T).((sc3 g a d w) \to (\forall (is: PList).((drop1 is d c) -\to (sc3 g a0 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs (THead -(Bind b) v t)))))))))) (arity_appls_bind g b H c v a1 (sc3_arity_gen g c v a1 -H3) t vs (AHead a a0) H5) (\lambda (d: C).(\lambda (w: T).(\lambda (H7: (sc3 -g a d w)).(\lambda (is: PList).(\lambda (H8: (drop1 is d c)).(let H_y \def -(H1 (TCons w (lifts1 is vs))) in (eq_ind_r T (THeads (Flat Appl) (lifts1 is -vs) (lift1 is (THead (Bind b) v t))) (\lambda (t0: T).(sc3 g a0 d (THead -(Flat Appl) w t0))) (eq_ind_r T (THead (Bind b) (lift1 is v) (lift1 (Ss is) -t)) (\lambda (t0: T).(sc3 g a0 d (THead (Flat Appl) w (THeads (Flat Appl) -(lifts1 is vs) t0)))) (H_y d (lift1 is v) (lift1 (Ss is) t) (eq_ind TList -(lifts1 (Ss is) (lifts (S O) O vs)) (\lambda (t0: TList).(sc3 g a0 (CHead d -(Bind b) (lift1 is v)) (THead (Flat Appl) (lift (S O) O w) (THeads (Flat -Appl) t0 (lift1 (Ss is) t))))) (eq_ind T (lift1 (Ss is) (THeads (Flat Appl) -(lifts (S O) O vs) t)) (\lambda (t0: T).(sc3 g a0 (CHead d (Bind b) (lift1 is -v)) (THead (Flat Appl) (lift (S O) O w) t0))) (H6 (CHead d (Bind b) (lift1 is -v)) (lift (S O) O w) (sc3_lift g a d w H7 (CHead d (Bind b) (lift1 is v)) (S -O) O (drop_drop (Bind b) O d d (drop_refl d) (lift1 is v))) (Ss is) -(drop1_skip_bind b c is d v H8)) (THeads (Flat Appl) (lifts1 (Ss is) (lifts -(S O) O vs)) (lift1 (Ss is) t)) (lifts1_flat Appl (Ss is) t (lifts (S O) O -vs))) (lifts (S O) O (lifts1 is vs)) (lifts1_xhg is vs)) (sc3_lift1 g c a1 is -d v H3 H8)) (lift1 is (THead (Bind b) v t)) (lift1_bind b is v t)) (lift1 is -(THeads (Flat Appl) vs (THead (Bind b) v t))) (lifts1_flat Appl is (THead -(Bind b) v t) vs))))))))))) H4)))))))))))) a2))))). -(* COMMENTS -Initial nodes: 1797 -END *) - -theorem sc3_appl: - \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (vs: -TList).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sc3 g a2 c (THeads -(Flat Appl) vs (THead (Bind Abbr) v t))) \to ((sc3 g a1 c v) \to (\forall (w: -T).((sc3 g (asucc g a1) c w) \to (sc3 g a2 c (THeads (Flat Appl) vs (THead -(Flat Appl) v (THead (Bind Abst) w t)))))))))))))) -\def - \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(A_ind (\lambda (a: -A).(\forall (vs: TList).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sc3 -g a c (THeads (Flat Appl) vs (THead (Bind Abbr) v t))) \to ((sc3 g a1 c v) -\to (\forall (w: T).((sc3 g (asucc g a1) c w) \to (sc3 g a c (THeads (Flat -Appl) vs (THead (Flat Appl) v (THead (Bind Abst) w t))))))))))))) (\lambda -(n: nat).(\lambda (n0: nat).(\lambda (vs: TList).(\lambda (c: C).(\lambda (v: -T).(\lambda (t: T).(\lambda (H: (land (arity g c (THeads (Flat Appl) vs -(THead (Bind Abbr) v t)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (THead -(Bind Abbr) v t))))).(\lambda (H0: (sc3 g a1 c v)).(\lambda (w: T).(\lambda -(H1: (sc3 g (asucc g a1) c w)).(let H2 \def H in (land_ind (arity g c (THeads -(Flat Appl) vs (THead (Bind Abbr) v t)) (ASort n n0)) (sn3 c (THeads (Flat -Appl) vs (THead (Bind Abbr) v t))) (land (arity g c (THeads (Flat Appl) vs -(THead (Flat Appl) v (THead (Bind Abst) w t))) (ASort n n0)) (sn3 c (THeads -(Flat Appl) vs (THead (Flat Appl) v (THead (Bind Abst) w t))))) (\lambda (H3: -(arity g c (THeads (Flat Appl) vs (THead (Bind Abbr) v t)) (ASort n -n0))).(\lambda (H4: (sn3 c (THeads (Flat Appl) vs (THead (Bind Abbr) v -t)))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat Appl) v (THead -(Bind Abst) w t))) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (THead (Flat -Appl) v (THead (Bind Abst) w t)))) (arity_appls_appl g c v a1 (sc3_arity_gen -g c v a1 H0) w (sc3_arity_gen g c w (asucc g a1) H1) t vs (ASort n n0) H3) -(sn3_appls_beta c v t vs H4 w (sc3_sn3 g (asucc g a1) c w H1))))) -H2)))))))))))) (\lambda (a: A).(\lambda (_: ((\forall (vs: TList).(\forall -(c: C).(\forall (v: T).(\forall (t: T).((sc3 g a c (THeads (Flat Appl) vs -(THead (Bind Abbr) v t))) \to ((sc3 g a1 c v) \to (\forall (w: T).((sc3 g -(asucc g a1) c w) \to (sc3 g a c (THeads (Flat Appl) vs (THead (Flat Appl) v -(THead (Bind Abst) w t)))))))))))))).(\lambda (a0: A).(\lambda (H0: ((\forall -(vs: TList).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sc3 g a0 c -(THeads (Flat Appl) vs (THead (Bind Abbr) v t))) \to ((sc3 g a1 c v) \to -(\forall (w: T).((sc3 g (asucc g a1) c w) \to (sc3 g a0 c (THeads (Flat Appl) -vs (THead (Flat Appl) v (THead (Bind Abst) w t)))))))))))))).(\lambda (vs: -TList).(\lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (H1: (land -(arity g c (THeads (Flat Appl) vs (THead (Bind Abbr) v t)) (AHead a a0)) -(\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall (is: -PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w (lift1 is (THeads -(Flat Appl) vs (THead (Bind Abbr) v t)))))))))))).(\lambda (H2: (sc3 g a1 c -v)).(\lambda (w: T).(\lambda (H3: (sc3 g (asucc g a1) c w)).(let H4 \def H1 -in (land_ind (arity g c (THeads (Flat Appl) vs (THead (Bind Abbr) v t)) -(AHead a a0)) (\forall (d: C).(\forall (w0: T).((sc3 g a d w0) \to (\forall -(is: PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w0 (lift1 is -(THeads (Flat Appl) vs (THead (Bind Abbr) v t)))))))))) (land (arity g c -(THeads (Flat Appl) vs (THead (Flat Appl) v (THead (Bind Abst) w t))) (AHead -a a0)) (\forall (d: C).(\forall (w0: T).((sc3 g a d w0) \to (\forall (is: -PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w0 (lift1 is -(THeads (Flat Appl) vs (THead (Flat Appl) v (THead (Bind Abst) w -t)))))))))))) (\lambda (H5: (arity g c (THeads (Flat Appl) vs (THead (Bind -Abbr) v t)) (AHead a a0))).(\lambda (H6: ((\forall (d: C).(\forall (w0: -T).((sc3 g a d w0) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a0 d -(THead (Flat Appl) w0 (lift1 is (THeads (Flat Appl) vs (THead (Bind Abbr) v -t)))))))))))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat Appl) v -(THead (Bind Abst) w t))) (AHead a a0)) (\forall (d: C).(\forall (w0: -T).((sc3 g a d w0) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a0 d -(THead (Flat Appl) w0 (lift1 is (THeads (Flat Appl) vs (THead (Flat Appl) v -(THead (Bind Abst) w t))))))))))) (arity_appls_appl g c v a1 (sc3_arity_gen g -c v a1 H2) w (sc3_arity_gen g c w (asucc g a1) H3) t vs (AHead a a0) H5) -(\lambda (d: C).(\lambda (w0: T).(\lambda (H7: (sc3 g a d w0)).(\lambda (is: -PList).(\lambda (H8: (drop1 is d c)).(eq_ind_r T (THeads (Flat Appl) (lifts1 -is vs) (lift1 is (THead (Flat Appl) v (THead (Bind Abst) w t)))) (\lambda -(t0: T).(sc3 g a0 d (THead (Flat Appl) w0 t0))) (eq_ind_r T (THead (Flat -Appl) (lift1 is v) (lift1 is (THead (Bind Abst) w t))) (\lambda (t0: T).(sc3 -g a0 d (THead (Flat Appl) w0 (THeads (Flat Appl) (lifts1 is vs) t0)))) -(eq_ind_r T (THead (Bind Abst) (lift1 is w) (lift1 (Ss is) t)) (\lambda (t0: -T).(sc3 g a0 d (THead (Flat Appl) w0 (THeads (Flat Appl) (lifts1 is vs) -(THead (Flat Appl) (lift1 is v) t0))))) (let H_y \def (H0 (TCons w0 (lifts1 -is vs))) in (H_y d (lift1 is v) (lift1 (Ss is) t) (eq_ind T (lift1 is (THead -(Bind Abbr) v t)) (\lambda (t0: T).(sc3 g a0 d (THead (Flat Appl) w0 (THeads -(Flat Appl) (lifts1 is vs) t0)))) (eq_ind T (lift1 is (THeads (Flat Appl) vs -(THead (Bind Abbr) v t))) (\lambda (t0: T).(sc3 g a0 d (THead (Flat Appl) w0 -t0))) (H6 d w0 H7 is H8) (THeads (Flat Appl) (lifts1 is vs) (lift1 is (THead -(Bind Abbr) v t))) (lifts1_flat Appl is (THead (Bind Abbr) v t) vs)) (THead -(Bind Abbr) (lift1 is v) (lift1 (Ss is) t)) (lift1_bind Abbr is v t)) -(sc3_lift1 g c a1 is d v H2 H8) (lift1 is w) (sc3_lift1 g c (asucc g a1) is d -w H3 H8))) (lift1 is (THead (Bind Abst) w t)) (lift1_bind Abst is w t)) -(lift1 is (THead (Flat Appl) v (THead (Bind Abst) w t))) (lift1_flat Appl is -v (THead (Bind Abst) w t))) (lift1 is (THeads (Flat Appl) vs (THead (Flat -Appl) v (THead (Bind Abst) w t)))) (lifts1_flat Appl is (THead (Flat Appl) v -(THead (Bind Abst) w t)) vs)))))))))) H4)))))))))))))) a2))). -(* COMMENTS -Initial nodes: 1901 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/sn3/defs.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/sn3/defs.ma deleted file mode 100644 index 686e6c673..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/sn3/defs.ma +++ /dev/null @@ -1,29 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr3/defs.ma". - -inductive sn3 (c: C): T \to Prop \def -| sn3_sing: \forall (t1: T).(((\forall (t2: T).((((eq T t1 t2) \to (\forall -(P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2))))) \to (sn3 c t1)). - -definition sns3: - C \to (TList \to Prop) -\def - let rec sns3 (c: C) (ts: TList) on ts: Prop \def (match ts with [TNil -\Rightarrow True | (TCons t ts0) \Rightarrow (land (sn3 c t) (sns3 c ts0))]) -in sns3. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/sn3/fwd.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/sn3/fwd.ma deleted file mode 100644 index 68276fe9f..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/sn3/fwd.ma +++ /dev/null @@ -1,197 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/sn3/defs.ma". - -include "Basic-1/pr3/props.ma". - -theorem sn3_gen_bind: - \forall (b: B).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 c -(THead (Bind b) u t)) \to (land (sn3 c u) (sn3 (CHead c (Bind b) u) t)))))) -\def - \lambda (b: B).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: -(sn3 c (THead (Bind b) u t))).(insert_eq T (THead (Bind b) u t) (\lambda (t0: -T).(sn3 c t0)) (\lambda (_: T).(land (sn3 c u) (sn3 (CHead c (Bind b) u) t))) -(\lambda (y: T).(\lambda (H0: (sn3 c y)).(unintro T t (\lambda (t0: T).((eq T -y (THead (Bind b) u t0)) \to (land (sn3 c u) (sn3 (CHead c (Bind b) u) t0)))) -(unintro T u (\lambda (t0: T).(\forall (x: T).((eq T y (THead (Bind b) t0 x)) -\to (land (sn3 c t0) (sn3 (CHead c (Bind b) t0) x))))) (sn3_ind c (\lambda -(t0: T).(\forall (x: T).(\forall (x0: T).((eq T t0 (THead (Bind b) x x0)) \to -(land (sn3 c x) (sn3 (CHead c (Bind b) x) x0)))))) (\lambda (t1: T).(\lambda -(H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 -c t1 t2) \to (sn3 c t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2) -\to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (\forall (x: T).(\forall -(x0: T).((eq T t2 (THead (Bind b) x x0)) \to (land (sn3 c x) (sn3 (CHead c -(Bind b) x) x0)))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: (eq T -t1 (THead (Bind b) x x0))).(let H4 \def (eq_ind T t1 (\lambda (t0: -T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c -t0 t2) \to (\forall (x1: T).(\forall (x2: T).((eq T t2 (THead (Bind b) x1 -x2)) \to (land (sn3 c x1) (sn3 (CHead c (Bind b) x1) x2))))))))) H2 (THead -(Bind b) x x0) H3) in (let H5 \def (eq_ind T t1 (\lambda (t0: T).(\forall -(t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to -(sn3 c t2))))) H1 (THead (Bind b) x x0) H3) in (conj (sn3 c x) (sn3 (CHead c -(Bind b) x) x0) (sn3_sing c x (\lambda (t2: T).(\lambda (H6: (((eq T x t2) -\to (\forall (P: Prop).P)))).(\lambda (H7: (pr3 c x t2)).(let H8 \def (H4 -(THead (Bind b) t2 x0) (\lambda (H8: (eq T (THead (Bind b) x x0) (THead (Bind -b) t2 x0))).(\lambda (P: Prop).(let H9 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x | -(TLRef _) \Rightarrow x | (THead _ t0 _) \Rightarrow t0])) (THead (Bind b) x -x0) (THead (Bind b) t2 x0) H8) in (let H10 \def (eq_ind_r T t2 (\lambda (t0: -T).(pr3 c x t0)) H7 x H9) in (let H11 \def (eq_ind_r T t2 (\lambda (t0: -T).((eq T x t0) \to (\forall (P0: Prop).P0))) H6 x H9) in (H11 (refl_equal T -x) P)))))) (pr3_head_12 c x t2 H7 (Bind b) x0 x0 (pr3_refl (CHead c (Bind b) -t2) x0)) t2 x0 (refl_equal T (THead (Bind b) t2 x0))) in (land_ind (sn3 c t2) -(sn3 (CHead c (Bind b) t2) x0) (sn3 c t2) (\lambda (H9: (sn3 c t2)).(\lambda -(_: (sn3 (CHead c (Bind b) t2) x0)).H9)) H8)))))) (sn3_sing (CHead c (Bind b) -x) x0 (\lambda (t2: T).(\lambda (H6: (((eq T x0 t2) \to (\forall (P: -Prop).P)))).(\lambda (H7: (pr3 (CHead c (Bind b) x) x0 t2)).(let H8 \def (H4 -(THead (Bind b) x t2) (\lambda (H8: (eq T (THead (Bind b) x x0) (THead (Bind -b) x t2))).(\lambda (P: Prop).(let H9 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | -(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind b) x -x0) (THead (Bind b) x t2) H8) in (let H10 \def (eq_ind_r T t2 (\lambda (t0: -T).(pr3 (CHead c (Bind b) x) x0 t0)) H7 x0 H9) in (let H11 \def (eq_ind_r T -t2 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H6 x0 H9) in -(H11 (refl_equal T x0) P)))))) (pr3_head_12 c x x (pr3_refl c x) (Bind b) x0 -t2 H7) x t2 (refl_equal T (THead (Bind b) x t2))) in (land_ind (sn3 c x) (sn3 -(CHead c (Bind b) x) t2) (sn3 (CHead c (Bind b) x) t2) (\lambda (_: (sn3 c -x)).(\lambda (H10: (sn3 (CHead c (Bind b) x) t2)).H10)) H8))))))))))))))) y -H0))))) H))))). -(* COMMENTS -Initial nodes: 1055 -END *) - -theorem sn3_gen_flat: - \forall (f: F).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 c -(THead (Flat f) u t)) \to (land (sn3 c u) (sn3 c t)))))) -\def - \lambda (f: F).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: -(sn3 c (THead (Flat f) u t))).(insert_eq T (THead (Flat f) u t) (\lambda (t0: -T).(sn3 c t0)) (\lambda (_: T).(land (sn3 c u) (sn3 c t))) (\lambda (y: -T).(\lambda (H0: (sn3 c y)).(unintro T t (\lambda (t0: T).((eq T y (THead -(Flat f) u t0)) \to (land (sn3 c u) (sn3 c t0)))) (unintro T u (\lambda (t0: -T).(\forall (x: T).((eq T y (THead (Flat f) t0 x)) \to (land (sn3 c t0) (sn3 -c x))))) (sn3_ind c (\lambda (t0: T).(\forall (x: T).(\forall (x0: T).((eq T -t0 (THead (Flat f) x x0)) \to (land (sn3 c x) (sn3 c x0)))))) (\lambda (t1: -T).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: -Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H2: ((\forall -(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to -(\forall (x: T).(\forall (x0: T).((eq T t2 (THead (Flat f) x x0)) \to (land -(sn3 c x) (sn3 c x0)))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: -(eq T t1 (THead (Flat f) x x0))).(let H4 \def (eq_ind T t1 (\lambda (t0: -T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c -t0 t2) \to (\forall (x1: T).(\forall (x2: T).((eq T t2 (THead (Flat f) x1 -x2)) \to (land (sn3 c x1) (sn3 c x2))))))))) H2 (THead (Flat f) x x0) H3) in -(let H5 \def (eq_ind T t1 (\lambda (t0: T).(\forall (t2: T).((((eq T t0 t2) -\to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (sn3 c t2))))) H1 (THead -(Flat f) x x0) H3) in (conj (sn3 c x) (sn3 c x0) (sn3_sing c x (\lambda (t2: -T).(\lambda (H6: (((eq T x t2) \to (\forall (P: Prop).P)))).(\lambda (H7: -(pr3 c x t2)).(let H8 \def (H4 (THead (Flat f) t2 x0) (\lambda (H8: (eq T -(THead (Flat f) x x0) (THead (Flat f) t2 x0))).(\lambda (P: Prop).(let H9 -\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t0 _) -\Rightarrow t0])) (THead (Flat f) x x0) (THead (Flat f) t2 x0) H8) in (let -H10 \def (eq_ind_r T t2 (\lambda (t0: T).(pr3 c x t0)) H7 x H9) in (let H11 -\def (eq_ind_r T t2 (\lambda (t0: T).((eq T x t0) \to (\forall (P0: -Prop).P0))) H6 x H9) in (H11 (refl_equal T x) P)))))) (pr3_head_12 c x t2 H7 -(Flat f) x0 x0 (pr3_refl (CHead c (Flat f) t2) x0)) t2 x0 (refl_equal T -(THead (Flat f) t2 x0))) in (land_ind (sn3 c t2) (sn3 c x0) (sn3 c t2) -(\lambda (H9: (sn3 c t2)).(\lambda (_: (sn3 c x0)).H9)) H8)))))) (sn3_sing c -x0 (\lambda (t2: T).(\lambda (H6: (((eq T x0 t2) \to (\forall (P: -Prop).P)))).(\lambda (H7: (pr3 c x0 t2)).(let H8 \def (H4 (THead (Flat f) x -t2) (\lambda (H8: (eq T (THead (Flat f) x x0) (THead (Flat f) x -t2))).(\lambda (P: Prop).(let H9 \def (f_equal T T (\lambda (e: T).(match e -in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) -\Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat f) x x0) -(THead (Flat f) x t2) H8) in (let H10 \def (eq_ind_r T t2 (\lambda (t0: -T).(pr3 c x0 t0)) H7 x0 H9) in (let H11 \def (eq_ind_r T t2 (\lambda (t0: -T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H6 x0 H9) in (H11 (refl_equal -T x0) P)))))) (pr3_thin_dx c x0 t2 H7 x f) x t2 (refl_equal T (THead (Flat f) -x t2))) in (land_ind (sn3 c x) (sn3 c t2) (sn3 c t2) (\lambda (_: (sn3 c -x)).(\lambda (H10: (sn3 c t2)).H10)) H8))))))))))))))) y H0))))) H))))). -(* COMMENTS -Initial nodes: 925 -END *) - -theorem sn3_gen_head: - \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 c -(THead k u t)) \to (sn3 c u))))) -\def - \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (c: C).(\forall (u: -T).(\forall (t: T).((sn3 c (THead k0 u t)) \to (sn3 c u)))))) (\lambda (b: -B).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (sn3 c (THead -(Bind b) u t))).(let H_x \def (sn3_gen_bind b c u t H) in (let H0 \def H_x in -(land_ind (sn3 c u) (sn3 (CHead c (Bind b) u) t) (sn3 c u) (\lambda (H1: (sn3 -c u)).(\lambda (_: (sn3 (CHead c (Bind b) u) t)).H1)) H0)))))))) (\lambda (f: -F).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (sn3 c (THead -(Flat f) u t))).(let H_x \def (sn3_gen_flat f c u t H) in (let H0 \def H_x in -(land_ind (sn3 c u) (sn3 c t) (sn3 c u) (\lambda (H1: (sn3 c u)).(\lambda (_: -(sn3 c t)).H1)) H0)))))))) k). -(* COMMENTS -Initial nodes: 191 -END *) - -theorem sn3_gen_cflat: - \forall (f: F).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 (CHead -c (Flat f) u) t) \to (sn3 c t))))) -\def - \lambda (f: F).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: -(sn3 (CHead c (Flat f) u) t)).(sn3_ind (CHead c (Flat f) u) (\lambda (t0: -T).(sn3 c t0)) (\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 -t2) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Flat f) u) t1 t2) \to -(sn3 (CHead c (Flat f) u) t2)))))).(\lambda (H1: ((\forall (t2: T).((((eq T -t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Flat f) u) t1 t2) \to -(sn3 c t2)))))).(sn3_sing c t1 (\lambda (t2: T).(\lambda (H2: (((eq T t1 t2) -\to (\forall (P: Prop).P)))).(\lambda (H3: (pr3 c t1 t2)).(H1 t2 H2 -(pr3_cflat c t1 t2 H3 f u))))))))) t H))))). -(* COMMENTS -Initial nodes: 175 -END *) - -theorem sn3_gen_lift: - \forall (c1: C).(\forall (t: T).(\forall (h: nat).(\forall (d: nat).((sn3 c1 -(lift h d t)) \to (\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 t))))))) -\def - \lambda (c1: C).(\lambda (t: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda -(H: (sn3 c1 (lift h d t))).(insert_eq T (lift h d t) (\lambda (t0: T).(sn3 c1 -t0)) (\lambda (_: T).(\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 t)))) -(\lambda (y: T).(\lambda (H0: (sn3 c1 y)).(unintro T t (\lambda (t0: T).((eq -T y (lift h d t0)) \to (\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 t0))))) -(sn3_ind c1 (\lambda (t0: T).(\forall (x: T).((eq T t0 (lift h d x)) \to -(\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 x)))))) (\lambda (t1: -T).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: -Prop).P))) \to ((pr3 c1 t1 t2) \to (sn3 c1 t2)))))).(\lambda (H2: ((\forall -(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c1 t1 t2) \to -(\forall (x: T).((eq T t2 (lift h d x)) \to (\forall (c2: C).((drop h d c1 -c2) \to (sn3 c2 x)))))))))).(\lambda (x: T).(\lambda (H3: (eq T t1 (lift h d -x))).(\lambda (c2: C).(\lambda (H4: (drop h d c1 c2)).(let H5 \def (eq_ind T -t1 (\lambda (t0: T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P: -Prop).P))) \to ((pr3 c1 t0 t2) \to (\forall (x0: T).((eq T t2 (lift h d x0)) -\to (\forall (c3: C).((drop h d c1 c3) \to (sn3 c3 x0))))))))) H2 (lift h d -x) H3) in (let H6 \def (eq_ind T t1 (\lambda (t0: T).(\forall (t2: T).((((eq -T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c1 t0 t2) \to (sn3 c1 t2))))) -H1 (lift h d x) H3) in (sn3_sing c2 x (\lambda (t2: T).(\lambda (H7: (((eq T -x t2) \to (\forall (P: Prop).P)))).(\lambda (H8: (pr3 c2 x t2)).(H5 (lift h d -t2) (\lambda (H9: (eq T (lift h d x) (lift h d t2))).(\lambda (P: Prop).(let -H10 \def (eq_ind_r T t2 (\lambda (t0: T).(pr3 c2 x t0)) H8 x (lift_inj x t2 h -d H9)) in (let H11 \def (eq_ind_r T t2 (\lambda (t0: T).((eq T x t0) \to -(\forall (P0: Prop).P0))) H7 x (lift_inj x t2 h d H9)) in (H11 (refl_equal T -x) P))))) (pr3_lift c1 c2 h d H4 x t2 H8) t2 (refl_equal T (lift h d t2)) c2 -H4)))))))))))))) y H0)))) H))))). -(* COMMENTS -Initial nodes: 565 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/sn3/lift1.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/sn3/lift1.ma deleted file mode 100644 index 1b64c22bb..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/sn3/lift1.ma +++ /dev/null @@ -1,46 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/sn3/props.ma". - -include "Basic-1/drop1/fwd.ma". - -include "Basic-1/lift1/fwd.ma". - -theorem sns3_lifts1: - \forall (e: C).(\forall (hds: PList).(\forall (c: C).((drop1 hds c e) \to -(\forall (ts: TList).((sns3 e ts) \to (sns3 c (lifts1 hds ts))))))) -\def - \lambda (e: C).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall -(c: C).((drop1 p c e) \to (\forall (ts: TList).((sns3 e ts) \to (sns3 c -(lifts1 p ts))))))) (\lambda (c: C).(\lambda (H: (drop1 PNil c e)).(\lambda -(ts: TList).(\lambda (H0: (sns3 e ts)).(let H_y \def (drop1_gen_pnil c e H) -in (eq_ind_r C e (\lambda (c0: C).(sns3 c0 (lifts1 PNil ts))) (eq_ind_r TList -ts (\lambda (t: TList).(sns3 e t)) H0 (lifts1 PNil ts) (lifts1_nil ts)) c -H_y)))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda -(H: ((\forall (c: C).((drop1 p c e) \to (\forall (ts: TList).((sns3 e ts) \to -(sns3 c (lifts1 p ts)))))))).(\lambda (c: C).(\lambda (H0: (drop1 (PCons n n0 -p) c e)).(\lambda (ts: TList).(\lambda (H1: (sns3 e ts)).(let H_x \def -(drop1_gen_pcons c e p n n0 H0) in (let H2 \def H_x in (ex2_ind C (\lambda -(c2: C).(drop n n0 c c2)) (\lambda (c2: C).(drop1 p c2 e)) (sns3 c (lifts1 -(PCons n n0 p) ts)) (\lambda (x: C).(\lambda (H3: (drop n n0 c x)).(\lambda -(H4: (drop1 p x e)).(eq_ind_r TList (lifts n n0 (lifts1 p ts)) (\lambda (t: -TList).(sns3 c t)) (sns3_lifts c x n n0 H3 (lifts1 p ts) (H x H4 ts H1)) -(lifts1 (PCons n n0 p) ts) (lifts1_cons n n0 p ts))))) H2))))))))))) hds)). -(* COMMENTS -Initial nodes: 323 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/sn3/nf2.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/sn3/nf2.ma deleted file mode 100644 index 824428084..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/sn3/nf2.ma +++ /dev/null @@ -1,66 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/sn3/defs.ma". - -include "Basic-1/nf2/dec.ma". - -include "Basic-1/nf2/pr3.ma". - -theorem sn3_nf2: - \forall (c: C).(\forall (t: T).((nf2 c t) \to (sn3 c t))) -\def - \lambda (c: C).(\lambda (t: T).(\lambda (H: (nf2 c t)).(sn3_sing c t -(\lambda (t2: T).(\lambda (H0: (((eq T t t2) \to (\forall (P: -Prop).P)))).(\lambda (H1: (pr3 c t t2)).(let H_y \def (nf2_pr3_unfold c t t2 -H1 H) in (let H2 \def (eq_ind_r T t2 (\lambda (t0: T).(pr3 c t t0)) H1 t H_y) -in (let H3 \def (eq_ind_r T t2 (\lambda (t0: T).((eq T t t0) \to (\forall (P: -Prop).P))) H0 t H_y) in (eq_ind T t (\lambda (t0: T).(sn3 c t0)) (H3 -(refl_equal T t) (sn3 c t)) t2 H_y)))))))))). -(* COMMENTS -Initial nodes: 129 -END *) - -theorem nf2_sn3: - \forall (c: C).(\forall (t: T).((sn3 c t) \to (ex2 T (\lambda (u: T).(pr3 c -t u)) (\lambda (u: T).(nf2 c u))))) -\def - \lambda (c: C).(\lambda (t: T).(\lambda (H: (sn3 c t)).(sn3_ind c (\lambda -(t0: T).(ex2 T (\lambda (u: T).(pr3 c t0 u)) (\lambda (u: T).(nf2 c u)))) -(\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) \to (\forall -(P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H1: ((\forall -(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to -(ex2 T (\lambda (u: T).(pr3 c t2 u)) (\lambda (u: T).(nf2 c u)))))))).(let -H_x \def (nf2_dec c t1) in (let H2 \def H_x in (or_ind (nf2 c t1) (ex2 T -(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: -T).(pr2 c t1 t2))) (ex2 T (\lambda (u: T).(pr3 c t1 u)) (\lambda (u: T).(nf2 -c u))) (\lambda (H3: (nf2 c t1)).(ex_intro2 T (\lambda (u: T).(pr3 c t1 u)) -(\lambda (u: T).(nf2 c u)) t1 (pr3_refl c t1) H3)) (\lambda (H3: (ex2 T -(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: -T).(pr2 c t1 t2)))).(ex2_ind T (\lambda (t2: T).((eq T t1 t2) \to (\forall -(P: Prop).P))) (\lambda (t2: T).(pr2 c t1 t2)) (ex2 T (\lambda (u: T).(pr3 c -t1 u)) (\lambda (u: T).(nf2 c u))) (\lambda (x: T).(\lambda (H4: (((eq T t1 -x) \to (\forall (P: Prop).P)))).(\lambda (H5: (pr2 c t1 x)).(let H_y \def (H1 -x H4) in (let H6 \def (H_y (pr3_pr2 c t1 x H5)) in (ex2_ind T (\lambda (u: -T).(pr3 c x u)) (\lambda (u: T).(nf2 c u)) (ex2 T (\lambda (u: T).(pr3 c t1 -u)) (\lambda (u: T).(nf2 c u))) (\lambda (x0: T).(\lambda (H7: (pr3 c x -x0)).(\lambda (H8: (nf2 c x0)).(ex_intro2 T (\lambda (u: T).(pr3 c t1 u)) -(\lambda (u: T).(nf2 c u)) x0 (pr3_sing c x t1 H5 x0 H7) H8)))) H6)))))) H3)) -H2)))))) t H))). -(* COMMENTS -Initial nodes: 443 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/sn3/props.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/sn3/props.ma deleted file mode 100644 index ea72c8869..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/sn3/props.ma +++ /dev/null @@ -1,2575 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/sn3/nf2.ma". - -include "Basic-1/sn3/fwd.ma". - -include "Basic-1/nf2/iso.ma". - -include "Basic-1/pr3/iso.ma". - -theorem sn3_pr3_trans: - \forall (c: C).(\forall (t1: T).((sn3 c t1) \to (\forall (t2: T).((pr3 c t1 -t2) \to (sn3 c t2))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (H: (sn3 c t1)).(sn3_ind c (\lambda -(t: T).(\forall (t2: T).((pr3 c t t2) \to (sn3 c t2)))) (\lambda (t2: -T).(\lambda (H0: ((\forall (t3: T).((((eq T t2 t3) \to (\forall (P: -Prop).P))) \to ((pr3 c t2 t3) \to (sn3 c t3)))))).(\lambda (H1: ((\forall -(t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to -(\forall (t4: T).((pr3 c t3 t4) \to (sn3 c t4)))))))).(\lambda (t3: -T).(\lambda (H2: (pr3 c t2 t3)).(sn3_sing c t3 (\lambda (t0: T).(\lambda (H3: -(((eq T t3 t0) \to (\forall (P: Prop).P)))).(\lambda (H4: (pr3 c t3 t0)).(let -H_x \def (term_dec t2 t3) in (let H5 \def H_x in (or_ind (eq T t2 t3) ((eq T -t2 t3) \to (\forall (P: Prop).P)) (sn3 c t0) (\lambda (H6: (eq T t2 t3)).(let -H7 \def (eq_ind_r T t3 (\lambda (t: T).(pr3 c t t0)) H4 t2 H6) in (let H8 -\def (eq_ind_r T t3 (\lambda (t: T).((eq T t t0) \to (\forall (P: Prop).P))) -H3 t2 H6) in (let H9 \def (eq_ind_r T t3 (\lambda (t: T).(pr3 c t2 t)) H2 t2 -H6) in (H0 t0 H8 H7))))) (\lambda (H6: (((eq T t2 t3) \to (\forall (P: -Prop).P)))).(H1 t3 H6 H2 t0 H4)) H5)))))))))))) t1 H))). -(* COMMENTS -Initial nodes: 289 -END *) - -theorem sn3_pr2_intro: - \forall (c: C).(\forall (t1: T).(((\forall (t2: T).((((eq T t1 t2) \to -(\forall (P: Prop).P))) \to ((pr2 c t1 t2) \to (sn3 c t2))))) \to (sn3 c t1))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (H: ((\forall (t2: T).((((eq T t1 -t2) \to (\forall (P: Prop).P))) \to ((pr2 c t1 t2) \to (sn3 c -t2)))))).(sn3_sing c t1 (\lambda (t2: T).(\lambda (H0: (((eq T t1 t2) \to -(\forall (P: Prop).P)))).(\lambda (H1: (pr3 c t1 t2)).(let H2 \def H0 in -((let H3 \def H in (pr3_ind c (\lambda (t: T).(\lambda (t0: T).(((\forall -(t3: T).((((eq T t t3) \to (\forall (P: Prop).P))) \to ((pr2 c t t3) \to (sn3 -c t3))))) \to ((((eq T t t0) \to (\forall (P: Prop).P))) \to (sn3 c t0))))) -(\lambda (t: T).(\lambda (H4: ((\forall (t3: T).((((eq T t t3) \to (\forall -(P: Prop).P))) \to ((pr2 c t t3) \to (sn3 c t3)))))).(\lambda (H5: (((eq T t -t) \to (\forall (P: Prop).P)))).(H4 t H5 (pr2_free c t t (pr0_refl t)))))) -(\lambda (t3: T).(\lambda (t4: T).(\lambda (H4: (pr2 c t4 t3)).(\lambda (t5: -T).(\lambda (H5: (pr3 c t3 t5)).(\lambda (H6: ((((\forall (t6: T).((((eq T t3 -t6) \to (\forall (P: Prop).P))) \to ((pr2 c t3 t6) \to (sn3 c t6))))) \to -((((eq T t3 t5) \to (\forall (P: Prop).P))) \to (sn3 c t5))))).(\lambda (H7: -((\forall (t6: T).((((eq T t4 t6) \to (\forall (P: Prop).P))) \to ((pr2 c t4 -t6) \to (sn3 c t6)))))).(\lambda (H8: (((eq T t4 t5) \to (\forall (P: -Prop).P)))).(let H_x \def (term_dec t4 t3) in (let H9 \def H_x in (or_ind (eq -T t4 t3) ((eq T t4 t3) \to (\forall (P: Prop).P)) (sn3 c t5) (\lambda (H10: -(eq T t4 t3)).(let H11 \def (eq_ind T t4 (\lambda (t: T).((eq T t t5) \to -(\forall (P: Prop).P))) H8 t3 H10) in (let H12 \def (eq_ind T t4 (\lambda (t: -T).(\forall (t6: T).((((eq T t t6) \to (\forall (P: Prop).P))) \to ((pr2 c t -t6) \to (sn3 c t6))))) H7 t3 H10) in (let H13 \def (eq_ind T t4 (\lambda (t: -T).(pr2 c t t3)) H4 t3 H10) in (H6 H12 H11))))) (\lambda (H10: (((eq T t4 t3) -\to (\forall (P: Prop).P)))).(sn3_pr3_trans c t3 (H7 t3 H10 H4) t5 H5)) -H9))))))))))) t1 t2 H1 H3)) H2)))))))). -(* COMMENTS -Initial nodes: 467 -END *) - -theorem sn3_cast: - \forall (c: C).(\forall (u: T).((sn3 c u) \to (\forall (t: T).((sn3 c t) \to -(sn3 c (THead (Flat Cast) u t)))))) -\def - \lambda (c: C).(\lambda (u: T).(\lambda (H: (sn3 c u)).(sn3_ind c (\lambda -(t: T).(\forall (t0: T).((sn3 c t0) \to (sn3 c (THead (Flat Cast) t t0))))) -(\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) \to (\forall -(P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H1: ((\forall -(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to -(\forall (t: T).((sn3 c t) \to (sn3 c (THead (Flat Cast) t2 -t))))))))).(\lambda (t: T).(\lambda (H2: (sn3 c t)).(sn3_ind c (\lambda (t0: -T).(sn3 c (THead (Flat Cast) t1 t0))) (\lambda (t0: T).(\lambda (H3: -((\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t0 -t2) \to (sn3 c t2)))))).(\lambda (H4: ((\forall (t2: T).((((eq T t0 t2) \to -(\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (sn3 c (THead (Flat Cast) t1 -t2))))))).(sn3_pr2_intro c (THead (Flat Cast) t1 t0) (\lambda (t2: -T).(\lambda (H5: (((eq T (THead (Flat Cast) t1 t0) t2) \to (\forall (P: -Prop).P)))).(\lambda (H6: (pr2 c (THead (Flat Cast) t1 t0) t2)).(let H7 \def -(pr2_gen_cast c t1 t0 t2 H6) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda -(t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c t1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t0 t3)))) (pr2 c -t0 t2) (sn3 c t2) (\lambda (H8: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: -T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c t1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t0 -t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead -(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2))) -(\lambda (_: T).(\lambda (t3: T).(pr2 c t0 t3))) (sn3 c t2) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H9: (eq T t2 (THead (Flat Cast) x0 -x1))).(\lambda (H10: (pr2 c t1 x0)).(\lambda (H11: (pr2 c t0 x1)).(let H12 -\def (eq_ind T t2 (\lambda (t3: T).((eq T (THead (Flat Cast) t1 t0) t3) \to -(\forall (P: Prop).P))) H5 (THead (Flat Cast) x0 x1) H9) in (eq_ind_r T -(THead (Flat Cast) x0 x1) (\lambda (t3: T).(sn3 c t3)) (let H_x \def -(term_dec x0 t1) in (let H13 \def H_x in (or_ind (eq T x0 t1) ((eq T x0 t1) -\to (\forall (P: Prop).P)) (sn3 c (THead (Flat Cast) x0 x1)) (\lambda (H14: -(eq T x0 t1)).(let H15 \def (eq_ind T x0 (\lambda (t3: T).((eq T (THead (Flat -Cast) t1 t0) (THead (Flat Cast) t3 x1)) \to (\forall (P: Prop).P))) H12 t1 -H14) in (let H16 \def (eq_ind T x0 (\lambda (t3: T).(pr2 c t1 t3)) H10 t1 -H14) in (eq_ind_r T t1 (\lambda (t3: T).(sn3 c (THead (Flat Cast) t3 x1))) -(let H_x0 \def (term_dec t0 x1) in (let H17 \def H_x0 in (or_ind (eq T t0 x1) -((eq T t0 x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Cast) t1 x1)) -(\lambda (H18: (eq T t0 x1)).(let H19 \def (eq_ind_r T x1 (\lambda (t3: -T).((eq T (THead (Flat Cast) t1 t0) (THead (Flat Cast) t1 t3)) \to (\forall -(P: Prop).P))) H15 t0 H18) in (let H20 \def (eq_ind_r T x1 (\lambda (t3: -T).(pr2 c t0 t3)) H11 t0 H18) in (eq_ind T t0 (\lambda (t3: T).(sn3 c (THead -(Flat Cast) t1 t3))) (H19 (refl_equal T (THead (Flat Cast) t1 t0)) (sn3 c -(THead (Flat Cast) t1 t0))) x1 H18)))) (\lambda (H18: (((eq T t0 x1) \to -(\forall (P: Prop).P)))).(H4 x1 H18 (pr3_pr2 c t0 x1 H11))) H17))) x0 H14)))) -(\lambda (H14: (((eq T x0 t1) \to (\forall (P: Prop).P)))).(H1 x0 (\lambda -(H15: (eq T t1 x0)).(\lambda (P: Prop).(let H16 \def (eq_ind_r T x0 (\lambda -(t3: T).((eq T t3 t1) \to (\forall (P0: Prop).P0))) H14 t1 H15) in (let H17 -\def (eq_ind_r T x0 (\lambda (t3: T).((eq T (THead (Flat Cast) t1 t0) (THead -(Flat Cast) t3 x1)) \to (\forall (P0: Prop).P0))) H12 t1 H15) in (let H18 -\def (eq_ind_r T x0 (\lambda (t3: T).(pr2 c t1 t3)) H10 t1 H15) in (H16 -(refl_equal T t1) P)))))) (pr3_pr2 c t1 x0 H10) x1 (let H_x0 \def (term_dec -t0 x1) in (let H15 \def H_x0 in (or_ind (eq T t0 x1) ((eq T t0 x1) \to -(\forall (P: Prop).P)) (sn3 c x1) (\lambda (H16: (eq T t0 x1)).(let H17 \def -(eq_ind_r T x1 (\lambda (t3: T).((eq T (THead (Flat Cast) t1 t0) (THead (Flat -Cast) x0 t3)) \to (\forall (P: Prop).P))) H12 t0 H16) in (let H18 \def -(eq_ind_r T x1 (\lambda (t3: T).(pr2 c t0 t3)) H11 t0 H16) in (eq_ind T t0 -(\lambda (t3: T).(sn3 c t3)) (sn3_sing c t0 H3) x1 H16)))) (\lambda (H16: -(((eq T t0 x1) \to (\forall (P: Prop).P)))).(H3 x1 H16 (pr3_pr2 c t0 x1 -H11))) H15))))) H13))) t2 H9))))))) H8)) (\lambda (H8: (pr2 c t0 -t2)).(sn3_pr3_trans c t0 (sn3_sing c t0 H3) t2 (pr3_pr2 c t0 t2 H8))) -H7))))))))) t H2)))))) u H))). -(* COMMENTS -Initial nodes: 1239 -END *) - -theorem sn3_cflat: - \forall (c: C).(\forall (t: T).((sn3 c t) \to (\forall (f: F).(\forall (u: -T).(sn3 (CHead c (Flat f) u) t))))) -\def - \lambda (c: C).(\lambda (t: T).(\lambda (H: (sn3 c t)).(\lambda (f: -F).(\lambda (u: T).(sn3_ind c (\lambda (t0: T).(sn3 (CHead c (Flat f) u) t0)) -(\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) \to (\forall -(P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H1: ((\forall -(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to -(sn3 (CHead c (Flat f) u) t2)))))).(sn3_pr2_intro (CHead c (Flat f) u) t1 -(\lambda (t2: T).(\lambda (H2: (((eq T t1 t2) \to (\forall (P: -Prop).P)))).(\lambda (H3: (pr2 (CHead c (Flat f) u) t1 t2)).(H1 t2 H2 -(pr3_pr2 c t1 t2 (pr2_gen_cflat f c u t1 t2 H3)))))))))) t H))))). -(* COMMENTS -Initial nodes: 175 -END *) - -theorem sn3_shift: - \forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sn3 c -(THead (Bind b) v t)) \to (sn3 (CHead c (Bind b) v) t))))) -\def - \lambda (b: B).(\lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (H: -(sn3 c (THead (Bind b) v t))).(let H_x \def (sn3_gen_bind b c v t H) in (let -H0 \def H_x in (land_ind (sn3 c v) (sn3 (CHead c (Bind b) v) t) (sn3 (CHead c -(Bind b) v) t) (\lambda (_: (sn3 c v)).(\lambda (H2: (sn3 (CHead c (Bind b) -v) t)).H2)) H0))))))). -(* COMMENTS -Initial nodes: 95 -END *) - -theorem sn3_change: - \forall (b: B).((not (eq B b Abbr)) \to (\forall (c: C).(\forall (v1: -T).(\forall (t: T).((sn3 (CHead c (Bind b) v1) t) \to (\forall (v2: T).(sn3 -(CHead c (Bind b) v2) t))))))) -\def - \lambda (b: B).(\lambda (H: (not (eq B b Abbr))).(\lambda (c: C).(\lambda -(v1: T).(\lambda (t: T).(\lambda (H0: (sn3 (CHead c (Bind b) v1) t)).(\lambda -(v2: T).(sn3_ind (CHead c (Bind b) v1) (\lambda (t0: T).(sn3 (CHead c (Bind -b) v2) t0)) (\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) -\to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) v1) t1 t2) \to (sn3 -(CHead c (Bind b) v1) t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 -t2) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) v1) t1 t2) \to -(sn3 (CHead c (Bind b) v2) t2)))))).(sn3_pr2_intro (CHead c (Bind b) v2) t1 -(\lambda (t2: T).(\lambda (H3: (((eq T t1 t2) \to (\forall (P: -Prop).P)))).(\lambda (H4: (pr2 (CHead c (Bind b) v2) t1 t2)).(H2 t2 H3 -(pr3_pr2 (CHead c (Bind b) v1) t1 t2 (pr2_change b H c v2 t1 t2 H4 -v1)))))))))) t H0))))))). -(* COMMENTS -Initial nodes: 239 -END *) - -theorem sn3_gen_def: - \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c -(CHead d (Bind Abbr) v)) \to ((sn3 c (TLRef i)) \to (sn3 d v)))))) -\def - \lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda -(H: (getl i c (CHead d (Bind Abbr) v))).(\lambda (H0: (sn3 c (TLRef -i))).(sn3_gen_lift c v (S i) O (sn3_pr3_trans c (TLRef i) H0 (lift (S i) O v) -(pr3_pr2 c (TLRef i) (lift (S i) O v) (pr2_delta c d v i H (TLRef i) (TLRef -i) (pr0_refl (TLRef i)) (lift (S i) O v) (subst0_lref v i)))) d (getl_drop -Abbr c d v i H))))))). -(* COMMENTS -Initial nodes: 139 -END *) - -theorem sn3_cdelta: - \forall (v: T).(\forall (t: T).(\forall (i: nat).(((\forall (w: T).(ex T -(\lambda (u: T).(subst0 i w t u))))) \to (\forall (c: C).(\forall (d: -C).((getl i c (CHead d (Bind Abbr) v)) \to ((sn3 c t) \to (sn3 d v)))))))) -\def - \lambda (v: T).(\lambda (t: T).(\lambda (i: nat).(\lambda (H: ((\forall (w: -T).(ex T (\lambda (u: T).(subst0 i w t u)))))).(let H_x \def (H v) in (let H0 -\def H_x in (ex_ind T (\lambda (u: T).(subst0 i v t u)) (\forall (c: -C).(\forall (d: C).((getl i c (CHead d (Bind Abbr) v)) \to ((sn3 c t) \to -(sn3 d v))))) (\lambda (x: T).(\lambda (H1: (subst0 i v t x)).(subst0_ind -(\lambda (n: nat).(\lambda (t0: T).(\lambda (t1: T).(\lambda (_: T).(\forall -(c: C).(\forall (d: C).((getl n c (CHead d (Bind Abbr) t0)) \to ((sn3 c t1) -\to (sn3 d t0))))))))) (\lambda (v0: T).(\lambda (i0: nat).(\lambda (c: -C).(\lambda (d: C).(\lambda (H2: (getl i0 c (CHead d (Bind Abbr) -v0))).(\lambda (H3: (sn3 c (TLRef i0))).(sn3_gen_def c d v0 i0 H2 H3))))))) -(\lambda (v0: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i0: -nat).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (H3: ((\forall (c: -C).(\forall (d: C).((getl i0 c (CHead d (Bind Abbr) v0)) \to ((sn3 c u1) \to -(sn3 d v0))))))).(\lambda (t0: T).(\lambda (k: K).(\lambda (c: C).(\lambda -(d: C).(\lambda (H4: (getl i0 c (CHead d (Bind Abbr) v0))).(\lambda (H5: (sn3 -c (THead k u1 t0))).(let H_y \def (sn3_gen_head k c u1 t0 H5) in (H3 c d H4 -H_y)))))))))))))) (\lambda (k: K).(\lambda (v0: T).(\lambda (t2: T).(\lambda -(t1: T).(\lambda (i0: nat).(\lambda (H2: (subst0 (s k i0) v0 t1 t2)).(\lambda -(H3: ((\forall (c: C).(\forall (d: C).((getl (s k i0) c (CHead d (Bind Abbr) -v0)) \to ((sn3 c t1) \to (sn3 d v0))))))).(\lambda (u: T).(\lambda (c: -C).(\lambda (d: C).(\lambda (H4: (getl i0 c (CHead d (Bind Abbr) -v0))).(\lambda (H5: (sn3 c (THead k u t1))).(K_ind (\lambda (k0: K).((subst0 -(s k0 i0) v0 t1 t2) \to (((\forall (c0: C).(\forall (d0: C).((getl (s k0 i0) -c0 (CHead d0 (Bind Abbr) v0)) \to ((sn3 c0 t1) \to (sn3 d0 v0)))))) \to ((sn3 -c (THead k0 u t1)) \to (sn3 d v0))))) (\lambda (b: B).(\lambda (_: (subst0 (s -(Bind b) i0) v0 t1 t2)).(\lambda (H7: ((\forall (c0: C).(\forall (d0: -C).((getl (s (Bind b) i0) c0 (CHead d0 (Bind Abbr) v0)) \to ((sn3 c0 t1) \to -(sn3 d0 v0))))))).(\lambda (H8: (sn3 c (THead (Bind b) u t1))).(let H_x0 \def -(sn3_gen_bind b c u t1 H8) in (let H9 \def H_x0 in (land_ind (sn3 c u) (sn3 -(CHead c (Bind b) u) t1) (sn3 d v0) (\lambda (_: (sn3 c u)).(\lambda (H11: -(sn3 (CHead c (Bind b) u) t1)).(H7 (CHead c (Bind b) u) d (getl_clear_bind b -(CHead c (Bind b) u) c u (clear_bind b c u) (CHead d (Bind Abbr) v0) i0 H4) -H11))) H9))))))) (\lambda (f: F).(\lambda (_: (subst0 (s (Flat f) i0) v0 t1 -t2)).(\lambda (H7: ((\forall (c0: C).(\forall (d0: C).((getl (s (Flat f) i0) -c0 (CHead d0 (Bind Abbr) v0)) \to ((sn3 c0 t1) \to (sn3 d0 v0))))))).(\lambda -(H8: (sn3 c (THead (Flat f) u t1))).(let H_x0 \def (sn3_gen_flat f c u t1 H8) -in (let H9 \def H_x0 in (land_ind (sn3 c u) (sn3 c t1) (sn3 d v0) (\lambda -(_: (sn3 c u)).(\lambda (H11: (sn3 c t1)).(H7 c d H4 H11))) H9))))))) k H2 H3 -H5))))))))))))) (\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda -(i0: nat).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (H3: ((\forall (c: -C).(\forall (d: C).((getl i0 c (CHead d (Bind Abbr) v0)) \to ((sn3 c u1) \to -(sn3 d v0))))))).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(_: (subst0 (s k i0) v0 t1 t2)).(\lambda (_: ((\forall (c: C).(\forall (d: -C).((getl (s k i0) c (CHead d (Bind Abbr) v0)) \to ((sn3 c t1) \to (sn3 d -v0))))))).(\lambda (c: C).(\lambda (d: C).(\lambda (H6: (getl i0 c (CHead d -(Bind Abbr) v0))).(\lambda (H7: (sn3 c (THead k u1 t1))).(let H_y \def -(sn3_gen_head k c u1 t1 H7) in (H3 c d H6 H_y))))))))))))))))) i v t x H1))) -H0)))))). -(* COMMENTS -Initial nodes: 949 -END *) - -theorem sn3_cpr3_trans: - \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall -(k: K).(\forall (t: T).((sn3 (CHead c k u1) t) \to (sn3 (CHead c k u2) -t))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1 -u2)).(\lambda (k: K).(\lambda (t: T).(\lambda (H0: (sn3 (CHead c k u1) -t)).(sn3_ind (CHead c k u1) (\lambda (t0: T).(sn3 (CHead c k u2) t0)) -(\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) \to (\forall -(P: Prop).P))) \to ((pr3 (CHead c k u1) t1 t2) \to (sn3 (CHead c k u1) -t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: -Prop).P))) \to ((pr3 (CHead c k u1) t1 t2) \to (sn3 (CHead c k u2) -t2)))))).(sn3_sing (CHead c k u2) t1 (\lambda (t2: T).(\lambda (H3: (((eq T -t1 t2) \to (\forall (P: Prop).P)))).(\lambda (H4: (pr3 (CHead c k u2) t1 -t2)).(H2 t2 H3 (pr3_pr3_pr3_t c u1 u2 H t1 t2 k H4))))))))) t H0))))))). -(* COMMENTS -Initial nodes: 203 -END *) - -theorem sn3_bind: - \forall (b: B).(\forall (c: C).(\forall (u: T).((sn3 c u) \to (\forall (t: -T).((sn3 (CHead c (Bind b) u) t) \to (sn3 c (THead (Bind b) u t))))))) -\def - \lambda (b: B).(\lambda (c: C).(\lambda (u: T).(\lambda (H: (sn3 c -u)).(sn3_ind c (\lambda (t: T).(\forall (t0: T).((sn3 (CHead c (Bind b) t) -t0) \to (sn3 c (THead (Bind b) t t0))))) (\lambda (t1: T).(\lambda (_: -((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 -t2) \to (sn3 c t2)))))).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to -(\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (\forall (t: T).((sn3 (CHead c -(Bind b) t2) t) \to (sn3 c (THead (Bind b) t2 t))))))))).(\lambda (t: -T).(\lambda (H2: (sn3 (CHead c (Bind b) t1) t)).(sn3_ind (CHead c (Bind b) -t1) (\lambda (t0: T).(sn3 c (THead (Bind b) t1 t0))) (\lambda (t2: -T).(\lambda (H3: ((\forall (t3: T).((((eq T t2 t3) \to (\forall (P: -Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to (sn3 (CHead c (Bind b) -t1) t3)))))).(\lambda (H4: ((\forall (t3: T).((((eq T t2 t3) \to (\forall (P: -Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to (sn3 c (THead (Bind b) -t1 t3))))))).(sn3_sing c (THead (Bind b) t1 t2) (\lambda (t3: T).(\lambda -(H5: (((eq T (THead (Bind b) t1 t2) t3) \to (\forall (P: Prop).P)))).(\lambda -(H6: (pr3 c (THead (Bind b) t1 t2) t3)).(let H_x \def (bind_dec_not b Abst) -in (let H7 \def H_x in (or_ind (eq B b Abst) (not (eq B b Abst)) (sn3 c t3) -(\lambda (H8: (eq B b Abst)).(let H9 \def (eq_ind B b (\lambda (b0: B).(pr3 c -(THead (Bind b0) t1 t2) t3)) H6 Abst H8) in (let H10 \def (eq_ind B b -(\lambda (b0: B).((eq T (THead (Bind b0) t1 t2) t3) \to (\forall (P: -Prop).P))) H5 Abst H8) in (let H11 \def (eq_ind B b (\lambda (b0: B).(\forall -(t4: T).((((eq T t2 t4) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind -b0) t1) t2 t4) \to (sn3 c (THead (Bind b0) t1 t4)))))) H4 Abst H8) in (let -H12 \def (eq_ind B b (\lambda (b0: B).(\forall (t4: T).((((eq T t2 t4) \to -(\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b0) t1) t2 t4) \to (sn3 -(CHead c (Bind b0) t1) t4))))) H3 Abst H8) in (let H13 \def (eq_ind B b -(\lambda (b0: B).(\forall (t4: T).((((eq T t1 t4) \to (\forall (P: Prop).P))) -\to ((pr3 c t1 t4) \to (\forall (t0: T).((sn3 (CHead c (Bind b0) t4) t0) \to -(sn3 c (THead (Bind b0) t4 t0)))))))) H1 Abst H8) in (let H14 \def -(pr3_gen_abst c t1 t2 t3 H9) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: -T).(eq T t3 (THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b0: B).(\forall -(u0: T).(pr3 (CHead c (Bind b0) u0) t2 t4))))) (sn3 c t3) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H15: (eq T t3 (THead (Bind Abst) x0 -x1))).(\lambda (H16: (pr3 c t1 x0)).(\lambda (H17: ((\forall (b0: B).(\forall -(u0: T).(pr3 (CHead c (Bind b0) u0) t2 x1))))).(let H18 \def (eq_ind T t3 -(\lambda (t0: T).((eq T (THead (Bind Abst) t1 t2) t0) \to (\forall (P: -Prop).P))) H10 (THead (Bind Abst) x0 x1) H15) in (eq_ind_r T (THead (Bind -Abst) x0 x1) (\lambda (t0: T).(sn3 c t0)) (let H_x0 \def (term_dec t1 x0) in -(let H19 \def H_x0 in (or_ind (eq T t1 x0) ((eq T t1 x0) \to (\forall (P: -Prop).P)) (sn3 c (THead (Bind Abst) x0 x1)) (\lambda (H20: (eq T t1 x0)).(let -H21 \def (eq_ind_r T x0 (\lambda (t0: T).((eq T (THead (Bind Abst) t1 t2) -(THead (Bind Abst) t0 x1)) \to (\forall (P: Prop).P))) H18 t1 H20) in (let -H22 \def (eq_ind_r T x0 (\lambda (t0: T).(pr3 c t1 t0)) H16 t1 H20) in -(eq_ind T t1 (\lambda (t0: T).(sn3 c (THead (Bind Abst) t0 x1))) (let H_x1 -\def (term_dec t2 x1) in (let H23 \def H_x1 in (or_ind (eq T t2 x1) ((eq T t2 -x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Bind Abst) t1 x1)) (\lambda -(H24: (eq T t2 x1)).(let H25 \def (eq_ind_r T x1 (\lambda (t0: T).((eq T -(THead (Bind Abst) t1 t2) (THead (Bind Abst) t1 t0)) \to (\forall (P: -Prop).P))) H21 t2 H24) in (let H26 \def (eq_ind_r T x1 (\lambda (t0: -T).(\forall (b0: B).(\forall (u0: T).(pr3 (CHead c (Bind b0) u0) t2 t0)))) -H17 t2 H24) in (eq_ind T t2 (\lambda (t0: T).(sn3 c (THead (Bind Abst) t1 -t0))) (H25 (refl_equal T (THead (Bind Abst) t1 t2)) (sn3 c (THead (Bind Abst) -t1 t2))) x1 H24)))) (\lambda (H24: (((eq T t2 x1) \to (\forall (P: -Prop).P)))).(H11 x1 H24 (H17 Abst t1))) H23))) x0 H20)))) (\lambda (H20: -(((eq T t1 x0) \to (\forall (P: Prop).P)))).(let H_x1 \def (term_dec t2 x1) -in (let H21 \def H_x1 in (or_ind (eq T t2 x1) ((eq T t2 x1) \to (\forall (P: -Prop).P)) (sn3 c (THead (Bind Abst) x0 x1)) (\lambda (H22: (eq T t2 x1)).(let -H23 \def (eq_ind_r T x1 (\lambda (t0: T).(\forall (b0: B).(\forall (u0: -T).(pr3 (CHead c (Bind b0) u0) t2 t0)))) H17 t2 H22) in (eq_ind T t2 (\lambda -(t0: T).(sn3 c (THead (Bind Abst) x0 t0))) (H13 x0 H20 H16 t2 (sn3_cpr3_trans -c t1 x0 H16 (Bind Abst) t2 (sn3_sing (CHead c (Bind Abst) t1) t2 H12))) x1 -H22))) (\lambda (H22: (((eq T t2 x1) \to (\forall (P: Prop).P)))).(H13 x0 H20 -H16 x1 (sn3_cpr3_trans c t1 x0 H16 (Bind Abst) x1 (H12 x1 H22 (H17 Abst -t1))))) H21)))) H19))) t3 H15))))))) H14)))))))) (\lambda (H8: (not (eq B b -Abst))).(let H_x0 \def (pr3_gen_bind b H8 c t1 t2 t3 H6) in (let H9 \def H_x0 -in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind -b) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_: -T).(\lambda (t4: T).(pr3 (CHead c (Bind b) t1) t2 t4)))) (pr3 (CHead c (Bind -b) t1) t2 (lift (S O) O t3)) (sn3 c t3) (\lambda (H10: (ex3_2 T T (\lambda -(u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr3 -(CHead c (Bind b) t1) t2 t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda -(t4: T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr3 (CHead c (Bind b) -t1) t2 t4))) (sn3 c t3) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq -T t3 (THead (Bind b) x0 x1))).(\lambda (H12: (pr3 c t1 x0)).(\lambda (H13: -(pr3 (CHead c (Bind b) t1) t2 x1)).(let H14 \def (eq_ind T t3 (\lambda (t0: -T).((eq T (THead (Bind b) t1 t2) t0) \to (\forall (P: Prop).P))) H5 (THead -(Bind b) x0 x1) H11) in (eq_ind_r T (THead (Bind b) x0 x1) (\lambda (t0: -T).(sn3 c t0)) (let H_x1 \def (term_dec t1 x0) in (let H15 \def H_x1 in -(or_ind (eq T t1 x0) ((eq T t1 x0) \to (\forall (P: Prop).P)) (sn3 c (THead -(Bind b) x0 x1)) (\lambda (H16: (eq T t1 x0)).(let H17 \def (eq_ind_r T x0 -(\lambda (t0: T).((eq T (THead (Bind b) t1 t2) (THead (Bind b) t0 x1)) \to -(\forall (P: Prop).P))) H14 t1 H16) in (let H18 \def (eq_ind_r T x0 (\lambda -(t0: T).(pr3 c t1 t0)) H12 t1 H16) in (eq_ind T t1 (\lambda (t0: T).(sn3 c -(THead (Bind b) t0 x1))) (let H_x2 \def (term_dec t2 x1) in (let H19 \def -H_x2 in (or_ind (eq T t2 x1) ((eq T t2 x1) \to (\forall (P: Prop).P)) (sn3 c -(THead (Bind b) t1 x1)) (\lambda (H20: (eq T t2 x1)).(let H21 \def (eq_ind_r -T x1 (\lambda (t0: T).((eq T (THead (Bind b) t1 t2) (THead (Bind b) t1 t0)) -\to (\forall (P: Prop).P))) H17 t2 H20) in (let H22 \def (eq_ind_r T x1 -(\lambda (t0: T).(pr3 (CHead c (Bind b) t1) t2 t0)) H13 t2 H20) in (eq_ind T -t2 (\lambda (t0: T).(sn3 c (THead (Bind b) t1 t0))) (H21 (refl_equal T (THead -(Bind b) t1 t2)) (sn3 c (THead (Bind b) t1 t2))) x1 H20)))) (\lambda (H20: -(((eq T t2 x1) \to (\forall (P: Prop).P)))).(H4 x1 H20 H13)) H19))) x0 -H16)))) (\lambda (H16: (((eq T t1 x0) \to (\forall (P: Prop).P)))).(let H_x2 -\def (term_dec t2 x1) in (let H17 \def H_x2 in (or_ind (eq T t2 x1) ((eq T t2 -x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Bind b) x0 x1)) (\lambda (H18: -(eq T t2 x1)).(let H19 \def (eq_ind_r T x1 (\lambda (t0: T).(pr3 (CHead c -(Bind b) t1) t2 t0)) H13 t2 H18) in (eq_ind T t2 (\lambda (t0: T).(sn3 c -(THead (Bind b) x0 t0))) (H1 x0 H16 H12 t2 (sn3_cpr3_trans c t1 x0 H12 (Bind -b) t2 (sn3_sing (CHead c (Bind b) t1) t2 H3))) x1 H18))) (\lambda (H18: (((eq -T t2 x1) \to (\forall (P: Prop).P)))).(H1 x0 H16 H12 x1 (sn3_cpr3_trans c t1 -x0 H12 (Bind b) x1 (H3 x1 H18 H13)))) H17)))) H15))) t3 H11))))))) H10)) -(\lambda (H10: (pr3 (CHead c (Bind b) t1) t2 (lift (S O) O -t3))).(sn3_gen_lift (CHead c (Bind b) t1) t3 (S O) O (sn3_pr3_trans (CHead c -(Bind b) t1) t2 (sn3_sing (CHead c (Bind b) t1) t2 H3) (lift (S O) O t3) H10) -c (drop_drop (Bind b) O c c (drop_refl c) t1))) H9)))) H7)))))))))) t -H2)))))) u H)))). -(* COMMENTS -Initial nodes: 2401 -END *) - -theorem sn3_beta: - \forall (c: C).(\forall (v: T).(\forall (t: T).((sn3 c (THead (Bind Abbr) v -t)) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) v (THead -(Bind Abst) w t)))))))) -\def - \lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (H: (sn3 c (THead -(Bind Abbr) v t))).(insert_eq T (THead (Bind Abbr) v t) (\lambda (t0: T).(sn3 -c t0)) (\lambda (_: T).(\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat -Appl) v (THead (Bind Abst) w t)))))) (\lambda (y: T).(\lambda (H0: (sn3 c -y)).(unintro T t (\lambda (t0: T).((eq T y (THead (Bind Abbr) v t0)) \to -(\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) v (THead (Bind Abst) -w t0))))))) (unintro T v (\lambda (t0: T).(\forall (x: T).((eq T y (THead -(Bind Abbr) t0 x)) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat -Appl) t0 (THead (Bind Abst) w x)))))))) (sn3_ind c (\lambda (t0: T).(\forall -(x: T).(\forall (x0: T).((eq T t0 (THead (Bind Abbr) x x0)) \to (\forall (w: -T).((sn3 c w) \to (sn3 c (THead (Flat Appl) x (THead (Bind Abst) w -x0))))))))) (\lambda (t1: T).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) -\to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda -(H2: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 -c t1 t2) \to (\forall (x: T).(\forall (x0: T).((eq T t2 (THead (Bind Abbr) x -x0)) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) x (THead -(Bind Abst) w x0))))))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: -(eq T t1 (THead (Bind Abbr) x x0))).(\lambda (w: T).(\lambda (H4: (sn3 c -w)).(let H5 \def (eq_ind T t1 (\lambda (t0: T).(\forall (t2: T).((((eq T t0 -t2) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (\forall (x1: -T).(\forall (x2: T).((eq T t2 (THead (Bind Abbr) x1 x2)) \to (\forall (w0: -T).((sn3 c w0) \to (sn3 c (THead (Flat Appl) x1 (THead (Bind Abst) w0 -x2)))))))))))) H2 (THead (Bind Abbr) x x0) H3) in (let H6 \def (eq_ind T t1 -(\lambda (t0: T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) -\to ((pr3 c t0 t2) \to (sn3 c t2))))) H1 (THead (Bind Abbr) x x0) H3) in -(sn3_ind c (\lambda (t0: T).(sn3 c (THead (Flat Appl) x (THead (Bind Abst) t0 -x0)))) (\lambda (t2: T).(\lambda (H7: ((\forall (t3: T).((((eq T t2 t3) \to -(\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to (sn3 c t3)))))).(\lambda (H8: -((\forall (t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t2 -t3) \to (sn3 c (THead (Flat Appl) x (THead (Bind Abst) t3 -x0)))))))).(sn3_pr2_intro c (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) -(\lambda (t3: T).(\lambda (H9: (((eq T (THead (Flat Appl) x (THead (Bind -Abst) t2 x0)) t3) \to (\forall (P: Prop).P)))).(\lambda (H10: (pr2 c (THead -(Flat Appl) x (THead (Bind Abst) t2 x0)) t3)).(let H11 \def (pr2_gen_appl c x -(THead (Bind Abst) t2 x0) t3 H10) in (or3_ind (ex3_2 T T (\lambda (u2: -T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c -(THead (Bind Abst) t2 x0) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) t2 x0) -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda -(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: -B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4)))))))) (ex6_6 B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind Abst) t2 x0) (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) -z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) -y2) z1 z2)))))))) (sn3 c t3) (\lambda (H12: (ex3_2 T T (\lambda (u2: -T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c -(THead (Bind Abst) t2 x0) t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda -(t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind Abst) -t2 x0) t4))) (sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (H13: (eq -T t3 (THead (Flat Appl) x1 x2))).(\lambda (H14: (pr2 c x x1)).(\lambda (H15: -(pr2 c (THead (Bind Abst) t2 x0) x2)).(let H16 \def (eq_ind T t3 (\lambda -(t0: T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) t0) \to -(\forall (P: Prop).P))) H9 (THead (Flat Appl) x1 x2) H13) in (eq_ind_r T -(THead (Flat Appl) x1 x2) (\lambda (t0: T).(sn3 c t0)) (let H17 \def -(pr2_gen_abst c t2 x0 x2 H15) in (ex3_2_ind T T (\lambda (u2: T).(\lambda -(t4: T).(eq T x2 (THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c t2 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall -(u: T).(pr2 (CHead c (Bind b) u) x0 t4))))) (sn3 c (THead (Flat Appl) x1 x2)) -(\lambda (x3: T).(\lambda (x4: T).(\lambda (H18: (eq T x2 (THead (Bind Abst) -x3 x4))).(\lambda (H19: (pr2 c t2 x3)).(\lambda (H20: ((\forall (b: -B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x0 x4))))).(let H21 \def (eq_ind -T x2 (\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) -(THead (Flat Appl) x1 t0)) \to (\forall (P: Prop).P))) H16 (THead (Bind Abst) -x3 x4) H18) in (eq_ind_r T (THead (Bind Abst) x3 x4) (\lambda (t0: T).(sn3 c -(THead (Flat Appl) x1 t0))) (let H_x \def (term_dec t2 x3) in (let H22 \def -H_x in (or_ind (eq T t2 x3) ((eq T t2 x3) \to (\forall (P: Prop).P)) (sn3 c -(THead (Flat Appl) x1 (THead (Bind Abst) x3 x4))) (\lambda (H23: (eq T t2 -x3)).(let H24 \def (eq_ind_r T x3 (\lambda (t0: T).((eq T (THead (Flat Appl) -x (THead (Bind Abst) t2 x0)) (THead (Flat Appl) x1 (THead (Bind Abst) t0 -x4))) \to (\forall (P: Prop).P))) H21 t2 H23) in (let H25 \def (eq_ind_r T x3 -(\lambda (t0: T).(pr2 c t2 t0)) H19 t2 H23) in (eq_ind T t2 (\lambda (t0: -T).(sn3 c (THead (Flat Appl) x1 (THead (Bind Abst) t0 x4)))) (let H_x0 \def -(term_dec x x1) in (let H26 \def H_x0 in (or_ind (eq T x x1) ((eq T x x1) \to -(\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Bind Abst) t2 -x4))) (\lambda (H27: (eq T x x1)).(let H28 \def (eq_ind_r T x1 (\lambda (t0: -T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) (THead (Flat Appl) -t0 (THead (Bind Abst) t2 x4))) \to (\forall (P: Prop).P))) H24 x H27) in (let -H29 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14 x H27) in (eq_ind -T x (\lambda (t0: T).(sn3 c (THead (Flat Appl) t0 (THead (Bind Abst) t2 -x4)))) (let H_x1 \def (term_dec x0 x4) in (let H30 \def H_x1 in (or_ind (eq T -x0 x4) ((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x -(THead (Bind Abst) t2 x4))) (\lambda (H31: (eq T x0 x4)).(let H32 \def -(eq_ind_r T x4 (\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind -Abst) t2 x0)) (THead (Flat Appl) x (THead (Bind Abst) t2 t0))) \to (\forall -(P: Prop).P))) H28 x0 H31) in (let H33 \def (eq_ind_r T x4 (\lambda (t0: -T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 -H31) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Flat Appl) x (THead -(Bind Abst) t2 t0)))) (H32 (refl_equal T (THead (Flat Appl) x (THead (Bind -Abst) t2 x0))) (sn3 c (THead (Flat Appl) x (THead (Bind Abst) t2 x0)))) x4 -H31)))) (\lambda (H31: (((eq T x0 x4) \to (\forall (P: Prop).P)))).(H5 (THead -(Bind Abbr) x x4) (\lambda (H32: (eq T (THead (Bind Abbr) x x0) (THead (Bind -Abbr) x x4))).(\lambda (P: Prop).(let H33 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | -(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind -Abbr) x x0) (THead (Bind Abbr) x x4) H32) in (let H34 \def (eq_ind_r T x4 -(\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H31 x0 H33) in -(let H35 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: -T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H33) in (H34 (refl_equal T x0) -P)))))) (pr3_pr2 c (THead (Bind Abbr) x x0) (THead (Bind Abbr) x x4) -(pr2_head_2 c x x0 x4 (Bind Abbr) (H20 Abbr x))) x x4 (refl_equal T (THead -(Bind Abbr) x x4)) t2 (sn3_sing c t2 H7))) H30))) x1 H27)))) (\lambda (H27: -(((eq T x x1) \to (\forall (P: Prop).P)))).(H5 (THead (Bind Abbr) x1 x4) -(\lambda (H28: (eq T (THead (Bind Abbr) x x0) (THead (Bind Abbr) x1 -x4))).(\lambda (P: Prop).(let H29 \def (f_equal T T (\lambda (e: T).(match e -in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) -\Rightarrow x | (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abbr) x x0) -(THead (Bind Abbr) x1 x4) H28) in ((let H30 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | -(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind -Abbr) x x0) (THead (Bind Abbr) x1 x4) H28) in (\lambda (H31: (eq T x -x1)).(let H32 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall -(u: T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H30) in (let H33 \def -(eq_ind_r T x1 (\lambda (t0: T).((eq T x t0) \to (\forall (P0: Prop).P0))) -H27 x H31) in (let H34 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14 -x H31) in (H33 (refl_equal T x) P)))))) H29)))) (pr3_head_12 c x x1 (pr3_pr2 -c x x1 H14) (Bind Abbr) x0 x4 (pr3_pr2 (CHead c (Bind Abbr) x1) x0 x4 (H20 -Abbr x1))) x1 x4 (refl_equal T (THead (Bind Abbr) x1 x4)) t2 (sn3_sing c t2 -H7))) H26))) x3 H23)))) (\lambda (H23: (((eq T t2 x3) \to (\forall (P: -Prop).P)))).(let H_x0 \def (term_dec x x1) in (let H24 \def H_x0 in (or_ind -(eq T x x1) ((eq T x x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) -x1 (THead (Bind Abst) x3 x4))) (\lambda (H25: (eq T x x1)).(let H26 \def -(eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14 x H25) in (eq_ind T x -(\lambda (t0: T).(sn3 c (THead (Flat Appl) t0 (THead (Bind Abst) x3 x4)))) -(let H_x1 \def (term_dec x0 x4) in (let H27 \def H_x1 in (or_ind (eq T x0 x4) -((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x (THead -(Bind Abst) x3 x4))) (\lambda (H28: (eq T x0 x4)).(let H29 \def (eq_ind_r T -x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) -x0 t0)))) H20 x0 H28) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Flat -Appl) x (THead (Bind Abst) x3 t0)))) (H8 x3 H23 (pr3_pr2 c t2 x3 H19)) x4 -H28))) (\lambda (H28: (((eq T x0 x4) \to (\forall (P: Prop).P)))).(H5 (THead -(Bind Abbr) x x4) (\lambda (H29: (eq T (THead (Bind Abbr) x x0) (THead (Bind -Abbr) x x4))).(\lambda (P: Prop).(let H30 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | -(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind -Abbr) x x0) (THead (Bind Abbr) x x4) H29) in (let H31 \def (eq_ind_r T x4 -(\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H28 x0 H30) in -(let H32 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: -T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H30) in (H31 (refl_equal T x0) -P)))))) (pr3_pr2 c (THead (Bind Abbr) x x0) (THead (Bind Abbr) x x4) -(pr2_head_2 c x x0 x4 (Bind Abbr) (H20 Abbr x))) x x4 (refl_equal T (THead -(Bind Abbr) x x4)) x3 (H7 x3 H23 (pr3_pr2 c t2 x3 H19)))) H27))) x1 H25))) -(\lambda (H25: (((eq T x x1) \to (\forall (P: Prop).P)))).(H5 (THead (Bind -Abbr) x1 x4) (\lambda (H26: (eq T (THead (Bind Abbr) x x0) (THead (Bind Abbr) -x1 x4))).(\lambda (P: Prop).(let H27 \def (f_equal T T (\lambda (e: T).(match -e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) -\Rightarrow x | (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abbr) x x0) -(THead (Bind Abbr) x1 x4) H26) in ((let H28 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | -(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind -Abbr) x x0) (THead (Bind Abbr) x1 x4) H26) in (\lambda (H29: (eq T x -x1)).(let H30 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall -(u: T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H28) in (let H31 \def -(eq_ind_r T x1 (\lambda (t0: T).((eq T x t0) \to (\forall (P0: Prop).P0))) -H25 x H29) in (let H32 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14 -x H29) in (H31 (refl_equal T x) P)))))) H27)))) (pr3_head_12 c x x1 (pr3_pr2 -c x x1 H14) (Bind Abbr) x0 x4 (pr3_pr2 (CHead c (Bind Abbr) x1) x0 x4 (H20 -Abbr x1))) x1 x4 (refl_equal T (THead (Bind Abbr) x1 x4)) x3 (H7 x3 H23 -(pr3_pr2 c t2 x3 H19)))) H24)))) H22))) x2 H18))))))) H17)) t3 H13))))))) -H12)) (\lambda (H12: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) t2 x0) (THead -(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda -(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: -B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4))))))))).(ex4_4_ind T T T -T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind Abst) t2 x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind -Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t4: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) -z1 t4))))))) (sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: -T).(\lambda (x4: T).(\lambda (H13: (eq T (THead (Bind Abst) t2 x0) (THead -(Bind Abst) x1 x2))).(\lambda (H14: (eq T t3 (THead (Bind Abbr) x3 -x4))).(\lambda (H15: (pr2 c x x3)).(\lambda (H16: ((\forall (b: B).(\forall -(u: T).(pr2 (CHead c (Bind b) u) x2 x4))))).(let H17 \def (eq_ind T t3 -(\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) t0) -\to (\forall (P: Prop).P))) H9 (THead (Bind Abbr) x3 x4) H14) in (eq_ind_r T -(THead (Bind Abbr) x3 x4) (\lambda (t0: T).(sn3 c t0)) (let H18 \def (f_equal -T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow t2 | (TLRef _) \Rightarrow t2 | (THead _ t0 _) \Rightarrow t0])) -(THead (Bind Abst) t2 x0) (THead (Bind Abst) x1 x2) H13) in ((let H19 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ t0) -\Rightarrow t0])) (THead (Bind Abst) t2 x0) (THead (Bind Abst) x1 x2) H13) in -(\lambda (_: (eq T t2 x1)).(let H21 \def (eq_ind_r T x2 (\lambda (t0: -T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t0 x4)))) H16 x0 -H19) in (let H_x \def (term_dec x x3) in (let H22 \def H_x in (or_ind (eq T x -x3) ((eq T x x3) \to (\forall (P: Prop).P)) (sn3 c (THead (Bind Abbr) x3 x4)) -(\lambda (H23: (eq T x x3)).(let H24 \def (eq_ind_r T x3 (\lambda (t0: -T).(pr2 c x t0)) H15 x H23) in (eq_ind T x (\lambda (t0: T).(sn3 c (THead -(Bind Abbr) t0 x4))) (let H_x0 \def (term_dec x0 x4) in (let H25 \def H_x0 in -(or_ind (eq T x0 x4) ((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead -(Bind Abbr) x x4)) (\lambda (H26: (eq T x0 x4)).(let H27 \def (eq_ind_r T x4 -(\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x0 -t0)))) H21 x0 H26) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Bind Abbr) -x t0))) (sn3_sing c (THead (Bind Abbr) x x0) H6) x4 H26))) (\lambda (H26: -(((eq T x0 x4) \to (\forall (P: Prop).P)))).(H6 (THead (Bind Abbr) x x4) -(\lambda (H27: (eq T (THead (Bind Abbr) x x0) (THead (Bind Abbr) x -x4))).(\lambda (P: Prop).(let H28 \def (f_equal T T (\lambda (e: T).(match e -in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) -\Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind Abbr) x x0) -(THead (Bind Abbr) x x4) H27) in (let H29 \def (eq_ind_r T x4 (\lambda (t0: -T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H26 x0 H28) in (let H30 \def -(eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c -(Bind b) u) x0 t0)))) H21 x0 H28) in (H29 (refl_equal T x0) P)))))) (pr3_pr2 -c (THead (Bind Abbr) x x0) (THead (Bind Abbr) x x4) (pr2_head_2 c x x0 x4 -(Bind Abbr) (H21 Abbr x))))) H25))) x3 H23))) (\lambda (H23: (((eq T x x3) -\to (\forall (P: Prop).P)))).(H6 (THead (Bind Abbr) x3 x4) (\lambda (H24: (eq -T (THead (Bind Abbr) x x0) (THead (Bind Abbr) x3 x4))).(\lambda (P: -Prop).(let H25 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | -(THead _ t0 _) \Rightarrow t0])) (THead (Bind Abbr) x x0) (THead (Bind Abbr) -x3 x4) H24) in ((let H26 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) -\Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind Abbr) x x0) -(THead (Bind Abbr) x3 x4) H24) in (\lambda (H27: (eq T x x3)).(let H28 \def -(eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c -(Bind b) u) x0 t0)))) H21 x0 H26) in (let H29 \def (eq_ind_r T x3 (\lambda -(t0: T).((eq T x t0) \to (\forall (P0: Prop).P0))) H23 x H27) in (let H30 -\def (eq_ind_r T x3 (\lambda (t0: T).(pr2 c x t0)) H15 x H27) in (H29 -(refl_equal T x) P)))))) H25)))) (pr3_head_12 c x x3 (pr3_pr2 c x x3 H15) -(Bind Abbr) x0 x4 (pr3_pr2 (CHead c (Bind Abbr) x3) x0 x4 (H21 Abbr x3))))) -H22)))))) H18)) t3 H14)))))))))) H12)) (\lambda (H12: (ex6_6 B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Bind Abst) t2 x0) (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) -z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) -y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B -b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) t2 x0) (THead -(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) (sn3 c t3) -(\lambda (x1: B).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda -(x5: T).(\lambda (x6: T).(\lambda (H13: (not (eq B x1 Abst))).(\lambda (H14: -(eq T (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3))).(\lambda (H15: (eq -T t3 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))).(\lambda -(_: (pr2 c x x5)).(\lambda (H17: (pr2 c x2 x6)).(\lambda (H18: (pr2 (CHead c -(Bind x1) x6) x3 x4)).(let H19 \def (eq_ind T t3 (\lambda (t0: T).((eq T -(THead (Flat Appl) x (THead (Bind Abst) t2 x0)) t0) \to (\forall (P: -Prop).P))) H9 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)) -H15) in (eq_ind_r T (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) -x4)) (\lambda (t0: T).(sn3 c t0)) (let H20 \def (f_equal T B (\lambda (e: -T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow Abst | -(TLRef _) \Rightarrow Abst | (THead k _ _) \Rightarrow (match k in K return -(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow -Abst])])) (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3) H14) in ((let H21 -\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow t2 | (TLRef _) \Rightarrow t2 | (THead _ t0 _) -\Rightarrow t0])) (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3) H14) in -((let H22 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: -T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ -t0) \Rightarrow t0])) (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3) H14) -in (\lambda (H23: (eq T t2 x2)).(\lambda (H24: (eq B Abst x1)).(let H25 \def -(eq_ind_r T x3 (\lambda (t0: T).(pr2 (CHead c (Bind x1) x6) t0 x4)) H18 x0 -H22) in (let H26 \def (eq_ind_r T x2 (\lambda (t0: T).(pr2 c t0 x6)) H17 t2 -H23) in (let H27 \def (eq_ind_r B x1 (\lambda (b: B).(pr2 (CHead c (Bind b) -x6) x0 x4)) H25 Abst H24) in (let H28 \def (eq_ind_r B x1 (\lambda (b: -B).(not (eq B b Abst))) H13 Abst H24) in (eq_ind B Abst (\lambda (b: B).(sn3 -c (THead (Bind b) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))) (let H29 -\def (match (H28 (refl_equal B Abst)) in False return (\lambda (_: -False).(sn3 c (THead (Bind Abst) x6 (THead (Flat Appl) (lift (S O) O x5) -x4)))) with []) in H29) x1 H24)))))))) H21)) H20)) t3 H15)))))))))))))) H12)) -H11))))))))) w H4))))))))))) y H0))))) H)))). -(* COMMENTS -Initial nodes: 5699 -END *) - -theorem sn3_appl_lref: - \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (v: -T).((sn3 c v) \to (sn3 c (THead (Flat Appl) v (TLRef i))))))) -\def - \lambda (c: C).(\lambda (i: nat).(\lambda (H: (nf2 c (TLRef i))).(\lambda -(v: T).(\lambda (H0: (sn3 c v)).(sn3_ind c (\lambda (t: T).(sn3 c (THead -(Flat Appl) t (TLRef i)))) (\lambda (t1: T).(\lambda (_: ((\forall (t2: -T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c -t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: -Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c (THead (Flat Appl) t2 (TLRef -i)))))))).(sn3_pr2_intro c (THead (Flat Appl) t1 (TLRef i)) (\lambda (t2: -T).(\lambda (H3: (((eq T (THead (Flat Appl) t1 (TLRef i)) t2) \to (\forall -(P: Prop).P)))).(\lambda (H4: (pr2 c (THead (Flat Appl) t1 (TLRef i)) -t2)).(let H5 \def (pr2_gen_appl c t1 (TLRef i) t2 H4) in (or3_ind (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2))) (\lambda (_: T).(\lambda -(t3: T).(pr2 c (TLRef i) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i) (THead (Bind Abst) y1 -z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: -T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2))))) (\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall -(u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T -(TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T -t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c t1 u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 -y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) -(sn3 c t2) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T -t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c t1 -u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (TLRef i) t3))))).(ex3_2_ind T -T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2))) (\lambda (_: T).(\lambda -(t3: T).(pr2 c (TLRef i) t3))) (sn3 c t2) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H7: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H8: (pr2 c t1 -x0)).(\lambda (H9: (pr2 c (TLRef i) x1)).(let H10 \def (eq_ind T t2 (\lambda -(t: T).((eq T (THead (Flat Appl) t1 (TLRef i)) t) \to (\forall (P: Prop).P))) -H3 (THead (Flat Appl) x0 x1) H7) in (eq_ind_r T (THead (Flat Appl) x0 x1) -(\lambda (t: T).(sn3 c t)) (let H11 \def (eq_ind_r T x1 (\lambda (t: T).((eq -T (THead (Flat Appl) t1 (TLRef i)) (THead (Flat Appl) x0 t)) \to (\forall (P: -Prop).P))) H10 (TLRef i) (H x1 H9)) in (let H12 \def (eq_ind_r T x1 (\lambda -(t: T).(pr2 c (TLRef i) t)) H9 (TLRef i) (H x1 H9)) in (eq_ind T (TLRef i) -(\lambda (t: T).(sn3 c (THead (Flat Appl) x0 t))) (let H_x \def (term_dec t1 -x0) in (let H13 \def H_x in (or_ind (eq T t1 x0) ((eq T t1 x0) \to (\forall -(P: Prop).P)) (sn3 c (THead (Flat Appl) x0 (TLRef i))) (\lambda (H14: (eq T -t1 x0)).(let H15 \def (eq_ind_r T x0 (\lambda (t: T).((eq T (THead (Flat -Appl) t1 (TLRef i)) (THead (Flat Appl) t (TLRef i))) \to (\forall (P: -Prop).P))) H11 t1 H14) in (let H16 \def (eq_ind_r T x0 (\lambda (t: T).(pr2 c -t1 t)) H8 t1 H14) in (eq_ind T t1 (\lambda (t: T).(sn3 c (THead (Flat Appl) t -(TLRef i)))) (H15 (refl_equal T (THead (Flat Appl) t1 (TLRef i))) (sn3 c -(THead (Flat Appl) t1 (TLRef i)))) x0 H14)))) (\lambda (H14: (((eq T t1 x0) -\to (\forall (P: Prop).P)))).(H2 x0 H14 (pr3_pr2 c t1 x0 H8))) H13))) x1 (H -x1 H9)))) t2 H7))))))) H6)) (\lambda (H6: (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i) (THead -(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall -(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3))))))))).(ex4_4_ind T -T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T -(TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 -u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: -T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3))))))) -(sn3 c t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: -T).(\lambda (H7: (eq T (TLRef i) (THead (Bind Abst) x0 x1))).(\lambda (H8: -(eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (_: (pr2 c t1 x2)).(\lambda (_: -((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 x3))))).(let -H11 \def (eq_ind T t2 (\lambda (t: T).((eq T (THead (Flat Appl) t1 (TLRef i)) -t) \to (\forall (P: Prop).P))) H3 (THead (Bind Abbr) x2 x3) H8) in (eq_ind_r -T (THead (Bind Abbr) x2 x3) (\lambda (t: T).(sn3 c t)) (let H12 \def (eq_ind -T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) -\Rightarrow False])) I (THead (Bind Abst) x0 x1) H7) in (False_ind (sn3 c -(THead (Bind Abbr) x2 x3)) H12)) t2 H8)))))))))) H6)) (\lambda (H6: (ex6_6 B -T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T (TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda -(_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq -T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c t1 u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 -y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 -z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i) (THead (Bind b) y1 -z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: -T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat -Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 -(CHead c (Bind b) y2) z1 z2))))))) (sn3 c t2) (\lambda (x0: B).(\lambda (x1: -T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: -T).(\lambda (_: (not (eq B x0 Abst))).(\lambda (H8: (eq T (TLRef i) (THead -(Bind x0) x1 x2))).(\lambda (H9: (eq T t2 (THead (Bind x0) x5 (THead (Flat -Appl) (lift (S O) O x4) x3)))).(\lambda (_: (pr2 c t1 x4)).(\lambda (_: (pr2 -c x1 x5)).(\lambda (_: (pr2 (CHead c (Bind x0) x5) x2 x3)).(let H13 \def -(eq_ind T t2 (\lambda (t: T).((eq T (THead (Flat Appl) t1 (TLRef i)) t) \to -(\forall (P: Prop).P))) H3 (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) -O x4) x3)) H9) in (eq_ind_r T (THead (Bind x0) x5 (THead (Flat Appl) (lift (S -O) O x4) x3)) (\lambda (t: T).(sn3 c t)) (let H14 \def (eq_ind T (TLRef i) -(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow -False])) I (THead (Bind x0) x1 x2) H8) in (False_ind (sn3 c (THead (Bind x0) -x5 (THead (Flat Appl) (lift (S O) O x4) x3))) H14)) t2 H9)))))))))))))) H6)) -H5))))))))) v H0))))). -(* COMMENTS -Initial nodes: 2125 -END *) - -theorem sn3_appl_abbr: - \forall (c: C).(\forall (d: C).(\forall (w: T).(\forall (i: nat).((getl i c -(CHead d (Bind Abbr) w)) \to (\forall (v: T).((sn3 c (THead (Flat Appl) v -(lift (S i) O w))) \to (sn3 c (THead (Flat Appl) v (TLRef i))))))))) -\def - \lambda (c: C).(\lambda (d: C).(\lambda (w: T).(\lambda (i: nat).(\lambda -(H: (getl i c (CHead d (Bind Abbr) w))).(\lambda (v: T).(\lambda (H0: (sn3 c -(THead (Flat Appl) v (lift (S i) O w)))).(insert_eq T (THead (Flat Appl) v -(lift (S i) O w)) (\lambda (t: T).(sn3 c t)) (\lambda (_: T).(sn3 c (THead -(Flat Appl) v (TLRef i)))) (\lambda (y: T).(\lambda (H1: (sn3 c y)).(unintro -T v (\lambda (t: T).((eq T y (THead (Flat Appl) t (lift (S i) O w))) \to (sn3 -c (THead (Flat Appl) t (TLRef i))))) (sn3_ind c (\lambda (t: T).(\forall (x: -T).((eq T t (THead (Flat Appl) x (lift (S i) O w))) \to (sn3 c (THead (Flat -Appl) x (TLRef i)))))) (\lambda (t1: T).(\lambda (H2: ((\forall (t2: -T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c -t2)))))).(\lambda (H3: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: -Prop).P))) \to ((pr3 c t1 t2) \to (\forall (x: T).((eq T t2 (THead (Flat -Appl) x (lift (S i) O w))) \to (sn3 c (THead (Flat Appl) x (TLRef -i)))))))))).(\lambda (x: T).(\lambda (H4: (eq T t1 (THead (Flat Appl) x (lift -(S i) O w)))).(let H5 \def (eq_ind T t1 (\lambda (t: T).(\forall (t2: -T).((((eq T t t2) \to (\forall (P: Prop).P))) \to ((pr3 c t t2) \to (\forall -(x0: T).((eq T t2 (THead (Flat Appl) x0 (lift (S i) O w))) \to (sn3 c (THead -(Flat Appl) x0 (TLRef i))))))))) H3 (THead (Flat Appl) x (lift (S i) O w)) -H4) in (let H6 \def (eq_ind T t1 (\lambda (t: T).(\forall (t2: T).((((eq T t -t2) \to (\forall (P: Prop).P))) \to ((pr3 c t t2) \to (sn3 c t2))))) H2 -(THead (Flat Appl) x (lift (S i) O w)) H4) in (sn3_pr2_intro c (THead (Flat -Appl) x (TLRef i)) (\lambda (t2: T).(\lambda (H7: (((eq T (THead (Flat Appl) -x (TLRef i)) t2) \to (\forall (P: Prop).P)))).(\lambda (H8: (pr2 c (THead -(Flat Appl) x (TLRef i)) t2)).(let H9 \def (pr2_gen_appl c x (TLRef i) t2 H8) -in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead -(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) -(\lambda (_: T).(\lambda (t3: T).(pr2 c (TLRef i) t3)))) (ex4_4 T T T T -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T -(TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x -u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: -T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) -(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T (TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda -(_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq -T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 -y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) -(sn3 c t2) (\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T -t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x -u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (TLRef i) t3))))).(ex3_2_ind T -T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t3: -T).(pr2 c (TLRef i) t3))) (sn3 c t2) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H11: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H12: (pr2 c -x x0)).(\lambda (H13: (pr2 c (TLRef i) x1)).(let H14 \def (eq_ind T t2 -(\lambda (t: T).((eq T (THead (Flat Appl) x (TLRef i)) t) \to (\forall (P: -Prop).P))) H7 (THead (Flat Appl) x0 x1) H11) in (eq_ind_r T (THead (Flat -Appl) x0 x1) (\lambda (t: T).(sn3 c t)) (let H15 \def (pr2_gen_lref c x1 i -H13) in (or_ind (eq T x1 (TLRef i)) (ex2_2 C T (\lambda (d0: C).(\lambda (u: -T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq -T x1 (lift (S i) O u))))) (sn3 c (THead (Flat Appl) x0 x1)) (\lambda (H16: -(eq T x1 (TLRef i))).(let H17 \def (eq_ind T x1 (\lambda (t: T).((eq T (THead -(Flat Appl) x (TLRef i)) (THead (Flat Appl) x0 t)) \to (\forall (P: -Prop).P))) H14 (TLRef i) H16) in (eq_ind_r T (TLRef i) (\lambda (t: T).(sn3 c -(THead (Flat Appl) x0 t))) (let H_x \def (term_dec x x0) in (let H18 \def H_x -in (or_ind (eq T x x0) ((eq T x x0) \to (\forall (P: Prop).P)) (sn3 c (THead -(Flat Appl) x0 (TLRef i))) (\lambda (H19: (eq T x x0)).(let H20 \def -(eq_ind_r T x0 (\lambda (t: T).((eq T (THead (Flat Appl) x (TLRef i)) (THead -(Flat Appl) t (TLRef i))) \to (\forall (P: Prop).P))) H17 x H19) in (let H21 -\def (eq_ind_r T x0 (\lambda (t: T).(pr2 c x t)) H12 x H19) in (eq_ind T x -(\lambda (t: T).(sn3 c (THead (Flat Appl) t (TLRef i)))) (H20 (refl_equal T -(THead (Flat Appl) x (TLRef i))) (sn3 c (THead (Flat Appl) x (TLRef i)))) x0 -H19)))) (\lambda (H19: (((eq T x x0) \to (\forall (P: Prop).P)))).(H5 (THead -(Flat Appl) x0 (lift (S i) O w)) (\lambda (H20: (eq T (THead (Flat Appl) x -(lift (S i) O w)) (THead (Flat Appl) x0 (lift (S i) O w)))).(\lambda (P: -Prop).(let H21 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | -(THead _ t _) \Rightarrow t])) (THead (Flat Appl) x (lift (S i) O w)) (THead -(Flat Appl) x0 (lift (S i) O w)) H20) in (let H22 \def (eq_ind_r T x0 -(\lambda (t: T).((eq T x t) \to (\forall (P0: Prop).P0))) H19 x H21) in (let -H23 \def (eq_ind_r T x0 (\lambda (t: T).(pr2 c x t)) H12 x H21) in (H22 -(refl_equal T x) P)))))) (pr3_pr2 c (THead (Flat Appl) x (lift (S i) O w)) -(THead (Flat Appl) x0 (lift (S i) O w)) (pr2_head_1 c x x0 H12 (Flat Appl) -(lift (S i) O w))) x0 (refl_equal T (THead (Flat Appl) x0 (lift (S i) O -w))))) H18))) x1 H16))) (\lambda (H16: (ex2_2 C T (\lambda (d0: C).(\lambda -(u: T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: -T).(eq T x1 (lift (S i) O u)))))).(ex2_2_ind C T (\lambda (d0: C).(\lambda -(u: T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: -T).(eq T x1 (lift (S i) O u)))) (sn3 c (THead (Flat Appl) x0 x1)) (\lambda -(x2: C).(\lambda (x3: T).(\lambda (H17: (getl i c (CHead x2 (Bind Abbr) -x3))).(\lambda (H18: (eq T x1 (lift (S i) O x3))).(let H19 \def (eq_ind T x1 -(\lambda (t: T).((eq T (THead (Flat Appl) x (TLRef i)) (THead (Flat Appl) x0 -t)) \to (\forall (P: Prop).P))) H14 (lift (S i) O x3) H18) in (eq_ind_r T -(lift (S i) O x3) (\lambda (t: T).(sn3 c (THead (Flat Appl) x0 t))) (let H20 -\def (eq_ind C (CHead d (Bind Abbr) w) (\lambda (c0: C).(getl i c c0)) H -(CHead x2 (Bind Abbr) x3) (getl_mono c (CHead d (Bind Abbr) w) i H (CHead x2 -(Bind Abbr) x3) H17)) in (let H21 \def (f_equal C C (\lambda (e: C).(match e -in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) -\Rightarrow c0])) (CHead d (Bind Abbr) w) (CHead x2 (Bind Abbr) x3) -(getl_mono c (CHead d (Bind Abbr) w) i H (CHead x2 (Bind Abbr) x3) H17)) in -((let H22 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: -C).T) with [(CSort _) \Rightarrow w | (CHead _ _ t) \Rightarrow t])) (CHead d -(Bind Abbr) w) (CHead x2 (Bind Abbr) x3) (getl_mono c (CHead d (Bind Abbr) w) -i H (CHead x2 (Bind Abbr) x3) H17)) in (\lambda (H23: (eq C d x2)).(let H24 -\def (eq_ind_r T x3 (\lambda (t: T).(getl i c (CHead x2 (Bind Abbr) t))) H20 -w H22) in (eq_ind T w (\lambda (t: T).(sn3 c (THead (Flat Appl) x0 (lift (S -i) O t)))) (let H25 \def (eq_ind_r C x2 (\lambda (c0: C).(getl i c (CHead c0 -(Bind Abbr) w))) H24 d H23) in (let H_x \def (term_dec x x0) in (let H26 \def -H_x in (or_ind (eq T x x0) ((eq T x x0) \to (\forall (P: Prop).P)) (sn3 c -(THead (Flat Appl) x0 (lift (S i) O w))) (\lambda (H27: (eq T x x0)).(let H28 -\def (eq_ind_r T x0 (\lambda (t: T).(pr2 c x t)) H12 x H27) in (eq_ind T x -(\lambda (t: T).(sn3 c (THead (Flat Appl) t (lift (S i) O w)))) (sn3_sing c -(THead (Flat Appl) x (lift (S i) O w)) H6) x0 H27))) (\lambda (H27: (((eq T x -x0) \to (\forall (P: Prop).P)))).(H6 (THead (Flat Appl) x0 (lift (S i) O w)) -(\lambda (H28: (eq T (THead (Flat Appl) x (lift (S i) O w)) (THead (Flat -Appl) x0 (lift (S i) O w)))).(\lambda (P: Prop).(let H29 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t _) \Rightarrow t])) -(THead (Flat Appl) x (lift (S i) O w)) (THead (Flat Appl) x0 (lift (S i) O -w)) H28) in (let H30 \def (eq_ind_r T x0 (\lambda (t: T).((eq T x t) \to -(\forall (P0: Prop).P0))) H27 x H29) in (let H31 \def (eq_ind_r T x0 (\lambda -(t: T).(pr2 c x t)) H12 x H29) in (H30 (refl_equal T x) P)))))) (pr3_pr2 c -(THead (Flat Appl) x (lift (S i) O w)) (THead (Flat Appl) x0 (lift (S i) O -w)) (pr2_head_1 c x x0 H12 (Flat Appl) (lift (S i) O w))))) H26)))) x3 -H22)))) H21))) x1 H18)))))) H16)) H15)) t2 H11))))))) H10)) (\lambda (H10: -(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(eq T (TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda -(_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 -t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: -T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 -t3))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T (TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda -(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead -(Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind -b) u) z1 t3))))))) (sn3 c t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: -T).(\lambda (x3: T).(\lambda (H11: (eq T (TLRef i) (THead (Bind Abst) x0 -x1))).(\lambda (H12: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (_: (pr2 c -x x2)).(\lambda (_: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) -u) x1 x3))))).(let H15 \def (eq_ind T t2 (\lambda (t: T).((eq T (THead (Flat -Appl) x (TLRef i)) t) \to (\forall (P: Prop).P))) H7 (THead (Bind Abbr) x2 -x3) H12) in (eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda (t: T).(sn3 c t)) -(let H16 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) x0 -x1) H11) in (False_ind (sn3 c (THead (Bind Abbr) x2 x3)) H16)) t2 -H12)))))))))) H10)) (\lambda (H10: (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i) -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind -B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T (TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda -(_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq -T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 -y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) -(sn3 c t2) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: -T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (not (eq B x0 -Abst))).(\lambda (H12: (eq T (TLRef i) (THead (Bind x0) x1 x2))).(\lambda -(H13: (eq T t2 (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) -x3)))).(\lambda (_: (pr2 c x x4)).(\lambda (_: (pr2 c x1 x5)).(\lambda (_: -(pr2 (CHead c (Bind x0) x5) x2 x3)).(let H17 \def (eq_ind T t2 (\lambda (t: -T).((eq T (THead (Flat Appl) x (TLRef i)) t) \to (\forall (P: Prop).P))) H7 -(THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) H13) in -(eq_ind_r T (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) -(\lambda (t: T).(sn3 c t)) (let H18 \def (eq_ind T (TLRef i) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I -(THead (Bind x0) x1 x2) H12) in (False_ind (sn3 c (THead (Bind x0) x5 (THead -(Flat Appl) (lift (S O) O x4) x3))) H18)) t2 H13)))))))))))))) H10)) -H9))))))))))))) y H1)))) H0))))))). -(* COMMENTS -Initial nodes: 3727 -END *) - -theorem sn3_appl_cast: - \forall (c: C).(\forall (v: T).(\forall (u: T).((sn3 c (THead (Flat Appl) v -u)) \to (\forall (t: T).((sn3 c (THead (Flat Appl) v t)) \to (sn3 c (THead -(Flat Appl) v (THead (Flat Cast) u t)))))))) -\def - \lambda (c: C).(\lambda (v: T).(\lambda (u: T).(\lambda (H: (sn3 c (THead -(Flat Appl) v u))).(insert_eq T (THead (Flat Appl) v u) (\lambda (t: T).(sn3 -c t)) (\lambda (_: T).(\forall (t0: T).((sn3 c (THead (Flat Appl) v t0)) \to -(sn3 c (THead (Flat Appl) v (THead (Flat Cast) u t0)))))) (\lambda (y: -T).(\lambda (H0: (sn3 c y)).(unintro T u (\lambda (t: T).((eq T y (THead -(Flat Appl) v t)) \to (\forall (t0: T).((sn3 c (THead (Flat Appl) v t0)) \to -(sn3 c (THead (Flat Appl) v (THead (Flat Cast) t t0))))))) (unintro T v -(\lambda (t: T).(\forall (x: T).((eq T y (THead (Flat Appl) t x)) \to -(\forall (t0: T).((sn3 c (THead (Flat Appl) t t0)) \to (sn3 c (THead (Flat -Appl) t (THead (Flat Cast) x t0)))))))) (sn3_ind c (\lambda (t: T).(\forall -(x: T).(\forall (x0: T).((eq T t (THead (Flat Appl) x x0)) \to (\forall (t0: -T).((sn3 c (THead (Flat Appl) x t0)) \to (sn3 c (THead (Flat Appl) x (THead -(Flat Cast) x0 t0))))))))) (\lambda (t1: T).(\lambda (H1: ((\forall (t2: -T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c -t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: -Prop).P))) \to ((pr3 c t1 t2) \to (\forall (x: T).(\forall (x0: T).((eq T t2 -(THead (Flat Appl) x x0)) \to (\forall (t: T).((sn3 c (THead (Flat Appl) x -t)) \to (sn3 c (THead (Flat Appl) x (THead (Flat Cast) x0 -t))))))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: (eq T t1 (THead -(Flat Appl) x x0))).(\lambda (t: T).(\lambda (H4: (sn3 c (THead (Flat Appl) x -t))).(insert_eq T (THead (Flat Appl) x t) (\lambda (t0: T).(sn3 c t0)) -(\lambda (_: T).(sn3 c (THead (Flat Appl) x (THead (Flat Cast) x0 t)))) -(\lambda (y0: T).(\lambda (H5: (sn3 c y0)).(unintro T t (\lambda (t0: T).((eq -T y0 (THead (Flat Appl) x t0)) \to (sn3 c (THead (Flat Appl) x (THead (Flat -Cast) x0 t0))))) (sn3_ind c (\lambda (t0: T).(\forall (x1: T).((eq T t0 -(THead (Flat Appl) x x1)) \to (sn3 c (THead (Flat Appl) x (THead (Flat Cast) -x0 x1)))))) (\lambda (t0: T).(\lambda (H6: ((\forall (t2: T).((((eq T t0 t2) -\to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (sn3 c t2)))))).(\lambda -(H7: ((\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 -c t0 t2) \to (\forall (x1: T).((eq T t2 (THead (Flat Appl) x x1)) \to (sn3 c -(THead (Flat Appl) x (THead (Flat Cast) x0 x1)))))))))).(\lambda (x1: -T).(\lambda (H8: (eq T t0 (THead (Flat Appl) x x1))).(let H9 \def (eq_ind T -t0 (\lambda (t2: T).(\forall (t3: T).((((eq T t2 t3) \to (\forall (P: -Prop).P))) \to ((pr3 c t2 t3) \to (\forall (x2: T).((eq T t3 (THead (Flat -Appl) x x2)) \to (sn3 c (THead (Flat Appl) x (THead (Flat Cast) x0 -x2))))))))) H7 (THead (Flat Appl) x x1) H8) in (let H10 \def (eq_ind T t0 -(\lambda (t2: T).(\forall (t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) -\to ((pr3 c t2 t3) \to (sn3 c t3))))) H6 (THead (Flat Appl) x x1) H8) in (let -H11 \def (eq_ind T t1 (\lambda (t2: T).(\forall (t3: T).((((eq T t2 t3) \to -(\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to (\forall (x2: T).(\forall (x3: -T).((eq T t3 (THead (Flat Appl) x2 x3)) \to (\forall (t4: T).((sn3 c (THead -(Flat Appl) x2 t4)) \to (sn3 c (THead (Flat Appl) x2 (THead (Flat Cast) x3 -t4)))))))))))) H2 (THead (Flat Appl) x x0) H3) in (let H12 \def (eq_ind T t1 -(\lambda (t2: T).(\forall (t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) -\to ((pr3 c t2 t3) \to (sn3 c t3))))) H1 (THead (Flat Appl) x x0) H3) in -(sn3_pr2_intro c (THead (Flat Appl) x (THead (Flat Cast) x0 x1)) (\lambda -(t2: T).(\lambda (H13: (((eq T (THead (Flat Appl) x (THead (Flat Cast) x0 -x1)) t2) \to (\forall (P: Prop).P)))).(\lambda (H14: (pr2 c (THead (Flat -Appl) x (THead (Flat Cast) x0 x1)) t2)).(let H15 \def (pr2_gen_appl c x -(THead (Flat Cast) x0 x1) t2 H14) in (or3_ind (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c -(THead (Flat Cast) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Cast) x0 x1) -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda -(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: -B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T -T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq -T (THead (Flat Cast) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) -z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) -y2) z1 z2)))))))) (sn3 c t2) (\lambda (H16: (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c -(THead (Flat Cast) x0 x1) t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda -(t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Flat Cast) -x0 x1) t3))) (sn3 c t2) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H17: (eq -T t2 (THead (Flat Appl) x2 x3))).(\lambda (H18: (pr2 c x x2)).(\lambda (H19: -(pr2 c (THead (Flat Cast) x0 x1) x3)).(let H20 \def (eq_ind T t2 (\lambda -(t3: T).((eq T (THead (Flat Appl) x (THead (Flat Cast) x0 x1)) t3) \to -(\forall (P: Prop).P))) H13 (THead (Flat Appl) x2 x3) H17) in (eq_ind_r T -(THead (Flat Appl) x2 x3) (\lambda (t3: T).(sn3 c t3)) (let H21 \def -(pr2_gen_cast c x0 x1 x3 H19) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda -(t3: T).(eq T x3 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c x1 t3)))) (pr2 c -x1 x3) (sn3 c (THead (Flat Appl) x2 x3)) (\lambda (H22: (ex3_2 T T (\lambda -(u2: T).(\lambda (t3: T).(eq T x3 (THead (Flat Cast) u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c x1 -t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x3 (THead -(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) -(\lambda (_: T).(\lambda (t3: T).(pr2 c x1 t3))) (sn3 c (THead (Flat Appl) x2 -x3)) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H23: (eq T x3 (THead (Flat -Cast) x4 x5))).(\lambda (H24: (pr2 c x0 x4)).(\lambda (H25: (pr2 c x1 -x5)).(let H26 \def (eq_ind T x3 (\lambda (t3: T).((eq T (THead (Flat Appl) x -(THead (Flat Cast) x0 x1)) (THead (Flat Appl) x2 t3)) \to (\forall (P: -Prop).P))) H20 (THead (Flat Cast) x4 x5) H23) in (eq_ind_r T (THead (Flat -Cast) x4 x5) (\lambda (t3: T).(sn3 c (THead (Flat Appl) x2 t3))) (let H_x -\def (term_dec (THead (Flat Appl) x x0) (THead (Flat Appl) x2 x4)) in (let -H27 \def H_x in (or_ind (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x2 -x4)) ((eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x2 x4)) \to (\forall -(P: Prop).P)) (sn3 c (THead (Flat Appl) x2 (THead (Flat Cast) x4 x5))) -(\lambda (H28: (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x2 -x4))).(let H29 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | -(THead _ t3 _) \Rightarrow t3])) (THead (Flat Appl) x x0) (THead (Flat Appl) -x2 x4) H28) in ((let H30 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) -\Rightarrow x0 | (THead _ _ t3) \Rightarrow t3])) (THead (Flat Appl) x x0) -(THead (Flat Appl) x2 x4) H28) in (\lambda (H31: (eq T x x2)).(let H32 \def -(eq_ind_r T x4 (\lambda (t3: T).((eq T (THead (Flat Appl) x (THead (Flat -Cast) x0 x1)) (THead (Flat Appl) x2 (THead (Flat Cast) t3 x5))) \to (\forall -(P: Prop).P))) H26 x0 H30) in (let H33 \def (eq_ind_r T x4 (\lambda (t3: -T).(pr2 c x0 t3)) H24 x0 H30) in (eq_ind T x0 (\lambda (t3: T).(sn3 c (THead -(Flat Appl) x2 (THead (Flat Cast) t3 x5)))) (let H34 \def (eq_ind_r T x2 -(\lambda (t3: T).((eq T (THead (Flat Appl) x (THead (Flat Cast) x0 x1)) -(THead (Flat Appl) t3 (THead (Flat Cast) x0 x5))) \to (\forall (P: Prop).P))) -H32 x H31) in (let H35 \def (eq_ind_r T x2 (\lambda (t3: T).(pr2 c x t3)) H18 -x H31) in (eq_ind T x (\lambda (t3: T).(sn3 c (THead (Flat Appl) t3 (THead -(Flat Cast) x0 x5)))) (let H_x0 \def (term_dec (THead (Flat Appl) x x1) -(THead (Flat Appl) x x5)) in (let H36 \def H_x0 in (or_ind (eq T (THead (Flat -Appl) x x1) (THead (Flat Appl) x x5)) ((eq T (THead (Flat Appl) x x1) (THead -(Flat Appl) x x5)) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x -(THead (Flat Cast) x0 x5))) (\lambda (H37: (eq T (THead (Flat Appl) x x1) -(THead (Flat Appl) x x5))).(let H38 \def (f_equal T T (\lambda (e: T).(match -e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x1 | (TLRef _) -\Rightarrow x1 | (THead _ _ t3) \Rightarrow t3])) (THead (Flat Appl) x x1) -(THead (Flat Appl) x x5) H37) in (let H39 \def (eq_ind_r T x5 (\lambda (t3: -T).((eq T (THead (Flat Appl) x (THead (Flat Cast) x0 x1)) (THead (Flat Appl) -x (THead (Flat Cast) x0 t3))) \to (\forall (P: Prop).P))) H34 x1 H38) in (let -H40 \def (eq_ind_r T x5 (\lambda (t3: T).(pr2 c x1 t3)) H25 x1 H38) in -(eq_ind T x1 (\lambda (t3: T).(sn3 c (THead (Flat Appl) x (THead (Flat Cast) -x0 t3)))) (H39 (refl_equal T (THead (Flat Appl) x (THead (Flat Cast) x0 x1))) -(sn3 c (THead (Flat Appl) x (THead (Flat Cast) x0 x1)))) x5 H38))))) (\lambda -(H37: (((eq T (THead (Flat Appl) x x1) (THead (Flat Appl) x x5)) \to (\forall -(P: Prop).P)))).(H9 (THead (Flat Appl) x x5) H37 (pr3_pr2 c (THead (Flat -Appl) x x1) (THead (Flat Appl) x x5) (pr2_thin_dx c x1 x5 H25 x Appl)) x5 -(refl_equal T (THead (Flat Appl) x x5)))) H36))) x2 H31))) x4 H30))))) H29))) -(\lambda (H28: (((eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x2 x4)) -\to (\forall (P: Prop).P)))).(let H_x0 \def (term_dec (THead (Flat Appl) x -x1) (THead (Flat Appl) x2 x5)) in (let H29 \def H_x0 in (or_ind (eq T (THead -(Flat Appl) x x1) (THead (Flat Appl) x2 x5)) ((eq T (THead (Flat Appl) x x1) -(THead (Flat Appl) x2 x5)) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat -Appl) x2 (THead (Flat Cast) x4 x5))) (\lambda (H30: (eq T (THead (Flat Appl) -x x1) (THead (Flat Appl) x2 x5))).(let H31 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x | -(TLRef _) \Rightarrow x | (THead _ t3 _) \Rightarrow t3])) (THead (Flat Appl) -x x1) (THead (Flat Appl) x2 x5) H30) in ((let H32 \def (f_equal T T (\lambda -(e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x1 -| (TLRef _) \Rightarrow x1 | (THead _ _ t3) \Rightarrow t3])) (THead (Flat -Appl) x x1) (THead (Flat Appl) x2 x5) H30) in (\lambda (H33: (eq T x -x2)).(let H34 \def (eq_ind_r T x5 (\lambda (t3: T).(pr2 c x1 t3)) H25 x1 H32) -in (eq_ind T x1 (\lambda (t3: T).(sn3 c (THead (Flat Appl) x2 (THead (Flat -Cast) x4 t3)))) (let H35 \def (eq_ind_r T x2 (\lambda (t3: T).((eq T (THead -(Flat Appl) x x0) (THead (Flat Appl) t3 x4)) \to (\forall (P: Prop).P))) H28 -x H33) in (let H36 \def (eq_ind_r T x2 (\lambda (t3: T).(pr2 c x t3)) H18 x -H33) in (eq_ind T x (\lambda (t3: T).(sn3 c (THead (Flat Appl) t3 (THead -(Flat Cast) x4 x1)))) (H11 (THead (Flat Appl) x x4) H35 (pr3_pr2 c (THead -(Flat Appl) x x0) (THead (Flat Appl) x x4) (pr2_thin_dx c x0 x4 H24 x Appl)) -x x4 (refl_equal T (THead (Flat Appl) x x4)) x1 (sn3_sing c (THead (Flat -Appl) x x1) H10)) x2 H33))) x5 H32)))) H31))) (\lambda (H30: (((eq T (THead -(Flat Appl) x x1) (THead (Flat Appl) x2 x5)) \to (\forall (P: -Prop).P)))).(H11 (THead (Flat Appl) x2 x4) H28 (pr3_flat c x x2 (pr3_pr2 c x -x2 H18) x0 x4 (pr3_pr2 c x0 x4 H24) Appl) x2 x4 (refl_equal T (THead (Flat -Appl) x2 x4)) x5 (H10 (THead (Flat Appl) x2 x5) H30 (pr3_flat c x x2 (pr3_pr2 -c x x2 H18) x1 x5 (pr3_pr2 c x1 x5 H25) Appl)))) H29)))) H27))) x3 H23))))))) -H22)) (\lambda (H22: (pr2 c x1 x3)).(let H_x \def (term_dec (THead (Flat -Appl) x x1) (THead (Flat Appl) x2 x3)) in (let H23 \def H_x in (or_ind (eq T -(THead (Flat Appl) x x1) (THead (Flat Appl) x2 x3)) ((eq T (THead (Flat Appl) -x x1) (THead (Flat Appl) x2 x3)) \to (\forall (P: Prop).P)) (sn3 c (THead -(Flat Appl) x2 x3)) (\lambda (H24: (eq T (THead (Flat Appl) x x1) (THead -(Flat Appl) x2 x3))).(let H25 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) -\Rightarrow x | (THead _ t3 _) \Rightarrow t3])) (THead (Flat Appl) x x1) -(THead (Flat Appl) x2 x3) H24) in ((let H26 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x1 | -(TLRef _) \Rightarrow x1 | (THead _ _ t3) \Rightarrow t3])) (THead (Flat -Appl) x x1) (THead (Flat Appl) x2 x3) H24) in (\lambda (H27: (eq T x -x2)).(let H28 \def (eq_ind_r T x3 (\lambda (t3: T).(pr2 c x1 t3)) H22 x1 H26) -in (let H29 \def (eq_ind_r T x3 (\lambda (t3: T).((eq T (THead (Flat Appl) x -(THead (Flat Cast) x0 x1)) (THead (Flat Appl) x2 t3)) \to (\forall (P: -Prop).P))) H20 x1 H26) in (eq_ind T x1 (\lambda (t3: T).(sn3 c (THead (Flat -Appl) x2 t3))) (let H30 \def (eq_ind_r T x2 (\lambda (t3: T).((eq T (THead -(Flat Appl) x (THead (Flat Cast) x0 x1)) (THead (Flat Appl) t3 x1)) \to -(\forall (P: Prop).P))) H29 x H27) in (let H31 \def (eq_ind_r T x2 (\lambda -(t3: T).(pr2 c x t3)) H18 x H27) in (eq_ind T x (\lambda (t3: T).(sn3 c -(THead (Flat Appl) t3 x1))) (sn3_sing c (THead (Flat Appl) x x1) H10) x2 -H27))) x3 H26))))) H25))) (\lambda (H24: (((eq T (THead (Flat Appl) x x1) -(THead (Flat Appl) x2 x3)) \to (\forall (P: Prop).P)))).(H10 (THead (Flat -Appl) x2 x3) H24 (pr3_flat c x x2 (pr3_pr2 c x x2 H18) x1 x3 (pr3_pr2 c x1 x3 -H22) Appl))) H23)))) H21)) t2 H17))))))) H16)) (\lambda (H16: (ex4_4 T T T T -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Flat Cast) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind -Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) -u0) z1 t3))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Cast) x0 x1) (THead -(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda -(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: -B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3))))))) (sn3 c t2) -(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda -(H17: (eq T (THead (Flat Cast) x0 x1) (THead (Bind Abst) x2 x3))).(\lambda -(H18: (eq T t2 (THead (Bind Abbr) x4 x5))).(\lambda (_: (pr2 c x -x4)).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) -u0) x3 x5))))).(let H21 \def (eq_ind T t2 (\lambda (t3: T).((eq T (THead -(Flat Appl) x (THead (Flat Cast) x0 x1)) t3) \to (\forall (P: Prop).P))) H13 -(THead (Bind Abbr) x4 x5) H18) in (eq_ind_r T (THead (Bind Abbr) x4 x5) -(\lambda (t3: T).(sn3 c t3)) (let H22 \def (eq_ind T (THead (Flat Cast) x0 -x1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort -_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) -\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) x2 -x3) H17) in (False_ind (sn3 c (THead (Bind Abbr) x4 x5)) H22)) t2 -H18)))))))))) H16)) (\lambda (H16: (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat -Cast) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T -t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 -y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 -z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b -Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Cast) x0 x1) (THead -(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) (sn3 c t2) -(\lambda (x2: B).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda -(x6: T).(\lambda (x7: T).(\lambda (_: (not (eq B x2 Abst))).(\lambda (H18: -(eq T (THead (Flat Cast) x0 x1) (THead (Bind x2) x3 x4))).(\lambda (H19: (eq -T t2 (THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) O x6) x5)))).(\lambda -(_: (pr2 c x x6)).(\lambda (_: (pr2 c x3 x7)).(\lambda (_: (pr2 (CHead c -(Bind x2) x7) x4 x5)).(let H23 \def (eq_ind T t2 (\lambda (t3: T).((eq T -(THead (Flat Appl) x (THead (Flat Cast) x0 x1)) t3) \to (\forall (P: -Prop).P))) H13 (THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) O x6) x5)) -H19) in (eq_ind_r T (THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) O x6) -x5)) (\lambda (t3: T).(sn3 c t3)) (let H24 \def (eq_ind T (THead (Flat Cast) -x0 x1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) -\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind x2) x3 x4) -H18) in (False_ind (sn3 c (THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) -O x6) x5))) H24)) t2 H19)))))))))))))) H16)) H15))))))))))))))) y0 H5)))) -H4))))))))) y H0))))) H)))). -(* COMMENTS -Initial nodes: 5149 -END *) - -theorem sn3_appl_bind: - \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (u: -T).((sn3 c u) \to (\forall (t: T).(\forall (v: T).((sn3 (CHead c (Bind b) u) -(THead (Flat Appl) (lift (S O) O v) t)) \to (sn3 c (THead (Flat Appl) v -(THead (Bind b) u t)))))))))) -\def - \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (c: C).(\lambda -(u: T).(\lambda (H0: (sn3 c u)).(sn3_ind c (\lambda (t: T).(\forall (t0: -T).(\forall (v: T).((sn3 (CHead c (Bind b) t) (THead (Flat Appl) (lift (S O) -O v) t0)) \to (sn3 c (THead (Flat Appl) v (THead (Bind b) t t0))))))) -(\lambda (t1: T).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall -(P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H2: ((\forall -(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to -(\forall (t: T).(\forall (v: T).((sn3 (CHead c (Bind b) t2) (THead (Flat -Appl) (lift (S O) O v) t)) \to (sn3 c (THead (Flat Appl) v (THead (Bind b) t2 -t))))))))))).(\lambda (t: T).(\lambda (v: T).(\lambda (H3: (sn3 (CHead c -(Bind b) t1) (THead (Flat Appl) (lift (S O) O v) t))).(insert_eq T (THead -(Flat Appl) (lift (S O) O v) t) (\lambda (t0: T).(sn3 (CHead c (Bind b) t1) -t0)) (\lambda (_: T).(sn3 c (THead (Flat Appl) v (THead (Bind b) t1 t)))) -(\lambda (y: T).(\lambda (H4: (sn3 (CHead c (Bind b) t1) y)).(unintro T t -(\lambda (t0: T).((eq T y (THead (Flat Appl) (lift (S O) O v) t0)) \to (sn3 c -(THead (Flat Appl) v (THead (Bind b) t1 t0))))) (unintro T v (\lambda (t0: -T).(\forall (x: T).((eq T y (THead (Flat Appl) (lift (S O) O t0) x)) \to (sn3 -c (THead (Flat Appl) t0 (THead (Bind b) t1 x)))))) (sn3_ind (CHead c (Bind b) -t1) (\lambda (t0: T).(\forall (x: T).(\forall (x0: T).((eq T t0 (THead (Flat -Appl) (lift (S O) O x) x0)) \to (sn3 c (THead (Flat Appl) x (THead (Bind b) -t1 x0))))))) (\lambda (t2: T).(\lambda (H5: ((\forall (t3: T).((((eq T t2 t3) -\to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to (sn3 -(CHead c (Bind b) t1) t3)))))).(\lambda (H6: ((\forall (t3: T).((((eq T t2 -t3) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to -(\forall (x: T).(\forall (x0: T).((eq T t3 (THead (Flat Appl) (lift (S O) O -x) x0)) \to (sn3 c (THead (Flat Appl) x (THead (Bind b) t1 -x0))))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H7: (eq T t2 (THead -(Flat Appl) (lift (S O) O x) x0))).(let H8 \def (eq_ind T t2 (\lambda (t0: -T).(\forall (t3: T).((((eq T t0 t3) \to (\forall (P: Prop).P))) \to ((pr3 -(CHead c (Bind b) t1) t0 t3) \to (\forall (x1: T).(\forall (x2: T).((eq T t3 -(THead (Flat Appl) (lift (S O) O x1) x2)) \to (sn3 c (THead (Flat Appl) x1 -(THead (Bind b) t1 x2)))))))))) H6 (THead (Flat Appl) (lift (S O) O x) x0) -H7) in (let H9 \def (eq_ind T t2 (\lambda (t0: T).(\forall (t3: T).((((eq T -t0 t3) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t0 t3) \to -(sn3 (CHead c (Bind b) t1) t3))))) H5 (THead (Flat Appl) (lift (S O) O x) x0) -H7) in (sn3_pr2_intro c (THead (Flat Appl) x (THead (Bind b) t1 x0)) (\lambda -(t3: T).(\lambda (H10: (((eq T (THead (Flat Appl) x (THead (Bind b) t1 x0)) -t3) \to (\forall (P: Prop).P)))).(\lambda (H11: (pr2 c (THead (Flat Appl) x -(THead (Bind b) t1 x0)) t3)).(let H12 \def (pr2_gen_appl c x (THead (Bind b) -t1 x0) t3 H11) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T -t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x -u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind b) t1 x0) t4)))) -(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(eq T (THead (Bind b) t1 x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind -Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t4: T).(\forall (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind b0) -u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B -b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) t1 x0) (THead -(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind -b0) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) -(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b0) y2) z1 z2)))))))) (sn3 c t3) -(\lambda (H13: (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead -(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) -(\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind b) t1 x0) -t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead -(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) -(\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind b) t1 x0) t4))) (sn3 c -t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (H14: (eq T t3 (THead (Flat -Appl) x1 x2))).(\lambda (H15: (pr2 c x x1)).(\lambda (H16: (pr2 c (THead -(Bind b) t1 x0) x2)).(let H17 \def (eq_ind T t3 (\lambda (t0: T).((eq T -(THead (Flat Appl) x (THead (Bind b) t1 x0)) t0) \to (\forall (P: Prop).P))) -H10 (THead (Flat Appl) x1 x2) H14) in (eq_ind_r T (THead (Flat Appl) x1 x2) -(\lambda (t0: T).(sn3 c t0)) (let H_x \def (pr3_gen_bind b H c t1 x0 x2) in -(let H18 \def (H_x (pr3_pr2 c (THead (Bind b) t1 x0) x2 H16)) in (or_ind -(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead (Bind b) u2 -t4)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_: -T).(\lambda (t4: T).(pr3 (CHead c (Bind b) t1) x0 t4)))) (pr3 (CHead c (Bind -b) t1) x0 (lift (S O) O x2)) (sn3 c (THead (Flat Appl) x1 x2)) (\lambda (H19: -(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead (Bind b) u2 -t4)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_: -T).(\lambda (t4: T).(pr3 (CHead c (Bind b) t1) x0 t4))))).(ex3_2_ind T T -(\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead (Bind b) u2 t4)))) (\lambda -(u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr3 -(CHead c (Bind b) t1) x0 t4))) (sn3 c (THead (Flat Appl) x1 x2)) (\lambda -(x3: T).(\lambda (x4: T).(\lambda (H20: (eq T x2 (THead (Bind b) x3 -x4))).(\lambda (H21: (pr3 c t1 x3)).(\lambda (H22: (pr3 (CHead c (Bind b) t1) -x0 x4)).(let H23 \def (eq_ind T x2 (\lambda (t0: T).((eq T (THead (Flat Appl) -x (THead (Bind b) t1 x0)) (THead (Flat Appl) x1 t0)) \to (\forall (P: -Prop).P))) H17 (THead (Bind b) x3 x4) H20) in (eq_ind_r T (THead (Bind b) x3 -x4) (\lambda (t0: T).(sn3 c (THead (Flat Appl) x1 t0))) (let H_x0 \def -(term_dec t1 x3) in (let H24 \def H_x0 in (or_ind (eq T t1 x3) ((eq T t1 x3) -\to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Bind b) x3 -x4))) (\lambda (H25: (eq T t1 x3)).(let H26 \def (eq_ind_r T x3 (\lambda (t0: -T).((eq T (THead (Flat Appl) x (THead (Bind b) t1 x0)) (THead (Flat Appl) x1 -(THead (Bind b) t0 x4))) \to (\forall (P: Prop).P))) H23 t1 H25) in (let H27 -\def (eq_ind_r T x3 (\lambda (t0: T).(pr3 c t1 t0)) H21 t1 H25) in (eq_ind T -t1 (\lambda (t0: T).(sn3 c (THead (Flat Appl) x1 (THead (Bind b) t0 x4)))) -(let H_x1 \def (term_dec x0 x4) in (let H28 \def H_x1 in (or_ind (eq T x0 x4) -((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead -(Bind b) t1 x4))) (\lambda (H29: (eq T x0 x4)).(let H30 \def (eq_ind_r T x4 -(\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind b) t1 x0)) (THead -(Flat Appl) x1 (THead (Bind b) t1 t0))) \to (\forall (P: Prop).P))) H26 x0 -H29) in (let H31 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 (CHead c (Bind b) -t1) x0 t0)) H22 x0 H29) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Flat -Appl) x1 (THead (Bind b) t1 t0)))) (let H_x2 \def (term_dec x x1) in (let H32 -\def H_x2 in (or_ind (eq T x x1) ((eq T x x1) \to (\forall (P: Prop).P)) (sn3 -c (THead (Flat Appl) x1 (THead (Bind b) t1 x0))) (\lambda (H33: (eq T x -x1)).(let H34 \def (eq_ind_r T x1 (\lambda (t0: T).((eq T (THead (Flat Appl) -x (THead (Bind b) t1 x0)) (THead (Flat Appl) t0 (THead (Bind b) t1 x0))) \to -(\forall (P: Prop).P))) H30 x H33) in (let H35 \def (eq_ind_r T x1 (\lambda -(t0: T).(pr2 c x t0)) H15 x H33) in (eq_ind T x (\lambda (t0: T).(sn3 c -(THead (Flat Appl) t0 (THead (Bind b) t1 x0)))) (H34 (refl_equal T (THead -(Flat Appl) x (THead (Bind b) t1 x0))) (sn3 c (THead (Flat Appl) x (THead -(Bind b) t1 x0)))) x1 H33)))) (\lambda (H33: (((eq T x x1) \to (\forall (P: -Prop).P)))).(H8 (THead (Flat Appl) (lift (S O) O x1) x0) (\lambda (H34: (eq T -(THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) -x0))).(\lambda (P: Prop).(let H35 \def (f_equal T T (\lambda (e: T).(match e -in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map -(f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n) -\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with -[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4) -\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in -lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow -((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match -t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef -(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | -(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) -t4))]) in lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _) -\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) -(lift (S O) O x1) x0) H34) in (let H36 \def (eq_ind_r T x1 (\lambda (t0: -T).((eq T x t0) \to (\forall (P0: Prop).P0))) H33 x (lift_inj x x1 (S O) O -H35)) in (let H37 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x -(lift_inj x x1 (S O) O H35)) in (H36 (refl_equal T x) P)))))) (pr3_flat -(CHead c (Bind b) t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c -(Bind b) t1) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1 -(pr3_pr2 c x x1 H15)) x0 x0 (pr3_refl (CHead c (Bind b) t1) x0) Appl) x1 x0 -(refl_equal T (THead (Flat Appl) (lift (S O) O x1) x0)))) H32))) x4 H29)))) -(\lambda (H29: (((eq T x0 x4) \to (\forall (P: Prop).P)))).(H8 (THead (Flat -Appl) (lift (S O) O x1) x4) (\lambda (H30: (eq T (THead (Flat Appl) (lift (S -O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) x4))).(\lambda (P: -Prop).(let H31 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat -\to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n) -\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with -[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4) -\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in -lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow -((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match -t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef -(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | -(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) -t4))]) in lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _) -\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) -(lift (S O) O x1) x4) H30) in ((let H32 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | -(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat -Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) x4) H30) in -(\lambda (H33: (eq T (lift (S O) O x) (lift (S O) O x1))).(let H34 \def -(eq_ind_r T x4 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) -H29 x0 H32) in (let H35 \def (eq_ind_r T x4 (\lambda (t0: T).((eq T (THead -(Flat Appl) x (THead (Bind b) t1 x0)) (THead (Flat Appl) x1 (THead (Bind b) -t1 t0))) \to (\forall (P0: Prop).P0))) H26 x0 H32) in (let H36 \def (eq_ind_r -T x4 (\lambda (t0: T).(pr3 (CHead c (Bind b) t1) x0 t0)) H22 x0 H32) in (let -H37 \def (eq_ind_r T x1 (\lambda (t0: T).((eq T (THead (Flat Appl) x (THead -(Bind b) t1 x0)) (THead (Flat Appl) t0 (THead (Bind b) t1 x0))) \to (\forall -(P0: Prop).P0))) H35 x (lift_inj x x1 (S O) O H33)) in (let H38 \def -(eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x (lift_inj x x1 (S O) O -H33)) in (H34 (refl_equal T x0) P)))))))) H31)))) (pr3_flat (CHead c (Bind b) -t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c (Bind b) t1) c (S -O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1 (pr3_pr2 c x x1 H15)) -x0 x4 H22 Appl) x1 x4 (refl_equal T (THead (Flat Appl) (lift (S O) O x1) -x4)))) H28))) x3 H25)))) (\lambda (H25: (((eq T t1 x3) \to (\forall (P: -Prop).P)))).(H2 x3 H25 H21 x4 x1 (sn3_cpr3_trans c t1 x3 H21 (Bind b) (THead -(Flat Appl) (lift (S O) O x1) x4) (let H_x1 \def (term_dec x0 x4) in (let H26 -\def H_x1 in (or_ind (eq T x0 x4) ((eq T x0 x4) \to (\forall (P: Prop).P)) -(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x1) x4)) (\lambda -(H27: (eq T x0 x4)).(let H28 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 (CHead -c (Bind b) t1) x0 t0)) H22 x0 H27) in (eq_ind T x0 (\lambda (t0: T).(sn3 -(CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x1) t0))) (let H_x2 -\def (term_dec x x1) in (let H29 \def H_x2 in (or_ind (eq T x x1) ((eq T x -x1) \to (\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat Appl) -(lift (S O) O x1) x0)) (\lambda (H30: (eq T x x1)).(let H31 \def (eq_ind_r T -x1 (\lambda (t0: T).(pr2 c x t0)) H15 x H30) in (eq_ind T x (\lambda (t0: -T).(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O t0) x0))) -(sn3_sing (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x) x0) H9) -x1 H30))) (\lambda (H30: (((eq T x x1) \to (\forall (P: Prop).P)))).(H9 -(THead (Flat Appl) (lift (S O) O x1) x0) (\lambda (H31: (eq T (THead (Flat -Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) -x0))).(\lambda (P: Prop).(let H32 \def (f_equal T T (\lambda (e: T).(match e -in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map -(f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n) -\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with -[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4) -\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in -lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow -((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match -t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef -(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | -(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) -t4))]) in lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _) -\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) -(lift (S O) O x1) x0) H31) in (let H33 \def (eq_ind_r T x1 (\lambda (t0: -T).((eq T x t0) \to (\forall (P0: Prop).P0))) H30 x (lift_inj x x1 (S O) O -H32)) in (let H34 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x -(lift_inj x x1 (S O) O H32)) in (H33 (refl_equal T x) P)))))) (pr3_flat -(CHead c (Bind b) t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c -(Bind b) t1) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1 -(pr3_pr2 c x x1 H15)) x0 x0 (pr3_refl (CHead c (Bind b) t1) x0) Appl))) -H29))) x4 H27))) (\lambda (H27: (((eq T x0 x4) \to (\forall (P: -Prop).P)))).(H9 (THead (Flat Appl) (lift (S O) O x1) x4) (\lambda (H28: (eq T -(THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) -x4))).(\lambda (P: Prop).(let H29 \def (f_equal T T (\lambda (e: T).(match e -in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map -(f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n) -\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with -[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4) -\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in -lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow -((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match -t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef -(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | -(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) -t4))]) in lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _) -\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) -(lift (S O) O x1) x4) H28) in ((let H30 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | -(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat -Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) x4) H28) in -(\lambda (H31: (eq T (lift (S O) O x) (lift (S O) O x1))).(let H32 \def -(eq_ind_r T x4 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) -H27 x0 H30) in (let H33 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 (CHead c -(Bind b) t1) x0 t0)) H22 x0 H30) in (let H34 \def (eq_ind_r T x1 (\lambda -(t0: T).(pr2 c x t0)) H15 x (lift_inj x x1 (S O) O H31)) in (H32 (refl_equal -T x0) P)))))) H29)))) (pr3_flat (CHead c (Bind b) t1) (lift (S O) O x) (lift -(S O) O x1) (pr3_lift (CHead c (Bind b) t1) c (S O) O (drop_drop (Bind b) O c -c (drop_refl c) t1) x x1 (pr3_pr2 c x x1 H15)) x0 x4 H22 Appl))) H26)))))) -H24))) x2 H20))))))) H19)) (\lambda (H19: (pr3 (CHead c (Bind b) t1) x0 (lift -(S O) O x2))).(sn3_gen_lift (CHead c (Bind b) t1) (THead (Flat Appl) x1 x2) -(S O) O (eq_ind_r T (THead (Flat Appl) (lift (S O) O x1) (lift (S O) (s (Flat -Appl) O) x2)) (\lambda (t0: T).(sn3 (CHead c (Bind b) t1) t0)) (sn3_pr3_trans -(CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x1) x0) (let H_x0 \def -(term_dec x x1) in (let H20 \def H_x0 in (or_ind (eq T x x1) ((eq T x x1) \to -(\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S -O) O x1) x0)) (\lambda (H21: (eq T x x1)).(let H22 \def (eq_ind_r T x1 -(\lambda (t0: T).(pr2 c x t0)) H15 x H21) in (eq_ind T x (\lambda (t0: -T).(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O t0) x0))) -(sn3_sing (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x) x0) H9) -x1 H21))) (\lambda (H21: (((eq T x x1) \to (\forall (P: Prop).P)))).(H9 -(THead (Flat Appl) (lift (S O) O x1) x0) (\lambda (H22: (eq T (THead (Flat -Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) -x0))).(\lambda (P: Prop).(let H23 \def (f_equal T T (\lambda (e: T).(match e -in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map -(f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n) -\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with -[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4) -\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in -lref_map) (\lambda (x3: nat).(plus x3 (S O))) O x) | (TLRef _) \Rightarrow -((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match -t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef -(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | -(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) -t4))]) in lref_map) (\lambda (x3: nat).(plus x3 (S O))) O x) | (THead _ t0 _) -\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) -(lift (S O) O x1) x0) H22) in (let H24 \def (eq_ind_r T x1 (\lambda (t0: -T).((eq T x t0) \to (\forall (P0: Prop).P0))) H21 x (lift_inj x x1 (S O) O -H23)) in (let H25 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x -(lift_inj x x1 (S O) O H23)) in (H24 (refl_equal T x) P)))))) (pr3_flat -(CHead c (Bind b) t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c -(Bind b) t1) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1 -(pr3_pr2 c x x1 H15)) x0 x0 (pr3_refl (CHead c (Bind b) t1) x0) Appl))) -H20))) (THead (Flat Appl) (lift (S O) O x1) (lift (S O) O x2)) (pr3_thin_dx -(CHead c (Bind b) t1) x0 (lift (S O) O x2) H19 (lift (S O) O x1) Appl)) (lift -(S O) O (THead (Flat Appl) x1 x2)) (lift_head (Flat Appl) x1 x2 (S O) O)) c -(drop_drop (Bind b) O c c (drop_refl c) t1))) H18))) t3 H14))))))) H13)) -(\lambda (H13: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T (THead (Bind b) t1 x0) (THead (Bind Abst) y1 -z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: -T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b0: B).(\forall (u0: -T).(pr2 (CHead c (Bind b0) u0) z1 t4))))))))).(ex4_4_ind T T T T (\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind -b) t1 x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x -u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: -T).(\forall (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind b0) u0) z1 t4))))))) -(sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: -T).(\lambda (H14: (eq T (THead (Bind b) t1 x0) (THead (Bind Abst) x1 -x2))).(\lambda (H15: (eq T t3 (THead (Bind Abbr) x3 x4))).(\lambda (_: (pr2 c -x x3)).(\lambda (H17: ((\forall (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind -b0) u0) x2 x4))))).(let H18 \def (eq_ind T t3 (\lambda (t0: T).((eq T (THead -(Flat Appl) x (THead (Bind b) t1 x0)) t0) \to (\forall (P: Prop).P))) H10 -(THead (Bind Abbr) x3 x4) H15) in (eq_ind_r T (THead (Bind Abbr) x3 x4) -(\lambda (t0: T).(sn3 c t0)) (let H19 \def (f_equal T B (\lambda (e: -T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | -(TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k in K return -(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow -b])])) (THead (Bind b) t1 x0) (THead (Bind Abst) x1 x2) H14) in ((let H20 -\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ t0 _) -\Rightarrow t0])) (THead (Bind b) t1 x0) (THead (Bind Abst) x1 x2) H14) in -((let H21 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: -T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ -t0) \Rightarrow t0])) (THead (Bind b) t1 x0) (THead (Bind Abst) x1 x2) H14) -in (\lambda (_: (eq T t1 x1)).(\lambda (H23: (eq B b Abst)).(let H24 \def -(eq_ind_r T x2 (\lambda (t0: T).(\forall (b0: B).(\forall (u0: T).(pr2 (CHead -c (Bind b0) u0) t0 x4)))) H17 x0 H21) in (let H25 \def (eq_ind B b (\lambda -(b0: B).((eq T (THead (Flat Appl) x (THead (Bind b0) t1 x0)) (THead (Bind -Abbr) x3 x4)) \to (\forall (P: Prop).P))) H18 Abst H23) in (let H26 \def -(eq_ind B b (\lambda (b0: B).(\forall (t4: T).((((eq T (THead (Flat Appl) -(lift (S O) O x) x0) t4) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind -b0) t1) (THead (Flat Appl) (lift (S O) O x) x0) t4) \to (sn3 (CHead c (Bind -b0) t1) t4))))) H9 Abst H23) in (let H27 \def (eq_ind B b (\lambda (b0: -B).(\forall (t4: T).((((eq T (THead (Flat Appl) (lift (S O) O x) x0) t4) \to -(\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b0) t1) (THead (Flat Appl) -(lift (S O) O x) x0) t4) \to (\forall (x5: T).(\forall (x6: T).((eq T t4 -(THead (Flat Appl) (lift (S O) O x5) x6)) \to (sn3 c (THead (Flat Appl) x5 -(THead (Bind b0) t1 x6)))))))))) H8 Abst H23) in (let H28 \def (eq_ind B b -(\lambda (b0: B).(\forall (t4: T).((((eq T t1 t4) \to (\forall (P: Prop).P))) -\to ((pr3 c t1 t4) \to (\forall (t0: T).(\forall (v0: T).((sn3 (CHead c (Bind -b0) t4) (THead (Flat Appl) (lift (S O) O v0) t0)) \to (sn3 c (THead (Flat -Appl) v0 (THead (Bind b0) t4 t0)))))))))) H2 Abst H23) in (let H29 \def -(eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H Abst H23) in (let H30 -\def (match (H29 (refl_equal B Abst)) in False return (\lambda (_: -False).(sn3 c (THead (Bind Abbr) x3 x4))) with []) in H30)))))))))) H20)) -H19)) t3 H15)))))))))) H13)) (\lambda (H13: (ex6_6 B T T T T T (\lambda (b0: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) -t1 x0) (THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T -t3 (THead (Bind b0) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 -y2))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b0) y2) z1 -z2))))))))).(ex6_6_ind B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 -Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) t1 x0) (THead (Bind -b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind b0) y2 (THead -(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x -u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b0: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 (CHead c (Bind b0) y2) z1 z2))))))) (sn3 c t3) (\lambda (x1: -B).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: -T).(\lambda (x6: T).(\lambda (_: (not (eq B x1 Abst))).(\lambda (H15: (eq T -(THead (Bind b) t1 x0) (THead (Bind x1) x2 x3))).(\lambda (H16: (eq T t3 -(THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))).(\lambda -(H17: (pr2 c x x5)).(\lambda (H18: (pr2 c x2 x6)).(\lambda (H19: (pr2 (CHead -c (Bind x1) x6) x3 x4)).(let H20 \def (eq_ind T t3 (\lambda (t0: T).((eq T -(THead (Flat Appl) x (THead (Bind b) t1 x0)) t0) \to (\forall (P: Prop).P))) -H10 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)) H16) in -(eq_ind_r T (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)) -(\lambda (t0: T).(sn3 c t0)) (let H21 \def (f_equal T B (\lambda (e: -T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | -(TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k in K return -(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow -b])])) (THead (Bind b) t1 x0) (THead (Bind x1) x2 x3) H15) in ((let H22 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ t0 _) -\Rightarrow t0])) (THead (Bind b) t1 x0) (THead (Bind x1) x2 x3) H15) in -((let H23 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: -T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ -t0) \Rightarrow t0])) (THead (Bind b) t1 x0) (THead (Bind x1) x2 x3) H15) in -(\lambda (H24: (eq T t1 x2)).(\lambda (H25: (eq B b x1)).(let H26 \def -(eq_ind_r T x3 (\lambda (t0: T).(pr2 (CHead c (Bind x1) x6) t0 x4)) H19 x0 -H23) in (let H27 \def (eq_ind_r T x2 (\lambda (t0: T).(pr2 c t0 x6)) H18 t1 -H24) in (let H28 \def (eq_ind_r B x1 (\lambda (b0: B).(pr2 (CHead c (Bind b0) -x6) x0 x4)) H26 b H25) in (eq_ind B b (\lambda (b0: B).(sn3 c (THead (Bind -b0) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))) (sn3_pr3_trans c (THead -(Bind b) t1 (THead (Flat Appl) (lift (S O) O x5) x4)) (sn3_bind b c t1 -(sn3_sing c t1 H1) (THead (Flat Appl) (lift (S O) O x5) x4) (let H_x \def -(term_dec x x5) in (let H29 \def H_x in (or_ind (eq T x x5) ((eq T x x5) \to -(\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S -O) O x5) x4)) (\lambda (H30: (eq T x x5)).(let H31 \def (eq_ind_r T x5 -(\lambda (t0: T).(pr2 c x t0)) H17 x H30) in (eq_ind T x (\lambda (t0: -T).(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O t0) x4))) (let -H_x0 \def (term_dec x0 x4) in (let H32 \def H_x0 in (or_ind (eq T x0 x4) ((eq -T x0 x4) \to (\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat -Appl) (lift (S O) O x) x4)) (\lambda (H33: (eq T x0 x4)).(let H34 \def -(eq_ind_r T x4 (\lambda (t0: T).(pr2 (CHead c (Bind b) x6) x0 t0)) H28 x0 -H33) in (eq_ind T x0 (\lambda (t0: T).(sn3 (CHead c (Bind b) t1) (THead (Flat -Appl) (lift (S O) O x) t0))) (sn3_sing (CHead c (Bind b) t1) (THead (Flat -Appl) (lift (S O) O x) x0) H9) x4 H33))) (\lambda (H33: (((eq T x0 x4) \to -(\forall (P: Prop).P)))).(H9 (THead (Flat Appl) (lift (S O) O x) x4) (\lambda -(H34: (eq T (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift -(S O) O x) x4))).(\lambda (P: Prop).(let H35 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | -(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat -Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x) x4) H34) in -(let H36 \def (eq_ind_r T x4 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: -Prop).P0))) H33 x0 H35) in (let H37 \def (eq_ind_r T x4 (\lambda (t0: T).(pr2 -(CHead c (Bind b) x6) x0 t0)) H28 x0 H35) in (H36 (refl_equal T x0) P)))))) -(pr3_pr3_pr3_t c t1 x6 (pr3_pr2 c t1 x6 H27) (THead (Flat Appl) (lift (S O) O -x) x0) (THead (Flat Appl) (lift (S O) O x) x4) (Bind b) (pr3_pr2 (CHead c -(Bind b) x6) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift -(S O) O x) x4) (pr2_thin_dx (CHead c (Bind b) x6) x0 x4 H28 (lift (S O) O x) -Appl))))) H32))) x5 H30))) (\lambda (H30: (((eq T x x5) \to (\forall (P: -Prop).P)))).(H9 (THead (Flat Appl) (lift (S O) O x5) x4) (\lambda (H31: (eq T -(THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x5) -x4))).(\lambda (P: Prop).(let H32 \def (f_equal T T (\lambda (e: T).(match e -in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map -(f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n) -\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with -[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4) -\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in -lref_map) (\lambda (x7: nat).(plus x7 (S O))) O x) | (TLRef _) \Rightarrow -((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match -t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef -(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | -(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) -t4))]) in lref_map) (\lambda (x7: nat).(plus x7 (S O))) O x) | (THead _ t0 _) -\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) -(lift (S O) O x5) x4) H31) in ((let H33 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | -(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat -Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x5) x4) H31) in -(\lambda (H34: (eq T (lift (S O) O x) (lift (S O) O x5))).(let H35 \def -(eq_ind_r T x5 (\lambda (t0: T).((eq T x t0) \to (\forall (P0: Prop).P0))) -H30 x (lift_inj x x5 (S O) O H34)) in (let H36 \def (eq_ind_r T x5 (\lambda -(t0: T).(pr2 c x t0)) H17 x (lift_inj x x5 (S O) O H34)) in (let H37 \def -(eq_ind_r T x4 (\lambda (t0: T).(pr2 (CHead c (Bind b) x6) x0 t0)) H28 x0 -H33) in (H35 (refl_equal T x) P)))))) H32)))) (pr3_pr3_pr3_t c t1 x6 (pr3_pr2 -c t1 x6 H27) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift -(S O) O x5) x4) (Bind b) (pr3_flat (CHead c (Bind b) x6) (lift (S O) O x) -(lift (S O) O x5) (pr3_lift (CHead c (Bind b) x6) c (S O) O (drop_drop (Bind -b) O c c (drop_refl c) x6) x x5 (pr3_pr2 c x x5 H17)) x0 x4 (pr3_pr2 (CHead c -(Bind b) x6) x0 x4 H28) Appl)))) H29)))) (THead (Bind b) x6 (THead (Flat -Appl) (lift (S O) O x5) x4)) (pr3_pr2 c (THead (Bind b) t1 (THead (Flat Appl) -(lift (S O) O x5) x4)) (THead (Bind b) x6 (THead (Flat Appl) (lift (S O) O -x5) x4)) (pr2_head_1 c t1 x6 H27 (Bind b) (THead (Flat Appl) (lift (S O) O -x5) x4)))) x1 H25))))))) H22)) H21)) t3 H16)))))))))))))) H13)) -H12)))))))))))))) y H4))))) H3))))))) u H0))))). -(* COMMENTS -Initial nodes: 9191 -END *) - -theorem sn3_appl_appl: - \forall (v1: T).(\forall (t1: T).(let u1 \def (THead (Flat Appl) v1 t1) in -(\forall (c: C).((sn3 c u1) \to (\forall (v2: T).((sn3 c v2) \to (((\forall -(u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to -(sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c (THead (Flat Appl) v2 -u1))))))))) -\def - \lambda (v1: T).(\lambda (t1: T).(let u1 \def (THead (Flat Appl) v1 t1) in -(\lambda (c: C).(\lambda (H: (sn3 c (THead (Flat Appl) v1 t1))).(insert_eq T -(THead (Flat Appl) v1 t1) (\lambda (t: T).(sn3 c t)) (\lambda (t: T).(\forall -(v2: T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c t u2) \to ((((iso t u2) -\to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to -(sn3 c (THead (Flat Appl) v2 t)))))) (\lambda (y: T).(\lambda (H0: (sn3 c -y)).(unintro T t1 (\lambda (t: T).((eq T y (THead (Flat Appl) v1 t)) \to -(\forall (v2: T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c y u2) \to ((((iso -y u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) -\to (sn3 c (THead (Flat Appl) v2 y))))))) (unintro T v1 (\lambda (t: -T).(\forall (x: T).((eq T y (THead (Flat Appl) t x)) \to (\forall (v2: -T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c y u2) \to ((((iso y u2) \to -(\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c -(THead (Flat Appl) v2 y)))))))) (sn3_ind c (\lambda (t: T).(\forall (x: -T).(\forall (x0: T).((eq T t (THead (Flat Appl) x x0)) \to (\forall (v2: -T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c t u2) \to ((((iso t u2) \to -(\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c -(THead (Flat Appl) v2 t))))))))) (\lambda (t2: T).(\lambda (H1: ((\forall -(t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to -(sn3 c t3)))))).(\lambda (H2: ((\forall (t3: T).((((eq T t2 t3) \to (\forall -(P: Prop).P))) \to ((pr3 c t2 t3) \to (\forall (x: T).(\forall (x0: T).((eq T -t3 (THead (Flat Appl) x x0)) \to (\forall (v2: T).((sn3 c v2) \to (((\forall -(u2: T).((pr3 c t3 u2) \to ((((iso t3 u2) \to (\forall (P: Prop).P))) \to -(sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c (THead (Flat Appl) v2 -t3))))))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: (eq T t2 -(THead (Flat Appl) x x0))).(\lambda (v2: T).(\lambda (H4: (sn3 c -v2)).(sn3_ind c (\lambda (t: T).(((\forall (u2: T).((pr3 c t2 u2) \to ((((iso -t2 u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) t u2)))))) -\to (sn3 c (THead (Flat Appl) t t2)))) (\lambda (t0: T).(\lambda (H5: -((\forall (t3: T).((((eq T t0 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t0 -t3) \to (sn3 c t3)))))).(\lambda (H6: ((\forall (t3: T).((((eq T t0 t3) \to -(\forall (P: Prop).P))) \to ((pr3 c t0 t3) \to (((\forall (u2: T).((pr3 c t2 -u2) \to ((((iso t2 u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat -Appl) t3 u2)))))) \to (sn3 c (THead (Flat Appl) t3 t2)))))))).(\lambda (H7: -((\forall (u2: T).((pr3 c t2 u2) \to ((((iso t2 u2) \to (\forall (P: -Prop).P))) \to (sn3 c (THead (Flat Appl) t0 u2))))))).(let H8 \def (eq_ind T -t2 (\lambda (t: T).(\forall (u2: T).((pr3 c t u2) \to ((((iso t u2) \to -(\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) t0 u2)))))) H7 (THead -(Flat Appl) x x0) H3) in (let H9 \def (eq_ind T t2 (\lambda (t: T).(\forall -(t3: T).((((eq T t0 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t3) \to -(((\forall (u2: T).((pr3 c t u2) \to ((((iso t u2) \to (\forall (P: -Prop).P))) \to (sn3 c (THead (Flat Appl) t3 u2)))))) \to (sn3 c (THead (Flat -Appl) t3 t))))))) H6 (THead (Flat Appl) x x0) H3) in (let H10 \def (eq_ind T -t2 (\lambda (t: T).(\forall (t3: T).((((eq T t t3) \to (\forall (P: -Prop).P))) \to ((pr3 c t t3) \to (\forall (x1: T).(\forall (x2: T).((eq T t3 -(THead (Flat Appl) x1 x2)) \to (\forall (v3: T).((sn3 c v3) \to (((\forall -(u2: T).((pr3 c t3 u2) \to ((((iso t3 u2) \to (\forall (P: Prop).P))) \to -(sn3 c (THead (Flat Appl) v3 u2)))))) \to (sn3 c (THead (Flat Appl) v3 -t3)))))))))))) H2 (THead (Flat Appl) x x0) H3) in (let H11 \def (eq_ind T t2 -(\lambda (t: T).(\forall (t3: T).((((eq T t t3) \to (\forall (P: Prop).P))) -\to ((pr3 c t t3) \to (sn3 c t3))))) H1 (THead (Flat Appl) x x0) H3) in -(eq_ind_r T (THead (Flat Appl) x x0) (\lambda (t: T).(sn3 c (THead (Flat -Appl) t0 t))) (sn3_pr2_intro c (THead (Flat Appl) t0 (THead (Flat Appl) x -x0)) (\lambda (t3: T).(\lambda (H12: (((eq T (THead (Flat Appl) t0 (THead -(Flat Appl) x x0)) t3) \to (\forall (P: Prop).P)))).(\lambda (H13: (pr2 c -(THead (Flat Appl) t0 (THead (Flat Appl) x x0)) t3)).(let H14 \def -(pr2_gen_appl c t0 (THead (Flat Appl) x x0) t3 H13) in (or3_ind (ex3_2 T T -(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))) (\lambda (_: T).(\lambda -(t4: T).(pr2 c (THead (Flat Appl) x x0) t4)))) (ex4_4 T T T T (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Appl) -x x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall -(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4)))))))) (ex6_6 B T T T -T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq -T (THead (Flat Appl) x x0) (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) -z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) -y2) z1 z2)))))))) (sn3 c t3) (\lambda (H15: (ex3_2 T T (\lambda (u2: -T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c t0 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c -(THead (Flat Appl) x x0) t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda -(t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c t0 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Flat Appl) -x x0) t4))) (sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (H16: (eq T -t3 (THead (Flat Appl) x1 x2))).(\lambda (H17: (pr2 c t0 x1)).(\lambda (H18: -(pr2 c (THead (Flat Appl) x x0) x2)).(let H19 \def (eq_ind T t3 (\lambda (t: -T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) t) \to (\forall (P: -Prop).P))) H12 (THead (Flat Appl) x1 x2) H16) in (eq_ind_r T (THead (Flat -Appl) x1 x2) (\lambda (t: T).(sn3 c t)) (let H20 \def (pr2_gen_appl c x x0 x2 -H18) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead -(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) -(\lambda (_: T).(\lambda (t4: T).(pr2 c x0 t4)))) (ex4_4 T T T T (\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x0 (THead -(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t4: T).(eq T x2 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda -(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: -B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4)))))))) (ex6_6 B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T x0 -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x2 (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (sn3 c -(THead (Flat Appl) x1 x2)) (\lambda (H21: (ex3_2 T T (\lambda (u2: -T).(\lambda (t4: T).(eq T x2 (THead (Flat Appl) u2 t4)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c x0 -t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead -(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) -(\lambda (_: T).(\lambda (t4: T).(pr2 c x0 t4))) (sn3 c (THead (Flat Appl) x1 -x2)) (\lambda (x3: T).(\lambda (x4: T).(\lambda (H22: (eq T x2 (THead (Flat -Appl) x3 x4))).(\lambda (H23: (pr2 c x x3)).(\lambda (H24: (pr2 c x0 -x4)).(let H25 \def (eq_ind T x2 (\lambda (t: T).((eq T (THead (Flat Appl) t0 -(THead (Flat Appl) x x0)) (THead (Flat Appl) x1 t)) \to (\forall (P: -Prop).P))) H19 (THead (Flat Appl) x3 x4) H22) in (eq_ind_r T (THead (Flat -Appl) x3 x4) (\lambda (t: T).(sn3 c (THead (Flat Appl) x1 t))) (let H_x \def -(term_dec (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4)) in (let H26 -\def H_x in (or_ind (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4)) -((eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4)) \to (\forall (P: -Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Flat Appl) x3 x4))) (\lambda -(H27: (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4))).(let H28 -\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t _) -\Rightarrow t])) (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4) H27) in -((let H29 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: -T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ -t) \Rightarrow t])) (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4) H27) -in (\lambda (H30: (eq T x x3)).(let H31 \def (eq_ind_r T x4 (\lambda (t: -T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) (THead (Flat Appl) -x1 (THead (Flat Appl) x3 t))) \to (\forall (P: Prop).P))) H25 x0 H29) in (let -H32 \def (eq_ind_r T x4 (\lambda (t: T).(pr2 c x0 t)) H24 x0 H29) in (eq_ind -T x0 (\lambda (t: T).(sn3 c (THead (Flat Appl) x1 (THead (Flat Appl) x3 t)))) -(let H33 \def (eq_ind_r T x3 (\lambda (t: T).((eq T (THead (Flat Appl) t0 -(THead (Flat Appl) x x0)) (THead (Flat Appl) x1 (THead (Flat Appl) t x0))) -\to (\forall (P: Prop).P))) H31 x H30) in (let H34 \def (eq_ind_r T x3 -(\lambda (t: T).(pr2 c x t)) H23 x H30) in (eq_ind T x (\lambda (t: T).(sn3 c -(THead (Flat Appl) x1 (THead (Flat Appl) t x0)))) (let H_x0 \def (term_dec t0 -x1) in (let H35 \def H_x0 in (or_ind (eq T t0 x1) ((eq T t0 x1) \to (\forall -(P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Flat Appl) x x0))) -(\lambda (H36: (eq T t0 x1)).(let H37 \def (eq_ind_r T x1 (\lambda (t: -T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) (THead (Flat Appl) -t (THead (Flat Appl) x x0))) \to (\forall (P: Prop).P))) H33 t0 H36) in (let -H38 \def (eq_ind_r T x1 (\lambda (t: T).(pr2 c t0 t)) H17 t0 H36) in (eq_ind -T t0 (\lambda (t: T).(sn3 c (THead (Flat Appl) t (THead (Flat Appl) x x0)))) -(H37 (refl_equal T (THead (Flat Appl) t0 (THead (Flat Appl) x x0))) (sn3 c -(THead (Flat Appl) t0 (THead (Flat Appl) x x0)))) x1 H36)))) (\lambda (H36: -(((eq T t0 x1) \to (\forall (P: Prop).P)))).(H9 x1 H36 (pr3_pr2 c t0 x1 H17) -(\lambda (u2: T).(\lambda (H37: (pr3 c (THead (Flat Appl) x x0) u2)).(\lambda -(H38: (((iso (THead (Flat Appl) x x0) u2) \to (\forall (P: -Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) t0 u2) (H8 u2 H37 H38) (THead -(Flat Appl) x1 u2) (pr3_pr2 c (THead (Flat Appl) t0 u2) (THead (Flat Appl) x1 -u2) (pr2_head_1 c t0 x1 H17 (Flat Appl) u2)))))))) H35))) x3 H30))) x4 -H29))))) H28))) (\lambda (H27: (((eq T (THead (Flat Appl) x x0) (THead (Flat -Appl) x3 x4)) \to (\forall (P: Prop).P)))).(H10 (THead (Flat Appl) x3 x4) H27 -(pr3_flat c x x3 (pr3_pr2 c x x3 H23) x0 x4 (pr3_pr2 c x0 x4 H24) Appl) x3 x4 -(refl_equal T (THead (Flat Appl) x3 x4)) x1 (sn3_pr3_trans c t0 (sn3_sing c -t0 H5) x1 (pr3_pr2 c t0 x1 H17)) (\lambda (u2: T).(\lambda (H28: (pr3 c -(THead (Flat Appl) x3 x4) u2)).(\lambda (H29: (((iso (THead (Flat Appl) x3 -x4) u2) \to (\forall (P: Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) t0 -u2) (H8 u2 (pr3_sing c (THead (Flat Appl) x x4) (THead (Flat Appl) x x0) -(pr2_thin_dx c x0 x4 H24 x Appl) u2 (pr3_sing c (THead (Flat Appl) x3 x4) -(THead (Flat Appl) x x4) (pr2_head_1 c x x3 H23 (Flat Appl) x4) u2 H28)) -(\lambda (H30: (iso (THead (Flat Appl) x x0) u2)).(\lambda (P: Prop).(H29 -(iso_trans (THead (Flat Appl) x3 x4) (THead (Flat Appl) x x0) (iso_head x3 x -x4 x0 (Flat Appl)) u2 H30) P)))) (THead (Flat Appl) x1 u2) (pr3_pr2 c (THead -(Flat Appl) t0 u2) (THead (Flat Appl) x1 u2) (pr2_head_1 c t0 x1 H17 (Flat -Appl) u2)))))))) H26))) x2 H22))))))) H21)) (\lambda (H21: (ex4_4 T T T T -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x0 -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t4: T).(eq T x2 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda -(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: -B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4))))))))).(ex4_4_ind T T T -T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x0 -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t4: T).(eq T x2 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda -(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: -B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4))))))) (sn3 c (THead (Flat -Appl) x1 x2)) (\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda -(x6: T).(\lambda (H22: (eq T x0 (THead (Bind Abst) x3 x4))).(\lambda (H23: -(eq T x2 (THead (Bind Abbr) x5 x6))).(\lambda (H24: (pr2 c x x5)).(\lambda -(H25: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x4 -x6))))).(let H26 \def (eq_ind T x2 (\lambda (t: T).((eq T (THead (Flat Appl) -t0 (THead (Flat Appl) x x0)) (THead (Flat Appl) x1 t)) \to (\forall (P: -Prop).P))) H19 (THead (Bind Abbr) x5 x6) H23) in (eq_ind_r T (THead (Bind -Abbr) x5 x6) (\lambda (t: T).(sn3 c (THead (Flat Appl) x1 t))) (let H27 \def -(eq_ind T x0 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) -x t)) (THead (Flat Appl) x1 (THead (Bind Abbr) x5 x6))) \to (\forall (P: -Prop).P))) H26 (THead (Bind Abst) x3 x4) H22) in (let H28 \def (eq_ind T x0 -(\lambda (t: T).(\forall (t4: T).((((eq T (THead (Flat Appl) x t) t4) \to -(\forall (P: Prop).P))) \to ((pr3 c (THead (Flat Appl) x t) t4) \to (sn3 c -t4))))) H11 (THead (Bind Abst) x3 x4) H22) in (let H29 \def (eq_ind T x0 -(\lambda (t: T).(\forall (t4: T).((((eq T (THead (Flat Appl) x t) t4) \to -(\forall (P: Prop).P))) \to ((pr3 c (THead (Flat Appl) x t) t4) \to (\forall -(x7: T).(\forall (x8: T).((eq T t4 (THead (Flat Appl) x7 x8)) \to (\forall -(v3: T).((sn3 c v3) \to (((\forall (u2: T).((pr3 c t4 u2) \to ((((iso t4 u2) -\to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v3 u2)))))) \to -(sn3 c (THead (Flat Appl) v3 t4)))))))))))) H10 (THead (Bind Abst) x3 x4) -H22) in (let H30 \def (eq_ind T x0 (\lambda (t: T).(\forall (u2: T).((pr3 c -(THead (Flat Appl) x t) u2) \to ((((iso (THead (Flat Appl) x t) u2) \to -(\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) t0 u2)))))) H8 (THead -(Bind Abst) x3 x4) H22) in (let H31 \def (eq_ind T x0 (\lambda (t: -T).(\forall (t4: T).((((eq T t0 t4) \to (\forall (P: Prop).P))) \to ((pr3 c -t0 t4) \to (((\forall (u2: T).((pr3 c (THead (Flat Appl) x t) u2) \to ((((iso -(THead (Flat Appl) x t) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead -(Flat Appl) t4 u2)))))) \to (sn3 c (THead (Flat Appl) t4 (THead (Flat Appl) x -t)))))))) H9 (THead (Bind Abst) x3 x4) H22) in (sn3_pr3_trans c (THead (Flat -Appl) t0 (THead (Bind Abbr) x5 x6)) (H30 (THead (Bind Abbr) x5 x6) (pr3_sing -c (THead (Bind Abbr) x x4) (THead (Flat Appl) x (THead (Bind Abst) x3 x4)) -(pr2_free c (THead (Flat Appl) x (THead (Bind Abst) x3 x4)) (THead (Bind -Abbr) x x4) (pr0_beta x3 x x (pr0_refl x) x4 x4 (pr0_refl x4))) (THead (Bind -Abbr) x5 x6) (pr3_head_12 c x x5 (pr3_pr2 c x x5 H24) (Bind Abbr) x4 x6 -(pr3_pr2 (CHead c (Bind Abbr) x5) x4 x6 (H25 Abbr x5)))) (\lambda (H32: (iso -(THead (Flat Appl) x (THead (Bind Abst) x3 x4)) (THead (Bind Abbr) x5 -x6))).(\lambda (P: Prop).(let H33 \def (match H32 in iso return (\lambda (t: -T).(\lambda (t4: T).(\lambda (_: (iso t t4)).((eq T t (THead (Flat Appl) x -(THead (Bind Abst) x3 x4))) \to ((eq T t4 (THead (Bind Abbr) x5 x6)) \to -P))))) with [(iso_sort n1 n2) \Rightarrow (\lambda (H33: (eq T (TSort n1) -(THead (Flat Appl) x (THead (Bind Abst) x3 x4)))).(\lambda (H34: (eq T (TSort -n2) (THead (Bind Abbr) x5 x6))).((let H35 \def (eq_ind T (TSort n1) (\lambda -(e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I -(THead (Flat Appl) x (THead (Bind Abst) x3 x4)) H33) in (False_ind ((eq T -(TSort n2) (THead (Bind Abbr) x5 x6)) \to P) H35)) H34))) | (iso_lref i1 i2) -\Rightarrow (\lambda (H33: (eq T (TLRef i1) (THead (Flat Appl) x (THead (Bind -Abst) x3 x4)))).(\lambda (H34: (eq T (TLRef i2) (THead (Bind Abbr) x5 -x6))).((let H35 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) x -(THead (Bind Abst) x3 x4)) H33) in (False_ind ((eq T (TLRef i2) (THead (Bind -Abbr) x5 x6)) \to P) H35)) H34))) | (iso_head v4 v5 t4 t5 k) \Rightarrow -(\lambda (H33: (eq T (THead k v4 t4) (THead (Flat Appl) x (THead (Bind Abst) -x3 x4)))).(\lambda (H34: (eq T (THead k v5 t5) (THead (Bind Abbr) x5 -x6))).((let H35 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow t4 | (TLRef _) \Rightarrow t4 -| (THead _ _ t) \Rightarrow t])) (THead k v4 t4) (THead (Flat Appl) x (THead -(Bind Abst) x3 x4)) H33) in ((let H36 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v4 | -(TLRef _) \Rightarrow v4 | (THead _ t _) \Rightarrow t])) (THead k v4 t4) -(THead (Flat Appl) x (THead (Bind Abst) x3 x4)) H33) in ((let H37 \def -(f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with -[(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) -\Rightarrow k0])) (THead k v4 t4) (THead (Flat Appl) x (THead (Bind Abst) x3 -x4)) H33) in (eq_ind K (Flat Appl) (\lambda (k0: K).((eq T v4 x) \to ((eq T -t4 (THead (Bind Abst) x3 x4)) \to ((eq T (THead k0 v5 t5) (THead (Bind Abbr) -x5 x6)) \to P)))) (\lambda (H38: (eq T v4 x)).(eq_ind T x (\lambda (_: -T).((eq T t4 (THead (Bind Abst) x3 x4)) \to ((eq T (THead (Flat Appl) v5 t5) -(THead (Bind Abbr) x5 x6)) \to P))) (\lambda (H39: (eq T t4 (THead (Bind -Abst) x3 x4))).(eq_ind T (THead (Bind Abst) x3 x4) (\lambda (_: T).((eq T -(THead (Flat Appl) v5 t5) (THead (Bind Abbr) x5 x6)) \to P)) (\lambda (H40: -(eq T (THead (Flat Appl) v5 t5) (THead (Bind Abbr) x5 x6))).(let H41 \def -(eq_ind T (THead (Flat Appl) v5 t5) (\lambda (e: T).(match e in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind Abbr) x5 x6) H40) in (False_ind P H41))) t4 (sym_eq -T t4 (THead (Bind Abst) x3 x4) H39))) v4 (sym_eq T v4 x H38))) k (sym_eq K k -(Flat Appl) H37))) H36)) H35)) H34)))]) in (H33 (refl_equal T (THead (Flat -Appl) x (THead (Bind Abst) x3 x4))) (refl_equal T (THead (Bind Abbr) x5 -x6))))))) (THead (Flat Appl) x1 (THead (Bind Abbr) x5 x6)) (pr3_pr2 c (THead -(Flat Appl) t0 (THead (Bind Abbr) x5 x6)) (THead (Flat Appl) x1 (THead (Bind -Abbr) x5 x6)) (pr2_head_1 c t0 x1 H17 (Flat Appl) (THead (Bind Abbr) x5 -x6))))))))) x2 H23)))))))))) H21)) (\lambda (H21: (ex6_6 B T T T T T (\lambda -(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T x0 -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x2 (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind -B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T x0 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T -x2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 -y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) -(sn3 c (THead (Flat Appl) x1 x2)) (\lambda (x3: B).(\lambda (x4: T).(\lambda -(x5: T).(\lambda (x6: T).(\lambda (x7: T).(\lambda (x8: T).(\lambda (H22: -(not (eq B x3 Abst))).(\lambda (H23: (eq T x0 (THead (Bind x3) x4 -x5))).(\lambda (H24: (eq T x2 (THead (Bind x3) x8 (THead (Flat Appl) (lift (S -O) O x7) x6)))).(\lambda (H25: (pr2 c x x7)).(\lambda (H26: (pr2 c x4 -x8)).(\lambda (H27: (pr2 (CHead c (Bind x3) x8) x5 x6)).(let H28 \def (eq_ind -T x2 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) -(THead (Flat Appl) x1 t)) \to (\forall (P: Prop).P))) H19 (THead (Bind x3) x8 -(THead (Flat Appl) (lift (S O) O x7) x6)) H24) in (eq_ind_r T (THead (Bind -x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)) (\lambda (t: T).(sn3 c -(THead (Flat Appl) x1 t))) (let H29 \def (eq_ind T x0 (\lambda (t: T).((eq T -(THead (Flat Appl) t0 (THead (Flat Appl) x t)) (THead (Flat Appl) x1 (THead -(Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))) \to (\forall (P: -Prop).P))) H28 (THead (Bind x3) x4 x5) H23) in (let H30 \def (eq_ind T x0 -(\lambda (t: T).(\forall (t4: T).((((eq T (THead (Flat Appl) x t) t4) \to -(\forall (P: Prop).P))) \to ((pr3 c (THead (Flat Appl) x t) t4) \to (sn3 c -t4))))) H11 (THead (Bind x3) x4 x5) H23) in (let H31 \def (eq_ind T x0 -(\lambda (t: T).(\forall (t4: T).((((eq T (THead (Flat Appl) x t) t4) \to -(\forall (P: Prop).P))) \to ((pr3 c (THead (Flat Appl) x t) t4) \to (\forall -(x9: T).(\forall (x10: T).((eq T t4 (THead (Flat Appl) x9 x10)) \to (\forall -(v3: T).((sn3 c v3) \to (((\forall (u2: T).((pr3 c t4 u2) \to ((((iso t4 u2) -\to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v3 u2)))))) \to -(sn3 c (THead (Flat Appl) v3 t4)))))))))))) H10 (THead (Bind x3) x4 x5) H23) -in (let H32 \def (eq_ind T x0 (\lambda (t: T).(\forall (u2: T).((pr3 c (THead -(Flat Appl) x t) u2) \to ((((iso (THead (Flat Appl) x t) u2) \to (\forall (P: -Prop).P))) \to (sn3 c (THead (Flat Appl) t0 u2)))))) H8 (THead (Bind x3) x4 -x5) H23) in (let H33 \def (eq_ind T x0 (\lambda (t: T).(\forall (t4: -T).((((eq T t0 t4) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t4) \to -(((\forall (u2: T).((pr3 c (THead (Flat Appl) x t) u2) \to ((((iso (THead -(Flat Appl) x t) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat -Appl) t4 u2)))))) \to (sn3 c (THead (Flat Appl) t4 (THead (Flat Appl) x -t)))))))) H9 (THead (Bind x3) x4 x5) H23) in (sn3_pr3_trans c (THead (Flat -Appl) t0 (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) (H32 -(THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)) (pr3_sing c -(THead (Bind x3) x4 (THead (Flat Appl) (lift (S O) O x) x5)) (THead (Flat -Appl) x (THead (Bind x3) x4 x5)) (pr2_free c (THead (Flat Appl) x (THead -(Bind x3) x4 x5)) (THead (Bind x3) x4 (THead (Flat Appl) (lift (S O) O x) -x5)) (pr0_upsilon x3 H22 x x (pr0_refl x) x4 x4 (pr0_refl x4) x5 x5 (pr0_refl -x5))) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)) -(pr3_head_12 c x4 x8 (pr3_pr2 c x4 x8 H26) (Bind x3) (THead (Flat Appl) (lift -(S O) O x) x5) (THead (Flat Appl) (lift (S O) O x7) x6) (pr3_head_12 (CHead c -(Bind x3) x8) (lift (S O) O x) (lift (S O) O x7) (pr3_lift (CHead c (Bind x3) -x8) c (S O) O (drop_drop (Bind x3) O c c (drop_refl c) x8) x x7 (pr3_pr2 c x -x7 H25)) (Flat Appl) x5 x6 (pr3_pr2 (CHead (CHead c (Bind x3) x8) (Flat Appl) -(lift (S O) O x7)) x5 x6 (pr2_cflat (CHead c (Bind x3) x8) x5 x6 H27 Appl -(lift (S O) O x7)))))) (\lambda (H34: (iso (THead (Flat Appl) x (THead (Bind -x3) x4 x5)) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) -x6)))).(\lambda (P: Prop).(let H35 \def (match H34 in iso return (\lambda (t: -T).(\lambda (t4: T).(\lambda (_: (iso t t4)).((eq T t (THead (Flat Appl) x -(THead (Bind x3) x4 x5))) \to ((eq T t4 (THead (Bind x3) x8 (THead (Flat -Appl) (lift (S O) O x7) x6))) \to P))))) with [(iso_sort n1 n2) \Rightarrow -(\lambda (H35: (eq T (TSort n1) (THead (Flat Appl) x (THead (Bind x3) x4 -x5)))).(\lambda (H36: (eq T (TSort n2) (THead (Bind x3) x8 (THead (Flat Appl) -(lift (S O) O x7) x6)))).((let H37 \def (eq_ind T (TSort n1) (\lambda (e: -T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I -(THead (Flat Appl) x (THead (Bind x3) x4 x5)) H35) in (False_ind ((eq T -(TSort n2) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to -P) H37)) H36))) | (iso_lref i1 i2) \Rightarrow (\lambda (H35: (eq T (TLRef -i1) (THead (Flat Appl) x (THead (Bind x3) x4 x5)))).(\lambda (H36: (eq T -(TLRef i2) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) -x6)))).((let H37 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) x -(THead (Bind x3) x4 x5)) H35) in (False_ind ((eq T (TLRef i2) (THead (Bind -x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to P) H37)) H36))) | -(iso_head v4 v5 t4 t5 k) \Rightarrow (\lambda (H35: (eq T (THead k v4 t4) -(THead (Flat Appl) x (THead (Bind x3) x4 x5)))).(\lambda (H36: (eq T (THead k -v5 t5) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))).((let -H37 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow t4 | (TLRef _) \Rightarrow t4 | (THead _ _ t) -\Rightarrow t])) (THead k v4 t4) (THead (Flat Appl) x (THead (Bind x3) x4 -x5)) H35) in ((let H38 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow v4 | (TLRef _) \Rightarrow v4 -| (THead _ t _) \Rightarrow t])) (THead k v4 t4) (THead (Flat Appl) x (THead -(Bind x3) x4 x5)) H35) in ((let H39 \def (f_equal T K (\lambda (e: T).(match -e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) -\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k v4 t4) (THead (Flat -Appl) x (THead (Bind x3) x4 x5)) H35) in (eq_ind K (Flat Appl) (\lambda (k0: -K).((eq T v4 x) \to ((eq T t4 (THead (Bind x3) x4 x5)) \to ((eq T (THead k0 -v5 t5) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to -P)))) (\lambda (H40: (eq T v4 x)).(eq_ind T x (\lambda (_: T).((eq T t4 -(THead (Bind x3) x4 x5)) \to ((eq T (THead (Flat Appl) v5 t5) (THead (Bind -x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to P))) (\lambda (H41: (eq -T t4 (THead (Bind x3) x4 x5))).(eq_ind T (THead (Bind x3) x4 x5) (\lambda (_: -T).((eq T (THead (Flat Appl) v5 t5) (THead (Bind x3) x8 (THead (Flat Appl) -(lift (S O) O x7) x6))) \to P)) (\lambda (H42: (eq T (THead (Flat Appl) v5 -t5) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))).(let H43 -\def (eq_ind T (THead (Flat Appl) v5 t5) (\lambda (e: T).(match e in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)) -H42) in (False_ind P H43))) t4 (sym_eq T t4 (THead (Bind x3) x4 x5) H41))) v4 -(sym_eq T v4 x H40))) k (sym_eq K k (Flat Appl) H39))) H38)) H37)) H36)))]) -in (H35 (refl_equal T (THead (Flat Appl) x (THead (Bind x3) x4 x5))) -(refl_equal T (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) -x6)))))))) (THead (Flat Appl) x1 (THead (Bind x3) x8 (THead (Flat Appl) (lift -(S O) O x7) x6))) (pr3_pr2 c (THead (Flat Appl) t0 (THead (Bind x3) x8 (THead -(Flat Appl) (lift (S O) O x7) x6))) (THead (Flat Appl) x1 (THead (Bind x3) x8 -(THead (Flat Appl) (lift (S O) O x7) x6))) (pr2_head_1 c t0 x1 H17 (Flat -Appl) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))))))))) -x2 H24)))))))))))))) H21)) H20)) t3 H16))))))) H15)) (\lambda (H15: (ex4_4 T -T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Flat Appl) x x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind -Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c t0 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t4: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) -z1 t4))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (_: T).(eq T (THead (Flat Appl) x x0) (THead (Bind Abst) y1 -z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: -T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))))) (\lambda (_: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall -(u: T).(pr2 (CHead c (Bind b) u) z1 t4))))))) (sn3 c t3) (\lambda (x1: -T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H16: (eq T -(THead (Flat Appl) x x0) (THead (Bind Abst) x1 x2))).(\lambda (H17: (eq T t3 -(THead (Bind Abbr) x3 x4))).(\lambda (_: (pr2 c t0 x3)).(\lambda (_: -((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x2 x4))))).(let -H20 \def (eq_ind T t3 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead -(Flat Appl) x x0)) t) \to (\forall (P: Prop).P))) H12 (THead (Bind Abbr) x3 -x4) H17) in (eq_ind_r T (THead (Bind Abbr) x3 x4) (\lambda (t: T).(sn3 c t)) -(let H21 \def (eq_ind T (THead (Flat Appl) x x0) (\lambda (ee: T).(match ee -in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef -_) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return -(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind Abst) x1 x2) H16) in (False_ind (sn3 c (THead (Bind -Abbr) x3 x4)) H21)) t3 H17)))))))))) H15)) (\lambda (H15: (ex6_6 B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T -(THead (Flat Appl) x x0) (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) -z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) -y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B -b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Appl) x x0) (THead -(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c t0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) (sn3 c t3) -(\lambda (x1: B).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda -(x5: T).(\lambda (x6: T).(\lambda (_: (not (eq B x1 Abst))).(\lambda (H17: -(eq T (THead (Flat Appl) x x0) (THead (Bind x1) x2 x3))).(\lambda (H18: (eq T -t3 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))).(\lambda -(_: (pr2 c t0 x5)).(\lambda (_: (pr2 c x2 x6)).(\lambda (_: (pr2 (CHead c -(Bind x1) x6) x3 x4)).(let H22 \def (eq_ind T t3 (\lambda (t: T).((eq T -(THead (Flat Appl) t0 (THead (Flat Appl) x x0)) t) \to (\forall (P: -Prop).P))) H12 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)) -H18) in (eq_ind_r T (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) -x4)) (\lambda (t: T).(sn3 c t)) (let H23 \def (eq_ind T (THead (Flat Appl) x -x0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort -_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) -\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind x1) x2 x3) -H17) in (False_ind (sn3 c (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) -O x5) x4))) H23)) t3 H18)))))))))))))) H15)) H14)))))) t2 H3))))))))) v2 -H4))))))))) y H0))))) H))))). -(* COMMENTS -Initial nodes: 9317 -END *) - -theorem sn3_appl_beta: - \forall (c: C).(\forall (u: T).(\forall (v: T).(\forall (t: T).((sn3 c -(THead (Flat Appl) u (THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w) -\to (sn3 c (THead (Flat Appl) u (THead (Flat Appl) v (THead (Bind Abst) w -t)))))))))) -\def - \lambda (c: C).(\lambda (u: T).(\lambda (v: T).(\lambda (t: T).(\lambda (H: -(sn3 c (THead (Flat Appl) u (THead (Bind Abbr) v t)))).(\lambda (w: -T).(\lambda (H0: (sn3 c w)).(let H_x \def (sn3_gen_flat Appl c u (THead (Bind -Abbr) v t) H) in (let H1 \def H_x in (land_ind (sn3 c u) (sn3 c (THead (Bind -Abbr) v t)) (sn3 c (THead (Flat Appl) u (THead (Flat Appl) v (THead (Bind -Abst) w t)))) (\lambda (H2: (sn3 c u)).(\lambda (H3: (sn3 c (THead (Bind -Abbr) v t))).(sn3_appl_appl v (THead (Bind Abst) w t) c (sn3_beta c v t H3 w -H0) u H2 (\lambda (u2: T).(\lambda (H4: (pr3 c (THead (Flat Appl) v (THead -(Bind Abst) w t)) u2)).(\lambda (H5: (((iso (THead (Flat Appl) v (THead (Bind -Abst) w t)) u2) \to (\forall (P: Prop).P)))).(sn3_pr3_trans c (THead (Flat -Appl) u (THead (Bind Abbr) v t)) H (THead (Flat Appl) u u2) (pr3_thin_dx c -(THead (Bind Abbr) v t) u2 (pr3_iso_beta v w t c u2 H4 H5) u Appl)))))))) -H1))))))))). -(* COMMENTS -Initial nodes: 289 -END *) - -theorem sn3_appl_appls: - \forall (v1: T).(\forall (t1: T).(\forall (vs: TList).(let u1 \def (THeads -(Flat Appl) (TCons v1 vs) t1) in (\forall (c: C).((sn3 c u1) \to (\forall -(v2: T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) -\to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to -(sn3 c (THead (Flat Appl) v2 u1)))))))))) -\def - \lambda (v1: T).(\lambda (t1: T).(\lambda (vs: TList).(let u1 \def (THeads -(Flat Appl) (TCons v1 vs) t1) in (\lambda (c: C).(\lambda (H: (sn3 c (THead -(Flat Appl) v1 (THeads (Flat Appl) vs t1)))).(\lambda (v2: T).(\lambda (H0: -(sn3 c v2)).(\lambda (H1: ((\forall (u2: T).((pr3 c (THead (Flat Appl) v1 -(THeads (Flat Appl) vs t1)) u2) \to ((((iso (THead (Flat Appl) v1 (THeads -(Flat Appl) vs t1)) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat -Appl) v2 u2))))))).(sn3_appl_appl v1 (THeads (Flat Appl) vs t1) c H v2 H0 -H1))))))))). -(* COMMENTS -Initial nodes: 141 -END *) - -theorem sn3_appls_lref: - \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (us: -TList).((sns3 c us) \to (sn3 c (THeads (Flat Appl) us (TLRef i))))))) -\def - \lambda (c: C).(\lambda (i: nat).(\lambda (H: (nf2 c (TLRef i))).(\lambda -(us: TList).(TList_ind (\lambda (t: TList).((sns3 c t) \to (sn3 c (THeads -(Flat Appl) t (TLRef i))))) (\lambda (_: True).(sn3_nf2 c (TLRef i) H)) -(\lambda (t: T).(\lambda (t0: TList).(TList_ind (\lambda (t1: TList).((((sns3 -c t1) \to (sn3 c (THeads (Flat Appl) t1 (TLRef i))))) \to ((land (sn3 c t) -(sns3 c t1)) \to (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t1 (TLRef -i))))))) (\lambda (_: (((sns3 c TNil) \to (sn3 c (THeads (Flat Appl) TNil -(TLRef i)))))).(\lambda (H1: (land (sn3 c t) (sns3 c TNil))).(let H2 \def H1 -in (land_ind (sn3 c t) True (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) -TNil (TLRef i)))) (\lambda (H3: (sn3 c t)).(\lambda (_: True).(sn3_appl_lref -c i H t H3))) H2)))) (\lambda (t1: T).(\lambda (t2: TList).(\lambda (_: -(((((sns3 c t2) \to (sn3 c (THeads (Flat Appl) t2 (TLRef i))))) \to ((land -(sn3 c t) (sns3 c t2)) \to (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t2 -(TLRef i)))))))).(\lambda (H1: (((sns3 c (TCons t1 t2)) \to (sn3 c (THeads -(Flat Appl) (TCons t1 t2) (TLRef i)))))).(\lambda (H2: (land (sn3 c t) (sns3 -c (TCons t1 t2)))).(let H3 \def H2 in (land_ind (sn3 c t) (land (sn3 c t1) -(sns3 c t2)) (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) (TCons t1 t2) -(TLRef i)))) (\lambda (H4: (sn3 c t)).(\lambda (H5: (land (sn3 c t1) (sns3 c -t2))).(land_ind (sn3 c t1) (sns3 c t2) (sn3 c (THead (Flat Appl) t (THeads -(Flat Appl) (TCons t1 t2) (TLRef i)))) (\lambda (H6: (sn3 c t1)).(\lambda -(H7: (sns3 c t2)).(sn3_appl_appls t1 (TLRef i) t2 c (H1 (conj (sn3 c t1) -(sns3 c t2) H6 H7)) t H4 (\lambda (u2: T).(\lambda (H8: (pr3 c (THeads (Flat -Appl) (TCons t1 t2) (TLRef i)) u2)).(\lambda (H9: (((iso (THeads (Flat Appl) -(TCons t1 t2) (TLRef i)) u2) \to (\forall (P: Prop).P)))).(H9 -(nf2_iso_appls_lref c i H (TCons t1 t2) u2 H8) (sn3 c (THead (Flat Appl) t -u2))))))))) H5))) H3))))))) t0))) us)))). -(* COMMENTS -Initial nodes: 577 -END *) - -theorem sn3_appls_cast: - \forall (c: C).(\forall (vs: TList).(\forall (u: T).((sn3 c (THeads (Flat -Appl) vs u)) \to (\forall (t: T).((sn3 c (THeads (Flat Appl) vs t)) \to (sn3 -c (THeads (Flat Appl) vs (THead (Flat Cast) u t)))))))) -\def - \lambda (c: C).(\lambda (vs: TList).(TList_ind (\lambda (t: TList).(\forall -(u: T).((sn3 c (THeads (Flat Appl) t u)) \to (\forall (t0: T).((sn3 c (THeads -(Flat Appl) t t0)) \to (sn3 c (THeads (Flat Appl) t (THead (Flat Cast) u -t0)))))))) (\lambda (u: T).(\lambda (H: (sn3 c u)).(\lambda (t: T).(\lambda -(H0: (sn3 c t)).(sn3_cast c u H t H0))))) (\lambda (t: T).(\lambda (t0: -TList).(TList_ind (\lambda (t1: TList).(((\forall (u: T).((sn3 c (THeads -(Flat Appl) t1 u)) \to (\forall (t2: T).((sn3 c (THeads (Flat Appl) t1 t2)) -\to (sn3 c (THeads (Flat Appl) t1 (THead (Flat Cast) u t2)))))))) \to -(\forall (u: T).((sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t1 u))) \to -(\forall (t2: T).((sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t1 t2))) -\to (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t1 (THead (Flat Cast) u -t2)))))))))) (\lambda (_: ((\forall (u: T).((sn3 c (THeads (Flat Appl) TNil -u)) \to (\forall (t1: T).((sn3 c (THeads (Flat Appl) TNil t1)) \to (sn3 c -(THeads (Flat Appl) TNil (THead (Flat Cast) u t1))))))))).(\lambda (u: -T).(\lambda (H0: (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) TNil -u)))).(\lambda (t1: T).(\lambda (H1: (sn3 c (THead (Flat Appl) t (THeads -(Flat Appl) TNil t1)))).(sn3_appl_cast c t u H0 t1 H1)))))) (\lambda (t1: -T).(\lambda (t2: TList).(\lambda (_: ((((\forall (u: T).((sn3 c (THeads (Flat -Appl) t2 u)) \to (\forall (t3: T).((sn3 c (THeads (Flat Appl) t2 t3)) \to -(sn3 c (THeads (Flat Appl) t2 (THead (Flat Cast) u t3)))))))) \to (\forall -(u: T).((sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t2 u))) \to (\forall -(t3: T).((sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t2 t3))) \to (sn3 c -(THead (Flat Appl) t (THeads (Flat Appl) t2 (THead (Flat Cast) u -t3))))))))))).(\lambda (H0: ((\forall (u: T).((sn3 c (THeads (Flat Appl) -(TCons t1 t2) u)) \to (\forall (t3: T).((sn3 c (THeads (Flat Appl) (TCons t1 -t2) t3)) \to (sn3 c (THeads (Flat Appl) (TCons t1 t2) (THead (Flat Cast) u -t3))))))))).(\lambda (u: T).(\lambda (H1: (sn3 c (THead (Flat Appl) t (THeads -(Flat Appl) (TCons t1 t2) u)))).(\lambda (t3: T).(\lambda (H2: (sn3 c (THead -(Flat Appl) t (THeads (Flat Appl) (TCons t1 t2) t3)))).(let H_x \def -(sn3_gen_flat Appl c t (THeads (Flat Appl) (TCons t1 t2) t3) H2) in (let H3 -\def H_x in (land_ind (sn3 c t) (sn3 c (THead (Flat Appl) t1 (THeads (Flat -Appl) t2 t3))) (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) (TCons t1 t2) -(THead (Flat Cast) u t3)))) (\lambda (_: (sn3 c t)).(\lambda (H5: (sn3 c -(THead (Flat Appl) t1 (THeads (Flat Appl) t2 t3)))).(let H6 \def H5 in (let -H_x0 \def (sn3_gen_flat Appl c t (THeads (Flat Appl) (TCons t1 t2) u) H1) in -(let H7 \def H_x0 in (land_ind (sn3 c t) (sn3 c (THead (Flat Appl) t1 (THeads -(Flat Appl) t2 u))) (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) (TCons t1 -t2) (THead (Flat Cast) u t3)))) (\lambda (H8: (sn3 c t)).(\lambda (H9: (sn3 c -(THead (Flat Appl) t1 (THeads (Flat Appl) t2 u)))).(let H10 \def H9 in -(sn3_appl_appls t1 (THead (Flat Cast) u t3) t2 c (H0 u H10 t3 H6) t H8 -(\lambda (u2: T).(\lambda (H11: (pr3 c (THeads (Flat Appl) (TCons t1 t2) -(THead (Flat Cast) u t3)) u2)).(\lambda (H12: (((iso (THeads (Flat Appl) -(TCons t1 t2) (THead (Flat Cast) u t3)) u2) \to (\forall (P: -Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) t (THeads (Flat Appl) (TCons -t1 t2) t3)) H2 (THead (Flat Appl) t u2) (pr3_thin_dx c (THeads (Flat Appl) -(TCons t1 t2) t3) u2 (pr3_iso_appls_cast c u t3 (TCons t1 t2) u2 H11 H12) t -Appl))))))))) H7)))))) H3))))))))))) t0))) vs)). -(* COMMENTS -Initial nodes: 1025 -END *) - -theorem sn3_appls_bind: - \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (u: -T).((sn3 c u) \to (\forall (vs: TList).(\forall (t: T).((sn3 (CHead c (Bind -b) u) (THeads (Flat Appl) (lifts (S O) O vs) t)) \to (sn3 c (THeads (Flat -Appl) vs (THead (Bind b) u t)))))))))) -\def - \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (c: C).(\lambda -(u: T).(\lambda (H0: (sn3 c u)).(\lambda (vs: TList).(TList_ind (\lambda (t: -TList).(\forall (t0: T).((sn3 (CHead c (Bind b) u) (THeads (Flat Appl) (lifts -(S O) O t) t0)) \to (sn3 c (THeads (Flat Appl) t (THead (Bind b) u t0)))))) -(\lambda (t: T).(\lambda (H1: (sn3 (CHead c (Bind b) u) t)).(sn3_bind b c u -H0 t H1))) (\lambda (v: T).(\lambda (vs0: TList).(TList_ind (\lambda (t: -TList).(((\forall (t0: T).((sn3 (CHead c (Bind b) u) (THeads (Flat Appl) -(lifts (S O) O t) t0)) \to (sn3 c (THeads (Flat Appl) t (THead (Bind b) u -t0)))))) \to (\forall (t0: T).((sn3 (CHead c (Bind b) u) (THead (Flat Appl) -(lift (S O) O v) (THeads (Flat Appl) (lifts (S O) O t) t0))) \to (sn3 c -(THead (Flat Appl) v (THeads (Flat Appl) t (THead (Bind b) u t0)))))))) -(\lambda (_: ((\forall (t: T).((sn3 (CHead c (Bind b) u) (THeads (Flat Appl) -(lifts (S O) O TNil) t)) \to (sn3 c (THeads (Flat Appl) TNil (THead (Bind b) -u t))))))).(\lambda (t: T).(\lambda (H2: (sn3 (CHead c (Bind b) u) (THead -(Flat Appl) (lift (S O) O v) (THeads (Flat Appl) (lifts (S O) O TNil) -t)))).(sn3_appl_bind b H c u H0 t v H2)))) (\lambda (t: T).(\lambda (t0: -TList).(\lambda (_: ((((\forall (t1: T).((sn3 (CHead c (Bind b) u) (THeads -(Flat Appl) (lifts (S O) O t0) t1)) \to (sn3 c (THeads (Flat Appl) t0 (THead -(Bind b) u t1)))))) \to (\forall (t1: T).((sn3 (CHead c (Bind b) u) (THead -(Flat Appl) (lift (S O) O v) (THeads (Flat Appl) (lifts (S O) O t0) t1))) \to -(sn3 c (THead (Flat Appl) v (THeads (Flat Appl) t0 (THead (Bind b) u -t1))))))))).(\lambda (H2: ((\forall (t1: T).((sn3 (CHead c (Bind b) u) -(THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1)) \to (sn3 c (THeads -(Flat Appl) (TCons t t0) (THead (Bind b) u t1))))))).(\lambda (t1: -T).(\lambda (H3: (sn3 (CHead c (Bind b) u) (THead (Flat Appl) (lift (S O) O -v) (THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1)))).(let H_x \def -(sn3_gen_flat Appl (CHead c (Bind b) u) (lift (S O) O v) (THeads (Flat Appl) -(lifts (S O) O (TCons t t0)) t1) H3) in (let H4 \def H_x in (land_ind (sn3 -(CHead c (Bind b) u) (lift (S O) O v)) (sn3 (CHead c (Bind b) u) (THead (Flat -Appl) (lift (S O) O t) (THeads (Flat Appl) (lifts (S O) O t0) t1))) (sn3 c -(THead (Flat Appl) v (THeads (Flat Appl) (TCons t t0) (THead (Bind b) u -t1)))) (\lambda (H5: (sn3 (CHead c (Bind b) u) (lift (S O) O v))).(\lambda -(H6: (sn3 (CHead c (Bind b) u) (THead (Flat Appl) (lift (S O) O t) (THeads -(Flat Appl) (lifts (S O) O t0) t1)))).(let H_y \def (sn3_gen_lift (CHead c -(Bind b) u) v (S O) O H5 c) in (sn3_appl_appls t (THead (Bind b) u t1) t0 c -(H2 t1 H6) v (H_y (drop_drop (Bind b) O c c (drop_refl c) u)) (\lambda (u2: -T).(\lambda (H7: (pr3 c (THeads (Flat Appl) (TCons t t0) (THead (Bind b) u -t1)) u2)).(\lambda (H8: (((iso (THeads (Flat Appl) (TCons t t0) (THead (Bind -b) u t1)) u2) \to (\forall (P: Prop).P)))).(let H9 \def (pr3_iso_appls_bind b -H (TCons t t0) u t1 c u2 H7 H8) in (sn3_pr3_trans c (THead (Flat Appl) v -(THead (Bind b) u (THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1))) -(sn3_appl_bind b H c u H0 (THeads (Flat Appl) (lifts (S O) O (TCons t t0)) -t1) v H3) (THead (Flat Appl) v u2) (pr3_flat c v v (pr3_refl c v) (THead -(Bind b) u (THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1)) u2 H9 -Appl)))))))))) H4))))))))) vs0))) vs)))))). -(* COMMENTS -Initial nodes: 1143 -END *) - -theorem sn3_appls_beta: - \forall (c: C).(\forall (v: T).(\forall (t: T).(\forall (us: TList).((sn3 c -(THeads (Flat Appl) us (THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c -w) \to (sn3 c (THeads (Flat Appl) us (THead (Flat Appl) v (THead (Bind Abst) -w t)))))))))) -\def - \lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (us: -TList).(TList_ind (\lambda (t0: TList).((sn3 c (THeads (Flat Appl) t0 (THead -(Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THeads (Flat -Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t)))))))) (\lambda (H: -(sn3 c (THead (Bind Abbr) v t))).(\lambda (w: T).(\lambda (H0: (sn3 c -w)).(sn3_beta c v t H w H0)))) (\lambda (u: T).(\lambda (us0: -TList).(TList_ind (\lambda (t0: TList).((((sn3 c (THeads (Flat Appl) t0 -(THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THeads -(Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t)))))))) \to ((sn3 -c (THead (Flat Appl) u (THeads (Flat Appl) t0 (THead (Bind Abbr) v t)))) \to -(\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) u (THeads (Flat -Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t)))))))))) (\lambda (_: -(((sn3 c (THeads (Flat Appl) TNil (THead (Bind Abbr) v t))) \to (\forall (w: -T).((sn3 c w) \to (sn3 c (THeads (Flat Appl) TNil (THead (Flat Appl) v (THead -(Bind Abst) w t))))))))).(\lambda (H0: (sn3 c (THead (Flat Appl) u (THeads -(Flat Appl) TNil (THead (Bind Abbr) v t))))).(\lambda (w: T).(\lambda (H1: -(sn3 c w)).(sn3_appl_beta c u v t H0 w H1))))) (\lambda (t0: T).(\lambda (t1: -TList).(\lambda (_: (((((sn3 c (THeads (Flat Appl) t1 (THead (Bind Abbr) v -t))) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THeads (Flat Appl) t1 (THead -(Flat Appl) v (THead (Bind Abst) w t)))))))) \to ((sn3 c (THead (Flat Appl) u -(THeads (Flat Appl) t1 (THead (Bind Abbr) v t)))) \to (\forall (w: T).((sn3 c -w) \to (sn3 c (THead (Flat Appl) u (THeads (Flat Appl) t1 (THead (Flat Appl) -v (THead (Bind Abst) w t))))))))))).(\lambda (H0: (((sn3 c (THeads (Flat -Appl) (TCons t0 t1) (THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w) -\to (sn3 c (THeads (Flat Appl) (TCons t0 t1) (THead (Flat Appl) v (THead -(Bind Abst) w t))))))))).(\lambda (H1: (sn3 c (THead (Flat Appl) u (THeads -(Flat Appl) (TCons t0 t1) (THead (Bind Abbr) v t))))).(\lambda (w: -T).(\lambda (H2: (sn3 c w)).(let H_x \def (sn3_gen_flat Appl c u (THeads -(Flat Appl) (TCons t0 t1) (THead (Bind Abbr) v t)) H1) in (let H3 \def H_x in -(land_ind (sn3 c u) (sn3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 -(THead (Bind Abbr) v t)))) (sn3 c (THead (Flat Appl) u (THeads (Flat Appl) -(TCons t0 t1) (THead (Flat Appl) v (THead (Bind Abst) w t))))) (\lambda (H4: -(sn3 c u)).(\lambda (H5: (sn3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 -(THead (Bind Abbr) v t))))).(sn3_appl_appls t0 (THead (Flat Appl) v (THead -(Bind Abst) w t)) t1 c (H0 H5 w H2) u H4 (\lambda (u2: T).(\lambda (H6: (pr3 -c (THeads (Flat Appl) (TCons t0 t1) (THead (Flat Appl) v (THead (Bind Abst) w -t))) u2)).(\lambda (H7: (((iso (THeads (Flat Appl) (TCons t0 t1) (THead (Flat -Appl) v (THead (Bind Abst) w t))) u2) \to (\forall (P: Prop).P)))).(let H8 -\def (pr3_iso_appls_beta (TCons t0 t1) v w t c u2 H6 H7) in (sn3_pr3_trans c -(THead (Flat Appl) u (THeads (Flat Appl) (TCons t0 t1) (THead (Bind Abbr) v -t))) H1 (THead (Flat Appl) u u2) (pr3_thin_dx c (THeads (Flat Appl) (TCons t0 -t1) (THead (Bind Abbr) v t)) u2 H8 u Appl))))))))) H3)))))))))) us0))) us)))). -(* COMMENTS -Initial nodes: 987 -END *) - -theorem sn3_lift: - \forall (d: C).(\forall (t: T).((sn3 d t) \to (\forall (c: C).(\forall (h: -nat).(\forall (i: nat).((drop h i c d) \to (sn3 c (lift h i t)))))))) -\def - \lambda (d: C).(\lambda (t: T).(\lambda (H: (sn3 d t)).(sn3_ind d (\lambda -(t0: T).(\forall (c: C).(\forall (h: nat).(\forall (i: nat).((drop h i c d) -\to (sn3 c (lift h i t0))))))) (\lambda (t1: T).(\lambda (_: ((\forall (t2: -T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 d t1 t2) \to (sn3 d -t2)))))).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: -Prop).P))) \to ((pr3 d t1 t2) \to (\forall (c: C).(\forall (h: nat).(\forall -(i: nat).((drop h i c d) \to (sn3 c (lift h i t2))))))))))).(\lambda (c: -C).(\lambda (h: nat).(\lambda (i: nat).(\lambda (H2: (drop h i c -d)).(sn3_pr2_intro c (lift h i t1) (\lambda (t2: T).(\lambda (H3: (((eq T -(lift h i t1) t2) \to (\forall (P: Prop).P)))).(\lambda (H4: (pr2 c (lift h i -t1) t2)).(let H5 \def (pr2_gen_lift c t1 t2 h i H4 d H2) in (ex2_ind T -(\lambda (t3: T).(eq T t2 (lift h i t3))) (\lambda (t3: T).(pr2 d t1 t3)) -(sn3 c t2) (\lambda (x: T).(\lambda (H6: (eq T t2 (lift h i x))).(\lambda -(H7: (pr2 d t1 x)).(let H8 \def (eq_ind T t2 (\lambda (t0: T).((eq T (lift h -i t1) t0) \to (\forall (P: Prop).P))) H3 (lift h i x) H6) in (eq_ind_r T -(lift h i x) (\lambda (t0: T).(sn3 c t0)) (H1 x (\lambda (H9: (eq T t1 -x)).(\lambda (P: Prop).(let H10 \def (eq_ind_r T x (\lambda (t0: T).((eq T -(lift h i t1) (lift h i t0)) \to (\forall (P0: Prop).P0))) H8 t1 H9) in (let -H11 \def (eq_ind_r T x (\lambda (t0: T).(pr2 d t1 t0)) H7 t1 H9) in (H10 -(refl_equal T (lift h i t1)) P))))) (pr3_pr2 d t1 x H7) c h i H2) t2 H6))))) -H5))))))))))))) t H))). -(* COMMENTS -Initial nodes: 439 -END *) - -theorem sn3_abbr: - \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c -(CHead d (Bind Abbr) v)) \to ((sn3 d v) \to (sn3 c (TLRef i))))))) -\def - \lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda -(H: (getl i c (CHead d (Bind Abbr) v))).(\lambda (H0: (sn3 d -v)).(sn3_pr2_intro c (TLRef i) (\lambda (t2: T).(\lambda (H1: (((eq T (TLRef -i) t2) \to (\forall (P: Prop).P)))).(\lambda (H2: (pr2 c (TLRef i) t2)).(let -H3 \def (pr2_gen_lref c t2 i H2) in (or_ind (eq T t2 (TLRef i)) (ex2_2 C T -(\lambda (d0: C).(\lambda (u: T).(getl i c (CHead d0 (Bind Abbr) u)))) -(\lambda (_: C).(\lambda (u: T).(eq T t2 (lift (S i) O u))))) (sn3 c t2) -(\lambda (H4: (eq T t2 (TLRef i))).(let H5 \def (eq_ind T t2 (\lambda (t: -T).((eq T (TLRef i) t) \to (\forall (P: Prop).P))) H1 (TLRef i) H4) in -(eq_ind_r T (TLRef i) (\lambda (t: T).(sn3 c t)) (H5 (refl_equal T (TLRef i)) -(sn3 c (TLRef i))) t2 H4))) (\lambda (H4: (ex2_2 C T (\lambda (d0: -C).(\lambda (u: T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_: -C).(\lambda (u: T).(eq T t2 (lift (S i) O u)))))).(ex2_2_ind C T (\lambda -(d0: C).(\lambda (u: T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_: -C).(\lambda (u: T).(eq T t2 (lift (S i) O u)))) (sn3 c t2) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (H5: (getl i c (CHead x0 (Bind Abbr) -x1))).(\lambda (H6: (eq T t2 (lift (S i) O x1))).(let H7 \def (eq_ind T t2 -(\lambda (t: T).((eq T (TLRef i) t) \to (\forall (P: Prop).P))) H1 (lift (S -i) O x1) H6) in (eq_ind_r T (lift (S i) O x1) (\lambda (t: T).(sn3 c t)) (let -H8 \def (eq_ind C (CHead d (Bind Abbr) v) (\lambda (c0: C).(getl i c c0)) H -(CHead x0 (Bind Abbr) x1) (getl_mono c (CHead d (Bind Abbr) v) i H (CHead x0 -(Bind Abbr) x1) H5)) in (let H9 \def (f_equal C C (\lambda (e: C).(match e in -C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) -\Rightarrow c0])) (CHead d (Bind Abbr) v) (CHead x0 (Bind Abbr) x1) -(getl_mono c (CHead d (Bind Abbr) v) i H (CHead x0 (Bind Abbr) x1) H5)) in -((let H10 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: -C).T) with [(CSort _) \Rightarrow v | (CHead _ _ t) \Rightarrow t])) (CHead d -(Bind Abbr) v) (CHead x0 (Bind Abbr) x1) (getl_mono c (CHead d (Bind Abbr) v) -i H (CHead x0 (Bind Abbr) x1) H5)) in (\lambda (H11: (eq C d x0)).(let H12 -\def (eq_ind_r T x1 (\lambda (t: T).(getl i c (CHead x0 (Bind Abbr) t))) H8 v -H10) in (eq_ind T v (\lambda (t: T).(sn3 c (lift (S i) O t))) (let H13 \def -(eq_ind_r C x0 (\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr) v))) H12 d -H11) in (sn3_lift d v H0 c (S i) O (getl_drop Abbr c d v i H13))) x1 H10)))) -H9))) t2 H6)))))) H4)) H3))))))))))). -(* COMMENTS -Initial nodes: 743 -END *) - -theorem sn3_appls_abbr: - \forall (c: C).(\forall (d: C).(\forall (w: T).(\forall (i: nat).((getl i c -(CHead d (Bind Abbr) w)) \to (\forall (vs: TList).((sn3 c (THeads (Flat Appl) -vs (lift (S i) O w))) \to (sn3 c (THeads (Flat Appl) vs (TLRef i))))))))) -\def - \lambda (c: C).(\lambda (d: C).(\lambda (w: T).(\lambda (i: nat).(\lambda -(H: (getl i c (CHead d (Bind Abbr) w))).(\lambda (vs: TList).(TList_ind -(\lambda (t: TList).((sn3 c (THeads (Flat Appl) t (lift (S i) O w))) \to (sn3 -c (THeads (Flat Appl) t (TLRef i))))) (\lambda (H0: (sn3 c (lift (S i) O -w))).(let H_y \def (sn3_gen_lift c w (S i) O H0 d (getl_drop Abbr c d w i H)) -in (sn3_abbr c d w i H H_y))) (\lambda (v: T).(\lambda (vs0: -TList).(TList_ind (\lambda (t: TList).((((sn3 c (THeads (Flat Appl) t (lift -(S i) O w))) \to (sn3 c (THeads (Flat Appl) t (TLRef i))))) \to ((sn3 c -(THead (Flat Appl) v (THeads (Flat Appl) t (lift (S i) O w)))) \to (sn3 c -(THead (Flat Appl) v (THeads (Flat Appl) t (TLRef i))))))) (\lambda (_: -(((sn3 c (THeads (Flat Appl) TNil (lift (S i) O w))) \to (sn3 c (THeads (Flat -Appl) TNil (TLRef i)))))).(\lambda (H1: (sn3 c (THead (Flat Appl) v (THeads -(Flat Appl) TNil (lift (S i) O w))))).(sn3_appl_abbr c d w i H v H1))) -(\lambda (t: T).(\lambda (t0: TList).(\lambda (_: (((((sn3 c (THeads (Flat -Appl) t0 (lift (S i) O w))) \to (sn3 c (THeads (Flat Appl) t0 (TLRef i))))) -\to ((sn3 c (THead (Flat Appl) v (THeads (Flat Appl) t0 (lift (S i) O w)))) -\to (sn3 c (THead (Flat Appl) v (THeads (Flat Appl) t0 (TLRef -i)))))))).(\lambda (H1: (((sn3 c (THeads (Flat Appl) (TCons t t0) (lift (S i) -O w))) \to (sn3 c (THeads (Flat Appl) (TCons t t0) (TLRef i)))))).(\lambda -(H2: (sn3 c (THead (Flat Appl) v (THeads (Flat Appl) (TCons t t0) (lift (S i) -O w))))).(let H_x \def (sn3_gen_flat Appl c v (THeads (Flat Appl) (TCons t -t0) (lift (S i) O w)) H2) in (let H3 \def H_x in (land_ind (sn3 c v) (sn3 c -(THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O w)))) (sn3 c (THead -(Flat Appl) v (THeads (Flat Appl) (TCons t t0) (TLRef i)))) (\lambda (H4: -(sn3 c v)).(\lambda (H5: (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 -(lift (S i) O w))))).(sn3_appl_appls t (TLRef i) t0 c (H1 H5) v H4 (\lambda -(u2: T).(\lambda (H6: (pr3 c (THeads (Flat Appl) (TCons t t0) (TLRef i)) -u2)).(\lambda (H7: (((iso (THeads (Flat Appl) (TCons t t0) (TLRef i)) u2) \to -(\forall (P: Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) v (THeads (Flat -Appl) (TCons t t0) (lift (S i) O w))) H2 (THead (Flat Appl) v u2) -(pr3_thin_dx c (THeads (Flat Appl) (TCons t t0) (lift (S i) O w)) u2 -(pr3_iso_appls_abbr c d w i H (TCons t t0) u2 H6 H7) v Appl)))))))) -H3)))))))) vs0))) vs)))))). -(* COMMENTS -Initial nodes: 797 -END *) - -theorem sns3_lifts: - \forall (c: C).(\forall (d: C).(\forall (h: nat).(\forall (i: nat).((drop h -i c d) \to (\forall (ts: TList).((sns3 d ts) \to (sns3 c (lifts h i ts)))))))) -\def - \lambda (c: C).(\lambda (d: C).(\lambda (h: nat).(\lambda (i: nat).(\lambda -(H: (drop h i c d)).(\lambda (ts: TList).(TList_ind (\lambda (t: -TList).((sns3 d t) \to (sns3 c (lifts h i t)))) (\lambda (H0: True).H0) -(\lambda (t: T).(\lambda (t0: TList).(\lambda (H0: (((sns3 d t0) \to (sns3 c -(lifts h i t0))))).(\lambda (H1: (land (sn3 d t) (sns3 d t0))).(let H2 \def -H1 in (land_ind (sn3 d t) (sns3 d t0) (land (sn3 c (lift h i t)) (sns3 c -(lifts h i t0))) (\lambda (H3: (sn3 d t)).(\lambda (H4: (sns3 d t0)).(conj -(sn3 c (lift h i t)) (sns3 c (lifts h i t0)) (sn3_lift d t H3 c h i H) (H0 -H4)))) H2)))))) ts)))))). -(* COMMENTS -Initial nodes: 185 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/spare.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/spare.ma deleted file mode 100644 index 95d2c24ee..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/spare.ma +++ /dev/null @@ -1,38 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/theory.ma". - -axiom pc3_gen_appls_sort_abst: - \forall (c: C).(\forall (vs: TList).(\forall (w: T).(\forall (u: T).(\forall -(n: nat).((pc3 c (THeads (Flat Appl) vs (TSort n)) (THead (Bind Abst) w u)) -\to False))))) -. - -axiom pc3_gen_appls_lref_abst: - \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c -(CHead d (Bind Abst) v)) \to (\forall (vs: TList).(\forall (w: T).(\forall -(u: T).((pc3 c (THeads (Flat Appl) vs (TLRef i)) (THead (Bind Abst) w u)) \to -False)))))))) -. - -axiom pc3_gen_appls_lref_sort: - \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c -(CHead d (Bind Abst) v)) \to (\forall (vs: TList).(\forall (ws: -TList).(\forall (n: nat).((pc3 c (THeads (Flat Appl) vs (TLRef i)) (THeads -(Flat Appl) ws (TSort n))) \to False)))))))) -. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/sty0/defs.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/sty0/defs.ma deleted file mode 100644 index 6468d9dbb..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/sty0/defs.ma +++ /dev/null @@ -1,39 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/G/defs.ma". - -include "Basic-1/getl/defs.ma". - -inductive sty0 (g: G): C \to (T \to (T \to Prop)) \def -| sty0_sort: \forall (c: C).(\forall (n: nat).(sty0 g c (TSort n) (TSort -(next g n)))) -| sty0_abbr: \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: -nat).((getl i c (CHead d (Bind Abbr) v)) \to (\forall (w: T).((sty0 g d v w) -\to (sty0 g c (TLRef i) (lift (S i) O w)))))))) -| sty0_abst: \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: -nat).((getl i c (CHead d (Bind Abst) v)) \to (\forall (w: T).((sty0 g d v w) -\to (sty0 g c (TLRef i) (lift (S i) O v)))))))) -| sty0_bind: \forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t1: -T).(\forall (t2: T).((sty0 g (CHead c (Bind b) v) t1 t2) \to (sty0 g c (THead -(Bind b) v t1) (THead (Bind b) v t2))))))) -| sty0_appl: \forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall (t2: -T).((sty0 g c t1 t2) \to (sty0 g c (THead (Flat Appl) v t1) (THead (Flat -Appl) v t2)))))) -| sty0_cast: \forall (c: C).(\forall (v1: T).(\forall (v2: T).((sty0 g c v1 -v2) \to (\forall (t1: T).(\forall (t2: T).((sty0 g c t1 t2) \to (sty0 g c -(THead (Flat Cast) v1 t1) (THead (Flat Cast) v2 t2)))))))). - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/sty0/fwd.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/sty0/fwd.ma deleted file mode 100644 index 134ec3c10..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/sty0/fwd.ma +++ /dev/null @@ -1,562 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/sty0/defs.ma". - -theorem sty0_gen_sort: - \forall (g: G).(\forall (c: C).(\forall (x: T).(\forall (n: nat).((sty0 g c -(TSort n) x) \to (eq T x (TSort (next g n))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda -(H: (sty0 g c (TSort n) x)).(insert_eq T (TSort n) (\lambda (t: T).(sty0 g c -t x)) (\lambda (_: T).(eq T x (TSort (next g n)))) (\lambda (y: T).(\lambda -(H0: (sty0 g c y x)).(sty0_ind g (\lambda (_: C).(\lambda (t: T).(\lambda -(t0: T).((eq T t (TSort n)) \to (eq T t0 (TSort (next g n))))))) (\lambda (_: -C).(\lambda (n0: nat).(\lambda (H1: (eq T (TSort n0) (TSort n))).(let H2 \def -(f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with -[(TSort n1) \Rightarrow n1 | (TLRef _) \Rightarrow n0 | (THead _ _ _) -\Rightarrow n0])) (TSort n0) (TSort n) H1) in (eq_ind_r nat n (\lambda (n1: -nat).(eq T (TSort (next g n1)) (TSort (next g n)))) (refl_equal T (TSort -(next g n))) n0 H2))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (v: -T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind Abbr) -v))).(\lambda (w: T).(\lambda (_: (sty0 g d v w)).(\lambda (_: (((eq T v -(TSort n)) \to (eq T w (TSort (next g n)))))).(\lambda (H4: (eq T (TLRef i) -(TSort n))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (TSort n) H4) in -(False_ind (eq T (lift (S i) O w) (TSort (next g n))) H5))))))))))) (\lambda -(c0: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (_: (getl -i c0 (CHead d (Bind Abst) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v -w)).(\lambda (_: (((eq T v (TSort n)) \to (eq T w (TSort (next g -n)))))).(\lambda (H4: (eq T (TLRef i) (TSort n))).(let H5 \def (eq_ind T -(TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) -\Rightarrow False])) I (TSort n) H4) in (False_ind (eq T (lift (S i) O v) -(TSort (next g n))) H5))))))))))) (\lambda (b: B).(\lambda (c0: C).(\lambda -(v: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (sty0 g (CHead c0 (Bind -b) v) t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2 (TSort (next g -n)))))).(\lambda (H3: (eq T (THead (Bind b) v t1) (TSort n))).(let H4 \def -(eq_ind T (THead (Bind b) v t1) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H3) in -(False_ind (eq T (THead (Bind b) v t2) (TSort (next g n))) H4)))))))))) -(\lambda (c0: C).(\lambda (v: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(_: (sty0 g c0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2 (TSort -(next g n)))))).(\lambda (H3: (eq T (THead (Flat Appl) v t1) (TSort n))).(let -H4 \def (eq_ind T (THead (Flat Appl) v t1) (\lambda (ee: T).(match ee in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H3) in -(False_ind (eq T (THead (Flat Appl) v t2) (TSort (next g n))) H4))))))))) -(\lambda (c0: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (sty0 g c0 v1 -v2)).(\lambda (_: (((eq T v1 (TSort n)) \to (eq T v2 (TSort (next g -n)))))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (sty0 g c0 t1 -t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2 (TSort (next g -n)))))).(\lambda (H5: (eq T (THead (Flat Cast) v1 t1) (TSort n))).(let H6 -\def (eq_ind T (THead (Flat Cast) v1 t1) (\lambda (ee: T).(match ee in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H5) in -(False_ind (eq T (THead (Flat Cast) v2 t2) (TSort (next g n))) H6)))))))))))) -c y x H0))) H))))). -(* COMMENTS -Initial nodes: 869 -END *) - -theorem sty0_gen_lref: - \forall (g: G).(\forall (c: C).(\forall (x: T).(\forall (n: nat).((sty0 g c -(TLRef n) x) \to (or (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda -(_: T).(getl n c (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: -T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(t: T).(eq T x (lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda -(u: T).(\lambda (_: T).(getl n c (CHead e (Bind Abst) u))))) (\lambda (e: -C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq T x (lift (S n) O u))))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda -(H: (sty0 g c (TLRef n) x)).(insert_eq T (TLRef n) (\lambda (t: T).(sty0 g c -t x)) (\lambda (_: T).(or (ex3_3 C T T (\lambda (e: C).(\lambda (u: -T).(\lambda (_: T).(getl n c (CHead e (Bind Abbr) u))))) (\lambda (e: -C).(\lambda (u: T).(\lambda (t0: T).(sty0 g e u t0)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (t0: T).(eq T x (lift (S n) O t0)))))) (ex3_3 C T -T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead e (Bind -Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t0: T).(sty0 g e u -t0)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T x (lift (S n) O -u)))))))) (\lambda (y: T).(\lambda (H0: (sty0 g c y x)).(sty0_ind g (\lambda -(c0: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (TLRef n)) \to (or (ex3_3 C -T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind -Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t1: T).(sty0 g e u -t1)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t1: T).(eq T t0 (lift (S n) -O t1)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl -n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda -(t1: T).(sty0 g e u t1)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: -T).(eq T t0 (lift (S n) O u))))))))))) (\lambda (c0: C).(\lambda (n0: -nat).(\lambda (H1: (eq T (TSort n0) (TLRef n))).(let H2 \def (eq_ind T (TSort -n0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort -_) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow -False])) I (TLRef n) H1) in (False_ind (or (ex3_3 C T T (\lambda (e: -C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u))))) -(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (t: T).(eq T (TSort (next g n0)) (lift (S n) -O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl -n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: -T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T -(TSort (next g n0)) (lift (S n) O u))))))) H2))))) (\lambda (c0: C).(\lambda -(d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d -(Bind Abbr) v))).(\lambda (w: T).(\lambda (H2: (sty0 g d v w)).(\lambda (_: -(((eq T v (TLRef n)) \to (or (ex3_3 C T T (\lambda (e: C).(\lambda (u: -T).(\lambda (_: T).(getl n d (CHead e (Bind Abbr) u))))) (\lambda (e: -C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda -(_: T).(\lambda (t: T).(eq T w (lift (S n) O t)))))) (ex3_3 C T T (\lambda -(e: C).(\lambda (u: T).(\lambda (_: T).(getl n d (CHead e (Bind Abst) u))))) -(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda -(_: C).(\lambda (u: T).(\lambda (_: T).(eq T w (lift (S n) O -u)))))))))).(\lambda (H4: (eq T (TLRef i) (TLRef n))).(let H5 \def (f_equal T -nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort _) -\Rightarrow i | (TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow i])) -(TLRef i) (TLRef n) H4) in (let H6 \def (eq_ind nat i (\lambda (n0: -nat).(getl n0 c0 (CHead d (Bind Abbr) v))) H1 n H5) in (eq_ind_r nat n -(\lambda (n0: nat).(or (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda -(_: T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: -T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(t: T).(eq T (lift (S n0) O w) (lift (S n) O t)))))) (ex3_3 C T T (\lambda -(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) -(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda -(_: C).(\lambda (u: T).(\lambda (_: T).(eq T (lift (S n0) O w) (lift (S n) O -u)))))))) (or_introl (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda -(_: T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: -T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(t: T).(eq T (lift (S n) O w) (lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: -C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) -(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda -(_: C).(\lambda (u: T).(\lambda (_: T).(eq T (lift (S n) O w) (lift (S n) O -u)))))) (ex3_3_intro C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: -T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: -T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(t: T).(eq T (lift (S n) O w) (lift (S n) O t))))) d v w H6 H2 (refl_equal T -(lift (S n) O w)))) i H5)))))))))))) (\lambda (c0: C).(\lambda (d: -C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d (Bind -Abst) v))).(\lambda (w: T).(\lambda (H2: (sty0 g d v w)).(\lambda (_: (((eq T -v (TLRef n)) \to (or (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda -(_: T).(getl n d (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: -T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(t: T).(eq T w (lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda -(u: T).(\lambda (_: T).(getl n d (CHead e (Bind Abst) u))))) (\lambda (e: -C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq T w (lift (S n) O u)))))))))).(\lambda (H4: (eq T -(TLRef i) (TLRef n))).(let H5 \def (f_equal T nat (\lambda (e: T).(match e in -T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow i | (TLRef n0) -\Rightarrow n0 | (THead _ _ _) \Rightarrow i])) (TLRef i) (TLRef n) H4) in -(let H6 \def (eq_ind nat i (\lambda (n0: nat).(getl n0 c0 (CHead d (Bind -Abst) v))) H1 n H5) in (eq_ind_r nat n (\lambda (n0: nat).(or (ex3_3 C T T -(\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind -Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u -t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T (lift (S n0) O v) -(lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda -(_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: -T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda -(_: T).(eq T (lift (S n0) O v) (lift (S n) O u)))))))) (or_intror (ex3_3 C T -T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind -Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u -t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T (lift (S n) O v) -(lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda -(_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: -T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda -(_: T).(eq T (lift (S n) O v) (lift (S n) O u)))))) (ex3_3_intro C T T -(\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind -Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u -t)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T (lift (S n) O v) -(lift (S n) O u))))) d v w H6 H2 (refl_equal T (lift (S n) O v)))) i -H5)))))))))))) (\lambda (b: B).(\lambda (c0: C).(\lambda (v: T).(\lambda (t1: -T).(\lambda (t2: T).(\lambda (_: (sty0 g (CHead c0 (Bind b) v) t1 -t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (or (ex3_3 C T T (\lambda (e: -C).(\lambda (u: T).(\lambda (_: T).(getl n (CHead c0 (Bind b) v) (CHead e -(Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e -u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T t2 (lift (S n) -O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl -n (CHead c0 (Bind b) v) (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda -(u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(eq T t2 (lift (S n) O u)))))))))).(\lambda (H3: (eq T -(THead (Bind b) v t1) (TLRef n))).(let H4 \def (eq_ind T (THead (Bind b) v -t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort -_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) -\Rightarrow True])) I (TLRef n) H3) in (False_ind (or (ex3_3 C T T (\lambda -(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u))))) -(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda -(_: C).(\lambda (_: T).(\lambda (t: T).(eq T (THead (Bind b) v t2) (lift (S -n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: -T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: -T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda -(_: T).(eq T (THead (Bind b) v t2) (lift (S n) O u))))))) H4)))))))))) -(\lambda (c0: C).(\lambda (v: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(_: (sty0 g c0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (or (ex3_3 C T -T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind -Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u -t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T t2 (lift (S n) O -t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n -c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: -T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T t2 -(lift (S n) O u)))))))))).(\lambda (H3: (eq T (THead (Flat Appl) v t1) (TLRef -n))).(let H4 \def (eq_ind T (THead (Flat Appl) v t1) (\lambda (ee: T).(match -ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | -(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) -H3) in (False_ind (or (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda -(_: T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: -T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(t: T).(eq T (THead (Flat Appl) v t2) (lift (S n) O t)))))) (ex3_3 C T T -(\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind -Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u -t)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T (THead (Flat -Appl) v t2) (lift (S n) O u))))))) H4))))))))) (\lambda (c0: C).(\lambda (v1: -T).(\lambda (v2: T).(\lambda (_: (sty0 g c0 v1 v2)).(\lambda (_: (((eq T v1 -(TLRef n)) \to (or (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: -T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: -T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(t: T).(eq T v2 (lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda -(u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: -C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq T v2 (lift (S n) O u)))))))))).(\lambda (t1: -T).(\lambda (t2: T).(\lambda (_: (sty0 g c0 t1 t2)).(\lambda (_: (((eq T t1 -(TLRef n)) \to (or (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: -T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: -T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(t: T).(eq T t2 (lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda -(u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: -C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq T t2 (lift (S n) O u)))))))))).(\lambda (H5: (eq T -(THead (Flat Cast) v1 t1) (TLRef n))).(let H6 \def (eq_ind T (THead (Flat -Cast) v1 t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) -with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ -_) \Rightarrow True])) I (TLRef n) H5) in (False_ind (or (ex3_3 C T T -(\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind -Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u -t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T (THead (Flat -Cast) v2 t2) (lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: -T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: -C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(eq T (THead (Flat Cast) v2 t2) (lift (S n) O u))))))) -H6)))))))))))) c y x H0))) H))))). -(* COMMENTS -Initial nodes: 3231 -END *) - -theorem sty0_gen_bind: - \forall (g: G).(\forall (b: B).(\forall (c: C).(\forall (u: T).(\forall (t1: -T).(\forall (x: T).((sty0 g c (THead (Bind b) u t1) x) \to (ex2 T (\lambda -(t2: T).(sty0 g (CHead c (Bind b) u) t1 t2)) (\lambda (t2: T).(eq T x (THead -(Bind b) u t2)))))))))) -\def - \lambda (g: G).(\lambda (b: B).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: -T).(\lambda (x: T).(\lambda (H: (sty0 g c (THead (Bind b) u t1) -x)).(insert_eq T (THead (Bind b) u t1) (\lambda (t: T).(sty0 g c t x)) -(\lambda (_: T).(ex2 T (\lambda (t2: T).(sty0 g (CHead c (Bind b) u) t1 t2)) -(\lambda (t2: T).(eq T x (THead (Bind b) u t2))))) (\lambda (y: T).(\lambda -(H0: (sty0 g c y x)).(sty0_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda -(t0: T).((eq T t (THead (Bind b) u t1)) \to (ex2 T (\lambda (t2: T).(sty0 g -(CHead c0 (Bind b) u) t1 t2)) (\lambda (t2: T).(eq T t0 (THead (Bind b) u -t2)))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (H1: (eq T (TSort n) -(THead (Bind b) u t1))).(let H2 \def (eq_ind T (TSort n) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I -(THead (Bind b) u t1) H1) in (False_ind (ex2 T (\lambda (t2: T).(sty0 g -(CHead c0 (Bind b) u) t1 t2)) (\lambda (t2: T).(eq T (TSort (next g n)) -(THead (Bind b) u t2)))) H2))))) (\lambda (c0: C).(\lambda (d: C).(\lambda -(v: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind Abbr) -v))).(\lambda (w: T).(\lambda (_: (sty0 g d v w)).(\lambda (_: (((eq T v -(THead (Bind b) u t1)) \to (ex2 T (\lambda (t2: T).(sty0 g (CHead d (Bind b) -u) t1 t2)) (\lambda (t2: T).(eq T w (THead (Bind b) u t2))))))).(\lambda (H4: -(eq T (TLRef i) (THead (Bind b) u t1))).(let H5 \def (eq_ind T (TLRef i) -(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow -False])) I (THead (Bind b) u t1) H4) in (False_ind (ex2 T (\lambda (t2: -T).(sty0 g (CHead c0 (Bind b) u) t1 t2)) (\lambda (t2: T).(eq T (lift (S i) O -w) (THead (Bind b) u t2)))) H5))))))))))) (\lambda (c0: C).(\lambda (d: -C).(\lambda (v: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind -Abst) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v w)).(\lambda (_: (((eq T -v (THead (Bind b) u t1)) \to (ex2 T (\lambda (t2: T).(sty0 g (CHead d (Bind -b) u) t1 t2)) (\lambda (t2: T).(eq T w (THead (Bind b) u t2))))))).(\lambda -(H4: (eq T (TLRef i) (THead (Bind b) u t1))).(let H5 \def (eq_ind T (TLRef i) -(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow -False])) I (THead (Bind b) u t1) H4) in (False_ind (ex2 T (\lambda (t2: -T).(sty0 g (CHead c0 (Bind b) u) t1 t2)) (\lambda (t2: T).(eq T (lift (S i) O -v) (THead (Bind b) u t2)))) H5))))))))))) (\lambda (b0: B).(\lambda (c0: -C).(\lambda (v: T).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H1: (sty0 g -(CHead c0 (Bind b0) v) t0 t2)).(\lambda (H2: (((eq T t0 (THead (Bind b) u -t1)) \to (ex2 T (\lambda (t3: T).(sty0 g (CHead (CHead c0 (Bind b0) v) (Bind -b) u) t1 t3)) (\lambda (t3: T).(eq T t2 (THead (Bind b) u t3))))))).(\lambda -(H3: (eq T (THead (Bind b0) v t0) (THead (Bind b) u t1))).(let H4 \def -(f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with -[(TSort _) \Rightarrow b0 | (TLRef _) \Rightarrow b0 | (THead k _ _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b1) -\Rightarrow b1 | (Flat _) \Rightarrow b0])])) (THead (Bind b0) v t0) (THead -(Bind b) u t1) H3) in ((let H5 \def (f_equal T T (\lambda (e: T).(match e in -T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v | (TLRef _) -\Rightarrow v | (THead _ t _) \Rightarrow t])) (THead (Bind b0) v t0) (THead -(Bind b) u t1) H3) in ((let H6 \def (f_equal T T (\lambda (e: T).(match e in -T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) -\Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Bind b0) v t0) (THead -(Bind b) u t1) H3) in (\lambda (H7: (eq T v u)).(\lambda (H8: (eq B b0 -b)).(let H9 \def (eq_ind T t0 (\lambda (t: T).((eq T t (THead (Bind b) u t1)) -\to (ex2 T (\lambda (t3: T).(sty0 g (CHead (CHead c0 (Bind b0) v) (Bind b) u) -t1 t3)) (\lambda (t3: T).(eq T t2 (THead (Bind b) u t3)))))) H2 t1 H6) in -(let H10 \def (eq_ind T t0 (\lambda (t: T).(sty0 g (CHead c0 (Bind b0) v) t -t2)) H1 t1 H6) in (let H11 \def (eq_ind T v (\lambda (t: T).((eq T t1 (THead -(Bind b) u t1)) \to (ex2 T (\lambda (t3: T).(sty0 g (CHead (CHead c0 (Bind -b0) t) (Bind b) u) t1 t3)) (\lambda (t3: T).(eq T t2 (THead (Bind b) u -t3)))))) H9 u H7) in (let H12 \def (eq_ind T v (\lambda (t: T).(sty0 g (CHead -c0 (Bind b0) t) t1 t2)) H10 u H7) in (eq_ind_r T u (\lambda (t: T).(ex2 T -(\lambda (t3: T).(sty0 g (CHead c0 (Bind b) u) t1 t3)) (\lambda (t3: T).(eq T -(THead (Bind b0) t t2) (THead (Bind b) u t3))))) (let H13 \def (eq_ind B b0 -(\lambda (b1: B).((eq T t1 (THead (Bind b) u t1)) \to (ex2 T (\lambda (t3: -T).(sty0 g (CHead (CHead c0 (Bind b1) u) (Bind b) u) t1 t3)) (\lambda (t3: -T).(eq T t2 (THead (Bind b) u t3)))))) H11 b H8) in (let H14 \def (eq_ind B -b0 (\lambda (b1: B).(sty0 g (CHead c0 (Bind b1) u) t1 t2)) H12 b H8) in -(eq_ind_r B b (\lambda (b1: B).(ex2 T (\lambda (t3: T).(sty0 g (CHead c0 -(Bind b) u) t1 t3)) (\lambda (t3: T).(eq T (THead (Bind b1) u t2) (THead -(Bind b) u t3))))) (ex_intro2 T (\lambda (t3: T).(sty0 g (CHead c0 (Bind b) -u) t1 t3)) (\lambda (t3: T).(eq T (THead (Bind b) u t2) (THead (Bind b) u -t3))) t2 H14 (refl_equal T (THead (Bind b) u t2))) b0 H8))) v H7)))))))) H5)) -H4)))))))))) (\lambda (c0: C).(\lambda (v: T).(\lambda (t0: T).(\lambda (t2: -T).(\lambda (_: (sty0 g c0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind b) u -t1)) \to (ex2 T (\lambda (t3: T).(sty0 g (CHead c0 (Bind b) u) t1 t3)) -(\lambda (t3: T).(eq T t2 (THead (Bind b) u t3))))))).(\lambda (H3: (eq T -(THead (Flat Appl) v t0) (THead (Bind b) u t1))).(let H4 \def (eq_ind T -(THead (Flat Appl) v t0) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind -b) u t1) H3) in (False_ind (ex2 T (\lambda (t3: T).(sty0 g (CHead c0 (Bind b) -u) t1 t3)) (\lambda (t3: T).(eq T (THead (Flat Appl) v t2) (THead (Bind b) u -t3)))) H4))))))))) (\lambda (c0: C).(\lambda (v1: T).(\lambda (v2: -T).(\lambda (_: (sty0 g c0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind b) u -t1)) \to (ex2 T (\lambda (t2: T).(sty0 g (CHead c0 (Bind b) u) t1 t2)) -(\lambda (t2: T).(eq T v2 (THead (Bind b) u t2))))))).(\lambda (t0: -T).(\lambda (t2: T).(\lambda (_: (sty0 g c0 t0 t2)).(\lambda (_: (((eq T t0 -(THead (Bind b) u t1)) \to (ex2 T (\lambda (t3: T).(sty0 g (CHead c0 (Bind b) -u) t1 t3)) (\lambda (t3: T).(eq T t2 (THead (Bind b) u t3))))))).(\lambda -(H5: (eq T (THead (Flat Cast) v1 t0) (THead (Bind b) u t1))).(let H6 \def -(eq_ind T (THead (Flat Cast) v1 t0) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind b) u t1) H5) in (False_ind (ex2 T (\lambda (t3: -T).(sty0 g (CHead c0 (Bind b) u) t1 t3)) (\lambda (t3: T).(eq T (THead (Flat -Cast) v2 t2) (THead (Bind b) u t3)))) H6)))))))))))) c y x H0))) H))))))). -(* COMMENTS -Initial nodes: 1975 -END *) - -theorem sty0_gen_appl: - \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (x: -T).((sty0 g c (THead (Flat Appl) u t1) x) \to (ex2 T (\lambda (t2: T).(sty0 g -c t1 t2)) (\lambda (t2: T).(eq T x (THead (Flat Appl) u t2))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (x: -T).(\lambda (H: (sty0 g c (THead (Flat Appl) u t1) x)).(insert_eq T (THead -(Flat Appl) u t1) (\lambda (t: T).(sty0 g c t x)) (\lambda (_: T).(ex2 T -(\lambda (t2: T).(sty0 g c t1 t2)) (\lambda (t2: T).(eq T x (THead (Flat -Appl) u t2))))) (\lambda (y: T).(\lambda (H0: (sty0 g c y x)).(sty0_ind g -(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Flat Appl) -u t1)) \to (ex2 T (\lambda (t2: T).(sty0 g c0 t1 t2)) (\lambda (t2: T).(eq T -t0 (THead (Flat Appl) u t2)))))))) (\lambda (c0: C).(\lambda (n: -nat).(\lambda (H1: (eq T (TSort n) (THead (Flat Appl) u t1))).(let H2 \def -(eq_ind T (TSort n) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | -(THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) u t1) H1) in -(False_ind (ex2 T (\lambda (t2: T).(sty0 g c0 t1 t2)) (\lambda (t2: T).(eq T -(TSort (next g n)) (THead (Flat Appl) u t2)))) H2))))) (\lambda (c0: -C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (_: (getl i c0 -(CHead d (Bind Abbr) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v -w)).(\lambda (_: (((eq T v (THead (Flat Appl) u t1)) \to (ex2 T (\lambda (t2: -T).(sty0 g d t1 t2)) (\lambda (t2: T).(eq T w (THead (Flat Appl) u -t2))))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat Appl) u t1))).(let H5 -\def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | -(THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) u t1) H4) in -(False_ind (ex2 T (\lambda (t2: T).(sty0 g c0 t1 t2)) (\lambda (t2: T).(eq T -(lift (S i) O w) (THead (Flat Appl) u t2)))) H5))))))))))) (\lambda (c0: -C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (_: (getl i c0 -(CHead d (Bind Abst) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v -w)).(\lambda (_: (((eq T v (THead (Flat Appl) u t1)) \to (ex2 T (\lambda (t2: -T).(sty0 g d t1 t2)) (\lambda (t2: T).(eq T w (THead (Flat Appl) u -t2))))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat Appl) u t1))).(let H5 -\def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | -(THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) u t1) H4) in -(False_ind (ex2 T (\lambda (t2: T).(sty0 g c0 t1 t2)) (\lambda (t2: T).(eq T -(lift (S i) O v) (THead (Flat Appl) u t2)))) H5))))))))))) (\lambda (b: -B).(\lambda (c0: C).(\lambda (v: T).(\lambda (t0: T).(\lambda (t2: -T).(\lambda (_: (sty0 g (CHead c0 (Bind b) v) t0 t2)).(\lambda (_: (((eq T t0 -(THead (Flat Appl) u t1)) \to (ex2 T (\lambda (t3: T).(sty0 g (CHead c0 (Bind -b) v) t1 t3)) (\lambda (t3: T).(eq T t2 (THead (Flat Appl) u -t3))))))).(\lambda (H3: (eq T (THead (Bind b) v t0) (THead (Flat Appl) u -t1))).(let H4 \def (eq_ind T (THead (Bind b) v t0) (\lambda (ee: T).(match ee -in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef -_) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return -(\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow -False])])) I (THead (Flat Appl) u t1) H3) in (False_ind (ex2 T (\lambda (t3: -T).(sty0 g c0 t1 t3)) (\lambda (t3: T).(eq T (THead (Bind b) v t2) (THead -(Flat Appl) u t3)))) H4)))))))))) (\lambda (c0: C).(\lambda (v: T).(\lambda -(t0: T).(\lambda (t2: T).(\lambda (H1: (sty0 g c0 t0 t2)).(\lambda (H2: (((eq -T t0 (THead (Flat Appl) u t1)) \to (ex2 T (\lambda (t3: T).(sty0 g c0 t1 t3)) -(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u t3))))))).(\lambda (H3: (eq T -(THead (Flat Appl) v t0) (THead (Flat Appl) u t1))).(let H4 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow v | (TLRef _) \Rightarrow v | (THead _ t _) \Rightarrow t])) -(THead (Flat Appl) v t0) (THead (Flat Appl) u t1) H3) in ((let H5 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) -\Rightarrow t])) (THead (Flat Appl) v t0) (THead (Flat Appl) u t1) H3) in -(\lambda (H6: (eq T v u)).(let H7 \def (eq_ind T t0 (\lambda (t: T).((eq T t -(THead (Flat Appl) u t1)) \to (ex2 T (\lambda (t3: T).(sty0 g c0 t1 t3)) -(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u t3)))))) H2 t1 H5) in (let H8 -\def (eq_ind T t0 (\lambda (t: T).(sty0 g c0 t t2)) H1 t1 H5) in (eq_ind_r T -u (\lambda (t: T).(ex2 T (\lambda (t3: T).(sty0 g c0 t1 t3)) (\lambda (t3: -T).(eq T (THead (Flat Appl) t t2) (THead (Flat Appl) u t3))))) (ex_intro2 T -(\lambda (t3: T).(sty0 g c0 t1 t3)) (\lambda (t3: T).(eq T (THead (Flat Appl) -u t2) (THead (Flat Appl) u t3))) t2 H8 (refl_equal T (THead (Flat Appl) u -t2))) v H6))))) H4))))))))) (\lambda (c0: C).(\lambda (v1: T).(\lambda (v2: -T).(\lambda (_: (sty0 g c0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Flat Appl) -u t1)) \to (ex2 T (\lambda (t2: T).(sty0 g c0 t1 t2)) (\lambda (t2: T).(eq T -v2 (THead (Flat Appl) u t2))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda -(_: (sty0 g c0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u t1)) \to -(ex2 T (\lambda (t3: T).(sty0 g c0 t1 t3)) (\lambda (t3: T).(eq T t2 (THead -(Flat Appl) u t3))))))).(\lambda (H5: (eq T (THead (Flat Cast) v1 t0) (THead -(Flat Appl) u t1))).(let H6 \def (eq_ind T (THead (Flat Cast) v1 t0) (\lambda -(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow -(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | -(Flat f) \Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl -\Rightarrow False | Cast \Rightarrow True])])])) I (THead (Flat Appl) u t1) -H5) in (False_ind (ex2 T (\lambda (t3: T).(sty0 g c0 t1 t3)) (\lambda (t3: -T).(eq T (THead (Flat Cast) v2 t2) (THead (Flat Appl) u t3)))) H6)))))))))))) -c y x H0))) H)))))). -(* COMMENTS -Initial nodes: 1489 -END *) - -theorem sty0_gen_cast: - \forall (g: G).(\forall (c: C).(\forall (v1: T).(\forall (t1: T).(\forall -(x: T).((sty0 g c (THead (Flat Cast) v1 t1) x) \to (ex3_2 T T (\lambda (v2: -T).(\lambda (_: T).(sty0 g c v1 v2))) (\lambda (_: T).(\lambda (t2: T).(sty0 -g c t1 t2))) (\lambda (v2: T).(\lambda (t2: T).(eq T x (THead (Flat Cast) v2 -t2)))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (v1: T).(\lambda (t1: T).(\lambda -(x: T).(\lambda (H: (sty0 g c (THead (Flat Cast) v1 t1) x)).(insert_eq T -(THead (Flat Cast) v1 t1) (\lambda (t: T).(sty0 g c t x)) (\lambda (_: -T).(ex3_2 T T (\lambda (v2: T).(\lambda (_: T).(sty0 g c v1 v2))) (\lambda -(_: T).(\lambda (t2: T).(sty0 g c t1 t2))) (\lambda (v2: T).(\lambda (t2: -T).(eq T x (THead (Flat Cast) v2 t2)))))) (\lambda (y: T).(\lambda (H0: (sty0 -g c y x)).(sty0_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq -T t (THead (Flat Cast) v1 t1)) \to (ex3_2 T T (\lambda (v2: T).(\lambda (_: -T).(sty0 g c0 v1 v2))) (\lambda (_: T).(\lambda (t2: T).(sty0 g c0 t1 t2))) -(\lambda (v2: T).(\lambda (t2: T).(eq T t0 (THead (Flat Cast) v2 t2))))))))) -(\lambda (c0: C).(\lambda (n: nat).(\lambda (H1: (eq T (TSort n) (THead (Flat -Cast) v1 t1))).(let H2 \def (eq_ind T (TSort n) (\lambda (ee: T).(match ee in -T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) -v1 t1) H1) in (False_ind (ex3_2 T T (\lambda (v2: T).(\lambda (_: T).(sty0 g -c0 v1 v2))) (\lambda (_: T).(\lambda (t2: T).(sty0 g c0 t1 t2))) (\lambda -(v2: T).(\lambda (t2: T).(eq T (TSort (next g n)) (THead (Flat Cast) v2 -t2))))) H2))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: -nat).(\lambda (_: (getl i c0 (CHead d (Bind Abbr) v))).(\lambda (w: -T).(\lambda (_: (sty0 g d v w)).(\lambda (_: (((eq T v (THead (Flat Cast) v1 -t1)) \to (ex3_2 T T (\lambda (v2: T).(\lambda (_: T).(sty0 g d v1 v2))) -(\lambda (_: T).(\lambda (t2: T).(sty0 g d t1 t2))) (\lambda (v2: T).(\lambda -(t2: T).(eq T w (THead (Flat Cast) v2 t2)))))))).(\lambda (H4: (eq T (TLRef -i) (THead (Flat Cast) v1 t1))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I -(THead (Flat Cast) v1 t1) H4) in (False_ind (ex3_2 T T (\lambda (v2: -T).(\lambda (_: T).(sty0 g c0 v1 v2))) (\lambda (_: T).(\lambda (t2: T).(sty0 -g c0 t1 t2))) (\lambda (v2: T).(\lambda (t2: T).(eq T (lift (S i) O w) (THead -(Flat Cast) v2 t2))))) H5))))))))))) (\lambda (c0: C).(\lambda (d: -C).(\lambda (v: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind -Abst) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v w)).(\lambda (_: (((eq T -v (THead (Flat Cast) v1 t1)) \to (ex3_2 T T (\lambda (v2: T).(\lambda (_: -T).(sty0 g d v1 v2))) (\lambda (_: T).(\lambda (t2: T).(sty0 g d t1 t2))) -(\lambda (v2: T).(\lambda (t2: T).(eq T w (THead (Flat Cast) v2 -t2)))))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat Cast) v1 t1))).(let H5 -\def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | -(THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) v1 t1) H4) in -(False_ind (ex3_2 T T (\lambda (v2: T).(\lambda (_: T).(sty0 g c0 v1 v2))) -(\lambda (_: T).(\lambda (t2: T).(sty0 g c0 t1 t2))) (\lambda (v2: -T).(\lambda (t2: T).(eq T (lift (S i) O v) (THead (Flat Cast) v2 t2))))) -H5))))))))))) (\lambda (b: B).(\lambda (c0: C).(\lambda (v: T).(\lambda (t0: -T).(\lambda (t2: T).(\lambda (_: (sty0 g (CHead c0 (Bind b) v) t0 -t2)).(\lambda (_: (((eq T t0 (THead (Flat Cast) v1 t1)) \to (ex3_2 T T -(\lambda (v2: T).(\lambda (_: T).(sty0 g (CHead c0 (Bind b) v) v1 v2))) -(\lambda (_: T).(\lambda (t3: T).(sty0 g (CHead c0 (Bind b) v) t1 t3))) -(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) v2 -t3)))))))).(\lambda (H3: (eq T (THead (Bind b) v t0) (THead (Flat Cast) v1 -t1))).(let H4 \def (eq_ind T (THead (Bind b) v t0) (\lambda (ee: T).(match ee -in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef -_) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return -(\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow -False])])) I (THead (Flat Cast) v1 t1) H3) in (False_ind (ex3_2 T T (\lambda -(v2: T).(\lambda (_: T).(sty0 g c0 v1 v2))) (\lambda (_: T).(\lambda (t3: -T).(sty0 g c0 t1 t3))) (\lambda (v2: T).(\lambda (t3: T).(eq T (THead (Bind -b) v t2) (THead (Flat Cast) v2 t3))))) H4)))))))))) (\lambda (c0: C).(\lambda -(v: T).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (sty0 g c0 t0 -t2)).(\lambda (_: (((eq T t0 (THead (Flat Cast) v1 t1)) \to (ex3_2 T T -(\lambda (v2: T).(\lambda (_: T).(sty0 g c0 v1 v2))) (\lambda (_: T).(\lambda -(t3: T).(sty0 g c0 t1 t3))) (\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead -(Flat Cast) v2 t3)))))))).(\lambda (H3: (eq T (THead (Flat Appl) v t0) (THead -(Flat Cast) v1 t1))).(let H4 \def (eq_ind T (THead (Flat Appl) v t0) (\lambda -(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow -(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | -(Flat f) \Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl -\Rightarrow True | Cast \Rightarrow False])])])) I (THead (Flat Cast) v1 t1) -H3) in (False_ind (ex3_2 T T (\lambda (v2: T).(\lambda (_: T).(sty0 g c0 v1 -v2))) (\lambda (_: T).(\lambda (t3: T).(sty0 g c0 t1 t3))) (\lambda (v2: -T).(\lambda (t3: T).(eq T (THead (Flat Appl) v t2) (THead (Flat Cast) v2 -t3))))) H4))))))))) (\lambda (c0: C).(\lambda (v0: T).(\lambda (v2: -T).(\lambda (H1: (sty0 g c0 v0 v2)).(\lambda (H2: (((eq T v0 (THead (Flat -Cast) v1 t1)) \to (ex3_2 T T (\lambda (v3: T).(\lambda (_: T).(sty0 g c0 v1 -v3))) (\lambda (_: T).(\lambda (t2: T).(sty0 g c0 t1 t2))) (\lambda (v3: -T).(\lambda (t2: T).(eq T v2 (THead (Flat Cast) v3 t2)))))))).(\lambda (t0: -T).(\lambda (t2: T).(\lambda (H3: (sty0 g c0 t0 t2)).(\lambda (H4: (((eq T t0 -(THead (Flat Cast) v1 t1)) \to (ex3_2 T T (\lambda (v3: T).(\lambda (_: -T).(sty0 g c0 v1 v3))) (\lambda (_: T).(\lambda (t3: T).(sty0 g c0 t1 t3))) -(\lambda (v3: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) v3 -t3)))))))).(\lambda (H5: (eq T (THead (Flat Cast) v0 t0) (THead (Flat Cast) -v1 t1))).(let H6 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow v0 | (TLRef _) \Rightarrow v0 -| (THead _ t _) \Rightarrow t])) (THead (Flat Cast) v0 t0) (THead (Flat Cast) -v1 t1) H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 -| (THead _ _ t) \Rightarrow t])) (THead (Flat Cast) v0 t0) (THead (Flat Cast) -v1 t1) H5) in (\lambda (H8: (eq T v0 v1)).(let H9 \def (eq_ind T t0 (\lambda -(t: T).((eq T t (THead (Flat Cast) v1 t1)) \to (ex3_2 T T (\lambda (v3: -T).(\lambda (_: T).(sty0 g c0 v1 v3))) (\lambda (_: T).(\lambda (t3: T).(sty0 -g c0 t1 t3))) (\lambda (v3: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) -v3 t3))))))) H4 t1 H7) in (let H10 \def (eq_ind T t0 (\lambda (t: T).(sty0 g -c0 t t2)) H3 t1 H7) in (let H11 \def (eq_ind T v0 (\lambda (t: T).((eq T t -(THead (Flat Cast) v1 t1)) \to (ex3_2 T T (\lambda (v3: T).(\lambda (_: -T).(sty0 g c0 v1 v3))) (\lambda (_: T).(\lambda (t3: T).(sty0 g c0 t1 t3))) -(\lambda (v3: T).(\lambda (t3: T).(eq T v2 (THead (Flat Cast) v3 t3))))))) H2 -v1 H8) in (let H12 \def (eq_ind T v0 (\lambda (t: T).(sty0 g c0 t v2)) H1 v1 -H8) in (ex3_2_intro T T (\lambda (v3: T).(\lambda (_: T).(sty0 g c0 v1 v3))) -(\lambda (_: T).(\lambda (t3: T).(sty0 g c0 t1 t3))) (\lambda (v3: -T).(\lambda (t3: T).(eq T (THead (Flat Cast) v2 t2) (THead (Flat Cast) v3 -t3)))) v2 t2 H12 H10 (refl_equal T (THead (Flat Cast) v2 t2))))))))) -H6)))))))))))) c y x H0))) H)))))). -(* COMMENTS -Initial nodes: 1855 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/sty0/props.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/sty0/props.ma deleted file mode 100644 index 00c9f6818..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/sty0/props.ma +++ /dev/null @@ -1,217 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/sty0/defs.ma". - -include "Basic-1/getl/drop.ma". - -theorem sty0_lift: - \forall (g: G).(\forall (e: C).(\forall (t1: T).(\forall (t2: T).((sty0 g e -t1 t2) \to (\forall (c: C).(\forall (h: nat).(\forall (d: nat).((drop h d c -e) \to (sty0 g c (lift h d t1) (lift h d t2)))))))))) -\def - \lambda (g: G).(\lambda (e: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H: (sty0 g e t1 t2)).(sty0_ind g (\lambda (c: C).(\lambda (t: T).(\lambda -(t0: T).(\forall (c0: C).(\forall (h: nat).(\forall (d: nat).((drop h d c0 c) -\to (sty0 g c0 (lift h d t) (lift h d t0))))))))) (\lambda (c: C).(\lambda -(n: nat).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (_: -(drop h d c0 c)).(eq_ind_r T (TSort n) (\lambda (t: T).(sty0 g c0 t (lift h d -(TSort (next g n))))) (eq_ind_r T (TSort (next g n)) (\lambda (t: T).(sty0 g -c0 (TSort n) t)) (sty0_sort g c0 n) (lift h d (TSort (next g n))) (lift_sort -(next g n) h d)) (lift h d (TSort n)) (lift_sort n h d)))))))) (\lambda (c: -C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H0: (getl i c -(CHead d (Bind Abbr) v))).(\lambda (w: T).(\lambda (H1: (sty0 g d v -w)).(\lambda (H2: ((\forall (c0: C).(\forall (h: nat).(\forall (d0: -nat).((drop h d0 c0 d) \to (sty0 g c0 (lift h d0 v) (lift h d0 -w)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d0: nat).(\lambda (H3: -(drop h d0 c0 c)).(lt_le_e i d0 (sty0 g c0 (lift h d0 (TLRef i)) (lift h d0 -(lift (S i) O w))) (\lambda (H4: (lt i d0)).(let H5 \def (drop_getl_trans_le -i d0 (le_S_n i d0 (le_S (S i) d0 H4)) c0 c h H3 (CHead d (Bind Abbr) v) H0) -in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop i O c0 e0))) -(\lambda (e0: C).(\lambda (e1: C).(drop h (minus d0 i) e0 e1))) (\lambda (_: -C).(\lambda (e1: C).(clear e1 (CHead d (Bind Abbr) v)))) (sty0 g c0 (lift h -d0 (TLRef i)) (lift h d0 (lift (S i) O w))) (\lambda (x0: C).(\lambda (x1: -C).(\lambda (H6: (drop i O c0 x0)).(\lambda (H7: (drop h (minus d0 i) x0 -x1)).(\lambda (H8: (clear x1 (CHead d (Bind Abbr) v))).(let H9 \def (eq_ind -nat (minus d0 i) (\lambda (n: nat).(drop h n x0 x1)) H7 (S (minus d0 (S i))) -(minus_x_Sy d0 i H4)) in (let H10 \def (drop_clear_S x1 x0 h (minus d0 (S i)) -H9 Abbr d v H8) in (ex2_ind C (\lambda (c1: C).(clear x0 (CHead c1 (Bind -Abbr) (lift h (minus d0 (S i)) v)))) (\lambda (c1: C).(drop h (minus d0 (S -i)) c1 d)) (sty0 g c0 (lift h d0 (TLRef i)) (lift h d0 (lift (S i) O w))) -(\lambda (x: C).(\lambda (H11: (clear x0 (CHead x (Bind Abbr) (lift h (minus -d0 (S i)) v)))).(\lambda (H12: (drop h (minus d0 (S i)) x d)).(eq_ind_r T -(TLRef i) (\lambda (t: T).(sty0 g c0 t (lift h d0 (lift (S i) O w)))) (eq_ind -nat (plus (S i) (minus d0 (S i))) (\lambda (n: nat).(sty0 g c0 (TLRef i) -(lift h n (lift (S i) O w)))) (eq_ind_r T (lift (S i) O (lift h (minus d0 (S -i)) w)) (\lambda (t: T).(sty0 g c0 (TLRef i) t)) (eq_ind nat d0 (\lambda (_: -nat).(sty0 g c0 (TLRef i) (lift (S i) O (lift h (minus d0 (S i)) w)))) -(sty0_abbr g c0 x (lift h (minus d0 (S i)) v) i (getl_intro i c0 (CHead x -(Bind Abbr) (lift h (minus d0 (S i)) v)) x0 H6 H11) (lift h (minus d0 (S i)) -w) (H2 x h (minus d0 (S i)) H12)) (plus (S i) (minus d0 (S i))) -(le_plus_minus (S i) d0 H4)) (lift h (plus (S i) (minus d0 (S i))) (lift (S -i) O w)) (lift_d w h (S i) (minus d0 (S i)) O (le_O_n (minus d0 (S i))))) d0 -(le_plus_minus_r (S i) d0 H4)) (lift h d0 (TLRef i)) (lift_lref_lt i h d0 -H4))))) H10)))))))) H5))) (\lambda (H4: (le d0 i)).(eq_ind_r T (TLRef (plus i -h)) (\lambda (t: T).(sty0 g c0 t (lift h d0 (lift (S i) O w)))) (eq_ind nat -(S i) (\lambda (_: nat).(sty0 g c0 (TLRef (plus i h)) (lift h d0 (lift (S i) -O w)))) (eq_ind_r T (lift (plus h (S i)) O w) (\lambda (t: T).(sty0 g c0 -(TLRef (plus i h)) t)) (eq_ind_r nat (plus (S i) h) (\lambda (n: nat).(sty0 g -c0 (TLRef (plus i h)) (lift n O w))) (sty0_abbr g c0 d v (plus i h) -(drop_getl_trans_ge i c0 c d0 h H3 (CHead d (Bind Abbr) v) H0 H4) w H1) (plus -h (S i)) (plus_sym h (S i))) (lift h d0 (lift (S i) O w)) (lift_free w (S i) -h O d0 (le_S d0 i H4) (le_O_n d0))) (plus i (S O)) (eq_ind_r nat (plus (S O) -i) (\lambda (n: nat).(eq nat (S i) n)) (refl_equal nat (plus (S O) i)) (plus -i (S O)) (plus_sym i (S O)))) (lift h d0 (TLRef i)) (lift_lref_ge i h d0 -H4)))))))))))))))) (\lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda -(i: nat).(\lambda (H0: (getl i c (CHead d (Bind Abst) v))).(\lambda (w: -T).(\lambda (H1: (sty0 g d v w)).(\lambda (H2: ((\forall (c0: C).(\forall (h: -nat).(\forall (d0: nat).((drop h d0 c0 d) \to (sty0 g c0 (lift h d0 v) (lift -h d0 w)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d0: nat).(\lambda -(H3: (drop h d0 c0 c)).(lt_le_e i d0 (sty0 g c0 (lift h d0 (TLRef i)) (lift h -d0 (lift (S i) O v))) (\lambda (H4: (lt i d0)).(let H5 \def -(drop_getl_trans_le i d0 (le_S_n i d0 (le_S (S i) d0 H4)) c0 c h H3 (CHead d -(Bind Abst) v) H0) in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop i -O c0 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d0 i) e0 e1))) -(\lambda (_: C).(\lambda (e1: C).(clear e1 (CHead d (Bind Abst) v)))) (sty0 g -c0 (lift h d0 (TLRef i)) (lift h d0 (lift (S i) O v))) (\lambda (x0: -C).(\lambda (x1: C).(\lambda (H6: (drop i O c0 x0)).(\lambda (H7: (drop h -(minus d0 i) x0 x1)).(\lambda (H8: (clear x1 (CHead d (Bind Abst) v))).(let -H9 \def (eq_ind nat (minus d0 i) (\lambda (n: nat).(drop h n x0 x1)) H7 (S -(minus d0 (S i))) (minus_x_Sy d0 i H4)) in (let H10 \def (drop_clear_S x1 x0 -h (minus d0 (S i)) H9 Abst d v H8) in (ex2_ind C (\lambda (c1: C).(clear x0 -(CHead c1 (Bind Abst) (lift h (minus d0 (S i)) v)))) (\lambda (c1: C).(drop h -(minus d0 (S i)) c1 d)) (sty0 g c0 (lift h d0 (TLRef i)) (lift h d0 (lift (S -i) O v))) (\lambda (x: C).(\lambda (H11: (clear x0 (CHead x (Bind Abst) (lift -h (minus d0 (S i)) v)))).(\lambda (H12: (drop h (minus d0 (S i)) x -d)).(eq_ind_r T (TLRef i) (\lambda (t: T).(sty0 g c0 t (lift h d0 (lift (S i) -O v)))) (eq_ind nat (plus (S i) (minus d0 (S i))) (\lambda (n: nat).(sty0 g -c0 (TLRef i) (lift h n (lift (S i) O v)))) (eq_ind_r T (lift (S i) O (lift h -(minus d0 (S i)) v)) (\lambda (t: T).(sty0 g c0 (TLRef i) t)) (eq_ind nat d0 -(\lambda (_: nat).(sty0 g c0 (TLRef i) (lift (S i) O (lift h (minus d0 (S i)) -v)))) (sty0_abst g c0 x (lift h (minus d0 (S i)) v) i (getl_intro i c0 (CHead -x (Bind Abst) (lift h (minus d0 (S i)) v)) x0 H6 H11) (lift h (minus d0 (S -i)) w) (H2 x h (minus d0 (S i)) H12)) (plus (S i) (minus d0 (S i))) -(le_plus_minus (S i) d0 H4)) (lift h (plus (S i) (minus d0 (S i))) (lift (S -i) O v)) (lift_d v h (S i) (minus d0 (S i)) O (le_O_n (minus d0 (S i))))) d0 -(le_plus_minus_r (S i) d0 H4)) (lift h d0 (TLRef i)) (lift_lref_lt i h d0 -H4))))) H10)))))))) H5))) (\lambda (H4: (le d0 i)).(eq_ind_r T (TLRef (plus i -h)) (\lambda (t: T).(sty0 g c0 t (lift h d0 (lift (S i) O v)))) (eq_ind nat -(S i) (\lambda (_: nat).(sty0 g c0 (TLRef (plus i h)) (lift h d0 (lift (S i) -O v)))) (eq_ind_r T (lift (plus h (S i)) O v) (\lambda (t: T).(sty0 g c0 -(TLRef (plus i h)) t)) (eq_ind_r nat (plus (S i) h) (\lambda (n: nat).(sty0 g -c0 (TLRef (plus i h)) (lift n O v))) (sty0_abst g c0 d v (plus i h) -(drop_getl_trans_ge i c0 c d0 h H3 (CHead d (Bind Abst) v) H0 H4) w H1) (plus -h (S i)) (plus_sym h (S i))) (lift h d0 (lift (S i) O v)) (lift_free v (S i) -h O d0 (le_S d0 i H4) (le_O_n d0))) (plus i (S O)) (eq_ind_r nat (plus (S O) -i) (\lambda (n: nat).(eq nat (S i) n)) (refl_equal nat (plus (S O) i)) (plus -i (S O)) (plus_sym i (S O)))) (lift h d0 (TLRef i)) (lift_lref_ge i h d0 -H4)))))))))))))))) (\lambda (b: B).(\lambda (c: C).(\lambda (v: T).(\lambda -(t3: T).(\lambda (t4: T).(\lambda (_: (sty0 g (CHead c (Bind b) v) t3 -t4)).(\lambda (H1: ((\forall (c0: C).(\forall (h: nat).(\forall (d: -nat).((drop h d c0 (CHead c (Bind b) v)) \to (sty0 g c0 (lift h d t3) (lift h -d t4)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda -(H2: (drop h d c0 c)).(eq_ind_r T (THead (Bind b) (lift h d v) (lift h (s -(Bind b) d) t3)) (\lambda (t: T).(sty0 g c0 t (lift h d (THead (Bind b) v -t4)))) (eq_ind_r T (THead (Bind b) (lift h d v) (lift h (s (Bind b) d) t4)) -(\lambda (t: T).(sty0 g c0 (THead (Bind b) (lift h d v) (lift h (s (Bind b) -d) t3)) t)) (sty0_bind g b c0 (lift h d v) (lift h (S d) t3) (lift h (S d) -t4) (H1 (CHead c0 (Bind b) (lift h d v)) h (S d) (drop_skip_bind h d c0 c H2 -b v))) (lift h d (THead (Bind b) v t4)) (lift_head (Bind b) v t4 h d)) (lift -h d (THead (Bind b) v t3)) (lift_head (Bind b) v t3 h d))))))))))))) (\lambda -(c: C).(\lambda (v: T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (sty0 g -c t3 t4)).(\lambda (H1: ((\forall (c0: C).(\forall (h: nat).(\forall (d: -nat).((drop h d c0 c) \to (sty0 g c0 (lift h d t3) (lift h d -t4)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2: -(drop h d c0 c)).(eq_ind_r T (THead (Flat Appl) (lift h d v) (lift h (s (Flat -Appl) d) t3)) (\lambda (t: T).(sty0 g c0 t (lift h d (THead (Flat Appl) v -t4)))) (eq_ind_r T (THead (Flat Appl) (lift h d v) (lift h (s (Flat Appl) d) -t4)) (\lambda (t: T).(sty0 g c0 (THead (Flat Appl) (lift h d v) (lift h (s -(Flat Appl) d) t3)) t)) (sty0_appl g c0 (lift h d v) (lift h (s (Flat Appl) -d) t3) (lift h (s (Flat Appl) d) t4) (H1 c0 h (s (Flat Appl) d) H2)) (lift h -d (THead (Flat Appl) v t4)) (lift_head (Flat Appl) v t4 h d)) (lift h d -(THead (Flat Appl) v t3)) (lift_head (Flat Appl) v t3 h d)))))))))))) -(\lambda (c: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (sty0 g c v1 -v2)).(\lambda (H1: ((\forall (c0: C).(\forall (h: nat).(\forall (d: -nat).((drop h d c0 c) \to (sty0 g c0 (lift h d v1) (lift h d -v2)))))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (sty0 g c t3 -t4)).(\lambda (H3: ((\forall (c0: C).(\forall (h: nat).(\forall (d: -nat).((drop h d c0 c) \to (sty0 g c0 (lift h d t3) (lift h d -t4)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H4: -(drop h d c0 c)).(eq_ind_r T (THead (Flat Cast) (lift h d v1) (lift h (s -(Flat Cast) d) t3)) (\lambda (t: T).(sty0 g c0 t (lift h d (THead (Flat Cast) -v2 t4)))) (eq_ind_r T (THead (Flat Cast) (lift h d v2) (lift h (s (Flat Cast) -d) t4)) (\lambda (t: T).(sty0 g c0 (THead (Flat Cast) (lift h d v1) (lift h -(s (Flat Cast) d) t3)) t)) (sty0_cast g c0 (lift h d v1) (lift h d v2) (H1 c0 -h d H4) (lift h (s (Flat Cast) d) t3) (lift h (s (Flat Cast) d) t4) (H3 c0 h -(s (Flat Cast) d) H4)) (lift h d (THead (Flat Cast) v2 t4)) (lift_head (Flat -Cast) v2 t4 h d)) (lift h d (THead (Flat Cast) v1 t3)) (lift_head (Flat Cast) -v1 t3 h d))))))))))))))) e t1 t2 H))))). -(* COMMENTS -Initial nodes: 3677 -END *) - -theorem sty0_correct: - \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((sty0 g c -t1 t) \to (ex T (\lambda (t2: T).(sty0 g c t t2))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H: -(sty0 g c t1 t)).(sty0_ind g (\lambda (c0: C).(\lambda (_: T).(\lambda (t2: -T).(ex T (\lambda (t3: T).(sty0 g c0 t2 t3)))))) (\lambda (c0: C).(\lambda -(n: nat).(ex_intro T (\lambda (t2: T).(sty0 g c0 (TSort (next g n)) t2)) -(TSort (next g (next g n))) (sty0_sort g c0 (next g n))))) (\lambda (c0: -C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 -(CHead d (Bind Abbr) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v -w)).(\lambda (H2: (ex T (\lambda (t2: T).(sty0 g d w t2)))).(let H3 \def H2 -in (ex_ind T (\lambda (t2: T).(sty0 g d w t2)) (ex T (\lambda (t2: T).(sty0 g -c0 (lift (S i) O w) t2))) (\lambda (x: T).(\lambda (H4: (sty0 g d w -x)).(ex_intro T (\lambda (t2: T).(sty0 g c0 (lift (S i) O w) t2)) (lift (S i) -O x) (sty0_lift g d w x H4 c0 (S i) O (getl_drop Abbr c0 d v i H0))))) -H3)))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: -nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abst) v))).(\lambda (w: -T).(\lambda (H1: (sty0 g d v w)).(\lambda (H2: (ex T (\lambda (t2: T).(sty0 g -d w t2)))).(let H3 \def H2 in (ex_ind T (\lambda (t2: T).(sty0 g d w t2)) (ex -T (\lambda (t2: T).(sty0 g c0 (lift (S i) O v) t2))) (\lambda (x: T).(\lambda -(_: (sty0 g d w x)).(ex_intro T (\lambda (t2: T).(sty0 g c0 (lift (S i) O v) -t2)) (lift (S i) O w) (sty0_lift g d v w H1 c0 (S i) O (getl_drop Abst c0 d v -i H0))))) H3)))))))))) (\lambda (b: B).(\lambda (c0: C).(\lambda (v: -T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (sty0 g (CHead c0 (Bind b) -v) t2 t3)).(\lambda (H1: (ex T (\lambda (t4: T).(sty0 g (CHead c0 (Bind b) v) -t3 t4)))).(let H2 \def H1 in (ex_ind T (\lambda (t4: T).(sty0 g (CHead c0 -(Bind b) v) t3 t4)) (ex T (\lambda (t4: T).(sty0 g c0 (THead (Bind b) v t3) -t4))) (\lambda (x: T).(\lambda (H3: (sty0 g (CHead c0 (Bind b) v) t3 -x)).(ex_intro T (\lambda (t4: T).(sty0 g c0 (THead (Bind b) v t3) t4)) (THead -(Bind b) v x) (sty0_bind g b c0 v t3 x H3)))) H2))))))))) (\lambda (c0: -C).(\lambda (v: T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (sty0 g c0 -t2 t3)).(\lambda (H1: (ex T (\lambda (t4: T).(sty0 g c0 t3 t4)))).(let H2 -\def H1 in (ex_ind T (\lambda (t4: T).(sty0 g c0 t3 t4)) (ex T (\lambda (t4: -T).(sty0 g c0 (THead (Flat Appl) v t3) t4))) (\lambda (x: T).(\lambda (H3: -(sty0 g c0 t3 x)).(ex_intro T (\lambda (t4: T).(sty0 g c0 (THead (Flat Appl) -v t3) t4)) (THead (Flat Appl) v x) (sty0_appl g c0 v t3 x H3)))) H2)))))))) -(\lambda (c0: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (sty0 g c0 v1 -v2)).(\lambda (H1: (ex T (\lambda (t2: T).(sty0 g c0 v2 t2)))).(\lambda (t2: -T).(\lambda (t3: T).(\lambda (_: (sty0 g c0 t2 t3)).(\lambda (H3: (ex T -(\lambda (t4: T).(sty0 g c0 t3 t4)))).(let H4 \def H1 in (ex_ind T (\lambda -(t4: T).(sty0 g c0 v2 t4)) (ex T (\lambda (t4: T).(sty0 g c0 (THead (Flat -Cast) v2 t3) t4))) (\lambda (x: T).(\lambda (H5: (sty0 g c0 v2 x)).(let H6 -\def H3 in (ex_ind T (\lambda (t4: T).(sty0 g c0 t3 t4)) (ex T (\lambda (t4: -T).(sty0 g c0 (THead (Flat Cast) v2 t3) t4))) (\lambda (x0: T).(\lambda (H7: -(sty0 g c0 t3 x0)).(ex_intro T (\lambda (t4: T).(sty0 g c0 (THead (Flat Cast) -v2 t3) t4)) (THead (Flat Cast) x x0) (sty0_cast g c0 v2 x H5 t3 x0 H7)))) -H6)))) H4))))))))))) c t1 t H))))). -(* COMMENTS -Initial nodes: 991 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/sty1/cnt.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/sty1/cnt.ma deleted file mode 100644 index 0f39bdfd9..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/sty1/cnt.ma +++ /dev/null @@ -1,89 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/sty1/props.ma". - -include "Basic-1/cnt/props.ma". - -theorem sty1_cnt: - \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((sty0 g c -t1 t) \to (ex2 T (\lambda (t2: T).(sty1 g c t1 t2)) (\lambda (t2: T).(cnt -t2))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H: -(sty0 g c t1 t)).(sty0_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (_: -T).(ex2 T (\lambda (t3: T).(sty1 g c0 t0 t3)) (\lambda (t3: T).(cnt t3)))))) -(\lambda (c0: C).(\lambda (n: nat).(ex_intro2 T (\lambda (t2: T).(sty1 g c0 -(TSort n) t2)) (\lambda (t2: T).(cnt t2)) (TSort (next g n)) (sty1_sty0 g c0 -(TSort n) (TSort (next g n)) (sty0_sort g c0 n)) (cnt_sort (next g n))))) -(\lambda (c0: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda -(H0: (getl i c0 (CHead d (Bind Abbr) v))).(\lambda (w: T).(\lambda (_: (sty0 -g d v w)).(\lambda (H2: (ex2 T (\lambda (t2: T).(sty1 g d v t2)) (\lambda -(t2: T).(cnt t2)))).(let H3 \def H2 in (ex2_ind T (\lambda (t2: T).(sty1 g d -v t2)) (\lambda (t2: T).(cnt t2)) (ex2 T (\lambda (t2: T).(sty1 g c0 (TLRef -i) t2)) (\lambda (t2: T).(cnt t2))) (\lambda (x: T).(\lambda (H4: (sty1 g d v -x)).(\lambda (H5: (cnt x)).(ex_intro2 T (\lambda (t2: T).(sty1 g c0 (TLRef i) -t2)) (\lambda (t2: T).(cnt t2)) (lift (S i) O x) (sty1_abbr g c0 d v i H0 x -H4) (cnt_lift x H5 (S i) O))))) H3)))))))))) (\lambda (c0: C).(\lambda (d: -C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind -Abst) v))).(\lambda (w: T).(\lambda (H1: (sty0 g d v w)).(\lambda (H2: (ex2 T -(\lambda (t2: T).(sty1 g d v t2)) (\lambda (t2: T).(cnt t2)))).(let H3 \def -H2 in (ex2_ind T (\lambda (t2: T).(sty1 g d v t2)) (\lambda (t2: T).(cnt t2)) -(ex2 T (\lambda (t2: T).(sty1 g c0 (TLRef i) t2)) (\lambda (t2: T).(cnt t2))) -(\lambda (x: T).(\lambda (H4: (sty1 g d v x)).(\lambda (H5: (cnt -x)).(ex_intro2 T (\lambda (t2: T).(sty1 g c0 (TLRef i) t2)) (\lambda (t2: -T).(cnt t2)) (lift (S i) O x) (sty1_trans g c0 (TLRef i) (lift (S i) O v) -(sty1_sty0 g c0 (TLRef i) (lift (S i) O v) (sty0_abst g c0 d v i H0 w H1)) -(lift (S i) O x) (sty1_lift g d v x H4 c0 (S i) O (getl_drop Abst c0 d v i -H0))) (cnt_lift x H5 (S i) O))))) H3)))))))))) (\lambda (b: B).(\lambda (c0: -C).(\lambda (v: T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (sty0 g -(CHead c0 (Bind b) v) t2 t3)).(\lambda (H1: (ex2 T (\lambda (t4: T).(sty1 g -(CHead c0 (Bind b) v) t2 t4)) (\lambda (t4: T).(cnt t4)))).(let H2 \def H1 in -(ex2_ind T (\lambda (t4: T).(sty1 g (CHead c0 (Bind b) v) t2 t4)) (\lambda -(t4: T).(cnt t4)) (ex2 T (\lambda (t4: T).(sty1 g c0 (THead (Bind b) v t2) -t4)) (\lambda (t4: T).(cnt t4))) (\lambda (x: T).(\lambda (H3: (sty1 g (CHead -c0 (Bind b) v) t2 x)).(\lambda (H4: (cnt x)).(ex_intro2 T (\lambda (t4: -T).(sty1 g c0 (THead (Bind b) v t2) t4)) (\lambda (t4: T).(cnt t4)) (THead -(Bind b) v x) (sty1_bind g b c0 v t2 x H3) (cnt_head x H4 (Bind b) v))))) -H2))))))))) (\lambda (c0: C).(\lambda (v: T).(\lambda (t2: T).(\lambda (t3: -T).(\lambda (_: (sty0 g c0 t2 t3)).(\lambda (H1: (ex2 T (\lambda (t4: -T).(sty1 g c0 t2 t4)) (\lambda (t4: T).(cnt t4)))).(let H2 \def H1 in -(ex2_ind T (\lambda (t4: T).(sty1 g c0 t2 t4)) (\lambda (t4: T).(cnt t4)) -(ex2 T (\lambda (t4: T).(sty1 g c0 (THead (Flat Appl) v t2) t4)) (\lambda -(t4: T).(cnt t4))) (\lambda (x: T).(\lambda (H3: (sty1 g c0 t2 x)).(\lambda -(H4: (cnt x)).(ex_intro2 T (\lambda (t4: T).(sty1 g c0 (THead (Flat Appl) v -t2) t4)) (\lambda (t4: T).(cnt t4)) (THead (Flat Appl) v x) (sty1_appl g c0 v -t2 x H3) (cnt_head x H4 (Flat Appl) v))))) H2)))))))) (\lambda (c0: -C).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H0: (sty0 g c0 v1 -v2)).(\lambda (_: (ex2 T (\lambda (t2: T).(sty1 g c0 v1 t2)) (\lambda (t2: -T).(cnt t2)))).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (sty0 g c0 t2 -t3)).(\lambda (H3: (ex2 T (\lambda (t4: T).(sty1 g c0 t2 t4)) (\lambda (t4: -T).(cnt t4)))).(let H4 \def H3 in (ex2_ind T (\lambda (t4: T).(sty1 g c0 t2 -t4)) (\lambda (t4: T).(cnt t4)) (ex2 T (\lambda (t4: T).(sty1 g c0 (THead -(Flat Cast) v1 t2) t4)) (\lambda (t4: T).(cnt t4))) (\lambda (x: T).(\lambda -(H5: (sty1 g c0 t2 x)).(\lambda (H6: (cnt x)).(let H_x \def (sty1_cast2 g c0 -t2 x H5 v1 v2 H0) in (let H7 \def H_x in (ex2_ind T (\lambda (v3: T).(sty1 g -c0 v1 v3)) (\lambda (v3: T).(sty1 g c0 (THead (Flat Cast) v1 t2) (THead (Flat -Cast) v3 x))) (ex2 T (\lambda (t4: T).(sty1 g c0 (THead (Flat Cast) v1 t2) -t4)) (\lambda (t4: T).(cnt t4))) (\lambda (x0: T).(\lambda (_: (sty1 g c0 v1 -x0)).(\lambda (H9: (sty1 g c0 (THead (Flat Cast) v1 t2) (THead (Flat Cast) x0 -x))).(ex_intro2 T (\lambda (t4: T).(sty1 g c0 (THead (Flat Cast) v1 t2) t4)) -(\lambda (t4: T).(cnt t4)) (THead (Flat Cast) x0 x) H9 (cnt_head x H6 (Flat -Cast) x0))))) H7)))))) H4))))))))))) c t1 t H))))). -(* COMMENTS -Initial nodes: 1313 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/sty1/defs.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/sty1/defs.ma deleted file mode 100644 index 8fd219524..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/sty1/defs.ma +++ /dev/null @@ -1,23 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/sty0/defs.ma". - -inductive sty1 (g: G) (c: C) (t1: T): T \to Prop \def -| sty1_sty0: \forall (t2: T).((sty0 g c t1 t2) \to (sty1 g c t1 t2)) -| sty1_sing: \forall (t: T).((sty1 g c t1 t) \to (\forall (t2: T).((sty0 g c -t t2) \to (sty1 g c t1 t2)))). - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/sty1/props.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/sty1/props.ma deleted file mode 100644 index 53061952e..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/sty1/props.ma +++ /dev/null @@ -1,163 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/sty1/defs.ma". - -include "Basic-1/sty0/props.ma". - -theorem sty1_trans: - \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((sty1 g c -t1 t) \to (\forall (t2: T).((sty1 g c t t2) \to (sty1 g c t1 t2))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H: -(sty1 g c t1 t)).(\lambda (t2: T).(\lambda (H0: (sty1 g c t t2)).(sty1_ind g -c t (\lambda (t0: T).(sty1 g c t1 t0)) (\lambda (t3: T).(\lambda (H1: (sty0 g -c t t3)).(sty1_sing g c t1 t H t3 H1))) (\lambda (t0: T).(\lambda (_: (sty1 g -c t t0)).(\lambda (H2: (sty1 g c t1 t0)).(\lambda (t3: T).(\lambda (H3: (sty0 -g c t0 t3)).(sty1_sing g c t1 t0 H2 t3 H3)))))) t2 H0))))))). -(* COMMENTS -Initial nodes: 131 -END *) - -theorem sty1_bind: - \forall (g: G).(\forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t1: -T).(\forall (t2: T).((sty1 g (CHead c (Bind b) v) t1 t2) \to (sty1 g c (THead -(Bind b) v t1) (THead (Bind b) v t2)))))))) -\def - \lambda (g: G).(\lambda (b: B).(\lambda (c: C).(\lambda (v: T).(\lambda (t1: -T).(\lambda (t2: T).(\lambda (H: (sty1 g (CHead c (Bind b) v) t1 -t2)).(sty1_ind g (CHead c (Bind b) v) t1 (\lambda (t: T).(sty1 g c (THead -(Bind b) v t1) (THead (Bind b) v t))) (\lambda (t3: T).(\lambda (H0: (sty0 g -(CHead c (Bind b) v) t1 t3)).(sty1_sty0 g c (THead (Bind b) v t1) (THead -(Bind b) v t3) (sty0_bind g b c v t1 t3 H0)))) (\lambda (t: T).(\lambda (_: -(sty1 g (CHead c (Bind b) v) t1 t)).(\lambda (H1: (sty1 g c (THead (Bind b) v -t1) (THead (Bind b) v t))).(\lambda (t3: T).(\lambda (H2: (sty0 g (CHead c -(Bind b) v) t t3)).(sty1_sing g c (THead (Bind b) v t1) (THead (Bind b) v t) -H1 (THead (Bind b) v t3) (sty0_bind g b c v t t3 H2))))))) t2 H))))))). -(* COMMENTS -Initial nodes: 259 -END *) - -theorem sty1_appl: - \forall (g: G).(\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall -(t2: T).((sty1 g c t1 t2) \to (sty1 g c (THead (Flat Appl) v t1) (THead (Flat -Appl) v t2))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (v: T).(\lambda (t1: T).(\lambda -(t2: T).(\lambda (H: (sty1 g c t1 t2)).(sty1_ind g c t1 (\lambda (t: T).(sty1 -g c (THead (Flat Appl) v t1) (THead (Flat Appl) v t))) (\lambda (t3: -T).(\lambda (H0: (sty0 g c t1 t3)).(sty1_sty0 g c (THead (Flat Appl) v t1) -(THead (Flat Appl) v t3) (sty0_appl g c v t1 t3 H0)))) (\lambda (t: -T).(\lambda (_: (sty1 g c t1 t)).(\lambda (H1: (sty1 g c (THead (Flat Appl) v -t1) (THead (Flat Appl) v t))).(\lambda (t3: T).(\lambda (H2: (sty0 g c t -t3)).(sty1_sing g c (THead (Flat Appl) v t1) (THead (Flat Appl) v t) H1 -(THead (Flat Appl) v t3) (sty0_appl g c v t t3 H2))))))) t2 H)))))). -(* COMMENTS -Initial nodes: 213 -END *) - -theorem sty1_lift: - \forall (g: G).(\forall (e: C).(\forall (t1: T).(\forall (t2: T).((sty1 g e -t1 t2) \to (\forall (c: C).(\forall (h: nat).(\forall (d: nat).((drop h d c -e) \to (sty1 g c (lift h d t1) (lift h d t2)))))))))) -\def - \lambda (g: G).(\lambda (e: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H: (sty1 g e t1 t2)).(sty1_ind g e t1 (\lambda (t: T).(\forall (c: -C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to (sty1 g c (lift h -d t1) (lift h d t))))))) (\lambda (t3: T).(\lambda (H0: (sty0 g e t1 -t3)).(\lambda (c: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: (drop -h d c e)).(sty1_sty0 g c (lift h d t1) (lift h d t3) (sty0_lift g e t1 t3 H0 -c h d H1)))))))) (\lambda (t: T).(\lambda (_: (sty1 g e t1 t)).(\lambda (H1: -((\forall (c: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to -(sty1 g c (lift h d t1) (lift h d t)))))))).(\lambda (t3: T).(\lambda (H2: -(sty0 g e t t3)).(\lambda (c: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda -(H3: (drop h d c e)).(sty1_sing g c (lift h d t1) (lift h d t) (H1 c h d H3) -(lift h d t3) (sty0_lift g e t t3 H2 c h d H3))))))))))) t2 H))))). -(* COMMENTS -Initial nodes: 277 -END *) - -theorem sty1_correct: - \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((sty1 g c -t1 t) \to (ex T (\lambda (t2: T).(sty0 g c t t2))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H: -(sty1 g c t1 t)).(sty1_ind g c t1 (\lambda (t0: T).(ex T (\lambda (t2: -T).(sty0 g c t0 t2)))) (\lambda (t2: T).(\lambda (H0: (sty0 g c t1 -t2)).(sty0_correct g c t1 t2 H0))) (\lambda (t0: T).(\lambda (_: (sty1 g c t1 -t0)).(\lambda (_: (ex T (\lambda (t2: T).(sty0 g c t0 t2)))).(\lambda (t2: -T).(\lambda (H2: (sty0 g c t0 t2)).(sty0_correct g c t0 t2 H2)))))) t H))))). -(* COMMENTS -Initial nodes: 123 -END *) - -theorem sty1_abbr: - \forall (g: G).(\forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: -nat).((getl i c (CHead d (Bind Abbr) v)) \to (\forall (w: T).((sty1 g d v w) -\to (sty1 g c (TLRef i) (lift (S i) O w))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: -nat).(\lambda (H: (getl i c (CHead d (Bind Abbr) v))).(\lambda (w: -T).(\lambda (H0: (sty1 g d v w)).(sty1_ind g d v (\lambda (t: T).(sty1 g c -(TLRef i) (lift (S i) O t))) (\lambda (t2: T).(\lambda (H1: (sty0 g d v -t2)).(sty1_sty0 g c (TLRef i) (lift (S i) O t2) (sty0_abbr g c d v i H t2 -H1)))) (\lambda (t: T).(\lambda (_: (sty1 g d v t)).(\lambda (H2: (sty1 g c -(TLRef i) (lift (S i) O t))).(\lambda (t2: T).(\lambda (H3: (sty0 g d t -t2)).(sty1_sing g c (TLRef i) (lift (S i) O t) H2 (lift (S i) O t2) -(sty0_lift g d t t2 H3 c (S i) O (getl_drop Abbr c d v i H)))))))) w -H0)))))))). -(* COMMENTS -Initial nodes: 231 -END *) - -theorem sty1_cast2: - \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((sty1 g c -t1 t2) \to (\forall (v1: T).(\forall (v2: T).((sty0 g c v1 v2) \to (ex2 T -(\lambda (v3: T).(sty1 g c v1 v3)) (\lambda (v3: T).(sty1 g c (THead (Flat -Cast) v1 t1) (THead (Flat Cast) v3 t2))))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H: (sty1 g c t1 t2)).(sty1_ind g c t1 (\lambda (t: T).(\forall (v1: -T).(\forall (v2: T).((sty0 g c v1 v2) \to (ex2 T (\lambda (v3: T).(sty1 g c -v1 v3)) (\lambda (v3: T).(sty1 g c (THead (Flat Cast) v1 t1) (THead (Flat -Cast) v3 t)))))))) (\lambda (t3: T).(\lambda (H0: (sty0 g c t1 t3)).(\lambda -(v1: T).(\lambda (v2: T).(\lambda (H1: (sty0 g c v1 v2)).(ex_intro2 T -(\lambda (v3: T).(sty1 g c v1 v3)) (\lambda (v3: T).(sty1 g c (THead (Flat -Cast) v1 t1) (THead (Flat Cast) v3 t3))) v2 (sty1_sty0 g c v1 v2 H1) -(sty1_sty0 g c (THead (Flat Cast) v1 t1) (THead (Flat Cast) v2 t3) (sty0_cast -g c v1 v2 H1 t1 t3 H0)))))))) (\lambda (t: T).(\lambda (_: (sty1 g c t1 -t)).(\lambda (H1: ((\forall (v1: T).(\forall (v2: T).((sty0 g c v1 v2) \to -(ex2 T (\lambda (v3: T).(sty1 g c v1 v3)) (\lambda (v3: T).(sty1 g c (THead -(Flat Cast) v1 t1) (THead (Flat Cast) v3 t))))))))).(\lambda (t3: T).(\lambda -(H2: (sty0 g c t t3)).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H3: (sty0 g -c v1 v2)).(let H_x \def (H1 v1 v2 H3) in (let H4 \def H_x in (ex2_ind T -(\lambda (v3: T).(sty1 g c v1 v3)) (\lambda (v3: T).(sty1 g c (THead (Flat -Cast) v1 t1) (THead (Flat Cast) v3 t))) (ex2 T (\lambda (v3: T).(sty1 g c v1 -v3)) (\lambda (v3: T).(sty1 g c (THead (Flat Cast) v1 t1) (THead (Flat Cast) -v3 t3)))) (\lambda (x: T).(\lambda (H5: (sty1 g c v1 x)).(\lambda (H6: (sty1 -g c (THead (Flat Cast) v1 t1) (THead (Flat Cast) x t))).(let H_x0 \def -(sty1_correct g c v1 x H5) in (let H7 \def H_x0 in (ex_ind T (\lambda (t4: -T).(sty0 g c x t4)) (ex2 T (\lambda (v3: T).(sty1 g c v1 v3)) (\lambda (v3: -T).(sty1 g c (THead (Flat Cast) v1 t1) (THead (Flat Cast) v3 t3)))) (\lambda -(x0: T).(\lambda (H8: (sty0 g c x x0)).(ex_intro2 T (\lambda (v3: T).(sty1 g -c v1 v3)) (\lambda (v3: T).(sty1 g c (THead (Flat Cast) v1 t1) (THead (Flat -Cast) v3 t3))) x0 (sty1_sing g c v1 x H5 x0 H8) (sty1_sing g c (THead (Flat -Cast) v1 t1) (THead (Flat Cast) x t) H6 (THead (Flat Cast) x0 t3) (sty0_cast -g c x x0 H8 t t3 H2))))) H7)))))) H4))))))))))) t2 H))))). -(* COMMENTS -Initial nodes: 657 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/subst/defs.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/subst/defs.ma deleted file mode 100644 index e0ed86b52..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/subst/defs.ma +++ /dev/null @@ -1,27 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/lift/defs.ma". - -definition subst: - nat \to (T \to (T \to T)) -\def - let rec subst (d: nat) (v: T) (t: T) on t: T \def (match t with [(TSort n) -\Rightarrow (TSort n) | (TLRef i) \Rightarrow (match (blt i d) with [true -\Rightarrow (TLRef i) | false \Rightarrow (match (blt d i) with [true -\Rightarrow (TLRef (pred i)) | false \Rightarrow (lift d O v)])]) | (THead k -u t0) \Rightarrow (THead k (subst d v u) (subst (s k d) v t0))]) in subst. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/subst/fwd.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/subst/fwd.ma deleted file mode 100644 index a0678e5fb..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/subst/fwd.ma +++ /dev/null @@ -1,79 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/subst/defs.ma". - -theorem subst_sort: - \forall (v: T).(\forall (d: nat).(\forall (k: nat).(eq T (subst d v (TSort -k)) (TSort k)))) -\def - \lambda (_: T).(\lambda (_: nat).(\lambda (k: nat).(refl_equal T (TSort -k)))). -(* COMMENTS -Initial nodes: 13 -END *) - -theorem subst_lref_lt: - \forall (v: T).(\forall (d: nat).(\forall (i: nat).((lt i d) \to (eq T -(subst d v (TLRef i)) (TLRef i))))) -\def - \lambda (v: T).(\lambda (d: nat).(\lambda (i: nat).(\lambda (H: (lt i -d)).(eq_ind_r bool true (\lambda (b: bool).(eq T (match b with [true -\Rightarrow (TLRef i) | false \Rightarrow (match (blt d i) with [true -\Rightarrow (TLRef (pred i)) | false \Rightarrow (lift d O v)])]) (TLRef i))) -(refl_equal T (TLRef i)) (blt i d) (lt_blt d i H))))). -(* COMMENTS -Initial nodes: 73 -END *) - -theorem subst_lref_eq: - \forall (v: T).(\forall (i: nat).(eq T (subst i v (TLRef i)) (lift i O v))) -\def - \lambda (v: T).(\lambda (i: nat).(eq_ind_r bool false (\lambda (b: bool).(eq -T (match b with [true \Rightarrow (TLRef i) | false \Rightarrow (match b with -[true \Rightarrow (TLRef (pred i)) | false \Rightarrow (lift i O v)])]) (lift -i O v))) (refl_equal T (lift i O v)) (blt i i) (le_bge i i (le_n i)))). -(* COMMENTS -Initial nodes: 71 -END *) - -theorem subst_lref_gt: - \forall (v: T).(\forall (d: nat).(\forall (i: nat).((lt d i) \to (eq T -(subst d v (TLRef i)) (TLRef (pred i)))))) -\def - \lambda (v: T).(\lambda (d: nat).(\lambda (i: nat).(\lambda (H: (lt d -i)).(eq_ind_r bool false (\lambda (b: bool).(eq T (match b with [true -\Rightarrow (TLRef i) | false \Rightarrow (match (blt d i) with [true -\Rightarrow (TLRef (pred i)) | false \Rightarrow (lift d O v)])]) (TLRef -(pred i)))) (eq_ind_r bool true (\lambda (b: bool).(eq T (match b with [true -\Rightarrow (TLRef (pred i)) | false \Rightarrow (lift d O v)]) (TLRef (pred -i)))) (refl_equal T (TLRef (pred i))) (blt d i) (lt_blt i d H)) (blt i d) -(le_bge d i (lt_le_weak d i H)))))). -(* COMMENTS -Initial nodes: 130 -END *) - -theorem subst_head: - \forall (k: K).(\forall (w: T).(\forall (u: T).(\forall (t: T).(\forall (d: -nat).(eq T (subst d w (THead k u t)) (THead k (subst d w u) (subst (s k d) w -t))))))) -\def - \lambda (k: K).(\lambda (w: T).(\lambda (u: T).(\lambda (t: T).(\lambda (d: -nat).(refl_equal T (THead k (subst d w u) (subst (s k d) w t))))))). -(* COMMENTS -Initial nodes: 37 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/subst/props.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/subst/props.ma deleted file mode 100644 index 3bad044e3..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/subst/props.ma +++ /dev/null @@ -1,116 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/subst/fwd.ma". - -include "Basic-1/subst0/defs.ma". - -include "Basic-1/lift/props.ma". - -theorem subst_lift_SO: - \forall (v: T).(\forall (t: T).(\forall (d: nat).(eq T (subst d v (lift (S -O) d t)) t))) -\def - \lambda (v: T).(\lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d: nat).(eq -T (subst d v (lift (S O) d t0)) t0))) (\lambda (n: nat).(\lambda (d: -nat).(eq_ind_r T (TSort n) (\lambda (t0: T).(eq T (subst d v t0) (TSort n))) -(eq_ind_r T (TSort n) (\lambda (t0: T).(eq T t0 (TSort n))) (refl_equal T -(TSort n)) (subst d v (TSort n)) (subst_sort v d n)) (lift (S O) d (TSort n)) -(lift_sort n (S O) d)))) (\lambda (n: nat).(\lambda (d: nat).(lt_le_e n d (eq -T (subst d v (lift (S O) d (TLRef n))) (TLRef n)) (\lambda (H: (lt n -d)).(eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (subst d v t0) (TLRef n))) -(eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 (TLRef n))) (refl_equal T -(TLRef n)) (subst d v (TLRef n)) (subst_lref_lt v d n H)) (lift (S O) d -(TLRef n)) (lift_lref_lt n (S O) d H))) (\lambda (H: (le d n)).(eq_ind_r T -(TLRef (plus n (S O))) (\lambda (t0: T).(eq T (subst d v t0) (TLRef n))) -(eq_ind nat (S (plus n O)) (\lambda (n0: nat).(eq T (subst d v (TLRef n0)) -(TLRef n))) (eq_ind_r T (TLRef (pred (S (plus n O)))) (\lambda (t0: T).(eq T -t0 (TLRef n))) (eq_ind nat (plus n O) (\lambda (n0: nat).(eq T (TLRef n0) -(TLRef n))) (f_equal nat T TLRef (plus n O) n (sym_eq nat n (plus n O) -(plus_n_O n))) (pred (S (plus n O))) (pred_Sn (plus n O))) (subst d v (TLRef -(S (plus n O)))) (subst_lref_gt v d (S (plus n O)) (le_n_S d (plus n O) -(le_plus_trans d n O H)))) (plus n (S O)) (plus_n_Sm n O)) (lift (S O) d -(TLRef n)) (lift_lref_ge n (S O) d H)))))) (\lambda (k: K).(\lambda (t0: -T).(\lambda (H: ((\forall (d: nat).(eq T (subst d v (lift (S O) d t0)) -t0)))).(\lambda (t1: T).(\lambda (H0: ((\forall (d: nat).(eq T (subst d v -(lift (S O) d t1)) t1)))).(\lambda (d: nat).(eq_ind_r T (THead k (lift (S O) -d t0) (lift (S O) (s k d) t1)) (\lambda (t2: T).(eq T (subst d v t2) (THead k -t0 t1))) (eq_ind_r T (THead k (subst d v (lift (S O) d t0)) (subst (s k d) v -(lift (S O) (s k d) t1))) (\lambda (t2: T).(eq T t2 (THead k t0 t1))) -(f_equal3 K T T T THead k k (subst d v (lift (S O) d t0)) t0 (subst (s k d) v -(lift (S O) (s k d) t1)) t1 (refl_equal K k) (H d) (H0 (s k d))) (subst d v -(THead k (lift (S O) d t0) (lift (S O) (s k d) t1))) (subst_head k v (lift (S -O) d t0) (lift (S O) (s k d) t1) d)) (lift (S O) d (THead k t0 t1)) -(lift_head k t0 t1 (S O) d)))))))) t)). -(* COMMENTS -Initial nodes: 879 -END *) - -theorem subst_subst0: - \forall (v: T).(\forall (t1: T).(\forall (t2: T).(\forall (d: nat).((subst0 -d v t1 t2) \to (eq T (subst d v t1) (subst d v t2)))))) -\def - \lambda (v: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (d: nat).(\lambda -(H: (subst0 d v t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t: -T).(\lambda (t0: T).(\lambda (t3: T).(eq T (subst n t t0) (subst n t t3)))))) -(\lambda (v0: T).(\lambda (i: nat).(eq_ind_r T (lift i O v0) (\lambda (t: -T).(eq T t (subst i v0 (lift (S i) O v0)))) (eq_ind nat (plus (S O) i) -(\lambda (n: nat).(eq T (lift i O v0) (subst i v0 (lift n O v0)))) (eq_ind T -(lift (S O) i (lift i O v0)) (\lambda (t: T).(eq T (lift i O v0) (subst i v0 -t))) (eq_ind_r T (lift i O v0) (\lambda (t: T).(eq T (lift i O v0) t)) -(refl_equal T (lift i O v0)) (subst i v0 (lift (S O) i (lift i O v0))) -(subst_lift_SO v0 (lift i O v0) i)) (lift (plus (S O) i) O v0) (lift_free v0 -i (S O) O i (le_n (plus O i)) (le_O_n i))) (S i) (refl_equal nat (S i))) -(subst i v0 (TLRef i)) (subst_lref_eq v0 i)))) (\lambda (v0: T).(\lambda (u2: -T).(\lambda (u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v0 u1 -u2)).(\lambda (H1: (eq T (subst i v0 u1) (subst i v0 u2))).(\lambda (t: -T).(\lambda (k: K).(eq_ind_r T (THead k (subst i v0 u1) (subst (s k i) v0 t)) -(\lambda (t0: T).(eq T t0 (subst i v0 (THead k u2 t)))) (eq_ind_r T (THead k -(subst i v0 u2) (subst (s k i) v0 t)) (\lambda (t0: T).(eq T (THead k (subst -i v0 u1) (subst (s k i) v0 t)) t0)) (eq_ind_r T (subst i v0 u2) (\lambda (t0: -T).(eq T (THead k t0 (subst (s k i) v0 t)) (THead k (subst i v0 u2) (subst (s -k i) v0 t)))) (refl_equal T (THead k (subst i v0 u2) (subst (s k i) v0 t))) -(subst i v0 u1) H1) (subst i v0 (THead k u2 t)) (subst_head k v0 u2 t i)) -(subst i v0 (THead k u1 t)) (subst_head k v0 u1 t i)))))))))) (\lambda (k: -K).(\lambda (v0: T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (i: -nat).(\lambda (_: (subst0 (s k i) v0 t4 t3)).(\lambda (H1: (eq T (subst (s k -i) v0 t4) (subst (s k i) v0 t3))).(\lambda (u: T).(eq_ind_r T (THead k (subst -i v0 u) (subst (s k i) v0 t4)) (\lambda (t: T).(eq T t (subst i v0 (THead k u -t3)))) (eq_ind_r T (THead k (subst i v0 u) (subst (s k i) v0 t3)) (\lambda -(t: T).(eq T (THead k (subst i v0 u) (subst (s k i) v0 t4)) t)) (eq_ind_r T -(subst (s k i) v0 t3) (\lambda (t: T).(eq T (THead k (subst i v0 u) t) (THead -k (subst i v0 u) (subst (s k i) v0 t3)))) (refl_equal T (THead k (subst i v0 -u) (subst (s k i) v0 t3))) (subst (s k i) v0 t4) H1) (subst i v0 (THead k u -t3)) (subst_head k v0 u t3 i)) (subst i v0 (THead k u t4)) (subst_head k v0 u -t4 i)))))))))) (\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda -(i: nat).(\lambda (_: (subst0 i v0 u1 u2)).(\lambda (H1: (eq T (subst i v0 -u1) (subst i v0 u2))).(\lambda (k: K).(\lambda (t3: T).(\lambda (t4: -T).(\lambda (_: (subst0 (s k i) v0 t3 t4)).(\lambda (H3: (eq T (subst (s k i) -v0 t3) (subst (s k i) v0 t4))).(eq_ind_r T (THead k (subst i v0 u1) (subst (s -k i) v0 t3)) (\lambda (t: T).(eq T t (subst i v0 (THead k u2 t4)))) (eq_ind_r -T (THead k (subst i v0 u2) (subst (s k i) v0 t4)) (\lambda (t: T).(eq T -(THead k (subst i v0 u1) (subst (s k i) v0 t3)) t)) (eq_ind_r T (subst i v0 -u2) (\lambda (t: T).(eq T (THead k t (subst (s k i) v0 t3)) (THead k (subst i -v0 u2) (subst (s k i) v0 t4)))) (eq_ind_r T (subst (s k i) v0 t4) (\lambda -(t: T).(eq T (THead k (subst i v0 u2) t) (THead k (subst i v0 u2) (subst (s k -i) v0 t4)))) (refl_equal T (THead k (subst i v0 u2) (subst (s k i) v0 t4))) -(subst (s k i) v0 t3) H3) (subst i v0 u1) H1) (subst i v0 (THead k u2 t4)) -(subst_head k v0 u2 t4 i)) (subst i v0 (THead k u1 t3)) (subst_head k v0 u1 -t3 i))))))))))))) d v t1 t2 H))))). -(* COMMENTS -Initial nodes: 1363 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/subst0/dec.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/subst0/dec.ma deleted file mode 100644 index 0234ff06c..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/subst0/dec.ma +++ /dev/null @@ -1,182 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/subst0/defs.ma". - -include "Basic-1/lift/props.ma". - -theorem dnf_dec2: - \forall (t: T).(\forall (d: nat).(or (\forall (w: T).(ex T (\lambda (v: -T).(subst0 d w t (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T t (lift (S -O) d v)))))) -\def - \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d: nat).(or (\forall (w: -T).(ex T (\lambda (v: T).(subst0 d w t0 (lift (S O) d v))))) (ex T (\lambda -(v: T).(eq T t0 (lift (S O) d v))))))) (\lambda (n: nat).(\lambda (d: -nat).(or_intror (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (TSort n) -(lift (S O) d v))))) (ex T (\lambda (v: T).(eq T (TSort n) (lift (S O) d -v)))) (ex_intro T (\lambda (v: T).(eq T (TSort n) (lift (S O) d v))) (TSort -n) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq T (TSort n) t0)) (refl_equal T -(TSort n)) (lift (S O) d (TSort n)) (lift_sort n (S O) d)))))) (\lambda (n: -nat).(\lambda (d: nat).(lt_eq_gt_e n d (or (\forall (w: T).(ex T (\lambda (v: -T).(subst0 d w (TLRef n) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T -(TLRef n) (lift (S O) d v))))) (\lambda (H: (lt n d)).(or_intror (\forall (w: -T).(ex T (\lambda (v: T).(subst0 d w (TLRef n) (lift (S O) d v))))) (ex T -(\lambda (v: T).(eq T (TLRef n) (lift (S O) d v)))) (ex_intro T (\lambda (v: -T).(eq T (TLRef n) (lift (S O) d v))) (TLRef n) (eq_ind_r T (TLRef n) -(\lambda (t0: T).(eq T (TLRef n) t0)) (refl_equal T (TLRef n)) (lift (S O) d -(TLRef n)) (lift_lref_lt n (S O) d H))))) (\lambda (H: (eq nat n d)).(eq_ind -nat n (\lambda (n0: nat).(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 n0 -w (TLRef n) (lift (S O) n0 v))))) (ex T (\lambda (v: T).(eq T (TLRef n) (lift -(S O) n0 v)))))) (or_introl (\forall (w: T).(ex T (\lambda (v: T).(subst0 n w -(TLRef n) (lift (S O) n v))))) (ex T (\lambda (v: T).(eq T (TLRef n) (lift (S -O) n v)))) (\lambda (w: T).(ex_intro T (\lambda (v: T).(subst0 n w (TLRef n) -(lift (S O) n v))) (lift n O w) (eq_ind_r T (lift (plus (S O) n) O w) -(\lambda (t0: T).(subst0 n w (TLRef n) t0)) (subst0_lref w n) (lift (S O) n -(lift n O w)) (lift_free w n (S O) O n (le_n (plus O n)) (le_O_n n)))))) d -H)) (\lambda (H: (lt d n)).(or_intror (\forall (w: T).(ex T (\lambda (v: -T).(subst0 d w (TLRef n) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T -(TLRef n) (lift (S O) d v)))) (ex_intro T (\lambda (v: T).(eq T (TLRef n) -(lift (S O) d v))) (TLRef (pred n)) (eq_ind_r T (TLRef n) (\lambda (t0: -T).(eq T (TLRef n) t0)) (refl_equal T (TLRef n)) (lift (S O) d (TLRef (pred -n))) (lift_lref_gt d n H)))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda -(H: ((\forall (d: nat).(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w -t0 (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T t0 (lift (S O) d -v)))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (d: nat).(or (\forall (w: -T).(ex T (\lambda (v: T).(subst0 d w t1 (lift (S O) d v))))) (ex T (\lambda -(v: T).(eq T t1 (lift (S O) d v)))))))).(\lambda (d: nat).(let H_x \def (H d) -in (let H1 \def H_x in (or_ind (\forall (w: T).(ex T (\lambda (v: T).(subst0 -d w t0 (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T t0 (lift (S O) d -v)))) (or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) -(lift (S O) d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 t1) (lift (S O) -d v))))) (\lambda (H2: ((\forall (w: T).(ex T (\lambda (v: T).(subst0 d w t0 -(lift (S O) d v))))))).(let H_x0 \def (H0 (s k d)) in (let H3 \def H_x0 in -(or_ind (\forall (w: T).(ex T (\lambda (v: T).(subst0 (s k d) w t1 (lift (S -O) (s k d) v))))) (ex T (\lambda (v: T).(eq T t1 (lift (S O) (s k d) v)))) -(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) (lift -(S O) d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 t1) (lift (S O) d -v))))) (\lambda (H4: ((\forall (w: T).(ex T (\lambda (v: T).(subst0 (s k d) w -t1 (lift (S O) (s k d) v))))))).(or_introl (\forall (w: T).(ex T (\lambda (v: -T).(subst0 d w (THead k t0 t1) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq -T (THead k t0 t1) (lift (S O) d v)))) (\lambda (w: T).(let H_x1 \def (H4 w) -in (let H5 \def H_x1 in (ex_ind T (\lambda (v: T).(subst0 (s k d) w t1 (lift -(S O) (s k d) v))) (ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) (lift (S -O) d v)))) (\lambda (x: T).(\lambda (H6: (subst0 (s k d) w t1 (lift (S O) (s -k d) x))).(let H_x2 \def (H2 w) in (let H7 \def H_x2 in (ex_ind T (\lambda -(v: T).(subst0 d w t0 (lift (S O) d v))) (ex T (\lambda (v: T).(subst0 d w -(THead k t0 t1) (lift (S O) d v)))) (\lambda (x0: T).(\lambda (H8: (subst0 d -w t0 (lift (S O) d x0))).(ex_intro T (\lambda (v: T).(subst0 d w (THead k t0 -t1) (lift (S O) d v))) (THead k x0 x) (eq_ind_r T (THead k (lift (S O) d x0) -(lift (S O) (s k d) x)) (\lambda (t2: T).(subst0 d w (THead k t0 t1) t2)) -(subst0_both w t0 (lift (S O) d x0) d H8 k t1 (lift (S O) (s k d) x) H6) -(lift (S O) d (THead k x0 x)) (lift_head k x0 x (S O) d))))) H7))))) H5)))))) -(\lambda (H4: (ex T (\lambda (v: T).(eq T t1 (lift (S O) (s k d) -v))))).(ex_ind T (\lambda (v: T).(eq T t1 (lift (S O) (s k d) v))) (or -(\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) (lift (S O) -d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 t1) (lift (S O) d v))))) -(\lambda (x: T).(\lambda (H5: (eq T t1 (lift (S O) (s k d) x))).(eq_ind_r T -(lift (S O) (s k d) x) (\lambda (t2: T).(or (\forall (w: T).(ex T (\lambda -(v: T).(subst0 d w (THead k t0 t2) (lift (S O) d v))))) (ex T (\lambda (v: -T).(eq T (THead k t0 t2) (lift (S O) d v)))))) (or_introl (\forall (w: T).(ex -T (\lambda (v: T).(subst0 d w (THead k t0 (lift (S O) (s k d) x)) (lift (S O) -d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 (lift (S O) (s k d) x)) -(lift (S O) d v)))) (\lambda (w: T).(let H_x1 \def (H2 w) in (let H6 \def -H_x1 in (ex_ind T (\lambda (v: T).(subst0 d w t0 (lift (S O) d v))) (ex T -(\lambda (v: T).(subst0 d w (THead k t0 (lift (S O) (s k d) x)) (lift (S O) d -v)))) (\lambda (x0: T).(\lambda (H7: (subst0 d w t0 (lift (S O) d -x0))).(ex_intro T (\lambda (v: T).(subst0 d w (THead k t0 (lift (S O) (s k d) -x)) (lift (S O) d v))) (THead k x0 x) (eq_ind_r T (THead k (lift (S O) d x0) -(lift (S O) (s k d) x)) (\lambda (t2: T).(subst0 d w (THead k t0 (lift (S O) -(s k d) x)) t2)) (subst0_fst w (lift (S O) d x0) t0 d H7 (lift (S O) (s k d) -x) k) (lift (S O) d (THead k x0 x)) (lift_head k x0 x (S O) d))))) H6))))) t1 -H5))) H4)) H3)))) (\lambda (H2: (ex T (\lambda (v: T).(eq T t0 (lift (S O) d -v))))).(ex_ind T (\lambda (v: T).(eq T t0 (lift (S O) d v))) (or (\forall (w: -T).(ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) (lift (S O) d v))))) (ex -T (\lambda (v: T).(eq T (THead k t0 t1) (lift (S O) d v))))) (\lambda (x: -T).(\lambda (H3: (eq T t0 (lift (S O) d x))).(let H_x0 \def (H0 (s k d)) in -(let H4 \def H_x0 in (or_ind (\forall (w: T).(ex T (\lambda (v: T).(subst0 (s -k d) w t1 (lift (S O) (s k d) v))))) (ex T (\lambda (v: T).(eq T t1 (lift (S -O) (s k d) v)))) (or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead -k t0 t1) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 t1) -(lift (S O) d v))))) (\lambda (H5: ((\forall (w: T).(ex T (\lambda (v: -T).(subst0 (s k d) w t1 (lift (S O) (s k d) v))))))).(eq_ind_r T (lift (S O) -d x) (\lambda (t2: T).(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w -(THead k t2 t1) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T (THead k t2 -t1) (lift (S O) d v)))))) (or_introl (\forall (w: T).(ex T (\lambda (v: -T).(subst0 d w (THead k (lift (S O) d x) t1) (lift (S O) d v))))) (ex T -(\lambda (v: T).(eq T (THead k (lift (S O) d x) t1) (lift (S O) d v)))) -(\lambda (w: T).(let H_x1 \def (H5 w) in (let H6 \def H_x1 in (ex_ind T -(\lambda (v: T).(subst0 (s k d) w t1 (lift (S O) (s k d) v))) (ex T (\lambda -(v: T).(subst0 d w (THead k (lift (S O) d x) t1) (lift (S O) d v)))) (\lambda -(x0: T).(\lambda (H7: (subst0 (s k d) w t1 (lift (S O) (s k d) -x0))).(ex_intro T (\lambda (v: T).(subst0 d w (THead k (lift (S O) d x) t1) -(lift (S O) d v))) (THead k x x0) (eq_ind_r T (THead k (lift (S O) d x) (lift -(S O) (s k d) x0)) (\lambda (t2: T).(subst0 d w (THead k (lift (S O) d x) t1) -t2)) (subst0_snd k w (lift (S O) (s k d) x0) t1 d H7 (lift (S O) d x)) (lift -(S O) d (THead k x x0)) (lift_head k x x0 (S O) d))))) H6))))) t0 H3)) -(\lambda (H5: (ex T (\lambda (v: T).(eq T t1 (lift (S O) (s k d) -v))))).(ex_ind T (\lambda (v: T).(eq T t1 (lift (S O) (s k d) v))) (or -(\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) (lift (S O) -d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 t1) (lift (S O) d v))))) -(\lambda (x0: T).(\lambda (H6: (eq T t1 (lift (S O) (s k d) x0))).(eq_ind_r T -(lift (S O) (s k d) x0) (\lambda (t2: T).(or (\forall (w: T).(ex T (\lambda -(v: T).(subst0 d w (THead k t0 t2) (lift (S O) d v))))) (ex T (\lambda (v: -T).(eq T (THead k t0 t2) (lift (S O) d v)))))) (eq_ind_r T (lift (S O) d x) -(\lambda (t2: T).(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead -k t2 (lift (S O) (s k d) x0)) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq -T (THead k t2 (lift (S O) (s k d) x0)) (lift (S O) d v)))))) (or_intror -(\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead k (lift (S O) d x) -(lift (S O) (s k d) x0)) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T -(THead k (lift (S O) d x) (lift (S O) (s k d) x0)) (lift (S O) d v)))) -(ex_intro T (\lambda (v: T).(eq T (THead k (lift (S O) d x) (lift (S O) (s k -d) x0)) (lift (S O) d v))) (THead k x x0) (eq_ind_r T (THead k (lift (S O) d -x) (lift (S O) (s k d) x0)) (\lambda (t2: T).(eq T (THead k (lift (S O) d x) -(lift (S O) (s k d) x0)) t2)) (refl_equal T (THead k (lift (S O) d x) (lift -(S O) (s k d) x0))) (lift (S O) d (THead k x x0)) (lift_head k x x0 (S O) -d)))) t0 H3) t1 H6))) H5)) H4))))) H2)) H1))))))))) t). -(* COMMENTS -Initial nodes: 3549 -END *) - -theorem dnf_dec: - \forall (w: T).(\forall (t: T).(\forall (d: nat).(ex T (\lambda (v: T).(or -(subst0 d w t (lift (S O) d v)) (eq T t (lift (S O) d v))))))) -\def - \lambda (w: T).(\lambda (t: T).(\lambda (d: nat).(let H_x \def (dnf_dec2 t -d) in (let H \def H_x in (or_ind (\forall (w0: T).(ex T (\lambda (v: -T).(subst0 d w0 t (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T t (lift (S -O) d v)))) (ex T (\lambda (v: T).(or (subst0 d w t (lift (S O) d v)) (eq T t -(lift (S O) d v))))) (\lambda (H0: ((\forall (w0: T).(ex T (\lambda (v: -T).(subst0 d w0 t (lift (S O) d v))))))).(let H_x0 \def (H0 w) in (let H1 -\def H_x0 in (ex_ind T (\lambda (v: T).(subst0 d w t (lift (S O) d v))) (ex T -(\lambda (v: T).(or (subst0 d w t (lift (S O) d v)) (eq T t (lift (S O) d -v))))) (\lambda (x: T).(\lambda (H2: (subst0 d w t (lift (S O) d -x))).(ex_intro T (\lambda (v: T).(or (subst0 d w t (lift (S O) d v)) (eq T t -(lift (S O) d v)))) x (or_introl (subst0 d w t (lift (S O) d x)) (eq T t -(lift (S O) d x)) H2)))) H1)))) (\lambda (H0: (ex T (\lambda (v: T).(eq T t -(lift (S O) d v))))).(ex_ind T (\lambda (v: T).(eq T t (lift (S O) d v))) (ex -T (\lambda (v: T).(or (subst0 d w t (lift (S O) d v)) (eq T t (lift (S O) d -v))))) (\lambda (x: T).(\lambda (H1: (eq T t (lift (S O) d x))).(eq_ind_r T -(lift (S O) d x) (\lambda (t0: T).(ex T (\lambda (v: T).(or (subst0 d w t0 -(lift (S O) d v)) (eq T t0 (lift (S O) d v)))))) (ex_intro T (\lambda (v: -T).(or (subst0 d w (lift (S O) d x) (lift (S O) d v)) (eq T (lift (S O) d x) -(lift (S O) d v)))) x (or_intror (subst0 d w (lift (S O) d x) (lift (S O) d -x)) (eq T (lift (S O) d x) (lift (S O) d x)) (refl_equal T (lift (S O) d -x)))) t H1))) H0)) H))))). -(* COMMENTS -Initial nodes: 603 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/subst0/defs.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/subst0/defs.ma deleted file mode 100644 index a493a7ac2..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/subst0/defs.ma +++ /dev/null @@ -1,32 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/lift/defs.ma". - -inductive subst0: nat \to (T \to (T \to (T \to Prop))) \def -| subst0_lref: \forall (v: T).(\forall (i: nat).(subst0 i v (TLRef i) (lift -(S i) O v))) -| subst0_fst: \forall (v: T).(\forall (u2: T).(\forall (u1: T).(\forall (i: -nat).((subst0 i v u1 u2) \to (\forall (t: T).(\forall (k: K).(subst0 i v -(THead k u1 t) (THead k u2 t)))))))) -| subst0_snd: \forall (k: K).(\forall (v: T).(\forall (t2: T).(\forall (t1: -T).(\forall (i: nat).((subst0 (s k i) v t1 t2) \to (\forall (u: T).(subst0 i -v (THead k u t1) (THead k u t2)))))))) -| subst0_both: \forall (v: T).(\forall (u1: T).(\forall (u2: T).(\forall (i: -nat).((subst0 i v u1 u2) \to (\forall (k: K).(\forall (t1: T).(\forall (t2: -T).((subst0 (s k i) v t1 t2) \to (subst0 i v (THead k u1 t1) (THead k u2 -t2)))))))))). - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/subst0/fwd.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/subst0/fwd.ma deleted file mode 100644 index 165555fe2..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/subst0/fwd.ma +++ /dev/null @@ -1,728 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/subst0/defs.ma". - -include "Basic-1/lift/props.ma". - -theorem subst0_gen_sort: - \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst0 -i v (TSort n) x) \to (\forall (P: Prop).P))))) -\def - \lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda -(H: (subst0 i v (TSort n) x)).(\lambda (P: Prop).(insert_eq T (TSort n) -(\lambda (t: T).(subst0 i v t x)) (\lambda (_: T).P) (\lambda (y: T).(\lambda -(H0: (subst0 i v y x)).(subst0_ind (\lambda (_: nat).(\lambda (_: T).(\lambda -(t0: T).(\lambda (_: T).((eq T t0 (TSort n)) \to P))))) (\lambda (_: -T).(\lambda (i0: nat).(\lambda (H1: (eq T (TLRef i0) (TSort n))).(let H2 \def -(eq_ind T (TLRef i0) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | -(THead _ _ _) \Rightarrow False])) I (TSort n) H1) in (False_ind P H2))))) -(\lambda (v0: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i0: -nat).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (_: (((eq T u1 (TSort n)) -\to P))).(\lambda (t: T).(\lambda (k: K).(\lambda (H3: (eq T (THead k u1 t) -(TSort n))).(let H4 \def (eq_ind T (THead k u1 t) (\lambda (ee: T).(match ee -in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef -_) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H3) in -(False_ind P H4))))))))))) (\lambda (k: K).(\lambda (v0: T).(\lambda (t2: -T).(\lambda (t1: T).(\lambda (i0: nat).(\lambda (_: (subst0 (s k i0) v0 t1 -t2)).(\lambda (_: (((eq T t1 (TSort n)) \to P))).(\lambda (u: T).(\lambda -(H3: (eq T (THead k u t1) (TSort n))).(let H4 \def (eq_ind T (THead k u t1) -(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow -True])) I (TSort n) H3) in (False_ind P H4))))))))))) (\lambda (v0: -T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i0: nat).(\lambda (_: (subst0 -i0 v0 u1 u2)).(\lambda (_: (((eq T u1 (TSort n)) \to P))).(\lambda (k: -K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (subst0 (s k i0) v0 t1 -t2)).(\lambda (_: (((eq T t1 (TSort n)) \to P))).(\lambda (H5: (eq T (THead k -u1 t1) (TSort n))).(let H6 \def (eq_ind T (THead k u1 t1) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I -(TSort n) H5) in (False_ind P H6)))))))))))))) i v y x H0))) H)))))). -(* COMMENTS -Initial nodes: 445 -END *) - -theorem subst0_gen_lref: - \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst0 -i v (TLRef n) x) \to (land (eq nat n i) (eq T x (lift (S n) O v))))))) -\def - \lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda -(H: (subst0 i v (TLRef n) x)).(insert_eq T (TLRef n) (\lambda (t: T).(subst0 -i v t x)) (\lambda (_: T).(land (eq nat n i) (eq T x (lift (S n) O v)))) -(\lambda (y: T).(\lambda (H0: (subst0 i v y x)).(subst0_ind (\lambda (n0: -nat).(\lambda (t: T).(\lambda (t0: T).(\lambda (t1: T).((eq T t0 (TLRef n)) -\to (land (eq nat n n0) (eq T t1 (lift (S n) O t)))))))) (\lambda (v0: -T).(\lambda (i0: nat).(\lambda (H1: (eq T (TLRef i0) (TLRef n))).(let H2 \def -(f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with -[(TSort _) \Rightarrow i0 | (TLRef n0) \Rightarrow n0 | (THead _ _ _) -\Rightarrow i0])) (TLRef i0) (TLRef n) H1) in (eq_ind_r nat n (\lambda (n0: -nat).(land (eq nat n n0) (eq T (lift (S n0) O v0) (lift (S n) O v0)))) (conj -(eq nat n n) (eq T (lift (S n) O v0) (lift (S n) O v0)) (refl_equal nat n) -(refl_equal T (lift (S n) O v0))) i0 H2))))) (\lambda (v0: T).(\lambda (u2: -T).(\lambda (u1: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v0 u1 -u2)).(\lambda (_: (((eq T u1 (TLRef n)) \to (land (eq nat n i0) (eq T u2 -(lift (S n) O v0)))))).(\lambda (t: T).(\lambda (k: K).(\lambda (H3: (eq T -(THead k u1 t) (TLRef n))).(let H4 \def (eq_ind T (THead k u1 t) (\lambda -(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow -True])) I (TLRef n) H3) in (False_ind (land (eq nat n i0) (eq T (THead k u2 -t) (lift (S n) O v0))) H4))))))))))) (\lambda (k: K).(\lambda (v0: -T).(\lambda (t2: T).(\lambda (t1: T).(\lambda (i0: nat).(\lambda (_: (subst0 -(s k i0) v0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (land (eq nat n (s -k i0)) (eq T t2 (lift (S n) O v0)))))).(\lambda (u: T).(\lambda (H3: (eq T -(THead k u t1) (TLRef n))).(let H4 \def (eq_ind T (THead k u t1) (\lambda -(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow -True])) I (TLRef n) H3) in (False_ind (land (eq nat n i0) (eq T (THead k u -t2) (lift (S n) O v0))) H4))))))))))) (\lambda (v0: T).(\lambda (u1: -T).(\lambda (u2: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v0 u1 -u2)).(\lambda (_: (((eq T u1 (TLRef n)) \to (land (eq nat n i0) (eq T u2 -(lift (S n) O v0)))))).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (_: (subst0 (s k i0) v0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef -n)) \to (land (eq nat n (s k i0)) (eq T t2 (lift (S n) O v0)))))).(\lambda -(H5: (eq T (THead k u1 t1) (TLRef n))).(let H6 \def (eq_ind T (THead k u1 t1) -(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow -True])) I (TLRef n) H5) in (False_ind (land (eq nat n i0) (eq T (THead k u2 -t2) (lift (S n) O v0))) H6)))))))))))))) i v y x H0))) H))))). -(* COMMENTS -Initial nodes: 779 -END *) - -theorem subst0_gen_head: - \forall (k: K).(\forall (v: T).(\forall (u1: T).(\forall (t1: T).(\forall -(x: T).(\forall (i: nat).((subst0 i v (THead k u1 t1) x) \to (or3 (ex2 T -(\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 -u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k u1 t2))) (\lambda (t2: -T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1 -u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1 t2))))))))))) -\def - \lambda (k: K).(\lambda (v: T).(\lambda (u1: T).(\lambda (t1: T).(\lambda -(x: T).(\lambda (i: nat).(\lambda (H: (subst0 i v (THead k u1 t1) -x)).(insert_eq T (THead k u1 t1) (\lambda (t: T).(subst0 i v t x)) (\lambda -(_: T).(or3 (ex2 T (\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2: -T).(subst0 i v u1 u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k u1 t2))) -(\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1 -t2)))))) (\lambda (y: T).(\lambda (H0: (subst0 i v y x)).(subst0_ind (\lambda -(n: nat).(\lambda (t: T).(\lambda (t0: T).(\lambda (t2: T).((eq T t0 (THead k -u1 t1)) \to (or3 (ex2 T (\lambda (u2: T).(eq T t2 (THead k u2 t1))) (\lambda -(u2: T).(subst0 n t u1 u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead k u1 -t3))) (\lambda (t3: T).(subst0 (s k n) t t1 t3))) (ex3_2 T T (\lambda (u2: -T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 n t u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k n) t t1 -t3)))))))))) (\lambda (v0: T).(\lambda (i0: nat).(\lambda (H1: (eq T (TLRef -i0) (THead k u1 t1))).(let H2 \def (eq_ind T (TLRef i0) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I -(THead k u1 t1) H1) in (False_ind (or3 (ex2 T (\lambda (u2: T).(eq T (lift (S -i0) O v0) (THead k u2 t1))) (\lambda (u2: T).(subst0 i0 v0 u1 u2))) (ex2 T -(\lambda (t2: T).(eq T (lift (S i0) O v0) (THead k u1 t2))) (\lambda (t2: -T).(subst0 (s k i0) v0 t1 t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T (lift (S i0) O v0) (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 i0 v0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i0) -v0 t1 t2))))) H2))))) (\lambda (v0: T).(\lambda (u2: T).(\lambda (u0: -T).(\lambda (i0: nat).(\lambda (H1: (subst0 i0 v0 u0 u2)).(\lambda (H2: (((eq -T u0 (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T u2 (THead k u3 -t1))) (\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t2: T).(eq T -u2 (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))) (ex3_2 T T -(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead k u3 t2)))) (\lambda (u3: -T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t2: -T).(subst0 (s k i0) v0 t1 t2)))))))).(\lambda (t: T).(\lambda (k0: -K).(\lambda (H3: (eq T (THead k0 u0 t) (THead k u1 t1))).(let H4 \def -(f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with -[(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _ _) -\Rightarrow k1])) (THead k0 u0 t) (THead k u1 t1) H3) in ((let H5 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t0 _) -\Rightarrow t0])) (THead k0 u0 t) (THead k u1 t1) H3) in ((let H6 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t0) -\Rightarrow t0])) (THead k0 u0 t) (THead k u1 t1) H3) in (\lambda (H7: (eq T -u0 u1)).(\lambda (H8: (eq K k0 k)).(eq_ind_r K k (\lambda (k1: K).(or3 (ex2 T -(\lambda (u3: T).(eq T (THead k1 u2 t) (THead k u3 t1))) (\lambda (u3: -T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t2: T).(eq T (THead k1 u2 t) -(THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))) (ex3_2 T T -(\lambda (u3: T).(\lambda (t2: T).(eq T (THead k1 u2 t) (THead k u3 t2)))) -(\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: -T).(\lambda (t2: T).(subst0 (s k i0) v0 t1 t2)))))) (eq_ind_r T t1 (\lambda -(t0: T).(or3 (ex2 T (\lambda (u3: T).(eq T (THead k u2 t0) (THead k u3 t1))) -(\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t2: T).(eq T (THead -k u2 t0) (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))) -(ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead k u2 t0) (THead k -u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda -(_: T).(\lambda (t2: T).(subst0 (s k i0) v0 t1 t2)))))) (let H9 \def (eq_ind -T u0 (\lambda (t0: T).((eq T t0 (THead k u1 t1)) \to (or3 (ex2 T (\lambda -(u3: T).(eq T u2 (THead k u3 t1))) (\lambda (u3: T).(subst0 i0 v0 u1 u3))) -(ex2 T (\lambda (t2: T).(eq T u2 (THead k u1 t2))) (\lambda (t2: T).(subst0 -(s k i0) v0 t1 t2))) (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 -(THead k u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) -(\lambda (_: T).(\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))))))) H2 u1 H7) -in (let H10 \def (eq_ind T u0 (\lambda (t0: T).(subst0 i0 v0 t0 u2)) H1 u1 -H7) in (or3_intro0 (ex2 T (\lambda (u3: T).(eq T (THead k u2 t1) (THead k u3 -t1))) (\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t2: T).(eq T -(THead k u2 t1) (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1 -t2))) (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead k u2 t1) -(THead k u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) -(\lambda (_: T).(\lambda (t2: T).(subst0 (s k i0) v0 t1 t2)))) (ex_intro2 T -(\lambda (u3: T).(eq T (THead k u2 t1) (THead k u3 t1))) (\lambda (u3: -T).(subst0 i0 v0 u1 u3)) u2 (refl_equal T (THead k u2 t1)) H10)))) t H6) k0 -H8)))) H5)) H4))))))))))) (\lambda (k0: K).(\lambda (v0: T).(\lambda (t2: -T).(\lambda (t0: T).(\lambda (i0: nat).(\lambda (H1: (subst0 (s k0 i0) v0 t0 -t2)).(\lambda (H2: (((eq T t0 (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u2: -T).(eq T t2 (THead k u2 t1))) (\lambda (u2: T).(subst0 (s k0 i0) v0 u1 u2))) -(ex2 T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 -(s k (s k0 i0)) v0 t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq -T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s k0 i0) v0 -u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k0 i0)) v0 t1 -t3)))))))).(\lambda (u: T).(\lambda (H3: (eq T (THead k0 u t0) (THead k u1 -t1))).(let H4 \def (f_equal T K (\lambda (e: T).(match e in T return (\lambda -(_: T).K) with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead -k1 _ _) \Rightarrow k1])) (THead k0 u t0) (THead k u1 t1) H3) in ((let H5 -\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) -\Rightarrow t])) (THead k0 u t0) (THead k u1 t1) H3) in ((let H6 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) -\Rightarrow t])) (THead k0 u t0) (THead k u1 t1) H3) in (\lambda (H7: (eq T u -u1)).(\lambda (H8: (eq K k0 k)).(eq_ind_r T u1 (\lambda (t: T).(or3 (ex2 T -(\lambda (u2: T).(eq T (THead k0 t t2) (THead k u2 t1))) (\lambda (u2: -T).(subst0 i0 v0 u1 u2))) (ex2 T (\lambda (t3: T).(eq T (THead k0 t t2) -(THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T (THead k0 t t2) (THead k u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(subst0 (s k i0) v0 t1 t3)))))) (let H9 \def (eq_ind T t0 -(\lambda (t: T).((eq T t (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u2: -T).(eq T t2 (THead k u2 t1))) (\lambda (u2: T).(subst0 (s k0 i0) v0 u1 u2))) -(ex2 T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 -(s k (s k0 i0)) v0 t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq -T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s k0 i0) v0 -u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k0 i0)) v0 t1 -t3))))))) H2 t1 H6) in (let H10 \def (eq_ind T t0 (\lambda (t: T).(subst0 (s -k0 i0) v0 t t2)) H1 t1 H6) in (let H11 \def (eq_ind K k0 (\lambda (k1: -K).((eq T t1 (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u2: T).(eq T t2 -(THead k u2 t1))) (\lambda (u2: T).(subst0 (s k1 i0) v0 u1 u2))) (ex2 T -(\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k (s -k1 i0)) v0 t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 -(THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s k1 i0) v0 u1 -u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k1 i0)) v0 t1 -t3))))))) H9 k H8) in (let H12 \def (eq_ind K k0 (\lambda (k1: K).(subst0 (s -k1 i0) v0 t1 t2)) H10 k H8) in (eq_ind_r K k (\lambda (k1: K).(or3 (ex2 T -(\lambda (u2: T).(eq T (THead k1 u1 t2) (THead k u2 t1))) (\lambda (u2: -T).(subst0 i0 v0 u1 u2))) (ex2 T (\lambda (t3: T).(eq T (THead k1 u1 t2) -(THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T (THead k1 u1 t2) (THead k u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(subst0 (s k i0) v0 t1 t3)))))) (or3_intro1 (ex2 T -(\lambda (u2: T).(eq T (THead k u1 t2) (THead k u2 t1))) (\lambda (u2: -T).(subst0 i0 v0 u1 u2))) (ex2 T (\lambda (t3: T).(eq T (THead k u1 t2) -(THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T -(\lambda (u2: T).(\lambda (t3: T).(eq T (THead k u1 t2) (THead k u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(subst0 (s k i0) v0 t1 t3)))) (ex_intro2 T (\lambda (t3: -T).(eq T (THead k u1 t2) (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) -v0 t1 t3)) t2 (refl_equal T (THead k u1 t2)) H12)) k0 H8))))) u H7)))) H5)) -H4))))))))))) (\lambda (v0: T).(\lambda (u0: T).(\lambda (u2: T).(\lambda -(i0: nat).(\lambda (H1: (subst0 i0 v0 u0 u2)).(\lambda (H2: (((eq T u0 (THead -k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T u2 (THead k u3 t1))) -(\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t2: T).(eq T u2 -(THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))) (ex3_2 T T -(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead k u3 t2)))) (\lambda (u3: -T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t2: -T).(subst0 (s k i0) v0 t1 t2)))))))).(\lambda (k0: K).(\lambda (t0: -T).(\lambda (t2: T).(\lambda (H3: (subst0 (s k0 i0) v0 t0 t2)).(\lambda (H4: -(((eq T t0 (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T t2 (THead -k u3 t1))) (\lambda (u3: T).(subst0 (s k0 i0) v0 u1 u3))) (ex2 T (\lambda -(t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k (s k0 i0)) -v0 t1 t3))) (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead k u3 -t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 (s k0 i0) v0 u1 u3))) -(\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k0 i0)) v0 t1 -t3)))))))).(\lambda (H5: (eq T (THead k0 u0 t0) (THead k u1 t1))).(let H6 -\def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) -with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _ _) -\Rightarrow k1])) (THead k0 u0 t0) (THead k u1 t1) H5) in ((let H7 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) -\Rightarrow t])) (THead k0 u0 t0) (THead k u1 t1) H5) in ((let H8 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) -\Rightarrow t])) (THead k0 u0 t0) (THead k u1 t1) H5) in (\lambda (H9: (eq T -u0 u1)).(\lambda (H10: (eq K k0 k)).(let H11 \def (eq_ind T t0 (\lambda (t: -T).((eq T t (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T t2 (THead -k u3 t1))) (\lambda (u3: T).(subst0 (s k0 i0) v0 u1 u3))) (ex2 T (\lambda -(t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k (s k0 i0)) -v0 t1 t3))) (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead k u3 -t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 (s k0 i0) v0 u1 u3))) -(\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k0 i0)) v0 t1 t3))))))) H4 -t1 H8) in (let H12 \def (eq_ind T t0 (\lambda (t: T).(subst0 (s k0 i0) v0 t -t2)) H3 t1 H8) in (let H13 \def (eq_ind K k0 (\lambda (k1: K).((eq T t1 -(THead k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T t2 (THead k u3 t1))) -(\lambda (u3: T).(subst0 (s k1 i0) v0 u1 u3))) (ex2 T (\lambda (t3: T).(eq T -t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k (s k1 i0)) v0 t1 t3))) -(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead k u3 t3)))) -(\lambda (u3: T).(\lambda (_: T).(subst0 (s k1 i0) v0 u1 u3))) (\lambda (_: -T).(\lambda (t3: T).(subst0 (s k (s k1 i0)) v0 t1 t3))))))) H11 k H10) in -(let H14 \def (eq_ind K k0 (\lambda (k1: K).(subst0 (s k1 i0) v0 t1 t2)) H12 -k H10) in (eq_ind_r K k (\lambda (k1: K).(or3 (ex2 T (\lambda (u3: T).(eq T -(THead k1 u2 t2) (THead k u3 t1))) (\lambda (u3: T).(subst0 i0 v0 u1 u3))) -(ex2 T (\lambda (t3: T).(eq T (THead k1 u2 t2) (THead k u1 t3))) (\lambda -(t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T (\lambda (u3: T).(\lambda -(t3: T).(eq T (THead k1 u2 t2) (THead k u3 t3)))) (\lambda (u3: T).(\lambda -(_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k -i0) v0 t1 t3)))))) (let H15 \def (eq_ind T u0 (\lambda (t: T).((eq T t (THead -k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T u2 (THead k u3 t1))) -(\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t3: T).(eq T u2 -(THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T -(\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead k u3 t3)))) (\lambda (u3: -T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t3: -T).(subst0 (s k i0) v0 t1 t3))))))) H2 u1 H9) in (let H16 \def (eq_ind T u0 -(\lambda (t: T).(subst0 i0 v0 t u2)) H1 u1 H9) in (or3_intro2 (ex2 T (\lambda -(u3: T).(eq T (THead k u2 t2) (THead k u3 t1))) (\lambda (u3: T).(subst0 i0 -v0 u1 u3))) (ex2 T (\lambda (t3: T).(eq T (THead k u2 t2) (THead k u1 t3))) -(\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T (\lambda (u3: -T).(\lambda (t3: T).(eq T (THead k u2 t2) (THead k u3 t3)))) (\lambda (u3: -T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t3: -T).(subst0 (s k i0) v0 t1 t3)))) (ex3_2_intro T T (\lambda (u3: T).(\lambda -(t3: T).(eq T (THead k u2 t2) (THead k u3 t3)))) (\lambda (u3: T).(\lambda -(_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k -i0) v0 t1 t3))) u2 t2 (refl_equal T (THead k u2 t2)) H16 H14)))) k0 -H10)))))))) H7)) H6)))))))))))))) i v y x H0))) H))))))). -(* COMMENTS -Initial nodes: 4255 -END *) - -theorem subst0_gen_lift_lt: - \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall -(h: nat).(\forall (d: nat).((subst0 i (lift h d u) (lift h (S (plus i d)) t1) -x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda -(t2: T).(subst0 i u t1 t2))))))))) -\def - \lambda (u: T).(\lambda (t1: T).(T_ind (\lambda (t: T).(\forall (x: -T).(\forall (i: nat).(\forall (h: nat).(\forall (d: nat).((subst0 i (lift h d -u) (lift h (S (plus i d)) t) x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h -(S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u t t2))))))))) (\lambda (n: -nat).(\lambda (x: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda (d: -nat).(\lambda (H: (subst0 i (lift h d u) (lift h (S (plus i d)) (TSort n)) -x)).(let H0 \def (eq_ind T (lift h (S (plus i d)) (TSort n)) (\lambda (t: -T).(subst0 i (lift h d u) t x)) H (TSort n) (lift_sort n h (S (plus i d)))) -in (subst0_gen_sort (lift h d u) x i n H0 (ex2 T (\lambda (t2: T).(eq T x -(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TSort n) -t2))))))))))) (\lambda (n: nat).(\lambda (x: T).(\lambda (i: nat).(\lambda -(h: nat).(\lambda (d: nat).(\lambda (H: (subst0 i (lift h d u) (lift h (S -(plus i d)) (TLRef n)) x)).(lt_le_e n (S (plus i d)) (ex2 T (\lambda (t2: -T).(eq T x (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef -n) t2))) (\lambda (H0: (lt n (S (plus i d)))).(let H1 \def (eq_ind T (lift h -(S (plus i d)) (TLRef n)) (\lambda (t: T).(subst0 i (lift h d u) t x)) H -(TLRef n) (lift_lref_lt n h (S (plus i d)) H0)) in (land_ind (eq nat n i) (eq -T x (lift (S n) O (lift h d u))) (ex2 T (\lambda (t2: T).(eq T x (lift h (S -(plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef n) t2))) (\lambda (H2: -(eq nat n i)).(\lambda (H3: (eq T x (lift (S n) O (lift h d u)))).(eq_ind_r T -(lift (S n) O (lift h d u)) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t -(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef n) t2)))) -(eq_ind_r nat i (\lambda (n0: nat).(ex2 T (\lambda (t2: T).(eq T (lift (S n0) -O (lift h d u)) (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u -(TLRef n0) t2)))) (eq_ind T (lift h (plus (S i) d) (lift (S i) O u)) (\lambda -(t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h (S (plus i d)) t2))) (\lambda -(t2: T).(subst0 i u (TLRef i) t2)))) (ex_intro2 T (\lambda (t2: T).(eq T -(lift h (S (plus i d)) (lift (S i) O u)) (lift h (S (plus i d)) t2))) -(\lambda (t2: T).(subst0 i u (TLRef i) t2)) (lift (S i) O u) (refl_equal T -(lift h (S (plus i d)) (lift (S i) O u))) (subst0_lref u i)) (lift (S i) O -(lift h d u)) (lift_d u h (S i) d O (le_O_n d))) n H2) x H3))) -(subst0_gen_lref (lift h d u) x i n H1)))) (\lambda (H0: (le (S (plus i d)) -n)).(let H1 \def (eq_ind T (lift h (S (plus i d)) (TLRef n)) (\lambda (t: -T).(subst0 i (lift h d u) t x)) H (TLRef (plus n h)) (lift_lref_ge n h (S -(plus i d)) H0)) in (land_ind (eq nat (plus n h) i) (eq T x (lift (S (plus n -h)) O (lift h d u))) (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) -t2))) (\lambda (t2: T).(subst0 i u (TLRef n) t2))) (\lambda (H2: (eq nat -(plus n h) i)).(\lambda (_: (eq T x (lift (S (plus n h)) O (lift h d -u)))).(let H4 \def (eq_ind_r nat i (\lambda (n0: nat).(le (S (plus n0 d)) n)) -H0 (plus n h) H2) in (le_false n (plus (plus n h) d) (ex2 T (\lambda (t2: -T).(eq T x (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef -n) t2))) (le_plus_trans n (plus n h) d (le_plus_l n h)) H4)))) -(subst0_gen_lref (lift h d u) x i (plus n h) H1))))))))))) (\lambda (k: -K).(\lambda (t: T).(\lambda (H: ((\forall (x: T).(\forall (i: nat).(\forall -(h: nat).(\forall (d: nat).((subst0 i (lift h d u) (lift h (S (plus i d)) t) -x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda -(t2: T).(subst0 i u t t2)))))))))).(\lambda (t0: T).(\lambda (H0: ((\forall -(x: T).(\forall (i: nat).(\forall (h: nat).(\forall (d: nat).((subst0 i (lift -h d u) (lift h (S (plus i d)) t0) x) \to (ex2 T (\lambda (t2: T).(eq T x -(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u t0 -t2)))))))))).(\lambda (x: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda (d: -nat).(\lambda (H1: (subst0 i (lift h d u) (lift h (S (plus i d)) (THead k t -t0)) x)).(let H2 \def (eq_ind T (lift h (S (plus i d)) (THead k t t0)) -(\lambda (t2: T).(subst0 i (lift h d u) t2 x)) H1 (THead k (lift h (S (plus i -d)) t) (lift h (s k (S (plus i d))) t0)) (lift_head k t t0 h (S (plus i d)))) -in (or3_ind (ex2 T (\lambda (u2: T).(eq T x (THead k u2 (lift h (s k (S (plus -i d))) t0)))) (\lambda (u2: T).(subst0 i (lift h d u) (lift h (S (plus i d)) -t) u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h (S (plus i d)) t) -t2))) (\lambda (t2: T).(subst0 (s k i) (lift h d u) (lift h (s k (S (plus i -d))) t0) t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k -u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i (lift h d u) (lift h (S -(plus i d)) t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) (lift h -d u) (lift h (s k (S (plus i d))) t0) t2)))) (ex2 T (\lambda (t2: T).(eq T x -(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) -t2))) (\lambda (H3: (ex2 T (\lambda (u2: T).(eq T x (THead k u2 (lift h (s k -(S (plus i d))) t0)))) (\lambda (u2: T).(subst0 i (lift h d u) (lift h (S -(plus i d)) t) u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead k u2 (lift h -(s k (S (plus i d))) t0)))) (\lambda (u2: T).(subst0 i (lift h d u) (lift h -(S (plus i d)) t) u2)) (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) -t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2))) (\lambda (x0: -T).(\lambda (H4: (eq T x (THead k x0 (lift h (s k (S (plus i d))) -t0)))).(\lambda (H5: (subst0 i (lift h d u) (lift h (S (plus i d)) t) -x0)).(eq_ind_r T (THead k x0 (lift h (s k (S (plus i d))) t0)) (\lambda (t2: -T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda -(t3: T).(subst0 i u (THead k t t0) t3)))) (ex2_ind T (\lambda (t2: T).(eq T -x0 (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u t t2)) (ex2 T -(\lambda (t2: T).(eq T (THead k x0 (lift h (s k (S (plus i d))) t0)) (lift h -(S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2))) -(\lambda (x1: T).(\lambda (H6: (eq T x0 (lift h (S (plus i d)) x1))).(\lambda -(H7: (subst0 i u t x1)).(eq_ind_r T (lift h (S (plus i d)) x1) (\lambda (t2: -T).(ex2 T (\lambda (t3: T).(eq T (THead k t2 (lift h (s k (S (plus i d))) -t0)) (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u (THead k t t0) -t3)))) (eq_ind T (lift h (S (plus i d)) (THead k x1 t0)) (\lambda (t2: -T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda -(t3: T).(subst0 i u (THead k t t0) t3)))) (ex_intro2 T (\lambda (t2: T).(eq T -(lift h (S (plus i d)) (THead k x1 t0)) (lift h (S (plus i d)) t2))) (\lambda -(t2: T).(subst0 i u (THead k t t0) t2)) (THead k x1 t0) (refl_equal T (lift h -(S (plus i d)) (THead k x1 t0))) (subst0_fst u x1 t i H7 t0 k)) (THead k -(lift h (S (plus i d)) x1) (lift h (s k (S (plus i d))) t0)) (lift_head k x1 -t0 h (S (plus i d)))) x0 H6)))) (H x0 i h d H5)) x H4)))) H3)) (\lambda (H3: -(ex2 T (\lambda (t2: T).(eq T x (THead k (lift h (S (plus i d)) t) t2))) -(\lambda (t2: T).(subst0 (s k i) (lift h d u) (lift h (s k (S (plus i d))) -t0) t2)))).(ex2_ind T (\lambda (t2: T).(eq T x (THead k (lift h (S (plus i -d)) t) t2))) (\lambda (t2: T).(subst0 (s k i) (lift h d u) (lift h (s k (S -(plus i d))) t0) t2)) (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) -t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2))) (\lambda (x0: -T).(\lambda (H4: (eq T x (THead k (lift h (S (plus i d)) t) x0))).(\lambda -(H5: (subst0 (s k i) (lift h d u) (lift h (s k (S (plus i d))) t0) -x0)).(eq_ind_r T (THead k (lift h (S (plus i d)) t) x0) (\lambda (t2: T).(ex2 -T (\lambda (t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda (t3: -T).(subst0 i u (THead k t t0) t3)))) (let H6 \def (eq_ind nat (s k (S (plus i -d))) (\lambda (n: nat).(subst0 (s k i) (lift h d u) (lift h n t0) x0)) H5 (S -(s k (plus i d))) (s_S k (plus i d))) in (let H7 \def (eq_ind nat (s k (plus -i d)) (\lambda (n: nat).(subst0 (s k i) (lift h d u) (lift h (S n) t0) x0)) -H6 (plus (s k i) d) (s_plus k i d)) in (ex2_ind T (\lambda (t2: T).(eq T x0 -(lift h (S (plus (s k i) d)) t2))) (\lambda (t2: T).(subst0 (s k i) u t0 t2)) -(ex2 T (\lambda (t2: T).(eq T (THead k (lift h (S (plus i d)) t) x0) (lift h -(S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2))) -(\lambda (x1: T).(\lambda (H8: (eq T x0 (lift h (S (plus (s k i) d)) -x1))).(\lambda (H9: (subst0 (s k i) u t0 x1)).(eq_ind_r T (lift h (S (plus (s -k i) d)) x1) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k (lift h -(S (plus i d)) t) t2) (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i -u (THead k t t0) t3)))) (eq_ind nat (s k (plus i d)) (\lambda (n: nat).(ex2 T -(\lambda (t2: T).(eq T (THead k (lift h (S (plus i d)) t) (lift h (S n) x1)) -(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) -t2)))) (eq_ind nat (s k (S (plus i d))) (\lambda (n: nat).(ex2 T (\lambda -(t2: T).(eq T (THead k (lift h (S (plus i d)) t) (lift h n x1)) (lift h (S -(plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2)))) (eq_ind -T (lift h (S (plus i d)) (THead k t x1)) (\lambda (t2: T).(ex2 T (\lambda -(t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u -(THead k t t0) t3)))) (ex_intro2 T (\lambda (t2: T).(eq T (lift h (S (plus i -d)) (THead k t x1)) (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u -(THead k t t0) t2)) (THead k t x1) (refl_equal T (lift h (S (plus i d)) -(THead k t x1))) (subst0_snd k u x1 t0 i H9 t)) (THead k (lift h (S (plus i -d)) t) (lift h (s k (S (plus i d))) x1)) (lift_head k t x1 h (S (plus i d)))) -(S (s k (plus i d))) (s_S k (plus i d))) (plus (s k i) d) (s_plus k i d)) x0 -H8)))) (H0 x0 (s k i) h d H7)))) x H4)))) H3)) (\lambda (H3: (ex3_2 T T -(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i (lift h d u) (lift h (S (plus i d)) t) u2))) -(\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) (lift h d u) (lift h (s k (S -(plus i d))) t0) t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq -T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i (lift h d -u) (lift h (S (plus i d)) t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 -(s k i) (lift h d u) (lift h (s k (S (plus i d))) t0) t2))) (ex2 T (\lambda -(t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u -(THead k t t0) t2))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T x -(THead k x0 x1))).(\lambda (H5: (subst0 i (lift h d u) (lift h (S (plus i d)) -t) x0)).(\lambda (H6: (subst0 (s k i) (lift h d u) (lift h (s k (S (plus i -d))) t0) x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t2: T).(ex2 T (\lambda -(t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u -(THead k t t0) t3)))) (let H7 \def (eq_ind nat (s k (S (plus i d))) (\lambda -(n: nat).(subst0 (s k i) (lift h d u) (lift h n t0) x1)) H6 (S (s k (plus i -d))) (s_S k (plus i d))) in (let H8 \def (eq_ind nat (s k (plus i d)) -(\lambda (n: nat).(subst0 (s k i) (lift h d u) (lift h (S n) t0) x1)) H7 -(plus (s k i) d) (s_plus k i d)) in (ex2_ind T (\lambda (t2: T).(eq T x1 -(lift h (S (plus (s k i) d)) t2))) (\lambda (t2: T).(subst0 (s k i) u t0 t2)) -(ex2 T (\lambda (t2: T).(eq T (THead k x0 x1) (lift h (S (plus i d)) t2))) -(\lambda (t2: T).(subst0 i u (THead k t t0) t2))) (\lambda (x2: T).(\lambda -(H9: (eq T x1 (lift h (S (plus (s k i) d)) x2))).(\lambda (H10: (subst0 (s k -i) u t0 x2)).(ex2_ind T (\lambda (t2: T).(eq T x0 (lift h (S (plus i d)) -t2))) (\lambda (t2: T).(subst0 i u t t2)) (ex2 T (\lambda (t2: T).(eq T -(THead k x0 x1) (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u -(THead k t t0) t2))) (\lambda (x3: T).(\lambda (H11: (eq T x0 (lift h (S -(plus i d)) x3))).(\lambda (H12: (subst0 i u t x3)).(eq_ind_r T (lift h (S -(plus i d)) x3) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k t2 -x1) (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u (THead k t t0) -t3)))) (eq_ind_r T (lift h (S (plus (s k i) d)) x2) (\lambda (t2: T).(ex2 T -(\lambda (t3: T).(eq T (THead k (lift h (S (plus i d)) x3) t2) (lift h (S -(plus i d)) t3))) (\lambda (t3: T).(subst0 i u (THead k t t0) t3)))) (eq_ind -nat (s k (plus i d)) (\lambda (n: nat).(ex2 T (\lambda (t2: T).(eq T (THead k -(lift h (S (plus i d)) x3) (lift h (S n) x2)) (lift h (S (plus i d)) t2))) -(\lambda (t2: T).(subst0 i u (THead k t t0) t2)))) (eq_ind nat (s k (S (plus -i d))) (\lambda (n: nat).(ex2 T (\lambda (t2: T).(eq T (THead k (lift h (S -(plus i d)) x3) (lift h n x2)) (lift h (S (plus i d)) t2))) (\lambda (t2: -T).(subst0 i u (THead k t t0) t2)))) (eq_ind T (lift h (S (plus i d)) (THead -k x3 x2)) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h (S (plus -i d)) t3))) (\lambda (t3: T).(subst0 i u (THead k t t0) t3)))) (ex_intro2 T -(\lambda (t2: T).(eq T (lift h (S (plus i d)) (THead k x3 x2)) (lift h (S -(plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2)) (THead k -x3 x2) (refl_equal T (lift h (S (plus i d)) (THead k x3 x2))) (subst0_both u -t x3 i H12 k t0 x2 H10)) (THead k (lift h (S (plus i d)) x3) (lift h (s k (S -(plus i d))) x2)) (lift_head k x3 x2 h (S (plus i d)))) (S (s k (plus i d))) -(s_S k (plus i d))) (plus (s k i) d) (s_plus k i d)) x1 H9) x0 H11)))) (H x0 -i h d H5))))) (H0 x1 (s k i) h d H8)))) x H4)))))) H3)) (subst0_gen_head k -(lift h d u) (lift h (S (plus i d)) t) (lift h (s k (S (plus i d))) t0) x i -H2))))))))))))) t1)). -(* COMMENTS -Initial nodes: 5157 -END *) - -theorem subst0_gen_lift_false: - \forall (t: T).(\forall (u: T).(\forall (x: T).(\forall (h: nat).(\forall -(d: nat).(\forall (i: nat).((le d i) \to ((lt i (plus d h)) \to ((subst0 i u -(lift h d t) x) \to (\forall (P: Prop).P))))))))) -\def - \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (u: T).(\forall (x: -T).(\forall (h: nat).(\forall (d: nat).(\forall (i: nat).((le d i) \to ((lt i -(plus d h)) \to ((subst0 i u (lift h d t0) x) \to (\forall (P: -Prop).P)))))))))) (\lambda (n: nat).(\lambda (u: T).(\lambda (x: T).(\lambda -(h: nat).(\lambda (d: nat).(\lambda (i: nat).(\lambda (_: (le d i)).(\lambda -(_: (lt i (plus d h))).(\lambda (H1: (subst0 i u (lift h d (TSort n)) -x)).(\lambda (P: Prop).(let H2 \def (eq_ind T (lift h d (TSort n)) (\lambda -(t0: T).(subst0 i u t0 x)) H1 (TSort n) (lift_sort n h d)) in -(subst0_gen_sort u x i n H2 P)))))))))))) (\lambda (n: nat).(\lambda (u: -T).(\lambda (x: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (i: -nat).(\lambda (H: (le d i)).(\lambda (H0: (lt i (plus d h))).(\lambda (H1: -(subst0 i u (lift h d (TLRef n)) x)).(\lambda (P: Prop).(lt_le_e n d P -(\lambda (H2: (lt n d)).(let H3 \def (eq_ind T (lift h d (TLRef n)) (\lambda -(t0: T).(subst0 i u t0 x)) H1 (TLRef n) (lift_lref_lt n h d H2)) in (land_ind -(eq nat n i) (eq T x (lift (S n) O u)) P (\lambda (H4: (eq nat n i)).(\lambda -(_: (eq T x (lift (S n) O u))).(let H6 \def (eq_ind nat n (\lambda (n0: -nat).(lt n0 d)) H2 i H4) in (le_false d i P H H6)))) (subst0_gen_lref u x i n -H3)))) (\lambda (H2: (le d n)).(let H3 \def (eq_ind T (lift h d (TLRef n)) -(\lambda (t0: T).(subst0 i u t0 x)) H1 (TLRef (plus n h)) (lift_lref_ge n h d -H2)) in (land_ind (eq nat (plus n h) i) (eq T x (lift (S (plus n h)) O u)) P -(\lambda (H4: (eq nat (plus n h) i)).(\lambda (_: (eq T x (lift (S (plus n -h)) O u))).(let H6 \def (eq_ind_r nat i (\lambda (n0: nat).(lt n0 (plus d -h))) H0 (plus n h) H4) in (le_false d n P H2 (lt_le_S n d (simpl_lt_plus_r h -n d H6)))))) (subst0_gen_lref u x i (plus n h) H3))))))))))))))) (\lambda (k: -K).(\lambda (t0: T).(\lambda (H: ((\forall (u: T).(\forall (x: T).(\forall -(h: nat).(\forall (d: nat).(\forall (i: nat).((le d i) \to ((lt i (plus d h)) -\to ((subst0 i u (lift h d t0) x) \to (\forall (P: -Prop).P))))))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (u: T).(\forall -(x: T).(\forall (h: nat).(\forall (d: nat).(\forall (i: nat).((le d i) \to -((lt i (plus d h)) \to ((subst0 i u (lift h d t1) x) \to (\forall (P: -Prop).P))))))))))).(\lambda (u: T).(\lambda (x: T).(\lambda (h: nat).(\lambda -(d: nat).(\lambda (i: nat).(\lambda (H1: (le d i)).(\lambda (H2: (lt i (plus -d h))).(\lambda (H3: (subst0 i u (lift h d (THead k t0 t1)) x)).(\lambda (P: -Prop).(let H4 \def (eq_ind T (lift h d (THead k t0 t1)) (\lambda (t2: -T).(subst0 i u t2 x)) H3 (THead k (lift h d t0) (lift h (s k d) t1)) -(lift_head k t0 t1 h d)) in (or3_ind (ex2 T (\lambda (u2: T).(eq T x (THead k -u2 (lift h (s k d) t1)))) (\lambda (u2: T).(subst0 i u (lift h d t0) u2))) -(ex2 T (\lambda (t2: T).(eq T x (THead k (lift h d t0) t2))) (\lambda (t2: -T).(subst0 (s k i) u (lift h (s k d) t1) t2))) (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 i u (lift h d t0) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 -(s k i) u (lift h (s k d) t1) t2)))) P (\lambda (H5: (ex2 T (\lambda (u2: -T).(eq T x (THead k u2 (lift h (s k d) t1)))) (\lambda (u2: T).(subst0 i u -(lift h d t0) u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead k u2 (lift h -(s k d) t1)))) (\lambda (u2: T).(subst0 i u (lift h d t0) u2)) P (\lambda -(x0: T).(\lambda (_: (eq T x (THead k x0 (lift h (s k d) t1)))).(\lambda (H7: -(subst0 i u (lift h d t0) x0)).(H u x0 h d i H1 H2 H7 P)))) H5)) (\lambda -(H5: (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h d t0) t2))) (\lambda -(t2: T).(subst0 (s k i) u (lift h (s k d) t1) t2)))).(ex2_ind T (\lambda (t2: -T).(eq T x (THead k (lift h d t0) t2))) (\lambda (t2: T).(subst0 (s k i) u -(lift h (s k d) t1) t2)) P (\lambda (x0: T).(\lambda (_: (eq T x (THead k -(lift h d t0) x0))).(\lambda (H7: (subst0 (s k i) u (lift h (s k d) t1) -x0)).(H0 u x0 h (s k d) (s k i) (s_le k d i H1) (eq_ind nat (s k (plus d h)) -(\lambda (n: nat).(lt (s k i) n)) (lt_le_S (s k i) (s k (plus d h)) (s_lt k i -(plus d h) H2)) (plus (s k d) h) (s_plus k d h)) H7 P)))) H5)) (\lambda (H5: -(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 i u (lift h d t0) u2))) (\lambda (_: -T).(\lambda (t2: T).(subst0 (s k i) u (lift h (s k d) t1) t2))))).(ex3_2_ind -T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda -(u2: T).(\lambda (_: T).(subst0 i u (lift h d t0) u2))) (\lambda (_: -T).(\lambda (t2: T).(subst0 (s k i) u (lift h (s k d) t1) t2))) P (\lambda -(x0: T).(\lambda (x1: T).(\lambda (_: (eq T x (THead k x0 x1))).(\lambda (H7: -(subst0 i u (lift h d t0) x0)).(\lambda (_: (subst0 (s k i) u (lift h (s k d) -t1) x1)).(H u x0 h d i H1 H2 H7 P)))))) H5)) (subst0_gen_head k u (lift h d -t0) (lift h (s k d) t1) x i H4))))))))))))))))) t). -(* COMMENTS -Initial nodes: 1621 -END *) - -theorem subst0_gen_lift_ge: - \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall -(h: nat).(\forall (d: nat).((subst0 i u (lift h d t1) x) \to ((le (plus d h) -i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: -T).(subst0 (minus i h) u t1 t2)))))))))) -\def - \lambda (u: T).(\lambda (t1: T).(T_ind (\lambda (t: T).(\forall (x: -T).(\forall (i: nat).(\forall (h: nat).(\forall (d: nat).((subst0 i u (lift h -d t) x) \to ((le (plus d h) i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d -t2))) (\lambda (t2: T).(subst0 (minus i h) u t t2)))))))))) (\lambda (n: -nat).(\lambda (x: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda (d: -nat).(\lambda (H: (subst0 i u (lift h d (TSort n)) x)).(\lambda (_: (le (plus -d h) i)).(let H1 \def (eq_ind T (lift h d (TSort n)) (\lambda (t: T).(subst0 -i u t x)) H (TSort n) (lift_sort n h d)) in (subst0_gen_sort u x i n H1 (ex2 -T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i -h) u (TSort n) t2)))))))))))) (\lambda (n: nat).(\lambda (x: T).(\lambda (i: -nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (subst0 i u (lift h d -(TLRef n)) x)).(\lambda (H0: (le (plus d h) i)).(lt_le_e n d (ex2 T (\lambda -(t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (TLRef -n) t2))) (\lambda (H1: (lt n d)).(let H2 \def (eq_ind T (lift h d (TLRef n)) -(\lambda (t: T).(subst0 i u t x)) H (TLRef n) (lift_lref_lt n h d H1)) in -(land_ind (eq nat n i) (eq T x (lift (S n) O u)) (ex2 T (\lambda (t2: T).(eq -T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (TLRef n) t2))) -(\lambda (H3: (eq nat n i)).(\lambda (_: (eq T x (lift (S n) O u))).(let H5 -\def (eq_ind nat n (\lambda (n0: nat).(lt n0 d)) H1 i H3) in (le_false (plus -d h) i (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: -T).(subst0 (minus i h) u (TLRef n) t2))) H0 (le_plus_trans (S i) d h H5))))) -(subst0_gen_lref u x i n H2)))) (\lambda (H1: (le d n)).(let H2 \def (eq_ind -T (lift h d (TLRef n)) (\lambda (t: T).(subst0 i u t x)) H (TLRef (plus n h)) -(lift_lref_ge n h d H1)) in (land_ind (eq nat (plus n h) i) (eq T x (lift (S -(plus n h)) O u)) (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda -(t2: T).(subst0 (minus i h) u (TLRef n) t2))) (\lambda (H3: (eq nat (plus n -h) i)).(\lambda (H4: (eq T x (lift (S (plus n h)) O u))).(eq_ind nat (plus n -h) (\lambda (n0: nat).(ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) -(\lambda (t2: T).(subst0 (minus n0 h) u (TLRef n) t2)))) (eq_ind_r T (lift (S -(plus n h)) O u) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h d -t2))) (\lambda (t2: T).(subst0 (minus (plus n h) h) u (TLRef n) t2)))) -(eq_ind_r nat n (\lambda (n0: nat).(ex2 T (\lambda (t2: T).(eq T (lift (S -(plus n h)) O u) (lift h d t2))) (\lambda (t2: T).(subst0 n0 u (TLRef n) -t2)))) (ex_intro2 T (\lambda (t2: T).(eq T (lift (S (plus n h)) O u) (lift h -d t2))) (\lambda (t2: T).(subst0 n u (TLRef n) t2)) (lift (S n) O u) -(eq_ind_r T (lift (plus h (S n)) O u) (\lambda (t: T).(eq T (lift (S (plus n -h)) O u) t)) (eq_ind_r nat (plus h n) (\lambda (n0: nat).(eq T (lift (S n0) O -u) (lift (plus h (S n)) O u))) (eq_ind_r nat (plus h (S n)) (\lambda (n0: -nat).(eq T (lift n0 O u) (lift (plus h (S n)) O u))) (refl_equal T (lift -(plus h (S n)) O u)) (S (plus h n)) (plus_n_Sm h n)) (plus n h) (plus_sym n -h)) (lift h d (lift (S n) O u)) (lift_free u (S n) h O d (le_trans_plus_r O d -(plus O (S n)) (le_plus_plus O O d (S n) (le_n O) (le_S d n H1))) (le_O_n -d))) (subst0_lref u n)) (minus (plus n h) h) (minus_plus_r n h)) x H4) i -H3))) (subst0_gen_lref u x i (plus n h) H2)))))))))))) (\lambda (k: -K).(\lambda (t: T).(\lambda (H: ((\forall (x: T).(\forall (i: nat).(\forall -(h: nat).(\forall (d: nat).((subst0 i u (lift h d t) x) \to ((le (plus d h) -i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: -T).(subst0 (minus i h) u t t2))))))))))).(\lambda (t0: T).(\lambda (H0: -((\forall (x: T).(\forall (i: nat).(\forall (h: nat).(\forall (d: -nat).((subst0 i u (lift h d t0) x) \to ((le (plus d h) i) \to (ex2 T (\lambda -(t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u t0 -t2))))))))))).(\lambda (x: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda -(d: nat).(\lambda (H1: (subst0 i u (lift h d (THead k t t0)) x)).(\lambda -(H2: (le (plus d h) i)).(let H3 \def (eq_ind T (lift h d (THead k t t0)) -(\lambda (t2: T).(subst0 i u t2 x)) H1 (THead k (lift h d t) (lift h (s k d) -t0)) (lift_head k t t0 h d)) in (or3_ind (ex2 T (\lambda (u2: T).(eq T x -(THead k u2 (lift h (s k d) t0)))) (\lambda (u2: T).(subst0 i u (lift h d t) -u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h d t) t2))) (\lambda -(t2: T).(subst0 (s k i) u (lift h (s k d) t0) t2))) (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 i u (lift h d t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s -k i) u (lift h (s k d) t0) t2)))) (ex2 T (\lambda (t2: T).(eq T x (lift h d -t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda -(H4: (ex2 T (\lambda (u2: T).(eq T x (THead k u2 (lift h (s k d) t0)))) -(\lambda (u2: T).(subst0 i u (lift h d t) u2)))).(ex2_ind T (\lambda (u2: -T).(eq T x (THead k u2 (lift h (s k d) t0)))) (\lambda (u2: T).(subst0 i u -(lift h d t) u2)) (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda -(t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (x0: T).(\lambda -(H5: (eq T x (THead k x0 (lift h (s k d) t0)))).(\lambda (H6: (subst0 i u -(lift h d t) x0)).(eq_ind_r T (THead k x0 (lift h (s k d) t0)) (\lambda (t2: -T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst0 -(minus i h) u (THead k t t0) t3)))) (ex2_ind T (\lambda (t2: T).(eq T x0 -(lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u t t2)) (ex2 T (\lambda -(t2: T).(eq T (THead k x0 (lift h (s k d) t0)) (lift h d t2))) (\lambda (t2: -T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (x1: T).(\lambda (H7: -(eq T x0 (lift h d x1))).(\lambda (H8: (subst0 (minus i h) u t x1)).(eq_ind_r -T (lift h d x1) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k t2 -(lift h (s k d) t0)) (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u -(THead k t t0) t3)))) (eq_ind T (lift h d (THead k x1 t0)) (\lambda (t2: -T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst0 -(minus i h) u (THead k t t0) t3)))) (ex_intro2 T (\lambda (t2: T).(eq T (lift -h d (THead k x1 t0)) (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u -(THead k t t0) t2)) (THead k x1 t0) (refl_equal T (lift h d (THead k x1 t0))) -(subst0_fst u x1 t (minus i h) H8 t0 k)) (THead k (lift h d x1) (lift h (s k -d) t0)) (lift_head k x1 t0 h d)) x0 H7)))) (H x0 i h d H6 H2)) x H5)))) H4)) -(\lambda (H4: (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h d t) t2))) -(\lambda (t2: T).(subst0 (s k i) u (lift h (s k d) t0) t2)))).(ex2_ind T -(\lambda (t2: T).(eq T x (THead k (lift h d t) t2))) (\lambda (t2: T).(subst0 -(s k i) u (lift h (s k d) t0) t2)) (ex2 T (\lambda (t2: T).(eq T x (lift h d -t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda -(x0: T).(\lambda (H5: (eq T x (THead k (lift h d t) x0))).(\lambda (H6: -(subst0 (s k i) u (lift h (s k d) t0) x0)).(eq_ind_r T (THead k (lift h d t) -x0) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3))) -(\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) (ex2_ind T -(\lambda (t2: T).(eq T x0 (lift h (s k d) t2))) (\lambda (t2: T).(subst0 -(minus (s k i) h) u t0 t2)) (ex2 T (\lambda (t2: T).(eq T (THead k (lift h d -t) x0) (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead k t t0) -t2))) (\lambda (x1: T).(\lambda (H7: (eq T x0 (lift h (s k d) x1))).(\lambda -(H8: (subst0 (minus (s k i) h) u t0 x1)).(eq_ind_r T (lift h (s k d) x1) -(\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k (lift h d t) t2) -(lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) -(eq_ind T (lift h d (THead k t x1)) (\lambda (t2: T).(ex2 T (\lambda (t3: -T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t -t0) t3)))) (let H9 \def (eq_ind_r nat (minus (s k i) h) (\lambda (n: -nat).(subst0 n u t0 x1)) H8 (s k (minus i h)) (s_minus k i h (le_trans_plus_r -d h i H2))) in (ex_intro2 T (\lambda (t2: T).(eq T (lift h d (THead k t x1)) -(lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead k t t0) t2)) -(THead k t x1) (refl_equal T (lift h d (THead k t x1))) (subst0_snd k u x1 t0 -(minus i h) H9 t))) (THead k (lift h d t) (lift h (s k d) x1)) (lift_head k t -x1 h d)) x0 H7)))) (H0 x0 (s k i) h (s k d) H6 (eq_ind nat (s k (plus d h)) -(\lambda (n: nat).(le n (s k i))) (s_le k (plus d h) i H2) (plus (s k d) h) -(s_plus k d h)))) x H5)))) H4)) (\lambda (H4: (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 i u (lift h d t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s -k i) u (lift h (s k d) t0) t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda -(t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i -u (lift h d t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) u (lift -h (s k d) t0) t2))) (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda -(t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (x0: T).(\lambda -(x1: T).(\lambda (H5: (eq T x (THead k x0 x1))).(\lambda (H6: (subst0 i u -(lift h d t) x0)).(\lambda (H7: (subst0 (s k i) u (lift h (s k d) t0) -x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq -T t2 (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) -t3)))) (ex2_ind T (\lambda (t2: T).(eq T x1 (lift h (s k d) t2))) (\lambda -(t2: T).(subst0 (minus (s k i) h) u t0 t2)) (ex2 T (\lambda (t2: T).(eq T -(THead k x0 x1) (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead -k t t0) t2))) (\lambda (x2: T).(\lambda (H8: (eq T x1 (lift h (s k d) -x2))).(\lambda (H9: (subst0 (minus (s k i) h) u t0 x2)).(ex2_ind T (\lambda -(t2: T).(eq T x0 (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u t -t2)) (ex2 T (\lambda (t2: T).(eq T (THead k x0 x1) (lift h d t2))) (\lambda -(t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (x3: T).(\lambda -(H10: (eq T x0 (lift h d x3))).(\lambda (H11: (subst0 (minus i h) u t -x3)).(eq_ind_r T (lift h d x3) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T -(THead k t2 x1) (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead -k t t0) t3)))) (eq_ind_r T (lift h (s k d) x2) (\lambda (t2: T).(ex2 T -(\lambda (t3: T).(eq T (THead k (lift h d x3) t2) (lift h d t3))) (\lambda -(t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) (eq_ind T (lift h d -(THead k x3 x2)) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d -t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) (let H12 -\def (eq_ind_r nat (minus (s k i) h) (\lambda (n: nat).(subst0 n u t0 x2)) H9 -(s k (minus i h)) (s_minus k i h (le_trans_plus_r d h i H2))) in (ex_intro2 T -(\lambda (t2: T).(eq T (lift h d (THead k x3 x2)) (lift h d t2))) (\lambda -(t2: T).(subst0 (minus i h) u (THead k t t0) t2)) (THead k x3 x2) (refl_equal -T (lift h d (THead k x3 x2))) (subst0_both u t x3 (minus i h) H11 k t0 x2 -H12))) (THead k (lift h d x3) (lift h (s k d) x2)) (lift_head k x3 x2 h d)) -x1 H8) x0 H10)))) (H x0 i h d H6 H2))))) (H0 x1 (s k i) h (s k d) H7 (eq_ind -nat (s k (plus d h)) (\lambda (n: nat).(le n (s k i))) (s_le k (plus d h) i -H2) (plus (s k d) h) (s_plus k d h)))) x H5)))))) H4)) (subst0_gen_head k u -(lift h d t) (lift h (s k d) t0) x i H3)))))))))))))) t1)). -(* COMMENTS -Initial nodes: 4191 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/subst0/props.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/subst0/props.ma deleted file mode 100644 index 5da05fa2a..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/subst0/props.ma +++ /dev/null @@ -1,241 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/subst0/fwd.ma". - -theorem subst0_refl: - \forall (u: T).(\forall (t: T).(\forall (d: nat).((subst0 d u t t) \to -(\forall (P: Prop).P)))) -\def - \lambda (u: T).(\lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d: -nat).((subst0 d u t0 t0) \to (\forall (P: Prop).P)))) (\lambda (n: -nat).(\lambda (d: nat).(\lambda (H: (subst0 d u (TSort n) (TSort -n))).(\lambda (P: Prop).(subst0_gen_sort u (TSort n) d n H P))))) (\lambda -(n: nat).(\lambda (d: nat).(\lambda (H: (subst0 d u (TLRef n) (TLRef -n))).(\lambda (P: Prop).(land_ind (eq nat n d) (eq T (TLRef n) (lift (S n) O -u)) P (\lambda (_: (eq nat n d)).(\lambda (H1: (eq T (TLRef n) (lift (S n) O -u))).(lift_gen_lref_false (S n) O n (le_O_n n) (le_n (plus O (S n))) u H1 -P))) (subst0_gen_lref u (TLRef n) d n H)))))) (\lambda (k: K).(\lambda (t0: -T).(\lambda (H: ((\forall (d: nat).((subst0 d u t0 t0) \to (\forall (P: -Prop).P))))).(\lambda (t1: T).(\lambda (H0: ((\forall (d: nat).((subst0 d u -t1 t1) \to (\forall (P: Prop).P))))).(\lambda (d: nat).(\lambda (H1: (subst0 -d u (THead k t0 t1) (THead k t0 t1))).(\lambda (P: Prop).(or3_ind (ex2 T -(\lambda (u2: T).(eq T (THead k t0 t1) (THead k u2 t1))) (\lambda (u2: -T).(subst0 d u t0 u2))) (ex2 T (\lambda (t2: T).(eq T (THead k t0 t1) (THead -k t0 t2))) (\lambda (t2: T).(subst0 (s k d) u t1 t2))) (ex3_2 T T (\lambda -(u2: T).(\lambda (t2: T).(eq T (THead k t0 t1) (THead k u2 t2)))) (\lambda -(u2: T).(\lambda (_: T).(subst0 d u t0 u2))) (\lambda (_: T).(\lambda (t2: -T).(subst0 (s k d) u t1 t2)))) P (\lambda (H2: (ex2 T (\lambda (u2: T).(eq T -(THead k t0 t1) (THead k u2 t1))) (\lambda (u2: T).(subst0 d u t0 -u2)))).(ex2_ind T (\lambda (u2: T).(eq T (THead k t0 t1) (THead k u2 t1))) -(\lambda (u2: T).(subst0 d u t0 u2)) P (\lambda (x: T).(\lambda (H3: (eq T -(THead k t0 t1) (THead k x t1))).(\lambda (H4: (subst0 d u t0 x)).(let H5 -\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ t2 _) -\Rightarrow t2])) (THead k t0 t1) (THead k x t1) H3) in (let H6 \def -(eq_ind_r T x (\lambda (t2: T).(subst0 d u t0 t2)) H4 t0 H5) in (H d H6 -P)))))) H2)) (\lambda (H2: (ex2 T (\lambda (t2: T).(eq T (THead k t0 t1) -(THead k t0 t2))) (\lambda (t2: T).(subst0 (s k d) u t1 t2)))).(ex2_ind T -(\lambda (t2: T).(eq T (THead k t0 t1) (THead k t0 t2))) (\lambda (t2: -T).(subst0 (s k d) u t1 t2)) P (\lambda (x: T).(\lambda (H3: (eq T (THead k -t0 t1) (THead k t0 x))).(\lambda (H4: (subst0 (s k d) u t1 x)).(let H5 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ _ t2) -\Rightarrow t2])) (THead k t0 t1) (THead k t0 x) H3) in (let H6 \def -(eq_ind_r T x (\lambda (t2: T).(subst0 (s k d) u t1 t2)) H4 t1 H5) in (H0 (s -k d) H6 P)))))) H2)) (\lambda (H2: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T (THead k t0 t1) (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 d u t0 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k d) u t1 -t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead k t0 -t1) (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 d u t0 u2))) -(\lambda (_: T).(\lambda (t2: T).(subst0 (s k d) u t1 t2))) P (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H3: (eq T (THead k t0 t1) (THead k x0 -x1))).(\lambda (H4: (subst0 d u t0 x0)).(\lambda (H5: (subst0 (s k d) u t1 -x1)).(let H6 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda -(_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead -_ t2 _) \Rightarrow t2])) (THead k t0 t1) (THead k x0 x1) H3) in ((let H7 -\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ _ t2) -\Rightarrow t2])) (THead k t0 t1) (THead k x0 x1) H3) in (\lambda (H8: (eq T -t0 x0)).(let H9 \def (eq_ind_r T x1 (\lambda (t2: T).(subst0 (s k d) u t1 -t2)) H5 t1 H7) in (let H10 \def (eq_ind_r T x0 (\lambda (t2: T).(subst0 d u -t0 t2)) H4 t0 H8) in (H d H10 P))))) H6))))))) H2)) (subst0_gen_head k u t0 -t1 (THead k t0 t1) d H1)))))))))) t)). -(* COMMENTS -Initial nodes: 1119 -END *) - -theorem subst0_lift_lt: - \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).((subst0 -i u t1 t2) \to (\forall (d: nat).((lt i d) \to (\forall (h: nat).(subst0 i -(lift h (minus d (S i)) u) (lift h d t1) (lift h d t2))))))))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H: (subst0 i u t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t: -T).(\lambda (t0: T).(\lambda (t3: T).(\forall (d: nat).((lt n d) \to (\forall -(h: nat).(subst0 n (lift h (minus d (S n)) t) (lift h d t0) (lift h d -t3))))))))) (\lambda (v: T).(\lambda (i0: nat).(\lambda (d: nat).(\lambda -(H0: (lt i0 d)).(\lambda (h: nat).(eq_ind_r T (TLRef i0) (\lambda (t: -T).(subst0 i0 (lift h (minus d (S i0)) v) t (lift h d (lift (S i0) O v)))) -(let w \def (minus d (S i0)) in (eq_ind nat (plus (S i0) (minus d (S i0))) -(\lambda (n: nat).(subst0 i0 (lift h w v) (TLRef i0) (lift h n (lift (S i0) O -v)))) (eq_ind_r T (lift (S i0) O (lift h (minus d (S i0)) v)) (\lambda (t: -T).(subst0 i0 (lift h w v) (TLRef i0) t)) (subst0_lref (lift h (minus d (S -i0)) v) i0) (lift h (plus (S i0) (minus d (S i0))) (lift (S i0) O v)) (lift_d -v h (S i0) (minus d (S i0)) O (le_O_n (minus d (S i0))))) d (le_plus_minus_r -(S i0) d H0))) (lift h d (TLRef i0)) (lift_lref_lt i0 h d H0))))))) (\lambda -(v: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i0: nat).(\lambda (_: -(subst0 i0 v u1 u2)).(\lambda (H1: ((\forall (d: nat).((lt i0 d) \to (\forall -(h: nat).(subst0 i0 (lift h (minus d (S i0)) v) (lift h d u1) (lift h d -u2))))))).(\lambda (t: T).(\lambda (k: K).(\lambda (d: nat).(\lambda (H2: (lt -i0 d)).(\lambda (h: nat).(eq_ind_r T (THead k (lift h d u1) (lift h (s k d) -t)) (\lambda (t0: T).(subst0 i0 (lift h (minus d (S i0)) v) t0 (lift h d -(THead k u2 t)))) (eq_ind_r T (THead k (lift h d u2) (lift h (s k d) t)) -(\lambda (t0: T).(subst0 i0 (lift h (minus d (S i0)) v) (THead k (lift h d -u1) (lift h (s k d) t)) t0)) (subst0_fst (lift h (minus d (S i0)) v) (lift h -d u2) (lift h d u1) i0 (H1 d H2 h) (lift h (s k d) t) k) (lift h d (THead k -u2 t)) (lift_head k u2 t h d)) (lift h d (THead k u1 t)) (lift_head k u1 t h -d))))))))))))) (\lambda (k: K).(\lambda (v: T).(\lambda (t0: T).(\lambda (t3: -T).(\lambda (i0: nat).(\lambda (_: (subst0 (s k i0) v t3 t0)).(\lambda (H1: -((\forall (d: nat).((lt (s k i0) d) \to (\forall (h: nat).(subst0 (s k i0) -(lift h (minus d (S (s k i0))) v) (lift h d t3) (lift h d t0))))))).(\lambda -(u0: T).(\lambda (d: nat).(\lambda (H2: (lt i0 d)).(\lambda (h: nat).(let H3 -\def (eq_ind_r nat (S (s k i0)) (\lambda (n: nat).(\forall (d0: nat).((lt (s -k i0) d0) \to (\forall (h0: nat).(subst0 (s k i0) (lift h0 (minus d0 n) v) -(lift h0 d0 t3) (lift h0 d0 t0)))))) H1 (s k (S i0)) (s_S k i0)) in (eq_ind_r -T (THead k (lift h d u0) (lift h (s k d) t3)) (\lambda (t: T).(subst0 i0 -(lift h (minus d (S i0)) v) t (lift h d (THead k u0 t0)))) (eq_ind_r T (THead -k (lift h d u0) (lift h (s k d) t0)) (\lambda (t: T).(subst0 i0 (lift h -(minus d (S i0)) v) (THead k (lift h d u0) (lift h (s k d) t3)) t)) (eq_ind -nat (minus (s k d) (s k (S i0))) (\lambda (n: nat).(subst0 i0 (lift h n v) -(THead k (lift h d u0) (lift h (s k d) t3)) (THead k (lift h d u0) (lift h (s -k d) t0)))) (subst0_snd k (lift h (minus (s k d) (s k (S i0))) v) (lift h (s -k d) t0) (lift h (s k d) t3) i0 (H3 (s k d) (s_lt k i0 d H2) h) (lift h d -u0)) (minus d (S i0)) (minus_s_s k d (S i0))) (lift h d (THead k u0 t0)) -(lift_head k u0 t0 h d)) (lift h d (THead k u0 t3)) (lift_head k u0 t3 h -d)))))))))))))) (\lambda (v: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda -(i0: nat).(\lambda (_: (subst0 i0 v u1 u2)).(\lambda (H1: ((\forall (d: -nat).((lt i0 d) \to (\forall (h: nat).(subst0 i0 (lift h (minus d (S i0)) v) -(lift h d u1) (lift h d u2))))))).(\lambda (k: K).(\lambda (t0: T).(\lambda -(t3: T).(\lambda (_: (subst0 (s k i0) v t0 t3)).(\lambda (H3: ((\forall (d: -nat).((lt (s k i0) d) \to (\forall (h: nat).(subst0 (s k i0) (lift h (minus d -(S (s k i0))) v) (lift h d t0) (lift h d t3))))))).(\lambda (d: nat).(\lambda -(H4: (lt i0 d)).(\lambda (h: nat).(let H5 \def (eq_ind_r nat (S (s k i0)) -(\lambda (n: nat).(\forall (d0: nat).((lt (s k i0) d0) \to (\forall (h0: -nat).(subst0 (s k i0) (lift h0 (minus d0 n) v) (lift h0 d0 t0) (lift h0 d0 -t3)))))) H3 (s k (S i0)) (s_S k i0)) in (eq_ind_r T (THead k (lift h d u1) -(lift h (s k d) t0)) (\lambda (t: T).(subst0 i0 (lift h (minus d (S i0)) v) t -(lift h d (THead k u2 t3)))) (eq_ind_r T (THead k (lift h d u2) (lift h (s k -d) t3)) (\lambda (t: T).(subst0 i0 (lift h (minus d (S i0)) v) (THead k (lift -h d u1) (lift h (s k d) t0)) t)) (subst0_both (lift h (minus d (S i0)) v) -(lift h d u1) (lift h d u2) i0 (H1 d H4 h) k (lift h (s k d) t0) (lift h (s k -d) t3) (eq_ind nat (minus (s k d) (s k (S i0))) (\lambda (n: nat).(subst0 (s -k i0) (lift h n v) (lift h (s k d) t0) (lift h (s k d) t3))) (H5 (s k d) -(s_lt k i0 d H4) h) (minus d (S i0)) (minus_s_s k d (S i0)))) (lift h d -(THead k u2 t3)) (lift_head k u2 t3 h d)) (lift h d (THead k u1 t0)) -(lift_head k u1 t0 h d))))))))))))))))) i u t1 t2 H))))). -(* COMMENTS -Initial nodes: 1805 -END *) - -theorem subst0_lift_ge: - \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).(\forall -(h: nat).((subst0 i u t1 t2) \to (\forall (d: nat).((le d i) \to (subst0 -(plus i h) u (lift h d t1) (lift h d t2))))))))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda -(h: nat).(\lambda (H: (subst0 i u t1 t2)).(subst0_ind (\lambda (n: -nat).(\lambda (t: T).(\lambda (t0: T).(\lambda (t3: T).(\forall (d: nat).((le -d n) \to (subst0 (plus n h) t (lift h d t0) (lift h d t3)))))))) (\lambda (v: -T).(\lambda (i0: nat).(\lambda (d: nat).(\lambda (H0: (le d i0)).(eq_ind_r T -(TLRef (plus i0 h)) (\lambda (t: T).(subst0 (plus i0 h) v t (lift h d (lift -(S i0) O v)))) (eq_ind_r T (lift (plus h (S i0)) O v) (\lambda (t: T).(subst0 -(plus i0 h) v (TLRef (plus i0 h)) t)) (eq_ind nat (S (plus h i0)) (\lambda -(n: nat).(subst0 (plus i0 h) v (TLRef (plus i0 h)) (lift n O v))) (eq_ind_r -nat (plus h i0) (\lambda (n: nat).(subst0 n v (TLRef n) (lift (S (plus h i0)) -O v))) (subst0_lref v (plus h i0)) (plus i0 h) (plus_sym i0 h)) (plus h (S -i0)) (plus_n_Sm h i0)) (lift h d (lift (S i0) O v)) (lift_free v (S i0) h O d -(le_S d i0 H0) (le_O_n d))) (lift h d (TLRef i0)) (lift_lref_ge i0 h d -H0)))))) (\lambda (v: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i0: -nat).(\lambda (_: (subst0 i0 v u1 u2)).(\lambda (H1: ((\forall (d: nat).((le -d i0) \to (subst0 (plus i0 h) v (lift h d u1) (lift h d u2)))))).(\lambda (t: -T).(\lambda (k: K).(\lambda (d: nat).(\lambda (H2: (le d i0)).(eq_ind_r T -(THead k (lift h d u1) (lift h (s k d) t)) (\lambda (t0: T).(subst0 (plus i0 -h) v t0 (lift h d (THead k u2 t)))) (eq_ind_r T (THead k (lift h d u2) (lift -h (s k d) t)) (\lambda (t0: T).(subst0 (plus i0 h) v (THead k (lift h d u1) -(lift h (s k d) t)) t0)) (subst0_fst v (lift h d u2) (lift h d u1) (plus i0 -h) (H1 d H2) (lift h (s k d) t) k) (lift h d (THead k u2 t)) (lift_head k u2 -t h d)) (lift h d (THead k u1 t)) (lift_head k u1 t h d)))))))))))) (\lambda -(k: K).(\lambda (v: T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (i0: -nat).(\lambda (_: (subst0 (s k i0) v t3 t0)).(\lambda (H1: ((\forall (d: -nat).((le d (s k i0)) \to (subst0 (plus (s k i0) h) v (lift h d t3) (lift h d -t0)))))).(\lambda (u0: T).(\lambda (d: nat).(\lambda (H2: (le d i0)).(let H3 -\def (eq_ind_r nat (plus (s k i0) h) (\lambda (n: nat).(\forall (d0: -nat).((le d0 (s k i0)) \to (subst0 n v (lift h d0 t3) (lift h d0 t0))))) H1 -(s k (plus i0 h)) (s_plus k i0 h)) in (eq_ind_r T (THead k (lift h d u0) -(lift h (s k d) t3)) (\lambda (t: T).(subst0 (plus i0 h) v t (lift h d (THead -k u0 t0)))) (eq_ind_r T (THead k (lift h d u0) (lift h (s k d) t0)) (\lambda -(t: T).(subst0 (plus i0 h) v (THead k (lift h d u0) (lift h (s k d) t3)) t)) -(subst0_snd k v (lift h (s k d) t0) (lift h (s k d) t3) (plus i0 h) (H3 (s k -d) (s_le k d i0 H2)) (lift h d u0)) (lift h d (THead k u0 t0)) (lift_head k -u0 t0 h d)) (lift h d (THead k u0 t3)) (lift_head k u0 t3 h d))))))))))))) -(\lambda (v: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i0: nat).(\lambda -(_: (subst0 i0 v u1 u2)).(\lambda (H1: ((\forall (d: nat).((le d i0) \to -(subst0 (plus i0 h) v (lift h d u1) (lift h d u2)))))).(\lambda (k: -K).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (subst0 (s k i0) v t0 -t3)).(\lambda (H3: ((\forall (d: nat).((le d (s k i0)) \to (subst0 (plus (s k -i0) h) v (lift h d t0) (lift h d t3)))))).(\lambda (d: nat).(\lambda (H4: (le -d i0)).(let H5 \def (eq_ind_r nat (plus (s k i0) h) (\lambda (n: -nat).(\forall (d0: nat).((le d0 (s k i0)) \to (subst0 n v (lift h d0 t0) -(lift h d0 t3))))) H3 (s k (plus i0 h)) (s_plus k i0 h)) in (eq_ind_r T -(THead k (lift h d u1) (lift h (s k d) t0)) (\lambda (t: T).(subst0 (plus i0 -h) v t (lift h d (THead k u2 t3)))) (eq_ind_r T (THead k (lift h d u2) (lift -h (s k d) t3)) (\lambda (t: T).(subst0 (plus i0 h) v (THead k (lift h d u1) -(lift h (s k d) t0)) t)) (subst0_both v (lift h d u1) (lift h d u2) (plus i0 -h) (H1 d H4) k (lift h (s k d) t0) (lift h (s k d) t3) (H5 (s k d) (s_le k d -i0 H4))) (lift h d (THead k u2 t3)) (lift_head k u2 t3 h d)) (lift h d (THead -k u1 t0)) (lift_head k u1 t0 h d)))))))))))))))) i u t1 t2 H)))))). -(* COMMENTS -Initial nodes: 1449 -END *) - -theorem subst0_lift_ge_S: - \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).((subst0 -i u t1 t2) \to (\forall (d: nat).((le d i) \to (subst0 (S i) u (lift (S O) d -t1) (lift (S O) d t2)))))))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H: (subst0 i u t1 t2)).(\lambda (d: nat).(\lambda (H0: (le d i)).(eq_ind nat -(plus i (S O)) (\lambda (n: nat).(subst0 n u (lift (S O) d t1) (lift (S O) d -t2))) (subst0_lift_ge t1 t2 u i (S O) H d H0) (S i) (eq_ind_r nat (plus (S O) -i) (\lambda (n: nat).(eq nat n (S i))) (refl_equal nat (S i)) (plus i (S O)) -(plus_sym i (S O)))))))))). -(* COMMENTS -Initial nodes: 137 -END *) - -theorem subst0_lift_ge_s: - \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).((subst0 -i u t1 t2) \to (\forall (d: nat).((le d i) \to (\forall (b: B).(subst0 (s -(Bind b) i) u (lift (S O) d t1) (lift (S O) d t2))))))))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H: (subst0 i u t1 t2)).(\lambda (d: nat).(\lambda (H0: (le d i)).(\lambda -(_: B).(subst0_lift_ge_S t1 t2 u i H d H0)))))))). -(* COMMENTS -Initial nodes: 43 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/subst0/subst0.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/subst0/subst0.ma deleted file mode 100644 index 66c167d2b..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/subst0/subst0.ma +++ /dev/null @@ -1,1407 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/subst0/props.ma". - -theorem subst0_subst0: - \forall (t1: T).(\forall (t2: T).(\forall (u2: T).(\forall (j: nat).((subst0 -j u2 t1 t2) \to (\forall (u1: T).(\forall (u: T).(\forall (i: nat).((subst0 i -u u1 u2) \to (ex2 T (\lambda (t: T).(subst0 j u1 t1 t)) (\lambda (t: -T).(subst0 (S (plus i j)) u t t2))))))))))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (u2: T).(\lambda (j: nat).(\lambda -(H: (subst0 j u2 t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t: -T).(\lambda (t0: T).(\lambda (t3: T).(\forall (u1: T).(\forall (u: -T).(\forall (i: nat).((subst0 i u u1 t) \to (ex2 T (\lambda (t4: T).(subst0 n -u1 t0 t4)) (\lambda (t4: T).(subst0 (S (plus i n)) u t4 t3))))))))))) -(\lambda (v: T).(\lambda (i: nat).(\lambda (u1: T).(\lambda (u: T).(\lambda -(i0: nat).(\lambda (H0: (subst0 i0 u u1 v)).(eq_ind nat (plus i0 (S i)) -(\lambda (n: nat).(ex2 T (\lambda (t: T).(subst0 i u1 (TLRef i) t)) (\lambda -(t: T).(subst0 n u t (lift (S i) O v))))) (ex_intro2 T (\lambda (t: -T).(subst0 i u1 (TLRef i) t)) (\lambda (t: T).(subst0 (plus i0 (S i)) u t -(lift (S i) O v))) (lift (S i) O u1) (subst0_lref u1 i) (subst0_lift_ge u1 v -u i0 (S i) H0 O (le_O_n i0))) (S (plus i0 i)) (sym_eq nat (S (plus i0 i)) -(plus i0 (S i)) (plus_n_Sm i0 i))))))))) (\lambda (v: T).(\lambda (u0: -T).(\lambda (u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1 -u0)).(\lambda (H1: ((\forall (u3: T).(\forall (u: T).(\forall (i0: -nat).((subst0 i0 u u3 v) \to (ex2 T (\lambda (t: T).(subst0 i u3 u1 t)) -(\lambda (t: T).(subst0 (S (plus i0 i)) u t u0))))))))).(\lambda (t: -T).(\lambda (k: K).(\lambda (u3: T).(\lambda (u: T).(\lambda (i0: -nat).(\lambda (H2: (subst0 i0 u u3 v)).(ex2_ind T (\lambda (t0: T).(subst0 i -u3 u1 t0)) (\lambda (t0: T).(subst0 (S (plus i0 i)) u t0 u0)) (ex2 T (\lambda -(t0: T).(subst0 i u3 (THead k u1 t) t0)) (\lambda (t0: T).(subst0 (S (plus i0 -i)) u t0 (THead k u0 t)))) (\lambda (x: T).(\lambda (H3: (subst0 i u3 u1 -x)).(\lambda (H4: (subst0 (S (plus i0 i)) u x u0)).(ex_intro2 T (\lambda (t0: -T).(subst0 i u3 (THead k u1 t) t0)) (\lambda (t0: T).(subst0 (S (plus i0 i)) -u t0 (THead k u0 t))) (THead k x t) (subst0_fst u3 x u1 i H3 t k) (subst0_fst -u u0 x (S (plus i0 i)) H4 t k))))) (H1 u3 u i0 H2)))))))))))))) (\lambda (k: -K).(\lambda (v: T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (i: -nat).(\lambda (_: (subst0 (s k i) v t3 t0)).(\lambda (H1: ((\forall (u1: -T).(\forall (u: T).(\forall (i0: nat).((subst0 i0 u u1 v) \to (ex2 T (\lambda -(t: T).(subst0 (s k i) u1 t3 t)) (\lambda (t: T).(subst0 (S (plus i0 (s k -i))) u t t0))))))))).(\lambda (u: T).(\lambda (u1: T).(\lambda (u0: -T).(\lambda (i0: nat).(\lambda (H2: (subst0 i0 u0 u1 v)).(ex2_ind T (\lambda -(t: T).(subst0 (s k i) u1 t3 t)) (\lambda (t: T).(subst0 (S (plus i0 (s k -i))) u0 t t0)) (ex2 T (\lambda (t: T).(subst0 i u1 (THead k u t3) t)) -(\lambda (t: T).(subst0 (S (plus i0 i)) u0 t (THead k u t0)))) (\lambda (x: -T).(\lambda (H3: (subst0 (s k i) u1 t3 x)).(\lambda (H4: (subst0 (S (plus i0 -(s k i))) u0 x t0)).(let H5 \def (eq_ind_r nat (plus i0 (s k i)) (\lambda (n: -nat).(subst0 (S n) u0 x t0)) H4 (s k (plus i0 i)) (s_plus_sym k i0 i)) in -(let H6 \def (eq_ind_r nat (S (s k (plus i0 i))) (\lambda (n: nat).(subst0 n -u0 x t0)) H5 (s k (S (plus i0 i))) (s_S k (plus i0 i))) in (ex_intro2 T -(\lambda (t: T).(subst0 i u1 (THead k u t3) t)) (\lambda (t: T).(subst0 (S -(plus i0 i)) u0 t (THead k u t0))) (THead k u x) (subst0_snd k u1 x t3 i H3 -u) (subst0_snd k u0 t0 x (S (plus i0 i)) H6 u))))))) (H1 u1 u0 i0 -H2)))))))))))))) (\lambda (v: T).(\lambda (u1: T).(\lambda (u0: T).(\lambda -(i: nat).(\lambda (_: (subst0 i v u1 u0)).(\lambda (H1: ((\forall (u3: -T).(\forall (u: T).(\forall (i0: nat).((subst0 i0 u u3 v) \to (ex2 T (\lambda -(t: T).(subst0 i u3 u1 t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u t -u0))))))))).(\lambda (k: K).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: -(subst0 (s k i) v t0 t3)).(\lambda (H3: ((\forall (u3: T).(\forall (u: -T).(\forall (i0: nat).((subst0 i0 u u3 v) \to (ex2 T (\lambda (t: T).(subst0 -(s k i) u3 t0 t)) (\lambda (t: T).(subst0 (S (plus i0 (s k i))) u t -t3))))))))).(\lambda (u3: T).(\lambda (u: T).(\lambda (i0: nat).(\lambda (H4: -(subst0 i0 u u3 v)).(ex2_ind T (\lambda (t: T).(subst0 (s k i) u3 t0 t)) -(\lambda (t: T).(subst0 (S (plus i0 (s k i))) u t t3)) (ex2 T (\lambda (t: -T).(subst0 i u3 (THead k u1 t0) t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u -t (THead k u0 t3)))) (\lambda (x: T).(\lambda (H5: (subst0 (s k i) u3 t0 -x)).(\lambda (H6: (subst0 (S (plus i0 (s k i))) u x t3)).(ex2_ind T (\lambda -(t: T).(subst0 i u3 u1 t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u t u0)) -(ex2 T (\lambda (t: T).(subst0 i u3 (THead k u1 t0) t)) (\lambda (t: -T).(subst0 (S (plus i0 i)) u t (THead k u0 t3)))) (\lambda (x0: T).(\lambda -(H7: (subst0 i u3 u1 x0)).(\lambda (H8: (subst0 (S (plus i0 i)) u x0 -u0)).(let H9 \def (eq_ind_r nat (plus i0 (s k i)) (\lambda (n: nat).(subst0 -(S n) u x t3)) H6 (s k (plus i0 i)) (s_plus_sym k i0 i)) in (let H10 \def -(eq_ind_r nat (S (s k (plus i0 i))) (\lambda (n: nat).(subst0 n u x t3)) H9 -(s k (S (plus i0 i))) (s_S k (plus i0 i))) in (ex_intro2 T (\lambda (t: -T).(subst0 i u3 (THead k u1 t0) t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u -t (THead k u0 t3))) (THead k x0 x) (subst0_both u3 u1 x0 i H7 k t0 x H5) -(subst0_both u x0 u0 (S (plus i0 i)) H8 k x t3 H10))))))) (H1 u3 u i0 H4))))) -(H3 u3 u i0 H4))))))))))))))))) j u2 t1 t2 H))))). -(* COMMENTS -Initial nodes: 1613 -END *) - -theorem subst0_subst0_back: - \forall (t1: T).(\forall (t2: T).(\forall (u2: T).(\forall (j: nat).((subst0 -j u2 t1 t2) \to (\forall (u1: T).(\forall (u: T).(\forall (i: nat).((subst0 i -u u2 u1) \to (ex2 T (\lambda (t: T).(subst0 j u1 t1 t)) (\lambda (t: -T).(subst0 (S (plus i j)) u t2 t))))))))))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (u2: T).(\lambda (j: nat).(\lambda -(H: (subst0 j u2 t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t: -T).(\lambda (t0: T).(\lambda (t3: T).(\forall (u1: T).(\forall (u: -T).(\forall (i: nat).((subst0 i u t u1) \to (ex2 T (\lambda (t4: T).(subst0 n -u1 t0 t4)) (\lambda (t4: T).(subst0 (S (plus i n)) u t3 t4))))))))))) -(\lambda (v: T).(\lambda (i: nat).(\lambda (u1: T).(\lambda (u: T).(\lambda -(i0: nat).(\lambda (H0: (subst0 i0 u v u1)).(eq_ind nat (plus i0 (S i)) -(\lambda (n: nat).(ex2 T (\lambda (t: T).(subst0 i u1 (TLRef i) t)) (\lambda -(t: T).(subst0 n u (lift (S i) O v) t)))) (ex_intro2 T (\lambda (t: -T).(subst0 i u1 (TLRef i) t)) (\lambda (t: T).(subst0 (plus i0 (S i)) u (lift -(S i) O v) t)) (lift (S i) O u1) (subst0_lref u1 i) (subst0_lift_ge v u1 u i0 -(S i) H0 O (le_O_n i0))) (S (plus i0 i)) (sym_eq nat (S (plus i0 i)) (plus i0 -(S i)) (plus_n_Sm i0 i))))))))) (\lambda (v: T).(\lambda (u0: T).(\lambda -(u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1 u0)).(\lambda (H1: -((\forall (u3: T).(\forall (u: T).(\forall (i0: nat).((subst0 i0 u v u3) \to -(ex2 T (\lambda (t: T).(subst0 i u3 u1 t)) (\lambda (t: T).(subst0 (S (plus -i0 i)) u u0 t))))))))).(\lambda (t: T).(\lambda (k: K).(\lambda (u3: -T).(\lambda (u: T).(\lambda (i0: nat).(\lambda (H2: (subst0 i0 u v -u3)).(ex2_ind T (\lambda (t0: T).(subst0 i u3 u1 t0)) (\lambda (t0: -T).(subst0 (S (plus i0 i)) u u0 t0)) (ex2 T (\lambda (t0: T).(subst0 i u3 -(THead k u1 t) t0)) (\lambda (t0: T).(subst0 (S (plus i0 i)) u (THead k u0 t) -t0))) (\lambda (x: T).(\lambda (H3: (subst0 i u3 u1 x)).(\lambda (H4: (subst0 -(S (plus i0 i)) u u0 x)).(ex_intro2 T (\lambda (t0: T).(subst0 i u3 (THead k -u1 t) t0)) (\lambda (t0: T).(subst0 (S (plus i0 i)) u (THead k u0 t) t0)) -(THead k x t) (subst0_fst u3 x u1 i H3 t k) (subst0_fst u x u0 (S (plus i0 -i)) H4 t k))))) (H1 u3 u i0 H2)))))))))))))) (\lambda (k: K).(\lambda (v: -T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (i: nat).(\lambda (_: (subst0 -(s k i) v t3 t0)).(\lambda (H1: ((\forall (u1: T).(\forall (u: T).(\forall -(i0: nat).((subst0 i0 u v u1) \to (ex2 T (\lambda (t: T).(subst0 (s k i) u1 -t3 t)) (\lambda (t: T).(subst0 (S (plus i0 (s k i))) u t0 t))))))))).(\lambda -(u: T).(\lambda (u1: T).(\lambda (u0: T).(\lambda (i0: nat).(\lambda (H2: -(subst0 i0 u0 v u1)).(ex2_ind T (\lambda (t: T).(subst0 (s k i) u1 t3 t)) -(\lambda (t: T).(subst0 (S (plus i0 (s k i))) u0 t0 t)) (ex2 T (\lambda (t: -T).(subst0 i u1 (THead k u t3) t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u0 -(THead k u t0) t))) (\lambda (x: T).(\lambda (H3: (subst0 (s k i) u1 t3 -x)).(\lambda (H4: (subst0 (S (plus i0 (s k i))) u0 t0 x)).(let H5 \def -(eq_ind_r nat (plus i0 (s k i)) (\lambda (n: nat).(subst0 (S n) u0 t0 x)) H4 -(s k (plus i0 i)) (s_plus_sym k i0 i)) in (let H6 \def (eq_ind_r nat (S (s k -(plus i0 i))) (\lambda (n: nat).(subst0 n u0 t0 x)) H5 (s k (S (plus i0 i))) -(s_S k (plus i0 i))) in (ex_intro2 T (\lambda (t: T).(subst0 i u1 (THead k u -t3) t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u0 (THead k u t0) t)) (THead -k u x) (subst0_snd k u1 x t3 i H3 u) (subst0_snd k u0 x t0 (S (plus i0 i)) H6 -u))))))) (H1 u1 u0 i0 H2)))))))))))))) (\lambda (v: T).(\lambda (u1: -T).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1 -u0)).(\lambda (H1: ((\forall (u3: T).(\forall (u: T).(\forall (i0: -nat).((subst0 i0 u v u3) \to (ex2 T (\lambda (t: T).(subst0 i u3 u1 t)) -(\lambda (t: T).(subst0 (S (plus i0 i)) u u0 t))))))))).(\lambda (k: -K).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (subst0 (s k i) v t0 -t3)).(\lambda (H3: ((\forall (u3: T).(\forall (u: T).(\forall (i0: -nat).((subst0 i0 u v u3) \to (ex2 T (\lambda (t: T).(subst0 (s k i) u3 t0 t)) -(\lambda (t: T).(subst0 (S (plus i0 (s k i))) u t3 t))))))))).(\lambda (u3: -T).(\lambda (u: T).(\lambda (i0: nat).(\lambda (H4: (subst0 i0 u v -u3)).(ex2_ind T (\lambda (t: T).(subst0 (s k i) u3 t0 t)) (\lambda (t: -T).(subst0 (S (plus i0 (s k i))) u t3 t)) (ex2 T (\lambda (t: T).(subst0 i u3 -(THead k u1 t0) t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u (THead k u0 t3) -t))) (\lambda (x: T).(\lambda (H5: (subst0 (s k i) u3 t0 x)).(\lambda (H6: -(subst0 (S (plus i0 (s k i))) u t3 x)).(ex2_ind T (\lambda (t: T).(subst0 i -u3 u1 t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u u0 t)) (ex2 T (\lambda -(t: T).(subst0 i u3 (THead k u1 t0) t)) (\lambda (t: T).(subst0 (S (plus i0 -i)) u (THead k u0 t3) t))) (\lambda (x0: T).(\lambda (H7: (subst0 i u3 u1 -x0)).(\lambda (H8: (subst0 (S (plus i0 i)) u u0 x0)).(let H9 \def (eq_ind_r -nat (plus i0 (s k i)) (\lambda (n: nat).(subst0 (S n) u t3 x)) H6 (s k (plus -i0 i)) (s_plus_sym k i0 i)) in (let H10 \def (eq_ind_r nat (S (s k (plus i0 -i))) (\lambda (n: nat).(subst0 n u t3 x)) H9 (s k (S (plus i0 i))) (s_S k -(plus i0 i))) in (ex_intro2 T (\lambda (t: T).(subst0 i u3 (THead k u1 t0) -t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u (THead k u0 t3) t)) (THead k x0 -x) (subst0_both u3 u1 x0 i H7 k t0 x H5) (subst0_both u u0 x0 (S (plus i0 i)) -H8 k t3 x H10))))))) (H1 u3 u i0 H4))))) (H3 u3 u i0 H4))))))))))))))))) j u2 -t1 t2 H))))). -(* COMMENTS -Initial nodes: 1613 -END *) - -theorem subst0_trans: - \forall (t2: T).(\forall (t1: T).(\forall (v: T).(\forall (i: nat).((subst0 -i v t1 t2) \to (\forall (t3: T).((subst0 i v t2 t3) \to (subst0 i v t1 -t3))))))) -\def - \lambda (t2: T).(\lambda (t1: T).(\lambda (v: T).(\lambda (i: nat).(\lambda -(H: (subst0 i v t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t: -T).(\lambda (t0: T).(\lambda (t3: T).(\forall (t4: T).((subst0 n t t3 t4) \to -(subst0 n t t0 t4))))))) (\lambda (v0: T).(\lambda (i0: nat).(\lambda (t3: -T).(\lambda (H0: (subst0 i0 v0 (lift (S i0) O v0) t3)).(subst0_gen_lift_false -v0 v0 t3 (S i0) O i0 (le_O_n i0) (le_n (plus O (S i0))) H0 (subst0 i0 v0 -(TLRef i0) t3)))))) (\lambda (v0: T).(\lambda (u2: T).(\lambda (u1: -T).(\lambda (i0: nat).(\lambda (H0: (subst0 i0 v0 u1 u2)).(\lambda (H1: -((\forall (t3: T).((subst0 i0 v0 u2 t3) \to (subst0 i0 v0 u1 t3))))).(\lambda -(t: T).(\lambda (k: K).(\lambda (t3: T).(\lambda (H2: (subst0 i0 v0 (THead k -u2 t) t3)).(or3_ind (ex2 T (\lambda (u3: T).(eq T t3 (THead k u3 t))) -(\lambda (u3: T).(subst0 i0 v0 u2 u3))) (ex2 T (\lambda (t4: T).(eq T t3 -(THead k u2 t4))) (\lambda (t4: T).(subst0 (s k i0) v0 t t4))) (ex3_2 T T -(\lambda (u3: T).(\lambda (t4: T).(eq T t3 (THead k u3 t4)))) (\lambda (u3: -T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) (\lambda (_: T).(\lambda (t4: -T).(subst0 (s k i0) v0 t t4)))) (subst0 i0 v0 (THead k u1 t) t3) (\lambda -(H3: (ex2 T (\lambda (u3: T).(eq T t3 (THead k u3 t))) (\lambda (u3: -T).(subst0 i0 v0 u2 u3)))).(ex2_ind T (\lambda (u3: T).(eq T t3 (THead k u3 -t))) (\lambda (u3: T).(subst0 i0 v0 u2 u3)) (subst0 i0 v0 (THead k u1 t) t3) -(\lambda (x: T).(\lambda (H4: (eq T t3 (THead k x t))).(\lambda (H5: (subst0 -i0 v0 u2 x)).(eq_ind_r T (THead k x t) (\lambda (t0: T).(subst0 i0 v0 (THead -k u1 t) t0)) (subst0_fst v0 x u1 i0 (H1 x H5) t k) t3 H4)))) H3)) (\lambda -(H3: (ex2 T (\lambda (t4: T).(eq T t3 (THead k u2 t4))) (\lambda (t4: -T).(subst0 (s k i0) v0 t t4)))).(ex2_ind T (\lambda (t4: T).(eq T t3 (THead k -u2 t4))) (\lambda (t4: T).(subst0 (s k i0) v0 t t4)) (subst0 i0 v0 (THead k -u1 t) t3) (\lambda (x: T).(\lambda (H4: (eq T t3 (THead k u2 x))).(\lambda -(H5: (subst0 (s k i0) v0 t x)).(eq_ind_r T (THead k u2 x) (\lambda (t0: -T).(subst0 i0 v0 (THead k u1 t) t0)) (subst0_both v0 u1 u2 i0 H0 k t x H5) t3 -H4)))) H3)) (\lambda (H3: (ex3_2 T T (\lambda (u3: T).(\lambda (t4: T).(eq T -t3 (THead k u3 t4)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) -(\lambda (_: T).(\lambda (t4: T).(subst0 (s k i0) v0 t t4))))).(ex3_2_ind T T -(\lambda (u3: T).(\lambda (t4: T).(eq T t3 (THead k u3 t4)))) (\lambda (u3: -T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) (\lambda (_: T).(\lambda (t4: -T).(subst0 (s k i0) v0 t t4))) (subst0 i0 v0 (THead k u1 t) t3) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H4: (eq T t3 (THead k x0 x1))).(\lambda (H5: -(subst0 i0 v0 u2 x0)).(\lambda (H6: (subst0 (s k i0) v0 t x1)).(eq_ind_r T -(THead k x0 x1) (\lambda (t0: T).(subst0 i0 v0 (THead k u1 t) t0)) -(subst0_both v0 u1 x0 i0 (H1 x0 H5) k t x1 H6) t3 H4)))))) H3)) -(subst0_gen_head k v0 u2 t t3 i0 H2)))))))))))) (\lambda (k: K).(\lambda (v0: -T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (i0: nat).(\lambda (H0: (subst0 -(s k i0) v0 t3 t0)).(\lambda (H1: ((\forall (t4: T).((subst0 (s k i0) v0 t0 -t4) \to (subst0 (s k i0) v0 t3 t4))))).(\lambda (u: T).(\lambda (t4: -T).(\lambda (H2: (subst0 i0 v0 (THead k u t0) t4)).(or3_ind (ex2 T (\lambda -(u2: T).(eq T t4 (THead k u2 t0))) (\lambda (u2: T).(subst0 i0 v0 u u2))) -(ex2 T (\lambda (t5: T).(eq T t4 (THead k u t5))) (\lambda (t5: T).(subst0 (s -k i0) v0 t0 t5))) (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 -(THead k u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u u2))) -(\lambda (_: T).(\lambda (t5: T).(subst0 (s k i0) v0 t0 t5)))) (subst0 i0 v0 -(THead k u t3) t4) (\lambda (H3: (ex2 T (\lambda (u2: T).(eq T t4 (THead k u2 -t0))) (\lambda (u2: T).(subst0 i0 v0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq -T t4 (THead k u2 t0))) (\lambda (u2: T).(subst0 i0 v0 u u2)) (subst0 i0 v0 -(THead k u t3) t4) (\lambda (x: T).(\lambda (H4: (eq T t4 (THead k x -t0))).(\lambda (H5: (subst0 i0 v0 u x)).(eq_ind_r T (THead k x t0) (\lambda -(t: T).(subst0 i0 v0 (THead k u t3) t)) (subst0_both v0 u x i0 H5 k t3 t0 H0) -t4 H4)))) H3)) (\lambda (H3: (ex2 T (\lambda (t5: T).(eq T t4 (THead k u -t5))) (\lambda (t5: T).(subst0 (s k i0) v0 t0 t5)))).(ex2_ind T (\lambda (t5: -T).(eq T t4 (THead k u t5))) (\lambda (t5: T).(subst0 (s k i0) v0 t0 t5)) -(subst0 i0 v0 (THead k u t3) t4) (\lambda (x: T).(\lambda (H4: (eq T t4 -(THead k u x))).(\lambda (H5: (subst0 (s k i0) v0 t0 x)).(eq_ind_r T (THead k -u x) (\lambda (t: T).(subst0 i0 v0 (THead k u t3) t)) (subst0_snd k v0 x t3 -i0 (H1 x H5) u) t4 H4)))) H3)) (\lambda (H3: (ex3_2 T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead k u2 t5)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 i0 v0 u u2))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s k i0) v0 -t0 t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead k -u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u u2))) (\lambda (_: -T).(\lambda (t5: T).(subst0 (s k i0) v0 t0 t5))) (subst0 i0 v0 (THead k u t3) -t4) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t4 (THead k x0 -x1))).(\lambda (H5: (subst0 i0 v0 u x0)).(\lambda (H6: (subst0 (s k i0) v0 t0 -x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t: T).(subst0 i0 v0 (THead k u t3) -t)) (subst0_both v0 u x0 i0 H5 k t3 x1 (H1 x1 H6)) t4 H4)))))) H3)) -(subst0_gen_head k v0 u t0 t4 i0 H2)))))))))))) (\lambda (v0: T).(\lambda -(u1: T).(\lambda (u2: T).(\lambda (i0: nat).(\lambda (H0: (subst0 i0 v0 u1 -u2)).(\lambda (H1: ((\forall (t3: T).((subst0 i0 v0 u2 t3) \to (subst0 i0 v0 -u1 t3))))).(\lambda (k: K).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H2: -(subst0 (s k i0) v0 t0 t3)).(\lambda (H3: ((\forall (t4: T).((subst0 (s k i0) -v0 t3 t4) \to (subst0 (s k i0) v0 t0 t4))))).(\lambda (t4: T).(\lambda (H4: -(subst0 i0 v0 (THead k u2 t3) t4)).(or3_ind (ex2 T (\lambda (u3: T).(eq T t4 -(THead k u3 t3))) (\lambda (u3: T).(subst0 i0 v0 u2 u3))) (ex2 T (\lambda -(t5: T).(eq T t4 (THead k u2 t5))) (\lambda (t5: T).(subst0 (s k i0) v0 t3 -t5))) (ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T t4 (THead k u3 -t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) (\lambda (_: -T).(\lambda (t5: T).(subst0 (s k i0) v0 t3 t5)))) (subst0 i0 v0 (THead k u1 -t0) t4) (\lambda (H5: (ex2 T (\lambda (u3: T).(eq T t4 (THead k u3 t3))) -(\lambda (u3: T).(subst0 i0 v0 u2 u3)))).(ex2_ind T (\lambda (u3: T).(eq T t4 -(THead k u3 t3))) (\lambda (u3: T).(subst0 i0 v0 u2 u3)) (subst0 i0 v0 (THead -k u1 t0) t4) (\lambda (x: T).(\lambda (H6: (eq T t4 (THead k x t3))).(\lambda -(H7: (subst0 i0 v0 u2 x)).(eq_ind_r T (THead k x t3) (\lambda (t: T).(subst0 -i0 v0 (THead k u1 t0) t)) (subst0_both v0 u1 x i0 (H1 x H7) k t0 t3 H2) t4 -H6)))) H5)) (\lambda (H5: (ex2 T (\lambda (t5: T).(eq T t4 (THead k u2 t5))) -(\lambda (t5: T).(subst0 (s k i0) v0 t3 t5)))).(ex2_ind T (\lambda (t5: -T).(eq T t4 (THead k u2 t5))) (\lambda (t5: T).(subst0 (s k i0) v0 t3 t5)) -(subst0 i0 v0 (THead k u1 t0) t4) (\lambda (x: T).(\lambda (H6: (eq T t4 -(THead k u2 x))).(\lambda (H7: (subst0 (s k i0) v0 t3 x)).(eq_ind_r T (THead -k u2 x) (\lambda (t: T).(subst0 i0 v0 (THead k u1 t0) t)) (subst0_both v0 u1 -u2 i0 H0 k t0 x (H3 x H7)) t4 H6)))) H5)) (\lambda (H5: (ex3_2 T T (\lambda -(u3: T).(\lambda (t5: T).(eq T t4 (THead k u3 t5)))) (\lambda (u3: -T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) (\lambda (_: T).(\lambda (t5: -T).(subst0 (s k i0) v0 t3 t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda -(t5: T).(eq T t4 (THead k u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 -i0 v0 u2 u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s k i0) v0 t3 t5))) -(subst0 i0 v0 (THead k u1 t0) t4) (\lambda (x0: T).(\lambda (x1: T).(\lambda -(H6: (eq T t4 (THead k x0 x1))).(\lambda (H7: (subst0 i0 v0 u2 x0)).(\lambda -(H8: (subst0 (s k i0) v0 t3 x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t: -T).(subst0 i0 v0 (THead k u1 t0) t)) (subst0_both v0 u1 x0 i0 (H1 x0 H7) k t0 -x1 (H3 x1 H8)) t4 H6)))))) H5)) (subst0_gen_head k v0 u2 t3 t4 i0 -H4))))))))))))))) i v t1 t2 H))))). -(* COMMENTS -Initial nodes: 2555 -END *) - -theorem subst0_confluence_neq: - \forall (t0: T).(\forall (t1: T).(\forall (u1: T).(\forall (i1: -nat).((subst0 i1 u1 t0 t1) \to (\forall (t2: T).(\forall (u2: T).(\forall -(i2: nat).((subst0 i2 u2 t0 t2) \to ((not (eq nat i1 i2)) \to (ex2 T (\lambda -(t: T).(subst0 i2 u2 t1 t)) (\lambda (t: T).(subst0 i1 u1 t2 t)))))))))))) -\def - \lambda (t0: T).(\lambda (t1: T).(\lambda (u1: T).(\lambda (i1: -nat).(\lambda (H: (subst0 i1 u1 t0 t1)).(subst0_ind (\lambda (n: -nat).(\lambda (t: T).(\lambda (t2: T).(\lambda (t3: T).(\forall (t4: -T).(\forall (u2: T).(\forall (i2: nat).((subst0 i2 u2 t2 t4) \to ((not (eq -nat n i2)) \to (ex2 T (\lambda (t5: T).(subst0 i2 u2 t3 t5)) (\lambda (t5: -T).(subst0 n t t4 t5)))))))))))) (\lambda (v: T).(\lambda (i: nat).(\lambda -(t2: T).(\lambda (u2: T).(\lambda (i2: nat).(\lambda (H0: (subst0 i2 u2 -(TLRef i) t2)).(\lambda (H1: (not (eq nat i i2))).(land_ind (eq nat i i2) (eq -T t2 (lift (S i) O u2)) (ex2 T (\lambda (t: T).(subst0 i2 u2 (lift (S i) O v) -t)) (\lambda (t: T).(subst0 i v t2 t))) (\lambda (H2: (eq nat i i2)).(\lambda -(H3: (eq T t2 (lift (S i) O u2))).(let H4 \def (eq_ind nat i (\lambda (n: -nat).(not (eq nat n i2))) H1 i2 H2) in (eq_ind_r T (lift (S i) O u2) (\lambda -(t: T).(ex2 T (\lambda (t3: T).(subst0 i2 u2 (lift (S i) O v) t3)) (\lambda -(t3: T).(subst0 i v t t3)))) (let H5 \def (match (H4 (refl_equal nat i2)) in -False return (\lambda (_: False).(ex2 T (\lambda (t: T).(subst0 i2 u2 (lift -(S i) O v) t)) (\lambda (t: T).(subst0 i v (lift (S i) O u2) t)))) with []) -in H5) t2 H3)))) (subst0_gen_lref u2 t2 i2 i H0))))))))) (\lambda (v: -T).(\lambda (u2: T).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H0: (subst0 -i v u0 u2)).(\lambda (H1: ((\forall (t2: T).(\forall (u3: T).(\forall (i2: -nat).((subst0 i2 u3 u0 t2) \to ((not (eq nat i i2)) \to (ex2 T (\lambda (t: -T).(subst0 i2 u3 u2 t)) (\lambda (t: T).(subst0 i v t2 t)))))))))).(\lambda -(t: T).(\lambda (k: K).(\lambda (t2: T).(\lambda (u3: T).(\lambda (i2: -nat).(\lambda (H2: (subst0 i2 u3 (THead k u0 t) t2)).(\lambda (H3: (not (eq -nat i i2))).(or3_ind (ex2 T (\lambda (u4: T).(eq T t2 (THead k u4 t))) -(\lambda (u4: T).(subst0 i2 u3 u0 u4))) (ex2 T (\lambda (t3: T).(eq T t2 -(THead k u0 t3))) (\lambda (t3: T).(subst0 (s k i2) u3 t t3))) (ex3_2 T T -(\lambda (u4: T).(\lambda (t3: T).(eq T t2 (THead k u4 t3)))) (\lambda (u4: -T).(\lambda (_: T).(subst0 i2 u3 u0 u4))) (\lambda (_: T).(\lambda (t3: -T).(subst0 (s k i2) u3 t t3)))) (ex2 T (\lambda (t3: T).(subst0 i2 u3 (THead -k u2 t) t3)) (\lambda (t3: T).(subst0 i v t2 t3))) (\lambda (H4: (ex2 T -(\lambda (u4: T).(eq T t2 (THead k u4 t))) (\lambda (u4: T).(subst0 i2 u3 u0 -u4)))).(ex2_ind T (\lambda (u4: T).(eq T t2 (THead k u4 t))) (\lambda (u4: -T).(subst0 i2 u3 u0 u4)) (ex2 T (\lambda (t3: T).(subst0 i2 u3 (THead k u2 t) -t3)) (\lambda (t3: T).(subst0 i v t2 t3))) (\lambda (x: T).(\lambda (H5: (eq -T t2 (THead k x t))).(\lambda (H6: (subst0 i2 u3 u0 x)).(eq_ind_r T (THead k -x t) (\lambda (t3: T).(ex2 T (\lambda (t4: T).(subst0 i2 u3 (THead k u2 t) -t4)) (\lambda (t4: T).(subst0 i v t3 t4)))) (ex2_ind T (\lambda (t3: -T).(subst0 i2 u3 u2 t3)) (\lambda (t3: T).(subst0 i v x t3)) (ex2 T (\lambda -(t3: T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i v (THead -k x t) t3))) (\lambda (x0: T).(\lambda (H7: (subst0 i2 u3 u2 x0)).(\lambda -(H8: (subst0 i v x x0)).(ex_intro2 T (\lambda (t3: T).(subst0 i2 u3 (THead k -u2 t) t3)) (\lambda (t3: T).(subst0 i v (THead k x t) t3)) (THead k x0 t) -(subst0_fst u3 x0 u2 i2 H7 t k) (subst0_fst v x0 x i H8 t k))))) (H1 x u3 i2 -H6 H3)) t2 H5)))) H4)) (\lambda (H4: (ex2 T (\lambda (t3: T).(eq T t2 (THead -k u0 t3))) (\lambda (t3: T).(subst0 (s k i2) u3 t t3)))).(ex2_ind T (\lambda -(t3: T).(eq T t2 (THead k u0 t3))) (\lambda (t3: T).(subst0 (s k i2) u3 t -t3)) (ex2 T (\lambda (t3: T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda (t3: -T).(subst0 i v t2 t3))) (\lambda (x: T).(\lambda (H5: (eq T t2 (THead k u0 -x))).(\lambda (H6: (subst0 (s k i2) u3 t x)).(eq_ind_r T (THead k u0 x) -(\lambda (t3: T).(ex2 T (\lambda (t4: T).(subst0 i2 u3 (THead k u2 t) t4)) -(\lambda (t4: T).(subst0 i v t3 t4)))) (ex_intro2 T (\lambda (t3: T).(subst0 -i2 u3 (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i v (THead k u0 x) t3)) -(THead k u2 x) (subst0_snd k u3 x t i2 H6 u2) (subst0_fst v u2 u0 i H0 x k)) -t2 H5)))) H4)) (\lambda (H4: (ex3_2 T T (\lambda (u4: T).(\lambda (t3: T).(eq -T t2 (THead k u4 t3)))) (\lambda (u4: T).(\lambda (_: T).(subst0 i2 u3 u0 -u4))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i2) u3 t -t3))))).(ex3_2_ind T T (\lambda (u4: T).(\lambda (t3: T).(eq T t2 (THead k u4 -t3)))) (\lambda (u4: T).(\lambda (_: T).(subst0 i2 u3 u0 u4))) (\lambda (_: -T).(\lambda (t3: T).(subst0 (s k i2) u3 t t3))) (ex2 T (\lambda (t3: -T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i v t2 t3))) -(\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T t2 (THead k x0 -x1))).(\lambda (H6: (subst0 i2 u3 u0 x0)).(\lambda (H7: (subst0 (s k i2) u3 t -x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t3: T).(ex2 T (\lambda (t4: -T).(subst0 i2 u3 (THead k u2 t) t4)) (\lambda (t4: T).(subst0 i v t3 t4)))) -(ex2_ind T (\lambda (t3: T).(subst0 i2 u3 u2 t3)) (\lambda (t3: T).(subst0 i -v x0 t3)) (ex2 T (\lambda (t3: T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda -(t3: T).(subst0 i v (THead k x0 x1) t3))) (\lambda (x: T).(\lambda (H8: -(subst0 i2 u3 u2 x)).(\lambda (H9: (subst0 i v x0 x)).(ex_intro2 T (\lambda -(t3: T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i v (THead -k x0 x1) t3)) (THead k x x1) (subst0_both u3 u2 x i2 H8 k t x1 H7) -(subst0_fst v x x0 i H9 x1 k))))) (H1 x0 u3 i2 H6 H3)) t2 H5)))))) H4)) -(subst0_gen_head k u3 u0 t t2 i2 H2))))))))))))))) (\lambda (k: K).(\lambda -(v: T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (i: nat).(\lambda (H0: -(subst0 (s k i) v t3 t2)).(\lambda (H1: ((\forall (t4: T).(\forall (u2: -T).(\forall (i2: nat).((subst0 i2 u2 t3 t4) \to ((not (eq nat (s k i) i2)) -\to (ex2 T (\lambda (t: T).(subst0 i2 u2 t2 t)) (\lambda (t: T).(subst0 (s k -i) v t4 t)))))))))).(\lambda (u: T).(\lambda (t4: T).(\lambda (u2: -T).(\lambda (i2: nat).(\lambda (H2: (subst0 i2 u2 (THead k u t3) -t4)).(\lambda (H3: (not (eq nat i i2))).(or3_ind (ex2 T (\lambda (u3: T).(eq -T t4 (THead k u3 t3))) (\lambda (u3: T).(subst0 i2 u2 u u3))) (ex2 T (\lambda -(t5: T).(eq T t4 (THead k u t5))) (\lambda (t5: T).(subst0 (s k i2) u2 t3 -t5))) (ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T t4 (THead k u3 -t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i2 u2 u u3))) (\lambda (_: -T).(\lambda (t5: T).(subst0 (s k i2) u2 t3 t5)))) (ex2 T (\lambda (t: -T).(subst0 i2 u2 (THead k u t2) t)) (\lambda (t: T).(subst0 i v t4 t))) -(\lambda (H4: (ex2 T (\lambda (u3: T).(eq T t4 (THead k u3 t3))) (\lambda -(u3: T).(subst0 i2 u2 u u3)))).(ex2_ind T (\lambda (u3: T).(eq T t4 (THead k -u3 t3))) (\lambda (u3: T).(subst0 i2 u2 u u3)) (ex2 T (\lambda (t: T).(subst0 -i2 u2 (THead k u t2) t)) (\lambda (t: T).(subst0 i v t4 t))) (\lambda (x: -T).(\lambda (H5: (eq T t4 (THead k x t3))).(\lambda (H6: (subst0 i2 u2 u -x)).(eq_ind_r T (THead k x t3) (\lambda (t: T).(ex2 T (\lambda (t5: -T).(subst0 i2 u2 (THead k u t2) t5)) (\lambda (t5: T).(subst0 i v t t5)))) -(ex_intro2 T (\lambda (t: T).(subst0 i2 u2 (THead k u t2) t)) (\lambda (t: -T).(subst0 i v (THead k x t3) t)) (THead k x t2) (subst0_fst u2 x u i2 H6 t2 -k) (subst0_snd k v t2 t3 i H0 x)) t4 H5)))) H4)) (\lambda (H4: (ex2 T -(\lambda (t5: T).(eq T t4 (THead k u t5))) (\lambda (t5: T).(subst0 (s k i2) -u2 t3 t5)))).(ex2_ind T (\lambda (t5: T).(eq T t4 (THead k u t5))) (\lambda -(t5: T).(subst0 (s k i2) u2 t3 t5)) (ex2 T (\lambda (t: T).(subst0 i2 u2 -(THead k u t2) t)) (\lambda (t: T).(subst0 i v t4 t))) (\lambda (x: -T).(\lambda (H5: (eq T t4 (THead k u x))).(\lambda (H6: (subst0 (s k i2) u2 -t3 x)).(eq_ind_r T (THead k u x) (\lambda (t: T).(ex2 T (\lambda (t5: -T).(subst0 i2 u2 (THead k u t2) t5)) (\lambda (t5: T).(subst0 i v t t5)))) -(ex2_ind T (\lambda (t: T).(subst0 (s k i2) u2 t2 t)) (\lambda (t: T).(subst0 -(s k i) v x t)) (ex2 T (\lambda (t: T).(subst0 i2 u2 (THead k u t2) t)) -(\lambda (t: T).(subst0 i v (THead k u x) t))) (\lambda (x0: T).(\lambda (H7: -(subst0 (s k i2) u2 t2 x0)).(\lambda (H8: (subst0 (s k i) v x x0)).(ex_intro2 -T (\lambda (t: T).(subst0 i2 u2 (THead k u t2) t)) (\lambda (t: T).(subst0 i -v (THead k u x) t)) (THead k u x0) (subst0_snd k u2 x0 t2 i2 H7 u) -(subst0_snd k v x0 x i H8 u))))) (H1 x u2 (s k i2) H6 (ex2_ind T (\lambda (t: -T).(subst0 (s k i2) u2 t2 t)) (\lambda (t: T).(subst0 (s k i) v x t)) ((eq -nat (s k i) (s k i2)) \to False) (\lambda (x0: T).(\lambda (_: (subst0 (s k -i2) u2 t2 x0)).(\lambda (_: (subst0 (s k i) v x x0)).(\lambda (H9: (eq nat (s -k i) (s k i2))).(H3 (s_inj k i i2 H9)))))) (H1 x u2 (s k i2) H6 (\lambda (H7: -(eq nat (s k i) (s k i2))).(H3 (s_inj k i i2 H7))))))) t4 H5)))) H4)) -(\lambda (H4: (ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T t4 (THead k -u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i2 u2 u u3))) (\lambda (_: -T).(\lambda (t5: T).(subst0 (s k i2) u2 t3 t5))))).(ex3_2_ind T T (\lambda -(u3: T).(\lambda (t5: T).(eq T t4 (THead k u3 t5)))) (\lambda (u3: -T).(\lambda (_: T).(subst0 i2 u2 u u3))) (\lambda (_: T).(\lambda (t5: -T).(subst0 (s k i2) u2 t3 t5))) (ex2 T (\lambda (t: T).(subst0 i2 u2 (THead k -u t2) t)) (\lambda (t: T).(subst0 i v t4 t))) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H5: (eq T t4 (THead k x0 x1))).(\lambda (H6: (subst0 i2 u2 u -x0)).(\lambda (H7: (subst0 (s k i2) u2 t3 x1)).(eq_ind_r T (THead k x0 x1) -(\lambda (t: T).(ex2 T (\lambda (t5: T).(subst0 i2 u2 (THead k u t2) t5)) -(\lambda (t5: T).(subst0 i v t t5)))) (ex2_ind T (\lambda (t: T).(subst0 (s k -i2) u2 t2 t)) (\lambda (t: T).(subst0 (s k i) v x1 t)) (ex2 T (\lambda (t: -T).(subst0 i2 u2 (THead k u t2) t)) (\lambda (t: T).(subst0 i v (THead k x0 -x1) t))) (\lambda (x: T).(\lambda (H8: (subst0 (s k i2) u2 t2 x)).(\lambda -(H9: (subst0 (s k i) v x1 x)).(ex_intro2 T (\lambda (t: T).(subst0 i2 u2 -(THead k u t2) t)) (\lambda (t: T).(subst0 i v (THead k x0 x1) t)) (THead k -x0 x) (subst0_both u2 u x0 i2 H6 k t2 x H8) (subst0_snd k v x x1 i H9 x0))))) -(H1 x1 u2 (s k i2) H7 (ex2_ind T (\lambda (t: T).(subst0 (s k i2) u2 t2 t)) -(\lambda (t: T).(subst0 (s k i) v x1 t)) ((eq nat (s k i) (s k i2)) \to -False) (\lambda (x: T).(\lambda (_: (subst0 (s k i2) u2 t2 x)).(\lambda (_: -(subst0 (s k i) v x1 x)).(\lambda (H10: (eq nat (s k i) (s k i2))).(H3 (s_inj -k i i2 H10)))))) (H1 x1 u2 (s k i2) H7 (\lambda (H8: (eq nat (s k i) (s k -i2))).(H3 (s_inj k i i2 H8))))))) t4 H5)))))) H4)) (subst0_gen_head k u2 u t3 -t4 i2 H2))))))))))))))) (\lambda (v: T).(\lambda (u0: T).(\lambda (u2: -T).(\lambda (i: nat).(\lambda (H0: (subst0 i v u0 u2)).(\lambda (H1: -((\forall (t2: T).(\forall (u3: T).(\forall (i2: nat).((subst0 i2 u3 u0 t2) -\to ((not (eq nat i i2)) \to (ex2 T (\lambda (t: T).(subst0 i2 u3 u2 t)) -(\lambda (t: T).(subst0 i v t2 t)))))))))).(\lambda (k: K).(\lambda (t2: -T).(\lambda (t3: T).(\lambda (H2: (subst0 (s k i) v t2 t3)).(\lambda (H3: -((\forall (t4: T).(\forall (u3: T).(\forall (i2: nat).((subst0 i2 u3 t2 t4) -\to ((not (eq nat (s k i) i2)) \to (ex2 T (\lambda (t: T).(subst0 i2 u3 t3 -t)) (\lambda (t: T).(subst0 (s k i) v t4 t)))))))))).(\lambda (t4: -T).(\lambda (u3: T).(\lambda (i2: nat).(\lambda (H4: (subst0 i2 u3 (THead k -u0 t2) t4)).(\lambda (H5: (not (eq nat i i2))).(or3_ind (ex2 T (\lambda (u4: -T).(eq T t4 (THead k u4 t2))) (\lambda (u4: T).(subst0 i2 u3 u0 u4))) (ex2 T -(\lambda (t5: T).(eq T t4 (THead k u0 t5))) (\lambda (t5: T).(subst0 (s k i2) -u3 t2 t5))) (ex3_2 T T (\lambda (u4: T).(\lambda (t5: T).(eq T t4 (THead k u4 -t5)))) (\lambda (u4: T).(\lambda (_: T).(subst0 i2 u3 u0 u4))) (\lambda (_: -T).(\lambda (t5: T).(subst0 (s k i2) u3 t2 t5)))) (ex2 T (\lambda (t: -T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v t4 t))) -(\lambda (H6: (ex2 T (\lambda (u4: T).(eq T t4 (THead k u4 t2))) (\lambda -(u4: T).(subst0 i2 u3 u0 u4)))).(ex2_ind T (\lambda (u4: T).(eq T t4 (THead k -u4 t2))) (\lambda (u4: T).(subst0 i2 u3 u0 u4)) (ex2 T (\lambda (t: -T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v t4 t))) -(\lambda (x: T).(\lambda (H7: (eq T t4 (THead k x t2))).(\lambda (H8: (subst0 -i2 u3 u0 x)).(eq_ind_r T (THead k x t2) (\lambda (t: T).(ex2 T (\lambda (t5: -T).(subst0 i2 u3 (THead k u2 t3) t5)) (\lambda (t5: T).(subst0 i v t t5)))) -(ex2_ind T (\lambda (t: T).(subst0 i2 u3 u2 t)) (\lambda (t: T).(subst0 i v x -t)) (ex2 T (\lambda (t: T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: -T).(subst0 i v (THead k x t2) t))) (\lambda (x0: T).(\lambda (H9: (subst0 i2 -u3 u2 x0)).(\lambda (H10: (subst0 i v x x0)).(ex_intro2 T (\lambda (t: -T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v (THead k x -t2) t)) (THead k x0 t3) (subst0_fst u3 x0 u2 i2 H9 t3 k) (subst0_both v x x0 -i H10 k t2 t3 H2))))) (H1 x u3 i2 H8 H5)) t4 H7)))) H6)) (\lambda (H6: (ex2 T -(\lambda (t5: T).(eq T t4 (THead k u0 t5))) (\lambda (t5: T).(subst0 (s k i2) -u3 t2 t5)))).(ex2_ind T (\lambda (t5: T).(eq T t4 (THead k u0 t5))) (\lambda -(t5: T).(subst0 (s k i2) u3 t2 t5)) (ex2 T (\lambda (t: T).(subst0 i2 u3 -(THead k u2 t3) t)) (\lambda (t: T).(subst0 i v t4 t))) (\lambda (x: -T).(\lambda (H7: (eq T t4 (THead k u0 x))).(\lambda (H8: (subst0 (s k i2) u3 -t2 x)).(eq_ind_r T (THead k u0 x) (\lambda (t: T).(ex2 T (\lambda (t5: -T).(subst0 i2 u3 (THead k u2 t3) t5)) (\lambda (t5: T).(subst0 i v t t5)))) -(ex2_ind T (\lambda (t: T).(subst0 (s k i2) u3 t3 t)) (\lambda (t: T).(subst0 -(s k i) v x t)) (ex2 T (\lambda (t: T).(subst0 i2 u3 (THead k u2 t3) t)) -(\lambda (t: T).(subst0 i v (THead k u0 x) t))) (\lambda (x0: T).(\lambda -(H9: (subst0 (s k i2) u3 t3 x0)).(\lambda (H10: (subst0 (s k i) v x -x0)).(ex_intro2 T (\lambda (t: T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda -(t: T).(subst0 i v (THead k u0 x) t)) (THead k u2 x0) (subst0_snd k u3 x0 t3 -i2 H9 u2) (subst0_both v u0 u2 i H0 k x x0 H10))))) (H3 x u3 (s k i2) H8 -(ex2_ind T (\lambda (t: T).(subst0 (s k i2) u3 t3 t)) (\lambda (t: T).(subst0 -(s k i) v x t)) ((eq nat (s k i) (s k i2)) \to False) (\lambda (x0: -T).(\lambda (_: (subst0 (s k i2) u3 t3 x0)).(\lambda (_: (subst0 (s k i) v x -x0)).(\lambda (H11: (eq nat (s k i) (s k i2))).(H5 (s_inj k i i2 H11)))))) -(H3 x u3 (s k i2) H8 (\lambda (H9: (eq nat (s k i) (s k i2))).(H5 (s_inj k i -i2 H9))))))) t4 H7)))) H6)) (\lambda (H6: (ex3_2 T T (\lambda (u4: -T).(\lambda (t5: T).(eq T t4 (THead k u4 t5)))) (\lambda (u4: T).(\lambda (_: -T).(subst0 i2 u3 u0 u4))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s k i2) -u3 t2 t5))))).(ex3_2_ind T T (\lambda (u4: T).(\lambda (t5: T).(eq T t4 -(THead k u4 t5)))) (\lambda (u4: T).(\lambda (_: T).(subst0 i2 u3 u0 u4))) -(\lambda (_: T).(\lambda (t5: T).(subst0 (s k i2) u3 t2 t5))) (ex2 T (\lambda -(t: T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v t4 t))) -(\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: (eq T t4 (THead k x0 -x1))).(\lambda (H8: (subst0 i2 u3 u0 x0)).(\lambda (H9: (subst0 (s k i2) u3 -t2 x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t: T).(ex2 T (\lambda (t5: -T).(subst0 i2 u3 (THead k u2 t3) t5)) (\lambda (t5: T).(subst0 i v t t5)))) -(ex2_ind T (\lambda (t: T).(subst0 i2 u3 u2 t)) (\lambda (t: T).(subst0 i v -x0 t)) (ex2 T (\lambda (t: T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: -T).(subst0 i v (THead k x0 x1) t))) (\lambda (x: T).(\lambda (H10: (subst0 i2 -u3 u2 x)).(\lambda (H11: (subst0 i v x0 x)).(ex2_ind T (\lambda (t: -T).(subst0 (s k i2) u3 t3 t)) (\lambda (t: T).(subst0 (s k i) v x1 t)) (ex2 T -(\lambda (t: T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v -(THead k x0 x1) t))) (\lambda (x2: T).(\lambda (H12: (subst0 (s k i2) u3 t3 -x2)).(\lambda (H13: (subst0 (s k i) v x1 x2)).(ex_intro2 T (\lambda (t: -T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v (THead k x0 -x1) t)) (THead k x x2) (subst0_both u3 u2 x i2 H10 k t3 x2 H12) (subst0_both -v x0 x i H11 k x1 x2 H13))))) (H3 x1 u3 (s k i2) H9 (ex2_ind T (\lambda (t: -T).(subst0 (s k i2) u3 t3 t)) (\lambda (t: T).(subst0 (s k i) v x1 t)) ((eq -nat (s k i) (s k i2)) \to False) (\lambda (x2: T).(\lambda (_: (subst0 (s k -i2) u3 t3 x2)).(\lambda (_: (subst0 (s k i) v x1 x2)).(\lambda (H14: (eq nat -(s k i) (s k i2))).(H5 (s_inj k i i2 H14)))))) (H3 x1 u3 (s k i2) H9 (\lambda -(H12: (eq nat (s k i) (s k i2))).(H5 (s_inj k i i2 H12)))))))))) (H1 x0 u3 i2 -H8 H5)) t4 H7)))))) H6)) (subst0_gen_head k u3 u0 t2 t4 i2 -H4)))))))))))))))))) i1 u1 t0 t1 H))))). -(* COMMENTS -Initial nodes: 5375 -END *) - -theorem subst0_confluence_eq: - \forall (t0: T).(\forall (t1: T).(\forall (u: T).(\forall (i: nat).((subst0 -i u t0 t1) \to (\forall (t2: T).((subst0 i u t0 t2) \to (or4 (eq T t1 t2) -(ex2 T (\lambda (t: T).(subst0 i u t1 t)) (\lambda (t: T).(subst0 i u t2 t))) -(subst0 i u t1 t2) (subst0 i u t2 t1)))))))) -\def - \lambda (t0: T).(\lambda (t1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H: (subst0 i u t0 t1)).(subst0_ind (\lambda (n: nat).(\lambda (t: -T).(\lambda (t2: T).(\lambda (t3: T).(\forall (t4: T).((subst0 n t t2 t4) \to -(or4 (eq T t3 t4) (ex2 T (\lambda (t5: T).(subst0 n t t3 t5)) (\lambda (t5: -T).(subst0 n t t4 t5))) (subst0 n t t3 t4) (subst0 n t t4 t3)))))))) (\lambda -(v: T).(\lambda (i0: nat).(\lambda (t2: T).(\lambda (H0: (subst0 i0 v (TLRef -i0) t2)).(land_ind (eq nat i0 i0) (eq T t2 (lift (S i0) O v)) (or4 (eq T -(lift (S i0) O v) t2) (ex2 T (\lambda (t: T).(subst0 i0 v (lift (S i0) O v) -t)) (\lambda (t: T).(subst0 i0 v t2 t))) (subst0 i0 v (lift (S i0) O v) t2) -(subst0 i0 v t2 (lift (S i0) O v))) (\lambda (_: (eq nat i0 i0)).(\lambda -(H2: (eq T t2 (lift (S i0) O v))).(or4_intro0 (eq T (lift (S i0) O v) t2) -(ex2 T (\lambda (t: T).(subst0 i0 v (lift (S i0) O v) t)) (\lambda (t: -T).(subst0 i0 v t2 t))) (subst0 i0 v (lift (S i0) O v) t2) (subst0 i0 v t2 -(lift (S i0) O v)) (sym_eq T t2 (lift (S i0) O v) H2)))) (subst0_gen_lref v -t2 i0 i0 H0)))))) (\lambda (v: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda -(i0: nat).(\lambda (H0: (subst0 i0 v u1 u2)).(\lambda (H1: ((\forall (t2: -T).((subst0 i0 v u1 t2) \to (or4 (eq T u2 t2) (ex2 T (\lambda (t: T).(subst0 -i0 v u2 t)) (\lambda (t: T).(subst0 i0 v t2 t))) (subst0 i0 v u2 t2) (subst0 -i0 v t2 u2)))))).(\lambda (t: T).(\lambda (k: K).(\lambda (t2: T).(\lambda -(H2: (subst0 i0 v (THead k u1 t) t2)).(or3_ind (ex2 T (\lambda (u3: T).(eq T -t2 (THead k u3 t))) (\lambda (u3: T).(subst0 i0 v u1 u3))) (ex2 T (\lambda -(t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v t -t3))) (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead k u3 -t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v u1 u3))) (\lambda (_: -T).(\lambda (t3: T).(subst0 (s k i0) v t t3)))) (or4 (eq T (THead k u2 t) t2) -(ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: -T).(subst0 i0 v t2 t3))) (subst0 i0 v (THead k u2 t) t2) (subst0 i0 v t2 -(THead k u2 t))) (\lambda (H3: (ex2 T (\lambda (u3: T).(eq T t2 (THead k u3 -t))) (\lambda (u3: T).(subst0 i0 v u1 u3)))).(ex2_ind T (\lambda (u3: T).(eq -T t2 (THead k u3 t))) (\lambda (u3: T).(subst0 i0 v u1 u3)) (or4 (eq T (THead -k u2 t) t2) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda -(t3: T).(subst0 i0 v t2 t3))) (subst0 i0 v (THead k u2 t) t2) (subst0 i0 v t2 -(THead k u2 t))) (\lambda (x: T).(\lambda (H4: (eq T t2 (THead k x -t))).(\lambda (H5: (subst0 i0 v u1 x)).(eq_ind_r T (THead k x t) (\lambda -(t3: T).(or4 (eq T (THead k u2 t) t3) (ex2 T (\lambda (t4: T).(subst0 i0 v -(THead k u2 t) t4)) (\lambda (t4: T).(subst0 i0 v t3 t4))) (subst0 i0 v -(THead k u2 t) t3) (subst0 i0 v t3 (THead k u2 t)))) (or4_ind (eq T u2 x) -(ex2 T (\lambda (t3: T).(subst0 i0 v u2 t3)) (\lambda (t3: T).(subst0 i0 v x -t3))) (subst0 i0 v u2 x) (subst0 i0 v x u2) (or4 (eq T (THead k u2 t) (THead -k x t)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda -(t3: T).(subst0 i0 v (THead k x t) t3))) (subst0 i0 v (THead k u2 t) (THead k -x t)) (subst0 i0 v (THead k x t) (THead k u2 t))) (\lambda (H6: (eq T u2 -x)).(eq_ind_r T x (\lambda (t3: T).(or4 (eq T (THead k t3 t) (THead k x t)) -(ex2 T (\lambda (t4: T).(subst0 i0 v (THead k t3 t) t4)) (\lambda (t4: -T).(subst0 i0 v (THead k x t) t4))) (subst0 i0 v (THead k t3 t) (THead k x -t)) (subst0 i0 v (THead k x t) (THead k t3 t)))) (or4_intro0 (eq T (THead k x -t) (THead k x t)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k x t) t3)) -(\lambda (t3: T).(subst0 i0 v (THead k x t) t3))) (subst0 i0 v (THead k x t) -(THead k x t)) (subst0 i0 v (THead k x t) (THead k x t)) (refl_equal T (THead -k x t))) u2 H6)) (\lambda (H6: (ex2 T (\lambda (t3: T).(subst0 i0 v u2 t3)) -(\lambda (t3: T).(subst0 i0 v x t3)))).(ex2_ind T (\lambda (t3: T).(subst0 i0 -v u2 t3)) (\lambda (t3: T).(subst0 i0 v x t3)) (or4 (eq T (THead k u2 t) -(THead k x t)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) -(\lambda (t3: T).(subst0 i0 v (THead k x t) t3))) (subst0 i0 v (THead k u2 t) -(THead k x t)) (subst0 i0 v (THead k x t) (THead k u2 t))) (\lambda (x0: -T).(\lambda (H7: (subst0 i0 v u2 x0)).(\lambda (H8: (subst0 i0 v x -x0)).(or4_intro1 (eq T (THead k u2 t) (THead k x t)) (ex2 T (\lambda (t3: -T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k x -t) t3))) (subst0 i0 v (THead k u2 t) (THead k x t)) (subst0 i0 v (THead k x -t) (THead k u2 t)) (ex_intro2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) -t3)) (\lambda (t3: T).(subst0 i0 v (THead k x t) t3)) (THead k x0 t) -(subst0_fst v x0 u2 i0 H7 t k) (subst0_fst v x0 x i0 H8 t k)))))) H6)) -(\lambda (H6: (subst0 i0 v u2 x)).(or4_intro2 (eq T (THead k u2 t) (THead k x -t)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: -T).(subst0 i0 v (THead k x t) t3))) (subst0 i0 v (THead k u2 t) (THead k x -t)) (subst0 i0 v (THead k x t) (THead k u2 t)) (subst0_fst v x u2 i0 H6 t -k))) (\lambda (H6: (subst0 i0 v x u2)).(or4_intro3 (eq T (THead k u2 t) -(THead k x t)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) -(\lambda (t3: T).(subst0 i0 v (THead k x t) t3))) (subst0 i0 v (THead k u2 t) -(THead k x t)) (subst0 i0 v (THead k x t) (THead k u2 t)) (subst0_fst v u2 x -i0 H6 t k))) (H1 x H5)) t2 H4)))) H3)) (\lambda (H3: (ex2 T (\lambda (t3: -T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v t -t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: -T).(subst0 (s k i0) v t t3)) (or4 (eq T (THead k u2 t) t2) (ex2 T (\lambda -(t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0 v t2 -t3))) (subst0 i0 v (THead k u2 t) t2) (subst0 i0 v t2 (THead k u2 t))) -(\lambda (x: T).(\lambda (H4: (eq T t2 (THead k u1 x))).(\lambda (H5: (subst0 -(s k i0) v t x)).(eq_ind_r T (THead k u1 x) (\lambda (t3: T).(or4 (eq T -(THead k u2 t) t3) (ex2 T (\lambda (t4: T).(subst0 i0 v (THead k u2 t) t4)) -(\lambda (t4: T).(subst0 i0 v t3 t4))) (subst0 i0 v (THead k u2 t) t3) -(subst0 i0 v t3 (THead k u2 t)))) (or4_ind (eq T u2 u2) (ex2 T (\lambda (t3: -T).(subst0 i0 v u2 t3)) (\lambda (t3: T).(subst0 i0 v u2 t3))) (subst0 i0 v -u2 u2) (subst0 i0 v u2 u2) (or4 (eq T (THead k u2 t) (THead k u1 x)) (ex2 T -(\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0 -v (THead k u1 x) t3))) (subst0 i0 v (THead k u2 t) (THead k u1 x)) (subst0 i0 -v (THead k u1 x) (THead k u2 t))) (\lambda (_: (eq T u2 u2)).(or4_intro1 (eq -T (THead k u2 t) (THead k u1 x)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead -k u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k u1 x) t3))) (subst0 i0 v -(THead k u2 t) (THead k u1 x)) (subst0 i0 v (THead k u1 x) (THead k u2 t)) -(ex_intro2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: -T).(subst0 i0 v (THead k u1 x) t3)) (THead k u2 x) (subst0_snd k v x t i0 H5 -u2) (subst0_fst v u2 u1 i0 H0 x k)))) (\lambda (H6: (ex2 T (\lambda (t3: -T).(subst0 i0 v u2 t3)) (\lambda (t3: T).(subst0 i0 v u2 t3)))).(ex2_ind T -(\lambda (t3: T).(subst0 i0 v u2 t3)) (\lambda (t3: T).(subst0 i0 v u2 t3)) -(or4 (eq T (THead k u2 t) (THead k u1 x)) (ex2 T (\lambda (t3: T).(subst0 i0 -v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k u1 x) t3))) -(subst0 i0 v (THead k u2 t) (THead k u1 x)) (subst0 i0 v (THead k u1 x) -(THead k u2 t))) (\lambda (x0: T).(\lambda (_: (subst0 i0 v u2 x0)).(\lambda -(_: (subst0 i0 v u2 x0)).(or4_intro1 (eq T (THead k u2 t) (THead k u1 x)) -(ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: -T).(subst0 i0 v (THead k u1 x) t3))) (subst0 i0 v (THead k u2 t) (THead k u1 -x)) (subst0 i0 v (THead k u1 x) (THead k u2 t)) (ex_intro2 T (\lambda (t3: -T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k u1 -x) t3)) (THead k u2 x) (subst0_snd k v x t i0 H5 u2) (subst0_fst v u2 u1 i0 -H0 x k)))))) H6)) (\lambda (_: (subst0 i0 v u2 u2)).(or4_intro1 (eq T (THead -k u2 t) (THead k u1 x)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) -t3)) (\lambda (t3: T).(subst0 i0 v (THead k u1 x) t3))) (subst0 i0 v (THead k -u2 t) (THead k u1 x)) (subst0 i0 v (THead k u1 x) (THead k u2 t)) (ex_intro2 -T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 -i0 v (THead k u1 x) t3)) (THead k u2 x) (subst0_snd k v x t i0 H5 u2) -(subst0_fst v u2 u1 i0 H0 x k)))) (\lambda (_: (subst0 i0 v u2 -u2)).(or4_intro1 (eq T (THead k u2 t) (THead k u1 x)) (ex2 T (\lambda (t3: -T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k u1 -x) t3))) (subst0 i0 v (THead k u2 t) (THead k u1 x)) (subst0 i0 v (THead k u1 -x) (THead k u2 t)) (ex_intro2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) -t3)) (\lambda (t3: T).(subst0 i0 v (THead k u1 x) t3)) (THead k u2 x) -(subst0_snd k v x t i0 H5 u2) (subst0_fst v u2 u1 i0 H0 x k)))) (H1 u2 H0)) -t2 H4)))) H3)) (\lambda (H3: (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq -T t2 (THead k u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v u1 -u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i0) v t -t3))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead k u3 -t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v u1 u3))) (\lambda (_: -T).(\lambda (t3: T).(subst0 (s k i0) v t t3))) (or4 (eq T (THead k u2 t) t2) -(ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: -T).(subst0 i0 v t2 t3))) (subst0 i0 v (THead k u2 t) t2) (subst0 i0 v t2 -(THead k u2 t))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t2 -(THead k x0 x1))).(\lambda (H5: (subst0 i0 v u1 x0)).(\lambda (H6: (subst0 (s -k i0) v t x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t3: T).(or4 (eq T (THead -k u2 t) t3) (ex2 T (\lambda (t4: T).(subst0 i0 v (THead k u2 t) t4)) (\lambda -(t4: T).(subst0 i0 v t3 t4))) (subst0 i0 v (THead k u2 t) t3) (subst0 i0 v t3 -(THead k u2 t)))) (or4_ind (eq T u2 x0) (ex2 T (\lambda (t3: T).(subst0 i0 v -u2 t3)) (\lambda (t3: T).(subst0 i0 v x0 t3))) (subst0 i0 v u2 x0) (subst0 i0 -v x0 u2) (or4 (eq T (THead k u2 t) (THead k x0 x1)) (ex2 T (\lambda (t3: -T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k x0 -x1) t3))) (subst0 i0 v (THead k u2 t) (THead k x0 x1)) (subst0 i0 v (THead k -x0 x1) (THead k u2 t))) (\lambda (H7: (eq T u2 x0)).(eq_ind_r T x0 (\lambda -(t3: T).(or4 (eq T (THead k t3 t) (THead k x0 x1)) (ex2 T (\lambda (t4: -T).(subst0 i0 v (THead k t3 t) t4)) (\lambda (t4: T).(subst0 i0 v (THead k x0 -x1) t4))) (subst0 i0 v (THead k t3 t) (THead k x0 x1)) (subst0 i0 v (THead k -x0 x1) (THead k t3 t)))) (or4_intro2 (eq T (THead k x0 t) (THead k x0 x1)) -(ex2 T (\lambda (t3: T).(subst0 i0 v (THead k x0 t) t3)) (\lambda (t3: -T).(subst0 i0 v (THead k x0 x1) t3))) (subst0 i0 v (THead k x0 t) (THead k x0 -x1)) (subst0 i0 v (THead k x0 x1) (THead k x0 t)) (subst0_snd k v x1 t i0 H6 -x0)) u2 H7)) (\lambda (H7: (ex2 T (\lambda (t3: T).(subst0 i0 v u2 t3)) -(\lambda (t3: T).(subst0 i0 v x0 t3)))).(ex2_ind T (\lambda (t3: T).(subst0 -i0 v u2 t3)) (\lambda (t3: T).(subst0 i0 v x0 t3)) (or4 (eq T (THead k u2 t) -(THead k x0 x1)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) -(\lambda (t3: T).(subst0 i0 v (THead k x0 x1) t3))) (subst0 i0 v (THead k u2 -t) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t))) (\lambda -(x: T).(\lambda (H8: (subst0 i0 v u2 x)).(\lambda (H9: (subst0 i0 v x0 -x)).(or4_intro1 (eq T (THead k u2 t) (THead k x0 x1)) (ex2 T (\lambda (t3: -T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k x0 -x1) t3))) (subst0 i0 v (THead k u2 t) (THead k x0 x1)) (subst0 i0 v (THead k -x0 x1) (THead k u2 t)) (ex_intro2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 -t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k x0 x1) t3)) (THead k x x1) -(subst0_both v u2 x i0 H8 k t x1 H6) (subst0_fst v x x0 i0 H9 x1 k)))))) H7)) -(\lambda (H7: (subst0 i0 v u2 x0)).(or4_intro2 (eq T (THead k u2 t) (THead k -x0 x1)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda -(t3: T).(subst0 i0 v (THead k x0 x1) t3))) (subst0 i0 v (THead k u2 t) (THead -k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t)) (subst0_both v u2 x0 -i0 H7 k t x1 H6))) (\lambda (H7: (subst0 i0 v x0 u2)).(or4_intro1 (eq T -(THead k u2 t) (THead k x0 x1)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k -u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k x0 x1) t3))) (subst0 i0 v -(THead k u2 t) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t)) -(ex_intro2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: -T).(subst0 i0 v (THead k x0 x1) t3)) (THead k u2 x1) (subst0_snd k v x1 t i0 -H6 u2) (subst0_fst v u2 x0 i0 H7 x1 k)))) (H1 x0 H5)) t2 H4)))))) H3)) -(subst0_gen_head k v u1 t t2 i0 H2)))))))))))) (\lambda (k: K).(\lambda (v: -T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (i0: nat).(\lambda (H0: (subst0 -(s k i0) v t3 t2)).(\lambda (H1: ((\forall (t4: T).((subst0 (s k i0) v t3 t4) -\to (or4 (eq T t2 t4) (ex2 T (\lambda (t: T).(subst0 (s k i0) v t2 t)) -(\lambda (t: T).(subst0 (s k i0) v t4 t))) (subst0 (s k i0) v t2 t4) (subst0 -(s k i0) v t4 t2)))))).(\lambda (u0: T).(\lambda (t4: T).(\lambda (H2: -(subst0 i0 v (THead k u0 t3) t4)).(or3_ind (ex2 T (\lambda (u2: T).(eq T t4 -(THead k u2 t3))) (\lambda (u2: T).(subst0 i0 v u0 u2))) (ex2 T (\lambda (t5: -T).(eq T t4 (THead k u0 t5))) (\lambda (t5: T).(subst0 (s k i0) v t3 t5))) -(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead k u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 i0 v u0 u2))) (\lambda (_: -T).(\lambda (t5: T).(subst0 (s k i0) v t3 t5)))) (or4 (eq T (THead k u0 t2) -t4) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u0 t2) t)) (\lambda (t: -T).(subst0 i0 v t4 t))) (subst0 i0 v (THead k u0 t2) t4) (subst0 i0 v t4 -(THead k u0 t2))) (\lambda (H3: (ex2 T (\lambda (u2: T).(eq T t4 (THead k u2 -t3))) (\lambda (u2: T).(subst0 i0 v u0 u2)))).(ex2_ind T (\lambda (u2: T).(eq -T t4 (THead k u2 t3))) (\lambda (u2: T).(subst0 i0 v u0 u2)) (or4 (eq T -(THead k u0 t2) t4) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u0 t2) t)) -(\lambda (t: T).(subst0 i0 v t4 t))) (subst0 i0 v (THead k u0 t2) t4) (subst0 -i0 v t4 (THead k u0 t2))) (\lambda (x: T).(\lambda (H4: (eq T t4 (THead k x -t3))).(\lambda (H5: (subst0 i0 v u0 x)).(eq_ind_r T (THead k x t3) (\lambda -(t: T).(or4 (eq T (THead k u0 t2) t) (ex2 T (\lambda (t5: T).(subst0 i0 v -(THead k u0 t2) t5)) (\lambda (t5: T).(subst0 i0 v t t5))) (subst0 i0 v -(THead k u0 t2) t) (subst0 i0 v t (THead k u0 t2)))) (or4_ind (eq T t2 t2) -(ex2 T (\lambda (t: T).(subst0 (s k i0) v t2 t)) (\lambda (t: T).(subst0 (s k -i0) v t2 t))) (subst0 (s k i0) v t2 t2) (subst0 (s k i0) v t2 t2) (or4 (eq T -(THead k u0 t2) (THead k x t3)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k -u0 t2) t)) (\lambda (t: T).(subst0 i0 v (THead k x t3) t))) (subst0 i0 v -(THead k u0 t2) (THead k x t3)) (subst0 i0 v (THead k x t3) (THead k u0 t2))) -(\lambda (_: (eq T t2 t2)).(or4_intro1 (eq T (THead k u0 t2) (THead k x t3)) -(ex2 T (\lambda (t: T).(subst0 i0 v (THead k u0 t2) t)) (\lambda (t: -T).(subst0 i0 v (THead k x t3) t))) (subst0 i0 v (THead k u0 t2) (THead k x -t3)) (subst0 i0 v (THead k x t3) (THead k u0 t2)) (ex_intro2 T (\lambda (t: -T).(subst0 i0 v (THead k u0 t2) t)) (\lambda (t: T).(subst0 i0 v (THead k x -t3) t)) (THead k x t2) (subst0_fst v x u0 i0 H5 t2 k) (subst0_snd k v t2 t3 -i0 H0 x)))) (\lambda (H6: (ex2 T (\lambda (t: T).(subst0 (s k i0) v t2 t)) -(\lambda (t: T).(subst0 (s k i0) v t2 t)))).(ex2_ind T (\lambda (t: -T).(subst0 (s k i0) v t2 t)) (\lambda (t: T).(subst0 (s k i0) v t2 t)) (or4 -(eq T (THead k u0 t2) (THead k x t3)) (ex2 T (\lambda (t: T).(subst0 i0 v -(THead k u0 t2) t)) (\lambda (t: T).(subst0 i0 v (THead k x t3) t))) (subst0 -i0 v (THead k u0 t2) (THead k x t3)) (subst0 i0 v (THead k x t3) (THead k u0 -t2))) (\lambda (x0: T).(\lambda (_: (subst0 (s k i0) v t2 x0)).(\lambda (_: -(subst0 (s k i0) v t2 x0)).(or4_intro1 (eq T (THead k u0 t2) (THead k x t3)) -(ex2 T (\lambda (t: T).(subst0 i0 v (THead k u0 t2) t)) (\lambda (t: -T).(subst0 i0 v (THead k x t3) t))) (subst0 i0 v (THead k u0 t2) (THead k x -t3)) (subst0 i0 v (THead k x t3) (THead k u0 t2)) (ex_intro2 T (\lambda (t: -T).(subst0 i0 v (THead k u0 t2) t)) (\lambda (t: T).(subst0 i0 v (THead k x -t3) t)) (THead k x t2) (subst0_fst v x u0 i0 H5 t2 k) (subst0_snd k v t2 t3 -i0 H0 x)))))) H6)) (\lambda (_: (subst0 (s k i0) v t2 t2)).(or4_intro1 (eq T -(THead k u0 t2) (THead k x t3)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k -u0 t2) t)) (\lambda (t: T).(subst0 i0 v (THead k x t3) t))) (subst0 i0 v -(THead k u0 t2) (THead k x t3)) (subst0 i0 v (THead k x t3) (THead k u0 t2)) -(ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k u0 t2) t)) (\lambda (t: -T).(subst0 i0 v (THead k x t3) t)) (THead k x t2) (subst0_fst v x u0 i0 H5 t2 -k) (subst0_snd k v t2 t3 i0 H0 x)))) (\lambda (_: (subst0 (s k i0) v t2 -t2)).(or4_intro1 (eq T (THead k u0 t2) (THead k x t3)) (ex2 T (\lambda (t: -T).(subst0 i0 v (THead k u0 t2) t)) (\lambda (t: T).(subst0 i0 v (THead k x -t3) t))) (subst0 i0 v (THead k u0 t2) (THead k x t3)) (subst0 i0 v (THead k x -t3) (THead k u0 t2)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k u0 -t2) t)) (\lambda (t: T).(subst0 i0 v (THead k x t3) t)) (THead k x t2) -(subst0_fst v x u0 i0 H5 t2 k) (subst0_snd k v t2 t3 i0 H0 x)))) (H1 t2 H0)) -t4 H4)))) H3)) (\lambda (H3: (ex2 T (\lambda (t5: T).(eq T t4 (THead k u0 -t5))) (\lambda (t5: T).(subst0 (s k i0) v t3 t5)))).(ex2_ind T (\lambda (t5: -T).(eq T t4 (THead k u0 t5))) (\lambda (t5: T).(subst0 (s k i0) v t3 t5)) -(or4 (eq T (THead k u0 t2) t4) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k -u0 t2) t)) (\lambda (t: T).(subst0 i0 v t4 t))) (subst0 i0 v (THead k u0 t2) -t4) (subst0 i0 v t4 (THead k u0 t2))) (\lambda (x: T).(\lambda (H4: (eq T t4 -(THead k u0 x))).(\lambda (H5: (subst0 (s k i0) v t3 x)).(eq_ind_r T (THead k -u0 x) (\lambda (t: T).(or4 (eq T (THead k u0 t2) t) (ex2 T (\lambda (t5: -T).(subst0 i0 v (THead k u0 t2) t5)) (\lambda (t5: T).(subst0 i0 v t t5))) -(subst0 i0 v (THead k u0 t2) t) (subst0 i0 v t (THead k u0 t2)))) (or4_ind -(eq T t2 x) (ex2 T (\lambda (t: T).(subst0 (s k i0) v t2 t)) (\lambda (t: -T).(subst0 (s k i0) v x t))) (subst0 (s k i0) v t2 x) (subst0 (s k i0) v x -t2) (or4 (eq T (THead k u0 t2) (THead k u0 x)) (ex2 T (\lambda (t: T).(subst0 -i0 v (THead k u0 t2) t)) (\lambda (t: T).(subst0 i0 v (THead k u0 x) t))) -(subst0 i0 v (THead k u0 t2) (THead k u0 x)) (subst0 i0 v (THead k u0 x) -(THead k u0 t2))) (\lambda (H6: (eq T t2 x)).(eq_ind_r T x (\lambda (t: -T).(or4 (eq T (THead k u0 t) (THead k u0 x)) (ex2 T (\lambda (t5: T).(subst0 -i0 v (THead k u0 t) t5)) (\lambda (t5: T).(subst0 i0 v (THead k u0 x) t5))) -(subst0 i0 v (THead k u0 t) (THead k u0 x)) (subst0 i0 v (THead k u0 x) -(THead k u0 t)))) (or4_intro0 (eq T (THead k u0 x) (THead k u0 x)) (ex2 T -(\lambda (t: T).(subst0 i0 v (THead k u0 x) t)) (\lambda (t: T).(subst0 i0 v -(THead k u0 x) t))) (subst0 i0 v (THead k u0 x) (THead k u0 x)) (subst0 i0 v -(THead k u0 x) (THead k u0 x)) (refl_equal T (THead k u0 x))) t2 H6)) -(\lambda (H6: (ex2 T (\lambda (t: T).(subst0 (s k i0) v t2 t)) (\lambda (t: -T).(subst0 (s k i0) v x t)))).(ex2_ind T (\lambda (t: T).(subst0 (s k i0) v -t2 t)) (\lambda (t: T).(subst0 (s k i0) v x t)) (or4 (eq T (THead k u0 t2) -(THead k u0 x)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u0 t2) t)) -(\lambda (t: T).(subst0 i0 v (THead k u0 x) t))) (subst0 i0 v (THead k u0 t2) -(THead k u0 x)) (subst0 i0 v (THead k u0 x) (THead k u0 t2))) (\lambda (x0: -T).(\lambda (H7: (subst0 (s k i0) v t2 x0)).(\lambda (H8: (subst0 (s k i0) v -x x0)).(or4_intro1 (eq T (THead k u0 t2) (THead k u0 x)) (ex2 T (\lambda (t: -T).(subst0 i0 v (THead k u0 t2) t)) (\lambda (t: T).(subst0 i0 v (THead k u0 -x) t))) (subst0 i0 v (THead k u0 t2) (THead k u0 x)) (subst0 i0 v (THead k u0 -x) (THead k u0 t2)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k u0 t2) -t)) (\lambda (t: T).(subst0 i0 v (THead k u0 x) t)) (THead k u0 x0) -(subst0_snd k v x0 t2 i0 H7 u0) (subst0_snd k v x0 x i0 H8 u0)))))) H6)) -(\lambda (H6: (subst0 (s k i0) v t2 x)).(or4_intro2 (eq T (THead k u0 t2) -(THead k u0 x)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u0 t2) t)) -(\lambda (t: T).(subst0 i0 v (THead k u0 x) t))) (subst0 i0 v (THead k u0 t2) -(THead k u0 x)) (subst0 i0 v (THead k u0 x) (THead k u0 t2)) (subst0_snd k v -x t2 i0 H6 u0))) (\lambda (H6: (subst0 (s k i0) v x t2)).(or4_intro3 (eq T -(THead k u0 t2) (THead k u0 x)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k -u0 t2) t)) (\lambda (t: T).(subst0 i0 v (THead k u0 x) t))) (subst0 i0 v -(THead k u0 t2) (THead k u0 x)) (subst0 i0 v (THead k u0 x) (THead k u0 t2)) -(subst0_snd k v t2 x i0 H6 u0))) (H1 x H5)) t4 H4)))) H3)) (\lambda (H3: -(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead k u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 i0 v u0 u2))) (\lambda (_: -T).(\lambda (t5: T).(subst0 (s k i0) v t3 t5))))).(ex3_2_ind T T (\lambda -(u2: T).(\lambda (t5: T).(eq T t4 (THead k u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i0 v u0 u2))) (\lambda (_: T).(\lambda (t5: -T).(subst0 (s k i0) v t3 t5))) (or4 (eq T (THead k u0 t2) t4) (ex2 T (\lambda -(t: T).(subst0 i0 v (THead k u0 t2) t)) (\lambda (t: T).(subst0 i0 v t4 t))) -(subst0 i0 v (THead k u0 t2) t4) (subst0 i0 v t4 (THead k u0 t2))) (\lambda -(x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t4 (THead k x0 x1))).(\lambda -(H5: (subst0 i0 v u0 x0)).(\lambda (H6: (subst0 (s k i0) v t3 x1)).(eq_ind_r -T (THead k x0 x1) (\lambda (t: T).(or4 (eq T (THead k u0 t2) t) (ex2 T -(\lambda (t5: T).(subst0 i0 v (THead k u0 t2) t5)) (\lambda (t5: T).(subst0 -i0 v t t5))) (subst0 i0 v (THead k u0 t2) t) (subst0 i0 v t (THead k u0 -t2)))) (or4_ind (eq T t2 x1) (ex2 T (\lambda (t: T).(subst0 (s k i0) v t2 t)) -(\lambda (t: T).(subst0 (s k i0) v x1 t))) (subst0 (s k i0) v t2 x1) (subst0 -(s k i0) v x1 t2) (or4 (eq T (THead k u0 t2) (THead k x0 x1)) (ex2 T (\lambda -(t: T).(subst0 i0 v (THead k u0 t2) t)) (\lambda (t: T).(subst0 i0 v (THead k -x0 x1) t))) (subst0 i0 v (THead k u0 t2) (THead k x0 x1)) (subst0 i0 v (THead -k x0 x1) (THead k u0 t2))) (\lambda (H7: (eq T t2 x1)).(eq_ind_r T x1 -(\lambda (t: T).(or4 (eq T (THead k u0 t) (THead k x0 x1)) (ex2 T (\lambda -(t5: T).(subst0 i0 v (THead k u0 t) t5)) (\lambda (t5: T).(subst0 i0 v (THead -k x0 x1) t5))) (subst0 i0 v (THead k u0 t) (THead k x0 x1)) (subst0 i0 v -(THead k x0 x1) (THead k u0 t)))) (or4_intro2 (eq T (THead k u0 x1) (THead k -x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u0 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(ex_intro2 T (\lambda (t: -T).(subst0 i0 v (THead k u0 t2) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 -x1) t)) (THead k x0 x) (subst0_both v u0 x0 i0 H5 k t2 x H8) (subst0_snd k v -x x1 i0 H9 x0)))))) H7)) (\lambda (H7: (subst0 (s k i0) v t2 x1)).(or4_intro2 -(eq T (THead k u0 t2) (THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v -(THead k u0 t2) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 -i0 v (THead k u0 t2) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k -u0 t2)) (subst0_both v u0 x0 i0 H5 k t2 x1 H7))) (\lambda (H7: (subst0 (s k -i0) v x1 t2)).(or4_intro1 (eq T (THead k u0 t2) (THead k x0 x1)) (ex2 T -(\lambda (t: T).(subst0 i0 v (THead k u0 t2) t)) (\lambda (t: T).(subst0 i0 v -(THead k x0 x1) t))) (subst0 i0 v (THead k u0 t2) (THead k x0 x1)) (subst0 i0 -v (THead k x0 x1) (THead k u0 t2)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v -(THead k u0 t2) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t)) (THead k -x0 t2) (subst0_fst v x0 u0 i0 H5 t2 k) (subst0_snd k v t2 x1 i0 H7 x0)))) (H1 -x1 H6)) t4 H4)))))) H3)) (subst0_gen_head k v u0 t3 t4 i0 H2)))))))))))) -(\lambda (v: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i0: nat).(\lambda -(H0: (subst0 i0 v u1 u2)).(\lambda (H1: ((\forall (t2: T).((subst0 i0 v u1 -t2) \to (or4 (eq T u2 t2) (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda -(t: T).(subst0 i0 v t2 t))) (subst0 i0 v u2 t2) (subst0 i0 v t2 -u2)))))).(\lambda (k: K).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H2: -(subst0 (s k i0) v t2 t3)).(\lambda (H3: ((\forall (t4: T).((subst0 (s k i0) -v t2 t4) \to (or4 (eq T t3 t4) (ex2 T (\lambda (t: T).(subst0 (s k i0) v t3 -t)) (\lambda (t: T).(subst0 (s k i0) v t4 t))) (subst0 (s k i0) v t3 t4) -(subst0 (s k i0) v t4 t3)))))).(\lambda (t4: T).(\lambda (H4: (subst0 i0 v -(THead k u1 t2) t4)).(or3_ind (ex2 T (\lambda (u3: T).(eq T t4 (THead k u3 -t2))) (\lambda (u3: T).(subst0 i0 v u1 u3))) (ex2 T (\lambda (t5: T).(eq T t4 -(THead k u1 t5))) (\lambda (t5: T).(subst0 (s k i0) v t2 t5))) (ex3_2 T T -(\lambda (u3: T).(\lambda (t5: T).(eq T t4 (THead k u3 t5)))) (\lambda (u3: -T).(\lambda (_: T).(subst0 i0 v u1 u3))) (\lambda (_: T).(\lambda (t5: -T).(subst0 (s k i0) v t2 t5)))) (or4 (eq T (THead k u2 t3) t4) (ex2 T -(\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v -t4 t))) (subst0 i0 v (THead k u2 t3) t4) (subst0 i0 v t4 (THead k u2 t3))) -(\lambda (H5: (ex2 T (\lambda (u3: T).(eq T t4 (THead k u3 t2))) (\lambda -(u3: T).(subst0 i0 v u1 u3)))).(ex2_ind T (\lambda (u3: T).(eq T t4 (THead k -u3 t2))) (\lambda (u3: T).(subst0 i0 v u1 u3)) (or4 (eq T (THead k u2 t3) t4) -(ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: -T).(subst0 i0 v t4 t))) (subst0 i0 v (THead k u2 t3) t4) (subst0 i0 v t4 -(THead k u2 t3))) (\lambda (x: T).(\lambda (H6: (eq T t4 (THead k x -t2))).(\lambda (H7: (subst0 i0 v u1 x)).(eq_ind_r T (THead k x t2) (\lambda -(t: T).(or4 (eq T (THead k u2 t3) t) (ex2 T (\lambda (t5: T).(subst0 i0 v -(THead 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-T).(or4 (eq T (THead k t t3) (THead k x t2)) (ex2 T (\lambda (t5: T).(subst0 -i0 v (THead k t t3) t5)) (\lambda (t5: T).(subst0 i0 v (THead k x t2) t5))) -(subst0 i0 v (THead k t t3) (THead k x t2)) (subst0 i0 v (THead k x t2) -(THead k t t3)))) (or4_intro3 (eq T (THead k x t3) (THead k x t2)) (ex2 T -(\lambda (t: T).(subst0 i0 v (THead k x t3) t)) (\lambda (t: T).(subst0 i0 v -(THead k x t2) t))) (subst0 i0 v (THead k x t3) (THead k x t2)) (subst0 i0 v -(THead k x t2) (THead k x t3)) (subst0_snd k v t3 t2 i0 H2 x)) u2 H9)) -(\lambda (H9: (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: -T).(subst0 i0 v x t)))).(ex2_ind T (\lambda (t: T).(subst0 i0 v u2 t)) -(\lambda (t: T).(subst0 i0 v x t)) (or4 (eq T (THead k u2 t3) (THead k x t2)) -(ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: -T).(subst0 i0 v (THead k x t2) t))) (subst0 i0 v (THead k u2 t3) (THead k x -t2)) (subst0 i0 v (THead k x t2) (THead k u2 t3))) (\lambda (x0: T).(\lambda -(H10: (subst0 i0 v u2 x0)).(\lambda (H11: (subst0 i0 v x x0)).(or4_intro1 (eq -T (THead k u2 t3) (THead k x t2)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead -k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x t2) t))) (subst0 i0 v -(THead k u2 t3) (THead k x t2)) (subst0 i0 v (THead k x t2) (THead k u2 t3)) -(ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: -T).(subst0 i0 v (THead k x t2) t)) (THead k x0 t3) (subst0_fst v x0 u2 i0 H10 -t3 k) (subst0_both v x x0 i0 H11 k t2 t3 H2)))))) H9)) (\lambda (H9: (subst0 -i0 v u2 x)).(or4_intro1 (eq T (THead k u2 t3) (THead k x t2)) (ex2 T (\lambda -(t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k -x t2) t))) (subst0 i0 v (THead k u2 t3) (THead k x t2)) (subst0 i0 v (THead k -x t2) (THead k u2 t3)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k u2 -t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x t2) t)) (THead k x t3) -(subst0_fst v x u2 i0 H9 t3 k) (subst0_snd k v t3 t2 i0 H2 x)))) (\lambda -(H9: (subst0 i0 v x u2)).(or4_intro3 (eq T (THead k u2 t3) (THead k x t2)) -(ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: -T).(subst0 i0 v (THead k x t2) t))) (subst0 i0 v (THead k u2 t3) (THead k x -t2)) (subst0 i0 v (THead k x t2) (THead k u2 t3)) (subst0_both v x u2 i0 H9 k -t2 t3 H2))) (H1 x H7))) (\lambda (H8: (ex2 T (\lambda (t: T).(subst0 (s k i0) -v t3 t)) (\lambda (t: T).(subst0 (s k i0) v t3 t)))).(ex2_ind T (\lambda (t: -T).(subst0 (s k i0) v t3 t)) (\lambda (t: T).(subst0 (s k i0) v t3 t)) (or4 -(eq T (THead k u2 t3) (THead k x t2)) (ex2 T (\lambda (t: T).(subst0 i0 v -(THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x t2) t))) (subst0 -i0 v (THead k u2 t3) (THead k x t2)) (subst0 i0 v (THead k x t2) (THead k u2 -t3))) (\lambda (x0: T).(\lambda (_: (subst0 (s k i0) v t3 x0)).(\lambda (_: -(subst0 (s k i0) v t3 x0)).(or4_ind (eq T u2 x) (ex2 T (\lambda (t: -T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0 i0 v x t))) (subst0 i0 v u2 x) -(subst0 i0 v x u2) (or4 (eq T (THead k u2 t3) (THead k x t2)) (ex2 T (\lambda -(t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k -x t2) t))) (subst0 i0 v (THead k u2 t3) (THead k x t2)) (subst0 i0 v (THead k -x t2) (THead k u2 t3))) (\lambda (H11: (eq T u2 x)).(eq_ind_r T x (\lambda -(t: T).(or4 (eq T (THead k t t3) (THead k x t2)) (ex2 T (\lambda (t5: -T).(subst0 i0 v (THead k t t3) t5)) (\lambda (t5: T).(subst0 i0 v (THead k x -t2) t5))) (subst0 i0 v (THead k t t3) (THead k x t2)) (subst0 i0 v (THead k x -t2) (THead k t t3)))) (or4_intro3 (eq T (THead k x t3) (THead k x t2)) (ex2 T -(\lambda (t: T).(subst0 i0 v (THead k x t3) t)) (\lambda (t: T).(subst0 i0 v -(THead k x t2) t))) (subst0 i0 v (THead k x t3) (THead k x t2)) (subst0 i0 v -(THead k x t2) (THead k x t3)) (subst0_snd k v t3 t2 i0 H2 x)) u2 H11)) -(\lambda (H11: (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: -T).(subst0 i0 v x t)))).(ex2_ind T (\lambda (t: T).(subst0 i0 v u2 t)) -(\lambda (t: T).(subst0 i0 v x t)) (or4 (eq T (THead k u2 t3) (THead k x t2)) -(ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: -T).(subst0 i0 v (THead k x t2) t))) (subst0 i0 v (THead k u2 t3) (THead k x -t2)) (subst0 i0 v (THead k x t2) (THead k u2 t3))) (\lambda (x1: T).(\lambda -(H12: (subst0 i0 v u2 x1)).(\lambda (H13: (subst0 i0 v x x1)).(or4_intro1 (eq -T (THead k u2 t3) (THead k x t2)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead -k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x t2) t))) (subst0 i0 v -(THead k u2 t3) (THead k x t2)) (subst0 i0 v (THead k x t2) (THead k u2 t3)) -(ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: -T).(subst0 i0 v (THead k x t2) t)) (THead k x1 t3) (subst0_fst v x1 u2 i0 H12 -t3 k) (subst0_both v x x1 i0 H13 k t2 t3 H2)))))) H11)) (\lambda (H11: -(subst0 i0 v u2 x)).(or4_intro1 (eq T (THead k u2 t3) (THead k x t2)) (ex2 T -(\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v -(THead k x t2) t))) (subst0 i0 v 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(subst0 i0 v (THead k x -t2) (THead k u2 t3))) (\lambda (x0: T).(\lambda (H10: (subst0 i0 v u2 -x0)).(\lambda (H11: (subst0 i0 v x x0)).(or4_intro1 (eq T (THead k u2 t3) -(THead k x t2)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) -(\lambda (t: T).(subst0 i0 v (THead k x t2) t))) (subst0 i0 v (THead k u2 t3) -(THead k x t2)) (subst0 i0 v (THead k x t2) (THead k u2 t3)) (ex_intro2 T -(\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v -(THead k x t2) t)) (THead k x0 t3) (subst0_fst v x0 u2 i0 H10 t3 k) -(subst0_both v x x0 i0 H11 k t2 t3 H2)))))) H9)) (\lambda (H9: (subst0 i0 v -u2 x)).(or4_intro1 (eq T (THead k u2 t3) (THead k x t2)) (ex2 T (\lambda (t: -T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x -t2) t))) (subst0 i0 v (THead k u2 t3) (THead k x t2)) (subst0 i0 v (THead k x -t2) (THead k u2 t3)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k u2 -t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x t2) t)) (THead k x t3) -(subst0_fst v x u2 i0 H9 t3 k) (subst0_snd k v t3 t2 i0 H2 x)))) (\lambda -(H9: (subst0 i0 v x u2)).(or4_intro3 (eq T (THead k u2 t3) (THead k x t2)) -(ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: -T).(subst0 i0 v (THead k x t2) t))) (subst0 i0 v (THead k u2 t3) (THead k x -t2)) (subst0 i0 v (THead k x t2) (THead k u2 t3)) (subst0_both v x u2 i0 H9 k -t2 t3 H2))) (H1 x H7))) (\lambda (_: (subst0 (s k i0) v t3 t3)).(or4_ind (eq -T u2 x) (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0 i0 -v x t))) (subst0 i0 v u2 x) (subst0 i0 v x u2) (or4 (eq T (THead k u2 t3) -(THead k x t2)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) -(\lambda (t: T).(subst0 i0 v (THead k x t2) t))) (subst0 i0 v (THead k u2 t3) -(THead k x t2)) (subst0 i0 v (THead k x t2) (THead k u2 t3))) (\lambda (H9: -(eq T u2 x)).(eq_ind_r T x (\lambda (t: T).(or4 (eq T (THead k t t3) (THead k -x t2)) (ex2 T (\lambda (t5: T).(subst0 i0 v (THead k t t3) t5)) (\lambda (t5: -T).(subst0 i0 v (THead k x t2) t5))) (subst0 i0 v (THead k t t3) (THead k x -t2)) (subst0 i0 v (THead k x t2) (THead k t t3)))) (or4_intro3 (eq T (THead k -x t3) (THead k x t2)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k x t3) t)) -(\lambda (t: T).(subst0 i0 v (THead k x t2) t))) (subst0 i0 v (THead k x t3) -(THead k x t2)) (subst0 i0 v (THead k x t2) (THead k x t3)) (subst0_snd k v -t3 t2 i0 H2 x)) u2 H9)) (\lambda (H9: (ex2 T (\lambda (t: T).(subst0 i0 v u2 -t)) (\lambda (t: T).(subst0 i0 v x t)))).(ex2_ind T (\lambda (t: T).(subst0 -i0 v u2 t)) (\lambda (t: T).(subst0 i0 v x t)) (or4 (eq T (THead k u2 t3) -(THead k x t2)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) -(\lambda (t: T).(subst0 i0 v (THead k x t2) t))) (subst0 i0 v (THead k u2 t3) -(THead k x t2)) (subst0 i0 v (THead k x t2) (THead k u2 t3))) (\lambda (x0: -T).(\lambda (H10: (subst0 i0 v u2 x0)).(\lambda (H11: (subst0 i0 v x -x0)).(or4_intro1 (eq T (THead k u2 t3) (THead k x t2)) (ex2 T (\lambda (t: -T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x -t2) t))) (subst0 i0 v (THead k u2 t3) (THead k x t2)) (subst0 i0 v (THead k x -t2) (THead k u2 t3)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k u2 -t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x t2) t)) (THead k x0 t3) -(subst0_fst v x0 u2 i0 H10 t3 k) (subst0_both v x x0 i0 H11 k t2 t3 H2)))))) -H9)) (\lambda (H9: (subst0 i0 v u2 x)).(or4_intro1 (eq T (THead k u2 t3) -(THead k x t2)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) -(\lambda (t: T).(subst0 i0 v (THead k x t2) t))) (subst0 i0 v (THead k u2 t3) -(THead k x t2)) (subst0 i0 v (THead k x t2) (THead k u2 t3)) (ex_intro2 T -(\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v -(THead k x t2) t)) (THead k x t3) (subst0_fst v x u2 i0 H9 t3 k) (subst0_snd -k v t3 t2 i0 H2 x)))) (\lambda (H9: (subst0 i0 v x u2)).(or4_intro3 (eq T -(THead k u2 t3) (THead k x t2)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k -u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x t2) t))) (subst0 i0 v -(THead k u2 t3) (THead k x t2)) (subst0 i0 v (THead k x t2) (THead k u2 t3)) -(subst0_both v x u2 i0 H9 k t2 t3 H2))) (H1 x H7))) (H3 t3 H2)) t4 H6)))) -H5)) (\lambda (H5: (ex2 T (\lambda (t5: T).(eq T t4 (THead k u1 t5))) -(\lambda (t5: T).(subst0 (s k i0) v t2 t5)))).(ex2_ind T (\lambda (t5: T).(eq -T t4 (THead k u1 t5))) (\lambda (t5: T).(subst0 (s k i0) v t2 t5)) (or4 (eq T -(THead k u2 t3) t4) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) -(\lambda (t: T).(subst0 i0 v t4 t))) (subst0 i0 v (THead k u2 t3) t4) (subst0 -i0 v t4 (THead k u2 t3))) (\lambda (x: T).(\lambda (H6: (eq T t4 (THead k u1 -x))).(\lambda (H7: (subst0 (s k i0) v t2 x)).(eq_ind_r T (THead k u1 x) -(\lambda (t: T).(or4 (eq T (THead k u2 t3) t) (ex2 T (\lambda (t5: T).(subst0 -i0 v (THead k u2 t3) t5)) (\lambda (t5: T).(subst0 i0 v t t5))) (subst0 i0 v -(THead k u2 t3) t) (subst0 i0 v t (THead k u2 t3)))) (or4_ind (eq T t3 x) -(ex2 T 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t)) (or4 (eq T (THead k u2 t3) -(THead k u1 x)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) -(\lambda (t: T).(subst0 i0 v (THead k u1 x) t))) (subst0 i0 v (THead k u2 t3) -(THead k u1 x)) (subst0 i0 v (THead k u1 x) (THead k u2 t3))) (\lambda (x0: -T).(\lambda (H9: (subst0 (s k i0) v t3 x0)).(\lambda (H10: (subst0 (s k i0) v -x x0)).(or4_ind (eq T u2 u2) (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) -(\lambda (t: T).(subst0 i0 v u2 t))) (subst0 i0 v u2 u2) (subst0 i0 v u2 u2) -(or4 (eq T (THead k u2 t3) (THead k u1 x)) (ex2 T (\lambda (t: T).(subst0 i0 -v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k u1 x) t))) -(subst0 i0 v (THead k u2 t3) (THead k u1 x)) (subst0 i0 v (THead k u1 x) -(THead k u2 t3))) (\lambda (_: (eq T u2 u2)).(or4_intro1 (eq T (THead k u2 -t3) (THead k u1 x)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) -(\lambda (t: T).(subst0 i0 v (THead k u1 x) t))) (subst0 i0 v (THead k u2 t3) -(THead k u1 x)) (subst0 i0 v (THead k u1 x) (THead k u2 t3)) (ex_intro2 T -(\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v -(THead k u1 x) t)) (THead k u2 x0) (subst0_snd k v x0 t3 i0 H9 u2) -(subst0_both v u1 u2 i0 H0 k x x0 H10)))) (\lambda (H11: (ex2 T (\lambda (t: -T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0 i0 v u2 t)))).(ex2_ind T -(\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0 i0 v u2 t)) (or4 -(eq T (THead k u2 t3) (THead k u1 x)) (ex2 T (\lambda (t: T).(subst0 i0 v -(THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k u1 x) t))) (subst0 -i0 v (THead k u2 t3) (THead k u1 x)) (subst0 i0 v (THead k u1 x) (THead k u2 -t3))) (\lambda (x1: T).(\lambda (_: (subst0 i0 v u2 x1)).(\lambda (_: (subst0 -i0 v u2 x1)).(or4_intro1 (eq T (THead k u2 t3) (THead k u1 x)) (ex2 T -(\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v -(THead k u1 x) t))) (subst0 i0 v (THead k u2 t3) (THead k u1 x)) (subst0 i0 v -(THead k u1 x) (THead k u2 t3)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v -(THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k u1 x) t)) (THead k -u2 x0) (subst0_snd k v x0 t3 i0 H9 u2) (subst0_both v u1 u2 i0 H0 k x x0 -H10)))))) H11)) (\lambda (_: (subst0 i0 v u2 u2)).(or4_intro1 (eq T (THead k -u2 t3) (THead k u1 x)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) -t)) (\lambda (t: T).(subst0 i0 v (THead k u1 x) t))) (subst0 i0 v (THead k u2 -t3) (THead k u1 x)) (subst0 i0 v (THead k u1 x) (THead k u2 t3)) (ex_intro2 T -(\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v -(THead k u1 x) t)) (THead k u2 x0) (subst0_snd k v x0 t3 i0 H9 u2) -(subst0_both v u1 u2 i0 H0 k x x0 H10)))) (\lambda (_: (subst0 i0 v u2 -u2)).(or4_intro1 (eq T (THead k u2 t3) (THead k u1 x)) (ex2 T (\lambda (t: -T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k u1 -x) t))) (subst0 i0 v (THead k u2 t3) (THead k u1 x)) (subst0 i0 v (THead k u1 -x) (THead k u2 t3)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) -t)) (\lambda (t: T).(subst0 i0 v (THead k u1 x) t)) (THead k u2 x0) -(subst0_snd k v x0 t3 i0 H9 u2) (subst0_both v u1 u2 i0 H0 k x x0 H10)))) (H1 -u2 H0))))) H8)) (\lambda (H8: (subst0 (s k i0) v t3 x)).(or4_ind (eq T u2 u2) -(ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0 i0 v u2 -t))) (subst0 i0 v u2 u2) (subst0 i0 v u2 u2) (or4 (eq T (THead k u2 t3) -(THead k u1 x)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) -(\lambda (t: T).(subst0 i0 v (THead k u1 x) t))) (subst0 i0 v (THead k u2 t3) -(THead k u1 x)) (subst0 i0 v (THead k u1 x) (THead k u2 t3))) (\lambda (_: -(eq T u2 u2)).(or4_intro1 (eq T (THead k u2 t3) (THead k u1 x)) (ex2 T -(\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v -(THead k u1 x) t))) (subst0 i0 v (THead k u2 t3) (THead k u1 x)) (subst0 i0 v -(THead k u1 x) (THead k u2 t3)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v -(THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k u1 x) t)) (THead k -u2 x) (subst0_snd k v x t3 i0 H8 u2) (subst0_fst v u2 u1 i0 H0 x k)))) -(\lambda (H9: (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: -T).(subst0 i0 v u2 t)))).(ex2_ind T (\lambda (t: T).(subst0 i0 v u2 t)) -(\lambda (t: T).(subst0 i0 v u2 t)) (or4 (eq T (THead k u2 t3) (THead k u1 -x)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: -T).(subst0 i0 v (THead k u1 x) t))) (subst0 i0 v (THead k u2 t3) (THead k u1 -x)) (subst0 i0 v (THead k u1 x) (THead k u2 t3))) (\lambda (x0: T).(\lambda -(_: (subst0 i0 v u2 x0)).(\lambda (_: (subst0 i0 v u2 x0)).(or4_intro1 (eq T -(THead k u2 t3) (THead k u1 x)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k -u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k u1 x) t))) (subst0 i0 v -(THead k u2 t3) (THead k u1 x)) (subst0 i0 v (THead k u1 x) (THead k u2 t3)) -(ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: -T).(subst0 i0 v (THead k u1 x) t)) (THead k u2 x) (subst0_snd k v x t3 i0 H8 -u2) (subst0_fst v u2 u1 i0 H0 x k)))))) H9)) (\lambda (_: (subst0 i0 v u2 -u2)).(or4_intro1 (eq T (THead k u2 t3) (THead k u1 x)) (ex2 T (\lambda (t: -T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k u1 -x) t))) (subst0 i0 v (THead k u2 t3) (THead k u1 x)) (subst0 i0 v (THead k u1 -x) (THead k u2 t3)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) -t)) (\lambda (t: T).(subst0 i0 v (THead k u1 x) t)) (THead k u2 x) -(subst0_snd k v x t3 i0 H8 u2) (subst0_fst v u2 u1 i0 H0 x k)))) (\lambda (_: -(subst0 i0 v u2 u2)).(or4_intro1 (eq T (THead k u2 t3) (THead k u1 x)) (ex2 T -(\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v -(THead k u1 x) t))) (subst0 i0 v (THead k u2 t3) (THead k u1 x)) (subst0 i0 v -(THead k u1 x) (THead k u2 t3)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v -(THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k u1 x) t)) (THead k -u2 x) (subst0_snd k v x t3 i0 H8 u2) (subst0_fst v u2 u1 i0 H0 x k)))) (H1 u2 -H0))) (\lambda (H8: (subst0 (s k i0) v x t3)).(or4_ind (eq T u2 u2) (ex2 T -(\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0 i0 v u2 t))) -(subst0 i0 v u2 u2) (subst0 i0 v u2 u2) (or4 (eq T (THead k u2 t3) (THead k -u1 x)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: -T).(subst0 i0 v (THead k u1 x) t))) (subst0 i0 v (THead k u2 t3) (THead k u1 -x)) (subst0 i0 v (THead k u1 x) (THead k u2 t3))) (\lambda (_: (eq T u2 -u2)).(or4_intro3 (eq T (THead k u2 t3) (THead k u1 x)) (ex2 T (\lambda (t: -T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k u1 -x) t))) (subst0 i0 v (THead k u2 t3) (THead k u1 x)) (subst0 i0 v (THead k u1 -x) (THead k u2 t3)) (subst0_both v u1 u2 i0 H0 k x t3 H8))) (\lambda (H9: -(ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0 i0 v u2 -t)))).(ex2_ind T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0 -i0 v u2 t)) (or4 (eq T (THead k u2 t3) (THead k u1 x)) (ex2 T (\lambda (t: -T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k u1 -x) t))) (subst0 i0 v (THead k u2 t3) (THead k u1 x)) (subst0 i0 v (THead k u1 -x) (THead k u2 t3))) (\lambda (x0: T).(\lambda (_: (subst0 i0 v u2 -x0)).(\lambda (_: (subst0 i0 v u2 x0)).(or4_intro3 (eq T (THead k u2 t3) -(THead k u1 x)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) -(\lambda (t: T).(subst0 i0 v (THead k u1 x) t))) (subst0 i0 v (THead k u2 t3) -(THead k u1 x)) (subst0 i0 v (THead k u1 x) (THead k u2 t3)) (subst0_both v -u1 u2 i0 H0 k x t3 H8))))) H9)) (\lambda (_: (subst0 i0 v u2 u2)).(or4_intro3 -(eq T (THead k u2 t3) (THead k u1 x)) (ex2 T (\lambda (t: T).(subst0 i0 v -(THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k u1 x) t))) (subst0 -i0 v (THead k u2 t3) (THead k u1 x)) (subst0 i0 v (THead k u1 x) (THead k u2 -t3)) (subst0_both v u1 u2 i0 H0 k x t3 H8))) (\lambda (_: (subst0 i0 v u2 -u2)).(or4_intro3 (eq T (THead k u2 t3) (THead k u1 x)) (ex2 T (\lambda (t: -T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k u1 -x) t))) (subst0 i0 v (THead k u2 t3) (THead k u1 x)) (subst0 i0 v (THead k u1 -x) (THead k u2 t3)) (subst0_both v u1 u2 i0 H0 k x t3 H8))) (H1 u2 H0))) (H3 -x H7)) t4 H6)))) H5)) (\lambda (H5: (ex3_2 T T (\lambda (u3: T).(\lambda (t5: -T).(eq T t4 (THead k u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v -u1 u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s k i0) v t2 -t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t5: T).(eq T t4 (THead k u3 -t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v u1 u3))) (\lambda (_: -T).(\lambda (t5: T).(subst0 (s k i0) v t2 t5))) (or4 (eq T (THead k u2 t3) -t4) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: -T).(subst0 i0 v t4 t))) (subst0 i0 v (THead k u2 t3) t4) (subst0 i0 v t4 -(THead k u2 t3))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H6: (eq T t4 -(THead k x0 x1))).(\lambda (H7: (subst0 i0 v u1 x0)).(\lambda (H8: (subst0 (s -k i0) v t2 x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t: T).(or4 (eq T (THead -k u2 t3) t) (ex2 T (\lambda (t5: T).(subst0 i0 v (THead k u2 t3) t5)) -(\lambda (t5: T).(subst0 i0 v t t5))) (subst0 i0 v (THead k u2 t3) t) (subst0 -i0 v t (THead k u2 t3)))) (or4_ind (eq T t3 x1) (ex2 T (\lambda (t: -T).(subst0 (s k i0) v t3 t)) (\lambda (t: T).(subst0 (s k i0) v x1 t))) -(subst0 (s k i0) v t3 x1) (subst0 (s k i0) v x1 t3) (or4 (eq T (THead k u2 -t3) (THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) -(\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 -t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t3))) (\lambda -(H9: (eq T t3 x1)).(eq_ind_r T x1 (\lambda (t: T).(or4 (eq T (THead k u2 t) -(THead k x0 x1)) (ex2 T (\lambda (t5: T).(subst0 i0 v (THead k u2 t) t5)) -(\lambda (t5: T).(subst0 i0 v (THead k x0 x1) t5))) (subst0 i0 v (THead k u2 -t) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t)))) (or4_ind -(eq T u2 x0) (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: -T).(subst0 i0 v x0 t))) (subst0 i0 v u2 x0) (subst0 i0 v x0 u2) (or4 (eq T -(THead k u2 x1) (THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k -u2 x1) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v -(THead k u2 x1) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 -x1))) (\lambda (H10: (eq T u2 x0)).(eq_ind_r T x0 (\lambda (t: T).(or4 (eq T -(THead k t x1) (THead k x0 x1)) (ex2 T (\lambda (t5: T).(subst0 i0 v (THead k -t x1) t5)) (\lambda (t5: T).(subst0 i0 v (THead k x0 x1) t5))) (subst0 i0 v -(THead k t x1) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k t -x1)))) (or4_intro0 (eq T (THead k x0 x1) (THead k x0 x1)) (ex2 T (\lambda (t: -T).(subst0 i0 v (THead k x0 x1) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 -x1) t))) (subst0 i0 v (THead k x0 x1) (THead k x0 x1)) (subst0 i0 v (THead k -x0 x1) (THead k x0 x1)) (refl_equal T (THead k x0 x1))) u2 H10)) (\lambda -(H10: (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0 i0 v -x0 t)))).(ex2_ind T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: -T).(subst0 i0 v x0 t)) (or4 (eq T (THead k u2 x1) (THead k x0 x1)) (ex2 T -(\lambda (t: T).(subst0 i0 v (THead k u2 x1) t)) (\lambda (t: T).(subst0 i0 v -(THead k x0 x1) t))) (subst0 i0 v (THead k u2 x1) (THead k x0 x1)) (subst0 i0 -v (THead k x0 x1) (THead k u2 x1))) (\lambda (x: T).(\lambda (H11: (subst0 i0 -v u2 x)).(\lambda (H12: (subst0 i0 v x0 x)).(or4_intro1 (eq T (THead k u2 x1) -(THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 x1) t)) -(\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 -x1) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 x1)) (ex_intro2 -T (\lambda (t: T).(subst0 i0 v (THead k u2 x1) t)) (\lambda (t: T).(subst0 i0 -v (THead k x0 x1) t)) (THead k x x1) (subst0_fst v x u2 i0 H11 x1 k) -(subst0_fst v x x0 i0 H12 x1 k)))))) H10)) (\lambda (H10: (subst0 i0 v u2 -x0)).(or4_intro2 (eq T (THead k u2 x1) (THead k x0 x1)) (ex2 T (\lambda (t: -T).(subst0 i0 v (THead k u2 x1) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 -x1) t))) (subst0 i0 v (THead k u2 x1) (THead k x0 x1)) (subst0 i0 v (THead k -x0 x1) (THead k u2 x1)) (subst0_fst v x0 u2 i0 H10 x1 k))) (\lambda (H10: -(subst0 i0 v x0 u2)).(or4_intro3 (eq T (THead k u2 x1) (THead k x0 x1)) (ex2 -T (\lambda (t: T).(subst0 i0 v (THead k u2 x1) t)) (\lambda (t: T).(subst0 i0 -v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 x1) (THead k x0 x1)) (subst0 -i0 v (THead k x0 x1) (THead k u2 x1)) (subst0_fst v u2 x0 i0 H10 x1 k))) (H1 -x0 H7)) t3 H9)) (\lambda (H9: (ex2 T (\lambda (t: T).(subst0 (s k i0) v t3 -t)) (\lambda (t: T).(subst0 (s k i0) v x1 t)))).(ex2_ind T (\lambda (t: -T).(subst0 (s k i0) v t3 t)) (\lambda (t: T).(subst0 (s k i0) v x1 t)) (or4 -(eq T (THead k u2 t3) (THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v -(THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 -i0 v (THead k u2 t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k -u2 t3))) (\lambda (x: T).(\lambda (H10: (subst0 (s k i0) v t3 x)).(\lambda -(H11: (subst0 (s k i0) v x1 x)).(or4_ind (eq T u2 x0) (ex2 T (\lambda (t: -T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0 i0 v x0 t))) (subst0 i0 v u2 -x0) (subst0 i0 v x0 u2) (or4 (eq T (THead k u2 t3) (THead k x0 x1)) (ex2 T -(\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v -(THead k x0 x1) t))) (subst0 i0 v (THead k u2 t3) (THead k x0 x1)) (subst0 i0 -v (THead k x0 x1) (THead k u2 t3))) (\lambda (H12: (eq T u2 x0)).(eq_ind_r T -x0 (\lambda (t: T).(or4 (eq T (THead k t t3) (THead k x0 x1)) (ex2 T (\lambda -(t5: T).(subst0 i0 v (THead k t t3) t5)) (\lambda (t5: T).(subst0 i0 v (THead -k x0 x1) t5))) (subst0 i0 v (THead k t t3) (THead k x0 x1)) (subst0 i0 v -(THead k x0 x1) (THead k t t3)))) (or4_intro1 (eq T (THead k x0 t3) (THead k -x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k x0 t3) t)) (\lambda (t: -T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k x0 t3) (THead k x0 -x1)) (subst0 i0 v (THead k x0 x1) (THead k x0 t3)) (ex_intro2 T (\lambda (t: -T).(subst0 i0 v (THead k x0 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 -x1) t)) (THead k x0 x) (subst0_snd k v x t3 i0 H10 x0) (subst0_snd k v x x1 -i0 H11 x0))) u2 H12)) (\lambda (H12: (ex2 T (\lambda (t: T).(subst0 i0 v u2 -t)) (\lambda (t: T).(subst0 i0 v x0 t)))).(ex2_ind T (\lambda (t: T).(subst0 -i0 v u2 t)) (\lambda (t: T).(subst0 i0 v x0 t)) (or4 (eq T (THead k u2 t3) -(THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) -(\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 -t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t3))) (\lambda -(x2: T).(\lambda (H13: (subst0 i0 v u2 x2)).(\lambda (H14: (subst0 i0 v x0 -x2)).(or4_intro1 (eq T (THead k u2 t3) (THead k x0 x1)) (ex2 T (\lambda (t: -T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 -x1) t))) (subst0 i0 v (THead k u2 t3) (THead k x0 x1)) (subst0 i0 v (THead k -x0 x1) (THead k u2 t3)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k u2 -t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t)) (THead k x2 x) -(subst0_both v u2 x2 i0 H13 k t3 x H10) (subst0_both v x0 x2 i0 H14 k x1 x -H11)))))) H12)) (\lambda (H12: (subst0 i0 v u2 x0)).(or4_intro1 (eq T (THead -k u2 t3) (THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) -t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k -u2 t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t3)) -(ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: -T).(subst0 i0 v (THead k x0 x1) t)) (THead k x0 x) (subst0_both v u2 x0 i0 -H12 k t3 x H10) (subst0_snd k v x x1 i0 H11 x0)))) (\lambda (H12: (subst0 i0 -v x0 u2)).(or4_intro1 (eq T (THead k u2 t3) (THead k x0 x1)) (ex2 T (\lambda -(t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k -x0 x1) t))) (subst0 i0 v (THead k u2 t3) (THead k x0 x1)) (subst0 i0 v (THead -k x0 x1) (THead k u2 t3)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k -u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t)) (THead k u2 x) -(subst0_snd k v x t3 i0 H10 u2) (subst0_both v x0 u2 i0 H12 k x1 x H11)))) -(H1 x0 H7))))) H9)) (\lambda (H9: (subst0 (s k i0) v t3 x1)).(or4_ind (eq T -u2 x0) (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0 i0 -v x0 t))) (subst0 i0 v u2 x0) (subst0 i0 v x0 u2) (or4 (eq T (THead k u2 t3) -(THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) -(\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 -t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t3))) (\lambda -(H10: (eq T u2 x0)).(eq_ind_r T x0 (\lambda (t: T).(or4 (eq T (THead k t t3) -(THead k x0 x1)) (ex2 T (\lambda (t5: T).(subst0 i0 v (THead k t t3) t5)) -(\lambda (t5: T).(subst0 i0 v (THead k x0 x1) t5))) (subst0 i0 v (THead k t -t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k t t3)))) -(or4_intro2 (eq T (THead k x0 t3) (THead k x0 x1)) (ex2 T (\lambda (t: -T).(subst0 i0 v (THead k x0 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 -x1) t))) (subst0 i0 v (THead k x0 t3) (THead k x0 x1)) (subst0 i0 v (THead k -x0 x1) (THead k x0 t3)) (subst0_snd k v x1 t3 i0 H9 x0)) u2 H10)) (\lambda -(H10: (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0 i0 v -x0 t)))).(ex2_ind T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: -T).(subst0 i0 v x0 t)) (or4 (eq T (THead k u2 t3) (THead k x0 x1)) (ex2 T -(\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v -(THead k x0 x1) t))) (subst0 i0 v (THead k u2 t3) (THead k x0 x1)) (subst0 i0 -v (THead k x0 x1) (THead k u2 t3))) (\lambda (x: T).(\lambda (H11: (subst0 i0 -v u2 x)).(\lambda (H12: (subst0 i0 v x0 x)).(or4_intro1 (eq T (THead k u2 t3) -(THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) -(\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 -t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t3)) (ex_intro2 -T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 -v (THead k x0 x1) t)) (THead k x x1) (subst0_both v u2 x i0 H11 k t3 x1 H9) -(subst0_fst v x x0 i0 H12 x1 k)))))) H10)) (\lambda (H10: (subst0 i0 v u2 -x0)).(or4_intro2 (eq T (THead k u2 t3) (THead k x0 x1)) (ex2 T (\lambda (t: -T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 -x1) t))) (subst0 i0 v (THead k u2 t3) (THead k x0 x1)) (subst0 i0 v (THead k -x0 x1) (THead k u2 t3)) (subst0_both v u2 x0 i0 H10 k t3 x1 H9))) (\lambda -(H10: (subst0 i0 v x0 u2)).(or4_intro1 (eq T (THead k u2 t3) (THead k x0 x1)) -(ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: -T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 t3) (THead k x0 -x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t3)) (ex_intro2 T (\lambda (t: -T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 -x1) t)) (THead k u2 x1) (subst0_snd k v x1 t3 i0 H9 u2) (subst0_fst v u2 x0 -i0 H10 x1 k)))) (H1 x0 H7))) (\lambda (H9: (subst0 (s k i0) v x1 -t3)).(or4_ind (eq T u2 x0) (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) -(\lambda (t: T).(subst0 i0 v x0 t))) (subst0 i0 v u2 x0) (subst0 i0 v x0 u2) -(or4 (eq T (THead k u2 t3) (THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 -v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) -(subst0 i0 v (THead k u2 t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) -(THead k u2 t3))) (\lambda (H10: (eq T u2 x0)).(eq_ind_r T x0 (\lambda (t: -T).(or4 (eq T (THead k t t3) (THead k x0 x1)) (ex2 T (\lambda (t5: T).(subst0 -i0 v (THead k t t3) t5)) (\lambda (t5: T).(subst0 i0 v (THead k x0 x1) t5))) -(subst0 i0 v (THead k t t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) -(THead k t t3)))) (or4_intro3 (eq T (THead k x0 t3) (THead k x0 x1)) (ex2 T -(\lambda (t: T).(subst0 i0 v (THead k x0 t3) t)) (\lambda (t: T).(subst0 i0 v -(THead k x0 x1) t))) (subst0 i0 v (THead k x0 t3) (THead k x0 x1)) (subst0 i0 -v (THead k x0 x1) (THead k x0 t3)) (subst0_snd k v t3 x1 i0 H9 x0)) u2 H10)) -(\lambda (H10: (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: -T).(subst0 i0 v x0 t)))).(ex2_ind T (\lambda (t: T).(subst0 i0 v u2 t)) -(\lambda (t: T).(subst0 i0 v x0 t)) (or4 (eq T (THead k u2 t3) (THead k x0 -x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: -T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 t3) (THead k x0 -x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t3))) (\lambda (x: T).(\lambda -(H11: (subst0 i0 v u2 x)).(\lambda (H12: (subst0 i0 v x0 x)).(or4_intro1 (eq -T (THead k u2 t3) (THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead -k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v -(THead k u2 t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 -t3)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda -(t: T).(subst0 i0 v (THead k x0 x1) t)) (THead k x t3) (subst0_fst v x u2 i0 -H11 t3 k) (subst0_both v x0 x i0 H12 k x1 t3 H9)))))) H10)) (\lambda (H10: -(subst0 i0 v u2 x0)).(or4_intro1 (eq T (THead k u2 t3) (THead k x0 x1)) (ex2 -T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 -v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 t3) (THead k x0 x1)) (subst0 -i0 v (THead k x0 x1) (THead k u2 t3)) (ex_intro2 T (\lambda (t: T).(subst0 i0 -v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t)) (THead -k x0 t3) (subst0_fst v x0 u2 i0 H10 t3 k) (subst0_snd k v t3 x1 i0 H9 x0)))) -(\lambda (H10: (subst0 i0 v x0 u2)).(or4_intro3 (eq T (THead k u2 t3) (THead -k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda -(t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 t3) (THead -k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t3)) (subst0_both v x0 u2 -i0 H10 k x1 t3 H9))) (H1 x0 H7))) (H3 x1 H8)) t4 H6)))))) H5)) -(subst0_gen_head k v u1 t2 t4 i0 H4))))))))))))))) i u t0 t1 H))))). -(* COMMENTS -Initial nodes: 25595 -END *) - -theorem subst0_confluence_lift: - \forall (t0: T).(\forall (t1: T).(\forall (u: T).(\forall (i: nat).((subst0 -i u t0 (lift (S O) i t1)) \to (\forall (t2: T).((subst0 i u t0 (lift (S O) i -t2)) \to (eq T t1 t2))))))) -\def - \lambda (t0: T).(\lambda (t1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H: (subst0 i u t0 (lift (S O) i t1))).(\lambda (t2: T).(\lambda (H0: (subst0 -i u t0 (lift (S O) i t2))).(or4_ind (eq T (lift (S O) i t2) (lift (S O) i -t1)) (ex2 T (\lambda (t: T).(subst0 i u (lift (S O) i t2) t)) (\lambda (t: -T).(subst0 i u (lift (S O) i t1) t))) (subst0 i u (lift (S O) i t2) (lift (S -O) i t1)) (subst0 i u (lift (S O) i t1) (lift (S O) i t2)) (eq T t1 t2) -(\lambda (H1: (eq T (lift (S O) i t2) (lift (S O) i t1))).(let H2 \def -(sym_eq T (lift (S O) i t2) (lift (S O) i t1) H1) in (lift_inj t1 t2 (S O) i -H2))) (\lambda (H1: (ex2 T (\lambda (t: T).(subst0 i u (lift (S O) i t2) t)) -(\lambda (t: T).(subst0 i u (lift (S O) i t1) t)))).(ex2_ind T (\lambda (t: -T).(subst0 i u (lift (S O) i t2) t)) (\lambda (t: T).(subst0 i u (lift (S O) -i t1) t)) (eq T t1 t2) (\lambda (x: T).(\lambda (_: (subst0 i u (lift (S O) i -t2) x)).(\lambda (H3: (subst0 i u (lift (S O) i t1) -x)).(subst0_gen_lift_false t1 u x (S O) i i (le_n i) (eq_ind_r nat (plus (S -O) i) (\lambda (n: nat).(lt i n)) (le_n (plus (S O) i)) (plus i (S O)) -(plus_sym i (S O))) H3 (eq T t1 t2))))) H1)) (\lambda (H1: (subst0 i u (lift -(S O) i t2) (lift (S O) i t1))).(subst0_gen_lift_false t2 u (lift (S O) i t1) -(S O) i i (le_n i) (eq_ind_r nat (plus (S O) i) (\lambda (n: nat).(lt i n)) -(le_n (plus (S O) i)) (plus i (S O)) (plus_sym i (S O))) H1 (eq T t1 t2))) -(\lambda (H1: (subst0 i u (lift (S O) i t1) (lift (S O) i -t2))).(subst0_gen_lift_false t1 u (lift (S O) i t2) (S O) i i (le_n i) -(eq_ind_r nat (plus (S O) i) (\lambda (n: nat).(lt i n)) (le_n (plus (S O) -i)) (plus i (S O)) (plus_sym i (S O))) H1 (eq T t1 t2))) -(subst0_confluence_eq t0 (lift (S O) i t2) u i H0 (lift (S O) i t1) H)))))))). -(* COMMENTS -Initial nodes: 703 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/subst0/tlt.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/subst0/tlt.ma deleted file mode 100644 index c8e8420bb..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/subst0/tlt.ma +++ /dev/null @@ -1,468 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/subst0/defs.ma". - -include "Basic-1/lift/props.ma". - -include "Basic-1/lift/tlt.ma". - -theorem subst0_weight_le: - \forall (u: T).(\forall (t: T).(\forall (z: T).(\forall (d: nat).((subst0 d -u t z) \to (\forall (f: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S -d) O u)) (g d)) \to (le (weight_map f z) (weight_map g t)))))))))) -\def - \lambda (u: T).(\lambda (t: T).(\lambda (z: T).(\lambda (d: nat).(\lambda -(H: (subst0 d u t z)).(subst0_ind (\lambda (n: nat).(\lambda (t0: T).(\lambda -(t1: T).(\lambda (t2: T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S -n) O t0)) (g n)) \to (le (weight_map f t2) (weight_map g t1)))))))))) -(\lambda (v: T).(\lambda (i: nat).(\lambda (f: ((nat \to nat))).(\lambda (g: -((nat \to nat))).(\lambda (_: ((\forall (m: nat).(le (f m) (g m))))).(\lambda -(H1: (lt (weight_map f (lift (S i) O v)) (g i))).(le_S_n (weight_map f (lift -(S i) O v)) (weight_map g (TLRef i)) (le_S (S (weight_map f (lift (S i) O -v))) (weight_map g (TLRef i)) H1)))))))) (\lambda (v: T).(\lambda (u2: -T).(\lambda (u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1 -u2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S -i) O v)) (g i)) \to (le (weight_map f u2) (weight_map g u1)))))))).(\lambda -(t0: T).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (f: ((nat \to -nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) -\to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (le (weight_map f (THead -k0 u2 t0)) (weight_map g (THead k0 u1 t0)))))))) (\lambda (b: B).(B_ind -(\lambda (b0: B).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S -i) O v)) (g i)) \to (le (weight_map f (THead (Bind b0) u2 t0)) (weight_map g -(THead (Bind b0) u1 t0)))))))) (\lambda (f: ((nat \to nat))).(\lambda (g: -((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g -m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g i))).(le_n_S -(plus (weight_map f u2) (weight_map (wadd f (S (weight_map f u2))) t0)) (plus -(weight_map g u1) (weight_map (wadd g (S (weight_map g u1))) t0)) -(le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f (S -(weight_map f u2))) t0) (weight_map (wadd g (S (weight_map g u1))) t0) (H1 f -g H2 H3) (weight_le t0 (wadd f (S (weight_map f u2))) (wadd g (S (weight_map -g u1))) (\lambda (n: nat).(wadd_le f g H2 (S (weight_map f u2)) (S -(weight_map g u1)) (le_n_S (weight_map f u2) (weight_map g u1) (H1 f g H2 -H3)) n))))))))) (\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to -nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3: (lt -(weight_map f (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map f u2) -(weight_map (wadd f O) t0)) (plus (weight_map g u1) (weight_map (wadd g O) -t0)) (le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f O) -t0) (weight_map (wadd g O) t0) (H1 f g H2 H3) (weight_le t0 (wadd f O) (wadd -g O) (\lambda (n: nat).(wadd_le f g H2 O O (le_n O) n))))))))) (\lambda (f: -((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: -nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g -i))).(le_n_S (plus (weight_map f u2) (weight_map (wadd f O) t0)) (plus -(weight_map g u1) (weight_map (wadd g O) t0)) (le_plus_plus (weight_map f u2) -(weight_map g u1) (weight_map (wadd f O) t0) (weight_map (wadd g O) t0) (H1 f -g H2 H3) (weight_le t0 (wadd f O) (wadd g O) (\lambda (n: nat).(wadd_le f g -H2 O O (le_n O) n))))))))) b)) (\lambda (_: F).(\lambda (f0: ((nat \to -nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f0 -m) (g m))))).(\lambda (H3: (lt (weight_map f0 (lift (S i) O v)) (g -i))).(le_n_S (plus (weight_map f0 u2) (weight_map f0 t0)) (plus (weight_map g -u1) (weight_map g t0)) (le_plus_plus (weight_map f0 u2) (weight_map g u1) -(weight_map f0 t0) (weight_map g t0) (H1 f0 g H2 H3) (weight_le t0 f0 g -H2)))))))) k))))))))) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (v: -T).(\forall (t2: T).(\forall (t1: T).(\forall (i: nat).((subst0 (s k0 i) v t1 -t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S -(s k0 i)) O v)) (g (s k0 i))) \to (le (weight_map f t2) (weight_map g -t1))))))) \to (\forall (u0: T).(\forall (f: ((nat \to nat))).(\forall (g: -((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map -f (lift (S i) O v)) (g i)) \to (le (weight_map f (THead k0 u0 t2)) -(weight_map g (THead k0 u0 t1))))))))))))))) (\lambda (b: B).(B_ind (\lambda -(b0: B).(\forall (v: T).(\forall (t2: T).(\forall (t1: T).(\forall (i: -nat).((subst0 (s (Bind b0) i) v t1 t2) \to (((\forall (f: ((nat \to -nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) -\to ((lt (weight_map f (lift (S (s (Bind b0) i)) O v)) (g (s (Bind b0) i))) -\to (le (weight_map f t2) (weight_map g t1))))))) \to (\forall (u0: -T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: -nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to -(le (weight_map f (THead (Bind b0) u0 t2)) (weight_map g (THead (Bind b0) u0 -t1))))))))))))))) (\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda -(i: nat).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H1: ((\forall (f: -((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) -(g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (le -(weight_map f t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f: -((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: -nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g -i))).(le_n_S (plus (weight_map f u0) (weight_map (wadd f (S (weight_map f -u0))) t2)) (plus (weight_map g u0) (weight_map (wadd g (S (weight_map g u0))) -t1)) (le_plus_plus (weight_map f u0) (weight_map g u0) (weight_map (wadd f (S -(weight_map f u0))) t2) (weight_map (wadd g (S (weight_map g u0))) t1) -(weight_le u0 f g H2) (H1 (wadd f (S (weight_map f u0))) (wadd g (S -(weight_map g u0))) (\lambda (m: nat).(wadd_le f g H2 (S (weight_map f u0)) -(S (weight_map g u0)) (le_n_S (weight_map f u0) (weight_map g u0) (weight_le -u0 f g H2)) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda (n: -nat).(lt n (g i))) H3 (weight_map (wadd f (S (weight_map f u0))) (lift (S (S -i)) O v)) (lift_weight_add_O (S (weight_map f u0)) v (S i) f)))))))))))))))) -(\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda (i: nat).(\lambda -(_: (subst0 (S i) v t1 t2)).(\lambda (H1: ((\forall (f: ((nat \to -nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) -\to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (le (weight_map f -t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f: ((nat \to -nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f -m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g -i))).(le_n_S (plus (weight_map f u0) (weight_map (wadd f O) t2)) (plus -(weight_map g u0) (weight_map (wadd g O) t1)) (le_plus_plus (weight_map f u0) -(weight_map g u0) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1) -(weight_le u0 f g H2) (H1 (wadd f O) (wadd g O) (\lambda (m: nat).(wadd_le f -g H2 O O (le_n O) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda -(n: nat).(lt n (g i))) H3 (weight_map (wadd f O) (lift (S (S i)) O v)) -(lift_weight_add_O O v (S i) f)))))))))))))))) (\lambda (v: T).(\lambda (t2: -T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (_: (subst0 (S i) v t1 -t2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S -(S i)) O v)) (g (S i))) \to (le (weight_map f t2) (weight_map g -t1)))))))).(\lambda (u0: T).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat -\to nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3: -(lt (weight_map f (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map f u0) -(weight_map (wadd f O) t2)) (plus (weight_map g u0) (weight_map (wadd g O) -t1)) (le_plus_plus (weight_map f u0) (weight_map g u0) (weight_map (wadd f O) -t2) (weight_map (wadd g O) t1) (weight_le u0 f g H2) (H1 (wadd f O) (wadd g -O) (\lambda (m: nat).(wadd_le f g H2 O O (le_n O) m)) (eq_ind nat (weight_map -f (lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H3 (weight_map (wadd f O) -(lift (S (S i)) O v)) (lift_weight_add_O O v (S i) f)))))))))))))))) b)) -(\lambda (_: F).(\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda -(i: nat).(\lambda (_: (subst0 i v t1 t2)).(\lambda (H1: ((\forall (f0: ((nat -\to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f0 m) (g -m)))) \to ((lt (weight_map f0 (lift (S i) O v)) (g i)) \to (le (weight_map f0 -t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f0: ((nat \to -nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f0 -m) (g m))))).(\lambda (H3: (lt (weight_map f0 (lift (S i) O v)) (g -i))).(le_n_S (plus (weight_map f0 u0) (weight_map f0 t2)) (plus (weight_map g -u0) (weight_map g t1)) (le_plus_plus (weight_map f0 u0) (weight_map g u0) -(weight_map f0 t2) (weight_map g t1) (weight_le u0 f0 g H2) (H1 f0 g H2 -H3))))))))))))))) k)) (\lambda (v: T).(\lambda (u1: T).(\lambda (u2: -T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1 u2)).(\lambda (H1: ((\forall -(f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f -m) (g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (le -(weight_map f u2) (weight_map g u1)))))))).(\lambda (k: K).(K_ind (\lambda -(k0: K).(\forall (t1: T).(\forall (t2: T).((subst0 (s k0 i) v t1 t2) \to -(((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: -nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S (s k0 i)) O v)) (g (s -k0 i))) \to (le (weight_map f t2) (weight_map g t1))))))) \to (\forall (f: -((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) -(g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (le (weight_map -f (THead k0 u2 t2)) (weight_map g (THead k0 u1 t1)))))))))))) (\lambda (b: -B).(B_ind (\lambda (b0: B).(\forall (t1: T).(\forall (t2: T).((subst0 (s -(Bind b0) i) v t1 t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat -\to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f -(lift (S (s (Bind b0) i)) O v)) (g (s (Bind b0) i))) \to (le (weight_map f -t2) (weight_map g t1))))))) \to (\forall (f: ((nat \to nat))).(\forall (g: -((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map -f (lift (S i) O v)) (g i)) \to (le (weight_map f (THead (Bind b0) u2 t2)) -(weight_map g (THead (Bind b0) u1 t1)))))))))))) (\lambda (t1: T).(\lambda -(t2: T).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H3: ((\forall (f: -((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) -(g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (le -(weight_map f t2) (weight_map g t1)))))))).(\lambda (f: ((nat \to -nat))).(\lambda (g: ((nat \to nat))).(\lambda (H4: ((\forall (m: nat).(le (f -m) (g m))))).(\lambda (H5: (lt (weight_map f (lift (S i) O v)) (g -i))).(le_n_S (plus (weight_map f u2) (weight_map (wadd f (S (weight_map f -u2))) t2)) (plus (weight_map g u1) (weight_map (wadd g (S (weight_map g u1))) -t1)) (le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f (S -(weight_map f u2))) t2) (weight_map (wadd g (S (weight_map g u1))) t1) (H1 f -g H4 H5) (H3 (wadd f (S (weight_map f u2))) (wadd g (S (weight_map g u1))) -(\lambda (m: nat).(wadd_le f g H4 (S (weight_map f u2)) (S (weight_map g u1)) -(le_n_S (weight_map f u2) (weight_map g u1) (H1 f g H4 H5)) m)) (eq_ind nat -(weight_map f (lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H5 -(weight_map (wadd f (S (weight_map f u2))) (lift (S (S i)) O v)) -(lift_weight_add_O (S (weight_map f u2)) v (S i) f))))))))))))) (\lambda (t1: -T).(\lambda (t2: T).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H3: -((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: -nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S -i))) \to (le (weight_map f t2) (weight_map g t1)))))))).(\lambda (f: ((nat -\to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H4: ((\forall (m: nat).(le -(f m) (g m))))).(\lambda (H5: (lt (weight_map f (lift (S i) O v)) (g -i))).(le_n_S (plus (weight_map f u2) (weight_map (wadd f O) t2)) (plus -(weight_map g u1) (weight_map (wadd g O) t1)) (le_plus_plus (weight_map f u2) -(weight_map g u1) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1) (H1 f -g H4 H5) (H3 (wadd f O) (wadd g O) (\lambda (m: nat).(wadd_le f g H4 O O -(le_n O) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda (n: -nat).(lt n (g i))) H5 (weight_map (wadd f O) (lift (S (S i)) O v)) -(lift_weight_add_O O v (S i) f))))))))))))) (\lambda (t1: T).(\lambda (t2: -T).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H3: ((\forall (f: ((nat \to -nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) -\to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (le (weight_map f -t2) (weight_map g t1)))))))).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat -\to nat))).(\lambda (H4: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H5: -(lt (weight_map f (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map f u2) -(weight_map (wadd f O) t2)) (plus (weight_map g u1) (weight_map (wadd g O) -t1)) (le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f O) -t2) (weight_map (wadd g O) t1) (H1 f g H4 H5) (H3 (wadd f O) (wadd g O) -(\lambda (m: nat).(wadd_le f g H4 O O (le_n O) m)) (eq_ind nat (weight_map f -(lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H5 (weight_map (wadd f O) -(lift (S (S i)) O v)) (lift_weight_add_O O v (S i) f))))))))))))) b)) -(\lambda (_: F).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (subst0 i v t1 -t2)).(\lambda (H3: ((\forall (f0: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (m: nat).(le (f0 m) (g m)))) \to ((lt (weight_map f0 (lift -(S i) O v)) (g i)) \to (le (weight_map f0 t2) (weight_map g -t1)))))))).(\lambda (f0: ((nat \to nat))).(\lambda (g: ((nat \to -nat))).(\lambda (H4: ((\forall (m: nat).(le (f0 m) (g m))))).(\lambda (H5: -(lt (weight_map f0 (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map f0 u2) -(weight_map f0 t2)) (plus (weight_map g u1) (weight_map g t1)) (le_plus_plus -(weight_map f0 u2) (weight_map g u1) (weight_map f0 t2) (weight_map g t1) (H1 -f0 g H4 H5) (H3 f0 g H4 H5)))))))))))) k)))))))) d u t z H))))). -(* COMMENTS -Initial nodes: 4101 -END *) - -theorem subst0_weight_lt: - \forall (u: T).(\forall (t: T).(\forall (z: T).(\forall (d: nat).((subst0 d -u t z) \to (\forall (f: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S -d) O u)) (g d)) \to (lt (weight_map f z) (weight_map g t)))))))))) -\def - \lambda (u: T).(\lambda (t: T).(\lambda (z: T).(\lambda (d: nat).(\lambda -(H: (subst0 d u t z)).(subst0_ind (\lambda (n: nat).(\lambda (t0: T).(\lambda -(t1: T).(\lambda (t2: T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S -n) O t0)) (g n)) \to (lt (weight_map f t2) (weight_map g t1)))))))))) -(\lambda (v: T).(\lambda (i: nat).(\lambda (f: ((nat \to nat))).(\lambda (g: -((nat \to nat))).(\lambda (_: ((\forall (m: nat).(le (f m) (g m))))).(\lambda -(H1: (lt (weight_map f (lift (S i) O v)) (g i))).H1)))))) (\lambda (v: -T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i -v u1 u2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S -i) O v)) (g i)) \to (lt (weight_map f u2) (weight_map g u1)))))))).(\lambda -(t0: T).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (f: ((nat \to -nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) -\to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (lt (weight_map f (THead -k0 u2 t0)) (weight_map g (THead k0 u1 t0)))))))) (\lambda (b: B).(B_ind -(\lambda (b0: B).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S -i) O v)) (g i)) \to (lt (weight_map f (THead (Bind b0) u2 t0)) (weight_map g -(THead (Bind b0) u1 t0)))))))) (\lambda (f: ((nat \to nat))).(\lambda (g: -((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g -m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g i))).(lt_n_S -(plus (weight_map f u2) (weight_map (wadd f (S (weight_map f u2))) t0)) (plus -(weight_map g u1) (weight_map (wadd g (S (weight_map g u1))) t0)) -(lt_le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f (S -(weight_map f u2))) t0) (weight_map (wadd g (S (weight_map g u1))) t0) (H1 f -g H2 H3) (weight_le t0 (wadd f (S (weight_map f u2))) (wadd g (S (weight_map -g u1))) (\lambda (n: nat).(wadd_lt f g H2 (S (weight_map f u2)) (S -(weight_map g u1)) (lt_n_S (weight_map f u2) (weight_map g u1) (H1 f g H2 -H3)) n))))))))) (\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to -nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3: (lt -(weight_map f (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map f u2) -(weight_map (wadd f O) t0)) (plus (weight_map g u1) (weight_map (wadd g O) -t0)) (lt_le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f -O) t0) (weight_map (wadd g O) t0) (H1 f g H2 H3) (weight_le t0 (wadd f O) -(wadd g O) (\lambda (n: nat).(le_S_n (wadd f O n) (wadd g O n) (le_n_S (wadd -f O n) (wadd g O n) (wadd_le f g H2 O O (le_n O) n))))))))))) (\lambda (f: -((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: -nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g -i))).(lt_n_S (plus (weight_map f u2) (weight_map (wadd f O) t0)) (plus -(weight_map g u1) (weight_map (wadd g O) t0)) (lt_le_plus_plus (weight_map f -u2) (weight_map g u1) (weight_map (wadd f O) t0) (weight_map (wadd g O) t0) -(H1 f g H2 H3) (weight_le t0 (wadd f O) (wadd g O) (\lambda (n: nat).(le_S_n -(wadd f O n) (wadd g O n) (le_n_S (wadd f O n) (wadd g O n) (wadd_le f g H2 O -O (le_n O) n))))))))))) b)) (\lambda (_: F).(\lambda (f0: ((nat \to -nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f0 -m) (g m))))).(\lambda (H3: (lt (weight_map f0 (lift (S i) O v)) (g -i))).(lt_n_S (plus (weight_map f0 u2) (weight_map f0 t0)) (plus (weight_map g -u1) (weight_map g t0)) (lt_le_plus_plus (weight_map f0 u2) (weight_map g u1) -(weight_map f0 t0) (weight_map g t0) (H1 f0 g H2 H3) (weight_le t0 f0 g -H2)))))))) k))))))))) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (v: -T).(\forall (t2: T).(\forall (t1: T).(\forall (i: nat).((subst0 (s k0 i) v t1 -t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S -(s k0 i)) O v)) (g (s k0 i))) \to (lt (weight_map f t2) (weight_map g -t1))))))) \to (\forall (u0: T).(\forall (f: ((nat \to nat))).(\forall (g: -((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map -f (lift (S i) O v)) (g i)) \to (lt (weight_map f (THead k0 u0 t2)) -(weight_map g (THead k0 u0 t1))))))))))))))) (\lambda (b: B).(B_ind (\lambda -(b0: B).(\forall (v: T).(\forall (t2: T).(\forall (t1: T).(\forall (i: -nat).((subst0 (s (Bind b0) i) v t1 t2) \to (((\forall (f: ((nat \to -nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) -\to ((lt (weight_map f (lift (S (s (Bind b0) i)) O v)) (g (s (Bind b0) i))) -\to (lt (weight_map f t2) (weight_map g t1))))))) \to (\forall (u0: -T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: -nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to -(lt (weight_map f (THead (Bind b0) u0 t2)) (weight_map g (THead (Bind b0) u0 -t1))))))))))))))) (\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda -(i: nat).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H1: ((\forall (f: -((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) -(g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (lt -(weight_map f t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f: -((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: -nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g -i))).(lt_n_S (plus (weight_map f u0) (weight_map (wadd f (S (weight_map f -u0))) t2)) (plus (weight_map g u0) (weight_map (wadd g (S (weight_map g u0))) -t1)) (le_lt_plus_plus (weight_map f u0) (weight_map g u0) (weight_map (wadd f -(S (weight_map f u0))) t2) (weight_map (wadd g (S (weight_map g u0))) t1) -(weight_le u0 f g H2) (H1 (wadd f (S (weight_map f u0))) (wadd g (S -(weight_map g u0))) (\lambda (m: nat).(wadd_le f g H2 (S (weight_map f u0)) -(S (weight_map g u0)) (le_n_S (weight_map f u0) (weight_map g u0) (weight_le -u0 f g H2)) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda (n: -nat).(lt n (g i))) H3 (weight_map (wadd f (S (weight_map f u0))) (lift (S (S -i)) O v)) (lift_weight_add_O (S (weight_map f u0)) v (S i) f)))))))))))))))) -(\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda (i: nat).(\lambda -(_: (subst0 (S i) v t1 t2)).(\lambda (H1: ((\forall (f: ((nat \to -nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) -\to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (lt (weight_map f -t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f: ((nat \to -nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f -m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g -i))).(lt_n_S (plus (weight_map f u0) (weight_map (wadd f O) t2)) (plus -(weight_map g u0) (weight_map (wadd g O) t1)) (le_lt_plus_plus (weight_map f -u0) (weight_map g u0) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1) -(weight_le u0 f g H2) (H1 (wadd f O) (wadd g O) (\lambda (m: nat).(wadd_le f -g H2 O O (le_n O) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda -(n: nat).(lt n (g i))) H3 (weight_map (wadd f O) (lift (S (S i)) O v)) -(lift_weight_add_O O v (S i) f)))))))))))))))) (\lambda (v: T).(\lambda (t2: -T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (_: (subst0 (S i) v t1 -t2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S -(S i)) O v)) (g (S i))) \to (lt (weight_map f t2) (weight_map g -t1)))))))).(\lambda (u0: T).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat -\to nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3: -(lt (weight_map f (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map f u0) -(weight_map (wadd f O) t2)) (plus (weight_map g u0) (weight_map (wadd g O) -t1)) (le_lt_plus_plus (weight_map f u0) (weight_map g u0) (weight_map (wadd f -O) t2) (weight_map (wadd g O) t1) (weight_le u0 f g H2) (H1 (wadd f O) (wadd -g O) (\lambda (m: nat).(wadd_le f g H2 O O (le_n O) m)) (eq_ind nat -(weight_map f (lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H3 -(weight_map (wadd f O) (lift (S (S i)) O v)) (lift_weight_add_O O v (S i) -f)))))))))))))))) b)) (\lambda (_: F).(\lambda (v: T).(\lambda (t2: -T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v t1 -t2)).(\lambda (H1: ((\forall (f0: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (m: nat).(le (f0 m) (g m)))) \to ((lt (weight_map f0 (lift -(S i) O v)) (g i)) \to (lt (weight_map f0 t2) (weight_map g -t1)))))))).(\lambda (u0: T).(\lambda (f0: ((nat \to nat))).(\lambda (g: ((nat -\to nat))).(\lambda (H2: ((\forall (m: nat).(le (f0 m) (g m))))).(\lambda -(H3: (lt (weight_map f0 (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map -f0 u0) (weight_map f0 t2)) (plus (weight_map g u0) (weight_map g t1)) -(le_lt_plus_plus (weight_map f0 u0) (weight_map g u0) (weight_map f0 t2) -(weight_map g t1) (weight_le u0 f0 g H2) (H1 f0 g H2 H3))))))))))))))) k)) -(\lambda (v: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i: nat).(\lambda -(_: (subst0 i v u1 u2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall -(g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt -(weight_map f (lift (S i) O v)) (g i)) \to (lt (weight_map f u2) (weight_map -g u1)))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t1: -T).(\forall (t2: T).((subst0 (s k0 i) v t1 t2) \to (((\forall (f: ((nat \to -nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) -\to ((lt (weight_map f (lift (S (s k0 i)) O v)) (g (s k0 i))) \to (lt -(weight_map f t2) (weight_map g t1))))))) \to (\forall (f: ((nat \to -nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) -\to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (lt (weight_map f (THead -k0 u2 t2)) (weight_map g (THead k0 u1 t1)))))))))))) (\lambda (b: B).(B_ind -(\lambda (b0: B).(\forall (t1: T).(\forall (t2: T).((subst0 (s (Bind b0) i) v -t1 t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S -(s (Bind b0) i)) O v)) (g (s (Bind b0) i))) \to (lt (weight_map f t2) -(weight_map g t1))))))) \to (\forall (f: ((nat \to nat))).(\forall (g: ((nat -\to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f -(lift (S i) O v)) (g i)) \to (lt (weight_map f (THead (Bind b0) u2 t2)) -(weight_map g (THead (Bind b0) u1 t1)))))))))))) (\lambda (t1: T).(\lambda -(t2: T).(\lambda (H2: (subst0 (S i) v t1 t2)).(\lambda (_: ((\forall (f: -((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) -(g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (lt -(weight_map f t2) (weight_map g t1)))))))).(\lambda (f: ((nat \to -nat))).(\lambda (g: ((nat \to nat))).(\lambda (H4: ((\forall (m: nat).(le (f -m) (g m))))).(\lambda (H5: (lt (weight_map f (lift (S i) O v)) (g -i))).(lt_n_S (plus (weight_map f u2) (weight_map (wadd f (S (weight_map f -u2))) t2)) (plus (weight_map g u1) (weight_map (wadd g (S (weight_map g u1))) -t1)) (lt_le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f -(S (weight_map f u2))) t2) (weight_map (wadd g (S (weight_map g u1))) t1) (H1 -f g H4 H5) (subst0_weight_le v t1 t2 (S i) H2 (wadd f (S (weight_map f u2))) -(wadd g (S (weight_map g u1))) (\lambda (m: nat).(wadd_lt f g H4 (S -(weight_map f u2)) (S (weight_map g u1)) (lt_n_S (weight_map f u2) -(weight_map g u1) (H1 f g H4 H5)) m)) (eq_ind nat (weight_map f (lift (S i) O -v)) (\lambda (n: nat).(lt n (g i))) H5 (weight_map (wadd f (S (weight_map f -u2))) (lift (S (S i)) O v)) (lift_weight_add_O (S (weight_map f u2)) v (S i) -f))))))))))))) (\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (subst0 (S i) v -t1 t2)).(\lambda (H3: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S -(S i)) O v)) (g (S i))) \to (lt (weight_map f t2) (weight_map g -t1)))))))).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to -nat))).(\lambda (H4: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H5: (lt -(weight_map f (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map f u2) -(weight_map (wadd f O) t2)) (plus (weight_map g u1) (weight_map (wadd g O) -t1)) (lt_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f O) -t2) (weight_map (wadd g O) t1) (H1 f g H4 H5) (H3 (wadd f O) (wadd g O) -(\lambda (m: nat).(le_S_n (wadd f O m) (wadd g O m) (le_n_S (wadd f O m) -(wadd g O m) (wadd_le f g H4 O O (le_n O) m)))) (eq_ind nat (weight_map f -(lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H5 (weight_map (wadd f O) -(lift (S (S i)) O v)) (lift_weight_add_O O v (S i) f))))))))))))) (\lambda -(t1: T).(\lambda (t2: T).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H3: -((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: -nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S -i))) \to (lt (weight_map f t2) (weight_map g t1)))))))).(\lambda (f: ((nat -\to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H4: ((\forall (m: nat).(le -(f m) (g m))))).(\lambda (H5: (lt (weight_map f (lift (S i) O v)) (g -i))).(lt_n_S (plus (weight_map f u2) (weight_map (wadd f O) t2)) (plus -(weight_map g u1) (weight_map (wadd g O) t1)) (lt_plus_plus (weight_map f u2) -(weight_map g u1) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1) (H1 f -g H4 H5) (H3 (wadd f O) (wadd g O) (\lambda (m: nat).(le_S_n (wadd f O m) -(wadd g O m) (le_n_S (wadd f O m) (wadd g O m) (wadd_le f g H4 O O (le_n O) -m)))) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda (n: nat).(lt n (g -i))) H5 (weight_map (wadd f O) (lift (S (S i)) O v)) (lift_weight_add_O O v -(S i) f))))))))))))) b)) (\lambda (_: F).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (_: (subst0 i v t1 t2)).(\lambda (H3: ((\forall (f0: ((nat \to -nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f0 m) (g m)))) -\to ((lt (weight_map f0 (lift (S i) O v)) (g i)) \to (lt (weight_map f0 t2) -(weight_map g t1)))))))).(\lambda (f0: ((nat \to nat))).(\lambda (g: ((nat -\to nat))).(\lambda (H4: ((\forall (m: nat).(le (f0 m) (g m))))).(\lambda -(H5: (lt (weight_map f0 (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map -f0 u2) (weight_map f0 t2)) (plus (weight_map g u1) (weight_map g t1)) -(lt_plus_plus (weight_map f0 u2) (weight_map g u1) (weight_map f0 t2) -(weight_map g t1) (H1 f0 g H4 H5) (H3 f0 g H4 H5)))))))))))) k)))))))) d u t -z H))))). -(* COMMENTS -Initial nodes: 4207 -END *) - -theorem subst0_tlt_head: - \forall (u: T).(\forall (t: T).(\forall (z: T).((subst0 O u t z) \to (tlt -(THead (Bind Abbr) u z) (THead (Bind Abbr) u t))))) -\def - \lambda (u: T).(\lambda (t: T).(\lambda (z: T).(\lambda (H: (subst0 O u t -z)).(lt_n_S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd -(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) z)) (plus -(weight_map (\lambda (_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) (S -(weight_map (\lambda (_: nat).O) u))) t)) (le_lt_plus_plus (weight_map -(\lambda (_: nat).O) u) (weight_map (\lambda (_: nat).O) u) (weight_map (wadd -(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) z) (weight_map -(wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t) (le_n -(weight_map (\lambda (_: nat).O) u)) (subst0_weight_lt u t z O H (wadd -(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) (wadd (\lambda -(_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) (\lambda (m: nat).(le_n -(wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u)) m))) -(eq_ind nat (weight_map (\lambda (_: nat).O) (lift O O u)) (\lambda (n: -nat).(lt n (S (weight_map (\lambda (_: nat).O) u)))) (eq_ind_r T u (\lambda -(t0: T).(lt (weight_map (\lambda (_: nat).O) t0) (S (weight_map (\lambda (_: -nat).O) u)))) (le_n (S (weight_map (\lambda (_: nat).O) u))) (lift O O u) -(lift_r u O)) (weight_map (wadd (\lambda (_: nat).O) (S (weight_map (\lambda -(_: nat).O) u))) (lift (S O) O u)) (lift_weight_add_O (S (weight_map (\lambda -(_: nat).O) u)) u O (\lambda (_: nat).O))))))))). -(* COMMENTS -Initial nodes: 347 -END *) - -theorem subst0_tlt: - \forall (u: T).(\forall (t: T).(\forall (z: T).((subst0 O u t z) \to (tlt z -(THead (Bind Abbr) u t))))) -\def - \lambda (u: T).(\lambda (t: T).(\lambda (z: T).(\lambda (H: (subst0 O u t -z)).(tlt_trans (THead (Bind Abbr) u z) z (THead (Bind Abbr) u t) (tlt_head_dx -(Bind Abbr) u z) (subst0_tlt_head u t z H))))). -(* COMMENTS -Initial nodes: 59 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/subst1/defs.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/subst1/defs.ma deleted file mode 100644 index 6a51bcfb5..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/subst1/defs.ma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/subst0/defs.ma". - -inductive subst1 (i: nat) (v: T) (t1: T): T \to Prop \def -| subst1_refl: subst1 i v t1 t1 -| subst1_single: \forall (t2: T).((subst0 i v t1 t2) \to (subst1 i v t1 t2)). - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/subst1/fwd.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/subst1/fwd.ma deleted file mode 100644 index a2bc1edd6..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/subst1/fwd.ma +++ /dev/null @@ -1,182 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/subst1/defs.ma". - -include "Basic-1/subst0/props.ma". - -theorem subst1_gen_sort: - \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst1 -i v (TSort n) x) \to (eq T x (TSort n)))))) -\def - \lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda -(H: (subst1 i v (TSort n) x)).(subst1_ind i v (TSort n) (\lambda (t: T).(eq T -t (TSort n))) (refl_equal T (TSort n)) (\lambda (t2: T).(\lambda (H0: (subst0 -i v (TSort n) t2)).(subst0_gen_sort v t2 i n H0 (eq T t2 (TSort n))))) x -H))))). -(* COMMENTS -Initial nodes: 89 -END *) - -theorem subst1_gen_lref: - \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst1 -i v (TLRef n) x) \to (or (eq T x (TLRef n)) (land (eq nat n i) (eq T x (lift -(S n) O v)))))))) -\def - \lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda -(H: (subst1 i v (TLRef n) x)).(subst1_ind i v (TLRef n) (\lambda (t: T).(or -(eq T t (TLRef n)) (land (eq nat n i) (eq T t (lift (S n) O v))))) (or_introl -(eq T (TLRef n) (TLRef n)) (land (eq nat n i) (eq T (TLRef n) (lift (S n) O -v))) (refl_equal T (TLRef n))) (\lambda (t2: T).(\lambda (H0: (subst0 i v -(TLRef n) t2)).(land_ind (eq nat n i) (eq T t2 (lift (S n) O v)) (or (eq T t2 -(TLRef n)) (land (eq nat n i) (eq T t2 (lift (S n) O v)))) (\lambda (H1: (eq -nat n i)).(\lambda (H2: (eq T t2 (lift (S n) O v))).(or_intror (eq T t2 -(TLRef n)) (land (eq nat n i) (eq T t2 (lift (S n) O v))) (conj (eq nat n i) -(eq T t2 (lift (S n) O v)) H1 H2)))) (subst0_gen_lref v t2 i n H0)))) x -H))))). -(* COMMENTS -Initial nodes: 305 -END *) - -theorem subst1_gen_head: - \forall (k: K).(\forall (v: T).(\forall (u1: T).(\forall (t1: T).(\forall -(x: T).(\forall (i: nat).((subst1 i v (THead k u1 t1) x) \to (ex3_2 T T -(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (t2: -T).(subst1 (s k i) v t1 t2)))))))))) -\def - \lambda (k: K).(\lambda (v: T).(\lambda (u1: T).(\lambda (t1: T).(\lambda -(x: T).(\lambda (i: nat).(\lambda (H: (subst1 i v (THead k u1 t1) -x)).(subst1_ind i v (THead k u1 t1) (\lambda (t: T).(ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T t (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (t2: T).(subst1 (s k i) v t1 -t2))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead k u1 -t1) (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2))) -(\lambda (_: T).(\lambda (t2: T).(subst1 (s k i) v t1 t2))) u1 t1 (refl_equal -T (THead k u1 t1)) (subst1_refl i v u1) (subst1_refl (s k i) v t1)) (\lambda -(t2: T).(\lambda (H0: (subst0 i v (THead k u1 t1) t2)).(or3_ind (ex2 T -(\lambda (u2: T).(eq T t2 (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 -u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: -T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: -T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v -u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3)))) (ex3_2 -T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) (\lambda -(u2: T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (t3: -T).(subst1 (s k i) v t1 t3)))) (\lambda (H1: (ex2 T (\lambda (u2: T).(eq T t2 -(THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2)))).(ex2_ind T (\lambda -(u2: T).(eq T t2 (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2)) -(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) -(\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(subst1 (s k i) v t1 t3)))) (\lambda (x0: T).(\lambda -(H2: (eq T t2 (THead k x0 t1))).(\lambda (H3: (subst0 i v u1 -x0)).(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 -t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(subst1 (s k i) v t1 t3))) x0 t1 H2 (subst1_single i v u1 -x0 H3) (subst1_refl (s k i) v t1))))) H1)) (\lambda (H1: (ex2 T (\lambda (t3: -T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 -t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: -T).(subst0 (s k i) v t1 t3)) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq -T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2))) -(\lambda (_: T).(\lambda (t3: T).(subst1 (s k i) v t1 t3)))) (\lambda (x0: -T).(\lambda (H2: (eq T t2 (THead k u1 x0))).(\lambda (H3: (subst0 (s k i) v -t1 x0)).(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_: -T).(\lambda (t3: T).(subst1 (s k i) v t1 t3))) u1 x0 H2 (subst1_refl i v u1) -(subst1_single (s k i) v t1 x0 H3))))) H1)) (\lambda (H1: (ex3_2 T T (\lambda -(u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t3: -T).(subst0 (s k i) v t1 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: -T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v -u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T -T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: -T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (t3: -T).(subst1 (s k i) v t1 t3)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda -(H2: (eq T t2 (THead k x0 x1))).(\lambda (H3: (subst0 i v u1 x0)).(\lambda -(H4: (subst0 (s k i) v t1 x1)).(ex3_2_intro T T (\lambda (u2: T).(\lambda -(t3: T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1 -i v u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst1 (s k i) v t1 t3))) x0 -x1 H2 (subst1_single i v u1 x0 H3) (subst1_single (s k i) v t1 x1 H4))))))) -H1)) (subst0_gen_head k v u1 t1 t2 i H0)))) x H))))))). -(* COMMENTS -Initial nodes: 1199 -END *) - -theorem subst1_gen_lift_lt: - \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall -(h: nat).(\forall (d: nat).((subst1 i (lift h d u) (lift h (S (plus i d)) t1) -x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda -(t2: T).(subst1 i u t1 t2))))))))) -\def - \lambda (u: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (i: nat).(\lambda -(h: nat).(\lambda (d: nat).(\lambda (H: (subst1 i (lift h d u) (lift h (S -(plus i d)) t1) x)).(subst1_ind i (lift h d u) (lift h (S (plus i d)) t1) -(\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h (S (plus i d)) t2))) -(\lambda (t2: T).(subst1 i u t1 t2)))) (ex_intro2 T (\lambda (t2: T).(eq T -(lift h (S (plus i d)) t1) (lift h (S (plus i d)) t2))) (\lambda (t2: -T).(subst1 i u t1 t2)) t1 (refl_equal T (lift h (S (plus i d)) t1)) -(subst1_refl i u t1)) (\lambda (t2: T).(\lambda (H0: (subst0 i (lift h d u) -(lift h (S (plus i d)) t1) t2)).(ex2_ind T (\lambda (t3: T).(eq T t2 (lift h -(S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u t1 t3)) (ex2 T (\lambda -(t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst1 i u t1 -t3))) (\lambda (x0: T).(\lambda (H1: (eq T t2 (lift h (S (plus i d)) -x0))).(\lambda (H2: (subst0 i u t1 x0)).(ex_intro2 T (\lambda (t3: T).(eq T -t2 (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst1 i u t1 t3)) x0 H1 -(subst1_single i u t1 x0 H2))))) (subst0_gen_lift_lt u t1 t2 i h d H0)))) x -H))))))). -(* COMMENTS -Initial nodes: 395 -END *) - -theorem subst1_gen_lift_eq: - \forall (t: T).(\forall (u: T).(\forall (x: T).(\forall (h: nat).(\forall -(d: nat).(\forall (i: nat).((le d i) \to ((lt i (plus d h)) \to ((subst1 i u -(lift h d t) x) \to (eq T x (lift h d t)))))))))) -\def - \lambda (t: T).(\lambda (u: T).(\lambda (x: T).(\lambda (h: nat).(\lambda -(d: nat).(\lambda (i: nat).(\lambda (H: (le d i)).(\lambda (H0: (lt i (plus d -h))).(\lambda (H1: (subst1 i u (lift h d t) x)).(subst1_ind i u (lift h d t) -(\lambda (t0: T).(eq T t0 (lift h d t))) (refl_equal T (lift h d t)) (\lambda -(t2: T).(\lambda (H2: (subst0 i u (lift h d t) t2)).(subst0_gen_lift_false t -u t2 h d i H H0 H2 (eq T t2 (lift h d t))))) x H1))))))))). -(* COMMENTS -Initial nodes: 141 -END *) - -theorem subst1_gen_lift_ge: - \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall -(h: nat).(\forall (d: nat).((subst1 i u (lift h d t1) x) \to ((le (plus d h) -i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: -T).(subst1 (minus i h) u t1 t2)))))))))) -\def - \lambda (u: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (i: nat).(\lambda -(h: nat).(\lambda (d: nat).(\lambda (H: (subst1 i u (lift h d t1) -x)).(\lambda (H0: (le (plus d h) i)).(subst1_ind i u (lift h d t1) (\lambda -(t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h d t2))) (\lambda (t2: -T).(subst1 (minus i h) u t1 t2)))) (ex_intro2 T (\lambda (t2: T).(eq T (lift -h d t1) (lift h d t2))) (\lambda (t2: T).(subst1 (minus i h) u t1 t2)) t1 -(refl_equal T (lift h d t1)) (subst1_refl (minus i h) u t1)) (\lambda (t2: -T).(\lambda (H1: (subst0 i u (lift h d t1) t2)).(ex2_ind T (\lambda (t3: -T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u t1 t3)) -(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst1 -(minus i h) u t1 t3))) (\lambda (x0: T).(\lambda (H2: (eq T t2 (lift h d -x0))).(\lambda (H3: (subst0 (minus i h) u t1 x0)).(ex_intro2 T (\lambda (t3: -T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst1 (minus i h) u t1 t3)) x0 -H2 (subst1_single (minus i h) u t1 x0 H3))))) (subst0_gen_lift_ge u t1 t2 i h -d H1 H0)))) x H)))))))). -(* COMMENTS -Initial nodes: 355 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/subst1/props.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/subst1/props.ma deleted file mode 100644 index ac8dea954..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/subst1/props.ma +++ /dev/null @@ -1,179 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/subst1/defs.ma". - -include "Basic-1/subst0/props.ma". - -theorem subst1_head: - \forall (v: T).(\forall (u1: T).(\forall (u2: T).(\forall (i: nat).((subst1 -i v u1 u2) \to (\forall (k: K).(\forall (t1: T).(\forall (t2: T).((subst1 (s -k i) v t1 t2) \to (subst1 i v (THead k u1 t1) (THead k u2 t2)))))))))) -\def - \lambda (v: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i: nat).(\lambda -(H: (subst1 i v u1 u2)).(subst1_ind i v u1 (\lambda (t: T).(\forall (k: -K).(\forall (t1: T).(\forall (t2: T).((subst1 (s k i) v t1 t2) \to (subst1 i -v (THead k u1 t1) (THead k t t2))))))) (\lambda (k: K).(\lambda (t1: -T).(\lambda (t2: T).(\lambda (H0: (subst1 (s k i) v t1 t2)).(subst1_ind (s k -i) v t1 (\lambda (t: T).(subst1 i v (THead k u1 t1) (THead k u1 t))) -(subst1_refl i v (THead k u1 t1)) (\lambda (t3: T).(\lambda (H1: (subst0 (s k -i) v t1 t3)).(subst1_single i v (THead k u1 t1) (THead k u1 t3) (subst0_snd k -v t3 t1 i H1 u1)))) t2 H0))))) (\lambda (t2: T).(\lambda (H0: (subst0 i v u1 -t2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t0: T).(\lambda (H1: (subst1 -(s k i) v t1 t0)).(subst1_ind (s k i) v t1 (\lambda (t: T).(subst1 i v (THead -k u1 t1) (THead k t2 t))) (subst1_single i v (THead k u1 t1) (THead k t2 t1) -(subst0_fst v t2 u1 i H0 t1 k)) (\lambda (t3: T).(\lambda (H2: (subst0 (s k -i) v t1 t3)).(subst1_single i v (THead k u1 t1) (THead k t2 t3) (subst0_both -v u1 t2 i H0 k t1 t3 H2)))) t0 H1))))))) u2 H))))). -(* COMMENTS -Initial nodes: 369 -END *) - -theorem subst1_lift_lt: - \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).((subst1 -i u t1 t2) \to (\forall (d: nat).((lt i d) \to (\forall (h: nat).(subst1 i -(lift h (minus d (S i)) u) (lift h d t1) (lift h d t2))))))))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H: (subst1 i u t1 t2)).(subst1_ind i u t1 (\lambda (t: T).(\forall (d: -nat).((lt i d) \to (\forall (h: nat).(subst1 i (lift h (minus d (S i)) u) -(lift h d t1) (lift h d t)))))) (\lambda (d: nat).(\lambda (_: (lt i -d)).(\lambda (h: nat).(subst1_refl i (lift h (minus d (S i)) u) (lift h d -t1))))) (\lambda (t3: T).(\lambda (H0: (subst0 i u t1 t3)).(\lambda (d: -nat).(\lambda (H1: (lt i d)).(\lambda (h: nat).(subst1_single i (lift h -(minus d (S i)) u) (lift h d t1) (lift h d t3) (subst0_lift_lt t1 t3 u i H0 d -H1 h))))))) t2 H))))). -(* COMMENTS -Initial nodes: 185 -END *) - -theorem subst1_lift_ge: - \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).(\forall -(h: nat).((subst1 i u t1 t2) \to (\forall (d: nat).((le d i) \to (subst1 -(plus i h) u (lift h d t1) (lift h d t2))))))))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda -(h: nat).(\lambda (H: (subst1 i u t1 t2)).(subst1_ind i u t1 (\lambda (t: -T).(\forall (d: nat).((le d i) \to (subst1 (plus i h) u (lift h d t1) (lift h -d t))))) (\lambda (d: nat).(\lambda (_: (le d i)).(subst1_refl (plus i h) u -(lift h d t1)))) (\lambda (t3: T).(\lambda (H0: (subst0 i u t1 t3)).(\lambda -(d: nat).(\lambda (H1: (le d i)).(subst1_single (plus i h) u (lift h d t1) -(lift h d t3) (subst0_lift_ge t1 t3 u i h H0 d H1)))))) t2 H)))))). -(* COMMENTS -Initial nodes: 157 -END *) - -theorem subst1_ex: - \forall (u: T).(\forall (t1: T).(\forall (d: nat).(ex T (\lambda (t2: -T).(subst1 d u t1 (lift (S O) d t2)))))) -\def - \lambda (u: T).(\lambda (t1: T).(T_ind (\lambda (t: T).(\forall (d: nat).(ex -T (\lambda (t2: T).(subst1 d u t (lift (S O) d t2)))))) (\lambda (n: -nat).(\lambda (d: nat).(ex_intro T (\lambda (t2: T).(subst1 d u (TSort n) -(lift (S O) d t2))) (TSort n) (eq_ind_r T (TSort n) (\lambda (t: T).(subst1 d -u (TSort n) t)) (subst1_refl d u (TSort n)) (lift (S O) d (TSort n)) -(lift_sort n (S O) d))))) (\lambda (n: nat).(\lambda (d: nat).(lt_eq_gt_e n d -(ex T (\lambda (t2: T).(subst1 d u (TLRef n) (lift (S O) d t2)))) (\lambda -(H: (lt n d)).(ex_intro T (\lambda (t2: T).(subst1 d u (TLRef n) (lift (S O) -d t2))) (TLRef n) (eq_ind_r T (TLRef n) (\lambda (t: T).(subst1 d u (TLRef n) -t)) (subst1_refl d u (TLRef n)) (lift (S O) d (TLRef n)) (lift_lref_lt n (S -O) d H)))) (\lambda (H: (eq nat n d)).(eq_ind nat n (\lambda (n0: nat).(ex T -(\lambda (t2: T).(subst1 n0 u (TLRef n) (lift (S O) n0 t2))))) (ex_intro T -(\lambda (t2: T).(subst1 n u (TLRef n) (lift (S O) n t2))) (lift n O u) -(eq_ind_r T (lift (plus (S O) n) O u) (\lambda (t: T).(subst1 n u (TLRef n) -t)) (subst1_single n u (TLRef n) (lift (S n) O u) (subst0_lref u n)) (lift (S -O) n (lift n O u)) (lift_free u n (S O) O n (le_n (plus O n)) (le_O_n n)))) d -H)) (\lambda (H: (lt d n)).(ex_intro T (\lambda (t2: T).(subst1 d u (TLRef n) -(lift (S O) d t2))) (TLRef (pred n)) (eq_ind_r T (TLRef n) (\lambda (t: -T).(subst1 d u (TLRef n) t)) (subst1_refl d u (TLRef n)) (lift (S O) d (TLRef -(pred n))) (lift_lref_gt d n H))))))) (\lambda (k: K).(\lambda (t: -T).(\lambda (H: ((\forall (d: nat).(ex T (\lambda (t2: T).(subst1 d u t (lift -(S O) d t2))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (d: nat).(ex T -(\lambda (t2: T).(subst1 d u t0 (lift (S O) d t2))))))).(\lambda (d: -nat).(let H_x \def (H d) in (let H1 \def H_x in (ex_ind T (\lambda (t2: -T).(subst1 d u t (lift (S O) d t2))) (ex T (\lambda (t2: T).(subst1 d u -(THead k t t0) (lift (S O) d t2)))) (\lambda (x: T).(\lambda (H2: (subst1 d u -t (lift (S O) d x))).(let H_x0 \def (H0 (s k d)) in (let H3 \def H_x0 in -(ex_ind T (\lambda (t2: T).(subst1 (s k d) u t0 (lift (S O) (s k d) t2))) (ex -T (\lambda (t2: T).(subst1 d u (THead k t t0) (lift (S O) d t2)))) (\lambda -(x0: T).(\lambda (H4: (subst1 (s k d) u t0 (lift (S O) (s k d) -x0))).(ex_intro T (\lambda (t2: T).(subst1 d u (THead k t t0) (lift (S O) d -t2))) (THead k x x0) (eq_ind_r T (THead k (lift (S O) d x) (lift (S O) (s k -d) x0)) (\lambda (t2: T).(subst1 d u (THead k t t0) t2)) (subst1_head u t -(lift (S O) d x) d H2 k t0 (lift (S O) (s k d) x0) H4) (lift (S O) d (THead k -x x0)) (lift_head k x x0 (S O) d))))) H3))))) H1))))))))) t1)). -(* COMMENTS -Initial nodes: 925 -END *) - -theorem subst1_lift_S: - \forall (u: T).(\forall (i: nat).(\forall (h: nat).((le h i) \to (subst1 i -(TLRef h) (lift (S h) (S i) u) (lift (S h) i u))))) -\def - \lambda (u: T).(T_ind (\lambda (t: T).(\forall (i: nat).(\forall (h: -nat).((le h i) \to (subst1 i (TLRef h) (lift (S h) (S i) t) (lift (S h) i -t)))))) (\lambda (n: nat).(\lambda (i: nat).(\lambda (h: nat).(\lambda (_: -(le h i)).(eq_ind_r T (TSort n) (\lambda (t: T).(subst1 i (TLRef h) t (lift -(S h) i (TSort n)))) (eq_ind_r T (TSort n) (\lambda (t: T).(subst1 i (TLRef -h) (TSort n) t)) (subst1_refl i (TLRef h) (TSort n)) (lift (S h) i (TSort n)) -(lift_sort n (S h) i)) (lift (S h) (S i) (TSort n)) (lift_sort n (S h) (S -i))))))) (\lambda (n: nat).(\lambda (i: nat).(\lambda (h: nat).(\lambda (H: -(le h i)).(lt_eq_gt_e n i (subst1 i (TLRef h) (lift (S h) (S i) (TLRef n)) -(lift (S h) i (TLRef n))) (\lambda (H0: (lt n i)).(eq_ind_r T (TLRef n) -(\lambda (t: T).(subst1 i (TLRef h) t (lift (S h) i (TLRef n)))) (eq_ind_r T -(TLRef n) (\lambda (t: T).(subst1 i (TLRef h) (TLRef n) t)) (subst1_refl i -(TLRef h) (TLRef n)) (lift (S h) i (TLRef n)) (lift_lref_lt n (S h) i H0)) -(lift (S h) (S i) (TLRef n)) (lift_lref_lt n (S h) (S i) (le_S (S n) i H0)))) -(\lambda (H0: (eq nat n i)).(let H1 \def (eq_ind_r nat i (\lambda (n0: -nat).(le h n0)) H n H0) in (eq_ind nat n (\lambda (n0: nat).(subst1 n0 (TLRef -h) (lift (S h) (S n0) (TLRef n)) (lift (S h) n0 (TLRef n)))) (eq_ind_r T -(TLRef n) (\lambda (t: T).(subst1 n (TLRef h) t (lift (S h) n (TLRef n)))) -(eq_ind_r T (TLRef (plus n (S h))) (\lambda (t: T).(subst1 n (TLRef h) (TLRef -n) t)) (eq_ind nat (S (plus n h)) (\lambda (n0: nat).(subst1 n (TLRef h) -(TLRef n) (TLRef n0))) (eq_ind_r nat (plus h n) (\lambda (n0: nat).(subst1 n -(TLRef h) (TLRef n) (TLRef (S n0)))) (eq_ind nat (plus h (S n)) (\lambda (n0: -nat).(subst1 n (TLRef h) (TLRef n) (TLRef n0))) (eq_ind T (lift (S n) O -(TLRef h)) (\lambda (t: T).(subst1 n (TLRef h) (TLRef n) t)) (subst1_single n -(TLRef h) (TLRef n) (lift (S n) O (TLRef h)) (subst0_lref (TLRef h) n)) -(TLRef (plus h (S n))) (lift_lref_ge h (S n) O (le_O_n h))) (S (plus h n)) -(sym_eq nat (S (plus h n)) (plus h (S n)) (plus_n_Sm h n))) (plus n h) -(plus_sym n h)) (plus n (S h)) (plus_n_Sm n h)) (lift (S h) n (TLRef n)) -(lift_lref_ge n (S h) n (le_n n))) (lift (S h) (S n) (TLRef n)) (lift_lref_lt -n (S h) (S n) (le_n (S n)))) i H0))) (\lambda (H0: (lt i n)).(eq_ind_r T -(TLRef (plus n (S h))) (\lambda (t: T).(subst1 i (TLRef h) t (lift (S h) i -(TLRef n)))) (eq_ind_r T (TLRef (plus n (S h))) (\lambda (t: T).(subst1 i -(TLRef h) (TLRef (plus n (S h))) t)) (subst1_refl i (TLRef h) (TLRef (plus n -(S h)))) (lift (S h) i (TLRef n)) (lift_lref_ge n (S h) i (le_S_n i n (le_S -(S i) n H0)))) (lift (S h) (S i) (TLRef n)) (lift_lref_ge n (S h) (S i) -H0)))))))) (\lambda (k: K).(\lambda (t: T).(\lambda (H: ((\forall (i: -nat).(\forall (h: nat).((le h i) \to (subst1 i (TLRef h) (lift (S h) (S i) t) -(lift (S h) i t))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (i: -nat).(\forall (h: nat).((le h i) \to (subst1 i (TLRef h) (lift (S h) (S i) -t0) (lift (S h) i t0))))))).(\lambda (i: nat).(\lambda (h: nat).(\lambda (H1: -(le h i)).(eq_ind_r T (THead k (lift (S h) (S i) t) (lift (S h) (s k (S i)) -t0)) (\lambda (t1: T).(subst1 i (TLRef h) t1 (lift (S h) i (THead k t t0)))) -(eq_ind_r T (THead k (lift (S h) i t) (lift (S h) (s k i) t0)) (\lambda (t1: -T).(subst1 i (TLRef h) (THead k (lift (S h) (S i) t) (lift (S h) (s k (S i)) -t0)) t1)) (subst1_head (TLRef h) (lift (S h) (S i) t) (lift (S h) i t) i (H i -h H1) k (lift (S h) (s k (S i)) t0) (lift (S h) (s k i) t0) (eq_ind_r nat (S -(s k i)) (\lambda (n: nat).(subst1 (s k i) (TLRef h) (lift (S h) n t0) (lift -(S h) (s k i) t0))) (H0 (s k i) h (le_trans h i (s k i) H1 (s_inc k i))) (s k -(S i)) (s_S k i))) (lift (S h) i (THead k t t0)) (lift_head k t t0 (S h) i)) -(lift (S h) (S i) (THead k t t0)) (lift_head k t t0 (S h) (S i))))))))))) u). -(* COMMENTS -Initial nodes: 1421 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/subst1/subst1.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/subst1/subst1.ma deleted file mode 100644 index e6dae0dc9..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/subst1/subst1.ma +++ /dev/null @@ -1,214 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/subst1/fwd.ma". - -include "Basic-1/subst0/subst0.ma". - -theorem subst1_subst1: - \forall (t1: T).(\forall (t2: T).(\forall (u2: T).(\forall (j: nat).((subst1 -j u2 t1 t2) \to (\forall (u1: T).(\forall (u: T).(\forall (i: nat).((subst1 i -u u1 u2) \to (ex2 T (\lambda (t: T).(subst1 j u1 t1 t)) (\lambda (t: -T).(subst1 (S (plus i j)) u t t2))))))))))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (u2: T).(\lambda (j: nat).(\lambda -(H: (subst1 j u2 t1 t2)).(subst1_ind j u2 t1 (\lambda (t: T).(\forall (u1: -T).(\forall (u: T).(\forall (i: nat).((subst1 i u u1 u2) \to (ex2 T (\lambda -(t0: T).(subst1 j u1 t1 t0)) (\lambda (t0: T).(subst1 (S (plus i j)) u t0 -t)))))))) (\lambda (u1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda (_: -(subst1 i u u1 u2)).(ex_intro2 T (\lambda (t: T).(subst1 j u1 t1 t)) (\lambda -(t: T).(subst1 (S (plus i j)) u t t1)) t1 (subst1_refl j u1 t1) (subst1_refl -(S (plus i j)) u t1)))))) (\lambda (t3: T).(\lambda (H0: (subst0 j u2 t1 -t3)).(\lambda (u1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda (H1: (subst1 -i u u1 u2)).(insert_eq T u2 (\lambda (t: T).(subst1 i u u1 t)) (\lambda (_: -T).(ex2 T (\lambda (t0: T).(subst1 j u1 t1 t0)) (\lambda (t0: T).(subst1 (S -(plus i j)) u t0 t3)))) (\lambda (y: T).(\lambda (H2: (subst1 i u u1 -y)).(subst1_ind i u u1 (\lambda (t: T).((eq T t u2) \to (ex2 T (\lambda (t0: -T).(subst1 j u1 t1 t0)) (\lambda (t0: T).(subst1 (S (plus i j)) u t0 t3))))) -(\lambda (H3: (eq T u1 u2)).(eq_ind_r T u2 (\lambda (t: T).(ex2 T (\lambda -(t0: T).(subst1 j t t1 t0)) (\lambda (t0: T).(subst1 (S (plus i j)) u t0 -t3)))) (ex_intro2 T (\lambda (t: T).(subst1 j u2 t1 t)) (\lambda (t: -T).(subst1 (S (plus i j)) u t t3)) t3 (subst1_single j u2 t1 t3 H0) -(subst1_refl (S (plus i j)) u t3)) u1 H3)) (\lambda (t0: T).(\lambda (H3: -(subst0 i u u1 t0)).(\lambda (H4: (eq T t0 u2)).(let H5 \def (eq_ind T t0 -(\lambda (t: T).(subst0 i u u1 t)) H3 u2 H4) in (ex2_ind T (\lambda (t: -T).(subst0 j u1 t1 t)) (\lambda (t: T).(subst0 (S (plus i j)) u t t3)) (ex2 T -(\lambda (t: T).(subst1 j u1 t1 t)) (\lambda (t: T).(subst1 (S (plus i j)) u -t t3))) (\lambda (x: T).(\lambda (H6: (subst0 j u1 t1 x)).(\lambda (H7: -(subst0 (S (plus i j)) u x t3)).(ex_intro2 T (\lambda (t: T).(subst1 j u1 t1 -t)) (\lambda (t: T).(subst1 (S (plus i j)) u t t3)) x (subst1_single j u1 t1 -x H6) (subst1_single (S (plus i j)) u x t3 H7))))) (subst0_subst0 t1 t3 u2 j -H0 u1 u i H5)))))) y H2))) H1))))))) t2 H))))). -(* COMMENTS -Initial nodes: 649 -END *) - -theorem subst1_subst1_back: - \forall (t1: T).(\forall (t2: T).(\forall (u2: T).(\forall (j: nat).((subst1 -j u2 t1 t2) \to (\forall (u1: T).(\forall (u: T).(\forall (i: nat).((subst1 i -u u2 u1) \to (ex2 T (\lambda (t: T).(subst1 j u1 t1 t)) (\lambda (t: -T).(subst1 (S (plus i j)) u t2 t))))))))))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (u2: T).(\lambda (j: nat).(\lambda -(H: (subst1 j u2 t1 t2)).(subst1_ind j u2 t1 (\lambda (t: T).(\forall (u1: -T).(\forall (u: T).(\forall (i: nat).((subst1 i u u2 u1) \to (ex2 T (\lambda -(t0: T).(subst1 j u1 t1 t0)) (\lambda (t0: T).(subst1 (S (plus i j)) u t -t0)))))))) (\lambda (u1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda (_: -(subst1 i u u2 u1)).(ex_intro2 T (\lambda (t: T).(subst1 j u1 t1 t)) (\lambda -(t: T).(subst1 (S (plus i j)) u t1 t)) t1 (subst1_refl j u1 t1) (subst1_refl -(S (plus i j)) u t1)))))) (\lambda (t3: T).(\lambda (H0: (subst0 j u2 t1 -t3)).(\lambda (u1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda (H1: (subst1 -i u u2 u1)).(subst1_ind i u u2 (\lambda (t: T).(ex2 T (\lambda (t0: -T).(subst1 j t t1 t0)) (\lambda (t0: T).(subst1 (S (plus i j)) u t3 t0)))) -(ex_intro2 T (\lambda (t: T).(subst1 j u2 t1 t)) (\lambda (t: T).(subst1 (S -(plus i j)) u t3 t)) t3 (subst1_single j u2 t1 t3 H0) (subst1_refl (S (plus i -j)) u t3)) (\lambda (t0: T).(\lambda (H2: (subst0 i u u2 t0)).(ex2_ind T -(\lambda (t: T).(subst0 j t0 t1 t)) (\lambda (t: T).(subst0 (S (plus i j)) u -t3 t)) (ex2 T (\lambda (t: T).(subst1 j t0 t1 t)) (\lambda (t: T).(subst1 (S -(plus i j)) u t3 t))) (\lambda (x: T).(\lambda (H3: (subst0 j t0 t1 -x)).(\lambda (H4: (subst0 (S (plus i j)) u t3 x)).(ex_intro2 T (\lambda (t: -T).(subst1 j t0 t1 t)) (\lambda (t: T).(subst1 (S (plus i j)) u t3 t)) x -(subst1_single j t0 t1 x H3) (subst1_single (S (plus i j)) u t3 x H4))))) -(subst0_subst0_back t1 t3 u2 j H0 t0 u i H2)))) u1 H1))))))) t2 H))))). -(* COMMENTS -Initial nodes: 487 -END *) - -theorem subst1_trans: - \forall (t2: T).(\forall (t1: T).(\forall (v: T).(\forall (i: nat).((subst1 -i v t1 t2) \to (\forall (t3: T).((subst1 i v t2 t3) \to (subst1 i v t1 -t3))))))) -\def - \lambda (t2: T).(\lambda (t1: T).(\lambda (v: T).(\lambda (i: nat).(\lambda -(H: (subst1 i v t1 t2)).(subst1_ind i v t1 (\lambda (t: T).(\forall (t3: -T).((subst1 i v t t3) \to (subst1 i v t1 t3)))) (\lambda (t3: T).(\lambda -(H0: (subst1 i v t1 t3)).H0)) (\lambda (t3: T).(\lambda (H0: (subst0 i v t1 -t3)).(\lambda (t4: T).(\lambda (H1: (subst1 i v t3 t4)).(subst1_ind i v t3 -(\lambda (t: T).(subst1 i v t1 t)) (subst1_single i v t1 t3 H0) (\lambda (t0: -T).(\lambda (H2: (subst0 i v t3 t0)).(subst1_single i v t1 t0 (subst0_trans -t3 t1 v i H0 t0 H2)))) t4 H1))))) t2 H))))). -(* COMMENTS -Initial nodes: 165 -END *) - -theorem subst1_confluence_neq: - \forall (t0: T).(\forall (t1: T).(\forall (u1: T).(\forall (i1: -nat).((subst1 i1 u1 t0 t1) \to (\forall (t2: T).(\forall (u2: T).(\forall -(i2: nat).((subst1 i2 u2 t0 t2) \to ((not (eq nat i1 i2)) \to (ex2 T (\lambda -(t: T).(subst1 i2 u2 t1 t)) (\lambda (t: T).(subst1 i1 u1 t2 t)))))))))))) -\def - \lambda (t0: T).(\lambda (t1: T).(\lambda (u1: T).(\lambda (i1: -nat).(\lambda (H: (subst1 i1 u1 t0 t1)).(subst1_ind i1 u1 t0 (\lambda (t: -T).(\forall (t2: T).(\forall (u2: T).(\forall (i2: nat).((subst1 i2 u2 t0 t2) -\to ((not (eq nat i1 i2)) \to (ex2 T (\lambda (t3: T).(subst1 i2 u2 t t3)) -(\lambda (t3: T).(subst1 i1 u1 t2 t3))))))))) (\lambda (t2: T).(\lambda (u2: -T).(\lambda (i2: nat).(\lambda (H0: (subst1 i2 u2 t0 t2)).(\lambda (_: (not -(eq nat i1 i2))).(ex_intro2 T (\lambda (t: T).(subst1 i2 u2 t0 t)) (\lambda -(t: T).(subst1 i1 u1 t2 t)) t2 H0 (subst1_refl i1 u1 t2))))))) (\lambda (t2: -T).(\lambda (H0: (subst0 i1 u1 t0 t2)).(\lambda (t3: T).(\lambda (u2: -T).(\lambda (i2: nat).(\lambda (H1: (subst1 i2 u2 t0 t3)).(\lambda (H2: (not -(eq nat i1 i2))).(subst1_ind i2 u2 t0 (\lambda (t: T).(ex2 T (\lambda (t4: -T).(subst1 i2 u2 t2 t4)) (\lambda (t4: T).(subst1 i1 u1 t t4)))) (ex_intro2 T -(\lambda (t: T).(subst1 i2 u2 t2 t)) (\lambda (t: T).(subst1 i1 u1 t0 t)) t2 -(subst1_refl i2 u2 t2) (subst1_single i1 u1 t0 t2 H0)) (\lambda (t4: -T).(\lambda (H3: (subst0 i2 u2 t0 t4)).(ex2_ind T (\lambda (t: T).(subst0 i1 -u1 t4 t)) (\lambda (t: T).(subst0 i2 u2 t2 t)) (ex2 T (\lambda (t: T).(subst1 -i2 u2 t2 t)) (\lambda (t: T).(subst1 i1 u1 t4 t))) (\lambda (x: T).(\lambda -(H4: (subst0 i1 u1 t4 x)).(\lambda (H5: (subst0 i2 u2 t2 x)).(ex_intro2 T -(\lambda (t: T).(subst1 i2 u2 t2 t)) (\lambda (t: T).(subst1 i1 u1 t4 t)) x -(subst1_single i2 u2 t2 x H5) (subst1_single i1 u1 t4 x H4))))) -(subst0_confluence_neq t0 t4 u2 i2 H3 t2 u1 i1 H0 (sym_not_eq nat i1 i2 -H2))))) t3 H1)))))))) t1 H))))). -(* COMMENTS -Initial nodes: 455 -END *) - -theorem subst1_confluence_eq: - \forall (t0: T).(\forall (t1: T).(\forall (u: T).(\forall (i: nat).((subst1 -i u t0 t1) \to (\forall (t2: T).((subst1 i u t0 t2) \to (ex2 T (\lambda (t: -T).(subst1 i u t1 t)) (\lambda (t: T).(subst1 i u t2 t))))))))) -\def - \lambda (t0: T).(\lambda (t1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H: (subst1 i u t0 t1)).(subst1_ind i u t0 (\lambda (t: T).(\forall (t2: -T).((subst1 i u t0 t2) \to (ex2 T (\lambda (t3: T).(subst1 i u t t3)) -(\lambda (t3: T).(subst1 i u t2 t3)))))) (\lambda (t2: T).(\lambda (H0: -(subst1 i u t0 t2)).(ex_intro2 T (\lambda (t: T).(subst1 i u t0 t)) (\lambda -(t: T).(subst1 i u t2 t)) t2 H0 (subst1_refl i u t2)))) (\lambda (t2: -T).(\lambda (H0: (subst0 i u t0 t2)).(\lambda (t3: T).(\lambda (H1: (subst1 i -u t0 t3)).(subst1_ind i u t0 (\lambda (t: T).(ex2 T (\lambda (t4: T).(subst1 -i u t2 t4)) (\lambda (t4: T).(subst1 i u t t4)))) (ex_intro2 T (\lambda (t: -T).(subst1 i u t2 t)) (\lambda (t: T).(subst1 i u t0 t)) t2 (subst1_refl i u -t2) (subst1_single i u t0 t2 H0)) (\lambda (t4: T).(\lambda (H2: (subst0 i u -t0 t4)).(or4_ind (eq T t4 t2) (ex2 T (\lambda (t: T).(subst0 i u t4 t)) -(\lambda (t: T).(subst0 i u t2 t))) (subst0 i u t4 t2) (subst0 i u t2 t4) -(ex2 T (\lambda (t: T).(subst1 i u t2 t)) (\lambda (t: T).(subst1 i u t4 t))) -(\lambda (H3: (eq T t4 t2)).(eq_ind_r T t2 (\lambda (t: T).(ex2 T (\lambda -(t5: T).(subst1 i u t2 t5)) (\lambda (t5: T).(subst1 i u t t5)))) (ex_intro2 -T (\lambda (t: T).(subst1 i u t2 t)) (\lambda (t: T).(subst1 i u t2 t)) t2 -(subst1_refl i u t2) (subst1_refl i u t2)) t4 H3)) (\lambda (H3: (ex2 T -(\lambda (t: T).(subst0 i u t4 t)) (\lambda (t: T).(subst0 i u t2 -t)))).(ex2_ind T (\lambda (t: T).(subst0 i u t4 t)) (\lambda (t: T).(subst0 i -u t2 t)) (ex2 T (\lambda (t: T).(subst1 i u t2 t)) (\lambda (t: T).(subst1 i -u t4 t))) (\lambda (x: T).(\lambda (H4: (subst0 i u t4 x)).(\lambda (H5: -(subst0 i u t2 x)).(ex_intro2 T (\lambda (t: T).(subst1 i u t2 t)) (\lambda -(t: T).(subst1 i u t4 t)) x (subst1_single i u t2 x H5) (subst1_single i u t4 -x H4))))) H3)) (\lambda (H3: (subst0 i u t4 t2)).(ex_intro2 T (\lambda (t: -T).(subst1 i u t2 t)) (\lambda (t: T).(subst1 i u t4 t)) t2 (subst1_refl i u -t2) (subst1_single i u t4 t2 H3))) (\lambda (H3: (subst0 i u t2 -t4)).(ex_intro2 T (\lambda (t: T).(subst1 i u t2 t)) (\lambda (t: T).(subst1 -i u t4 t)) t4 (subst1_single i u t2 t4 H3) (subst1_refl i u t4))) -(subst0_confluence_eq t0 t4 u i H2 t2 H0)))) t3 H1))))) t1 H))))). -(* COMMENTS -Initial nodes: 729 -END *) - -theorem subst1_confluence_lift: - \forall (t0: T).(\forall (t1: T).(\forall (u: T).(\forall (i: nat).((subst1 -i u t0 (lift (S O) i t1)) \to (\forall (t2: T).((subst1 i u t0 (lift (S O) i -t2)) \to (eq T t1 t2))))))) -\def - \lambda (t0: T).(\lambda (t1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H: (subst1 i u t0 (lift (S O) i t1))).(insert_eq T (lift (S O) i t1) -(\lambda (t: T).(subst1 i u t0 t)) (\lambda (_: T).(\forall (t2: T).((subst1 -i u t0 (lift (S O) i t2)) \to (eq T t1 t2)))) (\lambda (y: T).(\lambda (H0: -(subst1 i u t0 y)).(subst1_ind i u t0 (\lambda (t: T).((eq T t (lift (S O) i -t1)) \to (\forall (t2: T).((subst1 i u t0 (lift (S O) i t2)) \to (eq T t1 -t2))))) (\lambda (H1: (eq T t0 (lift (S O) i t1))).(\lambda (t2: T).(\lambda -(H2: (subst1 i u t0 (lift (S O) i t2))).(let H3 \def (eq_ind T t0 (\lambda -(t: T).(subst1 i u t (lift (S O) i t2))) H2 (lift (S O) i t1) H1) in (let H4 -\def (sym_eq T (lift (S O) i t2) (lift (S O) i t1) (subst1_gen_lift_eq t1 u -(lift (S O) i t2) (S O) i i (le_n i) (eq_ind_r nat (plus (S O) i) (\lambda -(n: nat).(lt i n)) (le_n (plus (S O) i)) (plus i (S O)) (plus_sym i (S O))) -H3)) in (lift_inj t1 t2 (S O) i H4)))))) (\lambda (t2: T).(\lambda (H1: -(subst0 i u t0 t2)).(\lambda (H2: (eq T t2 (lift (S O) i t1))).(\lambda (t3: -T).(\lambda (H3: (subst1 i u t0 (lift (S O) i t3))).(let H4 \def (eq_ind T t2 -(\lambda (t: T).(subst0 i u t0 t)) H1 (lift (S O) i t1) H2) in (insert_eq T -(lift (S O) i t3) (\lambda (t: T).(subst1 i u t0 t)) (\lambda (_: T).(eq T t1 -t3)) (\lambda (y0: T).(\lambda (H5: (subst1 i u t0 y0)).(subst1_ind i u t0 -(\lambda (t: T).((eq T t (lift (S O) i t3)) \to (eq T t1 t3))) (\lambda (H6: -(eq T t0 (lift (S O) i t3))).(let H7 \def (eq_ind T t0 (\lambda (t: -T).(subst0 i u t (lift (S O) i t1))) H4 (lift (S O) i t3) H6) in -(subst0_gen_lift_false t3 u (lift (S O) i t1) (S O) i i (le_n i) (eq_ind_r -nat (plus (S O) i) (\lambda (n: nat).(lt i n)) (le_n (plus (S O) i)) (plus i -(S O)) (plus_sym i (S O))) H7 (eq T t1 t3)))) (\lambda (t4: T).(\lambda (H6: -(subst0 i u t0 t4)).(\lambda (H7: (eq T t4 (lift (S O) i t3))).(let H8 \def -(eq_ind T t4 (\lambda (t: T).(subst0 i u t0 t)) H6 (lift (S O) i t3) H7) in -(sym_eq T t3 t1 (subst0_confluence_lift t0 t3 u i H8 t1 H4)))))) y0 H5))) -H3))))))) y H0))) H))))). -(* COMMENTS -Initial nodes: 735 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/theory.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/theory.ma deleted file mode 100644 index f405be588..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/theory.ma +++ /dev/null @@ -1,42 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/subst0/tlt.ma". - -include "Basic-1/subst/props.ma". - -include "Basic-1/sty1/cnt.ma". - -include "Basic-1/ex0/props.ma". - -include "Basic-1/wcpr0/fwd.ma". - -include "Basic-1/pr3/wcpr0.ma". - -include "Basic-1/ex2/props.ma". - -include "Basic-1/ex1/props.ma". - -include "Basic-1/ty3/sty0.ma". - -include "Basic-1/csubt/csuba.ma". - -include "Basic-1/ty3/fwd_nf2.ma". - -include "Basic-1/ty3/nf2.ma". - -include "Basic-1/wf3/props.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/tlist/defs.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/tlist/defs.ma deleted file mode 100644 index 9445013d4..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/tlist/defs.ma +++ /dev/null @@ -1,47 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/T/defs.ma". - -inductive TList: Set \def -| TNil: TList -| TCons: T \to (TList \to TList). - -definition THeads: - K \to (TList \to (T \to T)) -\def - let rec THeads (k: K) (us: TList) on us: (T \to T) \def (\lambda (t: -T).(match us with [TNil \Rightarrow t | (TCons u ul) \Rightarrow (THead k u -(THeads k ul t))])) in THeads. - -definition TApp: - TList \to (T \to TList) -\def - let rec TApp (ts: TList) on ts: (T \to TList) \def (\lambda (v: T).(match ts -with [TNil \Rightarrow (TCons v TNil) | (TCons t ts0) \Rightarrow (TCons t -(TApp ts0 v))])) in TApp. - -definition tslen: - TList \to nat -\def - let rec tslen (ts: TList) on ts: nat \def (match ts with [TNil \Rightarrow O -| (TCons _ ts0) \Rightarrow (S (tslen ts0))]) in tslen. - -definition tslt: - TList \to (TList \to Prop) -\def - \lambda (ts1: TList).(\lambda (ts2: TList).(lt (tslen ts1) (tslen ts2))). - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/tlist/props.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/tlist/props.ma deleted file mode 100644 index 92b0d05ff..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/tlist/props.ma +++ /dev/null @@ -1,131 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/tlist/defs.ma". - -theorem tslt_wf__q_ind: - \forall (P: ((TList \to Prop))).(((\forall (n: nat).((\lambda (P0: ((TList -\to Prop))).(\lambda (n0: nat).(\forall (ts: TList).((eq nat (tslen ts) n0) -\to (P0 ts))))) P n))) \to (\forall (ts: TList).(P ts))) -\def - let Q \def (\lambda (P: ((TList \to Prop))).(\lambda (n: nat).(\forall (ts: -TList).((eq nat (tslen ts) n) \to (P ts))))) in (\lambda (P: ((TList \to -Prop))).(\lambda (H: ((\forall (n: nat).(\forall (ts: TList).((eq nat (tslen -ts) n) \to (P ts)))))).(\lambda (ts: TList).(H (tslen ts) ts (refl_equal nat -(tslen ts)))))). -(* COMMENTS -Initial nodes: 61 -END *) - -theorem tslt_wf_ind: - \forall (P: ((TList \to Prop))).(((\forall (ts2: TList).(((\forall (ts1: -TList).((tslt ts1 ts2) \to (P ts1)))) \to (P ts2)))) \to (\forall (ts: -TList).(P ts))) -\def - let Q \def (\lambda (P: ((TList \to Prop))).(\lambda (n: nat).(\forall (ts: -TList).((eq nat (tslen ts) n) \to (P ts))))) in (\lambda (P: ((TList \to -Prop))).(\lambda (H: ((\forall (ts2: TList).(((\forall (ts1: TList).((lt -(tslen ts1) (tslen ts2)) \to (P ts1)))) \to (P ts2))))).(\lambda (ts: -TList).(tslt_wf__q_ind (\lambda (t: TList).(P t)) (\lambda (n: -nat).(lt_wf_ind n (Q (\lambda (t: TList).(P t))) (\lambda (n0: nat).(\lambda -(H0: ((\forall (m: nat).((lt m n0) \to (Q (\lambda (t: TList).(P t)) -m))))).(\lambda (ts0: TList).(\lambda (H1: (eq nat (tslen ts0) n0)).(let H2 -\def (eq_ind_r nat n0 (\lambda (n1: nat).(\forall (m: nat).((lt m n1) \to -(\forall (ts1: TList).((eq nat (tslen ts1) m) \to (P ts1)))))) H0 (tslen ts0) -H1) in (H ts0 (\lambda (ts1: TList).(\lambda (H3: (lt (tslen ts1) (tslen -ts0))).(H2 (tslen ts1) H3 ts1 (refl_equal nat (tslen ts1))))))))))))) ts)))). -(* COMMENTS -Initial nodes: 179 -END *) - -theorem theads_tapp: - \forall (k: K).(\forall (v: T).(\forall (t: T).(\forall (vs: TList).(eq T -(THeads k (TApp vs v) t) (THeads k vs (THead k v t)))))) -\def - \lambda (k: K).(\lambda (v: T).(\lambda (t: T).(\lambda (vs: -TList).(TList_ind (\lambda (t0: TList).(eq T (THeads k (TApp t0 v) t) (THeads -k t0 (THead k v t)))) (refl_equal T (THead k v t)) (\lambda (t0: T).(\lambda -(t1: TList).(\lambda (H: (eq T (THeads k (TApp t1 v) t) (THeads k t1 (THead k -v t)))).(eq_ind T (THeads k (TApp t1 v) t) (\lambda (t2: T).(eq T (THead k t0 -(THeads k (TApp t1 v) t)) (THead k t0 t2))) (refl_equal T (THead k t0 (THeads -k (TApp t1 v) t))) (THeads k t1 (THead k v t)) H)))) vs)))). -(* COMMENTS -Initial nodes: 175 -END *) - -theorem tcons_tapp_ex: - \forall (ts1: TList).(\forall (t1: T).(ex2_2 TList T (\lambda (ts2: -TList).(\lambda (t2: T).(eq TList (TCons t1 ts1) (TApp ts2 t2)))) (\lambda -(ts2: TList).(\lambda (_: T).(eq nat (tslen ts1) (tslen ts2)))))) -\def - \lambda (ts1: TList).(TList_ind (\lambda (t: TList).(\forall (t1: T).(ex2_2 -TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons t1 t) (TApp -ts2 t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (tslen t) (tslen -ts2))))))) (\lambda (t1: T).(ex2_2_intro TList T (\lambda (ts2: -TList).(\lambda (t2: T).(eq TList (TCons t1 TNil) (TApp ts2 t2)))) (\lambda -(ts2: TList).(\lambda (_: T).(eq nat O (tslen ts2)))) TNil t1 (refl_equal -TList (TApp TNil t1)) (refl_equal nat (tslen TNil)))) (\lambda (t: -T).(\lambda (t0: TList).(\lambda (H: ((\forall (t1: T).(ex2_2 TList T -(\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons t1 t0) (TApp ts2 -t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (tslen t0) (tslen -ts2)))))))).(\lambda (t1: T).(let H_x \def (H t) in (let H0 \def H_x in -(ex2_2_ind TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons t -t0) (TApp ts2 t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (tslen t0) -(tslen ts2)))) (ex2_2 TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq -TList (TCons t1 (TCons t t0)) (TApp ts2 t2)))) (\lambda (ts2: TList).(\lambda -(_: T).(eq nat (S (tslen t0)) (tslen ts2))))) (\lambda (x0: TList).(\lambda -(x1: T).(\lambda (H1: (eq TList (TCons t t0) (TApp x0 x1))).(\lambda (H2: (eq -nat (tslen t0) (tslen x0))).(eq_ind_r TList (TApp x0 x1) (\lambda (t2: -TList).(ex2_2 TList T (\lambda (ts2: TList).(\lambda (t3: T).(eq TList (TCons -t1 t2) (TApp ts2 t3)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (S -(tslen t0)) (tslen ts2)))))) (eq_ind_r nat (tslen x0) (\lambda (n: -nat).(ex2_2 TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons -t1 (TApp x0 x1)) (TApp ts2 t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq -nat (S n) (tslen ts2)))))) (ex2_2_intro TList T (\lambda (ts2: -TList).(\lambda (t2: T).(eq TList (TCons t1 (TApp x0 x1)) (TApp ts2 t2)))) -(\lambda (ts2: TList).(\lambda (_: T).(eq nat (S (tslen x0)) (tslen ts2)))) -(TCons t1 x0) x1 (refl_equal TList (TApp (TCons t1 x0) x1)) (refl_equal nat -(tslen (TCons t1 x0)))) (tslen t0) H2) (TCons t t0) H1))))) H0))))))) ts1). -(* COMMENTS -Initial nodes: 503 -END *) - -theorem tlist_ind_rev: - \forall (P: ((TList \to Prop))).((P TNil) \to (((\forall (ts: -TList).(\forall (t: T).((P ts) \to (P (TApp ts t)))))) \to (\forall (ts: -TList).(P ts)))) -\def - \lambda (P: ((TList \to Prop))).(\lambda (H: (P TNil)).(\lambda (H0: -((\forall (ts: TList).(\forall (t: T).((P ts) \to (P (TApp ts -t))))))).(\lambda (ts: TList).(tslt_wf_ind (\lambda (t: TList).(P t)) -(\lambda (ts2: TList).(TList_ind (\lambda (t: TList).(((\forall (ts1: -TList).((tslt ts1 t) \to (P ts1)))) \to (P t))) (\lambda (_: ((\forall (ts1: -TList).((tslt ts1 TNil) \to (P ts1))))).H) (\lambda (t: T).(\lambda (t0: -TList).(\lambda (_: ((((\forall (ts1: TList).((tslt ts1 t0) \to (P ts1)))) -\to (P t0)))).(\lambda (H2: ((\forall (ts1: TList).((tslt ts1 (TCons t t0)) -\to (P ts1))))).(let H_x \def (tcons_tapp_ex t0 t) in (let H3 \def H_x in -(ex2_2_ind TList T (\lambda (ts3: TList).(\lambda (t2: T).(eq TList (TCons t -t0) (TApp ts3 t2)))) (\lambda (ts3: TList).(\lambda (_: T).(eq nat (tslen t0) -(tslen ts3)))) (P (TCons t t0)) (\lambda (x0: TList).(\lambda (x1: -T).(\lambda (H4: (eq TList (TCons t t0) (TApp x0 x1))).(\lambda (H5: (eq nat -(tslen t0) (tslen x0))).(eq_ind_r TList (TApp x0 x1) (\lambda (t1: TList).(P -t1)) (H0 x0 x1 (H2 x0 (eq_ind nat (tslen t0) (\lambda (n: nat).(lt n (tslen -(TCons t t0)))) (le_n (tslen (TCons t t0))) (tslen x0) H5))) (TCons t t0) -H4))))) H3))))))) ts2)) ts)))). -(* COMMENTS -Initial nodes: 273 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/tlt/defs.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/tlt/defs.ma deleted file mode 100644 index 546bdb78d..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/tlt/defs.ma +++ /dev/null @@ -1,46 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/T/defs.ma". - -definition wadd: - ((nat \to nat)) \to (nat \to (nat \to nat)) -\def - \lambda (f: ((nat \to nat))).(\lambda (w: nat).(\lambda (n: nat).(match n -with [O \Rightarrow w | (S m) \Rightarrow (f m)]))). - -definition weight_map: - ((nat \to nat)) \to (T \to nat) -\def - let rec weight_map (f: ((nat \to nat))) (t: T) on t: nat \def (match t with -[(TSort _) \Rightarrow O | (TLRef n) \Rightarrow (f n) | (THead k u t0) -\Rightarrow (match k with [(Bind b) \Rightarrow (match b with [Abbr -\Rightarrow (S (plus (weight_map f u) (weight_map (wadd f (S (weight_map f -u))) t0))) | Abst \Rightarrow (S (plus (weight_map f u) (weight_map (wadd f -O) t0))) | Void \Rightarrow (S (plus (weight_map f u) (weight_map (wadd f O) -t0)))]) | (Flat _) \Rightarrow (S (plus (weight_map f u) (weight_map f -t0)))])]) in weight_map. - -definition weight: - T \to nat -\def - weight_map (\lambda (_: nat).O). - -definition tlt: - T \to (T \to Prop) -\def - \lambda (t1: T).(\lambda (t2: T).(lt (weight t1) (weight t2))). - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/tlt/props.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/tlt/props.ma deleted file mode 100644 index 0b4f16d41..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/tlt/props.ma +++ /dev/null @@ -1,300 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/tlt/defs.ma". - -theorem wadd_le: - \forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (n: -nat).(le (f n) (g n)))) \to (\forall (v: nat).(\forall (w: nat).((le v w) \to -(\forall (n: nat).(le (wadd f v n) (wadd g w n)))))))) -\def - \lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H: -((\forall (n: nat).(le (f n) (g n))))).(\lambda (v: nat).(\lambda (w: -nat).(\lambda (H0: (le v w)).(\lambda (n: nat).(nat_ind (\lambda (n0: -nat).(le (wadd f v n0) (wadd g w n0))) H0 (\lambda (n0: nat).(\lambda (_: (le -(wadd f v n0) (wadd g w n0))).(H n0))) n))))))). -(* COMMENTS -Initial nodes: 81 -END *) - -theorem wadd_lt: - \forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (n: -nat).(le (f n) (g n)))) \to (\forall (v: nat).(\forall (w: nat).((lt v w) \to -(\forall (n: nat).(le (wadd f v n) (wadd g w n)))))))) -\def - \lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H: -((\forall (n: nat).(le (f n) (g n))))).(\lambda (v: nat).(\lambda (w: -nat).(\lambda (H0: (lt v w)).(\lambda (n: nat).(nat_ind (\lambda (n0: -nat).(le (wadd f v n0) (wadd g w n0))) (le_S_n v w (le_S (S v) w H0)) -(\lambda (n0: nat).(\lambda (_: (le (wadd f v n0) (wadd g w n0))).(H n0))) -n))))))). -(* COMMENTS -Initial nodes: 95 -END *) - -theorem wadd_O: - \forall (n: nat).(eq nat (wadd (\lambda (_: nat).O) O n) O) -\def - \lambda (n: nat).(nat_ind (\lambda (n0: nat).(eq nat (wadd (\lambda (_: -nat).O) O n0) O)) (refl_equal nat O) (\lambda (n0: nat).(\lambda (_: (eq nat -(wadd (\lambda (_: nat).O) O n0) O)).(refl_equal nat O))) n). -(* COMMENTS -Initial nodes: 53 -END *) - -theorem weight_le: - \forall (t: T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (n: nat).(le (f n) (g n)))) \to (le (weight_map f t) -(weight_map g t))))) -\def - \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (f: ((nat \to -nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g n)))) -\to (le (weight_map f t0) (weight_map g t0)))))) (\lambda (n: nat).(\lambda -(f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (_: ((\forall -(n0: nat).(le (f n0) (g n0))))).(le_n (weight_map g (TSort n))))))) (\lambda -(n: nat).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda -(H: ((\forall (n0: nat).(le (f n0) (g n0))))).(H n))))) (\lambda (k: -K).(K_ind (\lambda (k0: K).(\forall (t0: T).(((\forall (f: ((nat \to -nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g n)))) -\to (le (weight_map f t0) (weight_map g t0)))))) \to (\forall (t1: -T).(((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall -(n: nat).(le (f n) (g n)))) \to (le (weight_map f t1) (weight_map g t1)))))) -\to (\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall -(n: nat).(le (f n) (g n)))) \to (le (weight_map f (THead k0 t0 t1)) -(weight_map g (THead k0 t0 t1))))))))))) (\lambda (b: B).(B_ind (\lambda (b0: -B).(\forall (t0: T).(((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (n: nat).(le (f n) (g n)))) \to (le (weight_map f t0) -(weight_map g t0)))))) \to (\forall (t1: T).(((\forall (f: ((nat \to -nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g n)))) -\to (le (weight_map f t1) (weight_map g t1)))))) \to (\forall (f: ((nat \to -nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g n)))) -\to (le (match b0 with [Abbr \Rightarrow (S (plus (weight_map f t0) -(weight_map (wadd f (S (weight_map f t0))) t1))) | Abst \Rightarrow (S (plus -(weight_map f t0) (weight_map (wadd f O) t1))) | Void \Rightarrow (S (plus -(weight_map f t0) (weight_map (wadd f O) t1)))]) (match b0 with [Abbr -\Rightarrow (S (plus (weight_map g t0) (weight_map (wadd g (S (weight_map g -t0))) t1))) | Abst \Rightarrow (S (plus (weight_map g t0) (weight_map (wadd g -O) t1))) | Void \Rightarrow (S (plus (weight_map g t0) (weight_map (wadd g O) -t1)))])))))))))) (\lambda (t0: T).(\lambda (H: ((\forall (f: ((nat \to -nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g n)))) -\to (le (weight_map f t0) (weight_map g t0))))))).(\lambda (t1: T).(\lambda -(H0: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall -(n: nat).(le (f n) (g n)))) \to (le (weight_map f t1) (weight_map g -t1))))))).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to -nat))).(\lambda (H1: ((\forall (n: nat).(le (f n) (g n))))).(le_n_S (plus -(weight_map f t0) (weight_map (wadd f (S (weight_map f t0))) t1)) (plus -(weight_map g t0) (weight_map (wadd g (S (weight_map g t0))) t1)) -(le_plus_plus (weight_map f t0) (weight_map g t0) (weight_map (wadd f (S -(weight_map f t0))) t1) (weight_map (wadd g (S (weight_map g t0))) t1) (H f g -H1) (H0 (wadd f (S (weight_map f t0))) (wadd g (S (weight_map g t0))) -(\lambda (n: nat).(wadd_le f g H1 (S (weight_map f t0)) (S (weight_map g t0)) -(le_n_S (weight_map f t0) (weight_map g t0) (H f g H1)) n)))))))))))) -(\lambda (t0: T).(\lambda (H: ((\forall (f: ((nat \to nat))).(\forall (g: -((nat \to nat))).(((\forall (n: nat).(le (f n) (g n)))) \to (le (weight_map f -t0) (weight_map g t0))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (f: -((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) -(g n)))) \to (le (weight_map f t1) (weight_map g t1))))))).(\lambda (f: ((nat -\to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H1: ((\forall (n: nat).(le -(f n) (g n))))).(le_n_S (plus (weight_map f t0) (weight_map (wadd f O) t1)) -(plus (weight_map g t0) (weight_map (wadd g O) t1)) (le_plus_plus (weight_map -f t0) (weight_map g t0) (weight_map (wadd f O) t1) (weight_map (wadd g O) t1) -(H f g H1) (H0 (wadd f O) (wadd g O) (\lambda (n: nat).(wadd_le f g H1 O O -(le_n O) n)))))))))))) (\lambda (t0: T).(\lambda (H: ((\forall (f: ((nat \to -nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g n)))) -\to (le (weight_map f t0) (weight_map g t0))))))).(\lambda (t1: T).(\lambda -(H0: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall -(n: nat).(le (f n) (g n)))) \to (le (weight_map f t1) (weight_map g -t1))))))).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to -nat))).(\lambda (H1: ((\forall (n: nat).(le (f n) (g n))))).(le_n_S (plus -(weight_map f t0) (weight_map (wadd f O) t1)) (plus (weight_map g t0) -(weight_map (wadd g O) t1)) (le_plus_plus (weight_map f t0) (weight_map g t0) -(weight_map (wadd f O) t1) (weight_map (wadd g O) t1) (H f g H1) (H0 (wadd f -O) (wadd g O) (\lambda (n: nat).(wadd_le f g H1 O O (le_n O) n)))))))))))) -b)) (\lambda (_: F).(\lambda (t0: T).(\lambda (H: ((\forall (f0: ((nat \to -nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f0 n) (g n)))) -\to (le (weight_map f0 t0) (weight_map g t0))))))).(\lambda (t1: T).(\lambda -(H0: ((\forall (f0: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall -(n: nat).(le (f0 n) (g n)))) \to (le (weight_map f0 t1) (weight_map g -t1))))))).(\lambda (f0: ((nat \to nat))).(\lambda (g: ((nat \to -nat))).(\lambda (H1: ((\forall (n: nat).(le (f0 n) (g n))))).(le_n_S (plus -(weight_map f0 t0) (weight_map f0 t1)) (plus (weight_map g t0) (weight_map g -t1)) (le_plus_plus (weight_map f0 t0) (weight_map g t0) (weight_map f0 t1) -(weight_map g t1) (H f0 g H1) (H0 f0 g H1))))))))))) k)) t). -(* COMMENTS -Initial nodes: 1309 -END *) - -theorem weight_eq: - \forall (t: T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to -nat))).(((\forall (n: nat).(eq nat (f n) (g n)))) \to (eq nat (weight_map f -t) (weight_map g t))))) -\def - \lambda (t: T).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to -nat))).(\lambda (H: ((\forall (n: nat).(eq nat (f n) (g n))))).(le_antisym -(weight_map f t) (weight_map g t) (weight_le t f g (\lambda (n: -nat).(eq_ind_r nat (g n) (\lambda (n0: nat).(le n0 (g n))) (le_n (g n)) (f n) -(H n)))) (weight_le t g f (\lambda (n: nat).(eq_ind_r nat (g n) (\lambda (n0: -nat).(le (g n) n0)) (le_n (g n)) (f n) (H n)))))))). -(* COMMENTS -Initial nodes: 121 -END *) - -theorem weight_add_O: - \forall (t: T).(eq nat (weight_map (wadd (\lambda (_: nat).O) O) t) -(weight_map (\lambda (_: nat).O) t)) -\def - \lambda (t: T).(weight_eq t (wadd (\lambda (_: nat).O) O) (\lambda (_: -nat).O) (\lambda (n: nat).(wadd_O n))). -(* COMMENTS -Initial nodes: 23 -END *) - -theorem weight_add_S: - \forall (t: T).(\forall (m: nat).(le (weight_map (wadd (\lambda (_: nat).O) -O) t) (weight_map (wadd (\lambda (_: nat).O) (S m)) t))) -\def - \lambda (t: T).(\lambda (m: nat).(weight_le t (wadd (\lambda (_: nat).O) O) -(wadd (\lambda (_: nat).O) (S m)) (\lambda (n: nat).(wadd_le (\lambda (_: -nat).O) (\lambda (_: nat).O) (\lambda (_: nat).(le_n O)) O (S m) (le_S O m -(le_O_n m)) n)))). -(* COMMENTS -Initial nodes: 61 -END *) - -theorem tlt_trans: - \forall (v: T).(\forall (u: T).(\forall (t: T).((tlt u v) \to ((tlt v t) \to -(tlt u t))))) -\def - \lambda (v: T).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (lt (weight u) -(weight v))).(\lambda (H0: (lt (weight v) (weight t))).(lt_trans (weight u) -(weight v) (weight t) H H0))))). -(* COMMENTS -Initial nodes: 43 -END *) - -theorem tlt_head_sx: - \forall (k: K).(\forall (u: T).(\forall (t: T).(tlt u (THead k u t)))) -\def - \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (u: T).(\forall (t: T).(lt -(weight_map (\lambda (_: nat).O) u) (weight_map (\lambda (_: nat).O) (THead -k0 u t)))))) (\lambda (b: B).(B_ind (\lambda (b0: B).(\forall (u: T).(\forall -(t: T).(lt (weight_map (\lambda (_: nat).O) u) (match b0 with [Abbr -\Rightarrow (S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd -(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t))) | Abst -\Rightarrow (S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd -(\lambda (_: nat).O) O) t))) | Void \Rightarrow (S (plus (weight_map (\lambda -(_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) O) t)))]))))) (\lambda -(u: T).(\lambda (t: T).(le_n_S (weight_map (\lambda (_: nat).O) u) (plus -(weight_map (\lambda (_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) (S -(weight_map (\lambda (_: nat).O) u))) t)) (le_plus_l (weight_map (\lambda (_: -nat).O) u) (weight_map (wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: -nat).O) u))) t))))) (\lambda (u: T).(\lambda (t: T).(le_n_S (weight_map -(\lambda (_: nat).O) u) (plus (weight_map (\lambda (_: nat).O) u) (weight_map -(wadd (\lambda (_: nat).O) O) t)) (le_plus_l (weight_map (\lambda (_: nat).O) -u) (weight_map (wadd (\lambda (_: nat).O) O) t))))) (\lambda (u: T).(\lambda -(t: T).(le_n_S (weight_map (\lambda (_: nat).O) u) (plus (weight_map (\lambda -(_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) O) t)) (le_plus_l -(weight_map (\lambda (_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) O) -t))))) b)) (\lambda (_: F).(\lambda (u: T).(\lambda (t: T).(le_n_S -(weight_map (\lambda (_: nat).O) u) (plus (weight_map (\lambda (_: nat).O) u) -(weight_map (\lambda (_: nat).O) t)) (le_plus_l (weight_map (\lambda (_: -nat).O) u) (weight_map (\lambda (_: nat).O) t)))))) k). -(* COMMENTS -Initial nodes: 379 -END *) - -theorem tlt_head_dx: - \forall (k: K).(\forall (u: T).(\forall (t: T).(tlt t (THead k u t)))) -\def - \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (u: T).(\forall (t: T).(lt -(weight_map (\lambda (_: nat).O) t) (weight_map (\lambda (_: nat).O) (THead -k0 u t)))))) (\lambda (b: B).(B_ind (\lambda (b0: B).(\forall (u: T).(\forall -(t: T).(lt (weight_map (\lambda (_: nat).O) t) (match b0 with [Abbr -\Rightarrow (S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd -(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t))) | Abst -\Rightarrow (S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd -(\lambda (_: nat).O) O) t))) | Void \Rightarrow (S (plus (weight_map (\lambda -(_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) O) t)))]))))) (\lambda -(u: T).(\lambda (t: T).(lt_le_trans (weight_map (\lambda (_: nat).O) t) (S -(weight_map (\lambda (_: nat).O) t)) (S (plus (weight_map (\lambda (_: -nat).O) u) (weight_map (wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: -nat).O) u))) t))) (lt_n_Sn (weight_map (\lambda (_: nat).O) t)) (le_n_S -(weight_map (\lambda (_: nat).O) t) (plus (weight_map (\lambda (_: nat).O) u) -(weight_map (wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) -u))) t)) (le_trans (weight_map (\lambda (_: nat).O) t) (weight_map (wadd -(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t) (plus -(weight_map (\lambda (_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) (S -(weight_map (\lambda (_: nat).O) u))) t)) (eq_ind nat (weight_map (wadd -(\lambda (_: nat).O) O) t) (\lambda (n: nat).(le n (weight_map (wadd (\lambda -(_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t))) (weight_add_S t -(weight_map (\lambda (_: nat).O) u)) (weight_map (\lambda (_: nat).O) t) -(weight_add_O t)) (le_plus_r (weight_map (\lambda (_: nat).O) u) (weight_map -(wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t))))))) -(\lambda (u: T).(\lambda (t: T).(eq_ind_r nat (weight_map (\lambda (_: -nat).O) t) (\lambda (n: nat).(lt (weight_map (\lambda (_: nat).O) t) (S (plus -(weight_map (\lambda (_: nat).O) u) n)))) (le_n_S (weight_map (\lambda (_: -nat).O) t) (plus (weight_map (\lambda (_: nat).O) u) (weight_map (\lambda (_: -nat).O) t)) (le_plus_r (weight_map (\lambda (_: nat).O) u) (weight_map -(\lambda (_: nat).O) t))) (weight_map (wadd (\lambda (_: nat).O) O) t) -(weight_add_O t)))) (\lambda (u: T).(\lambda (t: T).(eq_ind_r nat (weight_map -(\lambda (_: nat).O) t) (\lambda (n: nat).(lt (weight_map (\lambda (_: -nat).O) t) (S (plus (weight_map (\lambda (_: nat).O) u) n)))) (le_n_S -(weight_map (\lambda (_: nat).O) t) (plus (weight_map (\lambda (_: nat).O) u) -(weight_map (\lambda (_: nat).O) t)) (le_plus_r (weight_map (\lambda (_: -nat).O) u) (weight_map (\lambda (_: nat).O) t))) (weight_map (wadd (\lambda -(_: nat).O) O) t) (weight_add_O t)))) b)) (\lambda (_: F).(\lambda (u: -T).(\lambda (t: T).(le_n_S (weight_map (\lambda (_: nat).O) t) (plus -(weight_map (\lambda (_: nat).O) u) (weight_map (\lambda (_: nat).O) t)) -(le_plus_r (weight_map (\lambda (_: nat).O) u) (weight_map (\lambda (_: -nat).O) t)))))) k). -(* COMMENTS -Initial nodes: 659 -END *) - -theorem tlt_wf__q_ind: - \forall (P: ((T \to Prop))).(((\forall (n: nat).((\lambda (P0: ((T \to -Prop))).(\lambda (n0: nat).(\forall (t: T).((eq nat (weight t) n0) \to (P0 -t))))) P n))) \to (\forall (t: T).(P t))) -\def - let Q \def (\lambda (P: ((T \to Prop))).(\lambda (n: nat).(\forall (t: -T).((eq nat (weight t) n) \to (P t))))) in (\lambda (P: ((T \to -Prop))).(\lambda (H: ((\forall (n: nat).(\forall (t: T).((eq nat (weight t) -n) \to (P t)))))).(\lambda (t: T).(H (weight t) t (refl_equal nat (weight -t)))))). -(* COMMENTS -Initial nodes: 61 -END *) - -theorem tlt_wf_ind: - \forall (P: ((T \to Prop))).(((\forall (t: T).(((\forall (v: T).((tlt v t) -\to (P v)))) \to (P t)))) \to (\forall (t: T).(P t))) -\def - let Q \def (\lambda (P: ((T \to Prop))).(\lambda (n: nat).(\forall (t: -T).((eq nat (weight t) n) \to (P t))))) in (\lambda (P: ((T \to -Prop))).(\lambda (H: ((\forall (t: T).(((\forall (v: T).((lt (weight v) -(weight t)) \to (P v)))) \to (P t))))).(\lambda (t: T).(tlt_wf__q_ind -(\lambda (t0: T).(P t0)) (\lambda (n: nat).(lt_wf_ind n (Q (\lambda (t0: -T).(P t0))) (\lambda (n0: nat).(\lambda (H0: ((\forall (m: nat).((lt m n0) -\to (Q (\lambda (t0: T).(P t0)) m))))).(\lambda (t0: T).(\lambda (H1: (eq nat -(weight t0) n0)).(let H2 \def (eq_ind_r nat n0 (\lambda (n1: nat).(\forall -(m: nat).((lt m n1) \to (\forall (t1: T).((eq nat (weight t1) m) \to (P -t1)))))) H0 (weight t0) H1) in (H t0 (\lambda (v: T).(\lambda (H3: (lt -(weight v) (weight t0))).(H2 (weight v) H3 v (refl_equal nat (weight -v))))))))))))) t)))). -(* COMMENTS -Initial nodes: 179 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/arity.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/arity.ma deleted file mode 100644 index 4400e1540..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/arity.ma +++ /dev/null @@ -1,186 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/ty3/pr3_props.ma". - -include "Basic-1/arity/pr3.ma". - -include "Basic-1/asucc/fwd.ma". - -theorem ty3_arity: - \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c -t1 t2) \to (ex2 A (\lambda (a1: A).(arity g c t1 a1)) (\lambda (a1: A).(arity -g c t2 (asucc g a1)))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H: (ty3 g c t1 t2)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda -(t0: T).(ex2 A (\lambda (a1: A).(arity g c0 t a1)) (\lambda (a1: A).(arity g -c0 t0 (asucc g a1))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t: -T).(\lambda (_: (ty3 g c0 t3 t)).(\lambda (H1: (ex2 A (\lambda (a1: A).(arity -g c0 t3 a1)) (\lambda (a1: A).(arity g c0 t (asucc g a1))))).(\lambda (u: -T).(\lambda (t4: T).(\lambda (_: (ty3 g c0 u t4)).(\lambda (H3: (ex2 A -(\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t4 (asucc g -a1))))).(\lambda (H4: (pc3 c0 t4 t3)).(let H5 \def H1 in (ex2_ind A (\lambda -(a1: A).(arity g c0 t3 a1)) (\lambda (a1: A).(arity g c0 t (asucc g a1))) -(ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t3 -(asucc g a1)))) (\lambda (x: A).(\lambda (H6: (arity g c0 t3 x)).(\lambda (_: -(arity g c0 t (asucc g x))).(let H8 \def H3 in (ex2_ind A (\lambda (a1: -A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t4 (asucc g a1))) (ex2 A -(\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t3 (asucc g -a1)))) (\lambda (x0: A).(\lambda (H9: (arity g c0 u x0)).(\lambda (H10: -(arity g c0 t4 (asucc g x0))).(let H11 \def H4 in (ex2_ind T (\lambda (t0: -T).(pr3 c0 t4 t0)) (\lambda (t0: T).(pr3 c0 t3 t0)) (ex2 A (\lambda (a1: -A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t3 (asucc g a1)))) -(\lambda (x1: T).(\lambda (H12: (pr3 c0 t4 x1)).(\lambda (H13: (pr3 c0 t3 -x1)).(ex_intro2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity -g c0 t3 (asucc g a1))) x0 H9 (arity_repl g c0 t3 x H6 (asucc g x0) (leq_sym g -(asucc g x0) x (arity_mono g c0 x1 (asucc g x0) (arity_sred_pr3 c0 t4 x1 H12 -g (asucc g x0) H10) x (arity_sred_pr3 c0 t3 x1 H13 g x H6)))))))) H11))))) -H8))))) H5)))))))))))) (\lambda (c0: C).(\lambda (m: nat).(ex_intro2 A -(\lambda (a1: A).(arity g c0 (TSort m) a1)) (\lambda (a1: A).(arity g c0 -(TSort (next g m)) (asucc g a1))) (ASort O m) (arity_sort g c0 m) (arity_sort -g c0 (next g m))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: -C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abbr) -u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2: (ex2 A -(\lambda (a1: A).(arity g d u a1)) (\lambda (a1: A).(arity g d t (asucc g -a1))))).(let H3 \def H2 in (ex2_ind A (\lambda (a1: A).(arity g d u a1)) -(\lambda (a1: A).(arity g d t (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g -c0 (TLRef n) a1)) (\lambda (a1: A).(arity g c0 (lift (S n) O t) (asucc g -a1)))) (\lambda (x: A).(\lambda (H4: (arity g d u x)).(\lambda (H5: (arity g -d t (asucc g x))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (TLRef n) a1)) -(\lambda (a1: A).(arity g c0 (lift (S n) O t) (asucc g a1))) x (arity_abbr g -c0 d u n H0 x H4) (arity_lift g d t (asucc g x) H5 c0 (S n) O (getl_drop Abbr -c0 d u n H0)))))) H3)))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda -(d: C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abst) -u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2: (ex2 A -(\lambda (a1: A).(arity g d u a1)) (\lambda (a1: A).(arity g d t (asucc g -a1))))).(let H3 \def H2 in (ex2_ind A (\lambda (a1: A).(arity g d u a1)) -(\lambda (a1: A).(arity g d t (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g -c0 (TLRef n) a1)) (\lambda (a1: A).(arity g c0 (lift (S n) O u) (asucc g -a1)))) (\lambda (x: A).(\lambda (H4: (arity g d u x)).(\lambda (_: (arity g d -t (asucc g x))).(let H_x \def (leq_asucc g x) in (let H6 \def H_x in (ex_ind -A (\lambda (a0: A).(leq g x (asucc g a0))) (ex2 A (\lambda (a1: A).(arity g -c0 (TLRef n) a1)) (\lambda (a1: A).(arity g c0 (lift (S n) O u) (asucc g -a1)))) (\lambda (x0: A).(\lambda (H7: (leq g x (asucc g x0))).(ex_intro2 A -(\lambda (a1: A).(arity g c0 (TLRef n) a1)) (\lambda (a1: A).(arity g c0 -(lift (S n) O u) (asucc g a1))) x0 (arity_abst g c0 d u n H0 x0 (arity_repl g -d u x H4 (asucc g x0) H7)) (arity_repl g c0 (lift (S n) O u) x (arity_lift g -d u x H4 c0 (S n) O (getl_drop Abst c0 d u n H0)) (asucc g x0) H7)))) -H6)))))) H3)))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (t: -T).(\lambda (_: (ty3 g c0 u t)).(\lambda (H1: (ex2 A (\lambda (a1: A).(arity -g c0 u a1)) (\lambda (a1: A).(arity g c0 t (asucc g a1))))).(\lambda (b: -B).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) -u) t3 t4)).(\lambda (H3: (ex2 A (\lambda (a1: A).(arity g (CHead c0 (Bind b) -u) t3 a1)) (\lambda (a1: A).(arity g (CHead c0 (Bind b) u) t4 (asucc g -a1))))).(let H4 \def H1 in (ex2_ind A (\lambda (a1: A).(arity g c0 u a1)) -(\lambda (a1: A).(arity g c0 t (asucc g a1))) (ex2 A (\lambda (a1: A).(arity -g c0 (THead (Bind b) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind b) -u t4) (asucc g a1)))) (\lambda (x: A).(\lambda (H5: (arity g c0 u -x)).(\lambda (_: (arity g c0 t (asucc g x))).(let H7 \def H3 in (ex2_ind A -(\lambda (a1: A).(arity g (CHead c0 (Bind b) u) t3 a1)) (\lambda (a1: -A).(arity g (CHead c0 (Bind b) u) t4 (asucc g a1))) (ex2 A (\lambda (a1: -A).(arity g c0 (THead (Bind b) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead -(Bind b) u t4) (asucc g a1)))) (\lambda (x0: A).(\lambda (H8: (arity g (CHead -c0 (Bind b) u) t3 x0)).(\lambda (H9: (arity g (CHead c0 (Bind b) u) t4 (asucc -g x0))).(let H_x \def (leq_asucc g x) in (let H10 \def H_x in (ex_ind A -(\lambda (a0: A).(leq g x (asucc g a0))) (ex2 A (\lambda (a1: A).(arity g c0 -(THead (Bind b) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind b) u t4) -(asucc g a1)))) (\lambda (x1: A).(\lambda (H11: (leq g x (asucc g -x1))).(B_ind (\lambda (b0: B).((arity g (CHead c0 (Bind b0) u) t3 x0) \to -((arity g (CHead c0 (Bind b0) u) t4 (asucc g x0)) \to (ex2 A (\lambda (a1: -A).(arity g c0 (THead (Bind b0) u t3) a1)) (\lambda (a1: A).(arity g c0 -(THead (Bind b0) u t4) (asucc g a1))))))) (\lambda (H12: (arity g (CHead c0 -(Bind Abbr) u) t3 x0)).(\lambda (H13: (arity g (CHead c0 (Bind Abbr) u) t4 -(asucc g x0))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Bind Abbr) u -t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind Abbr) u t4) (asucc g a1))) -x0 (arity_bind g Abbr not_abbr_abst c0 u x H5 t3 x0 H12) (arity_bind g Abbr -not_abbr_abst c0 u x H5 t4 (asucc g x0) H13)))) (\lambda (H12: (arity g -(CHead c0 (Bind Abst) u) t3 x0)).(\lambda (H13: (arity g (CHead c0 (Bind -Abst) u) t4 (asucc g x0))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (THead -(Bind Abst) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind Abst) u t4) -(asucc g a1))) (AHead x1 x0) (arity_head g c0 u x1 (arity_repl g c0 u x H5 -(asucc g x1) H11) t3 x0 H12) (arity_repl g c0 (THead (Bind Abst) u t4) (AHead -x1 (asucc g x0)) (arity_head g c0 u x1 (arity_repl g c0 u x H5 (asucc g x1) -H11) t4 (asucc g x0) H13) (asucc g (AHead x1 x0)) (leq_refl g (asucc g (AHead -x1 x0))))))) (\lambda (H12: (arity g (CHead c0 (Bind Void) u) t3 -x0)).(\lambda (H13: (arity g (CHead c0 (Bind Void) u) t4 (asucc g -x0))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Bind Void) u t3) a1)) -(\lambda (a1: A).(arity g c0 (THead (Bind Void) u t4) (asucc g a1))) x0 -(arity_bind g Void (sym_not_eq B Abst Void not_abst_void) c0 u x H5 t3 x0 -H12) (arity_bind g Void (sym_not_eq B Abst Void not_abst_void) c0 u x H5 t4 -(asucc g x0) H13)))) b H8 H9))) H10)))))) H7))))) H4)))))))))))) (\lambda -(c0: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda -(H1: (ex2 A (\lambda (a1: A).(arity g c0 w a1)) (\lambda (a1: A).(arity g c0 -u (asucc g a1))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v -(THead (Bind Abst) u t))).(\lambda (H3: (ex2 A (\lambda (a1: A).(arity g c0 v -a1)) (\lambda (a1: A).(arity g c0 (THead (Bind Abst) u t) (asucc g -a1))))).(let H4 \def H1 in (ex2_ind A (\lambda (a1: A).(arity g c0 w a1)) -(\lambda (a1: A).(arity g c0 u (asucc g a1))) (ex2 A (\lambda (a1: A).(arity -g c0 (THead (Flat Appl) w v) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat -Appl) w (THead (Bind Abst) u t)) (asucc g a1)))) (\lambda (x: A).(\lambda -(H5: (arity g c0 w x)).(\lambda (H6: (arity g c0 u (asucc g x))).(let H7 \def -H3 in (ex2_ind A (\lambda (a1: A).(arity g c0 v a1)) (\lambda (a1: A).(arity -g c0 (THead (Bind Abst) u t) (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g -c0 (THead (Flat Appl) w v) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat -Appl) w (THead (Bind Abst) u t)) (asucc g a1)))) (\lambda (x0: A).(\lambda -(H8: (arity g c0 v x0)).(\lambda (H9: (arity g c0 (THead (Bind Abst) u t) -(asucc g x0))).(let H10 \def (arity_gen_abst g c0 u t (asucc g x0) H9) in -(ex3_2_ind A A (\lambda (a1: A).(\lambda (a2: A).(eq A (asucc g x0) (AHead a1 -a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g c0 u (asucc g a1)))) -(\lambda (_: A).(\lambda (a2: A).(arity g (CHead c0 (Bind Abst) u) t a2))) -(ex2 A (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w v) a1)) (\lambda -(a1: A).(arity g c0 (THead (Flat Appl) w (THead (Bind Abst) u t)) (asucc g -a1)))) (\lambda (x1: A).(\lambda (x2: A).(\lambda (H11: (eq A (asucc g x0) -(AHead x1 x2))).(\lambda (H12: (arity g c0 u (asucc g x1))).(\lambda (H13: -(arity g (CHead c0 (Bind Abst) u) t x2)).(let H14 \def (sym_eq A (asucc g x0) -(AHead x1 x2) H11) in (let H15 \def (asucc_gen_head g x1 x2 x0 H14) in -(ex2_ind A (\lambda (a0: A).(eq A x0 (AHead x1 a0))) (\lambda (a0: A).(eq A -x2 (asucc g a0))) (ex2 A (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w v) -a1)) (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w (THead (Bind Abst) u -t)) (asucc g a1)))) (\lambda (x3: A).(\lambda (H16: (eq A x0 (AHead x1 -x3))).(\lambda (H17: (eq A x2 (asucc g x3))).(let H18 \def (eq_ind A x2 -(\lambda (a: A).(arity g (CHead c0 (Bind Abst) u) t a)) H13 (asucc g x3) H17) -in (let H19 \def (eq_ind A x0 (\lambda (a: A).(arity g c0 v a)) H8 (AHead x1 -x3) H16) in (ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w v) -a1)) (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w (THead (Bind Abst) u -t)) (asucc g a1))) x3 (arity_appl g c0 w x1 (arity_repl g c0 w x H5 x1 -(leq_sym g x1 x (asucc_inj g x1 x (arity_mono g c0 u (asucc g x1) H12 (asucc -g x) H6)))) v x3 H19) (arity_appl g c0 w x H5 (THead (Bind Abst) u t) (asucc -g x3) (arity_head g c0 u x H6 t (asucc g x3) H18)))))))) H15)))))))) H10))))) -H7))))) H4))))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: -T).(\lambda (_: (ty3 g c0 t3 t4)).(\lambda (H1: (ex2 A (\lambda (a1: -A).(arity g c0 t3 a1)) (\lambda (a1: A).(arity g c0 t4 (asucc g -a1))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t4 t0)).(\lambda (H3: (ex2 A -(\lambda (a1: A).(arity g c0 t4 a1)) (\lambda (a1: A).(arity g c0 t0 (asucc g -a1))))).(let H4 \def H1 in (ex2_ind A (\lambda (a1: A).(arity g c0 t3 a1)) -(\lambda (a1: A).(arity g c0 t4 (asucc g a1))) (ex2 A (\lambda (a1: A).(arity -g c0 (THead (Flat Cast) t4 t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat -Cast) t0 t4) (asucc g a1)))) (\lambda (x: A).(\lambda (H5: (arity g c0 t3 -x)).(\lambda (H6: (arity g c0 t4 (asucc g x))).(let H7 \def H3 in (ex2_ind A -(\lambda (a1: A).(arity g c0 t4 a1)) (\lambda (a1: A).(arity g c0 t0 (asucc g -a1))) (ex2 A (\lambda (a1: A).(arity g c0 (THead (Flat Cast) t4 t3) a1)) -(\lambda (a1: A).(arity g c0 (THead (Flat Cast) t0 t4) (asucc g a1)))) -(\lambda (x0: A).(\lambda (H8: (arity g c0 t4 x0)).(\lambda (H9: (arity g c0 -t0 (asucc g x0))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Flat -Cast) t4 t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat Cast) t0 t4) -(asucc g a1))) x (arity_cast g c0 t4 x H6 t3 H5) (arity_cast g c0 t0 (asucc g -x) (arity_repl g c0 t0 (asucc g x0) H9 (asucc g (asucc g x)) (asucc_repl g x0 -(asucc g x) (arity_mono g c0 t4 x0 H8 (asucc g x) H6))) t4 H6))))) H7))))) -H4)))))))))) c t1 t2 H))))). -(* COMMENTS -Initial nodes: 3761 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/arity_props.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/arity_props.ma deleted file mode 100644 index 49d6c0572..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/arity_props.ma +++ /dev/null @@ -1,117 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/ty3/arity.ma". - -include "Basic-1/sc3/arity.ma". - -theorem ty3_predicative: - \forall (g: G).(\forall (c: C).(\forall (v: T).(\forall (t: T).(\forall (u: -T).((ty3 g c (THead (Bind Abst) v t) u) \to ((pc3 c u v) \to (\forall (P: -Prop).P))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (u: -T).(\lambda (H: (ty3 g c (THead (Bind Abst) v t) u)).(\lambda (H0: (pc3 c u -v)).(\lambda (P: Prop).(let H1 \def H in (ex3_2_ind T T (\lambda (t2: -T).(\lambda (_: T).(pc3 c (THead (Bind Abst) v t2) u))) (\lambda (_: -T).(\lambda (t0: T).(ty3 g c v t0))) (\lambda (t2: T).(\lambda (_: T).(ty3 g -(CHead c (Bind Abst) v) t t2))) P (\lambda (x0: T).(\lambda (x1: T).(\lambda -(_: (pc3 c (THead (Bind Abst) v x0) u)).(\lambda (H3: (ty3 g c v -x1)).(\lambda (_: (ty3 g (CHead c (Bind Abst) v) t x0)).(let H_y \def -(ty3_conv g c v x1 H3 (THead (Bind Abst) v t) u H H0) in (let H_x \def -(ty3_arity g c (THead (Bind Abst) v t) v H_y) in (let H5 \def H_x in (ex2_ind -A (\lambda (a1: A).(arity g c (THead (Bind Abst) v t) a1)) (\lambda (a1: -A).(arity g c v (asucc g a1))) P (\lambda (x: A).(\lambda (H6: (arity g c -(THead (Bind Abst) v t) x)).(\lambda (H7: (arity g c v (asucc g x))).(let H8 -\def (arity_gen_abst g c v t x H6) in (ex3_2_ind A A (\lambda (a1: -A).(\lambda (a2: A).(eq A x (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: -A).(arity g c v (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g -(CHead c (Bind Abst) v) t a2))) P (\lambda (x2: A).(\lambda (x3: A).(\lambda -(H9: (eq A x (AHead x2 x3))).(\lambda (H10: (arity g c v (asucc g -x2))).(\lambda (_: (arity g (CHead c (Bind Abst) v) t x3)).(let H12 \def -(eq_ind A x (\lambda (a: A).(arity g c v (asucc g a))) H7 (AHead x2 x3) H9) -in (leq_ahead_asucc_false g x2 (asucc g x3) (arity_mono g c v (asucc g (AHead -x2 x3)) H12 (asucc g x2) H10) P))))))) H8))))) H5))))))))) (ty3_gen_bind g -Abst c v t u H1)))))))))). -(* COMMENTS -Initial nodes: 497 -END *) - -theorem ty3_repellent: - \forall (g: G).(\forall (c: C).(\forall (w: T).(\forall (t: T).(\forall (u1: -T).((ty3 g c (THead (Bind Abst) w t) u1) \to (\forall (u2: T).((ty3 g (CHead -c (Bind Abst) w) t (lift (S O) O u2)) \to ((pc3 c u1 u2) \to (\forall (P: -Prop).P))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (w: T).(\lambda (t: T).(\lambda (u1: -T).(\lambda (H: (ty3 g c (THead (Bind Abst) w t) u1)).(\lambda (u2: -T).(\lambda (H0: (ty3 g (CHead c (Bind Abst) w) t (lift (S O) O -u2))).(\lambda (H1: (pc3 c u1 u2)).(\lambda (P: Prop).(ex_ind T (\lambda (t0: -T).(ty3 g (CHead c (Bind Abst) w) (lift (S O) O u2) t0)) P (\lambda (x: -T).(\lambda (H2: (ty3 g (CHead c (Bind Abst) w) (lift (S O) O u2) x)).(let H3 -\def (ty3_gen_lift g (CHead c (Bind Abst) w) u2 x (S O) O H2 c (drop_drop -(Bind Abst) O c c (drop_refl c) w)) in (ex2_ind T (\lambda (t2: T).(pc3 -(CHead c (Bind Abst) w) (lift (S O) O t2) x)) (\lambda (t2: T).(ty3 g c u2 -t2)) P (\lambda (x0: T).(\lambda (_: (pc3 (CHead c (Bind Abst) w) (lift (S O) -O x0) x)).(\lambda (H5: (ty3 g c u2 x0)).(let H_y \def (ty3_conv g c u2 x0 H5 -(THead (Bind Abst) w t) u1 H H1) in (let H_x \def (ty3_arity g (CHead c (Bind -Abst) w) t (lift (S O) O u2) H0) in (let H6 \def H_x in (ex2_ind A (\lambda -(a1: A).(arity g (CHead c (Bind Abst) w) t a1)) (\lambda (a1: A).(arity g -(CHead c (Bind Abst) w) (lift (S O) O u2) (asucc g a1))) P (\lambda (x1: -A).(\lambda (H7: (arity g (CHead c (Bind Abst) w) t x1)).(\lambda (H8: (arity -g (CHead c (Bind Abst) w) (lift (S O) O u2) (asucc g x1))).(let H_x0 \def -(ty3_arity g c (THead (Bind Abst) w t) u2 H_y) in (let H9 \def H_x0 in -(ex2_ind A (\lambda (a1: A).(arity g c (THead (Bind Abst) w t) a1)) (\lambda -(a1: A).(arity g c u2 (asucc g a1))) P (\lambda (x2: A).(\lambda (H10: (arity -g c (THead (Bind Abst) w t) x2)).(\lambda (H11: (arity g c u2 (asucc g -x2))).(arity_repellent g c w t x1 H7 x2 H10 (asucc_inj g x1 x2 (arity_mono g -c u2 (asucc g x1) (arity_gen_lift g (CHead c (Bind Abst) w) u2 (asucc g x1) -(S O) O H8 c (drop_drop (Bind Abst) O c c (drop_refl c) w)) (asucc g x2) -H11)) P)))) H9)))))) H6))))))) H3)))) (ty3_correct g (CHead c (Bind Abst) w) -t (lift (S O) O u2) H0))))))))))). -(* COMMENTS -Initial nodes: 651 -END *) - -theorem ty3_acyclic: - \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t -u) \to ((pc3 c u t) \to (\forall (P: Prop).P)))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H: -(ty3 g c t u)).(\lambda (H0: (pc3 c u t)).(\lambda (P: Prop).(let H_y \def -(ty3_conv g c t u H t u H H0) in (let H_x \def (ty3_arity g c t t H_y) in -(let H1 \def H_x in (ex2_ind A (\lambda (a1: A).(arity g c t a1)) (\lambda -(a1: A).(arity g c t (asucc g a1))) P (\lambda (x: A).(\lambda (H2: (arity g -c t x)).(\lambda (H3: (arity g c t (asucc g x))).(leq_asucc_false g x -(arity_mono g c t (asucc g x) H3 x H2) P)))) H1)))))))))). -(* COMMENTS -Initial nodes: 151 -END *) - -theorem ty3_sn3: - \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t -u) \to (sn3 c t))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H: -(ty3 g c t u)).(let H_x \def (ty3_arity g c t u H) in (let H0 \def H_x in -(ex2_ind A (\lambda (a1: A).(arity g c t a1)) (\lambda (a1: A).(arity g c u -(asucc g a1))) (sn3 c t) (\lambda (x: A).(\lambda (H1: (arity g c t -x)).(\lambda (_: (arity g c u (asucc g x))).(sc3_sn3 g x c t (sc3_arity g c t -x H1))))) H0))))))). -(* COMMENTS -Initial nodes: 119 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/dec.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/dec.ma deleted file mode 100644 index c176d635e..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/dec.ma +++ /dev/null @@ -1,438 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pc3/dec.ma". - -include "Basic-1/getl/flt.ma". - -include "Basic-1/getl/dec.ma". - -theorem ty3_inference: - \forall (g: G).(\forall (c: C).(\forall (t1: T).(or (ex T (\lambda (t2: -T).(ty3 g c t1 t2))) (\forall (t2: T).((ty3 g c t1 t2) \to False))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(flt_wf_ind (\lambda (c0: -C).(\lambda (t: T).(or (ex T (\lambda (t2: T).(ty3 g c0 t t2))) (\forall (t2: -T).((ty3 g c0 t t2) \to False))))) (\lambda (c2: C).(\lambda (t2: T).(T_ind -(\lambda (t: T).(((\forall (c1: C).(\forall (t3: T).((flt c1 t3 c2 t) \to (or -(ex T (\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4) -\to False))))))) \to (or (ex T (\lambda (t3: T).(ty3 g c2 t t3))) (\forall -(t3: T).((ty3 g c2 t t3) \to False))))) (\lambda (n: nat).(\lambda (_: -((\forall (c1: C).(\forall (t3: T).((flt c1 t3 c2 (TSort n)) \to (or (ex T -(\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to -False)))))))).(or_introl (ex T (\lambda (t3: T).(ty3 g c2 (TSort n) t3))) -(\forall (t3: T).((ty3 g c2 (TSort n) t3) \to False)) (ex_intro T (\lambda -(t3: T).(ty3 g c2 (TSort n) t3)) (TSort (next g n)) (ty3_sort g c2 n))))) -(\lambda (n: nat).(\lambda (H: ((\forall (c1: C).(\forall (t3: T).((flt c1 t3 -c2 (TLRef n)) \to (or (ex T (\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: -T).((ty3 g c1 t3 t4) \to False)))))))).(let H_x \def (getl_dec c2 n) in (let -H0 \def H_x in (or_ind (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda -(v: T).(getl n c2 (CHead e (Bind b) v)))))) (\forall (d: C).((getl n c2 d) -\to (\forall (P: Prop).P))) (or (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) -t3))) (\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to False))) (\lambda (H1: -(ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl n c2 (CHead -e (Bind b) v))))))).(ex_3_ind C B T (\lambda (e: C).(\lambda (b: B).(\lambda -(v: T).(getl n c2 (CHead e (Bind b) v))))) (or (ex T (\lambda (t3: T).(ty3 g -c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to False))) -(\lambda (x0: C).(\lambda (x1: B).(\lambda (x2: T).(\lambda (H2: (getl n c2 -(CHead x0 (Bind x1) x2))).(let H3 \def (H x0 x2 (getl_flt x1 c2 x0 x2 n H2)) -in (or_ind (ex T (\lambda (t3: T).(ty3 g x0 x2 t3))) (\forall (t3: T).((ty3 g -x0 x2 t3) \to False)) (or (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) -(\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to False))) (\lambda (H4: (ex T -(\lambda (t3: T).(ty3 g x0 x2 t3)))).(ex_ind T (\lambda (t3: T).(ty3 g x0 x2 -t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: -T).((ty3 g c2 (TLRef n) t3) \to False))) (\lambda (x: T).(\lambda (H5: (ty3 g -x0 x2 x)).(B_ind (\lambda (b: B).((getl n c2 (CHead x0 (Bind b) x2)) \to (or -(ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 -(TLRef n) t3) \to False))))) (\lambda (H6: (getl n c2 (CHead x0 (Bind Abbr) -x2))).(or_introl (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall -(t3: T).((ty3 g c2 (TLRef n) t3) \to False)) (ex_intro T (\lambda (t3: -T).(ty3 g c2 (TLRef n) t3)) (lift (S n) O x) (ty3_abbr g n c2 x0 x2 H6 x -H5)))) (\lambda (H6: (getl n c2 (CHead x0 (Bind Abst) x2))).(or_introl (ex T -(\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef -n) t3) \to False)) (ex_intro T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3)) -(lift (S n) O x2) (ty3_abst g n c2 x0 x2 H6 x H5)))) (\lambda (H6: (getl n c2 -(CHead x0 (Bind Void) x2))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 -(TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to False)) -(\lambda (t3: T).(\lambda (H7: (ty3 g c2 (TLRef n) t3)).(or_ind (ex3_3 C T T -(\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) -t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e -(Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u -t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 -(lift (S n) O u) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl -n c2 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: -T).(ty3 g e u t))))) False (\lambda (H8: (ex3_3 C T T (\lambda (_: -C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda -(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u))))) -(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind -C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O -t) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e -(Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u -t)))) False (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: -(pc3 c2 (lift (S n) O x5) t3)).(\lambda (H10: (getl n c2 (CHead x3 (Bind -Abbr) x4))).(\lambda (_: (ty3 g x3 x4 x5)).(let H12 \def (eq_ind C (CHead x0 -(Bind Void) x2) (\lambda (c0: C).(getl n c2 c0)) H6 (CHead x3 (Bind Abbr) x4) -(getl_mono c2 (CHead x0 (Bind Void) x2) n H6 (CHead x3 (Bind Abbr) x4) H10)) -in (let H13 \def (eq_ind C (CHead x0 (Bind Void) x2) (\lambda (ee: C).(match -ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | -(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr -\Rightarrow False | Abst \Rightarrow False | Void \Rightarrow True]) | (Flat -_) \Rightarrow False])])) I (CHead x3 (Bind Abbr) x4) (getl_mono c2 (CHead x0 -(Bind Void) x2) n H6 (CHead x3 (Bind Abbr) x4) H10)) in (False_ind False -H13))))))))) H8)) (\lambda (H8: (ex3_3 C T T (\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e: C).(\lambda -(u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u))))) (\lambda (e: -C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T -(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) -t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e -(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u -t)))) False (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: -(pc3 c2 (lift (S n) O x4) t3)).(\lambda (H10: (getl n c2 (CHead x3 (Bind -Abst) x4))).(\lambda (_: (ty3 g x3 x4 x5)).(let H12 \def (eq_ind C (CHead x0 -(Bind Void) x2) (\lambda (c0: C).(getl n c2 c0)) H6 (CHead x3 (Bind Abst) x4) -(getl_mono c2 (CHead x0 (Bind Void) x2) n H6 (CHead x3 (Bind Abst) x4) H10)) -in (let H13 \def (eq_ind C (CHead x0 (Bind Void) x2) (\lambda (ee: C).(match -ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | -(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr -\Rightarrow False | Abst \Rightarrow False | Void \Rightarrow True]) | (Flat -_) \Rightarrow False])])) I (CHead x3 (Bind Abst) x4) (getl_mono c2 (CHead x0 -(Bind Void) x2) n H6 (CHead x3 (Bind Abst) x4) H10)) in (False_ind False -H13))))))))) H8)) (ty3_gen_lref g c2 t3 n H7)))))) x1 H2))) H4)) (\lambda -(H4: ((\forall (t3: T).((ty3 g x0 x2 t3) \to False)))).(or_intror (ex T -(\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef -n) t3) \to False)) (\lambda (t3: T).(\lambda (H5: (ty3 g c2 (TLRef n) -t3)).(or_ind (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: -T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda -(_: T).(getl n c2 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: -T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e: -C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u))))) -(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))) False -(\lambda (H6: (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: -T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda -(_: T).(getl n c2 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: -T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T (\lambda (_: -C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda -(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u))))) -(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))) False -(\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (pc3 c2 (lift -(S n) O x5) t3)).(\lambda (H8: (getl n c2 (CHead x3 (Bind Abbr) -x4))).(\lambda (H9: (ty3 g x3 x4 x5)).(let H10 \def (eq_ind C (CHead x0 (Bind -x1) x2) (\lambda (c0: C).(getl n c2 c0)) H2 (CHead x3 (Bind Abbr) x4) -(getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind Abbr) x4) H8)) in -(let H11 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: -C).C) with [(CSort _) \Rightarrow x0 | (CHead c0 _ _) \Rightarrow c0])) -(CHead x0 (Bind x1) x2) (CHead x3 (Bind Abbr) x4) (getl_mono c2 (CHead x0 -(Bind x1) x2) n H2 (CHead x3 (Bind Abbr) x4) H8)) in ((let H12 \def (f_equal -C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with [(CSort _) -\Rightarrow x1 | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: -K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow x1])])) (CHead x0 -(Bind x1) x2) (CHead x3 (Bind Abbr) x4) (getl_mono c2 (CHead x0 (Bind x1) x2) -n H2 (CHead x3 (Bind Abbr) x4) H8)) in ((let H13 \def (f_equal C T (\lambda -(e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow x2 -| (CHead _ _ t) \Rightarrow t])) (CHead x0 (Bind x1) x2) (CHead x3 (Bind -Abbr) x4) (getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind Abbr) -x4) H8)) in (\lambda (_: (eq B x1 Abbr)).(\lambda (H15: (eq C x0 x3)).(let -H16 \def (eq_ind_r T x4 (\lambda (t: T).(getl n c2 (CHead x3 (Bind Abbr) t))) -H10 x2 H13) in (let H17 \def (eq_ind_r T x4 (\lambda (t: T).(ty3 g x3 t x5)) -H9 x2 H13) in (let H18 \def (eq_ind_r C x3 (\lambda (c0: C).(getl n c2 (CHead -c0 (Bind Abbr) x2))) H16 x0 H15) in (let H19 \def (eq_ind_r C x3 (\lambda -(c0: C).(ty3 g c0 x2 x5)) H17 x0 H15) in (H4 x5 H19)))))))) H12)) -H11))))))))) H6)) (\lambda (H6: (ex3_3 C T T (\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e: C).(\lambda -(u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u))))) (\lambda (e: -C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T -(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) -t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e -(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u -t)))) False (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H7: -(pc3 c2 (lift (S n) O x4) t3)).(\lambda (H8: (getl n c2 (CHead x3 (Bind Abst) -x4))).(\lambda (H9: (ty3 g x3 x4 x5)).(let H10 \def (eq_ind C (CHead x0 (Bind -x1) x2) (\lambda (c0: C).(getl n c2 c0)) H2 (CHead x3 (Bind Abst) x4) -(getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind Abst) x4) H8)) in -(let H11 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: -C).C) with [(CSort _) \Rightarrow x0 | (CHead c0 _ _) \Rightarrow c0])) -(CHead x0 (Bind x1) x2) (CHead x3 (Bind Abst) x4) (getl_mono c2 (CHead x0 -(Bind x1) x2) n H2 (CHead x3 (Bind Abst) x4) H8)) in ((let H12 \def (f_equal -C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with [(CSort _) -\Rightarrow x1 | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: -K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow x1])])) (CHead x0 -(Bind x1) x2) (CHead x3 (Bind Abst) x4) (getl_mono c2 (CHead x0 (Bind x1) x2) -n H2 (CHead x3 (Bind Abst) x4) H8)) in ((let H13 \def (f_equal C T (\lambda -(e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow x2 -| (CHead _ _ t) \Rightarrow t])) (CHead x0 (Bind x1) x2) (CHead x3 (Bind -Abst) x4) (getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind Abst) -x4) H8)) in (\lambda (_: (eq B x1 Abst)).(\lambda (H15: (eq C x0 x3)).(let -H16 \def (eq_ind_r T x4 (\lambda (t: T).(getl n c2 (CHead x3 (Bind Abst) t))) -H10 x2 H13) in (let H17 \def (eq_ind_r T x4 (\lambda (t: T).(ty3 g x3 t x5)) -H9 x2 H13) in (let H18 \def (eq_ind_r T x4 (\lambda (t: T).(pc3 c2 (lift (S -n) O t) t3)) H7 x2 H13) in (let H19 \def (eq_ind_r C x3 (\lambda (c0: -C).(getl n c2 (CHead c0 (Bind Abst) x2))) H16 x0 H15) in (let H20 \def -(eq_ind_r C x3 (\lambda (c0: C).(ty3 g c0 x2 x5)) H17 x0 H15) in (H4 x5 -H20))))))))) H12)) H11))))))))) H6)) (ty3_gen_lref g c2 t3 n H5)))))) -H3)))))) H1)) (\lambda (H1: ((\forall (d: C).((getl n c2 d) \to (\forall (P: -Prop).P))))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) -(\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to False)) (\lambda (t3: -T).(\lambda (H2: (ty3 g c2 (TLRef n) t3)).(or_ind (ex3_3 C T T (\lambda (_: -C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda -(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u))))) -(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T -T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) -t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e -(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u -t))))) False (\lambda (H3: (ex3_3 C T T (\lambda (_: C).(\lambda (_: -T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda (e: C).(\lambda -(u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u))))) (\lambda (e: -C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T -(\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) -t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e -(Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u -t)))) False (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: -(pc3 c2 (lift (S n) O x2) t3)).(\lambda (H5: (getl n c2 (CHead x0 (Bind Abbr) -x1))).(\lambda (_: (ty3 g x0 x1 x2)).(H1 (CHead x0 (Bind Abbr) x1) H5 -False))))))) H3)) (\lambda (H3: (ex3_3 C T T (\lambda (_: C).(\lambda (u: -T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e: C).(\lambda -(u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u))))) (\lambda (e: -C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T -(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) -t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e -(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u -t)))) False (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: -(pc3 c2 (lift (S n) O x1) t3)).(\lambda (H5: (getl n c2 (CHead x0 (Bind Abst) -x1))).(\lambda (_: (ty3 g x0 x1 x2)).(H1 (CHead x0 (Bind Abst) x1) H5 -False))))))) H3)) (ty3_gen_lref g c2 t3 n H2)))))) H0))))) (\lambda (k: -K).(\lambda (t: T).(\lambda (_: ((((\forall (c1: C).(\forall (t3: T).((flt c1 -t3 c2 t) \to (or (ex T (\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: -T).((ty3 g c1 t3 t4) \to False))))))) \to (or (ex T (\lambda (t3: T).(ty3 g -c2 t t3))) (\forall (t3: T).((ty3 g c2 t t3) \to False)))))).(\lambda (t0: -T).(\lambda (_: ((((\forall (c1: C).(\forall (t3: T).((flt c1 t3 c2 t0) \to -(or (ex T (\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3 g c1 t3 -t4) \to False))))))) \to (or (ex T (\lambda (t3: T).(ty3 g c2 t0 t3))) -(\forall (t3: T).((ty3 g c2 t0 t3) \to False)))))).(\lambda (H1: ((\forall -(c1: C).(\forall (t3: T).((flt c1 t3 c2 (THead k t t0)) \to (or (ex T -(\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to -False)))))))).(K_ind (\lambda (k0: K).(((\forall (c1: C).(\forall (t3: -T).((flt c1 t3 c2 (THead k0 t t0)) \to (or (ex T (\lambda (t4: T).(ty3 g c1 -t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to False))))))) \to (or (ex T -(\lambda (t3: T).(ty3 g c2 (THead k0 t t0) t3))) (\forall (t3: T).((ty3 g c2 -(THead k0 t t0) t3) \to False))))) (\lambda (b: B).(\lambda (H2: ((\forall -(c1: C).(\forall (t3: T).((flt c1 t3 c2 (THead (Bind b) t t0)) \to (or (ex T -(\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to -False)))))))).(let H3 \def (H2 c2 t (flt_thead_sx (Bind b) c2 t t0)) in -(or_ind (ex T (\lambda (t3: T).(ty3 g c2 t t3))) (\forall (t3: T).((ty3 g c2 -t t3) \to False)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Bind b) t t0) -t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0) t3) \to False))) -(\lambda (H4: (ex T (\lambda (t3: T).(ty3 g c2 t t3)))).(ex_ind T (\lambda -(t3: T).(ty3 g c2 t t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Bind b) -t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0) t3) \to -False))) (\lambda (x: T).(\lambda (H5: (ty3 g c2 t x)).(let H6 \def (H2 -(CHead c2 (Bind b) t) t0 (flt_shift (Bind b) c2 t t0)) in (or_ind (ex T -(\lambda (t3: T).(ty3 g (CHead c2 (Bind b) t) t0 t3))) (\forall (t3: T).((ty3 -g (CHead c2 (Bind b) t) t0 t3) \to False)) (or (ex T (\lambda (t3: T).(ty3 g -c2 (THead (Bind b) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t -t0) t3) \to False))) (\lambda (H7: (ex T (\lambda (t3: T).(ty3 g (CHead c2 -(Bind b) t) t0 t3)))).(ex_ind T (\lambda (t3: T).(ty3 g (CHead c2 (Bind b) t) -t0 t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Bind b) t t0) t3))) -(\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0) t3) \to False))) (\lambda -(x0: T).(\lambda (H8: (ty3 g (CHead c2 (Bind b) t) t0 x0)).(or_introl (ex T -(\lambda (t3: T).(ty3 g c2 (THead (Bind b) t t0) t3))) (\forall (t3: T).((ty3 -g c2 (THead (Bind b) t t0) t3) \to False)) (ex_intro T (\lambda (t3: T).(ty3 -g c2 (THead (Bind b) t t0) t3)) (THead (Bind b) t x0) (ty3_bind g c2 t x H5 b -t0 x0 H8))))) H7)) (\lambda (H7: ((\forall (t3: T).((ty3 g (CHead c2 (Bind b) -t) t0 t3) \to False)))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead -(Bind b) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0) t3) -\to False)) (\lambda (t3: T).(\lambda (H8: (ty3 g c2 (THead (Bind b) t t0) -t3)).(ex3_2_ind T T (\lambda (t4: T).(\lambda (_: T).(pc3 c2 (THead (Bind b) -t t4) t3))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c2 t t5))) (\lambda (t4: -T).(\lambda (_: T).(ty3 g (CHead c2 (Bind b) t) t0 t4))) False (\lambda (x0: -T).(\lambda (x1: T).(\lambda (_: (pc3 c2 (THead (Bind b) t x0) t3)).(\lambda -(_: (ty3 g c2 t x1)).(\lambda (H11: (ty3 g (CHead c2 (Bind b) t) t0 x0)).(H7 -x0 H11)))))) (ty3_gen_bind g b c2 t t0 t3 H8)))))) H6)))) H4)) (\lambda (H4: -((\forall (t3: T).((ty3 g c2 t t3) \to False)))).(or_intror (ex T (\lambda -(t3: T).(ty3 g c2 (THead (Bind b) t t0) t3))) (\forall (t3: T).((ty3 g c2 -(THead (Bind b) t t0) t3) \to False)) (\lambda (t3: T).(\lambda (H5: (ty3 g -c2 (THead (Bind b) t t0) t3)).(ex3_2_ind T T (\lambda (t4: T).(\lambda (_: -T).(pc3 c2 (THead (Bind b) t t4) t3))) (\lambda (_: T).(\lambda (t5: T).(ty3 -g c2 t t5))) (\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead c2 (Bind b) t) t0 -t4))) False (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 c2 (THead -(Bind b) t x0) t3)).(\lambda (H7: (ty3 g c2 t x1)).(\lambda (_: (ty3 g (CHead -c2 (Bind b) t) t0 x0)).(H4 x1 H7)))))) (ty3_gen_bind g b c2 t t0 t3 H5)))))) -H3)))) (\lambda (f: F).(\lambda (H2: ((\forall (c1: C).(\forall (t3: T).((flt -c1 t3 c2 (THead (Flat f) t t0)) \to (or (ex T (\lambda (t4: T).(ty3 g c1 t3 -t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to False)))))))).(F_ind (\lambda -(f0: F).(((\forall (c1: C).(\forall (t3: T).((flt c1 t3 c2 (THead (Flat f0) t -t0)) \to (or (ex T (\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3 -g c1 t3 t4) \to False))))))) \to (or (ex T (\lambda (t3: T).(ty3 g c2 (THead -(Flat f0) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat f0) t t0) t3) -\to False))))) (\lambda (H3: ((\forall (c1: C).(\forall (t3: T).((flt c1 t3 -c2 (THead (Flat Appl) t t0)) \to (or (ex T (\lambda (t4: T).(ty3 g c1 t3 -t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to False)))))))).(let H4 \def (H3 -c2 t (flt_thead_sx (Flat Appl) c2 t t0)) in (or_ind (ex T (\lambda (t3: -T).(ty3 g c2 t t3))) (\forall (t3: T).((ty3 g c2 t t3) \to False)) (or (ex T -(\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: -T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False))) (\lambda (H5: (ex T -(\lambda (t3: T).(ty3 g c2 t t3)))).(ex_ind T (\lambda (t3: T).(ty3 g c2 t -t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) -(\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False))) -(\lambda (x: T).(\lambda (H6: (ty3 g c2 t x)).(let H7 \def (H3 c2 t0 -(flt_thead_dx (Flat Appl) c2 t t0)) in (or_ind (ex T (\lambda (t3: T).(ty3 g -c2 t0 t3))) (\forall (t3: T).((ty3 g c2 t0 t3) \to False)) (or (ex T (\lambda -(t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 -(THead (Flat Appl) t t0) t3) \to False))) (\lambda (H8: (ex T (\lambda (t3: -T).(ty3 g c2 t0 t3)))).(ex_ind T (\lambda (t3: T).(ty3 g c2 t0 t3)) (or (ex T -(\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: -T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False))) (\lambda (x0: -T).(\lambda (H9: (ty3 g c2 t0 x0)).(ex_ind T (\lambda (t3: T).(ty3 g c2 x0 -t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) -(\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False))) -(\lambda (x1: T).(\lambda (H10: (ty3 g c2 x0 x1)).(ex_ind T (\lambda (t3: -T).(ty3 g c2 x t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t -t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to -False))) (\lambda (x2: T).(\lambda (H11: (ty3 g c2 x x2)).(let H12 \def -(ty3_sn3 g c2 x x2 H11) in (let H_x \def (nf2_sn3 c2 x H12) in (let H13 \def -H_x in (ex2_ind T (\lambda (u: T).(pr3 c2 x u)) (\lambda (u: T).(nf2 c2 u)) -(or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall -(t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False))) (\lambda (x3: -T).(\lambda (H14: (pr3 c2 x x3)).(\lambda (H15: (nf2 c2 x3)).(let H16 \def -(ty3_sred_pr3 c2 x x3 H14 g x2 H11) in (let H_x0 \def (pc3_abst_dec g c2 x0 -x1 H10 x3 x2 H16) in (let H17 \def H_x0 in (or_ind (ex4_2 T T (\lambda (u: -T).(\lambda (_: T).(pc3 c2 x0 (THead (Bind Abst) x3 u)))) (\lambda (u: -T).(\lambda (v2: T).(ty3 g c2 (THead (Bind Abst) v2 u) x1))) (\lambda (_: -T).(\lambda (v2: T).(pr3 c2 x3 v2))) (\lambda (_: T).(\lambda (v2: T).(nf2 c2 -v2)))) (\forall (u: T).((pc3 c2 x0 (THead (Bind Abst) x3 u)) \to False)) (or -(ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: -T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False))) (\lambda (H18: (ex4_2 -T T (\lambda (u: T).(\lambda (_: T).(pc3 c2 x0 (THead (Bind Abst) x3 u)))) -(\lambda (u: T).(\lambda (v2: T).(ty3 g c2 (THead (Bind Abst) v2 u) x1))) -(\lambda (_: T).(\lambda (v2: T).(pr3 c2 x3 v2))) (\lambda (_: T).(\lambda -(v2: T).(nf2 c2 v2))))).(ex4_2_ind T T (\lambda (u: T).(\lambda (_: T).(pc3 -c2 x0 (THead (Bind Abst) x3 u)))) (\lambda (u: T).(\lambda (v2: T).(ty3 g c2 -(THead (Bind Abst) v2 u) x1))) (\lambda (_: T).(\lambda (v2: T).(pr3 c2 x3 -v2))) (\lambda (_: T).(\lambda (v2: T).(nf2 c2 v2))) (or (ex T (\lambda (t3: -T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 -(THead (Flat Appl) t t0) t3) \to False))) (\lambda (x4: T).(\lambda (x5: -T).(\lambda (H19: (pc3 c2 x0 (THead (Bind Abst) x3 x4))).(\lambda (H20: (ty3 -g c2 (THead (Bind Abst) x5 x4) x1)).(\lambda (H21: (pr3 c2 x3 x5)).(\lambda -(_: (nf2 c2 x5)).(let H_y \def (nf2_pr3_unfold c2 x3 x5 H21 H15) in (let H23 -\def (eq_ind_r T x5 (\lambda (t3: T).(pr3 c2 x3 t3)) H21 x3 H_y) in (let H24 -\def (eq_ind_r T x5 (\lambda (t3: T).(ty3 g c2 (THead (Bind Abst) t3 x4) x1)) -H20 x3 H_y) in (or_introl (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) -t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to -False)) (ex_intro T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3)) -(THead (Flat Appl) t (THead (Bind Abst) x3 x4)) (ty3_appl g c2 t x3 (ty3_tred -g c2 t x H6 x3 H14) t0 x4 (ty3_conv g c2 (THead (Bind Abst) x3 x4) x1 H24 t0 -x0 H9 H19))))))))))))) H18)) (\lambda (H18: ((\forall (u: T).((pc3 c2 x0 -(THead (Bind Abst) x3 u)) \to False)))).(or_intror (ex T (\lambda (t3: -T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 -(THead (Flat Appl) t t0) t3) \to False)) (\lambda (t3: T).(\lambda (H19: (ty3 -g c2 (THead (Flat Appl) t t0) t3)).(ex3_2_ind T T (\lambda (u: T).(\lambda -(t4: T).(pc3 c2 (THead (Flat Appl) t (THead (Bind Abst) u t4)) t3))) (\lambda -(u: T).(\lambda (t4: T).(ty3 g c2 t0 (THead (Bind Abst) u t4)))) (\lambda (u: -T).(\lambda (_: T).(ty3 g c2 t u))) False (\lambda (x4: T).(\lambda (x5: -T).(\lambda (_: (pc3 c2 (THead (Flat Appl) t (THead (Bind Abst) x4 x5)) -t3)).(\lambda (H21: (ty3 g c2 t0 (THead (Bind Abst) x4 x5))).(\lambda (H22: -(ty3 g c2 t x4)).(let H_y \def (ty3_unique g c2 t x4 H22 x H6) in (let H_y0 -\def (ty3_unique g c2 t0 (THead (Bind Abst) x4 x5) H21 x0 H9) in (H18 x5 -(pc3_t (THead (Bind Abst) x4 x5) c2 x0 (pc3_s c2 x0 (THead (Bind Abst) x4 x5) -H_y0) (THead (Bind Abst) x3 x5) (pc3_head_1 c2 x4 x3 (pc3_t x c2 x4 H_y x3 -(pc3_pr3_r c2 x x3 H14)) (Bind Abst) x5)))))))))) (ty3_gen_appl g c2 t t0 t3 -H19)))))) H17))))))) H13)))))) (ty3_correct g c2 t x H6)))) (ty3_correct g c2 -t0 x0 H9)))) H8)) (\lambda (H8: ((\forall (t3: T).((ty3 g c2 t0 t3) \to -False)))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t -t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to -False)) (\lambda (t3: T).(\lambda (H9: (ty3 g c2 (THead (Flat Appl) t t0) -t3)).(ex3_2_ind T T (\lambda (u: T).(\lambda (t4: T).(pc3 c2 (THead (Flat -Appl) t (THead (Bind Abst) u t4)) t3))) (\lambda (u: T).(\lambda (t4: T).(ty3 -g c2 t0 (THead (Bind Abst) u t4)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c2 -t u))) False (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 c2 (THead -(Flat Appl) t (THead (Bind Abst) x0 x1)) t3)).(\lambda (H11: (ty3 g c2 t0 -(THead (Bind Abst) x0 x1))).(\lambda (_: (ty3 g c2 t x0)).(H8 (THead (Bind -Abst) x0 x1) H11)))))) (ty3_gen_appl g c2 t t0 t3 H9)))))) H7)))) H5)) -(\lambda (H5: ((\forall (t3: T).((ty3 g c2 t t3) \to False)))).(or_intror (ex -T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: -T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False)) (\lambda (t3: -T).(\lambda (H6: (ty3 g c2 (THead (Flat Appl) t t0) t3)).(ex3_2_ind T T -(\lambda (u: T).(\lambda (t4: T).(pc3 c2 (THead (Flat Appl) t (THead (Bind -Abst) u t4)) t3))) (\lambda (u: T).(\lambda (t4: T).(ty3 g c2 t0 (THead (Bind -Abst) u t4)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c2 t u))) False -(\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 c2 (THead (Flat Appl) t -(THead (Bind Abst) x0 x1)) t3)).(\lambda (_: (ty3 g c2 t0 (THead (Bind Abst) -x0 x1))).(\lambda (H9: (ty3 g c2 t x0)).(H5 x0 H9)))))) (ty3_gen_appl g c2 t -t0 t3 H6)))))) H4))) (\lambda (H3: ((\forall (c1: C).(\forall (t3: T).((flt -c1 t3 c2 (THead (Flat Cast) t t0)) \to (or (ex T (\lambda (t4: T).(ty3 g c1 -t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to False)))))))).(let H4 \def -(H3 c2 t (flt_thead_sx (Flat Cast) c2 t t0)) in (or_ind (ex T (\lambda (t3: -T).(ty3 g c2 t t3))) (\forall (t3: T).((ty3 g c2 t t3) \to False)) (or (ex T -(\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3: -T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False))) (\lambda (H5: (ex T -(\lambda (t3: T).(ty3 g c2 t t3)))).(ex_ind T (\lambda (t3: T).(ty3 g c2 t -t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) -(\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False))) -(\lambda (x: T).(\lambda (H6: (ty3 g c2 t x)).(let H7 \def (H3 c2 t0 -(flt_thead_dx (Flat Cast) c2 t t0)) in (or_ind (ex T (\lambda (t3: T).(ty3 g -c2 t0 t3))) (\forall (t3: T).((ty3 g c2 t0 t3) \to False)) (or (ex T (\lambda -(t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3: T).((ty3 g c2 -(THead (Flat Cast) t t0) t3) \to False))) (\lambda (H8: (ex T (\lambda (t3: -T).(ty3 g c2 t0 t3)))).(ex_ind T (\lambda (t3: T).(ty3 g c2 t0 t3)) (or (ex T -(\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3: -T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False))) (\lambda (x0: -T).(\lambda (H9: (ty3 g c2 t0 x0)).(ex_ind T (\lambda (t3: T).(ty3 g c2 x0 -t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) -(\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False))) -(\lambda (x1: T).(\lambda (H10: (ty3 g c2 x0 x1)).(let H_x \def (pc3_dec g c2 -x0 x1 H10 t x H6) in (let H11 \def H_x in (or_ind (pc3 c2 x0 t) ((pc3 c2 x0 -t) \to False) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) -t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False))) -(\lambda (H12: (pc3 c2 x0 t)).(or_introl (ex T (\lambda (t3: T).(ty3 g c2 -(THead (Flat Cast) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) -t t0) t3) \to False)) (ex_intro T (\lambda (t3: T).(ty3 g c2 (THead (Flat -Cast) t t0) t3)) (THead (Flat Cast) x t) (ty3_cast g c2 t0 t (ty3_conv g c2 t -x H6 t0 x0 H9 H12) x H6)))) (\lambda (H12: (((pc3 c2 x0 t) \to -False))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) -t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False)) -(\lambda (t3: T).(\lambda (H13: (ty3 g c2 (THead (Flat Cast) t t0) -t3)).(ex3_ind T (\lambda (t4: T).(pc3 c2 (THead (Flat Cast) t4 t) t3)) -(\lambda (_: T).(ty3 g c2 t0 t)) (\lambda (t4: T).(ty3 g c2 t t4)) False -(\lambda (x2: T).(\lambda (_: (pc3 c2 (THead (Flat Cast) x2 t) t3)).(\lambda -(H15: (ty3 g c2 t0 t)).(\lambda (H16: (ty3 g c2 t x2)).(let H_y \def -(ty3_unique g c2 t x2 H16 x H6) in (let H_y0 \def (ty3_unique g c2 t0 t H15 -x0 H9) in (H12 (ex2_sym T (pr3 c2 t) (pr3 c2 x0) H_y0)))))))) (ty3_gen_cast g -c2 t0 t t3 H13)))))) H11))))) (ty3_correct g c2 t0 x0 H9)))) H8)) (\lambda -(H8: ((\forall (t3: T).((ty3 g c2 t0 t3) \to False)))).(or_intror (ex T -(\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3: -T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False)) (\lambda (t3: -T).(\lambda (H9: (ty3 g c2 (THead (Flat Cast) t t0) t3)).(ex3_ind T (\lambda -(t4: T).(pc3 c2 (THead (Flat Cast) t4 t) t3)) (\lambda (_: T).(ty3 g c2 t0 -t)) (\lambda (t4: T).(ty3 g c2 t t4)) False (\lambda (x0: T).(\lambda (_: -(pc3 c2 (THead (Flat Cast) x0 t) t3)).(\lambda (H11: (ty3 g c2 t0 -t)).(\lambda (_: (ty3 g c2 t x0)).(H8 t H11))))) (ty3_gen_cast g c2 t0 t t3 -H9)))))) H7)))) H5)) (\lambda (H5: ((\forall (t3: T).((ty3 g c2 t t3) \to -False)))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t -t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to -False)) (\lambda (t3: T).(\lambda (H6: (ty3 g c2 (THead (Flat Cast) t t0) -t3)).(ex3_ind T (\lambda (t4: T).(pc3 c2 (THead (Flat Cast) t4 t) t3)) -(\lambda (_: T).(ty3 g c2 t0 t)) (\lambda (t4: T).(ty3 g c2 t t4)) False -(\lambda (x0: T).(\lambda (_: (pc3 c2 (THead (Flat Cast) x0 t) t3)).(\lambda -(_: (ty3 g c2 t0 t)).(\lambda (H9: (ty3 g c2 t x0)).(ex_ind T (\lambda (t4: -T).(ty3 g c2 x0 t4)) False (\lambda (x: T).(\lambda (_: (ty3 g c2 x0 x)).(H5 -x0 H9))) (ty3_correct g c2 t x0 H9)))))) (ty3_gen_cast g c2 t0 t t3 H6)))))) -H4))) f H2))) k H1))))))) t2))) c t1))). -(* COMMENTS -Initial nodes: 9001 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/defs.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/defs.ma deleted file mode 100644 index d1f500cdf..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/defs.ma +++ /dev/null @@ -1,49 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/G/defs.ma". - -include "Basic-1/pc3/defs.ma". - -inductive ty3 (g: G): C \to (T \to (T \to Prop)) \def -| ty3_conv: \forall (c: C).(\forall (t2: T).(\forall (t: T).((ty3 g c t2 t) -\to (\forall (u: T).(\forall (t1: T).((ty3 g c u t1) \to ((pc3 c t1 t2) \to -(ty3 g c u t2)))))))) -| ty3_sort: \forall (c: C).(\forall (m: nat).(ty3 g c (TSort m) (TSort (next -g m)))) -| ty3_abbr: \forall (n: nat).(\forall (c: C).(\forall (d: C).(\forall (u: -T).((getl n c (CHead d (Bind Abbr) u)) \to (\forall (t: T).((ty3 g d u t) \to -(ty3 g c (TLRef n) (lift (S n) O t)))))))) -| ty3_abst: \forall (n: nat).(\forall (c: C).(\forall (d: C).(\forall (u: -T).((getl n c (CHead d (Bind Abst) u)) \to (\forall (t: T).((ty3 g d u t) \to -(ty3 g c (TLRef n) (lift (S n) O u)))))))) -| ty3_bind: \forall (c: C).(\forall (u: T).(\forall (t: T).((ty3 g c u t) \to -(\forall (b: B).(\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c (Bind b) -u) t1 t2) \to (ty3 g c (THead (Bind b) u t1) (THead (Bind b) u t2))))))))) -| ty3_appl: \forall (c: C).(\forall (w: T).(\forall (u: T).((ty3 g c w u) \to -(\forall (v: T).(\forall (t: T).((ty3 g c v (THead (Bind Abst) u t)) \to (ty3 -g c (THead (Flat Appl) w v) (THead (Flat Appl) w (THead (Bind Abst) u -t))))))))) -| ty3_cast: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c t1 t2) -\to (\forall (t0: T).((ty3 g c t2 t0) \to (ty3 g c (THead (Flat Cast) t2 t1) -(THead (Flat Cast) t0 t2))))))). - -inductive tys3 (g: G) (c: C): TList \to (T \to Prop) \def -| tys3_nil: \forall (u: T).(\forall (u0: T).((ty3 g c u u0) \to (tys3 g c -TNil u))) -| tys3_cons: \forall (t: T).(\forall (u: T).((ty3 g c t u) \to (\forall (ts: -TList).((tys3 g c ts u) \to (tys3 g c (TCons t ts) u))))). - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/fsubst0.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/fsubst0.ma deleted file mode 100644 index f92366a26..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/fsubst0.ma +++ /dev/null @@ -1,995 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/ty3/props.ma". - -include "Basic-1/pc3/fsubst0.ma". - -include "Basic-1/getl/getl.ma". - -theorem ty3_fsubst0: - \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t: T).((ty3 g c1 -t1 t) \to (\forall (i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t2: -T).((fsubst0 i u c1 t1 c2 t2) \to (\forall (e: C).((getl i c1 (CHead e (Bind -Abbr) u)) \to (ty3 g c2 t2 t)))))))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t: T).(\lambda -(H: (ty3 g c1 t1 t)).(ty3_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda -(t2: T).(\forall (i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t3: -T).((fsubst0 i u c t0 c2 t3) \to (\forall (e: C).((getl i c (CHead e (Bind -Abbr) u)) \to (ty3 g c2 t3 t2))))))))))) (\lambda (c: C).(\lambda (t2: -T).(\lambda (t0: T).(\lambda (H0: (ty3 g c t2 t0)).(\lambda (H1: ((\forall -(i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t3: T).((fsubst0 i u c t2 -c2 t3) \to (\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) \to (ty3 g c2 -t3 t0)))))))))).(\lambda (u: T).(\lambda (t3: T).(\lambda (_: (ty3 g c u -t3)).(\lambda (H3: ((\forall (i: nat).(\forall (u0: T).(\forall (c2: -C).(\forall (t4: T).((fsubst0 i u0 c u c2 t4) \to (\forall (e: C).((getl i c -(CHead e (Bind Abbr) u0)) \to (ty3 g c2 t4 t3)))))))))).(\lambda (H4: (pc3 c -t3 t2)).(\lambda (i: nat).(\lambda (u0: T).(\lambda (c2: C).(\lambda (t4: -T).(\lambda (H5: (fsubst0 i u0 c u c2 t4)).(fsubst0_ind i u0 c u (\lambda -(c0: C).(\lambda (t5: T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u0)) -\to (ty3 g c0 t5 t2))))) (\lambda (t5: T).(\lambda (H6: (subst0 i u0 u -t5)).(\lambda (e: C).(\lambda (H7: (getl i c (CHead e (Bind Abbr) -u0))).(ty3_conv g c t2 t0 H0 t5 t3 (H3 i u0 c t5 (fsubst0_snd i u0 c u t5 H6) -e H7) H4))))) (\lambda (c3: C).(\lambda (H6: (csubst0 i u0 c c3)).(\lambda -(e: C).(\lambda (H7: (getl i c (CHead e (Bind Abbr) u0))).(ty3_conv g c3 t2 -t0 (H1 i u0 c3 t2 (fsubst0_fst i u0 c t2 c3 H6) e H7) u t3 (H3 i u0 c3 u -(fsubst0_fst i u0 c u c3 H6) e H7) (pc3_fsubst0 c t3 t2 H4 i u0 c3 t3 -(fsubst0_fst i u0 c t3 c3 H6) e H7)))))) (\lambda (t5: T).(\lambda (H6: -(subst0 i u0 u t5)).(\lambda (c3: C).(\lambda (H7: (csubst0 i u0 c -c3)).(\lambda (e: C).(\lambda (H8: (getl i c (CHead e (Bind Abbr) -u0))).(ty3_conv g c3 t2 t0 (H1 i u0 c3 t2 (fsubst0_fst i u0 c t2 c3 H7) e H8) -t5 t3 (H3 i u0 c3 t5 (fsubst0_both i u0 c u t5 H6 c3 H7) e H8) (pc3_fsubst0 c -t3 t2 H4 i u0 c3 t3 (fsubst0_fst i u0 c t3 c3 H7) e H8)))))))) c2 t4 -H5)))))))))))))))) (\lambda (c: C).(\lambda (m: nat).(\lambda (i: -nat).(\lambda (u: T).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H0: (fsubst0 -i u c (TSort m) c2 t2)).(fsubst0_ind i u c (TSort m) (\lambda (c0: -C).(\lambda (t0: T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) \to -(ty3 g c0 t0 (TSort (next g m))))))) (\lambda (t3: T).(\lambda (H1: (subst0 i -u (TSort m) t3)).(\lambda (e: C).(\lambda (_: (getl i c (CHead e (Bind Abbr) -u))).(subst0_gen_sort u t3 i m H1 (ty3 g c t3 (TSort (next g m)))))))) -(\lambda (c3: C).(\lambda (_: (csubst0 i u c c3)).(\lambda (e: C).(\lambda -(_: (getl i c (CHead e (Bind Abbr) u))).(ty3_sort g c3 m))))) (\lambda (t3: -T).(\lambda (H1: (subst0 i u (TSort m) t3)).(\lambda (c3: C).(\lambda (_: -(csubst0 i u c c3)).(\lambda (e: C).(\lambda (_: (getl i c (CHead e (Bind -Abbr) u))).(subst0_gen_sort u t3 i m H1 (ty3 g c3 t3 (TSort (next g -m)))))))))) c2 t2 H0)))))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d: -C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind Abbr) u))).(\lambda -(t0: T).(\lambda (H1: (ty3 g d u t0)).(\lambda (H2: ((\forall (i: -nat).(\forall (u0: T).(\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 d u c2 -t2) \to (\forall (e: C).((getl i d (CHead e (Bind Abbr) u0)) \to (ty3 g c2 t2 -t0)))))))))).(\lambda (i: nat).(\lambda (u0: T).(\lambda (c2: C).(\lambda -(t2: T).(\lambda (H3: (fsubst0 i u0 c (TLRef n) c2 t2)).(fsubst0_ind i u0 c -(TLRef n) (\lambda (c0: C).(\lambda (t3: T).(\forall (e: C).((getl i c (CHead -e (Bind Abbr) u0)) \to (ty3 g c0 t3 (lift (S n) O t0)))))) (\lambda (t3: -T).(\lambda (H4: (subst0 i u0 (TLRef n) t3)).(\lambda (e: C).(\lambda (H5: -(getl i c (CHead e (Bind Abbr) u0))).(land_ind (eq nat n i) (eq T t3 (lift (S -n) O u0)) (ty3 g c t3 (lift (S n) O t0)) (\lambda (H6: (eq nat n i)).(\lambda -(H7: (eq T t3 (lift (S n) O u0))).(eq_ind_r T (lift (S n) O u0) (\lambda (t4: -T).(ty3 g c t4 (lift (S n) O t0))) (let H8 \def (eq_ind_r nat i (\lambda (n0: -nat).(getl n0 c (CHead e (Bind Abbr) u0))) H5 n H6) in (let H9 \def (eq_ind C -(CHead d (Bind Abbr) u) (\lambda (c0: C).(getl n c c0)) H0 (CHead e (Bind -Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) n H0 (CHead e (Bind Abbr) u0) -H8)) in (let H10 \def (f_equal C C (\lambda (e0: C).(match e0 in C return -(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow -c0])) (CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono c (CHead d -(Bind Abbr) u) n H0 (CHead e (Bind Abbr) u0) H8)) in ((let H11 \def (f_equal -C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) -\Rightarrow u | (CHead _ _ t4) \Rightarrow t4])) (CHead d (Bind Abbr) u) -(CHead e (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) n H0 (CHead e -(Bind Abbr) u0) H8)) in (\lambda (H12: (eq C d e)).(let H13 \def (eq_ind_r C -e (\lambda (c0: C).(getl n c (CHead c0 (Bind Abbr) u0))) H9 d H12) in (let -H14 \def (eq_ind_r T u0 (\lambda (t4: T).(getl n c (CHead d (Bind Abbr) t4))) -H13 u H11) in (eq_ind T u (\lambda (t4: T).(ty3 g c (lift (S n) O t4) (lift -(S n) O t0))) (ty3_lift g d u t0 H1 c O (S n) (getl_drop Abbr c d u n H14)) -u0 H11))))) H10)))) t3 H7))) (subst0_gen_lref u0 t3 i n H4)))))) (\lambda -(c3: C).(\lambda (H4: (csubst0 i u0 c c3)).(\lambda (e: C).(\lambda (H5: -(getl i c (CHead e (Bind Abbr) u0))).(lt_le_e n i (ty3 g c3 (TLRef n) (lift -(S n) O t0)) (\lambda (H6: (lt n i)).(let H7 \def (csubst0_getl_lt i n H6 c -c3 u0 H4 (CHead d (Bind Abbr) u) H0) in (or4_ind (getl n c3 (CHead d (Bind -Abbr) u)) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b) u1)))))) -(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 -(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: -T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w)))))) (ex3_4 B C C T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C -(CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda -(_: C).(\lambda (e2: C).(\lambda (u1: T).(getl n c3 (CHead e2 (Bind b) -u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: -T).(csubst0 (minus i (S n)) u0 e1 e2)))))) (ex4_5 B C C T T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C -(CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e2 -(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -(minus i (S n)) u0 e1 e2))))))) (ty3 g c3 (TLRef n) (lift (S n) O t0)) -(\lambda (H8: (getl n c3 (CHead d (Bind Abbr) u))).(ty3_abbr g n c3 d u H8 t0 -H1)) (\lambda (H8: (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda -(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b) -u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: -T).(getl n c3 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 -w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b) u1)))))) -(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 -(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: -T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w))))) (ty3 g c3 (TLRef n) -(lift (S n) O t0)) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: -T).(\lambda (x3: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead x1 -(Bind x0) x2))).(\lambda (H10: (getl n c3 (CHead x1 (Bind x0) x3))).(\lambda -(H11: (subst0 (minus i (S n)) u0 x2 x3)).(let H12 \def (f_equal C C (\lambda -(e0: C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow -d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abbr) u) (CHead x1 (Bind -x0) x2) H9) in ((let H13 \def (f_equal C B (\lambda (e0: C).(match e0 in C -return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) -\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead -x1 (Bind x0) x2) H9) in ((let H14 \def (f_equal C T (\lambda (e0: C).(match -e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ -t3) \Rightarrow t3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H9) in -(\lambda (H15: (eq B Abbr x0)).(\lambda (H16: (eq C d x1)).(let H17 \def -(eq_ind_r T x2 (\lambda (t3: T).(subst0 (minus i (S n)) u0 t3 x3)) H11 u H14) -in (let H18 \def (eq_ind_r C x1 (\lambda (c0: C).(getl n c3 (CHead c0 (Bind -x0) x3))) H10 d H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b: B).(getl n -c3 (CHead d (Bind b) x3))) H18 Abbr H15) in (let H20 \def (eq_ind nat (minus -i n) (\lambda (n0: nat).(getl n0 (CHead d (Bind Abbr) x3) (CHead e (Bind -Abbr) u0))) (getl_conf_le i (CHead e (Bind Abbr) u0) c3 (csubst0_getl_ge i i -(le_n i) c c3 u0 H4 (CHead e (Bind Abbr) u0) H5) (CHead d (Bind Abbr) x3) n -H19 (le_S_n n i (le_S (S n) i H6))) (S (minus i (S n))) (minus_x_Sy i n H6)) -in (ty3_abbr g n c3 d x3 H19 t0 (H2 (minus i (S n)) u0 d x3 (fsubst0_snd -(minus i (S n)) u0 d u x3 H17) e (getl_gen_S (Bind Abbr) d (CHead e (Bind -Abbr) u0) x3 (minus i (S n)) H20)))))))))) H13)) H12))))))))) H8)) (\lambda -(H8: (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda -(u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda -(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl n c3 (CHead e2 -(Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 (minus i (S n)) u0 e1 e2))))))).(ex3_4_ind B C C T (\lambda -(b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind -Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u1: T).(getl n c3 (CHead e2 (Bind b) u1)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) -u0 e1 e2))))) (ty3 g c3 (TLRef n) (lift (S n) O t0)) (\lambda (x0: -B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H9: (eq C -(CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3))).(\lambda (H10: (getl n c3 -(CHead x2 (Bind x0) x3))).(\lambda (H11: (csubst0 (minus i (S n)) u0 x1 -x2)).(let H12 \def (f_equal C C (\lambda (e0: C).(match e0 in C return -(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow -c0])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H13 \def -(f_equal C B (\lambda (e0: C).(match e0 in C return (\lambda (_: C).B) with -[(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k in K return -(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow -Abbr])])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H14 -\def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow u | (CHead _ _ t3) \Rightarrow t3])) (CHead d -(Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in (\lambda (H15: (eq B Abbr -x0)).(\lambda (H16: (eq C d x1)).(let H17 \def (eq_ind_r T x3 (\lambda (t3: -T).(getl n c3 (CHead x2 (Bind x0) t3))) H10 u H14) in (let H18 \def (eq_ind_r -C x1 (\lambda (c0: C).(csubst0 (minus i (S n)) u0 c0 x2)) H11 d H16) in (let -H19 \def (eq_ind_r B x0 (\lambda (b: B).(getl n c3 (CHead x2 (Bind b) u))) -H17 Abbr H15) in (let H20 \def (eq_ind nat (minus i n) (\lambda (n0: -nat).(getl n0 (CHead x2 (Bind Abbr) u) (CHead e (Bind Abbr) u0))) -(getl_conf_le i (CHead e (Bind Abbr) u0) c3 (csubst0_getl_ge i i (le_n i) c -c3 u0 H4 (CHead e (Bind Abbr) u0) H5) (CHead x2 (Bind Abbr) u) n H19 (le_S_n -n i (le_S (S n) i H6))) (S (minus i (S n))) (minus_x_Sy i n H6)) in (ty3_abbr -g n c3 x2 u H19 t0 (H2 (minus i (S n)) u0 x2 u (fsubst0_fst (minus i (S n)) -u0 d u x2 H18) e (csubst0_getl_ge_back (minus i (S n)) (minus i (S n)) (le_n -(minus i (S n))) d x2 u0 H18 (CHead e (Bind Abbr) u0) (getl_gen_S (Bind Abbr) -x2 (CHead e (Bind Abbr) u0) u (minus i (S n)) H20))))))))))) H13)) -H12))))))))) H8)) (\lambda (H8: (ex4_5 B C C T T (\lambda (b: B).(\lambda -(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind -Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: -C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e2 -(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -(minus i (S n)) u0 e1 e2)))))))).(ex4_5_ind B C C T T (\lambda (b: -B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C -(CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda -(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e2 -(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda -(u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 -(minus i (S n)) u0 e1 e2)))))) (ty3 g c3 (TLRef n) (lift (S n) O t0)) -(\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda -(x4: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) -x3))).(\lambda (H10: (getl n c3 (CHead x2 (Bind x0) x4))).(\lambda (H11: -(subst0 (minus i (S n)) u0 x3 x4)).(\lambda (H12: (csubst0 (minus i (S n)) u0 -x1 x2)).(let H13 \def (f_equal C C (\lambda (e0: C).(match e0 in C return -(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow -c0])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H14 \def -(f_equal C B (\lambda (e0: C).(match e0 in C return (\lambda (_: C).B) with -[(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k in K return -(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow -Abbr])])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H15 -\def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow u | (CHead _ _ t3) \Rightarrow t3])) (CHead d -(Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in (\lambda (H16: (eq B Abbr -x0)).(\lambda (H17: (eq C d x1)).(let H18 \def (eq_ind_r T x3 (\lambda (t3: -T).(subst0 (minus i (S n)) u0 t3 x4)) H11 u H15) in (let H19 \def (eq_ind_r C -x1 (\lambda (c0: C).(csubst0 (minus i (S n)) u0 c0 x2)) H12 d H17) in (let -H20 \def (eq_ind_r B x0 (\lambda (b: B).(getl n c3 (CHead x2 (Bind b) x4))) -H10 Abbr H16) in (let H21 \def (eq_ind nat (minus i n) (\lambda (n0: -nat).(getl n0 (CHead x2 (Bind Abbr) x4) (CHead e (Bind Abbr) u0))) -(getl_conf_le i (CHead e (Bind Abbr) u0) c3 (csubst0_getl_ge i i (le_n i) c -c3 u0 H4 (CHead e (Bind Abbr) u0) H5) (CHead x2 (Bind Abbr) x4) n H20 (le_S_n -n i (le_S (S n) i H6))) (S (minus i (S n))) (minus_x_Sy i n H6)) in (ty3_abbr -g n c3 x2 x4 H20 t0 (H2 (minus i (S n)) u0 x2 x4 (fsubst0_both (minus i (S -n)) u0 d u x4 H18 x2 H19) e (csubst0_getl_ge_back (minus i (S n)) (minus i (S -n)) (le_n (minus i (S n))) d x2 u0 H19 (CHead e (Bind Abbr) u0) (getl_gen_S -(Bind Abbr) x2 (CHead e (Bind Abbr) u0) x4 (minus i (S n)) H21))))))))))) -H14)) H13))))))))))) H8)) H7))) (\lambda (H6: (le i n)).(ty3_abbr g n c3 d u -(csubst0_getl_ge i n H6 c c3 u0 H4 (CHead d (Bind Abbr) u) H0) t0 H1))))))) -(\lambda (t3: T).(\lambda (H4: (subst0 i u0 (TLRef n) t3)).(\lambda (c3: -C).(\lambda (H5: (csubst0 i u0 c c3)).(\lambda (e: C).(\lambda (H6: (getl i c -(CHead e (Bind Abbr) u0))).(land_ind (eq nat n i) (eq T t3 (lift (S n) O u0)) -(ty3 g c3 t3 (lift (S n) O t0)) (\lambda (H7: (eq nat n i)).(\lambda (H8: (eq -T t3 (lift (S n) O u0))).(eq_ind_r T (lift (S n) O u0) (\lambda (t4: T).(ty3 -g c3 t4 (lift (S n) O t0))) (let H9 \def (eq_ind_r nat i (\lambda (n0: -nat).(getl n0 c (CHead e (Bind Abbr) u0))) H6 n H7) in (let H10 \def -(eq_ind_r nat i (\lambda (n0: nat).(csubst0 n0 u0 c c3)) H5 n H7) in (let H11 -\def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c0: C).(getl n c c0)) H0 -(CHead e (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) n H0 (CHead e -(Bind Abbr) u0) H9)) in (let H12 \def (f_equal C C (\lambda (e0: C).(match e0 -in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) -\Rightarrow c0])) (CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono -c (CHead d (Bind Abbr) u) n H0 (CHead e (Bind Abbr) u0) H9)) in ((let H13 -\def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow u | (CHead _ _ t4) \Rightarrow t4])) (CHead d -(Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) -n H0 (CHead e (Bind Abbr) u0) H9)) in (\lambda (H14: (eq C d e)).(let H15 -\def (eq_ind_r C e (\lambda (c0: C).(getl n c (CHead c0 (Bind Abbr) u0))) H11 -d H14) in (let H16 \def (eq_ind_r T u0 (\lambda (t4: T).(getl n c (CHead d -(Bind Abbr) t4))) H15 u H13) in (let H17 \def (eq_ind_r T u0 (\lambda (t4: -T).(csubst0 n t4 c c3)) H10 u H13) in (eq_ind T u (\lambda (t4: T).(ty3 g c3 -(lift (S n) O t4) (lift (S n) O t0))) (ty3_lift g d u t0 H1 c3 O (S n) -(getl_drop Abbr c3 d u n (csubst0_getl_ge n n (le_n n) c c3 u H17 (CHead d -(Bind Abbr) u) H16))) u0 H13)))))) H12))))) t3 H8))) (subst0_gen_lref u0 t3 i -n H4)))))))) c2 t2 H3)))))))))))))) (\lambda (n: nat).(\lambda (c: -C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind -Abst) u))).(\lambda (t0: T).(\lambda (H1: (ty3 g d u t0)).(\lambda (H2: -((\forall (i: nat).(\forall (u0: T).(\forall (c2: C).(\forall (t2: -T).((fsubst0 i u0 d u c2 t2) \to (\forall (e: C).((getl i d (CHead e (Bind -Abbr) u0)) \to (ty3 g c2 t2 t0)))))))))).(\lambda (i: nat).(\lambda (u0: -T).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H3: (fsubst0 i u0 c (TLRef n) -c2 t2)).(fsubst0_ind i u0 c (TLRef n) (\lambda (c0: C).(\lambda (t3: -T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u0)) \to (ty3 g c0 t3 -(lift (S n) O u)))))) (\lambda (t3: T).(\lambda (H4: (subst0 i u0 (TLRef n) -t3)).(\lambda (e: C).(\lambda (H5: (getl i c (CHead e (Bind Abbr) -u0))).(land_ind (eq nat n i) (eq T t3 (lift (S n) O u0)) (ty3 g c t3 (lift (S -n) O u)) (\lambda (H6: (eq nat n i)).(\lambda (H7: (eq T t3 (lift (S n) O -u0))).(eq_ind_r T (lift (S n) O u0) (\lambda (t4: T).(ty3 g c t4 (lift (S n) -O u))) (let H8 \def (eq_ind_r nat i (\lambda (n0: nat).(getl n0 c (CHead e -(Bind Abbr) u0))) H5 n H6) in (let H9 \def (eq_ind C (CHead d (Bind Abst) u) -(\lambda (c0: C).(getl n c c0)) H0 (CHead e (Bind Abbr) u0) (getl_mono c -(CHead d (Bind Abst) u) n H0 (CHead e (Bind Abbr) u0) H8)) in (let H10 \def -(eq_ind C (CHead d (Bind Abst) u) (\lambda (ee: C).(match ee in C return -(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b) -\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow -False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) -\Rightarrow False])])) I (CHead e (Bind Abbr) u0) (getl_mono c (CHead d (Bind -Abst) u) n H0 (CHead e (Bind Abbr) u0) H8)) in (False_ind (ty3 g c (lift (S -n) O u0) (lift (S n) O u)) H10)))) t3 H7))) (subst0_gen_lref u0 t3 i n -H4)))))) (\lambda (c3: C).(\lambda (H4: (csubst0 i u0 c c3)).(\lambda (e: -C).(\lambda (H5: (getl i c (CHead e (Bind Abbr) u0))).(lt_le_e n i (ty3 g c3 -(TLRef n) (lift (S n) O u)) (\lambda (H6: (lt n i)).(let H7 \def -(csubst0_getl_lt i n H6 c c3 u0 H4 (CHead d (Bind Abst) u) H0) in (or4_ind -(getl n c3 (CHead d (Bind Abst) u)) (ex3_4 B C T T (\lambda (b: B).(\lambda -(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead -e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: -T).(\lambda (w: T).(getl n c3 (CHead e0 (Bind b) w)))))) (\lambda (_: -B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i (S n)) -u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: -C).(\lambda (u1: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1)))))) -(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl n c3 -(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: -C).(\lambda (_: T).(csubst0 (minus i (S n)) u0 e1 e2)))))) (ex4_5 B C C T T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1))))))) (\lambda -(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl -n c3 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus i (S n)) u0 e1 e2))))))) (ty3 g c3 (TLRef n) (lift (S -n) O u)) (\lambda (H8: (getl n c3 (CHead d (Bind Abst) u))).(ty3_abst g n c3 -d u H8 t0 H1)) (\lambda (H8: (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: -C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead e0 -(Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda -(w: T).(getl n c3 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: -C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 -w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: -T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead e0 (Bind b) u1)))))) -(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 -(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: -T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w))))) (ty3 g c3 (TLRef n) -(lift (S n) O u)) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda -(x3: T).(\lambda (H9: (eq C (CHead d (Bind Abst) u) (CHead x1 (Bind x0) -x2))).(\lambda (H10: (getl n c3 (CHead x1 (Bind x0) x3))).(\lambda (H11: -(subst0 (minus i (S n)) u0 x2 x3)).(let H12 \def (f_equal C C (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | -(CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) -x2) H9) in ((let H13 \def (f_equal C B (\lambda (e0: C).(match e0 in C return -(\lambda (_: C).B) with [(CSort _) \Rightarrow Abst | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) -\Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead d (Bind Abst) u) (CHead -x1 (Bind x0) x2) H9) in ((let H14 \def (f_equal C T (\lambda (e0: C).(match -e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ -t3) \Rightarrow t3])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x2) H9) in -(\lambda (H15: (eq B Abst x0)).(\lambda (H16: (eq C d x1)).(let H17 \def -(eq_ind_r T x2 (\lambda (t3: T).(subst0 (minus i (S n)) u0 t3 x3)) H11 u H14) -in (let H18 \def (eq_ind_r C x1 (\lambda (c0: C).(getl n c3 (CHead c0 (Bind -x0) x3))) H10 d H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b: B).(getl n -c3 (CHead d (Bind b) x3))) H18 Abst H15) in (let H20 \def (eq_ind nat (minus -i n) (\lambda (n0: nat).(getl n0 (CHead d (Bind Abst) x3) (CHead e (Bind -Abbr) u0))) (getl_conf_le i (CHead e (Bind Abbr) u0) c3 (csubst0_getl_ge i i -(le_n i) c c3 u0 H4 (CHead e (Bind Abbr) u0) H5) (CHead d (Bind Abst) x3) n -H19 (le_S_n n i (le_S (S n) i H6))) (S (minus i (S n))) (minus_x_Sy i n H6)) -in (ty3_conv g c3 (lift (S n) O u) (lift (S n) O t0) (ty3_lift g d u t0 H1 c3 -O (S n) (getl_drop Abst c3 d x3 n H19)) (TLRef n) (lift (S n) O x3) (ty3_abst -g n c3 d x3 H19 t0 (H2 (minus i (S n)) u0 d x3 (fsubst0_snd (minus i (S n)) -u0 d u x3 H17) e (getl_gen_S (Bind Abst) d (CHead e (Bind Abbr) u0) x3 (minus -i (S n)) H20))) (pc3_lift c3 d (S n) O (getl_drop Abst c3 d x3 n H19) x3 u -(pc3_pr2_x d x3 u (pr2_delta d e u0 (r (Bind Abst) (minus i (S n))) -(getl_gen_S (Bind Abst) d (CHead e (Bind Abbr) u0) x3 (minus i (S n)) H20) u -u (pr0_refl u) x3 H17))))))))))) H13)) H12))))))))) H8)) (\lambda (H8: (ex3_4 -B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq -C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: -B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl n c3 (CHead e2 -(Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda -(_: T).(csubst0 (minus i (S n)) u0 e1 e2))))))).(ex3_4_ind B C C T (\lambda -(b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind -Abst) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda -(e2: C).(\lambda (u1: T).(getl n c3 (CHead e2 (Bind b) u1)))))) (\lambda (_: -B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) -u0 e1 e2))))) (ty3 g c3 (TLRef n) (lift (S n) O u)) (\lambda (x0: B).(\lambda -(x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H9: (eq C (CHead d (Bind -Abst) u) (CHead x1 (Bind x0) x3))).(\lambda (H10: (getl n c3 (CHead x2 (Bind -x0) x3))).(\lambda (H11: (csubst0 (minus i (S n)) u0 x1 x2)).(let H12 \def -(f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with -[(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind -Abst) u) (CHead x1 (Bind x0) x3) H9) in ((let H13 \def (f_equal C B (\lambda -(e0: C).(match e0 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow -Abst | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with -[(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead d (Bind Abst) -u) (CHead x1 (Bind x0) x3) H9) in ((let H14 \def (f_equal C T (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | -(CHead _ _ t3) \Rightarrow t3])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) -x3) H9) in (\lambda (H15: (eq B Abst x0)).(\lambda (H16: (eq C d x1)).(let -H17 \def (eq_ind_r T x3 (\lambda (t3: T).(getl n c3 (CHead x2 (Bind x0) t3))) -H10 u H14) in (let H18 \def (eq_ind_r C x1 (\lambda (c0: C).(csubst0 (minus i -(S n)) u0 c0 x2)) H11 d H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b: -B).(getl n c3 (CHead x2 (Bind b) u))) H17 Abst H15) in (let H20 \def (eq_ind -nat (minus i n) (\lambda (n0: nat).(getl n0 (CHead x2 (Bind Abst) u) (CHead e -(Bind Abbr) u0))) (getl_conf_le i (CHead e (Bind Abbr) u0) c3 -(csubst0_getl_ge i i (le_n i) c c3 u0 H4 (CHead e (Bind Abbr) u0) H5) (CHead -x2 (Bind Abst) u) n H19 (le_S_n n i (le_S (S n) i H6))) (S (minus i (S n))) -(minus_x_Sy i n H6)) in (ty3_abst g n c3 x2 u H19 t0 (H2 (minus i (S n)) u0 -x2 u (fsubst0_fst (minus i (S n)) u0 d u x2 H18) e (csubst0_getl_ge_back -(minus i (S n)) (minus i (S n)) (le_n (minus i (S n))) d x2 u0 H18 (CHead e -(Bind Abbr) u0) (getl_gen_S (Bind Abst) x2 (CHead e (Bind Abbr) u0) u (minus -i (S n)) H20))))))))))) H13)) H12))))))))) H8)) (\lambda (H8: (ex4_5 B C C T -T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1))))))) (\lambda -(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl -n c3 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus i (S n)) u0 e1 e2)))))))).(ex4_5_ind B C C T T -(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda -(_: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1))))))) (\lambda -(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl -n c3 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: -C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w)))))) -(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda -(_: T).(csubst0 (minus i (S n)) u0 e1 e2)))))) (ty3 g c3 (TLRef n) (lift (S -n) O u)) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: -T).(\lambda (x4: T).(\lambda (H9: (eq C (CHead d (Bind Abst) u) (CHead x1 -(Bind x0) x3))).(\lambda (H10: (getl n c3 (CHead x2 (Bind x0) x4))).(\lambda -(H11: (subst0 (minus i (S n)) u0 x3 x4)).(\lambda (H12: (csubst0 (minus i (S -n)) u0 x1 x2)).(let H13 \def (f_equal C C (\lambda (e0: C).(match e0 in C -return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) -\Rightarrow c0])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3) H9) in -((let H14 \def (f_equal C B (\lambda (e0: C).(match e0 in C return (\lambda -(_: C).B) with [(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match -k in K return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) -\Rightarrow Abst])])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3) H9) in -((let H15 \def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda -(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t3) \Rightarrow t3])) -(CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3) H9) in (\lambda (H16: (eq B -Abst x0)).(\lambda (H17: (eq C d x1)).(let H18 \def (eq_ind_r T x3 (\lambda -(t3: T).(subst0 (minus i (S n)) u0 t3 x4)) H11 u H15) in (let H19 \def -(eq_ind_r C x1 (\lambda (c0: C).(csubst0 (minus i (S n)) u0 c0 x2)) H12 d -H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b: B).(getl n c3 (CHead x2 -(Bind b) x4))) H10 Abst H16) in (let H21 \def (eq_ind nat (minus i n) -(\lambda (n0: nat).(getl n0 (CHead x2 (Bind Abst) x4) (CHead e (Bind Abbr) -u0))) (getl_conf_le i (CHead e (Bind Abbr) u0) c3 (csubst0_getl_ge i i (le_n -i) c c3 u0 H4 (CHead e (Bind Abbr) u0) H5) (CHead x2 (Bind Abst) x4) n H20 -(le_S_n n i (le_S (S n) i H6))) (S (minus i (S n))) (minus_x_Sy i n H6)) in -(ty3_conv g c3 (lift (S n) O u) (lift (S n) O t0) (ty3_lift g x2 u t0 (H2 -(minus i (S n)) u0 x2 u (fsubst0_fst (minus i (S n)) u0 d u x2 H19) e -(csubst0_getl_ge_back (minus i (S n)) (minus i (S n)) (le_n (minus i (S n))) -d x2 u0 H19 (CHead e (Bind Abbr) u0) (getl_gen_S (Bind Abst) x2 (CHead e -(Bind Abbr) u0) x4 (minus i (S n)) H21))) c3 O (S n) (getl_drop Abst c3 x2 x4 -n H20)) (TLRef n) (lift (S n) O x4) (ty3_abst g n c3 x2 x4 H20 t0 (H2 (minus -i (S n)) u0 x2 x4 (fsubst0_both (minus i (S n)) u0 d u x4 H18 x2 H19) e -(csubst0_getl_ge_back (minus i (S n)) (minus i (S n)) (le_n (minus i (S n))) -d x2 u0 H19 (CHead e (Bind Abbr) u0) (getl_gen_S (Bind Abst) x2 (CHead e -(Bind Abbr) u0) x4 (minus i (S n)) H21)))) (pc3_lift c3 x2 (S n) O (getl_drop -Abst c3 x2 x4 n H20) x4 u (pc3_fsubst0 d u u (pc3_refl d u) (minus i (S n)) -u0 x2 x4 (fsubst0_both (minus i (S n)) u0 d u x4 H18 x2 H19) e -(csubst0_getl_ge_back (minus i (S n)) (minus i (S n)) (le_n (minus i (S n))) -d x2 u0 H19 (CHead e (Bind Abbr) u0) (getl_gen_S (Bind Abst) x2 (CHead e -(Bind Abbr) u0) x4 (minus i (S n)) H21)))))))))))) H14)) H13))))))))))) H8)) -H7))) (\lambda (H6: (le i n)).(ty3_abst g n c3 d u (csubst0_getl_ge i n H6 c -c3 u0 H4 (CHead d (Bind Abst) u) H0) t0 H1))))))) (\lambda (t3: T).(\lambda -(H4: (subst0 i u0 (TLRef n) t3)).(\lambda (c3: C).(\lambda (H5: (csubst0 i u0 -c c3)).(\lambda (e: C).(\lambda (H6: (getl i c (CHead e (Bind Abbr) -u0))).(land_ind (eq nat n i) (eq T t3 (lift (S n) O u0)) (ty3 g c3 t3 (lift -(S n) O u)) (\lambda (H7: (eq nat n i)).(\lambda (H8: (eq T t3 (lift (S n) O -u0))).(eq_ind_r T (lift (S n) O u0) (\lambda (t4: T).(ty3 g c3 t4 (lift (S n) -O u))) (let H9 \def (eq_ind_r nat i (\lambda (n0: nat).(getl n0 c (CHead e -(Bind Abbr) u0))) H6 n H7) in (let H10 \def (eq_ind_r nat i (\lambda (n0: -nat).(csubst0 n0 u0 c c3)) H5 n H7) in (let H11 \def (eq_ind C (CHead d (Bind -Abst) u) (\lambda (c0: C).(getl n c c0)) H0 (CHead e (Bind Abbr) u0) -(getl_mono c (CHead d (Bind Abst) u) n H0 (CHead e (Bind Abbr) u0) H9)) in -(let H12 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (ee: C).(match ee in -C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k -_) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b) -\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow -False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) -\Rightarrow False])])) I (CHead e (Bind Abbr) u0) (getl_mono c (CHead d (Bind -Abst) u) n H0 (CHead e (Bind Abbr) u0) H9)) in (False_ind (ty3 g c3 (lift (S -n) O u0) (lift (S n) O u)) H12))))) t3 H8))) (subst0_gen_lref u0 t3 i n -H4)))))))) c2 t2 H3)))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda -(t0: T).(\lambda (H0: (ty3 g c u t0)).(\lambda (H1: ((\forall (i: -nat).(\forall (u0: T).(\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 c u c2 -t2) \to (\forall (e: C).((getl i c (CHead e (Bind Abbr) u0)) \to (ty3 g c2 t2 -t0)))))))))).(\lambda (b: B).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H2: -(ty3 g (CHead c (Bind b) u) t2 t3)).(\lambda (H3: ((\forall (i: nat).(\forall -(u0: T).(\forall (c2: C).(\forall (t4: T).((fsubst0 i u0 (CHead c (Bind b) u) -t2 c2 t4) \to (\forall (e: C).((getl i (CHead c (Bind b) u) (CHead e (Bind -Abbr) u0)) \to (ty3 g c2 t4 t3)))))))))).(\lambda (i: nat).(\lambda (u0: -T).(\lambda (c2: C).(\lambda (t4: T).(\lambda (H4: (fsubst0 i u0 c (THead -(Bind b) u t2) c2 t4)).(fsubst0_ind i u0 c (THead (Bind b) u t2) (\lambda -(c0: C).(\lambda (t5: T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u0)) -\to (ty3 g c0 t5 (THead (Bind b) u t3)))))) (\lambda (t5: T).(\lambda (H5: -(subst0 i u0 (THead (Bind b) u t2) t5)).(\lambda (e: C).(\lambda (H6: (getl i -c (CHead e (Bind Abbr) u0))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t5 (THead -(Bind b) u2 t2))) (\lambda (u2: T).(subst0 i u0 u u2))) (ex2 T (\lambda (t6: -T).(eq T t5 (THead (Bind b) u t6))) (\lambda (t6: T).(subst0 (s (Bind b) i) -u0 t2 t6))) (ex3_2 T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead -(Bind b) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) -(\lambda (_: T).(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 t6)))) (ty3 g c -t5 (THead (Bind b) u t3)) (\lambda (H7: (ex2 T (\lambda (u2: T).(eq T t5 -(THead (Bind b) u2 t2))) (\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T -(\lambda (u2: T).(eq T t5 (THead (Bind b) u2 t2))) (\lambda (u2: T).(subst0 i -u0 u u2)) (ty3 g c t5 (THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H8: -(eq T t5 (THead (Bind b) x t2))).(\lambda (H9: (subst0 i u0 u x)).(eq_ind_r T -(THead (Bind b) x t2) (\lambda (t6: T).(ty3 g c t6 (THead (Bind b) u t3))) -(ex_ind T (\lambda (t6: T).(ty3 g (CHead c (Bind b) u) t3 t6)) (ty3 g c -(THead (Bind b) x t2) (THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (H10: -(ty3 g (CHead c (Bind b) u) t3 x0)).(ex_ind T (\lambda (t6: T).(ty3 g (CHead -c (Bind b) x) t3 t6)) (ty3 g c (THead (Bind b) x t2) (THead (Bind b) u t3)) -(\lambda (x1: T).(\lambda (_: (ty3 g (CHead c (Bind b) x) t3 x1)).(ty3_conv g -c (THead (Bind b) u t3) (THead (Bind b) u x0) (ty3_bind g c u t0 H0 b t3 x0 -H10) (THead (Bind b) x t2) (THead (Bind b) x t3) (ty3_bind g c x t0 (H1 i u0 -c x (fsubst0_snd i u0 c u x H9) e H6) b t2 t3 (H3 (S i) u0 (CHead c (Bind b) -x) t2 (fsubst0_fst (S i) u0 (CHead c (Bind b) u) t2 (CHead c (Bind b) x) -(csubst0_snd_bind b i u0 u x H9 c)) e (getl_head (Bind b) i c (CHead e (Bind -Abbr) u0) H6 u))) (pc3_fsubst0 c (THead (Bind b) u t3) (THead (Bind b) u t3) -(pc3_refl c (THead (Bind b) u t3)) i u0 c (THead (Bind b) x t3) (fsubst0_snd -i u0 c (THead (Bind b) u t3) (THead (Bind b) x t3) (subst0_fst u0 x u i H9 t3 -(Bind b))) e H6)))) (ty3_correct g (CHead c (Bind b) x) t2 t3 (H3 (S i) u0 -(CHead c (Bind b) x) t2 (fsubst0_fst (S i) u0 (CHead c (Bind b) u) t2 (CHead -c (Bind b) x) (csubst0_snd_bind b i u0 u x H9 c)) e (getl_head (Bind b) i c -(CHead e (Bind Abbr) u0) H6 u)))))) (ty3_correct g (CHead c (Bind b) u) t2 t3 -H2)) t5 H8)))) H7)) (\lambda (H7: (ex2 T (\lambda (t6: T).(eq T t5 (THead -(Bind b) u t6))) (\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 -t6)))).(ex2_ind T (\lambda (t6: T).(eq T t5 (THead (Bind b) u t6))) (\lambda -(t6: T).(subst0 (s (Bind b) i) u0 t2 t6)) (ty3 g c t5 (THead (Bind b) u t3)) -(\lambda (x: T).(\lambda (H8: (eq T t5 (THead (Bind b) u x))).(\lambda (H9: -(subst0 (s (Bind b) i) u0 t2 x)).(eq_ind_r T (THead (Bind b) u x) (\lambda -(t6: T).(ty3 g c t6 (THead (Bind b) u t3))) (ex_ind T (\lambda (t6: T).(ty3 g -(CHead c (Bind b) u) t3 t6)) (ty3 g c (THead (Bind b) u x) (THead (Bind b) u -t3)) (\lambda (x0: T).(\lambda (_: (ty3 g (CHead c (Bind b) u) t3 -x0)).(ty3_bind g c u t0 H0 b x t3 (H3 (S i) u0 (CHead c (Bind b) u) x -(fsubst0_snd (S i) u0 (CHead c (Bind b) u) t2 x H9) e (getl_head (Bind b) i c -(CHead e (Bind Abbr) u0) H6 u))))) (ty3_correct g (CHead c (Bind b) u) x t3 -(H3 (S i) u0 (CHead c (Bind b) u) x (fsubst0_snd (S i) u0 (CHead c (Bind b) -u) t2 x H9) e (getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H6 u)))) t5 -H8)))) H7)) (\lambda (H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t6: T).(eq T -t5 (THead (Bind b) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u -u2))) (\lambda (_: T).(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 -t6))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead -(Bind b) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) -(\lambda (_: T).(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 t6))) (ty3 g c -t5 (THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H8: (eq -T t5 (THead (Bind b) x0 x1))).(\lambda (H9: (subst0 i u0 u x0)).(\lambda -(H10: (subst0 (s (Bind b) i) u0 t2 x1)).(eq_ind_r T (THead (Bind b) x0 x1) -(\lambda (t6: T).(ty3 g c t6 (THead (Bind b) u t3))) (ex_ind T (\lambda (t6: -T).(ty3 g (CHead c (Bind b) u) t3 t6)) (ty3 g c (THead (Bind b) x0 x1) (THead -(Bind b) u t3)) (\lambda (x: T).(\lambda (H11: (ty3 g (CHead c (Bind b) u) t3 -x)).(ex_ind T (\lambda (t6: T).(ty3 g (CHead c (Bind b) x0) t3 t6)) (ty3 g c -(THead (Bind b) x0 x1) (THead (Bind b) u t3)) (\lambda (x2: T).(\lambda (_: -(ty3 g (CHead c (Bind b) x0) t3 x2)).(ty3_conv g c (THead (Bind b) u t3) -(THead (Bind b) u x) (ty3_bind g c u t0 H0 b t3 x H11) (THead (Bind b) x0 x1) -(THead (Bind b) x0 t3) (ty3_bind g c x0 t0 (H1 i u0 c x0 (fsubst0_snd i u0 c -u x0 H9) e H6) b x1 t3 (H3 (S i) u0 (CHead c (Bind b) x0) x1 (fsubst0_both (S -i) u0 (CHead c (Bind b) u) t2 x1 H10 (CHead c (Bind b) x0) (csubst0_snd_bind -b i u0 u x0 H9 c)) e (getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H6 u))) -(pc3_fsubst0 c (THead (Bind b) u t3) (THead (Bind b) u t3) (pc3_refl c (THead -(Bind b) u t3)) i u0 c (THead (Bind b) x0 t3) (fsubst0_snd i u0 c (THead -(Bind b) u t3) (THead (Bind b) x0 t3) (subst0_fst u0 x0 u i H9 t3 (Bind b))) -e H6)))) (ty3_correct g (CHead c (Bind b) x0) x1 t3 (H3 (S i) u0 (CHead c -(Bind b) x0) x1 (fsubst0_both (S i) u0 (CHead c (Bind b) u) t2 x1 H10 (CHead -c (Bind b) x0) (csubst0_snd_bind b i u0 u x0 H9 c)) e (getl_head (Bind b) i c -(CHead e (Bind Abbr) u0) H6 u)))))) (ty3_correct g (CHead c (Bind b) u) t2 t3 -H2)) t5 H8)))))) H7)) (subst0_gen_head (Bind b) u0 u t2 t5 i H5)))))) -(\lambda (c3: C).(\lambda (H5: (csubst0 i u0 c c3)).(\lambda (e: C).(\lambda -(H6: (getl i c (CHead e (Bind Abbr) u0))).(ex_ind T (\lambda (t5: T).(ty3 g -(CHead c3 (Bind b) u) t3 t5)) (ty3 g c3 (THead (Bind b) u t2) (THead (Bind b) -u t3)) (\lambda (x: T).(\lambda (_: (ty3 g (CHead c3 (Bind b) u) t3 -x)).(ty3_bind g c3 u t0 (H1 i u0 c3 u (fsubst0_fst i u0 c u c3 H5) e H6) b t2 -t3 (H3 (S i) u0 (CHead c3 (Bind b) u) t2 (fsubst0_fst (S i) u0 (CHead c (Bind -b) u) t2 (CHead c3 (Bind b) u) (csubst0_fst_bind b i c c3 u0 H5 u)) e -(getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H6 u))))) (ty3_correct g -(CHead c3 (Bind b) u) t2 t3 (H3 (S i) u0 (CHead c3 (Bind b) u) t2 -(fsubst0_fst (S i) u0 (CHead c (Bind b) u) t2 (CHead c3 (Bind b) u) -(csubst0_fst_bind b i c c3 u0 H5 u)) e (getl_head (Bind b) i c (CHead e (Bind -Abbr) u0) H6 u)))))))) (\lambda (t5: T).(\lambda (H5: (subst0 i u0 (THead -(Bind b) u t2) t5)).(\lambda (c3: C).(\lambda (H6: (csubst0 i u0 c -c3)).(\lambda (e: C).(\lambda (H7: (getl i c (CHead e (Bind Abbr) -u0))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t5 (THead (Bind b) u2 t2))) -(\lambda (u2: T).(subst0 i u0 u u2))) (ex2 T (\lambda (t6: T).(eq T t5 (THead -(Bind b) u t6))) (\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 t6))) (ex3_2 T -T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead (Bind b) u2 t6)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: -T).(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 t6)))) (ty3 g c3 t5 (THead -(Bind b) u t3)) (\lambda (H8: (ex2 T (\lambda (u2: T).(eq T t5 (THead (Bind -b) u2 t2))) (\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T (\lambda (u2: -T).(eq T t5 (THead (Bind b) u2 t2))) (\lambda (u2: T).(subst0 i u0 u u2)) -(ty3 g c3 t5 (THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H9: (eq T t5 -(THead (Bind b) x t2))).(\lambda (H10: (subst0 i u0 u x)).(eq_ind_r T (THead -(Bind b) x t2) (\lambda (t6: T).(ty3 g c3 t6 (THead (Bind b) u t3))) (ex_ind -T (\lambda (t6: T).(ty3 g (CHead c3 (Bind b) u) t3 t6)) (ty3 g c3 (THead -(Bind b) x t2) (THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (H11: (ty3 g -(CHead c3 (Bind b) u) t3 x0)).(ex_ind T (\lambda (t6: T).(ty3 g (CHead c3 -(Bind b) u) x0 t6)) (ty3 g c3 (THead (Bind b) x t2) (THead (Bind b) u t3)) -(\lambda (x1: T).(\lambda (_: (ty3 g (CHead c3 (Bind b) u) x0 x1)).(ex_ind T -(\lambda (t6: T).(ty3 g (CHead c3 (Bind b) x) t3 t6)) (ty3 g c3 (THead (Bind -b) x t2) (THead (Bind b) u t3)) (\lambda (x2: T).(\lambda (_: (ty3 g (CHead -c3 (Bind b) x) t3 x2)).(ty3_conv g c3 (THead (Bind b) u t3) (THead (Bind b) u -x0) (ty3_bind g c3 u t0 (H1 i u0 c3 u (fsubst0_fst i u0 c u c3 H6) e H7) b t3 -x0 H11) (THead (Bind b) x t2) (THead (Bind b) x t3) (ty3_bind g c3 x t0 (H1 i -u0 c3 x (fsubst0_both i u0 c u x H10 c3 H6) e H7) b t2 t3 (H3 (S i) u0 (CHead -c3 (Bind b) x) t2 (fsubst0_fst (S i) u0 (CHead c (Bind b) u) t2 (CHead c3 -(Bind b) x) (csubst0_both_bind b i u0 u x H10 c c3 H6)) e (getl_head (Bind b) -i c (CHead e (Bind Abbr) u0) H7 u))) (pc3_fsubst0 c (THead (Bind b) u t3) -(THead (Bind b) u t3) (pc3_refl c (THead (Bind b) u t3)) i u0 c3 (THead (Bind -b) x t3) (fsubst0_both i u0 c (THead (Bind b) u t3) (THead (Bind b) x t3) -(subst0_fst u0 x u i H10 t3 (Bind b)) c3 H6) e H7)))) (ty3_correct g (CHead -c3 (Bind b) x) t2 t3 (H3 (S i) u0 (CHead c3 (Bind b) x) t2 (fsubst0_fst (S i) -u0 (CHead c (Bind b) u) t2 (CHead c3 (Bind b) x) (csubst0_both_bind b i u0 u -x H10 c c3 H6)) e (getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H7 u)))))) -(ty3_correct g (CHead c3 (Bind b) u) t3 x0 H11)))) (ty3_correct g (CHead c3 -(Bind b) u) t2 t3 (H3 (S i) u0 (CHead c3 (Bind b) u) t2 (fsubst0_fst (S i) u0 -(CHead c (Bind b) u) t2 (CHead c3 (Bind b) u) (csubst0_fst_bind b i c c3 u0 -H6 u)) e (getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H7 u)))) t5 H9)))) -H8)) (\lambda (H8: (ex2 T (\lambda (t6: T).(eq T t5 (THead (Bind b) u t6))) -(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 t6)))).(ex2_ind T (\lambda (t6: -T).(eq T t5 (THead (Bind b) u t6))) (\lambda (t6: T).(subst0 (s (Bind b) i) -u0 t2 t6)) (ty3 g c3 t5 (THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H9: -(eq T t5 (THead (Bind b) u x))).(\lambda (H10: (subst0 (s (Bind b) i) u0 t2 -x)).(eq_ind_r T (THead (Bind b) u x) (\lambda (t6: T).(ty3 g c3 t6 (THead -(Bind b) u t3))) (ex_ind T (\lambda (t6: T).(ty3 g (CHead c3 (Bind b) u) t3 -t6)) (ty3 g c3 (THead (Bind b) u x) (THead (Bind b) u t3)) (\lambda (x0: -T).(\lambda (_: (ty3 g (CHead c3 (Bind b) u) t3 x0)).(ty3_bind g c3 u t0 (H1 -i u0 c3 u (fsubst0_fst i u0 c u c3 H6) e H7) b x t3 (H3 (S i) u0 (CHead c3 -(Bind b) u) x (fsubst0_both (S i) u0 (CHead c (Bind b) u) t2 x H10 (CHead c3 -(Bind b) u) (csubst0_fst_bind b i c c3 u0 H6 u)) e (getl_head (Bind b) i c -(CHead e (Bind Abbr) u0) H7 u))))) (ty3_correct g (CHead c3 (Bind b) u) x t3 -(H3 (S i) u0 (CHead c3 (Bind b) u) x (fsubst0_both (S i) u0 (CHead c (Bind b) -u) t2 x H10 (CHead c3 (Bind b) u) (csubst0_fst_bind b i c c3 u0 H6 u)) e -(getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H7 u)))) t5 H9)))) H8)) -(\lambda (H8: (ex3_2 T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead -(Bind b) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) -(\lambda (_: T).(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 -t6))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead -(Bind b) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) -(\lambda (_: T).(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 t6))) (ty3 g c3 -t5 (THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H9: (eq -T t5 (THead (Bind b) x0 x1))).(\lambda (H10: (subst0 i u0 u x0)).(\lambda -(H11: (subst0 (s (Bind b) i) u0 t2 x1)).(eq_ind_r T (THead (Bind b) x0 x1) -(\lambda (t6: T).(ty3 g c3 t6 (THead (Bind b) u t3))) (ex_ind T (\lambda (t6: -T).(ty3 g (CHead c3 (Bind b) u) t3 t6)) (ty3 g c3 (THead (Bind b) x0 x1) -(THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H12: (ty3 g (CHead c3 (Bind -b) u) t3 x)).(ex_ind T (\lambda (t6: T).(ty3 g (CHead c3 (Bind b) u) x t6)) -(ty3 g c3 (THead (Bind b) x0 x1) (THead (Bind b) u t3)) (\lambda (x2: -T).(\lambda (_: (ty3 g (CHead c3 (Bind b) u) x x2)).(ex_ind T (\lambda (t6: -T).(ty3 g (CHead c3 (Bind b) x0) t3 t6)) (ty3 g c3 (THead (Bind b) x0 x1) -(THead (Bind b) u t3)) (\lambda (x3: T).(\lambda (_: (ty3 g (CHead c3 (Bind -b) x0) t3 x3)).(ty3_conv g c3 (THead (Bind b) u t3) (THead (Bind b) u x) -(ty3_bind g c3 u t0 (H1 i u0 c3 u (fsubst0_fst i u0 c u c3 H6) e H7) b t3 x -H12) (THead (Bind b) x0 x1) (THead (Bind b) x0 t3) (ty3_bind g c3 x0 t0 (H1 i -u0 c3 x0 (fsubst0_both i u0 c u x0 H10 c3 H6) e H7) b x1 t3 (H3 (S i) u0 -(CHead c3 (Bind b) x0) x1 (fsubst0_both (S i) u0 (CHead c (Bind b) u) t2 x1 -H11 (CHead c3 (Bind b) x0) (csubst0_both_bind b i u0 u x0 H10 c c3 H6)) e -(getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H7 u))) (pc3_fsubst0 c -(THead (Bind b) u t3) (THead (Bind b) u t3) (pc3_refl c (THead (Bind b) u -t3)) i u0 c3 (THead (Bind b) x0 t3) (fsubst0_both i u0 c (THead (Bind b) u -t3) (THead (Bind b) x0 t3) (subst0_fst u0 x0 u i H10 t3 (Bind b)) c3 H6) e -H7)))) (ty3_correct g (CHead c3 (Bind b) x0) x1 t3 (H3 (S i) u0 (CHead c3 -(Bind b) x0) x1 (fsubst0_both (S i) u0 (CHead c (Bind b) u) t2 x1 H11 (CHead -c3 (Bind b) x0) (csubst0_both_bind b i u0 u x0 H10 c c3 H6)) e (getl_head -(Bind b) i c (CHead e (Bind Abbr) u0) H7 u)))))) (ty3_correct g (CHead c3 -(Bind b) u) t3 x H12)))) (ty3_correct g (CHead c3 (Bind b) u) t2 t3 (H3 (S i) -u0 (CHead c3 (Bind b) u) t2 (fsubst0_fst (S i) u0 (CHead c (Bind b) u) t2 -(CHead c3 (Bind b) u) (csubst0_fst_bind b i c c3 u0 H6 u)) e (getl_head (Bind -b) i c (CHead e (Bind Abbr) u0) H7 u)))) t5 H9)))))) H8)) (subst0_gen_head -(Bind b) u0 u t2 t5 i H5)))))))) c2 t4 H4)))))))))))))))) (\lambda (c: -C).(\lambda (w: T).(\lambda (u: T).(\lambda (H0: (ty3 g c w u)).(\lambda (H1: -((\forall (i: nat).(\forall (u0: T).(\forall (c2: C).(\forall (t2: -T).((fsubst0 i u0 c w c2 t2) \to (\forall (e: C).((getl i c (CHead e (Bind -Abbr) u0)) \to (ty3 g c2 t2 u)))))))))).(\lambda (v: T).(\lambda (t0: -T).(\lambda (H2: (ty3 g c v (THead (Bind Abst) u t0))).(\lambda (H3: -((\forall (i: nat).(\forall (u0: T).(\forall (c2: C).(\forall (t2: -T).((fsubst0 i u0 c v c2 t2) \to (\forall (e: C).((getl i c (CHead e (Bind -Abbr) u0)) \to (ty3 g c2 t2 (THead (Bind Abst) u t0))))))))))).(\lambda (i: -nat).(\lambda (u0: T).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H4: -(fsubst0 i u0 c (THead (Flat Appl) w v) c2 t2)).(fsubst0_ind i u0 c (THead -(Flat Appl) w v) (\lambda (c0: C).(\lambda (t3: T).(\forall (e: C).((getl i c -(CHead e (Bind Abbr) u0)) \to (ty3 g c0 t3 (THead (Flat Appl) w (THead (Bind -Abst) u t0))))))) (\lambda (t3: T).(\lambda (H5: (subst0 i u0 (THead (Flat -Appl) w v) t3)).(\lambda (e: C).(\lambda (H6: (getl i c (CHead e (Bind Abbr) -u0))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t3 (THead (Flat Appl) u2 v))) -(\lambda (u2: T).(subst0 i u0 w u2))) (ex2 T (\lambda (t4: T).(eq T t3 (THead -(Flat Appl) w t4))) (\lambda (t4: T).(subst0 (s (Flat Appl) i) u0 v t4))) -(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 -t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 w u2))) (\lambda (_: -T).(\lambda (t4: T).(subst0 (s (Flat Appl) i) u0 v t4)))) (ty3 g c t3 (THead -(Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (H7: (ex2 T (\lambda (u2: -T).(eq T t3 (THead (Flat Appl) u2 v))) (\lambda (u2: T).(subst0 i u0 w -u2)))).(ex2_ind T (\lambda (u2: T).(eq T t3 (THead (Flat Appl) u2 v))) -(\lambda (u2: T).(subst0 i u0 w u2)) (ty3 g c t3 (THead (Flat Appl) w (THead -(Bind Abst) u t0))) (\lambda (x: T).(\lambda (H8: (eq T t3 (THead (Flat Appl) -x v))).(\lambda (H9: (subst0 i u0 w x)).(eq_ind_r T (THead (Flat Appl) x v) -(\lambda (t4: T).(ty3 g c t4 (THead (Flat Appl) w (THead (Bind Abst) u t0)))) -(ex_ind T (\lambda (t4: T).(ty3 g c (THead (Bind Abst) u t0) t4)) (ty3 g c -(THead (Flat Appl) x v) (THead (Flat Appl) w (THead (Bind Abst) u t0))) -(\lambda (x0: T).(\lambda (H10: (ty3 g c (THead (Bind Abst) u t0) -x0)).(ex3_2_ind T T (\lambda (t4: T).(\lambda (_: T).(pc3 c (THead (Bind -Abst) u t4) x0))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c u t5))) (\lambda -(t4: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t0 t4))) (ty3 g c -(THead (Flat Appl) x v) (THead (Flat Appl) w (THead (Bind Abst) u t0))) -(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 c (THead (Bind Abst) u -x1) x0)).(\lambda (_: (ty3 g c u x2)).(\lambda (H13: (ty3 g (CHead c (Bind -Abst) u) t0 x1)).(ex_ind T (\lambda (t4: T).(ty3 g c u t4)) (ty3 g c (THead -(Flat Appl) x v) (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda -(x3: T).(\lambda (H14: (ty3 g c u x3)).(ty3_conv g c (THead (Flat Appl) w -(THead (Bind Abst) u t0)) (THead (Flat Appl) w (THead (Bind Abst) u x1)) -(ty3_appl g c w u H0 (THead (Bind Abst) u t0) x1 (ty3_bind g c u x3 H14 Abst -t0 x1 H13)) (THead (Flat Appl) x v) (THead (Flat Appl) x (THead (Bind Abst) u -t0)) (ty3_appl g c x u (H1 i u0 c x (fsubst0_snd i u0 c w x H9) e H6) v t0 -H2) (pc3_fsubst0 c (THead (Flat Appl) w (THead (Bind Abst) u t0)) (THead -(Flat Appl) w (THead (Bind Abst) u t0)) (pc3_refl c (THead (Flat Appl) w -(THead (Bind Abst) u t0))) i u0 c (THead (Flat Appl) x (THead (Bind Abst) u -t0)) (fsubst0_snd i u0 c (THead (Flat Appl) w (THead (Bind Abst) u t0)) -(THead (Flat Appl) x (THead (Bind Abst) u t0)) (subst0_fst u0 x w i H9 (THead -(Bind Abst) u t0) (Flat Appl))) e H6)))) (ty3_correct g c x u (H1 i u0 c x -(fsubst0_snd i u0 c w x H9) e H6)))))))) (ty3_gen_bind g Abst c u t0 x0 -H10)))) (ty3_correct g c v (THead (Bind Abst) u t0) H2)) t3 H8)))) H7)) -(\lambda (H7: (ex2 T (\lambda (t4: T).(eq T t3 (THead (Flat Appl) w t4))) -(\lambda (t4: T).(subst0 (s (Flat Appl) i) u0 v t4)))).(ex2_ind T (\lambda -(t4: T).(eq T t3 (THead (Flat Appl) w t4))) (\lambda (t4: T).(subst0 (s (Flat -Appl) i) u0 v t4)) (ty3 g c t3 (THead (Flat Appl) w (THead (Bind Abst) u -t0))) (\lambda (x: T).(\lambda (H8: (eq T t3 (THead (Flat Appl) w -x))).(\lambda (H9: (subst0 (s (Flat Appl) i) u0 v x)).(eq_ind_r T (THead -(Flat Appl) w x) (\lambda (t4: T).(ty3 g c t4 (THead (Flat Appl) w (THead -(Bind Abst) u t0)))) (ty3_appl g c w u H0 x t0 (H3 (s (Flat Appl) i) u0 c x -(fsubst0_snd (s (Flat Appl) i) u0 c v x H9) e H6)) t3 H8)))) H7)) (\lambda -(H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) -u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 w u2))) (\lambda (_: -T).(\lambda (t4: T).(subst0 (s (Flat Appl) i) u0 v t4))))).(ex3_2_ind T T -(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead 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(x0: T).(\lambda (H11: (ty3 g c3 (THead -(Bind Abst) u t0) x0)).(ex3_2_ind T T (\lambda (t4: T).(\lambda (_: T).(pc3 -c3 (THead (Bind Abst) u t4) x0))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c3 -u t5))) (\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead c3 (Bind Abst) u) t0 -t4))) (ty3 g c3 (THead (Flat Appl) x v) (THead (Flat Appl) w (THead (Bind -Abst) u t0))) (\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 c3 (THead -(Bind Abst) u x1) x0)).(\lambda (H13: (ty3 g c3 u x2)).(\lambda (H14: (ty3 g -(CHead c3 (Bind Abst) u) t0 x1)).(ty3_conv g c3 (THead (Flat Appl) w (THead -(Bind Abst) u t0)) (THead (Flat Appl) w (THead (Bind Abst) u x1)) (ty3_appl g -c3 w u (H1 i u0 c3 w (fsubst0_fst i u0 c w c3 H6) e H7) (THead (Bind Abst) u -t0) x1 (ty3_bind g c3 u x2 H13 Abst t0 x1 H14)) (THead (Flat Appl) x v) -(THead (Flat Appl) x (THead (Bind Abst) u t0)) (ty3_appl g c3 x u (H1 i u0 c3 -x (fsubst0_both i u0 c w x H10 c3 H6) e H7) v t0 (H3 i u0 c3 v (fsubst0_fst i -u0 c v c3 H6) e H7)) (pc3_fsubst0 c (THead (Flat Appl) w (THead (Bind Abst) u -t0)) (THead (Flat Appl) w (THead (Bind Abst) u t0)) (pc3_refl c (THead (Flat -Appl) w (THead (Bind Abst) u t0))) i u0 c3 (THead (Flat Appl) x (THead (Bind -Abst) u t0)) (fsubst0_both i u0 c (THead (Flat Appl) w (THead (Bind Abst) u -t0)) (THead (Flat Appl) x (THead (Bind Abst) u t0)) (subst0_fst u0 x w i H10 -(THead (Bind Abst) u t0) (Flat Appl)) c3 H6) e H7))))))) (ty3_gen_bind g Abst -c3 u t0 x0 H11)))) (ty3_correct g c3 v (THead (Bind Abst) u t0) (H3 i u0 c3 v -(fsubst0_fst i u0 c v c3 H6) e H7))) t3 H9)))) H8)) (\lambda (H8: (ex2 T -(\lambda (t4: T).(eq T t3 (THead (Flat Appl) w t4))) (\lambda (t4: T).(subst0 -(s (Flat Appl) i) u0 v t4)))).(ex2_ind T (\lambda (t4: T).(eq T t3 (THead -(Flat Appl) w t4))) (\lambda (t4: T).(subst0 (s (Flat Appl) i) u0 v t4)) (ty3 -g c3 t3 (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x: -T).(\lambda (H9: (eq T t3 (THead (Flat Appl) w x))).(\lambda (H10: (subst0 (s -(Flat Appl) i) u0 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t5 (THead (Flat Cast) t3 t6))) (\lambda (t6: -T).(subst0 (s (Flat Cast) i) u t2 t6)))).(ex2_ind T (\lambda (t6: T).(eq T t5 -(THead (Flat Cast) t3 t6))) (\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 -t6)) (ty3 g c t5 (THead (Flat Cast) t0 t3)) (\lambda (x: T).(\lambda (H8: (eq -T t5 (THead (Flat Cast) t3 x))).(\lambda (H9: (subst0 (s (Flat Cast) i) u t2 -x)).(eq_ind_r T (THead (Flat Cast) t3 x) (\lambda (t6: T).(ty3 g c t6 (THead -(Flat Cast) t0 t3))) (ty3_cast g c x t3 (H1 (s (Flat Cast) i) u c x -(fsubst0_snd (s (Flat Cast) i) u c t2 x H9) e H6) t0 H2) t5 H8)))) H7)) -(\lambda (H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead -(Flat Cast) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u t3 u2))) -(\lambda (_: T).(\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 -t6))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead -(Flat Cast) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u t3 u2))) -(\lambda (_: T).(\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 t6))) (ty3 g -c t5 (THead (Flat Cast) t0 t3)) (\lambda (x0: T).(\lambda (x1: T).(\lambda -(H8: (eq T t5 (THead (Flat Cast) x0 x1))).(\lambda (H9: (subst0 i u t3 -x0)).(\lambda (H10: (subst0 (s (Flat Cast) i) u t2 x1)).(eq_ind_r T (THead -(Flat Cast) x0 x1) (\lambda (t6: T).(ty3 g c t6 (THead (Flat Cast) t0 t3))) -(ex_ind T (\lambda (t6: T).(ty3 g c t0 t6)) (ty3 g c (THead (Flat Cast) x0 -x1) (THead (Flat Cast) t0 t3)) (\lambda (x: T).(\lambda (H11: (ty3 g c t0 -x)).(ty3_conv g c (THead (Flat Cast) t0 t3) (THead (Flat Cast) x t0) -(ty3_cast g c t3 t0 H2 x H11) (THead (Flat Cast) x0 x1) (THead (Flat Cast) t0 -x0) (ty3_cast g c x1 x0 (ty3_conv g c x0 t0 (H3 i u c x0 (fsubst0_snd i u c -t3 x0 H9) e H6) x1 t3 (H1 (s (Flat Cast) i) u c x1 (fsubst0_snd (s (Flat -Cast) i) u c t2 x1 H10) e H6) (pc3_s c t3 x0 (pc3_fsubst0 c t3 t3 (pc3_refl c -t3) i u c x0 (fsubst0_snd i u c t3 x0 H9) e H6))) t0 (H3 i u c x0 -(fsubst0_snd i u c t3 x0 H9) e H6)) (pc3_fsubst0 c (THead (Flat Cast) t0 t3) -(THead (Flat Cast) t0 t3) (pc3_refl c (THead (Flat Cast) t0 t3)) i u c (THead -(Flat Cast) t0 x0) (fsubst0_snd i u c (THead (Flat Cast) t0 t3) (THead (Flat -Cast) t0 x0) (subst0_snd (Flat Cast) u x0 t3 i H9 t0)) e H6)))) (ty3_correct -g c x0 t0 (H3 i u c x0 (fsubst0_snd i u c t3 x0 H9) e H6))) t5 H8)))))) H7)) -(subst0_gen_head (Flat Cast) u t3 t2 t5 i H5)))))) (\lambda (c3: C).(\lambda -(H5: (csubst0 i u c c3)).(\lambda (e: C).(\lambda (H6: (getl i c (CHead e -(Bind Abbr) u))).(ty3_cast g c3 t2 t3 (H1 i u c3 t2 (fsubst0_fst i u c t2 c3 -H5) e H6) t0 (H3 i u c3 t3 (fsubst0_fst i u c t3 c3 H5) e H6)))))) (\lambda -(t5: T).(\lambda (H5: (subst0 i u (THead (Flat Cast) t3 t2) t5)).(\lambda -(c3: C).(\lambda (H6: (csubst0 i u c c3)).(\lambda (e: C).(\lambda (H7: (getl -i c (CHead e (Bind Abbr) u))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t5 -(THead (Flat Cast) u2 t2))) (\lambda (u2: T).(subst0 i u t3 u2))) (ex2 T -(\lambda (t6: T).(eq T t5 (THead (Flat Cast) t3 t6))) (\lambda (t6: -T).(subst0 (s (Flat Cast) i) u t2 t6))) (ex3_2 T T (\lambda (u2: T).(\lambda -(t6: T).(eq T t5 (THead (Flat Cast) u2 t6)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 i u t3 u2))) (\lambda (_: T).(\lambda (t6: T).(subst0 (s (Flat -Cast) i) u t2 t6)))) (ty3 g c3 t5 (THead (Flat Cast) t0 t3)) (\lambda (H8: -(ex2 T (\lambda (u2: T).(eq T t5 (THead (Flat Cast) u2 t2))) (\lambda (u2: -T).(subst0 i u t3 u2)))).(ex2_ind T (\lambda (u2: T).(eq T t5 (THead (Flat -Cast) u2 t2))) (\lambda (u2: T).(subst0 i u t3 u2)) (ty3 g c3 t5 (THead (Flat -Cast) t0 t3)) (\lambda (x: T).(\lambda (H9: (eq T t5 (THead (Flat Cast) x -t2))).(\lambda (H10: (subst0 i u t3 x)).(eq_ind_r T (THead (Flat Cast) x t2) -(\lambda (t6: T).(ty3 g c3 t6 (THead (Flat Cast) t0 t3))) (ex_ind T (\lambda -(t6: T).(ty3 g c3 t0 t6)) (ty3 g c3 (THead (Flat Cast) x t2) (THead (Flat -Cast) t0 t3)) (\lambda (x0: T).(\lambda (H11: (ty3 g c3 t0 x0)).(ty3_conv g -c3 (THead (Flat Cast) t0 t3) (THead (Flat Cast) x0 t0) (ty3_cast g c3 t3 t0 -(H3 i u c3 t3 (fsubst0_fst i u c t3 c3 H6) e H7) x0 H11) (THead (Flat Cast) x -t2) (THead (Flat Cast) t0 x) (ty3_cast g c3 t2 x (ty3_conv g c3 x t0 (H3 i u -c3 x (fsubst0_both i u c t3 x H10 c3 H6) e H7) t2 t3 (H1 i u c3 t2 -(fsubst0_fst i u c t2 c3 H6) e H7) (pc3_s c3 t3 x (pc3_fsubst0 c t3 t3 -(pc3_refl c t3) i u c3 x (fsubst0_both i u c t3 x H10 c3 H6) e H7))) t0 (H3 i -u c3 x (fsubst0_both i u c t3 x H10 c3 H6) e H7)) (pc3_fsubst0 c (THead (Flat -Cast) t0 t3) (THead (Flat Cast) t0 t3) (pc3_refl c (THead (Flat Cast) t0 t3)) -i u c3 (THead (Flat Cast) t0 x) (fsubst0_both i u c (THead (Flat Cast) t0 t3) -(THead (Flat Cast) t0 x) (subst0_snd (Flat Cast) u x t3 i H10 t0) c3 H6) e -H7)))) (ty3_correct g c3 t3 t0 (H3 i u c3 t3 (fsubst0_fst i u c t3 c3 H6) e -H7))) t5 H9)))) H8)) (\lambda (H8: (ex2 T (\lambda (t6: T).(eq T t5 (THead -(Flat Cast) t3 t6))) (\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 -t6)))).(ex2_ind T (\lambda (t6: T).(eq T t5 (THead (Flat Cast) t3 t6))) -(\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 t6)) (ty3 g c3 t5 (THead -(Flat Cast) t0 t3)) (\lambda (x: T).(\lambda (H9: (eq T t5 (THead (Flat Cast) -t3 x))).(\lambda (H10: (subst0 (s (Flat Cast) i) u t2 x)).(eq_ind_r T (THead -(Flat Cast) t3 x) (\lambda (t6: T).(ty3 g c3 t6 (THead (Flat Cast) t0 t3))) -(ty3_cast g c3 x t3 (H1 i u c3 x (fsubst0_both i u c t2 x H10 c3 H6) e H7) t0 -(H3 i u c3 t3 (fsubst0_fst i u c t3 c3 H6) e H7)) t5 H9)))) H8)) (\lambda -(H8: (ex3_2 T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead (Flat Cast) -u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u t3 u2))) (\lambda (_: -T).(\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 t6))))).(ex3_2_ind T T -(\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead (Flat Cast) u2 t6)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 i u t3 u2))) (\lambda (_: -T).(\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 t6))) (ty3 g c3 t5 (THead -(Flat Cast) t0 t3)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H9: (eq T t5 -(THead (Flat Cast) x0 x1))).(\lambda (H10: (subst0 i u t3 x0)).(\lambda (H11: -(subst0 (s (Flat Cast) i) u t2 x1)).(eq_ind_r T (THead (Flat Cast) x0 x1) -(\lambda (t6: T).(ty3 g c3 t6 (THead (Flat Cast) t0 t3))) (ex_ind T (\lambda -(t6: T).(ty3 g c3 t0 t6)) (ty3 g c3 (THead (Flat Cast) x0 x1) (THead (Flat -Cast) t0 t3)) (\lambda (x: T).(\lambda (H12: (ty3 g c3 t0 x)).(ty3_conv g c3 -(THead (Flat Cast) t0 t3) (THead (Flat Cast) x t0) (ty3_cast g c3 t3 t0 (H3 i -u c3 t3 (fsubst0_fst i u c t3 c3 H6) e H7) x H12) (THead (Flat Cast) x0 x1) -(THead (Flat Cast) t0 x0) (ty3_cast g c3 x1 x0 (ty3_conv g c3 x0 t0 (H3 i u -c3 x0 (fsubst0_both i u c t3 x0 H10 c3 H6) e H7) x1 t3 (H1 i u c3 x1 -(fsubst0_both i u c t2 x1 H11 c3 H6) e H7) (pc3_s c3 t3 x0 (pc3_fsubst0 c t3 -t3 (pc3_refl c t3) i u c3 x0 (fsubst0_both i u c t3 x0 H10 c3 H6) e H7))) t0 -(H3 i u c3 x0 (fsubst0_both i u c t3 x0 H10 c3 H6) e H7)) (pc3_fsubst0 c -(THead (Flat Cast) t0 t3) (THead (Flat Cast) t0 t3) (pc3_refl c (THead (Flat -Cast) t0 t3)) i u c3 (THead (Flat Cast) t0 x0) (fsubst0_both i u c (THead -(Flat Cast) t0 t3) (THead (Flat Cast) t0 x0) (subst0_snd (Flat Cast) u x0 t3 -i H10 t0) c3 H6) e H7)))) (ty3_correct g c3 t3 t0 (H3 i u c3 t3 (fsubst0_fst -i u c t3 c3 H6) e H7))) t5 H9)))))) H8)) (subst0_gen_head (Flat Cast) u t3 t2 -t5 i H5)))))))) c2 t4 H4)))))))))))))) c1 t1 t H))))). -(* COMMENTS -Initial nodes: 23439 -END *) - -theorem ty3_csubst0: - \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c1 -t1 t2) \to (\forall (e: C).(\forall (u: T).(\forall (i: nat).((getl i c1 -(CHead e (Bind Abbr) u)) \to (\forall (c2: C).((csubst0 i u c1 c2) \to (ty3 g -c2 t1 t2))))))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H: (ty3 g c1 t1 t2)).(\lambda (e: C).(\lambda (u: T).(\lambda (i: -nat).(\lambda (H0: (getl i c1 (CHead e (Bind Abbr) u))).(\lambda (c2: -C).(\lambda (H1: (csubst0 i u c1 c2)).(ty3_fsubst0 g c1 t1 t2 H i u c2 t1 -(fsubst0_fst i u c1 t1 c2 H1) e H0))))))))))). -(* COMMENTS -Initial nodes: 89 -END *) - -theorem ty3_subst0: - \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((ty3 g c t1 -t) \to (\forall (e: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead e -(Bind Abbr) u)) \to (\forall (t2: T).((subst0 i u t1 t2) \to (ty3 g c t2 -t))))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H: -(ty3 g c t1 t)).(\lambda (e: C).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H0: (getl i c (CHead e (Bind Abbr) u))).(\lambda (t2: T).(\lambda (H1: -(subst0 i u t1 t2)).(ty3_fsubst0 g c t1 t H i u c t2 (fsubst0_snd i u c t1 t2 -H1) e H0))))))))))). -(* COMMENTS -Initial nodes: 89 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/fwd.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/fwd.ma deleted file mode 100644 index bf6634e45..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/fwd.ma +++ /dev/null @@ -1,922 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/ty3/defs.ma". - -include "Basic-1/pc3/props.ma". - -theorem ty3_gen_sort: - \forall (g: G).(\forall (c: C).(\forall (x: T).(\forall (n: nat).((ty3 g c -(TSort n) x) \to (pc3 c (TSort (next g n)) x))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda -(H: (ty3 g c (TSort n) x)).(insert_eq T (TSort n) (\lambda (t: T).(ty3 g c t -x)) (\lambda (_: T).(pc3 c (TSort (next g n)) x)) (\lambda (y: T).(\lambda -(H0: (ty3 g c y x)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: -T).((eq T t (TSort n)) \to (pc3 c0 (TSort (next g n)) t0))))) (\lambda (c0: -C).(\lambda (t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda -(_: (((eq T t2 (TSort n)) \to (pc3 c0 (TSort (next g n)) t)))).(\lambda (u: -T).(\lambda (t1: T).(\lambda (H3: (ty3 g c0 u t1)).(\lambda (H4: (((eq T u -(TSort n)) \to (pc3 c0 (TSort (next g n)) t1)))).(\lambda (H5: (pc3 c0 t1 -t2)).(\lambda (H6: (eq T u (TSort n))).(let H7 \def (f_equal T T (\lambda (e: -T).e) u (TSort n) H6) in (let H8 \def (eq_ind T u (\lambda (t0: T).((eq T t0 -(TSort n)) \to (pc3 c0 (TSort (next g n)) t1))) H4 (TSort n) H7) in (let H9 -\def (eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 t1)) H3 (TSort n) H7) in -(pc3_t t1 c0 (TSort (next g n)) (H8 (refl_equal T (TSort n))) t2 -H5))))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq T -(TSort m) (TSort n))).(let H2 \def (f_equal T nat (\lambda (e: T).(match e in -T return (\lambda (_: T).nat) with [(TSort n0) \Rightarrow n0 | (TLRef _) -\Rightarrow m | (THead _ _ _) \Rightarrow m])) (TSort m) (TSort n) H1) in -(eq_ind_r nat n (\lambda (n0: nat).(pc3 c0 (TSort (next g n)) (TSort (next g -n0)))) (pc3_refl c0 (TSort (next g n))) m H2))))) (\lambda (n0: nat).(\lambda -(c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (_: (getl n0 c0 (CHead d -(Bind Abbr) u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (_: -(((eq T u (TSort n)) \to (pc3 d (TSort (next g n)) t)))).(\lambda (H4: (eq T -(TLRef n0) (TSort n))).(let H5 \def (eq_ind T (TLRef n0) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I -(TSort n) H4) in (False_ind (pc3 c0 (TSort (next g n)) (lift (S n0) O t)) -H5))))))))))) (\lambda (n0: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda -(u: T).(\lambda (_: (getl n0 c0 (CHead d (Bind Abst) u))).(\lambda (t: -T).(\lambda (_: (ty3 g d u t)).(\lambda (_: (((eq T u (TSort n)) \to (pc3 d -(TSort (next g n)) t)))).(\lambda (H4: (eq T (TLRef n0) (TSort n))).(let H5 -\def (eq_ind T (TLRef n0) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | -(THead _ _ _) \Rightarrow False])) I (TSort n) H4) in (False_ind (pc3 c0 -(TSort (next g n)) (lift (S n0) O u)) H5))))))))))) (\lambda (c0: C).(\lambda -(u: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((eq T u -(TSort n)) \to (pc3 c0 (TSort (next g n)) t)))).(\lambda (b: B).(\lambda (t1: -T).(\lambda (t2: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t1 -t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (pc3 (CHead c0 (Bind b) u) (TSort -(next g n)) t2)))).(\lambda (H5: (eq T (THead (Bind b) u t1) (TSort n))).(let -H6 \def (eq_ind T (THead (Bind b) u t1) (\lambda (ee: T).(match ee in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H5) in -(False_ind (pc3 c0 (TSort (next g n)) (THead (Bind b) u t2)) H6))))))))))))) -(\lambda (c0: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w -u)).(\lambda (_: (((eq T w (TSort n)) \to (pc3 c0 (TSort (next g n)) -u)))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead (Bind -Abst) u t))).(\lambda (_: (((eq T v (TSort n)) \to (pc3 c0 (TSort (next g n)) -(THead (Bind Abst) u t))))).(\lambda (H5: (eq T (THead (Flat Appl) w v) -(TSort n))).(let H6 \def (eq_ind T (THead (Flat Appl) w v) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I -(TSort n) H5) in (False_ind (pc3 c0 (TSort (next g n)) (THead (Flat Appl) w -(THead (Bind Abst) u t))) H6)))))))))))) (\lambda (c0: C).(\lambda (t1: -T).(\lambda (t2: T).(\lambda (_: (ty3 g c0 t1 t2)).(\lambda (_: (((eq T t1 -(TSort n)) \to (pc3 c0 (TSort (next g n)) t2)))).(\lambda (t0: T).(\lambda -(_: (ty3 g c0 t2 t0)).(\lambda (_: (((eq T t2 (TSort n)) \to (pc3 c0 (TSort -(next g n)) t0)))).(\lambda (H5: (eq T (THead (Flat Cast) t2 t1) (TSort -n))).(let H6 \def (eq_ind T (THead (Flat Cast) t2 t1) (\lambda (ee: T).(match -ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | -(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) -H5) in (False_ind (pc3 c0 (TSort (next g n)) (THead (Flat Cast) t0 t2)) -H6))))))))))) c y x H0))) H))))). -(* COMMENTS -Initial nodes: 1179 -END *) - -theorem ty3_gen_lref: - \forall (g: G).(\forall (c: C).(\forall (x: T).(\forall (n: nat).((ty3 g c -(TLRef n) x) \to (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda -(t: T).(pc3 c (lift (S n) O t) x)))) (\lambda (e: C).(\lambda (u: T).(\lambda -(_: T).(getl n c (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: -T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(pc3 c (lift (S n) O u) x)))) (\lambda (e: C).(\lambda -(u: T).(\lambda (_: T).(getl n c (CHead e (Bind Abst) u))))) (\lambda (e: -C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda -(H: (ty3 g c (TLRef n) x)).(insert_eq T (TLRef n) (\lambda (t: T).(ty3 g c t -x)) (\lambda (_: T).(or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda -(t0: T).(pc3 c (lift (S n) O t0) x)))) (\lambda (e: C).(\lambda (u: -T).(\lambda (_: T).(getl n c (CHead e (Bind Abbr) u))))) (\lambda (e: -C).(\lambda (u: T).(\lambda (t0: T).(ty3 g e u t0))))) (ex3_3 C T T (\lambda -(_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c (lift (S n) O u) x)))) (\lambda -(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead e (Bind Abst) u))))) -(\lambda (e: C).(\lambda (u: T).(\lambda (t0: T).(ty3 g e u t0))))))) -(\lambda (y: T).(\lambda (H0: (ty3 g c y x)).(ty3_ind g (\lambda (c0: -C).(\lambda (t: T).(\lambda (t0: T).((eq T t (TLRef n)) \to (or (ex3_3 C T T -(\lambda (_: C).(\lambda (_: T).(\lambda (t1: T).(pc3 c0 (lift (S n) O t1) -t0)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e -(Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t1: T).(ty3 g e -u t1))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 -c0 (lift (S n) O u) t0)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: -T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: -T).(\lambda (t1: T).(ty3 g e u t1)))))))))) (\lambda (c0: C).(\lambda (t2: -T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda (_: (((eq T t2 -(TLRef n)) \to (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: -T).(pc3 c0 (lift (S n) O t0) t)))) (\lambda (e: C).(\lambda (u: T).(\lambda -(_: T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: -T).(\lambda (t0: T).(ty3 g e u t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u) t)))) (\lambda (e: -C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) -(\lambda (e: C).(\lambda (u: T).(\lambda (t0: T).(ty3 g e u -t0))))))))).(\lambda (u: T).(\lambda (t1: T).(\lambda (H3: (ty3 g c0 u -t1)).(\lambda (H4: (((eq T u (TLRef n)) \to (or (ex3_3 C T T (\lambda (_: -C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t1)))) (\lambda -(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) -u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) -(ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift -(S n) O u0) t1)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n -c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(t0: T).(ty3 g e u0 t0))))))))).(\lambda (H5: (pc3 c0 t1 t2)).(\lambda (H6: -(eq T u (TLRef n))).(let H7 \def (f_equal T T (\lambda (e: T).e) u (TLRef n) -H6) in (let H8 \def (eq_ind T u (\lambda (t0: T).((eq T t0 (TLRef n)) \to (or -(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t3: T).(pc3 c0 (lift -(S n) O t3) t1)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n -c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(t3: T).(ty3 g e u0 t3))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: -T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) t1)))) (\lambda (e: C).(\lambda -(u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: -C).(\lambda (u0: T).(\lambda (t3: T).(ty3 g e u0 t3)))))))) H4 (TLRef n) H7) -in (let H9 \def (eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 t1)) H3 (TLRef n) -H7) in (let H10 \def (H8 (refl_equal T (TLRef n))) in (or_ind (ex3_3 C T T -(\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) -t1)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e -(Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g -e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: -T).(pc3 c0 (lift (S n) O u0) t1)))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (t0: T).(ty3 g e u0 t0))))) (or (ex3_3 C T T (\lambda (_: -C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda -(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) -u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) -(ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift -(S n) O u0) t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n -c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(t0: T).(ty3 g e u0 t0)))))) (\lambda (H11: (ex3_3 C T T (\lambda (_: -C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t1)))) (\lambda -(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) -u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 -t0)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: -T).(pc3 c0 (lift (S n) O t0) t1)))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(_: T).(getl n c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (t0: T).(ty3 g e u0 t0)))) (or (ex3_3 C T T (\lambda (_: -C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda -(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) -u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) -(ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift -(S n) O u0) t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n -c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(t0: T).(ty3 g e u0 t0)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: -T).(\lambda (H12: (pc3 c0 (lift (S n) O x2) t1)).(\lambda (H13: (getl n c0 -(CHead x0 (Bind Abbr) x1))).(\lambda (H14: (ty3 g x0 x1 x2)).(or_introl -(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift -(S n) O t0) t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n -c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: -T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) t2)))) (\lambda (e: C).(\lambda -(u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: -C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3_intro C T T -(\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) -t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e -(Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g -e u0 t0)))) x0 x1 x2 (pc3_t t1 c0 (lift (S n) O x2) H12 t2 H5) H13 -H14)))))))) H11)) (\lambda (H11: (ex3_3 C T T (\lambda (_: C).(\lambda (u0: -T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) t1)))) (\lambda (e: C).(\lambda -(u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: -C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))))).(ex3_3_ind C T T -(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) -t1)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e -(Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g -e u0 t0)))) (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: -T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(_: T).(getl n c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda -(u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) t2)))) (\lambda (e: -C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) -(\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))))) -(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H12: (pc3 c0 -(lift (S n) O x1) t1)).(\lambda (H13: (getl n c0 (CHead x0 (Bind Abst) -x1))).(\lambda (H14: (ty3 g x0 x1 x2)).(or_intror (ex3_3 C T T (\lambda (_: -C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda -(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) -u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) -(ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift -(S n) O u0) t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n -c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(t0: T).(ty3 g e u0 t0))))) (ex3_3_intro C T T (\lambda (_: C).(\lambda (u0: -T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) t2)))) (\lambda (e: C).(\lambda -(u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: -C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))) x0 x1 x2 (pc3_t t1 c0 -(lift (S n) O x1) H12 t2 H5) H13 H14)))))))) H11)) H10)))))))))))))))) -(\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq T (TSort m) (TLRef -n))).(let H2 \def (eq_ind T (TSort m) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef n) H1) in -(False_ind (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: -T).(pc3 c0 (lift (S n) O t) (TSort (next g m)))))) (\lambda (e: C).(\lambda -(u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: -C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T T (\lambda -(_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u) (TSort (next -g m)))))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e -(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u -t)))))) H2))))) (\lambda (n0: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda -(u: T).(\lambda (H1: (getl n0 c0 (CHead d (Bind Abbr) u))).(\lambda (t: -T).(\lambda (H2: (ty3 g d u t)).(\lambda (_: (((eq T u (TLRef n)) \to (or -(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 d (lift (S -n) O t0) t)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n d -(CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: -T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda -(_: T).(pc3 d (lift (S n) O u0) t)))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (_: T).(getl n d (CHead e (Bind Abst) u0))))) (\lambda (e: -C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))))))).(\lambda (H4: -(eq T (TLRef n0) (TLRef n))).(let H5 \def (f_equal T nat (\lambda (e: -T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow n0 | -(TLRef n1) \Rightarrow n1 | (THead _ _ _) \Rightarrow n0])) (TLRef n0) (TLRef -n) H4) in (let H6 \def (eq_ind nat n0 (\lambda (n1: nat).(getl n1 c0 (CHead d -(Bind Abbr) u))) H1 n H5) in (eq_ind_r nat n (\lambda (n1: nat).(or (ex3_3 C -T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O -t0) (lift (S n1) O t))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: -T).(getl n c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda -(u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) (lift (S n1) O t))))) -(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind -Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 -t0))))))) (or_introl (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda -(t0: T).(pc3 c0 (lift (S n) O t0) (lift (S n) O t))))) (\lambda (e: -C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u0))))) -(\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3 -C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O -u0) (lift (S n) O t))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: -T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3_intro C T T (\lambda (_: -C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) (lift (S n) O -t))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e -(Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g -e u0 t0)))) d u t (pc3_refl c0 (lift (S n) O t)) H6 H2)) n0 H5)))))))))))) -(\lambda (n0: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda -(H1: (getl n0 c0 (CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (H2: (ty3 -g d u t)).(\lambda (_: (((eq T u (TLRef n)) \to (or (ex3_3 C T T (\lambda (_: -C).(\lambda (_: T).(\lambda (t0: T).(pc3 d (lift (S n) O t0) t)))) (\lambda -(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n d (CHead e (Bind Abbr) -u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) -(ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 d (lift (S -n) O u0) t)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n d -(CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: -T).(ty3 g e u0 t0))))))))).(\lambda (H4: (eq T (TLRef n0) (TLRef n))).(let H5 -\def (f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) -with [(TSort _) \Rightarrow n0 | (TLRef n1) \Rightarrow n1 | (THead _ _ _) -\Rightarrow n0])) (TLRef n0) (TLRef n) H4) in (let H6 \def (eq_ind nat n0 -(\lambda (n1: nat).(getl n1 c0 (CHead d (Bind Abst) u))) H1 n H5) in -(eq_ind_r nat n (\lambda (n1: nat).(or (ex3_3 C T T (\lambda (_: C).(\lambda -(_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) (lift (S n1) O u))))) -(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind -Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 -t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 -(lift (S n) O u0) (lift (S n1) O u))))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: -C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))))) (or_intror (ex3_3 -C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O -t0) (lift (S n) O u))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: -T).(getl n c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda -(u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) (lift (S n) O u))))) -(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind -Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 -t0))))) (ex3_3_intro C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: -T).(pc3 c0 (lift (S n) O u0) (lift (S n) O u))))) (\lambda (e: C).(\lambda -(u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: -C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))) d u t (pc3_refl c0 -(lift (S n) O u)) H6 H2)) n0 H5)))))))))))) (\lambda (c0: C).(\lambda (u: -T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((eq T u (TLRef -n)) \to (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: -T).(pc3 c0 (lift (S n) O t0) t)))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(_: T).(getl n c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda -(u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) t)))) (\lambda (e: -C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) -(\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 -t0))))))))).(\lambda (b: B).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: -(ty3 g (CHead c0 (Bind b) u) t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to -(or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 (CHead -c0 (Bind b) u) (lift (S n) O t0) t2)))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (_: T).(getl n (CHead c0 (Bind b) u) (CHead e (Bind Abbr) u0))))) -(\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3 -C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 (CHead c0 (Bind -b) u) (lift (S n) O u0) t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: -T).(getl n (CHead c0 (Bind b) u) (CHead e (Bind Abst) u0))))) (\lambda (e: -C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))))))).(\lambda (H5: -(eq T (THead (Bind b) u t1) (TLRef n))).(let H6 \def (eq_ind T (THead (Bind -b) u t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) -\Rightarrow True])) I (TLRef n) H5) in (False_ind (or (ex3_3 C T T (\lambda -(_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) (THead -(Bind b) u t2))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n -c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: -T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) (THead (Bind b) u t2))))) -(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind -Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 -t0)))))) H6))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda (u: -T).(\lambda (_: (ty3 g c0 w u)).(\lambda (_: (((eq T w (TLRef n)) \to (or -(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c0 (lift (S -n) O t) u)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 -(CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t: -T).(ty3 g e u0 t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda -(_: T).(pc3 c0 (lift (S n) O u0) u)))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: -C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t))))))))).(\lambda (v: -T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead (Bind Abst) u -t))).(\lambda (_: (((eq T v (TLRef n)) \to (or (ex3_3 C T T (\lambda (_: -C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) (THead (Bind -Abst) u t))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 -(CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: -T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda -(_: T).(pc3 c0 (lift (S n) O u0) (THead (Bind Abst) u t))))) (\lambda (e: -C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) -(\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 -t0))))))))).(\lambda (H5: (eq T (THead (Flat Appl) w v) (TLRef n))).(let H6 -\def (eq_ind T (THead (Flat Appl) w v) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H5) in -(False_ind (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: -T).(pc3 c0 (lift (S n) O t0) (THead (Flat Appl) w (THead (Bind Abst) u -t)))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e -(Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g -e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: -T).(pc3 c0 (lift (S n) O u0) (THead (Flat Appl) w (THead (Bind Abst) u -t)))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e -(Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g -e u0 t0)))))) H6)))))))))))) (\lambda (c0: C).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (_: (ty3 g c0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (or -(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c0 (lift (S -n) O t) t2)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 -(CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: -T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda -(_: T).(pc3 c0 (lift (S n) O u) t2)))) (\lambda (e: C).(\lambda (u: -T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: -C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))))))).(\lambda (t0: -T).(\lambda (_: (ty3 g c0 t2 t0)).(\lambda (_: (((eq T t2 (TLRef n)) \to (or -(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c0 (lift (S -n) O t) t0)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 -(CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: -T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda -(_: T).(pc3 c0 (lift (S n) O u) t0)))) (\lambda (e: C).(\lambda (u: -T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: -C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))))))).(\lambda (H5: (eq T -(THead (Flat Cast) t2 t1) (TLRef n))).(let H6 \def (eq_ind T (THead (Flat -Cast) t2 t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) -with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ -_) \Rightarrow True])) I (TLRef n) H5) in (False_ind (or (ex3_3 C T T -(\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c0 (lift (S n) O t) -(THead (Flat Cast) t0 t2))))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: -T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: -T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda -(u: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u) (THead (Flat Cast) t0 t2))))) -(\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind -Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u -t)))))) H6))))))))))) c y x H0))) H))))). -(* COMMENTS -Initial nodes: 5569 -END *) - -theorem ty3_gen_bind: - \forall (g: G).(\forall (b: B).(\forall (c: C).(\forall (u: T).(\forall (t1: -T).(\forall (x: T).((ty3 g c (THead (Bind b) u t1) x) \to (ex3_2 T T (\lambda -(t2: T).(\lambda (_: T).(pc3 c (THead (Bind b) u t2) x))) (\lambda (_: -T).(\lambda (t: T).(ty3 g c u t))) (\lambda (t2: T).(\lambda (_: T).(ty3 g -(CHead c (Bind b) u) t1 t2)))))))))) -\def - \lambda (g: G).(\lambda (b: B).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: -T).(\lambda (x: T).(\lambda (H: (ty3 g c (THead (Bind b) u t1) x)).(insert_eq -T (THead (Bind b) u t1) (\lambda (t: T).(ty3 g c t x)) (\lambda (_: T).(ex3_2 -T T (\lambda (t2: T).(\lambda (_: T).(pc3 c (THead (Bind b) u t2) x))) -(\lambda (_: T).(\lambda (t0: T).(ty3 g c u t0))) (\lambda (t2: T).(\lambda -(_: T).(ty3 g (CHead c (Bind b) u) t1 t2))))) (\lambda (y: T).(\lambda (H0: -(ty3 g c y x)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: -T).((eq T t (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda (t2: T).(\lambda -(_: T).(pc3 c0 (THead (Bind b) u t2) t0))) (\lambda (_: T).(\lambda (t3: -T).(ty3 g c0 u t3))) (\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind -b) u) t1 t2)))))))) (\lambda (c0: C).(\lambda (t2: T).(\lambda (t: -T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda (_: (((eq T t2 (THead (Bind b) u -t1)) \to (ex3_2 T T (\lambda (t3: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) -u t3) t))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u t0))) (\lambda (t3: -T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3))))))).(\lambda (u0: -T).(\lambda (t0: T).(\lambda (H3: (ty3 g c0 u0 t0)).(\lambda (H4: (((eq T u0 -(THead (Bind b) u t1)) \to (ex3_2 T T (\lambda (t3: T).(\lambda (_: T).(pc3 -c0 (THead (Bind b) u t3) t0))) (\lambda (_: T).(\lambda (t4: T).(ty3 g c0 u -t4))) (\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 -t3))))))).(\lambda (H5: (pc3 c0 t0 t2)).(\lambda (H6: (eq T u0 (THead (Bind -b) u t1))).(let H7 \def (f_equal T T (\lambda (e: T).e) u0 (THead (Bind b) u -t1) H6) in (let H8 \def (eq_ind T u0 (\lambda (t3: T).((eq T t3 (THead (Bind -b) u t1)) \to (ex3_2 T T (\lambda (t4: T).(\lambda (_: T).(pc3 c0 (THead -(Bind b) u t4) t0))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c0 u t5))) -(\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t4)))))) H4 -(THead (Bind b) u t1) H7) in (let H9 \def (eq_ind T u0 (\lambda (t3: T).(ty3 -g c0 t3 t0)) H3 (THead (Bind b) u t1) H7) in (let H10 \def (H8 (refl_equal T -(THead (Bind b) u t1))) in (ex3_2_ind T T (\lambda (t3: T).(\lambda (_: -T).(pc3 c0 (THead (Bind b) u t3) t0))) (\lambda (_: T).(\lambda (t4: T).(ty3 -g c0 u t4))) (\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 -t3))) (ex3_2 T T (\lambda (t3: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u -t3) t2))) (\lambda (_: T).(\lambda (t4: T).(ty3 g c0 u t4))) (\lambda (t3: -T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3)))) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H11: (pc3 c0 (THead (Bind b) u x0) -t0)).(\lambda (H12: (ty3 g c0 u x1)).(\lambda (H13: (ty3 g (CHead c0 (Bind b) -u) t1 x0)).(ex3_2_intro T T (\lambda (t3: T).(\lambda (_: T).(pc3 c0 (THead -(Bind b) u t3) t2))) (\lambda (_: T).(\lambda (t4: T).(ty3 g c0 u t4))) -(\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3))) x0 x1 -(pc3_t t0 c0 (THead (Bind b) u x0) H11 t2 H5) H12 H13)))))) -H10)))))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq T -(TSort m) (THead (Bind b) u t1))).(let H2 \def (eq_ind T (TSort m) (\lambda -(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow -False])) I (THead (Bind b) u t1) H1) in (False_ind (ex3_2 T T (\lambda (t2: -T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t2) (TSort (next g m))))) -(\lambda (_: T).(\lambda (t: T).(ty3 g c0 u t))) (\lambda (t2: T).(\lambda -(_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2)))) H2))))) (\lambda (n: -nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (_: (getl n -c0 (CHead d (Bind Abbr) u0))).(\lambda (t: T).(\lambda (_: (ty3 g d u0 -t)).(\lambda (_: (((eq T u0 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda -(t2: T).(\lambda (_: T).(pc3 d (THead (Bind b) u t2) t))) (\lambda (_: -T).(\lambda (t0: T).(ty3 g d u t0))) (\lambda (t2: T).(\lambda (_: T).(ty3 g -(CHead d (Bind b) u) t1 t2))))))).(\lambda (H4: (eq T (TLRef n) (THead (Bind -b) u t1))).(let H5 \def (eq_ind T (TLRef n) (\lambda (ee: T).(match ee in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t1) -H4) in (False_ind (ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3 c0 (THead -(Bind b) u t2) (lift (S n) O t)))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c0 -u t0))) (\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 -t2)))) H5))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: -C).(\lambda (u0: T).(\lambda (_: (getl n c0 (CHead d (Bind Abst) -u0))).(\lambda (t: T).(\lambda (_: (ty3 g d u0 t)).(\lambda (_: (((eq T u0 -(THead (Bind b) u t1)) \to (ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3 d -(THead (Bind b) u t2) t))) (\lambda (_: T).(\lambda (t0: T).(ty3 g d u t0))) -(\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead d (Bind b) u) t1 -t2))))))).(\lambda (H4: (eq T (TLRef n) (THead (Bind b) u t1))).(let H5 \def -(eq_ind T (TLRef n) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | -(THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t1) H4) in (False_ind -(ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t2) -(lift (S n) O u0)))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u t0))) -(\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2)))) -H5))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (t: T).(\lambda (H1: -(ty3 g c0 u0 t)).(\lambda (H2: (((eq T u0 (THead (Bind b) u t1)) \to (ex3_2 T -T (\lambda (t2: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t2) t))) -(\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u t0))) (\lambda (t2: T).(\lambda -(_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2))))))).(\lambda (b0: B).(\lambda -(t0: T).(\lambda (t2: T).(\lambda (H3: (ty3 g (CHead c0 (Bind b0) u0) t0 -t2)).(\lambda (H4: (((eq T t0 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda -(t3: T).(\lambda (_: T).(pc3 (CHead c0 (Bind b0) u0) (THead (Bind b) u t3) -t2))) (\lambda (_: T).(\lambda (t4: T).(ty3 g (CHead c0 (Bind b0) u0) u t4))) -(\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead (CHead c0 (Bind b0) u0) (Bind -b) u) t1 t3))))))).(\lambda (H5: (eq T (THead (Bind b0) u0 t0) (THead (Bind -b) u t1))).(let H6 \def (f_equal T B (\lambda (e: T).(match e in T return -(\lambda (_: T).B) with [(TSort _) \Rightarrow b0 | (TLRef _) \Rightarrow b0 -| (THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).B) with -[(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow b0])])) (THead (Bind b0) u0 -t0) (THead (Bind b) u t1) H5) in ((let H7 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | -(TLRef _) \Rightarrow u0 | (THead _ t3 _) \Rightarrow t3])) (THead (Bind b0) -u0 t0) (THead (Bind b) u t1) H5) in ((let H8 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | -(TLRef _) \Rightarrow t0 | (THead _ _ t3) \Rightarrow t3])) (THead (Bind b0) -u0 t0) (THead (Bind b) u t1) H5) in (\lambda (H9: (eq T u0 u)).(\lambda (H10: -(eq B b0 b)).(let H11 \def (eq_ind T t0 (\lambda (t3: T).((eq T t3 (THead -(Bind b) u t1)) \to (ex3_2 T T (\lambda (t4: T).(\lambda (_: T).(pc3 (CHead -c0 (Bind b0) u0) (THead (Bind b) u t4) t2))) (\lambda (_: T).(\lambda (t5: -T).(ty3 g (CHead c0 (Bind b0) u0) u t5))) (\lambda (t4: T).(\lambda (_: -T).(ty3 g (CHead (CHead c0 (Bind b0) u0) (Bind b) u) t1 t4)))))) H4 t1 H8) in -(let H12 \def (eq_ind T t0 (\lambda (t3: T).(ty3 g (CHead c0 (Bind b0) u0) t3 -t2)) H3 t1 H8) in (let H13 \def (eq_ind B b0 (\lambda (b1: B).((eq T t1 -(THead (Bind b) u t1)) \to (ex3_2 T T (\lambda (t3: T).(\lambda (_: T).(pc3 -(CHead c0 (Bind b1) u0) (THead (Bind b) u t3) t2))) (\lambda (_: T).(\lambda -(t4: T).(ty3 g (CHead c0 (Bind b1) u0) u t4))) (\lambda (t3: T).(\lambda (_: -T).(ty3 g (CHead (CHead c0 (Bind b1) u0) (Bind b) u) t1 t3)))))) H11 b H10) -in (let H14 \def (eq_ind B b0 (\lambda (b1: B).(ty3 g (CHead c0 (Bind b1) u0) -t1 t2)) H12 b H10) in (eq_ind_r B b (\lambda (b1: B).(ex3_2 T T (\lambda (t3: -T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t3) (THead (Bind b1) u0 t2)))) -(\lambda (_: T).(\lambda (t4: T).(ty3 g c0 u t4))) (\lambda (t3: T).(\lambda -(_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3))))) (let H15 \def (eq_ind T u0 -(\lambda (t3: T).((eq T t1 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda -(t4: T).(\lambda (_: T).(pc3 (CHead c0 (Bind b) t3) (THead (Bind b) u t4) -t2))) (\lambda (_: T).(\lambda (t5: T).(ty3 g (CHead c0 (Bind b) t3) u t5))) -(\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead (CHead c0 (Bind b) t3) (Bind -b) u) t1 t4)))))) H13 u H9) in (let H16 \def (eq_ind T u0 (\lambda (t3: -T).(ty3 g (CHead c0 (Bind b) t3) t1 t2)) H14 u H9) in (let H17 \def (eq_ind T -u0 (\lambda (t3: T).((eq T t3 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda -(t4: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t4) t))) (\lambda (_: -T).(\lambda (t5: T).(ty3 g c0 u t5))) (\lambda (t4: T).(\lambda (_: T).(ty3 g -(CHead c0 (Bind b) u) t1 t4)))))) H2 u H9) in (let H18 \def (eq_ind T u0 -(\lambda (t3: T).(ty3 g c0 t3 t)) H1 u H9) in (eq_ind_r T u (\lambda (t3: -T).(ex3_2 T T (\lambda (t4: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t4) -(THead (Bind b) t3 t2)))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c0 u t5))) -(\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t4))))) -(ex3_2_intro T T (\lambda (t3: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u -t3) (THead (Bind b) u t2)))) (\lambda (_: T).(\lambda (t4: T).(ty3 g c0 u -t4))) (\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3))) -t2 t (pc3_refl c0 (THead (Bind b) u t2)) H18 H16) u0 H9))))) b0 H10)))))))) -H7)) H6))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda (u0: -T).(\lambda (_: (ty3 g c0 w u0)).(\lambda (_: (((eq T w (THead (Bind b) u -t1)) \to (ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) -u t2) u0))) (\lambda (_: T).(\lambda (t: T).(ty3 g c0 u t))) (\lambda (t2: -T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2))))))).(\lambda (v: -T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead (Bind Abst) u0 -t))).(\lambda (_: (((eq T v (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda -(t2: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t2) (THead (Bind Abst) u0 -t)))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u t0))) (\lambda (t2: -T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2))))))).(\lambda (H5: -(eq T (THead (Flat Appl) w v) (THead (Bind b) u t1))).(let H6 \def (eq_ind T -(THead (Flat Appl) w v) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind -b) u t1) H5) in (False_ind (ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3 -c0 (THead (Bind b) u t2) (THead (Flat Appl) w (THead (Bind Abst) u0 t))))) -(\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u t0))) (\lambda (t2: T).(\lambda -(_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2)))) H6)))))))))))) (\lambda (c0: -C).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (ty3 g c0 t0 t2)).(\lambda -(_: (((eq T t0 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda (t3: -T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t3) t2))) (\lambda (_: -T).(\lambda (t: T).(ty3 g c0 u t))) (\lambda (t3: T).(\lambda (_: T).(ty3 g -(CHead c0 (Bind b) u) t1 t3))))))).(\lambda (t3: T).(\lambda (_: (ty3 g c0 t2 -t3)).(\lambda (_: (((eq T t2 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda -(t4: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t4) t3))) (\lambda (_: -T).(\lambda (t: T).(ty3 g c0 u t))) (\lambda (t4: T).(\lambda (_: T).(ty3 g -(CHead c0 (Bind b) u) t1 t4))))))).(\lambda (H5: (eq T (THead (Flat Cast) t2 -t0) (THead (Bind b) u t1))).(let H6 \def (eq_ind T (THead (Flat Cast) t2 t0) -(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow -(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | -(Flat _) \Rightarrow True])])) I (THead (Bind b) u t1) H5) in (False_ind -(ex3_2 T T (\lambda (t4: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t4) -(THead (Flat Cast) t3 t2)))) (\lambda (_: T).(\lambda (t: T).(ty3 g c0 u t))) -(\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t4)))) -H6))))))))))) c y x H0))) H))))))). -(* COMMENTS -Initial nodes: 3389 -END *) - -theorem ty3_gen_appl: - \forall (g: G).(\forall (c: C).(\forall (w: T).(\forall (v: T).(\forall (x: -T).((ty3 g c (THead (Flat Appl) w v) x) \to (ex3_2 T T (\lambda (u: -T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x))) -(\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) -(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (w: T).(\lambda (v: T).(\lambda (x: -T).(\lambda (H: (ty3 g c (THead (Flat Appl) w v) x)).(insert_eq T (THead -(Flat Appl) w v) (\lambda (t: T).(ty3 g c t x)) (\lambda (_: T).(ex3_2 T T -(\lambda (u: T).(\lambda (t0: T).(pc3 c (THead (Flat Appl) w (THead (Bind -Abst) u t0)) x))) (\lambda (u: T).(\lambda (t0: T).(ty3 g c v (THead (Bind -Abst) u t0)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c w u))))) (\lambda (y: -T).(\lambda (H0: (ty3 g c y x)).(ty3_ind g (\lambda (c0: C).(\lambda (t: -T).(\lambda (t0: T).((eq T t (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda -(u: T).(\lambda (t1: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u -t1)) t0))) (\lambda (u: T).(\lambda (t1: T).(ty3 g c0 v (THead (Bind Abst) u -t1)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c0 w u)))))))) (\lambda (c0: -C).(\lambda (t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda -(_: (((eq T t2 (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda (u: -T).(\lambda (t0: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u t0)) -t))) (\lambda (u: T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u t0)))) -(\lambda (u: T).(\lambda (_: T).(ty3 g c0 w u))))))).(\lambda (u: T).(\lambda -(t1: T).(\lambda (H3: (ty3 g c0 u t1)).(\lambda (H4: (((eq T u (THead (Flat -Appl) w v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0 (THead -(Flat Appl) w (THead (Bind Abst) u0 t0)) t1))) (\lambda (u0: T).(\lambda (t0: -T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_: -T).(ty3 g c0 w u0))))))).(\lambda (H5: (pc3 c0 t1 t2)).(\lambda (H6: (eq T u -(THead (Flat Appl) w v))).(let H7 \def (f_equal T T (\lambda (e: T).e) u -(THead (Flat Appl) w v) H6) in (let H8 \def (eq_ind T u (\lambda (t0: T).((eq -T t0 (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t3: -T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t3)) t1))) (\lambda -(u0: T).(\lambda (t3: T).(ty3 g c0 v (THead (Bind Abst) u0 t3)))) (\lambda -(u0: T).(\lambda (_: T).(ty3 g c0 w u0)))))) H4 (THead (Flat Appl) w v) H7) -in (let H9 \def (eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 t1)) H3 (THead -(Flat Appl) w v) H7) in (let H10 \def (H8 (refl_equal T (THead (Flat Appl) w -v))) in (ex3_2_ind T T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0 (THead (Flat -Appl) w (THead (Bind Abst) u0 t0)) t1))) (\lambda (u0: T).(\lambda (t0: -T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_: -T).(ty3 g c0 w u0))) (ex3_2 T T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0 -(THead (Flat Appl) w (THead (Bind Abst) u0 t0)) t2))) (\lambda (u0: -T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: -T).(\lambda (_: T).(ty3 g c0 w u0)))) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H11: (pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) x0 x1)) -t1)).(\lambda (H12: (ty3 g c0 v (THead (Bind Abst) x0 x1))).(\lambda (H13: -(ty3 g c0 w x0)).(ex3_2_intro T T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0 -(THead (Flat Appl) w (THead (Bind Abst) u0 t0)) t2))) (\lambda (u0: -T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: -T).(\lambda (_: T).(ty3 g c0 w u0))) x0 x1 (pc3_t t1 c0 (THead (Flat Appl) w -(THead (Bind Abst) x0 x1)) H11 t2 H5) H12 H13)))))) H10)))))))))))))))) -(\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq T (TSort m) (THead (Flat -Appl) w v))).(let H2 \def (eq_ind T (TSort m) (\lambda (ee: T).(match ee in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) w -v) H1) in (False_ind (ex3_2 T T (\lambda (u: T).(\lambda (t: T).(pc3 c0 -(THead (Flat Appl) w (THead (Bind Abst) u t)) (TSort (next g m))))) (\lambda -(u: T).(\lambda (t: T).(ty3 g c0 v (THead (Bind Abst) u t)))) (\lambda (u: -T).(\lambda (_: T).(ty3 g c0 w u)))) H2))))) (\lambda (n: nat).(\lambda (c0: -C).(\lambda (d: C).(\lambda (u: T).(\lambda (_: (getl n c0 (CHead d (Bind -Abbr) u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (_: (((eq T u -(THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t0: -T).(pc3 d (THead (Flat Appl) w (THead (Bind Abst) u0 t0)) t))) (\lambda (u0: -T).(\lambda (t0: T).(ty3 g d v (THead (Bind Abst) u0 t0)))) (\lambda (u0: -T).(\lambda (_: T).(ty3 g d w u0))))))).(\lambda (H4: (eq T (TLRef n) (THead -(Flat Appl) w v))).(let H5 \def (eq_ind T (TLRef n) (\lambda (ee: T).(match -ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | -(TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead -(Flat Appl) w v) H4) in (False_ind (ex3_2 T T (\lambda (u0: T).(\lambda (t0: -T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t0)) (lift (S n) O -t)))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u0 -t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w u0)))) H5))))))))))) -(\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda -(_: (getl n c0 (CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (_: (ty3 g -d u t)).(\lambda (_: (((eq T u (THead (Flat Appl) w v)) \to (ex3_2 T T -(\lambda (u0: T).(\lambda (t0: T).(pc3 d (THead (Flat Appl) w (THead (Bind -Abst) u0 t0)) t))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g d v (THead (Bind -Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g d w -u0))))))).(\lambda (H4: (eq T (TLRef n) (THead (Flat Appl) w v))).(let H5 -\def (eq_ind T (TLRef n) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | -(THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) w v) H4) in -(False_ind (ex3_2 T T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0 (THead (Flat -Appl) w (THead (Bind Abst) u0 t0)) (lift (S n) O u)))) (\lambda (u0: -T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: -T).(\lambda (_: T).(ty3 g c0 w u0)))) H5))))))))))) (\lambda (c0: C).(\lambda -(u: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((eq T u -(THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t0: -T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t0)) t))) (\lambda (u0: -T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: -T).(\lambda (_: T).(ty3 g c0 w u0))))))).(\lambda (b: B).(\lambda (t1: -T).(\lambda (t2: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t1 -t2)).(\lambda (_: (((eq T t1 (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda -(u0: T).(\lambda (t0: T).(pc3 (CHead c0 (Bind b) u) (THead (Flat Appl) w -(THead (Bind Abst) u0 t0)) t2))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g -(CHead c0 (Bind b) u) v (THead (Bind Abst) u0 t0)))) (\lambda (u0: -T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) w u0))))))).(\lambda (H5: (eq -T (THead (Bind b) u t1) (THead (Flat Appl) w v))).(let H6 \def (eq_ind T -(THead (Bind b) u t1) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat -Appl) w v) H5) in (False_ind (ex3_2 T T (\lambda (u0: T).(\lambda (t0: -T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t0)) (THead (Bind b) u -t2)))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u0 -t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w u0)))) H6))))))))))))) -(\lambda (c0: C).(\lambda (w0: T).(\lambda (u: T).(\lambda (H1: (ty3 g c0 w0 -u)).(\lambda (H2: (((eq T w0 (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda -(u0: T).(\lambda (t: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 -t)) u))) (\lambda (u0: T).(\lambda (t: T).(ty3 g c0 v (THead (Bind Abst) u0 -t)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w u0))))))).(\lambda (v0: -T).(\lambda (t: T).(\lambda (H3: (ty3 g c0 v0 (THead (Bind Abst) u -t))).(\lambda (H4: (((eq T v0 (THead (Flat Appl) w v)) \to (ex3_2 T T -(\lambda (u0: T).(\lambda (t0: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind -Abst) u0 t0)) (THead (Bind Abst) u t)))) (\lambda (u0: T).(\lambda (t0: -T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_: -T).(ty3 g c0 w u0))))))).(\lambda (H5: (eq T (THead (Flat Appl) w0 v0) (THead -(Flat Appl) w v))).(let H6 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow w0 | (TLRef _) -\Rightarrow w0 | (THead _ t0 _) \Rightarrow t0])) (THead (Flat Appl) w0 v0) -(THead (Flat Appl) w v) H5) in ((let H7 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v0 | -(TLRef _) \Rightarrow v0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat -Appl) w0 v0) (THead (Flat Appl) w v) H5) in (\lambda (H8: (eq T w0 w)).(let -H9 \def (eq_ind T v0 (\lambda (t0: T).((eq T t0 (THead (Flat Appl) w v)) \to -(ex3_2 T T (\lambda (u0: T).(\lambda (t1: T).(pc3 c0 (THead (Flat Appl) w -(THead (Bind Abst) u0 t1)) (THead (Bind Abst) u t)))) (\lambda (u0: -T).(\lambda (t1: T).(ty3 g c0 v (THead (Bind Abst) u0 t1)))) (\lambda (u0: -T).(\lambda (_: T).(ty3 g c0 w u0)))))) H4 v H7) in (let H10 \def (eq_ind T -v0 (\lambda (t0: T).(ty3 g c0 t0 (THead (Bind Abst) u t))) H3 v H7) in (let -H11 \def (eq_ind T w0 (\lambda (t0: T).((eq T t0 (THead (Flat Appl) w v)) \to -(ex3_2 T T (\lambda (u0: T).(\lambda (t1: T).(pc3 c0 (THead (Flat Appl) w -(THead (Bind Abst) u0 t1)) u))) (\lambda (u0: T).(\lambda (t1: T).(ty3 g c0 v -(THead (Bind Abst) u0 t1)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w -u0)))))) H2 w H8) in (let H12 \def (eq_ind T w0 (\lambda (t0: T).(ty3 g c0 t0 -u)) H1 w H8) in (eq_ind_r T w (\lambda (t0: T).(ex3_2 T T (\lambda (u0: -T).(\lambda (t1: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t1)) -(THead (Flat Appl) t0 (THead (Bind Abst) u t))))) (\lambda (u0: T).(\lambda -(t1: T).(ty3 g c0 v (THead (Bind Abst) u0 t1)))) (\lambda (u0: T).(\lambda -(_: T).(ty3 g c0 w u0))))) (ex3_2_intro T T (\lambda (u0: T).(\lambda (t0: -T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t0)) (THead (Flat Appl) -w (THead (Bind Abst) u t))))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g c0 v -(THead (Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w -u0))) u t (pc3_refl c0 (THead (Flat Appl) w (THead (Bind Abst) u t))) H10 -H12) w0 H8))))))) H6)))))))))))) (\lambda (c0: C).(\lambda (t1: T).(\lambda -(t2: T).(\lambda (_: (ty3 g c0 t1 t2)).(\lambda (_: (((eq T t1 (THead (Flat -Appl) w v)) \to (ex3_2 T T (\lambda (u: T).(\lambda (t: T).(pc3 c0 (THead -(Flat Appl) w (THead (Bind Abst) u t)) t2))) (\lambda (u: T).(\lambda (t: -T).(ty3 g c0 v (THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (_: -T).(ty3 g c0 w u))))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t2 -t0)).(\lambda (_: (((eq T t2 (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda -(u: T).(\lambda (t: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u t)) -t0))) (\lambda (u: T).(\lambda (t: T).(ty3 g c0 v (THead (Bind Abst) u t)))) -(\lambda (u: T).(\lambda (_: T).(ty3 g c0 w u))))))).(\lambda (H5: (eq T -(THead (Flat Cast) t2 t1) (THead (Flat Appl) w v))).(let H6 \def (eq_ind T -(THead (Flat Cast) t2 t1) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f in F return -(\lambda (_: F).Prop) with [Appl \Rightarrow False | Cast \Rightarrow -True])])])) I (THead (Flat Appl) w v) H5) in (False_ind (ex3_2 T T (\lambda -(u: T).(\lambda (t: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u t)) -(THead (Flat Cast) t0 t2)))) (\lambda (u: T).(\lambda (t: T).(ty3 g c0 v -(THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c0 w u)))) -H6))))))))))) c y x H0))) H)))))). -(* COMMENTS -Initial nodes: 3171 -END *) - -theorem ty3_gen_cast: - \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).(\forall -(x: T).((ty3 g c (THead (Flat Cast) t2 t1) x) \to (ex3 T (\lambda (t0: -T).(pc3 c (THead (Flat Cast) t0 t2) x)) (\lambda (_: T).(ty3 g c t1 t2)) -(\lambda (t0: T).(ty3 g c t2 t0)))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(x: T).(\lambda (H: (ty3 g c (THead (Flat Cast) t2 t1) x)).(insert_eq T -(THead (Flat Cast) t2 t1) (\lambda (t: T).(ty3 g c t x)) (\lambda (_: T).(ex3 -T (\lambda (t0: T).(pc3 c (THead (Flat Cast) t0 t2) x)) (\lambda (_: T).(ty3 -g c t1 t2)) (\lambda (t0: T).(ty3 g c t2 t0)))) (\lambda (y: T).(\lambda (H0: -(ty3 g c y x)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: -T).((eq T t (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t3: T).(pc3 c0 -(THead (Flat Cast) t3 t2) t0)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda -(t3: T).(ty3 g c0 t2 t3))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t: -T).(\lambda (_: (ty3 g c0 t0 t)).(\lambda (_: (((eq T t0 (THead (Flat Cast) -t2 t1)) \to (ex3 T (\lambda (t3: T).(pc3 c0 (THead (Flat Cast) t3 t2) t)) -(\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t3: T).(ty3 g c0 t2 -t3)))))).(\lambda (u: T).(\lambda (t3: T).(\lambda (H3: (ty3 g c0 u -t3)).(\lambda (H4: (((eq T u (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda -(t4: T).(pc3 c0 (THead (Flat Cast) t4 t2) t3)) (\lambda (_: T).(ty3 g c0 t1 -t2)) (\lambda (t4: T).(ty3 g c0 t2 t4)))))).(\lambda (H5: (pc3 c0 t3 -t0)).(\lambda (H6: (eq T u (THead (Flat Cast) t2 t1))).(let H7 \def (f_equal -T T (\lambda (e: T).e) u (THead (Flat Cast) t2 t1) H6) in (let H8 \def -(eq_ind T u (\lambda (t4: T).((eq T t4 (THead (Flat Cast) t2 t1)) \to (ex3 T -(\lambda (t5: T).(pc3 c0 (THead (Flat Cast) t5 t2) t3)) (\lambda (_: T).(ty3 -g c0 t1 t2)) (\lambda (t5: T).(ty3 g c0 t2 t5))))) H4 (THead (Flat Cast) t2 -t1) H7) in (let H9 \def (eq_ind T u (\lambda (t4: T).(ty3 g c0 t4 t3)) H3 -(THead (Flat Cast) t2 t1) H7) in (let H10 \def (H8 (refl_equal T (THead (Flat -Cast) t2 t1))) in (ex3_ind T (\lambda (t4: T).(pc3 c0 (THead (Flat Cast) t4 -t2) t3)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t4: T).(ty3 g c0 t2 t4)) -(ex3 T (\lambda (t4: T).(pc3 c0 (THead (Flat Cast) t4 t2) t0)) (\lambda (_: -T).(ty3 g c0 t1 t2)) (\lambda (t4: T).(ty3 g c0 t2 t4))) (\lambda (x0: -T).(\lambda (H11: (pc3 c0 (THead (Flat Cast) x0 t2) t3)).(\lambda (H12: (ty3 -g c0 t1 t2)).(\lambda (H13: (ty3 g c0 t2 x0)).(ex3_intro T (\lambda (t4: -T).(pc3 c0 (THead (Flat Cast) t4 t2) t0)) (\lambda (_: T).(ty3 g c0 t1 t2)) -(\lambda (t4: T).(ty3 g c0 t2 t4)) x0 (pc3_t t3 c0 (THead (Flat Cast) x0 t2) -H11 t0 H5) H12 H13))))) H10)))))))))))))))) (\lambda (c0: C).(\lambda (m: -nat).(\lambda (H1: (eq T (TSort m) (THead (Flat Cast) t2 t1))).(let H2 \def -(eq_ind T (TSort m) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | -(THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) t2 t1) H1) in -(False_ind (ex3 T (\lambda (t0: T).(pc3 c0 (THead (Flat Cast) t0 t2) (TSort -(next g m)))) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0 t2 -t0))) H2))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: -T).(\lambda (_: (getl n c0 (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda -(_: (ty3 g d u t)).(\lambda (_: (((eq T u (THead (Flat Cast) t2 t1)) \to (ex3 -T (\lambda (t0: T).(pc3 d (THead (Flat Cast) t0 t2) t)) (\lambda (_: T).(ty3 -g d t1 t2)) (\lambda (t0: T).(ty3 g d t2 t0)))))).(\lambda (H4: (eq T (TLRef -n) (THead (Flat Cast) t2 t1))).(let H5 \def (eq_ind T (TLRef n) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I -(THead (Flat Cast) t2 t1) H4) in (False_ind (ex3 T (\lambda (t0: T).(pc3 c0 -(THead (Flat Cast) t0 t2) (lift (S n) O t))) (\lambda (_: T).(ty3 g c0 t1 -t2)) (\lambda (t0: T).(ty3 g c0 t2 t0))) H5))))))))))) (\lambda (n: -nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (_: (getl n c0 -(CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (_: (ty3 g d u -t)).(\lambda (_: (((eq T u (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda -(t0: T).(pc3 d (THead (Flat Cast) t0 t2) t)) (\lambda (_: T).(ty3 g d t1 t2)) -(\lambda (t0: T).(ty3 g d t2 t0)))))).(\lambda (H4: (eq T (TLRef n) (THead -(Flat Cast) t2 t1))).(let H5 \def (eq_ind T (TLRef n) (\lambda (ee: T).(match -ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | -(TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead -(Flat Cast) t2 t1) H4) in (False_ind (ex3 T (\lambda (t0: T).(pc3 c0 (THead -(Flat Cast) t0 t2) (lift (S n) O u))) (\lambda (_: T).(ty3 g c0 t1 t2)) -(\lambda (t0: T).(ty3 g c0 t2 t0))) H5))))))))))) (\lambda (c0: C).(\lambda -(u: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((eq T u -(THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t0: T).(pc3 c0 (THead (Flat -Cast) t0 t2) t)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0 -t2 t0)))))).(\lambda (b: B).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: -(ty3 g (CHead c0 (Bind b) u) t0 t3)).(\lambda (_: (((eq T t0 (THead (Flat -Cast) t2 t1)) \to (ex3 T (\lambda (t4: T).(pc3 (CHead c0 (Bind b) u) (THead -(Flat Cast) t4 t2) t3)) (\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2)) -(\lambda (t4: T).(ty3 g (CHead c0 (Bind b) u) t2 t4)))))).(\lambda (H5: (eq T -(THead (Bind b) u t0) (THead (Flat Cast) t2 t1))).(let H6 \def (eq_ind T -(THead (Bind b) u t0) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat -Cast) t2 t1) H5) in (False_ind (ex3 T (\lambda (t4: T).(pc3 c0 (THead (Flat -Cast) t4 t2) (THead (Bind b) u t3))) (\lambda (_: T).(ty3 g c0 t1 t2)) -(\lambda (t4: T).(ty3 g c0 t2 t4))) H6))))))))))))) (\lambda (c0: C).(\lambda -(w: T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda (_: (((eq T w -(THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t0: T).(pc3 c0 (THead (Flat -Cast) t0 t2) u)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0 -t2 t0)))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead -(Bind Abst) u t))).(\lambda (_: (((eq T v (THead (Flat Cast) t2 t1)) \to (ex3 -T (\lambda (t0: T).(pc3 c0 (THead (Flat Cast) t0 t2) (THead (Bind Abst) u -t))) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0 t2 -t0)))))).(\lambda (H5: (eq T (THead (Flat Appl) w v) (THead (Flat Cast) t2 -t1))).(let H6 \def (eq_ind T (THead (Flat Appl) w v) (\lambda (ee: T).(match -ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | -(TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return -(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow -(match f in F return (\lambda (_: F).Prop) with [Appl \Rightarrow True | Cast -\Rightarrow False])])])) I (THead (Flat Cast) t2 t1) H5) in (False_ind (ex3 T -(\lambda (t0: T).(pc3 c0 (THead (Flat Cast) t0 t2) (THead (Flat Appl) w -(THead (Bind Abst) u t)))) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0: -T).(ty3 g c0 t2 t0))) H6)))))))))))) (\lambda (c0: C).(\lambda (t0: -T).(\lambda (t3: T).(\lambda (H1: (ty3 g c0 t0 t3)).(\lambda (H2: (((eq T t0 -(THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t4: T).(pc3 c0 (THead (Flat -Cast) t4 t2) t3)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t4: T).(ty3 g -c0 t2 t4)))))).(\lambda (t4: T).(\lambda (H3: (ty3 g c0 t3 t4)).(\lambda (H4: -(((eq T t3 (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t5: T).(pc3 c0 -(THead (Flat Cast) t5 t2) t4)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda -(t5: T).(ty3 g c0 t2 t5)))))).(\lambda (H5: (eq T (THead (Flat Cast) t3 t0) -(THead (Flat Cast) t2 t1))).(let H6 \def (f_equal T T (\lambda (e: T).(match -e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) -\Rightarrow t3 | (THead _ t _) \Rightarrow t])) (THead (Flat Cast) t3 t0) -(THead (Flat Cast) t2 t1) H5) in ((let H7 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | -(TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Flat Cast) -t3 t0) (THead (Flat Cast) t2 t1) H5) in (\lambda (H8: (eq T t3 t2)).(let H9 -\def (eq_ind T t3 (\lambda (t: T).((eq T t (THead (Flat Cast) t2 t1)) \to -(ex3 T (\lambda (t5: T).(pc3 c0 (THead (Flat Cast) t5 t2) t4)) (\lambda (_: -T).(ty3 g c0 t1 t2)) (\lambda (t5: T).(ty3 g c0 t2 t5))))) H4 t2 H8) in (let -H10 \def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 t t4)) H3 t2 H8) in (let H11 -\def (eq_ind T t3 (\lambda (t: T).((eq T t0 (THead (Flat Cast) t2 t1)) \to -(ex3 T (\lambda (t5: T).(pc3 c0 (THead (Flat Cast) t5 t2) t)) (\lambda (_: -T).(ty3 g c0 t1 t2)) (\lambda (t5: T).(ty3 g c0 t2 t5))))) H2 t2 H8) in (let -H12 \def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 t0 t)) H1 t2 H8) in (eq_ind_r -T t2 (\lambda (t: T).(ex3 T (\lambda (t5: T).(pc3 c0 (THead (Flat Cast) t5 -t2) (THead (Flat Cast) t4 t))) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda -(t5: T).(ty3 g c0 t2 t5)))) (let H13 \def (eq_ind T t0 (\lambda (t: T).((eq T -t (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t5: T).(pc3 c0 (THead (Flat -Cast) t5 t2) t2)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t5: T).(ty3 g -c0 t2 t5))))) H11 t1 H7) in (let H14 \def (eq_ind T t0 (\lambda (t: T).(ty3 g -c0 t t2)) H12 t1 H7) in (ex3_intro T (\lambda (t5: T).(pc3 c0 (THead (Flat -Cast) t5 t2) (THead (Flat Cast) t4 t2))) (\lambda (_: T).(ty3 g c0 t1 t2)) -(\lambda (t5: T).(ty3 g c0 t2 t5)) t4 (pc3_refl c0 (THead (Flat Cast) t4 t2)) -H14 H10))) t3 H8))))))) H6))))))))))) c y x H0))) H)))))). -(* COMMENTS -Initial nodes: 2609 -END *) - -theorem tys3_gen_nil: - \forall (g: G).(\forall (c: C).(\forall (u: T).((tys3 g c TNil u) \to (ex T -(\lambda (u0: T).(ty3 g c u u0)))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (H: (tys3 g c TNil -u)).(insert_eq TList TNil (\lambda (t: TList).(tys3 g c t u)) (\lambda (_: -TList).(ex T (\lambda (u0: T).(ty3 g c u u0)))) (\lambda (y: TList).(\lambda -(H0: (tys3 g c y u)).(tys3_ind g c (\lambda (t: TList).(\lambda (t0: T).((eq -TList t TNil) \to (ex T (\lambda (u0: T).(ty3 g c t0 u0)))))) (\lambda (u0: -T).(\lambda (u1: T).(\lambda (H1: (ty3 g c u0 u1)).(\lambda (_: (eq TList -TNil TNil)).(ex_intro T (\lambda (u2: T).(ty3 g c u0 u2)) u1 H1))))) (\lambda -(t: T).(\lambda (u0: T).(\lambda (_: (ty3 g c t u0)).(\lambda (ts: -TList).(\lambda (_: (tys3 g c ts u0)).(\lambda (_: (((eq TList ts TNil) \to -(ex T (\lambda (u1: T).(ty3 g c u0 u1)))))).(\lambda (H4: (eq TList (TCons t -ts) TNil)).(let H5 \def (eq_ind TList (TCons t ts) (\lambda (ee: -TList).(match ee in TList return (\lambda (_: TList).Prop) with [TNil -\Rightarrow False | (TCons _ _) \Rightarrow True])) I TNil H4) in (False_ind -(ex T (\lambda (u1: T).(ty3 g c u0 u1))) H5))))))))) y u H0))) H)))). -(* COMMENTS -Initial nodes: 255 -END *) - -theorem tys3_gen_cons: - \forall (g: G).(\forall (c: C).(\forall (ts: TList).(\forall (t: T).(\forall -(u: T).((tys3 g c (TCons t ts) u) \to (land (ty3 g c t u) (tys3 g c ts -u))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (ts: TList).(\lambda (t: T).(\lambda -(u: T).(\lambda (H: (tys3 g c (TCons t ts) u)).(insert_eq TList (TCons t ts) -(\lambda (t0: TList).(tys3 g c t0 u)) (\lambda (_: TList).(land (ty3 g c t u) -(tys3 g c ts u))) (\lambda (y: TList).(\lambda (H0: (tys3 g c y u)).(tys3_ind -g c (\lambda (t0: TList).(\lambda (t1: T).((eq TList t0 (TCons t ts)) \to -(land (ty3 g c t t1) (tys3 g c ts t1))))) (\lambda (u0: T).(\lambda (u1: -T).(\lambda (_: (ty3 g c u0 u1)).(\lambda (H2: (eq TList TNil (TCons t -ts))).(let H3 \def (eq_ind TList TNil (\lambda (ee: TList).(match ee in TList -return (\lambda (_: TList).Prop) with [TNil \Rightarrow True | (TCons _ _) -\Rightarrow False])) I (TCons t ts) H2) in (False_ind (land (ty3 g c t u0) -(tys3 g c ts u0)) H3)))))) (\lambda (t0: T).(\lambda (u0: T).(\lambda (H1: -(ty3 g c t0 u0)).(\lambda (ts0: TList).(\lambda (H2: (tys3 g c ts0 -u0)).(\lambda (H3: (((eq TList ts0 (TCons t ts)) \to (land (ty3 g c t u0) -(tys3 g c ts u0))))).(\lambda (H4: (eq TList (TCons t0 ts0) (TCons t -ts))).(let H5 \def (f_equal TList T (\lambda (e: TList).(match e in TList -return (\lambda (_: TList).T) with [TNil \Rightarrow t0 | (TCons t1 _) -\Rightarrow t1])) (TCons t0 ts0) (TCons t ts) H4) in ((let H6 \def (f_equal -TList TList (\lambda (e: TList).(match e in TList return (\lambda (_: -TList).TList) with [TNil \Rightarrow ts0 | (TCons _ t1) \Rightarrow t1])) -(TCons t0 ts0) (TCons t ts) H4) in (\lambda (H7: (eq T t0 t)).(let H8 \def -(eq_ind TList ts0 (\lambda (t1: TList).((eq TList t1 (TCons t ts)) \to (land -(ty3 g c t u0) (tys3 g c ts u0)))) H3 ts H6) in (let H9 \def (eq_ind TList -ts0 (\lambda (t1: TList).(tys3 g c t1 u0)) H2 ts H6) in (let H10 \def (eq_ind -T t0 (\lambda (t1: T).(ty3 g c t1 u0)) H1 t H7) in (conj (ty3 g c t u0) (tys3 -g c ts u0) H10 H9)))))) H5))))))))) y u H0))) H)))))). -(* COMMENTS -Initial nodes: 479 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/fwd_nf2.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/fwd_nf2.ma deleted file mode 100644 index ca4e40c7b..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/fwd_nf2.ma +++ /dev/null @@ -1,301 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/ty3/arity_props.ma". - -include "Basic-1/pc3/nf2.ma". - -include "Basic-1/nf2/fwd.ma". - -theorem ty3_gen_appl_nf2: - \forall (g: G).(\forall (c: C).(\forall (w: T).(\forall (v: T).(\forall (x: -T).((ty3 g c (THead (Flat Appl) w v) x) \to (ex4_2 T T (\lambda (u: -T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x))) -(\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) -(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t: -T).(nf2 c (THead (Bind Abst) u t)))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (w: T).(\lambda (v: T).(\lambda (x: -T).(\lambda (H: (ty3 g c (THead (Flat Appl) w v) x)).(ex3_2_ind T T (\lambda -(u: T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) -x))) (\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) -(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (ex4_2 T T (\lambda (u: -T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x))) -(\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) -(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t: -T).(nf2 c (THead (Bind Abst) u t))))) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H0: (pc3 c (THead (Flat Appl) w (THead (Bind Abst) x0 x1)) -x)).(\lambda (H1: (ty3 g c v (THead (Bind Abst) x0 x1))).(\lambda (H2: (ty3 g -c w x0)).(let H_x \def (ty3_correct g c v (THead (Bind Abst) x0 x1) H1) in -(let H3 \def H_x in (ex_ind T (\lambda (t: T).(ty3 g c (THead (Bind Abst) x0 -x1) t)) (ex4_2 T T (\lambda (u: T).(\lambda (t: T).(pc3 c (THead (Flat Appl) -w (THead (Bind Abst) u t)) x))) (\lambda (u: T).(\lambda (t: T).(ty3 g c v -(THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) -(\lambda (u: T).(\lambda (t: T).(nf2 c (THead (Bind Abst) u t))))) (\lambda -(x2: T).(\lambda (H4: (ty3 g c (THead (Bind Abst) x0 x1) x2)).(let H_x0 \def -(ty3_correct g c w x0 H2) in (let H5 \def H_x0 in (ex_ind T (\lambda (t: -T).(ty3 g c x0 t)) (ex4_2 T T (\lambda (u: T).(\lambda (t: T).(pc3 c (THead -(Flat Appl) w (THead (Bind Abst) u t)) x))) (\lambda (u: T).(\lambda (t: -T).(ty3 g c v (THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3 -g c w u))) (\lambda (u: T).(\lambda (t: T).(nf2 c (THead (Bind Abst) u t))))) -(\lambda (x3: T).(\lambda (H6: (ty3 g c x0 x3)).(let H7 \def (ty3_sn3 g c -(THead (Bind Abst) x0 x1) x2 H4) in (let H_x1 \def (nf2_sn3 c (THead (Bind -Abst) x0 x1) H7) in (let H8 \def H_x1 in (ex2_ind T (\lambda (u: T).(pr3 c -(THead (Bind Abst) x0 x1) u)) (\lambda (u: T).(nf2 c u)) (ex4_2 T T (\lambda -(u: T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) -x))) (\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) -(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t: -T).(nf2 c (THead (Bind Abst) u t))))) (\lambda (x4: T).(\lambda (H9: (pr3 c -(THead (Bind Abst) x0 x1) x4)).(\lambda (H10: (nf2 c x4)).(let H11 \def -(pr3_gen_abst c x0 x1 x4 H9) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: -T).(eq T x4 (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall -(u: T).(pr3 (CHead c (Bind b) u) x1 t2))))) (ex4_2 T T (\lambda (u: -T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x))) -(\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) -(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t: -T).(nf2 c (THead (Bind Abst) u t))))) (\lambda (x5: T).(\lambda (x6: -T).(\lambda (H12: (eq T x4 (THead (Bind Abst) x5 x6))).(\lambda (H13: (pr3 c -x0 x5)).(\lambda (H14: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind -b) u) x1 x6))))).(let H15 \def (eq_ind T x4 (\lambda (t: T).(nf2 c t)) H10 -(THead (Bind Abst) x5 x6) H12) in (let H16 \def (pr3_head_12 c x0 x5 H13 -(Bind Abst) x1 x6 (H14 Abst x5)) in (ex4_2_intro T T (\lambda (u: T).(\lambda -(t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x))) (\lambda (u: -T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) (\lambda (u: -T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t: T).(nf2 c -(THead (Bind Abst) u t)))) x5 x6 (pc3_pr3_conf c (THead (Flat Appl) w (THead -(Bind Abst) x0 x1)) x H0 (THead (Flat Appl) w (THead (Bind Abst) x5 x6)) -(pr3_thin_dx c (THead (Bind Abst) x0 x1) (THead (Bind Abst) x5 x6) H16 w -Appl)) (ty3_conv g c (THead (Bind Abst) x5 x6) x2 (ty3_sred_pr3 c (THead -(Bind Abst) x0 x1) (THead (Bind Abst) x5 x6) H16 g x2 H4) v (THead (Bind -Abst) x0 x1) H1 (pc3_pr3_r c (THead (Bind Abst) x0 x1) (THead (Bind Abst) x5 -x6) H16)) (ty3_conv g c x5 x3 (ty3_sred_pr3 c x0 x5 H13 g x3 H6) w x0 H2 -(pc3_pr3_r c x0 x5 H13)) H15)))))))) H11))))) H8)))))) H5))))) H3)))))))) -(ty3_gen_appl g c w v x H))))))). -(* COMMENTS -Initial nodes: 1289 -END *) - -theorem ty3_inv_lref_nf2_pc3: - \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (i: nat).((ty3 g c -(TLRef i) u1) \to ((nf2 c (TLRef i)) \to (\forall (u2: T).((nf2 c u2) \to -((pc3 c u1 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift (S i) O u)))))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (u1: T).(\lambda (i: nat).(\lambda -(H: (ty3 g c (TLRef i) u1)).(insert_eq T (TLRef i) (\lambda (t: T).(ty3 g c t -u1)) (\lambda (t: T).((nf2 c t) \to (\forall (u2: T).((nf2 c u2) \to ((pc3 c -u1 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift (S i) O u))))))))) (\lambda -(y: T).(\lambda (H0: (ty3 g c y u1)).(ty3_ind g (\lambda (c0: C).(\lambda (t: -T).(\lambda (t0: T).((eq T t (TLRef i)) \to ((nf2 c0 t) \to (\forall (u2: -T).((nf2 c0 u2) \to ((pc3 c0 t0 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift -(S i) O u)))))))))))) (\lambda (c0: C).(\lambda (t2: T).(\lambda (t: -T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda (_: (((eq T t2 (TLRef i)) \to ((nf2 -c0 t2) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 t u2) \to (ex T -(\lambda (u: T).(eq T u2 (lift (S i) O u))))))))))).(\lambda (u: T).(\lambda -(t1: T).(\lambda (H3: (ty3 g c0 u t1)).(\lambda (H4: (((eq T u (TLRef i)) \to -((nf2 c0 u) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 t1 u2) \to (ex T -(\lambda (u0: T).(eq T u2 (lift (S i) O u0))))))))))).(\lambda (H5: (pc3 c0 -t1 t2)).(\lambda (H6: (eq T u (TLRef i))).(\lambda (H7: (nf2 c0 u)).(\lambda -(u2: T).(\lambda (H8: (nf2 c0 u2)).(\lambda (H9: (pc3 c0 t2 u2)).(let H10 -\def (eq_ind T u (\lambda (t0: T).(nf2 c0 t0)) H7 (TLRef i) H6) in (let H11 -\def (eq_ind T u (\lambda (t0: T).((eq T t0 (TLRef i)) \to ((nf2 c0 t0) \to -(\forall (u3: T).((nf2 c0 u3) \to ((pc3 c0 t1 u3) \to (ex T (\lambda (u0: -T).(eq T u3 (lift (S i) O u0)))))))))) H4 (TLRef i) H6) in (let H12 \def -(eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 t1)) H3 (TLRef i) H6) in (let H_y -\def (H11 (refl_equal T (TLRef i)) H10 u2 H8) in (H_y (pc3_t t2 c0 t1 H5 u2 -H9))))))))))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq -T (TSort m) (TLRef i))).(\lambda (_: (nf2 c0 (TSort m))).(\lambda (u2: -T).(\lambda (_: (nf2 c0 u2)).(\lambda (_: (pc3 c0 (TSort (next g m)) -u2)).(let H5 \def (eq_ind T (TSort m) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) -\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef i) H1) in -(False_ind (ex T (\lambda (u: T).(eq T u2 (lift (S i) O u)))) H5))))))))) -(\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda -(H1: (getl n c0 (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda (_: (ty3 g -d u t)).(\lambda (_: (((eq T u (TLRef i)) \to ((nf2 d u) \to (\forall (u2: -T).((nf2 d u2) \to ((pc3 d t u2) \to (ex T (\lambda (u0: T).(eq T u2 (lift (S -i) O u0))))))))))).(\lambda (H4: (eq T (TLRef n) (TLRef i))).(\lambda (H5: -(nf2 c0 (TLRef n))).(\lambda (u2: T).(\lambda (_: (nf2 c0 u2)).(\lambda (H7: -(pc3 c0 (lift (S n) O t) u2)).(let H8 \def (f_equal T nat (\lambda (e: -T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow n | -(TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef -i) H4) in (let H9 \def (eq_ind nat n (\lambda (n0: nat).(pc3 c0 (lift (S n0) -O t) u2)) H7 i H8) in (let H10 \def (eq_ind nat n (\lambda (n0: nat).(nf2 c0 -(TLRef n0))) H5 i H8) in (let H11 \def (eq_ind nat n (\lambda (n0: nat).(getl -n0 c0 (CHead d (Bind Abbr) u))) H1 i H8) in (nf2_gen_lref c0 d u i H11 H10 -(ex T (\lambda (u0: T).(eq T u2 (lift (S i) O u0)))))))))))))))))))))) -(\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda -(H1: (getl n c0 (CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (_: (ty3 g -d u t)).(\lambda (_: (((eq T u (TLRef i)) \to ((nf2 d u) \to (\forall (u2: -T).((nf2 d u2) \to ((pc3 d t u2) \to (ex T (\lambda (u0: T).(eq T u2 (lift (S -i) O u0))))))))))).(\lambda (H4: (eq T (TLRef n) (TLRef i))).(\lambda (H5: -(nf2 c0 (TLRef n))).(\lambda (u2: T).(\lambda (H6: (nf2 c0 u2)).(\lambda (H7: -(pc3 c0 (lift (S n) O u) u2)).(let H8 \def (f_equal T nat (\lambda (e: -T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow n | -(TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef -i) H4) in (let H9 \def (eq_ind nat n (\lambda (n0: nat).(pc3 c0 (lift (S n0) -O u) u2)) H7 i H8) in (let H10 \def (eq_ind nat n (\lambda (n0: nat).(nf2 c0 -(TLRef n0))) H5 i H8) in (let H11 \def (eq_ind nat n (\lambda (n0: nat).(getl -n0 c0 (CHead d (Bind Abst) u))) H1 i H8) in (let H_y \def (pc3_nf2_unfold c0 -(lift (S i) O u) u2 H9 H6) in (let H12 \def (pr3_gen_lift c0 u u2 (S i) O H_y -d (getl_drop Abst c0 d u i H11)) in (ex2_ind T (\lambda (t2: T).(eq T u2 -(lift (S i) O t2))) (\lambda (t2: T).(pr3 d u t2)) (ex T (\lambda (u0: T).(eq -T u2 (lift (S i) O u0)))) (\lambda (x: T).(\lambda (H13: (eq T u2 (lift (S i) -O x))).(\lambda (_: (pr3 d u x)).(eq_ind_r T (lift (S i) O x) (\lambda (t0: -T).(ex T (\lambda (u0: T).(eq T t0 (lift (S i) O u0))))) (ex_intro T (\lambda -(u0: T).(eq T (lift (S i) O x) (lift (S i) O u0))) x (refl_equal T (lift (S -i) O x))) u2 H13)))) H12)))))))))))))))))))) (\lambda (c0: C).(\lambda (u: -T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((eq T u (TLRef -i)) \to ((nf2 c0 u) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 t u2) \to -(ex T (\lambda (u0: T).(eq T u2 (lift (S i) O u0))))))))))).(\lambda (b: -B).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) -u) t1 t2)).(\lambda (_: (((eq T t1 (TLRef i)) \to ((nf2 (CHead c0 (Bind b) u) -t1) \to (\forall (u2: T).((nf2 (CHead c0 (Bind b) u) u2) \to ((pc3 (CHead c0 -(Bind b) u) t2 u2) \to (ex T (\lambda (u0: T).(eq T u2 (lift (S i) O -u0))))))))))).(\lambda (H5: (eq T (THead (Bind b) u t1) (TLRef i))).(\lambda -(_: (nf2 c0 (THead (Bind b) u t1))).(\lambda (u2: T).(\lambda (_: (nf2 c0 -u2)).(\lambda (_: (pc3 c0 (THead (Bind b) u t2) u2)).(let H9 \def (eq_ind T -(THead (Bind b) u t1) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead _ _ _) \Rightarrow True])) I (TLRef i) H5) in (False_ind (ex T -(\lambda (u0: T).(eq T u2 (lift (S i) O u0)))) H9))))))))))))))))) (\lambda -(c0: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda -(_: (((eq T w (TLRef i)) \to ((nf2 c0 w) \to (\forall (u2: T).((nf2 c0 u2) -\to ((pc3 c0 u u2) \to (ex T (\lambda (u0: T).(eq T u2 (lift (S i) O -u0))))))))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead -(Bind Abst) u t))).(\lambda (_: (((eq T v (TLRef i)) \to ((nf2 c0 v) \to -(\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 (THead (Bind Abst) u t) u2) \to -(ex T (\lambda (u0: T).(eq T u2 (lift (S i) O u0))))))))))).(\lambda (H5: (eq -T (THead (Flat Appl) w v) (TLRef i))).(\lambda (_: (nf2 c0 (THead (Flat Appl) -w v))).(\lambda (u2: T).(\lambda (_: (nf2 c0 u2)).(\lambda (_: (pc3 c0 (THead -(Flat Appl) w (THead (Bind Abst) u t)) u2)).(let H9 \def (eq_ind T (THead -(Flat Appl) w v) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) -with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ -_) \Rightarrow True])) I (TLRef i) H5) in (False_ind (ex T (\lambda (u0: -T).(eq T u2 (lift (S i) O u0)))) H9)))))))))))))))) (\lambda (c0: C).(\lambda -(t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g c0 t1 t2)).(\lambda (_: (((eq T -t1 (TLRef i)) \to ((nf2 c0 t1) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 -t2 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift (S i) O u))))))))))).(\lambda -(t0: T).(\lambda (_: (ty3 g c0 t2 t0)).(\lambda (_: (((eq T t2 (TLRef i)) \to -((nf2 c0 t2) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 t0 u2) \to (ex T -(\lambda (u: T).(eq T u2 (lift (S i) O u))))))))))).(\lambda (H5: (eq T -(THead (Flat Cast) t2 t1) (TLRef i))).(\lambda (_: (nf2 c0 (THead (Flat Cast) -t2 t1))).(\lambda (u2: T).(\lambda (_: (nf2 c0 u2)).(\lambda (_: (pc3 c0 -(THead (Flat Cast) t0 t2) u2)).(let H9 \def (eq_ind T (THead (Flat Cast) t2 -t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort -_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) -\Rightarrow True])) I (TLRef i) H5) in (False_ind (ex T (\lambda (u: T).(eq T -u2 (lift (S i) O u)))) H9))))))))))))))) c y u1 H0))) H))))). -(* COMMENTS -Initial nodes: 2175 -END *) - -theorem ty3_inv_lref_nf2: - \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (i: nat).((ty3 g c -(TLRef i) u) \to ((nf2 c (TLRef i)) \to ((nf2 c u) \to (ex T (\lambda (u0: -T).(eq T u (lift (S i) O u0)))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (i: nat).(\lambda -(H: (ty3 g c (TLRef i) u)).(\lambda (H0: (nf2 c (TLRef i))).(\lambda (H1: -(nf2 c u)).(ty3_inv_lref_nf2_pc3 g c u i H H0 u H1 (pc3_refl c u)))))))). -(* COMMENTS -Initial nodes: 57 -END *) - -theorem ty3_inv_appls_lref_nf2: - \forall (g: G).(\forall (c: C).(\forall (vs: TList).(\forall (u1: -T).(\forall (i: nat).((ty3 g c (THeads (Flat Appl) vs (TLRef i)) u1) \to -((nf2 c (TLRef i)) \to ((nf2 c u1) \to (ex2 T (\lambda (u: T).(nf2 c (lift (S -i) O u))) (\lambda (u: T).(pc3 c (THeads (Flat Appl) vs (lift (S i) O u)) -u1)))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (vs: TList).(TList_ind (\lambda (t: -TList).(\forall (u1: T).(\forall (i: nat).((ty3 g c (THeads (Flat Appl) t -(TLRef i)) u1) \to ((nf2 c (TLRef i)) \to ((nf2 c u1) \to (ex2 T (\lambda (u: -T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3 c (THeads (Flat Appl) t -(lift (S i) O u)) u1))))))))) (\lambda (u1: T).(\lambda (i: nat).(\lambda (H: -(ty3 g c (TLRef i) u1)).(\lambda (H0: (nf2 c (TLRef i))).(\lambda (H1: (nf2 c -u1)).(let H_x \def (ty3_inv_lref_nf2 g c u1 i H H0 H1) in (let H2 \def H_x in -(ex_ind T (\lambda (u0: T).(eq T u1 (lift (S i) O u0))) (ex2 T (\lambda (u: -T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3 c (lift (S i) O u) u1))) -(\lambda (x: T).(\lambda (H3: (eq T u1 (lift (S i) O x))).(let H4 \def -(eq_ind T u1 (\lambda (t: T).(nf2 c t)) H1 (lift (S i) O x) H3) in (eq_ind_r -T (lift (S i) O x) (\lambda (t: T).(ex2 T (\lambda (u: T).(nf2 c (lift (S i) -O u))) (\lambda (u: T).(pc3 c (lift (S i) O u) t)))) (ex_intro2 T (\lambda -(u: T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3 c (lift (S i) O u) -(lift (S i) O x))) x H4 (pc3_refl c (lift (S i) O x))) u1 H3)))) H2)))))))) -(\lambda (t: T).(\lambda (t0: TList).(\lambda (H: ((\forall (u1: T).(\forall -(i: nat).((ty3 g c (THeads (Flat Appl) t0 (TLRef i)) u1) \to ((nf2 c (TLRef -i)) \to ((nf2 c u1) \to (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O u))) -(\lambda (u: T).(pc3 c (THeads (Flat Appl) t0 (lift (S i) O u)) -u1)))))))))).(\lambda (u1: T).(\lambda (i: nat).(\lambda (H0: (ty3 g c (THead -(Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) u1)).(\lambda (H1: (nf2 c -(TLRef i))).(\lambda (_: (nf2 c u1)).(let H_x \def (ty3_gen_appl_nf2 g c t -(THeads (Flat Appl) t0 (TLRef i)) u1 H0) in (let H3 \def H_x in (ex4_2_ind T -T (\lambda (u: T).(\lambda (t1: T).(pc3 c (THead (Flat Appl) t (THead (Bind -Abst) u t1)) u1))) (\lambda (u: T).(\lambda (t1: T).(ty3 g c (THeads (Flat -Appl) t0 (TLRef i)) (THead (Bind Abst) u t1)))) (\lambda (u: T).(\lambda (_: -T).(ty3 g c t u))) (\lambda (u: T).(\lambda (t1: T).(nf2 c (THead (Bind Abst) -u t1)))) (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O u))) (\lambda (u: -T).(pc3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O u))) -u1))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (pc3 c (THead (Flat -Appl) t (THead (Bind Abst) x0 x1)) u1)).(\lambda (H5: (ty3 g c (THeads (Flat -Appl) t0 (TLRef i)) (THead (Bind Abst) x0 x1))).(\lambda (_: (ty3 g c t -x0)).(\lambda (H7: (nf2 c (THead (Bind Abst) x0 x1))).(let H8 \def -(nf2_gen_abst c x0 x1 H7) in (land_ind (nf2 c x0) (nf2 (CHead c (Bind Abst) -x0) x1) (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3 -c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O u))) u1))) -(\lambda (H9: (nf2 c x0)).(\lambda (H10: (nf2 (CHead c (Bind Abst) x0) -x1)).(let H_y \def (H (THead (Bind Abst) x0 x1) i H5 H1) in (let H11 \def -(H_y (nf2_abst_shift c x0 H9 x1 H10)) in (ex2_ind T (\lambda (u: T).(nf2 c -(lift (S i) O u))) (\lambda (u: T).(pc3 c (THeads (Flat Appl) t0 (lift (S i) -O u)) (THead (Bind Abst) x0 x1))) (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O -u))) (\lambda (u: T).(pc3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift -(S i) O u))) u1))) (\lambda (x: T).(\lambda (H12: (nf2 c (lift (S i) O -x))).(\lambda (H13: (pc3 c (THeads (Flat Appl) t0 (lift (S i) O x)) (THead -(Bind Abst) x0 x1))).(ex_intro2 T (\lambda (u: T).(nf2 c (lift (S i) O u))) -(\lambda (u: T).(pc3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S -i) O u))) u1)) x H12 (pc3_t (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) c -(THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O x))) (pc3_thin_dx c -(THeads (Flat Appl) t0 (lift (S i) O x)) (THead (Bind Abst) x0 x1) H13 t -Appl) u1 H4))))) H11))))) H8)))))))) H3))))))))))) vs))). -(* COMMENTS -Initial nodes: 1213 -END *) - -theorem ty3_inv_lref_lref_nf2: - \forall (g: G).(\forall (c: C).(\forall (i: nat).(\forall (j: nat).((ty3 g c -(TLRef i) (TLRef j)) \to ((nf2 c (TLRef i)) \to ((nf2 c (TLRef j)) \to (lt i -j))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (i: nat).(\lambda (j: nat).(\lambda -(H: (ty3 g c (TLRef i) (TLRef j))).(\lambda (H0: (nf2 c (TLRef i))).(\lambda -(H1: (nf2 c (TLRef j))).(let H_x \def (ty3_inv_lref_nf2 g c (TLRef j) i H H0 -H1) in (let H2 \def H_x in (ex_ind T (\lambda (u0: T).(eq T (TLRef j) (lift -(S i) O u0))) (lt i j) (\lambda (x: T).(\lambda (H3: (eq T (TLRef j) (lift (S -i) O x))).(let H_x0 \def (lift_gen_lref x O (S i) j H3) in (let H4 \def H_x0 -in (or_ind (land (lt j O) (eq T x (TLRef j))) (land (le (S i) j) (eq T x -(TLRef (minus j (S i))))) (lt i j) (\lambda (H5: (land (lt j O) (eq T x -(TLRef j)))).(land_ind (lt j O) (eq T x (TLRef j)) (lt i j) (\lambda (H6: (lt -j O)).(\lambda (_: (eq T x (TLRef j))).(lt_x_O j H6 (lt i j)))) H5)) (\lambda -(H5: (land (le (S i) j) (eq T x (TLRef (minus j (S i)))))).(land_ind (le (S -i) j) (eq T x (TLRef (minus j (S i)))) (lt i j) (\lambda (H6: (le (S i) -j)).(\lambda (_: (eq T x (TLRef (minus j (S i))))).H6)) H5)) H4))))) -H2))))))))). -(* COMMENTS -Initial nodes: 337 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/nf2.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/nf2.ma deleted file mode 100644 index 47b675663..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/nf2.ma +++ /dev/null @@ -1,472 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/ty3/arity.ma". - -include "Basic-1/pc3/nf2.ma". - -include "Basic-1/nf2/arity.ma". - -definition ty3_nf2_inv_abst_premise: - C \to (T \to (T \to Prop)) -\def - \lambda (c: C).(\lambda (w: T).(\lambda (u: T).(\forall (d: C).(\forall (wi: -T).(\forall (i: nat).((getl i c (CHead d (Bind Abst) wi)) \to (\forall (vs: -TList).((pc3 c (THeads (Flat Appl) vs (lift (S i) O wi)) (THead (Bind Abst) w -u)) \to False)))))))). - -theorem ty3_nf2_inv_abst_premise_csort: - \forall (w: T).(\forall (u: T).(\forall (m: nat).(ty3_nf2_inv_abst_premise -(CSort m) w u))) -\def - \lambda (w: T).(\lambda (u: T).(\lambda (m: nat).(\lambda (d: C).(\lambda -(wi: T).(\lambda (i: nat).(\lambda (H: (getl i (CSort m) (CHead d (Bind Abst) -wi))).(\lambda (vs: TList).(\lambda (_: (pc3 (CSort m) (THeads (Flat Appl) vs -(lift (S i) O wi)) (THead (Bind Abst) w u))).(getl_gen_sort m i (CHead d -(Bind Abst) wi) H False))))))))). -(* COMMENTS -Initial nodes: 85 -END *) - -theorem ty3_nf2_inv_all: - \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t -u) \to ((nf2 c t) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T -t (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c w))) -(\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c (Bind Abst) w) u0)))) (ex nat -(\lambda (n: nat).(eq T t (TSort n)))) (ex3_2 TList nat (\lambda (ws: -TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i))))) -(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c (TLRef i))))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H: -(ty3 g c t u)).(\lambda (H0: (nf2 c t)).(let H_x \def (ty3_arity g c t u H) -in (let H1 \def H_x in (ex2_ind A (\lambda (a1: A).(arity g c t a1)) (\lambda -(a1: A).(arity g c u (asucc g a1))) (or3 (ex3_2 T T (\lambda (w: T).(\lambda -(u0: T).(eq T t (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: -T).(nf2 c w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c (Bind Abst) w) -u0)))) (ex nat (\lambda (n: nat).(eq T t (TSort n)))) (ex3_2 TList nat -(\lambda (ws: TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef -i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c (TLRef i)))))) (\lambda (x: A).(\lambda (H2: -(arity g c t x)).(\lambda (_: (arity g c u (asucc g x))).(arity_nf2_inv_all g -c t x H2 H0)))) H1)))))))). -(* COMMENTS -Initial nodes: 233 -END *) - -theorem ty3_nf2_inv_sort: - \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (m: nat).((ty3 g c t -(TSort m)) \to ((nf2 c t) \to (or (ex2 nat (\lambda (n: nat).(eq T t (TSort -n))) (\lambda (n: nat).(eq nat m (next g n)))) (ex3_2 TList nat (\lambda (ws: -TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i))))) -(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c (TLRef i))))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (m: nat).(\lambda -(H: (ty3 g c t (TSort m))).(\lambda (H0: (nf2 c t)).(let H_x \def -(ty3_nf2_inv_all g c t (TSort m) H H0) in (let H1 \def H_x in (or3_ind (ex3_2 -T T (\lambda (w: T).(\lambda (u: T).(eq T t (THead (Bind Abst) w u)))) -(\lambda (w: T).(\lambda (_: T).(nf2 c w))) (\lambda (w: T).(\lambda (u: -T).(nf2 (CHead c (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t -(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t -(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef i))))) -(or (ex2 nat (\lambda (n: nat).(eq T t (TSort n))) (\lambda (n: nat).(eq nat -m (next g n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T -t (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef -i)))))) (\lambda (H2: (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t -(THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c w))) -(\lambda (w: T).(\lambda (u: T).(nf2 (CHead c (Bind Abst) w) -u))))).(ex3_2_ind T T (\lambda (w: T).(\lambda (u: T).(eq T t (THead (Bind -Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c w))) (\lambda (w: -T).(\lambda (u: T).(nf2 (CHead c (Bind Abst) w) u))) (or (ex2 nat (\lambda -(n: nat).(eq T t (TSort n))) (\lambda (n: nat).(eq nat m (next g n)))) (ex3_2 -TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) -ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) -(\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef i)))))) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H3: (eq T t (THead (Bind Abst) x0 -x1))).(\lambda (_: (nf2 c x0)).(\lambda (_: (nf2 (CHead c (Bind Abst) x0) -x1)).(let H6 \def (eq_ind T t (\lambda (t0: T).(ty3 g c t0 (TSort m))) H -(THead (Bind Abst) x0 x1) H3) in (eq_ind_r T (THead (Bind Abst) x0 x1) -(\lambda (t0: T).(or (ex2 nat (\lambda (n: nat).(eq T t0 (TSort n))) (\lambda -(n: nat).(eq nat m (next g n)))) (ex3_2 TList nat (\lambda (ws: -TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i))))) -(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c (TLRef i))))))) (ex3_2_ind T T (\lambda (t2: -T).(\lambda (_: T).(pc3 c (THead (Bind Abst) x0 t2) (TSort m)))) (\lambda (_: -T).(\lambda (t0: T).(ty3 g c x0 t0))) (\lambda (t2: T).(\lambda (_: T).(ty3 g -(CHead c (Bind Abst) x0) x1 t2))) (or (ex2 nat (\lambda (n: nat).(eq T (THead -(Bind Abst) x0 x1) (TSort n))) (\lambda (n: nat).(eq nat m (next g n)))) -(ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Bind -Abst) x0 x1) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: -TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: -nat).(nf2 c (TLRef i)))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: -(pc3 c (THead (Bind Abst) x0 x2) (TSort m))).(\lambda (_: (ty3 g c x0 -x3)).(\lambda (_: (ty3 g (CHead c (Bind Abst) x0) x1 x2)).(pc3_gen_sort_abst -c x0 x2 m (pc3_s c (TSort m) (THead (Bind Abst) x0 x2) H7) (or (ex2 nat -(\lambda (n: nat).(eq T (THead (Bind Abst) x0 x1) (TSort n))) (\lambda (n: -nat).(eq nat m (next g n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda -(i: nat).(eq T (THead (Bind Abst) x0 x1) (THeads (Flat Appl) ws (TLRef i))))) -(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c (TLRef i)))))))))))) (ty3_gen_bind g Abst c -x0 x1 (TSort m) H6)) t H3))))))) H2)) (\lambda (H2: (ex nat (\lambda (n: -nat).(eq T t (TSort n))))).(ex_ind nat (\lambda (n: nat).(eq T t (TSort n))) -(or (ex2 nat (\lambda (n: nat).(eq T t (TSort n))) (\lambda (n: nat).(eq nat -m (next g n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T -t (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef -i)))))) (\lambda (x: nat).(\lambda (H3: (eq T t (TSort x))).(let H4 \def -(eq_ind T t (\lambda (t0: T).(ty3 g c t0 (TSort m))) H (TSort x) H3) in -(eq_ind_r T (TSort x) (\lambda (t0: T).(or (ex2 nat (\lambda (n: nat).(eq T -t0 (TSort n))) (\lambda (n: nat).(eq nat m (next g n)))) (ex3_2 TList nat -(\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef -i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c (TLRef i))))))) (eq_ind nat (next g x) -(\lambda (n: nat).(or (ex2 nat (\lambda (n0: nat).(eq T (TSort x) (TSort -n0))) (\lambda (n0: nat).(eq nat n (next g n0)))) (ex3_2 TList nat (\lambda -(ws: TList).(\lambda (i: nat).(eq T (TSort x) (THeads (Flat Appl) ws (TLRef -i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c (TLRef i))))))) (or_introl (ex2 nat (\lambda -(n: nat).(eq T (TSort x) (TSort n))) (\lambda (n: nat).(eq nat (next g x) -(next g n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T -(TSort x) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda -(_: nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef -i))))) (ex_intro2 nat (\lambda (n: nat).(eq T (TSort x) (TSort n))) (\lambda -(n: nat).(eq nat (next g x) (next g n))) x (refl_equal T (TSort x)) -(refl_equal nat (next g x)))) m (pc3_gen_sort c (next g x) m (ty3_gen_sort g -c (TSort m) x H4))) t H3)))) H2)) (\lambda (H2: (ex3_2 TList nat (\lambda -(ws: TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i))))) -(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c (TLRef i)))))).(ex3_2_ind TList nat (\lambda -(ws: TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i))))) -(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c (TLRef i)))) (or (ex2 nat (\lambda (n: -nat).(eq T t (TSort n))) (\lambda (n: nat).(eq nat m (next g n)))) (ex3_2 -TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) -ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) -(\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef i)))))) (\lambda (x0: -TList).(\lambda (x1: nat).(\lambda (H3: (eq T t (THeads (Flat Appl) x0 (TLRef -x1)))).(\lambda (H4: (nfs2 c x0)).(\lambda (H5: (nf2 c (TLRef x1))).(let H6 -\def (eq_ind T t (\lambda (t0: T).(ty3 g c t0 (TSort m))) H (THeads (Flat -Appl) x0 (TLRef x1)) H3) in (eq_ind_r T (THeads (Flat Appl) x0 (TLRef x1)) -(\lambda (t0: T).(or (ex2 nat (\lambda (n: nat).(eq T t0 (TSort n))) (\lambda -(n: nat).(eq nat m (next g n)))) (ex3_2 TList nat (\lambda (ws: -TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i))))) -(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c (TLRef i))))))) (or_intror (ex2 nat (\lambda -(n: nat).(eq T (THeads (Flat Appl) x0 (TLRef x1)) (TSort n))) (\lambda (n: -nat).(eq nat m (next g n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda -(i: nat).(eq T (THeads (Flat Appl) x0 (TLRef x1)) (THeads (Flat Appl) ws -(TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda -(_: TList).(\lambda (i: nat).(nf2 c (TLRef i))))) (ex3_2_intro TList nat -(\lambda (ws: TList).(\lambda (i: nat).(eq T (THeads (Flat Appl) x0 (TLRef -x1)) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: -nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef i)))) -x0 x1 (refl_equal T (THeads (Flat Appl) x0 (TLRef x1))) H4 H5)) t H3))))))) -H2)) H1)))))))). -(* COMMENTS -Initial nodes: 2045 -END *) - -theorem ty3_nf2_gen__ty3_nf2_inv_abst_aux: - \forall (c: C).(\forall (w1: T).(\forall (u1: T).((ty3_nf2_inv_abst_premise -c w1 u1) \to (\forall (t: T).(\forall (w2: T).(\forall (u2: T).((pc3 c (THead -(Flat Appl) t (THead (Bind Abst) w2 u2)) (THead (Bind Abst) w1 u1)) \to -(ty3_nf2_inv_abst_premise c w2 u2)))))))) -\def - \lambda (c: C).(\lambda (w1: T).(\lambda (u1: T).(\lambda (H: ((\forall (d: -C).(\forall (wi: T).(\forall (i: nat).((getl i c (CHead d (Bind Abst) wi)) -\to (\forall (vs: TList).((pc3 c (THeads (Flat Appl) vs (lift (S i) O wi)) -(THead (Bind Abst) w1 u1)) \to False)))))))).(\lambda (t: T).(\lambda (w2: -T).(\lambda (u2: T).(\lambda (H0: (pc3 c (THead (Flat Appl) t (THead (Bind -Abst) w2 u2)) (THead (Bind Abst) w1 u1))).(\lambda (d: C).(\lambda (wi: -T).(\lambda (i: nat).(\lambda (H1: (getl i c (CHead d (Bind Abst) -wi))).(\lambda (vs: TList).(\lambda (H2: (pc3 c (THeads (Flat Appl) vs (lift -(S i) O wi)) (THead (Bind Abst) w2 u2))).(H d wi i H1 (TCons t vs) (pc3_t -(THead (Flat Appl) t (THead (Bind Abst) w2 u2)) c (THead (Flat Appl) t -(THeads (Flat Appl) vs (lift (S i) O wi))) (pc3_thin_dx c (THeads (Flat Appl) -vs (lift (S i) O wi)) (THead (Bind Abst) w2 u2) H2 t Appl) (THead (Bind Abst) -w1 u1) H0))))))))))))))). -(* COMMENTS -Initial nodes: 271 -END *) - -theorem ty3_nf2_inv_abst: - \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (w: T).(\forall (u: -T).((ty3 g c t (THead (Bind Abst) w u)) \to ((nf2 c t) \to ((nf2 c w) \to -((ty3_nf2_inv_abst_premise c w u) \to (ex4_2 T T (\lambda (v: T).(\lambda (_: -T).(eq T t (THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g -c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v -u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) -v)))))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (w: T).(\lambda (u: -T).(\lambda (H: (ty3 g c t (THead (Bind Abst) w u))).(\lambda (H0: (nf2 c -t)).(\lambda (H1: (nf2 c w)).(\lambda (H2: (ty3_nf2_inv_abst_premise c w -u)).(let H_x \def (ty3_nf2_inv_all g c t (THead (Bind Abst) w u) H H0) in -(let H3 \def H_x in (or3_ind (ex3_2 T T (\lambda (w0: T).(\lambda (u0: T).(eq -T t (THead (Bind Abst) w0 u0)))) (\lambda (w0: T).(\lambda (_: T).(nf2 c -w0))) (\lambda (w0: T).(\lambda (u0: T).(nf2 (CHead c (Bind Abst) w0) u0)))) -(ex nat (\lambda (n: nat).(eq T t (TSort n)))) (ex3_2 TList nat (\lambda (ws: -TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i))))) -(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: -TList).(\lambda (i: nat).(nf2 c (TLRef i))))) (ex4_2 T T (\lambda (v: -T).(\lambda (_: T).(eq T t (THead (Bind Abst) w v)))) (\lambda (_: -T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g -(CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c -(Bind Abst) w) v)))) (\lambda (H4: (ex3_2 T T (\lambda (w0: T).(\lambda (u0: -T).(eq T t (THead (Bind Abst) w0 u0)))) (\lambda (w0: T).(\lambda (_: T).(nf2 -c w0))) (\lambda (w0: T).(\lambda (u0: T).(nf2 (CHead c (Bind Abst) w0) -u0))))).(ex3_2_ind T T (\lambda (w0: T).(\lambda (u0: T).(eq T t (THead (Bind -Abst) w0 u0)))) (\lambda (w0: T).(\lambda (_: T).(nf2 c w0))) (\lambda (w0: -T).(\lambda (u0: T).(nf2 (CHead c (Bind Abst) w0) u0))) (ex4_2 T T (\lambda -(v: T).(\lambda (_: T).(eq T t (THead (Bind Abst) w v)))) (\lambda (_: -T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g -(CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c -(Bind Abst) w) v)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T t -(THead (Bind Abst) x0 x1))).(\lambda (H6: (nf2 c x0)).(\lambda (H7: (nf2 -(CHead c (Bind Abst) x0) x1)).(let H8 \def (eq_ind T t (\lambda (t0: T).(ty3 -g c t0 (THead (Bind Abst) w u))) H (THead (Bind Abst) x0 x1) H5) in (eq_ind_r -T (THead (Bind Abst) x0 x1) (\lambda (t0: T).(ex4_2 T T (\lambda (v: -T).(\lambda (_: T).(eq T t0 (THead (Bind Abst) w v)))) (\lambda (_: -T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g -(CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c -(Bind Abst) w) v))))) (ex_ind T (\lambda (t0: T).(ty3 g c (THead (Bind Abst) -w u) t0)) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (THead (Bind Abst) -x0 x1) (THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w -w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u))) -(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v)))) (\lambda -(x: T).(\lambda (H9: (ty3 g c (THead (Bind Abst) w u) x)).(ex3_2_ind T T -(\lambda (t2: T).(\lambda (_: T).(pc3 c (THead (Bind Abst) w t2) x))) -(\lambda (_: T).(\lambda (t0: T).(ty3 g c w t0))) (\lambda (t2: T).(\lambda -(_: T).(ty3 g (CHead c (Bind Abst) w) u t2))) (ex4_2 T T (\lambda (v: -T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) w v)))) -(\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda -(_: T).(ty3 g (CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: -T).(nf2 (CHead c (Bind Abst) w) v)))) (\lambda (x2: T).(\lambda (x3: -T).(\lambda (_: (pc3 c (THead (Bind Abst) w x2) x)).(\lambda (H11: (ty3 g c w -x3)).(\lambda (H12: (ty3 g (CHead c (Bind Abst) w) u x2)).(ex3_2_ind T T -(\lambda (t2: T).(\lambda (_: T).(pc3 c (THead (Bind Abst) x0 t2) (THead -(Bind Abst) w u)))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c x0 t0))) -(\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x0) x1 t2))) -(ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) -(THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) -(\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u))) -(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v)))) (\lambda -(x4: T).(\lambda (x5: T).(\lambda (H13: (pc3 c (THead (Bind Abst) x0 x4) -(THead (Bind Abst) w u))).(\lambda (_: (ty3 g c x0 x5)).(\lambda (H15: (ty3 g -(CHead c (Bind Abst) x0) x1 x4)).(land_ind (pc3 c x0 w) (\forall (b: -B).(\forall (u0: T).(pc3 (CHead c (Bind b) u0) x4 u))) (ex4_2 T T (\lambda -(v: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) w -v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: -T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u))) (\lambda (v: -T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v)))) (\lambda (H16: (pc3 c -x0 w)).(\lambda (H17: ((\forall (b: B).(\forall (u0: T).(pc3 (CHead c (Bind -b) u0) x4 u))))).(let H_y \def (pc3_nf2 c x0 w H16 H6 H1) in (let H18 \def -(eq_ind T x0 (\lambda (t0: T).(ty3 g (CHead c (Bind Abst) t0) x1 x4)) H15 w -H_y) in (let H19 \def (eq_ind T x0 (\lambda (t0: T).(nf2 (CHead c (Bind Abst) -t0) x1)) H7 w H_y) in (eq_ind_r T w (\lambda (t0: T).(ex4_2 T T (\lambda (v: -T).(\lambda (_: T).(eq T (THead (Bind Abst) t0 x1) (THead (Bind Abst) w v)))) -(\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda -(_: T).(ty3 g (CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: -T).(nf2 (CHead c (Bind Abst) w) v))))) (ex4_2_intro T T (\lambda (v: -T).(\lambda (_: T).(eq T (THead (Bind Abst) w x1) (THead (Bind Abst) w v)))) -(\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda -(_: T).(ty3 g (CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: -T).(nf2 (CHead c (Bind Abst) w) v))) x1 x3 (refl_equal T (THead (Bind Abst) w -x1)) H11 (ty3_conv g (CHead c (Bind Abst) w) u x2 H12 x1 x4 H18 (H17 Abst w)) -H19) x0 H_y)))))) (pc3_gen_abst c x0 w x4 u H13))))))) (ty3_gen_bind g Abst c -x0 x1 (THead (Bind Abst) w u) H8))))))) (ty3_gen_bind g Abst c w u x H9)))) -(ty3_correct g c (THead (Bind Abst) x0 x1) (THead (Bind Abst) w u) H8)) t -H5))))))) H4)) (\lambda (H4: (ex nat (\lambda (n: nat).(eq T t (TSort -n))))).(ex_ind nat (\lambda (n: nat).(eq T t (TSort n))) (ex4_2 T T (\lambda -(v: T).(\lambda (_: T).(eq T t (THead (Bind Abst) w v)))) (\lambda (_: -T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g -(CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c -(Bind Abst) w) v)))) (\lambda (x: nat).(\lambda (H5: (eq T t (TSort x))).(let -H6 \def (eq_ind T t (\lambda (t0: T).(ty3 g c t0 (THead (Bind Abst) w u))) H -(TSort x) H5) in (eq_ind_r T (TSort x) (\lambda (t0: T).(ex4_2 T T (\lambda -(v: T).(\lambda (_: T).(eq T t0 (THead (Bind Abst) w v)))) (\lambda (_: -T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g -(CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c -(Bind Abst) w) v))))) (pc3_gen_sort_abst c w u (next g x) (ty3_gen_sort g c -(THead (Bind Abst) w u) x H6) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq -T (TSort x) (THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 -g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v -u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v))))) t -H5)))) H4)) (\lambda (H4: (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: -nat).(eq T t (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: -TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: -nat).(nf2 c (TLRef i)))))).(ex3_2_ind TList nat (\lambda (ws: TList).(\lambda -(i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: -TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: -nat).(nf2 c (TLRef i)))) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T t -(THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) -(\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u))) -(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v)))) (\lambda -(x0: TList).(\lambda (x1: nat).(\lambda (H5: (eq T t (THeads (Flat Appl) x0 -(TLRef x1)))).(\lambda (_: (nfs2 c x0)).(\lambda (H7: (nf2 c (TLRef -x1))).(let H8 \def (eq_ind T t (\lambda (t0: T).(ty3 g c t0 (THead (Bind -Abst) w u))) H (THeads (Flat Appl) x0 (TLRef x1)) H5) in (eq_ind_r T (THeads -(Flat Appl) x0 (TLRef x1)) (\lambda (t0: T).(ex4_2 T T (\lambda (v: -T).(\lambda (_: T).(eq T t0 (THead (Bind Abst) w v)))) (\lambda (_: -T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g -(CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c -(Bind Abst) w) v))))) (let H9 \def H2 in ((let H10 \def H8 in (unintro T u -(\lambda (t0: T).((ty3 g c (THeads (Flat Appl) x0 (TLRef x1)) (THead (Bind -Abst) w t0)) \to ((ty3_nf2_inv_abst_premise c w t0) \to (ex4_2 T T (\lambda -(v: T).(\lambda (_: T).(eq T (THeads (Flat Appl) x0 (TLRef x1)) (THead (Bind -Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: -T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v t0))) (\lambda (v: -T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v))))))) (unintro T w -(\lambda (t0: T).(\forall (x: T).((ty3 g c (THeads (Flat Appl) x0 (TLRef x1)) -(THead (Bind Abst) t0 x)) \to ((ty3_nf2_inv_abst_premise c t0 x) \to (ex4_2 T -T (\lambda (v: T).(\lambda (_: T).(eq T (THeads (Flat Appl) x0 (TLRef x1)) -(THead (Bind Abst) t0 v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c t0 -w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) t0) v x))) -(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) t0) v)))))))) -(TList_ind (\lambda (t0: TList).(\forall (x: T).(\forall (x2: T).((ty3 g c -(THeads (Flat Appl) t0 (TLRef x1)) (THead (Bind Abst) x x2)) \to -((ty3_nf2_inv_abst_premise c x x2) \to (ex4_2 T T (\lambda (v: T).(\lambda -(_: T).(eq T (THeads (Flat Appl) t0 (TLRef x1)) (THead (Bind Abst) x v)))) -(\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda -(_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_: -T).(nf2 (CHead c (Bind Abst) x) v))))))))) (\lambda (x: T).(\lambda (x2: -T).(\lambda (H11: (ty3 g c (TLRef x1) (THead (Bind Abst) x x2))).(\lambda -(H12: (ty3_nf2_inv_abst_premise c x x2)).(or_ind (ex3_3 C T T (\lambda (_: -C).(\lambda (_: T).(\lambda (t0: T).(pc3 c (lift (S x1) O t0) (THead (Bind -Abst) x x2))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl x1 c -(CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: -T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda -(_: T).(pc3 c (lift (S x1) O u0) (THead (Bind Abst) x x2))))) (\lambda (e: -C).(\lambda (u0: T).(\lambda (_: T).(getl x1 c (CHead e (Bind Abst) u0))))) -(\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex4_2 -T T (\lambda (v: T).(\lambda (_: T).(eq T (TLRef x1) (THead (Bind Abst) x -v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: -T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v: -T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) v)))) (\lambda (H13: (ex3_3 C -T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c (lift (S x1) O -t0) (THead (Bind Abst) x x2))))) (\lambda (e: C).(\lambda (u0: T).(\lambda -(_: T).(getl x1 c (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (t0: T).(ty3 g e u0 t0)))))).(ex3_3_ind C T T (\lambda (_: -C).(\lambda (_: T).(\lambda (t0: T).(pc3 c (lift (S x1) O t0) (THead (Bind -Abst) x x2))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl x1 c -(CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: -T).(ty3 g e u0 t0)))) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (TLRef -x1) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c x -w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) -(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) v)))) (\lambda -(x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (pc3 c (lift (S x1) O -x5) (THead (Bind Abst) x x2))).(\lambda (H15: (getl x1 c (CHead x3 (Bind -Abbr) x4))).(\lambda (_: (ty3 g x3 x4 x5)).(nf2_gen_lref c x3 x4 x1 H15 H7 -(ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (TLRef x1) (THead (Bind -Abst) x v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: -T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v: -T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) v))))))))))) H13)) (\lambda -(H13: (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c -(lift (S x1) O u0) (THead (Bind Abst) x x2))))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (_: T).(getl x1 c (CHead e (Bind Abst) u0))))) (\lambda (e: -C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))))).(ex3_3_ind C T T -(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c (lift (S x1) O u0) -(THead (Bind Abst) x x2))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: -T).(getl x1 c (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: -T).(\lambda (t0: T).(ty3 g e u0 t0)))) (ex4_2 T T (\lambda (v: T).(\lambda -(_: T).(eq T (TLRef x1) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda -(w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c -(Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind -Abst) x) v)))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda -(H14: (pc3 c (lift (S x1) O x4) (THead (Bind Abst) x x2))).(\lambda (H15: -(getl x1 c (CHead x3 (Bind Abst) x4))).(\lambda (_: (ty3 g x3 x4 x5)).(let -H_x0 \def (H12 x3 x4 x1 H15 TNil H14) in (let H17 \def H_x0 in (False_ind -(ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (TLRef x1) (THead (Bind -Abst) x v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: -T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v: -T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) v)))) H17))))))))) H13)) -(ty3_gen_lref g c (THead (Bind Abst) x x2) x1 H11)))))) (\lambda (t0: -T).(\lambda (t1: TList).(\lambda (H11: ((\forall (x: T).(\forall (x2: -T).((ty3 g c (THeads (Flat Appl) t1 (TLRef x1)) (THead (Bind Abst) x x2)) \to -((ty3_nf2_inv_abst_premise c x x2) \to (ex4_2 T T (\lambda (v: T).(\lambda -(_: T).(eq T (THeads (Flat Appl) t1 (TLRef x1)) (THead (Bind Abst) x v)))) -(\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda -(_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_: -T).(nf2 (CHead c (Bind Abst) x) v)))))))))).(\lambda (x: T).(\lambda (x2: -T).(\lambda (H12: (ty3 g c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 -(TLRef x1))) (THead (Bind Abst) x x2))).(\lambda (H13: -(ty3_nf2_inv_abst_premise c x x2)).(ex3_2_ind T T (\lambda (u0: T).(\lambda -(t2: T).(pc3 c (THead (Flat Appl) t0 (THead (Bind Abst) u0 t2)) (THead (Bind -Abst) x x2)))) (\lambda (u0: T).(\lambda (t2: T).(ty3 g c (THeads (Flat Appl) -t1 (TLRef x1)) (THead (Bind Abst) u0 t2)))) (\lambda (u0: T).(\lambda (_: -T).(ty3 g c t0 u0))) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (THead -(Flat Appl) t0 (THeads (Flat Appl) t1 (TLRef x1))) (THead (Bind Abst) x v)))) -(\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda -(_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_: -T).(nf2 (CHead c (Bind Abst) x) v)))) (\lambda (x3: T).(\lambda (x4: -T).(\lambda (H14: (pc3 c (THead (Flat Appl) t0 (THead (Bind Abst) x3 x4)) -(THead (Bind Abst) x x2))).(\lambda (H15: (ty3 g c (THeads (Flat Appl) t1 -(TLRef x1)) (THead (Bind Abst) x3 x4))).(\lambda (_: (ty3 g c t0 x3)).(let -H_y \def (ty3_nf2_gen__ty3_nf2_inv_abst_aux c x x2 H13 t0 x3 x4 H14) in (let -H_x0 \def (H11 x3 x4 H15 H_y) in (let H17 \def H_x0 in (ex4_2_ind T T -(\lambda (v: T).(\lambda (_: T).(eq T (THeads (Flat Appl) t1 (TLRef x1)) -(THead (Bind Abst) x3 v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c x3 -w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x3) v x4))) -(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x3) v))) (ex4_2 T T -(\lambda (v: T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (THeads (Flat -Appl) t1 (TLRef x1))) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda -(w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c -(Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind -Abst) x) v)))) (\lambda (x5: T).(\lambda (x6: T).(\lambda (H18: (eq T (THeads -(Flat Appl) t1 (TLRef x1)) (THead (Bind Abst) x3 x5))).(\lambda (_: (ty3 g c -x3 x6)).(\lambda (_: (ty3 g (CHead c (Bind Abst) x3) x5 x4)).(\lambda (_: -(nf2 (CHead c (Bind Abst) x3) x5)).(TList_ind (\lambda (t2: TList).((eq T -(THeads (Flat Appl) t2 (TLRef x1)) (THead (Bind Abst) x3 x5)) \to (ex4_2 T T -(\lambda (v: T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (THeads (Flat -Appl) t2 (TLRef x1))) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda -(w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c -(Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind -Abst) x) v)))))) (\lambda (H22: (eq T (THeads (Flat Appl) TNil (TLRef x1)) -(THead (Bind Abst) x3 x5))).(let H23 \def (eq_ind T (TLRef x1) (\lambda (ee: -T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I -(THead (Bind Abst) x3 x5) H22) in (False_ind (ex4_2 T T (\lambda (v: -T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (THeads (Flat Appl) TNil -(TLRef x1))) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 -g c x w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v -x2))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) v)))) -H23))) (\lambda (t2: T).(\lambda (t3: TList).(\lambda (_: (((eq T (THeads -(Flat Appl) t3 (TLRef x1)) (THead (Bind Abst) x3 x5)) \to (ex4_2 T T (\lambda -(v: T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (THeads (Flat Appl) t3 -(TLRef x1))) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 -g c x w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v -x2))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) -v))))))).(\lambda (H22: (eq T (THeads (Flat Appl) (TCons t2 t3) (TLRef x1)) -(THead (Bind Abst) x3 x5))).(let H23 \def (eq_ind T (THead (Flat Appl) t2 -(THeads (Flat Appl) t3 (TLRef x1))) (\lambda (ee: T).(match ee in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind Abst) x3 x5) H22) in (False_ind (ex4_2 T T (\lambda -(v: T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (THeads (Flat Appl) (TCons -t2 t3) (TLRef x1))) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda (w0: -T).(ty3 g c x w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind -Abst) x) v x2))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) -v)))) H23)))))) t1 H18))))))) H17))))))))) (ty3_gen_appl g c t0 (THeads (Flat -Appl) t1 (TLRef x1)) (THead (Bind Abst) x x2) H12))))))))) x0)) H10)) H9)) t -H5))))))) H4)) H3))))))))))). -(* COMMENTS -Initial nodes: 5333 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/pr3.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/pr3.ma deleted file mode 100644 index 8d184aa58..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/pr3.ma +++ /dev/null @@ -1,728 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/csubt/ty3.ma". - -include "Basic-1/ty3/subst1.ma". - -include "Basic-1/ty3/fsubst0.ma". - -include "Basic-1/pc3/pc1.ma". - -include "Basic-1/pc3/wcpr0.ma". - -include "Basic-1/pc1/props.ma". - -theorem ty3_sred_wcpr0_pr0: - \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t: T).((ty3 g c1 -t1 t) \to (\forall (c2: C).((wcpr0 c1 c2) \to (\forall (t2: T).((pr0 t1 t2) -\to (ty3 g c2 t2 t))))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t: T).(\lambda -(H: (ty3 g c1 t1 t)).(ty3_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda -(t2: T).(\forall (c2: C).((wcpr0 c c2) \to (\forall (t3: T).((pr0 t0 t3) \to -(ty3 g c2 t3 t2)))))))) (\lambda (c: C).(\lambda (t2: T).(\lambda (t0: -T).(\lambda (_: (ty3 g c t2 t0)).(\lambda (H1: ((\forall (c2: C).((wcpr0 c -c2) \to (\forall (t3: T).((pr0 t2 t3) \to (ty3 g c2 t3 t0))))))).(\lambda (u: -T).(\lambda (t3: T).(\lambda (_: (ty3 g c u t3)).(\lambda (H3: ((\forall (c2: -C).((wcpr0 c c2) \to (\forall (t4: T).((pr0 u t4) \to (ty3 g c2 t4 -t3))))))).(\lambda (H4: (pc3 c t3 t2)).(\lambda (c2: C).(\lambda (H5: (wcpr0 -c c2)).(\lambda (t4: T).(\lambda (H6: (pr0 u t4)).(ty3_conv g c2 t2 t0 (H1 c2 -H5 t2 (pr0_refl t2)) t4 t3 (H3 c2 H5 t4 H6) (pc3_wcpr0 c c2 H5 t3 t2 -H4)))))))))))))))) (\lambda (c: C).(\lambda (m: nat).(\lambda (c2: -C).(\lambda (_: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H1: (pr0 (TSort m) -t2)).(eq_ind_r T (TSort m) (\lambda (t0: T).(ty3 g c2 t0 (TSort (next g m)))) -(ty3_sort g c2 m) t2 (pr0_gen_sort t2 m H1)))))))) (\lambda (n: nat).(\lambda -(c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind -Abbr) u))).(\lambda (t0: T).(\lambda (_: (ty3 g d u t0)).(\lambda (H2: -((\forall (c2: C).((wcpr0 d c2) \to (\forall (t2: T).((pr0 u t2) \to (ty3 g -c2 t2 t0))))))).(\lambda (c2: C).(\lambda (H3: (wcpr0 c c2)).(\lambda (t2: -T).(\lambda (H4: (pr0 (TLRef n) t2)).(eq_ind_r T (TLRef n) (\lambda (t3: -T).(ty3 g c2 t3 (lift (S n) O t0))) (ex3_2_ind C T (\lambda (e2: C).(\lambda -(u2: T).(getl n c2 (CHead e2 (Bind Abbr) u2)))) (\lambda (e2: C).(\lambda (_: -T).(wcpr0 d e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u u2))) (ty3 g c2 -(TLRef n) (lift (S n) O t0)) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: -(getl n c2 (CHead x0 (Bind Abbr) x1))).(\lambda (H6: (wcpr0 d x0)).(\lambda -(H7: (pr0 u x1)).(ty3_abbr g n c2 x0 x1 H5 t0 (H2 x0 H6 x1 H7))))))) -(wcpr0_getl c c2 H3 n d u (Bind Abbr) H0)) t2 (pr0_gen_lref t2 n -H4)))))))))))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d: C).(\lambda -(u: T).(\lambda (H0: (getl n c (CHead d (Bind Abst) u))).(\lambda (t0: -T).(\lambda (_: (ty3 g d u t0)).(\lambda (H2: ((\forall (c2: C).((wcpr0 d c2) -\to (\forall (t2: T).((pr0 u t2) \to (ty3 g c2 t2 t0))))))).(\lambda (c2: -C).(\lambda (H3: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H4: (pr0 (TLRef n) -t2)).(eq_ind_r T (TLRef n) (\lambda (t3: T).(ty3 g c2 t3 (lift (S n) O u))) -(ex3_2_ind C T (\lambda (e2: C).(\lambda (u2: T).(getl n c2 (CHead e2 (Bind -Abst) u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 d e2))) (\lambda (_: -C).(\lambda (u2: T).(pr0 u u2))) (ty3 g c2 (TLRef n) (lift (S n) O u)) -(\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: (getl n c2 (CHead x0 (Bind -Abst) x1))).(\lambda (H6: (wcpr0 d x0)).(\lambda (H7: (pr0 u x1)).(ty3_conv g -c2 (lift (S n) O u) (lift (S n) O t0) (ty3_lift g x0 u t0 (H2 x0 H6 u -(pr0_refl u)) c2 O (S n) (getl_drop Abst c2 x0 x1 n H5)) (TLRef n) (lift (S -n) O x1) (ty3_abst g n c2 x0 x1 H5 t0 (H2 x0 H6 x1 H7)) (pc3_lift c2 x0 (S n) -O (getl_drop Abst c2 x0 x1 n H5) x1 u (pc3_pr2_x x0 x1 u (pr2_free x0 u x1 -H7))))))))) (wcpr0_getl c c2 H3 n d u (Bind Abst) H0)) t2 (pr0_gen_lref t2 n -H4)))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (t0: T).(\lambda -(_: (ty3 g c u t0)).(\lambda (H1: ((\forall (c2: C).((wcpr0 c c2) \to -(\forall (t2: T).((pr0 u t2) \to (ty3 g c2 t2 t0))))))).(\lambda (b: -B).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H2: (ty3 g (CHead c (Bind b) -u) t2 t3)).(\lambda (H3: ((\forall (c2: C).((wcpr0 (CHead c (Bind b) u) c2) -\to (\forall (t4: T).((pr0 t2 t4) \to (ty3 g c2 t4 t3))))))).(\lambda (c2: -C).(\lambda (H4: (wcpr0 c c2)).(\lambda (t4: T).(\lambda (H5: (pr0 (THead -(Bind b) u t2) t4)).(let H6 \def (match H5 in pr0 return (\lambda (t5: -T).(\lambda (t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 (THead (Bind b) u -t2)) \to ((eq T t6 t4) \to (ty3 g c2 t4 (THead (Bind b) u t3))))))) with -[(pr0_refl t5) \Rightarrow (\lambda (H6: (eq T t5 (THead (Bind b) u -t2))).(\lambda (H7: (eq T t5 t4)).(eq_ind T (THead (Bind b) u t2) (\lambda -(t6: T).((eq T t6 t4) \to (ty3 g c2 t4 (THead (Bind b) u t3)))) (\lambda (H8: -(eq T (THead (Bind b) u t2) t4)).(eq_ind T (THead (Bind b) u t2) (\lambda -(t6: T).(ty3 g c2 t6 (THead (Bind b) u t3))) (ty3_bind g c2 u t0 (H1 c2 H4 u -(pr0_refl u)) b t2 t3 (H3 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H4 u u -(pr0_refl u) (Bind b)) t2 (pr0_refl t2))) t4 H8)) t5 (sym_eq T t5 (THead -(Bind b) u t2) H6) H7))) | (pr0_comp u1 u2 H6 t5 t6 H7 k) \Rightarrow -(\lambda (H8: (eq T (THead k u1 t5) (THead (Bind b) u t2))).(\lambda (H9: (eq -T (THead k u2 t6) t4)).((let H10 \def (f_equal T T (\lambda (e: T).(match e -in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t5 | (TLRef _) -\Rightarrow t5 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t5) (THead -(Bind b) u t2) H8) in ((let H11 \def (f_equal T T (\lambda (e: T).(match e in -T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) -\Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t5) (THead -(Bind b) u t2) H8) in ((let H12 \def (f_equal T K (\lambda (e: T).(match e in -T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) -\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t5) (THead (Bind -b) u t2) H8) in (eq_ind K (Bind b) (\lambda (k0: K).((eq T u1 u) \to ((eq T -t5 t2) \to ((eq T (THead k0 u2 t6) t4) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to -(ty3 g c2 t4 (THead (Bind b) u t3)))))))) (\lambda (H13: (eq T u1 u)).(eq_ind -T u (\lambda (t7: T).((eq T t5 t2) \to ((eq T (THead (Bind b) u2 t6) t4) \to -((pr0 t7 u2) \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3))))))) -(\lambda (H14: (eq T t5 t2)).(eq_ind T t2 (\lambda (t7: T).((eq T (THead -(Bind b) u2 t6) t4) \to ((pr0 u u2) \to ((pr0 t7 t6) \to (ty3 g c2 t4 (THead -(Bind b) u t3)))))) (\lambda (H15: (eq T (THead (Bind b) u2 t6) t4)).(eq_ind -T (THead (Bind b) u2 t6) (\lambda (t7: T).((pr0 u u2) \to ((pr0 t2 t6) \to -(ty3 g c2 t7 (THead (Bind b) u t3))))) (\lambda (H16: (pr0 u u2)).(\lambda -(H17: (pr0 t2 t6)).(ex_ind T (\lambda (t7: T).(ty3 g (CHead c2 (Bind b) u) t3 -t7)) (ty3 g c2 (THead (Bind b) u2 t6) (THead (Bind b) u t3)) (\lambda (x: -T).(\lambda (H18: (ty3 g (CHead c2 (Bind b) u) t3 x)).(ex_ind T (\lambda (t7: -T).(ty3 g (CHead c2 (Bind b) u2) t3 t7)) (ty3 g c2 (THead (Bind b) u2 t6) -(THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (_: (ty3 g (CHead c2 (Bind -b) u2) t3 x0)).(ty3_conv g c2 (THead (Bind b) u t3) (THead (Bind b) u x) -(ty3_bind g c2 u t0 (H1 c2 H4 u (pr0_refl u)) b t3 x H18) (THead (Bind b) u2 -t6) (THead (Bind b) u2 t3) (ty3_bind g c2 u2 t0 (H1 c2 H4 u2 H16) b t6 t3 (H3 -(CHead c2 (Bind b) u2) (wcpr0_comp c c2 H4 u u2 H16 (Bind b)) t6 H17)) -(pc3_pr2_x c2 (THead (Bind b) u2 t3) (THead (Bind b) u t3) (pr2_head_1 c2 u -u2 (pr2_free c2 u u2 H16) (Bind b) t3))))) (ty3_correct g (CHead c2 (Bind b) -u2) t6 t3 (H3 (CHead c2 (Bind b) u2) (wcpr0_comp c c2 H4 u u2 H16 (Bind b)) -t6 H17))))) (ty3_correct g (CHead c2 (Bind b) u) t2 t3 (H3 (CHead c2 (Bind b) -u) (wcpr0_comp c c2 H4 u u (pr0_refl u) (Bind b)) t2 (pr0_refl t2)))))) t4 -H15)) t5 (sym_eq T t5 t2 H14))) u1 (sym_eq T u1 u H13))) k (sym_eq K k (Bind -b) H12))) H11)) H10)) H9 H6 H7))) | (pr0_beta u0 v1 v2 H6 t5 t6 H7) -\Rightarrow (\lambda (H8: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u0 -t5)) (THead (Bind b) u t2))).(\lambda (H9: (eq T (THead (Bind Abbr) v2 t6) -t4)).((let H10 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u0 -t5)) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort -_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) -\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u t2) -H8) in (False_ind ((eq T (THead (Bind Abbr) v2 t6) t4) \to ((pr0 v1 v2) \to -((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3))))) H10)) H9 H6 H7))) | -(pr0_upsilon b0 H6 v1 v2 H7 u1 u2 H8 t5 t6 H9) \Rightarrow (\lambda (H10: (eq -T (THead (Flat Appl) v1 (THead (Bind b0) u1 t5)) (THead (Bind b) u -t2))).(\lambda (H11: (eq T (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) -O v2) t6)) t4)).((let H12 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind -b0) u1 t5)) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) -\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u t2) -H10) in (False_ind ((eq T (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) -O v2) t6)) t4) \to ((not (eq B b0 Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) -\to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3))))))) H12)) H11 H6 H7 -H8 H9))) | (pr0_delta u1 u2 H6 t5 t6 H7 w H8) \Rightarrow (\lambda (H9: (eq T -(THead (Bind Abbr) u1 t5) (THead (Bind b) u t2))).(\lambda (H10: (eq T (THead -(Bind Abbr) u2 w) t4)).((let H11 \def (f_equal T T (\lambda (e: T).(match e -in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t5 | (TLRef _) -\Rightarrow t5 | (THead _ _ t7) \Rightarrow t7])) (THead (Bind Abbr) u1 t5) -(THead (Bind b) u t2) H9) in ((let H12 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | -(TLRef _) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead (Bind -Abbr) u1 t5) (THead (Bind b) u t2) H9) in ((let H13 \def (f_equal T B -(\lambda (e: T).(match e in T return (\lambda (_: T).B) with [(TSort _) -\Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | (THead k _ _) \Rightarrow -(match k in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | -(Flat _) \Rightarrow Abbr])])) (THead (Bind Abbr) u1 t5) (THead (Bind b) u -t2) H9) in (eq_ind B Abbr (\lambda (b0: B).((eq T u1 u) \to ((eq T t5 t2) \to -((eq T (THead (Bind Abbr) u2 w) t4) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to -((subst0 O u2 t6 w) \to (ty3 g c2 t4 (THead (Bind b0) u t3))))))))) (\lambda -(H14: (eq T u1 u)).(eq_ind T u (\lambda (t7: T).((eq T t5 t2) \to ((eq T -(THead (Bind Abbr) u2 w) t4) \to ((pr0 t7 u2) \to ((pr0 t5 t6) \to ((subst0 O -u2 t6 w) \to (ty3 g c2 t4 (THead (Bind Abbr) u t3)))))))) (\lambda (H15: (eq -T t5 t2)).(eq_ind T t2 (\lambda (t7: T).((eq T (THead (Bind Abbr) u2 w) t4) -\to ((pr0 u u2) \to ((pr0 t7 t6) \to ((subst0 O u2 t6 w) \to (ty3 g c2 t4 -(THead (Bind Abbr) u t3))))))) (\lambda (H16: (eq T (THead (Bind Abbr) u2 w) -t4)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t7: T).((pr0 u u2) \to -((pr0 t2 t6) \to ((subst0 O u2 t6 w) \to (ty3 g c2 t7 (THead (Bind Abbr) u -t3)))))) (\lambda (H17: (pr0 u u2)).(\lambda (H18: (pr0 t2 t6)).(\lambda -(H19: (subst0 O u2 t6 w)).(let H20 \def (eq_ind_r B b (\lambda (b0: -B).(\forall (c3: C).((wcpr0 (CHead c (Bind b0) u) c3) \to (\forall (t7: -T).((pr0 t2 t7) \to (ty3 g c3 t7 t3)))))) H3 Abbr H13) in (let H21 \def -(eq_ind_r B b (\lambda (b0: B).(ty3 g (CHead c (Bind b0) u) t2 t3)) H2 Abbr -H13) in (ex_ind T (\lambda (t7: T).(ty3 g (CHead c2 (Bind Abbr) u) t3 t7)) -(ty3 g c2 (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u t3)) (\lambda (x: -T).(\lambda (H22: (ty3 g (CHead c2 (Bind Abbr) u) t3 x)).(ex_ind T (\lambda -(t7: T).(ty3 g (CHead c2 (Bind Abbr) u2) t3 t7)) (ty3 g c2 (THead (Bind Abbr) -u2 w) (THead (Bind Abbr) u t3)) (\lambda (x0: T).(\lambda (_: (ty3 g (CHead -c2 (Bind Abbr) u2) t3 x0)).(ty3_conv g c2 (THead (Bind Abbr) u t3) (THead -(Bind Abbr) u x) (ty3_bind g c2 u t0 (H1 c2 H4 u (pr0_refl u)) Abbr t3 x H22) -(THead (Bind Abbr) u2 w) (THead (Bind Abbr) u2 t3) (ty3_bind g c2 u2 t0 (H1 -c2 H4 u2 H17) Abbr w t3 (ty3_subst0 g (CHead c2 (Bind Abbr) u2) t6 t3 (H20 -(CHead c2 (Bind Abbr) u2) (wcpr0_comp c c2 H4 u u2 H17 (Bind Abbr)) t6 H18) -c2 u2 O (getl_refl Abbr c2 u2) w H19)) (pc3_pr2_x c2 (THead (Bind Abbr) u2 -t3) (THead (Bind Abbr) u t3) (pr2_head_1 c2 u u2 (pr2_free c2 u u2 H17) (Bind -Abbr) t3))))) (ty3_correct g (CHead c2 (Bind Abbr) u2) t6 t3 (H20 (CHead c2 -(Bind Abbr) u2) (wcpr0_comp c c2 H4 u u2 H17 (Bind Abbr)) t6 H18))))) -(ty3_correct g (CHead c2 (Bind Abbr) u) t2 t3 (H20 (CHead c2 (Bind Abbr) u) -(wcpr0_comp c c2 H4 u u (pr0_refl u) (Bind Abbr)) t2 (pr0_refl t2))))))))) t4 -H16)) t5 (sym_eq T t5 t2 H15))) u1 (sym_eq T u1 u H14))) b H13)) H12)) H11)) -H10 H6 H7 H8))) | (pr0_zeta b0 H6 t5 t6 H7 u0) \Rightarrow (\lambda (H8: (eq -T (THead (Bind b0) u0 (lift (S O) O t5)) (THead (Bind b) u t2))).(\lambda -(H9: (eq T t6 t4)).((let H10 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: -((nat \to nat))) (d: nat) (t7: T) on t7: T \def (match t7 with [(TSort n) -\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with -[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u1 t8) -\Rightarrow (THead k (lref_map f d u1) (lref_map f (s k d) t8))]) in -lref_map) (\lambda (x: nat).(plus x (S O))) O t5) | (TLRef _) \Rightarrow -((let rec lref_map (f: ((nat \to nat))) (d: nat) (t7: T) on t7: T \def (match -t7 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef -(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | -(THead k u1 t8) \Rightarrow (THead k (lref_map f d u1) (lref_map f (s k d) -t8))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t5) | (THead _ _ t7) -\Rightarrow t7])) (THead (Bind b0) u0 (lift (S O) O t5)) (THead (Bind b) u -t2) H8) in ((let H11 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 -| (THead _ t7 _) \Rightarrow t7])) (THead (Bind b0) u0 (lift (S O) O t5)) -(THead (Bind b) u t2) H8) in ((let H12 \def (f_equal T B (\lambda (e: -T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b0 | -(TLRef _) \Rightarrow b0 | (THead k _ _) \Rightarrow (match k in K return -(\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow -b0])])) (THead (Bind b0) u0 (lift (S O) O t5)) (THead (Bind b) u t2) H8) in -(eq_ind B b (\lambda (b1: B).((eq T u0 u) \to ((eq T (lift (S O) O t5) t2) -\to ((eq T t6 t4) \to ((not (eq B b1 Abst)) \to ((pr0 t5 t6) \to (ty3 g c2 t4 -(THead (Bind b) u t3)))))))) (\lambda (H13: (eq T u0 u)).(eq_ind T u (\lambda -(_: T).((eq T (lift (S O) O t5) t2) \to ((eq T t6 t4) \to ((not (eq B b -Abst)) \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3))))))) (\lambda -(H14: (eq T (lift (S O) O t5) t2)).(eq_ind T (lift (S O) O t5) (\lambda (_: -T).((eq T t6 t4) \to ((not (eq B b Abst)) \to ((pr0 t5 t6) \to (ty3 g c2 t4 -(THead (Bind b) u t3)))))) (\lambda (H15: (eq T t6 t4)).(eq_ind T t4 (\lambda -(t7: T).((not (eq B b Abst)) \to ((pr0 t5 t7) \to (ty3 g c2 t4 (THead (Bind -b) u t3))))) (\lambda (H16: (not (eq B b Abst))).(\lambda (H17: (pr0 t5 -t4)).(let H18 \def (eq_ind_r T t2 (\lambda (t7: T).(\forall (c3: C).((wcpr0 -(CHead c (Bind b) u) c3) \to (\forall (t8: T).((pr0 t7 t8) \to (ty3 g c3 t8 -t3)))))) H3 (lift (S O) O t5) H14) in (let H19 \def (eq_ind_r T t2 (\lambda -(t7: T).(ty3 g (CHead c (Bind b) u) t7 t3)) H2 (lift (S O) O t5) H14) in -(ex_ind T (\lambda (t7: T).(ty3 g (CHead c2 (Bind b) u) t3 t7)) (ty3 g c2 t4 -(THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H20: (ty3 g (CHead c2 (Bind -b) u) t3 x)).(B_ind (\lambda (b1: B).((not (eq B b1 Abst)) \to ((ty3 g (CHead -c2 (Bind b1) u) t3 x) \to ((ty3 g (CHead c2 (Bind b1) u) (lift (S O) O t4) -t3) \to (ty3 g c2 t4 (THead (Bind b1) u t3)))))) (\lambda (H21: (not (eq B -Abbr Abst))).(\lambda (H22: (ty3 g (CHead c2 (Bind Abbr) u) t3 x)).(\lambda -(H23: (ty3 g (CHead c2 (Bind Abbr) u) (lift (S O) O t4) t3)).(let H24 \def -(ty3_gen_cabbr g (CHead c2 (Bind Abbr) u) (lift (S O) O t4) t3 H23 c2 u O -(getl_refl Abbr c2 u) (CHead c2 (Bind Abbr) u) (csubst1_refl O u (CHead c2 -(Bind Abbr) u)) c2 (drop_drop (Bind Abbr) O c2 c2 (drop_refl c2) u)) in -(ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(subst1 O u (lift (S O) O t4) -(lift (S O) O y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 O u t3 (lift (S -O) O y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g c2 y1 y2))) (ty3 g c2 t4 -(THead (Bind Abbr) u t3)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H25: -(subst1 O u (lift (S O) O t4) (lift (S O) O x0))).(\lambda (H26: (subst1 O u -t3 (lift (S O) O x1))).(\lambda (H27: (ty3 g c2 x0 x1)).(let H28 \def (eq_ind -T x0 (\lambda (t7: T).(ty3 g c2 t7 x1)) H27 t4 (lift_inj x0 t4 (S O) O -(subst1_gen_lift_eq t4 u (lift (S O) O x0) (S O) O O (le_n O) (eq_ind_r nat -(plus (S O) O) (\lambda (n: nat).(lt O n)) (le_n (plus (S O) O)) (plus O (S -O)) (plus_sym O (S O))) H25))) in (ty3_conv g c2 (THead (Bind Abbr) u t3) -(THead (Bind Abbr) u x) (ty3_bind g c2 u t0 (H1 c2 H4 u (pr0_refl u)) Abbr t3 -x H22) t4 x1 H28 (pc3_pr3_x c2 x1 (THead (Bind Abbr) u t3) (pr3_t (THead -(Bind Abbr) u (lift (S O) O x1)) (THead (Bind Abbr) u t3) c2 (pr3_pr2 c2 -(THead (Bind Abbr) u t3) (THead (Bind Abbr) u (lift (S O) O x1)) (pr2_free c2 -(THead (Bind Abbr) u t3) (THead (Bind Abbr) u (lift (S O) O x1)) (pr0_delta1 -u u (pr0_refl u) t3 t3 (pr0_refl t3) (lift (S O) O x1) H26))) x1 (pr3_pr2 c2 -(THead (Bind Abbr) u (lift (S O) O x1)) x1 (pr2_free c2 (THead (Bind Abbr) u -(lift (S O) O x1)) x1 (pr0_zeta Abbr H21 x1 x1 (pr0_refl x1) u)))))))))))) -H24))))) (\lambda (H21: (not (eq B Abst Abst))).(\lambda (_: (ty3 g (CHead c2 -(Bind Abst) u) t3 x)).(\lambda (_: (ty3 g (CHead c2 (Bind Abst) u) (lift (S -O) O t4) t3)).(let H24 \def (match (H21 (refl_equal B Abst)) in False return -(\lambda (_: False).(ty3 g c2 t4 (THead (Bind Abst) u t3))) with []) in -H24)))) (\lambda (H21: (not (eq B Void Abst))).(\lambda (H22: (ty3 g (CHead -c2 (Bind Void) u) t3 x)).(\lambda (H23: (ty3 g (CHead c2 (Bind Void) u) (lift -(S O) O t4) t3)).(let H24 \def (ty3_gen_cvoid g (CHead c2 (Bind Void) u) -(lift (S O) O t4) t3 H23 c2 u O (getl_refl Void c2 u) c2 (drop_drop (Bind -Void) O c2 c2 (drop_refl c2) u)) in (ex3_2_ind T T (\lambda (y1: T).(\lambda -(_: T).(eq T (lift (S O) O t4) (lift (S O) O y1)))) (\lambda (_: T).(\lambda -(y2: T).(eq T t3 (lift (S O) O y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 -g c2 y1 y2))) (ty3 g c2 t4 (THead (Bind Void) u t3)) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H25: (eq T (lift (S O) O t4) (lift (S O) O -x0))).(\lambda (H26: (eq T t3 (lift (S O) O x1))).(\lambda (H27: (ty3 g c2 x0 -x1)).(let H28 \def (eq_ind T t3 (\lambda (t7: T).(ty3 g (CHead c2 (Bind Void) -u) t7 x)) H22 (lift (S O) O x1) H26) in (eq_ind_r T (lift (S O) O x1) -(\lambda (t7: T).(ty3 g c2 t4 (THead (Bind Void) u t7))) (let H29 \def -(eq_ind_r T x0 (\lambda (t7: T).(ty3 g c2 t7 x1)) H27 t4 (lift_inj t4 x0 (S -O) O H25)) in (ty3_conv g c2 (THead (Bind Void) u (lift (S O) O x1)) (THead -(Bind Void) u x) (ty3_bind g c2 u t0 (H1 c2 H4 u (pr0_refl u)) Void (lift (S -O) O x1) x H28) t4 x1 H29 (pc3_s c2 x1 (THead (Bind Void) u (lift (S O) O -x1)) (pc3_pr2_r c2 (THead (Bind Void) u (lift (S O) O x1)) x1 (pr2_free c2 -(THead (Bind Void) u (lift (S O) O x1)) x1 (pr0_zeta Void H21 x1 x1 (pr0_refl -x1) u)))))) t3 H26))))))) H24))))) b H16 H20 (H18 (CHead c2 (Bind b) u) -(wcpr0_comp c c2 H4 u u (pr0_refl u) (Bind b)) (lift (S O) O t4) (pr0_lift t5 -t4 H17 (S O) O))))) (ty3_correct g (CHead c2 (Bind b) u) (lift (S O) O t4) t3 -(H18 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H4 u u (pr0_refl u) (Bind b)) -(lift (S O) O t4) (pr0_lift t5 t4 H17 (S O) O)))))))) t6 (sym_eq T t6 t4 -H15))) t2 H14)) u0 (sym_eq T u0 u H13))) b0 (sym_eq B b0 b H12))) H11)) H10)) -H9 H6 H7))) | (pr0_tau t5 t6 H6 u0) \Rightarrow (\lambda (H7: (eq T (THead -(Flat Cast) u0 t5) (THead (Bind b) u t2))).(\lambda (H8: (eq T t6 t4)).((let -H9 \def (eq_ind T (THead (Flat Cast) u0 t5) (\lambda (e: T).(match e in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow -True])])) I (THead (Bind b) u t2) H7) in (False_ind ((eq T t6 t4) \to ((pr0 -t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3)))) H9)) H8 H6)))]) in (H6 -(refl_equal T (THead (Bind b) u t2)) (refl_equal T t4))))))))))))))))) -(\lambda (c: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c w -u)).(\lambda (H1: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 -w t2) \to (ty3 g c2 t2 u))))))).(\lambda (v: T).(\lambda (t0: T).(\lambda -(H2: (ty3 g c v (THead (Bind Abst) u t0))).(\lambda (H3: ((\forall (c2: -C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 v t2) \to (ty3 g c2 t2 (THead -(Bind Abst) u t0)))))))).(\lambda (c2: C).(\lambda (H4: (wcpr0 c -c2)).(\lambda (t2: T).(\lambda (H5: (pr0 (THead (Flat Appl) w v) t2)).(let H6 -\def (match H5 in pr0 return (\lambda (t3: T).(\lambda (t4: T).(\lambda (_: -(pr0 t3 t4)).((eq T t3 (THead (Flat Appl) w v)) \to ((eq T t4 t2) \to (ty3 g -c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u t0)))))))) with [(pr0_refl -t3) \Rightarrow (\lambda (H6: (eq T t3 (THead (Flat Appl) w v))).(\lambda -(H7: (eq T t3 t2)).(eq_ind T (THead (Flat Appl) w v) (\lambda (t4: T).((eq T -t4 t2) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u t0))))) -(\lambda (H8: (eq T (THead (Flat Appl) w v) t2)).(eq_ind T (THead (Flat Appl) -w v) (\lambda (t4: T).(ty3 g c2 t4 (THead (Flat Appl) w (THead (Bind Abst) u -t0)))) (ty3_appl g c2 w u (H1 c2 H4 w (pr0_refl w)) v t0 (H3 c2 H4 v -(pr0_refl v))) t2 H8)) t3 (sym_eq T t3 (THead (Flat Appl) w v) H6) H7))) | -(pr0_comp u1 u2 H6 t3 t4 H7 k) \Rightarrow (\lambda (H8: (eq T (THead k u1 -t3) (THead (Flat Appl) w v))).(\lambda (H9: (eq T (THead k u2 t4) t2)).((let -H10 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t5) -\Rightarrow t5])) (THead k u1 t3) (THead (Flat Appl) w v) H8) in ((let H11 -\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t5 _) -\Rightarrow t5])) (THead k u1 t3) (THead (Flat Appl) w v) H8) in ((let H12 -\def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) -with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) -\Rightarrow k0])) (THead k u1 t3) (THead (Flat Appl) w v) H8) in (eq_ind K -(Flat Appl) (\lambda (k0: K).((eq T u1 w) \to ((eq T t3 v) \to ((eq T (THead -k0 u2 t4) t2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat -Appl) w (THead (Bind Abst) u t0))))))))) (\lambda (H13: (eq T u1 w)).(eq_ind -T w (\lambda (t5: T).((eq T t3 v) \to ((eq T (THead (Flat Appl) u2 t4) t2) -\to ((pr0 t5 u2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w -(THead (Bind Abst) u t0)))))))) (\lambda (H14: (eq T t3 v)).(eq_ind T v -(\lambda (t5: T).((eq T (THead (Flat Appl) u2 t4) t2) \to ((pr0 w u2) \to -((pr0 t5 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u -t0))))))) (\lambda (H15: (eq T (THead (Flat Appl) u2 t4) t2)).(eq_ind T -(THead (Flat Appl) u2 t4) (\lambda (t5: T).((pr0 w u2) \to ((pr0 v t4) \to -(ty3 g c2 t5 (THead (Flat Appl) w (THead (Bind Abst) u t0)))))) (\lambda -(H16: (pr0 w u2)).(\lambda (H17: (pr0 v t4)).(ex_ind T (\lambda (t5: T).(ty3 -g c2 (THead (Bind Abst) u t0) t5)) (ty3 g c2 (THead (Flat Appl) u2 t4) (THead -(Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x: T).(\lambda (H18: (ty3 -g c2 (THead (Bind Abst) u t0) x)).(ex3_2_ind T T (\lambda (t5: T).(\lambda -(_: T).(pc3 c2 (THead (Bind Abst) u t5) x))) (\lambda (_: T).(\lambda (t6: -T).(ty3 g c2 u t6))) (\lambda (t5: T).(\lambda (_: T).(ty3 g (CHead c2 (Bind -Abst) u) t0 t5))) (ty3 g c2 (THead (Flat Appl) u2 t4) (THead (Flat Appl) w -(THead (Bind Abst) u t0))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: -(pc3 c2 (THead (Bind Abst) u x0) x)).(\lambda (H20: (ty3 g c2 u x1)).(\lambda -(H21: (ty3 g (CHead c2 (Bind Abst) u) t0 x0)).(ty3_conv g c2 (THead (Flat -Appl) w (THead (Bind Abst) u t0)) (THead (Flat Appl) w (THead (Bind Abst) u -x0)) (ty3_appl g c2 w u (H1 c2 H4 w (pr0_refl w)) (THead (Bind Abst) u t0) x0 -(ty3_bind g c2 u x1 H20 Abst t0 x0 H21)) (THead (Flat Appl) u2 t4) (THead -(Flat Appl) u2 (THead (Bind Abst) u t0)) (ty3_appl g c2 u2 u (H1 c2 H4 u2 -H16) t4 t0 (H3 c2 H4 t4 H17)) (pc3_pr2_x c2 (THead (Flat Appl) u2 (THead -(Bind Abst) u t0)) (THead (Flat Appl) w (THead (Bind Abst) u t0)) (pr2_head_1 -c2 w u2 (pr2_free c2 w u2 H16) (Flat Appl) (THead (Bind Abst) u t0))))))))) -(ty3_gen_bind g Abst c2 u t0 x H18)))) (ty3_correct g c2 v (THead (Bind Abst) -u t0) (H3 c2 H4 v (pr0_refl v)))))) t2 H15)) t3 (sym_eq T t3 v H14))) u1 -(sym_eq T u1 w H13))) k (sym_eq K k (Flat Appl) H12))) H11)) H10)) H9 H6 -H7))) | (pr0_beta u0 v1 v2 H6 t3 t4 H7) \Rightarrow (\lambda (H8: (eq T -(THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) (THead (Flat Appl) w -v))).(\lambda (H9: (eq T (THead (Bind Abbr) v2 t4) t2)).((let H10 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow (THead (Bind Abst) u0 t3) | (TLRef _) \Rightarrow -(THead (Bind Abst) u0 t3) | (THead _ _ t5) \Rightarrow t5])) (THead (Flat -Appl) v1 (THead (Bind Abst) u0 t3)) (THead (Flat Appl) w v) H8) in ((let H11 -\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t5 _) -\Rightarrow t5])) (THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) (THead -(Flat Appl) w v) H8) in (eq_ind T w (\lambda (t5: T).((eq T (THead (Bind -Abst) u0 t3) v) \to ((eq T (THead (Bind Abbr) v2 t4) t2) \to ((pr0 t5 v2) \to -((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u -t0)))))))) (\lambda (H12: (eq T (THead (Bind Abst) u0 t3) v)).(eq_ind T -(THead (Bind Abst) u0 t3) (\lambda (_: T).((eq T (THead (Bind Abbr) v2 t4) -t2) \to ((pr0 w v2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w -(THead (Bind Abst) u t0))))))) (\lambda (H13: (eq T (THead (Bind Abbr) v2 t4) -t2)).(eq_ind T (THead (Bind Abbr) v2 t4) (\lambda (t5: T).((pr0 w v2) \to -((pr0 t3 t4) \to (ty3 g c2 t5 (THead (Flat Appl) w (THead (Bind Abst) u -t0)))))) (\lambda (H14: (pr0 w v2)).(\lambda (H15: (pr0 t3 t4)).(let H16 \def -(eq_ind_r T v (\lambda (t5: T).(\forall (c3: C).((wcpr0 c c3) \to (\forall -(t6: T).((pr0 t5 t6) \to (ty3 g c3 t6 (THead (Bind Abst) u t0))))))) H3 -(THead (Bind Abst) u0 t3) H12) in (let H17 \def (eq_ind_r T v (\lambda (t5: -T).(ty3 g c t5 (THead (Bind Abst) u t0))) H2 (THead (Bind Abst) u0 t3) H12) -in (ex_ind T (\lambda (t5: T).(ty3 g c2 (THead (Bind Abst) u t0) t5)) (ty3 g -c2 (THead (Bind Abbr) v2 t4) (THead (Flat Appl) w (THead (Bind Abst) u t0))) -(\lambda (x: T).(\lambda (H18: (ty3 g c2 (THead (Bind Abst) u t0) -x)).(ex3_2_ind T T (\lambda (t5: T).(\lambda (_: T).(pc3 c2 (THead (Bind -Abst) u t5) x))) (\lambda (_: T).(\lambda (t6: T).(ty3 g c2 u t6))) (\lambda -(t5: T).(\lambda (_: T).(ty3 g (CHead c2 (Bind Abst) u) t0 t5))) (ty3 g c2 -(THead (Bind Abbr) v2 t4) (THead (Flat Appl) w (THead (Bind Abst) u t0))) -(\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 c2 (THead (Bind Abst) u -x0) x)).(\lambda (H20: (ty3 g c2 u x1)).(\lambda (H21: (ty3 g (CHead c2 (Bind -Abst) u) t0 x0)).(ex3_2_ind T T (\lambda (t5: T).(\lambda (_: T).(pc3 c2 -(THead (Bind Abst) u0 t5) (THead (Bind Abst) u t0)))) (\lambda (_: -T).(\lambda (t6: T).(ty3 g c2 u0 t6))) (\lambda (t5: T).(\lambda (_: T).(ty3 -g (CHead c2 (Bind Abst) u0) t4 t5))) (ty3 g c2 (THead (Bind Abbr) v2 t4) -(THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x2: T).(\lambda -(x3: T).(\lambda (H22: (pc3 c2 (THead (Bind Abst) u0 x2) (THead (Bind Abst) u -t0))).(\lambda (H23: (ty3 g c2 u0 x3)).(\lambda (H24: (ty3 g (CHead c2 (Bind -Abst) u0) t4 x2)).(land_ind (pc3 c2 u0 u) (\forall (b: B).(\forall (u1: -T).(pc3 (CHead c2 (Bind b) u1) x2 t0))) (ty3 g c2 (THead (Bind Abbr) v2 t4) -(THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (H25: (pc3 c2 u0 -u)).(\lambda (H26: ((\forall (b: B).(\forall (u1: T).(pc3 (CHead c2 (Bind b) -u1) x2 t0))))).(ty3_conv g c2 (THead (Flat Appl) w (THead (Bind Abst) u t0)) -(THead (Flat Appl) w (THead (Bind Abst) u x0)) (ty3_appl g c2 w u (H1 c2 H4 w -(pr0_refl w)) (THead (Bind Abst) u t0) x0 (ty3_bind g c2 u x1 H20 Abst t0 x0 -H21)) (THead (Bind Abbr) v2 t4) (THead (Bind Abbr) v2 x2) (ty3_bind g c2 v2 u -(H1 c2 H4 v2 H14) Abbr t4 x2 (csubt_ty3_ld g c2 v2 u0 (ty3_conv g c2 u0 x3 -H23 v2 u (H1 c2 H4 v2 H14) (pc3_s c2 u u0 H25)) t4 x2 H24)) (pc3_t (THead -(Bind Abbr) v2 t0) c2 (THead (Bind Abbr) v2 x2) (pc3_head_2 c2 v2 x2 t0 (Bind -Abbr) (H26 Abbr v2)) (THead (Flat Appl) w (THead (Bind Abst) u t0)) -(pc3_pr2_x c2 (THead (Bind Abbr) v2 t0) (THead (Flat Appl) w (THead (Bind -Abst) u t0)) (pr2_free c2 (THead (Flat Appl) w (THead (Bind Abst) u t0)) -(THead (Bind Abbr) v2 t0) (pr0_beta u w v2 H14 t0 t0 (pr0_refl t0)))))))) -(pc3_gen_abst c2 u0 u x2 t0 H22))))))) (ty3_gen_bind g Abst c2 u0 t4 (THead -(Bind Abst) u t0) (H16 c2 H4 (THead (Bind Abst) u0 t4) (pr0_comp u0 u0 -(pr0_refl u0) t3 t4 H15 (Bind Abst)))))))))) (ty3_gen_bind g Abst c2 u t0 x -H18)))) (ty3_correct g c2 (THead (Bind Abst) u0 t3) (THead (Bind Abst) u t0) -(H16 c2 H4 (THead (Bind Abst) u0 t3) (pr0_refl (THead (Bind Abst) u0 -t3))))))))) t2 H13)) v H12)) v1 (sym_eq T v1 w H11))) H10)) H9 H6 H7))) | -(pr0_upsilon b H6 v1 v2 H7 u1 u2 H8 t3 t4 H9) \Rightarrow (\lambda (H10: (eq -T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) w -v))).(\lambda (H11: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O -v2) t4)) t2)).((let H12 \def (f_equal T T (\lambda (e: T).(match e in T -return (\lambda (_: T).T) with [(TSort _) \Rightarrow (THead (Bind b) u1 t3) -| (TLRef _) \Rightarrow (THead (Bind b) u1 t3) | (THead _ _ t5) \Rightarrow -t5])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) w v) -H10) in ((let H13 \def (f_equal T T (\lambda (e: T).(match e in T return -(\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 -| (THead _ t5 _) \Rightarrow t5])) (THead (Flat Appl) v1 (THead (Bind b) u1 -t3)) (THead (Flat Appl) w v) H10) in (eq_ind T w (\lambda (t5: T).((eq T -(THead (Bind b) u1 t3) v) \to ((eq T (THead (Bind b) u2 (THead (Flat Appl) -(lift (S O) O v2) t4)) t2) \to ((not (eq B b Abst)) \to ((pr0 t5 v2) \to -((pr0 u1 u2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead -(Bind Abst) u t0)))))))))) (\lambda (H14: (eq T (THead (Bind b) u1 t3) -v)).(eq_ind T (THead (Bind b) u1 t3) (\lambda (_: T).((eq T (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t2) \to ((not (eq B b Abst)) \to -((pr0 w v2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat -Appl) w (THead (Bind Abst) u t0))))))))) (\lambda (H15: (eq T (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t2)).(eq_ind T (THead (Bind b) -u2 (THead (Flat Appl) (lift (S O) O v2) t4)) (\lambda (t5: T).((not (eq B b -Abst)) \to ((pr0 w v2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ty3 g c2 t5 -(THead (Flat Appl) w (THead (Bind Abst) u t0)))))))) (\lambda (H16: (not (eq -B b Abst))).(\lambda (H17: (pr0 w v2)).(\lambda (H18: (pr0 u1 u2)).(\lambda -(H19: (pr0 t3 t4)).(let H20 \def (eq_ind_r T v (\lambda (t5: T).(\forall (c3: -C).((wcpr0 c c3) \to (\forall (t6: T).((pr0 t5 t6) \to (ty3 g c3 t6 (THead -(Bind Abst) u t0))))))) H3 (THead (Bind b) u1 t3) H14) in (let H21 \def -(eq_ind_r T v (\lambda (t5: T).(ty3 g c t5 (THead (Bind Abst) u t0))) H2 -(THead (Bind b) u1 t3) H14) in (ex_ind T (\lambda (t5: T).(ty3 g c2 (THead -(Bind Abst) u t0) t5)) (ty3 g c2 (THead (Bind b) u2 (THead (Flat Appl) (lift -(S O) O v2) t4)) (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x: -T).(\lambda (H22: (ty3 g c2 (THead (Bind Abst) u t0) x)).(let H23 \def H22 in -(ex3_2_ind T T (\lambda (t5: T).(\lambda (_: T).(pc3 c2 (THead (Bind Abst) u -t5) x))) (\lambda (_: T).(\lambda (t6: T).(ty3 g c2 u t6))) (\lambda (t5: -T).(\lambda (_: T).(ty3 g (CHead c2 (Bind Abst) u) t0 t5))) (ty3 g c2 (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) (THead (Flat Appl) w -(THead (Bind Abst) u t0))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: -(pc3 c2 (THead (Bind Abst) u x0) x)).(\lambda (H25: (ty3 g c2 u x1)).(\lambda -(H26: (ty3 g (CHead c2 (Bind Abst) u) t0 x0)).(ex3_2_ind T T (\lambda (t5: -T).(\lambda (_: T).(pc3 c2 (THead (Bind b) u2 t5) (THead (Bind Abst) u t0)))) -(\lambda (_: T).(\lambda (t6: T).(ty3 g c2 u2 t6))) (\lambda (t5: T).(\lambda -(_: T).(ty3 g (CHead c2 (Bind b) u2) t4 t5))) (ty3 g c2 (THead (Bind b) u2 -(THead (Flat Appl) (lift (S O) O v2) t4)) (THead (Flat Appl) w (THead (Bind -Abst) u t0))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H27: (pc3 c2 (THead -(Bind b) u2 x2) (THead (Bind Abst) u t0))).(\lambda (H28: (ty3 g c2 u2 -x3)).(\lambda (H29: (ty3 g (CHead c2 (Bind b) u2) t4 x2)).(let H30 \def -(eq_ind T (lift (S O) O (THead (Bind Abst) u t0)) (\lambda (t5: T).(pc3 -(CHead c2 (Bind b) u2) x2 t5)) (pc3_gen_not_abst b H16 c2 x2 t0 u2 u H27) -(THead (Bind Abst) (lift (S O) O u) (lift (S O) (S O) t0)) (lift_bind Abst u -t0 (S O) O)) in (let H31 \def (eq_ind T (lift (S O) O (THead (Bind Abst) u -t0)) (\lambda (t5: T).(ty3 g (CHead c2 (Bind b) u2) t5 (lift (S O) O x))) -(ty3_lift g c2 (THead (Bind Abst) u t0) x H22 (CHead c2 (Bind b) u2) O (S O) -(drop_drop (Bind b) O c2 c2 (drop_refl c2) u2)) (THead (Bind Abst) (lift (S -O) O u) (lift (S O) (S O) t0)) (lift_bind Abst u t0 (S O) O)) in (ex3_2_ind T -T (\lambda (t5: T).(\lambda (_: T).(pc3 (CHead c2 (Bind b) u2) (THead (Bind -Abst) (lift (S O) O u) t5) (lift (S O) O x)))) (\lambda (_: T).(\lambda (t6: -T).(ty3 g (CHead c2 (Bind b) u2) (lift (S O) O u) t6))) (\lambda (t5: -T).(\lambda (_: T).(ty3 g (CHead (CHead c2 (Bind b) u2) (Bind Abst) (lift (S -O) O u)) (lift (S O) (S O) t0) t5))) (ty3 g c2 (THead (Bind b) u2 (THead -(Flat Appl) (lift (S O) O v2) t4)) (THead (Flat Appl) w (THead (Bind Abst) u -t0))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (pc3 (CHead c2 (Bind b) -u2) (THead (Bind Abst) (lift (S O) O u) x4) (lift (S O) O x))).(\lambda (H33: -(ty3 g (CHead c2 (Bind b) u2) (lift (S O) O u) x5)).(\lambda (H34: (ty3 g -(CHead (CHead c2 (Bind b) u2) (Bind Abst) (lift (S O) O u)) (lift (S O) (S O) -t0) x4)).(ty3_conv g c2 (THead (Flat Appl) w (THead (Bind Abst) u t0)) (THead -(Flat Appl) w (THead (Bind Abst) u x0)) (ty3_appl g c2 w u (H1 c2 H4 w -(pr0_refl w)) (THead (Bind Abst) u t0) x0 (ty3_bind g c2 u x1 H25 Abst t0 x0 -H26)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) (THead -(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (THead (Bind Abst) (lift (S -O) O u) (lift (S O) (S O) t0)))) (ty3_bind g c2 u2 x3 H28 b (THead (Flat -Appl) (lift (S O) O v2) t4) (THead (Flat Appl) (lift (S O) O v2) (THead (Bind -Abst) (lift (S O) O u) (lift (S O) (S O) t0))) (ty3_appl g (CHead c2 (Bind b) -u2) (lift (S O) O v2) (lift (S O) O u) (ty3_lift g c2 v2 u (H1 c2 H4 v2 H17) -(CHead c2 (Bind b) u2) O (S O) (drop_drop (Bind b) O c2 c2 (drop_refl c2) -u2)) t4 (lift (S O) (S O) t0) (ty3_conv g (CHead c2 (Bind b) u2) (THead (Bind -Abst) (lift (S O) O u) (lift (S O) (S O) t0)) (THead (Bind Abst) (lift (S O) -O u) x4) (ty3_bind g (CHead c2 (Bind b) u2) (lift (S O) O u) x5 H33 Abst -(lift (S O) (S O) t0) x4 H34) t4 x2 H29 H30))) (eq_ind T (lift (S O) O (THead -(Bind Abst) u t0)) (\lambda (t5: T).(pc3 c2 (THead (Bind b) u2 (THead (Flat -Appl) (lift (S O) O v2) t5)) (THead (Flat Appl) w (THead (Bind Abst) u t0)))) -(pc3_pc1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (lift (S O) -O (THead (Bind Abst) u t0)))) (THead (Flat Appl) w (THead (Bind Abst) u t0)) -(pc1_pr0_u2 (THead (Flat Appl) v2 (THead (Bind b) u2 (lift (S O) O (THead -(Bind Abst) u t0)))) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) -(lift (S O) O (THead (Bind Abst) u t0)))) (pr0_upsilon b H16 v2 v2 (pr0_refl -v2) u2 u2 (pr0_refl u2) (lift (S O) O (THead (Bind Abst) u t0)) (lift (S O) O -(THead (Bind Abst) u t0)) (pr0_refl (lift (S O) O (THead (Bind Abst) u t0)))) -(THead (Flat Appl) w (THead (Bind Abst) u t0)) (pc1_head v2 w (pc1_pr0_x v2 w -H17) (THead (Bind b) u2 (lift (S O) O (THead (Bind Abst) u t0))) (THead (Bind -Abst) u t0) (pc1_pr0_r (THead (Bind b) u2 (lift (S O) O (THead (Bind Abst) u -t0))) (THead (Bind Abst) u t0) (pr0_zeta b H16 (THead (Bind Abst) u t0) -(THead (Bind Abst) u t0) (pr0_refl (THead (Bind Abst) u t0)) u2)) (Flat -Appl))) c2) (THead (Bind Abst) (lift (S O) O u) (lift (S O) (S O) t0)) -(lift_bind Abst u t0 (S O) O)))))))) (ty3_gen_bind g Abst (CHead c2 (Bind b) -u2) (lift (S O) O u) (lift (S O) (S O) t0) (lift (S O) O x) H31))))))))) -(ty3_gen_bind g b c2 u2 t4 (THead (Bind Abst) u t0) (H20 c2 H4 (THead (Bind -b) u2 t4) (pr0_comp u1 u2 H18 t3 t4 H19 (Bind b)))))))))) (ty3_gen_bind g -Abst c2 u t0 x H23))))) (ty3_correct g c2 (THead (Bind b) u2 t4) (THead (Bind -Abst) u t0) (H20 c2 H4 (THead (Bind b) u2 t4) (pr0_comp u1 u2 H18 t3 t4 H19 -(Bind b))))))))))) t2 H15)) v H14)) v1 (sym_eq T v1 w H13))) H12)) H11 H6 H7 -H8 H9))) | (pr0_delta u1 u2 H6 t3 t4 H7 w0 H8) \Rightarrow (\lambda (H9: (eq -T (THead (Bind Abbr) u1 t3) (THead (Flat Appl) w v))).(\lambda (H10: (eq T -(THead (Bind Abbr) u2 w0) t2)).((let H11 \def (eq_ind T (THead (Bind Abbr) u1 -t3) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort -_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) -\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) w v) -H9) in (False_ind ((eq T (THead (Bind Abbr) u2 w0) t2) \to ((pr0 u1 u2) \to -((pr0 t3 t4) \to ((subst0 O u2 t4 w0) \to (ty3 g c2 t2 (THead (Flat Appl) w -(THead (Bind Abst) u t0))))))) H11)) H10 H6 H7 H8))) | (pr0_zeta b H6 t3 t4 -H7 u0) \Rightarrow (\lambda (H8: (eq T (THead (Bind b) u0 (lift (S O) O t3)) -(THead (Flat Appl) w v))).(\lambda (H9: (eq T t4 t2)).((let H10 \def (eq_ind -T (THead (Bind b) u0 (lift (S O) O t3)) (\lambda (e: T).(match e in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow -False])])) I (THead (Flat Appl) w v) H8) in (False_ind ((eq T t4 t2) \to -((not (eq B b Abst)) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w -(THead (Bind Abst) u t0)))))) H10)) H9 H6 H7))) | (pr0_tau t3 t4 H6 u0) -\Rightarrow (\lambda (H7: (eq T (THead (Flat Cast) u0 t3) (THead (Flat Appl) -w v))).(\lambda (H8: (eq T t4 t2)).((let H9 \def (eq_ind T (THead (Flat Cast) -u0 t3) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) -\Rightarrow False | (Flat f) \Rightarrow (match f in F return (\lambda (_: -F).Prop) with [Appl \Rightarrow False | Cast \Rightarrow True])])])) I (THead -(Flat Appl) w v) H7) in (False_ind ((eq T t4 t2) \to ((pr0 t3 t4) \to (ty3 g -c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u t0))))) H9)) H8 H6)))]) in -(H6 (refl_equal T (THead (Flat Appl) w v)) (refl_equal T t2)))))))))))))))) -(\lambda (c: C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (ty3 g c t2 -t3)).(\lambda (H1: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t4: T).((pr0 -t2 t4) \to (ty3 g c2 t4 t3))))))).(\lambda (t0: T).(\lambda (_: (ty3 g c t3 -t0)).(\lambda (H3: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t4: T).((pr0 -t3 t4) \to (ty3 g c2 t4 t0))))))).(\lambda (c2: C).(\lambda (H4: (wcpr0 c -c2)).(\lambda (t4: T).(\lambda (H5: (pr0 (THead (Flat Cast) t3 t2) t4)).(let -H6 \def (match H5 in pr0 return (\lambda (t5: T).(\lambda (t6: T).(\lambda -(_: (pr0 t5 t6)).((eq T t5 (THead (Flat Cast) t3 t2)) \to ((eq T t6 t4) \to -(ty3 g c2 t4 (THead (Flat Cast) t0 t3))))))) with [(pr0_refl t5) \Rightarrow -(\lambda (H6: (eq T t5 (THead (Flat Cast) t3 t2))).(\lambda (H7: (eq T t5 -t4)).(eq_ind T (THead (Flat Cast) t3 t2) (\lambda (t6: T).((eq T t6 t4) \to -(ty3 g c2 t4 (THead (Flat Cast) t0 t3)))) (\lambda (H8: (eq T (THead (Flat -Cast) t3 t2) t4)).(eq_ind T (THead (Flat Cast) t3 t2) (\lambda (t6: T).(ty3 g -c2 t6 (THead (Flat Cast) t0 t3))) (ty3_cast g c2 t2 t3 (H1 c2 H4 t2 (pr0_refl -t2)) t0 (H3 c2 H4 t3 (pr0_refl t3))) t4 H8)) t5 (sym_eq T t5 (THead (Flat -Cast) t3 t2) H6) H7))) | (pr0_comp u1 u2 H6 t5 t6 H7 k) \Rightarrow (\lambda -(H8: (eq T (THead k u1 t5) (THead (Flat Cast) t3 t2))).(\lambda (H9: (eq T -(THead k u2 t6) t4)).((let H10 \def (f_equal T T (\lambda (e: T).(match e in -T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t5 | (TLRef _) -\Rightarrow t5 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t5) (THead -(Flat Cast) t3 t2) H8) in ((let H11 \def (f_equal T T (\lambda (e: T).(match -e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) -\Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t5) (THead -(Flat Cast) t3 t2) H8) in ((let H12 \def (f_equal T K (\lambda (e: T).(match -e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) -\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t5) (THead (Flat -Cast) t3 t2) H8) in (eq_ind K (Flat Cast) (\lambda (k0: K).((eq T u1 t3) \to -((eq T t5 t2) \to ((eq T (THead k0 u2 t6) t4) \to ((pr0 u1 u2) \to ((pr0 t5 -t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3)))))))) (\lambda (H13: (eq T u1 -t3)).(eq_ind T t3 (\lambda (t7: T).((eq T t5 t2) \to ((eq T (THead (Flat -Cast) u2 t6) t4) \to ((pr0 t7 u2) \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead -(Flat Cast) t0 t3))))))) (\lambda (H14: (eq T t5 t2)).(eq_ind T t2 (\lambda -(t7: T).((eq T (THead (Flat Cast) u2 t6) t4) \to ((pr0 t3 u2) \to ((pr0 t7 -t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3)))))) (\lambda (H15: (eq T -(THead (Flat Cast) u2 t6) t4)).(eq_ind T (THead (Flat Cast) u2 t6) (\lambda -(t7: T).((pr0 t3 u2) \to ((pr0 t2 t6) \to (ty3 g c2 t7 (THead (Flat Cast) t0 -t3))))) (\lambda (H16: (pr0 t3 u2)).(\lambda (H17: (pr0 t2 t6)).(ex_ind T -(\lambda (t7: T).(ty3 g c2 t0 t7)) (ty3 g c2 (THead (Flat Cast) u2 t6) (THead -(Flat Cast) t0 t3)) (\lambda (x: T).(\lambda (H18: (ty3 g c2 t0 x)).(ty3_conv -g c2 (THead (Flat Cast) t0 t3) (THead (Flat Cast) x t0) (ty3_cast g c2 t3 t0 -(H3 c2 H4 t3 (pr0_refl t3)) x H18) (THead (Flat Cast) u2 t6) (THead (Flat -Cast) t0 u2) (ty3_cast g c2 t6 u2 (ty3_conv g c2 u2 t0 (H3 c2 H4 u2 H16) t6 -t3 (H1 c2 H4 t6 H17) (pc3_pr2_r c2 t3 u2 (pr2_free c2 t3 u2 H16))) t0 (H3 c2 -H4 u2 H16)) (pc3_s c2 (THead (Flat Cast) t0 u2) (THead (Flat Cast) t0 t3) -(pc3_pr2_r c2 (THead (Flat Cast) t0 t3) (THead (Flat Cast) t0 u2) -(pr2_thin_dx c2 t3 u2 (pr2_free c2 t3 u2 H16) t0 Cast)))))) (ty3_correct g c2 -t3 t0 (H3 c2 H4 t3 (pr0_refl t3)))))) t4 H15)) t5 (sym_eq T t5 t2 H14))) u1 -(sym_eq T u1 t3 H13))) k (sym_eq K k (Flat Cast) H12))) H11)) H10)) H9 H6 -H7))) | (pr0_beta u v1 v2 H6 t5 t6 H7) \Rightarrow (\lambda (H8: (eq T (THead -(Flat Appl) v1 (THead (Bind Abst) u t5)) (THead (Flat Cast) t3 t2))).(\lambda -(H9: (eq T (THead (Bind Abbr) v2 t6) t4)).((let H10 \def (eq_ind T (THead -(Flat Appl) v1 (THead (Bind Abst) u t5)) (\lambda (e: T).(match e in T return -(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f -in F return (\lambda (_: F).Prop) with [Appl \Rightarrow True | Cast -\Rightarrow False])])])) I (THead (Flat Cast) t3 t2) H8) in (False_ind ((eq T -(THead (Bind Abbr) v2 t6) t4) \to ((pr0 v1 v2) \to ((pr0 t5 t6) \to (ty3 g c2 -t4 (THead (Flat Cast) t0 t3))))) H10)) H9 H6 H7))) | (pr0_upsilon b H6 v1 v2 -H7 u1 u2 H8 t5 t6 H9) \Rightarrow (\lambda (H10: (eq T (THead (Flat Appl) v1 -(THead (Bind b) u1 t5)) (THead (Flat Cast) t3 t2))).(\lambda (H11: (eq T -(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t6)) t4)).((let H12 -\def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t5)) (\lambda (e: -T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K -return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) -\Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl \Rightarrow -True | Cast \Rightarrow False])])])) I (THead (Flat Cast) t3 t2) H10) in -(False_ind ((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) -t6)) t4) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 -t5 t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3))))))) H12)) H11 H6 H7 H8 -H9))) | (pr0_delta u1 u2 H6 t5 t6 H7 w H8) \Rightarrow (\lambda (H9: (eq T -(THead (Bind Abbr) u1 t5) (THead (Flat Cast) t3 t2))).(\lambda (H10: (eq T -(THead (Bind Abbr) u2 w) t4)).((let H11 \def (eq_ind T (THead (Bind Abbr) u1 -t5) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort -_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) -\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Cast) t3 -t2) H9) in (False_ind ((eq T (THead (Bind Abbr) u2 w) t4) \to ((pr0 u1 u2) -\to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ty3 g c2 t4 (THead (Flat Cast) -t0 t3)))))) H11)) H10 H6 H7 H8))) | (pr0_zeta b H6 t5 t6 H7 u) \Rightarrow -(\lambda (H8: (eq T (THead (Bind b) u (lift (S O) O t5)) (THead (Flat Cast) -t3 t2))).(\lambda (H9: (eq T t6 t4)).((let H10 \def (eq_ind T (THead (Bind b) -u (lift (S O) O t5)) (\lambda (e: T).(match e in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with -[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat -Cast) t3 t2) H8) in (False_ind ((eq T t6 t4) \to ((not (eq B b Abst)) \to -((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3))))) H10)) H9 H6 H7))) -| (pr0_tau t5 t6 H6 u) \Rightarrow (\lambda (H7: (eq T (THead (Flat Cast) u -t5) (THead (Flat Cast) t3 t2))).(\lambda (H8: (eq T t6 t4)).((let H9 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow t5 | (TLRef _) \Rightarrow t5 | (THead _ _ t7) -\Rightarrow t7])) (THead (Flat Cast) u t5) (THead (Flat Cast) t3 t2) H7) in -((let H10 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: -T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t7 -_) \Rightarrow t7])) (THead (Flat Cast) u t5) (THead (Flat Cast) t3 t2) H7) -in (eq_ind T t3 (\lambda (_: T).((eq T t5 t2) \to ((eq T t6 t4) \to ((pr0 t5 -t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3)))))) (\lambda (H11: (eq T t5 -t2)).(eq_ind T t2 (\lambda (t7: T).((eq T t6 t4) \to ((pr0 t7 t6) \to (ty3 g -c2 t4 (THead (Flat Cast) t0 t3))))) (\lambda (H12: (eq T t6 t4)).(eq_ind T t4 -(\lambda (t7: T).((pr0 t2 t7) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3)))) -(\lambda (H13: (pr0 t2 t4)).(ex_ind T (\lambda (t7: T).(ty3 g c2 t0 t7)) (ty3 -g c2 t4 (THead (Flat Cast) t0 t3)) (\lambda (x: T).(\lambda (H14: (ty3 g c2 -t0 x)).(ty3_conv g c2 (THead (Flat Cast) t0 t3) (THead (Flat Cast) x t0) -(ty3_cast g c2 t3 t0 (H3 c2 H4 t3 (pr0_refl t3)) x H14) t4 t3 (H1 c2 H4 t4 -H13) (pc3_pr2_x c2 t3 (THead (Flat Cast) t0 t3) (pr2_free c2 (THead (Flat -Cast) t0 t3) t3 (pr0_tau t3 t3 (pr0_refl t3) t0)))))) (ty3_correct g c2 t3 t0 -(H3 c2 H4 t3 (pr0_refl t3))))) t6 (sym_eq T t6 t4 H12))) t5 (sym_eq T t5 t2 -H11))) u (sym_eq T u t3 H10))) H9)) H8 H6)))]) in (H6 (refl_equal T (THead -(Flat Cast) t3 t2)) (refl_equal T t4))))))))))))))) c1 t1 t H))))). -(* COMMENTS -Initial nodes: 14710 -END *) - -theorem ty3_sred_pr0: - \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (g: G).(\forall -(c: C).(\forall (t: T).((ty3 g c t1 t) \to (ty3 g c t2 t))))))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(\lambda (g: -G).(\lambda (c: C).(\lambda (t: T).(\lambda (H0: (ty3 g c t1 -t)).(ty3_sred_wcpr0_pr0 g c t1 t H0 c (wcpr0_refl c) t2 H))))))). -(* COMMENTS -Initial nodes: 47 -END *) - -theorem ty3_sred_pr1: - \forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall (g: G).(\forall -(c: C).(\forall (t: T).((ty3 g c t1 t) \to (ty3 g c t2 t))))))) -\def - \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr1 t1 t2)).(pr1_ind (\lambda -(t: T).(\lambda (t0: T).(\forall (g: G).(\forall (c: C).(\forall (t3: -T).((ty3 g c t t3) \to (ty3 g c t0 t3))))))) (\lambda (t: T).(\lambda (g: -G).(\lambda (c: C).(\lambda (t0: T).(\lambda (H0: (ty3 g c t t0)).H0))))) -(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t4 t3)).(\lambda (t5: -T).(\lambda (_: (pr1 t3 t5)).(\lambda (H2: ((\forall (g: G).(\forall (c: -C).(\forall (t: T).((ty3 g c t3 t) \to (ty3 g c t5 t))))))).(\lambda (g: -G).(\lambda (c: C).(\lambda (t: T).(\lambda (H3: (ty3 g c t4 t)).(H2 g c t -(ty3_sred_pr0 t4 t3 H0 g c t H3)))))))))))) t1 t2 H))). -(* COMMENTS -Initial nodes: 151 -END *) - -theorem ty3_sred_pr2: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall -(g: G).(\forall (t: T).((ty3 g c t1 t) \to (ty3 g c t2 t))))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 -t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (g: -G).(\forall (t3: T).((ty3 g c0 t t3) \to (ty3 g c0 t0 t3))))))) (\lambda (c0: -C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(\lambda (g: -G).(\lambda (t: T).(\lambda (H1: (ty3 g c0 t3 t)).(ty3_sred_wcpr0_pr0 g c0 t3 -t H1 c0 (wcpr0_refl c0) t4 H0)))))))) (\lambda (c0: C).(\lambda (d: -C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind -Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 -t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (g: -G).(\lambda (t0: T).(\lambda (H3: (ty3 g c0 t3 t0)).(ty3_subst0 g c0 t4 t0 -(ty3_sred_wcpr0_pr0 g c0 t3 t0 H3 c0 (wcpr0_refl c0) t4 H1) d u i H0 t -H2)))))))))))))) c t1 t2 H)))). -(* COMMENTS -Initial nodes: 205 -END *) - -theorem ty3_sred_pr3: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall -(g: G).(\forall (t: T).((ty3 g c t1 t) \to (ty3 g c t2 t))))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 -t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (g: G).(\forall -(t3: T).((ty3 g c t t3) \to (ty3 g c t0 t3)))))) (\lambda (t: T).(\lambda (g: -G).(\lambda (t0: T).(\lambda (H0: (ty3 g c t t0)).H0)))) (\lambda (t3: -T).(\lambda (t4: T).(\lambda (H0: (pr2 c t4 t3)).(\lambda (t5: T).(\lambda -(_: (pr3 c t3 t5)).(\lambda (H2: ((\forall (g: G).(\forall (t: T).((ty3 g c -t3 t) \to (ty3 g c t5 t)))))).(\lambda (g: G).(\lambda (t: T).(\lambda (H3: -(ty3 g c t4 t)).(H2 g t (ty3_sred_pr2 c t4 t3 H0 g t H3))))))))))) t1 t2 -H)))). -(* COMMENTS -Initial nodes: 151 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/pr3_props.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/pr3_props.ma deleted file mode 100644 index b5bb1fb69..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/pr3_props.ma +++ /dev/null @@ -1,513 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/ty3/pr3.ma". - -theorem ty3_cred_pr2: - \forall (g: G).(\forall (c: C).(\forall (v1: T).(\forall (v2: T).((pr2 c v1 -v2) \to (\forall (b: B).(\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c -(Bind b) v1) t1 t2) \to (ty3 g (CHead c (Bind b) v2) t1 t2))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda -(H: (pr2 c v1 v2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: -T).(\forall (b: B).(\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c0 (Bind -b) t) t1 t2) \to (ty3 g (CHead c0 (Bind b) t0) t1 t2)))))))) (\lambda (c0: -C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr0 t1 t2)).(\lambda (b: -B).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H1: (ty3 g (CHead c0 (Bind b) -t1) t0 t3)).(ty3_sred_wcpr0_pr0 g (CHead c0 (Bind b) t1) t0 t3 H1 (CHead c0 -(Bind b) t2) (wcpr0_comp c0 c0 (wcpr0_refl c0) t1 t2 H0 (Bind b)) t0 -(pr0_refl t0)))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: -T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr) -u))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H1: (pr0 t1 t2)).(\lambda -(t: T).(\lambda (H2: (subst0 i u t2 t)).(\lambda (b: B).(\lambda (t0: -T).(\lambda (t3: T).(\lambda (H3: (ty3 g (CHead c0 (Bind b) t1) t0 -t3)).(ty3_csubst0 g (CHead c0 (Bind b) t2) t0 t3 (ty3_sred_wcpr0_pr0 g (CHead -c0 (Bind b) t1) t0 t3 H3 (CHead c0 (Bind b) t2) (wcpr0_comp c0 c0 (wcpr0_refl -c0) t1 t2 H1 (Bind b)) t0 (pr0_refl t0)) d u (S i) (getl_clear_bind b (CHead -c0 (Bind b) t2) c0 t2 (clear_bind b c0 t2) (CHead d (Bind Abbr) u) i H0) -(CHead c0 (Bind b) t) (csubst0_snd_bind b i u t2 t H2 c0)))))))))))))))) c v1 -v2 H))))). -(* COMMENTS -Initial nodes: 383 -END *) - -theorem ty3_cred_pr3: - \forall (g: G).(\forall (c: C).(\forall (v1: T).(\forall (v2: T).((pr3 c v1 -v2) \to (\forall (b: B).(\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c -(Bind b) v1) t1 t2) \to (ty3 g (CHead c (Bind b) v2) t1 t2))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda -(H: (pr3 c v1 v2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (b: -B).(\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c (Bind b) t) t1 t2) \to -(ty3 g (CHead c (Bind b) t0) t1 t2))))))) (\lambda (t: T).(\lambda (b: -B).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (ty3 g (CHead c (Bind b) -t) t1 t2)).H0))))) (\lambda (t2: T).(\lambda (t1: T).(\lambda (H0: (pr2 c t1 -t2)).(\lambda (t3: T).(\lambda (_: (pr3 c t2 t3)).(\lambda (H2: ((\forall (b: -B).(\forall (t4: T).(\forall (t5: T).((ty3 g (CHead c (Bind b) t2) t4 t5) \to -(ty3 g (CHead c (Bind b) t3) t4 t5))))))).(\lambda (b: B).(\lambda (t0: -T).(\lambda (t4: T).(\lambda (H3: (ty3 g (CHead c (Bind b) t1) t0 t4)).(H2 b -t0 t4 (ty3_cred_pr2 g c t1 t2 H0 b t0 t4 H3)))))))))))) v1 v2 H))))). -(* COMMENTS -Initial nodes: 215 -END *) - -theorem ty3_gen_lift: - \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (x: T).(\forall (h: -nat).(\forall (d: nat).((ty3 g c (lift h d t1) x) \to (\forall (e: C).((drop -h d c e) \to (ex2 T (\lambda (t2: T).(pc3 c (lift h d t2) x)) (\lambda (t2: -T).(ty3 g e t1 t2))))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (x: T).(\lambda (h: -nat).(\lambda (d: nat).(\lambda (H: (ty3 g c (lift h d t1) x)).(insert_eq T -(lift h d t1) (\lambda (t: T).(ty3 g c t x)) (\lambda (_: T).(\forall (e: -C).((drop h d c e) \to (ex2 T (\lambda (t2: T).(pc3 c (lift h d t2) x)) -(\lambda (t2: T).(ty3 g e t1 t2)))))) (\lambda (y: T).(\lambda (H0: (ty3 g c -y x)).(unintro nat d (\lambda (n: nat).((eq T y (lift h n t1)) \to (\forall -(e: C).((drop h n c e) \to (ex2 T (\lambda (t2: T).(pc3 c (lift h n t2) x)) -(\lambda (t2: T).(ty3 g e t1 t2))))))) (unintro T t1 (\lambda (t: T).(\forall -(x0: nat).((eq T y (lift h x0 t)) \to (\forall (e: C).((drop h x0 c e) \to -(ex2 T (\lambda (t2: T).(pc3 c (lift h x0 t2) x)) (\lambda (t2: T).(ty3 g e t -t2)))))))) (ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: -T).(\forall (x0: T).(\forall (x1: nat).((eq T t (lift h x1 x0)) \to (\forall -(e: C).((drop h x1 c0 e) \to (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) -t0)) (\lambda (t2: T).(ty3 g e x0 t2))))))))))) (\lambda (c0: C).(\lambda -(t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda (_: ((\forall -(x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to (\forall (e: -C).((drop h x1 c0 e) \to (ex2 T (\lambda (t3: T).(pc3 c0 (lift h x1 t3) t)) -(\lambda (t3: T).(ty3 g e x0 t3)))))))))).(\lambda (u: T).(\lambda (t3: -T).(\lambda (H3: (ty3 g c0 u t3)).(\lambda (H4: ((\forall (x0: T).(\forall -(x1: nat).((eq T u (lift h x1 x0)) \to (\forall (e: C).((drop h x1 c0 e) \to -(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) t3)) (\lambda (t4: T).(ty3 g e -x0 t4)))))))))).(\lambda (H5: (pc3 c0 t3 t2)).(\lambda (x0: T).(\lambda (x1: -nat).(\lambda (H6: (eq T u (lift h x1 x0))).(\lambda (e: C).(\lambda (H7: -(drop h x1 c0 e)).(let H8 \def (eq_ind T u (\lambda (t0: T).(\forall (x2: -T).(\forall (x3: nat).((eq T t0 (lift h x3 x2)) \to (\forall (e0: C).((drop h -x3 c0 e0) \to (ex2 T (\lambda (t4: T).(pc3 c0 (lift h x3 t4) t3)) (\lambda -(t4: T).(ty3 g e0 x2 t4))))))))) H4 (lift h x1 x0) H6) in (let H9 \def -(eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 t3)) H3 (lift h x1 x0) H6) in (let -H10 \def (H8 x0 x1 (refl_equal T (lift h x1 x0)) e H7) in (ex2_ind T (\lambda -(t4: T).(pc3 c0 (lift h x1 t4) t3)) (\lambda (t4: T).(ty3 g e x0 t4)) (ex2 T -(\lambda (t4: T).(pc3 c0 (lift h x1 t4) t2)) (\lambda (t4: T).(ty3 g e x0 -t4))) (\lambda (x2: T).(\lambda (H11: (pc3 c0 (lift h x1 x2) t3)).(\lambda -(H12: (ty3 g e x0 x2)).(ex_intro2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) -t2)) (\lambda (t4: T).(ty3 g e x0 t4)) x2 (pc3_t t3 c0 (lift h x1 x2) H11 t2 -H5) H12)))) H10))))))))))))))))))) (\lambda (c0: C).(\lambda (m: -nat).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H1: (eq T (TSort m) (lift -h x1 x0))).(\lambda (e: C).(\lambda (_: (drop h x1 c0 e)).(eq_ind_r T (TSort -m) (\lambda (t: T).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (TSort -(next g m)))) (\lambda (t2: T).(ty3 g e t t2)))) (ex_intro2 T (\lambda (t2: -T).(pc3 c0 (lift h x1 t2) (TSort (next g m)))) (\lambda (t2: T).(ty3 g e -(TSort m) t2)) (TSort (next g m)) (eq_ind_r T (TSort (next g m)) (\lambda (t: -T).(pc3 c0 t (TSort (next g m)))) (pc3_refl c0 (TSort (next g m))) (lift h x1 -(TSort (next g m))) (lift_sort (next g m) h x1)) (ty3_sort g e m)) x0 -(lift_gen_sort h x1 m x0 H1))))))))) (\lambda (n: nat).(\lambda (c0: -C).(\lambda (d0: C).(\lambda (u: T).(\lambda (H1: (getl n c0 (CHead d0 (Bind -Abbr) u))).(\lambda (t: T).(\lambda (H2: (ty3 g d0 u t)).(\lambda (H3: -((\forall (x0: T).(\forall (x1: nat).((eq T u (lift h x1 x0)) \to (\forall -(e: C).((drop h x1 d0 e) \to (ex2 T (\lambda (t2: T).(pc3 d0 (lift h x1 t2) -t)) (\lambda (t2: T).(ty3 g e x0 t2)))))))))).(\lambda (x0: T).(\lambda (x1: -nat).(\lambda (H4: (eq T (TLRef n) (lift h x1 x0))).(\lambda (e: C).(\lambda -(H5: (drop h x1 c0 e)).(let H_x \def (lift_gen_lref x0 x1 h n H4) in (let H6 -\def H_x in (or_ind (land (lt n x1) (eq T x0 (TLRef n))) (land (le (plus x1 -h) n) (eq T x0 (TLRef (minus n h)))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h -x1 t2) (lift (S n) O t))) (\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (H7: -(land (lt n x1) (eq T x0 (TLRef n)))).(land_ind (lt n x1) (eq T x0 (TLRef n)) -(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t))) (\lambda -(t2: T).(ty3 g e x0 t2))) (\lambda (H8: (lt n x1)).(\lambda (H9: (eq T x0 -(TLRef n))).(eq_ind_r T (TLRef n) (\lambda (t0: T).(ex2 T (\lambda (t2: -T).(pc3 c0 (lift h x1 t2) (lift (S n) O t))) (\lambda (t2: T).(ty3 g e t0 -t2)))) (let H10 \def (eq_ind nat x1 (\lambda (n0: nat).(drop h n0 c0 e)) H5 -(S (plus n (minus x1 (S n)))) (lt_plus_minus n x1 H8)) in (ex3_2_ind T C -(\lambda (v: T).(\lambda (_: C).(eq T u (lift h (minus x1 (S n)) v)))) -(\lambda (v: T).(\lambda (e0: C).(getl n e (CHead e0 (Bind Abbr) v)))) -(\lambda (_: T).(\lambda (e0: C).(drop h (minus x1 (S n)) d0 e0))) (ex2 T -(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t))) (\lambda (t2: -T).(ty3 g e (TLRef n) t2))) (\lambda (x2: T).(\lambda (x3: C).(\lambda (H11: -(eq T u (lift h (minus x1 (S n)) x2))).(\lambda (H12: (getl n e (CHead x3 -(Bind Abbr) x2))).(\lambda (H13: (drop h (minus x1 (S n)) d0 x3)).(let H14 -\def (eq_ind T u (\lambda (t0: T).(\forall (x4: T).(\forall (x5: nat).((eq T -t0 (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 d0 e0) \to (ex2 T -(\lambda (t2: T).(pc3 d0 (lift h x5 t2) t)) (\lambda (t2: T).(ty3 g e0 x4 -t2))))))))) H3 (lift h (minus x1 (S n)) x2) H11) in (let H15 \def (eq_ind T u -(\lambda (t0: T).(ty3 g d0 t0 t)) H2 (lift h (minus x1 (S n)) x2) H11) in -(let H16 \def (H14 x2 (minus x1 (S n)) (refl_equal T (lift h (minus x1 (S n)) -x2)) x3 H13) in (ex2_ind T (\lambda (t2: T).(pc3 d0 (lift h (minus x1 (S n)) -t2) t)) (\lambda (t2: T).(ty3 g x3 x2 t2)) (ex2 T (\lambda (t2: T).(pc3 c0 -(lift h x1 t2) (lift (S n) O t))) (\lambda (t2: T).(ty3 g e (TLRef n) t2))) -(\lambda (x4: T).(\lambda (H17: (pc3 d0 (lift h (minus x1 (S n)) x4) -t)).(\lambda (H18: (ty3 g x3 x2 x4)).(eq_ind_r nat (plus (S n) (minus x1 (S -n))) (\lambda (n0: nat).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h n0 t2) (lift -(S n) O t))) (\lambda (t2: T).(ty3 g e (TLRef n) t2)))) (ex_intro2 T (\lambda -(t2: T).(pc3 c0 (lift h (plus (S n) (minus x1 (S n))) t2) (lift (S n) O t))) -(\lambda (t2: T).(ty3 g e (TLRef n) t2)) (lift (S n) O x4) (eq_ind_r T (lift -(S n) O (lift h (minus x1 (S n)) x4)) (\lambda (t0: T).(pc3 c0 t0 (lift (S n) -O t))) (pc3_lift c0 d0 (S n) O (getl_drop Abbr c0 d0 u n H1) (lift h (minus -x1 (S n)) x4) t H17) (lift h (plus (S n) (minus x1 (S n))) (lift (S n) O x4)) -(lift_d x4 h (S n) (minus x1 (S n)) O (le_O_n (minus x1 (S n))))) (ty3_abbr g -n e x3 x2 H12 x4 H18)) x1 (le_plus_minus (S n) x1 H8))))) H16))))))))) -(getl_drop_conf_lt Abbr c0 d0 u n H1 e h (minus x1 (S n)) H10))) x0 H9))) -H7)) (\lambda (H7: (land (le (plus x1 h) n) (eq T x0 (TLRef (minus n -h))))).(land_ind (le (plus x1 h) n) (eq T x0 (TLRef (minus n h))) (ex2 T -(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t))) (\lambda (t2: -T).(ty3 g e x0 t2))) (\lambda (H8: (le (plus x1 h) n)).(\lambda (H9: (eq T x0 -(TLRef (minus n h)))).(eq_ind_r T (TLRef (minus n h)) (\lambda (t0: T).(ex2 T -(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t))) (\lambda (t2: -T).(ty3 g e t0 t2)))) (ex_intro2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) -(lift (S n) O t))) (\lambda (t2: T).(ty3 g e (TLRef (minus n h)) t2)) (lift -(S (minus n h)) O t) (eq_ind_r T (lift (plus h (S (minus n h))) O t) (\lambda -(t0: T).(pc3 c0 t0 (lift (S n) O t))) (eq_ind nat (S (plus h (minus n h))) -(\lambda (n0: nat).(pc3 c0 (lift n0 O t) (lift (S n) O t))) (eq_ind nat n -(\lambda (n0: nat).(pc3 c0 (lift (S n0) O t) (lift (S n) O t))) (pc3_refl c0 -(lift (S n) O t)) (plus h (minus n h)) (le_plus_minus h n (le_trans h (plus -x1 h) n (le_plus_r x1 h) H8))) (plus h (S (minus n h))) (plus_n_Sm h (minus n -h))) (lift h x1 (lift (S (minus n h)) O t)) (lift_free t (S (minus n h)) h O -x1 (le_trans x1 (S (minus n h)) (plus O (S (minus n h))) (le_S_minus x1 h n -H8) (le_n (plus O (S (minus n h))))) (le_O_n x1))) (ty3_abbr g (minus n h) e -d0 u (getl_drop_conf_ge n (CHead d0 (Bind Abbr) u) c0 H1 e h x1 H5 H8) t H2)) -x0 H9))) H7)) H6)))))))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda -(d0: C).(\lambda (u: T).(\lambda (H1: (getl n c0 (CHead d0 (Bind Abst) -u))).(\lambda (t: T).(\lambda (H2: (ty3 g d0 u t)).(\lambda (H3: ((\forall -(x0: T).(\forall (x1: nat).((eq T u (lift h x1 x0)) \to (\forall (e: -C).((drop h x1 d0 e) \to (ex2 T (\lambda (t2: T).(pc3 d0 (lift h x1 t2) t)) -(\lambda (t2: T).(ty3 g e x0 t2)))))))))).(\lambda (x0: T).(\lambda (x1: -nat).(\lambda (H4: (eq T (TLRef n) (lift h x1 x0))).(\lambda (e: C).(\lambda -(H5: (drop h x1 c0 e)).(let H_x \def (lift_gen_lref x0 x1 h n H4) in (let H6 -\def H_x in (or_ind (land (lt n x1) (eq T x0 (TLRef n))) (land (le (plus x1 -h) n) (eq T x0 (TLRef (minus n h)))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h -x1 t2) (lift (S n) O u))) (\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (H7: -(land (lt n x1) (eq T x0 (TLRef n)))).(land_ind (lt n x1) (eq T x0 (TLRef n)) -(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O u))) (\lambda -(t2: T).(ty3 g e x0 t2))) (\lambda (H8: (lt n x1)).(\lambda (H9: (eq T x0 -(TLRef n))).(eq_ind_r T (TLRef n) (\lambda (t0: T).(ex2 T (\lambda (t2: -T).(pc3 c0 (lift h x1 t2) (lift (S n) O u))) (\lambda (t2: T).(ty3 g e t0 -t2)))) (let H10 \def (eq_ind nat x1 (\lambda (n0: nat).(drop h n0 c0 e)) H5 -(S (plus n (minus x1 (S n)))) (lt_plus_minus n x1 H8)) in (ex3_2_ind T C -(\lambda (v: T).(\lambda (_: C).(eq T u (lift h (minus x1 (S n)) v)))) -(\lambda (v: T).(\lambda (e0: C).(getl n e (CHead e0 (Bind Abst) v)))) -(\lambda (_: T).(\lambda (e0: C).(drop h (minus x1 (S n)) d0 e0))) (ex2 T -(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O u))) (\lambda (t2: -T).(ty3 g e (TLRef n) t2))) (\lambda (x2: T).(\lambda (x3: C).(\lambda (H11: -(eq T u (lift h (minus x1 (S n)) x2))).(\lambda (H12: (getl n e (CHead x3 -(Bind Abst) x2))).(\lambda (H13: (drop h (minus x1 (S n)) d0 x3)).(let H14 -\def (eq_ind T u (\lambda (t0: T).(\forall (x4: T).(\forall (x5: nat).((eq T -t0 (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 d0 e0) \to (ex2 T -(\lambda (t2: T).(pc3 d0 (lift h x5 t2) t)) (\lambda (t2: T).(ty3 g e0 x4 -t2))))))))) H3 (lift h (minus x1 (S n)) x2) H11) in (let H15 \def (eq_ind T u -(\lambda (t0: T).(ty3 g d0 t0 t)) H2 (lift h (minus x1 (S n)) x2) H11) in -(eq_ind_r T (lift h (minus x1 (S n)) x2) (\lambda (t0: T).(ex2 T (\lambda -(t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t0))) (\lambda (t2: T).(ty3 g e -(TLRef n) t2)))) (let H16 \def (H14 x2 (minus x1 (S n)) (refl_equal T (lift h -(minus x1 (S n)) x2)) x3 H13) in (ex2_ind T (\lambda (t2: T).(pc3 d0 (lift h -(minus x1 (S n)) t2) t)) (\lambda (t2: T).(ty3 g x3 x2 t2)) (ex2 T (\lambda -(t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O (lift h (minus x1 (S n)) x2)))) -(\lambda (t2: T).(ty3 g e (TLRef n) t2))) (\lambda (x4: T).(\lambda (_: (pc3 -d0 (lift h (minus x1 (S n)) x4) t)).(\lambda (H18: (ty3 g x3 x2 -x4)).(eq_ind_r nat (plus (S n) (minus x1 (S n))) (\lambda (n0: nat).(ex2 T -(\lambda (t2: T).(pc3 c0 (lift h n0 t2) (lift (S n) O (lift h (minus n0 (S -n)) x2)))) (\lambda (t2: T).(ty3 g e (TLRef n) t2)))) (ex_intro2 T (\lambda -(t2: T).(pc3 c0 (lift h (plus (S n) (minus x1 (S n))) t2) (lift (S n) O (lift -h (minus (plus (S n) (minus x1 (S n))) (S n)) x2)))) (\lambda (t2: T).(ty3 g -e (TLRef n) t2)) (lift (S n) O x2) (eq_ind_r T (lift (S n) O (lift h (minus -x1 (S n)) x2)) (\lambda (t0: T).(pc3 c0 t0 (lift (S n) O (lift h (minus (plus -(S n) (minus x1 (S n))) (S n)) x2)))) (eq_ind nat x1 (\lambda (n0: nat).(pc3 -c0 (lift (S n) O (lift h (minus x1 (S n)) x2)) (lift (S n) O (lift h (minus -n0 (S n)) x2)))) (pc3_refl c0 (lift (S n) O (lift h (minus x1 (S n)) x2))) -(plus (S n) (minus x1 (S n))) (le_plus_minus (S n) x1 H8)) (lift h (plus (S -n) (minus x1 (S n))) (lift (S n) O x2)) (lift_d x2 h (S n) (minus x1 (S n)) O -(le_O_n (minus x1 (S n))))) (ty3_abst g n e x3 x2 H12 x4 H18)) x1 -(le_plus_minus (S n) x1 H8))))) H16)) u H11)))))))) (getl_drop_conf_lt Abst -c0 d0 u n H1 e h (minus x1 (S n)) H10))) x0 H9))) H7)) (\lambda (H7: (land -(le (plus x1 h) n) (eq T x0 (TLRef (minus n h))))).(land_ind (le (plus x1 h) -n) (eq T x0 (TLRef (minus n h))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 -t2) (lift (S n) O u))) (\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (H8: (le -(plus x1 h) n)).(\lambda (H9: (eq T x0 (TLRef (minus n h)))).(eq_ind_r T -(TLRef (minus n h)) (\lambda (t0: T).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h -x1 t2) (lift (S n) O u))) (\lambda (t2: T).(ty3 g e t0 t2)))) (ex_intro2 T -(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O u))) (\lambda (t2: -T).(ty3 g e (TLRef (minus n h)) t2)) (lift (S (minus n h)) O u) (eq_ind_r T -(lift (plus h (S (minus n h))) O u) (\lambda (t0: T).(pc3 c0 t0 (lift (S n) O -u))) (eq_ind nat (S (plus h (minus n h))) (\lambda (n0: nat).(pc3 c0 (lift n0 -O u) (lift (S n) O u))) (eq_ind nat n (\lambda (n0: nat).(pc3 c0 (lift (S n0) -O u) (lift (S n) O u))) (pc3_refl c0 (lift (S n) O u)) (plus h (minus n h)) -(le_plus_minus h n (le_trans h (plus x1 h) n (le_plus_r x1 h) H8))) (plus h -(S (minus n h))) (plus_n_Sm h (minus n h))) (lift h x1 (lift (S (minus n h)) -O u)) (lift_free u (S (minus n h)) h O x1 (le_trans x1 (S (minus n h)) (plus -O (S (minus n h))) (le_S_minus x1 h n H8) (le_n (plus O (S (minus n h))))) -(le_O_n x1))) (ty3_abst g (minus n h) e d0 u (getl_drop_conf_ge n (CHead d0 -(Bind Abst) u) c0 H1 e h x1 H5 H8) t H2)) x0 H9))) H7)) H6)))))))))))))))) -(\lambda (c0: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H1: (ty3 g c0 u -t)).(\lambda (H2: ((\forall (x0: T).(\forall (x1: nat).((eq T u (lift h x1 -x0)) \to (\forall (e: C).((drop h x1 c0 e) \to (ex2 T (\lambda (t2: T).(pc3 -c0 (lift h x1 t2) t)) (\lambda (t2: T).(ty3 g e x0 t2)))))))))).(\lambda (b: -B).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H3: (ty3 g (CHead c0 (Bind b) -u) t2 t3)).(\lambda (H4: ((\forall (x0: T).(\forall (x1: nat).((eq T t2 (lift -h x1 x0)) \to (\forall (e: C).((drop h x1 (CHead c0 (Bind b) u) e) \to (ex2 T -(\lambda (t4: T).(pc3 (CHead c0 (Bind b) u) (lift h x1 t4) t3)) (\lambda (t4: -T).(ty3 g e x0 t4)))))))))).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H5: -(eq T (THead (Bind b) u t2) (lift h x1 x0))).(\lambda (e: C).(\lambda (H6: -(drop h x1 c0 e)).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 -(THead (Bind b) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u (lift h x1 -y0)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h (S x1) z)))) (ex2 T -(\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b) u t3))) (\lambda (t4: -T).(ty3 g e x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq T x0 -(THead (Bind b) x2 x3))).(\lambda (H8: (eq T u (lift h x1 x2))).(\lambda (H9: -(eq T t2 (lift h (S x1) x3))).(eq_ind_r T (THead (Bind b) x2 x3) (\lambda -(t0: T).(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b) u -t3))) (\lambda (t4: T).(ty3 g e t0 t4)))) (let H10 \def (eq_ind T t2 (\lambda -(t0: T).(\forall (x4: T).(\forall (x5: nat).((eq T t0 (lift h x5 x4)) \to -(\forall (e0: C).((drop h x5 (CHead c0 (Bind b) u) e0) \to (ex2 T (\lambda -(t4: T).(pc3 (CHead c0 (Bind b) u) (lift h x5 t4) t3)) (\lambda (t4: T).(ty3 -g e0 x4 t4))))))))) H4 (lift h (S x1) x3) H9) in (let H11 \def (eq_ind T t2 -(\lambda (t0: T).(ty3 g (CHead c0 (Bind b) u) t0 t3)) H3 (lift h (S x1) x3) -H9) in (let H12 \def (eq_ind T u (\lambda (t0: T).(ty3 g (CHead c0 (Bind b) -t0) (lift h (S x1) x3) t3)) H11 (lift h x1 x2) H8) in (let H13 \def (eq_ind T -u (\lambda (t0: T).(\forall (x4: T).(\forall (x5: nat).((eq T (lift h (S x1) -x3) (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 (CHead c0 (Bind b) t0) -e0) \to (ex2 T (\lambda (t4: T).(pc3 (CHead c0 (Bind b) t0) (lift h x5 t4) -t3)) (\lambda (t4: T).(ty3 g e0 x4 t4))))))))) H10 (lift h x1 x2) H8) in (let -H14 \def (eq_ind T u (\lambda (t0: T).(\forall (x4: T).(\forall (x5: -nat).((eq T t0 (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 c0 e0) \to -(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x5 t4) t)) (\lambda (t4: T).(ty3 g e0 -x4 t4))))))))) H2 (lift h x1 x2) H8) in (let H15 \def (eq_ind T u (\lambda -(t0: T).(ty3 g c0 t0 t)) H1 (lift h x1 x2) H8) in (eq_ind_r T (lift h x1 x2) -(\lambda (t0: T).(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind -b) t0 t3))) (\lambda (t4: T).(ty3 g e (THead (Bind b) x2 x3) t4)))) (let H16 -\def (H14 x2 x1 (refl_equal T (lift h x1 x2)) e H6) in (ex2_ind T (\lambda -(t4: T).(pc3 c0 (lift h x1 t4) t)) (\lambda (t4: T).(ty3 g e x2 t4)) (ex2 T -(\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b) (lift h x1 x2) t3))) -(\lambda (t4: T).(ty3 g e (THead (Bind b) x2 x3) t4))) (\lambda (x4: -T).(\lambda (_: (pc3 c0 (lift h x1 x4) t)).(\lambda (H18: (ty3 g e x2 -x4)).(let H19 \def (H13 x3 (S x1) (refl_equal T (lift h (S x1) x3)) (CHead e -(Bind b) x2) (drop_skip_bind h x1 c0 e H6 b x2)) in (ex2_ind T (\lambda (t4: -T).(pc3 (CHead c0 (Bind b) (lift h x1 x2)) (lift h (S x1) t4) t3)) (\lambda -(t4: T).(ty3 g (CHead e (Bind b) x2) x3 t4)) (ex2 T (\lambda (t4: T).(pc3 c0 -(lift h x1 t4) (THead (Bind b) (lift h x1 x2) t3))) (\lambda (t4: T).(ty3 g e -(THead (Bind b) x2 x3) t4))) (\lambda (x5: T).(\lambda (H20: (pc3 (CHead c0 -(Bind b) (lift h x1 x2)) (lift h (S x1) x5) t3)).(\lambda (H21: (ty3 g (CHead -e (Bind b) x2) x3 x5)).(ex_ind T (\lambda (t0: T).(ty3 g (CHead e (Bind b) -x2) x5 t0)) (ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b) -(lift h x1 x2) t3))) (\lambda (t4: T).(ty3 g e (THead (Bind b) x2 x3) t4))) -(\lambda (x6: T).(\lambda (_: (ty3 g (CHead e (Bind b) x2) x5 x6)).(ex_intro2 -T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b) (lift h x1 x2) -t3))) (\lambda (t4: T).(ty3 g e (THead (Bind b) x2 x3) t4)) (THead (Bind b) -x2 x5) (eq_ind_r T (THead (Bind b) (lift h x1 x2) (lift h (S x1) x5)) -(\lambda (t0: T).(pc3 c0 t0 (THead (Bind b) (lift h x1 x2) t3))) (pc3_head_2 -c0 (lift h x1 x2) (lift h (S x1) x5) t3 (Bind b) H20) (lift h x1 (THead (Bind -b) x2 x5)) (lift_bind b x2 x5 h x1)) (ty3_bind g e x2 x4 H18 b x3 x5 H21)))) -(ty3_correct g (CHead e (Bind b) x2) x3 x5 H21))))) H19))))) H16)) u -H8))))))) x0 H7)))))) (lift_gen_bind b u t2 x0 h x1 H5))))))))))))))))) -(\lambda (c0: C).(\lambda (w: T).(\lambda (u: T).(\lambda (H1: (ty3 g c0 w -u)).(\lambda (H2: ((\forall (x0: T).(\forall (x1: nat).((eq T w (lift h x1 -x0)) \to (\forall (e: C).((drop h x1 c0 e) \to (ex2 T (\lambda (t2: T).(pc3 -c0 (lift h x1 t2) u)) (\lambda (t2: T).(ty3 g e x0 t2)))))))))).(\lambda (v: -T).(\lambda (t: T).(\lambda (H3: (ty3 g c0 v (THead (Bind Abst) u -t))).(\lambda (H4: ((\forall (x0: T).(\forall (x1: nat).((eq T v (lift h x1 -x0)) \to (\forall (e: C).((drop h x1 c0 e) \to (ex2 T (\lambda (t2: T).(pc3 -c0 (lift h x1 t2) (THead (Bind Abst) u t))) (\lambda (t2: T).(ty3 g e x0 -t2)))))))))).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H5: (eq T (THead -(Flat Appl) w v) (lift h x1 x0))).(\lambda (e: C).(\lambda (H6: (drop h x1 c0 -e)).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Flat -Appl) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T w (lift h x1 y0)))) -(\lambda (_: T).(\lambda (z: T).(eq T v (lift h x1 z)))) (ex2 T (\lambda (t2: -T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) w (THead (Bind Abst) u t)))) -(\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (x2: T).(\lambda (x3: T).(\lambda -(H7: (eq T x0 (THead (Flat Appl) x2 x3))).(\lambda (H8: (eq T w (lift h x1 -x2))).(\lambda (H9: (eq T v (lift h x1 x3))).(eq_ind_r T (THead (Flat Appl) -x2 x3) (\lambda (t0: T).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead -(Flat Appl) w (THead (Bind Abst) u t)))) (\lambda (t2: T).(ty3 g e t0 t2)))) -(let H10 \def (eq_ind T v (\lambda (t0: T).(\forall (x4: T).(\forall (x5: -nat).((eq T t0 (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 c0 e0) \to -(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x5 t2) (THead (Bind Abst) u t))) -(\lambda (t2: T).(ty3 g e0 x4 t2))))))))) H4 (lift h x1 x3) H9) in (let H11 -\def (eq_ind T v (\lambda (t0: T).(ty3 g c0 t0 (THead (Bind Abst) u t))) H3 -(lift h x1 x3) H9) in (let H12 \def (eq_ind T w (\lambda (t0: T).(\forall -(x4: T).(\forall (x5: nat).((eq T t0 (lift h x5 x4)) \to (\forall (e0: -C).((drop h x5 c0 e0) \to (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x5 t2) u)) -(\lambda (t2: T).(ty3 g e0 x4 t2))))))))) H2 (lift h x1 x2) H8) in (let H13 -\def (eq_ind T w (\lambda (t0: T).(ty3 g c0 t0 u)) H1 (lift h x1 x2) H8) in -(eq_ind_r T (lift h x1 x2) (\lambda (t0: T).(ex2 T (\lambda (t2: T).(pc3 c0 -(lift h x1 t2) (THead (Flat Appl) t0 (THead (Bind Abst) u t)))) (\lambda (t2: -T).(ty3 g e (THead (Flat Appl) x2 x3) t2)))) (let H14 \def (H12 x2 x1 -(refl_equal T (lift h x1 x2)) e H6) in (ex2_ind T (\lambda (t2: T).(pc3 c0 -(lift h x1 t2) u)) (\lambda (t2: T).(ty3 g e x2 t2)) (ex2 T (\lambda (t2: -T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) (lift h x1 x2) (THead (Bind -Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead (Flat Appl) x2 x3) t2))) -(\lambda (x4: T).(\lambda (H15: (pc3 c0 (lift h x1 x4) u)).(\lambda (H16: -(ty3 g e x2 x4)).(let H17 \def (H10 x3 x1 (refl_equal T (lift h x1 x3)) e H6) -in (ex2_ind T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead (Bind Abst) u -t))) (\lambda (t2: T).(ty3 g e x3 t2)) (ex2 T (\lambda (t2: T).(pc3 c0 (lift -h x1 t2) (THead (Flat Appl) (lift h x1 x2) (THead (Bind Abst) u t)))) -(\lambda (t2: T).(ty3 g e (THead (Flat Appl) x2 x3) t2))) (\lambda (x5: -T).(\lambda (H18: (pc3 c0 (lift h x1 x5) (THead (Bind Abst) u t))).(\lambda -(H19: (ty3 g e x3 x5)).(ex3_2_ind T T (\lambda (u1: T).(\lambda (t2: T).(pr3 -e x5 (THead (Bind Abst) u1 t2)))) (\lambda (u1: T).(\lambda (_: T).(pr3 c0 u -(lift h x1 u1)))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall -(u0: T).(pr3 (CHead c0 (Bind b) u0) t (lift h (S x1) t2)))))) (ex2 T (\lambda -(t2: T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) (lift h x1 x2) (THead (Bind -Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead (Flat Appl) x2 x3) t2))) -(\lambda (x6: T).(\lambda (x7: T).(\lambda (H20: (pr3 e x5 (THead (Bind Abst) -x6 x7))).(\lambda (H21: (pr3 c0 u (lift h x1 x6))).(\lambda (H22: ((\forall -(b: B).(\forall (u0: T).(pr3 (CHead c0 (Bind b) u0) t (lift h (S x1) -x7)))))).(ex_ind T (\lambda (t0: T).(ty3 g e x5 t0)) (ex2 T (\lambda (t2: -T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) (lift h x1 x2) (THead (Bind -Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead (Flat Appl) x2 x3) t2))) -(\lambda (x8: T).(\lambda (H23: (ty3 g e x5 x8)).(let H_y \def (ty3_sred_pr3 -e x5 (THead (Bind Abst) x6 x7) H20 g x8 H23) in (ex3_2_ind T T (\lambda (t2: -T).(\lambda (_: T).(pc3 e (THead (Bind Abst) x6 t2) x8))) (\lambda (_: -T).(\lambda (t0: T).(ty3 g e x6 t0))) (\lambda (t2: T).(\lambda (_: T).(ty3 g -(CHead e (Bind Abst) x6) x7 t2))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 -t2) (THead (Flat Appl) (lift h x1 x2) (THead (Bind Abst) u t)))) (\lambda -(t2: T).(ty3 g e (THead (Flat Appl) x2 x3) t2))) (\lambda (x9: T).(\lambda -(x10: T).(\lambda (_: (pc3 e (THead (Bind Abst) x6 x9) x8)).(\lambda (H25: -(ty3 g e x6 x10)).(\lambda (H26: (ty3 g (CHead e (Bind Abst) x6) x7 -x9)).(ex_intro2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) -(lift h x1 x2) (THead (Bind Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead -(Flat Appl) x2 x3) t2)) (THead (Flat Appl) x2 (THead (Bind Abst) x6 x7)) -(eq_ind_r T (THead (Flat Appl) (lift h x1 x2) (lift h x1 (THead (Bind Abst) -x6 x7))) (\lambda (t0: T).(pc3 c0 t0 (THead (Flat Appl) (lift h x1 x2) (THead -(Bind Abst) u t)))) (pc3_thin_dx c0 (lift h x1 (THead (Bind Abst) x6 x7)) -(THead (Bind Abst) u t) (eq_ind_r T (THead (Bind Abst) (lift h x1 x6) (lift h -(S x1) x7)) (\lambda (t0: T).(pc3 c0 t0 (THead (Bind Abst) u t))) -(pc3_head_21 c0 (lift h x1 x6) u (pc3_pr3_x c0 (lift h x1 x6) u H21) (Bind -Abst) (lift h (S x1) x7) t (pc3_pr3_x (CHead c0 (Bind Abst) (lift h x1 x6)) -(lift h (S x1) x7) t (H22 Abst (lift h x1 x6)))) (lift h x1 (THead (Bind -Abst) x6 x7)) (lift_bind Abst x6 x7 h x1)) (lift h x1 x2) Appl) (lift h x1 -(THead (Flat Appl) x2 (THead (Bind Abst) x6 x7))) (lift_flat Appl x2 (THead -(Bind Abst) x6 x7) h x1)) (ty3_appl g e x2 x6 (ty3_conv g e x6 x10 H25 x2 x4 -H16 (pc3_gen_lift c0 x4 x6 h x1 (pc3_t u c0 (lift h x1 x4) H15 (lift h x1 x6) -(pc3_pr3_r c0 u (lift h x1 x6) H21)) e H6)) x3 x7 (ty3_conv g e (THead (Bind -Abst) x6 x7) (THead (Bind Abst) x6 x9) (ty3_bind g e x6 x10 H25 Abst x7 x9 -H26) x3 x5 H19 (pc3_pr3_r e x5 (THead (Bind Abst) x6 x7) H20))))))))) -(ty3_gen_bind g Abst e x6 x7 x8 H_y))))) (ty3_correct g e x3 x5 H19))))))) -(pc3_gen_lift_abst c0 x5 t u h x1 H18 e H6))))) H17))))) H14)) w H8))))) x0 -H7)))))) (lift_gen_flat Appl w v x0 h x1 H5)))))))))))))))) (\lambda (c0: -C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (ty3 g c0 t2 t3)).(\lambda -(H2: ((\forall (x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to -(\forall (e: C).((drop h x1 c0 e) \to (ex2 T (\lambda (t4: T).(pc3 c0 (lift h -x1 t4) t3)) (\lambda (t4: T).(ty3 g e x0 t4)))))))))).(\lambda (t0: -T).(\lambda (H3: (ty3 g c0 t3 t0)).(\lambda (H4: ((\forall (x0: T).(\forall -(x1: nat).((eq T t3 (lift h x1 x0)) \to (\forall (e: C).((drop h x1 c0 e) \to -(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) t0)) (\lambda (t4: T).(ty3 g e -x0 t4)))))))))).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H5: (eq T -(THead (Flat Cast) t3 t2) (lift h x1 x0))).(\lambda (e: C).(\lambda (H6: -(drop h x1 c0 e)).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 -(THead (Flat Cast) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T t3 (lift h -x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h x1 z)))) (ex2 T -(\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Flat Cast) t0 t3))) (\lambda -(t4: T).(ty3 g e x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq -T x0 (THead (Flat Cast) x2 x3))).(\lambda (H8: (eq T t3 (lift h x1 -x2))).(\lambda (H9: (eq T t2 (lift h x1 x3))).(eq_ind_r T (THead (Flat Cast) -x2 x3) (\lambda (t: T).(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead -(Flat Cast) t0 t3))) (\lambda (t4: T).(ty3 g e t t4)))) (let H10 \def (eq_ind -T t3 (\lambda (t: T).(\forall (x4: T).(\forall (x5: nat).((eq T t (lift h x5 -x4)) \to (\forall (e0: C).((drop h x5 c0 e0) \to (ex2 T (\lambda (t4: T).(pc3 -c0 (lift h x5 t4) t0)) (\lambda (t4: T).(ty3 g e0 x4 t4))))))))) H4 (lift h -x1 x2) H8) in (let H11 \def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 t t0)) H3 -(lift h x1 x2) H8) in (let H12 \def (eq_ind T t3 (\lambda (t: T).(\forall -(x4: T).(\forall (x5: nat).((eq T t2 (lift h x5 x4)) \to (\forall (e0: -C).((drop h x5 c0 e0) \to (ex2 T (\lambda (t4: T).(pc3 c0 (lift h x5 t4) t)) -(\lambda (t4: T).(ty3 g e0 x4 t4))))))))) H2 (lift h x1 x2) H8) in (let H13 -\def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 t2 t)) H1 (lift h x1 x2) H8) in -(eq_ind_r T (lift h x1 x2) (\lambda (t: T).(ex2 T (\lambda (t4: T).(pc3 c0 -(lift h x1 t4) (THead (Flat Cast) t0 t))) (\lambda (t4: T).(ty3 g e (THead -(Flat Cast) x2 x3) t4)))) (let H14 \def (eq_ind T t2 (\lambda (t: T).(ty3 g -c0 t (lift h x1 x2))) H13 (lift h x1 x3) H9) in (let H15 \def (eq_ind T t2 -(\lambda (t: T).(\forall (x4: T).(\forall (x5: nat).((eq T t (lift h x5 x4)) -\to (\forall (e0: C).((drop h x5 c0 e0) \to (ex2 T (\lambda (t4: T).(pc3 c0 -(lift h x5 t4) (lift h x1 x2))) (\lambda (t4: T).(ty3 g e0 x4 t4))))))))) H12 -(lift h x1 x3) H9) in (let H16 \def (H15 x3 x1 (refl_equal T (lift h x1 x3)) -e H6) in (ex2_ind T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (lift h x1 x2))) -(\lambda (t4: T).(ty3 g e x3 t4)) (ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 -t4) (THead (Flat Cast) t0 (lift h x1 x2)))) (\lambda (t4: T).(ty3 g e (THead -(Flat Cast) x2 x3) t4))) (\lambda (x4: T).(\lambda (H17: (pc3 c0 (lift h x1 -x4) (lift h x1 x2))).(\lambda (H18: (ty3 g e x3 x4)).(let H19 \def (H10 x2 x1 -(refl_equal T (lift h x1 x2)) e H6) in (ex2_ind T (\lambda (t4: T).(pc3 c0 -(lift h x1 t4) t0)) (\lambda (t4: T).(ty3 g e x2 t4)) (ex2 T (\lambda (t4: -T).(pc3 c0 (lift h x1 t4) (THead (Flat Cast) t0 (lift h x1 x2)))) (\lambda -(t4: T).(ty3 g e (THead (Flat Cast) x2 x3) t4))) (\lambda (x5: T).(\lambda -(H20: (pc3 c0 (lift h x1 x5) t0)).(\lambda (H21: (ty3 g e x2 x5)).(ex_intro2 -T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Flat Cast) t0 (lift h x1 -x2)))) (\lambda (t4: T).(ty3 g e (THead (Flat Cast) x2 x3) t4)) (THead (Flat -Cast) x5 x2) (eq_ind_r T (THead (Flat Cast) (lift h x1 x5) (lift h x1 x2)) -(\lambda (t: T).(pc3 c0 t (THead (Flat Cast) t0 (lift h x1 x2)))) (pc3_head_1 -c0 (lift h x1 x5) t0 H20 (Flat Cast) (lift h x1 x2)) (lift h x1 (THead (Flat -Cast) x5 x2)) (lift_flat Cast x5 x2 h x1)) (ty3_cast g e x3 x2 (ty3_conv g e -x2 x5 H21 x3 x4 H18 (pc3_gen_lift c0 x4 x2 h x1 H17 e H6)) x5 H21))))) -H19))))) H16)))) t3 H8))))) x0 H7)))))) (lift_gen_flat Cast t3 t2 x0 h x1 -H5))))))))))))))) c y x H0))))) H))))))). -(* COMMENTS -Initial nodes: 9781 -END *) - -theorem ty3_tred: - \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).((ty3 g c u -t1) \to (\forall (t2: T).((pr3 c t1 t2) \to (ty3 g c u t2))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (H: -(ty3 g c u t1)).(\lambda (t2: T).(\lambda (H0: (pr3 c t1 t2)).(ex_ind T -(\lambda (t: T).(ty3 g c t1 t)) (ty3 g c u t2) (\lambda (x: T).(\lambda (H1: -(ty3 g c t1 x)).(let H_y \def (ty3_sred_pr3 c t1 t2 H0 g x H1) in (ty3_conv g -c t2 x H_y u t1 H (pc3_pr3_r c t1 t2 H0))))) (ty3_correct g c u t1 H)))))))). -(* COMMENTS -Initial nodes: 121 -END *) - -theorem ty3_sconv_pc3: - \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (t1: T).((ty3 g c -u1 t1) \to (\forall (u2: T).(\forall (t2: T).((ty3 g c u2 t2) \to ((pc3 c u1 -u2) \to (pc3 c t1 t2))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda -(H: (ty3 g c u1 t1)).(\lambda (u2: T).(\lambda (t2: T).(\lambda (H0: (ty3 g c -u2 t2)).(\lambda (H1: (pc3 c u1 u2)).(let H2 \def H1 in (ex2_ind T (\lambda -(t: T).(pr3 c u1 t)) (\lambda (t: T).(pr3 c u2 t)) (pc3 c t1 t2) (\lambda (x: -T).(\lambda (H3: (pr3 c u1 x)).(\lambda (H4: (pr3 c u2 x)).(let H_y \def -(ty3_sred_pr3 c u2 x H4 g t2 H0) in (let H_y0 \def (ty3_sred_pr3 c u1 x H3 g -t1 H) in (ty3_unique g c x t1 H_y0 t2 H_y)))))) H2)))))))))). -(* COMMENTS -Initial nodes: 141 -END *) - -theorem ty3_sred_back: - \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t0: T).((ty3 g c -t1 t0) \to (\forall (t2: T).((pr3 c t1 t2) \to (\forall (t: T).((ty3 g c t2 -t) \to (ty3 g c t1 t))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t0: T).(\lambda -(H: (ty3 g c t1 t0)).(\lambda (t2: T).(\lambda (H0: (pr3 c t1 t2)).(\lambda -(t: T).(\lambda (H1: (ty3 g c t2 t)).(ex_ind T (\lambda (t3: T).(ty3 g c t -t3)) (ty3 g c t1 t) (\lambda (x: T).(\lambda (H2: (ty3 g c t x)).(ty3_conv g -c t x H2 t1 t0 H (ty3_unique g c t2 t0 (ty3_sred_pr3 c t1 t2 H0 g t0 H) t -H1)))) (ty3_correct g c t2 t H1)))))))))). -(* COMMENTS -Initial nodes: 137 -END *) - -theorem ty3_sconv: - \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (t1: T).((ty3 g c -u1 t1) \to (\forall (u2: T).(\forall (t2: T).((ty3 g c u2 t2) \to ((pc3 c u1 -u2) \to (ty3 g c u1 t2))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda -(H: (ty3 g c u1 t1)).(\lambda (u2: T).(\lambda (t2: T).(\lambda (H0: (ty3 g c -u2 t2)).(\lambda (H1: (pc3 c u1 u2)).(let H2 \def H1 in (ex2_ind T (\lambda -(t: T).(pr3 c u1 t)) (\lambda (t: T).(pr3 c u2 t)) (ty3 g c u1 t2) (\lambda -(x: T).(\lambda (H3: (pr3 c u1 x)).(\lambda (H4: (pr3 c u2 x)).(ty3_sred_back -g c u1 t1 H x H3 t2 (ty3_sred_pr3 c u2 x H4 g t2 H0))))) H2)))))))))). -(* COMMENTS -Initial nodes: 129 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/props.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/props.ma deleted file mode 100644 index 579a8dd69..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/props.ma +++ /dev/null @@ -1,691 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/ty3/fwd.ma". - -include "Basic-1/pc3/fwd.ma". - -theorem ty3_lift: - \forall (g: G).(\forall (e: C).(\forall (t1: T).(\forall (t2: T).((ty3 g e -t1 t2) \to (\forall (c: C).(\forall (d: nat).(\forall (h: nat).((drop h d c -e) \to (ty3 g c (lift h d t1) (lift h d t2)))))))))) -\def - \lambda (g: G).(\lambda (e: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H: (ty3 g e t1 t2)).(ty3_ind g (\lambda (c: C).(\lambda (t: T).(\lambda (t0: -T).(\forall (c0: C).(\forall (d: nat).(\forall (h: nat).((drop h d c0 c) \to -(ty3 g c0 (lift h d t) (lift h d t0))))))))) (\lambda (c: C).(\lambda (t0: -T).(\lambda (t: T).(\lambda (_: (ty3 g c t0 t)).(\lambda (H1: ((\forall (c0: -C).(\forall (d: nat).(\forall (h: nat).((drop h d c0 c) \to (ty3 g c0 (lift h -d t0) (lift h d t)))))))).(\lambda (u: T).(\lambda (t3: T).(\lambda (_: (ty3 -g c u t3)).(\lambda (H3: ((\forall (c0: C).(\forall (d: nat).(\forall (h: -nat).((drop h d c0 c) \to (ty3 g c0 (lift h d u) (lift h d -t3)))))))).(\lambda (H4: (pc3 c t3 t0)).(\lambda (c0: C).(\lambda (d: -nat).(\lambda (h: nat).(\lambda (H5: (drop h d c0 c)).(ty3_conv g c0 (lift h -d t0) (lift h d t) (H1 c0 d h H5) (lift h d u) (lift h d t3) (H3 c0 d h H5) -(pc3_lift c0 c h d H5 t3 t0 H4)))))))))))))))) (\lambda (c: C).(\lambda (m: -nat).(\lambda (c0: C).(\lambda (d: nat).(\lambda (h: nat).(\lambda (_: (drop -h d c0 c)).(eq_ind_r T (TSort m) (\lambda (t: T).(ty3 g c0 t (lift h d (TSort -(next g m))))) (eq_ind_r T (TSort (next g m)) (\lambda (t: T).(ty3 g c0 -(TSort m) t)) (ty3_sort g c0 m) (lift h d (TSort (next g m))) (lift_sort -(next g m) h d)) (lift h d (TSort m)) (lift_sort m h d)))))))) (\lambda (n: -nat).(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c -(CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u -t)).(\lambda (H2: ((\forall (c0: C).(\forall (d0: nat).(\forall (h: -nat).((drop h d0 c0 d) \to (ty3 g c0 (lift h d0 u) (lift h d0 -t)))))))).(\lambda (c0: C).(\lambda (d0: nat).(\lambda (h: nat).(\lambda (H3: -(drop h d0 c0 c)).(lt_le_e n d0 (ty3 g c0 (lift h d0 (TLRef n)) (lift h d0 -(lift (S n) O t))) (\lambda (H4: (lt n d0)).(let H5 \def (drop_getl_trans_le -n d0 (le_S_n n d0 (le_S (S n) d0 H4)) c0 c h H3 (CHead d (Bind Abbr) u) H0) -in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop n O c0 e0))) -(\lambda (e0: C).(\lambda (e1: C).(drop h (minus d0 n) e0 e1))) (\lambda (_: -C).(\lambda (e1: C).(clear e1 (CHead d (Bind Abbr) u)))) (ty3 g c0 (lift h d0 -(TLRef n)) (lift h d0 (lift (S n) O t))) (\lambda (x0: C).(\lambda (x1: -C).(\lambda (H6: (drop n O c0 x0)).(\lambda (H7: (drop h (minus d0 n) x0 -x1)).(\lambda (H8: (clear x1 (CHead d (Bind Abbr) u))).(let H9 \def (eq_ind -nat (minus d0 n) (\lambda (n0: nat).(drop h n0 x0 x1)) H7 (S (minus d0 (S -n))) (minus_x_Sy d0 n H4)) in (let H10 \def (drop_clear_S x1 x0 h (minus d0 -(S n)) H9 Abbr d u H8) in (ex2_ind C (\lambda (c1: C).(clear x0 (CHead c1 -(Bind Abbr) (lift h (minus d0 (S n)) u)))) (\lambda (c1: C).(drop h (minus d0 -(S n)) c1 d)) (ty3 g c0 (lift h d0 (TLRef n)) (lift h d0 (lift (S n) O t))) -(\lambda (x: C).(\lambda (H11: (clear x0 (CHead x (Bind Abbr) (lift h (minus -d0 (S n)) u)))).(\lambda (H12: (drop h (minus d0 (S n)) x d)).(eq_ind_r T -(TLRef n) (\lambda (t0: T).(ty3 g c0 t0 (lift h d0 (lift (S n) O t)))) -(eq_ind nat (plus (S n) (minus d0 (S n))) (\lambda (n0: nat).(ty3 g c0 (TLRef -n) (lift h n0 (lift (S n) O t)))) (eq_ind_r T (lift (S n) O (lift h (minus d0 -(S n)) t)) (\lambda (t0: T).(ty3 g c0 (TLRef n) t0)) (eq_ind nat d0 (\lambda -(_: nat).(ty3 g c0 (TLRef n) (lift (S n) O (lift h (minus d0 (S n)) t)))) -(ty3_abbr g n c0 x (lift h (minus d0 (S n)) u) (getl_intro n c0 (CHead x -(Bind Abbr) (lift h (minus d0 (S n)) u)) x0 H6 H11) (lift h (minus d0 (S n)) -t) (H2 x (minus d0 (S n)) h H12)) (plus (S n) (minus d0 (S n))) -(le_plus_minus (S n) d0 H4)) (lift h (plus (S n) (minus d0 (S n))) (lift (S -n) O t)) (lift_d t h (S n) (minus d0 (S n)) O (le_O_n (minus d0 (S n))))) d0 -(le_plus_minus_r (S n) d0 H4)) (lift h d0 (TLRef n)) (lift_lref_lt n h d0 -H4))))) H10)))))))) H5))) (\lambda (H4: (le d0 n)).(eq_ind_r T (TLRef (plus n -h)) (\lambda (t0: T).(ty3 g c0 t0 (lift h d0 (lift (S n) O t)))) (eq_ind nat -(S n) (\lambda (_: nat).(ty3 g c0 (TLRef (plus n h)) (lift h d0 (lift (S n) O -t)))) (eq_ind_r T (lift (plus h (S n)) O t) (\lambda (t0: T).(ty3 g c0 (TLRef -(plus n h)) t0)) (eq_ind_r nat (plus (S n) h) (\lambda (n0: nat).(ty3 g c0 -(TLRef (plus n h)) (lift n0 O t))) (ty3_abbr g (plus n h) c0 d u -(drop_getl_trans_ge n c0 c d0 h H3 (CHead d (Bind Abbr) u) H0 H4) t H1) (plus -h (S n)) (plus_sym h (S n))) (lift h d0 (lift (S n) O t)) (lift_free t (S n) -h O d0 (le_S d0 n H4) (le_O_n d0))) (plus n (S O)) (eq_ind_r nat (plus (S O) -n) (\lambda (n0: nat).(eq nat (S n) n0)) (refl_equal nat (plus (S O) n)) -(plus n (S O)) (plus_sym n (S O)))) (lift h d0 (TLRef n)) (lift_lref_ge n h -d0 H4)))))))))))))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d: -C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind Abst) u))).(\lambda -(t: T).(\lambda (H1: (ty3 g d u t)).(\lambda (H2: ((\forall (c0: C).(\forall -(d0: nat).(\forall (h: nat).((drop h d0 c0 d) \to (ty3 g c0 (lift h d0 u) -(lift h d0 t)))))))).(\lambda (c0: C).(\lambda (d0: nat).(\lambda (h: -nat).(\lambda (H3: (drop h d0 c0 c)).(lt_le_e n d0 (ty3 g c0 (lift h d0 -(TLRef n)) (lift h d0 (lift (S n) O u))) (\lambda (H4: (lt n d0)).(let H5 -\def (drop_getl_trans_le n d0 (le_S_n n d0 (le_S (S n) d0 H4)) c0 c h H3 -(CHead d (Bind Abst) u) H0) in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: -C).(drop n O c0 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d0 n) -e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear e1 (CHead d (Bind Abst) -u)))) (ty3 g c0 (lift h d0 (TLRef n)) (lift h d0 (lift (S n) O u))) (\lambda -(x0: C).(\lambda (x1: C).(\lambda (H6: (drop n O c0 x0)).(\lambda (H7: (drop -h (minus d0 n) x0 x1)).(\lambda (H8: (clear x1 (CHead d (Bind Abst) u))).(let -H9 \def (eq_ind nat (minus d0 n) (\lambda (n0: nat).(drop h n0 x0 x1)) H7 (S -(minus d0 (S n))) (minus_x_Sy d0 n H4)) in (let H10 \def (drop_clear_S x1 x0 -h (minus d0 (S n)) H9 Abst d u H8) in (ex2_ind C (\lambda (c1: C).(clear x0 -(CHead c1 (Bind Abst) (lift h (minus d0 (S n)) u)))) (\lambda (c1: C).(drop h -(minus d0 (S n)) c1 d)) (ty3 g c0 (lift h d0 (TLRef n)) (lift h d0 (lift (S -n) O u))) (\lambda (x: C).(\lambda (H11: (clear x0 (CHead x (Bind Abst) (lift -h (minus d0 (S n)) u)))).(\lambda (H12: (drop h (minus d0 (S n)) x -d)).(eq_ind_r T (TLRef n) (\lambda (t0: T).(ty3 g c0 t0 (lift h d0 (lift (S -n) O u)))) (eq_ind nat (plus (S n) (minus d0 (S n))) (\lambda (n0: nat).(ty3 -g c0 (TLRef n) (lift h n0 (lift (S n) O u)))) (eq_ind_r T (lift (S n) O (lift -h (minus d0 (S n)) u)) (\lambda (t0: T).(ty3 g c0 (TLRef n) t0)) (eq_ind nat -d0 (\lambda (_: nat).(ty3 g c0 (TLRef n) (lift (S n) O (lift h (minus d0 (S -n)) u)))) (ty3_abst g n c0 x (lift h (minus d0 (S n)) u) (getl_intro n c0 -(CHead x (Bind Abst) (lift h (minus d0 (S n)) u)) x0 H6 H11) (lift h (minus -d0 (S n)) t) (H2 x (minus d0 (S n)) h H12)) (plus (S n) (minus d0 (S n))) -(le_plus_minus (S n) d0 H4)) (lift h (plus (S n) (minus d0 (S n))) (lift (S -n) O u)) (lift_d u h (S n) (minus d0 (S n)) O (le_O_n (minus d0 (S n))))) d0 -(le_plus_minus_r (S n) d0 H4)) (lift h d0 (TLRef n)) (lift_lref_lt n h d0 -H4))))) H10)))))))) H5))) (\lambda (H4: (le d0 n)).(eq_ind_r T (TLRef (plus n -h)) (\lambda (t0: T).(ty3 g c0 t0 (lift h d0 (lift (S n) O u)))) (eq_ind nat -(S n) (\lambda (_: nat).(ty3 g c0 (TLRef (plus n h)) (lift h d0 (lift (S n) O -u)))) (eq_ind_r T (lift (plus h (S n)) O u) (\lambda (t0: T).(ty3 g c0 (TLRef -(plus n h)) t0)) (eq_ind_r nat (plus (S n) h) (\lambda (n0: nat).(ty3 g c0 -(TLRef (plus n h)) (lift n0 O u))) (ty3_abst g (plus n h) c0 d u -(drop_getl_trans_ge n c0 c d0 h H3 (CHead d (Bind Abst) u) H0 H4) t H1) (plus -h (S n)) (plus_sym h (S n))) (lift h d0 (lift (S n) O u)) (lift_free u (S n) -h O d0 (le_S d0 n H4) (le_O_n d0))) (plus n (S O)) (eq_ind_r nat (plus (S O) -n) (\lambda (n0: nat).(eq nat (S n) n0)) (refl_equal nat (plus (S O) n)) -(plus n (S O)) (plus_sym n (S O)))) (lift h d0 (TLRef n)) (lift_lref_ge n h -d0 H4)))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (t: -T).(\lambda (_: (ty3 g c u t)).(\lambda (H1: ((\forall (c0: C).(\forall (d: -nat).(\forall (h: nat).((drop h d c0 c) \to (ty3 g c0 (lift h d u) (lift h d -t)))))))).(\lambda (b: B).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (ty3 -g (CHead c (Bind b) u) t0 t3)).(\lambda (H3: ((\forall (c0: C).(\forall (d: -nat).(\forall (h: nat).((drop h d c0 (CHead c (Bind b) u)) \to (ty3 g c0 -(lift h d t0) (lift h d t3)))))))).(\lambda (c0: C).(\lambda (d: -nat).(\lambda (h: nat).(\lambda (H4: (drop h d c0 c)).(eq_ind_r T (THead -(Bind b) (lift h d u) (lift h (s (Bind b) d) t0)) (\lambda (t4: T).(ty3 g c0 -t4 (lift h d (THead (Bind b) u t3)))) (eq_ind_r T (THead (Bind b) (lift h d -u) (lift h (s (Bind b) d) t3)) (\lambda (t4: T).(ty3 g c0 (THead (Bind b) -(lift h d u) (lift h (s (Bind b) d) t0)) t4)) (ty3_bind g c0 (lift h d u) -(lift h d t) (H1 c0 d h H4) b (lift h (S d) t0) (lift h (S d) t3) (H3 (CHead -c0 (Bind b) (lift h d u)) (S d) h (drop_skip_bind h d c0 c H4 b u))) (lift h -d (THead (Bind b) u t3)) (lift_head (Bind b) u t3 h d)) (lift h d (THead -(Bind b) u t0)) (lift_head (Bind b) u t0 h d)))))))))))))))) (\lambda (c: -C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c w u)).(\lambda (H1: -((\forall (c0: C).(\forall (d: nat).(\forall (h: nat).((drop h d c0 c) \to -(ty3 g c0 (lift h d w) (lift h d u)))))))).(\lambda (v: T).(\lambda (t: -T).(\lambda (_: (ty3 g c v (THead (Bind Abst) u t))).(\lambda (H3: ((\forall -(c0: C).(\forall (d: nat).(\forall (h: nat).((drop h d c0 c) \to (ty3 g c0 -(lift h d v) (lift h d (THead (Bind Abst) u t))))))))).(\lambda (c0: -C).(\lambda (d: nat).(\lambda (h: nat).(\lambda (H4: (drop h d c0 -c)).(eq_ind_r T (THead (Flat Appl) (lift h d w) (lift h (s (Flat Appl) d) v)) -(\lambda (t0: T).(ty3 g c0 t0 (lift h d (THead (Flat Appl) w (THead (Bind -Abst) u t))))) (eq_ind_r T (THead (Flat Appl) (lift h d w) (lift h (s (Flat -Appl) d) (THead (Bind Abst) u t))) (\lambda (t0: T).(ty3 g c0 (THead (Flat -Appl) (lift h d w) (lift h (s (Flat Appl) d) v)) t0)) (eq_ind_r T (THead -(Bind Abst) (lift h (s (Flat Appl) d) u) (lift h (s (Bind Abst) (s (Flat -Appl) d)) t)) (\lambda (t0: T).(ty3 g c0 (THead (Flat Appl) (lift h d w) -(lift h (s (Flat Appl) d) v)) (THead (Flat Appl) (lift h d w) t0))) (ty3_appl -g c0 (lift h d w) (lift h d u) (H1 c0 d h H4) (lift h d v) (lift h (S d) t) -(eq_ind T (lift h d (THead (Bind Abst) u t)) (\lambda (t0: T).(ty3 g c0 (lift -h d v) t0)) (H3 c0 d h H4) (THead (Bind Abst) (lift h d u) (lift h (S d) t)) -(lift_bind Abst u t h d))) (lift h (s (Flat Appl) d) (THead (Bind Abst) u t)) -(lift_head (Bind Abst) u t h (s (Flat Appl) d))) (lift h d (THead (Flat Appl) -w (THead (Bind Abst) u t))) (lift_head (Flat Appl) w (THead (Bind Abst) u t) -h d)) (lift h d (THead (Flat Appl) w v)) (lift_head (Flat Appl) w v h -d))))))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda (t3: T).(\lambda -(_: (ty3 g c t0 t3)).(\lambda (H1: ((\forall (c0: C).(\forall (d: -nat).(\forall (h: nat).((drop h d c0 c) \to (ty3 g c0 (lift h d t0) (lift h d -t3)))))))).(\lambda (t4: T).(\lambda (_: (ty3 g c t3 t4)).(\lambda (H3: -((\forall (c0: C).(\forall (d: nat).(\forall (h: nat).((drop h d c0 c) \to -(ty3 g c0 (lift h d t3) (lift h d t4)))))))).(\lambda (c0: C).(\lambda (d: -nat).(\lambda (h: nat).(\lambda (H4: (drop h d c0 c)).(eq_ind_r T (THead -(Flat Cast) (lift h d t3) (lift h (s (Flat Cast) d) t0)) (\lambda (t: T).(ty3 -g c0 t (lift h d (THead (Flat Cast) t4 t3)))) (eq_ind_r T (THead (Flat Cast) -(lift h d t4) (lift h (s (Flat Cast) d) t3)) (\lambda (t: T).(ty3 g c0 (THead -(Flat Cast) (lift h d t3) (lift h (s (Flat Cast) d) t0)) t)) (ty3_cast g c0 -(lift h (s (Flat Cast) d) t0) (lift h (s (Flat Cast) d) t3) (H1 c0 (s (Flat -Cast) d) h H4) (lift h d t4) (H3 c0 d h H4)) (lift h d (THead (Flat Cast) t4 -t3)) (lift_head (Flat Cast) t4 t3 h d)) (lift h d (THead (Flat Cast) t3 t0)) -(lift_head (Flat Cast) t3 t0 h d)))))))))))))) e t1 t2 H))))). -(* COMMENTS -Initial nodes: 4253 -END *) - -theorem ty3_correct: - \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c -t1 t2) \to (ex T (\lambda (t: T).(ty3 g c t2 t))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H: (ty3 g c t1 t2)).(ty3_ind g (\lambda (c0: C).(\lambda (_: T).(\lambda -(t0: T).(ex T (\lambda (t3: T).(ty3 g c0 t0 t3)))))) (\lambda (c0: -C).(\lambda (t0: T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 t0 t)).(\lambda -(_: (ex T (\lambda (t3: T).(ty3 g c0 t t3)))).(\lambda (u: T).(\lambda (t3: -T).(\lambda (_: (ty3 g c0 u t3)).(\lambda (_: (ex T (\lambda (t4: T).(ty3 g -c0 t3 t4)))).(\lambda (_: (pc3 c0 t3 t0)).(ex_intro T (\lambda (t4: T).(ty3 g -c0 t0 t4)) t H0))))))))))) (\lambda (c0: C).(\lambda (m: nat).(ex_intro T -(\lambda (t: T).(ty3 g c0 (TSort (next g m)) t)) (TSort (next g (next g m))) -(ty3_sort g c0 (next g m))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: -C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abbr) -u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2: (ex T (\lambda -(t0: T).(ty3 g d t t0)))).(let H3 \def H2 in (ex_ind T (\lambda (t0: T).(ty3 -g d t t0)) (ex T (\lambda (t0: T).(ty3 g c0 (lift (S n) O t) t0))) (\lambda -(x: T).(\lambda (H4: (ty3 g d t x)).(ex_intro T (\lambda (t0: T).(ty3 g c0 -(lift (S n) O t) t0)) (lift (S n) O x) (ty3_lift g d t x H4 c0 O (S n) -(getl_drop Abbr c0 d u n H0))))) H3)))))))))) (\lambda (n: nat).(\lambda (c0: -C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind -Abst) u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u t)).(\lambda (_: (ex T -(\lambda (t0: T).(ty3 g d t t0)))).(ex_intro T (\lambda (t0: T).(ty3 g c0 -(lift (S n) O u) t0)) (lift (S n) O t) (ty3_lift g d u t H1 c0 O (S n) -(getl_drop Abst c0 d u n H0))))))))))) (\lambda (c0: C).(\lambda (u: -T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 u t)).(\lambda (_: (ex T (\lambda -(t0: T).(ty3 g c0 t t0)))).(\lambda (b: B).(\lambda (t0: T).(\lambda (t3: -T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t0 t3)).(\lambda (H3: (ex T -(\lambda (t4: T).(ty3 g (CHead c0 (Bind b) u) t3 t4)))).(let H4 \def H3 in -(ex_ind T (\lambda (t4: T).(ty3 g (CHead c0 (Bind b) u) t3 t4)) (ex T -(\lambda (t4: T).(ty3 g c0 (THead (Bind b) u t3) t4))) (\lambda (x: -T).(\lambda (H5: (ty3 g (CHead c0 (Bind b) u) t3 x)).(ex_intro T (\lambda -(t4: T).(ty3 g c0 (THead (Bind b) u t3) t4)) (THead (Bind b) u x) (ty3_bind g -c0 u t H0 b t3 x H5)))) H4)))))))))))) (\lambda (c0: C).(\lambda (w: -T).(\lambda (u: T).(\lambda (H0: (ty3 g c0 w u)).(\lambda (H1: (ex T (\lambda -(t: T).(ty3 g c0 u t)))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g -c0 v (THead (Bind Abst) u t))).(\lambda (H3: (ex T (\lambda (t0: T).(ty3 g c0 -(THead (Bind Abst) u t) t0)))).(let H4 \def H1 in (ex_ind T (\lambda (t0: -T).(ty3 g c0 u t0)) (ex T (\lambda (t0: T).(ty3 g c0 (THead (Flat Appl) w -(THead (Bind Abst) u t)) t0))) (\lambda (x: T).(\lambda (_: (ty3 g c0 u -x)).(let H6 \def H3 in (ex_ind T (\lambda (t0: T).(ty3 g c0 (THead (Bind -Abst) u t) t0)) (ex T (\lambda (t0: T).(ty3 g c0 (THead (Flat Appl) w (THead -(Bind Abst) u t)) t0))) (\lambda (x0: T).(\lambda (H7: (ty3 g c0 (THead (Bind -Abst) u t) x0)).(ex3_2_ind T T (\lambda (t3: T).(\lambda (_: T).(pc3 c0 -(THead (Bind Abst) u t3) x0))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u -t0))) (\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind Abst) u) t -t3))) (ex T (\lambda (t0: T).(ty3 g c0 (THead (Flat Appl) w (THead (Bind -Abst) u t)) t0))) (\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 c0 -(THead (Bind Abst) u x1) x0)).(\lambda (H9: (ty3 g c0 u x2)).(\lambda (H10: -(ty3 g (CHead c0 (Bind Abst) u) t x1)).(ex_intro T (\lambda (t0: T).(ty3 g c0 -(THead (Flat Appl) w (THead (Bind Abst) u t)) t0)) (THead (Flat Appl) w -(THead (Bind Abst) u x1)) (ty3_appl g c0 w u H0 (THead (Bind Abst) u t) x1 -(ty3_bind g c0 u x2 H9 Abst t x1 H10)))))))) (ty3_gen_bind g Abst c0 u t x0 -H7)))) H6)))) H4))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t3: -T).(\lambda (_: (ty3 g c0 t0 t3)).(\lambda (_: (ex T (\lambda (t: T).(ty3 g -c0 t3 t)))).(\lambda (t4: T).(\lambda (H2: (ty3 g c0 t3 t4)).(\lambda (H3: -(ex T (\lambda (t: T).(ty3 g c0 t4 t)))).(let H4 \def H3 in (ex_ind T -(\lambda (t: T).(ty3 g c0 t4 t)) (ex T (\lambda (t: T).(ty3 g c0 (THead (Flat -Cast) t4 t3) t))) (\lambda (x: T).(\lambda (H5: (ty3 g c0 t4 x)).(ex_intro T -(\lambda (t: T).(ty3 g c0 (THead (Flat Cast) t4 t3) t)) (THead (Flat Cast) x -t4) (ty3_cast g c0 t3 t4 H2 x H5)))) H4)))))))))) c t1 t2 H))))). -(* COMMENTS -Initial nodes: 1333 -END *) - -theorem ty3_unique: - \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).((ty3 g c u -t1) \to (\forall (t2: T).((ty3 g c u t2) \to (pc3 c t1 t2))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (H: -(ty3 g c u t1)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: -T).(\forall (t2: T).((ty3 g c0 t t2) \to (pc3 c0 t0 t2)))))) (\lambda (c0: -C).(\lambda (t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda -(_: ((\forall (t3: T).((ty3 g c0 t2 t3) \to (pc3 c0 t t3))))).(\lambda (u0: -T).(\lambda (t0: T).(\lambda (_: (ty3 g c0 u0 t0)).(\lambda (H3: ((\forall -(t3: T).((ty3 g c0 u0 t3) \to (pc3 c0 t0 t3))))).(\lambda (H4: (pc3 c0 t0 -t2)).(\lambda (t3: T).(\lambda (H5: (ty3 g c0 u0 t3)).(pc3_t t0 c0 t2 (pc3_s -c0 t2 t0 H4) t3 (H3 t3 H5)))))))))))))) (\lambda (c0: C).(\lambda (m: -nat).(\lambda (t2: T).(\lambda (H0: (ty3 g c0 (TSort m) t2)).(ty3_gen_sort g -c0 t2 m H0))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda -(u0: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abbr) u0))).(\lambda (t: -T).(\lambda (_: (ty3 g d u0 t)).(\lambda (H2: ((\forall (t2: T).((ty3 g d u0 -t2) \to (pc3 d t t2))))).(\lambda (t2: T).(\lambda (H3: (ty3 g c0 (TLRef n) -t2)).(or_ind (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: -T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda -(_: T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: C).(\lambda (u1: -T).(\lambda (t0: T).(ty3 g e u1 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda -(u1: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u1) t2)))) (\lambda (e: -C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u1))))) -(\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 t0))))) (pc3 c0 -(lift (S n) O t) t2) (\lambda (H4: (ex3_3 C T T (\lambda (_: C).(\lambda (_: -T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda (e: C).(\lambda -(u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: -C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 t0)))))).(ex3_3_ind C T T -(\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) -t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e -(Bind Abbr) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g -e u1 t0)))) (pc3 c0 (lift (S n) O t) t2) (\lambda (x0: C).(\lambda (x1: -T).(\lambda (x2: T).(\lambda (H5: (pc3 c0 (lift (S n) O x2) t2)).(\lambda -(H6: (getl n c0 (CHead x0 (Bind Abbr) x1))).(\lambda (H7: (ty3 g x0 x1 -x2)).(let H8 \def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda (c1: C).(getl n -c0 c1)) H0 (CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind Abbr) u0) n -H0 (CHead x0 (Bind Abbr) x1) H6)) in (let H9 \def (f_equal C C (\lambda (e: -C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | -(CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u0) (CHead x0 (Bind -Abbr) x1) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead x0 (Bind Abbr) -x1) H6)) in ((let H10 \def (f_equal C T (\lambda (e: C).(match e in C return -(\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t0) -\Rightarrow t0])) (CHead d (Bind Abbr) u0) (CHead x0 (Bind Abbr) x1) -(getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in -(\lambda (H11: (eq C d x0)).(let H12 \def (eq_ind_r T x1 (\lambda (t0: -T).(getl n c0 (CHead x0 (Bind Abbr) t0))) H8 u0 H10) in (let H13 \def -(eq_ind_r T x1 (\lambda (t0: T).(ty3 g x0 t0 x2)) H7 u0 H10) in (let H14 \def -(eq_ind_r C x0 (\lambda (c1: C).(getl n c0 (CHead c1 (Bind Abbr) u0))) H12 d -H11) in (let H15 \def (eq_ind_r C x0 (\lambda (c1: C).(ty3 g c1 u0 x2)) H13 d -H11) in (pc3_t (lift (S n) O x2) c0 (lift (S n) O t) (pc3_lift c0 d (S n) O -(getl_drop Abbr c0 d u0 n H14) t x2 (H2 x2 H15)) t2 H5))))))) H9))))))))) -H4)) (\lambda (H4: (ex3_3 C T T (\lambda (_: C).(\lambda (u1: T).(\lambda (_: -T).(pc3 c0 (lift (S n) O u1) t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda -(_: T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1: -T).(\lambda (t0: T).(ty3 g e u1 t0)))))).(ex3_3_ind C T T (\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u1) t2)))) (\lambda -(e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) -u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 t0)))) -(pc3 c0 (lift (S n) O t) t2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: -T).(\lambda (_: (pc3 c0 (lift (S n) O x1) t2)).(\lambda (H6: (getl n c0 -(CHead x0 (Bind Abst) x1))).(\lambda (_: (ty3 g x0 x1 x2)).(let H8 \def -(eq_ind C (CHead d (Bind Abbr) u0) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead -x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead x0 -(Bind Abst) x1) H6)) in (let H9 \def (eq_ind C (CHead d (Bind Abbr) u0) -(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind b) \Rightarrow (match b in B return (\lambda (_: -B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | Void -\Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead x0 (Bind Abst) -x1) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead x0 (Bind Abst) x1) -H6)) in (False_ind (pc3 c0 (lift (S n) O t) t2) H9))))))))) H4)) -(ty3_gen_lref g c0 t2 n H3)))))))))))) (\lambda (n: nat).(\lambda (c0: -C).(\lambda (d: C).(\lambda (u0: T).(\lambda (H0: (getl n c0 (CHead d (Bind -Abst) u0))).(\lambda (t: T).(\lambda (_: (ty3 g d u0 t)).(\lambda (_: -((\forall (t2: T).((ty3 g d u0 t2) \to (pc3 d t t2))))).(\lambda (t2: -T).(\lambda (H3: (ty3 g c0 (TLRef n) t2)).(or_ind (ex3_3 C T T (\lambda (_: -C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda -(e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) -u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 t0))))) -(ex3_3 C T T (\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(pc3 c0 (lift -(S n) O u1) t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n -c0 (CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda -(t0: T).(ty3 g e u1 t0))))) (pc3 c0 (lift (S n) O u0) t2) (\lambda (H4: -(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift -(S n) O t0) t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n -c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda -(t0: T).(ty3 g e u1 t0)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (_: -T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda (e: C).(\lambda -(u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: -C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 t0)))) (pc3 c0 (lift (S n) O -u0) t2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 -c0 (lift (S n) O x2) t2)).(\lambda (H6: (getl n c0 (CHead x0 (Bind Abbr) -x1))).(\lambda (_: (ty3 g x0 x1 x2)).(let H8 \def (eq_ind C (CHead d (Bind -Abst) u0) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead x0 (Bind Abbr) x1) -(getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in -(let H9 \def (eq_ind C (CHead d (Bind Abst) u0) (\lambda (ee: C).(match ee in -C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k -_) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b) -\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow -False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) -\Rightarrow False])])) I (CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d -(Bind Abst) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in (False_ind (pc3 c0 -(lift (S n) O u0) t2) H9))))))))) H4)) (\lambda (H4: (ex3_3 C T T (\lambda -(_: C).(\lambda (u1: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u1) t2)))) -(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind -Abst) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 -t0)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (u1: T).(\lambda (_: -T).(pc3 c0 (lift (S n) O u1) t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda -(_: T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1: -T).(\lambda (t0: T).(ty3 g e u1 t0)))) (pc3 c0 (lift (S n) O u0) t2) (\lambda -(x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H5: (pc3 c0 (lift (S n) O -x1) t2)).(\lambda (H6: (getl n c0 (CHead x0 (Bind Abst) x1))).(\lambda (H7: -(ty3 g x0 x1 x2)).(let H8 \def (eq_ind C (CHead d (Bind Abst) u0) (\lambda -(c1: C).(getl n c0 c1)) H0 (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d -(Bind Abst) u0) n H0 (CHead x0 (Bind Abst) x1) H6)) in (let H9 \def (f_equal -C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abst) u0) -(CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead -x0 (Bind Abst) x1) H6)) in ((let H10 \def (f_equal C T (\lambda (e: C).(match -e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ -t0) \Rightarrow t0])) (CHead d (Bind Abst) u0) (CHead x0 (Bind Abst) x1) -(getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead x0 (Bind Abst) x1) H6)) in -(\lambda (H11: (eq C d x0)).(let H12 \def (eq_ind_r T x1 (\lambda (t0: -T).(getl n c0 (CHead x0 (Bind Abst) t0))) H8 u0 H10) in (let H13 \def -(eq_ind_r T x1 (\lambda (t0: T).(ty3 g x0 t0 x2)) H7 u0 H10) in (let H14 \def -(eq_ind_r T x1 (\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)) H5 u0 H10) in -(let H15 \def (eq_ind_r C x0 (\lambda (c1: C).(getl n c0 (CHead c1 (Bind -Abst) u0))) H12 d H11) in (let H16 \def (eq_ind_r C x0 (\lambda (c1: C).(ty3 -g c1 u0 x2)) H13 d H11) in H14))))))) H9))))))))) H4)) (ty3_gen_lref g c0 t2 -n H3)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (t: T).(\lambda -(_: (ty3 g c0 u0 t)).(\lambda (_: ((\forall (t2: T).((ty3 g c0 u0 t2) \to -(pc3 c0 t t2))))).(\lambda (b: B).(\lambda (t0: T).(\lambda (t2: T).(\lambda -(_: (ty3 g (CHead c0 (Bind b) u0) t0 t2)).(\lambda (H3: ((\forall (t3: -T).((ty3 g (CHead c0 (Bind b) u0) t0 t3) \to (pc3 (CHead c0 (Bind b) u0) t2 -t3))))).(\lambda (t3: T).(\lambda (H4: (ty3 g c0 (THead (Bind b) u0 t0) -t3)).(ex3_2_ind T T (\lambda (t4: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) -u0 t4) t3))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c0 u0 t5))) (\lambda -(t4: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u0) t0 t4))) (pc3 c0 (THead -(Bind b) u0 t2) t3) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (pc3 c0 -(THead (Bind b) u0 x0) t3)).(\lambda (_: (ty3 g c0 u0 x1)).(\lambda (H7: (ty3 -g (CHead c0 (Bind b) u0) t0 x0)).(pc3_t (THead (Bind b) u0 x0) c0 (THead -(Bind b) u0 t2) (pc3_head_2 c0 u0 t2 x0 (Bind b) (H3 x0 H7)) t3 H5)))))) -(ty3_gen_bind g b c0 u0 t0 t3 H4)))))))))))))) (\lambda (c0: C).(\lambda (w: -T).(\lambda (u0: T).(\lambda (_: (ty3 g c0 w u0)).(\lambda (_: ((\forall (t2: -T).((ty3 g c0 w t2) \to (pc3 c0 u0 t2))))).(\lambda (v: T).(\lambda (t: -T).(\lambda (_: (ty3 g c0 v (THead (Bind Abst) u0 t))).(\lambda (H3: -((\forall (t2: T).((ty3 g c0 v t2) \to (pc3 c0 (THead (Bind Abst) u0 t) -t2))))).(\lambda (t2: T).(\lambda (H4: (ty3 g c0 (THead (Flat Appl) w v) -t2)).(ex3_2_ind T T (\lambda (u1: T).(\lambda (t0: T).(pc3 c0 (THead (Flat -Appl) w (THead (Bind Abst) u1 t0)) t2))) (\lambda (u1: T).(\lambda (t0: -T).(ty3 g c0 v (THead (Bind Abst) u1 t0)))) (\lambda (u1: T).(\lambda (_: -T).(ty3 g c0 w u1))) (pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t)) -t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (pc3 c0 (THead (Flat -Appl) w (THead (Bind Abst) x0 x1)) t2)).(\lambda (H6: (ty3 g c0 v (THead -(Bind Abst) x0 x1))).(\lambda (_: (ty3 g c0 w x0)).(pc3_t (THead (Flat Appl) -w (THead (Bind Abst) x0 x1)) c0 (THead (Flat Appl) w (THead (Bind Abst) u0 -t)) (pc3_thin_dx c0 (THead (Bind Abst) u0 t) (THead (Bind Abst) x0 x1) (H3 -(THead (Bind Abst) x0 x1) H6) w Appl) t2 H5)))))) (ty3_gen_appl g c0 w v t2 -H4))))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: T).(\lambda -(_: (ty3 g c0 t0 t2)).(\lambda (_: ((\forall (t3: T).((ty3 g c0 t0 t3) \to -(pc3 c0 t2 t3))))).(\lambda (t3: T).(\lambda (_: (ty3 g c0 t2 t3)).(\lambda -(H3: ((\forall (t4: T).((ty3 g c0 t2 t4) \to (pc3 c0 t3 t4))))).(\lambda (t4: -T).(\lambda (H4: (ty3 g c0 (THead (Flat Cast) t2 t0) t4)).(ex3_ind T (\lambda -(t5: T).(pc3 c0 (THead (Flat Cast) t5 t2) t4)) (\lambda (_: T).(ty3 g c0 t0 -t2)) (\lambda (t5: T).(ty3 g c0 t2 t5)) (pc3 c0 (THead (Flat Cast) t3 t2) t4) -(\lambda (x0: T).(\lambda (H5: (pc3 c0 (THead (Flat Cast) x0 t2) -t4)).(\lambda (_: (ty3 g c0 t0 t2)).(\lambda (H7: (ty3 g c0 t2 x0)).(pc3_t -(THead (Flat Cast) x0 t2) c0 (THead (Flat Cast) t3 t2) (pc3_head_1 c0 t3 x0 -(H3 x0 H7) (Flat Cast) t2) t4 H5))))) (ty3_gen_cast g c0 t0 t2 t4 -H4)))))))))))) c u t1 H))))). -(* COMMENTS -Initial nodes: 3459 -END *) - -theorem ty3_gen_abst_abst: - \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall -(t2: T).((ty3 g c (THead (Bind Abst) u t1) (THead (Bind Abst) u t2)) \to (ex2 -T (\lambda (w: T).(ty3 g c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst) -u) t1 t2)))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda -(t2: T).(\lambda (H: (ty3 g c (THead (Bind Abst) u t1) (THead (Bind Abst) u -t2))).(ex_ind T (\lambda (t: T).(ty3 g c (THead (Bind Abst) u t2) t)) (ex2 T -(\lambda (w: T).(ty3 g c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) -t1 t2))) (\lambda (x: T).(\lambda (H0: (ty3 g c (THead (Bind Abst) u t2) -x)).(ex3_2_ind T T (\lambda (t3: T).(\lambda (_: T).(pc3 c (THead (Bind Abst) -u t3) x))) (\lambda (_: T).(\lambda (t: T).(ty3 g c u t))) (\lambda (t3: -T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t2 t3))) (ex2 T (\lambda -(w: T).(ty3 g c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t1 t2))) -(\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 c (THead (Bind Abst) u -x0) x)).(\lambda (_: (ty3 g c u x1)).(\lambda (H3: (ty3 g (CHead c (Bind -Abst) u) t2 x0)).(ex3_2_ind T T (\lambda (t3: T).(\lambda (_: T).(pc3 c -(THead (Bind Abst) u t3) (THead (Bind Abst) u t2)))) (\lambda (_: T).(\lambda -(t: T).(ty3 g c u t))) (\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c (Bind -Abst) u) t1 t3))) (ex2 T (\lambda (w: T).(ty3 g c u w)) (\lambda (_: T).(ty3 -g (CHead c (Bind Abst) u) t1 t2))) (\lambda (x2: T).(\lambda (x3: T).(\lambda -(H4: (pc3 c (THead (Bind Abst) u x2) (THead (Bind Abst) u t2))).(\lambda (H5: -(ty3 g c u x3)).(\lambda (H6: (ty3 g (CHead c (Bind Abst) u) t1 x2)).(let H_y -\def (pc3_gen_abst_shift c u x2 t2 H4) in (ex_intro2 T (\lambda (w: T).(ty3 g -c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t1 t2)) x3 H5 -(ty3_conv g (CHead c (Bind Abst) u) t2 x0 H3 t1 x2 H6 H_y)))))))) -(ty3_gen_bind g Abst c u t1 (THead (Bind Abst) u t2) H))))))) (ty3_gen_bind g -Abst c u t2 x H0)))) (ty3_correct g c (THead (Bind Abst) u t1) (THead (Bind -Abst) u t2) H))))))). -(* COMMENTS -Initial nodes: 571 -END *) - -theorem ty3_typecheck: - \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (v: T).((ty3 g c t -v) \to (ex T (\lambda (u: T).(ty3 g c (THead (Flat Cast) v t) u))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (v: T).(\lambda (H: -(ty3 g c t v)).(ex_ind T (\lambda (t0: T).(ty3 g c v t0)) (ex T (\lambda (u: -T).(ty3 g c (THead (Flat Cast) v t) u))) (\lambda (x: T).(\lambda (H0: (ty3 g -c v x)).(ex_intro T (\lambda (u: T).(ty3 g c (THead (Flat Cast) v t) u)) -(THead (Flat Cast) x v) (ty3_cast g c t v H x H0)))) (ty3_correct g c t v -H)))))). -(* COMMENTS -Initial nodes: 131 -END *) - -theorem ty3_getl_subst0: - \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t -u) \to (\forall (v0: T).(\forall (t0: T).(\forall (i: nat).((subst0 i v0 t -t0) \to (\forall (b: B).(\forall (d: C).(\forall (v: T).((getl i c (CHead d -(Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v w))))))))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H: -(ty3 g c t u)).(ty3_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (_: -T).(\forall (v0: T).(\forall (t2: T).(\forall (i: nat).((subst0 i v0 t0 t2) -\to (\forall (b: B).(\forall (d: C).(\forall (v: T).((getl i c0 (CHead d -(Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v w)))))))))))))) (\lambda -(c0: C).(\lambda (t2: T).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t2 -t0)).(\lambda (_: ((\forall (v0: T).(\forall (t1: T).(\forall (i: -nat).((subst0 i v0 t2 t1) \to (\forall (b: B).(\forall (d: C).(\forall (v: -T).((getl i c0 (CHead d (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v -w))))))))))))).(\lambda (u0: T).(\lambda (t1: T).(\lambda (_: (ty3 g c0 u0 -t1)).(\lambda (H3: ((\forall (v0: T).(\forall (t3: T).(\forall (i: -nat).((subst0 i v0 u0 t3) \to (\forall (b: B).(\forall (d: C).(\forall (v: -T).((getl i c0 (CHead d (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v -w))))))))))))).(\lambda (_: (pc3 c0 t1 t2)).(\lambda (v0: T).(\lambda (t3: -T).(\lambda (i: nat).(\lambda (H5: (subst0 i v0 u0 t3)).(\lambda (b: -B).(\lambda (d: C).(\lambda (v: T).(\lambda (H6: (getl i c0 (CHead d (Bind b) -v))).(H3 v0 t3 i H5 b d v H6))))))))))))))))))) (\lambda (c0: C).(\lambda (m: -nat).(\lambda (v0: T).(\lambda (t0: T).(\lambda (i: nat).(\lambda (H0: -(subst0 i v0 (TSort m) t0)).(\lambda (b: B).(\lambda (d: C).(\lambda (v: -T).(\lambda (_: (getl i c0 (CHead d (Bind b) v))).(subst0_gen_sort v0 t0 i m -H0 (ex T (\lambda (w: T).(ty3 g d v w)))))))))))))) (\lambda (n: -nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (H0: (getl n -c0 (CHead d (Bind Abbr) u0))).(\lambda (t0: T).(\lambda (H1: (ty3 g d u0 -t0)).(\lambda (_: ((\forall (v0: T).(\forall (t1: T).(\forall (i: -nat).((subst0 i v0 u0 t1) \to (\forall (b: B).(\forall (d0: C).(\forall (v: -T).((getl i d (CHead d0 (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d0 v -w))))))))))))).(\lambda (v0: T).(\lambda (t1: T).(\lambda (i: nat).(\lambda -(H3: (subst0 i v0 (TLRef n) t1)).(\lambda (b: B).(\lambda (d0: C).(\lambda -(v: T).(\lambda (H4: (getl i c0 (CHead d0 (Bind b) v))).(land_ind (eq nat n -i) (eq T t1 (lift (S n) O v0)) (ex T (\lambda (w: T).(ty3 g d0 v w))) -(\lambda (H5: (eq nat n i)).(\lambda (_: (eq T t1 (lift (S n) O v0))).(let H7 -\def (eq_ind_r nat i (\lambda (n0: nat).(getl n0 c0 (CHead d0 (Bind b) v))) -H4 n H5) in (let H8 \def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda (c1: -C).(getl n c0 c1)) H0 (CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind -Abbr) u0) n H0 (CHead d0 (Bind b) v) H7)) in (let H9 \def (f_equal C C -(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u0) -(CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead d0 -(Bind b) v) H7)) in ((let H10 \def (f_equal C B (\lambda (e: C).(match e in C -return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b0) -\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u0) -(CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead d0 -(Bind b) v) H7)) in ((let H11 \def (f_equal C T (\lambda (e: C).(match e in C -return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t2) -\Rightarrow t2])) (CHead d (Bind Abbr) u0) (CHead d0 (Bind b) v) (getl_mono -c0 (CHead d (Bind Abbr) u0) n H0 (CHead d0 (Bind b) v) H7)) in (\lambda (H12: -(eq B Abbr b)).(\lambda (H13: (eq C d d0)).(let H14 \def (eq_ind_r T v -(\lambda (t2: T).(getl n c0 (CHead d0 (Bind b) t2))) H8 u0 H11) in (eq_ind T -u0 (\lambda (t2: T).(ex T (\lambda (w: T).(ty3 g d0 t2 w)))) (let H15 \def -(eq_ind_r C d0 (\lambda (c1: C).(getl n c0 (CHead c1 (Bind b) u0))) H14 d -H13) in (eq_ind C d (\lambda (c1: C).(ex T (\lambda (w: T).(ty3 g c1 u0 w)))) -(let H16 \def (eq_ind_r B b (\lambda (b0: B).(getl n c0 (CHead d (Bind b0) -u0))) H15 Abbr H12) in (ex_intro T (\lambda (w: T).(ty3 g d u0 w)) t0 H1)) d0 -H13)) v H11))))) H10)) H9)))))) (subst0_gen_lref v0 t1 i n -H3)))))))))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: -C).(\lambda (u0: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abst) -u0))).(\lambda (t0: T).(\lambda (H1: (ty3 g d u0 t0)).(\lambda (_: ((\forall -(v0: T).(\forall (t1: T).(\forall (i: nat).((subst0 i v0 u0 t1) \to (\forall -(b: B).(\forall (d0: C).(\forall (v: T).((getl i d (CHead d0 (Bind b) v)) \to -(ex T (\lambda (w: T).(ty3 g d0 v w))))))))))))).(\lambda (v0: T).(\lambda -(t1: T).(\lambda (i: nat).(\lambda (H3: (subst0 i v0 (TLRef n) t1)).(\lambda -(b: B).(\lambda (d0: C).(\lambda (v: T).(\lambda (H4: (getl i c0 (CHead d0 -(Bind b) v))).(land_ind (eq nat n i) (eq T t1 (lift (S n) O v0)) (ex T -(\lambda (w: T).(ty3 g d0 v w))) (\lambda (H5: (eq nat n i)).(\lambda (_: (eq -T t1 (lift (S n) O v0))).(let H7 \def (eq_ind_r nat i (\lambda (n0: -nat).(getl n0 c0 (CHead d0 (Bind b) v))) H4 n H5) in (let H8 \def (eq_ind C -(CHead d (Bind Abst) u0) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead d0 (Bind -b) v) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead d0 (Bind b) v) H7)) -in (let H9 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda -(_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) -(CHead d (Bind Abst) u0) (CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind -Abst) u0) n H0 (CHead d0 (Bind b) v) H7)) in ((let H10 \def (f_equal C B -(\lambda (e: C).(match e in C return (\lambda (_: C).B) with [(CSort _) -\Rightarrow Abst | (CHead _ k _) \Rightarrow (match k in K return (\lambda -(_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abst])])) -(CHead d (Bind Abst) u0) (CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind -Abst) u0) n H0 (CHead d0 (Bind b) v) H7)) in ((let H11 \def (f_equal C T -(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) -\Rightarrow u0 | (CHead _ _ t2) \Rightarrow t2])) (CHead d (Bind Abst) u0) -(CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead d0 -(Bind b) v) H7)) in (\lambda (H12: (eq B Abst b)).(\lambda (H13: (eq C d -d0)).(let H14 \def (eq_ind_r T v (\lambda (t2: T).(getl n c0 (CHead d0 (Bind -b) t2))) H8 u0 H11) in (eq_ind T u0 (\lambda (t2: T).(ex T (\lambda (w: -T).(ty3 g d0 t2 w)))) (let H15 \def (eq_ind_r C d0 (\lambda (c1: C).(getl n -c0 (CHead c1 (Bind b) u0))) H14 d H13) in (eq_ind C d (\lambda (c1: C).(ex T -(\lambda (w: T).(ty3 g c1 u0 w)))) (let H16 \def (eq_ind_r B b (\lambda (b0: -B).(getl n c0 (CHead d (Bind b0) u0))) H15 Abst H12) in (ex_intro T (\lambda -(w: T).(ty3 g d u0 w)) t0 H1)) d0 H13)) v H11))))) H10)) H9)))))) -(subst0_gen_lref v0 t1 i n H3)))))))))))))))))) (\lambda (c0: C).(\lambda -(u0: T).(\lambda (t0: T).(\lambda (_: (ty3 g c0 u0 t0)).(\lambda (H1: -((\forall (v0: T).(\forall (t1: T).(\forall (i: nat).((subst0 i v0 u0 t1) \to -(\forall (b: B).(\forall (d: C).(\forall (v: T).((getl i c0 (CHead d (Bind b) -v)) \to (ex T (\lambda (w: T).(ty3 g d v w))))))))))))).(\lambda (b: -B).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) -u0) t1 t2)).(\lambda (H3: ((\forall (v0: T).(\forall (t3: T).(\forall (i: -nat).((subst0 i v0 t1 t3) \to (\forall (b0: B).(\forall (d: C).(\forall (v: -T).((getl i (CHead c0 (Bind b) u0) (CHead d (Bind b0) v)) \to (ex T (\lambda -(w: T).(ty3 g d v w))))))))))))).(\lambda (v0: T).(\lambda (t3: T).(\lambda -(i: nat).(\lambda (H4: (subst0 i v0 (THead (Bind b) u0 t1) t3)).(\lambda (b0: -B).(\lambda (d: C).(\lambda (v: T).(\lambda (H5: (getl i c0 (CHead d (Bind -b0) v))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t3 (THead (Bind b) u2 t1))) -(\lambda (u2: T).(subst0 i v0 u0 u2))) (ex2 T (\lambda (t4: T).(eq T t3 -(THead (Bind b) u0 t4))) (\lambda (t4: T).(subst0 (s (Bind b) i) v0 t1 t4))) -(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind b) u2 -t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 u0 u2))) (\lambda (_: -T).(\lambda (t4: T).(subst0 (s (Bind b) i) v0 t1 t4)))) (ex T (\lambda (w: -T).(ty3 g d v w))) (\lambda (H6: (ex2 T (\lambda (u2: T).(eq T t3 (THead -(Bind b) u2 t1))) (\lambda (u2: T).(subst0 i v0 u0 u2)))).(ex2_ind T (\lambda -(u2: T).(eq T t3 (THead (Bind b) u2 t1))) (\lambda (u2: T).(subst0 i v0 u0 -u2)) (ex T (\lambda (w: T).(ty3 g d v w))) (\lambda (x: T).(\lambda (_: (eq T -t3 (THead (Bind b) x t1))).(\lambda (H8: (subst0 i v0 u0 x)).(H1 v0 x i H8 b0 -d v H5)))) H6)) (\lambda (H6: (ex2 T (\lambda (t4: T).(eq T t3 (THead (Bind -b) u0 t4))) (\lambda (t4: T).(subst0 (s (Bind b) i) v0 t1 t4)))).(ex2_ind T -(\lambda (t4: T).(eq T t3 (THead (Bind b) u0 t4))) (\lambda (t4: T).(subst0 -(s (Bind b) i) v0 t1 t4)) (ex T (\lambda (w: T).(ty3 g d v w))) (\lambda (x: -T).(\lambda (_: (eq T t3 (THead (Bind b) u0 x))).(\lambda (H8: (subst0 (s -(Bind b) i) v0 t1 x)).(H3 v0 x (S i) H8 b0 d v (getl_head (Bind b) i c0 -(CHead d (Bind b0) v) H5 u0))))) H6)) (\lambda (H6: (ex3_2 T T (\lambda (u2: -T).(\lambda (t4: T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i v0 u0 u2))) (\lambda (_: T).(\lambda (t4: -T).(subst0 (s (Bind b) i) v0 t1 t4))))).(ex3_2_ind T T (\lambda (u2: -T).(\lambda (t4: T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda (u2: -T).(\lambda (_: T).(subst0 i v0 u0 u2))) (\lambda (_: T).(\lambda (t4: -T).(subst0 (s (Bind b) i) v0 t1 t4))) (ex T (\lambda (w: T).(ty3 g d v w))) -(\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (eq T t3 (THead (Bind b) x0 -x1))).(\lambda (H8: (subst0 i v0 u0 x0)).(\lambda (_: (subst0 (s (Bind b) i) -v0 t1 x1)).(H1 v0 x0 i H8 b0 d v H5)))))) H6)) (subst0_gen_head (Bind b) v0 -u0 t1 t3 i H4)))))))))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda -(u0: T).(\lambda (_: (ty3 g c0 w u0)).(\lambda (H1: ((\forall (v0: -T).(\forall (t0: T).(\forall (i: nat).((subst0 i v0 w t0) \to (\forall (b: -B).(\forall (d: C).(\forall (v: T).((getl i c0 (CHead d (Bind b) v)) \to (ex -T (\lambda (w0: T).(ty3 g d v w0))))))))))))).(\lambda (v: T).(\lambda (t0: -T).(\lambda (_: (ty3 g c0 v (THead (Bind Abst) u0 t0))).(\lambda (H3: -((\forall (v0: T).(\forall (t1: T).(\forall (i: nat).((subst0 i v0 v t1) \to -(\forall (b: B).(\forall (d: C).(\forall (v1: T).((getl i c0 (CHead d (Bind -b) v1)) \to (ex T (\lambda (w0: T).(ty3 g d v1 w0))))))))))))).(\lambda (v0: -T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (H4: (subst0 i v0 (THead (Flat -Appl) w v) t1)).(\lambda (b: B).(\lambda (d: C).(\lambda (v1: T).(\lambda -(H5: (getl i c0 (CHead d (Bind b) v1))).(or3_ind (ex2 T (\lambda (u2: T).(eq -T t1 (THead (Flat Appl) u2 v))) (\lambda (u2: T).(subst0 i v0 w u2))) (ex2 T -(\lambda (t2: T).(eq T t1 (THead (Flat Appl) w t2))) (\lambda (t2: T).(subst0 -(s (Flat Appl) i) v0 v t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq -T t1 (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i -v0 w u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s (Flat Appl) i) v0 v -t2)))) (ex T (\lambda (w0: T).(ty3 g d v1 w0))) (\lambda (H6: (ex2 T (\lambda -(u2: T).(eq T t1 (THead (Flat Appl) u2 v))) (\lambda (u2: T).(subst0 i v0 w -u2)))).(ex2_ind T (\lambda (u2: T).(eq T t1 (THead (Flat Appl) u2 v))) -(\lambda (u2: T).(subst0 i v0 w u2)) (ex T (\lambda (w0: T).(ty3 g d v1 w0))) -(\lambda (x: T).(\lambda (_: (eq T t1 (THead (Flat Appl) x v))).(\lambda (H8: -(subst0 i v0 w x)).(H1 v0 x i H8 b d v1 H5)))) H6)) (\lambda (H6: (ex2 T -(\lambda (t2: T).(eq T t1 (THead (Flat Appl) w t2))) (\lambda (t2: T).(subst0 -(s (Flat Appl) i) v0 v t2)))).(ex2_ind T (\lambda (t2: T).(eq T t1 (THead -(Flat Appl) w t2))) (\lambda (t2: T).(subst0 (s (Flat Appl) i) v0 v t2)) (ex -T (\lambda (w0: T).(ty3 g d v1 w0))) (\lambda (x: T).(\lambda (_: (eq T t1 -(THead (Flat Appl) w x))).(\lambda (H8: (subst0 (s (Flat Appl) i) v0 v -x)).(H3 v0 x (s (Flat Appl) i) H8 b d v1 H5)))) H6)) (\lambda (H6: (ex3_2 T T -(\lambda (u2: T).(\lambda (t2: T).(eq T t1 (THead (Flat Appl) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 i v0 w u2))) (\lambda (_: -T).(\lambda (t2: T).(subst0 (s (Flat Appl) i) v0 v t2))))).(ex3_2_ind T T -(\lambda (u2: T).(\lambda (t2: T).(eq T t1 (THead (Flat Appl) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(subst0 i v0 w u2))) (\lambda (_: -T).(\lambda (t2: T).(subst0 (s (Flat Appl) i) v0 v t2))) (ex T (\lambda (w0: -T).(ty3 g d v1 w0))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (eq T t1 -(THead (Flat Appl) x0 x1))).(\lambda (_: (subst0 i v0 w x0)).(\lambda (H9: -(subst0 (s (Flat Appl) i) v0 v x1)).(H3 v0 x1 (s (Flat Appl) i) H9 b d v1 -H5)))))) H6)) (subst0_gen_head (Flat Appl) v0 w v t1 i H4))))))))))))))))))) -(\lambda (c0: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g c0 t1 -t2)).(\lambda (H1: ((\forall (v0: T).(\forall (t0: T).(\forall (i: -nat).((subst0 i v0 t1 t0) \to (\forall (b: B).(\forall (d: C).(\forall (v: -T).((getl i c0 (CHead d (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v -w))))))))))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t2 t0)).(\lambda (H3: -((\forall (v0: T).(\forall (t3: T).(\forall (i: nat).((subst0 i v0 t2 t3) \to -(\forall (b: B).(\forall (d: C).(\forall (v: T).((getl i c0 (CHead d (Bind b) -v)) \to (ex T (\lambda (w: T).(ty3 g d v w))))))))))))).(\lambda (v0: -T).(\lambda (t3: T).(\lambda (i: nat).(\lambda (H4: (subst0 i v0 (THead (Flat -Cast) t2 t1) t3)).(\lambda (b: B).(\lambda (d: C).(\lambda (v: T).(\lambda -(H5: (getl i c0 (CHead d (Bind b) v))).(or3_ind (ex2 T (\lambda (u2: T).(eq T -t3 (THead (Flat Cast) u2 t1))) (\lambda (u2: T).(subst0 i v0 t2 u2))) (ex2 T -(\lambda (t4: T).(eq T t3 (THead (Flat Cast) t2 t4))) (\lambda (t4: -T).(subst0 (s (Flat Cast) i) v0 t1 t4))) (ex3_2 T T (\lambda (u2: T).(\lambda -(t4: T).(eq T t3 (THead (Flat Cast) u2 t4)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 i v0 t2 u2))) (\lambda (_: T).(\lambda (t4: T).(subst0 (s (Flat -Cast) i) v0 t1 t4)))) (ex T (\lambda (w: T).(ty3 g d v w))) (\lambda (H6: -(ex2 T (\lambda (u2: T).(eq T t3 (THead (Flat Cast) u2 t1))) (\lambda (u2: -T).(subst0 i v0 t2 u2)))).(ex2_ind T (\lambda (u2: T).(eq T t3 (THead (Flat -Cast) u2 t1))) (\lambda (u2: T).(subst0 i v0 t2 u2)) (ex T (\lambda (w: -T).(ty3 g d v w))) (\lambda (x: T).(\lambda (_: (eq T t3 (THead (Flat Cast) x -t1))).(\lambda (H8: (subst0 i v0 t2 x)).(H3 v0 x i H8 b d v H5)))) H6)) -(\lambda (H6: (ex2 T (\lambda (t4: T).(eq T t3 (THead (Flat Cast) t2 t4))) -(\lambda (t4: T).(subst0 (s (Flat Cast) i) v0 t1 t4)))).(ex2_ind T (\lambda -(t4: T).(eq T t3 (THead (Flat Cast) t2 t4))) (\lambda (t4: T).(subst0 (s -(Flat Cast) i) v0 t1 t4)) (ex T (\lambda (w: T).(ty3 g d v w))) (\lambda (x: -T).(\lambda (_: (eq T t3 (THead (Flat Cast) t2 x))).(\lambda (H8: (subst0 (s -(Flat Cast) i) v0 t1 x)).(H1 v0 x (s (Flat Cast) i) H8 b d v H5)))) H6)) -(\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead -(Flat Cast) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 t2 u2))) -(\lambda (_: T).(\lambda (t4: T).(subst0 (s (Flat Cast) i) v0 t1 -t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead -(Flat Cast) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 t2 u2))) -(\lambda (_: T).(\lambda (t4: T).(subst0 (s (Flat Cast) i) v0 t1 t4))) (ex T -(\lambda (w: T).(ty3 g d v w))) (\lambda (x0: T).(\lambda (x1: T).(\lambda -(_: (eq T t3 (THead (Flat Cast) x0 x1))).(\lambda (H8: (subst0 i v0 t2 -x0)).(\lambda (_: (subst0 (s (Flat Cast) i) v0 t1 x1)).(H3 v0 x0 i H8 b d v -H5)))))) H6)) (subst0_gen_head (Flat Cast) v0 t2 t1 t3 i H4)))))))))))))))))) -c t u H))))). -(* COMMENTS -Initial nodes: 4343 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/sty0.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/sty0.ma deleted file mode 100644 index 3c630662c..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/sty0.ma +++ /dev/null @@ -1,237 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/ty3/pr3_props.ma". - -include "Basic-1/sty0/fwd.ma". - -theorem ty3_sty0: - \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).((ty3 g c u -t1) \to (\forall (t2: T).((sty0 g c u t2) \to (ty3 g c u t2))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (H: -(ty3 g c u t1)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (_: -T).(\forall (t2: T).((sty0 g c0 t t2) \to (ty3 g c0 t t2)))))) (\lambda (c0: -C).(\lambda (t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda -(_: ((\forall (t3: T).((sty0 g c0 t2 t3) \to (ty3 g c0 t2 t3))))).(\lambda -(u0: T).(\lambda (t3: T).(\lambda (_: (ty3 g c0 u0 t3)).(\lambda (H3: -((\forall (t4: T).((sty0 g c0 u0 t4) \to (ty3 g c0 u0 t4))))).(\lambda (_: -(pc3 c0 t3 t2)).(\lambda (t0: T).(\lambda (H5: (sty0 g c0 u0 t0)).(H3 t0 -H5))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (t2: T).(\lambda -(H0: (sty0 g c0 (TSort m) t2)).(let H_y \def (sty0_gen_sort g c0 t2 m H0) in -(let H1 \def (f_equal T T (\lambda (e: T).e) t2 (TSort (next g m)) H_y) in -(eq_ind_r T (TSort (next g m)) (\lambda (t: T).(ty3 g c0 (TSort m) t)) -(ty3_sort g c0 m) t2 H1))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda -(d: C).(\lambda (u0: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abbr) -u0))).(\lambda (t: T).(\lambda (_: (ty3 g d u0 t)).(\lambda (H2: ((\forall -(t2: T).((sty0 g d u0 t2) \to (ty3 g d u0 t2))))).(\lambda (t2: T).(\lambda -(H3: (sty0 g c0 (TLRef n) t2)).(let H_x \def (sty0_gen_lref g c0 t2 n H3) in -(let H4 \def H_x in (or_ind (ex3_3 C T T (\lambda (e: C).(\lambda (u1: -T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: -C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (t0: T).(eq T t2 (lift (S n) O t0)))))) (ex3_3 C -T T (\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e -(Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g -e u1 t0)))) (\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq T t2 (lift -(S n) O u1)))))) (ty3 g c0 (TLRef n) t2) (\lambda (H5: (ex3_3 C T T (\lambda -(e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) -u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g e u1 t0)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(eq T t2 (lift (S n) O -t0))))))).(ex3_3_ind C T T (\lambda (e: C).(\lambda (u1: T).(\lambda (_: -T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: C).(\lambda (u1: -T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (t0: T).(eq T t2 (lift (S n) O t0))))) (ty3 g c0 (TLRef n) t2) -(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H6: (getl n c0 -(CHead x0 (Bind Abbr) x1))).(\lambda (H7: (sty0 g x0 x1 x2)).(\lambda (H8: -(eq T t2 (lift (S n) O x2))).(let H9 \def (f_equal T T (\lambda (e: T).e) t2 -(lift (S n) O x2) H8) in (eq_ind_r T (lift (S n) O x2) (\lambda (t0: T).(ty3 -g c0 (TLRef n) t0)) (let H10 \def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda -(c1: C).(getl n c0 c1)) H0 (CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d -(Bind Abbr) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in (let H11 \def (f_equal -C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u0) -(CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead -x0 (Bind Abbr) x1) H6)) in ((let H12 \def (f_equal C T (\lambda (e: C).(match -e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ -t0) \Rightarrow t0])) (CHead d (Bind Abbr) u0) (CHead x0 (Bind Abbr) x1) -(getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in -(\lambda (H13: (eq C d x0)).(let H14 \def (eq_ind_r T x1 (\lambda (t0: -T).(getl n c0 (CHead x0 (Bind Abbr) t0))) H10 u0 H12) in (let H15 \def -(eq_ind_r T x1 (\lambda (t0: T).(sty0 g x0 t0 x2)) H7 u0 H12) in (let H16 -\def (eq_ind_r C x0 (\lambda (c1: C).(getl n c0 (CHead c1 (Bind Abbr) u0))) -H14 d H13) in (let H17 \def (eq_ind_r C x0 (\lambda (c1: C).(sty0 g c1 u0 -x2)) H15 d H13) in (ty3_abbr g n c0 d u0 H16 x2 (H2 x2 H17)))))))) H11))) t2 -H9)))))))) H5)) (\lambda (H5: (ex3_3 C T T (\lambda (e: C).(\lambda (u1: -T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e: -C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq T t2 (lift (S n) O -u1))))))).(ex3_3_ind C T T (\lambda (e: C).(\lambda (u1: T).(\lambda (_: -T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1: -T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_: C).(\lambda (u1: -T).(\lambda (_: T).(eq T t2 (lift (S n) O u1))))) (ty3 g c0 (TLRef n) t2) -(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H6: (getl n c0 -(CHead x0 (Bind Abst) x1))).(\lambda (_: (sty0 g x0 x1 x2)).(\lambda (H8: (eq -T t2 (lift (S n) O x1))).(let H9 \def (f_equal T T (\lambda (e: T).e) t2 -(lift (S n) O x1) H8) in (eq_ind_r T (lift (S n) O x1) (\lambda (t0: T).(ty3 -g c0 (TLRef n) t0)) (let H10 \def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda -(c1: C).(getl n c0 c1)) H0 (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d -(Bind Abbr) u0) n H0 (CHead x0 (Bind Abst) x1) H6)) in (let H11 \def (eq_ind -C (CHead d (Bind Abbr) u0) (\lambda (ee: C).(match ee in C return (\lambda -(_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow -(match k in K return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match -b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst -\Rightarrow False | Void \Rightarrow False]) | (Flat _) \Rightarrow -False])])) I (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abbr) u0) -n H0 (CHead x0 (Bind Abst) x1) H6)) in (False_ind (ty3 g c0 (TLRef n) (lift -(S n) O x1)) H11))) t2 H9)))))))) H5)) H4))))))))))))) (\lambda (n: -nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (H0: (getl n -c0 (CHead d (Bind Abst) u0))).(\lambda (t: T).(\lambda (H1: (ty3 g d u0 -t)).(\lambda (_: ((\forall (t2: T).((sty0 g d u0 t2) \to (ty3 g d u0 -t2))))).(\lambda (t2: T).(\lambda (H3: (sty0 g c0 (TLRef n) t2)).(let H_x -\def (sty0_gen_lref g c0 t2 n H3) in (let H4 \def H_x in (or_ind (ex3_3 C T T -(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind -Abbr) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g e u1 -t0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(eq T t2 (lift (S n) -O t0)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u1: T).(\lambda (_: -T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1: -T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_: C).(\lambda (u1: -T).(\lambda (_: T).(eq T t2 (lift (S n) O u1)))))) (ty3 g c0 (TLRef n) t2) -(\lambda (H5: (ex3_3 C T T (\lambda (e: C).(\lambda (u1: T).(\lambda (_: -T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: C).(\lambda (u1: -T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_: C).(\lambda (_: -T).(\lambda (t0: T).(eq T t2 (lift (S n) O t0))))))).(ex3_3_ind C T T -(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind -Abbr) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g e u1 -t0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(eq T t2 (lift (S n) -O t0))))) (ty3 g c0 (TLRef n) t2) (\lambda (x0: C).(\lambda (x1: T).(\lambda -(x2: T).(\lambda (H6: (getl n c0 (CHead x0 (Bind Abbr) x1))).(\lambda (_: -(sty0 g x0 x1 x2)).(\lambda (H8: (eq T t2 (lift (S n) O x2))).(let H9 \def -(f_equal T T (\lambda (e: T).e) t2 (lift (S n) O x2) H8) in (eq_ind_r T (lift -(S n) O x2) (\lambda (t0: T).(ty3 g c0 (TLRef n) t0)) (let H10 \def (eq_ind C -(CHead d (Bind Abst) u0) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead x0 (Bind -Abbr) x1) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead x0 (Bind Abbr) -x1) H6)) in (let H11 \def (eq_ind C (CHead d (Bind Abst) u0) (\lambda (ee: -C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow -False | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) -with [(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with -[Abbr \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | -(Flat _) \Rightarrow False])])) I (CHead x0 (Bind Abbr) x1) (getl_mono c0 -(CHead d (Bind Abst) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in (False_ind -(ty3 g c0 (TLRef n) (lift (S n) O x2)) H11))) t2 H9)))))))) H5)) (\lambda -(H5: (ex3_3 C T T (\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 -(CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: -T).(sty0 g e u1 t0)))) (\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq T -t2 (lift (S n) O u1))))))).(ex3_3_ind C T T (\lambda (e: C).(\lambda (u1: -T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e: -C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_: -C).(\lambda (u1: T).(\lambda (_: T).(eq T t2 (lift (S n) O u1))))) (ty3 g c0 -(TLRef n) t2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda -(H6: (getl n c0 (CHead x0 (Bind Abst) x1))).(\lambda (H7: (sty0 g x0 x1 -x2)).(\lambda (H8: (eq T t2 (lift (S n) O x1))).(let H9 \def (f_equal T T -(\lambda (e: T).e) t2 (lift (S n) O x1) H8) in (eq_ind_r T (lift (S n) O x1) -(\lambda (t0: T).(ty3 g c0 (TLRef n) t0)) (let H10 \def (eq_ind C (CHead d -(Bind Abst) u0) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead x0 (Bind Abst) x1) -(getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead x0 (Bind Abst) x1) H6)) in -(let H11 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: -C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead -d (Bind Abst) u0) (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind -Abst) u0) n H0 (CHead x0 (Bind Abst) x1) H6)) in ((let H12 \def (f_equal C T -(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) -\Rightarrow u0 | (CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind Abst) u0) -(CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead -x0 (Bind Abst) x1) H6)) in (\lambda (H13: (eq C d x0)).(let H14 \def -(eq_ind_r T x1 (\lambda (t0: T).(getl n c0 (CHead x0 (Bind Abst) t0))) H10 u0 -H12) in (let H15 \def (eq_ind_r T x1 (\lambda (t0: T).(sty0 g x0 t0 x2)) H7 -u0 H12) in (eq_ind T u0 (\lambda (t0: T).(ty3 g c0 (TLRef n) (lift (S n) O -t0))) (let H16 \def (eq_ind_r C x0 (\lambda (c1: C).(getl n c0 (CHead c1 -(Bind Abst) u0))) H14 d H13) in (let H17 \def (eq_ind_r C x0 (\lambda (c1: -C).(sty0 g c1 u0 x2)) H15 d H13) in (ty3_abst g n c0 d u0 H16 t H1))) x1 -H12))))) H11))) t2 H9)))))))) H5)) H4))))))))))))) (\lambda (c0: C).(\lambda -(u0: T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 u0 t)).(\lambda (_: ((\forall -(t2: T).((sty0 g c0 u0 t2) \to (ty3 g c0 u0 t2))))).(\lambda (b: B).(\lambda -(t2: T).(\lambda (t3: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u0) t2 -t3)).(\lambda (H3: ((\forall (t4: T).((sty0 g (CHead c0 (Bind b) u0) t2 t4) -\to (ty3 g (CHead c0 (Bind b) u0) t2 t4))))).(\lambda (t0: T).(\lambda (H4: -(sty0 g c0 (THead (Bind b) u0 t2) t0)).(let H_x \def (sty0_gen_bind g b c0 u0 -t2 t0 H4) in (let H5 \def H_x in (ex2_ind T (\lambda (t4: T).(sty0 g (CHead -c0 (Bind b) u0) t2 t4)) (\lambda (t4: T).(eq T t0 (THead (Bind b) u0 t4))) -(ty3 g c0 (THead (Bind b) u0 t2) t0) (\lambda (x: T).(\lambda (H6: (sty0 g -(CHead c0 (Bind b) u0) t2 x)).(\lambda (H7: (eq T t0 (THead (Bind b) u0 -x))).(let H8 \def (f_equal T T (\lambda (e: T).e) t0 (THead (Bind b) u0 x) -H7) in (eq_ind_r T (THead (Bind b) u0 x) (\lambda (t4: T).(ty3 g c0 (THead -(Bind b) u0 t2) t4)) (ty3_bind g c0 u0 t H0 b t2 x (H3 x H6)) t0 H8))))) -H5))))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda (u0: T).(\lambda -(H0: (ty3 g c0 w u0)).(\lambda (_: ((\forall (t2: T).((sty0 g c0 w t2) \to -(ty3 g c0 w t2))))).(\lambda (v: T).(\lambda (t: T).(\lambda (H2: (ty3 g c0 v -(THead (Bind Abst) u0 t))).(\lambda (H3: ((\forall (t2: T).((sty0 g c0 v t2) -\to (ty3 g c0 v t2))))).(\lambda (t2: T).(\lambda (H4: (sty0 g c0 (THead -(Flat Appl) w v) t2)).(let H_x \def (sty0_gen_appl g c0 w v t2 H4) in (let H5 -\def H_x in (ex2_ind T (\lambda (t3: T).(sty0 g c0 v t3)) (\lambda (t3: -T).(eq T t2 (THead (Flat Appl) w t3))) (ty3 g c0 (THead (Flat Appl) w v) t2) -(\lambda (x: T).(\lambda (H6: (sty0 g c0 v x)).(\lambda (H7: (eq T t2 (THead -(Flat Appl) w x))).(let H8 \def (f_equal T T (\lambda (e: T).e) t2 (THead -(Flat Appl) w x) H7) in (eq_ind_r T (THead (Flat Appl) w x) (\lambda (t0: -T).(ty3 g c0 (THead (Flat Appl) w v) t0)) (let H_y \def (H3 x H6) in (let H9 -\def (ty3_unique g c0 v x H_y (THead (Bind Abst) u0 t) H2) in (ex_ind T -(\lambda (t0: T).(ty3 g c0 x t0)) (ty3 g c0 (THead (Flat Appl) w v) (THead -(Flat Appl) w x)) (\lambda (x0: T).(\lambda (H10: (ty3 g c0 x x0)).(ex_ind T -(\lambda (t0: T).(ty3 g c0 u0 t0)) (ty3 g c0 (THead (Flat Appl) w v) (THead -(Flat Appl) w x)) (\lambda (x1: T).(\lambda (_: (ty3 g c0 u0 x1)).(ex_ind T -(\lambda (t0: T).(ty3 g c0 (THead (Bind Abst) u0 t) t0)) (ty3 g c0 (THead -(Flat Appl) w v) (THead (Flat Appl) w x)) (\lambda (x2: T).(\lambda (H12: -(ty3 g c0 (THead (Bind Abst) u0 t) x2)).(ex3_2_ind T T (\lambda (t3: -T).(\lambda (_: T).(pc3 c0 (THead (Bind Abst) u0 t3) x2))) (\lambda (_: -T).(\lambda (t0: T).(ty3 g c0 u0 t0))) (\lambda (t3: T).(\lambda (_: T).(ty3 -g (CHead c0 (Bind Abst) u0) t t3))) (ty3 g c0 (THead (Flat Appl) w v) (THead -(Flat Appl) w x)) (\lambda (x3: T).(\lambda (x4: T).(\lambda (_: (pc3 c0 -(THead (Bind Abst) u0 x3) x2)).(\lambda (H14: (ty3 g c0 u0 x4)).(\lambda -(H15: (ty3 g (CHead c0 (Bind Abst) u0) t x3)).(ty3_conv g c0 (THead (Flat -Appl) w x) (THead (Flat Appl) w (THead (Bind Abst) u0 x3)) (ty3_appl g c0 w -u0 H0 x x3 (ty3_sconv g c0 x x0 H10 (THead (Bind Abst) u0 t) (THead (Bind -Abst) u0 x3) (ty3_bind g c0 u0 x4 H14 Abst t x3 H15) H9)) (THead (Flat Appl) -w v) (THead (Flat Appl) w (THead (Bind Abst) u0 t)) (ty3_appl g c0 w u0 H0 v -t H2) (pc3_thin_dx c0 (THead (Bind Abst) u0 t) x (ty3_unique g c0 v (THead -(Bind Abst) u0 t) H2 x H_y) w Appl))))))) (ty3_gen_bind g Abst c0 u0 t x2 -H12)))) (ty3_correct g c0 v (THead (Bind Abst) u0 t) H2)))) (ty3_correct g c0 -w u0 H0)))) (ty3_correct g c0 v x H_y)))) t2 H8))))) H5)))))))))))))) -(\lambda (c0: C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H0: (ty3 g c0 t2 -t3)).(\lambda (H1: ((\forall (t4: T).((sty0 g c0 t2 t4) \to (ty3 g c0 t2 -t4))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t3 t0)).(\lambda (H3: -((\forall (t4: T).((sty0 g c0 t3 t4) \to (ty3 g c0 t3 t4))))).(\lambda (t4: -T).(\lambda (H4: (sty0 g c0 (THead (Flat Cast) t3 t2) t4)).(let H_x \def -(sty0_gen_cast g c0 t3 t2 t4 H4) in (let H5 \def H_x in (ex3_2_ind T T -(\lambda (v2: T).(\lambda (_: T).(sty0 g c0 t3 v2))) (\lambda (_: T).(\lambda -(t5: T).(sty0 g c0 t2 t5))) (\lambda (v2: T).(\lambda (t5: T).(eq T t4 (THead -(Flat Cast) v2 t5)))) (ty3 g c0 (THead (Flat Cast) t3 t2) t4) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H6: (sty0 g c0 t3 x0)).(\lambda (H7: (sty0 g c0 -t2 x1)).(\lambda (H8: (eq T t4 (THead (Flat Cast) x0 x1))).(let H9 \def -(f_equal T T (\lambda (e: T).e) t4 (THead (Flat Cast) x0 x1) H8) in (eq_ind_r -T (THead (Flat Cast) x0 x1) (\lambda (t: T).(ty3 g c0 (THead (Flat Cast) t3 -t2) t)) (let H_y \def (H1 x1 H7) in (let H_y0 \def (H3 x0 H6) in (let H10 -\def (ty3_unique g c0 t2 x1 H_y t3 H0) in (ex_ind T (\lambda (t: T).(ty3 g c0 -x0 t)) (ty3 g c0 (THead (Flat Cast) t3 t2) (THead (Flat Cast) x0 x1)) -(\lambda (x: T).(\lambda (H11: (ty3 g c0 x0 x)).(ex_ind T (\lambda (t: -T).(ty3 g c0 x1 t)) (ty3 g c0 (THead (Flat Cast) t3 t2) (THead (Flat Cast) x0 -x1)) (\lambda (x2: T).(\lambda (H12: (ty3 g c0 x1 x2)).(ty3_conv g c0 (THead -(Flat Cast) x0 x1) (THead (Flat Cast) x x0) (ty3_cast g c0 x1 x0 (ty3_sconv g -c0 x1 x2 H12 t3 x0 H_y0 H10) x H11) (THead (Flat Cast) t3 t2) (THead (Flat -Cast) x0 t3) (ty3_cast g c0 t2 t3 H0 x0 H_y0) (pc3_thin_dx c0 t3 x1 -(ty3_unique g c0 t2 t3 H0 x1 H_y) x0 Cast)))) (ty3_correct g c0 t2 x1 H_y)))) -(ty3_correct g c0 t3 x0 H_y0))))) t4 H9))))))) H5))))))))))))) c u t1 H))))). -(* COMMENTS -Initial nodes: 4539 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/subst1.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/subst1.ma deleted file mode 100644 index ca9516e58..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/ty3/subst1.ma +++ /dev/null @@ -1,1102 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/ty3/props.ma". - -include "Basic-1/pc3/subst1.ma". - -include "Basic-1/getl/getl.ma". - -theorem ty3_gen_cabbr: - \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c -t1 t2) \to (\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c -(CHead e (Bind Abbr) u)) \to (\forall (a0: C).((csubst1 d u c a0) \to -(\forall (a: C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda -(_: T).(subst1 d u t1 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: -T).(subst1 d u t2 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 -g a y1 y2)))))))))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H: (ty3 g c t1 t2)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda -(t0: T).(\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead -e (Bind Abbr) u)) \to (\forall (a0: C).((csubst1 d u c0 a0) \to (\forall (a: -C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: -T).(subst1 d u t (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: -T).(subst1 d u t0 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 -g a y1 y2))))))))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t: -T).(\lambda (_: (ty3 g c0 t3 t)).(\lambda (H1: ((\forall (e: C).(\forall (u: -T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u)) \to (\forall (a0: -C).((csubst1 d u c0 a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (ex3_2 T -T (\lambda (y1: T).(\lambda (_: T).(subst1 d u t3 (lift (S O) d y1)))) -(\lambda (_: T).(\lambda (y2: T).(subst1 d u t (lift (S O) d y2)))) (\lambda -(y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))))).(\lambda (u: -T).(\lambda (t4: T).(\lambda (_: (ty3 g c0 u t4)).(\lambda (H3: ((\forall (e: -C).(\forall (u0: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u0)) -\to (\forall (a0: C).((csubst1 d u0 c0 a0) \to (\forall (a: C).((drop (S O) d -a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u (lift (S -O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t4 (lift (S O) d -y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 -y2)))))))))))))).(\lambda (H4: (pc3 c0 t4 t3)).(\lambda (e: C).(\lambda (u0: -T).(\lambda (d: nat).(\lambda (H5: (getl d c0 (CHead e (Bind Abbr) -u0))).(\lambda (a0: C).(\lambda (H6: (csubst1 d u0 c0 a0)).(\lambda (a: -C).(\lambda (H7: (drop (S O) d a0 a)).(let H8 \def (H3 e u0 d H5 a0 H6 a H7) -in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u (lift (S O) -d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t4 (lift (S O) d -y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T -(\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u (lift (S O) d y1)))) (\lambda -(_: T).(\lambda (y2: T).(subst1 d u0 t3 (lift (S O) d y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H9: (subst1 d u0 u (lift (S O) d x0))).(\lambda (H10: (subst1 d -u0 t4 (lift (S O) d x1))).(\lambda (H11: (ty3 g a x0 x1)).(let H12 \def (H1 e -u0 d H5 a0 H6 a H7) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: -T).(subst1 d u0 t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: -T).(subst1 d u0 t (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 -g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u (lift -(S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t3 (lift (S O) d -y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: -T).(\lambda (x3: T).(\lambda (H13: (subst1 d u0 t3 (lift (S O) d -x2))).(\lambda (_: (subst1 d u0 t (lift (S O) d x3))).(\lambda (H15: (ty3 g a -x2 x3)).(ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u -(lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t3 (lift -(S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) x0 x2 H9 -H13 (ty3_conv g a x2 x3 H15 x0 x1 H11 (pc3_gen_cabbr c0 t4 t3 H4 e u0 d H5 a0 -H6 a H7 x1 H10 x2 H13)))))))) H12))))))) H8)))))))))))))))))))) (\lambda (c0: -C).(\lambda (m: nat).(\lambda (e: C).(\lambda (u: T).(\lambda (d: -nat).(\lambda (_: (getl d c0 (CHead e (Bind Abbr) u))).(\lambda (a0: -C).(\lambda (_: (csubst1 d u c0 a0)).(\lambda (a: C).(\lambda (_: (drop (S O) -d a0 a)).(ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u (TSort -m) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u (TSort -(next g m)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a -y1 y2))) (TSort m) (TSort (next g m)) (eq_ind_r T (TSort m) (\lambda (t: -T).(subst1 d u (TSort m) t)) (subst1_refl d u (TSort m)) (lift (S O) d (TSort -m)) (lift_sort m (S O) d)) (eq_ind_r T (TSort (next g m)) (\lambda (t: -T).(subst1 d u (TSort (next g m)) t)) (subst1_refl d u (TSort (next g m))) -(lift (S O) d (TSort (next g m))) (lift_sort (next g m) (S O) d)) (ty3_sort g -a m)))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda -(u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abbr) u))).(\lambda (t: -T).(\lambda (H1: (ty3 g d u t)).(\lambda (H2: ((\forall (e: C).(\forall (u0: -T).(\forall (d0: nat).((getl d0 d (CHead e (Bind Abbr) u0)) \to (\forall (a0: -C).((csubst1 d0 u0 d a0) \to (\forall (a: C).((drop (S O) d0 a0 a) \to (ex3_2 -T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 u (lift (S O) d0 y1)))) -(\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 t (lift (S O) d0 y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))))).(\lambda (e: -C).(\lambda (u0: T).(\lambda (d0: nat).(\lambda (H3: (getl d0 c0 (CHead e -(Bind Abbr) u0))).(\lambda (a0: C).(\lambda (H4: (csubst1 d0 u0 c0 -a0)).(\lambda (a: C).(\lambda (H5: (drop (S O) d0 a0 a)).(lt_eq_gt_e n d0 -(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S -O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O t) -(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) -(\lambda (H6: (lt n d0)).(let H7 \def (eq_ind nat (minus d0 n) (\lambda (n0: -nat).(getl n0 (CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0))) -(getl_conf_le d0 (CHead e (Bind Abbr) u0) c0 H3 (CHead d (Bind Abbr) u) n H0 -(le_S_n n d0 (le_S (S n) d0 H6))) (S (minus d0 (S n))) (minus_x_Sy d0 n H6)) -in (ex2_ind C (\lambda (e2: C).(csubst1 (minus d0 n) u0 (CHead d (Bind Abbr) -u) e2)) (\lambda (e2: C).(getl n a0 e2)) (ex3_2 T T (\lambda (y1: T).(\lambda -(_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda -(y2: T).(subst1 d0 u0 (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x: C).(\lambda (H8: (csubst1 -(minus d0 n) u0 (CHead d (Bind Abbr) u) x)).(\lambda (H9: (getl n a0 x)).(let -H10 \def (eq_ind nat (minus d0 n) (\lambda (n0: nat).(csubst1 n0 u0 (CHead d -(Bind Abbr) u) x)) H8 (S (minus d0 (S n))) (minus_x_Sy d0 n H6)) in (let H11 -\def (csubst1_gen_head (Bind Abbr) d x u u0 (minus d0 (S n)) H10) in -(ex3_2_ind T C (\lambda (u2: T).(\lambda (c2: C).(eq C x (CHead c2 (Bind -Abbr) u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 (minus d0 (S n)) u0 u -u2))) (\lambda (_: T).(\lambda (c2: C).(csubst1 (minus d0 (S n)) u0 d c2))) -(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S -O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O t) -(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) -(\lambda (x0: T).(\lambda (x1: C).(\lambda (H12: (eq C x (CHead x1 (Bind -Abbr) x0))).(\lambda (H13: (subst1 (minus d0 (S n)) u0 u x0)).(\lambda (H14: -(csubst1 (minus d0 (S n)) u0 d x1)).(let H15 \def (eq_ind C x (\lambda (c1: -C).(getl n a0 c1)) H9 (CHead x1 (Bind Abbr) x0) H12) in (let H16 \def (eq_ind -nat d0 (\lambda (n0: nat).(drop (S O) n0 a0 a)) H5 (S (plus n (minus d0 (S -n)))) (lt_plus_minus n d0 H6)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: -C).(eq T x0 (lift (S O) (minus d0 (S n)) v)))) (\lambda (v: T).(\lambda (e0: -C).(getl n a (CHead e0 (Bind Abbr) v)))) (\lambda (_: T).(\lambda (e0: -C).(drop (S O) (minus d0 (S n)) x1 e0))) (ex3_2 T T (\lambda (y1: T).(\lambda -(_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda -(y2: T).(subst1 d0 u0 (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: T).(\lambda (x3: -C).(\lambda (H17: (eq T x0 (lift (S O) (minus d0 (S n)) x2))).(\lambda (H18: -(getl n a (CHead x3 (Bind Abbr) x2))).(\lambda (H19: (drop (S O) (minus d0 (S -n)) x1 x3)).(let H20 \def (eq_ind T x0 (\lambda (t0: T).(subst1 (minus d0 (S -n)) u0 u t0)) H13 (lift (S O) (minus d0 (S n)) x2) H17) in (let H21 \def (H2 -e u0 (minus d0 (S n)) (getl_gen_S (Bind Abbr) d (CHead e (Bind Abbr) u0) u -(minus d0 (S n)) H7) x1 H14 x3 H19) in (ex3_2_ind T T (\lambda (y1: -T).(\lambda (_: T).(subst1 (minus d0 (S n)) u0 u (lift (S O) (minus d0 (S n)) -y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 (minus d0 (S n)) u0 t (lift -(S O) (minus d0 (S n)) y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g x3 y1 -y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) -(lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S -n) O t) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 -y2)))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H22: (subst1 (minus d0 (S -n)) u0 u (lift (S O) (minus d0 (S n)) x4))).(\lambda (H23: (subst1 (minus d0 -(S n)) u0 t (lift (S O) (minus d0 (S n)) x5))).(\lambda (H24: (ty3 g x3 x4 -x5)).(let H25 \def (eq_ind T x4 (\lambda (t0: T).(ty3 g x3 t0 x5)) H24 x2 -(subst1_confluence_lift u x4 u0 (minus d0 (S n)) H22 x2 H20)) in (eq_ind_r -nat (plus (minus d0 (S n)) (S n)) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: -T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: -T).(\lambda (y2: T).(subst1 n0 u0 (lift (S n) O t) (lift (S O) d0 y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r nat (plus (S -n) (minus d0 (S n))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda -(_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda -(y2: T).(subst1 (plus (minus d0 (S n)) (S n)) u0 (lift (S n) O t) (lift (S O) -n0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro -T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 -y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 (plus (minus d0 (S n)) (S n)) -u0 (lift (S n) O t) (lift (S O) (plus (S n) (minus d0 (S n))) y2)))) (\lambda -(y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (TLRef n) (lift (S n) O x5) -(eq_ind_r T (TLRef n) (\lambda (t0: T).(subst1 d0 u0 (TLRef n) t0)) -(subst1_refl d0 u0 (TLRef n)) (lift (S O) d0 (TLRef n)) (lift_lref_lt n (S O) -d0 H6)) (eq_ind_r T (lift (S n) O (lift (S O) (minus d0 (S n)) x5)) (\lambda -(t0: T).(subst1 (plus (minus d0 (S n)) (S n)) u0 (lift (S n) O t) t0)) -(subst1_lift_ge t (lift (S O) (minus d0 (S n)) x5) u0 (minus d0 (S n)) (S n) -H23 O (le_O_n (minus d0 (S n)))) (lift (S O) (plus (S n) (minus d0 (S n))) -(lift (S n) O x5)) (lift_d x5 (S O) (S n) (minus d0 (S n)) O (le_O_n (minus -d0 (S n))))) (ty3_abbr g n a x3 x2 H18 x5 H25)) d0 (le_plus_minus (S n) d0 -H6)) d0 (le_plus_minus_sym (S n) d0 H6)))))))) H21)))))))) (getl_drop_conf_lt -Abbr a0 x1 x0 n H15 a (S O) (minus d0 (S n)) H16))))))))) H11)))))) -(csubst1_getl_lt d0 n H6 c0 a0 u0 H4 (CHead d (Bind Abbr) u) H0)))) (\lambda -(H6: (eq nat n d0)).(let H7 \def (eq_ind_r nat d0 (\lambda (n0: nat).(drop (S -O) n0 a0 a)) H5 n H6) in (let H8 \def (eq_ind_r nat d0 (\lambda (n0: -nat).(csubst1 n0 u0 c0 a0)) H4 n H6) in (let H9 \def (eq_ind_r nat d0 -(\lambda (n0: nat).(getl n0 c0 (CHead e (Bind Abbr) u0))) H3 n H6) in (eq_ind -nat n (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 -n0 u0 (TLRef n) (lift (S O) n0 y1)))) (\lambda (_: T).(\lambda (y2: -T).(subst1 n0 u0 (lift (S n) O t) (lift (S O) n0 y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H10 \def (eq_ind C (CHead d -(Bind Abbr) u) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead e (Bind Abbr) u0) -(getl_mono c0 (CHead d (Bind Abbr) u) n H0 (CHead e (Bind Abbr) u0) H9)) in -(let H11 \def (f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda -(_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) -(CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d (Bind -Abbr) u) n H0 (CHead e (Bind Abbr) u0) H9)) in ((let H12 \def (f_equal C T -(\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) -\Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind Abbr) u) -(CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d (Bind Abbr) u) n H0 (CHead e -(Bind Abbr) u0) H9)) in (\lambda (H13: (eq C d e)).(let H14 \def (eq_ind_r T -u0 (\lambda (t0: T).(getl n c0 (CHead e (Bind Abbr) t0))) H10 u H12) in (let -H15 \def (eq_ind_r T u0 (\lambda (t0: T).(csubst1 n t0 c0 a0)) H8 u H12) in -(eq_ind T u (\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: -T).(subst1 n t0 (TLRef n) (lift (S O) n y1)))) (\lambda (_: T).(\lambda (y2: -T).(subst1 n t0 (lift (S n) O t) (lift (S O) n y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H16 \def (eq_ind_r C e (\lambda -(c1: C).(getl n c0 (CHead c1 (Bind Abbr) u))) H14 d H13) in (ex3_2_intro T T -(\lambda (y1: T).(\lambda (_: T).(subst1 n u (TLRef n) (lift (S O) n y1)))) -(\lambda (_: T).(\lambda (y2: T).(subst1 n u (lift (S n) O t) (lift (S O) n -y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (lift n O u) (lift -n O t) (subst1_single n u (TLRef n) (lift (S O) n (lift n O u)) (eq_ind_r T -(lift (plus (S O) n) O u) (\lambda (t0: T).(subst0 n u (TLRef n) t0)) -(subst0_lref u n) (lift (S O) n (lift n O u)) (lift_free u n (S O) O n (le_n -(plus O n)) (le_O_n n)))) (eq_ind_r T (lift (plus (S O) n) O t) (\lambda (t0: -T).(subst1 n u (lift (S n) O t) t0)) (subst1_refl n u (lift (S n) O t)) (lift -(S O) n (lift n O t)) (lift_free t n (S O) O n (le_n (plus O n)) (le_O_n n))) -(ty3_lift g d u t H1 a O n (getl_conf_ge_drop Abbr a0 d u n (csubst1_getl_ge -n n (le_n n) c0 a0 u H15 (CHead d (Bind Abbr) u) H16) a H7)))) u0 H12))))) -H11))) d0 H6))))) (\lambda (H6: (lt d0 n)).(eq_ind_r nat (S (plus O (minus n -(S O)))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: -T).(subst1 d0 u0 (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda -(y2: T).(subst1 d0 u0 (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind nat (plus (S O) (minus n (S -O))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 -d0 u0 (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: -T).(subst1 d0 u0 (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r nat (plus (minus n (S O)) -(S O)) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 -d0 u0 (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: -T).(subst1 d0 u0 (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1: -T).(\lambda (_: T).(subst1 d0 u0 (TLRef (plus (minus n (S O)) (S O))) (lift -(S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O -t) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) -(TLRef (minus n (S O))) (lift n O t) (eq_ind_r T (TLRef (plus (minus n (S O)) -(S O))) (\lambda (t0: T).(subst1 d0 u0 (TLRef (plus (minus n (S O)) (S O))) -t0)) (subst1_refl d0 u0 (TLRef (plus (minus n (S O)) (S O)))) (lift (S O) d0 -(TLRef (minus n (S O)))) (lift_lref_ge (minus n (S O)) (S O) d0 (lt_le_minus -d0 n H6))) (eq_ind_r T (lift (plus (S O) n) O t) (\lambda (t0: T).(subst1 d0 -u0 (lift (S n) O t) t0)) (subst1_refl d0 u0 (lift (S n) O t)) (lift (S O) d0 -(lift n O t)) (lift_free t n (S O) O d0 (le_S_n d0 (plus O n) (le_S (S d0) -(plus O n) H6)) (le_O_n d0))) (eq_ind_r nat (S (minus n (S O))) (\lambda (n0: -nat).(ty3 g a (TLRef (minus n (S O))) (lift n0 O t))) (ty3_abbr g (minus n (S -O)) a d u (getl_drop_conf_ge n (CHead d (Bind Abbr) u) a0 (csubst1_getl_ge d0 -n (le_S_n d0 n (le_S (S d0) n H6)) c0 a0 u0 H4 (CHead d (Bind Abbr) u) H0) a -(S O) d0 H5 (eq_ind_r nat (plus (S O) d0) (\lambda (n0: nat).(le n0 n)) H6 -(plus d0 (S O)) (plus_sym d0 (S O)))) t H1) n (minus_x_SO n (le_lt_trans O d0 -n (le_O_n d0) H6)))) (plus (S O) (minus n (S O))) (plus_sym (S O) (minus n (S -O)))) (S (plus O (minus n (S O)))) (refl_equal nat (S (plus O (minus n (S -O)))))) n (lt_plus_minus O n (le_lt_trans O d0 n (le_O_n d0) -H6))))))))))))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: -C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abst) -u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u t)).(\lambda (H2: ((\forall (e: -C).(\forall (u0: T).(\forall (d0: nat).((getl d0 d (CHead e (Bind Abbr) u0)) -\to (\forall (a0: C).((csubst1 d0 u0 d a0) \to (\forall (a: C).((drop (S O) -d0 a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 u -(lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 t (lift -(S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 -y2)))))))))))))).(\lambda (e: C).(\lambda (u0: T).(\lambda (d0: nat).(\lambda -(H3: (getl d0 c0 (CHead e (Bind Abbr) u0))).(\lambda (a0: C).(\lambda (H4: -(csubst1 d0 u0 c0 a0)).(\lambda (a: C).(\lambda (H5: (drop (S O) d0 a0 -a)).(lt_eq_gt_e n d0 (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 -u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 -d0 u0 (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: -T).(ty3 g a y1 y2)))) (\lambda (H6: (lt n d0)).(let H7 \def (eq_ind nat -(minus d0 n) (\lambda (n0: nat).(getl n0 (CHead d (Bind Abst) u) (CHead e -(Bind Abbr) u0))) (getl_conf_le d0 (CHead e (Bind Abbr) u0) c0 H3 (CHead d -(Bind Abst) u) n H0 (le_S_n n d0 (le_S (S n) d0 H6))) (S (minus d0 (S n))) -(minus_x_Sy d0 n H6)) in (ex2_ind C (\lambda (e2: C).(csubst1 (minus d0 n) u0 -(CHead d (Bind Abst) u) e2)) (\lambda (e2: C).(getl n a0 e2)) (ex3_2 T T -(\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 -y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O u) (lift -(S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda -(x: C).(\lambda (H8: (csubst1 (minus d0 n) u0 (CHead d (Bind Abst) u) -x)).(\lambda (H9: (getl n a0 x)).(let H10 \def (eq_ind nat (minus d0 n) -(\lambda (n0: nat).(csubst1 n0 u0 (CHead d (Bind Abst) u) x)) H8 (S (minus d0 -(S n))) (minus_x_Sy d0 n H6)) in (let H11 \def (csubst1_gen_head (Bind Abst) -d x u u0 (minus d0 (S n)) H10) in (ex3_2_ind T C (\lambda (u2: T).(\lambda -(c2: C).(eq C x (CHead c2 (Bind Abst) u2)))) (\lambda (u2: T).(\lambda (_: -C).(subst1 (minus d0 (S n)) u0 u u2))) (\lambda (_: T).(\lambda (c2: -C).(csubst1 (minus d0 (S n)) u0 d c2))) (ex3_2 T T (\lambda (y1: T).(\lambda -(_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda -(y2: T).(subst1 d0 u0 (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1: -C).(\lambda (H12: (eq C x (CHead x1 (Bind Abst) x0))).(\lambda (H13: (subst1 -(minus d0 (S n)) u0 u x0)).(\lambda (H14: (csubst1 (minus d0 (S n)) u0 d -x1)).(let H15 \def (eq_ind C x (\lambda (c1: C).(getl n a0 c1)) H9 (CHead x1 -(Bind Abst) x0) H12) in (let H16 \def (eq_ind nat d0 (\lambda (n0: nat).(drop -(S O) n0 a0 a)) H5 (S (plus n (minus d0 (S n)))) (lt_plus_minus n d0 H6)) in -(ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T x0 (lift (S O) (minus d0 -(S n)) v)))) (\lambda (v: T).(\lambda (e0: C).(getl n a (CHead e0 (Bind Abst) -v)))) (\lambda (_: T).(\lambda (e0: C).(drop (S O) (minus d0 (S n)) x1 e0))) -(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S -O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O u) -(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) -(\lambda (x2: T).(\lambda (x3: C).(\lambda (H17: (eq T x0 (lift (S O) (minus -d0 (S n)) x2))).(\lambda (H18: (getl n a (CHead x3 (Bind Abst) x2))).(\lambda -(H19: (drop (S O) (minus d0 (S n)) x1 x3)).(let H20 \def (eq_ind T x0 -(\lambda (t0: T).(subst1 (minus d0 (S n)) u0 u t0)) H13 (lift (S O) (minus d0 -(S n)) x2) H17) in (let H21 \def (H2 e u0 (minus d0 (S n)) (getl_gen_S (Bind -Abst) d (CHead e (Bind Abbr) u0) u (minus d0 (S n)) H7) x1 H14 x3 H19) in -(ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(subst1 (minus d0 (S n)) u0 u -(lift (S O) (minus d0 (S n)) y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 -(minus d0 (S n)) u0 t (lift (S O) (minus d0 (S n)) y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g x3 y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda -(_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda -(y2: T).(subst1 d0 u0 (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x4: T).(\lambda (x5: -T).(\lambda (H22: (subst1 (minus d0 (S n)) u0 u (lift (S O) (minus d0 (S n)) -x4))).(\lambda (_: (subst1 (minus d0 (S n)) u0 t (lift (S O) (minus d0 (S n)) -x5))).(\lambda (H24: (ty3 g x3 x4 x5)).(let H25 \def (eq_ind T x4 (\lambda -(t0: T).(ty3 g x3 t0 x5)) H24 x2 (subst1_confluence_lift u x4 u0 (minus d0 (S -n)) H22 x2 H20)) in (eq_ind_r nat (plus (minus d0 (S n)) (S n)) (\lambda (n0: -nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) -(lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 n0 u0 (lift (S -n) O u) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 -y2))))) (eq_ind_r nat (plus (S n) (minus d0 (S n))) (\lambda (n0: nat).(ex3_2 -T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 -y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 (plus (minus d0 (S n)) (S n)) -u0 (lift (S n) O u) (lift (S O) n0 y2)))) (\lambda (y1: T).(\lambda (y2: -T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: -T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda -(y2: T).(subst1 (plus (minus d0 (S n)) (S n)) u0 (lift (S n) O u) (lift (S O) -(plus (S n) (minus d0 (S n))) y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g -a y1 y2))) (TLRef n) (lift (S n) O x2) (eq_ind_r T (TLRef n) (\lambda (t0: -T).(subst1 d0 u0 (TLRef n) t0)) (subst1_refl d0 u0 (TLRef n)) (lift (S O) d0 -(TLRef n)) (lift_lref_lt n (S O) d0 H6)) (eq_ind_r T (lift (S n) O (lift (S -O) (minus d0 (S n)) x2)) (\lambda (t0: T).(subst1 (plus (minus d0 (S n)) (S -n)) u0 (lift (S n) O u) t0)) (subst1_lift_ge u (lift (S O) (minus d0 (S n)) -x2) u0 (minus d0 (S n)) (S n) H20 O (le_O_n (minus d0 (S n)))) (lift (S O) -(plus (S n) (minus d0 (S n))) (lift (S n) O x2)) (lift_d x2 (S O) (S n) -(minus d0 (S n)) O (le_O_n (minus d0 (S n))))) (ty3_abst g n a x3 x2 H18 x5 -H25)) d0 (le_plus_minus (S n) d0 H6)) d0 (le_plus_minus_sym (S n) d0 -H6)))))))) H21)))))))) (getl_drop_conf_lt Abst a0 x1 x0 n H15 a (S O) (minus -d0 (S n)) H16))))))))) H11)))))) (csubst1_getl_lt d0 n H6 c0 a0 u0 H4 (CHead -d (Bind Abst) u) H0)))) (\lambda (H6: (eq nat n d0)).(let H7 \def (eq_ind_r -nat d0 (\lambda (n0: nat).(drop (S O) n0 a0 a)) H5 n H6) in (let H8 \def -(eq_ind_r nat d0 (\lambda (n0: nat).(csubst1 n0 u0 c0 a0)) H4 n H6) in (let -H9 \def (eq_ind_r nat d0 (\lambda (n0: nat).(getl n0 c0 (CHead e (Bind Abbr) -u0))) H3 n H6) in (eq_ind nat n (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: -T).(\lambda (_: T).(subst1 n0 u0 (TLRef n) (lift (S O) n0 y1)))) (\lambda (_: -T).(\lambda (y2: T).(subst1 n0 u0 (lift (S n) O u) (lift (S O) n0 y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H10 \def (eq_ind C -(CHead d (Bind Abst) u) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead e (Bind -Abbr) u0) (getl_mono c0 (CHead d (Bind Abst) u) n H0 (CHead e (Bind Abbr) u0) -H9)) in (let H11 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (ee: -C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow -False | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) -with [(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with -[Abbr \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | -(Flat _) \Rightarrow False])])) I (CHead e (Bind Abbr) u0) (getl_mono c0 -(CHead d (Bind Abst) u) n H0 (CHead e (Bind Abbr) u0) H9)) in (False_ind -(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 n u0 (TLRef n) (lift (S -O) n y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 n u0 (lift (S n) O u) -(lift (S O) n y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) -H11))) d0 H6))))) (\lambda (H6: (lt d0 n)).(eq_ind_r nat (S (plus O (minus n -(S O)))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: -T).(subst1 d0 u0 (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda -(y2: T).(subst1 d0 u0 (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind nat (plus (S O) (minus n (S -O))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 -d0 u0 (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: -T).(subst1 d0 u0 (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r nat (plus (minus n (S O)) -(S O)) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 -d0 u0 (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: -T).(subst1 d0 u0 (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1: -T).(\lambda (_: T).(subst1 d0 u0 (TLRef (plus (minus n (S O)) (S O))) (lift -(S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O -u) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) -(TLRef (minus n (S O))) (lift n O u) (eq_ind_r T (TLRef (plus (minus n (S O)) -(S O))) (\lambda (t0: T).(subst1 d0 u0 (TLRef (plus (minus n (S O)) (S O))) -t0)) (subst1_refl d0 u0 (TLRef (plus (minus n (S O)) (S O)))) (lift (S O) d0 -(TLRef (minus n (S O)))) (lift_lref_ge (minus n (S O)) (S O) d0 (lt_le_minus -d0 n H6))) (eq_ind_r T (lift (plus (S O) n) O u) (\lambda (t0: T).(subst1 d0 -u0 (lift (S n) O u) t0)) (subst1_refl d0 u0 (lift (S n) O u)) (lift (S O) d0 -(lift n O u)) (lift_free u n (S O) O d0 (le_S_n d0 (plus O n) (le_S (S d0) -(plus O n) H6)) (le_O_n d0))) (eq_ind_r nat (S (minus n (S O))) (\lambda (n0: -nat).(ty3 g a (TLRef (minus n (S O))) (lift n0 O u))) (ty3_abst g (minus n (S -O)) a d u (getl_drop_conf_ge n (CHead d (Bind Abst) u) a0 (csubst1_getl_ge d0 -n (le_S_n d0 n (le_S (S d0) n H6)) c0 a0 u0 H4 (CHead d (Bind Abst) u) H0) a -(S O) d0 H5 (eq_ind_r nat (plus (S O) d0) (\lambda (n0: nat).(le n0 n)) H6 -(plus d0 (S O)) (plus_sym d0 (S O)))) t H1) n (minus_x_SO n (le_lt_trans O d0 -n (le_O_n d0) H6)))) (plus (S O) (minus n (S O))) (plus_sym (S O) (minus n (S -O)))) (S (plus O (minus n (S O)))) (refl_equal nat (S (plus O (minus n (S -O)))))) n (lt_plus_minus O n (le_lt_trans O d0 n (le_O_n d0) -H6))))))))))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (t: -T).(\lambda (_: (ty3 g c0 u t)).(\lambda (H1: ((\forall (e: C).(\forall (u0: -T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u0)) \to (\forall (a0: -C).((csubst1 d u0 c0 a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (ex3_2 -T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u (lift (S O) d y1)))) -(\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t (lift (S O) d y2)))) (\lambda -(y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))))).(\lambda (b: -B).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) -u) t3 t4)).(\lambda (H3: ((\forall (e: C).(\forall (u0: T).(\forall (d: -nat).((getl d (CHead c0 (Bind b) u) (CHead e (Bind Abbr) u0)) \to (\forall -(a0: C).((csubst1 d u0 (CHead c0 (Bind b) u) a0) \to (\forall (a: C).((drop -(S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 t3 -(lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t4 (lift -(S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 -y2)))))))))))))).(\lambda (e: C).(\lambda (u0: T).(\lambda (d: nat).(\lambda -(H4: (getl d c0 (CHead e (Bind Abbr) u0))).(\lambda (a0: C).(\lambda (H5: -(csubst1 d u0 c0 a0)).(\lambda (a: C).(\lambda (H6: (drop (S O) d a0 a)).(let -H7 \def (H1 e u0 d H4 a0 H5 a H6) in (ex3_2_ind T T (\lambda (y1: T).(\lambda -(_: T).(subst1 d u0 u (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: -T).(subst1 d u0 t (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 -g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 (THead -(Bind b) u t3) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 -d u0 (THead (Bind b) u t4) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda -(y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H8: -(subst1 d u0 u (lift (S O) d x0))).(\lambda (_: (subst1 d u0 t (lift (S O) d -x1))).(\lambda (H10: (ty3 g a x0 x1)).(let H11 \def (H3 e u0 (S d) (getl_head -(Bind b) d c0 (CHead e (Bind Abbr) u0) H4 u) (CHead a0 (Bind b) (lift (S O) d -x0)) (csubst1_bind b d u0 u (lift (S O) d x0) H8 c0 a0 H5) (CHead a (Bind b) -x0) (drop_skip_bind (S O) d a0 a H6 b x0)) in (ex3_2_ind T T (\lambda (y1: -T).(\lambda (_: T).(subst1 (S d) u0 t3 (lift (S O) (S d) y1)))) (\lambda (_: -T).(\lambda (y2: T).(subst1 (S d) u0 t4 (lift (S O) (S d) y2)))) (\lambda -(y1: T).(\lambda (y2: T).(ty3 g (CHead a (Bind b) x0) y1 y2))) (ex3_2 T T -(\lambda (y1: T).(\lambda (_: T).(subst1 d u0 (THead (Bind b) u t3) (lift (S -O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Bind b) u -t4) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) -(\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (subst1 (S d) u0 t3 (lift (S -O) (S d) x2))).(\lambda (H13: (subst1 (S d) u0 t4 (lift (S O) (S d) -x3))).(\lambda (H14: (ty3 g (CHead a (Bind b) x0) x2 x3)).(ex3_2_intro T T -(\lambda (y1: T).(\lambda (_: T).(subst1 d u0 (THead (Bind b) u t3) (lift (S -O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Bind b) u -t4) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) -(THead (Bind b) x0 x2) (THead (Bind b) x0 x3) (eq_ind_r T (THead (Bind b) -(lift (S O) d x0) (lift (S O) (S d) x2)) (\lambda (t0: T).(subst1 d u0 (THead -(Bind b) u t3) t0)) (subst1_head u0 u (lift (S O) d x0) d H8 (Bind b) t3 -(lift (S O) (S d) x2) H12) (lift (S O) d (THead (Bind b) x0 x2)) (lift_bind b -x0 x2 (S O) d)) (eq_ind_r T (THead (Bind b) (lift (S O) d x0) (lift (S O) (S -d) x3)) (\lambda (t0: T).(subst1 d u0 (THead (Bind b) u t4) t0)) (subst1_head -u0 u (lift (S O) d x0) d H8 (Bind b) t4 (lift (S O) (S d) x3) H13) (lift (S -O) d (THead (Bind b) x0 x3)) (lift_bind b x0 x3 (S O) d)) (ty3_bind g a x0 x1 -H10 b x2 x3 H14))))))) H11))))))) H7)))))))))))))))))))) (\lambda (c0: -C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda (H1: -((\forall (e: C).(\forall (u0: T).(\forall (d: nat).((getl d c0 (CHead e -(Bind Abbr) u0)) \to (\forall (a0: C).((csubst1 d u0 c0 a0) \to (\forall (a: -C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: -T).(subst1 d u0 w (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: -T).(subst1 d u0 u (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 -g a y1 y2)))))))))))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g -c0 v (THead (Bind Abst) u t))).(\lambda (H3: ((\forall (e: C).(\forall (u0: -T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u0)) \to (\forall (a0: -C).((csubst1 d u0 c0 a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (ex3_2 -T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 v (lift (S O) d y1)))) -(\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Bind Abst) u t) (lift -(S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 -y2)))))))))))))).(\lambda (e: C).(\lambda (u0: T).(\lambda (d: nat).(\lambda -(H4: (getl d c0 (CHead e (Bind Abbr) u0))).(\lambda (a0: C).(\lambda (H5: -(csubst1 d u0 c0 a0)).(\lambda (a: C).(\lambda (H6: (drop (S O) d a0 a)).(let -H7 \def (H3 e u0 d H4 a0 H5 a H6) in (ex3_2_ind T T (\lambda (y1: T).(\lambda -(_: T).(subst1 d u0 v (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: -T).(subst1 d u0 (THead (Bind Abst) u t) (lift (S O) d y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda -(_: T).(subst1 d u0 (THead (Flat Appl) w v) (lift (S O) d y1)))) (\lambda (_: -T).(\lambda (y2: T).(subst1 d u0 (THead (Flat Appl) w (THead (Bind Abst) u -t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) -(\lambda (x0: T).(\lambda (x1: T).(\lambda (H8: (subst1 d u0 v (lift (S O) d -x0))).(\lambda (H9: (subst1 d u0 (THead (Bind Abst) u t) (lift (S O) d -x1))).(\lambda (H10: (ty3 g a x0 x1)).(let H11 \def (H1 e u0 d H4 a0 H5 a H6) -in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 w (lift (S O) -d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 u (lift (S O) d y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: -T).(\lambda (_: T).(subst1 d u0 (THead (Flat Appl) w v) (lift (S O) d y1)))) -(\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Flat Appl) w (THead -(Bind Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 -g a y1 y2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (subst1 d u0 w -(lift (S O) d x2))).(\lambda (H13: (subst1 d u0 u (lift (S O) d -x3))).(\lambda (H14: (ty3 g a x2 x3)).(let H_x \def (subst1_gen_head (Bind -Abst) u0 u t (lift (S O) d x1) d H9) in (let H15 \def H_x in (ex3_2_ind T T -(\lambda (u2: T).(\lambda (t3: T).(eq T (lift (S O) d x1) (THead (Bind Abst) -u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1 d u0 u u2))) (\lambda (_: -T).(\lambda (t3: T).(subst1 (S d) u0 t t3))) (ex3_2 T T (\lambda (y1: -T).(\lambda (_: T).(subst1 d u0 (THead (Flat Appl) w v) (lift (S O) d y1)))) -(\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Flat Appl) w (THead -(Bind Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 -g a y1 y2)))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H16: (eq T (lift (S -O) d x1) (THead (Bind Abst) x4 x5))).(\lambda (H17: (subst1 d u0 u -x4)).(\lambda (H18: (subst1 (S d) u0 t x5)).(let H19 \def (sym_eq T (lift (S -O) d x1) (THead (Bind Abst) x4 x5) H16) in (ex3_2_ind T T (\lambda (y: -T).(\lambda (z: T).(eq T x1 (THead (Bind Abst) y z)))) (\lambda (y: -T).(\lambda (_: T).(eq T x4 (lift (S O) d y)))) (\lambda (_: T).(\lambda (z: -T).(eq T x5 (lift (S O) (S d) z)))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: -T).(subst1 d u0 (THead (Flat Appl) w v) (lift (S O) d y1)))) (\lambda (_: -T).(\lambda (y2: T).(subst1 d u0 (THead (Flat Appl) w (THead (Bind Abst) u -t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) -(\lambda (x6: T).(\lambda (x7: T).(\lambda (H20: (eq T x1 (THead (Bind Abst) -x6 x7))).(\lambda (H21: (eq T x4 (lift (S O) d x6))).(\lambda (H22: (eq T x5 -(lift (S O) (S d) x7))).(let H23 \def (eq_ind T x5 (\lambda (t0: T).(subst1 -(S d) u0 t t0)) H18 (lift (S O) (S d) x7) H22) in (let H24 \def (eq_ind T x4 -(\lambda (t0: T).(subst1 d u0 u t0)) H17 (lift (S O) d x6) H21) in (let H25 -\def (eq_ind T x1 (\lambda (t0: T).(ty3 g a x0 t0)) H10 (THead (Bind Abst) x6 -x7) H20) in (let H26 \def (eq_ind T x6 (\lambda (t0: T).(ty3 g a x0 (THead -(Bind Abst) t0 x7))) H25 x3 (subst1_confluence_lift u x6 u0 d H24 x3 H13)) in -(ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 (THead (Flat -Appl) w v) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 -(THead (Flat Appl) w (THead (Bind Abst) u t)) (lift (S O) d y2)))) (\lambda -(y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (THead (Flat Appl) x2 x0) (THead -(Flat Appl) x2 (THead (Bind Abst) x3 x7)) (eq_ind_r T (THead (Flat Appl) -(lift (S O) d x2) (lift (S O) d x0)) (\lambda (t0: T).(subst1 d u0 (THead -(Flat Appl) w v) t0)) (subst1_head u0 w (lift (S O) d x2) d H12 (Flat Appl) v -(lift (S O) d x0) H8) (lift (S O) d (THead (Flat Appl) x2 x0)) (lift_flat -Appl x2 x0 (S O) d)) (eq_ind_r T (THead (Flat Appl) (lift (S O) d x2) (lift -(S O) d (THead (Bind Abst) x3 x7))) (\lambda (t0: T).(subst1 d u0 (THead -(Flat Appl) w (THead (Bind Abst) u t)) t0)) (subst1_head u0 w (lift (S O) d -x2) d H12 (Flat Appl) (THead (Bind Abst) u t) (lift (S O) d (THead (Bind -Abst) x3 x7)) (eq_ind_r T (THead (Bind Abst) (lift (S O) d x3) (lift (S O) (S -d) x7)) (\lambda (t0: T).(subst1 (s (Flat Appl) d) u0 (THead (Bind Abst) u t) -t0)) (subst1_head u0 u (lift (S O) d x3) (s (Flat Appl) d) H13 (Bind Abst) t -(lift (S O) (S d) x7) H23) (lift (S O) d (THead (Bind Abst) x3 x7)) -(lift_bind Abst x3 x7 (S O) d))) (lift (S O) d (THead (Flat Appl) x2 (THead -(Bind Abst) x3 x7))) (lift_flat Appl x2 (THead (Bind Abst) x3 x7) (S O) d)) -(ty3_appl g a x2 x3 H14 x0 x7 H26))))))))))) (lift_gen_bind Abst x4 x5 x1 (S -O) d H19)))))))) H15)))))))) H11))))))) H7))))))))))))))))))) (\lambda (c0: -C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (ty3 g c0 t3 t4)).(\lambda -(H1: ((\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead e -(Bind Abbr) u)) \to (\forall (a0: C).((csubst1 d u c0 a0) \to (\forall (a: -C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: -T).(subst1 d u t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: -T).(subst1 d u t4 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 -g a y1 y2)))))))))))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t4 -t0)).(\lambda (H3: ((\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl -d c0 (CHead e (Bind Abbr) u)) \to (\forall (a0: C).((csubst1 d u c0 a0) \to -(\forall (a: C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda -(_: T).(subst1 d u t4 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: -T).(subst1 d u t0 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 -g a y1 y2)))))))))))))).(\lambda (e: C).(\lambda (u: T).(\lambda (d: -nat).(\lambda (H4: (getl d c0 (CHead e (Bind Abbr) u))).(\lambda (a0: -C).(\lambda (H5: (csubst1 d u c0 a0)).(\lambda (a: C).(\lambda (H6: (drop (S -O) d a0 a)).(let H7 \def (H3 e u d H4 a0 H5 a H6) in (ex3_2_ind T T (\lambda -(y1: T).(\lambda (_: T).(subst1 d u t4 (lift (S O) d y1)))) (\lambda (_: -T).(\lambda (y2: T).(subst1 d u t0 (lift (S O) d y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda -(_: T).(subst1 d u (THead (Flat Cast) t4 t3) (lift (S O) d y1)))) (\lambda -(_: T).(\lambda (y2: T).(subst1 d u (THead (Flat Cast) t0 t4) (lift (S O) d -y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H8: (subst1 d u t4 (lift (S O) d x0))).(\lambda -(H9: (subst1 d u t0 (lift (S O) d x1))).(\lambda (H10: (ty3 g a x0 x1)).(let -H11 \def (H1 e u d H4 a0 H5 a H6) in (ex3_2_ind T T (\lambda (y1: T).(\lambda -(_: T).(subst1 d u t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: -T).(subst1 d u t4 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 -g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u (THead -(Flat Cast) t4 t3) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: -T).(subst1 d u (THead (Flat Cast) t0 t4) (lift (S O) d y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: T).(\lambda (x3: -T).(\lambda (H12: (subst1 d u t3 (lift (S O) d x2))).(\lambda (H13: (subst1 d -u t4 (lift (S O) d x3))).(\lambda (H14: (ty3 g a x2 x3)).(let H15 \def -(eq_ind T x3 (\lambda (t: T).(ty3 g a x2 t)) H14 x0 (subst1_confluence_lift -t4 x3 u d H13 x0 H8)) in (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: -T).(subst1 d u (THead (Flat Cast) t4 t3) (lift (S O) d y1)))) (\lambda (_: -T).(\lambda (y2: T).(subst1 d u (THead (Flat Cast) t0 t4) (lift (S O) d -y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (THead (Flat Cast) -x0 x2) (THead (Flat Cast) x1 x0) (eq_ind_r T (THead (Flat Cast) (lift (S O) d -x0) (lift (S O) d x2)) (\lambda (t: T).(subst1 d u (THead (Flat Cast) t4 t3) -t)) (subst1_head u t4 (lift (S O) d x0) d H8 (Flat Cast) t3 (lift (S O) d x2) -H12) (lift (S O) d (THead (Flat Cast) x0 x2)) (lift_flat Cast x0 x2 (S O) d)) -(eq_ind_r T (THead (Flat Cast) (lift (S O) d x1) (lift (S O) d x0)) (\lambda -(t: T).(subst1 d u (THead (Flat Cast) t0 t4) t)) (subst1_head u t0 (lift (S -O) d x1) d H9 (Flat Cast) t4 (lift (S O) d x0) H8) (lift (S O) d (THead (Flat -Cast) x1 x0)) (lift_flat Cast x1 x0 (S O) d)) (ty3_cast g a x2 x0 H15 x1 -H10)))))))) H11))))))) H7)))))))))))))))))) c t1 t2 H))))). -(* COMMENTS -Initial nodes: 12848 -END *) - -theorem ty3_gen_cvoid: - \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c -t1 t2) \to (\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c -(CHead e (Bind Void) u)) \to (\forall (a: C).((drop (S O) d c a) \to (ex3_2 T -T (\lambda (y1: T).(\lambda (_: T).(eq T t1 (lift (S O) d y1)))) (\lambda (_: -T).(\lambda (y2: T).(eq T t2 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda -(y2: T).(ty3 g a y1 y2)))))))))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H: (ty3 g c t1 t2)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda -(t0: T).(\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead -e (Bind Void) u)) \to (\forall (a: C).((drop (S O) d c0 a) \to (ex3_2 T T -(\lambda (y1: T).(\lambda (_: T).(eq T t (lift (S O) d y1)))) (\lambda (_: -T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda -(y2: T).(ty3 g a y1 y2))))))))))))) (\lambda (c0: C).(\lambda (t3: -T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 t3 t)).(\lambda (H1: ((\forall (e: -C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Void) u)) \to -(\forall (a: C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda -(_: T).(eq T t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t -(lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 -y2)))))))))))).(\lambda (u: T).(\lambda (t4: T).(\lambda (H2: (ty3 g c0 u -t4)).(\lambda (H3: ((\forall (e: C).(\forall (u0: T).(\forall (d: nat).((getl -d c0 (CHead e (Bind Void) u0)) \to (\forall (a: C).((drop (S O) d c0 a) \to -(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T u (lift (S O) d y1)))) -(\lambda (_: T).(\lambda (y2: T).(eq T t4 (lift (S O) d y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (H4: (pc3 c0 t4 -t3)).(\lambda (e: C).(\lambda (u0: T).(\lambda (d: nat).(\lambda (H5: (getl d -c0 (CHead e (Bind Void) u0))).(\lambda (a: C).(\lambda (H6: (drop (S O) d c0 -a)).(let H7 \def (H3 e u0 d H5 a H6) in (ex3_2_ind T T (\lambda (y1: -T).(\lambda (_: T).(eq T u (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: -T).(eq T t4 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a -y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T u (lift (S O) d -y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t3 (lift (S O) d y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H8: (eq T u (lift (S O) d x0))).(\lambda (H9: -(eq T t4 (lift (S O) d x1))).(\lambda (H10: (ty3 g a x0 x1)).(let H11 \def -(eq_ind T t4 (\lambda (t0: T).(pc3 c0 t0 t3)) H4 (lift (S O) d x1) H9) in -(let H12 \def (eq_ind T t4 (\lambda (t0: T).(ty3 g c0 u t0)) H2 (lift (S O) d -x1) H9) in (let H13 \def (eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 (lift (S -O) d x1))) H12 (lift (S O) d x0) H8) in (eq_ind_r T (lift (S O) d x0) -(\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t0 (lift -(S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t3 (lift (S O) d y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H14 \def (H1 e u0 -d H5 a H6) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T t3 (lift -(S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: -T).(\lambda (_: T).(eq T (lift (S O) d x0) (lift (S O) d y1)))) (\lambda (_: -T).(\lambda (y2: T).(eq T t3 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda -(y2: T).(ty3 g a y1 y2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H15: -(eq T t3 (lift (S O) d x2))).(\lambda (H16: (eq T t (lift (S O) d -x3))).(\lambda (H17: (ty3 g a x2 x3)).(let H18 \def (eq_ind T t (\lambda (t0: -T).(ty3 g c0 t3 t0)) H0 (lift (S O) d x3) H16) in (let H19 \def (eq_ind T t3 -(\lambda (t0: T).(ty3 g c0 t0 (lift (S O) d x3))) H18 (lift (S O) d x2) H15) -in (let H20 \def (eq_ind T t3 (\lambda (t0: T).(pc3 c0 (lift (S O) d x1) t0)) -H11 (lift (S O) d x2) H15) in (eq_ind_r T (lift (S O) d x2) (\lambda (t0: -T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (lift (S O) d x0) (lift -(S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T -(\lambda (y1: T).(\lambda (_: T).(eq T (lift (S O) d x0) (lift (S O) d y1)))) -(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d x2) (lift (S O) d y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) x0 x2 (refl_equal T (lift -(S O) d x0)) (refl_equal T (lift (S O) d x2)) (ty3_conv g a x2 x3 H17 x0 x1 -H10 (pc3_gen_lift c0 x1 x2 (S O) d H20 a H6))) t3 H15))))))))) H14)) u -H8))))))))) H7)))))))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda -(e: C).(\lambda (u: T).(\lambda (d: nat).(\lambda (_: (getl d c0 (CHead e -(Bind Void) u))).(\lambda (a: C).(\lambda (_: (drop (S O) d c0 -a)).(ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(eq T (TSort m) (lift -(S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (TSort (next g m)) -(lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) -(TSort m) (TSort (next g m)) (eq_ind_r T (TSort m) (\lambda (t: T).(eq T -(TSort m) t)) (refl_equal T (TSort m)) (lift (S O) d (TSort m)) (lift_sort m -(S O) d)) (eq_ind_r T (TSort (next g m)) (\lambda (t: T).(eq T (TSort (next g -m)) t)) (refl_equal T (TSort (next g m))) (lift (S O) d (TSort (next g m))) -(lift_sort (next g m) (S O) d)) (ty3_sort g a m)))))))))) (\lambda (n: -nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n -c0 (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u -t)).(\lambda (H2: ((\forall (e: C).(\forall (u0: T).(\forall (d0: nat).((getl -d0 d (CHead e (Bind Void) u0)) \to (\forall (a: C).((drop (S O) d0 d a) \to -(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T u (lift (S O) d0 y1)))) -(\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d0 y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (e: C).(\lambda (u0: -T).(\lambda (d0: nat).(\lambda (H3: (getl d0 c0 (CHead e (Bind Void) -u0))).(\lambda (a: C).(\lambda (H4: (drop (S O) d0 c0 a)).(lt_eq_gt_e n d0 -(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 -y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O t) (lift (S O) d0 -y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (H5: (lt -n d0)).(let H6 \def (eq_ind nat (minus d0 n) (\lambda (n0: nat).(getl n0 -(CHead d (Bind Abbr) u) (CHead e (Bind Void) u0))) (getl_conf_le d0 (CHead e -(Bind Void) u0) c0 H3 (CHead d (Bind Abbr) u) n H0 (le_S_n n d0 (le_S (S n) -d0 H5))) (S (minus d0 (S n))) (minus_x_Sy d0 n H5)) in (let H7 \def (eq_ind -nat d0 (\lambda (n0: nat).(drop (S O) n0 c0 a)) H4 (S (plus n (minus d0 (S -n)))) (lt_plus_minus n d0 H5)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: -C).(eq T u (lift (S O) (minus d0 (S n)) v)))) (\lambda (v: T).(\lambda (e0: -C).(getl n a (CHead e0 (Bind Abbr) v)))) (\lambda (_: T).(\lambda (e0: -C).(drop (S O) (minus d0 (S n)) d e0))) (ex3_2 T T (\lambda (y1: T).(\lambda -(_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: -T).(eq T (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda -(y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1: C).(\lambda (H8: -(eq T u (lift (S O) (minus d0 (S n)) x0))).(\lambda (H9: (getl n a (CHead x1 -(Bind Abbr) x0))).(\lambda (H10: (drop (S O) (minus d0 (S n)) d x1)).(let H11 -\def (eq_ind T u (\lambda (t0: T).(\forall (e0: C).(\forall (u1: T).(\forall -(d1: nat).((getl d1 d (CHead e0 (Bind Void) u1)) \to (\forall (a0: C).((drop -(S O) d1 d a0) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t0 (lift -(S O) d1 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d1 -y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a0 y1 y2))))))))))) H2 (lift -(S O) (minus d0 (S n)) x0) H8) in (let H12 \def (eq_ind T u (\lambda (t0: -T).(ty3 g d t0 t)) H1 (lift (S O) (minus d0 (S n)) x0) H8) in (let H13 \def -(H11 e u0 (minus d0 (S n)) (getl_gen_S (Bind Abbr) d (CHead e (Bind Void) u0) -u (minus d0 (S n)) H6) x1 H10) in (ex3_2_ind T T (\lambda (y1: T).(\lambda -(_: T).(eq T (lift (S O) (minus d0 (S n)) x0) (lift (S O) (minus d0 (S n)) -y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) (minus d0 (S n)) -y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g x1 y1 y2))) (ex3_2 T T -(\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) -(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O t) (lift (S O) d0 y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: -T).(\lambda (x3: T).(\lambda (H14: (eq T (lift (S O) (minus d0 (S n)) x0) -(lift (S O) (minus d0 (S n)) x2))).(\lambda (H15: (eq T t (lift (S O) (minus -d0 (S n)) x3))).(\lambda (H16: (ty3 g x1 x2 x3)).(let H17 \def (eq_ind T t -(\lambda (t0: T).(ty3 g d (lift (S O) (minus d0 (S n)) x0) t0)) H12 (lift (S -O) (minus d0 (S n)) x3) H15) in (eq_ind_r T (lift (S O) (minus d0 (S n)) x3) -(\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) -(lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O -t0) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 -y2))))) (let H18 \def (eq_ind_r T x2 (\lambda (t0: T).(ty3 g x1 t0 x3)) H16 -x0 (lift_inj x0 x2 (S O) (minus d0 (S n)) H14)) in (eq_ind T (lift (S O) -(plus (S n) (minus d0 (S n))) (lift (S n) O x3)) (\lambda (t0: T).(ex3_2 T T -(\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) -(\lambda (_: T).(\lambda (y2: T).(eq T t0 (lift (S O) d0 y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind nat d0 (\lambda (n0: -nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) -d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) n0 (lift (S n) O -x3)) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 -y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) -(lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d0 -(lift (S n) O x3)) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: -T).(ty3 g a y1 y2))) (TLRef n) (lift (S n) O x3) (eq_ind_r T (TLRef n) -(\lambda (t0: T).(eq T (TLRef n) t0)) (refl_equal T (TLRef n)) (lift (S O) d0 -(TLRef n)) (lift_lref_lt n (S O) d0 H5)) (refl_equal T (lift (S O) d0 (lift -(S n) O x3))) (ty3_abbr g n a x1 x0 H9 x3 H18)) (plus (S n) (minus d0 (S n))) -(le_plus_minus (S n) d0 H5)) (lift (S n) O (lift (S O) (minus d0 (S n)) x3)) -(lift_d x3 (S O) (S n) (minus d0 (S n)) O (le_O_n (minus d0 (S n)))))) t -H15))))))) H13))))))))) (getl_drop_conf_lt Abbr c0 d u n H0 a (S O) (minus d0 -(S n)) H7))))) (\lambda (H5: (eq nat n d0)).(let H6 \def (eq_ind_r nat d0 -(\lambda (n0: nat).(drop (S O) n0 c0 a)) H4 n H5) in (let H7 \def (eq_ind_r -nat d0 (\lambda (n0: nat).(getl n0 c0 (CHead e (Bind Void) u0))) H3 n H5) in -(eq_ind nat n (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: -T).(eq T (TLRef n) (lift (S O) n0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq -T (lift (S n) O t) (lift (S O) n0 y2)))) (\lambda (y1: T).(\lambda (y2: -T).(ty3 g a y1 y2))))) (let H8 \def (eq_ind C (CHead d (Bind Abbr) u) -(\lambda (c1: C).(getl n c0 c1)) H0 (CHead e (Bind Void) u0) (getl_mono c0 -(CHead d (Bind Abbr) u) n H0 (CHead e (Bind Void) u0) H7)) in (let H9 \def -(eq_ind C (CHead d (Bind Abbr) u) (\lambda (ee: C).(match ee in C return -(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b) -\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow -True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _) -\Rightarrow False])])) I (CHead e (Bind Void) u0) (getl_mono c0 (CHead d -(Bind Abbr) u) n H0 (CHead e (Bind Void) u0) H7)) in (False_ind (ex3_2 T T -(\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) n y1)))) -(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O t) (lift (S O) n y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) H9))) d0 H5)))) (\lambda -(H5: (lt d0 n)).(eq_ind_r nat (S (plus O (minus n (S O)))) (\lambda (n0: -nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n0) (lift (S O) -d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O t) (lift (S O) -d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind nat -(plus (S O) (minus n (S O))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: -T).(\lambda (_: T).(eq T (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: -T).(\lambda (y2: T).(eq T (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda -(y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r nat (plus (minus n (S -O)) (S O)) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq -T (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T -(lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 -g a y1 y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef -(plus (minus n (S O)) (S O))) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda -(y2: T).(eq T (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2))) (TLRef (minus n (S O))) (lift n O t) -(eq_ind_r T (TLRef (plus (minus n (S O)) (S O))) (\lambda (t0: T).(eq T -(TLRef (plus (minus n (S O)) (S O))) t0)) (refl_equal T (TLRef (plus (minus n -(S O)) (S O)))) (lift (S O) d0 (TLRef (minus n (S O)))) (lift_lref_ge (minus -n (S O)) (S O) d0 (lt_le_minus d0 n H5))) (eq_ind_r T (lift (plus (S O) n) O -t) (\lambda (t0: T).(eq T (lift (S n) O t) t0)) (refl_equal T (lift (S n) O -t)) (lift (S O) d0 (lift n O t)) (lift_free t n (S O) O d0 (le_S_n d0 (plus O -n) (le_S (S d0) (plus O n) H5)) (le_O_n d0))) (eq_ind_r nat (S (minus n (S -O))) (\lambda (n0: nat).(ty3 g a (TLRef (minus n (S O))) (lift n0 O t))) -(ty3_abbr g (minus n (S O)) a d u (getl_drop_conf_ge n (CHead d (Bind Abbr) -u) c0 H0 a (S O) d0 H4 (eq_ind_r nat (plus (S O) d0) (\lambda (n0: nat).(le -n0 n)) H5 (plus d0 (S O)) (plus_sym d0 (S O)))) t H1) n (minus_x_SO n -(le_lt_trans O d0 n (le_O_n d0) H5)))) (plus (S O) (minus n (S O))) (plus_sym -(S O) (minus n (S O)))) (S (plus O (minus n (S O)))) (refl_equal nat (S (plus -O (minus n (S O)))))) n (lt_plus_minus O n (le_lt_trans O d0 n (le_O_n d0) -H5))))))))))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: -C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abst) -u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u t)).(\lambda (H2: ((\forall (e: -C).(\forall (u0: T).(\forall (d0: nat).((getl d0 d (CHead e (Bind Void) u0)) -\to (\forall (a: C).((drop (S O) d0 d a) \to (ex3_2 T T (\lambda (y1: -T).(\lambda (_: T).(eq T u (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda -(y2: T).(eq T t (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 -g a y1 y2)))))))))))).(\lambda (e: C).(\lambda (u0: T).(\lambda (d0: -nat).(\lambda (H3: (getl d0 c0 (CHead e (Bind Void) u0))).(\lambda (a: -C).(\lambda (H4: (drop (S O) d0 c0 a)).(lt_eq_gt_e n d0 (ex3_2 T T (\lambda -(y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: -T).(\lambda (y2: T).(eq T (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda -(y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (H5: (lt n d0)).(let H6 -\def (eq_ind nat (minus d0 n) (\lambda (n0: nat).(getl n0 (CHead d (Bind -Abst) u) (CHead e (Bind Void) u0))) (getl_conf_le d0 (CHead e (Bind Void) u0) -c0 H3 (CHead d (Bind Abst) u) n H0 (le_S_n n d0 (le_S (S n) d0 H5))) (S -(minus d0 (S n))) (minus_x_Sy d0 n H5)) in (let H7 \def (eq_ind nat d0 -(\lambda (n0: nat).(drop (S O) n0 c0 a)) H4 (S (plus n (minus d0 (S n)))) -(lt_plus_minus n d0 H5)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: -C).(eq T u (lift (S O) (minus d0 (S n)) v)))) (\lambda (v: T).(\lambda (e0: -C).(getl n a (CHead e0 (Bind Abst) v)))) (\lambda (_: T).(\lambda (e0: -C).(drop (S O) (minus d0 (S n)) d e0))) (ex3_2 T T (\lambda (y1: T).(\lambda -(_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: -T).(eq T (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda -(y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1: C).(\lambda (H8: -(eq T u (lift (S O) (minus d0 (S n)) x0))).(\lambda (H9: (getl n a (CHead x1 -(Bind Abst) x0))).(\lambda (H10: (drop (S O) (minus d0 (S n)) d x1)).(let H11 -\def (eq_ind T u (\lambda (t0: T).(\forall (e0: C).(\forall (u1: T).(\forall -(d1: nat).((getl d1 d (CHead e0 (Bind Void) u1)) \to (\forall (a0: C).((drop -(S O) d1 d a0) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t0 (lift -(S O) d1 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d1 -y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a0 y1 y2))))))))))) H2 (lift -(S O) (minus d0 (S n)) x0) H8) in (let H12 \def (eq_ind T u (\lambda (t0: -T).(ty3 g d t0 t)) H1 (lift (S O) (minus d0 (S n)) x0) H8) in (eq_ind_r T -(lift (S O) (minus d0 (S n)) x0) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: -T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: -T).(\lambda (y2: T).(eq T (lift (S n) O t0) (lift (S O) d0 y2)))) (\lambda -(y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H13 \def (H11 e u0 (minus -d0 (S n)) (getl_gen_S (Bind Abst) d (CHead e (Bind Void) u0) u (minus d0 (S -n)) H6) x1 H10) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T -(lift (S O) (minus d0 (S n)) x0) (lift (S O) (minus d0 (S n)) y1)))) (\lambda -(_: T).(\lambda (y2: T).(eq T t (lift (S O) (minus d0 (S n)) y2)))) (\lambda -(y1: T).(\lambda (y2: T).(ty3 g x1 y1 y2))) (ex3_2 T T (\lambda (y1: -T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: -T).(\lambda (y2: T).(eq T (lift (S n) O (lift (S O) (minus d0 (S n)) x0)) -(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) -(\lambda (x2: T).(\lambda (x3: T).(\lambda (H14: (eq T (lift (S O) (minus d0 -(S n)) x0) (lift (S O) (minus d0 (S n)) x2))).(\lambda (H15: (eq T t (lift (S -O) (minus d0 (S n)) x3))).(\lambda (H16: (ty3 g x1 x2 x3)).(let H17 \def -(eq_ind T t (\lambda (t0: T).(ty3 g d (lift (S O) (minus d0 (S n)) x0) t0)) -H12 (lift (S O) (minus d0 (S n)) x3) H15) in (let H18 \def (eq_ind_r T x2 -(\lambda (t0: T).(ty3 g x1 t0 x3)) H16 x0 (lift_inj x0 x2 (S O) (minus d0 (S -n)) H14)) in (eq_ind T (lift (S O) (plus (S n) (minus d0 (S n))) (lift (S n) -O x0)) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T -(TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t0 -(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) -(eq_ind nat d0 (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: -T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq -T (lift (S O) n0 (lift (S n) O x0)) (lift (S O) d0 y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1: -T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: -T).(\lambda (y2: T).(eq T (lift (S O) d0 (lift (S n) O x0)) (lift (S O) d0 -y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (TLRef n) (lift (S -n) O x0) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (TLRef n) t0)) -(refl_equal T (TLRef n)) (lift (S O) d0 (TLRef n)) (lift_lref_lt n (S O) d0 -H5)) (refl_equal T (lift (S O) d0 (lift (S n) O x0))) (ty3_abst g n a x1 x0 -H9 x3 H18)) (plus (S n) (minus d0 (S n))) (le_plus_minus (S n) d0 H5)) (lift -(S n) O (lift (S O) (minus d0 (S n)) x0)) (lift_d x0 (S O) (S n) (minus d0 (S -n)) O (le_O_n (minus d0 (S n)))))))))))) H13)) u H8)))))))) -(getl_drop_conf_lt Abst c0 d u n H0 a (S O) (minus d0 (S n)) H7))))) (\lambda -(H5: (eq nat n d0)).(let H6 \def (eq_ind_r nat d0 (\lambda (n0: nat).(drop (S -O) n0 c0 a)) H4 n H5) in (let H7 \def (eq_ind_r nat d0 (\lambda (n0: -nat).(getl n0 c0 (CHead e (Bind Void) u0))) H3 n H5) in (eq_ind nat n -(\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef -n) (lift (S O) n0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O -u) (lift (S O) n0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 -y2))))) (let H8 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (c1: C).(getl -n c0 c1)) H0 (CHead e (Bind Void) u0) (getl_mono c0 (CHead d (Bind Abst) u) n -H0 (CHead e (Bind Void) u0) H7)) in (let H9 \def (eq_ind C (CHead d (Bind -Abst) u) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with -[(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return -(\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B return -(\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow True | -Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead e (Bind -Void) u0) (getl_mono c0 (CHead d (Bind Abst) u) n H0 (CHead e (Bind Void) u0) -H7)) in (False_ind (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef -n) (lift (S O) n y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O -u) (lift (S O) n y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) -H9))) d0 H5)))) (\lambda (H5: (lt d0 n)).(eq_ind_r nat (S (plus O (minus n (S -O)))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T -(TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift -(S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a -y1 y2))))) (eq_ind nat (plus (S O) (minus n (S O))) (\lambda (n0: nat).(ex3_2 -T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n0) (lift (S O) d0 y1)))) -(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O u) (lift (S O) d0 y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r nat (plus -(minus n (S O)) (S O)) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: -T).(\lambda (_: T).(eq T (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: -T).(\lambda (y2: T).(eq T (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda -(y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1: -T).(\lambda (_: T).(eq T (TLRef (plus (minus n (S O)) (S O))) (lift (S O) d0 -y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O u) (lift (S O) d0 -y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (TLRef (minus n (S -O))) (lift n O u) (eq_ind_r T (TLRef (plus (minus n (S O)) (S O))) (\lambda -(t0: T).(eq T (TLRef (plus (minus n (S O)) (S O))) t0)) (refl_equal T (TLRef -(plus (minus n (S O)) (S O)))) (lift (S O) d0 (TLRef (minus n (S O)))) -(lift_lref_ge (minus n (S O)) (S O) d0 (lt_le_minus d0 n H5))) (eq_ind_r T -(lift (plus (S O) n) O u) (\lambda (t0: T).(eq T (lift (S n) O u) t0)) -(refl_equal T (lift (S n) O u)) (lift (S O) d0 (lift n O u)) (lift_free u n -(S O) O d0 (le_S_n d0 (plus O n) (le_S (S d0) (plus O n) H5)) (le_O_n d0))) -(eq_ind_r nat (S (minus n (S O))) (\lambda (n0: nat).(ty3 g a (TLRef (minus n -(S O))) (lift n0 O u))) (ty3_abst g (minus n (S O)) a d u (getl_drop_conf_ge -n (CHead d (Bind Abst) u) c0 H0 a (S O) d0 H4 (eq_ind_r nat (plus (S O) d0) -(\lambda (n0: nat).(le n0 n)) H5 (plus d0 (S O)) (plus_sym d0 (S O)))) t H1) -n (minus_x_SO n (le_lt_trans O d0 n (le_O_n d0) H5)))) (plus (S O) (minus n -(S O))) (plus_sym (S O) (minus n (S O)))) (S (plus O (minus n (S O)))) -(refl_equal nat (S (plus O (minus n (S O)))))) n (lt_plus_minus O n -(le_lt_trans O d0 n (le_O_n d0) H5))))))))))))))))))) (\lambda (c0: -C).(\lambda (u: T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 u t)).(\lambda -(H1: ((\forall (e: C).(\forall (u0: T).(\forall (d: nat).((getl d c0 (CHead e -(Bind Void) u0)) \to (\forall (a: C).((drop (S O) d c0 a) \to (ex3_2 T T -(\lambda (y1: T).(\lambda (_: T).(eq T u (lift (S O) d y1)))) (\lambda (_: -T).(\lambda (y2: T).(eq T t (lift (S O) d y2)))) (\lambda (y1: T).(\lambda -(y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (b: B).(\lambda (t3: T).(\lambda -(t4: T).(\lambda (H2: (ty3 g (CHead c0 (Bind b) u) t3 t4)).(\lambda (H3: -((\forall (e: C).(\forall (u0: T).(\forall (d: nat).((getl d (CHead c0 (Bind -b) u) (CHead e (Bind Void) u0)) \to (\forall (a: C).((drop (S O) d (CHead c0 -(Bind b) u) a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t3 (lift -(S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t4 (lift (S O) d y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (e: -C).(\lambda (u0: T).(\lambda (d: nat).(\lambda (H4: (getl d c0 (CHead e (Bind -Void) u0))).(\lambda (a: C).(\lambda (H5: (drop (S O) d c0 a)).(let H6 \def -(H1 e u0 d H4 a H5) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T -u (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d -y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T -(\lambda (y1: T).(\lambda (_: T).(eq T (THead (Bind b) u t3) (lift (S O) d -y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Bind b) u t4) (lift (S -O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda -(x0: T).(\lambda (x1: T).(\lambda (H7: (eq T u (lift (S O) d x0))).(\lambda -(H8: (eq T t (lift (S O) d x1))).(\lambda (H9: (ty3 g a x0 x1)).(let H10 \def -(eq_ind T t (\lambda (t0: T).(ty3 g c0 u t0)) H0 (lift (S O) d x1) H8) in -(let H11 \def (eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 (lift (S O) d x1))) -H10 (lift (S O) d x0) H7) in (let H12 \def (eq_ind T u (\lambda (t0: -T).(\forall (e0: C).(\forall (u1: T).(\forall (d0: nat).((getl d0 (CHead c0 -(Bind b) t0) (CHead e0 (Bind Void) u1)) \to (\forall (a0: C).((drop (S O) d0 -(CHead c0 (Bind b) t0) a0) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: -T).(eq T t3 (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t4 -(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a0 y1 -y2))))))))))) H3 (lift (S O) d x0) H7) in (let H13 \def (eq_ind T u (\lambda -(t0: T).(ty3 g (CHead c0 (Bind b) t0) t3 t4)) H2 (lift (S O) d x0) H7) in -(eq_ind_r T (lift (S O) d x0) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: -T).(\lambda (_: T).(eq T (THead (Bind b) t0 t3) (lift (S O) d y1)))) (\lambda -(_: T).(\lambda (y2: T).(eq T (THead (Bind b) t0 t4) (lift (S O) d y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H14 \def (H12 e u0 -(S d) (getl_head (Bind b) d c0 (CHead e (Bind Void) u0) H4 (lift (S O) d x0)) -(CHead a (Bind b) x0) (drop_skip_bind (S O) d c0 a H5 b x0)) in (ex3_2_ind T -T (\lambda (y1: T).(\lambda (_: T).(eq T t3 (lift (S O) (S d) y1)))) (\lambda -(_: T).(\lambda (y2: T).(eq T t4 (lift (S O) (S d) y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g (CHead a (Bind b) x0) y1 y2))) (ex3_2 T T (\lambda -(y1: T).(\lambda (_: T).(eq T (THead (Bind b) (lift (S O) d x0) t3) (lift (S -O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Bind b) (lift (S -O) d x0) t4) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a -y1 y2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H15: (eq T t3 (lift (S -O) (S d) x2))).(\lambda (H16: (eq T t4 (lift (S O) (S d) x3))).(\lambda (H17: -(ty3 g (CHead a (Bind b) x0) x2 x3)).(eq_ind_r T (lift (S O) (S d) x3) -(\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead -(Bind b) (lift (S O) d x0) t3) (lift (S O) d y1)))) (\lambda (_: T).(\lambda -(y2: T).(eq T (THead (Bind b) (lift (S O) d x0) t0) (lift (S O) d y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r T (lift (S O) -(S d) x2) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T -(THead (Bind b) (lift (S O) d x0) t0) (lift (S O) d y1)))) (\lambda (_: -T).(\lambda (y2: T).(eq T (THead (Bind b) (lift (S O) d x0) (lift (S O) (S d) -x3)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 -y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Bind -b) (lift (S O) d x0) (lift (S O) (S d) x2)) (lift (S O) d y1)))) (\lambda (_: -T).(\lambda (y2: T).(eq T (THead (Bind b) (lift (S O) d x0) (lift (S O) (S d) -x3)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) -(THead (Bind b) x0 x2) (THead (Bind b) x0 x3) (sym_eq T (lift (S O) d (THead -(Bind b) x0 x2)) (THead (Bind b) (lift (S O) d x0) (lift (S O) (S d) x2)) -(lift_bind b x0 x2 (S O) d)) (sym_eq T (lift (S O) d (THead (Bind b) x0 x3)) -(THead (Bind b) (lift (S O) d x0) (lift (S O) (S d) x3)) (lift_bind b x0 x3 -(S O) d)) (ty3_bind g a x0 x1 H9 b x2 x3 H17)) t3 H15) t4 H16)))))) H14)) u -H7)))))))))) H6)))))))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda -(u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda (H1: ((\forall (e: C).(\forall -(u0: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Void) u0)) \to (\forall -(a: C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: -T).(eq T w (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T u -(lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 -y2)))))))))))).(\lambda (v: T).(\lambda (t: T).(\lambda (H2: (ty3 g c0 v -(THead (Bind Abst) u t))).(\lambda (H3: ((\forall (e: C).(\forall (u0: -T).(\forall (d: nat).((getl d c0 (CHead e (Bind Void) u0)) \to (\forall (a: -C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T -v (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Bind -Abst) u t) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 -y2)))))))))))).(\lambda (e: C).(\lambda (u0: T).(\lambda (d: nat).(\lambda -(H4: (getl d c0 (CHead e (Bind Void) u0))).(\lambda (a: C).(\lambda (H5: -(drop (S O) d c0 a)).(let H6 \def (H3 e u0 d H4 a H5) in (ex3_2_ind T T -(\lambda (y1: T).(\lambda (_: T).(eq T v (lift (S O) d y1)))) (\lambda (_: -T).(\lambda (y2: T).(eq T (THead (Bind Abst) u t) (lift (S O) d y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: -T).(\lambda (_: T).(eq T (THead (Flat Appl) w v) (lift (S O) d y1)))) -(\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) w (THead (Bind -Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a -y1 y2)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: (eq T v (lift (S O) -d x0))).(\lambda (H8: (eq T (THead (Bind Abst) u t) (lift (S O) d -x1))).(\lambda (H9: (ty3 g a x0 x1)).(let H10 \def (eq_ind T v (\lambda (t0: -T).(ty3 g c0 t0 (THead (Bind Abst) u t))) H2 (lift (S O) d x0) H7) in -(eq_ind_r T (lift (S O) d x0) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: -T).(\lambda (_: T).(eq T (THead (Flat Appl) w t0) (lift (S O) d y1)))) -(\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) w (THead (Bind -Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a -y1 y2))))) (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x1 (THead -(Bind Abst) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift (S O) d -y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift (S O) (S d) z)))) (ex3_2 -T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Appl) w (lift (S O) d -x0)) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat -Appl) w (THead (Bind Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: T).(\lambda (x3: -T).(\lambda (H11: (eq T x1 (THead (Bind Abst) x2 x3))).(\lambda (H12: (eq T u -(lift (S O) d x2))).(\lambda (H13: (eq T t (lift (S O) (S d) x3))).(let H14 -\def (eq_ind T x1 (\lambda (t0: T).(ty3 g a x0 t0)) H9 (THead (Bind Abst) x2 -x3) H11) in (eq_ind_r T (lift (S O) (S d) x3) (\lambda (t0: T).(ex3_2 T T -(\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Appl) w (lift (S O) d -x0)) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat -Appl) w (THead (Bind Abst) u t0)) (lift (S O) d y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H15 \def (eq_ind T u (\lambda -(t0: T).(\forall (e0: C).(\forall (u1: T).(\forall (d0: nat).((getl d0 c0 -(CHead e0 (Bind Void) u1)) \to (\forall (a0: C).((drop (S O) d0 c0 a0) \to -(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T w (lift (S O) d0 y1)))) -(\lambda (_: T).(\lambda (y2: T).(eq T t0 (lift (S O) d0 y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a0 y1 y2))))))))))) H1 (lift (S O) d x2) H12) in -(eq_ind_r T (lift (S O) d x2) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: -T).(\lambda (_: T).(eq T (THead (Flat Appl) w (lift (S O) d x0)) (lift (S O) -d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) w (THead -(Bind Abst) t0 (lift (S O) (S d) x3))) (lift (S O) d y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H16 \def (H15 e u0 d H4 a H5) in -(ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T w (lift (S O) d y1)))) -(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d x2) (lift (S O) d y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: -T).(\lambda (_: T).(eq T (THead (Flat Appl) w (lift (S O) d x0)) (lift (S O) -d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) w (THead -(Bind Abst) (lift (S O) d x2) (lift (S O) (S d) x3))) (lift (S O) d y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x4: -T).(\lambda (x5: T).(\lambda (H17: (eq T w (lift (S O) d x4))).(\lambda (H18: -(eq T (lift (S O) d x2) (lift (S O) d x5))).(\lambda (H19: (ty3 g a x4 -x5)).(eq_ind_r T (lift (S O) d x4) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: -T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (lift (S O) d x0)) (lift (S O) -d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) t0 (THead -(Bind Abst) (lift (S O) d x2) (lift (S O) (S d) x3))) (lift (S O) d y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H20 \def (eq_ind_r -T x5 (\lambda (t0: T).(ty3 g a x4 t0)) H19 x2 (lift_inj x2 x5 (S O) d H18)) -in (eq_ind T (lift (S O) d (THead (Bind Abst) x2 x3)) (\lambda (t0: T).(ex3_2 -T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Appl) (lift (S O) d -x4) (lift (S O) d x0)) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: -T).(eq T (THead (Flat Appl) (lift (S O) d x4) t0) (lift (S O) d y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind T (lift (S O) d -(THead (Flat Appl) x4 x0)) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: -T).(\lambda (_: T).(eq T t0 (lift (S O) d y1)))) (\lambda (_: T).(\lambda -(y2: T).(eq T (THead (Flat Appl) (lift (S O) d x4) (lift (S O) d (THead (Bind -Abst) x2 x3))) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g -a y1 y2))))) (eq_ind T (lift (S O) d (THead (Flat Appl) x4 (THead (Bind Abst) -x2 x3))) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T -(lift (S O) d (THead (Flat Appl) x4 x0)) (lift (S O) d y1)))) (\lambda (_: -T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda -(y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: -T).(eq T (lift (S O) d (THead (Flat Appl) x4 x0)) (lift (S O) d y1)))) -(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d (THead (Flat Appl) x4 -(THead (Bind Abst) x2 x3))) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda -(y2: T).(ty3 g a y1 y2))) (THead (Flat Appl) x4 x0) (THead (Flat Appl) x4 -(THead (Bind Abst) x2 x3)) (refl_equal T (lift (S O) d (THead (Flat Appl) x4 -x0))) (refl_equal T (lift (S O) d (THead (Flat Appl) x4 (THead (Bind Abst) x2 -x3)))) (ty3_appl g a x4 x2 H20 x0 x3 H14)) (THead (Flat Appl) (lift (S O) d -x4) (lift (S O) d (THead (Bind Abst) x2 x3))) (lift_flat Appl x4 (THead (Bind -Abst) x2 x3) (S O) d)) (THead (Flat Appl) (lift (S O) d x4) (lift (S O) d -x0)) (lift_flat Appl x4 x0 (S O) d)) (THead (Bind Abst) (lift (S O) d x2) -(lift (S O) (S d) x3)) (lift_bind Abst x2 x3 (S O) d))) w H17)))))) H16)) u -H12)) t H13))))))) (lift_gen_bind Abst u t x1 (S O) d H8)) v H7))))))) -H6))))))))))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: -T).(\lambda (H0: (ty3 g c0 t3 t4)).(\lambda (H1: ((\forall (e: C).(\forall -(u: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Void) u)) \to (\forall -(a: C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: -T).(eq T t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t4 -(lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 -y2)))))))))))).(\lambda (t0: T).(\lambda (H2: (ty3 g c0 t4 t0)).(\lambda (H3: -((\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead e (Bind -Void) u)) \to (\forall (a: C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda -(y1: T).(\lambda (_: T).(eq T t4 (lift (S O) d y1)))) (\lambda (_: -T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda -(y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (e: C).(\lambda (u: T).(\lambda -(d: nat).(\lambda (H4: (getl d c0 (CHead e (Bind Void) u))).(\lambda (a: -C).(\lambda (H5: (drop (S O) d c0 a)).(let H6 \def (H3 e u d H4 a H5) in -(ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T t4 (lift (S O) d y1)))) -(\lambda (_: T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2)))) (\lambda (y1: -T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda -(_: T).(eq T (THead (Flat Cast) t4 t3) (lift (S O) d y1)))) (\lambda (_: -T).(\lambda (y2: T).(eq T (THead (Flat Cast) t0 t4) (lift (S O) d y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H7: (eq T t4 (lift (S O) d x0))).(\lambda (H8: -(eq T t0 (lift (S O) d x1))).(\lambda (H9: (ty3 g a x0 x1)).(let H10 \def -(eq_ind T t0 (\lambda (t: T).(ty3 g c0 t4 t)) H2 (lift (S O) d x1) H8) in -(eq_ind_r T (lift (S O) d x1) (\lambda (t: T).(ex3_2 T T (\lambda (y1: -T).(\lambda (_: T).(eq T (THead (Flat Cast) t4 t3) (lift (S O) d y1)))) -(\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Cast) t t4) (lift (S O) d -y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H11 \def -(eq_ind T t4 (\lambda (t: T).(ty3 g c0 t (lift (S O) d x1))) H10 (lift (S O) -d x0) H7) in (let H12 \def (eq_ind T t4 (\lambda (t: T).(\forall (e0: -C).(\forall (u0: T).(\forall (d0: nat).((getl d0 c0 (CHead e0 (Bind Void) -u0)) \to (\forall (a0: C).((drop (S O) d0 c0 a0) \to (ex3_2 T T (\lambda (y1: -T).(\lambda (_: T).(eq T t3 (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda -(y2: T).(eq T t (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 -g a0 y1 y2))))))))))) H1 (lift (S O) d x0) H7) in (let H13 \def (eq_ind T t4 -(\lambda (t: T).(ty3 g c0 t3 t)) H0 (lift (S O) d x0) H7) in (eq_ind_r T -(lift (S O) d x0) (\lambda (t: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: -T).(eq T (THead (Flat Cast) t t3) (lift (S O) d y1)))) (\lambda (_: -T).(\lambda (y2: T).(eq T (THead (Flat Cast) (lift (S O) d x1) t) (lift (S O) -d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H14 \def -(H12 e u d H4 a H5) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T -t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d -x0) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) -(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Cast) (lift (S -O) d x0) t3) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T -(THead (Flat Cast) (lift (S O) d x1) (lift (S O) d x0)) (lift (S O) d y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: -T).(\lambda (x3: T).(\lambda (H15: (eq T t3 (lift (S O) d x2))).(\lambda -(H16: (eq T (lift (S O) d x0) (lift (S O) d x3))).(\lambda (H17: (ty3 g a x2 -x3)).(let H18 \def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 t (lift (S O) d -x0))) H13 (lift (S O) d x2) H15) in (eq_ind_r T (lift (S O) d x2) (\lambda -(t: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Cast) -(lift (S O) d x0) t) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: -T).(eq T (THead (Flat Cast) (lift (S O) d x1) (lift (S O) d x0)) (lift (S O) -d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H19 \def -(eq_ind_r T x3 (\lambda (t: T).(ty3 g a x2 t)) H17 x0 (lift_inj x0 x3 (S O) d -H16)) in (eq_ind T (lift (S O) d (THead (Flat Cast) x0 x2)) (\lambda (t: -T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t (lift (S O) d y1)))) -(\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Cast) (lift (S O) d x1) -(lift (S O) d x0)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: -T).(ty3 g a y1 y2))))) (eq_ind T (lift (S O) d (THead (Flat Cast) x1 x0)) -(\lambda (t: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (lift (S O) -d (THead (Flat Cast) x0 x2)) (lift (S O) d y1)))) (\lambda (_: T).(\lambda -(y2: T).(eq T t (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g -a y1 y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(eq T (lift (S -O) d (THead (Flat Cast) x0 x2)) (lift (S O) d y1)))) (\lambda (_: T).(\lambda -(y2: T).(eq T (lift (S O) d (THead (Flat Cast) x1 x0)) (lift (S O) d y2)))) -(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (THead (Flat Cast) x0 x2) -(THead (Flat Cast) x1 x0) (refl_equal T (lift (S O) d (THead (Flat Cast) x0 -x2))) (refl_equal T (lift (S O) d (THead (Flat Cast) x1 x0))) (ty3_cast g a -x2 x0 H19 x1 H9)) (THead (Flat Cast) (lift (S O) d x1) (lift (S O) d x0)) -(lift_flat Cast x1 x0 (S O) d)) (THead (Flat Cast) (lift (S O) d x0) (lift (S -O) d x2)) (lift_flat Cast x0 x2 (S O) d))) t3 H15))))))) H14)) t4 H7)))) t0 -H8))))))) H6)))))))))))))))) c t1 t2 H))))). -(* COMMENTS -Initial nodes: 13105 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/wcpr0/defs.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/wcpr0/defs.ma deleted file mode 100644 index cdc9bf3a4..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/wcpr0/defs.ma +++ /dev/null @@ -1,26 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/pr0/defs.ma". - -include "Basic-1/C/defs.ma". - -inductive wcpr0: C \to (C \to Prop) \def -| wcpr0_refl: \forall (c: C).(wcpr0 c c) -| wcpr0_comp: \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall -(u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (k: K).(wcpr0 (CHead c1 k -u1) (CHead c2 k u2)))))))). - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/wcpr0/fwd.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/wcpr0/fwd.ma deleted file mode 100644 index 2b0531a8a..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/wcpr0/fwd.ma +++ /dev/null @@ -1,105 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/wcpr0/defs.ma". - -theorem wcpr0_gen_sort: - \forall (x: C).(\forall (n: nat).((wcpr0 (CSort n) x) \to (eq C x (CSort -n)))) -\def - \lambda (x: C).(\lambda (n: nat).(\lambda (H: (wcpr0 (CSort n) -x)).(insert_eq C (CSort n) (\lambda (c: C).(wcpr0 c x)) (\lambda (c: C).(eq C -x c)) (\lambda (y: C).(\lambda (H0: (wcpr0 y x)).(wcpr0_ind (\lambda (c: -C).(\lambda (c0: C).((eq C c (CSort n)) \to (eq C c0 c)))) (\lambda (c: -C).(\lambda (H1: (eq C c (CSort n))).(let H2 \def (f_equal C C (\lambda (e: -C).e) c (CSort n) H1) in (eq_ind_r C (CSort n) (\lambda (c0: C).(eq C c0 c0)) -(refl_equal C (CSort n)) c H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda -(_: (wcpr0 c1 c2)).(\lambda (_: (((eq C c1 (CSort n)) \to (eq C c2 -c1)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda -(k: K).(\lambda (H4: (eq C (CHead c1 k u1) (CSort n))).(let H5 \def (eq_ind C -(CHead c1 k u1) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) -with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I -(CSort n) H4) in (False_ind (eq C (CHead c2 k u2) (CHead c1 k u1)) -H5))))))))))) y x H0))) H))). -(* COMMENTS -Initial nodes: 249 -END *) - -theorem wcpr0_gen_head: - \forall (k: K).(\forall (c1: C).(\forall (x: C).(\forall (u1: T).((wcpr0 -(CHead c1 k u1) x) \to (or (eq C x (CHead c1 k u1)) (ex3_2 C T (\lambda (c2: -C).(\lambda (u2: T).(eq C x (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: -T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2))))))))) -\def - \lambda (k: K).(\lambda (c1: C).(\lambda (x: C).(\lambda (u1: T).(\lambda -(H: (wcpr0 (CHead c1 k u1) x)).(insert_eq C (CHead c1 k u1) (\lambda (c: -C).(wcpr0 c x)) (\lambda (c: C).(or (eq C x c) (ex3_2 C T (\lambda (c2: -C).(\lambda (u2: T).(eq C x (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: -T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))) (\lambda -(y: C).(\lambda (H0: (wcpr0 y x)).(wcpr0_ind (\lambda (c: C).(\lambda (c0: -C).((eq C c (CHead c1 k u1)) \to (or (eq C c0 c) (ex3_2 C T (\lambda (c2: -C).(\lambda (u2: T).(eq C c0 (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: -T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))))) -(\lambda (c: C).(\lambda (H1: (eq C c (CHead c1 k u1))).(let H2 \def (f_equal -C C (\lambda (e: C).e) c (CHead c1 k u1) H1) in (eq_ind_r C (CHead c1 k u1) -(\lambda (c0: C).(or (eq C c0 c0) (ex3_2 C T (\lambda (c2: C).(\lambda (u2: -T).(eq C c0 (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: T).(wcpr0 c1 -c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))) (or_introl (eq C -(CHead c1 k u1) (CHead c1 k u1)) (ex3_2 C T (\lambda (c2: C).(\lambda (u2: -T).(eq C (CHead c1 k u1) (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: -T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))) -(refl_equal C (CHead c1 k u1))) c H2)))) (\lambda (c0: C).(\lambda (c2: -C).(\lambda (H1: (wcpr0 c0 c2)).(\lambda (H2: (((eq C c0 (CHead c1 k u1)) \to -(or (eq C c2 c0) (ex3_2 C T (\lambda (c3: C).(\lambda (u2: T).(eq C c2 (CHead -c3 k u2)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: -C).(\lambda (u2: T).(pr0 u1 u2)))))))).(\lambda (u0: T).(\lambda (u2: -T).(\lambda (H3: (pr0 u0 u2)).(\lambda (k0: K).(\lambda (H4: (eq C (CHead c0 -k0 u0) (CHead c1 k u1))).(let H5 \def (f_equal C C (\lambda (e: C).(match e -in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) -\Rightarrow c])) (CHead c0 k0 u0) (CHead c1 k u1) H4) in ((let H6 \def -(f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with -[(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead c0 k0 u0) -(CHead c1 k u1) H4) in ((let H7 \def (f_equal C T (\lambda (e: C).(match e in -C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) -\Rightarrow t])) (CHead c0 k0 u0) (CHead c1 k u1) H4) in (\lambda (H8: (eq K -k0 k)).(\lambda (H9: (eq C c0 c1)).(eq_ind_r K k (\lambda (k1: K).(or (eq C -(CHead c2 k1 u2) (CHead c0 k1 u0)) (ex3_2 C T (\lambda (c3: C).(\lambda (u3: -T).(eq C (CHead c2 k1 u2) (CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_: -T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3)))))) (let H10 -\def (eq_ind T u0 (\lambda (t: T).(pr0 t u2)) H3 u1 H7) in (eq_ind_r T u1 -(\lambda (t: T).(or (eq C (CHead c2 k u2) (CHead c0 k t)) (ex3_2 C T (\lambda -(c3: C).(\lambda (u3: T).(eq C (CHead c2 k u2) (CHead c3 k u3)))) (\lambda -(c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 -u1 u3)))))) (let H11 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 k -u1)) \to (or (eq C c2 c) (ex3_2 C T (\lambda (c3: C).(\lambda (u3: T).(eq C -c2 (CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) -(\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))))))) H2 c1 H9) in (let H12 \def -(eq_ind C c0 (\lambda (c: C).(wcpr0 c c2)) H1 c1 H9) in (eq_ind_r C c1 -(\lambda (c: C).(or (eq C (CHead c2 k u2) (CHead c k u1)) (ex3_2 C T (\lambda -(c3: C).(\lambda (u3: T).(eq C (CHead c2 k u2) (CHead c3 k u3)))) (\lambda -(c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 -u1 u3)))))) (or_intror (eq C (CHead c2 k u2) (CHead c1 k u1)) (ex3_2 C T -(\lambda (c3: C).(\lambda (u3: T).(eq C (CHead c2 k u2) (CHead c3 k u3)))) -(\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda -(u3: T).(pr0 u1 u3)))) (ex3_2_intro C T (\lambda (c3: C).(\lambda (u3: T).(eq -C (CHead c2 k u2) (CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 -c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))) c2 u2 (refl_equal C -(CHead c2 k u2)) H12 H10)) c0 H9))) u0 H7)) k0 H8)))) H6)) H5))))))))))) y x -H0))) H))))). -(* COMMENTS -Initial nodes: 1133 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/wcpr0/getl.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/wcpr0/getl.ma deleted file mode 100644 index d3a109e0d..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/wcpr0/getl.ma +++ /dev/null @@ -1,464 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/wcpr0/defs.ma". - -include "Basic-1/getl/props.ma". - -theorem wcpr0_drop: - \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (h: -nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((drop h O c1 (CHead -e1 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c2 -(CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda -(_: C).(\lambda (u2: T).(pr0 u1 u2))))))))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(wcpr0_ind -(\lambda (c: C).(\lambda (c0: C).(\forall (h: nat).(\forall (e1: C).(\forall -(u1: T).(\forall (k: K).((drop h O c (CHead e1 k u1)) \to (ex3_2 C T (\lambda -(e2: C).(\lambda (u2: T).(drop h O c0 (CHead e2 k u2)))) (\lambda (e2: -C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 -u2))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda -(u1: T).(\lambda (k: K).(\lambda (H0: (drop h O c (CHead e1 k -u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c (CHead -e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: -C).(\lambda (u2: T).(pr0 u1 u2))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl -u1)))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wcpr0 c3 -c4)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1: -T).(\forall (k: K).((drop h O c3 (CHead e1 k u1)) \to (ex3_2 C T (\lambda -(e2: C).(\lambda (u2: T).(drop h O c4 (CHead e2 k u2)))) (\lambda (e2: -C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 -u2))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 -u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall -(e1: C).(\forall (u3: T).(\forall (k0: K).((drop n O (CHead c3 k u1) (CHead -e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead -c4 k u2) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) -(\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))) (\lambda (e1: -C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (drop O O (CHead c3 k u1) -(CHead e1 k0 u0))).(let H4 \def (f_equal C C (\lambda (e: C).(match e in C -return (\lambda (_: C).C) with [(CSort _) \Rightarrow c3 | (CHead c _ _) -\Rightarrow c])) (CHead c3 k u1) (CHead e1 k0 u0) (drop_gen_refl (CHead c3 k -u1) (CHead e1 k0 u0) H3)) in ((let H5 \def (f_equal C K (\lambda (e: -C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | -(CHead _ k1 _) \Rightarrow k1])) (CHead c3 k u1) (CHead e1 k0 u0) -(drop_gen_refl (CHead c3 k u1) (CHead e1 k0 u0) H3)) in ((let H6 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead c3 k u1) -(CHead e1 k0 u0) (drop_gen_refl (CHead c3 k u1) (CHead e1 k0 u0) H3)) in -(\lambda (H7: (eq K k k0)).(\lambda (H8: (eq C c3 e1)).(eq_ind K k (\lambda -(k1: K).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c4 k -u2) (CHead e2 k1 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) -(\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))))) (eq_ind T u1 (\lambda (t: -T).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c4 k u2) -(CHead e2 k u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda -(_: C).(\lambda (u3: T).(pr0 t u3))))) (eq_ind C c3 (\lambda (c: C).(ex3_2 C -T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c4 k u2) (CHead e2 k -u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 c e2))) (\lambda (_: -C).(\lambda (u3: T).(pr0 u1 u3))))) (ex3_2_intro C T (\lambda (e2: -C).(\lambda (u3: T).(drop O O (CHead c4 k u2) (CHead e2 k u3)))) (\lambda -(e2: C).(\lambda (_: T).(wcpr0 c3 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 -u1 u3))) c4 u2 (drop_refl (CHead c4 k u2)) H0 H2) e1 H8) u0 H6) k0 H7)))) -H5)) H4)))))) (K_ind (\lambda (k0: K).(\forall (n: nat).(((\forall (e1: -C).(\forall (u3: T).(\forall (k1: K).((drop n O (CHead c3 k0 u1) (CHead e1 k1 -u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c4 k0 -u2) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) -(\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))) \to (\forall (e1: -C).(\forall (u3: T).(\forall (k1: K).((drop (S n) O (CHead c3 k0 u1) (CHead -e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop (S n) O -(CHead c4 k0 u2) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 -e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))))) (\lambda (b: -B).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall -(k0: K).((drop n O (CHead c3 (Bind b) u1) (CHead e1 k0 u3)) \to (ex3_2 C T -(\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c4 (Bind b) u2) (CHead e2 -k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: -C).(\lambda (u4: T).(pr0 u3 u4)))))))))).(\lambda (e1: C).(\lambda (u0: -T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c3 (Bind b) u1) (CHead -e1 k0 u0))).(let H5 \def (H1 n e1 u0 k0 (drop_gen_drop (Bind b) c3 (CHead e1 -k0 u0) u1 n H4)) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop n -O c4 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) -(\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) (ex3_2 C T (\lambda (e2: -C).(\lambda (u3: T).(drop (S n) O (CHead c4 (Bind b) u2) (CHead e2 k0 u3)))) -(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda -(u3: T).(pr0 u0 u3)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (drop -n O c4 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 e1 x0)).(\lambda (H8: (pr0 u0 -x1)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead -c4 (Bind b) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 -e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) x0 x1 (drop_drop -(Bind b) n c4 (CHead x0 k0 x1) H6 u2) H7 H8)))))) H5))))))))) (\lambda (f: -F).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall -(k0: K).((drop n O (CHead c3 (Flat f) u1) (CHead e1 k0 u3)) \to (ex3_2 C T -(\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c4 (Flat f) u2) (CHead e2 -k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: -C).(\lambda (u4: T).(pr0 u3 u4)))))))))).(\lambda (e1: C).(\lambda (u0: -T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c3 (Flat f) u1) (CHead -e1 k0 u0))).(let H5 \def (H1 (S n) e1 u0 k0 (drop_gen_drop (Flat f) c3 (CHead -e1 k0 u0) u1 n H4)) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop -(S n) O c4 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 -e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) (ex3_2 C T (\lambda (e2: -C).(\lambda (u3: T).(drop (S n) O (CHead c4 (Flat f) u2) (CHead e2 k0 u3)))) -(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda -(u3: T).(pr0 u0 u3)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (drop -(S n) O c4 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 e1 x0)).(\lambda (H8: (pr0 -u0 x1)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O -(CHead c4 (Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: -T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) x0 x1 -(drop_drop (Flat f) n c4 (CHead x0 k0 x1) H6 u2) H7 H8)))))) H5))))))))) k) -h)))))))))) c1 c2 H))). -(* COMMENTS -Initial nodes: 1755 -END *) - -theorem wcpr0_drop_back: - \forall (c1: C).(\forall (c2: C).((wcpr0 c2 c1) \to (\forall (h: -nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((drop h O c1 (CHead -e1 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c2 -(CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda -(_: C).(\lambda (u2: T).(pr0 u2 u1))))))))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c2 c1)).(wcpr0_ind -(\lambda (c: C).(\lambda (c0: C).(\forall (h: nat).(\forall (e1: C).(\forall -(u1: T).(\forall (k: K).((drop h O c0 (CHead e1 k u1)) \to (ex3_2 C T -(\lambda (e2: C).(\lambda (u2: T).(drop h O c (CHead e2 k u2)))) (\lambda -(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 -u2 u1))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda -(u1: T).(\lambda (k: K).(\lambda (H0: (drop h O c (CHead e1 k -u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c (CHead -e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: -C).(\lambda (u2: T).(pr0 u2 u1))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl -u1)))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wcpr0 c3 -c4)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1: -T).(\forall (k: K).((drop h O c4 (CHead e1 k u1)) \to (ex3_2 C T (\lambda -(e2: C).(\lambda (u2: T).(drop h O c3 (CHead e2 k u2)))) (\lambda (e2: -C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 -u1))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 -u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall -(e1: C).(\forall (u3: T).(\forall (k0: K).((drop n O (CHead c4 k u2) (CHead -e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead -c3 k u1) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) -(\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))) (\lambda (e1: -C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (drop O O (CHead c4 k u2) -(CHead e1 k0 u0))).(let H4 \def (f_equal C C (\lambda (e: C).(match e in C -return (\lambda (_: C).C) with [(CSort _) \Rightarrow c4 | (CHead c _ _) -\Rightarrow c])) (CHead c4 k u2) (CHead e1 k0 u0) (drop_gen_refl (CHead c4 k -u2) (CHead e1 k0 u0) H3)) in ((let H5 \def (f_equal C K (\lambda (e: -C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | -(CHead _ k1 _) \Rightarrow k1])) (CHead c4 k u2) (CHead e1 k0 u0) -(drop_gen_refl (CHead c4 k u2) (CHead e1 k0 u0) H3)) in ((let H6 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c4 k u2) -(CHead e1 k0 u0) (drop_gen_refl (CHead c4 k u2) (CHead e1 k0 u0) H3)) in -(\lambda (H7: (eq K k k0)).(\lambda (H8: (eq C c4 e1)).(eq_ind K k (\lambda -(k1: K).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c3 k -u1) (CHead e2 k1 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) -(\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))))) (eq_ind T u2 (\lambda (t: -T).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c3 k u1) -(CHead e2 k u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda -(_: C).(\lambda (u3: T).(pr0 u3 t))))) (eq_ind C c4 (\lambda (c: C).(ex3_2 C -T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c3 k u1) (CHead e2 k -u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 c))) (\lambda (_: -C).(\lambda (u3: T).(pr0 u3 u2))))) (ex3_2_intro C T (\lambda (e2: -C).(\lambda (u3: T).(drop O O (CHead c3 k u1) (CHead e2 k u3)))) (\lambda -(e2: C).(\lambda (_: T).(wcpr0 e2 c4))) (\lambda (_: C).(\lambda (u3: T).(pr0 -u3 u2))) c3 u1 (drop_refl (CHead c3 k u1)) H0 H2) e1 H8) u0 H6) k0 H7)))) -H5)) H4)))))) (K_ind (\lambda (k0: K).(\forall (n: nat).(((\forall (e1: -C).(\forall (u3: T).(\forall (k1: K).((drop n O (CHead c4 k0 u2) (CHead e1 k1 -u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c3 k0 -u1) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) -(\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))) \to (\forall (e1: -C).(\forall (u3: T).(\forall (k1: K).((drop (S n) O (CHead c4 k0 u2) (CHead -e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop (S n) O -(CHead c3 k0 u1) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 -e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))))) (\lambda (b: -B).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall -(k0: K).((drop n O (CHead c4 (Bind b) u2) (CHead e1 k0 u3)) \to (ex3_2 C T -(\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c3 (Bind b) u1) (CHead e2 -k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: -C).(\lambda (u4: T).(pr0 u4 u3)))))))))).(\lambda (e1: C).(\lambda (u0: -T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c4 (Bind b) u2) (CHead -e1 k0 u0))).(let H5 \def (H1 n e1 u0 k0 (drop_gen_drop (Bind b) c4 (CHead e1 -k0 u0) u2 n H4)) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop n -O c3 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) -(\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) (ex3_2 C T (\lambda (e2: -C).(\lambda (u3: T).(drop (S n) O (CHead c3 (Bind b) u1) (CHead e2 k0 u3)))) -(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda -(u3: T).(pr0 u3 u0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (drop -n O c3 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 x0 e1)).(\lambda (H8: (pr0 x1 -u0)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead -c3 (Bind b) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 -e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) x0 x1 (drop_drop -(Bind b) n c3 (CHead x0 k0 x1) H6 u1) H7 H8)))))) H5))))))))) (\lambda (f: -F).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall -(k0: K).((drop n O (CHead c4 (Flat f) u2) (CHead e1 k0 u3)) \to (ex3_2 C T -(\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c3 (Flat f) u1) (CHead e2 -k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: -C).(\lambda (u4: T).(pr0 u4 u3)))))))))).(\lambda (e1: C).(\lambda (u0: -T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c4 (Flat f) u2) (CHead -e1 k0 u0))).(let H5 \def (H1 (S n) e1 u0 k0 (drop_gen_drop (Flat f) c4 (CHead -e1 k0 u0) u2 n H4)) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop -(S n) O c3 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 -e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) (ex3_2 C T (\lambda (e2: -C).(\lambda (u3: T).(drop (S n) O (CHead c3 (Flat f) u1) (CHead e2 k0 u3)))) -(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda -(u3: T).(pr0 u3 u0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (drop -(S n) O c3 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 x0 e1)).(\lambda (H8: (pr0 -x1 u0)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O -(CHead c3 (Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: -T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) x0 x1 -(drop_drop (Flat f) n c3 (CHead x0 k0 x1) H6 u1) H7 H8)))))) H5))))))))) k) -h)))))))))) c2 c1 H))). -(* COMMENTS -Initial nodes: 1755 -END *) - -theorem wcpr0_getl: - \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (h: -nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((getl h c1 (CHead e1 -k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(getl h c2 (CHead e2 -k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: -C).(\lambda (u2: T).(pr0 u1 u2))))))))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(wcpr0_ind -(\lambda (c: C).(\lambda (c0: C).(\forall (h: nat).(\forall (e1: C).(\forall -(u1: T).(\forall (k: K).((getl h c (CHead e1 k u1)) \to (ex3_2 C T (\lambda -(e2: C).(\lambda (u2: T).(getl h c0 (CHead e2 k u2)))) (\lambda (e2: -C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 -u2))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda -(u1: T).(\lambda (k: K).(\lambda (H0: (getl h c (CHead e1 k -u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(getl h c (CHead e2 -k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: -C).(\lambda (u2: T).(pr0 u1 u2))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl -u1)))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wcpr0 c3 -c4)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1: -T).(\forall (k: K).((getl h c3 (CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2: -C).(\lambda (u2: T).(getl h c4 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda -(_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 -u2))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 -u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall -(e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c3 k u1) (CHead e1 -k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c4 k -u2) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) -(\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))) (\lambda (e1: -C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (getl O (CHead c3 k u1) -(CHead e1 k0 u0))).(K_ind (\lambda (k1: K).((clear (CHead c3 k1 u1) (CHead e1 -k0 u0)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c4 k1 -u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) -(\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))))) (\lambda (b: B).(\lambda -(H4: (clear (CHead c3 (Bind b) u1) (CHead e1 k0 u0))).(let H5 \def (f_equal C -C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow e1 | (CHead c _ _) \Rightarrow c])) (CHead e1 k0 u0) (CHead c3 -(Bind b) u1) (clear_gen_bind b c3 (CHead e1 k0 u0) u1 H4)) in ((let H6 \def -(f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with -[(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead e1 k0 u0) -(CHead c3 (Bind b) u1) (clear_gen_bind b c3 (CHead e1 k0 u0) u1 H4)) in ((let -H7 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead e1 k0 -u0) (CHead c3 (Bind b) u1) (clear_gen_bind b c3 (CHead e1 k0 u0) u1 H4)) in -(\lambda (H8: (eq K k0 (Bind b))).(\lambda (H9: (eq C e1 c3)).(eq_ind_r K -(Bind b) (\lambda (k1: K).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl -O (CHead c4 (Bind b) u2) (CHead e2 k1 u3)))) (\lambda (e2: C).(\lambda (_: -T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))))) (eq_ind_r -T u1 (\lambda (t: T).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O -(CHead c4 (Bind b) u2) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda -(_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 t u3))))) -(eq_ind_r C c3 (\lambda (c: C).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: -T).(getl O (CHead c4 (Bind b) u2) (CHead e2 (Bind b) u3)))) (\lambda (e2: -C).(\lambda (_: T).(wcpr0 c e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 -u3))))) (ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c4 -(Bind b) u2) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda (_: -T).(wcpr0 c3 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))) c4 u2 -(getl_refl b c4 u2) H0 H2) e1 H9) u0 H7) k0 H8)))) H6)) H5)))) (\lambda (f: -F).(\lambda (H4: (clear (CHead c3 (Flat f) u1) (CHead e1 k0 u0))).(let H5 -\def (H1 O e1 u0 k0 (getl_intro O c3 (CHead e1 k0 u0) c3 (drop_refl c3) -(clear_gen_flat f c3 (CHead e1 k0 u0) u1 H4))) in (ex3_2_ind C T (\lambda -(e2: C).(\lambda (u3: T).(getl O c4 (CHead e2 k0 u3)))) (\lambda (e2: -C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 -u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c4 (Flat f) -u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) -(\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))) (\lambda (x0: C).(\lambda -(x1: T).(\lambda (H6: (getl O c4 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 e1 -x0)).(\lambda (H8: (pr0 u0 x1)).(ex3_2_intro C T (\lambda (e2: C).(\lambda -(u3: T).(getl O (CHead c4 (Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2: -C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 -u3))) x0 x1 (getl_flat c4 (CHead x0 k0 x1) O H6 f u2) H7 H8)))))) H5)))) k -(getl_gen_O (CHead c3 k u1) (CHead e1 k0 u0) H3)))))) (K_ind (\lambda (k0: -K).(\forall (n: nat).(((\forall (e1: C).(\forall (u3: T).(\forall (k1: -K).((getl n (CHead c3 k0 u1) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: -C).(\lambda (u4: T).(getl n (CHead c4 k0 u2) (CHead e2 k1 u4)))) (\lambda -(e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 -u3 u4))))))))) \to (\forall (e1: C).(\forall (u3: T).(\forall (k1: K).((getl -(S n) (CHead c3 k0 u1) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: -C).(\lambda (u4: T).(getl (S n) (CHead c4 k0 u2) (CHead e2 k1 u4)))) (\lambda -(e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 -u3 u4))))))))))) (\lambda (b: B).(\lambda (n: nat).(\lambda (_: ((\forall -(e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c3 (Bind b) u1) -(CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n -(CHead c4 (Bind b) u2) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: -T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 u3 -u4)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4: -(getl (S n) (CHead c3 (Bind b) u1) (CHead e1 k0 u0))).(let H5 \def (H1 n e1 -u0 k0 (getl_gen_S (Bind b) c3 (CHead e1 k0 u0) u1 n H4)) in (ex3_2_ind C T -(\lambda (e2: C).(\lambda (u3: T).(getl n c4 (CHead e2 k0 u3)))) (\lambda -(e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 -u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c4 -(Bind b) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 -e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (H6: (getl n c4 (CHead x0 k0 x1))).(\lambda (H7: -(wcpr0 e1 x0)).(\lambda (H8: (pr0 u0 x1)).(ex3_2_intro C T (\lambda (e2: -C).(\lambda (u3: T).(getl (S n) (CHead c4 (Bind b) u2) (CHead e2 k0 u3)))) -(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda -(u3: T).(pr0 u0 u3))) x0 x1 (getl_head (Bind b) n c4 (CHead x0 k0 x1) H6 u2) -H7 H8)))))) H5))))))))) (\lambda (f: F).(\lambda (n: nat).(\lambda (_: -((\forall (e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c3 (Flat -f) u1) (CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: -T).(getl n (CHead c4 (Flat f) u2) (CHead e2 k0 u4)))) (\lambda (e2: -C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 u3 -u4)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4: -(getl (S n) (CHead c3 (Flat f) u1) (CHead e1 k0 u0))).(let H5 \def (H1 (S n) -e1 u0 k0 (getl_gen_S (Flat f) c3 (CHead e1 k0 u0) u1 n H4)) in (ex3_2_ind C T -(\lambda (e2: C).(\lambda (u3: T).(getl (S n) c4 (CHead e2 k0 u3)))) (\lambda -(e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 -u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c4 -(Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 -e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (H6: (getl (S n) c4 (CHead x0 k0 x1))).(\lambda -(H7: (wcpr0 e1 x0)).(\lambda (H8: (pr0 u0 x1)).(ex3_2_intro C T (\lambda (e2: -C).(\lambda (u3: T).(getl (S n) (CHead c4 (Flat f) u2) (CHead e2 k0 u3)))) -(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda -(u3: T).(pr0 u0 u3))) x0 x1 (getl_head (Flat f) n c4 (CHead x0 k0 x1) H6 u2) -H7 H8)))))) H5))))))))) k) h)))))))))) c1 c2 H))). -(* COMMENTS -Initial nodes: 2103 -END *) - -theorem wcpr0_getl_back: - \forall (c1: C).(\forall (c2: C).((wcpr0 c2 c1) \to (\forall (h: -nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((getl h c1 (CHead e1 -k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(getl h c2 (CHead e2 -k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: -C).(\lambda (u2: T).(pr0 u2 u1))))))))))) -\def - \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c2 c1)).(wcpr0_ind -(\lambda (c: C).(\lambda (c0: C).(\forall (h: nat).(\forall (e1: C).(\forall -(u1: T).(\forall (k: K).((getl h c0 (CHead e1 k u1)) \to (ex3_2 C T (\lambda -(e2: C).(\lambda (u2: T).(getl h c (CHead e2 k u2)))) (\lambda (e2: -C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 -u1))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda -(u1: T).(\lambda (k: K).(\lambda (H0: (getl h c (CHead e1 k -u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(getl h c (CHead e2 -k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: -C).(\lambda (u2: T).(pr0 u2 u1))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl -u1)))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wcpr0 c3 -c4)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1: -T).(\forall (k: K).((getl h c4 (CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2: -C).(\lambda (u2: T).(getl h c3 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda -(_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 -u1))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 -u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall -(e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c4 k u2) (CHead e1 -k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c3 k -u1) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) -(\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))) (\lambda (e1: -C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (getl O (CHead c4 k u2) -(CHead e1 k0 u0))).(K_ind (\lambda (k1: K).((clear (CHead c4 k1 u2) (CHead e1 -k0 u0)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 k1 -u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) -(\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))))) (\lambda (b: B).(\lambda -(H4: (clear (CHead c4 (Bind b) u2) (CHead e1 k0 u0))).(let H5 \def (f_equal C -C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow e1 | (CHead c _ _) \Rightarrow c])) (CHead e1 k0 u0) (CHead c4 -(Bind b) u2) (clear_gen_bind b c4 (CHead e1 k0 u0) u2 H4)) in ((let H6 \def -(f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with -[(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead e1 k0 u0) -(CHead c4 (Bind b) u2) (clear_gen_bind b c4 (CHead e1 k0 u0) u2 H4)) in ((let -H7 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead e1 k0 -u0) (CHead c4 (Bind b) u2) (clear_gen_bind b c4 (CHead e1 k0 u0) u2 H4)) in -(\lambda (H8: (eq K k0 (Bind b))).(\lambda (H9: (eq C e1 c4)).(eq_ind_r K -(Bind b) (\lambda (k1: K).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl -O (CHead c3 (Bind b) u1) (CHead e2 k1 u3)))) (\lambda (e2: C).(\lambda (_: -T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))))) (eq_ind_r -T u2 (\lambda (t: T).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O -(CHead c3 (Bind b) u1) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda -(_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 t))))) -(eq_ind_r C c4 (\lambda (c: C).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: -T).(getl O (CHead c3 (Bind b) u1) (CHead e2 (Bind b) u3)))) (\lambda (e2: -C).(\lambda (_: T).(wcpr0 e2 c))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 -u2))))) (ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 -(Bind b) u1) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda (_: -T).(wcpr0 e2 c4))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u2))) c3 u1 -(getl_refl b c3 u1) H0 H2) e1 H9) u0 H7) k0 H8)))) H6)) H5)))) (\lambda (f: -F).(\lambda (H4: (clear (CHead c4 (Flat f) u2) (CHead e1 k0 u0))).(let H5 -\def (H1 O e1 u0 k0 (getl_intro O c4 (CHead e1 k0 u0) c4 (drop_refl c4) -(clear_gen_flat f c4 (CHead e1 k0 u0) u2 H4))) in (ex3_2_ind C T (\lambda -(e2: C).(\lambda (u3: T).(getl O c3 (CHead e2 k0 u3)))) (\lambda (e2: -C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 -u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 (Flat f) -u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) -(\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))) (\lambda (x0: C).(\lambda -(x1: T).(\lambda (H6: (getl O c3 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 x0 -e1)).(\lambda (H8: (pr0 x1 u0)).(ex3_2_intro C T (\lambda (e2: C).(\lambda -(u3: T).(getl O (CHead c3 (Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2: -C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 -u0))) x0 x1 (getl_flat c3 (CHead x0 k0 x1) O H6 f u1) H7 H8)))))) H5)))) k -(getl_gen_O (CHead c4 k u2) (CHead e1 k0 u0) H3)))))) (K_ind (\lambda (k0: -K).(\forall (n: nat).(((\forall (e1: C).(\forall (u3: T).(\forall (k1: -K).((getl n (CHead c4 k0 u2) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: -C).(\lambda (u4: T).(getl n (CHead c3 k0 u1) (CHead e2 k1 u4)))) (\lambda -(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 -u4 u3))))))))) \to (\forall (e1: C).(\forall (u3: T).(\forall (k1: K).((getl -(S n) (CHead c4 k0 u2) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: -C).(\lambda (u4: T).(getl (S n) (CHead c3 k0 u1) (CHead e2 k1 u4)))) (\lambda -(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 -u4 u3))))))))))) (\lambda (b: B).(\lambda (n: nat).(\lambda (_: ((\forall -(e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c4 (Bind b) u2) -(CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n -(CHead c3 (Bind b) u1) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: -T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 u4 -u3)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4: -(getl (S n) (CHead c4 (Bind b) u2) (CHead e1 k0 u0))).(let H5 \def (H1 n e1 -u0 k0 (getl_gen_S (Bind b) c4 (CHead e1 k0 u0) u2 n H4)) in (ex3_2_ind C T -(\lambda (e2: C).(\lambda (u3: T).(getl n c3 (CHead e2 k0 u3)))) (\lambda -(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 -u3 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c3 -(Bind b) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 -e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (H6: (getl n c3 (CHead x0 k0 x1))).(\lambda (H7: -(wcpr0 x0 e1)).(\lambda (H8: (pr0 x1 u0)).(ex3_2_intro C T (\lambda (e2: -C).(\lambda (u3: T).(getl (S n) (CHead c3 (Bind b) u1) (CHead e2 k0 u3)))) -(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda -(u3: T).(pr0 u3 u0))) x0 x1 (getl_head (Bind b) n c3 (CHead x0 k0 x1) H6 u1) -H7 H8)))))) H5))))))))) (\lambda (f: F).(\lambda (n: nat).(\lambda (_: -((\forall (e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c4 (Flat -f) u2) (CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: -T).(getl n (CHead c3 (Flat f) u1) (CHead e2 k0 u4)))) (\lambda (e2: -C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 u4 -u3)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4: -(getl (S n) (CHead c4 (Flat f) u2) (CHead e1 k0 u0))).(let H5 \def (H1 (S n) -e1 u0 k0 (getl_gen_S (Flat f) c4 (CHead e1 k0 u0) u2 n H4)) in (ex3_2_ind C T -(\lambda (e2: C).(\lambda (u3: T).(getl (S n) c3 (CHead e2 k0 u3)))) (\lambda -(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 -u3 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c3 -(Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 -e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (H6: (getl (S n) c3 (CHead x0 k0 x1))).(\lambda -(H7: (wcpr0 x0 e1)).(\lambda (H8: (pr0 x1 u0)).(ex3_2_intro C T (\lambda (e2: -C).(\lambda (u3: T).(getl (S n) (CHead c3 (Flat f) u1) (CHead e2 k0 u3)))) -(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda -(u3: T).(pr0 u3 u0))) x0 x1 (getl_head (Flat f) n c3 (CHead x0 k0 x1) H6 u1) -H7 H8)))))) H5))))))))) k) h)))))))))) c2 c1 H))). -(* COMMENTS -Initial nodes: 2103 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/wf3/clear.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/wf3/clear.ma deleted file mode 100644 index b2ddd47af..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/wf3/clear.ma +++ /dev/null @@ -1,91 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/wf3/fwd.ma". - -theorem wf3_clear_conf: - \forall (c1: C).(\forall (c: C).((clear c1 c) \to (\forall (g: G).(\forall -(c2: C).((wf3 g c1 c2) \to (wf3 g c c2)))))) -\def - \lambda (c1: C).(\lambda (c: C).(\lambda (H: (clear c1 c)).(clear_ind -(\lambda (c0: C).(\lambda (c2: C).(\forall (g: G).(\forall (c3: C).((wf3 g c0 -c3) \to (wf3 g c2 c3)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (u: -T).(\lambda (g: G).(\lambda (c2: C).(\lambda (H0: (wf3 g (CHead e (Bind b) u) -c2)).H0)))))) (\lambda (e: C).(\lambda (c0: C).(\lambda (_: (clear e -c0)).(\lambda (H1: ((\forall (g: G).(\forall (c2: C).((wf3 g e c2) \to (wf3 g -c0 c2)))))).(\lambda (f: F).(\lambda (u: T).(\lambda (g: G).(\lambda (c2: -C).(\lambda (H2: (wf3 g (CHead e (Flat f) u) c2)).(let H_y \def -(wf3_gen_flat1 g e c2 u f H2) in (H1 g c2 H_y))))))))))) c1 c H))). -(* COMMENTS -Initial nodes: 145 -END *) - -theorem clear_wf3_trans: - \forall (c1: C).(\forall (d1: C).((clear c1 d1) \to (\forall (g: G).(\forall -(d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda -(c2: C).(clear c2 d2)))))))) -\def - \lambda (c1: C).(\lambda (d1: C).(\lambda (H: (clear c1 d1)).(clear_ind -(\lambda (c: C).(\lambda (c0: C).(\forall (g: G).(\forall (d2: C).((wf3 g c0 -d2) \to (ex2 C (\lambda (c2: C).(wf3 g c c2)) (\lambda (c2: C).(clear c2 -d2)))))))) (\lambda (b: B).(\lambda (e: C).(\lambda (u: T).(\lambda (g: -G).(\lambda (d2: C).(\lambda (H0: (wf3 g (CHead e (Bind b) u) d2)).(let H_x -\def (wf3_gen_bind1 g e d2 u b H0) in (let H1 \def H_x in (or_ind (ex3_2 C T -(\lambda (c2: C).(\lambda (_: T).(eq C d2 (CHead c2 (Bind b) u)))) (\lambda -(c2: C).(\lambda (_: T).(wf3 g e c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g -e u w)))) (ex3 C (\lambda (c2: C).(eq C d2 (CHead c2 (Bind Void) (TSort O)))) -(\lambda (c2: C).(wf3 g e c2)) (\lambda (_: C).(\forall (w: T).((ty3 g e u w) -\to False)))) (ex2 C (\lambda (c2: C).(wf3 g (CHead e (Bind b) u) c2)) -(\lambda (c2: C).(clear c2 d2))) (\lambda (H2: (ex3_2 C T (\lambda (c2: -C).(\lambda (_: T).(eq C d2 (CHead c2 (Bind b) u)))) (\lambda (c2: -C).(\lambda (_: T).(wf3 g e c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g e u -w))))).(ex3_2_ind C T (\lambda (c2: C).(\lambda (_: T).(eq C d2 (CHead c2 -(Bind b) u)))) (\lambda (c2: C).(\lambda (_: T).(wf3 g e c2))) (\lambda (_: -C).(\lambda (w: T).(ty3 g e u w))) (ex2 C (\lambda (c2: C).(wf3 g (CHead e -(Bind b) u) c2)) (\lambda (c2: C).(clear c2 d2))) (\lambda (x0: C).(\lambda -(x1: T).(\lambda (H3: (eq C d2 (CHead x0 (Bind b) u))).(\lambda (H4: (wf3 g e -x0)).(\lambda (H5: (ty3 g e u x1)).(eq_ind_r C (CHead x0 (Bind b) u) (\lambda -(c: C).(ex2 C (\lambda (c2: C).(wf3 g (CHead e (Bind b) u) c2)) (\lambda (c2: -C).(clear c2 c)))) (ex_intro2 C (\lambda (c2: C).(wf3 g (CHead e (Bind b) u) -c2)) (\lambda (c2: C).(clear c2 (CHead x0 (Bind b) u))) (CHead x0 (Bind b) u) -(wf3_bind g e x0 H4 u x1 H5 b) (clear_bind b x0 u)) d2 H3)))))) H2)) (\lambda -(H2: (ex3 C (\lambda (c2: C).(eq C d2 (CHead c2 (Bind Void) (TSort O)))) -(\lambda (c2: C).(wf3 g e c2)) (\lambda (_: C).(\forall (w: T).((ty3 g e u w) -\to False))))).(ex3_ind C (\lambda (c2: C).(eq C d2 (CHead c2 (Bind Void) -(TSort O)))) (\lambda (c2: C).(wf3 g e c2)) (\lambda (_: C).(\forall (w: -T).((ty3 g e u w) \to False))) (ex2 C (\lambda (c2: C).(wf3 g (CHead e (Bind -b) u) c2)) (\lambda (c2: C).(clear c2 d2))) (\lambda (x0: C).(\lambda (H3: -(eq C d2 (CHead x0 (Bind Void) (TSort O)))).(\lambda (H4: (wf3 g e -x0)).(\lambda (H5: ((\forall (w: T).((ty3 g e u w) \to False)))).(eq_ind_r C -(CHead x0 (Bind Void) (TSort O)) (\lambda (c: C).(ex2 C (\lambda (c2: C).(wf3 -g (CHead e (Bind b) u) c2)) (\lambda (c2: C).(clear c2 c)))) (ex_intro2 C -(\lambda (c2: C).(wf3 g (CHead e (Bind b) u) c2)) (\lambda (c2: C).(clear c2 -(CHead x0 (Bind Void) (TSort O)))) (CHead x0 (Bind Void) (TSort O)) (wf3_void -g e x0 H4 u H5 b) (clear_bind Void x0 (TSort O))) d2 H3))))) H2)) H1))))))))) -(\lambda (e: C).(\lambda (c: C).(\lambda (_: (clear e c)).(\lambda (H1: -((\forall (g: G).(\forall (d2: C).((wf3 g c d2) \to (ex2 C (\lambda (c2: -C).(wf3 g e c2)) (\lambda (c2: C).(clear c2 d2)))))))).(\lambda (f: -F).(\lambda (u: T).(\lambda (g: G).(\lambda (d2: C).(\lambda (H2: (wf3 g c -d2)).(let H_x \def (H1 g d2 H2) in (let H3 \def H_x in (ex2_ind C (\lambda -(c2: C).(wf3 g e c2)) (\lambda (c2: C).(clear c2 d2)) (ex2 C (\lambda (c2: -C).(wf3 g (CHead e (Flat f) u) c2)) (\lambda (c2: C).(clear c2 d2))) (\lambda -(x: C).(\lambda (H4: (wf3 g e x)).(\lambda (H5: (clear x d2)).(ex_intro2 C -(\lambda (c2: C).(wf3 g (CHead e (Flat f) u) c2)) (\lambda (c2: C).(clear c2 -d2)) x (wf3_flat g e x H4 u f) H5)))) H3)))))))))))) c1 d1 H))). -(* COMMENTS -Initial nodes: 1023 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/wf3/defs.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/wf3/defs.ma deleted file mode 100644 index a99661941..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/wf3/defs.ma +++ /dev/null @@ -1,29 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/ty3/defs.ma". - -inductive wf3 (g: G): C \to (C \to Prop) \def -| wf3_sort: \forall (m: nat).(wf3 g (CSort m) (CSort m)) -| wf3_bind: \forall (c1: C).(\forall (c2: C).((wf3 g c1 c2) \to (\forall (u: -T).(\forall (t: T).((ty3 g c1 u t) \to (\forall (b: B).(wf3 g (CHead c1 (Bind -b) u) (CHead c2 (Bind b) u)))))))) -| wf3_void: \forall (c1: C).(\forall (c2: C).((wf3 g c1 c2) \to (\forall (u: -T).(((\forall (t: T).((ty3 g c1 u t) \to False))) \to (\forall (b: B).(wf3 g -(CHead c1 (Bind b) u) (CHead c2 (Bind Void) (TSort O)))))))) -| wf3_flat: \forall (c1: C).(\forall (c2: C).((wf3 g c1 c2) \to (\forall (u: -T).(\forall (f: F).(wf3 g (CHead c1 (Flat f) u) c2))))). - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/wf3/fwd.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/wf3/fwd.ma deleted file mode 100644 index 71c903f06..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/wf3/fwd.ma +++ /dev/null @@ -1,311 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/wf3/defs.ma". - -theorem wf3_gen_sort1: - \forall (g: G).(\forall (x: C).(\forall (m: nat).((wf3 g (CSort m) x) \to -(eq C x (CSort m))))) -\def - \lambda (g: G).(\lambda (x: C).(\lambda (m: nat).(\lambda (H: (wf3 g (CSort -m) x)).(insert_eq C (CSort m) (\lambda (c: C).(wf3 g c x)) (\lambda (c: -C).(eq C x c)) (\lambda (y: C).(\lambda (H0: (wf3 g y x)).(wf3_ind g (\lambda -(c: C).(\lambda (c0: C).((eq C c (CSort m)) \to (eq C c0 c)))) (\lambda (m0: -nat).(\lambda (H1: (eq C (CSort m0) (CSort m))).(let H2 \def (f_equal C nat -(\lambda (e: C).(match e in C return (\lambda (_: C).nat) with [(CSort n) -\Rightarrow n | (CHead _ _ _) \Rightarrow m0])) (CSort m0) (CSort m) H1) in -(eq_ind_r nat m (\lambda (n: nat).(eq C (CSort n) (CSort n))) (refl_equal C -(CSort m)) m0 H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (wf3 g c1 -c2)).(\lambda (_: (((eq C c1 (CSort m)) \to (eq C c2 c1)))).(\lambda (u: -T).(\lambda (t: T).(\lambda (_: (ty3 g c1 u t)).(\lambda (b: B).(\lambda (H4: -(eq C (CHead c1 (Bind b) u) (CSort m))).(let H5 \def (eq_ind C (CHead c1 -(Bind b) u) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with -[(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort m) -H4) in (False_ind (eq C (CHead c2 (Bind b) u) (CHead c1 (Bind b) u)) -H5))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (wf3 g c1 -c2)).(\lambda (_: (((eq C c1 (CSort m)) \to (eq C c2 c1)))).(\lambda (u: -T).(\lambda (_: ((\forall (t: T).((ty3 g c1 u t) \to False)))).(\lambda (b: -B).(\lambda (H4: (eq C (CHead c1 (Bind b) u) (CSort m))).(let H5 \def (eq_ind -C (CHead c1 (Bind b) u) (\lambda (ee: C).(match ee in C return (\lambda (_: -C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow -True])) I (CSort m) H4) in (False_ind (eq C (CHead c2 (Bind Void) (TSort O)) -(CHead c1 (Bind b) u)) H5)))))))))) (\lambda (c1: C).(\lambda (c2: -C).(\lambda (_: (wf3 g c1 c2)).(\lambda (_: (((eq C c1 (CSort m)) \to (eq C -c2 c1)))).(\lambda (u: T).(\lambda (f: F).(\lambda (H3: (eq C (CHead c1 (Flat -f) u) (CSort m))).(let H4 \def (eq_ind C (CHead c1 (Flat f) u) (\lambda (ee: -C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow -False | (CHead _ _ _) \Rightarrow True])) I (CSort m) H3) in (False_ind (eq C -c2 (CHead c1 (Flat f) u)) H4))))))))) y x H0))) H)))). -(* COMMENTS -Initial nodes: 523 -END *) - -theorem wf3_gen_bind1: - \forall (g: G).(\forall (c1: C).(\forall (x: C).(\forall (v: T).(\forall (b: -B).((wf3 g (CHead c1 (Bind b) v) x) \to (or (ex3_2 C T (\lambda (c2: -C).(\lambda (_: T).(eq C x (CHead c2 (Bind b) v)))) (\lambda (c2: C).(\lambda -(_: T).(wf3 g c1 c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 -C (\lambda (c2: C).(eq C x (CHead c2 (Bind Void) (TSort O)))) (\lambda (c2: -C).(wf3 g c1 c2)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to -False)))))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (x: C).(\lambda (v: T).(\lambda (b: -B).(\lambda (H: (wf3 g (CHead c1 (Bind b) v) x)).(insert_eq C (CHead c1 (Bind -b) v) (\lambda (c: C).(wf3 g c x)) (\lambda (_: C).(or (ex3_2 C T (\lambda -(c2: C).(\lambda (_: T).(eq C x (CHead c2 (Bind b) v)))) (\lambda (c2: -C).(\lambda (_: T).(wf3 g c1 c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 -v w)))) (ex3 C (\lambda (c2: C).(eq C x (CHead c2 (Bind Void) (TSort O)))) -(\lambda (c2: C).(wf3 g c1 c2)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v -w) \to False)))))) (\lambda (y: C).(\lambda (H0: (wf3 g y x)).(wf3_ind g -(\lambda (c: C).(\lambda (c0: C).((eq C c (CHead c1 (Bind b) v)) \to (or -(ex3_2 C T (\lambda (c2: C).(\lambda (_: T).(eq C c0 (CHead c2 (Bind b) v)))) -(\lambda (c2: C).(\lambda (_: T).(wf3 g c1 c2))) (\lambda (_: C).(\lambda (w: -T).(ty3 g c1 v w)))) (ex3 C (\lambda (c2: C).(eq C c0 (CHead c2 (Bind Void) -(TSort O)))) (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (_: C).(\forall (w: -T).((ty3 g c1 v w) \to False)))))))) (\lambda (m: nat).(\lambda (H1: (eq C -(CSort m) (CHead c1 (Bind b) v))).(let H2 \def (eq_ind C (CSort m) (\lambda -(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead c1 (Bind b) v) -H1) in (False_ind (or (ex3_2 C T (\lambda (c2: C).(\lambda (_: T).(eq C -(CSort m) (CHead c2 (Bind b) v)))) (\lambda (c2: C).(\lambda (_: T).(wf3 g c1 -c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c2: -C).(eq C (CSort m) (CHead c2 (Bind Void) (TSort O)))) (\lambda (c2: C).(wf3 g -c1 c2)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to False))))) H2)))) -(\lambda (c0: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c0 c2)).(\lambda (H2: -(((eq C c0 (CHead c1 (Bind b) v)) \to (or (ex3_2 C T (\lambda (c3: -C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda (c3: -C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 -v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort O)))) -(\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v -w) \to False)))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H3: (ty3 g c0 -u t)).(\lambda (b0: B).(\lambda (H4: (eq C (CHead c0 (Bind b0) u) (CHead c1 -(Bind b) v))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return -(\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow -c])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) H4) in ((let H6 \def -(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with -[(CSort _) \Rightarrow b0 | (CHead _ k _) \Rightarrow (match k in K return -(\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow -b0])])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) H4) in ((let H7 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead c0 (Bind -b0) u) (CHead c1 (Bind b) v) H4) in (\lambda (H8: (eq B b0 b)).(\lambda (H9: -(eq C c0 c1)).(eq_ind_r B b (\lambda (b1: B).(or (ex3_2 C T (\lambda (c3: -C).(\lambda (_: T).(eq C (CHead c2 (Bind b1) u) (CHead c3 (Bind b) v)))) -(\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: -T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C (CHead c2 (Bind b1) u) -(CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda -(_: C).(\forall (w: T).((ty3 g c1 v w) \to False)))))) (let H10 \def (eq_ind -T u (\lambda (t0: T).(ty3 g c0 t0 t)) H3 v H7) in (eq_ind_r T v (\lambda (t0: -T).(or (ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C (CHead c2 (Bind b) -t0) (CHead c3 (Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) -(\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq -C (CHead c2 (Bind b) t0) (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: -C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to -False)))))) (let H11 \def (eq_ind C c0 (\lambda (c: C).(ty3 g c v t)) H10 c1 -H9) in (let H12 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 (Bind b) -v)) \to (or (ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 -(Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: -C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead -c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: -C).(\forall (w: T).((ty3 g c1 v w) \to False))))))) H2 c1 H9) in (let H13 -\def (eq_ind C c0 (\lambda (c: C).(wf3 g c c2)) H1 c1 H9) in (or_introl -(ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C (CHead c2 (Bind b) v) -(CHead c3 (Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) -(\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq -C (CHead c2 (Bind b) v) (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: -C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to -False)))) (ex3_2_intro C T (\lambda (c3: C).(\lambda (_: T).(eq C (CHead c2 -(Bind b) v) (CHead c3 (Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g -c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w))) c2 t (refl_equal C -(CHead c2 (Bind b) v)) H13 H11))))) u H7)) b0 H8)))) H6)) H5))))))))))) -(\lambda (c0: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c0 c2)).(\lambda (H2: -(((eq C c0 (CHead c1 (Bind b) v)) \to (or (ex3_2 C T (\lambda (c3: -C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda (c3: -C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 -v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort O)))) -(\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v -w) \to False)))))))).(\lambda (u: T).(\lambda (H3: ((\forall (t: T).((ty3 g -c0 u t) \to False)))).(\lambda (b0: B).(\lambda (H4: (eq C (CHead c0 (Bind -b0) u) (CHead c1 (Bind b) v))).(let H5 \def (f_equal C C (\lambda (e: -C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | -(CHead c _ _) \Rightarrow c])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) -H4) in ((let H6 \def (f_equal C B (\lambda (e: C).(match e in C return -(\lambda (_: C).B) with [(CSort _) \Rightarrow b0 | (CHead _ k _) \Rightarrow -(match k in K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | -(Flat _) \Rightarrow b0])])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) H4) -in ((let H7 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda -(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) -(CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) H4) in (\lambda (_: (eq B b0 -b)).(\lambda (H9: (eq C c0 c1)).(let H10 \def (eq_ind T u (\lambda (t: -T).(\forall (t0: T).((ty3 g c0 t t0) \to False))) H3 v H7) in (let H11 \def -(eq_ind C c0 (\lambda (c: C).(\forall (t: T).((ty3 g c v t) \to False))) H10 -c1 H9) in (let H12 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 (Bind -b) v)) \to (or (ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 -(Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: -C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead -c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: -C).(\forall (w: T).((ty3 g c1 v w) \to False))))))) H2 c1 H9) in (let H13 -\def (eq_ind C c0 (\lambda (c: C).(wf3 g c c2)) H1 c1 H9) in (or_intror -(ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C (CHead c2 (Bind Void) -(TSort O)) (CHead c3 (Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g -c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda -(c3: C).(eq C (CHead c2 (Bind Void) (TSort O)) (CHead c3 (Bind Void) (TSort -O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g -c1 v w) \to False)))) (ex3_intro C (\lambda (c3: C).(eq C (CHead c2 (Bind -Void) (TSort O)) (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g -c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to False))) c2 -(refl_equal C (CHead c2 (Bind Void) (TSort O))) H13 H11))))))))) H6)) -H5)))))))))) (\lambda (c0: C).(\lambda (c2: C).(\lambda (_: (wf3 g c0 -c2)).(\lambda (_: (((eq C c0 (CHead c1 (Bind b) v)) \to (or (ex3_2 C T -(\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda -(c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 -g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort -O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g -c1 v w) \to False)))))))).(\lambda (u: T).(\lambda (f: F).(\lambda (H3: (eq C -(CHead c0 (Flat f) u) (CHead c1 (Bind b) v))).(let H4 \def (eq_ind C (CHead -c0 (Flat f) u) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) -with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K -return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) -\Rightarrow True])])) I (CHead c1 (Bind b) v) H3) in (False_ind (or (ex3_2 C -T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda -(c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 -g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort -O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g -c1 v w) \to False))))) H4))))))))) y x H0))) H)))))). -(* COMMENTS -Initial nodes: 2507 -END *) - -theorem wf3_gen_flat1: - \forall (g: G).(\forall (c1: C).(\forall (x: C).(\forall (v: T).(\forall (f: -F).((wf3 g (CHead c1 (Flat f) v) x) \to (wf3 g c1 x)))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (x: C).(\lambda (v: T).(\lambda (f: -F).(\lambda (H: (wf3 g (CHead c1 (Flat f) v) x)).(insert_eq C (CHead c1 (Flat -f) v) (\lambda (c: C).(wf3 g c x)) (\lambda (_: C).(wf3 g c1 x)) (\lambda (y: -C).(\lambda (H0: (wf3 g y x)).(wf3_ind g (\lambda (c: C).(\lambda (c0: -C).((eq C c (CHead c1 (Flat f) v)) \to (wf3 g c1 c0)))) (\lambda (m: -nat).(\lambda (H1: (eq C (CSort m) (CHead c1 (Flat f) v))).(let H2 \def -(eq_ind C (CSort m) (\lambda (ee: C).(match ee in C return (\lambda (_: -C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow -False])) I (CHead c1 (Flat f) v) H1) in (False_ind (wf3 g c1 (CSort m)) -H2)))) (\lambda (c0: C).(\lambda (c2: C).(\lambda (_: (wf3 g c0 c2)).(\lambda -(_: (((eq C c0 (CHead c1 (Flat f) v)) \to (wf3 g c1 c2)))).(\lambda (u: -T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (b: B).(\lambda (H4: -(eq C (CHead c0 (Bind b) u) (CHead c1 (Flat f) v))).(let H5 \def (eq_ind C -(CHead c0 (Bind b) u) (\lambda (ee: C).(match ee in C return (\lambda (_: -C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match -k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat -_) \Rightarrow False])])) I (CHead c1 (Flat f) v) H4) in (False_ind (wf3 g c1 -(CHead c2 (Bind b) u)) H5))))))))))) (\lambda (c0: C).(\lambda (c2: -C).(\lambda (_: (wf3 g c0 c2)).(\lambda (_: (((eq C c0 (CHead c1 (Flat f) v)) -\to (wf3 g c1 c2)))).(\lambda (u: T).(\lambda (_: ((\forall (t: T).((ty3 g c0 -u t) \to False)))).(\lambda (b: B).(\lambda (H4: (eq C (CHead c0 (Bind b) u) -(CHead c1 (Flat f) v))).(let H5 \def (eq_ind C (CHead c0 (Bind b) u) (\lambda -(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow -False])])) I (CHead c1 (Flat f) v) H4) in (False_ind (wf3 g c1 (CHead c2 -(Bind Void) (TSort O))) H5)))))))))) (\lambda (c0: C).(\lambda (c2: -C).(\lambda (H1: (wf3 g c0 c2)).(\lambda (H2: (((eq C c0 (CHead c1 (Flat f) -v)) \to (wf3 g c1 c2)))).(\lambda (u: T).(\lambda (f0: F).(\lambda (H3: (eq C -(CHead c0 (Flat f0) u) (CHead c1 (Flat f) v))).(let H4 \def (f_equal C C -(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 (Flat f0) u) (CHead -c1 (Flat f) v) H3) in ((let H5 \def (f_equal C F (\lambda (e: C).(match e in -C return (\lambda (_: C).F) with [(CSort _) \Rightarrow f0 | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).F) with [(Bind _) -\Rightarrow f0 | (Flat f1) \Rightarrow f1])])) (CHead c0 (Flat f0) u) (CHead -c1 (Flat f) v) H3) in ((let H6 \def (f_equal C T (\lambda (e: C).(match e in -C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) -\Rightarrow t])) (CHead c0 (Flat f0) u) (CHead c1 (Flat f) v) H3) in (\lambda -(_: (eq F f0 f)).(\lambda (H8: (eq C c0 c1)).(let H9 \def (eq_ind C c0 -(\lambda (c: C).((eq C c (CHead c1 (Flat f) v)) \to (wf3 g c1 c2))) H2 c1 H8) -in (let H10 \def (eq_ind C c0 (\lambda (c: C).(wf3 g c c2)) H1 c1 H8) in -H10))))) H5)) H4))))))))) y x H0))) H)))))). -(* COMMENTS -Initial nodes: 737 -END *) - -theorem wf3_gen_head2: - \forall (g: G).(\forall (x: C).(\forall (c: C).(\forall (v: T).(\forall (k: -K).((wf3 g x (CHead c k v)) \to (ex B (\lambda (b: B).(eq K k (Bind b))))))))) -\def - \lambda (g: G).(\lambda (x: C).(\lambda (c: C).(\lambda (v: T).(\lambda (k: -K).(\lambda (H: (wf3 g x (CHead c k v))).(insert_eq C (CHead c k v) (\lambda -(c0: C).(wf3 g x c0)) (\lambda (_: C).(ex B (\lambda (b: B).(eq K k (Bind -b))))) (\lambda (y: C).(\lambda (H0: (wf3 g x y)).(wf3_ind g (\lambda (_: -C).(\lambda (c1: C).((eq C c1 (CHead c k v)) \to (ex B (\lambda (b: B).(eq K -k (Bind b))))))) (\lambda (m: nat).(\lambda (H1: (eq C (CSort m) (CHead c k -v))).(let H2 \def (eq_ind C (CSort m) (\lambda (ee: C).(match ee in C return -(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) -\Rightarrow False])) I (CHead c k v) H1) in (False_ind (ex B (\lambda (b: -B).(eq K k (Bind b)))) H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: -(wf3 g c1 c2)).(\lambda (H2: (((eq C c2 (CHead c k v)) \to (ex B (\lambda (b: -B).(eq K k (Bind b))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H3: (ty3 -g c1 u t)).(\lambda (b: B).(\lambda (H4: (eq C (CHead c2 (Bind b) u) (CHead c -k v))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return -(\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ _) -\Rightarrow c0])) (CHead c2 (Bind b) u) (CHead c k v) H4) in ((let H6 \def -(f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with -[(CSort _) \Rightarrow (Bind b) | (CHead _ k0 _) \Rightarrow k0])) (CHead c2 -(Bind b) u) (CHead c k v) H4) in ((let H7 \def (f_equal C T (\lambda (e: -C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | -(CHead _ _ t0) \Rightarrow t0])) (CHead c2 (Bind b) u) (CHead c k v) H4) in -(\lambda (H8: (eq K (Bind b) k)).(\lambda (H9: (eq C c2 c)).(let H10 \def -(eq_ind T u (\lambda (t0: T).(ty3 g c1 t0 t)) H3 v H7) in (let H11 \def -(eq_ind C c2 (\lambda (c0: C).((eq C c0 (CHead c k v)) \to (ex B (\lambda -(b0: B).(eq K k (Bind b0)))))) H2 c H9) in (let H12 \def (eq_ind C c2 -(\lambda (c0: C).(wf3 g c1 c0)) H1 c H9) in (let H13 \def (eq_ind_r K k -(\lambda (k0: K).((eq C c (CHead c k0 v)) \to (ex B (\lambda (b0: B).(eq K k0 -(Bind b0)))))) H11 (Bind b) H8) in (eq_ind K (Bind b) (\lambda (k0: K).(ex B -(\lambda (b0: B).(eq K k0 (Bind b0))))) (ex_intro B (\lambda (b0: B).(eq K -(Bind b) (Bind b0))) b (refl_equal K (Bind b))) k H8)))))))) H6)) -H5))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c1 -c2)).(\lambda (H2: (((eq C c2 (CHead c k v)) \to (ex B (\lambda (b: B).(eq K -k (Bind b))))))).(\lambda (u: T).(\lambda (_: ((\forall (t: T).((ty3 g c1 u -t) \to False)))).(\lambda (_: B).(\lambda (H4: (eq C (CHead c2 (Bind Void) -(TSort O)) (CHead c k v))).(let H5 \def (f_equal C C (\lambda (e: C).(match e -in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ -_) \Rightarrow c0])) (CHead c2 (Bind Void) (TSort O)) (CHead c k v) H4) in -((let H6 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: -C).K) with [(CSort _) \Rightarrow (Bind Void) | (CHead _ k0 _) \Rightarrow -k0])) (CHead c2 (Bind Void) (TSort O)) (CHead c k v) H4) in ((let H7 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow (TSort O) | (CHead _ _ t) \Rightarrow t])) (CHead c2 -(Bind Void) (TSort O)) (CHead c k v) H4) in (\lambda (H8: (eq K (Bind Void) -k)).(\lambda (H9: (eq C c2 c)).(let H10 \def (eq_ind C c2 (\lambda (c0: -C).((eq C c0 (CHead c k v)) \to (ex B (\lambda (b0: B).(eq K k (Bind b0)))))) -H2 c H9) in (let H11 \def (eq_ind C c2 (\lambda (c0: C).(wf3 g c1 c0)) H1 c -H9) in (let H12 \def (eq_ind_r K k (\lambda (k0: K).((eq C c (CHead c k0 v)) -\to (ex B (\lambda (b0: B).(eq K k0 (Bind b0)))))) H10 (Bind Void) H8) in -(eq_ind K (Bind Void) (\lambda (k0: K).(ex B (\lambda (b0: B).(eq K k0 (Bind -b0))))) (let H13 \def (eq_ind_r T v (\lambda (t: T).((eq C c (CHead c (Bind -Void) t)) \to (ex B (\lambda (b0: B).(eq K (Bind Void) (Bind b0)))))) H12 -(TSort O) H7) in (ex_intro B (\lambda (b0: B).(eq K (Bind Void) (Bind b0))) -Void (refl_equal K (Bind Void)))) k H8))))))) H6)) H5)))))))))) (\lambda (c1: -C).(\lambda (c2: C).(\lambda (H1: (wf3 g c1 c2)).(\lambda (H2: (((eq C c2 -(CHead c k v)) \to (ex B (\lambda (b: B).(eq K k (Bind b))))))).(\lambda (_: -T).(\lambda (_: F).(\lambda (H3: (eq C c2 (CHead c k v))).(let H4 \def -(f_equal C C (\lambda (e: C).e) c2 (CHead c k v) H3) in (let H5 \def (eq_ind -C c2 (\lambda (c0: C).((eq C c0 (CHead c k v)) \to (ex B (\lambda (b: B).(eq -K k (Bind b)))))) H2 (CHead c k v) H4) in (let H6 \def (eq_ind C c2 (\lambda -(c0: C).(wf3 g c1 c0)) H1 (CHead c k v) H4) in (H5 (refl_equal C (CHead c k -v))))))))))))) x y H0))) H)))))). -(* COMMENTS -Initial nodes: 1225 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/wf3/getl.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/wf3/getl.ma deleted file mode 100644 index 8a8f13780..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/wf3/getl.ma +++ /dev/null @@ -1,205 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/wf3/clear.ma". - -include "Basic-1/ty3/dec.ma". - -theorem wf3_getl_conf: - \forall (b: B).(\forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall -(v: T).((getl i c1 (CHead d1 (Bind b) v)) \to (\forall (g: G).(\forall (c2: -C).((wf3 g c1 c2) \to (\forall (w: T).((ty3 g d1 v w) \to (ex2 C (\lambda -(d2: C).(getl i c2 (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 -d2))))))))))))) -\def - \lambda (b: B).(\lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: -C).(\forall (d1: C).(\forall (v: T).((getl n c1 (CHead d1 (Bind b) v)) \to -(\forall (g: G).(\forall (c2: C).((wf3 g c1 c2) \to (\forall (w: T).((ty3 g -d1 v w) \to (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind b) v))) -(\lambda (d2: C).(wf3 g d1 d2))))))))))))) (\lambda (c1: C).(\lambda (d1: -C).(\lambda (v: T).(\lambda (H: (getl O c1 (CHead d1 (Bind b) v))).(\lambda -(g: G).(\lambda (c2: C).(\lambda (H0: (wf3 g c1 c2)).(\lambda (w: T).(\lambda -(H1: (ty3 g d1 v w)).(let H_y \def (wf3_clear_conf c1 (CHead d1 (Bind b) v) -(getl_gen_O c1 (CHead d1 (Bind b) v) H) g c2 H0) in (let H_x \def -(wf3_gen_bind1 g d1 c2 v b H_y) in (let H2 \def H_x in (or_ind (ex3_2 C T -(\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda -(c3: C).(\lambda (_: T).(wf3 g d1 c3))) (\lambda (_: C).(\lambda (w0: T).(ty3 -g d1 v w0)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort -O)))) (\lambda (c3: C).(wf3 g d1 c3)) (\lambda (_: C).(\forall (w0: T).((ty3 -g d1 v w0) \to False)))) (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind -b) v))) (\lambda (d2: C).(wf3 g d1 d2))) (\lambda (H3: (ex3_2 C T (\lambda -(c3: C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda (c3: -C).(\lambda (_: T).(wf3 g d1 c3))) (\lambda (_: C).(\lambda (w0: T).(ty3 g d1 -v w0))))).(ex3_2_ind C T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 -(Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g d1 c3))) (\lambda (_: -C).(\lambda (w0: T).(ty3 g d1 v w0))) (ex2 C (\lambda (d2: C).(getl O c2 -(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (H4: (eq C c2 (CHead x0 (Bind b) v))).(\lambda -(H5: (wf3 g d1 x0)).(\lambda (_: (ty3 g d1 v x1)).(eq_ind_r C (CHead x0 (Bind -b) v) (\lambda (c: C).(ex2 C (\lambda (d2: C).(getl O c (CHead d2 (Bind b) -v))) (\lambda (d2: C).(wf3 g d1 d2)))) (ex_intro2 C (\lambda (d2: C).(getl O -(CHead x0 (Bind b) v) (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)) -x0 (getl_refl b x0 v) H5) c2 H4)))))) H3)) (\lambda (H3: (ex3 C (\lambda (c3: -C).(eq C c2 (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g d1 -c3)) (\lambda (_: C).(\forall (w0: T).((ty3 g d1 v w0) \to -False))))).(ex3_ind C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort -O)))) (\lambda (c3: C).(wf3 g d1 c3)) (\lambda (_: C).(\forall (w0: T).((ty3 -g d1 v w0) \to False))) (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind b) -v))) (\lambda (d2: C).(wf3 g d1 d2))) (\lambda (x0: C).(\lambda (H4: (eq C c2 -(CHead x0 (Bind Void) (TSort O)))).(\lambda (_: (wf3 g d1 x0)).(\lambda (H6: -((\forall (w0: T).((ty3 g d1 v w0) \to False)))).(eq_ind_r C (CHead x0 (Bind -Void) (TSort O)) (\lambda (c: C).(ex2 C (\lambda (d2: C).(getl O c (CHead d2 -(Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)))) (let H_x0 \def (H6 w H1) in -(let H7 \def H_x0 in (False_ind (ex2 C (\lambda (d2: C).(getl O (CHead x0 -(Bind Void) (TSort O)) (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 -d2))) H7))) c2 H4))))) H3)) H2))))))))))))) (\lambda (n: nat).(\lambda (H: -((\forall (c1: C).(\forall (d1: C).(\forall (v: T).((getl n c1 (CHead d1 -(Bind b) v)) \to (\forall (g: G).(\forall (c2: C).((wf3 g c1 c2) \to (\forall -(w: T).((ty3 g d1 v w) \to (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind -b) v))) (\lambda (d2: C).(wf3 g d1 d2)))))))))))))).(\lambda (c1: C).(C_ind -(\lambda (c: C).(\forall (d1: C).(\forall (v: T).((getl (S n) c (CHead d1 -(Bind b) v)) \to (\forall (g: G).(\forall (c2: C).((wf3 g c c2) \to (\forall -(w: T).((ty3 g d1 v w) \to (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 -(Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)))))))))))) (\lambda (n0: -nat).(\lambda (d1: C).(\lambda (v: T).(\lambda (H0: (getl (S n) (CSort n0) -(CHead d1 (Bind b) v))).(\lambda (g: G).(\lambda (c2: C).(\lambda (_: (wf3 g -(CSort n0) c2)).(\lambda (w: T).(\lambda (_: (ty3 g d1 v w)).(getl_gen_sort -n0 (S n) (CHead d1 (Bind b) v) H0 (ex2 C (\lambda (d2: C).(getl (S n) c2 -(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2))))))))))))) (\lambda -(c: C).(\lambda (H0: ((\forall (d1: C).(\forall (v: T).((getl (S n) c (CHead -d1 (Bind b) v)) \to (\forall (g: G).(\forall (c2: C).((wf3 g c c2) \to -(\forall (w: T).((ty3 g d1 v w) \to (ex2 C (\lambda (d2: C).(getl (S n) c2 -(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2))))))))))))).(\lambda -(k: K).(\lambda (t: T).(\lambda (d1: C).(\lambda (v: T).(\lambda (H1: (getl -(S n) (CHead c k t) (CHead d1 (Bind b) v))).(\lambda (g: G).(\lambda (c2: -C).(\lambda (H2: (wf3 g (CHead c k t) c2)).(\lambda (w: T).(\lambda (H3: (ty3 -g d1 v w)).(K_ind (\lambda (k0: K).((wf3 g (CHead c k0 t) c2) \to ((getl (r -k0 n) c (CHead d1 (Bind b) v)) \to (ex2 C (\lambda (d2: C).(getl (S n) c2 -(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)))))) (\lambda (b0: -B).(\lambda (H4: (wf3 g (CHead c (Bind b0) t) c2)).(\lambda (H5: (getl (r -(Bind b0) n) c (CHead d1 (Bind b) v))).(let H_x \def (wf3_gen_bind1 g c c2 t -b0 H4) in (let H6 \def H_x in (or_ind (ex3_2 C T (\lambda (c3: C).(\lambda -(_: T).(eq C c2 (CHead c3 (Bind b0) t)))) (\lambda (c3: C).(\lambda (_: -T).(wf3 g c c3))) (\lambda (_: C).(\lambda (w0: T).(ty3 g c t w0)))) (ex3 C -(\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: -C).(wf3 g c c3)) (\lambda (_: C).(\forall (w0: T).((ty3 g c t w0) \to -False)))) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind b) v))) -(\lambda (d2: C).(wf3 g d1 d2))) (\lambda (H7: (ex3_2 C T (\lambda (c3: -C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b0) t)))) (\lambda (c3: -C).(\lambda (_: T).(wf3 g c c3))) (\lambda (_: C).(\lambda (w0: T).(ty3 g c t -w0))))).(ex3_2_ind C T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 -(Bind b0) t)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c c3))) (\lambda (_: -C).(\lambda (w0: T).(ty3 g c t w0))) (ex2 C (\lambda (d2: C).(getl (S n) c2 -(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2))) (\lambda (x0: -C).(\lambda (x1: T).(\lambda (H8: (eq C c2 (CHead x0 (Bind b0) t))).(\lambda -(H9: (wf3 g c x0)).(\lambda (_: (ty3 g c t x1)).(eq_ind_r C (CHead x0 (Bind -b0) t) (\lambda (c0: C).(ex2 C (\lambda (d2: C).(getl (S n) c0 (CHead d2 -(Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)))) (let H_x0 \def (H c d1 v H5 g -x0 H9 w H3) in (let H11 \def H_x0 in (ex2_ind C (\lambda (d2: C).(getl n x0 -(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)) (ex2 C (\lambda (d2: -C).(getl (S n) (CHead x0 (Bind b0) t) (CHead d2 (Bind b) v))) (\lambda (d2: -C).(wf3 g d1 d2))) (\lambda (x: C).(\lambda (H12: (getl n x0 (CHead x (Bind -b) v))).(\lambda (H13: (wf3 g d1 x)).(ex_intro2 C (\lambda (d2: C).(getl (S -n) (CHead x0 (Bind b0) t) (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 -d2)) x (getl_head (Bind b0) n x0 (CHead x (Bind b) v) H12 t) H13)))) H11))) -c2 H8)))))) H7)) (\lambda (H7: (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 -(Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c c3)) (\lambda (_: -C).(\forall (w0: T).((ty3 g c t w0) \to False))))).(ex3_ind C (\lambda (c3: -C).(eq C c2 (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c c3)) -(\lambda (_: C).(\forall (w0: T).((ty3 g c t w0) \to False))) (ex2 C (\lambda -(d2: C).(getl (S n) c2 (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 -d2))) (\lambda (x0: C).(\lambda (H8: (eq C c2 (CHead x0 (Bind Void) (TSort -O)))).(\lambda (H9: (wf3 g c x0)).(\lambda (_: ((\forall (w0: T).((ty3 g c t -w0) \to False)))).(eq_ind_r C (CHead x0 (Bind Void) (TSort O)) (\lambda (c0: -C).(ex2 C (\lambda (d2: C).(getl (S n) c0 (CHead d2 (Bind b) v))) (\lambda -(d2: C).(wf3 g d1 d2)))) (let H_x0 \def (H c d1 v H5 g x0 H9 w H3) in (let -H11 \def H_x0 in (ex2_ind C (\lambda (d2: C).(getl n x0 (CHead d2 (Bind b) -v))) (\lambda (d2: C).(wf3 g d1 d2)) (ex2 C (\lambda (d2: C).(getl (S n) -(CHead x0 (Bind Void) (TSort O)) (CHead d2 (Bind b) v))) (\lambda (d2: -C).(wf3 g d1 d2))) (\lambda (x: C).(\lambda (H12: (getl n x0 (CHead x (Bind -b) v))).(\lambda (H13: (wf3 g d1 x)).(ex_intro2 C (\lambda (d2: C).(getl (S -n) (CHead x0 (Bind Void) (TSort O)) (CHead d2 (Bind b) v))) (\lambda (d2: -C).(wf3 g d1 d2)) x (getl_head (Bind Void) n x0 (CHead x (Bind b) v) H12 -(TSort O)) H13)))) H11))) c2 H8))))) H7)) H6)))))) (\lambda (f: F).(\lambda -(H4: (wf3 g (CHead c (Flat f) t) c2)).(\lambda (H5: (getl (r (Flat f) n) c -(CHead d1 (Bind b) v))).(let H_y \def (wf3_gen_flat1 g c c2 t f H4) in (H0 d1 -v H5 g c2 H_y w H3))))) k H2 (getl_gen_S k c (CHead d1 (Bind b) v) t n -H1)))))))))))))) c1)))) i)). -(* COMMENTS -Initial nodes: 2531 -END *) - -theorem getl_wf3_trans: - \forall (i: nat).(\forall (c1: C).(\forall (d1: C).((getl i c1 d1) \to -(\forall (g: G).(\forall (d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda (c2: -C).(wf3 g c1 c2)) (\lambda (c2: C).(getl i c2 d2))))))))) -\def - \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (d1: -C).((getl n c1 d1) \to (\forall (g: G).(\forall (d2: C).((wf3 g d1 d2) \to -(ex2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(getl n c2 -d2)))))))))) (\lambda (c1: C).(\lambda (d1: C).(\lambda (H: (getl O c1 -d1)).(\lambda (g: G).(\lambda (d2: C).(\lambda (H0: (wf3 g d1 d2)).(let H_x -\def (clear_wf3_trans c1 d1 (getl_gen_O c1 d1 H) g d2 H0) in (let H1 \def H_x -in (ex2_ind C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(clear c2 d2)) -(ex2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(getl O c2 d2))) -(\lambda (x: C).(\lambda (H2: (wf3 g c1 x)).(\lambda (H3: (clear x -d2)).(ex_intro2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(getl O c2 -d2)) x H2 (getl_intro O x d2 x (drop_refl x) H3))))) H1))))))))) (\lambda (n: -nat).(\lambda (H: ((\forall (c1: C).(\forall (d1: C).((getl n c1 d1) \to -(\forall (g: G).(\forall (d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda (c2: -C).(wf3 g c1 c2)) (\lambda (c2: C).(getl n c2 d2))))))))))).(\lambda (c1: -C).(C_ind (\lambda (c: C).(\forall (d1: C).((getl (S n) c d1) \to (\forall -(g: G).(\forall (d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda (c2: C).(wf3 g c -c2)) (\lambda (c2: C).(getl (S n) c2 d2))))))))) (\lambda (n0: nat).(\lambda -(d1: C).(\lambda (H0: (getl (S n) (CSort n0) d1)).(\lambda (g: G).(\lambda -(d2: C).(\lambda (_: (wf3 g d1 d2)).(getl_gen_sort n0 (S n) d1 H0 (ex2 C -(\lambda (c2: C).(wf3 g (CSort n0) c2)) (\lambda (c2: C).(getl (S n) c2 -d2)))))))))) (\lambda (c: C).(\lambda (H0: ((\forall (d1: C).((getl (S n) c -d1) \to (\forall (g: G).(\forall (d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda -(c2: C).(wf3 g c c2)) (\lambda (c2: C).(getl (S n) c2 d2)))))))))).(\lambda -(k: K).(\lambda (t: T).(\lambda (d1: C).(\lambda (H1: (getl (S n) (CHead c k -t) d1)).(\lambda (g: G).(\lambda (d2: C).(\lambda (H2: (wf3 g d1 d2)).(K_ind -(\lambda (k0: K).((getl (r k0 n) c d1) \to (ex2 C (\lambda (c2: C).(wf3 g -(CHead c k0 t) c2)) (\lambda (c2: C).(getl (S n) c2 d2))))) (\lambda (b: -B).(\lambda (H3: (getl (r (Bind b) n) c d1)).(let H_x \def (H c d1 H3 g d2 -H2) in (let H4 \def H_x in (ex2_ind C (\lambda (c2: C).(wf3 g c c2)) (\lambda -(c2: C).(getl n c2 d2)) (ex2 C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) -c2)) (\lambda (c2: C).(getl (S n) c2 d2))) (\lambda (x: C).(\lambda (H5: (wf3 -g c x)).(\lambda (H6: (getl n x d2)).(let H_x0 \def (ty3_inference g c t) in -(let H7 \def H_x0 in (or_ind (ex T (\lambda (t2: T).(ty3 g c t t2))) (\forall -(t2: T).((ty3 g c t t2) \to False)) (ex2 C (\lambda (c2: C).(wf3 g (CHead c -(Bind b) t) c2)) (\lambda (c2: C).(getl (S n) c2 d2))) (\lambda (H8: (ex T -(\lambda (t2: T).(ty3 g c t t2)))).(ex_ind T (\lambda (t2: T).(ty3 g c t t2)) -(ex2 C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2)) (\lambda (c2: -C).(getl (S n) c2 d2))) (\lambda (x0: T).(\lambda (H9: (ty3 g c t -x0)).(ex_intro2 C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2)) (\lambda -(c2: C).(getl (S n) c2 d2)) (CHead x (Bind b) t) (wf3_bind g c x H5 t x0 H9 -b) (getl_head (Bind b) n x d2 H6 t)))) H8)) (\lambda (H8: ((\forall (t2: -T).((ty3 g c t t2) \to False)))).(ex_intro2 C (\lambda (c2: C).(wf3 g (CHead -c (Bind b) t) c2)) (\lambda (c2: C).(getl (S n) c2 d2)) (CHead x (Bind Void) -(TSort O)) (wf3_void g c x H5 t H8 b) (getl_head (Bind Void) n x d2 H6 (TSort -O)))) H7)))))) H4))))) (\lambda (f: F).(\lambda (H3: (getl (r (Flat f) n) c -d1)).(let H_x \def (H0 d1 H3 g d2 H2) in (let H4 \def H_x in (ex2_ind C -(\lambda (c2: C).(wf3 g c c2)) (\lambda (c2: C).(getl (S n) c2 d2)) (ex2 C -(\lambda (c2: C).(wf3 g (CHead c (Flat f) t) c2)) (\lambda (c2: C).(getl (S -n) c2 d2))) (\lambda (x: C).(\lambda (H5: (wf3 g c x)).(\lambda (H6: (getl (S -n) x d2)).(ex_intro2 C (\lambda (c2: C).(wf3 g (CHead c (Flat f) t) c2)) -(\lambda (c2: C).(getl (S n) c2 d2)) x (wf3_flat g c x H5 t f) H6)))) H4))))) -k (getl_gen_S k c d1 t n H1))))))))))) c1)))) i). -(* COMMENTS -Initial nodes: 1139 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/wf3/props.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/wf3/props.ma deleted file mode 100644 index 98a05c637..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/wf3/props.ma +++ /dev/null @@ -1,248 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/wf3/ty3.ma". - -include "Basic-1/app/defs.ma". - -theorem wf3_mono: - \forall (g: G).(\forall (c: C).(\forall (c1: C).((wf3 g c c1) \to (\forall -(c2: C).((wf3 g c c2) \to (eq C c1 c2)))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (c1: C).(\lambda (H: (wf3 g c -c1)).(wf3_ind g (\lambda (c0: C).(\lambda (c2: C).(\forall (c3: C).((wf3 g c0 -c3) \to (eq C c2 c3))))) (\lambda (m: nat).(\lambda (c2: C).(\lambda (H0: -(wf3 g (CSort m) c2)).(let H_y \def (wf3_gen_sort1 g c2 m H0) in (eq_ind_r C -(CSort m) (\lambda (c0: C).(eq C (CSort m) c0)) (refl_equal C (CSort m)) c2 -H_y))))) (\lambda (c2: C).(\lambda (c3: C).(\lambda (_: (wf3 g c2 -c3)).(\lambda (H1: ((\forall (c4: C).((wf3 g c2 c4) \to (eq C c3 -c4))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H2: (ty3 g c2 u -t)).(\lambda (b: B).(\lambda (c0: C).(\lambda (H3: (wf3 g (CHead c2 (Bind b) -u) c0)).(let H_x \def (wf3_gen_bind1 g c2 c0 u b H3) in (let H4 \def H_x in -(or_ind (ex3_2 C T (\lambda (c4: C).(\lambda (_: T).(eq C c0 (CHead c4 (Bind -b) u)))) (\lambda (c4: C).(\lambda (_: T).(wf3 g c2 c4))) (\lambda (_: -C).(\lambda (w: T).(ty3 g c2 u w)))) (ex3 C (\lambda (c4: C).(eq C c0 (CHead -c4 (Bind Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda (_: -C).(\forall (w: T).((ty3 g c2 u w) \to False)))) (eq C (CHead c3 (Bind b) u) -c0) (\lambda (H5: (ex3_2 C T (\lambda (c4: C).(\lambda (_: T).(eq C c0 (CHead -c4 (Bind b) u)))) (\lambda (c4: C).(\lambda (_: T).(wf3 g c2 c4))) (\lambda -(_: C).(\lambda (w: T).(ty3 g c2 u w))))).(ex3_2_ind C T (\lambda (c4: -C).(\lambda (_: T).(eq C c0 (CHead c4 (Bind b) u)))) (\lambda (c4: -C).(\lambda (_: T).(wf3 g c2 c4))) (\lambda (_: C).(\lambda (w: T).(ty3 g c2 -u w))) (eq C (CHead c3 (Bind b) u) c0) (\lambda (x0: C).(\lambda (x1: -T).(\lambda (H6: (eq C c0 (CHead x0 (Bind b) u))).(\lambda (H7: (wf3 g c2 -x0)).(\lambda (_: (ty3 g c2 u x1)).(eq_ind_r C (CHead x0 (Bind b) u) (\lambda -(c4: C).(eq C (CHead c3 (Bind b) u) c4)) (f_equal3 C K T C CHead c3 x0 (Bind -b) (Bind b) u u (H1 x0 H7) (refl_equal K (Bind b)) (refl_equal T u)) c0 -H6)))))) H5)) (\lambda (H5: (ex3 C (\lambda (c4: C).(eq C c0 (CHead c4 (Bind -Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda (_: C).(\forall -(w: T).((ty3 g c2 u w) \to False))))).(ex3_ind C (\lambda (c4: C).(eq C c0 -(CHead c4 (Bind Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda -(_: C).(\forall (w: T).((ty3 g c2 u w) \to False))) (eq C (CHead c3 (Bind b) -u) c0) (\lambda (x0: C).(\lambda (H6: (eq C c0 (CHead x0 (Bind Void) (TSort -O)))).(\lambda (_: (wf3 g c2 x0)).(\lambda (H8: ((\forall (w: T).((ty3 g c2 u -w) \to False)))).(eq_ind_r C (CHead x0 (Bind Void) (TSort O)) (\lambda (c4: -C).(eq C (CHead c3 (Bind b) u) c4)) (let H_x0 \def (H8 t H2) in (let H9 \def -H_x0 in (False_ind (eq C (CHead c3 (Bind b) u) (CHead x0 (Bind Void) (TSort -O))) H9))) c0 H6))))) H5)) H4))))))))))))) (\lambda (c2: C).(\lambda (c3: -C).(\lambda (_: (wf3 g c2 c3)).(\lambda (H1: ((\forall (c4: C).((wf3 g c2 c4) -\to (eq C c3 c4))))).(\lambda (u: T).(\lambda (H2: ((\forall (t: T).((ty3 g -c2 u t) \to False)))).(\lambda (b: B).(\lambda (c0: C).(\lambda (H3: (wf3 g -(CHead c2 (Bind b) u) c0)).(let H_x \def (wf3_gen_bind1 g c2 c0 u b H3) in -(let H4 \def H_x in (or_ind (ex3_2 C T (\lambda (c4: C).(\lambda (_: T).(eq C -c0 (CHead c4 (Bind b) u)))) (\lambda (c4: C).(\lambda (_: T).(wf3 g c2 c4))) -(\lambda (_: C).(\lambda (w: T).(ty3 g c2 u w)))) (ex3 C (\lambda (c4: C).(eq -C c0 (CHead c4 (Bind Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) -(\lambda (_: C).(\forall (w: T).((ty3 g c2 u w) \to False)))) (eq C (CHead c3 -(Bind Void) (TSort O)) c0) (\lambda (H5: (ex3_2 C T (\lambda (c4: C).(\lambda -(_: T).(eq C c0 (CHead c4 (Bind b) u)))) (\lambda (c4: C).(\lambda (_: -T).(wf3 g c2 c4))) (\lambda (_: C).(\lambda (w: T).(ty3 g c2 u -w))))).(ex3_2_ind C T (\lambda (c4: C).(\lambda (_: T).(eq C c0 (CHead c4 -(Bind b) u)))) (\lambda (c4: C).(\lambda (_: T).(wf3 g c2 c4))) (\lambda (_: -C).(\lambda (w: T).(ty3 g c2 u w))) (eq C (CHead c3 (Bind Void) (TSort O)) -c0) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C c0 (CHead x0 (Bind -b) u))).(\lambda (_: (wf3 g c2 x0)).(\lambda (H8: (ty3 g c2 u x1)).(eq_ind_r -C (CHead x0 (Bind b) u) (\lambda (c4: C).(eq C (CHead c3 (Bind Void) (TSort -O)) c4)) (let H_x0 \def (H2 x1 H8) in (let H9 \def H_x0 in (False_ind (eq C -(CHead c3 (Bind Void) (TSort O)) (CHead x0 (Bind b) u)) H9))) c0 H6)))))) -H5)) (\lambda (H5: (ex3 C (\lambda (c4: C).(eq C c0 (CHead c4 (Bind Void) -(TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda (_: C).(\forall (w: -T).((ty3 g c2 u w) \to False))))).(ex3_ind C (\lambda (c4: C).(eq C c0 (CHead -c4 (Bind Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda (_: -C).(\forall (w: T).((ty3 g c2 u w) \to False))) (eq C (CHead c3 (Bind Void) -(TSort O)) c0) (\lambda (x0: C).(\lambda (H6: (eq C c0 (CHead x0 (Bind Void) -(TSort O)))).(\lambda (H7: (wf3 g c2 x0)).(\lambda (_: ((\forall (w: T).((ty3 -g c2 u w) \to False)))).(eq_ind_r C (CHead x0 (Bind Void) (TSort O)) (\lambda -(c4: C).(eq C (CHead c3 (Bind Void) (TSort O)) c4)) (f_equal3 C K T C CHead -c3 x0 (Bind Void) (Bind Void) (TSort O) (TSort O) (H1 x0 H7) (refl_equal K -(Bind Void)) (refl_equal T (TSort O))) c0 H6))))) H5)) H4)))))))))))) -(\lambda (c2: C).(\lambda (c3: C).(\lambda (_: (wf3 g c2 c3)).(\lambda (H1: -((\forall (c4: C).((wf3 g c2 c4) \to (eq C c3 c4))))).(\lambda (u: -T).(\lambda (f: F).(\lambda (c0: C).(\lambda (H2: (wf3 g (CHead c2 (Flat f) -u) c0)).(let H_y \def (wf3_gen_flat1 g c2 c0 u f H2) in (H1 c0 H_y)))))))))) -c c1 H)))). -(* COMMENTS -Initial nodes: 1555 -END *) - -theorem wf3_total: - \forall (g: G).(\forall (c1: C).(ex C (\lambda (c2: C).(wf3 g c1 c2)))) -\def - \lambda (g: G).(\lambda (c1: C).(C_ind (\lambda (c: C).(ex C (\lambda (c2: -C).(wf3 g c c2)))) (\lambda (n: nat).(ex_intro C (\lambda (c2: C).(wf3 g -(CSort n) c2)) (CSort n) (wf3_sort g n))) (\lambda (c: C).(\lambda (H: (ex C -(\lambda (c2: C).(wf3 g c c2)))).(\lambda (k: K).(\lambda (t: T).(let H0 \def -H in (ex_ind C (\lambda (c2: C).(wf3 g c c2)) (ex C (\lambda (c2: C).(wf3 g -(CHead c k t) c2))) (\lambda (x: C).(\lambda (H1: (wf3 g c x)).(K_ind -(\lambda (k0: K).(ex C (\lambda (c2: C).(wf3 g (CHead c k0 t) c2)))) (\lambda -(b: B).(let H_x \def (ty3_inference g c t) in (let H2 \def H_x in (or_ind (ex -T (\lambda (t2: T).(ty3 g c t t2))) (\forall (t2: T).((ty3 g c t t2) \to -False)) (ex C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2))) (\lambda -(H3: (ex T (\lambda (t2: T).(ty3 g c t t2)))).(ex_ind T (\lambda (t2: T).(ty3 -g c t t2)) (ex C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2))) (\lambda -(x0: T).(\lambda (H4: (ty3 g c t x0)).(ex_intro C (\lambda (c2: C).(wf3 g -(CHead c (Bind b) t) c2)) (CHead x (Bind b) t) (wf3_bind g c x H1 t x0 H4 -b)))) H3)) (\lambda (H3: ((\forall (t2: T).((ty3 g c t t2) \to -False)))).(ex_intro C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2)) -(CHead x (Bind Void) (TSort O)) (wf3_void g c x H1 t H3 b))) H2)))) (\lambda -(f: F).(ex_intro C (\lambda (c2: C).(wf3 g (CHead c (Flat f) t) c2)) x -(wf3_flat g c x H1 t f))) k))) H0)))))) c1)). -(* COMMENTS -Initial nodes: 435 -END *) - -theorem ty3_shift1: - \forall (g: G).(\forall (c: C).((wf3 g c c) \to (\forall (t1: T).(\forall -(t2: T).((ty3 g c t1 t2) \to (ty3 g (CSort (cbk c)) (app1 c t1) (app1 c -t2))))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (H: (wf3 g c c)).(insert_eq C c -(\lambda (c0: C).(wf3 g c0 c)) (\lambda (c0: C).(\forall (t1: T).(\forall -(t2: T).((ty3 g c0 t1 t2) \to (ty3 g (CSort (cbk c0)) (app1 c0 t1) (app1 c0 -t2)))))) (\lambda (y: C).(\lambda (H0: (wf3 g y c)).(wf3_ind g (\lambda (c0: -C).(\lambda (c1: C).((eq C c0 c1) \to (\forall (t1: T).(\forall (t2: T).((ty3 -g c0 t1 t2) \to (ty3 g (CSort (cbk c0)) (app1 c0 t1) (app1 c0 t2)))))))) -(\lambda (m: nat).(\lambda (_: (eq C (CSort m) (CSort m))).(\lambda (t1: -T).(\lambda (t2: T).(\lambda (H2: (ty3 g (CSort m) t1 t2)).H2))))) (\lambda -(c1: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c1 c2)).(\lambda (H2: (((eq C -c1 c2) \to (\forall (t1: T).(\forall (t2: T).((ty3 g c1 t1 t2) \to (ty3 g -(CSort (cbk c1)) (app1 c1 t1) (app1 c1 t2)))))))).(\lambda (u: T).(\lambda -(t: T).(\lambda (H3: (ty3 g c1 u t)).(\lambda (b: B).(\lambda (H4: (eq C -(CHead c1 (Bind b) u) (CHead c2 (Bind b) u))).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (H5: (ty3 g (CHead c1 (Bind b) u) t1 t2)).(let H6 \def (f_equal C -C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow c1 | (CHead c0 _ _) \Rightarrow c0])) (CHead c1 (Bind b) u) -(CHead c2 (Bind b) u) H4) in (let H7 \def (eq_ind_r C c2 (\lambda (c0: -C).((eq C c1 c0) \to (\forall (t3: T).(\forall (t4: T).((ty3 g c1 t3 t4) \to -(ty3 g (CSort (cbk c1)) (app1 c1 t3) (app1 c1 t4))))))) H2 c1 H6) in (let H8 -\def (eq_ind_r C c2 (\lambda (c0: C).(wf3 g c1 c0)) H1 c1 H6) in (ex_ind T -(\lambda (t0: T).(ty3 g (CHead c1 (Bind b) u) t2 t0)) (ty3 g (CSort (cbk c1)) -(app1 c1 (THead (Bind b) u t1)) (app1 c1 (THead (Bind b) u t2))) (\lambda (x: -T).(\lambda (_: (ty3 g (CHead c1 (Bind b) u) t2 x)).(H7 (refl_equal C c1) -(THead (Bind b) u t1) (THead (Bind b) u t2) (ty3_bind g c1 u t H3 b t1 t2 -H5)))) (ty3_correct g (CHead c1 (Bind b) u) t1 t2 H5))))))))))))))))) -(\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c1 c2)).(\lambda (H2: -(((eq C c1 c2) \to (\forall (t1: T).(\forall (t2: T).((ty3 g c1 t1 t2) \to -(ty3 g (CSort (cbk c1)) (app1 c1 t1) (app1 c1 t2)))))))).(\lambda (u: -T).(\lambda (H3: ((\forall (t: T).((ty3 g c1 u t) \to False)))).(\lambda (b: -B).(\lambda (H4: (eq C (CHead c1 (Bind b) u) (CHead c2 (Bind Void) (TSort -O)))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H5: (ty3 g (CHead c1 (Bind -b) u) t1 t2)).(let H6 \def (f_equal C C (\lambda (e: C).(match e in C return -(\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _ _) -\Rightarrow c0])) (CHead c1 (Bind b) u) (CHead c2 (Bind Void) (TSort O)) H4) -in ((let H7 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda -(_: C).B) with [(CSort _) \Rightarrow b | (CHead _ k _) \Rightarrow (match k -in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) -\Rightarrow b])])) (CHead c1 (Bind b) u) (CHead c2 (Bind Void) (TSort O)) H4) -in ((let H8 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda -(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) -(CHead c1 (Bind b) u) (CHead c2 (Bind Void) (TSort O)) H4) in (\lambda (H9: -(eq B b Void)).(\lambda (H10: (eq C c1 c2)).(let H11 \def (eq_ind B b -(\lambda (b0: B).(ty3 g (CHead c1 (Bind b0) u) t1 t2)) H5 Void H9) in -(eq_ind_r B Void (\lambda (b0: B).(ty3 g (CSort (cbk (CHead c1 (Bind b0) u))) -(app1 (CHead c1 (Bind b0) u) t1) (app1 (CHead c1 (Bind b0) u) t2))) (let H12 -\def (eq_ind T u (\lambda (t: T).(ty3 g (CHead c1 (Bind Void) t) t1 t2)) H11 -(TSort O) H8) in (let H13 \def (eq_ind T u (\lambda (t: T).(\forall (t0: -T).((ty3 g c1 t t0) \to False))) H3 (TSort O) H8) in (eq_ind_r T (TSort O) -(\lambda (t: T).(ty3 g (CSort (cbk (CHead c1 (Bind Void) t))) (app1 (CHead c1 -(Bind Void) t) t1) (app1 (CHead c1 (Bind Void) t) t2))) (let H14 \def -(eq_ind_r C c2 (\lambda (c0: C).((eq C c1 c0) \to (\forall (t3: T).(\forall -(t4: T).((ty3 g c1 t3 t4) \to (ty3 g (CSort (cbk c1)) (app1 c1 t3) (app1 c1 -t4))))))) H2 c1 H10) in (let H15 \def (eq_ind_r C c2 (\lambda (c0: C).(wf3 g -c1 c0)) H1 c1 H10) in (ex_ind T (\lambda (t: T).(ty3 g (CHead c1 (Bind Void) -(TSort O)) t2 t)) (ty3 g (CSort (cbk c1)) (app1 c1 (THead (Bind Void) (TSort -O) t1)) (app1 c1 (THead (Bind Void) (TSort O) t2))) (\lambda (x: T).(\lambda -(_: (ty3 g (CHead c1 (Bind Void) (TSort O)) t2 x)).(H14 (refl_equal C c1) -(THead (Bind Void) (TSort O) t1) (THead (Bind Void) (TSort O) t2) (ty3_bind g -c1 (TSort O) (TSort (next g O)) (ty3_sort g c1 O) Void t1 t2 H12)))) -(ty3_correct g (CHead c1 (Bind Void) (TSort O)) t1 t2 H12)))) u H8))) b -H9))))) H7)) H6))))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: -(wf3 g c1 c2)).(\lambda (H2: (((eq C c1 c2) \to (\forall (t1: T).(\forall -(t2: T).((ty3 g c1 t1 t2) \to (ty3 g (CSort (cbk c1)) (app1 c1 t1) (app1 c1 -t2)))))))).(\lambda (u: T).(\lambda (f: F).(\lambda (H3: (eq C (CHead c1 -(Flat f) u) c2)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g (CHead -c1 (Flat f) u) t1 t2)).(let H5 \def (f_equal C C (\lambda (e: C).e) (CHead c1 -(Flat f) u) c2 H3) in (let H6 \def (eq_ind_r C c2 (\lambda (c0: C).((eq C c1 -c0) \to (\forall (t3: T).(\forall (t4: T).((ty3 g c1 t3 t4) \to (ty3 g (CSort -(cbk c1)) (app1 c1 t3) (app1 c1 t4))))))) H2 (CHead c1 (Flat f) u) H5) in -(let H7 \def (eq_ind_r C c2 (\lambda (c0: C).(wf3 g c1 c0)) H1 (CHead c1 -(Flat f) u) H5) in (let H_x \def (wf3_gen_head2 g c1 c1 u (Flat f) H7) in -(let H8 \def H_x in (ex_ind B (\lambda (b: B).(eq K (Flat f) (Bind b))) (ty3 -g (CSort (cbk c1)) (app1 c1 (THead (Flat f) u t1)) (app1 c1 (THead (Flat f) u -t2))) (\lambda (x: B).(\lambda (H9: (eq K (Flat f) (Bind x))).(let H10 \def -(eq_ind K (Flat f) (\lambda (ee: K).(match ee in K return (\lambda (_: -K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])) I -(Bind x) H9) in (False_ind (ty3 g (CSort (cbk c1)) (app1 c1 (THead (Flat f) u -t1)) (app1 c1 (THead (Flat f) u t2))) H10)))) H8)))))))))))))))) y c H0))) -H))). -(* COMMENTS -Initial nodes: 1677 -END *) - -theorem wf3_idem: - \forall (g: G).(\forall (c1: C).(\forall (c2: C).((wf3 g c1 c2) \to (wf3 g -c2 c2)))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wf3 g c1 -c2)).(wf3_ind g (\lambda (_: C).(\lambda (c0: C).(wf3 g c0 c0))) (\lambda (m: -nat).(wf3_sort g m)) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wf3 g -c3 c4)).(\lambda (H1: (wf3 g c4 c4)).(\lambda (u: T).(\lambda (t: T).(\lambda -(H2: (ty3 g c3 u t)).(\lambda (b: B).(wf3_bind g c4 c4 H1 u t (wf3_ty3_conf g -c3 u t H2 c4 H0) b))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (_: -(wf3 g c3 c4)).(\lambda (H1: (wf3 g c4 c4)).(\lambda (u: T).(\lambda (_: -((\forall (t: T).((ty3 g c3 u t) \to False)))).(\lambda (_: B).(wf3_bind g c4 -c4 H1 (TSort O) (TSort (next g O)) (ty3_sort g c4 O) Void)))))))) (\lambda -(c3: C).(\lambda (c4: C).(\lambda (_: (wf3 g c3 c4)).(\lambda (H1: (wf3 g c4 -c4)).(\lambda (_: T).(\lambda (_: F).H1)))))) c1 c2 H)))). -(* COMMENTS -Initial nodes: 207 -END *) - -theorem wf3_ty3: - \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (u: T).((ty3 g c1 t -u) \to (ex2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(ty3 g c2 t -u))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H: -(ty3 g c1 t u)).(let H_x \def (wf3_total g c1) in (let H0 \def H_x in (ex_ind -C (\lambda (c2: C).(wf3 g c1 c2)) (ex2 C (\lambda (c2: C).(wf3 g c1 c2)) -(\lambda (c2: C).(ty3 g c2 t u))) (\lambda (x: C).(\lambda (H1: (wf3 g c1 -x)).(ex_intro2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(ty3 g c2 t -u)) x H1 (wf3_ty3_conf g c1 t u H x H1)))) H0))))))). -(* COMMENTS -Initial nodes: 123 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/wf3/ty3.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/wf3/ty3.ma deleted file mode 100644 index e47d4a2e0..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/wf3/ty3.ma +++ /dev/null @@ -1,138 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/wf3/getl.ma". - -theorem wf3_pr2_conf: - \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pr2 c1 -t1 t2) \to (\forall (c2: C).((wf3 g c1 c2) \to (\forall (u: T).((ty3 g c1 t1 -u) \to (pr2 c2 t1 t2))))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H: (pr2 c1 t1 t2)).(pr2_ind (\lambda (c: C).(\lambda (t: T).(\lambda (t0: -T).(\forall (c2: C).((wf3 g c c2) \to (\forall (u: T).((ty3 g c t u) \to (pr2 -c2 t t0)))))))) (\lambda (c: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda -(H0: (pr0 t3 t4)).(\lambda (c2: C).(\lambda (_: (wf3 g c c2)).(\lambda (u: -T).(\lambda (_: (ty3 g c t3 u)).(pr2_free c2 t3 t4 H0))))))))) (\lambda (c: -C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c -(CHead d (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: -(pr0 t3 t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (c2: -C).(\lambda (H3: (wf3 g c c2)).(\lambda (u0: T).(\lambda (H4: (ty3 g c t3 -u0)).(let H_y \def (ty3_sred_pr0 t3 t4 H1 g c u0 H4) in (let H_x \def -(ty3_getl_subst0 g c t4 u0 H_y u t i H2 Abbr d u H0) in (let H5 \def H_x in -(ex_ind T (\lambda (w: T).(ty3 g d u w)) (pr2 c2 t3 t) (\lambda (x: -T).(\lambda (H6: (ty3 g d u x)).(let H_x0 \def (wf3_getl_conf Abbr i c d u H0 -g c2 H3 x H6) in (let H7 \def H_x0 in (ex2_ind C (\lambda (d2: C).(getl i c2 -(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(wf3 g d d2)) (pr2 c2 t3 t) -(\lambda (x0: C).(\lambda (H8: (getl i c2 (CHead x0 (Bind Abbr) u))).(\lambda -(_: (wf3 g d x0)).(pr2_delta c2 x0 u i H8 t3 t4 H1 t H2)))) H7))))) -H5)))))))))))))))))) c1 t1 t2 H))))). -(* COMMENTS -Initial nodes: 373 -END *) - -theorem wf3_pr3_conf: - \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pr3 c1 -t1 t2) \to (\forall (c2: C).((wf3 g c1 c2) \to (\forall (u: T).((ty3 g c1 t1 -u) \to (pr3 c2 t1 t2))))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H: (pr3 c1 t1 t2)).(pr3_ind c1 (\lambda (t: T).(\lambda (t0: T).(\forall -(c2: C).((wf3 g c1 c2) \to (\forall (u: T).((ty3 g c1 t u) \to (pr3 c2 t -t0))))))) (\lambda (t: T).(\lambda (c2: C).(\lambda (_: (wf3 g c1 -c2)).(\lambda (u: T).(\lambda (_: (ty3 g c1 t u)).(pr3_refl c2 t)))))) -(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr2 c1 t4 t3)).(\lambda (t5: -T).(\lambda (_: (pr3 c1 t3 t5)).(\lambda (H2: ((\forall (c2: C).((wf3 g c1 -c2) \to (\forall (u: T).((ty3 g c1 t3 u) \to (pr3 c2 t3 t5))))))).(\lambda -(c2: C).(\lambda (H3: (wf3 g c1 c2)).(\lambda (u: T).(\lambda (H4: (ty3 g c1 -t4 u)).(pr3_sing c2 t3 t4 (wf3_pr2_conf g c1 t4 t3 H0 c2 H3 u H4) t5 (H2 c2 -H3 u (ty3_sred_pr2 c1 t4 t3 H0 g u H4))))))))))))) t1 t2 H))))). -(* COMMENTS -Initial nodes: 217 -END *) - -theorem wf3_pc3_conf: - \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pc3 c1 -t1 t2) \to (\forall (c2: C).((wf3 g c1 c2) \to (\forall (u1: T).((ty3 g c1 t1 -u1) \to (\forall (u2: T).((ty3 g c1 t2 u2) \to (pc3 c2 t1 t2))))))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H: (pc3 c1 t1 t2)).(\lambda (c2: C).(\lambda (H0: (wf3 g c1 c2)).(\lambda -(u1: T).(\lambda (H1: (ty3 g c1 t1 u1)).(\lambda (u2: T).(\lambda (H2: (ty3 g -c1 t2 u2)).(let H3 \def H in (ex2_ind T (\lambda (t: T).(pr3 c1 t1 t)) -(\lambda (t: T).(pr3 c1 t2 t)) (pc3 c2 t1 t2) (\lambda (x: T).(\lambda (H4: -(pr3 c1 t1 x)).(\lambda (H5: (pr3 c1 t2 x)).(pc3_pr3_t c2 t1 x (wf3_pr3_conf -g c1 t1 x H4 c2 H0 u1 H1) t2 (wf3_pr3_conf g c1 t2 x H5 c2 H0 u2 H2))))) -H3)))))))))))). -(* COMMENTS -Initial nodes: 153 -END *) - -theorem wf3_ty3_conf: - \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c1 -t1 t2) \to (\forall (c2: C).((wf3 g c1 c2) \to (ty3 g c2 t1 t2))))))) -\def - \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda -(H: (ty3 g c1 t1 t2)).(ty3_ind g (\lambda (c: C).(\lambda (t: T).(\lambda -(t0: T).(\forall (c2: C).((wf3 g c c2) \to (ty3 g c2 t t0)))))) (\lambda (c: -C).(\lambda (t3: T).(\lambda (t: T).(\lambda (H0: (ty3 g c t3 t)).(\lambda -(H1: ((\forall (c2: C).((wf3 g c c2) \to (ty3 g c2 t3 t))))).(\lambda (u: -T).(\lambda (t4: T).(\lambda (H2: (ty3 g c u t4)).(\lambda (H3: ((\forall -(c2: C).((wf3 g c c2) \to (ty3 g c2 u t4))))).(\lambda (H4: (pc3 c t4 -t3)).(\lambda (c2: C).(\lambda (H5: (wf3 g c c2)).(ex_ind T (\lambda (t0: -T).(ty3 g c t4 t0)) (ty3 g c2 u t3) (\lambda (x: T).(\lambda (H6: (ty3 g c t4 -x)).(ty3_conv g c2 t3 t (H1 c2 H5) u t4 (H3 c2 H5) (wf3_pc3_conf g c t4 t3 H4 -c2 H5 x H6 t H0)))) (ty3_correct g c u t4 H2)))))))))))))) (\lambda (c: -C).(\lambda (m: nat).(\lambda (c2: C).(\lambda (_: (wf3 g c c2)).(ty3_sort g -c2 m))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d: C).(\lambda (u: -T).(\lambda (H0: (getl n c (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda -(H1: (ty3 g d u t)).(\lambda (H2: ((\forall (c2: C).((wf3 g d c2) \to (ty3 g -c2 u t))))).(\lambda (c2: C).(\lambda (H3: (wf3 g c c2)).(let H_x \def -(wf3_getl_conf Abbr n c d u H0 g c2 H3 t H1) in (let H4 \def H_x in (ex2_ind -C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: -C).(wf3 g d d2)) (ty3 g c2 (TLRef n) (lift (S n) O t)) (\lambda (x: -C).(\lambda (H5: (getl n c2 (CHead x (Bind Abbr) u))).(\lambda (H6: (wf3 g d -x)).(ty3_abbr g n c2 x u H5 t (H2 x H6))))) H4))))))))))))) (\lambda (n: -nat).(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c -(CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u -t)).(\lambda (H2: ((\forall (c2: C).((wf3 g d c2) \to (ty3 g c2 u -t))))).(\lambda (c2: C).(\lambda (H3: (wf3 g c c2)).(let H_x \def -(wf3_getl_conf Abst n c d u H0 g c2 H3 t H1) in (let H4 \def H_x in (ex2_ind -C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: -C).(wf3 g d d2)) (ty3 g c2 (TLRef n) (lift (S n) O u)) (\lambda (x: -C).(\lambda (H5: (getl n c2 (CHead x (Bind Abst) u))).(\lambda (H6: (wf3 g d -x)).(ty3_abst g n c2 x u H5 t (H2 x H6))))) H4))))))))))))) (\lambda (c: -C).(\lambda (u: T).(\lambda (t: T).(\lambda (H0: (ty3 g c u t)).(\lambda (H1: -((\forall (c2: C).((wf3 g c c2) \to (ty3 g c2 u t))))).(\lambda (b: -B).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (ty3 g (CHead c (Bind b) u) -t3 t4)).(\lambda (H3: ((\forall (c2: C).((wf3 g (CHead c (Bind b) u) c2) \to -(ty3 g c2 t3 t4))))).(\lambda (c2: C).(\lambda (H4: (wf3 g c c2)).(ty3_bind g -c2 u t (H1 c2 H4) b t3 t4 (H3 (CHead c2 (Bind b) u) (wf3_bind g c c2 H4 u t -H0 b))))))))))))))) (\lambda (c: C).(\lambda (w: T).(\lambda (u: T).(\lambda -(_: (ty3 g c w u)).(\lambda (H1: ((\forall (c2: C).((wf3 g c c2) \to (ty3 g -c2 w u))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c v (THead -(Bind Abst) u t))).(\lambda (H3: ((\forall (c2: C).((wf3 g c c2) \to (ty3 g -c2 v (THead (Bind Abst) u t)))))).(\lambda (c2: C).(\lambda (H4: (wf3 g c -c2)).(ty3_appl g c2 w u (H1 c2 H4) v t (H3 c2 H4))))))))))))) (\lambda (c: -C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (ty3 g c t3 t4)).(\lambda -(H1: ((\forall (c2: C).((wf3 g c c2) \to (ty3 g c2 t3 t4))))).(\lambda (t0: -T).(\lambda (_: (ty3 g c t4 t0)).(\lambda (H3: ((\forall (c2: C).((wf3 g c -c2) \to (ty3 g c2 t4 t0))))).(\lambda (c2: C).(\lambda (H4: (wf3 g c -c2)).(ty3_cast g c2 t3 t4 (H1 c2 H4) t0 (H3 c2 H4)))))))))))) c1 t1 t2 H))))). -(* COMMENTS -Initial nodes: 1027 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/A/defs.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/A/defs.mma deleted file mode 100644 index 626dbb793..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/A/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/preamble.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/C/defs.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/C/defs.mma deleted file mode 100644 index e86d73b04..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/C/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/T/defs.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/C/props.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/C/props.mma deleted file mode 100644 index 7cbed8e15..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/C/props.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/C/defs.ma". - -include "Basic-2/T/props.ma". - -inline "Basic-1/C/props.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/G/defs.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/G/defs.mma deleted file mode 100644 index 626dbb793..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/G/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/preamble.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/T/dec.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/T/dec.mma deleted file mode 100644 index 85ff33c1b..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/T/dec.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/T/defs.ma". - -inline "Basic-1/T/dec.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/T/defs.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/T/defs.mma deleted file mode 100644 index 626dbb793..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/T/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/preamble.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/T/props.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/T/props.mma deleted file mode 100644 index 58ffc668a..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/T/props.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/T/defs.ma". - -inline "Basic-1/T/props.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/aplus/defs.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/aplus/defs.mma deleted file mode 100644 index c0cf937b3..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/aplus/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/asucc/defs.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/aplus/props.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/aplus/props.mma deleted file mode 100644 index 9e2234b3b..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/aplus/props.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/aplus/defs.ma". - -include "Basic-2/next_plus/props.ma". - -inline "Basic-1/aplus/props.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/app/defs.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/app/defs.mma deleted file mode 100644 index 8132fde52..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/app/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/C/defs.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/aprem/defs.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/aprem/defs.mma deleted file mode 100644 index 8e2b3a9f6..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/aprem/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/A/defs.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/aprem/fwd.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/aprem/fwd.mma deleted file mode 100644 index 757c62bde..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/aprem/fwd.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/aprem/defs.ma". - -inline "Basic-1/aprem/fwd.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/aprem/props.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/aprem/props.mma deleted file mode 100644 index 11509ac4b..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/aprem/props.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/aprem/fwd.ma". - -include "Basic-2/leq/defs.ma". - -inline "Basic-1/aprem/props.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/arity/aprem.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/arity/aprem.mma deleted file mode 100644 index 7a368efa9..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/arity/aprem.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/arity/props.ma". - -include "Basic-2/arity/cimp.ma". - -include "Basic-2/aprem/props.ma". - -inline "Basic-1/arity/aprem.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/arity/cimp.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/arity/cimp.mma deleted file mode 100644 index ca1bf3cad..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/arity/cimp.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/arity/defs.ma". - -include "Basic-2/cimp/props.ma". - -inline "Basic-1/arity/cimp.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/arity/defs.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/arity/defs.mma deleted file mode 100644 index 4619a73fb..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/arity/defs.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/leq/defs.ma". - -include "Basic-2/getl/defs.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/arity/fwd.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/arity/fwd.mma deleted file mode 100644 index 7ef60bef2..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/arity/fwd.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/arity/defs.ma". - -include "Basic-2/leq/asucc.ma". - -include "Basic-2/getl/drop.ma". - -inline "Basic-1/arity/fwd.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/arity/lift1.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/arity/lift1.mma deleted file mode 100644 index adde30893..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/arity/lift1.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/arity/props.ma". - -include "Basic-2/drop1/fwd.ma". - -inline "Basic-1/arity/lift1.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/arity/pr3.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/arity/pr3.mma deleted file mode 100644 index 8459c2a35..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/arity/pr3.mma +++ /dev/null @@ -1,30 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csuba/arity.ma". - -include "Basic-2/pr3/defs.ma". - -include "Basic-2/pr1/defs.ma". - -include "Basic-2/wcpr0/getl.ma". - -include "Basic-2/pr0/fwd.ma". - -include "Basic-2/arity/subst0.ma". - -inline "Basic-1/arity/pr3.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/arity/props.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/arity/props.mma deleted file mode 100644 index 2d5c30d0d..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/arity/props.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/arity/fwd.ma". - -inline "Basic-1/arity/props.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/arity/subst0.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/arity/subst0.mma deleted file mode 100644 index ff090fc82..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/arity/subst0.mma +++ /dev/null @@ -1,30 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/arity/props.ma". - -include "Basic-2/fsubst0/fwd.ma". - -include "Basic-2/csubst0/getl.ma". - -include "Basic-2/subst0/dec.ma". - -include "Basic-2/subst0/fwd.ma". - -include "Basic-2/getl/getl.ma". - -inline "Basic-1/arity/subst0.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/asucc/defs.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/asucc/defs.mma deleted file mode 100644 index 067d1947d..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/asucc/defs.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/A/defs.ma". - -include "Basic-2/G/defs.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/asucc/fwd.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/asucc/fwd.mma deleted file mode 100644 index d0e22947a..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/asucc/fwd.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/asucc/defs.ma". - -inline "Basic-1/asucc/fwd.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/cimp/defs.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/cimp/defs.mma deleted file mode 100644 index 0d4047fa3..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/cimp/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/getl/defs.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/cimp/props.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/cimp/props.mma deleted file mode 100644 index 288bce2f2..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/cimp/props.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/cimp/defs.ma". - -include "Basic-2/getl/getl.ma". - -inline "Basic-1/cimp/props.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/clear/defs.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/clear/defs.mma deleted file mode 100644 index 8132fde52..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/clear/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/C/defs.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/clear/drop.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/clear/drop.mma deleted file mode 100644 index d78b3941a..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/clear/drop.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/clear/fwd.ma". - -include "Basic-2/drop/fwd.ma". - -inline "Basic-1/clear/drop.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/clear/fwd.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/clear/fwd.mma deleted file mode 100644 index e56fb51b1..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/clear/fwd.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/clear/defs.ma". - -inline "Basic-1/clear/fwd.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/clear/props.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/clear/props.mma deleted file mode 100644 index 0bd37aee0..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/clear/props.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/clear/fwd.ma". - -inline "Basic-1/clear/props.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/clen/defs.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/clen/defs.mma deleted file mode 100644 index d5e42a147..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/clen/defs.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/C/defs.ma". - -include "Basic-2/s/defs.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/clen/getl.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/clen/getl.mma deleted file mode 100644 index 3d2d8f86c..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/clen/getl.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/clen/defs.ma". - -include "Basic-2/getl/props.ma". - -inline "Basic-1/clen/getl.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/cnt/defs.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/cnt/defs.mma deleted file mode 100644 index e86d73b04..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/cnt/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/T/defs.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/cnt/props.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/cnt/props.mma deleted file mode 100644 index 8b288f51f..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/cnt/props.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/cnt/defs.ma". - -include "Basic-2/lift/fwd.ma". - -inline "Basic-1/cnt/props.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csuba/arity.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csuba/arity.mma deleted file mode 100644 index 2d7cdb86d..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csuba/arity.mma +++ /dev/null @@ -1,26 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csuba/getl.ma". - -include "Basic-2/csuba/props.ma". - -include "Basic-2/arity/props.ma". - -include "Basic-2/csubv/getl.ma". - -inline "Basic-1/csuba/arity.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csuba/clear.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csuba/clear.mma deleted file mode 100644 index 15098af70..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csuba/clear.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csuba/defs.ma". - -include "Basic-2/clear/fwd.ma". - -inline "Basic-1/csuba/clear.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csuba/defs.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csuba/defs.mma deleted file mode 100644 index b9af9219e..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csuba/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/arity/defs.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csuba/drop.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csuba/drop.mma deleted file mode 100644 index 4aec6957d..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csuba/drop.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csuba/fwd.ma". - -include "Basic-2/drop/fwd.ma". - -inline "Basic-1/csuba/drop.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csuba/fwd.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csuba/fwd.mma deleted file mode 100644 index 2c5a16f23..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csuba/fwd.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csuba/defs.ma". - -inline "Basic-1/csuba/fwd.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csuba/getl.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csuba/getl.mma deleted file mode 100644 index e52f58df9..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csuba/getl.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csuba/drop.ma". - -include "Basic-2/csuba/clear.ma". - -include "Basic-2/getl/clear.ma". - -inline "Basic-1/csuba/getl.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csuba/props.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csuba/props.mma deleted file mode 100644 index d1ab31945..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csuba/props.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csuba/defs.ma". - -inline "Basic-1/csuba/props.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubc/arity.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubc/arity.mma deleted file mode 100644 index 63fff2532..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubc/arity.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubc/csuba.ma". - -inline "Basic-1/csubc/arity.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubc/clear.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubc/clear.mma deleted file mode 100644 index 1a0cbcf35..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubc/clear.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubc/fwd.ma". - -inline "Basic-1/csubc/clear.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubc/csuba.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubc/csuba.mma deleted file mode 100644 index f3465bf35..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubc/csuba.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubc/defs.ma". - -include "Basic-2/sc3/props.ma". - -inline "Basic-1/csubc/csuba.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubc/defs.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubc/defs.mma deleted file mode 100644 index 537c3eb52..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubc/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/sc3/defs.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubc/drop.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubc/drop.mma deleted file mode 100644 index 2a5d715f5..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubc/drop.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubc/fwd.ma". - -include "Basic-2/sc3/props.ma". - -inline "Basic-1/csubc/drop.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubc/drop1.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubc/drop1.mma deleted file mode 100644 index 9863063cf..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubc/drop1.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubc/drop.ma". - -inline "Basic-1/csubc/drop1.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubc/fwd.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubc/fwd.mma deleted file mode 100644 index 1d0dd32ab..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubc/fwd.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubc/defs.ma". - -inline "Basic-1/csubc/fwd.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubc/getl.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubc/getl.mma deleted file mode 100644 index 60462e153..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubc/getl.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubc/drop.ma". - -include "Basic-2/csubc/clear.ma". - -inline "Basic-1/csubc/getl.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubc/props.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubc/props.mma deleted file mode 100644 index ff7a8f4e5..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubc/props.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubc/defs.ma". - -include "Basic-2/sc3/props.ma". - -inline "Basic-1/csubc/props.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubst0/clear.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubst0/clear.mma deleted file mode 100644 index 0b3c214d7..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubst0/clear.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubst0/props.ma". - -include "Basic-2/csubst0/fwd.ma". - -include "Basic-2/clear/fwd.ma". - -inline "Basic-1/csubst0/clear.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubst0/defs.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubst0/defs.mma deleted file mode 100644 index 912ad35a1..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubst0/defs.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/subst0/defs.ma". - -include "Basic-2/C/defs.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubst0/drop.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubst0/drop.mma deleted file mode 100644 index 8ec4de71c..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubst0/drop.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubst0/fwd.ma". - -include "Basic-2/drop/fwd.ma". - -include "Basic-2/s/props.ma". - -inline "Basic-1/csubst0/drop.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubst0/fwd.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubst0/fwd.mma deleted file mode 100644 index 85ddb329e..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubst0/fwd.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubst0/defs.ma". - -inline "Basic-1/csubst0/fwd.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubst0/getl.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubst0/getl.mma deleted file mode 100644 index 679c49e9e..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubst0/getl.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubst0/clear.ma". - -include "Basic-2/csubst0/drop.ma". - -include "Basic-2/getl/fwd.ma". - -inline "Basic-1/csubst0/getl.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubst0/props.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubst0/props.mma deleted file mode 100644 index fee62c651..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubst0/props.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubst0/defs.ma". - -inline "Basic-1/csubst0/props.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubst1/defs.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubst1/defs.mma deleted file mode 100644 index 16b76c403..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubst1/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubst0/defs.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubst1/fwd.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubst1/fwd.mma deleted file mode 100644 index 2c9d5bedb..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubst1/fwd.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubst1/defs.ma". - -include "Basic-2/csubst0/fwd.ma". - -include "Basic-2/subst1/props.ma". - -inline "Basic-1/csubst1/fwd.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubst1/getl.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubst1/getl.mma deleted file mode 100644 index 05b80d444..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubst1/getl.mma +++ /dev/null @@ -1,26 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubst1/props.ma". - -include "Basic-2/csubst0/getl.ma". - -include "Basic-2/subst1/props.ma". - -include "Basic-2/drop/props.ma". - -inline "Basic-1/csubst1/getl.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubst1/props.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubst1/props.mma deleted file mode 100644 index 7480c122d..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubst1/props.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubst1/defs.ma". - -include "Basic-2/subst1/defs.ma". - -inline "Basic-1/csubst1/props.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubt/clear.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubt/clear.mma deleted file mode 100644 index c35e253b4..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubt/clear.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubt/defs.ma". - -include "Basic-2/clear/fwd.ma". - -inline "Basic-1/csubt/clear.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubt/csuba.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubt/csuba.mma deleted file mode 100644 index 3145d010e..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubt/csuba.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/ty3/arity.ma". - -inline "Basic-1/csubt/csuba.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubt/defs.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubt/defs.mma deleted file mode 100644 index 2dde8c752..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubt/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/ty3/defs.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubt/drop.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubt/drop.mma deleted file mode 100644 index 7d8d90b33..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubt/drop.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubt/fwd.ma". - -include "Basic-2/drop/fwd.ma". - -inline "Basic-1/csubt/drop.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubt/fwd.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubt/fwd.mma deleted file mode 100644 index 195de62cb..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubt/fwd.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubt/defs.ma". - -inline "Basic-1/csubt/fwd.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubt/getl.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubt/getl.mma deleted file mode 100644 index 14299efa2..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubt/getl.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubt/clear.ma". - -include "Basic-2/csubt/drop.ma". - -include "Basic-2/getl/clear.ma". - -inline "Basic-1/csubt/getl.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubt/pc3.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubt/pc3.mma deleted file mode 100644 index 48152a367..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubt/pc3.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubt/getl.ma". - -include "Basic-2/pc3/left.ma". - -inline "Basic-1/csubt/pc3.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubt/props.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubt/props.mma deleted file mode 100644 index 59d3c5188..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubt/props.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubt/defs.ma". - -inline "Basic-1/csubt/props.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubt/ty3.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubt/ty3.mma deleted file mode 100644 index 8bb74616a..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubt/ty3.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubt/pc3.ma". - -include "Basic-2/csubt/props.ma". - -inline "Basic-1/csubt/ty3.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubv/clear.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubv/clear.mma deleted file mode 100644 index 77768e993..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubv/clear.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubv/defs.ma". - -include "Basic-2/clear/fwd.ma". - -inline "Basic-1/csubv/clear.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubv/defs.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubv/defs.mma deleted file mode 100644 index 8132fde52..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubv/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/C/defs.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubv/drop.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubv/drop.mma deleted file mode 100644 index d0d2b3f76..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubv/drop.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubv/props.ma". - -include "Basic-2/drop/fwd.ma". - -inline "Basic-1/csubv/drop.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubv/getl.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubv/getl.mma deleted file mode 100644 index 66c63a639..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubv/getl.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubv/clear.ma". - -include "Basic-2/csubv/drop.ma". - -include "Basic-2/getl/fwd.ma". - -inline "Basic-1/csubv/getl.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubv/props.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubv/props.mma deleted file mode 100644 index 44d2edf27..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/csubv/props.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubv/defs.ma". - -include "Basic-2/T/props.ma". - -inline "Basic-1/csubv/props.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/drop/defs.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/drop/defs.mma deleted file mode 100644 index 8837d010f..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/drop/defs.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/C/defs.ma". - -include "Basic-2/lift/defs.ma". - -include "Basic-2/r/defs.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/drop/fwd.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/drop/fwd.mma deleted file mode 100644 index d88936b9c..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/drop/fwd.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/drop/defs.ma". - -inline "Basic-1/drop/fwd.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/drop/props.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/drop/props.mma deleted file mode 100644 index 9af06df85..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/drop/props.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/drop/fwd.ma". - -include "Basic-2/lift/props.ma". - -include "Basic-2/r/props.ma". - -inline "Basic-1/drop/props.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/drop1/defs.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/drop1/defs.mma deleted file mode 100644 index 6d92e123e..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/drop1/defs.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/drop/defs.ma". - -include "Basic-2/lift1/defs.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/drop1/fwd.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/drop1/fwd.mma deleted file mode 100644 index 78b91b349..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/drop1/fwd.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/drop1/defs.ma". - -inline "Basic-1/drop1/fwd.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/drop1/getl.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/drop1/getl.mma deleted file mode 100644 index 0b9e358d2..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/drop1/getl.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/drop1/fwd.ma". - -include "Basic-2/getl/drop.ma". - -inline "Basic-1/drop1/getl.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/drop1/props.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/drop1/props.mma deleted file mode 100644 index f4f2204ac..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/drop1/props.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/drop1/fwd.ma". - -include "Basic-2/drop/props.ma". - -include "Basic-2/getl/defs.ma". - -inline "Basic-1/drop1/props.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/ex0/defs.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/ex0/defs.mma deleted file mode 100644 index 067d1947d..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/ex0/defs.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/A/defs.ma". - -include "Basic-2/G/defs.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/ex0/props.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/ex0/props.mma deleted file mode 100644 index 80204275f..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/ex0/props.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/ex0/defs.ma". - -include "Basic-2/leq/defs.ma". - -include "Basic-2/aplus/props.ma". - -inline "Basic-1/ex0/props.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/ex1/defs.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/ex1/defs.mma deleted file mode 100644 index 8132fde52..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/ex1/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/C/defs.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/ex1/props.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/ex1/props.mma deleted file mode 100644 index 8ec7ebe3e..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/ex1/props.mma +++ /dev/null @@ -1,32 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/ex1/defs.ma". - -include "Basic-2/ty3/fwd.ma". - -include "Basic-2/pc3/fwd.ma". - -include "Basic-2/nf2/pr3.ma". - -include "Basic-2/nf2/props.ma". - -include "Basic-2/arity/defs.ma". - -include "Basic-2/leq/props.ma". - -inline "Basic-1/ex1/props.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/ex2/defs.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/ex2/defs.mma deleted file mode 100644 index 8132fde52..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/ex2/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/C/defs.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/ex2/props.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/ex2/props.mma deleted file mode 100644 index 56d0d8557..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/ex2/props.mma +++ /dev/null @@ -1,26 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/ex2/defs.ma". - -include "Basic-2/nf2/defs.ma". - -include "Basic-2/pr2/fwd.ma". - -include "Basic-2/arity/fwd.ma". - -inline "Basic-1/ex2/props.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/flt/defs.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/flt/defs.mma deleted file mode 100644 index 8132fde52..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/flt/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/C/defs.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/flt/props.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/flt/props.mma deleted file mode 100644 index e5b76348f..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/flt/props.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/flt/defs.ma". - -include "Basic-2/C/props.ma". - -inline "Basic-1/flt/props.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/fsubst0/defs.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/fsubst0/defs.mma deleted file mode 100644 index 16b76c403..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/fsubst0/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubst0/defs.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/fsubst0/fwd.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/fsubst0/fwd.mma deleted file mode 100644 index 4a60d7191..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/fsubst0/fwd.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/fsubst0/defs.ma". - -inline "Basic-1/fsubst0/fwd.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/getl/clear.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/getl/clear.mma deleted file mode 100644 index 3f69bde5d..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/getl/clear.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/getl/props.ma". - -include "Basic-2/clear/drop.ma". - -inline "Basic-1/getl/clear.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/getl/dec.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/getl/dec.mma deleted file mode 100644 index 4085b140e..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/getl/dec.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/getl/props.ma". - -inline "Basic-1/getl/dec.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/getl/defs.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/getl/defs.mma deleted file mode 100644 index 818f4f3ae..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/getl/defs.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/drop/defs.ma". - -include "Basic-2/clear/defs.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/getl/drop.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/getl/drop.mma deleted file mode 100644 index ff4bb646f..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/getl/drop.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/getl/props.ma". - -include "Basic-2/clear/drop.ma". - -inline "Basic-1/getl/drop.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/getl/flt.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/getl/flt.mma deleted file mode 100644 index c030693f5..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/getl/flt.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/getl/fwd.ma". - -include "Basic-2/clear/props.ma". - -include "Basic-2/flt/props.ma". - -inline "Basic-1/getl/flt.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/getl/fwd.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/getl/fwd.mma deleted file mode 100644 index dfcee9576..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/getl/fwd.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/getl/defs.ma". - -include "Basic-2/drop/fwd.ma". - -include "Basic-2/clear/fwd.ma". - -inline "Basic-1/getl/fwd.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/getl/getl.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/getl/getl.mma deleted file mode 100644 index b5e5163d7..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/getl/getl.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/getl/drop.ma". - -include "Basic-2/getl/clear.ma". - -inline "Basic-1/getl/getl.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/getl/props.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/getl/props.mma deleted file mode 100644 index 00ff97539..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/getl/props.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/getl/fwd.ma". - -include "Basic-2/drop/props.ma". - -include "Basic-2/clear/props.ma". - -inline "Basic-1/getl/props.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/iso/defs.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/iso/defs.mma deleted file mode 100644 index e86d73b04..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/iso/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/T/defs.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/iso/fwd.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/iso/fwd.mma deleted file mode 100644 index 8ae8aaad1..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/iso/fwd.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/iso/defs.ma". - -include "Basic-2/tlist/defs.ma". - -inline "Basic-1/iso/fwd.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/iso/props.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/iso/props.mma deleted file mode 100644 index c689ea1fc..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/iso/props.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/iso/fwd.ma". - -inline "Basic-1/iso/props.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/leq/asucc.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/leq/asucc.mma deleted file mode 100644 index a89191b08..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/leq/asucc.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/leq/props.ma". - -inline "Basic-1/leq/asucc.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/leq/defs.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/leq/defs.mma deleted file mode 100644 index dadf5769f..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/leq/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/aplus/defs.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/leq/fwd.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/leq/fwd.mma deleted file mode 100644 index 750437135..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/leq/fwd.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/leq/defs.ma". - -inline "Basic-1/leq/fwd.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/leq/props.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/leq/props.mma deleted file mode 100644 index b2509dd31..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/leq/props.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/leq/fwd.ma". - -include "Basic-2/aplus/props.ma". - -inline "Basic-1/leq/props.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/lift/defs.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/lift/defs.mma deleted file mode 100644 index 85138ec00..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/lift/defs.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/tlist/defs.ma". - -include "Basic-2/s/defs.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/lift/fwd.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/lift/fwd.mma deleted file mode 100644 index 95a454dbe..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/lift/fwd.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/lift/defs.ma". - -inline "Basic-1/lift/fwd.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/lift/props.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/lift/props.mma deleted file mode 100644 index 88f85f5cb..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/lift/props.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/lift/fwd.ma". - -include "Basic-2/s/props.ma". - -inline "Basic-1/lift/props.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/lift/tlt.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/lift/tlt.mma deleted file mode 100644 index 51c972987..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/lift/tlt.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/lift/fwd.ma". - -include "Basic-2/tlt/props.ma". - -inline "Basic-1/lift/tlt.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/lift1/defs.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/lift1/defs.mma deleted file mode 100644 index 2948c111d..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/lift1/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/lift/defs.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/lift1/fwd.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/lift1/fwd.mma deleted file mode 100644 index 3aabe676f..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/lift1/fwd.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/lift1/defs.ma". - -include "Basic-2/lift/fwd.ma". - -inline "Basic-1/lift1/fwd.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/lift1/props.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/lift1/props.mma deleted file mode 100644 index 89bed2166..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/lift1/props.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/lift/props.ma". - -include "Basic-2/drop1/defs.ma". - -inline "Basic-1/lift1/props.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/llt/defs.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/llt/defs.mma deleted file mode 100644 index 8e2b3a9f6..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/llt/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/A/defs.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/llt/props.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/llt/props.mma deleted file mode 100644 index 462e5c575..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/llt/props.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/llt/defs.ma". - -include "Basic-2/leq/defs.ma". - -inline "Basic-1/llt/props.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/next_plus/defs.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/next_plus/defs.mma deleted file mode 100644 index 33d7fdddc..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/next_plus/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/G/defs.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/next_plus/props.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/next_plus/props.mma deleted file mode 100644 index 8e6ff559c..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/next_plus/props.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/next_plus/defs.ma". - -inline "Basic-1/next_plus/props.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/nf2/arity.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/nf2/arity.mma deleted file mode 100644 index 356dfa591..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/nf2/arity.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/nf2/fwd.ma". - -include "Basic-2/arity/subst0.ma". - -inline "Basic-1/nf2/arity.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/nf2/dec.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/nf2/dec.mma deleted file mode 100644 index 179da42ff..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/nf2/dec.mma +++ /dev/null @@ -1,28 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/nf2/defs.ma". - -include "Basic-2/pr2/clen.ma". - -include "Basic-2/pr2/fwd.ma". - -include "Basic-2/pr0/dec.ma". - -include "Basic-2/C/props.ma". - -inline "Basic-1/nf2/dec.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/nf2/defs.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/nf2/defs.mma deleted file mode 100644 index 4f97474c7..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/nf2/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pr2/defs.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/nf2/fwd.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/nf2/fwd.mma deleted file mode 100644 index 0f941386e..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/nf2/fwd.mma +++ /dev/null @@ -1,26 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/nf2/defs.ma". - -include "Basic-2/pr2/clen.ma". - -include "Basic-2/subst0/dec.ma". - -include "Basic-2/T/props.ma". - -inline "Basic-1/nf2/fwd.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/nf2/iso.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/nf2/iso.mma deleted file mode 100644 index 0c6309e1c..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/nf2/iso.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/nf2/pr3.ma". - -include "Basic-2/pr3/fwd.ma". - -include "Basic-2/iso/props.ma". - -inline "Basic-1/nf2/iso.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/nf2/lift1.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/nf2/lift1.mma deleted file mode 100644 index 250c8dd78..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/nf2/lift1.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/nf2/props.ma". - -include "Basic-2/drop1/fwd.ma". - -inline "Basic-1/nf2/lift1.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/nf2/pr3.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/nf2/pr3.mma deleted file mode 100644 index 00ba3ffab..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/nf2/pr3.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/nf2/defs.ma". - -include "Basic-2/pr3/pr3.ma". - -inline "Basic-1/nf2/pr3.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/nf2/props.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/nf2/props.mma deleted file mode 100644 index 6977bd069..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/nf2/props.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/nf2/defs.ma". - -include "Basic-2/pr2/fwd.ma". - -inline "Basic-1/nf2/props.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pc1/defs.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pc1/defs.mma deleted file mode 100644 index 9818026a3..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pc1/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pr1/defs.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pc1/props.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pc1/props.mma deleted file mode 100644 index f031a2362..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pc1/props.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pc1/defs.ma". - -include "Basic-2/pr1/pr1.ma". - -inline "Basic-1/pc1/props.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pc3/dec.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pc3/dec.mma deleted file mode 100644 index d5ed4097b..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pc3/dec.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/ty3/arity_props.ma". - -include "Basic-2/nf2/fwd.ma". - -inline "Basic-1/pc3/dec.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pc3/defs.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pc3/defs.mma deleted file mode 100644 index ac791ab6d..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pc3/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pr3/defs.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pc3/fsubst0.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pc3/fsubst0.mma deleted file mode 100644 index 478c285af..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pc3/fsubst0.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pc3/left.ma". - -include "Basic-2/fsubst0/defs.ma". - -include "Basic-2/csubst0/getl.ma". - -inline "Basic-1/pc3/fsubst0.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pc3/fwd.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pc3/fwd.mma deleted file mode 100644 index 83816ceac..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pc3/fwd.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pc3/props.ma". - -include "Basic-2/pr3/fwd.ma". - -inline "Basic-1/pc3/fwd.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pc3/left.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pc3/left.mma deleted file mode 100644 index a0102b201..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pc3/left.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pc3/props.ma". - -inline "Basic-1/pc3/left.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pc3/nf2.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pc3/nf2.mma deleted file mode 100644 index 38a00ff31..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pc3/nf2.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pc3/defs.ma". - -include "Basic-2/nf2/pr3.ma". - -inline "Basic-1/pc3/nf2.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pc3/pc1.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pc3/pc1.mma deleted file mode 100644 index 0a82a8adc..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pc3/pc1.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pc3/defs.ma". - -include "Basic-2/pc1/defs.ma". - -include "Basic-2/pr3/pr1.ma". - -inline "Basic-1/pc3/pc1.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pc3/props.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pc3/props.mma deleted file mode 100644 index 01c5ee4e8..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pc3/props.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pc3/defs.ma". - -include "Basic-2/pr3/pr3.ma". - -inline "Basic-1/pc3/props.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pc3/subst1.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pc3/subst1.mma deleted file mode 100644 index 1ec4a10d5..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pc3/subst1.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pc3/props.ma". - -include "Basic-2/pr3/subst1.ma". - -inline "Basic-1/pc3/subst1.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pc3/wcpr0.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pc3/wcpr0.mma deleted file mode 100644 index 98ae77c9b..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pc3/wcpr0.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pc3/props.ma". - -include "Basic-2/wcpr0/getl.ma". - -inline "Basic-1/pc3/wcpr0.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr0/dec.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr0/dec.mma deleted file mode 100644 index c88b2b725..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr0/dec.mma +++ /dev/null @@ -1,26 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pr0/fwd.ma". - -include "Basic-2/subst0/dec.ma". - -include "Basic-2/T/dec.ma". - -include "Basic-2/T/props.ma". - -inline "Basic-1/pr0/dec.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr0/defs.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr0/defs.mma deleted file mode 100644 index e94eab620..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr0/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/subst0/defs.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr0/fwd.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr0/fwd.mma deleted file mode 100644 index 2349a5127..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr0/fwd.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pr0/props.ma". - -inline "Basic-1/pr0/fwd.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr0/pr0.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr0/pr0.mma deleted file mode 100644 index cc88c47e0..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr0/pr0.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pr0/fwd.ma". - -include "Basic-2/lift/tlt.ma". - -inline "Basic-1/pr0/pr0.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr0/props.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr0/props.mma deleted file mode 100644 index 9b8e83a73..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr0/props.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pr0/defs.ma". - -include "Basic-2/subst0/subst0.ma". - -inline "Basic-1/pr0/props.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr0/subst1.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr0/subst1.mma deleted file mode 100644 index a4af1a695..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr0/subst1.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pr0/props.ma". - -include "Basic-2/subst1/defs.ma". - -inline "Basic-1/pr0/subst1.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr1/defs.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr1/defs.mma deleted file mode 100644 index 9fb12774d..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr1/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pr0/defs.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr1/pr1.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr1/pr1.mma deleted file mode 100644 index f05ca1cd3..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr1/pr1.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pr1/props.ma". - -include "Basic-2/pr0/pr0.ma". - -inline "Basic-1/pr1/pr1.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr1/props.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr1/props.mma deleted file mode 100644 index 5ec9c7f3e..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr1/props.mma +++ /dev/null @@ -1,26 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pr1/defs.ma". - -include "Basic-2/pr0/subst1.ma". - -include "Basic-2/subst1/props.ma". - -include "Basic-2/T/props.ma". - -inline "Basic-1/pr1/props.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr2/clen.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr2/clen.mma deleted file mode 100644 index e3ea2f078..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr2/clen.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pr2/props.ma". - -include "Basic-2/clen/getl.ma". - -inline "Basic-1/pr2/clen.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr2/defs.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr2/defs.mma deleted file mode 100644 index b43baa983..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr2/defs.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pr0/defs.ma". - -include "Basic-2/getl/defs.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr2/fwd.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr2/fwd.mma deleted file mode 100644 index 9ec452e99..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr2/fwd.mma +++ /dev/null @@ -1,26 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pr2/defs.ma". - -include "Basic-2/pr0/fwd.ma". - -include "Basic-2/getl/drop.ma". - -include "Basic-2/getl/clear.ma". - -inline "Basic-1/pr2/fwd.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr2/pr2.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr2/pr2.mma deleted file mode 100644 index 8879c71e8..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr2/pr2.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pr2/defs.ma". - -include "Basic-2/pr0/pr0.ma". - -include "Basic-2/getl/props.ma". - -inline "Basic-1/pr2/pr2.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr2/props.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr2/props.mma deleted file mode 100644 index b0e27c8fc..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr2/props.mma +++ /dev/null @@ -1,26 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pr2/defs.ma". - -include "Basic-2/pr0/props.ma". - -include "Basic-2/getl/drop.ma". - -include "Basic-2/getl/clear.ma". - -inline "Basic-1/pr2/props.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr2/subst1.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr2/subst1.mma deleted file mode 100644 index 61b161222..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr2/subst1.mma +++ /dev/null @@ -1,32 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pr2/defs.ma". - -include "Basic-2/pr0/subst1.ma". - -include "Basic-2/pr0/fwd.ma". - -include "Basic-2/csubst1/getl.ma". - -include "Basic-2/csubst1/fwd.ma". - -include "Basic-2/subst1/subst1.ma". - -include "Basic-2/getl/drop.ma". - -inline "Basic-1/pr2/subst1.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr3/defs.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr3/defs.mma deleted file mode 100644 index 4f97474c7..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr3/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pr2/defs.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr3/fwd.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr3/fwd.mma deleted file mode 100644 index a29ddcee0..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr3/fwd.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pr3/props.ma". - -include "Basic-2/pr2/fwd.ma". - -inline "Basic-1/pr3/fwd.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr3/iso.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr3/iso.mma deleted file mode 100644 index a401d94dc..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr3/iso.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pr3/fwd.ma". - -include "Basic-2/iso/props.ma". - -include "Basic-2/tlist/props.ma". - -inline "Basic-1/pr3/iso.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr3/pr1.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr3/pr1.mma deleted file mode 100644 index 6827dfddc..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr3/pr1.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pr3/defs.ma". - -include "Basic-2/pr1/defs.ma". - -inline "Basic-1/pr3/pr1.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr3/pr3.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr3/pr3.mma deleted file mode 100644 index a393080b6..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr3/pr3.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pr3/props.ma". - -include "Basic-2/pr2/pr2.ma". - -inline "Basic-1/pr3/pr3.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr3/props.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr3/props.mma deleted file mode 100644 index c87af259a..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr3/props.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pr3/pr1.ma". - -include "Basic-2/pr2/props.ma". - -include "Basic-2/pr1/props.ma". - -inline "Basic-1/pr3/props.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr3/subst1.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr3/subst1.mma deleted file mode 100644 index 73288e883..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr3/subst1.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pr3/defs.ma". - -include "Basic-2/pr2/subst1.ma". - -inline "Basic-1/pr3/subst1.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr3/wcpr0.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr3/wcpr0.mma deleted file mode 100644 index 6c30039f1..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/pr3/wcpr0.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pr3/props.ma". - -include "Basic-2/wcpr0/getl.ma". - -inline "Basic-1/pr3/wcpr0.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/preamble.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/preamble.ma deleted file mode 100644 index d0ddca34a..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/preamble.ma +++ /dev/null @@ -1,16 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "Ground-2/theory.ma". -include "Basic-1/definitions.ma". diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/r/defs.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/r/defs.mma deleted file mode 100644 index e86d73b04..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/r/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/T/defs.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/r/props.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/r/props.mma deleted file mode 100644 index 79edc95f4..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/r/props.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/r/defs.ma". - -include "Basic-2/s/defs.ma". - -inline "Basic-1/r/props.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/s/defs.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/s/defs.mma deleted file mode 100644 index e86d73b04..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/s/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/T/defs.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/s/props.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/s/props.mma deleted file mode 100644 index 35f1ad31b..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/s/props.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/s/defs.ma". - -inline "Basic-1/s/props.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/sc3/arity.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/sc3/arity.mma deleted file mode 100644 index 9fa3eaf93..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/sc3/arity.mma +++ /dev/null @@ -1,26 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubc/arity.ma". - -include "Basic-2/csubc/getl.ma". - -include "Basic-2/csubc/drop1.ma". - -include "Basic-2/csubc/props.ma". - -inline "Basic-1/sc3/arity.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/sc3/defs.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/sc3/defs.mma deleted file mode 100644 index d1dde1884..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/sc3/defs.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/sn3/defs.ma". - -include "Basic-2/arity/defs.ma". - -include "Basic-2/drop1/defs.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/sc3/props.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/sc3/props.mma deleted file mode 100644 index 859458266..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/sc3/props.mma +++ /dev/null @@ -1,38 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/sc3/defs.ma". - -include "Basic-2/sn3/lift1.ma". - -include "Basic-2/nf2/lift1.ma". - -include "Basic-2/csuba/arity.ma". - -include "Basic-2/arity/lift1.ma". - -include "Basic-2/arity/aprem.ma". - -include "Basic-2/llt/props.ma". - -include "Basic-2/drop1/getl.ma". - -include "Basic-2/drop1/props.ma". - -include "Basic-2/lift1/props.ma". - -inline "Basic-1/sc3/props.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/sn3/defs.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/sn3/defs.mma deleted file mode 100644 index ac791ab6d..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/sn3/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pr3/defs.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/sn3/fwd.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/sn3/fwd.mma deleted file mode 100644 index a4662eefa..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/sn3/fwd.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/sn3/defs.ma". - -include "Basic-2/pr3/props.ma". - -inline "Basic-1/sn3/fwd.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/sn3/lift1.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/sn3/lift1.mma deleted file mode 100644 index e8c0c4de2..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/sn3/lift1.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/sn3/props.ma". - -include "Basic-2/drop1/fwd.ma". - -include "Basic-2/lift1/fwd.ma". - -inline "Basic-1/sn3/lift1.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/sn3/nf2.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/sn3/nf2.mma deleted file mode 100644 index 81de1d480..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/sn3/nf2.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/sn3/defs.ma". - -include "Basic-2/nf2/dec.ma". - -include "Basic-2/nf2/pr3.ma". - -inline "Basic-1/sn3/nf2.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/sn3/props.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/sn3/props.mma deleted file mode 100644 index 3244a5237..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/sn3/props.mma +++ /dev/null @@ -1,26 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/sn3/nf2.ma". - -include "Basic-2/sn3/fwd.ma". - -include "Basic-2/nf2/iso.ma". - -include "Basic-2/pr3/iso.ma". - -inline "Basic-1/sn3/props.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/sty0/defs.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/sty0/defs.mma deleted file mode 100644 index 24df71ad7..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/sty0/defs.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/G/defs.ma". - -include "Basic-2/getl/defs.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/sty0/fwd.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/sty0/fwd.mma deleted file mode 100644 index a2600b078..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/sty0/fwd.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/sty0/defs.ma". - -inline "Basic-1/sty0/fwd.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/sty0/props.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/sty0/props.mma deleted file mode 100644 index 403332ea0..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/sty0/props.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/sty0/defs.ma". - -include "Basic-2/getl/drop.ma". - -inline "Basic-1/sty0/props.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/sty1/cnt.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/sty1/cnt.mma deleted file mode 100644 index 062ba4446..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/sty1/cnt.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/sty1/props.ma". - -include "Basic-2/cnt/props.ma". - -inline "Basic-1/sty1/cnt.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/sty1/defs.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/sty1/defs.mma deleted file mode 100644 index d2245411f..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/sty1/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/sty0/defs.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/sty1/props.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/sty1/props.mma deleted file mode 100644 index f764f229a..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/sty1/props.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/sty1/defs.ma". - -include "Basic-2/sty0/props.ma". - -inline "Basic-1/sty1/props.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/subst/defs.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/subst/defs.mma deleted file mode 100644 index 2948c111d..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/subst/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/lift/defs.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/subst/fwd.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/subst/fwd.mma deleted file mode 100644 index 27b37067f..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/subst/fwd.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/subst/defs.ma". - -inline "Basic-1/subst/fwd.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/subst/props.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/subst/props.mma deleted file mode 100644 index 1a45f22b5..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/subst/props.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/subst/fwd.ma". - -include "Basic-2/subst0/defs.ma". - -include "Basic-2/lift/props.ma". - -inline "Basic-1/subst/props.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/subst0/dec.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/subst0/dec.mma deleted file mode 100644 index b6595349c..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/subst0/dec.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/subst0/defs.ma". - -include "Basic-2/lift/props.ma". - -inline "Basic-1/subst0/dec.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/subst0/defs.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/subst0/defs.mma deleted file mode 100644 index 2948c111d..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/subst0/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/lift/defs.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/subst0/fwd.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/subst0/fwd.mma deleted file mode 100644 index 3da14bb5e..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/subst0/fwd.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/subst0/defs.ma". - -include "Basic-2/lift/props.ma". - -inline "Basic-1/subst0/fwd.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/subst0/props.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/subst0/props.mma deleted file mode 100644 index 155ba9e4f..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/subst0/props.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/subst0/fwd.ma". - -inline "Basic-1/subst0/props.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/subst0/subst0.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/subst0/subst0.mma deleted file mode 100644 index 3e87d4560..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/subst0/subst0.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/subst0/props.ma". - -inline "Basic-1/subst0/subst0.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/subst0/tlt.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/subst0/tlt.mma deleted file mode 100644 index 6583c3c92..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/subst0/tlt.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/subst0/defs.ma". - -include "Basic-2/lift/props.ma". - -include "Basic-2/lift/tlt.ma". - -inline "Basic-1/subst0/tlt.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/subst1/defs.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/subst1/defs.mma deleted file mode 100644 index e94eab620..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/subst1/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/subst0/defs.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/subst1/fwd.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/subst1/fwd.mma deleted file mode 100644 index 78669f892..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/subst1/fwd.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/subst1/defs.ma". - -include "Basic-2/subst0/props.ma". - -inline "Basic-1/subst1/fwd.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/subst1/props.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/subst1/props.mma deleted file mode 100644 index 35ae5d43d..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/subst1/props.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/subst1/defs.ma". - -include "Basic-2/subst0/props.ma". - -inline "Basic-1/subst1/props.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/subst1/subst1.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/subst1/subst1.mma deleted file mode 100644 index af3f35761..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/subst1/subst1.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/subst1/fwd.ma". - -include "Basic-2/subst0/subst0.ma". - -inline "Basic-1/subst1/subst1.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/theory.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/theory.ma deleted file mode 100644 index 9ab93dc9b..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/theory.ma +++ /dev/null @@ -1,42 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/subst0/tlt.ma". - -include "Basic-2/subst/props.ma". - -include "Basic-2/sty1/cnt.ma". - -include "Basic-2/ex0/props.ma". - -include "Basic-2/wcpr0/fwd.ma". - -include "Basic-2/pr3/wcpr0.ma". - -include "Basic-2/ex2/props.ma". - -include "Basic-2/ex1/props.ma". - -include "Basic-2/ty3/sty0.ma". - -include "Basic-2/csubt/csuba.ma". - -include "Basic-2/ty3/fwd_nf2.ma". - -include "Basic-2/ty3/nf2.ma". - -include "Basic-2/wf3/props.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/tlist/defs.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/tlist/defs.mma deleted file mode 100644 index e86d73b04..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/tlist/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/T/defs.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/tlist/props.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/tlist/props.mma deleted file mode 100644 index 4fcfcb1fb..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/tlist/props.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/tlist/defs.ma". - -inline "Basic-1/tlist/props.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/tlt/defs.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/tlt/defs.mma deleted file mode 100644 index e86d73b04..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/tlt/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/T/defs.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/tlt/props.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/tlt/props.mma deleted file mode 100644 index 75aff34af..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/tlt/props.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/tlt/defs.ma". - -inline "Basic-1/tlt/props.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/arity.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/arity.mma deleted file mode 100644 index a50d25340..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/arity.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/ty3/pr3_props.ma". - -include "Basic-2/arity/pr3.ma". - -include "Basic-2/asucc/fwd.ma". - -inline "Basic-1/ty3/arity.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/arity_props.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/arity_props.mma deleted file mode 100644 index 65475e81c..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/arity_props.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/ty3/arity.ma". - -include "Basic-2/sc3/arity.ma". - -inline "Basic-1/ty3/arity_props.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/dec.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/dec.mma deleted file mode 100644 index da5291990..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/dec.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pc3/dec.ma". - -include "Basic-2/getl/flt.ma". - -include "Basic-2/getl/dec.ma". - -inline "Basic-1/ty3/dec.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/defs.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/defs.mma deleted file mode 100644 index a1a47d554..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/defs.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/G/defs.ma". - -include "Basic-2/pc3/defs.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/fsubst0.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/fsubst0.mma deleted file mode 100644 index 11b8873da..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/fsubst0.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/ty3/props.ma". - -include "Basic-2/pc3/fsubst0.ma". - -include "Basic-2/getl/getl.ma". - -inline "Basic-1/ty3/fsubst0.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/fwd.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/fwd.mma deleted file mode 100644 index c019d001a..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/fwd.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/ty3/defs.ma". - -include "Basic-2/pc3/props.ma". - -inline "Basic-1/ty3/fwd.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/fwd_nf2.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/fwd_nf2.mma deleted file mode 100644 index 6894ae466..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/fwd_nf2.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/ty3/arity_props.ma". - -include "Basic-2/pc3/nf2.ma". - -include "Basic-2/nf2/fwd.ma". - -inline "Basic-1/ty3/fwd_nf2.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/nf2.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/nf2.mma deleted file mode 100644 index af7960855..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/nf2.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/ty3/arity.ma". - -include "Basic-2/pc3/nf2.ma". - -include "Basic-2/nf2/arity.ma". - -inline "Basic-1/ty3/nf2.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/pr3.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/pr3.mma deleted file mode 100644 index dd43847bd..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/pr3.mma +++ /dev/null @@ -1,30 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/csubt/ty3.ma". - -include "Basic-2/ty3/subst1.ma". - -include "Basic-2/ty3/fsubst0.ma". - -include "Basic-2/pc3/pc1.ma". - -include "Basic-2/pc3/wcpr0.ma". - -include "Basic-2/pc1/props.ma". - -inline "Basic-1/ty3/pr3.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/pr3_props.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/pr3_props.mma deleted file mode 100644 index a352ace49..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/pr3_props.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/ty3/pr3.ma". - -inline "Basic-1/ty3/pr3_props.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/props.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/props.mma deleted file mode 100644 index c5e6cb33e..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/props.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/ty3/fwd.ma". - -include "Basic-2/pc3/fwd.ma". - -inline "Basic-1/ty3/props.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/sty0.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/sty0.mma deleted file mode 100644 index 0c5ebc151..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/sty0.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/ty3/pr3_props.ma". - -include "Basic-2/sty0/fwd.ma". - -inline "Basic-1/ty3/sty0.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/subst1.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/subst1.mma deleted file mode 100644 index 97f45a1ec..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/ty3/subst1.mma +++ /dev/null @@ -1,24 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/ty3/props.ma". - -include "Basic-2/pc3/subst1.ma". - -include "Basic-2/getl/getl.ma". - -inline "Basic-1/ty3/subst1.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/wcpr0/defs.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/wcpr0/defs.mma deleted file mode 100644 index 65ae9a1d5..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/wcpr0/defs.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/pr0/defs.ma". - -include "Basic-2/C/defs.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/wcpr0/fwd.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/wcpr0/fwd.mma deleted file mode 100644 index 0a9a87638..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/wcpr0/fwd.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/wcpr0/defs.ma". - -inline "Basic-1/wcpr0/fwd.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/wcpr0/getl.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/wcpr0/getl.mma deleted file mode 100644 index e412692e5..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/wcpr0/getl.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/wcpr0/defs.ma". - -include "Basic-2/getl/props.ma". - -inline "Basic-1/wcpr0/getl.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/wf3/clear.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/wf3/clear.mma deleted file mode 100644 index e63bb3e82..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/wf3/clear.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/wf3/fwd.ma". - -inline "Basic-1/wf3/clear.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/wf3/defs.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/wf3/defs.mma deleted file mode 100644 index 2dde8c752..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/wf3/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/ty3/defs.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/wf3/fwd.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/wf3/fwd.mma deleted file mode 100644 index 73bf245a7..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/wf3/fwd.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/wf3/defs.ma". - -inline "Basic-1/wf3/fwd.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/wf3/getl.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/wf3/getl.mma deleted file mode 100644 index fe6e74042..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/wf3/getl.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/wf3/clear.ma". - -include "Basic-2/ty3/dec.ma". - -inline "Basic-1/wf3/getl.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/wf3/props.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/wf3/props.mma deleted file mode 100644 index 35a6a6474..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/wf3/props.mma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/wf3/ty3.ma". - -include "Basic-2/app/defs.ma". - -inline "Basic-1/wf3/props.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/wf3/ty3.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/wf3/ty3.mma deleted file mode 100644 index 8af324db5..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-2/wf3/ty3.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-2/wf3/getl.ma". - -inline "Basic-1/wf3/ty3.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Ground-1/blt/defs.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Ground-1/blt/defs.ma deleted file mode 100644 index 009627a3b..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Ground-1/blt/defs.ma +++ /dev/null @@ -1,25 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Ground-1/preamble.ma". - -definition blt: - nat \to (nat \to bool) -\def - let rec blt (m: nat) (n: nat) on n: bool \def (match n with [O \Rightarrow -false | (S n0) \Rightarrow (match m with [O \Rightarrow true | (S m0) -\Rightarrow (blt m0 n0)])]) in blt. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Ground-1/blt/props.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Ground-1/blt/props.ma deleted file mode 100644 index e5b569925..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Ground-1/blt/props.ma +++ /dev/null @@ -1,112 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Ground-1/blt/defs.ma". - -theorem lt_blt: - \forall (x: nat).(\forall (y: nat).((lt y x) \to (eq bool (blt y x) true))) -\def - \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((lt y n) \to -(eq bool (blt y n) true)))) (\lambda (y: nat).(\lambda (H: (lt y O)).(let H0 -\def (match H in le return (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat -n O) \to (eq bool (blt y O) true)))) with [le_n \Rightarrow (\lambda (H0: (eq -nat (S y) O)).(let H1 \def (eq_ind nat (S y) (\lambda (e: nat).(match e in -nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) -\Rightarrow True])) I O H0) in (False_ind (eq bool (blt y O) true) H1))) | -(le_S m H0) \Rightarrow (\lambda (H1: (eq nat (S m) O)).((let H2 \def (eq_ind -nat (S m) (\lambda (e: nat).(match e in nat return (\lambda (_: nat).Prop) -with [O \Rightarrow False | (S _) \Rightarrow True])) I O H1) in (False_ind -((le (S y) m) \to (eq bool (blt y O) true)) H2)) H0))]) in (H0 (refl_equal -nat O))))) (\lambda (n: nat).(\lambda (H: ((\forall (y: nat).((lt y n) \to -(eq bool (blt y n) true))))).(\lambda (y: nat).(nat_ind (\lambda (n0: -nat).((lt n0 (S n)) \to (eq bool (blt n0 (S n)) true))) (\lambda (_: (lt O (S -n))).(refl_equal bool true)) (\lambda (n0: nat).(\lambda (_: (((lt n0 (S n)) -\to (eq bool (match n0 with [O \Rightarrow true | (S m) \Rightarrow (blt m -n)]) true)))).(\lambda (H1: (lt (S n0) (S n))).(H n0 (le_S_n (S n0) n H1))))) -y)))) x). -(* COMMENTS -Initial nodes: 291 -END *) - -theorem le_bge: - \forall (x: nat).(\forall (y: nat).((le x y) \to (eq bool (blt y x) false))) -\def - \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((le n y) \to -(eq bool (blt y n) false)))) (\lambda (y: nat).(\lambda (_: (le O -y)).(refl_equal bool false))) (\lambda (n: nat).(\lambda (H: ((\forall (y: -nat).((le n y) \to (eq bool (blt y n) false))))).(\lambda (y: nat).(nat_ind -(\lambda (n0: nat).((le (S n) n0) \to (eq bool (blt n0 (S n)) false))) -(\lambda (H0: (le (S n) O)).(let H1 \def (match H0 in le return (\lambda (n0: -nat).(\lambda (_: (le ? n0)).((eq nat n0 O) \to (eq bool (blt O (S n)) -false)))) with [le_n \Rightarrow (\lambda (H1: (eq nat (S n) O)).(let H2 \def -(eq_ind nat (S n) (\lambda (e: nat).(match e in nat return (\lambda (_: -nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H1) in -(False_ind (eq bool (blt O (S n)) false) H2))) | (le_S m H1) \Rightarrow -(\lambda (H2: (eq nat (S m) O)).((let H3 \def (eq_ind nat (S m) (\lambda (e: -nat).(match e in nat return (\lambda (_: nat).Prop) with [O \Rightarrow False -| (S _) \Rightarrow True])) I O H2) in (False_ind ((le (S n) m) \to (eq bool -(blt O (S n)) false)) H3)) H1))]) in (H1 (refl_equal nat O)))) (\lambda (n0: -nat).(\lambda (_: (((le (S n) n0) \to (eq bool (blt n0 (S n)) -false)))).(\lambda (H1: (le (S n) (S n0))).(H n0 (le_S_n n n0 H1))))) y)))) -x). -(* COMMENTS -Initial nodes: 293 -END *) - -theorem blt_lt: - \forall (x: nat).(\forall (y: nat).((eq bool (blt y x) true) \to (lt y x))) -\def - \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((eq bool (blt -y n) true) \to (lt y n)))) (\lambda (y: nat).(\lambda (H: (eq bool (blt y O) -true)).(let H0 \def (match H in eq return (\lambda (b: bool).(\lambda (_: (eq -? ? b)).((eq bool b true) \to (lt y O)))) with [refl_equal \Rightarrow -(\lambda (H0: (eq bool (blt y O) true)).(let H1 \def (eq_ind bool (blt y O) -(\lambda (e: bool).(match e in bool return (\lambda (_: bool).Prop) with -[true \Rightarrow False | false \Rightarrow True])) I true H0) in (False_ind -(lt y O) H1)))]) in (H0 (refl_equal bool true))))) (\lambda (n: nat).(\lambda -(H: ((\forall (y: nat).((eq bool (blt y n) true) \to (lt y n))))).(\lambda -(y: nat).(nat_ind (\lambda (n0: nat).((eq bool (blt n0 (S n)) true) \to (lt -n0 (S n)))) (\lambda (_: (eq bool true true)).(le_S_n (S O) (S n) (le_n_S (S -O) (S n) (le_n_S O n (le_O_n n))))) (\lambda (n0: nat).(\lambda (_: (((eq -bool (match n0 with [O \Rightarrow true | (S m) \Rightarrow (blt m n)]) true) -\to (lt n0 (S n))))).(\lambda (H1: (eq bool (blt n0 n) true)).(lt_n_S n0 n (H -n0 H1))))) y)))) x). -(* COMMENTS -Initial nodes: 252 -END *) - -theorem bge_le: - \forall (x: nat).(\forall (y: nat).((eq bool (blt y x) false) \to (le x y))) -\def - \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((eq bool (blt -y n) false) \to (le n y)))) (\lambda (y: nat).(\lambda (_: (eq bool (blt y O) -false)).(le_O_n y))) (\lambda (n: nat).(\lambda (H: ((\forall (y: nat).((eq -bool (blt y n) false) \to (le n y))))).(\lambda (y: nat).(nat_ind (\lambda -(n0: nat).((eq bool (blt n0 (S n)) false) \to (le (S n) n0))) (\lambda (H0: -(eq bool (blt O (S n)) false)).(let H1 \def (match H0 in eq return (\lambda -(b: bool).(\lambda (_: (eq ? ? b)).((eq bool b false) \to (le (S n) O)))) -with [refl_equal \Rightarrow (\lambda (H1: (eq bool (blt O (S n)) -false)).(let H2 \def (eq_ind bool (blt O (S n)) (\lambda (e: bool).(match e -in bool return (\lambda (_: bool).Prop) with [true \Rightarrow True | false -\Rightarrow False])) I false H1) in (False_ind (le (S n) O) H2)))]) in (H1 -(refl_equal bool false)))) (\lambda (n0: nat).(\lambda (_: (((eq bool (blt n0 -(S n)) false) \to (le (S n) n0)))).(\lambda (H1: (eq bool (blt (S n0) (S n)) -false)).(le_S_n (S n) (S n0) (le_n_S (S n) (S n0) (le_n_S n n0 (H n0 -H1))))))) y)))) x). -(* COMMENTS -Initial nodes: 262 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Ground-1/definitions.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Ground-1/definitions.ma deleted file mode 100644 index df31468aa..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Ground-1/definitions.ma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Ground-1/types/defs.ma". - -include "Ground-1/blt/defs.ma". - -include "Ground-1/plist/defs.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Ground-1/ext/arith.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Ground-1/ext/arith.ma deleted file mode 100644 index f9796e7fd..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Ground-1/ext/arith.ma +++ /dev/null @@ -1,737 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Ground-1/preamble.ma". - -theorem nat_dec: - \forall (n1: nat).(\forall (n2: nat).(or (eq nat n1 n2) ((eq nat n1 n2) \to -(\forall (P: Prop).P)))) -\def - \lambda (n1: nat).(nat_ind (\lambda (n: nat).(\forall (n2: nat).(or (eq nat -n n2) ((eq nat n n2) \to (\forall (P: Prop).P))))) (\lambda (n2: -nat).(nat_ind (\lambda (n: nat).(or (eq nat O n) ((eq nat O n) \to (\forall -(P: Prop).P)))) (or_introl (eq nat O O) ((eq nat O O) \to (\forall (P: -Prop).P)) (refl_equal nat O)) (\lambda (n: nat).(\lambda (_: (or (eq nat O n) -((eq nat O n) \to (\forall (P: Prop).P)))).(or_intror (eq nat O (S n)) ((eq -nat O (S n)) \to (\forall (P: Prop).P)) (\lambda (H0: (eq nat O (S -n))).(\lambda (P: Prop).(let H1 \def (eq_ind nat O (\lambda (ee: nat).(match -ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) -\Rightarrow False])) I (S n) H0) in (False_ind P H1))))))) n2)) (\lambda (n: -nat).(\lambda (H: ((\forall (n2: nat).(or (eq nat n n2) ((eq nat n n2) \to -(\forall (P: Prop).P)))))).(\lambda (n2: nat).(nat_ind (\lambda (n0: nat).(or -(eq nat (S n) n0) ((eq nat (S n) n0) \to (\forall (P: Prop).P)))) (or_intror -(eq nat (S n) O) ((eq nat (S n) O) \to (\forall (P: Prop).P)) (\lambda (H0: -(eq nat (S n) O)).(\lambda (P: Prop).(let H1 \def (eq_ind nat (S n) (\lambda -(ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow -False | (S _) \Rightarrow True])) I O H0) in (False_ind P H1))))) (\lambda -(n0: nat).(\lambda (H0: (or (eq nat (S n) n0) ((eq nat (S n) n0) \to (\forall -(P: Prop).P)))).(or_ind (eq nat n n0) ((eq nat n n0) \to (\forall (P: -Prop).P)) (or (eq nat (S n) (S n0)) ((eq nat (S n) (S n0)) \to (\forall (P: -Prop).P))) (\lambda (H1: (eq nat n n0)).(let H2 \def (eq_ind_r nat n0 -(\lambda (n3: nat).(or (eq nat (S n) n3) ((eq nat (S n) n3) \to (\forall (P: -Prop).P)))) H0 n H1) in (eq_ind nat n (\lambda (n3: nat).(or (eq nat (S n) (S -n3)) ((eq nat (S n) (S n3)) \to (\forall (P: Prop).P)))) (or_introl (eq nat -(S n) (S n)) ((eq nat (S n) (S n)) \to (\forall (P: Prop).P)) (refl_equal nat -(S n))) n0 H1))) (\lambda (H1: (((eq nat n n0) \to (\forall (P: -Prop).P)))).(or_intror (eq nat (S n) (S n0)) ((eq nat (S n) (S n0)) \to -(\forall (P: Prop).P)) (\lambda (H2: (eq nat (S n) (S n0))).(\lambda (P: -Prop).(let H3 \def (f_equal nat nat (\lambda (e: nat).(match e in nat return -(\lambda (_: nat).nat) with [O \Rightarrow n | (S n3) \Rightarrow n3])) (S n) -(S n0) H2) in (let H4 \def (eq_ind_r nat n0 (\lambda (n3: nat).((eq nat n n3) -\to (\forall (P0: Prop).P0))) H1 n H3) in (let H5 \def (eq_ind_r nat n0 -(\lambda (n3: nat).(or (eq nat (S n) n3) ((eq nat (S n) n3) \to (\forall (P0: -Prop).P0)))) H0 n H3) in (H4 (refl_equal nat n) P)))))))) (H n0)))) n2)))) -n1). -(* COMMENTS -Initial nodes: 676 -END *) - -theorem simpl_plus_r: - \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((eq nat (plus m n) -(plus p n)) \to (eq nat m p)))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (eq nat -(plus m n) (plus p n))).(simpl_plus_l n m p (eq_ind_r nat (plus m n) (\lambda -(n0: nat).(eq nat n0 (plus n p))) (eq_ind_r nat (plus p n) (\lambda (n0: -nat).(eq nat n0 (plus n p))) (sym_eq nat (plus n p) (plus p n) (plus_sym n -p)) (plus m n) H) (plus n m) (plus_sym n m)))))). -(* COMMENTS -Initial nodes: 119 -END *) - -theorem minus_Sx_Sy: - \forall (x: nat).(\forall (y: nat).(eq nat (minus (S x) (S y)) (minus x y))) -\def - \lambda (x: nat).(\lambda (y: nat).(refl_equal nat (minus x y))). -(* COMMENTS -Initial nodes: 13 -END *) - -theorem minus_plus_r: - \forall (m: nat).(\forall (n: nat).(eq nat (minus (plus m n) n) m)) -\def - \lambda (m: nat).(\lambda (n: nat).(eq_ind_r nat (plus n m) (\lambda (n0: -nat).(eq nat (minus n0 n) m)) (minus_plus n m) (plus m n) (plus_sym m n))). -(* COMMENTS -Initial nodes: 45 -END *) - -theorem plus_permute_2_in_3: - \forall (x: nat).(\forall (y: nat).(\forall (z: nat).(eq nat (plus (plus x -y) z) (plus (plus x z) y)))) -\def - \lambda (x: nat).(\lambda (y: nat).(\lambda (z: nat).(eq_ind_r nat (plus x -(plus y z)) (\lambda (n: nat).(eq nat n (plus (plus x z) y))) (eq_ind_r nat -(plus z y) (\lambda (n: nat).(eq nat (plus x n) (plus (plus x z) y))) (eq_ind -nat (plus (plus x z) y) (\lambda (n: nat).(eq nat n (plus (plus x z) y))) -(refl_equal nat (plus (plus x z) y)) (plus x (plus z y)) (plus_assoc_r x z -y)) (plus y z) (plus_sym y z)) (plus (plus x y) z) (plus_assoc_r x y z)))). -(* COMMENTS -Initial nodes: 163 -END *) - -theorem plus_permute_2_in_3_assoc: - \forall (n: nat).(\forall (h: nat).(\forall (k: nat).(eq nat (plus (plus n -h) k) (plus n (plus k h))))) -\def - \lambda (n: nat).(\lambda (h: nat).(\lambda (k: nat).(eq_ind_r nat (plus -(plus n k) h) (\lambda (n0: nat).(eq nat n0 (plus n (plus k h)))) (eq_ind_r -nat (plus (plus n k) h) (\lambda (n0: nat).(eq nat (plus (plus n k) h) n0)) -(refl_equal nat (plus (plus n k) h)) (plus n (plus k h)) (plus_assoc_l n k -h)) (plus (plus n h) k) (plus_permute_2_in_3 n h k)))). -(* COMMENTS -Initial nodes: 119 -END *) - -theorem plus_O: - \forall (x: nat).(\forall (y: nat).((eq nat (plus x y) O) \to (land (eq nat -x O) (eq nat y O)))) -\def - \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((eq nat (plus -n y) O) \to (land (eq nat n O) (eq nat y O))))) (\lambda (y: nat).(\lambda -(H: (eq nat (plus O y) O)).(conj (eq nat O O) (eq nat y O) (refl_equal nat O) -H))) (\lambda (n: nat).(\lambda (_: ((\forall (y: nat).((eq nat (plus n y) O) -\to (land (eq nat n O) (eq nat y O)))))).(\lambda (y: nat).(\lambda (H0: (eq -nat (plus (S n) y) O)).(let H1 \def (match H0 in eq return (\lambda (n0: -nat).(\lambda (_: (eq ? ? n0)).((eq nat n0 O) \to (land (eq nat (S n) O) (eq -nat y O))))) with [refl_equal \Rightarrow (\lambda (H1: (eq nat (plus (S n) -y) O)).(let H2 \def (eq_ind nat (plus (S n) y) (\lambda (e: nat).(match e in -nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) -\Rightarrow True])) I O H1) in (False_ind (land (eq nat (S n) O) (eq nat y -O)) H2)))]) in (H1 (refl_equal nat O))))))) x). -(* COMMENTS -Initial nodes: 233 -END *) - -theorem minus_Sx_SO: - \forall (x: nat).(eq nat (minus (S x) (S O)) x) -\def - \lambda (x: nat).(eq_ind nat x (\lambda (n: nat).(eq nat n x)) (refl_equal -nat x) (minus x O) (minus_n_O x)). -(* COMMENTS -Initial nodes: 33 -END *) - -theorem eq_nat_dec: - \forall (i: nat).(\forall (j: nat).(or (not (eq nat i j)) (eq nat i j))) -\def - \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (j: nat).(or (not (eq -nat n j)) (eq nat n j)))) (\lambda (j: nat).(nat_ind (\lambda (n: nat).(or -(not (eq nat O n)) (eq nat O n))) (or_intror (not (eq nat O O)) (eq nat O O) -(refl_equal nat O)) (\lambda (n: nat).(\lambda (_: (or (not (eq nat O n)) (eq -nat O n))).(or_introl (not (eq nat O (S n))) (eq nat O (S n)) (O_S n)))) j)) -(\lambda (n: nat).(\lambda (H: ((\forall (j: nat).(or (not (eq nat n j)) (eq -nat n j))))).(\lambda (j: nat).(nat_ind (\lambda (n0: nat).(or (not (eq nat -(S n) n0)) (eq nat (S n) n0))) (or_introl (not (eq nat (S n) O)) (eq nat (S -n) O) (sym_not_eq nat O (S n) (O_S n))) (\lambda (n0: nat).(\lambda (_: (or -(not (eq nat (S n) n0)) (eq nat (S n) n0))).(or_ind (not (eq nat n n0)) (eq -nat n n0) (or (not (eq nat (S n) (S n0))) (eq nat (S n) (S n0))) (\lambda -(H1: (not (eq nat n n0))).(or_introl (not (eq nat (S n) (S n0))) (eq nat (S -n) (S n0)) (not_eq_S n n0 H1))) (\lambda (H1: (eq nat n n0)).(or_intror (not -(eq nat (S n) (S n0))) (eq nat (S n) (S n0)) (f_equal nat nat S n n0 H1))) (H -n0)))) j)))) i). -(* COMMENTS -Initial nodes: 401 -END *) - -theorem neq_eq_e: - \forall (i: nat).(\forall (j: nat).(\forall (P: Prop).((((not (eq nat i j)) -\to P)) \to ((((eq nat i j) \to P)) \to P)))) -\def - \lambda (i: nat).(\lambda (j: nat).(\lambda (P: Prop).(\lambda (H: (((not -(eq nat i j)) \to P))).(\lambda (H0: (((eq nat i j) \to P))).(let o \def -(eq_nat_dec i j) in (or_ind (not (eq nat i j)) (eq nat i j) P H H0 o)))))). -(* COMMENTS -Initial nodes: 61 -END *) - -theorem le_false: - \forall (m: nat).(\forall (n: nat).(\forall (P: Prop).((le m n) \to ((le (S -n) m) \to P)))) -\def - \lambda (m: nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).(\forall (P: -Prop).((le n n0) \to ((le (S n0) n) \to P))))) (\lambda (n: nat).(\lambda (P: -Prop).(\lambda (_: (le O n)).(\lambda (H0: (le (S n) O)).(let H1 \def (match -H0 in le return (\lambda (n0: nat).(\lambda (_: (le ? n0)).((eq nat n0 O) \to -P))) with [le_n \Rightarrow (\lambda (H1: (eq nat (S n) O)).(let H2 \def -(eq_ind nat (S n) (\lambda (e: nat).(match e in nat return (\lambda (_: -nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H1) in -(False_ind P H2))) | (le_S m0 H1) \Rightarrow (\lambda (H2: (eq nat (S m0) -O)).((let H3 \def (eq_ind nat (S m0) (\lambda (e: nat).(match e in nat return -(\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) -I O H2) in (False_ind ((le (S n) m0) \to P) H3)) H1))]) in (H1 (refl_equal -nat O))))))) (\lambda (n: nat).(\lambda (H: ((\forall (n0: nat).(\forall (P: -Prop).((le n n0) \to ((le (S n0) n) \to P)))))).(\lambda (n0: nat).(nat_ind -(\lambda (n1: nat).(\forall (P: Prop).((le (S n) n1) \to ((le (S n1) (S n)) -\to P)))) (\lambda (P: Prop).(\lambda (H0: (le (S n) O)).(\lambda (_: (le (S -O) (S n))).(let H2 \def (match H0 in le return (\lambda (n1: nat).(\lambda -(_: (le ? n1)).((eq nat n1 O) \to P))) with [le_n \Rightarrow (\lambda (H2: -(eq nat (S n) O)).(let H3 \def (eq_ind nat (S n) (\lambda (e: nat).(match e -in nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) -\Rightarrow True])) I O H2) in (False_ind P H3))) | (le_S m0 H2) \Rightarrow -(\lambda (H3: (eq nat (S m0) O)).((let H4 \def (eq_ind nat (S m0) (\lambda -(e: nat).(match e in nat return (\lambda (_: nat).Prop) with [O \Rightarrow -False | (S _) \Rightarrow True])) I O H3) in (False_ind ((le (S n) m0) \to P) -H4)) H2))]) in (H2 (refl_equal nat O)))))) (\lambda (n1: nat).(\lambda (_: -((\forall (P: Prop).((le (S n) n1) \to ((le (S n1) (S n)) \to P))))).(\lambda -(P: Prop).(\lambda (H1: (le (S n) (S n1))).(\lambda (H2: (le (S (S n1)) (S -n))).(H n1 P (le_S_n n n1 H1) (le_S_n (S n1) n H2))))))) n0)))) m). -(* COMMENTS -Initial nodes: 409 -END *) - -theorem le_Sx_x: - \forall (x: nat).((le (S x) x) \to (\forall (P: Prop).P)) -\def - \lambda (x: nat).(\lambda (H: (le (S x) x)).(\lambda (P: Prop).(let H0 \def -le_Sn_n in (False_ind P (H0 x H))))). -(* COMMENTS -Initial nodes: 23 -END *) - -theorem le_n_pred: - \forall (n: nat).(\forall (m: nat).((le n m) \to (le (pred n) (pred m)))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(le_ind n (\lambda -(n0: nat).(le (pred n) (pred n0))) (le_n (pred n)) (\lambda (m0: -nat).(\lambda (_: (le n m0)).(\lambda (H1: (le (pred n) (pred m0))).(le_trans -(pred n) (pred m0) m0 H1 (le_pred_n m0))))) m H))). -(* COMMENTS -Initial nodes: 71 -END *) - -theorem minus_le: - \forall (x: nat).(\forall (y: nat).(le (minus x y) x)) -\def - \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).(le (minus n -y) n))) (\lambda (_: nat).(le_n O)) (\lambda (n: nat).(\lambda (H: ((\forall -(y: nat).(le (minus n y) n)))).(\lambda (y: nat).(nat_ind (\lambda (n0: -nat).(le (minus (S n) n0) (S n))) (le_n (S n)) (\lambda (n0: nat).(\lambda -(_: (le (match n0 with [O \Rightarrow (S n) | (S l) \Rightarrow (minus n l)]) -(S n))).(le_S (minus n n0) n (H n0)))) y)))) x). -(* COMMENTS -Initial nodes: 101 -END *) - -theorem le_plus_minus_sym: - \forall (n: nat).(\forall (m: nat).((le n m) \to (eq nat m (plus (minus m n) -n)))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(eq_ind_r nat -(plus n (minus m n)) (\lambda (n0: nat).(eq nat m n0)) (le_plus_minus n m H) -(plus (minus m n) n) (plus_sym (minus m n) n)))). -(* COMMENTS -Initial nodes: 61 -END *) - -theorem le_minus_minus: - \forall (x: nat).(\forall (y: nat).((le x y) \to (\forall (z: nat).((le y z) -\to (le (minus y x) (minus z x)))))) -\def - \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (le x y)).(\lambda (z: -nat).(\lambda (H0: (le y z)).(simpl_le_plus_l x (minus y x) (minus z x) -(eq_ind_r nat y (\lambda (n: nat).(le n (plus x (minus z x)))) (eq_ind_r nat -z (\lambda (n: nat).(le y n)) H0 (plus x (minus z x)) (le_plus_minus_r x z -(le_trans x y z H H0))) (plus x (minus y x)) (le_plus_minus_r x y H))))))). -(* COMMENTS -Initial nodes: 117 -END *) - -theorem le_minus_plus: - \forall (z: nat).(\forall (x: nat).((le z x) \to (\forall (y: nat).(eq nat -(minus (plus x y) z) (plus (minus x z) y))))) -\def - \lambda (z: nat).(nat_ind (\lambda (n: nat).(\forall (x: nat).((le n x) \to -(\forall (y: nat).(eq nat (minus (plus x y) n) (plus (minus x n) y)))))) -(\lambda (x: nat).(\lambda (H: (le O x)).(let H0 \def (match H in le return -(\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat n x) \to (\forall (y: -nat).(eq nat (minus (plus x y) O) (plus (minus x O) y)))))) with [le_n -\Rightarrow (\lambda (H0: (eq nat O x)).(eq_ind nat O (\lambda (n: -nat).(\forall (y: nat).(eq nat (minus (plus n y) O) (plus (minus n O) y)))) -(\lambda (y: nat).(sym_eq nat (plus (minus O O) y) (minus (plus O y) O) -(minus_n_O (plus O y)))) x H0)) | (le_S m H0) \Rightarrow (\lambda (H1: (eq -nat (S m) x)).(eq_ind nat (S m) (\lambda (n: nat).((le O m) \to (\forall (y: -nat).(eq nat (minus (plus n y) O) (plus (minus n O) y))))) (\lambda (_: (le O -m)).(\lambda (y: nat).(refl_equal nat (plus (minus (S m) O) y)))) x H1 H0))]) -in (H0 (refl_equal nat x))))) (\lambda (z0: nat).(\lambda (H: ((\forall (x: -nat).((le z0 x) \to (\forall (y: nat).(eq nat (minus (plus x y) z0) (plus -(minus x z0) y))))))).(\lambda (x: nat).(nat_ind (\lambda (n: nat).((le (S -z0) n) \to (\forall (y: nat).(eq nat (minus (plus n y) (S z0)) (plus (minus n -(S z0)) y))))) (\lambda (H0: (le (S z0) O)).(\lambda (y: nat).(let H1 \def -(match H0 in le return (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat n O) -\to (eq nat (minus (plus O y) (S z0)) (plus (minus O (S z0)) y))))) with -[le_n \Rightarrow (\lambda (H1: (eq nat (S z0) O)).(let H2 \def (eq_ind nat -(S z0) (\lambda (e: nat).(match e in nat return (\lambda (_: nat).Prop) with -[O \Rightarrow False | (S _) \Rightarrow True])) I O H1) in (False_ind (eq -nat (minus (plus O y) (S z0)) (plus (minus O (S z0)) y)) H2))) | (le_S m H1) -\Rightarrow (\lambda (H2: (eq nat (S m) O)).((let H3 \def (eq_ind nat (S m) -(\lambda (e: nat).(match e in nat return (\lambda (_: nat).Prop) with [O -\Rightarrow False | (S _) \Rightarrow True])) I O H2) in (False_ind ((le (S -z0) m) \to (eq nat (minus (plus O y) (S z0)) (plus (minus O (S z0)) y))) H3)) -H1))]) in (H1 (refl_equal nat O))))) (\lambda (n: nat).(\lambda (_: (((le (S -z0) n) \to (\forall (y: nat).(eq nat (minus (plus n y) (S z0)) (plus (minus n -(S z0)) y)))))).(\lambda (H1: (le (S z0) (S n))).(\lambda (y: nat).(H n -(le_S_n z0 n H1) y))))) x)))) z). -(* COMMENTS -Initial nodes: 603 -END *) - -theorem le_minus: - \forall (x: nat).(\forall (z: nat).(\forall (y: nat).((le (plus x y) z) \to -(le x (minus z y))))) -\def - \lambda (x: nat).(\lambda (z: nat).(\lambda (y: nat).(\lambda (H: (le (plus -x y) z)).(eq_ind nat (minus (plus x y) y) (\lambda (n: nat).(le n (minus z -y))) (le_minus_minus y (plus x y) (le_plus_r x y) z H) x (minus_plus_r x -y))))). -(* COMMENTS -Initial nodes: 69 -END *) - -theorem le_trans_plus_r: - \forall (x: nat).(\forall (y: nat).(\forall (z: nat).((le (plus x y) z) \to -(le y z)))) -\def - \lambda (x: nat).(\lambda (y: nat).(\lambda (z: nat).(\lambda (H: (le (plus -x y) z)).(le_trans y (plus x y) z (le_plus_r x y) H)))). -(* COMMENTS -Initial nodes: 35 -END *) - -theorem lt_x_O: - \forall (x: nat).((lt x O) \to (\forall (P: Prop).P)) -\def - \lambda (x: nat).(\lambda (H: (le (S x) O)).(\lambda (P: Prop).(let H_y \def -(le_n_O_eq (S x) H) in (let H0 \def (eq_ind nat O (\lambda (ee: nat).(match -ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) -\Rightarrow False])) I (S x) H_y) in (False_ind P H0))))). -(* COMMENTS -Initial nodes: 48 -END *) - -theorem le_gen_S: - \forall (m: nat).(\forall (x: nat).((le (S m) x) \to (ex2 nat (\lambda (n: -nat).(eq nat x (S n))) (\lambda (n: nat).(le m n))))) -\def - \lambda (m: nat).(\lambda (x: nat).(\lambda (H: (le (S m) x)).(let H0 \def -(match H in le return (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat n x) -\to (ex2 nat (\lambda (n0: nat).(eq nat x (S n0))) (\lambda (n0: nat).(le m -n0)))))) with [le_n \Rightarrow (\lambda (H0: (eq nat (S m) x)).(eq_ind nat -(S m) (\lambda (n: nat).(ex2 nat (\lambda (n0: nat).(eq nat n (S n0))) -(\lambda (n0: nat).(le m n0)))) (ex_intro2 nat (\lambda (n: nat).(eq nat (S -m) (S n))) (\lambda (n: nat).(le m n)) m (refl_equal nat (S m)) (le_n m)) x -H0)) | (le_S m0 H0) \Rightarrow (\lambda (H1: (eq nat (S m0) x)).(eq_ind nat -(S m0) (\lambda (n: nat).((le (S m) m0) \to (ex2 nat (\lambda (n0: nat).(eq -nat n (S n0))) (\lambda (n0: nat).(le m n0))))) (\lambda (H2: (le (S m) -m0)).(ex_intro2 nat (\lambda (n: nat).(eq nat (S m0) (S n))) (\lambda (n: -nat).(le m n)) m0 (refl_equal nat (S m0)) (le_S_n m m0 (le_S (S m) m0 H2)))) -x H1 H0))]) in (H0 (refl_equal nat x))))). -(* COMMENTS -Initial nodes: 261 -END *) - -theorem lt_x_plus_x_Sy: - \forall (x: nat).(\forall (y: nat).(lt x (plus x (S y)))) -\def - \lambda (x: nat).(\lambda (y: nat).(eq_ind_r nat (plus (S y) x) (\lambda (n: -nat).(lt x n)) (le_S_n (S x) (S (plus y x)) (le_n_S (S x) (S (plus y x)) -(le_n_S x (plus y x) (le_plus_r y x)))) (plus x (S y)) (plus_sym x (S y)))). -(* COMMENTS -Initial nodes: 83 -END *) - -theorem simpl_lt_plus_r: - \forall (p: nat).(\forall (n: nat).(\forall (m: nat).((lt (plus n p) (plus m -p)) \to (lt n m)))) -\def - \lambda (p: nat).(\lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt (plus -n p) (plus m p))).(simpl_lt_plus_l n m p (let H0 \def (eq_ind nat (plus n p) -(\lambda (n0: nat).(lt n0 (plus m p))) H (plus p n) (plus_sym n p)) in (let -H1 \def (eq_ind nat (plus m p) (\lambda (n0: nat).(lt (plus p n) n0)) H0 -(plus p m) (plus_sym m p)) in H1)))))). -(* COMMENTS -Initial nodes: 101 -END *) - -theorem minus_x_Sy: - \forall (x: nat).(\forall (y: nat).((lt y x) \to (eq nat (minus x y) (S -(minus x (S y)))))) -\def - \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((lt y n) \to -(eq nat (minus n y) (S (minus n (S y))))))) (\lambda (y: nat).(\lambda (H: -(lt y O)).(let H0 \def (match H in le return (\lambda (n: nat).(\lambda (_: -(le ? n)).((eq nat n O) \to (eq nat (minus O y) (S (minus O (S y))))))) with -[le_n \Rightarrow (\lambda (H0: (eq nat (S y) O)).(let H1 \def (eq_ind nat (S -y) (\lambda (e: nat).(match e in nat return (\lambda (_: nat).Prop) with [O -\Rightarrow False | (S _) \Rightarrow True])) I O H0) in (False_ind (eq nat -(minus O y) (S (minus O (S y)))) H1))) | (le_S m H0) \Rightarrow (\lambda -(H1: (eq nat (S m) O)).((let H2 \def (eq_ind nat (S m) (\lambda (e: -nat).(match e in nat return (\lambda (_: nat).Prop) with [O \Rightarrow False -| (S _) \Rightarrow True])) I O H1) in (False_ind ((le (S y) m) \to (eq nat -(minus O y) (S (minus O (S y))))) H2)) H0))]) in (H0 (refl_equal nat O))))) -(\lambda (n: nat).(\lambda (H: ((\forall (y: nat).((lt y n) \to (eq nat -(minus n y) (S (minus n (S y)))))))).(\lambda (y: nat).(nat_ind (\lambda (n0: -nat).((lt n0 (S n)) \to (eq nat (minus (S n) n0) (S (minus (S n) (S n0)))))) -(\lambda (_: (lt O (S n))).(eq_ind nat n (\lambda (n0: nat).(eq nat (S n) (S -n0))) (refl_equal nat (S n)) (minus n O) (minus_n_O n))) (\lambda (n0: -nat).(\lambda (_: (((lt n0 (S n)) \to (eq nat (minus (S n) n0) (S (minus (S -n) (S n0))))))).(\lambda (H1: (lt (S n0) (S n))).(let H2 \def (le_S_n (S n0) -n H1) in (H n0 H2))))) y)))) x). -(* COMMENTS -Initial nodes: 383 -END *) - -theorem lt_plus_minus: - \forall (x: nat).(\forall (y: nat).((lt x y) \to (eq nat y (S (plus x (minus -y (S x))))))) -\def - \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(le_plus_minus (S -x) y H))). -(* COMMENTS -Initial nodes: 19 -END *) - -theorem lt_plus_minus_r: - \forall (x: nat).(\forall (y: nat).((lt x y) \to (eq nat y (S (plus (minus y -(S x)) x))))) -\def - \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(eq_ind_r nat -(plus x (minus y (S x))) (\lambda (n: nat).(eq nat y (S n))) (lt_plus_minus x -y H) (plus (minus y (S x)) x) (plus_sym (minus y (S x)) x)))). -(* COMMENTS -Initial nodes: 69 -END *) - -theorem minus_x_SO: - \forall (x: nat).((lt O x) \to (eq nat x (S (minus x (S O))))) -\def - \lambda (x: nat).(\lambda (H: (lt O x)).(eq_ind nat (minus x O) (\lambda (n: -nat).(eq nat x n)) (eq_ind nat x (\lambda (n: nat).(eq nat x n)) (refl_equal -nat x) (minus x O) (minus_n_O x)) (S (minus x (S O))) (minus_x_Sy x O H))). -(* COMMENTS -Initial nodes: 77 -END *) - -theorem le_x_pred_y: - \forall (y: nat).(\forall (x: nat).((lt x y) \to (le x (pred y)))) -\def - \lambda (y: nat).(nat_ind (\lambda (n: nat).(\forall (x: nat).((lt x n) \to -(le x (pred n))))) (\lambda (x: nat).(\lambda (H: (lt x O)).(let H0 \def -(match H in le return (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat n O) -\to (le x O)))) with [le_n \Rightarrow (\lambda (H0: (eq nat (S x) O)).(let -H1 \def (eq_ind nat (S x) (\lambda (e: nat).(match e in nat return (\lambda -(_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H0) -in (False_ind (le x O) H1))) | (le_S m H0) \Rightarrow (\lambda (H1: (eq nat -(S m) O)).((let H2 \def (eq_ind nat (S m) (\lambda (e: nat).(match e in nat -return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow -True])) I O H1) in (False_ind ((le (S x) m) \to (le x O)) H2)) H0))]) in (H0 -(refl_equal nat O))))) (\lambda (n: nat).(\lambda (_: ((\forall (x: nat).((lt -x n) \to (le x (pred n)))))).(\lambda (x: nat).(\lambda (H0: (lt x (S -n))).(le_S_n x n H0))))) y). -(* COMMENTS -Initial nodes: 189 -END *) - -theorem lt_le_minus: - \forall (x: nat).(\forall (y: nat).((lt x y) \to (le x (minus y (S O))))) -\def - \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(le_minus x y (S -O) (eq_ind_r nat (plus (S O) x) (\lambda (n: nat).(le n y)) H (plus x (S O)) -(plus_sym x (S O)))))). -(* COMMENTS -Initial nodes: 57 -END *) - -theorem lt_le_e: - \forall (n: nat).(\forall (d: nat).(\forall (P: Prop).((((lt n d) \to P)) -\to ((((le d n) \to P)) \to P)))) -\def - \lambda (n: nat).(\lambda (d: nat).(\lambda (P: Prop).(\lambda (H: (((lt n -d) \to P))).(\lambda (H0: (((le d n) \to P))).(let H1 \def (le_or_lt d n) in -(or_ind (le d n) (lt n d) P H0 H H1)))))). -(* COMMENTS -Initial nodes: 49 -END *) - -theorem lt_eq_e: - \forall (x: nat).(\forall (y: nat).(\forall (P: Prop).((((lt x y) \to P)) -\to ((((eq nat x y) \to P)) \to ((le x y) \to P))))) -\def - \lambda (x: nat).(\lambda (y: nat).(\lambda (P: Prop).(\lambda (H: (((lt x -y) \to P))).(\lambda (H0: (((eq nat x y) \to P))).(\lambda (H1: (le x -y)).(or_ind (lt x y) (eq nat x y) P H H0 (le_lt_or_eq x y H1))))))). -(* COMMENTS -Initial nodes: 59 -END *) - -theorem lt_eq_gt_e: - \forall (x: nat).(\forall (y: nat).(\forall (P: Prop).((((lt x y) \to P)) -\to ((((eq nat x y) \to P)) \to ((((lt y x) \to P)) \to P))))) -\def - \lambda (x: nat).(\lambda (y: nat).(\lambda (P: Prop).(\lambda (H: (((lt x -y) \to P))).(\lambda (H0: (((eq nat x y) \to P))).(\lambda (H1: (((lt y x) -\to P))).(lt_le_e x y P H (\lambda (H2: (le y x)).(lt_eq_e y x P H1 (\lambda -(H3: (eq nat y x)).(H0 (sym_eq nat y x H3))) H2)))))))). -(* COMMENTS -Initial nodes: 79 -END *) - -theorem lt_gen_xS: - \forall (x: nat).(\forall (n: nat).((lt x (S n)) \to (or (eq nat x O) (ex2 -nat (\lambda (m: nat).(eq nat x (S m))) (\lambda (m: nat).(lt m n)))))) -\def - \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).((lt n (S -n0)) \to (or (eq nat n O) (ex2 nat (\lambda (m: nat).(eq nat n (S m))) -(\lambda (m: nat).(lt m n0))))))) (\lambda (n: nat).(\lambda (_: (lt O (S -n))).(or_introl (eq nat O O) (ex2 nat (\lambda (m: nat).(eq nat O (S m))) -(\lambda (m: nat).(lt m n))) (refl_equal nat O)))) (\lambda (n: nat).(\lambda -(_: ((\forall (n0: nat).((lt n (S n0)) \to (or (eq nat n O) (ex2 nat (\lambda -(m: nat).(eq nat n (S m))) (\lambda (m: nat).(lt m n0)))))))).(\lambda (n0: -nat).(\lambda (H0: (lt (S n) (S n0))).(or_intror (eq nat (S n) O) (ex2 nat -(\lambda (m: nat).(eq nat (S n) (S m))) (\lambda (m: nat).(lt m n0))) -(ex_intro2 nat (\lambda (m: nat).(eq nat (S n) (S m))) (\lambda (m: nat).(lt -m n0)) n (refl_equal nat (S n)) (le_S_n (S n) n0 H0))))))) x). -(* COMMENTS -Initial nodes: 243 -END *) - -theorem le_lt_false: - \forall (x: nat).(\forall (y: nat).((le x y) \to ((lt y x) \to (\forall (P: -Prop).P)))) -\def - \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (le x y)).(\lambda (H0: (lt -y x)).(\lambda (P: Prop).(False_ind P (le_not_lt x y H H0)))))). -(* COMMENTS -Initial nodes: 31 -END *) - -theorem lt_neq: - \forall (x: nat).(\forall (y: nat).((lt x y) \to (not (eq nat x y)))) -\def - \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(\lambda (H0: (eq -nat x y)).(let H1 \def (eq_ind nat x (\lambda (n: nat).(lt n y)) H y H0) in -(lt_n_n y H1))))). -(* COMMENTS -Initial nodes: 43 -END *) - -theorem arith0: - \forall (h2: nat).(\forall (d2: nat).(\forall (n: nat).((le (plus d2 h2) n) -\to (\forall (h1: nat).(le (plus d2 h1) (minus (plus n h1) h2)))))) -\def - \lambda (h2: nat).(\lambda (d2: nat).(\lambda (n: nat).(\lambda (H: (le -(plus d2 h2) n)).(\lambda (h1: nat).(eq_ind nat (minus (plus h2 (plus d2 h1)) -h2) (\lambda (n0: nat).(le n0 (minus (plus n h1) h2))) (le_minus_minus h2 -(plus h2 (plus d2 h1)) (le_plus_l h2 (plus d2 h1)) (plus n h1) (eq_ind_r nat -(plus (plus h2 d2) h1) (\lambda (n0: nat).(le n0 (plus n h1))) (eq_ind_r nat -(plus d2 h2) (\lambda (n0: nat).(le (plus n0 h1) (plus n h1))) (le_S_n (plus -(plus d2 h2) h1) (plus n h1) (le_n_S (plus (plus d2 h2) h1) (plus n h1) -(le_plus_plus (plus d2 h2) n h1 h1 H (le_n h1)))) (plus h2 d2) (plus_sym h2 -d2)) (plus h2 (plus d2 h1)) (plus_assoc_l h2 d2 h1))) (plus d2 h1) -(minus_plus h2 (plus d2 h1))))))). -(* COMMENTS -Initial nodes: 235 -END *) - -theorem O_minus: - \forall (x: nat).(\forall (y: nat).((le x y) \to (eq nat (minus x y) O))) -\def - \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((le n y) \to -(eq nat (minus n y) O)))) (\lambda (y: nat).(\lambda (_: (le O -y)).(refl_equal nat O))) (\lambda (x0: nat).(\lambda (H: ((\forall (y: -nat).((le x0 y) \to (eq nat (minus x0 y) O))))).(\lambda (y: nat).(nat_ind -(\lambda (n: nat).((le (S x0) n) \to (eq nat (match n with [O \Rightarrow (S -x0) | (S l) \Rightarrow (minus x0 l)]) O))) (\lambda (H0: (le (S x0) -O)).(ex2_ind nat (\lambda (n: nat).(eq nat O (S n))) (\lambda (n: nat).(le x0 -n)) (eq nat (S x0) O) (\lambda (x1: nat).(\lambda (H1: (eq nat O (S -x1))).(\lambda (_: (le x0 x1)).(let H3 \def (eq_ind nat O (\lambda (ee: -nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True -| (S _) \Rightarrow False])) I (S x1) H1) in (False_ind (eq nat (S x0) O) -H3))))) (le_gen_S x0 O H0))) (\lambda (n: nat).(\lambda (_: (((le (S x0) n) -\to (eq nat (match n with [O \Rightarrow (S x0) | (S l) \Rightarrow (minus x0 -l)]) O)))).(\lambda (H1: (le (S x0) (S n))).(H n (le_S_n x0 n H1))))) y)))) -x). -(* COMMENTS -Initial nodes: 252 -END *) - -theorem minus_minus: - \forall (z: nat).(\forall (x: nat).(\forall (y: nat).((le z x) \to ((le z y) -\to ((eq nat (minus x z) (minus y z)) \to (eq nat x y)))))) -\def - \lambda (z: nat).(nat_ind (\lambda (n: nat).(\forall (x: nat).(\forall (y: -nat).((le n x) \to ((le n y) \to ((eq nat (minus x n) (minus y n)) \to (eq -nat x y))))))) (\lambda (x: nat).(\lambda (y: nat).(\lambda (_: (le O -x)).(\lambda (_: (le O y)).(\lambda (H1: (eq nat (minus x O) (minus y -O))).(let H2 \def (eq_ind_r nat (minus x O) (\lambda (n: nat).(eq nat n -(minus y O))) H1 x (minus_n_O x)) in (let H3 \def (eq_ind_r nat (minus y O) -(\lambda (n: nat).(eq nat x n)) H2 y (minus_n_O y)) in H3))))))) (\lambda -(z0: nat).(\lambda (IH: ((\forall (x: nat).(\forall (y: nat).((le z0 x) \to -((le z0 y) \to ((eq nat (minus x z0) (minus y z0)) \to (eq nat x -y)))))))).(\lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((le -(S z0) n) \to ((le (S z0) y) \to ((eq nat (minus n (S z0)) (minus y (S z0))) -\to (eq nat n y)))))) (\lambda (y: nat).(\lambda (H: (le (S z0) O)).(\lambda -(_: (le (S z0) y)).(\lambda (_: (eq nat (minus O (S z0)) (minus y (S -z0)))).(ex2_ind nat (\lambda (n: nat).(eq nat O (S n))) (\lambda (n: nat).(le -z0 n)) (eq nat O y) (\lambda (x0: nat).(\lambda (H2: (eq nat O (S -x0))).(\lambda (_: (le z0 x0)).(let H4 \def (eq_ind nat O (\lambda (ee: -nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True -| (S _) \Rightarrow False])) I (S x0) H2) in (False_ind (eq nat O y) H4))))) -(le_gen_S z0 O H)))))) (\lambda (x0: nat).(\lambda (_: ((\forall (y: -nat).((le (S z0) x0) \to ((le (S z0) y) \to ((eq nat (minus x0 (S z0)) (minus -y (S z0))) \to (eq nat x0 y))))))).(\lambda (y: nat).(nat_ind (\lambda (n: -nat).((le (S z0) (S x0)) \to ((le (S z0) n) \to ((eq nat (minus (S x0) (S -z0)) (minus n (S z0))) \to (eq nat (S x0) n))))) (\lambda (H: (le (S z0) (S -x0))).(\lambda (H0: (le (S z0) O)).(\lambda (_: (eq nat (minus (S x0) (S z0)) -(minus O (S z0)))).(let H_y \def (le_S_n z0 x0 H) in (ex2_ind nat (\lambda -(n: nat).(eq nat O (S n))) (\lambda (n: nat).(le z0 n)) (eq nat (S x0) O) -(\lambda (x1: nat).(\lambda (H2: (eq nat O (S x1))).(\lambda (_: (le z0 -x1)).(let H4 \def (eq_ind nat O (\lambda (ee: nat).(match ee in nat return -(\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) -I (S x1) H2) in (False_ind (eq nat (S x0) O) H4))))) (le_gen_S z0 O H0)))))) -(\lambda (y0: nat).(\lambda (_: (((le (S z0) (S x0)) \to ((le (S z0) y0) \to -((eq nat (minus (S x0) (S z0)) (minus y0 (S z0))) \to (eq nat (S x0) -y0)))))).(\lambda (H: (le (S z0) (S x0))).(\lambda (H0: (le (S z0) (S -y0))).(\lambda (H1: (eq nat (minus (S x0) (S z0)) (minus (S y0) (S -z0)))).(f_equal nat nat S x0 y0 (IH x0 y0 (le_S_n z0 x0 H) (le_S_n z0 y0 H0) -H1))))))) y)))) x)))) z). -(* COMMENTS -Initial nodes: 751 -END *) - -theorem plus_plus: - \forall (z: nat).(\forall (x1: nat).(\forall (x2: nat).(\forall (y1: -nat).(\forall (y2: nat).((le x1 z) \to ((le x2 z) \to ((eq nat (plus (minus z -x1) y1) (plus (minus z x2) y2)) \to (eq nat (plus x1 y2) (plus x2 y1))))))))) -\def - \lambda (z: nat).(nat_ind (\lambda (n: nat).(\forall (x1: nat).(\forall (x2: -nat).(\forall (y1: nat).(\forall (y2: nat).((le x1 n) \to ((le x2 n) \to ((eq -nat (plus (minus n x1) y1) (plus (minus n x2) y2)) \to (eq nat (plus x1 y2) -(plus x2 y1)))))))))) (\lambda (x1: nat).(\lambda (x2: nat).(\lambda (y1: -nat).(\lambda (y2: nat).(\lambda (H: (le x1 O)).(\lambda (H0: (le x2 -O)).(\lambda (H1: (eq nat y1 y2)).(eq_ind nat y1 (\lambda (n: nat).(eq nat -(plus x1 n) (plus x2 y1))) (let H_y \def (le_n_O_eq x2 H0) in (eq_ind nat O -(\lambda (n: nat).(eq nat (plus x1 y1) (plus n y1))) (let H_y0 \def -(le_n_O_eq x1 H) in (eq_ind nat O (\lambda (n: nat).(eq nat (plus n y1) (plus -O y1))) (refl_equal nat (plus O y1)) x1 H_y0)) x2 H_y)) y2 H1)))))))) -(\lambda (z0: nat).(\lambda (IH: ((\forall (x1: nat).(\forall (x2: -nat).(\forall (y1: nat).(\forall (y2: nat).((le x1 z0) \to ((le x2 z0) \to -((eq nat (plus (minus z0 x1) y1) (plus (minus z0 x2) y2)) \to (eq nat (plus -x1 y2) (plus x2 y1))))))))))).(\lambda (x1: nat).(nat_ind (\lambda (n: -nat).(\forall (x2: nat).(\forall (y1: nat).(\forall (y2: nat).((le n (S z0)) -\to ((le x2 (S z0)) \to ((eq nat (plus (minus (S z0) n) y1) (plus (minus (S -z0) x2) y2)) \to (eq nat (plus n y2) (plus x2 y1))))))))) (\lambda (x2: -nat).(nat_ind (\lambda (n: nat).(\forall (y1: nat).(\forall (y2: nat).((le O -(S z0)) \to ((le n (S z0)) \to ((eq nat (plus (minus (S z0) O) y1) (plus -(minus (S z0) n) y2)) \to (eq nat (plus O y2) (plus n y1)))))))) (\lambda -(y1: nat).(\lambda (y2: nat).(\lambda (_: (le O (S z0))).(\lambda (_: (le O -(S z0))).(\lambda (H1: (eq nat (S (plus z0 y1)) (S (plus z0 y2)))).(let H_y -\def (IH O O) in (let H2 \def (eq_ind_r nat (minus z0 O) (\lambda (n: -nat).(\forall (y3: nat).(\forall (y4: nat).((le O z0) \to ((le O z0) \to ((eq -nat (plus n y3) (plus n y4)) \to (eq nat y4 y3))))))) H_y z0 (minus_n_O z0)) -in (H2 y1 y2 (le_O_n z0) (le_O_n z0) (eq_add_S (plus z0 y1) (plus z0 y2) -H1))))))))) (\lambda (x3: nat).(\lambda (_: ((\forall (y1: nat).(\forall (y2: -nat).((le O (S z0)) \to ((le x3 (S z0)) \to ((eq nat (S (plus z0 y1)) (plus -(match x3 with [O \Rightarrow (S z0) | (S l) \Rightarrow (minus z0 l)]) y2)) -\to (eq nat y2 (plus x3 y1))))))))).(\lambda (y1: nat).(\lambda (y2: -nat).(\lambda (_: (le O (S z0))).(\lambda (H0: (le (S x3) (S z0))).(\lambda -(H1: (eq nat (S (plus z0 y1)) (plus (minus z0 x3) y2))).(let H_y \def (IH O -x3 (S y1)) in (let H2 \def (eq_ind_r nat (minus z0 O) (\lambda (n: -nat).(\forall (y3: nat).((le O z0) \to ((le x3 z0) \to ((eq nat (plus n (S -y1)) (plus (minus z0 x3) y3)) \to (eq nat y3 (plus x3 (S y1)))))))) H_y z0 -(minus_n_O z0)) in (let H3 \def (eq_ind_r nat (plus z0 (S y1)) (\lambda (n: -nat).(\forall (y3: nat).((le O z0) \to ((le x3 z0) \to ((eq nat n (plus -(minus z0 x3) y3)) \to (eq nat y3 (plus x3 (S y1)))))))) H2 (S (plus z0 y1)) -(plus_n_Sm z0 y1)) in (let H4 \def (eq_ind_r nat (plus x3 (S y1)) (\lambda -(n: nat).(\forall (y3: nat).((le O z0) \to ((le x3 z0) \to ((eq nat (S (plus -z0 y1)) (plus (minus z0 x3) y3)) \to (eq nat y3 n)))))) H3 (S (plus x3 y1)) -(plus_n_Sm x3 y1)) in (H4 y2 (le_O_n z0) (le_S_n x3 z0 H0) H1)))))))))))) -x2)) (\lambda (x2: nat).(\lambda (_: ((\forall (x3: nat).(\forall (y1: -nat).(\forall (y2: nat).((le x2 (S z0)) \to ((le x3 (S z0)) \to ((eq nat -(plus (minus (S z0) x2) y1) (plus (minus (S z0) x3) y2)) \to (eq nat (plus x2 -y2) (plus x3 y1)))))))))).(\lambda (x3: nat).(nat_ind (\lambda (n: -nat).(\forall (y1: nat).(\forall (y2: nat).((le (S x2) (S z0)) \to ((le n (S -z0)) \to ((eq nat (plus (minus (S z0) (S x2)) y1) (plus (minus (S z0) n) y2)) -\to (eq nat (plus (S x2) y2) (plus n y1)))))))) (\lambda (y1: nat).(\lambda -(y2: nat).(\lambda (H: (le (S x2) (S z0))).(\lambda (_: (le O (S -z0))).(\lambda (H1: (eq nat (plus (minus z0 x2) y1) (S (plus z0 y2)))).(let -H_y \def (IH x2 O y1 (S y2)) in (let H2 \def (eq_ind_r nat (minus z0 O) -(\lambda (n: nat).((le x2 z0) \to ((le O z0) \to ((eq nat (plus (minus z0 x2) -y1) (plus n (S y2))) \to (eq nat (plus x2 (S y2)) y1))))) H_y z0 (minus_n_O -z0)) in (let H3 \def (eq_ind_r nat (plus z0 (S y2)) (\lambda (n: nat).((le x2 -z0) \to ((le O z0) \to ((eq nat (plus (minus z0 x2) y1) n) \to (eq nat (plus -x2 (S y2)) y1))))) H2 (S (plus z0 y2)) (plus_n_Sm z0 y2)) in (let H4 \def -(eq_ind_r nat (plus x2 (S y2)) (\lambda (n: nat).((le x2 z0) \to ((le O z0) -\to ((eq nat (plus (minus z0 x2) y1) (S (plus z0 y2))) \to (eq nat n y1))))) -H3 (S (plus x2 y2)) (plus_n_Sm x2 y2)) in (H4 (le_S_n x2 z0 H) (le_O_n z0) -H1)))))))))) (\lambda (x4: nat).(\lambda (_: ((\forall (y1: nat).(\forall -(y2: nat).((le (S x2) (S z0)) \to ((le x4 (S z0)) \to ((eq nat (plus (minus -z0 x2) y1) (plus (match x4 with [O \Rightarrow (S z0) | (S l) \Rightarrow -(minus z0 l)]) y2)) \to (eq nat (S (plus x2 y2)) (plus x4 -y1))))))))).(\lambda (y1: nat).(\lambda (y2: nat).(\lambda (H: (le (S x2) (S -z0))).(\lambda (H0: (le (S x4) (S z0))).(\lambda (H1: (eq nat (plus (minus z0 -x2) y1) (plus (minus z0 x4) y2))).(f_equal nat nat S (plus x2 y2) (plus x4 -y1) (IH x2 x4 y1 y2 (le_S_n x2 z0 H) (le_S_n x4 z0 H0) H1))))))))) x3)))) -x1)))) z). -(* COMMENTS -Initial nodes: 1495 -END *) - -theorem le_S_minus: - \forall (d: nat).(\forall (h: nat).(\forall (n: nat).((le (plus d h) n) \to -(le d (S (minus n h)))))) -\def - \lambda (d: nat).(\lambda (h: nat).(\lambda (n: nat).(\lambda (H: (le (plus -d h) n)).(let H0 \def (le_trans d (plus d h) n (le_plus_l d h) H) in (let H1 -\def (eq_ind nat n (\lambda (n0: nat).(le d n0)) H0 (plus (minus n h) h) -(le_plus_minus_sym h n (le_trans h (plus d h) n (le_plus_r d h) H))) in (le_S -d (minus n h) (le_minus d n h H))))))). -(* COMMENTS -Initial nodes: 107 -END *) - -theorem lt_x_pred_y: - \forall (x: nat).(\forall (y: nat).((lt x (pred y)) \to (lt (S x) y))) -\def - \lambda (x: nat).(\lambda (y: nat).(nat_ind (\lambda (n: nat).((lt x (pred -n)) \to (lt (S x) n))) (\lambda (H: (lt x O)).(lt_x_O x H (lt (S x) O))) -(\lambda (n: nat).(\lambda (_: (((lt x (pred n)) \to (lt (S x) n)))).(\lambda -(H0: (lt x n)).(le_S_n (S (S x)) (S n) (le_n_S (S (S x)) (S n) (le_n_S (S x) -n H0)))))) y)). -(* COMMENTS -Initial nodes: 103 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Ground-1/ext/tactics.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Ground-1/ext/tactics.ma deleted file mode 100644 index 766db9f1f..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Ground-1/ext/tactics.ma +++ /dev/null @@ -1,50 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Ground-1/preamble.ma". - -theorem insert_eq: - \forall (S: Set).(\forall (x: S).(\forall (P: ((S \to Prop))).(\forall (G: -((S \to Prop))).(((\forall (y: S).((P y) \to ((eq S y x) \to (G y))))) \to -((P x) \to (G x)))))) -\def - \lambda (S: Set).(\lambda (x: S).(\lambda (P: ((S \to Prop))).(\lambda (G: -((S \to Prop))).(\lambda (H: ((\forall (y: S).((P y) \to ((eq S y x) \to (G -y)))))).(\lambda (H0: (P x)).(H x H0 (refl_equal S x))))))). -(* COMMENTS -Initial nodes: 45 -END *) - -theorem unintro: - \forall (A: Set).(\forall (a: A).(\forall (P: ((A \to Prop))).(((\forall (x: -A).(P x))) \to (P a)))) -\def - \lambda (A: Set).(\lambda (a: A).(\lambda (P: ((A \to Prop))).(\lambda (H: -((\forall (x: A).(P x)))).(H a)))). -(* COMMENTS -Initial nodes: 17 -END *) - -theorem xinduction: - \forall (A: Set).(\forall (t: A).(\forall (P: ((A \to Prop))).(((\forall (x: -A).((eq A t x) \to (P x)))) \to (P t)))) -\def - \lambda (A: Set).(\lambda (t: A).(\lambda (P: ((A \to Prop))).(\lambda (H: -((\forall (x: A).((eq A t x) \to (P x))))).(H t (refl_equal A t))))). -(* COMMENTS -Initial nodes: 31 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Ground-1/plist/defs.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Ground-1/plist/defs.ma deleted file mode 100644 index 8f9c1d3cc..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Ground-1/plist/defs.ma +++ /dev/null @@ -1,43 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Ground-1/preamble.ma". - -inductive PList: Set \def -| PNil: PList -| PCons: nat \to (nat \to (PList \to PList)). - -definition PConsTail: - PList \to (nat \to (nat \to PList)) -\def - let rec PConsTail (hds: PList) on hds: (nat \to (nat \to PList)) \def -(\lambda (h0: nat).(\lambda (d0: nat).(match hds with [PNil \Rightarrow -(PCons h0 d0 PNil) | (PCons h d hds0) \Rightarrow (PCons h d (PConsTail hds0 -h0 d0))]))) in PConsTail. - -definition Ss: - PList \to PList -\def - let rec Ss (hds: PList) on hds: PList \def (match hds with [PNil \Rightarrow -PNil | (PCons h d hds0) \Rightarrow (PCons h (S d) (Ss hds0))]) in Ss. - -definition papp: - PList \to (PList \to PList) -\def - let rec papp (a: PList) on a: (PList \to PList) \def (\lambda (b: -PList).(match a with [PNil \Rightarrow b | (PCons h d a0) \Rightarrow (PCons -h d (papp a0 b))])) in papp. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Ground-1/plist/props.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Ground-1/plist/props.ma deleted file mode 100644 index 990397229..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Ground-1/plist/props.ma +++ /dev/null @@ -1,34 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Ground-1/plist/defs.ma". - -theorem papp_ss: - \forall (is1: PList).(\forall (is2: PList).(eq PList (papp (Ss is1) (Ss -is2)) (Ss (papp is1 is2)))) -\def - \lambda (is1: PList).(PList_ind (\lambda (p: PList).(\forall (is2: -PList).(eq PList (papp (Ss p) (Ss is2)) (Ss (papp p is2))))) (\lambda (is2: -PList).(refl_equal PList (Ss is2))) (\lambda (n: nat).(\lambda (n0: -nat).(\lambda (p: PList).(\lambda (H: ((\forall (is2: PList).(eq PList (papp -(Ss p) (Ss is2)) (Ss (papp p is2)))))).(\lambda (is2: PList).(eq_ind_r PList -(Ss (papp p is2)) (\lambda (p0: PList).(eq PList (PCons n (S n0) p0) (PCons n -(S n0) (Ss (papp p is2))))) (refl_equal PList (PCons n (S n0) (Ss (papp p -is2)))) (papp (Ss p) (Ss is2)) (H is2))))))) is1). -(* COMMENTS -Initial nodes: 151 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Ground-1/preamble.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Ground-1/preamble.ma deleted file mode 100644 index 16ff2dc44..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Ground-1/preamble.ma +++ /dev/null @@ -1,15 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "Legacy-1/theory.ma". diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Ground-1/spare.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Ground-1/spare.ma deleted file mode 100644 index a966c2ae2..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Ground-1/spare.ma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Ground-1/theory.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Ground-1/theory.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Ground-1/theory.ma deleted file mode 100644 index 3b59dc6e8..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Ground-1/theory.ma +++ /dev/null @@ -1,26 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Ground-1/ext/tactics.ma". - -include "Ground-1/ext/arith.ma". - -include "Ground-1/types/props.ma". - -include "Ground-1/blt/props.ma". - -include "Ground-1/plist/props.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Ground-1/types/defs.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Ground-1/types/defs.ma deleted file mode 100644 index f94969d88..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Ground-1/types/defs.ma +++ /dev/null @@ -1,172 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Ground-1/preamble.ma". - -inductive and3 (P0: Prop) (P1: Prop) (P2: Prop): Prop \def -| and3_intro: P0 \to (P1 \to (P2 \to (and3 P0 P1 P2))). - -inductive and4 (P0: Prop) (P1: Prop) (P2: Prop) (P3: Prop): Prop \def -| and4_intro: P0 \to (P1 \to (P2 \to (P3 \to (and4 P0 P1 P2 P3)))). - -inductive and5 (P0: Prop) (P1: Prop) (P2: Prop) (P3: Prop) (P4: Prop): Prop -\def -| and5_intro: P0 \to (P1 \to (P2 \to (P3 \to (P4 \to (and5 P0 P1 P2 P3 -P4))))). - -inductive or3 (P0: Prop) (P1: Prop) (P2: Prop): Prop \def -| or3_intro0: P0 \to (or3 P0 P1 P2) -| or3_intro1: P1 \to (or3 P0 P1 P2) -| or3_intro2: P2 \to (or3 P0 P1 P2). - -inductive or4 (P0: Prop) (P1: Prop) (P2: Prop) (P3: Prop): Prop \def -| or4_intro0: P0 \to (or4 P0 P1 P2 P3) -| or4_intro1: P1 \to (or4 P0 P1 P2 P3) -| or4_intro2: P2 \to (or4 P0 P1 P2 P3) -| or4_intro3: P3 \to (or4 P0 P1 P2 P3). - -inductive or5 (P0: Prop) (P1: Prop) (P2: Prop) (P3: Prop) (P4: Prop): Prop -\def -| or5_intro0: P0 \to (or5 P0 P1 P2 P3 P4) -| or5_intro1: P1 \to (or5 P0 P1 P2 P3 P4) -| or5_intro2: P2 \to (or5 P0 P1 P2 P3 P4) -| or5_intro3: P3 \to (or5 P0 P1 P2 P3 P4) -| or5_intro4: P4 \to (or5 P0 P1 P2 P3 P4). - -inductive ex3 (A0: Set) (P0: A0 \to Prop) (P1: A0 \to Prop) (P2: A0 \to -Prop): Prop \def -| ex3_intro: \forall (x0: A0).((P0 x0) \to ((P1 x0) \to ((P2 x0) \to (ex3 A0 -P0 P1 P2)))). - -inductive ex4 (A0: Set) (P0: A0 \to Prop) (P1: A0 \to Prop) (P2: A0 \to Prop) -(P3: A0 \to Prop): Prop \def -| ex4_intro: \forall (x0: A0).((P0 x0) \to ((P1 x0) \to ((P2 x0) \to ((P3 x0) -\to (ex4 A0 P0 P1 P2 P3))))). - -inductive ex_2 (A0: Set) (A1: Set) (P0: A0 \to (A1 \to Prop)): Prop \def -| ex_2_intro: \forall (x0: A0).(\forall (x1: A1).((P0 x0 x1) \to (ex_2 A0 A1 -P0))). - -inductive ex2_2 (A0: Set) (A1: Set) (P0: A0 \to (A1 \to Prop)) (P1: A0 \to -(A1 \to Prop)): Prop \def -| ex2_2_intro: \forall (x0: A0).(\forall (x1: A1).((P0 x0 x1) \to ((P1 x0 x1) -\to (ex2_2 A0 A1 P0 P1)))). - -inductive ex3_2 (A0: Set) (A1: Set) (P0: A0 \to (A1 \to Prop)) (P1: A0 \to -(A1 \to Prop)) (P2: A0 \to (A1 \to Prop)): Prop \def -| ex3_2_intro: \forall (x0: A0).(\forall (x1: A1).((P0 x0 x1) \to ((P1 x0 x1) -\to ((P2 x0 x1) \to (ex3_2 A0 A1 P0 P1 P2))))). - -inductive ex4_2 (A0: Set) (A1: Set) (P0: A0 \to (A1 \to Prop)) (P1: A0 \to -(A1 \to Prop)) (P2: A0 \to (A1 \to Prop)) (P3: A0 \to (A1 \to Prop)): Prop -\def -| ex4_2_intro: \forall (x0: A0).(\forall (x1: A1).((P0 x0 x1) \to ((P1 x0 x1) -\to ((P2 x0 x1) \to ((P3 x0 x1) \to (ex4_2 A0 A1 P0 P1 P2 P3)))))). - -inductive ex_3 (A0: Set) (A1: Set) (A2: Set) (P0: A0 \to (A1 \to (A2 \to -Prop))): Prop \def -| ex_3_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).((P0 x0 x1 -x2) \to (ex_3 A0 A1 A2 P0)))). - -inductive ex2_3 (A0: Set) (A1: Set) (A2: Set) (P0: A0 \to (A1 \to (A2 \to -Prop))) (P1: A0 \to (A1 \to (A2 \to Prop))): Prop \def -| ex2_3_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).((P0 x0 -x1 x2) \to ((P1 x0 x1 x2) \to (ex2_3 A0 A1 A2 P0 P1))))). - -inductive ex3_3 (A0: Set) (A1: Set) (A2: Set) (P0: A0 \to (A1 \to (A2 \to -Prop))) (P1: A0 \to (A1 \to (A2 \to Prop))) (P2: A0 \to (A1 \to (A2 \to -Prop))): Prop \def -| ex3_3_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).((P0 x0 -x1 x2) \to ((P1 x0 x1 x2) \to ((P2 x0 x1 x2) \to (ex3_3 A0 A1 A2 P0 P1 -P2)))))). - -inductive ex4_3 (A0: Set) (A1: Set) (A2: Set) (P0: A0 \to (A1 \to (A2 \to -Prop))) (P1: A0 \to (A1 \to (A2 \to Prop))) (P2: A0 \to (A1 \to (A2 \to -Prop))) (P3: A0 \to (A1 \to (A2 \to Prop))): Prop \def -| ex4_3_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).((P0 x0 -x1 x2) \to ((P1 x0 x1 x2) \to ((P2 x0 x1 x2) \to ((P3 x0 x1 x2) \to (ex4_3 A0 -A1 A2 P0 P1 P2 P3))))))). - -inductive ex5_3 (A0: Set) (A1: Set) (A2: Set) (P0: A0 \to (A1 \to (A2 \to -Prop))) (P1: A0 \to (A1 \to (A2 \to Prop))) (P2: A0 \to (A1 \to (A2 \to -Prop))) (P3: A0 \to (A1 \to (A2 \to Prop))) (P4: A0 \to (A1 \to (A2 \to -Prop))): Prop \def -| ex5_3_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).((P0 x0 -x1 x2) \to ((P1 x0 x1 x2) \to ((P2 x0 x1 x2) \to ((P3 x0 x1 x2) \to ((P4 x0 -x1 x2) \to (ex5_3 A0 A1 A2 P0 P1 P2 P3 P4)))))))). - -inductive ex3_4 (A0: Set) (A1: Set) (A2: Set) (A3: Set) (P0: A0 \to (A1 \to -(A2 \to (A3 \to Prop)))) (P1: A0 \to (A1 \to (A2 \to (A3 \to Prop)))) (P2: A0 -\to (A1 \to (A2 \to (A3 \to Prop)))): Prop \def -| ex3_4_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall -(x3: A3).((P0 x0 x1 x2 x3) \to ((P1 x0 x1 x2 x3) \to ((P2 x0 x1 x2 x3) \to -(ex3_4 A0 A1 A2 A3 P0 P1 P2))))))). - -inductive ex4_4 (A0: Set) (A1: Set) (A2: Set) (A3: Set) (P0: A0 \to (A1 \to -(A2 \to (A3 \to Prop)))) (P1: A0 \to (A1 \to (A2 \to (A3 \to Prop)))) (P2: A0 -\to (A1 \to (A2 \to (A3 \to Prop)))) (P3: A0 \to (A1 \to (A2 \to (A3 \to -Prop)))): Prop \def -| ex4_4_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall -(x3: A3).((P0 x0 x1 x2 x3) \to ((P1 x0 x1 x2 x3) \to ((P2 x0 x1 x2 x3) \to -((P3 x0 x1 x2 x3) \to (ex4_4 A0 A1 A2 A3 P0 P1 P2 P3)))))))). - -inductive ex4_5 (A0: Set) (A1: Set) (A2: Set) (A3: Set) (A4: Set) (P0: A0 \to -(A1 \to (A2 \to (A3 \to (A4 \to Prop))))) (P1: A0 \to (A1 \to (A2 \to (A3 \to -(A4 \to Prop))))) (P2: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to Prop))))) (P3: -A0 \to (A1 \to (A2 \to (A3 \to (A4 \to Prop))))): Prop \def -| ex4_5_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall -(x3: A3).(\forall (x4: A4).((P0 x0 x1 x2 x3 x4) \to ((P1 x0 x1 x2 x3 x4) \to -((P2 x0 x1 x2 x3 x4) \to ((P3 x0 x1 x2 x3 x4) \to (ex4_5 A0 A1 A2 A3 A4 P0 P1 -P2 P3))))))))). - -inductive ex5_5 (A0: Set) (A1: Set) (A2: Set) (A3: Set) (A4: Set) (P0: A0 \to -(A1 \to (A2 \to (A3 \to (A4 \to Prop))))) (P1: A0 \to (A1 \to (A2 \to (A3 \to -(A4 \to Prop))))) (P2: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to Prop))))) (P3: -A0 \to (A1 \to (A2 \to (A3 \to (A4 \to Prop))))) (P4: A0 \to (A1 \to (A2 \to -(A3 \to (A4 \to Prop))))): Prop \def -| ex5_5_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall -(x3: A3).(\forall (x4: A4).((P0 x0 x1 x2 x3 x4) \to ((P1 x0 x1 x2 x3 x4) \to -((P2 x0 x1 x2 x3 x4) \to ((P3 x0 x1 x2 x3 x4) \to ((P4 x0 x1 x2 x3 x4) \to -(ex5_5 A0 A1 A2 A3 A4 P0 P1 P2 P3 P4)))))))))). - -inductive ex6_6 (A0: Set) (A1: Set) (A2: Set) (A3: Set) (A4: Set) (A5: Set) -(P0: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to Prop)))))) (P1: A0 \to -(A1 \to (A2 \to (A3 \to (A4 \to (A5 \to Prop)))))) (P2: A0 \to (A1 \to (A2 -\to (A3 \to (A4 \to (A5 \to Prop)))))) (P3: A0 \to (A1 \to (A2 \to (A3 \to -(A4 \to (A5 \to Prop)))))) (P4: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 -\to Prop)))))) (P5: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to -Prop)))))): Prop \def -| ex6_6_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall -(x3: A3).(\forall (x4: A4).(\forall (x5: A5).((P0 x0 x1 x2 x3 x4 x5) \to ((P1 -x0 x1 x2 x3 x4 x5) \to ((P2 x0 x1 x2 x3 x4 x5) \to ((P3 x0 x1 x2 x3 x4 x5) -\to ((P4 x0 x1 x2 x3 x4 x5) \to ((P5 x0 x1 x2 x3 x4 x5) \to (ex6_6 A0 A1 A2 -A3 A4 A5 P0 P1 P2 P3 P4 P5)))))))))))). - -inductive ex6_7 (A0: Set) (A1: Set) (A2: Set) (A3: Set) (A4: Set) (A5: Set) -(A6: Set) (P0: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 \to -Prop))))))) (P1: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 \to -Prop))))))) (P2: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 \to -Prop))))))) (P3: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 \to -Prop))))))) (P4: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 \to -Prop))))))) (P5: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 \to -Prop))))))): Prop \def -| ex6_7_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall -(x3: A3).(\forall (x4: A4).(\forall (x5: A5).(\forall (x6: A6).((P0 x0 x1 x2 -x3 x4 x5 x6) \to ((P1 x0 x1 x2 x3 x4 x5 x6) \to ((P2 x0 x1 x2 x3 x4 x5 x6) -\to ((P3 x0 x1 x2 x3 x4 x5 x6) \to ((P4 x0 x1 x2 x3 x4 x5 x6) \to ((P5 x0 x1 -x2 x3 x4 x5 x6) \to (ex6_7 A0 A1 A2 A3 A4 A5 A6 P0 P1 P2 P3 P4 -P5))))))))))))). - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Ground-1/types/props.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Ground-1/types/props.ma deleted file mode 100644 index 9c326f44c..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Ground-1/types/props.ma +++ /dev/null @@ -1,33 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Ground-1/types/defs.ma". - -theorem ex2_sym: - \forall (A: Set).(\forall (P: ((A \to Prop))).(\forall (Q: ((A \to -Prop))).((ex2 A (\lambda (x: A).(P x)) (\lambda (x: A).(Q x))) \to (ex2 A -(\lambda (x: A).(Q x)) (\lambda (x: A).(P x)))))) -\def - \lambda (A: Set).(\lambda (P: ((A \to Prop))).(\lambda (Q: ((A \to -Prop))).(\lambda (H: (ex2 A (\lambda (x: A).(P x)) (\lambda (x: A).(Q -x)))).(ex2_ind A (\lambda (x: A).(P x)) (\lambda (x: A).(Q x)) (ex2 A -(\lambda (x: A).(Q x)) (\lambda (x: A).(P x))) (\lambda (x: A).(\lambda (H0: -(P x)).(\lambda (H1: (Q x)).(ex_intro2 A (\lambda (x0: A).(Q x0)) (\lambda -(x0: A).(P x0)) x H1 H0)))) H)))). -(* COMMENTS -Initial nodes: 91 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Ground-2/blt/defs.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Ground-2/blt/defs.mma deleted file mode 100644 index 028f98a42..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Ground-2/blt/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Ground-2/preamble.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Ground-2/blt/props.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Ground-2/blt/props.mma deleted file mode 100644 index 24fd85f96..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Ground-2/blt/props.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Ground-2/blt/defs.ma". - -inline "Ground-1/blt/props.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Ground-2/ext/arith.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Ground-2/ext/arith.mma deleted file mode 100644 index ab8a4c76e..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Ground-2/ext/arith.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Ground-2/preamble.ma". - -inline "Ground-1/ext/arith.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Ground-2/ext/tactics.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Ground-2/ext/tactics.mma deleted file mode 100644 index 72cacebfc..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Ground-2/ext/tactics.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Ground-2/preamble.ma". - -inline "Ground-1/ext/tactics.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Ground-2/plist/defs.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Ground-2/plist/defs.mma deleted file mode 100644 index 028f98a42..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Ground-2/plist/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Ground-2/preamble.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Ground-2/plist/props.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Ground-2/plist/props.mma deleted file mode 100644 index 196cf33a7..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Ground-2/plist/props.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Ground-2/plist/defs.ma". - -inline "Ground-1/plist/props.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Ground-2/preamble.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Ground-2/preamble.ma deleted file mode 100644 index ffe3dce3a..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Ground-2/preamble.ma +++ /dev/null @@ -1,16 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "Ground-1/definitions.ma". -include "Legacy-2/theory.ma". diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Ground-2/theory.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Ground-2/theory.ma deleted file mode 100644 index 25de9e4c3..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Ground-2/theory.ma +++ /dev/null @@ -1,26 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Ground-2/ext/tactics.ma". - -include "Ground-2/ext/arith.ma". - -include "Ground-2/types/props.ma". - -include "Ground-2/blt/props.ma". - -include "Ground-2/plist/props.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Ground-2/types/defs.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Ground-2/types/defs.mma deleted file mode 100644 index 028f98a42..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Ground-2/types/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Ground-2/preamble.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Ground-2/types/props.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Ground-2/types/props.mma deleted file mode 100644 index 49444e73b..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Ground-2/types/props.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Ground-2/types/defs.ma". - -inline "Ground-1/types/props.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Legacy-1/coq/defs.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Legacy-1/coq/defs.ma deleted file mode 100644 index 7d4696229..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Legacy-1/coq/defs.ma +++ /dev/null @@ -1,99 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Legacy-1/preamble.ma". - -inductive eq (A: Set) (x: A): A \to Prop \def -| refl_equal: eq A x x. - -inductive True: Prop \def -| I: True. - -inductive land (A: Prop) (B: Prop): Prop \def -| conj: A \to (B \to (land A B)). - -inductive or (A: Prop) (B: Prop): Prop \def -| or_introl: A \to (or A B) -| or_intror: B \to (or A B). - -inductive ex (A: Set) (P: A \to Prop): Prop \def -| ex_intro: \forall (x: A).((P x) \to (ex A P)). - -inductive ex2 (A: Set) (P: A \to Prop) (Q: A \to Prop): Prop \def -| ex_intro2: \forall (x: A).((P x) \to ((Q x) \to (ex2 A P Q))). - -definition not: - Prop \to Prop -\def - \lambda (A: Prop).(A \to False). - -inductive bool: Set \def -| true: bool -| false: bool. - -inductive nat: Set \def -| O: nat -| S: nat \to nat. - -inductive le (n: nat): nat \to Prop \def -| le_n: le n n -| le_S: \forall (m: nat).((le n m) \to (le n (S m))). - -definition lt: - nat \to (nat \to Prop) -\def - \lambda (n: nat).(\lambda (m: nat).(le (S n) m)). - -definition IsSucc: - nat \to Prop -\def - \lambda (n: nat).(match n with [O \Rightarrow False | (S _) \Rightarrow -True]). - -definition pred: - nat \to nat -\def - \lambda (n: nat).(match n with [O \Rightarrow O | (S u) \Rightarrow u]). - -definition plus: - nat \to (nat \to nat) -\def - let rec plus (n: nat) on n: (nat \to nat) \def (\lambda (m: nat).(match n -with [O \Rightarrow m | (S p) \Rightarrow (S (plus p m))])) in plus. - -definition minus: - nat \to (nat \to nat) -\def - let rec minus (n: nat) on n: (nat \to nat) \def (\lambda (m: nat).(match n -with [O \Rightarrow O | (S k) \Rightarrow (match m with [O \Rightarrow (S k) -| (S l) \Rightarrow (minus k l)])])) in minus. - -inductive Acc (A: Set) (R: A \to (A \to Prop)): A \to Prop \def -| Acc_intro: \forall (x: A).(((\forall (y: A).((R y x) \to (Acc A R y)))) \to -(Acc A R x)). - -definition well_founded: - \forall (A: Set).(((A \to (A \to Prop))) \to Prop) -\def - \lambda (A: Set).(\lambda (R: ((A \to (A \to Prop)))).(\forall (a: A).(Acc A -R a))). - -definition ltof: - \forall (A: Set).(((A \to nat)) \to (A \to (A \to Prop))) -\def - \lambda (A: Set).(\lambda (f: ((A \to nat))).(\lambda (a: A).(\lambda (b: -A).(lt (f a) (f b))))). - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Legacy-1/coq/props.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Legacy-1/coq/props.ma deleted file mode 100644 index 0b9d97b42..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Legacy-1/coq/props.ma +++ /dev/null @@ -1,805 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Legacy-1/coq/defs.ma". - -theorem f_equal: - \forall (A: Set).(\forall (B: Set).(\forall (f: ((A \to B))).(\forall (x: -A).(\forall (y: A).((eq A x y) \to (eq B (f x) (f y))))))) -\def - \lambda (A: Set).(\lambda (B: Set).(\lambda (f: ((A \to B))).(\lambda (x: -A).(\lambda (y: A).(\lambda (H: (eq A x y)).(eq_ind A x (\lambda (a: A).(eq B -(f x) (f a))) (refl_equal B (f x)) y H)))))). -(* COMMENTS -Initial nodes: 51 -END *) - -theorem f_equal2: - \forall (A1: Set).(\forall (A2: Set).(\forall (B: Set).(\forall (f: ((A1 \to -(A2 \to B)))).(\forall (x1: A1).(\forall (y1: A1).(\forall (x2: A2).(\forall -(y2: A2).((eq A1 x1 y1) \to ((eq A2 x2 y2) \to (eq B (f x1 x2) (f y1 -y2))))))))))) -\def - \lambda (A1: Set).(\lambda (A2: Set).(\lambda (B: Set).(\lambda (f: ((A1 \to -(A2 \to B)))).(\lambda (x1: A1).(\lambda (y1: A1).(\lambda (x2: A2).(\lambda -(y2: A2).(\lambda (H: (eq A1 x1 y1)).(eq_ind A1 x1 (\lambda (a: A1).((eq A2 -x2 y2) \to (eq B (f x1 x2) (f a y2)))) (\lambda (H0: (eq A2 x2 y2)).(eq_ind -A2 x2 (\lambda (a: A2).(eq B (f x1 x2) (f x1 a))) (refl_equal B (f x1 x2)) y2 -H0)) y1 H))))))))). -(* COMMENTS -Initial nodes: 109 -END *) - -theorem f_equal3: - \forall (A1: Set).(\forall (A2: Set).(\forall (A3: Set).(\forall (B: -Set).(\forall (f: ((A1 \to (A2 \to (A3 \to B))))).(\forall (x1: A1).(\forall -(y1: A1).(\forall (x2: A2).(\forall (y2: A2).(\forall (x3: A3).(\forall (y3: -A3).((eq A1 x1 y1) \to ((eq A2 x2 y2) \to ((eq A3 x3 y3) \to (eq B (f x1 x2 -x3) (f y1 y2 y3))))))))))))))) -\def - \lambda (A1: Set).(\lambda (A2: Set).(\lambda (A3: Set).(\lambda (B: -Set).(\lambda (f: ((A1 \to (A2 \to (A3 \to B))))).(\lambda (x1: A1).(\lambda -(y1: A1).(\lambda (x2: A2).(\lambda (y2: A2).(\lambda (x3: A3).(\lambda (y3: -A3).(\lambda (H: (eq A1 x1 y1)).(eq_ind A1 x1 (\lambda (a: A1).((eq A2 x2 y2) -\to ((eq A3 x3 y3) \to (eq B (f x1 x2 x3) (f a y2 y3))))) (\lambda (H0: (eq -A2 x2 y2)).(eq_ind A2 x2 (\lambda (a: A2).((eq A3 x3 y3) \to (eq B (f x1 x2 -x3) (f x1 a y3)))) (\lambda (H1: (eq A3 x3 y3)).(eq_ind A3 x3 (\lambda (a: -A3).(eq B (f x1 x2 x3) (f x1 x2 a))) (refl_equal B (f x1 x2 x3)) y3 H1)) y2 -H0)) y1 H)))))))))))). -(* COMMENTS -Initial nodes: 183 -END *) - -theorem sym_eq: - \forall (A: Set).(\forall (x: A).(\forall (y: A).((eq A x y) \to (eq A y -x)))) -\def - \lambda (A: Set).(\lambda (x: A).(\lambda (y: A).(\lambda (H: (eq A x -y)).(eq_ind A x (\lambda (a: A).(eq A a x)) (refl_equal A x) y H)))). -(* COMMENTS -Initial nodes: 39 -END *) - -theorem eq_ind_r: - \forall (A: Set).(\forall (x: A).(\forall (P: ((A \to Prop))).((P x) \to -(\forall (y: A).((eq A y x) \to (P y)))))) -\def - \lambda (A: Set).(\lambda (x: A).(\lambda (P: ((A \to Prop))).(\lambda (H: -(P x)).(\lambda (y: A).(\lambda (H0: (eq A y x)).(match (sym_eq A y x H0) in -eq return (\lambda (a: A).(\lambda (_: (eq ? ? a)).(P a))) with [refl_equal -\Rightarrow H])))))). -(* COMMENTS -Initial nodes: 38 -END *) - -theorem trans_eq: - \forall (A: Set).(\forall (x: A).(\forall (y: A).(\forall (z: A).((eq A x y) -\to ((eq A y z) \to (eq A x z)))))) -\def - \lambda (A: Set).(\lambda (x: A).(\lambda (y: A).(\lambda (z: A).(\lambda -(H: (eq A x y)).(\lambda (H0: (eq A y z)).(eq_ind A y (\lambda (a: A).(eq A x -a)) H z H0)))))). -(* COMMENTS -Initial nodes: 45 -END *) - -theorem sym_not_eq: - \forall (A: Set).(\forall (x: A).(\forall (y: A).((not (eq A x y)) \to (not -(eq A y x))))) -\def - \lambda (A: Set).(\lambda (x: A).(\lambda (y: A).(\lambda (h1: (not (eq A x -y))).(\lambda (h2: (eq A y x)).(h1 (eq_ind A y (\lambda (a: A).(eq A a y)) -(refl_equal A y) x h2)))))). -(* COMMENTS -Initial nodes: 51 -END *) - -theorem nat_double_ind: - \forall (R: ((nat \to (nat \to Prop)))).(((\forall (n: nat).(R O n))) \to -(((\forall (n: nat).(R (S n) O))) \to (((\forall (n: nat).(\forall (m: -nat).((R n m) \to (R (S n) (S m)))))) \to (\forall (n: nat).(\forall (m: -nat).(R n m)))))) -\def - \lambda (R: ((nat \to (nat \to Prop)))).(\lambda (H: ((\forall (n: nat).(R O -n)))).(\lambda (H0: ((\forall (n: nat).(R (S n) O)))).(\lambda (H1: ((\forall -(n: nat).(\forall (m: nat).((R n m) \to (R (S n) (S m))))))).(\lambda (n: -nat).(nat_ind (\lambda (n0: nat).(\forall (m: nat).(R n0 m))) H (\lambda (n0: -nat).(\lambda (H2: ((\forall (m: nat).(R n0 m)))).(\lambda (m: nat).(nat_ind -(\lambda (n1: nat).(R (S n0) n1)) (H0 n0) (\lambda (n1: nat).(\lambda (_: (R -(S n0) n1)).(H1 n0 n1 (H2 n1)))) m)))) n))))). -(* COMMENTS -Initial nodes: 111 -END *) - -theorem eq_add_S: - \forall (n: nat).(\forall (m: nat).((eq nat (S n) (S m)) \to (eq nat n m))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (eq nat (S n) (S -m))).(f_equal nat nat pred (S n) (S m) H))). -(* COMMENTS -Initial nodes: 33 -END *) - -theorem O_S: - \forall (n: nat).(not (eq nat O (S n))) -\def - \lambda (n: nat).(\lambda (H: (eq nat O (S n))).(eq_ind nat (S n) (\lambda -(n0: nat).(IsSucc n0)) I O (sym_eq nat O (S n) H))). -(* COMMENTS -Initial nodes: 41 -END *) - -theorem not_eq_S: - \forall (n: nat).(\forall (m: nat).((not (eq nat n m)) \to (not (eq nat (S -n) (S m))))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (not (eq nat n m))).(\lambda -(H0: (eq nat (S n) (S m))).(H (eq_add_S n m H0))))). -(* COMMENTS -Initial nodes: 35 -END *) - -theorem pred_Sn: - \forall (m: nat).(eq nat m (pred (S m))) -\def - \lambda (m: nat).(refl_equal nat (pred (S m))). -(* COMMENTS -Initial nodes: 11 -END *) - -theorem S_pred: - \forall (n: nat).(\forall (m: nat).((lt m n) \to (eq nat n (S (pred n))))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt m n)).(le_ind (S m) -(\lambda (n0: nat).(eq nat n0 (S (pred n0)))) (refl_equal nat (S (pred (S -m)))) (\lambda (m0: nat).(\lambda (_: (le (S m) m0)).(\lambda (_: (eq nat m0 -(S (pred m0)))).(refl_equal nat (S (pred (S m0))))))) n H))). -(* COMMENTS -Initial nodes: 79 -END *) - -theorem le_trans: - \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((le n m) \to ((le m p) -\to (le n p))))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (le n -m)).(\lambda (H0: (le m p)).(le_ind m (\lambda (n0: nat).(le n n0)) H -(\lambda (m0: nat).(\lambda (_: (le m m0)).(\lambda (IHle: (le n m0)).(le_S n -m0 IHle)))) p H0))))). -(* COMMENTS -Initial nodes: 57 -END *) - -theorem le_trans_S: - \forall (n: nat).(\forall (m: nat).((le (S n) m) \to (le n m))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le (S n) m)).(le_trans n (S -n) m (le_S n n (le_n n)) H))). -(* COMMENTS -Initial nodes: 33 -END *) - -theorem le_n_S: - \forall (n: nat).(\forall (m: nat).((le n m) \to (le (S n) (S m)))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(le_ind n (\lambda -(n0: nat).(le (S n) (S n0))) (le_n (S n)) (\lambda (m0: nat).(\lambda (_: (le -n m0)).(\lambda (IHle: (le (S n) (S m0))).(le_S (S n) (S m0) IHle)))) m H))). -(* COMMENTS -Initial nodes: 65 -END *) - -theorem le_O_n: - \forall (n: nat).(le O n) -\def - \lambda (n: nat).(nat_ind (\lambda (n0: nat).(le O n0)) (le_n O) (\lambda -(n0: nat).(\lambda (IHn: (le O n0)).(le_S O n0 IHn))) n). -(* COMMENTS -Initial nodes: 33 -END *) - -theorem le_S_n: - \forall (n: nat).(\forall (m: nat).((le (S n) (S m)) \to (le n m))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le (S n) (S m))).(le_ind (S -n) (\lambda (n0: nat).(le (pred (S n)) (pred n0))) (le_n n) (\lambda (m0: -nat).(\lambda (H0: (le (S n) m0)).(\lambda (_: (le n (pred m0))).(le_trans_S -n m0 H0)))) (S m) H))). -(* COMMENTS -Initial nodes: 69 -END *) - -theorem le_Sn_O: - \forall (n: nat).(not (le (S n) O)) -\def - \lambda (n: nat).(\lambda (H: (le (S n) O)).(le_ind (S n) (\lambda (n0: -nat).(IsSucc n0)) I (\lambda (m: nat).(\lambda (_: (le (S n) m)).(\lambda (_: -(IsSucc m)).I))) O H)). -(* COMMENTS -Initial nodes: 43 -END *) - -theorem le_Sn_n: - \forall (n: nat).(not (le (S n) n)) -\def - \lambda (n: nat).(nat_ind (\lambda (n0: nat).(not (le (S n0) n0))) (le_Sn_O -O) (\lambda (n0: nat).(\lambda (IHn: (not (le (S n0) n0))).(\lambda (H: (le -(S (S n0)) (S n0))).(IHn (le_S_n (S n0) n0 H))))) n). -(* COMMENTS -Initial nodes: 57 -END *) - -theorem le_antisym: - \forall (n: nat).(\forall (m: nat).((le n m) \to ((le m n) \to (eq nat n -m)))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (h: (le n m)).(le_ind n (\lambda -(n0: nat).((le n0 n) \to (eq nat n n0))) (\lambda (_: (le n n)).(refl_equal -nat n)) (\lambda (m0: nat).(\lambda (H: (le n m0)).(\lambda (_: (((le m0 n) -\to (eq nat n m0)))).(\lambda (H1: (le (S m0) n)).(False_ind (eq nat n (S -m0)) (let H2 \def (le_trans (S m0) n m0 H1 H) in ((let H3 \def (le_Sn_n m0) -in (\lambda (H4: (le (S m0) m0)).(H3 H4))) H2))))))) m h))). -(* COMMENTS -Initial nodes: 119 -END *) - -theorem le_n_O_eq: - \forall (n: nat).((le n O) \to (eq nat O n)) -\def - \lambda (n: nat).(\lambda (H: (le n O)).(le_antisym O n (le_O_n n) H)). -(* COMMENTS -Initial nodes: 19 -END *) - -theorem le_elim_rel: - \forall (P: ((nat \to (nat \to Prop)))).(((\forall (p: nat).(P O p))) \to -(((\forall (p: nat).(\forall (q: nat).((le p q) \to ((P p q) \to (P (S p) (S -q))))))) \to (\forall (n: nat).(\forall (m: nat).((le n m) \to (P n m)))))) -\def - \lambda (P: ((nat \to (nat \to Prop)))).(\lambda (H: ((\forall (p: nat).(P O -p)))).(\lambda (H0: ((\forall (p: nat).(\forall (q: nat).((le p q) \to ((P p -q) \to (P (S p) (S q)))))))).(\lambda (n: nat).(nat_ind (\lambda (n0: -nat).(\forall (m: nat).((le n0 m) \to (P n0 m)))) (\lambda (m: nat).(\lambda -(_: (le O m)).(H m))) (\lambda (n0: nat).(\lambda (IHn: ((\forall (m: -nat).((le n0 m) \to (P n0 m))))).(\lambda (m: nat).(\lambda (Le: (le (S n0) -m)).(le_ind (S n0) (\lambda (n1: nat).(P (S n0) n1)) (H0 n0 n0 (le_n n0) (IHn -n0 (le_n n0))) (\lambda (m0: nat).(\lambda (H1: (le (S n0) m0)).(\lambda (_: -(P (S n0) m0)).(H0 n0 m0 (le_trans_S n0 m0 H1) (IHn m0 (le_trans_S n0 m0 -H1)))))) m Le))))) n)))). -(* COMMENTS -Initial nodes: 181 -END *) - -theorem lt_n_n: - \forall (n: nat).(not (lt n n)) -\def - le_Sn_n. -(* COMMENTS -Initial nodes: 1 -END *) - -theorem lt_n_S: - \forall (n: nat).(\forall (m: nat).((lt n m) \to (lt (S n) (S m)))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt n m)).(le_n_S (S n) m -H))). -(* COMMENTS -Initial nodes: 19 -END *) - -theorem lt_n_Sn: - \forall (n: nat).(lt n (S n)) -\def - \lambda (n: nat).(le_n (S n)). -(* COMMENTS -Initial nodes: 7 -END *) - -theorem lt_S_n: - \forall (n: nat).(\forall (m: nat).((lt (S n) (S m)) \to (lt n m))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt (S n) (S m))).(le_S_n (S -n) m H))). -(* COMMENTS -Initial nodes: 23 -END *) - -theorem lt_n_O: - \forall (n: nat).(not (lt n O)) -\def - le_Sn_O. -(* COMMENTS -Initial nodes: 1 -END *) - -theorem lt_trans: - \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((lt n m) \to ((lt m p) -\to (lt n p))))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (lt n -m)).(\lambda (H0: (lt m p)).(le_ind (S m) (\lambda (n0: nat).(lt n n0)) (le_S -(S n) m H) (\lambda (m0: nat).(\lambda (_: (le (S m) m0)).(\lambda (IHle: (lt -n m0)).(le_S (S n) m0 IHle)))) p H0))))). -(* COMMENTS -Initial nodes: 71 -END *) - -theorem lt_O_Sn: - \forall (n: nat).(lt O (S n)) -\def - \lambda (n: nat).(le_n_S O n (le_O_n n)). -(* COMMENTS -Initial nodes: 11 -END *) - -theorem lt_le_S: - \forall (n: nat).(\forall (p: nat).((lt n p) \to (le (S n) p))) -\def - \lambda (n: nat).(\lambda (p: nat).(\lambda (H: (lt n p)).H)). -(* COMMENTS -Initial nodes: 11 -END *) - -theorem le_not_lt: - \forall (n: nat).(\forall (m: nat).((le n m) \to (not (lt m n)))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(le_ind n (\lambda -(n0: nat).(not (lt n0 n))) (lt_n_n n) (\lambda (m0: nat).(\lambda (_: (le n -m0)).(\lambda (IHle: (not (lt m0 n))).(\lambda (H1: (lt (S m0) n)).(IHle -(le_trans_S (S m0) n H1)))))) m H))). -(* COMMENTS -Initial nodes: 67 -END *) - -theorem le_lt_n_Sm: - \forall (n: nat).(\forall (m: nat).((le n m) \to (lt n (S m)))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(le_n_S n m H))). -(* COMMENTS -Initial nodes: 17 -END *) - -theorem le_lt_trans: - \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((le n m) \to ((lt m p) -\to (lt n p))))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (le n -m)).(\lambda (H0: (lt m p)).(le_ind (S m) (\lambda (n0: nat).(lt n n0)) -(le_n_S n m H) (\lambda (m0: nat).(\lambda (_: (le (S m) m0)).(\lambda (IHle: -(lt n m0)).(le_S (S n) m0 IHle)))) p H0))))). -(* COMMENTS -Initial nodes: 69 -END *) - -theorem lt_le_trans: - \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((lt n m) \to ((le m p) -\to (lt n p))))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (lt n -m)).(\lambda (H0: (le m p)).(le_ind m (\lambda (n0: nat).(lt n n0)) H -(\lambda (m0: nat).(\lambda (_: (le m m0)).(\lambda (IHle: (lt n m0)).(le_S -(S n) m0 IHle)))) p H0))))). -(* COMMENTS -Initial nodes: 59 -END *) - -theorem lt_le_weak: - \forall (n: nat).(\forall (m: nat).((lt n m) \to (le n m))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt n m)).(le_trans_S n m -H))). -(* COMMENTS -Initial nodes: 17 -END *) - -theorem lt_n_Sm_le: - \forall (n: nat).(\forall (m: nat).((lt n (S m)) \to (le n m))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt n (S m))).(le_S_n n m -H))). -(* COMMENTS -Initial nodes: 19 -END *) - -theorem le_lt_or_eq: - \forall (n: nat).(\forall (m: nat).((le n m) \to (or (lt n m) (eq nat n m)))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(le_ind n (\lambda -(n0: nat).(or (lt n n0) (eq nat n n0))) (or_intror (lt n n) (eq nat n n) -(refl_equal nat n)) (\lambda (m0: nat).(\lambda (H0: (le n m0)).(\lambda (_: -(or (lt n m0) (eq nat n m0))).(or_introl (lt n (S m0)) (eq nat n (S m0)) -(le_n_S n m0 H0))))) m H))). -(* COMMENTS -Initial nodes: 109 -END *) - -theorem le_or_lt: - \forall (n: nat).(\forall (m: nat).(or (le n m) (lt m n))) -\def - \lambda (n: nat).(\lambda (m: nat).(nat_double_ind (\lambda (n0: -nat).(\lambda (n1: nat).(or (le n0 n1) (lt n1 n0)))) (\lambda (n0: -nat).(or_introl (le O n0) (lt n0 O) (le_O_n n0))) (\lambda (n0: -nat).(or_intror (le (S n0) O) (lt O (S n0)) (lt_le_S O (S n0) (lt_O_Sn n0)))) -(\lambda (n0: nat).(\lambda (m0: nat).(\lambda (H: (or (le n0 m0) (lt m0 -n0))).(or_ind (le n0 m0) (lt m0 n0) (or (le (S n0) (S m0)) (lt (S m0) (S -n0))) (\lambda (H0: (le n0 m0)).(or_introl (le (S n0) (S m0)) (lt (S m0) (S -n0)) (le_n_S n0 m0 H0))) (\lambda (H0: (lt m0 n0)).(or_intror (le (S n0) (S -m0)) (lt (S m0) (S n0)) (le_n_S (S m0) n0 H0))) H)))) n m)). -(* COMMENTS -Initial nodes: 209 -END *) - -theorem plus_n_O: - \forall (n: nat).(eq nat n (plus n O)) -\def - \lambda (n: nat).(nat_ind (\lambda (n0: nat).(eq nat n0 (plus n0 O))) -(refl_equal nat O) (\lambda (n0: nat).(\lambda (H: (eq nat n0 (plus n0 -O))).(f_equal nat nat S n0 (plus n0 O) H))) n). -(* COMMENTS -Initial nodes: 57 -END *) - -theorem plus_n_Sm: - \forall (n: nat).(\forall (m: nat).(eq nat (S (plus n m)) (plus n (S m)))) -\def - \lambda (m: nat).(\lambda (n: nat).(nat_ind (\lambda (n0: nat).(eq nat (S -(plus n0 n)) (plus n0 (S n)))) (refl_equal nat (S n)) (\lambda (n0: -nat).(\lambda (H: (eq nat (S (plus n0 n)) (plus n0 (S n)))).(f_equal nat nat -S (S (plus n0 n)) (plus n0 (S n)) H))) m)). -(* COMMENTS -Initial nodes: 85 -END *) - -theorem plus_sym: - \forall (n: nat).(\forall (m: nat).(eq nat (plus n m) (plus m n))) -\def - \lambda (n: nat).(\lambda (m: nat).(nat_ind (\lambda (n0: nat).(eq nat (plus -n0 m) (plus m n0))) (plus_n_O m) (\lambda (y: nat).(\lambda (H: (eq nat (plus -y m) (plus m y))).(eq_ind nat (S (plus m y)) (\lambda (n0: nat).(eq nat (S -(plus y m)) n0)) (f_equal nat nat S (plus y m) (plus m y) H) (plus m (S y)) -(plus_n_Sm m y)))) n)). -(* COMMENTS -Initial nodes: 111 -END *) - -theorem plus_Snm_nSm: - \forall (n: nat).(\forall (m: nat).(eq nat (plus (S n) m) (plus n (S m)))) -\def - \lambda (n: nat).(\lambda (m: nat).(eq_ind_r nat (plus m n) (\lambda (n0: -nat).(eq nat (S n0) (plus n (S m)))) (eq_ind_r nat (plus (S m) n) (\lambda -(n0: nat).(eq nat (S (plus m n)) n0)) (refl_equal nat (plus (S m) n)) (plus n -(S m)) (plus_sym n (S m))) (plus n m) (plus_sym n m))). -(* COMMENTS -Initial nodes: 99 -END *) - -theorem plus_assoc_l: - \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(eq nat (plus n (plus m -p)) (plus (plus n m) p)))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(nat_ind (\lambda (n0: -nat).(eq nat (plus n0 (plus m p)) (plus (plus n0 m) p))) (refl_equal nat -(plus m p)) (\lambda (n0: nat).(\lambda (H: (eq nat (plus n0 (plus m p)) -(plus (plus n0 m) p))).(f_equal nat nat S (plus n0 (plus m p)) (plus (plus n0 -m) p) H))) n))). -(* COMMENTS -Initial nodes: 101 -END *) - -theorem plus_assoc_r: - \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(eq nat (plus (plus n -m) p) (plus n (plus m p))))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(sym_eq nat (plus n -(plus m p)) (plus (plus n m) p) (plus_assoc_l n m p)))). -(* COMMENTS -Initial nodes: 37 -END *) - -theorem simpl_plus_l: - \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((eq nat (plus n m) -(plus n p)) \to (eq nat m p)))) -\def - \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (m: nat).(\forall (p: -nat).((eq nat (plus n0 m) (plus n0 p)) \to (eq nat m p))))) (\lambda (m: -nat).(\lambda (p: nat).(\lambda (H: (eq nat m p)).H))) (\lambda (n0: -nat).(\lambda (IHn: ((\forall (m: nat).(\forall (p: nat).((eq nat (plus n0 m) -(plus n0 p)) \to (eq nat m p)))))).(\lambda (m: nat).(\lambda (p: -nat).(\lambda (H: (eq nat (S (plus n0 m)) (S (plus n0 p)))).(IHn m p (IHn -(plus n0 m) (plus n0 p) (f_equal nat nat (plus n0) (plus n0 m) (plus n0 p) -(eq_add_S (plus n0 m) (plus n0 p) H))))))))) n). -(* COMMENTS -Initial nodes: 161 -END *) - -theorem minus_n_O: - \forall (n: nat).(eq nat n (minus n O)) -\def - \lambda (n: nat).(nat_ind (\lambda (n0: nat).(eq nat n0 (minus n0 O))) -(refl_equal nat O) (\lambda (n0: nat).(\lambda (_: (eq nat n0 (minus n0 -O))).(refl_equal nat (S n0)))) n). -(* COMMENTS -Initial nodes: 47 -END *) - -theorem minus_n_n: - \forall (n: nat).(eq nat O (minus n n)) -\def - \lambda (n: nat).(nat_ind (\lambda (n0: nat).(eq nat O (minus n0 n0))) -(refl_equal nat O) (\lambda (n0: nat).(\lambda (IHn: (eq nat O (minus n0 -n0))).IHn)) n). -(* COMMENTS -Initial nodes: 41 -END *) - -theorem minus_Sn_m: - \forall (n: nat).(\forall (m: nat).((le m n) \to (eq nat (S (minus n m)) -(minus (S n) m)))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (Le: (le m n)).(le_elim_rel -(\lambda (n0: nat).(\lambda (n1: nat).(eq nat (S (minus n1 n0)) (minus (S n1) -n0)))) (\lambda (p: nat).(f_equal nat nat S (minus p O) p (sym_eq nat p -(minus p O) (minus_n_O p)))) (\lambda (p: nat).(\lambda (q: nat).(\lambda (_: -(le p q)).(\lambda (H0: (eq nat (S (minus q p)) (match p with [O \Rightarrow -(S q) | (S l) \Rightarrow (minus q l)]))).H0)))) m n Le))). -(* COMMENTS -Initial nodes: 111 -END *) - -theorem plus_minus: - \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((eq nat n (plus m p)) -\to (eq nat p (minus n m))))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(nat_double_ind -(\lambda (n0: nat).(\lambda (n1: nat).((eq nat n1 (plus n0 p)) \to (eq nat p -(minus n1 n0))))) (\lambda (n0: nat).(\lambda (H: (eq nat n0 p)).(eq_ind nat -n0 (\lambda (n1: nat).(eq nat p n1)) (sym_eq nat n0 p H) (minus n0 O) -(minus_n_O n0)))) (\lambda (n0: nat).(\lambda (H: (eq nat O (S (plus n0 -p)))).(False_ind (eq nat p O) (let H0 \def H in ((let H1 \def (O_S (plus n0 -p)) in (\lambda (H2: (eq nat O (S (plus n0 p)))).(H1 H2))) H0))))) (\lambda -(n0: nat).(\lambda (m0: nat).(\lambda (H: (((eq nat m0 (plus n0 p)) \to (eq -nat p (minus m0 n0))))).(\lambda (H0: (eq nat (S m0) (S (plus n0 p)))).(H -(eq_add_S m0 (plus n0 p) H0)))))) m n))). -(* COMMENTS -Initial nodes: 199 -END *) - -theorem minus_plus: - \forall (n: nat).(\forall (m: nat).(eq nat (minus (plus n m) n) m)) -\def - \lambda (n: nat).(\lambda (m: nat).(sym_eq nat m (minus (plus n m) n) -(plus_minus (plus n m) n m (refl_equal nat (plus n m))))). -(* COMMENTS -Initial nodes: 41 -END *) - -theorem le_pred_n: - \forall (n: nat).(le (pred n) n) -\def - \lambda (n: nat).(nat_ind (\lambda (n0: nat).(le (pred n0) n0)) (le_n O) -(\lambda (n0: nat).(\lambda (_: (le (pred n0) n0)).(le_S (pred (S n0)) n0 -(le_n n0)))) n). -(* COMMENTS -Initial nodes: 43 -END *) - -theorem le_plus_l: - \forall (n: nat).(\forall (m: nat).(le n (plus n m))) -\def - \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (m: nat).(le n0 (plus -n0 m)))) (\lambda (m: nat).(le_O_n m)) (\lambda (n0: nat).(\lambda (IHn: -((\forall (m: nat).(le n0 (plus n0 m))))).(\lambda (m: nat).(le_n_S n0 (plus -n0 m) (IHn m))))) n). -(* COMMENTS -Initial nodes: 55 -END *) - -theorem le_plus_r: - \forall (n: nat).(\forall (m: nat).(le m (plus n m))) -\def - \lambda (n: nat).(\lambda (m: nat).(nat_ind (\lambda (n0: nat).(le m (plus -n0 m))) (le_n m) (\lambda (n0: nat).(\lambda (H: (le m (plus n0 m))).(le_S m -(plus n0 m) H))) n)). -(* COMMENTS -Initial nodes: 47 -END *) - -theorem simpl_le_plus_l: - \forall (p: nat).(\forall (n: nat).(\forall (m: nat).((le (plus p n) (plus p -m)) \to (le n m)))) -\def - \lambda (p: nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).(\forall (m: -nat).((le (plus n n0) (plus n m)) \to (le n0 m))))) (\lambda (n: -nat).(\lambda (m: nat).(\lambda (H: (le n m)).H))) (\lambda (p0: -nat).(\lambda (IHp: ((\forall (n: nat).(\forall (m: nat).((le (plus p0 n) -(plus p0 m)) \to (le n m)))))).(\lambda (n: nat).(\lambda (m: nat).(\lambda -(H: (le (S (plus p0 n)) (S (plus p0 m)))).(IHp n m (le_S_n (plus p0 n) (plus -p0 m) H))))))) p). -(* COMMENTS -Initial nodes: 113 -END *) - -theorem le_plus_trans: - \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((le n m) \to (le n -(plus m p))))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (le n -m)).(le_trans n m (plus m p) H (le_plus_l m p))))). -(* COMMENTS -Initial nodes: 31 -END *) - -theorem le_reg_l: - \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((le n m) \to (le (plus -p n) (plus p m))))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(nat_ind (\lambda (n0: -nat).((le n m) \to (le (plus n0 n) (plus n0 m)))) (\lambda (H: (le n m)).H) -(\lambda (p0: nat).(\lambda (IHp: (((le n m) \to (le (plus p0 n) (plus p0 -m))))).(\lambda (H: (le n m)).(le_n_S (plus p0 n) (plus p0 m) (IHp H))))) -p))). -(* COMMENTS -Initial nodes: 85 -END *) - -theorem le_plus_plus: - \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(\forall (q: nat).((le -n m) \to ((le p q) \to (le (plus n p) (plus m q))))))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (q: -nat).(\lambda (H: (le n m)).(\lambda (H0: (le p q)).(le_ind n (\lambda (n0: -nat).(le (plus n p) (plus n0 q))) (le_reg_l p q n H0) (\lambda (m0: -nat).(\lambda (_: (le n m0)).(\lambda (H2: (le (plus n p) (plus m0 q))).(le_S -(plus n p) (plus m0 q) H2)))) m H)))))). -(* COMMENTS -Initial nodes: 91 -END *) - -theorem le_plus_minus: - \forall (n: nat).(\forall (m: nat).((le n m) \to (eq nat m (plus n (minus m -n))))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (Le: (le n m)).(le_elim_rel -(\lambda (n0: nat).(\lambda (n1: nat).(eq nat n1 (plus n0 (minus n1 n0))))) -(\lambda (p: nat).(minus_n_O p)) (\lambda (p: nat).(\lambda (q: nat).(\lambda -(_: (le p q)).(\lambda (H0: (eq nat q (plus p (minus q p)))).(f_equal nat nat -S q (plus p (minus q p)) H0))))) n m Le))). -(* COMMENTS -Initial nodes: 91 -END *) - -theorem le_plus_minus_r: - \forall (n: nat).(\forall (m: nat).((le n m) \to (eq nat (plus n (minus m -n)) m))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(sym_eq nat m -(plus n (minus m n)) (le_plus_minus n m H)))). -(* COMMENTS -Initial nodes: 33 -END *) - -theorem simpl_lt_plus_l: - \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((lt (plus p n) (plus p -m)) \to (lt n m)))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(nat_ind (\lambda (n0: -nat).((lt (plus n0 n) (plus n0 m)) \to (lt n m))) (\lambda (H: (lt n m)).H) -(\lambda (p0: nat).(\lambda (IHp: (((lt (plus p0 n) (plus p0 m)) \to (lt n -m)))).(\lambda (H: (lt (S (plus p0 n)) (S (plus p0 m)))).(IHp (le_S_n (S -(plus p0 n)) (plus p0 m) H))))) p))). -(* COMMENTS -Initial nodes: 99 -END *) - -theorem lt_reg_l: - \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((lt n m) \to (lt (plus -p n) (plus p m))))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(nat_ind (\lambda (n0: -nat).((lt n m) \to (lt (plus n0 n) (plus n0 m)))) (\lambda (H: (lt n m)).H) -(\lambda (p0: nat).(\lambda (IHp: (((lt n m) \to (lt (plus p0 n) (plus p0 -m))))).(\lambda (H: (lt n m)).(lt_n_S (plus p0 n) (plus p0 m) (IHp H))))) -p))). -(* COMMENTS -Initial nodes: 85 -END *) - -theorem lt_reg_r: - \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((lt n m) \to (lt (plus -n p) (plus m p))))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (lt n -m)).(eq_ind_r nat (plus p n) (\lambda (n0: nat).(lt n0 (plus m p))) (eq_ind_r -nat (plus p m) (\lambda (n0: nat).(lt (plus p n) n0)) (nat_ind (\lambda (n0: -nat).(lt (plus n0 n) (plus n0 m))) H (\lambda (n0: nat).(\lambda (_: (lt -(plus n0 n) (plus n0 m))).(lt_reg_l n m (S n0) H))) p) (plus m p) (plus_sym m -p)) (plus n p) (plus_sym n p))))). -(* COMMENTS -Initial nodes: 129 -END *) - -theorem le_lt_plus_plus: - \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(\forall (q: nat).((le -n m) \to ((lt p q) \to (lt (plus n p) (plus m q))))))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (q: -nat).(\lambda (H: (le n m)).(\lambda (H0: (le (S p) q)).(eq_ind_r nat (plus n -(S p)) (\lambda (n0: nat).(le n0 (plus m q))) (le_plus_plus n m (S p) q H H0) -(plus (S n) p) (plus_Snm_nSm n p))))))). -(* COMMENTS -Initial nodes: 75 -END *) - -theorem lt_le_plus_plus: - \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(\forall (q: nat).((lt -n m) \to ((le p q) \to (lt (plus n p) (plus m q))))))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (q: -nat).(\lambda (H: (le (S n) m)).(\lambda (H0: (le p q)).(le_plus_plus (S n) m -p q H H0)))))). -(* COMMENTS -Initial nodes: 37 -END *) - -theorem lt_plus_plus: - \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(\forall (q: nat).((lt -n m) \to ((lt p q) \to (lt (plus n p) (plus m q))))))) -\def - \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (q: -nat).(\lambda (H: (lt n m)).(\lambda (H0: (lt p q)).(lt_le_plus_plus n m p q -H (lt_le_weak p q H0))))))). -(* COMMENTS -Initial nodes: 39 -END *) - -theorem well_founded_ltof: - \forall (A: Set).(\forall (f: ((A \to nat))).(well_founded A (ltof A f))) -\def - \lambda (A: Set).(\lambda (f: ((A \to nat))).(let H \def (\lambda (n: -nat).(nat_ind (\lambda (n0: nat).(\forall (a: A).((lt (f a) n0) \to (Acc A -(ltof A f) a)))) (\lambda (a: A).(\lambda (H: (lt (f a) O)).(False_ind (Acc A -(ltof A f) a) (let H0 \def H in ((let H1 \def (lt_n_O (f a)) in (\lambda (H2: -(lt (f a) O)).(H1 H2))) H0))))) (\lambda (n0: nat).(\lambda (IHn: ((\forall -(a: A).((lt (f a) n0) \to (Acc A (ltof A f) a))))).(\lambda (a: A).(\lambda -(ltSma: (lt (f a) (S n0))).(Acc_intro A (ltof A f) a (\lambda (b: A).(\lambda -(ltfafb: (lt (f b) (f a))).(IHn b (lt_le_trans (f b) (f a) n0 ltfafb -(lt_n_Sm_le (f a) n0 ltSma)))))))))) n)) in (\lambda (a: A).(H (S (f a)) a -(le_n (S (f a))))))). -(* COMMENTS -Initial nodes: 189 -END *) - -theorem lt_wf: - well_founded nat lt -\def - well_founded_ltof nat (\lambda (m: nat).m). -(* COMMENTS -Initial nodes: 7 -END *) - -theorem lt_wf_ind: - \forall (p: nat).(\forall (P: ((nat \to Prop))).(((\forall (n: -nat).(((\forall (m: nat).((lt m n) \to (P m)))) \to (P n)))) \to (P p))) -\def - \lambda (p: nat).(\lambda (P: ((nat \to Prop))).(\lambda (H: ((\forall (n: -nat).(((\forall (m: nat).((lt m n) \to (P m)))) \to (P n))))).(Acc_ind nat lt -(\lambda (n: nat).(P n)) (\lambda (x: nat).(\lambda (_: ((\forall (y: -nat).((lt y x) \to (Acc nat lt y))))).(\lambda (H1: ((\forall (y: nat).((lt y -x) \to (P y))))).(H x H1)))) p (lt_wf p)))). -(* COMMENTS -Initial nodes: 77 -END *) - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Legacy-1/definitions.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Legacy-1/definitions.ma deleted file mode 100644 index 63fc85890..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Legacy-1/definitions.ma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Legacy-1/coq/defs.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Legacy-1/preamble.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Legacy-1/preamble.ma deleted file mode 100644 index 96c1bc1fa..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Legacy-1/preamble.ma +++ /dev/null @@ -1,15 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -inductive False: Prop \def . diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Legacy-1/spare.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Legacy-1/spare.ma deleted file mode 100644 index 77939a1b6..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Legacy-1/spare.ma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Legacy-1/theory.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Legacy-1/theory.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Legacy-1/theory.ma deleted file mode 100644 index 4ee597e09..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Legacy-1/theory.ma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Legacy-1/coq/props.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Legacy-2/coq/defs.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Legacy-2/coq/defs.mma deleted file mode 100644 index 175a39cdd..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Legacy-2/coq/defs.mma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Legacy-2/preamble.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Legacy-2/coq/props.mma b/matitaB/matita/contribs/LAMBDA-TYPES/Legacy-2/coq/props.mma deleted file mode 100644 index e214da177..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Legacy-2/coq/props.mma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Legacy-2/coq/defs.ma". - -inline "Legacy-1/coq/props.ma" procedural. - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Legacy-2/preamble.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Legacy-2/preamble.ma deleted file mode 100644 index 00c38151d..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Legacy-2/preamble.ma +++ /dev/null @@ -1,62 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "Legacy-1/theory.ma". - -default "equality" - cic:/matita/LAMBDA-TYPES/Legacy-1/coq/defs/eq.ind - cic:/matita/LAMBDA-TYPES/Legacy-1/coq/props/sym_eq.con - cic:/matita/LAMBDA-TYPES/Legacy-1/coq/props/trans_eq.con - cic:/matita/LAMBDA-TYPES/Legacy-1/coq/defs/eq_ind.con - cic:/matita/LAMBDA-TYPES/Legacy-1/coq/props/eq_ind_r.con - cic:/matita/LAMBDA-TYPES/Legacy-1/coq/defs/eq_rec.con - cic:/matita/LAMBDA-TYPES/Legacy-1/coq/props/eq_rec_r.con - cic:/matita/LAMBDA-TYPES/Legacy-1/coq/defs/eq_rect.con - cic:/matita/LAMBDA-TYPES/Legacy-1/coq/props/eq_rect_r.con - cic:/matita/LAMBDA-TYPES/Legacy-1/coq/props/f_equal.con - cic:/matita/LAMBDA-TYPES/Legacy-2/preamble/f_equal_sym.con. - -default "true" - cic:/matita/LAMBDA-TYPES/Legacy-1/coq/defs/True.ind. -default "false" - cic:/matita/LAMBDA-TYPES/Legacy-1/preamble/False.ind. -default "absurd" - cic:/matita/LAMBDA-TYPES/Legacy-1/coq/props/absurd.con. - -interpretation "Coq 7.3.1 natural plus" 'plus x y = (cic:/matita/LAMBDA-TYPES/Legacy-1/coq/defs/plus.con x y). -interpretation "Coq 7.3.1 natural minus" 'minus x y = (cic:/matita/LAMBDA-TYPES/Legacy-1/coq/defs/minus.con x y). -interpretation "Coq 7.3.1 logical and" 'and x y = (cic:/matita/LAMBDA-TYPES/Legacy-1/coq/defs/land.ind#xpointer(1/1) x y). -interpretation "Coq 7.3.1 logical or" 'or x y = (cic:/matita/LAMBDA-TYPES/Legacy-1/coq/defs/or.ind#xpointer(1/1) x y). -interpretation "Coq 7.3.1 logical not" 'not x = (cic:/matita/LAMBDA-TYPES/Legacy-1/coq/defs/not.con x). -interpretation "Coq 7.3.1 exists" 'exists \eta.x = (cic:/matita/LAMBDA-TYPES/Legacy-1/coq/defs/ex.ind#xpointer(1/1) ? x). -interpretation "Coq 7.3.1 natural 'less or equal to'" 'leq x y = (cic:/matita/LAMBDA-TYPES/Legacy-1/coq/defs/le.ind#xpointer(1/1) x y). -interpretation "Coq 7.3.1 natural 'less than'" 'lt x y = (cic:/matita/LAMBDA-TYPES/Legacy-1/coq/defs/lt.con x y). -interpretation "Coq 7.3.1 leibnitz's equality" 'eq t x y = (cic:/matita/LAMBDA-TYPES/Legacy-1/coq/defs/eq.ind#xpointer(1/1) t x y). - -alias symbol "plus" = "Coq 7.3.1 natural plus". -alias symbol "minus" = "Coq 7.3.1 natural minus". -alias symbol "and" = "Coq 7.3.1 logical and". -alias symbol "or" = "Coq 7.3.1 logical or". -alias symbol "not" = "Coq 7.3.1 logical not". -alias symbol "exists" = "Coq 7.3.1 exists". -alias symbol "leq" = "Coq 7.3.1 natural 'less or equal to'". -alias symbol "lt" = "Coq 7.3.1 natural 'less than'". -alias symbol "eq" = "Coq 7.3.1 leibnitz's equality". - -theorem f_equal_sym: \forall A,B:Set.\forall f:A\to B.\forall x,y. - x = y \to (f y) = (f x). - intros; symmetry. - apply cic:/matita/LAMBDA-TYPES/Legacy-1/coq/props/f_equal.con. - assumption. -qed. diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Legacy-2/theory.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Legacy-2/theory.ma deleted file mode 100644 index 5cd562260..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Legacy-2/theory.ma +++ /dev/null @@ -1,18 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Legacy-2/coq/props.ma". - diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Makefile b/matitaB/matita/contribs/LAMBDA-TYPES/Makefile deleted file mode 100644 index 3e42a4ea1..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Makefile +++ /dev/null @@ -1,50 +0,0 @@ -include ../Makefile.defs - -DIR=$(shell basename $$PWD) - -H=@ - -MATITAOPTIONS=$(MATITAUSEROPTIONS) -onepass - -LOG = log.txt - -MMAS = $(shell find -name "*.mma") -MAS = $(MMAS:%.mma=%.ma) - -$(DIR) all: - $(H)$(RM) $(LOG) - $(H)$(BIN)matitac $(MATITAOPTIONS) 2>> $(LOG) -$(DIR).opt opt all.opt: - $(H)$(RM) $(LOG) - $(H)$(BIN)matitac.opt $(MATITAOPTIONS) 2>> $(LOG) - -%.ma %.mma: - $(H)$(RM) $(LOG) - $(H)$(BIN)matitac $(MATITAOPTIONS) $@ 2>> $(LOG) -%.ma.opt: - $(H)$(RM) $(LOG) - $(H)$(BIN)matitac.opt $(MATITAOPTIONS) $*.ma 2>> $(LOG) -%.mma.opt: - $(H)$(RM) $(LOG) - $(H)$(BIN)matitac.opt $(MATITAOPTIONS) $*.mma 2>> $(LOG) - -clean: - $(H)$(BIN)matitaclean $(MATITAOPTIONS) - $(H)$(RM) $(MAS) -clean.opt: - $(H)$(BIN)matitaclean.opt $(MATITAOPTIONS) - $(H)$(RM) $(MAS) - -depend: - $$(H)(BIN)matitadep $(MATITAOPTIONS) -depend.opt: - $(H)$(BIN)matitadep.opt $(MATITAOPTIONS) - -ifneq ($(strip $(MAS)),) -clean.ma: - $(H)$(BIN)matitaclean.opt $(MATITAOPTIONS) $(MAS) - $(H)$(RM) $(MAS) -else -clean.ma: - $(H)echo no files to clean -endif diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/depends b/matitaB/matita/contribs/LAMBDA-TYPES/depends deleted file mode 100644 index d205172eb..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/depends +++ /dev/null @@ -1,696 +0,0 @@ -Basic-2/clear/defs.ma Basic-2/clear/defs.mma -Basic-1/lift/defs.ma Basic-1/s/defs.ma Basic-1/tlist/defs.ma -Basic-1/subst0/dec.ma Basic-1/lift/props.ma Basic-1/subst0/defs.ma -Basic-2/sc3/props.mma Basic-1/sc3/props.ma Basic-2/arity/aprem.ma Basic-2/arity/lift1.ma Basic-2/csuba/arity.ma Basic-2/drop1/getl.ma Basic-2/drop1/props.ma Basic-2/lift1/props.ma Basic-2/llt/props.ma Basic-2/nf2/lift1.ma Basic-2/sc3/defs.ma Basic-2/sn3/lift1.ma -Basic-2/pr3/subst1.ma Basic-2/pr3/subst1.mma -Basic-2/csubt/csuba.mma Basic-1/csubt/csuba.ma Basic-2/ty3/arity.ma -Basic-2/sn3/props.mma Basic-1/sn3/props.ma Basic-2/nf2/iso.ma Basic-2/pr3/iso.ma Basic-2/sn3/fwd.ma Basic-2/sn3/nf2.ma -Basic-2/pr1/defs.ma Basic-2/pr1/defs.mma -Basic-2/arity/props.mma Basic-1/arity/props.ma Basic-2/arity/fwd.ma -Basic-2/flt/defs.mma Basic-2/C/defs.ma -Basic-2/clen/getl.ma Basic-2/clen/getl.mma -Basic-2/drop/defs.ma Basic-2/drop/defs.mma -Legacy-1/theory.ma Legacy-1/coq/props.ma -Basic-1/csubt/defs.ma Basic-1/ty3/defs.ma -Basic-1/clear/drop.ma Basic-1/clear/fwd.ma Basic-1/drop/fwd.ma -Basic-1/lift1/props.ma Basic-1/drop1/defs.ma Basic-1/lift/props.ma -Basic-1/nf2/lift1.ma Basic-1/drop1/fwd.ma Basic-1/nf2/props.ma -Basic-1/flt/defs.ma Basic-1/C/defs.ma -Basic-2/getl/props.ma Basic-2/getl/props.mma -Basic-1/pc3/subst1.ma Basic-1/pc3/props.ma Basic-1/pr3/subst1.ma -Basic-2/pr0/defs.mma Basic-2/subst0/defs.ma -Basic-1/drop1/fwd.ma Basic-1/drop1/defs.ma -Basic-2/arity/cimp.mma Basic-1/arity/cimp.ma Basic-2/arity/defs.ma Basic-2/cimp/props.ma -Basic-2/ty3/props.ma Basic-2/ty3/props.mma -Basic-2/drop1/getl.mma Basic-1/drop1/getl.ma Basic-2/drop1/fwd.ma Basic-2/getl/drop.ma -Basic-2/subst/defs.ma Basic-2/subst/defs.mma -Basic-2/lift1/defs.mma Basic-2/lift/defs.ma -Basic-2/A/defs.mma Basic-2/preamble.ma -Basic-2/csubt/clear.ma Basic-2/csubt/clear.mma -Basic-1/csuba/defs.ma Basic-1/arity/defs.ma -Basic-2/csuba/props.ma Basic-2/csuba/props.mma -Basic-1/csubc/fwd.ma Basic-1/csubc/defs.ma -Basic-2/pc3/wcpr0.mma Basic-1/pc3/wcpr0.ma Basic-2/pc3/props.ma Basic-2/wcpr0/getl.ma -Basic-2/subst/props.mma Basic-1/subst/props.ma Basic-2/lift/props.ma Basic-2/subst/fwd.ma Basic-2/subst0/defs.ma -Legacy-1/coq/defs.ma Legacy-1/preamble.ma -Basic-1/tlt/props.ma Basic-1/tlt/defs.ma -Basic-2/pr1/pr1.ma Basic-2/pr1/pr1.mma -Ground-1/ext/arith.ma Ground-1/preamble.ma -Basic-2/subst1/props.ma Basic-2/subst1/props.mma -Basic-2/llt/props.mma Basic-1/llt/props.ma Basic-2/leq/defs.ma Basic-2/llt/defs.ma -Basic-2/sn3/nf2.mma Basic-1/sn3/nf2.ma Basic-2/nf2/dec.ma Basic-2/nf2/pr3.ma Basic-2/sn3/defs.ma -Basic-1/pr3/pr3.ma Basic-1/pr2/pr2.ma Basic-1/pr3/props.ma -Basic-1/lift1/defs.ma Basic-1/lift/defs.ma -Basic-2/sty1/cnt.ma Basic-2/sty1/cnt.mma -Basic-1/wf3/props.ma Basic-1/app/defs.ma Basic-1/wf3/ty3.ma -Basic-2/wf3/getl.ma Basic-2/wf3/getl.mma -Basic-2/arity/subst0.ma Basic-2/arity/subst0.mma -Basic-2/pr0/props.mma Basic-1/pr0/props.ma Basic-2/pr0/defs.ma Basic-2/subst0/subst0.ma -Basic-2/csubt/csuba.ma Basic-2/csubt/csuba.mma -Basic-2/csubst0/clear.mma Basic-1/csubst0/clear.ma Basic-2/clear/fwd.ma Basic-2/csubst0/fwd.ma Basic-2/csubst0/props.ma -Basic-2/nf2/arity.mma Basic-1/nf2/arity.ma Basic-2/arity/subst0.ma Basic-2/nf2/fwd.ma -Basic-1/fsubst0/defs.ma Basic-1/csubst0/defs.ma -Basic-1/G/defs.ma Basic-1/preamble.ma -Basic-2/tlist/props.ma Basic-2/tlist/props.mma -Basic-2/subst0/props.ma Basic-2/subst0/props.mma -Basic-2/ex2/defs.mma Basic-2/C/defs.ma -Basic-1/ty3/defs.ma Basic-1/G/defs.ma Basic-1/pc3/defs.ma -Basic-2/clear/props.ma Basic-2/clear/props.mma -Basic-2/nf2/iso.ma Basic-2/nf2/iso.mma -Basic-1/theory.ma Basic-1/csubt/csuba.ma Basic-1/ex0/props.ma Basic-1/ex1/props.ma Basic-1/ex2/props.ma Basic-1/pr3/wcpr0.ma Basic-1/sty1/cnt.ma Basic-1/subst/props.ma Basic-1/subst0/tlt.ma Basic-1/ty3/fwd_nf2.ma Basic-1/ty3/nf2.ma Basic-1/ty3/sty0.ma Basic-1/wcpr0/fwd.ma Basic-1/wf3/props.ma -Basic-2/wf3/getl.mma Basic-1/wf3/getl.ma Basic-2/ty3/dec.ma Basic-2/wf3/clear.ma -Basic-2/csubst1/fwd.mma Basic-1/csubst1/fwd.ma Basic-2/csubst0/fwd.ma Basic-2/csubst1/defs.ma Basic-2/subst1/props.ma -Basic-2/app/defs.mma Basic-2/C/defs.ma -Basic-1/pr0/defs.ma Basic-1/subst0/defs.ma -Basic-1/nf2/defs.ma Basic-1/pr2/defs.ma -Basic-2/asucc/defs.mma Basic-2/A/defs.ma Basic-2/G/defs.ma -Basic-2/lift/fwd.ma Basic-2/lift/fwd.mma -Basic-2/csubst1/props.ma Basic-2/csubst1/props.mma -Basic-2/sty1/props.ma Basic-2/sty1/props.mma -Basic-2/ex1/defs.mma Basic-2/C/defs.ma -Basic-2/subst1/subst1.mma Basic-1/subst1/subst1.ma Basic-2/subst0/subst0.ma Basic-2/subst1/fwd.ma -Basic-2/pr2/props.mma Basic-1/pr2/props.ma Basic-2/getl/clear.ma Basic-2/getl/drop.ma Basic-2/pr0/props.ma Basic-2/pr2/defs.ma -Basic-2/arity/pr3.ma Basic-2/arity/pr3.mma -Basic-1/pr2/clen.ma Basic-1/clen/getl.ma Basic-1/pr2/props.ma -Basic-2/wcpr0/fwd.ma Basic-2/wcpr0/fwd.mma -Basic-2/ty3/dec.mma Basic-1/ty3/dec.ma Basic-2/getl/dec.ma Basic-2/getl/flt.ma Basic-2/pc3/dec.ma -Basic-2/sty0/props.mma Basic-1/sty0/props.ma Basic-2/getl/drop.ma Basic-2/sty0/defs.ma -Basic-2/T/props.mma Basic-1/T/props.ma Basic-2/T/defs.ma -Basic-1/wf3/defs.ma Basic-1/ty3/defs.ma -Basic-2/csubst0/props.ma Basic-2/csubst0/props.mma -Basic-2/pr0/dec.mma Basic-1/pr0/dec.ma Basic-2/T/dec.ma Basic-2/T/props.ma Basic-2/pr0/fwd.ma Basic-2/subst0/dec.ma -Basic-1/drop1/getl.ma Basic-1/drop1/fwd.ma Basic-1/getl/drop.ma -Basic-2/arity/props.ma Basic-2/arity/props.mma -Basic-2/C/defs.ma Basic-2/C/defs.mma -Basic-1/getl/flt.ma Basic-1/clear/props.ma Basic-1/flt/props.ma Basic-1/getl/fwd.ma -Basic-2/llt/props.ma Basic-2/llt/props.mma -Basic-2/drop/defs.mma Basic-2/C/defs.ma Basic-2/lift/defs.ma Basic-2/r/defs.ma -Basic-2/leq/asucc.mma Basic-1/leq/asucc.ma Basic-2/leq/props.ma -Basic-2/csubt/getl.ma Basic-2/csubt/getl.mma -Basic-2/clen/getl.mma Basic-1/clen/getl.ma Basic-2/clen/defs.ma Basic-2/getl/props.ma -Basic-2/sty0/props.ma Basic-2/sty0/props.mma -Basic-2/tlist/props.mma Basic-1/tlist/props.ma Basic-2/tlist/defs.ma -Ground-1/blt/defs.ma Ground-1/preamble.ma -Basic-1/sc3/defs.ma Basic-1/arity/defs.ma Basic-1/drop1/defs.ma Basic-1/sn3/defs.ma -Basic-2/ex0/defs.mma Basic-2/A/defs.ma Basic-2/G/defs.ma -Basic-2/nf2/dec.mma Basic-1/nf2/dec.ma Basic-2/C/props.ma Basic-2/nf2/defs.ma Basic-2/pr0/dec.ma Basic-2/pr2/clen.ma Basic-2/pr2/fwd.ma -Basic-1/subst/fwd.ma Basic-1/subst/defs.ma -Basic-2/iso/defs.ma Basic-2/iso/defs.mma -Basic-2/aplus/props.ma Basic-2/aplus/props.mma -Basic-2/s/props.mma Basic-1/s/props.ma Basic-2/s/defs.ma -Basic-2/pr2/subst1.mma Basic-1/pr2/subst1.ma Basic-2/csubst1/fwd.ma Basic-2/csubst1/getl.ma Basic-2/getl/drop.ma Basic-2/pr0/fwd.ma Basic-2/pr0/subst1.ma Basic-2/pr2/defs.ma Basic-2/subst1/subst1.ma -Basic-1/llt/defs.ma Basic-1/A/defs.ma -Basic-2/ty3/fsubst0.mma Basic-1/ty3/fsubst0.ma Basic-2/getl/getl.ma Basic-2/pc3/fsubst0.ma Basic-2/ty3/props.ma -Basic-2/pc3/defs.ma Basic-2/pc3/defs.mma -Basic-1/ty3/fsubst0.ma Basic-1/getl/getl.ma Basic-1/pc3/fsubst0.ma Basic-1/ty3/props.ma -Basic-1/subst1/props.ma Basic-1/subst0/props.ma Basic-1/subst1/defs.ma -Basic-1/wcpr0/defs.ma Basic-1/C/defs.ma Basic-1/pr0/defs.ma -Basic-2/csuba/getl.ma Basic-2/csuba/getl.mma -Basic-2/nf2/defs.mma Basic-2/pr2/defs.ma -Ground-1/preamble.ma Legacy-1/theory.ma -Basic-2/asucc/fwd.mma Basic-1/asucc/fwd.ma Basic-2/asucc/defs.ma -Basic-2/pr3/props.ma Basic-2/pr3/props.mma -Basic-2/nf2/props.mma Basic-1/nf2/props.ma Basic-2/nf2/defs.ma Basic-2/pr2/fwd.ma -Basic-1/s/props.ma Basic-1/s/defs.ma -Basic-1/tlist/defs.ma Basic-1/T/defs.ma -Basic-1/arity/subst0.ma Basic-1/arity/props.ma Basic-1/csubst0/getl.ma Basic-1/fsubst0/fwd.ma Basic-1/getl/getl.ma Basic-1/subst0/dec.ma Basic-1/subst0/fwd.ma -Basic-2/csubv/drop.ma Basic-2/csubv/drop.mma -Basic-2/nf2/iso.mma Basic-1/nf2/iso.ma Basic-2/iso/props.ma Basic-2/nf2/pr3.ma Basic-2/pr3/fwd.ma -Legacy-1/definitions.ma Legacy-1/coq/defs.ma -Basic-2/getl/defs.ma Basic-2/getl/defs.mma -Basic-2/cimp/defs.mma Basic-2/getl/defs.ma -Basic-2/wf3/clear.ma Basic-2/wf3/clear.mma -Basic-1/csubv/getl.ma Basic-1/csubv/clear.ma Basic-1/csubv/drop.ma Basic-1/getl/fwd.ma -Basic-2/clear/fwd.ma Basic-2/clear/fwd.mma -Basic-1/subst0/props.ma Basic-1/subst0/fwd.ma -Basic-1/cnt/props.ma Basic-1/cnt/defs.ma Basic-1/lift/fwd.ma -Basic-2/getl/fwd.ma Basic-2/getl/fwd.mma -Basic-1/r/props.ma Basic-1/r/defs.ma Basic-1/s/defs.ma -Basic-2/arity/aprem.mma Basic-1/arity/aprem.ma Basic-2/aprem/props.ma Basic-2/arity/cimp.ma Basic-2/arity/props.ma -Basic-2/leq/asucc.ma Basic-2/leq/asucc.mma -Basic-2/ex1/defs.ma Basic-2/ex1/defs.mma -Basic-2/arity/cimp.ma Basic-2/arity/cimp.mma -Basic-2/ty3/fwd_nf2.mma Basic-1/ty3/fwd_nf2.ma Basic-2/nf2/fwd.ma Basic-2/pc3/nf2.ma Basic-2/ty3/arity_props.ma -Basic-1/drop/fwd.ma Basic-1/drop/defs.ma -Basic-2/subst1/subst1.ma Basic-2/subst1/subst1.mma -Basic-2/pr2/props.ma Basic-2/pr2/props.mma -Basic-2/aplus/props.mma Basic-1/aplus/props.ma Basic-2/aplus/defs.ma Basic-2/next_plus/props.ma -Basic-1/csubst0/clear.ma Basic-1/clear/fwd.ma Basic-1/csubst0/fwd.ma Basic-1/csubst0/props.ma -Basic-1/csubc/getl.ma Basic-1/csubc/clear.ma Basic-1/csubc/drop.ma -Basic-1/csubt/pc3.ma Basic-1/csubt/getl.ma Basic-1/pc3/left.ma -Basic-2/csubc/drop1.mma Basic-1/csubc/drop1.ma Basic-2/csubc/drop.ma -Basic-2/pr3/wcpr0.ma Basic-2/pr3/wcpr0.mma -Basic-1/sc3/arity.ma Basic-1/csubc/arity.ma Basic-1/csubc/drop1.ma Basic-1/csubc/getl.ma Basic-1/csubc/props.ma -Basic-1/csubc/drop1.ma Basic-1/csubc/drop.ma -Basic-2/csubc/drop.ma Basic-2/csubc/drop.mma -Basic-2/pr3/pr1.ma Basic-2/pr3/pr1.mma -Basic-2/C/props.ma Basic-2/C/props.mma -Basic-2/wf3/fwd.ma Basic-2/wf3/fwd.mma 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-Basic-2/pr2/clen.ma Basic-2/pr2/clen.mma -Basic-2/nf2/defs.ma Basic-2/nf2/defs.mma -Basic-1/ex2/props.ma Basic-1/arity/fwd.ma Basic-1/ex2/defs.ma Basic-1/nf2/defs.ma Basic-1/pr2/fwd.ma -Basic-2/subst1/fwd.mma Basic-1/subst1/fwd.ma Basic-2/subst0/props.ma Basic-2/subst1/defs.ma -Ground-2/types/props.ma Ground-2/types/props.mma -Basic-2/lift/props.ma Basic-2/lift/props.mma -Basic-1/nf2/arity.ma Basic-1/arity/subst0.ma Basic-1/nf2/fwd.ma -Basic-2/csubc/fwd.ma Basic-2/csubc/fwd.mma -Basic-2/wf3/defs.ma Basic-2/wf3/defs.mma -Basic-2/sc3/defs.mma Basic-2/arity/defs.ma Basic-2/drop1/defs.ma Basic-2/sn3/defs.ma -Basic-2/csubst1/getl.ma Basic-2/csubst1/getl.mma -Basic-2/tlt/props.ma Basic-2/tlt/props.mma -Basic-2/cnt/props.mma Basic-1/cnt/props.ma Basic-2/cnt/defs.ma Basic-2/lift/fwd.ma -Ground-2/ext/arith.ma Ground-2/ext/arith.mma -Basic-2/getl/flt.ma Basic-2/getl/flt.mma -Basic-2/subst1/fwd.ma Basic-2/subst1/fwd.mma -Basic-1/pr3/iso.ma Basic-1/iso/props.ma Basic-1/pr3/fwd.ma 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-Basic-2/ty3/pr3_props.ma Basic-2/ty3/pr3_props.mma -Basic-1/csubc/arity.ma Basic-1/csubc/csuba.ma -Basic-2/arity/pr3.mma Basic-1/arity/pr3.ma Basic-2/arity/subst0.ma Basic-2/csuba/arity.ma Basic-2/pr0/fwd.ma Basic-2/pr1/defs.ma Basic-2/pr3/defs.ma Basic-2/wcpr0/getl.ma -Basic-2/aprem/defs.mma Basic-2/A/defs.ma -Basic-2/pc3/subst1.ma Basic-2/pc3/subst1.mma -Basic-1/csubt/drop.ma Basic-1/csubt/fwd.ma Basic-1/drop/fwd.ma -Basic-2/subst0/tlt.mma Basic-1/subst0/tlt.ma Basic-2/lift/props.ma Basic-2/lift/tlt.ma Basic-2/subst0/defs.ma -Ground-2/blt/defs.mma Ground-2/preamble.ma -Basic-2/csubc/clear.mma Basic-1/csubc/clear.ma Basic-2/csubc/fwd.ma -Legacy-1/preamble.ma -Basic-2/subst0/subst0.mma Basic-1/subst0/subst0.ma Basic-2/subst0/props.ma -Basic-2/drop/fwd.ma Basic-2/drop/fwd.mma -Basic-2/pc3/nf2.ma Basic-2/pc3/nf2.mma -Basic-2/s/defs.ma Basic-2/s/defs.mma -Basic-2/csuba/defs.ma Basic-2/csuba/defs.mma -Basic-2/subst/fwd.ma Basic-2/subst/fwd.mma -Basic-2/next_plus/props.ma 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-Basic-1/subst1/subst1.ma Basic-1/subst0/subst0.ma Basic-1/subst1/fwd.ma -Basic-2/arity/fwd.mma Basic-1/arity/fwd.ma Basic-2/arity/defs.ma Basic-2/getl/drop.ma Basic-2/leq/asucc.ma -Ground-2/ext/tactics.mma Ground-1/ext/tactics.ma Ground-2/preamble.ma -Basic-2/leq/props.mma Basic-1/leq/props.ma Basic-2/aplus/props.ma Basic-2/leq/fwd.ma -Basic-2/subst1/props.mma Basic-1/subst1/props.ma Basic-2/subst0/props.ma Basic-2/subst1/defs.ma -Basic-2/pr3/fwd.ma Basic-2/pr3/fwd.mma -Basic-1/asucc/fwd.ma Basic-1/asucc/defs.ma -Basic-1/csuba/arity.ma Basic-1/arity/props.ma Basic-1/csuba/getl.ma Basic-1/csuba/props.ma Basic-1/csubv/getl.ma -Basic-1/pc3/props.ma Basic-1/pc3/defs.ma Basic-1/pr3/pr3.ma -Basic-1/csubc/defs.ma Basic-1/sc3/defs.ma -Basic-2/cnt/props.ma Basic-2/cnt/props.mma -Basic-2/pr0/pr0.mma Basic-1/pr0/pr0.ma Basic-2/lift/tlt.ma Basic-2/pr0/fwd.ma -Basic-2/pc3/nf2.mma Basic-1/pc3/nf2.ma Basic-2/nf2/pr3.ma Basic-2/pc3/defs.ma -Basic-1/ty3/sty0.ma Basic-1/sty0/fwd.ma 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Basic-2/sty1/props.ma -Basic-2/csubv/clear.ma Basic-2/csubv/clear.mma -Basic-2/fsubst0/defs.ma Basic-2/fsubst0/defs.mma -Basic-2/C/defs.mma Basic-2/T/defs.ma -Basic-2/csubc/props.ma Basic-2/csubc/props.mma -Basic-2/cnt/defs.ma Basic-2/cnt/defs.mma -Basic-2/arity/fwd.ma Basic-2/arity/fwd.mma -Basic-2/pr1/defs.mma Basic-2/pr0/defs.ma -Basic-1/clen/defs.ma Basic-1/C/defs.ma Basic-1/s/defs.ma -Basic-2/arity/subst0.mma Basic-1/arity/subst0.ma Basic-2/arity/props.ma Basic-2/csubst0/getl.ma Basic-2/fsubst0/fwd.ma Basic-2/getl/getl.ma Basic-2/subst0/dec.ma Basic-2/subst0/fwd.ma diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/root b/matitaB/matita/contribs/LAMBDA-TYPES/root deleted file mode 100644 index ca1729d65..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/root +++ /dev/null @@ -1 +0,0 @@ -baseuri=cic:/matita/LAMBDA-TYPES